
// Make newform 2548.2.j.r in Magma, downloaded from the LMFDB on 18 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2548_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2548_j_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2548_2_j_r();".
// 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_2548_2_j_r();".
// 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, -32, 216, -336, 484, -240, 220, -56, 80, -8, 10, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12684696, -1728720, 10721360, 147974882, -25453960, 68974400, 4327072, 28166440, 152800, 3778645, 322300, 379000], [1118396056, 32244016, -207884308, -335361882, 43991346, -602456824, -290522622, -264964348, -37653720, -41548809, -7830811, -3905120], [270746912, -21835548, 138125424, 882781546, -174283368, 637429512, 120064991, 266775954, 13824960, 38183017, 4692768, 3705810], [510976, -55654, 347652, 3045328, -683794, 2005116, 187358, 824777, 15880, 114461, 10949, 11205], [103206288, -695694944, 917539582, -1509394256, 692107360, -701985352, 149419304, -253846720, 21590747, -32034040, -379000, -3171174], [-826492776, 5597355288, -10902283748, 13509795300, -7801088064, 5742246638, -2056338684, 2147759760, -293907002, 262540674, -7411620, 27192621], [139153820, -4561916576, 7672711674, -12671186152, 6423390232, -6008662636, 1320236584, -2284521748, 197195993, -291648120, -4899106, -29498324], [-608382084, 73440136, -455869368, -4914958702, 1059533356, -2895561024, -178289157, -1183392788, -8253520, -161115064, -13784186, -15940620], [201658580, -8916475484, 11096043566, -18207025768, 7596062558, -8358574604, 1740177496, -2920168012, 245386747, -364956680, -2624119, -35536656], [-1617413908, 10957605904, -23001774114, 26645163240, -15595168232, 11491856774, -4149316202, 4219807680, -582294921, 514656112, -16024810, 52697053]];
    Rf_basisdens := [1, 1, 86544668, 432723340, 216361670, 347290, 86544668, 432723340, 432723340, 216361670, 432723340, 432723340];
    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_2548_j();"
function MakeCharacter_2548_j()
    N := 2548;
    order := 3;
    char_gens := [1275, 885, 197];
    v := [3, 2, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2548_j_Hecke();"
function MakeCharacter_2548_j_Hecke(Kf)
    N := 2548;
    order := 3;
    char_gens := [1275, 885, 197];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 2, -1, 0, 0, 1, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, -1, 0, -2, 0, -1, 0, 0, -1], [-1, -1, 0, 0, 0, 2, 1, 2, 1, 1, 2, -1], [0, -1, 0, 0, 0, 0, 0, 0, 0, 2, 1, -2], [1, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, -1, 0, 0, -1, 1], [-3, 0, 0, 0, 0, -2, 3, -2, -2, -2, -1, 2], [3, 0, 2, -2, -2, -1, 0, 0, 0, -1, 0, 0], [1, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0], [-5, -1, 0, 0, 0, 2, 5, 2, 1, 1, 0, -1], [0, 3, 3, -3, 0, 0, 2, -2, 0, 0, -3, 0], [0, 1, 1, -1, 3, 0, -2, -1, 3, 0, -1, 0], [-2, 0, 0, 0, 0, -2, 2, -2, -1, -1, 2, 1], [0, 2, 2, 0, 1, 0, 0, -3, 1, 0, 0, -1], [-2, 0, 0, 3, 2, -2, 0, 0, 0, 1, 0, 0], [0, 3, 3, 0, -1, 0, -3, 2, -1, 0, 0, 3], [-2, 0, 0, 0, 0, -4, 2, -4, 1, 0, 0, 0], [-4, 0, 2, 3, 2, -2, 0, 0, 0, 0, 0, 0], [-4, -3, 0, 0, 0, -5, 4, -5, -1, -2, 2, 2], [4, 0, 1, 0, -3, -1, 0, 0, 0, 2, 0, 0], [0, -3, -3, 0, -1, 0, -8, -2, -1, 0, 0, -1], [-4, 2, 0, 0, 0, 1, 4, 1, 3, -1, 0, 1], [1, 3, 0, 0, 0, 0, -1, 0, 4, 2, 1, -2], [0, 0, 0, -1, 1, 0, 1, -5, 1, 0, -1, 1], [-1, 0, 5, -1, -3, -2, 0, 0, 0, 0, 0, 0], [-3, 0, 3, 3, -2, 0, 0, 0, 0, 2, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, -2], [0, -4, -4, 3, 3, 0, 1, 1, 3, 0, 3, -3], [-6, 0, -3, 2, 4, 3, 0, 0, 0, -2, 0, 0], [8, 2, 0, 0, 0, 1, -8, 1, -1, 0, 1, 0], [0, 0, 0, -3, -5, 0, 5, -5, -5, 0, -3, -1], [0, 1, 1, 0, -2, 0, -4, -6, -2, 0, 0, 0], [-6, 0, 0, 0, 0, -3, 6, -3, -1, 1, -6, -1], [1, 0, 3, 2, 3, -2, 0, 0, 0, -3, 0, 0], [-6, 1, 0, 0, 0, 5, 6, 5, -2, 2, 2, -2], [0, -4, -4, 4, 2, 0, -3, 2, 2, 0, 4, -4], [-4, 0, 1, 4, 3, -1, 0, 0, 0, 1, 0, 0], [-5, 1, 0, 0, 0, -10, 5, -10, 0, -4, 3, 4], [0, -4, -4, 0, 2, 0, -8, -6, 2, 0, 0, 0], [-1, 0, 0, 4, 5, -3, 0, 0, 0, -2, 0, 0], [0, -7, -7, 2, 4, 0, -4, -6, 4, 0, 2, 0], [-1, 0, 7, -3, -3, -2, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0], [0, -2, -2, -4, 0, 0, 2, 0, 0, 0, -4, 2], [-6, -3, 0, 0, 0, -3, 6, -3, -3, -2, -5, 2], [9, 2, 0, 0, 0, -4, -9, -4, 2, -4, -2, 4], [7, 0, 2, -8, -1, -7, 0, 0, 0, 2, 0, 0], [0, -6, -6, 1, 2, 0, 3, -1, 2, 0, 1, -3], [-6, 0, 5, -6, 0, 5, 0, 0, 0, 6, 0, 0], [-12, -3, 0, 0, 0, 5, 12, 5, 1, 3, 2, -3], [0, 2, 2, -4, -3, 0, 0, 2, -3, 0, -4, 2], [0, 10, 10, -6, 2, 0, 0, -3, 2, 0, -6, 2], [0, 0, 0, 0, 0, -4, 0, -4, 2, 2, -6, -2], [0, -6, -6, 2, 3, 0, 0, 2, 3, 0, 2, -2], [-6, 0, 6, -1, -7, -5, 0, 0, 0, 2, 0, 0], [0, 4, 4, 6, 1, 0, 4, -1, 1, 0, 6, 1], [11, 0, 1, 2, -3, -6, 0, 0, 0, -1, 0, 0], [5, 0, -2, -9, 0, 3, 0, 0, 0, -1, 0, 0], [0, -1, -1, 2, -1, 0, -2, 13, -1, 0, 2, 1], [0, 3, 0, 0, 0, 3, 0, 3, 7, 3, 4, -3], [3, 8, 0, 0, 0, 4, -3, 4, 0, -5, -4, 5], [1, -4, 0, 0, 0, 0, -1, 0, 0, 5, -2, -5], [-1, 0, 1, -5, -5, 6, 0, 0, 0, 0, 0, 0], [0, 4, 4, -2, -1, 0, -11, 6, -1, 0, -2, 0], [5, 0, 5, -6, -3, 10, 0, 0, 0, 5, 0, 0], [0, -1, -1, -1, 3, 0, -10, -3, 3, 0, -1, -2], [10, -6, 0, 0, 0, 1, -10, 1, 3, 6, 3, -6], [0, 9, 9, -10, -3, 0, 6, 0, -3, 0, -10, 1], [-7, -7, 0, 0, 0, 4, 7, 4, 4, 2, 3, -2], [-4, 0, -6, 5, 0, 10, 0, 0, 0, 5, 0, 0], [6, -4, 0, 0, 0, -2, -6, -2, 2, -6, -4, 6], [0, -1, -1, -3, -11, 0, 8, 2, -11, 0, -3, 6], [-4, 3, 0, 0, 0, -1, 4, -1, -3, 0, -8, 0], [-9, -2, 0, 0, 0, 1, 9, 1, -1, 4, 4, -4], [0, -5, -5, 4, 7, 0, 13, 2, 7, 0, 4, 1], [-2, 0, -7, 2, -1, -1, 0, 0, 0, 1, 0, 0], [-5, 0, 0, 9, 3, -7, 0, 0, 0, -3, 0, 0], [0, 0, 0, 2, 4, 0, -1, 12, 4, 0, 2, -1], [4, 0, 14, -4, -1, -2, 0, 0, 0, 3, 0, 0], [6, -2, 0, 0, 0, -11, -6, -11, 4, 0, -2, 0], [-20, 7, 0, 0, 0, 2, 20, 2, -2, -4, -1, 4], [-14, 0, -4, 10, 5, -1, 0, 0, 0, 1, 0, 0], [2, 4, 0, 0, 0, 18, -2, 18, 2, -3, 1, 3], [8, 0, 4, -8, 0, -4, 0, 0, 0, 0, 0, 0], [-13, 0, 4, -1, -7, -3, 0, 0, 0, 3, 0, 0], [14, -3, 0, 0, 0, 3, -14, 3, -2, 0, 2, 0], [0, -2, -2, 3, 4, 0, -4, -10, 4, 0, 3, -6], [0, -6, -6, 0, 6, 0, 8, 2, 6, 0, 0, 0], [2, 0, -5, -7, 5, -4, 0, 0, 0, -2, 0, 0], [22, -4, 0, 0, 0, -5, -22, -5, 3, 0, 1, 0], [12, 0, 0, -6, -3, 10, 0, 0, 0, 1, 0, 0], [-5, -1, 0, 0, 0, 0, 5, 0, 7, 1, 0, -1], [0, -6, -6, 8, 5, 0, -16, -11, 5, 0, 8, -5], [10, 2, 0, 0, 0, -3, -10, -3, 6, -6, -2, 6], [12, 0, 0, 0, 0, 2, -12, 2, 4, -1, -5, 1], [-3, 0, 7, 3, -1, 6, 0, 0, 0, 2, 0, 0], [0, -4, -4, -6, -1, 0, -2, 7, -1, 0, -6, 3], [0, -4, -4, -2, 7, 0, 4, -10, 7, 0, -2, -3], [-8, -8, 0, 0, 0, 4, 8, 4, -3, 2, 0, -2], [0, -5, -5, 5, 2, 0, 2, -14, 2, 0, 5, 4], [0, -2, -2, 2, -4, 0, 5, 7, -4, 0, 2, 5], [-2, 0, 0, 1, -4, 0, 0, 0, 0, -4, 0, 0], [-5, -6, 0, 0, 0, 11, 5, 11, 4, 5, 5, -5], [0, -4, -4, 2, 2, 0, -9, 14, 2, 0, 2, 2], [-2, 0, 4, 0, 1, 1, 0, 0, 0, -1, 0, 0], [10, -1, 0, 0, 0, 0, -10, 0, -4, 2, 8, -2], [0, 0, 0, 12, 4, 0, -6, 14, 4, 0, 12, -2], [-12, 0, 4, 2, 0, -10, 0, 0, 0, 0, 0, 0], [0, 2, 2, -4, 2, 0, 10, 12, 2, 0, -4, 6], [1, 0, -10, 6, 0, -6, 0, 0, 0, 1, 0, 0], [0, -2, -2, 2, -4, 0, -1, -4, -4, 0, 2, 8], [-23, 0, 2, 4, 8, -7, 0, 0, 0, -3, 0, 0], [0, 9, 9, -4, -7, 0, -10, -6, -7, 0, -4, 3], [1, -14, 0, 0, 0, -6, -1, -6, 3, 2, 4, -2], [16, 0, 5, -3, -8, -12, 0, 0, 0, 4, 0, 0], [0, 8, 8, 1, 6, 0, 8, -10, 6, 0, 1, -6], [12, 0, 9, -3, 0, 2, 0, 0, 0, 2, 0, 0], [-10, 7, 0, 0, 0, -13, 10, -13, -8, -6, -10, 6], [0, -4, -4, -6, 6, 0, 13, 6, 6, 0, -6, -7], [5, 15, 0, 0, 0, 6, -5, 6, -5, -5, -4, 5], [0, 0, 2, -2, -1, -14, 0, 0, 0, 2, 0, 0], [11, -8, 0, 0, 0, 2, -11, 2, -4, -4, 3, 4], [-4, -3, 0, 0, 0, 0, 4, 0, -2, 6, -8, -6], [14, 0, -6, -2, -2, 0, 0, 0, 0, 4, 0, 0], [2, -2, 0, 0, 0, 12, -2, 12, -2, 2, -9, -2], [0, -2, -2, -11, -4, 0, -11, -14, -4, 0, -11, 2], [7, 0, 6, -10, 2, -8, 0, 0, 0, 3, 0, 0], [-7, 4, 0, 0, 0, -14, 7, -14, 0, 2, -2, -2], [8, 0, 6, -10, -9, 12, 0, 0, 0, 2, 0, 0], [12, -10, 0, 0, 0, 2, -12, 2, 6, 0, 17, 0], [3, 0, -2, -1, 8, -9, 0, 0, 0, -5, 0, 0], [0, -8, -8, 2, 2, 0, -25, -5, 2, 0, 2, 1], [0, -6, -6, 11, 0, 0, -2, 6, 0, 0, 11, -2], [18, 0, 10, 6, 5, -1, 0, 0, 0, -3, 0, 0], [0, 6, 6, -2, 7, 0, -2, 4, 7, 0, -2, 2], [-9, 0, -4, 0, 6, -11, 0, 0, 0, -3, 0, 0], [7, -6, 0, 0, 0, -17, -7, -17, 3, 4, 2, -4], [0, 3, 3, -5, -1, 0, 10, -2, -1, 0, -5, 0], [-2, 0, 4, -2, 1, -7, 0, 0, 0, -7, 0, 0], [0, -5, -5, 6, 5, 0, 10, 9, 5, 0, 6, -13], [11, 0, 16, 0, 2, 3, 0, 0, 0, 3, 0, 0], [-11, 0, -16, 5, 0, 3, 0, 0, 0, -3, 0, 0], [0, -6, -6, -8, -1, 0, -12, 10, -1, 0, -8, 1], [-6, 0, 0, 0, 0, -23, 6, -23, -5, -3, -10, 3], [-2, 8, 0, 0, 0, 0, 2, 0, -2, 2, -14, -2], [17, 10, 0, 0, 0, 10, -17, 10, -4, 4, -9, -4], [-8, 0, 0, 6, 2, 13, 0, 0, 0, 4, 0, 0], [-3, 0, 5, -7, -2, 20, 0, 0, 0, 6, 0, 0], [-8, 4, 0, 0, 0, 1, 8, 1, -9, -3, -2, 3], [0, 7, 7, -1, 10, 0, 4, -2, 10, 0, -1, 4], [-10, 0, 0, -2, -3, -4, 0, 0, 0, 8, 0, 0], [-11, -14, 0, 0, 0, 2, 11, 2, -3, 4, 16, -4], [-17, 10, 0, 0, 0, 3, 17, 3, -4, -9, -3, 9], [6, 0, -5, -4, -10, 9, 0, 0, 0, -4, 0, 0], [0, 0, 0, -7, 8, 0, -2, 14, 8, 0, -7, 0], [-25, -8, 0, 0, 0, 1, 25, 1, -9, -2, 4, 2], [1, 0, 4, -2, 4, -2, 0, 0, 0, 0, 0, 0], [-25, 0, -4, 9, 8, 2, 0, 0, 0, 0, 0, 0], [6, -12, 0, 0, 0, -12, -6, -12, -1, 1, -4, -1], [0, -12, -12, 9, 0, 0, 2, -4, 0, 0, 9, -9], [0, 3, 3, -4, 3, 0, -21, 20, 3, 0, -4, 1], [0, -7, -7, -1, -1, 0, 12, -2, -1, 0, -1, -12], [0, -4, -4, -2, -9, 0, 8, -19, -9, 0, -2, -5], [0, 0, 10, -1, 2, -10, 0, 0, 0, -11, 0, 0], [-14, 9, 0, 0, 0, -8, 14, -8, -13, -11, -8, 11], [0, 6, 6, 3, -7, 0, 18, 11, -7, 0, 3, 2], [-28, 0, 0, -2, 2, -10, 0, 0, 0, -2, 0, 0], [2, 7, 0, 0, 0, -22, -2, -22, 0, -4, -9, 4], [0, 10, 10, -7, -12, 0, 22, 10, -12, 0, -7, 1], [0, 4, 4, 4, 8, 0, -24, 12, 8, 0, 4, 4], [10, 0, -6, 4, 0, -13, 0, 0, 0, -8, 0, 0], [4, 0, 2, 6, 0, 16, 0, 0, 0, -2, 0, 0], [0, 0, 0, -2, 3, 0, 8, -13, 3, 0, -2, -5], [-6, 0, -3, 2, -5, 2, 0, 0, 0, -1, 0, 0], [8, -7, 0, 0, 0, 8, -8, 8, 11, 11, 6, -11], [27, 0, 0, 0, 0, -9, -27, -9, -5, -10, -6, 10], [4, 0, -5, -6, 1, 2, 0, 0, 0, 7, 0, 0], [7, 0, -11, 7, -4, -4, 0, 0, 0, -6, 0, 0], [-25, -1, 0, 0, 0, 8, 25, 8, -1, -5, 0, 5], [0, 0, 0, 5, -2, 0, -40, 2, -2, 0, 5, 3], [0, -1, -1, 8, -4, 0, -26, 0, -4, 0, 8, 2], [0, 13, 13, -11, -8, 0, 16, -8, -8, 0, -11, 12], [0, 4, 4, 1, 0, 0, 12, 8, 0, 0, 1, 2], [-13, -4, 0, 0, 0, -13, 13, -13, -11, -6, 2, 6], [0, 1, 1, -6, 5, 0, -20, -5, 5, 0, -6, 2], [-10, 1, 0, 0, 0, -9, 10, -9, -8, 4, 2, -4], [0, 0, 0, 4, 9, 1, 0, 0, 0, -1, 0, 0], [-27, -11, 0, 0, 0, -8, 27, -8, -3, 4, -1, -4], [-8, 0, 0, 0, 0, -22, 8, -22, 2, 4, -9, -4], [0, -9, -9, -11, 4, 0, 26, -2, 4, 0, -11, -4], [0, 1, 1, -6, 1, 0, -4, -9, 1, 0, -6, -1], [0, 16, 16, 1, 1, 0, 2, -7, 1, 0, 1, -4], [-4, -8, 0, 0, 0, 14, 4, 14, 11, 10, 10, -10], [17, 0, 14, -7, -4, -19, 0, 0, 0, 5, 0, 0], [-14, 2, 0, 0, 0, 1, 14, 1, 13, 3, 6, -3], [0, 18, 18, 13, 3, 0, 5, 17, 3, 0, 13, 3], [-2, 0, 3, 8, 13, 1, 0, 0, 0, -9, 0, 0], [-20, -13, 0, 0, 0, -19, 20, -19, -9, -6, -4, 6], [0, 2, 2, 18, 4, 0, -12, -4, 4, 0, 18, -6], [10, 0, 5, -10, 1, 27, 0, 0, 0, 0, 0, 0], [0, 10, 10, -10, -9, 0, -26, 3, -9, 0, -10, 1], [-27, -6, 0, 0, 0, -15, 27, -15, -6, -1, 1, 1], [-26, -6, 0, 0, 0, 2, 26, 2, 5, 10, 0, -10], [-5, 0, -2, 5, -6, -6, 0, 0, 0, 0, 0, 0], [0, 12, 12, -8, -3, 0, 24, 6, -3, 0, -8, -5], [7, 6, 0, 0, 0, -6, -7, -6, -10, -8, 6, 8], [30, 0, 15, -22, -8, 3, 0, 0, 0, 0, 0, 0], [-4, 0, -2, -10, 0, 10, 0, 0, 0, -4, 0, 0], [1, 14, 0, 0, 0, 13, -1, 13, 0, 1, -4, -1], [0, 2, 2, -1, -6, 0, -16, 22, -6, 0, -1, 2], [24, -6, 0, 0, 0, 4, -24, 4, -2, 0, 1, 0], [0, 6, 6, -4, 2, 0, -1, -2, 2, 0, -4, -4], [0, -8, -8, 6, 0, 0, -14, -8, 0, 0, 6, -8], [-6, 10, 0, 0, 0, 11, 6, 11, -1, 1, -6, -1], [20, 0, -22, 3, 9, -3, 0, 0, 0, -6, 0, 0], [-2, 15, 0, 0, 0, -12, 2, -12, 6, 4, 1, -4], [-2, 0, 7, 1, -7, 3, 0, 0, 0, 8, 0, 0], [7, -20, 0, 0, 0, -13, -7, -13, 3, 6, 2, -6], [4, 2, 0, 0, 0, 20, -4, 20, 12, 8, 14, -8], [-2, 0, -14, 6, 9, 3, 0, 0, 0, -3, 0, 0], [-18, 0, 1, 5, 10, 16, 0, 0, 0, -6, 0, 0], [-30, 10, 0, 0, 0, -8, 30, -8, -1, -7, -6, 7], [0, -4, -4, 0, 4, 0, -21, 12, 4, 0, 0, 4], [-22, -1, 0, 0, 0, -13, 22, -13, -6, 0, 2, 0], [22, 6, 0, 0, 0, -10, -22, -10, 6, 3, -7, -3], [0, 3, 3, 11, -2, 0, -25, 4, -2, 0, 11, -2], [0, -27, -27, 12, 6, 0, 6, -4, 6, 0, 12, -2], [3, 0, -14, 12, -2, -16, 0, 0, 0, -7, 0, 0], [0, -7, -7, 3, -1, 0, 4, 0, -1, 0, 3, -18], [18, 0, -2, 8, -3, -7, 0, 0, 0, -1, 0, 0], [0, -5, -5, 10, 5, 0, -14, 13, 5, 0, 10, 0], [-10, 8, 0, 0, 0, 6, 10, 6, -8, 6, 0, -6], [-19, 0, 0, 0, 0, -1, 19, -1, -3, 6, 14, -6], [-15, 0, 7, -11, -7, 2, 0, 0, 0, 6, 0, 0], [-2, 0, 12, 10, 5, -11, 0, 0, 0, 5, 0, 0], [0, 18, 18, -8, -5, 0, 6, -3, -5, 0, -8, 9], [-6, -14, 0, 0, 0, -13, 6, -13, 2, 6, 6, -6], [0, 10, 10, -8, 13, 0, -14, -16, 13, 0, -8, -1], [-17, -2, 0, 0, 0, 1, 17, 1, 1, -2, -6, 2], [2, 0, 6, -3, 0, -6, 0, 0, 0, -2, 0, 0], [10, -17, 0, 0, 0, 0, -10, 0, -4, 0, 4, 0], [21, 0, -7, 14, 9, -2, 0, 0, 0, -3, 0, 0], [0, 6, 6, -7, 2, 0, 18, -12, 2, 0, -7, -8], [0, -4, -4, 7, 0, 0, 1, -4, 0, 0, 7, -10], [-1, 0, -11, 5, -1, -12, 0, 0, 0, -6, 0, 0], [0, 0, 26, -4, -4, -10, 0, 0, 0, 12, 0, 0], [10, -20, 0, 0, 0, 4, -10, 4, 2, 8, 6, -8], [0, 0, 0, 8, 2, 0, -23, 8, 2, 0, 8, 13], [-18, 0, 8, 10, 14, -1, 0, 0, 0, -10, 0, 0], [0, 10, 10, 0, -4, 0, -12, -13, -4, 0, 0, 0], [11, 0, 0, 0, 0, 18, -11, 18, 16, 5, 18, -5], [0, -4, -4, 6, 13, 0, 10, 8, 13, 0, 6, -12], [0, -2, -2, 6, 0, 0, 3, 13, 0, 0, 6, -11], [2, 0, -20, -4, 0, 2, 0, 0, 0, 2, 0, 0], [17, 34, 0, 0, 0, 5, -17, 5, -2, -3, -9, 3], [0, -23, -23, 19, 8, 0, -2, -12, 8, 0, 19, 4], [10, 0, 10, 4, 11, 10, 0, 0, 0, 4, 0, 0], [0, -1, -1, -10, -7, 0, -7, 6, -7, 0, -10, 3], [2, 0, 1, 6, -4, 6, 0, 0, 0, 12, 0, 0], [0, 4, 4, -2, 0, 0, -18, 10, 0, 0, -2, -12], [-10, 0, 0, 0, 0, -11, 10, -11, -12, -16, -2, 16], [-19, 0, 0, -14, -9, 5, 0, 0, 0, 4, 0, 0], [-19, 0, 3, 8, 11, -12, 0, 0, 0, -5, 0, 0], [16, 0, 6, 1, -9, 9, 0, 0, 0, 4, 0, 0], [0, 5, 5, 3, 4, 0, -1, -12, 4, 0, 3, -2], [6, -1, 0, 0, 0, 12, -6, 12, 1, -5, -14, 5], [0, -10, -10, 11, 7, 0, 7, -1, 7, 0, 11, -1], [0, 8, 8, 12, 0, 0, 8, -6, 0, 0, 12, -6], [-12, -3, 0, 0, 0, -14, 12, -14, 8, 2, -13, -2], [4, 0, 7, -22, -4, 19, 0, 0, 0, 6, 0, 0], [5, 0, 0, 0, 0, -23, -5, -23, -4, -11, 8, 11], [8, -6, 0, 0, 0, 12, -8, 12, -4, -5, -1, 5], [0, -6, -6, 9, 14, 0, 14, 8, 14, 0, 9, -5], [7, -2, 0, 0, 0, -14, -7, -14, 18, 1, 8, -1], [24, 18, 0, 0, 0, -4, -24, -4, -5, -9, -2, 9], [0, -14, -14, 4, 14, 0, -12, 2, 14, 0, 4, -16], [0, 2, 2, -14, -5, 0, -25, -25, -5, 0, -14, 18], [16, 0, -12, 16, 0, 6, 0, 0, 0, 0, 0, 0], [2, 0, -9, -4, -1, -1, 0, 0, 0, -8, 0, 0], [14, -5, 0, 0, 0, 7, -14, 7, -5, -6, 11, 6], [20, -14, 0, 0, 0, 8, -20, 8, 7, 6, 12, -6], [0, 6, 6, -20, -1, 0, 8, 10, -1, 0, -20, -2], [4, 0, 10, 18, 6, -8, 0, 0, 0, -4, 0, 0], [0, 0, 0, 16, -6, 0, -26, 4, -6, 0, 16, 6], [24, 0, -9, 0, 4, 19, 0, 0, 0, -8, 0, 0], [0, -7, -7, 15, 12, 0, -16, -8, 12, 0, 15, 0], [0, -24, -24, 14, 10, 0, 20, 0, 10, 0, 14, -4], [31, -4, 0, 0, 0, -18, -31, -18, 2, -5, 4, 5], [-34, 0, -7, -8, 5, 13, 0, 0, 0, 4, 0, 0], [-16, 0, -11, 3, -2, 4, 0, 0, 0, 14, 0, 0], [0, -8, -8, 10, -3, 0, 16, 16, -3, 0, 10, -5], [10, 30, 0, 0, 0, -12, -10, -12, 4, -2, -14, 2], [-4, 0, 8, 6, -4, 0, 0, 0, 0, -6, 0, 0], [-6, 19, 0, 0, 0, 5, 6, 5, -7, 5, -12, -5], [39, 0, 12, -1, 8, 7, 0, 0, 0, -9, 0, 0], [56, 3, 0, 0, 0, 1, -56, 1, 7, 3, -4, -3], [29, -4, 0, 0, 0, -5, -29, -5, -9, -7, -5, 7], [0, -13, -13, -3, 1, 0, -3, 14, 1, 0, -3, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2548_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2548_2_j_r();".
// 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_2548_2_j_r(:prec:=12)
    chi := MakeCharacter_2548_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2548_2_j_r();".
// 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_2548_2_j_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2548_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![196, -112, 456, 112, 648, 80, 324, 56, 116, 8, 12, 0, 1]>,<5,R![49, 42, 365, -394, 1958, -654, 1381, 1178, 808, 274, 71, 10, 1]>],Snew);
    return Vf;
end function;