Properties

Label 294.103
Modulus $294$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,29]))
 
pari: [g,chi] = znchar(Mod(103,294))
 

Basic properties

Modulus: \(294\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 294.o

\(\chi_{294}(61,\cdot)\) \(\chi_{294}(73,\cdot)\) \(\chi_{294}(103,\cdot)\) \(\chi_{294}(115,\cdot)\) \(\chi_{294}(145,\cdot)\) \(\chi_{294}(157,\cdot)\) \(\chi_{294}(187,\cdot)\) \(\chi_{294}(199,\cdot)\) \(\chi_{294}(229,\cdot)\) \(\chi_{294}(241,\cdot)\) \(\chi_{294}(271,\cdot)\) \(\chi_{294}(283,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((197,199)\) → \((1,e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 294 }(103, a) \) \(-1\)\(1\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 294 }(103,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 294 }(103,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 294 }(103,·),\chi_{ 294 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 294 }(103,·)) \;\) at \(\; a,b = \) e.g. 1,2