Properties

Label 8027.1772
Modulus $8027$
Conductor $349$
Order $58$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13]))
 
pari: [g,chi] = znchar(Mod(1772,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(349\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{349}(27,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8027.v

\(\chi_{8027}(139,\cdot)\) \(\chi_{8027}(231,\cdot)\) \(\chi_{8027}(530,\cdot)\) \(\chi_{8027}(921,\cdot)\) \(\chi_{8027}(1496,\cdot)\) \(\chi_{8027}(1657,\cdot)\) \(\chi_{8027}(1772,\cdot)\) \(\chi_{8027}(2163,\cdot)\) \(\chi_{8027}(2186,\cdot)\) \(\chi_{8027}(2209,\cdot)\) \(\chi_{8027}(2761,\cdot)\) \(\chi_{8027}(3221,\cdot)\) \(\chi_{8027}(3773,\cdot)\) \(\chi_{8027}(4233,\cdot)\) \(\chi_{8027}(4601,\cdot)\) \(\chi_{8027}(4923,\cdot)\) \(\chi_{8027}(4946,\cdot)\) \(\chi_{8027}(5360,\cdot)\) \(\chi_{8027}(5659,\cdot)\) \(\chi_{8027}(5705,\cdot)\) \(\chi_{8027}(5866,\cdot)\) \(\chi_{8027}(5981,\cdot)\) \(\chi_{8027}(6648,\cdot)\) \(\chi_{8027}(6717,\cdot)\) \(\chi_{8027}(6809,\cdot)\) \(\chi_{8027}(7016,\cdot)\) \(\chi_{8027}(7568,\cdot)\) \(\chi_{8027}(7637,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((350,5935)\) → \((1,e\left(\frac{13}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(1772, a) \) \(1\)\(1\)\(e\left(\frac{13}{58}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{3}{58}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{35}{58}\right)\)\(e\left(\frac{41}{58}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(1772,a) \;\) at \(\;a = \) e.g. 2