Properties

Label 4011.556
Modulus $4011$
Conductor $1337$
Order $570$
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(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,95,9]))
 
pari: [g,chi] = znchar(Mod(556,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(570\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1337}(556,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.cf

\(\chi_{4011}(19,\cdot)\) \(\chi_{4011}(61,\cdot)\) \(\chi_{4011}(73,\cdot)\) \(\chi_{4011}(94,\cdot)\) \(\chi_{4011}(124,\cdot)\) \(\chi_{4011}(145,\cdot)\) \(\chi_{4011}(157,\cdot)\) \(\chi_{4011}(178,\cdot)\) \(\chi_{4011}(187,\cdot)\) \(\chi_{4011}(220,\cdot)\) \(\chi_{4011}(262,\cdot)\) \(\chi_{4011}(292,\cdot)\) \(\chi_{4011}(304,\cdot)\) \(\chi_{4011}(334,\cdot)\) \(\chi_{4011}(355,\cdot)\) \(\chi_{4011}(367,\cdot)\) \(\chi_{4011}(439,\cdot)\) \(\chi_{4011}(481,\cdot)\) \(\chi_{4011}(493,\cdot)\) \(\chi_{4011}(514,\cdot)\) \(\chi_{4011}(523,\cdot)\) \(\chi_{4011}(556,\cdot)\) \(\chi_{4011}(565,\cdot)\) \(\chi_{4011}(649,\cdot)\) \(\chi_{4011}(661,\cdot)\) \(\chi_{4011}(724,\cdot)\) \(\chi_{4011}(754,\cdot)\) \(\chi_{4011}(808,\cdot)\) \(\chi_{4011}(817,\cdot)\) \(\chi_{4011}(838,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 570 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{190}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(556, a) \) \(1\)\(1\)\(e\left(\frac{8}{285}\right)\)\(e\left(\frac{16}{285}\right)\)\(e\left(\frac{71}{114}\right)\)\(e\left(\frac{8}{95}\right)\)\(e\left(\frac{371}{570}\right)\)\(e\left(\frac{1}{114}\right)\)\(e\left(\frac{51}{190}\right)\)\(e\left(\frac{32}{285}\right)\)\(e\left(\frac{407}{570}\right)\)\(e\left(\frac{242}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(556,a) \;\) at \(\;a = \) e.g. 2