Properties

Label 7200.169
Modulus $7200$
Conductor $3600$
Order $60$
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(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,45,40,54]))
 
pari: [g,chi] = znchar(Mod(169,7200))
 

Basic properties

Modulus: \(7200\)
Conductor: \(3600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3600}(1069,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7200.hf

\(\chi_{7200}(169,\cdot)\) \(\chi_{7200}(409,\cdot)\) \(\chi_{7200}(889,\cdot)\) \(\chi_{7200}(1129,\cdot)\) \(\chi_{7200}(1609,\cdot)\) \(\chi_{7200}(2329,\cdot)\) \(\chi_{7200}(2569,\cdot)\) \(\chi_{7200}(3289,\cdot)\) \(\chi_{7200}(3769,\cdot)\) \(\chi_{7200}(4009,\cdot)\) \(\chi_{7200}(4489,\cdot)\) \(\chi_{7200}(4729,\cdot)\) \(\chi_{7200}(5209,\cdot)\) \(\chi_{7200}(5929,\cdot)\) \(\chi_{7200}(6169,\cdot)\) \(\chi_{7200}(6889,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((6751,901,6401,577)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 7200 }(169, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7200 }(169,a) \;\) at \(\;a = \) e.g. 2