Properties

Label 3456.25
Modulus $3456$
Conductor $1728$
Order $144$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3456, base_ring=CyclotomicField(144))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,80]))
 
pari: [g,chi] = znchar(Mod(25,3456))
 

Basic properties

Modulus: \(3456\)
Conductor: \(1728\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(144\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1728}(133,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3456.cm

\(\chi_{3456}(25,\cdot)\) \(\chi_{3456}(121,\cdot)\) \(\chi_{3456}(169,\cdot)\) \(\chi_{3456}(265,\cdot)\) \(\chi_{3456}(313,\cdot)\) \(\chi_{3456}(409,\cdot)\) \(\chi_{3456}(457,\cdot)\) \(\chi_{3456}(553,\cdot)\) \(\chi_{3456}(601,\cdot)\) \(\chi_{3456}(697,\cdot)\) \(\chi_{3456}(745,\cdot)\) \(\chi_{3456}(841,\cdot)\) \(\chi_{3456}(889,\cdot)\) \(\chi_{3456}(985,\cdot)\) \(\chi_{3456}(1033,\cdot)\) \(\chi_{3456}(1129,\cdot)\) \(\chi_{3456}(1177,\cdot)\) \(\chi_{3456}(1273,\cdot)\) \(\chi_{3456}(1321,\cdot)\) \(\chi_{3456}(1417,\cdot)\) \(\chi_{3456}(1465,\cdot)\) \(\chi_{3456}(1561,\cdot)\) \(\chi_{3456}(1609,\cdot)\) \(\chi_{3456}(1705,\cdot)\) \(\chi_{3456}(1753,\cdot)\) \(\chi_{3456}(1849,\cdot)\) \(\chi_{3456}(1897,\cdot)\) \(\chi_{3456}(1993,\cdot)\) \(\chi_{3456}(2041,\cdot)\) \(\chi_{3456}(2137,\cdot)\) ...

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

\((2431,2053,2945)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 3456 }(25, a) \) \(1\)\(1\)\(e\left(\frac{121}{144}\right)\)\(e\left(\frac{37}{72}\right)\)\(e\left(\frac{77}{144}\right)\)\(e\left(\frac{55}{144}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{71}{72}\right)\)\(e\left(\frac{49}{72}\right)\)\(e\left(\frac{35}{144}\right)\)\(e\left(\frac{11}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3456 }(25,a) \;\) at \(\;a = \) e.g. 2