Properties

Label 2738.75
Modulus $2738$
Conductor $1369$
Order $37$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(74))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(75,2738))
 

Basic properties

Modulus: \(2738\)
Conductor: \(1369\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(37\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1369}(75,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2738.j

\(\chi_{2738}(75,\cdot)\) \(\chi_{2738}(149,\cdot)\) \(\chi_{2738}(223,\cdot)\) \(\chi_{2738}(297,\cdot)\) \(\chi_{2738}(371,\cdot)\) \(\chi_{2738}(445,\cdot)\) \(\chi_{2738}(519,\cdot)\) \(\chi_{2738}(593,\cdot)\) \(\chi_{2738}(667,\cdot)\) \(\chi_{2738}(741,\cdot)\) \(\chi_{2738}(815,\cdot)\) \(\chi_{2738}(889,\cdot)\) \(\chi_{2738}(963,\cdot)\) \(\chi_{2738}(1037,\cdot)\) \(\chi_{2738}(1111,\cdot)\) \(\chi_{2738}(1185,\cdot)\) \(\chi_{2738}(1259,\cdot)\) \(\chi_{2738}(1333,\cdot)\) \(\chi_{2738}(1407,\cdot)\) \(\chi_{2738}(1481,\cdot)\) \(\chi_{2738}(1555,\cdot)\) \(\chi_{2738}(1629,\cdot)\) \(\chi_{2738}(1703,\cdot)\) \(\chi_{2738}(1777,\cdot)\) \(\chi_{2738}(1851,\cdot)\) \(\chi_{2738}(1925,\cdot)\) \(\chi_{2738}(1999,\cdot)\) \(\chi_{2738}(2073,\cdot)\) \(\chi_{2738}(2147,\cdot)\) \(\chi_{2738}(2221,\cdot)\) ...

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

\(1371\) → \(e\left(\frac{2}{37}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2738 }(75, a) \) \(1\)\(1\)\(e\left(\frac{36}{37}\right)\)\(e\left(\frac{17}{37}\right)\)\(e\left(\frac{23}{37}\right)\)\(e\left(\frac{35}{37}\right)\)\(e\left(\frac{24}{37}\right)\)\(e\left(\frac{2}{37}\right)\)\(e\left(\frac{16}{37}\right)\)\(e\left(\frac{10}{37}\right)\)\(e\left(\frac{33}{37}\right)\)\(e\left(\frac{22}{37}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2738 }(75,a) \;\) at \(\;a = \) e.g. 2