from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,0,22,9]))
pari: [g,chi] = znchar(Mod(79,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(644\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{644}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.fy
\(\chi_{7728}(79,\cdot)\) \(\chi_{7728}(319,\cdot)\) \(\chi_{7728}(655,\cdot)\) \(\chi_{7728}(751,\cdot)\) \(\chi_{7728}(1423,\cdot)\) \(\chi_{7728}(1663,\cdot)\) \(\chi_{7728}(1759,\cdot)\) \(\chi_{7728}(1999,\cdot)\) \(\chi_{7728}(2767,\cdot)\) \(\chi_{7728}(3007,\cdot)\) \(\chi_{7728}(3103,\cdot)\) \(\chi_{7728}(4111,\cdot)\) \(\chi_{7728}(5695,\cdot)\) \(\chi_{7728}(6031,\cdot)\) \(\chi_{7728}(6367,\cdot)\) \(\chi_{7728}(6703,\cdot)\) \(\chi_{7728}(6799,\cdot)\) \(\chi_{7728}(7135,\cdot)\) \(\chi_{7728}(7375,\cdot)\) \(\chi_{7728}(7471,\cdot)\)
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
\((4831,5797,5153,6625,6721)\) → \((-1,1,1,e\left(\frac{1}{3}\right),e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)