from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2000, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,41]))
pari: [g,chi] = znchar(Mod(79,2000))
Basic properties
Modulus: | \(2000\) | |
Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{500}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2000.bp
\(\chi_{2000}(79,\cdot)\) \(\chi_{2000}(159,\cdot)\) \(\chi_{2000}(239,\cdot)\) \(\chi_{2000}(319,\cdot)\) \(\chi_{2000}(479,\cdot)\) \(\chi_{2000}(559,\cdot)\) \(\chi_{2000}(639,\cdot)\) \(\chi_{2000}(719,\cdot)\) \(\chi_{2000}(879,\cdot)\) \(\chi_{2000}(959,\cdot)\) \(\chi_{2000}(1039,\cdot)\) \(\chi_{2000}(1119,\cdot)\) \(\chi_{2000}(1279,\cdot)\) \(\chi_{2000}(1359,\cdot)\) \(\chi_{2000}(1439,\cdot)\) \(\chi_{2000}(1519,\cdot)\) \(\chi_{2000}(1679,\cdot)\) \(\chi_{2000}(1759,\cdot)\) \(\chi_{2000}(1839,\cdot)\) \(\chi_{2000}(1919,\cdot)\)
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
\((751,501,1377)\) → \((-1,1,e\left(\frac{41}{50}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 2000 }(79, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) |
sage: chi.jacobi_sum(n)