
// Make newform 324.3.d.b in Magma, downloaded from the LMFDB on 20 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_b();".
// 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_b();".
// 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;
    Rf_num := [[1, 0], [-1, 2]];
    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 := [[-1, -1], [0, 0], [4, 0], [0, 2], [0, 7], [-22, 0], [-11, 0], [0, 9], [0, 14], [34, 0], [0, 4], [-16, 0], [13, 0], [0, 29], [0, -2], [52, 0], [0, -31], [-16, 0], [0, 67], [0, 0], [-25, 0], [0, -16], [0, -20], [-2, 0], [-43, 0], [-20, 0], [0, -14], [0, 9], [-88, 0], [-50, 0], [0, 126], [0, -112], [169, 0], [0, -113], [130, 0], [0, -70], [-4, 0], [0, -180], [0, 104], [-2, 0], [0, 108], [254, 0], [0, -2], [-67, 0], [268, 0], [0, -18], [0, 76], [0, -34], [0, 259], [410, 0], [-65, 0], [0, -22], [-223, 0], [0, 63], [-437, 0], [0, -182], [304, 0], [0, -180], [-34, 0], [-218, 0], [0, 4], [202, 0], [0, 63], [0, 158], [-79, 0], [502, 0], [0, 236], [-337, 0], [0, -157], [272, 0], [-461, 0], [0, -306], [0, 56], [-346, 0], [0, 189], [0, -364], [-146, 0], [488, 0], [445, 0], [-67, 0], [0, 356], [272, 0], [0, 234], [-439, 0], [0, 488], [0, -191], [-47, 0], [-331, 0], [538, 0], [0, 328], [0, -369], [0, 70], [0, 234], [0, 419], [0, 301], [0, 504], [760, 0], [745, 0], [0, 324], [-520, 0], [0, -223], [934, 0], [0, -409], [-695, 0], [0, -311], [227, 0], [0, -83], [-506, 0], [0, 32], [335, 0], [0, 364], [-340, 0], [391, 0], [0, -7], [0, -252], [421, 0], [0, -239], [0, 234], [886, 0], [0, 484], [248, 0], [1154, 0], [1132, 0], [0, -459], [0, -520], [142, 0], [740, 0], [0, 72], [0, 470], [914, 0], [0, -207], [0, 230], [0, -644], [758, 0], [-374, 0], [-22, 0], [-1334, 0], [0, 508], [832, 0], [493, 0], [0, -189], [-758, 0], [0, -500], [0, 252], [-718, 0], [0, -524], [146, 0], [-146, 0], [0, 49], [0, -684], [-1480, 0], [142, 0], [0, -693], [0, -364], [0, 371], [0, 232], [0, 450], [-758, 0], [-754, 0], [-1796, 0], [0, 61], [1213, 0], [0, 202], [0, 828], [157, 0], [0, -812], [0, -144], [-412, 0]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-4, 1]>],Snew);
    return Vf;
end function;