from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2760, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,11,4]))
pari: [g,chi] = znchar(Mod(349,2760))
Basic properties
| Modulus: | \(2760\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(349,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2760.cx
\(\chi_{2760}(349,\cdot)\) \(\chi_{2760}(469,\cdot)\) \(\chi_{2760}(949,\cdot)\) \(\chi_{2760}(1189,\cdot)\) \(\chi_{2760}(1429,\cdot)\) \(\chi_{2760}(1549,\cdot)\) \(\chi_{2760}(1669,\cdot)\) \(\chi_{2760}(1789,\cdot)\) \(\chi_{2760}(2509,\cdot)\) \(\chi_{2760}(2749,\cdot)\)
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
\((2071,1381,1841,1657,1201)\) → \((1,-1,1,-1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2760 }(349, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)