Properties

Label 751.207
Modulus $751$
Conductor $751$
Order $125$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([178]))
 
pari: [g,chi] = znchar(Mod(207,751))
 

Basic properties

Modulus: \(751\)
Conductor: \(751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(125\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 751.l

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

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 125 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{89}{125}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 751 }(207, a) \) \(1\)\(1\)\(e\left(\frac{24}{125}\right)\)\(e\left(\frac{89}{125}\right)\)\(e\left(\frac{48}{125}\right)\)\(e\left(\frac{4}{125}\right)\)\(e\left(\frac{113}{125}\right)\)\(e\left(\frac{64}{125}\right)\)\(e\left(\frac{72}{125}\right)\)\(e\left(\frac{53}{125}\right)\)\(e\left(\frac{28}{125}\right)\)\(e\left(\frac{2}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 751 }(207,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 751 }(207,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 751 }(207,·),\chi_{ 751 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 751 }(207,·)) \;\) at \(\; a,b = \) e.g. 1,2