Properties

Label 5544.79
Modulus $5544$
Conductor $2772$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5544, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,0,20,10,3]))
 
pari: [g,chi] = znchar(Mod(79,5544))
 

Basic properties

Modulus: \(5544\)
Conductor: \(2772\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2772}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5544.hq

\(\chi_{5544}(79,\cdot)\) \(\chi_{5544}(1327,\cdot)\) \(\chi_{5544}(1591,\cdot)\) \(\chi_{5544}(3847,\cdot)\) \(\chi_{5544}(4111,\cdot)\) \(\chi_{5544}(4351,\cdot)\) \(\chi_{5544}(4615,\cdot)\) \(\chi_{5544}(5359,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((4159,2773,4313,1585,2521)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5544 }(79, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(-1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5544 }(79,a) \;\) at \(\;a = \) e.g. 2