Properties

Label 5600.289
Modulus $5600$
Conductor $175$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5600, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,9,10]))
 
pari: [g,chi] = znchar(Mod(289,5600))
 

Basic properties

Modulus: \(5600\)
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 \(\chi_{175}(114,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5600.ge

\(\chi_{5600}(289,\cdot)\) \(\chi_{5600}(1089,\cdot)\) \(\chi_{5600}(1409,\cdot)\) \(\chi_{5600}(2209,\cdot)\) \(\chi_{5600}(2529,\cdot)\) \(\chi_{5600}(3329,\cdot)\) \(\chi_{5600}(4769,\cdot)\) \(\chi_{5600}(5569,\cdot)\)

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: 30.30.35434884492252294752034913472016341984272003173828125.1

Values on generators

\((351,4901,5377,801)\) → \((1,1,e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 5600 }(289, a) \) \(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5600 }(289,a) \;\) at \(\;a = \) e.g. 2