Properties

Label 2.34.a.a
Level 2
Weight 34
Character orbit 2.a
Self dual Yes
Analytic conductor 13.797
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(13.7965657762\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 65536q^{2} \) \(\mathstrut -\mathstrut 133005564q^{3} \) \(\mathstrut +\mathstrut 4294967296q^{4} \) \(\mathstrut +\mathstrut 538799132550q^{5} \) \(\mathstrut +\mathstrut 8716652642304q^{6} \) \(\mathstrut -\mathstrut 33347311051768q^{7} \) \(\mathstrut -\mathstrut 281474976710656q^{8} \) \(\mathstrut +\mathstrut 12131419488402573q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 65536q^{2} \) \(\mathstrut -\mathstrut 133005564q^{3} \) \(\mathstrut +\mathstrut 4294967296q^{4} \) \(\mathstrut +\mathstrut 538799132550q^{5} \) \(\mathstrut +\mathstrut 8716652642304q^{6} \) \(\mathstrut -\mathstrut 33347311051768q^{7} \) \(\mathstrut -\mathstrut 281474976710656q^{8} \) \(\mathstrut +\mathstrut 12131419488402573q^{9} \) \(\mathstrut -\mathstrut 35310739950796800q^{10} \) \(\mathstrut -\mathstrut 85882263625386228q^{11} \) \(\mathstrut -\mathstrut 571254547566034944q^{12} \) \(\mathstrut +\mathstrut 1144054008875905166q^{13} \) \(\mathstrut +\mathstrut 2185449377088667648q^{14} \) \(\mathstrut -\mathstrut 71663282507523508200q^{15} \) \(\mathstrut +\mathstrut 18446744073709551616q^{16} \) \(\mathstrut -\mathstrut 139113675669385621998q^{17} \) \(\mathstrut -\mathstrut 795044707591951024128q^{18} \) \(\mathstrut +\mathstrut 80695000174130231060q^{19} \) \(\mathstrut +\mathstrut 2314124653415419084800q^{20} \) \(\mathstrut +\mathstrut 4435377914323836037152q^{21} \) \(\mathstrut +\mathstrut 5628380028953311838208q^{22} \) \(\mathstrut -\mathstrut 14120372378143910765544q^{23} \) \(\mathstrut +\mathstrut 37437738029287666089984q^{24} \) \(\mathstrut +\mathstrut 173889183409697655049375q^{25} \) \(\mathstrut -\mathstrut 74976723525691320958976q^{26} \) \(\mathstrut -\mathstrut 874160305210698806986200q^{27} \) \(\mathstrut -\mathstrut 143225610376882922979328q^{28} \) \(\mathstrut -\mathstrut 1632686905195131326709090q^{29} \) \(\mathstrut +\mathstrut 4696524882413060633395200q^{30} \) \(\mathstrut -\mathstrut 1894078958241443951861728q^{31} \) \(\mathstrut -\mathstrut 1208925819614629174706176q^{32} \) \(\mathstrut +\mathstrut 11422818911091179972972592q^{33} \) \(\mathstrut +\mathstrut 9116953848668856123260928q^{34} \) \(\mathstrut -\mathstrut 17967502267567626543848400q^{35} \) \(\mathstrut +\mathstrut 52104049956746102317252608q^{36} \) \(\mathstrut -\mathstrut 96444218751358368990635098q^{37} \) \(\mathstrut -\mathstrut 5288427531411798822748160q^{38} \) \(\mathstrut -\mathstrut 152165548697000772614343624q^{39} \) \(\mathstrut -\mathstrut 151658473286232905141452800q^{40} \) \(\mathstrut +\mathstrut 641768233498553833164038442q^{41} \) \(\mathstrut -\mathstrut 290676926993126918530793472q^{42} \) \(\mathstrut -\mathstrut 817975597351211427387164884q^{43} \) \(\mathstrut -\mathstrut 368861513577484244628799488q^{44} \) \(\mathstrut +\mathstrut 6536398296951471117588051150q^{45} \) \(\mathstrut +\mathstrut 925392724174039335930691584q^{46} \) \(\mathstrut -\mathstrut 6229246687280441243201826768q^{47} \) \(\mathstrut -\mathstrut 2453519599487396484873191424q^{48} \) \(\mathstrut -\mathstrut 6618950565324076329761168583q^{49} \) \(\mathstrut -\mathstrut 11396001523937945521315840000q^{50} \) \(\mathstrut +\mathstrut 18502892892519712187334796872q^{51} \) \(\mathstrut +\mathstrut 4913674552979706410367451136q^{52} \) \(\mathstrut -\mathstrut 21322120079333214208388446794q^{53} \) \(\mathstrut +\mathstrut 57288969762288357014647603200q^{54} \) \(\mathstrut -\mathstrut 46273289142788517805116521400q^{55} \) \(\mathstrut +\mathstrut 9386433601659399240373239808q^{56} \) \(\mathstrut -\mathstrut 10732884010140289591585617840q^{57} \) \(\mathstrut +\mathstrut 106999769018868126627206922240q^{58} \) \(\mathstrut +\mathstrut 298987905886407341741567881020q^{59} \) \(\mathstrut -\mathstrut 307791454693822341670187827200q^{60} \) \(\mathstrut -\mathstrut 455881915835062287556960014658q^{61} \) \(\mathstrut +\mathstrut 124130358607311270829210206208q^{62} \) \(\mathstrut -\mathstrut 404550219179240819106827399064q^{63} \) \(\mathstrut +\mathstrut 79228162514264337593543950336q^{64} \) \(\mathstrut +\mathstrut 616415307572687704036863753300q^{65} \) \(\mathstrut -\mathstrut 748605860157271570708731789312q^{66} \) \(\mathstrut +\mathstrut 1172332419477563429554964377412q^{67} \) \(\mathstrut -\mathstrut 597488687426362154894028177408q^{68} \) \(\mathstrut +\mathstrut 1878088092045052124536851486816q^{69} \) \(\mathstrut +\mathstrut 1177518228607311973177648742400q^{70} \) \(\mathstrut +\mathstrut 2591524145775150288511661030472q^{71} \) \(\mathstrut -\mathstrut 3414691017965312561463466917888q^{72} \) \(\mathstrut -\mathstrut 2825174388069163226217247688374q^{73} \) \(\mathstrut +\mathstrut 6320568320089022070170261782528q^{74} \) \(\mathstrut -\mathstrut 23128228912906279679319569722500q^{75} \) \(\mathstrut +\mathstrut 346582386698603647647623413760q^{76} \) \(\mathstrut +\mathstrut 2863942558945695063591902251104q^{77} \) \(\mathstrut +\mathstrut 9972321399406642634053623742464q^{78} \) \(\mathstrut +\mathstrut 920688453939087595198198640720q^{79} \) \(\mathstrut +\mathstrut 9939089705286559671350250700800q^{80} \) \(\mathstrut +\mathstrut 48828888726639213211634490656121q^{81} \) \(\mathstrut -\mathstrut 42058922950561224010238423334912q^{82} \) \(\mathstrut -\mathstrut 16199219945453134166417678661804q^{83} \) \(\mathstrut +\mathstrut 19049803087421565732834080980992q^{84} \) \(\mathstrut -\mathstrut 74954327776507013723964597834900q^{85} \) \(\mathstrut +\mathstrut 53606848748008992105245237837824q^{86} \) \(\mathstrut +\mathstrut 217156442660892972163010779376760q^{87} \) \(\mathstrut +\mathstrut 24173708153814007455993003245568q^{88} \) \(\mathstrut -\mathstrut 203491630107372946965013220025510q^{89} \) \(\mathstrut -\mathstrut 428369398789011611162250520166400q^{90} \) \(\mathstrut -\mathstrut 38151124894006957908596484633488q^{91} \) \(\mathstrut -\mathstrut 60646537571469841919553803649024q^{92} \) \(\mathstrut +\mathstrut 251923040101435700991757982654592q^{93} \) \(\mathstrut +\mathstrut 408239910897610997314474919067648q^{94} \) \(\mathstrut +\mathstrut 43478396094943467445859067003000q^{95} \) \(\mathstrut +\mathstrut 160793860472006016032649473163264q^{96} \) \(\mathstrut +\mathstrut 226806680667600950875216250271842q^{97} \) \(\mathstrut +\mathstrut 433779544249078666347227944255488q^{98} \) \(\mathstrut -\mathstrut 1041873766653137898400473873964644q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−65536.0 −1.33006e8 4.29497e9 5.38799e11 8.71665e12 −3.33473e13 −2.81475e14 1.21314e16 −3.53107e16
\(n\): 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 \(T_{3} \) \(\mathstrut +\mathstrut 133005564 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\).