from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9408, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,21,13]))
pari: [g,chi] = znchar(Mod(353,9408))
Basic properties
| Modulus: | \(9408\) | |
| Conductor: | \(1176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1176}(941,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9408.dy
\(\chi_{9408}(353,\cdot)\) \(\chi_{9408}(929,\cdot)\) \(\chi_{9408}(3041,\cdot)\) \(\chi_{9408}(3617,\cdot)\) \(\chi_{9408}(4385,\cdot)\) \(\chi_{9408}(4961,\cdot)\) \(\chi_{9408}(5729,\cdot)\) \(\chi_{9408}(6305,\cdot)\) \(\chi_{9408}(7073,\cdot)\) \(\chi_{9408}(7649,\cdot)\) \(\chi_{9408}(8417,\cdot)\) \(\chi_{9408}(8993,\cdot)\)
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: | 42.42.11402108177106104552822037830207017370882719938852769609842060849495301710065603498954056531968.1 |
Values on generators
\((1471,6469,3137,4609)\) → \((1,-1,-1,e\left(\frac{13}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9408 }(353, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{42}\right)\) |
sage: chi.jacobi_sum(n)