Properties

Label 191.41
Modulus $191$
Conductor $191$
Order $38$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(191)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33]))
 
pari: [g,chi] = znchar(Mod(41,191))
 

Basic properties

Modulus: \(191\)
Conductor: \(191\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 191.f

\(\chi_{191}(11,\cdot)\) \(\chi_{191}(14,\cdot)\) \(\chi_{191}(31,\cdot)\) \(\chi_{191}(37,\cdot)\) \(\chi_{191}(38,\cdot)\) \(\chi_{191}(41,\cdot)\) \(\chi_{191}(55,\cdot)\) \(\chi_{191}(66,\cdot)\) \(\chi_{191}(70,\cdot)\) \(\chi_{191}(84,\cdot)\) \(\chi_{191}(122,\cdot)\) \(\chi_{191}(139,\cdot)\) \(\chi_{191}(155,\cdot)\) \(\chi_{191}(159,\cdot)\) \(\chi_{191}(161,\cdot)\) \(\chi_{191}(166,\cdot)\) \(\chi_{191}(185,\cdot)\) \(\chi_{191}(186,\cdot)\)

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

Values on generators

\(19\) → \(e\left(\frac{33}{38}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{18}{19}\right)\)\(-1\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{31}{38}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{19})\)
Fixed field: 38.0.2501696311112367702213593384284049957523786814936027691413131289153060507631187229631.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 191 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{191}(41,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(41,r) e\left(\frac{2r}{191}\right) = 10.4990524173+8.9872074828i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 191 }(41,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{191}(41,\cdot),\chi_{191}(1,\cdot)) = \sum_{r\in \Z/191\Z} \chi_{191}(41,r) \chi_{191}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 191 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{191}(41,·)) = \sum_{r \in \Z/191\Z} \chi_{191}(41,r) e\left(\frac{1 r + 2 r^{-1}}{191}\right) = 3.622320982+-4.6539580928i \)