Properties

Label 8040.223
Modulus $8040$
Conductor $1340$
Order $44$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8040, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,0,0,33,40]))
 
pari: [g,chi] = znchar(Mod(223,8040))
 

Basic properties

Modulus: \(8040\)
Conductor: \(1340\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1340}(223,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8040.fh

\(\chi_{8040}(223,\cdot)\) \(\chi_{8040}(1087,\cdot)\) \(\chi_{8040}(1447,\cdot)\) \(\chi_{8040}(2287,\cdot)\) \(\chi_{8040}(2407,\cdot)\) \(\chi_{8040}(2503,\cdot)\) \(\chi_{8040}(3007,\cdot)\) \(\chi_{8040}(3967,\cdot)\) \(\chi_{8040}(4303,\cdot)\) \(\chi_{8040}(4447,\cdot)\) \(\chi_{8040}(4663,\cdot)\) \(\chi_{8040}(4687,\cdot)\) \(\chi_{8040}(5047,\cdot)\) \(\chi_{8040}(5503,\cdot)\) \(\chi_{8040}(5623,\cdot)\) \(\chi_{8040}(6223,\cdot)\) \(\chi_{8040}(7183,\cdot)\) \(\chi_{8040}(7327,\cdot)\) \(\chi_{8040}(7663,\cdot)\) \(\chi_{8040}(7903,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((6031,4021,2681,3217,5161)\) → \((-1,1,1,-i,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8040 }(223, a) \) \(1\)\(1\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{44}\right)\)\(-1\)\(e\left(\frac{5}{22}\right)\)\(-i\)\(e\left(\frac{2}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8040 }(223,a) \;\) at \(\;a = \) e.g. 2