Properties

Label 2800.2577
Modulus $2800$
Conductor $25$
Order $20$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1,0]))
 
pari: [g,chi] = znchar(Mod(2577,2800))
 

Basic properties

Modulus: \(2800\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2800.ea

\(\chi_{2800}(113,\cdot)\) \(\chi_{2800}(337,\cdot)\) \(\chi_{2800}(673,\cdot)\) \(\chi_{2800}(897,\cdot)\) \(\chi_{2800}(1233,\cdot)\) \(\chi_{2800}(2017,\cdot)\) \(\chi_{2800}(2353,\cdot)\) \(\chi_{2800}(2577,\cdot)\)

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: Number field defined by a degree 20 polynomial

Values on generators

\((351,2101,2577,801)\) → \((1,1,e\left(\frac{1}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2800 }(2577, a) \) \(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2800 }(2577,a) \;\) at \(\;a = \) e.g. 2