
// Make newform 3525.2.a.bb in Magma, downloaded from the LMFDB on 24 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_bb();".
// 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_bb();".
// 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 := [3, -12, -9, 20, 6, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-16, -24, 1, 13, 7, -1, -1], [-19, 9, 34, -8, -12, 1, 1], [2, -27, 13, 19, -14, -3, 2], [-18, 28, 38, -21, -19, 2, 2], [22, -2, -52, -6, 26, 2, -3]];
    Rf_basisdens := [1, 1, 5, 5, 5, 5, 5];
    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], [-1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, -1, 0, 0], [-2, 1, 0, 0, 0, -1, -1], [0, 0, 1, -1, 0, -1, -1], [1, 1, 0, 0, 1, 0, -2], [0, 0, -1, -1, 1, -1, -1], [1, -1, -1, 0, 2, 0, 0], [-2, -1, 0, -1, -2, 0, 1], [1, 1, 3, -2, -1, 1, 1], [1, 1, 0, -1, 0, -1, 1], [-3, 3, -1, 0, 0, 2, -1], [4, 0, 1, -1, -2, 4, 1], [-1, 0, 0, 0, 0, 0, 0], [3, -3, 0, 3, 1, 0, -1], [2, 2, -1, 2, 3, -2, -2], [-1, 2, -1, -3, 1, 3, -2], [3, 3, 1, 0, -3, 2, 0], [-5, 0, 1, -3, -1, -3, -3], [4, 1, 2, -3, 0, -1, 2], [-1, -2, -3, 5, 2, -4, -2], [7, -3, -1, 1, 1, -2, -2], [0, 0, 2, -1, -1, 4, 1], [0, 0, -4, 0, 3, -6, -2], [-7, -2, -3, 0, 0, -4, -1], [3, -1, -2, 2, -2, 1, 1], [0, 4, 0, -3, 2, 0, 2], [2, 1, -3, 5, -1, -2, 0], [0, 4, -4, 0, 1, 0, 0], [-2, -1, -4, 7, 5, -5, -5], [2, 4, -1, 2, -2, 5, 4], [6, 7, -1, 1, -3, 2, 2], [6, 3, 3, -2, -2, 8, 4], [-9, -3, -3, -3, -1, -1, 2], [1, -7, -1, 6, 0, -6, -2], [-3, 0, -4, 1, 0, -1, 1], [6, -5, -2, 2, 1, -3, -4], [-2, 4, -2, 2, -4, 3, 3], [-1, -4, 1, 3, -2, -5, 0], [-1, -6, -1, -3, 3, -2, 1], [9, -4, 1, 4, 3, -1, 4], [-4, 3, 3, -6, 1, 0, 0], [-2, -3, 0, 7, 3, -8, -3], [3, 4, 2, -5, -3, 6, 0], [6, -3, -2, 7, -1, -2, 1], [5, 3, 0, -5, 6, 4, -3], [6, -5, 3, -3, -4, 2, 6], [7, -4, -2, 0, 1, -4, -2], [2, -4, -9, 6, 6, -4, -2], [-3, -1, -5, 0, 3, -2, -8], [-1, -5, 4, -4, -2, -3, 0], [1, -4, -3, 4, 1, 2, 4], [5, -3, -2, 9, -4, -3, 1], [-2, 0, -4, -4, -1, -1, -1], [-4, -6, -8, 5, 4, -4, 2], [-4, -4, 3, 1, 0, -1, 3], [4, 2, -4, 1, 2, 1, 4], [9, -4, -1, 2, 3, -1, -3], [-4, 1, 2, 6, 1, -3, -2], [10, 3, 0, -2, -2, 1, 1], [-5, -4, -1, 4, -4, -3, -3], [13, 2, 2, 1, 1, 7, 10], [2, 4, 0, 0, 5, 2, -1], [-7, 0, 0, -1, 4, -1, -6], [4, 0, 1, -7, 0, 3, 1], [-7, -5, -3, 0, 3, -5, -9], [4, 2, 3, -4, -2, -3, 2], [3, -2, -4, 9, 1, -2, 3], [9, 0, 4, -4, -3, 4, 5], [-4, -5, 2, -1, -6, -3, 1], [-14, -2, -7, 6, 6, -7, -5], [3, 5, 5, -5, 0, 4, 3], [-7, 11, 3, -5, -2, 1, -1], [0, -6, 1, -1, 1, 4, 1], [10, 4, 7, -9, -8, 7, 6], [-8, -2, -2, -5, 1, 4, -4], [9, 2, 6, 4, 2, 0, 5], [-8, 0, -5, 3, -1, -4, -4], [-2, -3, 2, -3, -5, -2, -2], [-9, 1, -2, 4, 5, -4, -1], [0, -3, -7, 1, 2, 2, 0], [3, -11, 1, 9, -2, -10, -1], [-13, -6, -8, 9, 11, -6, -7], [10, -4, 1, 2, 4, -1, 9], [7, 4, -3, -4, -1, 3, 2], [-3, 4, 7, -9, 0, 13, 0], [-2, -5, -5, 4, -4, 3, -4], [19, -3, 1, 0, 6, 6, 2], [5, 8, -1, 2, 4, 4, -5], [-11, 5, 2, -5, -8, 1, -1], [-5, 0, 8, -5, 1, 5, 2], [0, -5, 1, -2, -6, -3, -1], [-11, 3, 0, 3, 10, -13, -5], [9, -5, 0, 6, 2, 8, -1], [3, -13, 1, 2, -3, 4, 1], [-7, -8, -4, 2, -5, -12, 4], [7, 14, 1, -2, -2, 10, -1], [10, -1, 9, -4, 5, 3, -1], [2, 5, 5, 6, -7, 5, 7], [6, 2, 1, -6, -1, -5, -9], [6, 2, 5, 3, 2, 11, 10], [6, 7, 4, -10, -2, 7, -2], [-27, -1, -4, 1, 2, 0, -1], [12, -13, -7, 4, 2, -4, -3], [1, 4, -2, 2, 5, 6, 7], [1, 2, -8, 4, -1, -12, -5], [4, -7, 2, 1, 7, 0, 2], [4, 2, 3, -5, -3, 5, 1], [11, -6, 5, 8, 1, -9, 2], [-2, -5, -11, 13, 6, -16, -10], [6, 1, 9, -10, -13, 9, 12], [-5, 3, -6, -1, 8, -5, -4], [-2, 5, -3, 12, 2, -8, -4], [11, -2, 3, -4, 9, 0, -1], [-11, 2, 8, -7, 6, 1, -4], [7, 4, 2, -7, -5, 4, -4], [-3, 0, 6, -11, 7, 2, -3], [3, 0, 6, -1, -2, 11, 13], [-2, 16, 1, -4, -2, 7, 1], [-4, 5, -1, 0, -6, -3, -7], [-4, -5, 4, 2, 0, 4, 10], [4, -10, 3, 1, -7, 0, -5], [-7, -14, 3, 1, 10, -6, -4], [12, -8, -8, 3, 0, -6, 1], [9, -6, -2, -3, -7, -6, 1], [10, 13, -2, 1, 4, 7, -3], [9, 3, 6, -14, -11, 14, 11], [-3, 7, -6, 3, -9, -5, 4], [-8, 16, 6, -2, 1, 5, -1], [-9, -3, -3, 10, 14, -5, -7], [14, -17, 0, 1, 3, 0, 9], [8, 4, -4, 9, 11, -8, -1], [11, -3, 4, 7, 1, -3, 4], [-22, -4, -4, 4, 6, -16, -9], [3, -11, 1, -4, 6, 3, 4], [-9, -8, -2, -3, 6, -9, -2], [13, 2, -7, 7, 2, -5, 1], [4, -4, -3, 7, 1, 6, 4], [-10, 1, 4, -2, 0, 1, -2], [2, 12, -4, -8, -10, -2, -5], [11, -13, 3, 0, -3, -1, -4], [-15, 0, -8, -1, 1, -2, 3], [13, -10, 6, 7, 1, 2, 1], [8, 14, 10, -12, -7, 14, 4], [-14, -4, -7, 3, -5, -13, -8], [-20, 0, -4, 1, -2, -11, -12], [2, -4, 3, 1, -7, 10, 15], [-7, -1, -17, 12, 3, -13, -6], [-9, -10, -8, 18, 6, -7, -6], [7, -2, -2, 4, 11, 2, 13], [0, -1, -3, -3, 1, 10, 5], [-8, 7, 2, -1, 11, 2, -4], [-4, 5, -11, 6, 6, -1, -5], [1, -6, 0, -9, -1, 10, 3], [-11, -7, 6, -5, -2, -3, -7], [-2, 6, 7, -6, 0, -7, -4], [-9, 1, -3, 12, 0, 4, 6], [-17, -5, -14, 3, 7, -11, -2], [11, 8, -5, 9, 13, 3, -2], [4, -10, -8, 6, 8, 4, -4], [-7, 13, 5, -17, -2, -2, 5], [-12, 18, 6, -11, 0, 8, -2], [2, 12, 9, 0, 1, 5, 3], [1, -6, 6, 4, 8, -9, -7], [0, 2, 1, 0, 0, 12, -2], [-12, 6, -4, -6, -10, 5, -9], [2, -1, 3, 5, 12, -6, -4], [3, 2, 0, -8, 4, -1, 0], [3, -7, -7, 7, 0, -8, -10], [-2, 19, 2, -15, 0, 8, 6], [2, 4, -12, -4, 9, 0, -7], [-1, 1, -12, -3, 2, 5, 3], [14, 13, -1, 6, 0, 5, 5], [7, 13, 4, 1, 5, 16, 2], [-2, -8, 1, 0, 14, 2, -10], [4, 5, -9, 5, 0, -2, 5], [30, -2, -6, 7, 3, 9, 7], [8, 8, 1, -2, 1, 11, 4], [-7, -4, 4, -12, -5, 8, -3], [-39, 10, -8, 2, 1, 3, -6], [11, -1, 3, 7, -5, -7, 7], [-8, -4, 2, -6, 8, -12, -5], [-5, -6, 16, -12, -10, 11, 3], [-2, 10, 4, -14, -4, 3, 4], [7, 0, 3, -1, 4, -1, -9], [-15, 10, -3, 9, 7, 0, -5], [11, 3, -6, 10, 6, 6, -1], [-3, -7, 1, -8, -11, 8, 12], [-4, 7, -5, 4, -9, 10, 7], [1, -10, 5, 3, 5, -2, 1], [-11, -10, 6, -13, -8, 4, 7], [26, 18, 4, -16, -9, 5, -1], [5, 2, -1, 4, 3, -3, 0], [13, -2, 17, -9, 2, 10, 1], [-14, -3, -10, 11, 3, -1, -11], [24, -2, 2, 0, 10, -2, -3], [6, 6, 1, 13, 8, -5, -9], [43, 1, 2, -1, 4, -4, 0], [27, 2, -3, 2, 2, 5, 7], [-23, -4, -11, 11, 2, 4, 0], [9, 4, 13, -10, -4, 23, 7], [-5, 13, -5, -5, -8, 2, -3], [-7, 15, -1, 2, 10, -7, -6], [-9, 1, 7, -14, -5, -1, -6], [-15, -14, -5, 1, -2, -8, 12], [12, -4, 5, 8, -1, -3, -7], [14, 2, -14, 5, 9, -17, -10], [-20, 3, -3, 2, 1, -8, -5], [14, -5, -2, 16, 2, -2, 5], [-1, -10, -11, 13, 10, -8, -2], [-2, -4, 3, -2, -5, -5, 2], [14, -5, 8, -8, 1, -3, -5], [21, 8, 5, 2, -7, 0, 4], [27, 0, -6, 17, 2, -3, 6], [19, 1, 9, -9, 2, -5, -5], [-8, -14, -2, -2, 0, 3, 4], [-10, 22, 11, -7, -8, 6, 1], [-22, 3, -2, -9, -11, -5, -5], [-1, -16, 7, 8, -2, -9, 2], [2, 9, 12, -20, -6, 8, 2], [1, -9, -9, 7, 3, -7, -3], [-12, 0, -17, 6, -5, 3, 8], [22, 0, -4, 4, -6, -8, -5], [9, -7, 6, 4, 0, 10, 1], [3, 10, -1, -10, -4, 2, -5], [-1, 9, 8, -9, 16, 8, -8], [-2, 23, 13, -5, -8, 8, 4], [-2, -1, 2, -7, -8, -6, -4], [-21, 6, 2, -4, -4, 3, 2], [-18, 11, -13, -4, -7, -7, -1], [6, -5, -4, 8, -8, 9, 3], [23, 3, 5, 2, -12, 8, 0], [-12, 10, -2, 3, 13, 8, 1], [14, -2, 7, -4, 8, -9, 2], [18, -3, 3, 3, -1, 9, -2], [-4, -1, 9, 4, -5, -7, 11], [16, -11, -1, -8, -11, 13, 14], [11, -6, 4, -11, -12, 8, 1], [9, -2, 1, -2, -1, 15, 9], [-7, 11, 8, 10, -13, 7, -1], [-11, 8, -7, 6, -7, -5, -12], [14, 3, -1, 9, 7, -13, 1], [23, -7, 11, 3, 0, 3, 12], [-21, -1, -7, 10, 4, -11, -15], [11, 7, 5, -10, 15, -7, -20], [-15, 0, 7, -2, -7, 7, -2], [-24, -15, 1, 11, 4, -17, -7], [4, -2, 0, 0, 16, -2, 10], [8, 22, 13, -16, -8, 22, 3], [-7, 8, 3, -6, 4, -9, -4], [-23, -7, -8, 11, -4, -7, -2], [2, -2, -14, 6, 12, -9, -11], [24, 6, -2, -2, 4, 10, 6], [-16, 15, -13, -6, 0, 1, 0], [19, -13, 4, -12, -3, -11, -4], [4, 9, 17, -22, -5, 9, 4], [10, -17, 13, -3, 2, -1, -2], [7, -15, -4, 6, -1, -8, 1], [-29, -6, 8, -5, -2, 14, 8], [28, 9, 2, 0, 10, 6, -12], [-9, 7, -5, 1, 1, 5, -4], [-5, 11, 0, -3, -12, 12, 10], [5, -13, -14, 16, 8, -4, 0], [11, -2, 0, -17, -7, 1, 1], [-24, 9, -5, -11, -10, -13, 0], [-6, -21, -4, 23, 8, -1, 1], [2, -5, -1, -6, -11, 9, 14], [35, 5, 8, 5, 2, 17, -1], [2, -20, 5, 1, 4, -1, 10], [-12, -7, 9, -10, -6, 16, 7], [4, -10, -12, 21, -2, -14, -5], [4, -10, -13, 30, -1, -6, -3], [33, 12, 5, -2, -14, 14, 17], [2, -2, -1, 18, 12, -3, -5], [19, 3, 3, 3, -6, 4, -5], [1, 6, -13, 11, 19, -7, -2], [-18, 1, 11, -9, -3, -1, -2], [-9, 30, 3, -13, -1, 6, -7], [-23, 23, -1, 1, 9, 1, -4], [-9, 18, -3, 0, 1, 6, 1], [-20, 21, 5, -6, 8, -10, -2], [41, -13, -4, 0, -1, -6, 3], [2, 3, 4, -11, 17, 1, 7], [9, -2, -6, -5, 14, -13, -10], [-4, -1, -12, 14, 10, -18, -9], [24, -10, 4, 15, 0, -11, 5], [-19, -10, 0, 9, -1, 1, 10], [-9, 10, 2, -12, 7, 7, -8], [1, 5, 5, -11, 5, 20, 1], [-19, 17, -5, 6, -2, 2, -4], [13, -10, 4, -7, -24, 3, 0], [11, -4, -4, 19, 5, -17, 3], [-8, -13, 4, -3, -2, -17, 2], [-20, -1, -7, 4, 8, 4, 1], [6, 3, -2, -14, 9, -2, 0], [-31, -13, -7, -4, 11, -21, -3], [-6, 2, 1, -4, -9, -15, 7], [3, -15, -11, 25, -6, -17, 5], [-31, -11, -13, 9, 8, -25, -9], [-16, 19, 0, -6, -14, -3, -9], [-21, 2, -12, 7, 8, -7, -6], [-4, 11, 5, -5, 9, 0, -6], [-26, -6, -9, 17, 2, -13, -4], [-6, 10, 15, -19, -5, 23, 23], [-25, -10, -7, 10, -4, -4, 1], [9, 0, -7, 7, 10, 1, 12], [-29, 16, -10, 8, -6, 1, -5], [-3, 7, -4, 12, 4, 4, 1], [-13, 0, 8, -28, -4, -10, -6], [8, -6, 1, -5, 0, -1, 9], [-21, -17, -15, 14, 12, -28, -12], [-1, -10, 7, 7, 2, 4, 13], [15, -13, -1, 3, 2, -7, 1], [16, -3, 8, -14, -16, 5, 13], [-36, -9, -9, 5, 14, -19, -13], [12, -14, 6, -1, -9, 14, 24], [2, -17, 13, -5, -4, -11, -15], [40, -3, -1, -8, 10, 12, 4], [20, -1, -11, 25, 19, -2, -5], [14, 22, 11, 8, -7, 9, 8], [-33, -3, 9, -7, -14, -1, 4], [10, -15, -12, 19, 2, -6, 9], [15, -13, 18, 1, -6, 0, 3], [-23, -5, -11, 13, 10, -4, -20], [19, -13, 9, -8, -1, -5, 8], [13, 13, -3, -9, 5, 16, 7], [53, -15, 6, -6, 2, 4, 8], [-17, 0, 13, 6, 1, -8, 2], [8, -10, -6, -7, 0, 3, -2], [-6, -4, -4, 11, -10, -8, 8], [-1, -8, 3, -11, -14, -13, 8], [-38, 5, 2, 7, -10, -7, -2], [13, 6, 8, 2, 14, -5, -12], [0, 13, 4, 2, 1, 18, -8], [12, 22, 6, -23, 1, 1, -10], [24, 10, 10, 1, -18, 16, 4], [-3, 14, 18, -25, -7, 20, 9], [29, 12, -3, 11, 5, 28, 8], [-22, -19, 6, 4, -7, 12, 6], [-25, 7, -3, 0, 4, -21, -8], [-2, -5, 8, -28, -17, 3, 9], [6, -8, -12, -6, -5, -10, -5], [3, -16, 5, -15, -3, 16, 7], [1, -11, -1, -1, -6, -15, 3], [3, 1, -13, 7, 12, -18, 2], [21, -1, -6, -4, -9, -11, -16], [-20, 4, 2, -10, -6, -5, -19], [13, 2, -2, -12, 6, -6, 3], [-32, 12, -2, 7, 12, 10, -4], [-18, -3, -2, -3, -23, -3, 8], [-10, -3, -7, -5, 5, 7, 8], [5, -8, -4, 6, -4, 13, -8], [6, -22, -6, 12, 3, -11, 18], [6, 7, 3, 20, 0, 14, 7], [-2, -1, 4, 13, -2, -7, 7], [18, -8, 5, 5, 12, -11, -9], [-11, -1, -26, 14, 1, -10, -1], [1, 30, 15, 1, -5, 13, -1], [9, -18, -19, 24, 4, -10, 3], [-15, 31, 7, -19, 3, 13, -10], [-8, 0, 9, -21, -16, 10, 4], [16, 33, -5, 0, -8, 19, 15], [-4, 9, 3, 0, 3, 11, 9], [13, 3, 12, 0, 2, 16, 19], [-28, -15, 1, -11, -9, -20, -5], [-15, -9, -6, -17, 3, -8, -3], [25, -23, -13, 20, 4, -3, -3], [6, -21, -6, 2, -5, 9, 16], [7, -5, -2, 0, -2, -3, -8], [-15, -20, 6, -18, 9, -1, -7], [-6, -8, -2, -9, -1, -19, -20], [18, 15, 5, -13, -15, 7, -2], [-38, -13, -10, -13, 4, -11, -14], [-7, 3, 1, -5, 18, -13, -10], [-1, -10, 6, -23, -18, -8, 5], [-13, 12, 5, -18, -11, 17, 1], [-18, -9, -28, 17, 16, -13, -14], [15, 28, 17, -12, -13, 25, 2], [-3, -20, -20, 16, 3, -3, 2], [1, 16, -14, 13, -1, -7, -7], [5, 4, 1, 13, 2, 1, 1], [-1, -5, -2, -12, -23, 10, -1], [-4, -16, -7, 21, -15, -13, 15], [-25, -15, -1, 2, 1, -19, -15], [37, -13, 7, -3, -1, 4, 2], [46, -12, -11, 18, 7, -14, 2], [-6, -10, 14, 8, -3, -17, -7], [32, -6, 5, -7, -15, 13, 13], [-25, 18, -12, 6, -10, -11, -14], [9, 17, -6, 0, 16, 0, -25], [23, -16, 22, -6, 1, 1, 1], [11, -3, -12, -1, 3, -9, -20], [-14, -10, 6, -12, -10, 22, 2], [-59, 15, -6, -3, -2, 0, 2], [-17, 8, 12, -8, -18, 7, 21], [27, 12, -4, -10, 7, 8, 2], [-30, 14, 13, -21, -15, -2, 4], [-5, -31, -21, 30, 7, -26, 9], [-43, -4, 9, -5, -8, -14, -3], [10, -16, 1, -17, 6, 7, 4], [19, 13, 11, -12, -16, 9, 7], [-28, -2, 8, -11, -10, 20, 23], [-5, 15, -4, 8, 8, -11, -1], [-21, 5, 2, 0, -12, -3, -15], [-20, -8, 5, 1, 2, -8, -7], [8, 11, 5, 3, -1, -7, -10], [-20, 9, 11, -29, 1, 16, 4], [-14, 14, -18, 3, 7, -12, -2], [-1, -5, -7, 15, 11, -9, 9], [8, 3, -2, -3, -7, 12, 10], [1, -6, 25, 4, -4, 13, 14], [-16, -7, -2, -10, -10, 7, -1], [-6, -15, 9, -9, -14, -1, -6], [-4, 21, 0, 2, 2, 24, 2], [-2, 33, -10, -12, 1, 9, -8], [14, -23, -14, 1, 17, -10, 0], [-29, -17, 0, 11, 10, 1, 12], [-14, 19, 7, -2, -7, 14, -5], [23, -6, 2, 3, -1, 24, 14], [-31, 14, 2, -2, 3, -4, -5], [11, 11, -10, 12, 3, 11, 12], [0, 6, 21, -37, -19, 3, 0], [-12, 17, 18, -11, -21, 3, 16], [52, -13, 20, -13, -1, 10, 19], [13, -8, -16, 38, 13, -17, -8], [-26, 19, 18, -6, 9, 9, 3], [-70, -2, 3, 0, -5, -10, -5], [-16, 14, -22, 11, 9, 3, -6], [52, -2, -14, 22, 19, 7, 7]];
    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_bb();".
// 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_bb(:prec:=7)
    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_bb();".
// 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_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -12, -9, 20, 6, -9, -1, 1]>,<7,R![23, 206, -31, -201, 45, 29, -11, 1]>,<11,R![5, 21, -216, -365, -152, -3, 8, 1]>],Snew);
    return Vf;
end function;