Properties

Label 8021.157
Modulus $8021$
Conductor $617$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8021, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,35]))
 
pari: [g,chi] = znchar(Mod(157,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{617}(157,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8021.bv

\(\chi_{8021}(157,\cdot)\) \(\chi_{8021}(352,\cdot)\) \(\chi_{8021}(1249,\cdot)\) \(\chi_{8021}(1522,\cdot)\) \(\chi_{8021}(1561,\cdot)\) \(\chi_{8021}(1782,\cdot)\) \(\chi_{8021}(1951,\cdot)\) \(\chi_{8021}(2003,\cdot)\) \(\chi_{8021}(2692,\cdot)\) \(\chi_{8021}(2861,\cdot)\) \(\chi_{8021}(3550,\cdot)\) \(\chi_{8021}(3602,\cdot)\) \(\chi_{8021}(3771,\cdot)\) \(\chi_{8021}(3992,\cdot)\) \(\chi_{8021}(4031,\cdot)\) \(\chi_{8021}(4304,\cdot)\) \(\chi_{8021}(5201,\cdot)\) \(\chi_{8021}(5396,\cdot)\) \(\chi_{8021}(6631,\cdot)\) \(\chi_{8021}(6943,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((6788,2471)\) → \((1,e\left(\frac{35}{44}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(157, a) \) \(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(i\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(157,a) \;\) at \(\;a = \) e.g. 2