
// Make newform 5625.2.a.bc in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5625_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_5625_2_a_bc();".
// 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_5625_2_a_bc();".
// 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, 8, -48, -20, 41, 10, -12, -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], [8, -40, -20, 41, 10, -12, -1, 1], [-56, 8, 94, -21, -34, 10, 3, -1], [20, -12, -62, 21, 30, -10, -3, 1], [-32, 12, 66, -21, -30, 10, 3, -1], [-24, 88, 60, -115, -30, 36, 3, -3], [-28, 30, 61, -52, -30, 21, 3, -2]];
    Rf_basisdens := [1, 1, 8, 8, 4, 4, 8, 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_5625_a();"
function MakeCharacter_5625_a()
    N := 5625;
    order := 1;
    char_gens := [4376, 1252];
    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_5625_a_Hecke(Kf)
    return MakeCharacter_5625_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], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 1, 0, 0, 0], [-3, 0, -1, -2, -1, -1, -1, -2], [-2, 1, 0, 1, 0, 1, 0, 0], [1, -2, 0, 2, -1, 1, 0, 2], [3, 0, -1, 1, 0, -1, -1, 1], [-1, 3, -3, 0, -1, -2, -1, -2], [1, -1, 0, 0, -1, 1, 2, 2], [2, 1, 0, -2, 0, -3, 1, -1], [1, 0, 1, 0, 0, -1, 1, -1], [1, 0, -1, 0, 1, -1, -1, -2], [-2, -4, 4, 1, 1, 2, 2, 1], [-3, 0, 1, -4, -1, 0, 1, 0], [-2, -1, -1, 0, 2, 0, -1, -4], [-1, -1, 2, 2, -1, 1, -2, 4], [3, -3, -1, 2, -1, 1, 2, 1], [-2, -1, 3, -2, 0, -2, -2, 3], [-4, -3, 4, -2, 0, 0, 2, -2], [4, 0, 2, 4, 0, 1, 0, 3], [1, -5, 3, 2, 1, 4, 3, -1], [-4, -1, 4, -2, 0, -2, 0, 0], [3, -1, 4, 4, 1, 4, 0, 6], [-2, 2, -1, 0, -2, 0, 0, 3], [-5, 3, -3, 4, 1, -1, -1, -4], [-10, 2, -3, -2, -3, -3, -1, -5], [-4, 1, 4, -2, 0, 0, -2, -2], [4, -2, 2, -1, -1, 2, 0, -3], [-3, -2, 3, -6, 1, 1, -1, 2], [-2, 0, 4, 0, -2, -2, 2, 7], [4, -1, 6, 4, 2, 2, 4, 10], [-11, 1, -2, -4, -1, -3, 0, -4], [-1, 3, -2, -6, -1, 0, 0, -1], [-6, 0, 2, 6, 0, 2, 0, 2], [6, 2, 1, 4, 2, 1, 2, 4], [-2, 5, -1, -6, 1, -2, 1, -4], [1, 5, -10, -2, 2, -2, -4, -6], [-1, -3, -3, 2, 5, 2, -1, -6], [-4, -6, 2, 4, 0, 1, 4, 4], [1, 3, -3, -4, -1, -6, -1, -10], [7, 2, 0, 1, -3, -1, 0, 4], [4, 0, -3, -4, 0, -3, -1, 10], [2, 3, 5, 2, -3, 1, 1, 6], [-1, -3, -6, -2, 3, 2, 4, -6], [-2, 6, -5, 0, 2, 0, -6, -6], [-5, 3, 2, -7, 1, -1, 0, -8], [-5, 5, -7, 1, 1, 4, -1, -6], [9, -2, 4, 2, -1, -1, 0, 14], [8, 1, 3, 3, -2, 1, 5, 12], [1, 2, -4, 4, 5, 1, -4, 6], [-1, 0, 1, -2, -3, 2, 1, -8], [-2, -2, 1, 3, -3, -1, -1, 3], [9, 1, -8, 4, 1, 1, 0, 4], [-11, -1, -1, 2, 3, 0, -1, -2], [-11, -4, -4, -2, 1, 2, -2, -2], [-7, 4, 4, -2, -1, 2, 2, 2], [7, -2, 7, 10, 2, 7, 3, 11], [1, -2, 2, 4, 3, 5, 3, 0], [-2, -4, 0, 6, 0, 1, -2, 4], [1, -3, 4, -2, 2, -3, -4, 0], [-5, 3, 1, 6, 5, 2, -1, 0], [-11, 0, -8, -3, -3, -3, -4, -6], [1, 12, -5, 2, -1, -5, -1, 2], [-5, 3, 1, 6, 3, 0, -4, 9], [-13, -4, 12, 2, 1, 5, 4, 2], [-6, -1, -9, 4, 3, -2, 1, -9], [2, -8, 3, -4, -2, 9, 4, 6], [-4, 7, -11, 2, 2, 1, -3, 2], [3, 1, -1, -10, 2, 3, 1, 7], [6, 1, -2, -2, 0, 0, 0, 0], [-7, 3, -1, 8, 3, -7, -3, -10], [-8, -4, 5, 3, 1, 4, 1, -1], [0, -4, -7, -4, -3, -3, 3, -8], [-6, 9, -12, -6, -4, -5, -5, -2], [-5, -2, 1, -10, -1, -7, 1, -6], [2, 1, 0, -2, -2, -3, 2, 6], [4, 0, 6, -1, 3, -2, 0, 1], [-9, 9, -5, -4, -3, 1, 1, -6], [2, -4, 6, 8, -2, 6, 5, 4], [1, 8, -5, -2, 1, 0, -3, 2], [4, 6, 1, -4, 0, -5, -3, -3], [3, 4, 5, -4, -3, -5, -3, 6], [-3, -4, 1, -4, -1, 4, -8, -2], [-3, -3, 4, -7, 2, -2, -2, -1], [-8, 5, 2, 4, 0, -5, -6, -6], [3, -12, 13, 6, 1, 0, 3, 12], [-3, -2, -6, 0, 2, 0, -6, -13], [-7, 5, -4, 6, 3, 6, -4, 6], [10, -8, -4, 2, -1, 3, 4, 1], [-4, 1, -10, 2, -8, -2, -6, 2], [-6, -4, -2, 0, 0, 6, 4, -10], [0, -9, 10, -14, -5, 0, 6, 0], [5, 2, -7, -6, 3, -7, 1, 0], [8, -13, 10, 10, 1, 11, 8, 18], [-14, -8, 7, 0, 0, 8, 5, 6], [-7, 1, -4, -6, -1, 0, 2, -4], [-2, -3, 7, 0, 2, -1, 7, -10], [-5, -6, 10, 8, -3, 0, 5, 1], [9, 11, -12, -2, 2, -6, -6, -6], [-14, 2, 0, -6, 3, 0, -6, -21], [-11, 3, 1, 0, 3, 8, 5, 4], [-3, 6, -4, 6, -1, -3, -2, -8], [7, -8, 6, 2, -3, 2, -2, 10], [-6, 4, -4, 4, -2, -1, 1, -1], [-4, -6, -5, 1, 4, 5, 3, -2], [-8, -3, 8, 0, -2, 5, 2, -2], [-5, 4, -6, -2, -9, -1, -2, -2], [-2, 5, -8, 0, 6, 5, 0, -4], [20, 5, 0, 10, 1, 2, 4, 20], [-9, -1, 4, 2, -9, -4, -2, 13], [14, 1, 11, -6, 4, -2, 7, 5], [-5, 0, -2, 2, 5, -1, 6, 0], [-5, -4, 1, -9, 3, 0, -3, 0], [4, 8, 0, 3, 1, -3, 0, -7], [2, 12, -3, 0, -2, -1, 3, -2], [-2, 2, -6, 8, -6, 0, 0, 15], [-15, 8, -1, 0, 3, -5, 1, 2], [4, -3, 10, 4, -4, -2, 2, 10], [8, 5, 9, 6, 4, 5, 9, 10], [4, 6, 8, 8, 0, 7, 2, 3], [10, 3, -8, -6, -2, -11, 0, 4], [5, 2, 14, -2, -1, -4, 2, 10], [-5, -1, -5, 2, 5, 8, -7, -8], [1, -4, -1, 8, 3, 4, -7, 9], [-26, -5, 0, -14, 6, 5, -2, -24], [10, 8, -2, 0, -1, -9, 2, -12], [-10, -11, 13, -10, 2, -5, 5, -14], [-6, -8, 11, -4, 2, 8, 4, -3], [8, 4, -12, 12, -6, -6, 4, 8], [0, 13, -13, -12, 5, 1, -1, -3], [7, -6, 18, 10, 1, 13, 10, 22], [-15, 2, 3, -2, 2, 2, -3, -1], [4, -8, 12, -12, -4, 7, 5, 13], [-7, -5, -1, -2, -1, 0, -7, -8], [3, -7, 3, -6, -1, 0, -2, 0], [12, -7, 7, -8, -8, -8, 1, 14], [-3, -14, 5, 8, -8, -1, -7, -10], [12, -5, -4, 2, -6, -3, -8, 4], [2, 9, -7, -20, 2, -9, -1, -8], [-8, 0, -1, 16, -5, -2, -1, 2], [-23, 8, -7, 4, 1, -6, -5, -6], [-9, -6, 18, -7, -8, -3, 0, 13], [-9, 13, -13, -6, -1, -7, 1, 4], [7, -5, 2, 6, -4, -8, -8, -1], [-18, 0, 2, -6, 6, 10, -4, -12], [9, -5, 2, 6, 9, 12, 5, 16], [-8, 4, -12, 4, 0, -13, -8, -28], [-8, 9, -18, 6, 2, 7, -8, -3], [-1, 12, 1, 10, 3, -5, 1, 8], [5, 7, 11, 10, -2, 0, -1, 22], [-5, 4, -11, 12, 3, -6, -7, -14], [4, 5, 10, -3, 8, -7, 6, 2], [-21, 7, -2, 0, -1, -5, -2, 2], [0, -1, -11, 12, -1, 7, -3, -5], [1, 2, -5, -4, 5, -6, -1, -14], [21, -5, 12, 6, 1, 8, 5, 18], [6, 10, -6, 16, 6, 8, -2, 2], [-13, 11, 1, -3, 2, -13, -9, -17], [-5, -2, -5, -6, 9, -4, 1, -20], [-27, -1, 6, -6, -7, -9, -6, -6], [-14, 0, 4, 6, -2, 2, -2, -6], [-1, 4, -6, -4, -7, -4, -10, -3], [2, -14, 15, 10, 2, -2, 3, 14], [-5, -14, 11, -2, -1, 0, 5, -10], [-12, -3, 6, 4, -2, 9, 0, 20], [-5, 11, -21, -9, 1, -10, -3, -18], [-3, -9, 10, -2, -2, 3, -6, 17], [-23, 13, -4, -2, -3, -1, -3, -2], [22, -8, -12, 8, -8, 0, 6, 2], [-8, -6, -7, 8, -4, 9, -5, 14], [10, 1, -14, -2, -1, 1, -4, -14], [-7, 5, 2, -14, 5, 5, 4, 2], [15, 19, -11, 1, -1, -9, 1, 6], [-9, -4, 1, -4, -11, 0, 4, 3], [-4, 11, -6, -8, 2, 5, -4, -12], [-2, -4, 20, -13, -4, 4, 8, 6], [-21, -4, -10, -4, 1, 2, 6, -14], [-7, -3, -4, 2, -7, 1, -3, -5], [11, 6, 6, -12, -4, -9, 0, -6], [-13, -4, -3, 0, 5, 10, 0, -13], [-7, -4, 1, 18, -5, 2, 5, -4], [9, 0, 2, 8, 11, 2, 5, -2], [13, -14, 4, 18, -1, 2, -2, 4], [14, 6, -9, -2, 4, 3, 5, 20], [9, -23, 0, 0, -1, 1, 4, 8], [4, -7, -8, -1, -5, -2, -8, -23], [-5, 8, -12, -8, -3, -2, 5, -4], [11, 1, 8, 2, 7, -2, 1, -11], [5, -1, 8, 12, -5, 5, -2, 24], [-17, 9, -33, -5, -4, 0, -1, -13], [-18, 3, -8, 6, 0, 11, -2, -12], [-9, -13, 9, -2, 4, -3, -5, 0], [25, 3, -17, -6, 5, -11, -7, -16], [-25, -7, -1, -6, 5, -8, -1, -18], [3, 8, 3, -10, 5, -2, -1, 8], [17, 0, -1, 4, -2, -6, -3, 11], [-22, 4, -15, 0, 4, 9, -7, -26], [13, 2, 8, 0, -7, 7, -2, 12], [17, 0, 1, -2, -7, 4, 5, 14], [-4, -13, 19, 18, -6, 0, 5, 28], [-10, -1, 1, 4, -1, 6, -1, -1], [24, -9, 5, -6, 6, -6, 1, 15], [-13, -4, -24, -12, 1, -8, -10, -31], [16, 4, 1, -4, 6, 2, -5, -14], [-9, 10, -15, -10, 3, 1, -13, -12], [-7, -18, 19, 3, 9, 17, 9, 2], [5, 0, 4, 2, 11, 6, 14, -6], [21, 3, 11, -8, -7, -8, -9, 16], [8, -11, 10, 7, 0, 7, 6, 2], [6, -9, 1, 2, -7, 13, 9, 6], [-1, 7, -2, -10, -3, -17, -2, 8], [-12, -7, 23, 1, 4, -2, 3, -4], [-3, -8, 22, -4, 1, 2, 4, 22], [-3, -1, 7, 6, -5, 4, 3, -4], [-12, -1, 6, -18, -3, 7, 0, -3], [-5, 8, -16, -9, 7, 3, 0, -10], [5, 14, 10, -6, -3, -14, -4, -2], [1, 1, -1, 10, -5, 11, -3, 4], [-6, 9, -15, 6, 6, 10, -9, -4], [25, -3, 24, 7, 1, -2, 8, 0], [-14, 8, -13, 0, -7, 0, -5, -22], [-1, 10, -16, 2, -9, 6, 4, -10], [0, -11, 7, 0, 1, -1, -5, -11], [2, -11, -24, 2, 2, 6, 6, -4], [32, 4, 4, 24, 4, 7, -1, 14], [24, 0, -17, 4, 2, -11, -1, -4], [19, 11, -13, 0, -3, -7, -5, -8], [-20, -20, 14, -11, -2, -7, 0, -8], [-7, -4, -23, -4, 3, -6, 5, -2], [-2, 2, 6, -16, -6, -13, -1, -7], [-19, -8, -5, -18, 0, -6, 1, -19], [-16, 14, -5, 13, 3, 4, -9, -11], [15, -13, 18, 10, -9, 5, 20, 14], [21, 1, 2, 6, -3, 5, 6, 43], [-21, -9, -10, -8, 3, 2, -10, -30], [-8, -18, 17, -8, 0, 17, -3, -11], [17, 1, 10, -20, -1, -3, -4, 20], [8, 16, -9, -2, 8, -1, -1, -6], [23, -14, 3, 30, -3, 11, 3, 14], [27, 1, 18, -4, -5, -11, 0, 10], [-6, 0, 15, 0, -2, -11, 6, 4], [-10, 1, -24, -2, 5, 2, -8, -24], [-25, 8, -10, 1, -6, -4, 6, 5], [-9, 12, -9, -10, -1, -22, -5, -6], [20, -2, 10, 16, -2, -7, -8, 18], [23, -4, 6, 1, -1, -1, 14, 10], [17, -1, -7, 4, 1, -6, -7, -22], [-25, -8, -1, -9, -2, -3, -1, -17], [-26, 8, -6, -10, -6, 2, 8, 0], [2, 7, 2, -22, -2, -6, -4, -6], [2, -10, -3, 12, -4, 22, 7, 10], [-5, 5, 4, -6, -1, 4, 2, -16], [6, 1, 6, -1, 12, 9, -2, -8], [15, 9, -20, -8, 9, 0, -2, -36], [-15, -16, 4, 2, 7, 5, 2, -20], [8, 7, 1, -19, 13, -1, -3, 5], [1, 9, -19, 2, 5, -3, -5, 5], [11, -1, 10, 26, 7, 3, 8, 22], [-7, 9, 1, -26, -11, -11, -3, -3], [-13, 7, 5, -9, -10, 1, -1, 3], [7, 1, -7, 6, -1, -11, 3, 6], [-11, -6, -6, 4, 4, 5, 12, -5], [10, 1, -6, 14, -12, 5, 7, 13], [16, 0, -4, 12, -2, 6, 6, -6], [-34, 9, 3, 0, 5, 1, 3, 11], [0, -4, -30, 4, 4, -9, -4, -26], [-6, -3, 4, -12, -6, 2, 14, -12], [4, 1, 3, -6, 10, -2, -3, -5], [-20, -3, 7, -14, -2, 3, 11, -16], [-13, -6, 0, -20, -3, -6, -1, -22], [30, -2, 2, 24, -6, 7, 6, 31], [-12, -14, 10, 0, -8, 9, 0, 15], [-8, -10, 22, 3, -7, 2, 8, 27], [23, 17, -19, 11, -4, -10, 1, -1], [-7, -6, -2, 13, -2, -1, 4, -19], [13, 0, -6, 12, -17, -3, 2, 26], [-11, -7, -4, 6, 5, -10, -7, -16], [4, 2, 3, 16, 3, 9, -11, 0], [17, 0, 3, 2, -9, -5, 7, 2], [1, -5, 1, -2, -7, -5, -9, -14], [-21, 3, -1, -2, -5, 1, -3, 15], [-25, -21, 22, 2, -9, -5, 6, -12], [6, 16, -14, 8, -10, -8, -11, 23], [10, -20, -5, 29, 4, 27, 3, 4], [14, -7, 7, -6, -12, 1, 5, 14], [8, 20, -24, -4, 4, -8, -15, -4], [-14, 16, -4, -14, -10, -2, -6, -20], [-18, 21, -21, 2, 0, -9, -7, 12], [-5, -15, 9, -10, 5, 16, 3, 8], [24, -3, 2, 2, 0, 0, 0, 24], [13, 4, 8, 8, 11, 7, -2, 12], [-26, -13, 6, -18, -2, 5, -10, -4], [-18, 20, -4, -28, -2, -13, -2, -4], [-18, -4, -3, -4, -6, -5, 10, 7], [-21, 6, -4, 18, 7, -3, 4, -4], [-38, -1, -15, -14, 1, 10, -5, -20], [16, -20, 14, 6, -4, 18, 14, 20], [-5, 13, -15, 2, -7, -6, 3, -6], [-44, -3, -2, -4, 5, 3, 12, -23], [-3, 5, -12, -6, -14, -11, -4, -19], [-23, -11, -15, -6, 5, -5, 3, -32], [2, -1, 3, -1, -2, 7, -3, 10], [-38, 8, -20, -22, 0, -7, -16, -32], [-11, 8, -32, -10, -2, -11, -14, -39], [-35, 14, -31, -4, -13, -15, -14, -37], [-29, -3, -17, -10, 1, 11, -7, -8], [18, -18, -14, -4, -5, 7, 10, 4], [13, 2, 5, -14, -9, -25, 1, 0], [26, 26, -35, 8, 2, -15, -3, -23], [-9, -13, 16, 18, -9, -10, 4, 8], [4, -13, 3, -4, 0, 6, 3, 22], [-24, -1, -6, 22, 4, -1, 6, 0], [-8, 23, -1, 6, 2, 9, -8, 1], [-25, 4, 0, -16, -5, -10, -4, -8], [-50, -10, 15, -11, -2, -18, 1, -28], [25, -28, 5, 16, -5, 16, 11, 18], [24, 2, -1, 6, -6, 11, 9, 6], [19, 9, 4, -1, 4, -9, 8, 17], [18, 13, -3, -2, 6, -8, 1, 22], [-19, 16, 1, 12, 15, 14, -3, -10], [19, 20, 1, -17, -12, -14, -5, 3], [-26, 19, -3, -24, 12, 1, -3, -8], [-13, -13, 1, -9, -2, -3, -5, -13], [6, 14, -15, 20, 16, 6, -11, -6], [4, 9, -26, 10, 7, -4, -14, -3], [47, 3, -9, 2, 3, 0, -4, -3], [17, -5, 25, -2, 1, -8, 2, 10], [5, 13, -16, -10, 11, 1, 6, 12], [-20, 13, -15, 12, 6, 1, -1, 0], [14, -5, -11, 2, -4, 14, 8, -6], [12, -31, 20, 10, -6, -9, 6, 12], [10, 6, -18, 0, -6, -6, -3, 2], [-35, 11, -8, 8, -3, 2, -12, 2], [-15, -2, -8, -15, 6, 2, 8, -17], [18, -3, 14, 0, 10, 15, -4, 40], [20, 0, -26, 3, 8, 6, 2, 0], [0, 19, -4, -18, -6, -20, 0, -16], [24, -3, -11, 2, -10, 1, -2, 28], [-6, 20, -23, -4, -2, -26, -15, -19], [2, -21, 19, 17, 3, 29, 3, 29], [10, 21, -18, -2, 4, 2, 2, 16], [5, 16, -23, 12, 1, 0, -11, 24], [3, 23, -14, -38, 7, -8, -9, -18], [10, 5, -10, -16, -2, 3, -6, -20], [-20, -9, 14, -6, 0, -2, 0, 0], [-9, 27, -24, -6, -1, -11, -9, -5], [10, 10, 5, -16, 2, -1, 6, 25], [6, -22, 18, 4, 2, 18, 0, 8], [-13, -21, 7, -4, 1, 12, -1, -16], [13, 3, 4, 18, -11, 10, 9, 18], [20, 0, -7, -4, 7, 1, 7, 15], [11, 2, -21, 18, 7, 10, -5, -16], [-6, 15, -22, -3, -7, -10, -6, 5], [47, -5, -1, 17, 3, 6, 13, 28], [-10, 2, 3, -10, 12, -6, 3, -14], [30, -2, 33, -4, 2, 1, 11, 26], [1, -1, -12, -10, 7, 5, -10, -8], [31, -16, 14, 28, 5, 11, 0, 26], [-16, -3, -4, 10, 14, 2, -6, -4], [16, -17, -26, 15, -6, 5, 2, -4], [-15, 15, -23, -20, -5, -22, -17, -8], [28, -1, 19, 18, 2, 9, 11, 28], [20, -25, 3, 18, 0, 21, 5, -12], [24, -9, 8, -10, 2, 3, 10, 31], [9, 6, 3, -3, -13, -18, -9, 10], [29, 2, -3, 26, 11, 6, 1, 20], [18, -7, -6, 14, -12, -8, 0, -10], [7, 1, 26, 7, 0, 1, -14, 37], [-9, 14, -9, -8, 11, 15, -11, -20], [8, -7, 19, -10, -9, -10, 1, 16], [13, 12, -33, 12, -7, -14, -9, -14], [-28, 9, -19, 0, 6, 8, -3, -14], [-1, -6, 1, 8, -19, 10, 1, 22], [17, -20, 14, -4, 15, 5, -1, -12], [-11, 27, -28, 0, 3, -17, -8, -6], [-19, -7, -4, 0, -3, -10, 2, -28], [-43, 19, 3, -18, 14, -7, -7, -8], [15, 5, -4, 0, 21, 14, 2, 16], [-34, 0, 10, -24, 18, 1, -11, -26], [-12, -30, 15, 20, -4, 6, 9, 16], [28, -11, -11, -8, 0, 7, 1, -40], [19, -4, -8, 24, 1, -1, -4, 16], [3, 8, -23, 16, 17, 20, -5, -16], [15, 11, 18, -14, -1, -17, 8, 30], [1, -6, 17, 6, 23, 22, 9, 8], [-13, 10, -29, -8, -11, -9, -12, -26], [-3, 13, 9, 18, 13, 0, -1, -3], [-32, -1, -4, -8, -7, 12, -8, -25], [-33, 26, 8, -22, 7, -6, 2, -14], [5, 9, -7, -1, -11, -7, 5, -14], [0, 13, -12, -10, -2, 14, 0, 8], [6, -6, 18, 18, 18, 7, 14, 8], [3, 23, -22, 2, 7, -8, -5, -18], [0, -9, 10, 6, 16, 12, -10, 12], [23, -3, -15, 18, -1, -1, -2, -2], [18, -8, -20, -16, -10, -13, 3, -35], [13, 15, -22, -14, 7, -8, 0, 16], [37, 2, 14, 8, 7, 12, 9, 13], [-57, -10, 1, -28, 7, 17, -3, -32], [11, -11, 17, -6, -5, -11, -3, 19], [7, -2, -2, -6, 5, 14, -6, -18], [-17, -6, 5, -16, 10, -6, 15, -24], [7, 3, 34, 16, 7, -2, 0, 22], [24, 11, 11, 24, 0, 12, 5, 26], [21, -22, 20, 0, -5, 10, 12, 33], [-35, 21, -1, 6, 5, -4, -6, -32], [13, -2, -6, 26, 1, -4, 10, 14], [-1, -8, 8, -29, 2, -4, 12, -23], [16, 2, 11, -28, -4, -7, 9, -2], [-31, 11, -9, -14, -15, 5, 2, -26], [17, -4, -2, 16, -5, 6, -4, 14], [38, 18, -27, 28, 0, -1, 3, 6], [20, 3, 20, 6, -14, -10, -9, 16], [6, -31, 26, 11, 1, 26, 6, 25], [-17, 20, -26, -20, 7, -17, -10, 14], [-15, 3, -6, -16, -5, 9, -2, -12], [-8, -5, -18, -6, -6, -5, -13, 7], [-10, 26, -32, -26, 2, -16, -14, -42], [-13, -22, 26, -6, -5, 6, 10, -32], [-22, -13, -13, 0, 12, 12, -1, 2], [13, 18, -8, 9, 3, -10, -2, 34], [21, -4, 2, -6, 3, 0, 2, 50], [18, -17, 21, -12, 12, 9, 9, 12], [4, -1, -26, 6, 7, -5, 12, -1], [29, -3, 35, 8, -1, -3, 9, 46], [-29, -12, 5, -20, 7, 14, 15, -8], [-26, 20, 0, 0, -14, -21, -22, -8], [21, -4, 3, 8, -13, 11, 8, 18], [-1, -13, 3, -10, 1, -7, -1, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5625_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5625_2_a_bc();".
// 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_5625_2_a_bc(:prec:=8)
    chi := MakeCharacter_5625_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_5625_2_a_bc();".
// 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_5625_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 8, -48, -20, 41, 10, -12, -1, 1]>,<7,R![-1055, 320, 1650, -160, -654, -142, 29, 12, 1]>],Snew);
    return Vf;
end function;