from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,22,42]))
pari: [g,chi] = znchar(Mod(1717,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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 7623.dl
\(\chi_{7623}(67,\cdot)\) \(\chi_{7623}(331,\cdot)\) \(\chi_{7623}(760,\cdot)\) \(\chi_{7623}(1024,\cdot)\) \(\chi_{7623}(1717,\cdot)\) \(\chi_{7623}(2146,\cdot)\) \(\chi_{7623}(2410,\cdot)\) \(\chi_{7623}(2839,\cdot)\) \(\chi_{7623}(3103,\cdot)\) \(\chi_{7623}(3532,\cdot)\) \(\chi_{7623}(3796,\cdot)\) \(\chi_{7623}(4225,\cdot)\) \(\chi_{7623}(4489,\cdot)\) \(\chi_{7623}(4918,\cdot)\) \(\chi_{7623}(5182,\cdot)\) \(\chi_{7623}(5611,\cdot)\) \(\chi_{7623}(5875,\cdot)\) \(\chi_{7623}(6304,\cdot)\) \(\chi_{7623}(6568,\cdot)\) \(\chi_{7623}(6997,\cdot)\)
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
\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(1717, a) \) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)