from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2169, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,7]))
pari: [g,chi] = znchar(Mod(38,2169))
Basic properties
| Modulus: | \(2169\) | |
| Conductor: | \(2169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 2169.cq
\(\chi_{2169}(38,\cdot)\) \(\chi_{2169}(65,\cdot)\) \(\chi_{2169}(230,\cdot)\) \(\chi_{2169}(263,\cdot)\) \(\chi_{2169}(329,\cdot)\) \(\chi_{2169}(635,\cdot)\) \(\chi_{2169}(734,\cdot)\) \(\chi_{2169}(1424,\cdot)\) \(\chi_{2169}(1427,\cdot)\) \(\chi_{2169}(1535,\cdot)\) \(\chi_{2169}(1598,\cdot)\) \(\chi_{2169}(1622,\cdot)\) \(\chi_{2169}(1649,\cdot)\) \(\chi_{2169}(1706,\cdot)\) \(\chi_{2169}(1865,\cdot)\) \(\chi_{2169}(1991,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((965,730)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2169 }(38, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(-i\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)