from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6776, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,6]))
pari: [g,chi] = znchar(Mod(177,6776))
Basic properties
| Modulus: | \(6776\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{847}(177,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6776.dc
\(\chi_{6776}(177,\cdot)\) \(\chi_{6776}(529,\cdot)\) \(\chi_{6776}(793,\cdot)\) \(\chi_{6776}(1145,\cdot)\) \(\chi_{6776}(1409,\cdot)\) \(\chi_{6776}(1761,\cdot)\) \(\chi_{6776}(2025,\cdot)\) \(\chi_{6776}(2377,\cdot)\) \(\chi_{6776}(2641,\cdot)\) \(\chi_{6776}(2993,\cdot)\) \(\chi_{6776}(3257,\cdot)\) \(\chi_{6776}(3609,\cdot)\) \(\chi_{6776}(4225,\cdot)\) \(\chi_{6776}(4489,\cdot)\) \(\chi_{6776}(5105,\cdot)\) \(\chi_{6776}(5457,\cdot)\) \(\chi_{6776}(5721,\cdot)\) \(\chi_{6776}(6073,\cdot)\) \(\chi_{6776}(6337,\cdot)\) \(\chi_{6776}(6689,\cdot)\)
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 33 polynomial |
Values on generators
\((1695,3389,969,3753)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 6776 }(177, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)