Properties

Label 108.29
Modulus $108$
Conductor $27$
Order $18$
Real no
Primitive no
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(108)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(29,108))
 

Basic properties

Modulus: \(108\)
Conductor: \(27\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{27}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 108.k

\(\chi_{108}(5,\cdot)\) \(\chi_{108}(29,\cdot)\) \(\chi_{108}(41,\cdot)\) \(\chi_{108}(65,\cdot)\) \(\chi_{108}(77,\cdot)\) \(\chi_{108}(101,\cdot)\)

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

\((55,29)\) → \((1,e\left(\frac{1}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{27})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 108 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{108}(29,\cdot)) = \sum_{r\in \Z/108\Z} \chi_{108}(29,r) e\left(\frac{r}{54}\right) = 9.5423816967+-4.1161816718i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 108 }(29,·),\chi_{ 108 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{108}(29,\cdot),\chi_{108}(1,\cdot)) = \sum_{r\in \Z/108\Z} \chi_{108}(29,r) \chi_{108}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 108 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{108}(29,·)) = \sum_{r \in \Z/108\Z} \chi_{108}(29,r) e\left(\frac{1 r + 2 r^{-1}}{108}\right) = 0.0 \)