
// Make newform 728.2.ds.b in Magma, downloaded from the LMFDB on 13 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_728_ds();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_728_ds_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_728_2_ds_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_728_2_ds_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 := [6561, 0, -2916, 0, 648, 0, 360, 0, -161, 0, 40, 0, 8, 0, -4, 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], [9893259, 0, -35721, 0, -2268, 0, 520738, 0, -1988, 0, -7, 0, 13823, 0, -56, 0], [-52488, 0, 2621, 0, -81, 0, -2556, 0, -71, 0, -36, 0, -63, 0, -2, 0], [-55404, 0, -193914, 0, -12312, 0, -2698, 0, -10792, 0, -38, 0, 577, 0, -304, 0], [0, 729, 0, -648, 0, -360, 0, 161, 0, -40, 0, -8, 0, 4, 0, -1], [0, -79461, 0, 46737, 0, -342, 0, -10792, 0, 473, 0, -152, 0, -266, 0, 71], [0, -138510, 0, 40014, 0, 3078, 0, -7790, 0, 2698, 0, 455, 0, -367, 0, 76], [0, 69255, 0, 9396, 0, 144, 0, 4544, 0, 257, 0, 64, 0, 112, 0, 17], [-886464, 0, 186948, 0, -1368, 0, -43168, 0, 11369, 0, -608, 0, -1064, 0, 284, 0], [-180792, 0, 1243674, 0, -279, 0, -8804, 0, 62596, 0, -124, 0, -217, 0, 1582, 0], [-2618568, 0, 1236546, 0, -4041, 0, -127516, 0, 67799, 0, -1796, 0, -3143, 0, 2363, 0], [6232221, 0, -2447820, 0, -237969, 0, 342457, 0, -137249, 0, -2518, 0, 7820, 0, -3413, 0], [0, 847098, 0, -274833, 0, 837, 0, 42296, 0, -13768, 0, 31, 0, 1069, 0, -346], [0, 18422559, 0, -2488158, 0, 30843, 0, 973268, 0, -169987, 0, 13708, 0, 23989, 0, -4879], [0, -760347, 0, -353565, 0, -837, 0, -45184, 0, -17872, 0, 31, 0, -1145, 0, -454], [0, 1834164, 0, -2016711, 0, 15057, 0, 89318, 0, -112483, 0, 1258, 0, 1558, 0, -2806]];
    Rf_basisdens := [1, 521235, 6435, 521235, 2187, 142155, 120285, 24057, 104247, 521235, 521235, 521235, 120285, 1563705, 142155, 521235];
    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_728_ds();"
function MakeCharacter_728_ds()
    N := 728;
    order := 12;
    char_gens := [183, 365, 521, 561];
    v := [12, 6, 6, 11];
    // 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_728_ds_Hecke();"
function MakeCharacter_728_ds_Hecke( : Kf := false)
    N := 728;
    order := 12;
    char_gens := [183, 365, 521, 561];
    char_values := [[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], [-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, 1, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 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, 0, -1, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, -1, -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, 0, 1, 1, -1, 0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, -1, 0, 1, 0, 0, 0, 0, 0, -1, -2, 1], [0, 2, 1, 0, 0, 0, 0, 0, 0, -3, 1, -2, 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, 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, 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, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -1, 3, -1, 0, 0, 0, 0, 2, -2, 1, -3], [0, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 1, 4, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 3, -3, 2, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 1, -1, 0, 0, 0, 0, 3, 0, 2, -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, 0, 0, 0, 1, 2, -3, -2, 0, 0, 0, 0, -1, 0, -2, -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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, -3, -3, 0, 0, 0, 0, 0, -10, 0, 0, 3, 0, 0, 0, 0], [-2, 0, 3, 0, 0, 0, 0, 0, -1, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, -4, -4, 0, 0, 0, 0, 0, -2, 0, -2, 0], [-9, -2, -2, 0, 0, 0, 0, 0, 0, -9, 4, -2, 0, 0, 0, 0], [0, 0, 0, 0, 2, -3, -5, 3, 0, 0, 0, 0, -4, 2, 8, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, -3, 0, 0, 0, 0, 0, 0, -5, 5, -3, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, -7, -6, 1, 0, 0, 0, 0, -4, -6, -3, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 2, 1, -1, 0, 0, 0, 0, 5, 8, 5, -2], [16, -3, -3, 0, 0, 0, 0, 0, 16, 0, 0, 3, 0, 0, 0, 0], [6, 0, -5, -6, 0, 0, 0, 0, 3, -3, 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, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -5, -4, 0, 0, 0, 0, -6, -9, -8, 7], [0, 0, 0, 0, 0, 3, 5, 9, 0, 0, 0, 0, 11, 0, 6, -5], [3, -4, 0, 0, 0, 0, 0, 0, 6, 0, 4, -4, 0, 0, 0, 0], [15, 1, 2, 0, 0, 0, 0, 0, 15, 15, -1, 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, 3, 4, 5, 0, 0, 0, 0, 1, 4, 2, -8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 3, -20, 0, 0, 0, 0, 10, -10, -3, 3, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, -6, -1, 0, 0, 0, 0, -9, -6, -3, -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], [-15, 4, 0, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 6, 3, 5, 0, 0, 0, 0, 1, -4, 2, -3], [0, 0, 0, 0, -16, -4, -10, 15, 0, 0, 0, 0, -2, -5, 3, -6], [0, 0, 0, 0, -8, -3, 6, 2, 0, 0, 0, 0, -6, -9, 3, 2], [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, 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, -5, 0, 18, 0, 0, 0, 0, 0, 0, 5, -5, 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, -6, -3, -1, 0, 0, 0, 0, 6, 0, -2, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 3, 0, 0, 0, 0, -3, 3, -5, -5, 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, 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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -7, 8, -3, 0, 0, 0, 0, -1, -5, 4, -3], [0, 0, 6, 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, -7, -2, -1, 2, 0, 0, 0, 0, -5, 0, 4, -7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 10, 5, 9, 0, 0, 0, 0, 9, 0, 5, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, -4, 15, 0, 0, 0, 0, 15, 0, -8, 4, 0, 0, 0, 0], [-30, 0, 0, -30, 0, 0, 0, 0, -15, -15, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13, -4, -3, 2, 0, 0, 0, 0, -10, -1, -7, 2], [-12, 0, 0, -12, 0, 0, 0, 0, 0, 0, -5, -5, 0, 0, 0, 0], [0, 0, 0, 0, -10, 1, -3, -4, 0, 0, 0, 0, -8, -3, -8, 6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, -3, 19, 0, 0, 0, 0, -19, 0, 0, -3, 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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, -15, -10, 10, 0, 0, 0, 0, -16, -12, 1, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, 4, 2, -3, 0, 0, 0, 0, -2, 2, -4, 8], [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 12, 1, -11, 0, 0, 0, 0, 16, 10, 5, -11], [-2, 9, 9, 0, 0, 0, 0, 0, -1, 1, 9, -9, 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, 0, 0, 0], [0, 0, 0, 0, 10, -1, 6, -19, 0, 0, 0, 0, -9, 3, -5, 7], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, -9, 4, 0, 0, 0, 0, 0, 8, 0, -9, 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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 5, 0, 0, 0, 0, -5, -5, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, -13, -7, 3, 10, 0, 0, 0, 0, 2, -5, -9, 10], [12, -7, -7, -12, 0, 0, 0, 0, -12, 12, 0, 0, 0, 0, 0, 0], [-17, -3, 0, 17, 0, 0, 0, 0, 17, 17, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 10, 7, 8, 2, 0, 0, 0, 0, 13, 10, 15, -9], [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, -5, 4, -4, 0, 0, 0, 0, 16, 2, 1, -4], [0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, -3, 12, -2, 0, 0, 0, 0, -4, -5, 6, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 10, 1, 3, 0, 0, 0, 0, 16, 9, -2, -5], [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, 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, 0, 0, -5, -12, -6, 17, 0, 0, 0, 0, -9, 5, 13, 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, 27, -27, 2, 0, 0, 0, 0, 0], [50, 0, 3, 0, 0, 0, 0, 0, 25, 0, 3, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12, 8, -2, -9, 0, 0, 0, 0, 4, 11, 17, -10], [0, 0, 0, 0, 17, 8, 4, -11, 0, 0, 0, 0, -1, 7, -5, 6], [0, 0, 0, 0, -13, -11, 1, -2, 0, 0, 0, 0, -15, -6, -3, -1], [0, 0, 9, 0, 0, 0, 0, 0, -22, 0, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 3, 4, -12, 0, 0, 0, 0, 10, 4, 2, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-18, 0, 5, -18, 0, 0, 0, 0, 18, -18, -5, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, -6, 10, -7, 0, 0, 0, 0, 6, 5, -4, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, -6, -15, 1, 0, 0, 0, 0, -3, 0, -12, 10], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 6, -6, 0, 0, 0, 0, 2, 3, -4, -6], [-20, 0, 0, -20, 0, 0, 0, 0, -40, -40, 3, -3, 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, 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, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, -15, 0, 0, 0, 0, 0, -30, -7, 7, 0, 0, 0, 0], [-6, 0, 11, -6, 0, 0, 0, 0, -6, 6, 11, -11, 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, 0, 0, 0], [0, 0, 0, 0, -16, -12, -14, 13, 0, 0, 0, 0, -5, 2, 10, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, -4, 27, 0, 0, 0, 0, 0, 27, 8, -4, 0, 0, 0, 0], [29, 0, 0, -29, 0, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0], [0, 0, 0, 0, 14, 6, -4, -15, 0, 0, 0, 0, -6, -4, -16, 4], [9, -16, -8, 9, 0, 0, 0, 0, 9, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 36, 0, 0, 0, 0, 18, 18, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, -13, 3, -22, 1, 0, 0, 0, 0, -20, 4, -11, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_728_ds_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_728_2_ds_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_728_2_ds_b(:prec:=16)
    chi := MakeCharacter_728_ds();
    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_728_2_ds_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_728_2_ds_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_728_ds();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![28561, 0, 41236, 0, 39932, 0, 21544, 0, 8407, 0, 1832, 0, 284, 0, 20, 0, 1]>],Snew);
    return Vf;
end function;