from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2842, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([2,12]))
pari: [g,chi] = znchar(Mod(401,2842))
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 \(\chi_{1421}(401,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2842.bt
\(\chi_{2842}(401,\cdot)\) \(\chi_{2842}(683,\cdot)\) \(\chi_{2842}(835,\cdot)\) \(\chi_{2842}(893,\cdot)\) \(\chi_{2842}(935,\cdot)\) \(\chi_{2842}(1649,\cdot)\) \(\chi_{2842}(1689,\cdot)\) \(\chi_{2842}(1731,\cdot)\) \(\chi_{2842}(1997,\cdot)\) \(\chi_{2842}(2083,\cdot)\) \(\chi_{2842}(2195,\cdot)\) \(\chi_{2842}(2249,\cdot)\)
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: | 21.21.27345727561662191028601731515684872548776286491521654395289.3 |
Values on generators
\((1277,785)\) → \((e\left(\frac{1}{21}\right),e\left(\frac{2}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 2842 }(401, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)