Properties

Label 121.89
Modulus $121$
Conductor $121$
Order $11$
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(121, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(89,121))
 

Basic properties

Modulus: \(121\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
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 121.e

\(\chi_{121}(12,\cdot)\) \(\chi_{121}(23,\cdot)\) \(\chi_{121}(34,\cdot)\) \(\chi_{121}(45,\cdot)\) \(\chi_{121}(56,\cdot)\) \(\chi_{121}(67,\cdot)\) \(\chi_{121}(78,\cdot)\) \(\chi_{121}(89,\cdot)\) \(\chi_{121}(100,\cdot)\) \(\chi_{121}(111,\cdot)\)

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

\(2\) → \(e\left(\frac{6}{11}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(1\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 11.11.672749994932560009201.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 121 }(89,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{121}(89,\cdot)) = \sum_{r\in \Z/121\Z} \chi_{121}(89,r) e\left(\frac{2r}{121}\right) = -10.3793433316+3.6426957057i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 121 }(89,·),\chi_{ 121 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{121}(89,\cdot),\chi_{121}(1,\cdot)) = \sum_{r\in \Z/121\Z} \chi_{121}(89,r) \chi_{121}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 121 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{121}(89,·)) = \sum_{r \in \Z/121\Z} \chi_{121}(89,r) e\left(\frac{1 r + 2 r^{-1}}{121}\right) = -0.0406438811+0.2826844026i \)