
// Make newform 825.2.bx.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_bx();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_bx_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_bx_h();".
// 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_825_2_bx_h();".
// 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, 0, -1, 0, 15, 0, -59, 0, 104, 0, -59, 0, 15, 0, -1, 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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-63, 0, -622, 0, 1045, 0, 772, 0, -908, 0, 289, 0, 2, 0, 21, 0], [-119, 0, 1498, 0, -1995, 0, 2436, 0, -1260, 0, 329, 0, -22, 0, 21, 0], [0, 119, 0, -1498, 0, 1995, 0, -2436, 0, 1260, 0, -329, 0, 22, 0, -21], [-40, 0, 137, 0, -671, 0, 804, 0, -384, 0, 76, 0, -11, 0, 5, 0], [91, 0, -246, 0, 475, 0, -1604, 0, 1084, 0, -309, 0, 10, 0, -21, 0], [30, 0, 155, 0, -1107, 0, 1332, 0, -648, 0, 138, 0, -17, 0, 9, 0], [0, 30, 0, 155, 0, -1107, 0, 1332, 0, -648, 0, 138, 0, -17, 0, 9], [0, -16, 0, -685, 0, 1867, 0, -1748, 0, 736, 0, -148, 0, 23, 0, -9], [77, 0, 466, 0, -1511, 0, 4564, 0, -2972, 0, 845, 0, -30, 0, 57, 0], [0, 685, 0, -1990, 0, 2945, 0, -5644, 0, 3428, 0, -947, 0, 42, 0, -63], [-58, 0, 201, 0, -101, 0, -1108, 0, 824, 0, -246, 0, 5, 0, -17, 0], [0, -58, 0, 201, 0, -101, 0, -1108, 0, 824, 0, -246, 0, 5, 0, -17], [0, -543, 0, 2666, 0, -9107, 0, 10980, 0, -5388, 0, 1185, 0, -134, 0, 77], [0, 97, 0, -71, 0, -1556, 0, 3776, 0, -2284, 0, 589, 0, -35, 0, 40]];
    Rf_basisdens := [1, 1, 384, 384, 384, 96, 192, 96, 96, 96, 384, 384, 96, 96, 384, 96];
    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_825_bx();"
function MakeCharacter_825_bx()
    N := 825;
    order := 10;
    char_gens := [551, 727, 376];
    v := [10, 5, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_825_bx_Hecke();"
function MakeCharacter_825_bx_Hecke(Kf)
    N := 825;
    order := 10;
    char_gens := [551, 727, 376];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1], [0, 0, -3, 1, 0, -1, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [-1, 0, 5, -2, 0, 2, 0, -2, 0, 0, 2, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 1], [0, 0, 4, -1, 0, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, -2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, -5], [-6, 0, 3, 1, 0, -1, 0, 1, 0, 0, -1, 0, -6, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, -3, 0, 1, -3, -3], [0, 1, 0, 0, -4, 0, 0, 0, -1, -3, 0, -3, 0, 0, -6, 0], [0, -3, 0, 0, 0, 0, 0, 0, -3, 5, 0, 0, 0, 3, 5, -4], [-6, 0, 1, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -6, -6, -1, 0, 0, -3, 0, -2, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, 0, 5, 4, 0, 5, 0, 5, 5, -4], [0, 0, 0, 4, 0, -2, -2, 8, 0, 0, 3, 0, -4, 0, 0, 0], [0, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0, -2, 0, 8, 4, -4], [-7, 0, -2, 7, 0, 0, -2, 0, 0, 0, -2, 0, -2, 0, 0, 0], [0, 3, 0, 0, 5, 0, 0, 0, -5, -2, 0, 8, 0, 3, 0, -8], [-2, 0, 2, 0, 0, 2, 2, -3, 0, 0, 2, 0, 3, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 3, -3, 0, 0, 0, -3, -3, 4], [0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 1, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, -5, 5, -5], [0, 6, 0, 0, 7, 0, 0, 0, -6, 5, 0, 5, 0, 0, 1, 0], [4, 0, -1, 0, 0, -1, 6, 2, 0, 0, 6, 0, -2, 0, 0, 0], [0, -4, 0, 0, -5, 0, 0, 0, 4, 6, 0, 6, 0, 0, 6, 0], [0, 7, 0, 0, 4, 0, 0, 0, -4, -4, 0, 4, 0, 7, 0, -4], [4, 0, 5, 0, 0, 5, 11, 5, 0, 0, 11, 0, -5, 0, 0, 0], [0, 2, 0, 0, 3, 0, 0, 0, -3, 10, 0, 1, 0, 2, 0, -1], [6, 0, -3, -3, 0, -7, -3, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, -1, -1, 4, 0, 0, 3, 0, -4, 0, 0, 0], [-1, 0, 4, -2, 0, 4, 0, -2, 0, 0, 4, 0, -1, 0, 0, 0], [0, -6, 0, 0, -11, 0, 0, 0, 6, -10, 0, -10, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, 1, 0, 0, 0, 8, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, -6, 8, 0, 0, 0, 6, 8, -16], [0, 1, 0, 0, -7, 0, 0, 0, -1, -6, 0, -6, 0, 0, -6, 0], [-3, 0, 3, -2, 0, 0, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, -5, 0, 6, 6, -11, 0, 0, -3, 0, 5, 0, 0, 0], [-12, 0, -6, 5, 0, 5, -6, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, -2, 0, 0, 0, 2, -15, 0, -6, 0, 8, 0, 6], [0, 0, 0, 0, -3, 0, 0, 0, 4, -19, 0, -17, 0, -3, -17, 19], [4, 0, 9, 0, 0, 9, 9, 7, 0, 0, 9, 0, -7, 0, 0, 0], [-11, 0, 3, 11, 0, 0, 6, 0, 0, 0, 3, 0, -14, 0, 0, 0], [0, -4, 0, 0, -6, 0, 0, 0, 4, 11, 0, 11, 0, 0, 13, 0], [0, -5, 0, 0, -5, 0, 0, 0, 0, 0, 0, 7, 0, -4, -2, 2], [2, 0, -3, -2, 0, 0, 10, 0, 0, 0, -3, 0, 10, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, -5, 5, 0, 0, 0, 5, 5, -3], [4, 0, 9, -7, 0, 3, 0, -7, 0, 0, 3, 0, 4, 0, 0, 0], [1, 0, -8, 0, 0, -8, -4, -2, 0, 0, -4, 0, 2, 0, 0, 0], [6, 0, 6, -6, 0, 0, -6, 0, 0, 0, 6, 0, -9, 0, 0, 0], [0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 11, 0, -1, -3, 3], [0, 0, 0, 0, -9, 0, 0, 0, -4, 3, 0, -7, 0, -9, -7, -3], [0, 0, 0, 6, 0, -7, -7, 8, 0, 0, 0, 0, -6, 0, 0, 0], [0, 0, -1, -11, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -4, 0, 8], [0, 0, 0, -2, 0, -8, -8, -7, 0, 0, -8, 0, 2, 0, 0, 0], [0, -3, 0, 0, 3, 0, 0, 0, 3, -10, 0, -10, 0, 0, -5, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -11, 3, -3], [0, 0, 0, 0, -9, 0, 0, 0, 2, -4, 0, -9, 0, -9, -9, 4], [-17, 0, 13, 18, 0, 15, 0, 18, 0, 0, 15, 0, -17, 0, 0, 0], [0, 4, 0, 0, 10, 0, 0, 0, -10, 15, 0, 7, 0, 4, 0, -7], [0, -10, 0, 0, 0, 0, 0, 0, 6, -2, 0, 0, 0, -6, -2, -4], [-6, 0, 7, 0, 0, 7, 10, 7, 0, 0, 10, 0, -7, 0, 0, 0], [0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 5, 0, 17, -7, 7], [0, -6, 0, 0, -5, 0, 0, 0, 5, 7, 0, -11, 0, -6, 0, 11], [18, 0, -7, -16, 0, -16, 0, -16, 0, 0, -16, 0, 18, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 4, -17, 0, -12, 0, 6, -12, 17], [1, 0, 11, -9, 0, -1, 11, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 11, 0, 0, 11, 0, 0, 0, 0, 0, 0, 2, 0, 13, -1, 1], [0, 0, 0, 0, 4, 0, 0, 0, -5, -4, 0, -2, 0, 4, -2, 4], [0, 0, 0, 2, 0, 11, 11, 3, 0, 0, 2, 0, -2, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 11, 5, 0, 0, 0, -11, 5, 16], [5, 0, -13, -4, 0, -7, -13, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13, 0, 0, 0, 9, -19, 0, -5, 0, -13, -5, 19], [0, 0, 0, -9, 0, -19, -19, -14, 0, 0, -11, 0, 9, 0, 0, 0], [-20, 0, 0, 20, 0, 0, -4, 0, 0, 0, 0, 0, -18, 0, 0, 0], [8, 0, -5, 0, 0, -5, 4, 17, 0, 0, 4, 0, -17, 0, 0, 0], [11, 0, -13, -10, 0, -21, 0, -10, 0, 0, -21, 0, 11, 0, 0, 0], [-15, 0, -16, 15, 0, 0, -2, 0, 0, 0, -16, 0, 4, 0, 0, 0], [0, -4, 0, 0, -4, 0, 0, 0, 0, 0, 0, 10, 0, -13, 0, 0], [13, 0, 8, 0, 0, 8, 5, -3, 0, 0, 5, 0, 3, 0, 0, 0], [0, -6, 0, 0, -8, 0, 0, 0, 6, -7, 0, -7, 0, 0, -8, 0], [-7, 0, -6, 7, 0, 0, -7, 0, 0, 0, -6, 0, -4, 0, 0, 0], [0, -5, 0, 0, -21, 0, 0, 0, 21, -16, 0, -12, 0, -5, 0, 12], [-9, 0, 10, 0, 0, 10, 16, 21, 0, 0, 16, 0, -21, 0, 0, 0], [0, 0, 0, 0, -8, 0, 0, 0, 14, -17, 0, 6, 0, -8, 6, 17], [0, 12, 0, 0, 17, 0, 0, 0, -17, 8, 0, -4, 0, 12, 0, 4], [0, 0, 0, -27, 0, 8, 8, -14, 0, 0, -1, 0, 27, 0, 0, 0], [0, 0, 0, 0, -8, 0, 0, 0, 0, -13, 0, -13, 0, 0, 4, 0], [-12, 0, -5, 7, 0, -7, -5, 12, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, -14, 19, 0, -3, -14, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 0, -6, 0, -6, 0, 0, -7, 0], [5, 0, -10, 10, 0, -7, 0, 10, 0, 0, -7, 0, 5, 0, 0, 0], [18, 0, 17, -10, 0, 6, 17, -18, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, -3, 0, 0, 0, 3, 23, 0, 2, 0, -7, 0, -2], [-1, 0, -17, 1, 0, 0, -7, 0, 0, 0, -17, 0, 0, 0, 0, 0], [0, 18, 0, 0, 4, 0, 0, 0, -18, 11, 0, 11, 0, 0, -15, 0], [0, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0, 4, 0, 9, 7, -7], [0, 9, 0, 0, 0, 0, 0, 0, -9, -5, 0, 0, 0, 9, -5, 18], [-3, 0, -17, -4, 0, -14, 0, -4, 0, 0, -14, 0, -3, 0, 0, 0], [-7, 0, -17, 0, 0, -17, -7, -19, 0, 0, -7, 0, 19, 0, 0, 0], [0, -5, 0, 0, -3, 0, 0, 0, 3, 10, 0, 3, 0, -5, 0, -3], [0, 13, 0, 0, 13, 0, 0, 0, 0, 0, 0, 10, 0, 7, 3, -3], [0, 0, 0, 0, 8, 0, 0, 0, 12, 12, 0, 11, 0, 8, 11, -12], [0, 0, 0, 15, 0, -8, -8, 12, 0, 0, 10, 0, -15, 0, 0, 0], [22, 0, 11, -18, 0, 8, 11, -22, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 7, 8, 0, 0, 0, -7, 8, 25], [0, -1, 0, 0, -5, 0, 0, 0, 1, 7, 0, 7, 0, 0, 4, 0], [0, 0, 0, 0, 8, 0, 0, 0, 23, -22, 0, -4, 0, 8, -4, 22], [13, 0, -22, 13, 0, 6, 0, 13, 0, 0, 6, 0, 13, 0, 0, 0], [4, 0, -5, 9, 0, 12, -5, -4, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -3, -4, 0, -7, 0, -4, 0, 0, -7, 0, 2, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, -1], [0, -3, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, -2, -1, -15], [0, -10, 0, 0, -10, 0, 0, 0, 0, 0, 0, -7, 0, -14, -9, 9], [19, 0, 10, 0, 0, 10, 4, -16, 0, 0, 4, 0, 16, 0, 0, 0], [-13, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, -14, 0, 0, 0], [0, -27, 0, 0, 0, 0, 0, 0, 4, 9, 0, 0, 0, -4, 9, -13], [0, -25, 0, 0, -22, 0, 0, 0, 22, 7, 0, -4, 0, -25, 0, 4], [0, 0, 0, 0, 6, 0, 0, 0, -14, 12, 0, 17, 0, 6, 17, -12], [3, 0, 23, -3, 0, 0, 7, 0, 0, 0, 23, 0, -28, 0, 0, 0], [25, 0, -20, -21, 0, -7, 0, -21, 0, 0, -7, 0, 25, 0, 0, 0], [0, 0, 0, 25, 0, 6, 6, 33, 0, 0, 2, 0, -25, 0, 0, 0], [-14, 0, -3, 32, 0, 19, -3, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0, -10, 14, 0, 10, 0, -10, 10, -14], [0, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, -23, 0, 13, 8, -8], [-11, 0, -15, 11, 0, 0, 4, 0, 0, 0, -15, 0, 7, 0, 0, 0], [0, 5, 0, 0, -11, 0, 0, 0, 11, -7, 0, -3, 0, 5, 0, 3], [-3, 0, 20, -9, 0, 1, 0, -9, 0, 0, 1, 0, -3, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0, -28, 8, 0, 0, 0, 28, 8, -24], [2, 0, 8, -2, 0, 0, 5, 0, 0, 0, 8, 0, 16, 0, 0, 0], [-21, 0, -6, 0, 0, -6, -6, -10, 0, 0, -6, 0, 10, 0, 0, 0], [0, 8, 0, 0, -5, 0, 0, 0, -8, -11, 0, -11, 0, 0, 6, 0], [0, -3, 0, 0, 1, 0, 0, 0, -1, -13, 0, 3, 0, -3, 0, -3], [0, 3, 0, 0, -3, 0, 0, 0, 3, -15, 0, -9, 0, 3, 0, 9], [0, 0, 0, -11, 0, 2, 2, -6, 0, 0, -13, 0, 11, 0, 0, 0], [2, 0, 18, 12, 0, 15, 0, 12, 0, 0, 15, 0, 2, 0, 0, 0], [17, 0, 19, -4, 0, 6, 19, -17, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 25, -8, 0, 0, 0, -25, -8, 20], [0, -11, 0, 0, 0, 0, 0, 0, -2, 11, 0, 0, 0, 2, 11, -3], [24, 0, 6, -22, 0, 15, 6, -24, 0, 0, 0, 0, 0, 0, 0, 0], [-25, 0, 14, 11, 0, 10, 0, 11, 0, 0, 10, 0, -25, 0, 0, 0], [0, -4, 0, 0, -7, 0, 0, 0, 7, -6, 0, -10, 0, -4, 0, 10], [0, 0, 0, 0, 15, 0, 0, 0, -6, -7, 0, -12, 0, 15, -12, 7], [-28, 0, 18, 0, 0, 18, 21, 12, 0, 0, 21, 0, -12, 0, 0, 0], [0, -22, 0, 0, -11, 0, 0, 0, 11, 13, 0, 15, 0, -22, 0, -15], [0, 0, 0, 0, 13, 0, 0, 0, 0, 12, 0, 12, 0, 0, 9, 0], [0, 0, -9, 0, 0, -9, -17, -12, 0, 0, -17, 0, 12, 0, 0, 0], [0, -12, 0, 0, 34, 0, 0, 0, 12, 7, 0, 7, 0, 0, 10, 0], [0, -3, 0, 0, -3, 0, 0, 0, 0, 0, 0, -7, 0, -32, -14, 14], [0, -6, 0, 0, -6, 0, 0, 0, 6, 15, 0, 24, 0, -6, 0, -24], [-7, 0, -16, 7, 0, 0, 4, 0, 0, 0, -16, 0, -7, 0, 0, 0], [0, 0, 0, 13, 0, -6, -6, -6, 0, 0, -15, 0, -13, 0, 0, 0], [0, 0, 0, -25, 0, -2, -2, -17, 0, 0, -7, 0, 25, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, -9, 14, 0, 0, 0, 9, 14, -39], [0, 0, 0, -4, 0, 4, 4, 13, 0, 0, 18, 0, 4, 0, 0, 0], [0, 0, 0, 0, -15, 0, 0, 0, 12, 3, 0, 15, 0, -15, 15, -3], [0, 23, 0, 0, 23, 0, 0, 0, 0, 0, 0, 11, 0, 8, -9, 9], [0, 0, 0, 0, 1, 0, 0, 0, -9, 26, 0, 3, 0, 1, 3, -26], [-20, 0, -20, 12, 0, 2, 0, 12, 0, 0, 2, 0, -20, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, -11, 16, 0, 0, 0, 11, 16, -36], [0, 5, 0, 0, 5, 0, 0, 0, 0, 0, 0, -20, 0, 15, 13, -13], [-30, 0, 7, 0, 0, 7, 11, 21, 0, 0, 11, 0, -21, 0, 0, 0], [0, -6, 0, 0, -6, 0, 0, 0, 0, 0, 0, 18, 0, -6, 6, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_bx_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_2_bx_h();".
// 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_825_2_bx_h(:prec:=16)
    chi := MakeCharacter_825_bx();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_825_2_bx_h();".
// 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_825_2_bx_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_bx();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![14641, 0, -15609, 0, 9087, 0, -3647, 0, 2730, 0, -473, 0, 57, 0, -6, 0, 1]>,<13,R![1, 0, -13, 0, 75, 0, -103, 0, 2874, 0, 683, 0, 665, 0, -42, 0, 1]>],Snew);
    return Vf;
end function;