Properties

Label 7200.241
Modulus $7200$
Conductor $1800$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,20,6]))
 
pari: [g,chi] = znchar(Mod(241,7200))
 

Basic properties

Modulus: \(7200\)
Conductor: \(1800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1800}(1141,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7200.fx

\(\chi_{7200}(241,\cdot)\) \(\chi_{7200}(1681,\cdot)\) \(\chi_{7200}(2641,\cdot)\) \(\chi_{7200}(3121,\cdot)\) \(\chi_{7200}(4081,\cdot)\) \(\chi_{7200}(4561,\cdot)\) \(\chi_{7200}(5521,\cdot)\) \(\chi_{7200}(6961,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((6751,901,6401,577)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 7200 }(241, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{2}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7200 }(241,a) \;\) at \(\;a = \) e.g. 2