from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3020, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,11]))
pari: [g,chi] = znchar(Mod(211,3020))
Basic properties
| Modulus: | \(3020\) | |
| Conductor: | \(604\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{604}(211,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3020.bx
\(\chi_{3020}(131,\cdot)\) \(\chi_{3020}(211,\cdot)\) \(\chi_{3020}(631,\cdot)\) \(\chi_{3020}(671,\cdot)\) \(\chi_{3020}(711,\cdot)\) \(\chi_{3020}(971,\cdot)\) \(\chi_{3020}(1211,\cdot)\) \(\chi_{3020}(1291,\cdot)\) \(\chi_{3020}(1551,\cdot)\) \(\chi_{3020}(1611,\cdot)\) \(\chi_{3020}(1731,\cdot)\) \(\chi_{3020}(1891,\cdot)\) \(\chi_{3020}(1991,\cdot)\) \(\chi_{3020}(2171,\cdot)\) \(\chi_{3020}(2291,\cdot)\) \(\chi_{3020}(2591,\cdot)\) \(\chi_{3020}(2771,\cdot)\) \(\chi_{3020}(2791,\cdot)\) \(\chi_{3020}(2991,\cdot)\) \(\chi_{3020}(3011,\cdot)\)
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 50 polynomial |
Values on generators
\((1511,2417,761)\) → \((-1,1,e\left(\frac{11}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3020 }(211, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{24}{25}\right)\) |
sage: chi.jacobi_sum(n)