Properties

Label 5.20.a.b
Level 5
Weight 20
Character orbit 5.a
Self dual Yes
Analytic conductor 11.441
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.4408348278\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -105 - \beta_{1} ) q^{2} \) \( + ( 770 + 14 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 521248 - 18 \beta_{1} - 26 \beta_{2} - \beta_{3} ) q^{4} \) \( -1953125 q^{5} \) \( + ( -14517938 - 24482 \beta_{1} + 256 \beta_{2} + 96 \beta_{3} ) q^{6} \) \( + ( 53505350 - 49970 \beta_{1} - 81 \beta_{2} - 720 \beta_{3} ) q^{7} \) \( + ( 20434440 - 698512 \beta_{1} - 968 \beta_{2} + 2660 \beta_{3} ) q^{8} \) \( + ( 38516137 + 508024 \beta_{1} + 10108 \beta_{2} - 5472 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -105 - \beta_{1} ) q^{2} \) \( + ( 770 + 14 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 521248 - 18 \beta_{1} - 26 \beta_{2} - \beta_{3} ) q^{4} \) \( -1953125 q^{5} \) \( + ( -14517938 - 24482 \beta_{1} + 256 \beta_{2} + 96 \beta_{3} ) q^{6} \) \( + ( 53505350 - 49970 \beta_{1} - 81 \beta_{2} - 720 \beta_{3} ) q^{7} \) \( + ( 20434440 - 698512 \beta_{1} - 968 \beta_{2} + 2660 \beta_{3} ) q^{8} \) \( + ( 38516137 + 508024 \beta_{1} + 10108 \beta_{2} - 5472 \beta_{3} ) q^{9} \) \( + ( 205078125 + 1953125 \beta_{1} ) q^{10} \) \( + ( 2896428192 + 1619940 \beta_{1} - 77230 \beta_{2} + 40240 \beta_{3} ) q^{11} \) \( + ( 26407339840 + 14330908 \beta_{1} - 306164 \beta_{2} - 97890 \beta_{3} ) q^{12} \) \( + ( 3703083290 + 11351688 \beta_{1} + 1026500 \beta_{2} - 19040 \beta_{3} ) q^{13} \) \( + ( 46293538986 - 89110694 \beta_{1} + 353792 \beta_{2} + 349792 \beta_{3} ) q^{14} \) \( + ( -1503906250 - 27343750 \beta_{1} + 1953125 \beta_{2} ) q^{15} \) \( + ( 446445024736 - 20310704 \beta_{1} - 10773648 \beta_{2} - 1547208 \beta_{3} ) q^{16} \) \( + ( 212258435610 - 121164168 \beta_{1} - 7550788 \beta_{2} + 1498720 \beta_{3} ) q^{17} \) \( + ( -528444749905 + 72826199 \beta_{1} + 26929664 \beta_{2} + 2666880 \beta_{3} ) q^{18} \) \( + ( 494541927140 + 1219060368 \beta_{1} + 13782296 \beta_{2} - 6204944 \beta_{3} ) q^{19} \) \( + ( -1018062500000 + 35156250 \beta_{1} + 50781250 \beta_{2} + 1953125 \beta_{3} ) q^{20} \) \( + ( -371755046388 + 1698233136 \beta_{1} - 135214248 \beta_{2} + 10013952 \beta_{3} ) q^{21} \) \( + ( -1988337190560 - 3130150432 \beta_{1} - 59096320 \beta_{2} - 14018240 \beta_{3} ) q^{22} \) \( + ( -6642476738190 + 4000888282 \beta_{1} - 123482299 \beta_{2} - 14097200 \beta_{3} ) q^{23} \) \( + ( -9943560868080 - 23321100576 \beta_{1} + 431250288 \beta_{2} + 42552648 \beta_{3} ) q^{24} \) \( + 3814697265625 q^{25} \) \( + ( -12173914551978 + 23076627174 \beta_{1} + 449950208 \beta_{2} - 62425472 \beta_{3} ) q^{26} \) \( + ( -1889401482340 + 10033684532 \beta_{1} + 229379498 \beta_{2} + 35144640 \beta_{3} ) q^{27} \) \( + ( 59153008379840 - 8746152492 \beta_{1} - 3043521116 \beta_{2} + 68379290 \beta_{3} ) q^{28} \) \( + ( 29066793584910 + 37650049232 \beta_{1} + 1412036264 \beta_{2} - 102722816 \beta_{3} ) q^{29} \) \( + ( 28355347656250 + 47816406250 \beta_{1} - 500000000 \beta_{2} - 187500000 \beta_{3} ) q^{30} \) \( + ( 62762418097172 + 143885209220 \beta_{1} + 1856329010 \beta_{2} + 396745120 \beta_{3} ) q^{31} \) \( + ( -35623835743680 - 415616198336 \beta_{1} + 2386834560 \beta_{2} + 313297920 \beta_{3} ) q^{32} \) \( + ( 89976420159040 - 25172828792 \beta_{1} + 2909475268 \beta_{2} - 784570080 \beta_{3} ) q^{33} \) \( + ( 102962253760086 - 362176079386 \beta_{1} - 7424799232 \beta_{2} - 320300672 \beta_{3} ) q^{34} \) \( + ( -104502636718750 + 97597656250 \beta_{1} + 158203125 \beta_{2} + 1406250000 \beta_{3} ) q^{35} \) \( + ( -42077161433824 + 1057466011166 \beta_{1} - 6652777258 \beta_{2} - 722618793 \beta_{3} ) q^{36} \) \( + ( 13367914429130 - 243628254544 \beta_{1} - 12933056552 \beta_{2} - 2445430720 \beta_{3} ) q^{37} \) \( + ( -1311854471721540 - 230181524388 \beta_{1} + 47505068288 \beta_{2} + 3476811520 \beta_{3} ) q^{38} \) \( + ( -851758128854476 + 188486367356 \beta_{1} - 21141587218 \beta_{2} + 3294428352 \beta_{3} ) q^{39} \) \( + ( -39911015625000 + 1364281250000 \beta_{1} + 1890625000 \beta_{2} - 5195312500 \beta_{3} ) q^{40} \) \( + ( 1401555474487062 + 712181442920 \beta_{1} - 23031632140 \beta_{2} - 4324587680 \beta_{3} ) q^{41} \) \( + ( -1714545194677740 - 2477330532108 \beta_{1} + 6438721536 \beta_{2} + 7318183680 \beta_{3} ) q^{42} \) \( + ( 981048278190650 - 2731979985890 \beta_{1} + 21833437543 \beta_{2} - 766535200 \beta_{3} ) q^{43} \) \( + ( 1935264214703616 - 1282492323136 \beta_{1} - 14921453632 \beta_{2} - 11727642272 \beta_{3} ) q^{44} \) \( + ( -75226830078125 - 992234375000 \beta_{1} - 19742187500 \beta_{2} + 10687500000 \beta_{3} ) q^{45} \) \( + ( -3431637177517218 + 3457482355310 \beta_{1} + 123223344640 \beta_{2} + 21823508000 \beta_{3} ) q^{46} \) \( + ( 2493111081747390 - 1921361759922 \beta_{1} + 43968973391 \beta_{2} - 14751076400 \beta_{3} ) q^{47} \) \( + ( 11292482147354560 + 12149260695712 \beta_{1} - 497466984224 \beta_{2} - 30594817680 \beta_{3} ) q^{48} \) \( + ( 10045788995007693 - 1457710478360 \beta_{1} + 276503710900 \beta_{2} + 22539227360 \beta_{3} ) q^{49} \) \( + ( -400543212890625 - 3814697265625 \beta_{1} ) q^{50} \) \( + ( 5476550405314612 - 2834457510308 \beta_{1} + 92115646174 \beta_{2} - 31093023936 \beta_{3} ) q^{51} \) \( + ( -24538474874257600 + 18129280575020 \beta_{1} + 254483286364 \beta_{2} + 30247461350 \beta_{3} ) q^{52} \) \( + ( -1865569067700030 - 15326779667656 \beta_{1} - 31983718116 \beta_{2} + 67159889760 \beta_{3} ) q^{53} \) \( + ( -10202550729785660 + 10294976962468 \beta_{1} + 204534954496 \beta_{2} - 27964407744 \beta_{3} ) q^{54} \) \( + ( -5657086312500000 - 3163945312500 \beta_{1} + 150839843750 \beta_{2} - 78593750000 \beta_{3} ) q^{55} \) \( + ( -21314281604882640 - 88315928163168 \beta_{1} - 899408546736 \beta_{2} + 20095738584 \beta_{3} ) q^{56} \) \( + ( 6186502338827720 + 50531162336024 \beta_{1} - 1635940938484 \beta_{2} - 12296359200 \beta_{3} ) q^{57} \) \( + ( -42036010828824510 + 7568878769138 \beta_{1} + 1368485455872 \beta_{2} - 22050266880 \beta_{3} ) q^{58} \) \( + ( 46538735490709620 + 12678949138904 \beta_{1} + 1608356028 \beta_{2} + 42781997328 \beta_{3} ) q^{59} \) \( + ( -51576835625000000 - 27990054687500 \beta_{1} + 597976562500 \beta_{2} + 191191406250 \beta_{3} ) q^{60} \) \( + ( 62129398277661242 + 36325386740800 \beta_{1} - 360031877600 \beta_{2} - 98525459200 \beta_{3} ) q^{61} \) \( + ( -155643965122198260 + 17247073473708 \beta_{1} + 3025811235840 \beta_{2} - 224956605120 \beta_{3} ) q^{62} \) \( + ( 93830902747359030 + 86628984993486 \beta_{1} + 3222178475247 \beta_{2} + 38425889520 \beta_{3} ) q^{63} \) \( + ( 199431472340789248 + 67780690973824 \beta_{1} - 5622840260992 \beta_{2} + 28785291328 \beta_{3} ) q^{64} \) \( + ( -7232584550781250 - 22171265625000 \beta_{1} - 2004882812500 \beta_{2} + 37187500000 \beta_{3} ) q^{65} \) \( + ( 16692504795571904 - 48929113958464 \beta_{1} + 1470517524992 \beta_{2} + 164625462912 \beta_{3} ) q^{66} \) \( + ( -20994236520169090 - 100961063907158 \beta_{1} - 4551579249123 \beta_{2} + 520771602720 \beta_{3} ) q^{67} \) \( + ( 253016494946137920 - 285016057651156 \beta_{1} - 5520414891172 \beta_{2} - 364219286810 \beta_{3} ) q^{68} \) \( + ( 180346636055539044 + 61513477497648 \beta_{1} + 5874469229016 \beta_{2} - 785130793344 \beta_{3} ) q^{69} \) \( + ( -90417068332031250 + 174044324218750 \beta_{1} - 691000000000 \beta_{2} - 683187500000 \beta_{3} ) q^{70} \) \( + ( -81713627050007268 - 165342580099100 \beta_{1} + 12014191896450 \beta_{2} + 563398838400 \beta_{3} ) q^{71} \) \( + ( -811964280114984120 - 62741693742864 \beta_{1} + 14324520841464 \beta_{2} + 599330425860 \beta_{3} ) q^{72} \) \( + ( 176517759515566610 + 114653489212472 \beta_{1} - 8035911594084 \beta_{2} + 533085805920 \beta_{3} ) q^{73} \) \( + ( 251952897108168486 - 465331970700490 \beta_{1} - 2087050624000 \beta_{2} + 2152087555840 \beta_{3} ) q^{74} \) \( + ( 2937316894531250 + 53405761718750 \beta_{1} - 3814697265625 \beta_{2} ) q^{75} \) \( + ( 113369002484309120 + 1984953914422904 \beta_{1} - 16104545652392 \beta_{2} - 2770758534052 \beta_{3} ) q^{76} \) \( + ( -797005811864635200 - 289371117595880 \beta_{1} - 26179749673972 \beta_{2} - 3998719592480 \beta_{3} ) q^{77} \) \( + ( -105558933269153300 + 462592999243660 \beta_{1} - 4986186504704 \beta_{2} + 123338639040 \beta_{3} ) q^{78} \) \( + ( 325058972626185560 - 916151083256168 \beta_{1} + 25529181697644 \beta_{2} + 518296355904 \beta_{3} ) q^{79} \) \( + ( -871962938937500000 + 39669343750000 \beta_{1} + 21042281250000 \beta_{2} + 3021890625000 \beta_{3} ) q^{80} \) \( + ( -141410005235895899 - 527939527382072 \beta_{1} - 9945870611804 \beta_{2} + 6072574425696 \beta_{3} ) q^{81} \) \( + ( -881580398582728710 - 2064311548431382 \beta_{1} + 26010449610240 \beta_{2} + 4962000151680 \beta_{3} ) q^{82} \) \( + ( -223679067331191270 - 302602901639754 \beta_{1} + 35028167349747 \beta_{2} - 3909329328000 \beta_{3} ) q^{83} \) \( + ( 2935294467059976576 + 961718890126632 \beta_{1} - 9714372186936 \beta_{2} - 12251228853516 \beta_{3} ) q^{84} \) \( + ( -414567257050781250 + 236648765625000 \beta_{1} + 14747632812500 \beta_{2} - 2927187500000 \beta_{3} ) q^{85} \) \( + ( 2722474643148850902 - 790939896487258 \beta_{1} - 66904305136896 \beta_{2} - 4103793645216 \beta_{3} ) q^{86} \) \( + ( -810890913618894820 + 1323576375103076 \beta_{1} - 65981238080446 \beta_{2} + 3006473882880 \beta_{3} ) q^{87} \) \( + ( 2170173579127092480 - 1277706411451904 \beta_{1} + 23094572390144 \beta_{2} + 13693954568320 \beta_{3} ) q^{88} \) \( + ( -1098182629867912470 + 2022193747562256 \beta_{1} - 37763877572088 \beta_{2} + 19031306062272 \beta_{3} ) q^{89} \) \( + ( 1032118652158203125 - 142238669921875 \beta_{1} - 52597000000000 \beta_{2} - 5208750000000 \beta_{3} ) q^{90} \) \( + ( -291156347212121828 + 615305257900572 \beta_{1} + 136801207369694 \beta_{2} - 19345994464736 \beta_{3} ) q^{91} \) \( + ( 253945939451568960 + 5723824112632892 \beta_{1} + 117574293573292 \beta_{2} - 11171474479570 \beta_{3} ) q^{92} \) \( + ( 147443934397088040 + 2715916308075168 \beta_{1} - 58844357766192 \beta_{2} - 9208958807040 \beta_{3} ) q^{93} \) \( + ( 1728238864772268786 - 2172469170316702 \beta_{1} - 11161272300544 \beta_{2} + 2527270136416 \beta_{3} ) q^{94} \) \( + ( -965902201445312500 - 2380977281250000 \beta_{1} - 26918546875000 \beta_{2} + 12119031250000 \beta_{3} ) q^{95} \) \( + ( -8508982462610912128 - 11387980371740032 \beta_{1} + 106667832002816 \beta_{2} + 47336481140736 \beta_{3} ) q^{96} \) \( + ( 359467624487945090 + 2690599733940328 \beta_{1} - 33285019698124 \beta_{2} - 217413745760 \beta_{3} ) q^{97} \) \( + ( 433793301169875435 - 2334784242857933 \beta_{1} - 60058608407040 \beta_{2} - 36437432910720 \beta_{3} ) q^{98} \) \( + ( -6316356913070425696 + 560080364166788 \beta_{1} - 132619259450974 \beta_{2} - 26293460644944 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 420q^{2} \) \(\mathstrut +\mathstrut 3080q^{3} \) \(\mathstrut +\mathstrut 2084992q^{4} \) \(\mathstrut -\mathstrut 7812500q^{5} \) \(\mathstrut -\mathstrut 58071752q^{6} \) \(\mathstrut +\mathstrut 214021400q^{7} \) \(\mathstrut +\mathstrut 81737760q^{8} \) \(\mathstrut +\mathstrut 154064548q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 420q^{2} \) \(\mathstrut +\mathstrut 3080q^{3} \) \(\mathstrut +\mathstrut 2084992q^{4} \) \(\mathstrut -\mathstrut 7812500q^{5} \) \(\mathstrut -\mathstrut 58071752q^{6} \) \(\mathstrut +\mathstrut 214021400q^{7} \) \(\mathstrut +\mathstrut 81737760q^{8} \) \(\mathstrut +\mathstrut 154064548q^{9} \) \(\mathstrut +\mathstrut 820312500q^{10} \) \(\mathstrut +\mathstrut 11585712768q^{11} \) \(\mathstrut +\mathstrut 105629359360q^{12} \) \(\mathstrut +\mathstrut 14812333160q^{13} \) \(\mathstrut +\mathstrut 185174155944q^{14} \) \(\mathstrut -\mathstrut 6015625000q^{15} \) \(\mathstrut +\mathstrut 1785780098944q^{16} \) \(\mathstrut +\mathstrut 849033742440q^{17} \) \(\mathstrut -\mathstrut 2113778999620q^{18} \) \(\mathstrut +\mathstrut 1978167708560q^{19} \) \(\mathstrut -\mathstrut 4072250000000q^{20} \) \(\mathstrut -\mathstrut 1487020185552q^{21} \) \(\mathstrut -\mathstrut 7953348762240q^{22} \) \(\mathstrut -\mathstrut 26569906952760q^{23} \) \(\mathstrut -\mathstrut 39774243472320q^{24} \) \(\mathstrut +\mathstrut 15258789062500q^{25} \) \(\mathstrut -\mathstrut 48695658207912q^{26} \) \(\mathstrut -\mathstrut 7557605929360q^{27} \) \(\mathstrut +\mathstrut 236612033519360q^{28} \) \(\mathstrut +\mathstrut 116267174339640q^{29} \) \(\mathstrut +\mathstrut 113421390625000q^{30} \) \(\mathstrut +\mathstrut 251049672388688q^{31} \) \(\mathstrut -\mathstrut 142495342974720q^{32} \) \(\mathstrut +\mathstrut 359905680636160q^{33} \) \(\mathstrut +\mathstrut 411849015040344q^{34} \) \(\mathstrut -\mathstrut 418010546875000q^{35} \) \(\mathstrut -\mathstrut 168308645735296q^{36} \) \(\mathstrut +\mathstrut 53471657716520q^{37} \) \(\mathstrut -\mathstrut 5247417886886160q^{38} \) \(\mathstrut -\mathstrut 3407032515417904q^{39} \) \(\mathstrut -\mathstrut 159644062500000q^{40} \) \(\mathstrut +\mathstrut 5606221897948248q^{41} \) \(\mathstrut -\mathstrut 6858180778710960q^{42} \) \(\mathstrut +\mathstrut 3924193112762600q^{43} \) \(\mathstrut +\mathstrut 7741056858814464q^{44} \) \(\mathstrut -\mathstrut 300907320312500q^{45} \) \(\mathstrut -\mathstrut 13726548710068872q^{46} \) \(\mathstrut +\mathstrut 9972444326989560q^{47} \) \(\mathstrut +\mathstrut 45169928589418240q^{48} \) \(\mathstrut +\mathstrut 40183155980030772q^{49} \) \(\mathstrut -\mathstrut 1602172851562500q^{50} \) \(\mathstrut +\mathstrut 21906201621258448q^{51} \) \(\mathstrut -\mathstrut 98153899497030400q^{52} \) \(\mathstrut -\mathstrut 7462276270800120q^{53} \) \(\mathstrut -\mathstrut 40810202919142640q^{54} \) \(\mathstrut -\mathstrut 22628345250000000q^{55} \) \(\mathstrut -\mathstrut 85257126419530560q^{56} \) \(\mathstrut +\mathstrut 24746009355310880q^{57} \) \(\mathstrut -\mathstrut 168144043315298040q^{58} \) \(\mathstrut +\mathstrut 186154941962838480q^{59} \) \(\mathstrut -\mathstrut 206307342500000000q^{60} \) \(\mathstrut +\mathstrut 248517593110644968q^{61} \) \(\mathstrut -\mathstrut 622575860488793040q^{62} \) \(\mathstrut +\mathstrut 375323610989436120q^{63} \) \(\mathstrut +\mathstrut 797725889363156992q^{64} \) \(\mathstrut -\mathstrut 28930338203125000q^{65} \) \(\mathstrut +\mathstrut 66770019182287616q^{66} \) \(\mathstrut -\mathstrut 83976946080676360q^{67} \) \(\mathstrut +\mathstrut 1012065979784551680q^{68} \) \(\mathstrut +\mathstrut 721386544222156176q^{69} \) \(\mathstrut -\mathstrut 361668273328125000q^{70} \) \(\mathstrut -\mathstrut 326854508200029072q^{71} \) \(\mathstrut -\mathstrut 3247857120459936480q^{72} \) \(\mathstrut +\mathstrut 706071038062266440q^{73} \) \(\mathstrut +\mathstrut 1007811588432673944q^{74} \) \(\mathstrut +\mathstrut 11749267578125000q^{75} \) \(\mathstrut +\mathstrut 453476009937236480q^{76} \) \(\mathstrut -\mathstrut 3188023247458540800q^{77} \) \(\mathstrut -\mathstrut 422235733076613200q^{78} \) \(\mathstrut +\mathstrut 1300235890504742240q^{79} \) \(\mathstrut -\mathstrut 3487851755750000000q^{80} \) \(\mathstrut -\mathstrut 565640020943583596q^{81} \) \(\mathstrut -\mathstrut 3526321594330914840q^{82} \) \(\mathstrut -\mathstrut 894716269324765080q^{83} \) \(\mathstrut +\mathstrut 11741177868239906304q^{84} \) \(\mathstrut -\mathstrut 1658269028203125000q^{85} \) \(\mathstrut +\mathstrut 10889898572595403608q^{86} \) \(\mathstrut -\mathstrut 3243563654475579280q^{87} \) \(\mathstrut +\mathstrut 8680694316508369920q^{88} \) \(\mathstrut -\mathstrut 4392730519471649880q^{89} \) \(\mathstrut +\mathstrut 4128474608632812500q^{90} \) \(\mathstrut -\mathstrut 1164625388848487312q^{91} \) \(\mathstrut +\mathstrut 1015783757806275840q^{92} \) \(\mathstrut +\mathstrut 589775737588352160q^{93} \) \(\mathstrut +\mathstrut 6912955459089075144q^{94} \) \(\mathstrut -\mathstrut 3863608805781250000q^{95} \) \(\mathstrut -\mathstrut 34035929850443648512q^{96} \) \(\mathstrut +\mathstrut 1437870497951780360q^{97} \) \(\mathstrut +\mathstrut 1735173204679501740q^{98} \) \(\mathstrut -\mathstrut 25265427652281702784q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(2\) \(x^{3}\mathstrut -\mathstrut \) \(323561\) \(x^{2}\mathstrut -\mathstrut \) \(26738538\) \(x\mathstrut +\mathstrut \) \(10870990650\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -50 \nu^{3} + 12712 \nu^{2} + 10576678 \nu - 1034892417 \)\()/1360737\)
\(\beta_{2}\)\(=\)\((\)\( -100 \nu^{3} + 25424 \nu^{2} + 57439676 \nu - 2087927994 \)\()/453579\)
\(\beta_{3}\)\(=\)\((\)\( -340 \nu^{3} - 224584 \nu^{2} + 120441404 \nu + 43256659674 \)\()/194391\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(40\)\()/80\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(78\) \(\beta_{2}\mathstrut +\mathstrut \) \(722\) \(\beta_{1}\mathstrut +\mathstrut \) \(6471260\)\()/40\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(12712\) \(\beta_{3}\mathstrut +\mathstrut \) \(251195\) \(\beta_{2}\mathstrut -\mathstrut \) \(3079258\) \(\beta_{1}\mathstrut +\mathstrut \) \(1643139760\)\()/80\)

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
−462.664
150.720
581.867
−267.923
−1387.10 48079.9 1.39976e6 −1.95313e6 −6.66917e7 9.13351e7 −1.21437e9 1.14942e9 2.70918e9
1.2 −602.379 −7268.55 −161427. −1.95313e6 4.37842e6 −1.76809e8 4.13061e8 −1.10943e9 1.17652e9
1.3 208.721 −48249.1 −480724. −1.95313e6 −1.00706e7 1.75500e8 −2.09767e8 1.16572e9 −4.07657e8
1.4 1360.76 10517.7 1.32738e6 −1.95313e6 1.43121e7 1.23996e8 1.09282e9 −1.05164e9 −2.65773e9
\(n\): 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
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 420 T_{2}^{3} \) \(\mathstrut -\mathstrut 2002872 T_{2}^{2} \) \(\mathstrut -\mathstrut 746347520 T_{2} \) \(\mathstrut +\mathstrut 237314973696 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\).