Properties

Label 89.58
Modulus $89$
Conductor $89$
Order $88$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(89)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([75]))
 
pari: [g,chi] = znchar(Mod(58,89))
 

Basic properties

Modulus: \(89\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 89.h

\(\chi_{89}(3,\cdot)\) \(\chi_{89}(6,\cdot)\) \(\chi_{89}(7,\cdot)\) \(\chi_{89}(13,\cdot)\) \(\chi_{89}(14,\cdot)\) \(\chi_{89}(15,\cdot)\) \(\chi_{89}(19,\cdot)\) \(\chi_{89}(23,\cdot)\) \(\chi_{89}(24,\cdot)\) \(\chi_{89}(26,\cdot)\) \(\chi_{89}(27,\cdot)\) \(\chi_{89}(28,\cdot)\) \(\chi_{89}(29,\cdot)\) \(\chi_{89}(30,\cdot)\) \(\chi_{89}(31,\cdot)\) \(\chi_{89}(33,\cdot)\) \(\chi_{89}(35,\cdot)\) \(\chi_{89}(38,\cdot)\) \(\chi_{89}(41,\cdot)\) \(\chi_{89}(43,\cdot)\) \(\chi_{89}(46,\cdot)\) \(\chi_{89}(48,\cdot)\) \(\chi_{89}(51,\cdot)\) \(\chi_{89}(54,\cdot)\) \(\chi_{89}(56,\cdot)\) \(\chi_{89}(58,\cdot)\) \(\chi_{89}(59,\cdot)\) \(\chi_{89}(60,\cdot)\) \(\chi_{89}(61,\cdot)\) \(\chi_{89}(62,\cdot)\) ...

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

\(3\) → \(e\left(\frac{75}{88}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{75}{88}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{43}{88}\right)\)\(e\left(\frac{3}{88}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{13}{22}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 89 }(58,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{89}(58,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(58,r) e\left(\frac{2r}{89}\right) = 8.5745484105+3.9340970445i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 89 }(58,·),\chi_{ 89 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{89}(58,\cdot),\chi_{89}(1,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(58,r) \chi_{89}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 89 }(58,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{89}(58,·)) = \sum_{r \in \Z/89\Z} \chi_{89}(58,r) e\left(\frac{1 r + 2 r^{-1}}{89}\right) = -6.727406184+-3.0723034608i \)