Properties

Label 4026.31
Modulus $4026$
Conductor $671$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,36,59]))
 
pari: [g,chi] = znchar(Mod(31,4026))
 

Basic properties

Modulus: \(4026\)
Conductor: \(671\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{671}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4026.fw

\(\chi_{4026}(31,\cdot)\) \(\chi_{4026}(181,\cdot)\) \(\chi_{4026}(421,\cdot)\) \(\chi_{4026}(823,\cdot)\) \(\chi_{4026}(1027,\cdot)\) \(\chi_{4026}(1039,\cdot)\) \(\chi_{4026}(1081,\cdot)\) \(\chi_{4026}(1291,\cdot)\) \(\chi_{4026}(1945,\cdot)\) \(\chi_{4026}(2161,\cdot)\) \(\chi_{4026}(2335,\cdot)\) \(\chi_{4026}(2935,\cdot)\) \(\chi_{4026}(2995,\cdot)\) \(\chi_{4026}(3007,\cdot)\) \(\chi_{4026}(3085,\cdot)\) \(\chi_{4026}(3337,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1343,1465,3235)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{59}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4026 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4026 }(31,a) \;\) at \(\;a = \) e.g. 2