Properties

Label 512.2.a.f
Level $512$
Weight $2$
Character orbit 512.a
Self dual yes
Analytic conductor $4.088$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 0.585786 0 0 0 0 0 −2.65685 0
1.2 0 3.41421 0 0 0 0 0 8.65685 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.f yes 2
3.b odd 2 1 4608.2.a.i 2
4.b odd 2 1 512.2.a.a 2
8.b even 2 1 512.2.a.a 2
8.d odd 2 1 CM 512.2.a.f yes 2
12.b even 2 1 4608.2.a.k 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.i 2
24.h odd 2 1 4608.2.a.k 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 4.b odd 2 1
512.2.a.a 2 8.b even 2 1
512.2.a.f yes 2 1.a even 1 1 trivial
512.2.a.f yes 2 8.d odd 2 1 CM
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 3.b odd 2 1
4608.2.a.i 2 24.f even 2 1
4608.2.a.k 2 12.b even 2 1
4608.2.a.k 2 24.h odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\):

\( T_{3}^{2} - 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

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