from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2366, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
pari: [g,chi] = znchar(Mod(155,2366))
Basic properties
Modulus: | \(2366\) | |
Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{169}(155,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2366.bf
\(\chi_{2366}(155,\cdot)\) \(\chi_{2366}(519,\cdot)\) \(\chi_{2366}(701,\cdot)\) \(\chi_{2366}(883,\cdot)\) \(\chi_{2366}(1065,\cdot)\) \(\chi_{2366}(1247,\cdot)\) \(\chi_{2366}(1429,\cdot)\) \(\chi_{2366}(1611,\cdot)\) \(\chi_{2366}(1793,\cdot)\) \(\chi_{2366}(1975,\cdot)\) \(\chi_{2366}(2157,\cdot)\) \(\chi_{2366}(2339,\cdot)\)
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: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((339,2199)\) → \((1,e\left(\frac{5}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 2366 }(155, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
sage: chi.jacobi_sum(n)