
// Make newform 605.2.e.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_605_e_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_605_2_e_a();".
// 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_605_2_e_a();".
// 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, 0, 0, 0, 16040, 0, 0, 0, 9193, 0, 0, 0, 1315, 0, 0, 0, 67, 0, 0, 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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [14018812, 0, 0, 0, 3732211, 0, 0, 0, 371423, 0, 0, 0, 17445, 0, 0, 0, 253, 0, 0, 0], [0, 68818580, 0, 0, 0, 38655603, 0, 0, 0, 3691749, 0, 0, 0, 94553, 0, 0, 0, 627, 0, 0], [-17207320, 0, 553705724, 0, -27038662, 0, 298958639, 0, -1440730, 0, 40723037, 0, 90014, 0, 1994661, 0, 2794, 0, 29131, 0], [17207320, 0, 553705724, 0, 27038662, 0, 298958639, 0, 1440730, 0, 40723037, 0, -90014, 0, 1994661, 0, -2794, 0, 29131, 0], [0, -352512664, 0, -1088277028, 0, -239341410, 0, -652386595, 0, -37504502, 0, -92769865, 0, -2028502, 0, -4654673, 0, -31178, 0, -68791], [34912152, 0, -1088277028, 0, -54861310, 0, -652386595, 0, -19766122, 0, -92769865, 0, -1510730, 0, -4654673, 0, -26646, 0, -68791, 0], [-34912152, 0, -1088277028, 0, 54861310, 0, -652386595, 0, 19766122, 0, -92769865, 0, 1510730, 0, -4654673, 0, 26646, 0, -68791, 0], [0, 352512664, 0, -1088277028, 0, 239341410, 0, -652386595, 0, 37504502, 0, -92769865, 0, 2028502, 0, -4654673, 0, 31178, 0, -68791], [0, 0, 598127204, 0, 0, 0, 335733979, 0, 0, 0, 47881865, 0, 0, 0, 2437065, 0, 0, 0, 36359, 0], [0, 0, 0, 598127204, 0, 0, 0, 335733979, 0, 0, 0, 47881865, 0, 0, 0, 2437065, 0, 0, 0, 36359], [-62676248, 0, -1983090404, 0, -62382362, 0, -1158063823, 0, -5897806, 0, -165346749, 0, -119326, 0, -8385109, 0, -242, 0, -124787, 0], [62676248, 0, -1983090404, 0, 62382362, 0, -1158063823, 0, 5897806, 0, -165346749, 0, 119326, 0, -8385109, 0, 242, 0, -124787, 0], [28261608, 0, -4426670852, 0, 8305038, 0, -2574550419, 0, 3016346, 0, -371191865, 0, 299354, 0, -19018177, 0, 5830, 0, -284679, 0], [0, 212598072, 0, 6730336708, 0, 92240050, 0, 3843935655, 0, 8869190, 0, 548401669, 0, 258886, 0, 27881629, 0, 2266, 0, 415659], [0, 212598072, 0, -6730336708, 0, 92240050, 0, -3843935655, 0, 8869190, 0, -548401669, 0, 258886, 0, -27881629, 0, 2266, 0, -415659], [0, 0, 0, -4356713556, 0, 0, 0, -2500999739, 0, 0, 0, -356874209, 0, 0, 0, -18133369, 0, 0, 0, -270223], [0, -491828168, 0, -5578503780, 0, -289613954, 0, -3209243037, 0, -43447270, 0, -459796767, 0, -2307622, 0, -23449903, 0, -35226, 0, -350169], [0, -593372344, 0, 18782157644, 0, -339886498, 0, 10768098957, 0, -49390038, 0, 1540572087, 0, -2586742, 0, 78511871, 0, -39274, 0, 1171993]];
    Rf_basisdens := [1, 1, 4721416, 9442832, 18885664, 18885664, 37771328, 37771328, 37771328, 37771328, 18885664, 18885664, 37771328, 37771328, 37771328, 37771328, 37771328, 18885664, 18885664, 37771328];
    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_605_e();"
function MakeCharacter_605_e()
    N := 605;
    order := 4;
    char_gens := [122, 486];
    v := [1, 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_605_e_Hecke();"
function MakeCharacter_605_e_Hecke(Kf)
    N := 605;
    order := 4;
    char_gens := [122, 486];
    char_values := [[0, 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, 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, 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, 0, 0, 0, 0, -1, 0, 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, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [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, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, -1], [0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0], [0, 0, 1, 0, -1, 0, 0, -2, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, 0, 0, -1, -1, 1, -2], [0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-4, 0, 2, 0, 0, 1, 0, -2, 1, 0, 4, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, -2, 1], [0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, -1, 1], [1, 0, 1, 0, -3, 1, 0, 2, 1, 0, -1, 0, 0, -4, 1, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -3, 0, 2, 2, 0, 2, 0, -2, 2, -4, 0, 0, 0, 0, 0], [0, -2, 0, -2, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, -1, 0], [2, 0, -1, 0, 1, -2, 0, 0, -2, 0, -2, 0, 0, 2, -1, 0, 0, 0, 0, 0], [2, 0, 0, 0, 2, -2, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 4, 1, -1], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, -3, 1, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1, -1, 0, 0, 0, 2, 4, -2, 0, 0, 0, 0, 0], [2, 0, -1, 0, 3, -1, 0, -2, -1, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, 0], [0, -1, 0, -2, 0, 0, 5, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, -3, -3, -1], [-3, 0, -2, 0, 0, -1, 0, 2, 1, 0, -3, 0, -1, -2, 2, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, -4, 0, 0, 2, 0, 0, 0, 0, 0, -2, -2, 2, 2, 2], [0, -2, 0, -2, 0, 0, 2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 3, 0, -2, 2], [-4, 0, -1, 0, 0, 0, 0, 1, 0, 0, -4, 0, 0, -1, 1, 0, 0, 0, 0, 0], [0, 3, 0, 3, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 2, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, 0, -2, 0, 0, -4, 3], [2, 0, -1, 0, 0, 2, 0, 1, 2, 0, -2, 0, 0, -3, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, -2, 0, 0, 4, 0, 2, 0, 0, 0, 0, 4, -1, 2, -1], [0, 2, 0, 2, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 2, 2], [0, 4, 0, 0, 0, 0, -3, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 3, 3, 0], [-2, 0, -1, 0, 1, -1, 0, 0, -1, 0, 2, 0, 0, -1, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -3, 0, 0, 3, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -2, -2, 1, 1, 1], [0, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, -9, 0, 0, 0, 1, -1, 2, 5, -5], [0, 0, 0, 0, -4, 0, 0, -1, -1, 0, 4, 0, 4, 0, 4, 0, 0, 0, 0, 0], [-4, 0, 6, 0, 4, -4, 0, -7, 7, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -3, 3, 0, 2, -2, 0, 0, 0, 5, -5, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 1, 0, 0, 3, 0, 5, 0, 0, 0, 4, -4, -3, 2, -2], [0, -2, 0, 3, 0, 0, -5, 0, 0, 2, 0, 0, 0, 0, 0, 1, 1, 3, 3, 3], [0, 0, 0, 0, -1, -1, 0, 3, 3, 0, -2, 0, -5, -5, 0, 0, 0, 0, 0, 0], [0, -1, 0, 3, 0, 0, -4, 0, 0, 0, 0, -5, 0, 0, 0, -2, 0, 1, 4, 0], [6, 0, -1, 0, 1, -2, 0, 2, 2, 0, 6, 0, -5, -1, 1, 0, 0, 0, 0, 0], [0, -1, 0, 1, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0, -5, -5, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 1, 1, 0, 8, 0, 1, 7, -6, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, -1, 0, 5, 0, 0, 0, 5, -5, 1, -2, 2], [0, 2, 0, -2, 0, 0, 1, 0, 0, -2, 0, -3, 0, 0, 0, 0, 2, -1, -1, 4], [0, 3, 0, -2, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, -7, 0, 0, 0, -1], [-2, 0, -6, 0, -4, 4, 0, 5, -5, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 4, 0, -6, 2, 0, 2, 2, 0, 1, 0, 0, -6, 4, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, -2, 0, 0, 6, 0, -3, 0, 0, 0, -1, 1, -7, 8, -8], [0, 0, 0, 0, -2, 0, 0, -2, -2, 0, -2, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, -3, 0, 0, 0, -4, 0, 2, 3, -1], [0, 3, 0, 1, 0, 0, -6, 0, 0, 6, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0], [0, 2, 0, -1, 0, 0, -5, 0, 0, 0, 0, 3, 0, 0, 0, -3, 0, 6, -1, 3], [0, -3, 0, 0, 0, 0, 2, 0, 0, -1, 0, 3, 0, 0, 0, -2, 2, 0, -3, 3], [0, -7, 0, 0, 0, 0, 4, 0, 0, -3, 0, 3, 0, 0, 0, -5, 5, -4, -7, 7], [0, -3, 0, -1, 0, 0, 3, 0, 0, -3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0], [-12, 0, 10, 0, 2, -2, 0, -8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 0, 3, 6, 0, 1, -6, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 0], [-4, 0, 5, 0, -1, 1, 0, -4, 1, 0, 4, 0, 0, 1, 5, 0, 0, 0, 0, 0], [0, 0, 4, 0, -3, 3, 0, -4, 4, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [0, -2, 0, -7, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 3, 3, 1, 1, 1], [0, -6, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 5, -1, -1, -1], [0, 6, 0, -3, 0, 0, 1, 0, 0, -2, 0, -7, 0, 0, 0, 0, 9, -2, -1, -5], [-2, 0, -4, 0, 0, -3, 0, 4, 3, 0, -2, 0, -6, -4, 4, 0, 0, 0, 0, 0], [0, 6, 0, 3, 0, 0, -3, 0, 0, 6, 0, -3, 0, 0, 0, 0, -2, 0, 3, 1], [-2, 0, -1, 0, 0, -3, 0, 1, -3, 0, 2, 0, 0, 12, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, -2, 0, -7, 0, 0, 0, -5, 5, 2, -1, 1], [0, 0, 0, 0, 5, 1, 0, 4, 4, 0, 6, 0, -6, -2, -4, 0, 0, 0, 0, 0], [2, 0, 2, 0, 6, 0, 0, 4, 0, 0, 2, 0, -5, 2, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 6, 0, 3, 3, 0, 4, 0, -5, -9, 4, 0, 0, 0, 0, 0], [-5, 0, -3, 0, -2, 1, 0, 5, 1, 0, 5, 0, 0, 1, -3, 0, 0, 0, 0, 0], [-10, 0, 0, 0, -7, 7, 0, -1, 1, 0, 0, 0, 7, -7, 0, 0, 0, 0, 0, 0], [0, -4, 0, 6, 0, 0, -2, 0, 0, 4, 0, 6, 0, 0, 0, 0, 7, 4, 2, 6], [0, 0, 0, 0, 3, -1, 0, 5, 5, 0, -6, 0, -9, -5, -4, 0, 0, 0, 0, 0], [0, 0, 4, 0, 5, -5, 0, 5, -5, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0], [0, -4, 0, 7, 0, 0, -12, 0, 0, 0, 0, -2, 0, 0, 0, 6, 0, 5, -2, 7], [6, 0, -3, 0, -6, 6, 0, -3, -6, 0, 6, 0, 6, -3, 3, 0, 0, 0, 0, 0], [0, -1, 0, -6, 0, 0, 3, 0, 0, -6, 0, -2, 0, 0, 0, 0, -6, -3, -3, 1], [-2, 0, 0, 0, 1, 5, 0, 1, -5, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, 0, 1, 1, 0, 6, 0, -6, -10, 4, 0, 0, 0, 0, 0], [0, 10, 0, -9, 0, 0, 4, 0, 0, -5, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 2, 0, 1, 0, 0, 6, 0, 0, 0, 0, -2, 0, 0, 0, 1, 0, -7, 4, -5], [6, 0, -9, 0, -6, 9, 0, 3, -9, 0, 6, 0, 7, -9, 9, 0, 0, 0, 0, 0], [4, 0, 1, 0, -7, 6, 0, 6, 6, 0, -4, 0, 0, 2, 1, 0, 0, 0, 0, 0], [0, 2, 0, 1, 0, 0, 1, 0, 0, -2, 0, -3, 0, 0, 0, 0, -12, 2, -1, 1], [0, 0, -9, 0, 6, -3, 0, 3, -3, 0, 0, 0, 0, 10, -9, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, -8, -6, 0], [0, 0, 0, 0, -5, 1, 0, 7, 7, 0, -2, 0, -5, -11, 6, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 3, 0, 0, 6, 0, -3, 0, 0, 0, 4, -4, 4, 3, -3], [0, 0, 0, 0, -4, 2, 0, 1, 1, 0, 28, 0, 3, -3, 6, 0, 0, 0, 0, 0], [-14, 0, 14, 0, -2, 2, 0, -7, 7, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, -9, 0, 0, 0, 0, 2, 0, 0, -7, 0, 1, 0, 0, 0, -7, 7, 6, -9, 9], [0, -5, 0, -1, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 0, -3, 0, 0, -9, 4], [0, 2, 0, -3, 0, 0, 8, 0, 0, -7, 0, 0, 0, 0, 0, -4, -4, -1, -1, -1], [0, -5, 0, -1, 0, 0, 4, 0, 0, -2, 0, 0, 0, 0, 0, -11, -11, -2, -2, -2], [0, 4, 0, 0, 0, 0, 1, 0, 0, 5, 0, -9, 0, 0, 0, 9, -9, -1, 4, -4], [0, -7, 0, 7, 0, 0, -1, 0, 0, 2, 0, 8, 0, 0, 0, 0, 3, 6, 1, 6], [0, -1, 0, 4, 0, 0, -11, 0, 0, 0, 0, -2, 0, 0, 0, 4, 0, 7, 1, 5], [8, 0, 9, 0, -8, 0, 0, -1, 0, 0, -8, 0, 0, -9, 9, 0, 0, 0, 0, 0], [9, 0, -3, 0, -3, 3, 0, 6, 3, 0, -9, 0, 0, -6, -3, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 5, 0, 0, -3, 0, 5, 0, 0, 0, 4, -4, 3, -8, 8], [0, 0, 0, 0, 5, -5, 0, -5, -5, 0, -6, 0, 5, 15, -10, 0, 0, 0, 0, 0], [0, -9, 0, 1, 0, 0, -4, 0, 0, 0, 0, -13, 0, 0, 0, -5, 0, 3, 4, 0], [0, 14, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 1, 1, 1], [0, -6, 0, 0, 0, 0, -3, 0, 0, -9, 0, -7, 0, 0, 0, 10, -10, 1, -6, 6], [7, 0, -1, 0, -6, -1, 0, 7, -1, 0, -7, 0, 0, -5, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 4, 0, 5, 5, 0, -4, 0, -8, -8, 0, 0, 0, 0, 0, 0], [6, 0, -12, 0, 0, 0, 0, 5, -5, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [6, 0, -12, 0, -6, 6, 0, -1, 1, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [-6, 0, -6, 0, -5, 9, 0, 1, -9, 0, -6, 0, 1, -6, 6, 0, 0, 0, 0, 0], [8, 0, 2, 0, 12, -4, 0, -14, -4, 0, -8, 0, 0, 9, 2, 0, 0, 0, 0, 0], [4, 0, 2, 0, 3, 2, 0, 1, -2, 0, 4, 0, 10, 2, -2, 0, 0, 0, 0, 0], [0, -7, 0, 3, 0, 0, -3, 0, 0, 6, 0, 10, 0, 0, 0, 0, 8, 0, 3, 0], [-8, 0, 2, 0, 4, -4, 0, 1, -1, 0, 0, 0, -13, 13, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 3, 0, 0, -4, 0, -3, 0, 0, 0, -7, 7, 8, -7, 7], [0, 1, 0, -1, 0, 0, 8, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -1, 0, -4, -2, 0, -3, 2, 0, -7, 0, -5, -1, 1, 0, 0, 0, 0, 0], [-26, 0, -6, 0, 0, 0, 0, 1, -1, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0], [0, -2, 0, -7, 0, 0, 7, 0, 0, 0, 0, -5, 0, 0, 0, -1, 0, 0, 3, -5], [0, 0, 0, 0, -10, 8, 0, -7, -7, 0, -4, 0, 6, -12, 18, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, -4, 0, 5, 5, 0, -28, 0, -6, -10, 4, 0, 0, 0, 0, 0], [-6, 0, -4, 0, 1, 7, 0, 3, 7, 0, 6, 0, 0, 1, -4, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 4, 1, -1], [0, -11, 0, -7, 0, 0, 3, 0, 0, -6, 0, 8, 0, 0, 0, 0, -8, -4, -3, 2], [0, 3, 0, 0, 0, 0, -6, 0, 0, -3, 0, 3, 0, 0, 0, -14, 14, 2, 3, -3], [10, 0, -10, 0, -6, 6, 0, 11, -11, 0, 0, 0, 9, -9, 0, 0, 0, 0, 0, 0], [3, 0, -1, 0, -12, -1, 0, 13, -1, 0, -3, 0, 0, -5, -1, 0, 0, 0, 0, 0], [0, 14, 0, 5, 0, 0, -6, 0, 0, 0, 0, 2, 0, 0, 0, -3, 0, 1, 12, -3], [0, 3, 0, 4, 0, 0, -4, 0, 0, 8, 0, 1, 0, 0, 0, 0, -13, 0, 4, -7], [-13, 0, 0, 0, 2, -4, 0, 2, 4, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 11, 0, 0, -7, 0, 0, 9, 0, 0, 0, 0, 0, -7, -7, -2, -2, -2], [18, 0, -5, 0, -1, -1, 0, 6, -1, 0, -18, 0, 0, -7, -5, 0, 0, 0, 0, 0], [0, 14, 0, 0, 0, 0, -7, 0, 0, 14, 0, -7, 0, 0, 0, 0, -9, -7, 7, -8], [0, 5, 0, 3, 0, 0, 5, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, -8, 3, -4], [0, 8, 0, 1, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 0, 7, 0, -4, 3, -3], [2, 0, 2, 0, 2, -8, 0, 0, 8, 0, 2, 0, 13, 2, -2, 0, 0, 0, 0, 0], [0, 3, 0, -5, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -3, -3, -2, -2, -2], [0, 0, 0, 0, 2, -2, 0, -1, -1, 0, 4, 0, -8, -4, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 3, 0, -10, -10, 0, -18, 0, -2, 8, -10, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, -1, 0, 0, -4, 0, 3, 0, 0, 0, 5, -5, -2, -3, 3], [0, 11, 0, -5, 0, 0, 5, 0, 0, -5, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0], [0, 0, 0, 0, -6, 2, 0, -11, -11, 0, -12, 0, 6, -2, 8, 0, 0, 0, 0, 0], [-5, 0, 8, 0, 8, -9, 0, 0, 9, 0, -5, 0, -11, 8, -8, 0, 0, 0, 0, 0], [0, 6, 0, 7, 0, 0, -4, 0, 0, -1, 0, 0, 0, 0, 0, 10, 10, 5, 5, 5], [-8, 0, 10, 0, 0, 0, 0, -7, 7, 0, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0], [5, 0, -4, 0, -4, -1, 0, 0, 1, 0, 5, 0, -7, -4, 4, 0, 0, 0, 0, 0], [0, -3, 0, 13, 0, 0, -6, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2], [1, 0, 3, 0, -7, 9, 0, 4, 9, 0, -1, 0, 0, -8, 3, 0, 0, 0, 0, 0], [-12, 0, 12, 0, 6, -6, 0, -8, 8, 0, 0, 0, -10, 10, 0, 0, 0, 0, 0, 0], [0, -11, 0, 3, 0, 0, 1, 0, 0, -2, 0, 10, 0, 0, 0, 0, -6, 4, -1, 4], [0, 0, 0, 0, -6, -4, 0, 8, 8, 0, 8, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 9, 0, -13, 0, 0, 5, 0, 0, -3, 0, 0, 0, 0, 0, 1, 1, -2, -2, -2], [0, -4, 0, 2, 0, 0, -14, 0, 0, 0, 0, -6, 0, 0, 0, 1, 0, 12, 2, 6], [20, 0, 5, 0, -15, -2, 0, 10, -2, 0, -20, 0, 0, -12, 5, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, -1, 0, 0, 4, 0, -5, 0, 0, 0, -8, 8, 2, 5, -5], [0, 0, 0, -3, 0, 0, 10, 0, 0, -7, 0, 0, 0, 0, 0, 4, 4, -3, -3, -3], [-32, 0, -2, 0, -5, 5, 0, 2, -2, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [-4, 0, -13, 0, 8, -4, 0, 5, -4, 0, 4, 0, 0, 11, -13, 0, 0, 0, 0, 0], [12, 0, -6, 0, -8, 8, 0, -2, -8, 0, 12, 0, 5, -6, 6, 0, 0, 0, 0, 0], [12, 0, -8, 0, 10, -10, 0, -6, 6, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 8, 0, 7, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 9, 9, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_2_e_a();".
// 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_605_2_e_a(:prec:=20)
    chi := MakeCharacter_605_e();
    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_605_2_e_a();".
// 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_605_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 0, 0, 0, 16040, 0, 0, 0, 9193, 0, 0, 0, 1315, 0, 0, 0, 67, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;