from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(697, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,3]))
pari: [g,chi] = znchar(Mod(339,697))
Basic properties
| Modulus: | \(697\) | |
| Conductor: | \(697\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 697.bw
\(\chi_{697}(67,\cdot)\) \(\chi_{697}(101,\cdot)\) \(\chi_{697}(135,\cdot)\) \(\chi_{697}(152,\cdot)\) \(\chi_{697}(186,\cdot)\) \(\chi_{697}(220,\cdot)\) \(\chi_{697}(322,\cdot)\) \(\chi_{697}(339,\cdot)\) \(\chi_{697}(356,\cdot)\) \(\chi_{697}(458,\cdot)\) \(\chi_{697}(475,\cdot)\) \(\chi_{697}(509,\cdot)\) \(\chi_{697}(526,\cdot)\) \(\chi_{697}(628,\cdot)\) \(\chi_{697}(645,\cdot)\) \(\chi_{697}(662,\cdot)\)
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: | 40.0.3217724369317177596907031731223092219803204403094256869508157994902742845244016168900761.1 |
Values on generators
\((411,375)\) → \((-1,e\left(\frac{3}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 697 }(339, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) |
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)