Properties

Label 3872.2591
Modulus $3872$
Conductor $484$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3872, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,0,39]))
 
pari: [g,chi] = znchar(Mod(2591,3872))
 

Basic properties

Modulus: \(3872\)
Conductor: \(484\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{484}(171,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3872.cd

\(\chi_{3872}(63,\cdot)\) \(\chi_{3872}(95,\cdot)\) \(\chi_{3872}(127,\cdot)\) \(\chi_{3872}(255,\cdot)\) \(\chi_{3872}(415,\cdot)\) \(\chi_{3872}(447,\cdot)\) \(\chi_{3872}(479,\cdot)\) \(\chi_{3872}(607,\cdot)\) \(\chi_{3872}(767,\cdot)\) \(\chi_{3872}(799,\cdot)\) \(\chi_{3872}(831,\cdot)\) \(\chi_{3872}(1119,\cdot)\) \(\chi_{3872}(1151,\cdot)\) \(\chi_{3872}(1311,\cdot)\) \(\chi_{3872}(1471,\cdot)\) \(\chi_{3872}(1503,\cdot)\) \(\chi_{3872}(1535,\cdot)\) \(\chi_{3872}(1663,\cdot)\) \(\chi_{3872}(1823,\cdot)\) \(\chi_{3872}(1887,\cdot)\) \(\chi_{3872}(2015,\cdot)\) \(\chi_{3872}(2207,\cdot)\) \(\chi_{3872}(2239,\cdot)\) \(\chi_{3872}(2367,\cdot)\) \(\chi_{3872}(2527,\cdot)\) \(\chi_{3872}(2559,\cdot)\) \(\chi_{3872}(2591,\cdot)\) \(\chi_{3872}(2719,\cdot)\) \(\chi_{3872}(2879,\cdot)\) \(\chi_{3872}(2911,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((1695,485,2785)\) → \((-1,1,e\left(\frac{39}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 3872 }(2591, a) \) \(1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3872 }(2591,a) \;\) at \(\;a = \) e.g. 2