
// Make newform 4025.2.a.bc in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4025_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4025_2_a_bc();".
// 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_4025_2_a_bc();".
// 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 := [-9, -133, -218, 891, 539, -1853, -27, 1330, -244, -414, 111, 58, -18, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [142730, -75581, -1900192, 1597821, 4556819, -3155676, -3407257, 2103680, 1080275, -615369, -152987, 81881, 7975, -4055], [-316562, -3505754, -4309323, 14277936, 14830170, -20539065, -12373995, 12330084, 4180444, -3443457, -619227, 446796, 33409, -21776], [89726, 2002366, 6121945, -10653101, -19453226, 19231422, 15921004, -12816147, -5321389, 3774769, 782303, -504992, -41963, 25090], [-320996, -4174085, -7714479, 18540140, 25007434, -29115247, -20566123, 18260121, 6900067, -5217903, -1017543, 686159, 54713, -33730], [-436758, -4230503, -7961510, 18099684, 26520409, -28918074, -21833829, 18364740, 7310378, -5289707, -1075557, 699434, 57715, -34523], [131732, 3009015, 10894262, -16164725, -34949868, 30567527, 28645999, -20836696, -9565583, 6219514, 1403897, -839833, -75160, 42027], [-264833, -5051254, -11779454, 22798483, 36707581, -37450710, -29850195, 24081761, 9941128, -6982144, -1457654, 926777, 78020, -45854], [-326507, -4784149, -11527225, 22338951, 36720610, -37494256, -30025957, 24294777, 10030001, -7064153, -1473873, 938797, 79019, -46477], [362877, 6282190, 13388234, -27425832, -41769609, 44003702, 33964264, -27962096, -11315041, 8052400, 1659877, -1064311, -88882, 52513], [291069, 7829560, 22247589, -38037463, -67878225, 65368699, 54897262, -42782683, -18243758, 12509311, 2672763, -1668584, -143014, 82827]];
    Rf_basisdens := [1, 1, 1, 1, 22937, 22937, 22937, 22937, 22937, 22937, 22937, 22937, 22937, 22937];
    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_4025_a();"
function MakeCharacter_4025_a()
    N := 4025;
    order := 1;
    char_gens := [2577, 1151, 3501];
    v := [1, 1, 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;

function MakeCharacter_4025_a_Hecke(Kf)
    return MakeCharacter_4025_a();
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, -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, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 1, -1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, -1, 0, 0, 1, -1, -1, -1, -1, -1, 0], [1, 0, 0, 0, -1, 0, 0, -1, -1, 0, -1, 0, -1, -1], [2, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 0], [-1, 0, -1, 0, 0, -1, 0, -2, 1, 1, 1, 2, 1, 0], [2, 0, -1, 1, -1, 0, 0, -2, 0, 1, 1, 1, 1, -1], [2, -1, 0, -1, 1, 0, 0, 1, 1, 0, -1, 0, 0, 0], [0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 1, 0, 1, -1], [-1, 2, -2, 0, -1, 1, 0, -2, -1, 1, 0, 0, 1, -1], [0, 0, 1, 0, -1, 0, 1, -1, -1, 0, 1, 1, 1, 0], [4, 1, 2, 0, -1, 0, 0, 1, -1, 0, 0, 0, -2, -1], [-1, -1, 1, -1, 0, -2, 0, 4, 1, -2, 0, -2, -2, 1], [3, -1, -1, -1, -1, -1, 0, -1, -1, 1, 0, 0, -1, 0], [1, 2, -1, -1, 0, -1, 1, 0, -1, 0, 1, 0, -1, 1], [1, -3, -1, 0, 1, 1, -1, 1, 1, 0, 0, -1, 0, 2], [1, 2, 1, 0, 1, 0, 1, -2, -1, -1, 0, 1, -1, 0], [3, -1, 2, 0, 1, 0, -1, 1, -1, 0, -2, -1, -1, 0], [-3, 2, -1, 0, 1, -2, 1, -1, 0, 0, 1, 1, 1, 0], [4, 0, 2, 2, 2, 0, 1, 1, 0, -1, 1, -1, 1, -1], [-2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 1, -3, 0, 0, 0, 0, 1, -2], [-2, 0, -3, 1, 1, -3, 0, 1, 2, -1, 1, 2, 0, 0], [5, -1, -1, -1, -1, 2, -2, -2, -1, 1, 0, 1, 0, 1], [-1, 0, 1, 1, -1, 3, 1, -2, -1, 1, 1, 1, 1, -1], [0, -1, 1, -1, 0, 0, -1, 4, 0, 0, -2, 0, -2, 1], [-1, -2, 0, -2, -2, 0, 0, 1, 1, 1, -1, -1, -1, -1], [2, -1, -3, -1, -1, 2, -2, -2, 3, 2, 1, 0, 1, 0], [2, 0, -1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 2], [0, 0, -2, 2, 1, 2, 0, -5, 1, 1, 0, 3, 2, 0], [7, -2, 1, 0, -2, 0, 0, 1, -2, -1, -1, 2, 1, 0], [3, -1, -1, 1, -2, 1, -2, -4, 0, 1, 1, 2, 0, -1], [-2, 1, 1, -2, -2, 2, 0, -2, 1, 2, 1, 0, 2, -2], [-2, 0, 1, 1, 2, -3, -1, 2, 2, -1, 0, -1, -2, 0], [2, -3, 0, 1, 2, 0, 0, -2, 0, 1, 1, 4, 3, 1], [1, -1, -2, 0, -1, 0, -1, 3, 0, 0, -2, 1, -1, 0], [5, -3, -2, 0, 0, -1, 0, 1, 0, 0, 1, 2, -1, 0], [0, -2, -1, 3, 1, -1, -2, 2, 1, -1, -1, -2, 0, 0], [-3, -4, 2, -1, 2, 0, 0, 3, 0, -1, -4, -3, -1, 0], [-3, -4, -1, 1, 2, 1, -2, 2, 3, 0, -1, -1, -2, 0], [6, -4, 0, 1, 1, 1, 1, 0, 0, -1, 1, 0, 1, 0], [0, 0, -3, 1, 1, -1, 2, -1, -1, -3, -1, 0, 0, 0], [-1, 0, 0, 2, -1, -1, -1, 0, -1, -1, 2, -1, 0, 0], [-1, 3, -1, -2, 0, -2, 1, 5, 0, -1, -2, -2, 0, 0], [-1, 3, 0, -2, -1, 0, -2, 1, 2, 1, 1, -2, -2, 1], [2, 1, -3, 0, -2, -1, 0, 0, -1, 1, 0, -3, -1, -1], [3, -1, 1, 0, -3, 3, -2, -4, -2, 1, 1, 2, -1, -3], [5, -2, -1, 0, -1, 1, -2, 1, 1, 0, 1, -1, -1, 0], [6, 0, -3, 1, -2, 1, -2, -4, -1, 0, -1, 3, -2, -2], [-6, 0, -2, 1, 0, 2, -2, -6, -1, 2, 1, 0, 0, -1], [-2, -5, -1, -1, -1, 2, 1, 0, -1, 0, -1, 1, 0, -1], [1, 1, -1, -1, -1, -1, -4, 5, 2, 1, 0, -3, -2, 1], [3, -4, 0, 1, -1, 1, -2, -1, -1, 0, -2, -1, -2, -1], [-1, 4, 1, 1, 2, -1, 1, -2, 0, 0, -1, -2, 3, -1], [-2, 1, 0, 3, 1, 2, 0, -1, 1, 2, 1, -1, 3, -1], [0, 0, -1, 2, 2, -2, 1, -4, 2, -1, 3, 0, 3, 0], [-4, 3, 2, 0, 2, -4, -1, 4, 2, 0, 1, -1, -2, 1], [-4, 3, 4, -1, -1, 0, 2, 1, 0, -2, 1, -3, -1, 1], [6, -3, 1, 2, -2, 1, 0, 0, 1, 1, 0, 1, -1, -1], [3, -8, -2, 2, 2, 2, 0, 2, 2, -1, -1, 2, 2, 2], [1, 8, -2, -2, -2, -2, 3, -2, -2, 2, 2, 1, 1, -1], [1, -2, -2, 0, -2, -1, -1, 0, 0, 1, -1, -2, -2, -2], [-4, -6, -1, 3, 1, -2, 0, 2, 4, -1, 1, 2, 0, 0], [-2, -3, 1, -1, 0, -1, -2, -3, 2, 1, 1, 0, 0, -3], [2, -1, -3, -1, -5, 1, 1, 0, -3, -1, 0, 2, -1, -2], [1, 0, 0, -3, -2, 0, 0, -1, -1, -1, -2, -1, -1, 0], [1, -2, -1, -4, 0, 0, -4, 8, 3, 1, -3, -3, -2, 1], [1, -2, 3, -2, 4, -2, 2, 6, 1, -1, -3, -3, 2, 1], [3, 5, -1, -1, -1, -1, 3, 1, 1, 2, 1, 0, 1, -2], [5, 2, 0, 2, 0, 3, 3, -3, -3, 0, -2, -2, 0, 0], [6, 6, 0, 1, 1, -4, -1, 7, 1, -2, -2, -2, -1, -2], [6, -1, 3, -2, 1, 0, 1, 1, -1, -1, -1, -2, -1, -3], [-6, -7, 0, 0, -1, 5, -3, 0, 1, 1, -1, 0, -2, 1], [-3, 1, 2, 4, 1, 2, 2, 4, -2, -1, 1, -2, 0, -3], [3, -1, -1, -1, -6, 1, -1, -6, -2, -1, -1, -1, -3, -4], [4, -1, 1, -4, -2, -2, 1, 4, -2, -2, 1, 1, 1, -1], [0, -1, 3, 2, 1, -5, 2, 1, 0, -2, 2, -2, -3, -1], [-3, -1, -2, -2, 1, -1, 1, -5, 1, 2, 0, 2, 2, -3], [0, 8, 1, -2, -2, -2, 0, 5, -2, 0, 0, -2, -5, -2], [-3, 0, -3, -2, -2, -1, -4, 0, 3, 4, 2, 0, -1, 0], [6, 2, -5, 1, -2, 1, 0, 0, -3, 2, -1, 1, 2, -2], [-3, 5, 3, -2, 2, -1, 1, -4, -1, -3, 0, -1, -1, 1], [7, 4, -1, -2, -2, -1, -1, -1, 2, 3, 3, 1, 0, -1], [7, -1, 4, 1, -1, 1, 0, -3, -2, -1, 2, 2, 0, 2], [-2, -3, 0, -2, -1, -1, 0, 0, 2, -2, 0, -1, -1, 0], [5, -1, 7, -3, 1, -1, 1, 7, 3, 0, -1, -3, 0, -1], [-5, 0, 2, 3, 2, 0, 3, -2, 0, 2, 3, 1, 7, 0], [-4, 2, 2, 1, -3, -2, 1, 1, -2, 0, 2, 3, 0, -1], [0, -3, 1, 3, 0, -1, 3, -5, 2, 2, 2, 3, 5, 0], [-12, -4, 2, 1, 3, 0, 2, -4, 2, 1, 3, 4, 3, 0], [4, 8, -1, -1, 1, -4, 3, 3, -1, 0, 0, 0, 2, 2], [-5, 1, 1, -3, 1, -3, -4, 6, 5, 1, -2, -1, -2, 1], [6, -3, 3, -3, 0, -4, 1, 1, 2, 2, 0, -2, 2, -1], [10, -4, 0, -4, 1, -4, 1, 0, 5, 0, 1, 1, 0, 2], [-8, -4, 6, -4, 0, 1, 1, 2, -1, -1, -1, -4, 0, 2], [5, -1, 0, -3, 0, -2, 0, -3, 1, 0, 2, 2, 3, 1], [4, -10, 2, -1, 2, 1, -3, 10, 1, -1, -6, -3, -5, 2], [6, -1, 1, 4, 4, 4, 1, 4, 2, 0, 1, -3, 3, 1], [4, 2, -3, 3, 1, -1, 0, 0, 0, -1, 1, 2, 2, 2], [0, -6, -4, 5, 2, 2, -1, -1, 2, 1, 0, 1, -1, -2], [-12, 1, -1, 4, 0, 1, 0, -6, -1, -3, 0, -1, 1, 1], [-8, -2, -1, 2, 0, 1, -3, 0, 3, 2, 0, -1, -2, 1], [2, -1, 0, -2, -2, -2, -3, -2, 0, -2, -5, -1, -6, -1], [5, 6, -6, 1, 1, -3, 3, 0, 2, 1, -1, 0, 3, 0], [2, -1, -1, -4, -1, -3, 1, -2, 0, -1, -1, -1, 0, 0], [0, -9, 1, 0, -1, 4, 0, 0, 0, 2, 0, -3, 0, 0], [-5, 5, 2, 3, 3, 2, 0, -4, 1, -1, 2, 1, 3, 1], [-4, -1, 1, 0, 3, 2, -1, 3, 3, -1, 3, -4, 1, 3], [16, 1, 2, -5, -3, -3, -1, 5, -1, 1, 0, -1, -3, -1], [-3, -6, 0, 1, 5, 0, 3, -1, -2, 1, 1, 1, 4, 3], [3, 4, 1, -1, -2, -1, 2, -6, -5, 1, 2, 6, -1, -1], [1, -2, -2, -2, 0, 2, 0, 2, -3, 1, 1, 3, 1, 1], [-10, 4, 4, -1, -2, -4, 2, 4, -1, -4, 1, -3, -3, 0], [-3, 2, -5, 3, -1, 0, -1, -6, 4, 3, 1, 4, 2, 0], [-3, 5, 2, -1, -1, 5, 2, -8, -4, 3, 0, 2, 2, -2], [6, 2, -1, -3, -4, -1, 2, 2, -1, 0, 1, -1, -4, 0], [7, 3, -3, 5, 1, -1, 1, -2, -1, 0, 5, 1, 0, 1], [-11, 9, 0, 1, 3, -3, 3, -3, 3, 1, 0, -3, 3, -1], [6, 3, 3, 3, 6, -1, 3, -1, 1, -2, 0, -3, 1, 0], [-5, 9, 6, -1, 0, 1, 1, -9, 0, 1, 1, 2, -1, -1], [1, 0, 0, 1, 3, 6, -1, -3, 2, 1, 3, 1, 4, 3], [19, -2, -2, 1, 0, 0, -2, 5, 2, -1, 0, -3, -3, 0], [-3, 7, 7, 2, 2, -1, 2, -7, -3, 0, 3, 1, -3, 1], [-1, 1, -2, 2, 0, -1, 1, 6, -1, -1, -2, 0, 1, 0], [-9, 8, 2, -1, -4, 1, 1, -7, -3, -2, -1, -1, -3, 2], [11, 3, 3, 1, -4, -4, -3, 7, -1, 1, -1, -2, -2, 1], [1, 4, -3, -1, 0, -2, 1, 5, 2, -2, -3, -5, 0, 2], [1, -1, 2, 2, -2, -3, 1, 0, -3, 1, 0, 4, -1, -2], [-12, 4, -1, 3, 2, 1, 4, -4, 1, 2, 1, -1, 4, 0], [-14, 3, 9, 0, -2, 5, 1, -7, -3, 0, -1, 0, -2, -2], [7, 0, 1, -2, -5, 0, 0, 1, -4, 2, -1, 2, -4, 1], [-13, 4, -4, 3, 3, 0, 5, 1, -1, -3, 3, 3, 2, 1], [-4, 1, 1, 1, -2, 7, -1, -2, -3, -1, -1, -3, -1, 0], [-8, 9, -7, -1, 0, 0, 1, -4, -4, 2, 2, 1, -1, 0], [1, 4, -9, 1, 0, 2, -1, 0, -1, 1, -4, -3, 0, 2], [2, 2, -6, 0, -2, 2, 0, -5, -6, 0, -2, 0, 0, 0], [-7, 0, 7, -3, 3, -6, -1, 14, 2, -5, -3, -6, -2, 0], [-11, 0, 7, -2, 0, 1, 0, 0, -1, 0, 2, -4, -1, 0], [10, -5, -2, 1, -3, 4, -4, -1, -6, 0, -1, 7, -1, 1], [-11, -6, 0, 2, 1, 4, 1, -3, 4, 1, 2, -4, 3, 1], [4, 4, 0, -2, 0, 3, 1, 3, -1, -3, 1, 1, 1, 1], [5, 0, 5, -1, 3, 4, 0, 8, -4, 0, -6, 0, -2, 2], [-1, 5, 9, -4, 4, 5, 0, 4, -2, 0, -1, -5, -3, 5], [1, 3, -2, -8, -5, 5, -5, -5, -1, 5, -3, 0, -4, -3], [5, 1, -2, -2, 4, 2, -1, 0, 8, 3, 0, 0, 5, 0], [1, 0, 2, 1, -1, 3, 3, -1, 0, -2, 2, 4, 3, 2], [9, -8, 0, 4, 2, 1, 2, 3, -3, 1, -3, 1, -3, -3], [2, 3, 0, 2, -1, 0, 4, -5, -1, 2, -2, 0, 2, -3], [-3, -6, 0, 0, 5, 6, -2, 0, -1, 4, -1, 5, 4, 2], [-7, -3, 3, -1, 6, -4, -5, 15, 3, -1, -3, -8, -4, 3], [-7, 3, -3, -3, 0, -4, 0, 3, 2, 2, 0, 2, 4, -3], [4, 6, 3, -4, -4, 5, -7, 3, -4, 3, -5, -3, -4, 1], [-3, 5, -4, -2, -2, -2, -2, 5, -1, -2, -3, -3, -4, 3], [0, 5, -3, -1, -3, -3, -1, -3, -3, 5, 0, 1, 0, -3], [-3, 6, 0, 0, -2, 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4, 2, 7, 6, -2], [10, -6, -11, 4, -3, 3, -5, 1, -4, 0, -3, -5, -1, 2], [-8, 7, 4, -6, 6, -9, 5, 11, 9, -2, 5, -8, 1, 4], [10, 17, -6, 4, -4, 5, 4, -13, -9, 1, 2, 0, 5, -2], [-1, -3, 1, 7, 7, 5, 0, -14, 4, 3, 2, 1, 6, 1], [6, 9, -5, -8, -1, 1, 1, -8, -8, 1, -1, 1, 0, 2], [-5, 0, -3, -3, 1, 2, 2, 1, 1, 6, -4, -4, -4, 0], [-2, 1, 6, 5, 1, 2, 0, 5, 6, 0, -1, -3, 1, -1], [-14, -3, -8, 6, 1, 4, -2, -6, 1, 2, -4, -6, 0, -3], [-18, 18, -2, 5, -1, 3, 0, -16, -2, 5, 5, 3, 8, -1], [-5, -3, -5, 4, -2, -2, 1, -3, -5, -1, 4, 7, 1, -3], [-3, -4, -5, 11, 2, -4, 4, -2, 0, 1, 6, 4, 1, 1], [-10, 2, 3, 0, 5, 6, 2, 14, -9, -1, -4, -4, 0, 5], [-10, 8, -5, -4, 2, -11, 4, -2, 3, 2, 0, -1, 4, -5], [-28, -7, 9, 10, 7, -6, -4, 12, 4, -6, 0, -7, 2, 4], [-7, 4, -8, 5, 2, 5, -1, -11, 2, 4, 1, 3, 9, -2], [6, -4, -4, -3, -2, -3, -1, 0, -4, -3, -2, 6, -2, -3], [1, -8, 2, -6, -6, 4, -6, 5, -1, 3, -1, -4, -6, -2], [-16, 2, -4, -1, 2, -1, 3, -10, 3, 5, 12, 9, 11, 4], [4, 1, -12, -1, 1, -4, 6, -4, 7, -2, 3, 8, 10, 0], [-21, 8, -2, 6, -1, 4, -1, -1, -3, 1, -4, -3, -10, 0], [14, 0, -6, -4, -8, 2, -1, -3, 2, 3, -3, 7, 1, 1], [-22, -2, -1, 4, 8, 1, -3, 4, 0, 1, -3, -3, 6, 3], [-9, 8, 3, 2, 0, 6, -6, -10, -1, 3, -5, 1, 0, -3], [7, -4, 8, -3, 2, -3, 0, 20, -6, -7, -4, -4, -8, 3], [-3, -8, -3, 5, 8, 1, 3, 3, 8, -2, 7, -8, 5, -1], [19, 0, -4, 4, 5, 5, 4, -2, -5, 1, 2, 4, 3, 3], [-11, -8, -3, 1, 5, -5, -3, 1, 3, -1, -1, 1, -1, -1], [1, 1, 11, -12, -7, -1, 0, 18, -4, -3, -7, -6, -3, 1], [6, -6, 5, -1, 5, 7, -4, -1, 2, 5, -1, 2, 10, 6], [9, -11, -2, 4, 3, 3, 0, -10, 7, 3, 7, 4, 2, 1], [9, 20, -6, 1, -7, 0, 1, -19, -6, 3, 7, -3, -2, 1], [-5, -15, 7, -2, 5, 4, -4, 11, -1, -1, -7, -7, -4, 6], [0, -2, 6, -3, -5, -6, -4, -3, 2, 1, -3, -1, -4, -1], [9, 3, 3, -2, 1, 8, 0, 1, -4, -5, -3, -4, -3, 3], [13, 12, -5, -1, 5, -4, 3, 2, -5, 1, 3, 7, 5, 3], [7, 10, -4, -3, -2, -2, 4, -8, -8, 5, 0, 0, -6, -1], [2, -9, -2, 2, 2, 6, -2, 8, -1, -3, 0, -5, -4, 1], [-20, -4, -5, 7, -5, 7, -1, -6, -5, -2, 4, 0, -4, 0], [1, -6, 1, 0, 2, -7, -1, -4, -1, 0, 4, 13, 2, -1], [3, -13, 2, 1, 1, 6, -2, 6, -7, -5, -4, -5, -1, 3], [-10, -5, -10, -1, -2, 2, 3, -2, -3, 4, -4, 0, -1, -3], [19, 6, -11, 3, -1, 15, 0, -22, -5, 2, 0, 10, 2, 0], [17, -5, 3, 5, 0, 4, 3, -12, -1, -1, 5, -1, 3, -2], [-19, -1, -3, 7, 7, -9, 0, 2, 4, -5, 2, -3, -2, 5], [-3, 12, -10, -1, -9, 4, -3, -17, -3, 9, 1, 9, 0, -7], [-13, 17, 0, -2, 8, -2, 0, -1, -6, 0, 1, -5, 2, 3], [-18, 5, -7, 0, -1, -5, -1, 6, 0, -1, -5, -5, 0, -6], [-11, -6, -2, 5, 5, -4, -3, -11, 12, 7, 5, 11, 12, 3], [-27, -9, 4, -5, 7, 2, 3, 1, 2, -4, 3, 1, 5, 1], [-4, 12, -6, 7, -1, 8, -6, -10, -6, 2, 0, 6, 3, -4], [8, -13, 9, 1, 6, -2, 4, 5, 4, -7, 1, 1, 1, 8], [3, -6, -10, -2, 3, -1, 1, -2, 4, -3, -6, 1, 0, 2], [2, 10, 0, -2, -3, -1, 1, -2, -2, -8, -5, -5, -2, 2], [-2, 0, -13, 7, 7, -3, 2, -3, 3, -5, -3, -3, -3, 3], [-5, 4, -4, 6, -1, -6, 5, -13, -6, -1, 8, 6, -3, 5], [-6, 4, 4, 5, 3, -3, -6, 12, 7, -3, 3, -5, -6, 1], [16, -1, -10, -2, -10, 0, -1, -13, -6, 6, -1, 2, -3, -2], [0, 12, 7, -3, -2, -2, 9, -6, 0, 1, 2, 5, 8, -4], [31, -17, 0, -6, -1, -1, -4, 9, 5, 3, -3, -3, -5, 4], [10, -11, 9, -4, -3, 8, 0, 10, -5, -4, 0, -9, -4, 2], [-1, -5, 10, -4, 2, -2, 3, 17, -3, -5, -10, -9, -4, 3], [20, 4, 1, 3, 5, -6, 11, 12, -4, -5, -1, -4, 0, 0], [1, -2, 0, -6, 6, -3, -1, -1, 8, 4, 2, 4, 6, 0], [-12, 5, -7, 0, -6, 5, 3, -9, 0, 2, 1, -7, -1, -1], [3, -1, 8, 9, 6, -9, 2, -1, 0, -6, 6, 3, 0, 0], [20, -2, -4, -2, 5, -1, -6, 0, 1, 1, -2, -4, -5, 7], [4, -5, 6, -7, -5, 9, -10, 6, 1, 6, -7, -3, -9, 1], [13, -21, -2, 0, 2, 8, -5, 12, 3, 1, -6, 1, -2, -1], [-11, -5, 10, 0, -4, 0, 3, -5, -12, -7, -4, 5, -2, -1], [12, -12, 11, 5, 9, 1, -3, 8, -2, -4, -6, -9, 1, -3], [11, -10, 11, -11, -7, 4, -6, 9, -2, 2, -2, -1, -1, -1], [-25, 21, 6, -3, -4, 2, 7, -3, -8, -1, -3, -13, -6, -2], [12, 14, 3, -4, -5, -1, 2, 0, -1, 2, 5, 5, -3, 2], [21, -15, 2, 7, 4, 5, -3, 0, 5, -1, -1, 8, 7, -3], [4, 5, 4, -1, -3, -1, 2, -4, 2, 0, -1, 0, -2, 4], [7, 13, -9, 0, -1, -2, 0, -7, 1, 5, -1, 1, -4, -6], [-21, 11, 6, -3, -1, -2, 3, 8, 0, -6, -7, -11, 1, 3], [-4, 4, -7, -2, -3, 0, 0, -16, -4, 5, 4, 1, 3, 2], [2, 4, -7, -4, 0, -8, -1, 6, 7, 9, 5, 4, -1, 0], [1, 5, 6, -8, 1, 4, -5, -9, -2, -2, -6, -5, -9, 4], [-11, 11, 11, -8, -1, 0, -8, 7, 1, -1, -1, 3, 0, 4], [-4, -2, -2, 3, 3, 14, 3, -7, -10, 2, 2, 2, 5, 2], [3, 5, 2, 1, -2, -2, 9, -6, -7, -5, 3, 7, 4, -4], [2, -12, -2, 8, 5, -5, 0, 6, 5, -7, 2, 0, -7, 1], [37, 4, -6, 1, -12, 1, 4, -9, 3, 4, 7, 8, 4, -5], [6, 0, 8, 4, 1, -2, -1, -3, 0, -4, 1, 0, -5, -5], [15, 1, -12, -3, 2, -5, 1, -11, 0, 5, 5, 2, 3, 3], [27, 1, -2, -1, -5, 5, -4, 0, -6, 1, 6, -1, -1, -5], [-24, 2, 14, -5, 5, -7, -2, 31, -5, -13, -5, -16, -11, 6], [-15, 17, 3, 3, 1, -4, 9, -9, -10, -1, 4, 1, 2, 1], [27, 2, 7, -3, -4, 2, -4, 6, -6, 3, -6, 0, -11, -3], [9, -4, 7, -2, -2, 2, 4, -6, -1, 7, 5, 5, 8, -5], [25, -4, -3, 5, 5, 0, 0, -7, -1, 0, 4, 13, 9, 1], [-24, -1, 2, -1, 0, 5, -5, 7, -4, -5, -7, -6, -9, 3], [9, 6, -4, 4, 7, 1, -3, 2, 7, 7, 2, 0, 7, 1], [-17, 8, -6, -2, -2, -5, 7, -15, 3, 4, 0, 4, 4, -6], [0, -6, 5, 5, -1, 4, 12, -16, -4, -5, 5, 13, 11, -3], [-8, -5, -5, -10, -4, 1, -4, 0, 1, 3, -4, -3, -3, 7], [-7, -7, 0, -4, 3, -1, 3, -6, 7, -4, 0, 12, 4, 5], [-5, -3, 7, -6, 0, -4, 6, 6, 5, -2, 3, 0, 4, -2], [-9, 11, 6, -5, -11, -3, 1, -4, -13, -2, -5, 1, -9, -5], [-6, 1, 13, 0, -4, 5, -6, -4, -3, 1, 4, 7, 3, 1], [-7, -1, 2, 2, 8, 0, -9, 7, 3, -1, -4, -8, -1, 2], [-14, -12, -3, 7, 3, 3, -5, 0, 1, 0, 0, 2, -2, 4], [7, 3, -13, 2, 4, -10, 4, -18, 4, 4, 3, 12, 10, 0], [-17, -1, 5, -1, 1, -10, -9, 2, 9, 4, 1, 1, -2, 1], [4, -8, -1, -1, -10, 11, 4, -18, -8, -4, -3, -2, -7, -7], [-5, -11, 2, -4, 2, 0, 1, -9, 5, -1, -8, -3, 2, 5], [4, -5, -5, -8, 5, -5, 0, -13, 1, 2, 6, 6, 1, 3], [12, -13, 10, -3, -4, 11, -7, 13, -5, -2, -8, -2, -5, 3], [48, 5, -4, -9, -8, 3, 3, 8, -1, -2, -4, 2, -3, 1], [-7, -6, -10, 2, 6, 3, 6, -9, 3, 0, 8, 9, 9, 1], [-4, -5, -2, -9, 1, 9, -3, -9, 5, 7, -2, -2, 0, -2], [5, -6, -8, 1, 4, 0, -2, -9, 8, 3, 6, 10, 12, 1], [-21, -20, 14, -9, -2, -1, 1, 13, -1, -6, -5, -5, -5, 2], [3, 3, -2, -10, 0, 1, -8, -2, 1, 8, 1, -2, -1, -2], [-21, -5, -9, 14, 4, -5, -5, -4, 3, -5, 2, 2, 2, 2], [7, -8, 6, 2, -8, -2, 4, 2, -9, -1, 7, 7, -1, -3], [20, -10, 7, 5, -4, 2, 3, 6, -5, -3, -6, -1, -8, -4], [19, -3, 8, 4, 11, 0, -1, 18, -2, 1, -3, -5, 1, 4], [-19, 9, -12, 1, -4, 9, -8, -8, 1, 8, -2, -1, 2, 4], [3, -17, -2, 2, 5, 3, -2, -9, -1, -3, 5, 1, 3, 4], [-3, 4, -12, 1, -7, 3, 6, -1, -2, 10, 0, 1, 6, -3], [-9, 3, 6, 7, 2, -6, 4, 9, 1, -6, 0, -2, -5, 1], [14, 11, -4, -4, -4, -9, -3, 17, -8, -5, 2, 1, -6, 3], [4, 23, 13, -7, 0, -6, 5, -3, 1, -1, 9, 2, 0, 3], [-9, 15, -2, -4, 0, -8, -1, 4, 9, 13, 0, -1, 4, 1], [21, 10, -10, -2, -8, 5, 1, -17, -4, 2, 2, 5, -5, -5], [-12, -15, -1, 2, 4, 11, 4, 4, -5, -3, 2, 3, 3, 3], [6, -5, -2, -5, -2, -8, -4, 5, 5, -2, 2, 8, -3, -3], [-3, -3, 0, -12, -1, 2, 4, 9, 1, 5, 3, -2, 4, 5], [-16, -9, -1, 6, 5, 2, 1, -10, 3, -1, 3, 3, 4, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4025_2_a_bc();".
// 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_4025_2_a_bc(:prec:=14)
    chi := MakeCharacter_4025_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_4025_2_a_bc();".
// 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_4025_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, -133, -218, 891, 539, -1853, -27, 1330, -244, -414, 111, 58, -18, -3, 1]>,<3,R![7, -923, -182, 4982, -950, -6420, 1350, 3338, -712, -815, 182, 93, -22, -4, 1]>,<11,R![4800, -55520, 246496, -522000, 505664, -92976, -189963, 113578, -380, -12318, 1609, 443, -78, -5, 1]>],Snew);
    return Vf;
end function;