from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,45]))
pari: [g,chi] = znchar(Mod(65,414))
Basic properties
| Modulus: | \(414\) | |
| Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{207}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 414.p
\(\chi_{414}(5,\cdot)\) \(\chi_{414}(11,\cdot)\) \(\chi_{414}(65,\cdot)\) \(\chi_{414}(83,\cdot)\) \(\chi_{414}(113,\cdot)\) \(\chi_{414}(149,\cdot)\) \(\chi_{414}(155,\cdot)\) \(\chi_{414}(191,\cdot)\) \(\chi_{414}(203,\cdot)\) \(\chi_{414}(221,\cdot)\) \(\chi_{414}(227,\cdot)\) \(\chi_{414}(245,\cdot)\) \(\chi_{414}(263,\cdot)\) \(\chi_{414}(281,\cdot)\) \(\chi_{414}(293,\cdot)\) \(\chi_{414}(329,\cdot)\) \(\chi_{414}(365,\cdot)\) \(\chi_{414}(383,\cdot)\) \(\chi_{414}(389,\cdot)\) \(\chi_{414}(401,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((47,235)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 414 }(65, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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)