Properties

Label 3136.17
Modulus $3136$
Conductor $784$
Order $84$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,50]))
 
pari: [g,chi] = znchar(Mod(17,3136))
 

Basic properties

Modulus: \(3136\)
Conductor: \(784\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{784}(605,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3136.cm

\(\chi_{3136}(17,\cdot)\) \(\chi_{3136}(145,\cdot)\) \(\chi_{3136}(241,\cdot)\) \(\chi_{3136}(369,\cdot)\) \(\chi_{3136}(465,\cdot)\) \(\chi_{3136}(593,\cdot)\) \(\chi_{3136}(689,\cdot)\) \(\chi_{3136}(817,\cdot)\) \(\chi_{3136}(1041,\cdot)\) \(\chi_{3136}(1137,\cdot)\) \(\chi_{3136}(1265,\cdot)\) \(\chi_{3136}(1361,\cdot)\) \(\chi_{3136}(1585,\cdot)\) \(\chi_{3136}(1713,\cdot)\) \(\chi_{3136}(1809,\cdot)\) \(\chi_{3136}(1937,\cdot)\) \(\chi_{3136}(2033,\cdot)\) \(\chi_{3136}(2161,\cdot)\) \(\chi_{3136}(2257,\cdot)\) \(\chi_{3136}(2385,\cdot)\) \(\chi_{3136}(2609,\cdot)\) \(\chi_{3136}(2705,\cdot)\) \(\chi_{3136}(2833,\cdot)\) \(\chi_{3136}(2929,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1471,197,1473)\) → \((1,-i,e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3136 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{1}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3136 }(17,a) \;\) at \(\;a = \) e.g. 2