Properties

Label 2880.307
Modulus $2880$
Conductor $320$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,15,0,4]))
 
pari: [g,chi] = znchar(Mod(307,2880))
 

Basic properties

Modulus: \(2880\)
Conductor: \(320\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{320}(307,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2880.du

\(\chi_{2880}(307,\cdot)\) \(\chi_{2880}(523,\cdot)\) \(\chi_{2880}(1027,\cdot)\) \(\chi_{2880}(1243,\cdot)\) \(\chi_{2880}(1747,\cdot)\) \(\chi_{2880}(1963,\cdot)\) \(\chi_{2880}(2467,\cdot)\) \(\chi_{2880}(2683,\cdot)\)

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: 16.16.147573952589676412928000000000000.1

Values on generators

\((2431,901,641,577)\) → \((-1,e\left(\frac{15}{16}\right),1,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2880 }(307, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(-1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(1\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2880 }(307,a) \;\) at \(\;a = \) e.g. 2