from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31200, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,35,20,38,30]))
pari: [g,chi] = znchar(Mod(12563,31200))
Basic properties
Modulus: | \(31200\) | |
Conductor: | \(31200\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 31200.xz
\(\chi_{31200}(83,\cdot)\) \(\chi_{31200}(827,\cdot)\) \(\chi_{31200}(3203,\cdot)\) \(\chi_{31200}(3947,\cdot)\) \(\chi_{31200}(6323,\cdot)\) \(\chi_{31200}(7067,\cdot)\) \(\chi_{31200}(10187,\cdot)\) \(\chi_{31200}(12563,\cdot)\) \(\chi_{31200}(15683,\cdot)\) \(\chi_{31200}(16427,\cdot)\) \(\chi_{31200}(18803,\cdot)\) \(\chi_{31200}(19547,\cdot)\) \(\chi_{31200}(21923,\cdot)\) \(\chi_{31200}(22667,\cdot)\) \(\chi_{31200}(25787,\cdot)\) \(\chi_{31200}(28163,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((1951,27301,20801,14977,12001)\) → \((-1,e\left(\frac{7}{8}\right),-1,e\left(\frac{19}{20}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 31200 }(12563, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)