from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3312, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,4]))
pari: [g,chi] = znchar(Mod(73,3312))
Basic properties
Modulus: | \(3312\) | |
Conductor: | \(184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{184}(165,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3312.ch
\(\chi_{3312}(73,\cdot)\) \(\chi_{3312}(361,\cdot)\) \(\chi_{3312}(1225,\cdot)\) \(\chi_{3312}(1369,\cdot)\) \(\chi_{3312}(1513,\cdot)\) \(\chi_{3312}(1945,\cdot)\) \(\chi_{3312}(2233,\cdot)\) \(\chi_{3312}(2377,\cdot)\) \(\chi_{3312}(2809,\cdot)\) \(\chi_{3312}(2953,\cdot)\)
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: | 22.22.14741666340843480753092741810452692992.1 |
Values on generators
\((415,2485,2945,2305)\) → \((1,-1,1,e\left(\frac{2}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 3312 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)