Properties

Label 276.85
Modulus $276$
Conductor $23$
Order $11$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(276, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,8]))
 
pari: [g,chi] = znchar(Mod(85,276))
 

Basic properties

Modulus: \(276\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 276.i

\(\chi_{276}(13,\cdot)\) \(\chi_{276}(25,\cdot)\) \(\chi_{276}(49,\cdot)\) \(\chi_{276}(73,\cdot)\) \(\chi_{276}(85,\cdot)\) \(\chi_{276}(121,\cdot)\) \(\chi_{276}(133,\cdot)\) \(\chi_{276}(169,\cdot)\) \(\chi_{276}(193,\cdot)\) \(\chi_{276}(265,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{23})^+\)

Values on generators

\((139,185,97)\) → \((1,1,e\left(\frac{4}{11}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{3}{11}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 276 }(85,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{276}(85,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(85,r) e\left(\frac{r}{138}\right) = -2.593718761+9.2343176786i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 276 }(85,·),\chi_{ 276 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{276}(85,\cdot),\chi_{276}(1,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(85,r) \chi_{276}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 276 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{276}(85,·)) = \sum_{r \in \Z/276\Z} \chi_{276}(85,r) e\left(\frac{1 r + 2 r^{-1}}{276}\right) = -0.0 \)