Properties

Label 9800.1077
Modulus $9800$
Conductor $1400$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,1,10]))
 
pari: [g,chi] = znchar(Mod(1077,9800))
 

Basic properties

Modulus: \(9800\)
Conductor: \(1400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1400}(1077,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9800.dj

\(\chi_{9800}(1077,\cdot)\) \(\chi_{9800}(2253,\cdot)\) \(\chi_{9800}(3037,\cdot)\) \(\chi_{9800}(4213,\cdot)\) \(\chi_{9800}(4997,\cdot)\) \(\chi_{9800}(6173,\cdot)\) \(\chi_{9800}(8133,\cdot)\) \(\chi_{9800}(8917,\cdot)\)

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: 20.20.882735153125000000000000000000000000000000.1

Values on generators

\((7351,4901,1177,5001)\) → \((1,-1,e\left(\frac{1}{20}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 9800 }(1077, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9800 }(1077,a) \;\) at \(\;a = \) e.g. 2