
// Make newform 3750.2.a.v in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3750_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_3750_2_a_v();".
// 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_3750_2_a_v();".
// 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, -96, -132, 40, 101, 10, -18, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [106, 208, 21, -145, -70, 32, 11, -3], [86, 10, -247, 48, 144, -26, -19, 4], [18, -340, -361, 319, 222, -78, -27, 7], [22, 998, 569, -862, -386, 186, 47, -14], [-162, -866, -587, 840, 450, -194, -57, 16], [-42, -578, -364, 502, 256, -111, -32, 9], [-18, -1650, -1309, 1446, 838, -332, -103, 28]];
    Rf_basisdens := [1, 20, 20, 20, 20, 20, 10, 20];
    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_3750_a();"
function MakeCharacter_3750_a()
    N := 3750;
    order := 1;
    char_gens := [2501, 3127];
    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_3750_a_Hecke(Kf)
    return MakeCharacter_3750_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 := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, -1, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, -1, 0, 0, -1], [1, 1, 1, 0, 0, 0, 1, 0], [2, 0, 0, 0, 1, 1, 0, 0], [1, 0, 0, 1, 1, 0, 0, 1], [2, 0, -1, -1, 0, 0, 1, 0], [-1, 0, 0, 1, 0, 1, 0, 1], [2, 1, 0, -2, -1, 1, 0, 0], [1, 1, -1, 0, 0, 0, -1, 0], [1, -1, -1, 2, -1, 1, -1, 2], [3, 1, 0, -2, 1, 0, 0, 1], [-2, -1, -1, 3, 0, 1, 1, 0], [3, 0, 1, -2, 1, 1, 1, 2], [-1, 1, 1, 0, 1, -1, 1, -2], [1, -1, 1, -2, 0, 0, 1, 0], [-1, 0, -1, -2, -1, -1, 1, -2], [-1, 0, 2, 4, -2, 1, 0, 1], [3, -1, 2, -1, -1, 1, 0, 0], [0, 1, 1, 5, 1, 0, 1, -1], [-3, 1, -1, 1, 1, 0, 2, 1], [5, 1, 1, -6, 1, 1, -1, -1], [-4, -1, 1, 0, -2, 0, 2, 0], [1, -2, -2, 6, 1, 0, 0, 1], [3, 0, -1, 0, 1, -1, -1, 0], [3, 0, 0, -1, -1, 0, 0, 1], [-5, 0, 0, 3, -2, 1, 0, -2], [1, 1, 1, -4, -1, 1, -1, 0], [-1, 0, -1, 2, -2, -2, -1, -2], [0, 0, 0, 2, -1, -1, 2, 0], [-3, 1, 2, 7, 0, 1, 0, 0], [3, -1, -2, -5, 1, -1, 0, 2], [-13, -2, 0, 1, 0, -1, 2, 1], [-4, 2, 2, -2, 1, 1, 0, 0], [0, 0, 2, 4, 0, 0, 2, -2], [-1, -2, -3, -2, -1, -2, -3, -1], [-4, -1, -1, 2, 0, -2, 2, 0], [2, -3, -3, -1, 0, -1, -1, 1], [7, -1, -3, -6, 3, -1, -1, 0], [-8, 0, -1, -2, 2, 0, 2, 0], [-1, 2, 0, 1, 1, -1, -2, 2], [0, -1, -1, 2, 0, 3, -1, -1], [2, 0, 2, 0, 0, 0, 0, -2], [-5, 0, -4, 3, 1, 0, -2, 0], [-3, 0, 2, 6, -2, 1, -2, -1], [5, 0, 1, 0, -1, -1, -1, 0], [3, 0, 1, -6, 2, 0, -1, 0], [-2, 0, 0, 6, 0, 0, -2, 2], [8, 3, 2, -4, 0, -1, -2, -3], [1, 0, 0, -6, 0, 0, 3, 0], [1, 4, 1, -2, -2, 0, -3, 0], [6, 2, -2, -10, 3, -3, 2, 0], [2, 0, 1, 4, 1, -1, 2, -2], [-2, 0, 1, 1, 3, 5, 1, 4], [-11, 0, -2, -1, 0, -3, 0, -3], [12, 1, 2, -14, 1, -1, 2, -2], [-11, -1, -1, 6, -3, -1, -1, 0], [-5, 3, 1, 7, -3, 1, -1, -2], [-5, 3, 3, -2, -2, 0, 1, -2], [3, 3, 3, -6, 1, -1, 3, -4], [-16, 1, 3, 5, 0, 1, 1, 2], [-11, 0, -2, 4, 2, 1, -2, 1], [4, 2, 0, 4, -1, 1, -2, 2], [-9, 0, 0, -6, 1, 1, 1, 0], [-4, -1, 1, 8, -2, -3, 3, 1], [4, 1, -1, -3, 2, -1, 1, -3], [-10, -4, -2, 2, -2, 0, -2, 0], [4, -2, -2, 2, 1, 1, 0, 2], [-3, 0, 0, -4, -2, -2, -1, -4], [-20, 0, 0, 4, 0, 2, 0, 0], [0, -4, -2, -4, 6, 0, 0, 4], [0, 0, -2, -4, 3, -3, 2, -4], [-1, -3, -2, 5, 0, -1, 0, 0], [-12, 0, 2, -8, 2, -2, -2, -2], [4, -3, -2, 0, -3, -1, 0, -4], [12, -4, -2, -5, 0, -2, -2, 2], [-3, 2, -2, -4, 1, 3, 3, 0], [2, -1, 1, 1, -2, -1, -3, 1], [1, -3, -3, 2, 2, 0, -3, 0], [-5, -3, -4, 9, 3, 2, 0, 2], [-2, 3, 1, -8, 5, -4, -1, -3], [0, 0, 1, -5, 3, 1, 5, 2], [-2, -1, 1, 3, 1, -2, 5, -1], [6, -1, -1, -21, 4, -1, 1, 0], [2, 0, 4, -3, 4, 3, -1, 5], [-11, -3, -1, 10, 2, 2, 3, 0], [-4, -1, -1, 4, 1, -2, -1, 1], [4, 2, -2, -4, -5, -3, -2, -2], [-5, 2, 2, 1, 1, 2, 0, 0], [3, 0, 0, 0, -3, -1, 1, 0], [8, 2, -2, -10, 1, -1, 0, -2], [6, 2, 2, -6, 2, 0, 4, 0], [-5, 1, 7, -2, -1, 3, 1, 1], [6, 1, 2, -2, -3, -1, 0, -2], [-1, -1, 1, 4, 5, -5, 3, 0], [-6, -1, -3, 7, 4, -3, -1, -3], [12, 0, 0, -10, -1, -1, -4, 0], [-5, -2, -3, 2, 1, -2, 5, 1], [3, -3, -6, -1, -2, 0, -2, -2], [4, -3, 1, 3, 1, 2, -1, -1], [-11, -1, -1, 22, -3, -3, 3, 0], [-1, -4, -4, -2, -3, 1, -3, 4], [7, -2, -2, 6, -1, -4, 0, -5], [13, 2, 4, -5, -1, 2, -4, 3], [10, 2, 2, -14, 1, -1, 0, 4], [3, 3, 0, -10, 6, 3, 4, 3], [5, 0, 0, 13, 0, 1, -4, -2], [1, 2, 0, 3, -4, -3, -2, -3], [13, -2, -5, -16, 0, -2, -1, 2], [-4, -2, 0, -2, 2, 0, 0, -2], [-15, 3, 1, -6, 0, 0, -1, 2], [-4, 2, -4, 6, 2, 0, -2, 2], [10, 4, 2, -8, 6, -2, 0, 2], [-4, 4, 2, 14, 1, 5, 0, 0], [1, 0, 1, 12, -3, 2, 1, -3], [19, 2, -2, -18, 4, -4, 1, 0], [0, -4, 0, -2, 2, 2, 6, 4], [-5, 1, 1, 17, -5, 2, 0, 1], [-9, 3, 2, -5, 2, 2, 0, 0], [17, -2, -4, -14, 0, 1, 0, -3], [-15, -5, 1, 14, -3, 5, 1, -2], [-7, 4, 7, 0, -2, 0, -1, -6], [-3, -2, 1, 6, 5, 3, 3, 4], [6, 2, -2, -10, 7, -1, 2, -2], [17, 1, -1, -6, 3, -1, -3, 2], [-12, 2, 2, -6, 0, 0, 4, -4], [-9, -1, 3, -4, -2, 2, -1, 0], [5, -1, 3, 6, 1, -1, -1, 2], [-2, 4, -3, 5, 0, 0, -5, 0], [-4, -3, -2, -4, -6, 4, -2, -2], [-10, -3, -8, -2, -1, -1, -4, 2], [1, 3, 6, -3, 4, 6, 0, 4], [1, 1, 4, -2, 3, 0, 0, 7], [-1, -5, -3, 2, -6, -2, -3, -2], [15, -3, -2, -8, -1, 4, -6, -1], [-6, 1, -1, -7, 5, -2, 1, 1], [-11, 1, 1, -2, -1, -1, -5, -4], [1, 0, 6, 4, 1, 1, -1, 0], [21, -4, 0, -15, -1, 4, -2, 4], [8, 2, -1, 9, 2, -4, 1, -4], [-11, 0, 3, 0, -1, -7, 3, -2], [-1, -1, 3, 14, -3, 3, -5, 4], [2, 1, -2, 4, 0, 4, 0, 0], [-2, 2, 4, 10, 4, 0, -2, 6], [6, 0, 0, -20, 0, 2, 4, -2], [25, -3, -5, -16, -1, -1, -1, 0], [-6, 3, -2, -8, -4, -4, -4, 4], [-10, 2, 4, -8, -4, 2, 2, 4], [3, 1, -7, -4, 4, 2, -3, 0], [1, -1, 7, -6, -2, 0, 1, 2], [-13, 1, -1, 16, -2, -2, -3, -4], [25, 5, 3, -18, 2, -2, -3, 0], [-28, -4, -2, 2, -5, 1, 0, 4], [-12, -3, -5, -3, 3, 2, -7, 7], [19, 3, 1, -1, 1, 2, 2, -1], [0, 0, 4, -3, 4, -1, 1, 5], [-1, -5, -2, 7, -1, -3, 4, -4], [5, 4, 6, -13, -1, -4, -2, -5], [-3, 0, -4, 8, -3, -3, 1, -2], [-9, -2, -2, 18, -2, -2, 1, -4], [25, 2, -1, -14, 0, 2, -3, 0], [13, 3, -3, -22, 0, 4, 5, -2], [-6, 0, 3, -9, -7, -5, -1, -4], [-24, -4, -4, 4, 0, -2, 2, 0], [-22, 1, 0, 8, -10, -2, -2, -2], [-7, -3, 2, 6, -1, 2, 10, -3], [24, -1, -1, -11, -5, -2, -1, -4], [1, -2, -3, -6, 3, -3, -3, -2], [15, -3, 3, -2, -2, 6, -1, 4], [-11, -7, 1, 11, 1, -2, 0, -1], [-3, -4, 4, -10, 1, -4, 0, 3], [-4, 1, 1, 15, 3, 0, 5, 1], [11, 1, -1, 4, -2, -4, 3, -4], [-1, 1, -5, 2, -1, -1, 1, 3], [-4, 3, 5, -7, -6, 1, 7, -2], [2, 3, 1, 11, -6, 3, -5, 5], [1, 0, 0, 13, 1, 6, 2, 4], [-2, 4, 4, -4, -5, -1, -4, -8], [-7, 4, 2, 3, 4, -3, 4, -1], [-1, -2, -4, 9, -7, 2, -2, -1], [8, 4, -2, 8, 3, 1, -2, 4], [-2, -2, 0, 10, -4, 6, 4, 2], [-13, 0, 2, 5, 0, 3, -4, 7], [-3, 3, 1, -18, -1, 1, 1, -2], [2, 5, 4, 0, 2, -1, 4, -5], [16, 2, 2, 2, -1, 5, -2, 4], [2, 0, -4, -10, -4, -2, -4, 2], [-12, 3, 3, -4, -3, 2, 7, -7], [-5, -3, -3, 8, -1, 3, -7, 2], [6, 8, 3, -6, 1, 5, 0, 0], [17, 3, 6, 5, 0, 2, 2, -2], [-3, -6, -7, 16, 2, -2, -5, 0], [-13, -5, 0, 13, -2, 4, -2, 0], [-14, -1, -2, 20, -2, -2, -4, 4], [-7, 2, -5, -4, -2, -6, 1, -6], [-1, -5, 1, -2, 5, 1, 5, -2], [-4, -6, 1, -4, 1, -1, 4, -6], [-11, 0, 0, 2, -5, 2, 0, 3], [-30, 2, 2, 10, -2, -2, 0, 0], [19, 4, 2, -9, 2, -4, 2, -2], [-2, -4, -8, -4, -1, -1, 0, -4], [-18, -1, 3, -1, 3, -6, 1, -1], [11, -1, -4, -19, 4, 0, -4, -2], [-10, 0, 3, 19, 4, 2, -3, 4], [24, 5, -4, -2, 1, -7, -4, -8], [0, -2, 4, 14, 1, 5, -8, 6], [18, 2, 5, -24, -2, -6, 6, -6], [-19, 0, 4, 18, 1, 3, 1, 6], [-5, -5, 1, 10, 5, -3, -1, 2], [-7, -1, -2, 13, -6, 4, -2, 2], [11, -5, -7, 6, 0, -4, -5, 4], [3, 1, 0, 26, -2, 7, 0, -1], [-6, -3, -1, 20, -1, -3, 2, -2], [-2, 5, -2, 12, -1, -5, -2, -4], [24, -2, 4, -18, 2, 2, 2, 4], [-21, -1, -1, 19, 1, -5, 1, 0], [4, -1, -7, 3, 0, -7, -1, -9], [-30, -8, -1, 19, -2, 2, -3, 4], [22, 0, 10, -10, 2, 0, 2, -2], [14, 5, 3, 1, -3, -4, 3, -4], [13, -4, -2, -18, -6, -3, -6, -5], [-12, -3, -1, 1, -2, 3, -3, -7], [-11, 0, 1, 16, -2, 4, -5, 4], [-3, 1, 7, 10, -8, 6, 3, 2], [21, 1, -5, 10, 1, 1, -7, 6], [6, -5, -9, 11, 4, 1, 1, 4], [1, 0, -4, -15, 0, 1, 4, 7], [1, 3, -1, 14, 5, -3, -1, -4], [-5, 10, 0, -4, 8, -3, -2, 3], [-12, -5, 3, 17, -1, -4, -9, 1], [14, 0, 6, -6, -1, 5, -2, 8], [-10, 2, 0, 28, -1, -1, -2, 2], [-7, -4, -2, -11, -2, 1, 0, -5], [-24, 1, -2, 20, 1, 5, 2, 0], [14, -3, 1, -3, -3, 8, 1, 8], [6, 0, 4, -22, -7, -1, 0, -8], [0, 6, 5, 3, -2, -4, -5, -12], [8, 4, 4, -22, -7, -5, -4, -6], [-4, -5, 1, 5, 2, -3, 3, -6], [13, 6, 9, 6, 1, 7, -5, 6], [-3, 0, 7, 18, 0, -6, 3, -6], [-9, 3, 1, -8, -6, -4, 5, 0], [12, -7, -5, 7, 0, 1, -3, 6], [12, 5, -7, 12, -3, 1, 2, 0], [21, -1, -3, 4, -6, -4, 1, -8], [3, 5, 3, 24, 5, 1, 1, 0], [-13, -6, -6, 21, -2, 5, -8, 5], [8, 1, -1, -27, 3, -10, 7, -5], [5, -5, -7, -2, 1, -3, 7, -2], [13, 4, 6, -7, 8, 0, 2, 0], [-6, 0, -2, -14, -2, 0, -2, -6], [3, 0, 0, 9, 6, -1, 0, 4], [12, -1, -3, -17, -6, 3, 5, -3], [4, 2, 12, 6, 2, 4, -4, 2], [-7, 0, -5, -14, -3, 0, -5, 7], [-18, 4, -4, 4, 0, -8, 6, -9], [19, -1, 1, -14, 1, 5, -1, 6], [3, -1, 0, 5, -3, -1, -10, 0], [-11, 2, -5, 8, -3, -1, 3, 0], [10, -5, -2, -22, -2, -2, -2, 2], [31, 8, -2, -23, -1, 0, -2, 1], [28, -6, 0, -12, 0, 0, -2, 10], [-5, -1, 2, 37, 2, 5, -4, 2], [-10, -1, 3, 5, -3, 6, -5, 9], [-19, -9, -9, 18, -1, -1, 5, 2], [7, 3, 6, -8, 5, 2, 2, 3], [7, -4, -7, 2, -5, 1, -11, 6], [-12, 4, 4, 12, 1, 1, -4, 2], [-7, 5, 6, 21, -7, 4, 2, 0], [-18, -3, 3, 4, -3, -6, 1, -1], [12, 4, 0, 2, 2, 0, -2, 1], [-6, 3, 1, 15, -12, -3, -1, -2], [23, 6, 2, -26, -1, 0, -4, -5], [38, 1, -1, 3, -6, 7, -3, 3], [-3, -4, 0, 7, 2, -1, -6, -7], [-4, -4, 8, 2, 8, -2, 6, 0], [8, 10, 2, 2, -3, 1, -6, 2], [5, -3, 4, 15, -4, 4, 0, -6], [2, -2, -4, 2, -1, -1, -2, 8], [25, 2, 1, 6, 5, -3, -5, 0], [2, 4, 0, 10, -5, -1, 6, -8], [23, 5, 5, -18, 2, -4, -5, 0], [-7, -4, 3, 0, 3, -3, 5, -4], [-8, -1, 7, 1, 5, 2, 9, -3], [-4, -6, -5, 33, 7, -3, -1, 0], [-36, 3, 6, 12, 5, -1, 6, -2], [9, 5, 5, 22, -5, 7, -5, 3], [-3, 9, 1, -14, 6, -2, -5, 2], [-15, -2, 0, 31, 3, -2, -6, 1], [3, 2, 2, -21, 4, 3, 6, 4], [2, 3, 4, 2, -6, -2, -8, 0], [-1, 1, -8, -1, 4, -3, -6, 6], [6, 1, 3, -4, 3, -4, 1, -5], [-22, -1, -5, -13, -1, 2, 5, 7], [-4, -3, -3, -5, -2, 11, -9, 8], [-19, -3, -3, 14, 2, 0, 7, 2], [6, 8, 3, 2, 0, 6, -4, -4], [-15, -1, -1, 3, -7, -3, 5, -2], [15, 0, 4, 10, 9, -6, 4, 5], [-6, -8, -6, 28, -7, 1, 0, 6], [15, 11, 7, -20, -5, -3, 3, 0], [25, -1, 2, -23, 3, -6, 2, -2], [-2, -2, -7, 1, 5, -5, -7, -10], [26, -7, -5, -21, 0, 3, -3, -3], [20, -6, -2, 4, -2, 2, 8, 4], [-14, 2, 6, 26, -2, 6, -2, 6], [-10, -8, -4, -12, 5, 5, -4, 16], [6, 8, 4, 6, 2, 4, 4, 6], [14, -1, 2, 14, -3, 7, 6, 6], [24, 2, 4, -4, 6, 6, 6, 2], [21, 1, 5, -26, -3, -3, 5, 0], [16, 1, 6, -8, 8, 1, 6, -5], [-2, 0, -4, -4, -5, -13, 4, -12], [-7, -1, -3, 26, 0, 10, -1, -2], [22, -1, 7, 1, -2, -1, 11, -1], [-22, -5, 0, 28, -5, -3, -8, -6], [17, -7, -3, -10, 9, 7, -3, 2], [21, -7, -2, -23, 1, 4, 6, 6], [-17, -10, -5, 6, -6, -6, -9, -8], [21, 6, 6, -16, -3, 5, 1, 2], [9, -6, 2, -7, 11, 2, 8, 1], [-5, 2, -8, -15, 10, 1, -2, 3], [-12, 4, 0, 2, -13, -1, 2, -8], [9, -1, -3, -22, -4, -4, -5, -2], [14, 10, -2, -4, -1, 1, 2, -6], [-32, -3, 2, 24, -3, -3, 6, 0], [-15, -2, 3, 10, 1, -1, -7, -2], [-7, 0, -6, 13, -3, -3, 4, 0], [-2, 4, 3, -1, -1, -7, 3, 4], [2, -5, -7, 4, 0, 5, -5, -1], [30, 0, 4, -2, 8, 10, 0, 4], [21, 7, 5, 10, 5, -3, 1, -4], [22, 3, -3, -3, 12, 1, -3, 3], [-9, -6, -3, 28, -8, 2, 7, -2], [11, 5, 12, -20, 2, -1, 0, 3], [16, -1, 1, 0, 4, -2, 2, -4], [5, 10, -4, -7, 6, 5, -2, -3], [-4, -4, -4, -24, 2, -6, -6, 8], [-40, -4, 5, -12, 3, -3, 0, -6], [37, 0, -6, -12, 6, -5, 0, -1], [-18, -3, -1, -1, 3, -10, 5, -1], [-7, -4, -4, 12, -9, -11, -5, 0], [-8, 3, 1, -7, 4, 3, 7, 9], [21, 5, 3, 16, -4, 8, -5, 4], [-8, -8, -14, -12, -8, -6, -4, -4], [-25, 8, 2, 27, 1, -1, 8, -2], [9, -3, 1, -6, 1, 7, -3, 2], [2, 2, -11, 4, 0, -4, 2, -6], [-8, -6, -10, -9, -12, -3, -9, -1], [14, -3, 3, -24, -1, -3, -2, 10], [-2, 2, -6, 20, -6, -2, -4, 6], [11, 6, 1, -18, 2, -2, 7, -8], [-10, -6, -10, 8, -6, -8, -8, -6], [30, -5, -5, -8, -1, -1, -4, -4], [21, 4, -2, -9, -9, 4, -10, -3], [1, 1, -1, -8, 14, -4, 7, 2], [-1, -4, -4, 27, -6, 1, -6, 1], [-24, 3, 3, 15, -9, 4, -9, -3], [-14, -6, 4, 12, -2, -8, -6, 0], [4, 3, 3, 14, -6, 4, 6, 4], [2, 8, 0, -8, 4, -6, 8, 2], [-10, 6, 0, 28, -4, -4, -6, -9], [5, 1, 8, 4, 1, -2, 12, 7], [21, 2, -2, -9, 1, -8, 0, 0], [12, -3, -2, -12, 0, -5, -2, 1], [8, 3, -5, 10, 0, -2, 6, 4], [8, -4, 2, 4, -11, -3, -4, -4], [7, 1, 5, 4, 6, -2, 5, 2], [-7, -12, -4, 26, 5, 3, 1, -10], [-16, -10, 1, 7, -1, -3, -3, 0], [4, -2, -8, -6, 2, -8, -10, -4], [19, 0, 12, -7, -11, 2, 2, 0], [1, -1, -7, -14, 0, 6, -9, 2], [-4, 9, 7, 15, 12, 3, 3, 13], [-10, 5, -1, -5, -5, -12, -1, 1], [8, -8, -6, 24, 6, 2, -4, -3], [-34, -2, 4, 6, -12, 2, 0, -2], [5, 6, 2, 23, -1, 6, 8, 3], [-19, -8, 4, -4, -8, 4, 7, -4], [-37, -6, -1, 12, -6, 2, -7, -4], [21, 5, -9, 8, -2, 2, -5, 2], [-29, 5, 3, -16, 0, -8, 3, 0], [34, 8, 1, -7, -3, 3, -3, 0], [5, -2, 0, 9, -4, 11, -8, 7], [24, 4, -6, -14, 1, 9, 6, 6], [38, -4, -8, -8, 5, -9, -6, 6], [-3, 3, 10, -20, 4, 5, 2, -1], [-23, 1, 2, -23, -2, 2, 0, 4], [20, -2, -2, -20, -3, 13, -6, 4], [3, -1, 3, -18, -4, -2, -3, -2], [9, -1, -1, 22, -10, -4, -11, 4], [25, -1, -10, -3, -1, 6, -2, 6], [7, 7, 0, -3, -1, 5, 2, -10], [-18, 1, -1, -27, 5, 0, 1, 11], [-17, 1, 9, 30, 1, -5, -1, -6], [35, 10, 12, -26, 5, 3, -1, 2], [30, -5, 1, -13, 4, 3, 9, 5], [11, 4, -3, 12, 2, 6, -5, 0], [11, -1, 1, -10, 2, 2, -7, 4], [9, 4, 5, -6, 0, 0, -9, 4], [4, -3, -7, 17, -4, 9, -9, 10], [-31, 3, 3, 44, 1, 3, 3, 2], [14, 7, 2, -2, 10, 4, -2, 6], [-50, 3, 2, 22, 1, 7, 10, -6], [-32, 0, -6, 20, -4, 4, 2, -10], [7, -2, 10, -13, 5, 0, 8, -8], [10, 3, 3, -17, -4, 3, 1, -4], [-4, 4, -8, 2, -2, 4, 0, -6], [-3, -11, -7, -10, -2, 2, -9, 2], [25, 5, 1, -34, 3, -1, -1, -4], [10, 4, 9, 10, -6, -4, 8, 0], [-25, -6, 4, 38, -3, -1, -9, 6], [15, -7, -3, 14, -2, -2, -7, -2], [-1, 8, 0, -9, 3, -2, -6, -8], [12, 6, 2, 10, -6, -4, 0, -12], [14, 2, -10, -10, 3, -11, 2, 0], [26, 2, -9, -30, 5, -7, 8, -4], [-11, -4, 4, 27, 0, -5, -6, 7], [-38, -4, -8, 8, 7, 3, -8, 8], [-6, 9, -3, -8, 1, 1, -12, 4], [11, -9, -5, -7, 1, 4, -4, 1], [-7, 4, 12, 4, -2, -5, 12, -5], [-10, 0, -1, 15, -12, 2, -7, 0], [-34, -5, 6, 26, -7, 3, -10, -4], [-31, 5, 5, 0, -3, 3, 1, -8], [7, 1, 3, -24, 7, 9, 1, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3750_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3750_2_a_v();".
// 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_3750_2_a_v(:prec:=8)
    chi := MakeCharacter_3750_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_3750_2_a_v();".
// 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_3750_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3750_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![316, 1304, 592, -1328, 205, 148, -33, -4, 1]>],Snew);
    return Vf;
end function;