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Properties

Label 925.508
Modulus $925$
Conductor $925$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([9,10]))
 
Copy content pari:[g,chi] = znchar(Mod(508,925))
 

Basic properties

Modulus: \(925\)
Conductor: \(925\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 925.bw

\(\chi_{925}(27,\cdot)\) \(\chi_{925}(48,\cdot)\) \(\chi_{925}(122,\cdot)\) \(\chi_{925}(138,\cdot)\) \(\chi_{925}(212,\cdot)\) \(\chi_{925}(233,\cdot)\) \(\chi_{925}(323,\cdot)\) \(\chi_{925}(397,\cdot)\) \(\chi_{925}(492,\cdot)\) \(\chi_{925}(508,\cdot)\) \(\chi_{925}(603,\cdot)\) \(\chi_{925}(677,\cdot)\) \(\chi_{925}(767,\cdot)\) \(\chi_{925}(788,\cdot)\) \(\chi_{925}(862,\cdot)\) \(\chi_{925}(878,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((852,76)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 925 }(508, a) \) \(-1\)\(1\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{41}{60}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 925 }(508,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 925 }(508,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 925 }(508,·),\chi_{ 925 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 925 }(508,·)) \;\) at \(\; a,b = \) e.g. 1,2