Properties

Label 241.201
Modulus $241$
Conductor $241$
Order $20$
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(241, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9]))
 
pari: [g,chi] = znchar(Mod(201,241))
 

Basic properties

Modulus: \(241\)
Conductor: \(241\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 241.l

\(\chi_{241}(6,\cdot)\) \(\chi_{241}(25,\cdot)\) \(\chi_{241}(40,\cdot)\) \(\chi_{241}(106,\cdot)\) \(\chi_{241}(135,\cdot)\) \(\chi_{241}(201,\cdot)\) \(\chi_{241}(216,\cdot)\) \(\chi_{241}(235,\cdot)\)

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

\(7\) → \(e\left(\frac{9}{20}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(-1\)\(e\left(\frac{9}{10}\right)\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(-1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.20.1812691127043107566808878608684122677382270161.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 241 }(201,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{241}(201,\cdot)) = \sum_{r\in \Z/241\Z} \chi_{241}(201,r) e\left(\frac{2r}{241}\right) = -14.8600506141+-4.4920925798i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 241 }(201,·),\chi_{ 241 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{241}(201,\cdot),\chi_{241}(1,\cdot)) = \sum_{r\in \Z/241\Z} \chi_{241}(201,r) \chi_{241}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 241 }(201,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{241}(201,·)) = \sum_{r \in \Z/241\Z} \chi_{241}(201,r) e\left(\frac{1 r + 2 r^{-1}}{241}\right) = -17.3815869366i \)