Properties

Label 307.269
Modulus $307$
Conductor $307$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(307, base_ring=CyclotomicField(34))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(269,307))
 

Basic properties

Modulus: \(307\)
Conductor: \(307\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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 307.f

\(\chi_{307}(9,\cdot)\) \(\chi_{307}(24,\cdot)\) \(\chi_{307}(64,\cdot)\) \(\chi_{307}(81,\cdot)\) \(\chi_{307}(102,\cdot)\) \(\chi_{307}(105,\cdot)\) \(\chi_{307}(114,\cdot)\) \(\chi_{307}(115,\cdot)\) \(\chi_{307}(216,\cdot)\) \(\chi_{307}(235,\cdot)\) \(\chi_{307}(269,\cdot)\) \(\chi_{307}(272,\cdot)\) \(\chi_{307}(273,\cdot)\) \(\chi_{307}(280,\cdot)\) \(\chi_{307}(299,\cdot)\) \(\chi_{307}(304,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: 17.17.6226070121392010397563990173530787496001.1

Values on generators

\(5\) → \(e\left(\frac{6}{17}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{7}{17}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 307 }(269,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{307}(269,\cdot)) = \sum_{r\in \Z/307\Z} \chi_{307}(269,r) e\left(\frac{2r}{307}\right) = 9.6786326121+14.6056177808i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 307 }(269,·),\chi_{ 307 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{307}(269,\cdot),\chi_{307}(1,\cdot)) = \sum_{r\in \Z/307\Z} \chi_{307}(269,r) \chi_{307}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 307 }(269,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{307}(269,·)) = \sum_{r \in \Z/307\Z} \chi_{307}(269,r) e\left(\frac{1 r + 2 r^{-1}}{307}\right) = 5.8036229496+-3.5934511944i \)