Properties

Conductor 861
Order 120
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 861.cj

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(861)
 
sage: chi = H[26]
 
pari: [g,chi] = znchar(Mod(26,861))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 861
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 120
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 861.cj
Orbit index = 62

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{861}(17,\cdot)\) \(\chi_{861}(26,\cdot)\) \(\chi_{861}(47,\cdot)\) \(\chi_{861}(89,\cdot)\) \(\chi_{861}(101,\cdot)\) \(\chi_{861}(110,\cdot)\) \(\chi_{861}(152,\cdot)\) \(\chi_{861}(194,\cdot)\) \(\chi_{861}(227,\cdot)\) \(\chi_{861}(257,\cdot)\) \(\chi_{861}(299,\cdot)\) \(\chi_{861}(311,\cdot)\) \(\chi_{861}(341,\cdot)\) \(\chi_{861}(362,\cdot)\) \(\chi_{861}(395,\cdot)\) \(\chi_{861}(404,\cdot)\) \(\chi_{861}(416,\cdot)\) \(\chi_{861}(425,\cdot)\) \(\chi_{861}(458,\cdot)\) \(\chi_{861}(479,\cdot)\) \(\chi_{861}(509,\cdot)\) \(\chi_{861}(521,\cdot)\) \(\chi_{861}(563,\cdot)\) \(\chi_{861}(593,\cdot)\) \(\chi_{861}(626,\cdot)\) \(\chi_{861}(668,\cdot)\) \(\chi_{861}(710,\cdot)\) \(\chi_{861}(719,\cdot)\) \(\chi_{861}(731,\cdot)\) \(\chi_{861}(773,\cdot)\) ...

Values on generators

\((575,493,211)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{17}{40}\right))\)

Values

-112458101113161719
\(-1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{13}{120}\right)\)\(e\left(\frac{27}{40}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{119}{120}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{120})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(26,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{861}(26,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(26,r) e\left(\frac{2r}{861}\right) = -29.3280907856+0.9290268426i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 861 }(26,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{861}(26,\cdot),\chi_{861}(1,\cdot)) = \sum_{r\in \Z/861\Z} \chi_{861}(26,r) \chi_{861}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(26,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{861}(26,·)) = \sum_{r \in \Z/861\Z} \chi_{861}(26,r) e\left(\frac{1 r + 2 r^{-1}}{861}\right) = -0.0 \)