Properties

Label 33.13
Modulus $33$
Conductor $11$
Order $10$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(13,33))
 

Basic properties

Modulus: \(33\)
Conductor: \(11\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{11}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 33.g

\(\chi_{33}(7,\cdot)\) \(\chi_{33}(13,\cdot)\) \(\chi_{33}(19,\cdot)\) \(\chi_{33}(28,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{11})\)

Values on generators

\((23,13)\) → \((1,e\left(\frac{1}{10}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 33 }(13, a) \) \(-1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(-1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 33 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 33 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 33 }(13,·),\chi_{ 33 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 33 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2