
// Make newform 324.3.d.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_d_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_3_d_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_324_3_d_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 := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 3]];
    Rf_basisdens := [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_d();"
function MakeCharacter_324_d()
    N := 324;
    order := 2;
    char_gens := [163, 245];
    v := [1, 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_324_d_Hecke();"
function MakeCharacter_324_d_Hecke(Kf)
    N := 324;
    order := 2;
    char_gens := [163, 245];
    char_values := [[-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 := 3;
    raw_aps := [[-2, 0], [0, 0], [-4, 1], [0, 0], [0, 0], [5, 4], [8, -5], [0, 0], [0, 0], [20, 7], [0, 0], [35, 4], [80, 0], [0, 0], [0, 0], [56, 0], [0, 0], [11, -20], [0, 0], [0, 0], [-55, 16], [0, 0], [0, 0], [80, 13], [-130, 0], [-40, 0], [0, 0], [0, 0], [-91, -20], [-112, -5], [0, 0], [0, 0], [-88, -35], [0, 0], [140, -17], [0, 0], [-85, -44], [0, 0], [0, 0], [-52, 55], [0, 0], [38, 0], [0, 0], [95, -56], [-28, -65], [0, 0], [0, 0], [0, 0], [0, 0], [221, -20], [-208, -35], [0, 0], [209, 40], [0, 0], [32, 85], [0, 0], [260, -23], [0, 0], [230, 0], [-160, -77], [0, 0], [68, -95], [0, 0], [0, 0], [-25, -104], [308, 25], [0, 0], [350, 0], [0, 0], [-598, 0], [-544, 0], [0, 0], [0, 0], [-550, 0], [0, 0], [0, 0], [680, 0], [-325, 76], [-40, 133], [-391, 40], [0, 0], [29, -140], [0, 0], [-145, 136], [0, 0], [0, 0], [560, 0], [425, -56], [-760, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [440, 0], [-880, 0], [0, 0], [341, -140], [0, 0], [-532, 55], [0, 0], [-520, -77], [0, 0], [575, 16], [0, 0], [368, -155], [0, 0], [551, -80], [0, 0], [-70, 0], [608, -35], [0, 0], [0, 0], [200, -203], [0, 0], [0, 0], [-1144, 0], [0, 0], [-589, 100], [-385, 184], [104, 0], [0, 0], [0, 0], [260, 217], [-259, 220], [0, 0], [0, 0], [-1450, 0], [0, 0], [0, 0], [0, 0], [1190, 0], [-760, 13], [-481, -200], [-748, -65], [0, 0], [572, -185], [-280, 253], [0, 0], [-700, -143], [0, 0], [0, 0], [-1258, 0], [0, 0], [410, 0], [-232, -275], [0, 0], [0, 0], [-805, -116], [-1600, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [920, 43], [215, 304], [-580, 247], [0, 0], [728, 205], [0, 0], [0, 0], [-496, 0], [0, 0], [0, 0], [-925, 124]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_3_d_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_324_3_d_a(:prec:=2)
    chi := MakeCharacter_324_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_d_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_324_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-11, 8, 1]>],Snew);
    return Vf;
end function;