from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,42]))
pari: [g,chi] = znchar(Mod(23,1694))
Basic properties
Modulus: | \(1694\) | |
Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{847}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1694.u
\(\chi_{1694}(23,\cdot)\) \(\chi_{1694}(67,\cdot)\) \(\chi_{1694}(177,\cdot)\) \(\chi_{1694}(221,\cdot)\) \(\chi_{1694}(331,\cdot)\) \(\chi_{1694}(375,\cdot)\) \(\chi_{1694}(529,\cdot)\) \(\chi_{1694}(639,\cdot)\) \(\chi_{1694}(683,\cdot)\) \(\chi_{1694}(793,\cdot)\) \(\chi_{1694}(837,\cdot)\) \(\chi_{1694}(947,\cdot)\) \(\chi_{1694}(991,\cdot)\) \(\chi_{1694}(1101,\cdot)\) \(\chi_{1694}(1145,\cdot)\) \(\chi_{1694}(1255,\cdot)\) \(\chi_{1694}(1299,\cdot)\) \(\chi_{1694}(1409,\cdot)\) \(\chi_{1694}(1563,\cdot)\) \(\chi_{1694}(1607,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((969,365)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 1694 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)