Properties

Label 690.587
Modulus $690$
Conductor $345$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,11,40]))
 
pari: [g,chi] = znchar(Mod(587,690))
 

Basic properties

Modulus: \(690\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{345}(242,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 690.x

\(\chi_{690}(77,\cdot)\) \(\chi_{690}(167,\cdot)\) \(\chi_{690}(173,\cdot)\) \(\chi_{690}(197,\cdot)\) \(\chi_{690}(233,\cdot)\) \(\chi_{690}(257,\cdot)\) \(\chi_{690}(317,\cdot)\) \(\chi_{690}(347,\cdot)\) \(\chi_{690}(353,\cdot)\) \(\chi_{690}(377,\cdot)\) \(\chi_{690}(407,\cdot)\) \(\chi_{690}(443,\cdot)\) \(\chi_{690}(473,\cdot)\) \(\chi_{690}(533,\cdot)\) \(\chi_{690}(587,\cdot)\) \(\chi_{690}(593,\cdot)\) \(\chi_{690}(623,\cdot)\) \(\chi_{690}(647,\cdot)\) \(\chi_{690}(653,\cdot)\) \(\chi_{690}(683,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((461,277,511)\) → \((-1,i,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 690 }(587, a) \) \(1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{13}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 690 }(587,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 690 }(587,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 690 }(587,·),\chi_{ 690 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 690 }(587,·)) \;\) at \(\; a,b = \) e.g. 1,2