from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1875, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,42]))
pari: [g,chi] = znchar(Mod(76,1875))
Basic properties
| Modulus: | \(1875\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{125}(66,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1875.m
\(\chi_{1875}(76,\cdot)\) \(\chi_{1875}(151,\cdot)\) \(\chi_{1875}(226,\cdot)\) \(\chi_{1875}(301,\cdot)\) \(\chi_{1875}(451,\cdot)\) \(\chi_{1875}(526,\cdot)\) \(\chi_{1875}(601,\cdot)\) \(\chi_{1875}(676,\cdot)\) \(\chi_{1875}(826,\cdot)\) \(\chi_{1875}(901,\cdot)\) \(\chi_{1875}(976,\cdot)\) \(\chi_{1875}(1051,\cdot)\) \(\chi_{1875}(1201,\cdot)\) \(\chi_{1875}(1276,\cdot)\) \(\chi_{1875}(1351,\cdot)\) \(\chi_{1875}(1426,\cdot)\) \(\chi_{1875}(1576,\cdot)\) \(\chi_{1875}(1651,\cdot)\) \(\chi_{1875}(1726,\cdot)\) \(\chi_{1875}(1801,\cdot)\)
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 25 polynomial |
Values on generators
\((626,1252)\) → \((1,e\left(\frac{21}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1875 }(76, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) |
sage: chi.jacobi_sum(n)