Properties

Label 1216.217
Modulus $1216$
Conductor $608$
Order $24$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,4]))
 
pari: [g,chi] = znchar(Mod(217,1216))
 

Basic properties

Modulus: \(1216\)
Conductor: \(608\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{608}(293,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1216.bp

\(\chi_{1216}(217,\cdot)\) \(\chi_{1216}(297,\cdot)\) \(\chi_{1216}(521,\cdot)\) \(\chi_{1216}(601,\cdot)\) \(\chi_{1216}(825,\cdot)\) \(\chi_{1216}(905,\cdot)\) \(\chi_{1216}(1129,\cdot)\) \(\chi_{1216}(1209,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.0.372273065750166762311522006998539295982015765641428992.1

Values on generators

\((191,837,705)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 1216 }(217, a) \) \(-1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1216 }(217,a) \;\) at \(\;a = \) e.g. 2