from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([1,0,4]))
pari: [g,chi] = znchar(Mod(379,3850))
Basic properties
| Modulus: | \(3850\) | |
| Conductor: | \(275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{275}(104,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.cm
\(\chi_{3850}(379,\cdot)\) \(\chi_{3850}(939,\cdot)\) \(\chi_{3850}(1219,\cdot)\) \(\chi_{3850}(3809,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.10.163542847442626953125.4 |
Values on generators
\((2927,2201,1751)\) → \((e\left(\frac{1}{10}\right),1,e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3850 }(379, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)