
// Make newform 336.4.bl.c in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_bl();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_bl_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_bl_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_bl_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_bl();"
function MakeCharacter_336_bl()
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    v := [3, 6, 6, 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_bl_Hecke();"
function MakeCharacter_336_bl_Hecke( : Kf := false)
    N := 336;
    order := 6;
    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], [-4, 2], [-14, -7], [38, 38], [21, -42], [-36, -36], [0, -115], [-224, 112], [-96, 0], [-209, 209], [0, -265], [202, -404], [-317, 634], [0, -66], [288, -288], [-720, 720], [-560, 280], [-339, -339], [314, -628], [37, 37], [198, -99], [66, 0], [904, -452], [1012, -2024], [-402, -402], [0, -545], [2520, -1260], [-1181, 1181], [-1002, 0], [221, -442], [0, 102], [636, -636], [2009, 0], [0, -1188], [58, 58], [-240, -240], [-1844, 922], [3054, 0], [1120, -560], [462, 462], [-1105, 2210], [172, -86], [-925, 925], [-4164, 0], [-4264, 4264], [1502, -3004], [1552, 0], [294, -294], [-1070, 535], [0, 258], [2082, -4164], [1880, 1880], [-2808, 0], [-1516, 758], [-320, -320], [-146, -146], [0, 7048], [2009, -2009], [60, 0], [665, -665], [4092, -8184], [-4619, 0], [2550, -2550], [1762, -881], [0, 10668], [-494, 247], [-2059, 0], [-2396, -2396], [5968, -11936], [2266, 2266], [-7512, 3756], [-10681, 10681], [0, 6037], [3339, -6678], [0, -13140], [-9006, 9006], [3142, -1571], [0, 24], [-5877, -5877], [6330, 0], [-9533, 0], [6898, 6898], [5545, -11090], [0, 2648], [-14392, 7196], [-3222, 0], [0, 2491], [-248, 496], [-2491, 4982], [0, -9138], [9720, -9720], [-5237, -5237], [-692, 1384], [-6518, 3259], [-12342, 0], [22044, -11022], [5624, 5624], [0, -5747], [0, 7481], [2386, -4772], [5694, -5694], [-12378, 12378], [0, 23910], [-11579, -11579], [-3321, -3321], [-19668, 0], [-16204, 8102], [12004, 12004], [6239, -12478], [0, -4609], [-4898, 4898], [-9018, 0], [-21119, 21119], [-3778, 7556], [10692, -10692], [-7481, 0], [20202, -20202], [0, -17166], [-9484, 18968], [-11117, -11117], [-2807, 0], [23816, -11908], [-3496, -3496], [0, 14401], [24672, 0], [0, -8030], [0, -34518], [-10621, 0], [21542, -10771], [-10447, -10447], [-14650, 29300], [-20158, 10079], [-10126, 0], [-4536, 2268], [-17455, 34910], [15470, 15470], [17416, -17416], [7928, -15856], [-13398, 13398], [-35968, 0], [0, -762], [10070, 10070], [-7394, 14788], [-10117, -10117], [30312, 0], [-26705, 53410], [-12320, -12320], [0, 23240], [52236, -26118], [0, 14734], [4432, -8864], [-13853, 27706], [0, -37578], [-12837, -12837], [13548, -27096], [32090, -16045], [44868, -22434], [-30201, 60402], [3452, 3452], [22012, -11006], [26820, 0], [30753, -61506], [0, 14388], [42282, -42282], [51168, -51168], [12747, 12747], [13201, 13201]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_bl_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_4_bl_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_bl_c(:prec:=2)
    chi := MakeCharacter_336_bl();
    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_bl_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_bl_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_bl();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![12, 6, 1]>,<11,R![4332, -114, 1]>],Snew);
    return Vf;
end function;