Properties

Label 149.82
Modulus $149$
Conductor $149$
Order $74$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(149, base_ring=CyclotomicField(74))
 
M = H._module
 
chi = DirichletCharacter(H, M([21]))
 
pari: [g,chi] = znchar(Mod(82,149))
 

Basic properties

Modulus: \(149\)
Conductor: \(149\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(74\)
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 149.e

\(\chi_{149}(4,\cdot)\) \(\chi_{149}(7,\cdot)\) \(\chi_{149}(9,\cdot)\) \(\chi_{149}(20,\cdot)\) \(\chi_{149}(22,\cdot)\) \(\chi_{149}(24,\cdot)\) \(\chi_{149}(26,\cdot)\) \(\chi_{149}(35,\cdot)\) \(\chi_{149}(42,\cdot)\) \(\chi_{149}(45,\cdot)\) \(\chi_{149}(47,\cdot)\) \(\chi_{149}(53,\cdot)\) \(\chi_{149}(54,\cdot)\) \(\chi_{149}(61,\cdot)\) \(\chi_{149}(64,\cdot)\) \(\chi_{149}(68,\cdot)\) \(\chi_{149}(69,\cdot)\) \(\chi_{149}(76,\cdot)\) \(\chi_{149}(82,\cdot)\) \(\chi_{149}(86,\cdot)\) \(\chi_{149}(100,\cdot)\) \(\chi_{149}(103,\cdot)\) \(\chi_{149}(110,\cdot)\) \(\chi_{149}(112,\cdot)\) \(\chi_{149}(113,\cdot)\) \(\chi_{149}(116,\cdot)\) \(\chi_{149}(118,\cdot)\) \(\chi_{149}(119,\cdot)\) \(\chi_{149}(120,\cdot)\) \(\chi_{149}(121,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{37})$
Fixed field: Number field defined by a degree 74 polynomial

Values on generators

\(2\) → \(e\left(\frac{21}{74}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 149 }(82, a) \) \(1\)\(1\)\(e\left(\frac{21}{74}\right)\)\(e\left(\frac{51}{74}\right)\)\(e\left(\frac{21}{37}\right)\)\(e\left(\frac{19}{37}\right)\)\(e\left(\frac{36}{37}\right)\)\(e\left(\frac{11}{37}\right)\)\(e\left(\frac{63}{74}\right)\)\(e\left(\frac{14}{37}\right)\)\(e\left(\frac{59}{74}\right)\)\(e\left(\frac{69}{74}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 149 }(82,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 149 }(82,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 149 }(82,·),\chi_{ 149 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 149 }(82,·)) \;\) at \(\; a,b = \) e.g. 1,2