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

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

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 1048576q^{2} \) \(\mathstrut +\mathstrut 5043516516q^{3} \) \(\mathstrut +\mathstrut 1099511627776q^{4} \) \(\mathstrut -\mathstrut 48504195130650q^{5} \) \(\mathstrut -\mathstrut 5288510374281216q^{6} \) \(\mathstrut -\mathstrut 119392445696650168q^{7} \) \(\mathstrut -\mathstrut 1152921504606846976q^{8} \) \(\mathstrut -\mathstrut 11035937530006008147q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 1048576q^{2} \) \(\mathstrut +\mathstrut 5043516516q^{3} \) \(\mathstrut +\mathstrut 1099511627776q^{4} \) \(\mathstrut -\mathstrut 48504195130650q^{5} \) \(\mathstrut -\mathstrut 5288510374281216q^{6} \) \(\mathstrut -\mathstrut 119392445696650168q^{7} \) \(\mathstrut -\mathstrut 1152921504606846976q^{8} \) \(\mathstrut -\mathstrut 11035937530006008147q^{9} \) \(\mathstrut +\mathstrut 50860334913316454400q^{10} \) \(\mathstrut +\mathstrut 3153808852281809358252q^{11} \) \(\mathstrut +\mathstrut 5545405054222300348416q^{12} \) \(\mathstrut -\mathstrut 11410299686425943429074q^{13} \) \(\mathstrut +\mathstrut 125192053138810646560768q^{14} \) \(\mathstrut -\mathstrut 244631709236720052815400q^{15} \) \(\mathstrut +\mathstrut 1208925819614629174706176q^{16} \) \(\mathstrut -\mathstrut 26723760622401267203746158q^{17} \) \(\mathstrut +\mathstrut 11572019231463579998748672q^{18} \) \(\mathstrut +\mathstrut 67975218671585815673353460q^{19} \) \(\mathstrut -\mathstrut 53330926542065714488934400q^{20} \) \(\mathstrut -\mathstrut 602157771756688248182174688q^{21} \) \(\mathstrut -\mathstrut 3307008271090250529638449152q^{22} \) \(\mathstrut -\mathstrut 13505073711965391061204062504q^{23} \) \(\mathstrut -\mathstrut 5814778650136202810140655616q^{24} \) \(\mathstrut -\mathstrut 43122078143374240725182530625q^{25} \) \(\mathstrut +\mathstrut 11964566403993770057084698624q^{26} \) \(\mathstrut -\mathstrut 239612092818398574244863787800q^{27} \) \(\mathstrut -\mathstrut 131273382312081512528103866368q^{28} \) \(\mathstrut +\mathstrut 136715473782261984486517330110q^{29} \) \(\mathstrut +\mathstrut 256514939144602966100960870400q^{30} \) \(\mathstrut +\mathstrut 3061422960031285210493644903712q^{31} \) \(\mathstrut -\mathstrut 1267650600228229401496703205376q^{32} \) \(\mathstrut +\mathstrut 15906287034790309784707322890032q^{33} \) \(\mathstrut +\mathstrut 28021894018395031159435331371008q^{34} \) \(\mathstrut +\mathstrut 5791034483195853625722104449200q^{35} \) \(\mathstrut -\mathstrut 12134141637651154860767887491072q^{36} \) \(\mathstrut -\mathstrut 221949310451710778755633026909178q^{37} \) \(\mathstrut -\mathstrut 71277182893776768255502277672960q^{38} \) \(\mathstrut -\mathstrut 57548034920998866695416393586184q^{39} \) \(\mathstrut +\mathstrut 55921529629773098635948877414400q^{40} \) \(\mathstrut -\mathstrut 501985001900072034529449934630038q^{41} \) \(\mathstrut +\mathstrut 631408187677541136525872005644288q^{42} \) \(\mathstrut -\mathstrut 3118478207799237814439067265458484q^{43} \) \(\mathstrut +\mathstrut 3467649504866730539366166458007552q^{44} \) \(\mathstrut +\mathstrut 535289267405075008628961629405550q^{45} \) \(\mathstrut +\mathstrut 14161096172597821897393111044194304q^{46} \) \(\mathstrut +\mathstrut 13155518377394721101101992302953392q^{47} \) \(\mathstrut +\mathstrut 6097237337845218997846048103202816q^{48} \) \(\mathstrut -\mathstrut 30313084236935636680507069400139783q^{49} \) \(\mathstrut +\mathstrut 45216776211266787842648997232640000q^{50} \) \(\mathstrut -\mathstrut 134781728068711230721422884944545528q^{51} \) \(\mathstrut -\mathstrut 12545757181634171431377644944359424q^{52} \) \(\mathstrut -\mathstrut 323998598696161709213951685217879914q^{53} \) \(\mathstrut +\mathstrut 251251489839145103387382291156172800q^{54} \) \(\mathstrut -\mathstrut 152972959975848202616098259795623800q^{55} \) \(\mathstrut +\mathstrut 137650118131273184080669039780691968q^{56} \) \(\mathstrut +\mathstrut 342834138048854641259889836615745360q^{57} \) \(\mathstrut -\mathstrut 143356564636709142644934395937423360q^{58} \) \(\mathstrut +\mathstrut 3459574233993702447512816107676048220q^{59} \) \(\mathstrut -\mathstrut 268975408828491199782281145640550400q^{60} \) \(\mathstrut -\mathstrut 978043389864200648530692366749996578q^{61} \) \(\mathstrut -\mathstrut 3210134641737764920878584198554714112q^{62} \) \(\mathstrut +\mathstrut 1317607572262865911676440031616918696q^{63} \) \(\mathstrut +\mathstrut 1329227995784915872903807060280344576q^{64} \) \(\mathstrut +\mathstrut 553447402489598467174323474438518100q^{65} \) \(\mathstrut -\mathstrut 16678950833792283872809265806738194432q^{66} \) \(\mathstrut +\mathstrut 16627547762843789524411885842685183652q^{67} \) \(\mathstrut -\mathstrut 29383085542232588193036062027686084608q^{68} \) \(\mathstrut -\mathstrut 68113062316094876637581456085220316064q^{69} \) \(\mathstrut -\mathstrut 6072339774251575411445181394924339200q^{70} \) \(\mathstrut +\mathstrut 116968857862178710569082285660379434632q^{71} \) \(\mathstrut +\mathstrut 12723569701841697359284548393838313472q^{72} \) \(\mathstrut +\mathstrut 190708537939407193015711554853622755466q^{73} \) \(\mathstrut +\mathstrut 232730720156213081544466656824318230528q^{74} \) \(\mathstrut -\mathstrut 217486913320350599066417910321863302500q^{75} \) \(\mathstrut +\mathstrut 74739543330024868550281556313201704960q^{76} \) \(\mathstrut -\mathstrut 376540952133670515087596878910227986336q^{77} \) \(\mathstrut +\mathstrut 60343488265321307644012940321026473984q^{78} \) \(\mathstrut -\mathstrut 561362454250092673715323867778422642480q^{79} \) \(\mathstrut -\mathstrut 58637973853068956675288730083681894400q^{80} \) \(\mathstrut -\mathstrut 805973838012325945866576827820804479559q^{81} \) \(\mathstrut +\mathstrut 526369425352369933678752494654626725888q^{82} \) \(\mathstrut -\mathstrut 605770926784301808468900622973886535884q^{83} \) \(\mathstrut -\mathstrut 662079471802165374773752764190464933888q^{84} \) \(\mathstrut +\mathstrut 1296214499853731758014333938302245542700q^{85} \) \(\mathstrut +\mathstrut 3269961405221293590513259396945395318784q^{86} \) \(\mathstrut +\mathstrut 689526750013603306596685933730009096760q^{87} \) \(\mathstrut -\mathstrut 3636094047215136842046417359871726845952q^{88} \) \(\mathstrut +\mathstrut 11915426835300121180468989518247227755290q^{89} \) \(\mathstrut -\mathstrut 561291478858543932248122069515553996800q^{90} \) \(\mathstrut +\mathstrut 1362303585694113890907809755352898184432q^{91} \) \(\mathstrut -\mathstrut 14848985580277933693880878806277086511104q^{92} \) \(\mathstrut +\mathstrut 15440337261379394835831234584910701707392q^{93} \) \(\mathstrut -\mathstrut 13794560838095047073309122681061655969792q^{94} \) \(\mathstrut -\mathstrut 3297083270495201682097079345488329549000q^{95} \) \(\mathstrut -\mathstrut 6393416738768388355885417735863995990016q^{96} \) \(\mathstrut -\mathstrut 63576010574535484047856854187104758743198q^{97} \) \(\mathstrut +\mathstrut 31785572616829022167899380803320973099008q^{98} \) \(\mathstrut -\mathstrut 34805237475361994580232872393610353679044q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−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.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut -\mathstrut 5043516516 \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(2))\).