from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(935, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,8,5]))
pari: [g,chi] = znchar(Mod(26,935))
Basic properties
| Modulus: | \(935\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{187}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 935.ci
\(\chi_{935}(26,\cdot)\) \(\chi_{935}(36,\cdot)\) \(\chi_{935}(196,\cdot)\) \(\chi_{935}(236,\cdot)\) \(\chi_{935}(246,\cdot)\) \(\chi_{935}(291,\cdot)\) \(\chi_{935}(366,\cdot)\) \(\chi_{935}(416,\cdot)\) \(\chi_{935}(576,\cdot)\) \(\chi_{935}(586,\cdot)\) \(\chi_{935}(621,\cdot)\) \(\chi_{935}(631,\cdot)\) \(\chi_{935}(746,\cdot)\) \(\chi_{935}(801,\cdot)\) \(\chi_{935}(841,\cdot)\) \(\chi_{935}(916,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((562,596,496)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{1}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 935 }(26, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) |
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)