
// Make newform 3525.2.a.bh in Magma, downloaded from the LMFDB on 23 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_bh();".
// 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_bh();".
// 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, 12, 293, -107, -923, 309, 852, -288, -316, 106, 51, -17, -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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -5, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 4, -6, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 32, 68, -79, -72, 55, 22, -13, -2, 1, 0, 0, 0], [10, 134, -7, -336, 99, 326, -158, -142, 78, 28, -15, -2, 1], [4, 96, 13, -308, -62, 281, 46, -105, -12, 17, 1, -1, 0], [-50, -294, 379, 824, -627, -694, 430, 238, -134, -36, 19, 2, -1], [-2, -98, 63, 440, -257, -466, 268, 186, -104, -32, 17, 2, -1], [-2, 130, -47, -468, 253, 470, -268, -186, 104, 32, -17, -2, 1], [6, 390, -169, -1182, 403, 1106, -326, -418, 110, 68, -17, -4, 1], [-2, 166, -113, -552, 327, 506, -290, -190, 106, 32, -17, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 2, 8, 2, 8, 4, 4, 8, 4];
    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, 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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0], [0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 1, 0, 0], [2, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [0, -1, 1, 0, -1, 0, 2, 0, 0, 1, -1, 0, 0], [2, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, -1, 1, 1, -1, -1, 1, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, -1, 0, 0, -1, 1, 1, -1, -1, 0, 0], [3, -2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0], [2, 1, 1, 1, -1, 0, 0, 0, -2, 1, 0, 0, -1], [1, 1, 1, 0, 1, 0, -1, 0, 0, -2, 0, 0, 1], [2, 0, -2, -2, 0, 0, -2, 1, 0, -1, 1, 0, 1], [1, -2, 2, 1, 0, 1, -1, 1, -1, 1, 1, 0, -1], [1, 1, 1, 2, -1, -1, 1, -1, -1, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 1, -2, 1, 0, 0, 2, 0, -1], [2, -2, 1, 1, 2, 1, 1, -1, -1, -1, -1, 0, 0], [2, -1, 1, -1, 1, 0, -1, 0, 1, -1, 1, 0, 0], [0, 0, 0, 1, 0, 0, -2, 0, -2, -2, 0, -2, -1], [2, 0, 0, -1, -2, 0, 1, 0, 1, 0, 1, 0, 0], [1, 1, 0, 1, -3, 0, 1, 0, 1, 1, 1, 2, -1], [2, -1, -2, -1, 1, 0, 2, 0, 0, 1, -1, 0, 1], [-1, -1, 1, 2, 1, 0, 1, 0, 1, 0, 0, -1, 0], [3, 4, 0, -1, -2, -2, 0, 0, -1, 0, 0, 0, 1], [6, -4, 0, 1, 0, 0, 2, 0, 2, 2, 0, 2, 1], [-3, 1, 1, 1, 1, 0, -2, 0, -2, 0, 2, 1, 0], [4, 0, -2, 0, 0, 1, 0, 0, 2, 0, -2, 3, 1], [4, 0, -1, 2, 0, 1, 1, 1, 1, 1, -1, 2, 1], [2, 2, -5, -1, -2, 0, -1, 1, 1, -1, -1, 1, 1], [1, -2, 0, -1, 0, 3, -1, 1, -1, 2, 0, 2, 1], [3, -2, -1, 1, 0, 1, 1, -1, 2, 1, 1, 2, 2], [2, 1, 0, 1, 1, 0, 1, 2, 1, -3, -2, 0, 0], [-3, 0, -1, 0, 0, 1, 1, 2, 1, 1, -1, 0, 1], [1, -2, -4, -1, 0, 1, -1, 1, 1, 0, 0, 0, 1], [0, -1, -3, -1, 1, 1, -1, -1, 1, 0, 0, 2, 2], [3, 0, 0, 3, 2, 0, 1, -2, -1, 1, 1, 0, 1], [3, 4, -2, -1, -2, -1, 1, 1, 1, -1, -1, -2, -1], [-3, 2, -2, -4, -2, 0, 0, 1, 0, 2, 2, 0, 0], [2, 1, 0, -1, 3, 3, -4, 2, -2, 1, 3, -1, 0], [2, -2, 1, 0, 2, 1, 1, -1, -1, -2, 0, -3, 1], [3, 2, -1, -2, 0, 0, -3, 0, -2, 1, 3, 0, -1], [0, -1, 3, 2, 1, 0, 5, -2, -1, 2, 1, -2, -1], [0, 2, -2, -1, -2, -4, 2, 0, 0, -2, 0, 0, -1], [-4, 2, 5, 0, -2, 2, 1, 0, -3, 1, 1, -1, -1], [-1, 3, 3, 5, -1, 3, 0, -1, -2, 1, -1, 2, 0], [2, -1, -1, -2, 3, 1, -6, 3, 0, -2, 5, 0, 0], [6, 5, 1, 3, -1, 1, 1, 0, -1, 3, 3, 4, 1], [-4, 2, -6, -3, 0, 0, 1, 2, 3, -2, -4, 1, 0], [-4, 3, 2, 1, 1, 1, -4, -1, -2, -1, 3, 0, -1], [4, 2, 1, 2, -2, 1, 1, -1, 3, -1, -1, 2, 1], [1, 1, 1, -3, 1, 0, 0, -2, 0, 0, 0, -2, 0], [-1, 5, 1, 3, 3, 1, 1, -1, -1, 1, 1, 2, -1], [1, 0, 5, 1, 0, 2, 0, -2, -1, 3, 0, -2, 1], [3, 2, 2, -1, 4, 1, 1, -1, -3, 2, 2, -2, 1], [-2, -2, 0, -2, 2, 2, -2, 0, 0, 4, 0, -1, 2], [0, 0, -3, -3, -2, -2, 2, 0, 2, -1, -2, -4, -1], [3, 2, 3, 0, 2, 0, 1, 2, -3, 1, 1, -2, 0], [3, 3, -2, 3, -3, 0, 0, 1, 0, -3, -3, 2, -1], [-6, 4, 1, -2, -4, 0, 1, 0, 1, -1, -1, 1, 1], [4, -4, 5, 0, 0, 0, 1, 0, -5, 1, 3, -2, 3], [-3, 6, -5, -2, -2, -3, -1, -1, 1, -3, -1, -2, 0], [0, 7, -5, -2, 1, 1, -3, 1, 1, -4, 0, 2, -1], [0, 2, 4, 3, -2, 2, 0, 0, 0, 0, 2, 0, 1], [-5, 1, 1, 1, -3, -2, 6, -4, 2, 1, -3, 2, 2], [7, -2, 0, -1, 0, -3, -3, -1, 1, -2, 0, 0, 1], [7, 5, 1, 1, -3, -1, -1, 3, -2, 0, 2, 0, 0], [3, 5, -7, -1, 5, 2, -4, 4, 2, -1, 1, 0, 0], [5, 1, 0, 2, -1, -3, 0, -1, 0, 3, -1, 2, -1], [-1, 4, -3, 1, 0, -2, 2, 0, 2, 2, 2, 0, -2], [-2, -4, 4, -2, -2, 1, 2, 1, -1, 2, -1, -2, -1], [4, 0, 1, 6, 0, 2, 1, -2, -1, 1, 1, 4, 1], [-2, 3, -3, -2, 1, -1, -1, 1, 1, -2, 2, -2, 1], [4, 1, 2, 3, 3, 2, -2, -2, -4, 3, 3, 2, -1], [11, -2, 4, 1, 0, 3, -1, 1, 1, 2, 0, -4, -1], [-2, 2, 3, 0, -2, -1, 3, -1, 1, 1, -5, 0, 1], [3, 2, -1, 2, 0, -3, 2, 0, 2, -3, -3, 4, 0], [1, 0, -1, -2, 0, 0, 3, -1, 1, -3, -1, 0, 1], [-1, 1, 4, 4, -1, -1, 6, -1, 2, -1, -5, -2, -1], [1, 4, 4, 0, 0, 0, -2, -2, 0, 0, 1, 2, 1], [-1, 4, 0, 1, 0, 0, -3, 0, 0, -2, 1, -2, 0], [-6, -1, 2, 5, 5, 2, 4, -2, 0, 2, 0, 0, 1], [-4, -2, -1, 2, 2, 4, -2, 4, 1, 1, 1, 4, -1], [5, -1, 1, 4, -1, 0, 3, 0, -2, 2, 4, 2, -1], [-2, -1, 3, 6, -3, 0, 7, -2, -1, 4, -1, -2, -3], [-1, 2, -1, 1, 0, -1, 4, 0, 4, -3, -7, -2, 1], [5, 0, -4, -6, 0, 1, -2, 3, -2, 4, 7, -2, 1], [-2, 1, -1, -4, -1, -3, 5, 0, -1, -1, -1, -2, 0], [0, -3, 0, -4, 3, -1, -4, 0, 0, -1, -1, -1, 1], [-2, -5, 5, -6, 1, 0, -1, 2, -1, 0, 3, -4, 1], [6, -2, 0, 0, 0, 1, 2, -3, -1, -2, -1, -4, 1], [-4, 6, -6, -6, -2, -2, -4, 2, 2, -2, 4, 0, 2], [2, 4, -8, -2, 0, 0, 0, 4, 0, 2, 3, 0, 0], [-7, 4, 2, 2, 2, 0, 1, 0, -3, 2, 1, -2, 0], [3, 1, -5, -5, 5, 1, -7, 1, 2, -4, 0, -2, 2], [0, 0, -3, -2, 4, 0, 1, 0, -1, -3, -1, -3, 1], [6, -4, -3, 6, 0, 2, 3, 0, 3, 1, -1, 6, 1], [6, -2, 2, -6, -2, 0, -1, 0, -2, 2, 4, 2, -2], [10, -8, 8, 5, -2, 3, 3, -1, 1, 2, 0, -2, -2], [-1, 2, 0, 1, -2, 0, -2, -1, -2, -4, 4, -2, 1], [-10, 0, 7, -4, 4, 4, -1, -2, -3, 1, 1, -6, 1], [-1, 3, 2, 2, -3, 2, 4, 0, 2, 3, -3, 3, 3], [-1, 4, -5, -7, 2, 1, -2, 3, 0, 3, 5, -2, 2], [-1, -2, 0, 1, 2, -3, -1, -1, -1, -5, 0, 2, -3], [5, -8, 0, 3, -2, 2, 4, 0, 5, 0, -2, 6, 1], [13, 4, 0, -2, -4, -1, -2, -1, 2, 0, 0, -2, -1], [-3, 6, 1, 1, 0, -2, -2, 0, -2, 4, 4, 0, 0], [13, 3, 3, 1, -5, -4, 0, 0, -4, -2, -4, -2, -2], [-4, -4, 5, 0, -4, 1, 1, 1, -1, 2, 0, 1, 1], [-8, -4, 3, 2, -4, 0, 8, -4, -1, 5, -1, -2, 1], [-1, 0, -4, -5, 2, -1, 3, 1, -3, 0, 1, -2, 1], [7, -5, -4, -2, 5, 4, -3, 4, 3, -2, 0, 2, 4], [-7, -1, 8, 2, -3, 0, -2, 1, -2, -1, -1, -2, -4], [12, 8, -5, -3, 0, 1, -7, 1, -3, -1, 1, 0, -2], [-2, -5, 0, 4, -5, -1, 6, 1, 4, 0, -3, 2, 0], [11, -9, 7, 0, 1, 0, 3, -5, 1, 4, 2, 0, 1], [-9, -3, 6, -1, 1, -2, 1, 0, -3, -1, 3, 1, 1], [-1, -1, 2, 0, 1, 2, -3, 0, 3, -2, -2, 2, -4], [4, 2, 4, 3, -2, -1, 6, 0, 4, -2, -2, -1, -2], [-5, 0, 3, 4, 2, 1, 1, 1, 0, -5, 0, 2, 1], [2, 6, 3, 9, 0, 1, -1, -1, 1, -2, -2, 0, 0], [10, 0, -4, -2, 4, -2, 2, -1, 0, 5, 3, 0, 1], [7, 6, -5, -3, 0, 1, 1, -1, -2, 3, 3, 0, 2], [3, 2, -6, -2, 2, -3, 2, -1, 2, 4, -2, -2, 3], [-7, 3, 2, 5, -1, 4, -1, -2, -3, -1, -3, 0, -3], [5, 6, -6, -2, 2, 0, -4, 2, -2, 2, -1, 0, 5], [-2, -7, -1, -1, -1, 2, 1, 1, 1, 0, -6, -5, -1], [-3, 10, 0, -3, 0, -1, -7, 1, -5, -1, -1, 0, -3], [-4, 3, 6, 2, 1, 0, 0, -2, -2, 3, -1, 0, 0], [3, -5, 5, 6, 5, 1, -2, -1, -2, 1, 3, -4, -1], [-1, -6, 0, 3, 4, 1, 1, -3, -3, -1, 1, 0, 1], [-5, -3, 0, 5, -3, -4, 4, -2, 3, -3, -7, 0, -1], [2, -2, 0, 1, -8, -2, 1, 4, -1, 0, -6, 1, -2], [6, 0, -5, -2, 0, -2, 4, 2, 3, 3, 3, 8, -1], [-1, 9, -2, -7, 1, 0, -3, 4, -1, 3, 5, -2, -1], [-2, -2, 7, 3, 6, 2, 2, -2, -3, 1, -2, -2, 4], [1, -10, 0, 1, 4, -1, -1, 1, -1, -2, -2, -2, 3], [-6, 2, -3, -1, 2, -2, 5, 0, 3, -5, -3, -6, 0], [-4, -2, -1, 2, 2, -3, -1, -3, -3, 1, 3, 0, 1], [-1, -4, 1, -5, 2, 1, -4, 1, 2, -1, 5, -2, 0], [14, -6, 0, 4, 6, 2, 0, 0, -4, 6, 4, -4, 0], [0, -10, 3, 3, 4, 4, -2, -2, 4, 1, -3, 1, -1], [6, -8, -3, -3, 4, 2, -5, -1, 1, 5, 1, 3, 3], [6, 1, 0, 4, -5, -2, -2, 2, 2, 1, 5, 4, 0], [3, 9, 0, 2, -1, 1, -4, 4, 0, -5, -1, -2, -2], [-9, 14, 1, -1, -2, 0, 2, -4, -1, -1, -2, -4, -3], [7, 0, 10, 2, -4, -1, 4, -3, -2, 6, 3, 2, -1], [4, -6, -3, 2, 2, 6, -1, 4, 1, 1, -3, 4, 5], [14, -4, -1, -3, -2, -2, 0, 0, -2, 1, 0, -4, 1], [0, -5, 1, 1, -3, 3, -1, 1, 3, 0, 4, 4, -2], [2, 4, 0, 1, 0, -3, 4, -2, 2, -6, -10, -3, 0], [-12, 7, -4, -4, -3, -2, 2, 2, 4, -5, -9, 0, -2], [11, -1, -5, -1, -1, 0, -6, 2, 2, -5, 1, 4, -4], [-6, 6, -7, -6, -2, -3, 1, 1, 3, 3, -3, -6, 1], [5, -4, -3, 1, -4, -2, -2, 0, 2, -2, -2, 0, 0], [-18, 0, 0, 4, 4, 2, 0, 0, -2, -2, 0, -1, 2], [7, -4, 10, 9, 2, 3, -1, -3, -3, -2, -2, -2, -1], [13, 8, 3, -3, 2, 1, -2, -1, -2, 1, 3, 0, 2], [-9, 0, -1, 0, 0, 3, -3, 2, 5, 3, 5, 2, 1], [-1, 4, -3, 0, -4, -4, -1, 0, -1, 3, -4, 0, -4], [-5, 9, -7, -1, 1, -2, -4, 6, 4, -8, -4, 0, -2], [17, -7, 0, 2, -1, 1, 4, -6, 0, 1, -3, 2, 2], [6, -3, -4, 4, -1, 2, 2, 2, 4, -3, -5, 4, 4], [13, -3, -1, -2, -5, -6, 3, -2, 3, -2, -4, -3, 0], [5, -1, 7, 0, 7, 3, -2, -4, -6, 2, 4, 0, 6], [-3, 0, 2, 2, -2, -4, 2, 4, 4, -3, -2, 4, 1], [-4, 4, 2, 0, 0, -2, 4, 0, 0, 0, -2, -8, 2], [3, 2, -4, -6, 6, -3, -6, 0, 0, 2, 6, -4, -2], [-6, -9, -6, 0, 5, 5, 2, 0, 0, 1, 1, 7, 5], [7, 2, -1, 5, -2, -2, 4, -6, -3, -1, -4, 6, 1], [7, 9, -3, -3, 1, -2, 4, 0, 0, -2, 2, 2, 2], [2, -5, 5, 0, -3, 1, -1, -1, -3, 6, 2, -2, -1], [-3, -1, 1, -9, -1, 1, -1, 1, 0, -2, -2, 0, -2], [0, 2, 4, 2, 2, -2, -2, -4, -2, 2, 2, -2, 0], [1, -8, -5, 1, 2, 3, 4, -1, 4, -5, -3, 2, 2], [5, 0, 4, -3, 6, 3, -1, 1, 1, -4, 4, -6, 1], [9, 8, 6, 0, 0, 3, -8, -3, -4, 0, 6, -2, 1], [11, 5, 4, 5, -1, -5, -2, -2, -8, 1, 3, -2, -1], [7, 0, 1, 2, 0, 6, -5, -2, -5, -1, 6, 0, 2], [2, 1, -9, -3, 3, 3, -3, 3, 5, -4, -6, 4, 4], [11, -3, -4, 0, 1, 1, 1, 1, 5, 4, 0, 2, 2], [-5, 4, 2, -3, 2, 1, -9, 3, 1, -2, -3, -2, -1], [-1, -8, 5, 8, -2, 0, 5, -8, -3, 2, 1, 4, 2], [13, 0, 4, 11, 2, 5, -1, -3, -1, 4, 2, 2, -1], [5, -3, -4, -7, 3, -3, -6, 3, -2, -4, 6, -2, 2], [1, -2, -2, -7, 4, 4, -1, 0, 3, 0, 7, -2, 1], [10, 0, -4, 2, 0, 5, -4, -2, 0, -2, 4, 9, 1], [-6, 0, 0, 0, -8, 0, 6, -2, 2, -2, -4, 4, 4], [12, -5, -4, 1, 5, 3, 0, -5, -2, 3, -1, 0, 1], [-3, -3, -7, -1, 1, 0, 4, 4, 0, -5, 3, 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-3, 0, 0], [-5, 4, 17, 8, 0, -3, 0, -9, -10, 4, 2, -4, 3], [13, -6, 2, -1, 4, 5, -7, -1, -1, 6, -6, 2, 5], [-13, -10, 4, -4, -4, -3, -6, 7, 5, -4, 2, 0, -6], [-11, 6, 1, -1, -4, -4, 6, 4, 6, 6, -8, 0, -4], [0, 12, 3, -7, -2, -2, -8, -6, -4, -3, 3, -11, -5], [-9, -2, 0, 3, -8, 1, -3, 3, 3, -2, -5, -6, -1], [-10, 1, 2, 5, -5, 4, 8, -6, 0, 9, -7, 2, -3], [1, -2, -5, 2, 6, 0, -13, 4, 1, -9, -5, 2, -6], [-2, -4, 8, 3, 6, 7, -7, 1, -1, 4, 8, 4, 4], [1, -11, 0, 1, 5, 0, 1, 0, -5, 5, 3, 0, 5], [-22, -2, 5, -2, -2, 0, -2, 0, 1, -9, -5, 0, 5], [0, 2, -7, 0, -2, -4, -7, -3, -9, 1, -1, -3, 0], [-1, 3, 14, 11, -3, 0, -2, -1, -4, 1, 1, -4, 1], [-1, 14, 8, -3, -4, 1, -7, -7, -9, 4, 8, 0, -1], [-6, -15, 16, 14, 3, 4, 2, -7, -8, 0, 0, 0, 1], [-2, 13, 5, 9, -9, 0, 9, -8, -1, 9, -3, 8, -8], [0, 0, -17, 2, 0, 3, -3, 1, 5, -3, -3, 2, 11], [-17, 4, 5, 2, 0, -1, 3, -5, -11, -1, -7, -6, -4], [-26, 13, 0, -3, -5, 2, -2, 4, -6, 1, -7, -6, -5], [5, -3, -9, -11, -1, 2, -3, 5, 7, -2, 2, 0, -4], [-3, -2, 2, -2, 0, 3, -2, 1, 1, 0, -2, 4, -6], [-35, -1, 16, 14, 1, 2, 4, -7, -4, -1, -5, 2, 0], [31, -13, -5, 3, 3, 6, -2, 2, -2, 2, -10, 4, 0], [-8, -1, -13, 2, 5, -2, -3, 4, 9, -10, -13, 0, 5], [-16, 3, -11, -4, 5, 2, -7, 7, -5, -8, -12, 1, 0], [15, 2, -13, -13, 10, -3, -5, 4, 5, -1, 1, 2, 2], [-8, 12, 2, -7, 2, -14, 1, -2, -1, -2, 3, 0, -2], [11, -8, 1, 2, 0, 3, 0, -3, -2, -2, 8, 4, 5], [-22, 1, -6, -11, -1, 0, -6, 4, 4, -3, 3, 2, 3], [3, 8, 6, 13, 4, 3, -2, -4, -4, 0, 6, 0, 1], [1, -4, -15, -8, 12, -2, -9, 6, 5, -9, -1, -8, 0], [-2, 16, 1, -6, 0, -6, -1, 4, -3, 1, -2, -4, 3], [-5, -13, 23, 11, -1, 1, 2, -2, 2, 2, 4, -2, 1], [-6, -20, -2, -4, 4, -2, -4, 6, 4, -2, 0, -8, 6], [-12, 17, -10, 3, -9, -1, -6, 2, 2, -9, -3, 7, 2], [4, -16, 4, -11, 4, 2, 0, -2, 0, 8, 0, -6, 1], [-4, -7, 3, 0, -1, 1, -9, -1, 1, 3, 7, 3, 9], [-22, -4, -2, -5, 4, 2, 3, 2, 6, -6, -5, -10, 1], [-6, -17, 1, 4, 1, 5, -3, 1, 9, -4, -4, 8, 5], [-5, -16, 6, -6, 4, -2, 6, -7, -2, 2, -2, -10, 0], [3, 3, -3, -1, -1, -1, -11, 11, -2, -10, 8, 2, -6], [13, 6, 6, -3, -2, -1, 2, 0, -4, 2, 12, 2, 3], [-10, -8, 12, 3, 8, -4, 6, -4, -6, 6, 8, -10, -5], [-13, -15, 9, -2, -1, 9, 5, -3, 3, 8, -6, 4, 6], [2, 3, -4, -3, -3, 6, -4, 6, 4, -6, -8, 0, 1], [0, -7, 10, 10, -1, -6, 18, -6, 2, 7, -7, 4, 6], [9, -6, 12, 4, 4, 1, 8, -3, -1, -2, -6, -12, -4], [-1, -10, 15, 12, -2, 0, 3, -2, 1, 3, 1, 0, 4], [5, -1, -11, 4, 1, 4, -15, 7, 3, -6, 0, 12, 5], [5, 12, 2, 7, 0, 3, -8, 4, -2, 2, 0, 8, 7], [-21, 1, -2, -5, -3, -2, 3, 2, -1, -1, -5, -6, -3], [4, 11, -12, 2, 9, 6, 4, 8, 6, -4, -4, 5, 4], [-15, 1, 6, -12, 1, -1, 5, 5, 1, 6, 0, -10, -2], [-7, 6, 1, -15, 4, -3, 1, -7, 2, 3, 5, -4, 8], [-28, 2, 14, -7, 2, 2, 4, -3, -10, -3, 3, -10, 4], [-7, -25, 5, -1, 3, 3, 9, -1, 6, 10, 0, 6, 4], [-14, 2, -3, 5, 0, 6, -4, 4, 2, -5, -7, -5, -3], [8, -10, -12, -6, -2, 0, 2, 2, 6, -4, -2, 0, 2], [6, 7, -18, 0, -7, -2, -8, 4, 0, -3, -3, 12, -2], [-17, -1, 3, -6, -1, -3, -6, -3, 4, 11, 7, 2, 3], [-14, 17, -7, -10, 3, 0, -7, 6, 3, -6, 7, 0, 5], [11, 5, 5, 5, 1, -5, -2, -3, -8, 3, 13, -6, 0], [7, -1, 18, 19, 9, 0, -4, -5, -16, 3, 7, -2, -3], [-1, 1, -3, 0, 11, -4, -3, 2, -1, 0, 4, -13, -2], [-13, -9, 6, 1, 5, -11, 13, -1, 1, 4, 2, -2, 5], [-22, 0, -10, 4, 12, -2, -8, 4, -4, -2, -2, 0, 2], [12, 5, -3, 4, -1, 6, 5, 1, 1, -4, -8, 1, 8], [7, -6, -1, 8, -2, -2, 3, 0, -5, 3, 3, -2, 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_bh();".
// 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_bh(:prec:=13)
    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_bh();".
// 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_bh( : 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, 12, -293, -107, 923, 309, -852, -288, 316, 106, -51, -17, 3, 1]>,<7,R![24064, 56192, -1792, -72464, -22104, 33677, 12580, -7272, -2388, 842, 172, -48, -4, 1]>,<11,R![-2451456, -6953472, -505856, 4668608, -98752, -1014460, 104072, 94552, -15976, -3548, 892, 17, -16, 1]>],Snew);
    return Vf;
end function;