Properties

Conductor 149
Order 37
Real No
Primitive Yes
Parity Even
Orbit Label 149.d

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(149)
sage: chi = H[17]
pari: [g,chi] = znchar(Mod(17,149))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 149
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 37
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even
Orbit label = 149.d
Orbit index = 4

Galois orbit

sage: chi.sage_character().galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

\(\chi_{149}(5,\cdot)\) \(\chi_{149}(6,\cdot)\) \(\chi_{149}(16,\cdot)\) \(\chi_{149}(17,\cdot)\) \(\chi_{149}(19,\cdot)\) \(\chi_{149}(25,\cdot)\) \(\chi_{149}(28,\cdot)\) \(\chi_{149}(29,\cdot)\) \(\chi_{149}(30,\cdot)\) \(\chi_{149}(31,\cdot)\) \(\chi_{149}(33,\cdot)\) \(\chi_{149}(36,\cdot)\) \(\chi_{149}(37,\cdot)\) \(\chi_{149}(39,\cdot)\) \(\chi_{149}(46,\cdot)\) \(\chi_{149}(49,\cdot)\) \(\chi_{149}(63,\cdot)\) \(\chi_{149}(67,\cdot)\) \(\chi_{149}(73,\cdot)\) \(\chi_{149}(80,\cdot)\) \(\chi_{149}(81,\cdot)\) \(\chi_{149}(85,\cdot)\) \(\chi_{149}(88,\cdot)\) \(\chi_{149}(95,\cdot)\) \(\chi_{149}(96,\cdot)\) \(\chi_{149}(102,\cdot)\) \(\chi_{149}(104,\cdot)\) \(\chi_{149}(107,\cdot)\) \(\chi_{149}(114,\cdot)\) \(\chi_{149}(123,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{31}{37}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{31}{37}\right)\)\(e\left(\frac{33}{37}\right)\)\(e\left(\frac{25}{37}\right)\)\(e\left(\frac{5}{37}\right)\)\(e\left(\frac{27}{37}\right)\)\(e\left(\frac{36}{37}\right)\)\(e\left(\frac{19}{37}\right)\)\(e\left(\frac{29}{37}\right)\)\(e\left(\frac{36}{37}\right)\)\(e\left(\frac{12}{37}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{37})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 149 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{149}(17,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(17,r) e\left(\frac{2r}{149}\right) = 10.6375030141+-5.9869466027i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 149 }(17,·),\chi_{ 149 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{149}(17,\cdot),\chi_{149}(1,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(17,r) \chi_{149}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 149 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{149}(17,·)) = \sum_{r \in \Z/149\Z} \chi_{149}(17,r) e\left(\frac{1 r + 2 r^{-1}}{149}\right) = 14.001383834+-7.8216422843i \)