
// Make newform 570.2.m.a in Magma, downloaded from the LMFDB on 23 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_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_570_2_m_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 := [16, 0, 0, 0, 19120, 0, 0, 0, 78648, 0, 0, 0, 6416, 0, 0, 0, 153, 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], [-599186688, 0, 0, 0, -1515721480, 0, 0, 0, -115242314, 0, 0, 0, -2625903, 0, 0, 0, -16765, 0, 0, 0], [0, 12328816792, 0, 0, 0, 63121710948, 0, 0, 0, 5168924442, 0, 0, 0, 123405389, 0, 0, 0, 806911, 0, 0], [0, -10577293948, 0, 0, 0, -47719026672, 0, 0, 0, -3887923677, 0, 0, 0, -92597906, 0, 0, 0, -604825, 0, 0], [0, 0, 1096207364, 0, 0, 0, 4261561356, 0, 0, 0, 347394797, 0, 0, 0, 8282406, 0, 0, 0, 54129, 0], [-7006091376, 0, 216220477192, 0, -61610737104, 0, 933673009644, 0, -5124003060, 0, 76223127354, 0, -123229932, 0, 1818129965, 0, -808344, 0, 11884471, 0], [-7006091376, 0, -216220477192, 0, -61610737104, 0, -933673009644, 0, -5124003060, 0, -76223127354, 0, -123229932, 0, -1818129965, 0, -808344, 0, -11884471, 0], [-6281901056, 0, 240878110776, 0, -46444027032, 0, 1059916431540, 0, -3830938056, 0, 86560976238, 0, -92070010, 0, 2064940743, 0, -604298, 0, 13498293, 0], [-7190240256, 64444570240, -248442762312, 0, -18188657760, 137351806272, -1002348912780, 0, -1382907768, 11011628496, -81728941602, 0, -31510836, 260775248, -1948517589, 0, -201180, 1699264, -12734103, 0], [-6281901056, 0, -240878110776, 0, -46444027032, 0, -1059916431540, 0, -3830938056, 0, -86560976238, 0, -92070010, 0, -2064940743, 0, -604298, 0, -13498293, 0], [-7190240256, -64444570240, -248442762312, 0, -18188657760, -137351806272, -1002348912780, 0, -1382907768, -11011628496, -81728941602, 0, -31510836, -260775248, -1948517589, 0, -201180, -1699264, -12734103, 0], [0, 0, 0, -850240722328, 0, 0, 0, -3558464452956, 0, 0, 0, -290369158398, 0, 0, 0, -6924865889, 0, 0, 0, -45261895], [0, 13651993928, 0, -423306773744, 0, 40901095068, 0, -1727640346328, 0, 3327761602, 0, -140922857856, 0, 79345865, 0, -3360431092, 0, 518651, 0, -21963356], [0, 13651993928, 0, 423306773744, 0, 40901095068, 0, 1727640346328, 0, 3327761602, 0, 140922857856, 0, 79345865, 0, 3360431092, 0, 518651, 0, 21963356], [0, -12258860296, 0, 471539897760, 0, -40901095068, 0, 1915149045992, 0, -3327761602, 0, 156208228924, 0, -79345865, 0, 3724856956, 0, -518651, 0, 24345032], [0, -12258860296, 0, -471539897760, 0, -40901095068, 0, -1915149045992, 0, -3327761602, 0, -156208228924, 0, -79345865, 0, -3724856956, 0, -518651, 0, -24345032], [11065189808, -42309175792, 0, -1459603333880, 136345642968, -190876106688, 0, -6057012474372, 11136819804, -15551694708, 0, -494177340558, 264708058, -370391624, 0, -11784791035, 1726694, -2419300, 0, -77025497], [0, -42309175792, -379366025296, -1459603333880, 0, -190876106688, -1748182870776, -6057012474372, 0, -15551694708, -142817533980, -494177340558, 0, -370391624, -3406995518, -11784791035, 0, -2419300, -22270858, -77025497], [7190240256, 0, 248442762312, 2227441426544, 18188657760, 0, 1002348912780, 8934778735776, 1382907768, 0, 81728941602, 728609156748, 31510836, 0, 1948517589, 17372793436, 201180, 0, 12734103, 113542400], [0, 0, 0, 1459603333880, 0, 0, 0, 6057012474372, 0, 0, 0, 494177340558, 0, 0, 0, 11784791035, 0, 0, 0, 77025497]];
    Rf_basisdens := [1, 174141704, 2089700448, 522425112, 31662128, 4179400896, 4179400896, 4179400896, 4179400896, 4179400896, 4179400896, 4179400896, 1393133632, 1393133632, 1393133632, 1393133632, 4179400896, 4179400896, 4179400896, 2089700448];
    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, 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, 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, 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, 1, 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, -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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1], [0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [2, 0, 0, 0, 2, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 1, 1, 0, 0, -1, 1], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 1, 1, 0, 0, 1, -1], [0, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, -2, -2, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, -1], [3, 0, 0, 0, 3, 1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 2, 0, 0, 0, 2, -1, 2, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -2, -2, 2, 0, 0, 0, 0, -2, 0], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 0, 1, 0, 0, 0, -1, 1], [2, 0, 0, 0, 0, 1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, -2, 0, -1, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1], [-6, 0, 0, 0, -6, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -5, 2, 1, -2, 1, 0, 0, 0, 0], [0, 0, -2, -3, 0, 0, 0, 0, -1, 0, 1, 0, 1, 1, -3, -3, 0, 0, 0, 0], [0, 0, 0, -1, 0, -2, -2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, -2, 3, -3, 0, 0, 0, -2], [0, 0, -6, -2, 0, 0, 0, 0, -1, 0, 1, 0, 1, 1, -3, -3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1, -1, 0, 0, -1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -3, 3, -1, 1, 0, 0, 0, 0], [0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0], [0, 1, 0, -1, 0, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [-4, 2, 0, 1, 4, 1, 2, 1, -1, -2, -1, 0, 0, 0, 0, 0, -1, -1, 0, -1], [0, -2, 0, 0, 2, -3, 3, 1, 2, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 1, 0, 2, -2, -1, 2, 1, 2, 0, 0, 0, 0, 0, 0, -2, 0, -1], [0, 0, -2, 1, 0, 0, 0, 0, -1, 0, 0, 2, 2, -2, 2, 2, 0, 0, -1, -1], [2, 2, 0, 1, -2, 3, -2, 3, -1, 2, -1, 0, 0, 0, 0, 0, -1, -1, 0, -1], [5, 0, 0, 0, 5, -1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 4, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 2, 2, -2, 0, 0, 0, 2], [0, 0, 5, -2, 0, 0, 0, 0, 0, 0, -1, 5, 1, 0, -1, 0, 0, 0, -1, -2], [0, 0, 0, 1, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -1], [2, 1, 0, 0, 0, -4, -4, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -2, -3, 3, -1, 1, 0, 0, 2, -1], [-2, 0, 0, 2, 2, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, -2], [0, 0, 0, 1, -4, -1, 1, 3, 0, -3, 0, 0, 0, 0, 0, 0, 0, -2, 0, -1], [0, 0, -2, 1, 0, 0, 0, 0, -2, 0, 0, 2, 2, 0, 2, 0, 0, 0, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -14, 0, 0, 1, -1, 0, 0, 2, 0], [0, 0, -2, -4, 0, 0, 0, 0, -3, 0, 3, 0, 1, 1, -3, -3, 0, 0, 0, 0], [0, 2, 0, -1, 2, 0, 0, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 2, 0, 1], [-2, 0, 0, 0, -2, 4, -3, -4, -3, -3, -3, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, -1, 0, 1, -2, -6, 6, -1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -2, 0, -1], [0, 0, 0, -1, 0, 0, 0, 0, -4, 0, 0, 0, -2, 2, -2, -2, 0, 0, -4, 1], [4, -1, 0, -1, 0, 3, 3, -3, 0, -3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 3, 3, 0, 0, 0, 0], [4, 0, 0, 0, 4, 1, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 1, 0, 0, 0, 0, 0, 0, -1, -4, -3, -3, 3, -3, 0, 0, -1, 1], [-4, 0, 0, 1, 0, 5, 5, -3, 0, -3, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [-2, -2, 0, 3, 2, -7, 0, -7, 1, 0, 1, 0, 0, 0, 0, 0, -3, -3, 0, -3], [0, 0, -5, -1, 0, 0, 0, 0, -2, 0, 0, 5, -3, 0, -3, 0, 0, 0, -2, 1], [1, 0, 0, 0, 1, -3, 1, 3, -3, 1, -3, 0, 0, 0, 0, 0, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, -2, -2, 2, 0, 0, 2, 0], [0, 0, 8, 0, 0, 0, 0, 0, -2, 0, 2, 0, -2, -2, -6, -6, 0, 0, 0, 0], [2, -1, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 5, 6, -4, -6, -1, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, -2, 0, 0, 0, 0, -2, 0, 2, 0, 6, 6, -4, -4, 0, 0, 0, 0], [0, 0, 8, -2, 0, 0, 0, 0, -2, 0, 0, -8, 2, 4, 2, -4, 0, 0, -2, 2], [0, 0, -2, 3, 0, 0, 0, 0, 3, 0, -3, 0, -1, -1, 3, 3, 0, 0, 0, 0], [4, 0, 0, -1, -4, 5, 4, 5, 0, -4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1], [0, 2, 0, 1, 18, -1, 1, -1, -2, 1, -2, 0, 0, 0, 0, 0, 0, -2, 0, -1], [6, 0, 0, 0, 6, -6, 5, 6, 3, 5, 3, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 5, 0, -3, -2, 1, -1, 2, -5, -2, -5, 0, 0, 0, 0, 0, 0, 6, 0, 3], [4, -4, 0, 0, -4, -2, -3, -2, 2, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 4, -3, 3, -2, 2, 0, 0, 2, 1], [0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 16, 0, 0, 0, 0, 0, 0, 2, -4], [0, 0, 0, -1, -4, 1, -1, 8, 0, -8, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1], [6, 2, 0, -3, -6, -1, -2, -1, -1, 2, -1, 0, 0, 0, 0, 0, 3, 3, 0, 3], [0, 0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 3, -1, 2, -1, -2, 0, 0, 0, -1], [0, 0, -6, -1, 0, 0, 0, 0, 0, 0, 2, -6, 4, 0, -4, 0, 0, 0, 2, -1], [0, -3, 0, 0, 4, -6, 6, -2, 3, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 1, -1, 1, 1, 0, 0, -3, 2], [0, 0, -4, 2, 0, 0, 0, 0, 4, 0, 0, 4, 6, -2, 6, 2, 0, 0, 4, -2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -10, 1, -1, 7, -7, 0, 0, 2, 1], [0, 0, 2, -1, 0, 0, 0, 0, 0, 0, -1, 2, -3, -3, 3, -3, 0, 0, -1, -1], [-14, 0, 0, 0, -14, 2, 1, -2, 2, 1, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 9, -2, 0, 0, 0, 0, 0, 0, 2, 9, -2, 1, 2, 1, 0, 0, 2, -2], [-13, 0, 0, -4, 13, 4, -6, 4, 0, 6, 0, 0, 0, 0, 0, 0, 4, 4, 0, 4], [-4, 2, 0, -2, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2], [-10, 0, 0, 0, -10, -3, 0, 3, 2, 0, 2, 0, 0, 0, 0, 0, -2, 2, 0, 0], [-4, 2, 0, -1, 4, -3, -6, -3, -1, 6, -1, 0, 0, 0, 0, 0, 1, 1, 0, 1], [0, -5, 0, 2, -2, -1, 1, -2, 5, 2, 5, 0, 0, 0, 0, 0, 0, -4, 0, -2], [0, 0, -14, 2, 0, 0, 0, 0, 1, 0, -1, 0, 3, 3, 2, 2, 0, 0, 0, 0], [8, -1, 0, 1, 0, 1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [0, 0, 0, 2, -4, 6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -2], [10, 0, 0, 0, 10, 4, -5, -4, 0, -5, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3, 0, -1, 5, 1, 5, 0, 0, 3, 1], [0, 0, 7, 0, 0, 0, 0, 0, 2, 0, 0, -7, -3, -8, -3, 8, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, -8, 0, 0, -4, 4, 0, 0, 4, 4], [-18, 2, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [0, 0, 16, 2, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, -4, -4, 0, 0, 0, 0], [4, 0, 0, 4, -4, -4, -3, -4, 0, 3, 0, 0, 0, 0, 0, 0, -4, -4, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 10, -7, 7, 3, -3, 0, 0, -6, 6], [0, 0, 5, 2, 0, 0, 0, 0, 0, 0, -2, 5, 2, -3, -2, -3, 0, 0, -2, 2], [8, 0, 0, -1, 0, -1, -1, 7, 0, 7, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [-1, -4, 0, -4, 1, -2, 0, -2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 4], [-4, 6, 0, -1, 4, -3, -4, -3, -3, 4, -3, 0, 0, 0, 0, 0, 1, 1, 0, 1], [0, 0, 0, 0, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, -6, 2, 0, 0, 0, 0, 0, 0, -4, -6, -4, 4, 4, 4, 0, 0, -4, 2], [0, 0, 6, -2, 0, 0, 0, 0, -2, 0, 0, -6, -6, 2, -6, -2, 0, 0, -2, 2], [0, 0, -26, 2, 0, 0, 0, 0, 1, 0, -1, 0, 1, 1, 2, 2, 0, 0, 0, 0], [8, 0, 0, 0, 8, 2, 3, -2, 5, 3, 5, 0, 0, 0, 0, 0, -1, 1, 0, 0], [20, -4, 0, -1, -20, 5, 2, 5, 2, -2, 2, 0, 0, 0, 0, 0, 1, 1, 0, 1], [0, -6, 0, 2, -14, -5, 5, 1, 6, -1, 6, 0, 0, 0, 0, 0, 0, -4, 0, -2], [4, -2, 0, 2, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2], [0, 0, -3, 4, 0, 0, 0, 0, 2, 0, 0, 3, 3, 2, 3, -2, 0, 0, 2, -4], [-9, 0, 0, 0, -9, 3, -7, -3, -1, -7, -1, 0, 0, 0, 0, 0, -1, 1, 0, 0], [2, 0, 0, -2, -2, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2], [-18, 0, 0, 0, -18, -1, -2, 1, -2, -2, -2, 0, 0, 0, 0, 0, -2, 2, 0, 0], [0, 0, 1, 6, 0, 0, 0, 0, 0, 0, -5, 1, -1, 2, 1, 2, 0, 0, -5, 6], [0, 0, -12, 5, 0, 0, 0, 0, 1, 0, 0, 12, 4, 2, 4, -2, 0, 0, 1, -5], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 10, 9, -9, 1, -1, 0, 0, 6, -1], [0, 0, -6, 2, 0, 0, 0, 0, 0, 0, 0, 0, -10, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -10, 3, -3, -3, 3, 0, 0, -2, -4], [0, 0, 0, -1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [4, -6, 0, 2, 0, -6, -6, -5, 0, -5, 0, 0, 0, 0, 0, 0, -4, 0, 0, -2], [0, 0, 0, -2, 18, -1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2], [0, 5, 0, -3, -10, 5, -5, 2, -5, -2, -5, 0, 0, 0, 0, 0, 0, 6, 0, 3], [8, 4, 0, 0, -8, 4, -1, 4, -2, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, 0, 0, 12, 0, -3, 0, -5, -3, -5, 0, 0, 0, 0, 0, -3, 3, 0, 0], [0, -8, 0, 3, -2, -6, 6, -4, 8, 4, 8, 0, 0, 0, 0, 0, 0, -6, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -4, -4, 4, -4, 4, 0, 0, -2, -6], [0, 0, -8, 3, 0, 0, 0, 0, 3, 0, 0, 8, 1, 1, 1, -1, 0, 0, 3, -3], [6, 2, 0, -3, -6, 3, 6, 3, -1, -6, -1, 0, 0, 0, 0, 0, 3, 3, 0, 3], [2, -2, 0, 1, 0, -1, -1, -3, 0, -3, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [0, 2, 0, -2, -10, 4, -4, 2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 4, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 6, 8, -8, 6, -6, 0, 0, 4, 0], [0, 0, 8, -4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -4, -4, 0, 0, 0, 0], [0, 0, 6, 4, 0, 0, 0, 0, 2, 0, -2, 0, 8, 8, 4, 4, 0, 0, 0, 0], [0, 6, 0, -3, -14, 1, -1, -1, -6, 1, -6, 0, 0, 0, 0, 0, 0, 6, 0, 3], [0, 0, 14, -1, 0, 0, 0, 0, 2, 0, 0, -14, -6, -6, -6, 6, 0, 0, 2, 1], [20, 4, 0, 1, 0, 3, 3, -4, 0, -4, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [-14, 0, 0, 0, -14, -5, 4, 5, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 4, 0, 0, 0, 0, -3, 0, 3, 0, 1, 1, 1, 1, 0, 0, 0, 0], [0, 0, 16, -4, 0, 0, 0, 0, 0, 0, 1, 16, -3, -3, 3, -3, 0, 0, 1, -4], [0, 0, -6, 0, 0, 0, 0, 0, 0, 0, -2, -6, 4, -8, -4, -8, 0, 0, -2, 0], [14, 0, 0, 0, 14, 4, 7, -4, -1, 7, -1, 0, 0, 0, 0, 0, 3, -3, 0, 0], [0, 0, 4, 0, 0, 0, 0, 0, -4, 0, 4, 0, 1, 1, -3, -3, 0, 0, 0, 0], [0, -2, 0, -1, 16, 3, -3, -1, 2, 1, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, -4, -6, 6, -8, 8, 0, 0, -6, 0], [0, 0, 0, 3, 0, 0, 0, 0, 4, 0, -4, 0, -12, -12, 3, 3, 0, 0, 0, 0], [10, 4, 0, 1, 0, 3, 3, -1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1], [11, 0, 0, 0, 11, -7, 3, 7, 4, 3, 4, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 12, 2, 0, 0, 0, 0, 3, 0, -3, 0, -2, -2, -4, -4, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, -4, 0, 4, 0, -6, -6, -4, -4, 0, 0, 0, 0], [0, 0, -12, -4, 0, 0, 0, 0, 0, 0, 0, 12, 4, 4, 4, -4, 0, 0, 0, 4], [0, 2, 0, 2, 8, -3, 3, 1, -2, -1, -2, 0, 0, 0, 0, 0, 0, -4, 0, -2], [0, -6, 0, 1, -6, -9, 9, 1, 6, -1, 6, 0, 0, 0, 0, 0, 0, -2, 0, -1], [-9, -2, 0, -2, 9, -4, 2, -4, 1, -2, 1, 0, 0, 0, 0, 0, 2, 2, 0, 2], [0, 0, 4, -7, 0, 0, 0, 0, -9, 0, 0, -4, -1, 7, -1, -7, 0, 0, -9, 7], [0, -4, 0, 3, 0, -7, -2, -7, 2, 2, 2, 0, 0, 0, 0, 0, -3, -3, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -4, 2, -8, 8, 6, -6, 0, 0, -8, 4], [10, -12, 0, 6, -10, -6, -3, -6, 6, 3, 6, 0, 0, 0, 0, 0, -6, -6, 0, -6], [0, 0, 5, -2, 0, 0, 0, 0, 1, 0, 0, -5, -8, 1, -8, -1, 0, 0, 1, 2], [0, 0, -10, -2, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 10, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 2, -2, 2, -2, 0, 0, 0, -2], [0, 0, 14, 1, 0, 0, 0, 0, 5, 0, 0, -14, 7, 1, 7, -1, 0, 0, 5, -1], [-2, -10, 0, 5, 2, -7, 2, -7, 5, -2, 5, 0, 0, 0, 0, 0, -5, -5, 0, -5]];
    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_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_570_2_m_a(: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_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_570_2_m_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![128, 640, 1600, -1600, 1120, 864, 320, 8, 2, 2, 1]>],Snew);
    return Vf;
end function;