
// Make newform 520.2.q.f in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_520_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_520_q_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_520_2_q_f();".
// 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_520_2_q_f();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_520_q();"
function MakeCharacter_520_q()
    N := 520;
    order := 3;
    char_gens := [391, 261, 417, 41];
    v := [3, 3, 3, 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_520_q_Hecke();"
function MakeCharacter_520_q_Hecke( : Kf := false)
    N := 520;
    order := 3;
    char_gens := [391, 261, 417, 41];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 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, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0], [0, 0, 3, 0], [1, 1, -1, 2], [4, 1, -4, 1], [1, 0, -1, 0], [0, 1, 0, 0], [0, 1, -4, 0], [2, 0, 0, -2], [0, -2, -7, 0], [0, -1, 6, 0], [6, 3, -6, 3], [-1, 0, 0, 3], [-1, 0, 0, 1], [8, -1, -8, -1], [-2, 1, 2, 1], [0, 1, 0, 0], [-2, -7, 2, -7], [0, 0, 0, -2], [6, 0, 0, -2], [4, 0, 0, 0], [0, -6, 7, 0], [-8, -7, 8, -7], [0, -7, 0, 0], [-9, 0, 0, -5], [0, -1, 10, 0], [-2, 0, 0, -4], [4, 5, -4, 5], [0, -2, 7, 0], [-15, 0, 0, 1], [4, -3, -4, -3], [-3, 0, 3, 0], [-6, -3, 6, -3], [-6, 0, 0, 2], [5, 0, 0, -5], [8, 9, -8, 9], [0, -4, 1, 0], [-7, -2, 7, -2], [0, -1, 8, 0], [-10, 0, 0, 4], [-10, -7, 10, -7], [0, -1, -20, 0], [0, -4, -1, 0], [-2, 5, 2, 5], [0, 8, -13, 0], [0, 4, 17, 0], [-8, 5, 8, 5], [-2, 0, 0, -4], [-22, 0, 0, 0], [14, 0, 0, 6], [1, -8, -1, -8], [9, 4, -9, 4], [0, 3, 20, 0], [0, -12, 9, 0], [8, -5, -8, -5], [0, 11, -4, 0], [-15, -2, 15, -2], [10, 0, 0, 12], [0, 11, 2, 0], [-5, 8, 5, 8], [2, 0, 0, 14], [-12, 0, 0, 4], [-2, 0, 0, 4], [-17, 0, 0, -3], [0, -1, 0, -1], [-14, 0, 0, 4], [16, 1, -16, 1], [0, 9, -16, 0], [0, -1, -24, 0], [-14, 0, 0, 2], [0, -3, -20, 0], [-22, -9, 22, -9], [0, -6, -3, 0], [-16, -3, 16, -3], [8, 0, 0, 14], [1, -2, -1, -2], [0, 2, -1, 0], [-1, -10, 1, -10], [0, -5, 0, 0], [4, 0, 0, 2], [0, -9, 8, 0], [-4, -15, 4, -15], [0, 13, -2, 0], [-18, 0, 0, -2], [15, 2, -15, 2], [0, -11, 14, 0], [-8, 11, 8, 11], [8, 0, 0, -12], [-20, 0, 0, 0], [0, 3, 28, 0], [17, 6, -17, 6], [0, 4, -5, 0], [12, 0, 0, -4], [-27, -10, 27, -10], [0, -9, 10, 0], [29, 0, 0, -5], [0, -9, 30, 0], [6, 0, 0, 12], [-22, 0, 0, 2], [0, 20, -5, 0], [-4, -19, 4, -19], [0, 2, 19, 0], [-1, 0, 0, 11], [18, 0, 0, 8], [0, 3, 6, 0], [4, 0, 0, -6], [-14, 0, 0, -6], [0, -6, 19, 0], [-33, 4, 33, 4], [0, -6, -7, 0], [-6, -5, 6, -5], [-23, 0, 0, -7], [24, 15, -24, 15], [1, 4, -1, 4], [24, 5, -24, 5], [0, 2, -29, 0], [0, 0, -25, 0], [40, 3, -40, 3], [0, 3, -34, 0], [0, 11, -12, 0], [22, 0, 0, 8], [22, 11, -22, 11], [0, -4, -13, 0], [-12, 0, 0, 2], [-24, 3, 24, 3], [40, 7, -40, 7], [35, 0, 0, 3], [-9, 0, 0, -11], [0, -10, 33, 0], [0, -3, 4, 0], [0, 3, -24, 0], [0, 2, 17, 0], [-29, -10, 29, -10], [0, 11, 22, 0], [-3, 2, 3, 2], [-4, -15, 4, -15], [18, 11, -18, 11], [0, -1, 14, 0], [17, 0, 0, 13], [0, 7, -14, 0], [1, -2, -1, -2], [18, 0, 0, -2], [0, 5, 24, 0], [18, 13, -18, 13], [18, 0, 0, 8], [22, 0, 0, -12], [-13, 0, 0, -13], [-8, 0, 0, -8], [-2, -13, 2, -13], [0, 18, -13, 0], [10, 0, 0, 6], [0, 8, 17, 0], [0, -13, 26, 0], [20, 0, 0, 16], [34, 13, -34, 13], [-6, -23, 6, -23], [-2, 0, 0, 20], [-16, 0, 0, -18], [0, 3, 18, 0], [22, 3, -22, 3], [-1, 0, 0, -5], [-23, 4, 23, 4], [0, 3, -20, 0], [37, 0, 0, -3], [0, 13, 6, 0], [-1, -8, 1, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_520_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_520_2_q_f();".
// 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_520_2_q_f(:prec:=4)
    chi := MakeCharacter_520_q();
    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_520_2_q_f();".
// 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_520_2_q_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_520_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 4, 5, -1, 1]>,<11,R![9, -3, 1]>],Snew);
    return Vf;
end function;