Properties

Label 8512.3805
Modulus $8512$
Conductor $8512$
Order $144$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, base_ring=CyclotomicField(144))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,99,96,128]))
 
pari: [g,chi] = znchar(Mod(3805,8512))
 

Basic properties

Modulus: \(8512\)
Conductor: \(8512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(144\)
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 8512.lm

\(\chi_{8512}(541,\cdot)\) \(\chi_{8512}(613,\cdot)\) \(\chi_{8512}(709,\cdot)\) \(\chi_{8512}(821,\cdot)\) \(\chi_{8512}(1005,\cdot)\) \(\chi_{8512}(1061,\cdot)\) \(\chi_{8512}(1605,\cdot)\) \(\chi_{8512}(1677,\cdot)\) \(\chi_{8512}(1773,\cdot)\) \(\chi_{8512}(1885,\cdot)\) \(\chi_{8512}(2069,\cdot)\) \(\chi_{8512}(2125,\cdot)\) \(\chi_{8512}(2669,\cdot)\) \(\chi_{8512}(2741,\cdot)\) \(\chi_{8512}(2837,\cdot)\) \(\chi_{8512}(2949,\cdot)\) \(\chi_{8512}(3133,\cdot)\) \(\chi_{8512}(3189,\cdot)\) \(\chi_{8512}(3733,\cdot)\) \(\chi_{8512}(3805,\cdot)\) \(\chi_{8512}(3901,\cdot)\) \(\chi_{8512}(4013,\cdot)\) \(\chi_{8512}(4197,\cdot)\) \(\chi_{8512}(4253,\cdot)\) \(\chi_{8512}(4797,\cdot)\) \(\chi_{8512}(4869,\cdot)\) \(\chi_{8512}(4965,\cdot)\) \(\chi_{8512}(5077,\cdot)\) \(\chi_{8512}(5261,\cdot)\) \(\chi_{8512}(5317,\cdot)\) ...

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: $\Q(\zeta_{144})$
Fixed field: Number field defined by a degree 144 polynomial (not computed)

Values on generators

\((5055,6917,7297,3137)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{2}{3}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 8512 }(3805, a) \) \(1\)\(1\)\(e\left(\frac{41}{144}\right)\)\(e\left(\frac{35}{144}\right)\)\(e\left(\frac{41}{72}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{109}{144}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{53}{72}\right)\)\(e\left(\frac{35}{72}\right)\)\(e\left(\frac{41}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8512 }(3805,a) \;\) at \(\;a = \) e.g. 2