Properties

Label 60.13
Modulus $60$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(60)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,3]))
 
pari: [g,chi] = znchar(Mod(13,60))
 

Basic properties

Modulus: \(60\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 60.k

\(\chi_{60}(13,\cdot)\) \(\chi_{60}(37,\cdot)\)

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

\((31,41,37)\) → \((1,1,-i)\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(-i\)\(1\)\(i\)\(-i\)\(-1\)\(i\)\(-1\)\(1\)\(-i\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 60 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{60}(13,\cdot)) = \sum_{r\in \Z/60\Z} \chi_{60}(13,r) e\left(\frac{r}{30}\right) = 2.3511410092+3.8042260652i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 60 }(13,·),\chi_{ 60 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{60}(13,\cdot),\chi_{60}(1,\cdot)) = \sum_{r\in \Z/60\Z} \chi_{60}(13,r) \chi_{60}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 60 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{60}(13,·)) = \sum_{r \in \Z/60\Z} \chi_{60}(13,r) e\left(\frac{1 r + 2 r^{-1}}{60}\right) = -0.0 \)