Properties

Label 8024.41
Modulus $8024$
Conductor $1003$
Order $464$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8024, base_ring=CyclotomicField(464))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,319,112]))
 
pari: [g,chi] = znchar(Mod(41,8024))
 

Basic properties

Modulus: \(8024\)
Conductor: \(1003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(464\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1003}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8024.cy

\(\chi_{8024}(41,\cdot)\) \(\chi_{8024}(57,\cdot)\) \(\chi_{8024}(105,\cdot)\) \(\chi_{8024}(193,\cdot)\) \(\chi_{8024}(241,\cdot)\) \(\chi_{8024}(265,\cdot)\) \(\chi_{8024}(369,\cdot)\) \(\chi_{8024}(449,\cdot)\) \(\chi_{8024}(481,\cdot)\) \(\chi_{8024}(513,\cdot)\) \(\chi_{8024}(521,\cdot)\) \(\chi_{8024}(609,\cdot)\) \(\chi_{8024}(617,\cdot)\) \(\chi_{8024}(641,\cdot)\) \(\chi_{8024}(737,\cdot)\) \(\chi_{8024}(753,\cdot)\) \(\chi_{8024}(793,\cdot)\) \(\chi_{8024}(889,\cdot)\) \(\chi_{8024}(913,\cdot)\) \(\chi_{8024}(921,\cdot)\) \(\chi_{8024}(993,\cdot)\) \(\chi_{8024}(1025,\cdot)\) \(\chi_{8024}(1049,\cdot)\) \(\chi_{8024}(1065,\cdot)\) \(\chi_{8024}(1081,\cdot)\) \(\chi_{8024}(1185,\cdot)\) \(\chi_{8024}(1201,\cdot)\) \(\chi_{8024}(1265,\cdot)\) \(\chi_{8024}(1425,\cdot)\) \(\chi_{8024}(1433,\cdot)\) ...

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: $\Q(\zeta_{464})$
Fixed field: Number field defined by a degree 464 polynomial (not computed)

Values on generators

\((2007,4013,3777,3129)\) → \((1,1,e\left(\frac{11}{16}\right),e\left(\frac{7}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 8024 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{351}{464}\right)\)\(e\left(\frac{411}{464}\right)\)\(e\left(\frac{421}{464}\right)\)\(e\left(\frac{119}{232}\right)\)\(e\left(\frac{393}{464}\right)\)\(e\left(\frac{71}{116}\right)\)\(e\left(\frac{149}{232}\right)\)\(e\left(\frac{185}{232}\right)\)\(e\left(\frac{77}{116}\right)\)\(e\left(\frac{433}{464}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8024 }(41,a) \;\) at \(\;a = \) e.g. 2