Properties

Label 705.38
Modulus $705$
Conductor $705$
Order $92$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(705, base_ring=CyclotomicField(92))
 
M = H._module
 
chi = DirichletCharacter(H, M([46,69,34]))
 
pari: [g,chi] = znchar(Mod(38,705))
 

Basic properties

Modulus: \(705\)
Conductor: \(705\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
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 705.u

\(\chi_{705}(23,\cdot)\) \(\chi_{705}(38,\cdot)\) \(\chi_{705}(62,\cdot)\) \(\chi_{705}(77,\cdot)\) \(\chi_{705}(92,\cdot)\) \(\chi_{705}(107,\cdot)\) \(\chi_{705}(113,\cdot)\) \(\chi_{705}(137,\cdot)\) \(\chi_{705}(152,\cdot)\) \(\chi_{705}(167,\cdot)\) \(\chi_{705}(182,\cdot)\) \(\chi_{705}(203,\cdot)\) \(\chi_{705}(218,\cdot)\) \(\chi_{705}(227,\cdot)\) \(\chi_{705}(233,\cdot)\) \(\chi_{705}(248,\cdot)\) \(\chi_{705}(257,\cdot)\) \(\chi_{705}(278,\cdot)\) \(\chi_{705}(287,\cdot)\) \(\chi_{705}(293,\cdot)\) \(\chi_{705}(302,\cdot)\) \(\chi_{705}(308,\cdot)\) \(\chi_{705}(317,\cdot)\) \(\chi_{705}(323,\cdot)\) \(\chi_{705}(362,\cdot)\) \(\chi_{705}(368,\cdot)\) \(\chi_{705}(398,\cdot)\) \(\chi_{705}(407,\cdot)\) \(\chi_{705}(428,\cdot)\) \(\chi_{705}(443,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((236,142,616)\) → \((-1,-i,e\left(\frac{17}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 705 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{83}{92}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{65}{92}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{29}{92}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{15}{92}\right)\)\(e\left(\frac{3}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 705 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 705 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 705 }(38,·),\chi_{ 705 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 705 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2