Properties

Label 967.916
Modulus $967$
Conductor $967$
Order $23$
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(967, base_ring=CyclotomicField(46))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12]))
 
pari: [g,chi] = znchar(Mod(916,967))
 

Basic properties

Modulus: \(967\)
Conductor: \(967\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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 967.h

\(\chi_{967}(69,\cdot)\) \(\chi_{967}(72,\cdot)\) \(\chi_{967}(133,\cdot)\) \(\chi_{967}(157,\cdot)\) \(\chi_{967}(187,\cdot)\) \(\chi_{967}(196,\cdot)\) \(\chi_{967}(283,\cdot)\) \(\chi_{967}(332,\cdot)\) \(\chi_{967}(349,\cdot)\) \(\chi_{967}(474,\cdot)\) \(\chi_{967}(574,\cdot)\) \(\chi_{967}(641,\cdot)\) \(\chi_{967}(667,\cdot)\) \(\chi_{967}(696,\cdot)\) \(\chi_{967}(703,\cdot)\) \(\chi_{967}(714,\cdot)\) \(\chi_{967}(795,\cdot)\) \(\chi_{967}(873,\cdot)\) \(\chi_{967}(893,\cdot)\) \(\chi_{967}(916,\cdot)\) \(\chi_{967}(926,\cdot)\) \(\chi_{967}(953,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: 23.23.477949960252640343082669666642744217085836889062320413931701625489.1

Values on generators

\(5\) → \(e\left(\frac{6}{23}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{15}{23}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 967 }(916,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{967}(916,\cdot)) = \sum_{r\in \Z/967\Z} \chi_{967}(916,r) e\left(\frac{2r}{967}\right) = -27.1235732679+-15.2089372799i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 967 }(916,·),\chi_{ 967 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{967}(916,\cdot),\chi_{967}(1,\cdot)) = \sum_{r\in \Z/967\Z} \chi_{967}(916,r) \chi_{967}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 967 }(916,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{967}(916,·)) = \sum_{r \in \Z/967\Z} \chi_{967}(916,r) e\left(\frac{1 r + 2 r^{-1}}{967}\right) = -2.213790002+10.6533423709i \)