from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,18]))
pari: [g,chi] = znchar(Mod(11,368))
Basic properties
| Modulus: | \(368\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 368.x
\(\chi_{368}(11,\cdot)\) \(\chi_{368}(19,\cdot)\) \(\chi_{368}(43,\cdot)\) \(\chi_{368}(51,\cdot)\) \(\chi_{368}(67,\cdot)\) \(\chi_{368}(83,\cdot)\) \(\chi_{368}(99,\cdot)\) \(\chi_{368}(107,\cdot)\) \(\chi_{368}(155,\cdot)\) \(\chi_{368}(171,\cdot)\) \(\chi_{368}(195,\cdot)\) \(\chi_{368}(203,\cdot)\) \(\chi_{368}(227,\cdot)\) \(\chi_{368}(235,\cdot)\) \(\chi_{368}(251,\cdot)\) \(\chi_{368}(267,\cdot)\) \(\chi_{368}(283,\cdot)\) \(\chi_{368}(291,\cdot)\) \(\chi_{368}(339,\cdot)\) \(\chi_{368}(355,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608.1 |
Values on generators
\((47,277,97)\) → \((-1,i,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
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)