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: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.34.a.a 1
3.b odd 2 1 18.34.a.c 1
4.b odd 2 1 16.34.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.34.a.a 1 1.a even 1 1 trivial
16.34.a.a 1 4.b odd 2 1
18.34.a.c 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 133005564 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 65536 T \)
$3$ \( 1 + 133005564 T + 5559060566555523 T^{2} \)
$5$ \( 1 - 538799132550 T + \)\(11\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 + 33347311051768 T + \)\(77\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 + 85882263625386228 T + \)\(23\!\cdots\!31\)\( T^{2} \)
$13$ \( 1 - 1144054008875905166 T + \)\(57\!\cdots\!53\)\( T^{2} \)
$17$ \( 1 + \)\(13\!\cdots\!98\)\( T + \)\(40\!\cdots\!37\)\( T^{2} \)
$19$ \( 1 - 80695000174130231060 T + \)\(15\!\cdots\!59\)\( T^{2} \)
$23$ \( 1 + \)\(14\!\cdots\!44\)\( T + \)\(86\!\cdots\!83\)\( T^{2} \)
$29$ \( 1 + \)\(16\!\cdots\!90\)\( T + \)\(18\!\cdots\!89\)\( T^{2} \)
$31$ \( 1 + \)\(18\!\cdots\!28\)\( T + \)\(16\!\cdots\!91\)\( T^{2} \)
$37$ \( 1 + \)\(96\!\cdots\!98\)\( T + \)\(56\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 - \)\(64\!\cdots\!42\)\( T + \)\(16\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 + \)\(81\!\cdots\!84\)\( T + \)\(80\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + \)\(62\!\cdots\!68\)\( T + \)\(15\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 + \)\(21\!\cdots\!94\)\( T + \)\(79\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(27\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + \)\(45\!\cdots\!58\)\( T + \)\(82\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 - \)\(11\!\cdots\!12\)\( T + \)\(18\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 - \)\(25\!\cdots\!72\)\( T + \)\(12\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 + \)\(28\!\cdots\!74\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 - \)\(92\!\cdots\!20\)\( T + \)\(41\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 + \)\(16\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 + \)\(20\!\cdots\!10\)\( T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 - \)\(22\!\cdots\!42\)\( T + \)\(36\!\cdots\!77\)\( T^{2} \)
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