Properties

Label 2760.301
Modulus $2760$
Conductor $184$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,0,0,2]))
 
pari: [g,chi] = znchar(Mod(301,2760))
 

Basic properties

Modulus: \(2760\)
Conductor: \(184\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{184}(117,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2760.cj

\(\chi_{2760}(301,\cdot)\) \(\chi_{2760}(541,\cdot)\) \(\chi_{2760}(901,\cdot)\) \(\chi_{2760}(1021,\cdot)\) \(\chi_{2760}(1501,\cdot)\) \(\chi_{2760}(1741,\cdot)\) \(\chi_{2760}(1981,\cdot)\) \(\chi_{2760}(2101,\cdot)\) \(\chi_{2760}(2221,\cdot)\) \(\chi_{2760}(2341,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.22.14741666340843480753092741810452692992.1

Values on generators

\((2071,1381,1841,1657,1201)\) → \((1,-1,1,1,e\left(\frac{1}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2760 }(301, a) \) \(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2760 }(301,a) \;\) at \(\;a = \) e.g. 2