Properties

Modulus 235
Conductor 235
Order 92
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 235.k

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(235)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([23,64]))
 
pari: [g,chi] = znchar(Mod(7,235))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 235
Conductor = 235
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 92
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 235.k
Orbit index = 11

Galois orbit

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

\(\chi_{235}(2,\cdot)\) \(\chi_{235}(3,\cdot)\) \(\chi_{235}(7,\cdot)\) \(\chi_{235}(8,\cdot)\) \(\chi_{235}(12,\cdot)\) \(\chi_{235}(17,\cdot)\) \(\chi_{235}(18,\cdot)\) \(\chi_{235}(27,\cdot)\) \(\chi_{235}(28,\cdot)\) \(\chi_{235}(32,\cdot)\) \(\chi_{235}(37,\cdot)\) \(\chi_{235}(42,\cdot)\) \(\chi_{235}(53,\cdot)\) \(\chi_{235}(63,\cdot)\) \(\chi_{235}(68,\cdot)\) \(\chi_{235}(72,\cdot)\) \(\chi_{235}(83,\cdot)\) \(\chi_{235}(97,\cdot)\) \(\chi_{235}(98,\cdot)\) \(\chi_{235}(102,\cdot)\) \(\chi_{235}(103,\cdot)\) \(\chi_{235}(108,\cdot)\) \(\chi_{235}(112,\cdot)\) \(\chi_{235}(118,\cdot)\) \(\chi_{235}(122,\cdot)\) \(\chi_{235}(128,\cdot)\) \(\chi_{235}(143,\cdot)\) \(\chi_{235}(147,\cdot)\) \(\chi_{235}(148,\cdot)\) \(\chi_{235}(153,\cdot)\) ...

Values on generators

\((142,146)\) → \((i,e\left(\frac{16}{23}\right))\)

Values

-112346789111213
\(-1\)\(1\)\(e\left(\frac{71}{92}\right)\)\(e\left(\frac{61}{92}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{47}{92}\right)\)\(e\left(\frac{29}{92}\right)\)\(e\left(\frac{15}{46}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{19}{92}\right)\)\(e\left(\frac{37}{92}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{92})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 235 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{235}(7,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(7,r) e\left(\frac{2r}{235}\right) = -10.1026596802+11.5297991043i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 235 }(7,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{235}(7,\cdot),\chi_{235}(1,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(7,r) \chi_{235}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 235 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{235}(7,·)) = \sum_{r \in \Z/235\Z} \chi_{235}(7,r) e\left(\frac{1 r + 2 r^{-1}}{235}\right) = 12.4207791922+14.2447514577i \)