Properties

Label 3312.73
Modulus $3312$
Conductor $184$
Order $22$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
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)
 
\( \chi_{ 3312 }(73,a) \;\) at \(\;a = \) e.g. 2