
// Make newform 192.3.i.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_192_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_192_i_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_192_3_i_b();".
// 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_192_3_i_b();".
// 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 := [1048576, 0, -131072, 0, 24576, 0, -6144, 0, -384, 0, 1216, 0, -24, 0, -24, 0, 6, 0, -2, 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, 0, 0, 0, 0, 0, 0, 0], [-16842752, 0, 1482752, 0, 1111040, 0, -64512, 0, 6496, 0, -11816, 0, -528, 0, 1090, 0, -42, 0, -33, 0], [0, -1638400, 0, 2621440, 0, 278528, 0, -19328, 0, -3904, 0, -5816, 0, 1288, 0, 430, 0, -90, 0, -3], [2064384, 0, -942080, 0, 11520, 0, 45664, 0, 5240, 0, 2736, 0, -790, 0, -74, 0, 51, 0, 4, 0], [18481152, 753664, -7708672, 1181696, 750080, -398080, 29632, -17696, -64976, 24248, 13216, -368, -5372, 1946, -980, -410, 542, -101, -112, -4], [18481152, -753664, -7708672, -1181696, 750080, 398080, 29632, 17696, -64976, -24248, 13216, 368, -5372, -1946, -980, 410, 542, 101, -112, 4], [-139460608, 0, 4169728, 0, 3809280, 0, -782208, 0, -159936, 0, -90104, 0, -21592, 0, 6590, 0, -858, 0, -187, 0], [1966080, 0, -1171456, 0, 706560, 0, -125824, 0, -17984, 0, 7432, 0, -1400, 0, -258, 0, -106, 0, -43, 0], [-96468992, 10878976, 46989312, -6332416, -8714240, 10240, -1806592, 78336, 518336, 68512, -77056, -12392, 18832, -4896, 2320, -2030, -3112, -34, 592, 79], [-96468992, -10878976, 46989312, 6332416, -8714240, -10240, -1806592, -78336, 518336, -68512, -77056, 12392, 18832, 4896, 2320, 2030, -3112, 34, 592, -79], [0, -18546688, 0, -7176192, 0, 3051520, 0, -161408, 0, 45504, 0, -18184, 0, -8552, 0, 2210, 0, 282, 0, -117], [-327680, 0, 3276800, 0, -372224, 0, 6976, 0, 11696, 0, -280, 0, 1180, 0, -22, 0, -100, 0, 29, 0], [15990784, 31326208, -7733248, -7356416, -2379776, 1164288, 655360, 184960, 55040, -31936, 2944, 32824, -2624, -392, -2272, -670, 320, 394, -48, -53], [0, -3457024, 0, 845824, 0, 22016, 0, -7808, 0, 8448, 0, -1640, 0, 136, 0, 154, 0, -38, 0, 7], [15990784, -31326208, -7733248, 7356416, -2379776, -1164288, 655360, -184960, 55040, 31936, 2944, -32824, -2624, 392, -2272, 670, 320, -394, -48, 53], [-17825792, 47972352, 20643840, -14385152, -1581056, -1160192, -337408, 772736, 4864, -69568, -23008, 22296, 11040, -5800, 504, -4022, -808, 1234, 84, -25], [-399769600, -48889856, 18513920, -4603904, 23490560, 5811200, -3965440, -523136, -995840, -311552, -101280, 28232, -22240, -27464, 35560, 2190, -2840, -386, -1700, -539], [0, 2428928, 0, 686592, 0, -250240, 0, 6368, 0, 8936, 0, 904, 0, 302, 0, -80, 0, -37, 0, 17], [-76021760, -27000832, -27852800, 41500672, 6983680, -4244480, -1966080, -2165632, -443520, 562176, -103840, -39736, 8000, 33272, 3400, -530, -40, -962, -740, 597], [399769600, -48889856, -18513920, -4603904, -23490560, 5811200, 3965440, -523136, 995840, -311552, 101280, 28232, 22240, -27464, -35560, 2190, 2840, -386, 1700, -539]];
    Rf_basisdens := [1, 10158080, 4063232, 507904, 10158080, 10158080, 10158080, 2031616, 40632320, 40632320, 20316160, 1015808, 8126464, 1015808, 8126464, 8126464, 40632320, 1269760, 40632320, 40632320];
    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_192_i();"
function MakeCharacter_192_i()
    N := 192;
    order := 4;
    char_gens := [127, 133, 65];
    v := [4, 3, 2];
    // 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_192_i_Hecke();"
function MakeCharacter_192_i_Hecke(Kf)
    N := 192;
    order := 4;
    char_gens := [127, 133, 65];
    char_values := [[1, 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, 0, 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, 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+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 := 3;
    raw_aps := [[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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, 1, 0, 0, -1, 0], [5, -5, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 1, -1, 0, 0, -1, 1, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0], [4, -4, 0, 0, -3, -3, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1], [0, 0, -3, 0, 1, -1, 0, 0, 1, -1, -3, 0, 0, 3, 0, 0, 0, 3, 0, 0], [0, 0, 2, -2, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 1, -2, -1, 1, -2, -1], [4, 0, 0, 3, 0, 0, 0, -1, 0, 0, 0, -3, -2, 0, -2, 0, 0, 0, 0, 0], [-7, -7, 0, 0, 0, 0, -2, 2, -1, -1, 0, -3, 1, 0, 1, 0, 1, 0, 0, -1], [0, 0, -1, 2, 1, -1, 0, 0, 1, -1, 2, 2, 0, 1, 0, 0, 2, -2, 4, 2], [-12, -12, 0, 0, 0, 0, 1, -1, -3, -3, 0, -2, 1, 0, 1, 0, 1, 0, 0, -1], [0, 0, -5, -1, 1, -1, 0, 0, -1, 1, 1, 1, -2, -5, 2, -2, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 1, 0, 2, -6, -2, 0, -2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 12, 0, 2, 0, -2, 0, -2, 0, 0, -2], [-7, 7, 0, 3, -7, -7, 2, 2, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, -1], [-16, 16, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 2, 0, 0, -2], [0, 0, 0, 1, -4, 4, 0, 0, -4, 4, -4, 1, 0, 0, 0, 0, -2, 4, 2, -2], [0, -20, 0, -3, -7, -7, 4, 0, -7, -7, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-20, 0, 0, 1, -2, -2, 0, 1, 2, 2, 0, -1, 4, 0, 4, 0, 0, 0, 0, 0], [0, 0, 9, -1, 8, -8, 0, 0, 0, 0, 0, 0, -4, 0, 4, -1, -4, 1, -1, -4], [0, 0, -14, -4, -8, 8, 0, 0, -8, 8, 6, -4, 0, 14, 0, 0, -3, -6, -8, -3], [18, 0, 0, 2, 6, 6, 0, 8, -6, -6, 0, -2, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, -9, 17, -2, -3, -12, 3, 2, 3, 0, -2, 3], [0, -8, 0, 2, -6, -6, 5, 0, -6, -6, 0, 2, 0, 0, 0, 0, -6, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, 0, 11, -11, 0, 6, -6, 4, 6, -6, 6, 0, 6, 6], [-3, 3, 0, -7, 15, 15, -6, -6, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1], [0, 0, 18, 2, 8, -8, 0, 0, -8, 8, -12, -2, 2, 18, -2, 4, 0, -12, 0, 0], [36, 0, 0, -9, -4, -4, 0, 5, 4, 4, 0, 9, 2, 0, 2, 0, 0, 0, 0, 0], [0, 0, -4, -6, -5, 5, 0, 0, 0, 0, 0, 0, -4, 0, 4, -6, -4, -4, -6, -4], [0, 0, -15, -4, 19, -19, 0, 0, 19, -19, -8, -4, 0, 15, 0, 0, -6, 8, -8, -6], [72, 72, 0, 0, 0, 0, -6, 6, 7, 7, 0, 8, 2, 0, 2, 0, 2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, -27, 27, 3, 4, -8, -20, 8, -4, 8, 0, 4, 8], [0, -24, 0, 10, 10, 10, 1, 0, 10, 10, 0, 10, 0, 0, 0, 0, 2, 0, 0, -2], [13, -13, 0, -7, -17, -17, 4, 4, 0, 0, 0, 0, 6, 0, 6, 0, -6, 0, 0, 6], [12, -12, 0, -14, 3, 3, -1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, -1], [0, 0, -15, -6, -9, 9, 0, 0, -9, 9, 13, -6, 0, 15, 0, 0, -4, -13, -12, -4], [0, 0, 6, -4, -23, 23, 0, 0, 0, 0, 0, 0, -6, 0, 6, -4, -6, -1, -4, -6], [0, 0, 14, 6, 11, -11, 0, 0, 0, 0, 0, 0, 6, 0, -6, 6, 6, 18, 6, 6], [-7, -7, 0, 0, 0, 0, 4, -4, 39, 39, 0, 5, -2, 0, -2, 0, -2, 0, 0, 2], [0, 0, 1, 1, -17, 17, 0, 0, 17, -17, 3, -1, 2, 1, -2, 2, 0, 3, 0, 0], [-82, 0, 0, -7, 33, 33, 0, 0, -33, -33, 0, 7, -4, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 33, -33, -13, 2, 11, -10, -11, -2, -11, 0, 2, -11], [0, 120, 0, -8, 20, 20, -5, 0, 20, 20, 0, -8, 0, 0, 0, 0, 6, 0, 0, -6], [-24, 24, 0, -4, 13, 13, 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [28, 0, 0, -1, 8, 8, 0, -5, -8, -8, 0, 1, -4, 0, -4, 0, 0, 0, 0, 0], [0, 0, 5, 11, -20, 20, 0, 0, 0, 0, 0, 0, 8, 0, -8, 11, 8, -35, 11, 8], [17, 17, 0, 0, 0, 0, -4, 4, -25, -25, 0, 13, 12, 0, 12, 0, 12, 0, 0, -12], [0, 0, 16, 0, 10, -10, 0, 0, 10, -10, 26, 0, 0, -16, 0, 0, -1, -26, 0, -1], [0, 0, -5, -11, -5, 5, 0, 0, 5, -5, 5, 11, 2, -5, -2, -22, 0, 5, 0, 0], [-58, 0, 0, -1, -1, -1, 0, -16, 1, 1, 0, 1, -6, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -20, 20, 11, -5, -4, -29, 4, 5, 4, 0, -5, 4], [0, 0, 6, 6, -2, 2, 0, 0, 2, -2, 46, -6, -8, 6, 8, 12, 0, 46, 0, 0], [0, 0, 22, 11, -2, 2, 0, 0, -2, 2, -2, 11, 0, -22, 0, 0, 6, 2, 22, 6], [0, 0, -28, -20, -43, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, -20, 0, -35, -20, 0], [28, 0, 0, -1, -40, -40, 0, 5, 40, 40, 0, 1, -8, 0, -8, 0, 0, 0, 0, 0], [-7, -7, 0, 0, 0, 0, -14, 14, 35, 35, 0, 9, 11, 0, 11, 0, 11, 0, 0, -11], [0, 0, 6, 8, -36, 36, 0, 0, -36, 36, -22, 8, 0, -6, 0, 0, 5, 22, 16, 5], [-120, -120, 0, 0, 0, 0, -12, 12, -3, -3, 0, 18, 10, 0, 10, 0, 10, 0, 0, -10], [0, 0, 0, 0, 0, 0, 0, 0, 41, -41, -57, 2, -9, 42, 9, -2, 9, 0, 2, 9], [160, -160, 0, -34, -29, -29, -10, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -30, -11, 48, -48, 0, 0, 48, -48, 18, -11, 0, 30, 0, 0, -26, -18, -22, -26], [0, -68, 0, 15, -1, -1, 12, 0, -1, -1, 0, 15, 0, 0, 0, 0, -11, 0, 0, 11], [0, 0, 16, -4, 33, -33, 0, 0, 0, 0, 0, 0, 10, 0, -10, -4, 10, -51, -4, 10], [96, 96, 0, 0, 0, 0, -10, 10, -39, -39, 0, -2, -10, 0, -10, 0, -10, 0, 0, 10], [-10, 0, 0, 17, -27, -27, 0, 8, 27, 27, 0, -17, -5, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 30, -30, -39, 5, -8, -55, 8, -5, 8, 0, 5, 8], [-7, 7, 0, 39, -43, -43, -16, -16, 0, 0, 0, 0, 14, 0, 14, 0, -14, 0, 0, 14], [0, 0, -22, 12, -56, 56, 0, 0, 56, -56, -14, -12, 12, -22, -12, 24, 0, -14, 0, 0], [0, 0, 16, -3, 16, -16, 0, 0, 16, -16, 20, -3, 0, -16, 0, 0, 22, -20, -6, 22], [-188, 0, 0, 15, 36, 36, 0, 9, -36, -36, 0, -15, -6, 0, -6, 0, 0, 0, 0, 0], [-19, -19, 0, 0, 0, 0, -4, 4, -11, -11, 0, -21, -18, 0, -18, 0, -18, 0, 0, 18], [-36, -36, 0, 0, 0, 0, 9, -9, 33, 33, 0, -46, 5, 0, 5, 0, 5, 0, 0, -5], [0, 0, 19, 5, -25, 25, 0, 0, 25, -25, -23, -5, -10, 19, 10, 10, 0, -23, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, -17, 73, 0, 4, 46, -4, 0, -4, 0, 0, -4], [45, -45, 0, -55, -17, -17, 2, 2, 0, 0, 0, 0, 5, 0, 5, 0, -5, 0, 0, 5], [0, 0, 49, -8, 19, -19, 0, 0, -19, 19, 20, 8, -11, 49, 11, -16, 0, 20, 0, 0], [0, 104, 0, 20, 16, 16, 0, 0, 16, 16, 0, 20, 0, 0, 0, 0, 18, 0, 0, -18], [0, 0, -59, -7, 50, -50, 0, 0, 0, 0, 0, 0, -8, 0, 8, -7, -8, 101, -7, -8], [-199, -199, 0, 0, 0, 0, 40, -40, -85, -85, 0, -15, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 39, 1, -81, 81, 0, 0, 81, -81, -55, -1, 26, 39, -26, 2, 0, -55, 0, 0], [2, 0, 0, -6, -58, -58, 0, 60, 58, 58, 0, 6, 9, 0, 9, 0, 0, 0, 0, 0], [0, -120, 0, 28, -72, -72, 13, 0, -72, -72, 0, 28, 0, 0, 0, 0, -6, 0, 0, 6], [0, 0, 0, 0, 0, 0, 0, 0, 34, -34, -85, 9, 4, 91, -4, -9, -4, 0, 9, -4], [0, 0, -5, -16, 19, -19, 0, 0, -19, 19, 26, 16, 21, -5, -21, -32, 0, 26, 0, 0], [0, -72, 0, 16, -4, -4, -20, 0, -4, -4, 0, 16, 0, 0, 0, 0, 26, 0, 0, -26], [0, 0, 90, 34, -7, 7, 0, 0, 0, 0, 0, 0, 13, 0, -13, 34, 13, -43, 34, 13], [-524, 0, 0, -3, -18, -18, 0, -9, 18, 18, 0, 3, 2, 0, 2, 0, 0, 0, 0, 0], [0, 0, -3, 13, -10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 53, 13, 0], [0, 0, -6, 8, 56, -56, 0, 0, -56, 56, 98, -8, 24, -6, -24, 16, 0, 98, 0, 0], [0, 80, 0, -48, -8, -8, 21, 0, -8, -8, 0, -48, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -45, 45, 94, 10, 0, -66, 0, -10, 0, 0, 10, 0], [40, -40, 0, 40, 33, 33, -42, -42, 0, 0, 0, 0, -8, 0, -8, 0, 8, 0, 0, -8], [0, 0, -10, 7, 22, -22, 0, 0, 22, -22, 74, 7, 0, 10, 0, 0, 26, -74, 14, 26], [0, 0, -58, -16, 9, -9, 0, 0, 0, 0, 0, 0, 12, 0, -12, -16, 12, 3, -16, 12], [0, 0, 37, 28, 45, -45, 0, 0, 45, -45, -42, 28, 0, -37, 0, 0, -8, 42, 56, -8], [124, 124, 0, 0, 0, 0, -41, 41, 71, 71, 0, 12, 13, 0, 13, 0, 13, 0, 0, -13], [113, -113, 0, -73, -19, -19, 26, 26, 0, 0, 0, 0, 23, 0, 23, 0, -23, 0, 0, 23], [-332, 332, 0, -4, 33, 33, -7, -7, 0, 0, 0, 0, -9, 0, -9, 0, 9, 0, 0, -9], [0, 0, -24, -14, -11, 11, 0, 0, 0, 0, 0, 0, -5, 0, 5, -14, -5, -13, -14, -5], [0, 0, 103, -1, -104, 104, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -17, -1, 0], [0, 0, -59, -4, 41, -41, 0, 0, 41, -41, -110, -4, 0, 59, 0, 0, -4, 110, -8, -4], [-256, -256, 0, 0, 0, 0, 30, -30, -55, -55, 0, 30, -20, 0, -20, 0, -20, 0, 0, 20], [70, 0, 0, -35, -31, -31, 0, -56, 31, 31, 0, 35, -25, 0, -25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 6, -4, -24, 94, 24, 4, 24, 0, -4, 24], [0, 0, 72, 40, 4, -4, 0, 0, -4, 4, 20, -40, 20, 72, -20, 80, 0, 20, 0, 0], [0, 0, -51, 10, -69, 69, 0, 0, -69, 69, -43, 10, 0, 51, 0, 0, 24, 43, 20, 24], [0, 52, 0, 27, -1, -1, -44, 0, -1, -1, 0, 27, 0, 0, 0, 0, -16, 0, 0, 16], [660, 0, 0, -15, -70, -70, 0, 25, 70, 70, 0, 15, -10, 0, -10, 0, 0, 0, 0, 0], [61, 61, 0, 0, 0, 0, -50, 50, 117, 117, 0, 11, 17, 0, 17, 0, 17, 0, 0, -17], [0, 0, -44, -8, -60, 60, 0, 0, -60, 60, 112, -8, 0, 44, 0, 0, -11, -112, -16, -11], [288, 288, 0, 0, 0, 0, 22, -22, -39, -39, 0, 94, 18, 0, 18, 0, 18, 0, 0, -18], [0, -448, 0, -18, -62, -62, -25, 0, -62, -62, 0, -18, 0, 0, 0, 0, 20, 0, 0, -20], [0, 0, -67, 0, 95, -95, 0, 0, -95, 95, 28, 0, -47, -67, 47, 0, 0, 28, 0, 0], [28, -28, 0, 36, 17, 17, 25, 25, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1], [0, 0, -73, 2, 47, -47, 0, 0, 47, -47, -97, 2, 0, 73, 0, 0, -28, 97, 4, -28], [0, 0, -134, -22, 49, -49, 0, 0, 0, 0, 0, 0, -27, 0, 27, -22, -27, 117, -22, -27], [0, 0, 2, -2, 15, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -198, -2, 0], [-67, -67, 0, 0, 0, 0, -26, 26, -31, -31, 0, 79, 19, 0, 19, 0, 19, 0, 0, -19], [-270, 0, 0, -26, 18, 18, 0, 28, -18, -18, 0, 26, 5, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -7, 7, 97, -62, 27, 104, -27, 62, -27, 0, -62, -27], [0, 0, 0, 0, 0, 0, 0, 0, 20, -20, 43, -45, 28, -21, -28, 45, -28, 0, -45, -28], [164, -164, 0, -10, 23, 23, -1, -1, 0, 0, 0, 0, 35, 0, 35, 0, -35, 0, 0, 35], [0, 0, 92, 42, -99, 99, 0, 0, 0, 0, 0, 0, 35, 0, -35, 42, 35, 71, 42, 35], [297, 297, 0, 0, 0, 0, 6, -6, 15, 15, 0, 61, 1, 0, 1, 0, 1, 0, 0, -1], [0, 0, -92, -40, 52, -52, 0, 0, -52, 52, -64, 40, -60, -92, 60, -80, 0, -64, 0, 0], [0, -344, 0, 8, 124, 124, -41, 0, 124, 124, 0, 8, 0, 0, 0, 0, 22, 0, 0, -22], [177, -177, 0, 51, 145, 145, -24, -24, 0, 0, 0, 0, -26, 0, -26, 0, 26, 0, 0, -26], [-336, 336, 0, -8, -7, -7, 14, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -35, 50, -13, 13, 0, 0, -13, 13, 5, 50, 0, 35, 0, 0, 24, -5, 100, 24], [-420, 0, 0, -15, 86, 86, 0, 59, -86, -86, 0, 15, 20, 0, 20, 0, 0, 0, 0, 0], [-119, -119, 0, 0, 0, 0, 64, -64, -25, -25, 0, -43, 2, 0, 2, 0, 2, 0, 0, -2], [0, 0, -65, 14, 81, -81, 0, 0, 81, -81, 46, 14, 0, 65, 0, 0, 14, -46, 28, 14], [38, 0, 0, 15, -49, -49, 0, 40, 49, 49, 0, -15, 26, 0, 26, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -95, 95, -91, 34, 7, -52, -7, -34, -7, 0, 34, -7], [140, -140, 0, -94, -29, -29, 35, 35, 0, 0, 0, 0, 25, 0, 25, 0, -25, 0, 0, 25], [0, 0, -156, -18, -119, 119, 0, 0, 0, 0, 0, 0, 7, 0, -7, -18, 7, -1, -18, 7], [0, 0, -9, 18, -123, 123, 0, 0, -123, 123, 46, 18, 0, 9, 0, 0, -2, -46, 36, -2], [356, 356, 0, 0, 0, 0, 31, -31, 23, 23, 0, -28, -1, 0, -1, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 209, 18, -35, -108, 35, -18, 35, 0, 18, 35], [0, -192, 0, -16, -144, -144, 43, 0, -144, -144, 0, -16, 0, 0, 0, 0, -4, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, -35, 35, 142, 48, -2, -94, 2, -48, 2, 0, 48, 2], [-203, 203, 0, -3, 99, 99, 56, 56, 0, 0, 0, 0, 6, 0, 6, 0, -6, 0, 0, 6], [0, 0, 150, -11, -8, 8, 0, 0, -8, 8, -58, -11, 0, -150, 0, 0, -2, 58, -22, -2], [53, 53, 0, 0, 0, 0, 20, -20, -167, -167, 0, -81, -34, 0, -34, 0, -34, 0, 0, 34], [0, 0, -47, 24, 35, -35, 0, 0, 35, -35, -76, 24, 0, 47, 0, 0, -42, 76, 48, -42], [-268, -268, 0, 0, 0, 0, 1, -1, -119, -119, 0, 2, -13, 0, -13, 0, -13, 0, 0, 13], [0, 0, -37, 47, 113, -113, 0, 0, -113, 113, -23, -47, 62, -37, -62, 94, 0, -23, 0, 0], [81, -81, 0, -81, -91, -91, -16, -16, 0, 0, 0, 0, 12, 0, 12, 0, -12, 0, 0, 12], [0, 0, 124, -10, -60, 60, 0, 0, 60, -60, -6, 10, -46, 124, 46, -20, 0, -6, 0, 0], [124, -124, 0, 60, -23, -23, -11, -11, 0, 0, 0, 0, -37, 0, -37, 0, 37, 0, 0, -37], [0, 0, -18, 29, 188, -188, 0, 0, 188, -188, -86, 29, 0, 18, 0, 0, 34, 86, 58, 34], [220, 220, 0, 0, 0, 0, -7, 7, 77, 77, 0, -106, -5, 0, -5, 0, -5, 0, 0, 5], [0, 0, -51, 1, 59, -59, 0, 0, -59, 59, -5, -1, 86, -51, -86, 2, 0, -5, 0, 0], [0, -592, 0, 50, 54, 54, 87, 0, 54, 54, 0, 50, 0, 0, 0, 0, -24, 0, 0, 24], [0, 0, -103, 12, 29, -29, 0, 0, -29, 29, -122, -12, 7, -103, -7, 24, 0, -122, 0, 0], [0, -296, 0, -22, 66, 66, -84, 0, 66, 66, 0, -22, 0, 0, 0, 0, 23, 0, 0, -23], [0, 0, -98, -2, 33, -33, 0, 0, 0, 0, 0, 0, 15, 0, -15, -2, 15, -51, -2, 15], [0, 0, -124, -2, 67, -67, 0, 0, 0, 0, 0, 0, -30, 0, 30, -2, -30, 24, -2, -30], [0, 0, 85, -24, 41, -41, 0, 0, 41, -41, 94, -24, 0, -85, 0, 0, -24, -94, -48, -24], [0, 96, 0, -48, -32, -32, 53, 0, -32, -32, 0, -48, 0, 0, 0, 0, 12, 0, 0, -12], [0, 0, 0, 0, 0, 0, 0, 0, 174, -174, 9, 5, -24, -95, 24, -5, 24, 0, 5, 24], [0, 0, -17, -64, -69, 69, 0, 0, 69, -69, -26, 64, -19, -17, 19, -128, 0, -26, 0, 0], [0, 0, 46, 5, -152, 152, 0, 0, -152, 152, 30, 5, 0, -46, 0, 0, 46, -30, 10, 46], [588, 0, 0, 21, -118, -118, 0, 77, 118, 118, 0, -21, 60, 0, 60, 0, 0, 0, 0, 0], [-255, -255, 0, 0, 0, 0, 2, -2, 163, 163, 0, -31, -15, 0, -15, 0, -15, 0, 0, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_192_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_192_3_i_b();".
// 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_192_3_i_b(:prec:=20)
    chi := MakeCharacter_192_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_192_3_i_b();".
// 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_192_3_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_192_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4096, 0, 0, 0, 108375444480, 0, 0, 0, 1068787520, 0, 0, 0, 3190384, 0, 0, 0, 3404, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;