Properties

Label 2450.19
Modulus $2450$
Conductor $175$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([27,25]))
 
pari: [g,chi] = znchar(Mod(19,2450))
 

Basic properties

Modulus: \(2450\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2450.bc

\(\chi_{2450}(19,\cdot)\) \(\chi_{2450}(129,\cdot)\) \(\chi_{2450}(509,\cdot)\) \(\chi_{2450}(619,\cdot)\) \(\chi_{2450}(1109,\cdot)\) \(\chi_{2450}(1489,\cdot)\) \(\chi_{2450}(1979,\cdot)\) \(\chi_{2450}(2089,\cdot)\)

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: 30.0.595554103661284317897450790724178659729659557342529296875.1

Values on generators

\((1177,101)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2450 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2450 }(19,a) \;\) at \(\;a = \) e.g. 2