Properties

Label 91.41
Modulus $91$
Conductor $91$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(91, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,1]))
 
pari: [g,chi] = znchar(Mod(41,91))
 

Basic properties

Modulus: \(91\)
Conductor: \(91\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 91.bc

\(\chi_{91}(6,\cdot)\) \(\chi_{91}(20,\cdot)\) \(\chi_{91}(41,\cdot)\) \(\chi_{91}(76,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.210845878198059013.1

Values on generators

\((66,15)\) → \((-1,e\left(\frac{1}{12}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 91 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{91}(41,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(41,r) e\left(\frac{2r}{91}\right) = 9.5362197315+-0.245994374i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 91 }(41,·),\chi_{ 91 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{91}(41,\cdot),\chi_{91}(1,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(41,r) \chi_{91}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 91 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{91}(41,·)) = \sum_{r \in \Z/91\Z} \chi_{91}(41,r) e\left(\frac{1 r + 2 r^{-1}}{91}\right) = -8.7563435888+-2.3462551933i \)