Properties

Label 300.119
Modulus $300$
Conductor $300$
Order $10$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(300, base_ring=CyclotomicField(10))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([5,5,9]))
 
pari: [g,chi] = znchar(Mod(119,300))
 

Basic properties

Modulus: \(300\)
Conductor: \(300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
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 300.r

\(\chi_{300}(59,\cdot)\) \(\chi_{300}(119,\cdot)\) \(\chi_{300}(179,\cdot)\) \(\chi_{300}(239,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: 10.10.189843750000000000.1

Values on generators

\((151,101,277)\) → \((-1,-1,e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 300 }(119,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{300}(119,\cdot)) = \sum_{r\in \Z/300\Z} \chi_{300}(119,r) e\left(\frac{r}{150}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 300 }(119,·),\chi_{ 300 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{300}(119,\cdot),\chi_{300}(1,\cdot)) = \sum_{r\in \Z/300\Z} \chi_{300}(119,r) \chi_{300}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 300 }(119,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{300}(119,·)) = \sum_{r \in \Z/300\Z} \chi_{300}(119,r) e\left(\frac{1 r + 2 r^{-1}}{300}\right) = 0.0 \)