Properties

Label 1078.291
Modulus $1078$
Conductor $539$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,84]))
 
pari: [g,chi] = znchar(Mod(291,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(291,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1078.bc

\(\chi_{1078}(9,\cdot)\) \(\chi_{1078}(25,\cdot)\) \(\chi_{1078}(37,\cdot)\) \(\chi_{1078}(53,\cdot)\) \(\chi_{1078}(81,\cdot)\) \(\chi_{1078}(93,\cdot)\) \(\chi_{1078}(135,\cdot)\) \(\chi_{1078}(137,\cdot)\) \(\chi_{1078}(163,\cdot)\) \(\chi_{1078}(179,\cdot)\) \(\chi_{1078}(191,\cdot)\) \(\chi_{1078}(207,\cdot)\) \(\chi_{1078}(235,\cdot)\) \(\chi_{1078}(247,\cdot)\) \(\chi_{1078}(289,\cdot)\) \(\chi_{1078}(291,\cdot)\) \(\chi_{1078}(317,\cdot)\) \(\chi_{1078}(333,\cdot)\) \(\chi_{1078}(345,\cdot)\) \(\chi_{1078}(389,\cdot)\) \(\chi_{1078}(401,\cdot)\) \(\chi_{1078}(443,\cdot)\) \(\chi_{1078}(445,\cdot)\) \(\chi_{1078}(487,\cdot)\) \(\chi_{1078}(499,\cdot)\) \(\chi_{1078}(515,\cdot)\) \(\chi_{1078}(543,\cdot)\) \(\chi_{1078}(555,\cdot)\) \(\chi_{1078}(597,\cdot)\) \(\chi_{1078}(599,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((199,981)\) → \((e\left(\frac{11}{21}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1078 }(291, a) \) \(1\)\(1\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{83}{105}\right)\)\(e\left(\frac{47}{105}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{73}{105}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{61}{105}\right)\)\(e\left(\frac{6}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(291,a) \;\) at \(\;a = \) e.g. 2