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

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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:

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

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