from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2320, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,7,2]))
pari: [g,chi] = znchar(Mod(1077,2320))
Basic properties
| Modulus: | \(2320\) | |
| Conductor: | \(2320\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2320.ds
\(\chi_{2320}(13,\cdot)\) \(\chi_{2320}(93,\cdot)\) \(\chi_{2320}(357,\cdot)\) \(\chi_{2320}(573,\cdot)\) \(\chi_{2320}(1053,\cdot)\) \(\chi_{2320}(1077,\cdot)\) \(\chi_{2320}(1397,\cdot)\) \(\chi_{2320}(1637,\cdot)\) \(\chi_{2320}(1717,\cdot)\) \(\chi_{2320}(1773,\cdot)\) \(\chi_{2320}(2093,\cdot)\) \(\chi_{2320}(2197,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((2031,581,1857,321)\) → \((1,i,i,e\left(\frac{1}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2320 }(1077, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)