
// Make newform 76.2.i.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_76_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_76_i_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_76_2_i_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_76_2_i_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 := [4161, 4446, -2088, -3969, 267, 1659, 64, -426, -15, 70, -3, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-104999089, -98039407, 42461956, 62490886, -6600456, -19762014, 1223776, 3512934, -459237, -261706, 59764, -1861], [91337490, 94201422, -38613723, -59109957, 5399484, 18054126, -652331, -2839567, 84243, 250822, -25876, -3014], [-39192874, -29279114, 32215080, 27062496, -10833438, -10863570, 2513886, 2422014, -590557, -242771, 92642, -7410], [-26703616, -44659861, 6822812, 33428312, 2408876, -11344066, -1428976, 2370458, 170744, -276099, 11132, 7410], [-11328251, -40761460, -4569699, 31536935, 9113704, -11636805, -3363517, 2627431, 455105, -346679, 8018, 8405], [1553197, -22774844, -27341345, 8455587, 17338641, -277950, -5646270, -105905, 1088115, -80323, -89341, 15815], [33629265, 68308502, 5511793, -37334292, -7009805, 10730801, 1083449, -2014841, 357074, 124775, -88198, 16036], [-745423, 1084425, 2375103, -388357, -1477233, 4698, 485184, -1422, -90477, 6301, 8283, -1506], [20539212, 16526830, -17409849, -5650797, 11702325, 1737702, -4743060, -536869, 1222749, 8466, -175660, 32367], [-476163, -273582, 142404, 203904, 48159, -39078, -42146, 605, 12670, 162, -2101, 382]];
    Rf_basisdens := [1, 1, 8740601, 8740601, 8740601, 8740601, 8740601, 8740601, 8740601, 514153, 8740601, 80189];
    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_76_i();"
function MakeCharacter_76_i()
    N := 76;
    order := 9;
    char_gens := [39, 21];
    v := [9, 8];
    // 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_76_i_Hecke();"
function MakeCharacter_76_i_Hecke(Kf)
    N := 76;
    order := 9;
    char_gens := [39, 21];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 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, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 1], [0, 1, 1, -1, 1, 0, -1, -1, 0, -1, 0, 0], [0, 0, 0, 1, 0, -1, 1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 1, -1, -1, 0], [-1, -1, 0, 0, -1, -2, 1, 2, 1, 0, -1, 0], [-3, 1, -1, -1, 1, 3, -1, -2, -1, 3, -1, 0], [1, -1, 0, 0, 0, 2, -1, -3, -1, -1, 1, -1], [-2, 1, 1, 0, 3, 0, -1, -1, 0, 2, 0, -1], [3, 0, 0, -1, -2, 4, 2, 0, 0, -1, 1, 0], [-2, 0, -1, -1, 0, -2, 2, 1, -2, 3, -1, 2], [-1, 1, 1, 0, -2, -1, 4, -1, 2, -1, 1, 1], [1, 0, 1, 1, -1, 0, -6, 0, 0, 0, 0, -1], [4, -1, -1, 1, 2, -1, -3, 4, 0, -1, 1, -1], [-2, 2, 1, -1, 4, 3, 1, 0, 1, -3, -1, 2], [-1, 0, 1, -1, -2, 0, 3, -4, 0, -3, -1, 1], [-3, 0, 0, -2, 5, -1, -2, 3, -2, -3, -1, 0], [-4, -1, -3, -1, 4, 0, 0, 0, 0, 4, -1, 2], [3, -1, 1, 3, -3, 2, 4, 0, 1, -1, -2, -1], [3, -1, 0, 0, 0, 1, -3, 0, 1, -1, 3, -3], [3, -1, 0, 1, -1, 3, 4, 0, -1, 1, 0, 0], [1, -1, -2, -1, 1, 3, -4, -1, -1, 0, 0, -3], [1, 0, -2, -1, -2, -1, 1, 4, 1, 0, 3, -1], [-2, -1, 0, 2, -2, -9, 1, 9, 0, 1, -1, 2], [-3, 4, -4, -2, 7, 1, -2, -1, -2, 3, 1, 0], [4, -2, 0, 2, 4, 1, -2, -1, -3, 1, -2, 2], [2, 3, 1, -3, 1, 1, -5, -5, -2, -3, 2, 1], [2, 0, 2, 3, 0, -1, 1, -2, 2, -1, 0, -2], [2, 2, 3, -3, 1, -6, 0, 0, 1, 0, -1, 1], [5, -1, 2, 0, -6, 0, 1, -3, 0, -2, 2, -2], [-4, -4, 0, 1, -4, 6, 5, -6, -3, 8, -4, 1], [6, 0, 0, 1, -7, -4, 1, 0, 1, -3, 2, 0], [3, 0, 1, 3, -6, 0, 3, 0, 0, -3, 3, -3], [-5, 1, 0, 1, -5, -3, 6, 3, 1, 5, 1, 1], [-3, -2, -1, 1, -1, -2, -2, 4, -2, 6, 0, -3], [-5, 3, 1, 0, 6, -1, -2, 5, 2, -1, 1, 1], [-2, 1, -1, 1, -1, 6, -1, -3, -2, -1, 2, -2], [-4, -4, -2, 4, -6, 4, 2, -1, 1, 7, -1, -2], [-6, -3, -3, 3, 0, 0, 6, 6, 0, 6, 3, 0], [6, -2, 2, 1, 2, -5, 1, -2, 1, -6, -3, 0], [-1, -2, 1, 2, 1, -4, -3, 0, 0, 3, 0, 1], [1, -1, 1, -1, -5, 2, -2, 0, 1, 3, 2, -1], [-10, 0, -1, 0, -6, -7, 1, 0, 0, 1, -1, 1], [7, 0, 0, 0, 0, 7, 10, -1, 0, -1, 0, 1], [-4, 0, -4, 3, -6, -1, 1, 10, -4, 11, 0, 4], [11, -1, 1, 1, -5, -2, 10, 0, 1, -7, 0, -1], [4, -3, 3, -2, -4, 11, -2, -11, -2, 6, 1, 0], [-2, 1, -3, 3, 9, 4, -11, -1, -4, 3, 2, -2], [10, 1, 4, 0, -3, -9, -7, 9, 3, -4, 4, -1], [-9, -4, -4, 0, 7, 5, 0, 2, -1, 4, -4, 3], [-7, 0, 4, -4, -5, 0, 6, -10, 4, 2, -1, 1], [5, 3, -2, -3, -6, 11, 4, -4, -1, -7, 1, -2], [4, 0, 0, 3, 4, -6, -3, 6, 5, -8, 0, 3], [-5, 2, -4, -4, 4, 0, 1, 6, 0, -2, -4, 2], [9, -6, 0, 7, -4, 2, 4, 0, 0, -5, -1, -6], [-11, 1, 1, 0, 0, -12, 2, 11, 1, 10, 0, 2], [5, 2, 2, 0, 0, 3, -8, -8, 2, -10, 0, -2], [-7, -2, 2, -2, 3, 0, -2, -8, 4, 4, -1, 1], [5, 3, -4, -3, 12, -5, -4, 6, -1, -7, 1, -4], [-6, 7, 2, -1, 13, 0, -7, -11, 0, 5, -1, -6], [0, 2, -2, -3, -1, -8, -3, 4, -3, -9, -2, 0], [-11, 0, 5, 0, -3, -2, 11, 4, 4, 7, -4, 5], [-7, -1, 1, 2, -2, -8, 10, 3, -1, 4, 0, -2], [1, 0, 4, -6, 0, 16, -16, -4, 1, -2, -3, -1], [-6, 3, -2, -4, -2, -9, -4, 0, -2, 10, 1, 3], [13, 0, 0, -4, 13, 8, -20, -8, -4, -16, 0, -4], [11, 0, -2, 0, 6, -4, 0, 4, 2, -13, -2, -2], [0, -1, 1, -1, 1, 0, -3, -3, 2, 1, 2, -2], [5, 1, -2, 2, -11, -4, 9, 5, -3, 3, 2, -2], [0, 0, -4, 1, -2, 1, -1, 6, 0, -5, 4, 0], [-7, 3, 1, -3, -3, 5, 7, 2, -1, 5, 1, 1], [12, 1, -1, 0, -3, 2, 0, -11, 0, 3, 4, 0], [0, -3, 0, 3, 0, -12, 2, 12, 2, 0, -3, 3], [1, 0, 1, 8, -14, -13, 13, 13, 1, 4, 0, -1], [11, 1, 6, 0, -4, 10, -1, -9, -2, -6, 6, -8], [1, 6, 4, -2, 2, -3, 6, -12, 6, -18, 0, 2], [-14, 2, -1, 3, 6, -7, -7, 0, -1, 9, -5, 2], [5, -2, 2, -2, 2, 2, -2, 1, -2, -2, 6, 0], [-6, 6, -6, -1, 7, -2, -1, 14, -1, 3, 12, 0], [-2, 2, 1, 2, 1, -1, 4, 0, 1, 0, -4, 2], [-4, 4, -1, 0, 0, 7, 1, -12, -6, 1, -1, -5], [-5, 5, 0, -5, 5, -5, 4, 6, 5, 1, 0, 2], [-10, -3, 0, 4, -10, 7, -1, -7, -2, 1, -3, 4], [6, -1, -4, -2, 4, 0, -10, 2, 0, 3, -2, 3], [-13, 2, -2, 0, -6, 8, 0, -1, 0, 10, -4, 0], [0, -7, 0, 3, 0, 16, 6, -16, 2, 4, -7, 3], [4, 0, 5, 1, 2, 12, -12, -7, 4, -13, -1, -4], [1, -2, -3, 0, 3, 2, -11, 0, -7, 3, -3, -4], [5, -3, 0, 0, -3, -4, 7, 9, 3, -5, -4, 4], [-2, 0, -6, 0, 6, -4, 4, 0, -2, 1, 4, 2], [1, -4, -1, 4, -13, 10, 15, 10, -4, 7, 4, -1], [-4, -3, -6, 5, -5, 0, 9, 5, 0, 3, 5, -2], [7, 0, 0, 0, -12, -2, 2, 12, 7, -16, 7, -7], [9, -7, 2, 9, -9, 7, -12, 3, -7, 10, 0, -3], [-16, 4, 3, -3, 15, -4, -4, -8, -1, 6, -11, 11], [-5, 2, 4, 1, 8, 10, 0, 0, 4, -7, -3, 2], [-6, -5, -2, 2, -2, 0, 8, -8, 0, 5, 2, 3], [-10, 2, -5, -2, -1, 8, -1, 3, -7, 15, 7, -5], [-4, -4, 1, 1, 4, 13, -11, 0, 1, -1, 3, -4], [-3, 5, -3, -10, 4, 0, -3, 0, -3, 2, 5, 5], [8, -1, -3, -6, -7, 0, -1, 1, 0, -12, -6, 7], [14, 0, 0, 4, 14, -11, 2, 11, 5, -3, 0, 4], [-2, 0, -5, -7, 4, -4, 4, 1, -2, 3, 3, 2], [9, 1, -1, 0, 0, -13, 6, 11, 8, 1, -1, 9], [14, 2, 6, 0, 4, -2, -8, 10, 4, -6, 6, -2], [-3, 0, -1, 5, 4, -2, 2, -3, -3, 6, -2, 3], [3, 4, -4, -6, -6, 8, -6, -11, -6, 6, 4, 0], [-6, -7, -3, 4, 11, 0, -5, 5, 0, 22, 4, 3], [14, -5, -2, 9, -15, -5, 13, 0, -2, 3, -4, -5], [-2, 0, -1, 5, 10, -6, 6, -9, -2, 9, -1, 2], [0, -5, -2, 0, 9, 13, -7, -1, -5, 2, -2, -3], [3, 1, -2, 2, -4, -3, 1, 2, -3, 5, 3, -3], [-5, -1, -3, 2, 10, 18, -5, 0, -3, -2, -1, -1], [-14, 10, -1, -10, 15, -4, -7, 1, -3, 7, 3, -1], [-8, 11, 1, -6, 1, 0, 7, -11, 0, -16, -6, -5], [2, -8, -9, 9, -18, 1, 16, 9, -1, 8, 2, -2], [3, 6, 2, -4, 4, 1, 0, 3, 6, -3, 0, 1], [-2, -11, -1, 0, -15, -6, 2, 1, 1, 1, -1, 2], [-8, 0, 0, -8, 8, -12, 12, -8, -8, 5, -8, 8], [4, -7, 0, -4, 4, -3, -5, 3, -4, -1, -7, -4], [-11, 4, -2, 0, -6, 0, 17, -6, 0, -10, 0, -4], [16, -10, -4, 10, -20, 14, -2, -15, 3, -9, -3, -4], [1, 0, -3, 2, -2, 5, -5, 5, 1, -12, 4, -1], [-3, 3, 1, 0, -24, -15, 18, -11, 4, -1, 1, 3], [6, 0, 9, 4, 2, -7, 7, -11, 6, 5, -3, -6], [6, -1, 1, 6, 0, -11, 6, -3, 6, -6, -9, 0], [2, -4, 0, -3, 2, -2, -17, 2, -1, -16, -4, -3], [-11, 7, -3, -10, 10, -8, -2, 7, 7, 0, 0, 2], [-9, 10, 8, -8, 17, 2, -10, -10, -2, -8, -7, 7], [-7, 2, 2, -2, -10, 10, 6, -8, 2, 3, -2, 2], [-9, -1, 1, -3, 6, 0, -3, 6, -3, -4, -4, 0], [15, -8, 8, 0, -3, -19, 0, 17, 0, -27, 5, 0], [-3, 9, 7, -10, 2, -2, 5, 0, 7, -6, 1, 9], [5, 4, 9, 5, -5, -4, 2, 6, 4, 2, 0, 9], [13, 7, 6, 0, -18, -12, -7, -7, -4, -6, 6, -10], [7, -1, -7, -2, 6, 10, 3, 0, -7, -6, 3, -1], [17, -3, 0, -2, 17, -2, 5, 2, 1, 4, -3, -2], [-24, 4, 0, -4, 8, -4, -16, -5, -7, 27, 7, 0], [20, -8, -1, 0, 6, 13, 1, 0, 0, 1, -1, 1], [-1, 1, 7, -1, -5, -1, -3, -12, 3, -3, -3, 7], [-21, 5, 0, -3, -21, -19, 6, 19, -7, 13, 5, -3], [3, 0, -3, -3, 0, 0, -3, 18, 0, -3, -3, 3], [13, -2, 0, -4, 13, 14, -20, -14, -2, -18, -2, -4], [-4, 10, 1, -8, -7, 3, 8, 0, 1, 10, -2, 10], [3, 0, 2, -6, -14, -12, 12, 12, 3, -18, 1, -3], [7, -10, -11, -1, 1, 18, -2, 1, -10, 11, 0, 8], [-14, 1, -1, 0, -12, 5, 0, 11, 0, 6, 3, 0], [-21, -4, 4, 6, 0, -2, 6, 17, 6, 0, -10, 0], [17, -6, -5, 8, -9, 0, -8, 17, 0, 5, 8, -2], [-10, 5, 1, -5, 5, -1, -13, -17, 4, 1, -4, 1], [-5, 0, -2, -2, 2, -3, -2, -5, 0, -5, 0, 3], [3, 0, 2, -4, -2, 10, -10, 0, 3, 2, 1, -3], [-16, -3, -6, 7, 17, -7, -15, 0, -6, 5, -4, -3], [-10, 6, 0, 1, -10, 4, -1, -4, 5, -6, 6, 1], [-13, 8, 8, -8, -3, 2, 4, -20, 0, 2, -1, 1], [5, 13, 5, 0, -6, -5, -8, -9, -6, -5, 5, -11], [-2, 0, -7, -9, 20, 12, -12, -13, -2, -5, 5, 2], [6, 2, -2, -9, -3, 4, -9, -6, -9, -3, 6, 0], [-6, -4, 0, 7, -6, -16, 17, 16, 11, 6, -4, 7], [-2, -4, -9, 4, 5, 31, -6, 0, -9, 6, 0, -4], [1, 6, 2, -4, 4, -1, -14, -8, 6, -14, 0, -4], [-21, 3, -5, -8, 8, -16, -2, 10, 3, 7, 0, 3], [4, -4, 4, 13, 2, -15, 13, 18, 13, -6, 3, 0], [-6, 4, -4, 3, 12, -7, 3, 8, 3, 0, -1, 0], [-1, 4, 4, -4, -8, 8, 12, -10, 10, -13, -10, 4], [23, -2, 12, -12, 3, -8, -2, 12, 14, -28, -1, 1], [3, -7, -7, 0, 0, 10, -12, 14, -7, 21, 0, -9], [6, 1, 4, 4, -20, 3, 12, 0, 4, 10, -5, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_76_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_76_2_i_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_76_2_i_a(:prec:=12)
    chi := MakeCharacter_76_i();
    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_76_2_i_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_76_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_76_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;