Properties

Label 861.145
Modulus $861$
Conductor $287$
Order $120$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,100,87]))
 
pari: [g,chi] = znchar(Mod(145,861))
 

Basic properties

Modulus: \(861\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(145,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 861.ci

\(\chi_{861}(19,\cdot)\) \(\chi_{861}(52,\cdot)\) \(\chi_{861}(94,\cdot)\) \(\chi_{861}(136,\cdot)\) \(\chi_{861}(145,\cdot)\) \(\chi_{861}(157,\cdot)\) \(\chi_{861}(199,\cdot)\) \(\chi_{861}(220,\cdot)\) \(\chi_{861}(229,\cdot)\) \(\chi_{861}(304,\cdot)\) \(\chi_{861}(313,\cdot)\) \(\chi_{861}(334,\cdot)\) \(\chi_{861}(376,\cdot)\) \(\chi_{861}(388,\cdot)\) \(\chi_{861}(397,\cdot)\) \(\chi_{861}(439,\cdot)\) \(\chi_{861}(481,\cdot)\) \(\chi_{861}(514,\cdot)\) \(\chi_{861}(544,\cdot)\) \(\chi_{861}(586,\cdot)\) \(\chi_{861}(598,\cdot)\) \(\chi_{861}(628,\cdot)\) \(\chi_{861}(649,\cdot)\) \(\chi_{861}(682,\cdot)\) \(\chi_{861}(691,\cdot)\) \(\chi_{861}(703,\cdot)\) \(\chi_{861}(712,\cdot)\) \(\chi_{861}(745,\cdot)\) \(\chi_{861}(766,\cdot)\) \(\chi_{861}(796,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((575,493,211)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{29}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 861 }(145, a) \) \(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{61}{120}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{91}{120}\right)\)\(e\left(\frac{83}{120}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 861 }(145,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 861 }(145,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 861 }(145,·),\chi_{ 861 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 861 }(145,·)) \;\) at \(\; a,b = \) e.g. 1,2