from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,3,0]))
pari: [g,chi] = znchar(Mod(1712,2001))
Basic properties
| Modulus: | \(2001\) | |
| Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{69}(56,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2001.z
\(\chi_{2001}(320,\cdot)\) \(\chi_{2001}(494,\cdot)\) \(\chi_{2001}(755,\cdot)\) \(\chi_{2001}(842,\cdot)\) \(\chi_{2001}(1190,\cdot)\) \(\chi_{2001}(1364,\cdot)\) \(\chi_{2001}(1538,\cdot)\) \(\chi_{2001}(1625,\cdot)\) \(\chi_{2001}(1712,\cdot)\) \(\chi_{2001}(1799,\cdot)\)
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: | \(\Q(\zeta_{69})^+\) |
Values on generators
\((668,1132,553)\) → \((-1,e\left(\frac{3}{22}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(1712, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) |
sage: chi.jacobi_sum(n)