Properties

Conductor 113
Order 56
Real No
Primitive Yes
Parity Even
Orbit Label 113.i

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(113)
sage: chi = H[87]
pari: [g,chi] = znchar(Mod(87,113))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 113
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 56
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 = 113.i
Orbit index = 9

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 ]

\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{45}{56}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{45}{56}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{39}{56}\right)\)\(e\left(\frac{25}{56}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{13}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{56})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 113 }(87,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{113}(87,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(87,r) e\left(\frac{2r}{113}\right) = 3.4027294859+-10.0708208229i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 113 }(87,·),\chi_{ 113 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{113}(87,\cdot),\chi_{113}(1,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(87,r) \chi_{113}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 113 }(87,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{113}(87,·)) = \sum_{r \in \Z/113\Z} \chi_{113}(87,r) e\left(\frac{1 r + 2 r^{-1}}{113}\right) = 3.9200157393+-8.1399965575i \)