
// Make newform 765.2.t.e in Magma, downloaded from the LMFDB on 03 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_765_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_765_t_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_765_2_t_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_765_2_t_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 := [100, 0, 464, 0, 776, 0, 572, 0, 188, 0, 24, 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], [-920, -1052, -2692, -3296, -2770, -3016, -1136, -1078, -160, -142, -7, -6], [1080, 0, 3438, 0, 3440, 0, 1344, 0, 185, 0, 8, 0], [-1320, 0, -4204, 0, -4110, 0, -1552, 0, -210, 0, -9, 0], [0, -1666, 0, -5096, 0, -4948, 0, -1883, 0, -256, 0, -11], [0, -1646, 0, -5096, 0, -4948, 0, -1883, 0, -256, 0, -11], [3560, -1052, 11100, -3296, 10990, -3016, 4240, -1078, 580, -142, 25, -6], [4420, -1284, 13236, -4456, 12380, -4642, 4558, -1838, 610, -254, 26, -11], [4420, 1284, 13236, 4456, 12380, 4642, 4558, 1838, 610, 254, 26, 11], [-5560, 2372, -17056, 7500, -16460, 7126, -6208, 2630, -840, 352, -36, 15], [5560, 2372, 17056, 7500, 16460, 7126, 6208, 2630, 840, 352, 36, 15], [-3560, 5596, -11100, 16804, -10990, 15928, -4240, 5922, -580, 796, -25, 34]];
    Rf_basisdens := [1, 40, 20, 20, 20, 20, 40, 40, 40, 40, 40, 40];
    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_765_t();"
function MakeCharacter_765_t()
    N := 765;
    order := 4;
    char_gens := [596, 307, 496];
    v := [4, 2, 1];
    // 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_765_t_Hecke();"
function MakeCharacter_765_t_Hecke( : Kf := false)
    N := 765;
    order := 4;
    char_gens := [596, 307, 496];
    char_values := [[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, 1, 0, 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, 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, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, -1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 1], [0, 0, 0, -2, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, -2, 0, -1, 0, 0, -3, 0, 0, 1], [-1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -2, -1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1], [0, 1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1], [-2, 1, 0, 0, 2, 0, -2, 0, 0, 2, 0, -1], [0, 0, 2, -1, 0, 0, 0, -2, -2, 0, 0, 0], [0, -1, 0, -1, 0, 2, 0, 1, -1, 0, 0, 1], [0, 0, 4, -2, 0, 0, 0, -1, -1, 0, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 0, 4, 0, 1, -1, 0, 0, 1], [3, 1, 0, 0, 3, 0, 0, 0, 0, 0, -1, 1], [0, -2, -2, 0, 0, -2, 0, 4, 0, 0, 0, 2], [0, -2, 0, 0, 0, 0, 4, 0, 0, -1, 0, 2], [0, 0, 2, 1, 0, 0, 0, -1, -1, 0, 0, 0], [0, -5, 0, 0, 0, 0, 5, 0, 0, 2, -2, 0], [0, 5, 1, 0, 0, 1, 0, -4, 0, 0, 0, -5], [2, 1, 0, 0, 0, 0, -1, 0, 0, -2, 2, 0], [0, -5, 0, -5, 0, 0, 0, 4, -4, 0, 0, 5], [0, 0, 3, 0, 0, 3, 0, 3, 0, 0, 0, 0], [-6, -1, 0, 0, 6, 0, 2, 0, 0, 2, 0, 1], [0, 0, -2, 0, 0, 2, 0, 0, -2, 0, 0, 0], [0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4], [0, -3, 0, -3, 0, 0, 0, -1, 1, 0, 0, 3], [5, -3, 0, 0, 5, 0, 0, 0, 0, 0, -1, -3], [-10, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0], [0, 0, 0, 0, -8, 0, -2, 0, 0, -2, -2, -2], [0, -2, 0, -2, 0, -6, 0, 3, -3, 0, 0, 2], [0, -4, 0, -8, 0, 0, 0, 0, -1, 0, 0, 4], [0, 2, 0, 0, 0, 0, 0, -3, 0, 0, 0, -2], [0, 1, -5, 0, 0, -5, 0, 8, 0, 0, 0, -1], [0, 0, 0, 0, 4, 0, -8, 0, 0, -3, -3, -8], [5, 1, 0, 0, -5, 0, -2, 0, 0, -4, 0, -1], [8, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [0, 3, 2, 6, 0, -2, 0, 0, 4, 0, 0, -3], [0, -2, 1, -4, 0, -1, 0, 0, 2, 0, 0, 2], [-2, 4, 0, 0, -2, 0, 0, 0, 0, 0, 1, 4], [-8, -1, 0, 0, 8, 0, 2, 0, 0, 3, 0, 1], [0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 4, 0, 0, -4, 0, 0, -7, 0, 0, 0], [0, 0, 0, 0, -12, 0, 1, 0, 0, 0, 0, 1], [0, -4, 5, 0, 0, 5, 0, -2, 0, 0, 0, 4], [16, 4, 0, 0, 0, 0, -4, 0, 0, -2, 2, 0], [6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0], [0, 0, -10, 9, 0, 0, 0, 5, 5, 0, 0, 0], [0, 0, 2, 1, 0, 0, 0, 4, 4, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, -4, 3], [-6, 8, 0, 0, 0, 0, -8, 0, 0, -2, 2, 0], [0, 0, 8, 0, 0, 8, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0, 0, -3, -3, 0], [0, -5, 8, -10, 0, -8, 0, 0, -7, 0, 0, 5], [0, 8, 0, 8, 0, -2, 0, -6, 6, 0, 0, -8], [0, -5, 0, -5, 0, -10, 0, 2, -2, 0, 0, 5], [11, 3, 0, 0, 11, 0, 0, 0, 0, 0, 3, 3], [0, 0, -4, 0, 0, 4, 0, 0, -4, 0, 0, 0], [0, -3, -2, -6, 0, 2, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, -4, 0, -6, 0, 0, 0, 0, -6], [0, 5, -5, 0, 0, -5, 0, 0, 0, 0, 0, -5], [0, -1, 4, -2, 0, -4, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, 6, 0, -2, 0, 0, -1, -1, -2], [0, 4, 0, 4, 0, -4, 0, 6, -6, 0, 0, -4], [0, 0, 0, 0, 4, 0, 2, 0, 0, 5, 5, 2], [0, -6, -2, -12, 0, 2, 0, 0, -5, 0, 0, 6], [0, 7, 0, 7, 0, 2, 0, 1, -1, 0, 0, -7], [-2, -1, 0, 0, -2, 0, 0, 0, 0, 0, -1, -1], [0, 0, -6, -5, 0, 0, 0, -7, -7, 0, 0, 0], [0, 0, 0, 0, -14, 0, -5, 0, 0, 3, 3, -5], [0, -3, -6, -6, 0, 6, 0, 0, -2, 0, 0, 3], [11, 3, 0, 0, -11, 0, -6, 0, 0, -4, 0, -3], [-2, 2, 0, 0, 0, 0, -2, 0, 0, -2, 2, 0], [-6, -7, 0, 0, 6, 0, 14, 0, 0, -3, 0, 7], [10, 7, 0, 0, 0, 0, -7, 0, 0, -1, 1, 0], [4, 0, 0, 0, -4, 0, 0, 0, 0, -3, 0, 0], [0, 0, 4, 3, 0, 0, 0, -3, -3, 0, 0, 0], [3, 4, 0, 0, 3, 0, 0, 0, 0, 0, 1, 4], [0, -9, 0, -9, 0, -6, 0, 6, -6, 0, 0, 9], [6, -4, 0, 0, -6, 0, 8, 0, 0, -4, 0, 4], [0, 0, -12, -3, 0, 0, 0, 11, 11, 0, 0, 0], [0, 0, 0, 0, 2, 0, -2, 0, 0, 7, 7, -2], [0, 3, 0, 3, 0, -4, 0, -4, 4, 0, 0, -3], [0, 0, 0, -5, 0, 0, 0, -8, -8, 0, 0, 0], [-12, -5, 0, 0, -12, 0, 0, 0, 0, 0, -1, -5], [0, -8, -6, -16, 0, 6, 0, 0, 3, 0, 0, 8], [0, 0, 0, 0, 26, 0, 4, 0, 0, 1, 1, 4], [2, 1, 0, 0, -2, 0, -2, 0, 0, -3, 0, -1], [0, 5, -4, 10, 0, 4, 0, 0, 1, 0, 0, -5], [0, 10, 0, 0, 0, 0, -10, 0, 0, 2, -2, 0], [1, 3, 0, 0, -1, 0, -6, 0, 0, 6, 0, -3], [0, 11, 0, 11, 0, 4, 0, -1, 1, 0, 0, -11], [-9, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, -1, 0, -7, 0, 0, 0, 1], [0, 11, 0, 11, 0, 0, 0, -9, 9, 0, 0, -11], [0, 0, 12, 7, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, -6, 0, 2, 0, 0, 0, 0, 2], [-2, -2, 0, 0, 2, 0, 4, 0, 0, 5, 0, 2], [0, -9, 0, -9, 0, 10, 0, -5, 5, 0, 0, 9], [0, 0, 4, -11, 0, 0, 0, -3, -3, 0, 0, 0], [0, 0, -12, 6, 0, 0, 0, 7, 7, 0, 0, 0], [-12, -8, 0, 0, 0, 0, 8, 0, 0, 1, -1, 0], [0, -4, 0, 0, 0, 0, 8, 0, 0, -4, 0, 4], [0, -1, -9, 0, 0, -9, 0, 5, 0, 0, 0, 1], [0, 10, 0, 10, 0, 10, 0, -6, 6, 0, 0, -10], [0, -4, -2, 0, 0, -2, 0, 4, 0, 0, 0, 4], [11, -3, 0, 0, -11, 0, 6, 0, 0, -9, 0, 3], [0, 0, 0, 0, -4, 0, -6, 0, 0, 6, 6, -6], [-28, 2, 0, 0, -28, 0, 0, 0, 0, 0, 2, 2], [0, 4, -3, 0, 0, -3, 0, -1, 0, 0, 0, -4], [0, 5, 0, 5, 0, -6, 0, 5, -5, 0, 0, -5], [0, -3, -9, -6, 0, 9, 0, 0, -6, 0, 0, 3], [22, -2, 0, 0, 0, 0, 2, 0, 0, -6, 6, 0], [0, 0, 0, 0, -10, 0, 10, 0, 0, 7, 7, 10], [0, 12, -6, 24, 0, 6, 0, 0, 12, 0, 0, -12], [0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, -4], [0, -3, 3, 0, 0, 3, 0, -13, 0, 0, 0, 3], [-11, 2, 0, 0, 11, 0, -4, 0, 0, 1, 0, -2], [12, -9, 0, 0, 0, 0, 9, 0, 0, 5, -5, 0], [-13, 0, 0, 0, -13, 0, 0, 0, 0, 0, -8, 0], [-11, 4, 0, 0, -11, 0, 0, 0, 0, 0, -5, 4], [0, 5, 0, 5, 0, -2, 0, 6, -6, 0, 0, -5], [0, 0, -2, 18, 0, 0, 0, 4, 4, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 13, -1, 0, 0, -1, 0, -1, 0, 0, 0, -13], [-5, 1, 0, 0, -5, 0, 0, 0, 0, 0, -11, 1], [0, 0, -18, 5, 0, 0, 0, 9, 9, 0, 0, 0], [-6, -1, 0, 0, 0, 0, 1, 0, 0, 8, -8, 0], [-16, -9, 0, 0, 0, 0, 9, 0, 0, 6, -6, 0], [0, 0, -14, 1, 0, 0, 0, 11, 11, 0, 0, 0], [0, -3, 10, 0, 0, 10, 0, 7, 0, 0, 0, 3], [0, 0, -24, 0, 0, 0, 0, 10, 10, 0, 0, 0], [-7, -6, 0, 0, 7, 0, 12, 0, 0, -2, 0, 6], [-5, 1, 0, 0, -5, 0, 0, 0, 0, 0, -1, 1], [8, -4, 0, 0, 8, 0, 0, 0, 0, 0, 6, -4], [0, 6, -11, 12, 0, 11, 0, 0, 13, 0, 0, -6], [0, -2, -1, -4, 0, 1, 0, 0, -15, 0, 0, 2], [2, -2, 0, 0, 0, 0, 2, 0, 0, 6, -6, 0], [-15, -3, 0, 0, 15, 0, 6, 0, 0, 7, 0, 3], [0, 7, 15, 0, 0, 15, 0, -10, 0, 0, 0, -7], [0, 0, -8, 0, 0, 8, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 22, 0, 2, 0, 0, 9, 9, 2], [0, 5, 0, 5, 0, 16, 0, -1, 1, 0, 0, -5], [0, -10, 0, -20, 0, 0, 0, 0, -14, 0, 0, 10], [4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0], [0, -3, 0, -3, 0, 22, 0, -4, 4, 0, 0, 3], [0, -12, -7, 0, 0, -7, 0, 11, 0, 0, 0, 12], [0, -6, 14, -12, 0, -14, 0, 0, -9, 0, 0, 6], [-19, -2, 0, 0, 19, 0, 4, 0, 0, 9, 0, 2], [12, -12, 0, 0, 0, 0, 12, 0, 0, 7, -7, 0], [-2, -1, 0, 0, 2, 0, 2, 0, 0, -6, 0, 1], [0, 0, 4, 2, 0, 0, 0, -11, -11, 0, 0, 0], [9, 0, 0, 0, -9, 0, 0, 0, 0, -14, 0, 0], [0, 0, 8, 0, 0, 8, 0, 3, 0, 0, 0, 0], [0, 13, 0, 13, 0, 4, 0, -13, 13, 0, 0, -13], [0, 0, -2, -3, 0, 0, 0, -4, -4, 0, 0, 0], [0, 0, 0, 0, 12, 0, 8, 0, 0, -1, -1, 8], [0, 0, 2, 0, 0, 0, 0, -7, -7, 0, 0, 0], [0, -1, -4, 0, 0, -4, 0, 7, 0, 0, 0, 1], [19, 2, 0, 0, 19, 0, 0, 0, 0, 0, -7, 2], [0, 5, -3, 10, 0, 3, 0, 0, 8, 0, 0, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_765_t_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_765_2_t_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_765_2_t_e(:prec:=12)
    chi := MakeCharacter_765_t();
    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_765_2_t_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_765_2_t_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_765_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 0, 19, 0, -9, 0, 1]>],Snew);
    return Vf;
end function;