# Properties

 Label 25.34.a.b Level $25$ Weight $34$ Character orbit 25.a Self dual yes Analytic conductor $172.457$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$34$$ Character orbit: $$[\chi]$$ $$=$$ 25.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$172.457072203$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2 x^{4} - 1372039866 x^{3} - 648067657640 x^{2} + 285631173782445856 x - 33409741805340964224$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{21}\cdot 3^{5}\cdot 5^{4}\cdot 11$$ Twist minimal: no (minimal twist has level 5) 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.

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

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
 −33002.4 −16460.4 117.007 15553.7 33794.2
−138106. 1.45282e7 1.04832e10 0 −2.00642e12 −3.75584e13 −2.61472e14 −5.34799e15 0
1.2 −71937.8 −1.37573e8 −3.41489e9 0 9.89671e12 1.07684e14 8.63600e14 1.33673e16 0
1.3 −5627.97 1.24605e8 −8.55826e9 0 −7.01272e11 −7.01560e13 9.65096e13 9.96727e15 0
1.4 56118.7 −5.48591e7 −5.44063e9 0 −3.07862e12 −7.38557e13 −7.87377e14 −2.54954e15 0
1.5 129081. 6.82881e7 8.07187e9 0 8.81467e12 1.39338e14 −6.68716e13 −8.95800e14 0
 $$n$$: 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

## Atkin-Lehner signs

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

## 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 25.34.a.b 5
5.b even 2 1 5.34.a.a 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 5.b even 2 1
25.34.a.b 5 1.a even 1 1 trivial
25.34.b.b 10 5.c odd 4 2

## 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 $$S_{34}^{\mathrm{new}}(\Gamma_0(25))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$40\!\cdots\!32$$$$+ 70175438451212025856 T - 440682607210496 T^{2} - 21581220768 T^{3} + 30472 T^{4} + T^{5}$$
$3$ $$-$$$$93\!\cdots\!24$$$$+$$$$61\!\cdots\!16$$$$T +$$$$48\!\cdots\!88$$$$T^{2} - 21055946137906392 T^{3} - 14988714 T^{4} + T^{5}$$
$5$ $$T^{5}$$
$7$ $$29\!\cdots\!32$$$$+$$$$11\!\cdots\!96$$$$T +$$$$30\!\cdots\!24$$$$T^{2} -$$$$19\!\cdots\!28$$$$T^{3} - 65452561787158 T^{4} + T^{5}$$
$11$ $$62\!\cdots\!68$$$$-$$$$73\!\cdots\!20$$$$T -$$$$33\!\cdots\!80$$$$T^{2} +$$$$26\!\cdots\!40$$$$T^{3} + 287801801877976640 T^{4} + T^{5}$$
$13$ $$12\!\cdots\!76$$$$+$$$$11\!\cdots\!36$$$$T -$$$$11\!\cdots\!92$$$$T^{2} -$$$$90\!\cdots\!12$$$$T^{3} + 2397201150889907466 T^{4} + T^{5}$$
$17$ $$-$$$$24\!\cdots\!68$$$$+$$$$49\!\cdots\!76$$$$T +$$$$18\!\cdots\!64$$$$T^{2} -$$$$50\!\cdots\!48$$$$T^{3} -$$$$30\!\cdots\!18$$$$T^{4} + T^{5}$$
$19$ $$-$$$$97\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T +$$$$20\!\cdots\!00$$$$T^{2} -$$$$25\!\cdots\!00$$$$T^{3} -$$$$91\!\cdots\!00$$$$T^{4} + T^{5}$$
$23$ $$72\!\cdots\!76$$$$+$$$$18\!\cdots\!56$$$$T -$$$$13\!\cdots\!72$$$$T^{2} -$$$$26\!\cdots\!32$$$$T^{3} +$$$$55\!\cdots\!46$$$$T^{4} + T^{5}$$
$29$ $$-$$$$58\!\cdots\!00$$$$-$$$$13\!\cdots\!00$$$$T -$$$$78\!\cdots\!00$$$$T^{2} -$$$$72\!\cdots\!00$$$$T^{3} +$$$$92\!\cdots\!50$$$$T^{4} + T^{5}$$
$31$ $$-$$$$18\!\cdots\!32$$$$-$$$$82\!\cdots\!20$$$$T -$$$$10\!\cdots\!80$$$$T^{2} -$$$$38\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!40$$$$T^{4} + T^{5}$$
$37$ $$14\!\cdots\!32$$$$-$$$$20\!\cdots\!64$$$$T -$$$$87\!\cdots\!56$$$$T^{2} +$$$$55\!\cdots\!12$$$$T^{3} -$$$$14\!\cdots\!38$$$$T^{4} + T^{5}$$
$41$ $$14\!\cdots\!68$$$$+$$$$74\!\cdots\!80$$$$T +$$$$10\!\cdots\!20$$$$T^{2} +$$$$55\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!90$$$$T^{4} + T^{5}$$
$43$ $$-$$$$73\!\cdots\!24$$$$-$$$$33\!\cdots\!04$$$$T +$$$$26\!\cdots\!68$$$$T^{2} -$$$$20\!\cdots\!72$$$$T^{3} -$$$$23\!\cdots\!94$$$$T^{4} + T^{5}$$
$47$ $$-$$$$52\!\cdots\!68$$$$-$$$$16\!\cdots\!84$$$$T +$$$$38\!\cdots\!84$$$$T^{2} +$$$$54\!\cdots\!92$$$$T^{3} -$$$$77\!\cdots\!98$$$$T^{4} + T^{5}$$
$53$ $$16\!\cdots\!76$$$$-$$$$16\!\cdots\!84$$$$T -$$$$58\!\cdots\!12$$$$T^{2} +$$$$25\!\cdots\!08$$$$T^{3} +$$$$10\!\cdots\!86$$$$T^{4} + T^{5}$$
$59$ $$-$$$$50\!\cdots\!00$$$$+$$$$71\!\cdots\!00$$$$T +$$$$41\!\cdots\!00$$$$T^{2} -$$$$16\!\cdots\!00$$$$T^{3} -$$$$25\!\cdots\!00$$$$T^{4} + T^{5}$$
$61$ $$-$$$$88\!\cdots\!32$$$$-$$$$50\!\cdots\!20$$$$T +$$$$73\!\cdots\!20$$$$T^{2} -$$$$21\!\cdots\!60$$$$T^{3} -$$$$74\!\cdots\!10$$$$T^{4} + T^{5}$$
$67$ $$10\!\cdots\!32$$$$-$$$$77\!\cdots\!24$$$$T -$$$$19\!\cdots\!36$$$$T^{2} +$$$$46\!\cdots\!52$$$$T^{3} -$$$$37\!\cdots\!18$$$$T^{4} + T^{5}$$
$71$ $$-$$$$43\!\cdots\!32$$$$-$$$$64\!\cdots\!20$$$$T -$$$$27\!\cdots\!80$$$$T^{2} -$$$$30\!\cdots\!60$$$$T^{3} +$$$$55\!\cdots\!40$$$$T^{4} + T^{5}$$
$73$ $$47\!\cdots\!76$$$$-$$$$40\!\cdots\!44$$$$T +$$$$95\!\cdots\!28$$$$T^{2} -$$$$20\!\cdots\!32$$$$T^{3} -$$$$13\!\cdots\!54$$$$T^{4} + T^{5}$$
$79$ $$25\!\cdots\!00$$$$+$$$$70\!\cdots\!00$$$$T -$$$$25\!\cdots\!00$$$$T^{2} -$$$$13\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!00$$$$T^{4} + T^{5}$$
$83$ $$10\!\cdots\!76$$$$+$$$$10\!\cdots\!76$$$$T -$$$$36\!\cdots\!52$$$$T^{2} +$$$$95\!\cdots\!48$$$$T^{3} +$$$$92\!\cdots\!26$$$$T^{4} + T^{5}$$
$89$ $$-$$$$20\!\cdots\!00$$$$+$$$$54\!\cdots\!00$$$$T +$$$$41\!\cdots\!00$$$$T^{2} -$$$$50\!\cdots\!00$$$$T^{3} +$$$$60\!\cdots\!50$$$$T^{4} + T^{5}$$
$97$ $$42\!\cdots\!32$$$$+$$$$22\!\cdots\!16$$$$T +$$$$93\!\cdots\!84$$$$T^{2} -$$$$26\!\cdots\!08$$$$T^{3} -$$$$86\!\cdots\!98$$$$T^{4} + T^{5}$$