Properties

Label 1280.3.e.j
Level $1280$
Weight $3$
Character orbit 1280.e
Analytic conductor $34.877$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{7} + q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{3} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{7} + q^{9} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{11} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{13} + ( -8 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{15} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{17} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{19} + 24 \zeta_{24}^{6} q^{21} + ( 18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} ) q^{23} + ( 23 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{25} + ( 20 \zeta_{24} - 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} ) q^{27} + 22 \zeta_{24}^{6} q^{29} + ( 32 - 64 \zeta_{24}^{4} ) q^{31} + ( -16 \zeta_{24} + 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{33} + ( -6 \zeta_{24} + 48 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 24 \zeta_{24}^{6} ) q^{35} + ( -20 \zeta_{24} - 20 \zeta_{24}^{3} - 20 \zeta_{24}^{5} + 40 \zeta_{24}^{7} ) q^{37} + ( 16 - 32 \zeta_{24}^{4} ) q^{39} -22 q^{41} + ( -42 \zeta_{24} + 42 \zeta_{24}^{3} + 42 \zeta_{24}^{5} ) q^{43} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{45} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{47} + 23 q^{49} + ( -64 \zeta_{24}^{2} + 32 \zeta_{24}^{6} ) q^{51} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{53} + ( 8 - 48 \zeta_{24} - 48 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 48 \zeta_{24}^{5} ) q^{55} + ( -16 \zeta_{24} + 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{57} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{59} -46 \zeta_{24}^{6} q^{61} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{63} + ( -48 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{65} + ( 42 \zeta_{24} - 42 \zeta_{24}^{3} - 42 \zeta_{24}^{5} ) q^{67} -72 \zeta_{24}^{6} q^{69} + ( 16 - 32 \zeta_{24}^{4} ) q^{71} + ( -32 \zeta_{24} + 32 \zeta_{24}^{3} - 32 \zeta_{24}^{5} - 64 \zeta_{24}^{7} ) q^{73} + ( 46 \zeta_{24} + 32 \zeta_{24}^{2} - 46 \zeta_{24}^{3} - 46 \zeta_{24}^{5} - 16 \zeta_{24}^{6} ) q^{75} + ( -48 \zeta_{24} - 48 \zeta_{24}^{3} - 48 \zeta_{24}^{5} + 96 \zeta_{24}^{7} ) q^{77} -71 q^{81} + ( -54 \zeta_{24} + 54 \zeta_{24}^{3} + 54 \zeta_{24}^{5} ) q^{83} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 96 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{85} + ( 44 \zeta_{24} + 44 \zeta_{24}^{3} - 44 \zeta_{24}^{5} ) q^{87} -146 q^{89} + ( -96 \zeta_{24}^{2} + 48 \zeta_{24}^{6} ) q^{91} + ( -64 \zeta_{24} - 64 \zeta_{24}^{3} - 64 \zeta_{24}^{5} + 128 \zeta_{24}^{7} ) q^{93} + ( 8 - 48 \zeta_{24} - 48 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 48 \zeta_{24}^{5} ) q^{95} + ( -24 \zeta_{24} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 48 \zeta_{24}^{7} ) q^{97} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} + 184 q^{25} - 176 q^{41} + 184 q^{49} - 384 q^{65} - 568 q^{81} - 1168 q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
639.1
0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0 2.82843i 0 −4.89898 1.00000i 0 −8.48528 0 1.00000 0
639.2 0 2.82843i 0 −4.89898 + 1.00000i 0 8.48528 0 1.00000 0
639.3 0 2.82843i 0 4.89898 1.00000i 0 −8.48528 0 1.00000 0
639.4 0 2.82843i 0 4.89898 + 1.00000i 0 8.48528 0 1.00000 0
639.5 0 2.82843i 0 −4.89898 1.00000i 0 8.48528 0 1.00000 0
639.6 0 2.82843i 0 −4.89898 + 1.00000i 0 −8.48528 0 1.00000 0
639.7 0 2.82843i 0 4.89898 1.00000i 0 8.48528 0 1.00000 0
639.8 0 2.82843i 0 4.89898 + 1.00000i 0 −8.48528 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 639.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.j 8
4.b odd 2 1 inner 1280.3.e.j 8
5.b even 2 1 inner 1280.3.e.j 8
8.b even 2 1 inner 1280.3.e.j 8
8.d odd 2 1 inner 1280.3.e.j 8
16.e even 4 1 80.3.h.b 4
16.e even 4 1 320.3.h.e 4
16.f odd 4 1 80.3.h.b 4
16.f odd 4 1 320.3.h.e 4
20.d odd 2 1 inner 1280.3.e.j 8
40.e odd 2 1 inner 1280.3.e.j 8
40.f even 2 1 inner 1280.3.e.j 8
48.i odd 4 1 720.3.j.e 4
48.k even 4 1 720.3.j.e 4
80.i odd 4 1 400.3.b.h 4
80.i odd 4 1 1600.3.b.t 4
80.j even 4 1 400.3.b.h 4
80.j even 4 1 1600.3.b.t 4
80.k odd 4 1 80.3.h.b 4
80.k odd 4 1 320.3.h.e 4
80.q even 4 1 80.3.h.b 4
80.q even 4 1 320.3.h.e 4
80.s even 4 1 400.3.b.h 4
80.s even 4 1 1600.3.b.t 4
80.t odd 4 1 400.3.b.h 4
80.t odd 4 1 1600.3.b.t 4
240.t even 4 1 720.3.j.e 4
240.z odd 4 1 3600.3.e.bd 4
240.bb even 4 1 3600.3.e.bd 4
240.bd odd 4 1 3600.3.e.bd 4
240.bf even 4 1 3600.3.e.bd 4
240.bm odd 4 1 720.3.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.h.b 4 16.e even 4 1
80.3.h.b 4 16.f odd 4 1
80.3.h.b 4 80.k odd 4 1
80.3.h.b 4 80.q even 4 1
320.3.h.e 4 16.e even 4 1
320.3.h.e 4 16.f odd 4 1
320.3.h.e 4 80.k odd 4 1
320.3.h.e 4 80.q even 4 1
400.3.b.h 4 80.i odd 4 1
400.3.b.h 4 80.j even 4 1
400.3.b.h 4 80.s even 4 1
400.3.b.h 4 80.t odd 4 1
720.3.j.e 4 48.i odd 4 1
720.3.j.e 4 48.k even 4 1
720.3.j.e 4 240.t even 4 1
720.3.j.e 4 240.bm odd 4 1
1280.3.e.j 8 1.a even 1 1 trivial
1280.3.e.j 8 4.b odd 2 1 inner
1280.3.e.j 8 5.b even 2 1 inner
1280.3.e.j 8 8.b even 2 1 inner
1280.3.e.j 8 8.d odd 2 1 inner
1280.3.e.j 8 20.d odd 2 1 inner
1280.3.e.j 8 40.e odd 2 1 inner
1280.3.e.j 8 40.f even 2 1 inner
1600.3.b.t 4 80.i odd 4 1
1600.3.b.t 4 80.j even 4 1
1600.3.b.t 4 80.s even 4 1
1600.3.b.t 4 80.t odd 4 1
3600.3.e.bd 4 240.z odd 4 1
3600.3.e.bd 4 240.bb even 4 1
3600.3.e.bd 4 240.bd odd 4 1
3600.3.e.bd 4 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3}^{2} + 8 \)
\( T_{7}^{2} - 72 \)
\( T_{11}^{2} - 192 \)
\( T_{13}^{2} - 96 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 8 + T^{2} )^{4} \)
$5$ \( ( 625 - 46 T^{2} + T^{4} )^{2} \)
$7$ \( ( -72 + T^{2} )^{4} \)
$11$ \( ( -192 + T^{2} )^{4} \)
$13$ \( ( -96 + T^{2} )^{4} \)
$17$ \( ( 384 + T^{2} )^{4} \)
$19$ \( ( -192 + T^{2} )^{4} \)
$23$ \( ( -648 + T^{2} )^{4} \)
$29$ \( ( 484 + T^{2} )^{4} \)
$31$ \( ( 3072 + T^{2} )^{4} \)
$37$ \( ( -2400 + T^{2} )^{4} \)
$41$ \( ( 22 + T )^{8} \)
$43$ \( ( 3528 + T^{2} )^{4} \)
$47$ \( ( -72 + T^{2} )^{4} \)
$53$ \( ( -864 + T^{2} )^{4} \)
$59$ \( ( -192 + T^{2} )^{4} \)
$61$ \( ( 2116 + T^{2} )^{4} \)
$67$ \( ( 3528 + T^{2} )^{4} \)
$71$ \( ( 768 + T^{2} )^{4} \)
$73$ \( ( 6144 + T^{2} )^{4} \)
$79$ \( T^{8} \)
$83$ \( ( 5832 + T^{2} )^{4} \)
$89$ \( ( 146 + T )^{8} \)
$97$ \( ( 3456 + T^{2} )^{4} \)
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