Properties

Label 1560.301
Modulus $1560$
Conductor $104$
Order $12$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,6,0,0,1]))
 
pari: [g,chi] = znchar(Mod(301,1560))
 

Basic properties

Modulus: \(1560\)
Conductor: \(104\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{104}(93,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1560.fm

\(\chi_{1560}(301,\cdot)\) \(\chi_{1560}(661,\cdot)\) \(\chi_{1560}(1021,\cdot)\) \(\chi_{1560}(1501,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.469804094334435328.1

Values on generators

\((391,781,521,937,1081)\) → \((1,-1,1,1,e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1560 }(301, a) \) \(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1560 }(301,a) \;\) at \(\;a = \) e.g. 2