from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1352, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,2]))
pari: [g,chi] = znchar(Mod(209,1352))
Basic properties
| Modulus: | \(1352\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1352.y
\(\chi_{1352}(105,\cdot)\) \(\chi_{1352}(209,\cdot)\) \(\chi_{1352}(313,\cdot)\) \(\chi_{1352}(417,\cdot)\) \(\chi_{1352}(521,\cdot)\) \(\chi_{1352}(625,\cdot)\) \(\chi_{1352}(729,\cdot)\) \(\chi_{1352}(833,\cdot)\) \(\chi_{1352}(937,\cdot)\) \(\chi_{1352}(1041,\cdot)\) \(\chi_{1352}(1145,\cdot)\) \(\chi_{1352}(1249,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 13.13.542800770374370512771595361.1 |
Values on generators
\((1015,677,1185)\) → \((1,1,e\left(\frac{1}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1352 }(209, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)