Properties

Label 8034.1663
Modulus $8034$
Conductor $1339$
Order $102$
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([0,51,40]))
 
pari: [g,chi] = znchar(Mod(1663,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(324,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.de

\(\chi_{8034}(25,\cdot)\) \(\chi_{8034}(337,\cdot)\) \(\chi_{8034}(883,\cdot)\) \(\chi_{8034}(1663,\cdot)\) \(\chi_{8034}(1819,\cdot)\) \(\chi_{8034}(1975,\cdot)\) \(\chi_{8034}(2521,\cdot)\) \(\chi_{8034}(2833,\cdot)\) \(\chi_{8034}(2989,\cdot)\) \(\chi_{8034}(3145,\cdot)\) \(\chi_{8034}(3379,\cdot)\) \(\chi_{8034}(3457,\cdot)\) \(\chi_{8034}(3535,\cdot)\) \(\chi_{8034}(3847,\cdot)\) \(\chi_{8034}(4315,\cdot)\) \(\chi_{8034}(4549,\cdot)\) \(\chi_{8034}(4939,\cdot)\) \(\chi_{8034}(5485,\cdot)\) \(\chi_{8034}(5797,\cdot)\) \(\chi_{8034}(5875,\cdot)\) \(\chi_{8034}(5953,\cdot)\) \(\chi_{8034}(6109,\cdot)\) \(\chi_{8034}(6187,\cdot)\) \(\chi_{8034}(6343,\cdot)\) \(\chi_{8034}(6655,\cdot)\) \(\chi_{8034}(6733,\cdot)\) \(\chi_{8034}(6889,\cdot)\) \(\chi_{8034}(7045,\cdot)\) \(\chi_{8034}(7123,\cdot)\) \(\chi_{8034}(7435,\cdot)\) ...

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 102 polynomial (not computed)

Values on generators

\((5357,1237,5773)\) → \((1,-1,e\left(\frac{20}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(1663, a) \) \(1\)\(1\)\(e\left(\frac{91}{102}\right)\)\(e\left(\frac{7}{102}\right)\)\(e\left(\frac{43}{102}\right)\)\(e\left(\frac{23}{51}\right)\)\(e\left(\frac{89}{102}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{37}{51}\right)\)\(e\left(\frac{29}{34}\right)\)\(e\left(\frac{49}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(1663,a) \;\) at \(\;a = \) e.g. 2