# Properties

 Label 7.9.d.a Level 7 Weight 9 Character orbit 7.d Analytic conductor 2.852 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 7.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.85165027043$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3\cdot 7^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{2} + ( -7 + 7 \beta_{2} - \beta_{5} + \beta_{6} ) q^{3} + ( -41 + 3 \beta_{1} - 41 \beta_{2} + \beta_{4} - \beta_{6} ) q^{4} + ( -140 + \beta_{1} - 70 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5} + ( -12 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( 182 - 21 \beta_{1} + 399 \beta_{2} + 42 \beta_{3} - 7 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{7} + ( 818 - 38 \beta_{3} + 14 \beta_{5} - 22 \beta_{6} - 6 \beta_{7} ) q^{8} + ( -93 \beta_{1} - 99 \beta_{2} + 93 \beta_{3} - 15 \beta_{4} + 84 \beta_{5} - 42 \beta_{6} + 15 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{2} + ( -7 + 7 \beta_{2} - \beta_{5} + \beta_{6} ) q^{3} + ( -41 + 3 \beta_{1} - 41 \beta_{2} + \beta_{4} - \beta_{6} ) q^{4} + ( -140 + \beta_{1} - 70 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{5} + ( -12 \beta_{1} + 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( 182 - 21 \beta_{1} + 399 \beta_{2} + 42 \beta_{3} - 7 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{7} + ( 818 - 38 \beta_{3} + 14 \beta_{5} - 22 \beta_{6} - 6 \beta_{7} ) q^{8} + ( -93 \beta_{1} - 99 \beta_{2} + 93 \beta_{3} - 15 \beta_{4} + 84 \beta_{5} - 42 \beta_{6} + 15 \beta_{7} ) q^{9} + ( 483 + 267 \beta_{1} - 483 \beta_{2} - 534 \beta_{3} + 8 \beta_{4} + 34 \beta_{5} - 26 \beta_{6} - 16 \beta_{7} ) q^{10} + ( 446 + 721 \beta_{1} + 446 \beta_{2} - 9 \beta_{4} - 77 \beta_{5} - 68 \beta_{6} ) q^{11} + ( -6818 - 261 \beta_{1} - 3409 \beta_{2} - 261 \beta_{3} + 5 \beta_{4} + 200 \beta_{6} + 5 \beta_{7} ) q^{12} + ( 5915 - 1602 \beta_{1} + 11830 \beta_{2} + 801 \beta_{3} - 6 \beta_{4} - 98 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{13} + ( -7434 - 154 \beta_{1} - 12404 \beta_{2} + 1960 \beta_{3} + 77 \beta_{4} - 126 \beta_{5} + 112 \beta_{6} - 7 \beta_{7} ) q^{14} + ( 16476 - 447 \beta_{3} - 35 \beta_{5} - 11 \beta_{6} + 81 \beta_{7} ) q^{15} + ( -1062 \beta_{1} + 2146 \beta_{2} + 1062 \beta_{3} + 158 \beta_{4} - 112 \beta_{5} + 56 \beta_{6} - 158 \beta_{7} ) q^{16} + ( -11788 + 2369 \beta_{1} + 11788 \beta_{2} - 4738 \beta_{3} - 81 \beta_{4} - 78 \beta_{5} - 3 \beta_{6} + 162 \beta_{7} ) q^{17} + ( -30486 + 5430 \beta_{1} - 30486 \beta_{2} + 12 \beta_{4} + 294 \beta_{5} + 282 \beta_{6} ) q^{18} + ( -42924 - 2019 \beta_{1} - 21462 \beta_{2} - 2019 \beta_{3} - 157 \beta_{4} - 865 \beta_{6} - 157 \beta_{7} ) q^{19} + ( 62517 - 4590 \beta_{1} + 125034 \beta_{2} + 2295 \beta_{3} - 138 \beta_{4} + 756 \beta_{5} + 69 \beta_{6} + 69 \beta_{7} ) q^{20} + ( 1792 - 378 \beta_{1} - 114919 \beta_{2} + 3633 \beta_{3} - 308 \beta_{4} + 1316 \beta_{5} - 1477 \beta_{6} + 91 \beta_{7} ) q^{21} + ( 213282 - 1776 \beta_{3} + 28 \beta_{5} + 407 \beta_{6} - 463 \beta_{7} ) q^{22} + ( 296 \beta_{1} + 87235 \beta_{2} - 296 \beta_{3} - 522 \beta_{4} - 1778 \beta_{5} + 889 \beta_{6} + 522 \beta_{7} ) q^{23} + ( -74592 - 144 \beta_{1} + 74592 \beta_{2} + 288 \beta_{3} + 220 \beta_{4} - 722 \beta_{5} + 942 \beta_{6} - 440 \beta_{7} ) q^{24} + ( -139438 - 7080 \beta_{1} - 139438 \beta_{2} + 156 \beta_{4} + 952 \beta_{5} + 796 \beta_{6} ) q^{25} + ( -485520 + 6984 \beta_{1} - 242760 \beta_{2} + 6984 \beta_{3} + 908 \beta_{4} - 1886 \beta_{6} + 908 \beta_{7} ) q^{26} + ( 284088 + 15066 \beta_{1} + 568176 \beta_{2} - 7533 \beta_{3} + 1206 \beta_{4} - 1857 \beta_{5} - 603 \beta_{6} - 603 \beta_{7} ) q^{27} + ( -77077 - 1407 \beta_{1} - 525427 \beta_{2} - 19866 \beta_{3} + 455 \beta_{4} - 1274 \beta_{5} + 3437 \beta_{6} - 462 \beta_{7} ) q^{28} + ( 622897 + 11195 \beta_{3} - 1834 \beta_{5} + 2231 \beta_{6} + 1437 \beta_{7} ) q^{29} + ( 1272 \beta_{1} + 137778 \beta_{2} - 1272 \beta_{3} - 99 \beta_{4} + 2128 \beta_{5} - 1064 \beta_{6} + 99 \beta_{7} ) q^{30} + ( -198058 - 27357 \beta_{1} + 198058 \beta_{2} + 54714 \beta_{3} + 475 \beta_{4} + 3569 \beta_{5} - 3094 \beta_{6} - 950 \beta_{7} ) q^{31} + ( -83004 - 46668 \beta_{1} - 83004 \beta_{2} - 780 \beta_{4} - 5908 \beta_{5} - 5128 \beta_{6} ) q^{32} + ( -953190 - 324 \beta_{1} - 476595 \beta_{2} - 324 \beta_{3} - 1948 \beta_{4} + 12412 \beta_{6} - 1948 \beta_{7} ) q^{33} + ( 678237 + 52086 \beta_{1} + 1356474 \beta_{2} - 26043 \beta_{3} - 4096 \beta_{4} + 3246 \beta_{5} + 2048 \beta_{6} + 2048 \beta_{7} ) q^{34} + ( -170303 + 23947 \beta_{1} - 963424 \beta_{2} - 36512 \beta_{3} - 35 \beta_{4} - 6608 \beta_{5} - 924 \beta_{6} + 1036 \beta_{7} ) q^{35} + ( 1658676 - 9036 \beta_{3} + 10332 \beta_{5} - 18156 \beta_{6} - 2508 \beta_{7} ) q^{36} + ( 59724 \beta_{1} - 123185 \beta_{2} - 59724 \beta_{3} + 4082 \beta_{4} + 21028 \beta_{5} - 10514 \beta_{6} - 4082 \beta_{7} ) q^{37} + ( -590674 - 6028 \beta_{1} + 590674 \beta_{2} + 12056 \beta_{3} - 3669 \beta_{4} - 4236 \beta_{5} + 567 \beta_{6} + 7338 \beta_{7} ) q^{38} + ( -712593 - 21021 \beta_{1} - 712593 \beta_{2} + 861 \beta_{4} + 6216 \beta_{5} + 5355 \beta_{6} ) q^{39} + ( -1266972 + 25626 \beta_{1} - 633486 \beta_{2} + 25626 \beta_{3} - 302 \beta_{4} - 3690 \beta_{6} - 302 \beta_{7} ) q^{40} + ( 817719 - 49242 \beta_{1} + 1635438 \beta_{2} + 24621 \beta_{3} + 5722 \beta_{4} - 14602 \beta_{5} - 2861 \beta_{6} - 2861 \beta_{7} ) q^{41} + ( -48195 - 47838 \beta_{1} - 992922 \beta_{2} + 21945 \beta_{3} + 896 \beta_{4} + 1624 \beta_{5} + 6230 \beta_{6} - 532 \beta_{7} ) q^{42} + ( 556054 + 35340 \beta_{3} - 18074 \beta_{5} + 34436 \beta_{6} + 1712 \beta_{7} ) q^{43} + ( -145527 \beta_{1} + 919701 \beta_{2} + 145527 \beta_{3} - 5553 \beta_{4} - 52276 \beta_{5} + 26138 \beta_{6} + 5553 \beta_{7} ) q^{44} + ( 277347 + 28587 \beta_{1} - 277347 \beta_{2} - 57174 \beta_{3} + 4941 \beta_{4} - 4818 \beta_{5} + 9759 \beta_{6} - 9882 \beta_{7} ) q^{45} + ( -56640 + 180906 \beta_{1} - 56640 \beta_{2} + 3937 \beta_{4} + 5530 \beta_{5} + 1593 \beta_{6} ) q^{46} + ( 450688 - 100769 \beta_{1} + 225344 \beta_{2} - 100769 \beta_{3} + 7619 \beta_{4} - 36815 \beta_{6} + 7619 \beta_{7} ) q^{47} + ( -952966 - 111324 \beta_{1} - 1905932 \beta_{2} + 55662 \beta_{3} + 2972 \beta_{4} + 43076 \beta_{5} - 1486 \beta_{6} - 1486 \beta_{7} ) q^{48} + ( -98 + 51744 \beta_{1} + 952560 \beta_{2} + 220353 \beta_{3} - 5096 \beta_{4} + 40670 \beta_{5} - 39935 \beta_{6} - 833 \beta_{7} ) q^{49} + ( -1953614 - 126682 \beta_{3} + 280 \beta_{5} - 4316 \beta_{6} + 3756 \beta_{7} ) q^{50} + ( 103185 \beta_{1} + 87714 \beta_{2} - 103185 \beta_{3} - 10221 \beta_{4} + 4046 \beta_{5} - 2023 \beta_{6} + 10221 \beta_{7} ) q^{51} + ( 927934 + 276858 \beta_{1} - 927934 \beta_{2} - 553716 \beta_{3} + 8870 \beta_{4} + 13188 \beta_{5} - 4318 \beta_{6} - 17740 \beta_{7} ) q^{52} + ( 570365 + 179944 \beta_{1} + 570365 \beta_{2} - 15834 \beta_{4} - 4690 \beta_{5} + 11144 \beta_{6} ) q^{53} + ( 4092228 + 95940 \beta_{1} + 2046114 \beta_{2} + 95940 \beta_{3} - 7485 \beta_{4} - 14070 \beta_{6} - 7485 \beta_{7} ) q^{54} + ( -1414581 - 290868 \beta_{1} - 2829162 \beta_{2} + 145434 \beta_{3} - 24784 \beta_{4} - 43489 \beta_{5} + 12392 \beta_{6} + 12392 \beta_{7} ) q^{55} + ( 1441076 - 201012 \beta_{1} + 4525500 \beta_{2} + 79240 \beta_{3} + 1372 \beta_{4} + 910 \beta_{5} + 32928 \beta_{6} - 4004 \beta_{7} ) q^{56} + ( -5454801 - 14580 \beta_{3} + 24248 \beta_{5} - 33100 \beta_{6} - 15396 \beta_{7} ) q^{57} + ( -317094 \beta_{1} - 2860674 \beta_{2} + 317094 \beta_{3} + 21008 \beta_{4} + 32900 \beta_{5} - 16450 \beta_{6} - 21008 \beta_{7} ) q^{58} + ( 2107595 - 227362 \beta_{1} - 2107595 \beta_{2} + 454724 \beta_{3} - 27050 \beta_{4} + 11059 \beta_{5} - 38109 \beta_{6} + 54100 \beta_{7} ) q^{59} + ( 4419891 + 86751 \beta_{1} + 4419891 \beta_{2} + 16569 \beta_{4} + 12474 \beta_{5} - 4095 \beta_{6} ) q^{60} + ( 9894794 - 294960 \beta_{1} + 4947397 \beta_{2} - 294960 \beta_{3} - 5762 \beta_{4} + 111362 \beta_{6} - 5762 \beta_{7} ) q^{61} + ( -8251838 + 363688 \beta_{1} - 16503676 \beta_{2} - 181844 \beta_{3} + 44726 \beta_{4} - 12812 \beta_{5} - 22363 \beta_{6} - 22363 \beta_{7} ) q^{62} + ( -2378628 + 485205 \beta_{1} + 10263183 \beta_{2} - 218841 \beta_{3} + 20559 \beta_{4} - 163128 \beta_{5} + 44310 \beta_{6} + 12621 \beta_{7} ) q^{63} + ( -14269132 + 244452 \beta_{3} - 13440 \beta_{5} + 596 \beta_{6} + 26284 \beta_{7} ) q^{64} + ( 711795 \beta_{1} - 4230849 \beta_{2} - 711795 \beta_{3} + 24633 \beta_{4} + 76972 \beta_{5} - 38486 \beta_{6} - 24633 \beta_{7} ) q^{65} + ( 38619 - 115917 \beta_{1} - 38619 \beta_{2} + 231834 \beta_{3} + 2348 \beta_{4} - 98848 \beta_{5} + 101196 \beta_{6} - 4696 \beta_{7} ) q^{66} + ( -26777 - 819354 \beta_{1} - 26777 \beta_{2} + 30738 \beta_{4} - 67417 \beta_{5} - 98155 \beta_{6} ) q^{67} + ( 7386162 + 567741 \beta_{1} + 3693081 \beta_{2} + 567741 \beta_{3} - 2409 \beta_{4} + 117294 \beta_{6} - 2409 \beta_{7} ) q^{68} + ( -4484424 + 129474 \beta_{1} - 8968848 \beta_{2} - 64737 \beta_{3} - 35614 \beta_{4} + 55144 \beta_{5} + 17807 \beta_{6} + 17807 \beta_{7} ) q^{69} + ( 8138634 - 697158 \beta_{1} + 11497248 \beta_{2} - 476322 \beta_{3} - 26719 \beta_{4} + 28364 \beta_{5} - 15883 \beta_{6} + 16016 \beta_{7} ) q^{70} + ( -10351220 + 104234 \beta_{3} + 20930 \beta_{5} - 21718 \beta_{6} - 20142 \beta_{7} ) q^{71} + ( -503196 \beta_{1} - 3324396 \beta_{2} + 503196 \beta_{3} - 44484 \beta_{4} + 121632 \beta_{5} - 60816 \beta_{6} + 44484 \beta_{7} ) q^{72} + ( 9729867 - 595146 \beta_{1} - 9729867 \beta_{2} + 1190292 \beta_{3} + 39502 \beta_{4} + 156708 \beta_{5} - 117206 \beta_{6} - 79004 \beta_{7} ) q^{73} + ( 18172585 - 840841 \beta_{1} + 18172585 \beta_{2} - 103512 \beta_{4} - 36120 \beta_{5} + 67392 \beta_{6} ) q^{74} + ( 13305404 + 2532 \beta_{1} + 6652702 \beta_{2} + 2532 \beta_{3} + 22608 \beta_{4} - 268102 \beta_{6} + 22608 \beta_{7} ) q^{75} + ( -8371587 + 1874082 \beta_{1} - 16743174 \beta_{2} - 937041 \beta_{3} - 37842 \beta_{4} - 156198 \beta_{5} + 18921 \beta_{6} + 18921 \beta_{7} ) q^{76} + ( -7691761 + 931735 \beta_{1} - 822472 \beta_{2} - 967197 \beta_{3} + 5915 \beta_{4} + 312690 \beta_{5} + 71092 \beta_{6} - 94171 \beta_{7} ) q^{77} + ( -5515314 - 640794 \beta_{3} - 378 \beta_{5} + 966 \beta_{6} - 210 \beta_{7} ) q^{78} + ( 757308 \beta_{1} + 12657023 \beta_{2} - 757308 \beta_{3} - 80346 \beta_{4} - 500458 \beta_{5} + 250229 \beta_{6} + 80346 \beta_{7} ) q^{79} + ( -7799302 + 78446 \beta_{1} + 7799302 \beta_{2} - 156892 \beta_{3} + 2762 \beta_{4} + 189440 \beta_{5} - 186678 \beta_{6} - 5524 \beta_{7} ) q^{80} + ( -24217218 + 68355 \beta_{1} - 24217218 \beta_{2} + 62433 \beta_{4} + 242046 \beta_{5} + 179613 \beta_{6} ) q^{81} + ( -15176952 - 29532 \beta_{1} - 7588476 \beta_{2} - 29532 \beta_{3} + 30640 \beta_{4} - 210646 \beta_{6} + 30640 \beta_{7} ) q^{82} + ( 8989554 - 921852 \beta_{1} + 17979108 \beta_{2} + 460926 \beta_{3} + 137108 \beta_{4} + 347704 \beta_{5} - 68554 \beta_{6} - 68554 \beta_{7} ) q^{83} + ( 16771916 - 263571 \beta_{1} - 4980311 \beta_{2} + 203007 \beta_{3} - 30233 \beta_{4} - 112308 \beta_{5} - 396704 \beta_{6} + 88837 \beta_{7} ) q^{84} + ( 14903901 + 1468062 \beta_{3} - 110642 \beta_{5} + 200644 \beta_{6} + 20640 \beta_{7} ) q^{85} + ( -767396 \beta_{1} - 9773272 \beta_{2} + 767396 \beta_{3} + 94698 \beta_{4} - 24360 \beta_{5} + 12180 \beta_{6} - 94698 \beta_{7} ) q^{86} + ( 2225223 + 163191 \beta_{1} - 2225223 \beta_{2} - 326382 \beta_{3} - 27811 \beta_{4} - 811666 \beta_{5} + 783855 \beta_{6} + 55622 \beta_{7} ) q^{87} + ( 10348212 + 1108308 \beta_{1} + 10348212 \beta_{2} + 122072 \beta_{4} + 18298 \beta_{5} - 103774 \beta_{6} ) q^{88} + ( -387086 - 245408 \beta_{1} - 193543 \beta_{2} - 245408 \beta_{3} - 63324 \beta_{4} + 82020 \beta_{6} - 63324 \beta_{7} ) q^{89} + ( 9489270 - 3235164 \beta_{1} + 18978540 \beta_{2} + 1617582 \beta_{3} - 77184 \beta_{4} - 217158 \beta_{5} + 38592 \beta_{6} + 38592 \beta_{7} ) q^{90} + ( -30745001 - 1690500 \beta_{1} - 23541364 \beta_{2} + 3357039 \beta_{3} + 26068 \beta_{4} - 50372 \beta_{5} + 140287 \beta_{6} + 91777 \beta_{7} ) q^{91} + ( 31727391 - 1250997 \beta_{3} - 183526 \beta_{5} + 442727 \beta_{6} - 75675 \beta_{7} ) q^{92} + ( -558084 \beta_{1} + 14423301 \beta_{2} + 558084 \beta_{3} + 147006 \beta_{4} + 212044 \beta_{5} - 106022 \beta_{6} - 147006 \beta_{7} ) q^{93} + ( -28797174 + 1735512 \beta_{1} + 28797174 \beta_{2} - 3471024 \beta_{3} - 99489 \beta_{4} + 363152 \beta_{5} - 462641 \beta_{6} + 198978 \beta_{7} ) q^{94} + ( -43196763 + 2734806 \beta_{1} - 43196763 \beta_{2} - 138948 \beta_{4} - 258125 \beta_{5} - 119177 \beta_{6} ) q^{95} + ( -69409200 - 611832 \beta_{1} - 34704600 \beta_{2} - 611832 \beta_{3} - 48192 \beta_{4} + 304956 \beta_{6} - 48192 \beta_{7} ) q^{96} + ( 48586783 + 999222 \beta_{1} + 97173566 \beta_{2} - 499611 \beta_{3} - 59766 \beta_{4} + 45962 \beta_{5} + 29883 \beta_{6} + 29883 \beta_{7} ) q^{97} + ( 13376412 + 2312555 \beta_{1} - 66662099 \beta_{2} + 256417 \beta_{3} + 131320 \beta_{4} - 2940 \beta_{5} + 215894 \beta_{6} - 203252 \beta_{7} ) q^{98} + ( 84030501 - 2313849 \beta_{3} + 741846 \beta_{5} - 1624701 \beta_{6} + 141009 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} - 84q^{3} - 164q^{4} - 840q^{5} - 140q^{7} + 6544q^{8} + 396q^{9} + O(q^{10})$$ $$8q - 4q^{2} - 84q^{3} - 164q^{4} - 840q^{5} - 140q^{7} + 6544q^{8} + 396q^{9} + 5796q^{10} + 1784q^{11} - 40908q^{12} - 9856q^{14} + 131808q^{15} - 8584q^{16} - 141456q^{17} - 121944q^{18} - 257544q^{19} + 474012q^{21} + 1706256q^{22} - 348940q^{23} - 895104q^{24} - 557752q^{25} - 2913120q^{26} + 1485092q^{28} + 4983176q^{29} - 551112q^{30} - 2376696q^{31} - 332016q^{32} - 5719140q^{33} + 2491272q^{35} + 13269408q^{36} + 492740q^{37} - 7088088q^{38} - 2850372q^{39} - 7601832q^{40} + 3586128q^{42} + 4448432q^{43} - 3678804q^{44} + 3328164q^{45} - 226560q^{46} + 2704128q^{47} - 3811024q^{49} - 15628912q^{50} - 350856q^{51} + 11135208q^{52} + 2281460q^{53} + 24553368q^{54} - 6573392q^{56} - 43638408q^{57} + 11442696q^{58} + 25291140q^{59} + 17679564q^{60} + 59368764q^{61} - 60081756q^{63} - 114153056q^{64} + 16923396q^{65} + 463428q^{66} - 107108q^{67} + 44316972q^{68} + 19120080q^{70} - 82809760q^{71} + 13297584q^{72} + 116758404q^{73} + 72690340q^{74} + 79832424q^{75} - 58244200q^{77} - 44122512q^{78} - 50628092q^{79} - 93591624q^{80} - 96868872q^{81} - 91061712q^{82} + 154096572q^{84} + 119231208q^{85} + 39093088q^{86} + 26702676q^{87} + 41392848q^{88} - 2322516q^{89} - 151794552q^{91} + 253819128q^{92} - 57693204q^{93} - 345566088q^{94} - 172787052q^{95} - 416455200q^{96} + 373659692q^{98} + 672244008q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 592 x^{6} - 1176 x^{5} + 336397 x^{4} - 348096 x^{3} + 8673408 x^{2} + 8271396 x + 197880489$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-40665038848 \nu^{7} + 933800395536 \nu^{6} - 23107427488768 \nu^{5} + 578443118889849 \nu^{4} - 14228671705925824 \nu^{3} + 327714819020519376 \nu^{2} - 339657458304359916 \nu - 24226776665406171$$$$)/ 7800602713756314075$$ $$\beta_{3}$$ $$=$$ $$($$$$-199147024 \nu^{7} - 206072832 \nu^{6} - 113162941609 \nu^{5} + 117098450112 \nu^{4} - 66871290607312 \nu^{3} - 2782472800212 \nu^{2} - 66566402858133 \nu - 1716105304424448$$$$)/ 1663596228141675$$ $$\beta_{4}$$ $$=$$ $$($$$$4986016178489 \nu^{7} + 729234428373612 \nu^{6} + 1954798485604859 \nu^{5} + 440139351248781243 \nu^{4} + 208410678677305067 \nu^{3} + 242602731949311036057 \nu^{2} - 549685532050544077272 \nu + 5840361335857569042933$$$$)/ 21841687598517679410$$ $$\beta_{5}$$ $$=$$ $$($$$$-142450897959062 \nu^{7} - 415028421151641 \nu^{6} - 89730500895110567 \nu^{5} - 81489430662144594 \nu^{4} - 50276218913682852731 \nu^{3} - 79052178550067250006 \nu^{2} - 2640294308877978734829 \nu - 2293927437451668499449$$$$)/$$$$10\!\cdots\!50$$ $$\beta_{6}$$ $$=$$ $$($$$$35983628949817 \nu^{7} + 17429303220966 \nu^{6} + 19568825417085307 \nu^{5} - 37683429688699701 \nu^{4} + 11054456035703486131 \nu^{3} - 7203424572374123829 \nu^{2} - 247240100774204675046 \nu - 87312579891615556011$$$$)/ 21841687598517679410$$ $$\beta_{7}$$ $$=$$ $$($$$$206519207416061 \nu^{7} - 209968840103652 \nu^{6} + 112959866406343301 \nu^{5} - 204058573291680393 \nu^{4} + 64395501685950743543 \nu^{3} - 35645454072445999857 \nu^{2} - 1227308896964839498488 \nu + 29414857890603535700697$$$$)/$$$$10\!\cdots\!50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} + \beta_{3} + 296 \beta_{2} - \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{7} - 25 \beta_{6} + 14 \beta_{5} - 544 \beta_{3} + 441$$ $$\nu^{4}$$ $$=$$ $$-592 \beta_{6} + 592 \beta_{4} - 161165 \beta_{2} + 1180 \beta_{1} - 161165$$ $$\nu^{5}$$ $$=$$ $$2364 \beta_{7} + 8288 \beta_{6} - 16576 \beta_{5} - 2364 \beta_{4} + 308569 \beta_{3} + 435120 \beta_{2} - 308569 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-338161 \beta_{7} + 321697 \beta_{6} + 8232 \beta_{5} - 1004365 \beta_{3} + 91505156$$ $$\nu^{7}$$ $$=$$ $$3004175 \beta_{6} + 4709558 \beta_{5} + 1705383 \beta_{4} - 346152513 \beta_{2} + 175714240 \beta_{1} - 346152513$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −12.1698 + 21.0787i −2.23583 + 3.87257i 2.77916 − 4.81365i 11.6264 − 20.1376i −12.1698 − 21.0787i −2.23583 − 3.87257i 2.77916 + 4.81365i 11.6264 + 20.1376i
−12.6698 21.9447i −7.68317 4.43588i −193.046 + 334.366i −538.352 + 310.818i 224.806i 207.497 2392.02i 3296.47 −3241.15 5613.83i 13641.6 + 7875.98i
3.2 −2.73583 4.73860i 58.8837 + 33.9965i 113.030 195.774i 586.541 338.640i 372.035i −168.652 + 2395.07i −2637.67 −968.977 1678.32i −3209.35 1852.92i
3.3 2.27916 + 3.94762i −124.416 71.8315i 117.611 203.708i −163.006 + 94.1113i 654.862i −2345.65 512.580i 2239.15 7039.03 + 12192.0i −743.031 428.989i
3.4 11.1264 + 19.2716i 31.2153 + 18.0222i −119.595 + 207.145i −305.183 + 176.198i 802.090i 2236.80 872.649i 374.058 −2630.90 4556.86i −6791.21 3920.91i
5.1 −12.6698 + 21.9447i −7.68317 + 4.43588i −193.046 334.366i −538.352 310.818i 224.806i 207.497 + 2392.02i 3296.47 −3241.15 + 5613.83i 13641.6 7875.98i
5.2 −2.73583 + 4.73860i 58.8837 33.9965i 113.030 + 195.774i 586.541 + 338.640i 372.035i −168.652 2395.07i −2637.67 −968.977 + 1678.32i −3209.35 + 1852.92i
5.3 2.27916 3.94762i −124.416 + 71.8315i 117.611 + 203.708i −163.006 94.1113i 654.862i −2345.65 + 512.580i 2239.15 7039.03 12192.0i −743.031 + 428.989i
5.4 11.1264 19.2716i 31.2153 18.0222i −119.595 207.145i −305.183 176.198i 802.090i 2236.80 + 872.649i 374.058 −2630.90 + 4556.86i −6791.21 + 3920.91i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.9.d.a 8
3.b odd 2 1 63.9.m.b 8
4.b odd 2 1 112.9.s.a 8
7.b odd 2 1 49.9.d.c 8
7.c even 3 1 49.9.b.a 8
7.c even 3 1 49.9.d.c 8
7.d odd 6 1 inner 7.9.d.a 8
7.d odd 6 1 49.9.b.a 8
21.g even 6 1 63.9.m.b 8
28.f even 6 1 112.9.s.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.d.a 8 1.a even 1 1 trivial
7.9.d.a 8 7.d odd 6 1 inner
49.9.b.a 8 7.c even 3 1
49.9.b.a 8 7.d odd 6 1
49.9.d.c 8 7.b odd 2 1
49.9.d.c 8 7.c even 3 1
63.9.m.b 8 3.b odd 2 1
63.9.m.b 8 21.g even 6 1
112.9.s.a 8 4.b odd 2 1
112.9.s.a 8 28.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(7, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T - 422 T^{2} - 3208 T^{3} + 74676 T^{4} + 863136 T^{5} + 14045792 T^{6} - 108251904 T^{7} - 6403153664 T^{8} - 27712487424 T^{9} + 920505024512 T^{10} + 14481019109376 T^{11} + 320730977796096 T^{12} - 3527233301905408 T^{13} - 118782440171896832 T^{14} + 288230376151711744 T^{15} + 18446744073709551616 T^{16}$$
$3$ $$1 + 84 T + 16452 T^{2} + 1184400 T^{3} + 154752903 T^{4} + 15614686332 T^{5} + 1274438663244 T^{6} + 136656865682064 T^{7} + 8056424602211184 T^{8} + 896605695740021904 T^{9} + 54860405568277422924 T^{10} +$$$$44\!\cdots\!92$$$$T^{11} +$$$$28\!\cdots\!23$$$$T^{12} +$$$$14\!\cdots\!00$$$$T^{13} +$$$$13\!\cdots\!72$$$$T^{14} +$$$$43\!\cdots\!64$$$$T^{15} +$$$$34\!\cdots\!81$$$$T^{16}$$
$5$ $$1 + 840 T + 1412926 T^{2} + 989289840 T^{3} + 963491077201 T^{4} + 459468571579920 T^{5} + 381647807924437150 T^{6} +$$$$14\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!00$$$$T^{8} +$$$$55\!\cdots\!00$$$$T^{9} +$$$$58\!\cdots\!50$$$$T^{10} +$$$$27\!\cdots\!00$$$$T^{11} +$$$$22\!\cdots\!25$$$$T^{12} +$$$$89\!\cdots\!00$$$$T^{13} +$$$$50\!\cdots\!50$$$$T^{14} +$$$$11\!\cdots\!00$$$$T^{15} +$$$$54\!\cdots\!25$$$$T^{16}$$
$7$ $$1 + 140 T + 1915312 T^{2} + 4021242820 T^{3} - 41445333483778 T^{4} + 23181664629978820 T^{5} + 63651430715123630512 T^{6} +$$$$26\!\cdots\!40$$$$T^{7} +$$$$11\!\cdots\!01$$$$T^{8}$$
$11$ $$1 - 1784 T - 333848516 T^{2} + 1417907526152 T^{3} - 3397673438524281 T^{4} -$$$$14\!\cdots\!80$$$$T^{5} -$$$$72\!\cdots\!96$$$$T^{6} -$$$$15\!\cdots\!72$$$$T^{7} +$$$$57\!\cdots\!72$$$$T^{8} -$$$$33\!\cdots\!32$$$$T^{9} -$$$$33\!\cdots\!56$$$$T^{10} -$$$$14\!\cdots\!80$$$$T^{11} -$$$$71\!\cdots\!01$$$$T^{12} +$$$$64\!\cdots\!52$$$$T^{13} -$$$$32\!\cdots\!96$$$$T^{14} -$$$$37\!\cdots\!24$$$$T^{15} +$$$$44\!\cdots\!41$$$$T^{16}$$
$13$ $$1 - 3599704928 T^{2} + 6293879827435246396 T^{4} -$$$$73\!\cdots\!48$$$$T^{6} +$$$$65\!\cdots\!34$$$$T^{8} -$$$$48\!\cdots\!68$$$$T^{10} +$$$$27\!\cdots\!76$$$$T^{12} -$$$$10\!\cdots\!88$$$$T^{14} +$$$$19\!\cdots\!61$$$$T^{16}$$
$17$ $$1 + 141456 T + 24752467462 T^{2} + 2557882950722400 T^{3} +$$$$31\!\cdots\!53$$$$T^{4} +$$$$34\!\cdots\!68$$$$T^{5} +$$$$33\!\cdots\!54$$$$T^{6} +$$$$32\!\cdots\!76$$$$T^{7} +$$$$25\!\cdots\!24$$$$T^{8} +$$$$22\!\cdots\!16$$$$T^{9} +$$$$16\!\cdots\!74$$$$T^{10} +$$$$11\!\cdots\!28$$$$T^{11} +$$$$74\!\cdots\!33$$$$T^{12} +$$$$42\!\cdots\!00$$$$T^{13} +$$$$28\!\cdots\!42$$$$T^{14} +$$$$11\!\cdots\!36$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$19$ $$1 + 257544 T + 70009367908 T^{2} + 12336288216616224 T^{3} +$$$$20\!\cdots\!39$$$$T^{4} +$$$$30\!\cdots\!20$$$$T^{5} +$$$$43\!\cdots\!52$$$$T^{6} +$$$$61\!\cdots\!88$$$$T^{7} +$$$$82\!\cdots\!52$$$$T^{8} +$$$$10\!\cdots\!08$$$$T^{9} +$$$$12\!\cdots\!12$$$$T^{10} +$$$$14\!\cdots\!20$$$$T^{11} +$$$$16\!\cdots\!79$$$$T^{12} +$$$$17\!\cdots\!24$$$$T^{13} +$$$$16\!\cdots\!28$$$$T^{14} +$$$$10\!\cdots\!64$$$$T^{15} +$$$$69\!\cdots\!21$$$$T^{16}$$
$23$ $$1 + 348940 T - 186978426644 T^{2} - 32676900622844200 T^{3} +$$$$35\!\cdots\!19$$$$T^{4} +$$$$26\!\cdots\!40$$$$T^{5} -$$$$40\!\cdots\!80$$$$T^{6} -$$$$10\!\cdots\!80$$$$T^{7} +$$$$34\!\cdots\!72$$$$T^{8} -$$$$80\!\cdots\!80$$$$T^{9} -$$$$24\!\cdots\!80$$$$T^{10} +$$$$12\!\cdots\!40$$$$T^{11} +$$$$13\!\cdots\!99$$$$T^{12} -$$$$96\!\cdots\!00$$$$T^{13} -$$$$43\!\cdots\!64$$$$T^{14} +$$$$63\!\cdots\!40$$$$T^{15} +$$$$14\!\cdots\!41$$$$T^{16}$$
$29$ $$( 1 - 2491588 T + 3930639559056 T^{2} - 4191570786881848748 T^{3} +$$$$34\!\cdots\!34$$$$T^{4} -$$$$20\!\cdots\!28$$$$T^{5} +$$$$98\!\cdots\!76$$$$T^{6} -$$$$31\!\cdots\!28$$$$T^{7} +$$$$62\!\cdots\!41$$$$T^{8} )^{2}$$
$31$ $$1 + 2376696 T + 4495851115156 T^{2} + 6210203237206004064 T^{3} +$$$$70\!\cdots\!63$$$$T^{4} +$$$$75\!\cdots\!64$$$$T^{5} +$$$$78\!\cdots\!20$$$$T^{6} +$$$$78\!\cdots\!92$$$$T^{7} +$$$$76\!\cdots\!08$$$$T^{8} +$$$$66\!\cdots\!72$$$$T^{9} +$$$$57\!\cdots\!20$$$$T^{10} +$$$$47\!\cdots\!44$$$$T^{11} +$$$$37\!\cdots\!43$$$$T^{12} +$$$$28\!\cdots\!64$$$$T^{13} +$$$$17\!\cdots\!96$$$$T^{14} +$$$$78\!\cdots\!76$$$$T^{15} +$$$$27\!\cdots\!21$$$$T^{16}$$
$37$ $$1 - 492740 T - 6839200540314 T^{2} + 5004808393655458680 T^{3} +$$$$16\!\cdots\!69$$$$T^{4} -$$$$13\!\cdots\!40$$$$T^{5} -$$$$40\!\cdots\!30$$$$T^{6} +$$$$11\!\cdots\!20$$$$T^{7} +$$$$20\!\cdots\!12$$$$T^{8} +$$$$39\!\cdots\!20$$$$T^{9} -$$$$50\!\cdots\!30$$$$T^{10} -$$$$58\!\cdots\!40$$$$T^{11} +$$$$24\!\cdots\!89$$$$T^{12} +$$$$26\!\cdots\!80$$$$T^{13} -$$$$12\!\cdots\!94$$$$T^{14} -$$$$32\!\cdots\!40$$$$T^{15} +$$$$23\!\cdots\!61$$$$T^{16}$$
$41$ $$1 - 37601617521056 T^{2} +$$$$67\!\cdots\!60$$$$T^{4} -$$$$78\!\cdots\!52$$$$T^{6} +$$$$69\!\cdots\!34$$$$T^{8} -$$$$50\!\cdots\!32$$$$T^{10} +$$$$27\!\cdots\!60$$$$T^{12} -$$$$97\!\cdots\!76$$$$T^{14} +$$$$16\!\cdots\!61$$$$T^{16}$$
$43$ $$( 1 - 2224216 T + 36058463751268 T^{2} - 59838646062515811848 T^{3} +$$$$58\!\cdots\!34$$$$T^{4} -$$$$69\!\cdots\!48$$$$T^{5} +$$$$49\!\cdots\!68$$$$T^{6} -$$$$35\!\cdots\!16$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$1 - 2704128 T + 43354016357404 T^{2} -$$$$11\!\cdots\!28$$$$T^{3} +$$$$86\!\cdots\!35$$$$T^{4} +$$$$26\!\cdots\!60$$$$T^{5} -$$$$76\!\cdots\!24$$$$T^{6} +$$$$20\!\cdots\!24$$$$T^{7} -$$$$69\!\cdots\!32$$$$T^{8} +$$$$48\!\cdots\!64$$$$T^{9} -$$$$43\!\cdots\!04$$$$T^{10} +$$$$35\!\cdots\!60$$$$T^{11} +$$$$27\!\cdots\!35$$$$T^{12} -$$$$84\!\cdots\!28$$$$T^{13} +$$$$79\!\cdots\!44$$$$T^{14} -$$$$11\!\cdots\!88$$$$T^{15} +$$$$10\!\cdots\!81$$$$T^{16}$$
$53$ $$1 - 2281460 T - 194027845315466 T^{2} +$$$$48\!\cdots\!80$$$$T^{3} +$$$$21\!\cdots\!21$$$$T^{4} -$$$$44\!\cdots\!40$$$$T^{5} -$$$$17\!\cdots\!38$$$$T^{6} +$$$$13\!\cdots\!20$$$$T^{7} +$$$$11\!\cdots\!08$$$$T^{8} +$$$$81\!\cdots\!20$$$$T^{9} -$$$$66\!\cdots\!98$$$$T^{10} -$$$$10\!\cdots\!40$$$$T^{11} +$$$$32\!\cdots\!61$$$$T^{12} +$$$$45\!\cdots\!80$$$$T^{13} -$$$$11\!\cdots\!26$$$$T^{14} -$$$$82\!\cdots\!60$$$$T^{15} +$$$$22\!\cdots\!81$$$$T^{16}$$
$59$ $$1 - 25291140 T + 442662123701212 T^{2} -$$$$58\!\cdots\!80$$$$T^{3} +$$$$47\!\cdots\!71$$$$T^{4} -$$$$19\!\cdots\!16$$$$T^{5} -$$$$50\!\cdots\!36$$$$T^{6} +$$$$14\!\cdots\!92$$$$T^{7} -$$$$20\!\cdots\!44$$$$T^{8} +$$$$21\!\cdots\!32$$$$T^{9} -$$$$10\!\cdots\!76$$$$T^{10} -$$$$62\!\cdots\!76$$$$T^{11} +$$$$21\!\cdots\!51$$$$T^{12} -$$$$39\!\cdots\!80$$$$T^{13} +$$$$44\!\cdots\!52$$$$T^{14} -$$$$37\!\cdots\!40$$$$T^{15} +$$$$21\!\cdots\!61$$$$T^{16}$$
$61$ $$1 - 59368764 T + 2052606459885622 T^{2} -$$$$52\!\cdots\!60$$$$T^{3} +$$$$10\!\cdots\!05$$$$T^{4} -$$$$18\!\cdots\!72$$$$T^{5} +$$$$29\!\cdots\!94$$$$T^{6} -$$$$44\!\cdots\!08$$$$T^{7} +$$$$62\!\cdots\!84$$$$T^{8} -$$$$84\!\cdots\!48$$$$T^{9} +$$$$10\!\cdots\!34$$$$T^{10} -$$$$13\!\cdots\!52$$$$T^{11} +$$$$14\!\cdots\!05$$$$T^{12} -$$$$13\!\cdots\!60$$$$T^{13} +$$$$10\!\cdots\!82$$$$T^{14} -$$$$56\!\cdots\!04$$$$T^{15} +$$$$18\!\cdots\!41$$$$T^{16}$$
$67$ $$1 + 107108 T - 839204332190988 T^{2} -$$$$18\!\cdots\!44$$$$T^{3} +$$$$37\!\cdots\!55$$$$T^{4} +$$$$11\!\cdots\!96$$$$T^{5} +$$$$84\!\cdots\!16$$$$T^{6} -$$$$32\!\cdots\!36$$$$T^{7} -$$$$68\!\cdots\!16$$$$T^{8} -$$$$13\!\cdots\!76$$$$T^{9} +$$$$13\!\cdots\!96$$$$T^{10} +$$$$76\!\cdots\!16$$$$T^{11} +$$$$10\!\cdots\!55$$$$T^{12} -$$$$20\!\cdots\!44$$$$T^{13} -$$$$37\!\cdots\!08$$$$T^{14} +$$$$19\!\cdots\!48$$$$T^{15} +$$$$73\!\cdots\!21$$$$T^{16}$$
$71$ $$( 1 + 41404880 T + 3162773608457988 T^{2} +$$$$83\!\cdots\!72$$$$T^{3} +$$$$32\!\cdots\!98$$$$T^{4} +$$$$53\!\cdots\!92$$$$T^{5} +$$$$13\!\cdots\!48$$$$T^{6} +$$$$11\!\cdots\!80$$$$T^{7} +$$$$17\!\cdots\!41$$$$T^{8} )^{2}$$
$73$ $$1 - 116758404 T + 8148090587057374 T^{2} -$$$$42\!\cdots\!08$$$$T^{3} +$$$$17\!\cdots\!45$$$$T^{4} -$$$$65\!\cdots\!68$$$$T^{5} +$$$$21\!\cdots\!86$$$$T^{6} -$$$$66\!\cdots\!72$$$$T^{7} +$$$$19\!\cdots\!68$$$$T^{8} -$$$$53\!\cdots\!32$$$$T^{9} +$$$$14\!\cdots\!46$$$$T^{10} -$$$$34\!\cdots\!88$$$$T^{11} +$$$$75\!\cdots\!45$$$$T^{12} -$$$$14\!\cdots\!08$$$$T^{13} +$$$$22\!\cdots\!94$$$$T^{14} -$$$$25\!\cdots\!44$$$$T^{15} +$$$$17\!\cdots\!41$$$$T^{16}$$
$79$ $$1 + 50628092 T - 1644037081623204 T^{2} -$$$$20\!\cdots\!48$$$$T^{3} +$$$$47\!\cdots\!03$$$$T^{4} -$$$$41\!\cdots\!00$$$$T^{5} -$$$$54\!\cdots\!80$$$$T^{6} -$$$$48\!\cdots\!40$$$$T^{7} +$$$$31\!\cdots\!48$$$$T^{8} -$$$$74\!\cdots\!40$$$$T^{9} -$$$$12\!\cdots\!80$$$$T^{10} -$$$$14\!\cdots\!00$$$$T^{11} +$$$$25\!\cdots\!23$$$$T^{12} -$$$$16\!\cdots\!48$$$$T^{13} -$$$$20\!\cdots\!44$$$$T^{14} +$$$$93\!\cdots\!32$$$$T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$83$ $$1 - 11214933300710504 T^{2} +$$$$52\!\cdots\!64$$$$T^{4} -$$$$14\!\cdots\!32$$$$T^{6} +$$$$31\!\cdots\!18$$$$T^{8} -$$$$72\!\cdots\!92$$$$T^{10} +$$$$13\!\cdots\!04$$$$T^{12} -$$$$14\!\cdots\!64$$$$T^{14} +$$$$66\!\cdots\!21$$$$T^{16}$$
$89$ $$1 + 2322516 T + 13958973359482918 T^{2} +$$$$32\!\cdots\!56$$$$T^{3} +$$$$11\!\cdots\!33$$$$T^{4} +$$$$13\!\cdots\!20$$$$T^{5} +$$$$67\!\cdots\!14$$$$T^{6} +$$$$46\!\cdots\!92$$$$T^{7} +$$$$30\!\cdots\!76$$$$T^{8} +$$$$18\!\cdots\!52$$$$T^{9} +$$$$10\!\cdots\!54$$$$T^{10} +$$$$85\!\cdots\!20$$$$T^{11} +$$$$27\!\cdots\!93$$$$T^{12} +$$$$30\!\cdots\!56$$$$T^{13} +$$$$51\!\cdots\!58$$$$T^{14} +$$$$34\!\cdots\!76$$$$T^{15} +$$$$57\!\cdots\!41$$$$T^{16}$$
$97$ $$1 - 32582515803200672 T^{2} +$$$$61\!\cdots\!96$$$$T^{4} -$$$$75\!\cdots\!32$$$$T^{6} +$$$$68\!\cdots\!18$$$$T^{8} -$$$$46\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!36$$$$T^{12} -$$$$75\!\cdots\!92$$$$T^{14} +$$$$14\!\cdots\!81$$$$T^{16}$$