Properties

Label 9025.4476
Modulus $9025$
Conductor $361$
Order $57$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,94]))
 
pari: [g,chi] = znchar(Mod(4476,9025))
 

Basic properties

Modulus: \(9025\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(144,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9025.bh

\(\chi_{9025}(26,\cdot)\) \(\chi_{9025}(201,\cdot)\) \(\chi_{9025}(501,\cdot)\) \(\chi_{9025}(676,\cdot)\) \(\chi_{9025}(976,\cdot)\) \(\chi_{9025}(1451,\cdot)\) \(\chi_{9025}(1626,\cdot)\) \(\chi_{9025}(1926,\cdot)\) \(\chi_{9025}(2101,\cdot)\) \(\chi_{9025}(2401,\cdot)\) \(\chi_{9025}(2576,\cdot)\) \(\chi_{9025}(2876,\cdot)\) \(\chi_{9025}(3051,\cdot)\) \(\chi_{9025}(3351,\cdot)\) \(\chi_{9025}(3526,\cdot)\) \(\chi_{9025}(3826,\cdot)\) \(\chi_{9025}(4001,\cdot)\) \(\chi_{9025}(4301,\cdot)\) \(\chi_{9025}(4476,\cdot)\) \(\chi_{9025}(4776,\cdot)\) \(\chi_{9025}(4951,\cdot)\) \(\chi_{9025}(5251,\cdot)\) \(\chi_{9025}(5426,\cdot)\) \(\chi_{9025}(5726,\cdot)\) \(\chi_{9025}(5901,\cdot)\) \(\chi_{9025}(6201,\cdot)\) \(\chi_{9025}(6376,\cdot)\) \(\chi_{9025}(6676,\cdot)\) \(\chi_{9025}(6851,\cdot)\) \(\chi_{9025}(7326,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\((5777,3251)\) → \((1,e\left(\frac{47}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 9025 }(4476, a) \) \(1\)\(1\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{35}{57}\right)\)\(e\left(\frac{37}{57}\right)\)\(e\left(\frac{25}{57}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{16}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9025 }(4476,a) \;\) at \(\;a = \) e.g. 2