from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7220, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,34]))
pari: [g,chi] = znchar(Mod(333,7220))
Basic properties
| Modulus: | \(7220\) | |
| Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{95}(48,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7220.bi
\(\chi_{7220}(333,\cdot)\) \(\chi_{7220}(477,\cdot)\) \(\chi_{7220}(1777,\cdot)\) \(\chi_{7220}(2293,\cdot)\) \(\chi_{7220}(2473,\cdot)\) \(\chi_{7220}(3737,\cdot)\) \(\chi_{7220}(3917,\cdot)\) \(\chi_{7220}(4233,\cdot)\) \(\chi_{7220}(5353,\cdot)\) \(\chi_{7220}(5677,\cdot)\) \(\chi_{7220}(6253,\cdot)\) \(\chi_{7220}(6797,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | \(\Q(\zeta_{95})^+\) |
Values on generators
\((3611,5777,6861)\) → \((1,-i,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 7220 }(333, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)