Properties

Label 1340.41
Modulus $1340$
Conductor $67$
Order $66$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,53]))
 
pari: [g,chi] = znchar(Mod(41,1340))
 

Basic properties

Modulus: \(1340\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{67}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1340.bo

\(\chi_{1340}(41,\cdot)\) \(\chi_{1340}(61,\cdot)\) \(\chi_{1340}(101,\cdot)\) \(\chi_{1340}(141,\cdot)\) \(\chi_{1340}(221,\cdot)\) \(\chi_{1340}(281,\cdot)\) \(\chi_{1340}(381,\cdot)\) \(\chi_{1340}(481,\cdot)\) \(\chi_{1340}(501,\cdot)\) \(\chi_{1340}(621,\cdot)\) \(\chi_{1340}(681,\cdot)\) \(\chi_{1340}(701,\cdot)\) \(\chi_{1340}(721,\cdot)\) \(\chi_{1340}(781,\cdot)\) \(\chi_{1340}(861,\cdot)\) \(\chi_{1340}(921,\cdot)\) \(\chi_{1340}(1001,\cdot)\) \(\chi_{1340}(1141,\cdot)\) \(\chi_{1340}(1301,\cdot)\) \(\chi_{1340}(1321,\cdot)\)

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

\((671,537,1141)\) → \((1,1,e\left(\frac{53}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1340 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1340 }(41,a) \;\) at \(\;a = \) e.g. 2