from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8664, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,19,19,36]))
pari: [g,chi] = znchar(Mod(6803,8664))
Basic properties
| Modulus: | \(8664\) | |
| Conductor: | \(8664\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8664.cd
\(\chi_{8664}(419,\cdot)\) \(\chi_{8664}(875,\cdot)\) \(\chi_{8664}(1331,\cdot)\) \(\chi_{8664}(1787,\cdot)\) \(\chi_{8664}(2243,\cdot)\) \(\chi_{8664}(2699,\cdot)\) \(\chi_{8664}(3155,\cdot)\) \(\chi_{8664}(4067,\cdot)\) \(\chi_{8664}(4523,\cdot)\) \(\chi_{8664}(4979,\cdot)\) \(\chi_{8664}(5435,\cdot)\) \(\chi_{8664}(5891,\cdot)\) \(\chi_{8664}(6347,\cdot)\) \(\chi_{8664}(6803,\cdot)\) \(\chi_{8664}(7259,\cdot)\) \(\chi_{8664}(7715,\cdot)\) \(\chi_{8664}(8171,\cdot)\) \(\chi_{8664}(8627,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((2167,4333,5777,8305)\) → \((-1,-1,-1,e\left(\frac{18}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8664 }(6803, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) |
sage: chi.jacobi_sum(n)