from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5166, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,23]))
pari: [g,chi] = znchar(Mod(71,5166))
Basic properties
| Modulus: | \(5166\) | |
| Conductor: | \(123\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{123}(71,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5166.ev
\(\chi_{5166}(71,\cdot)\) \(\chi_{5166}(827,\cdot)\) \(\chi_{5166}(1079,\cdot)\) \(\chi_{5166}(1331,\cdot)\) \(\chi_{5166}(1457,\cdot)\) \(\chi_{5166}(1709,\cdot)\) \(\chi_{5166}(1961,\cdot)\) \(\chi_{5166}(2717,\cdot)\) \(\chi_{5166}(2969,\cdot)\) \(\chi_{5166}(3347,\cdot)\) \(\chi_{5166}(3473,\cdot)\) \(\chi_{5166}(3725,\cdot)\) \(\chi_{5166}(4229,\cdot)\) \(\chi_{5166}(4481,\cdot)\) \(\chi_{5166}(4607,\cdot)\) \(\chi_{5166}(4985,\cdot)\)
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: | \(\Q(\zeta_{123})^+\) |
Values on generators
\((2297,2215,3655)\) → \((-1,1,e\left(\frac{23}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5166 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)