from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1144, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,3,20]))
pari: [g,chi] = znchar(Mod(35,1144))
Basic properties
| Modulus: | \(1144\) | |
| Conductor: | \(1144\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1144.di
\(\chi_{1144}(35,\cdot)\) \(\chi_{1144}(107,\cdot)\) \(\chi_{1144}(139,\cdot)\) \(\chi_{1144}(211,\cdot)\) \(\chi_{1144}(315,\cdot)\) \(\chi_{1144}(347,\cdot)\) \(\chi_{1144}(523,\cdot)\) \(\chi_{1144}(1075,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((287,573,937,353)\) → \((-1,-1,e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 1144 }(35, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)