from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,34,0]))
pari: [g,chi] = znchar(Mod(529,8085))
Basic properties
| Modulus: | \(8085\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{245}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.eo
\(\chi_{8085}(529,\cdot)\) \(\chi_{8085}(1024,\cdot)\) \(\chi_{8085}(2179,\cdot)\) \(\chi_{8085}(2839,\cdot)\) \(\chi_{8085}(3334,\cdot)\) \(\chi_{8085}(3994,\cdot)\) \(\chi_{8085}(5149,\cdot)\) \(\chi_{8085}(5644,\cdot)\) \(\chi_{8085}(6304,\cdot)\) \(\chi_{8085}(6799,\cdot)\) \(\chi_{8085}(7459,\cdot)\) \(\chi_{8085}(7954,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1 |
Values on generators
\((2696,4852,1816,3676)\) → \((1,-1,e\left(\frac{17}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 8085 }(529, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)