from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,27]))
pari: [g,chi] = znchar(Mod(296,7623))
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 \(\chi_{2541}(296,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7623.eg
\(\chi_{7623}(296,\cdot)\) \(\chi_{7623}(494,\cdot)\) \(\chi_{7623}(989,\cdot)\) \(\chi_{7623}(1187,\cdot)\) \(\chi_{7623}(1682,\cdot)\) \(\chi_{7623}(1880,\cdot)\) \(\chi_{7623}(2375,\cdot)\) \(\chi_{7623}(2573,\cdot)\) \(\chi_{7623}(3068,\cdot)\) \(\chi_{7623}(3761,\cdot)\) \(\chi_{7623}(3959,\cdot)\) \(\chi_{7623}(4454,\cdot)\) \(\chi_{7623}(4652,\cdot)\) \(\chi_{7623}(5147,\cdot)\) \(\chi_{7623}(5345,\cdot)\) \(\chi_{7623}(5840,\cdot)\) \(\chi_{7623}(6038,\cdot)\) \(\chi_{7623}(6731,\cdot)\) \(\chi_{7623}(7226,\cdot)\) \(\chi_{7623}(7424,\cdot)\)
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
\((848,4357,4600)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 7623 }(296, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)