Properties

Label 5610.3391
Modulus $5610$
Conductor $187$
Order $40$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,32,25]))
 
pari: [g,chi] = znchar(Mod(3391,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.ex

\(\chi_{5610}(631,\cdot)\) \(\chi_{5610}(841,\cdot)\) \(\chi_{5610}(961,\cdot)\) \(\chi_{5610}(1171,\cdot)\) \(\chi_{5610}(1351,\cdot)\) \(\chi_{5610}(1681,\cdot)\) \(\chi_{5610}(2161,\cdot)\) \(\chi_{5610}(2491,\cdot)\) \(\chi_{5610}(2671,\cdot)\) \(\chi_{5610}(3001,\cdot)\) \(\chi_{5610}(3391,\cdot)\) \(\chi_{5610}(3721,\cdot)\) \(\chi_{5610}(4711,\cdot)\) \(\chi_{5610}(4921,\cdot)\) \(\chi_{5610}(5041,\cdot)\) \(\chi_{5610}(5251,\cdot)\)

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: 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1

Values on generators

\((1871,3367,1531,3301)\) → \((1,1,e\left(\frac{4}{5}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(3391, a) \) \(1\)\(1\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(i\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(3391,a) \;\) at \(\;a = \) e.g. 2