from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1890, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([14,9,12]))
pari: [g,chi] = znchar(Mod(317,1890))
Basic properties
| Modulus: | \(1890\) | |
| Conductor: | \(945\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{945}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1890.dp
\(\chi_{1890}(317,\cdot)\) \(\chi_{1890}(347,\cdot)\) \(\chi_{1890}(443,\cdot)\) \(\chi_{1890}(473,\cdot)\) \(\chi_{1890}(947,\cdot)\) \(\chi_{1890}(977,\cdot)\) \(\chi_{1890}(1073,\cdot)\) \(\chi_{1890}(1103,\cdot)\) \(\chi_{1890}(1577,\cdot)\) \(\chi_{1890}(1607,\cdot)\) \(\chi_{1890}(1703,\cdot)\) \(\chi_{1890}(1733,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((1541,757,1081)\) → \((e\left(\frac{7}{18}\right),i,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1890 }(317, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) |
sage: chi.jacobi_sum(n)