Properties

Label 4005.8
Modulus $4005$
Conductor $1335$
Order $44$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,33,24]))
 
pari: [g,chi] = znchar(Mod(8,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(1335\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1335}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4005.ct

\(\chi_{4005}(8,\cdot)\) \(\chi_{4005}(242,\cdot)\) \(\chi_{4005}(512,\cdot)\) \(\chi_{4005}(728,\cdot)\) \(\chi_{4005}(1043,\cdot)\) \(\chi_{4005}(1313,\cdot)\) \(\chi_{4005}(1367,\cdot)\) \(\chi_{4005}(1502,\cdot)\) \(\chi_{4005}(1997,\cdot)\) \(\chi_{4005}(2168,\cdot)\) \(\chi_{4005}(2303,\cdot)\) \(\chi_{4005}(2537,\cdot)\) \(\chi_{4005}(2672,\cdot)\) \(\chi_{4005}(2798,\cdot)\) \(\chi_{4005}(2852,\cdot)\) \(\chi_{4005}(3212,\cdot)\) \(\chi_{4005}(3338,\cdot)\) \(\chi_{4005}(3473,\cdot)\) \(\chi_{4005}(3653,\cdot)\) \(\chi_{4005}(3932,\cdot)\)

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

\((3116,802,181)\) → \((-1,-i,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(8, a) \) \(1\)\(1\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{13}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(8,a) \;\) at \(\;a = \) e.g. 2