Properties

Label 950.61
Modulus $950$
Conductor $475$
Order $45$
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([72,10]))
 
pari: [g,chi] = znchar(Mod(61,950))
 

Basic properties

Modulus: \(950\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 950.bc

\(\chi_{950}(61,\cdot)\) \(\chi_{950}(81,\cdot)\) \(\chi_{950}(111,\cdot)\) \(\chi_{950}(131,\cdot)\) \(\chi_{950}(161,\cdot)\) \(\chi_{950}(271,\cdot)\) \(\chi_{950}(291,\cdot)\) \(\chi_{950}(321,\cdot)\) \(\chi_{950}(441,\cdot)\) \(\chi_{950}(461,\cdot)\) \(\chi_{950}(481,\cdot)\) \(\chi_{950}(491,\cdot)\) \(\chi_{950}(511,\cdot)\) \(\chi_{950}(541,\cdot)\) \(\chi_{950}(631,\cdot)\) \(\chi_{950}(671,\cdot)\) \(\chi_{950}(681,\cdot)\) \(\chi_{950}(731,\cdot)\) \(\chi_{950}(821,\cdot)\) \(\chi_{950}(841,\cdot)\) \(\chi_{950}(861,\cdot)\) \(\chi_{950}(871,\cdot)\) \(\chi_{950}(891,\cdot)\) \(\chi_{950}(921,\cdot)\)

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

\((77,401)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{22}{45}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: 45.45.299215681303998835585125432825671967739342947202402933846152911778748517690473818220198154449462890625.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 950 }(61,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{950}(61,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(61,r) e\left(\frac{r}{475}\right) = 7.798732016+-20.3514072964i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 950 }(61,·),\chi_{ 950 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{950}(61,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(61,r) \chi_{950}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 950 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{950}(61,·)) = \sum_{r \in \Z/950\Z} \chi_{950}(61,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right) = -4.116476721+1.1803807053i \)