from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,54]))
pari: [g,chi] = znchar(Mod(259,414))
Basic properties
Modulus: | \(414\) | |
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 \(\chi_{207}(52,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 414.m
\(\chi_{414}(13,\cdot)\) \(\chi_{414}(25,\cdot)\) \(\chi_{414}(31,\cdot)\) \(\chi_{414}(49,\cdot)\) \(\chi_{414}(85,\cdot)\) \(\chi_{414}(121,\cdot)\) \(\chi_{414}(133,\cdot)\) \(\chi_{414}(151,\cdot)\) \(\chi_{414}(169,\cdot)\) \(\chi_{414}(187,\cdot)\) \(\chi_{414}(193,\cdot)\) \(\chi_{414}(211,\cdot)\) \(\chi_{414}(223,\cdot)\) \(\chi_{414}(259,\cdot)\) \(\chi_{414}(265,\cdot)\) \(\chi_{414}(301,\cdot)\) \(\chi_{414}(331,\cdot)\) \(\chi_{414}(349,\cdot)\) \(\chi_{414}(403,\cdot)\) \(\chi_{414}(409,\cdot)\)
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
\((47,235)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 414 }(259, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) |
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)