Properties

Label 8034.7219
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([0,17,78]))
 
pari: [g,chi] = znchar(Mod(7219,8034))
 

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 \(\chi_{1339}(524,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.dx

\(\chi_{8034}(751,\cdot)\) \(\chi_{8034}(1141,\cdot)\) \(\chi_{8034}(1297,\cdot)\) \(\chi_{8034}(1369,\cdot)\) \(\chi_{8034}(1609,\cdot)\) \(\chi_{8034}(1759,\cdot)\) \(\chi_{8034}(1765,\cdot)\) \(\chi_{8034}(1915,\cdot)\) \(\chi_{8034}(2227,\cdot)\) \(\chi_{8034}(2383,\cdot)\) \(\chi_{8034}(2701,\cdot)\) \(\chi_{8034}(2857,\cdot)\) \(\chi_{8034}(3169,\cdot)\) \(\chi_{8034}(3319,\cdot)\) \(\chi_{8034}(3475,\cdot)\) \(\chi_{8034}(3787,\cdot)\) \(\chi_{8034}(4339,\cdot)\) \(\chi_{8034}(4495,\cdot)\) \(\chi_{8034}(4957,\cdot)\) \(\chi_{8034}(5113,\cdot)\) \(\chi_{8034}(5119,\cdot)\) \(\chi_{8034}(5353,\cdot)\) \(\chi_{8034}(5737,\cdot)\) \(\chi_{8034}(5971,\cdot)\) \(\chi_{8034}(6055,\cdot)\) \(\chi_{8034}(6523,\cdot)\) \(\chi_{8034}(6601,\cdot)\) \(\chi_{8034}(6673,\cdot)\) \(\chi_{8034}(7141,\cdot)\) \(\chi_{8034}(7219,\cdot)\) ...

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

\((5357,1237,5773)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{13}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(7219, a) \) \(1\)\(1\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{91}{102}\right)\)\(e\left(\frac{83}{102}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{1}{102}\right)\)\(e\left(\frac{1}{51}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{8}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(7219,a) \;\) at \(\;a = \) e.g. 2