from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2169, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,49]))
pari: [g,chi] = znchar(Mod(82,2169))
Basic properties
| Modulus: | \(2169\) | |
| Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{241}(82,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2169.dc
\(\chi_{2169}(82,\cdot)\) \(\chi_{2169}(118,\cdot)\) \(\chi_{2169}(145,\cdot)\) \(\chi_{2169}(640,\cdot)\) \(\chi_{2169}(820,\cdot)\) \(\chi_{2169}(874,\cdot)\) \(\chi_{2169}(955,\cdot)\) \(\chi_{2169}(973,\cdot)\) \(\chi_{2169}(1054,\cdot)\) \(\chi_{2169}(1108,\cdot)\) \(\chi_{2169}(1288,\cdot)\) \(\chi_{2169}(1783,\cdot)\) \(\chi_{2169}(1810,\cdot)\) \(\chi_{2169}(1846,\cdot)\) \(\chi_{2169}(2035,\cdot)\) \(\chi_{2169}(2062,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((965,730)\) → \((1,e\left(\frac{49}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2169 }(82, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(-1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)