Properties

Label 1024.2.a.i
Level $1024$
Weight $2$
Character orbit 1024.a
Self dual yes
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( 1 + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( 1 + 2 \beta_{2} ) q^{9} + ( -\beta_{1} - 2 \beta_{3} ) q^{11} + ( 2 - \beta_{2} ) q^{13} + ( 3 \beta_{1} + \beta_{3} ) q^{15} -2 \beta_{2} q^{17} + ( -\beta_{1} + 2 \beta_{3} ) q^{19} + 4 q^{21} + ( \beta_{1} + 3 \beta_{3} ) q^{23} + ( 1 + 4 \beta_{2} ) q^{25} + 2 \beta_{3} q^{27} + ( 6 - \beta_{2} ) q^{29} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -4 - 6 \beta_{2} ) q^{33} + 2 \beta_{1} q^{35} + ( 2 - 5 \beta_{2} ) q^{37} + ( \beta_{1} - \beta_{3} ) q^{39} -4 q^{41} -\beta_{1} q^{43} + ( 6 + 5 \beta_{2} ) q^{45} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{51} + ( 6 + \beta_{2} ) q^{53} + ( -5 \beta_{1} - 3 \beta_{3} ) q^{55} + ( -4 + 2 \beta_{2} ) q^{57} + \beta_{1} q^{59} + ( 6 + 5 \beta_{2} ) q^{61} + ( \beta_{1} + 3 \beta_{3} ) q^{63} + 2 q^{65} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{67} + ( 4 + 8 \beta_{2} ) q^{69} + ( -5 \beta_{1} + \beta_{3} ) q^{71} + ( 2 + 6 \beta_{2} ) q^{73} + ( 5 \beta_{1} + 4 \beta_{3} ) q^{75} + ( 4 - 8 \beta_{2} ) q^{77} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{79} + ( -3 - 2 \beta_{2} ) q^{81} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -4 - 4 \beta_{2} ) q^{85} + ( 5 \beta_{1} - \beta_{3} ) q^{87} + ( 2 - 2 \beta_{2} ) q^{89} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{91} + ( -8 - 8 \beta_{2} ) q^{93} + ( -\beta_{1} + \beta_{3} ) q^{95} + ( 8 + 2 \beta_{2} ) q^{97} -7 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 4 q^{9} + O(q^{10}) \) \( 4 q + 8 q^{5} + 4 q^{9} + 8 q^{13} + 16 q^{21} + 4 q^{25} + 24 q^{29} - 16 q^{33} + 8 q^{37} - 16 q^{41} + 24 q^{45} + 4 q^{49} + 24 q^{53} - 16 q^{57} + 24 q^{61} + 8 q^{65} + 16 q^{69} + 8 q^{73} + 16 q^{77} - 12 q^{81} - 16 q^{85} + 8 q^{89} - 32 q^{93} + 32 q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.84776
0.765367
−0.765367
1.84776
0 −2.61313 0 3.41421 0 −1.53073 0 3.82843 0
1.2 0 −1.08239 0 0.585786 0 −3.69552 0 −1.82843 0
1.3 0 1.08239 0 0.585786 0 3.69552 0 −1.82843 0
1.4 0 2.61313 0 3.41421 0 1.53073 0 3.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.a.i 4
3.b odd 2 1 9216.2.a.w 4
4.b odd 2 1 inner 1024.2.a.i 4
8.b even 2 1 1024.2.a.h 4
8.d odd 2 1 1024.2.a.h 4
12.b even 2 1 9216.2.a.w 4
16.e even 4 2 1024.2.b.g 8
16.f odd 4 2 1024.2.b.g 8
24.f even 2 1 9216.2.a.bp 4
24.h odd 2 1 9216.2.a.bp 4
32.g even 8 2 512.2.e.i 8
32.g even 8 2 512.2.e.j yes 8
32.h odd 8 2 512.2.e.i 8
32.h odd 8 2 512.2.e.j yes 8
96.o even 8 2 4608.2.k.bd 8
96.o even 8 2 4608.2.k.bi 8
96.p odd 8 2 4608.2.k.bd 8
96.p odd 8 2 4608.2.k.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 32.g even 8 2
512.2.e.i 8 32.h odd 8 2
512.2.e.j yes 8 32.g even 8 2
512.2.e.j yes 8 32.h odd 8 2
1024.2.a.h 4 8.b even 2 1
1024.2.a.h 4 8.d odd 2 1
1024.2.a.i 4 1.a even 1 1 trivial
1024.2.a.i 4 4.b odd 2 1 inner
1024.2.b.g 8 16.e even 4 2
1024.2.b.g 8 16.f odd 4 2
4608.2.k.bd 8 96.o even 8 2
4608.2.k.bd 8 96.p odd 8 2
4608.2.k.bi 8 96.o even 8 2
4608.2.k.bi 8 96.p odd 8 2
9216.2.a.w 4 3.b odd 2 1
9216.2.a.w 4 12.b even 2 1
9216.2.a.bp 4 24.f even 2 1
9216.2.a.bp 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1024))\):

\( T_{3}^{4} - 8 T_{3}^{2} + 8 \)
\( T_{5}^{2} - 4 T_{5} + 2 \)
\( T_{7}^{4} - 16 T_{7}^{2} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 8 - 8 T^{2} + T^{4} \)
$5$ \( ( 2 - 4 T + T^{2} )^{2} \)
$7$ \( 32 - 16 T^{2} + T^{4} \)
$11$ \( 392 - 40 T^{2} + T^{4} \)
$13$ \( ( 2 - 4 T + T^{2} )^{2} \)
$17$ \( ( -8 + T^{2} )^{2} \)
$19$ \( 8 - 40 T^{2} + T^{4} \)
$23$ \( 1568 - 80 T^{2} + T^{4} \)
$29$ \( ( 34 - 12 T + T^{2} )^{2} \)
$31$ \( 512 - 64 T^{2} + T^{4} \)
$37$ \( ( -46 - 4 T + T^{2} )^{2} \)
$41$ \( ( 4 + T )^{4} \)
$43$ \( 8 - 8 T^{2} + T^{4} \)
$47$ \( 512 - 64 T^{2} + T^{4} \)
$53$ \( ( 34 - 12 T + T^{2} )^{2} \)
$59$ \( 8 - 8 T^{2} + T^{4} \)
$61$ \( ( -14 - 12 T + T^{2} )^{2} \)
$67$ \( 392 - 104 T^{2} + T^{4} \)
$71$ \( 9248 - 208 T^{2} + T^{4} \)
$73$ \( ( -68 - 4 T + T^{2} )^{2} \)
$79$ \( 8192 - 256 T^{2} + T^{4} \)
$83$ \( 2312 - 200 T^{2} + T^{4} \)
$89$ \( ( -4 - 4 T + T^{2} )^{2} \)
$97$ \( ( 56 - 16 T + T^{2} )^{2} \)
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