from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(705, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,23,45]))
pari: [g,chi] = znchar(Mod(19,705))
Basic properties
| Modulus: | \(705\) | |
| Conductor: | \(235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{235}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 705.t
\(\chi_{705}(19,\cdot)\) \(\chi_{705}(109,\cdot)\) \(\chi_{705}(124,\cdot)\) \(\chi_{705}(139,\cdot)\) \(\chi_{705}(154,\cdot)\) \(\chi_{705}(184,\cdot)\) \(\chi_{705}(199,\cdot)\) \(\chi_{705}(214,\cdot)\) \(\chi_{705}(229,\cdot)\) \(\chi_{705}(274,\cdot)\) \(\chi_{705}(304,\cdot)\) \(\chi_{705}(334,\cdot)\) \(\chi_{705}(349,\cdot)\) \(\chi_{705}(364,\cdot)\) \(\chi_{705}(409,\cdot)\) \(\chi_{705}(454,\cdot)\) \(\chi_{705}(499,\cdot)\) \(\chi_{705}(514,\cdot)\) \(\chi_{705}(574,\cdot)\) \(\chi_{705}(604,\cdot)\) \(\chi_{705}(634,\cdot)\) \(\chi_{705}(649,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | 46.0.20927324417353262576794873573459132405730192352302940670222843367282253010356426239013671875.1 |
Values on generators
\((236,142,616)\) → \((1,-1,e\left(\frac{45}{46}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 705 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{1}{46}\right)\) |
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)