
// Make newform 5225.2.a.u in Magma, downloaded from the LMFDB on 01 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5225_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_5225_2_a_u();".
// 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_5225_2_a_u();".
// 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 := [-5, -93, -171, 485, 793, -1160, -1055, 1239, 606, -645, -165, 168, 21, -21, -1, 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, 1, 0, 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], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1420, 25426, -5275, -150693, 66496, 212754, -100496, -127760, 61223, 37341, -17984, -5165, 2498, 267, -130], [-7976, -36464, 46923, 193237, -171498, -244004, 205459, 128798, -106985, -32380, 26975, 3839, -3244, -172, 149], [2251, 20572, 77723, -16644, -245341, -61675, 293967, 74317, -168032, -28621, 47352, 4518, -6317, -249, 317], [986, 53839, 12953, -291708, 116686, 400116, -224116, -233327, 145839, 64844, -42547, -8462, 5697, 414, -284], [-6291, -15999, 88551, 119260, -319479, -172563, 388431, 99686, -211521, -26536, 56044, 3267, -7054, -149, 337], [500, 15060, 8868, -76104, 3384, 108573, -28168, -65671, 23670, 18534, -7659, -2422, 1077, 118, -55], [7731, 102628, 5549, -464865, 173948, 592275, -333755, -329418, 221841, 88387, -65615, -11217, 8851, 536, -443], [10275, 187784, 64809, -868956, 97156, 1158986, -320730, -667481, 239156, 182710, -73690, -23428, 10100, 1128, -508], [9958, 155945, -46982, -850267, 462541, 1169723, -704485, -672114, 420437, 180742, -115657, -22614, 14746, 1060, -705], [-4904, -100407, 70383, 683198, -573603, -991699, 853900, 587309, -510457, -162143, 142334, 20808, -18484, -997, 901], [897, 12550, -19596, -89620, 107227, 135738, -150718, -82869, 88893, 23174, -24657, -2979, 3190, 142, -155]];
    Rf_basisdens := [1, 1, 1, 1, 917, 917, 917, 917, 917, 131, 917, 917, 917, 917, 131];
    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_5225_a();"
function MakeCharacter_5225_a()
    N := 5225;
    order := 1;
    char_gens := [2927, 2851, 4676];
    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_5225_a_Hecke( : Kf := Kf)
    return MakeCharacter_5225_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]];
    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, 1, 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, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 1, 1, -1, 0, 0, 0, 0, 1, 1], [0, 1, 1, 1, 1, 1, 1, -1, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, -1, -1, -1, 1, 1, 1, -1, 1, 0, -1, -1, 0], [1, 1, 1, 1, 1, 0, 0, 1, -1, -1, -1, 1, 0, 1, 1], [3, -1, -1, 0, 0, -1, -1, 0, 1, -1, 0, 1, 1, 0, 0], [1, -1, 0, 0, 0, 1, 0, -2, 1, 1, 0, 1, 0, 0, 0], [2, 2, 1, 0, 0, 1, 1, -1, -1, 1, 0, 0, 0, 0, 0], [1, 1, -1, 0, 1, 0, 0, -1, -1, 1, -1, 0, -1, 0, 0], [2, -2, -3, 0, -2, -1, -3, -2, 4, 0, 2, 2, 1, -1, -1], [1, -4, -4, -1, -4, -3, -3, -2, 5, -1, 2, 1, -2, -2, 0], [-1, 0, 1, -2, -1, -1, 2, 2, 0, -1, -1, -1, -1, 0, 0], [2, -1, 3, 1, 3, 3, 2, 1, -5, 0, -1, -1, 2, 1, 0], [0, -2, 0, -1, -2, -1, -1, 1, 2, -1, 0, 0, 0, -1, 1], [0, 3, 2, -1, 0, 1, 2, 1, 0, 1, -1, -2, -1, 1, 1], [0, 2, 2, 1, 2, 2, 3, 1, -2, 1, 0, -1, 1, 2, 1], [-1, -1, 2, 0, 1, 2, 2, 1, -2, 1, 1, -3, 1, 0, -1], [0, 1, 0, 1, 3, 1, 1, 1, 0, 0, -2, 0, 0, 0, 0], [4, -1, -1, 3, 1, 1, -2, 0, -3, 1, 0, 0, 2, 2, 1], [-3, -2, 1, 0, 1, 2, 2, 1, -2, 2, 0, -2, 0, 0, -1], [4, 0, -2, 0, -2, -2, 0, -2, 2, 1, 1, 1, 1, -1, -2], [0, 0, 2, 2, 2, 2, 2, 0, -3, 2, 2, -1, 2, 0, 0], [-1, 1, 2, 1, 4, 2, 2, 2, -4, 1, 1, -2, 3, 1, 0], [2, 4, -1, 0, 0, -1, -1, 2, 2, -3, -2, 1, -1, 0, 3], [2, -1, 2, 0, -2, 1, 1, 0, 0, -1, 1, -2, -1, 0, 1], [-1, 3, 3, 2, 3, 3, 3, 0, -3, 0, -2, 0, 1, 1, 0], [1, 0, 0, -3, -2, -2, 0, 1, 1, -2, -3, 0, -2, 0, 1], [3, 1, 0, 2, 3, 0, 0, 1, -1, 1, 2, 0, 2, 2, -1], [2, 0, 3, 0, 3, 2, 1, 0, -1, 1, -1, 0, 0, -1, 2], [0, 1, 2, 0, 1, 2, 1, 1, -2, -2, -2, 0, 0, 0, 3], [9, -3, -5, 2, -1, -3, -5, -3, 3, -1, -2, 4, -1, -2, 0], [5, 3, 1, -1, 1, 1, 1, 0, -4, 2, -2, -2, 1, 0, -1], [0, -2, -4, -1, -3, -1, -1, 0, 3, 0, -1, 1, -2, 1, 1], [-2, 0, -2, 0, -2, -2, 0, -1, 1, 1, 0, -1, 0, -1, -2], [3, 0, -1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1], [2, 0, -1, 0, 1, 1, -1, 0, 2, 0, 0, 1, 1, 1, 1], [-5, 3, -2, 1, 2, -2, -1, -2, -1, 0, 0, 0, 1, 0, -1], [0, 3, 0, -1, -2, 1, 1, 1, 2, -2, -2, 1, -2, -1, 3], [0, -1, -3, 0, -2, 0, -3, 1, -1, 1, 0, -2, -1, 1, -1], [-1, 5, 1, 2, 0, 0, 0, 0, -2, 2, 0, -1, -3, 2, 0], [0, -3, -1, 1, 2, 2, 2, 2, -5, 1, -1, 0, 2, 3, -2], [5, -1, -7, -3, -7, -4, -4, -4, 7, -2, 1, 3, -4, -3, 2], [3, -2, 2, -2, 1, 2, 2, 0, 2, -4, -2, 0, -2, -1, 4], [0, -2, -4, -1, -5, -3, -4, 0, 3, 0, 2, -1, -3, 1, -1], [0, 1, -2, -3, 0, 0, 2, 1, -2, -2, -3, 0, 0, 0, 1], [-2, 4, 5, 0, 4, 3, 3, 0, -5, 0, 0, -1, 1, 1, 0], [4, -7, -5, -1, -6, -4, -5, -4, 3, 2, 4, 0, -1, -1, -5], [1, -2, -1, 2, 1, 0, -2, -1, -4, 2, 2, 2, 2, -1, -1], [-5, 0, -5, -2, -2, -1, 1, 6, -1, -1, -1, -1, -1, 1, 1], [1, 2, 2, 2, 1, 2, 1, -1, -3, 2, 0, 0, 2, 3, 2], [0, -6, -6, -2, -5, -2, -6, 1, 5, -2, 1, 3, 1, -3, 1], [0, 0, -3, 1, -3, -3, -1, -3, 5, 0, 0, 1, -1, -2, -1], [1, -5, -5, 0, -6, -2, 0, -3, 3, 2, 3, 0, 0, -3, -4], [0, 5, 3, -1, 1, 2, 4, 5, -6, 4, -1, -4, 0, 1, -2], [2, -2, -1, 1, 2, 0, -2, -3, -4, 1, 3, -1, 2, 0, -6], [-3, -4, -6, -1, -7, -7, -6, -4, 13, -3, 5, 4, -2, -4, 3], [2, 3, 4, -2, 3, 0, 4, 6, 2, -3, -5, -1, -3, 3, 2], [2, 5, -1, 4, -1, 3, -1, -7, -3, 2, -1, 1, -2, 0, 1], [3, 2, 3, -2, 0, -2, 1, 6, 1, -3, -1, 1, -4, 2, 5], [-2, 5, 4, -2, 0, 0, 4, -3, 2, 1, 0, -1, -4, 0, 2], [-1, 1, -1, -3, 0, -1, 1, 2, -2, 2, -2, -4, -1, -1, 0], [5, 7, 7, -1, 6, 4, 6, 4, -3, -2, -3, 0, -2, 3, 2], [-2, 4, -3, 0, 2, -5, 1, 2, 1, 0, 3, -1, -2, -1, 0], [-4, 5, 7, -3, 1, 1, 5, -5, 1, 2, -3, -2, -3, -2, 0], [6, -2, -4, 3, -3, -3, -7, -1, 4, 1, 1, 2, 1, -2, 1], [6, 3, 4, 2, 2, 3, 2, -1, -5, 0, -2, -3, -2, 2, 0], [8, -4, -3, 0, 0, 4, -3, -1, -2, 4, 0, -1, 2, 0, -2], [2, -3, 4, 2, -1, 4, -1, -4, -3, 3, 2, -1, 0, 0, -1], [0, 2, 0, -4, -2, -3, 0, 1, 2, -1, -1, 3, -2, -1, 2], [3, -2, -4, -1, -5, -5, -5, -6, 8, -2, 4, 5, -4, -1, 3], [3, -2, -4, 5, 1, 2, -3, -5, -4, 2, 3, 0, 5, 0, -3], [2, 0, 4, -4, -1, -3, -1, 3, 5, -3, -1, -1, -2, 0, 2], [4, -3, 8, 0, 4, 1, 2, 1, -5, 0, 0, -3, 3, 2, -1], [-5, 3, 3, -1, 5, 2, 5, 2, -4, -1, 0, -1, 2, 2, -3], [-6, 7, 3, -2, 2, 0, 6, 0, 1, 1, 0, -4, -1, -1, 0], [-2, 0, -1, -7, 1, -3, 2, 6, 3, -4, -4, -1, 2, 0, 1], [4, -8, -11, -1, -9, -4, -6, -4, 9, -3, 1, 5, -2, -4, -2], [7, 0, -1, 1, 0, -2, -1, -7, 2, 0, -3, 2, -5, 1, 1], [-2, 5, 4, 4, 9, 6, 4, -5, -11, 0, -3, -2, 0, 3, 1], [1, 0, -2, 0, -2, -4, 1, 1, 1, -1, 2, 2, 1, 0, 0], [-4, -12, -12, 0, -8, -8, -13, -2, 9, -1, 4, 3, 0, -2, 0], [8, -2, 1, -2, -3, -4, -5, -2, 7, -3, -2, 2, -3, -1, 2], [3, 4, 9, 0, -1, 3, 5, 1, 1, -1, 1, -4, -2, 0, 1], [15, 5, 5, 2, 6, 4, 3, 2, -1, 0, -3, -1, 2, 1, 3], [1, -1, -12, -1, -11, -7, -8, -6, 11, -4, 2, 7, -6, -2, 2], [-8, 8, -2, -2, -3, -5, -1, -2, 2, -2, 0, 1, -6, 0, 2], [11, 1, 7, 4, 9, 8, 4, 0, -15, 5, 0, -6, 6, 0, -5], [-3, 0, 3, 5, 0, 7, 0, -2, -5, 5, 2, -1, 2, 4, 0], [8, -1, -4, -1, -5, -3, -6, -6, 8, -2, 0, 3, 0, -3, 0], [4, 3, 8, 4, 10, 8, 5, 2, -9, 3, -5, -4, 3, 0, -2], [0, 0, 0, 2, 3, 1, 2, 2, -3, 4, -3, -2, -1, 1, 1], [-1, 8, 6, -1, 5, 3, 4, 0, -6, -2, -1, 2, 0, 3, 1], [6, -7, -3, 2, -2, 0, -3, 0, -2, -3, 0, 0, 3, 3, 0], [2, 0, 6, 1, 3, 4, 2, -4, -2, -1, -2, -1, -3, -1, -3], [-2, 5, 1, -2, 1, -2, 0, 8, 2, -5, -5, 1, -2, -1, 2], [-2, -6, -2, -3, 0, -2, 0, -2, 1, -5, -1, 3, 2, -3, 2], [-3, -4, -7, -6, -6, -7, -2, -1, 11, -3, 1, 3, -1, -6, -1], [-4, 0, -5, 0, 4, -1, -5, 0, -4, 2, -1, 0, 5, 1, 0], [5, 6, 1, 5, 0, 5, 1, -8, -6, 6, 1, 0, 1, 2, -3], [4, -4, -1, 4, -4, -1, -4, -14, 1, 3, 4, -1, -1, -3, -3], [0, 15, 7, -1, 9, 3, 5, 11, -2, -2, -2, 0, 2, 7, 4], [-1, 4, 6, 4, 0, 2, 2, -4, -2, 1, 4, 0, -1, 1, 1], [-2, 0, 3, 1, 4, 6, 4, 5, -8, -1, -6, -1, 5, 0, 0], [3, -3, -5, 6, -3, -1, -5, -11, 2, 1, 6, 5, 0, -2, 1], [7, -11, -3, 0, 2, 2, -4, -2, 0, 3, -2, 1, 4, -3, -3], [4, -7, -5, -4, -2, -4, -6, -3, 8, -1, 2, 2, -4, -5, 2], [-4, -1, -1, 0, -1, 2, -2, -5, -7, 6, 1, -5, 4, 0, -9], [-6, 12, -4, -1, 4, -3, -1, 3, 2, 2, 0, 2, 1, 2, 2], [-2, 2, -2, -7, 2, 0, -2, 2, -2, -2, 0, -1, 0, 2, 0], [0, 6, 0, -8, -3, -6, 2, 7, 8, -6, -5, 0, -5, 0, 7], [6, -1, 3, 2, 2, 3, 2, 2, 0, 2, 1, -1, 2, 5, 3], [3, 4, 9, 1, 6, 7, 7, -2, -4, 0, -3, 0, 3, 4, 5], [4, -8, -4, 4, 0, 1, -2, 1, 1, 2, 3, 0, 2, 1, 2], [-2, -10, -13, -5, -16, -15, -11, -12, 23, -7, 4, 7, -9, -9, 1], [22, -3, -6, 2, 1, -2, -9, 4, 4, -2, 0, 7, 4, 2, -2], [-3, 1, 1, 5, 5, 2, 2, 3, -6, 3, 2, -3, 3, 7, 0], [-3, 2, -5, -2, -8, -3, -4, -11, 9, 1, 4, 4, -2, -5, -5], [11, -3, 1, -1, -6, -2, -3, -8, 9, 1, 4, 2, -2, -6, -7], [10, -5, -1, 5, 3, -5, -7, -2, 2, 0, 4, 2, 5, -1, -1], [3, -1, -1, -3, -1, -3, -2, 8, 0, -2, 1, -1, 0, -2, 3], [-3, 4, 3, 1, 4, 3, 2, 5, -4, 3, -1, -5, 1, 1, 0], [-1, -6, -1, 4, 0, -1, 1, -8, 2, 0, 8, 3, 3, -3, 0], [-2, 7, 5, 4, 9, 1, 1, 3, -6, -3, -3, 2, -2, 6, 5], [11, 3, 0, -2, 1, -3, -3, 3, 1, -3, 2, 2, 4, -1, 1], [-1, -12, -3, -6, -8, -3, -2, 0, 10, -2, 1, -1, 0, -1, 0], [5, 2, -6, 8, -1, 2, -5, -6, -1, 4, -3, 5, 2, 0, 0], [4, 7, 3, -2, -3, 0, 2, 0, 2, 0, -5, 1, -5, -4, 7], [9, 4, 7, 2, 8, 2, 6, 5, -6, 2, -2, -4, 3, 7, 0], [-5, 0, -4, 2, -2, -4, 1, -3, 4, -4, 3, -1, -4, -1, 1], [-11, 4, 5, 2, 4, 5, 5, 0, -12, 3, -2, -5, 1, 5, 0], [5, -3, -4, 3, 1, 7, -1, 3, -5, 2, -8, -1, 0, 1, -1], [4, 0, 3, -4, 0, 1, 0, 3, 0, 4, -1, -1, 2, 4, -1], [1, 3, 8, 8, 10, 9, 6, -4, -16, 4, 2, -4, 11, 2, -6], [-2, 13, 6, 3, 8, 3, 0, 7, -6, 1, -6, 2, -1, 1, 10], [8, -1, -6, -4, -1, -10, -2, 6, 6, -5, -1, 0, 0, -2, 4], [9, 0, 1, 0, 1, 1, 3, -9, -3, 3, 2, -1, 2, 1, -4], [10, 0, -3, -9, -3, -10, -1, 1, 10, -5, -1, 1, -2, -4, 5], [-7, 9, 2, 2, 2, 5, 6, 6, -8, 2, -6, 0, 1, 2, 4], [-8, 7, 1, 0, 3, 5, 4, 4, -17, 4, -4, -4, 1, 6, 0], [-1, 10, 10, 3, 7, 11, 10, 8, -11, 3, -4, -3, 1, 6, 5], [-3, 2, 4, 4, 2, 3, 4, -1, -1, 3, 4, -2, 1, -4, -3], [-14, -3, -2, 0, -1, -1, 1, -8, 1, 3, 1, 0, -2, -6, 0], [-6, 10, 1, -4, 6, -2, 0, 6, -3, -5, -10, 5, -2, 6, 6], [0, -2, 5, -1, 6, 0, 4, 3, -4, 0, -3, 2, 2, 1, 4], [-1, -13, -9, 5, -6, -2, -12, 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-15, -9, -11, -4, -4, -12, 14, -3, 0, 4, -8, -8, -1], [-3, 17, -4, -5, 2, -2, 5, -3, 2, -9, -10, 2, -11, 2, 7], [-18, 1, 3, -2, 15, 10, 6, 12, -15, 2, -6, -3, 13, 6, -2], [-5, -3, -5, -6, -2, -3, -4, 12, 6, 5, -4, -3, -4, 4, 2], [0, -19, -12, -1, -11, -11, -13, -15, 15, 8, 12, 0, -4, -5, -7], [13, 1, -7, 2, 6, 6, 2, 5, -13, 10, 3, -11, 6, -1, -12], [13, 3, 1, -1, -6, -7, -2, -4, 7, -8, 1, -5, -4, -2, 4], [0, -1, 4, -6, -6, -15, 2, -8, 8, -5, 1, -1, -7, -3, 1], [-12, 4, -2, -3, -9, 2, -9, -7, 7, -2, -1, 3, -1, 0, 2], [-10, 4, 6, 6, 5, 12, 9, -6, -18, 10, 2, -6, 10, 0, -5], [-21, -11, 5, -13, -7, -8, 0, 5, 8, -3, 2, -4, -7, -8, 3], [10, 8, 10, 10, 16, 5, 9, -8, -7, 0, -3, 0, -1, -2, 0], [15, 6, -14, 2, -9, -5, -14, -13, 12, -5, -3, 6, -1, -7, 2], [2, -3, -2, -4, -2, -6, -6, 4, -8, -4, 0, -6, -2, -2, -1], [-4, -13, -3, -2, -11, -5, -6, 2, 8, -7, 3, 3, 1, -4, -3], [13, -5, 4, -8, -1, -7, 3, 7, 3, -3, -7, 0, -9, 2, 9], [0, 6, 11, 7, 17, 18, 5, 17, -32, 9, -9, -6, 16, 10, 1], [-2, 6, 15, 0, 1, 7, 3, 0, -6, 0, 1, -6, 1, 5, 11], [-9, -15, -1, -13, -11, 2, 2, -7, 15, -1, -2, -1, -6, -5, 6], [23, 8, -12, 4, -5, -5, -9, -11, 4, 2, -6, 10, 3, -3, 5], [1, 9, 6, 9, 11, 11, 10, -8, -6, 2, -5, -1, 5, 6, 0], [11, 4, 15, 0, 5, 11, 6, 6, -12, 3, -5, -4, 2, 8, 5], [1, -2, -13, -15, -10, -23, -13, 6, 19, -4, 6, 5, -6, -6, 1], [14, 5, 14, 4, 9, 7, 11, -4, -10, 2, 2, -5, 2, 2, 5], [10, -24, -14, 0, -17, -9, -23, -13, 20, -3, 0, 5, -1, -8, -10], [10, 7, -6, 11, 7, 6, -7, 4, -16, 5, 0, 0, 4, 3, -5], [-5, 9, 3, 5, 1, 7, -7, 1, -1, 3, 0, 0, 3, 3, 13], [-9, 10, 5, -1, -13, -5, 3, -3, 20, -6, 8, -4, -9, -7, 2], [13, 4, -5, 1, -8, -10, -13, 9, 9, -8, -6, 2, -9, 6, 9], [-5, 9, 17, 1, 8, 0, 5, 7, 7, -9, 1, 1, -9, 5, 10], [14, -5, -7, 6, -9, -3, -8, -6, -2, 1, 2, -1, -2, 3, 2], [-16, -8, 3, 1, -3, 0, -1, 0, 8, 4, 4, 0, 0, -5, -7], [15, -15, -22, -8, -17, -17, -19, -5, 17, -3, 1, 4, -1, -11, -6], [4, -15, -3, 7, 6, 6, -7, -5, -15, 8, -6, 4, 14, -3, -4], [14, 7, -9, 7, -5, 1, -12, -8, 0, -1, -5, 4, 3, -2, 5], [5, -11, -6, -8, -2, -2, 2, -1, 6, -1, -2, 1, -2, -4, 1], [-2, -15, 4, -1, -17, 2, -1, -10, 11, -3, 9, -2, -3, 0, 2], [-5, 6, 14, -2, 9, 8, 14, 14, -25, -4, -10, -1, 5, 13, 7], [16, 20, 4, 2, 16, 0, 4, 8, -10, 3, -1, -5, 3, 0, -5], [4, -9, -5, 0, -11, -7, -7, -14, 17, 0, 5, -2, -5, -9, -4], [-14, 4, -7, -5, -12, 2, -2, 0, 8, 2, 3, -1, -3, -1, -1], [-10, -8, 6, 6, 16, 9, 3, -8, -17, 6, 3, -3, 11, 2, -13], [9, 7, 10, 13, 9, 0, 0, 0, -9, 0, 1, 5, 3, -2, 2], [-3, 9, -7, -7, -8, -7, -3, -11, 8, -3, -4, 2, -5, -8, 4], [-8, -6, -13, 4, -18, -8, -8, -6, 25, -3, 6, 7, -4, -7, -1], [-13, -2, 3, -8, 2, 0, 0, -12, 3, 5, -2, -4, -3, -3, -6], [14, 26, -5, 0, -3, -7, 4, 6, -1, -11, -2, 7, -9, 2, 6], [-6, -20, -2, 3, -5, 5, 2, -11, 3, 3, 6, -2, -9, -2, -10], [-5, 4, 7, -7, 5, 4, 9, -1, 3, 3, 4, -1, 5, -1, 1], [6, 19, -1, -13, -9, -7, 1, 10, 9, -7, -9, 0, -6, 4, 6], [5, 6, -13, 3, -3, 0, -4, -2, 10, 4, 0, 6, -1, 3, 5], [-1, 1, -4, -6, -9, 2, 1, -9, -5, 1, -7, 0, -5, -5, -7], [22, -10, -18, -3, -8, -3, -8, 8, 7, -5, -2, 2, 3, 2, -1], [3, 4, 1, -10, -7, -11, -3, 5, 18, -10, -7, 4, -12, -10, 8], [-2, 17, 7, 2, 9, 1, 13, 16, -13, -4, -4, -5, 0, 15, 14], [1, 29, 14, -5, 11, 7, 13, 16, -8, 4, -5, -12, -4, 6, 1], [-1, -10, 8, -11, 3, 3, 14, 25, -3, -4, -3, -5, -4, 2, 3], [-3, 1, 3, -10, 1, -10, 0, 18, 16, -11, -3, 5, -7, 7, 10], [-10, 7, -6, -3, 6, 0, -5, -4, -13, 1, -2, 1, -1, 3, 2], [15, -14, -10, 3, -4, -6, -10, -7, -1, 8, 11, 5, 7, -2, 1], [-4, 9, 3, 1, 2, 10, 12, -7, -1, 5, -2, -5, 1, 0, -13], [10, -7, -2, -5, 5, 6, 4, -1, 2, 5, -3, -4, 6, -2, 0], [4, 14, 9, -6, 3, -4, 5, 14, -5, 4, -10, -7, -5, 3, -4], [-8, -9, 3, -9, -10, -4, -3, 3, 14, -1, 1, -4, -5, -5, 1], [-6, 0, 6, -10, 2, 6, 6, 3, -10, 4, 0, -4, -2, -1, 2], [-8, -11, -8, 14, 11, 3, -6, -1, -17, 8, 6, 2, 15, 5, -14], [35, -26, -1, 8, 3, 3, -11, -16, -9, 6, 6, -1, 10, -6, -6], [9, 1, 5, -8, 0, -6, 0, 14, -6, -1, 3, -3, 1, -1, 9], [-20, 14, 1, 0, 5, -1, 1, 3, -7, 1, 3, 0, -1, 8, 10], [19, 14, 18, 1, 19, 9, 14, 2, -15, 10, 0, -13, 4, 4, 3], [28, -4, -4, 1, -9, -3, -13, -2, 5, -5, 2, -3, -4, 0, 2], [20, -18, -13, 2, -14, -6, -8, -13, 12, 3, 9, 2, 1, -7, -5], [-5, -11, -3, -8, -11, -5, 9, 1, 4, -3, 4, -2, -4, -6, 2], [-6, 2, 2, -2, -5, -10, -9, -5, 17, 4, 8, 2, -6, 0, 6], [14, -4, -11, -3, -7, -6, -2, -4, 17, 1, 8, 0, -8, -4, -1], [-6, 5, 13, -11, -3, 5, 18, 19, -2, -7, -3, -9, -4, -1, 10], [15, 7, -10, -3, -3, -6, 1, 0, 7, -1, -1, 0, 1, -7, -11], [5, 7, 13, 6, 7, -4, 6, 11, -19, -4, -6, 0, -6, 8, 8], [13, 0, 7, 15, 13, 6, 1, -8, -14, 0, -1, 1, 4, 0, -9], [-19, 5, 0, -1, 3, 1, 0, 6, -1, 8, 7, -4, 2, -7, -13], [-18, -4, -2, -11, -6, -1, 9, 0, 6, 1, 2, -1, -2, -6, 2], [4, 4, 2, 7, 8, 12, 2, 3, -16, 9, -5, -3, 5, -1, -1], [3, 7, 4, 4, 6, 9, 0, -6, -9, 11, 0, -1, -1, 9, -2], [-12, 19, 19, 1, 13, 3, 13, 18, -17, 2, -5, -12, -2, 13, 8], [9, -14, 1, 3, 4, 10, -1, -13, -9, 11, 4, 3, 6, 0, -13], [2, -3, 0, 3, 1, 5, 0, 1, -6, 2, 2, -1, -7, -3, 4], [-1, -2, 4, 3, 11, 7, -1, 5, -12, -3, -1, -5, 7, 8, -5], [10, 4, -3, -4, -10, -5, -6, 18, 13, 1, -4, 1, -5, -1, 8], [-11, -3, -4, -14, -1, 3, 13, 14, 2, -8, -5, -4, -2, 4, 9], [32, 14, 4, 9, 14, 14, 10, -9, -23, 12, -4, -6, 5, 4, -4], [-22, 6, 9, -10, 0, -4, 14, 4, 6, -8, -2, -11, -4, 0, 5], [1, 20, 21, -2, 11, 3, 12, 23, -3, -6, 2, -1, 2, 5, 13], [-11, -9, -6, -6, -14, -16, -3, 5, 18, -12, 5, 8, -12, -2, -6], [23, 15, -6, 12, 6, 3, -11, -2, -4, 3, -4, 7, 2, 6, 6], [9, 4, -4, 2, 0, -8, -4, 19, 1, -11, -5, 3, -7, 1, 13], [-14, -4, -14, -14, -16, -14, -5, -13, 18, -9, -2, 1, -11, -13, 0], [2, -2, -11, 5, 0, 2, -6, -12, 15, 4, 12, 2, 4, -1, -9], [-8, -12, 4, -5, -15, -4, 0, -15, 16, -4, 0, -5, -7, -3, 1], [7, -16, -5, 6, 9, -5, -7, -5, -2, -2, 5, -1, 4, 1, -10], [39, 2, 4, 4, -4, 1, -9, -6, 5, 2, 1, 7, 3, -3, 5], [-4, -20, 5, -9, -10, -3, 1, 5, 20, -7, 0, -2, -9, -3, 6], [16, -21, -17, -3, 1, -7, -8, 4, 5, -4, 7, 5, 9, 1, -3], [8, -15, -6, 17, 4, 8, -13, -2, -2, 5, 13, 1, 11, -5, -9], [-14, -4, 5, -6, -2, -2, 5, 7, 1, -10, -3, -3, -11, 5, 14], [5, 10, -2, -1, -13, -7, -5, -4, 17, -4, -4, 2, -10, -6, 7], [-7, -7, -1, -2, -4, 0, -2, 11, -1, -1, 4, 1, 5, -3, 3], [7, 1, -2, -14, -3, -6, 8, 6, 20, -4, -4, 7, -2, -3, 6], [-15, -3, 15, -9, 6, 3, 17, 9, 0, -9, -1, -1, 2, 4, 14], [-14, 6, 9, 1, -1, 4, 8, 5, -7, 5, 1, -6, -4, 11, 5], [14, 1, 13, 3, 4, 9, 5, 10, -11, 7, 2, -7, 5, -2, 2], [-6, 15, 8, 8, 17, 10, 7, -11, -14, 0, 0, 3, 8, 3, 3], [2, 5, 4, 8, 21, 14, 4, 8, -34, 5, -3, -5, 14, 8, 3], [-16, -6, 11, -3, -5, 4, 2, -1, -2, 7, 8, 1, 0, 1, -4], [-7, -16, -14, 2, 3, 1, 2, 15, -8, 5, 0, -5, -1, 2, -4], [1, 6, 5, -1, 0, -3, -5, -4, -2, -2, -2, -4, 1, 0, -3], [3, 7, 1, -2, 5, 14, 4, -4, -13, 8, -8, 2, 4, 8, 3], [14, 9, -11, 5, -5, -4, -10, -3, 12, -14, -12, 12, -4, -5, 7], [-18, -4, 5, 0, 2, -3, 6, 2, -10, 5, 12, -11, -1, 2, -10], [-14, 17, -4, -1, -11, -7, -3, -5, 14, 3, 4, -3, -7, -4, -9], [-5, -4, -16, -3, -7, -4, -4, -4, 7, -9, -1, -1, -2, -2, -10], [0, 10, 1, 10, 9, 11, -2, 4, -13, 5, 7, 3, 6, 4, -5], [-5, 12, -17, -4, -6, -4, 3, -4, 1, 1, -11, 4, -5, 0, -2], [13, -25, -23, -6, -11, -14, -18, 12, 11, -4, 8, 3, 5, -9, -10], [-12, 26, 24, -15, 4, 7, 21, 5, -6, 8, -1, -10, -7, 4, 11], [25, 16, 4, 9, 10, 4, 2, -5, -14, 3, -2, 4, -1, 4, 3], [23, 8, -4, 3, 1, 9, -6, 1, -7, 5, -3, 4, 13, 7, 2], [15, -6, -11, 9, -3, 10, -7, -7, -4, 5, -2, 7, 5, -4, -1], [-7, -10, 4, 8, 8, 4, 7, 3, -19, 0, 3, -2, 7, 1, 8], [-7, 2, -13, -5, -7, -2, -4, 2, 22, -5, 1, 5, -3, -7, 8], [4, -1, 17, 1, 15, 6, 9, 7, -16, -4, -1, 1, 5, 5, 6], [0, -13, -8, 3, -11, -5, -8, 7, 11, -3, 4, 4, 2, 4, -6], [-17, -23, -22, -16, -19, -13, -14, -3, 26, -11, -3, 8, -4, -12, 5], [8, -3, 12, -4, 2, 7, 2, 14, -11, -12, -7, -1, 2, 6, 6], [4, -7, 1, -3, -1, -3, -2, 1, -7, 2, 3, 4, 2, -2, 4], [-19, 3, 10, -11, -1, 11, 15, 0, -2, 5, 2, -11, 0, -2, 9], [0, -4, -14, -4, -6, -1, -7, -8, 7, -2, 7, 1, -2, -2, -1], [9, 14, -2, -7, 6, -9, 6, 11, -5, -12, -11, 5, -6, 0, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5225_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5225_2_a_u();".
// 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_5225_2_a_u(:prec:=15)
    chi := MakeCharacter_5225_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_5225_2_a_u();".
// 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_5225_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![5, -93, 171, 485, -793, -1160, 1055, 1239, -606, -645, 165, 168, -21, -21, 1, 1]>,<7,R![78609, -284689, 99459, 576209, -509711, -200034, 329737, -37894, -65992, 21402, 3534, -2497, 197, 78, -17, 1]>],Snew);
    return Vf;
end function;