Properties

Label 8880.307
Modulus $8880$
Conductor $2960$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8880, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,9,0,3,10]))
 
pari: [g,chi] = znchar(Mod(307,8880))
 

Basic properties

Modulus: \(8880\)
Conductor: \(2960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2960}(307,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8880.jn

\(\chi_{8880}(307,\cdot)\) \(\chi_{8880}(1507,\cdot)\) \(\chi_{8880}(6523,\cdot)\) \(\chi_{8880}(7723,\cdot)\)

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: Number field defined by a degree 12 polynomial

Values on generators

\((5551,6661,5921,1777,8401)\) → \((-1,-i,1,i,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 8880 }(307, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(i\)\(i\)\(1\)\(e\left(\frac{1}{6}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8880 }(307,a) \;\) at \(\;a = \) e.g. 2