Properties

Label 8400.647
Modulus $8400$
Conductor $4200$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8400, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,30,30,51,10]))
 
pari: [g,chi] = znchar(Mod(647,8400))
 

Basic properties

Modulus: \(8400\)
Conductor: \(4200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4200}(2747,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8400.kt

\(\chi_{8400}(647,\cdot)\) \(\chi_{8400}(887,\cdot)\) \(\chi_{8400}(983,\cdot)\) \(\chi_{8400}(1223,\cdot)\) \(\chi_{8400}(2327,\cdot)\) \(\chi_{8400}(2567,\cdot)\) \(\chi_{8400}(2663,\cdot)\) \(\chi_{8400}(2903,\cdot)\) \(\chi_{8400}(4247,\cdot)\) \(\chi_{8400}(4583,\cdot)\) \(\chi_{8400}(5687,\cdot)\) \(\chi_{8400}(5927,\cdot)\) \(\chi_{8400}(6023,\cdot)\) \(\chi_{8400}(6263,\cdot)\) \(\chi_{8400}(7367,\cdot)\) \(\chi_{8400}(7703,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3151,2101,2801,5377,3601)\) → \((-1,-1,-1,e\left(\frac{17}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8400 }(647, a) \) \(1\)\(1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{2}{5}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8400 }(647,a) \;\) at \(\;a = \) e.g. 2