
// Make newform 336.4.q.c in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_q_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_q_c();".
// 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_q_c();".
// 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 := [1, -1, 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_q();"
function MakeCharacter_336_q()
    N := 336;
    order := 3;
    char_gens := [127, 85, 113, 241];
    v := [3, 3, 3, 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_336_q_Hecke();"
function MakeCharacter_336_q_Hecke( : Kf := false)
    N := 336;
    order := 3;
    char_gens := [127, 85, 113, 241];
    char_values := [[1, 0], [1, 0], [1, 0], [0, -1]];
    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, 3], [0, 11], [7, -21], [39, -39], [-32, 0], [-12, 12], [0, -88], [0, -92], [255, 0], [-35, 35], [0, 4], [16, 0], [330, 0], [0, -298], [717, -717], [-217, 217], [0, -386], [906, -906], [34, 0], [838, -838], [0, 1325], [-1163, 0], [0, 54], [7, 0], [46, -46], [0, 1656], [0, -367], [-1622, 1622], [-736, 0], [-1175, 0], [0, -2541], [802, -802], [-3110, 0], [0, -2598], [1333, -1333], [-1580, 1580], [0, 1268], [-878, 0], [0, 2882], [1476, -1476], [4368, 0], [0, 3016], [1563, -1563], [-1846, 0], [-3996, 3996], [-4182, 0], [-1159, 0], [1785, -1785], [0, -588], [0, -6828], [-2508, 0], [4297, -4297], [2489, 0], [0, -1786], [2318, -2318], [2749, -2749], [0, 1825], [304, -304], [4578, 0], [4302, -4302], [6727, 0], [-116, 0], [-3792, 3792], [0, 3687], [0, 8281], [0, -6668], [-8679, 0], [9092, -9092], [11642, 0], [-6808, 6808], [0, 1110], [959, -959], [0, -16], [736, 0], [0, -8414], [-694, 694], [0, -11476], [0, 12040], [-13711, 13711], [4128, 0], [-8442, 0], [-9564, 9564], [-13394, 0], [0, 15513], [0, 12379], [-8368, 0], [0, 1577], [8202, 0], [1936, 0], [0, 13156], [-6630, 6630], [4195, -4195], [-1107, 0], [0, 13630], [11760, 0], [0, 19101], [-10590, 10590], [0, 17400], [0, 6410], [24020, 0], [26139, -26139], [9171, -9171], [0, 19608], [-3674, 3674], [3873, -3873], [13743, 0], [0, -16356], [10782, -10782], [25547, 0], [0, 14299], [-23900, 23900], [-9246, 0], [13950, -13950], [-15787, 0], [8438, -8438], [-22522, 0], [17138, -17138], [0, 9963], [8664, 0], [-2450, 2450], [-6643, 0], [0, -15593], [-371, 371], [0, 8232], [30751, 0], [0, -12318], [0, 18022], [1279, 0], [0, 36910], [-31226, 31226], [17758, 0], [0, 2019], [-2398, 0], [0, -10440], [-4739, 0], [34030, -34030], [7226, -7226], [4881, 0], [13112, -13112], [-40982, 0], [0, -39843], [7256, -7256], [23005, 0], [-368, 368], [5556, 0], [-5674, 0], [7826, -7826], [0, 3370], [0, 27734], [0, -45280], [-47610, 0], [28328, 0], [0, -48612], [4100, -4100], [13902, 0], [0, 45824], [0, -38874], [-1829, 0], [5855, -5855], [0, 21120], [21650, 0], [-36597, 0], [0, -54187], [-36034, 36034], [-40628, 40628], [-40431, 40431], [33266, -33266]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_4_q_c();".
// 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_q_c(:prec:=2)
    chi := MakeCharacter_336_q();
    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_q_c();".
// 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_q_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![121, -11, 1]>],Snew);
    return Vf;
end function;