Properties

Label 301.76
Modulus $301$
Conductor $301$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(301, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,31]))
 
pari: [g,chi] = znchar(Mod(76,301))
 

Basic properties

Modulus: \(301\)
Conductor: \(301\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 301.bn

\(\chi_{301}(20,\cdot)\) \(\chi_{301}(34,\cdot)\) \(\chi_{301}(48,\cdot)\) \(\chi_{301}(55,\cdot)\) \(\chi_{301}(62,\cdot)\) \(\chi_{301}(69,\cdot)\) \(\chi_{301}(76,\cdot)\) \(\chi_{301}(104,\cdot)\) \(\chi_{301}(132,\cdot)\) \(\chi_{301}(202,\cdot)\) \(\chi_{301}(244,\cdot)\) \(\chi_{301}(286,\cdot)\)

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: 42.42.5239206534209069133889646882729090965733830111939046184442828436267483950448112600301.1

Values on generators

\((87,218)\) → \((-1,e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 301 }(76, a) \) \(1\)\(1\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 301 }(76,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 301 }(76,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 301 }(76,·),\chi_{ 301 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 301 }(76,·)) \;\) at \(\; a,b = \) e.g. 1,2