Properties

Label 121.20
Modulus $121$
Conductor $121$
Order $55$
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)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([38]))
 
pari: [g,chi] = znchar(Mod(20,121))
 

Basic properties

Modulus: \(121\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
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.g

\(\chi_{121}(4,\cdot)\) \(\chi_{121}(5,\cdot)\) \(\chi_{121}(14,\cdot)\) \(\chi_{121}(15,\cdot)\) \(\chi_{121}(16,\cdot)\) \(\chi_{121}(20,\cdot)\) \(\chi_{121}(25,\cdot)\) \(\chi_{121}(26,\cdot)\) \(\chi_{121}(31,\cdot)\) \(\chi_{121}(36,\cdot)\) \(\chi_{121}(37,\cdot)\) \(\chi_{121}(38,\cdot)\) \(\chi_{121}(42,\cdot)\) \(\chi_{121}(47,\cdot)\) \(\chi_{121}(48,\cdot)\) \(\chi_{121}(49,\cdot)\) \(\chi_{121}(53,\cdot)\) \(\chi_{121}(58,\cdot)\) \(\chi_{121}(59,\cdot)\) \(\chi_{121}(60,\cdot)\) \(\chi_{121}(64,\cdot)\) \(\chi_{121}(69,\cdot)\) \(\chi_{121}(70,\cdot)\) \(\chi_{121}(71,\cdot)\) \(\chi_{121}(75,\cdot)\) \(\chi_{121}(80,\cdot)\) \(\chi_{121}(82,\cdot)\) \(\chi_{121}(86,\cdot)\) \(\chi_{121}(91,\cdot)\) \(\chi_{121}(92,\cdot)\) ...

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

\(2\) → \(e\left(\frac{38}{55}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 121 }(20,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{121}(20,\cdot)) = \sum_{r\in \Z/121\Z} \chi_{121}(20,r) e\left(\frac{2r}{121}\right) = 10.9407318606+-1.1403448402i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 121 }(20,·),\chi_{ 121 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{121}(20,\cdot),\chi_{121}(1,\cdot)) = \sum_{r\in \Z/121\Z} \chi_{121}(20,r) \chi_{121}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 121 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{121}(20,·)) = \sum_{r \in \Z/121\Z} \chi_{121}(20,r) e\left(\frac{1 r + 2 r^{-1}}{121}\right) = 11.1150692074+-16.2552706044i \)