Properties

Conductor 751
Order 125
Real No
Primitive Yes
Parity Even
Orbit Label 751.l

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(751)
 
sage: chi = H[49]
 
pari: [g,chi] = znchar(Mod(49,751))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 751
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 125
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 = 751.l
Orbit index = 12

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 ]
 

\(\chi_{751}(8,\cdot)\) \(\chi_{751}(10,\cdot)\) \(\chi_{751}(26,\cdot)\) \(\chi_{751}(36,\cdot)\) \(\chi_{751}(38,\cdot)\) \(\chi_{751}(42,\cdot)\) \(\chi_{751}(43,\cdot)\) \(\chi_{751}(45,\cdot)\) \(\chi_{751}(46,\cdot)\) \(\chi_{751}(49,\cdot)\) \(\chi_{751}(64,\cdot)\) \(\chi_{751}(71,\cdot)\) \(\chi_{751}(93,\cdot)\) \(\chi_{751}(94,\cdot)\) \(\chi_{751}(100,\cdot)\) \(\chi_{751}(118,\cdot)\) \(\chi_{751}(125,\cdot)\) \(\chi_{751}(132,\cdot)\) \(\chi_{751}(148,\cdot)\) \(\chi_{751}(151,\cdot)\) \(\chi_{751}(154,\cdot)\) \(\chi_{751}(165,\cdot)\) \(\chi_{751}(185,\cdot)\) \(\chi_{751}(187,\cdot)\) \(\chi_{751}(189,\cdot)\) \(\chi_{751}(191,\cdot)\) \(\chi_{751}(207,\cdot)\) \(\chi_{751}(208,\cdot)\) \(\chi_{751}(237,\cdot)\) \(\chi_{751}(244,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{92}{125}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{22}{125}\right)\)\(e\left(\frac{92}{125}\right)\)\(e\left(\frac{44}{125}\right)\)\(e\left(\frac{87}{125}\right)\)\(e\left(\frac{114}{125}\right)\)\(e\left(\frac{17}{125}\right)\)\(e\left(\frac{66}{125}\right)\)\(e\left(\frac{59}{125}\right)\)\(e\left(\frac{109}{125}\right)\)\(e\left(\frac{6}{25}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{125})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 751 }(49,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{751}(49,\cdot)) = \sum_{r\in \Z/751\Z} \chi_{751}(49,r) e\left(\frac{2r}{751}\right) = 27.4038963455+0.1626809307i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 751 }(49,·),\chi_{ 751 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{751}(49,\cdot),\chi_{751}(1,\cdot)) = \sum_{r\in \Z/751\Z} \chi_{751}(49,r) \chi_{751}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 751 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{751}(49,·)) = \sum_{r \in \Z/751\Z} \chi_{751}(49,r) e\left(\frac{1 r + 2 r^{-1}}{751}\right) = 40.1772137019+24.7945830322i \)