Properties

Label 8.14.a.a
Level 8
Weight 14
Character orbit 8.a
Self dual Yes
Analytic conductor 8.578
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.57847431615\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 4330q^{5} \) \(\mathstrut -\mathstrut 139992q^{7} \) \(\mathstrut -\mathstrut 1594179q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 4330q^{5} \) \(\mathstrut -\mathstrut 139992q^{7} \) \(\mathstrut -\mathstrut 1594179q^{9} \) \(\mathstrut -\mathstrut 6484324q^{11} \) \(\mathstrut -\mathstrut 22588034q^{13} \) \(\mathstrut +\mathstrut 51960q^{15} \) \(\mathstrut -\mathstrut 23732270q^{17} \) \(\mathstrut +\mathstrut 325344836q^{19} \) \(\mathstrut +\mathstrut 1679904q^{21} \) \(\mathstrut +\mathstrut 921600632q^{23} \) \(\mathstrut -\mathstrut 1201954225q^{25} \) \(\mathstrut +\mathstrut 38262024q^{27} \) \(\mathstrut -\mathstrut 3865879218q^{29} \) \(\mathstrut -\mathstrut 2253401440q^{31} \) \(\mathstrut +\mathstrut 77811888q^{33} \) \(\mathstrut +\mathstrut 606165360q^{35} \) \(\mathstrut +\mathstrut 18250384566q^{37} \) \(\mathstrut +\mathstrut 271056408q^{39} \) \(\mathstrut +\mathstrut 34422845322q^{41} \) \(\mathstrut -\mathstrut 17192501444q^{43} \) \(\mathstrut +\mathstrut 6902795070q^{45} \) \(\mathstrut -\mathstrut 67371749904q^{47} \) \(\mathstrut -\mathstrut 77291250343q^{49} \) \(\mathstrut +\mathstrut 284787240q^{51} \) \(\mathstrut -\mathstrut 87281218426q^{53} \) \(\mathstrut +\mathstrut 28077122920q^{55} \) \(\mathstrut -\mathstrut 3904138032q^{57} \) \(\mathstrut +\mathstrut 540214518668q^{59} \) \(\mathstrut -\mathstrut 51276568850q^{61} \) \(\mathstrut +\mathstrut 223172306568q^{63} \) \(\mathstrut +\mathstrut 97806187220q^{65} \) \(\mathstrut +\mathstrut 25519930676q^{67} \) \(\mathstrut -\mathstrut 11059207584q^{69} \) \(\mathstrut -\mathstrut 1387500699032q^{71} \) \(\mathstrut -\mathstrut 819049441238q^{73} \) \(\mathstrut +\mathstrut 14423450700q^{75} \) \(\mathstrut +\mathstrut 907753485408q^{77} \) \(\mathstrut -\mathstrut 4030935615344q^{79} \) \(\mathstrut +\mathstrut 2541177101529q^{81} \) \(\mathstrut +\mathstrut 4180823831428q^{83} \) \(\mathstrut +\mathstrut 102760729100q^{85} \) \(\mathstrut +\mathstrut 46390550616q^{87} \) \(\mathstrut +\mathstrut 2677027798266q^{89} \) \(\mathstrut +\mathstrut 3162144055728q^{91} \) \(\mathstrut +\mathstrut 27040817280q^{93} \) \(\mathstrut -\mathstrut 1408743139880q^{95} \) \(\mathstrut -\mathstrut 14039464316446q^{97} \) \(\mathstrut +\mathstrut 10337173149996q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −12.0000 0 −4330.00 0 −139992. 0 −1.59418e6 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 \(T_{3} \) \(\mathstrut +\mathstrut 12 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(8))\).