from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,12]))
pari: [g,chi] = znchar(Mod(77,368))
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.w
\(\chi_{368}(13,\cdot)\) \(\chi_{368}(29,\cdot)\) \(\chi_{368}(77,\cdot)\) \(\chi_{368}(85,\cdot)\) \(\chi_{368}(101,\cdot)\) \(\chi_{368}(117,\cdot)\) \(\chi_{368}(133,\cdot)\) \(\chi_{368}(141,\cdot)\) \(\chi_{368}(165,\cdot)\) \(\chi_{368}(173,\cdot)\) \(\chi_{368}(197,\cdot)\) \(\chi_{368}(213,\cdot)\) \(\chi_{368}(261,\cdot)\) \(\chi_{368}(269,\cdot)\) \(\chi_{368}(285,\cdot)\) \(\chi_{368}(301,\cdot)\) \(\chi_{368}(317,\cdot)\) \(\chi_{368}(325,\cdot)\) \(\chi_{368}(349,\cdot)\) \(\chi_{368}(357,\cdot)\)
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: | 44.44.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1 |
Values on generators
\((47,277,97)\) → \((1,-i,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 368 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) |
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)