Properties

Label 4026.7
Modulus $4026$
Conductor $671$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,42,49]))
 
pari: [g,chi] = znchar(Mod(7,4026))
 

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 \(\chi_{671}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4026.fr

\(\chi_{4026}(7,\cdot)\) \(\chi_{4026}(193,\cdot)\) \(\chi_{4026}(547,\cdot)\) \(\chi_{4026}(871,\cdot)\) \(\chi_{4026}(1129,\cdot)\) \(\chi_{4026}(1405,\cdot)\) \(\chi_{4026}(1531,\cdot)\) \(\chi_{4026}(1909,\cdot)\) \(\chi_{4026}(1921,\cdot)\) \(\chi_{4026}(2239,\cdot)\) \(\chi_{4026}(2527,\cdot)\) \(\chi_{4026}(2983,\cdot)\) \(\chi_{4026}(3043,\cdot)\) \(\chi_{4026}(3451,\cdot)\) \(\chi_{4026}(3643,\cdot)\) \(\chi_{4026}(3955,\cdot)\)

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

\((1343,1465,3235)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{49}{60}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4026 }(7, a) \) \(1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{41}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4026 }(7,a) \;\) at \(\;a = \) e.g. 2