from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,0,11,7]))
pari: [g,chi] = znchar(Mod(799,5520))
Basic properties
Modulus: | \(5520\) | |
Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{460}(339,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.dw
\(\chi_{5520}(79,\cdot)\) \(\chi_{5520}(319,\cdot)\) \(\chi_{5520}(559,\cdot)\) \(\chi_{5520}(799,\cdot)\) \(\chi_{5520}(1279,\cdot)\) \(\chi_{5520}(1759,\cdot)\) \(\chi_{5520}(1999,\cdot)\) \(\chi_{5520}(2719,\cdot)\) \(\chi_{5520}(2959,\cdot)\) \(\chi_{5520}(4159,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | 22.22.8083780427918435509708715954790400000000000.1 |
Values on generators
\((4831,1381,1841,4417,1201)\) → \((-1,1,1,-1,e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5520 }(799, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)