Properties

Label 950.37
Modulus $950$
Conductor $475$
Order $20$
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,10]))
 
pari: [g,chi] = znchar(Mod(37,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.w

\(\chi_{950}(37,\cdot)\) \(\chi_{950}(113,\cdot)\) \(\chi_{950}(227,\cdot)\) \(\chi_{950}(303,\cdot)\) \(\chi_{950}(417,\cdot)\) \(\chi_{950}(683,\cdot)\) \(\chi_{950}(797,\cdot)\) \(\chi_{950}(873,\cdot)\)

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

\((77,401)\) → \((e\left(\frac{9}{20}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{2}{5}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.20.17843751288604107685387134552001953125.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(37,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{950}(37,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(37,r) e\left(\frac{r}{475}\right) = 15.8875028997+-14.9193582842i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(37,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{950}(37,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(37,r) \chi_{950}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{950}(37,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(37,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = -0.0 \)