
// Make newform 6025.2.a.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6025_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_6025_2_a_i();".
// 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_6025_2_a_i();".
// 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, -17, -57, 89, 394, -297, -699, 437, 519, -296, -184, 99, 31, -16, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -13, -9, -45, 198, 172, -385, -172, 284, 73, -98, -14, 16, 1, -1], [3, 11, -11, -143, 170, 295, -354, -233, 274, 86, -97, -15, 16, 1, -1], [10, 85, 9, -409, -119, 589, 153, -367, -71, 113, 14, -17, -1, 1, 0], [-1, 11, 85, 9, -409, -119, 589, 153, -367, -71, 113, 14, -17, -1, 1], [1, 34, -8, -272, 210, 546, -408, -448, 293, 170, -99, -30, 16, 2, -1], [6, 63, 17, -440, 55, 754, -223, -538, 212, 186, -84, -31, 15, 2, -1], [-2, -46, -3, 359, -175, -667, 376, 509, -283, -183, 98, 31, -16, -2, 1], [-5, -46, 2, 359, -176, -667, 376, 509, -283, -183, 98, 31, -16, -2, 1], [0, 57, 160, -306, -624, 397, 794, -193, -450, 40, 128, -3, -18, 0, 1], [-6, -67, -54, 564, -2, -1053, 204, 793, -210, -281, 84, 47, -15, -3, 1], [-4, -84, -147, 572, 413, -942, -386, 641, 157, -210, -29, 33, 2, -2, 0]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 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_6025_a();"
function MakeCharacter_6025_a()
    N := 6025;
    order := 1;
    char_gens := [2652, 2176];
    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_6025_a_Hecke(Kf)
    return MakeCharacter_6025_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, 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, 1, -1, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0], [1, 1, 0, 1, 0, -1, 0, 0, -1, 0, 1, 0, 1, 0, 0], [0, 0, 0, -1, -1, 0, 0, -1, 1, 0, 1, 1, 0, 0, -1], [0, 0, 0, 1, 0, -1, 1, 1, -1, 1, -1, 1, -1, -1, 0], [-2, 2, 0, 1, 1, 1, 0, 0, 0, -1, -1, 0, 0, -1, 2], [-3, -2, 1, 0, -1, 2, -1, 0, 1, -1, 1, 0, 1, 1, 0], [-1, 2, -1, 1, 1, -1, -1, 1, -1, 0, -2, 1, -1, 0, -1], [0, 1, 2, 0, 0, 0, 1, 0, -1, -2, 1, 0, -3, 0, -1], [-2, 1, 1, 1, 1, 0, 2, 0, -1, -3, -1, -1, -1, 0, 0], [2, 2, -1, 2, 2, 0, 1, 2, 0, -1, -3, 0, -2, -1, 2], [-2, 3, -1, 0, 2, -1, -1, 0, 2, 0, -1, -1, 1, -3, 2], [1, -3, 0, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 1, -1], [-4, 2, -1, 3, 0, 0, 0, 0, 0, -2, -2, 1, 0, -2, 2], [0, 1, 0, 2, -1, 0, 2, 0, -2, 2, 0, 1, 1, 0, 1], [3, -5, -1, -3, -1, 1, -2, 0, 2, 1, 2, 2, -1, 3, -4], [0, 0, 0, -3, 2, -1, 0, 1, 1, 1, 0, -1, 1, -1, 1], [1, -3, 1, -2, -2, -1, 0, 1, -1, -1, 4, -1, 0, 0, -1], [-3, 3, 1, 0, 1, -2, 0, 1, 0, 0, 1, -1, 0, -2, 1], [1, 1, -2, 3, 1, -1, 2, 0, -2, -1, -3, 0, 1, -1, 0], [-4, -4, 1, -3, -1, 3, -1, 1, 3, -4, 2, 0, -2, 3, -3], [1, 2, 3, 4, 0, 0, 3, 1, -3, -1, -2, 3, -1, -4, 0], [-5, 0, -1, -4, -1, 4, -1, -3, 3, -3, 1, 0, -1, 3, -1], [3, -4, -1, 1, -2, 0, -1, 4, 1, -1, 2, -1, -1, 3, 0], [-5, 0, 4, 0, 0, 4, 2, 2, 0, -3, -2, -1, -4, -2, 0], [-1, 2, 1, 0, 1, -2, 2, -3, 0, 1, 3, -3, 2, -1, 3], [-2, 0, -1, -2, 0, -1, -2, -3, 0, 0, 0, -1, 1, 2, 2], [2, 2, 1, 2, 0, 0, 0, -1, -2, 4, -4, 3, 0, -3, 0], [-5, 5, -2, 2, 1, 0, -1, 4, 3, -3, -2, -2, 0, -1, 3], [-5, -2, 3, -3, -1, 3, 3, -3, 0, -2, 0, 0, 0, 0, -1], [-8, 4, -3, 2, 3, -1, -2, 2, 0, -1, -2, -3, 2, 1, 3], [0, 3, -1, 2, 2, -4, 0, -2, -2, 1, -1, -1, 3, -3, 0], [1, 1, -4, 4, -2, 0, -1, 0, -3, 1, 1, 2, 4, 4, 0], [0, -3, 0, -5, -1, 0, -3, -1, 2, 0, 1, -1, 0, 4, -1], [7, 6, -4, 2, 7, -5, 1, 1, 0, 2, -4, -3, 3, -3, 5], [-4, -4, 0, -8, 1, 2, -3, -3, 1, 1, 0, -3, 0, 7, -4], [-4, -8, 5, -5, -5, 5, -2, 0, 1, -2, 5, 1, 0, 6, -6], [-4, 0, -3, -1, -4, 0, -3, -2, 1, 1, -2, 1, -1, 1, -1], [-2, -4, 1, -2, -1, 0, 0, -2, 1, -2, 2, 1, -1, 2, -2], [5, 5, -4, 4, 5, -2, 2, 4, 0, 1, -4, 3, -3, -2, 3], [1, 2, -5, 3, -5, -3, -3, -1, 0, 3, 3, 1, 7, 4, 2], [-6, -3, -2, -6, -1, 2, -2, -1, 3, -1, 2, -2, 1, 1, 0], [-8, -1, 3, -1, 0, 1, -2, 0, 0, -3, 2, -4, -2, 2, 4], [-3, -6, -1, -4, -2, 2, -2, 2, 1, -1, 4, 1, 0, 8, -3], [4, 3, -1, 0, 0, -3, -3, 0, -3, 3, -3, -2, 1, -1, 2], [-2, 1, -2, -1, 0, -2, -2, 2, 1, 1, 1, -4, 2, -1, 3], [-2, -3, 3, -4, -3, 0, -2, 0, -2, 3, -1, 3, 0, 2, -5], [-6, 1, 4, 0, 5, 4, 1, 0, 0, -2, -1, -3, 2, -1, 0], [1, -8, -1, 1, -2, 1, 0, 3, 0, 0, 1, 3, 0, 3, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -4, 5, -2, 0, 3, 0, -4, 0, 2, -4, 4, 2, 1, -1], [3, -2, -3, -1, -6, -1, -3, 2, 0, 1, -1, 5, -1, 4, -2], [-9, -5, 0, -7, -3, 3, -5, -2, 2, -3, 3, -6, 0, 8, 2], [-6, -5, 4, 0, -5, 3, -2, -1, 2, -4, 6, -3, 1, 4, 1], [-1, 0, 2, 1, -2, -3, -2, 3, -3, 6, -3, 9, -1, -3, -3], [-3, 0, 1, 1, -4, 0, -2, -1, 0, 0, 6, -1, 2, 6, -4], [2, 2, -5, -1, -1, -5, -3, -1, 2, 3, 3, -8, 9, 2, 6], [5, 4, 9, 2, 1, 1, 3, 0, -3, 2, -5, 1, -6, -7, 1], [3, 10, -6, 4, 3, -10, 0, 2, -3, 3, -5, -3, 3, -6, 6], [3, 9, 2, 5, 1, 0, 4, -3, -5, -1, -5, -2, -1, -5, 2], [2, -4, -5, -3, -5, 3, -2, -3, 0, 4, 2, 4, 1, 8, -3], [10, -3, -1, -1, -1, -2, 5, -4, -3, 5, -2, 6, -4, 0, -7], [1, 3, -2, 2, 1, -4, 4, 2, -3, 2, -3, 1, -3, -7, -2], [-6, 1, 6, -1, -2, -3, 1, -1, -2, -3, 0, 0, -7, -2, -1], [-1, -5, -2, 0, -2, -2, -7, 8, 3, 4, 5, 2, 1, 2, -2], [2, 1, -6, 4, 2, -4, 3, 2, 1, 2, 6, -1, 4, 1, 4], [2, 2, 2, -6, 2, 2, -4, -3, 3, 4, -7, 7, -3, -3, 0], [0, 3, 0, 2, 3, -5, 2, 0, 1, 5, -1, -1, -1, -6, 5], [-1, 2, 1, 6, 3, -1, 4, 2, 2, 3, -1, 0, -1, -12, 3], [1, -6, 2, 7, -8, 2, 1, -5, -2, -2, 6, -1, 5, 3, -1], [-3, -6, 1, -2, -7, 2, -3, -4, 2, -2, 2, -1, 7, 5, -1], [-14, 3, 3, 8, -1, 0, 5, 1, -1, -6, 4, -2, -1, -3, 1], [-8, 6, 0, 4, -2, -1, -5, -3, -1, 0, -4, -1, -1, -2, -3], [5, -1, -1, 3, 5, -1, 1, 2, -7, 3, -3, 1, 1, -2, 0], [-9, 0, 0, -5, 5, 2, -3, 0, 6, -6, 5, -7, -1, 3, 0], [4, 3, -2, 3, 2, -3, 3, -2, -1, 1, 4, -3, -1, -6, 0], [8, 2, -1, 6, 1, -3, 1, 4, 1, 4, 3, 6, 1, -6, -1], [-5, 1, -1, 3, 6, -1, 4, 2, 0, 2, -6, 1, -3, -5, 2], [-3, 6, -4, 0, 4, -5, -2, -3, 2, 1, 1, -8, 10, 3, 8], [4, -11, -4, -7, -4, 2, -6, -3, 2, 3, 5, 5, 5, 12, -7], [-2, 8, 2, 2, 2, 5, 2, -2, 0, -6, -4, -2, -6, 3, -1], [-5, -12, 6, -8, -10, 10, -2, -2, 0, 2, 6, 4, -3, 7, -10], [-1, -11, 8, -5, -3, 5, -4, 0, -2, 1, 7, 5, 1, 7, -9], [1, -2, 5, -2, -2, 8, 4, -1, 0, 1, 3, -1, -3, 0, -1], [-6, -9, 3, -3, -1, 5, -2, 0, 3, -4, 3, -4, 0, 11, -1], [-5, 0, -2, -2, -3, -4, -4, -2, 2, -1, 11, -4, 3, 5, -2], [10, -11, 1, -2, -8, 3, 1, -2, -8, 8, 2, -2, 4, 3, -5], [7, -5, 4, 2, -2, 4, 3, 0, -1, 5, -6, 8, -10, -7, -6], [-1, -10, -1, -5, 0, -1, 2, 2, -1, 4, 3, -4, 0, -1, -4], [4, 3, -1, -6, 4, -5, 3, 0, 1, 2, 3, -2, 2, -4, 5], [-2, 3, -3, -3, 2, -4, -3, 4, 2, 0, 0, 3, -7, 1, -4], [-3, 2, 3, 3, -2, -2, 4, -6, -4, 0, 1, 0, 6, 0, -1], [-1, -7, 3, -9, 4, 4, -2, 1, -1, 1, 0, -4, -7, 5, -2], [2, 12, -5, 8, 9, -2, 1, 1, 1, -2, -5, -1, -6, -7, 7], [-1, -11, 9, -4, 0, 6, 3, 4, -2, -2, 8, -6, 0, 5, -3], [11, 0, -9, -4, 4, 2, 0, 0, 3, 0, 2, -2, 2, -2, 5], [-4, -8, 2, -5, 2, 5, 0, -2, 5, -1, -2, 3, -3, 1, -6], [1, -4, 3, -4, 4, 3, 2, -7, 0, -4, 4, -3, -3, -2, -1], [0, 6, -6, 10, -2, -4, -4, 1, 0, 3, 2, -2, 10, -4, 8], [-15, -9, 7, -6, -4, 13, -6, -4, 8, -12, 6, -1, -2, 7, -4], [0, 1, -2, -9, 5, -2, 1, -4, -3, 0, -4, -6, -2, 2, 2], [-1, -6, 10, 4, 1, 6, 7, 3, -4, -2, -4, 8, -7, -6, -7], [11, 2, 1, 11, -2, -4, 7, -1, -1, 1, -2, -1, 6, -3, 4], [9, -1, -6, -1, 1, -4, -4, -1, 3, 5, -2, 6, 4, 2, -1], [2, 10, -3, 5, 7, -3, 4, 0, 1, 5, -6, 5, 0, -9, 6], [-2, -7, 5, 1, 0, 9, 5, -3, -1, -3, -2, -7, 1, 1, 1], [2, 8, -3, 10, 4, -3, 6, 2, 3, -2, -5, -1, -7, -12, 6], [6, 7, -5, 4, 10, -2, 0, 2, 3, 2, -10, -2, 3, -11, 12], [-8, 10, -4, 5, 10, 2, 1, 2, 5, -4, -5, 3, -2, -6, 4], [3, 2, -3, 9, -3, -4, 3, -3, -4, -1, 6, -6, 0, 1, 0], [-2, 0, -1, -3, 4, -4, 5, 1, 2, 4, 4, -9, 2, -3, 5], [-6, -10, 1, 4, -2, 4, -5, -3, 2, -2, 1, -1, 3, 5, -2], [-4, -3, 3, -4, -1, 0, 1, -4, -6, 6, 7, -7, 2, -2, -6], [7, 6, 4, -5, -3, 1, 0, -5, 0, -1, 3, -2, 4, -3, 0], [17, -12, -4, 6, -5, -7, 4, 8, -7, 12, -7, 11, -1, -3, -2], [-6, -1, -1, -2, 4, -4, 5, 5, 0, 7, 0, -1, -4, -1, 0], [1, -10, -9, -8, -4, -5, -9, -3, 5, 7, 10, 5, 6, 6, -6], [2, -4, 3, -2, -3, 1, 2, -2, -6, 2, 2, 3, -8, 1, -11], [0, -10, -8, -7, -3, -5, -2, 0, -2, 3, 8, -4, 7, 15, -5], [-5, 3, -3, -5, 4, 1, -4, -3, 9, -9, 3, -2, -3, 4, -1], [-2, -6, -8, 0, -6, -4, -2, 1, 0, 6, -3, 3, 3, 4, -3], [-6, 12, -7, 2, 1, -6, 1, -6, -5, 10, -6, -1, 3, -2, 4], [3, 4, -1, -1, 8, -5, 1, 0, -3, 1, -6, 3, -1, -8, 3], [8, 0, -1, 1, -5, 0, -1, 4, 3, 2, 4, 5, -3, 3, -3], [-3, 5, -4, -3, 4, -2, 0, 2, 2, -3, -6, -9, 0, -1, 8], [4, -1, 0, 5, -5, 2, -4, 6, 0, -5, 0, 6, -2, 4, -4], [15, -10, -8, -2, -4, -2, 1, 2, -5, 9, 11, 8, 1, 6, -8], [-11, 2, -2, 2, 4, -4, 1, -2, 0, -3, -4, -10, -1, -4, 5], [-5, -4, -5, 9, 4, -1, 2, 3, -3, -3, 0, 0, -1, 7, -1], [-6, -9, -1, -3, -10, 3, -10, 6, 7, -2, 10, 2, 1, 8, -7], [7, 2, -16, -1, 2, -10, -8, 3, 0, 11, -4, -5, 9, -1, 7], [9, 0, -4, 6, 0, -5, 3, 0, -8, 1, -1, 1, -2, -3, -7], [10, -4, 0, 2, 0, -8, 1, -4, -1, 6, 6, 1, 12, 1, -1], [-8, -10, 9, -2, -8, 6, -7, 4, -1, 0, 0, 8, -8, 11, -11], [15, 9, -2, 4, 13, -7, 7, -7, -5, 5, -9, -1, 1, -8, 8], [9, 8, 9, 2, 2, -2, 8, -2, -9, 1, -8, 7, -5, -7, 1], [10, -17, -2, -1, -10, 2, 1, -1, 4, 5, 5, 4, 0, 7, -5], [-8, -12, 8, -3, 0, 3, -6, -7, 6, 6, 0, 5, 7, -5, 2], [-3, 9, -3, -8, 2, -4, -2, 3, 5, 4, -2, -3, 7, 0, 8], [-7, 13, -5, 3, 0, 3, -5, 2, 5, -4, -9, 1, 6, 4, 9], [5, -17, -11, -4, -4, 0, -4, 6, 0, 7, 9, 2, 1, 3, -4], [14, -5, 0, 6, 2, 3, 5, 10, 0, 3, 0, 7, -2, -3, 7], [-1, 1, -5, -7, 2, -2, -7, -5, -2, 9, -11, -3, -3, 0, 0], [-8, 4, 1, 3, -3, 1, 2, 5, 3, -1, -4, 3, -3, -9, 1], [-5, 7, -4, 2, 15, -4, -4, 2, 11, -4, -12, -9, 2, -11, 13], [-11, 11, 2, 4, 6, -4, -2, 4, -1, -7, -9, -6, 5, -6, 13], [-11, -6, -3, -3, -3, -2, -8, 0, 8, -2, 4, 3, -3, 2, -3], [-7, -1, 6, 0, 3, 10, 7, 4, 0, -12, 4, -4, 1, 2, 4], [0, -3, 2, -8, -7, -4, -2, -2, -1, 3, 17, -2, 3, 10, -4], [10, 12, 9, 5, -3, -5, 0, 1, -6, 2, -8, 5, -1, -9, 5], [-14, -1, 4, 6, -7, 4, -3, -2, 6, -9, 10, 1, 11, 0, 6], [3, 9, -3, 7, 1, -1, 2, 2, 7, -2, -5, 6, 3, 0, 8], [-10, 8, 7, 2, -1, 2, 9, -7, -11, -4, -5, 2, -11, -4, -5], [-3, 17, -1, 4, 0, -7, 2, 3, 1, 0, -5, 0, -1, -17, 11], [17, 9, -16, 4, 11, -13, 3, 6, 3, 10, -6, 0, 6, -4, 10], [17, -11, 10, -4, -8, 3, 4, 6, 1, 5, 9, 4, -1, 2, -3], [3, 0, 0, -11, -5, 8, -3, -3, 0, -2, 2, 2, -9, 9, -7], [-6, -10, -5, -3, -5, 8, 1, 0, 6, -5, 8, -1, -2, 5, 0], [-10, -3, 3, 0, -4, 8, -2, 4, 2, 2, -8, 2, -5, 0, 0], [-5, 1, 6, 5, 8, 0, 0, 7, -1, -7, 1, 0, -12, -10, 0], [-11, 5, 8, 6, -2, 10, -1, 1, -4, -3, 1, -1, -6, -3, -4], [-5, 7, -3, 6, 7, -1, -2, 5, -2, -5, -6, -7, 6, 8, 7], [-17, -4, 12, -4, -7, -1, -4, -4, -1, 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9, -2, 6, -3, 6], [-1, 8, 3, 4, 0, 4, 6, -2, -10, 2, -1, 12, -12, -6, -11], [2, 6, 14, 10, 3, 8, 7, -2, 6, -8, -18, 1, -7, -6, 7], [-1, 11, -6, 18, 4, 0, 7, 2, -5, 10, -10, 15, 3, -16, 8], [12, 3, 2, 9, 3, -1, 1, 7, -13, 14, -12, 2, 10, -5, 7], [3, -1, -7, -14, 0, 5, -4, -1, 7, 1, -1, -15, -13, 2, 3], [4, 16, -16, 5, 9, -13, 2, 9, 5, 1, -8, -3, 3, -6, 11], [-35, -20, 12, -9, -12, 11, -3, 9, 11, -13, 10, -1, -11, 3, -13], [-4, 1, 13, -3, -2, 12, -1, -12, 5, 5, -9, 10, -2, -13, -3], [30, 7, -13, 0, -5, -15, -7, -3, -4, 16, 4, -9, 5, -1, 4], [10, 18, -4, 13, 3, -14, 13, 0, -2, -4, -1, 8, -13, -18, 5], [4, 7, 5, 12, 6, -8, 8, -3, -3, 4, -2, 0, -9, -17, 6], [-2, 0, -6, -7, -2, -9, -1, 5, 5, 8, -7, -14, 7, -9, 14], [0, -10, -4, 0, 0, 4, 1, -3, -4, -3, -12, -5, 10, 1, 12], [-20, -10, 4, -6, 3, 23, -7, -7, 7, -16, -8, 3, -8, 10, -5], [3, -5, 3, -6, -5, -4, 11, -8, -3, 6, 2, -6, -1, -16, -4], [4, -2, 2, 4, -6, -5, 1, 4, -5, 0, 17, -10, 4, 5, 6], [-3, -10, -6, -3, -8, 6, -6, 1, 12, 2, 0, 0, 2, 1, -5], [-27, 0, -1, -6, 4, 4, 2, 1, -1, -7, 1, 4, -7, 6, 3], [0, 14, 0, 5, -2, 7, 4, -7, -11, -1, -11, 12, -18, 1, -8], [18, 23, -17, 0, 12, -21, -8, -6, -4, 11, -8, -19, 12, -9, 20], [5, 3, -9, 18, 4, 2, 15, -3, -2, 6, -3, 5, 15, -10, 8], [-11, 7, 2, 17, -5, 0, -5, 0, -1, 4, -9, 7, 7, 7, -6], [-11, -9, 4, 17, 7, 8, 5, 5, -8, -3, -10, 3, -5, -14, 1], [-23, -11, 12, -22, 6, 6, -9, 7, 7, -10, 13, -8, -10, 14, -13], [16, -1, 9, 1, -8, -1, 2, -1, 12, -5, 10, 4, 7, -2, 6], [21, 12, -3, 14, 0, -16, 0, -2, -9, 6, 2, -6, 18, 4, 4], [-7, 2, -2, 22, -6, -7, -10, 3, 7, 0, -1, 5, 15, 7, 13], [-31, -11, 12, -18, 2, 7, -3, -2, 5, -3, 9, -21, 2, 9, -6], [0, -20, 8, -7, -7, 15, -3, 7, 6, -15, 11, 3, -3, 18, -15], [-9, 20, 1, 8, 5, 5, 3, 2, -6, -15, -12, -4, -22, -15, 9], [-4, -12, 17, 3, -9, 3, 2, 1, -6, -10, 8, 2, -12, 0, -14], [1, 6, -7, -4, 12, -5, 3, 4, 7, 6, -2, -16, -10, -9, 8], [8, 4, 2, 6, 11, -12, 6, 1, -7, 5, 1, -5, -12, -11, -2], [17, -11, 0, 7, 4, -3, 8, 8, -12, 6, -20, 8, -10, -10, -6], [-23, -1, 10, -1, -4, 15, 5, -7, 7, -19, 19, -12, -7, 4, 0], [6, 22, -7, 18, 9, -9, 6, 1, -4, 2, -10, -6, 11, -10, 5], [1, 20, -1, 8, 11, -13, 2, 6, 2, 3, -19, -1, -2, -18, 15], [-12, 2, 8, -2, -5, -2, -1, -3, -6, -8, 8, -10, -2, 15, -8], [15, 25, -19, 24, 2, -12, 9, 4, -9, -5, -14, 12, 1, -8, 5], [-1, 12, 13, 8, 2, -3, 12, 3, -9, 0, 2, 10, 1, -3, -4], [-10, 8, 3, -6, -3, -4, -2, -8, -6, 13, -12, 2, -14, -4, -17], [-23, -34, 13, -18, -15, 4, -6, 4, 4, -8, 20, -3, 3, 21, -20], [2, -12, 7, 6, 5, -3, 7, -5, -5, 0, 18, 9, 10, 2, -2], [3, 12, -10, 10, 15, 2, 14, 0, -8, -3, -15, 0, -8, -6, -2], [19, -9, 9, 14, -6, -2, 4, 4, -8, 9, -5, 14, -8, 2, -11], [-10, -25, 3, -13, -9, 12, -12, -1, 4, -12, 9, -7, -3, 17, -21], [-27, -4, -3, -8, 12, 0, -1, 0, 20, -3, 7, -10, -3, -12, -4], [-20, 12, 5, 0, 3, 2, 7, -1, -2, -25, -2, -11, -10, -17, 10], [-4, -7, 0, -8, -4, 4, -8, -8, 11, -6, 11, -15, 10, 6, -1], [11, 16, 3, -10, 7, -2, -5, 3, 3, 7, -3, -7, 10, 6, 17], [1, 17, -11, 10, 6, 4, 6, -16, 0, 9, -1, -8, 5, -7, 4], [-12, -3, 10, 13, -9, 1, 4, 17, -8, 6, -1, 10, -3, -5, 5], [-14, -6, -4, -10, 10, -1, -6, 10, 6, -8, -7, -27, 1, 9, 3], [4, -10, -3, -9, -6, -11, -11, 4, 11, 12, 14, -3, 16, 9, 2], [-22, -7, -6, -3, -5, 9, 6, -7, 10, 4, 11, -5, 14, -4, 5], [14, 1, -9, -11, 4, -7, 8, -7, -3, 5, 11, -6, 8, 2, 3], [-33, 5, -4, -8, 8, 1, -4, 2, 19, -5, 5, -10, -1, -7, 5], [-10, -13, 15, -8, -3, -1, 6, 1, -9, 8, -3, 6, -12, -3, -13], [5, 14, 5, 7, 24, -3, 1, 9, -7, -10, -10, -16, -9, -15, 16], [-9, 0, -4, 18, -3, -7, 1, 9, -5, -5, 9, -17, 10, -4, 19], [4, -6, 16, 7, -7, -2, 11, -7, -11, 8, -2, 14, -5, -5, -8], [6, -4, -3, 5, 2, -12, 14, 6, 3, 14, 4, 11, -2, -19, 9], [-16, 19, 0, 2, 5, 1, -2, 4, 5, -9, -17, -10, -4, 2, 5], [13, -2, 5, 13, 9, 5, 6, 4, -8, -7, -21, -1, -11, -17, 9], [-5, -33, -6, -17, -2, 2, -5, 3, 8, 4, 12, 5, 13, 12, -9], [-21, 12, 11, 7, 3, 10, 11, 5, 4, -18, -13, 2, -15, -16, 15], [-2, 12, -6, 6, -1, -7, 14, -6, -5, 8, 7, 7, 7, -15, 5], [-18, -12, -6, -11, -13, -4, -4, -3, -4, -3, 13, -17, 12, 15, -9], [2, -9, 24, 9, -8, 0, 11, -1, -1, -1, 7, 9, -6, -12, 2], [-32, -20, -6, -24, -9, 17, -13, 4, 18, -9, 3, 7, -12, 20, -5], [15, -2, 5, 8, -8, -10, -1, 11, 7, 12, 7, 11, 9, -5, 5], [3, -31, 0, 13, -4, 1, 6, 11, 0, 0, 6, 1, -14, -9, -1], [14, 14, -7, 9, -6, -22, 2, 3, 3, 13, 17, 10, 12, -4, 3], [1, 23, 0, 24, 13, -2, 16, -5, -1, -11, -8, -5, -8, -9, 9], [-28, -23, 21, 7, -6, 12, -8, 3, 9, -8, 11, 5, 3, 14, 3], [16, 13, 9, 16, 7, 2, 6, 0, -1, -7, -5, 6, -3, -7, -3], [16, 28, 0, -2, 3, -6, -2, 5, -13, 14, -18, 14, -6, -7, 2], [-12, 0, 4, 21, 6, 6, 15, -2, -1, 2, -6, 8, -2, -13, 4], [19, -17, -2, -10, -10, 4, 6, -5, 2, 14, 20, 11, -7, 3, -24], [9, -11, -3, -10, -6, 12, -9, 8, 1, 1, -8, 5, -15, 14, -5], [11, 2, 2, 8, -12, -15, 1, 1, -9, 14, -5, 19, 2, -17, 6], [-18, -16, -14, -15, -2, 14, -1, -5, 18, -9, 22, -3, 7, 10, -4], [10, 6, 2, -3, 2, 3, 6, -9, 4, -10, 12, 3, 5, 2, 0], [-1, 26, -21, 0, -3, -9, -5, -5, -1, 3, -2, -18, 8, 4, 9], [-37, -1, 25, 21, 5, 14, 12, 14, 1, -12, -10, 9, -8, -9, -5], [-2, 11, -21, -1, 11, -11, -8, -4, 1, -7, 5, -24, 27, 2, 18], [-3, -8, 14, -23, 1, 8, 5, -18, 5, -5, 0, 6, -7, 8, -7], [5, 13, -4, 20, 4, -15, 18, 5, -12, 7, 0, 1, 9, -10, 0], [13, -12, -12, 10, 1, -21, -6, 5, -3, 18, 13, -15, 29, 0, 12], [4, 0, 0, 11, 10, -3, 7, 11, -6, 9, -14, -4, -8, -6, 3], [-5, 0, 4, 12, -1, 2, 5, 4, 1, 0, -17, 23, 4, 9, -8], [20, -5, -16, -11, -12, -13, -6, -1, 2, 12, 30, -7, 6, 6, -14], [-6, 5, 10, -10, -2, 3, 4, -6, -4, 1, -12, 1, 1, -3, 3], [-7, -2, -4, -12, 18, 1, 5, 4, 8, 5, -14, -5, -5, -15, 11], [-29, -12, 14, 3, -2, 9, -11, -5, 9, -17, 13, -16, 9, 14, 10], [0, -14, 22, -3, -9, 7, 1, 1, -9, 0, 10, 20, -1, 8, -13], [-4, -15, -4, -8, -8, 6, -11, 9, 5, -6, 16, 8, 6, 15, -22], [21, 3, -19, -6, 3, -12, -10, 14, 2, 5, 1, 5, -8, -2, -1], [3, 11, -5, 1, -1, 2, -2, 14, 0, -7, 14, -8, 9, 10, 3], [-13, -24, 5, -10, -5, 16, 5, 3, 16, -4, 23, 16, -6, 16, -24], [6, -8, 8, -1, -5, 4, -3, 0, -4, 15, -8, 14, 0, 0, -12], [-8, -32, 10, -14, -15, 10, -9, -2, 2, 6, 14, 28, 2, 27, -23], [-4, -24, -5, -4, -4, 3, -7, 9, -2, -1, 1, -13, -10, 3, -11], [41, -2, -15, 13, 6, -13, 3, 5, -5, -8, -2, -2, 7, -6, 7], [-23, -25, 19, -4, -2, 20, 7, -4, 4, -16, 4, 4, 2, 18, -10], [-14, -4, -2, -6, -14, 14, -2, -7, -1, -6, 13, 24, 3, 17, -10], [22, 21, 9, 16, 10, -10, 13, 17, -7, -6, -28, 2, -6, -12, 17], [26, 39, 5, 4, 26, -16, 8, -3, -21, 3, -29, -7, -14, -13, 9], [26, 3, -8, 15, 15, -16, 15, 8, -9, 10, -17, 13, -13, -21, -8], [3, -5, -12, 4, -4, -2, 2, -6, -9, 16, -13, -20, 5, -9, 10], [-8, -36, 8, -24, -16, 7, -7, -12, 12, 11, 17, 15, 4, 16, -20], [15, 2, 6, 11, -12, 4, 8, -1, -9, -8, -3, 13, -5, 9, -12], [20, -16, -3, -3, -9, -12, -14, -3, -16, 18, 4, 1, 18, 15, -4], [24, -15, 9, -17, -6, 11, 7, 8, 2, -2, 1, 12, -4, 2, -4], [-47, 1, 0, -11, 5, 15, -10, -11, 19, -31, 17, -15, 3, 10, 3], [-11, -6, -11, -1, 3, -9, -9, 12, 0, -5, -11, -2, -13, 3, 10], [-8, -25, 0, 4, 3, 0, 7, 11, 12, 1, -1, 4, -5, 3, -1], [-18, 1, 2, -10, -3, 6, 3, -6, -7, -1, 2, 2, 9, 9, -5], [1, -9, 4, 4, -3, 8, 0, 6, -4, 12, -8, 4, -2, 12, 1], [-7, 25, 12, -1, 2, -5, -8, -5, -3, -5, -9, -5, 0, -13, 3], [12, -8, 8, -4, -2, -8, 8, -14, -1, 23, 19, 3, 13, -12, -2], [-4, -2, 9, 13, -5, 14, 8, 7, -12, -19, 4, 15, -14, -1, -17], [-40, 16, -5, 7, 11, -2, 7, -3, 11, -9, -7, 1, -5, -19, 24], [5, 6, 20, 2, 9, -6, 5, 13, -13, 7, -28, 11, -25, -17, -6], [-27, -1, 8, -11, -9, -2, -9, -8, 19, -1, 7, 2, 7, 3, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6025_2_a_i();".
// 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_6025_2_a_i(:prec:=15)
    chi := MakeCharacter_6025_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_6025_2_a_i();".
// 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_6025_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -17, 57, 89, -394, -297, 699, 437, -519, -296, 184, 99, -31, -16, 2, 1]>,<3,R![-191, 447, 721, -2131, -611, 3617, -499, -2749, 949, 970, -484, -133, 100, -1, -7, 1]>],Snew);
    return Vf;
end function;