Properties

Label 1648.417
Modulus $1648$
Conductor $103$
Order $102$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1648, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1]))
 
pari: [g,chi] = znchar(Mod(417,1648))
 

Basic properties

Modulus: \(1648\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1648.bo

\(\chi_{1648}(65,\cdot)\) \(\chi_{1648}(177,\cdot)\) \(\chi_{1648}(241,\cdot)\) \(\chi_{1648}(257,\cdot)\) \(\chi_{1648}(273,\cdot)\) \(\chi_{1648}(305,\cdot)\) \(\chi_{1648}(321,\cdot)\) \(\chi_{1648}(353,\cdot)\) \(\chi_{1648}(417,\cdot)\) \(\chi_{1648}(433,\cdot)\) \(\chi_{1648}(465,\cdot)\) \(\chi_{1648}(497,\cdot)\) \(\chi_{1648}(513,\cdot)\) \(\chi_{1648}(577,\cdot)\) \(\chi_{1648}(593,\cdot)\) \(\chi_{1648}(689,\cdot)\) \(\chi_{1648}(705,\cdot)\) \(\chi_{1648}(769,\cdot)\) \(\chi_{1648}(817,\cdot)\) \(\chi_{1648}(1041,\cdot)\) \(\chi_{1648}(1073,\cdot)\) \(\chi_{1648}(1105,\cdot)\) \(\chi_{1648}(1153,\cdot)\) \(\chi_{1648}(1217,\cdot)\) \(\chi_{1648}(1281,\cdot)\) \(\chi_{1648}(1313,\cdot)\) \(\chi_{1648}(1345,\cdot)\) \(\chi_{1648}(1393,\cdot)\) \(\chi_{1648}(1409,\cdot)\) \(\chi_{1648}(1425,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((207,1237,417)\) → \((1,1,e\left(\frac{1}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1648 }(417, a) \) \(-1\)\(1\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{1}{102}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{61}{102}\right)\)\(e\left(\frac{12}{17}\right)\)\(e\left(\frac{20}{51}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{43}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1648 }(417,a) \;\) at \(\;a = \) e.g. 2