from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,10]))
pari: [g,chi] = znchar(Mod(155,1694))
Basic properties
| Modulus: | \(1694\) | |
| Conductor: | \(121\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{121}(34,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1694.m
\(\chi_{1694}(155,\cdot)\) \(\chi_{1694}(309,\cdot)\) \(\chi_{1694}(463,\cdot)\) \(\chi_{1694}(617,\cdot)\) \(\chi_{1694}(771,\cdot)\) \(\chi_{1694}(925,\cdot)\) \(\chi_{1694}(1079,\cdot)\) \(\chi_{1694}(1233,\cdot)\) \(\chi_{1694}(1387,\cdot)\) \(\chi_{1694}(1541,\cdot)\)
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: | 11.11.672749994932560009201.1 |
Values on generators
\((969,365)\) → \((1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1694 }(155, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)