Properties

Label 861.22
Modulus $861$
Conductor $41$
Order $40$
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(861, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,29]))
 
pari: [g,chi] = znchar(Mod(22,861))
 

Basic properties

Modulus: \(861\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 861.cb

\(\chi_{861}(22,\cdot)\) \(\chi_{861}(106,\cdot)\) \(\chi_{861}(190,\cdot)\) \(\chi_{861}(211,\cdot)\) \(\chi_{861}(253,\cdot)\) \(\chi_{861}(274,\cdot)\) \(\chi_{861}(316,\cdot)\) \(\chi_{861}(358,\cdot)\) \(\chi_{861}(421,\cdot)\) \(\chi_{861}(463,\cdot)\) \(\chi_{861}(505,\cdot)\) \(\chi_{861}(526,\cdot)\) \(\chi_{861}(568,\cdot)\) \(\chi_{861}(589,\cdot)\) \(\chi_{861}(673,\cdot)\) \(\chi_{861}(757,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((575,493,211)\) → \((1,1,e\left(\frac{29}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 861 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{21}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 861 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 861 }(22,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2