from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1932, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,44,57]))
pari: [g,chi] = znchar(Mod(53,1932))
Basic properties
| Modulus: | \(1932\) | |
| Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{483}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1932.cj
\(\chi_{1932}(53,\cdot)\) \(\chi_{1932}(65,\cdot)\) \(\chi_{1932}(149,\cdot)\) \(\chi_{1932}(221,\cdot)\) \(\chi_{1932}(389,\cdot)\) \(\chi_{1932}(401,\cdot)\) \(\chi_{1932}(557,\cdot)\) \(\chi_{1932}(569,\cdot)\) \(\chi_{1932}(641,\cdot)\) \(\chi_{1932}(893,\cdot)\) \(\chi_{1932}(977,\cdot)\) \(\chi_{1932}(1073,\cdot)\) \(\chi_{1932}(1157,\cdot)\) \(\chi_{1932}(1229,\cdot)\) \(\chi_{1932}(1325,\cdot)\) \(\chi_{1932}(1397,\cdot)\) \(\chi_{1932}(1493,\cdot)\) \(\chi_{1932}(1661,\cdot)\) \(\chi_{1932}(1745,\cdot)\) \(\chi_{1932}(1901,\cdot)\)
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
\((967,1289,829,925)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1932 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)