from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2020, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,42]))
pari: [g,chi] = znchar(Mod(839,2020))
Basic properties
| Modulus: | \(2020\) | |
| Conductor: | \(2020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2020.bu
\(\chi_{2020}(19,\cdot)\) \(\chi_{2020}(79,\cdot)\) \(\chi_{2020}(159,\cdot)\) \(\chi_{2020}(179,\cdot)\) \(\chi_{2020}(239,\cdot)\) \(\chi_{2020}(299,\cdot)\) \(\chi_{2020}(319,\cdot)\) \(\chi_{2020}(359,\cdot)\) \(\chi_{2020}(559,\cdot)\) \(\chi_{2020}(759,\cdot)\) \(\chi_{2020}(799,\cdot)\) \(\chi_{2020}(839,\cdot)\) \(\chi_{2020}(879,\cdot)\) \(\chi_{2020}(1179,\cdot)\) \(\chi_{2020}(1199,\cdot)\) \(\chi_{2020}(1419,\cdot)\) \(\chi_{2020}(1439,\cdot)\) \(\chi_{2020}(1539,\cdot)\) \(\chi_{2020}(1899,\cdot)\) \(\chi_{2020}(1999,\cdot)\)
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
\((1011,1617,1921)\) → \((-1,-1,e\left(\frac{21}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2020 }(839, a) \) | \(-1\) | \(1\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
sage: chi.jacobi_sum(n)