Properties

Label 1340.43
Modulus $1340$
Conductor $1340$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,33,6]))
 
pari: [g,chi] = znchar(Mod(43,1340))
 

Basic properties

Modulus: \(1340\)
Conductor: \(1340\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1340.bh

\(\chi_{1340}(3,\cdot)\) \(\chi_{1340}(27,\cdot)\) \(\chi_{1340}(43,\cdot)\) \(\chi_{1340}(187,\cdot)\) \(\chi_{1340}(243,\cdot)\) \(\chi_{1340}(343,\cdot)\) \(\chi_{1340}(387,\cdot)\) \(\chi_{1340}(407,\cdot)\) \(\chi_{1340}(447,\cdot)\) \(\chi_{1340}(527,\cdot)\) \(\chi_{1340}(563,\cdot)\) \(\chi_{1340}(723,\cdot)\) \(\chi_{1340}(807,\cdot)\) \(\chi_{1340}(847,\cdot)\) \(\chi_{1340}(923,\cdot)\) \(\chi_{1340}(943,\cdot)\) \(\chi_{1340}(983,\cdot)\) \(\chi_{1340}(1047,\cdot)\) \(\chi_{1340}(1063,\cdot)\) \(\chi_{1340}(1147,\cdot)\)

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

\((671,537,1141)\) → \((-1,-i,e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1340 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{9}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1340 }(43,a) \;\) at \(\;a = \) e.g. 2