
// Make newform 7623.2.a.cw in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7623_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_7623_2_a_cw();".
// 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_7623_2_a_cw();".
// 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, 30, -30, -30, 35, 10, -11, -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, 1, 0, 0, 0, 0, 0, 0], [-140, 135, 130, -30, -30, -5, 2, 1], [-45, 180, -60, -165, 35, 35, -4, -2], [-20, -170, 165, 160, -65, -40, 6, 3], [-90, 210, 105, -280, -55, 95, 7, -9], [15, -210, -80, 280, 55, -95, -7, 9], [170, -405, -90, 440, 15, -135, -1, 12]];
    Rf_basisdens := [1, 1, 25, 25, 25, 25, 25, 25];
    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_7623_a();"
function MakeCharacter_7623_a()
    N := 7623;
    order := 1;
    char_gens := [848, 4357, 4600];
    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_7623_a_Hecke(Kf)
    return MakeCharacter_7623_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], [-2, 0, -1, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 1, 2, 1, 0], [-1, -1, 1, -1, -1, -1, 0, 0], [2, 1, -1, 0, 0, 1, -1, 0], [0, 0, 1, 2, 1, 1, -1, 0], [0, -1, 1, 0, 0, -2, -1, 2], [0, -1, 0, -3, -2, -1, -1, 0], [1, 1, 1, 1, 2, 0, 0, 1], [-1, 0, 0, 0, -1, 0, -2, 1], [2, 0, 1, 2, -2, -1, -1, -1], [-2, 2, -1, 1, 0, 0, 2, -2], [-4, -1, 0, 0, -2, -2, 0, -3], [1, -3, 1, 0, 1, 1, -1, 1], [-3, -3, 0, -1, 1, -1, -1, 1], [-3, 3, 1, 1, -1, -2, 0, 0], [1, 1, 3, 0, -3, 0, -3, -1], [-1, 1, -2, 0, -1, -5, 2, 0], [0, -2, 0, 1, 4, -1, -1, 1], [-4, -1, -2, -1, -2, -1, 1, -2], [-4, 3, 2, -1, -2, -1, -3, 0], [5, -3, 3, -2, -1, 3, -1, -1], [-2, -1, -3, 2, 2, 1, 1, -2], [-2, -4, 1, 1, 2, -4, 2, 2], [-1, -3, 0, -1, 0, -4, 3, 0], [8, 1, 1, 2, -1, 4, -1, -3], [-6, 2, 0, 0, -2, 1, 0, 0], [-4, -1, -3, 0, -1, 0, 5, -1], [-2, -2, 3, -2, 0, 2, -3, -1], [-7, -4, 1, 2, 0, -1, 1, 1], [-5, -2, 1, -3, 3, 0, 0, 3], [4, 0, 0, 5, 0, -2, 1, -1], [0, -2, 5, 3, -3, 3, 0, 1], [7, -1, -1, 6, 1, 9, 1, -1], [-8, 4, -6, -5, 3, 1, 3, 0], [9, -3, 0, 0, -3, 9, 0, -4], [1, 0, -3, 2, 1, -4, -1, -4], [3, -4, 1, 4, 2, -3, -1, -1], [-2, 2, 3, 4, 0, -4, -3, 5], [3, 1, -1, 0, 3, 6, 5, 1], [1, -1, 4, 8, 5, -2, -2, 0], [-9, 3, -4, -1, -4, -4, -1, 0], [-6, 0, -1, 3, 2, -8, 0, 4], [1, -3, 1, 3, 2, -2, 2, 5], [-4, 1, 1, -2, 0, -5, -3, 4], [-2, -1, -1, 0, 2, 5, 7, -2], [-2, -1, 2, 2, 4, 7, 4, 5], [8, 5, 5, 2, 2, 5, -3, 0], [7, 1, 1, 3, -3, -8, -2, -2], [-2, -4, -6, 2, 2, -2, 2, 2], [4, -1, 3, 0, 0, 9, 1, 4], [-12, 4, -2, 1, 0, 0, 3, -5], [1, 4, -2, -2, 1, 3, 4, 1], [4, -4, 2, 1, 0, -4, -1, -3], [6, -4, 6, 0, -2, -8, -6, -2], [-2, 2, -1, -3, -5, 1, 0, -7], [4, 2, -4, 5, 2, 7, 1, -5], [3, -4, -1, -4, 7, -2, 3, 2], [0, 5, 0, -4, -6, 3, 2, 3], [-12, 4, -7, -3, 2, 0, 4, 6], [-10, 1, -6, 3, -2, -5, 5, 4], [3, -4, -2, 2, 5, -2, 0, -1], [-2, 1, 5, -3, 0, 2, -4, -1], [0, -2, 3, 1, -7, -9, -4, -3], [9, 4, 1, -4, 4, 11, -1, 3], [-5, -4, 2, -1, 9, -3, 1, 0], [1, -5, -1, -4, -3, -1, -1, -1], [-14, 0, 3, -1, 0, 2, 2, 0], [11, 6, 4, 0, -2, -3, -4, -2], [-9, -3, -5, -1, 1, 5, 4, 2], [0, -1, 5, -2, -10, 3, -3, -2], [0, -1, 2, -2, -7, 4, -6, -6], [7, 2, -4, -1, -1, 16, 5, -2], [-1, -2, -2, -1, 2, -1, 1, -1], [4, 5, 0, -1, -6, 1, -1, 0], [-8, 10, -4, -2, 7, 13, 8, -1], [-10, 3, -2, 2, 0, 7, 0, -1], [-3, -9, -1, -10, 5, -3, 5, 1], [2, 9, -2, 5, -3, -6, 1, -1], [9, -2, 4, -9, -2, 11, -5, -5], [2, 2, -8, -4, -2, 0, 0, -10], [10, -3, 3, -4, -10, -9, -1, -6], [-16, 3, -2, -3, -2, 0, 5, 2], [-10, -5, 3, 0, 3, -4, 3, 7], [8, 0, 6, 4, 1, -3, 0, -5], [-12, 2, -7, -4, 2, -8, 1, 1], [1, 5, -4, -4, 7, 12, 0, -4], [7, 8, -1, 2, 3, 10, -5, 2], [-13, 6, -2, -2, -3, -2, 0, -1], [-5, 3, 4, 2, 1, -4, -6, 6], [13, 0, 3, -1, 3, -2, -2, 1], [-5, -10, 2, 5, -3, 3, 3, 0], [0, 12, -2, 1, -2, 6, 5, -7], [-2, 3, -5, -3, 6, 7, 8, 3], [-6, -6, 3, 8, -4, -6, 3, 3], [4, 3, 1, -5, -6, -3, 0, 1], [3, 3, 2, 8, 8, -4, 2, -3], [-4, 6, -3, 5, 8, -4, 4, 0], [-6, 4, 4, 10, 10, -3, 6, 8], [-11, 0, 5, 4, 1, 6, 3, 6], [-1, 2, -3, -7, -1, 4, 2, 7], [-6, -1, 12, -4, -1, -4, -6, 0], [-4, -3, -12, -3, 2, -3, 5, -10], [-6, -3, -3, -7, 2, -3, 2, -5], [0, 3, 4, 6, 0, -11, -8, -1], [-2, -6, -2, -2, -3, -5, 2, 1], [-1, 0, -3, -1, -5, 0, 10, 1], [20, 3, -2, -1, -4, 17, -3, 0], [-8, 6, -6, -4, -4, -5, -6, -4], [-17, 1, -3, 2, 5, -10, 7, -5], [-1, 0, -7, -1, 7, 10, 2, -9], [-10, 3, 1, -7, 3, -2, 6, 2], [-4, 11, 2, 4, -2, -4, -6, 3], [-5, 8, 3, 5, -3, -12, 2, -1], [7, -10, 4, 1, 3, 2, 5, -4], [-3, 0, 6, 4, -2, -13, -6, 12], [17, 4, 4, 3, 5, 10, 5, -6], [10, 4, -1, 4, 4, 14, 3, -7], [-12, -8, -1, -10, 2, -11, -1, 5], [2, -5, -1, 5, 0, 9, 6, 3], [-3, 8, -9, -3, -2, -1, 2, 0], [0, 12, -5, -5, -2, 12, -2, -6], [-6, -2, -3, -2, -3, -5, -5, -4], [15, -7, 11, 5, -5, 2, -8, 4], [11, -7, 4, 2, -5, 1, -6, -8], [-14, 0, 5, -2, 6, -12, -1, 1], [-8, 12, -6, 5, -8, -16, 1, -3], [7, 12, -3, 1, -9, 1, -4, -11], [5, 3, 5, 14, -3, -12, 1, 5], [-25, -7, 4, -4, -4, -8, 0, 7], [-2, -5, 8, 3, -2, -4, -5, -2], [-14, 5, 1, 0, -4, -1, -1, 4], [9, 6, 2, 5, 1, -6, -3, -8], [2, -3, 3, 5, 14, 15, -2, 5], [-17, 13, -8, -11, -11, -1, -1, -9], [-13, -1, -4, -2, 1, -11, 8, 6], [2, 9, -4, 2, -5, 4, 8, -6], [5, 7, -1, 3, -3, 15, -2, -4], [17, -8, 3, 14, 3, 4, -7, -2], [-14, -1, 2, 0, -2, -10, -4, 3], [-9, -2, 2, -3, 3, -8, 1, 2], [-6, -6, 2, 12, -2, -10, -4, -8], [8, 9, -10, -7, -8, 1, -1, -8], [1, 11, 3, 1, -5, 15, 2, -4], [13, -11, 2, 7, 7, -9, 5, 13], [-14, -2, -2, 2, 4, 10, 6, 2], [7, -7, 4, 7, -3, -1, -7, -5], [-4, 15, -4, -1, 0, -2, 3, 4], [-13, 2, 0, -15, -1, -14, -5, 4], [-11, 2, -5, 3, 3, -8, 2, -5], [-22, 2, -12, -5, 4, 2, -1, 7], [-4, 9, -11, 2, 2, 16, -1, 8], [19, 6, -7, -2, -13, 6, -1, -10], [-18, 9, -4, -1, 12, -4, 1, 18], [-5, 3, -7, -5, 1, 19, 2, 2], [-7, 10, -4, 3, 13, 12, 15, 0], [6, -7, 1, 6, -2, -13, -5, 0], [-19, -4, -2, -8, -1, -12, 0, 1], [0, 4, 3, -6, -7, 19, -3, -4], [1, 19, -6, -6, -6, -2, -4, -1], [-3, -9, 13, 0, -9, 3, -9, -1], [-28, -2, 7, 0, -3, -3, -5, 0], [-14, 0, -12, -13, -8, -6, 3, -3], [5, 14, -2, 2, 11, 2, -10, 13], [-16, 2, 12, 7, 0, -8, -13, 3], [10, 3, 1, 8, -9, 0, -7, 5], [-13, 8, 3, 10, 1, -12, -7, 10], [4, -9, -2, -3, -3, 8, -3, 3], [3, -2, -10, 8, 9, 14, 10, -7], [46, 1, -4, 6, 3, 10, 4, -4], [-14, 1, -3, -7, 4, -20, -4, 3], [-9, -2, -7, -13, 3, 4, 10, 13], [7, 1, -2, 5, -1, 19, -1, -5], [9, -4, -6, 3, 5, 6, 7, 4], [-8, 9, 1, 8, 0, -22, -1, 4], [11, -2, 2, 11, 5, -4, 1, 2], [5, 1, 9, 6, 9, 9, -1, 7], [-10, -12, -6, -5, 9, -11, -1, 14], [-14, 2, -2, 7, 6, 20, 5, -1], [15, -5, 4, -11, -7, 12, -5, -5], [-7, 5, 0, 1, 1, -1, 5, -3], [17, -8, 9, 3, -1, 2, -12, -5], [-27, -2, -10, 6, -3, -24, 8, 3], [-13, -5, 9, -6, -9, 3, -5, -5], [-20, -13, -3, -2, -6, -13, -5, 2], [24, -2, 2, 5, -8, 32, -1, -5], [-13, 12, -5, -2, 13, -4, 3, 12], [11, 5, 6, 17, -3, -17, -5, 9], [17, -2, 8, 3, 2, -1, 3, -5], [6, -18, 7, -2, 8, -2, 3, 1], [0, -1, 8, 8, 4, -21, -4, 5], [-11, 0, 6, -2, -6, -19, -4, 0], [-20, -7, 11, -9, 2, -3, -6, 3], [-37, 0, -2, -5, 13, -6, -1, 12], [1, -1, -9, 12, 4, -12, -3, 0], [6, -2, -11, -1, 10, 8, 12, 2], [-18, 9, -2, -9, 4, -11, -1, 6], [-11, 3, 2, 9, -2, 16, 11, -2], [5, -7, 1, -6, -5, -21, -7, 3], [-25, -1, -1, -11, -7, -11, 2, 0], [1, 12, -5, 6, 3, -12, 7, -4], [-12, 3, 3, 9, 4, -21, 4, 11], [4, 7, -9, -4, -6, 9, -1, -2], [9, -4, -8, 1, 3, 0, -5, -2], [8, -2, 1, -12, -20, 1, -7, -7], [4, -16, 19, 9, -8, -21, -10, -4], [7, -5, -5, 8, 7, 13, -5, -5], [28, -8, 6, 4, 0, 9, 2, -10], [27, 2, -7, 14, 7, 30, 3, -14], [-2, 6, -3, 2, -6, 9, -5, -15], [-17, 8, -1, -5, 1, -22, 4, 7], [-22, -8, -6, -1, 10, -20, 5, -5], [-9, 7, -4, -2, 11, 7, 0, 8], [4, -17, -4, 8, 0, 0, -2, -1], [-13, -1, -23, -12, 7, 3, 15, -7], [-14, -7, -9, 7, 10, 6, 2, -7], [4, -10, -2, 0, 7, -7, 0, -1], [-5, -10, 8, 2, -8, -19, 6, 0], [13, 0, 2, 13, 13, 6, -9, 2], [3, -6, 8, -3, 3, 19, -5, 8], [-12, 12, -11, -5, -8, -6, -10, -4], [-3, 9, -2, 15, 1, 15, -5, -7], [13, -2, -2, -7, -3, -11, 1, -8], [-13, 7, -10, 1, 11, 11, -5, 3], [6, -5, -4, -11, 8, 20, 19, -2], [5, -8, -8, -4, -9, -4, -8, -11], [-1, -9, -8, -17, -9, 4, -9, -1], [-26, -17, -3, -4, -10, -22, 11, -4], [-11, -4, 10, 3, -5, 2, 5, -6], [-8, -5, 8, 5, 2, -26, -1, 8], [21, -7, -3, -7, -12, 18, 4, -17], [-17, 6, -10, -6, 9, -2, 0, -1], [4, -4, 6, 10, -4, 6, -12, -6], [12, -3, -5, -11, -6, 9, 12, -5], [-19, 13, -1, 5, 3, -9, -8, 12], [-1, 15, 1, 11, 5, 2, -6, 8], [13, 11, 7, 9, -1, 1, -12, 4], [-30, 10, -9, -4, -8, -18, 1, -13], [-3, -3, -1, 19, -5, -11, 0, 4], [-12, -2, -7, 1, 16, 0, 10, 6], [16, -12, -1, -12, 6, 9, 1, -5], [4, -1, -14, 6, 10, 9, 8, 1], [-19, -10, -1, -10, 9, -4, -1, 18], [4, -1, -21, -3, 2, -5, 20, -7], [21, -7, 6, -2, 7, 3, -18, 8], [2, 15, -7, -5, -6, 0, -8, 1], [19, 16, -7, 6, -8, -1, -7, -1], [8, -6, -14, 4, -4, 5, 18, 0], [16, 4, 12, -4, -8, -10, -12, -4], [-7, 5, -4, 0, 16, -2, 4, -1], [15, 5, -15, 3, 1, 23, 2, 0], [11, -10, 6, 4, 3, -6, 0, 1], [0, -11, -7, -11, -5, -14, 6, -4], [-22, 12, 3, -4, -4, -29, -7, 9], [-25, -8, -3, 2, 7, -12, -9, 8], [-30, 4, -2, -11, 0, -6, -3, 9], [13, 9, -10, -16, 1, 25, 2, -12], [6, 3, 0, -16, 10, 6, 2, 13], [9, -11, 12, 5, -17, -33, -13, -5], [-12, -8, 4, -5, -2, 0, 13, -3], [8, -6, 19, -6, -16, -12, -9, 7], [-34, -14, -2, -2, -14, -8, 0, -4], [9, 1, -3, 5, 11, 17, 10, 0], [-5, 1, 1, -10, -18, -22, 1, -8], [-11, 8, -8, 4, 5, -30, 10, 5], [21, 1, 8, -1, 5, 4, -17, 11], [-11, -5, -7, -15, -14, -14, 8, -9], [-14, -5, 5, 4, 2, -11, -3, 10], [21, -12, -3, 6, -7, 4, 7, -6], [14, -15, 13, 3, 4, -11, 0, 7], [-21, -9, -15, -18, 6, 14, 1, 4], [5, -23, 8, -3, 11, 1, 9, 1], [24, 20, 5, -2, -6, 14, 3, -5], [28, -3, 22, 6, -3, 18, -2, 8], [-43, 2, -9, -14, -1, -10, 9, 0], [19, -2, -3, 1, -3, 31, 6, -11], [0, -6, 4, 12, 14, 8, 14, 8], [23, 7, 10, 3, 9, 27, -11, -5], [-6, -1, -9, 7, -5, -26, 2, -10], [-37, -11, -1, 5, -7, -41, 0, 6], [7, -9, 14, 12, -7, -3, 0, 4], [-2, -15, -11, 4, -12, -13, 7, -2], [-7, -4, -6, -1, 13, -14, 7, 8], [19, 0, -9, -7, -16, 13, 6, -12], [-10, 4, -14, -16, 14, 29, 14, -8], [18, 0, 2, -5, -2, 8, 11, 3], [11, -12, 0, 14, 9, 20, 6, 5], [9, -13, -9, 6, 4, 4, 11, -10], [6, -9, -5, 13, 16, 5, 2, -9], [-12, 9, -11, -3, 20, 31, 4, 3], [-27, -5, -4, 3, 17, 3, 7, 17], [21, -13, 0, -15, -1, 28, -9, -7], [-47, 1, 0, 7, 9, -13, 1, 13], [1, 5, -9, -4, -7, -15, 7, -5], [27, 6, 8, -1, 21, 28, 3, 8], [-6, 0, 17, 25, 16, -10, 12, 16], [23, -11, 10, 13, 11, 13, 7, 13], [-8, -5, 6, -13, -10, -3, -3, -12], [12, -1, -5, -10, 9, 0, 9, -1], [-19, 13, -5, -5, 1, 11, 10, 0], [-2, 6, 7, -3, -10, -16, -18, -16], [1, -16, 2, -5, 7, -7, -7, 6], [6, 8, 2, 4, -10, -1, 6, 4], [9, 1, -10, 0, 2, 0, -2, 3], [-23, -8, 24, -10, -5, -12, -8, 9], [15, 10, 6, 5, -7, 15, -9, -4], [15, 0, -2, 9, 3, -7, -3, 0], [-27, 2, -18, -26, 7, -6, 8, 1], [-7, -6, 11, 1, 2, -1, 2, 16], [-33, -12, 4, -10, 10, 13, 6, 16], [-3, 3, -9, 8, 6, 32, 9, -4], [7, 22, -4, -10, -25, 1, -6, -9], [6, -5, 5, -8, -4, -9, -1, -6], [-19, -10, 2, -20, -1, 10, 8, 1], [-8, 4, -22, -5, -6, -18, 13, -11], [6, -4, -9, -14, -2, 8, -3, -3], [24, -12, 0, -15, -9, 27, 3, -18], [13, -7, 2, 19, 5, 9, -15, -13], [-9, -9, -10, -22, -13, 18, 6, -10], [22, 3, 7, -15, -16, 3, -18, 11], [21, 21, -2, 0, -3, 5, 4, -4], [16, -7, 0, -16, -2, 18, -8, -17], [14, 13, -4, -20, -12, 3, -18, -3], [37, -2, -7, -9, 1, 17, 14, -1], [-13, 7, 2, 1, 7, 5, 13, -1], [12, -11, 12, 6, 8, 8, 4, -1], [-25, 6, -20, -10, 17, -7, 8, 13], [26, -8, 2, -3, -12, 13, -5, -3], [9, 16, 4, 1, -20, 1, -3, -17], [-26, -17, -1, -23, -2, -7, 6, 11], [13, 6, 10, 9, -5, 14, -9, -12], [-14, 3, 1, -12, -1, -2, -7, 23], [6, -3, 19, 16, 8, -11, -7, 12], [-16, 6, -5, -1, -4, -17, 12, -12], [51, 15, 0, 0, 15, 25, 2, 0], [-15, 2, 3, 12, 11, -10, -13, 6], [-24, -2, -1, 22, 14, -12, 5, 5], [8, 4, 8, 26, -9, 7, -12, -15], [-17, 2, 0, -6, -20, 5, -14, 0], [37, -10, 1, -17, 7, 2, 4, -1], [-13, 14, 6, -6, 0, -5, 4, 0], [21, -7, -3, -1, 1, 32, -4, -10], [25, 6, -9, 2, 11, 11, 11, -8], [-36, -7, -5, 3, -2, -7, 6, 13], [23, 21, 4, 24, -7, -14, -12, 2], [5, -19, -1, -8, 9, 9, 1, -1], [17, -25, 11, 9, -13, 11, 2, -12], [-12, -1, -13, -19, 6, -9, 6, 3], [19, -15, 16, 7, 4, -2, -9, -2], [30, 10, 11, 12, -4, 1, 1, -9], [-11, -6, 7, 7, 1, -14, 4, 29], [12, 10, 3, 0, 12, -10, -7, -1], [6, 11, -4, 9, 28, 15, 13, -2], [-13, 6, 2, 3, 19, 0, 5, -4], [-13, -15, -10, -19, -17, -5, -1, -9], [-11, -2, 13, 17, 19, -3, -10, 7], [-9, -14, 14, 2, 4, -21, -12, 4], [11, -10, -10, -16, -7, 5, 8, -11], [14, -10, 1, -1, -6, 44, 2, -8], [-18, 22, -7, 1, -6, 20, -4, 2], [14, -1, -14, -9, 0, 8, -1, 2], [-6, -10, -13, -4, 8, 20, 15, 17], [-16, 1, 3, -5, -5, -24, -18, 2], [-10, 3, 10, 3, -8, -33, -1, 0], [-39, 2, 2, 0, -5, -42, -4, 7], [-24, 19, -15, -2, -3, -16, -5, -3], [6, 9, -10, 0, -2, -14, 0, 1], [50, 7, -1, 8, 12, 15, 1, -4], [2, 9, 1, -7, 14, 2, 6, 21], [14, -11, 8, 2, 7, 8, -12, 6], [23, 8, 21, 14, -1, 34, -7, 8], [20, -3, 14, -4, 5, 16, -12, 0], [-32, -12, 2, 2, -4, -2, 10, 4], [0, -31, 12, -1, 7, -20, 5, 11], [2, 16, -5, -4, 6, 10, 15, 3], [-1, 7, -7, 4, 13, 13, 15, -7], [-16, 15, -3, 2, -12, -23, 5, -2], [26, -1, 24, 8, -10, -9, -16, 3], [-48, 8, -3, 16, 6, -21, 7, 21], [5, 1, 17, 5, -10, 18, -4, 7], [-21, 1, -8, 3, 0, 6, -13, -2], [-9, 10, 1, 2, 11, -4, 3, 28], [-8, -8, 2, -2, -2, -10, 10, 0], [-4, -17, 8, 11, -9, -14, -3, 1], [-12, 18, 6, 5, -14, 26, -1, -7], [16, 8, -12, 20, -5, -19, -10, -15], [8, -9, 16, 30, 12, -7, -2, 11], [16, -7, 2, 4, 10, -12, -10, 1], [18, 3, 19, 21, 10, 15, -12, 1], [12, 2, 4, 15, 20, 11, 1, -9], [5, 8, -8, 12, -5, -16, 8, -11], [16, -11, -4, 1, 6, 4, -13, 18], [10, -11, -6, 1, 1, 34, -3, -17], [12, 3, -4, 2, -16, 7, 12, -5], [0, 8, -9, 22, -18, -26, -21, -11], [-32, -2, 8, 2, -6, -33, 0, 2], [11, 7, -9, 14, 5, -14, -11, -5], [0, 11, 9, -6, 0, 15, -5, 14], [-31, -7, -7, -10, 16, -10, -3, 22], [25, -5, -14, 1, -4, 0, -5, -10], [-17, -22, 6, -6, -5, 6, 6, -7], [-40, 24, 5, 5, 2, -18, -2, 24], [-7, -20, -21, -9, -1, -26, 14, -11], [-11, -16, 11, -17, -13, -33, -18, 17], [22, -8, 14, -11, -28, -33, -17, -11], [12, 9, -1, -2, 14, 15, 3, -6], [-47, 15, -28, -23, 3, 1, 9, -7], [-6, -4, 12, -22, -14, 0, -8, 6], [35, 4, 3, -6, -12, 21, -5, 3], [23, 0, -2, 10, -16, 5, -4, 2], [-1, 7, 8, -12, -7, 4, 4, 12], [-10, -5, 7, -4, 5, -6, -9, 17], [48, -22, 9, -2, 4, 20, -9, -5], [34, 9, 12, -10, -2, 16, 4, 9], [10, -8, 8, -7, -10, 22, 3, -19], [-66, 3, -1, -5, -6, -13, 2, 17], [-27, 4, -11, -12, -5, -18, 3, 2], [2, 4, 3, -7, 2, 11, 18, 6], [-2, -25, 0, -1, 14, -29, 9, 20], [-45, 3, -28, -13, 11, 0, 11, 7], [27, -9, 4, -11, -7, 41, -3, -15], [1, -7, -4, 18, 7, 7, 8, -16], [-11, -10, -6, 6, -1, -12, 22, -9], [-8, 6, -13, -14, 2, -10, -21, 3], [1, 17, -31, -3, 6, 16, 12, -11], [48, -21, -5, -8, -14, 1, -3, -22], [-28, -1, 18, 6, -12, -3, -12, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7623_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7623_2_a_cw();".
// 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_7623_2_a_cw(:prec:=8)
    chi := MakeCharacter_7623_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_7623_2_a_cw();".
// 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_7623_2_a_cw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, 30, -30, -30, 35, 10, -11, -1, 1]>,<5,R![-16, 576, 680, -62, -285, -67, 22, 10, 1]>,<13,R![-13744, 10784, 2104, -3114, 195, 261, -36, -6, 1]>],Snew);
    return Vf;
end function;