Properties

Label 950.159
Modulus $950$
Conductor $475$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(950, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,10]))
 
pari: [g,chi] = znchar(Mod(159,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(159,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.x

\(\chi_{950}(159,\cdot)\) \(\chi_{950}(239,\cdot)\) \(\chi_{950}(429,\cdot)\) \(\chi_{950}(539,\cdot)\) \(\chi_{950}(619,\cdot)\) \(\chi_{950}(729,\cdot)\) \(\chi_{950}(809,\cdot)\) \(\chi_{950}(919,\cdot)\)

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

Values on generators

\((77,401)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{7}{30}\right)\)\(-1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{15}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.30.16693301611046410834296161329604046841268427670001983642578125.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(159,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{950}(159,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(159,r) e\left(\frac{r}{475}\right) = -12.3772476978+17.9388890243i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(159,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{950}(159,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(159,r) \chi_{950}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(159,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{950}(159,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(159,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = 57.1289349036+6.0044930146i \)