
// Make newform 3525.2.a.bc 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_bc();".
// 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_bc();".
// 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, 0, -15, 0, 18, -5, -3, 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], [-2, -1, 1, 0, 0, 0, 0], [4, 4, -6, -1, 1, 0, 0], [-2, 4, 5, -6, -1, 1, 0], [1, 6, -5, -11, 6, 2, -1], [0, -15, 0, 18, -5, -3, 1]];
    Rf_basisdens := [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], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0], [2, -1, -1, -1, 1, 0, -1], [1, 0, 0, 0, 1, 0, -1], [3, -2, 0, -1, 1, 0, 0], [1, -1, 0, -1, 0, 0, -1], [0, 0, 1, 0, -3, -1, -2], [0, 2, 1, 1, 0, 1, 1], [2, -1, -2, 0, 1, 1, 1], [3, -2, -1, -2, 0, -1, 1], [2, 0, 0, 0, -2, -1, 1], [1, 2, 0, -1, -1, -2, 1], [-1, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 2, 1, -1, -1], [3, 0, 1, -1, 0, 1, 0], [0, -1, -1, 1, -1, 0, 0], [0, -1, 1, 1, -2, -1, 0], [-2, 3, 2, 5, 0, 0, 1], [-2, 4, 0, 2, 1, -1, 2], [-4, 2, -1, 1, 2, -2, 0], [6, -2, -3, -2, 1, 0, 2], [0, 0, 1, 0, 1, -1, -1], [4, -5, 0, -1, -1, 0, -4], [0, 1, 0, 0, -4, -1, -3], [1, 1, 3, 1, -1, -2, -1], [6, -3, 0, 3, 5, 3, 2], [-7, 4, 3, 2, -3, 0, 0], [8, -1, -2, -3, 1, 2, -2], [0, -1, -3, 1, -1, -1, -1], [-3, 1, 1, -1, -3, -3, -2], [-3, -2, 1, 0, -3, -2, 2], [-7, 4, 1, 0, -1, -3, 2], [2, -4, 1, -1, -1, 2, -4], [0, -6, 1, -2, 1, -1, -4], [-1, -1, -1, 4, -2, -3, 1], [2, 4, 2, 6, 1, 0, 2], [6, -6, -1, 1, 5, 4, 1], [10, -8, -5, -4, 2, 0, -4], [4, -1, 2, 2, 2, -2, 5], [0, -4, -3, -2, 4, -1, -4], [5, 1, -1, -5, 0, 1, 2], [8, -5, -3, -8, -3, -3, -3], [-3, 0, -2, 0, -4, -1, 4], [-2, -1, 1, -2, -3, 1, -1], [-5, 0, 3, 3, 2, 3, 1], [9, -5, -1, -1, 0, 0, 0], [-1, 7, 4, 3, -1, 2, 4], [-1, 1, -1, 2, -1, 5, 0], [8, -1, -5, -1, 2, 5, 0], [9, -11, 0, -3, -2, 0, -6], [-8, 0, 1, 3, -4, -7, 2], [-5, 6, 3, 2, 0, 5, 7], [-4, -5, -1, -2, -4, -6, -9], [10, 1, 0, 5, 7, 7, 8], [-3, 0, 0, 4, -3, 2, 1], [-8, 3, 2, 4, 1, 1, 4], [0, 6, -2, 2, 3, 5, 5], [8, -10, -4, -2, 1, 0, -4], [4, -5, 1, -3, -3, 0, -5], [-6, 15, 8, 5, -4, -1, 5], [-3, -2, 0, -1, -4, -7, -8], [-8, 7, -1, 1, -4, -4, 5], [9, -5, 0, -4, 1, 5, 6], [-7, 6, 6, 2, -3, 2, 5], [-2, -9, -6, -8, -1, -5, -7], [1, -1, 1, -5, -5, 1, -4], [3, 0, 3, 2, -5, -7, -5], [3, -3, 3, -3, -4, 2, 3], [-6, 8, 5, 6, 0, 5, -3], [-5, 3, 6, 5, 2, -1, 1], [-4, 17, 4, 5, -1, 4, 11], [-12, 3, 0, 8, 1, 4, 7], [-3, -5, 2, 2, 0, -6, -9], [1, 8, -1, 2, 4, -2, 2], [12, -8, -2, -4, -2, 3, 2], [13, 0, 3, 2, 5, 4, -3], [3, -9, -3, -8, 1, -6, -8], [-8, 11, 12, 4, -2, 1, -4], [1, 0, 3, 3, -2, 4, 5], [3, -17, -3, -5, 2, 2, -2], [0, -7, -6, -1, 1, -4, -5], [-11, 6, 3, 0, 7, 1, 5], [-5, 3, 2, -1, -4, -3, -3], [-6, 4, -3, -2, -6, -7, 2], [16, -6, -1, -4, 6, -6, 2], [11, -2, 3, -5, 3, -3, -2], [-8, 4, -1, 10, -1, -5, 0], [8, -3, -4, -6, 10, 3, -1], [11, -10, 3, -8, -6, -3, -11], [-5, 8, 2, 5, 0, -5, 8], [-5, -6, 4, 1, -4, -9, -11], [-3, -2, 3, 4, 0, 1, 9], [1, 7, 3, 0, 1, -2, 3], [6, -1, 6, 1, 9, 7, 3], [1, 3, 0, 1, 1, -6, 2], [-2, 1, -4, 2, 8, 1, 5], [1, -1, -2, -2, -1, -7, 3], [-1, -3, 2, -6, -5, 1, -7], [9, 0, 0, 2, 3, -5, -3], [13, 2, -1, -2, 6, -2, 8], [8, 1, -4, 1, 2, 4, 6], [19, -6, -11, 1, 6, 9, 7], [7, -2, -2, 2, 12, 7, 15], [19, 7, -7, -5, 2, 2, 7], [17, -1, -1, -3, 0, -2, -1], [2, 1, 4, 2, -2, -1, -4], [5, 1, -2, -3, -2, 4, -7], [-24, 4, 7, 6, -2, -7, 6], [-13, 3, 5, 3, -4, -6, 4], [-7, -1, -5, 0, 0, 9, -4], [-9, 16, 4, 6, 1, -1, 12], [-3, -2, -9, -2, 9, 1, 4], [-14, 16, 6, 3, -5, 1, 3], [7, 11, -2, 2, 9, 9, 8], [17, 0, -2, 6, 12, 9, 4], [3, 6, -1, 4, -3, 3, 1], [-9, -7, 1, -4, -4, -3, 1], [3, -1, -2, -5, -4, -1, -7], [-5, 8, 8, 5, 2, 5, -3], [-10, 7, 6, 4, 2, 6, 2], [7, 5, 6, 4, -4, 4, -7], [-12, 12, 5, 1, -14, -12, 2], [-12, 13, 9, 7, -8, -3, 9], [-5, -2, -1, -2, 1, -3, 7], [-8, 14, 3, 4, -8, -3, 3], [-1, -6, -7, -3, 2, 2, -1], [-7, -5, -6, -7, -6, -7, -10], [-20, 9, 11, 2, -6, 6, 7], [-16, 4, 2, 5, -10, 5, 5], [-8, 11, 5, -6, -1, -5, -3], [-12, 2, 1, 4, 1, 1, -7], [5, 1, 0, -7, 10, 3, 2], [-14, 12, 13, -2, -9, -10, -7], [-4, 0, -5, 7, 1, -7, -2], [3, 1, -10, 2, 15, 5, 6], [4, 2, 0, 8, 11, 6, 13], [15, -15, -11, 0, 1, 2, 4], [4, 15, 8, -1, -4, -4, -10], [2, -4, -3, -8, -1, 10, -3], [26, -5, -13, -6, 2, 1, -4], [9, -12, -9, -4, 5, -1, 3], [-9, 12, -1, -4, 1, 1, 5], [2, -3, 8, -7, -7, 0, -10], [15, -7, -15, 5, 11, 8, 8], [-20, 3, -2, -4, -13, -9, 0], [1, 7, -10, 6, 8, -4, -3], [-20, 9, 5, 0, -10, -3, 6], [7, -8, 2, 5, -10, 2, 0], [-7, 1, 7, 7, -4, 4, 5], [-19, -2, 2, 4, 2, 2, -7], [-4, 25, 6, 2, -6, -5, -4], [9, -2, 2, 3, 8, 5, -1], [9, -16, -1, -2, 15, 7, -5], [3, 6, -5, -2, 0, -1, -1], [-9, -3, 11, -5, -19, -9, -10], [22, -12, -11, 0, 5, 5, 6], [3, -13, -4, 3, -8, -11, -8], [4, 5, -1, 1, -1, 0, 0], [12, -4, 4, -6, -2, -10, -4], [-10, 7, -2, -3, -5, 0, 5], [-12, -1, 10, 1, -11, 5, -8], [-25, 6, 0, -4, -3, -5, -1], [-1, 8, -1, -1, -1, 8, -5], [-7, 3, -5, -10, -5, -5, -12], [2, 2, -3, -5, 9, 0, 3], [9, 7, 1, 7, -6, 0, 2], [-9, -12, -4, -5, -8, -12, -4], [-14, 7, 11, 1, 6, -6, 0], [-14, 0, -6, 7, -5, -5, 8], [-10, -7, -7, -7, 8, 6, 1], [1, 0, -1, -4, 0, 3, -5], [-11, 14, 2, 15, -6, 11, 9], [-1, 5, -4, -2, 0, 5, 8], [29, 1, 5, -6, 7, 9, 4], [3, -19, -2, -11, -1, -2, -3], [12, 0, 7, -1, 3, 7, -5], [17, -2, 5, -4, -2, -7, -8], [-21, 11, -1, 3, -10, -10, 3], [-19, 15, 4, 0, -5, -2, 2], [26, -2, 2, -9, -6, -6, 3], [-18, -6, 9, 6, -9, -10, -1], [-5, 12, 11, 5, -11, -10, 1], [-2, 4, -2, -3, -4, 4, 0], [0, 2, 0, -6, -1, 8, -3], [26, -7, 0, -2, 2, 12, 11], [27, -5, 1, 6, 9, 10, 11], [0, 15, -3, -3, 4, -3, 2], [-23, 9, 0, -1, 13, -5, -1], [10, -15, -2, -8, 4, -2, 5], [25, -7, -7, 5, 14, 15, 13], [-8, 11, -7, 8, 0, 2, 5], [4, 4, -1, -6, -12, -1, 4], [18, -14, -4, -1, -5, -6, -5], [-16, 15, -1, -1, -13, 1, 3], [-8, -16, 1, -4, 3, -8, -3], [-17, 0, 12, 14, -9, 2, 3], [9, -8, 0, -9, 13, -1, -2], [0, 9, 12, 5, -4, 7, 9], [-6, 10, 13, 19, -2, 0, 2], [-10, 11, -4, -1, 18, 9, 15], [6, -13, -8, -5, -1, 14, 1], [-12, 14, 11, -1, -1, 5, 4], [-2, 7, -3, -10, 4, 13, 10], [-12, 20, 13, -5, -10, -12, -14], [21, 0, 2, -2, 9, 15, 9], [4, -22, 4, -9, -4, 5, -8], [7, -9, 6, -3, -17, -9, -17], [28, 9, -3, 2, 5, 8, 9], [-24, 10, 5, 7, 2, 10, 4], [-11, 12, 1, 2, 12, 9, 2], [16, -20, -7, 4, 2, -2, -1], [24, 4, 5, -9, 2, -7, -8], [1, 7, 6, -8, 13, 11, 10], [-16, 5, -6, -2, -11, -2, -11], [6, -6, -8, -3, 2, 6, 4], [7, 6, 2, 11, 7, -2, 11], [-8, -1, -3, 1, 7, -3, 1], [-12, -1, 1, -11, -1, -15, -8], [-10, 1, -10, -6, 0, -12, -6], [-6, 4, -4, 7, 0, 0, -3], [9, 2, 1, -2, 10, 5, 8], [16, -12, -5, -6, -3, 6, -15], [-3, 13, 3, 6, -5, -12, -7], [18, -27, -4, -4, 18, 13, -3], [-11, 10, 0, 3, 7, -1, 12], [-3, 13, 13, 9, 2, 0, 6], [-14, 12, -8, -1, -1, -7, 10], [-7, 10, 4, 19, 0, -6, 6], [-7, 5, -2, 10, 11, 3, 17], [-20, 17, 9, 19, -3, 10, 7], [26, -20, -13, -2, -3, -5, -8], [-4, 8, 5, 14, 7, 11, 19], [21, 1, 11, 1, -1, 11, 4], [-7, 12, -9, 7, -1, -6, -2], [1, -9, -6, 2, -5, -1, 1], [5, -7, -9, 0, 8, 9, 18], [-31, 9, 21, -1, -10, -1, -3], [26, -14, -16, -10, -5, 1, -9], [11, 16, -3, 4, 2, -2, 3], [10, 12, 1, 8, -2, -5, 0], [19, -6, -6, 7, 16, 18, 7], [-4, 3, 3, 4, 12, -2, 10], [16, 4, -2, -1, 12, 7, 11], [-16, -13, -3, 4, 8, 11, -4], [0, -22, -9, 0, 2, -1, 2], [-23, 25, 2, -6, -5, -4, -5], [8, 6, -6, -18, -2, -8, -6], [21, -25, -12, -6, -4, -11, -17], [2, 7, 11, 5, -19, -1, -6], [-27, 28, 12, 0, -13, -9, 2], [-14, 2, -13, 9, -11, -6, -5], [6, 14, 8, 12, -2, 4, 6], [13, -22, -3, -13, 1, 1, -6], [5, -2, -6, -8, -15, 4, -8], [-31, 17, 3, 6, 2, -1, 8], [19, -13, -17, -2, -6, -4, -8], [9, -2, 5, 5, 14, 16, -3], [19, -3, 2, -5, 5, 9, 2], [26, -3, -10, -12, 4, -6, 0], [-8, 24, 3, 17, -1, 2, 8], [-7, -18, 4, -1, -7, -7, 4], [17, -11, -18, -16, 6, -6, -20], [-19, 14, -7, 3, -3, -7, 17], [-3, -13, -13, -6, -2, -4, 8], [-6, 20, 1, 10, 0, 9, 11], [9, -9, -1, -8, 0, -7, -6], [-5, 8, 17, 20, -12, 13, 9], [-15, -12, -1, 10, 2, 4, 8], [11, 2, 4, -4, 16, 5, -7], [-10, 25, 1, 9, 10, 13, 17], [7, -22, -10, -3, 7, 5, -5], [2, -25, 0, -10, 4, -9, -5], [11, -3, -12, -3, -2, -1, -9], [38, -5, -6, -5, 7, 6, 11], [-18, 5, 5, 13, -11, 2, 0], [-29, 22, 7, 3, -19, -2, 8], [-24, -7, 2, -4, -8, -20, -15], [34, -2, 1, -1, 1, 4, 6], [6, 24, 8, 9, -3, -11, 3], [21, -14, -3, -4, 9, 7, -8], [19, -16, 1, -21, -16, -10, -3], [-6, 13, -3, 11, 9, -5, 19], [-11, 9, 10, -8, 1, 3, -6], [24, -23, -7, -4, -9, -16, -5], [-4, 9, -7, 4, 6, -3, -1], [13, 8, 6, 11, -2, 9, 8], [-3, -3, -16, 0, -5, -8, -10], [18, 6, 6, 0, -2, 4, -13], [-6, -18, -11, 2, 13, 13, -12], [11, 3, -14, 0, -3, 3, 0], [-7, -12, 7, 5, -5, 1, 7], [-12, -2, -4, -2, 7, -3, 2], [-23, 0, -6, -4, -2, 9, -7], [-18, 0, -4, -7, 5, 10, 10], [2, -1, 8, 4, 13, 3, 3], [-3, -22, -4, -10, 1, -15, -19], [4, -33, 0, 4, 9, 12, -5], [16, 9, -8, -6, 9, 6, -13], [24, 3, 1, -1, -21, -6, -11], [-17, 5, -4, 6, 17, 9, -2], [9, -12, -3, -10, 7, 4, 2], [23, -10, 3, -12, 4, 2, -2], [-25, 15, 6, 7, -12, -2, -13], [-40, 21, 4, -2, 2, 1, -3], [28, 4, 11, 2, -2, 0, -4], [-5, 4, -9, 1, -11, 0, -9], [9, -5, -13, 8, 23, 10, 3], [-11, 20, -6, 6, -2, 8, 10], [13, -30, 4, -22, -7, 6, -15], [2, 4, 3, 2, 13, -9, -8], [20, 0, 4, 3, -11, -14, -1], [26, -13, -14, -8, -1, 4, 9], [-20, 3, 7, 9, 3, -6, 11], [-7, 11, -8, 4, 7, -18, 21], [16, -6, -14, -8, 12, -1, 18], [27, -5, -2, -12, 3, -8, 3], [5, 12, 15, 6, -1, 5, -2], [5, 6, 4, 4, 2, -12, 13], [25, -18, -1, -12, -6, -13, -20], [-30, 15, 3, 8, 14, -2, 6], [8, 0, -7, -13, 4, 13, 1], [-11, 24, -5, 15, 4, -13, 13], [-24, 5, 13, -7, -4, 0, -6], [-24, 11, 9, 4, -18, -9, 4], [-2, 12, 8, 6, -6, 2, 1], [31, -21, 0, 4, -6, 14, 8], [-26, 8, -5, 19, 4, 5, 10], [26, -11, 2, -8, 13, 13, -2], [0, 17, 6, 3, 3, 11, 26], [-8, -20, -3, 2, 2, 0, -4], [-8, 12, 4, -4, -13, -17, -2], [21, -16, 0, 3, 16, 14, 2], [-32, 16, 14, 9, -9, 10, 18], [12, -2, 6, 11, -16, 9, 4], [24, -5, -4, 9, 15, 19, 24], [7, -12, 3, -2, -11, 13, -7], [-14, 3, 15, -2, 1, 4, 12], [-34, 20, 6, 7, 11, 4, 16], [4, 16, -7, 3, 5, -5, -6], [-6, 0, -3, 8, 12, 6, 13], [-10, 1, -13, -9, 14, 0, 15], [56, -5, 4, 0, 8, 16, 1], [-4, 1, -4, -4, 9, 12, 3], [10, 9, -11, 3, -12, 14, 4], [41, -2, -16, -12, 5, 12, -6], [8, 0, 5, 3, 9, -12, 11], [35, -16, -1, -4, -3, -8, -19], [-32, 7, 20, 5, 5, -9, -2], [4, -21, 6, -13, -2, 11, -10], [-17, 24, 5, 1, -1, 4, 18], [-31, 4, 12, 15, 6, -6, 4], [-32, 5, 16, 16, -7, -6, 12], [-5, 6, -6, 10, 20, 5, -3], [-4, 8, -9, -6, -14, -13, -7], [-55, 10, 6, -2, -9, 0, 1], [39, -6, -15, -3, 14, -6, 19], [6, -27, -2, -27, -6, -4, -15], [-34, 23, 10, -2, -6, -1, 3], [-26, 10, 5, -13, -29, -18, -30], [33, -6, -23, 5, 24, 0, -2], [-7, -14, -6, -1, 22, 11, 5], [-19, 27, 4, 6, 1, -7, -15], [-5, -9, 5, -26, -23, -12, -19], [7, -2, -12, -16, 2, -6, 1], [27, -27, -7, 2, 1, 9, 9], [0, 6, 12, 5, -4, -5, -17], [-18, -14, -6, 0, -3, -16, -30], [31, -15, -10, -3, -2, 10, 4], [29, 3, 3, -8, 8, -14, 3], [-18, 24, -8, 23, 2, 11, 14], [-11, -6, 11, 5, 5, -8, 2], [4, 24, -4, 2, 1, 3, 18], [-12, -1, -3, -6, 6, -4, -16], [-3, 3, -15, 7, 4, 3, 9], [16, -2, -20, 2, 29, 5, 11], [-22, 28, -4, 8, 3, 3, 2], [4, 0, 3, -2, 16, 1, 14], [-39, 13, 20, 14, 11, 4, 2], [-5, 0, 9, -7, -21, 1, -3], [-12, 12, 8, 8, 13, 8, 21], [-21, 18, 5, 3, -10, 8, -17], [-19, -21, 4, 7, -4, 0, 5], [30, -9, -2, 2, 19, 15, 7], [-34, 14, -1, 10, -7, 4, 18], [25, -28, -5, -9, -8, -19, 2], [18, -11, -1, -24, -18, -16, -23], [9, 17, 2, -7, -18, -8, -1], [-1, 12, 22, -11, 2, 10, 0], [-5, 11, 5, 8, -21, -6, -9], [-51, 17, 13, 6, 4, -2, 9], [23, -27, -4, -5, 20, 5, 10], [-36, -6, 20, 4, -10, -6, -10], [37, -10, 6, -15, 1, 11, -20], [19, -26, -11, 1, 11, 2, -1], [3, -9, -10, 9, 7, -6, 4], [-5, 13, 19, 16, -11, 0, 0], [-8, 31, 4, 11, -1, -5, 1], [16, -6, -14, -9, 17, -8, -1], [-5, 0, -5, -6, 0, 10, -18], [-14, 14, 10, -15, -2, 3, 1], [-8, -23, 13, -9, -2, 9, 9], [-49, 23, -1, 13, -17, -10, 13], [19, 21, 9, 3, -3, -12, -3], [-11, -10, 4, -3, -21, -24, -11], [-35, 8, 7, -7, -25, 0, -4], [15, 4, -3, 2, -5, 14, 8], [-14, 26, -4, 14, -2, -15, 8], [-24, -4, 2, 17, 2, 8, -7], [-10, -3, 8, -4, 0, -25, -16], [14, -17, -21, -23, 11, 3, 0], [-6, 23, 7, 1, 1, -12, 9], [-11, -3, -7, 0, 8, 14, -4], [-20, 5, 18, 12, 6, 6, -10], [-14, -16, -2, -12, -11, -6, -10], [6, -7, -14, -10, -2, -9, -4], [1, -22, -10, 2, 11, 9, -16], [-35, 19, 8, 9, -1, 7, 25], [-23, -8, 16, 12, -8, 11, 2], [-41, 17, 5, 9, -8, -14, -5], [-51, 30, 4, 16, -5, 2, 28], [19, -25, -17, 1, 7, -8, -20], [-38, 9, 6, 3, -8, -25, -22], [-42, 13, -1, 10, -5, -11, -1], [-17, -13, -6, 0, -5, 20, -14], [-22, 8, 19, 2, -14, -14, -9], [23, -16, 8, -3, 1, 9, 27], [16, 20, 10, 17, -6, 13, 4], [-12, 7, 19, -12, -20, -18, -15]];
    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_bc();".
// 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_bc(: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_bc();".
// 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_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -15, 0, 18, -5, -3, 1]>,<7,R![5, -6, -53, -17, 51, -7, -5, 1]>,<11,R![1137, 39, -844, 161, 126, -31, -4, 1]>],Snew);
    return Vf;
end function;