Properties

Label 7600.9
Modulus $7600$
Conductor $3800$
Order $90$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,45,63,40]))
 
pari: [g,chi] = znchar(Mod(9,7600))
 

Basic properties

Modulus: \(7600\)
Conductor: \(3800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3800}(1909,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7600.ic

\(\chi_{7600}(9,\cdot)\) \(\chi_{7600}(169,\cdot)\) \(\chi_{7600}(329,\cdot)\) \(\chi_{7600}(1289,\cdot)\) \(\chi_{7600}(1529,\cdot)\) \(\chi_{7600}(1689,\cdot)\) \(\chi_{7600}(2569,\cdot)\) \(\chi_{7600}(2809,\cdot)\) \(\chi_{7600}(2969,\cdot)\) \(\chi_{7600}(3209,\cdot)\) \(\chi_{7600}(3369,\cdot)\) \(\chi_{7600}(4089,\cdot)\) \(\chi_{7600}(4329,\cdot)\) \(\chi_{7600}(4489,\cdot)\) \(\chi_{7600}(4569,\cdot)\) \(\chi_{7600}(4729,\cdot)\) \(\chi_{7600}(4889,\cdot)\) \(\chi_{7600}(5609,\cdot)\) \(\chi_{7600}(6009,\cdot)\) \(\chi_{7600}(6089,\cdot)\) \(\chi_{7600}(6409,\cdot)\) \(\chi_{7600}(7129,\cdot)\) \(\chi_{7600}(7369,\cdot)\) \(\chi_{7600}(7529,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((4751,5701,5777,401)\) → \((1,-1,e\left(\frac{7}{10}\right),e\left(\frac{4}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 7600 }(9, a) \) \(1\)\(1\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{41}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7600 }(9,a) \;\) at \(\;a = \) e.g. 2