Properties

Label 8036.169
Modulus $8036$
Conductor $2009$
Order $140$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,80,77]))
 
pari: [g,chi] = znchar(Mod(169,8036))
 

Basic properties

Modulus: \(8036\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2009}(169,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8036.dy

\(\chi_{8036}(169,\cdot)\) \(\chi_{8036}(225,\cdot)\) \(\chi_{8036}(449,\cdot)\) \(\chi_{8036}(617,\cdot)\) \(\chi_{8036}(841,\cdot)\) \(\chi_{8036}(869,\cdot)\) \(\chi_{8036}(897,\cdot)\) \(\chi_{8036}(1317,\cdot)\) \(\chi_{8036}(1345,\cdot)\) \(\chi_{8036}(1597,\cdot)\) \(\chi_{8036}(1989,\cdot)\) \(\chi_{8036}(2017,\cdot)\) \(\chi_{8036}(2045,\cdot)\) \(\chi_{8036}(2465,\cdot)\) \(\chi_{8036}(2493,\cdot)\) \(\chi_{8036}(2521,\cdot)\) \(\chi_{8036}(2913,\cdot)\) \(\chi_{8036}(3165,\cdot)\) \(\chi_{8036}(3193,\cdot)\) \(\chi_{8036}(3613,\cdot)\) \(\chi_{8036}(3641,\cdot)\) \(\chi_{8036}(3669,\cdot)\) \(\chi_{8036}(3893,\cdot)\) \(\chi_{8036}(4061,\cdot)\) \(\chi_{8036}(4285,\cdot)\) \(\chi_{8036}(4341,\cdot)\) \(\chi_{8036}(4761,\cdot)\) \(\chi_{8036}(4789,\cdot)\) \(\chi_{8036}(4817,\cdot)\) \(\chi_{8036}(5041,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((4019,493,785)\) → \((1,e\left(\frac{4}{7}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 8036 }(169, a) \) \(1\)\(1\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{71}{140}\right)\)\(e\left(\frac{127}{140}\right)\)\(e\left(\frac{69}{140}\right)\)\(e\left(\frac{61}{140}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{12}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8036 }(169,a) \;\) at \(\;a = \) e.g. 2