from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,55,45]))
pari: [g,chi] = znchar(Mod(1825,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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.ed
\(\chi_{7623}(439,\cdot)\) \(\chi_{7623}(472,\cdot)\) \(\chi_{7623}(1132,\cdot)\) \(\chi_{7623}(1165,\cdot)\) \(\chi_{7623}(1825,\cdot)\) \(\chi_{7623}(1858,\cdot)\) \(\chi_{7623}(2518,\cdot)\) \(\chi_{7623}(2551,\cdot)\) \(\chi_{7623}(3211,\cdot)\) \(\chi_{7623}(3244,\cdot)\) \(\chi_{7623}(3904,\cdot)\) \(\chi_{7623}(3937,\cdot)\) \(\chi_{7623}(4630,\cdot)\) \(\chi_{7623}(5290,\cdot)\) \(\chi_{7623}(5983,\cdot)\) \(\chi_{7623}(6016,\cdot)\) \(\chi_{7623}(6676,\cdot)\) \(\chi_{7623}(6709,\cdot)\) \(\chi_{7623}(7369,\cdot)\) \(\chi_{7623}(7402,\cdot)\)
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 66 polynomial |
Values on generators
\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(1825, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) |
sage: chi.jacobi_sum(n)