from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1656, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,44,6]))
pari: [g,chi] = znchar(Mod(25,1656))
Basic properties
| Modulus: | \(1656\) | |
| Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{207}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1656.bw
\(\chi_{1656}(25,\cdot)\) \(\chi_{1656}(49,\cdot)\) \(\chi_{1656}(121,\cdot)\) \(\chi_{1656}(169,\cdot)\) \(\chi_{1656}(193,\cdot)\) \(\chi_{1656}(265,\cdot)\) \(\chi_{1656}(409,\cdot)\) \(\chi_{1656}(601,\cdot)\) \(\chi_{1656}(625,\cdot)\) \(\chi_{1656}(673,\cdot)\) \(\chi_{1656}(745,\cdot)\) \(\chi_{1656}(817,\cdot)\) \(\chi_{1656}(841,\cdot)\) \(\chi_{1656}(913,\cdot)\) \(\chi_{1656}(961,\cdot)\) \(\chi_{1656}(1129,\cdot)\) \(\chi_{1656}(1177,\cdot)\) \(\chi_{1656}(1273,\cdot)\) \(\chi_{1656}(1393,\cdot)\) \(\chi_{1656}(1465,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.70011645999218458416472683122408534303895571350166174758601569.1 |
Values on generators
\((415,829,1289,649)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1656 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)