Properties

Label 5520.4451
Modulus $5520$
Conductor $1104$
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(5520, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,33,22,0,40]))
 
pari: [g,chi] = znchar(Mod(4451,5520))
 

Basic properties

Modulus: \(5520\)
Conductor: \(1104\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1104}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5520.ep

\(\chi_{5520}(131,\cdot)\) \(\chi_{5520}(371,\cdot)\) \(\chi_{5520}(491,\cdot)\) \(\chi_{5520}(611,\cdot)\) \(\chi_{5520}(731,\cdot)\) \(\chi_{5520}(1451,\cdot)\) \(\chi_{5520}(1691,\cdot)\) \(\chi_{5520}(2051,\cdot)\) \(\chi_{5520}(2171,\cdot)\) \(\chi_{5520}(2651,\cdot)\) \(\chi_{5520}(2891,\cdot)\) \(\chi_{5520}(3131,\cdot)\) \(\chi_{5520}(3251,\cdot)\) \(\chi_{5520}(3371,\cdot)\) \(\chi_{5520}(3491,\cdot)\) \(\chi_{5520}(4211,\cdot)\) \(\chi_{5520}(4451,\cdot)\) \(\chi_{5520}(4811,\cdot)\) \(\chi_{5520}(4931,\cdot)\) \(\chi_{5520}(5411,\cdot)\)

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

\((4831,1381,1841,4417,1201)\) → \((-1,-i,-1,1,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5520 }(4451, a) \) \(1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{35}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5520 }(4451,a) \;\) at \(\;a = \) e.g. 2