Properties

Label 4011.670
Modulus $4011$
Conductor $1337$
Order $570$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,475,216]))
 
pari: [g,chi] = znchar(Mod(670,4011))
 

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 \(\chi_{1337}(670,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.cg

\(\chi_{4011}(10,\cdot)\) \(\chi_{4011}(40,\cdot)\) \(\chi_{4011}(103,\cdot)\) \(\chi_{4011}(115,\cdot)\) \(\chi_{4011}(199,\cdot)\) \(\chi_{4011}(208,\cdot)\) \(\chi_{4011}(241,\cdot)\) \(\chi_{4011}(250,\cdot)\) \(\chi_{4011}(271,\cdot)\) \(\chi_{4011}(283,\cdot)\) \(\chi_{4011}(325,\cdot)\) \(\chi_{4011}(397,\cdot)\) \(\chi_{4011}(409,\cdot)\) \(\chi_{4011}(430,\cdot)\) \(\chi_{4011}(460,\cdot)\) \(\chi_{4011}(472,\cdot)\) \(\chi_{4011}(502,\cdot)\) \(\chi_{4011}(544,\cdot)\) \(\chi_{4011}(577,\cdot)\) \(\chi_{4011}(586,\cdot)\) \(\chi_{4011}(607,\cdot)\) \(\chi_{4011}(619,\cdot)\) \(\chi_{4011}(640,\cdot)\) \(\chi_{4011}(670,\cdot)\) \(\chi_{4011}(691,\cdot)\) \(\chi_{4011}(703,\cdot)\) \(\chi_{4011}(745,\cdot)\) \(\chi_{4011}(766,\cdot)\) \(\chi_{4011}(787,\cdot)\) \(\chi_{4011}(829,\cdot)\) ...

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

\((2675,2866,2311)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{36}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(670, a) \) \(-1\)\(1\)\(e\left(\frac{97}{285}\right)\)\(e\left(\frac{194}{285}\right)\)\(e\left(\frac{13}{114}\right)\)\(e\left(\frac{2}{95}\right)\)\(e\left(\frac{259}{570}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{179}{190}\right)\)\(e\left(\frac{103}{285}\right)\)\(e\left(\frac{553}{570}\right)\)\(e\left(\frac{311}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(670,a) \;\) at \(\;a = \) e.g. 2