from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,7]))
pari: [g,chi] = znchar(Mod(201,1840))
Basic properties
Modulus: | \(1840\) | |
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 \(\chi_{184}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cb
\(\chi_{1840}(201,\cdot)\) \(\chi_{1840}(281,\cdot)\) \(\chi_{1840}(521,\cdot)\) \(\chi_{1840}(681,\cdot)\) \(\chi_{1840}(1161,\cdot)\) \(\chi_{1840}(1321,\cdot)\) \(\chi_{1840}(1401,\cdot)\) \(\chi_{1840}(1561,\cdot)\) \(\chi_{1840}(1721,\cdot)\) \(\chi_{1840}(1801,\cdot)\)
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: | 22.0.339058325839400057321133061640411938816.1 |
Values on generators
\((1151,1381,737,1201)\) → \((1,-1,1,e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(201, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) |
sage: chi.jacobi_sum(n)