
// Make newform 3525.2.a.bf in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_bf();".
// 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_3525_2_a_bf();".
// 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 := [-4, -30, 6, 101, -21, -91, 25, 29, -9, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [3, 3, -5, -1, 1, 0, 0, 0, 0, 0], [-12, -40, -3, 110, 43, -82, -23, 22, 3, -2], [4, 40, 19, -118, -47, 84, 23, -22, -3, 2], [6, 54, 11, -122, -45, 84, 23, -22, -3, 2], [8, 29, -9, -73, -9, 49, 7, -12, -1, 1], [2, 48, 63, -155, -105, 121, 43, -33, -5, 3], [-10, -78, -20, 201, 66, -139, -32, 35, 4, -3]];
    Rf_basisdens := [1, 1, 1, 1, 2, 2, 2, 1, 2, 2];
    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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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], [1, 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], [-2, 1, 0, -1, -1, 0, 0, 0, 0, 1], [-1, 1, 0, 1, -1, 0, -1, 0, -1, -1], [-1, 1, 1, 0, -1, 1, -1, 0, 1, 0], [-3, 0, -2, 2, 1, 0, 0, -1, -1, -2], [-1, 1, -1, -1, 0, 0, 1, 0, 1, 2], [0, -1, 0, -1, 0, -1, 1, 0, 3, 1], [-3, 0, 0, 0, 0, 0, 1, 1, -1, 1], [-1, -1, 0, 1, 1, 0, 1, 2, -1, 0], [-2, -1, -1, 1, 1, 0, -1, -1, 0, -1], [3, -1, 1, 1, -1, 0, 1, -2, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 1, -1, -1, -2], [-2, -4, 0, 0, 0, 1, 1, 0, -1, -1], [-2, -1, 0, -3, 1, -1, 0, 2, 1, -1], [1, -1, 1, -3, 0, -3, 3, 0, 0, 2], [-4, 1, 0, 2, -1, 2, -1, 2, -1, -1], [1, 0, -1, -1, 2, 1, 1, 1, 1, -2], [-2, 1, 2, 1, 0, -1, -4, 1, -1, 1], [-2, 1, 2, -2, -1, -1, -2, 2, -2, 0], [2, -1, 1, 1, -1, 2, 0, -1, 0, 1], [2, -1, 2, -2, 1, -2, -1, 5, -1, 0], [-1, -2, 2, 0, 1, -1, 1, -2, 0, -3], [1, 2, -4, -1, 1, 0, 2, -1, 0, 0], [0, 0, -1, -1, -2, 0, 3, -1, 1, 3], [0, -3, 3, -1, -2, -1, 0, -2, 1, 1], [-6, 3, -6, -1, 2, -2, 3, -4, 2, 1], [1, -3, -3, -1, 2, -1, 1, -3, 0, 0], [-7, 0, 0, -4, 3, -3, 3, 2, 2, 3], [-7, -1, 0, 2, -2, 0, -3, 1, 0, 3], [-5, -2, 1, -1, 0, -2, 2, -1, 0, -3], [2, -3, 2, 0, -2, 4, -2, -5, 1, 0], [0, -1, -1, 1, 1, 3, 1, -6, -1, -2], [0, 3, 4, 0, -7, 2, -5, -1, -1, 2], [-3, 0, -5, 3, 4, -1, 1, -2, -5, -4], [0, -3, -1, 0, 2, 0, -1, -1, 1, -2], [8, 0, 8, -1, -5, 2, -2, -3, 5, 0], [-8, 5, 2, 0, -3, 0, -5, 3, -1, 4], [-9, 2, -6, 6, 4, 2, -3, 1, -5, -5], [2, 2, -1, 1, 0, 3, 2, -3, 2, -1], [4, 0, 0, -2, -2, 2, -2, -1, 0, 0], [-3, 1, 1, 0, -1, -3, -3, 4, -1, 0], [-7, 5, 0, -4, 0, -1, 2, 1, -1, 0], [-6, -2, 3, -5, 3, -4, 2, 4, 3, 3], [-2, 2, 2, 0, -1, 1, -8, 4, 1, 2], [-4, -2, 3, -5, 2, 0, 0, 3, -3, 1], [4, 0, -1, -1, -3, 0, 4, -6, 3, 3], [-7, -3, -1, 5, 0, 4, -2, 2, -5, -4], [-8, -2, -3, -1, 3, -4, 2, 2, 1, 3], [3, 0, 3, -2, 1, -3, 0, 0, -2, -3], [-8, 3, 5, 7, -7, 5, -7, -3, -3, -2], [-9, 4, 1, 10, -2, 2, -5, 2, -1, -1], [3, -1, 3, 3, -2, 3, 1, -4, 2, 0], [2, -1, 6, -4, -2, -2, 2, 5, 1, 2], [-11, 3, -1, -3, -2, -2, -4, 4, -1, 4], [9, -6, -8, -1, 2, 2, 4, -3, 2, -2], [4, 8, 5, 4, -4, 6, -5, -1, 2, 0], [2, -3, -2, 4, 3, -4, 3, 1, 1, 0], [-4, -6, -1, 7, 2, 2, -1, 3, 1, 1], [9, -7, -3, 8, 1, 3, -2, -2, -3, -7], [-4, 2, 3, -4, 1, 1, -4, 1, -1, 0], [3, 4, -4, 3, -1, 2, -1, -4, -4, -5], [4, -3, -4, -2, 2, -1, 9, -6, 2, 0], [-1, 1, 6, 4, -6, 2, -3, -1, 0, 3], [5, 0, 3, 8, -3, 7, -5, -3, -2, 1], [5, 0, -5, -1, 5, 2, 3, -1, 0, 0], [-12, 2, -7, 3, 1, 0, 0, -2, -7, -3], [-3, -4, 5, -1, 1, -2, -5, 4, -4, -2], [0, -7, 0, -4, 1, -6, 3, -1, 1, 6], [-6, 9, 4, -10, -3, -1, 2, 2, 2, 6], [6, -4, 2, 6, -1, 1, -4, -2, -1, -4], [-5, 4, -6, -5, 4, -5, 7, 2, 3, 2], [-5, -1, 5, 1, -6, 3, -3, -2, 0, 8], [-10, 10, 4, 11, -1, 7, -12, 3, -6, -1], [2, -5, -3, -3, 2, -3, 0, 1, 1, 1], [-4, -2, 6, -2, 3, -3, -6, 7, -3, 0], [-4, -7, -1, 1, 4, -1, 2, 4, -5, -1], [-4, 1, -6, -2, 5, -6, 5, 3, 3, 0], [5, -3, -3, 7, 0, 5, 1, -3, -4, -4], [-1, 8, 1, -1, -4, 4, -2, -4, 6, 1], [-5, -4, -6, 7, 1, 2, -3, 1, 2, 3], [-2, -2, -3, -4, 2, -2, 6, 2, 4, -2], [-6, -3, 4, -7, -2, -4, 0, 4, 6, 5], [2, 3, -8, 1, -2, 3, 1, -6, 3, -3], [8, 0, -1, -9, 1, -5, 5, 2, 6, 9], [6, -1, -6, 2, 3, 4, 3, -1, 3, -2], [14, -13, -2, -10, 12, -8, 14, 2, 1, -2], [5, -2, -2, -2, 12, -5, 3, 1, 0, -5], [-8, 6, -1, 5, -3, 1, -3, -2, -6, -6], [3, -3, 0, -1, 0, -2, 1, 4, 5, 8], [8, -7, -4, 0, 2, 5, 3, -5, 2, -3], [-3, -7, 5, -7, 1, 1, 0, 1, 4, 1], [-8, 10, 5, 5, -8, 6, -10, -3, 1, 7], [-4, -2, -2, 5, 1, 3, -6, 5, -6, -9], [-1, 4, -7, -7, 0, -3, 11, 0, 3, 4], [6, 0, 0, -6, 5, -6, 8, 4, 2, 0], [8, 7, 2, 0, -5, 1, -3, 3, 4, 2], [8, 1, -4, -4, 7, -3, 4, 4, 0, -6], [-12, 1, 6, 4, -4, 3, -5, 4, 0, 2], [-3, -1, 2, 2, 0, -4, -3, 3, 0, 5], [2, 0, -5, 1, 3, 1, 3, -4, -2, 3], [-3, 2, 3, -4, 2, 3, -4, 3, 2, 3], [-8, 1, 1, 4, 4, 0, -6, 8, -5, 3], [11, 3, 4, -8, 0, -4, 3, -3, 0, -3], [-5, -6, -10, -1, 0, -2, 8, -5, 0, 0], [-8, 0, 5, -1, -5, -3, 1, 10, 0, 7], [7, 6, 1, -5, 1, 1, 3, -3, 1, 0], [2, -1, 10, 2, -10, 8, -6, -1, -1, 2], [19, 1, -2, -3, 3, 0, 5, 0, 1, 0], [-11, 9, 3, -3, -2, -3, 1, -6, 0, 2], [-2, 5, -2, 0, 2, 5, -1, -2, 4, 0], [-4, 4, -7, -5, 5, -8, 10, 0, 3, 3], [24, -1, -2, -8, 3, -6, 7, -1, 1, -2], [10, 3, 8, 4, -10, 6, -7, -3, -3, -4], [15, -1, 7, 1, -4, 5, 1, -5, 4, 4], [-5, 2, -7, 6, 4, 0, 1, 4, 1, -6], [-4, 5, -1, 5, 7, -6, 0, 1, -4, -3], [-6, 0, -9, 1, 5, 0, -1, -4, -7, -5], [-4, -1, 5, 0, -2, -2, -2, 3, 5, 9], [-5, -4, 5, 1, 0, -3, -7, 4, 1, -4], [9, 3, -1, -1, 0, 1, -1, -9, 4, 2], [-3, -4, -4, 2, -1, 8, -2, -1, -1, 2], [-2, 10, -7, 7, 2, -4, -6, -2, -5, -3], [-8, -2, 9, 9, -4, 2, -7, 3, -7, 3], [1, -5, -11, 7, -3, -3, 0, -4, -2, -1], [1, -6, 1, 5, -2, -1, 1, -8, 3, -6], [-5, 7, -10, 1, 7, -4, -1, 0, -5, -6], [-4, -2, -11, 3, 8, -4, 7, -3, -5, -3], [-8, -1, -4, -3, 4, 2, -2, 0, 0, -5], [-4, 0, 9, 1, -2, -4, 1, 12, -1, 2], [18, -6, 6, -6, 2, -2, 8, -1, 4, -2], [-3, 3, -7, 11, 1, 2, -1, 0, -3, -8], [13, -20, -5, -5, 4, -3, 13, -10, 7, -4], [14, 6, 6, 2, -4, 6, 4, -7, 4, -4], [6, 16, -2, 4, -1, -1, -6, 3, -5, -6], [0, 0, -2, 5, 2, 5, -4, 4, -2, -3], [6, 3, 16, -4, -3, 2, -7, 5, 7, 4], [-5, -8, 2, -2, 4, -5, 6, 3, 2, 11], [-12, 7, -9, 1, 5, -4, 7, 1, 2, 9], [12, -5, -10, 2, 5, 2, 7, -9, -3, -4], [4, -3, -2, -4, -5, 1, 4, -1, 6, 2], [-7, -1, -3, -7, 0, -9, 5, 5, -2, -2], [10, 12, 11, -3, -5, 9, -11, 4, 4, 9], [-2, 4, 8, 6, -2, 6, -10, 8, -10, 6], [-11, -1, 4, 11, 4, 11, -7, 3, -4, 0], [-2, -3, 5, 5, -3, 9, -7, -8, 1, -4], [0, 11, 2, -8, 1, -9, 4, 6, 0, 2], [0, 4, -1, -2, -4, -4, 0, 4, 6, 7], [-8, 2, 5, 2, -8, 8, -7, 9, 4, 6], [-1, 9, -10, 9, 0, 2, -3, -8, -1, -6], [3, -3, 0, 4, 0, -4, -5, -5, -6, -11], [7, -7, -1, -2, 7, -5, 4, -6, 1, -11], [3, 12, -2, -8, -2, -8, 3, 1, 5, 1], [1, 2, 6, -12, -3, -2, 4, 11, -3, 8], [5, -5, -1, 5, 5, 2, 9, -10, 3, -8], [4, -4, -6, -8, 5, -11, 14, -6, -1, -8], [5, 13, -3, -5, -10, 1, 3, -5, 0, 8], [-3, -3, -5, -7, 0, -3, 15, -6, 2, -10], [-6, 0, -1, -11, 2, -8, 11, 1, 9, 5], [-10, 5, 12, 6, -9, 2, -9, 13, -3, 2], [-6, 0, 3, 5, -1, 3, -9, 10, -6, 3], [0, -8, 0, 1, -1, 8, -6, -9, 5, 0], [7, -2, -8, 1, 6, 0, 5, -9, -6, -12], [18, 1, -6, -8, 10, -3, 13, 0, 4, 0], [-6, 12, -11, 2, 8, -2, 4, -8, 2, -5], [14, -3, -12, -10, 9, 3, 16, -4, 8, 0], [0, 3, -18, -11, 10, -6, 9, 0, 2, 1], [-15, -3, 3, -15, 1, -15, 6, 8, 6, 9], [2, -12, 0, 1, 1, 0, -5, 4, 3, 1], [-8, -4, -8, -2, 2, -6, 8, -8, -4, 0], [4, 5, 11, 4, -9, 15, -8, -6, 2, 7], [17, -7, -5, 7, 7, -1, -2, 1, -4, -11], [-12, 10, 5, 10, -10, 4, -11, -3, -4, -2], [-20, -5, 0, 6, 2, -1, -9, 10, -4, -6], [7, 6, 5, -9, -7, -6, 3, -3, -2, -2], [-11, 21, 5, -3, 1, 2, -9, 6, -5, -2], [-10, 4, 6, 2, -12, 2, -12, 4, 0, 4], [4, 10, 1, 1, -1, -3, -1, 10, -8, -1], [0, -3, 8, -2, -1, 8, -5, -3, 13, 4], [13, -15, -4, 11, 5, 0, -3, -6, -15, -18], [-8, 2, 8, 11, -9, 4, -6, 0, -5, 4], [-5, 14, -8, -7, -2, -2, 9, -1, 4, 4], [19, -2, 4, -4, -4, 8, 7, -13, 7, -5], [4, 3, 21, -3, -8, -1, -6, 7, 5, 7], [-2, 1, -14, 0, 8, -8, 11, 5, 1, 8], [1, -1, 4, -7, -3, -4, -4, 15, 3, 8], [4, -3, 2, 10, -1, 6, -1, -17, 7, -6], [15, -15, -11, 0, 6, 1, 10, -2, 1, -5], [-2, 3, -5, 9, 6, 3, -1, 7, -7, 1], [-7, 6, -1, -9, -6, -8, 8, 3, 0, 9], [-3, 10, -9, -5, 10, 1, 5, 0, 5, 2], [5, -1, 1, 3, -10, 7, -5, -4, -4, 4], [-1, 4, -7, 19, 1, 8, -9, -8, -4, -14], [-5, -11, -2, 0, -1, -7, 9, 1, -3, 7], [2, 14, 5, 4, -7, 7, -8, -1, 1, 3], [-6, -2, -10, 17, 7, 8, -8, 1, -7, -8], [16, -4, 10, 0, -8, 2, -8, 9, 4, 0], [-9, 7, -14, -6, 1, -7, 13, -11, 1, 5], [-11, 1, 10, 11, -14, 2, -14, 2, -3, 0], [5, 1, -12, -11, 7, -4, 13, -1, 11, 3], [-17, 12, -9, 7, -7, -5, -7, 1, 1, 12], [-13, 4, 11, -9, 6, -8, 0, 8, 2, 5], [13, -2, 5, -11, -7, -2, 5, -6, 4, 12], [-11, 1, -3, 9, -3, 3, -14, 1, -10, -5], [-6, -2, 6, -6, -6, -2, 6, -2, 4, 20], [6, 1, -8, -8, 11, -4, 7, 1, -1, 0], [-6, 3, 7, 3, -10, 3, -6, 10, 5, 5], [29, 16, 1, -6, -7, 1, 11, -15, 10, 7], [17, -2, 18, -6, -10, 4, 3, -7, 15, 9], [18, -1, 23, -3, -20, 5, -6, -5, 9, 3], [-28, 5, 7, -14, 0, -8, 3, 9, 3, 16], [0, 5, -2, -2, -3, 4, -1, -4, -3, 9], [-7, 6, -9, 3, 3, 2, 1, -13, 4, -8], [-2, -12, -2, -8, -4, -10, 4, 1, -2, 8], [10, -6, 11, -3, 0, 0, -4, 0, 5, -1], [-9, 1, -13, 1, 16, -1, 7, 11, 0, -2], [-17, 8, -9, 0, 0, 0, -7, 2, 3, 9], [3, 4, -6, 4, 2, 1, 0, -5, -12, -17], [-15, 2, 11, -9, -12, 3, -1, 2, 1, 6], [-8, 10, 12, 2, -14, 2, -14, 8, -8, -8], [-6, -1, 6, 6, -6, 12, -8, -3, -9, -4], [4, 5, 14, -4, -23, 3, -10, 0, -2, 6], [-9, -4, 1, -1, -1, -2, -3, 11, -4, -4], [-16, 5, -4, 5, 10, 6, -9, 2, -8, -1], [2, -8, 3, 18, -5, 3, -10, -1, -1, -10], [2, 2, -17, 3, -5, -5, 1, 0, 2, 7], [2, -23, 14, -2, 3, 0, -1, 1, -1, -6], [5, -9, 0, -5, 11, -4, 3, 11, 7, -5], [5, -6, -1, 10, -2, 5, 2, -11, -2, -13], [16, -24, -3, 0, 6, 0, 6, -16, 6, -14], [-5, 1, -8, 8, 4, -10, 3, -2, -8, -9], [-8, -20, -4, 2, 9, -7, 6, -2, 3, -4], [-20, 19, -8, 10, -2, 5, -1, 4, -6, -6], [-4, 0, 3, 9, -12, 4, -15, -1, 1, 3], [8, -14, 2, 9, 3, 4, 6, -3, -5, -4], [0, 5, -5, 11, 0, 8, -13, 2, -4, 0], [1, 8, -10, 2, 8, 5, 0, -3, -4, -18], [0, 6, -8, -11, -2, -7, 9, 4, -2, 7], [-7, 9, -7, 6, -6, 6, 9, -9, -2, -9], [5, 0, 3, -3, 0, 3, 11, -3, -1, -2], [-10, 6, -12, 4, 6, 2, -8, -4, -8, -16], [-14, 2, -8, 2, 16, -8, 2, -3, -6, -14], [-14, 8, -11, -5, -6, -1, -2, 5, 6, 13], [-17, -3, 3, -4, 1, 1, 16, 0, 5, 11], [-24, 22, -15, 0, -1, -3, 0, -5, -5, -6], [0, -9, -6, -9, -4, 1, 10, -11, 7, 15], [10, -18, 18, 6, -2, 4, -8, 13, -4, 6], [7, -1, 2, -17, 6, -13, 7, 6, -2, -3], [-2, -2, -7, 3, 2, -11, 2, -15, 2, -13], [0, -9, -10, -2, -1, -7, 12, 0, 0, 2], [-1, -2, 7, -9, 5, -8, 11, 5, 8, 6], [1, 25, 4, 1, -16, 9, -15, 3, -2, 6], [-20, 0, -5, 7, -1, -3, 1, -2, 2, 0], [-15, 11, -10, -8, 15, -7, 9, 3, 5, 11], [-1, 5, -7, -1, 3, 0, 11, 1, -1, -6], [-2, 1, 0, 12, 0, 5, -3, -2, 0, 4], [6, -3, 24, 14, -7, 14, -13, -1, -3, -4], [26, -15, 9, -7, 5, 1, 4, -7, -1, -5], [-47, 12, 5, 0, -4, 2, -6, 0, -3, -3], [4, 5, 13, 18, -9, 17, -11, 1, -14, -10], [1, 3, 6, -5, 1, 2, 1, -2, 5, 4], [10, 10, -6, 7, -4, -1, -6, -8, -2, 3], [-22, 2, 5, 2, 4, -14, -2, 18, -6, 6], [-4, 4, 19, -19, -6, -13, 2, 15, -2, 11], [-26, -2, -9, 4, 8, -2, 3, 11, -10, 10], [-5, 0, 1, -7, 4, 3, 1, -3, -3, -22], [15, -16, 1, -4, 12, 0, 5, 0, 3, -11], [15, -10, -9, 13, 3, 1, 1, 3, -1, 10], [4, 5, 11, -3, -3, -1, -20, 17, 3, 1], [-8, 16, 13, 3, -11, 13, -7, -2, 0, -3], [14, -9, 9, 3, 4, -3, 10, -6, 7, -1], [-10, 2, 11, 20, 7, 11, -18, 6, -5, -11], [-6, -2, 16, 4, -13, 1, -4, 2, -11, -2], [-17, 8, 5, -13, -1, -8, -1, 2, 6, 2], [9, 8, 5, 0, -11, -1, 2, 0, -4, -3], [9, 13, -6, -8, -3, -1, 15, -5, 9, 9], [-14, -10, 0, 4, 0, 7, 7, -12, -5, -5], [12, -3, -6, -2, 5, -3, 2, -6, -6, -8], [2, -20, 15, -1, -4, 4, -1, 2, 11, 12], [-15, 0, 7, 7, 5, 0, -9, -5, -10, -16], [-4, 10, 9, 5, -12, 2, -13, 3, 5, 19], [27, -4, 9, -4, -4, 1, 1, 3, 6, 13], [14, -5, -9, -12, 10, -13, 15, 0, 2, -4], [-6, -3, 5, -8, 6, -12, 2, 6, -3, -3], [-13, -4, 8, -10, 8, 5, -2, 9, 12, 1], [1, 11, 7, 5, -11, 11, -6, -3, 0, 5], [14, -7, 1, 13, 11, 10, 2, -15, 2, -13], [9, -17, -1, -1, 14, -1, 7, 5, -8, 2], [-17, 6, -5, -13, 9, -10, 7, 4, 6, 8], [-31, 1, -18, 7, 0, 3, 3, -1, -2, -8], [-14, 0, 7, -17, -1, -5, 1, 2, 6, 6], [16, -6, 11, -1, 1, -5, 3, -2, 0, 7], [2, 20, 9, 5, -2, 2, -6, -2, 5, -5], [-15, 15, -8, -11, 6, -14, 12, 0, -3, 2], [22, -15, 6, -10, -4, -3, 5, 6, 6, 16], [-4, 0, 13, 2, -3, -7, -12, 11, -9, -4], [-10, 13, 14, 0, -10, 10, -6, -8, 11, 0], [11, -6, 10, -5, 3, 4, -3, 7, 10, 15], [29, -1, -14, -5, 0, 11, 5, -25, 4, -8], [14, -16, 16, 11, 2, 10, -17, -1, -1, -5], [17, 5, 4, 10, -14, 14, -11, -1, -4, -3], [-10, 21, 0, 0, -6, 9, -11, -2, 0, 0], [13, -5, 4, 15, -1, 16, -17, -12, -3, -15], [10, -18, 18, 8, 2, 2, -6, -3, 4, -2], [19, -8, 10, 0, -7, -5, 0, 0, -8, -11], [14, 0, -5, 1, -1, -3, -3, 0, 2, 6], [16, 6, -5, -9, -3, -11, 13, -10, 6, 1], [19, -2, -9, 5, 1, 8, -1, -22, -2, -10], [4, -4, -14, 8, 17, -1, 8, 3, -5, -8], [-34, 13, 11, 22, -5, 8, -18, 2, -9, 0], [-19, -19, 7, 3, -5, 2, -13, 7, 3, -2], [3, -9, -15, 10, 6, -7, 7, 4, -9, -11], [7, 8, -3, -4, 9, -9, 12, -2, -6, -5], [4, -9, 14, -2, 3, 2, -9, 11, 1, -16], [-18, -3, -1, -12, 4, -17, 13, 10, 0, 8], [-9, 4, -2, 5, -6, 1, -1, 4, -15, 7], [3, 7, -22, -6, 4, -2, 5, 12, -4, 1], [-16, -20, -6, 3, 4, 0, 3, 5, -9, 3], [-4, 14, -7, -14, 5, 3, 4, 5, 13, 13], [32, -18, -10, 2, 6, -8, 6, 4, -2, 16], [-22, 5, -7, -21, 9, -9, 14, 5, -1, 5], [-30, -2, 5, -13, 6, -8, 5, 15, 1, -1], [22, 15, -3, -5, 2, -1, 6, -10, 5, -1], [-13, 20, 6, -4, -10, 3, -12, 9, -8, 1], [-14, 5, -2, 8, 14, 4, -8, 7, -13, -14], [-23, 0, 3, -1, -2, -5, -11, 4, -1, -6], [-39, 6, -19, 19, -3, -6, -9, -2, -10, -4], [1, -21, 8, 6, -3, 14, -1, -4, 6, 11], [16, 11, 22, 9, -12, 17, -26, 6, 1, -2], [3, 19, -22, -2, 8, -3, 12, 1, 3, 6], [17, 14, 1, -13, 1, -8, 5, 3, 8, 6], [-37, -3, 8, 6, -12, 4, -13, 9, -4, -3], [7, -6, 4, -19, -2, 0, 1, -5, 10, 4], [1, 12, -15, -1, 5, 4, -1, 3, 0, -6], [-19, 11, 11, -4, -15, -1, -5, 3, -7, 4], [5, -36, -21, 16, 15, -1, 10, -8, -12, -23], [-7, 0, 9, 3, -4, -1, -23, 19, -7, 0], [-1, 9, -5, -14, 10, -11, 10, 16, 3, -1], [-16, -3, 1, -16, 4, -6, 8, 7, 9, -2], [7, -13, -10, -2, 1, -1, 19, -9, 3, 1], [11, 11, -19, -6, 0, 2, 5, -9, -6, -3], [-13, 10, 9, 3, -6, 11, -7, 6, 7, 0], [-1, 1, -3, 13, -4, 6, -4, -1, -13, 5], [6, 10, 17, -1, -5, -4, -6, 14, -17, 5], [-3, -8, 1, -7, -2, -3, 11, 8, -5, 0], [-10, 2, -6, 20, 3, 12, -17, 9, -6, -1], [0, 7, -14, -5, 16, -1, 8, 13, -5, 0], [-17, -11, 17, 23, -10, 16, -22, 10, -1, 8], [17, -10, -19, 2, 17, -9, 9, 1, 6, 9], [38, -9, 14, 4, -11, 10, -11, -7, 3, -8], [-21, 16, 9, -5, -9, 10, -11, 12, 0, 10], [39, -16, -7, -1, 11, -4, 7, 3, 6, -6], [-1, -4, -18, -16, 20, -9, 14, -3, 0, -19], [-24, 0, 5, -5, -8, 3, -2, 9, 0, 21], [3, -2, 2, 2, -12, 3, -12, -7, -2, 8], [13, -4, 11, 20, -12, 10, -5, -6, -5, -7], [-2, -7, 2, 14, -8, -1, 0, -4, -10, -2], [-29, -5, 2, -7, -5, 2, -3, -1, 1, 5], [-10, -7, -22, -5, 10, -5, 5, -4, 3, -5], [8, 5, -16, 6, -3, -9, 6, -8, -8, 0], [-26, 11, -4, 18, -7, 12, -5, 11, -3, 2], [-2, 11, -2, -8, 7, -15, -2, 15, 2, 4], [-18, 23, -10, 2, -4, 2, 2, -6, 9, 8], [-3, -9, 2, 5, -2, 9, 1, -17, 8, -8], [0, 9, -4, -2, -6, -2, 9, -13, 11, 12], [-10, 25, 7, -13, -16, -3, -8, -4, 5, 9], [8, 10, -1, -18, 1, -5, 20, -3, 3, 16], [14, 8, 0, 13, -14, 14, -15, -15, -3, -7], [8, 8, 7, 9, -4, 4, -11, -13, 11, -3], [-10, 11, 4, -16, -17, -8, -3, -9, 9, 16], [-24, 3, -24, 0, 16, -5, 17, -2, 4, -13], [-1, 16, 8, 11, -3, 4, -25, 6, -10, -3], [8, -13, 16, -14, -17, 2, 5, -11, 5, 6], [-6, -12, -12, 6, 12, 1, 7, -2, -15, -13], [2, 9, 12, -2, 1, -2, -3, 7, -15, 0], [-10, -7, -20, 2, 12, -14, 6, -9, 5, 0], [19, 0, -11, 2, 9, -7, -3, 9, -4, -17], [4, -3, 9, 17, -6, 7, -16, -8, 7, -11], [-19, -15, -9, 12, 9, 5, 5, 1, -7, -8], [5, -7, 3, -33, -5, -18, 19, 9, -1, 12], [7, 2, 23, -22, -18, 0, -1, 2, 1, 12], [-42, 3, -4, 14, -2, 9, -9, 14, -4, 6], [1, 22, -18, -2, -8, 3, 1, -3, 0, -1], [0, -10, 2, 0, 9, -16, 5, -6, -10, -3], [5, -18, -16, 7, 2, -3, 15, -11, -7, -15], [-23, 6, -4, -11, -1, -2, 4, -3, -8, 6], [34, -13, 2, -12, 1, -9, 10, -16, 6, -12], [2, 1, -7, -11, -4, -9, 14, 8, 1, 13], [-4, -15, 8, -5, 6, -3, -10, 10, 5, 16], [-10, -10, -3, -14, 7, -7, 22, 3, -1, 14], [-8, 1, -5, 3, 1, -3, 0, 15, -13, -5], [-8, 1, 3, 13, -6, 18, -15, 8, -12, -8], [24, 0, 8, -4, 2, -10, -2, 8, -6, 4], [-13, 5, -2, -2, 18, -9, 8, 5, -1, 0], [-3, 28, -6, 14, -5, 1, -9, 4, -4, 5], [-18, 17, -9, -9, -18, -1, 2, 4, 3, 21], [40, -6, 2, 4, 5, 3, -1, 0, -7, -18], [-9, -4, 13, -5, -1, -6, 1, 5, 16, 12], [-5, 19, 9, -12, -11, -11, -11, 8, 7, 12], [-17, 15, -13, -5, 15, 0, 5, -2, -1, -6], [-18, 9, 0, -3, -10, 5, 2, -4, 1, 8], [-18, -4, 3, 21, 5, 22, -18, 16, -7, 7], [-13, 23, -19, 0, 12, -10, 7, 7, -12, -5], [40, -4, -7, -1, -3, 7, -3, 6, -6, -3], [7, -21, 23, 13, -3, 2, -11, 8, -7, 6], [-22, 11, -6, 0, 3, -5, 2, 5, -4, 6], [-20, 8, -17, -4, 3, 3, 6, -13, 11, 0], [-3, -1, -4, 7, 8, 13, -11, 3, 4, -5], [4, -3, -2, -7, -2, -5, 2, -4, 17, -2], [1, -20, -2, -6, 14, -11, 6, -5, 14, 5], [9, -14, 6, -6, -4, -8, 3, -5, -1, 1], [-6, -9, -17, -5, 6, -9, 14, 3, -15, -3], [24, -6, 4, 4, -12, 16, -8, -10, 8, -6], [4, -9, 1, -21, -4, -8, -1, 6, 6, 16], [-5, 1, -9, 9, -7, -8, 1, -6, 7, -4], [3, -9, 6, 15, -2, -3, -3, 7, 4, 2], [-8, -16, 6, 0, -2, -16, 4, 14, -16, 2], [27, -21, 1, -5, 11, -2, 15, 2, 3, -6], [26, -8, 10, -4, -8, 0, -6, -4, 2, 6], [0, -32, 3, -1, 2, -6, 12, 1, 1, -3], [6, -24, 9, -1, -5, 2, -2, -16, 3, 7], [0, 2, -7, 14, -3, -5, 0, -14, -1, -5], [13, 7, 5, 18, -8, 3, -6, 0, 1, -17], [-12, 17, -2, 6, 11, 9, -12, 19, -4, -2], [-11, 6, -17, 3, 2, -3, 15, 2, 1, -2], [10, 6, 10, 6, -11, 4, 5, 4, 6, -1], [0, -8, -15, 4, 13, 5, 12, 3, 9, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_bf();".
// 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_3525_2_a_bf(:prec:=10)
    chi := MakeCharacter_3525_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_3525_2_a_bf();".
// 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_3525_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, 30, 6, -101, -21, 91, 25, -29, -9, 3, 1]>,<7,R![-32, 812, 745, -1084, -980, 388, 382, -20, -36, 0, 1]>,<11,R![8192, -9888, -11148, 10088, 6776, -1832, -1652, -172, 65, 16, 1]>],Snew);
    return Vf;
end function;