Properties

Label 4400.301
Modulus $4400$
Conductor $176$
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(4400, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0,4]))
 
pari: [g,chi] = znchar(Mod(301,4400))
 

Basic properties

Modulus: \(4400\)
Conductor: \(176\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{176}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4400.je

\(\chi_{4400}(301,\cdot)\) \(\chi_{4400}(1301,\cdot)\) \(\chi_{4400}(1501,\cdot)\) \(\chi_{4400}(1901,\cdot)\) \(\chi_{4400}(2501,\cdot)\) \(\chi_{4400}(3501,\cdot)\) \(\chi_{4400}(3701,\cdot)\) \(\chi_{4400}(4101,\cdot)\)

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: 20.20.1655513490330868290261743826894848.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4400 }(301, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{20}\right)\)\(-i\)\(-1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{13}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4400 }(301,a) \;\) at \(\;a = \) e.g. 2