from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,25,18]))
pari: [g,chi] = znchar(Mod(111,4000))
Basic properties
Modulus: | \(4000\) | |
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 \(\chi_{1000}(611,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4000.ci
\(\chi_{4000}(111,\cdot)\) \(\chi_{4000}(271,\cdot)\) \(\chi_{4000}(431,\cdot)\) \(\chi_{4000}(591,\cdot)\) \(\chi_{4000}(911,\cdot)\) \(\chi_{4000}(1071,\cdot)\) \(\chi_{4000}(1231,\cdot)\) \(\chi_{4000}(1391,\cdot)\) \(\chi_{4000}(1711,\cdot)\) \(\chi_{4000}(1871,\cdot)\) \(\chi_{4000}(2031,\cdot)\) \(\chi_{4000}(2191,\cdot)\) \(\chi_{4000}(2511,\cdot)\) \(\chi_{4000}(2671,\cdot)\) \(\chi_{4000}(2831,\cdot)\) \(\chi_{4000}(2991,\cdot)\) \(\chi_{4000}(3311,\cdot)\) \(\chi_{4000}(3471,\cdot)\) \(\chi_{4000}(3631,\cdot)\) \(\chi_{4000}(3791,\cdot)\)
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
\((2751,2501,1377)\) → \((-1,-1,e\left(\frac{9}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 4000 }(111, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{31}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{14}{25}\right)\) |
sage: chi.jacobi_sum(n)