Properties

Label 8034.5461
Modulus $8034$
Conductor $103$
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([0,0,44]))
 
pari: [g,chi] = znchar(Mod(5461,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.ct

\(\chi_{8034}(235,\cdot)\) \(\chi_{8034}(313,\cdot)\) \(\chi_{8034}(391,\cdot)\) \(\chi_{8034}(547,\cdot)\) \(\chi_{8034}(625,\cdot)\) \(\chi_{8034}(781,\cdot)\) \(\chi_{8034}(1093,\cdot)\) \(\chi_{8034}(1171,\cdot)\) \(\chi_{8034}(1327,\cdot)\) \(\chi_{8034}(1483,\cdot)\) \(\chi_{8034}(1561,\cdot)\) \(\chi_{8034}(1873,\cdot)\) \(\chi_{8034}(1951,\cdot)\) \(\chi_{8034}(2419,\cdot)\) \(\chi_{8034}(2497,\cdot)\) \(\chi_{8034}(2809,\cdot)\) \(\chi_{8034}(3355,\cdot)\) \(\chi_{8034}(4135,\cdot)\) \(\chi_{8034}(4291,\cdot)\) \(\chi_{8034}(4447,\cdot)\) \(\chi_{8034}(4993,\cdot)\) \(\chi_{8034}(5305,\cdot)\) \(\chi_{8034}(5461,\cdot)\) \(\chi_{8034}(5617,\cdot)\) \(\chi_{8034}(5851,\cdot)\) \(\chi_{8034}(5929,\cdot)\) \(\chi_{8034}(6007,\cdot)\) \(\chi_{8034}(6319,\cdot)\) \(\chi_{8034}(6787,\cdot)\) \(\chi_{8034}(7021,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,1,e\left(\frac{22}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(5461, a) \) \(1\)\(1\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{37}{51}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{26}{51}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{5}{51}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{8}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(5461,a) \;\) at \(\;a = \) e.g. 2