Properties

Label 512.2.a.g
Level $512$
Weight $2$
Character orbit 512.a
Self dual yes
Analytic conductor $4.088$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + 3 q^{9} + \beta_{1} q^{11} + \beta_{2} q^{13} + 3 \beta_{3} q^{15} + 4 q^{17} -\beta_{1} q^{19} + 2 \beta_{2} q^{21} -\beta_{3} q^{23} + 7 q^{25} -\beta_{2} q^{29} -2 \beta_{3} q^{31} + 6 q^{33} -4 \beta_{1} q^{35} + \beta_{2} q^{37} -3 \beta_{3} q^{39} -2 q^{41} -5 \beta_{1} q^{43} -3 \beta_{2} q^{45} + 4 \beta_{3} q^{47} + q^{49} + 4 \beta_{1} q^{51} -3 \beta_{2} q^{53} + 3 \beta_{3} q^{55} -6 q^{57} -\beta_{1} q^{59} + \beta_{2} q^{61} -3 \beta_{3} q^{63} -12 q^{65} + 3 \beta_{1} q^{67} + 2 \beta_{2} q^{69} + \beta_{3} q^{71} + 8 q^{73} + 7 \beta_{1} q^{75} + 2 \beta_{2} q^{77} -2 \beta_{3} q^{79} -9 q^{81} -3 \beta_{1} q^{83} -4 \beta_{2} q^{85} + 3 \beta_{3} q^{87} -8 q^{89} + 4 \beta_{1} q^{91} + 4 \beta_{2} q^{93} -3 \beta_{3} q^{95} + 12 q^{97} + 3 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 16q^{17} + 28q^{25} + 24q^{33} - 8q^{41} + 4q^{49} - 24q^{57} - 48q^{65} + 32q^{73} - 36q^{81} - 32q^{89} + 48q^{97} + O(q^{100}) \)

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
1.93185
0.517638
−1.93185
−0.517638
0 −2.44949 0 −3.46410 0 −2.82843 0 3.00000 0
1.2 0 −2.44949 0 3.46410 0 2.82843 0 3.00000 0
1.3 0 2.44949 0 −3.46410 0 2.82843 0 3.00000 0
1.4 0 2.44949 0 3.46410 0 −2.82843 0 3.00000 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.a.g 4
3.b odd 2 1 4608.2.a.w 4
4.b odd 2 1 inner 512.2.a.g 4
8.b even 2 1 inner 512.2.a.g 4
8.d odd 2 1 inner 512.2.a.g 4
12.b even 2 1 4608.2.a.w 4
16.e even 4 2 512.2.b.d 4
16.f odd 4 2 512.2.b.d 4
24.f even 2 1 4608.2.a.w 4
24.h odd 2 1 4608.2.a.w 4
32.g even 8 4 1024.2.e.p 8
32.h odd 8 4 1024.2.e.p 8
48.i odd 4 2 4608.2.d.d 4
48.k even 4 2 4608.2.d.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.g 4 1.a even 1 1 trivial
512.2.a.g 4 4.b odd 2 1 inner
512.2.a.g 4 8.b even 2 1 inner
512.2.a.g 4 8.d odd 2 1 inner
512.2.b.d 4 16.e even 4 2
512.2.b.d 4 16.f odd 4 2
1024.2.e.p 8 32.g even 8 4
1024.2.e.p 8 32.h odd 8 4
4608.2.a.w 4 3.b odd 2 1
4608.2.a.w 4 12.b even 2 1
4608.2.a.w 4 24.f even 2 1
4608.2.a.w 4 24.h odd 2 1
4608.2.d.d 4 48.i odd 4 2
4608.2.d.d 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\):

\( T_{3}^{2} - 6 \)
\( T_{5}^{2} - 12 \)
\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -6 + T^{2} )^{2} \)
$5$ \( ( -12 + T^{2} )^{2} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( ( -6 + T^{2} )^{2} \)
$13$ \( ( -12 + T^{2} )^{2} \)
$17$ \( ( -4 + T )^{4} \)
$19$ \( ( -6 + T^{2} )^{2} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( ( -12 + T^{2} )^{2} \)
$31$ \( ( -32 + T^{2} )^{2} \)
$37$ \( ( -12 + T^{2} )^{2} \)
$41$ \( ( 2 + T )^{4} \)
$43$ \( ( -150 + T^{2} )^{2} \)
$47$ \( ( -128 + T^{2} )^{2} \)
$53$ \( ( -108 + T^{2} )^{2} \)
$59$ \( ( -6 + T^{2} )^{2} \)
$61$ \( ( -12 + T^{2} )^{2} \)
$67$ \( ( -54 + T^{2} )^{2} \)
$71$ \( ( -8 + T^{2} )^{2} \)
$73$ \( ( -8 + T )^{4} \)
$79$ \( ( -32 + T^{2} )^{2} \)
$83$ \( ( -54 + T^{2} )^{2} \)
$89$ \( ( 8 + T )^{4} \)
$97$ \( ( -12 + T )^{4} \)
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