from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([0,11,7]))
pari: [g,chi] = znchar(Mod(160,3381))
Basic properties
| Modulus: | \(3381\) | |
| Conductor: | \(1127\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1127}(160,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3381.s
\(\chi_{3381}(160,\cdot)\) \(\chi_{3381}(643,\cdot)\) \(\chi_{3381}(1609,\cdot)\) \(\chi_{3381}(2092,\cdot)\) \(\chi_{3381}(2575,\cdot)\) \(\chi_{3381}(3058,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 14.14.4566104562405032283182504435729.1 |
Values on generators
\((2255,346,442)\) → \((1,e\left(\frac{11}{14}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3381 }(160, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)