Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} + 4 q^{7} - q^{9} +O(q^{10})\) \( q + \beta q^{3} + 2 \beta q^{5} + 4 q^{7} - q^{9} + \beta q^{11} -2 \beta q^{13} + 4 q^{15} -4 q^{17} -5 \beta q^{19} + 4 \beta q^{21} + 4 q^{23} + 3 q^{25} -4 \beta q^{27} -6 \beta q^{29} + 8 q^{31} + 2 q^{33} + 8 \beta q^{35} -2 \beta q^{37} -4 q^{39} + 2 q^{41} + 3 \beta q^{43} -2 \beta q^{45} + 9 q^{49} -4 \beta q^{51} -2 \beta q^{53} + 4 q^{55} -10 q^{57} + 3 \beta q^{59} + 6 \beta q^{61} -4 q^{63} -8 q^{65} + 3 \beta q^{67} + 4 \beta q^{69} -4 q^{71} -4 q^{73} + 3 \beta q^{75} + 4 \beta q^{77} -8 q^{79} -5 q^{81} -7 \beta q^{83} -8 \beta q^{85} -12 q^{87} + 12 q^{89} -8 \beta q^{91} + 8 \beta q^{93} -20 q^{95} -4 q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 8q^{7} - 2q^{9} + 8q^{15} - 8q^{17} + 8q^{23} + 6q^{25} + 16q^{31} + 4q^{33} - 8q^{39} + 4q^{41} + 18q^{49} + 8q^{55} - 20q^{57} - 8q^{63} - 16q^{65} - 8q^{71} - 8q^{73} - 16q^{79} - 10q^{81} - 24q^{87} + 24q^{89} - 40q^{95} - 8q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 −1.41421 0 −2.82843 0 4.00000 0 −1.00000 0
1.2 0 1.41421 0 2.82843 0 4.00000 0 −1.00000 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
8.b even 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.d yes 2
3.b odd 2 1 4608.2.a.m 2
4.b odd 2 1 512.2.a.c 2
8.b even 2 1 inner 512.2.a.d yes 2
8.d odd 2 1 512.2.a.c 2
12.b even 2 1 4608.2.a.f 2
16.e even 4 2 512.2.b.a 2
16.f odd 4 2 512.2.b.b 2
24.f even 2 1 4608.2.a.f 2
24.h odd 2 1 4608.2.a.m 2
32.g even 8 2 1024.2.e.c 2
32.g even 8 2 1024.2.e.d 2
32.h odd 8 2 1024.2.e.b 2
32.h odd 8 2 1024.2.e.e 2
48.i odd 4 2 4608.2.d.a 2
48.k even 4 2 4608.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.c 2 4.b odd 2 1
512.2.a.c 2 8.d odd 2 1
512.2.a.d yes 2 1.a even 1 1 trivial
512.2.a.d yes 2 8.b even 2 1 inner
512.2.b.a 2 16.e even 4 2
512.2.b.b 2 16.f odd 4 2
1024.2.e.b 2 32.h odd 8 2
1024.2.e.c 2 32.g even 8 2
1024.2.e.d 2 32.g even 8 2
1024.2.e.e 2 32.h odd 8 2
4608.2.a.f 2 12.b even 2 1
4608.2.a.f 2 24.f even 2 1
4608.2.a.m 2 3.b odd 2 1
4608.2.a.m 2 24.h odd 2 1
4608.2.d.a 2 48.i odd 4 2
4608.2.d.b 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -2 + T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( -2 + T^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( ( 4 + T )^{2} \)
$19$ \( -50 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( -72 + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( -8 + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -18 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -8 + T^{2} \)
$59$ \( -18 + T^{2} \)
$61$ \( -72 + T^{2} \)
$67$ \( -18 + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( ( 4 + T )^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -98 + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( ( 4 + T )^{2} \)
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