Properties

Label 245.114
Modulus $245$
Conductor $245$
Order $42$
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(245, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,20]))
 
pari: [g,chi] = znchar(Mod(114,245))
 

Basic properties

Modulus: \(245\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 245.t

\(\chi_{245}(4,\cdot)\) \(\chi_{245}(9,\cdot)\) \(\chi_{245}(39,\cdot)\) \(\chi_{245}(44,\cdot)\) \(\chi_{245}(74,\cdot)\) \(\chi_{245}(109,\cdot)\) \(\chi_{245}(114,\cdot)\) \(\chi_{245}(144,\cdot)\) \(\chi_{245}(149,\cdot)\) \(\chi_{245}(179,\cdot)\) \(\chi_{245}(184,\cdot)\) \(\chi_{245}(219,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1

Values on generators

\((197,101)\) → \((-1,e\left(\frac{10}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{11}{21}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 245 }(114,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{245}(114,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(114,r) e\left(\frac{2r}{245}\right) = 13.1181538157+-8.5389718624i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 245 }(114,·),\chi_{ 245 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{245}(114,\cdot),\chi_{245}(1,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(114,r) \chi_{245}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 245 }(114,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{245}(114,·)) = \sum_{r \in \Z/245\Z} \chi_{245}(114,r) e\left(\frac{1 r + 2 r^{-1}}{245}\right) = -0.0 \)