from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,50,42]))
pari: [g,chi] = znchar(Mod(1195,1232))
Basic properties
| Modulus: | \(1232\) | |
| Conductor: | \(1232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1232.dr
\(\chi_{1232}(19,\cdot)\) \(\chi_{1232}(171,\cdot)\) \(\chi_{1232}(227,\cdot)\) \(\chi_{1232}(283,\cdot)\) \(\chi_{1232}(299,\cdot)\) \(\chi_{1232}(523,\cdot)\) \(\chi_{1232}(563,\cdot)\) \(\chi_{1232}(579,\cdot)\) \(\chi_{1232}(635,\cdot)\) \(\chi_{1232}(787,\cdot)\) \(\chi_{1232}(843,\cdot)\) \(\chi_{1232}(899,\cdot)\) \(\chi_{1232}(915,\cdot)\) \(\chi_{1232}(1139,\cdot)\) \(\chi_{1232}(1179,\cdot)\) \(\chi_{1232}(1195,\cdot)\)
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
\((463,309,353,673)\) → \((-1,i,e\left(\frac{5}{6}\right),e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1232 }(1195, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)