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Properties

Label 950.9
Modulus $950$
Conductor $475$
Order $90$
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(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,40]))
 
pari: [g,chi] = znchar(Mod(9,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.bg

\(\chi_{950}(9,\cdot)\) \(\chi_{950}(119,\cdot)\) \(\chi_{950}(139,\cdot)\) \(\chi_{950}(169,\cdot)\) \(\chi_{950}(289,\cdot)\) \(\chi_{950}(309,\cdot)\) \(\chi_{950}(329,\cdot)\) \(\chi_{950}(339,\cdot)\) \(\chi_{950}(359,\cdot)\) \(\chi_{950}(389,\cdot)\) \(\chi_{950}(479,\cdot)\) \(\chi_{950}(519,\cdot)\) \(\chi_{950}(529,\cdot)\) \(\chi_{950}(579,\cdot)\) \(\chi_{950}(669,\cdot)\) \(\chi_{950}(689,\cdot)\) \(\chi_{950}(709,\cdot)\) \(\chi_{950}(719,\cdot)\) \(\chi_{950}(739,\cdot)\) \(\chi_{950}(769,\cdot)\) \(\chi_{950}(859,\cdot)\) \(\chi_{950}(879,\cdot)\) \(\chi_{950}(909,\cdot)\) \(\chi_{950}(929,\cdot)\)

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

\((77,401)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{43}{45}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{950}(9,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(9,r) e\left(\frac{r}{475}\right) = -20.7621283969+-6.6282746194i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(9,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{950}(9,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(9,r) \chi_{950}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{950}(9,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(9,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = 27.1734851592+13.2533942571i \)