Properties

Label 1911.1166
Modulus $1911$
Conductor $1911$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,34,28]))
 
pari: [g,chi] = znchar(Mod(1166,1911))
 

Basic properties

Modulus: \(1911\)
Conductor: \(1911\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 1911.dx

\(\chi_{1911}(74,\cdot)\) \(\chi_{1911}(107,\cdot)\) \(\chi_{1911}(347,\cdot)\) \(\chi_{1911}(380,\cdot)\) \(\chi_{1911}(620,\cdot)\) \(\chi_{1911}(653,\cdot)\) \(\chi_{1911}(893,\cdot)\) \(\chi_{1911}(926,\cdot)\) \(\chi_{1911}(1166,\cdot)\) \(\chi_{1911}(1199,\cdot)\) \(\chi_{1911}(1472,\cdot)\) \(\chi_{1911}(1712,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((638,1522,1471)\) → \((-1,e\left(\frac{17}{21}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1911 }(1166, a) \) \(-1\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1911 }(1166,a) \;\) at \(\;a = \) e.g. 2