Properties

Label 6008.1201
Modulus $6008$
Conductor $751$
Order $25$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,26]))
 
pari: [g,chi] = znchar(Mod(1201,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{751}(450,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6008.z

\(\chi_{6008}(193,\cdot)\) \(\chi_{6008}(481,\cdot)\) \(\chi_{6008}(913,\cdot)\) \(\chi_{6008}(1201,\cdot)\) \(\chi_{6008}(1217,\cdot)\) \(\chi_{6008}(1417,\cdot)\) \(\chi_{6008}(1553,\cdot)\) \(\chi_{6008}(1673,\cdot)\) \(\chi_{6008}(1681,\cdot)\) \(\chi_{6008}(1977,\cdot)\) \(\chi_{6008}(2001,\cdot)\) \(\chi_{6008}(2601,\cdot)\) \(\chi_{6008}(2673,\cdot)\) \(\chi_{6008}(2809,\cdot)\) \(\chi_{6008}(3057,\cdot)\) \(\chi_{6008}(3121,\cdot)\) \(\chi_{6008}(3329,\cdot)\) \(\chi_{6008}(3489,\cdot)\) \(\chi_{6008}(4465,\cdot)\) \(\chi_{6008}(5209,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 25 polynomial

Values on generators

\((1503,3005,1505)\) → \((1,1,e\left(\frac{13}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(1201, a) \) \(1\)\(1\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{1}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(1201,a) \;\) at \(\;a = \) e.g. 2