Properties

Label 847.29
Modulus $847$
Conductor $121$
Order $110$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,17]))
 
pari: [g,chi] = znchar(Mod(29,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 847.z

\(\chi_{847}(8,\cdot)\) \(\chi_{847}(29,\cdot)\) \(\chi_{847}(50,\cdot)\) \(\chi_{847}(57,\cdot)\) \(\chi_{847}(85,\cdot)\) \(\chi_{847}(106,\cdot)\) \(\chi_{847}(127,\cdot)\) \(\chi_{847}(134,\cdot)\) \(\chi_{847}(162,\cdot)\) \(\chi_{847}(183,\cdot)\) \(\chi_{847}(204,\cdot)\) \(\chi_{847}(211,\cdot)\) \(\chi_{847}(260,\cdot)\) \(\chi_{847}(281,\cdot)\) \(\chi_{847}(288,\cdot)\) \(\chi_{847}(316,\cdot)\) \(\chi_{847}(337,\cdot)\) \(\chi_{847}(358,\cdot)\) \(\chi_{847}(365,\cdot)\) \(\chi_{847}(393,\cdot)\) \(\chi_{847}(414,\cdot)\) \(\chi_{847}(435,\cdot)\) \(\chi_{847}(442,\cdot)\) \(\chi_{847}(470,\cdot)\) \(\chi_{847}(491,\cdot)\) \(\chi_{847}(512,\cdot)\) \(\chi_{847}(519,\cdot)\) \(\chi_{847}(547,\cdot)\) \(\chi_{847}(568,\cdot)\) \(\chi_{847}(589,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((122,365)\) → \((1,e\left(\frac{17}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 847 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{24}{55}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{67}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(29,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2