from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(296, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,23]))
pari: [g,chi] = znchar(Mod(5,296))
Basic properties
| Modulus: | \(296\) | |
| Conductor: | \(296\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 296.bg
\(\chi_{296}(5,\cdot)\) \(\chi_{296}(13,\cdot)\) \(\chi_{296}(61,\cdot)\) \(\chi_{296}(69,\cdot)\) \(\chi_{296}(93,\cdot)\) \(\chi_{296}(109,\cdot)\) \(\chi_{296}(133,\cdot)\) \(\chi_{296}(165,\cdot)\) \(\chi_{296}(205,\cdot)\) \(\chi_{296}(237,\cdot)\) \(\chi_{296}(261,\cdot)\) \(\chi_{296}(277,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.138892919952333446776057851184385905517238171566853781889085447929331712.1 |
Values on generators
\((223,149,113)\) → \((1,-1,e\left(\frac{23}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 296 }(5, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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)