Properties

Label 5.8.a.a
Level 5
Weight 8
Character orbit 5.a
Self dual Yes
Analytic conductor 1.562
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.56192512742\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut +\mathstrut 68q^{4} \) \(\mathstrut +\mathstrut 125q^{5} \) \(\mathstrut +\mathstrut 672q^{6} \) \(\mathstrut -\mathstrut 1644q^{7} \) \(\mathstrut +\mathstrut 840q^{8} \) \(\mathstrut +\mathstrut 117q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 48q^{3} \) \(\mathstrut +\mathstrut 68q^{4} \) \(\mathstrut +\mathstrut 125q^{5} \) \(\mathstrut +\mathstrut 672q^{6} \) \(\mathstrut -\mathstrut 1644q^{7} \) \(\mathstrut +\mathstrut 840q^{8} \) \(\mathstrut +\mathstrut 117q^{9} \) \(\mathstrut -\mathstrut 1750q^{10} \) \(\mathstrut +\mathstrut 172q^{11} \) \(\mathstrut -\mathstrut 3264q^{12} \) \(\mathstrut +\mathstrut 3862q^{13} \) \(\mathstrut +\mathstrut 23016q^{14} \) \(\mathstrut -\mathstrut 6000q^{15} \) \(\mathstrut -\mathstrut 20464q^{16} \) \(\mathstrut -\mathstrut 12254q^{17} \) \(\mathstrut -\mathstrut 1638q^{18} \) \(\mathstrut -\mathstrut 25940q^{19} \) \(\mathstrut +\mathstrut 8500q^{20} \) \(\mathstrut +\mathstrut 78912q^{21} \) \(\mathstrut -\mathstrut 2408q^{22} \) \(\mathstrut +\mathstrut 12972q^{23} \) \(\mathstrut -\mathstrut 40320q^{24} \) \(\mathstrut +\mathstrut 15625q^{25} \) \(\mathstrut -\mathstrut 54068q^{26} \) \(\mathstrut +\mathstrut 99360q^{27} \) \(\mathstrut -\mathstrut 111792q^{28} \) \(\mathstrut -\mathstrut 81610q^{29} \) \(\mathstrut +\mathstrut 84000q^{30} \) \(\mathstrut -\mathstrut 156888q^{31} \) \(\mathstrut +\mathstrut 178976q^{32} \) \(\mathstrut -\mathstrut 8256q^{33} \) \(\mathstrut +\mathstrut 171556q^{34} \) \(\mathstrut -\mathstrut 205500q^{35} \) \(\mathstrut +\mathstrut 7956q^{36} \) \(\mathstrut +\mathstrut 110126q^{37} \) \(\mathstrut +\mathstrut 363160q^{38} \) \(\mathstrut -\mathstrut 185376q^{39} \) \(\mathstrut +\mathstrut 105000q^{40} \) \(\mathstrut +\mathstrut 467882q^{41} \) \(\mathstrut -\mathstrut 1104768q^{42} \) \(\mathstrut -\mathstrut 499208q^{43} \) \(\mathstrut +\mathstrut 11696q^{44} \) \(\mathstrut +\mathstrut 14625q^{45} \) \(\mathstrut -\mathstrut 181608q^{46} \) \(\mathstrut -\mathstrut 396884q^{47} \) \(\mathstrut +\mathstrut 982272q^{48} \) \(\mathstrut +\mathstrut 1879193q^{49} \) \(\mathstrut -\mathstrut 218750q^{50} \) \(\mathstrut +\mathstrut 588192q^{51} \) \(\mathstrut +\mathstrut 262616q^{52} \) \(\mathstrut -\mathstrut 1280498q^{53} \) \(\mathstrut -\mathstrut 1391040q^{54} \) \(\mathstrut +\mathstrut 21500q^{55} \) \(\mathstrut -\mathstrut 1380960q^{56} \) \(\mathstrut +\mathstrut 1245120q^{57} \) \(\mathstrut +\mathstrut 1142540q^{58} \) \(\mathstrut -\mathstrut 1337420q^{59} \) \(\mathstrut -\mathstrut 408000q^{60} \) \(\mathstrut -\mathstrut 923978q^{61} \) \(\mathstrut +\mathstrut 2196432q^{62} \) \(\mathstrut -\mathstrut 192348q^{63} \) \(\mathstrut +\mathstrut 113728q^{64} \) \(\mathstrut +\mathstrut 482750q^{65} \) \(\mathstrut +\mathstrut 115584q^{66} \) \(\mathstrut -\mathstrut 797304q^{67} \) \(\mathstrut -\mathstrut 833272q^{68} \) \(\mathstrut -\mathstrut 622656q^{69} \) \(\mathstrut +\mathstrut 2877000q^{70} \) \(\mathstrut +\mathstrut 5103392q^{71} \) \(\mathstrut +\mathstrut 98280q^{72} \) \(\mathstrut -\mathstrut 4267478q^{73} \) \(\mathstrut -\mathstrut 1541764q^{74} \) \(\mathstrut -\mathstrut 750000q^{75} \) \(\mathstrut -\mathstrut 1763920q^{76} \) \(\mathstrut -\mathstrut 282768q^{77} \) \(\mathstrut +\mathstrut 2595264q^{78} \) \(\mathstrut -\mathstrut 960q^{79} \) \(\mathstrut -\mathstrut 2558000q^{80} \) \(\mathstrut -\mathstrut 5025159q^{81} \) \(\mathstrut -\mathstrut 6550348q^{82} \) \(\mathstrut +\mathstrut 6140832q^{83} \) \(\mathstrut +\mathstrut 5366016q^{84} \) \(\mathstrut -\mathstrut 1531750q^{85} \) \(\mathstrut +\mathstrut 6988912q^{86} \) \(\mathstrut +\mathstrut 3917280q^{87} \) \(\mathstrut +\mathstrut 144480q^{88} \) \(\mathstrut +\mathstrut 2010570q^{89} \) \(\mathstrut -\mathstrut 204750q^{90} \) \(\mathstrut -\mathstrut 6349128q^{91} \) \(\mathstrut +\mathstrut 882096q^{92} \) \(\mathstrut +\mathstrut 7530624q^{93} \) \(\mathstrut +\mathstrut 5556376q^{94} \) \(\mathstrut -\mathstrut 3242500q^{95} \) \(\mathstrut -\mathstrut 8590848q^{96} \) \(\mathstrut -\mathstrut 4881934q^{97} \) \(\mathstrut -\mathstrut 26308702q^{98} \) \(\mathstrut +\mathstrut 20124q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−14.0000 −48.0000 68.0000 125.000 672.000 −1644.00 840.000 117.000 −1750.00
\(n\): 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
\(5\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 14 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\).