
// Make newform 425.4.b.a in Magma, downloaded from the LMFDB on 27 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_425_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_425_b_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_425_4_b_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_425_4_b_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_425_b();"
function MakeCharacter_425_b()
    N := 425;
    order := 2;
    char_gens := [52, 326];
    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_425_b_Hecke();"
function MakeCharacter_425_b_Hecke(Kf)
    N := 425;
    order := 2;
    char_gens := [52, 326];
    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]];
    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 := [[0, 3], [0, 7], [0, 0], [0, -22], [-64, 0], [0, -73], [0, -17], [49, 0], [0, -110], [-155, 0], [-197, 0], [0, -372], [-262, 0], [0, -258], [0, -13], [0, 653], [333, 0], [-355, 0], [0, 814], [47, 0], [0, 437], [384, 0], [0, 736], [-511, 0], [0, 537], [-550, 0], [0, -1512], [0, 708], [491, 0], [0, -921], [0, -163], [-1930, 0], [0, -1878], [-628, 0], [-1684, 0], [-218, 0], [0, -2182], [0, 392], [0, -306], [0, 98], [-324, 0], [-982, 0], [1240, 0], [0, -5058], [0, -2408], [35, 0], [3850, 0], [0, -601], [0, -2645], [2826, 0], [0, 831], [5622, 0], [58, 0], [-2308, 0], [0, -3918], [0, 3553], [-4305, 0], [132, 0], [0, 9110], [-1661, 0], [0, 4973], [0, 6533], [0, -1360], [9488, 0], [0, 410], [0, -8072], [-6119, 0], [0, 7643], [0, -8599], [-2296, 0], [0, -9098], [6450, 0], [0, 8914], [0, -7238], [3760, 0], [0, 11983], [4776, 0], [0, -14208], [5976, 0], [-7621, 0], [1482, 0], [-11280, 0], [1736, 0], [0, -14122], [-13368, 0], [0, -1512], [-15264, 0], [0, 6214], [-1294, 0], [0, -5659], [0, -7584], [16243, 0], [0, 15782], [-15375, 0], [218, 0], [0, -200], [16660, 0], [-12492, 0], [0, 11210], [-3258, 0], [0, 3545], [0, 19249], [0, 3582], [-6219, 0], [-7058, 0], [0, -9300], [0, 19790], [0, 19782], [-11514, 0], [-8268, 0], [0, -1046], [0, 23159], [0, -29967], [-14486, 0], [11972, 0], [-14802, 0], [0, 13868], [0, 14417], [0, 8502], [10543, 0], [5180, 0], [0, 25601], [0, -8238], [0, -23013], [8440, 0], [15540, 0], [-14311, 0], [12257, 0], [0, -23101], [0, 28350], [18243, 0], [0, -12742], [-2367, 0], [0, 22829], [12246, 0], [20743, 0], [0, -31710], [0, -39735], [0, -9566], [-32922, 0], [37022, 0], [22665, 0], [0, -18326], [0, -10228], [17980, 0], [31083, 0], [0, 5698], [0, 9305], [20273, 0], [0, -28112], [0, 4088], [-5490, 0], [0, 37714], [0, -32514], [0, -7913], [-28024, 0], [-9454, 0], [49134, 0], [0, 23960], [679, 0], [0, 41003], [0, -45228], [0, 22448], [-18197, 0], [0, 27444], [0, -39818], [-45445, 0], [0, 20962]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_425_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_425_4_b_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_425_4_b_a(:prec:=2)
    chi := MakeCharacter_425_b();
    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_425_4_b_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_425_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_425_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, 0, 1]>,<3,R![49, 0, 1]>],Snew);
    return Vf;
end function;