Properties

Label 512.2.a.e
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: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

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

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 2.00000 0 −2.82843 0 −1.00000 0
1.2 0 1.41421 0 2.00000 0 2.82843 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Atkin-Lehner signs

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

Hecke kernels

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

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