
// Make newform 324.5.g.d in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_g_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_324_5_g_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_324_5_g_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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 9, 0, 0], [0, 0, 1, 0], [0, 0, 0, 9]];
    Rf_basisdens := [1, 1, 1, 1];
    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_324_g();"
function MakeCharacter_324_g()
    N := 324;
    order := 6;
    char_gens := [163, 245];
    v := [6, 5];
    // 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_324_g_Hecke();"
function MakeCharacter_324_g_Hecke(Kf)
    N := 324;
    order := 6;
    char_gens := [163, 245];
    char_values := [[1, 0, 0, 0], [1, 0, -1, 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+1)]];
    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 := 5;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, -5, 0], [0, -13, 0, 13], [34, 0, -34, 0], [0, 0, 0, -50], [-64, 0, 0, 0], [0, 68, 0, 0], [0, -118, 0, 118], [697, 0, -697, 0], [-748, 0, 0, 0], [0, 76, 0, 0], [0, 0, -2618, 0], [0, 294, 0, -294], [0, 0, 0, 119], [0, -646, 0, 0], [0, 0, -6404, 0], [5218, 0, -5218, 0], [0, 0, 0, 730], [-4519, 0, 0, 0], [0, 0, -7502, 0], [0, 609, 0, -609], [0, 0, 0, -986], [0, 0, -10571, 0], [0, -1457, 0, 1457], [5830, 0, -5830, 0], [0, 0, 0, -121], [-5020, 0, 0, 0], [0, 2376, 0, 0], [9227, 0, 0, 0], [0, 475, 0, 0], [0, -1258, 0, 1258], [-9812, 0, 9812, 0], [0, -1831, 0, 0], [0, 0, 25555, 0], [-21164, 0, 21164, 0], [33830, 0, 0, 0], [0, -3042, 0, 0], [0, 5133, 0, -5133], [0, 0, 0, 2309], [-19504, 0, 0, 0], [0, -782, 0, 782], [-51527, 0, 51527, 0], [0, 0, 0, 6715], [45665, 0, 0, 0], [-13124, 0, 13124, 0], [0, 0, 78850, 0], [0, -2478, 0, 2478], [-8870, 0, 8870, 0], [0, 0, 0, -2890], [0, 7128, 0, 0], [0, 0, -33422, 0], [0, 0, 0, -12274], [0, 1686, 0, 0], [0, -7772, 0, 7772], [0, 0, 0, 5594], [108323, 0, 0, 0], [0, 0, -30716, 0], [0, -14362, 0, 14362], [63976, 0, -63976, 0], [0, -12070, 0, 0], [6410, 0, 0, 0], [0, 7888, 0, 0], [0, 0, -160961, 0], [0, 21637, 0, -21637], [0, 0, 33286, 0], [127690, 0, -127690, 0], [0, -15423, 0, 0], [0, 0, -203792, 0], [0, -3592, 0, 3592], [0, 0, 0, 20658], [0, 0, -151079, 0], [6478, 0, -6478, 0], [99008, 0, 0, 0], [0, -10118, 0, 0], [0, 10591, 0, -10591], [-163438, 0, 0, 0], [0, 31624, 0, 0], [-107525, 0, 107525, 0], [0, -16698, 0, 0], [0, 0, -134420, 0], [0, 0, 0, -26414], [112187, 0, 0, 0], [0, 0, 204643, 0], [0, -5610, 0, 5610], [0, 0, 0, -40392], [0, 0, -6677, 0], [0, -25483, 0, 25483], [-238799, 0, 238799, 0], [0, 0, 0, 29237], [0, -1038, 0, 1038], [331262, 0, 0, 0], [0, 44033, 0, 0], [-45050, 0, 45050, 0], [0, 0, 0, 25908], [0, -18161, 0, 0], [0, 0, 0, -42734], [-214642, 0, 0, 0], [545156, 0, 0, 0], [0, 0, 491422, 0], [0, 0, 0, 49453], [0, -17085, 0, 0], [0, 29968, 0, -29968], [-589718, 0, 589718, 0], [184094, 0, 0, 0], [0, 45727, 0, -45727], [0, 0, 0, 37774], [0, -33286, 0, 0], [0, 0, 516115, 0], [486574, 0, -486574, 0], [-189550, 0, 0, 0], [0, -16472, 0, 0], [0, 0, -11390, 0], [-6715, 0, 0, 0], [0, -49012, 0, 49012], [-547448, 0, 547448, 0], [0, 0, 0, -4692], [0, -28777, 0, 0], [0, -34787, 0, 34787], [686320, 0, -686320, 0], [0, 0, -214727, 0], [0, 38182, 0, -38182], [0, 0, 0, -69582], [0, 0, -684476, 0], [0, 0, 0, -89947], [0, 0, 319648, 0], [0, 0, 0, -6204], [0, 0, -863873, 0], [-207608, 0, 207608, 0], [-803590, 0, 0, 0], [0, 68850, 0, 0], [1065985, 0, -1065985, 0], [750410, 0, 0, 0], [0, -45912, 0, 0], [733381, 0, -733381, 0], [0, 0, 0, -25758], [-502724, 0, 502724, 0], [0, -66249, 0, 0], [0, 0, 0, 41782], [-331072, 0, 0, 0], [0, 30514, 0, -30514], [541195, 0, -541195, 0], [0, 0, 0, -39818], [39626, 0, 0, 0], [0, 117270, 0, -117270], [0, 0, 33034, 0], [0, 50694, 0, -50694], [-343604, 0, 343604, 0], [0, 0, 0, 8284], [696094, 0, -696094, 0], [0, 0, 0, 74714], [-1448128, 0, 0, 0], [0, -167738, 0, 0], [0, 0, -374828, 0], [0, -134610, 0, 134610], [656777, 0, 0, 0], [0, 64332, 0, -64332], [195173, 0, 0, 0], [0, 146591, 0, 0], [0, 108069, 0, -108069], [0, 0, 0, -59376], [783619, 0, -783619, 0], [0, 0, 0, -30753], [0, 72026, 0, 0], [0, -32318, 0, 32318], [881243, 0, 0, 0], [0, 0, 690166, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_5_g_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_324_5_g_d(:prec:=4)
    chi := MakeCharacter_324_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_324_5_g_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_324_5_g_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<5,R![6561, 0, -81, 0, 1]>,<7,R![25, 5, 1]>],Snew);
    return Vf;
end function;