Properties

Label 1911.1577
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([21,2,7]))
 
pari: [g,chi] = znchar(Mod(1577,1911))
 

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.ds

\(\chi_{1911}(179,\cdot)\) \(\chi_{1911}(212,\cdot)\) \(\chi_{1911}(452,\cdot)\) \(\chi_{1911}(485,\cdot)\) \(\chi_{1911}(725,\cdot)\) \(\chi_{1911}(758,\cdot)\) \(\chi_{1911}(1031,\cdot)\) \(\chi_{1911}(1271,\cdot)\) \(\chi_{1911}(1544,\cdot)\) \(\chi_{1911}(1577,\cdot)\) \(\chi_{1911}(1817,\cdot)\) \(\chi_{1911}(1850,\cdot)\)

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: 42.0.171796661872303139426286971553175729564906991908179786155620948893904232254867038476961967376575716647979404317071.1

Values on generators

\((638,1522,1471)\) → \((-1,e\left(\frac{1}{21}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 1911 }(1577, a) \) \(-1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(-1\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1911 }(1577,a) \;\) at \(\;a = \) e.g. 2