from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(235, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,34]))
pari: [g,chi] = znchar(Mod(34,235))
Basic properties
| Modulus: | \(235\) | |
| Conductor: | \(235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | 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 235.i
\(\chi_{235}(4,\cdot)\) \(\chi_{235}(9,\cdot)\) \(\chi_{235}(14,\cdot)\) \(\chi_{235}(24,\cdot)\) \(\chi_{235}(34,\cdot)\) \(\chi_{235}(49,\cdot)\) \(\chi_{235}(54,\cdot)\) \(\chi_{235}(59,\cdot)\) \(\chi_{235}(64,\cdot)\) \(\chi_{235}(74,\cdot)\) \(\chi_{235}(79,\cdot)\) \(\chi_{235}(84,\cdot)\) \(\chi_{235}(89,\cdot)\) \(\chi_{235}(119,\cdot)\) \(\chi_{235}(144,\cdot)\) \(\chi_{235}(149,\cdot)\) \(\chi_{235}(159,\cdot)\) \(\chi_{235}(169,\cdot)\) \(\chi_{235}(194,\cdot)\) \(\chi_{235}(204,\cdot)\) \(\chi_{235}(209,\cdot)\) \(\chi_{235}(224,\cdot)\)
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: | 46.46.445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125.1 |
Values on generators
\((142,146)\) → \((-1,e\left(\frac{17}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 235 }(34, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{29}{46}\right)\) |
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)