Properties

Label 4029.967
Modulus $4029$
Conductor $1343$
Order $312$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(312))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,117,128]))
 
pari: [g,chi] = znchar(Mod(967,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(1343\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(312\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1343}(967,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4029.cx

\(\chi_{4029}(19,\cdot)\) \(\chi_{4029}(25,\cdot)\) \(\chi_{4029}(49,\cdot)\) \(\chi_{4029}(76,\cdot)\) \(\chi_{4029}(121,\cdot)\) \(\chi_{4029}(151,\cdot)\) \(\chi_{4029}(178,\cdot)\) \(\chi_{4029}(202,\cdot)\) \(\chi_{4029}(253,\cdot)\) \(\chi_{4029}(325,\cdot)\) \(\chi_{4029}(400,\cdot)\) \(\chi_{4029}(406,\cdot)\) \(\chi_{4029}(427,\cdot)\) \(\chi_{4029}(478,\cdot)\) \(\chi_{4029}(604,\cdot)\) \(\chi_{4029}(637,\cdot)\) \(\chi_{4029}(682,\cdot)\) \(\chi_{4029}(784,\cdot)\) \(\chi_{4029}(835,\cdot)\) \(\chi_{4029}(841,\cdot)\) \(\chi_{4029}(961,\cdot)\) \(\chi_{4029}(967,\cdot)\) \(\chi_{4029}(988,\cdot)\) \(\chi_{4029}(1063,\cdot)\) \(\chi_{4029}(1069,\cdot)\) \(\chi_{4029}(1198,\cdot)\) \(\chi_{4029}(1216,\cdot)\) \(\chi_{4029}(1273,\cdot)\) \(\chi_{4029}(1300,\cdot)\) \(\chi_{4029}(1345,\cdot)\) ...

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: $\Q(\zeta_{312})$
Fixed field: Number field defined by a degree 312 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{16}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(967, a) \) \(1\)\(1\)\(e\left(\frac{139}{156}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{97}{312}\right)\)\(e\left(\frac{271}{312}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{21}{104}\right)\)\(e\left(\frac{163}{312}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{79}{104}\right)\)\(e\left(\frac{22}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(967,a) \;\) at \(\;a = \) e.g. 2