Properties

Label 5610.163
Modulus $5610$
Conductor $935$
Order $80$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,60,48,15]))
 
pari: [g,chi] = znchar(Mod(163,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(935\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{935}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.fa

\(\chi_{5610}(37,\cdot)\) \(\chi_{5610}(97,\cdot)\) \(\chi_{5610}(163,\cdot)\) \(\chi_{5610}(313,\cdot)\) \(\chi_{5610}(823,\cdot)\) \(\chi_{5610}(1213,\cdot)\) \(\chi_{5610}(1303,\cdot)\) \(\chi_{5610}(1357,\cdot)\) \(\chi_{5610}(1567,\cdot)\) \(\chi_{5610}(1807,\cdot)\) \(\chi_{5610}(1813,\cdot)\) \(\chi_{5610}(2077,\cdot)\) \(\chi_{5610}(2137,\cdot)\) \(\chi_{5610}(2203,\cdot)\) \(\chi_{5610}(2743,\cdot)\) \(\chi_{5610}(2863,\cdot)\) \(\chi_{5610}(2887,\cdot)\) \(\chi_{5610}(3337,\cdot)\) \(\chi_{5610}(3397,\cdot)\) \(\chi_{5610}(3667,\cdot)\) \(\chi_{5610}(3733,\cdot)\) \(\chi_{5610}(3853,\cdot)\) \(\chi_{5610}(4117,\cdot)\) \(\chi_{5610}(4273,\cdot)\) \(\chi_{5610}(4393,\cdot)\) \(\chi_{5610}(4783,\cdot)\) \(\chi_{5610}(4867,\cdot)\) \(\chi_{5610}(5197,\cdot)\) \(\chi_{5610}(5263,\cdot)\) \(\chi_{5610}(5377,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1871,3367,1531,3301)\) → \((1,-i,e\left(\frac{3}{5}\right),e\left(\frac{3}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(163, a) \) \(1\)\(1\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{11}{80}\right)\)\(e\left(\frac{23}{80}\right)\)\(e\left(\frac{11}{80}\right)\)\(e\left(\frac{69}{80}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(163,a) \;\) at \(\;a = \) e.g. 2