Properties

Label 8008.375
Modulus $8008$
Conductor $364$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,0,8,0,7]))
 
pari: [g,chi] = znchar(Mod(375,8008))
 

Basic properties

Modulus: \(8008\)
Conductor: \(364\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{364}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8008.ie

\(\chi_{8008}(375,\cdot)\) \(\chi_{8008}(1255,\cdot)\) \(\chi_{8008}(4071,\cdot)\) \(\chi_{8008}(4335,\cdot)\)

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: 12.12.42317611137863236145152.2

Values on generators

\((6007,4005,3433,4369,4929)\) → \((-1,1,e\left(\frac{2}{3}\right),1,e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 8008 }(375, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8008 }(375,a) \;\) at \(\;a = \) e.g. 2