from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5796, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,44,0,15]))
pari: [g,chi] = znchar(Mod(43,5796))
Basic properties
Modulus: | \(5796\) | |
Conductor: | \(828\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{828}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5796.ex
\(\chi_{5796}(43,\cdot)\) \(\chi_{5796}(295,\cdot)\) \(\chi_{5796}(799,\cdot)\) \(\chi_{5796}(1303,\cdot)\) \(\chi_{5796}(1555,\cdot)\) \(\chi_{5796}(1723,\cdot)\) \(\chi_{5796}(1975,\cdot)\) \(\chi_{5796}(2227,\cdot)\) \(\chi_{5796}(2311,\cdot)\) \(\chi_{5796}(2563,\cdot)\) \(\chi_{5796}(2731,\cdot)\) \(\chi_{5796}(3235,\cdot)\) \(\chi_{5796}(3319,\cdot)\) \(\chi_{5796}(3487,\cdot)\) \(\chi_{5796}(3823,\cdot)\) \(\chi_{5796}(4243,\cdot)\) \(\chi_{5796}(4495,\cdot)\) \(\chi_{5796}(5251,\cdot)\) \(\chi_{5796}(5587,\cdot)\) \(\chi_{5796}(5755,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((2899,1289,829,4789)\) → \((-1,e\left(\frac{2}{3}\right),1,e\left(\frac{5}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5796 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) |
sage: chi.jacobi_sum(n)