from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,38]))
pari: [g,chi] = znchar(Mod(53,368))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 368.v
\(\chi_{368}(5,\cdot)\) \(\chi_{368}(21,\cdot)\) \(\chi_{368}(37,\cdot)\) \(\chi_{368}(53,\cdot)\) \(\chi_{368}(61,\cdot)\) \(\chi_{368}(109,\cdot)\) \(\chi_{368}(125,\cdot)\) \(\chi_{368}(149,\cdot)\) \(\chi_{368}(157,\cdot)\) \(\chi_{368}(181,\cdot)\) \(\chi_{368}(189,\cdot)\) \(\chi_{368}(205,\cdot)\) \(\chi_{368}(221,\cdot)\) \(\chi_{368}(237,\cdot)\) \(\chi_{368}(245,\cdot)\) \(\chi_{368}(293,\cdot)\) \(\chi_{368}(309,\cdot)\) \(\chi_{368}(333,\cdot)\) \(\chi_{368}(341,\cdot)\) \(\chi_{368}(365,\cdot)\)
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: | 44.0.4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608.1 |
Values on generators
\((47,277,97)\) → \((1,i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) |
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)