from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4011, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,19,21]))
pari: [g,chi] = znchar(Mod(139,4011))
Basic properties
Modulus: | \(4011\) | |
Conductor: | \(1337\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1337}(139,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4011.bj
\(\chi_{4011}(55,\cdot)\) \(\chi_{4011}(139,\cdot)\) \(\chi_{4011}(202,\cdot)\) \(\chi_{4011}(643,\cdot)\) \(\chi_{4011}(1021,\cdot)\) \(\chi_{4011}(1378,\cdot)\) \(\chi_{4011}(1714,\cdot)\) \(\chi_{4011}(1756,\cdot)\) \(\chi_{4011}(1924,\cdot)\) \(\chi_{4011}(2071,\cdot)\) \(\chi_{4011}(2260,\cdot)\) \(\chi_{4011}(2323,\cdot)\) \(\chi_{4011}(2638,\cdot)\) \(\chi_{4011}(3331,\cdot)\) \(\chi_{4011}(3604,\cdot)\) \(\chi_{4011}(3667,\cdot)\) \(\chi_{4011}(3751,\cdot)\) \(\chi_{4011}(3814,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\((2675,2866,2311)\) → \((1,-1,e\left(\frac{21}{38}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 4011 }(139, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) |
sage: chi.jacobi_sum(n)