Properties

Label 4400.1119
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([5,0,9,3]))
 
pari: [g,chi] = znchar(Mod(1119,4400))
 

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 \(\chi_{1100}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4400.cu

\(\chi_{4400}(1119,\cdot)\) \(\chi_{4400}(1359,\cdot)\) \(\chi_{4400}(1679,\cdot)\) \(\chi_{4400}(1839,\cdot)\)

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: 10.10.1842146633593750000000000.2

Values on generators

\((2751,3301,177,1201)\) → \((-1,1,e\left(\frac{9}{10}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4400 }(1119, a) \) \(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4400 }(1119,a) \;\) at \(\;a = \) e.g. 2