from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,17,1]))
pari: [g,chi] = znchar(Mod(781,2312))
Basic properties
| Modulus: | \(2312\) | |
| Conductor: | \(2312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2312.v
\(\chi_{2312}(101,\cdot)\) \(\chi_{2312}(237,\cdot)\) \(\chi_{2312}(373,\cdot)\) \(\chi_{2312}(509,\cdot)\) \(\chi_{2312}(645,\cdot)\) \(\chi_{2312}(781,\cdot)\) \(\chi_{2312}(917,\cdot)\) \(\chi_{2312}(1053,\cdot)\) \(\chi_{2312}(1189,\cdot)\) \(\chi_{2312}(1325,\cdot)\) \(\chi_{2312}(1461,\cdot)\) \(\chi_{2312}(1597,\cdot)\) \(\chi_{2312}(1869,\cdot)\) \(\chi_{2312}(2005,\cdot)\) \(\chi_{2312}(2141,\cdot)\) \(\chi_{2312}(2277,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | 34.34.214639510116448210748631066635120907555924504566182022171002423006964754343412080865644260622336.1 |
Values on generators
\((1735,1157,1737)\) → \((1,-1,e\left(\frac{1}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(781, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) |
sage: chi.jacobi_sum(n)