from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,22,22,42]))
pari: [g,chi] = znchar(Mod(83,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(7728\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 7728.fj
\(\chi_{7728}(83,\cdot)\) \(\chi_{7728}(251,\cdot)\) \(\chi_{7728}(419,\cdot)\) \(\chi_{7728}(755,\cdot)\) \(\chi_{7728}(1091,\cdot)\) \(\chi_{7728}(1259,\cdot)\) \(\chi_{7728}(1763,\cdot)\) \(\chi_{7728}(2435,\cdot)\) \(\chi_{7728}(2771,\cdot)\) \(\chi_{7728}(3779,\cdot)\) \(\chi_{7728}(3947,\cdot)\) \(\chi_{7728}(4115,\cdot)\) \(\chi_{7728}(4283,\cdot)\) \(\chi_{7728}(4619,\cdot)\) \(\chi_{7728}(4955,\cdot)\) \(\chi_{7728}(5123,\cdot)\) \(\chi_{7728}(5627,\cdot)\) \(\chi_{7728}(6299,\cdot)\) \(\chi_{7728}(6635,\cdot)\) \(\chi_{7728}(7643,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((-1,-i,-1,-1,e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)