from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,16]))
pari: [g,chi] = znchar(Mod(601,9200))
Basic properties
Modulus: | \(9200\) | |
Conductor: | \(184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{184}(141,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.dn
\(\chi_{9200}(601,\cdot)\) \(\chi_{9200}(1001,\cdot)\) \(\chi_{9200}(2201,\cdot)\) \(\chi_{9200}(2601,\cdot)\) \(\chi_{9200}(3801,\cdot)\) \(\chi_{9200}(7001,\cdot)\) \(\chi_{9200}(7401,\cdot)\) \(\chi_{9200}(7801,\cdot)\) \(\chi_{9200}(8201,\cdot)\) \(\chi_{9200}(9001,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | 22.22.14741666340843480753092741810452692992.1 |
Values on generators
\((1151,6901,2577,1201)\) → \((1,-1,1,e\left(\frac{8}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 9200 }(601, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)