from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,0,54]))
pari: [g,chi] = znchar(Mod(121,8280))
Basic properties
Modulus: | \(8280\) | |
Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{207}(121,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8280.ey
\(\chi_{8280}(121,\cdot)\) \(\chi_{8280}(601,\cdot)\) \(\chi_{8280}(841,\cdot)\) \(\chi_{8280}(961,\cdot)\) \(\chi_{8280}(1681,\cdot)\) \(\chi_{8280}(1921,\cdot)\) \(\chi_{8280}(2281,\cdot)\) \(\chi_{8280}(2401,\cdot)\) \(\chi_{8280}(3121,\cdot)\) \(\chi_{8280}(3361,\cdot)\) \(\chi_{8280}(3481,\cdot)\) \(\chi_{8280}(3721,\cdot)\) \(\chi_{8280}(4441,\cdot)\) \(\chi_{8280}(5161,\cdot)\) \(\chi_{8280}(5641,\cdot)\) \(\chi_{8280}(5881,\cdot)\) \(\chi_{8280}(6241,\cdot)\) \(\chi_{8280}(6361,\cdot)\) \(\chi_{8280}(7441,\cdot)\) \(\chi_{8280}(7801,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | 33.33.70011645999218458416472683122408534303895571350166174758601569.1 |
Values on generators
\((2071,4141,4601,1657,3961)\) → \((1,1,e\left(\frac{1}{3}\right),1,e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8280 }(121, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) |
sage: chi.jacobi_sum(n)