Properties

Label 5.34.a.a
Level 5
Weight 34
Character orbit 5.a
Self dual Yes
Analytic conductor 34.491
Analytic rank 1
Dimension 5
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(34.4914144405\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5^{4}\cdot 11 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 6094 - \beta_{1} ) q^{2} \) \( + ( -2997861 - 296 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 228258528 - 9354 \beta_{1} + 14 \beta_{2} + \beta_{3} ) q^{4} \) \( -152587890625 q^{5} \) \( + ( 2585018216412 + 13970890 \beta_{1} - 24064 \beta_{2} + 204 \beta_{3} - 24 \beta_{4} ) q^{6} \) \( + ( -13090622515936 - 275204776 \beta_{1} + 370132 \beta_{2} - 6076 \beta_{3} + 343 \beta_{4} ) q^{7} \) \( + ( 31122858928672 + 1828919312 \beta_{1} + 1244240 \beta_{2} + 9352 \beta_{3} + 2464 \beta_{4} ) q^{8} \) \( + ( 2908246110459663 - 10225151520 \beta_{1} - 13421058 \beta_{2} - 641112 \beta_{3} - 26778 \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(6094 - \beta_{1}) q^{2}\) \(+(-2997861 - 296 \beta_{1} - \beta_{2}) q^{3}\) \(+(228258528 - 9354 \beta_{1} + 14 \beta_{2} + \beta_{3}) q^{4}\) \(-152587890625 q^{5}\) \(+(2585018216412 + 13970890 \beta_{1} - 24064 \beta_{2} + 204 \beta_{3} - 24 \beta_{4}) q^{6}\) \(+(-13090622515936 - 275204776 \beta_{1} + 370132 \beta_{2} - 6076 \beta_{3} + 343 \beta_{4}) q^{7}\) \(+(31122858928672 + 1828919312 \beta_{1} + 1244240 \beta_{2} + 9352 \beta_{3} + 2464 \beta_{4}) q^{8}\) \(+(2908246110459663 - 10225151520 \beta_{1} - 13421058 \beta_{2} - 641112 \beta_{3} - 26778 \beta_{4}) q^{9}\) \(+(-929870605468750 + 152587890625 \beta_{1}) q^{10}\) \(+(-57560251322868121 + 273035402352 \beta_{1} + 791220671 \beta_{2} - 7637996 \beta_{3} + 336003 \beta_{4}) q^{11}\) \(+(-81076371449901120 - 948459700780 \beta_{1} + 4860284836 \beta_{2} - 53157282 \beta_{3} - 4224 \beta_{4}) q^{12}\) \(+(479444227890890768 + 10002587132448 \beta_{1} + 16927525838 \beta_{2} + 157471432 \beta_{3} - 1432226 \beta_{4}) q^{13}\) \(+(2335308174110519412 + 45502707068518 \beta_{1} + 52829559552 \beta_{2} + 841732668 \beta_{3} - 6035960 \beta_{4}) q^{14}\) \(+(457437286376953125 + 45166015625000 \beta_{1} + 152587890625 \beta_{2}) q^{15}\) \(+(-17836027712587802368 - 24401194738624 \beta_{1} + 204137390400 \beta_{2} - 6547621920 \beta_{3} + 35471616 \beta_{4}) q^{16}\) \(+(-61744518479618496924 + 1520616075502176 \beta_{1} - 678974309078 \beta_{2} + 7844386136 \beta_{3} + 452458202 \beta_{4}) q^{17}\) \(+(\)\(10\!\cdots\!02\)\( + 1339843201269087 \beta_{1} - 4045253125632 \beta_{2} - 31141001904 \beta_{3} - 1531079328 \beta_{4}) q^{18}\) \(+(\)\(18\!\cdots\!53\)\( + 7656079374584928 \beta_{1} - 8069358999567 \beta_{2} + 51677080452 \beta_{3} - 318526969 \beta_{4}) q^{19}\) \(+(-34829487304687500000 + 1427307128906250 \beta_{1} - 2136230468750 \beta_{2} - 152587890625 \beta_{3}) q^{20}\) \(+(-\)\(21\!\cdots\!96\)\( + 58793044217339712 \beta_{1} + 16770950530488 \beta_{2} + 683931417792 \beta_{3} + 20758921680 \beta_{4}) q^{21}\) \(+(-\)\(27\!\cdots\!08\)\( + 96643688734934512 \beta_{1} + 55035020026112 \beta_{2} + 323545318888 \beta_{3} + 786823216 \beta_{4}) q^{22}\) \(+(\)\(11\!\cdots\!20\)\( + 73151646643535560 \beta_{1} + 41953628056496 \beta_{2} - 2466432910340 \beta_{3} - 168387969255 \beta_{4}) q^{23}\) \(+(-\)\(14\!\cdots\!64\)\( + 236721085707131616 \beta_{1} + 318370515316320 \beta_{2} - 294628701840 \beta_{3} + 209711069376 \beta_{4}) q^{24}\) \(+\)\(23\!\cdots\!25\)\( q^{25}\) \(+(-\)\(84\!\cdots\!08\)\( - 1643326842467950050 \beta_{1} + 271442359812608 \beta_{2} - 10839062774288 \beta_{3} + 749666738208 \beta_{4}) q^{26}\) \(+(\)\(13\!\cdots\!22\)\( - 1364715153513867888 \beta_{1} - 3724825957052130 \beta_{2} + 36308722359312 \beta_{3} - 2015706185316 \beta_{4}) q^{27}\) \(+(-\)\(27\!\cdots\!56\)\( - 5772692568086046036 \beta_{1} - 2476503850140964 \beta_{2} + 1231881015778 \beta_{3} + 147777549696 \beta_{4}) q^{28}\) \(+(-\)\(18\!\cdots\!06\)\( - 4953438057677613376 \beta_{1} - 1804843226208312 \beta_{2} + 45010439372352 \beta_{3} + 1337266709872 \beta_{4}) q^{29}\) \(+(-\)\(39\!\cdots\!00\)\( - 2131788635253906250 \beta_{1} + 3671875000000000 \beta_{2} - 31127929687500 \beta_{3} + 3662109375000 \beta_{4}) q^{30}\) \(+(-\)\(26\!\cdots\!52\)\( - 12325071667990845104 \beta_{1} + 22225900117419208 \beta_{2} - 433249088292808 \beta_{3} + 1903411871394 \beta_{4}) q^{31}\) \(+(-\)\(16\!\cdots\!68\)\( + 41010186297503890432 \beta_{1} - 4855020230030336 \beta_{2} + 25551021710336 \beta_{3} - 30405376283648 \beta_{4}) q^{32}\) \(+(-\)\(66\!\cdots\!10\)\( + 71147106764068400800 \beta_{1} + 67073908274686574 \beta_{2} + 969367359905544 \beta_{3} + 41884529604558 \beta_{4}) q^{33}\) \(+(-\)\(13\!\cdots\!40\)\( + 24103804454038413286 \beta_{1} + 21785413011228160 \beta_{2} - 901140812434480 \beta_{3} - 2226787292064 \beta_{4}) q^{34}\) \(+(\)\(19\!\cdots\!00\)\( + 41992916259765625000 \beta_{1} - 56477661132812500 \beta_{2} + 927124023437500 \beta_{3} - 52337646484375 \beta_{4}) q^{35}\) \(+(-\)\(36\!\cdots\!48\)\( + \)\(23\!\cdots\!66\)\( \beta_{1} - 231451367573556210 \beta_{2} + 1493403977156865 \beta_{3} + 75547401540096 \beta_{4}) q^{36}\) \(+(-\)\(28\!\cdots\!98\)\( + 63192379264926603712 \beta_{1} - 310572844736348292 \beta_{2} + 1309166929437264 \beta_{3} - 67184756977652 \beta_{4}) q^{37}\) \(+(-\)\(66\!\cdots\!64\)\( - \)\(38\!\cdots\!84\)\( \beta_{1} - 337522087390031616 \beta_{2} - 8952328672923936 \beta_{3} - 81848332367552 \beta_{4}) q^{38}\) \(+(-\)\(15\!\cdots\!78\)\( - \)\(38\!\cdots\!04\)\( \beta_{1} + 543208139075091626 \beta_{2} - 274687131302256 \beta_{3} + 481953507778812 \beta_{4}) q^{39}\) \(+(-\)\(47\!\cdots\!00\)\( - \)\(27\!\cdots\!00\)\( \beta_{1} - 189855957031250000 \beta_{2} - 1427001953125000 \beta_{3} - 375976562500000 \beta_{4}) q^{40}\) \(+(-\)\(26\!\cdots\!12\)\( - \)\(80\!\cdots\!44\)\( \beta_{1} + 1601744566635231538 \beta_{2} + 2315959677073112 \beta_{3} - 892735108284166 \beta_{4}) q^{41}\) \(+(-\)\(52\!\cdots\!88\)\( - \)\(22\!\cdots\!48\)\( \beta_{1} + 2737718472035635200 \beta_{2} - 26380455088312416 \beta_{3} + 1737507124049088 \beta_{4}) q^{42}\) \(+(-\)\(46\!\cdots\!19\)\( - \)\(18\!\cdots\!04\)\( \beta_{1} + 4724506331249339553 \beta_{2} + 113827889436286920 \beta_{3} - 1087365276923810 \beta_{4}) q^{43}\) \(+(-\)\(37\!\cdots\!80\)\( - \)\(19\!\cdots\!44\)\( \beta_{1} - 6266279539481441920 \beta_{2} - 25184464128968000 \beta_{3} - 881462448148224 \beta_{4}) q^{44}\) \(+(-\)\(44\!\cdots\!75\)\( + \)\(15\!\cdots\!00\)\( \beta_{1} + 2047890930175781250 \beta_{2} + 97825927734375000 \beta_{3} + 4085998535156250 \beta_{4}) q^{45}\) \(+(-\)\(57\!\cdots\!92\)\( + \)\(46\!\cdots\!86\)\( \beta_{1} - 22673691941281665792 \beta_{2} - 323669043109800108 \beta_{3} - 3265031938168616 \beta_{4}) q^{46}\) \(+(-\)\(15\!\cdots\!94\)\( + \)\(72\!\cdots\!36\)\( \beta_{1} - 2314738312949068126 \beta_{2} - 350318335022645540 \beta_{3} - 870596870258655 \beta_{4}) q^{47}\) \(+(-\)\(14\!\cdots\!08\)\( + \)\(20\!\cdots\!12\)\( \beta_{1} - 10065000765950784128 \beta_{2} + 570331472626319424 \beta_{3} + 5833882171411968 \beta_{4}) q^{48}\) \(+(\)\(82\!\cdots\!31\)\( + \)\(45\!\cdots\!84\)\( \beta_{1} - 17093305674722124358 \beta_{2} + 287587336313087608 \beta_{3} - 19428477831767214 \beta_{4}) q^{49}\) \(+(\)\(14\!\cdots\!50\)\( - \)\(23\!\cdots\!25\)\( \beta_{1}) q^{50}\) \(+(\)\(14\!\cdots\!10\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} + \)\(16\!\cdots\!78\)\( \beta_{2} - 1329935351003519568 \beta_{3} + 48752164266519156 \beta_{4}) q^{51}\) \(+(\)\(97\!\cdots\!52\)\( + \)\(57\!\cdots\!32\)\( \beta_{1} - 28764583903899727844 \beta_{2} + 1476269886987441410 \beta_{3} - 8596248368826880 \beta_{4}) q^{52}\) \(+(\)\(21\!\cdots\!80\)\( - \)\(37\!\cdots\!40\)\( \beta_{1} + \)\(29\!\cdots\!22\)\( \beta_{2} + 1643564349213352632 \beta_{3} - 31036630863163326 \beta_{4}) q^{53}\) \(+(\)\(12\!\cdots\!88\)\( - \)\(32\!\cdots\!12\)\( \beta_{1} - \)\(31\!\cdots\!92\)\( \beta_{2} - 2107942576365640248 \beta_{3} - 439765913802384 \beta_{4}) q^{54}\) \(+(\)\(87\!\cdots\!25\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!75\)\( \beta_{2} + 1165465698242187500 \beta_{3} - 51269989013671875 \beta_{4}) q^{55}\) \(+(\)\(28\!\cdots\!96\)\( - \)\(11\!\cdots\!84\)\( \beta_{1} - \)\(42\!\cdots\!88\)\( \beta_{2} - 1461129436904112752 \beta_{3} - 5823257790116032 \beta_{4}) q^{56}\) \(+(\)\(41\!\cdots\!82\)\( - \)\(40\!\cdots\!68\)\( \beta_{1} + 94973868093326173394 \beta_{2} - 10551361338090824520 \beta_{3} + 3800863423580610 \beta_{4}) q^{57}\) \(+(\)\(42\!\cdots\!48\)\( - \)\(97\!\cdots\!22\)\( \beta_{1} + \)\(21\!\cdots\!28\)\( \beta_{2} + 6774753333761091424 \beta_{3} + 44709830638107968 \beta_{4}) q^{58}\) \(+(\)\(51\!\cdots\!67\)\( - \)\(13\!\cdots\!88\)\( \beta_{1} - \)\(73\!\cdots\!37\)\( \beta_{2} + 4120978175839918492 \beta_{3} + 547033541408337905 \beta_{4}) q^{59}\) \(+(\)\(12\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( \beta_{1} - \)\(74\!\cdots\!00\)\( \beta_{2} + 8111157531738281250 \beta_{3} + 644531250000000 \beta_{4}) q^{60}\) \(+(\)\(14\!\cdots\!82\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(15\!\cdots\!00\)\( \beta_{2} + 1716415096131067200 \beta_{3} - 1217578806810299600 \beta_{4}) q^{61}\) \(+(\)\(10\!\cdots\!80\)\( + \)\(27\!\cdots\!20\)\( \beta_{1} + \)\(72\!\cdots\!76\)\( \beta_{2} + 17840120851053055624 \beta_{3} - 399572245923119632 \beta_{4}) q^{62}\) \(+(-\)\(20\!\cdots\!00\)\( + \)\(28\!\cdots\!20\)\( \beta_{1} + \)\(69\!\cdots\!68\)\( \beta_{2} - 12410058440127099276 \beta_{3} + 991600158507478443 \beta_{4}) q^{63}\) \(+(-\)\(20\!\cdots\!68\)\( + \)\(46\!\cdots\!60\)\( \beta_{1} - \)\(62\!\cdots\!88\)\( \beta_{2} - 31752890677639058432 \beta_{3} - 190495590188515328 \beta_{4}) q^{64}\) \(+(-\)\(73\!\cdots\!00\)\( - \)\(15\!\cdots\!00\)\( \beta_{1} - \)\(25\!\cdots\!50\)\( \beta_{2} - 24028233642578125000 \beta_{3} + 218540344238281250 \beta_{4}) q^{65}\) \(+(-\)\(66\!\cdots\!92\)\( - \)\(16\!\cdots\!36\)\( \beta_{1} + \)\(68\!\cdots\!84\)\( \beta_{2} - 2162057533246071504 \beta_{3} + 3429657268765039008 \beta_{4}) q^{66}\) \(+(-\)\(75\!\cdots\!49\)\( - \)\(58\!\cdots\!24\)\( \beta_{1} - \)\(22\!\cdots\!33\)\( \beta_{2} + 24396663538239925176 \beta_{3} + 710362442767852482 \beta_{4}) q^{67}\) \(+(\)\(23\!\cdots\!76\)\( + \)\(61\!\cdots\!96\)\( \beta_{1} + \)\(51\!\cdots\!00\)\( \beta_{2} - 91725834494896803462 \beta_{3} - 5268448731091049984 \beta_{4}) q^{68}\) \(+(-\)\(55\!\cdots\!84\)\( - \)\(18\!\cdots\!60\)\( \beta_{1} - \)\(39\!\cdots\!56\)\( \beta_{2} + \)\(18\!\cdots\!16\)\( \beta_{3} - 6977086754652503016 \beta_{4}) q^{69}\) \(+(-\)\(35\!\cdots\!00\)\( - \)\(69\!\cdots\!50\)\( \beta_{1} - \)\(80\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + 921014404296875000 \beta_{4}) q^{70}\) \(+(-\)\(11\!\cdots\!38\)\( - \)\(24\!\cdots\!00\)\( \beta_{1} + \)\(13\!\cdots\!50\)\( \beta_{2} + \)\(40\!\cdots\!00\)\( \beta_{3} + 5349512571233196200 \beta_{4}) q^{71}\) \(+(-\)\(31\!\cdots\!28\)\( + \)\(18\!\cdots\!32\)\( \beta_{1} + \)\(35\!\cdots\!08\)\( \beta_{2} + \)\(12\!\cdots\!08\)\( \beta_{3} + 10336827195375217056 \beta_{4}) q^{72}\) \(+(-\)\(27\!\cdots\!48\)\( - \)\(89\!\cdots\!28\)\( \beta_{1} + \)\(42\!\cdots\!82\)\( \beta_{2} - \)\(63\!\cdots\!96\)\( \beta_{3} + 5792348749848995578 \beta_{4}) q^{73}\) \(+(-\)\(72\!\cdots\!80\)\( + \)\(24\!\cdots\!46\)\( \beta_{1} - \)\(17\!\cdots\!52\)\( \beta_{2} - \)\(19\!\cdots\!48\)\( \beta_{3} - 4278169457359479616 \beta_{4}) q^{74}\) \(+(-\)\(69\!\cdots\!25\)\( - \)\(68\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!25\)\( \beta_{2}) q^{75}\) \(+(\)\(14\!\cdots\!72\)\( + \)\(59\!\cdots\!12\)\( \beta_{1} + \)\(48\!\cdots\!64\)\( \beta_{2} - \)\(19\!\cdots\!44\)\( \beta_{3} - 23940239356113081344 \beta_{4}) q^{76}\) \(+(\)\(94\!\cdots\!22\)\( + \)\(22\!\cdots\!52\)\( \beta_{1} - \)\(56\!\cdots\!94\)\( \beta_{2} - 97285217744744793064 \beta_{3} - 41153529239063866198 \beta_{4}) q^{77}\) \(+(\)\(24\!\cdots\!96\)\( + \)\(15\!\cdots\!36\)\( \beta_{1} + \)\(80\!\cdots\!68\)\( \beta_{2} + \)\(11\!\cdots\!08\)\( \beta_{3} + 9657130777273888656 \beta_{4}) q^{78}\) \(+(-\)\(20\!\cdots\!60\)\( - \)\(26\!\cdots\!40\)\( \beta_{1} - \)\(87\!\cdots\!24\)\( \beta_{2} - \)\(30\!\cdots\!36\)\( \beta_{3} + \)\(11\!\cdots\!36\)\( \beta_{4}) q^{79}\) \(+(\)\(27\!\cdots\!00\)\( + \)\(37\!\cdots\!00\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2} + \)\(99\!\cdots\!00\)\( \beta_{3} - 5412539062500000000 \beta_{4}) q^{80}\) \(+(\)\(15\!\cdots\!67\)\( - \)\(28\!\cdots\!44\)\( \beta_{1} - \)\(15\!\cdots\!66\)\( \beta_{2} - \)\(97\!\cdots\!44\)\( \beta_{3} - 75223021076941246470 \beta_{4}) q^{81}\) \(+(\)\(54\!\cdots\!20\)\( + \)\(23\!\cdots\!10\)\( \beta_{1} - \)\(53\!\cdots\!64\)\( \beta_{2} - \)\(41\!\cdots\!36\)\( \beta_{3} + 48548095420105302048 \beta_{4}) q^{82}\) \(+(\)\(18\!\cdots\!23\)\( - \)\(30\!\cdots\!12\)\( \beta_{1} - \)\(51\!\cdots\!13\)\( \beta_{2} + \)\(15\!\cdots\!00\)\( \beta_{3} + 24580259143340337000 \beta_{4}) q^{83}\) \(+(\)\(34\!\cdots\!68\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} + \)\(16\!\cdots\!88\)\( \beta_{2} - \)\(65\!\cdots\!48\)\( \beta_{3} - \)\(17\!\cdots\!08\)\( \beta_{4}) q^{84}\) \(+(\)\(94\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!50\)\( \beta_{2} - \)\(11\!\cdots\!00\)\( \beta_{3} - 69039642639160156250 \beta_{4}) q^{85}\) \(+(\)\(13\!\cdots\!84\)\( - \)\(31\!\cdots\!38\)\( \beta_{1} + \)\(10\!\cdots\!40\)\( \beta_{2} + \)\(44\!\cdots\!20\)\( \beta_{3} + \)\(36\!\cdots\!32\)\( \beta_{4}) q^{86}\) \(+(\)\(27\!\cdots\!10\)\( + \)\(19\!\cdots\!40\)\( \beta_{1} + \)\(44\!\cdots\!62\)\( \beta_{2} - \)\(32\!\cdots\!36\)\( \beta_{3} - 79886836424468661552 \beta_{4}) q^{87}\) \(+(\)\(38\!\cdots\!04\)\( - \)\(22\!\cdots\!16\)\( \beta_{1} - \)\(75\!\cdots\!20\)\( \beta_{2} - \)\(26\!\cdots\!36\)\( \beta_{3} - \)\(20\!\cdots\!52\)\( \beta_{4}) q^{88}\) \(+(-\)\(12\!\cdots\!98\)\( + \)\(42\!\cdots\!92\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2} + \)\(42\!\cdots\!76\)\( \beta_{3} - 10026730679801932284 \beta_{4}) q^{89}\) \(+(-\)\(16\!\cdots\!50\)\( - \)\(20\!\cdots\!75\)\( \beta_{1} + \)\(61\!\cdots\!00\)\( \beta_{2} + \)\(47\!\cdots\!00\)\( \beta_{3} + \)\(23\!\cdots\!00\)\( \beta_{4}) q^{90}\) \(+(-\)\(53\!\cdots\!32\)\( - \)\(22\!\cdots\!84\)\( \beta_{1} - \)\(66\!\cdots\!28\)\( \beta_{2} - \)\(19\!\cdots\!12\)\( \beta_{3} - 84176942221259978622 \beta_{4}) q^{91}\) \(+(-\)\(13\!\cdots\!00\)\( + \)\(22\!\cdots\!60\)\( \beta_{1} - \)\(17\!\cdots\!96\)\( \beta_{2} + \)\(98\!\cdots\!54\)\( \beta_{3} + \)\(23\!\cdots\!28\)\( \beta_{4}) q^{92}\) \(+(-\)\(13\!\cdots\!96\)\( + \)\(13\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!04\)\( \beta_{2} + \)\(46\!\cdots\!12\)\( \beta_{3} + \)\(27\!\cdots\!84\)\( \beta_{4}) q^{93}\) \(+(-\)\(72\!\cdots\!88\)\( + \)\(38\!\cdots\!18\)\( \beta_{1} - \)\(52\!\cdots\!32\)\( \beta_{2} - \)\(82\!\cdots\!08\)\( \beta_{3} - \)\(79\!\cdots\!44\)\( \beta_{4}) q^{94}\) \(+(-\)\(28\!\cdots\!25\)\( - \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!75\)\( \beta_{2} - \)\(78\!\cdots\!00\)\( \beta_{3} + 48603358306884765625 \beta_{4}) q^{95}\) \(+(-\)\(68\!\cdots\!40\)\( - \)\(40\!\cdots\!12\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(11\!\cdots\!60\)\( \beta_{3} - \)\(86\!\cdots\!92\)\( \beta_{4}) q^{96}\) \(+(-\)\(17\!\cdots\!68\)\( + \)\(39\!\cdots\!52\)\( \beta_{1} + \)\(30\!\cdots\!62\)\( \beta_{2} - \)\(40\!\cdots\!68\)\( \beta_{3} - 44673311295842708826 \beta_{4}) q^{97}\) \(+(-\)\(39\!\cdots\!14\)\( - \)\(22\!\cdots\!09\)\( \beta_{1} - \)\(33\!\cdots\!56\)\( \beta_{2} - \)\(77\!\cdots\!24\)\( \beta_{3} + \)\(31\!\cdots\!32\)\( \beta_{4}) q^{98}\) \(+(-\)\(35\!\cdots\!53\)\( + \)\(37\!\cdots\!56\)\( \beta_{1} + \)\(13\!\cdots\!27\)\( \beta_{2} + \)\(89\!\cdots\!08\)\( \beta_{3} + \)\(35\!\cdots\!23\)\( \beta_{4}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(5q \) \(\mathstrut +\mathstrut 30472q^{2} \) \(\mathstrut -\mathstrut 14988714q^{3} \) \(\mathstrut +\mathstrut 1141311360q^{4} \) \(\mathstrut -\mathstrut 762939453125q^{5} \) \(\mathstrut +\mathstrut 12925063115760q^{6} \) \(\mathstrut -\mathstrut 65452561787158q^{7} \) \(\mathstrut +\mathstrut 155610638035200q^{8} \) \(\mathstrut +\mathstrut 14541250990408965q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 30472q^{2} \) \(\mathstrut -\mathstrut 14988714q^{3} \) \(\mathstrut +\mathstrut 1141311360q^{4} \) \(\mathstrut -\mathstrut 762939453125q^{5} \) \(\mathstrut +\mathstrut 12925063115760q^{6} \) \(\mathstrut -\mathstrut 65452561787158q^{7} \) \(\mathstrut +\mathstrut 155610638035200q^{8} \) \(\mathstrut +\mathstrut 14541250990408965q^{9} \) \(\mathstrut -\mathstrut 4649658203125000q^{10} \) \(\mathstrut -\mathstrut 287801801877976640q^{11} \) \(\mathstrut -\mathstrut 405379955363513088q^{12} \) \(\mathstrut +\mathstrut 2397201150889907466q^{13} \) \(\mathstrut +\mathstrut 11676449916272482320q^{14} \) \(\mathstrut +\mathstrut 2287096252441406250q^{15} \) \(\mathstrut -\mathstrut 89180089543245957120q^{16} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!18\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!56\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(91\!\cdots\!00\)\(q^{19} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!96\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!46\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!25\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!40\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!36\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!40\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!08\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!48\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!80\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!50\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!20\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!38\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(78\!\cdots\!20\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!90\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!36\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!94\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!80\)\(q^{44} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!25\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!40\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(77\!\cdots\!98\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!04\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!85\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!60\)\(q^{51} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!72\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!86\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!00\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(74\!\cdots\!10\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!24\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!34\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!40\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!50\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!80\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!18\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!44\)\(q^{68} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!20\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!40\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!54\)\(q^{73} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!80\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!50\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!44\)\(q^{77} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!72\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!05\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!84\)\(q^{82} \) \(\mathstrut +\mathstrut \)\(92\!\cdots\!26\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!50\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(65\!\cdots\!60\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(60\!\cdots\!50\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(82\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!40\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!68\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!88\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!80\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!40\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!98\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!96\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!20\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5}\mathstrut -\mathstrut \) \(2\) \(x^{4}\mathstrut -\mathstrut \) \(1372039866\) \(x^{3}\mathstrut -\mathstrut \) \(648067657640\) \(x^{2}\mathstrut +\mathstrut \) \(285631173782445856\) \(x\mathstrut -\mathstrut \) \(33409741805340964224\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\( 77 \nu^{4} - 192270 \nu^{3} - 92052810618 \nu^{2} + 119506052126816 \nu + 10209085202875426224 \)\()/\)\(84151650384\)
\(\beta_{3}\)\(=\)\((\)\( -539 \nu^{4} + 1345890 \nu^{3} + 1317582877398 \nu^{2} - 1314187132467296 \nu - 440933546972744601072 \)\()/\)\(42075825192\)
\(\beta_{4}\)\(=\)\((\)\( -11597 \nu^{4} - 699627858 \nu^{3} + 15491770165050 \nu^{2} + 677163322606966624 \nu - 2146655115535860495168 \)\()/\)\(28050550128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(2838\) \(\beta_{1}\mathstrut +\mathstrut \) \(8781056288\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(308\) \(\beta_{4}\mathstrut +\mathstrut \) \(1117\) \(\beta_{3}\mathstrut -\mathstrut \) \(123526\) \(\beta_{2}\mathstrut +\mathstrut \) \(1911420946\) \(\beta_{1}\mathstrut +\mathstrut \) \(3124660933052\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(769080\) \(\beta_{4}\mathstrut +\mathstrut \) \(600534687\) \(\beta_{3}\mathstrut +\mathstrut \) \(16803019914\) \(\beta_{2}\mathstrut +\mathstrut \) \(3365191419490\) \(\beta_{1}\mathstrut +\mathstrut \) \(4195949021597623688\)\()/8\)

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
33794.2
15553.7
117.007
−16460.4
−33002.4
−129081. −6.82881e7 8.07187e9 −1.52588e11 8.81467e12 −1.39338e14 6.68716e13 −8.95800e14 1.96961e16
1.2 −56118.7 5.48591e7 −5.44063e9 −1.52588e11 −3.07862e12 7.38557e13 7.87377e14 −2.54954e15 8.56303e15
1.3 5627.97 −1.24605e8 −8.55826e9 −1.52588e11 −7.01272e11 7.01560e13 −9.65096e13 9.96727e15 −8.58760e14
1.4 71937.8 1.37573e8 −3.41489e9 −1.52588e11 9.89671e12 −1.07684e14 −8.63600e14 1.33673e16 −1.09768e16
1.5 138106. −1.45282e7 1.04832e10 −1.52588e11 −2.00642e12 3.75584e13 2.61472e14 −5.34799e15 −2.10732e16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(T_{2}^{5} \) \(\mathstrut -\mathstrut 30472 T_{2}^{4} \) \(\mathstrut -\mathstrut 21581220768 T_{2}^{3} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!96\)\( T_{2}^{2} \) \(\mathstrut +\mathstrut \)\(70\!\cdots\!56\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!32\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(5))\).