Properties

Label 2695.36
Modulus $2695$
Conductor $539$
Order $35$
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(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,56]))
 
pari: [g,chi] = znchar(Mod(36,2695))
 

Basic properties

Modulus: \(2695\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(36,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2695.cf

\(\chi_{2695}(36,\cdot)\) \(\chi_{2695}(71,\cdot)\) \(\chi_{2695}(141,\cdot)\) \(\chi_{2695}(421,\cdot)\) \(\chi_{2695}(456,\cdot)\) \(\chi_{2695}(526,\cdot)\) \(\chi_{2695}(631,\cdot)\) \(\chi_{2695}(806,\cdot)\) \(\chi_{2695}(841,\cdot)\) \(\chi_{2695}(911,\cdot)\) \(\chi_{2695}(1016,\cdot)\) \(\chi_{2695}(1191,\cdot)\) \(\chi_{2695}(1296,\cdot)\) \(\chi_{2695}(1401,\cdot)\) \(\chi_{2695}(1576,\cdot)\) \(\chi_{2695}(1611,\cdot)\) \(\chi_{2695}(1681,\cdot)\) \(\chi_{2695}(1786,\cdot)\) \(\chi_{2695}(1996,\cdot)\) \(\chi_{2695}(2066,\cdot)\) \(\chi_{2695}(2171,\cdot)\) \(\chi_{2695}(2346,\cdot)\) \(\chi_{2695}(2381,\cdot)\) \(\chi_{2695}(2556,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 35 polynomial

Values on generators

\((2157,1816,981)\) → \((1,e\left(\frac{2}{7}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 2695 }(36, a) \) \(1\)\(1\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{12}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2695 }(36,a) \;\) at \(\;a = \) e.g. 2