
// Make newform 513.2.y.c in Magma, downloaded from the LMFDB on 11 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_513_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_513_y_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_513_2_y_c();".
// 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_513_2_y_c();".
// 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, -1, 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_513_y();"
function MakeCharacter_513_y()
    N := 513;
    order := 9;
    char_gens := [191, 325];
    v := [9, 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_513_y_Hecke();"
function MakeCharacter_513_y_Hecke( : Kf := false)
    N := 513;
    order := 9;
    char_gens := [191, 325];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 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 := [[0, 1, 1, 1, 0, -1], [0, 0, 0, 0, 0, 0], [0, -1, -1, 1, 1, 0], [3, 1, 0, -3, -1, -1], [0, 2, -1, 3, -1, 2], [1, -1, 0, 0, 0, 3], [-4, 1, -4, 0, 0, 0], [-2, 2, -2, 0, -4, 1], [1, 0, 1, 0, 1, 0], [-3, -3, 2, 2, 1, -2], [0, 5, -2, 0, -3, -3], [-9, 2, 2, 0, 0, -2], [-1, 0, 1, 0, -7, -1], [2, -6, -1, 1, 6, -2], [-1, -1, 2, 4, -3, -2], [-1, 4, -2, 4, -1, 0], [3, 1, 1, -2, 0, 2], [-1, 4, -3, 4, -1, 0], [-1, -1, 3, 1, 0, -3], [-1, -1, 6, -6, 1, 1], [-3, 0, 3, -1, -6, -4], [9, 0, -9, -7, 0, 2], [0, 0, -3, 0, 3, 3], [0, -5, 0, 5, 5, 11], [8, 1, 7, -1, 0, 1], [-6, 4, 0, 2, 2, 5], [0, 4, -1, 0, -1, 4], [3, -3, 9, -3, -6, -6], [-1, -7, -5, 5, 7, 1], [-3, 1, 1, 0, -5, 4], [-2, -2, 0, 4, 4, 10], [6, 1, -5, -11, 0, 11], [-9, -4, 1, -4, -9, 0], [5, -9, 0, 4, 4, 12], [-1, 0, 1, -3, 11, -4], [5, 0, 0, 0, 3, -3], [9, 0, 1, -1, 0, -9], [0, 5, 0, -9, 0, 5], [11, -8, -5, -8, 11, 0], [7, 1, 12, 5, 0, -5], [0, -3, 6, 3, 6, -3], [1, 1, 7, -8, 7, -7], [0, -5, -5, 0, 1, 4], [1, 0, -1, 9, 0, 10], [-18, 4, 1, 18, -5, -5], [-12, -12, 6, 5, 7, -6], [-1, 3, 12, 13, 0, -13], [0, 9, 7, -7, -9, 0], [12, -15, -15, 0, 6, 9], [-12, 0, 0, 0, 4, -4], [5, 17, 12, -12, -17, -5], [0, -2, 4, 6, 4, -2], [0, 1, 0, -1, -1, 9], [-2, -3, -11, -3, -2, 0], [-4, -4, 14, 1, 3, -14], [4, 0, -4, -10, 5, -6], [8, 0, -8, 0, 5, 8], [2, -7, 4, -4, 7, -2], [0, 6, 0, 16, 0, 6], [9, 5, -14, 5, 9, 0], [5, 4, 1, -4, 0, 4], [0, -4, -1, -12, -1, -4], [-1, 0, 1, 2, -4, 1], [6, -1, 2, -6, -1, -1], [-5, -5, 22, -2, 7, -22], [8, -9, 0, 1, 1, 5], [0, 9, -11, -3, -11, 9], [14, 12, -4, 4, -12, -14], [12, 2, -1, 1, -2, -12], [3, 4, -9, -3, 5, 5], [0, 6, -12, -24, -12, 6], [7, -5, -6, -13, 0, 13], [3, -8, 0, 5, 5, -9], [16, -9, 12, -16, -3, -3], [-6, 9, 9, 0, -2, -7], [-9, 0, 9, 10, 8, 1], [-16, -16, 11, 13, 3, -11], [9, 0, -1, -10, 0, 10], [-13, 16, -7, 6, 0, -6], [1, 1, -8, -5, 4, 8], [18, -13, -13, 0, -1, 14], [-10, 0, 10, -7, -7, -17], [-8, 11, 0, -3, -3, 20], [2, -8, 3, -8, 2, 0], [6, 9, -7, -13, 0, 13], [17, -12, 0, -5, -5, -1], [9, -4, -1, -9, 5, 5], [15, 10, 10, 0, 5, -15], [-16, 14, -3, 3, -14, 16], [9, -9, 14, -9, -5, -5], [0, 18, -3, 3, -3, 18], [4, -9, 4, -9, 4, 0], [-5, -3, -15, 5, 18, 18], [21, 0, -21, -17, -4, 4], [-11, -18, -21, 21, 18, 11], [-18, -18, 14, 2, 16, -14], [0, -16, 1, -16, 0, 0], [0, 7, -2, 0, -2, 7], [-4, -4, 24, 1, 3, -24], [1, 1, 9, 12, -13, -9], [0, -16, -6, -16, 0, 0], [4, -1, 0, -3, -3, -28], [-18, 8, -10, 18, 2, 2], [-3, 12, 12, 0, -3, -9], [15, -7, -7, 0, -9, 16], [-17, 3, 6, 17, -9, -9], [3, 13, 13, 10, 0, -10], [-8, -6, -5, -6, -8, 0], [-5, -5, 23, 3, 2, -23], [-3, -11, 0, 3, 11, 11], [3, 23, 23, 0, -18, -5], [-13, 5, -14, 14, -5, 13], [-5, -5, 2, 24, -19, -2], [0, -14, 17, -9, 17, -14], [-8, -3, -15, -3, -8, 0], [-2, -19, 13, -13, 19, 2], [-14, 0, 14, 12, 3, -2], [-6, -19, -19, 0, 14, 5], [-3, 9, -24, 3, 15, 15], [-7, 3, 0, 4, 4, -19], [10, -8, -25, -8, 10, 0], [0, -12, -5, 6, -5, -12], [-3, 10, -2, 3, -8, -8], [9, -7, -7, 0, 14, -7], [31, -9, -6, -31, 15, 15], [-14, -14, -18, -4, 0, 4], [20, -15, 0, -5, -5, -15], [13, 0, -13, 2, -13, 15], [-20, -3, -12, 12, 3, 20], [0, 9, -15, -8, -15, 9], [-16, -6, -6, 10, 0, -10], [-23, -8, -15, 8, 0, -8], [9, 9, 9, 14, -23, -9], [2, 0, -2, 4, -9, 6], [3, -14, -14, 0, 10, 4], [0, 0, 6, -1, 1, -6], [-2, 11, 0, -9, -9, -16], [0, -19, 24, 6, 24, -19], [-27, -13, -13, 0, 17, -4], [0, -2, -5, 33, -5, -2], [4, 4, 0, -8, -8, 4], [-25, 22, 4, 22, -25, 0], [-5, -2, 0, 7, 7, 19], [15, 15, 5, 11, -26, -5], [-3, 13, -30, 3, 17, 17], [-4, 0, 4, 6, -28, 2], [-13, -42, -6, 7, 0, -7], [26, 25, 11, -15, 0, 15], [15, -16, 18, -16, 15, 0], [0, 20, -1, -9, -1, 20], [-6, 0, 6, 8, -9, 2], [-30, 7, -8, 30, 1, 1], [1, 1, 3, -12, 11, -3], [25, -10, 0, -15, -15, -13], [14, 2, 28, -28, -2, -14], [-18, -4, -4, 0, 8, -4], [15, -1, 4, -15, -3, -3], [-3, -8, -8, -5, 0, 5], [-28, 18, 0, 10, 10, 21], [23, 23, -25, -14, -9, 25], [7, 0, -7, 23, 8, 30], [-11, 0, 11, -7, 5, -18], [2, 9, 21, 19, 0, -19], [11, 7, 2, -9, 0, 9], [0, 9, -3, 30, -3, 9], [-15, 2, 2, -2, -2, 15], [-2, 0, 2, 0, -24, -2], [-20, -20, -3, 33, -13, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_513_y_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_513_2_y_c();".
// 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_513_2_y_c(:prec:=6)
    chi := MakeCharacter_513_y();
    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_513_2_y_c();".
// 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_513_2_y_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_513_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 9, 3, 0, -3, 1]>],Snew);
    return Vf;
end function;