Properties

Label 1216.737
Modulus $1216$
Conductor $152$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,11]))
 
pari: [g,chi] = znchar(Mod(737,1216))
 

Basic properties

Modulus: \(1216\)
Conductor: \(152\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{152}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1216.bi

\(\chi_{1216}(33,\cdot)\) \(\chi_{1216}(97,\cdot)\) \(\chi_{1216}(545,\cdot)\) \(\chi_{1216}(737,\cdot)\) \(\chi_{1216}(801,\cdot)\) \(\chi_{1216}(865,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.0.735565072612935262326166126592.1

Values on generators

\((191,837,705)\) → \((1,-1,e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 1216 }(737, a) \) \(-1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1216 }(737,a) \;\) at \(\;a = \) e.g. 2