from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1932, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,11,12]))
pari: [g,chi] = znchar(Mod(41,1932))
Basic properties
Modulus: | \(1932\) | |
Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{483}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1932.bn
\(\chi_{1932}(41,\cdot)\) \(\chi_{1932}(209,\cdot)\) \(\chi_{1932}(377,\cdot)\) \(\chi_{1932}(545,\cdot)\) \(\chi_{1932}(629,\cdot)\) \(\chi_{1932}(1133,\cdot)\) \(\chi_{1932}(1301,\cdot)\) \(\chi_{1932}(1553,\cdot)\) \(\chi_{1932}(1637,\cdot)\) \(\chi_{1932}(1889,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((967,1289,829,925)\) → \((1,-1,-1,e\left(\frac{6}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1932 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)