Properties

Label 8034.3671
Modulus $8034$
Conductor $4017$
Order $68$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(68))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,51,28]))
 
pari: [g,chi] = znchar(Mod(3671,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(68\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4017}(3671,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cv

\(\chi_{8034}(203,\cdot)\) \(\chi_{8034}(317,\cdot)\) \(\chi_{8034}(473,\cdot)\) \(\chi_{8034}(785,\cdot)\) \(\chi_{8034}(905,\cdot)\) \(\chi_{8034}(941,\cdot)\) \(\chi_{8034}(1373,\cdot)\) \(\chi_{8034}(1451,\cdot)\) \(\chi_{8034}(1877,\cdot)\) \(\chi_{8034}(2033,\cdot)\) \(\chi_{8034}(2153,\cdot)\) \(\chi_{8034}(2345,\cdot)\) \(\chi_{8034}(3515,\cdot)\) \(\chi_{8034}(3635,\cdot)\) \(\chi_{8034}(3671,\cdot)\) \(\chi_{8034}(4025,\cdot)\) \(\chi_{8034}(4181,\cdot)\) \(\chi_{8034}(4295,\cdot)\) \(\chi_{8034}(4493,\cdot)\) \(\chi_{8034}(4529,\cdot)\) \(\chi_{8034}(4649,\cdot)\) \(\chi_{8034}(5231,\cdot)\) \(\chi_{8034}(5585,\cdot)\) \(\chi_{8034}(5699,\cdot)\) \(\chi_{8034}(5741,\cdot)\) \(\chi_{8034}(5777,\cdot)\) \(\chi_{8034}(6053,\cdot)\) \(\chi_{8034}(6479,\cdot)\) \(\chi_{8034}(7223,\cdot)\) \(\chi_{8034}(7379,\cdot)\) ...

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: $\Q(\zeta_{68})$
Fixed field: Number field defined by a degree 68 polynomial

Values on generators

\((5357,1237,5773)\) → \((-1,-i,e\left(\frac{7}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(3671, a) \) \(1\)\(1\)\(e\left(\frac{45}{68}\right)\)\(e\left(\frac{61}{68}\right)\)\(e\left(\frac{59}{68}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{47}{68}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{31}{34}\right)\)\(e\left(\frac{15}{68}\right)\)\(e\left(\frac{19}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(3671,a) \;\) at \(\;a = \) e.g. 2