Properties

Label 3234.2099
Modulus $3234$
Conductor $1617$
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([35,25,42]))
 
pari: [g,chi] = znchar(Mod(2099,3234))
 

Basic properties

Modulus: \(3234\)
Conductor: \(1617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1617}(482,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3234.cc

\(\chi_{3234}(125,\cdot)\) \(\chi_{3234}(251,\cdot)\) \(\chi_{3234}(335,\cdot)\) \(\chi_{3234}(377,\cdot)\) \(\chi_{3234}(713,\cdot)\) \(\chi_{3234}(797,\cdot)\) \(\chi_{3234}(839,\cdot)\) \(\chi_{3234}(1049,\cdot)\) \(\chi_{3234}(1259,\cdot)\) \(\chi_{3234}(1301,\cdot)\) \(\chi_{3234}(1511,\cdot)\) \(\chi_{3234}(1637,\cdot)\) \(\chi_{3234}(1721,\cdot)\) \(\chi_{3234}(1973,\cdot)\) \(\chi_{3234}(2099,\cdot)\) \(\chi_{3234}(2183,\cdot)\) \(\chi_{3234}(2225,\cdot)\) \(\chi_{3234}(2435,\cdot)\) \(\chi_{3234}(2561,\cdot)\) \(\chi_{3234}(2687,\cdot)\) \(\chi_{3234}(2897,\cdot)\) \(\chi_{3234}(3023,\cdot)\) \(\chi_{3234}(3107,\cdot)\) \(\chi_{3234}(3149,\cdot)\)

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

\((1079,199,2059)\) → \((-1,e\left(\frac{5}{14}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(2099, a) \) \(1\)\(1\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{23}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3234 }(2099,a) \;\) at \(\;a = \) e.g. 2