Properties

Label 1122.59
Modulus $1122$
Conductor $561$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,8,25]))
 
pari: [g,chi] = znchar(Mod(59,1122))
 

Basic properties

Modulus: \(1122\)
Conductor: \(561\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{561}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1122.bj

\(\chi_{1122}(53,\cdot)\) \(\chi_{1122}(59,\cdot)\) \(\chi_{1122}(179,\cdot)\) \(\chi_{1122}(185,\cdot)\) \(\chi_{1122}(257,\cdot)\) \(\chi_{1122}(383,\cdot)\) \(\chi_{1122}(389,\cdot)\) \(\chi_{1122}(467,\cdot)\) \(\chi_{1122}(587,\cdot)\) \(\chi_{1122}(665,\cdot)\) \(\chi_{1122}(773,\cdot)\) \(\chi_{1122}(797,\cdot)\) \(\chi_{1122}(971,\cdot)\) \(\chi_{1122}(977,\cdot)\) \(\chi_{1122}(995,\cdot)\) \(\chi_{1122}(1103,\cdot)\)

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

\((749,409,1057)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 1122 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1122 }(59,a) \;\) at \(\;a = \) e.g. 2