from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,30]))
pari: [g,chi] = znchar(Mod(124,207))
Basic properties
| Modulus: | \(207\) | |
| Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 207.m
\(\chi_{207}(4,\cdot)\) \(\chi_{207}(13,\cdot)\) \(\chi_{207}(16,\cdot)\) \(\chi_{207}(25,\cdot)\) \(\chi_{207}(31,\cdot)\) \(\chi_{207}(49,\cdot)\) \(\chi_{207}(52,\cdot)\) \(\chi_{207}(58,\cdot)\) \(\chi_{207}(85,\cdot)\) \(\chi_{207}(94,\cdot)\) \(\chi_{207}(121,\cdot)\) \(\chi_{207}(124,\cdot)\) \(\chi_{207}(133,\cdot)\) \(\chi_{207}(142,\cdot)\) \(\chi_{207}(151,\cdot)\) \(\chi_{207}(169,\cdot)\) \(\chi_{207}(187,\cdot)\) \(\chi_{207}(193,\cdot)\) \(\chi_{207}(196,\cdot)\) \(\chi_{207}(202,\cdot)\)
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
\((47,28)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 207 }(124, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)