from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,25]))
pari: [g,chi] = znchar(Mod(2389,8450))
Basic properties
Modulus: | \(8450\) | |
Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{325}(114,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8450.bj
\(\chi_{8450}(2389,\cdot)\) \(\chi_{8450}(3189,\cdot)\) \(\chi_{8450}(4079,\cdot)\) \(\chi_{8450}(4879,\cdot)\) \(\chi_{8450}(5769,\cdot)\) \(\chi_{8450}(6569,\cdot)\) \(\chi_{8450}(7459,\cdot)\) \(\chi_{8450}(8259,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.3133675547880273211567620924800081638750270940363407135009765625.1 |
Values on generators
\((677,3551)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 8450 }(2389, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)