Properties

Label 8400.13
Modulus $8400$
Conductor $2800$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8400, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0,19,10]))
 
pari: [g,chi] = znchar(Mod(13,8400))
 

Basic properties

Modulus: \(8400\)
Conductor: \(2800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2800}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8400.hz

\(\chi_{8400}(13,\cdot)\) \(\chi_{8400}(517,\cdot)\) \(\chi_{8400}(2197,\cdot)\) \(\chi_{8400}(3373,\cdot)\) \(\chi_{8400}(3877,\cdot)\) \(\chi_{8400}(5053,\cdot)\) \(\chi_{8400}(6733,\cdot)\) \(\chi_{8400}(7237,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.29619676669542400000000000000000000000000000000000.2

Values on generators

\((3151,2101,2801,5377,3601)\) → \((1,-i,1,e\left(\frac{19}{20}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8400 }(13, a) \) \(1\)\(1\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8400 }(13,a) \;\) at \(\;a = \) e.g. 2