from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,28]))
pari: [g,chi] = znchar(Mod(49,2368))
Basic properties
Modulus: | \(2368\) | |
Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{592}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2368.dc
\(\chi_{2368}(49,\cdot)\) \(\chi_{2368}(81,\cdot)\) \(\chi_{2368}(145,\cdot)\) \(\chi_{2368}(305,\cdot)\) \(\chi_{2368}(497,\cdot)\) \(\chi_{2368}(625,\cdot)\) \(\chi_{2368}(1233,\cdot)\) \(\chi_{2368}(1265,\cdot)\) \(\chi_{2368}(1329,\cdot)\) \(\chi_{2368}(1489,\cdot)\) \(\chi_{2368}(1681,\cdot)\) \(\chi_{2368}(1809,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1407,1925,705)\) → \((1,i,e\left(\frac{7}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2368 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
sage: chi.jacobi_sum(n)