# 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

# Related objects

## 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: $$2^{21}\cdot 3^{5}\cdot 5^{4}\cdot 11$$ Twist minimal: yes 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.

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

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

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.34.a.a 5
5.b even 2 1 25.34.a.b 5
5.c odd 4 2 25.34.b.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.34.a.a 5 1.a even 1 1 trivial
25.34.a.b 5 5.b even 2 1
25.34.b.b 10 5.c odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 30472 T_{2}^{4} - 21581220768 T_{2}^{3} +$$$$44\!\cdots\!96$$$$T_{2}^{2} +$$$$70\!\cdots\!56$$$$T_{2} -$$$$40\!\cdots\!32$$ acting on $$S_{34}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 30472 T + 21368452192 T^{2} - 606327340339200 T^{3} +$$$$25\!\cdots\!28$$$$T^{4} -$$$$63\!\cdots\!16$$$$T^{5} +$$$$21\!\cdots\!76$$$$T^{6} -$$$$44\!\cdots\!00$$$$T^{7} +$$$$13\!\cdots\!96$$$$T^{8} -$$$$16\!\cdots\!12$$$$T^{9} +$$$$46\!\cdots\!32$$$$T^{10}$$
$3$ $$1 + 14988714 T + 6739356694871223 T^{2} -$$$$15\!\cdots\!00$$$$T^{3} +$$$$19\!\cdots\!58$$$$T^{4} -$$$$17\!\cdots\!88$$$$T^{5} +$$$$10\!\cdots\!34$$$$T^{6} -$$$$47\!\cdots\!00$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} +$$$$14\!\cdots\!74$$$$T^{9} +$$$$53\!\cdots\!43$$$$T^{10}$$
$5$ $$( 1 + 152587890625 T )^{5}$$
$7$ $$1 + 65452561787158 T +$$$$19\!\cdots\!07$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{3} +$$$$26\!\cdots\!98$$$$T^{4} +$$$$15\!\cdots\!84$$$$T^{5} +$$$$20\!\cdots\!86$$$$T^{6} +$$$$10\!\cdots\!00$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8} +$$$$23\!\cdots\!58$$$$T^{9} +$$$$27\!\cdots\!07$$$$T^{10}$$
$11$ $$1 + 287801801877976640 T +$$$$11\!\cdots\!95$$$$T^{2} +$$$$23\!\cdots\!80$$$$T^{3} +$$$$53\!\cdots\!10$$$$T^{4} +$$$$78\!\cdots\!48$$$$T^{5} +$$$$12\!\cdots\!10$$$$T^{6} +$$$$12\!\cdots\!80$$$$T^{7} +$$$$14\!\cdots\!45$$$$T^{8} +$$$$83\!\cdots\!40$$$$T^{9} +$$$$67\!\cdots\!51$$$$T^{10}$$
$13$ $$1 - 2397201150889907466 T +$$$$19\!\cdots\!53$$$$T^{2} -$$$$43\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!18$$$$T^{4} -$$$$35\!\cdots\!88$$$$T^{5} +$$$$10\!\cdots\!54$$$$T^{6} -$$$$14\!\cdots\!00$$$$T^{7} +$$$$37\!\cdots\!81$$$$T^{8} -$$$$26\!\cdots\!46$$$$T^{9} +$$$$63\!\cdots\!93$$$$T^{10}$$
$17$ $$1 +$$$$30\!\cdots\!18$$$$T +$$$$15\!\cdots\!37$$$$T^{2} +$$$$31\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!38$$$$T^{4} +$$$$17\!\cdots\!84$$$$T^{5} +$$$$42\!\cdots\!06$$$$T^{6} +$$$$50\!\cdots\!00$$$$T^{7} +$$$$98\!\cdots\!61$$$$T^{8} +$$$$81\!\cdots\!98$$$$T^{9} +$$$$10\!\cdots\!57$$$$T^{10}$$
$19$ $$1 -$$$$91\!\cdots\!00$$$$T +$$$$53\!\cdots\!95$$$$T^{2} -$$$$37\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!10$$$$T^{4} -$$$$82\!\cdots\!00$$$$T^{5} +$$$$22\!\cdots\!90$$$$T^{6} -$$$$93\!\cdots\!00$$$$T^{7} +$$$$21\!\cdots\!05$$$$T^{8} -$$$$57\!\cdots\!00$$$$T^{9} +$$$$98\!\cdots\!99$$$$T^{10}$$
$23$ $$1 -$$$$55\!\cdots\!46$$$$T +$$$$16\!\cdots\!83$$$$T^{2} -$$$$54\!\cdots\!00$$$$T^{3} +$$$$24\!\cdots\!78$$$$T^{4} -$$$$84\!\cdots\!88$$$$T^{5} +$$$$21\!\cdots\!74$$$$T^{6} -$$$$40\!\cdots\!00$$$$T^{7} +$$$$10\!\cdots\!21$$$$T^{8} -$$$$31\!\cdots\!66$$$$T^{9} +$$$$48\!\cdots\!43$$$$T^{10}$$
$29$ $$1 +$$$$92\!\cdots\!50$$$$T +$$$$83\!\cdots\!45$$$$T^{2} +$$$$59\!\cdots\!00$$$$T^{3} +$$$$28\!\cdots\!10$$$$T^{4} +$$$$15\!\cdots\!00$$$$T^{5} +$$$$52\!\cdots\!90$$$$T^{6} +$$$$19\!\cdots\!00$$$$T^{7} +$$$$50\!\cdots\!05$$$$T^{8} +$$$$10\!\cdots\!50$$$$T^{9} +$$$$19\!\cdots\!49$$$$T^{10}$$
$31$ $$1 +$$$$13\!\cdots\!40$$$$T +$$$$43\!\cdots\!95$$$$T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$70\!\cdots\!10$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$11\!\cdots\!10$$$$T^{6} -$$$$44\!\cdots\!20$$$$T^{7} +$$$$19\!\cdots\!45$$$$T^{8} +$$$$96\!\cdots\!40$$$$T^{9} +$$$$11\!\cdots\!51$$$$T^{10}$$
$37$ $$1 +$$$$14\!\cdots\!38$$$$T +$$$$33\!\cdots\!97$$$$T^{2} +$$$$32\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!18$$$$T^{4} +$$$$27\!\cdots\!84$$$$T^{5} +$$$$23\!\cdots\!46$$$$T^{6} +$$$$10\!\cdots\!00$$$$T^{7} +$$$$60\!\cdots\!81$$$$T^{8} +$$$$14\!\cdots\!78$$$$T^{9} +$$$$56\!\cdots\!57$$$$T^{10}$$
$41$ $$1 +$$$$13\!\cdots\!90$$$$T +$$$$13\!\cdots\!45$$$$T^{2} +$$$$98\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!10$$$$T^{4} +$$$$25\!\cdots\!48$$$$T^{5} +$$$$94\!\cdots\!10$$$$T^{6} +$$$$27\!\cdots\!80$$$$T^{7} +$$$$64\!\cdots\!45$$$$T^{8} +$$$$10\!\cdots\!90$$$$T^{9} +$$$$12\!\cdots\!01$$$$T^{10}$$
$43$ $$1 +$$$$23\!\cdots\!94$$$$T +$$$$38\!\cdots\!43$$$$T^{2} +$$$$49\!\cdots\!00$$$$T^{3} +$$$$56\!\cdots\!98$$$$T^{4} +$$$$55\!\cdots\!12$$$$T^{5} +$$$$45\!\cdots\!14$$$$T^{6} +$$$$31\!\cdots\!00$$$$T^{7} +$$$$19\!\cdots\!01$$$$T^{8} +$$$$97\!\cdots\!94$$$$T^{9} +$$$$33\!\cdots\!43$$$$T^{10}$$
$47$ $$1 +$$$$77\!\cdots\!98$$$$T +$$$$80\!\cdots\!27$$$$T^{2} +$$$$42\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!58$$$$T^{4} +$$$$94\!\cdots\!84$$$$T^{5} +$$$$37\!\cdots\!66$$$$T^{6} +$$$$97\!\cdots\!00$$$$T^{7} +$$$$27\!\cdots\!41$$$$T^{8} +$$$$40\!\cdots\!18$$$$T^{9} +$$$$78\!\cdots\!07$$$$T^{10}$$
$53$ $$1 -$$$$10\!\cdots\!86$$$$T +$$$$65\!\cdots\!73$$$$T^{2} -$$$$29\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!58$$$$T^{4} -$$$$33\!\cdots\!88$$$$T^{5} +$$$$86\!\cdots\!34$$$$T^{6} -$$$$18\!\cdots\!00$$$$T^{7} +$$$$33\!\cdots\!41$$$$T^{8} -$$$$44\!\cdots\!26$$$$T^{9} +$$$$32\!\cdots\!93$$$$T^{10}$$
$59$ $$1 -$$$$25\!\cdots\!00$$$$T +$$$$12\!\cdots\!95$$$$T^{2} -$$$$24\!\cdots\!00$$$$T^{3} +$$$$62\!\cdots\!10$$$$T^{4} -$$$$93\!\cdots\!00$$$$T^{5} +$$$$17\!\cdots\!90$$$$T^{6} -$$$$18\!\cdots\!00$$$$T^{7} +$$$$24\!\cdots\!05$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{9} +$$$$15\!\cdots\!99$$$$T^{10}$$
$61$ $$1 -$$$$74\!\cdots\!10$$$$T +$$$$40\!\cdots\!45$$$$T^{2} -$$$$17\!\cdots\!20$$$$T^{3} +$$$$62\!\cdots\!10$$$$T^{4} -$$$$18\!\cdots\!52$$$$T^{5} +$$$$51\!\cdots\!10$$$$T^{6} -$$$$11\!\cdots\!20$$$$T^{7} +$$$$22\!\cdots\!45$$$$T^{8} -$$$$34\!\cdots\!10$$$$T^{9} +$$$$37\!\cdots\!01$$$$T^{10}$$
$67$ $$1 +$$$$37\!\cdots\!18$$$$T +$$$$13\!\cdots\!87$$$$T^{2} +$$$$29\!\cdots\!00$$$$T^{3} +$$$$58\!\cdots\!38$$$$T^{4} +$$$$81\!\cdots\!84$$$$T^{5} +$$$$10\!\cdots\!06$$$$T^{6} +$$$$97\!\cdots\!00$$$$T^{7} +$$$$83\!\cdots\!61$$$$T^{8} +$$$$41\!\cdots\!98$$$$T^{9} +$$$$20\!\cdots\!07$$$$T^{10}$$
$71$ $$1 +$$$$55\!\cdots\!40$$$$T +$$$$31\!\cdots\!95$$$$T^{2} -$$$$69\!\cdots\!20$$$$T^{3} -$$$$24\!\cdots\!90$$$$T^{4} -$$$$22\!\cdots\!52$$$$T^{5} -$$$$29\!\cdots\!90$$$$T^{6} -$$$$10\!\cdots\!20$$$$T^{7} +$$$$59\!\cdots\!45$$$$T^{8} +$$$$12\!\cdots\!40$$$$T^{9} +$$$$28\!\cdots\!51$$$$T^{10}$$
$73$ $$1 +$$$$13\!\cdots\!54$$$$T +$$$$13\!\cdots\!33$$$$T^{2} +$$$$73\!\cdots\!00$$$$T^{3} +$$$$35\!\cdots\!78$$$$T^{4} +$$$$14\!\cdots\!12$$$$T^{5} +$$$$10\!\cdots\!74$$$$T^{6} +$$$$69\!\cdots\!00$$$$T^{7} +$$$$39\!\cdots\!21$$$$T^{8} +$$$$12\!\cdots\!34$$$$T^{9} +$$$$28\!\cdots\!93$$$$T^{10}$$
$79$ $$1 +$$$$10\!\cdots\!00$$$$T +$$$$74\!\cdots\!95$$$$T^{2} -$$$$86\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!10$$$$T^{4} -$$$$82\!\cdots\!00$$$$T^{5} +$$$$51\!\cdots\!90$$$$T^{6} -$$$$15\!\cdots\!00$$$$T^{7} +$$$$54\!\cdots\!05$$$$T^{8} +$$$$31\!\cdots\!00$$$$T^{9} +$$$$12\!\cdots\!99$$$$T^{10}$$
$83$ $$1 -$$$$92\!\cdots\!26$$$$T +$$$$11\!\cdots\!63$$$$T^{2} -$$$$75\!\cdots\!00$$$$T^{3} +$$$$51\!\cdots\!38$$$$T^{4} -$$$$23\!\cdots\!88$$$$T^{5} +$$$$11\!\cdots\!94$$$$T^{6} -$$$$34\!\cdots\!00$$$$T^{7} +$$$$11\!\cdots\!61$$$$T^{8} -$$$$19\!\cdots\!86$$$$T^{9} +$$$$44\!\cdots\!43$$$$T^{10}$$
$89$ $$1 +$$$$60\!\cdots\!50$$$$T +$$$$56\!\cdots\!45$$$$T^{2} +$$$$56\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!10$$$$T^{4} +$$$$16\!\cdots\!00$$$$T^{5} +$$$$39\!\cdots\!90$$$$T^{6} +$$$$25\!\cdots\!00$$$$T^{7} +$$$$54\!\cdots\!05$$$$T^{8} +$$$$12\!\cdots\!50$$$$T^{9} +$$$$44\!\cdots\!49$$$$T^{10}$$
$97$ $$1 +$$$$86\!\cdots\!98$$$$T +$$$$15\!\cdots\!77$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!58$$$$T^{4} +$$$$62\!\cdots\!84$$$$T^{5} +$$$$39\!\cdots\!66$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{7} +$$$$76\!\cdots\!41$$$$T^{8} +$$$$15\!\cdots\!18$$$$T^{9} +$$$$65\!\cdots\!57$$$$T^{10}$$