Properties

Label 4030.3697
Modulus $4030$
Conductor $2015$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,15,8]))
 
pari: [g,chi] = znchar(Mod(3697,4030))
 

Basic properties

Modulus: \(4030\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2015}(1682,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4030.er

\(\chi_{4030}(593,\cdot)\) \(\chi_{4030}(853,\cdot)\) \(\chi_{4030}(1893,\cdot)\) \(\chi_{4030}(2023,\cdot)\) \(\chi_{4030}(2267,\cdot)\) \(\chi_{4030}(2527,\cdot)\) \(\chi_{4030}(3567,\cdot)\) \(\chi_{4030}(3697,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((807,1861,2731)\) → \((i,-i,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4030 }(3697, a) \) \(1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4030 }(3697,a) \;\) at \(\;a = \) e.g. 2