Properties

Label 4031.136
Modulus $4031$
Conductor $4031$
Order $483$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, base_ring=CyclotomicField(966))
 
M = H._module
 
chi = DirichletCharacter(H, M([828,770]))
 
pari: [g,chi] = znchar(Mod(136,4031))
 

Basic properties

Modulus: \(4031\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(483\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4031.bo

\(\chi_{4031}(7,\cdot)\) \(\chi_{4031}(16,\cdot)\) \(\chi_{4031}(20,\cdot)\) \(\chi_{4031}(24,\cdot)\) \(\chi_{4031}(25,\cdot)\) \(\chi_{4031}(49,\cdot)\) \(\chi_{4031}(54,\cdot)\) \(\chi_{4031}(78,\cdot)\) \(\chi_{4031}(81,\cdot)\) \(\chi_{4031}(83,\cdot)\) \(\chi_{4031}(107,\cdot)\) \(\chi_{4031}(136,\cdot)\) \(\chi_{4031}(152,\cdot)\) \(\chi_{4031}(168,\cdot)\) \(\chi_{4031}(169,\cdot)\) \(\chi_{4031}(170,\cdot)\) \(\chi_{4031}(190,\cdot)\) \(\chi_{4031}(210,\cdot)\) \(\chi_{4031}(228,\cdot)\) \(\chi_{4031}(252,\cdot)\) \(\chi_{4031}(256,\cdot)\) \(\chi_{4031}(257,\cdot)\) \(\chi_{4031}(285,\cdot)\) \(\chi_{4031}(306,\cdot)\) \(\chi_{4031}(313,\cdot)\) \(\chi_{4031}(315,\cdot)\) \(\chi_{4031}(344,\cdot)\) \(\chi_{4031}(364,\cdot)\) \(\chi_{4031}(400,\cdot)\) \(\chi_{4031}(402,\cdot)\) ...

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: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 483 polynomial (not computed)

Values on generators

\((3337,697)\) → \((e\left(\frac{6}{7}\right),e\left(\frac{55}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4031 }(136, a) \) \(1\)\(1\)\(e\left(\frac{316}{483}\right)\)\(e\left(\frac{467}{483}\right)\)\(e\left(\frac{149}{483}\right)\)\(e\left(\frac{197}{483}\right)\)\(e\left(\frac{100}{161}\right)\)\(e\left(\frac{68}{483}\right)\)\(e\left(\frac{155}{161}\right)\)\(e\left(\frac{451}{483}\right)\)\(e\left(\frac{10}{161}\right)\)\(e\left(\frac{4}{483}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4031 }(136,a) \;\) at \(\;a = \) e.g. 2