Properties

Label 930.13
Modulus $930$
Conductor $155$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,45,22]))
 
pari: [g,chi] = znchar(Mod(13,930))
 

Basic properties

Modulus: \(930\)
Conductor: \(155\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{155}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 930.bt

\(\chi_{930}(13,\cdot)\) \(\chi_{930}(43,\cdot)\) \(\chi_{930}(73,\cdot)\) \(\chi_{930}(127,\cdot)\) \(\chi_{930}(313,\cdot)\) \(\chi_{930}(427,\cdot)\) \(\chi_{930}(487,\cdot)\) \(\chi_{930}(517,\cdot)\) \(\chi_{930}(613,\cdot)\) \(\chi_{930}(637,\cdot)\) \(\chi_{930}(673,\cdot)\) \(\chi_{930}(703,\cdot)\) \(\chi_{930}(757,\cdot)\) \(\chi_{930}(787,\cdot)\) \(\chi_{930}(817,\cdot)\) \(\chi_{930}(823,\cdot)\)

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

\((311,187,871)\) → \((1,-i,e\left(\frac{11}{30}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{60}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 930 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{930}(13,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(13,r) e\left(\frac{r}{465}\right) = 8.3065483571+9.2736861275i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 930 }(13,·),\chi_{ 930 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{930}(13,\cdot),\chi_{930}(1,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(13,r) \chi_{930}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 930 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{930}(13,·)) = \sum_{r \in \Z/930\Z} \chi_{930}(13,r) e\left(\frac{1 r + 2 r^{-1}}{930}\right) = 3.2689873334+-20.6395737278i \)