from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,35,39]))
pari: [g,chi] = znchar(Mod(167,4176))
Basic properties
| Modulus: | \(4176\) | |
| Conductor: | \(2088\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2088}(1211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.fb
\(\chi_{4176}(167,\cdot)\) \(\chi_{4176}(1463,\cdot)\) \(\chi_{4176}(1559,\cdot)\) \(\chi_{4176}(2039,\cdot)\) \(\chi_{4176}(2471,\cdot)\) \(\chi_{4176}(2615,\cdot)\) \(\chi_{4176}(2759,\cdot)\) \(\chi_{4176}(2855,\cdot)\) \(\chi_{4176}(3431,\cdot)\) \(\chi_{4176}(3863,\cdot)\) \(\chi_{4176}(4007,\cdot)\) \(\chi_{4176}(4151,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1567,1045,929,4033)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 4176 }(167, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) |
sage: chi.jacobi_sum(n)