Properties

Label 1.22.a.a
Level 1
Weight 22
Character orbit 1.a
Self dual Yes
Analytic conductor 2.795
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 22 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(2.79477344287\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 288q^{2} \) \(\mathstrut -\mathstrut 128844q^{3} \) \(\mathstrut -\mathstrut 2014208q^{4} \) \(\mathstrut +\mathstrut 21640950q^{5} \) \(\mathstrut +\mathstrut 37107072q^{6} \) \(\mathstrut -\mathstrut 768078808q^{7} \) \(\mathstrut +\mathstrut 1184071680q^{8} \) \(\mathstrut +\mathstrut 6140423133q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 288q^{2} \) \(\mathstrut -\mathstrut 128844q^{3} \) \(\mathstrut -\mathstrut 2014208q^{4} \) \(\mathstrut +\mathstrut 21640950q^{5} \) \(\mathstrut +\mathstrut 37107072q^{6} \) \(\mathstrut -\mathstrut 768078808q^{7} \) \(\mathstrut +\mathstrut 1184071680q^{8} \) \(\mathstrut +\mathstrut 6140423133q^{9} \) \(\mathstrut -\mathstrut 6232593600q^{10} \) \(\mathstrut -\mathstrut 94724929188q^{11} \) \(\mathstrut +\mathstrut 259518615552q^{12} \) \(\mathstrut -\mathstrut 80621789794q^{13} \) \(\mathstrut +\mathstrut 221206696704q^{14} \) \(\mathstrut -\mathstrut 2788306561800q^{15} \) \(\mathstrut +\mathstrut 3883087691776q^{16} \) \(\mathstrut +\mathstrut 3052282930002q^{17} \) \(\mathstrut -\mathstrut 1768441862304q^{18} \) \(\mathstrut -\mathstrut 7920788351740q^{19} \) \(\mathstrut -\mathstrut 43589374617600q^{20} \) \(\mathstrut +\mathstrut 98962345937952q^{21} \) \(\mathstrut +\mathstrut 27280779606144q^{22} \) \(\mathstrut -\mathstrut 73845437470344q^{23} \) \(\mathstrut -\mathstrut 152560531537920q^{24} \) \(\mathstrut -\mathstrut 8506441300625q^{25} \) \(\mathstrut +\mathstrut 23219075460672q^{26} \) \(\mathstrut +\mathstrut 556597069939080q^{27} \) \(\mathstrut +\mathstrut 1547070479704064q^{28} \) \(\mathstrut -\mathstrut 4253031736469010q^{29} \) \(\mathstrut +\mathstrut 803032289798400q^{30} \) \(\mathstrut +\mathstrut 1900541176310432q^{31} \) \(\mathstrut -\mathstrut 3601507547086848q^{32} \) \(\mathstrut +\mathstrut 12204738776298672q^{33} \) \(\mathstrut -\mathstrut 879057483840576q^{34} \) \(\mathstrut -\mathstrut 16621955079987600q^{35} \) \(\mathstrut -\mathstrut 12368089397873664q^{36} \) \(\mathstrut +\mathstrut 22191429912035222q^{37} \) \(\mathstrut +\mathstrut 2281187045301120q^{38} \) \(\mathstrut +\mathstrut 10387633884218136q^{39} \) \(\mathstrut +\mathstrut 25624436023296000q^{40} \) \(\mathstrut -\mathstrut 20622803144546358q^{41} \) \(\mathstrut -\mathstrut 28501155630130176q^{42} \) \(\mathstrut -\mathstrut 193605854685795844q^{43} \) \(\mathstrut +\mathstrut 190795710169903104q^{44} \) \(\mathstrut +\mathstrut 132884590000096350q^{45} \) \(\mathstrut +\mathstrut 21267485991459072q^{46} \) \(\mathstrut +\mathstrut 146960504315611632q^{47} \) \(\mathstrut -\mathstrut 500312550559186944q^{48} \) \(\mathstrut +\mathstrut 31399191215416857q^{49} \) \(\mathstrut +\mathstrut 2449855094580000q^{50} \) \(\mathstrut -\mathstrut 393268341833177688q^{51} \) \(\mathstrut +\mathstrut 162389053977393152q^{52} \) \(\mathstrut +\mathstrut 2038267110310687206q^{53} \) \(\mathstrut -\mathstrut 160299956142455040q^{54} \) \(\mathstrut -\mathstrut 2049937456311048600q^{55} \) \(\mathstrut -\mathstrut 909460364560957440q^{56} \) \(\mathstrut +\mathstrut 1020546054391588560q^{57} \) \(\mathstrut +\mathstrut 1224873140103074880q^{58} \) \(\mathstrut -\mathstrut 5975882742742352820q^{59} \) \(\mathstrut +\mathstrut 5616229383230054400q^{60} \) \(\mathstrut +\mathstrut 6190617154478149262q^{61} \) \(\mathstrut -\mathstrut 547355858777404416q^{62} \) \(\mathstrut -\mathstrut 4716328880610265464q^{63} \) \(\mathstrut -\mathstrut 7106190945422409728q^{64} \) \(\mathstrut -\mathstrut 1744732121842464300q^{65} \) \(\mathstrut -\mathstrut 3514964767574017536q^{66} \) \(\mathstrut +\mathstrut 16961315295446680052q^{67} \) \(\mathstrut -\mathstrut 6147932695873468416q^{68} \) \(\mathstrut +\mathstrut 9514541545429002336q^{69} \) \(\mathstrut +\mathstrut 4787123063036428800q^{70} \) \(\mathstrut -\mathstrut 5632758963952293528q^{71} \) \(\mathstrut +\mathstrut 7270701135002173440q^{72} \) \(\mathstrut -\mathstrut 43284759511102937494q^{73} \) \(\mathstrut -\mathstrut 6391131814666143936q^{74} \) \(\mathstrut +\mathstrut 1096003922937727500q^{75} \) \(\mathstrut +\mathstrut 15954115264381521920q^{76} \) \(\mathstrut +\mathstrut 72756210698603447904q^{77} \) \(\mathstrut -\mathstrut 2991638558654823168q^{78} \) \(\mathstrut -\mathstrut 51264938664949064560q^{79} \) \(\mathstrut +\mathstrut 84033706583339827200q^{80} \) \(\mathstrut -\mathstrut 135945187666282668519q^{81} \) \(\mathstrut +\mathstrut 5939367305629351104q^{82} \) \(\mathstrut +\mathstrut 48911854702961049156q^{83} \) \(\mathstrut -\mathstrut 199330748886990422016q^{84} \) \(\mathstrut +\mathstrut 66054302274026781900q^{85} \) \(\mathstrut +\mathstrut 55758486149509203072q^{86} \) \(\mathstrut +\mathstrut 547977621053613124440q^{87} \) \(\mathstrut -\mathstrut 112161106041516195840q^{88} \) \(\mathstrut -\mathstrut 504303489899844009030q^{89} \) \(\mathstrut -\mathstrut 38270761920027748800q^{90} \) \(\mathstrut +\mathstrut 61923888203802085552q^{91} \) \(\mathstrut +\mathstrut 148740070916266647552q^{92} \) \(\mathstrut -\mathstrut 244873327320541300608q^{93} \) \(\mathstrut -\mathstrut 42324625242896150016q^{94} \) \(\mathstrut -\mathstrut 171413384680587753000q^{95} \) \(\mathstrut +\mathstrut 464032638396857843712q^{96} \) \(\mathstrut +\mathstrut 808275058155029184482q^{97} \) \(\mathstrut -\mathstrut 9042967070040054816q^{98} \) \(\mathstrut -\mathstrut 581651146457782106004q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−288.000 −128844. −2.01421e6 2.16410e7 3.71071e7 −7.68079e8 1.18407e9 6.14042e9 −6.23259e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\).