from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,35,20,12]))
pari: [g,chi] = znchar(Mod(239,9600))
Basic properties
| Modulus: | \(9600\) | |
| Conductor: | \(2400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2400}(1139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9600.fj
\(\chi_{9600}(239,\cdot)\) \(\chi_{9600}(719,\cdot)\) \(\chi_{9600}(1679,\cdot)\) \(\chi_{9600}(2159,\cdot)\) \(\chi_{9600}(2639,\cdot)\) \(\chi_{9600}(3119,\cdot)\) \(\chi_{9600}(4079,\cdot)\) \(\chi_{9600}(4559,\cdot)\) \(\chi_{9600}(5039,\cdot)\) \(\chi_{9600}(5519,\cdot)\) \(\chi_{9600}(6479,\cdot)\) \(\chi_{9600}(6959,\cdot)\) \(\chi_{9600}(7439,\cdot)\) \(\chi_{9600}(7919,\cdot)\) \(\chi_{9600}(8879,\cdot)\) \(\chi_{9600}(9359,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.53955375229419889123391977410055372800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((4351,901,6401,5377)\) → \((-1,e\left(\frac{7}{8}\right),-1,e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 9600 }(239, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)