Properties

Label 231.116
Modulus $231$
Conductor $231$
Order $30$
Real no
Primitive yes
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(231, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,20,27]))
 
pari: [g,chi] = znchar(Mod(116,231))
 

Basic properties

Modulus: \(231\)
Conductor: \(231\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 231.be

\(\chi_{231}(2,\cdot)\) \(\chi_{231}(74,\cdot)\) \(\chi_{231}(95,\cdot)\) \(\chi_{231}(107,\cdot)\) \(\chi_{231}(116,\cdot)\) \(\chi_{231}(128,\cdot)\) \(\chi_{231}(149,\cdot)\) \(\chi_{231}(200,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.15010049358097810645468215169836895080663686428828497.1

Values on generators

\((155,199,211)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{9}{10}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 231 }(116,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{231}(116,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(116,r) e\left(\frac{2r}{231}\right) = 10.6285486481+10.8643432215i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 231 }(116,·),\chi_{ 231 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{231}(116,\cdot),\chi_{231}(1,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(116,r) \chi_{231}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 231 }(116,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{231}(116,·)) = \sum_{r \in \Z/231\Z} \chi_{231}(116,r) e\left(\frac{1 r + 2 r^{-1}}{231}\right) = 0.0 \)