Properties

Label 1280.2.f.a
Level $1280$
Weight $2$
Character orbit 1280.f
Analytic conductor $10.221$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{3} + ( -2 - i ) q^{5} -2 i q^{7} + q^{9} +O(q^{10})\) \( q -2 q^{3} + ( -2 - i ) q^{5} -2 i q^{7} + q^{9} -4 i q^{11} -4 q^{13} + ( 4 + 2 i ) q^{15} -4 i q^{19} + 4 i q^{21} -2 i q^{23} + ( 3 + 4 i ) q^{25} + 4 q^{27} + 2 i q^{29} + 8 i q^{33} + ( -2 + 4 i ) q^{35} -4 q^{37} + 8 q^{39} -2 q^{41} -6 q^{43} + ( -2 - i ) q^{45} -6 i q^{47} + 3 q^{49} -4 q^{53} + ( -4 + 8 i ) q^{55} + 8 i q^{57} + 12 i q^{59} + 10 i q^{61} -2 i q^{63} + ( 8 + 4 i ) q^{65} + 14 q^{67} + 4 i q^{69} -8 q^{71} + 8 i q^{73} + ( -6 - 8 i ) q^{75} -8 q^{77} -16 q^{79} -11 q^{81} -2 q^{83} -4 i q^{87} + 6 q^{89} + 8 i q^{91} + ( -4 + 8 i ) q^{95} + 16 i q^{97} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 4q^{5} + 2q^{9} - 8q^{13} + 8q^{15} + 6q^{25} + 8q^{27} - 4q^{35} - 8q^{37} + 16q^{39} - 4q^{41} - 12q^{43} - 4q^{45} + 6q^{49} - 8q^{53} - 8q^{55} + 16q^{65} + 28q^{67} - 16q^{71} - 12q^{75} - 16q^{77} - 32q^{79} - 22q^{81} - 4q^{83} + 12q^{89} - 8q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 −2.00000 0 −2.00000 1.00000i 0 2.00000i 0 1.00000 0
129.2 0 −2.00000 0 −2.00000 + 1.00000i 0 2.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1280.2.f.a 2
4.b odd 2 1 1280.2.f.e 2
5.b even 2 1 1280.2.f.f 2
8.b even 2 1 1280.2.f.f 2
8.d odd 2 1 1280.2.f.b 2
16.e even 4 1 40.2.c.a 2
16.e even 4 1 320.2.c.c 2
16.f odd 4 1 80.2.c.a 2
16.f odd 4 1 320.2.c.b 2
20.d odd 2 1 1280.2.f.b 2
40.e odd 2 1 1280.2.f.e 2
40.f even 2 1 inner 1280.2.f.a 2
48.i odd 4 1 360.2.f.c 2
48.i odd 4 1 2880.2.f.h 2
48.k even 4 1 720.2.f.e 2
48.k even 4 1 2880.2.f.i 2
80.i odd 4 1 200.2.a.d 1
80.i odd 4 1 1600.2.a.v 1
80.j even 4 1 400.2.a.g 1
80.j even 4 1 1600.2.a.u 1
80.k odd 4 1 80.2.c.a 2
80.k odd 4 1 320.2.c.b 2
80.q even 4 1 40.2.c.a 2
80.q even 4 1 320.2.c.c 2
80.s even 4 1 400.2.a.b 1
80.s even 4 1 1600.2.a.d 1
80.t odd 4 1 200.2.a.b 1
80.t odd 4 1 1600.2.a.f 1
112.l odd 4 1 1960.2.g.b 2
240.t even 4 1 720.2.f.e 2
240.t even 4 1 2880.2.f.i 2
240.z odd 4 1 3600.2.a.k 1
240.bb even 4 1 1800.2.a.s 1
240.bd odd 4 1 3600.2.a.bb 1
240.bf even 4 1 1800.2.a.j 1
240.bm odd 4 1 360.2.f.c 2
240.bm odd 4 1 2880.2.f.h 2
560.r even 4 1 9800.2.a.bf 1
560.bf odd 4 1 1960.2.g.b 2
560.bn even 4 1 9800.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 16.e even 4 1
40.2.c.a 2 80.q even 4 1
80.2.c.a 2 16.f odd 4 1
80.2.c.a 2 80.k odd 4 1
200.2.a.b 1 80.t odd 4 1
200.2.a.d 1 80.i odd 4 1
320.2.c.b 2 16.f odd 4 1
320.2.c.b 2 80.k odd 4 1
320.2.c.c 2 16.e even 4 1
320.2.c.c 2 80.q even 4 1
360.2.f.c 2 48.i odd 4 1
360.2.f.c 2 240.bm odd 4 1
400.2.a.b 1 80.s even 4 1
400.2.a.g 1 80.j even 4 1
720.2.f.e 2 48.k even 4 1
720.2.f.e 2 240.t even 4 1
1280.2.f.a 2 1.a even 1 1 trivial
1280.2.f.a 2 40.f even 2 1 inner
1280.2.f.b 2 8.d odd 2 1
1280.2.f.b 2 20.d odd 2 1
1280.2.f.e 2 4.b odd 2 1
1280.2.f.e 2 40.e odd 2 1
1280.2.f.f 2 5.b even 2 1
1280.2.f.f 2 8.b even 2 1
1600.2.a.d 1 80.s even 4 1
1600.2.a.f 1 80.t odd 4 1
1600.2.a.u 1 80.j even 4 1
1600.2.a.v 1 80.i odd 4 1
1800.2.a.j 1 240.bf even 4 1
1800.2.a.s 1 240.bb even 4 1
1960.2.g.b 2 112.l odd 4 1
1960.2.g.b 2 560.bf odd 4 1
2880.2.f.h 2 48.i odd 4 1
2880.2.f.h 2 240.bm odd 4 1
2880.2.f.i 2 48.k even 4 1
2880.2.f.i 2 240.t even 4 1
3600.2.a.k 1 240.z odd 4 1
3600.2.a.bb 1 240.bd odd 4 1
9800.2.a.d 1 560.bn even 4 1
9800.2.a.bf 1 560.r even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1280, [\chi])\):

\( T_{3} + 2 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 2 + T )^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 16 + T^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( ( 6 + T )^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( 100 + T^{2} \)
$67$ \( ( -14 + T )^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 64 + T^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( ( 2 + T )^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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