
// Make newform 296.2.g.a in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_296_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_296_g_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_296_2_g_a();".
// 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_296_2_g_a();".
// 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 := [32, -112, 196, 12, 18, -46, 53, -6, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [50678928, 1163840, -246192, -7139728, 23521585, -3911172, 170662, 372086, 522953, -8394], [-31958752, -342952, 163578, 1785272, -9877902, 1186653, 916, -240502, -249836, -14595], [18017600, -360664, -48058, 2647768, 1155951, 1165407, -122834, 63448, 39435, 25917], [-6239328, 22596720, 9237001, 3015400, -3274916, 4718228, -128815, 92346, 17066, 82826], [17347792, 1694912, 469680, -13296400, -404339, -5385300, 714406, -648682, -321579, -167514], [18030008, -64988236, 2450206, -6742090, 9613982, -14066739, 2204940, 26952, -69536, -287075], [-33324560, 119806552, 5158156, 10328414, -17878973, 27406513, -2805784, 205098, 128413, 517347], [31656552, -111752580, 28736242, 3662562, 17880066, -31317849, 5583404, 393408, -205680, -656817], [32344528, -114554504, -12244675, 1358452, 18090832, -31405794, 3764909, 89418, -187342, -618220]];
    Rf_basisdens := [1, 10819708, 10819708, 5409854, 5409854, 10819708, 10819708, 10819708, 10819708, 5409854];
    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_296_g();"
function MakeCharacter_296_g()
    N := 296;
    order := 2;
    char_gens := [223, 149, 113];
    v := [2, 2, 1];
    // 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_296_g_Hecke();"
function MakeCharacter_296_g_Hecke( : Kf := false)
    N := 296;
    order := 2;
    char_gens := [223, 149, 113];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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<X, Y> := 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<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, -1], [0, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 2, 1, -1], [0, 0, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 0, 0, -1, 0, -1, -1, 0, -2], [-1, 0, -1, 0, 0, 1, 1, -1, 0, 0], [-2, -1, 1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 2, 0, 1, 0, 0, 0, 0, 0, 0], [-2, -1, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -1, 0, 0, 1], [0, 0, 0, 0, -1, 0, 0, 3, 0, 1], [0, 2, 2, -1, 0, 1, 0, 0, 0, 0], [0, -2, 0, 1, 0, -2, 0, 0, 0, 0], [-2, 1, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 1, -2], [-4, 0, 2, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 2, 1, 2], [0, 0, 0, 0, -3, 0, -1, -2, 3, -1], [-2, 1, 2, -3, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 1, -1, 2, -1], [0, -2, -3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -1, -2, -3, 2], [0, 0, 0, 0, 3, 0, 1, 2, -3, 1], [8, 0, 2, -3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, -1, -1], [-2, 0, -6, -1, 0, -1, 0, 0, 0, 0], [0, -2, -3, 1, 0, 0, 0, 0, 0, 0], [6, 1, 2, 3, 0, -1, 0, 0, 0, 0], [0, 2, -2, -2, 0, -2, 0, 0, 0, 0], [6, 1, -6, -3, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 1, -2, 2, 0], [0, 0, 0, 0, -1, 0, -2, -3, 3, -2], [6, 1, 4, 1, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -3, 0, 2, -3], [-2, 1, 4, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -2, -3, 3, -3], [0, 0, 0, 0, 0, 0, 3, 4, -1, 0], [6, -1, 0, -1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -2, -2, 1], [8, 0, 4, 2, 0, 1, 0, 0, 0, 0], [-8, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -4, -2, 1, 0], [6, -1, 2, -1, 0, -3, 0, 0, 0, 0], [-2, 2, 1, 3, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 2, 5, -1, 3], [0, 0, 0, 0, 1, 0, -1, -4, 3, 1], [0, 0, 0, 0, -3, 0, -3, 6, 2, 0], [0, 0, 0, 0, 3, 0, 3, -2, -1, -3], [0, -4, -6, 1, 0, -2, 0, 0, 0, 0], [6, 0, -2, 2, 0, -4, 0, 0, 0, 0], [-8, 4, 4, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 5, 1, -3, 2], [0, 0, 0, 0, 0, 0, -2, 0, 2, -4], [0, 0, 0, 0, 4, 0, 1, -6, -2, 3], [-2, 0, 4, 4, 0, -4, 0, 0, 0, 0], [8, -2, 1, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, -2, 2, -1], [0, 0, 0, 0, 3, 0, -2, -2, -2, 3], [6, 2, 2, -6, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 4, 4, -3, 4], [-10, 3, 4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -1, 2, 0, -1], [-10, 4, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -6, 0, 1], [0, 2, 6, -3, 0, 4, 0, 0, 0, 0], [8, 0, -2, -2, 0, 4, 0, 0, 0, 0], [6, 3, 4, 3, 0, 3, 0, 0, 0, 0], [-8, 0, 3, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 2, 2, -3, 0], [0, 0, 0, 0, -6, 0, -2, 0, 6, -7], [-10, -3, -2, -3, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -4, 2, 4, -3], [0, 0, 0, 0, 1, 0, 3, 4, -1, -1], [-8, -2, 1, 2, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 3, 0, -1, 5], [0, 0, 0, 0, -4, 0, 0, 6, -1, -3], [-10, 1, -11, -4, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 3, 10, 0, 2], [24, -2, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -3, -6, 1, -2], [0, 0, 0, 0, -4, 0, -3, 0, -3, 0], [0, 0, 0, 0, -1, 0, 0, -2, -2, 1], [0, 0, 0, 0, -2, 0, -3, -2, 2, 0], [0, 0, 0, 0, -3, 0, -2, -6, 3, -4], [0, 0, 0, 0, -4, 0, -1, -6, 0, 0], [0, 0, 0, 0, 4, 0, -2, -2, 1, -3], [-8, -6, 3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 4, -2, -7, 6], [0, 0, 0, 0, 1, 0, 0, -1, 1, 4], [-2, 5, 6, -1, 0, 3, 0, 0, 0, 0], [10, 3, -2, -5, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 3, 2, 4, -4], [0, 0, 0, 0, -4, 0, -7, 5, 5, -4], [0, 0, 0, 0, 7, 0, 4, -2, 1, -2], [0, 0, 0, 0, 8, 0, 5, 3, -3, 8], [0, 0, 0, 0, 2, 0, 2, -2, -5, 3], [0, 0, 0, 0, -5, 0, -3, -2, 1, -1], [0, -4, 5, 2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -3, -2, 5, -8], [0, 0, 0, 0, -5, 0, 3, 4, 0, 4], [6, -3, -8, 0, 0, 4, 0, 0, 0, 0], [-16, -2, -2, 9, 0, 0, 0, 0, 0, 0], [6, -4, 6, 7, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, -3, 2, -2, 3], [6, 3, 4, 3, 0, 1, 0, 0, 0, 0], [6, 1, 1, 2, 0, 1, 0, 0, 0, 0], [0, -2, 4, -4, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 5, 2, -2, -1], [6, -5, -5, -4, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 5, 12, -2, 3], [0, 0, 0, 0, 2, 0, 4, -6, 1, 0], [0, 0, 0, 0, 2, 0, -4, -4, -4, -5], [-8, 0, -3, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, -4, 1, 3], [6, -1, 7, 0, 0, -5, 0, 0, 0, 0], [-2, 5, -6, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -6, -3, -5], [-12, 2, -4, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 2, 11, 2, 1], [0, 0, 0, 0, 3, 0, -1, 1, -5, -3], [24, 2, 0, 3, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, -3, -3, -6, 7], [-2, 3, -10, -1, 0, -1, 0, 0, 0, 0], [24, 2, 4, 0, 0, 3, 0, 0, 0, 0], [-16, 4, 10, -1, 0, -2, 0, 0, 0, 0], [-8, 0, -2, 3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -2, 3, 2, -3], [-2, -1, -13, 4, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 5, -6, -3, -3], [-10, -3, -4, 3, 0, 1, 0, 0, 0, 0], [-4, 0, -6, -5, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 5, 0, 1, 4], [0, 0, 0, 0, -6, 0, 4, 0, 4, -8], [-8, 0, -8, 4, 0, -1, 0, 0, 0, 0], [22, 1, -4, -9, 0, 3, 0, 0, 0, 0], [-8, 6, 0, 0, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 2, 0, -1, -4], [0, 0, 0, 0, 3, 0, 1, -12, -3, 0], [-8, 2, -2, -8, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 1, -7, 1, -4], [0, 0, 0, 0, -5, 0, -8, -2, 0, -5], [0, 0, 0, 0, 0, 0, -2, -8, -1, 1], [-8, -2, -4, 2, 0, -8, 0, 0, 0, 0], [-10, -2, 10, 2, 0, -2, 0, 0, 0, 0], [-2, 0, 10, -1, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 3, 0, 0, -3], [0, -2, 8, 3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 3, 10, 0, 11], [0, 0, 0, 0, -8, 0, 0, -2, -1, -3], [0, 0, 0, 0, 1, 0, 1, -6, 4, 0], [-10, -2, -2, 3, 0, -3, 0, 0, 0, 0], [22, 3, -1, 2, 0, -1, 0, 0, 0, 0], [-10, 6, 10, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -1, -6, 0, -6], [-10, -1, -8, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -2, 13, -1, -1], [-24, 0, -10, -5, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -6, 4, -2, 0], [16, 0, 2, 3, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -1, -6, -4, -5], [0, 0, 0, 0, 9, 0, 7, -2, -3, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_296_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_296_2_g_a();".
// 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_296_2_g_a(:prec:=10)
    chi := MakeCharacter_296_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_296_2_g_a();".
// 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_296_2_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_296_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;