from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9660, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,0,55,63]))
pari: [g,chi] = znchar(Mod(6431,9660))
Basic properties
| Modulus: | \(9660\) | |
| Conductor: | \(1932\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1932}(635,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9660.fy
\(\chi_{9660}(971,\cdot)\) \(\chi_{9660}(1391,\cdot)\) \(\chi_{9660}(1571,\cdot)\) \(\chi_{9660}(1811,\cdot)\) \(\chi_{9660}(2411,\cdot)\) \(\chi_{9660}(3671,\cdot)\) \(\chi_{9660}(4091,\cdot)\) \(\chi_{9660}(4331,\cdot)\) \(\chi_{9660}(5171,\cdot)\) \(\chi_{9660}(5351,\cdot)\) \(\chi_{9660}(5771,\cdot)\) \(\chi_{9660}(6431,\cdot)\) \(\chi_{9660}(6611,\cdot)\) \(\chi_{9660}(6851,\cdot)\) \(\chi_{9660}(7871,\cdot)\) \(\chi_{9660}(8111,\cdot)\) \(\chi_{9660}(8291,\cdot)\) \(\chi_{9660}(8531,\cdot)\) \(\chi_{9660}(8711,\cdot)\) \(\chi_{9660}(9371,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((4831,3221,5797,2761,6721)\) → \((-1,-1,1,e\left(\frac{5}{6}\right),e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 9660 }(6431, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)