from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9408, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,7,0,1]))
pari: [g,chi] = znchar(Mod(223,9408))
Basic properties
| Modulus: | \(9408\) | |
| Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{392}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9408.cf
\(\chi_{9408}(223,\cdot)\) \(\chi_{9408}(2911,\cdot)\) \(\chi_{9408}(4255,\cdot)\) \(\chi_{9408}(5599,\cdot)\) \(\chi_{9408}(6943,\cdot)\) \(\chi_{9408}(8287,\cdot)\)
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: | 14.14.2812424737865523319657201664.1 |
Values on generators
\((1471,6469,3137,4609)\) → \((-1,-1,1,e\left(\frac{1}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9408 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)