
// Make newform 126.4.d.a in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_126_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_126_d_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_126_4_d_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_126_4_d_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 := [50625, 0, 0, 0, 0, 0, 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, 0, 0, 2, 0, 0, 0], [0, 0, 225, 0, 0, 0, 1, 0], [0, 0, 0, -60, 0, -4, 0, 0], [0, 0, 450, 0, 0, 0, -2, 0], [0, 6750, 0, 0, 0, 0, 0, -2], [0, 0, 0, 30, 0, -2, 0, 0], [0, 3375, 225, 75, 0, 5, -1, 1]];
    Rf_basisdens := [1, 225, 1125, 225, 1125, 3375, 75, 1125];
    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_126_d();"
function MakeCharacter_126_d()
    N := 126;
    order := 2;
    char_gens := [29, 73];
    v := [1, 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_126_d_Hecke();"
function MakeCharacter_126_d_Hecke(Kf)
    N := 126;
    order := 2;
    char_gens := [29, 73];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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<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 := 4;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [5, 0, 0, 0, 0, 0, 0, 1], [0, -15, -7, 0, 0, 0, 0, 0], [0, 0, 0, 3, -1, 0, 0, 2], [0, 0, 0, 0, 0, -7, -2, 0], [0, 0, 0, 4, -1, 0, 0, 2], [0, 21, -5, 0, 0, 0, 0, 0], [0, -30, 29, 0, 0, 0, 0, 0], [0, 0, 0, -2, -2, 0, 0, 4], [-20, 0, 0, 0, -36, 0, 0, 0], [0, 0, 0, 0, 0, 7, 6, 0], [130, 0, 0, 0, 17, 0, 0, 0], [0, 0, 0, 0, 0, -28, 6, 0], [0, 6, -45, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -8, 16, 0], [0, 0, 0, -16, 12, 0, 0, -24], [-460, 0, 0, 0, -70, 0, 0, 0], [0, -255, 127, 0, 0, 0, 0, 0], [0, 0, 0, 9, 7, 0, 0, -14], [56, 0, 0, 0, -90, 0, 0, 0], [0, 0, 0, 0, 0, -32, -18, 0], [0, 0, 0, 0, 0, 33, -26, 0], [0, 0, 0, 31, 21, 0, 0, -42], [0, 0, 0, 0, 0, 105, -28, 0], [0, 0, 0, 42, -20, 0, 0, 40], [0, 333, -235, 0, 0, 0, 0, 0], [1300, 0, 0, 0, -10, 0, 0, 0], [0, -336, 135, 0, 0, 0, 0, 0], [-400, 0, 0, 0, 130, 0, 0, 0], [0, 0, 0, 0, 0, 36, 34, 0], [0, 72, 365, 0, 0, 0, 0, 0], [0, 0, 0, 8, -30, 0, 0, 60], [0, 1290, 81, 0, 0, 0, 0, 0], [970, 0, 0, 0, -75, 0, 0, 0], [0, 0, 0, -86, 10, 0, 0, -20], [-1060, 0, 0, 0, -330, 0, 0, 0], [0, 0, 0, 0, 0, 80, 106, 0], [0, 0, 0, 0, 0, 165, -4, 0], [0, -645, 559, 0, 0, 0, 0, 0], [0, 0, 0, -17, 111, 0, 0, -222], [0, 735, -647, 0, 0, 0, 0, 0], [1600, 0, 0, 0, 18, 0, 0, 0], [0, -618, -415, 0, 0, 0, 0, 0], [0, 0, 0, 36, 5, 0, 0, -10], [550, 0, 0, 0, 115, 0, 0, 0], [0, 0, 0, 36, -147, 0, 0, 294], [0, 0, 0, 0, 0, -300, 48, 0], [0, 0, 0, 187, -5, 0, 0, 10], [0, -876, -925, 0, 0, 0, 0, 0], [0, -945, -179, 0, 0, 0, 0, 0], [0, 0, 0, 45, -113, 0, 0, 226], [0, 0, 0, 0, 0, -208, -106, 0], [0, 0, 0, 0, 0, -351, 70, 0], [0, -639, 295, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 85, -88, 0], [0, 0, 0, 178, 78, 0, 0, -156], [950, 0, 0, 0, -230, 0, 0, 0], [0, 2040, -77, 0, 0, 0, 0, 0], [0, 0, 0, 152, 47, 0, 0, -94], [0, 0, 0, 0, 0, 151, -56, 0], [0, 0, 0, -188, 91, 0, 0, -182], [0, 0, 0, 0, 0, 344, -88, 0], [0, 0, 0, -92, -60, 0, 0, 120], [0, -1998, 975, 0, 0, 0, 0, 0], [-2270, 0, 0, 0, 645, 0, 0, 0], [520, 0, 0, 0, -1176, 0, 0, 0], [0, 1503, -85, 0, 0, 0, 0, 0], [0, 0, 0, 182, 30, 0, 0, -60], [0, 0, 0, 0, 0, -643, 38, 0], [0, -585, -539, 0, 0, 0, 0, 0], [0, 0, 0, -480, 53, 0, 0, -106], [-6910, 0, 0, 0, 540, 0, 0, 0], [4750, 0, 0, 0, -405, 0, 0, 0], [0, 0, 0, 0, 0, 16, -46, 0], [0, 1230, 731, 0, 0, 0, 0, 0], [0, 0, 0, -236, 208, 0, 0, -416], [0, -5040, -287, 0, 0, 0, 0, 0], [0, 0, 0, 73, -217, 0, 0, 434], [0, 0, 0, 0, 0, 140, -324, 0], [-262, 0, 0, 0, -130, 0, 0, 0], [0, -3945, -43, 0, 0, 0, 0, 0], [0, 0, 0, 198, -198, 0, 0, 396], [0, 0, 0, 428, -37, 0, 0, 74], [0, -3141, -1545, 0, 0, 0, 0, 0], [0, -2460, -1301, 0, 0, 0, 0, 0], [7360, 0, 0, 0, 756, 0, 0, 0], [0, 0, 0, 0, 0, -383, -80, 0], [16040, 0, 0, 0, -150, 0, 0, 0], [0, 0, 0, 0, 0, 1056, 118, 0], [0, 0, 0, 0, 0, 628, -232, 0], [-6650, 0, 0, 0, 151, 0, 0, 0], [0, -1395, -543, 0, 0, 0, 0, 0], [-10490, 0, 0, 0, 995, 0, 0, 0], [0, 0, 0, 0, 0, -1028, -106, 0], [0, 0, 0, 0, 0, -317, -388, 0], [0, 0, 0, 0, 0, -1937, 242, 0], [0, 0, 0, -12, -268, 0, 0, 536], [11918, 0, 0, 0, -1110, 0, 0, 0], [-13660, 0, 0, 0, -630, 0, 0, 0], [0, -5118, 485, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -336, -158, 0], [0, -5760, 1111, 0, 0, 0, 0, 0], [-8332, 0, 0, 0, -1170, 0, 0, 0], [0, 0, 0, -782, 98, 0, 0, -196], [0, 0, 0, 0, 0, -108, 290, 0], [0, 0, 0, 0, 0, -743, 354, 0], [0, 7335, 689, 0, 0, 0, 0, 0], [0, 0, 0, -534, -402, 0, 0, 804], [0, 0, 0, -832, 535, 0, 0, -1070], [11150, 0, 0, 0, 620, 0, 0, 0], [0, 2988, -3465, 0, 0, 0, 0, 0], [0, 0, 0, -140, 234, 0, 0, -468], [-11248, 0, 0, 0, 150, 0, 0, 0], [0, 8460, -553, 0, 0, 0, 0, 0], [0, 0, 0, 392, 317, 0, 0, -634], [0, 0, 0, 0, 0, 636, 172, 0], [0, 4446, 4685, 0, 0, 0, 0, 0], [0, -1425, 1351, 0, 0, 0, 0, 0], [0, 0, 0, -230, 294, 0, 0, -588], [-20600, 0, 0, 0, 58, 0, 0, 0], [0, 0, 0, 0, 0, 2103, -84, 0], [0, -12219, -2175, 0, 0, 0, 0, 0], [0, 0, 0, 864, -126, 0, 0, 252], [0, -7890, -4263, 0, 0, 0, 0, 0], [21580, 0, 0, 0, -570, 0, 0, 0], [0, 0, 0, 0, 0, -188, -436, 0], [0, 0, 0, -110, -176, 0, 0, 352], [0, 0, 0, 503, 107, 0, 0, -214], [-124, 0, 0, 0, -450, 0, 0, 0], [0, -4941, 1325, 0, 0, 0, 0, 0], [-8450, 0, 0, 0, -2645, 0, 0, 0], [-4700, 0, 0, 0, 2034, 0, 0, 0], [0, 0, 0, 0, 0, -581, 394, 0], [0, 0, 0, 154, -250, 0, 0, 500], [0, 0, 0, 0, 0, -247, -780, 0], [0, 0, 0, 296, 580, 0, 0, -1160], [0, 0, 0, 0, 0, 2379, -476, 0], [0, -11040, 2721, 0, 0, 0, 0, 0], [0, 0, 0, -172, -808, 0, 0, 1616], [0, -1110, 5493, 0, 0, 0, 0, 0], [-13120, 0, 0, 0, 2550, 0, 0, 0], [0, -5817, -45, 0, 0, 0, 0, 0], [0, 0, 0, 369, 677, 0, 0, -1354], [0, 0, 0, 0, 0, -1396, -634, 0], [0, 0, 0, -860, 564, 0, 0, -1128], [0, 0, 0, 0, 0, -289, 66, 0], [0, 0, 0, -448, 109, 0, 0, -218], [0, 19089, -1985, 0, 0, 0, 0, 0], [30890, 0, 0, 0, 240, 0, 0, 0], [0, 0, 0, 0, 0, -1461, -330, 0], [7450, 0, 0, 0, -1887, 0, 0, 0], [0, 0, 0, 0, 0, -1648, 514, 0], [-18110, 0, 0, 0, 861, 0, 0, 0], [0, -5835, 7907, 0, 0, 0, 0, 0], [-33530, 0, 0, 0, -985, 0, 0, 0], [0, 0, 0, 0, 0, 2693, 226, 0], [0, 0, 0, -632, 116, 0, 0, -232], [0, 0, 0, 0, 0, 1189, 796, 0], [0, -12363, 265, 0, 0, 0, 0, 0], [0, -3816, -1995, 0, 0, 0, 0, 0], [19880, 0, 0, 0, -3670, 0, 0, 0], [0, 0, 0, 0, 0, -764, 248, 0], [0, 4188, -7325, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1904, 754, 0], [-22430, 0, 0, 0, -1635, 0, 0, 0], [0, 0, 0, 604, 172, 0, 0, -344]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_126_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_126_4_d_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_126_4_d_a(:prec:=8)
    chi := MakeCharacter_126_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_126_4_d_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_126_4_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_126_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;