Properties

Conductor 731
Order 112
Real No
Primitive Yes
Parity Even
Orbit Label 731.bj

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(731)
 
sage: chi = H[22]
 
pari: [g,chi] = znchar(Mod(22,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 112
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 = 731.bj
Orbit index = 36

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 ]
 

\(\chi_{731}(22,\cdot)\) \(\chi_{731}(27,\cdot)\) \(\chi_{731}(39,\cdot)\) \(\chi_{731}(45,\cdot)\) \(\chi_{731}(65,\cdot)\) \(\chi_{731}(75,\cdot)\) \(\chi_{731}(82,\cdot)\) \(\chi_{731}(88,\cdot)\) \(\chi_{731}(108,\cdot)\) \(\chi_{731}(113,\cdot)\) \(\chi_{731}(125,\cdot)\) \(\chi_{731}(131,\cdot)\) \(\chi_{731}(156,\cdot)\) \(\chi_{731}(180,\cdot)\) \(\chi_{731}(194,\cdot)\) \(\chi_{731}(199,\cdot)\) \(\chi_{731}(211,\cdot)\) \(\chi_{731}(260,\cdot)\) \(\chi_{731}(266,\cdot)\) \(\chi_{731}(303,\cdot)\) \(\chi_{731}(309,\cdot)\) \(\chi_{731}(328,\cdot)\) \(\chi_{731}(333,\cdot)\) \(\chi_{731}(346,\cdot)\) \(\chi_{731}(352,\cdot)\) \(\chi_{731}(371,\cdot)\) \(\chi_{731}(414,\cdot)\) \(\chi_{731}(419,\cdot)\) \(\chi_{731}(432,\cdot)\) \(\chi_{731}(452,\cdot)\) ...

Values on generators

\((173,562)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{5}{14}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{75}{112}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{55}{112}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{56}\right)\)\(e\left(\frac{19}{56}\right)\)\(e\left(\frac{57}{112}\right)\)\(e\left(\frac{101}{112}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{112})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(22,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(22,r) e\left(\frac{2r}{731}\right) = 8.1752542575+-25.7714031016i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(22,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(22,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(22,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(22,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(22,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = -9.0911368847+-0.5105472964i \)