Properties

Label 8034.17
Modulus $8034$
Conductor $4017$
Order $102$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([51,17,70]))
 
pari: [g,chi] = znchar(Mod(17,8034))
 

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 \(\chi_{4017}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8034.do

\(\chi_{8034}(17,\cdot)\) \(\chi_{8034}(335,\cdot)\) \(\chi_{8034}(407,\cdot)\) \(\chi_{8034}(647,\cdot)\) \(\chi_{8034}(1193,\cdot)\) \(\chi_{8034}(1343,\cdot)\) \(\chi_{8034}(1421,\cdot)\) \(\chi_{8034}(1577,\cdot)\) \(\chi_{8034}(1655,\cdot)\) \(\chi_{8034}(1895,\cdot)\) \(\chi_{8034}(1973,\cdot)\) \(\chi_{8034}(2123,\cdot)\) \(\chi_{8034}(2201,\cdot)\) \(\chi_{8034}(2285,\cdot)\) \(\chi_{8034}(2357,\cdot)\) \(\chi_{8034}(2363,\cdot)\) \(\chi_{8034}(2909,\cdot)\) \(\chi_{8034}(3221,\cdot)\) \(\chi_{8034}(3449,\cdot)\) \(\chi_{8034}(3767,\cdot)\) \(\chi_{8034}(4703,\cdot)\) \(\chi_{8034}(4859,\cdot)\) \(\chi_{8034}(5165,\cdot)\) \(\chi_{8034}(5405,\cdot)\) \(\chi_{8034}(5873,\cdot)\) \(\chi_{8034}(6029,\cdot)\) \(\chi_{8034}(6263,\cdot)\) \(\chi_{8034}(6335,\cdot)\) \(\chi_{8034}(6959,\cdot)\) \(\chi_{8034}(7037,\cdot)\) ...

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

\((5357,1237,5773)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{35}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8034 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{59}{102}\right)\)\(e\left(\frac{9}{17}\right)\)\(e\left(\frac{89}{102}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{65}{102}\right)\)\(e\left(\frac{19}{51}\right)\)\(e\left(\frac{19}{102}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{9}{34}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8034 }(17,a) \;\) at \(\;a = \) e.g. 2