Properties

Conductor 25
Order 20
Real No
Primitive No
Parity Odd
Orbit Label 75.k

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(75)
 
sage: chi = H[52]
 
pari: [g,chi] = znchar(Mod(52,75))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 25
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 20
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 75.k
Orbit index = 11

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 ]
 

\(\chi_{75}(13,\cdot)\) \(\chi_{75}(22,\cdot)\) \(\chi_{75}(28,\cdot)\) \(\chi_{75}(37,\cdot)\) \(\chi_{75}(52,\cdot)\) \(\chi_{75}(58,\cdot)\) \(\chi_{75}(67,\cdot)\) \(\chi_{75}(73,\cdot)\)

Inducing primitive character

\(\chi_{25}(2,\cdot)\)

Values on generators

\((26,52)\) → \((1,e\left(\frac{1}{20}\right))\)

Values

-112478111314161719
\(-1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 75 }(52,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{75}(52,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(52,r) e\left(\frac{2r}{75}\right) = -0.6266661678+4.9605735066i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 75 }(52,·),\chi_{ 75 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{75}(52,\cdot),\chi_{75}(1,\cdot)) = \sum_{r\in \Z/75\Z} \chi_{75}(52,r) \chi_{75}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 75 }(52,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{75}(52,·)) = \sum_{r \in \Z/75\Z} \chi_{75}(52,r) e\left(\frac{1 r + 2 r^{-1}}{75}\right) = -0.294435673+1.8589936764i \)