
// Make newform 570.2.m.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_m_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_570_2_m_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_570_2_m_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 := [256, 0, 0, 0, 5057, 0, 0, 0, 4652, 0, 0, 0, 1318, 0, 0, 0, 108, 0, 0, 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, 0, 0, 9961151, 0, 0, 0, 6031344, 0, 0, 0, -289166, 0, 0, 0, -120336, 0, 0, 0, -1233], [0, 0, 11554103, 0, 0, 0, 8963426, 0, 0, 0, 2457754, 0, 0, 0, 199306, 0, 0, 0, 1843, 0], [0, 0, -932561, 0, 0, 0, 4366379, 0, 0, 0, 2661844, 0, 0, 0, 273087, 0, 0, 0, 2595, 0], [-13013696, 0, 42067188, 0, -24076543, 0, 32921622, 0, -8670841, 0, 7084096, 0, -770625, 0, 477582, 0, -7207, 0, 4296, 0], [0, 8864472, 0, -23108206, 0, 21144461, 0, -17926852, 0, 5923921, 0, -4915508, 0, 450983, 0, -398612, 0, 4131, 0, -3686], [-13013696, 0, -42067188, 0, -24076543, 0, -32921622, 0, -8670841, 0, -7084096, 0, -770625, 0, -477582, 0, -7207, 0, -4296, 0], [0, -12050376, 0, 0, 0, -27008625, 0, 0, 0, -11417761, 0, 0, 0, -1090267, 0, 0, 0, -10283, 0, 0], [9123648, 0, -54117564, 0, -3630112, 0, -59930247, 0, -5104050, 0, -18501857, 0, -610604, 0, -1567849, 0, -5890, 0, -14579, 0], [9123648, 0, 54117564, 0, -3630112, 0, 59930247, 0, -5104050, 0, 18501857, 0, -610604, 0, 1567849, 0, -5890, 0, 14579, 0], [0, 0, 0, 98273977, 0, 0, 0, 113829150, 0, 0, 0, 37292880, 0, 0, 0, 3256034, 0, 0, 0, 30391], [0, -24973328, 0, -88088723, 0, -9194094, 0, -78087767, 0, 408180, 0, -20551431, 0, 147562, 0, -1619593, 0, 1504, 0, -14918], [0, -24973328, 0, 88088723, 0, -9194094, 0, 78087767, 0, 408180, 0, 20551431, 0, 147562, 0, 1619593, 0, 1504, 0, 14918], [0, 19860784, 0, 111196929, 0, 9194094, 0, 96014619, 0, -408180, 0, 25466939, 0, -147562, 0, 2018205, 0, -1504, 0, 18604], [0, 19860784, 0, -111196929, 0, 9194094, 0, -96014619, 0, -408180, 0, -25466939, 0, -147562, 0, -2018205, 0, -1504, 0, -18604], [18284064, -963320, 0, 31009358, 32174759, 2932082, 0, 42003395, 14554471, 2746920, 0, 13586349, 1438589, 319642, 0, 1169237, 13621, 3076, 0, 10893], [-8423808, 0, -54117564, 0, 37863114, 0, -59930247, 0, 24041672, 0, -18501857, 0, 2504338, 0, -1567849, 0, 23836, 0, -14579, 0], [0, 963320, -76262450, -31009358, 0, -2932082, -80789181, -42003395, 0, -2746920, -26164285, -13586349, 0, -319642, -2286103, -1169237, 0, -3076, -21341, -10893], [-349920, -37373624, 12922952, 0, -17116501, -53319220, -17814531, 0, -9468811, -20478392, -11825941, 0, -946867, -1889572, -1237829, 0, -8973, -17752, -11787, 0], [699840, 0, 25845904, -316937591, 34233002, 0, -35629062, -323323904, 18937622, 0, -23651882, -98874134, 1893734, 0, -2475658, -8384792, 17946, 0, -23574, -77979]];
    Rf_basisdens := [1, 5112544, 2556272, 1278136, 5112544, 5112544, 5112544, 5112544, 5112544, 5112544, 10225088, 5112544, 5112544, 5112544, 5112544, 5112544, 5112544, 5112544, 5112544, 10225088];
    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_570_m();"
function MakeCharacter_570_m()
    N := 570;
    order := 4;
    char_gens := [191, 457, 211];
    v := [4, 1, 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_570_m_Hecke();"
function MakeCharacter_570_m_Hecke(Kf)
    N := 570;
    order := 4;
    char_gens := [191, 457, 211];
    char_values := [[1, 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, 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, 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, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, -1, 1, -1, 1], [1, 0, 1, 0, 2, 0, 1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1, 1, 1, -1, 0, -1, 0, -1, 1], [0, 1, 2, -1, 1, 1, -1, 2, 1, -1, -1, -1, 0, 1, 0, 0, 0, 0, -1, -1], [0, 0, -2, 1, -1, -1, 1, 1, -1, 2, 0, -1, 0, -1, 0, 0, 1, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, -1, 0, -1, 0, -1, 2, -1, 2, -1, 2, -1, -1, 1, 0, 0], [-3, 0, 3, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 2, 2, -1, 1, 0, -1, 0, 1, 0, 0, -2, 2, -2, 2, 0, 1, 0, 0, -2], [0, -1, 0, 0, 0, 1, 0, 0, 0, -1, -1, 0, -2, 0, 0, 0, -1, 0, -1, 1], [4, 0, 0, 0, 3, 0, 3, 0, 1, -3, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0], [0, 0, -4, 2, -2, 0, 2, 0, -2, 4, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0], [0, -2, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -3, 1, -5, 0, -1, 0, -5, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 4, 0, -1, 5, 0, 1, 0, 1, 0, -1, 0, -1, 1], [4, 1, 4, 1, 0, -1, 1, -1, 0, 0, 2, -1, 2, -1, 2, -1, 1, 1, 0, 0], [0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 2, -3, 1, -3, 1, 0, 0, 0, 0, 0], [0, 0, -2, 1, -1, -5, 1, -3, -1, 2, 0, -2, 3, -1, 4, 0, 1, 0, 2, 0], [-2, 1, 0, 0, 2, -1, 2, -1, 1, 1, 2, -1, 2, -1, 2, -2, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -3, 3, 0, 0, 0, 0, 0], [0, 0, -2, 1, -1, 0, 1, 2, -1, 2, 0, 1, 1, 3, 3, 0, 1, 0, 2, 0], [0, 2, 0, 0, 0, -2, 0, -3, 0, -1, -1, -2, 0, -2, 4, 0, -1, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, -1, -3, 3, 0, 0, 0, 0, 0], [0, 0, -4, 2, -2, -4, 2, 0, -2, 4, 0, -1, 3, 0, 4, 0, 2, 0, 4, 0], [8, 1, 0, 0, 0, -1, 0, -1, 3, 2, 2, -1, 2, -1, 2, -2, -1, 0, 0, 0], [2, 0, -2, 1, 1, 0, 2, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0], [0, 2, -8, 4, 1, -2, -1, -2, 1, -1, 4, -2, 4, -2, 4, 0, 0, 4, 0, 0], [0, 1, 4, -2, 0, -1, 0, -1, 3, -3, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [0, 1, 6, -3, 3, 1, -3, 2, 3, -3, -1, 4, 3, -4, 3, 0, 0, 0, -3, -3], [-6, 0, 6, 1, -3, 0, -4, 0, 3, -5, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0], [0, 1, 0, 2, 3, -1, -1, -1, 3, 3, 2, -1, 2, -1, 2, -1, 2, 1, 0, 0], [0, 0, -4, 2, -2, 2, 2, -2, -2, 4, 0, 2, 0, 2, 0, 0, 2, 0, 4, 0], [0, -2, 4, -2, 2, 0, -2, 0, 2, 0, -10, 4, -4, 6, -6, 0, 2, 0, 0, -4], [0, -2, 0, 0, 0, 2, 0, -1, 0, -1, -3, 3, -4, 3, 0, 0, -1, 0, -1, 1], [0, 1, 2, -1, 1, 1, -1, -6, 1, -1, 7, 2, 1, -2, 1, 0, 0, 0, -1, -1], [-6, 0, 0, 0, 1, 0, 1, 0, -1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0], [0, -5, -2, 1, -1, 0, 1, 0, -1, 0, -2, 3, -3, 6, -6, 0, -1, 0, 0, 2], [-3, 0, 3, 2, 1, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 1, 0, 4, -5, -1, 5, -1, -1, 1, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [0, 3, 0, 0, 0, 3, 0, -5, 0, 0, 8, -3, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 2, -1, 1, 0, -1, 0, 1, 0, -6, 2, -2, -1, 1, 0, 1, 0, 0, -2], [0, 0, 2, -1, 1, -2, -1, 8, 1, -2, 0, -7, -5, -7, -5, 0, -1, 0, -2, 0], [0, -1, -4, 6, 0, 1, 0, 1, -4, 4, -2, 1, -2, 1, -2, 0, 0, -2, 0, 0], [-6, -1, -6, -1, -4, 1, 1, 1, -4, -2, -2, 1, -2, 1, -2, 1, -1, -1, 0, 0], [0, -1, -2, -1, 2, 1, -2, 1, -2, 2, -2, 1, -2, 1, -2, 0, 0, -2, 0, 0], [0, -1, 0, 0, 0, -1, 0, 3, 0, 0, -4, -3, -6, 3, -6, 0, 0, 0, 0, 0], [-12, 1, 0, 0, -6, -1, -6, -1, -1, 0, 2, -1, 2, -1, 2, -2, 1, 0, 0, 0], [0, 0, 8, -4, 4, 4, -4, -2, 4, -8, 0, 3, -1, 1, -3, 0, -4, 0, -8, 0], [-3, 0, -3, -1, -2, 0, 1, 0, -2, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 9, 0, -1, 9, 1, 3, 1, -3, 0, -1, 0, -1, 1], [-4, -1, 0, 0, -1, 1, -1, 1, -5, -1, -2, 1, -2, 1, -2, 2, 4, 0, 0, 0], [4, 0, -4, -3, 3, 0, 2, 0, -3, 5, 0, 0, 0, 0, 0, -1, 3, -1, 0, 0], [0, 0, -4, 2, -2, 0, 2, -2, -2, 2, 2, -5, -4, 5, -4, 0, 0, 0, 2, 2], [-8, 3, -8, 4, 1, -3, -1, -3, 1, 5, 6, -3, 6, -3, 6, -3, 4, 3, 0, 0], [0, 6, 2, -1, 1, 0, -1, 0, 1, 0, -8, 0, 0, 6, -6, 0, 1, 0, 0, -2], [0, 0, -4, 2, -2, 4, 2, -12, -2, 4, 0, 4, 0, 4, 0, 0, 2, 0, 4, 0], [8, 0, 0, 0, 6, 0, 6, 0, 2, -1, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [-5, 0, -5, -1, -1, 0, 3, 0, -1, -4, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 4, -2, 2, 6, -2, -8, 2, -4, 0, 4, -2, 0, -6, 0, -2, 0, -4, 0], [0, -4, 4, -2, 2, -4, -2, 4, 2, -2, -8, 2, 0, -2, 0, 0, 0, 0, -2, -2], [0, 0, 2, -1, 1, 5, -1, 15, 1, -2, 0, -4, -9, 3, -2, 0, -1, 0, -2, 0], [-6, 0, 6, -4, -2, 0, -7, 0, 2, -3, 0, 0, 0, 0, 0, -1, 4, -1, 0, 0], [0, 1, -10, 2, -5, -1, 5, -1, -3, 3, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [6, -3, 6, -3, 6, 3, 1, 3, 6, -4, -6, 3, -6, 3, -6, 3, -3, -3, 0, 0], [0, 1, 4, -3, 5, -1, -5, -1, -3, 3, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [1, 0, -1, -7, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0, -4, 7, -4, 0, 0], [0, 1, -2, 1, -1, 0, 1, 0, -1, 0, 2, -3, 3, -5, 5, 0, -1, 0, 0, 2], [0, 2, 0, 0, 0, -2, 0, 0, 0, -4, 2, -6, 0, -6, 4, 0, -4, 0, -4, 4], [0, 4, 2, -1, 1, 0, -1, 0, 1, 0, 6, -4, 4, -4, 4, 0, 1, 0, 0, -2], [0, -1, 10, 0, -5, 1, 5, 1, 0, 0, -2, 1, -2, 1, -2, 0, 0, -2, 0, 0], [-2, 0, 2, -1, 5, 0, -4, 0, -5, -3, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 6, 1, 2, -1, 2, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 5, 0, 13, 0, -2, 8, -3, -8, -3, -2, 0, -2, 0, -2, 2], [0, 4, -8, 9, -2, -4, 2, -4, -5, 5, 8, -4, 8, -4, 8, 0, 0, 8, 0, 0], [0, -8, -6, 3, -3, -8, 3, -3, -3, 3, -5, -2, 4, 2, 4, 0, 0, 0, 3, 3], [0, -2, -8, 4, -4, -2, 4, -14, -4, 4, 12, 4, 2, -4, 2, 0, 0, 0, 4, 4], [0, 1, -2, 1, -1, 0, 1, 0, -1, 0, -2, -5, 5, -4, 4, 0, -1, 0, 0, 2], [0, 5, 0, 0, 0, -5, 0, 6, 0, 1, 11, -1, 8, -1, 2, 0, 1, 0, 1, -1], [2, -3, 2, 0, 1, 3, -4, 3, 1, 4, -6, 3, -6, 3, -6, 3, 0, -3, 0, 0], [0, 3, 0, 0, 0, -3, 0, -1, 0, 2, 2, -9, -3, -9, 9, 0, 2, 0, 2, -2], [9, 0, -9, 0, 6, 0, 8, 0, -6, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 2, 0, 0, -7, -2, -7, -2, 0, -2, 4, -2, 4, -2, 4, -4, -2, 0, 0, 0], [3, 2, 3, 3, -9, -2, -4, -2, -9, 7, 4, -2, 4, -2, 4, -2, 3, 2, 0, 0], [4, 0, -4, 1, 1, 0, 2, 0, -1, 1, 0, 0, 0, 0, 0, -5, -1, -5, 0, 0], [0, 0, -6, -9, 1, 0, -1, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 1, -1, -10, 1, 0, -1, 2, 0, -3, 7, -4, 6, 0, 1, 0, 2, 0], [-8, -1, 0, 0, -4, 1, -4, 1, -3, -6, -2, 1, -2, 1, -2, 2, -3, 0, 0, 0], [0, 4, 8, 6, -2, -4, 2, -4, 2, -2, 8, -4, 8, -4, 8, 0, 0, 8, 0, 0], [-5, -2, -5, -2, -4, 2, 3, 2, -4, -5, -4, 2, -4, 2, -4, 2, -2, -2, 0, 0], [0, 4, 0, 0, 0, -4, 0, -17, 0, -1, -13, 9, 9, 9, -1, 0, -1, 0, -1, 1], [0, -5, -12, 6, -6, -5, 6, 1, -6, 6, -6, -4, 4, 4, 4, 0, 0, 0, 6, 6], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 2, -2, -2, 2, 0, 0, 0, 0, 0], [6, -1, 0, 0, -3, 1, -3, 1, -5, -3, -2, 1, -2, 1, -2, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 2, -2, -4, 0, 0, 0, 0, 0], [13, 0, -13, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -10, -6, 3, -3, 0, 3, 0, -3, 0, -10, 7, -7, 9, -9, 0, -3, 0, 0, 6], [0, 7, 0, 0, 0, -7, 0, -5, 0, 6, 2, -1, 9, -1, 5, 0, 6, 0, 6, -6], [4, -3, 0, 0, 1, 3, 1, 3, -1, -3, -6, 3, -6, 3, -6, 6, -2, 0, 0, 0], [-7, 0, 7, 0, -4, 0, 6, 0, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 6, 3, 3, 0, 6, 0, -3, 3, 0, 0, 0, 0, 0, -1, -3, -1, 0, 0], [16, -3, 16, -1, -2, 3, 1, 3, -2, -2, -6, 3, -6, 3, -6, 3, -1, -3, 0, 0], [0, 4, 0, 0, 0, -4, 0, 14, 0, 4, 18, 2, -2, 2, 10, 0, 4, 0, 4, -4], [0, 8, 4, -2, 2, 8, -2, 0, 2, -2, 8, 2, -2, -2, -2, 0, 0, 0, -2, -2], [0, 0, 6, -3, 3, 6, -3, 0, 3, -6, 0, 7, 1, 4, -2, 0, -3, 0, -6, 0], [-4, -1, -4, -1, 0, 1, 5, 1, 0, -6, -2, 1, -2, 1, -2, 1, -1, -1, 0, 0], [-10, 0, 10, -2, -2, 0, -5, 0, 2, -3, 0, 0, 0, 0, 0, -3, 2, -3, 0, 0], [0, 2, -12, 2, 5, -2, -5, -2, -3, 3, 4, -2, 4, -2, 4, 0, 0, 4, 0, 0], [6, -2, 0, 0, 9, 2, 9, 2, 1, -7, -4, 2, -4, 2, -4, 4, -8, 0, 0, 0], [0, 1, -12, 6, -6, 1, 6, -7, -6, 6, 8, 4, 10, -4, 10, 0, 0, 0, 6, 6], [-20, 3, -20, 2, -5, -3, -5, -3, -5, 7, 6, -3, 6, -3, 6, -3, 2, 3, 0, 0], [15, 0, -15, -2, 3, 0, 4, 0, -3, 6, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0], [15, -4, 15, -4, 7, 4, 2, 4, 7, -6, -8, 4, -8, 4, -8, 4, -4, -4, 0, 0], [0, -4, 0, 0, 0, 4, 0, 21, 0, 3, 17, -5, -6, -5, -2, 0, 3, 0, 3, -3], [0, 1, -6, 3, -3, 1, 3, 4, -3, 3, -3, -4, 5, 4, 5, 0, 0, 0, 3, 3], [0, -5, -2, 1, -1, 0, 1, 0, -1, 0, 14, 1, -1, 2, -2, 0, -1, 0, 0, 2], [0, 0, 8, -4, 4, 2, -4, 0, 4, -8, 0, 8, 6, 0, -2, 0, -4, 0, -8, 0], [0, -8, -10, 5, -5, 0, 5, 0, -5, 0, -6, 3, -3, 3, -3, 0, -5, 0, 0, 10], [-8, -3, 0, 0, -1, 3, -1, 3, 1, 3, -6, 3, -6, 3, -6, 6, 2, 0, 0, 0], [24, 0, 0, 0, 4, 0, 4, 0, 1, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0], [0, -2, -16, -4, -1, 2, 1, 2, -1, 1, -4, 2, -4, 2, -4, 0, 0, -4, 0, 0], [0, 1, 12, -3, -3, -1, 3, -1, 1, -1, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [5, 0, -5, 1, 2, 0, 3, 0, -2, 2, 0, 0, 0, 0, 0, -4, -1, -4, 0, 0], [-6, 1, -6, 5, -6, -1, -5, -1, -6, 10, 2, -1, 2, -1, 2, -1, 5, 1, 0, 0], [0, -1, 8, -2, 4, 1, -4, 1, -4, 4, -2, 1, -2, 1, -2, 0, 0, -2, 0, 0], [0, -6, -6, 3, -3, 0, 3, 0, -3, 0, -14, -2, 2, 0, 0, 0, -3, 0, 0, 6], [0, 3, 6, -3, 3, 3, -3, -4, 3, -3, 7, 1, -10, -1, -10, 0, 0, 0, -3, -3], [10, 0, -10, -5, -1, 0, -6, 0, 1, -1, 0, 0, 0, 0, 0, 1, 5, 1, 0, 0], [-14, 1, 0, 0, 3, -1, 3, -1, 3, 5, 2, -1, 2, -1, 2, -2, 2, 0, 0, 0], [0, 0, 14, 4, -4, 0, 4, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, -14, 2, -2, -4, 4, 0, 0, 0, 0, 0], [0, 0, 4, -2, 2, 16, -2, -4, 2, -4, 0, 4, -12, 0, -16, 0, -2, 0, -4, 0], [0, 0, 4, -2, 2, 4, -2, 14, 2, -4, 0, -4, -8, 0, -4, 0, -2, 0, -4, 0], [0, 1, 14, -2, 1, -1, -1, -1, 5, -5, 2, -1, 2, -1, 2, 0, 0, 2, 0, 0], [0, 9, 8, -4, 4, 9, -4, 3, 4, -4, 6, 1, 0, -1, 0, 0, 0, 0, -4, -4], [4, -3, 0, 0, 7, 3, 7, 3, 6, 6, -6, 3, -6, 3, -6, 6, 0, 0, 0, 0], [15, 4, 15, 5, -1, -4, 6, -4, -1, -1, 8, -4, 8, -4, 8, -4, 5, 4, 0, 0], [0, 0, -6, 3, -3, 2, 3, 8, -3, 6, 0, -15, -17, 1, -1, 0, 3, 0, 6, 0], [0, 4, 0, 0, 0, -4, 0, -3, 0, 1, 1, -12, 8, -12, 0, 0, 1, 0, 1, -1], [0, 2, 0, 0, 0, -2, 0, 6, 0, 2, 8, -2, 4, -2, 0, 0, 2, 0, 2, -2], [4, -1, 4, -1, 6, 1, 5, 1, 6, -6, -2, 1, -2, 1, -2, 1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 7, 5, 7, 5, 0, 0, 0, 0, 0], [0, -1, -12, 12, -3, 1, 3, 1, 1, -1, -2, 1, -2, 1, -2, 0, 0, -2, 0, 0], [0, 0, -14, 7, -7, 0, 7, 0, -7, 0, -14, 2, -2, 4, -4, 0, -7, 0, 0, 14], [0, 0, 4, -2, 2, 9, -2, -11, 2, -4, 0, 1, -8, 5, -4, 0, -2, 0, -4, 0], [-14, 1, 0, 0, -7, -1, -7, -1, -7, 5, 2, -1, 2, -1, 2, -2, 12, 0, 0, 0], [-14, 5, -14, 1, -12, -5, -2, -5, -12, 3, 10, -5, 10, -5, 10, -5, 1, 5, 0, 0], [0, 0, 2, -1, 1, 2, -1, -12, 1, -2, 0, 6, 4, -6, -8, 0, -1, 0, -2, 0], [0, 0, 0, 0, 0, 6, 0, 18, 0, 0, 0, 12, 6, 4, -2, 0, 0, 0, 0, 0], [0, 0, 16, -8, 8, 0, -8, 12, 8, -8, -12, 2, -2, -2, -2, 0, 0, 0, -8, -8], [0, -2, -18, -4, -1, 2, 1, 2, -1, 1, -4, 2, -4, 2, -4, 0, 0, -4, 0, 0], [0, -5, 22, -2, -1, 5, 1, 5, 11, -11, -10, 5, -10, 5, -10, 0, 0, -10, 0, 0], [7, 0, -7, 3, 1, 0, 5, 0, -1, 2, 0, 0, 0, 0, 0, 6, -3, 6, 0, 0], [0, 4, -2, 1, -1, 4, 1, 15, -1, 1, -11, -1, 1, 1, 1, 0, 0, 0, 1, 1], [-8, 0, 8, -2, -4, 0, -7, 0, 4, -5, 0, 0, 0, 0, 0, 3, 2, 3, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 10, -6, 6, 2, -2, 0, 0, 0, 0, 0], [-3, 0, 3, -3, 2, 0, -5, 0, -2, -2, 0, 0, 0, 0, 0, 2, 3, 2, 0, 0], [0, 2, -14, 7, -7, 2, 7, -17, -7, 7, 19, -2, 1, 2, 1, 0, 0, 0, 7, 7], [0, 0, 0, 0, 0, -12, 0, -18, 0, 0, 0, -6, 6, -4, 8, 0, 0, 0, 0, 0], [0, 6, -12, 6, -6, 0, 6, 0, -6, 0, 20, -8, 8, -10, 10, 0, -6, 0, 0, 12], [0, 5, 14, -7, 7, 5, -7, -2, 7, -7, 7, -3, 2, 3, 2, 0, 0, 0, -7, -7], [-20, 0, 20, 7, -1, 0, 8, 0, 1, 1, 0, 0, 0, 0, 0, 5, -7, 5, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_570_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_570_2_m_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_570_2_m_b(:prec:=20)
    chi := MakeCharacter_570_m();
    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_570_2_m_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_570_2_m_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3200, 3200, 1600, -320, 1408, 1184, 496, 8, 2, 2, 1]>],Snew);
    return Vf;
end function;