Properties

Label 8034.4663
Modulus $8034$
Conductor $1339$
Order $51$
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(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,68,92]))
 
pari: [g,chi] = znchar(Mod(4663,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(646,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.cr

\(\chi_{8034}(367,\cdot)\) \(\chi_{8034}(445,\cdot)\) \(\chi_{8034}(757,\cdot)\) \(\chi_{8034}(1225,\cdot)\) \(\chi_{8034}(1459,\cdot)\) \(\chi_{8034}(1777,\cdot)\) \(\chi_{8034}(1849,\cdot)\) \(\chi_{8034}(2089,\cdot)\) \(\chi_{8034}(2635,\cdot)\) \(\chi_{8034}(2785,\cdot)\) \(\chi_{8034}(2863,\cdot)\) \(\chi_{8034}(3019,\cdot)\) \(\chi_{8034}(3097,\cdot)\) \(\chi_{8034}(3337,\cdot)\) \(\chi_{8034}(3415,\cdot)\) \(\chi_{8034}(3565,\cdot)\) \(\chi_{8034}(3643,\cdot)\) \(\chi_{8034}(3727,\cdot)\) \(\chi_{8034}(3799,\cdot)\) \(\chi_{8034}(3805,\cdot)\) \(\chi_{8034}(4351,\cdot)\) \(\chi_{8034}(4663,\cdot)\) \(\chi_{8034}(4891,\cdot)\) \(\chi_{8034}(5209,\cdot)\) \(\chi_{8034}(6145,\cdot)\) \(\chi_{8034}(6301,\cdot)\) \(\chi_{8034}(6607,\cdot)\) \(\chi_{8034}(6847,\cdot)\) \(\chi_{8034}(7315,\cdot)\) \(\chi_{8034}(7471,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 51 polynomial

Values on generators

\((5357,1237,5773)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{46}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(4663, a) \) \(1\)\(1\)\(e\left(\frac{46}{51}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{41}{51}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{43}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(4663,a) \;\) at \(\;a = \) e.g. 2