from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,0,17]))
pari: [g,chi] = znchar(Mod(176,2415))
Basic properties
Modulus: | \(2415\) | |
Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{69}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2415.cf
\(\chi_{2415}(176,\cdot)\) \(\chi_{2415}(281,\cdot)\) \(\chi_{2415}(596,\cdot)\) \(\chi_{2415}(701,\cdot)\) \(\chi_{2415}(911,\cdot)\) \(\chi_{2415}(1121,\cdot)\) \(\chi_{2415}(1226,\cdot)\) \(\chi_{2415}(1331,\cdot)\) \(\chi_{2415}(1436,\cdot)\) \(\chi_{2415}(2066,\cdot)\)
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: | \(\Q(\zeta_{69})^+\) |
Values on generators
\((806,967,346,1891)\) → \((-1,1,1,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(26\) |
\( \chi_{ 2415 }(176, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(-1\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)