Properties

Label 2.42.a.a
Level 2
Weight 42
Character orbit 2.a
Self dual yes
Analytic conductor 21.294
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.2943340913\)
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

\(f(q)\) \(=\) \( q - 1048576q^{2} + 5043516516q^{3} + 1099511627776q^{4} - 48504195130650q^{5} - 5288510374281216q^{6} - 119392445696650168q^{7} - 1152921504606846976q^{8} - 11035937530006008147q^{9} + O(q^{10}) \) \( q - 1048576q^{2} + 5043516516q^{3} + 1099511627776q^{4} - 48504195130650q^{5} - 5288510374281216q^{6} - 119392445696650168q^{7} - 1152921504606846976q^{8} - 11035937530006008147q^{9} + 50860334913316454400q^{10} + 3153808852281809358252q^{11} + 5545405054222300348416q^{12} - 11410299686425943429074q^{13} + 125192053138810646560768q^{14} - 244631709236720052815400q^{15} + 1208925819614629174706176q^{16} - 26723760622401267203746158q^{17} + 11572019231463579998748672q^{18} + 67975218671585815673353460q^{19} - 53330926542065714488934400q^{20} - 602157771756688248182174688q^{21} - 3307008271090250529638449152q^{22} - 13505073711965391061204062504q^{23} - 5814778650136202810140655616q^{24} - 43122078143374240725182530625q^{25} + 11964566403993770057084698624q^{26} - 239612092818398574244863787800q^{27} - 131273382312081512528103866368q^{28} + 136715473782261984486517330110q^{29} + 256514939144602966100960870400q^{30} + 3061422960031285210493644903712q^{31} - 1267650600228229401496703205376q^{32} + 15906287034790309784707322890032q^{33} + 28021894018395031159435331371008q^{34} + 5791034483195853625722104449200q^{35} - 12134141637651154860767887491072q^{36} - 221949310451710778755633026909178q^{37} - 71277182893776768255502277672960q^{38} - 57548034920998866695416393586184q^{39} + 55921529629773098635948877414400q^{40} - 501985001900072034529449934630038q^{41} + 631408187677541136525872005644288q^{42} - 3118478207799237814439067265458484q^{43} + 3467649504866730539366166458007552q^{44} + 535289267405075008628961629405550q^{45} + 14161096172597821897393111044194304q^{46} + 13155518377394721101101992302953392q^{47} + 6097237337845218997846048103202816q^{48} - 30313084236935636680507069400139783q^{49} + 45216776211266787842648997232640000q^{50} - 134781728068711230721422884944545528q^{51} - 12545757181634171431377644944359424q^{52} - 323998598696161709213951685217879914q^{53} + 251251489839145103387382291156172800q^{54} - 152972959975848202616098259795623800q^{55} + 137650118131273184080669039780691968q^{56} + 342834138048854641259889836615745360q^{57} - 143356564636709142644934395937423360q^{58} + 3459574233993702447512816107676048220q^{59} - 268975408828491199782281145640550400q^{60} - 978043389864200648530692366749996578q^{61} - 3210134641737764920878584198554714112q^{62} + 1317607572262865911676440031616918696q^{63} + 1329227995784915872903807060280344576q^{64} + 553447402489598467174323474438518100q^{65} - 16678950833792283872809265806738194432q^{66} + 16627547762843789524411885842685183652q^{67} - 29383085542232588193036062027686084608q^{68} - 68113062316094876637581456085220316064q^{69} - 6072339774251575411445181394924339200q^{70} + 116968857862178710569082285660379434632q^{71} + 12723569701841697359284548393838313472q^{72} + 190708537939407193015711554853622755466q^{73} + 232730720156213081544466656824318230528q^{74} - 217486913320350599066417910321863302500q^{75} + 74739543330024868550281556313201704960q^{76} - 376540952133670515087596878910227986336q^{77} + 60343488265321307644012940321026473984q^{78} - 561362454250092673715323867778422642480q^{79} - 58637973853068956675288730083681894400q^{80} - 805973838012325945866576827820804479559q^{81} + 526369425352369933678752494654626725888q^{82} - 605770926784301808468900622973886535884q^{83} - 662079471802165374773752764190464933888q^{84} + 1296214499853731758014333938302245542700q^{85} + 3269961405221293590513259396945395318784q^{86} + 689526750013603306596685933730009096760q^{87} - 3636094047215136842046417359871726845952q^{88} + 11915426835300121180468989518247227755290q^{89} - 561291478858543932248122069515553996800q^{90} + 1362303585694113890907809755352898184432q^{91} - 14848985580277933693880878806277086511104q^{92} + 15440337261379394835831234584910701707392q^{93} - 13794560838095047073309122681061655969792q^{94} - 3297083270495201682097079345488329549000q^{95} - 6393416738768388355885417735863995990016q^{96} - 63576010574535484047856854187104758743198q^{97} + 31785572616829022167899380803320973099008q^{98} - 34805237475361994580232872393610353679044q^{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
−1.04858e6 5.04352e9 1.09951e12 −4.85042e13 −5.28851e15 −1.19392e17 −1.15292e18 −1.10359e19 5.08603e19
\(n\): 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 2.42.a.a 1
3.b odd 2 1 18.42.a.b 1
4.b odd 2 1 16.42.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.42.a.a 1 1.a even 1 1 trivial
16.42.a.a 1 4.b odd 2 1
18.42.a.b 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 \( T_{3} - 5043516516 \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 1048576 T \)
$3$ \( 1 - 5043516516 T + 36472996377170786403 T^{2} \)
$5$ \( 1 + 48504195130650 T + \)\(45\!\cdots\!25\)\( T^{2} \)
$7$ \( 1 + 119392445696650168 T + \)\(44\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 - \)\(31\!\cdots\!52\)\( T + \)\(49\!\cdots\!11\)\( T^{2} \)
$13$ \( 1 + \)\(11\!\cdots\!74\)\( T + \)\(46\!\cdots\!13\)\( T^{2} \)
$17$ \( 1 + \)\(26\!\cdots\!58\)\( T + \)\(28\!\cdots\!17\)\( T^{2} \)
$19$ \( 1 - \)\(67\!\cdots\!60\)\( T + \)\(26\!\cdots\!19\)\( T^{2} \)
$23$ \( 1 + \)\(13\!\cdots\!04\)\( T + \)\(67\!\cdots\!23\)\( T^{2} \)
$29$ \( 1 - \)\(13\!\cdots\!10\)\( T + \)\(90\!\cdots\!29\)\( T^{2} \)
$31$ \( 1 - \)\(30\!\cdots\!12\)\( T + \)\(13\!\cdots\!31\)\( T^{2} \)
$37$ \( 1 + \)\(22\!\cdots\!78\)\( T + \)\(19\!\cdots\!37\)\( T^{2} \)
$41$ \( 1 + \)\(50\!\cdots\!38\)\( T + \)\(13\!\cdots\!41\)\( T^{2} \)
$43$ \( 1 + \)\(31\!\cdots\!84\)\( T + \)\(93\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - \)\(13\!\cdots\!92\)\( T + \)\(35\!\cdots\!47\)\( T^{2} \)
$53$ \( 1 + \)\(32\!\cdots\!14\)\( T + \)\(49\!\cdots\!53\)\( T^{2} \)
$59$ \( 1 - \)\(34\!\cdots\!20\)\( T + \)\(40\!\cdots\!59\)\( T^{2} \)
$61$ \( 1 + \)\(97\!\cdots\!78\)\( T + \)\(15\!\cdots\!61\)\( T^{2} \)
$67$ \( 1 - \)\(16\!\cdots\!52\)\( T + \)\(73\!\cdots\!67\)\( T^{2} \)
$71$ \( 1 - \)\(11\!\cdots\!32\)\( T + \)\(79\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - \)\(19\!\cdots\!66\)\( T + \)\(24\!\cdots\!73\)\( T^{2} \)
$79$ \( 1 + \)\(56\!\cdots\!80\)\( T + \)\(63\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + \)\(60\!\cdots\!84\)\( T + \)\(48\!\cdots\!83\)\( T^{2} \)
$89$ \( 1 - \)\(11\!\cdots\!90\)\( T + \)\(84\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + \)\(63\!\cdots\!98\)\( T + \)\(28\!\cdots\!97\)\( T^{2} \)
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