Properties

Label 309.142
Modulus $309$
Conductor $103$
Order $34$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(142,309))
 

Basic properties

Modulus: \(309\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 309.j

\(\chi_{309}(10,\cdot)\) \(\chi_{309}(22,\cdot)\) \(\chi_{309}(31,\cdot)\) \(\chi_{309}(37,\cdot)\) \(\chi_{309}(73,\cdot)\) \(\chi_{309}(94,\cdot)\) \(\chi_{309}(106,\cdot)\) \(\chi_{309}(127,\cdot)\) \(\chi_{309}(130,\cdot)\) \(\chi_{309}(142,\cdot)\) \(\chi_{309}(145,\cdot)\) \(\chi_{309}(172,\cdot)\) \(\chi_{309}(193,\cdot)\) \(\chi_{309}(286,\cdot)\) \(\chi_{309}(295,\cdot)\) \(\chi_{309}(301,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 34 polynomial

Values on generators

\((104,211)\) → \((1,e\left(\frac{3}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 309 }(142, a) \) \(-1\)\(1\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{33}{34}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{6}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{9}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 309 }(142,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 309 }(142,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 309 }(142,·),\chi_{ 309 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 309 }(142,·)) \;\) at \(\; a,b = \) e.g. 1,2