from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([17,10,8]))
pari: [g,chi] = znchar(Mod(797,3850))
Basic properties
| Modulus: | \(3850\) | |
| Conductor: | \(1925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1925}(797,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.dw
\(\chi_{3850}(797,\cdot)\) \(\chi_{3850}(867,\cdot)\) \(\chi_{3850}(1483,\cdot)\) \(\chi_{3850}(1763,\cdot)\) \(\chi_{3850}(1973,\cdot)\) \(\chi_{3850}(2127,\cdot)\) \(\chi_{3850}(2687,\cdot)\) \(\chi_{3850}(2953,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((2927,2201,1751)\) → \((e\left(\frac{17}{20}\right),-1,e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3850 }(797, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(-1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)