
// Make newform 400.2.s.d in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_s_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_400_2_s_d();".
// 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_400_2_s_d();".
// 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 := [512, 0, 256, -256, -160, -224, -80, 24, 74, 70, 37, 6, -10, -14, -5, -4, 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], [-24064, -16768, -23488, -3584, 4592, 13472, 13124, 7730, 1732, -2001, -3026, -2354, -1148, -123, 144, 286, 98, 71], [58624, 44800, 64960, 22176, 368, -25048, -29424, -21502, -9450, 123, 4402, 4526, 2778, 797, 116, -398, -124, -129], [6048, 4832, 7152, 2444, -44, -2972, -3408, -2471, -1033, 98, 580, 559, 331, 88, 2, -53, -19, -16], [28416, 21696, 31296, 9920, -848, -13500, -15406, -10894, -4379, 583, 2676, 2550, 1487, 375, -6, -248, -89, -75], [89600, 71296, 102976, 37376, 1200, -40528, -48500, -35846, -15784, 331, 7678, 7846, 4800, 1377, 120, -706, -258, -229], [75264, 58816, 83776, 28832, -1872, -35116, -40294, -28628, -11749, 1284, 6810, 6492, 3793, 954, 6, -614, -217, -178], [-222976, -175616, -245696, -79200, 10928, 108280, 120736, 84014, 32794, -5723, -21346, -19710, -11242, -2685, 156, 1918, 684, 545], [-133504, -110016, -154176, -57152, -1008, 59756, 71254, 52268, 22749, -604, -10970, -11092, -6713, -1874, -166, 974, 337, 298], [-54848, -36384, -50784, -6096, 13104, 32638, 30068, 17383, 3611, -5065, -7306, -5511, -2621, -257, 358, 665, 235, 159], [-351232, -268416, -381376, -117504, 21456, 171792, 188932, 130042, 49524, -10465, -34194, -31194, -17564, -4043, 352, 3110, 1090, 871], [-168960, -108032, -154432, -14672, 44320, 103336, 94050, 53232, 9903, -17047, -23616, -17512, -8175, -679, 1230, 2162, 761, 503], [46208, 32320, 45504, 8592, -8624, -26316, -25600, -15672, -4196, 3435, 5762, 4568, 2280, 321, -232, -524, -186, -129], [-33024, -23040, -32672, -6240, 6024, 18760, 18318, 11252, 3063, -2417, -4104, -3268, -1639, -237, 162, 374, 133, 93], [33024, 23168, 32672, 6240, -6024, -18760, -18318, -11252, -3063, 2417, 4104, 3268, 1639, 237, -162, -374, -133, -93], [363008, 259392, 368512, 82784, -53344, -196852, -198348, -125876, -38378, 21883, 41990, 34564, 17886, 3013, -1348, -3828, -1344, -981], [372864, 258560, 364800, 67376, -69792, -211368, -205144, -125222, -33350, 27653, 46182, 36566, 18238, 2547, -1844, -4198, -1484, -1039], [-1132032, -772224, -1092544, -172352, 240976, 663696, 630772, 375990, 90616, -95003, -147678, -114262, -55664, -6769, 6632, 13458, 4754, 3253]];
    Rf_basisdens := [1, 1280, 1280, 160, 640, 1280, 640, 1280, 640, 320, 1280, 640, 128, 64, 64, 640, 640, 1280];
    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_400_s();"
function MakeCharacter_400_s()
    N := 400;
    order := 4;
    char_gens := [351, 101, 177];
    v := [2, 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_400_s_Hecke();"
function MakeCharacter_400_s_Hecke(Kf)
    N := 400;
    order := 4;
    char_gens := [351, 101, 177];
    char_values := [[-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, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 0, 1, 1, 1, -1, 1, 2, 0, 0, 1, 0, 1, -1, 1, 0, -1, 1], [1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, -1, 0, -1, 0, 0, 0, 0], [1, 1, -1, 0, 1, -1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 1, -1], [0, 1, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, -2, 0, 0, -1, 0, 0], [1, 1, 0, 0, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 0, 1, -1, 1], [1, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1], [1, 0, -2, 1, 0, -1, -1, 0, -1, 1, -1, -1, -1, 0, 0, -1, 2, 0], [0, 0, 3, -1, 0, 0, 1, 0, 0, 1, 1, 0, 1, -1, 2, -1, 0, 0], [1, 1, 2, -2, 0, 0, 0, 1, -1, 0, 1, -1, 0, -1, 2, 0, 0, 1], [0, 0, 1, 1, 0, 0, -1, -1, 0, 2, 0, 0, 2, 1, 1, -2, 1, 0], [-1, 1, 0, 0, 0, -1, -1, 1, -1, -1, 1, -1, -3, -1, 0, -1, 0, 1], [-1, -1, 0, 0, 0, 0, 2, 1, -1, 0, 1, 1, 0, 1, 0, 0, 0, -1], [-2, 0, 2, 1, 0, 2, 1, -1, 1, -1, -2, 1, 0, 2, 0, -1, 0, -1], [1, 1, 3, -1, 1, -2, -1, -1, 0, 0, 1, -1, 0, 1, 0, -1, -1, 0], [0, 0, -3, 1, 0, 0, -1, 3, 0, 0, -2, 0, 0, 1, -1, 0, 1, 2], [3, -1, 1, 3, 2, -1, 1, 1, -1, 1, -1, 1, 1, 0, 2, 1, 0, 1], [1, -1, 1, 0, 1, -1, 0, 3, 0, 0, -1, 0, -1, 0, 0, 0, 1, 1], [3, 0, 1, 2, 0, -3, -2, 5, 1, 1, 1, -1, 1, -1, 1, 1, 0, 2], [-2, -1, -2, -1, -3, 2, -3, -2, 0, 0, 2, 0, 0, -3, -1, 0, 0, 1], [-1, -1, 4, -2, -2, -1, 0, -2, -1, -1, -1, 1, -1, -1, -1, -1, 0, -1], [0, -1, 2, -2, -3, 0, 0, 0, -1, 1, 2, -1, 2, -3, -1, 1, -3, 1], [-2, -1, 2, -4, -2, -1, 2, 0, 1, -1, 1, 1, 0, -3, -1, -1, -2, -1], [-3, -1, 0, 0, -3, 1, -3, -5, 3, 3, -3, -1, -1, 3, -2, -1, 3, -1], [-6, 2, 0, 0, -1, 2, 0, -4, 2, 0, -4, -2, 0, 2, 0, 0, 0, 0], [0, -2, -1, 1, 1, 1, 1, -3, -1, 1, 4, 0, 1, 0, -1, 0, 1, 1], [0, 0, -1, -2, -2, 1, -2, -3, -1, -1, -1, -1, -2, 1, -1, -1, 2, -2], [4, -4, 4, 4, 1, 0, 1, 1, 2, 2, 2, 2, 8, -2, -1, 2, -1, 0], [-4, 0, 2, -3, -3, 4, 1, -3, 3, -3, 2, -3, 4, 0, -4, 3, -3, -1], [0, 0, 3, 0, 4, -3, 4, 1, -1, -1, 1, 1, 0, -1, 3, 1, -4, 0], [5, 1, 2, -1, 0, 1, -5, -3, -4, 4, 3, -2, 1, -5, 3, 2, 0, 4], [0, 1, 2, -1, 3, 1, -1, 2, 1, 1, 2, 0, 1, 2, -1, 0, -3, 0], [-8, 1, 2, -3, -6, 0, -1, -2, 2, -2, -2, -2, -2, 1, -1, -2, 0, -1], [2, 1, -4, 1, 0, 0, -1, 1, -2, 1, 3, 2, 1, -3, 0, 1, 0, 1], [4, 4, 0, 0, 3, -2, 4, 0, 0, -2, -4, 0, -2, 4, 2, -2, 0, -2], [-5, -1, 3, -5, -3, 3, 4, -4, 5, -5, 1, 1, -1, 0, -3, -1, -3, -3], [-2, 0, 3, -3, 0, 0, 3, 0, 2, -1, 1, 2, -1, -1, 2, 1, 0, 0], [-3, 2, -5, -1, 2, 1, 5, 4, -2, -2, -1, 2, -3, -1, 3, 2, -2, 1], [-2, -1, -2, 0, 0, 1, 0, 0, 3, -3, 1, -1, 0, 3, -1, 1, 0, 1], [-2, 0, -3, 1, 0, 0, -1, 1, 2, 2, -4, 2, 2, 3, 1, -2, 2, 0], [-3, 3, -3, -4, -1, -1, -2, -3, -2, -2, -3, -4, -9, 2, 2, -4, 1, -1], [0, -1, 3, 0, 0, 3, 2, 4, 0, -3, 2, 0, 3, -2, -3, 3, 2, 1], [3, 1, 0, -2, 0, -5, 0, -2, -3, 1, 3, -3, -3, -3, 3, -1, -4, 3], [6, 2, 0, 1, 3, 0, -1, -1, -1, 1, -2, 1, -6, 0, 2, -1, 3, -1], [-2, -2, 1, -5, -3, 0, 2, -2, 2, -2, 4, 0, 0, -3, -2, 0, -3, 0], [4, 5, -3, -4, 0, 0, -6, -1, -4, 0, -2, -4, -2, 2, 4, 0, 3, -1], [5, -1, 3, 4, -2, -2, 2, 5, 2, 2, 2, 0, 7, -2, 2, 0, 2, 3], [-4, 3, 2, -2, 1, 2, 2, -4, 1, -1, -2, 1, 2, -1, -1, -1, 1, -5], [-5, -6, 3, -2, -2, 3, -2, -1, 1, 1, 3, -1, 1, -1, -3, 1, 0, 0], [-2, -1, 3, -2, -4, -1, 0, -2, 0, -3, -2, 0, -3, 0, 1, -3, 0, -1], [3, 4, -1, -1, -1, -1, -3, 0, 0, 0, 1, 0, 3, 1, -3, 0, 1, 3], [-4, 0, -5, 6, 2, -3, 4, 3, 1, -1, -5, 5, -2, 7, 1, -5, 2, -4], [0, -2, 2, 6, 0, -2, -1, 5, 2, 2, 2, -2, 0, -4, 1, -2, 0, 4], [-3, 4, -2, -1, 5, 0, -1, -2, 2, -2, -3, 1, 0, 1, 2, -1, 5, -3], [-1, 0, -2, -1, 0, -1, 1, -4, 1, -1, -1, 1, -3, 0, 0, 1, 4, -4], [-1, -1, 3, 1, 0, -3, 1, -2, 1, 2, 4, 1, 4, 0, 1, -2, -6, 5], [6, 2, 4, -2, 0, -1, -2, 1, -6, -1, 6, -6, -3, -4, 1, 1, -6, 4], [4, 3, -1, -4, 0, -2, -2, 1, -4, 0, 0, -4, -10, -2, 4, 0, -1, 1], [6, -1, 6, -3, -4, 0, -1, -1, 2, 1, -1, -2, 1, 1, -4, 1, 0, -1], [0, 0, 7, -5, 0, 2, 5, 1, 0, -2, 2, 0, -6, -5, 3, 2, -3, -4], [-8, -1, 0, -3, 0, 6, -5, -8, 2, 2, 0, -2, 2, -3, -5, 2, 0, 1], [7, 2, -10, 6, 2, -8, -6, 6, -4, 4, -2, 2, -5, 2, 8, -2, 2, 4], [0, -2, 4, 0, -8, 3, 0, -3, 6, -1, -6, -6, -1, 6, 1, -1, 0, 0], [-2, 3, -1, -1, 3, -2, 7, 4, 1, 1, -2, 1, 0, 2, 6, 1, -3, 2], [1, 0, -2, 4, 4, 0, -2, -2, -4, 4, 2, 0, 3, 0, 2, 0, 4, 2], [14, -1, 2, -1, 1, 2, -3, -2, 0, 4, 2, 0, 4, -3, -5, 4, 0, 1], [5, 1, -5, 0, -2, -6, -2, 3, -2, 2, 0, 2, -5, -2, 4, -2, -2, 1], [3, -4, 4, 2, 6, 4, 2, 4, 0, 0, 6, 6, 9, -2, -4, 6, -6, -2], [2, 3, -5, -2, 0, 0, -4, -1, -2, 0, -4, -2, -14, 2, 2, 0, 6, -3], [-6, 4, 2, -4, 3, -4, 1, -1, 0, 0, -2, -4, -10, 4, 5, -4, -3, 0], [1, 0, -6, 1, 0, 5, -3, -5, -1, 2, 2, 1, 2, -4, -1, 2, 0, 2], [5, 3, -6, 3, 2, -1, 1, -3, -2, 2, 1, 0, -3, 1, -1, 0, 2, -4], [8, -2, -1, 3, -3, 0, -2, 4, 0, 0, 0, -2, -6, 3, 0, 2, -3, 4], [-6, 1, -6, -1, -5, 3, -3, -6, 1, 1, -4, 2, -3, 2, -5, 2, 5, 0], [11, -1, -2, 8, 10, -6, -2, 11, -1, 2, 3, 1, 2, -3, 6, 2, 0, 5], [3, -3, 0, 2, -4, 1, 4, 4, -1, 3, 3, 1, 3, 3, -1, 3, 0, -1], [10, 1, -5, 8, 4, -5, 2, 10, 0, -1, 0, 0, -1, 2, 9, -1, 0, 3], [1, -4, 1, 5, -2, -1, 1, 2, 0, 0, 1, 0, 1, -5, 1, 0, 2, 1], [8, 2, -9, 7, 3, -3, -5, -3, -5, 5, 2, 2, -5, 2, 7, -2, 3, 3], [5, 2, 8, -3, 0, -1, -1, -2, -5, 3, 5, -5, -7, -4, 6, -3, -8, 0], [6, -1, 6, 6, 1, -6, 2, 4, 3, 3, 2, -1, 8, -3, 3, -1, -1, 5], [-2, 1, -5, 0, 0, 0, -2, 3, 2, 4, -8, 2, 2, 4, 4, -4, 10, -1], [0, -3, 1, 4, 0, -4, 2, -5, 0, 2, 4, 0, 0, -2, -2, -2, -3, 1], [-3, 1, 8, -6, 0, 0, 4, -5, 3, -4, 7, 3, 8, -1, 2, 4, -14, 1], [1, 6, -4, -6, 6, -2, 2, 0, -6, -6, 0, 0, -5, 0, 2, 0, -6, 0], [8, -1, 2, 2, 2, -3, 4, 4, -1, -1, 1, -1, 6, -3, 5, -1, -2, 1], [9, 5, -13, 5, 10, -7, -6, 2, -5, 5, 1, -1, -3, 0, 5, 1, 10, 1], [-4, 1, -9, 2, 0, -2, -4, -7, 4, 0, -4, 4, -14, 8, -2, 0, 5, -1], [6, 0, 0, -2, -6, 4, -2, -4, 6, -2, -4, -6, -2, 4, 0, -2, 0, 0], [6, -2, 5, 1, 2, 0, 4, -4, -6, 6, 2, 4, -4, -5, 0, -4, 2, -2], [-4, -3, 1, 3, -1, 0, -5, -4, -1, -1, 2, 1, -4, -4, -6, 1, 1, 0], [-7, -2, 3, 1, -4, -1, 1, 0, 4, 4, -3, -4, -7, 3, 5, -4, 4, -1], [3, 1, -8, 2, 3, 3, 1, -1, 3, 3, -5, 3, 9, 1, -2, 3, -3, 3], [-5, -1, 3, 4, -4, 2, 2, 3, 0, 0, -4, 0, 5, 4, -2, 0, -4, -1], [-6, -4, -2, 8, 0, 2, 0, -6, 6, 2, -2, 6, 6, 6, -6, -2, 8, -4], [-2, -3, 3, 0, 0, 6, 6, 9, 2, -4, -2, 2, 14, -4, -4, 4, -5, -3], [-6, 1, -6, -4, -4, 7, -2, 4, -3, -3, 1, -1, -10, 1, -1, -1, 4, -3], [0, -3, 11, 2, 0, 2, 4, 1, 0, 0, 8, 0, 18, -4, -2, 0, -1, 3], [-9, -2, 6, -5, 2, -3, -1, 1, 1, -4, 0, -1, -4, 0, -1, -4, 0, -2], [-8, 2, 2, 0, -1, -4, 0, 2, 4, -6, -6, -4, -6, 4, 6, -6, 0, 0], [-4, 3, -3, -10, -8, 3, -4, -4, 2, -3, 4, -2, -3, -2, -7, -3, 0, -3], [7, -6, -3, 1, -3, 5, -3, -6, 2, 2, -5, -2, 7, 1, -1, -2, 3, -5], [2, 0, 5, 6, 6, 1, 2, 1, -3, 3, -7, 5, -4, 3, 3, -5, 6, -2], [-6, 3, 0, 5, 9, -10, -1, 12, -4, 2, 4, 4, 2, -5, 3, 2, 0, 3], [1, 4, 0, -6, -4, 2, 0, 4, 0, 0, 0, -4, -3, 6, 4, -4, 4, -2], [1, -3, 6, 2, 0, 3, 4, 8, -1, -3, 5, -1, 13, -5, -5, 3, -12, 5], [4, 4, -8, -6, 0, -5, -2, -1, -4, -1, -4, -4, -11, -2, 5, 1, -2, -2], [5, -5, -8, 2, -8, 5, -5, -7, -1, -1, -5, -1, 3, -3, -4, -1, 8, -3], [14, 4, -8, 6, 6, -1, 6, 3, 0, -3, -4, 0, -3, 6, 9, -3, 0, 0], [-10, -5, 0, -2, -9, 10, -4, -2, 5, 5, -2, -1, -6, 7, -3, -1, 9, -7], [-4, 0, -12, 9, 6, -4, 1, 1, -1, 1, 0, 3, 2, 8, 2, -3, 6, -1], [-18, 1, 4, -3, 12, -6, 11, 8, 0, -6, -2, 0, -6, 11, 3, -6, 0, -7], [12, -3, 3, 2, 4, -5, -8, 12, 2, 5, 8, -2, 5, -6, -3, 5, 0, 5], [2, 5, 0, -9, -5, -2, -3, -4, 0, -6, 0, 0, -6, -3, -3, -6, 0, -3], [10, 0, -3, 1, 0, -4, -6, 0, -4, 4, 2, 2, -8, -3, 8, -2, 0, 6], [0, 3, -5, -6, 0, 1, 0, 0, 0, -5, -4, 0, 5, 0, 1, 5, -4, -5], [1, -4, 11, 5, 6, -5, -3, -6, 4, 4, 3, 2, 7, -1, -5, 2, -6, 1], [-9, -2, 14, -13, -7, 10, 3, -14, 4, -4, 7, -1, 6, -9, -4, 1, -7, -1], [2, -11, 6, 10, 1, 0, 0, -8, 5, 5, 0, 7, 14, -5, -7, 7, -1, -1], [-4, -3, -1, 2, 0, -5, 4, -10, 4, 3, 0, 4, 1, 0, 1, -3, 2, -5], [4, 4, -5, -1, 0, -2, -7, -3, -4, -4, 4, -4, 0, 3, -3, 4, -5, 6], [-8, -4, 10, 1, -5, 10, 7, -5, 5, -5, -2, 1, 2, 6, -6, -1, -5, -3], [2, -3, 4, -3, -1, 1, 5, -4, 3, 3, -4, 0, 5, 6, 5, 0, 1, -6], [-5, -5, 3, 6, 0, -4, 2, 1, -2, -2, 2, 0, -7, -8, 2, 0, 0, 1], [-11, 0, -9, 6, 2, -1, 6, 3, -5, -3, 1, 5, -3, 1, 9, -3, 0, 0], [-16, -2, 5, -3, -8, 2, 2, 8, 6, -6, -4, -4, 14, 3, -6, 4, -8, 0], [-4, -2, -3, 3, 0, 0, 1, 2, 4, 5, -7, 4, -11, 3, 2, -5, 8, -2], [5, 6, 3, -5, 8, -3, 5, 4, -6, -6, 7, 6, 5, -1, -1, 6, -8, 1], [-10, 0, -12, 0, 0, 4, -5, -7, -2, -2, -6, 0, -12, -2, -5, 0, 0, 0], [-6, -6, -3, 11, 0, -2, 1, -7, 6, 4, 0, 6, -8, 5, -7, -4, -6, 0], [-5, -3, 5, -3, 0, 1, 9, 0, 5, -6, 4, 5, 4, -4, -3, 6, -6, -1], [-7, -3, 6, -2, 0, 7, 8, 2, 7, -3, -3, 7, -1, -1, -1, 3, 0, -9], [-2, -2, 18, -8, 0, 8, 12, 4, 2, -8, 8, 2, 12, -10, 0, 8, -10, -6], [-20, 0, 2, -6, -6, 7, 2, -9, 2, 1, -2, -2, 1, 4, -7, 1, 0, -4], [8, 2, -1, 1, 0, -6, -5, -3, -8, 0, 8, -8, -16, -3, -1, 0, -7, 8], [4, -4, -2, 10, 2, -1, 2, 7, -6, 9, 6, 6, 9, -4, 1, 9, 0, 4], [18, 0, -2, 14, 12, -6, 6, 14, -4, 8, 2, 4, 8, 2, 6, 8, 0, 4], [-11, 8, -11, -5, 1, 1, -3, -2, -2, -2, -5, -2, -15, 3, -1, -2, -1, 3], [5, 6, 4, -1, -3, 4, -7, 0, 0, 0, -5, -5, 0, -1, 2, 5, -3, 1], [7, -3, 12, 0, 5, 3, -7, 3, 1, 1, 11, 1, 9, 1, -8, 1, -5, -3], [4, -3, -6, -5, -15, 10, -3, -10, 2, -2, 2, -2, -2, -1, -3, -2, 0, -1], [3, 4, -8, 1, 3, -6, -3, 12, 4, -4, -7, -3, -4, 5, 0, 3, 3, -1], [-8, -5, 3, 8, 0, 4, 2, 3, 8, 6, -4, 8, 0, 6, -2, -6, 0, 1], [-10, -12, 6, -6, -16, 11, -2, -9, 4, 1, 2, -4, 1, 2, -7, 1, 0, -2], [-2, -2, -1, -2, -8, 5, -10, -7, 5, 5, -5, -7, -4, 7, -3, -7, 8, -4], [2, 0, 0, -15, -4, -2, -3, -5, 5, -5, 8, -1, -6, -10, 2, 1, -4, 3], [7, -3, -5, 5, 8, -5, 2, 4, 1, -1, -3, 7, -15, 6, 5, -7, 8, 1], [-10, 3, 2, -7, 0, -2, 3, 5, 2, -1, 3, -2, -1, 5, -6, -1, 0, -5], [-5, 0, -9, 2, -6, 4, 2, -4, -5, 0, 3, 5, 0, -3, 2, 0, 0, 0], [-8, -4, 2, 6, 5, 2, 6, -2, -4, 4, -2, 4, 4, 2, 2, 4, 0, 0], [-14, -2, 5, 5, 4, 8, 0, -8, -2, 2, 2, 0, 16, 3, 0, 0, 4, 2], [-12, -10, 0, -2, -11, 14, -10, -12, 4, 6, 4, -4, 6, -6, -8, 6, 0, 4], [5, 2, -6, 7, 0, -1, -11, 0, -5, 5, -1, -5, -9, 6, -2, -5, 18, 6], [5, 3, -10, 0, 0, -5, -6, 0, -5, 5, -7, -5, -7, 1, 5, -5, 0, -1], [7, 5, -6, 6, 0, -2, -16, -11, -7, 10, -3, -7, -14, 9, 4, -10, 18, -3], [8, -7, 4, 10, -1, 4, -8, 6, 3, 3, 8, 1, 12, -7, -9, 1, 1, 3], [-14, 1, -8, -4, -2, 5, 0, -10, 3, 3, -9, 3, -8, 7, -3, 3, 2, -3], [2, 0, -1, 1, 0, -6, -1, 1, -2, 6, 0, -2, -22, -1, 5, -6, 3, 6], [7, -2, 10, -6, -4, 2, -2, -10, -8, 8, 8, 2, -1, -14, 4, -2, -4, 4], [9, -1, -7, 7, 5, -5, 2, 8, 5, -5, -9, 7, -21, 12, 5, -7, 5, -1], [-12, -3, 9, 1, -3, -2, 3, 8, 1, 1, 4, -5, -16, 0, 8, -5, 3, -2], [-4, -1, 6, 0, -1, 2, 4, 4, 1, -1, -4, -1, 4, 1, -5, 1, -1, -3], [10, -2, -8, 2, -12, 6, -9, -1, 0, 0, -2, -4, 6, -2, -5, -4, 12, 0], [6, 6, 3, -11, 0, -8, -1, -9, -6, -4, 10, -6, 12, -5, 7, 4, 0, 4], [7, 6, 3, -8, 0, -8, -4, -4, -7, 2, 7, -7, -22, -3, 10, -2, 2, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_2_s_d();".
// 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_400_2_s_d(:prec:=18)
    chi := MakeCharacter_400_s();
    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_400_2_s_d();".
// 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_400_2_s_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 20, -72, -104, 40, 76, -4, -16, 0, 1]>],Snew);
    return Vf;
end function;