from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2842, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([10,24]))
pari: [g,chi] = znchar(Mod(837,2842))
Basic properties
Modulus: | \(2842\) | |
Conductor: | \(1421\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1421}(837,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2842.bx
\(\chi_{2842}(837,\cdot)\) \(\chi_{2842}(977,\cdot)\) \(\chi_{2842}(1299,\cdot)\) \(\chi_{2842}(1437,\cdot)\) \(\chi_{2842}(1591,\cdot)\) \(\chi_{2842}(1619,\cdot)\) \(\chi_{2842}(1677,\cdot)\) \(\chi_{2842}(1747,\cdot)\) \(\chi_{2842}(1901,\cdot)\) \(\chi_{2842}(2095,\cdot)\) \(\chi_{2842}(2459,\cdot)\) \(\chi_{2842}(2543,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1277,785)\) → \((e\left(\frac{5}{21}\right),e\left(\frac{4}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 2842 }(837, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)