from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1012, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,14]))
pari: [g,chi] = znchar(Mod(197,1012))
Basic properties
| Modulus: | \(1012\) | |
| Conductor: | \(253\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{253}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1012.r
\(\chi_{1012}(197,\cdot)\) \(\chi_{1012}(285,\cdot)\) \(\chi_{1012}(417,\cdot)\) \(\chi_{1012}(593,\cdot)\) \(\chi_{1012}(637,\cdot)\) \(\chi_{1012}(725,\cdot)\) \(\chi_{1012}(813,\cdot)\) \(\chi_{1012}(857,\cdot)\) \(\chi_{1012}(901,\cdot)\) \(\chi_{1012}(945,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((507,277,925)\) → \((1,-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(25\) |
| \( \chi_{ 1012 }(197, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)