
// Make newform 3751.2.a.l in Magma, downloaded from the LMFDB on 25 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3751_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_3751_2_a_l();".
// 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_3751_2_a_l();".
// 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 := [4, -72, 37, 845, 313, -1980, -751, 1918, 535, -880, -160, 204, 21, -23, -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], [-1954, -9089, 120083, 117241, -478138, -258971, 600884, 191459, -310552, -58972, 76554, 7879, -8937, -379, 397], [-5444, -7421, 164959, 173149, -466654, -315185, 500280, 202563, -242568, -57494, 58118, 7329, -6691, -343, 295], [-14170, 202757, -154099, -1141657, 765802, 1748687, -1095396, -1048791, 620580, 287720, -163850, -36063, 20101, 1675, -925], [-3744, 75599, -205235, -577599, 711128, 958869, -872656, -596365, 460448, 166784, -116864, -21127, 14003, 987, -635], [33530, -512671, 736637, 3281183, -2944958, -5198053, 3875916, 3153645, -2110008, -868780, 543718, 109065, -65639, -5069, 2987], [-18590, 314929, -1160315, -2945849, 3931466, 5063227, -4801188, -3179403, 2531688, 892348, -642298, -113199, 76913, 5291, -3485], [-3238, 211373, -536855, -1529629, 1913474, 2551255, -2399404, -1599055, 1275688, 449780, -324426, -57195, 38869, 2679, -1761], [-13386, 782191, -2216293, -6097815, 7410358, 10202781, -9045212, -6352517, 4762672, 1775884, -1206238, -224897, 144199, 10509, -6523], [22178, -505349, 1164303, 3639797, -4176690, -5988775, 5229200, 3716723, -2779420, -1038908, 706958, 131639, -84705, -6155, 3837], [29482, -573395, 1162597, 3935035, -4341118, -6425909, 5537064, 3975405, -2968536, -1109060, 758498, 140325, -91123, -6553, 4135], [-66278, 1174593, -2565667, -8299889, 9513586, 13664443, -12054820, -8487643, 6436280, 2373180, -1640298, -300655, 196729, 14051, -8917]];
    Rf_basisdens := [1, 1, 1, 1, 1048, 786, 1572, 786, 3144, 3144, 1048, 3144, 1572, 1572, 3144];
    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_3751_a();"
function MakeCharacter_3751_a()
    N := 3751;
    order := 1;
    char_gens := [2543, 2421];
    v := [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_3751_a_Hecke( : Kf := Kf)
    return MakeCharacter_3751_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, 0, 0, 0, 0, 0, 1, 0, 0], [1, 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, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0], [-1, 0, -1, 0, 0, 1, -1, 0, -1, 0, 1, -1, 0, 0, 0], [-1, 0, 0, 0, -1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0], [1, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, -1, -2, 1, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 1, -1, 1, 0, 1, -2, 0, -1, 1, -1, 1, 0], [0, -1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 0, -1, 1, -1], [1, -1, 2, 0, -1, -1, 2, 1, -1, 1, 0, 1, 1, 1, -1], [-1, 1, 0, 1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 1, 0, -1, 1, 1, 0, -1, 0, -2, 1, -1, 0, -2], [0, 2, -2, -1, 1, 0, -1, 1, 0, 1, 1, 0, 2, 0, -1], [-2, 0, 0, 1, 0, 2, -1, 1, -3, 0, 1, 0, -1, 1, 0], [-2, -1, -2, -1, 1, 0, -1, 1, 1, 0, -1, -1, 0, 1, -1], [2, 0, 0, -1, -1, 0, 1, 1, 0, 0, -2, 0, -1, 1, -2], [0, 1, -3, 0, 2, 1, -1, 1, 1, 0, 1, -1, 1, 0, 0], [-1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, -1, 2, 1, 0], [-1, 1, 2, 0, 1, -1, 0, 1, -1, 0, 1, 1, 2, 0, 0], [4, -2, 1, 1, -2, 1, 2, 2, 0, 0, -1, 3, 2, 0, -3], [2, -2, 0, 2, 0, 0, 2, 1, 1, 0, -1, 1, 3, -2, -1], [2, 1, 2, 2, 2, 0, 0, -2, 1, -1, 2, -1, 0, -1, 3], [2, 1, 1, 1, 1, -1, 2, 0, 2, -1, -2, 0, 1, -1, 0], [1, 0, 1, 1, 0, 0, 0, -1, -2, 0, 2, -1, -2, 2, 3], [6, 0, 1, 1, 2, -2, 0, -2, 4, -3, 0, 0, 1, -1, 2], [2, 1, 2, 1, 1, 0, 1, 0, 0, 1, 1, -1, 1, 1, -1], [2, -3, 1, -1, -2, -1, 2, 0, 2, 0, -4, 1, 0, -1, 2], [-1, 1, 0, 1, 0, -1, 3, 1, 2, 0, -2, 1, 1, -2, 1], [6, -2, -1, 2, 0, 0, -1, 0, 0, 0, 1, 1, 1, 1, 2], [4, 1, -2, -3, 4, -2, -3, -2, 5, -1, 0, -3, -1, -3, 3], [-1, 0, -1, 1, 0, 0, -2, 0, -1, 0, 3, -1, 3, 1, 1], [-3, 2, -3, -4, 1, 1, -3, 1, -1, 1, 0, -2, -2, 1, -2], [0, 2, -1, 0, -1, -2, 1, 0, 1, -1, -3, 1, 0, 0, 3], [3, -1, -2, -1, 1, -1, -3, -2, 3, -1, -1, 0, 0, -1, 0], [0, -1, -1, -1, -1, 3, -2, 0, -4, 1, 0, -1, -1, 2, -1], [-1, 0, -1, 2, 0, -1, 1, -1, -1, 0, 2, 1, 0, 1, 1], [-1, 2, 1, -1, 1, -1, -2, 0, 0, 0, 2, -2, 1, -1, 1], [5, 1, 0, -1, -1, -2, 3, 2, 1, 0, -2, 2, 1, 1, 0], [0, -2, -1, 1, 1, 3, 0, 0, -1, 0, 2, 0, -3, 1, -2], [4, -3, 1, -3, 2, -3, 0, -3, 3, 0, 1, -3, 1, -2, 2], [2, 1, 4, 4, -1, -1, 2, 1, -3, -1, 0, 4, 3, 0, -1], [0, -4, 3, 3, -1, -1, 4, -2, 2, 0, 0, 1, 3, -2, -1], [-6, 0, 2, 0, -1, 0, 0, 1, -1, -1, -1, 2, -1, 0, -3], [5, 4, 3, 1, -1, -1, 2, 0, -2, 0, 0, 2, 2, 1, 3], [6, -1, 0, -2, -3, 0, 2, 3, -2, 0, -3, 1, -2, 1, -2], [3, 3, -3, 1, 1, 1, 0, 0, 0, 2, 1, -2, -1, 0, 2], [0, -1, 1, -1, -4, 2, 2, 2, -1, 0, 0, 1, 3, 0, -5], [-1, 1, -1, 0, 3, -1, 3, 0, 1, 0, -1, -1, 1, -1, 1], [-2, 1, 0, -1, -3, 0, 1, -2, -1, 0, -2, -1, -1, 2, 2], [4, 1, -1, 1, 0, -2, 0, -1, 5, 0, -1, -1, 3, -3, 1], [3, -4, 3, 4, -1, -2, 3, -1, 3, -1, -2, 2, 1, -2, 1], [-3, -2, 5, 1, -3, -2, 6, -1, -1, 1, -4, 2, 2, -2, -3], [9, -4, 0, -2, -1, 0, -2, 0, 0, -1, -2, 1, -6, 2, 1], [4, 1, -1, 1, 2, 1, 0, -2, 2, 0, 0, -1, 0, -1, 0], [-2, -4, 2, 2, 0, -2, 6, 1, 6, 1, -4, 3, 8, -3, -2], [-2, 2, -1, 0, 1, 0, 1, 0, 5, -2, -2, 1, 1, -3, 0], [-5, 1, 2, 0, -3, 0, 2, 2, -2, 2, -2, 3, 3, -2, -3], [-5, 1, -5, -1, 2, 4, 0, 1, -1, 2, 1, -1, 1, 1, -1], [-3, 3, -1, 1, -1, 4, 0, 3, -4, 0, 0, 1, -1, 1, -2], [-1, 0, 3, 2, 0, -3, 1, -3, 0, 0, 0, -1, 1, -1, 2], [1, 1, -5, 0, 3, 3, -7, 1, 1, -1, 1, 0, -1, 2, -1], [4, 1, 4, -2, -3, -1, 4, 3, -1, -1, -4, 3, 2, -1, -6], [-8, -2, -1, -1, -2, 4, 0, 4, -3, 0, -1, 0, 1, 0, -5], [2, -5, 0, 0, -2, 1, -2, 1, -3, -1, 1, 2, -3, 2, -3], [-3, -2, 1, -1, 1, -1, 0, 2, 4, -1, -3, -2, -1, -2, 0], [-6, 0, 7, 4, 2, -1, 1, -1, 1, -2, 2, 0, 2, -2, 1], [8, -1, -4, -3, 1, -1, 1, 4, 4, -1, -3, 0, 2, -1, -3], [-1, 0, 1, 2, 0, -3, 5, 0, 2, -1, -2, 2, 0, 2, 0], [3, 2, 2, 1, 0, 0, -1, 2, -3, 1, 1, 0, -4, 3, -2], [-4, -1, 2, 1, 0, 2, 1, -2, 0, 1, 2, 0, 1, -2, -2], [2, 5, -3, -2, 6, -2, -5, -5, 2, -1, 6, -7, -3, -2, 5], [0, 1, 3, -1, 1, -3, 4, -1, 4, 1, -5, 2, 4, -2, 0], [10, 2, 1, 3, 6, -5, -2, -5, 6, 0, 4, -4, 0, -4, 8], [-6, -1, -2, -3, 2, 2, -3, -2, 2, -2, 1, -4, -6, -1, 3], [6, -1, 2, 0, -4, -1, 2, -3, 1, 2, 2, 2, 3, -1, 0], [2, -4, 7, 4, 2, -4, 3, -3, 6, -1, -1, 0, 2, -2, 4], [-3, 3, 2, 2, 0, 1, 2, -1, -2, 3, 3, 0, 0, 0, 0], [6, 4, -3, -3, 1, 2, -6, 2, 3, -1, -1, -3, -3, -2, -1], [1, 3, 4, -2, 1, -5, 2, -2, 3, 0, -2, 1, 5, -1, 3], [1, 1, -3, -1, 5, -1, -4, -3, 2, 1, 5, -3, -1, -1, 6], [-2, -1, 2, 2, 6, -6, 0, -4, 9, -2, 1, 0, 3, -5, 1], [-1, -1, -2, 0, 4, 2, -2, 2, -2, 1, 0, 0, 1, 0, -3], [11, 2, 4, -1, -3, -1, 7, -1, -2, 1, -5, 2, -1, 1, -1], [7, 1, 2, 0, -4, 2, 0, -1, -2, 0, 0, -2, -5, -2, 2], [-5, 2, 2, -2, -1, 3, -2, 3, -3, -2, -2, 1, -3, 1, -3], [-10, -6, -5, 2, 0, 5, -5, 0, -3, 1, 2, -2, -3, 4, 1], [-6, -5, -1, 0, -7, 2, 1, 2, -3, 1, -2, 2, -3, 2, -3], [-2, -3, -1, -3, 0, 0, -4, -3, 5, -4, -1, -2, -3, -4, -2], [-2, 5, 3, 1, 2, 1, 0, -2, -2, 3, 3, -1, -2, 2, 5], [-3, 1, 2, -1, 3, -2, -3, -1, -1, 0, 5, -1, 6, -3, -1], [6, 0, 2, 4, 6, -4, 0, -6, 4, -1, 1, -2, 1, -5, 7], [8, 7, -7, 0, 2, 2, -1, 0, -3, 1, 3, -3, -3, 3, 2], [10, 8, -7, 2, 1, 3, -3, 2, -3, -1, 1, -2, -3, 2, 3], [0, 0, 0, 3, -1, 7, -3, 2, -6, -2, 1, -1, -4, 0, -1], [3, -10, 7, 4, -1, 0, 7, 2, 4, -2, -2, 6, 5, -4, -6], [-3, 5, -7, -2, 1, 4, -5, 4, -3, 2, 3, 0, 2, 3, -1], [3, 0, 0, 3, 5, -3, -3, -3, 6, -2, 4, -1, 5, -1, 7], [1, -1, -1, 2, 1, -1, 3, 3, 9, -2, -6, 4, 7, -5, -2], [-10, -3, -1, -3, -2, -1, 0, -1, 1, -1, -2, 1, -2, 1, -3], [-3, -4, 0, -2, -6, 1, 4, 5, -3, 1, -4, 5, 1, 4, -8], [-10, 0, -5, 3, -1, 1, -4, 0, -2, 1, 3, -1, -6, 5, 2], [1, 0, 0, -3, 2, 0, -5, -2, 5, -3, -1, -3, -4, 0, 4], [-5, -5, 3, -2, 1, -4, 1, -3, 0, 0, 0, -1, 4, -1, -4], [0, 9, -2, -3, -4, 5, -1, 4, -5, 1, -2, 0, -3, 7, 1], [-13, -6, -1, 0, 0, 1, -3, 2, -3, 1, 4, 0, -3, 0, 0], [-5, -4, 1, 1, -9, 5, 4, 5, -4, 1, -6, 5, 0, 2, -5], [3, -7, 5, 1, 5, -4, -4, -5, 5, -3, 1, -1, 3, -4, 2], [-1, 1, 4, 2, -2, -5, 12, 1, 3, 0, -4, 3, 6, 0, -6], [15, 1, 1, 0, 4, -4, -3, -8, 10, -2, 4, -4, -2, -5, 7], [1, 3, -4, 0, -1, 2, -2, 4, -2, 3, -1, 1, 4, 1, 0], [-4, 2, -6, -3, 1, 3, -7, -1, -2, 2, 6, -5, -4, 1, 2], [11, -7, -6, -4, 5, -1, -8, -2, 3, 1, 3, -6, 3, 1, 2], [-4, 6, 1, -1, 4, -3, 2, -2, 6, 1, 0, -1, 4, -5, 4], [-3, 5, 0, 2, -3, -2, 4, 1, -2, 2, 1, -2, 7, 0, -4], [0, 1, -2, -4, -3, 0, 0, 5, -4, 3, -2, 4, -1, 3, -3], [-7, 2, -4, -2, -5, 3, 0, 2, -7, 3, 0, -1, -4, 1, -2], [3, -12, 4, -2, -4, -4, 6, -1, 1, 0, -3, 3, 6, -2, -4], [5, -2, -3, -3, 1, 2, -2, 1, -1, 2, 1, -6, -5, -1, 6], [6, 0, 1, 0, -3, 0, 1, -2, -2, 0, -2, 5, 4, 3, 0], [-4, 3, -1, 5, -2, 0, -2, 1, -4, -2, 1, 5, 4, 3, -1], [-4, 1, -2, 3, -3, 2, -1, 2, -1, 3, -1, -2, -1, 2, -2], [6, 1, -11, 0, 3, 4, -9, 2, -1, 0, 1, -3, -6, 4, 1], [2, -3, -1, 0, -2, -1, 5, 2, 7, -2, -4, 2, 4, 0, -5], [-1, -4, 0, 3, -1, 2, 3, 3, -3, -1, -1, -1, 1, -1, 1], [-1, 2, 1, 0, 3, 1, -3, -3, -2, 1, 2, 0, -5, 1, 0], [-9, 2, 3, 2, -3, -1, 5, 2, 0, 0, 0, 3, 12, -2, -6], [-9, 5, -5, -1, 1, -1, 2, 0, 0, 1, 0, 0, 1, -1, 5], [5, 0, 9, -3, -7, -1, 8, 3, -2, -1, -8, 7, -1, -2, -3], [-2, 4, -5, -4, 1, -2, -1, 1, 4, 2, -1, 1, 0, 2, -2], [-2, -2, 5, -3, -1, -1, 4, 3, 2, -1, -2, 4, 1, -3, -1], [-8, 5, -6, 0, -1, 4, -6, 0, -8, 2, 2, -2, -6, 3, -1], [-7, -5, -2, -1, 2, -2, 3, 2, 5, 2, -5, 1, 7, -4, -1], [7, -4, 2, -4, -3, -2, 6, 3, 2, -2, -8, 2, 2, 1, -5], [-5, -8, -2, 0, 2, -1, -2, 0, 1, -2, 2, -4, -4, 3, 0], [-1, 4, -1, 1, 3, -1, -2, 2, 0, 0, 2, -4, -3, 3, 9], [-5, 6, 3, -2, 1, 0, -3, 2, -4, 3, 5, 0, 5, 1, -1], [-7, 8, 2, 1, -1, 1, 3, 1, -2, 3, 1, 1, 2, 2, -3], [-1, 0, 8, 1, -3, 0, 3, 3, -7, 3, 1, 3, 3, 1, 7], [-2, -4, 5, 8, 0, 3, 5, 1, 1, -1, -1, 1, 5, -2, -2], [-6, -3, 13, 5, 0, -4, 10, -3, 7, -2, -1, 3, 6, -7, -1], [-5, 0, -8, 0, -2, 2, 0, 3, 0, -1, -2, 0, -4, 0, 2], [3, -4, 2, 4, 5, -7, 8, -3, 9, -3, -1, 4, 5, -5, 4], [26, -2, 5, 2, 1, -6, 7, 0, 4, -1, -1, 3, 11, -2, 1], [-9, -2, -3, 2, 1, 0, -1, 4, 2, 1, -1, -1, 0, 0, -1], [-10, 1, 0, 4, -2, 4, -4, 3, -6, -1, 1, 3, -5, 2, 1], [-6, -5, 3, 4, -4, -4, 5, -1, 6, 0, -5, 2, 1, -6, -2], [-9, -2, -2, -1, -1, 3, -3, -3, 3, 0, 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6, -6, 4, 8], [-6, -13, 1, -4, -1, -2, -9, 0, 2, -6, -5, -1, -7, 0, 3], [-5, 5, 1, 0, 0, -3, 7, -2, 5, 3, -3, -1, 1, 0, -4], [-11, -4, -4, 1, -5, -2, 3, -5, -9, 5, -1, 0, -1, 0, 5], [7, 5, 0, -3, -6, 4, 3, 4, -11, 3, 4, 6, -4, 4, -1], [12, 10, -1, -3, 0, -5, 8, -4, -2, 1, 0, 1, 0, 3, 2], [0, -2, 1, 4, 0, 0, -1, 1, 8, -4, -8, 3, -4, -2, 7], [-11, -4, -7, -4, 1, 1, -1, 2, 5, 5, 3, -7, 4, 3, 9], [-2, -7, -1, -2, 2, 1, 3, 1, 7, -1, 0, 3, 4, -1, -1], [25, -6, -3, 1, 10, -11, 0, -8, 22, -5, -8, 0, 6, -10, 4], [0, -9, 12, 3, -1, 3, 7, 1, -8, 0, 1, 3, -4, 2, -3], [-24, 10, 0, 6, 6, 4, 0, -2, -2, 3, 5, -5, -1, 2, -5], [-1, -6, 6, -1, -8, -11, 13, 0, 16, -3, -15, 9, 8, -3, -6], [10, 11, -10, 7, 9, 4, -3, 2, -2, -1, 3, -2, -3, 2, 0], [16, 9, -1, 1, 4, -9, 10, -2, 14, -3, -3, 3, 17, -8, -3], [-26, -4, 2, -3, -6, 1, 1, 8, -9, 5, -5, 7, 12, 1, -15], [-2, 0, 3, -6, 1, -11, 1, -4, 5, 0, -2, -4, 13, -3, 0], [6, -6, -9, -1, 3, 3, -8, -5, -4, 5, 10, -2, -3, 3, -7], [14, 7, -7, -5, 3, -4, -2, -5, -6, 1, 3, -1, -5, 2, 10], [-18, 6, -9, 4, -1, 13, -11, 9, -15, 2, 5, -5, -8, 11, 1], [38, -5, 7, -1, -4, -1, 6, -7, 0, -1, -6, -3, -11, -5, 7], [0, -15, 9, 5, 1, -2, -4, -2, 1, -4, 7, 2, -9, 1, 2], [13, 8, 0, 4, -5, 4, -4, -4, -4, -5, 0, -1, -17, 2, 13], [5, 6, 8, -1, -8, 5, 11, 8, -16, 3, -7, 5, -7, 11, -12], [-1, -18, 3, -1, -4, -7, 4, -2, -1, -1, 2, -1, -2, -3, 7], [-1, -1, 11, 7, -4, 2, 6, -5, -3, -3, -2, 10, 4, 1, -2], [0, 7, 4, 4, 3, 2, 4, 7, 8, 0, -7, 5, 4, -1, -6], [-4, 9, -6, -1, 5, 9, -5, 3, -14, 6, 13, -9, 0, 4, 1], [-5, 0, 10, -6, -7, 2, 12, 5, -7, 5, -5, 2, 0, 6, -14], [-6, 8, 11, 1, -6, -1, 10, -8, 1, 1, -4, -2, -12, 4, 5], [10, -8, -1, -4, 5, 0, 1, 5, 10, -4, -5, 0, -6, -5, -13], [-6, -5, 15, -4, 2, -6, 7, 0, 8, -2, -6, 5, -3, -6, -5], [-3, -4, 7, -5, -2, -9, 6, -4, 6, 4, -5, 1, 7, -6, -2], [-8, -13, 4, -3, -13, 5, 5, 3, -13, 1, -10, 4, -13, 8, -3], [8, -7, 6, 5, 0, 4, -1, 0, -7, 0, 5, 1, -12, -2, -2], [8, -15, 7, 5, -1, -7, 8, 5, 2, -3, 1, 4, 9, -6, -8], [3, 11, 0, -1, 4, -3, 1, 0, -7, 2, 9, -9, -5, -1, 4], [-4, -1, 6, 2, -4, -3, 0, -4, -2, -3, -6, -4, -10, 5, 11], [2, -4, 1, 4, 3, 3, 1, 1, 7, -9, -8, 4, -5, 1, -5], [13, 0, 9, 4, 5, -14, 15, 0, 14, -1, -5, 2, 17, -11, -5], [10, -7, 12, 6, 5, -14, 0, -6, 9, -5, -3, 4, 4, -6, 6], [4, 16, 5, 5, -7, 6, 4, 2, -6, -5, -7, 4, 0, -1, 6], [5, 16, -1, -12, 1, -3, -5, -2, -1, 1, -7, -7, -9, 1, 7], [3, 0, 11, 7, 2, -7, 6, -8, 12, -1, -2, 5, -1, -5, 4], [2, 11, -5, -2, -3, 6, -7, -5, -5, -2, 8, -2, -14, 2, 10], [-30, 3, -7, -7, -5, 7, -10, 3, 0, 1, 3, -6, -5, -4, 2], [-4, -4, -12, -6, 3, -1, -6, 5, -1, 3, -1, -4, -3, 1, -6], [3, 8, -6, -2, -2, 3, -6, -2, -9, -6, -2, 0, 2, 5, 2], [-3, 9, -17, -16, 1, 2, -7, 5, -6, 6, 2, -9, -8, 1, 2], [-12, -20, 8, 7, -8, -4, 7, 4, -1, -2, -8, 5, 0, 3, -9], [-26, 13, 10, 5, 2, 2, -3, -1, -11, 2, 10, 1, 4, -1, -7], [-14, 1, -11, -1, -4, 9, -4, 0, -3, 8, 4, -1, -11, 7, -2], [-2, 9, 4, 7, 4, -8, 1, -7, 9, -1, 3, -3, 10, -6, 20], [-9, -1, 3, -8, 6, -3, -7, -15, 3, 0, 5, -12, -11, -1, 12], [24, -7, 11, 2, -1, -14, 13, -6, 8, -1, -10, 9, 7, -5, 4], [-28, 6, -5, -5, 2, 6, -8, 10, -1, -3, 2, -1, 0, 4, -5], [-9, -7, -9, 2, 4, -2, -7, -10, 16, -3, 7, -2, 3, -12, 6], [7, -4, 21, 8, -2, -5, 9, -6, 1, -2, -3, 1, 7, 0, -2], [16, -5, 6, -3, -5, -2, 7, 2, 2, 0, -12, 4, 3, -3, -11], [2, 1, -6, 1, -4, -3, 3, -6, -8, 2, 3, 0, -9, 6, 4], [12, -3, -3, -2, 1, 0, -11, -1, -3, -8, 0, -5, -5, -1, 11], [6, -9, 2, 9, 8, -2, -7, -9, -1, -8, 8, 0, -3, -4, 9], [12, -4, -2, 7, -2, -4, 9, 6, -5, 3, -3, 11, 13, -2, -10], [1, 14, -10, -2, 5, -5, 4, 3, 17, 4, -6, -3, 14, -5, 9], [5, -5, 1, 0, -8, 8, -1, 7, -12, 1, 4, 3, 0, 12, -9], [-6, 0, 7, 2, 0, -5, 5, -7, -2, 2, 6, 1, 0, 1, 9], [-10, 8, 8, 4, 2, 12, -2, 0, -16, -2, 2, 1, -19, 9, 3], [13, -8, -1, 4, 5, -2, -3, 1, 2, -1, 6, -5, 10, 1, 6], [-6, 5, 2, 6, 3, 6, 4, 8, 8, -5, -5, 4, 0, -2, -4], [-25, -1, 6, -3, -3, 0, 1, 6, -4, 2, -2, 4, 8, 1, -6], [-20, -6, 1, -4, 5, -3, 1, -11, 8, 0, 1, -8, 2, -6, 12], [10, -9, 9, 12, 2, -10, 9, -1, 12, 1, 0, 2, 13, -1, 7], [6, 5, -10, 2, 4, 8, -10, 8, -7, 1, 6, -12, -15, 6, 4], [-14, 12, -5, -9, 7, 1, -6, -6, 4, -2, 4, -13, -10, -4, 5], [46, 7, 12, 4, -7, -9, 12, -3, 1, -5, -10, 12, 1, 2, 11], [0, -2, 2, -14, -1, -9, 4, -11, 5, -2, -8, -5, -8, -5, 7], [-24, -9, 1, 0, -3, 3, -3, 4, 11, -4, -3, 1, -8, -7, 1], [6, -4, 8, -3, -8, -7, 21, 3, 10, -2, -15, 3, 11, -3, -2], [-5, -1, -13, -8, 9, 0, -17, -8, 12, -4, 9, -9, -10, 0, 10], [27, -1, 14, 7, 6, -9, 11, -2, 11, -2, -1, -2, 8, -4, 1], [-10, 4, -8, -6, -3, 6, -14, -1, -4, 4, 10, -5, 9, 0, -5], [6, 1, 17, 4, -5, -3, 11, 6, -5, 2, -11, 10, 2, 0, 1], [-15, -3, 9, 1, -3, 6, -4, 0, 3, -6, -7, 0, -2, -9, -3], [4, 2, 2, 11, 3, -4, 5, -3, -2, 0, 0, -6, 8, -3, 12], [11, -5, -1, 5, 6, -4, 4, -2, 7, 0, -2, -4, -2, -1, 8], [-30, -3, 2, 1, -9, 4, 7, 2, -5, 4, 4, -1, 5, 2, -2], [1, 7, 12, -3, -12, 0, 13, 0, -5, 0, -7, -2, 0, 1, -10], [5, -2, 10, 2, 2, 3, -6, 2, -9, -5, 3, 8, -5, 4, -9], [1, -12, 4, 5, 1, -13, 9, 4, 16, -3, -12, 5, 3, -11, 10], [-9, -20, 9, 10, -8, 2, 11, 3, -6, 1, -1, 5, 5, 1, -2], [2, 6, -10, 3, 4, -6, -7, -4, 1, 6, 10, -7, -1, 3, -3], [-18, 0, 4, -1, -7, 15, -9, 3, -24, 0, 6, 0, -28, 12, -7], [4, -3, 2, 6, -5, 3, -6, -3, -5, -3, 0, 3, -21, 3, 8], [-24, 13, -9, 0, 0, 10, -15, 2, -12, 4, 12, -9, -13, 8, 11], [23, 0, -2, 4, 0, -4, 2, 7, 4, 3, -8, 2, 8, -4, -8], [-12, 12, -3, 5, -6, 6, 8, 10, -11, 9, -2, 4, -5, 9, -2], [-7, 0, -6, 3, -5, -1, 1, -3, 4, -1, -3, -2, -6, 5, 3], [-14, 10, -6, -3, 1, 1, -3, 10, 0, -1, -6, 0, -2, 4, 2], [-1, 13, -11, -10, 2, -2, -5, 0, 3, 6, 0, -6, -3, 1, 5], [10, 2, 9, 6, -3, -5, 15, -6, -11, 5, -3, 1, 1, 1, 1], [-9, -1, 6, 0, -5, 2, 4, 3, -18, 4, 9, 0, -8, 10, 0], [-9, -16, 7, 6, -2, -2, 15, -4, 10, -3, -6, 5, 20, -8, 3], [6, -2, 0, 2, 6, 0, -4, -3, -8, 6, 9, -7, -1, 8, 5], [0, -7, 15, 1, -2, 1, 4, -9, -7, -1, 6, 4, 1, -4, -6], [8, 10, -2, 1, 0, -7, -1, 4, -2, -1, -3, 6, -2, 4, 6], [7, 12, -8, 5, -5, 8, 7, 5, -13, 3, -3, 5, 6, 7, -1], [31, 8, 9, -4, -1, -8, 17, -4, 14, -1, -9, 0, -2, -3, -1], [15, 0, 8, 9, -3, 2, 3, -5, -8, 4, 14, 5, 7, 4, -2], [24, 13, -8, 7, -1, 3, -1, 4, -2, -4, -2, 4, -1, -2, 10], [-7, -22, -3, -5, -8, -1, 6, -2, -9, 3, -2, -1, 7, 5, -7], [-4, 0, 6, 10, 7, -4, 8, 4, 6, -3, 0, 0, 14, -3, 3], [-15, 3, 3, -4, -8, 9, 5, 8, -15, 0, -8, 6, -16, 6, -8], [3, -8, 14, -4, -6, 2, 8, 1, 4, 0, -9, 9, -2, -6, -9], [-17, -3, -5, -4, -8, 6, -5, -3, -12, -2, -1, -7, -31, 9, 10], [18, -8, 5, 1, -6, -10, 12, -4, 3, -5, -10, 14, 6, -3, -2], [-3, 3, 6, -3, -4, 2, -7, -2, -3, 3, 0, 2, -2, 2, -3], [-6, 8, 4, 13, 0, 3, -1, 0, 0, 1, 1, 2, 11, -6, 4], [14, -3, 0, -1, 3, 2, -5, 3, 4, 4, 9, 2, 4, -2, -9], [-20, 8, -8, -12, 4, -4, -10, -3, 8, 1, -4, -7, 1, -5, 6], [16, 7, 1, 8, 2, -6, 1, -6, 6, -4, 6, 1, 12, -4, -5], [5, -7, 12, 18, 6, -2, 6, -3, 3, -3, 9, 9, 19, 0, -4], [-5, 1, 4, 1, -13, -1, 11, 2, 5, 3, -13, 0, 6, 5, -4], [15, -3, 0, 7, -2, -2, -7, -6, 3, -11, 0, -4, -15, -2, 5], [-19, -7, -18, -2, 2, -5, 4, 6, 18, 0, -13, -1, 12, -1, -6], [17, -24, 10, 6, 1, -6, 0, -3, 8, -5, 1, 3, 2, -1, 0], [10, -3, -5, -3, 1, -7, 4, 0, 13, -4, -9, 8, 13, -8, -5], [15, -7, 11, -3, 2, -7, 0, -5, 12, 1, -6, -1, -6, -3, 2], [9, 7, -2, -11, -2, -4, 7, 0, -5, 3, 4, 0, 6, 2, -9], [18, -9, -14, -3, 2, 11, -11, 3, -8, -2, 2, -7, -26, 4, 3], [-18, -8, -17, -5, 12, -1, -16, -1, 13, -4, 10, -12, -4, -2, 4], [-7, -9, 6, 10, 12, -3, -2, -3, 11, -6, 0, -8, 0, -7, -5], [-9, -1, 3, 10, 9, -4, 1, -1, 10, -6, -6, 9, 14, -9, 10], [-5, -9, 5, 11, -3, 1, 12, 8, 4, -1, -4, 7, 13, -6, -21], [-29, 7, -1, 1, -5, 8, 0, 9, 7, 0, -8, 0, -1, 0, -1], [2, -13, 0, -2, -5, -1, 6, 7, 5, 2, 1, 2, 14, -3, -8], [-9, -14, -2, -1, -1, 8, -9, -2, 5, -3, -2, -4, -5, -3, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3751_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3751_2_a_l();".
// 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_3751_2_a_l(:prec:=15)
    chi := MakeCharacter_3751_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_3751_2_a_l();".
// 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_3751_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3751_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, -72, -37, 845, -313, -1980, 751, 1918, -535, -880, 160, 204, -21, -23, 1, 1]>,<3,R![64, -608, -512, 5704, 2380, -12408, -5043, 8284, 2791, -2590, -621, 418, 59, -33, -2, 1]>],Snew);
    return Vf;
end function;