Properties

Label 4011.886
Modulus $4011$
Conductor $1337$
Order $114$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,76,93]))
 
pari: [g,chi] = znchar(Mod(886,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1337}(886,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.bw

\(\chi_{4011}(37,\cdot)\) \(\chi_{4011}(205,\cdot)\) \(\chi_{4011}(352,\cdot)\) \(\chi_{4011}(466,\cdot)\) \(\chi_{4011}(541,\cdot)\) \(\chi_{4011}(604,\cdot)\) \(\chi_{4011}(739,\cdot)\) \(\chi_{4011}(802,\cdot)\) \(\chi_{4011}(886,\cdot)\) \(\chi_{4011}(919,\cdot)\) \(\chi_{4011}(949,\cdot)\) \(\chi_{4011}(1201,\cdot)\) \(\chi_{4011}(1285,\cdot)\) \(\chi_{4011}(1348,\cdot)\) \(\chi_{4011}(1612,\cdot)\) \(\chi_{4011}(1789,\cdot)\) \(\chi_{4011}(1885,\cdot)\) \(\chi_{4011}(1948,\cdot)\) \(\chi_{4011}(2032,\cdot)\) \(\chi_{4011}(2095,\cdot)\) \(\chi_{4011}(2167,\cdot)\) \(\chi_{4011}(2347,\cdot)\) \(\chi_{4011}(2431,\cdot)\) \(\chi_{4011}(2494,\cdot)\) \(\chi_{4011}(2524,\cdot)\) \(\chi_{4011}(2860,\cdot)\) \(\chi_{4011}(2902,\cdot)\) \(\chi_{4011}(2935,\cdot)\) \(\chi_{4011}(3070,\cdot)\) \(\chi_{4011}(3217,\cdot)\) ...

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 114 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{31}{38}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(886, a) \) \(-1\)\(1\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{20}{57}\right)\)\(e\left(\frac{1}{114}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{35}{57}\right)\)\(e\left(\frac{17}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(886,a) \;\) at \(\;a = \) e.g. 2