Properties

Label 7800.53
Modulus $7800$
Conductor $600$
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(7800, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,10,10,7,0]))
 
pari: [g,chi] = znchar(Mod(53,7800))
 

Basic properties

Modulus: \(7800\)
Conductor: \(600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{600}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7800.hw

\(\chi_{7800}(53,\cdot)\) \(\chi_{7800}(677,\cdot)\) \(\chi_{7800}(1613,\cdot)\) \(\chi_{7800}(2237,\cdot)\) \(\chi_{7800}(3173,\cdot)\) \(\chi_{7800}(3797,\cdot)\) \(\chi_{7800}(4733,\cdot)\) \(\chi_{7800}(6917,\cdot)\)

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: 20.20.184528125000000000000000000000000000000.1

Values on generators

\((1951,3901,5201,7177,4201)\) → \((1,-1,-1,e\left(\frac{7}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 7800 }(53, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7800 }(53,a) \;\) at \(\;a = \) e.g. 2