Properties

Label 8040.407
Modulus $8040$
Conductor $4020$
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([22,0,22,11,10]))
 
pari: [g,chi] = znchar(Mod(407,8040))
 

Basic properties

Modulus: \(8040\)
Conductor: \(4020\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4020}(407,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8040.fk

\(\chi_{8040}(407,\cdot)\) \(\chi_{8040}(527,\cdot)\) \(\chi_{8040}(983,\cdot)\) \(\chi_{8040}(1343,\cdot)\) \(\chi_{8040}(1367,\cdot)\) \(\chi_{8040}(1583,\cdot)\) \(\chi_{8040}(1727,\cdot)\) \(\chi_{8040}(2063,\cdot)\) \(\chi_{8040}(3023,\cdot)\) \(\chi_{8040}(3527,\cdot)\) \(\chi_{8040}(3623,\cdot)\) \(\chi_{8040}(3743,\cdot)\) \(\chi_{8040}(4583,\cdot)\) \(\chi_{8040}(4943,\cdot)\) \(\chi_{8040}(5807,\cdot)\) \(\chi_{8040}(6167,\cdot)\) \(\chi_{8040}(6407,\cdot)\) \(\chi_{8040}(6743,\cdot)\) \(\chi_{8040}(6887,\cdot)\) \(\chi_{8040}(7847,\cdot)\)

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

\((6031,4021,2681,3217,5161)\) → \((-1,1,-1,i,e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8040 }(407, a) \) \(1\)\(1\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{5}{44}\right)\)\(1\)\(e\left(\frac{2}{11}\right)\)\(i\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8040 }(407,a) \;\) at \(\;a = \) e.g. 2