Properties

Modulus 149
Conductor 149
Order 37
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 149.d

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(149)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11]))
 
pari: [g,chi] = znchar(Mod(30,149))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 149
Conductor = 149
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 37
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 149.d
Orbit index = 4

Galois orbit

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

\(\chi_{149}(5,\cdot)\) \(\chi_{149}(6,\cdot)\) \(\chi_{149}(16,\cdot)\) \(\chi_{149}(17,\cdot)\) \(\chi_{149}(19,\cdot)\) \(\chi_{149}(25,\cdot)\) \(\chi_{149}(28,\cdot)\) \(\chi_{149}(29,\cdot)\) \(\chi_{149}(30,\cdot)\) \(\chi_{149}(31,\cdot)\) \(\chi_{149}(33,\cdot)\) \(\chi_{149}(36,\cdot)\) \(\chi_{149}(37,\cdot)\) \(\chi_{149}(39,\cdot)\) \(\chi_{149}(46,\cdot)\) \(\chi_{149}(49,\cdot)\) \(\chi_{149}(63,\cdot)\) \(\chi_{149}(67,\cdot)\) \(\chi_{149}(73,\cdot)\) \(\chi_{149}(80,\cdot)\) \(\chi_{149}(81,\cdot)\) \(\chi_{149}(85,\cdot)\) \(\chi_{149}(88,\cdot)\) \(\chi_{149}(95,\cdot)\) \(\chi_{149}(96,\cdot)\) \(\chi_{149}(102,\cdot)\) \(\chi_{149}(104,\cdot)\) \(\chi_{149}(107,\cdot)\) \(\chi_{149}(114,\cdot)\) \(\chi_{149}(123,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{11}{37}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{11}{37}\right)\)\(e\left(\frac{32}{37}\right)\)\(e\left(\frac{22}{37}\right)\)\(e\left(\frac{34}{37}\right)\)\(e\left(\frac{6}{37}\right)\)\(e\left(\frac{8}{37}\right)\)\(e\left(\frac{33}{37}\right)\)\(e\left(\frac{27}{37}\right)\)\(e\left(\frac{8}{37}\right)\)\(e\left(\frac{15}{37}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{37})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 149 }(30,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{149}(30,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(30,r) e\left(\frac{2r}{149}\right) = -1.8827585508+-12.0604817582i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 149 }(30,·),\chi_{ 149 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{149}(30,\cdot),\chi_{149}(1,\cdot)) = \sum_{r\in \Z/149\Z} \chi_{149}(30,r) \chi_{149}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 149 }(30,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{149}(30,·)) = \sum_{r \in \Z/149\Z} \chi_{149}(30,r) e\left(\frac{1 r + 2 r^{-1}}{149}\right) = 7.7308423882+10.4527889515i \)