Properties

Label 430.417
Modulus $430$
Conductor $215$
Order $84$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,22]))
 
pari: [g,chi] = znchar(Mod(417,430))
 

Basic properties

Modulus: \(430\)
Conductor: \(215\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{215}(202,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 430.x

\(\chi_{430}(3,\cdot)\) \(\chi_{430}(33,\cdot)\) \(\chi_{430}(63,\cdot)\) \(\chi_{430}(73,\cdot)\) \(\chi_{430}(77,\cdot)\) \(\chi_{430}(147,\cdot)\) \(\chi_{430}(157,\cdot)\) \(\chi_{430}(163,\cdot)\) \(\chi_{430}(177,\cdot)\) \(\chi_{430}(227,\cdot)\) \(\chi_{430}(233,\cdot)\) \(\chi_{430}(243,\cdot)\) \(\chi_{430}(263,\cdot)\) \(\chi_{430}(277,\cdot)\) \(\chi_{430}(287,\cdot)\) \(\chi_{430}(313,\cdot)\) \(\chi_{430}(327,\cdot)\) \(\chi_{430}(347,\cdot)\) \(\chi_{430}(363,\cdot)\) \(\chi_{430}(373,\cdot)\) \(\chi_{430}(377,\cdot)\) \(\chi_{430}(407,\cdot)\) \(\chi_{430}(413,\cdot)\) \(\chi_{430}(417,\cdot)\)

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

\((87,261)\) → \((i,e\left(\frac{11}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 430 }(417, a) \) \(1\)\(1\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{1}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 430 }(417,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 430 }(417,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 430 }(417,·),\chi_{ 430 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 430 }(417,·)) \;\) at \(\; a,b = \) e.g. 1,2