from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,14]))
pari: [g,chi] = znchar(Mod(301,6010))
Basic properties
Modulus: | \(6010\) | |
Conductor: | \(601\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{601}(301,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6010.bn
\(\chi_{6010}(301,\cdot)\) \(\chi_{6010}(451,\cdot)\) \(\chi_{6010}(1091,\cdot)\) \(\chi_{6010}(1581,\cdot)\) \(\chi_{6010}(1811,\cdot)\) \(\chi_{6010}(1911,\cdot)\) \(\chi_{6010}(1931,\cdot)\) \(\chi_{6010}(2431,\cdot)\) \(\chi_{6010}(2561,\cdot)\) \(\chi_{6010}(3021,\cdot)\) \(\chi_{6010}(3221,\cdot)\) \(\chi_{6010}(3261,\cdot)\) \(\chi_{6010}(3531,\cdot)\) \(\chi_{6010}(3851,\cdot)\) \(\chi_{6010}(4211,\cdot)\) \(\chi_{6010}(4261,\cdot)\) \(\chi_{6010}(4271,\cdot)\) \(\chi_{6010}(5071,\cdot)\) \(\chi_{6010}(5411,\cdot)\) \(\chi_{6010}(5921,\cdot)\)
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 25 polynomial |
Values on generators
\((3607,2411)\) → \((1,e\left(\frac{7}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 6010 }(301, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
sage: chi.jacobi_sum(n)