from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,44,60]))
pari: [g,chi] = znchar(Mod(3301,4830))
Basic properties
Modulus: | \(4830\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.cm
\(\chi_{4830}(121,\cdot)\) \(\chi_{4830}(151,\cdot)\) \(\chi_{4830}(331,\cdot)\) \(\chi_{4830}(361,\cdot)\) \(\chi_{4830}(541,\cdot)\) \(\chi_{4830}(961,\cdot)\) \(\chi_{4830}(991,\cdot)\) \(\chi_{4830}(1411,\cdot)\) \(\chi_{4830}(1591,\cdot)\) \(\chi_{4830}(2221,\cdot)\) \(\chi_{4830}(2431,\cdot)\) \(\chi_{4830}(2671,\cdot)\) \(\chi_{4830}(2881,\cdot)\) \(\chi_{4830}(3061,\cdot)\) \(\chi_{4830}(3091,\cdot)\) \(\chi_{4830}(3301,\cdot)\) \(\chi_{4830}(3481,\cdot)\) \(\chi_{4830}(3721,\cdot)\) \(\chi_{4830}(4351,\cdot)\) \(\chi_{4830}(4741,\cdot)\)
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: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((3221,967,2761,1891)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{10}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4830 }(3301, a) \) | \(1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)