Properties

Label 8040.329
Modulus $8040$
Conductor $1005$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,33,33,7]))
 
pari: [g,chi] = znchar(Mod(329,8040))
 

Basic properties

Modulus: \(8040\)
Conductor: \(1005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1005}(329,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8040.gt

\(\chi_{8040}(329,\cdot)\) \(\chi_{8040}(1049,\cdot)\) \(\chi_{8040}(1409,\cdot)\) \(\chi_{8040}(1649,\cdot)\) \(\chi_{8040}(1889,\cdot)\) \(\chi_{8040}(2609,\cdot)\) \(\chi_{8040}(3089,\cdot)\) \(\chi_{8040}(3329,\cdot)\) \(\chi_{8040}(3449,\cdot)\) \(\chi_{8040}(3569,\cdot)\) \(\chi_{8040}(3809,\cdot)\) \(\chi_{8040}(4769,\cdot)\) \(\chi_{8040}(5009,\cdot)\) \(\chi_{8040}(5609,\cdot)\) \(\chi_{8040}(5729,\cdot)\) \(\chi_{8040}(5849,\cdot)\) \(\chi_{8040}(6329,\cdot)\) \(\chi_{8040}(6929,\cdot)\) \(\chi_{8040}(7649,\cdot)\) \(\chi_{8040}(7889,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((6031,4021,2681,3217,5161)\) → \((1,1,-1,-1,e\left(\frac{7}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 8040 }(329, a) \) \(1\)\(1\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8040 }(329,a) \;\) at \(\;a = \) e.g. 2