
// Make newform 3525.2.a.be in Magma, downloaded from the LMFDB on 16 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_be();".
// 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_be();".
// 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 := [2, 13, 8, -47, 8, 24, -7, -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, 0, 1, 0, 0, 0, 0, 0], [9, 13, -46, 7, 24, -7, -3, 1], [11, 33, -61, -7, 32, -5, -4, 1], [14, 36, -87, 7, 47, -13, -6, 2], [26, 64, -148, 1, 79, -18, -10, 3], [-45, -109, 275, -16, -149, 38, 19, -6]];
    Rf_basisdens := [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_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], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, 0, -1, 0, 0, 0], [-3, 1, -1, 0, -1, -1, 1, -1], [1, -1, 0, 1, -1, 0, 0, 0], [-2, 3, -1, 0, 0, -2, 1, -1], [-2, 1, 0, 0, -2, -1, 0, 1], [2, 1, -2, -2, 2, 0, -1, -2], [-2, 2, 0, 0, 1, -1, -2, -3], [-2, 1, 0, 0, -1, -3, -1, 0], [-1, 0, 1, 0, 0, -3, 0, 0], [-1, -3, 1, 1, -1, 0, -1, 0], [5, -1, 0, 0, 1, 1, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0], [-2, 2, -1, 0, -1, -3, -1, 0], [0, 2, 1, 2, 4, 0, 0, -1], [4, -1, 0, -4, 0, 2, -1, -1], [5, -3, 1, 0, 0, 0, -2, -3], [3, -1, 0, 2, 2, 1, 2, 3], [1, 2, -3, 0, -2, 0, 1, 0], [-1, 3, 1, 1, 0, -1, -1, -5], [8, 4, 1, 2, 3, 2, 1, 0], [-1, -1, 0, -1, 0, 4, -2, 0], [5, -6, 1, -2, 0, 2, -1, 1], [-3, -2, 1, 1, -1, -2, 1, -2], [2, 3, -2, 1, 2, 0, -2, 1], [9, -5, 1, 1, -2, 5, 0, 1], [1, -4, -1, -2, -1, 2, -1, 4], [-1, 0, 4, 5, -3, -3, 2, 3], [3, 0, 4, 2, 0, -1, -1, -1], [3, -4, -1, 2, -3, 0, 5, 3], [-9, 0, -4, 1, -6, -3, 0, -2], [-5, 1, -1, 3, -3, -5, 0, 0], [-2, 2, 1, -1, 1, 1, 0, 3], [-2, -1, 5, 5, -5, -3, 0, 4], [-5, 1, 2, 5, -5, -4, 5, 2], [-1, 1, -3, -3, -2, -1, 4, 2], [11, 0, -1, -2, 5, 1, -1, -3], [7, -3, 1, -4, 4, 4, -4, -2], [1, -1, 2, 1, 2, 2, -3, -2], [3, -6, 3, -3, -2, 1, 1, 0], [6, -7, 1, -2, 2, 2, -4, -1], [1, -2, 1, -2, -5, 1, -1, 0], [5, 5, -4, -3, 2, -1, 1, -4], [-3, 2, -3, 2, -1, -5, -1, -2], [11, 0, 0, -2, 3, 3, -2, -1], [2, 2, 1, 1, 4, 1, -5, -5], [6, 0, 3, 6, 0, -2, 1, 5], [11, -4, 2, -2, 4, 0, 1, 2], [1, -3, 0, 3, -3, 5, 1, 7], [-11, 3, -3, 2, -3, -4, 4, 5], [-1, 6, -1, -2, 0, -2, -2, -3], [-1, 0, -1, -2, 4, 1, 4, -4], [-4, 10, 1, 3, 0, -2, 3, 0], [-3, -5, 1, -1, -4, -5, -2, 1], [3, 1, 1, 0, -2, -7, -3, -2], [1, -3, -3, -4, -2, 0, -3, 0], [8, -6, 4, 0, 3, 5, -5, 4], [-13, -3, -2, -4, -1, 0, 1, 2], [13, 3, 0, -1, 4, -2, 2, 1], [5, 0, -1, -6, 1, 3, 2, -1], [15, -3, 3, 1, 1, 1, 3, 5], [-10, -2, -3, 2, -8, -5, 2, 1], [0, -5, 5, 4, -6, 2, 3, 4], [-3, 7, -3, 6, -2, -5, 1, 2], [0, 5, -7, 1, 0, 0, 6, 2], [-11, 8, -2, 1, 6, -1, 4, -3], [4, -3, 3, 2, -7, -1, -1, 4], [3, 2, -6, -5, -5, 0, 1, -5], [4, -4, 2, 3, -1, 0, 5, -2], [4, -8, 0, -7, 2, 4, -5, -1], [4, -4, -1, -3, -4, 2, 2, 1], [2, 2, 1, 2, 0, -5, -1, 2], [4, -1, -2, -3, 0, 4, 1, 0], [-12, 5, -4, -1, -1, -7, -3, -6], [0, 1, 1, 2, -5, -4, 2, -4], [6, -2, 1, -3, 7, 4, -4, -4], [-7, -1, 4, 9, 0, -5, 0, 8], [2, -2, -2, 3, 4, -3, 5, -2], [-5, 5, -2, 3, 5, 0, 5, 3], [-16, 4, -6, 4, -7, -2, 8, -1], [-2, -6, -2, 4, -1, -1, 1, -3], [0, -3, -4, -5, -2, 0, -4, -10], [2, 0, -7, 0, 1, -3, 2, -1], [7, -10, 4, -4, 1, 6, 2, 8], [-15, 1, -1, 6, -8, -8, 2, 2], [-7, 9, -3, 8, 3, -6, 1, -3], [-24, 5, 2, 4, 2, -4, 3, -4], [5, -2, 5, -1, 1, -2, -3, -6], [-1, 4, -5, 0, -8, -7, 4, -2], [5, 1, 0, 6, -2, 4, 6, 3], [12, 3, 3, 0, 5, -1, 4, 5], [-5, 4, -3, -4, 2, -3, -2, -6], [2, 1, -5, -3, -1, 2, 0, 8], [-8, 5, -5, 2, -8, -3, 7, -1], [4, 2, 3, -2, 2, -4, -1, -5], [-13, 4, -3, -3, 3, 2, 1, 0], [9, -17, 9, -1, -8, 0, -8, 0], [9, 1, -5, -1, 6, -2, 4, -4], [5, -4, 2, -8, 9, 3, -9, -4], [-2, -11, -2, 1, -5, 3, -3, 8], [-4, -7, -6, 3, -6, -5, 5, -1], [3, 4, 1, -1, 4, 6, 3, -3], [-14, 1, -1, 5, -5, -6, 7, 8], [-3, 0, -1, 0, 0, 5, 4, 5], [3, -4, 0, -7, -9, -2, -3, -3], [-10, 8, -3, 2, -1, -8, 2, -2], [-4, -3, -1, 3, -13, 2, 3, -1], [-15, -2, 3, -1, -8, -1, -1, 2], [-16, 10, -4, 0, -7, -8, 6, -9], [-18, 3, -9, -5, 2, -5, -1, -12], [0, -8, 0, 1, 3, -3, 6, -6], [15, 3, 2, -2, 10, 2, -9, -6], [6, -2, -8, -5, 15, 6, 0, -7], [-8, -5, 6, 3, 1, 3, -2, 6], [5, 9, 6, -3, 10, 5, -3, 2], [4, -5, -3, 2, 9, 9, 1, 2], [-17, 1, -5, 0, 2, 5, -2, -2], [-2, -10, 4, -5, -2, 0, -1, 1], [16, 1, -4, -5, 4, 6, -3, 1], [-12, 5, 2, 2, -4, -4, 6, 12], [4, -7, 5, 4, -5, -1, 2, 4], [7, 3, 1, 1, -2, 3, -3, -3], [8, 3, 2, 4, 1, 3, 0, 1], [2, -11, 7, -4, 3, -5, -5, -4], [-6, 6, -3, 6, -8, 1, 2, -2], [9, -3, -2, 1, 9, 6, -3, -5], [-15, 10, 3, -1, -7, -7, 3, 3], [18, 2, 9, 3, 12, 8, -5, 4], [13, -5, 5, -2, -9, 1, -4, 1], [3, -8, 0, 1, 2, 6, -2, 0], [23, -7, 10, 1, 2, 12, 0, 8], [-1, -20, 3, -1, -5, 1, -1, 7], [5, -4, 3, -1, 3, 12, -4, 4], [-12, 9, -7, 0, -3, -14, 7, 1], [2, 7, 2, -3, 9, 3, 0, -2], [26, 9, -3, 0, 8, -1, 1, -2], [-9, -1, -2, 3, -2, -5, 2, -2], [-6, 5, -2, 5, 4, 5, 9, 9], [-7, -4, 4, -2, 2, -3, -1, 6], [7, 3, 0, -6, 13, 12, -8, 0], [0, 3, 0, -3, 4, 10, 4, 2], [8, -5, -1, 6, -5, 1, 5, 2], [15, 10, -4, 3, 1, -3, -2, -1], [13, -3, 4, 0, -2, 0, -7, 5], [9, 5, -6, -9, 5, 3, 2, 0], [2, 3, -3, 2, 3, -7, 0, -6], [1, -1, -6, -8, -7, -2, -4, -2], [-9, 2, 3, -2, -11, -2, 0, 1], [9, 8, 5, 0, 2, 11, -2, 1], [8, 10, -3, 4, 3, 1, 0, 4], [-6, -2, 2, -5, 0, -7, -5, -2], [-22, 3, -8, 1, -10, -4, 1, 1], [-12, 1, -5, 2, -3, 3, -1, 8], [3, -10, 9, 2, -10, 5, 4, 12], [-13, 6, -4, 5, 0, 1, 0, 5], [16, -5, 0, -6, 4, 4, -3, -4], [5, 2, -7, -5, -1, -1, 1, 1], [10, -5, -1, -1, 1, 13, -4, -3], [-10, 0, 0, 0, -2, 0, 10, -10], [2, -6, -8, -12, 5, 1, -7, -5], [-13, 17, -3, -11, 2, -11, 0, -9], [5, 12, 2, 4, 3, 3, -3, -4], [2, -16, 0, 1, -12, 0, 2, 9], [11, -20, -3, -7, -5, 1, -8, 0], [7, -10, -4, -7, -8, -2, -6, -5], [14, -6, 10, -2, 5, 16, -6, -1], [-5, 14, 0, 1, 10, -9, -1, -13], [1, -12, 3, -1, 10, 5, -1, 9], [7, 0, 9, 7, 2, -5, 2, 3], [-6, -2, 0, 1, -9, -8, 1, -3], [1, 2, 3, 4, -2, 7, -2, 2], [-2, 7, 5, 12, 3, -1, 6, 3], [25, 2, 9, -2, 7, 14, -6, 2], [9, 10, -7, 1, 5, -3, 4, -4], [19, 0, 8, 2, 5, 9, -3, -3], [-3, -9, -4, 8, -8, 1, 5, 1], [0, -4, -2, 2, -7, 8, 12, 16], [11, -10, -5, -6, 6, -1, -2, 9], [20, -20, 12, 6, -13, 12, 1, 12], [1, 8, 0, -8, 10, -3, 0, 1], [11, 14, 0, 2, -2, 7, -1, 12], [-6, 4, 8, -1, -10, -8, -2, -5], [20, -12, -6, -7, 4, 5, 9, -1], [10, -5, 6, -1, 3, 0, -10, -6], [19, -21, -3, -4, 3, 3, -2, 8], [4, 9, -7, 3, 9, 10, 0, -4], [-1, 9, 1, -10, 2, 4, -4, 3], [7, 17, -8, 3, -1, 0, 8, 9], [-7, -15, 13, 12, -5, -3, 4, 14], [35, -3, 9, -7, 6, 16, -7, 1], [12, -4, -4, 2, -7, -1, 14, 6], [-29, 14, 6, 4, -3, -14, -2, 2], [8, 1, -2, 4, 3, -5, 3, 1], [5, -20, 6, 0, -2, -7, -7, -3], [23, -16, 5, -11, 9, 14, -4, 2], [-5, -17, -1, 0, -8, 3, -3, 14], [18, -12, 4, -8, -1, 14, -3, 1], [7, -14, 11, 4, -3, 1, -8, 1], [13, 5, 7, -3, -4, -1, -7, -1], [-1, 4, -1, -2, -1, -3, 6, 11], [22, 8, 0, 8, 13, 7, -7, -9], [-10, 13, 6, 7, -4, -9, 8, 1], [-7, -7, 2, 10, -9, 2, 4, 6], [7, -1, 3, 1, -4, -1, 1, 3], [-5, -16, 3, 11, -2, 6, 8, 12], [6, 3, 1, -5, 7, 9, -13, -5], [1, 3, -2, 13, -7, 2, 8, 3], [-9, 3, 2, 6, -2, -1, -7, 0], [-1, 13, -6, -10, 11, 0, -8, -3], [10, -6, -7, -9, -4, 7, 1, -4], [-2, -1, 8, -6, 5, -7, -4, -10], [-1, 0, 18, 5, 3, 1, -9, -5], [-30, 13, -4, 11, -11, -15, 4, 4], [12, -12, -2, 7, -13, 2, 0, -4], [6, 8, 1, -4, -5, -4, 0, -1], [7, 7, -3, -3, 20, 12, 6, 1], [9, 2, 8, 0, 2, -1, 1, 7], [18, -14, -2, 5, -6, 3, -8, 1], [8, -21, 13, 7, -5, 3, -7, 6], [-7, -4, 11, 1, 3, 0, 1, -5], [-22, 0, 5, 7, -2, -3, 9, 4], [-5, -22, -9, 1, -5, -6, 6, -4], [16, 13, -6, 8, 2, 1, 5, 6], [-13, 4, -4, 0, -4, -7, 2, -4], [10, -18, 8, -16, 1, 10, -7, -3], [10, 10, -10, -4, 11, -2, 8, -3], [21, -22, 6, -4, 7, 8, -5, 1], [-7, 2, 11, 6, -5, 2, 2, 5], [-11, 3, 9, 7, 4, -10, -4, -4], [-5, 3, 11, 6, -5, -7, -11, -1], [18, -5, 9, 3, 13, 7, 0, 6], [-11, 1, -12, 1, 6, 6, 10, 6], [7, 4, -13, -4, 5, -2, 1, -5], [-37, 12, -6, -4, -4, 0, 13, -1], [-3, 5, -2, -4, -7, -9, -11, 0], [12, -7, 3, 1, 4, -7, -9, -22], [9, -15, -7, -5, -2, 8, 7, 7], [14, -24, 3, -7, 2, 6, -8, 0], [-23, 7, -14, -2, -17, -11, 16, 2], [-17, -14, 3, -1, -6, -15, -5, -4], [-2, 4, 6, 12, -12, -5, -2, 1], [17, -14, 2, -3, 3, 9, 9, 8], [-5, 7, -11, 8, 1, -9, 8, 7], [1, -13, 20, -2, 1, -4, -4, -4], [25, -10, -9, -5, 12, 7, -1, -4], [1, -2, -13, 2, 4, -1, 11, 2], [4, 6, 9, 3, 9, 13, 5, -2], [-2, -6, 2, 2, 2, -7, 6, 6], [-26, 8, -8, 5, 10, -1, 4, 5], [22, 2, -6, -9, 15, -3, -2, -4], [-7, 2, -5, -6, 3, -3, 1, -11], [-4, 8, -5, -7, -3, 3, -5, -4], [-11, -17, -9, -8, -13, -4, -3, -1], [12, -7, 1, -3, 4, 3, -2, -18], [-6, 3, 2, 6, -7, 0, 14, 14], [48, 2, 2, 5, 11, 5, -9, 0], [-7, -11, 4, 1, 1, 2, -1, -7], [10, 15, 2, 2, 11, 4, 5, -3], [-2, -19, 5, 1, -9, -5, 7, -4], [-10, -4, 7, 5, 9, -3, -10, -15], [8, -2, 4, 8, -9, -6, 5, -11], [13, -9, -2, 14, 2, 4, 10, 16], [-12, 5, -9, 1, 9, 10, 0, -1], [-32, 6, 2, 5, -19, -11, 7, 2], [-34, -12, 0, -9, -11, -6, 9, -2], [-20, -15, 11, 1, 6, 3, -3, 6], [9, 28, -5, 2, 14, -1, -2, -14], [-6, 29, -8, 5, 9, -1, -3, -8], [18, 1, 11, -3, 1, -8, -1, -2], [-22, -3, -16, -6, -9, -3, 8, -3], [-5, -10, 0, -7, 3, 8, -14, -3], [-25, -6, 0, 2, -2, 13, 2, 10], [8, -4, 15, 6, -3, 11, -6, 9], [48, -18, 7, -17, 6, 16, -16, 3], [-32, -7, -1, 11, -19, 1, 14, 17], [23, -10, 7, 3, 9, -1, 2, 9], [27, -21, 13, -4, 15, 31, -6, 10], [18, -16, -3, -9, -17, 7, 6, 7], [-14, 4, -17, -12, 0, -3, 13, 1], [-13, 7, -3, -3, -7, -9, -8, -8], [-7, 5, 10, -3, 11, 16, 7, 9], [11, 2, -3, -1, 3, 18, 2, 3], [20, -7, -5, 2, 0, 4, -9, 8], [-9, 5, -5, 3, -5, 2, -2, -2], [-14, 15, -13, -8, 8, -15, 12, -16], [-3, 1, 6, 14, -12, 4, 2, 13], [-42, 6, -7, -11, -2, -15, -4, -6], [-7, 12, -16, -13, 12, -12, -3, 0], [-2, 1, 14, 4, 10, 4, -5, -6], [-14, 1, -8, 1, 5, 1, -2, 0], [6, -1, -13, 7, 5, 14, 4, 5], [-13, 6, 7, -6, -2, 0, -3, 4], [-8, 3, -7, -15, -5, -4, 5, 10], [-37, 23, -12, 5, 5, -1, 2, -9], [4, 13, -14, 2, -3, -21, -7, -6], [21, -18, -2, 2, -5, 11, -1, 10], [-26, 0, -14, 1, -13, 0, 8, 12], [16, 10, 2, 3, 5, -8, -4, 0], [3, -17, -5, -6, 5, 11, -3, -5], [20, -5, 2, -5, 7, -3, -14, -14], [-15, -2, -1, -12, 5, -6, -1, -14], [24, -9, 0, -1, -5, -1, -7, -2], [10, 23, -7, 1, 3, 0, -5, 3], [1, -8, -6, -8, 6, 5, -2, 6], [-35, 1, -1, 8, -10, -12, 12, 0], [16, 2, 3, 8, 5, 9, 1, 13], [30, -7, 4, 0, 12, -1, -15, -12], [-43, 20, -3, -9, -7, -11, 9, -14], [7, -9, -1, -16, 0, 5, -3, 4], [-4, -9, 14, 20, -4, 4, 3, 6], [-14, 5, 8, 6, 3, 7, -7, -3], [-34, 10, -9, 12, -2, 3, 15, 2], [17, -19, -5, 4, -11, 3, -5, 1], [9, -20, 2, -3, 1, -4, 2, -2], [10, 9, 3, 0, 7, 2, 2, 0], [11, -4, -9, -8, -12, -3, -7, -8], [-21, 3, -4, -4, 0, 0, -3, -18], [-14, 2, 5, -2, 1, -3, 0, -2], [12, -10, 5, -5, 6, 25, -13, 4], [1, -4, 6, -7, -7, -7, 7, 4], [-10, 8, 0, 18, -11, 1, 16, 7], [-7, 2, 18, 6, 1, 7, 0, 3], [-31, 12, -5, 9, -2, -31, 3, -7], [-1, 19, -18, -8, 1, -4, -8, -14], [1, -24, 9, 0, 3, 3, -8, -8], [13, -1, 18, 12, -4, 6, -4, 14], [-17, 4, 10, -3, -9, 3, 1, 7], [13, -15, 3, 2, 14, 12, 3, 6], [1, 15, 2, 8, 9, 10, 13, 5], [-1, 11, 7, 2, 8, 6, 6, 4], [-5, 0, 6, -5, 1, 15, -2, 2], [19, 10, 7, 0, 9, 2, -12, -21], [13, 19, -9, 12, 0, -8, 7, -1], [40, -3, -5, -9, -5, -1, -3, -3], [1, 13, 8, -2, 5, -8, -11, -11], [8, 15, -9, 4, 11, 11, -5, -4], [14, -15, -8, -7, 10, 7, -4, 2], [21, -15, 22, 9, -15, -3, -8, 1], [-10, 5, 3, 6, 9, 12, 5, 1], [-14, 13, -3, 7, -4, 6, 15, 14], [40, -32, 11, -14, -2, 19, -10, 11], [41, -21, 15, -6, 7, 19, -14, 2], [-42, 14, -6, 1, -21, -12, 16, -4], [1, -28, 15, 9, -14, -7, -1, 15], [-26, 25, 0, 2, -11, -18, 5, 10], [-23, 16, -14, 15, 0, -6, 10, 7], [-24, 6, 0, 0, 10, 5, -1, 4], [11, 3, 3, -17, 12, 1, 0, 5], [2, 24, -6, -2, 12, 6, -4, -11], [-13, 2, 16, 12, 14, -6, 1, -13], [-23, 6, -5, -16, -11, 2, 6, -3], [9, 6, -7, -5, -10, -13, -4, -4], [-6, -15, 11, -5, -17, -2, -7, -4], [-6, 8, -4, -3, 5, 12, 5, -16], [31, 13, 11, 8, 20, 9, -1, 8], [-19, 5, -10, 5, -3, 10, 13, 7], [19, -19, 18, -2, 6, 15, -2, 3], [-5, 18, -7, -15, 15, 12, -13, -6], [-34, -10, 4, 0, -26, -16, 17, 17], [-24, 17, -6, 6, 3, -10, 16, -11], [-22, 3, -6, 1, -10, -14, 3, -7], [-23, 7, 4, 2, -5, -7, -5, -2], [26, -19, -10, 0, 8, -1, -1, 2], [-12, 6, 10, 9, 8, -2, -1, -4], [18, -6, 13, 0, 9, 19, -17, 8], [23, 15, 13, -4, 11, 13, -8, -11], [3, -9, -1, -3, 8, -9, -4, -4], [-43, 9, 3, -2, 5, -6, 4, -1], [5, -6, 15, 15, 4, 8, -15, 2], [-24, -7, -5, -3, -17, -9, -3, 13], [-35, 14, -6, 4, 6, -4, -1, -23], [-17, -6, -3, 0, 2, -4, -3, -4], [-5, -4, -22, 1, -17, -1, 11, 2], [9, 13, -13, -7, -10, -8, 15, 7], [46, -18, 9, -1, 22, 8, -16, 8], [-13, -8, 12, -3, 5, -3, -2, 0], [17, 4, -6, -16, 5, 16, 6, -6], [40, -20, 6, 4, 5, 14, 3, 10], [-42, 3, 13, 7, 1, -22, -4, 1], [16, -5, -4, -13, 11, 16, 0, 0], [-5, -3, 6, 19, -7, -4, 13, 17], [26, -7, 21, 9, -13, -16, -3, -1], [14, -3, 5, 5, 8, 10, -14, 16], [-20, 0, 11, -2, -5, -20, -7, -4], [26, -12, -4, -17, -1, 15, 1, 5], [-16, 8, 7, 1, 8, 10, -13, -14], [6, -11, -3, 5, 9, 16, 1, 1], [3, 10, -7, -4, 15, -4, -10, -11], [-36, -3, -7, 5, -11, -22, 6, 0], [-31, 22, 4, -2, -3, 1, 4, 6], [37, 6, -2, 8, 10, 10, -10, 13], [42, -22, 11, -17, 9, 13, -11, -8], [46, -4, 10, -12, 7, 0, -15, -1], [-36, 22, 1, -2, 15, 3, -11, 1], [-14, 14, -2, -7, 17, 15, -2, 9], [7, 1, -10, 7, 0, 1, 10, 8], [-4, 13, -20, -5, -9, -12, 14, 1], [28, -29, -2, -16, -13, 9, -5, 9], [0, -9, 4, 9, -3, 3, -1, -8], [-11, -13, 1, 4, -9, 13, 10, 13], [-8, 17, -9, -11, 0, -4, -1, -11], [28, 5, 10, 5, 30, 14, 2, 7], [-4, 5, -4, -3, 20, 6, 0, -11], [-11, 5, -13, -3, 6, -6, 10, -15], [-13, 8, -9, 7, 3, -13, -4, -21], [-21, 10, -20, -13, 10, 1, 4, -14], [20, -15, 6, -13, 14, 1, 3, -4], [29, -4, -2, 0, 0, 15, -4, -13], [-6, -4, 10, 5, 2, -21, 5, -6], [-32, 12, -18, 15, -2, -1, 8, 3], [-41, 17, 7, 8, -5, -3, 5, 5], [20, -10, 11, 8, -4, 24, 4, 18], [12, -13, 7, -7, 11, 5, 1, 8], [41, -17, 22, 0, 1, 14, -11, 8], [-38, 4, -7, 6, -5, -24, 0, -8], [42, -22, 5, -20, -8, 16, -1, 14], [27, -23, -4, 7, 1, 18, 2, 15], [13, 11, -10, -13, 6, 11, -5, -16], [-5, 10, -8, -5, 1, -20, 13, -13], [22, -37, 16, 8, -17, 2, -15, 12], [-49, 1, -4, -2, -12, -18, 7, -7], [16, -14, 15, 7, -1, -1, -9, -1], [-13, -4, 6, 1, -4, 14, 8, -2], [2, 10, -13, -3, 26, 7, 0, -12], [34, 1, 9, -3, -4, 10, -7, 0], [-19, 8, 14, 5, -11, -18, -10, 3], [-48, -9, 16, -6, -6, 4, 8, 5], [18, -6, 3, -19, 18, 10, -7, -2]];
    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_be();".
// 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_be(:prec:=8)
    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_be();".
// 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_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 13, 8, -47, 8, 24, -7, -3, 1]>,<7,R![-171, 723, 613, -550, -170, 144, -4, -8, 1]>,<11,R![-4420, -2423, 3505, 2140, -341, -334, -27, 8, 1]>],Snew);
    return Vf;
end function;