↑

Properties

Label 345.37
Modulus $345$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(44)) M = H._module chi = DirichletCharacter(H, M([0,11,42]))
 
Copy content pari:[g,chi] = znchar(Mod(37,345))
 

Basic properties

Modulus: \(345\)
Conductor: \(115\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(44\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(37,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 345.w

\(\chi_{345}(7,\cdot)\) \(\chi_{345}(28,\cdot)\) \(\chi_{345}(37,\cdot)\) \(\chi_{345}(43,\cdot)\) \(\chi_{345}(67,\cdot)\) \(\chi_{345}(88,\cdot)\) \(\chi_{345}(97,\cdot)\) \(\chi_{345}(103,\cdot)\) \(\chi_{345}(112,\cdot)\) \(\chi_{345}(148,\cdot)\) \(\chi_{345}(157,\cdot)\) \(\chi_{345}(172,\cdot)\) \(\chi_{345}(178,\cdot)\) \(\chi_{345}(217,\cdot)\) \(\chi_{345}(247,\cdot)\) \(\chi_{345}(268,\cdot)\) \(\chi_{345}(283,\cdot)\) \(\chi_{345}(313,\cdot)\) \(\chi_{345}(337,\cdot)\) \(\chi_{345}(343,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((116,277,166)\) → \((1,i,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 345 }(37, a) \) \(1\)\(1\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{9}{11}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 345 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 345 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 345 }(37,·),\chi_{ 345 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 345 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2