Properties

Label 5520.4493
Modulus $5520$
Conductor $5520$
Order $44$
Real no
Primitive yes
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([0,33,22,33,12]))
 
pari: [g,chi] = znchar(Mod(4493,5520))
 

Basic properties

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

Galois orbit 5520.fi

\(\chi_{5520}(173,\cdot)\) \(\chi_{5520}(197,\cdot)\) \(\chi_{5520}(653,\cdot)\) \(\chi_{5520}(1133,\cdot)\) \(\chi_{5520}(1373,\cdot)\) \(\chi_{5520}(1613,\cdot)\) \(\chi_{5520}(1637,\cdot)\) \(\chi_{5520}(1853,\cdot)\) \(\chi_{5520}(2837,\cdot)\) \(\chi_{5520}(3077,\cdot)\) \(\chi_{5520}(3293,\cdot)\) \(\chi_{5520}(3797,\cdot)\) \(\chi_{5520}(4037,\cdot)\) \(\chi_{5520}(4493,\cdot)\) \(\chi_{5520}(4517,\cdot)\) \(\chi_{5520}(4733,\cdot)\) \(\chi_{5520}(4997,\cdot)\) \(\chi_{5520}(5237,\cdot)\) \(\chi_{5520}(5453,\cdot)\) \(\chi_{5520}(5477,\cdot)\)

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

\((4831,1381,1841,4417,1201)\) → \((1,-i,-1,-i,e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5520 }(4493, a) \) \(1\)\(1\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{4}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5520 }(4493,a) \;\) at \(\;a = \) e.g. 2