from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,39]))
pari: [g,chi] = znchar(Mod(89,2624))
Basic properties
| Modulus: | \(2624\) | |
| Conductor: | \(1312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1312}(581,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2624.dz
\(\chi_{2624}(89,\cdot)\) \(\chi_{2624}(153,\cdot)\) \(\chi_{2624}(393,\cdot)\) \(\chi_{2624}(889,\cdot)\) \(\chi_{2624}(1001,\cdot)\) \(\chi_{2624}(1177,\cdot)\) \(\chi_{2624}(1529,\cdot)\) \(\chi_{2624}(1577,\cdot)\) \(\chi_{2624}(1737,\cdot)\) \(\chi_{2624}(1769,\cdot)\) \(\chi_{2624}(1817,\cdot)\) \(\chi_{2624}(2249,\cdot)\) \(\chi_{2624}(2281,\cdot)\) \(\chi_{2624}(2441,\cdot)\) \(\chi_{2624}(2553,\cdot)\) \(\chi_{2624}(2617,\cdot)\)
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
\((575,1477,129)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{39}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2624 }(89, a) \) | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) |
sage: chi.jacobi_sum(n)