Properties

Label 1911.1181
Modulus $1911$
Conductor $1911$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,58,49]))
 
pari: [g,chi] = znchar(Mod(1181,1911))
 

Basic properties

Modulus: \(1911\)
Conductor: \(1911\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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.ep

\(\chi_{1911}(59,\cdot)\) \(\chi_{1911}(89,\cdot)\) \(\chi_{1911}(236,\cdot)\) \(\chi_{1911}(332,\cdot)\) \(\chi_{1911}(500,\cdot)\) \(\chi_{1911}(605,\cdot)\) \(\chi_{1911}(635,\cdot)\) \(\chi_{1911}(773,\cdot)\) \(\chi_{1911}(782,\cdot)\) \(\chi_{1911}(878,\cdot)\) \(\chi_{1911}(908,\cdot)\) \(\chi_{1911}(1046,\cdot)\) \(\chi_{1911}(1055,\cdot)\) \(\chi_{1911}(1151,\cdot)\) \(\chi_{1911}(1181,\cdot)\) \(\chi_{1911}(1319,\cdot)\) \(\chi_{1911}(1328,\cdot)\) \(\chi_{1911}(1424,\cdot)\) \(\chi_{1911}(1454,\cdot)\) \(\chi_{1911}(1592,\cdot)\) \(\chi_{1911}(1601,\cdot)\) \(\chi_{1911}(1727,\cdot)\) \(\chi_{1911}(1865,\cdot)\) \(\chi_{1911}(1874,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((638,1522,1471)\) → \((-1,e\left(\frac{29}{42}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1911 }(1181, a) \) \(-1\)\(1\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{71}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1911 }(1181,a) \;\) at \(\;a = \) e.g. 2