Properties

Label 8.8.a.a
Level 8
Weight 8
Character orbit 8.a
Self dual Yes
Analytic conductor 2.499
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.49908020387\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 84q^{3} \) \(\mathstrut -\mathstrut 82q^{5} \) \(\mathstrut -\mathstrut 456q^{7} \) \(\mathstrut +\mathstrut 4869q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 84q^{3} \) \(\mathstrut -\mathstrut 82q^{5} \) \(\mathstrut -\mathstrut 456q^{7} \) \(\mathstrut +\mathstrut 4869q^{9} \) \(\mathstrut -\mathstrut 2524q^{11} \) \(\mathstrut -\mathstrut 10778q^{13} \) \(\mathstrut +\mathstrut 6888q^{15} \) \(\mathstrut -\mathstrut 11150q^{17} \) \(\mathstrut +\mathstrut 4124q^{19} \) \(\mathstrut +\mathstrut 38304q^{21} \) \(\mathstrut +\mathstrut 81704q^{23} \) \(\mathstrut -\mathstrut 71401q^{25} \) \(\mathstrut -\mathstrut 225288q^{27} \) \(\mathstrut +\mathstrut 99798q^{29} \) \(\mathstrut -\mathstrut 40480q^{31} \) \(\mathstrut +\mathstrut 212016q^{33} \) \(\mathstrut +\mathstrut 37392q^{35} \) \(\mathstrut -\mathstrut 419442q^{37} \) \(\mathstrut +\mathstrut 905352q^{39} \) \(\mathstrut +\mathstrut 141402q^{41} \) \(\mathstrut -\mathstrut 690428q^{43} \) \(\mathstrut -\mathstrut 399258q^{45} \) \(\mathstrut -\mathstrut 682032q^{47} \) \(\mathstrut -\mathstrut 615607q^{49} \) \(\mathstrut +\mathstrut 936600q^{51} \) \(\mathstrut +\mathstrut 1813118q^{53} \) \(\mathstrut +\mathstrut 206968q^{55} \) \(\mathstrut -\mathstrut 346416q^{57} \) \(\mathstrut -\mathstrut 966028q^{59} \) \(\mathstrut +\mathstrut 1887670q^{61} \) \(\mathstrut -\mathstrut 2220264q^{63} \) \(\mathstrut +\mathstrut 883796q^{65} \) \(\mathstrut +\mathstrut 2965868q^{67} \) \(\mathstrut -\mathstrut 6863136q^{69} \) \(\mathstrut -\mathstrut 2548232q^{71} \) \(\mathstrut -\mathstrut 1680326q^{73} \) \(\mathstrut +\mathstrut 5997684q^{75} \) \(\mathstrut +\mathstrut 1150944q^{77} \) \(\mathstrut +\mathstrut 4038064q^{79} \) \(\mathstrut +\mathstrut 8275689q^{81} \) \(\mathstrut -\mathstrut 5385764q^{83} \) \(\mathstrut +\mathstrut 914300q^{85} \) \(\mathstrut -\mathstrut 8383032q^{87} \) \(\mathstrut -\mathstrut 6473046q^{89} \) \(\mathstrut +\mathstrut 4914768q^{91} \) \(\mathstrut +\mathstrut 3400320q^{93} \) \(\mathstrut -\mathstrut 338168q^{95} \) \(\mathstrut -\mathstrut 6065758q^{97} \) \(\mathstrut -\mathstrut 12289356q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0
0 −84.0000 0 −82.0000 0 −456.000 0 4869.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 84 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(8))\).