
// Make newform 325.4.d.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_325_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_325_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_325_4_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_325_4_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 := [1, 0, 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_325_d();"
function MakeCharacter_325_d()
    N := 325;
    order := 2;
    char_gens := [27, 301];
    v := [1, 1];
    // 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_325_d_Hecke();"
function MakeCharacter_325_d_Hecke(Kf)
    N := 325;
    order := 2;
    char_gens := [27, 301];
    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 := 4;
    raw_aps := [[-3, 0], [0, -1], [0, 0], [15, 0], [0, -48], [-39, 26], [0, 45], [0, 6], [0, 162], [144, 0], [0, -264], [303, 0], [0, 192], [0, -97], [111, 0], [0, -414], [0, -522], [376, 0], [36, 0], [0, -357], [1098, 0], [830, 0], [-438, 0], [0, 438], [852, 0], [396, 0], [0, 182], [0, 612], [0, 1083], [0, 90], [0, -2086], [-1467, 0], [-414, 0], [2419, 0], [0, -930], [0, -1683], [0, -1874], [1194, 0], [-2388, 0], [0, -1566], [657, 0], [-1222, 0], [1260, 0], [-342, 0], [-81, 0], [-1996, 0], [2833, 0], [-3507, 0], [228, 0], [0, -5493], [0, 3627], [0, 6075], [0, 210], [7092, 0], [0, 5805], [0, 792], [-5472, 0], [0, 2331], [0, 1384], [0, -4062], [0, -3764], [-4227, 0], [306, 0], [-2106, 0], [0, 10051], [2154, 0], [0, -10770], [0, 2171], [0, 7047], [0, 6873], [-9318, 0], [0, -4128], [0, 2536], [0, -92], [0, 10182], [-579, 0], [-2106, 0], [-1974, 0], [0, 11886], [0, 1254], [-5823, 0], [0, 7341], [0, 7485], [0, -15203], [1762, 0], [0, -7317], [0, 5016], [-9870, 0], [0, 14541], [2112, 0], [0, -3276], [0, 15453], [3660, 0], [747, 0], [0, -15804], [0, -12078], [0, 16110], [3915, 0], [0, 16184], [0, -7923], [0, 14389], [-10383, 0], [0, -16425], [-12213, 0], [-6383, 0], [-6426, 0], [21330, 0], [-12084, 0], [-2394, 0], [-21971, 0], [0, 15406], [-9630, 0], [-14748, 0], [0, 3672], [0, -19875], [1710, 0], [-16452, 0], [0, -25902], [0, 18108], [32904, 0], [0, 15318], [0, -7729], [0, -19242], [-22518, 0], [0, -9168], [1170, 0], [0, 1662], [-30960, 0], [0, 8372], [-2739, 0], [0, 6756], [29643, 0], [18128, 0], [0, 6410], [0, 28290], [0, -27960], [-5649, 0], [756, 0], [0, -31194], [-17055, 0], [0, -35520], [0, -1095], [0, -2554], [21522, 0], [13124, 0], [0, 23424], [-31077, 0], [0, 19422], [-1744, 0], [19179, 0], [-29217, 0], [-15633, 0], [0, 30589], [0, 25884], [0, 12305], [29772, 0], [-47644, 0], [0, 21924], [0, -32398], [0, -2097], [-20016, 0], [0, 24993], [40959, 0], [-48933, 0], [-47388, 0], [-16803, 0], [-57526, 0], [0, 25000]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_325_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_325_4_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_325_4_d_a(:prec:=2)
    chi := MakeCharacter_325_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_325_4_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_325_4_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_325_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![3, 1]>],Snew);
    return Vf;
end function;