Properties

Label 8040.1651
Modulus $8040$
Conductor $536$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8040, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,0,0,3]))
 
pari: [g,chi] = znchar(Mod(1651,8040))
 

Basic properties

Modulus: \(8040\)
Conductor: \(536\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{536}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8040.ea

\(\chi_{8040}(1651,\cdot)\) \(\chi_{8040}(3931,\cdot)\) \(\chi_{8040}(4291,\cdot)\) \(\chi_{8040}(4531,\cdot)\) \(\chi_{8040}(5011,\cdot)\) \(\chi_{8040}(5971,\cdot)\) \(\chi_{8040}(6571,\cdot)\) \(\chi_{8040}(6691,\cdot)\) \(\chi_{8040}(7531,\cdot)\) \(\chi_{8040}(7891,\cdot)\)

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: 22.22.1912320645087103459758821027834145013889251672064.1

Values on generators

\((6031,4021,2681,3217,5161)\) → \((-1,-1,1,1,e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8040 }(1651, a) \) \(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(-1\)\(e\left(\frac{10}{11}\right)\)\(-1\)\(e\left(\frac{5}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8040 }(1651,a) \;\) at \(\;a = \) e.g. 2