Properties

Label 301.111
Modulus $301$
Conductor $301$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(301, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,8]))
 
pari: [g,chi] = znchar(Mod(111,301))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 301.bc

\(\chi_{301}(13,\cdot)\) \(\chi_{301}(83,\cdot)\) \(\chi_{301}(111,\cdot)\) \(\chi_{301}(139,\cdot)\) \(\chi_{301}(146,\cdot)\) \(\chi_{301}(153,\cdot)\) \(\chi_{301}(160,\cdot)\) \(\chi_{301}(167,\cdot)\) \(\chi_{301}(181,\cdot)\) \(\chi_{301}(195,\cdot)\) \(\chi_{301}(230,\cdot)\) \(\chi_{301}(272,\cdot)\)

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: 42.0.121842012423466724043945276342536999203112328184628981033554149680639161638328200007.1

Values on generators

\((87,218)\) → \((-1,e\left(\frac{4}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 301 }(111, a) \) \(-1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{41}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 301 }(111,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 301 }(111,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 301 }(111,·),\chi_{ 301 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 301 }(111,·)) \;\) at \(\; a,b = \) e.g. 1,2