from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2842, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([18,19]))
pari: [g,chi] = znchar(Mod(55,2842))
Basic properties
Modulus: | \(2842\) | |
Conductor: | \(1421\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1421}(55,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2842.ch
\(\chi_{2842}(55,\cdot)\) \(\chi_{2842}(279,\cdot)\) \(\chi_{2842}(475,\cdot)\) \(\chi_{2842}(601,\cdot)\) \(\chi_{2842}(1203,\cdot)\) \(\chi_{2842}(1245,\cdot)\) \(\chi_{2842}(1315,\cdot)\) \(\chi_{2842}(1539,\cdot)\) \(\chi_{2842}(1777,\cdot)\) \(\chi_{2842}(1917,\cdot)\) \(\chi_{2842}(1987,\cdot)\) \(\chi_{2842}(2687,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((1277,785)\) → \((e\left(\frac{9}{14}\right),e\left(\frac{19}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 2842 }(55, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(1\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)