// Make newform 2352.3.m.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2352_m();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2352_m_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_3_m_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2352_3_m_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [81, 9, 10, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0], [36, -10, 10, -1], [14, 0, 0, 1], [9, 38, -2, 2]]; Rf_basisdens := [1, 45, 5, 9]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2352_m();" function MakeCharacter_2352_m() N := 2352; order := 2; char_gens := [1471, 1765, 785, 2257]; v := [1, 2, 2, 2]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2352_m_Hecke();" function MakeCharacter_2352_m_Hecke(Kf) N := 2352; order := 2; char_gens := [1471, 1765, 785, 2257]; char_values := [[-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 3; raw_aps := [[0, 0, 0, 0], [0, 1, 0, 0], [-3, 0, -1, 0], [0, 0, 0, 0], [0, 1, 0, 1], [0, 0, 0, 0], [9, 0, -1, 0], [0, 2, 0, 2], [0, 11, 0, -1], [12, 0, 6, 0], [0, -10, 0, 2], [16, 0, -6, 0], [-33, 0, 1, 0], [0, -16, 0, 0], [0, 12, 0, -4], [48, 0, 6, 0], [0, 0, 0, -4], [-66, 0, -6, 0], [0, 6, 0, 6], [0, -7, 0, 5], [6, 0, -6, 0], [0, 50, 0, -6], [0, 24, 0, 4], [-69, 0, 13, 0], [-150, 0, 6, 0], [63, 0, 13, 0], [0, -22, 0, 14], [0, -15, 0, 9], [-70, 0, -12, 0], [-42, 0, -12, 0], [0, -14, 0, -6], [0, 108, 0, 0], [6, 0, -24, 0], [0, 48, 0, -12], [48, 0, 6, 0], [0, -8, 0, -24], [102, 0, -30, 0], [0, -62, 0, 18], [0, 108, 0, -4], [-213, 0, 1, 0], [0, 51, 0, 3], [-168, 0, 24, 0], [0, -59, 0, -23], [-40, 0, -18, 0], [216, 0, -18, 0], [0, -16, 0, 8], [0, 12, 0, 12], [0, 172, 0, 4], [0, -120, 0, -12], [-180, 0, 36, 0], [-306, 0, 0, 0], [0, 159, 0, 3], [102, 0, 42, 0], [0, -12, 0, 16], [-141, 0, -11, 0], [0, 131, 0, -1], [69, 0, -25, 0], [0, 102, 0, -6], [-104, 0, 30, 0], [162, 0, 36, 0], [0, -138, 0, -18], [-297, 0, 13, 0], [0, -10, 0, 14], [0, 180, 0, -4], [-120, 0, -24, 0], [36, 0, -54, 0], [0, 92, 0, -36], [-10, 0, 24, 0], [0, 223, 0, 7], [6, 0, 66, 0], [-75, 0, -13, 0], [0, -111, 0, 21], [0, 272, 0, -16], [26, 0, -48, 0], [0, -92, 0, -12], [0, -48, 0, 56], [492, 0, -42, 0], [294, 0, 66, 0], [-330, 0, 24, 0], [294, 0, -6, 0], [0, -132, 0, -56], [64, 0, -42, 0], [0, 325, 0, 25], [-84, 0, -60, 0], [0, -84, 0, -60], [0, -13, 0, -37], [306, 0, 0, 0], [298, 0, -24, 0], [255, 0, -59, 0], [0, -146, 0, -42], [0, 108, 0, 0], [0, 192, 0, -8], [0, -12, 0, -12], [0, 355, 0, -29], [0, 268, 0, 12], [0, 60, 0, 68], [117, 0, -49, 0], [-507, 0, -61, 0], [0, 20, 0, -40], [376, 0, 30, 0], [0, -298, 0, 6], [240, 0, -42, 0], [0, 48, 0, -60], [-150, 0, -84, 0], [0, -166, 0, -54], [60, 0, 84, 0], [0, -336, 0, -12], [753, 0, 47, 0], [0, 517, 0, 1], [228, 0, -84, 0], [0, -180, 0, -12], [-266, 0, -96, 0], [402, 0, -60, 0], [0, -296, 0, 4], [0, -110, 0, 42], [282, 0, -84, 0], [0, -406, 0, 2], [0, 84, 0, 84], [60, 0, -42, 0], [0, 109, 0, 85], [-882, 0, -54, 0], [-32, 0, 90, 0], [471, 0, -107, 0], [0, 575, 0, 23], [0, 156, 0, 48], [60, 0, -42, 0], [-886, 0, -12, 0], [0, -72, 0, -64], [0, 330, 0, -18], [768, 0, -48, 0], [0, -78, 0, -78], [0, 5, 0, 41], [0, 432, 0, 0], [170, 0, 180, 0], [387, 0, -59, 0], [1116, 0, 36, 0], [-465, 0, -83, 0], [0, -240, 0, 60], [615, 0, -11, 0], [-714, 0, 84, 0], [0, -356, 0, -32], [-216, 0, -162, 0], [0, -386, 0, 6], [0, 557, 0, -19], [168, 0, -24, 0], [0, 72, 0, -16], [678, 0, -30, 0], [75, 0, 157, 0], [0, -286, 0, 26], [0, -321, 0, -69], [-482, 0, -72, 0], [213, 0, -229, 0], [0, -620, 0, 36], [0, -612, 0, -4], [0, -416, 0, 0], [0, 525, 0, -15], [0, -140, 0, -60], [459, 0, 133, 0], [24, 0, -24, 0], [-147, 0, 215, 0], [0, -223, 0, -31], [522, 0, 72, 0], [0, 330, 0, 114], [0, 192, 0, -68], [-210, 0, -96, 0], [0, 564, 0, -28], [0, -172, 0, 36], [-234, 0, -270, 0], [124, 0, -222, 0], [183, 0, 109, 0], [0, 165, 0, 45], [-18, 0, 234, 0], [0, 213, 0, 33], [-436, 0, -138, 0], [0, 416, 0, 104], [969, 0, 71, 0], [0, -90, 0, -90], [348, 0, -186, 0], [0, 478, 0, -62], [-738, 0, -54, 0], [0, -130, 0, -42], [0, -252, 0, 16], [272, 0, -246, 0], [261, 0, -37, 0], [0, 427, 0, -113], [615, 0, 13, 0], [-866, 0, 36, 0], [0, 108, 0, 48], [1388, 0, 6, 0], [0, 372, 0, 148], [6, 0, -6, 0], [0, -183, 0, -39], [0, 136, 0, 72], [99, 0, -191, 0], [0, 133, 0, -107], [213, 0, 107, 0], [-344, 0, -222, 0], [-1024, 0, -54, 0], [-1365, 0, -107, 0], [0, -204, 0, -68], [132, 0, -150, 0], [0, 220, 0, -20], [-492, 0, 60, 0], [978, 0, 30, 0], [0, -396, 0, 152], [-333, 0, -287, 0], [0, 924, 0, 12], [0, -547, 0, 77], [150, 0, -60, 0], [0, -822, 0, 18], [358, 0, -168, 0], [-435, 0, 119, 0], [0, -1180, 0, -12], [0, -36, 0, 16], [0, -408, 0, -152], [408, 0, 312, 0], [0, 692, 0, -60], [825, 0, -73, 0], [0, 407, 0, -109], [-1548, 0, -18, 0], [-238, 0, -36, 0], [0, 642, 0, 78], [-1062, 0, -216, 0], [0, -160, 0, 48], [0, 384, 0, -116], [832, 0, 138, 0], [-1077, 0, 85, 0], [0, -209, 0, 43], [0, 254, 0, -190], [0, 1039, 0, 7], [-286, 0, 60, 0], [0, -98, 0, -146], [0, 692, 0, 84], [-1206, 0, 216, 0], [0, 926, 0, -10], [0, 456, 0, 24], [-726, 0, 150, 0], [552, 0, -354, 0], [0, -525, 0, -117], [0, 396, 0, 116], [0, -209, 0, 151], [0, 436, 0, -200], [0, -140, 0, -20], [-1408, 0, -138, 0], [-1545, 0, 97, 0], [0, 0, 0, 200], [0, 278, 0, -22], [0, 648, 0, 4], [0, -518, 0, 90], [0, -1025, 0, 19], [886, 0, 252, 0], [-675, 0, 83, 0], [0, -783, 0, 21], [-330, 0, 42, 0], [-1401, 0, -203, 0], [0, 737, 0, -55], [128, 0, -102, 0], [0, -110, 0, 10], [-909, 0, -47, 0], [-600, 0, 312, 0], [0, 978, 0, -102], [0, -253, 0, -205], [1812, 0, -84, 0], [-2106, 0, 18, 0], [-723, 0, -61, 0], [0, 1336, 0, -20], [48, 0, 150, 0], [-2295, 0, -49, 0], [0, 934, 0, 102], [156, 0, 6, 0], [-150, 0, -66, 0], [0, -168, 0, 264], [-1218, 0, -222, 0], [0, 68, 0, 180], [90, 0, 342, 0], [0, -542, 0, -98], [0, -403, 0, -43], [-304, 0, -378, 0], [-496, 0, -90, 0], [0, 456, 0, 4], [0, 684, 0, 4], [0, -1054, 0, -6], [0, 612, 0, 20], [-3162, 0, -78, 0], [0, -508, 0, -40], [0, -595, 0, 65], [1618, 0, -96, 0], [-1596, 0, -294, 0], [0, -1538, 0, 58], [543, 0, -263, 0], [-1212, 0, 330, 0], [0, 828, 0, 64], [-702, 0, 180, 0], [0, 696, 0, -92], [-238, 0, -264, 0], [2919, 0, -107, 0], [0, 1024, 0, -176], [-63, 0, 299, 0], [0, -360, 0, -116], [0, 56, 0, -4], [540, 0, 468, 0], [2424, 0, -174, 0], [0, 126, 0, -90]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2352_m_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_3_m_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_2352_3_m_e(:prec:=4) chi := MakeCharacter_2352_m(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2352_3_m_e();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_2352_3_m_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2352_m(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<5,R![-28, 6, 1]>],Snew); return Vf; end function;