Properties

Label 261.139
Modulus $261$
Conductor $261$
Order $21$
Real no
Primitive yes
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(261, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,30]))
 
pari: [g,chi] = znchar(Mod(139,261))
 

Basic properties

Modulus: \(261\)
Conductor: \(261\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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 261.q

\(\chi_{261}(7,\cdot)\) \(\chi_{261}(16,\cdot)\) \(\chi_{261}(25,\cdot)\) \(\chi_{261}(49,\cdot)\) \(\chi_{261}(52,\cdot)\) \(\chi_{261}(94,\cdot)\) \(\chi_{261}(103,\cdot)\) \(\chi_{261}(112,\cdot)\) \(\chi_{261}(139,\cdot)\) \(\chi_{261}(169,\cdot)\) \(\chi_{261}(223,\cdot)\) \(\chi_{261}(256,\cdot)\)

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

Values on generators

\((146,118)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 21.21.4814587615056751193058435502319478353721.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 261 }(139,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{261}(139,\cdot)) = \sum_{r\in \Z/261\Z} \chi_{261}(139,r) e\left(\frac{2r}{261}\right) = 10.8410759038+11.9779411105i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 261 }(139,·),\chi_{ 261 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{261}(139,\cdot),\chi_{261}(1,\cdot)) = \sum_{r\in \Z/261\Z} \chi_{261}(139,r) \chi_{261}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 261 }(139,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{261}(139,·)) = \sum_{r \in \Z/261\Z} \chi_{261}(139,r) e\left(\frac{1 r + 2 r^{-1}}{261}\right) = -9.0234762533+-1.3600702101i \)