Properties

Label 3.26.a.b
Level 3
Weight 26
Character orbit 3.a
Self dual Yes
Analytic conductor 11.880
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 26 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.8799033986\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 5^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -1226 + \beta_{1} ) q^{2} \) \( + 531441 q^{3} \) \( + ( 30249196 - 1499 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( -54384250 + 43676 \beta_{1} - 12 \beta_{2} ) q^{5} \) \( + ( -651546666 + 531441 \beta_{1} ) q^{6} \) \( + ( -3207524248 + 3973036 \beta_{1} + 676 \beta_{2} ) q^{7} \) \( + ( -89337610216 + 27346058 \beta_{1} - 3678 \beta_{2} ) q^{8} \) \( + 282429536481 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-1226 + \beta_{1}) q^{2}\) \(+531441 q^{3}\) \(+(30249196 - 1499 \beta_{1} + \beta_{2}) q^{4}\) \(+(-54384250 + 43676 \beta_{1} - 12 \beta_{2}) q^{5}\) \(+(-651546666 + 531441 \beta_{1}) q^{6}\) \(+(-3207524248 + 3973036 \beta_{1} + 676 \beta_{2}) q^{7}\) \(+(-89337610216 + 27346058 \beta_{1} - 3678 \beta_{2}) q^{8}\) \(+282429536481 q^{9}\) \(+(2787729624900 - 429212602 \beta_{1} + 69824 \beta_{2}) q^{10}\) \(+(-1982332710260 - 398988488 \beta_{1} - 290328 \beta_{2}) q^{11}\) \(+(16075662971436 - 796630059 \beta_{1} + 531441 \beta_{2}) q^{12}\) \(+(82712591469230 + 6520110088 \beta_{1} - 730664 \beta_{2}) q^{13}\) \(+(251453880421616 + 16151474216 \beta_{1} + 2500032 \beta_{2}) q^{14}\) \(+(-28902020204250 + 23211217116 \beta_{1} - 6377292 \beta_{2}) q^{15}\) \(+(798212617450672 - 157735312908 \beta_{1} + 1805988 \beta_{2}) q^{16}\) \(+(2213628400415058 - 120931939176 \beta_{1} + 14873352 \beta_{2}) q^{17}\) \(+(-346258611725706 + 282429536481 \beta_{1}) q^{18}\) \(+(-1427572986360172 - 839383716840 \beta_{1} - 17877624 \beta_{2}) q^{19}\) \(+(-28333196849697400 + 3551003379422 \beta_{1} - 178705914 \beta_{2}) q^{20}\) \(+(-1704609893881368 + 2111434224876 \beta_{1} + 359254116 \beta_{2}) q^{21}\) \(+(-22426485104016504 - 10653527681012 \beta_{1} + 233636224 \beta_{2}) q^{22}\) \(+(-18186318650521384 - 4247981439784 \beta_{1} - 255718968 \beta_{2}) q^{23}\) \(+(-47477668910801256 + 14532816409578 \beta_{1} - 1954639998 \beta_{2}) q^{24}\) \(+(95090275796389375 - 38783464180400 \beta_{1} - 143085200 \beta_{2}) q^{25}\) \(+(304801771842409652 + 58835811772718 \beta_{1} + 8112226944 \beta_{2}) q^{26}\) \(+150094635296999121 q^{27}\) \(+(805586700839865824 + 189337696911240 \beta_{1} - 11978891544 \beta_{2}) q^{28}\) \(+(-168783618890594 - 231877876918796 \beta_{1} - 27546130116 \beta_{2}) q^{29}\) \(+(1481513819586480900 - 228101174419482 \beta_{1} + 37107336384 \beta_{2}) q^{30}\) \(+(-40198486317117952 - 117865532741348 \beta_{1} + 76747695988 \beta_{2}) q^{31}\) \(+(-7807935317604570528 - 21690275657304 \beta_{1} - 38257359864 \beta_{2}) q^{32}\) \(+(-1053492877873284660 - 212038841051208 \beta_{1} - 154292202648 \beta_{2}) q^{33}\) \(+(-10248054350655218868 + 2696443727508690 \beta_{1} - 153340973184 \beta_{2}) q^{34}\) \(+(-4298570811467776400 - 505925378619032 \beta_{1} + 437874843384 \beta_{2}) q^{35}\) \(+(8543266405202919276 - 423361875185019 \beta_{1} + 282429536481 \beta_{2}) q^{36}\) \(+(23708009755490611142 - 2198827671598752 \beta_{1} - 259144083168 \beta_{2}) q^{37}\) \(+(-50543841139140403912 - 1739077534471660 \beta_{1} - 800428374144 \beta_{2}) q^{38}\) \(+(43956862322999060430 + 3465053825276808 \beta_{1} - 388304806824 \beta_{2}) q^{39}\) \(+(\)\(16\!\cdots\!00\)\( - 20305071929411780 \beta_{1} + 1597498906060 \beta_{2}) q^{40}\) \(+(-59412632811749142742 + 20814364543552520 \beta_{1} - 264410428968 \beta_{2}) q^{41}\) \(+(\)\(13\!\cdots\!56\)\( + 8583555608825256 \beta_{1} + 1328619506112 \beta_{2}) q^{42}\) \(+(77910715263985000748 + 12542497254509704 \beta_{1} - 1161139853480 \beta_{2}) q^{43}\) \(+(-\)\(56\!\cdots\!96\)\( + 935402382113596 \beta_{1} - 1420829879412 \beta_{2}) q^{44}\) \(+(-15359718519366824250 + 12335392435344156 \beta_{1} - 3389154437772 \beta_{2}) q^{45}\) \(+(-\)\(24\!\cdots\!12\)\( - 24760089880887208 \beta_{1} - 3690769808512 \beta_{2}) q^{46}\) \(+(-\)\(19\!\cdots\!64\)\( - 100308014136338360 \beta_{1} + 20298709187736 \beta_{2}) q^{47}\) \(+(\)\(42\!\cdots\!52\)\( - 83827012427140428 \beta_{1} + 959776068708 \beta_{2}) q^{48}\) \(+(\)\(51\!\cdots\!09\)\( + 138831556001738176 \beta_{1} - 7490538607808 \beta_{2}) q^{49}\) \(+(-\)\(25\!\cdots\!50\)\( + 101350969312530175 \beta_{1} - 38471681529600 \beta_{2}) q^{50}\) \(+(\)\(11\!\cdots\!78\)\( - 64268190687632616 \beta_{1} + 7904309060232 \beta_{2}) q^{51}\) \(+(\)\(51\!\cdots\!48\)\( + 315291515407800870 \beta_{1} + 65676284764590 \beta_{2}) q^{52}\) \(+(\)\(30\!\cdots\!86\)\( - 123056057069269020 \beta_{1} - 30109878830196 \beta_{2}) q^{53}\) \(+(-\)\(18\!\cdots\!46\)\( + 150094635296999121 \beta_{1}) q^{54}\) \(+(\)\(55\!\cdots\!00\)\( - 420588468838719328 \beta_{1} - 75688827890464 \beta_{2}) q^{55}\) \(+(\)\(23\!\cdots\!20\)\( - 150322474356809456 \beta_{1} + 131552547843792 \beta_{2}) q^{56}\) \(+(-\)\(75\!\cdots\!52\)\( - 446082921861166440 \beta_{1} - 9500902376184 \beta_{2}) q^{57}\) \(+(-\)\(14\!\cdots\!84\)\( - 769918035778849058 \beta_{1} - 171854859396032 \beta_{2}) q^{58}\) \(+(\)\(79\!\cdots\!32\)\( + 463557317096698784 \beta_{1} + 70020321310176 \beta_{2}) q^{59}\) \(+(-\)\(15\!\cdots\!00\)\( + 1887148786963407102 \beta_{1} - 94971649642074 \beta_{2}) q^{60}\) \(+(-\)\(37\!\cdots\!06\)\( - 422487516587392304 \beta_{1} + 430417432514032 \beta_{2}) q^{61}\) \(+(-\)\(72\!\cdots\!96\)\( + 2312987768285997248 \beta_{1} - 285098762299200 \beta_{2}) q^{62}\) \(+(-\)\(90\!\cdots\!88\)\( + 1122102715902326316 \beta_{1} + 190922366661156 \beta_{2}) q^{63}\) \(+(-\)\(18\!\cdots\!28\)\( - 3666276681630117168 \beta_{1} + 1073609947536 \beta_{2}) q^{64}\) \(+(\)\(29\!\cdots\!00\)\( - 54637084392366536 \beta_{1} - 828227058265368 \beta_{2}) q^{65}\) \(+(-\)\(11\!\cdots\!64\)\( - 5661721404324698292 \beta_{1} + 124163868518784 \beta_{2}) q^{66}\) \(+(\)\(17\!\cdots\!08\)\( + 7770440088092160928 \beta_{1} - 347254212217376 \beta_{2}) q^{67}\) \(+(\)\(10\!\cdots\!28\)\( - 11563728943431594534 \beta_{1} + 2531506829780562 \beta_{2}) q^{68}\) \(+(-\)\(96\!\cdots\!44\)\( - 2257551504340248744 \beta_{1} - 135899544072888 \beta_{2}) q^{69}\) \(+(-\)\(26\!\cdots\!00\)\( + 9081787168128654064 \beta_{1} - 1460054662352768 \beta_{2}) q^{70}\) \(+(\)\(87\!\cdots\!12\)\( + 2932710170227808376 \beta_{1} - 1364710659252312 \beta_{2}) q^{71}\) \(+(-\)\(25\!\cdots\!96\)\( + 7723334485522541898 \beta_{1} - 1038775835177118 \beta_{2}) q^{72}\) \(+(-\)\(39\!\cdots\!34\)\( + 18136518967415958912 \beta_{1} + 3282880162757760 \beta_{2}) q^{73}\) \(+(-\)\(16\!\cdots\!24\)\( + 16471236984016842182 \beta_{1} - 1634152714375680 \beta_{2}) q^{74}\) \(+(\)\(50\!\cdots\!75\)\( - 20611122987495956400 \beta_{1} - 76041341773200 \beta_{2}) q^{75}\) \(+(\)\(15\!\cdots\!72\)\( - 46110637643178521500 \beta_{1} + 604929411617684 \beta_{2}) q^{76}\) \(+(-\)\(46\!\cdots\!96\)\( - 45490655578567562032 \beta_{1} + 5297651311492080 \beta_{2}) q^{77}\) \(+(\)\(16\!\cdots\!32\)\( + 31267762644305026638 \beta_{1} + 4311169999346304 \beta_{2}) q^{78}\) \(+(-\)\(16\!\cdots\!20\)\( - 18460520919225292724 \beta_{1} - 9806787941675132 \beta_{2}) q^{79}\) \(+(-\)\(51\!\cdots\!00\)\( + 97128570749814883256 \beta_{1} - 17789646606405672 \beta_{2}) q^{80}\) \(+\)\(79\!\cdots\!61\)\( q^{81}\) \(+(\)\(13\!\cdots\!04\)\( - 73091272240487977558 \beta_{1} + 21390514868273792 \beta_{2}) q^{82}\) \(+(-\)\(19\!\cdots\!28\)\( - 29155446461563785512 \beta_{1} + 22771868367189960 \beta_{2}) q^{83}\) \(+(\)\(42\!\cdots\!84\)\( + \)\(10\!\cdots\!40\)\( \beta_{1} - 6366074101034904 \beta_{2}) q^{84}\) \(+(-\)\(78\!\cdots\!00\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} - 29258683959391656 \beta_{2}) q^{85}\) \(+(\)\(68\!\cdots\!80\)\( + 39371344268191508396 \beta_{1} + 15072620995242624 \beta_{2}) q^{86}\) \(+(-89698535206836165954 - \)\(12\!\cdots\!36\)\( \beta_{1} - 14639142934977156 \beta_{2}) q^{87}\) \(+(\)\(15\!\cdots\!64\)\( - \)\(25\!\cdots\!24\)\( \beta_{1} - 3808140101592424 \beta_{2}) q^{88}\) \(+(-\)\(11\!\cdots\!22\)\( - \)\(16\!\cdots\!32\)\( \beta_{1} - 38200822640016720 \beta_{2}) q^{89}\) \(+(\)\(78\!\cdots\!00\)\( - \)\(12\!\cdots\!62\)\( \beta_{1} + 19720359955249344 \beta_{2}) q^{90}\) \(+(\)\(41\!\cdots\!48\)\( + \)\(39\!\cdots\!08\)\( \beta_{1} + 97665474270713656 \beta_{2}) q^{91}\) \(+(-\)\(63\!\cdots\!68\)\( - \)\(20\!\cdots\!44\)\( \beta_{1} - 8137397745273384 \beta_{2}) q^{92}\) \(+(-\)\(21\!\cdots\!32\)\( - 62638576585594722468 \beta_{1} + 40786872303558708 \beta_{2}) q^{93}\) \(+(-\)\(60\!\cdots\!00\)\( + \)\(44\!\cdots\!28\)\( \beta_{1} - 144538901456415104 \beta_{2}) q^{94}\) \(+(-\)\(18\!\cdots\!00\)\( + \)\(22\!\cdots\!20\)\( \beta_{1} - 33900729606997440 \beta_{2}) q^{95}\) \(+(-\)\(41\!\cdots\!48\)\( - 11527101785593295064 \beta_{1} - 20331529583484024 \beta_{2}) q^{96}\) \(+(-\)\(40\!\cdots\!22\)\( + \)\(50\!\cdots\!28\)\( \beta_{1} + 18084146264215024 \beta_{2}) q^{97}\) \(+(\)\(80\!\cdots\!50\)\( + \)\(24\!\cdots\!25\)\( \beta_{1} + 155153439628151808 \beta_{2}) q^{98}\) \(+(-\)\(55\!\cdots\!60\)\( - \)\(11\!\cdots\!28\)\( \beta_{1} - 81997202467455768 \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 3678q^{2} \) \(\mathstrut +\mathstrut 1594323q^{3} \) \(\mathstrut +\mathstrut 90747588q^{4} \) \(\mathstrut -\mathstrut 163152750q^{5} \) \(\mathstrut -\mathstrut 1954639998q^{6} \) \(\mathstrut -\mathstrut 9622572744q^{7} \) \(\mathstrut -\mathstrut 268012830648q^{8} \) \(\mathstrut +\mathstrut 847288609443q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3678q^{2} \) \(\mathstrut +\mathstrut 1594323q^{3} \) \(\mathstrut +\mathstrut 90747588q^{4} \) \(\mathstrut -\mathstrut 163152750q^{5} \) \(\mathstrut -\mathstrut 1954639998q^{6} \) \(\mathstrut -\mathstrut 9622572744q^{7} \) \(\mathstrut -\mathstrut 268012830648q^{8} \) \(\mathstrut +\mathstrut 847288609443q^{9} \) \(\mathstrut +\mathstrut 8363188874700q^{10} \) \(\mathstrut -\mathstrut 5946998130780q^{11} \) \(\mathstrut +\mathstrut 48226988914308q^{12} \) \(\mathstrut +\mathstrut 248137774407690q^{13} \) \(\mathstrut +\mathstrut 754361641264848q^{14} \) \(\mathstrut -\mathstrut 86706060612750q^{15} \) \(\mathstrut +\mathstrut 2394637852352016q^{16} \) \(\mathstrut +\mathstrut 6640885201245174q^{17} \) \(\mathstrut -\mathstrut 1038775835177118q^{18} \) \(\mathstrut -\mathstrut 4282718959080516q^{19} \) \(\mathstrut -\mathstrut 84999590549092200q^{20} \) \(\mathstrut -\mathstrut 5113829681644104q^{21} \) \(\mathstrut -\mathstrut 67279455312049512q^{22} \) \(\mathstrut -\mathstrut 54558955951564152q^{23} \) \(\mathstrut -\mathstrut 142433006732403768q^{24} \) \(\mathstrut +\mathstrut 285270827389168125q^{25} \) \(\mathstrut +\mathstrut 914405315527228956q^{26} \) \(\mathstrut +\mathstrut 450283905890997363q^{27} \) \(\mathstrut +\mathstrut 2416760102519597472q^{28} \) \(\mathstrut -\mathstrut 506350856671782q^{29} \) \(\mathstrut +\mathstrut 4444541458759442700q^{30} \) \(\mathstrut -\mathstrut 120595458951353856q^{31} \) \(\mathstrut -\mathstrut 23423805952813711584q^{32} \) \(\mathstrut -\mathstrut 3160478633619853980q^{33} \) \(\mathstrut -\mathstrut 30744163051965656604q^{34} \) \(\mathstrut -\mathstrut 12895712434403329200q^{35} \) \(\mathstrut +\mathstrut 25629799215608757828q^{36} \) \(\mathstrut +\mathstrut 71124029266471833426q^{37} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!36\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!90\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!26\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!68\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!44\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!88\)\(q^{44} \) \(\mathstrut -\mathstrut 46079155558100472750q^{45} \) \(\mathstrut -\mathstrut \)\(72\!\cdots\!36\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!92\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!56\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!27\)\(q^{49} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!50\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!34\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!44\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(90\!\cdots\!58\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!38\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!60\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!56\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!52\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!96\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!18\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!88\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!64\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!84\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(89\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!92\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(52\!\cdots\!24\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!84\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!32\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!36\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!88\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!02\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!72\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!25\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!16\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!88\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!96\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!60\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!83\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!12\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!84\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!52\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!62\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!92\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!66\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!44\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!04\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!96\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!44\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!66\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!50\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(16\!\cdots\!80\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(783420\) \(x\mathstrut +\mathstrut \) \(148321440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + 455 \nu - 522432 \)\()/42\)
\(\beta_{2}\)\(=\)\((\)\( -2227 \nu^{2} + 1708315 \nu + 1162548864 \)\()/42\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(2227\) \(\beta_{1}\mathstrut +\mathstrut \) \(21600\)\()/64800\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(91\) \(\beta_{2}\mathstrut +\mathstrut \) \(341663\) \(\beta_{1}\mathstrut +\mathstrut \) \(6768753120\)\()/12960\)

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
199.394
−967.357
768.963
−10558.1 531441. 7.79199e7 −8.66156e8 −5.61103e9 −1.75156e10 −4.68416e11 2.82430e11 9.14500e12
1.2 −1864.10 531441. −3.00796e7 6.53169e8 −9.90659e8 −4.71716e10 1.18620e11 2.82430e11 −1.21757e12
1.3 8744.24 531441. 4.29073e7 4.98342e7 4.64705e9 5.50645e10 8.17835e10 2.82430e11 4.35762e11
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 3678 T_{2}^{2} \) \(\mathstrut -\mathstrut 88941600 T_{2} \) \(\mathstrut -\mathstrut 172099067904 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(3))\).