from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,23,21]))
pari: [g,chi] = znchar(Mod(285,2548))
Basic properties
| Modulus: | \(2548\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{637}(285,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2548.dj
\(\chi_{2548}(285,\cdot)\) \(\chi_{2548}(493,\cdot)\) \(\chi_{2548}(649,\cdot)\) \(\chi_{2548}(857,\cdot)\) \(\chi_{2548}(1013,\cdot)\) \(\chi_{2548}(1221,\cdot)\) \(\chi_{2548}(1377,\cdot)\) \(\chi_{2548}(1585,\cdot)\) \(\chi_{2548}(1741,\cdot)\) \(\chi_{2548}(1949,\cdot)\) \(\chi_{2548}(2105,\cdot)\) \(\chi_{2548}(2313,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.29198428620782310880522337720254845955751250559410488348634029682058779274295867292920491.1 |
Values on generators
\((1275,885,197)\) → \((1,e\left(\frac{23}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2548 }(285, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) |
sage: chi.jacobi_sum(n)