------ ------------ Legenda: The forms D and Q are predefined in pari/gp; D is called mfDelta; Q can be retrieved using mfEk(4); the product QD is done with F=mfmul(Q,D). I generated the first 1000 coefficients of both D and QD. The needed quadratic form is generated with Qfb(1,0,q). Then I iterated for every prime p up to 1000, computed tau(p) mod q, computed the kronecker symbol and, if p is a nontrivial solution of the quadratic form, also the quadratic form. ------ ------------ Last login: Mon May 10 17:41:31 on ttys001 languasc@languasc-pro ~ % gp Reading GPRC: /Users/languasc/.gprc GPRC Done. GP/PARI CALCULATOR Version 2.13.1 (released) arm64 running darwin (aarch64/GMP-6.2.1 kernel) 64-bit version compiled: Apr 16 2021, Apple clang version 12.0.0 (clang-1200.0.32.29) threading engine: single (readline v8.1 enabled, extended help enabled) Copyright (C) 2000-2020 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisizemax = 2048000000, primelimit = 50000 ------------ q=23 ------------ ? q=23;quadform = Qfb(1,0,q); D = mfDelta(); V = mfcoefs(D,1000); forprime(p=2,1000, if (qfbsolve(quadform,p) <> [],print1("p = ",p,"-> ",qfbsolve(quadform,p))); print(" p = ",p," -> ",kronecker(p,q), "; tau(p)(mod q) = ",V[p+1]%q)) p = 2 -> 1; tau(p)(mod q) = 22 p = 3 -> 1; tau(p)(mod q) = 22 p = 5 -> -1; tau(p)(mod q) = 0 p = 7 -> -1; tau(p)(mod q) = 0 p = 11 -> -1; tau(p)(mod q) = 0 p = 13 -> 1; tau(p)(mod q) = 22 p = 17 -> -1; tau(p)(mod q) = 0 p = 19 -> -1; tau(p)(mod q) = 0 p = 23-> [0, -1] p = 23 -> 0; tau(p)(mod q) = 1 p = 29 -> 1; tau(p)(mod q) = 22 p = 31 -> 1; tau(p)(mod q) = 22 p = 37 -> -1; tau(p)(mod q) = 0 p = 41 -> 1; tau(p)(mod q) = 22 p = 43 -> -1; tau(p)(mod q) = 0 p = 47 -> 1; tau(p)(mod q) = 22 p = 53 -> -1; tau(p)(mod q) = 0 p = 59-> [6, -1] p = 59 -> 1; tau(p)(mod q) = 2 p = 61 -> -1; tau(p)(mod q) = 0 p = 67 -> -1; tau(p)(mod q) = 0 p = 71 -> 1; tau(p)(mod q) = 22 p = 73 -> 1; tau(p)(mod q) = 22 p = 79 -> -1; tau(p)(mod q) = 0 p = 83 -> -1; tau(p)(mod q) = 0 p = 89 -> -1; tau(p)(mod q) = 0 p = 97 -> -1; tau(p)(mod q) = 0 p = 101-> [-3, 2] p = 101 -> 1; tau(p)(mod q) = 2 p = 103 -> -1; tau(p)(mod q) = 0 p = 107 -> -1; tau(p)(mod q) = 0 p = 109 -> -1; tau(p)(mod q) = 0 p = 113 -> -1; tau(p)(mod q) = 0 p = 127 -> 1; tau(p)(mod q) = 22 p = 131 -> 1; tau(p)(mod q) = 22 p = 137 -> -1; tau(p)(mod q) = 0 p = 139 -> 1; tau(p)(mod q) = 22 p = 149 -> -1; tau(p)(mod q) = 0 p = 151 -> 1; tau(p)(mod q) = 22 p = 157 -> -1; tau(p)(mod q) = 0 p = 163 -> 1; tau(p)(mod q) = 22 p = 167-> [12, -1] p = 167 -> 1; tau(p)(mod q) = 2 p = 173-> [-9, 2] p = 173 -> 1; tau(p)(mod q) = 2 p = 179 -> 1; tau(p)(mod q) = 22 p = 181 -> -1; tau(p)(mod q) = 0 p = 191 -> -1; tau(p)(mod q) = 0 p = 193 -> 1; tau(p)(mod q) = 22 p = 197 -> 1; tau(p)(mod q) = 22 p = 199 -> -1; tau(p)(mod q) = 0 p = 211-> [2, 3] p = 211 -> 1; tau(p)(mod q) = 2 p = 223-> [-4, 3] p = 223 -> 1; tau(p)(mod q) = 2 p = 227 -> -1; tau(p)(mod q) = 0 p = 229 -> -1; tau(p)(mod q) = 0 p = 233 -> 1; tau(p)(mod q) = 22 p = 239 -> 1; tau(p)(mod q) = 22 p = 241 -> -1; tau(p)(mod q) = 0 p = 251 -> -1; tau(p)(mod q) = 0 p = 257 -> 1; tau(p)(mod q) = 22 p = 263 -> -1; tau(p)(mod q) = 0 p = 269 -> 1; tau(p)(mod q) = 22 p = 271-> [8, 3] p = 271 -> 1; tau(p)(mod q) = 2 p = 277 -> 1; tau(p)(mod q) = 22 p = 281 -> -1; tau(p)(mod q) = 0 p = 283 -> -1; tau(p)(mod q) = 0 p = 293 -> -1; tau(p)(mod q) = 0 p = 307-> [-10, 3] p = 307 -> 1; tau(p)(mod q) = 2 p = 311 -> 1; tau(p)(mod q) = 22 p = 313 -> -1; tau(p)(mod q) = 0 p = 317-> [-15, 2] p = 317 -> 1; tau(p)(mod q) = 2 p = 331 -> 1; tau(p)(mod q) = 22 p = 337 -> -1; tau(p)(mod q) = 0 p = 347-> [18, -1] p = 347 -> 1; tau(p)(mod q) = 2 p = 349 -> 1; tau(p)(mod q) = 22 p = 353 -> 1; tau(p)(mod q) = 22 p = 359 -> -1; tau(p)(mod q) = 0 p = 367 -> -1; tau(p)(mod q) = 0 p = 373 -> -1; tau(p)(mod q) = 0 p = 379 -> -1; tau(p)(mod q) = 0 p = 383 -> -1; tau(p)(mod q) = 0 p = 389 -> -1; tau(p)(mod q) = 0 p = 397 -> 1; tau(p)(mod q) = 22 p = 401 -> -1; tau(p)(mod q) = 0 p = 409 -> 1; tau(p)(mod q) = 22 p = 419 -> -1; tau(p)(mod q) = 0 p = 421 -> -1; tau(p)(mod q) = 0 p = 431 -> -1; tau(p)(mod q) = 0 p = 433 -> -1; tau(p)(mod q) = 0 p = 439 -> 1; tau(p)(mod q) = 22 p = 443 -> 1; tau(p)(mod q) = 22 p = 449-> [-9, -4] p = 449 -> 1; tau(p)(mod q) = 2 p = 457 -> -1; tau(p)(mod q) = 0 p = 461 -> 1; tau(p)(mod q) = 22 p = 463-> [-16, 3] p = 463 -> 1; tau(p)(mod q) = 2 p = 467 -> -1; tau(p)(mod q) = 0 p = 479 -> -1; tau(p)(mod q) = 0 p = 487 -> 1; tau(p)(mod q) = 22 p = 491 -> 1; tau(p)(mod q) = 22 p = 499 -> 1; tau(p)(mod q) = 22 p = 503 -> -1; tau(p)(mod q) = 0 p = 509 -> 1; tau(p)(mod q) = 22 p = 521 -> -1; tau(p)(mod q) = 0 p = 523 -> -1; tau(p)(mod q) = 0 p = 541 -> 1; tau(p)(mod q) = 22 p = 547 -> 1; tau(p)(mod q) = 22 p = 557 -> -1; tau(p)(mod q) = 0 p = 563 -> -1; tau(p)(mod q) = 0 p = 569 -> -1; tau(p)(mod q) = 0 p = 571 -> -1; tau(p)(mod q) = 0 p = 577 -> 1; tau(p)(mod q) = 22 p = 587 -> 1; tau(p)(mod q) = 22 p = 593-> [15, 4] p = 593 -> 1; tau(p)(mod q) = 2 p = 599-> [24, -1] p = 599 -> 1; tau(p)(mod q) = 2 p = 601 -> 1; tau(p)(mod q) = 22 p = 607-> [20, 3] p = 607 -> 1; tau(p)(mod q) = 2 p = 613 -> -1; tau(p)(mod q) = 0 p = 617 -> -1; tau(p)(mod q) = 0 p = 619 -> -1; tau(p)(mod q) = 0 p = 631 -> -1; tau(p)(mod q) = 0 p = 641 -> -1; tau(p)(mod q) = 0 p = 643 -> -1; tau(p)(mod q) = 0 p = 647 -> 1; tau(p)(mod q) = 22 p = 653 -> 1; tau(p)(mod q) = 22 p = 659 -> -1; tau(p)(mod q) = 0 p = 661 -> -1; tau(p)(mod q) = 0 p = 673 -> 1; tau(p)(mod q) = 22 p = 677 -> -1; tau(p)(mod q) = 0 p = 683 -> 1; tau(p)(mod q) = 22 p = 691-> [-22, 3] p = 691 -> 1; tau(p)(mod q) = 2 p = 701 -> -1; tau(p)(mod q) = 0 p = 709 -> -1; tau(p)(mod q) = 0 p = 719-> [12, 5] p = 719 -> 1; tau(p)(mod q) = 2 p = 727 -> -1; tau(p)(mod q) = 0 p = 733 -> -1; tau(p)(mod q) = 0 p = 739 -> 1; tau(p)(mod q) = 22 p = 743 -> -1; tau(p)(mod q) = 0 p = 751 -> -1; tau(p)(mod q) = 0 p = 757 -> -1; tau(p)(mod q) = 0 p = 761 -> 1; tau(p)(mod q) = 22 p = 769 -> -1; tau(p)(mod q) = 0 p = 773 -> -1; tau(p)(mod q) = 0 p = 787 -> -1; tau(p)(mod q) = 0 p = 797 -> -1; tau(p)(mod q) = 0 p = 809-> [-21, -4] p = 809 -> 1; tau(p)(mod q) = 2 p = 811 -> 1; tau(p)(mod q) = 22 p = 821-> [-27, 2] p = 821 -> 1; tau(p)(mod q) = 2 p = 823 -> 1; tau(p)(mod q) = 22 p = 827 -> -1; tau(p)(mod q) = 0 p = 829-> [-1, -6] p = 829 -> 1; tau(p)(mod q) = 2 p = 839 -> -1; tau(p)(mod q) = 0 p = 853-> [5, -6] p = 853 -> 1; tau(p)(mod q) = 2 p = 857 -> 1; tau(p)(mod q) = 22 p = 859 -> 1; tau(p)(mod q) = 22 p = 863 -> 1; tau(p)(mod q) = 22 p = 877-> [-7, -6] p = 877 -> 1; tau(p)(mod q) = 2 p = 881 -> -1; tau(p)(mod q) = 0 p = 883-> [26, 3] p = 883 -> 1; tau(p)(mod q) = 2 p = 887 -> 1; tau(p)(mod q) = 22 p = 907 -> -1; tau(p)(mod q) = 0 p = 911 -> -1; tau(p)(mod q) = 0 p = 919 -> -1; tau(p)(mod q) = 0 p = 929 -> 1; tau(p)(mod q) = 22 p = 937 -> -1; tau(p)(mod q) = 0 p = 941 -> -1; tau(p)(mod q) = 0 p = 947 -> 1; tau(p)(mod q) = 22 p = 953 -> -1; tau(p)(mod q) = 0 p = 967 -> 1; tau(p)(mod q) = 22 p = 971 -> -1; tau(p)(mod q) = 0 p = 977 -> -1; tau(p)(mod q) = 0 p = 983 -> -1; tau(p)(mod q) = 0 p = 991-> [-28, 3] p = 991 -> 1; tau(p)(mod q) = 2 p = 997-> [-13, -6] p = 997 -> 1; tau(p)(mod q) = 2 ------------ q=31 ------------------------- ? q=31;quadform = Qfb(1,0,q); D = mfDelta(); Q = mfEk(4);F=mfmul(Q,D); V = mfcoefs(F,1000); forprime(p=2,1000, if (qfbsolve(quadform,p) <> [],print1("p = ",p,"-> ",qfbsolve(quadform,p))); print(" p = ",p," -> ",kronecker(p,q), "; tau(p)(mod q) = ",V[p+1]%q)) p = 2 -> 1; tau(p)(mod q) = 30 p = 3 -> -1; tau(p)(mod q) = 0 p = 5 -> 1; tau(p)(mod q) = 30 p = 7 -> 1; tau(p)(mod q) = 30 p = 11 -> -1; tau(p)(mod q) = 0 p = 13 -> -1; tau(p)(mod q) = 0 p = 17 -> -1; tau(p)(mod q) = 0 p = 19 -> 1; tau(p)(mod q) = 30 p = 23 -> -1; tau(p)(mod q) = 0 p = 29 -> -1; tau(p)(mod q) = 0 p = 31-> [0, -1] p = 31 -> 0; tau(p)(mod q) = 1 p = 37 -> -1; tau(p)(mod q) = 0 p = 41 -> 1; tau(p)(mod q) = 30 p = 43 -> -1; tau(p)(mod q) = 0 p = 47-> [4, -1] p = 47 -> 1; tau(p)(mod q) = 2 p = 53 -> -1; tau(p)(mod q) = 0 p = 59 -> 1; tau(p)(mod q) = 30 p = 61 -> -1; tau(p)(mod q) = 0 p = 67-> [6, -1] p = 67 -> 1; tau(p)(mod q) = 2 p = 71 -> 1; tau(p)(mod q) = 30 p = 73 -> -1; tau(p)(mod q) = 0 p = 79 -> -1; tau(p)(mod q) = 0 p = 83 -> -1; tau(p)(mod q) = 0 p = 89 -> -1; tau(p)(mod q) = 0 p = 97 -> 1; tau(p)(mod q) = 30 p = 101 -> 1; tau(p)(mod q) = 30 p = 103 -> 1; tau(p)(mod q) = 30 p = 107 -> 1; tau(p)(mod q) = 30 p = 109 -> 1; tau(p)(mod q) = 30 p = 113 -> 1; tau(p)(mod q) = 30 p = 127 -> -1; tau(p)(mod q) = 0 p = 131-> [10, -1] p = 131 -> 1; tau(p)(mod q) = 2 p = 137 -> -1; tau(p)(mod q) = 0 p = 139 -> -1; tau(p)(mod q) = 0 p = 149-> [-5, 2] p = 149 -> 1; tau(p)(mod q) = 2 p = 151 -> -1; tau(p)(mod q) = 0 p = 157 -> 1; tau(p)(mod q) = 30 p = 163 -> 1; tau(p)(mod q) = 30 p = 167 -> -1; tau(p)(mod q) = 0 p = 173-> [-7, 2] p = 173 -> 1; tau(p)(mod q) = 2 p = 179 -> -1; tau(p)(mod q) = 0 p = 181 -> -1; tau(p)(mod q) = 0 p = 191 -> 1; tau(p)(mod q) = 30 p = 193 -> 1; tau(p)(mod q) = 30 p = 197 -> -1; tau(p)(mod q) = 0 p = 199 -> -1; tau(p)(mod q) = 0 p = 211 -> 1; tau(p)(mod q) = 30 p = 223 -> -1; tau(p)(mod q) = 0 p = 227-> [14, -1] p = 227 -> 1; tau(p)(mod q) = 2 p = 229 -> -1; tau(p)(mod q) = 0 p = 233 -> 1; tau(p)(mod q) = 30 p = 239 -> -1; tau(p)(mod q) = 0 p = 241 -> -1; tau(p)(mod q) = 0 p = 251 -> -1; tau(p)(mod q) = 0 p = 257 -> 1; tau(p)(mod q) = 30 p = 263 -> -1; tau(p)(mod q) = 0 p = 269 -> -1; tau(p)(mod q) = 0 p = 271 -> -1; tau(p)(mod q) = 0 p = 277 -> -1; tau(p)(mod q) = 0 p = 281 -> 1; tau(p)(mod q) = 30 p = 283-> [2, 3] p = 283 -> 1; tau(p)(mod q) = 2 p = 293-> [-13, 2] p = 293 -> 1; tau(p)(mod q) = 2 p = 307 -> 1; tau(p)(mod q) = 30 p = 311 -> 1; tau(p)(mod q) = 30 p = 313 -> -1; tau(p)(mod q) = 0 p = 317 -> 1; tau(p)(mod q) = 30 p = 331 -> -1; tau(p)(mod q) = 0 p = 337 -> -1; tau(p)(mod q) = 0 p = 347 -> -1; tau(p)(mod q) = 0 p = 349-> [-15, 2] p = 349 -> 1; tau(p)(mod q) = 2 p = 353 -> -1; tau(p)(mod q) = 0 p = 359 -> 1; tau(p)(mod q) = 30 p = 367 -> -1; tau(p)(mod q) = 0 p = 373 -> 1; tau(p)(mod q) = 30 p = 379-> [-10, 3] p = 379 -> 1; tau(p)(mod q) = 2 p = 383 -> -1; tau(p)(mod q) = 0 p = 389 -> -1; tau(p)(mod q) = 0 p = 397 -> 1; tau(p)(mod q) = 30 p = 401 -> -1; tau(p)(mod q) = 0 p = 409 -> -1; tau(p)(mod q) = 0 p = 419 -> 1; tau(p)(mod q) = 30 p = 421 -> 1; tau(p)(mod q) = 30 p = 431-> [20, -1] p = 431 -> 1; tau(p)(mod q) = 2 p = 433 -> -1; tau(p)(mod q) = 0 p = 439 -> 1; tau(p)(mod q) = 30 p = 443 -> 1; tau(p)(mod q) = 30 p = 449 -> -1; tau(p)(mod q) = 0 p = 457 -> -1; tau(p)(mod q) = 0 p = 461 -> -1; tau(p)(mod q) = 0 p = 463 -> -1; tau(p)(mod q) = 0 p = 467 -> 1; tau(p)(mod q) = 30 p = 479 -> 1; tau(p)(mod q) = 30 p = 487 -> -1; tau(p)(mod q) = 0 p = 491 -> -1; tau(p)(mod q) = 0 p = 499 -> -1; tau(p)(mod q) = 0 p = 503 -> 1; tau(p)(mod q) = 30 p = 509 -> -1; tau(p)(mod q) = 0 p = 521-> [-5, -4] p = 521 -> 1; tau(p)(mod q) = 2 p = 523 -> -1; tau(p)(mod q) = 0 p = 541 -> 1; tau(p)(mod q) = 30 p = 547 -> 1; tau(p)(mod q) = 30 p = 557 -> -1; tau(p)(mod q) = 0 p = 563 -> 1; tau(p)(mod q) = 30 p = 569 -> -1; tau(p)(mod q) = 0 p = 571 -> -1; tau(p)(mod q) = 0 p = 577-> [-9, -4] p = 577 -> 1; tau(p)(mod q) = 2 p = 587 -> -1; tau(p)(mod q) = 0 p = 593 -> 1; tau(p)(mod q) = 30 p = 599 -> 1; tau(p)(mod q) = 30 p = 601 -> -1; tau(p)(mod q) = 0 p = 607-> [24, -1] p = 607 -> 1; tau(p)(mod q) = 2 p = 613 -> -1; tau(p)(mod q) = 0 p = 617-> [11, 4] p = 617 -> 1; tau(p)(mod q) = 2 p = 619 -> -1; tau(p)(mod q) = 0 p = 631 -> -1; tau(p)(mod q) = 0 p = 641 -> -1; tau(p)(mod q) = 0 p = 643 -> -1; tau(p)(mod q) = 0 p = 647 -> -1; tau(p)(mod q) = 0 p = 653-> [-23, 2] p = 653 -> 1; tau(p)(mod q) = 2 p = 659 -> 1; tau(p)(mod q) = 30 p = 661 -> 1; tau(p)(mod q) = 30 p = 673 -> -1; tau(p)(mod q) = 0 p = 677 -> -1; tau(p)(mod q) = 0 p = 683 -> 1; tau(p)(mod q) = 30 p = 691 -> 1; tau(p)(mod q) = 30 p = 701 -> 1; tau(p)(mod q) = 30 p = 709 -> -1; tau(p)(mod q) = 0 p = 719 -> -1; tau(p)(mod q) = 0 p = 727 -> 1; tau(p)(mod q) = 30 p = 733 -> 1; tau(p)(mod q) = 30 p = 739 -> -1; tau(p)(mod q) = 0 p = 743 -> -1; tau(p)(mod q) = 0 p = 751 -> 1; tau(p)(mod q) = 30 p = 757 -> -1; tau(p)(mod q) = 0 p = 761 -> -1; tau(p)(mod q) = 0 p = 769 -> 1; tau(p)(mod q) = 30 p = 773 -> -1; tau(p)(mod q) = 0 p = 787 -> -1; tau(p)(mod q) = 0 p = 797 -> -1; tau(p)(mod q) = 0 p = 809 -> -1; tau(p)(mod q) = 0 p = 811-> [-6, -5] p = 811 -> 1; tau(p)(mod q) = 2 p = 821 -> -1; tau(p)(mod q) = 0 p = 823 -> -1; tau(p)(mod q) = 0 p = 827 -> -1; tau(p)(mod q) = 0 p = 829 -> -1; tau(p)(mod q) = 0 p = 839-> [8, -5] p = 839 -> 1; tau(p)(mod q) = 2 p = 853-> [-27, 2] p = 853 -> 1; tau(p)(mod q) = 2 p = 857-> [19, 4] p = 857 -> 1; tau(p)(mod q) = 2 p = 859 -> -1; tau(p)(mod q) = 0 p = 863 -> -1; tau(p)(mod q) = 0 p = 877 -> 1; tau(p)(mod q) = 30 p = 881 -> -1; tau(p)(mod q) = 0 p = 883 -> -1; tau(p)(mod q) = 0 p = 887 -> 1; tau(p)(mod q) = 30 p = 907 -> 1; tau(p)(mod q) = 30 p = 911 -> -1; tau(p)(mod q) = 0 p = 919-> [12, 5] p = 919 -> 1; tau(p)(mod q) = 2 p = 929 -> -1; tau(p)(mod q) = 0 p = 937-> [-21, -4] p = 937 -> 1; tau(p)(mod q) = 2 p = 941 -> -1; tau(p)(mod q) = 0 p = 947 -> -1; tau(p)(mod q) = 0 p = 953 -> -1; tau(p)(mod q) = 0 p = 967 -> -1; tau(p)(mod q) = 0 p = 971-> [14, -5] p = 971 -> 1; tau(p)(mod q) = 2 p = 977 -> 1; tau(p)(mod q) = 30 p = 983 -> -1; tau(p)(mod q) = 0 p = 991 -> -1; tau(p)(mod q) = 0 p = 997 -> 1; tau(p)(mod q) = 30 --------- ? q=691; k =12; F = mfDelta(); V = mfcoefs(F,q+1);forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 1 q = 691; tau_k(q)(mod q) = 1 time = 6 ms. ? q=3617; k =16; D = mfDelta(); Q = mfEk(4); F=mfmul(Q,D); V = mfcoefs(F,q+1); forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 0 p = 13; tau_k(p)(mod p) = 0 q = 3617; tau_k(q)(mod q) = 1 time = 13 ms. ? q=43867; k =18; D = mfDelta(); R = mfEk(6); F=mfmul(R,D); V = mfcoefs(F,q+1); forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 0 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 1 q = 43867; tau_k(q)(mod q) = 1 time = 76 ms. ? q=283; k =20; D = mfDelta(); Q = mfEk(4); F=mfmul(mfpow(Q,2),D); V = mfcoefs(F,q+1); forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 0 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 0 p = 19; tau_k(p)(mod p) = 1 q = 283; tau_k(q)(mod q) = 1 time = 2 ms. ? q=617; k =20; D = mfDelta(); Q = mfEk(4); F=mfmul(mfpow(Q,2),D); V = mfcoefs(F,q+1); forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 0 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 0 p = 19; tau_k(p)(mod p) = 1 q = 617; tau_k(q)(mod q) = 1 time = 3 ms. ? q=131; k =22; D = mfDelta(); Q = mfEk(4); R = mfEk(6); F=mfmul(mfmul(Q,R),D); V = mfcoefs(F,q+1);forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 1 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 0 p = 19; tau_k(p)(mod p) = 0 q = 131; tau_k(q)(mod q) = 1 time = 1 ms. ? q=593; k =22; D = mfDelta(); Q = mfEk(4); R = mfEk(6); F=mfmul(mfmul(Q,R),D); V = mfcoefs(F,q+1);forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 1 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 0 p = 19; tau_k(p)(mod p) = 0 q = 593; tau_k(q)(mod q) = 1 time = 4 ms. ? q=657931; k =26; D = mfDelta(); Q = mfEk(4); R = mfEk(6); F=mfmul(mfmul(mfpow(Q,2),R),D); V = mfcoefs(F,q+1);forprime(p=2,k,print("p = ", p, "; tau_k(p)(mod p) = ", V[p+1]%p)); print("q = ", q, "; tau_k(q)(mod q) = ", V[q+1]%q) p = 2; tau_k(p)(mod p) = 0 p = 3; tau_k(p)(mod p) = 0 p = 5; tau_k(p)(mod p) = 0 p = 7; tau_k(p)(mod p) = 0 p = 11; tau_k(p)(mod p) = 0 p = 13; tau_k(p)(mod p) = 0 p = 17; tau_k(p)(mod p) = 0 p = 19; tau_k(p)(mod p) = 0 p = 23; tau_k(p)(mod p) = 0 q = 657931; tau_k(q)(mod q) = 1 time = 4,305 ms.