Properties

Label 4011.1301
Modulus $4011$
Conductor $4011$
Order $38$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([19,19,7]))
 
pari: [g,chi] = znchar(Mod(1301,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(4011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.bk

\(\chi_{4011}(41,\cdot)\) \(\chi_{4011}(377,\cdot)\) \(\chi_{4011}(419,\cdot)\) \(\chi_{4011}(587,\cdot)\) \(\chi_{4011}(734,\cdot)\) \(\chi_{4011}(923,\cdot)\) \(\chi_{4011}(986,\cdot)\) \(\chi_{4011}(1301,\cdot)\) \(\chi_{4011}(1994,\cdot)\) \(\chi_{4011}(2267,\cdot)\) \(\chi_{4011}(2330,\cdot)\) \(\chi_{4011}(2414,\cdot)\) \(\chi_{4011}(2477,\cdot)\) \(\chi_{4011}(2729,\cdot)\) \(\chi_{4011}(2813,\cdot)\) \(\chi_{4011}(2876,\cdot)\) \(\chi_{4011}(3317,\cdot)\) \(\chi_{4011}(3695,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: Number field defined by a degree 38 polynomial

Values on generators

\((2675,2866,2311)\) → \((-1,-1,e\left(\frac{7}{38}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(1301, a) \) \(-1\)\(1\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{5}{38}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{13}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(1301,a) \;\) at \(\;a = \) e.g. 2