from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,8]))
pari: [g,chi] = znchar(Mod(241,9600))
Basic properties
| Modulus: | \(9600\) | |
| Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{800}(741,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9600.fg
\(\chi_{9600}(241,\cdot)\) \(\chi_{9600}(721,\cdot)\) \(\chi_{9600}(1681,\cdot)\) \(\chi_{9600}(2161,\cdot)\) \(\chi_{9600}(2641,\cdot)\) \(\chi_{9600}(3121,\cdot)\) \(\chi_{9600}(4081,\cdot)\) \(\chi_{9600}(4561,\cdot)\) \(\chi_{9600}(5041,\cdot)\) \(\chi_{9600}(5521,\cdot)\) \(\chi_{9600}(6481,\cdot)\) \(\chi_{9600}(6961,\cdot)\) \(\chi_{9600}(7441,\cdot)\) \(\chi_{9600}(7921,\cdot)\) \(\chi_{9600}(8881,\cdot)\) \(\chi_{9600}(9361,\cdot)\)
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: | Number field defined by a degree 40 polynomial |
Values on generators
\((4351,901,6401,5377)\) → \((1,e\left(\frac{1}{8}\right),1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 9600 }(241, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)