from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([11,10,0]))
pari: [g,chi] = znchar(Mod(573,3850))
Basic properties
| Modulus: | \(3850\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{175}(48,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.dn
\(\chi_{3850}(573,\cdot)\) \(\chi_{3850}(727,\cdot)\) \(\chi_{3850}(1497,\cdot)\) \(\chi_{3850}(2113,\cdot)\) \(\chi_{3850}(2267,\cdot)\) \(\chi_{3850}(2883,\cdot)\) \(\chi_{3850}(3037,\cdot)\) \(\chi_{3850}(3653,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.822111175511963665485382080078125.1 |
Values on generators
\((2927,2201,1751)\) → \((e\left(\frac{11}{20}\right),-1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3850 }(573, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)