from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1312, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,29]))
pari: [g,chi] = znchar(Mod(63,1312))
Basic properties
| Modulus: | \(1312\) | |
| Conductor: | \(164\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{164}(63,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1312.cy
\(\chi_{1312}(63,\cdot)\) \(\chi_{1312}(95,\cdot)\) \(\chi_{1312}(479,\cdot)\) \(\chi_{1312}(511,\cdot)\) \(\chi_{1312}(639,\cdot)\) \(\chi_{1312}(671,\cdot)\) \(\chi_{1312}(703,\cdot)\) \(\chi_{1312}(767,\cdot)\) \(\chi_{1312}(831,\cdot)\) \(\chi_{1312}(895,\cdot)\) \(\chi_{1312}(991,\cdot)\) \(\chi_{1312}(1055,\cdot)\) \(\chi_{1312}(1119,\cdot)\) \(\chi_{1312}(1183,\cdot)\) \(\chi_{1312}(1215,\cdot)\) \(\chi_{1312}(1247,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | \(\Q(\zeta_{164})^+\) |
Values on generators
\((575,165,129)\) → \((-1,1,e\left(\frac{29}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1312 }(63, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(-i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) |
sage: chi.jacobi_sum(n)