Properties

Label 8004.307
Modulus $8004$
Conductor $2668$
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([22,0,12,33]))
 
pari: [g,chi] = znchar(Mod(307,8004))
 

Basic properties

Modulus: \(8004\)
Conductor: \(2668\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2668}(307,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8004.ct

\(\chi_{8004}(307,\cdot)\) \(\chi_{8004}(331,\cdot)\) \(\chi_{8004}(679,\cdot)\) \(\chi_{8004}(1375,\cdot)\) \(\chi_{8004}(2395,\cdot)\) \(\chi_{8004}(2419,\cdot)\) \(\chi_{8004}(2743,\cdot)\) \(\chi_{8004}(3091,\cdot)\) \(\chi_{8004}(3439,\cdot)\) \(\chi_{8004}(3463,\cdot)\) \(\chi_{8004}(3811,\cdot)\) \(\chi_{8004}(4135,\cdot)\) \(\chi_{8004}(4855,\cdot)\) \(\chi_{8004}(5179,\cdot)\) \(\chi_{8004}(5551,\cdot)\) \(\chi_{8004}(6223,\cdot)\) \(\chi_{8004}(6571,\cdot)\) \(\chi_{8004}(7615,\cdot)\) \(\chi_{8004}(7639,\cdot)\) \(\chi_{8004}(7987,\cdot)\)

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: Number field defined by a degree 44 polynomial

Values on generators

\((4003,2669,3133,553)\) → \((-1,1,e\left(\frac{3}{11}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 8004 }(307, a) \) \(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{43}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8004 }(307,a) \;\) at \(\;a = \) e.g. 2