Properties

Conductor 225
Order 60
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 225.x

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(225)
 
sage: chi = H[13]
 
pari: [g,chi] = znchar(Mod(13,225))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 225
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
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 = 225.x
Orbit index = 24

Galois orbit

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

\(\chi_{225}(13,\cdot)\) \(\chi_{225}(22,\cdot)\) \(\chi_{225}(52,\cdot)\) \(\chi_{225}(58,\cdot)\) \(\chi_{225}(67,\cdot)\) \(\chi_{225}(88,\cdot)\) \(\chi_{225}(97,\cdot)\) \(\chi_{225}(103,\cdot)\) \(\chi_{225}(112,\cdot)\) \(\chi_{225}(133,\cdot)\) \(\chi_{225}(142,\cdot)\) \(\chi_{225}(148,\cdot)\) \(\chi_{225}(178,\cdot)\) \(\chi_{225}(187,\cdot)\) \(\chi_{225}(202,\cdot)\) \(\chi_{225}(223,\cdot)\)

Values on generators

\((101,127)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right))\)

Values

-112478111314161719
\(-1\)\(1\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 225 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{225}(13,\cdot)) = \sum_{r\in \Z/225\Z} \chi_{225}(13,r) e\left(\frac{2r}{225}\right) = -14.9766114627+0.8373225749i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 225 }(13,·),\chi_{ 225 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{225}(13,\cdot),\chi_{225}(1,\cdot)) = \sum_{r\in \Z/225\Z} \chi_{225}(13,r) \chi_{225}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 225 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{225}(13,·)) = \sum_{r \in \Z/225\Z} \chi_{225}(13,r) e\left(\frac{1 r + 2 r^{-1}}{225}\right) = -0.0 \)