Properties

Label 9450.1187
Modulus $9450$
Conductor $525$
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(9450, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,27,40]))
 
pari: [g,chi] = znchar(Mod(1187,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(525\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{525}(137,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.fq

\(\chi_{9450}(53,\cdot)\) \(\chi_{9450}(863,\cdot)\) \(\chi_{9450}(1187,\cdot)\) \(\chi_{9450}(1997,\cdot)\) \(\chi_{9450}(2753,\cdot)\) \(\chi_{9450}(3077,\cdot)\) \(\chi_{9450}(3833,\cdot)\) \(\chi_{9450}(3887,\cdot)\) \(\chi_{9450}(4967,\cdot)\) \(\chi_{9450}(5723,\cdot)\) \(\chi_{9450}(5777,\cdot)\) \(\chi_{9450}(6533,\cdot)\) \(\chi_{9450}(7613,\cdot)\) \(\chi_{9450}(7667,\cdot)\) \(\chi_{9450}(8423,\cdot)\) \(\chi_{9450}(8747,\cdot)\)

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

\((9101,6427,6751)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(1187, a) \) \(1\)\(1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{3}{10}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(1187,a) \;\) at \(\;a = \) e.g. 2