Properties

Label 2.14.a.a
Level 2
Weight 14
Character orbit 2.a
Self dual Yes
Analytic conductor 2.145
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.14461857904\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut -\mathstrut 1836q^{3} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 3990q^{5} \) \(\mathstrut +\mathstrut 117504q^{6} \) \(\mathstrut -\mathstrut 433432q^{7} \) \(\mathstrut -\mathstrut 262144q^{8} \) \(\mathstrut +\mathstrut 1776573q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 64q^{2} \) \(\mathstrut -\mathstrut 1836q^{3} \) \(\mathstrut +\mathstrut 4096q^{4} \) \(\mathstrut +\mathstrut 3990q^{5} \) \(\mathstrut +\mathstrut 117504q^{6} \) \(\mathstrut -\mathstrut 433432q^{7} \) \(\mathstrut -\mathstrut 262144q^{8} \) \(\mathstrut +\mathstrut 1776573q^{9} \) \(\mathstrut -\mathstrut 255360q^{10} \) \(\mathstrut +\mathstrut 1619772q^{11} \) \(\mathstrut -\mathstrut 7520256q^{12} \) \(\mathstrut -\mathstrut 10878466q^{13} \) \(\mathstrut +\mathstrut 27739648q^{14} \) \(\mathstrut -\mathstrut 7325640q^{15} \) \(\mathstrut +\mathstrut 16777216q^{16} \) \(\mathstrut +\mathstrut 60569298q^{17} \) \(\mathstrut -\mathstrut 113700672q^{18} \) \(\mathstrut -\mathstrut 243131740q^{19} \) \(\mathstrut +\mathstrut 16343040q^{20} \) \(\mathstrut +\mathstrut 795781152q^{21} \) \(\mathstrut -\mathstrut 103665408q^{22} \) \(\mathstrut -\mathstrut 606096456q^{23} \) \(\mathstrut +\mathstrut 481296384q^{24} \) \(\mathstrut -\mathstrut 1204783025q^{25} \) \(\mathstrut +\mathstrut 696221824q^{26} \) \(\mathstrut -\mathstrut 334611000q^{27} \) \(\mathstrut -\mathstrut 1775337472q^{28} \) \(\mathstrut +\mathstrut 5258639310q^{29} \) \(\mathstrut +\mathstrut 468840960q^{30} \) \(\mathstrut -\mathstrut 1824312928q^{31} \) \(\mathstrut -\mathstrut 1073741824q^{32} \) \(\mathstrut -\mathstrut 2973901392q^{33} \) \(\mathstrut -\mathstrut 3876435072q^{34} \) \(\mathstrut -\mathstrut 1729393680q^{35} \) \(\mathstrut +\mathstrut 7276843008q^{36} \) \(\mathstrut -\mathstrut 3005875402q^{37} \) \(\mathstrut +\mathstrut 15560431360q^{38} \) \(\mathstrut +\mathstrut 19972863576q^{39} \) \(\mathstrut -\mathstrut 1045954560q^{40} \) \(\mathstrut -\mathstrut 49704880758q^{41} \) \(\mathstrut -\mathstrut 50929993728q^{42} \) \(\mathstrut +\mathstrut 58766693084q^{43} \) \(\mathstrut +\mathstrut 6634586112q^{44} \) \(\mathstrut +\mathstrut 7088526270q^{45} \) \(\mathstrut +\mathstrut 38790173184q^{46} \) \(\mathstrut -\mathstrut 42095878032q^{47} \) \(\mathstrut -\mathstrut 30802968576q^{48} \) \(\mathstrut +\mathstrut 90974288217q^{49} \) \(\mathstrut +\mathstrut 77106113600q^{50} \) \(\mathstrut -\mathstrut 111205231128q^{51} \) \(\mathstrut -\mathstrut 44558196736q^{52} \) \(\mathstrut -\mathstrut 181140755706q^{53} \) \(\mathstrut +\mathstrut 21415104000q^{54} \) \(\mathstrut +\mathstrut 6462890280q^{55} \) \(\mathstrut +\mathstrut 113621598208q^{56} \) \(\mathstrut +\mathstrut 446389874640q^{57} \) \(\mathstrut -\mathstrut 336552915840q^{58} \) \(\mathstrut +\mathstrut 206730587820q^{59} \) \(\mathstrut -\mathstrut 30005821440q^{60} \) \(\mathstrut -\mathstrut 124479015058q^{61} \) \(\mathstrut +\mathstrut 116756027392q^{62} \) \(\mathstrut -\mathstrut 770023588536q^{63} \) \(\mathstrut +\mathstrut 68719476736q^{64} \) \(\mathstrut -\mathstrut 43405079340q^{65} \) \(\mathstrut +\mathstrut 190329689088q^{66} \) \(\mathstrut +\mathstrut 95665133588q^{67} \) \(\mathstrut +\mathstrut 248091844608q^{68} \) \(\mathstrut +\mathstrut 1112793093216q^{69} \) \(\mathstrut +\mathstrut 110681195520q^{70} \) \(\mathstrut -\mathstrut 371436487128q^{71} \) \(\mathstrut -\mathstrut 465717952512q^{72} \) \(\mathstrut -\mathstrut 1800576064726q^{73} \) \(\mathstrut +\mathstrut 192376025728q^{74} \) \(\mathstrut +\mathstrut 2211981633900q^{75} \) \(\mathstrut -\mathstrut 995867607040q^{76} \) \(\mathstrut -\mathstrut 702061017504q^{77} \) \(\mathstrut -\mathstrut 1278263268864q^{78} \) \(\mathstrut +\mathstrut 1557932091920q^{79} \) \(\mathstrut +\mathstrut 66941091840q^{80} \) \(\mathstrut -\mathstrut 2218085399079q^{81} \) \(\mathstrut +\mathstrut 3181112368512q^{82} \) \(\mathstrut +\mathstrut 2492790917604q^{83} \) \(\mathstrut +\mathstrut 3259519598592q^{84} \) \(\mathstrut +\mathstrut 241671499020q^{85} \) \(\mathstrut -\mathstrut 3761068357376q^{86} \) \(\mathstrut -\mathstrut 9654861773160q^{87} \) \(\mathstrut -\mathstrut 424613511168q^{88} \) \(\mathstrut +\mathstrut 2994235754490q^{89} \) \(\mathstrut -\mathstrut 453665681280q^{90} \) \(\mathstrut +\mathstrut 4715075275312q^{91} \) \(\mathstrut -\mathstrut 2482571083776q^{92} \) \(\mathstrut +\mathstrut 3349438535808q^{93} \) \(\mathstrut +\mathstrut 2694136194048q^{94} \) \(\mathstrut -\mathstrut 970095642600q^{95} \) \(\mathstrut +\mathstrut 1971389988864q^{96} \) \(\mathstrut +\mathstrut 4382492665058q^{97} \) \(\mathstrut -\mathstrut 5822354445888q^{98} \) \(\mathstrut +\mathstrut 2877643201356q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−64.0000 −1836.00 4096.00 3990.00 117504. −433432. −262144. 1.77657e6 −255360.
\(n\): 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 \(T_{3} \) \(\mathstrut +\mathstrut 1836 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(2))\).