from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1932, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,11,30]))
pari: [g,chi] = znchar(Mod(101,1932))
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 \(\chi_{483}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1932.ch
\(\chi_{1932}(101,\cdot)\) \(\chi_{1932}(173,\cdot)\) \(\chi_{1932}(257,\cdot)\) \(\chi_{1932}(269,\cdot)\) \(\chi_{1932}(353,\cdot)\) \(\chi_{1932}(509,\cdot)\) \(\chi_{1932}(593,\cdot)\) \(\chi_{1932}(761,\cdot)\) \(\chi_{1932}(857,\cdot)\) \(\chi_{1932}(929,\cdot)\) \(\chi_{1932}(1025,\cdot)\) \(\chi_{1932}(1097,\cdot)\) \(\chi_{1932}(1181,\cdot)\) \(\chi_{1932}(1277,\cdot)\) \(\chi_{1932}(1361,\cdot)\) \(\chi_{1932}(1613,\cdot)\) \(\chi_{1932}(1685,\cdot)\) \(\chi_{1932}(1697,\cdot)\) \(\chi_{1932}(1853,\cdot)\) \(\chi_{1932}(1865,\cdot)\)
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
\((967,1289,829,925)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1932 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)