Properties

Label 930.11
Modulus $930$
Conductor $93$
Order $30$
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(930, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,0,23]))
 
pari: [g,chi] = znchar(Mod(11,930))
 

Basic properties

Modulus: \(930\)
Conductor: \(93\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{93}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 930.br

\(\chi_{930}(11,\cdot)\) \(\chi_{930}(251,\cdot)\) \(\chi_{930}(551,\cdot)\) \(\chi_{930}(611,\cdot)\) \(\chi_{930}(641,\cdot)\) \(\chi_{930}(761,\cdot)\) \(\chi_{930}(881,\cdot)\) \(\chi_{930}(911,\cdot)\)

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

\((311,187,871)\) → \((-1,1,e\left(\frac{23}{30}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{17}{30}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: \(\Q(\zeta_{93})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 930 }(11,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{930}(11,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(11,r) e\left(\frac{r}{465}\right) = 7.2941256078+6.3083858171i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 930 }(11,·),\chi_{ 930 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{930}(11,\cdot),\chi_{930}(1,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(11,r) \chi_{930}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 930 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{930}(11,·)) = \sum_{r \in \Z/930\Z} \chi_{930}(11,r) e\left(\frac{1 r + 2 r^{-1}}{930}\right) = 0.0 \)