
// Make newform 351.2.e.c in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_351_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_351_e_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_351_2_e_c();".
// 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_351_2_e_c();".
// 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 := [729, -243, -243, -27, -18, 27, 24, 9, -2, -1, -3, -1, 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], [-3321, -2430, -639, 144, 456, 345, 133, 0, -28, -31, -12, 1], [1944, -4536, -4752, -1710, -135, 660, 603, 299, 46, -42, -53, -28], [-2187, 14904, 12177, 3762, -342, -1965, -1605, -686, -52, 117, 152, 61], [-22356, -7290, 3564, 3231, 2763, 984, -174, -424, -245, -129, 13, 47], [30861, 486, -12393, -6948, -3690, -51, 1329, 1046, 364, 87, -116, -103], [33777, -6318, -17253, -8676, -3744, 795, 1941, 1358, 394, 51, -176, -133], [31104, 8505, -5832, -5040, -3843, -1083, 399, 673, 320, 156, -37, -68], [39609, 6237, -10881, -7524, -4698, -822, 1008, 1060, 440, 159, -91, -107], [8505, 810, -2709, -1674, -996, -117, 263, 252, 94, 31, -24, -25], [-37503, -5940, 10197, 6822, 4494, 795, -913, -986, -414, -155, 86, 99], [-118341, -20412, 31833, 21744, 14166, 2733, -2823, -3092, -1324, -501, 254, 307]];
    Rf_basisdens := [1, 162, 729, 1458, 729, 1458, 1458, 729, 729, 162, 486, 1458];
    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_351_e();"
function MakeCharacter_351_e()
    N := 351;
    order := 3;
    char_gens := [326, 28];
    v := [1, 3];
    // 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_351_e_Hecke();"
function MakeCharacter_351_e_Hecke( : Kf := false)
    N := 351;
    order := 3;
    char_gens := [326, 28];
    char_values := [[-1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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], [-1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [3, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1], [0, 0, 0, 2, -1, 0, 0, 1, 0, 0, -2, 0], [-2, -1, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0], [0, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 0], [-2, 2, 0, 0, 0, -2, 0, 0, 0, 1, 1, 0], [-2, 1, 1, -1, -1, 0, 2, 1, -1, -2, 1, 1], [0, 1, 0, 0, 0, 0, 0, -2, -1, -1, 0, 0], [0, 0, -1, 1, 1, 1, 1, 0, 0, 0, 0, 2], [0, 0, -2, 2, 1, 2, 0, 0, 0, 0, 0, 0], [4, 1, 0, 0, -1, 0, 2, 1, 0, -2, 0, 1], [-3, -3, 0, 0, 0, -3, 0, -2, 4, -4, -4, 0], [0, 0, 1, 0, -4, -1, 0, 0, 0, 0, 0, 2], [2, -1, 0, 0, 0, 2, 0, 0, 0, -2, 1, 0], [2, -2, 2, -1, -3, 0, -1, 3, -2, 1, 1, -2], [0, -3, 0, 1, 1, 0, 0, -1, 0, 0, -1, -3], [0, 0, -3, 1, -3, 0, 1, 0, 0, 0, 0, 1], [0, 0, 2, -5, -3, 5, -2, 0, 0, 0, 0, 2], [-4, 5, -1, -2, -3, 0, -1, 3, 1, 1, 2, 5], [0, 0, -4, 3, 3, 1, 1, 0, 0, 0, 0, -1], [0, 0, 0, -4, -3, 0, -3, 0, 0, 0, 0, 4], [-2, 1, 0, 0, 0, -2, 0, -2, 7, 0, -1, 0], [10, 1, 1, 2, 0, 0, -2, 0, -1, 2, -2, 1], [3, -3, 3, -2, 4, 0, 0, -4, -3, 0, 2, -3], [-1, -3, 0, 0, 0, -1, 0, 1, 4, -3, 2, 0], [-3, 5, -2, 0, -1, 0, 0, 1, 2, 0, 0, 5], [-4, 4, 0, 0, 0, -4, 0, 2, -1, -1, 2, 0], [0, 0, -2, 0, 4, 5, 3, 0, 0, 0, 0, -4], [-4, -1, 0, 0, 0, -4, 0, 1, 3, -2, 2, 0], [4, -1, 0, 0, 0, 4, 0, -3, -3, 2, -1, 0], [0, 0, 4, 1, 1, 3, -1, 0, 0, 0, 0, -5], [1, -5, 0, 0, 0, 1, 0, -1, 5, -5, -4, 0], [1, 1, -2, -1, -1, 0, -1, 1, 2, 1, 1, 1], [5, 5, 0, 0, 0, 5, 0, 5, 1, -1, -3, 0], [0, 0, 5, 0, -2, 2, 0, 0, 0, 0, 0, 5], [1, 0, 2, 3, 3, 0, 1, -3, -2, -1, -3, 0], [-4, 2, 3, -3, 4, 0, -4, -4, -3, 4, 3, 2], [0, 0, -1, -3, 0, 0, -6, 0, 0, 0, 0, 0], [-3, -2, 0, 0, 0, -3, 0, -1, 0, -4, -5, 0], [-2, 1, -3, 0, 2, 0, -1, -2, 3, 1, 0, 1], [-2, -3, -1, 1, 0, 0, -1, 0, 1, 1, -1, -3], [-3, 0, 0, 0, 0, -3, 0, 4, -4, -1, -1, 0], [0, 0, -1, -1, -3, -4, 0, 0, 0, 0, 0, -5], [0, 0, -1, 4, 3, 1, 2, 0, 0, 0, 0, -7], [8, 0, 0, 0, 0, 8, 0, 1, -8, 4, 0, 0], [5, -7, 6, 3, 4, 0, 4, -4, -6, -4, -3, -7], [4, -1, 0, 0, 0, 4, 0, -4, 0, 3, 6, 0], [0, 0, 9, -1, 1, -3, -2, 0, 0, 0, 0, -2], [6, -4, -5, 1, 3, 0, 6, -3, 5, -6, -1, -4], [9, 0, 0, 0, 0, 9, 0, 1, -5, -1, 5, 0], [0, 0, -4, 2, -1, 12, 4, 0, 0, 0, 0, -1], [-6, 1, 1, -8, -2, 0, -1, 2, -1, 1, 8, 1], [4, -3, -4, 4, -6, 0, 5, 6, 4, -5, -4, -3], [0, 0, 6, -3, 0, -5, 3, 0, 0, 0, 0, -3], [0, 0, 5, -2, -1, -8, 3, 0, 0, 0, 0, -3], [3, 4, 0, 0, 0, 3, 0, 1, -9, 5, 3, 0], [-4, -1, 0, 0, 0, -4, 0, -3, -2, 1, 2, 0], [-5, 5, -4, -5, -2, 0, -5, 2, 4, 5, 5, 5], [-5, -3, 0, 0, 0, -5, 0, 2, -4, 3, 1, 0], [0, 0, -7, -4, 0, 9, 0, 0, 0, 0, 0, 1], [0, 0, 7, 1, -5, -9, 2, 0, 0, 0, 0, 4], [0, 0, 0, 1, -1, 0, -1, 0, 0, 0, 0, -4], [14, 5, 0, 0, 0, 14, 0, 8, -5, 5, -1, 0], [-12, 9, 0, 0, 0, -12, 0, 0, 4, 3, -3, 0], [0, 0, -1, 3, 0, -2, 10, 0, 0, 0, 0, -1], [0, 0, -6, -3, -1, 4, 3, 0, 0, 0, 0, -2], [6, -12, -4, 5, 4, 0, 5, -4, 4, -5, -5, -12], [0, 0, 4, -8, -3, 15, -6, 0, 0, 0, 0, 8], [-1, 3, 0, 0, 0, -1, 0, -5, 2, 2, 5, 0], [7, 4, 8, -1, -3, 0, -6, 3, -8, 6, 1, 4], [1, 1, 0, 0, 0, 1, 0, 6, -5, -4, -2, 0], [0, 0, -2, 4, -3, 2, 4, 0, 0, 0, 0, -1], [-1, 3, -7, -1, -5, 0, -1, 5, 7, 1, 1, 3], [11, 4, 0, 0, 0, 11, 0, 7, -10, 9, 6, 0], [-15, 4, 0, 0, 0, -15, 0, 4, 6, 2, 3, 0], [-17, -4, 0, 0, 0, -17, 0, -2, 3, -5, -5, 0], [0, 0, -2, 0, 3, 5, 5, 0, 0, 0, 0, -2], [-2, 3, -2, -3, 3, 0, -2, -3, 2, 2, 3, 3], [-1, 1, 5, -5, 5, 0, 0, -5, -5, 0, 5, 1], [0, 0, 7, -5, 6, -6, -3, 0, 0, 0, 0, 5], [0, 0, 4, 4, -1, 14, 5, 0, 0, 0, 0, 0], [10, 0, 7, -1, -8, 0, -9, 8, -7, 9, 1, 0], [0, 0, 0, 5, 5, -11, -6, 0, 0, 0, 0, 0], [0, 0, 2, -8, 0, -3, -3, 0, 0, 0, 0, 2], [-2, -7, 0, 0, 0, -2, 0, -3, 8, -9, -4, 0], [16, 4, 2, -1, -6, 0, -8, 6, -2, 8, 1, 4], [0, 0, 7, 5, 0, -8, 2, 0, 0, 0, 0, -5], [-6, 5, 0, -4, -9, 0, -7, 9, 0, 7, 4, 5], [3, -1, 0, 0, 0, 3, 0, 1, 0, 5, -5, 0], [9, -3, 0, 0, 0, 9, 0, 1, 7, 4, 6, 0], [2, -3, -8, -1, 4, 0, 1, -4, 8, -1, 1, -3], [3, -4, 0, 0, 0, 3, 0, -3, 3, -9, 2, 0], [-5, 7, 5, -11, 4, 0, -15, -4, -5, 15, 11, 7], [2, 1, 2, -8, 2, 0, 3, -2, -2, -3, 8, 1], [1, 10, 5, -7, -3, 0, -6, 3, -5, 6, 7, 10], [0, 0, -3, 5, 1, -18, 7, 0, 0, 0, 0, -2], [1, -4, 6, 7, 3, 0, 4, -3, -6, -4, -7, -4], [-10, -4, 0, 0, 0, -10, 0, -12, 8, -8, -7, 0], [0, 0, -4, 5, -1, -17, 9, 0, 0, 0, 0, -7], [-7, 4, 0, 0, 0, -7, 0, 2, -7, 1, -4, 0], [5, -10, -3, 0, 1, 0, 5, -1, 3, -5, 0, -10], [0, 0, 11, -1, -2, 9, -4, 0, 0, 0, 0, 0], [-15, 15, 4, 1, -4, 0, -5, 4, -4, 5, -1, 15], [-1, -13, 0, 0, 0, -1, 0, -12, 13, -2, -7, 0], [0, 0, 7, -12, 3, 4, -13, 0, 0, 0, 0, 10], [-5, -4, 0, 0, 0, -5, 0, 3, -7, -12, 2, 0], [0, -9, -4, 0, 5, 0, 11, -5, 4, -11, 0, -9], [-11, 7, 0, 0, 0, -11, 0, 8, 7, 0, -4, 0], [0, 0, 1, 3, 0, -23, 8, 0, 0, 0, 0, -8], [3, 2, -5, 3, 2, 0, 6, -2, 5, -6, -3, 2], [0, 0, -6, 4, 6, 1, 14, 0, 0, 0, 0, -7], [4, -12, 0, 0, 0, 4, 0, -12, 16, -14, -16, 0], [-4, 7, -1, -8, 4, 0, -8, -4, 1, 8, 8, 7], [-5, 9, 0, 0, 0, -5, 0, 0, -10, -1, 3, 0], [0, 0, 5, -8, -1, 19, -6, 0, 0, 0, 0, 9], [12, 1, 0, 0, 0, 12, 0, -11, 3, -4, -9, 0], [0, 0, -2, -7, -9, 12, 1, 0, 0, 0, 0, -3], [0, 0, -11, 0, 4, -3, 2, 0, 0, 0, 0, -1], [17, 1, 4, -5, -2, 0, 7, 2, -4, -7, 5, 1], [0, 0, 9, -2, -10, 13, -1, 0, 0, 0, 0, 14], [-2, 8, -10, 2, -10, 0, -1, 10, 10, 1, -2, 8], [0, 0, -2, 6, -3, 0, -1, 0, 0, 0, 0, -5], [-5, -2, -3, 0, -7, 0, 5, 7, 3, -5, 0, -2], [0, 0, 8, -15, -7, -1, -7, 0, 0, 0, 0, 12], [-7, -9, 0, 0, 0, -7, 0, -8, -6, 0, 4, 0], [-7, 6, 2, -3, -1, 0, -10, 1, -2, 10, 3, 6], [-5, 5, 0, 0, 0, -5, 0, 1, -2, 4, -2, 0], [5, 0, 0, 0, 0, 5, 0, -3, -3, 0, -9, 0], [0, -8, -5, 10, -5, 0, 11, 5, 5, -11, -10, -8], [3, 12, 0, 0, 0, 3, 0, 6, 9, 6, -3, 0], [3, -4, 0, 0, 0, 3, 0, 0, 8, -6, -1, 0], [-3, 5, -13, 9, -2, 0, 5, 2, 13, -5, -9, 5], [-2, -3, 0, 0, 0, -2, 0, -5, -5, -5, 6, 0], [-5, -8, 0, 0, 0, -5, 0, -2, -2, -2, -3, 0], [4, -12, -11, 5, 1, 0, 0, -1, 11, 0, -5, -12], [-7, 0, -5, 2, 3, 0, -5, -3, 5, 5, -2, 0], [0, 0, 3, -7, -10, -9, -14, 0, 0, 0, 0, 8], [10, -14, 0, 0, 0, 10, 0, 5, -1, 1, -4, 0], [-2, 13, -4, -19, -9, 0, -8, 9, 4, 8, 19, 13], [6, -2, 9, -10, 1, 0, 4, -1, -9, -4, 10, -2], [0, 0, -12, 1, -8, -6, 5, 0, 0, 0, 0, -8], [0, 0, -8, -13, -8, 1, -3, 0, 0, 0, 0, 11], [0, 0, 12, 4, 8, -5, 3, 0, 0, 0, 0, -12], [18, -6, 0, 0, 0, 18, 0, -4, -1, 5, 6, 0], [-15, -3, -10, -3, 3, 0, 3, -3, 10, -3, 3, -3], [1, 5, 0, 0, 0, 1, 0, 9, -9, 7, 19, 0], [-20, 3, 5, 4, 2, 0, 2, -2, -5, -2, -4, 3], [-15, 4, 4, -9, -6, 0, 2, 6, -4, -2, 9, 4], [-4, -13, 0, 0, 0, -4, 0, -7, 9, -10, 0, 0], [0, 0, -6, 12, 15, 1, 12, 0, 0, 0, 0, -18], [0, 0, 10, 4, 2, -16, 2, 0, 0, 0, 0, -3], [-2, 8, -7, -3, 3, 0, -14, -3, 7, 14, 3, 8], [0, 0, 12, 10, 3, -18, -12, 0, 0, 0, 0, 2], [11, 6, 13, 6, 1, 0, -8, -1, -13, 8, -6, 6], [24, 4, 0, 0, 0, 24, 0, -10, 1, 1, -1, 0], [0, 0, -8, -8, 3, 4, 2, 0, 0, 0, 0, -16], [12, 6, 19, -6, 6, 0, -12, -6, -19, 12, 6, 6], [-7, 2, 0, 0, 0, -7, 0, -6, 1, -4, -15, 0], [12, -4, 2, 15, 7, 0, -1, -7, -2, 1, -15, -4], [-11, 9, 0, 0, 0, -11, 0, 7, -2, 16, 2, 0], [0, 0, 4, -4, 0, -5, -4, 0, 0, 0, 0, 13], [7, -8, -10, 13, 8, 0, 9, -8, 10, -9, -13, -8], [0, 0, 16, -4, 2, 3, -13, 0, 0, 0, 0, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_351_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_351_2_e_c();".
// 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_351_2_e_c(:prec:=12)
    chi := MakeCharacter_351_e();
    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_351_2_e_c();".
// 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_351_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_351_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 22, 101, 158, 246, 38, 196, 55, 63, 10, 11, 2, 1]>],Snew);
    return Vf;
end function;