Properties

Label 8001.104
Modulus $8001$
Conductor $8001$
Order $126$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,63,58]))
 
pari: [g,chi] = znchar(Mod(104,8001))
 

Basic properties

Modulus: \(8001\)
Conductor: \(8001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8001.mf

\(\chi_{8001}(104,\cdot)\) \(\chi_{8001}(209,\cdot)\) \(\chi_{8001}(272,\cdot)\) \(\chi_{8001}(398,\cdot)\) \(\chi_{8001}(797,\cdot)\) \(\chi_{8001}(1154,\cdot)\) \(\chi_{8001}(1217,\cdot)\) \(\chi_{8001}(1301,\cdot)\) \(\chi_{8001}(1406,\cdot)\) \(\chi_{8001}(1427,\cdot)\) \(\chi_{8001}(1469,\cdot)\) \(\chi_{8001}(1847,\cdot)\) \(\chi_{8001}(1931,\cdot)\) \(\chi_{8001}(3002,\cdot)\) \(\chi_{8001}(3254,\cdot)\) \(\chi_{8001}(3317,\cdot)\) \(\chi_{8001}(3569,\cdot)\) \(\chi_{8001}(4262,\cdot)\) \(\chi_{8001}(4367,\cdot)\) \(\chi_{8001}(4388,\cdot)\) \(\chi_{8001}(4997,\cdot)\) \(\chi_{8001}(5249,\cdot)\) \(\chi_{8001}(5375,\cdot)\) \(\chi_{8001}(5396,\cdot)\) \(\chi_{8001}(5585,\cdot)\) \(\chi_{8001}(5648,\cdot)\) \(\chi_{8001}(5963,\cdot)\) \(\chi_{8001}(6005,\cdot)\) \(\chi_{8001}(6089,\cdot)\) \(\chi_{8001}(6194,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((3557,1144,7750)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{29}{63}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8001 }(104, a) \) \(1\)\(1\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{17}{126}\right)\)\(e\left(\frac{55}{126}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{31}{63}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8001 }(104,a) \;\) at \(\;a = \) e.g. 2