Properties

Label 431.144
Modulus $431$
Conductor $431$
Order $43$
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(431, base_ring=CyclotomicField(86))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([30]))
 
pari: [g,chi] = znchar(Mod(144,431))
 

Basic properties

Modulus: \(431\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(43\)
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 431.e

\(\chi_{431}(2,\cdot)\) \(\chi_{431}(3,\cdot)\) \(\chi_{431}(4,\cdot)\) \(\chi_{431}(6,\cdot)\) \(\chi_{431}(8,\cdot)\) \(\chi_{431}(9,\cdot)\) \(\chi_{431}(12,\cdot)\) \(\chi_{431}(16,\cdot)\) \(\chi_{431}(18,\cdot)\) \(\chi_{431}(24,\cdot)\) \(\chi_{431}(27,\cdot)\) \(\chi_{431}(32,\cdot)\) \(\chi_{431}(36,\cdot)\) \(\chi_{431}(48,\cdot)\) \(\chi_{431}(54,\cdot)\) \(\chi_{431}(55,\cdot)\) \(\chi_{431}(64,\cdot)\) \(\chi_{431}(72,\cdot)\) \(\chi_{431}(81,\cdot)\) \(\chi_{431}(96,\cdot)\) \(\chi_{431}(108,\cdot)\) \(\chi_{431}(110,\cdot)\) \(\chi_{431}(128,\cdot)\) \(\chi_{431}(144,\cdot)\) \(\chi_{431}(145,\cdot)\) \(\chi_{431}(149,\cdot)\) \(\chi_{431}(162,\cdot)\) \(\chi_{431}(165,\cdot)\) \(\chi_{431}(192,\cdot)\) \(\chi_{431}(216,\cdot)\) ...

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: $\Q(\zeta_{43})$
Fixed field: 43.43.444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961.1

Values on generators

\(7\) → \(e\left(\frac{15}{43}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{32}{43}\right)\)\(e\left(\frac{29}{43}\right)\)\(e\left(\frac{21}{43}\right)\)\(e\left(\frac{13}{43}\right)\)\(e\left(\frac{18}{43}\right)\)\(e\left(\frac{15}{43}\right)\)\(e\left(\frac{10}{43}\right)\)\(e\left(\frac{15}{43}\right)\)\(e\left(\frac{2}{43}\right)\)\(e\left(\frac{35}{43}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 431 }(144,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{431}(144,\cdot)) = \sum_{r\in \Z/431\Z} \chi_{431}(144,r) e\left(\frac{2r}{431}\right) = 14.2494189897+-15.0981475174i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 431 }(144,·),\chi_{ 431 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{431}(144,\cdot),\chi_{431}(1,\cdot)) = \sum_{r\in \Z/431\Z} \chi_{431}(144,r) \chi_{431}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 431 }(144,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{431}(144,·)) = \sum_{r \in \Z/431\Z} \chi_{431}(144,r) e\left(\frac{1 r + 2 r^{-1}}{431}\right) = 26.3563604721+-27.3371861253i \)