Properties

Label 4400.431
Modulus $4400$
Conductor $1100$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,0,4,1]))
 
pari: [g,chi] = znchar(Mod(431,4400))
 

Basic properties

Modulus: \(4400\)
Conductor: \(1100\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1100}(431,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4400.ed

\(\chi_{4400}(431,\cdot)\) \(\chi_{4400}(591,\cdot)\) \(\chi_{4400}(2911,\cdot)\) \(\chi_{4400}(3471,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.368429326718750000000000.2

Values on generators

\((2751,3301,177,1201)\) → \((-1,1,e\left(\frac{2}{5}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4400 }(431, a) \) \(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4400 }(431,a) \;\) at \(\;a = \) e.g. 2