Properties

Label 8034.1979
Modulus $8034$
Conductor $4017$
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([51,34,3]))
 
pari: [g,chi] = znchar(Mod(1979,8034))
 

Basic properties

Modulus: \(8034\)
Conductor: \(4017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4017}(1979,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.da

\(\chi_{8034}(113,\cdot)\) \(\chi_{8034}(731,\cdot)\) \(\chi_{8034}(815,\cdot)\) \(\chi_{8034}(893,\cdot)\) \(\chi_{8034}(1361,\cdot)\) \(\chi_{8034}(1433,\cdot)\) \(\chi_{8034}(1511,\cdot)\) \(\chi_{8034}(1979,\cdot)\) \(\chi_{8034}(2063,\cdot)\) \(\chi_{8034}(2297,\cdot)\) \(\chi_{8034}(2681,\cdot)\) \(\chi_{8034}(2915,\cdot)\) \(\chi_{8034}(2921,\cdot)\) \(\chi_{8034}(3077,\cdot)\) \(\chi_{8034}(3539,\cdot)\) \(\chi_{8034}(3695,\cdot)\) \(\chi_{8034}(4247,\cdot)\) \(\chi_{8034}(4559,\cdot)\) \(\chi_{8034}(4715,\cdot)\) \(\chi_{8034}(4865,\cdot)\) \(\chi_{8034}(5177,\cdot)\) \(\chi_{8034}(5333,\cdot)\) \(\chi_{8034}(5651,\cdot)\) \(\chi_{8034}(5807,\cdot)\) \(\chi_{8034}(6119,\cdot)\) \(\chi_{8034}(6269,\cdot)\) \(\chi_{8034}(6275,\cdot)\) \(\chi_{8034}(6425,\cdot)\) \(\chi_{8034}(6665,\cdot)\) \(\chi_{8034}(6737,\cdot)\) ...

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

\((5357,1237,5773)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(1979, a) \) \(1\)\(1\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{40}{51}\right)\)\(e\left(\frac{32}{51}\right)\)\(e\left(\frac{23}{102}\right)\)\(e\left(\frac{1}{51}\right)\)\(e\left(\frac{55}{102}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{37}{102}\right)\)\(e\left(\frac{23}{34}\right)\)\(e\left(\frac{16}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(1979,a) \;\) at \(\;a = \) e.g. 2