
// Make newform 7623.2.a.cs in Magma, downloaded from the LMFDB on 18 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_cs();".
// 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_cs();".
// 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 := [-2, -6, 7, 8, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-2, -1, 1, 0, 0, 0], [1, -3, -1, 1, 0, 0], [2, 2, -4, -1, 1, 0], [-2, 4, 6, -4, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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 := [[1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [1, -1, -1, 0, 1, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 1, -1], [4, -1, -1, -1, 0, 0], [-1, -1, 0, 1, -2, -1], [-1, 2, 1, 1, -3, 0], [2, 1, 1, 1, -1, 1], [-1, 0, 1, -1, -2, -2], [4, -3, 2, 1, 1, 2], [4, 1, -1, -1, 1, 0], [2, -3, 1, -2, 3, 1], [3, -2, -3, 1, -2, -1], [0, -2, -1, -1, 3, 0], [3, -4, 3, -3, 0, 3], [4, -6, -4, 2, 1, 2], [-1, 4, -1, 1, 1, -2], [-4, -1, 1, 0, 0, -2], [-1, -1, 4, 1, 2, 3], [6, -3, -3, -2, -1, 1], [3, 3, 0, 3, -2, 1], [0, 1, 4, 1, 3, 1], [-1, 3, 1, -2, 1, 2], [7, -3, 4, 3, 3, 1], [-2, 0, -2, 2, -2, 1], [1, -2, -3, 3, -1, 1], [-5, 4, -3, 0, 3, -1], [3, -1, 1, 0, -4, 0], [-2, 2, -4, 2, -6, -4], [6, 4, 2, 0, 0, 4], [-4, -3, -3, -2, 0, 4], [7, 3, 4, -3, 0, 1], [7, -3, -4, 0, -2, 0], [3, 6, 3, 5, -5, -3], [4, -2, -4, -2, -2, -2], [-1, -1, -2, -1, -3, 2], [4, -4, -4, -2, -2, -6], [-3, 3, 0, 1, 3, -7], [-5, 3, -2, -5, 3, 4], [4, 1, -2, 7, 1, 2], [-5, -3, -4, 3, 5, -4], [-1, 7, -2, -4, -2, 2], [3, 4, -3, 0, 3, -3], [-13, -2, -5, -1, 2, -2], [2, -1, -1, 4, 3, -3], [-1, -4, 3, 5, -4, 3], [-1, -3, 4, -9, 3, 3], [-8, -3, -6, 3, 3, 0], [11, 0, -3, 4, 1, -1], [-1, 1, 0, 1, 2, 4], [2, -6, 6, -4, 1, 6], [5, 0, -1, 1, 0, 4], [10, -3, 6, -1, 3, 5], [8, 0, 4, -2, -2, 0], [0, 13, -2, -1, -1, -1], [-1, 5, -4, 5, -3, -1], [-3, -3, 6, -6, 8, 6], [7, 2, -1, 3, -1, -1], [7, -1, 4, -3, 0, 7], [14, -1, 3, 5, -1, 0], [-3, 7, -2, -5, 2, -3], [1, 0, -3, -9, 0, -4], [-1, 1, 7, 2, -5, -3], [1, -8, -1, -3, 3, 8], [-4, -2, 8, 2, 0, 6], [-8, -4, -8, -3, 2, 2], [7, 2, 7, 1, 3, -1], [4, -3, 5, -11, 6, 6], [4, 7, -2, 1, -1, -7], [-8, 5, -5, -4, -1, -3], [2, 4, 4, 4, 2, -2], [8, -7, -5, 4, -1, -5], [8, 0, -2, 2, -6, 2], [9, 0, 5, -1, 2, 8], [-2, -3, -12, -1, 1, -4], [-9, 3, 7, 0, 5, -2], [2, 7, 0, 1, 5, -2], [5, 2, -9, -4, -3, -9], [8, 4, 0, -4, 4, -6], [-2, -1, 0, -11, 3, 2], [7, -9, -6, -5, 10, 4], [-4, 5, 4, -3, -1, -7], [7, 5, 0, 5, -7, 1], [-1, 0, -15, -5, -3, -4], [-9, 5, 1, -6, -2, -6], [-10, -1, -5, 9, -9, -3], [6, -1, -7, -5, -4, 2], [-7, 7, 6, 7, 3, 4], [-5, 2, 11, 11, -6, 7], [-5, 7, 0, 9, -10, -3], [-3, 3, -8, 1, 5, -6], [-5, 3, -6, 1, -2, 4], [8, -1, -5, 6, 4, 8], [-10, 8, 6, -2, -10, -4], [6, 4, 14, 2, 2, 7], [-7, 4, 5, 7, 0, -2], [4, 2, 8, 2, 5, 10], [-9, -4, 5, -1, -1, 3], [-8, 9, 7, -6, -1, 5], [19, -6, -5, -1, 9, 1], [0, 2, 4, 8, -9, 2], [12, 6, -2, -8, 8, -4], [9, 8, 7, -3, -3, 1], [-5, 9, 3, -6, 9, 8], [7, 8, 17, 7, -14, 0], [21, -8, 1, 6, 5, 9], [-2, 12, 2, 2, -4, 6], [-7, 6, -1, 14, -9, -1], [12, -6, 4, 0, 9, 6], [18, -4, -6, 5, -6, -4], [6, -9, 4, 5, -7, -2], [-10, 12, 4, 4, -10, -2], [-9, -1, -2, -1, -3, 0], [-1, -12, -6, -10, 4, 0], [-7, 0, 1, -13, 4, 2], [-6, 6, -16, 2, -2, -4], [-11, 7, 16, 5, 1, 2], [1, 4, 3, 5, -5, 7], [2, 5, -12, 9, -7, -5], [-8, 6, 14, 0, -2, -4], [-9, -14, -5, -2, 6, 7], [-2, 13, -7, -6, 4, 0], [-10, -4, -14, 2, -6, -10], [-6, -9, 7, 3, -1, 3], [-2, -3, 11, 0, 10, 2], [3, 8, -5, 9, -2, 2], [-4, 6, -6, 0, 0, 5], [-15, 10, -1, -8, -5, -3], [19, 5, -4, -9, 4, -8], [1, -3, -6, -13, 0, -2], [16, 4, -8, 4, 8, 0], [-16, 5, 8, 3, -1, -2], [-1, 8, 3, -12, 1, -9], [15, 10, 7, -2, -2, 3], [-3, 6, 7, 3, -6, 14], [16, -12, -18, -2, 3, 2], [-15, 0, -11, 11, -6, -8], [1, 3, 0, -7, 6, 10], [-4, 4, 0, 2, 0, 6], [5, 2, 13, 1, -13, -7], [10, 0, -4, -4, 18, -6], [14, -4, -10, -8, 16, -4], [-8, 9, 6, 3, 1, -7], [9, -6, 5, 9, -12, -4], [8, 1, 5, 3, 6, -8], [29, -11, -6, -7, -5, 3], [11, -4, 9, -13, 4, 3], [-23, 11, 4, -3, 5, 0], [-2, 8, 18, 15, -6, 12], [17, 5, 5, -4, 1, 11], [5, 12, 19, -3, 1, -2], [2, 4, 4, -4, -5, 2], [-23, 3, -14, -1, 5, -8], [5, -7, 6, -3, 9, 8], [-11, 4, 3, 1, -9, 1], [10, -15, 2, 3, 1, 8], [2, 3, 7, -9, 1, 8], [-8, -5, -3, 13, 5, 2], [-16, -12, -10, -10, 8, -6], [8, 2, 14, -1, -10, -2], [15, 11, 2, 1, 0, 12], [22, -2, -4, -10, 2, -8], [15, -6, 6, -4, 4, 4], [12, 2, -6, 4, 6, 7], [13, -20, 3, -11, 15, 14], [11, 2, 7, -6, -10, 5], [-10, -12, 2, -3, 0, -6], [-13, 4, -1, -11, -6, -10], [11, -16, -3, 7, -1, -1], [-11, 6, 1, 5, 4, -9], [0, -12, 4, -2, 4, -8], [-12, 2, 2, 3, 2, 0], [-2, 20, 6, -18, 2, -3], [5, 5, -11, 0, -7, 5], [-21, 8, 13, 3, -5, 3], [-2, 0, -5, 5, 1, -16], [-3, -15, 6, 5, 3, 9], [2, 1, -5, 9, 8, -2], [-3, 7, 8, -3, -17, 4], [-14, 20, 14, 2, -8, -10], [-11, 8, -9, 5, -11, -18], [14, 7, 11, 7, -7, -6], [3, 11, 14, -13, 3, 2], [-3, -4, 17, 3, -4, 0], [-20, 17, 11, 9, -7, -5], [15, 14, 11, 7, -10, 0], [-4, -6, -14, -5, -2, 4], [-9, -1, 10, 13, 1, 3], [1, -6, -7, 1, 0, 13], [-10, 7, 23, 2, 5, 5], [13, 8, 3, -7, 9, 10], [17, 11, 7, -4, -5, -10], [5, -1, -4, -5, 10, -8], [-15, -1, 5, 6, 3, -3], [-15, 4, 11, -6, -11, 3], [3, -17, -6, -5, 7, 14], [9, 18, -3, 24, -18, -11], [-3, 5, -8, 5, -20, -15], [3, -12, 13, -1, -5, -3], [-32, 10, -8, -10, -5, -6], [-9, 1, -17, 6, -3, -8], [-9, -18, -5, -12, 8, -3], [17, 16, 5, -9, 4, 2], [-20, 6, 2, -20, -12, -7], [3, 0, -11, -13, 16, 4], [-5, 4, -11, -13, 3, 1], [-6, 1, -7, -9, 7, -1], [9, 4, 1, -1, 4, 1], [0, -3, 15, -7, -7, -1], [-37, 13, -1, 2, 5, -3], [6, 8, -12, 4, -14, -2], [1, 2, -7, 13, -10, -2], [-2, -8, -12, -10, 5, -6], [3, -1, -11, -6, -9, -3], [-14, -6, 12, 6, 2, 2], [-16, 8, 8, 0, -8, 6], [-9, -8, 3, 9, 7, 6], [-23, 18, 8, 18, -6, 0], [25, -3, 2, -4, -6, -2], [-3, 25, 8, 7, -6, 3], [-11, -10, -4, 6, -6, 4], [-12, -3, 15, -4, -2, -6], [-8, 8, -16, -18, 2, -4], [1, 3, 12, -4, -2, -10], [3, -1, 21, -2, -15, 5], [-12, 4, 12, 8, 8, 16], [-30, -6, 8, 12, -20, 4], [15, -17, -2, 3, -6, -14], [-10, 17, 18, 5, -5, 10], [-28, 6, -2, 0, 4, 4], [3, 5, 16, -23, 2, 4], [-16, -2, 2, 15, -4, -8], [6, 10, 8, -18, 10, 10], [17, 7, 0, -9, 5, -5], [-3, -14, 11, -5, -2, 0], [-4, -10, -10, 7, -16, 8], [-29, 5, 0, 15, 3, -10], [15, -8, -9, 5, 12, 2], [28, -11, -9, -4, -4, -4], [-21, 25, -10, 9, -14, -15], [25, -24, -11, -1, -2, -10], [-3, 4, -6, -14, 8, -10], [21, 3, 2, 3, 8, 17], [-32, 16, 20, 10, -6, 6], [-10, -16, -14, 6, 2, -13], [12, -24, -24, 0, 10, -3], [-7, 13, -6, 7, -12, -2], [14, 9, 5, 18, -1, -13], [12, 18, -6, 0, -8, 4], [20, 13, 19, 5, -7, -10], [7, 7, 14, -1, -3, 18], [5, -3, 3, 2, 23, -12], [5, 6, 7, -12, 7, 7], [31, 4, 5, 3, -9, -1], [9, 6, 8, 6, -16, -2], [-16, -14, 4, -16, 5, 4], [36, 3, -2, 9, -1, 13], [-7, 11, 18, -1, -4, 3], [7, -8, -11, -3, -5, -11], [-27, 17, 2, 3, -24, -14], [-9, 22, 3, 12, -14, -11], [19, -9, -8, 1, 4, 5], [8, -13, 4, -5, 7, 0], [36, -20, 6, 10, 8, 10], [-24, 10, 8, 14, -20, 8], [-21, -2, -3, -3, -2, 16], [-22, 34, 4, -4, -8, -4], [-16, -7, 3, 4, 1, -11], [-34, 3, 20, -9, 9, 2], [-22, 19, 19, 2, -16, -8], [-8, 3, 4, -1, 1, 20], [5, 4, -9, 19, 3, 7], [-23, -1, -14, 3, 4, -15], [-25, 22, 15, -5, -18, -4], [19, -16, -1, -19, 7, 2], [22, 5, 21, 3, -5, 21], [30, -19, 17, -2, 11, 19], [-37, 11, 4, -9, -20, -7], [-24, -12, -14, -14, 11, -4], [13, -33, -2, -1, 7, 16], [-21, 29, 10, 7, -10, -5], [10, -23, 5, -13, 5, 6], [-11, 1, -16, -5, -6, -5], [-27, 15, -2, -1, -5, -22], [-13, -16, -11, 23, -7, 8], [9, -4, -17, 5, -5, 1], [20, -22, -18, -16, 8, 0], [11, 2, -11, 16, -13, 7], [18, 6, 7, 7, -5, -14], [1, 4, -3, 3, 4, -7], [42, 2, 8, -3, 8, 8], [10, -12, -12, 8, 7, 2], [12, 17, 3, 24, -21, -7], [19, -27, -22, 9, 14, 3], [-35, 0, -1, 1, 0, -7], [-19, 23, 5, -6, 9, 0], [-3, -9, 20, -5, 1, 7], [-11, -5, -12, 3, 1, -15], [-13, 29, -6, 17, 0, -19], [10, -5, -13, -5, 3, -1], [-33, 12, -7, -9, 5, 3], [9, -8, -11, -9, 1, -14], [-3, -1, -2, 15, -1, -8], [24, -28, 6, -2, 6, 16], [17, -15, -12, 15, -1, -8], [24, 3, -10, 1, -17, -2], [5, -12, -21, -3, 13, -8], [12, 17, -1, -15, 11, -17], [13, -7, 0, -21, -4, 5], [13, 5, -25, -24, 2, -12], [1, -25, -8, -8, 2, -6], [-13, -1, -6, 3, -23, -4], [-5, 12, 9, 21, 1, 17], [4, -21, -21, -5, 10, 2], [1, 24, 17, -5, 4, 2], [39, 0, 5, 3, 7, 17], [11, 18, -3, -7, -2, -18], [6, -10, 8, -5, 8, 24], [3, -9, 12, 1, -12, -3], [4, 12, -5, 25, -9, -2], [-13, 4, 19, -4, 15, -9], [26, -16, -8, 16, -2, 8], [19, 3, 10, -12, 22, -4], [-17, -4, 3, -7, 4, 8], [-13, 12, -9, 9, -19, -2], [42, 4, 16, -24, -2, 10], [10, 11, -33, 8, -17, -9], [10, -14, -16, -9, -2, 10], [-17, 7, -6, 10, -8, -14], [-11, -1, -4, 1, -17, -4], [4, -26, 8, -18, 17, 8], [0, 0, 4, 8, -12, 2], [23, -33, -30, -9, 10, -4], [0, 28, -6, 22, -20, -11], [0, -25, -10, -13, 9, -12], [18, -8, -10, 2, 24, 14], [8, -21, -18, -15, 7, -6], [-1, -11, 2, 1, 22, -17], [10, 12, 11, -3, 11, 4], [-13, -13, 14, 11, 1, -2], [4, -9, -27, -29, 7, -6], [21, 10, -7, -7, -1, 0], [-2, -16, 0, -16, 6, 18], [-19, 40, 3, 15, -15, -23], [-2, -15, -20, -15, 25, 6], [25, -14, -7, 25, 1, 0], [-20, 12, -2, -4, -27, 4], [19, 9, 1, -12, 17, -18], [27, 13, 2, -9, -6, -9], [14, -5, -11, 24, -22, -14], [-15, -2, 8, 10, 2, 0], [-22, -4, 14, -8, 10, -10], [2, -5, -25, 13, 5, -3], [32, 5, 15, 7, -21, -6], [38, -36, -16, -16, 4, 6], [-8, 4, 22, 0, 7, 8], [44, -12, 28, -5, 10, 24], [-16, -16, -28, -4, 22, -4], [-18, 6, -32, -6, 4, -6], [-2, -20, -10, -2, 6, 0], [-17, -9, 28, -15, 21, 8], [-40, 33, 7, 0, -5, -7], [-13, 4, 3, 5, -10, -3], [-23, 21, 11, -12, -6, -12], [2, 19, 11, -17, 21, 12], [-16, 1, 7, -10, -1, -7], [-2, 11, 19, 9, -15, 9], [24, -13, -15, 12, -2, -2], [16, 12, 12, 0, -18, 3], [27, -25, 2, 7, 14, 14], [-22, 6, -4, 5, -10, -8], [34, -4, 20, -14, 7, 24], [-25, -21, -7, 4, 0, -10], [29, 12, 5, 3, -6, -6], [-4, -17, 9, -14, 27, 7], [6, -15, -3, -13, -3, 2], [-25, 18, 11, 2, -14, 1], [1, -7, 28, 11, -10, -3], [39, -44, -27, 3, 6, 11], [-9, -2, 12, -26, -4, 0], [-41, 10, -13, 9, 9, -9], [53, 6, -3, -15, 17, -3], [5, 19, 8, 3, -3, -13], [8, 32, -4, 20, -12, -22], [24, 1, 9, -10, 8, 2], [34, -1, 8, -1, -1, -10], [29, 20, -9, 13, -13, -13], [16, 2, -2, -8, 16, 8], [-22, -1, -23, -16, 12, 8], [-10, -6, 14, 0, 8, 4], [-13, 27, 4, -11, 1, 10], [11, -14, 7, 7, 14, -12], [-33, -11, -4, 3, -7, -8], [22, -28, 10, 6, 10, 16], [-54, 6, -6, -4, -3, -20], [21, -9, -21, 8, -23, -8], [-12, 2, -6, 4, -6, -8], [-8, -30, -10, -11, 20, -2], [-5, 24, -9, -4, 1, -9], [60, 14, 3, 3, -13, 14], [38, 14, -4, 8, 6, -2], [4, 18, -12, 18, -14, -4], [-26, -6, -10, -8, 7, 14], [-28, 34, 12, 14, -9, -6], [-15, 14, 0, 18, -6, 2], [-2, -38, 12, -1, -2, 20], [34, -6, 12, 2, 18, 26], [-19, 14, 9, -15, 20, 8], [-19, 0, 3, 8, 24, -15], [-12, -8, -12, 23, 4, -2], [-41, -11, 6, -23, 1, 0], [7, -14, 11, -7, 1, 27], [-11, 6, -3, 16, 3, -11], [-13, -11, -13, 2, 3, 0], [-12, 10, -14, 4, 14, -22], [4, -30, 22, -8, 2, 22], [26, -20, -4, -16, 24, 10], [-27, -25, 14, -7, 16, -3], [-17, 2, 18, 24, 8, 12], [2, 18, -28, -14, -4, -6], [-14, -8, 18, -2, 2, -8], [19, -29, -12, 13, -16, -13], [5, 9, 39, 18, -15, 8], [20, 5, 4, -3, -5, -8], [-1, 17, 10, 41, -23, -2], [-7, 25, -6, 25, -32, -16], [-3, -6, -13, -9, -8, 1], [-3, 5, -4, 15, -12, 13]];
    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_cs();".
// 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_cs(:prec:=6)
    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_cs();".
// 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_cs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -8, -4, 12, 0, -4, 1]>,<5,R![1, -8, -1, 36, -9, -4, 1]>,<13,R![16, 32, -36, -72, -19, 4, 1]>],Snew);
    return Vf;
end function;