
// Make newform 378.2.l.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_378_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_378_l_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_378_2_l_a();".
// 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_378_2_l_a();".
// 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 := [6561, -17496, 16767, -1944, -10611, 10260, -2601, -2640, 2785, -880, -289, 380, -131, -8, 23, -8, 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, 0, 0], [2825604, -4444713, 1318518, 2372247, -2730942, 756198, 724572, -757363, 206194, 107467, -107246, 27422, 7676, -6827, 1352, 50], [-244944, 629856, -524799, -55431, 432351, -313347, 13734, 128722, -90010, 12131, 19055, -13199, 2308, 1319, -866, 169], [-1285227, 2135727, -949725, -871830, 1355805, -566733, -226785, 382978, -159688, -23518, 55022, -22478, -224, 3608, -1325, 154], [-7117227, 13935564, -8165529, -3909951, 8827731, -4779069, -693301, 2419149, -1283242, 13019, 334353, -172766, 15627, 21081, -9836, 1445], [-14041269, 27464589, -16197165, -7519770, 17293545, -9430341, -1293175, 4717386, -2521996, 37829, 649239, -338066, 31392, 40866, -19265, 2858], [42998607, -89475273, 57644217, 20576268, -57047193, 33710346, 2544783, -15633236, 9027170, -494788, -2154880, 1215277, -145859, -136087, 70123, -11192], [-69445998, 142069707, -88915158, -35165097, 90692622, -52089894, -5100702, 24892859, -13978340, 565132, 3436840, -1883968, 208526, 217033, -108472, 16898], [14963454, -31653180, 20790351, 7028856, -20322468, 12187116, 796053, -5585140, 3269857, -200900, -771188, 441224, -55006, -48788, 25571, -4120], [-30865131, 65494089, -43148781, -14468679, 42144849, -25345758, -1602394, 11591968, -6806990, 428149, 1601225, -919066, 115567, 101261, -53254, 8586], [102032298, -212064642, 135776493, 50026572, -135722682, 79507809, 6581337, -37270684, 21324130, -1063457, -5145350, 2873363, -334861, -325403, 165707, -26068], [117273501, -244082322, 157282722, 56669868, -156501153, 92428227, 7063563, -43003982, 24841391, -1366837, -5936959, 3354958, -405716, -374983, 194263, -31130], [-121811526, 256798269, -167924421, -57791394, 165029724, -98586783, -6735609, 45384563, -26474600, 1580329, 6268915, -3574291, 441842, 396466, -207064, 33311], [-18826182, 40175190, -26672436, -8698959, 25851399, -15657912, -906329, 7108452, -4204439, 280045, 981351, -567820, 72930, 62025, -32959, 5368], [244346949, -507326409, 324340605, 120144465, -324842130, 190088901, 15905550, -89230126, 51016561, -2527469, -12322064, 6878696, -802747, -778856, 397070, -62668], [-18816948, 39638646, -25866621, -8949609, 25447554, -15172956, -1054329, 6992981, -4072523, 239686, 965497, -549514, 67583, 61039, -31804, 5105]];
    Rf_basisdens := [1, 142155, 47385, 47385, 47385, 47385, 142155, 142155, 28431, 47385, 142155, 142155, 142155, 15795, 142155, 10935];
    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_378_l();"
function MakeCharacter_378_l()
    N := 378;
    order := 6;
    char_gens := [29, 325];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_378_l_Hecke();"
function MakeCharacter_378_l_Hecke(Kf)
    N := 378;
    order := 6;
    char_gens := [29, 325];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 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+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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1], [1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0], [2, -1, -1, 0, 0, 0, -1, 0, -2, 0, 0, 1, 0, 1, 0, 1], [0, 1, 1, 0, 1, -1, 1, -1, 0, -1, 0, 0, -1, -1, -1, 1], [-1, 0, 1, -2, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, -1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 0], [0, 3, 1, 1, -2, 0, -1, 3, 1, 0, 0, 0, 0, 0, 0, 0], [1, 3, 1, -1, 1, -1, 2, 3, 0, 1, -2, 1, 1, -1, 1, -2], [1, 0, 0, 0, 1, 1, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0], [0, 0, -1, -1, 1, 0, 0, 0, -1, 1, -1, -2, -1, -1, 1, -2], [3, 0, -3, 2, 1, 1, 2, 0, -2, -1, -1, -1, 0, 1, 0, -1], [1, -1, 0, 2, -1, -1, -2, -2, 2, -2, 2, -1, -2, 1, -1, 1], [-3, 0, -1, 2, -1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 1, -2], [-3, 2, 1, -2, 2, 1, 1, 1, 7, -1, 1, 1, 2, 0, -2, 0], [-4, 0, 0, -1, -3, -1, -2, 0, 2, 0, 1, 0, 0, 0, 0, 0], [-1, -1, 1, -2, -1, 1, 0, 1, -2, 3, 1, -3, -1, -1, 1, -1], [2, 1, -1, 2, 1, 0, 1, -4, -1, 0, -1, 1, 1, 1, 1, -1], [1, 0, -4, 1, -1, 1, 1, 0, -1, -1, -1, -1, -1, 1, -1, -1], [0, 0, 0, 2, -2, -1, -1, 0, 1, -1, 0, 2, 0, 1, -1, 2], [1, 1, -1, 1, 1, -1, 4, -2, 2, 0, -1, 2, 0, 0, 1, -2], [0, -1, -1, 2, 0, -2, -1, 4, 0, -2, 0, -1, 0, 1, 2, 1], [2, 2, 4, -2, 0, -1, -2, 2, 8, 2, -3, 0, 2, -2, 1, -2], [2, 0, -3, 2, 1, 2, 1, 3, 1, -2, 1, 0, -1, 2, -3, 0], [-2, -2, -5, 0, -1, 3, -1, 2, -1, 2, -3, -1, 0, 2, 3, -3], [1, -3, -4, 0, 1, 1, 0, 0, -4, 2, -2, -1, -2, 1, 3, -1], [-4, 2, 1, 0, -3, -2, -3, 1, 4, 1, 1, 2, -1, 1, -2, 1], [-3, -6, -1, -2, -1, 1, 0, 6, 0, 1, -1, 1, 0, -1, 0, 1], [2, 5, 5, -2, 2, 0, 3, 0, 0, -2, 2, -1, 2, -3, -2, -1], [-10, 1, -1, 1, 0, 2, 1, 2, 4, 2, -1, 2, 1, 2, 2, 0], [-5, 7, -2, 0, 1, 1, 1, -1, 1, -1, -1, 0, 0, -1, 1, 1], [3, 1, 1, -1, -1, -1, 1, 5, 3, 2, -1, 1, 1, -1, 2, -1], [-2, 0, 5, -2, -5, 2, -1, 0, 0, 2, 4, -1, -2, -2, -2, -1], [-1, 1, 0, -2, 1, 4, 2, 2, -2, -1, 4, 1, 2, 2, -2, 2], [-2, 0, 4, -3, -3, -2, 0, 0, 4, 2, -1, -1, -1, -2, 2, -1], [-4, 1, -1, -1, -2, 1, 1, 4, 5, -2, 2, 0, -1, 2, -1, 2], [-4, 0, -2, -3, 1, 2, 0, -2, 0, -1, -2, -1, -1, -1, -1, 1], [-2, -3, 0, -3, 1, -1, -2, -6, 2, -2, -1, -3, 1, -1, 2, 3], [-2, -7, 5, -5, -2, 1, -3, -5, 3, 5, 1, -5, 1, -3, -1, -3], [2, 5, 4, -2, -4, 2, 0, 7, -8, 0, 2, 0, 1, 2, -1, 2], [3, -3, 4, -5, 3, 2, -3, 3, 3, 0, -2, 0, 1, 0, 1, 0], [7, 3, -2, 3, 2, -2, 2, 0, -10, -2, -2, 2, -1, -1, 1, -1], [-2, 0, 1, -4, -3, 4, -3, 3, 7, -2, 1, 0, -1, 2, -5, 0], [3, -3, 5, -5, 3, 0, 2, -9, -2, 7, -1, -2, 1, -7, 0, -5], [-3, 0, 3, -3, 0, -1, 2, 6, -5, 5, 1, 0, -1, -5, 0, 0], [2, 0, -3, 4, 1, 3, -1, 0, 7, -2, 3, 1, 0, 2, -3, 1], [-2, -2, -3, 1, 4, -1, 3, 5, -7, 0, 0, -2, 0, -2, 1, 2], [-5, -1, 2, -1, -2, -1, -3, 1, 5, 1, 2, 0, 1, 1, -1, -1], [-4, -2, -2, -2, 2, 1, 0, -4, 1, -4, 1, -3, 2, -4, 1, 1], [6, -4, 0, 4, 0, 2, 4, 4, -5, 4, -2, 2, 0, 4, 2, -2], [2, 6, 1, 0, -2, -1, 0, -5, 0, 2, 1, 2, -3, -1, -3, 1], [5, -4, -1, 0, 0, -5, -1, -3, -5, 0, -5, -2, -5, -2, 0, -2], [-6, -7, 0, -2, 2, 0, 0, -2, 3, -2, 0, -5, 0, -2, 0, -3], [4, -8, 1, -1, -2, 1, -3, -3, -3, 2, 2, -2, 2, 0, -1, -2], [-2, 7, 1, -1, 2, 3, 1, 2, 1, 2, 0, 0, 3, 2, 3, -2], [0, -6, -6, 6, 0, -3, 2, -12, 0, -4, -1, 2, -2, 4, 3, 2], [4, -1, 2, -2, -1, -5, 0, 1, 5, 2, -2, -1, 2, -3, 7, 1], [0, -4, -2, -1, 1, -3, -2, -2, -2, 2, -3, -2, -2, 0, 2, 0], [3, -5, -1, 1, 3, -1, 2, -11, -2, 3, 0, -1, -1, -3, 1, -2], [0, -12, -8, 3, 5, -3, 0, -12, 0, -2, -3, 2, -2, 2, 2, 2], [-4, 9, 4, 2, 2, 0, 2, -5, -2, 4, 0, 4, 5, 0, 5, 0], [4, -8, -2, 2, 2, -2, 2, -10, -6, 0, -2, 0, -2, -2, 2, -2], [5, -1, -2, 0, 3, 0, 0, -2, -8, 1, 0, -1, 0, -2, 0, -2], [-7, 0, 0, 2, -1, -3, -2, -3, 2, -4, 3, -4, -6, 1, -6, -1], [4, 0, -6, 6, 4, 2, 2, -6, 9, -6, -2, 6, 4, 6, -2, 0], [3, 0, -6, 5, 4, -6, 6, -3, -6, -2, -6, 2, -6, -1, 6, -1], [-3, 1, 1, -5, -3, 3, -1, 17, 5, 0, -1, 1, 1, 1, -2, -1], [1, 5, -2, 7, -4, 1, -1, 5, 7, -7, 0, 6, 1, 7, -1, 1], [-6, -1, 3, 4, 5, -1, 9, -2, 0, -2, -1, 1, -2, -2, -1, 3], [-4, -3, -4, -2, 0, 2, -2, 0, 1, 0, -2, 1, 0, 0, 2, 1], [-5, 6, -3, 3, -3, 0, -6, 3, 8, -3, 3, 6, 3, 3, -3, 6], [4, -6, 1, 2, 0, -3, 0, 3, 0, -2, 3, -2, 3, -1, 3, 1], [-2, 4, -5, 4, -7, 5, -7, 6, 1, 4, 4, 0, 4, 4, 1, 0], [5, 2, 3, 0, 2, 0, 2, 1, -2, 1, 0, -3, 0, 1, 0, -4], [6, 9, -4, 1, -3, 0, -2, 0, -4, 0, -1, 2, -1, 0, 0, 2], [5, -2, -4, 0, 7, 2, 4, -10, -2, 0, -1, -2, 1, -2, -1, 2], [-2, -1, -3, 6, -2, 4, 2, -2, 4, -1, 4, 1, -1, 0, 1, 0], [-2, -3, 0, 1, -3, -4, 0, -3, -15, 1, 0, -2, -4, -1, 4, 1], [-10, 18, 5, -3, 0, -1, 1, 9, 11, 2, 0, -4, 0, -2, -1, -4], [5, 2, 5, -5, 0, 3, -2, 10, 7, 3, -3, 2, 3, -1, 0, -2], [2, 3, 1, 2, 0, -2, 4, 0, -2, -7, -2, 7, 1, 4, -1, 4], [-2, -23, -4, 4, -4, 2, -4, -19, 0, 4, 2, -4, -3, 0, 3, 0], [-3, -18, -6, -3, 6, 0, 0, -15, 0, 0, 0, 0, 0, 3, 0, 3], [4, 1, -1, -2, 2, 0, -6, 2, -4, 1, 0, -1, 3, 0, -3, 0], [-3, -6, 6, -4, 0, -1, -3, 6, 3, 2, 1, 2, 4, -2, 4, 2], [1, -10, -1, 7, -5, 2, 4, -3, -4, 1, -1, 2, -1, 3, 3, 2], [-4, -6, -3, 2, -3, 2, -3, -6, -8, -4, 4, -1, -2, 4, -2, 5], [6, -8, -1, 8, -1, -1, -3, -20, -3, -2, -1, 3, 0, 2, 1, -1], [-4, -7, -11, 1, 6, 1, 7, -10, -1, -4, 2, 0, -1, 4, -1, 4], [0, 18, 7, -2, 1, -8, 1, 3, 5, -4, -3, -4, -3, -8, -5, -4], [6, -3, 3, -6, 3, -1, -3, 3, 0, 3, -4, 0, -5, 3, -1, -3], [-2, 6, 5, -8, 3, 3, -2, 6, 4, 3, 5, 0, 5, 3, -2, 0], [7, -6, -2, -3, 3, 3, 1, 4, -1, -1, -3, -1, 3, -1, 3, 1], [12, 1, -2, -4, 6, 2, 1, 0, -13, -1, 1, 1, 1, 0, 1, 1], [2, 13, 4, 0, 1, -1, 0, 3, 3, -6, 4, 5, 4, -1, -5, 5], [11, 0, -3, -1, 1, 6, 0, 6, -7, 6, -4, 7, 2, 6, 6, 1], [8, 0, 3, 1, 4, 3, -1, -3, -1, 7, 3, -2, 0, -7, -3, -5], [10, 12, 4, -5, 4, -8, -1, 6, -6, -3, -4, 6, -4, 3, -4, 6], [-3, 2, 0, -4, -5, 0, -3, 4, 9, -3, -6, 2, 6, 5, 6, -2], [1, -6, 2, -2, 1, -3, -2, -1, 2, -4, 3, -4, -4, -3, -4, 3], [3, 26, -1, 5, 3, 6, 5, 19, 3, -9, 6, 9, 6, 2, -6, 2], [12, 6, -3, 6, -2, -1, 2, -3, -2, 0, 1, 0, 1, 3, 1, -3], [3, -2, -1, 2, -2, -5, 3, -7, 1, 2, 5, -2, -5, -4, 0, 2], [-20, 16, 4, -3, 0, -4, -5, 6, 26, -3, 2, 2, 2, -1, -6, 2], [-4, -6, 0, -2, -2, -4, -4, 0, 1, 4, 2, -8, -6, -4, 4, 4], [-4, 9, 2, -2, 0, -8, 4, 9, 4, 4, -8, -4, -7, -4, 7, -4], [-8, -1, 4, -6, -3, 3, -2, 1, -1, 2, 4, -1, -4, -3, -7, 1], [-5, 2, 1, 4, -1, 3, -2, 1, -7, -1, 3, 2, -3, 3, -6, -2], [1, 6, 0, 4, 3, -2, 7, 0, -1, -7, -2, 8, -2, 1, 0, 8], [0, -6, -3, 6, 3, 2, 9, -3, -6, -3, -1, 0, 1, -3, 2, 3], [4, 2, 7, -6, 3, -1, -3, -2, 3, -2, 5, 1, 4, -2, -1, 3], [-7, 0, 1, -4, -3, -3, -6, -6, 6, -5, 3, -5, 0, -1, 0, 1], [10, -11, -5, 3, 2, 6, 5, 2, -10, 2, -3, 2, 3, 2, 6, 0], [7, 1, -2, 4, -3, 3, 1, 5, -5, 5, -1, -6, 2, 5, 3, -11], [8, -22, -1, 0, 3, 1, 5, -12, -11, 0, -2, -2, -2, -2, 3, -2], [-7, -5, -8, 2, 1, -3, 6, -16, -8, -2, 6, -5, -6, -3, -3, 5], [-6, -4, -5, 3, 1, 0, -2, -2, -3, -7, 3, -4, -3, 3, -3, 4], [6, 13, 3, 0, -1, 3, -5, 11, -3, -2, 3, 2, 6, 0, -6, 0], [-5, -12, -2, -2, 2, 6, 0, -6, 3, 2, 4, -4, 4, -2, 2, -4], [-24, -9, 3, -5, -4, -1, -5, 9, 5, 1, 1, 1, 1, -1, 1, 1], [-12, -1, -10, 4, 1, -2, 8, -8, -15, -2, 1, -1, -1, 1, 1, 1], [-2, -6, 0, 1, -7, -3, -4, 0, -4, 2, -3, -2, -4, 4, 4, 4], [-5, -6, 7, -8, -2, 11, 5, -3, -1, 3, 11, -3, 4, 0, -4, 0], [-11, -3, -4, 8, -1, -4, 4, 0, -4, -3, 4, -3, 4, 0, 4, 0], [4, 8, -2, 5, 1, 7, -6, 14, 12, -6, 2, 4, 5, 6, -7, 2], [3, -2, -9, 10, 7, 3, 4, -1, -9, -5, 5, -2, -5, 3, -8, 2], [-9, -2, 3, -5, -11, -6, -4, -1, 8, -1, 3, -2, -3, -1, 3, 2], [-8, 12, 0, -1, 4, -2, -1, 0, 12, -7, 2, 2, 2, -5, -4, 2], [6, 12, 2, 0, 2, 3, 2, 0, -3, 0, 3, 1, 6, 0, 3, 1], [-1, -12, -5, 3, 4, -6, 4, -9, -1, -5, -8, 4, -8, -1, 2, 4], [16, -3, -7, 2, 5, 5, 1, 3, -1, -6, -5, -6, -2, 6, -2, -6], [22, 11, 2, 0, -4, 3, 0, 9, -24, 6, 1, -5, 1, 1, 2, -5], [17, 9, -7, 9, -5, 1, 4, 0, -14, 0, -1, 2, 0, 0, 1, 2], [-16, 18, 0, -2, 4, -1, 2, 6, 16, -2, -1, -2, -1, -4, 0, -2], [5, -10, -4, -4, 9, -9, 8, -11, -16, 4, -9, -4, -8, -5, 8, -5], [2, 7, -1, 2, 1, 12, 1, 7, 11, -5, 7, 6, 5, 5, -12, -1], [-7, -2, -12, 4, 7, 4, 8, 8, -14, -4, 1, -2, -1, 2, -5, 2], [9, -2, -5, 2, -1, -2, -8, 2, -18, 2, -2, -2, -1, 2, 1, 2], [9, -3, 0, 0, 3, -10, 0, -6, -12, -3, -10, 3, -2, 0, 2, 0], [-4, -6, 1, 1, -2, -2, -1, 6, 1, 0, 2, 0, 5, 0, 5, 0], [-5, 8, 15, -4, -7, -4, -2, 4, 20, 4, -4, -4, 1, -8, -1, -8], [5, 3, -3, 1, -1, 3, 1, 9, 7, -2, -3, 3, 3, 5, 0, -3], [-5, -26, -5, -1, -2, 2, -6, -9, 3, 5, 0, -2, 0, 3, 2, -2], [5, 0, -9, 1, 5, 8, 4, -9, 4, -5, -1, 0, 1, 5, -7, 0], [-3, -12, 3, -2, -4, -3, -2, -15, -4, 7, 5, -8, -8, -7, 3, 1], [-4, -2, 0, -10, 6, 9, -3, 2, 3, -3, -6, -2, 6, 1, -3, 2], [10, 23, 7, -1, 0, 1, -1, 4, -1, 4, 4, -2, 5, 4, 1, -6], [4, -3, -2, 8, -6, -6, -3, -6, -1, -5, -3, 1, -3, -4, -3, 1], [-4, -3, 6, -3, -7, -5, -8, 3, 8, 0, 5, 0, -7, 0, -7, 0], [-10, 12, 0, 2, 2, 0, 4, -12, -4, 2, 0, 2, 6, -2, 6, 2], [1, -4, -7, 6, -1, 1, 0, -3, -3, -3, -1, 4, -1, 1, 2, 4], [15, 9, 4, 10, 1, -11, -1, 3, -9, 3, -13, 6, 2, -3, 11, -9], [-7, -2, 1, -1, -2, -1, -6, -4, 1, -5, -1, -2, 1, 3, 2, 2], [-10, -9, -9, -4, 3, 5, 1, -6, -6, -6, 5, -3, 0, 6, -5, 9], [-9, 9, 5, 2, 2, -3, -3, -10, 3, -3, 3, -3, -3, 2, -3, -2], [-7, -11, -6, -1, 3, -4, -3, -7, 9, 6, -4, -6, -5, -2, 5, -2], [4, -6, 2, 2, 0, -4, 2, 2, -2, 0, 4, 0, 2, -4, 2, 4], [7, 4, -13, 13, -4, -5, -6, 8, -16, -2, -5, 2, -7, 6, 7, 6], [10, 14, 0, 2, 2, 6, -2, 10, -10, -2, 6, 2, 6, -2, -6, -2], [4, 6, 2, -3, 5, 3, 2, 6, 10, 2, -2, 2, 5, -2, -3, -4], [7, 5, -5, 9, 3, -6, 4, 5, 2, 1, -3, 2, -3, -1, 6, -3], [4, -5, -7, 4, 2, -6, -6, -4, -4, 1, -6, -1, -3, 0, 3, 0], [-6, 4, 11, -11, -6, -4, -1, 10, 7, 6, -3, -4, -1, -6, 4, -2], [0, -24, -5, 2, -5, -1, -7, -6, -1, 4, -2, 4, -2, 8, 1, 4], [10, 10, -2, 0, -8, -3, -8, 2, -2, 2, 1, 2, -2, 2, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_378_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_378_2_l_a();".
// 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_378_2_l_a(:prec:=16)
    chi := MakeCharacter_378_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_378_2_l_a();".
// 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_378_2_l_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_378_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;