Properties

Label 735.37
Modulus $735$
Conductor $245$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,64]))
 
pari: [g,chi] = znchar(Mod(37,735))
 

Basic properties

Modulus: \(735\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 735.bv

\(\chi_{735}(37,\cdot)\) \(\chi_{735}(58,\cdot)\) \(\chi_{735}(88,\cdot)\) \(\chi_{735}(142,\cdot)\) \(\chi_{735}(163,\cdot)\) \(\chi_{735}(172,\cdot)\) \(\chi_{735}(193,\cdot)\) \(\chi_{735}(247,\cdot)\) \(\chi_{735}(268,\cdot)\) \(\chi_{735}(277,\cdot)\) \(\chi_{735}(298,\cdot)\) \(\chi_{735}(352,\cdot)\) \(\chi_{735}(382,\cdot)\) \(\chi_{735}(403,\cdot)\) \(\chi_{735}(457,\cdot)\) \(\chi_{735}(478,\cdot)\) \(\chi_{735}(487,\cdot)\) \(\chi_{735}(562,\cdot)\) \(\chi_{735}(583,\cdot)\) \(\chi_{735}(592,\cdot)\) \(\chi_{735}(613,\cdot)\) \(\chi_{735}(688,\cdot)\) \(\chi_{735}(697,\cdot)\) \(\chi_{735}(718,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((491,442,346)\) → \((1,i,e\left(\frac{16}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 735 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{59}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 735 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 735 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 735 }(37,·),\chi_{ 735 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 735 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2