
// Make newform 336.4.b.d in Magma, downloaded from the LMFDB on 28 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_b_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_336_4_b_d();".
// 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_336_4_b_d();".
// 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 := [6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_336_b();"
function MakeCharacter_336_b()
    N := 336;
    order := 2;
    char_gens := [127, 85, 113, 241];
    v := [1, 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_336_b_Hecke();"
function MakeCharacter_336_b_Hecke( : Kf := false)
    N := 336;
    order := 2;
    char_gens := [127, 85, 113, 241];
    char_values := [[-1, 0], [1, 0], [1, 0], [-1, 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 := 4;
    raw_aps := [[0, 0], [3, 0], [0, 1], [17, 3], [0, -23], [0, -30], [0, 21], [10, 0], [0, 47], [126, 0], [8, 0], [244, 0], [0, -151], [0, -66], [-180, 0], [594, 0], [-540, 0], [0, -144], [0, 432], [0, 73], [0, -270], [0, -420], [-864, 0], [0, 527], [0, 330], [0, 15], [1376, 0], [0, 483], [602, 0], [-2106, 0], [0, 420], [-576, 0], [-1674, 0], [308, 0], [-522, 0], [0, -558], [0, 1176], [0, 120], [-3564, 0], [0, -565], [0, 57], [0, -1266], [0, 275], [-2144, 0], [234, 0], [-2654, 0], [0, -942], [838, 0], [-4860, 0], [0, 6], [1458, 0], [0, 1311], [0, 330], [576, 0], [0, -1811], [0, 2213], [0, -2545], [5024, 0], [4024, 0], [630, 0], [-9002, 0], [0, 3735], [-5246, 0], [-8136, 0], [0, 204], [8838, 0], [0, 3702], [7982, 0], [0, -1759], [0, -1752], [0, 2555], [0, 3513], [-5570, 0], [-7282, 0], [0, -402], [11628, 0], [-4518, 0], [0, 2964], [10062, 0], [0, 2778], [12564, 0], [-3752, 0], [0, -2485], [0, 1356], [14506, 0], [0, -4841], [-7866, 0], [-6058, 0], [0, -4279], [0, 5604], [14472, 0], [-3816, 0], [0, 6042], [0, 551], [0, 1098], [-7596, 0], [0, 8031], [0, 4243], [-17956, 0], [-14192, 0], [0, -1116], [16254, 0], [3672, 0], [11466, 0], [0, 8520], [0, -8064], [18720, 0], [0, -671], [0, -4123], [0, 3048], [7198, 0], [-5662, 0], [-3546, 0], [8180, 0], [0, -984], [26010, 0], [4582, 0], [-5832, 0], [21474, 0], [0, -11405], [0, 7524], [14344, 0], [0, -10859], [0, -11189], [6716, 0], [-12546, 0], [-24562, 0], [-7704, 0], [22160, 0], [0, -6942], [0, -7500], [0, 9607], [0, 4422], [12206, 0], [0, -9651], [0, -11316], [0, 16441], [12188, 0], [0, -3719], [-19710, 0], [-8164, 0], [17226, 0], [0, -6792], [0, 16133], [0, -15870], [-33804, 0], [0, 2556], [0, 4997], [-40382, 0], [0, -3825], [578, 0], [0, 4859], [0, 19038], [-23724, 0], [0, -10614], [0, -18423], [0, 14418], [0, 11049], [0, -5424], [0, 3613], [0, -1387], [-41346, 0], [0, -17784], [-54252, 0], [13734, 0], [17172, 0], [0, -5598], [0, -5904]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_4_b_d();".
// 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_336_4_b_d(:prec:=2)
    chi := MakeCharacter_336_b();
    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_336_4_b_d();".
// 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_336_4_b_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![6, 0, 1]>,<19,R![-10, 1]>],Snew);
    return Vf;
end function;