Properties

Conductor 151
Order 75
Real No
Primitive Yes
Parity Even
Orbit Label 151.k

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(151)
sage: chi = H[62]
pari: [g,chi] = znchar(Mod(62,151))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 151
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 75
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 = 151.k
Orbit index = 11

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 ]

\(\chi_{151}(5,\cdot)\) \(\chi_{151}(10,\cdot)\) \(\chi_{151}(11,\cdot)\) \(\chi_{151}(17,\cdot)\) \(\chi_{151}(18,\cdot)\) \(\chi_{151}(21,\cdot)\) \(\chi_{151}(22,\cdot)\) \(\chi_{151}(25,\cdot)\) \(\chi_{151}(31,\cdot)\) \(\chi_{151}(34,\cdot)\) \(\chi_{151}(36,\cdot)\) \(\chi_{151}(37,\cdot)\) \(\chi_{151}(39,\cdot)\) \(\chi_{151}(40,\cdot)\) \(\chi_{151}(42,\cdot)\) \(\chi_{151}(43,\cdot)\) \(\chi_{151}(45,\cdot)\) \(\chi_{151}(47,\cdot)\) \(\chi_{151}(49,\cdot)\) \(\chi_{151}(55,\cdot)\) \(\chi_{151}(58,\cdot)\) \(\chi_{151}(62,\cdot)\) \(\chi_{151}(69,\cdot)\) \(\chi_{151}(74,\cdot)\) \(\chi_{151}(80,\cdot)\) \(\chi_{151}(88,\cdot)\) \(\chi_{151}(90,\cdot)\) \(\chi_{151}(95,\cdot)\) \(\chi_{151}(97,\cdot)\) \(\chi_{151}(99,\cdot)\) ...

Values on generators

\(6\) → \(e\left(\frac{49}{75}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{13}{75}\right)\)\(e\left(\frac{49}{75}\right)\)\(e\left(\frac{58}{75}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{68}{75}\right)\)\(e\left(\frac{31}{75}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{75})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 151 }(62,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{151}(62,\cdot)) = \sum_{r\in \Z/151\Z} \chi_{151}(62,r) e\left(\frac{2r}{151}\right) = 12.2828885395+-0.3614541844i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 151 }(62,·),\chi_{ 151 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{151}(62,\cdot),\chi_{151}(1,\cdot)) = \sum_{r\in \Z/151\Z} \chi_{151}(62,r) \chi_{151}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 151 }(62,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{151}(62,·)) = \sum_{r \in \Z/151\Z} \chi_{151}(62,r) e\left(\frac{1 r + 2 r^{-1}}{151}\right) = 3.828800643+-4.2523139109i \)