
// Make newform 324.3.g.d in Magma, downloaded from the LMFDB on 19 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_3_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_3_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, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 3, 0, 0, 0, 3, 0], [0, 0, 6, 0, 0, 0, -3, 0], [0, 0, 0, 3, 0, -3, 0, 0], [0, 3, 0, 0, 0, -3, 0, 3], [0, 6, 0, 9, 0, 3, 0, -3], [0, 9, 0, 6, 0, -6, 0, -9]];
    Rf_basisdens := [1, 1, 1, 1, 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, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 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 := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1], [1, -1, 2, -2, 0, 0, 0, 0], [0, 0, 0, 0, -5, 5, -2, 2], [-7, 0, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 1, 0], [0, 0, 0, 0, 5, 0, 0, 3], [-8, 8, 6, -6, 0, 0, 0, 0], [14, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, -1, 0], [0, 13, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 2], [0, 0, 0, 0, 4, -4, -3, 3], [0, 0, 0, 0, 0, -14, 2, 0], [0, 16, 9, 0, 0, 0, 0, 0], [-5, 5, 3, -3, 0, 0, 0, 0], [0, 0, 0, 0, -2, 2, 7, -7], [-64, 0, 0, -9, 0, 0, 0, 0], [0, -77, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, -10], [0, 0, 0, 0, 19, -19, 1, -1], [0, 40, 22, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -5], [112, -112, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 22, -22, 12, -12], [47, 0, 0, -24, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -10, 0], [-115, 0, 0, -9, 0, 0, 0, 0], [0, 0, 0, 0, 0, -26, 3, 0], [0, 0, 0, 0, 13, 0, 0, -1], [-146, 146, 10, -10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, -3, 0], [0, 16, 12, 0, 0, 0, 0, 0], [154, -154, 3, -3, 0, 0, 0, 0], [-4, 0, 0, 46, 0, 0, 0, 0], [0, 0, 0, 0, 0, 22, 13, 0], [0, 0, 0, 0, 33, 0, 0, 3], [0, 0, 0, 0, 4, -4, -6, 6], [-238, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -58, 0, 0, 25], [-131, 131, 30, -30, 0, 0, 0, 0], [0, 0, 0, 0, -27, 27, -13, 13], [164, 0, 0, 8, 0, 0, 0, 0], [253, -253, 23, -23, 0, 0, 0, 0], [0, 211, -21, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 7], [55, -55, 24, -24, 0, 0, 0, 0], [0, 0, 0, 0, 27, -27, -5, 5], [0, 0, 0, 0, 0, 48, 12, 0], [0, -296, -13, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 6, -11, 11], [0, 0, 0, 0, 0, 7, 13, 0], [0, 0, 0, 0, 50, 0, 0, 4], [0, 0, 0, 0, -37, 37, 8, -8], [17, 0, 0, -51, 0, 0, 0, 0], [0, 76, 24, 0, 0, 0, 0, 0], [0, 0, 0, 0, -41, 0, 0, 8], [10, -10, 66, -66, 0, 0, 0, 0], [0, 0, 0, 0, 0, -51, -14, 0], [110, 0, 0, -24, 0, 0, 0, 0], [0, 0, 0, 0, 0, -34, 34, 0], [0, 181, -34, 0, 0, 0, 0, 0], [0, 0, 0, 0, -105, 0, 0, -4], [0, 229, 37, 0, 0, 0, 0, 0], [-2, 2, -32, 32, 0, 0, 0, 0], [0, 0, 0, 0, 0, 24, -33, 0], [0, -152, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, -39], [0, 0, 0, 0, -20, 20, 17, -17], [0, 88, 66, 0, 0, 0, 0, 0], [-44, 44, 26, -26, 0, 0, 0, 0], [-4, 0, 0, -38, 0, 0, 0, 0], [0, 0, 0, 0, 0, -60, -13, 0], [0, 0, 0, 0, -12, 0, 0, -25], [-286, 0, 0, -21, 0, 0, 0, 0], [0, 0, 0, 0, 0, -67, 43, 0], [16, -16, 25, -25, 0, 0, 0, 0], [0, 0, 0, 0, 0, 14, -46, 0], [0, 127, 80, 0, 0, 0, 0, 0], [0, 0, 0, 0, -112, 112, -18, 18], [-43, 0, 0, 90, 0, 0, 0, 0], [0, 328, 78, 0, 0, 0, 0, 0], [0, 0, 0, 0, 122, 0, 0, -46], [0, 0, 0, 0, -92, 92, -7, 7], [0, 40, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -36, 0, 0, 37], [-332, 332, -46, 46, 0, 0, 0, 0], [0, 0, 0, 0, 16, -16, -18, 18], [0, 0, 0, 0, -116, 0, 0, 31], [-34, 0, 0, -76, 0, 0, 0, 0], [0, 0, 0, 0, 0, 114, -17, 0], [-377, 377, 27, -27, 0, 0, 0, 0], [0, 0, 0, 0, 22, -22, -28, 28], [0, 0, 0, 0, 0, 80, 25, 0], [0, 0, 0, 0, -144, 144, 35, -35], [107, 0, 0, 135, 0, 0, 0, 0], [365, 0, 0, 28, 0, 0, 0, 0], [0, 304, -54, 0, 0, 0, 0, 0], [0, 0, 0, 0, -95, 95, -41, 41], [0, 0, 0, 0, 0, -70, -40, 0], [0, 0, 0, 0, -93, 0, 0, 20], [142, -142, -20, 20, 0, 0, 0, 0], [-217, 0, 0, 102, 0, 0, 0, 0], [0, 0, 0, 0, 108, 0, 0, 23], [0, 0, 0, 0, -27, 27, -49, 49], [0, 0, 0, 0, 0, 62, 30, 0], [0, -311, 84, 0, 0, 0, 0, 0], [25, -25, 19, -19, 0, 0, 0, 0], [-424, 0, 0, 102, 0, 0, 0, 0], [0, 0, 0, 0, 0, 41, -11, 0], [0, 394, -120, 0, 0, 0, 0, 0], [-616, 0, 0, -104, 0, 0, 0, 0], [0, 0, 0, 0, 53, 0, 0, 57], [172, -172, 108, -108, 0, 0, 0, 0], [0, 0, 0, 0, -214, 214, 55, -55], [0, 0, 0, 0, 0, 100, -49, 0], [0, 0, 0, 0, 154, 0, 0, -21], [-86, 86, -63, 63, 0, 0, 0, 0], [0, 535, -38, 0, 0, 0, 0, 0], [0, 0, 0, 0, -104, 0, 0, -15], [0, 0, 0, 0, 190, -190, -16, 16], [0, -149, -63, 0, 0, 0, 0, 0], [0, 0, 0, 0, -105, 105, -26, 26], [0, -617, 40, 0, 0, 0, 0, 0], [0, 0, 0, 0, 116, -116, 67, -67], [0, 301, 63, 0, 0, 0, 0, 0], [730, -730, 66, -66, 0, 0, 0, 0], [-340, 0, 0, -96, 0, 0, 0, 0], [0, 0, 0, 0, 0, -44, -80, 0], [43, -43, -53, 53, 0, 0, 0, 0], [-172, 0, 0, -228, 0, 0, 0, 0], [0, 0, 0, 0, 0, -95, 25, 0], [631, -631, -70, 70, 0, 0, 0, 0], [0, 0, 0, 0, 221, -221, 67, -67], [-167, 167, 21, -21, 0, 0, 0, 0], [0, 0, 0, 0, 0, -31, -13, 0], [0, 0, 0, 0, 175, -175, -29, 29], [-664, 0, 0, -72, 0, 0, 0, 0], [0, 0, 0, 0, -33, 0, 0, -52], [-746, 746, 50, -50, 0, 0, 0, 0], [0, 0, 0, 0, -42, 42, -13, 13], [164, 0, 0, 104, 0, 0, 0, 0], [0, 0, 0, 0, -56, 0, 0, -29], [0, 430, 120, 0, 0, 0, 0, 0], [0, 0, 0, 0, -47, 0, 0, -71], [-812, 812, -74, 74, 0, 0, 0, 0], [0, 0, 0, 0, 102, -102, 89, -89], [772, -772, 49, -49, 0, 0, 0, 0], [0, 0, 0, 0, -156, 156, -71, 71], [1298, 0, 0, 50, 0, 0, 0, 0], [0, 0, 0, 0, 0, -22, 37, 0], [0, -680, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, -164, 0, 0, 31], [-637, 0, 0, 39, 0, 0, 0, 0], [0, 0, 0, 0, -87, 0, 0, 84], [41, 0, 0, -302, 0, 0, 0, 0], [0, 0, 0, 0, 0, 165, 67, 0], [0, 0, 0, 0, -158, 0, 0, 61], [0, 0, 0, 0, 161, -161, -77, 77], [949, -949, 15, -15, 0, 0, 0, 0], [0, 0, 0, 0, -66, 66, 39, -39], [0, 0, 0, 0, 0, 152, -27, 0], [0, 0, 0, 0, -170, 0, 0, -24], [1475, 0, 0, -57, 0, 0, 0, 0], [0, 886, -39, 0, 0, 0, 0, 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_3_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_3_g_d(:prec:=8)
    chi := MakeCharacter_324_g();
    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_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_3_g_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_324_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![6561, 0, -2916, 0, 1215, 0, -36, 0, 1]>,<7,R![676, 52, 30, -2, 1]>],Snew);
    return Vf;
end function;