
// Make newform 475.2.j.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_475_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_475_j_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_475_2_j_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_475_2_j_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 := [81, 0, -414, 0, 1765, 0, -1596, 0, 1006, 0, -337, 0, 82, 0, -11, 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, 0, 0, 0, 0, 0, 0], [-46638, 0, 198007, 0, -201735, 0, 96538, 0, -26683, 0, 3853, 0, -281, 0, -17, 0], [-284972250, 0, -12081573, 0, 51321701, 0, -71187724, 0, 25578470, 0, -7295401, 0, 1015482, 0, -105339, 0], [45708276, 0, 29973573, 0, -127325701, 0, 140708376, 0, -63458470, 0, 18099401, 0, -2789005, 0, 261339, 0], [-1141592625, 0, 4868604136, 0, -4179410832, 0, 2808027448, 0, -952119472, 0, 238509928, 0, -32273192, 0, 3020113, 0], [-184780641, 0, -43978536, 0, 186817832, 0, -239139277, 0, 93109040, 0, -26556232, 0, 3735013, 0, -383448, 0], [-733877370, 0, 3126614206, 0, -3097432050, 0, 1740211588, 0, -568442422, 0, 129078913, 0, -17040962, 0, 1440028, 0], [0, -931665523, 0, -86033682, 0, 365465234, 0, -451035377, 0, 182145980, 0, -51951034, 0, 7539500, 0, -750126], [0, -1786582273, 0, -122278401, 0, 519430337, 0, -664598549, 0, 258881390, 0, -73837237, 0, 10585946, 0, -1066143], [-1060049574, 0, 4520206150, 0, -4025445729, 0, 2594464276, 0, -875384062, 0, 216623725, 0, -29226746, 0, 2704096, 0], [0, -573803980, 0, -56060109, 0, 238139533, 0, -310327001, 0, 118687510, 0, -33851633, 0, 4750495, 0, -488787], [0, -1239024512, 0, -172067364, 0, 730930468, 0, -902070754, 0, 364291960, 0, -103902068, 0, 15079000, 0, -1500252], [0, 14234057118, 0, -60658992614, 0, 56231857968, 0, -34081774484, 0, 11246493662, 0, -2649325358, 0, 349306330, 0, -30472184], [0, -2156816727, 0, 45709611853, 0, -43943348586, 0, 29524287979, 0, -10118919067, 0, 2507751769, 0, -339328241, 0, 31047565], [0, 6523373457, 0, -27809065204, 0, 25548044169, 0, -15812055040, 0, 5283598924, 0, -1275617842, 0, 171076316, 0, -15452485], [0, -20082252600, 0, 85617164065, 0, -76883173041, 0, 48815148070, 0, -16358061805, 0, 3989805988, 0, -534513455, 0, 48633295]];
    Rf_basisdens := [1, 73569, 104051089, 104051089, 936459801, 104051089, 312153267, 312153267, 312153267, 312153267, 104051089, 312153267, 2809379403, 2809379403, 936459801, 2809379403];
    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_475_j();"
function MakeCharacter_475_j()
    N := 475;
    order := 6;
    char_gens := [77, 401];
    v := [3, 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_475_j_Hecke();"
function MakeCharacter_475_j_Hecke(Kf)
    N := 475;
    order := 6;
    char_gens := [77, 401];
    char_values := [[-1, 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]];
    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, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 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, -1], [0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, -2, 1, 0], [-1, -1, -1, -1, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-1, 0, -3, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -2, 1, 1, 0, -2, 1], [3, 2, 2, 2, 3, -3, 3, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -2, -4, -1, -1], [0, 0, 0, 0, 0, 0, 0, -2, 3, 0, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 3, 0, 0, 0, 0, 0], [0, 5, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -2, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0], [-6, -1, -3, -1, -6, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 3, 1, -1], [5, 1, 2, 1, 5, -1, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -1, 2, 2, 2, -1, 2], [0, -3, 0, 0, -3, 0, 3, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 4, 5, 0, 4], [0, 1, 0, 0, -3, 0, -3, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 2, 2, 3, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 5, -3, -3, -1, 5, 1], [8, -1, -1, -1, 8, 2, -2, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, -2, -2, -3, 2, 2], [0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 2, 6, 0, 0, 0, 0], [3, 0, 8, 0, 3, 1, -1, 0, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 7, 0, -1, -3, 0, 0, 0, 0], [0, -5, 0, 0, 4, 0, -1, 0, 0, 4, 0, 0, 0, 0, 0, 0], [-3, 0, 7, 0, -3, -4, 4, 0, 0, -7, 0, 0, 0, 0, 0, 0], [5, 0, -4, -4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 1, -4, -4, -4, 1, -6], [0, 0, 0, 0, 0, 0, 0, -6, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 2, -6, -4, 7], [3, 0, 5, -4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, -10, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0], [3, 0, -4, 1, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -5, -4, -3, 2], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -6, 10, 10, 2, -6, 3], [0, 3, 0, 0, -2, 0, 4, 0, 0, -3, 0, 0, 0, 0, 0, 0], [-11, -1, -2, -1, -11, 7, -7, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -8, 1, 6, -6], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -4, -9, -9, 2, -4, -2], [4, 0, -2, 4, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 5, -1, 6], [9, 0, -3, 9, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 5, 0, 7, 0, 0, -4, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, -1, 0, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 2, 0, -4], [-3, -6, -6, -6, -3, -4, 4, 0, 0, 6, 0, 0, 0, 0, 0, 0], [13, 0, 5, 0, 13, 6, -6, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -2, -2, -2, -2, -2, 6], [0, 7, 0, 0, -9, 0, 6, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 5, -4], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -3, -6, -6, -5, -3, 5], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 2, -3, -4, -4], [-9, 0, -2, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 7, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, -9, 0, 3, 2, 0, 0, 0, 0], [5, 0, 0, 13, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, -11, -14, 2, -4], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, -2, -5, 3], [-5, 0, -9, 3, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 5, 5, 5, 4, 5, 6], [0, -4, 5, -4, 0, 6, -6, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, -7, 0, -6, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 9, 5, 5, -6, 9, 5], [-8, 0, 3, 5, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 2, -3, 1], [0, 8, 0, 0, -12, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, -4, 8, 0, -3], [-6, -9, -3, -9, -6, -6, 6, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-3, 0, 5, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 14, -2, 14, 4, -7, 7, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, -18, 0, -4, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, -2, 0, 4, 5, 0, 0, 0, 0], [-10, -6, -4, -6, -10, -1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, -5, 0, 0, 12, 0, 0, 0, 0], [9, 0, -7, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 1, 1, -4, 5, -4], [6, -3, -4, -3, 6, -2, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 7, 6, 6, -11, 7, 3], [0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, -9, 0, -9], [0, 12, 0, 0, -6, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -13, -13, -12, -5, 0], [-3, 0, 5, 0, -3, -4, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0], [-13, 4, -7, 4, -13, -9, 9, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 2, 0, 8, 7, 0, 0, 0, 0], [0, 0, 0, 0, 18, 0, -6, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-12, 0, -1, -2, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, -2, 0, -10, 2, 0, 0, 0, 0], [0, 3, 0, 0, 32, 0, 6, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -6, 0, -5, 8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6, -11, 0, -3, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, -10, -10, -13, 2, -1], [-3, 0, -2, -13, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 6, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -1, 0, 0, 10, -1, -8], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 1, 0], [0, 0, 0, 0, 0, 0, 0, -4, 10, 0, -2, 0, 0, 0, 0, 0], [0, -5, 0, 0, -6, 0, 1, 0, 0, -8, 0, 0, 0, 0, 0, 0], [5, 0, -8, -3, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 7, -10, -10, 1, 7, -7], [0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, -5, 13, -5, -3], [0, 0, 0, 0, 0, 0, 0, -3, -1, 0, 5, -1, 0, 0, 0, 0], [19, 0, 2, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 17, 0, 9, 0, 0, -12, 0, 0, 0, 0, 0, 0], [9, -3, -5, -3, 9, -9, 9, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 6, -4, -10, 9], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -11, -7, -7, -4, -11, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 0], [0, 9, 0, 0, 18, 0, -3, 0, 0, 11, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -10, 0, -6, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -10, 5, 5, -3, -10, -10], [0, 0, 0, 0, 0, 0, 0, 0, -9, 0, -5, 3, 3, -6, -5, 9], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -10, 1, 1, -9, -10, 2], [14, 0, 2, 9, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 3, -2, 3, -3, -2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, -2, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 6, 8, 6, -3, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, -2, -5, 5], [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 2, -6, -6, 5, 2, -10], [-1, -17, -1, -17, -1, -1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, -1, -1], [0, -11, 0, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 3, 0, 9], [9, 0, 1, 2, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 9, 0, 0, -23, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 11, -16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 7, -11, -11, -6, 7, -14], [0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 4, 10, 10, -11, 4, 16], [-21, 0, -9, -11, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 12, 0, 0, -1, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 14, 0, 0, -6, 0, 1, 0, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 12, 0, -16, -6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 18, 5, 0, 0, 0, 0], [-11, 0, -19, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-30, 0, -7, 0, -30, 8, -8, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -3, -9, 17, -1], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -7, -2, -2, -15], [0, 11, 0, 0, 4, 0, 6, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 9, 12, 12, 14, 9, 1], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 11, -2, 8, 13], [-15, 0, -14, 1, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -10, -3, 0, 9, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 14, 0, -3, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 12, 10, 4, 6], [-21, 0, -4, -12, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17, 0, -7, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 15, 13, 15, 6, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 8, 0, 9, -3, 0, 0, 0, 0], [0, -11, 0, 0, -27, 0, -1, 0, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -4, -9, 7, -2], [0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, -13, -7, 2, 2], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 7, -4, 8, 4], [-3, -7, 9, -7, -3, -8, 8, 0, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -3, 2, 2, 1, -3, -3], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, -9, -6, 7, 3], [-14, -4, -16, -4, -14, 10, -10, 0, 0, 16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -13, 3, 0, 10, 3, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_475_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_475_2_j_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_475_2_j_c(:prec:=16)
    chi := MakeCharacter_475_j();
    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_475_2_j_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_475_2_j_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1296, 0, -2628, 0, 3529, 0, -2714, 0, 1515, 0, -504, 0, 119, 0, -13, 0, 1]>],Snew);
    return Vf;
end function;