Properties

Label 3.24.a.b
Level 3
Weight 24
Character orbit 3.a
Self dual Yes
Analytic conductor 10.056
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 24 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.0561211204\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{530401}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{530401}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -621 - \beta ) q^{2} -177147 q^{3} + ( -3229358 + 1242 \beta ) q^{4} + ( -23404410 - 62720 \beta ) q^{5} + ( 110008287 + 177147 \beta ) q^{6} + ( -105981952 - 3642624 \beta ) q^{7} + ( 1285934508 + 10846684 \beta ) q^{8} + 31381059609 q^{9} +O(q^{10})\) \( q + ( -621 - \beta ) q^{2} -177147 q^{3} + ( -3229358 + 1242 \beta ) q^{4} + ( -23404410 - 62720 \beta ) q^{5} + ( 110008287 + 177147 \beta ) q^{6} + ( -105981952 - 3642624 \beta ) q^{7} + ( 1285934508 + 10846684 \beta ) q^{8} + 31381059609 q^{9} + ( 313934895090 + 62353530 \beta ) q^{10} + ( 734486183244 - 135883264 \beta ) q^{11} + ( 572071081626 - 220016574 \beta ) q^{12} + ( 5245827132374 - 1301405184 \beta ) q^{13} + ( 17454277502208 + 2368051456 \beta ) q^{14} + ( 4146021018270 + 11110659840 \beta ) q^{15} + ( -25486575338360 - 18440376408 \beta ) q^{16} + ( -105444005760414 + 15197836800 \beta ) q^{17} + ( -19487638017189 - 31381059609 \beta ) q^{18} + ( -453691224268972 - 142703202816 \beta ) q^{19} + ( -296274520879380 + 173477056540 \beta ) q^{20} + ( 18774384850944 + 645279913728 \beta ) q^{21} + ( 192537652185252 - 650102676300 \beta ) q^{22} + ( 5058461661946056 + 114772628992 \beta ) q^{23} + ( -227799440288676 - 1921457530548 \beta ) q^{24} + ( 7405252898795575 + 2935849190400 \beta ) q^{25} + ( 2954740849784802 - 4437654513110 \beta ) q^{26} -5559060566555523 q^{27} + ( -21254217021293056 + 11631707371008 \beta ) q^{28} + ( 9239376772588446 + 10893499728128 \beta ) q^{29} + ( -55612624860508230 - 11045740778910 \beta ) q^{30} + ( -136396811296372744 - 25314451976448 \beta ) q^{31} + ( 93067109568453168 - 54050531088144 \beta ) q^{32} + ( -130112023903124868 + 24071312567808 \beta ) q^{33} + ( -7067802951794106 + 96006149107614 \beta ) q^{34} + ( 1093084826229411840 + 91900653601280 \beta ) q^{35} + ( -101340675896801022 + 38975276034378 \beta ) q^{36} + ( -239497518393682 - 546678772752384 \beta ) q^{37} + ( 962951543562314556 + 542309913217708 \beta ) q^{38} + ( -929282539018656978 + 230540024130048 \beta ) q^{39} + ( -3277601933357892600 - 334514051818200 \beta ) q^{40} + ( 2777857480654385706 - 1766616871075328 \beta ) q^{41} + ( -3091972896683640576 - 419493211276032 \beta ) q^{42} + ( -599161073804660020 + 4583930058975744 \beta ) q^{43} + ( -3177546568147359144 + 1351047545253560 \beta ) q^{44} + ( -734455185323475690 - 1968220058676480 \beta ) q^{45} + ( -3689184346778372904 - 5129735464550088 \beta ) q^{46} + ( 13154282836427888640 + 667845663375872 \beta ) q^{47} + ( 4514870361464458920 + 3266657359547976 \beta ) q^{48} + ( 35982116424678135945 + 772104803844096 \beta ) q^{49} + ( -18613258168088205675 - 9228415246033975 \beta ) q^{50} + ( 18679089288440058858 - 2692251195609600 \beta ) q^{51} + ( -24656453994293443444 + 10718020540600380 \beta ) q^{52} + ( -20563750949707612314 + 25226888867419392 \beta ) q^{53} + ( 3452176611830979783 + 5559060566555523 \beta ) q^{54} + ( 23493336262593844680 - 42886705790269440 \beta ) q^{55} + ( -188743446110629186560 - 5833728644316160 \beta ) q^{56} + ( 80370039305575582884 + 25279444269245952 \beta ) q^{57} + ( -57738961319466798918 - 16004240103755934 \beta ) q^{58} + ( -153768530954722143348 - 51173029829187584 \beta ) q^{59} + ( 52484142550219528860 - 30730940134891380 \beta ) q^{60} + ( 297480589549251875510 + 102017420065926144 \beta ) q^{61} + ( 205543715599887434856 + 152117085973746952 \beta ) q^{62} + ( -3325825953190176768 - 114309400877174016 \beta ) q^{63} + ( 414018316391103977248 + 95187359296444320 \beta ) q^{64} + ( 266866207581388222980 - 298559657240035840 \beta ) q^{65} + ( -34107467471660836044 + 115163738798516100 \beta ) q^{66} + ( -815112912563496632044 + 219887015935518720 \beta ) q^{67} + ( 430621718471470944612 - 180040711007208588 \beta ) q^{68} + ( -896091308028757982232 - 20331626908045824 \beta ) q^{69} + ( -1117503464225417372160 - 1150155132115806720 \beta ) q^{70} + ( -191971299290824636488 + 1000477109551480320 \beta ) q^{71} + ( 40353987448818087372 + 340380437163986556 \beta ) q^{72} + ( -1406101175720280846214 + 744202923877896192 \beta ) q^{73} + ( 2609779437678657510378 + 339727015397624146 \beta ) q^{74} + ( -1311818335262939724525 - 520076876531788800 \beta ) q^{75} + ( 619069442355025463528 - 102644770902591096 \beta ) q^{76} + ( 2284958789562030759936 - 2661055825190140928 \beta ) q^{77} + ( -523423477316828319894 + 786117184033897170 \beta ) q^{78} + ( 10534694287656642440 - 167160086945199360 \beta ) q^{79} + ( 6117560905046011291440 + 2030104135229098480 \beta ) q^{80} + 984770902183611232881 q^{81} + ( 6708088699830651895326 - 1680788403716607018 \beta ) q^{82} + ( -7408631040162201348012 + 1186031845274398208 \beta ) q^{83} + ( 3765120782671000991232 - 2060522065651954176 \beta ) q^{84} + ( -2082389091920491438260 + 6257751637712878080 \beta ) q^{85} + ( -21509810758064448467676 - 2247459492819277004 \beta ) q^{86} + ( -1636727877133725443562 - 1929750796338690816 \beta ) q^{87} + ( -6091239192053213678832 + 7792002553776488784 \beta ) q^{88} + ( -21724902630442228278582 - 7489945387427920896 \beta ) q^{89} + ( 9851609656164451419810 + 1956719841761569770 \beta ) q^{90} + ( 22073472513261952208896 - 18970650350493470208 \beta ) q^{91} + ( -15655117104549130329072 + 5911967476520654416 \beta ) q^{92} + ( 24162285930718542481368 + 4484379224271833856 \beta ) q^{93} + ( -11356843710723751807488 - 13569014993384305152 \beta ) q^{94} + ( 53343822301422237214200 + 31795397853168742400 \beta ) q^{95} + ( -16486559258722773351696 + 9574889430671445168 \beta ) q^{96} + ( 53568286522621084225346 + 19434897144083543040 \beta ) q^{97} + ( -26030620740298533684309 - 36461593507865319561 \beta ) q^{98} + ( 23048954698366860991596 - 4264160807449483776 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1242q^{2} - 354294q^{3} - 6458716q^{4} - 46808820q^{5} + 220016574q^{6} - 211963904q^{7} + 2571869016q^{8} + 62762119218q^{9} + O(q^{10}) \) \( 2q - 1242q^{2} - 354294q^{3} - 6458716q^{4} - 46808820q^{5} + 220016574q^{6} - 211963904q^{7} + 2571869016q^{8} + 62762119218q^{9} + 627869790180q^{10} + 1468972366488q^{11} + 1144142163252q^{12} + 10491654264748q^{13} + 34908555004416q^{14} + 8292042036540q^{15} - 50973150676720q^{16} - 210888011520828q^{17} - 38975276034378q^{18} - 907382448537944q^{19} - 592549041758760q^{20} + 37548769701888q^{21} + 385075304370504q^{22} + 10116923323892112q^{23} - 455598880577352q^{24} + 14810505797591150q^{25} + 5909481699569604q^{26} - 11118121133111046q^{27} - 42508434042586112q^{28} + 18478753545176892q^{29} - 111225249721016460q^{30} - 272793622592745488q^{31} + 186134219136906336q^{32} - 260224047806249736q^{33} - 14135605903588212q^{34} + 2186169652458823680q^{35} - 202681351793602044q^{36} - 478995036787364q^{37} + 1925903087124629112q^{38} - 1858565078037313956q^{39} - 6555203866715785200q^{40} + 5555714961308771412q^{41} - 6183945793367281152q^{42} - 1198322147609320040q^{43} - 6355093136294718288q^{44} - 1468910370646951380q^{45} - 7378368693556745808q^{46} + 26308565672855777280q^{47} + 9029740722928917840q^{48} + 71964232849356271890q^{49} - 37226516336176411350q^{50} + 37358178576880117716q^{51} - 49312907988586886888q^{52} - 41127501899415224628q^{53} + 6904353223661959566q^{54} + 46986672525187689360q^{55} - 377486892221258373120q^{56} + 160740078611151165768q^{57} - 115477922638933597836q^{58} - 307537061909444286696q^{59} + 104968285100439057720q^{60} + 594961179098503751020q^{61} + 411087431199774869712q^{62} - 6651651906380353536q^{63} + 828036632782207954496q^{64} + 533732415162776445960q^{65} - 68214934943321672088q^{66} - 1630225825126993264088q^{67} + 861243436942941889224q^{68} - 1792182616057515964464q^{69} - 2235006928450834744320q^{70} - 383942598581649272976q^{71} + 80707974897636174744q^{72} - 2812202351440561692428q^{73} + 5219558875357315020756q^{74} - 2623636670525879449050q^{75} + 1238138884710050927056q^{76} + 4569917579124061519872q^{77} - 1046846954633656639788q^{78} + 21069388575313284880q^{79} + 12235121810092022582880q^{80} + 1969541804367222465762q^{81} + 13416177399661303790652q^{82} - 14817262080324402696024q^{83} + 7530241565342001982464q^{84} - 4164778183840982876520q^{85} - 43019621516128896935352q^{86} - 3273455754267450887124q^{87} - 12182478384106427357664q^{88} - 43449805260884456557164q^{89} + 19703219312328902839620q^{90} + 44146945026523904417792q^{91} - 31310234209098260658144q^{92} + 48324571861437084962736q^{93} - 22713687421447503614976q^{94} + 106687644602844474428400q^{95} - 32973118517445546703392q^{96} + 107136573045242168450692q^{97} - 52061241480597067368618q^{98} + 46097909396733721983192q^{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
364.643
−363.643
−2805.86 −177147. −515763. −1.60439e8 4.97050e8 −8.06460e9 2.49844e10 3.13811e10 4.50169e11
1.2 1563.86 −177147. −5.94295e6 1.13630e8 −2.77033e8 7.85264e9 −2.24125e10 3.13811e10 1.77701e11
\(n\): 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
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1242 T_{2} - 4387968 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(3))\).