Properties

Label 8004.157
Modulus $8004$
Conductor $667$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,30,11]))
 
pari: [g,chi] = znchar(Mod(157,8004))
 

Basic properties

Modulus: \(8004\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(157,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8004.cp

\(\chi_{8004}(157,\cdot)\) \(\chi_{8004}(481,\cdot)\) \(\chi_{8004}(1201,\cdot)\) \(\chi_{8004}(1525,\cdot)\) \(\chi_{8004}(1873,\cdot)\) \(\chi_{8004}(1897,\cdot)\) \(\chi_{8004}(2245,\cdot)\) \(\chi_{8004}(2593,\cdot)\) \(\chi_{8004}(2917,\cdot)\) \(\chi_{8004}(2941,\cdot)\) \(\chi_{8004}(3961,\cdot)\) \(\chi_{8004}(4657,\cdot)\) \(\chi_{8004}(5005,\cdot)\) \(\chi_{8004}(5029,\cdot)\) \(\chi_{8004}(5353,\cdot)\) \(\chi_{8004}(5701,\cdot)\) \(\chi_{8004}(5725,\cdot)\) \(\chi_{8004}(6769,\cdot)\) \(\chi_{8004}(7117,\cdot)\) \(\chi_{8004}(7789,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2829456642779506738660199294300931896594438764890376623447387777021003184630861677846958804464951379972781.1

Values on generators

\((4003,2669,3133,553)\) → \((1,1,e\left(\frac{15}{22}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 8004 }(157, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{3}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8004 }(157,a) \;\) at \(\;a = \) e.g. 2