from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,40]))
pari: [g,chi] = znchar(Mod(223,1340))
Basic properties
Modulus: | \(1340\) | |
Conductor: | \(1340\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1340.bj
\(\chi_{1340}(107,\cdot)\) \(\chi_{1340}(143,\cdot)\) \(\chi_{1340}(223,\cdot)\) \(\chi_{1340}(263,\cdot)\) \(\chi_{1340}(283,\cdot)\) \(\chi_{1340}(327,\cdot)\) \(\chi_{1340}(427,\cdot)\) \(\chi_{1340}(483,\cdot)\) \(\chi_{1340}(627,\cdot)\) \(\chi_{1340}(643,\cdot)\) \(\chi_{1340}(667,\cdot)\) \(\chi_{1340}(863,\cdot)\) \(\chi_{1340}(947,\cdot)\) \(\chi_{1340}(963,\cdot)\) \(\chi_{1340}(1027,\cdot)\) \(\chi_{1340}(1067,\cdot)\) \(\chi_{1340}(1087,\cdot)\) \(\chi_{1340}(1163,\cdot)\) \(\chi_{1340}(1203,\cdot)\) \(\chi_{1340}(1287,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((671,537,1141)\) → \((-1,-i,e\left(\frac{10}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1340 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) |
sage: chi.jacobi_sum(n)