
// Make newform 2520.2.bi.r in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2520_bi();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2520_bi_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_2520_2_bi_r();".
// 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_2520_2_bi_r();".
// 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 := [588, 0, 1212, 0, 855, 0, 247, 0, 29, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-812, 642, 0, 4121, 0, 2607, 0, 467, 0, 19], [-8190, 3252, -13349, 4701, -5530, 316, -763, -171, -28, -10], [10360, 1794, 22386, -4121, 12124, -2607, 2086, -467, 84, -19], [-2400, 0, -2583, 0, -781, 0, -89, 0, -3, 0], [-35560, -28886, -39662, -34381, -9870, -11845, -686, -1495, -14, -53], [-25396, -11680, -31430, -16300, -10997, -6378, -1386, -872, -49, -32], [-4368, 0, -6089, 0, -2437, 0, -347, 0, -13, 0], [-35560, 28886, -39662, 34381, -9870, 11845, -686, 1495, -14, 53], [25396, -11680, 31430, -16300, 10997, -6378, 1386, -872, 49, -32]];
    Rf_basisdens := [1, 1624, 812, 1624, 58, 1624, 812, 58, 1624, 812];
    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_2520_bi();"
function MakeCharacter_2520_bi()
    N := 2520;
    order := 3;
    char_gens := [631, 1261, 281, 2017, 1081];
    v := [3, 3, 3, 3, 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_2520_bi_Hecke();"
function MakeCharacter_2520_bi_Hecke(Kf)
    N := 2520;
    order := 3;
    char_gens := [631, 1261, 281, 2017, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 1], [0, 0, 0, -1, 0, -2, 1, 0, 1, -1], [1, 0, -1, 0, -1, 0, 1, 0, 1, 0], [-1, -2, 1, 0, 1, 1, 0, 0, -2, 0], [0, 0, 0, 0, -1, 0, -1, 1, 0, 1], [1, 1, 1, 1, 0, 2, 0, 0, -2, 2], [0, 2, 0, 0, 0, 0, 0, 0, 1, 1], [2, 0, 0, 0, 0, 0, -1, -1, 0, 1], [0, 0, 0, 0, 1, 1, -3, 0, 1, 3], [1, 0, -1, -1, -1, 0, 2, -1, 0, -2], [-2, -2, -1, 1, 0, 0, 0, 0, 0, 0], [1, 1, -1, -1, 0, -1, 0, 0, 1, -1], [1, 1, -1, 0, -1, 2, 3, 0, -1, 0], [-2, -2, 0, 0, 0, 3, 0, 0, -3, 3], [-4, 0, 0, 0, 1, 0, -1, 1, 0, 1], [-1, -1, 1, -2, 0, 0, 1, 0, -1, 1], [-2, 2, 2, -1, 2, 2, 1, -1, -1, 2], [4, 0, 0, 0, 0, 1, 3, -1, 1, -3], [-1, 3, 1, -1, 1, -1, -1, -1, 1, 0], [6, 0, 0, 0, -1, 1, 0, -2, 1, 0], [-1, -1, 1, 1, 0, 1, 2, 0, -3, 3], [1, -1, -1, 0, -1, -1, 0, 0, 4, 2], [0, -2, 0, 1, 0, 1, 0, 1, -2, 0], [0, 0, 0, -1, 0, 1, 1, 0, -2, 2], [2, 0, 0, 0, 0, 0, 2, 0, 0, -2], [-2, 0, 0, 0, -1, 2, -3, 2, 2, 3], [-1, -1, 1, 0, 1, 1, 0, 0, -1, 1], [-8, -8, 2, 1, 0, 0, 1, 0, -1, 1], [6, 0, 0, 0, 0, 1, -2, -1, 1, 2], [2, 0, -2, 0, -2, -2, 0, 0, 6, 2], [-7, -7, -1, 2, 0, -1, 1, 0, 0, 0], [5, 5, -1, 0, 0, -1, 2, 0, -1, 1], [2, 10, -2, 2, -2, 2, 2, 2, -2, 0], [1, 0, 0, 0, -1, 1, -1, 0, 1, 1], [-1, 2, 1, -1, 1, 2, 2, -1, -2, 0], [1, 1, -1, 0, 0, 1, -2, 0, 1, -1], [8, 0, 0, 0, 0, 3, -6, 3, 3, 6], [0, 2, 0, -2, 0, -4, -2, -2, 6, 0], [-5, -5, -1, 0, 0, -2, 3, 0, -1, 1], [5, 0, 0, 0, 1, -1, -3, -2, -1, 3], [5, 5, -1, 2, 0, 3, -1, 0, -2, 2], [1, 0, 0, 0, 0, -3, 2, 1, -3, -2], [12, 0, 0, 0, -2, 0, 4, 0, 0, -4], [6, 6, -2, 3, 0, 2, -1, 0, -1, 1], [-1, -1, 1, 1, 1, -2, -4, 1, 4, 4], [-2, -2, 2, 0, 2, 0, -2, 0, -4, -2], [-2, 0, 0, 0, 0, 0, 4, -4, 0, -4], [3, 3, -2, 1, 0, 5, -3, 0, -2, 2], [12, 0, 0, 0, -2, -2, 1, 3, -2, -1], [4, 8, -4, 3, -4, -3, -2, 3, 4, 0], [6, 6, -2, -2, 0, 0, -4, 0, 4, -4], [0, 0, 2, 2, 0, 2, 2, 0, -4, 4], [-3, -9, 3, -2, 3, -2, -3, -2, -1, -2], [-11, -11, 1, -4, 0, -2, -1, 0, 3, -3], [-2, 0, 0, 0, -1, -2, -2, 0, -2, 2], [9, 9, 1, -2, 0, -4, 3, 0, 1, -1], [-1, 0, 0, 0, -3, 4, -4, 3, 4, 4], [2, 0, 0, 0, 2, 2, 3, -4, 2, -3], [-15, -15, 1, -1, 0, -1, 2, 0, -1, 1], [-1, -1, 1, -2, 1, -1, 0, -2, 0, -2], [0, -4, 0, -3, 0, -1, 2, -3, 6, 2], [0, 15, 0, 0, 0, -4, -4, 0, 2, -2], [6, 0, 0, 0, -2, -4, 1, 2, -4, -1], [0, 0, 4, -4, 0, -4, 2, 0, 2, -2], [8, 0, 0, 0, -2, -2, 6, 0, -2, -6], [10, 10, 0, 1, 0, -2, -3, 0, 5, -5], [-1, -13, 1, 1, 1, 7, 5, 1, -9, 0], [11, 11, -1, 0, 0, 4, -4, 0, 0, 0], [-4, 12, 4, 2, 4, 1, -5, 2, -5, 2], [-1, 0, 0, 0, -4, 0, 4, 2, 0, -4], [1, -4, -1, 3, -1, 2, 0, 3, -6, -2], [-9, -9, -3, -1, 0, 1, 2, 0, -3, 3], [1, -1, -1, 4, -1, -3, -6, 4, 0, 0], [-3, 11, 3, 4, 3, 1, -6, 4, -3, 5], [3, 3, 3, -1, 0, -2, 0, 0, 2, -2], [0, 0, 0, 0, 3, 2, -4, -2, 2, 4], [2, 0, 0, 0, 0, 1, 6, 1, 1, -6], [24, 24, -2, 2, 0, 2, 2, 0, -4, 4], [12, 0, 0, 0, 2, -4, 3, 0, -4, -3], [1, -5, -1, 2, -1, 2, 1, 2, -5, -2], [2, 8, -2, 0, -2, 2, 4, 0, -6, -6], [10, 0, 0, 0, -1, -2, 2, 2, -2, -2], [-2, 14, 2, 0, 2, -1, -3, 0, -5, -4], [0, 0, 0, 0, 1, 2, -7, -3, 2, 7], [-2, 0, 0, 0, 5, -2, -1, -2, -2, 1], [0, -10, 0, -4, 0, 0, 4, -4, 2, -2], [2, 2, 0, 0, 0, -8, 4, 0, 4, -4], [-6, -6, 2, -4, 0, -1, -2, 0, 3, -3], [-8, 0, 0, 0, -4, -2, 2, -4, -2, -2], [-1, 19, 1, 3, 1, 0, -4, 3, -2, 2], [-4, 0, 0, 0, 4, 4, 0, 0, 4, 0], [-2, 2, 2, -2, 2, 4, 4, -2, -2, 2], [-11, -11, -3, 0, 0, -7, 4, 0, 3, -3], [0, -14, 0, 0, 0, -1, -1, 0, -1, -2], [1, 13, -1, -1, -1, 4, 6, -1, -6, -4], [10, 0, 0, 0, 2, 2, 4, -2, 2, -4], [-16, -16, 3, 3, 0, 4, 2, 0, -6, 6], [-2, -2, 4, -4, 0, 2, -4, 0, 2, -2], [0, -2, 0, -1, 0, -4, -3, -1, 2, -3], [-10, -10, 0, 3, 0, -7, 5, 0, 2, -2], [0, 0, 2, 0, 0, -5, 0, 0, 5, -5], [-4, 0, 0, 0, -2, 1, -3, -1, 1, 3], [0, 8, 0, -1, 0, 1, 2, -1, -8, -8], [22, 22, 0, 2, 0, 0, 2, 0, -2, 2], [12, 0, 0, 0, 0, 3, 2, 1, 3, -2], [5, 3, -5, 2, -5, 0, 3, 2, 2, -1], [-2, -2, -4, -2, 0, -4, -1, 0, 5, -5], [18, 0, 0, 0, 4, -4, 8, -2, -4, -8], [11, 11, -2, -6, 0, 0, -2, 0, 2, -2], [2, 0, 0, 0, 0, -4, 6, -4, -4, -6], [-6, -6, 4, 1, 0, -6, 2, 0, 4, -4], [8, 0, 0, 0, -2, 2, -5, 0, 2, 5], [2, 2, 5, 2, 0, 4, 1, 0, -5, 5], [1, 2, -1, 2, -1, 7, 6, 2, -6, 2], [-4, 0, 0, 0, -2, -4, 4, 4, -4, -4], [8, 8, 2, 3, 0, 3, -1, 0, -2, 2], [4, 0, 0, 0, 0, -2, -1, 8, -2, 1], [-3, 2, 3, 6, 3, 7, -2, 6, -6, 10], [16, 16, 2, 5, 0, 8, -5, 0, -3, 3], [3, -5, -3, -1, -3, -6, -2, -1, 0, -10], [12, 0, 0, 0, 3, -2, -7, 1, -2, 7], [1, 27, -1, 0, -1, -6, -5, 0, 5, -2], [2, -6, -2, 0, -2, 0, 2, 0, 4, 2], [8, 0, 0, 0, -4, 5, -1, 2, 5, 1], [1, -7, -1, 4, -1, 0, -3, 4, -4, -1], [-4, -4, -3, -2, 0, 5, -3, 0, -2, 2], [-13, 0, 0, 0, 5, -1, 1, 0, -1, -1], [1, 17, -1, 3, -1, -6, -8, 3, 4, 0], [-8, 0, 0, 0, 6, -6, 2, 0, -6, -2], [-4, 6, 4, -7, 4, 0, 3, -7, -4, -7], [-1, 0, 0, 0, 3, -1, 0, -4, -1, 0], [6, 6, -5, 2, 0, 2, -3, 0, 1, -1], [4, 4, 4, -2, 0, 4, 0, 0, -4, 4], [-24, 0, 0, 0, 0, 3, -3, -1, 3, 3], [-17, -17, -1, 0, 0, -3, -2, 0, 5, -5], [17, 0, 0, 0, -4, -5, 4, 3, -5, -4], [1, 21, -1, -1, -1, 5, 7, -1, -1, 2], [-27, -27, 5, 2, 0, 5, 1, 0, -6, 6], [2, 0, 0, 0, 0, -2, -6, 2, -2, 6], [8, 8, 4, 1, 0, -3, 5, 0, -2, 2], [-6, 0, 0, 0, -7, 0, 1, 1, 0, -1], [10, 0, 0, 0, -2, 3, -1, 2, 3, 1], [26, 26, 4, 1, 0, 6, -5, 0, -1, 1], [-1, 5, 1, -2, 1, -2, -1, -2, 5, 2], [1, -22, -1, -5, -1, -2, 4, -5, 6, -2], [3, 7, -3, -4, -3, -11, -4, -4, 15, -3], [6, 0, 0, 0, -1, -4, 2, 4, -4, -2], [0, 0, 0, 0, 1, -1, 3, 0, -1, -3], [2, -16, -2, -7, -2, -5, 4, -7, 8, -6], [9, 9, -9, 2, 0, -2, -1, 0, 3, -3], [-24, 0, 0, 0, 1, -6, -1, 3, -6, 1], [1, 1, -1, 1, -1, 0, 0, 1, 2, 2], [6, 16, -6, -3, -6, -8, 1, -3, 4, -13], [36, 0, 0, 0, -1, -1, 1, 0, -1, -1], [-25, -25, -1, -1, 0, 9, -2, 0, -7, 7], [-2, -4, 2, -1, 2, 7, 6, -1, -4, 4], [24, 0, 0, 0, -2, 0, 6, 2, 0, -6], [14, 0, 0, 0, 3, 3, -7, 0, 3, 7], [-3, -11, 3, 2, 3, -1, -6, 2, 5, 9], [-14, -14, -6, 5, 0, 0, -5, 0, 5, -5], [-2, -2, -5, 6, 0, 4, -1, 0, -3, 3], [-4, -4, -6, -5, 0, -13, 5, 0, 8, -8], [-1, -1, -1, 2, 0, 2, -1, 0, -1, 1], [-3, 0, 0, 0, -2, 2, 0, 4, 2, 0], [-3, 14, 3, -6, 3, 5, 8, -6, -4, -2], [4, 4, 0, 2, 0, -8, 8, 0, 0, 0], [4, 0, 0, 0, -1, 7, -2, -4, 7, 2], [-5, -1, 5, -3, 5, -1, -3, -3, 5, 6], [-2, -2, 2, 4, 0, 7, 0, 0, -7, 7], [7, 7, -5, 1, 0, -4, 4, 0, 0, 0], [-24, 0, 0, 0, -6, 4, 4, 6, 4, -4], [3, 0, 0, 0, 4, 2, -14, 0, 2, 14], [2, 2, 0, 0, 0, 2, 0, 0, -2, 2], [30, 0, 0, 0, 3, 2, -2, -2, 2, 2], [-1, -21, 1, 3, 1, -2, -6, 3, 2, 4], [0, 16, 0, 10, 0, 12, 2, 10, -13, 9], [32, 0, 0, 0, 2, -2, 3, -1, -2, -3], [12, 0, 0, 0, -5, -4, 9, 2, -4, -9], [-6, 10, 6, 2, 6, -4, -12, 2, -2, 2], [18, 18, 3, 1, 0, -8, 2, 0, 6, -6], [32, 32, -2, 0, 0, -4, 0, 0, 4, -4], [-9, -9, 5, 4, 0, 1, 2, 0, -3, 3], [15, 15, -1, 4, 0, 8, -3, 0, -5, 5], [3, 24, -3, 6, -3, 8, 5, 6, -5, 6], [-37, -37, 1, 3, 0, -1, 2, 0, -1, 1], [0, -18, 0, 6, 0, 1, -5, 6, 3, 10], [-6, 0, 0, 0, 4, -3, -3, -5, -3, 3], [1, 18, -1, 3, -1, -2, -4, 3, 0, 0], [6, 20, -6, 2, -6, 4, 8, 2, 0, 0], [2, 2, -2, -3, 0, 4, 1, 0, -5, 5], [-4, -4, 4, -1, 0, 6, -7, 0, 1, -1], [-19, -19, -2, -8, 0, -4, 6, 0, -2, 2], [-1, 17, 1, 2, 1, 6, 3, 2, -3, 6], [-18, 0, 0, 0, 8, 2, -6, 2, 2, 6], [0, 6, 0, 3, 0, 11, 8, 3, -8, 6], [-22, -22, 6, -4, 0, 4, -2, 0, -2, 2], [14, 0, 0, 0, 7, -6, 10, -7, -6, -10], [-4, 8, 4, -2, 4, -4, -6, -2, 6, 4], [11, 11, -4, -1, 0, -3, -3, 0, 6, -6], [24, 0, 0, 0, -3, 2, 5, -5, 2, -5], [20, 20, 1, -1, 0, -10, 4, 0, 6, -6], [-2, 0, 2, -4, 2, 8, 10, -4, -4, 2], [2, 4, -2, 8, -2, 2, -4, 8, -2, 6], [14, 0, 0, 0, 3, -4, 4, -2, -4, -4], [-11, -11, -2, -3, 0, 1, 7, 0, -8, 8], [-4, -6, 4, 0, 4, -1, -5, 0, -3, 0], [20, 0, 0, 0, -1, 0, 9, -5, 0, -9], [4, 0, 0, 0, -4, 0, 2, 6, 0, -2], [-4, -40, 4, 0, 4, 0, -4, 0, -2, 2], [-3, -3, -5, -7, 0, -7, 4, 0, 3, -3], [-1, -8, 1, 2, 1, 6, 3, 2, 1, 10], [-5, -5, 7, 4, 0, 15, -2, 0, -13, 13], [-11, -11, 1, 6, 0, -3, 0, 0, 3, -3], [1, -46, -1, 1, -1, 0, 0, 1, 2, 2], [17, 0, 0, 0, 1, 4, -4, -5, 4, 4], [4, 14, -4, 1, -4, 4, 7, 1, -1, 0], [26, 0, 0, 0, -2, 3, -8, 5, 3, 8], [2, -12, -2, 3, -2, 6, 5, 3, -14, -7], [5, 7, -5, 4, -5, -3, -2, 4, 9, 5], [8, 0, 0, 0, 2, -2, -6, 8, -2, 6], [6, 0, 0, 0, 3, -3, 8, -8, -3, -8], [6, 20, -6, 10, -6, -8, -12, 10, 8, 4], [8, 8, 2, -6, 0, -2, 2, 0, 0, 0], [2, -6, -2, -4, -2, 4, 10, -4, -1, -3], [-3, 37, 3, 3, 3, -3, -9, 3, -5, -2], [-14, -14, 0, -2, 0, -1, -3, 0, 4, -4], [-8, -8, 1, 9, 0, 6, -2, 0, -4, 4], [14, 0, 0, 0, -8, -5, 4, 3, -5, -4], [-14, -14, -8, 7, 0, 12, -8, 0, -4, 4], [-6, 0, 0, 0, 3, 1, 1, 2, 1, -1], [6, 6, 0, -9, 0, 2, -1, 0, -1, 1], [-3, -9, 3, -9, 3, -4, 2, -9, 8, -2], [3, 6, -3, 0, -3, -5, -2, 0, 10, 2], [0, 0, 0, 0, 1, -10, 10, 2, -10, -10], [24, 0, 0, 0, -5, 12, -3, -5, 12, 3], [4, 4, -4, -5, 0, -8, 5, 0, 3, -3], [7, -8, -7, 4, -7, -2, 1, 4, 5, 0], [-8, -8, -4, 0, 0, -1, 4, 0, -3, 3], [-13, 5, 13, 0, 13, 0, -13, 0, -1, 12], [10, 0, 0, 0, 0, 0, 8, -8, 0, -8], [-4, -16, 4, 0, 4, -4, -8, 0, 2, 2], [-16, 0, 0, 0, -2, 7, -7, -4, 7, 7], [-28, -28, 0, 3, 0, 6, -2, 0, -4, 4], [-13, -13, -1, 0, 0, -11, 7, 0, 4, -4], [-7, 0, 0, 0, -5, -6, 12, 5, -6, -12], [0, 0, 0, 0, -4, 0, 3, -2, 0, -3], [-3, 27, 3, 4, 3, 5, -2, 4, -17, -5], [18, 18, 1, -1, 0, 8, -6, 0, -2, 2], [-7, 0, 0, 0, 1, -5, 10, 6, -5, -10], [16, 16, 10, -4, 0, 4, 2, 0, -6, 6], [4, 0, -4, 1, -4, 10, 13, 1, 2, 9], [26, 26, -4, 5, 0, 7, -3, 0, -4, 4], [6, 6, 0, 10, 0, 3, -4, 0, 1, -1], [-23, 0, 0, 0, -3, -3, 1, -4, -3, -1], [4, -6, -4, 2, -4, 0, 2, 2, -4, -6], [-6, -6, -10, -4, 0, -9, -1, 0, 10, -10], [16, 0, 0, 0, -2, 10, 6, -6, 10, -6], [13, 13, 1, 5, 0, 8, 4, 0, -12, 12], [0, 0, 0, 0, 2, 2, 9, 2, 2, -9], [30, 30, 2, -4, 0, 6, -6, 0, 0, 0], [0, 27, 0, 8, 0, 10, 2, 8, -8, 10], [34, 0, 0, 0, 1, -2, 7, -1, -2, -7], [18, 0, 0, 0, -5, 4, -14, 12, 4, 14], [14, 0, 0, 0, -1, 3, -10, 6, 3, 10], [-26, -26, 1, 4, 0, 4, 5, 0, -9, 9], [0, 10, 0, 5, 0, 2, -3, 5, -7, 0], [43, 43, 1, 0, 0, 7, -3, 0, -4, 4], [-13, -13, 5, 4, 0, 0, 7, 0, -7, 7], [-1, -13, 1, 3, 1, 6, 2, 3, -12, -2], [8, 0, 0, 0, -2, -6, 0, 14, -6, 0], [5, 19, -5, 8, -5, -9, -12, 8, 9, 3], [-8, -16, 8, -5, 8, -9, -12, -5, 6, 0], [23, 23, -11, 11, 0, -8, 6, 0, 2, -2], [5, -4, -5, -2, -5, 0, 7, -2, 1, -6], [7, -13, -7, -2, -7, 3, 12, -2, -1, -7], [-10, -10, 3, 5, 0, 0, 6, 0, -6, 6], [-12, -12, -6, 1, 0, 5, 1, 0, -6, 6], [-37, 0, 0, 0, 1, 8, 2, 1, 8, -2], [-2, 0, 0, 0, 4, 3, 12, -7, 3, -12], [-8, -6, 8, 4, 8, 10, -2, 4, -18, 4], [9, 9, -9, 1, -9, 9, 17, 1, -13, -12], [19, 19, 3, 2, 0, -4, 3, 0, 1, -1], [25, 0, 0, 0, -1, 5, -9, -2, 5, 9], [-14, -14, 4, -7, 0, -3, -1, 0, 4, -4], [-6, 0, 0, 0, -2, 4, -5, -7, 4, 5], [4, 4, -2, -8, 0, -8, -4, 0, 12, -12], [40, 40, 2, 0, 0, 2, -6, 0, 4, -4], [4, 10, -4, -7, -4, -1, 10, -7, 6, -6], [60, 0, 0, 0, 2, 0, -3, -1, 0, 3], [2, 0, 0, 0, 12, 1, -1, -2, 1, 1], [20, 20, -8, 4, 0, 0, 4, 0, -4, 4], [0, 2, 0, 3, 0, -6, -9, 3, -3, -6], [-8, 0, 0, 0, -2, -5, -7, 9, -5, 7], [2, -10, -2, 3, -2, 12, 11, 3, -22, -9], [-30, 0, 0, 0, -3, 7, -5, -2, 7, 5], [-2, 14, 2, -4, 2, -2, 0, -4, 4, 0], [1, -12, -1, 1, -1, -10, -10, 1, 6, -4], [-3, -3, 5, -8, 0, -13, 3, 0, 10, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2520_bi_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2520_2_bi_r();".
// 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_2520_2_bi_r(:prec:=10)
    chi := MakeCharacter_2520_bi();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2520_2_bi_r();".
// 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_2520_2_bi_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2520_bi();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![448900, 146060, 125244, 29652, 21054, 4554, 1695, 150, 45, 2, 1]>,<13,R![471, 525, 44, -42, -3, 1]>],Snew);
    return Vf;
end function;