Properties

Conductor 149
Order 74
Real No
Primitive Yes
Parity Even
Orbit Label 149.e

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[22]
pari: [g,chi] = znchar(Mod(22,149))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 149
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 74
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.e
Orbit index = 5

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 ]

\(\chi_{149}(4,\cdot)\) \(\chi_{149}(7,\cdot)\) \(\chi_{149}(9,\cdot)\) \(\chi_{149}(20,\cdot)\) \(\chi_{149}(22,\cdot)\) \(\chi_{149}(24,\cdot)\) \(\chi_{149}(26,\cdot)\) \(\chi_{149}(35,\cdot)\) \(\chi_{149}(42,\cdot)\) \(\chi_{149}(45,\cdot)\) \(\chi_{149}(47,\cdot)\) \(\chi_{149}(53,\cdot)\) \(\chi_{149}(54,\cdot)\) \(\chi_{149}(61,\cdot)\) \(\chi_{149}(64,\cdot)\) \(\chi_{149}(68,\cdot)\) \(\chi_{149}(69,\cdot)\) \(\chi_{149}(76,\cdot)\) \(\chi_{149}(82,\cdot)\) \(\chi_{149}(86,\cdot)\) \(\chi_{149}(100,\cdot)\) \(\chi_{149}(103,\cdot)\) \(\chi_{149}(110,\cdot)\) \(\chi_{149}(112,\cdot)\) \(\chi_{149}(113,\cdot)\) \(\chi_{149}(116,\cdot)\) \(\chi_{149}(118,\cdot)\) \(\chi_{149}(119,\cdot)\) \(\chi_{149}(120,\cdot)\) \(\chi_{149}(121,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{55}{74}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{55}{74}\right)\)\(e\left(\frac{49}{74}\right)\)\(e\left(\frac{18}{37}\right)\)\(e\left(\frac{11}{37}\right)\)\(e\left(\frac{15}{37}\right)\)\(e\left(\frac{20}{37}\right)\)\(e\left(\frac{17}{74}\right)\)\(e\left(\frac{12}{37}\right)\)\(e\left(\frac{3}{74}\right)\)\(e\left(\frac{1}{74}\right)\)
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)
\( \tau_{ a }( \chi_{ 149 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{149}(22,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(22,r) e\left(\frac{2r}{149}\right) = -7.4057213853+9.7033649197i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 149 }(22,·),\chi_{ 149 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{149}(22,\cdot),\chi_{149}(1,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(22,r) \chi_{149}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 149 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{149}(22,·)) = \sum_{r \in \Z/149\Z} \chi_{149}(22,r) e\left(\frac{1 r + 2 r^{-1}}{149}\right) = 15.8151283882+-16.5012096132i \)