from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,9,22]))
pari: [g,chi] = znchar(Mod(357,3800))
Basic properties
| Modulus: | \(3800\) | |
| Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{760}(357,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3800.eg
\(\chi_{3800}(357,\cdot)\) \(\chi_{3800}(1093,\cdot)\) \(\chi_{3800}(1693,\cdot)\) \(\chi_{3800}(2093,\cdot)\) \(\chi_{3800}(2157,\cdot)\) \(\chi_{3800}(2293,\cdot)\) \(\chi_{3800}(2693,\cdot)\) \(\chi_{3800}(2757,\cdot)\) \(\chi_{3800}(3093,\cdot)\) \(\chi_{3800}(3157,\cdot)\) \(\chi_{3800}(3357,\cdot)\) \(\chi_{3800}(3757,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1 |
Values on generators
\((951,1901,1977,401)\) → \((1,-1,i,e\left(\frac{11}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3800 }(357, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) |
sage: chi.jacobi_sum(n)