from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,11,63]))
pari: [g,chi] = znchar(Mod(3349,4830))
Basic properties
Modulus: | \(4830\) | |
Conductor: | \(805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{805}(129,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.cx
\(\chi_{4830}(19,\cdot)\) \(\chi_{4830}(199,\cdot)\) \(\chi_{4830}(619,\cdot)\) \(\chi_{4830}(649,\cdot)\) \(\chi_{4830}(1069,\cdot)\) \(\chi_{4830}(1249,\cdot)\) \(\chi_{4830}(1279,\cdot)\) \(\chi_{4830}(1459,\cdot)\) \(\chi_{4830}(1489,\cdot)\) \(\chi_{4830}(1699,\cdot)\) \(\chi_{4830}(2089,\cdot)\) \(\chi_{4830}(2719,\cdot)\) \(\chi_{4830}(2959,\cdot)\) \(\chi_{4830}(3139,\cdot)\) \(\chi_{4830}(3349,\cdot)\) \(\chi_{4830}(3379,\cdot)\) \(\chi_{4830}(3559,\cdot)\) \(\chi_{4830}(3769,\cdot)\) \(\chi_{4830}(4009,\cdot)\) \(\chi_{4830}(4219,\cdot)\)
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
\((3221,967,2761,1891)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4830 }(3349, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)