from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,16]))
pari: [g,chi] = znchar(Mod(109,430))
Basic properties
| Modulus: | \(430\) | |
| Conductor: | \(215\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{215}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 430.t
\(\chi_{430}(9,\cdot)\) \(\chi_{430}(99,\cdot)\) \(\chi_{430}(109,\cdot)\) \(\chi_{430}(139,\cdot)\) \(\chi_{430}(169,\cdot)\) \(\chi_{430}(189,\cdot)\) \(\chi_{430}(229,\cdot)\) \(\chi_{430}(239,\cdot)\) \(\chi_{430}(289,\cdot)\) \(\chi_{430}(339,\cdot)\) \(\chi_{430}(359,\cdot)\) \(\chi_{430}(369,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.104017955712751803355033526522081856753017553948018605377854818344593048095703125.1 |
Values on generators
\((87,261)\) → \((-1,e\left(\frac{8}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 430 }(109, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) |
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)