from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,22,27]))
pari: [g,chi] = znchar(Mod(149,4830))
Basic properties
| Modulus: | \(4830\) | |
| Conductor: | \(2415\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2415}(149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.dh
\(\chi_{4830}(149,\cdot)\) \(\chi_{4830}(359,\cdot)\) \(\chi_{4830}(389,\cdot)\) \(\chi_{4830}(569,\cdot)\) \(\chi_{4830}(779,\cdot)\) \(\chi_{4830}(1019,\cdot)\) \(\chi_{4830}(1229,\cdot)\) \(\chi_{4830}(1859,\cdot)\) \(\chi_{4830}(2039,\cdot)\) \(\chi_{4830}(2459,\cdot)\) \(\chi_{4830}(2489,\cdot)\) \(\chi_{4830}(2909,\cdot)\) \(\chi_{4830}(3089,\cdot)\) \(\chi_{4830}(3119,\cdot)\) \(\chi_{4830}(3299,\cdot)\) \(\chi_{4830}(3329,\cdot)\) \(\chi_{4830}(3539,\cdot)\) \(\chi_{4830}(3929,\cdot)\) \(\chi_{4830}(4559,\cdot)\) \(\chi_{4830}(4799,\cdot)\)
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
\((3221,967,2761,1891)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(149, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)