from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1143, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,41]))
pari: [g,chi] = znchar(Mod(80,1143))
Basic properties
Modulus: | \(1143\) | |
Conductor: | \(381\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{381}(80,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1143.by
\(\chi_{1143}(80,\cdot)\) \(\chi_{1143}(89,\cdot)\) \(\chi_{1143}(287,\cdot)\) \(\chi_{1143}(305,\cdot)\) \(\chi_{1143}(386,\cdot)\) \(\chi_{1143}(458,\cdot)\) \(\chi_{1143}(548,\cdot)\) \(\chi_{1143}(701,\cdot)\) \(\chi_{1143}(737,\cdot)\) \(\chi_{1143}(899,\cdot)\) \(\chi_{1143}(1043,\cdot)\) \(\chi_{1143}(1070,\cdot)\)
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: | 42.42.1885814229083612999756170685190350016159423006500366264928575446894255145089191907938237816767181.1 |
Values on generators
\((128,892)\) → \((-1,e\left(\frac{41}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 1143 }(80, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)