Properties

Label 3234.1987
Modulus $3234$
Conductor $539$
Order $70$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,49]))
 
pari: [g,chi] = znchar(Mod(1987,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(370,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.ca

\(\chi_{3234}(13,\cdot)\) \(\chi_{3234}(139,\cdot)\) \(\chi_{3234}(349,\cdot)\) \(\chi_{3234}(475,\cdot)\) \(\chi_{3234}(601,\cdot)\) \(\chi_{3234}(811,\cdot)\) \(\chi_{3234}(853,\cdot)\) \(\chi_{3234}(937,\cdot)\) \(\chi_{3234}(1063,\cdot)\) \(\chi_{3234}(1315,\cdot)\) \(\chi_{3234}(1399,\cdot)\) \(\chi_{3234}(1525,\cdot)\) \(\chi_{3234}(1735,\cdot)\) \(\chi_{3234}(1777,\cdot)\) \(\chi_{3234}(1987,\cdot)\) \(\chi_{3234}(2197,\cdot)\) \(\chi_{3234}(2239,\cdot)\) \(\chi_{3234}(2323,\cdot)\) \(\chi_{3234}(2659,\cdot)\) \(\chi_{3234}(2701,\cdot)\) \(\chi_{3234}(2785,\cdot)\) \(\chi_{3234}(2911,\cdot)\) \(\chi_{3234}(3121,\cdot)\) \(\chi_{3234}(3163,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1079,199,2059)\) → \((1,e\left(\frac{1}{14}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1987, a) \) \(1\)\(1\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{6}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(1987,a) \;\) at \(\;a = \) e.g. 2