Properties

Label 205.11
Modulus $205$
Conductor $41$
Order $40$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(205, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3]))
 
pari: [g,chi] = znchar(Mod(11,205))
 

Basic properties

Modulus: \(205\)
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 \(\chi_{41}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 205.z

\(\chi_{205}(6,\cdot)\) \(\chi_{205}(11,\cdot)\) \(\chi_{205}(26,\cdot)\) \(\chi_{205}(56,\cdot)\) \(\chi_{205}(71,\cdot)\) \(\chi_{205}(76,\cdot)\) \(\chi_{205}(101,\cdot)\) \(\chi_{205}(106,\cdot)\) \(\chi_{205}(111,\cdot)\) \(\chi_{205}(116,\cdot)\) \(\chi_{205}(136,\cdot)\) \(\chi_{205}(151,\cdot)\) \(\chi_{205}(171,\cdot)\) \(\chi_{205}(176,\cdot)\) \(\chi_{205}(181,\cdot)\) \(\chi_{205}(186,\cdot)\)

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

\((42,6)\) → \((1,e\left(\frac{3}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 205 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{17}{20}\right)\)\(i\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{13}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 205 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 205 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 205 }(11,·),\chi_{ 205 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 205 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2