from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,28]))
pari: [g,chi] = znchar(Mod(292,8007))
Basic properties
Modulus: | \(8007\) | |
Conductor: | \(2669\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2669}(292,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8007.cv
\(\chi_{8007}(292,\cdot)\) \(\chi_{8007}(1306,\cdot)\) \(\chi_{8007}(1435,\cdot)\) \(\chi_{8007}(1705,\cdot)\) \(\chi_{8007}(2305,\cdot)\) \(\chi_{8007}(2647,\cdot)\) \(\chi_{8007}(4060,\cdot)\) \(\chi_{8007}(4189,\cdot)\) \(\chi_{8007}(4261,\cdot)\) \(\chi_{8007}(4732,\cdot)\) \(\chi_{8007}(5545,\cdot)\) \(\chi_{8007}(6958,\cdot)\) \(\chi_{8007}(7015,\cdot)\) \(\chi_{8007}(7486,\cdot)\) \(\chi_{8007}(7558,\cdot)\) \(\chi_{8007}(7900,\cdot)\)
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
\((5339,1414,7855)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 8007 }(292, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)