from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1035, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,16]))
pari: [g,chi] = znchar(Mod(26,1035))
Basic properties
| Modulus: | \(1035\) | |
| Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{69}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1035.bc
\(\chi_{1035}(26,\cdot)\) \(\chi_{1035}(71,\cdot)\) \(\chi_{1035}(386,\cdot)\) \(\chi_{1035}(476,\cdot)\) \(\chi_{1035}(611,\cdot)\) \(\chi_{1035}(656,\cdot)\) \(\chi_{1035}(791,\cdot)\) \(\chi_{1035}(836,\cdot)\) \(\chi_{1035}(926,\cdot)\) \(\chi_{1035}(1016,\cdot)\)
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: | 22.0.304011857053427966889939263171547.1 |
Values on generators
\((461,622,856)\) → \((-1,1,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1035 }(26, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)