Properties

Label 5.34.a.b
Level 5
Weight 34
Character orbit 5.a
Self dual Yes
Analytic conductor 34.491
Analytic rank 0
Dimension 6
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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5^{6}\cdot 7\cdot 11^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 24558 + \beta_{1} ) q^{2} \) \( + ( 4418958 + 77 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( 4920092068 + 46623 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + 152587890625 q^{5} \) \( + ( 1102624263657 + 6864255 \beta_{1} + 30552 \beta_{2} + 468 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{6} \) \( + ( 3185671255582 - 97559148 \beta_{1} - 216147 \beta_{2} - 3312 \beta_{3} + 17 \beta_{4} - 5 \beta_{5} ) q^{7} \) \( + ( 511635642610246 + 3133914612 \beta_{1} + 3852490 \beta_{2} + 83674 \beta_{3} - 134 \beta_{4} - 40 \beta_{5} ) q^{8} \) \( + ( 2267015634483853 + 4382146760 \beta_{1} - 6075156 \beta_{2} + 193296 \beta_{3} - 28 \beta_{4} + 756 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(24558 + \beta_{1}) q^{2}\) \(+(4418958 + 77 \beta_{1} + \beta_{2}) q^{3}\) \(+(4920092068 + 46623 \beta_{1} + \beta_{2} + \beta_{3}) q^{4}\) \(+152587890625 q^{5}\) \(+(1102624263657 + 6864255 \beta_{1} + 30552 \beta_{2} + 468 \beta_{3} + \beta_{4} - 2 \beta_{5}) q^{6}\) \(+(3185671255582 - 97559148 \beta_{1} - 216147 \beta_{2} - 3312 \beta_{3} + 17 \beta_{4} - 5 \beta_{5}) q^{7}\) \(+(511635642610246 + 3133914612 \beta_{1} + 3852490 \beta_{2} + 83674 \beta_{3} - 134 \beta_{4} - 40 \beta_{5}) q^{8}\) \(+(2267015634483853 + 4382146760 \beta_{1} - 6075156 \beta_{2} + 193296 \beta_{3} - 28 \beta_{4} + 756 \beta_{5}) q^{9}\) \(+(3747253417968750 + 152587890625 \beta_{1}) q^{10}\) \(+(25126821644470344 - 15919797219 \beta_{1} + 735667448 \beta_{2} + 14411696 \beta_{3} + 15903 \beta_{4} + 1269 \beta_{5}) q^{11}\) \(+(77724158528649828 + 3321987710240 \beta_{1} + 6695550484 \beta_{2} + 35444916 \beta_{3} - 75556 \beta_{4} - 97360 \beta_{5}) q^{12}\) \(+(-233847859187027282 - 4227595028852 \beta_{1} + 13220112692 \beta_{2} - 154586416 \beta_{3} + 81656 \beta_{4} + 178960 \beta_{5}) q^{13}\) \(+(-1181012244499862693 - 18235901054591 \beta_{1} + 35093460488 \beta_{2} - 793087972 \beta_{3} + 311515 \beta_{4} + 842570 \beta_{5}) q^{14}\) \(+(674279479980468750 + 11749267578125 \beta_{1} + 152587890625 \beta_{2}) q^{15}\) \(+(10751698031801611492 + 666344497775648 \beta_{1} + 436988040948 \beta_{2} + 4293352404 \beta_{3} - 10297284 \beta_{4} - 14765232 \beta_{5}) q^{16}\) \(+(22396782737615808818 + 71978632509244 \beta_{1} - 1207850209132 \beta_{2} + 6749454032 \beta_{3} + 37738088 \beta_{4} + 24227680 \beta_{5}) q^{17}\) \(+(\)\(11\!\cdots\!50\)\( + 3473896119269977 \beta_{1} - 3655399616928 \beta_{2} - 2496038448 \beta_{3} - 11408732 \beta_{4} + 19767480 \beta_{5}) q^{18}\) \(+(\)\(28\!\cdots\!68\)\( + 10244681461820999 \beta_{1} - 6692355956710 \beta_{2} - 34909194064 \beta_{3} - 143281169 \beta_{4} - 18607387 \beta_{5}) q^{19}\) \(+(\)\(75\!\cdots\!00\)\( + 7114105224609375 \beta_{1} + 152587890625 \beta_{2} + 152587890625 \beta_{3}) q^{20}\) \(+(-\)\(17\!\cdots\!16\)\( + 22440764316024556 \beta_{1} - 11501053976368 \beta_{2} - 591355152288 \beta_{3} + 786288980 \beta_{4} - 127864660 \beta_{5}) q^{21}\) \(+(\)\(41\!\cdots\!22\)\( + 108546266480938682 \beta_{1} + 94869534047408 \beta_{2} + 243484405256 \beta_{3} - 968413846 \beta_{4} - 1630203860 \beta_{5}) q^{22}\) \(+(\)\(11\!\cdots\!90\)\( + 42933190737365684 \beta_{1} + 975037194779 \beta_{2} - 1358005856080 \beta_{3} + 2794904655 \beta_{4} + 5189585925 \beta_{5}) q^{23}\) \(+(\)\(35\!\cdots\!96\)\( + 301874229418111968 \beta_{1} + 495562416978408 \beta_{2} + 7377544160424 \beta_{3} - 10378784424 \beta_{4} - 1289936352 \beta_{5}) q^{24}\) \(+\)\(23\!\cdots\!25\)\( q^{25}\) \(+(-\)\(60\!\cdots\!48\)\( - 1210185937538331270 \beta_{1} - 1041229904012704 \beta_{2} - 10442017299376 \beta_{3} + 40147714308 \beta_{4} - 15011166216 \beta_{5}) q^{26}\) \(+(-\)\(56\!\cdots\!44\)\( - 2250451644212229998 \beta_{1} - 918896520555990 \beta_{2} - 204743125056 \beta_{3} - 25988358704 \beta_{4} - 22240248240 \beta_{5}) q^{27}\) \(+(-\)\(29\!\cdots\!64\)\( - 5300112711327556284 \beta_{1} - 4232529613391816 \beta_{2} - 29596449989864 \beta_{3} + 23710527924 \beta_{4} + 27389911440 \beta_{5}) q^{28}\) \(+(\)\(21\!\cdots\!94\)\( - 6137808882409723068 \beta_{1} - 3112687079867392 \beta_{2} - 9630610193440 \beta_{3} - 242403578228 \beta_{4} + 243089857556 \beta_{5}) q^{29}\) \(+(\)\(16\!\cdots\!25\)\( + 1047402191162109375 \beta_{1} + 4661865234375000 \beta_{2} + 71411132812500 \beta_{3} + 152587890625 \beta_{4} - 305175781250 \beta_{5}) q^{30}\) \(+(\)\(18\!\cdots\!68\)\( - 3388099694918505812 \beta_{1} + 4025742456653054 \beta_{2} - 123228527552992 \beta_{3} + 387650863794 \beta_{4} + 182108055462 \beta_{5}) q^{31}\) \(+(\)\(44\!\cdots\!40\)\( + 23674599322570287136 \beta_{1} + 62140876314710376 \beta_{2} + 867369316347432 \beta_{3} + 462454977688 \beta_{4} - 1232999492320 \beta_{5}) q^{32}\) \(+(\)\(58\!\cdots\!64\)\( + 54300701840174618364 \beta_{1} + 43618058243442636 \beta_{2} - 171779073856272 \beta_{3} - 1478052244448 \beta_{4} + 437837130520 \beta_{5}) q^{33}\) \(+(\)\(14\!\cdots\!20\)\( + 62028152501636042878 \beta_{1} - 85162746659265952 \beta_{2} - 1825311948550576 \beta_{3} - 953122000764 \beta_{4} + 3519611975928 \beta_{5}) q^{34}\) \(+(\)\(48\!\cdots\!50\)\( - 14886344604492187500 \beta_{1} - 32981414794921875 \beta_{2} - 505371093750000 \beta_{3} + 2593994140625 \beta_{4} - 762939453125 \beta_{5}) q^{35}\) \(+(\)\(28\!\cdots\!12\)\( + \)\(13\!\cdots\!43\)\( \beta_{1} - 191907891499071027 \beta_{2} + 349244529375309 \beta_{3} - 2673384863504 \beta_{4} + 1059460771008 \beta_{5}) q^{36}\) \(+(\)\(46\!\cdots\!78\)\( + \)\(17\!\cdots\!88\)\( \beta_{1} - 64934905550931848 \beta_{2} - 1366726671422432 \beta_{3} + 5897342122712 \beta_{4} - 18795609634280 \beta_{5}) q^{37}\) \(+(\)\(13\!\cdots\!56\)\( + \)\(30\!\cdots\!28\)\( \beta_{1} - 448701969129176224 \beta_{2} + 11481687976142768 \beta_{3} - 4355893191188 \beta_{4} + 11060468596520 \beta_{5}) q^{38}\) \(+(\)\(96\!\cdots\!52\)\( - \)\(92\!\cdots\!82\)\( \beta_{1} - 157334254410157330 \beta_{2} - 5493547831068288 \beta_{3} + 13641937276912 \beta_{4} + 18617989176976 \beta_{5}) q^{39}\) \(+(\)\(78\!\cdots\!50\)\( + \)\(47\!\cdots\!00\)\( \beta_{1} + 587843322753906250 \beta_{2} + 12767639160156250 \beta_{3} - 20446777343750 \beta_{4} - 6103515625000 \beta_{5}) q^{40}\) \(+(\)\(74\!\cdots\!58\)\( - \)\(21\!\cdots\!32\)\( \beta_{1} + 3245263265241030644 \beta_{2} - 17944925624479312 \beta_{3} - 54662526732316 \beta_{4} - 13379833591868 \beta_{5}) q^{41}\) \(+(\)\(24\!\cdots\!48\)\( - \)\(46\!\cdots\!64\)\( \beta_{1} - 618975393651013440 \beta_{2} - 28094202897223392 \beta_{3} + 74946257277672 \beta_{4} + 60762377359920 \beta_{5}) q^{42}\) \(+(\)\(37\!\cdots\!82\)\( - \)\(23\!\cdots\!49\)\( \beta_{1} + 1230021894277821697 \beta_{2} - 23796012831990560 \beta_{3} + 89622397359410 \beta_{4} - 26036900579450 \beta_{5}) q^{43}\) \(+(\)\(11\!\cdots\!80\)\( + \)\(44\!\cdots\!88\)\( \beta_{1} + 5045952962716245548 \beta_{2} + 90405825913235564 \beta_{3} - 132704516523624 \beta_{4} - 265612825035552 \beta_{5}) q^{44}\) \(+(\)\(34\!\cdots\!25\)\( + \)\(66\!\cdots\!00\)\( \beta_{1} - 926995239257812500 \beta_{2} + 29494628906250000 \beta_{3} - 4272460937500 \beta_{4} + 115356445312500 \beta_{5}) q^{45}\) \(+(\)\(58\!\cdots\!53\)\( - \)\(57\!\cdots\!17\)\( \beta_{1} - 29212393650096783176 \beta_{2} - 186615301401726332 \beta_{3} + 365524022392109 \beta_{4} + 220470269423782 \beta_{5}) q^{46}\) \(+(\)\(63\!\cdots\!62\)\( + \)\(17\!\cdots\!90\)\( \beta_{1} - 6154437256507651999 \beta_{2} - 136200415632987280 \beta_{3} - 867892046326545 \beta_{4} + 579907554579525 \beta_{5}) q^{47}\) \(+(\)\(40\!\cdots\!28\)\( + \)\(56\!\cdots\!24\)\( \beta_{1} - 23500066991230398512 \beta_{2} + 832522204704740688 \beta_{3} - 21895879190608 \beta_{4} - 698410891602880 \beta_{5}) q^{48}\) \(+(-\)\(11\!\cdots\!39\)\( + \)\(23\!\cdots\!52\)\( \beta_{1} + 43372989519856592836 \beta_{2} - 118196428042094288 \beta_{3} + 1143985608347436 \beta_{4} - 114523758942372 \beta_{5}) q^{49}\) \(+(\)\(57\!\cdots\!50\)\( + \)\(23\!\cdots\!25\)\( \beta_{1}) q^{50}\) \(+(-\)\(91\!\cdots\!00\)\( - \)\(28\!\cdots\!86\)\( \beta_{1} + \)\(11\!\cdots\!70\)\( \beta_{2} - 1323031152136627584 \beta_{3} + 74390991001456 \beta_{4} - 1935010016550512 \beta_{5}) q^{51}\) \(+(-\)\(15\!\cdots\!36\)\( - \)\(11\!\cdots\!98\)\( \beta_{1} - 40103374382504825906 \beta_{2} - 1775376851823262130 \beta_{3} - 232265487474320 \beta_{4} + 1547599231006400 \beta_{5}) q^{52}\) \(+(\)\(78\!\cdots\!62\)\( + \)\(41\!\cdots\!32\)\( \beta_{1} + 67069168248378721308 \beta_{2} + 3496079966838332784 \beta_{3} - 1637763997437144 \beta_{4} + 2435648622453360 \beta_{5}) q^{53}\) \(+(-\)\(30\!\cdots\!22\)\( - \)\(10\!\cdots\!46\)\( \beta_{1} + 50443714279879755120 \beta_{2} - 1373410896625130424 \beta_{3} - 1849398390364534 \beta_{4} + 794088938732268 \beta_{5}) q^{54}\) \(+(\)\(38\!\cdots\!00\)\( - \)\(24\!\cdots\!75\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2} + 2199050292968750000 \beta_{3} + 2426605224609375 \beta_{4} + 193634033203125 \beta_{5}) q^{55}\) \(+(-\)\(65\!\cdots\!04\)\( - \)\(42\!\cdots\!12\)\( \beta_{1} - \)\(66\!\cdots\!88\)\( \beta_{2} - 2479956419208447600 \beta_{3} - 1840379833947232 \beta_{4} + 3457921386360064 \beta_{5}) q^{56}\) \(+(-\)\(40\!\cdots\!36\)\( - \)\(11\!\cdots\!36\)\( \beta_{1} + \)\(30\!\cdots\!56\)\( \beta_{2} + 7588398299320953360 \beta_{3} + 11899491891624040 \beta_{4} - 24613480734732800 \beta_{5}) q^{57}\) \(+(-\)\(73\!\cdots\!40\)\( + \)\(23\!\cdots\!22\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2} - 3574520024274182112 \beta_{3} + 1839281336554792 \beta_{4} + 6080938064012720 \beta_{5}) q^{58}\) \(+(-\)\(29\!\cdots\!28\)\( - \)\(12\!\cdots\!19\)\( \beta_{1} + \)\(14\!\cdots\!82\)\( \beta_{2} - 5609198583504272688 \beta_{3} - 12275424379581295 \beta_{4} + 23068122370727515 \beta_{5}) q^{59}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(50\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2} + 5408464965820312500 \beta_{3} - 11528930664062500 \beta_{4} - 14855957031250000 \beta_{5}) q^{60}\) \(+(-\)\(47\!\cdots\!18\)\( + \)\(64\!\cdots\!00\)\( \beta_{1} - 58427209505596046000 \beta_{2} - 9337839871530008000 \beta_{3} - 19689438454463200 \beta_{4} + 27236975122556800 \beta_{5}) q^{61}\) \(+(\)\(57\!\cdots\!54\)\( + \)\(10\!\cdots\!42\)\( \beta_{1} - \)\(73\!\cdots\!16\)\( \beta_{2} - 22404603041652321912 \beta_{3} + 30772461553341242 \beta_{4} + 9227620660724620 \beta_{5}) q^{62}\) \(+(-\)\(93\!\cdots\!26\)\( + \)\(34\!\cdots\!40\)\( \beta_{1} - \)\(14\!\cdots\!83\)\( \beta_{2} - 3109775174438892912 \beta_{3} + 21542654788301517 \beta_{4} - 32189211391833105 \beta_{5}) q^{63}\) \(+(\)\(32\!\cdots\!72\)\( + \)\(43\!\cdots\!00\)\( \beta_{1} + \)\(89\!\cdots\!24\)\( \beta_{2} + 47755489026283332176 \beta_{3} + 5038109519575472 \beta_{4} - 45309613567146944 \beta_{5}) q^{64}\) \(+(-\)\(35\!\cdots\!50\)\( - \)\(64\!\cdots\!00\)\( \beta_{1} + \)\(20\!\cdots\!00\)\( \beta_{2} - 23588015136718750000 \beta_{3} + 12459716796875000 \beta_{4} + 27307128906250000 \beta_{5}) q^{65}\) \(+(\)\(84\!\cdots\!28\)\( + \)\(60\!\cdots\!12\)\( \beta_{1} - \)\(40\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!08\)\( \beta_{3} + 59892633435960908 \beta_{4} - 105372895828350616 \beta_{5}) q^{66}\) \(+(\)\(55\!\cdots\!58\)\( - \)\(41\!\cdots\!71\)\( \beta_{1} - \)\(70\!\cdots\!77\)\( \beta_{2} + 43285766613366926112 \beta_{3} - 231815093720766642 \beta_{4} + 180299083915239930 \beta_{5}) q^{67}\) \(+(\)\(64\!\cdots\!80\)\( - \)\(83\!\cdots\!30\)\( \beta_{1} - \)\(19\!\cdots\!30\)\( \beta_{2} - \)\(12\!\cdots\!94\)\( \beta_{3} - 76916991962596496 \beta_{4} + 82472691965378240 \beta_{5}) q^{68}\) \(+(\)\(52\!\cdots\!96\)\( - \)\(17\!\cdots\!40\)\( \beta_{1} + \)\(13\!\cdots\!48\)\( \beta_{2} - \)\(11\!\cdots\!08\)\( \beta_{3} + 230176942325623884 \beta_{4} - 159634180030227468 \beta_{5}) q^{69}\) \(+(-\)\(18\!\cdots\!25\)\( - \)\(27\!\cdots\!75\)\( \beta_{1} + \)\(53\!\cdots\!00\)\( \beta_{2} - \)\(12\!\cdots\!00\)\( \beta_{3} + 47533416748046875 \beta_{4} + 128565979003906250 \beta_{5}) q^{70}\) \(+(\)\(14\!\cdots\!12\)\( - \)\(12\!\cdots\!50\)\( \beta_{1} - \)\(26\!\cdots\!50\)\( \beta_{2} - 40820877937029072000 \beta_{3} + 207364024339354400 \beta_{4} - 90375567776933600 \beta_{5}) q^{71}\) \(+(\)\(14\!\cdots\!42\)\( + \)\(31\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!62\)\( \beta_{2} + \)\(16\!\cdots\!46\)\( \beta_{3} - 146088562735132686 \beta_{4} + 161652988215842040 \beta_{5}) q^{72}\) \(+(-\)\(28\!\cdots\!22\)\( - \)\(25\!\cdots\!36\)\( \beta_{1} - \)\(24\!\cdots\!72\)\( \beta_{2} + 88538608537798936848 \beta_{3} - 80987079575138968 \beta_{4} - 537894925079953280 \beta_{5}) q^{73}\) \(+(\)\(34\!\cdots\!40\)\( + \)\(42\!\cdots\!38\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} + \)\(21\!\cdots\!28\)\( \beta_{3} - 329207764768098616 \beta_{4} - 71601375399252368 \beta_{5}) q^{74}\) \(+(\)\(10\!\cdots\!50\)\( + \)\(17\!\cdots\!25\)\( \beta_{1} + \)\(23\!\cdots\!25\)\( \beta_{2}) q^{75}\) \(+(\)\(49\!\cdots\!72\)\( + \)\(12\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!44\)\( \beta_{2} + \)\(81\!\cdots\!40\)\( \beta_{3} - 515337314594221744 \beta_{4} + 749865497621658688 \beta_{5}) q^{76}\) \(+(-\)\(19\!\cdots\!00\)\( - \)\(40\!\cdots\!36\)\( \beta_{1} - \)\(63\!\cdots\!16\)\( \beta_{2} - \)\(36\!\cdots\!68\)\( \beta_{3} + 1140507339514434488 \beta_{4} - 408073919617488320 \beta_{5}) q^{77}\) \(+(-\)\(95\!\cdots\!90\)\( + \)\(44\!\cdots\!62\)\( \beta_{1} - \)\(10\!\cdots\!48\)\( \beta_{2} - \)\(19\!\cdots\!04\)\( \beta_{3} + 1281263495726796814 \beta_{4} + 1218660320970579940 \beta_{5}) q^{78}\) \(+(-\)\(36\!\cdots\!40\)\( - \)\(37\!\cdots\!60\)\( \beta_{1} - \)\(42\!\cdots\!08\)\( \beta_{2} + \)\(72\!\cdots\!68\)\( \beta_{3} - 805205716297778864 \beta_{4} + 1307751931934890928 \beta_{5}) q^{79}\) \(+(\)\(16\!\cdots\!00\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} + \)\(66\!\cdots\!00\)\( \beta_{2} + \)\(65\!\cdots\!00\)\( \beta_{3} - 1571240844726562500 \beta_{4} - 2252995605468750000 \beta_{5}) q^{80}\) \(+(-\)\(22\!\cdots\!43\)\( + \)\(51\!\cdots\!28\)\( \beta_{1} - \)\(87\!\cdots\!04\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} - 2405565674579150020 \beta_{4} - 30198118577362260 \beta_{5}) q^{81}\) \(+(-\)\(26\!\cdots\!76\)\( - \)\(74\!\cdots\!78\)\( \beta_{1} + \)\(16\!\cdots\!24\)\( \beta_{2} + \)\(51\!\cdots\!68\)\( \beta_{3} + 4585761217977876012 \beta_{4} - 7331031212250440280 \beta_{5}) q^{82}\) \(+(-\)\(78\!\cdots\!34\)\( - \)\(15\!\cdots\!35\)\( \beta_{1} + \)\(62\!\cdots\!13\)\( \beta_{2} + \)\(39\!\cdots\!00\)\( \beta_{3} - 2086873799279959500 \beta_{4} + 2166286256070697500 \beta_{5}) q^{83}\) \(+(-\)\(39\!\cdots\!32\)\( - \)\(21\!\cdots\!08\)\( \beta_{1} - \)\(24\!\cdots\!32\)\( \beta_{2} - \)\(44\!\cdots\!60\)\( \beta_{3} - 911602853655848608 \beta_{4} + 6366546648555065216 \beta_{5}) q^{84}\) \(+(\)\(34\!\cdots\!50\)\( + \)\(10\!\cdots\!00\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3} + 5758375244140625000 \beta_{4} + 3696850585937500000 \beta_{5}) q^{85}\) \(+(-\)\(21\!\cdots\!71\)\( + \)\(18\!\cdots\!51\)\( \beta_{1} + \)\(22\!\cdots\!56\)\( \beta_{2} - \)\(53\!\cdots\!32\)\( \beta_{3} + 4876108330794071357 \beta_{4} - 49702121502984314 \beta_{5}) q^{86}\) \(+(-\)\(29\!\cdots\!56\)\( - \)\(10\!\cdots\!14\)\( \beta_{1} + \)\(81\!\cdots\!98\)\( \beta_{2} + \)\(20\!\cdots\!68\)\( \beta_{3} - 12679467875975812288 \beta_{4} + 9872598734998961120 \beta_{5}) q^{87}\) \(+(\)\(82\!\cdots\!72\)\( + \)\(88\!\cdots\!84\)\( \beta_{1} + \)\(93\!\cdots\!80\)\( \beta_{2} + \)\(16\!\cdots\!68\)\( \beta_{3} - 7981295213666861288 \beta_{4} - 8030832015281795680 \beta_{5}) q^{88}\) \(+(-\)\(56\!\cdots\!98\)\( - \)\(91\!\cdots\!44\)\( \beta_{1} - \)\(33\!\cdots\!56\)\( \beta_{2} - \)\(15\!\cdots\!00\)\( \beta_{3} + 7021873421274053016 \beta_{4} - 24228933901779274632 \beta_{5}) q^{89}\) \(+(\)\(17\!\cdots\!50\)\( + \)\(53\!\cdots\!25\)\( \beta_{1} - \)\(55\!\cdots\!00\)\( \beta_{2} - \)\(38\!\cdots\!00\)\( \beta_{3} - 1740834350585937500 \beta_{4} + 3016278076171875000 \beta_{5}) q^{90}\) \(+(\)\(32\!\cdots\!68\)\( + \)\(13\!\cdots\!88\)\( \beta_{1} + \)\(56\!\cdots\!82\)\( \beta_{2} - \)\(73\!\cdots\!20\)\( \beta_{3} + 19731114985660701678 \beta_{4} - 13984987258151544006 \beta_{5}) q^{91}\) \(+(-\)\(69\!\cdots\!56\)\( - \)\(10\!\cdots\!56\)\( \beta_{1} - \)\(22\!\cdots\!44\)\( \beta_{2} - \)\(28\!\cdots\!52\)\( \beta_{3} - 17131266635838573268 \beta_{4} + 33821537288514599920 \beta_{5}) q^{92}\) \(+(\)\(35\!\cdots\!20\)\( - \)\(43\!\cdots\!16\)\( \beta_{1} + \)\(20\!\cdots\!84\)\( \beta_{2} - \)\(15\!\cdots\!56\)\( \beta_{3} + 19012349798843418296 \beta_{4} + 4943549919321157160 \beta_{5}) q^{93}\) \(+(\)\(24\!\cdots\!27\)\( + \)\(24\!\cdots\!69\)\( \beta_{1} - \)\(52\!\cdots\!20\)\( \beta_{2} + \)\(30\!\cdots\!76\)\( \beta_{3} + 17191986391800001331 \beta_{4} + 9663523947126672538 \beta_{5}) q^{94}\) \(+(\)\(43\!\cdots\!00\)\( + \)\(15\!\cdots\!75\)\( \beta_{1} - \)\(10\!\cdots\!50\)\( \beta_{2} - \)\(53\!\cdots\!00\)\( \beta_{3} - 21862971343994140625 \beta_{4} - 2839261932373046875 \beta_{5}) q^{95}\) \(+(\)\(52\!\cdots\!40\)\( + \)\(75\!\cdots\!24\)\( \beta_{1} + \)\(20\!\cdots\!24\)\( \beta_{2} + \)\(26\!\cdots\!52\)\( \beta_{3} - 66536165423703186592 \beta_{4} + 11533125890540845184 \beta_{5}) q^{96}\) \(+(\)\(42\!\cdots\!26\)\( - \)\(26\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!68\)\( \beta_{2} + \)\(38\!\cdots\!84\)\( \beta_{3} + 3898782859774391856 \beta_{4} + 11150994043844352360 \beta_{5}) q^{97}\) \(+(\)\(11\!\cdots\!58\)\( - \)\(17\!\cdots\!23\)\( \beta_{1} + \)\(33\!\cdots\!96\)\( \beta_{2} - \)\(21\!\cdots\!88\)\( \beta_{3} + 70950961974920944108 \beta_{4} - 58534955193326850520 \beta_{5}) q^{98}\) \(+(\)\(27\!\cdots\!72\)\( - \)\(23\!\cdots\!67\)\( \beta_{1} + \)\(23\!\cdots\!12\)\( \beta_{2} - \)\(95\!\cdots\!20\)\( \beta_{3} - 57462850055393817677 \beta_{4} - 62703625440425722671 \beta_{5}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut +\mathstrut 147350q^{2} \) \(\mathstrut +\mathstrut 26513900q^{3} \) \(\mathstrut +\mathstrut 29520645652q^{4} \) \(\mathstrut +\mathstrut 915527343750q^{5} \) \(\mathstrut +\mathstrut 6615759249352q^{6} \) \(\mathstrut +\mathstrut 19113832847500q^{7} \) \(\mathstrut +\mathstrut 3069820115785800q^{8} \) \(\mathstrut +\mathstrut 13602102583345438q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut 147350q^{2} \) \(\mathstrut +\mathstrut 26513900q^{3} \) \(\mathstrut +\mathstrut 29520645652q^{4} \) \(\mathstrut +\mathstrut 915527343750q^{5} \) \(\mathstrut +\mathstrut 6615759249352q^{6} \) \(\mathstrut +\mathstrut 19113832847500q^{7} \) \(\mathstrut +\mathstrut 3069820115785800q^{8} \) \(\mathstrut +\mathstrut 13602102583345438q^{9} \) \(\mathstrut +\mathstrut 22483825683593750q^{10} \) \(\mathstrut +\mathstrut 150760896555890192q^{11} \) \(\mathstrut +\mathstrut 466351581756413200q^{12} \) \(\mathstrut -\mathstrut 1403095636752804700q^{13} \) \(\mathstrut -\mathstrut 7086110008989891456q^{14} \) \(\mathstrut +\mathstrut 4045700073242187500q^{15} \) \(\mathstrut +\mathstrut 64511520005858668816q^{16} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!50\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!00\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!28\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!00\)\(q^{23} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!50\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!88\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!00\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!00\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!00\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!52\)\(q^{31} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{32} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(88\!\cdots\!24\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!96\)\(q^{36} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!00\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(83\!\cdots\!00\)\(q^{38} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!56\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!32\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!00\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{43} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!64\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!50\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!72\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!00\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!00\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(71\!\cdots\!58\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!50\)\(q^{50} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!88\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!00\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!00\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!00\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!00\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!00\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(71\!\cdots\!00\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(28\!\cdots\!08\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!00\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(55\!\cdots\!00\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!72\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(50\!\cdots\!64\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!00\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!36\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(84\!\cdots\!72\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(87\!\cdots\!00\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!84\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(57\!\cdots\!00\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!74\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(47\!\cdots\!00\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!76\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!08\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!00\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(49\!\cdots\!00\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(33\!\cdots\!00\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!50\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!32\)\(q^{91} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!00\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!64\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!72\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!00\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(69\!\cdots\!50\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!16\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut -\mathstrut \) \(9680197848\) \(x^{4}\mathstrut +\mathstrut \) \(8052423720422\) \(x^{3}\mathstrut +\mathstrut \) \(24239866893261762265\) \(x^{2}\mathstrut -\mathstrut \) \(69081627028404093368325\) \(x\mathstrut -\mathstrut \) \(10572274201725134136583265250\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(64759\) \(\nu^{5}\mathstrut +\mathstrut \) \(178753080\) \(\nu^{4}\mathstrut -\mathstrut \) \(443366379644016\) \(\nu^{3}\mathstrut -\mathstrut \) \(2219345891051097862\) \(\nu^{2}\mathstrut +\mathstrut \) \(497177359501705207052073\) \(\nu\mathstrut +\mathstrut \) \(3158210254432945978141147422\)\()/\)\(11\!\cdots\!72\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(64759\) \(\nu^{5}\mathstrut -\mathstrut \) \(178753080\) \(\nu^{4}\mathstrut +\mathstrut \) \(443366379644016\) \(\nu^{3}\mathstrut +\mathstrut \) \(467687168437307494150\) \(\nu^{2}\mathstrut +\mathstrut \) \(83028281302203390920919\) \(\nu\mathstrut -\mathstrut \) \(1505098511780245718200582504734\)\()/\)\(11\!\cdots\!72\)
\(\beta_{4}\)\(=\)\((\)\(693353137\) \(\nu^{5}\mathstrut +\mathstrut \) \(22711573265160\) \(\nu^{4}\mathstrut -\mathstrut \) \(7564015938151063056\) \(\nu^{3}\mathstrut -\mathstrut \) \(142065885137099881758346\) \(\nu^{2}\mathstrut +\mathstrut \) \(18384255498505988786988504111\) \(\nu\mathstrut +\mathstrut \) \(90683077682865758752943810163378\)\()/\)\(50910543090996793344\)
\(\beta_{5}\)\(=\)\((\)\(2773802021\) \(\nu^{5}\mathstrut -\mathstrut \) \(549722510060376\) \(\nu^{4}\mathstrut -\mathstrut \) \(24950793531285037392\) \(\nu^{3}\mathstrut +\mathstrut \) \(3482773264826083164386926\) \(\nu^{2}\mathstrut +\mathstrut \) \(52334398719922215930068580411\) \(\nu\mathstrut -\mathstrut \) \(3030648596667910248295146115485222\)\()/\)\(40\!\cdots\!52\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(2493\) \(\beta_{1}\mathstrut +\mathstrut \) \(12906931296\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(20\) \(\beta_{5}\mathstrut -\mathstrut \) \(67\) \(\beta_{4}\mathstrut +\mathstrut \) \(5000\) \(\beta_{3}\mathstrut +\mathstrut \) \(1889408\) \(\beta_{2}\mathstrut +\mathstrut \) \(9344083493\) \(\beta_{1}\mathstrut -\mathstrut \) \(16088601109849\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(2708988\) \(\beta_{5}\mathstrut +\mathstrut \) \(716451\) \(\beta_{4}\mathstrut +\mathstrut \) \(6365565999\) \(\beta_{3}\mathstrut +\mathstrut \) \(21984654807\) \(\beta_{2}\mathstrut -\mathstrut \) \(4547030871598\) \(\beta_{1}\mathstrut +\mathstrut \) \(60301975528888621041\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(64725270960\) \(\beta_{5}\mathstrut -\mathstrut \) \(230343390564\) \(\beta_{4}\mathstrut +\mathstrut \) \(25466060786969\) \(\beta_{3}\mathstrut +\mathstrut \) \(10048466840966201\) \(\beta_{2}\mathstrut +\mathstrut \) \(24272901966106437312\) \(\beta_{1}\mathstrut -\mathstrut \) \(14671356594135807142800\)\()/2\)

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
−75479.4
−60371.0
−21408.2
25925.6
54181.8
77152.3
−126401. −8.63211e7 7.38725e9 1.52588e11 1.09111e13 7.78650e13 1.52021e14 1.89228e15 −1.92872e16
1.2 −96184.0 1.07272e8 6.61431e8 1.52588e11 −1.03179e13 −8.61302e13 7.62595e14 5.94832e15 −1.46765e16
1.3 −18258.4 −3.67420e7 −8.25656e9 1.52588e11 6.70852e11 −1.51681e13 3.07591e14 −4.20908e15 −2.78602e15
1.4 76409.2 7.43191e7 −2.75157e9 1.52588e11 5.67866e12 1.36987e14 −8.66595e14 −3.57381e13 1.16591e16
1.5 132922. −1.17526e8 9.07819e9 1.52588e11 −1.56217e13 −1.34853e13 6.49000e13 8.25322e15 2.02822e16
1.6 178863. 8.55112e7 2.34019e10 1.52588e11 1.52948e13 −8.09549e13 2.64931e15 1.75311e15 2.72923e16
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 \(T_{2}^{6} \) \(\mathstrut -\mathstrut 147350 T_{2}^{5} \) \(\mathstrut -\mathstrut 29674115352 T_{2}^{4} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!00\)\( T_{2}^{3} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!68\)\( T_{2}^{2} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!00\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!04\)\( \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(5))\).