Properties

Label 345.287
Modulus $345$
Conductor $345$
Order $44$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,11,18]))
 
pari: [g,chi] = znchar(Mod(287,345))
 

Basic properties

Modulus: \(345\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 345.u

\(\chi_{345}(17,\cdot)\) \(\chi_{345}(38,\cdot)\) \(\chi_{345}(53,\cdot)\) \(\chi_{345}(83,\cdot)\) \(\chi_{345}(107,\cdot)\) \(\chi_{345}(113,\cdot)\) \(\chi_{345}(122,\cdot)\) \(\chi_{345}(143,\cdot)\) \(\chi_{345}(152,\cdot)\) \(\chi_{345}(158,\cdot)\) \(\chi_{345}(182,\cdot)\) \(\chi_{345}(203,\cdot)\) \(\chi_{345}(212,\cdot)\) \(\chi_{345}(218,\cdot)\) \(\chi_{345}(227,\cdot)\) \(\chi_{345}(263,\cdot)\) \(\chi_{345}(272,\cdot)\) \(\chi_{345}(287,\cdot)\) \(\chi_{345}(293,\cdot)\) \(\chi_{345}(332,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((116,277,166)\) → \((-1,i,e\left(\frac{9}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 345 }(287, a) \) \(-1\)\(1\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 345 }(287,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 345 }(287,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 345 }(287,·),\chi_{ 345 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 345 }(287,·)) \;\) at \(\; a,b = \) e.g. 1,2