
// Make newform 3627.2.a.u in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3627_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_3627_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_3627_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 := [16, 0, -128, 0, 316, 0, -275, 0, 103, 0, -17, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, -32, 0, 39, 0, -12, 0, 1, 0, 0, 0], [20, 0, -88, 0, 61, 0, -14, 0, 1, 0, 0, 0], [-32, 0, 92, 0, -61, 0, 14, 0, -1, 0, 0, 0], [0, -64, 0, 238, 0, -251, 0, 101, 0, -17, 0, 1], [-60, 0, 280, 0, -271, 0, 103, 0, -17, 0, 1, 0], [96, 0, -440, 0, 463, 0, -190, 0, 33, 0, -2, 0], [0, -176, 0, 650, 0, -675, 0, 279, 0, -49, 0, 3], [0, -208, 0, 826, 0, -797, 0, 307, 0, -51, 0, 3], [0, -68, 0, 422, 0, -461, 0, 190, 0, -33, 0, 2], [0, -96, 0, 440, 0, -463, 0, 190, 0, -33, 0, 2]];
    Rf_basisdens := [1, 1, 4, 4, 4, 8, 4, 4, 8, 8, 4, 4];
    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_3627_a();"
function MakeCharacter_3627_a()
    N := 3627;
    order := 1;
    char_gens := [1613, 3070, 1522];
    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_3627_a_Hecke( : Kf := Kf)
    return MakeCharacter_3627_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, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0], [-1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 0, -1, -1, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0], [-1, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, -1], [0, 4, 0, 0, 0, 3, 0, 0, -1, 0, 1, -2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2], [2, 0, -1, 2, 1, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0], [0, -2, 0, 0, 0, -2, 0, 0, 2, 0, -2, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, -2, -2, 2], [-1, 0, 3, 1, 1, 0, -1, -1, 0, 0, 0, 0], [-5, 0, 2, 1, 0, 0, -2, -2, 0, 0, 0, 0], [0, 3, 0, 0, 0, 1, 0, 0, 1, 2, -2, -1], [-2, 0, -1, -1, 0, 0, 1, 2, 0, 0, 0, 0], [-3, 0, 1, 4, -3, 0, 2, -3, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1], [0, 3, 0, 0, 0, 3, 0, 0, 2, 0, 2, 1], [-9, 0, 2, -2, -2, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -4, 0, 0, 0, 3, 1, 4], [-6, 0, -2, 3, 1, 0, 2, 2, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, 1, -4, -1, -3], [-1, 0, 4, -2, 4, 0, -3, -2, 0, 0, 0, 0], [0, 4, 0, 0, 0, 4, 0, 0, -1, 4, -2, -4], [2, 0, -1, 4, -1, 0, 3, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, -4, 0, 0, -1, 2, -1, 3], [0, -5, 0, 0, 0, 2, 0, 0, -3, -1, 1, 0], [6, 0, 0, -1, 2, 0, -1, 4, 0, 0, 0, 0], [0, -3, 0, 0, 0, -2, 0, 0, 2, -6, -1, 2], [-3, 0, -3, 0, -2, 0, -1, 1, 0, 0, 0, 0], [-5, 0, -3, -3, 0, 0, 2, -2, 0, 0, 0, 0], [-2, 0, 2, -2, -6, 0, -2, -2, 0, 0, 0, 0], [0, 2, 0, 0, 0, -7, 0, 0, 2, 2, -5, 0], [0, 3, 0, 0, 0, -2, 0, 0, 0, 5, -1, 2], [0, 4, 0, 0, 0, 3, 0, 0, 0, -1, 1, -3], [-3, 0, 3, -1, 3, 0, -3, 1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 4, 0, 0, -3, 6, 0, -2], [-5, 0, -4, -2, -2, 0, 3, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, -5, 0, 0, 0, 0, -2, -1], [-5, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0], [-6, 0, 1, 0, -2, 0, -2, 3, 0, 0, 0, 0], [3, 0, 6, -1, 1, 0, -3, 4, 0, 0, 0, 0], [0, 7, 0, 0, 0, 5, 0, 0, 1, 2, -2, -3], [-12, 0, 0, 0, 1, 0, 3, -1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 4, 0, 0, -5, -2, -2, -6], [0, 3, 0, 0, 0, 2, 0, 0, -1, 3, 3, -2], [-5, 0, -2, -3, 4, 0, -4, 1, 0, 0, 0, 0], [0, -10, 0, 0, 0, -8, 0, 0, 0, -6, -2, 2], [0, -3, 0, 0, 0, -6, 0, 0, -2, 3, -3, 6], [0, -8, 0, 0, 0, -5, 0, 0, 6, -1, 1, 7], [0, 13, 0, 0, 0, 7, 0, 0, 2, 7, 2, 1], [11, 0, -3, -4, 0, 0, -1, 1, 0, 0, 0, 0], [1, 0, -7, -5, -5, 0, 1, 1, 0, 0, 0, 0], [0, -7, 0, 0, 0, 1, 0, 0, -2, -7, 0, 6], [-8, 0, 3, 0, 4, 0, -6, -1, 0, 0, 0, 0], [0, 4, 0, 0, 0, 8, 0, 0, 3, -1, 0, -3], [0, 0, -4, 5, -2, 0, 7, 2, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, -5, 8, 1, 0], [-4, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, -8, -2, 1, 2], [4, 0, -1, 3, 13, 0, -2, 3, 0, 0, 0, 0], [-1, 0, -1, -1, -3, 0, -7, -3, 0, 0, 0, 0], [0, -10, 0, 0, 0, -11, 0, 0, 4, -5, -3, 3], [-6, 0, 0, 6, -4, 0, -2, -2, 0, 0, 0, 0], [0, -1, 0, 0, 0, -1, 0, 0, 6, -6, 2, 3], [0, 4, 0, 0, 0, 0, 0, 0, -2, 8, 0, 2], [10, 0, -3, -10, -7, 0, -1, 0, 0, 0, 0, 0], [-6, 0, -2, -2, -1, 0, 1, 2, 0, 0, 0, 0], [-6, 0, -4, 5, 0, 0, 3, -1, 0, 0, 0, 0], [0, 5, 0, 0, 0, -1, 0, 0, 2, 2, 2, -3], [0, 2, 0, 0, 0, 2, 0, 0, 1, -7, 0, 1], [-12, 0, 0, -10, -4, 0, 4, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, -1, 0, 0, -1, 7, 5, -1], [-1, 0, 0, 2, -1, 0, 4, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 4, 0, 0, 2, 6, -3, -2], [-1, 0, -4, 6, -4, 0, 5, -6, 0, 0, 0, 0], [0, 5, 0, 0, 0, -5, 0, 0, 3, -8, 6, -3], [-6, 0, 4, -6, 2, 0, -4, -2, 0, 0, 0, 0], [2, 0, -3, -7, -9, 0, 8, 3, 0, 0, 0, 0], [0, 6, 0, 0, 0, 2, 0, 0, 2, -7, 6, -6], [0, 5, 0, 0, 0, 2, 0, 0, -1, 9, -5, 0], [-6, 0, -2, 4, 1, 0, 5, -5, 0, 0, 0, 0], [0, -2, 0, 0, 0, 11, 0, 0, -7, 3, 5, 3], [7, 0, 2, -1, 5, 0, 5, 2, 0, 0, 0, 0], [0, 1, 0, 0, 0, 2, 0, 0, -2, -8, -1, 3], [0, 0, 0, 0, 0, -4, 0, 0, 2, -2, -4, 2], [2, 0, -2, -6, -6, 0, -2, -8, 0, 0, 0, 0], [0, 16, 0, 0, 0, 3, 0, 0, 0, 5, 5, -3], [-2, 0, 0, -6, -12, 0, 4, 2, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 0, 0, -4, -2, 0, -4], [0, 0, 0, 0, 0, -10, 0, 0, 0, -2, 2, -2], [0, 0, 0, 0, 0, 4, 0, 0, -1, 2, -2, -4], [4, 0, -2, -3, 6, 0, -5, 4, 0, 0, 0, 0], [-5, 0, -4, -4, -8, 0, 3, -2, 0, 0, 0, 0], [-6, 0, 1, 1, 11, 0, 0, -1, 0, 0, 0, 0], [0, 10, 0, 0, 0, -5, 0, 0, -1, 1, 5, 1], [0, -12, 0, 0, 0, -10, 0, 0, 1, 4, -6, 0], [0, 1, 0, 0, 0, -1, 0, 0, 2, 1, -6, 1], [-5, 0, 1, 4, -1, 0, -2, -5, 0, 0, 0, 0], [7, 0, -2, -2, -2, 0, -5, 4, 0, 0, 0, 0], [0, -7, 0, 0, 0, -7, 0, 0, -7, -11, -2, 8], [0, 4, 0, 0, 0, 1, 0, 0, -3, 6, 7, -11], [0, -3, 0, 0, 0, 4, 0, 0, -2, 2, 1, -4], [-17, 0, 5, 1, -15, 0, -1, -7, 0, 0, 0, 0], [10, 0, 0, 5, 9, 0, 0, 2, 0, 0, 0, 0], [-5, 0, -4, 5, -6, 0, 2, 3, 0, 0, 0, 0], [0, 10, 0, 0, 0, 4, 0, 0, 4, -2, 4, -2], [-15, 0, 0, -5, -11, 0, -1, 0, 0, 0, 0, 0], [-3, 0, 3, -4, 8, 0, 1, -3, 0, 0, 0, 0], [0, -5, 0, 0, 0, 1, 0, 0, 2, -3, -2, -9], [5, 0, 2, -1, -3, 0, 3, 4, 0, 0, 0, 0], [0, 2, 0, 0, 0, -14, 0, 0, 6, 10, -4, -2], [0, -6, 0, 0, 0, -10, 0, 0, 7, 4, 2, 6], [0, 3, 0, 0, 0, 6, 0, 0, -6, 0, 5, -2], [-1, 0, 2, -6, 2, 0, -9, 2, 0, 0, 0, 0], [3, 0, -3, -13, 5, 0, 1, -5, 0, 0, 0, 0], [0, -4, 0, 0, 0, -11, 0, 0, 3, 2, -1, 4], [0, -10, 0, 0, 0, -2, 0, 0, 6, -2, 4, 2], [-6, 0, 0, 5, 8, 0, -1, 1, 0, 0, 0, 0], [0, 12, 0, 0, 0, -2, 0, 0, 3, -4, 8, 0], [15, 0, 5, -3, -2, 0, 0, 3, 0, 0, 0, 0], [0, -2, 0, 0, 0, 3, 0, 0, -4, -5, -1, -11], [-2, 0, 3, -7, -3, 0, -2, -7, 0, 0, 0, 0], [0, 0, 2, 12, 2, 0, 4, 0, 0, 0, 0, 0], [3, 0, 4, 1, -3, 0, 3, 2, 0, 0, 0, 0], [0, 6, 0, 0, 0, -7, 0, 0, -1, -1, -3, 3], [2, 0, 2, 1, -5, 0, -6, -2, 0, 0, 0, 0], [6, 0, -2, -12, 2, 0, -2, 8, 0, 0, 0, 0], [0, -12, 0, 0, 0, 2, 0, 0, -4, -6, -6, 0], [-4, 0, 6, -4, 10, 0, 0, 4, 0, 0, 0, 0], [0, -7, 0, 0, 0, -1, 0, 0, 2, -10, -8, 3], [13, 0, 6, -3, -5, 0, 7, 0, 0, 0, 0, 0], [0, -12, 0, 0, 0, -4, 0, 0, 5, 1, 4, 7], [0, 3, 0, 0, 0, 4, 0, 0, 4, -8, 3, -14], [6, 0, 12, -5, -2, 0, -9, 5, 0, 0, 0, 0], [0, 13, 0, 0, 0, 12, 0, 0, -3, -3, 1, -12], [-10, 0, -2, -1, -6, 0, -1, -6, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, -3, -7, 0, 17], [0, 0, -4, 0, -6, 0, 4, 6, 0, 0, 0, 0], [0, -1, 0, 0, 0, 11, 0, 0, 5, 0, -4, -5], [-8, 0, -6, 10, 2, 0, -2, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 12, 0, 0, -2, 7, 5, -4], [-2, 0, -2, 13, 6, 0, -3, 2, 0, 0, 0, 0], [0, 15, 0, 0, 0, 4, 0, 0, 3, -3, -7, -8], [-2, 0, -4, 12, 0, 0, 4, 6, 0, 0, 0, 0], [0, 19, 0, 0, 0, 17, 0, 0, -10, 9, 6, -5], [17, 0, 6, 9, 8, 0, 0, -3, 0, 0, 0, 0], [0, -7, 0, 0, 0, -2, 0, 0, -12, -2, -1, 0], [-4, 0, 0, -3, 5, 0, 6, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0, 2, 3, 3, 1], [8, 0, 2, -9, -9, 0, -2, -2, 0, 0, 0, 0], [0, 2, 0, 0, 0, 7, 0, 0, -3, 7, -3, -5], [-22, 0, 2, 6, 5, 0, -1, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, -7, 0, 0, 10, -6, -6, 9], [0, -11, 0, 0, 0, -1, 0, 0, -7, -5, 4, -4], [0, -4, 0, 0, 0, 7, 0, 0, 1, -18, -3, -8], [11, 0, -3, 4, 6, 0, -3, -7, 0, 0, 0, 0], [0, -18, 0, 0, 0, -8, 0, 0, 1, -8, 2, 14], [0, 6, 0, 0, 0, -4, 0, 0, 7, 11, 2, -3], [0, -12, 0, 0, 0, 10, 0, 0, 1, -1, -6, -2], [23, 0, 1, 5, 7, 0, -1, -13, 0, 0, 0, 0], [-4, 0, 6, 0, -13, 0, 1, -4, 0, 0, 0, 0], [15, 0, 6, -3, 5, 0, -7, 6, 0, 0, 0, 0], [0, -8, 0, 0, 0, 7, 0, 0, -11, -2, 5, 2], [0, -7, 0, 0, 0, -1, 0, 0, 11, -3, 0, -3], [3, 0, -6, -6, 9, 0, -2, 2, 0, 0, 0, 0], [0, 9, 0, 0, 0, 6, 0, 0, 1, -10, 1, 3], [-3, 0, -4, 12, 8, 0, 5, -4, 0, 0, 0, 0], [10, 0, -3, -8, 0, 0, 4, 5, 0, 0, 0, 0], [0, 12, 0, 0, 0, 4, 0, 0, -5, -7, 0, -13], [-26, 0, -2, -11, -14, 0, 5, 2, 0, 0, 0, 0], [0, -2, 0, 0, 0, -8, 0, 0, -5, -3, -2, 5], [2, 0, -5, 16, 6, 0, -2, -1, 0, 0, 0, 0], [11, 0, -1, 11, -5, 0, 7, 1, 0, 0, 0, 0], [2, 0, -4, -9, -2, 0, -1, -1, 0, 0, 0, 0], [0, 2, 0, 0, 0, 11, 0, 0, 0, -15, 3, 3], [-29, 0, 4, 2, -5, 0, -8, -3, 0, 0, 0, 0], [0, -20, 0, 0, 0, -13, 0, 0, -1, -15, -5, 7], [0, -6, 0, 0, 0, 0, 0, 0, -10, 8, -2, -10], [0, 2, 0, 0, 0, 0, 0, 0, 13, 5, 6, 5], [-9, 0, -5, -1, 8, 0, 0, 2, 0, 0, 0, 0], [4, 0, 2, -3, -2, 0, 7, 13, 0, 0, 0, 0], [-29, 0, 7, -2, 3, 0, -6, -6, 0, 0, 0, 0], [0, -1, 0, 0, 0, 7, 0, 0, -1, 2, 4, 5], [-1, 0, -5, 1, -13, 0, 1, -9, 0, 0, 0, 0], [0, -11, 0, 0, 0, 1, 0, 0, -7, 0, 0, 3], [16, 0, -10, 8, -4, 0, 4, 6, 0, 0, 0, 0], [0, 16, 0, 0, 0, -5, 0, 0, 3, 15, -1, 1], [0, 5, 0, 0, 0, -4, 0, 0, -6, 6, -7, 4], [0, 6, 0, 0, 0, 1, 0, 0, 5, -4, -7, 6], [3, 0, 7, -3, 12, 0, -2, 9, 0, 0, 0, 0], [6, 0, -2, -11, 7, 0, 4, 8, 0, 0, 0, 0], [0, -14, 0, 0, 0, -12, 0, 0, 4, 6, 0, 2], [0, 5, 0, 0, 0, 14, 0, 0, 1, -10, 5, -11], [0, -6, 0, 0, 0, -1, 0, 0, 13, 6, 3, 3], [-15, 0, 14, 1, 8, 0, -8, -9, 0, 0, 0, 0], [-35, 0, 2, 6, -4, 0, 3, -10, 0, 0, 0, 0], [-22, 0, -3, 1, 8, 0, -5, -7, 0, 0, 0, 0], [0, -15, 0, 0, 0, -3, 0, 0, 5, -2, -8, 5], [0, 14, 0, 0, 0, 1, 0, 0, 5, -12, -3, 4], [7, 0, 2, 1, 12, 0, -2, 13, 0, 0, 0, 0], [0, -6, 0, 0, 0, -1, 0, 0, -8, 7, 11, -1], [0, 2, 0, 0, 0, -14, 0, 0, 2, 8, 2, 16], [-12, 0, -1, 4, 16, 0, -2, -1, 0, 0, 0, 0], [-4, 0, 4, -6, -2, 0, 6, -4, 0, 0, 0, 0], [0, -19, 0, 0, 0, -2, 0, 0, -8, -20, 7, 12], [28, 0, -11, 9, -9, 0, 6, 1, 0, 0, 0, 0], [0, -9, 0, 0, 0, 1, 0, 0, -5, 6, 10, -1], [0, 22, 0, 0, 0, 15, 0, 0, 6, 16, 9, -8], [-6, 0, 4, 22, 6, 0, -4, 0, 0, 0, 0, 0], [14, 0, -14, 11, -7, 0, 6, 4, 0, 0, 0, 0], [0, 2, 0, 0, 0, 12, 0, 0, -17, 6, 0, -4], [0, 10, 0, 0, 0, 3, 0, 0, -6, -4, -3, 3], [0, -6, 0, 0, 0, -13, 0, 0, -1, -12, -1, 9], [12, 0, -16, 12, 4, 0, 6, 10, 0, 0, 0, 0], [5, 0, 2, 4, 10, 0, -3, 4, 0, 0, 0, 0], [0, -12, 0, 0, 0, 0, 0, 0, 1, -1, 8, 7], [-18, 0, 12, -7, 0, 0, -9, -8, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 9, -19, 2, -5], [-6, 0, 2, -4, -4, 0, 8, 6, 0, 0, 0, 0], [0, 1, 0, 0, 0, 6, 0, 0, -10, -5, 5, -8], [0, -16, 0, 0, 0, -10, 0, 0, 2, 8, -10, 16], [-14, 0, -6, -13, 0, 0, -3, -14, 0, 0, 0, 0], [0, -12, 0, 0, 0, -8, 0, 0, 4, 0, 4, 14], [-1, 0, 1, -19, -9, 0, 1, -7, 0, 0, 0, 0], [-4, 0, 6, -3, -9, 0, 4, -14, 0, 0, 0, 0], [13, 0, -6, -4, -2, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, 0, 0, 6, -13, 1, -1], [-6, 0, -14, -7, -14, 0, 1, 2, 0, 0, 0, 0], [0, -12, 0, 0, 0, -17, 0, 0, 2, 4, 7, 5], [-22, 0, 10, -8, 4, 0, -4, -4, 0, 0, 0, 0], [0, -17, 0, 0, 0, -2, 0, 0, -2, -2, 9, 8], [0, 12, 0, 0, 0, 2, 0, 0, -6, 10, -2, -2], [0, -28, 0, 0, 0, -31, 0, 0, 4, 1, -13, 5], [0, 4, 0, 0, 0, 12, 0, 0, 4, -2, 0, 10], [19, 0, -3, -12, 7, 0, -6, 1, 0, 0, 0, 0], [18, 0, -2, -7, -2, 0, 5, 0, 0, 0, 0, 0], [7, 0, -3, 8, -3, 0, -2, 4, 0, 0, 0, 0], [0, 12, 0, 0, 0, 9, 0, 0, -3, -1, 9, -11], [0, 6, 0, 0, 0, -2, 0, 0, 3, 13, -12, -2], [19, 0, -2, 1, 7, 0, 5, -4, 0, 0, 0, 0], [0, 8, 0, 0, 0, 8, 0, 0, 3, 19, 0, 0], [6, 0, 6, -2, -14, 0, -6, -2, 0, 0, 0, 0], [0, -8, 0, 0, 0, -2, 0, 0, -4, 23, 0, 4], [-23, 0, -6, 12, 6, 0, 7, 14, 0, 0, 0, 0], [0, 6, 0, 0, 0, -1, 0, 0, 7, 4, -1, -5], [0, 12, 0, 0, 0, 0, 0, 0, -3, 21, -4, -3], [10, 0, 5, -6, 12, 0, -4, -13, 0, 0, 0, 0], [0, -23, 0, 0, 0, 6, 0, 0, -3, -17, -5, 0], [0, -26, 0, 0, 0, -10, 0, 0, -6, -20, -4, 14], [7, 0, 6, 3, 12, 0, 6, -5, 0, 0, 0, 0], [5, 0, -5, -8, 6, 0, -9, 3, 0, 0, 0, 0], [0, -7, 0, 0, 0, -18, 0, 0, -3, 7, -1, 10], [-12, 0, -4, 2, 4, 0, 4, -4, 0, 0, 0, 0], [0, 0, -6, 13, -1, 0, 12, 14, 0, 0, 0, 0], [0, -2, 0, 0, 0, -6, 0, 0, 0, 2, 0, 8], [-7, 0, -4, 5, -3, 0, 3, -4, 0, 0, 0, 0], [-18, 0, 9, -13, 7, 0, -8, 5, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, 1, 17, 1, 0], [-20, 0, -3, 9, -1, 0, -6, 1, 0, 0, 0, 0], [0, -19, 0, 0, 0, -19, 0, 0, 11, -10, -6, 3], [0, 18, 0, 0, 0, 10, 0, 0, -3, 15, -4, -9], [-11, 0, 14, 3, 0, 0, -2, -11, 0, 0, 0, 0], [0, -18, 0, 0, 0, 2, 0, 0, 0, -6, -4, 14], [-4, 0, -3, 3, 2, 0, -3, -3, 0, 0, 0, 0], [-21, 0, -8, -9, -27, 0, 9, -6, 0, 0, 0, 0], [8, 0, -15, 0, 5, 0, 1, 11, 0, 0, 0, 0], [8, 0, 4, -14, -10, 0, -4, -12, 0, 0, 0, 0], [15, 0, -1, 1, 14, 0, -4, -2, 0, 0, 0, 0], [9, 0, -2, -3, 2, 0, 14, 7, 0, 0, 0, 0], [0, 6, 0, 0, 0, 18, 0, 0, -18, 2, 0, -6], [16, 0, 0, -8, -1, 0, 5, 7, 0, 0, 0, 0], [-17, 0, 3, 7, 2, 0, -10, -1, 0, 0, 0, 0], [0, 30, 0, 0, 0, 9, 0, 0, 4, 25, -7, -9], [0, 24, 0, 0, 0, 0, 0, 0, -2, 13, 10, -15], [-4, 0, -8, 3, 12, 0, 7, -7, 0, 0, 0, 0], [0, 2, 0, 0, 0, -6, 0, 0, -1, -7, -12, 1], [-16, 0, -4, 2, 6, 0, -10, -6, 0, 0, 0, 0], [-15, 0, -2, 7, 4, 0, -2, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 1, 0, 0, 0, -23, 7, 1], [-16, 0, -8, 16, -4, 0, 18, 8, 0, 0, 0, 0], [0, 2, 0, 0, 0, 1, 0, 0, -9, -5, -5, -17], [10, 0, -10, -9, 2, 0, 3, 8, 0, 0, 0, 0], [0, 21, 0, 0, 0, 0, 0, 0, 6, 18, 9, -4], [0, -1, 0, 0, 0, 2, 0, 0, 5, 7, 5, -16], [0, 1, 0, 0, 0, 8, 0, 0, -5, -8, 7, -4], [0, -3, 0, 0, 0, 1, 0, 0, 0, -18, -6, 11], [0, -9, 0, 0, 0, -5, 0, 0, 11, -8, -8, -5], [13, 0, -5, 1, -1, 0, 7, 11, 0, 0, 0, 0], [0, -14, 0, 0, 0, -20, 0, 0, 2, -2, -8, 6], [-3, 0, -1, 4, 1, 0, 6, 5, 0, 0, 0, 0], [0, 4, 0, 0, 0, 8, 0, 0, 0, 16, 4, 8], [0, -2, 0, 0, 0, -12, 0, 0, 0, -12, 2, 10], [-30, 0, -7, 3, -5, 0, 2, 7, 0, 0, 0, 0], [18, 0, 4, -3, -1, 0, -12, 2, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 0, -1, -5, -5, -16], [12, 0, 11, -6, 17, 0, -11, 6, 0, 0, 0, 0], [0, -10, 0, 0, 0, -2, 0, 0, 9, 7, 4, 17], [14, 0, 5, 2, 1, 0, 3, -8, 0, 0, 0, 0], [-2, 0, 4, 14, 22, 0, -4, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 1, 0, 0, 8, -3, -9, 3], [-7, 0, -3, 9, 0, 0, 4, 2, 0, 0, 0, 0], [0, -21, 0, 0, 0, -7, 0, 0, 8, -10, -10, 15], [-3, 0, 3, -6, -9, 0, -4, 6, 0, 0, 0, 0], [0, -4, 0, 0, 0, -18, 0, 0, 4, -14, -14, 4], [0, 8, 0, 0, 0, -2, 0, 0, 4, 22, 2, -8], [0, -10, 0, 0, 0, -12, 0, 0, -9, -4, -4, 4], [30, 0, -1, -4, 9, 0, 5, -6, 0, 0, 0, 0], [0, -1, 0, 0, 0, 21, 0, 0, -1, -14, -2, -5], [2, 0, 8, -4, -8, 0, -8, -6, 0, 0, 0, 0], [0, 3, 0, 0, 0, 17, 0, 0, 3, -9, 16, -18], [0, -11, 0, 0, 0, -5, 0, 0, 11, 3, 4, 14], [11, 0, -10, -10, -10, 0, 3, 4, 0, 0, 0, 0], [0, -8, 0, 0, 0, -11, 0, 0, 15, -18, -1, 2], [18, 0, 0, 14, -10, 0, 6, 0, 0, 0, 0, 0], [-1, 0, 10, -3, 5, 0, -15, -6, 0, 0, 0, 0], [0, 10, 0, 0, 0, 6, 0, 0, 6, 0, -4, 2], [-3, 0, 8, -8, 14, 0, -23, -6, 0, 0, 0, 0], [0, -16, 0, 0, 0, 10, 0, 0, 1, -15, 2, -3], [3, 0, 11, 5, -1, 0, 5, -7, 0, 0, 0, 0], [-16, 0, 6, 18, 2, 0, 12, -10, 0, 0, 0, 0], [-16, 0, 4, 11, -2, 0, 5, 10, 0, 0, 0, 0], [0, 2, 0, 0, 0, -22, 0, 0, 4, 20, 8, 6], [0, -9, 0, 0, 0, -11, 0, 0, -8, -15, -14, 3], [-12, 0, 4, 6, -18, 0, 2, 2, 0, 0, 0, 0], [0, -8, 0, 0, 0, 16, 0, 0, -12, -20, 0, 8], [10, 0, -15, 20, -8, 0, 16, 3, 0, 0, 0, 0], [0, -4, 0, 0, 0, 1, 0, 0, -10, 12, 5, -11], [-16, 0, 22, -9, 14, 0, -19, -8, 0, 0, 0, 0], [0, 10, 0, 0, 0, 14, 0, 0, 2, -14, 12, -22], [8, 0, 4, -12, 17, 0, -1, -9, 0, 0, 0, 0], [0, -11, 0, 0, 0, 7, 0, 0, 4, -13, 10, -4], [-8, 0, -4, -2, -10, 0, 10, -2, 0, 0, 0, 0], [-3, 0, 5, -19, 15, 0, -3, -5, 0, 0, 0, 0], [-3, 0, -6, -14, -11, 0, 0, -14, 0, 0, 0, 0], [0, -20, 0, 0, 0, -22, 0, 0, 8, 6, -4, 16], [0, 7, 0, 0, 0, -4, 0, 0, 7, -25, 15, 0], [-3, 0, 8, -9, -8, 0, -6, 17, 0, 0, 0, 0], [0, -25, 0, 0, 0, -27, 0, 0, 14, 9, -14, 12], [0, 16, 0, 0, 0, 0, 0, 0, -6, 8, 6, -12], [-5, 0, 13, 4, 5, 0, 2, -6, 0, 0, 0, 0], [-9, 0, 8, -5, -25, 0, -5, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 0, 9, -5, 2, 12], [0, -22, 0, 0, 0, -18, 0, 0, 11, -14, -10, 8], [-23, 0, 4, -9, 1, 0, 7, 2, 0, 0, 0, 0], [27, 0, -3, 10, 2, 0, 9, 17, 0, 0, 0, 0], [0, 4, 0, 0, 0, -10, 0, 0, 11, 13, -6, -1], [-10, 0, -12, -3, -8, 0, 3, -2, 0, 0, 0, 0], [-26, 0, 8, -6, -1, 0, -15, 2, 0, 0, 0, 0], [0, -8, 0, 0, 0, -4, 0, 0, 0, -4, -2, -6], [0, 3, 0, 0, 0, 27, 0, 0, -7, -3, 2, -10], [0, 12, 0, 0, 0, 5, 0, 0, -6, -6, -7, 4], [0, -14, 0, 0, 0, -20, 0, 0, -1, 17, -14, 11], [0, 0, 0, 0, 0, 4, 0, 0, 4, -14, 8, -22], [-8, 0, -14, 2, -10, 0, -6, -6, 0, 0, 0, 0], [0, -22, 0, 0, 0, -25, 0, 0, -10, -13, -11, -3], [0, 10, 0, 0, 0, 9, 0, 0, -6, -11, 5, -7], [0, -11, 0, 0, 0, 13, 0, 0, -13, 0, -4, -9], [24, 0, -1, -16, 6, 0, -4, 5, 0, 0, 0, 0], [-17, 0, -5, 3, 15, 0, -5, 1, 0, 0, 0, 0], [0, 10, 0, 0, 0, 17, 0, 0, 7, -10, -7, -13], [49, 0, 4, 5, 17, 0, -7, -6, 0, 0, 0, 0], [-2, 0, -8, -6, -5, 0, 5, -4, 0, 0, 0, 0], [0, -2, 0, 0, 0, -6, 0, 0, -6, 13, -6, -15], [-16, 0, -2, 5, -15, 0, 2, -10, 0, 0, 0, 0], [0, 22, 0, 0, 0, 12, 0, 0, -6, 0, 2, -16], [0, 4, 0, 0, 0, 24, 0, 0, -9, -10, 10, 8], [36, 0, 2, -5, 3, 0, 2, 14, 0, 0, 0, 0], [19, 0, -11, 3, 1, 0, -3, 15, 0, 0, 0, 0], [0, -14, 0, 0, 0, -6, 0, 0, -13, 13, 4, 4], [0, -11, 0, 0, 0, -24, 0, 0, 7, 18, 3, 6], [-16, 0, -16, 4, -8, 0, 6, -12, 0, 0, 0, 0], [0, -22, 0, 0, 0, -14, 0, 0, 5, -2, -2, 14], [-8, 0, -12, -2, 0, 0, -4, 12, 0, 0, 0, 0], [0, -15, 0, 0, 0, 4, 0, 0, 3, -1, 9, -2], [0, -6, 0, 0, 0, -9, 0, 0, 7, 5, 1, 11], [-2, 0, 2, -2, 6, 0, 18, -4, 0, 0, 0, 0], [0, -29, 0, 0, 0, -11, 0, 0, 12, -32, -8, -3], [31, 0, -13, 2, 8, 0, 9, 3, 0, 0, 0, 0], [0, 6, 0, 0, 0, 2, 0, 0, 18, -2, 8, -12], [-11, 0, 2, -3, 14, 0, -2, 4, 0, 0, 0, 0], [-13, 0, -1, -9, -15, 0, -3, -19, 0, 0, 0, 0], [17, 0, 9, 8, 10, 0, -5, -11, 0, 0, 0, 0], [0, -20, 0, 0, 0, 22, 0, 0, -8, -28, -2, -12], [35, 0, 0, -7, 6, 0, -6, 11, 0, 0, 0, 0], [0, -17, 0, 0, 0, 14, 0, 0, -13, -23, 5, 0], [0, -12, 0, 0, 0, -2, 0, 0, 2, -10, 6, 22], [16, 0, 9, -15, 7, 0, -10, -13, 0, 0, 0, 0], [0, -20, 0, 0, 0, -24, 0, 0, 8, -20, 8, 12], [33, 0, 8, -4, 9, 0, 2, -9, 0, 0, 0, 0], [-8, 0, -10, 12, -6, 0, 18, 16, 0, 0, 0, 0], [0, 25, 0, 0, 0, 30, 0, 0, 4, -9, 13, -10], [-22, 0, -12, 7, -20, 0, 25, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0, -1, -11, -11, 3], [-3, 0, 0, -10, 3, 0, 2, 6, 0, 0, 0, 0], [0, -18, 0, 0, 0, 2, 0, 0, 1, 6, -2, -4], [14, 0, -4, 3, 28, 0, 1, 11, 0, 0, 0, 0], [0, 0, 0, 0, 0, -15, 0, 0, 5, -10, -5, -13], [0, -13, 0, 0, 0, -13, 0, 0, -6, -14, 10, 3], [10, 0, -9, 2, -18, 0, 16, 15, 0, 0, 0, 0], [-1, 0, 0, 16, -2, 0, 9, 14, 0, 0, 0, 0], [0, 28, 0, 0, 0, 27, 0, 0, 7, 15, -1, -9], [-9, 0, 11, 10, 18, 0, -5, 1, 0, 0, 0, 0], [0, -12, 0, 0, 0, -14, 0, 0, 17, -13, -6, 18], [-1, 0, -9, 1, 5, 0, -1, 5, 0, 0, 0, 0], [0, 2, 0, 0, 0, -10, 0, 0, -6, 4, 8, 4], [0, 16, 0, 0, 0, 24, 0, 0, -14, -8, 4, -12], [-21, 0, 13, 8, -5, 0, -6, -19, 0, 0, 0, 0], [45, 0, 7, 13, 2, 0, 2, 8, 0, 0, 0, 0], [0, 15, 0, 0, 0, 6, 0, 0, -10, -11, -5, -6], [0, 2, 0, 0, 0, -1, 0, 0, 0, 8, -7, 21], [-18, 0, 12, -5, 13, 0, -10, 8, 0, 0, 0, 0], [0, -34, 0, 0, 0, -1, 0, 0, 1, -14, -1, -9], [0, 1, 0, 0, 0, 0, 0, 0, 0, -16, 9, 8], [0, -5, 0, 0, 0, 8, 0, 0, -10, -16, 11, 12], [-25, 0, 0, 0, 7, 0, 2, 16, 0, 0, 0, 0], [0, 4, 0, 0, 0, -1, 0, 0, 6, -16, 7, -11], [0, 8, 0, 0, 0, -18, 0, 0, -2, 11, 4, -8], [7, 0, 2, 4, 14, 0, -7, 16, 0, 0, 0, 0], [0, -23, 0, 0, 0, -5, 0, 0, 0, 12, -12, 25], [0, 11, 0, 0, 0, -6, 0, 0, 9, 2, -13, 2], [0, -21, 0, 0, 0, -19, 0, 0, -2, -14, 12, 11], [27, 0, 3, -2, -6, 0, -5, -17, 0, 0, 0, 0], [0, 0, 0, 0, 0, 18, 0, 0, -6, -18, 10, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3627_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3627_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_3627_2_a_u(:prec:=12)
    chi := MakeCharacter_3627_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_3627_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_3627_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3627_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 0, -128, 0, 316, 0, -275, 0, 103, 0, -17, 0, 1]>,<5,R![64, 0, -720, 0, 1936, 0, -1672, 0, 403, 0, -35, 0, 1]>],Snew);
    return Vf;
end function;