Properties

Label 5.34.b.a
Level 5
Weight 34
Character orbit 5.b
Analytic conductor 34.491
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 34 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(34.4914144405\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{83}\cdot 3^{26}\cdot 5^{53}\cdot 7^{4}\cdot 11^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -52 \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -4553207930 + \beta_{2} ) q^{4} \) \( + ( -14510510019 + 105885 \beta_{1} + 3 \beta_{2} - 70 \beta_{3} - \beta_{4} ) q^{5} \) \( + ( 687611084267 - \beta_{1} - 176 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{6} \) \( + ( 2 + 49478166 \beta_{1} - 4 \beta_{2} - 168935 \beta_{3} + 20 \beta_{4} + \beta_{8} ) q^{7} \) \( + ( 46 - 3608565601 \beta_{1} - 90 \beta_{2} + 1699774 \beta_{3} + 473 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{10} ) q^{8} \) \( + ( -1720947285919957 + 607 \beta_{1} + 16412 \beta_{2} - 1361 \beta_{3} - 3119 \beta_{4} - 295 \beta_{5} - \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+\beta_{1} q^{2}\) \(+(-52 \beta_{1} + \beta_{3}) q^{3}\) \(+(-4553207930 + \beta_{2}) q^{4}\) \(+(-14510510019 + 105885 \beta_{1} + 3 \beta_{2} - 70 \beta_{3} - \beta_{4}) q^{5}\) \(+(687611084267 - \beta_{1} - 176 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5}) q^{6}\) \(+(2 + 49478166 \beta_{1} - 4 \beta_{2} - 168935 \beta_{3} + 20 \beta_{4} + \beta_{8}) q^{7}\) \(+(46 - 3608565601 \beta_{1} - 90 \beta_{2} + 1699774 \beta_{3} + 473 \beta_{4} - 2 \beta_{5} - \beta_{8} + \beta_{10}) q^{8}\) \(+(-1720947285919957 + 607 \beta_{1} + 16412 \beta_{2} - 1361 \beta_{3} - 3119 \beta_{4} - 295 \beta_{5} - \beta_{9}) q^{9}\) \(+(-1391949613974573 - 35959960852 \beta_{1} + 657797 \beta_{2} + 60765556 \beta_{3} + 1010 \beta_{4} - 136 \beta_{5} - 2 \beta_{7} + 44 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11}) q^{10}\) \(+(13748411922418897 + 20168 \beta_{1} - 2027449 \beta_{2} - 48906 \beta_{3} - 120984 \beta_{4} - 1985 \beta_{5} + \beta_{6} - 6 \beta_{7} + 24 \beta_{8} - 4 \beta_{9} + 12 \beta_{10}) q^{11}\) \(+(-13010 + 1573497516868 \beta_{1} + 25257 \beta_{2} - 4644109496 \beta_{3} - 134588 \beta_{4} + 761 \beta_{5} + 17 \beta_{7} + 1774 \beta_{8} + 102 \beta_{10} - 3 \beta_{11} - \beta_{14}) q^{12}\) \(+(-64580 - 106980514504 \beta_{1} + 126553 \beta_{2} + 8191506863 \beta_{3} - 662160 \beta_{4} + 2616 \beta_{5} + 33 \beta_{7} - 2273 \beta_{8} - 470 \beta_{10} + 11 \beta_{11} + \beta_{14} + \beta_{15}) q^{13}\) \(+(-651002723924728523 - 427050 \beta_{1} + 156158341 \beta_{2} + 1230269 \beta_{3} + 3477849 \beta_{4} - 371027 \beta_{5} - 17 \beta_{6} + 289 \beta_{7} - 1187 \beta_{8} - 138 \beta_{9} - 561 \beta_{10} - 70 \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14}) q^{14}\) \(+(577025968672594462 + 35355192165256 \beta_{1} - 173550593 \beta_{2} - 59198586675 \beta_{3} + 469991 \beta_{4} + 600207 \beta_{5} - 36 \beta_{6} + 801 \beta_{7} + 33042 \beta_{8} + 823 \beta_{9} - 302 \beta_{10} + 23 \beta_{11} - \beta_{12} - 19 \beta_{14} + 7 \beta_{15}) q^{15}\) \(+(8323216025193180534 + 2922188 \beta_{1} - 1638364767 \beta_{2} - 7294339 \beta_{3} - 18509684 \beta_{4} + 153948 \beta_{5} + 349 \beta_{6} - 1265 \beta_{7} + 4876 \beta_{8} + 1101 \beta_{9} + 2628 \beta_{10} - 420 \beta_{11} - \beta_{12} + 4 \beta_{13} + 4 \beta_{14}) q^{16}\) \(+(9356663 + 404343167718 \beta_{1} - 18326809 \beta_{2} - 324985511449 \beta_{3} + 95853479 \beta_{4} - 383910 \beta_{5} + 80 \beta_{6} - 7414 \beta_{7} + 443417 \beta_{8} + 100 \beta_{9} - 17360 \beta_{10} + 2393 \beta_{11} + 20 \beta_{13} - 65 \beta_{14} - 49 \beta_{15}) q^{17}\) \(+(39068194 - 1839702686565056 \beta_{1} - 76280224 \beta_{2} + 3949024309216 \beta_{3} + 401711005 \beta_{4} - 1845032 \beta_{5} + 260 \beta_{6} - 33600 \beta_{7} - 886727 \beta_{8} + 325 \beta_{9} - 39335 \beta_{10} + 11468 \beta_{11} + 65 \beta_{13} + 701 \beta_{14} - 40 \beta_{15}) q^{18}\) \(+(-\)\(18\!\cdots\!05\)\( - 70370652 \beta_{1} - 32447176765 \beta_{2} + 178691420 \beta_{3} + 460052296 \beta_{4} - 10139351 \beta_{5} - 1367 \beta_{6} + 34848 \beta_{7} - 145516 \beta_{8} - 73302 \beta_{9} - 66448 \beta_{10} - 14000 \beta_{11} - 66 \beta_{12} + 124 \beta_{13} + 124 \beta_{14}) q^{19}\) \(+(\)\(34\!\cdots\!62\)\( - 5512027296421550 \beta_{1} - 10030623117 \beta_{2} + 2244100715549 \beta_{3} + 4480809670 \beta_{4} + 51659227 \beta_{5} - 7463 \beta_{6} + 20430 \beta_{7} + 4878952 \beta_{8} + 330945 \beta_{9} + 779220 \beta_{10} + 6123 \beta_{11} + 67 \beta_{12} + 500 \beta_{13} + 273 \beta_{14} - 344 \beta_{15}) q^{20}\) \(+(\)\(12\!\cdots\!55\)\( - 566190188 \beta_{1} - 193682000646 \beta_{2} + 1334959050 \beta_{3} + 3217509541 \beta_{4} + 136319846 \beta_{5} + 20576 \beta_{6} + 79735 \beta_{7} - 359748 \beta_{8} + 549064 \beta_{9} - 137614 \beta_{10} - 92960 \beta_{11} + 68 \beta_{12} + 968 \beta_{13} + 968 \beta_{14}) q^{21}\) \(+(-110745764 + 29442714125744706 \beta_{1} + 215653084 \beta_{2} + 28601013970352 \beta_{3} - 1160093378 \beta_{4} + 6105636 \beta_{5} + 6120 \beta_{6} - 392436 \beta_{7} + 39541286 \beta_{8} + 7650 \beta_{9} - 14795322 \beta_{10} + 268188 \beta_{11} + 1530 \beta_{13} + 5790 \beta_{14} + 2856 \beta_{15}) q^{22}\) \(+(499450336 - 19185972451781698 \beta_{1} - 970416830 \beta_{2} - 61346351487833 \beta_{3} + 5174981110 \beta_{4} - 28224288 \beta_{5} + 8080 \beta_{6} - 17496 \beta_{7} - 86191187 \beta_{8} + 10100 \beta_{9} + 13052984 \beta_{10} + 222226 \beta_{11} + 2020 \beta_{13} - 27814 \beta_{14} + 586 \beta_{15}) q^{23}\) \(+(-\)\(14\!\cdots\!22\)\( + 5468144812 \beta_{1} + 249590287723 \beta_{2} - 11578210921 \beta_{3} - 24887663380 \beta_{4} - 4105077820 \beta_{5} + 63415 \beta_{6} + 473645 \beta_{7} - 1995732 \beta_{8} - 8926529 \beta_{9} - 892316 \beta_{10} - 229180 \beta_{11} + 2077 \beta_{12} + 2932 \beta_{13} + 2932 \beta_{14}) q^{24}\) \(+(\)\(16\!\cdots\!92\)\( - 304905739769977255 \beta_{1} - 430890719823 \beta_{2} + 210268828924432 \beta_{3} + 6508655204 \beta_{4} - 539786759 \beta_{5} + 328316 \beta_{6} - 1482350 \beta_{7} + 392204041 \beta_{8} + 6428065 \beta_{9} + 83391040 \beta_{10} + 167009 \beta_{11} - 2144 \beta_{12} - 5500 \beta_{13} + 29639 \beta_{14} + 7383 \beta_{15}) q^{25}\) \(+(\)\(14\!\cdots\!52\)\( - 28906904483 \beta_{1} + 1740604261594 \beta_{2} + 64038656116 \beta_{3} + 144905290127 \beta_{4} + 15700139840 \beta_{5} - 951840 \beta_{6} + 4338148 \beta_{7} - 16593225 \beta_{8} + 5024297 \beta_{9} - 9083395 \beta_{10} + 1729210 \beta_{11} - 2212 \beta_{12} - 18277 \beta_{13} - 18277 \beta_{14}) q^{26}\) \(+(-35231182860 + 1981249080053253468 \beta_{1} + 68773855704 \beta_{2} - 2223098092447842 \beta_{3} - 362761608360 \beta_{4} + 1681508088 \beta_{5} - 161760 \beta_{6} + 16186848 \beta_{7} + 1722277986 \beta_{8} - 202200 \beta_{9} - 374280000 \beta_{10} - 6989952 \beta_{11} - 40440 \beta_{13} - 111480 \beta_{14} - 78744 \beta_{15}) q^{27}\) \(+(-48945666422 - 1422414056615623080 \beta_{1} + 96564411847 \beta_{2} + 3332786636083504 \beta_{3} - 497160346944 \beta_{4} + 1315565223 \beta_{5} - 322400 \beta_{6} + 29931271 \beta_{7} - 10190043282 \beta_{8} - 403000 \beta_{9} + 186682974 \beta_{10} - 10421301 \beta_{11} - 80600 \beta_{13} + 532737 \beta_{14} - 1264 \beta_{15}) q^{28}\) \(+(\)\(13\!\cdots\!28\)\( - 47609204178 \beta_{1} + 23190688745212 \beta_{2} + 116201096382 \beta_{3} + 289071546908 \beta_{4} + 3089971354 \beta_{5} - 685768 \beta_{6} + 6615242 \beta_{7} - 20140688 \beta_{8} + 98635162 \beta_{9} - 16628404 \beta_{10} + 14377440 \beta_{11} - 41340 \beta_{12} - 158520 \beta_{13} - 158520 \beta_{14}) q^{29}\) \(+(-\)\(46\!\cdots\!87\)\( + 1898901842167734250 \beta_{1} + 47967277739967 \beta_{2} - 7917338583132549 \beta_{3} - 567703245495 \beta_{4} - 213723138327 \beta_{5} - 5465987 \beta_{6} + 134256895 \beta_{7} + 15226432573 \beta_{8} - 179506720 \beta_{9} + 237259255 \beta_{10} - 9579898 \beta_{11} + 43483 \beta_{12} - 99875 \beta_{13} - 1067823 \beta_{14} - 83256 \beta_{15}) q^{30}\) \(+(\)\(30\!\cdots\!00\)\( - 504157464248 \beta_{1} - 125510057195454 \beta_{2} + 1059637418758 \beta_{3} + 2258152109888 \beta_{4} + 395348367588 \beta_{5} + 15211514 \beta_{6} + 69293246 \beta_{7} - 276203428 \beta_{8} - 279493086 \beta_{9} - 139085964 \beta_{10} + 2239440 \beta_{11} + 45694 \beta_{12} - 9796 \beta_{13} - 9796 \beta_{14}) q^{31}\) \(+(-266812800456 - 10121918290459703260 \beta_{1} + 520034618912 \beta_{2} + 10911959852878152 \beta_{3} - 2749405566404 \beta_{4} + 13634481760 \beta_{5} + 852800 \beta_{6} + 330024312 \beta_{7} + 15413546356 \beta_{8} + 1066000 \beta_{9} + 1910714348 \beta_{10} + 44367800 \beta_{11} + 213200 \beta_{13} + 574264 \beta_{14} + 1359776 \beta_{15}) q^{32}\) \(+(-876554630979 + 14218625307231936450 \beta_{1} + 1717743666115 \beta_{2} - 17717589579550761 \beta_{3} - 8985095666235 \beta_{4} + 35481963934 \beta_{5} + 3368560 \beta_{6} + 515948108 \beta_{7} - 35372703459 \beta_{8} + 4210700 \beta_{9} - 4168159780 \beta_{10} + 105704629 \beta_{11} + 842140 \beta_{13} - 6369573 \beta_{14} - 83189 \beta_{15}) q^{33}\) \(+(-\)\(66\!\cdots\!24\)\( - 4494526954010 \beta_{1} + 140377142009924 \beta_{2} + 11517989459264 \beta_{3} + 29873764672918 \beta_{4} - 870762587476 \beta_{5} - 4752800 \beta_{6} + 1756170632 \beta_{7} - 7115153418 \beta_{8} + 1040865258 \beta_{9} - 3463706414 \beta_{10} - 205810780 \beta_{11} + 581560 \beta_{12} + 2266270 \beta_{13} + 2266270 \beta_{14}) q^{34}\) \(+(\)\(21\!\cdots\!35\)\( + 27985576641036466208 \beta_{1} - 437187748716015 \beta_{2} + 62828659426820269 \beta_{3} - 2222752963394 \beta_{4} - 1971829051957 \beta_{5} + 42181489 \beta_{6} + 1638680278 \beta_{7} - 9123476552 \beta_{8} + 675135124 \beta_{9} - 23619569876 \beta_{10} + 160000822 \beta_{11} - 624976 \beta_{12} + 2397500 \beta_{13} + 16674206 \beta_{14} + 333582 \beta_{15}) q^{35}\) \(+(\)\(94\!\cdots\!74\)\( - 23323059630192 \beta_{1} - 2244788929883847 \beta_{2} + 54442374964770 \beta_{3} + 129964356514320 \beta_{4} + 6782962784808 \beta_{5} - 117053598 \beta_{6} + 5510155062 \beta_{7} - 22176647856 \beta_{8} + 2707523370 \beta_{9} - 10947831888 \beta_{10} - 310453080 \beta_{11} - 670602 \beta_{12} + 2976288 \beta_{13} + 2976288 \beta_{14}) q^{36}\) \(+(-5042155996861 - \)\(20\!\cdots\!50\)\( \beta_{1} + 9870808902962 \beta_{2} + 84157327048134320 \beta_{3} - 51690524872337 \beta_{4} + 213896035806 \beta_{5} + 12702560 \beta_{6} + 5265621485 \beta_{7} - 194457426842 \beta_{8} + 15878200 \beta_{9} + 33565544462 \beta_{10} + 369990634 \beta_{11} + 3175640 \beta_{13} + 9314006 \beta_{14} - 16390218 \beta_{15}) q^{37}\) \(+(-14592469428180 + 63129807158549418850 \beta_{1} + 28475909959988 \beta_{2} + 47451307137852432 \beta_{3} - 150255926278250 \beta_{4} + 708960758764 \beta_{5} - 4548760 \beta_{6} + 11558515396 \beta_{7} + 674946692254 \beta_{8} - 5685950 \beta_{9} - 42895698946 \beta_{10} - 12431932 \beta_{11} - 1137190 \beta_{13} + 48629686 \beta_{14} + 1390040 \beta_{15}) q^{38}\) \(+(-\)\(59\!\cdots\!30\)\( - 29818961011096 \beta_{1} + 6870098524739982 \beta_{2} + 76655776595508 \beta_{3} + 199334302099472 \beta_{4} - 6279019836794 \beta_{5} + 189079378 \beta_{6} + 15274134140 \beta_{7} - 60560729136 \beta_{8} - 21531527824 \beta_{9} - 30837421688 \beta_{10} + 1217162240 \beta_{11} - 6112544 \beta_{12} - 14166464 \beta_{13} - 14166464 \beta_{14}) q^{39}\) \(+(\)\(60\!\cdots\!24\)\( + \)\(10\!\cdots\!65\)\( \beta_{1} - 6408619589681711 \beta_{2} - 393325329000688931 \beta_{3} - 20094136838817 \beta_{4} - 299551861178 \beta_{5} - 58587053 \beta_{6} + 34696893225 \beta_{7} - 1405234245903 \beta_{8} + 9753756955 \beta_{9} + 13955851455 \beta_{10} - 1382432772 \beta_{11} + 6735377 \beta_{12} - 17517500 \beta_{13} - 149602212 \beta_{14} + 4816736 \beta_{15}) q^{40}\) \(+(\)\(60\!\cdots\!04\)\( - 96089283914703 \beta_{1} - 5920145018836776 \beta_{2} + 239041480733797 \beta_{3} + 604793152649021 \beta_{4} - 3324007644501 \beta_{5} + 253437208 \beta_{6} + 36132996274 \beta_{7} - 143091565944 \beta_{8} - 5521332215 \beta_{9} - 73033350052 \beta_{10} + 3287636800 \beta_{11} + 7403768 \beta_{12} - 31431952 \beta_{13} - 31431952 \beta_{14}) q^{41}\) \(+(-141565697297982 + \)\(27\!\cdots\!01\)\( \beta_{1} + 276993212636036 \beta_{2} - 2240807028004130064 \beta_{3} - 1453218764327835 \beta_{4} + 6130181165444 \beta_{5} - 223559020 \beta_{6} + 90836290420 \beta_{7} - 1544087719263 \beta_{8} - 279448775 \beta_{9} - 578135582495 \beta_{10} - 7781127016 \beta_{11} - 55889755 \beta_{13} - 204939387 \beta_{14} + 145158080 \beta_{15}) q^{42}\) \(+(-113877837468512 + 88995403440358941776 \beta_{1} + 222378027435220 \beta_{2} - 1360150274972658795 \beta_{3} - 1171092623239028 \beta_{4} + 5369793895080 \beta_{5} - 192202240 \beta_{6} + 103574218160 \beta_{7} + 2582767536138 \beta_{8} - 240252800 \beta_{9} + 145513271920 \beta_{10} - 7815587596 \beta_{11} - 48050560 \beta_{13} - 192363044 \beta_{14} - 9870628 \beta_{15}) q^{43}\) \(+(-\)\(26\!\cdots\!40\)\( - 454441193891168 \beta_{1} + 102008826208855608 \beta_{2} + 1092701359263436 \beta_{3} + 2682385914376608 \beta_{4} + 64461729600112 \beta_{5} - 2097028532 \beta_{6} + 126649818692 \beta_{7} - 506683381984 \beta_{8} + 101291553724 \beta_{9} - 253233986592 \beta_{10} - 159543440 \beta_{11} + 49362116 \beta_{12} + 15731456 \beta_{13} + 15731456 \beta_{14}) q^{44}\) \(+(\)\(37\!\cdots\!95\)\( - \)\(38\!\cdots\!65\)\( \beta_{1} - 136290588006300896 \beta_{2} + 6389631488627654483 \beta_{3} + 1336338145268789 \beta_{4} + 108739886050994 \beta_{5} - 1763362736 \beta_{6} + 115500322185 \beta_{7} - 952795728181 \beta_{8} - 95753510410 \beta_{9} + 331133466890 \beta_{10} + 6494686831 \beta_{11} - 56054076 \beta_{12} + 12616000 \beta_{13} + 779631181 \beta_{14} - 103826843 \beta_{15}) q^{45}\) \(+(\)\(25\!\cdots\!77\)\( - 122244408199832 \beta_{1} - 102631163157550581 \beta_{2} + 435544068017671 \beta_{3} + 1387727263094075 \beta_{4} - 283059191887767 \beta_{5} + 3651845657 \beta_{6} + 149644083911 \beta_{7} - 603782176897 \beta_{8} - 71969679210 \beta_{9} - 296530192731 \beta_{10} - 11905767090 \beta_{11} - 63412217 \beta_{12} + 103369643 \beta_{13} + 103369643 \beta_{14}) q^{46}\) \(+(65286600920128 - \)\(51\!\cdots\!34\)\( \beta_{1} - 128273401806450 \beta_{2} - 1244992196789306669 \beta_{3} + 665496312735194 \beta_{4} - 2246282123576 \beta_{5} + 1459408080 \beta_{6} - 52321144472 \beta_{7} + 8812839989627 \beta_{8} + 1824260100 \beta_{9} - 529947375512 \beta_{10} + 52769612190 \beta_{11} + 364852020 \beta_{13} + 2040207054 \beta_{14} - 964244994 \beta_{15}) q^{47}\) \(+(-45440678263240 - \)\(31\!\cdots\!80\)\( \beta_{1} + 91803953718288 \beta_{2} + 14072102595668868520 \beta_{3} - 450575435209972 \beta_{4} - 854840343920 \beta_{5} + 1777379520 \beta_{6} - 36784301432 \beta_{7} - 30488075462140 \beta_{8} + 2221724400 \beta_{9} - 826695405924 \beta_{10} + 65188930632 \beta_{11} + 444344880 \beta_{13} - 350217272 \beta_{14} + 4024416 \beta_{15}) q^{48}\) \(+(-\)\(30\!\cdots\!81\)\( - 406625584755145 \beta_{1} + 635152046552402148 \beta_{2} + 834852369010695 \beta_{3} + 1728025150959429 \beta_{4} + 360749145158465 \beta_{5} + 11568715872 \beta_{6} - 19291070036 \beta_{7} + 58537241504 \beta_{8} + 273244289535 \beta_{9} + 48416167912 \beta_{10} - 42653410240 \beta_{11} - 308816200 \beta_{12} + 347005680 \beta_{13} + 347005680 \beta_{14}) q^{49}\) \(+(\)\(40\!\cdots\!32\)\( + \)\(20\!\cdots\!20\)\( \beta_{1} - 792688909670529358 \beta_{2} + 6414084288915967872 \beta_{3} - 720056715743991 \beta_{4} + 243163526546836 \beta_{5} + 15855946036 \beta_{6} - 205781197100 \beta_{7} + 56350805626761 \beta_{8} + 132019252115 \beta_{9} - 4128889054785 \beta_{10} - 10892516586 \beta_{11} + 364247376 \beta_{12} + 662312625 \beta_{13} - 1455490731 \beta_{14} + 1059211368 \beta_{15}) q^{50}\) \(+(\)\(23\!\cdots\!68\)\( + 921164957643716 \beta_{1} - 392350967305031942 \beta_{2} - 2090017908462406 \beta_{3} - 4849902152082608 \beta_{4} - 395893195656992 \beta_{5} - 40886491830 \beta_{6} - 252447932190 \beta_{7} + 990090636876 \beta_{8} + 563032107102 \beta_{9} + 515612398188 \beta_{10} - 44622122160 \beta_{11} + 426991134 \beta_{12} + 577325244 \beta_{13} + 577325244 \beta_{14}) q^{51}\) \(+(1329900629886100 - \)\(13\!\cdots\!68\)\( \beta_{1} - 2605937204467262 \beta_{2} - \)\(13\!\cdots\!08\)\( \beta_{3} + 13642392502181420 \beta_{4} - 54003237724590 \beta_{5} - 3782486240 \beta_{6} - 403322759174 \beta_{7} + 33580732510808 \beta_{8} - 4728107800 \beta_{9} + 16860566828272 \beta_{10} - 143166187198 \beta_{11} - 945621560 \beta_{13} - 13485285602 \beta_{14} + 4773410256 \beta_{15}) q^{52}\) \(+(1647593068667524 + \)\(27\!\cdots\!80\)\( \beta_{1} - 3213698306732033 \beta_{2} + 97171389657186527953 \beta_{3} + 16969218966638512 \beta_{4} - 81676941823392 \beta_{5} - 5558701280 \beta_{6} - 1655166387865 \beta_{7} - 81170406383063 \beta_{8} - 6948376600 \beta_{9} - 3267176577370 \beta_{10} - 171370195043 \beta_{11} - 1389675320 \beta_{13} + 10129952447 \beta_{14} + 662696287 \beta_{15}) q^{53}\) \(+(-\)\(26\!\cdots\!02\)\( + 13495561853889432 \beta_{1} + 4888427420104004802 \beta_{2} - 32148017729714406 \beta_{3} - 78237939073608654 \beta_{4} - 2554703233676706 \beta_{5} - 17985163146 \beta_{6} - 3334855086966 \beta_{7} + 13451794873002 \beta_{8} - 4531579099596 \beta_{9} + 6610220586606 \beta_{10} + 257070275700 \beta_{11} + 1475656842 \beta_{12} - 2201549838 \beta_{13} - 2201549838 \beta_{14}) q^{54}\) \(+(\)\(13\!\cdots\!12\)\( + \)\(12\!\cdots\!70\)\( \beta_{1} - 2117336421904368559 \beta_{2} + \)\(14\!\cdots\!30\)\( \beta_{3} - 21788954651459417 \beta_{4} - 1179400502902765 \beta_{5} - 44590102890 \beta_{6} - 4398715213625 \beta_{7} - 54600648799765 \beta_{8} + 2021609957025 \beta_{9} + 9641870546150 \beta_{10} - 50166543485 \beta_{11} - 1833210115 \beta_{12} - 4783492500 \beta_{13} - 11682788435 \beta_{14} - 7221355445 \beta_{15}) q^{55}\) \(+(\)\(13\!\cdots\!22\)\( + 18394729938006660 \beta_{1} - 2351222368981683399 \beta_{2} - 48320430894054019 \beta_{3} - 127819978317012668 \beta_{4} + 6068540298282540 \beta_{5} + 193926984413 \beta_{6} - 8404198560145 \beta_{7} + 33915405436932 \beta_{8} - 1456911754987 \beta_{9} + 16647731802796 \beta_{10} + 679088783660 \beta_{11} - 2252840897 \beta_{12} - 7573146212 \beta_{13} - 7573146212 \beta_{14}) q^{56}\) \(+(9028549216918747 + \)\(22\!\cdots\!74\)\( \beta_{1} - 17624984994443619 \beta_{2} - \)\(54\!\cdots\!15\)\( \beta_{3} + 92888253497091379 \beta_{4} - 432367752615718 \beta_{5} - 7013444880 \beta_{6} - 8720778057124 \beta_{7} - 269256975542525 \beta_{8} - 8766806100 \beta_{9} - 19282424916972 \beta_{10} - 195873021717 \beta_{11} - 1753361220 \beta_{13} + 66401870429 \beta_{14} - 16728476403 \beta_{15}) q^{57}\) \(+(12698937295207848 - \)\(39\!\cdots\!18\)\( \beta_{1} - 24918494309249832 \beta_{2} - 15811686943955724320 \beta_{3} + 129955255376290964 \beta_{4} - 480093931593624 \beta_{5} - 13680905680 \beta_{6} - 6893318279752 \beta_{7} + 897830893954308 \beta_{8} - 17101132100 \beta_{9} + 57778773869636 \beta_{10} - 770299620680 \beta_{11} - 3420226420 \beta_{13} - 65863343212 \beta_{14} - 7886981808 \beta_{15}) q^{58}\) \(+(-\)\(12\!\cdots\!07\)\( + 16212357896764124 \beta_{1} + 5528180011934457673 \beta_{2} - 40124178840507356 \beta_{3} - 101072918215278504 \beta_{4} + 118716817319691 \beta_{5} - 217445918565 \beta_{6} - 7446753904912 \beta_{7} + 29585560492412 \beta_{8} + 14722991201006 \beta_{9} + 14999137490736 \beta_{10} - 462436447760 \beta_{11} - 5077313734 \beta_{12} + 3268078196 \beta_{13} + 3268078196 \beta_{14}) q^{59}\) \(+(-\)\(20\!\cdots\!86\)\( - \)\(52\!\cdots\!72\)\( \beta_{1} + 827437550476838179 \beta_{2} + \)\(23\!\cdots\!08\)\( \beta_{3} - 18983739653805216 \beta_{4} - 14886640463685465 \beta_{5} - 205666071784 \beta_{6} - 7202523612017 \beta_{7} - 1039470616490610 \beta_{8} - 9816117069856 \beta_{9} + 75186547112894 \beta_{10} + 319323446155 \beta_{11} + 6855049256 \beta_{12} + 11369345000 \beta_{13} + 121134011489 \beta_{14} + 34913960208 \beta_{15}) q^{60}\) \(+(-\)\(39\!\cdots\!33\)\( - 14743792643519306 \beta_{1} - 4014704198374582074 \beta_{2} + 32248193847007444 \beta_{3} + 71965308625740197 \beta_{4} + 8884455741941268 \beta_{5} - 317910092296 \beta_{6} + 1599307895729 \beta_{7} - 7793481158404 \beta_{8} + 817631221182 \beta_{9} - 2447998301922 \beta_{10} - 3176260362240 \beta_{11} + 9045100768 \beta_{12} + 34995134528 \beta_{13} + 34995134528 \beta_{14}) q^{61}\) \(+(-5126480449291768 + \)\(12\!\cdots\!00\)\( \beta_{1} + 10017747137399872 \beta_{2} - \)\(53\!\cdots\!40\)\( \beta_{3} - 52868000099378508 \beta_{4} + 237469112895904 \beta_{5} + 65923367440 \beta_{6} - 1565538995520 \beta_{7} + 375972753738276 \beta_{8} + 82404209300 \beta_{9} - 165883947248988 \beta_{10} + 1947351564400 \beta_{11} + 16480841860 \beta_{13} - 259294727116 \beta_{14} + 33698788576 \beta_{15}) q^{62}\) \(+(11719619941169580 - \)\(11\!\cdots\!10\)\( \beta_{1} - 22945311535187790 \beta_{2} + \)\(37\!\cdots\!09\)\( \beta_{3} + 120019754014083366 \beta_{4} - 485896982603424 \beta_{5} + 188428677840 \beta_{6} - 9514050944856 \beta_{7} + 574069030948023 \beta_{8} + 235535847300 \beta_{9} - 48716257564584 \beta_{10} + 8048423473434 \beta_{11} + 47107169460 \beta_{13} + 198260591682 \beta_{14} + 54334181106 \beta_{15}) q^{63}\) \(+(\)\(20\!\cdots\!20\)\( - 39034636170303952 \beta_{1} - 34988889678880164204 \beta_{2} + 93507426532091012 \beta_{3} + 228749910959075632 \beta_{4} + 6283581136492464 \beta_{5} + 1959851143684 \beta_{6} + 9702293063788 \beta_{7} - 39422273513168 \beta_{8} + 15151772974980 \beta_{9} - 19072485921264 \beta_{10} - 1390825815120 \beta_{11} + 9943329836 \beta_{12} + 17033051536 \beta_{13} + 17033051536 \beta_{14}) q^{64}\) \(+(-\)\(72\!\cdots\!96\)\( + \)\(11\!\cdots\!31\)\( \beta_{1} + 11900227991981052444 \beta_{2} + \)\(29\!\cdots\!67\)\( \beta_{3} + 51382823301710465 \beta_{4} + 26084115188796263 \beta_{5} + 2507615662380 \beta_{6} + 20600379320206 \beta_{7} + 2876882891751598 \beta_{8} - 7508598010947 \beta_{9} - 37918410785572 \beta_{10} - 397899771858 \beta_{11} - 16440202920 \beta_{12} + 34163220000 \beta_{13} - 584585014230 \beta_{14} - 115328618310 \beta_{15}) q^{65}\) \(+(-\)\(18\!\cdots\!76\)\( - 95050340021375464 \beta_{1} + 41658318512966167848 \beta_{2} + 261496961740007592 \beta_{3} + 716041047525676484 \beta_{4} - 56409106340300924 \beta_{5} - 1675571863760 \beta_{6} + 55593986072384 \beta_{7} - 219525654090348 \beta_{8} - 11435247148324 \beta_{9} - 112723038299684 \beta_{10} + 6479688823160 \beta_{11} - 25065004448 \beta_{12} - 73280703548 \beta_{13} - 73280703548 \beta_{14}) q^{66}\) \(+(-61667105039407456 - \)\(22\!\cdots\!20\)\( \beta_{1} + 120429768061621964 \beta_{2} - \)\(12\!\cdots\!99\)\( \beta_{3} - 633830134925901548 \beta_{4} + 2907032923082472 \beta_{5} + 21239857600 \beta_{6} + 74596471102960 \beta_{7} + 645007183285190 \beta_{8} + 26549822000 \beta_{9} + 537442792301520 \beta_{10} + 3408075349356 \beta_{11} + 5309964400 \beta_{13} + 826572720180 \beta_{14} + 17146561652 \beta_{15}) q^{67}\) \(+(-245055111622241936 - \)\(11\!\cdots\!64\)\( \beta_{1} + 480281309177708928 \beta_{2} + \)\(68\!\cdots\!64\)\( \beta_{3} - 2509949659741048792 \beta_{4} + 9799961741672416 \beta_{5} - 709192063680 \beta_{6} + 164030267951376 \beta_{7} - 12895775959725736 \beta_{8} - 886490079600 \beta_{9} - 357835808819864 \beta_{10} - 28255207625232 \beta_{11} - 177298015920 \beta_{13} + 64564486944 \beta_{14} - 253937089248 \beta_{15}) q^{68}\) \(+(\)\(43\!\cdots\!85\)\( - 534120367259022730 \beta_{1} - \)\(17\!\cdots\!22\)\( \beta_{2} + 1307278546000358600 \beta_{3} + 3260993235499395025 \beta_{4} + 27035214625586320 \beta_{5} - 8483341223496 \beta_{6} + 202222425048753 \beta_{7} - 806920750586988 \beta_{8} - 202806823002054 \beta_{9} - 405494225587074 \beta_{10} + 4493259130080 \beta_{11} + 9038607396 \beta_{12} - 43267123704 \beta_{13} - 43267123704 \beta_{14}) q^{69}\) \(+(-\)\(36\!\cdots\!97\)\( + \)\(55\!\cdots\!95\)\( \beta_{1} + \)\(15\!\cdots\!10\)\( \beta_{2} + \)\(24\!\cdots\!12\)\( \beta_{3} + 1151204941619366868 \beta_{4} + 219547954891913521 \beta_{5} - 11223884317974 \beta_{6} + 256377392358890 \beta_{7} + 4331359107944696 \beta_{8} + 191337433307610 \beta_{9} - 1765825981313940 \beta_{10} - 2104192596096 \beta_{11} + 5617165966 \beta_{12} - 338313313500 \beta_{13} + 1685038874104 \beta_{14} + 200606900488 \beta_{15}) q^{70}\) \(+(-\)\(54\!\cdots\!22\)\( - 281996795212316816 \beta_{1} + \)\(10\!\cdots\!36\)\( \beta_{2} + 782291623185744542 \beta_{3} + 2154748137327735856 \beta_{4} - 181110094893347290 \beta_{5} + 12920906435164 \beta_{6} + 146760959347454 \beta_{7} - 587053200831804 \beta_{8} + 151258715861046 \beta_{9} - 293504986489772 \beta_{10} - 2791722640 \beta_{11} + 28484758618 \beta_{12} + 8166989428 \beta_{13} + 8166989428 \beta_{14}) q^{71}\) \(+(-432362538580679614 + \)\(10\!\cdots\!73\)\( \beta_{1} + 845144481431981050 \beta_{2} - \)\(63\!\cdots\!62\)\( \beta_{3} - 4440453021325589569 \beta_{4} + 19510546113600674 \beta_{5} - 1639605269120 \beta_{6} + 377261003206032 \beta_{7} - 505146536754295 \beta_{8} - 2049506586400 \beta_{9} + 739598268734903 \beta_{10} - 70479692881136 \beta_{11} - 409901317280 \beta_{13} - 2086454582768 \beta_{14} - 393129107264 \beta_{15}) q^{72}\) \(+(-330163595250456923 - \)\(74\!\cdots\!58\)\( \beta_{1} + 645298620201390607 \beta_{2} + \)\(11\!\cdots\!99\)\( \beta_{3} - 3394349227025242627 \beta_{4} + 15068177834098782 \beta_{5} + 1049444758960 \beta_{6} + 235627938591088 \beta_{7} + 5132298653020817 \beta_{8} + 1311805948700 \beta_{9} - 1280207449832156 \beta_{10} + 35934045249017 \beta_{11} + 262361189740 \beta_{13} - 3147139000073 \beta_{14} + 785455622247 \beta_{15}) q^{73}\) \(+(\)\(26\!\cdots\!48\)\( - 58742594014607203 \beta_{1} - \)\(42\!\cdots\!66\)\( \beta_{2} + 90536940997963592 \beta_{3} + 108019552148271675 \beta_{4} + 115941801128033444 \beta_{5} + 21996806003132 \beta_{6} - 40340902428248 \beta_{7} + 177768484206451 \beta_{8} + 220507926971993 \beta_{9} + 71843151094353 \beta_{10} + 37288370349650 \beta_{11} - 152777690144 \beta_{12} - 424144343609 \beta_{13} - 424144343609 \beta_{14}) q^{74}\) \(+(-\)\(17\!\cdots\!16\)\( + \)\(24\!\cdots\!40\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2} + \)\(46\!\cdots\!39\)\( \beta_{3} - 349535537759780842 \beta_{4} - 405970706131608518 \beta_{5} + 25336007176882 \beta_{6} - 30503446070200 \beta_{7} - 13263195130176918 \beta_{8} - 589002245629120 \beta_{9} + 3078421173383080 \beta_{10} + 8874568821518 \beta_{11} + 153712869712 \beta_{12} + 1085410526500 \beta_{13} - 2136223284722 \beta_{14} + 265759796766 \beta_{15}) q^{75}\) \(+(-\)\(24\!\cdots\!76\)\( - 418186649245231456 \beta_{1} + \)\(16\!\cdots\!76\)\( \beta_{2} + 931772896986495356 \beta_{3} + 2121934229839124896 \beta_{4} + 215853006877640496 \beta_{5} - 38396175191300 \beta_{6} + 159100159948756 \beta_{7} - 638119169281760 \beta_{8} - 793274401516884 \beta_{9} - 317225048716320 \beta_{10} - 3829447676880 \beta_{11} + 133915699796 \beta_{12} + 77337625216 \beta_{13} + 77337625216 \beta_{14}) q^{76}\) \(+(729435675951907263 + \)\(29\!\cdots\!10\)\( \beta_{1} - 1424855962629375531 \beta_{2} - \)\(47\!\cdots\!67\)\( \beta_{3} + 7488270794396242695 \beta_{4} - 33711465628874586 \beta_{5} + 7469896288800 \beta_{6} - 887145507016208 \beta_{7} + 5148012241550475 \beta_{8} + 9337370361000 \beta_{9} - 8349285971164308 \beta_{10} + 299407857161331 \beta_{11} + 1867474072200 \beta_{13} + 3483687051969 \beta_{14} + 1337894905953 \beta_{15}) q^{77}\) \(+(1642653380380425528 - \)\(11\!\cdots\!52\)\( \beta_{1} - 3219642805534186280 \beta_{2} + \)\(80\!\cdots\!92\)\( \beta_{3} + 16830866827551862908 \beta_{4} - 65643903765030488 \beta_{5} + 85753030480 \beta_{6} - 908220845966920 \beta_{7} + 76533465114343500 \beta_{8} + 107191288100 \beta_{9} + 8029863760349324 \beta_{10} + 31178604569752 \beta_{11} + 21438257620 \beta_{13} + 12378073920828 \beta_{14} - 1143739697264 \beta_{15}) q^{78}\) \(+(-\)\(66\!\cdots\!44\)\( + 562879936795002624 \beta_{1} + \)\(22\!\cdots\!94\)\( \beta_{2} - 1172440452708787386 \beta_{3} - 2471399560919256576 \beta_{4} - 464334211586355432 \beta_{5} - 31582457237922 \beta_{6} - 241896191258898 \beta_{7} + 842743837010652 \beta_{8} + 1569534523411914 \beta_{9} + 550811402697876 \beta_{10} - 284142189130800 \beta_{11} + 582029064990 \beta_{12} + 3065704111740 \beta_{13} + 3065704111740 \beta_{14}) q^{79}\) \(+(\)\(15\!\cdots\!18\)\( + \)\(62\!\cdots\!20\)\( \beta_{1} - \)\(19\!\cdots\!05\)\( \beta_{2} + \)\(11\!\cdots\!67\)\( \beta_{3} - 8775454628031614832 \beta_{4} - 850475085376739964 \beta_{5} - 11204366791609 \beta_{6} - 1602380727857435 \beta_{7} - 55062741322988664 \beta_{8} + 438321929825735 \beta_{9} + 7049459930279560 \beta_{10} - 13243949327236 \beta_{11} - 750755840819 \beta_{12} - 1583887818500 \beta_{13} - 5198173135436 \beta_{14} - 3022539631392 \beta_{15}) q^{80}\) \(+(\)\(78\!\cdots\!99\)\( + 1445505851300689305 \beta_{1} - \)\(13\!\cdots\!68\)\( \beta_{2} - 5010351085723413987 \beta_{3} - 15752257292383594767 \beta_{4} + 3050072632816358643 \beta_{5} + 37297049473656 \beta_{6} - 1911333796772994 \beta_{7} + 7563064098371976 \beta_{8} + 2714850259751241 \beta_{9} + 3866292983583108 \beta_{10} - 188092780404480 \beta_{11} - 907959386880 \beta_{12} + 1659897118080 \beta_{13} + 1659897118080 \beta_{14}) q^{81}\) \(+(1652438480993994666 + \)\(10\!\cdots\!79\)\( \beta_{1} - 3222668269845251976 \beta_{2} - \)\(20\!\cdots\!04\)\( \beta_{3} + 17033044042734997857 \beta_{4} - 82771456729083264 \beta_{5} - 14335168114700 \beta_{6} - 1518408511782728 \beta_{7} - 103049874063912019 \beta_{8} - 17918960143375 \beta_{9} + 2606084988816653 \beta_{10} - 546474423943292 \beta_{11} - 3583792028675 \beta_{13} - 108194051071 \beta_{14} - 1520603689688 \beta_{15}) q^{82}\) \(+(3647661271105686872 - \)\(19\!\cdots\!16\)\( \beta_{1} - 7136281834398994252 \beta_{2} - \)\(12\!\cdots\!23\)\( \beta_{3} + 37428528110615702156 \beta_{4} - 158911379740872360 \beta_{5} + 4580955654080 \beta_{6} - 3323230820948880 \beta_{7} + 72362828810129822 \beta_{8} + 5726194567600 \beta_{9} - 10560906076719824 \beta_{10} + 107126430721988 \beta_{11} + 1145238913520 \beta_{13} - 13657880469572 \beta_{14} - 2817565747972 \beta_{15}) q^{83}\) \(+(-\)\(25\!\cdots\!68\)\( + 17540042913833425488 \beta_{1} + \)\(53\!\cdots\!82\)\( \beta_{2} - 42608532788561749326 \beta_{3} - \)\(10\!\cdots\!20\)\( \beta_{4} - 1567723817867812632 \beta_{5} + 9934094168562 \beta_{6} - 5095895050374042 \beta_{7} + 20729688912533520 \beta_{8} - 4103227678469190 \beta_{9} + 10006254739518960 \beta_{10} + 788172094648680 \beta_{11} - 973343277402 \beta_{12} - 8320670473632 \beta_{13} - 8320670473632 \beta_{14}) q^{84}\) \(+(\)\(10\!\cdots\!81\)\( + \)\(26\!\cdots\!28\)\( \beta_{1} - \)\(23\!\cdots\!09\)\( \beta_{2} - \)\(29\!\cdots\!75\)\( \beta_{3} - 4418106529467790217 \beta_{4} - 1257890798023286484 \beta_{5} - 87311025358768 \beta_{6} - 2854912799783212 \beta_{7} + 7795469462536121 \beta_{8} + 52494000679574 \beta_{9} - 6911657306848676 \beta_{10} + 53500557020849 \beta_{11} + 1724947745312 \beta_{12} + 1030030940000 \beta_{13} + 34733811702803 \beta_{14} + 9413324564691 \beta_{15}) q^{85}\) \(+(-\)\(11\!\cdots\!77\)\( + 23122025995478217313 \beta_{1} + 98209131287629109774 \beta_{2} - 55358738920869606092 \beta_{3} - \)\(13\!\cdots\!58\)\( \beta_{4} - 3783714004557098415 \beta_{5} + 111803164721498 \beta_{6} - 5822621162554746 \beta_{7} + 23707725271084558 \beta_{8} - 8132451344114268 \beta_{9} + 11423178463223674 \beta_{10} + 952649257537180 \beta_{11} + 2654901045446 \beta_{12} - 8962367635354 \beta_{13} - 8962367635354 \beta_{14}) q^{86}\) \(+(8359841688651846456 - \)\(64\!\cdots\!36\)\( \beta_{1} - 16389930933996108200 \beta_{2} + \)\(65\!\cdots\!50\)\( \beta_{3} + 85670293866113306280 \beta_{4} - 329683327400079272 \beta_{5} + 3751621909600 \beta_{6} - 2950018330260976 \beta_{7} + 360774790041190752 \beta_{8} + 4689527387000 \beta_{9} + 85274198739853952 \beta_{10} + 37860224658280 \beta_{11} + 937905477400 \beta_{13} - 22128310441392 \beta_{14} - 4437845018768 \beta_{15}) q^{87}\) \(+(-5397778556742129368 - \)\(80\!\cdots\!28\)\( \beta_{1} + 10603987938993250408 \beta_{2} - \)\(55\!\cdots\!52\)\( \beta_{3} - 55079609103589492852 \beta_{4} + 189767991344267784 \beta_{5} - 47583800907520 \beta_{6} + 4484240543237216 \beta_{7} - 653465511781029164 \beta_{8} - 59479751134400 \beta_{9} + 34237403328530604 \beta_{10} - 1783450234313376 \beta_{11} - 11895950226880 \beta_{13} - 64604211829920 \beta_{14} + 22857373122688 \beta_{15}) q^{88}\) \(+(-\)\(51\!\cdots\!24\)\( - 3271074298652427650 \beta_{1} + \)\(70\!\cdots\!64\)\( \beta_{2} + 6946856116671032266 \beta_{3} + 14992332438799049604 \beta_{4} + 2413323100182200446 \beta_{5} + 58600515557560 \beta_{6} + 1067005498564478 \beta_{7} - 4528285380730408 \beta_{8} - 7592807035671722 \beta_{9} - 1995268618731324 \beta_{10} - 593886835962560 \beta_{11} - 1118645431624 \beta_{12} + 5740456774256 \beta_{13} + 5740456774256 \beta_{14}) q^{89}\) \(+(\)\(50\!\cdots\!07\)\( + \)\(14\!\cdots\!01\)\( \beta_{1} - \)\(76\!\cdots\!73\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3} + 4832221095507441371 \beta_{4} + 11758486207358782312 \beta_{5} + 148637571023184 \beta_{6} + 3423273806350746 \beta_{7} + 980931918363084597 \beta_{8} + 8231923861929658 \beta_{9} - 72694329508717842 \beta_{10} - 419039543781357 \beta_{11} - 444121189356 \beta_{12} - 5485845511875 \beta_{13} - 84187128152139 \beta_{14} - 10264619197008 \beta_{15}) q^{90}\) \(+(\)\(35\!\cdots\!34\)\( - 31824640214431020596 \beta_{1} - \)\(39\!\cdots\!32\)\( \beta_{2} + 83039327332907124402 \beta_{3} + \)\(21\!\cdots\!16\)\( \beta_{4} - 9315297706584760046 \beta_{5} - 376698924992964 \beta_{6} + 11847007564289954 \beta_{7} - 48104671846551308 \beta_{8} + 22650488174616498 \beta_{9} - 23311930301242804 \beta_{10} - 1635197435582480 \beta_{11} - 2950498917158 \beta_{12} + 15842688427572 \beta_{13} + 15842688427572 \beta_{14}) q^{91}\) \(+(-17593698182823639302 + \)\(87\!\cdots\!32\)\( \beta_{1} + 34455654266900008935 \beta_{2} + \)\(28\!\cdots\!96\)\( \beta_{3} - \)\(18\!\cdots\!16\)\( \beta_{4} + 732774810553441063 \beta_{5} + 18938095212000 \beta_{6} + 9585124574969847 \beta_{7} - 433702410050767642 \beta_{8} + 23672619015000 \beta_{9} - 112953388239735770 \beta_{10} + 1110142417292571 \beta_{11} + 4734523803000 \beta_{13} + 80825494088769 \beta_{14} + 20277736047408 \beta_{15}) q^{92}\) \(+(-5308389447797270768 - \)\(30\!\cdots\!60\)\( \beta_{1} + 10479520030015465416 \beta_{2} + \)\(28\!\cdots\!20\)\( \beta_{3} - 54159970933198359056 \beta_{4} + 142527383799628376 \beta_{5} + 133192045952160 \beta_{6} - 3183522135124408 \beta_{7} - 723276144257353064 \beta_{8} + 166490057440200 \beta_{9} - 182546295136310640 \beta_{10} + 5338843816688184 \beta_{11} + 33298011488040 \beta_{13} + 303417024727760 \beta_{14} - 66601371151632 \beta_{15}) q^{93}\) \(+(\)\(67\!\cdots\!61\)\( - \)\(10\!\cdots\!30\)\( \beta_{1} - \)\(14\!\cdots\!67\)\( \beta_{2} + \)\(26\!\cdots\!37\)\( \beta_{3} + \)\(64\!\cdots\!77\)\( \beta_{4} + 12763716215877564553 \beta_{5} - 294356122238825 \beta_{6} + 28673150572009065 \beta_{7} - 115599955477612903 \beta_{8} + 42951739667436354 \beta_{9} - 56856411995020189 \beta_{10} - 2060826802463870 \beta_{11} + 10895002591593 \beta_{12} + 24141702806413 \beta_{13} + 24141702806413 \beta_{14}) q^{94}\) \(+(-\)\(60\!\cdots\!24\)\( + \)\(13\!\cdots\!30\)\( \beta_{1} - \)\(42\!\cdots\!19\)\( \beta_{2} - \)\(13\!\cdots\!44\)\( \beta_{3} + \)\(20\!\cdots\!77\)\( \beta_{4} - 396605945732031207 \beta_{5} + 262271014514808 \beta_{6} + 18643061633497795 \beta_{7} - 1614589058975124307 \beta_{8} - 35822902393806595 \beta_{9} + 229812587387178630 \beta_{10} + 1511057134012857 \beta_{11} - 11187527436547 \beta_{12} + 24201193732000 \beta_{13} + 75164250679107 \beta_{14} - 29794239698671 \beta_{15}) q^{95}\) \(+(-\)\(86\!\cdots\!36\)\( - 66358809410153190864 \beta_{1} - \)\(64\!\cdots\!68\)\( \beta_{2} + \)\(17\!\cdots\!28\)\( \beta_{3} + \)\(45\!\cdots\!40\)\( \beta_{4} - 17963797610857968336 \beta_{5} + 75476403305572 \beta_{6} + 34692670272474764 \beta_{7} - 139218844619622096 \beta_{8} - 45561382563617180 \beta_{9} - 69149452787695088 \beta_{10} - 1027350025631440 \beta_{11} - 9143756760500 \beta_{12} + 7870661595280 \beta_{13} + 7870661595280 \beta_{14}) q^{96}\) \(+(-47995005622468654517 + \)\(30\!\cdots\!42\)\( \beta_{1} + 93740391620437100431 \beta_{2} + \)\(34\!\cdots\!71\)\( \beta_{3} - \)\(49\!\cdots\!01\)\( \beta_{4} + 2251307726783236506 \beta_{5} + 56083612622960 \beta_{6} + 32116874451808502 \beta_{7} + 1537730022138651121 \beta_{8} + 70104515778700 \beta_{9} - 262496661898986312 \beta_{10} + 1431935777151777 \beta_{11} + 14020903155740 \beta_{13} - 166263425545185 \beta_{14} - 19798564552241 \beta_{15}) q^{97}\) \(+(-32215839943042550550 - \)\(79\!\cdots\!20\)\( \beta_{1} + 62497417927664174448 \beta_{2} - \)\(36\!\cdots\!48\)\( \beta_{3} - \)\(33\!\cdots\!55\)\( \beta_{4} + 1932598843837021416 \beta_{5} - 34410630328620 \beta_{6} + 53721667708735376 \beta_{7} + 4550541640721408157 \beta_{8} - 43013287910775 \beta_{9} + 497087029125501885 \beta_{10} - 1965435294224724 \beta_{11} - 8602657582155 \beta_{13} - 415806534271695 \beta_{14} + 70472536041432 \beta_{15}) q^{98}\) \(+(\)\(21\!\cdots\!11\)\( - 43902112857546520764 \beta_{1} - \)\(50\!\cdots\!85\)\( \beta_{2} + \)\(10\!\cdots\!36\)\( \beta_{3} + \)\(25\!\cdots\!96\)\( \beta_{4} + 8837830495878222177 \beta_{5} + 1470863517453021 \beta_{6} + 15749437602977196 \beta_{7} - 61026832882264500 \beta_{8} - 25751727594914658 \beta_{9} - 32564116764820680 \beta_{10} + 4474701000656880 \beta_{11} - 26349102746214 \beta_{12} - 53188529362764 \beta_{13} - 53188529362764 \beta_{14}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 72851326872q^{4} \) \(\mathstrut -\mathstrut 232168160280q^{5} \) \(\mathstrut +\mathstrut 11001777346872q^{6} \) \(\mathstrut -\mathstrut 27535156574590368q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 72851326872q^{4} \) \(\mathstrut -\mathstrut 232168160280q^{5} \) \(\mathstrut +\mathstrut 11001777346872q^{6} \) \(\mathstrut -\mathstrut 27535156574590368q^{9} \) \(\mathstrut -\mathstrut 22271193818331880q^{10} \) \(\mathstrut +\mathstrut 219974590742466912q^{11} \) \(\mathstrut -\mathstrut 10416043581549356184q^{14} \) \(\mathstrut +\mathstrut 9232415497377901440q^{15} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!76\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(55\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!92\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!00\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!00\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!32\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(74\!\cdots\!60\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(48\!\cdots\!72\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!64\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!20\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!56\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!16\)\(q^{39} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(96\!\cdots\!52\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!04\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!40\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!92\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(48\!\cdots\!12\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(37\!\cdots\!32\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!40\)\(q^{55} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!80\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(62\!\cdots\!88\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!08\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!96\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!04\)\(q^{69} \) \(\mathstrut -\mathstrut \)\(58\!\cdots\!80\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!08\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!76\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!00\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!00\)\(q^{79} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!20\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!36\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!64\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{85} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!88\)\(q^{86} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!00\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!40\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!52\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!00\)\(q^{95} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!08\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!24\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(26286285043\) \(x^{14}\mathstrut +\mathstrut \) \(277176279803774573548\) \(x^{12}\mathstrut +\mathstrut \) \(1496006322727341104924267746816\) \(x^{10}\mathstrut +\mathstrut \) \(4376228902975645752284567792282218577920\) \(x^{8}\mathstrut +\mathstrut \) \(6785210496827316795155770011788917433647451078656\) \(x^{6}\mathstrut +\mathstrut \) \(5093319366938751307797338793282286802764384084972313509888\) \(x^{4}\mathstrut +\mathstrut \) \(1608229292592013531797229664939538769264311330074914543142623510528\) \(x^{2}\mathstrut +\mathstrut \) \(163532475457876517394407745867705361011086521692855999713211555842344615936\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} + 13143142522 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(37\!\cdots\!75\) \(\nu^{15}\mathstrut -\mathstrut \) \(83\!\cdots\!69\) \(\nu^{13}\mathstrut -\mathstrut \) \(73\!\cdots\!20\) \(\nu^{11}\mathstrut -\mathstrut \) \(32\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(76\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(91\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(48\!\cdots\!04\) \(\nu^{3}\mathstrut -\mathstrut \) \(74\!\cdots\!40\) \(\nu\)\()/\)\(22\!\cdots\!40\)
\(\beta_{4}\)\(=\)\((\)\(30\!\cdots\!91\) \(\nu^{15}\mathstrut +\mathstrut \) \(25\!\cdots\!68\) \(\nu^{14}\mathstrut +\mathstrut \) \(68\!\cdots\!77\) \(\nu^{13}\mathstrut +\mathstrut \) \(57\!\cdots\!96\) \(\nu^{12}\mathstrut +\mathstrut \) \(61\!\cdots\!76\) \(\nu^{11}\mathstrut +\mathstrut \) \(51\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(28\!\cdots\!60\) \(\nu^{9}\mathstrut +\mathstrut \) \(23\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(75\!\cdots\!60\) \(\nu^{7}\mathstrut +\mathstrut \) \(61\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(11\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(83\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(86\!\cdots\!52\) \(\nu^{3}\mathstrut +\mathstrut \) \(50\!\cdots\!96\) \(\nu^{2}\mathstrut +\mathstrut \) \(20\!\cdots\!56\) \(\nu\mathstrut +\mathstrut \) \(86\!\cdots\!88\)\()/\)\(51\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(60\!\cdots\!57\) \(\nu^{15}\mathstrut +\mathstrut \) \(33\!\cdots\!64\) \(\nu^{14}\mathstrut -\mathstrut \) \(13\!\cdots\!79\) \(\nu^{13}\mathstrut +\mathstrut \) \(69\!\cdots\!08\) \(\nu^{12}\mathstrut -\mathstrut \) \(12\!\cdots\!52\) \(\nu^{11}\mathstrut +\mathstrut \) \(53\!\cdots\!04\) \(\nu^{10}\mathstrut -\mathstrut \) \(56\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(19\!\cdots\!40\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!20\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!40\) \(\nu^{6}\mathstrut -\mathstrut \) \(22\!\cdots\!72\) \(\nu^{5}\mathstrut +\mathstrut \) \(31\!\cdots\!44\) \(\nu^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!04\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!08\) \(\nu^{2}\mathstrut -\mathstrut \) \(39\!\cdots\!12\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!24\)\()/\)\(25\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(42\!\cdots\!31\) \(\nu^{15}\mathstrut -\mathstrut \) \(19\!\cdots\!88\) \(\nu^{14}\mathstrut -\mathstrut \) \(96\!\cdots\!57\) \(\nu^{13}\mathstrut -\mathstrut \) \(35\!\cdots\!36\) \(\nu^{12}\mathstrut -\mathstrut \) \(85\!\cdots\!16\) \(\nu^{11}\mathstrut -\mathstrut \) \(23\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(39\!\cdots\!60\) \(\nu^{9}\mathstrut -\mathstrut \) \(57\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!76\) \(\nu^{5}\mathstrut +\mathstrut \) \(23\!\cdots\!52\) \(\nu^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!32\) \(\nu^{3}\mathstrut +\mathstrut \) \(36\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(28\!\cdots\!96\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!92\)\()/\)\(25\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(30\!\cdots\!81\) \(\nu^{15}\mathstrut -\mathstrut \) \(14\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(11\!\cdots\!93\) \(\nu^{13}\mathstrut -\mathstrut \) \(98\!\cdots\!64\) \(\nu^{12}\mathstrut -\mathstrut \) \(32\!\cdots\!84\) \(\nu^{11}\mathstrut +\mathstrut \) \(20\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(31\!\cdots\!40\) \(\nu^{9}\mathstrut +\mathstrut \) \(26\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(11\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(33\!\cdots\!24\) \(\nu^{5}\mathstrut +\mathstrut \) \(22\!\cdots\!48\) \(\nu^{4}\mathstrut -\mathstrut \) \(34\!\cdots\!68\) \(\nu^{3}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\nu^{2}\mathstrut -\mathstrut \) \(81\!\cdots\!04\) \(\nu\mathstrut +\mathstrut \) \(30\!\cdots\!08\)\()/\)\(17\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(45\!\cdots\!93\) \(\nu^{15}\mathstrut -\mathstrut \) \(51\!\cdots\!36\) \(\nu^{14}\mathstrut +\mathstrut \) \(93\!\cdots\!71\) \(\nu^{13}\mathstrut -\mathstrut \) \(11\!\cdots\!92\) \(\nu^{12}\mathstrut +\mathstrut \) \(73\!\cdots\!48\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!96\) \(\nu^{10}\mathstrut +\mathstrut \) \(28\!\cdots\!80\) \(\nu^{9}\mathstrut -\mathstrut \) \(47\!\cdots\!60\) \(\nu^{8}\mathstrut +\mathstrut \) \(58\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(61\!\cdots\!28\) \(\nu^{5}\mathstrut -\mathstrut \) \(16\!\cdots\!56\) \(\nu^{4}\mathstrut +\mathstrut \) \(28\!\cdots\!96\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!92\) \(\nu^{2}\mathstrut +\mathstrut \) \(42\!\cdots\!88\) \(\nu\mathstrut -\mathstrut \) \(17\!\cdots\!76\)\()/\)\(51\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(29\!\cdots\!37\) \(\nu^{15}\mathstrut +\mathstrut \) \(18\!\cdots\!24\) \(\nu^{14}\mathstrut -\mathstrut \) \(66\!\cdots\!39\) \(\nu^{13}\mathstrut +\mathstrut \) \(18\!\cdots\!28\) \(\nu^{12}\mathstrut -\mathstrut \) \(59\!\cdots\!32\) \(\nu^{11}\mathstrut -\mathstrut \) \(76\!\cdots\!36\) \(\nu^{10}\mathstrut -\mathstrut \) \(27\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(15\!\cdots\!60\) \(\nu^{8}\mathstrut -\mathstrut \) \(72\!\cdots\!20\) \(\nu^{7}\mathstrut -\mathstrut \) \(66\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!52\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(83\!\cdots\!64\) \(\nu^{3}\mathstrut -\mathstrut \) \(69\!\cdots\!72\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(10\!\cdots\!16\)\()/\)\(25\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(18\!\cdots\!09\) \(\nu^{15}\mathstrut +\mathstrut \) \(70\!\cdots\!32\) \(\nu^{14}\mathstrut +\mathstrut \) \(39\!\cdots\!23\) \(\nu^{13}\mathstrut -\mathstrut \) \(95\!\cdots\!96\) \(\nu^{12}\mathstrut +\mathstrut \) \(33\!\cdots\!24\) \(\nu^{11}\mathstrut -\mathstrut \) \(40\!\cdots\!48\) \(\nu^{10}\mathstrut +\mathstrut \) \(14\!\cdots\!40\) \(\nu^{9}\mathstrut -\mathstrut \) \(38\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(32\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(15\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(37\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(28\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(18\!\cdots\!48\) \(\nu^{3}\mathstrut -\mathstrut \) \(20\!\cdots\!96\) \(\nu^{2}\mathstrut +\mathstrut \) \(45\!\cdots\!44\) \(\nu\mathstrut -\mathstrut \) \(37\!\cdots\!88\)\()/\)\(51\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(17\!\cdots\!49\) \(\nu^{15}\mathstrut +\mathstrut \) \(30\!\cdots\!48\) \(\nu^{14}\mathstrut -\mathstrut \) \(36\!\cdots\!03\) \(\nu^{13}\mathstrut +\mathstrut \) \(58\!\cdots\!56\) \(\nu^{12}\mathstrut -\mathstrut \) \(31\!\cdots\!64\) \(\nu^{11}\mathstrut +\mathstrut \) \(40\!\cdots\!28\) \(\nu^{10}\mathstrut -\mathstrut \) \(15\!\cdots\!40\) \(\nu^{9}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(47\!\cdots\!40\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(83\!\cdots\!04\) \(\nu^{5}\mathstrut +\mathstrut \) \(84\!\cdots\!08\) \(\nu^{4}\mathstrut -\mathstrut \) \(66\!\cdots\!28\) \(\nu^{3}\mathstrut +\mathstrut \) \(29\!\cdots\!56\) \(\nu^{2}\mathstrut -\mathstrut \) \(13\!\cdots\!84\) \(\nu\mathstrut +\mathstrut \) \(41\!\cdots\!68\)\()/\)\(51\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(81\!\cdots\!53\) \(\nu^{15}\mathstrut -\mathstrut \) \(53\!\cdots\!56\) \(\nu^{14}\mathstrut +\mathstrut \) \(16\!\cdots\!91\) \(\nu^{13}\mathstrut -\mathstrut \) \(10\!\cdots\!32\) \(\nu^{12}\mathstrut +\mathstrut \) \(14\!\cdots\!08\) \(\nu^{11}\mathstrut -\mathstrut \) \(82\!\cdots\!16\) \(\nu^{10}\mathstrut +\mathstrut \) \(71\!\cdots\!80\) \(\nu^{9}\mathstrut -\mathstrut \) \(30\!\cdots\!60\) \(\nu^{8}\mathstrut +\mathstrut \) \(21\!\cdots\!80\) \(\nu^{7}\mathstrut -\mathstrut \) \(57\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(37\!\cdots\!88\) \(\nu^{5}\mathstrut -\mathstrut \) \(53\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(30\!\cdots\!16\) \(\nu^{3}\mathstrut -\mathstrut \) \(19\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(59\!\cdots\!48\) \(\nu\mathstrut -\mathstrut \) \(21\!\cdots\!96\)\()/\)\(81\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(37\!\cdots\!61\) \(\nu^{15}\mathstrut -\mathstrut \) \(22\!\cdots\!28\) \(\nu^{14}\mathstrut -\mathstrut \) \(19\!\cdots\!67\) \(\nu^{13}\mathstrut -\mathstrut \) \(42\!\cdots\!16\) \(\nu^{12}\mathstrut -\mathstrut \) \(29\!\cdots\!96\) \(\nu^{11}\mathstrut -\mathstrut \) \(29\!\cdots\!08\) \(\nu^{10}\mathstrut -\mathstrut \) \(20\!\cdots\!60\) \(\nu^{9}\mathstrut -\mathstrut \) \(91\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(73\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!56\) \(\nu^{5}\mathstrut -\mathstrut \) \(73\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(87\!\cdots\!92\) \(\nu^{3}\mathstrut -\mathstrut \) \(37\!\cdots\!16\) \(\nu^{2}\mathstrut -\mathstrut \) \(15\!\cdots\!76\) \(\nu\mathstrut -\mathstrut \) \(86\!\cdots\!48\)\()/\)\(25\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(53\!\cdots\!73\) \(\nu^{15}\mathstrut -\mathstrut \) \(45\!\cdots\!04\) \(\nu^{14}\mathstrut -\mathstrut \) \(33\!\cdots\!31\) \(\nu^{13}\mathstrut -\mathstrut \) \(87\!\cdots\!88\) \(\nu^{12}\mathstrut +\mathstrut \) \(13\!\cdots\!72\) \(\nu^{11}\mathstrut -\mathstrut \) \(61\!\cdots\!44\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!20\) \(\nu^{9}\mathstrut -\mathstrut \) \(19\!\cdots\!40\) \(\nu^{8}\mathstrut +\mathstrut \) \(49\!\cdots\!20\) \(\nu^{7}\mathstrut -\mathstrut \) \(26\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(82\!\cdots\!92\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(53\!\cdots\!44\) \(\nu^{3}\mathstrut -\mathstrut \) \(64\!\cdots\!88\) \(\nu^{2}\mathstrut +\mathstrut \) \(92\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(10\!\cdots\!64\)\()/\)\(25\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(21\!\cdots\!99\) \(\nu^{15}\mathstrut -\mathstrut \) \(48\!\cdots\!88\) \(\nu^{14}\mathstrut +\mathstrut \) \(57\!\cdots\!53\) \(\nu^{13}\mathstrut -\mathstrut \) \(93\!\cdots\!36\) \(\nu^{12}\mathstrut +\mathstrut \) \(61\!\cdots\!64\) \(\nu^{11}\mathstrut -\mathstrut \) \(65\!\cdots\!68\) \(\nu^{10}\mathstrut +\mathstrut \) \(34\!\cdots\!40\) \(\nu^{9}\mathstrut -\mathstrut \) \(20\!\cdots\!80\) \(\nu^{8}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\nu^{7}\mathstrut -\mathstrut \) \(27\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(19\!\cdots\!04\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(18\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(56\!\cdots\!36\) \(\nu^{2}\mathstrut +\mathstrut \) \(63\!\cdots\!84\) \(\nu\mathstrut -\mathstrut \) \(85\!\cdots\!08\)\()/\)\(10\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(13143142522\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(473\) \(\beta_{4}\mathstrut +\mathstrut \) \(1699774\) \(\beta_{3}\mathstrut -\mathstrut \) \(90\) \(\beta_{2}\mathstrut -\mathstrut \) \(20788434785\) \(\beta_{1}\mathstrut +\mathstrut \) \(46\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(4\) \(\beta_{14}\mathstrut +\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(420\) \(\beta_{11}\mathstrut +\mathstrut \) \(2628\) \(\beta_{10}\mathstrut +\mathstrut \) \(1101\) \(\beta_{9}\mathstrut +\mathstrut \) \(4876\) \(\beta_{8}\mathstrut -\mathstrut \) \(1265\) \(\beta_{7}\mathstrut +\mathstrut \) \(349\) \(\beta_{6}\mathstrut +\mathstrut \) \(153948\) \(\beta_{5}\mathstrut -\mathstrut \) \(18509684\) \(\beta_{4}\mathstrut -\mathstrut \) \(7294339\) \(\beta_{3}\mathstrut -\mathstrut \) \(27408168543\) \(\beta_{2}\mathstrut +\mathstrut \) \(2922188\) \(\beta_{1}\mathstrut +\mathstrut \) \(273232443522296737142\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(339944\) \(\beta_{15}\mathstrut +\mathstrut \) \(143566\) \(\beta_{14}\mathstrut +\mathstrut \) \(53300\) \(\beta_{13}\mathstrut +\mathstrut \) \(11091950\) \(\beta_{11}\mathstrut -\mathstrut \) \(8112256005\) \(\beta_{10}\mathstrut +\mathstrut \) \(266500\) \(\beta_{9}\mathstrut +\mathstrut \) \(12443321181\) \(\beta_{8}\mathstrut +\mathstrut \) \(82506078\) \(\beta_{7}\mathstrut +\mathstrut \) \(213200\) \(\beta_{6}\mathstrut +\mathstrut \) \(20588489624\) \(\beta_{5}\mathstrut -\mathstrut \) \(4750390453617\) \(\beta_{4}\mathstrut -\mathstrut \) \(11872957517962670\) \(\beta_{3}\mathstrut +\mathstrut \) \(903102768008\) \(\beta_{2}\mathstrut +\mathstrut \) \(120700583279464002057\) \(\beta_{1}\mathstrut -\mathstrut \) \(461840191346\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(38691410076\) \(\beta_{14}\mathstrut -\mathstrut \) \(38691410076\) \(\beta_{13}\mathstrut +\mathstrut \) \(13223250699\) \(\beta_{12}\mathstrut +\mathstrut \) \(4162009207020\) \(\beta_{11}\mathstrut -\mathstrut \) \(32986056615036\) \(\beta_{10}\mathstrut -\mathstrut \) \(8033954238495\) \(\beta_{9}\mathstrut -\mathstrut \) \(62211219716532\) \(\beta_{8}\mathstrut +\mathstrut \) \(16008407339547\) \(\beta_{7}\mathstrut -\mathstrut \) \(3257396179839\) \(\beta_{6}\mathstrut -\mathstrut \) \(82108779088404\) \(\beta_{5}\mathstrut +\mathstrut \) \(255933696338005068\) \(\beta_{4}\mathstrut +\mathstrut \) \(101699225260366113\) \(\beta_{3}\mathstrut +\mathstrut \) \(174865281976625073573\) \(\beta_{2}\mathstrut -\mathstrut \) \(41135413774485108\) \(\beta_{1}\mathstrut -\mathstrut \) \(1586434433353057643791277177810\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(3180059300568888\) \(\beta_{15}\mathstrut -\mathstrut \) \(1989562846558986\) \(\beta_{14}\mathstrut -\mathstrut \) \(682495578683100\) \(\beta_{13}\mathstrut -\mathstrut \) \(133101004168009962\) \(\beta_{11}\mathstrut +\mathstrut \) \(55867615673651554899\) \(\beta_{10}\mathstrut -\mathstrut \) \(3412477893415500\) \(\beta_{9}\mathstrut -\mathstrut \) \(110870247895935388635\) \(\beta_{8}\mathstrut -\mathstrut \) \(778203779586997818\) \(\beta_{7}\mathstrut -\mathstrut \) \(2729982314732400\) \(\beta_{6}\mathstrut -\mathstrut \) \(152650600346693921136\) \(\beta_{5}\mathstrut +\mathstrut \) \(34703566322342268813495\) \(\beta_{4}\mathstrut +\mathstrut \) \(70940770412212062890977026\) \(\beta_{3}\mathstrut -\mathstrut \) \(6593137442392485800736\) \(\beta_{2}\mathstrut -\mathstrut \) \(730927628979807215737910071967\) \(\beta_{1}\mathstrut +\mathstrut \) \(3372828349439632712142\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(264609344508375812196\) \(\beta_{14}\mathstrut +\mathstrut \) \(264609344508375812196\) \(\beta_{13}\mathstrut -\mathstrut \) \(119516119976303242629\) \(\beta_{12}\mathstrut -\mathstrut \) \(29278167121157320388820\) \(\beta_{11}\mathstrut +\mathstrut \) \(294271162303712449575684\) \(\beta_{10}\mathstrut +\mathstrut \) \(37763170620119501046129\) \(\beta_{9}\mathstrut +\mathstrut \) \(562123706066748174042828\) \(\beta_{8}\mathstrut -\mathstrut \) \(143734025330080993219701\) \(\beta_{7}\mathstrut +\mathstrut \) \(24068914271773728638673\) \(\beta_{6}\mathstrut -\mathstrut \) \(9473316504359568444683220\) \(\beta_{5}\mathstrut -\mathstrut \) \(2424199017627265921901428020\) \(\beta_{4}\mathstrut -\mathstrut \) \(967502227512887159280328335\) \(\beta_{3}\mathstrut -\mathstrut \) \(1105442735316356066918295259403\) \(\beta_{2}\mathstrut +\mathstrut \) \(393292499853826490331221004\) \(\beta_{1}\mathstrut +\mathstrut \) \(9606981528873488131698910265170973424014\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(22987125440706420849426504\) \(\beta_{15}\mathstrut +\mathstrut \) \(19009323057437253107077782\) \(\beta_{14}\mathstrut +\mathstrut \) \(6429781916420479973156580\) \(\beta_{13}\mathstrut +\mathstrut \) \(1198962478244614642116975798\) \(\beta_{11}\mathstrut -\mathstrut \) \(367089260383507723505490241325\) \(\beta_{10}\mathstrut +\mathstrut \) \(32148909582102399865782900\) \(\beta_{9}\mathstrut +\mathstrut \) \(868061735238426627390866509349\) \(\beta_{8}\mathstrut +\mathstrut \) \(5156950633741252150064882790\) \(\beta_{7}\mathstrut +\mathstrut \) \(25719127665681919892626320\) \(\beta_{6}\mathstrut +\mathstrut \) \(1002617710503556632140842715440\) \(\beta_{5}\mathstrut -\mathstrut \) \(225180218569333261335024791389321\) \(\beta_{4}\mathstrut -\mathstrut \) \(405340061473344909430516411732026494\) \(\beta_{3}\mathstrut +\mathstrut \) \(42752837531957321621604798140928\) \(\beta_{2}\mathstrut +\mathstrut \) \(4515521248584313509424046101458943315873\) \(\beta_{1}\mathstrut -\mathstrut \) \(21877133063760774840188807300978\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(15\!\cdots\!40\) \(\beta_{14}\mathstrut -\mathstrut \) \(15\!\cdots\!40\) \(\beta_{13}\mathstrut +\mathstrut \) \(93\!\cdots\!35\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\beta_{11}\mathstrut -\mathstrut \) \(23\!\cdots\!60\) \(\beta_{10}\mathstrut -\mathstrut \) \(94\!\cdots\!55\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\!\cdots\!20\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\!\cdots\!15\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\!\cdots\!15\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\!\cdots\!80\) \(\beta_{5}\mathstrut +\mathstrut \) \(20\!\cdots\!00\) \(\beta_{4}\mathstrut +\mathstrut \) \(81\!\cdots\!25\) \(\beta_{3}\mathstrut +\mathstrut \) \(69\!\cdots\!85\) \(\beta_{2}\mathstrut -\mathstrut \) \(33\!\cdots\!00\) \(\beta_{1}\mathstrut -\mathstrut \) \(59\!\cdots\!06\)\()/16\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(15\!\cdots\!40\) \(\beta_{15}\mathstrut -\mathstrut \) \(15\!\cdots\!90\) \(\beta_{14}\mathstrut -\mathstrut \) \(54\!\cdots\!00\) \(\beta_{13}\mathstrut -\mathstrut \) \(97\!\cdots\!90\) \(\beta_{11}\mathstrut +\mathstrut \) \(23\!\cdots\!75\) \(\beta_{10}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(63\!\cdots\!75\) \(\beta_{8}\mathstrut -\mathstrut \) \(29\!\cdots\!70\) \(\beta_{7}\mathstrut -\mathstrut \) \(21\!\cdots\!00\) \(\beta_{6}\mathstrut -\mathstrut \) \(62\!\cdots\!40\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\!\cdots\!75\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\!\cdots\!90\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\!\cdots\!40\) \(\beta_{2}\mathstrut -\mathstrut \) \(28\!\cdots\!79\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\!\cdots\!10\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(87\!\cdots\!60\) \(\beta_{14}\mathstrut +\mathstrut \) \(87\!\cdots\!60\) \(\beta_{13}\mathstrut -\mathstrut \) \(69\!\cdots\!65\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\!\cdots\!20\) \(\beta_{10}\mathstrut -\mathstrut \) \(51\!\cdots\!95\) \(\beta_{9}\mathstrut +\mathstrut \) \(35\!\cdots\!40\) \(\beta_{8}\mathstrut -\mathstrut \) \(88\!\cdots\!25\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\!\cdots\!65\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\!\cdots\!00\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\!\cdots\!20\) \(\beta_{4}\mathstrut -\mathstrut \) \(63\!\cdots\!75\) \(\beta_{3}\mathstrut -\mathstrut \) \(44\!\cdots\!39\) \(\beta_{2}\mathstrut +\mathstrut \) \(26\!\cdots\!60\) \(\beta_{1}\mathstrut +\mathstrut \) \(37\!\cdots\!78\)\()/16\)
\(\nu^{13}\)\(=\)\((\)\(98\!\cdots\!80\) \(\beta_{15}\mathstrut +\mathstrut \) \(12\!\cdots\!90\) \(\beta_{14}\mathstrut +\mathstrut \) \(42\!\cdots\!00\) \(\beta_{13}\mathstrut +\mathstrut \) \(75\!\cdots\!10\) \(\beta_{11}\mathstrut -\mathstrut \) \(15\!\cdots\!41\) \(\beta_{10}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\beta_{9}\mathstrut +\mathstrut \) \(45\!\cdots\!21\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\!\cdots\!50\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\beta_{6}\mathstrut +\mathstrut \) \(37\!\cdots\!32\) \(\beta_{5}\mathstrut -\mathstrut \) \(82\!\cdots\!13\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\!\cdots\!94\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\!\cdots\!25\) \(\beta_{1}\mathstrut -\mathstrut \) \(80\!\cdots\!46\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(45\!\cdots\!64\) \(\beta_{14}\mathstrut -\mathstrut \) \(45\!\cdots\!64\) \(\beta_{13}\mathstrut +\mathstrut \) \(49\!\cdots\!91\) \(\beta_{12}\mathstrut +\mathstrut \) \(58\!\cdots\!20\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\!\cdots\!48\) \(\beta_{10}\mathstrut +\mathstrut \) \(11\!\cdots\!09\) \(\beta_{9}\mathstrut -\mathstrut \) \(26\!\cdots\!16\) \(\beta_{8}\mathstrut +\mathstrut \) \(66\!\cdots\!15\) \(\beta_{7}\mathstrut -\mathstrut \) \(73\!\cdots\!59\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\!\cdots\!52\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\!\cdots\!24\) \(\beta_{4}\mathstrut +\mathstrut \) \(48\!\cdots\!89\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\!\cdots\!13\) \(\beta_{2}\mathstrut -\mathstrut \) \(19\!\cdots\!28\) \(\beta_{1}\mathstrut -\mathstrut \) \(23\!\cdots\!42\)\()/16\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(62\!\cdots\!04\) \(\beta_{15}\mathstrut -\mathstrut \) \(91\!\cdots\!86\) \(\beta_{14}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(\beta_{13}\mathstrut -\mathstrut \) \(56\!\cdots\!90\) \(\beta_{11}\mathstrut +\mathstrut \) \(97\!\cdots\!95\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\beta_{9}\mathstrut -\mathstrut \) \(31\!\cdots\!31\) \(\beta_{8}\mathstrut -\mathstrut \) \(64\!\cdots\!38\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\!\cdots\!00\) \(\beta_{6}\mathstrut -\mathstrut \) \(22\!\cdots\!64\) \(\beta_{5}\mathstrut +\mathstrut \) \(48\!\cdots\!47\) \(\beta_{4}\mathstrut +\mathstrut \) \(68\!\cdots\!50\) \(\beta_{3}\mathstrut -\mathstrut \) \(91\!\cdots\!68\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\!\cdots\!07\) \(\beta_{1}\mathstrut +\mathstrut \) \(47\!\cdots\!66\)\()/8\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
81004.3i
78261.9i
78034.2i
57471.0i
51228.2i
31348.8i
20202.3i
13863.7i
13863.7i
20202.3i
31348.8i
51228.2i
57471.0i
78034.2i
78261.9i
81004.3i
162009.i 1.50582e7i −1.76568e10 3.29148e11 + 8.98715e10i −2.43956e12 1.33679e14i 1.46891e15i 5.33231e15 1.45600e16 5.33248e16i
4.2 156524.i 1.41317e8i −1.59098e10 −2.87690e11 1.83439e11i 2.21195e13 8.25005e13i 1.14573e15i −1.44114e16 −2.87126e16 + 4.50303e16i
4.3 156068.i 6.96239e7i −1.57674e10 −2.14154e11 2.65619e11i −1.08661e13 1.55193e14i 1.12018e15i 7.11573e14 −4.14548e16 + 3.34226e16i
4.4 114942.i 6.13071e7i −4.62174e9 1.22948e11 + 3.18275e11i 7.04677e12 1.47106e14i 4.56113e14i 1.80050e15 3.65832e16 1.41318e16i
4.5 102456.i 8.77372e7i −1.90737e9 −2.51871e11 + 2.30165e11i −8.98924e12 1.11815e14i 6.84671e14i −2.13876e15 2.35819e16 + 2.58058e16i
4.6 62697.7i 3.24591e7i 4.65894e9 1.34579e11 3.13534e11i 2.03511e12 2.03055e13i 8.30673e14i 4.50547e15 −1.96579e16 8.43781e15i
4.7 40404.6i 1.28550e8i 6.95740e9 3.40199e11 2.60749e10i −5.19400e12 4.50276e13i 6.28184e14i −1.09659e16 −1.05355e15 1.37456e16i
4.8 27727.3i 6.45007e7i 7.82113e9 −2.89243e11 + 1.80980e11i 1.78843e12 3.09117e13i 4.55035e14i 1.39871e15 5.01809e15 + 8.01994e15i
4.9 27727.3i 6.45007e7i 7.82113e9 −2.89243e11 1.80980e11i 1.78843e12 3.09117e13i 4.55035e14i 1.39871e15 5.01809e15 8.01994e15i
4.10 40404.6i 1.28550e8i 6.95740e9 3.40199e11 + 2.60749e10i −5.19400e12 4.50276e13i 6.28184e14i −1.09659e16 −1.05355e15 + 1.37456e16i
4.11 62697.7i 3.24591e7i 4.65894e9 1.34579e11 + 3.13534e11i 2.03511e12 2.03055e13i 8.30673e14i 4.50547e15 −1.96579e16 + 8.43781e15i
4.12 102456.i 8.77372e7i −1.90737e9 −2.51871e11 2.30165e11i −8.98924e12 1.11815e14i 6.84671e14i −2.13876e15 2.35819e16 2.58058e16i
4.13 114942.i 6.13071e7i −4.62174e9 1.22948e11 3.18275e11i 7.04677e12 1.47106e14i 4.56113e14i 1.80050e15 3.65832e16 + 1.41318e16i
4.14 156068.i 6.96239e7i −1.57674e10 −2.14154e11 + 2.65619e11i −1.08661e13 1.55193e14i 1.12018e15i 7.11573e14 −4.14548e16 3.34226e16i
4.15 156524.i 1.41317e8i −1.59098e10 −2.87690e11 + 1.83439e11i 2.21195e13 8.25005e13i 1.14573e15i −1.44114e16 −2.87126e16 4.50303e16i
4.16 162009.i 1.50582e7i −1.76568e10 3.29148e11 8.98715e10i −2.43956e12 1.33679e14i 1.46891e15i 5.33231e15 1.45600e16 + 5.33248e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{34}^{\mathrm{new}}(5, [\chi])\).