Properties

Label 2240.1637
Modulus $2240$
Conductor $2240$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,4,8]))
 
pari: [g,chi] = znchar(Mod(1637,2240))
 

Basic properties

Modulus: \(2240\)
Conductor: \(2240\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2240.du

\(\chi_{2240}(13,\cdot)\) \(\chi_{2240}(517,\cdot)\) \(\chi_{2240}(573,\cdot)\) \(\chi_{2240}(1077,\cdot)\) \(\chi_{2240}(1133,\cdot)\) \(\chi_{2240}(1637,\cdot)\) \(\chi_{2240}(1693,\cdot)\) \(\chi_{2240}(2197,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.850734469462919174923747328000000000000.2

Values on generators

\((1471,1541,897,1921)\) → \((1,e\left(\frac{9}{16}\right),i,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2240 }(1637, a) \) \(1\)\(1\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(-1\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2240 }(1637,a) \;\) at \(\;a = \) e.g. 2