Properties

Label 3.19.b.b
Level 3
Weight 19
Character orbit 3.b
Analytic conductor 6.162
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.16158413129\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.601940665.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{11} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 3969 + 11 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -263384 + 7 \beta_{2} + 7 \beta_{3} ) q^{4} \) \( + ( 706 \beta_{1} + 45 \beta_{2} - 9 \beta_{3} ) q^{5} \) \( + ( 5674536 + 1467 \beta_{1} - 9 \beta_{2} - 297 \beta_{3} ) q^{6} \) \( + ( -23936038 - 287 \beta_{2} - 287 \beta_{3} ) q^{7} \) \( + ( 229552 \beta_{1} + 10080 \beta_{2} - 2016 \beta_{3} ) q^{8} \) \( + ( -221335335 - 294354 \beta_{1} + 13041 \beta_{2} + 567 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 3969 + 11 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -263384 + 7 \beta_{2} + 7 \beta_{3} ) q^{4} \) \( + ( 706 \beta_{1} + 45 \beta_{2} - 9 \beta_{3} ) q^{5} \) \( + ( 5674536 + 1467 \beta_{1} - 9 \beta_{2} - 297 \beta_{3} ) q^{6} \) \( + ( -23936038 - 287 \beta_{2} - 287 \beta_{3} ) q^{7} \) \( + ( 229552 \beta_{1} + 10080 \beta_{2} - 2016 \beta_{3} ) q^{8} \) \( + ( -221335335 - 294354 \beta_{1} + 13041 \beta_{2} + 567 \beta_{3} ) q^{9} \) \( + ( 365284080 - 14230 \beta_{2} - 14230 \beta_{3} ) q^{10} \) \( + ( 3151562 \beta_{1} - 72135 \beta_{2} + 14427 \beta_{3} ) q^{11} \) \( + ( 1780792776 - 10912336 \beta_{1} - 156221 \beta_{2} + 11907 \beta_{3} ) q^{12} \) \( + ( -1356555382 + 271492 \beta_{2} + 271492 \beta_{3} ) q^{13} \) \( + ( 14575246 \beta_{1} - 413280 \beta_{2} + 82656 \beta_{3} ) q^{14} \) \( + ( -17071955280 - 8111862 \beta_{1} + 353115 \beta_{2} + 48573 \beta_{3} ) q^{15} \) \( + ( 50306002048 - 1852368 \beta_{2} - 1852368 \beta_{3} ) q^{16} \) \( + ( -45023208 \beta_{1} + 5174460 \beta_{2} - 1034892 \beta_{3} ) q^{17} \) \( + ( -156016905840 + 307637271 \beta_{1} + 3725190 \beta_{2} - 847098 \beta_{3} ) q^{18} \) \( + ( 47854162370 + 5826567 \beta_{2} + 5826567 \beta_{3} ) q^{19} \) \( + ( -644336096 \beta_{1} - 8694720 \beta_{2} + 1738944 \beta_{3} ) q^{20} \) \( + ( -210874853574 + 65323174 \beta_{1} - 28329721 \beta_{2} - 488187 \beta_{3} ) q^{21} \) \( + ( 1665433191600 - 7172270 \beta_{2} - 7172270 \beta_{3} ) q^{22} \) \( + ( 253636396 \beta_{1} - 75062610 \beta_{2} + 15012522 \beta_{3} ) q^{23} \) \( + ( -4229325249408 - 1921812624 \beta_{1} + 79740432 \beta_{2} + 32088528 \beta_{3} ) q^{24} \) \( + ( 2851899863545 + 6957580 \beta_{2} + 6957580 \beta_{3} ) q^{25} \) \( + ( 10211538454 \beta_{1} + 390948480 \beta_{2} - 78189696 \beta_{3} ) q^{26} \) \( + ( -1158051922839 - 6667176663 \beta_{1} - 103375116 \beta_{2} - 120138471 \beta_{3} ) q^{27} \) \( + ( 1437715245008 - 91961058 \beta_{2} - 91961058 \beta_{3} ) q^{28} \) \( + ( -7721532154 \beta_{1} - 588630105 \beta_{2} + 117726021 \beta_{3} ) q^{29} \) \( + ( -4295374900560 + 20311702560 \beta_{1} + 147437010 \beta_{2} - 24205230 \beta_{3} ) q^{30} \) \( + ( 8932103771402 + 419221045 \beta_{2} + 419221045 \beta_{3} ) q^{31} \) \( + ( -50547157248 \beta_{1} - 24998400 \beta_{2} + 4999680 \beta_{3} ) q^{32} \) \( + ( 3060717755760 + 6719743026 \beta_{1} - 527493825 \beta_{2} + 1194271641 \beta_{3} ) q^{33} \) \( + ( -24320836709568 - 752846088 \beta_{2} - 752846088 \beta_{3} ) q^{34} \) \( + ( 1895023844 \beta_{1} - 1206581670 \beta_{2} + 241316334 \beta_{3} ) q^{35} \) \( + ( 103164363505512 + 76079780064 \beta_{1} + 356253471 \beta_{2} - 2766764385 \beta_{3} ) q^{36} \) \( + ( -118824958375558 + 147405468 \beta_{2} + 147405468 \beta_{3} ) q^{37} \) \( + ( 142185146902 \beta_{1} + 8390256480 \beta_{2} - 1678051296 \beta_{3} ) q^{38} \) \( + ( 104227386251274 - 325784793074 \beta_{1} + 2799715646 \beta_{2} + 461807892 \beta_{3} ) q^{39} \) \( + ( -241750823650560 + 2574633760 \beta_{2} + 2574633760 \beta_{3} ) q^{40} \) \( + ( 420913326988 \beta_{1} - 5946773490 \beta_{2} + 1189354698 \beta_{3} ) q^{41} \) \( + ( 37287932487120 + 43605567558 \beta_{1} - 3053475810 \beta_{2} + 5808473118 \beta_{3} ) q^{42} \) \( + ( 399901831143698 - 8960155673 \beta_{2} - 8960155673 \beta_{3} ) q^{43} \) \( + ( -1073200880992 \beta_{1} - 29237826240 \beta_{2} + 5847565248 \beta_{3} ) q^{44} \) \( + ( -55310650325280 - 329533029486 \beta_{1} - 13668702615 \beta_{2} - 3148863741 \beta_{3} ) q^{45} \) \( + ( 142865492344992 + 13717467932 \beta_{2} + 13717467932 \beta_{3} ) q^{46} \) \( + ( 564438621368 \beta_{1} + 43796122380 \beta_{2} - 8759224476 \beta_{3} ) q^{47} \) \( + ( -548206266335616 + 2674357020416 \beta_{1} + 21948100336 \beta_{2} - 3150877968 \beta_{3} ) q^{48} \) \( + ( -855946861082061 + 13739285812 \beta_{2} + 13739285812 \beta_{3} ) q^{49} \) \( + ( -2624971434265 \beta_{1} + 10018915200 \beta_{2} - 2003783040 \beta_{3} ) q^{50} \) \( + ( -1246916675934144 - 747624439944 \beta_{1} + 39468144996 \beta_{2} - 31897494468 \beta_{3} ) q^{51} \) \( + ( 4960980274354832 - 81002536602 \beta_{2} - 81002536602 \beta_{3} ) q^{52} \) \( + ( 5144386753962 \beta_{1} + 61399629585 \beta_{2} - 12279925917 \beta_{3} ) q^{53} \) \( + ( -3505569492615624 - 2669258849517 \beta_{1} - 125189253459 \beta_{2} + 77582178189 \beta_{3} ) q^{54} \) \( + ( -21249758648160 + 49798073660 \beta_{2} + 49798073660 \beta_{3} ) q^{55} \) \( + ( -616303825312 \beta_{1} - 240762795840 \beta_{2} + 48152559168 \beta_{3} ) q^{56} \) \( + ( 2542337906616162 - 6145116654002 \beta_{1} + 137053076573 \beta_{2} + 9910990467 \beta_{3} ) q^{57} \) \( + ( -3982815471605040 + 175543978750 \beta_{2} + 175543978750 \beta_{3} ) q^{58} \) \( + ( 509087468002 \beta_{1} - 118883830635 \beta_{2} + 23776766127 \beta_{3} ) q^{59} \) \( + ( 6180817013902080 + 2312468383392 \beta_{1} - 72798798240 \beta_{2} - 160238983968 \beta_{3} ) q^{60} \) \( + ( -551817186470998 - 222731765300 \beta_{2} - 222731765300 \beta_{3} ) q^{61} \) \( + ( 4741209832318 \beta_{1} + 603678304800 \beta_{2} - 120735660960 \beta_{3} ) q^{62} \) \( + ( 3458295706395978 + 7105051038204 \beta_{1} - 467582684373 \beta_{2} + 93742718391 \beta_{3} ) q^{63} \) \( + ( -13373341897398272 - 126597386496 \beta_{2} - 126597386496 \beta_{3} ) q^{64} \) \( + ( -18736071132844 \beta_{1} + 61419619170 \beta_{2} - 12283923834 \beta_{3} ) q^{65} \) \( + ( 3714388572570480 + 26532129013920 \beta_{1} + 1555632910170 \beta_{2} - 12200031270 \beta_{3} ) q^{66} \) \( + ( -10332884019732478 + 96122514523 \beta_{2} + 96122514523 \beta_{3} ) q^{67} \) \( + ( -12036555134592 \beta_{1} + 272355275520 \beta_{2} - 54471055104 \beta_{3} ) q^{68} \) \( + ( 20355131369645856 + 11431360704348 \beta_{1} - 576134732574 \beta_{2} + 344069165934 \beta_{3} ) q^{69} \) \( + ( 1149759107370720 + 235773289780 \beta_{2} + 235773289780 \beta_{3} ) q^{70} \) \( + ( -18502672209756 \beta_{1} - 2553691296870 \beta_{2} + 510738259374 \beta_{3} ) q^{71} \) \( + ( -1248730460640000 - 95783160852432 \beta_{1} - 3327797753280 \beta_{2} - 644855904000 \beta_{3} ) q^{72} \) \( + ( -11347488690614062 + 566482634352 \beta_{2} + 566482634352 \beta_{3} ) q^{73} \) \( + ( 123632735119846 \beta_{1} + 212263873920 \beta_{2} - 42452774784 \beta_{3} ) q^{74} \) \( + ( 14128227877345785 + 23404358077715 \beta_{1} + 2958413455765 \beta_{2} + 11834843580 \beta_{3} ) q^{75} \) \( + ( 86196978217464464 - 1199645386138 \beta_{2} - 1199645386138 \beta_{3} ) q^{76} \) \( + ( -65467582908812 \beta_{1} + 3704344375410 \beta_{2} - 740868875082 \beta_{3} ) q^{77} \) \( + ( -171457484867426160 - 76456193495058 \beta_{1} + 3104474643270 \beta_{2} + 1633153172742 \beta_{3} ) q^{78} \) \( + ( -44341346007473110 + 563620174117 \beta_{2} + 563620174117 \beta_{3} ) q^{79} \) \( + ( 156816236816896 \beta_{1} + 1428203934720 \beta_{2} - 285640786944 \beta_{3} ) q^{80} \) \( + ( -18195631968179439 + 129206926869948 \beta_{1} - 2906838780930 \beta_{2} - 2229768066582 \beta_{3} ) q^{81} \) \( + ( 221960109520144800 - 1718979240580 \beta_{2} - 1718979240580 \beta_{3} ) q^{82} \) \( + ( -110848762476802 \beta_{1} + 52215388155 \beta_{2} - 10443077631 \beta_{3} ) q^{83} \) \( + ( -31421853871205616 + 121111750482016 \beta_{1} + 29883408086 \beta_{2} - 156425759658 \beta_{3} ) q^{84} \) \( + ( -64609566788972160 - 995855184240 \beta_{2} - 995855184240 \beta_{3} ) q^{85} \) \( + ( -692146268574266 \beta_{1} - 12902624169120 \beta_{2} + 2580524833824 \beta_{3} ) q^{86} \) \( + ( 214724639019436560 + 103888394420958 \beta_{1} - 4605359652495 \beta_{2} - 185881547817 \beta_{3} ) q^{87} \) \( + ( -123685199282960640 + 11666925956000 \beta_{2} + 11666925956000 \beta_{3} ) q^{88} \) \( + ( 476882969842068 \beta_{1} + 6310433389410 \beta_{2} - 1262086677882 \beta_{3} ) q^{89} \) \( + ( -172060869602900880 - 104578533570240 \beta_{1} - 2942981624070 \beta_{2} + 5527968516090 \beta_{3} ) q^{90} \) \( + ( -156280535783851388 - 6109111434062 \beta_{2} - 6109111434062 \beta_{3} ) q^{91} \) \( + ( 371032701118144 \beta_{1} + 75940986240 \beta_{2} - 15188197248 \beta_{3} ) q^{92} \) \( + ( 204706858691706858 - 381761662576298 \beta_{1} + 15349958749307 \beta_{2} + 713094997545 \beta_{3} ) q^{93} \) \( + ( 291043137989201472 - 12990590008808 \beta_{2} - 12990590008808 \beta_{3} ) q^{94} \) \( + ( -347761035769132 \beta_{1} + 4781685149010 \beta_{2} - 956337029802 \beta_{3} ) q^{95} \) \( + ( 294089939422046208 + 78083628086016 \beta_{1} - 647557085952 \beta_{2} - 14923006430976 \beta_{3} ) q^{96} \) \( + ( -872891133628690078 - 2928442015292 \beta_{2} - 2928442015292 \beta_{3} ) q^{97} \) \( + ( 1304067407126253 \beta_{1} + 19784571569280 \beta_{2} - 3956914313856 \beta_{3} ) q^{98} \) \( + ( 756927064917779040 - 789455123862822 \beta_{1} + 5954898468645 \beta_{2} + 8675986498263 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 15876q^{3} \) \(\mathstrut -\mathstrut 1053536q^{4} \) \(\mathstrut +\mathstrut 22698144q^{6} \) \(\mathstrut -\mathstrut 95744152q^{7} \) \(\mathstrut -\mathstrut 885341340q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 15876q^{3} \) \(\mathstrut -\mathstrut 1053536q^{4} \) \(\mathstrut +\mathstrut 22698144q^{6} \) \(\mathstrut -\mathstrut 95744152q^{7} \) \(\mathstrut -\mathstrut 885341340q^{9} \) \(\mathstrut +\mathstrut 1461136320q^{10} \) \(\mathstrut +\mathstrut 7123171104q^{12} \) \(\mathstrut -\mathstrut 5426221528q^{13} \) \(\mathstrut -\mathstrut 68287821120q^{15} \) \(\mathstrut +\mathstrut 201224008192q^{16} \) \(\mathstrut -\mathstrut 624067623360q^{18} \) \(\mathstrut +\mathstrut 191416649480q^{19} \) \(\mathstrut -\mathstrut 843499414296q^{21} \) \(\mathstrut +\mathstrut 6661732766400q^{22} \) \(\mathstrut -\mathstrut 16917300997632q^{24} \) \(\mathstrut +\mathstrut 11407599454180q^{25} \) \(\mathstrut -\mathstrut 4632207691356q^{27} \) \(\mathstrut +\mathstrut 5750860980032q^{28} \) \(\mathstrut -\mathstrut 17181499602240q^{30} \) \(\mathstrut +\mathstrut 35728415085608q^{31} \) \(\mathstrut +\mathstrut 12242871023040q^{33} \) \(\mathstrut -\mathstrut 97283346838272q^{34} \) \(\mathstrut +\mathstrut 412657454022048q^{36} \) \(\mathstrut -\mathstrut 475299833502232q^{37} \) \(\mathstrut +\mathstrut 416909545005096q^{39} \) \(\mathstrut -\mathstrut 967003294602240q^{40} \) \(\mathstrut +\mathstrut 149151729948480q^{42} \) \(\mathstrut +\mathstrut 1599607324574792q^{43} \) \(\mathstrut -\mathstrut 221242601301120q^{45} \) \(\mathstrut +\mathstrut 571461969379968q^{46} \) \(\mathstrut -\mathstrut 2192825065342464q^{48} \) \(\mathstrut -\mathstrut 3423787444328244q^{49} \) \(\mathstrut -\mathstrut 4987666703736576q^{51} \) \(\mathstrut +\mathstrut 19843921097419328q^{52} \) \(\mathstrut -\mathstrut 14022277970462496q^{54} \) \(\mathstrut -\mathstrut 84999034592640q^{55} \) \(\mathstrut +\mathstrut 10169351626464648q^{57} \) \(\mathstrut -\mathstrut 15931261886420160q^{58} \) \(\mathstrut +\mathstrut 24723268055608320q^{60} \) \(\mathstrut -\mathstrut 2207268745883992q^{61} \) \(\mathstrut +\mathstrut 13833182825583912q^{63} \) \(\mathstrut -\mathstrut 53493367589593088q^{64} \) \(\mathstrut +\mathstrut 14857554290281920q^{66} \) \(\mathstrut -\mathstrut 41331536078929912q^{67} \) \(\mathstrut +\mathstrut 81420525478583424q^{69} \) \(\mathstrut +\mathstrut 4599036429482880q^{70} \) \(\mathstrut -\mathstrut 4994921842560000q^{72} \) \(\mathstrut -\mathstrut 45389954762456248q^{73} \) \(\mathstrut +\mathstrut 56512911509383140q^{75} \) \(\mathstrut +\mathstrut 344787912869857856q^{76} \) \(\mathstrut -\mathstrut 685829939469704640q^{78} \) \(\mathstrut -\mathstrut 177365384029892440q^{79} \) \(\mathstrut -\mathstrut 72782527872717756q^{81} \) \(\mathstrut +\mathstrut 887840438080579200q^{82} \) \(\mathstrut -\mathstrut 125687415484822464q^{84} \) \(\mathstrut -\mathstrut 258438267155888640q^{85} \) \(\mathstrut +\mathstrut 858898556077746240q^{87} \) \(\mathstrut -\mathstrut 494740797131842560q^{88} \) \(\mathstrut -\mathstrut 688243478411603520q^{90} \) \(\mathstrut -\mathstrut 625122143135405552q^{91} \) \(\mathstrut +\mathstrut 818827434766827432q^{93} \) \(\mathstrut +\mathstrut 1164172551956805888q^{94} \) \(\mathstrut +\mathstrut 1176359757688184832q^{96} \) \(\mathstrut -\mathstrut 3491564534514760312q^{97} \) \(\mathstrut +\mathstrut 3027708259671116160q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut +\mathstrut \) \(123\) \(x^{2}\mathstrut -\mathstrut \) \(1744\) \(x\mathstrut +\mathstrut \) \(16016\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -36 \nu^{3} + 216 \nu^{2} - 11592 \nu + 59904 \)\()/169\)
\(\beta_{2}\)\(=\)\((\)\( 882 \nu^{3} + 22086 \nu^{2} + 201870 \nu + 229788 \)\()/169\)
\(\beta_{3}\)\(=\)\((\)\( -4770 \nu^{3} + 1242 \nu^{2} - 139662 \nu + 5911308 \)\()/169\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(108\) \(\beta_{1}\mathstrut +\mathstrut \) \(1944\)\()/7776\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(284\) \(\beta_{1}\mathstrut -\mathstrut \) \(158760\)\()/2592\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(38\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(423\) \(\beta_{1}\mathstrut +\mathstrut \) \(1181952\)\()/972\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.07949 12.9551i
6.57949 5.90892i
6.57949 + 5.90892i
−6.07949 + 12.9551i
932.767i −4234.02 + 19222.2i −607911. 1.14247e6i 1.79299e7 + 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 1.62775e8i 1.06566e9
2.2 425.442i 12172.0 15468.1i 81143.0 787628.i −6.58078e6 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 3.76556e8i −3.35090e8
2.3 425.442i 12172.0 + 15468.1i 81143.0 787628.i −6.58078e6 + 5.17849e6i −3.80616e7 1.46049e8i −9.11041e7 + 3.76556e8i −3.35090e8
2.4 932.767i −4234.02 19222.2i −607911. 1.14247e6i 1.79299e7 3.94936e6i −9.81043e6 3.22520e8i −3.51567e8 + 1.62775e8i 1.06566e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 1051056 T_{2}^{2} \) \(\mathstrut +\mathstrut 157480796160 \) acting on \(S_{19}^{\mathrm{new}}(3, [\chi])\).