from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,11,10]))
pari: [g,chi] = znchar(Mod(2309,4830))
Basic properties
| Modulus: | \(4830\) | |
| Conductor: | \(2415\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2415}(2309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.ch
\(\chi_{4830}(209,\cdot)\) \(\chi_{4830}(629,\cdot)\) \(\chi_{4830}(1889,\cdot)\) \(\chi_{4830}(2099,\cdot)\) \(\chi_{4830}(2309,\cdot)\) \(\chi_{4830}(2519,\cdot)\) \(\chi_{4830}(2939,\cdot)\) \(\chi_{4830}(3569,\cdot)\) \(\chi_{4830}(4199,\cdot)\) \(\chi_{4830}(4409,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((3221,967,2761,1891)\) → \((-1,-1,-1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(2309, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)