from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(728, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,0,8,1]))
pari: [g,chi] = znchar(Mod(431,728))
Basic properties
| Modulus: | \(728\) | |
| Conductor: | \(364\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{364}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 728.dm
\(\chi_{728}(375,\cdot)\) \(\chi_{728}(431,\cdot)\) \(\chi_{728}(527,\cdot)\) \(\chi_{728}(695,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.42317611137863236145152.2 |
Values on generators
\((183,365,521,561)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 728 }(431, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) |
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)