Properties

Label 143.68
Modulus $143$
Conductor $143$
Order $30$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(143, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,10]))
 
pari: [g,chi] = znchar(Mod(68,143))
 

Basic properties

Modulus: \(143\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 143.t

\(\chi_{143}(29,\cdot)\) \(\chi_{143}(35,\cdot)\) \(\chi_{143}(61,\cdot)\) \(\chi_{143}(68,\cdot)\) \(\chi_{143}(74,\cdot)\) \(\chi_{143}(94,\cdot)\) \(\chi_{143}(107,\cdot)\) \(\chi_{143}(139,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.249154964698353870876083085574129912384252301255171.1

Values on generators

\((79,67)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 143 }(68,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{143}(68,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(68,r) e\left(\frac{2r}{143}\right) = -11.9501706309+-0.4397975594i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 143 }(68,·),\chi_{ 143 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{143}(68,\cdot),\chi_{143}(1,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(68,r) \chi_{143}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 143 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{143}(68,·)) = \sum_{r \in \Z/143\Z} \chi_{143}(68,r) e\left(\frac{1 r + 2 r^{-1}}{143}\right) = -12.4405022554+2.6443103849i \)