
// Make newform 425.2.n.b in Magma, downloaded from the LMFDB on 20 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_425_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_425_n_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_2_n_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_425_2_n_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, 0, 0, 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_n();"
function MakeCharacter_425_n()
    N := 425;
    order := 8;
    char_gens := [52, 326];
    v := [4, 3];
    // 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_n_Hecke();"
function MakeCharacter_425_n_Hecke( : Kf := false)
    N := 425;
    order := 8;
    char_gens := [52, 326];
    char_values := [[-1, 0, 0, 0], [0, 1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := 2;
    raw_aps := [[1, -1, 1, 0], [1, -1, -1, -1], [0, 0, 0, 0], [1, -1, 1, -1], [-1, 1, -1, 1], [0, 1, 0, -1], [3, 2, 0, 2], [-2, 0, 2, 2], [3, 1, 1, 3], [1, -1, -2, -2], [-3, -3, 3, -3], [-5, 5, 0, 0], [-1, -4, -4, -1], [2, 0, -2, -2], [8, 2, 0, -2], [-1, 0, -1, 0], [0, 6, 0, 0], [0, 5, 5, 0], [0, -2, -4, -2], [5, 5, 5, -5], [0, 0, -7, 7], [1, 3, -3, -1], [4, -6, 4, 0], [0, 1, 8, 1], [-1, -1, -6, 6], [-12, -1, 0, 1], [0, -6, 4, -6], [3, 3, 1, -1], [-8, 1, 1, -8], [-3, 10, 10, -3], [-8, 0, 8, 6], [-7, 7, -3, -3], [0, -9, 4, -9], [11, 11, -7, 7], [0, -12, 0, -12], [-2, 0, 2, 10], [4, -4, 0, 4], [1, 5, -5, -1], [1, -1, -5, -5], [2, -2, -1, -1], [0, -6, 0, 0], [-8, 3, -3, 8], [0, 0, -20, 0], [0, 3, 3, 0], [-6, -5, 5, 6], [-5, 5, -3, -3], [-7, 11, 11, -7], [2, -2, 2, 0], [9, -1, -1, 9], [-2, -20, -2, 0], [3, 3, 6, -6], [-12, 2, 0, -2], [5, 5, 4, -4], [0, -6, 12, -6], [10, 8, 10, 0], [6, 0, -6, 2], [-8, -8, 11, -11], [8, -10, 0, 10], [-14, -14, 5, -5], [-7, -8, -7, 0], [13, 7, 7, 13], [-18, -4, 0, 4], [0, -10, 12, -10], [11, -11, 7, 7], [4, 7, -7, -4], [10, -1, -1, 10], [-14, 2, -14, 0], [-4, 4, 17, 17], [-5, 7, -7, 5], [-3, 0, -3, 0], [14, 0, 0, 0], [2, 0, -2, 26], [13, 3, -3, -13], [0, -11, -4, -11], [1, -1, 1, 1], [6, 0, -6, 14], [10, 0, -10, -2], [3, -16, -16, 3], [3, 8, 8, 3], [-8, -8, 0, 8], [15, -1, 1, -15], [0, 1, 16, 1], [-17, -11, 11, 17], [-2, 0, 2, -18], [-3, -3, -7, 7], [0, 14, -4, 14], [-2, 11, 11, -2], [2, 16, 2, 0], [17, 0, -17, 0], [-8, -16, 0, 16], [16, 0, -16, 10], [15, 15, 9, -9], [13, -3, -3, 13], [22, 0, -22, 6], [-13, -17, 17, 13], [-11, 7, -7, 11], [20, -12, 0, 12], [5, -8, 8, -5], [-4, -2, 0, 2], [-12, -12, 5, -5], [1, -1, -13, -13], [-24, -3, 0, 3], [26, 2, 26, 0], [-17, -12, -17, 0], [-11, -21, -21, -11], [0, -5, 20, -5], [8, 0, -8, 34], [-5, -20, -5, 0], [0, 16, 12, 16], [-11, 0, 0, 11], [19, 19, -1, 1], [0, -8, -6, -8], [1, 1, 2, -2], [-15, 1, 1, -15], [-8, 0, 8, -18], [-22, 22, -1, -1], [21, -21, -13, -13], [0, -2, 0, -2], [10, -1, 1, -10], [0, -6, 0, -6], [15, 0, -15, -20], [1, -2, 2, -1], [26, -26, -13, -13], [-5, 5, -19, -19], [15, 17, -17, -15], [0, 21, -8, 21], [-9, 9, -10, -10], [13, -13, 13, 13], [-12, 22, 0, -22], [-17, 12, -17, 0], [-20, -6, -20, 0], [13, 13, 33, -33], [-3, -3, -27, 27], [18, 0, -18, -28], [0, -21, 8, -21], [0, -9, 0, -9], [2, 0, -2, -2], [19, 19, 19, -19], [15, -4, 15, 0], [22, 7, 7, 22], [-17, 17, -31, -31], [15, -15, 14, 14], [-9, -9, 9, 9], [-15, 25, 25, -15], [0, 24, -20, 24], [5, -7, 7, -5], [-17, 17, 16, 16], [0, -5, 5, 0], [-12, 18, -12, 0], [-12, -16, 0, 16], [-27, 16, -16, 27], [15, 15, -12, 12], [0, 0, 8, 0], [-11, 11, -31, -31], [11, 9, 9, 11], [1, -5, -5, 1], [-8, 8, 0, -8], [-13, 4, -4, 13], [-11, 12, -11, 0], [1, 22, -22, -1], [-1, -1, -17, 17], [0, -21, 20, -21], [4, -38, 4, 0], [-38, 0, 38, 2], [-13, 0, 13, -20], [9, -9, -9, -9], [25, -25, -7, -7], [-2, 5, 5, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_425_n_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_425_2_n_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_425_2_n_b(:prec:=4)
    chi := MakeCharacter_425_n();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_n_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_425_2_n_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_425_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -4, 8, -4, 1]>],Snew);
    return Vf;
end function;