Properties

Label 690.37
Modulus $690$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(690, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,42]))
 
pari: [g,chi] = znchar(Mod(37,690))
 

Basic properties

Modulus: \(690\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 690.w

\(\chi_{690}(7,\cdot)\) \(\chi_{690}(37,\cdot)\) \(\chi_{690}(43,\cdot)\) \(\chi_{690}(67,\cdot)\) \(\chi_{690}(97,\cdot)\) \(\chi_{690}(103,\cdot)\) \(\chi_{690}(157,\cdot)\) \(\chi_{690}(217,\cdot)\) \(\chi_{690}(247,\cdot)\) \(\chi_{690}(283,\cdot)\) \(\chi_{690}(313,\cdot)\) \(\chi_{690}(337,\cdot)\) \(\chi_{690}(343,\cdot)\) \(\chi_{690}(373,\cdot)\) \(\chi_{690}(433,\cdot)\) \(\chi_{690}(457,\cdot)\) \(\chi_{690}(493,\cdot)\) \(\chi_{690}(517,\cdot)\) \(\chi_{690}(523,\cdot)\) \(\chi_{690}(613,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((461,277,511)\) → \((1,i,e\left(\frac{21}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{23}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 690 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 690 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 690 }(37,·),\chi_{ 690 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 690 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2