from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,30]))
pari: [g,chi] = znchar(Mod(34,7623))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7623.ei
\(\chi_{7623}(34,\cdot)\) \(\chi_{7623}(265,\cdot)\) \(\chi_{7623}(958,\cdot)\) \(\chi_{7623}(1420,\cdot)\) \(\chi_{7623}(1651,\cdot)\) \(\chi_{7623}(2113,\cdot)\) \(\chi_{7623}(2344,\cdot)\) \(\chi_{7623}(2806,\cdot)\) \(\chi_{7623}(3037,\cdot)\) \(\chi_{7623}(3499,\cdot)\) \(\chi_{7623}(3730,\cdot)\) \(\chi_{7623}(4192,\cdot)\) \(\chi_{7623}(4423,\cdot)\) \(\chi_{7623}(4885,\cdot)\) \(\chi_{7623}(5116,\cdot)\) \(\chi_{7623}(5578,\cdot)\) \(\chi_{7623}(6271,\cdot)\) \(\chi_{7623}(6502,\cdot)\) \(\chi_{7623}(6964,\cdot)\) \(\chi_{7623}(7195,\cdot)\)
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
\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 7623 }(34, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{47}{66}\right)\) |
sage: chi.jacobi_sum(n)