Properties

Label 2.32.a.a
Level 2
Weight 32
Character orbit 2.a
Self dual Yes
Analytic conductor 12.175
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 32 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(12.1754265638\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 32768q^{2} \) \(\mathstrut -\mathstrut 19984212q^{3} \) \(\mathstrut +\mathstrut 1073741824q^{4} \) \(\mathstrut +\mathstrut 42951708750q^{5} \) \(\mathstrut -\mathstrut 654842658816q^{6} \) \(\mathstrut -\mathstrut 16835358997576q^{7} \) \(\mathstrut +\mathstrut 35184372088832q^{8} \) \(\mathstrut -\mathstrut 218304667023003q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 32768q^{2} \) \(\mathstrut -\mathstrut 19984212q^{3} \) \(\mathstrut +\mathstrut 1073741824q^{4} \) \(\mathstrut +\mathstrut 42951708750q^{5} \) \(\mathstrut -\mathstrut 654842658816q^{6} \) \(\mathstrut -\mathstrut 16835358997576q^{7} \) \(\mathstrut +\mathstrut 35184372088832q^{8} \) \(\mathstrut -\mathstrut 218304667023003q^{9} \) \(\mathstrut +\mathstrut 1407441592320000q^{10} \) \(\mathstrut -\mathstrut 7207832704992348q^{11} \) \(\mathstrut -\mathstrut 21457884244082688q^{12} \) \(\mathstrut -\mathstrut 270053634881821882q^{13} \) \(\mathstrut -\mathstrut 551661043632570368q^{14} \) \(\mathstrut -\mathstrut 858356053422255000q^{15} \) \(\mathstrut +\mathstrut 1152921504606846976q^{16} \) \(\mathstrut -\mathstrut 16275482960925966606q^{17} \) \(\mathstrut -\mathstrut 7153407329009762304q^{18} \) \(\mathstrut +\mathstrut 109087314160337984540q^{19} \) \(\mathstrut +\mathstrut 46119046097141760000q^{20} \) \(\mathstrut +\mathstrut 336441383303666270112q^{21} \) \(\mathstrut -\mathstrut 236186262077189259264q^{22} \) \(\mathstrut -\mathstrut 305935391155430403672q^{23} \) \(\mathstrut -\mathstrut 703131950910101520384q^{24} \) \(\mathstrut -\mathstrut 2811763588532566015625q^{25} \) \(\mathstrut -\mathstrut 8849117507807539429376q^{26} \) \(\mathstrut +\mathstrut 16706362844475509873400q^{27} \) \(\mathstrut -\mathstrut 18076829077752065818624q^{28} \) \(\mathstrut +\mathstrut 34216595637255487326390q^{29} \) \(\mathstrut -\mathstrut 28126611158540451840000q^{30} \) \(\mathstrut +\mathstrut 162779044025674703954912q^{31} \) \(\mathstrut +\mathstrut 37778931862957161709568q^{32} \) \(\mathstrut +\mathstrut 144042856837100540809776q^{33} \) \(\mathstrut -\mathstrut 533315025663622073745408q^{34} \) \(\mathstrut -\mathstrut 723107436365576307990000q^{35} \) \(\mathstrut -\mathstrut 234402851356991891177472q^{36} \) \(\mathstrut -\mathstrut 2508947534199378170868946q^{37} \) \(\mathstrut +\mathstrut 3574573110405955077406720q^{38} \) \(\mathstrut +\mathstrut 5396809090848923436126984q^{39} \) \(\mathstrut +\mathstrut 1511228902511141191680000q^{40} \) \(\mathstrut -\mathstrut 9294979700151843029406438q^{41} \) \(\mathstrut +\mathstrut 11024511248094536339030016q^{42} \) \(\mathstrut +\mathstrut 5860087384842373652051588q^{43} \) \(\mathstrut -\mathstrut 7739351435745337647562752q^{44} \) \(\mathstrut -\mathstrut 9376558476737754406376250q^{45} \) \(\mathstrut -\mathstrut 10024890897381143467524096q^{46} \) \(\mathstrut -\mathstrut 91636472270843977428811056q^{47} \) \(\mathstrut -\mathstrut 23040227767422206619942912q^{48} \) \(\mathstrut +\mathstrut 125653930542417372958833033q^{49} \) \(\mathstrut -\mathstrut 92135869269035123200000000q^{50} \) \(\mathstrut +\mathstrut 325252701893532232959224472q^{51} \) \(\mathstrut -\mathstrut 289967882495837452021792768q^{52} \) \(\mathstrut +\mathstrut 588383623092844605701712798q^{53} \) \(\mathstrut +\mathstrut 547434097687773507531571200q^{54} \) \(\mathstrut -\mathstrut 309588731063556002274645000q^{55} \) \(\mathstrut -\mathstrut 592341535219779692744671232q^{56} \) \(\mathstrut -\mathstrut 2180024012690796274700082480q^{57} \) \(\mathstrut +\mathstrut 1121209405841587808711147520q^{58} \) \(\mathstrut -\mathstrut 3224099468795819619777086220q^{59} \) \(\mathstrut -\mathstrut 921652794443053525893120000q^{60} \) \(\mathstrut +\mathstrut 4044242774173218113506748822q^{61} \) \(\mathstrut +\mathstrut 5333943714633308699194556416q^{62} \) \(\mathstrut +\mathstrut 3675237440178546250213240728q^{63} \) \(\mathstrut +\mathstrut 1237940039285380274899124224q^{64} \) \(\mathstrut -\mathstrut 11599265072322854145040867500q^{65} \) \(\mathstrut +\mathstrut 4719996332838110521254739968q^{66} \) \(\mathstrut +\mathstrut 16451964098682811241246675564q^{67} \) \(\mathstrut -\mathstrut 17475666760945568112489529344q^{68} \) \(\mathstrut +\mathstrut 6113877715153046138226826464q^{69} \) \(\mathstrut -\mathstrut 23694784474827204460216320000q^{70} \) \(\mathstrut -\mathstrut 6891776556518947499300114568q^{71} \) \(\mathstrut -\mathstrut 7680912633265910290103402496q^{72} \) \(\mathstrut -\mathstrut 23463506103704111776642483462q^{73} \) \(\mathstrut -\mathstrut 82213192800645223903033622528q^{74} \) \(\mathstrut +\mathstrut 56190879647115568160245312500q^{75} \) \(\mathstrut +\mathstrut 117131611681782335976463400960q^{76} \) \(\mathstrut +\mathstrut 121346451183015484356030548448q^{77} \) \(\mathstrut +\mathstrut 176842640288937523155009011712q^{78} \) \(\mathstrut -\mathstrut 63560008348497207229978461520q^{79} \) \(\mathstrut +\mathstrut 49519948677485074568970240000q^{80} \) \(\mathstrut -\mathstrut 199022511728187189702750127959q^{81} \) \(\mathstrut -\mathstrut 304577894814575592387590160384q^{82} \) \(\mathstrut -\mathstrut 280657037489676757114557835812q^{83} \) \(\mathstrut +\mathstrut 361251184577561766757335564288q^{84} \) \(\mathstrut -\mathstrut 699059803903279747973138002500q^{85} \) \(\mathstrut +\mathstrut 192023343426514899830426435584q^{86} \) \(\mathstrut -\mathstrut 683791701133188756893890954680q^{87} \) \(\mathstrut -\mathstrut 253603067846503224035336257536q^{88} \) \(\mathstrut +\mathstrut 1976245272429462521105993151210q^{89} \) \(\mathstrut -\mathstrut 307251068165742736388136960000q^{90} \) \(\mathstrut +\mathstrut 4546449891835783946572101758032q^{91} \) \(\mathstrut -\mathstrut 328495624925385309143829577728q^{92} \) \(\mathstrut -\mathstrut 3253010924966416726872199849344q^{93} \) \(\mathstrut -\mathstrut 3002743923371015452387280683008q^{94} \) \(\mathstrut +\mathstrut 4685486546134587913524082725000q^{95} \) \(\mathstrut -\mathstrut 754982183482890866522289340416q^{96} \) \(\mathstrut -\mathstrut 7780538805992504970550343918686q^{97} \) \(\mathstrut +\mathstrut 4117427996013932477115040825344q^{98} \) \(\mathstrut +\mathstrut 1573503518620865543401054981044q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
32768.0 −1.99842e7 1.07374e9 4.29517e10 −6.54843e11 −1.68354e13 3.51844e13 −2.18305e14 1.40744e15
\(n\): 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 \(T_{3} \) \(\mathstrut +\mathstrut 19984212 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(2))\).