Properties

Label 690.7
Modulus $690$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: 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([0,11,38]))
 
pari: [g,chi] = znchar(Mod(7,690))
 

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 \(\chi_{115}(7,\cdot)\)
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 ]
 

Values on generators

\((461,277,511)\) → \((1,i,e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{3}{44}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 690 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{690}(7,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) e\left(\frac{r}{345}\right) = 7.7979078568+-7.3615645794i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 690 }(7,·),\chi_{ 690 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{690}(7,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(7,r) \chi_{690}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 690 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{690}(7,·)) = \sum_{r \in \Z/690\Z} \chi_{690}(7,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right) = 41.5825976865+-2.9740460828i \)