Properties

Label 950.3
Modulus $950$
Conductor $475$
Order $180$
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(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,130]))
 
pari: [g,chi] = znchar(Mod(3,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.bi

\(\chi_{950}(3,\cdot)\) \(\chi_{950}(13,\cdot)\) \(\chi_{950}(33,\cdot)\) \(\chi_{950}(53,\cdot)\) \(\chi_{950}(67,\cdot)\) \(\chi_{950}(97,\cdot)\) \(\chi_{950}(117,\cdot)\) \(\chi_{950}(127,\cdot)\) \(\chi_{950}(147,\cdot)\) \(\chi_{950}(167,\cdot)\) \(\chi_{950}(173,\cdot)\) \(\chi_{950}(203,\cdot)\) \(\chi_{950}(223,\cdot)\) \(\chi_{950}(287,\cdot)\) \(\chi_{950}(317,\cdot)\) \(\chi_{950}(333,\cdot)\) \(\chi_{950}(337,\cdot)\) \(\chi_{950}(363,\cdot)\) \(\chi_{950}(383,\cdot)\) \(\chi_{950}(413,\cdot)\) \(\chi_{950}(433,\cdot)\) \(\chi_{950}(447,\cdot)\) \(\chi_{950}(477,\cdot)\) \(\chi_{950}(497,\cdot)\) \(\chi_{950}(523,\cdot)\) \(\chi_{950}(527,\cdot)\) \(\chi_{950}(547,\cdot)\) \(\chi_{950}(553,\cdot)\) \(\chi_{950}(573,\cdot)\) \(\chi_{950}(583,\cdot)\) ...

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

\((77,401)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{13}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{151}{180}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{47}{180}\right)\)\(e\left(\frac{139}{180}\right)\)\(e\left(\frac{83}{90}\right)\)\(e\left(\frac{53}{180}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{44}{45}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{950}(3,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(3,r) e\left(\frac{r}{475}\right) = 18.3940742754+11.6900826153i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(3,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{950}(3,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(3,r) \chi_{950}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{950}(3,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(3,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = 0.0 \)