from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61731, base_ring=CyclotomicField(2166))
M = H._module
chi = DirichletCharacter(H, M([1805,1127]))
pari: [g,chi] = znchar(Mod(29633,61731))
Basic properties
| Modulus: | \(61731\) | |
| Conductor: | \(61731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2166\) | 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 61731.cz
\(\chi_{61731}(50,\cdot)\) \(\chi_{61731}(65,\cdot)\) \(\chi_{61731}(221,\cdot)\) \(\chi_{61731}(236,\cdot)\) \(\chi_{61731}(392,\cdot)\) \(\chi_{61731}(407,\cdot)\) \(\chi_{61731}(563,\cdot)\) \(\chi_{61731}(578,\cdot)\) \(\chi_{61731}(734,\cdot)\) \(\chi_{61731}(749,\cdot)\) \(\chi_{61731}(905,\cdot)\) \(\chi_{61731}(920,\cdot)\) \(\chi_{61731}(1076,\cdot)\) \(\chi_{61731}(1091,\cdot)\) \(\chi_{61731}(1247,\cdot)\) \(\chi_{61731}(1262,\cdot)\) \(\chi_{61731}(1418,\cdot)\) \(\chi_{61731}(1433,\cdot)\)
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: | $\Q(\zeta_{1083})$ |
| Fixed field: | Number field defined by a degree 2166 polynomial (not computed) |
Values on generators
\((6860,54874)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1127}{2166}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 61731 }(29633, a) \) | \(1\) | \(1\) | \(e\left(\frac{383}{1083}\right)\) | \(e\left(\frac{766}{1083}\right)\) | \(e\left(\frac{103}{722}\right)\) | \(e\left(\frac{355}{1083}\right)\) | \(e\left(\frac{22}{361}\right)\) | \(e\left(\frac{1075}{2166}\right)\) | \(e\left(\frac{23}{2166}\right)\) | \(e\left(\frac{131}{2166}\right)\) | \(e\left(\frac{246}{361}\right)\) | \(e\left(\frac{449}{1083}\right)\) |
sage: chi.jacobi_sum(n)