
// Make newform 1045.2.b.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1045_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1045_b_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_1045_2_b_c();".
// 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_1045_2_b_c();".
// 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, 0, 276, 0, 2753, 0, 9095, 0, 13894, 0, 11551, 0, 5623, 0, 1640, 0, 281, 0, 26, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1604, 0, -131958, 0, -522039, 0, -592650, 0, -240718, 0, 5459, 0, 33228, 0, 10262, 0, 1299, 0, 61], [0, -27024, 0, -288912, 0, -866947, 0, -946122, 0, -428112, 0, -49969, 0, 23376, 0, 9104, 0, 1205, 0, 57], [-1602, 0, -59110, 0, -319949, 0, -735502, 0, -819526, 0, -489083, 0, -164582, 0, -31252, 0, -3119, 0, -127, 0], [0, -8898, 0, -78267, 0, 189834, 0, 1360162, 0, 2051391, 0, 1398132, 0, 504698, 0, 99883, 0, 10233, 0, 424], [0, -13202, 0, -79343, 0, 189834, 0, 1360162, 0, 2051391, 0, 1398132, 0, 504698, 0, 99883, 0, 10233, 0, 424], [-29, 0, -36260, 0, -282189, 0, -697023, 0, -783140, 0, -465652, 0, -155907, 0, -29474, 0, -2931, 0, -119, 0], [-1100, 0, -68197, 0, -516802, 0, -1400840, 0, -1682291, 0, -1043638, 0, -358938, 0, -69037, 0, -6943, 0, -284, 0], [0, 44494, 0, 772083, 0, 2655663, 0, 3799744, 0, 2874857, 0, 1269599, 0, 338506, 0, 53633, 0, 4644, 0, 169], [2676, -2730, -151708, 55629, -1815624, -280745, -5449516, -1827220, -6729844, -2756833, -4211192, -1878649, -1452564, -676778, -279704, -133533, -28148, -13634, -1152, -563], [-3848, 0, -131302, 0, -1246615, 0, -3666446, 0, -4530894, 0, -2846233, 0, -986020, 0, -190662, 0, -19261, 0, -791, 0], [-2676, -2730, 151708, 55629, 1815624, -280745, 5449516, -1827220, 6729844, -2756833, 4211192, -1878649, 1452564, -676778, 279704, -133533, 28148, -13634, 1152, -563], [1126, 74244, 76329, 740557, 802784, 2446555, 2419870, 3737486, 2997551, 3107219, 1873190, 1512587, 643550, 441160, 123271, 75589, 12335, 6994, 502, 269], [-1126, 74244, -76329, 740557, -802784, 2446555, -2419870, 3737486, -2997551, 3107219, -1873190, 1512587, -643550, 441160, -123271, 75589, -12335, 6994, -502, 269], [3674, 0, 153152, 0, 1214287, 0, 3254612, 0, 3838002, 0, 2343125, 0, 795708, 0, 151546, 0, 15125, 0, 615, 0], [0, -28512, 0, -434513, 0, -2033120, 0, -4207528, 0, -4382125, 0, -2505428, 0, -819456, 0, -152531, 0, -15001, 0, -604], [-460, 0, -311854, 0, -3034215, 0, -8704666, 0, -10567846, 0, -6554317, 0, -2248272, 0, -431250, 0, -43273, 0, -1767, 0], [0, 40090, 0, 803421, 0, 4138215, 0, 8896864, 0, 9417307, 0, 5425363, 0, 1781218, 0, 332179, 0, 32698, 0, 1317]];
    Rf_basisdens := [1, 1, 1, 1076, 1076, 538, 1076, 1076, 269, 538, 1076, 2152, 1076, 2152, 1076, 1076, 538, 538, 1076, 1076];
    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_1045_b();"
function MakeCharacter_1045_b()
    N := 1045;
    order := 2;
    char_gens := [837, 761, 496];
    v := [1, 2, 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_1045_b_Hecke();"
function MakeCharacter_1045_b_Hecke(Kf)
    N := 1045;
    order := 2;
    char_gens := [837, 761, 496];
    char_values := [[-1, 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], [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 := 2;
    raw_aps := [[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, 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, 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], [1, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 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, 2, 0, 2, -1, 0, -1, -2, 0, 0, 1, -1, 0, -1, -1, -1, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 1, 0, -1, 0], [-4, 0, 3, 0, 0, -2, 0, 0, 1, 0, 0, 1, 2, -1, -1, 1, 0, 0, -2, 0], [0, -2, 0, -1, 3, 0, -2, 3, 0, 0, -1, 2, 0, 2, 1, 1, 0, -1, 0, -1], [-1, 0, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, -1, 0, 0, -1, 1, 0, 1, 0, 0, 0, -2, 0, -2], [0, 1, 0, 2, -2, 0, 2, -4, 0, 0, 1, -2, 0, -2, -2, -2, 0, 1, 0, 3], [0, -1, 0, 0, -3, 0, 2, -4, 0, 0, 1, -3, 0, -3, -1, -1, 0, 0, 0, 2], [3, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 1, -1, -3, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, 1, 0, 0, 0, -1, 0, -1, 1, 0, 0, 1, 0], [0, 2, 0, 3, -5, 0, 0, -5, 0, 0, 1, -3, 0, -3, -2, -2, 0, 1, 0, 2], [-2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, -1, -1, -1, 1, 2, 0, -1, 0], [0, -2, 0, 1, -1, 0, -1, 2, 0, 0, 1, -1, 0, -1, 1, 1, 0, 4, 0, 1], [6, 0, -2, 0, 0, -1, 0, 0, -2, 1, 0, 0, 1, 0, 0, 0, -1, 0, 1, 0], [0, 3, 0, 1, -1, 0, 1, -4, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 0, -1], [1, 0, 3, 0, 0, 1, 0, 0, 0, -2, 0, 0, 1, 0, 1, -1, 1, 0, -1, 0], [0, -2, 0, -2, 2, 0, -1, 2, 0, 0, 1, -1, 0, -1, 0, 0, 0, 1, 0, 0], [-2, 0, 1, 0, 0, -1, 0, 0, -2, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, -1, 0, -1, -2, 0, 0, 0, 0, 0, 3, -1, 0, -1, 0, 0, 0, 1, 0, -2], [0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0, 1, -1, -1, 0, -1, 0, -1], [6, 0, -2, 0, 0, 0, 0, 0, -1, 3, 0, 1, -2, -1, -1, 1, 1, 0, 0, 0], [0, 1, 0, -1, 2, 0, -1, 5, 0, 0, -1, 4, 0, 4, 2, 2, 0, -2, 0, -2], [0, -5, 0, -1, 1, 0, 0, 4, 0, 0, -2, 0, 0, 0, 2, 2, 0, 2, 0, 2], [-2, 0, -1, 0, 0, 0, 0, 0, 1, 2, 0, -1, 3, 1, 0, 0, -2, 0, -1, 0], [0, -1, 0, -1, 2, 0, -3, 4, 0, 0, -2, 4, 0, 4, 1, 1, 0, 0, 0, -2], [3, 0, -3, 0, 0, 0, 0, 0, 0, -1, 0, 1, -2, -1, -2, 2, 2, 0, 2, 0], [2, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0, 0, -3, 0, 0, 0, 2, 0, 3, 0], [-11, 0, 4, 0, 0, -5, 0, 0, -2, 3, 0, -2, 5, 2, -1, 1, -2, 0, 1, 0], [0, -3, 0, -2, 3, 0, 0, 5, 0, 0, -1, 6, 0, 6, 3, 3, 0, 0, 0, -1], [0, 2, 0, 1, 4, 0, -3, 6, 0, 0, -3, 4, 0, 4, 2, 2, 0, -1, 0, -2], [0, 4, 0, -1, 2, 0, 0, 1, 0, 0, -3, 3, 0, 3, 1, 1, 0, 0, 0, -1], [0, -7, 0, -9, 5, 0, 0, 12, 0, 0, -1, 5, 0, 5, 5, 5, 0, 2, 0, -2], [0, 0, 2, 0, 0, 0, 0, 0, -1, 1, 0, -4, -2, 4, 4, -4, -3, 0, 0, 0], [-6, 0, 2, 0, 0, -4, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0], [-5, 0, 1, 0, 0, -2, 0, 0, -2, -2, 0, -1, 3, 1, 2, -2, -1, 0, 1, 0], [0, 0, 0, 1, 4, 0, 2, 3, 0, 0, -1, 2, 0, 2, 1, 1, 0, 2, 0, 0], [0, 1, 0, 0, -1, 0, 4, 2, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 1], [8, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 4, -2, -4, 1, -1, -1, 0, -4, 0], [-5, 0, -1, 0, 0, 3, 0, 0, -3, 1, 0, -1, -2, 1, 2, -2, 0, 0, 2, 0], [0, 0, 0, 3, -1, 0, 2, -7, 0, 0, 0, -3, 0, -3, -4, -4, 0, -5, 0, 1], [0, -3, 0, -4, 4, 0, 0, 4, 0, 0, -2, 3, 0, 3, 2, 2, 0, -3, 0, -3], [-1, 0, 3, 0, 0, -2, 0, 0, 1, 0, 0, 0, 3, 0, -2, 2, -1, 0, -1, 0], [0, -2, 0, -4, 2, 0, 1, 4, 0, 0, -2, 5, 0, 5, 1, 1, 0, 1, 0, 0], [5, 0, 0, 0, 0, 3, 0, 0, 4, -2, 0, -5, 0, 5, 0, 0, -1, 0, 0, 0], [-1, 0, -6, 0, 0, 1, 0, 0, -1, -1, 0, -2, -2, 2, 3, -3, 0, 0, 2, 0], [7, 0, -5, 0, 0, 4, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, -1, 0, 1, 0], [0, 1, 0, 3, 0, 0, -4, 0, 0, 0, 3, 2, 0, 2, -1, -1, 0, 1, 0, -1], [0, -5, 0, -2, 2, 0, 1, 2, 0, 0, -1, 0, 0, 0, 1, 1, 0, 2, 0, 0], [4, 0, 1, 0, 0, -1, 0, 0, -3, -2, 0, -2, 3, 2, 2, -2, -2, 0, 1, 0], [0, 0, -4, 0, 0, 3, 0, 0, 1, 1, 0, -1, -1, 1, -1, 1, -1, 0, 3, 0], [0, 0, 0, 1, 1, 0, 1, -4, 0, 0, 4, 0, 0, 0, 0, 0, 0, -2, 0, -1], [-5, 0, 2, 0, 0, -1, 0, 0, 2, 0, 0, 0, 1, 0, -2, 2, 1, 0, 1, 0], [0, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 2, 0, 2, -2, -2, 0, 0, 0, 0], [0, -1, 0, 2, 2, 0, -1, -5, 0, 0, 0, 1, 0, 1, -1, -1, 0, -2, 0, 5], [0, 1, 0, -2, 0, 0, -1, 5, 0, 0, 1, -1, 0, -1, 1, 1, 0, 2, 0, 2], [-1, 0, 4, 0, 0, -1, 0, 0, -1, 3, 0, -1, 1, 1, 1, -1, 0, 0, -1, 0], [0, -4, 0, 2, 0, 0, 3, -5, 0, 0, -2, -3, 0, -3, -1, -1, 0, -1, 0, 2], [0, 7, 0, 4, -1, 0, 3, -11, 0, 0, 2, -1, 0, -1, -4, -4, 0, -3, 0, 2], [-6, 0, -2, 0, 0, -2, 0, 0, 0, 4, 0, 2, 1, -2, -1, 1, 0, 0, -3, 0], [0, 6, 0, 0, -3, 0, 2, -7, 0, 0, 0, -1, 0, -1, -2, -2, 0, -1, 0, -4], [0, -1, 0, 4, -4, 0, 4, -11, 0, 0, 2, -3, 0, -3, -2, -2, 0, 1, 0, 2], [1, 0, 4, 0, 0, -1, 0, 0, 4, -4, 0, -1, 2, 1, -1, 1, -3, 0, -4, 0], [0, 6, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, -3, -3, 0, -1, 0, 2], [3, 0, -2, 0, 0, 2, 0, 0, -2, -3, 0, 3, -5, -3, 3, -3, 2, 0, 1, 0], [0, -1, 0, 4, -4, 0, 1, -5, 0, 0, 2, -3, 0, -3, -1, -1, 0, 6, 0, 4], [0, -7, 0, -2, 0, 0, -1, 5, 0, 0, -3, 5, 0, 5, 4, 4, 0, 2, 0, -2], [3, 0, -2, 0, 0, 0, 0, 0, 1, 2, 0, -1, 1, 1, 1, -1, -3, 0, 3, 0], [0, 1, 0, 2, -7, 0, 4, -2, 0, 0, 1, -6, 0, -6, 0, 0, 0, 4, 0, 4], [0, 0, -4, 0, 0, 3, 0, 0, 1, -2, 0, 3, 2, -3, -2, 2, 4, 0, 0, 0], [0, 2, 0, -3, 5, 0, 4, -2, 0, 0, -4, 5, 0, 5, 0, 0, 0, -3, 0, -3], [3, 0, -7, 0, 0, 2, 0, 0, -4, -1, 0, -3, -2, 3, 1, -1, -2, 0, 4, 0], [8, 0, -3, 0, 0, 5, 0, 0, 3, 3, 0, -2, 3, 2, -2, 2, -1, 0, -1, 0], [-1, 0, 2, 0, 0, 0, 0, 0, 5, 5, 0, 2, -1, -2, -5, 5, 2, 0, -1, 0], [-5, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0, -3, 4, 3, -2, 2, -1, 0, 2, 0], [2, 0, -4, 0, 0, 2, 0, 0, 1, -1, 0, 2, -11, -2, 1, -1, 1, 0, 1, 0], [0, -12, 0, -9, 3, 0, -4, 13, 0, 0, -2, 3, 0, 3, 6, 6, 0, 3, 0, -3], [9, 0, -2, 0, 0, -3, 0, 0, -4, -2, 0, 3, 0, -3, -1, 1, 1, 0, 2, 0], [0, -4, 0, -5, 0, 0, 3, 8, 0, 0, 0, -1, 0, -1, 0, 0, 0, 5, 0, 3], [-3, 0, 7, 0, 0, -2, 0, 0, 3, 1, 0, 5, 0, -5, -4, 4, 2, 0, -6, 0], [0, -2, 0, -1, -1, 0, 0, -1, 0, 0, 1, -5, 0, -5, -1, -1, 0, 3, 0, 1], [-4, 0, 9, 0, 0, -5, 0, 0, 0, -1, 0, 2, 2, -2, -3, 3, 1, 0, -6, 0], [0, -2, 0, 2, -7, 0, -3, 4, 0, 0, 4, -7, 0, -7, -1, -1, 0, 6, 0, 1], [0, -3, 0, -4, -2, 0, 2, 6, 0, 0, 1, 0, 0, 0, 3, 3, 0, 7, 0, 4], [-3, 0, 8, 0, 0, -4, 0, 0, 1, -4, 0, 5, 6, -5, -3, 3, 1, 0, -6, 0], [0, 9, 0, 0, -3, 0, 1, -7, 0, 0, 2, 0, 0, 0, -3, -3, 0, -1, 0, -3], [-4, 0, 2, 0, 0, -4, 0, 0, -2, 1, 0, 1, 5, -1, -2, 2, 1, 0, 3, 0], [3, 0, -7, 0, 0, -3, 0, 0, 1, 3, 0, -3, 0, 3, -1, 1, -2, 0, 4, 0], [0, 6, 0, 5, -12, 0, 1, -9, 0, 0, 4, -7, 0, -7, -3, -3, 0, 3, 0, 3], [1, 0, -10, 0, 0, 5, 0, 0, -6, -1, 0, -3, -5, 3, 4, -4, 4, 0, 9, 0], [-2, 0, 4, 0, 0, -5, 0, 0, 2, -5, 0, 1, 3, -1, -2, 2, -1, 0, -7, 0], [0, -3, 0, -5, -1, 0, 8, -6, 0, 0, 1, -5, 0, -5, -2, -2, 0, 4, 0, 6], [-8, 0, -1, 0, 0, 2, 0, 0, -2, 6, 0, -2, -1, 2, -1, 1, 2, 0, 5, 0], [0, 3, 0, -5, -1, 0, 0, -2, 0, 0, -2, 1, 0, 1, 1, 1, 0, 2, 0, -3], [0, 6, 0, 1, 1, 0, 2, -7, 0, 0, -1, -5, 0, -5, 0, 0, 0, -3, 0, 3], [0, 6, 0, 4, -5, 0, 1, -10, 0, 0, -2, 0, 0, 0, -6, -6, 0, -4, 0, 1], [4, 0, 0, 0, 0, -2, 0, 0, 5, -3, 0, 5, -4, -5, -4, 4, 5, 0, -6, 0], [-2, 0, -9, 0, 0, -1, 0, 0, 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2, 2, -2, -7, 0, -7, 0], [6, 0, 6, 0, 0, 0, 0, 0, 0, 3, 0, 1, -1, -1, -1, 1, 3, 0, -1, 0], [-12, 0, 1, 0, 0, 3, 0, 0, 1, 4, 0, 7, -3, -7, 5, -5, -2, 0, -5, 0], [0, -1, 0, 6, -4, 0, 10, -20, 0, 0, 1, -1, 0, -1, -8, -8, 0, -7, 0, 11], [0, 3, 0, -2, 12, 0, 2, 4, 0, 0, -7, 10, 0, 10, 2, 2, 0, -13, 0, -13], [13, 0, -14, 0, 0, 11, 0, 0, -4, 6, 0, -11, -6, 11, 2, -2, -5, 0, 12, 0], [0, 5, 0, 6, -7, 0, 7, -11, 0, 0, 2, -3, 0, -3, 0, 0, 0, 9, 0, 7], [13, 0, -7, 0, 0, 3, 0, 0, -7, -2, 0, 9, -6, -9, 0, 0, 1, 0, 2, 0], [0, -29, 0, -7, -1, 0, -5, 17, 0, 0, -3, -2, 0, -2, 6, 6, 0, 3, 0, -5], [5, 0, 1, 0, 0, -5, 0, 0, 0, 5, 0, 2, -2, -2, -4, 4, -6, 0, 0, 0], [0, -4, 0, -7, -3, 0, 9, -1, 0, 0, 6, -8, 0, -8, 1, 1, 0, 4, 0, 9], [-6, 0, -6, 0, 0, 5, 0, 0, 2, 10, 0, -7, -4, 7, 0, 0, 0, 0, 6, 0], [-19, 0, -15, 0, 0, 3, 0, 0, -1, 5, 0, 3, -10, -3, 3, -3, -4, 0, 0, 0], [1, 0, -7, 0, 0, -5, 0, 0, -4, 5, 0, 3, 1, -3, 0, 0, 6, 0, 5, 0], [0, 17, 0, 5, -13, 0, 5, -21, 0, 0, 9, -9, 0, -9, -6, -6, 0, 9, 0, 9], [0, 4, 0, -7, 0, 0, -6, 2, 0, 0, 2, 3, 0, 3, 2, 2, 0, 10, 0, -5], [-24, 0, -1, 0, 0, -5, 0, 0, -14, 3, 0, -7, 0, 7, 7, -7, -1, 0, 4, 0], [0, 22, 0, 7, -7, 0, 1, -6, 0, 0, 2, -9, 0, -9, -5, -5, 0, 6, 0, 8], [-3, 0, -3, 0, 0, -3, 0, 0, 3, 6, 0, -1, 3, 1, 0, 0, -5, 0, 3, 0], [7, 0, -13, 0, 0, 9, 0, 0, -13, 3, 0, -7, -7, 7, 8, -8, -2, 0, 19, 0], [0, 6, 0, 1, 5, 0, -7, 22, 0, 0, -4, 7, 0, 7, 7, 7, 0, 2, 0, -3], [10, 0, 6, 0, 0, -9, 0, 0, -1, 2, 0, -10, 14, 10, 2, -2, -12, 0, -2, 0], [0, -4, 0, -13, 24, 0, -5, 18, 0, 0, -5, 18, 0, 18, 10, 10, 0, -13, 0, -12], [6, 0, -6, 0, 0, 6, 0, 0, -2, 0, 0, 1, 0, -1, -3, 3, 6, 0, 0, 0], [0, 3, 0, -5, 1, 0, 7, -5, 0, 0, -6, 2, 0, 2, 0, 0, 0, -11, 0, -2], [-12, 0, 10, 0, 0, -1, 0, 0, 6, 0, 0, 1, 6, -1, -4, 4, 0, 0, 0, 0], [0, 1, 0, -3, 9, 0, 3, 9, 0, 0, -4, 2, 0, 2, 0, 0, 0, -9, 0, -12], [-6, 0, 15, 0, 0, -8, 0, 0, 2, 2, 0, 1, 4, -1, 0, 0, 0, 0, -4, 0], [16, 0, -4, 0, 0, 1, 0, 0, -3, -6, 0, 4, 2, -4, 1, -1, 6, 0, -2, 0], [0, 10, 0, 3, -8, 0, 8, -18, 0, 0, 3, -15, 0, -15, -9, -9, 0, -6, 0, 6], [0, 2, 0, -2, 0, 0, -5, 12, 0, 0, -4, 8, 0, 8, 3, 3, 0, 1, 0, 5], [-5, 0, 7, 0, 0, -8, 0, 0, 8, 7, 0, -1, 4, 1, -9, 9, 0, 0, -8, 0], [0, 3, 0, -2, -14, 0, 8, -14, 0, 0, 6, -10, 0, -10, -5, -5, 0, 13, 0, 9], [2, 0, -2, 0, 0, 6, 0, 0, -2, 0, 0, 12, -10, -12, -8, 8, 10, 0, -4, 0], [0, 4, 0, -3, -3, 0, 3, 2, 0, 0, 5, -1, 0, -1, 1, 1, 0, 4, 0, -5], [20, 0, -15, 0, 0, 10, 0, 0, -1, -3, 0, -10, -6, 10, 8, -8, -1, 0, 4, 0], [2, 0, -13, 0, 0, 12, 0, 0, 2, -2, 0, -4, -17, 4, 4, -4, 8, 0, 9, 0], [0, 0, 0, 5, 4, 0, -6, -5, 0, 0, 5, 1, 0, 1, -1, -1, 0, 0, 0, -1], [0, 7, 0, 15, -6, 0, 3, -21, 0, 0, 5, -16, 0, -16, -8, -8, 0, 8, 0, 6], [0, -6, 0, -12, 10, 0, 5, 18, 0, 0, -5, 14, 0, 14, 10, 10, 0, 1, 0, -2], [0, -8, 0, -4, 17, 0, -2, 23, 0, 0, -6, 3, 0, 3, 11, 11, 0, 1, 0, -2], [0, -11, 0, -5, -7, 0, 4, -6, 0, 0, -6, -3, 0, -3, -1, -1, 0, 2, 0, 5], [0, 0, 12, 0, 0, -7, 0, 0, -6, -6, 0, 1, 2, -1, 3, -3, 4, 0, 4, 0], [0, -6, 0, 5, 0, 0, -7, 2, 0, 0, 6, 1, 0, 1, 1, 1, 0, 7, 0, 6], [0, -10, 0, -10, 12, 0, 5, 3, 0, 0, -2, 8, 0, 8, 7, 7, 0, -9, 0, -8], [20, 0, 6, 0, 0, 6, 0, 0, 4, -6, 0, 3, -3, -3, -4, 4, 2, 0, -3, 0], [-30, 0, 1, 0, 0, -8, 0, 0, 4, -3, 0, 2, 0, -2, -4, 4, 3, 0, -6, 0], [-3, 0, 12, 0, 0, -12, 0, 0, 1, -2, 0, 4, -2, -4, -6, 6, 1, 0, 0, 0], [0, -26, 0, -3, 2, 0, -8, 9, 0, 0, 2, 3, 0, 3, 5, 5, 0, -4, 0, -2], [0, 26, 0, 11, 5, 0, 0, -5, 0, 0, 3, 1, 0, 1, 2, 2, 0, -5, 0, -3], [-14, 0, -3, 0, 0, 0, 0, 0, 4, -2, 0, -5, 11, 5, 1, -1, 0, 0, 3, 0], [0, -5, 0, -7, -2, 0, 12, -6, 0, 0, -2, -4, 0, -4, -3, -3, 0, 1, 0, 5], [0, -9, 0, -9, 8, 0, -7, 18, 0, 0, -3, 7, 0, 7, 1, 1, 0, -4, 0, -11], [19, 0, -12, 0, 0, 6, 0, 0, -3, -2, 0, -5, -2, 5, 6, -6, -3, 0, 6, 0], [0, -15, 0, -6, -4, 0, 3, -11, 0, 0, 6, -3, 0, -3, -3, -3, 0, 6, 0, 1], [0, 7, 0, -2, 0, 0, -6, -6, 0, 0, 4, -6, 0, -6, -5, -5, 0, -3, 0, 1], [0, -17, 0, -9, 3, 0, -2, 10, 0, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, 1], [-22, 0, 5, 0, 0, 2, 0, 0, 4, 1, 0, -12, 1, 12, 3, -3, -15, 0, 3, 0], [1, 0, -3, 0, 0, 3, 0, 0, -2, -10, 0, 0, 6, 0, 2, -2, 1, 0, -2, 0], [3, 0, 17, 0, 0, -12, 0, 0, -10, 2, 0, 8, 9, -8, 1, -1, 3, 0, -3, 0], [0, 0, 0, -1, 0, 0, 0, 9, 0, 0, 4, 4, 0, 4, -1, -1, 0, -4, 0, -1], [-10, 0, 0, 0, 0, 8, 0, 0, -1, -2, 0, -8, -4, 8, 2, -2, 0, 0, 12, 0], [0, -7, 0, -4, 6, 0, 6, 15, 0, 0, -5, 2, 0, 2, 4, 4, 0, -3, 0, 2], [12, 0, -12, 0, 0, 15, 0, 0, -2, -3, 0, 3, -16, -3, 1, -1, 15, 0, 8, 0], [0, -15, 0, -5, 9, 0, -6, 25, 0, 0, 0, 10, 0, 10, 12, 12, 0, 8, 0, 1], [-6, 0, 6, 0, 0, 3, 0, 0, 4, 6, 0, 2, 4, -2, -2, 2, 2, 0, -6, 0], [0, -1, 0, 1, 8, 0, 2, 9, 0, 0, -2, 0, 0, 0, 5, 5, 0, 1, 0, 1], [0, 12, 0, 7, -2, 0, 7, 0, 0, 0, 3, -4, 0, -4, -4, -4, 0, -3, 0, -2], [3, 0, 9, 0, 0, -3, 0, 0, 6, -2, 0, -4, 9, 4, 0, 0, -1, 0, -7, 0], [12, 0, 5, 0, 0, -1, 0, 0, 2, -2, 0, 12, 0, -12, -3, 3, 4, 0, -6, 0], [18, 0, -12, 0, 0, 5, 0, 0, -4, 1, 0, 1, -13, -1, -1, 1, 1, 0, 5, 0], [0, -17, 0, -24, 16, 0, 1, 27, 0, 0, -8, 15, 0, 15, 10, 10, 0, 4, 0, -7], [0, -3, 0, 15, -9, 0, -10, -2, 0, 0, 1, -6, 0, -6, -5, -5, 0, -2, 0, 3], [-9, 0, 4, 0, 0, -4, 0, 0, 10, -3, 0, -7, -1, 7, -1, 1, 6, 0, -7, 0], [0, -1, 0, -6, -2, 0, -2, -8, 0, 0, 3, -7, 0, -7, -3, -3, 0, 9, 0, -2], [-9, 0, 12, 0, 0, 4, 0, 0, -9, -10, 0, -3, -6, 3, 6, -6, 1, 0, 4, 0], [-16, 0, 7, 0, 0, -4, 0, 0, 6, -2, 0, 12, 6, -12, -5, 5, 4, 0, -6, 0], [0, -11, 0, -1, -12, 0, -4, 11, 0, 0, -3, -3, 0, -3, 5, 5, 0, 19, 0, 8], [7, 0, -1, 0, 0, -2, 0, 0, 4, -2, 0, 4, -5, -4, -1, 1, 5, 0, 1, 0], [0, 13, 0, 9, -7, 0, -4, -25, 0, 0, 1, -3, 0, -3, -12, -12, 0, -4, 0, 5], [0, -6, 0, 3, 3, 0, -6, 7, 0, 0, -2, 12, 0, 12, 5, 5, 0, -5, 0, -3], [0, 16, 0, 9, -8, 0, 6, -19, 0, 0, 8, -5, 0, -5, -5, -5, 0, 0, 0, 12], [16, 0, 18, 0, 0, -11, 0, 0, -7, 5, 0, -3, -3, 3, 0, 0, -3, 0, 3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1045_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1045_2_b_c();".
// 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_1045_2_b_c(:prec:=20)
    chi := MakeCharacter_1045_b();
    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_1045_2_b_c();".
// 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_1045_2_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1045_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 276, 0, 2753, 0, 9095, 0, 13894, 0, 11551, 0, 5623, 0, 1640, 0, 281, 0, 26, 0, 1]>],Snew);
    return Vf;
end function;