from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9408, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,8]))
pari: [g,chi] = znchar(Mod(191,9408))
Basic properties
| Modulus: | \(9408\) | |
| Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{588}(191,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9408.dx
\(\chi_{9408}(191,\cdot)\) \(\chi_{9408}(767,\cdot)\) \(\chi_{9408}(1535,\cdot)\) \(\chi_{9408}(2111,\cdot)\) \(\chi_{9408}(2879,\cdot)\) \(\chi_{9408}(3455,\cdot)\) \(\chi_{9408}(4223,\cdot)\) \(\chi_{9408}(4799,\cdot)\) \(\chi_{9408}(6911,\cdot)\) \(\chi_{9408}(7487,\cdot)\) \(\chi_{9408}(8255,\cdot)\) \(\chi_{9408}(8831,\cdot)\)
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
\((1471,6469,3137,4609)\) → \((-1,1,-1,e\left(\frac{4}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9408 }(191, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)