
// Make newform 5625.2.a.s in Magma, downloaded from the LMFDB on 20 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_s();".
// 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_s();".
// 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 := [1, 16, -27, -35, 26, 20, -8, -3, 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], [-3, -1, 1, 0, 0, 0, 0, 0], [1, -15, -1, 17, -2, -4, 1, 0], [0, 4, -15, -1, 17, -2, -4, 1], [-4, 36, -5, -26, 5, 4, -1, 0], [0, -8, 51, -4, -43, 7, 8, -2], [-6, 1, 57, -7, -43, 7, 8, -2]];
    Rf_basisdens := [1, 1, 1, 3, 3, 3, 3, 3];
    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 := [[-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, -1, 0, 1], [-2, -1, 0, -1, 2, 0, -2, 1], [3, 0, 0, 1, -1, 0, 2, -1], [-4, 0, -1, -3, 0, 1, -1, -1], [-2, -2, -1, -2, 1, 0, -1, -2], [-3, -1, 0, 0, -1, 1, 0, -1], [2, -1, 1, 3, -1, -1, 2, 1], [1, 0, 2, 4, 0, -1, 1, 4], [-2, 1, -2, -2, -1, -1, -1, -2], [4, -2, 1, 3, -1, 2, 4, -1], [-2, 0, 0, -1, 1, 2, -2, -1], [-2, 1, 0, 1, -1, -1, 1, -2], [-1, 3, 1, 2, -1, 1, 3, 3], [3, 0, 1, 4, -1, 1, 2, 2], [-1, -1, -1, 0, 2, 3, 0, -1], [1, -2, -1, -1, -1, 0, -4, -1], [1, -2, -2, -2, -1, 1, 0, -2], [2, 3, 1, 4, -3, 1, -1, 0], [0, -1, -1, 4, 0, 1, -1, -4], [-9, 1, -2, -1, 1, 3, -2, 2], [-3, 0, -4, -3, 1, 3, -1, -3], [-1, 0, 3, -4, -1, 2, 1, -1], [4, -1, 1, -1, 2, -3, 1, -2], [2, 0, -2, -1, 0, 2, -2, 3], [-2, -1, 1, 0, 0, -2, 1, -3], [0, 2, 2, 3, -3, 0, 6, 2], [0, 2, 2, -2, -3, -1, 3, -2], [-1, -3, 2, 1, 4, -1, -1, 0], [-3, 2, 3, -6, 1, 1, -2, -1], [9, 0, 0, 4, -1, -3, -2, -1], [-4, 2, -3, 0, 0, -2, -6, 2], [-6, 6, -3, 0, -2, -1, -2, 3], [0, 2, 0, 3, 1, -5, 4, 3], [-5, 4, 1, 1, -6, 1, 3, 3], [-6, 5, -1, 1, -1, -2, 1, 6], [-10, -2, 0, 3, 4, -2, -3, 6], [2, 5, 5, 2, -3, -4, 4, 2], [3, -2, 0, -4, 1, 2, -3, -10], [-4, -5, 0, -7, 5, 0, 0, -5], [-4, 1, -2, -9, 3, 1, 1, 0], [-4, -6, 2, 2, 3, -4, 1, 2], [6, -2, 2, 5, -5, 3, 1, -10], [0, 0, -3, 6, 0, -1, -4, 4], [-3, -4, 4, 3, 3, -2, 4, 2], [4, -4, 4, -5, 1, 2, -1, -7], [-20, -2, -1, -11, 6, 3, -8, -2], [6, -2, 0, -2, -8, 3, 0, -6], [5, -5, -1, -3, -2, 0, 5, -3], [-2, 4, 1, 8, 3, -2, 2, 3], [-2, 5, 0, 2, 0, 0, 5, 10], [-5, 3, 3, 1, -6, 1, 0, 3], [1, 1, 3, 4, 1, 2, 11, 4], [-9, -5, 1, -1, 6, -3, -1, -1], [6, -7, -1, 5, 2, -4, 7, -3], [-4, -5, 2, -1, 5, 4, 1, 4], [-6, -4, -3, -11, 1, 4, -5, -3], [10, -1, 2, 9, 2, -1, -6, 0], [-9, -1, -5, -14, 4, -2, -10, -3], [7, -6, 0, 9, -4, -2, -2, -2], [14, -4, 2, 15, 0, -2, 7, 7], [1, 6, -1, 4, -2, -2, -1, 12], [-7, -3, -5, -5, 0, 5, 0, -15], [8, -5, 7, -3, 0, -3, -3, 3], [-4, 5, 1, 1, 0, -3, -2, 12], [-10, 4, -4, 4, 3, 5, -1, 1], [-1, 3, 4, -5, -5, -3, 3, 1], [-12, -3, -1, -15, -4, 3, -9, -9], [-6, 1, 1, -7, -2, 1, -10, 2], [3, -5, 2, 2, 4, -2, 5, -1], [-10, -2, 0, -11, 9, -3, -8, 2], [6, 4, 2, 16, 0, -2, 4, 11], [4, 5, -1, -2, -2, 6, -4, 0], [-7, 3, 1, 3, -3, 4, -1, 2], [11, 6, 6, 6, -6, -7, 6, 11], [6, 6, 6, 8, -3, -7, 3, 1], [-9, 2, -2, -8, 1, 6, 3, -11], [0, 2, 3, 6, 7, 2, -1, 9], [-9, 8, 6, 1, -3, -5, 8, 1], [4, -3, 9, 13, 1, -2, 11, 1], [7, 2, -2, 9, 1, -9, 8, 9], [-5, -2, 1, -6, 1, 0, -3, 7], [12, 6, -1, -3, 2, -3, 3, -3], [-14, 8, 1, -3, -2, 0, -10, 3], [-7, 2, 6, 1, 1, 7, 2, 0], [-8, 1, 9, -10, 5, -8, 4, 3], [-7, -6, -4, -18, 7, 2, -4, -7], [-21, -4, -1, -12, 7, -2, -10, -1], [-10, 5, 0, -8, 0, -2, 0, -3], [15, 6, 2, 9, -5, -9, -1, 6], [0, -6, 1, -1, 3, 5, -6, 6], [-6, 3, -3, -4, 4, -1, -8, 1], [-2, -3, -5, 3, 6, 0, 4, -7], [-3, -1, -2, 4, -8, 0, 9, -8], [-8, 11, 5, 0, -8, 4, 6, -6], [-3, -8, 0, -5, 1, 5, -6, 3], [-9, -2, 2, -6, 7, 3, 10, 3], [5, 8, 2, 5, -6, -1, 12, -9], [-1, 2, 3, 10, -5, 2, 4, 0], [6, 6, -2, -6, -4, -3, 1, 9], [7, -3, 0, 18, -3, 0, 11, -8], [11, 3, -4, 11, -3, 5, 3, -9], [-23, 4, -5, -26, 7, 10, -17, -4], [-12, 9, -2, 5, -5, -4, -7, 4], [-6, -1, 0, -13, -4, 7, -6, -9], [-4, 2, 1, 10, 10, 0, 1, 15], [-4, -6, -8, -3, 2, -6, -16, -10], [1, 11, 6, 7, 3, -8, 0, 21], [3, -1, -7, 1, -6, 0, 10, -6], [-9, 0, -3, -4, 9, 4, 4, 0], [-13, 2, -12, -24, 1, 4, -19, -8], [-17, 13, -5, 1, 1, 7, -3, 10], [28, 3, 1, 8, -6, -3, 4, -1], [8, -11, 6, 2, 7, 2, 1, -2], [-4, 4, 2, 2, 3, -9, 6, 12], [-37, -6, -10, -18, 13, -1, -22, 2], [6, 4, 9, 4, -3, 2, -5, 0], [3, 5, 3, 5, 4, -5, -2, 21], [-5, 3, -5, 19, -1, -5, 1, 7], [19, -14, 9, 15, -4, -3, 9, -2], [5, -6, -1, 16, -6, -2, 11, 1], [8, 0, -4, 22, -2, -6, -3, 13], [-3, 11, -1, 8, 3, -2, -1, 7], [5, 3, -4, -5, -1, -8, -6, -11], [2, 5, 0, 9, 1, -9, -9, 5], [15, -1, 2, -2, -5, -8, -2, 7], [2, -12, 4, -7, -3, -2, 5, -1], [4, 7, -3, -3, -7, -3, -6, 7], [-27, 0, -3, -13, 4, 3, -5, 4], [-21, -1, -12, -16, 6, 4, -4, -3], [14, 8, 0, 2, -6, 5, 6, -13], [3, 2, -7, -3, 3, 0, 7, -2], [-9, 16, 8, 1, -7, 6, 6, 0], [4, 2, 6, 10, -7, -6, -2, 3], [15, -2, 7, 18, -4, 5, 8, -8], [2, 3, -1, -8, 0, 1, 3, -18], [-7, 15, 0, -6, -9, -6, 7, 5], [2, 1, -11, 2, 2, 7, -7, -8], [-15, 3, -6, -14, 4, -2, -7, 0], [-1, 0, -3, 8, 5, 9, -4, 9], [1, 12, -8, 7, -11, 3, -4, 0], [-6, 9, 0, -9, -6, 0, 12, 9], [9, 0, 4, 8, -2, -4, 6, 10], [7, 5, 4, 1, -4, 9, 13, -7], [-3, 4, 8, 17, -8, -3, 4, 10], [3, -3, -4, 11, 4, -7, 13, 1], [17, -10, 4, 1, 1, 4, 8, -2], [-19, 3, 1, -5, 5, 2, 1, -2], [-6, -1, -8, -15, 10, 9, -8, -14], [24, 0, 8, -4, -10, 1, 14, 1], [-15, 10, 1, -12, -3, 0, 0, 13], [-23, 15, 2, -11, -11, 6, 10, 0], [-2, 13, 3, 9, -9, 0, -4, 11], [-26, 8, -1, -3, 4, -2, 3, 15], [-4, 0, -1, -4, 7, -10, -7, 1], [-13, -6, -11, -13, 4, 5, -19, -3], [-2, -3, -7, -10, 7, 9, -7, -4], [-14, 6, 2, 8, -2, -1, 3, -7], [6, -6, 7, 7, 8, -7, -15, 23], [-9, -14, -11, -2, 13, -3, -17, -9], [17, -5, 3, -7, -5, -2, 3, -18], [-10, -11, 0, 13, 7, -10, -2, 11], [-7, 2, 1, -13, -7, 2, -10, -4], [-10, 3, 9, -12, 2, -5, -3, -4], [6, -3, 5, 0, -4, 0, -6, -12], [1, -10, 1, -14, -1, 3, 14, -7], [17, -5, 0, -1, 1, 6, 11, -10], [12, 9, -5, 2, -2, -10, 4, 13], [-43, -8, -9, -13, 14, -5, -19, 7], [-8, -11, 3, -7, 7, -4, -8, 7], [14, 0, -2, 16, 5, 11, 4, 1], [2, -15, 4, 8, 12, -4, -2, -3], [20, -12, 0, -2, -7, 9, 5, -28], [-9, 4, -4, -16, -3, -3, -4, -22], [-3, -10, -8, -4, 10, 4, -9, -6], [-4, -14, -4, 5, 3, -3, -15, 1], [9, 9, -6, 9, -9, -2, 7, -7], [-8, -6, -4, -2, 4, 4, 7, 1], [-13, 14, 9, -1, -12, -9, 2, 10], [0, 2, -11, 8, 1, 3, -9, -9], [24, 2, 4, 23, 2, 5, 10, 17], [-14, -4, 2, -4, 9, 8, -7, -11], [-25, -10, 0, -14, 10, -1, 0, 6], [27, -4, 4, 7, -8, -3, 18, -8], [14, 2, 1, 2, -9, -11, 4, -13], [2, -5, 1, 8, 8, 6, 7, -9], [6, -2, -2, 2, -3, -1, -11, 0], [7, -4, -9, 17, -12, 7, 10, -14], [18, 5, -8, -11, 0, -10, -8, -5], [-11, 1, -12, -24, 9, 6, -20, -22], [-6, -3, -3, 2, 1, -1, -15, 2], [-31, 5, -13, -22, 5, 14, -19, -1], [1, -2, -2, 10, 3, -2, -7, 7], [-16, -12, -7, -13, -2, 9, -3, -6], [-40, -13, -9, -39, 6, 12, -8, -20], [11, 4, 12, 19, 6, -11, 5, 22], [-14, -12, 4, -22, 4, 6, 1, -7], [-10, 3, 7, 13, 6, -10, 18, 22], [-21, -3, 1, -25, 3, -1, -22, -2], [-12, 3, -1, -18, 14, 3, 3, 0], [25, -3, 1, 0, -13, 0, 21, 0], [9, 1, 0, 24, -7, 0, 12, -1], [-9, -13, -1, -4, 10, 7, -5, 21], [-12, -4, 6, -25, 10, 7, 1, 3], [2, -3, -12, -12, -1, 9, 7, -19], [9, 5, 10, 22, -7, 2, 1, 10], [-11, 3, -4, 4, -10, -2, 10, -6], [16, -8, 2, -14, -9, -1, 5, -8], [2, -3, 5, -10, 0, -4, -14, -4], [-19, -3, -9, -24, 11, -3, -13, 5], [-3, 10, 3, 9, 10, -13, -2, 28], [-2, 4, -19, -15, 5, 3, -4, -13], [23, 6, -2, 39, 6, -1, 15, 17], [7, -11, -1, -14, -8, 3, 5, -11], [-2, 16, 13, -1, -4, -9, 6, 15], [19, -7, 6, 18, -6, -8, 4, -6], [38, 4, 2, 30, -8, -3, 8, 0], [28, 4, -1, 35, -1, -6, 14, 6], [13, -7, 6, 13, -1, 7, 4, -1], [14, 6, -4, 12, -7, 12, -5, 6], [10, -2, -5, -7, -2, -6, 0, -18], [18, 2, 5, 2, -1, -11, -11, 9], [7, 0, 7, 18, -8, 14, 18, -2], [-17, -19, 0, -5, 11, 3, -14, -9], [-5, 5, -1, -3, 3, -5, -20, 12], [7, -10, 0, -1, -4, -14, 11, -5], [-4, 1, 1, -22, -8, 13, -2, 1], [1, -8, -7, 7, -4, 1, 18, 4], [34, -13, 11, 8, 1, -7, 4, 3], [-14, 1, -18, -16, 4, 0, -1, 3], [17, -1, -3, 25, -3, 4, 14, 0], [10, 19, 13, 17, -3, -14, 2, 12], [0, -12, 1, -6, 1, 5, -23, -13], [-17, 5, 1, 4, 4, -11, 9, 17], [-2, -10, 8, 5, 4, 0, 8, -4], [-13, -11, 5, -16, 11, 13, 8, -2], [18, 7, -3, 29, -4, -1, -10, 2], [2, -20, -3, -16, 5, 14, 6, -11], [11, -4, -9, -26, 5, -2, -11, -12], [3, 3, -9, -2, -6, 12, 9, -21], [3, -19, -10, -14, -1, 7, -8, -33], [20, -5, -4, -12, -5, -8, 8, -3], [-4, -1, -6, 9, 17, -8, -5, -5], [30, -2, 0, 23, -7, -1, 5, -13], [4, 14, -9, -9, -13, -6, 17, -12], [-27, 6, -7, 10, -2, 1, -9, 5], [-15, 4, 1, 10, 0, 10, 5, 0], [-18, 3, -13, -7, -8, 1, 0, -20], [36, 5, 6, 11, -6, -7, -11, -7], [26, -13, 12, 9, 6, -11, 0, 7], [3, -14, 11, 19, 1, 3, 4, 18], [-12, 9, -9, -17, -2, 8, -3, -22], [-10, 5, 2, 14, 1, 14, 15, 2], [-5, 9, -2, 11, -15, 14, 16, -1], [-2, 8, -11, -3, -1, -7, -7, 15], [3, 20, 3, -3, -2, 1, 22, 7], [24, -1, 5, 12, -4, 2, 19, 11], [-11, 7, -13, 12, 8, -2, -9, -5], [12, -8, 6, 35, -15, -3, 6, -7], [-3, 25, 3, 12, -5, -6, -2, 1], [9, -8, 8, 6, 13, 3, 9, 22], [-9, 3, -3, 5, -15, -4, 17, -6], [2, 7, -4, -12, 2, -4, 6, 9], [-17, 12, -5, -27, 6, 10, -1, -17], [11, -3, -7, -27, 0, -5, 2, -16], [47, 7, 1, 28, -14, -3, 17, 20], [11, 11, 7, 5, -14, -12, 7, 7], [1, -4, 8, -1, 11, -7, -12, 7], [-17, 5, -21, -1, 5, 3, -8, -7], [-10, -11, 2, 24, 6, -10, -4, 23], [-30, -11, -5, -8, 12, -4, -29, 11], [5, -18, 1, 10, 7, -7, -2, 25], [28, 14, 3, 25, -10, -14, 3, 29], [-26, 3, -8, -6, 21, 7, -2, 25], [-7, 8, -4, -15, 8, -1, -19, 16], [23, 15, -7, 15, -11, 0, 3, 6], [15, -8, -1, 15, -4, 18, 11, -19], [-25, -3, -18, -38, 6, 9, -5, -23], [-49, 12, 1, -9, -7, -2, 3, -4], [-7, -2, 2, -14, -6, 9, -3, -9], [-25, 21, 13, 1, -1, 0, -1, 7], [25, -7, 6, 7, 14, -18, 4, -6], [-1, -3, -14, 17, -4, 4, -11, -3], [-10, -1, -13, -6, 2, -1, -31, -5], [-53, 1, -14, -30, 16, 1, -29, 19], [-3, -24, 2, -5, 10, -6, -13, 11], [7, -18, -3, -10, -1, -13, 4, -21], [2, -25, -12, -2, 6, 7, -15, -9], [12, 1, 7, 10, 3, 4, -12, 18], [40, -11, 9, 25, -5, -11, 4, 1], [36, -3, 11, 28, -13, -3, 37, 5], [8, 6, 7, 7, -17, 9, 3, -2], [4, 15, -18, -12, 7, 2, -9, -9], [-52, -8, -6, -30, 11, 13, -19, -9], [-9, -4, -4, -23, 13, 7, -30, 11], [-6, -6, -2, 15, 4, 2, -17, 18], [0, -19, 3, -3, 15, -17, 2, -7], [-9, -4, -5, 13, 9, -7, 1, 4], [26, -8, 2, 3, 7, 1, -17, 21], [-1, 2, -1, -2, -3, -8, 9, -12], [-5, 0, 14, -18, 5, 8, -8, -2], [21, 5, 4, 21, -10, 2, 19, 23], [8, -15, 1, 10, -10, 12, 13, -8], [17, -4, -7, -10, 8, 13, -8, -20], [-9, 9, -7, 1, 9, -14, -5, 8], [-30, 19, -5, -28, 10, -7, -13, 18], [-4, 1, 6, -29, 4, 8, -14, -19], [-34, -7, -1, -35, 14, 19, -14, -10], [-41, -24, -7, -24, 4, 11, -30, -17], [5, 4, -9, -15, 10, 10, -15, -15], [-32, 1, -7, 0, 3, -9, 1, 5], [-9, 12, 15, 22, -12, -2, 7, 16], [-10, -12, -11, -25, -2, 0, -6, -30], [6, -8, 11, -18, 0, -4, -9, -5], [8, -13, 11, -4, 6, -13, -23, 20], [39, -7, 13, 54, 4, -9, 15, 38], [-9, -16, 12, -1, 16, -1, 0, 12], [-3, 19, -8, 6, 4, 1, 5, -2], [6, -20, -1, 15, -10, 3, 17, -7], [13, 0, 13, -29, 1, 9, -10, -15], [12, 5, -6, 0, 0, 3, -23, 8], [-7, 3, 0, -5, -10, 7, 10, -17], [26, 10, 4, 11, -5, -8, 9, -17], [-41, 19, -2, -3, -8, 6, -13, 17], [0, -21, 1, -20, 10, 5, 10, 0], [-62, 1, -13, -14, 8, 10, -9, 3], [-9, -11, -7, 15, 10, -3, -17, 28], [20, 4, -13, 22, 3, 14, -4, -1], [-4, -5, 7, -6, -10, 14, 6, 4], [10, -4, 14, -14, 12, -15, 6, -1], [3, -10, -5, -14, -1, -6, 14, -10], [-15, 0, 2, -34, 4, 7, -21, 7], [-8, 0, 11, -17, 0, 7, 8, 12], [2, -3, -7, -19, 1, 14, -4, -18], [38, 8, 10, 16, -14, -23, 7, 0], [-16, 0, 3, 27, -3, -3, -8, 11], [-7, -7, -1, -28, -1, -7, -22, -30], [-21, 1, -3, 13, -1, 17, 8, 20], [17, 8, 1, 20, -10, -21, 1, 11], [-11, -3, -11, -12, 12, 7, -6, -17], [41, 2, 4, 26, -5, 7, 35, -4], [29, 9, 0, 18, -13, -5, 3, 6], [45, 21, 4, 29, -19, -5, 14, 25], [24, 13, 9, 8, -10, -1, 14, 5], [34, -21, -6, 15, -8, -2, 5, -1], [-25, 16, -8, -7, -13, 10, 14, -6], [-5, -22, -9, -5, 9, -8, -26, -16], [-28, 7, -13, -4, 9, 0, -6, -11], [16, 15, 1, 26, -10, -18, 3, 12], [-32, 12, -2, 1, 9, -4, -10, 33], [-19, -2, 11, -33, -1, 7, -16, -6], [12, -18, -6, 2, 3, 11, 6, -17], [4, -1, 4, -21, 1, 7, 14, -6], [21, 12, 26, 30, -20, -15, 26, 15], [-14, 24, 2, -8, 7, 4, 14, 15], [-2, -1, -3, 15, -13, -11, 14, -15], [-22, 10, 12, 2, 10, -13, -8, 28], [-11, -14, -2, -21, 0, -8, 11, -16], [-16, -3, 12, 5, 2, 17, 2, -12], [-29, 1, 1, 20, 8, -8, 15, 21], [12, -17, 6, 10, -15, -4, 10, -3], [-38, 1, -21, -43, 12, 12, -32, -3], [-35, 16, 15, 20, 3, 2, -6, 11], [-14, -1, 2, 26, -5, -7, 14, -14], [-13, -4, -8, -45, 5, -6, -18, 1], [6, -9, -1, 1, -2, -11, -2, -11], [42, 19, 16, 27, -8, -8, 31, 28], [0, -16, -1, 20, 7, -2, -5, 9], [10, -17, 5, -2, 18, 4, -5, 2], [-10, -11, -3, -11, 6, -3, -15, -16], [13, 22, 13, -13, -4, 1, 18, 24], [41, 1, 5, 29, -17, -25, 12, 14], [40, -35, -4, 3, 2, 11, 7, -29], [6, -15, -5, -16, 12, 2, 11, -17], [-16, 8, -10, -5, 9, 20, 1, -18], [10, 24, 13, 2, -2, -7, 22, 10], [-17, 0, 11, -1, 4, -9, 22, 17], [-37, 16, 12, 10, -17, 6, 11, 7], [29, -5, -7, -9, 10, 1, 9, -24], [-16, 5, 0, -8, 3, 22, -9, 0], [-1, -16, -7, -6, 16, -3, -28, -9], [0, -3, 9, -19, -16, -21, 1, -5], [-18, 15, -15, -11, -14, 1, -4, 0], [-38, 2, 0, -22, -3, -5, -14, 7], [32, -1, -1, -4, 2, 8, 5, -21], [30, -6, 6, 20, 0, -23, 1, 18], [-31, -10, 2, -37, 2, 2, 6, 11], [-29, -4, -7, 6, 7, 8, 10, -24], [-38, -4, -14, -42, 17, 11, -31, 6], [7, 25, -5, 1, -16, -22, 22, 1], [-22, 7, -6, 30, -9, 8, -4, 21], [-21, -2, 7, -1, 6, 18, -10, -2], [3, -1, 12, -15, -13, -11, 9, 20], [-1, 8, 9, 22, -12, 18, 23, -10], [-7, 8, 15, 11, -9, 22, 12, -15], [-38, 14, -7, -11, -1, 8, -12, 20], [-46, 17, -5, -14, 1, -10, -21, 13], [-36, 13, -15, -50, -11, 5, -9, -18], [-28, -12, -1, 4, 19, 3, -12, 5], [-8, -1, 13, -16, 7, -5, -21, 22], [-22, -10, 2, -22, 25, -19, -23, 19], [-10, 19, -1, 6, -6, 18, -9, 21], [-21, 18, -2, -35, 15, 2, -4, 23], [-23, 21, 11, -6, -12, 17, 10, -19], [-38, -18, -20, -50, 20, 21, -9, -34], [44, -26, 11, -1, -13, -3, 1, -17], [35, -6, 15, 22, 7, -9, 31, 1], [-1, -30, 2, 2, -2, 22, -5, -24], [10, -10, 6, -27, -13, 0, -20, -22], [60, -5, 9, 69, 4, -15, 22, 16], [-33, 0, -5, -39, -1, -17, -28, -9], [-4, -14, -19, 22, 3, -3, -30, -9], [15, 11, 1, -10, 0, 6, -4, -6], [1, 6, 0, 2, 8, -5, 7, -1], [-20, -14, -18, -16, 15, 11, -6, -24], [1, 27, 15, 20, 7, -22, 0, 37], [9, -15, 14, -11, 16, 17, 16, 2], [-23, 16, -12, 18, -12, 5, -3, 23], [-20, -7, -6, -25, 8, 7, -28, -5], [1, 3, -11, 24, 9, -20, 1, 4], [-23, 8, -13, -21, 6, 4, -5, -18], [19, 8, -9, 24, 10, -7, 10, -6], [20, -5, 0, -11, -15, 21, 10, -16], [-20, -9, 9, -21, -4, 19, -8, -23], [-10, 1, 2, -4, 4, -21, -16, 25], [51, -2, 0, 3, -23, 3, 10, -26], [-45, -19, -6, -26, 13, 13, -26, -13], [39, -4, -17, 8, 3, 1, -4, 0]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, 0, 69, 60, -29, -35, -1, 5, 1]>,<7,R![1, -45, 124, 25, -154, 35, 24, -10, 1]>],Snew);
    return Vf;
end function;