from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8032, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,37]))
pari: [g,chi] = znchar(Mod(47,8032))
Basic properties
| Modulus: | \(8032\) | |
| Conductor: | \(2008\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2008}(1051,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8032.bk
\(\chi_{8032}(47,\cdot)\) \(\chi_{8032}(879,\cdot)\) \(\chi_{8032}(1263,\cdot)\) \(\chi_{8032}(1295,\cdot)\) \(\chi_{8032}(1455,\cdot)\) \(\chi_{8032}(1807,\cdot)\) \(\chi_{8032}(2159,\cdot)\) \(\chi_{8032}(2255,\cdot)\) \(\chi_{8032}(2447,\cdot)\) \(\chi_{8032}(3183,\cdot)\) \(\chi_{8032}(3247,\cdot)\) \(\chi_{8032}(5679,\cdot)\) \(\chi_{8032}(5775,\cdot)\) \(\chi_{8032}(5999,\cdot)\) \(\chi_{8032}(6959,\cdot)\) \(\chi_{8032}(7023,\cdot)\) \(\chi_{8032}(7215,\cdot)\) \(\chi_{8032}(7407,\cdot)\) \(\chi_{8032}(7439,\cdot)\) \(\chi_{8032}(7791,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((6527,3013,257)\) → \((-1,-1,e\left(\frac{37}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8032 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) |
sage: chi.jacobi_sum(n)