
// Make newform 336.4.q.a in Magma, downloaded from the LMFDB on 02 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_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_336_4_q_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 := [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, -7], [21, -7], [7, -7], [-52, 0], [-72, 72], [0, 20], [0, -48], [-243, 0], [95, -95], [0, -352], [-296, 0], [-158, 0], [0, -142], [375, -375], [279, -279], [0, -246], [-730, 730], [-338, 0], [542, -542], [0, -305], [-1123, 0], [0, 426], [-369, 0], [-1270, 1270], [0, 1832], [0, -999], [1666, -1666], [-1832, 0], [1931, 0], [0, 979], [486, -486], [2630, 0], [0, 1878], [2103, -2103], [-1404, 1404], [0, -1536], [3634, 0], [0, -858], [-2420, 2420], [-2672, 0], [0, -1712], [3395, -3395], [510, 0], [276, -276], [-3198, 0], [5091, 0], [-3895, 3895], [0, 6428], [0, 3824], [-568, 0], [-3535, 3535], [-4335, 0], [0, -3574], [5466, -5466], [5815, -5815], [0, -3237], [3176, -3176], [-3282, 0], [2182, -2182], [3021, 0], [1304, 0], [6208, -6208], [0, -3337], [0, -8637], [0, 1940], [-5527, 0], [9564, -9564], [-918, 0], [-11480, 11480], [0, 3654], [-10067, 10067], [0, -8800], [-1136, 0], [0, 4290], [862, -862], [0, 9140], [0, 14232], [10001, -10001], [10256, 0], [-4502, 0], [12356, -12356], [862, 0], [0, -1365], [0, 8931], [11228, 0], [0, -12151], [18534, 0], [-17096, 0], [0, -2140], [-13266, 13266], [-19183, 19183], [21693, 0], [0, 19658], [-19436, 0], [0, -3089], [-1950, 1950], [0, 13132], [0, -2930], [-19824, 0], [19169, -19169], [7267, -7267], [0, 4892], [22126, -22126], [1097, -1097], [-9969, 0], [0, 6424], [17046, -17046], [-18877, 0], [0, -13783], [-18192, 18192], [-13054, 0], [7246, -7246], [-2817, 0], [-15786, 15786], [17426, 0], [-25834, 25834], [0, -27983], [28296, 0], [-21254, 21254], [-28259, 0], [0, -31779], [-9747, 9747], [0, -24512], [14325, 0], [0, -12078], [0, -10658], [-19099, 0], [0, -1406], [6730, -6730], [1166, 0], [0, -16967], [11878, 0], [0, -27324], [-11971, 0], [338, -338], [14114, -14114], [1563, 0], [31568, -31568], [-2626, 0], [0, -361], [-29544, 29544], [-24163, 0], [39196, -39196], [-6528, 0], [1046, 0], [-32910, 32910], [0, 22682], [0, -7450], [0, -10080], [-32958, 0], [19784, 0], [0, -11480], [-45352, 45352], [39906, 0], [0, 17528], [0, -17066], [-30821, 0], [-4179, 4179], [0, 9776], [14778, 0], [-8759, 0], [0, -33579], [-42634, 42634], [29148, -29148], [-44797, 44797], [-10654, 10654]];
    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_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_336_4_q_a(: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_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_336_4_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![49, 7, 1]>],Snew);
    return Vf;
end function;