Properties

Label 512.2.a.a
Level $512$
Weight $2$
Character orbit 512.a
Self dual yes
Analytic conductor $4.088$
Analytic rank $1$
Dimension $2$
CM discriminant -8
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: \(1\)
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: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 + \beta ) q^{3} + ( 3 - 4 \beta ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta ) q^{3} + ( 3 - 4 \beta ) q^{9} + ( -2 - 3 \beta ) q^{11} + 4 \beta q^{17} + ( -6 - \beta ) q^{19} -5 q^{25} + ( -8 + 8 \beta ) q^{27} + ( -2 + 4 \beta ) q^{33} -6 q^{41} + ( -6 - 5 \beta ) q^{43} -7 q^{49} + ( 8 - 8 \beta ) q^{51} + ( 10 - 4 \beta ) q^{57} + ( 10 + 3 \beta ) q^{59} + ( -6 + 7 \beta ) q^{67} + 12 \beta q^{73} + ( 10 - 5 \beta ) q^{75} + ( 23 - 12 \beta ) q^{81} + ( -2 + 9 \beta ) q^{83} -4 \beta q^{89} -12 \beta q^{97} + ( 18 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 6q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 6q^{9} - 4q^{11} - 12q^{19} - 10q^{25} - 16q^{27} - 4q^{33} - 12q^{41} - 12q^{43} - 14q^{49} + 16q^{51} + 20q^{57} + 20q^{59} - 12q^{67} + 20q^{75} + 46q^{81} - 4q^{83} + 36q^{99} + O(q^{100}) \)

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 −3.41421 0 0 0 0 0 8.65685 0
1.2 0 −0.585786 0 0 0 0 0 −2.65685 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.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2 + 4 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -14 + 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -32 + T^{2} \)
$19$ \( 34 + 12 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( -14 + 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 82 - 20 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( -62 + 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -288 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -158 + 4 T + T^{2} \)
$89$ \( -32 + T^{2} \)
$97$ \( -288 + T^{2} \)
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