Properties

Label 8035.6
Modulus $8035$
Conductor $1607$
Order $803$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8035, base_ring=CyclotomicField(1606))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,250]))
 
pari: [g,chi] = znchar(Mod(6,8035))
 

Basic properties

Modulus: \(8035\)
Conductor: \(1607\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(803\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1607}(6,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8035.s

\(\chi_{8035}(6,\cdot)\) \(\chi_{8035}(16,\cdot)\) \(\chi_{8035}(31,\cdot)\) \(\chi_{8035}(36,\cdot)\) \(\chi_{8035}(41,\cdot)\) \(\chi_{8035}(46,\cdot)\) \(\chi_{8035}(51,\cdot)\) \(\chi_{8035}(81,\cdot)\) \(\chi_{8035}(91,\cdot)\) \(\chi_{8035}(101,\cdot)\) \(\chi_{8035}(106,\cdot)\) \(\chi_{8035}(111,\cdot)\) \(\chi_{8035}(121,\cdot)\) \(\chi_{8035}(136,\cdot)\) \(\chi_{8035}(146,\cdot)\) \(\chi_{8035}(186,\cdot)\) \(\chi_{8035}(196,\cdot)\) \(\chi_{8035}(201,\cdot)\) \(\chi_{8035}(206,\cdot)\) \(\chi_{8035}(211,\cdot)\) \(\chi_{8035}(216,\cdot)\) \(\chi_{8035}(226,\cdot)\) \(\chi_{8035}(231,\cdot)\) \(\chi_{8035}(236,\cdot)\) \(\chi_{8035}(241,\cdot)\) \(\chi_{8035}(246,\cdot)\) \(\chi_{8035}(256,\cdot)\) \(\chi_{8035}(266,\cdot)\) \(\chi_{8035}(276,\cdot)\) \(\chi_{8035}(281,\cdot)\) ...

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: $\Q(\zeta_{803})$
Fixed field: Number field defined by a degree 803 polynomial (not computed)

Values on generators

\((4822,4826)\) → \((1,e\left(\frac{125}{803}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 8035 }(6, a) \) \(1\)\(1\)\(e\left(\frac{366}{803}\right)\)\(e\left(\frac{370}{803}\right)\)\(e\left(\frac{732}{803}\right)\)\(e\left(\frac{736}{803}\right)\)\(e\left(\frac{37}{803}\right)\)\(e\left(\frac{295}{803}\right)\)\(e\left(\frac{740}{803}\right)\)\(e\left(\frac{47}{803}\right)\)\(e\left(\frac{299}{803}\right)\)\(e\left(\frac{731}{803}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8035 }(6,a) \;\) at \(\;a = \) e.g. 2