Properties

Conductor 107
Order 53
Real No
Primitive Yes
Parity Even
Orbit Label 107.c

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(107)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,107))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 107
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 53
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 = 107.c
Orbit index = 3

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 ]

\(\chi_{107}(3,\cdot)\) \(\chi_{107}(4,\cdot)\) \(\chi_{107}(9,\cdot)\) \(\chi_{107}(10,\cdot)\) \(\chi_{107}(11,\cdot)\) \(\chi_{107}(12,\cdot)\) \(\chi_{107}(13,\cdot)\) \(\chi_{107}(14,\cdot)\) \(\chi_{107}(16,\cdot)\) \(\chi_{107}(19,\cdot)\) \(\chi_{107}(23,\cdot)\) \(\chi_{107}(25,\cdot)\) \(\chi_{107}(27,\cdot)\) \(\chi_{107}(29,\cdot)\) \(\chi_{107}(30,\cdot)\) \(\chi_{107}(33,\cdot)\) \(\chi_{107}(34,\cdot)\) \(\chi_{107}(35,\cdot)\) \(\chi_{107}(36,\cdot)\) \(\chi_{107}(37,\cdot)\) \(\chi_{107}(39,\cdot)\) \(\chi_{107}(40,\cdot)\) \(\chi_{107}(41,\cdot)\) \(\chi_{107}(42,\cdot)\) \(\chi_{107}(44,\cdot)\) \(\chi_{107}(47,\cdot)\) \(\chi_{107}(48,\cdot)\) \(\chi_{107}(49,\cdot)\) \(\chi_{107}(52,\cdot)\) \(\chi_{107}(53,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{35}{53}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{35}{53}\right)\)\(e\left(\frac{12}{53}\right)\)\(e\left(\frac{17}{53}\right)\)\(e\left(\frac{2}{53}\right)\)\(e\left(\frac{47}{53}\right)\)\(e\left(\frac{21}{53}\right)\)\(e\left(\frac{52}{53}\right)\)\(e\left(\frac{24}{53}\right)\)\(e\left(\frac{37}{53}\right)\)\(e\left(\frac{28}{53}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{53})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 107 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{107}(3,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(3,r) e\left(\frac{2r}{107}\right) = -10.2729683643+-1.2108348301i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 107 }(3,·),\chi_{ 107 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{107}(3,\cdot),\chi_{107}(1,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(3,r) \chi_{107}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 107 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{107}(3,·)) = \sum_{r \in \Z/107\Z} \chi_{107}(3,r) e\left(\frac{1 r + 2 r^{-1}}{107}\right) = -1.9062049857+3.457657981i \)