from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,36]))
pari: [g,chi] = znchar(Mod(71,2000))
Basic properties
| Modulus: | \(2000\) | |
| Conductor: | \(1000\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1000}(571,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2000.bs
\(\chi_{2000}(71,\cdot)\) \(\chi_{2000}(231,\cdot)\) \(\chi_{2000}(311,\cdot)\) \(\chi_{2000}(391,\cdot)\) \(\chi_{2000}(471,\cdot)\) \(\chi_{2000}(631,\cdot)\) \(\chi_{2000}(711,\cdot)\) \(\chi_{2000}(791,\cdot)\) \(\chi_{2000}(871,\cdot)\) \(\chi_{2000}(1031,\cdot)\) \(\chi_{2000}(1111,\cdot)\) \(\chi_{2000}(1191,\cdot)\) \(\chi_{2000}(1271,\cdot)\) \(\chi_{2000}(1431,\cdot)\) \(\chi_{2000}(1511,\cdot)\) \(\chi_{2000}(1591,\cdot)\) \(\chi_{2000}(1671,\cdot)\) \(\chi_{2000}(1831,\cdot)\) \(\chi_{2000}(1911,\cdot)\) \(\chi_{2000}(1991,\cdot)\)
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
\((751,501,1377)\) → \((-1,-1,e\left(\frac{18}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2000 }(71, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{3}{25}\right)\) |
sage: chi.jacobi_sum(n)