from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(232, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,14,17]))
pari: [g,chi] = znchar(Mod(21,232))
Basic properties
Modulus: | \(232\) | |
Conductor: | \(232\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 232.u
\(\chi_{232}(21,\cdot)\) \(\chi_{232}(37,\cdot)\) \(\chi_{232}(61,\cdot)\) \(\chi_{232}(69,\cdot)\) \(\chi_{232}(77,\cdot)\) \(\chi_{232}(85,\cdot)\) \(\chi_{232}(101,\cdot)\) \(\chi_{232}(189,\cdot)\) \(\chi_{232}(205,\cdot)\) \(\chi_{232}(213,\cdot)\) \(\chi_{232}(221,\cdot)\) \(\chi_{232}(229,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.0.13427827737836760536055607671312169337571202392129536.1 |
Values on generators
\((175,117,89)\) → \((1,-1,e\left(\frac{17}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 232 }(21, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) |
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)