Properties

Label 8008.711
Modulus $8008$
Conductor $4004$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,0,20,21,20]))
 
pari: [g,chi] = znchar(Mod(711,8008))
 

Basic properties

Modulus: \(8008\)
Conductor: \(4004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4004}(711,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8008.mz

\(\chi_{8008}(711,\cdot)\) \(\chi_{8008}(1927,\cdot)\) \(\chi_{8008}(2895,\cdot)\) \(\chi_{8008}(3383,\cdot)\) \(\chi_{8008}(4111,\cdot)\) \(\chi_{8008}(4351,\cdot)\) \(\chi_{8008}(5079,\cdot)\) \(\chi_{8008}(7751,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((6007,4005,3433,4369,4929)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 8008 }(711, a) \) \(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{15}\right)\)\(-1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8008 }(711,a) \;\) at \(\;a = \) e.g. 2