Properties

Label 2695.16
Modulus $2695$
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(2695, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,100,84]))
 
pari: [g,chi] = znchar(Mod(16,2695))
 

Basic properties

Modulus: \(2695\)
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 \(\chi_{539}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2695.dc

\(\chi_{2695}(16,\cdot)\) \(\chi_{2695}(81,\cdot)\) \(\chi_{2695}(86,\cdot)\) \(\chi_{2695}(191,\cdot)\) \(\chi_{2695}(256,\cdot)\) \(\chi_{2695}(291,\cdot)\) \(\chi_{2695}(366,\cdot)\) \(\chi_{2695}(401,\cdot)\) \(\chi_{2695}(466,\cdot)\) \(\chi_{2695}(576,\cdot)\) \(\chi_{2695}(641,\cdot)\) \(\chi_{2695}(676,\cdot)\) \(\chi_{2695}(746,\cdot)\) \(\chi_{2695}(751,\cdot)\) \(\chi_{2695}(786,\cdot)\) \(\chi_{2695}(856,\cdot)\) \(\chi_{2695}(1026,\cdot)\) \(\chi_{2695}(1061,\cdot)\) \(\chi_{2695}(1131,\cdot)\) \(\chi_{2695}(1136,\cdot)\) \(\chi_{2695}(1171,\cdot)\) \(\chi_{2695}(1236,\cdot)\) \(\chi_{2695}(1241,\cdot)\) \(\chi_{2695}(1346,\cdot)\) \(\chi_{2695}(1411,\cdot)\) \(\chi_{2695}(1446,\cdot)\) \(\chi_{2695}(1516,\cdot)\) \(\chi_{2695}(1521,\cdot)\) \(\chi_{2695}(1556,\cdot)\) \(\chi_{2695}(1621,\cdot)\) ...

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

\((2157,1816,981)\) → \((1,e\left(\frac{10}{21}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 2695 }(16, a) \) \(1\)\(1\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{71}{105}\right)\)\(e\left(\frac{59}{105}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{37}{105}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{13}{105}\right)\)\(e\left(\frac{53}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2695 }(16,a) \;\) at \(\;a = \) e.g. 2