↑

Properties

Label 451.413
Modulus $451$
Conductor $451$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(451, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([36,15]))
 
Copy content pari:[g,chi] = znchar(Mod(413,451))
 

Basic properties

Modulus: \(451\)
Conductor: \(451\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(40\)
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: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 451.bu

\(\chi_{451}(68,\cdot)\) \(\chi_{451}(79,\cdot)\) \(\chi_{451}(85,\cdot)\) \(\chi_{451}(96,\cdot)\) \(\chi_{451}(150,\cdot)\) \(\chi_{451}(161,\cdot)\) \(\chi_{451}(167,\cdot)\) \(\chi_{451}(178,\cdot)\) \(\chi_{451}(249,\cdot)\) \(\chi_{451}(260,\cdot)\) \(\chi_{451}(314,\cdot)\) \(\chi_{451}(325,\cdot)\) \(\chi_{451}(413,\cdot)\) \(\chi_{451}(424,\cdot)\) \(\chi_{451}(437,\cdot)\) \(\chi_{451}(448,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: 40.40.8661072039006834456417184171435002124680206277510464931530297507976049073847948875349119910761.1

Values on generators

\((288,375)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 451 }(413, a) \) \(1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(-1\)\(e\left(\frac{1}{8}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 451 }(413,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 451 }(413,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 451 }(413,·),\chi_{ 451 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 451 }(413,·)) \;\) at \(\; a,b = \) e.g. 1,2