from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,10]))
pari: [g,chi] = znchar(Mod(241,1216))
Basic properties
Modulus: | \(1216\) | |
Conductor: | \(304\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{304}(165,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1216.bv
\(\chi_{1216}(241,\cdot)\) \(\chi_{1216}(337,\cdot)\) \(\chi_{1216}(401,\cdot)\) \(\chi_{1216}(433,\cdot)\) \(\chi_{1216}(497,\cdot)\) \(\chi_{1216}(561,\cdot)\) \(\chi_{1216}(849,\cdot)\) \(\chi_{1216}(945,\cdot)\) \(\chi_{1216}(1009,\cdot)\) \(\chi_{1216}(1041,\cdot)\) \(\chi_{1216}(1105,\cdot)\) \(\chi_{1216}(1169,\cdot)\)
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: | 36.0.19036714782161565107424425435655777110146017378670996611401194085493506048.1 |
Values on generators
\((191,837,705)\) → \((1,i,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
\( \chi_{ 1216 }(241, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)