
// Make newform 324.3.f.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_324_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_324_f_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_f_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_f_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_324_f();"
function MakeCharacter_324_f()
    N := 324;
    order := 6;
    char_gens := [163, 245];
    v := [3, 4];
    // 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_f_Hecke();"
function MakeCharacter_324_f_Hecke(Kf)
    N := 324;
    order := 6;
    char_gens := [163, 245];
    char_values := [[-1, 0], [-1, 1]];
    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], [-2, 2], [8, -4], [8, -4], [-2, 2], [-10, 0], [12, -24], [16, 16], [0, -26], [4, 4], [26, 0], [58, -58], [56, -28], [80, -40], [74, 0], [52, 52], [0, -26], [4, 4], [0, 0], [-46, 0], [-136, 68], [56, -28], [-82, 0], [0, -2], [0, -74], [-44, -44], [-12, 24], [-46, 0], [-110, 110], [-84, 168], [-68, -68], [0, 10], [28, 28], [-2, 2], [104, -52], [214, -214], [12, -24], [-56, -56], [0, 334], [108, -216], [2, 0], [-256, 128], [-290, 290], [26, 0], [228, -456], [-140, -140], [392, -196], [-328, 164], [142, -142], [-82, 0], [-224, -224], [0, 46], [-84, 168], [-254, 254], [176, -88], [-262, 0], [12, -24], [0, -290], [0, 226], [172, 172], [-362, 362], [84, -168], [136, 136], [0, 478], [0, -170], [-472, 236], [-338, 338], [-116, -116], [0, -506], [0, 178], [-96, 192], [-232, 116], [310, -310], [-252, 504], [352, 352], [0, -578], [26, 0], [250, -250], [-290, 290], [196, 196], [0, -674], [-312, 624], [-334, 0], [-136, 68], [-88, 44], [-394, 0], [0, 478], [0, 142], [364, 364], [-12, 24], [848, -424], [60, -120], [532, 532], [-44, -44], [336, -672], [-842, 842], [326, 0], [180, -360], [530, 0], [392, -196], [-766, 0], [-284, -284], [0, -422], [-164, -164], [-46, 0], [728, -364], [-82, 0], [-32, -32], [0, 334], [-212, -212], [-214, 0], [-1118, 1118], [776, -388], [-84, 168], [0, 10], [700, 700], [192, -384], [670, -670], [-952, 476], [1222, -1222], [0, 334], [0, 1006], [108, -216], [-1144, 572], [1034, 0], [0, -530], [408, -816], [-280, 140], [-194, 194], [-780, 1560], [-392, -392], [-380, -380], [-1006, 0], [-758, 758], [-2, 2], [-262, 0], [-836, -836], [-866, 866], [-10, 0], [-252, 504], [0, 838], [508, 508], [420, -840], [1298, 0], [224, -112], [0, -506], [0, -998], [292, 292], [-96, 192], [646, -646], [-898, 0], [-420, 840], [-488, -488], [-1576, 788], [-448, 224], [348, -696], [0, 1594], [674, 0], [430, -430], [-88, 44], [-730, 0], [532, 532], [-852, 1704], [346, -346], [848, -424], [564, -1128], [0, -458]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_324_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_324_3_f_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_f_a(:prec:=2)
    chi := MakeCharacter_324_f();
    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_f_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_f_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, 2, 1]>,<7,R![48, -12, 1]>,<11,R![48, -12, 1]>],Snew);
    return Vf;
end function;