from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,18]))
pari: [g,chi] = znchar(Mod(3554,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(2541\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2541}(1013,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7623.dz
\(\chi_{7623}(89,\cdot)\) \(\chi_{7623}(584,\cdot)\) \(\chi_{7623}(782,\cdot)\) \(\chi_{7623}(1277,\cdot)\) \(\chi_{7623}(1475,\cdot)\) \(\chi_{7623}(1970,\cdot)\) \(\chi_{7623}(2168,\cdot)\) \(\chi_{7623}(2861,\cdot)\) \(\chi_{7623}(3356,\cdot)\) \(\chi_{7623}(3554,\cdot)\) \(\chi_{7623}(4049,\cdot)\) \(\chi_{7623}(4247,\cdot)\) \(\chi_{7623}(4742,\cdot)\) \(\chi_{7623}(4940,\cdot)\) \(\chi_{7623}(5435,\cdot)\) \(\chi_{7623}(5633,\cdot)\) \(\chi_{7623}(6128,\cdot)\) \(\chi_{7623}(6326,\cdot)\) \(\chi_{7623}(6821,\cdot)\) \(\chi_{7623}(7514,\cdot)\)
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
\((848,4357,4600)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(3554, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)