Properties

Label 4011.1879
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([0,19,48]))
 
pari: [g,chi] = znchar(Mod(1879,4011))
 

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 \(\chi_{1337}(542,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.bv

\(\chi_{4011}(52,\cdot)\) \(\chi_{4011}(136,\cdot)\) \(\chi_{4011}(388,\cdot)\) \(\chi_{4011}(418,\cdot)\) \(\chi_{4011}(451,\cdot)\) \(\chi_{4011}(535,\cdot)\) \(\chi_{4011}(598,\cdot)\) \(\chi_{4011}(733,\cdot)\) \(\chi_{4011}(796,\cdot)\) \(\chi_{4011}(871,\cdot)\) \(\chi_{4011}(985,\cdot)\) \(\chi_{4011}(1132,\cdot)\) \(\chi_{4011}(1300,\cdot)\) \(\chi_{4011}(1342,\cdot)\) \(\chi_{4011}(1564,\cdot)\) \(\chi_{4011}(1678,\cdot)\) \(\chi_{4011}(1879,\cdot)\) \(\chi_{4011}(1942,\cdot)\) \(\chi_{4011}(2035,\cdot)\) \(\chi_{4011}(2131,\cdot)\) \(\chi_{4011}(2278,\cdot)\) \(\chi_{4011}(2413,\cdot)\) \(\chi_{4011}(2446,\cdot)\) \(\chi_{4011}(2488,\cdot)\) \(\chi_{4011}(2824,\cdot)\) \(\chi_{4011}(2854,\cdot)\) \(\chi_{4011}(2917,\cdot)\) \(\chi_{4011}(3001,\cdot)\) \(\chi_{4011}(3181,\cdot)\) \(\chi_{4011}(3253,\cdot)\) ...

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

\((2675,2866,2311)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{8}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(1879, a) \) \(-1\)\(1\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{41}{57}\right)\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{85}{114}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{25}{57}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{29}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(1879,a) \;\) at \(\;a = \) e.g. 2