
// Make newform 770.2.m.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_770_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_770_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_770_2_m_e();".
// 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_770_2_m_e();".
// 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 := [529, 0, 6904, 0, 17180, 0, 17648, 0, 9170, 0, 2600, 0, 404, 0, 32, 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], [-104150, 0, 550826, 0, 580417, 0, 38697, 0, -88672, 0, -29156, 0, -3381, 0, -133, 0], [-44689, -26474442, -64101644, -54822474, -118862275, -41772858, -82357779, -15682752, -27494729, -3095298, -4711090, -306090, -396267, -12594, -12857, -96], [0, -136832, 0, -445273, 0, -404918, 0, -165183, 0, -35680, 0, -4277, 0, -270, 0, -7], [-688, 0, -59095, 0, -139743, 0, -119553, 0, -46994, 0, -9095, 0, -837, 0, -29, 0], [-5847244, -12716121, -117442646, 32560869, -238829056, 77777865, -183261516, 48319359, -67257980, 13382919, -12465034, 1853445, -1115040, 125037, -37904, 3267], [5518597, 13154201, 20447759, 69111166, 12797338, 91732691, -8382948, 48040770, -8173717, 12086401, -2192015, 1552478, -239016, 97503, -9154, 2362], [12567522, 23592469, 102463482, 50195657, 159122832, 12283723, 88097130, -23732847, 22601502, -15901507, 2925186, -3770627, 185196, -384441, 4554, -14119], [-15822896, -23592469, -165191152, -50195657, -320383238, -12283723, -248045892, 23732847, -91404484, 15901507, -16928552, 3770627, -1510962, 384441, -51244, 14119], [15822896, -23592469, 165191152, -50195657, 320383238, -12283723, 248045892, 23732847, 91404484, 15901507, 16928552, 3770627, 1510962, 384441, 51244, 14119], [-4272089, 7716027, -31162930, 27732903, -54678935, 46701897, -42555198, 35747514, -16168057, 13345695, -3082598, 2511069, -281451, 227205, -9706, 7788], [4272089, 7716027, 31162930, 27732903, 54678935, 46701897, 42555198, 35747514, 16168057, 13345695, 3082598, 2511069, 281451, 227205, 9706, 7788], [1044384, 9667109, 89706210, 24097111, 212129874, 55797785, 181481454, 56736477, 71336892, 24638785, 13806210, 5030831, 1270566, 477105, 44022, 16825], [-13463050, -44519689, -86437496, -224645303, -31697956, -348621037, 50952222, -237928413, 36435082, -81147701, 8818844, -14300983, 906936, -1235013, 33442, -40961], [23044850, -44519689, 35761504, -224645303, -21700408, -348621037, -54512346, -237928413, -28277258, -81147701, -6136492, -14300983, -595884, -1235013, -21206, -40961], [-1044384, 3632315, -89706210, -115000355, -212129874, -265075321, -181481454, -206562909, -71336892, -74957105, -13806210, -13688803, -1270566, -1209825, -44022, -40769]];
    Rf_basisdens := [1, 52140, 2398440, 30360, 1580, 4796880, 2398440, 4796880, 4796880, 4796880, 2398440, 2398440, 4796880, 4796880, 4796880, 4796880];
    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_770_m();"
function MakeCharacter_770_m()
    N := 770;
    order := 4;
    char_gens := [617, 661, 211];
    v := [3, 4, 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_770_m_Hecke();"
function MakeCharacter_770_m_Hecke(Kf)
    N := 770;
    order := 4;
    char_gens := [617, 661, 211];
    char_values := [[0, 0, 0, -1, 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]];
    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, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, -1, 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, 1, 0, 1, 0, 0, 0, -1, -1, -1, 0, 0, 1, 1, 1, 0], [0, -2, 0, -1, -1, 0, 0, 0, 1, 0, -1, 1, -1, -1, -1, 0], [0, 0, 0, 1, -1, 0, 2, 0, 1, -1, -1, 1, 2, 1, 1, 0], [0, 0, 0, 0, -1, -1, 1, 0, 1, 0, 0, -1, 0, 0, 0, 0], [3, 0, -2, 3, -1, 0, 0, 0, -1, -3, 0, 2, 0, 2, 0, 0], [0, 2, 0, 0, 0, 2, -2, 0, 0, -2, 0, 2, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 1, -1, 0], [-4, 0, 0, 4, -1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 2], [0, -2, 0, -2, 0, -2, -2, 0, 0, 2, -1, -3, -4, -2, -2, 0], [0, -4, 0, -2, -4, 0, 0, 0, 4, 0, -2, 2, -4, -2, -2, 0], [-1, 0, 0, 1, -2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2], [-2, 0, 4, -2, -3, 0, 0, -2, -3, 1, 0, -4, 0, -4, 0, 2], [1, 0, -2, 2, -1, 0, 0, 0, 6, 4, 0, 2, 0, 1, 1, 2], [0, 0, 0, 0, 1, 1, 1, 0, -1, -2, 5, 6, 4, 0, 0, 0], [4, 0, 0, -4, 4, 0, 0, -2, -4, -4, 0, 0, 0, 0, 0, -2], [-1, 0, 0, 0, 0, 0, 0, -4, -2, -2, 0, 0, 0, 1, -1, 0], [0, -2, 0, -1, 1, -2, 0, 0, -1, 1, -1, -5, 0, -1, -1, 0], [0, -2, 0, 0, -4, -2, 2, 0, 4, -2, 3, -5, 0, 0, 0, 0], [0, -4, 0, -2, 0, -6, 0, 0, 0, 3, -2, 2, -3, -2, -2, 0], [2, 0, -4, 0, 0, 0, 0, 0, -2, -6, 0, 4, 0, 2, 2, 0], [-2, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, -2, 0, 0, -2, 2], [0, 0, 0, 0, -3, 1, 1, 0, 3, 2, 5, 6, -4, 0, 0, 0], [-3, 0, 0, -3, -2, 0, 0, -2, -2, -2, 0, 0, 0, 0, 0, 2], [0, 0, 0, 2, -1, 0, 2, 0, 1, 1, -4, 2, 0, 2, 2, 0], [0, -2, 0, 0, -1, 2, -2, 0, 1, -3, -2, 4, 0, 0, 0, 0], [0, 0, -2, 0, 4, 0, 0, 4, 4, 2, 0, 2, 0, 2, 0, -4], [0, 0, 0, 1, 1, 0, -2, 0, -1, -3, -11, 1, 2, 1, 1, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, -1, -8, -7, 2, 0, 0, 0], [-1, 0, -2, 3, -2, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 2], [0, -4, 0, 0, 2, -1, 1, 0, -2, 3, 0, -1, 0, 0, 0, 0], [0, -2, 0, 0, -1, -4, 4, 0, 1, 3, 2, -6, 0, 0, 0, 0], [0, 4, 0, 4, 2, -2, -2, 0, -2, 0, 0, -2, 0, 4, 4, 0], [0, 0, 0, 0, 5, 0, 0, -3, -5, -5, 0, 0, 0, 0, 0, -3], [-2, 0, -4, -2, 1, 0, 0, -2, 1, -3, 0, 4, 0, 4, 0, 2], [0, 0, 0, 0, 2, 0, -4, 0, -2, -6, 2, 0, 4, 0, 0, 0], [0, -6, 0, -3, 4, -2, 0, 0, -4, 1, -3, -9, 3, -3, -3, 0], [4, 0, -8, -4, 4, 0, 0, 0, 2, -6, 0, 8, 0, 4, 4, -8], [-4, 0, 0, 0, -2, 0, 0, -2, -1, -1, 0, 0, 0, -2, 2, 0], [-9, 0, 0, 0, 0, 0, 0, -4, -2, -2, 0, 0, 0, 1, -1, 0], [0, -4, 0, -2, 4, 0, 0, 0, -4, 0, -2, -8, 4, -2, -2, 0], [0, 0, 0, 0, 5, 0, -10, 0, -5, -1, 6, 0, -4, 0, 0, 0], [1, 0, -2, 4, -1, 0, 0, 0, -1, -3, 0, 2, 0, 1, 1, 2], [0, 2, 0, 2, -1, 0, 0, 0, 1, 1, 12, 12, -2, 2, 2, 0], [-5, 0, -6, -5, 4, 0, 0, 2, 4, -2, 0, 6, 0, 6, 0, -2], [0, 0, 0, -2, 1, 0, -2, 0, -1, 2, -16, -2, -3, -2, -2, 0], [4, 0, -8, -2, 6, 0, 0, 0, -3, -11, 0, 8, 0, 4, 4, -12], [0, 2, 0, 1, -8, 4, 0, 0, 8, -2, 1, 1, -6, 1, 1, 0], [0, 2, 0, 0, -2, 0, 0, 0, 2, -2, 4, -4, 0, 0, 0, 0], [0, -2, 0, -2, -8, 6, 6, 0, 8, 2, -5, 1, -4, -2, -2, 0], [-1, 0, 0, 0, -2, 0, 0, 10, 5, 5, 0, 0, 0, 3, -3, 0], [10, 0, -2, -8, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 2, 0], [0, 0, 0, 0, 6, -4, 0, 0, -6, 2, 0, -12, 4, 0, 0, 0], [0, 0, 0, 2, -1, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0, 2], [0, -6, 0, -6, -4, 0, 0, 0, 4, 4, -6, -6, -8, -6, -6, 0], [0, 0, 0, -4, -2, 0, 4, 0, 2, 4, 6, -4, -2, -4, -4, 0], [0, -2, 0, -2, 2, -4, -4, 0, -2, 2, -6, -10, -4, -2, -2, 0], [0, -4, 0, -2, 2, -10, 0, 0, -2, 5, -2, 14, -3, -2, -2, 0], [0, 6, 0, 3, 2, 6, 0, 0, -2, -3, 3, -7, 5, 3, 3, 0], [0, 0, 0, 4, -1, 0, 2, 0, 1, -6, 6, 4, 7, 4, 4, 0], [-1, 0, 0, 0, -7, 0, 0, 6, 3, 3, 0, 0, 0, 3, -3, 0], [-2, 0, 4, -2, -4, 0, 0, -2, -4, 0, 0, -4, 0, -4, 0, 2], [-12, 0, 6, 6, -2, 0, 0, 6, 2, 8, 0, -6, 0, 0, -6, 6], [10, 0, 0, 0, 9, 0, 0, -14, -7, -7, 0, 0, 0, -2, 2, 0], [0, 0, 0, 2, -2, 0, 4, 0, 2, 6, 16, 2, -4, 2, 2, 0], [0, 0, 0, -2, 2, 0, -4, 0, -2, -4, -14, -2, 2, -2, -2, 0], [0, -6, 0, 0, -5, 1, -1, 0, 5, -6, -3, 4, 0, 0, 0, 0], [4, 0, 4, 4, -4, 0, 0, 0, -4, 0, 0, -4, 0, -4, 0, 0], [0, -2, 0, 0, 2, 0, 0, 0, -2, 2, -6, 6, 0, 0, 0, 0], [11, 0, -4, -7, 0, 0, 0, 0, 0, -4, 0, 4, 0, 0, 4, 0], [0, -4, 0, -2, 3, -6, 0, 0, -3, 3, -2, -2, 0, -2, -2, 0], [0, 0, 0, 0, 2, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, -4], [-3, 0, 4, -3, -4, 0, 0, 2, -4, 0, 0, -4, 0, -4, 0, -2], [4, 0, -8, 14, 0, 0, 0, 0, -4, -12, 0, 8, 0, 4, 4, 0], [-8, 0, 0, 8, 1, 0, 0, 3, -1, -1, 0, 0, 0, 0, 0, 3], [3, 0, 0, 0, 5, 0, 0, -6, -3, -3, 0, 0, 0, 3, -3, 0], [0, 2, 0, 0, 4, 2, -2, 0, -4, 2, 13, -11, 0, 0, 0, 0], [-3, 0, 6, 12, -4, 0, 0, 0, -3, 3, 0, -6, 0, -3, -3, 8], [-10, 0, 0, 0, 5, 0, 0, -2, -1, -1, 0, 0, 0, -2, 2, 0], [0, 6, 0, 6, 8, 0, 0, 0, -8, -8, 6, 6, 16, 6, 6, 0], [2, 0, -6, 2, 0, 0, 0, 6, 0, -6, 0, 6, 0, 6, 0, -6], [0, 2, 0, 0, 8, 4, -4, 0, -8, 4, -10, 14, 0, 0, 0, 0], [6, 0, -8, 6, 9, 0, 0, 6, 9, 1, 0, 8, 0, 8, 0, -6], [2, 0, -4, -10, 4, 0, 0, 0, 0, -4, 0, 4, 0, 2, 2, -8], [0, 0, 0, 4, -6, 0, 12, 0, 6, 4, -12, 4, 2, 4, 4, 0], [0, -2, 0, -2, -8, 3, 3, 0, 8, 5, 3, 6, -10, -2, -2, 0], [-8, 0, 2, -8, 4, 0, 0, 2, 4, 6, 0, -2, 0, -2, 0, -2], [5, 0, -4, -1, 9, 0, 0, -9, -9, -13, 0, 4, 0, 0, 4, -9], [0, -2, 0, 0, -8, 2, -2, 0, 8, -10, 2, 0, 0, 0, 0, 0], [5, 0, -6, 1, -1, 0, 0, -4, 1, -5, 0, 6, 0, 0, 6, -4], [0, 0, 0, 0, -4, -4, -4, 0, 4, 8, 6, 2, -16, 0, 0, 0], [-6, 0, 12, -12, -5, 0, 0, 0, -1, 11, 0, -12, 0, -6, -6, 10], [0, 4, 0, 2, -8, 8, 0, 0, 8, -4, 2, 10, -4, 2, 2, 0], [2, 0, -4, 10, 6, 0, 0, 0, 7, 3, 0, 4, 0, 2, 2, -12], [4, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0], [0, -4, 0, -2, -11, -4, 0, 0, 11, 2, -2, -4, -13, -2, -2, 0], [0, 10, 0, 10, -4, 2, 2, 0, 4, 2, 10, 12, -4, 10, 10, 0], [0, 0, 0, -8, 3, 0, -6, 0, -3, 7, 18, -8, -10, -8, -8, 0], [0, 0, 0, 0, 3, 0, -6, 0, -3, -9, 2, 0, 6, 0, 0, 0], [0, 12, 0, 6, -3, 8, 0, 0, 3, -4, 6, 8, 1, 6, 6, 0], [0, -4, 0, 0, -1, -10, 10, 0, 1, 9, -17, 7, 0, 0, 0, 0], [0, 10, 0, 10, -3, 12, 12, 0, 3, -9, 0, 12, 18, 10, 10, 0], [-8, 0, -2, 10, 0, 0, 0, -4, 0, -2, 0, 2, 0, 0, 2, -4], [17, 0, -4, -13, 11, 0, 0, -13, -11, -15, 0, 4, 0, 0, 4, -13], [0, 10, 0, 5, 11, 10, 0, 0, -11, -5, 5, 31, 16, 5, 5, 0], [1, 0, -2, -20, 0, 0, 0, 0, -2, -4, 0, 2, 0, 1, 1, 0], [0, 4, 0, 4, 2, 6, 6, 0, -2, -8, -1, 5, 16, 4, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 4, 0, 0, 0], [0, 4, 0, 2, -5, -6, 0, 0, 5, 3, 2, -6, -8, 2, 2, 0], [4, 0, 0, -4, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, -10], [-3, 0, 6, -2, 1, 0, 0, 0, 2, 8, 0, -6, 0, -3, -3, -2], [-5, 0, 0, 0, -8, 0, 0, 4, 2, 2, 0, 0, 0, -7, 7, 0], [-13, 0, 0, 0, 7, 0, 0, -2, -1, -1, 0, 0, 0, 5, -5, 0], [7, 0, -12, 7, 8, 0, 0, 11, 8, -4, 0, 12, 0, 12, 0, -11], [-9, 0, -2, 11, -10, 0, 0, 12, 10, 8, 0, 2, 0, 0, 2, 12], [-6, 0, -2, -6, 8, 0, 0, 12, 8, 6, 0, 2, 0, 2, 0, -12], [0, 0, 0, 0, -4, 2, -2, 0, 4, -6, 4, -2, 0, 0, 0, 0], [-6, 0, 0, 0, 8, 0, 0, 6, 3, 3, 0, 0, 0, 4, -4, 0], [0, -2, 0, -1, -10, 8, 0, 0, 10, -4, -1, -13, -6, -1, -1, 0], [0, 0, 0, -1, 3, 0, -6, 0, -3, 10, -1, -1, -13, -1, -1, 0], [-14, 0, -12, -14, 9, 0, 0, 14, 9, -3, 0, 12, 0, 12, 0, -14], [-11, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 0, -5, 5, 0], [0, 14, 0, 14, -7, 16, 16, 0, 7, -9, -4, 12, 18, 14, 14, 0], [8, 0, -16, 18, 2, 0, 0, 0, 4, -12, 0, 16, 0, 8, 8, -4], [-3, 0, 6, -16, -1, 0, 0, 0, -5, 1, 0, -6, 0, -3, -3, 2], [-3, 0, -8, 11, 8, 0, 0, -6, -8, -16, 0, 8, 0, 0, 8, -6], [0, -10, 0, -5, -2, 6, 0, 0, 2, -3, -5, 17, 1, -5, -5, 0], [0, -14, 0, 0, -1, -4, 4, 0, 1, 3, -8, 4, 0, 0, 0, 0], [0, 14, 0, 7, 5, 2, 0, 0, -5, -1, 7, 1, 6, 7, 7, 0], [-8, 0, 0, 0, 12, 0, 0, -4, -2, -2, 0, 0, 0, 2, -2, 0], [8, 0, -10, 2, -12, 0, 0, 6, 12, 2, 0, 10, 0, 0, 10, 6], [0, 10, 0, 10, -8, 6, 6, 0, 8, 2, 15, 21, -4, 10, 10, 0], [0, -2, 0, 0, 6, 6, -6, 0, -6, 0, 9, -3, 0, 0, 0, 0], [-10, 0, 4, -10, 3, 0, 0, -5, 3, 7, 0, -4, 0, -4, 0, 5], [0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 44, 2, 1, 2, 2, 0], [18, 0, 0, -18, 2, 0, 0, -1, -2, -2, 0, 0, 0, 0, 0, -1], [0, -8, 0, 0, 1, 2, -2, 0, -1, -1, -11, 13, 0, 0, 0, 0], [0, 4, 0, 4, 4, 1, 1, 0, -4, -5, 16, 17, 10, 4, 4, 0], [0, -6, 0, -6, -3, 2, 2, 0, 3, 1, -18, -16, -2, -6, -6, 0], [0, 0, 2, 0, 12, 0, 0, 10, 12, 14, 0, -2, 0, -2, 0, -10], [0, 0, 0, -4, 6, 0, -12, 0, -6, -2, -12, -4, -4, -4, -4, 0], [-6, 0, 12, -2, -6, 0, 0, 0, -3, 9, 0, -12, 0, -6, -6, 12], [-9, 0, 18, 2, 3, 0, 0, 0, -13, 5, 0, -18, 0, -9, -9, -6], [0, 10, 0, 5, 8, -2, 0, 0, -8, 1, 5, 7, 7, 5, 5, 0], [0, 0, 0, -3, 3, 0, -6, 0, -3, -9, -7, -3, 6, -3, -3, 0], [-3, 0, 6, 16, -4, 0, 0, 0, 7, 13, 0, -6, 0, -3, -3, 8], [14, 0, 0, 14, -2, 0, 0, -2, -2, -2, 0, 0, 0, 0, 0, 2], [0, 0, 0, -10, 4, 0, -8, 0, -4, 2, 16, -10, -6, -10, -10, 0], [6, 0, 0, 0, 6, 0, 0, -16, -8, -8, 0, 0, 0, -6, 6, 0], [14, 0, 0, 14, -3, 0, 0, -2, -3, -3, 0, 0, 0, 0, 0, 2], [0, 0, 0, 2, 0, 0, 0, 0, 0, -6, 10, 2, 6, 2, 2, 0], [-8, 0, 16, -8, -8, 0, 0, 12, 8, 24, 0, -16, 0, 0, -16, 12], [16, 0, 0, 0, -2, 0, 0, 8, 4, 4, 0, 0, 0, 6, -6, 0], [0, 12, 0, 0, 6, 12, -12, 0, -6, -6, 9, 3, 0, 0, 0, 0], [-6, 0, 12, 2, -4, 0, 0, 0, -14, -2, 0, -12, 0, -6, -6, 8], [0, 0, 0, 1, 5, 0, -10, 0, -5, -9, 3, 1, 4, 1, 1, 0], [0, 2, 0, 2, -4, 3, 3, 0, 4, 1, 3, 6, -2, 2, 2, 0], [-2, 0, 8, -6, 0, 0, 0, 2, 0, 8, 0, -8, 0, 0, -8, 2], [0, 18, 0, 9, -2, 24, 0, 0, 2, -12, 9, -3, 10, 9, 9, 0], [0, 0, 0, -5, 1, 0, -2, 0, -1, 9, 21, -5, -10, -5, -5, 0], [-21, 0, 0, 0, 2, 0, 0, -2, -1, -1, 0, 0, 0, 3, -3, 0], [-23, 0, 16, 7, -4, 0, 0, 16, 4, 20, 0, -16, 0, 0, -16, 16], [-3, 0, -4, -3, 4, 0, 0, 2, 4, 0, 0, 4, 0, 4, 0, -2], [29, 0, 0, 0, 10, 0, 0, -24, -12, -12, 0, 0, 0, -5, 5, 0], [0, 0, 0, -9, 8, 0, -16, 0, -8, -5, -15, -9, -3, -9, -9, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_770_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_770_2_m_e();".
// 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_770_2_m_e(:prec:=16)
    chi := MakeCharacter_770_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_770_2_m_e();".
// 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_770_2_m_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_770_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![324, 720, 800, 392, 100, 8, 8, 4, 1]>,<17,R![303595776, 0, 0, 0, 21971200, 0, 0, 0, 276576, 0, 0, 0, 1040, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;