from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2790, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,45,58]))
pari: [g,chi] = znchar(Mod(83,2790))
Basic properties
| Modulus: | \(2790\) | |
| Conductor: | \(1395\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1395}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2790.ea
\(\chi_{2790}(83,\cdot)\) \(\chi_{2790}(137,\cdot)\) \(\chi_{2790}(167,\cdot)\) \(\chi_{2790}(203,\cdot)\) \(\chi_{2790}(437,\cdot)\) \(\chi_{2790}(923,\cdot)\) \(\chi_{2790}(983,\cdot)\) \(\chi_{2790}(1127,\cdot)\) \(\chi_{2790}(1253,\cdot)\) \(\chi_{2790}(1283,\cdot)\) \(\chi_{2790}(1553,\cdot)\) \(\chi_{2790}(1757,\cdot)\) \(\chi_{2790}(1877,\cdot)\) \(\chi_{2790}(2243,\cdot)\) \(\chi_{2790}(2597,\cdot)\) \(\chi_{2790}(2657,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2171,1117,1801)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{29}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2790 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)