from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,4]))
pari: [g,chi] = znchar(Mod(103,1216))
Basic properties
| Modulus: | \(1216\) | |
| Conductor: | \(608\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{608}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1216.br
\(\chi_{1216}(103,\cdot)\) \(\chi_{1216}(183,\cdot)\) \(\chi_{1216}(407,\cdot)\) \(\chi_{1216}(487,\cdot)\) \(\chi_{1216}(711,\cdot)\) \(\chi_{1216}(791,\cdot)\) \(\chi_{1216}(1015,\cdot)\) \(\chi_{1216}(1095,\cdot)\)
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: | 24.24.372273065750166762311522006998539295982015765641428992.1 |
Values on generators
\((191,837,705)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1216 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(-i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)