Properties

Label 2112.71
Modulus $2112$
Conductor $1056$
Order $40$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,25,20,16]))
 
pari: [g,chi] = znchar(Mod(71,2112))
 

Basic properties

Modulus: \(2112\)
Conductor: \(1056\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1056}(203,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2112.cp

\(\chi_{2112}(71,\cdot)\) \(\chi_{2112}(119,\cdot)\) \(\chi_{2112}(311,\cdot)\) \(\chi_{2112}(455,\cdot)\) \(\chi_{2112}(599,\cdot)\) \(\chi_{2112}(647,\cdot)\) \(\chi_{2112}(839,\cdot)\) \(\chi_{2112}(983,\cdot)\) \(\chi_{2112}(1127,\cdot)\) \(\chi_{2112}(1175,\cdot)\) \(\chi_{2112}(1367,\cdot)\) \(\chi_{2112}(1511,\cdot)\) \(\chi_{2112}(1655,\cdot)\) \(\chi_{2112}(1703,\cdot)\) \(\chi_{2112}(1895,\cdot)\) \(\chi_{2112}(2039,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((2047,133,1409,1729)\) → \((-1,e\left(\frac{5}{8}\right),-1,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2112 }(71, a) \) \(1\)\(1\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{40}\right)\)\(-i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2112 }(71,a) \;\) at \(\;a = \) e.g. 2