Properties

Label 1369.408
Modulus $1369$
Conductor $1369$
Order $37$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1369, base_ring=CyclotomicField(74))
 
M = H._module
 
chi = DirichletCharacter(H, M([22]))
 
pari: [g,chi] = znchar(Mod(408,1369))
 

Basic properties

Modulus: \(1369\)
Conductor: \(1369\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(37\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1369.j

\(\chi_{1369}(38,\cdot)\) \(\chi_{1369}(75,\cdot)\) \(\chi_{1369}(112,\cdot)\) \(\chi_{1369}(149,\cdot)\) \(\chi_{1369}(186,\cdot)\) \(\chi_{1369}(223,\cdot)\) \(\chi_{1369}(260,\cdot)\) \(\chi_{1369}(297,\cdot)\) \(\chi_{1369}(334,\cdot)\) \(\chi_{1369}(371,\cdot)\) \(\chi_{1369}(408,\cdot)\) \(\chi_{1369}(445,\cdot)\) \(\chi_{1369}(482,\cdot)\) \(\chi_{1369}(519,\cdot)\) \(\chi_{1369}(556,\cdot)\) \(\chi_{1369}(593,\cdot)\) \(\chi_{1369}(630,\cdot)\) \(\chi_{1369}(667,\cdot)\) \(\chi_{1369}(704,\cdot)\) \(\chi_{1369}(741,\cdot)\) \(\chi_{1369}(778,\cdot)\) \(\chi_{1369}(815,\cdot)\) \(\chi_{1369}(852,\cdot)\) \(\chi_{1369}(889,\cdot)\) \(\chi_{1369}(926,\cdot)\) \(\chi_{1369}(963,\cdot)\) \(\chi_{1369}(1000,\cdot)\) \(\chi_{1369}(1037,\cdot)\) \(\chi_{1369}(1074,\cdot)\) \(\chi_{1369}(1111,\cdot)\) ...

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: $\Q(\zeta_{37})$
Fixed field: 37.37.81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681.1

Values on generators

\(2\) → \(e\left(\frac{11}{37}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1369 }(408, a) \) \(1\)\(1\)\(e\left(\frac{11}{37}\right)\)\(e\left(\frac{13}{37}\right)\)\(e\left(\frac{22}{37}\right)\)\(e\left(\frac{1}{37}\right)\)\(e\left(\frac{24}{37}\right)\)\(e\left(\frac{34}{37}\right)\)\(e\left(\frac{33}{37}\right)\)\(e\left(\frac{26}{37}\right)\)\(e\left(\frac{12}{37}\right)\)\(e\left(\frac{21}{37}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1369 }(408,a) \;\) at \(\;a = \) e.g. 2