Properties

Label 916.45
Modulus $916$
Conductor $229$
Order $114$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(916, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,29]))
 
pari: [g,chi] = znchar(Mod(45,916))
 

Basic properties

Modulus: \(916\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{229}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 916.v

\(\chi_{916}(5,\cdot)\) \(\chi_{916}(33,\cdot)\) \(\chi_{916}(45,\cdot)\) \(\chi_{916}(49,\cdot)\) \(\chi_{916}(85,\cdot)\) \(\chi_{916}(97,\cdot)\) \(\chi_{916}(181,\cdot)\) \(\chi_{916}(209,\cdot)\) \(\chi_{916}(241,\cdot)\) \(\chi_{916}(265,\cdot)\) \(\chi_{916}(285,\cdot)\) \(\chi_{916}(305,\cdot)\) \(\chi_{916}(309,\cdot)\) \(\chi_{916}(329,\cdot)\) \(\chi_{916}(377,\cdot)\) \(\chi_{916}(421,\cdot)\) \(\chi_{916}(433,\cdot)\) \(\chi_{916}(449,\cdot)\) \(\chi_{916}(529,\cdot)\) \(\chi_{916}(557,\cdot)\) \(\chi_{916}(561,\cdot)\) \(\chi_{916}(605,\cdot)\) \(\chi_{916}(673,\cdot)\) \(\chi_{916}(733,\cdot)\) \(\chi_{916}(745,\cdot)\) \(\chi_{916}(749,\cdot)\) \(\chi_{916}(757,\cdot)\) \(\chi_{916}(765,\cdot)\) \(\chi_{916}(805,\cdot)\) \(\chi_{916}(825,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((459,693)\) → \((1,e\left(\frac{29}{114}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 916 }(45, a) \) \(1\)\(1\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{25}{114}\right)\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{5}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 916 }(45,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 916 }(45,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 916 }(45,·),\chi_{ 916 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 916 }(45,·)) \;\) at \(\; a,b = \) e.g. 1,2