from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1476, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,4,15]))
pari: [g,chi] = znchar(Mod(137,1476))
Basic properties
| Modulus: | \(1476\) | |
| Conductor: | \(369\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{369}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1476.bq
\(\chi_{1476}(137,\cdot)\) \(\chi_{1476}(437,\cdot)\) \(\chi_{1476}(653,\cdot)\) \(\chi_{1476}(905,\cdot)\) \(\chi_{1476}(929,\cdot)\) \(\chi_{1476}(1121,\cdot)\) \(\chi_{1476}(1145,\cdot)\) \(\chi_{1476}(1397,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.1108730924679628633118445533811818265986494447636761.1 |
Values on generators
\((739,821,1441)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1476 }(137, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)