Properties

Label 882.41
Modulus $882$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,15]))
 
pari: [g,chi] = znchar(Mod(41,882))
 

Basic properties

Modulus: \(882\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 882.bj

\(\chi_{882}(41,\cdot)\) \(\chi_{882}(83,\cdot)\) \(\chi_{882}(167,\cdot)\) \(\chi_{882}(209,\cdot)\) \(\chi_{882}(335,\cdot)\) \(\chi_{882}(419,\cdot)\) \(\chi_{882}(461,\cdot)\) \(\chi_{882}(545,\cdot)\) \(\chi_{882}(671,\cdot)\) \(\chi_{882}(713,\cdot)\) \(\chi_{882}(797,\cdot)\) \(\chi_{882}(839,\cdot)\)

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

\((785,199)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 882 }(41, a) \) \(1\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{3}{7}\right)\)\(-1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 882 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 882 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 882 }(41,·),\chi_{ 882 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 882 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2