Properties

Label 4011.991
Modulus $4011$
Conductor $1337$
Order $57$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,76,78]))
 
pari: [g,chi] = znchar(Mod(991,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1337}(991,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.bo

\(\chi_{4011}(25,\cdot)\) \(\chi_{4011}(121,\cdot)\) \(\chi_{4011}(298,\cdot)\) \(\chi_{4011}(562,\cdot)\) \(\chi_{4011}(625,\cdot)\) \(\chi_{4011}(709,\cdot)\) \(\chi_{4011}(961,\cdot)\) \(\chi_{4011}(991,\cdot)\) \(\chi_{4011}(1024,\cdot)\) \(\chi_{4011}(1108,\cdot)\) \(\chi_{4011}(1171,\cdot)\) \(\chi_{4011}(1306,\cdot)\) \(\chi_{4011}(1369,\cdot)\) \(\chi_{4011}(1444,\cdot)\) \(\chi_{4011}(1558,\cdot)\) \(\chi_{4011}(1705,\cdot)\) \(\chi_{4011}(1873,\cdot)\) \(\chi_{4011}(1915,\cdot)\) \(\chi_{4011}(2137,\cdot)\) \(\chi_{4011}(2251,\cdot)\) \(\chi_{4011}(2452,\cdot)\) \(\chi_{4011}(2515,\cdot)\) \(\chi_{4011}(2608,\cdot)\) \(\chi_{4011}(2704,\cdot)\) \(\chi_{4011}(2851,\cdot)\) \(\chi_{4011}(2986,\cdot)\) \(\chi_{4011}(3019,\cdot)\) \(\chi_{4011}(3061,\cdot)\) \(\chi_{4011}(3397,\cdot)\) \(\chi_{4011}(3427,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\((2675,2866,2311)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{13}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(991, a) \) \(1\)\(1\)\(e\left(\frac{25}{57}\right)\)\(e\left(\frac{50}{57}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{56}{57}\right)\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{41}{57}\right)\)\(e\left(\frac{1}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(991,a) \;\) at \(\;a = \) e.g. 2