Properties

Label 936.635
Modulus $936$
Conductor $936$
Order $12$
Real no
Primitive yes
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(936, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,6,10,7]))
 
pari: [g,chi] = znchar(Mod(635,936))
 

Basic properties

Modulus: \(936\)
Conductor: \(936\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 936.do

\(\chi_{936}(227,\cdot)\) \(\chi_{936}(275,\cdot)\) \(\chi_{936}(371,\cdot)\) \(\chi_{936}(635,\cdot)\)

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

\((703,469,209,145)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{12}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.182011731961249064312635392.2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 936 }(635,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{936}(635,\cdot)) = \sum_{r\in \Z/936\Z} \chi_{936}(635,r) e\left(\frac{r}{468}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 936 }(635,·),\chi_{ 936 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{936}(635,\cdot),\chi_{936}(1,\cdot)) = \sum_{r\in \Z/936\Z} \chi_{936}(635,r) \chi_{936}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 936 }(635,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{936}(635,·)) = \sum_{r \in \Z/936\Z} \chi_{936}(635,r) e\left(\frac{1 r + 2 r^{-1}}{936}\right) = 7.7301966347+-2.0712999456i \)