Properties

Label 7.10.c.a
Level 7
Weight 10
Character orbit 7.c
Analytic conductor 3.605
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.60525085315\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{2} + ( 33 - \beta_{1} - 33 \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} + ( -191 + 7 \beta_{1} + 191 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{4} + ( 307 \beta_{3} - \beta_{9} ) q^{5} + ( -897 + 60 \beta_{2} + 7 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} ) q^{6} + ( -797 + 26 \beta_{1} - 67 \beta_{2} + 1430 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{7} + ( 3462 - 26 \beta_{2} - 10 \beta_{4} - 66 \beta_{5} ) q^{8} + ( -258 \beta_{1} + 258 \beta_{2} - 7284 \beta_{3} - 86 \beta_{7} + 16 \beta_{8} + 7 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{2} + ( 33 - \beta_{1} - 33 \beta_{3} - \beta_{5} - \beta_{7} ) q^{3} + ( -191 + 7 \beta_{1} + 191 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{4} + ( 307 \beta_{3} - \beta_{9} ) q^{5} + ( -897 + 60 \beta_{2} + 7 \beta_{4} + 11 \beta_{5} - 2 \beta_{6} ) q^{6} + ( -797 + 26 \beta_{1} - 67 \beta_{2} + 1430 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} + 3 \beta_{6} - 10 \beta_{7} - 10 \beta_{8} - \beta_{9} ) q^{7} + ( 3462 - 26 \beta_{2} - 10 \beta_{4} - 66 \beta_{5} ) q^{8} + ( -258 \beta_{1} + 258 \beta_{2} - 7284 \beta_{3} - 86 \beta_{7} + 16 \beta_{8} + 7 \beta_{9} ) q^{9} + ( 910 - 299 \beta_{1} - 910 \beta_{3} - 14 \beta_{4} + 170 \beta_{5} - 4 \beta_{6} + 170 \beta_{7} + 14 \beta_{8} + 4 \beta_{9} ) q^{10} + ( 8425 + 3 \beta_{1} - 8425 \beta_{3} - 86 \beta_{4} - 9 \beta_{5} + 7 \beta_{6} - 9 \beta_{7} + 86 \beta_{8} - 7 \beta_{9} ) q^{11} + ( 2959 \beta_{1} - 2959 \beta_{2} + 28419 \beta_{3} + 341 \beta_{7} - 77 \beta_{8} + 16 \beta_{9} ) q^{12} + ( -32504 + 1042 \beta_{2} + 28 \beta_{4} + 286 \beta_{5} + 27 \beta_{6} ) q^{13} + ( 21074 - 3242 \beta_{1} - 1038 \beta_{2} - 47991 \beta_{3} + 50 \beta_{4} - 246 \beta_{5} - 44 \beta_{6} - 789 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} ) q^{14} + ( 13101 + 6201 \beta_{2} + 86 \beta_{4} - 1055 \beta_{5} - 7 \beta_{6} ) q^{15} + ( -4560 \beta_{1} + 4560 \beta_{2} + 62344 \beta_{3} - 664 \beta_{7} - 120 \beta_{8} - 112 \beta_{9} ) q^{16} + ( 68163 - 8798 \beta_{1} - 68163 \beta_{3} + 168 \beta_{4} + 750 \beta_{5} + 65 \beta_{6} + 750 \beta_{7} - 168 \beta_{8} - 65 \beta_{9} ) q^{17} + ( -208608 + 14898 \beta_{1} + 208608 \beta_{3} + 824 \beta_{4} + 712 \beta_{5} - 112 \beta_{6} + 712 \beta_{7} - 824 \beta_{8} + 112 \beta_{9} ) q^{18} + ( 19463 \beta_{1} - 19463 \beta_{2} + 4211 \beta_{3} - 1397 \beta_{7} + 826 \beta_{8} - 109 \beta_{9} ) q^{19} + ( -35363 + 1923 \beta_{2} - 707 \beta_{4} - 1395 \beta_{5} - 128 \beta_{6} ) q^{20} + ( -25725 - 18130 \beta_{1} - 16758 \beta_{2} - 238875 \beta_{3} - 784 \beta_{4} + 245 \beta_{6} + 2450 \beta_{7} + 588 \beta_{8} ) q^{21} + ( -22441 + 34660 \beta_{2} - 305 \beta_{4} + 5491 \beta_{5} + 126 \beta_{6} ) q^{22} + ( -7791 \beta_{1} + 7791 \beta_{2} + 526371 \beta_{3} + 4941 \beta_{7} + 148 \beta_{8} + 728 \beta_{9} ) q^{23} + ( 1708074 - 38142 \beta_{1} - 1708074 \beta_{3} - 966 \beta_{4} - 7230 \beta_{5} - 432 \beta_{6} - 7230 \beta_{7} + 966 \beta_{8} + 432 \beta_{9} ) q^{24} + ( -259928 - 1520 \beta_{1} + 259928 \beta_{3} - 2840 \beta_{4} - 2840 \beta_{5} + 714 \beta_{6} - 2840 \beta_{7} + 2840 \beta_{8} - 714 \beta_{9} ) q^{25} + ( 50734 \beta_{1} - 50734 \beta_{2} + 855644 \beta_{3} + 684 \beta_{7} - 3052 \beta_{8} + 408 \beta_{9} ) q^{26} + ( -1835079 + 6339 \beta_{2} + 2282 \beta_{4} - 521 \beta_{5} + 131 \beta_{6} ) q^{27} + ( -2152215 + 33933 \beta_{1} + 11054 \beta_{2} - 152527 \beta_{3} + 3993 \beta_{4} + 10385 \beta_{5} - 496 \beta_{6} + 8455 \beta_{7} - 2591 \beta_{8} - 432 \beta_{9} ) q^{28} + ( 1530044 - 5930 \beta_{2} + 1268 \beta_{4} + 5538 \beta_{5} - 959 \beta_{6} ) q^{29} + ( 6210 \beta_{1} - 6210 \beta_{2} + 4115493 \beta_{3} + 1055 \beta_{7} + 485 \beta_{8} - 2254 \beta_{9} ) q^{30} + ( 3799953 + 90345 \beta_{1} - 3799953 \beta_{3} + 4158 \beta_{4} + 73 \beta_{5} + 1397 \beta_{6} + 73 \beta_{7} - 4158 \beta_{8} - 1397 \beta_{9} ) q^{31} + ( -1215128 - 81384 \beta_{1} + 1215128 \beta_{3} + 2216 \beta_{4} - 9432 \beta_{5} - 2016 \beta_{6} - 9432 \beta_{7} - 2216 \beta_{8} + 2016 \beta_{9} ) q^{32} + ( -245136 \beta_{1} + 245136 \beta_{2} + 237315 \beta_{3} - 8312 \beta_{7} + 2296 \beta_{8} - 908 \beta_{9} ) q^{33} + ( -6302736 + 26341 \beta_{2} + 3892 \beta_{4} + 4116 \beta_{5} + 904 \beta_{6} ) q^{34} + ( -2273306 + 80213 \beta_{1} + 54341 \beta_{2} - 4262559 \beta_{3} - 8526 \beta_{4} - 33810 \beta_{5} - 833 \beta_{6} - 18375 \beta_{7} - 1078 \beta_{8} + 2499 \beta_{9} ) q^{35} + ( 7314918 - 388038 \beta_{2} - 6938 \beta_{4} - 37402 \beta_{5} + 3808 \beta_{6} ) q^{36} + ( 132990 \beta_{1} - 132990 \beta_{2} + 7949757 \beta_{3} - 41494 \beta_{7} + 2868 \beta_{8} + 2030 \beta_{9} ) q^{37} + ( 13360763 + 221252 \beta_{1} - 13360763 \beta_{3} - 15085 \beta_{4} + 53511 \beta_{5} - 1578 \beta_{6} + 53511 \beta_{7} + 15085 \beta_{8} + 1578 \beta_{9} ) q^{38} + ( -8898414 + 25270 \beta_{1} + 8898414 \beta_{3} + 8078 \beta_{4} + 74690 \beta_{5} + 371 \beta_{6} + 74690 \beta_{7} - 8078 \beta_{8} - 371 \beta_{9} ) q^{39} + ( -357926 \beta_{1} + 357926 \beta_{2} + 1028214 \beta_{3} + 62130 \beta_{7} + 13286 \beta_{8} + 1184 \beta_{9} ) q^{40} + ( -5185700 - 435718 \beta_{2} - 32816 \beta_{4} + 26982 \beta_{5} - 2635 \beta_{6} ) q^{41} + ( -13136802 + 94374 \beta_{1} + 317667 \beta_{2} - 10695720 \beta_{3} + 1666 \beta_{4} + 22442 \beta_{5} + 6076 \beta_{6} - 92512 \beta_{7} + 27440 \beta_{8} - 5488 \beta_{9} ) q^{42} + ( 10269764 - 132344 \beta_{2} + 18832 \beta_{4} - 66624 \beta_{5} - 7028 \beta_{6} ) q^{43} + ( 166623 \beta_{1} - 166623 \beta_{2} + 20018819 \beta_{3} + 48597 \beta_{7} - 26253 \beta_{8} + 7504 \beta_{9} ) q^{44} + ( 16879164 + 391218 \beta_{1} - 16879164 \beta_{3} + 36148 \beta_{4} - 18730 \beta_{5} - 3209 \beta_{6} - 18730 \beta_{7} - 36148 \beta_{8} + 3209 \beta_{9} ) q^{45} + ( -2609163 - 582710 \beta_{1} + 2609163 \beta_{3} - 12107 \beta_{4} - 158127 \beta_{5} + 13090 \beta_{6} - 158127 \beta_{7} + 12107 \beta_{8} - 13090 \beta_{9} ) q^{46} + ( -88975 \beta_{1} + 88975 \beta_{2} + 6414639 \beta_{3} - 113391 \beta_{7} - 31990 \beta_{8} - 129 \beta_{9} ) q^{47} + ( -18775680 + 769288 \beta_{2} + 45248 \beta_{4} - 90256 \beta_{5} - 2608 \beta_{6} ) q^{48} + ( -6311697 - 301532 \beta_{1} - 139622 \beta_{2} + 3142748 \beta_{3} + 27328 \beta_{4} - 2534 \beta_{5} - 9037 \beta_{6} + 202244 \beta_{7} - 47264 \beta_{8} - 826 \beta_{9} ) q^{49} + ( -52660 + 676184 \beta_{2} + 44 \beta_{4} + 290460 \beta_{5} - 2856 \beta_{6} ) q^{50} + ( -486345 \beta_{1} + 486345 \beta_{2} + 8644815 \beta_{3} - 37389 \beta_{7} + 57234 \beta_{8} - 22953 \beta_{9} ) q^{51} + ( 21387550 - 1415726 \beta_{1} - 21387550 \beta_{3} - 25634 \beta_{4} - 130050 \beta_{5} + 10720 \beta_{6} - 130050 \beta_{7} + 25634 \beta_{8} - 10720 \beta_{9} ) q^{52} + ( -5847531 + 1630882 \beta_{1} + 5847531 \beta_{3} - 11844 \beta_{4} + 153342 \beta_{5} - 28672 \beta_{6} + 153342 \beta_{7} + 11844 \beta_{8} + 28672 \beta_{9} ) q^{53} + ( 2529708 \beta_{1} - 2529708 \beta_{2} + 11291151 \beta_{3} + 90677 \beta_{7} + 1799 \beta_{8} - 6130 \beta_{9} ) q^{54} + ( -14307503 - 495623 \beta_{2} + 41692 \beta_{4} + 91565 \beta_{5} + 19856 \beta_{6} ) q^{55} + ( 9077142 + 623692 \beta_{1} - 2086366 \beta_{2} + 6271140 \beta_{3} - 52730 \beta_{4} + 106014 \beta_{5} - 7408 \beta_{6} + 130860 \beta_{7} - 31036 \beta_{8} + 27296 \beta_{9} ) q^{56} + ( -12832221 + 1674308 \beta_{2} - 98600 \beta_{4} - 112292 \beta_{5} + 41650 \beta_{6} ) q^{57} + ( -1024314 \beta_{1} + 1024314 \beta_{2} - 9635948 \beta_{3} + 274612 \beta_{7} - 10068 \beta_{8} + 12376 \beta_{9} ) q^{58} + ( 10188249 - 1888789 \beta_{1} - 10188249 \beta_{3} - 107072 \beta_{4} - 244461 \beta_{5} - 2652 \beta_{6} - 244461 \beta_{7} + 107072 \beta_{8} + 2652 \beta_{9} ) q^{59} + ( 26869857 - 820029 \beta_{1} - 26869857 \beta_{3} - 1519 \beta_{4} - 149975 \beta_{5} - 9520 \beta_{6} - 149975 \beta_{7} + 1519 \beta_{8} + 9520 \beta_{9} ) q^{60} + ( 307914 \beta_{1} - 307914 \beta_{2} - 22604563 \beta_{3} - 296018 \beta_{7} + 75684 \beta_{8} + 23626 \beta_{9} ) q^{61} + ( 45842373 + 2282626 \beta_{2} - 97867 \beta_{4} - 57327 \beta_{5} - 13758 \beta_{6} ) q^{62} + ( 35103588 - 3933990 \beta_{1} + 1330818 \beta_{2} - 6105438 \beta_{3} + 32680 \beta_{4} + 10244 \beta_{5} + 35020 \beta_{6} - 98034 \beta_{7} + 122718 \beta_{8} - 40749 \beta_{9} ) q^{63} + ( -19537232 + 904752 \beta_{2} + 111408 \beta_{4} + 64368 \beta_{5} - 72576 \beta_{6} ) q^{64} + ( 1411242 \beta_{1} - 1411242 \beta_{2} - 67114040 \beta_{3} - 56490 \beta_{7} - 110432 \beta_{8} + 27853 \beta_{9} ) q^{65} + ( -168072516 + 1398237 \beta_{1} + 168072516 \beta_{3} + 275408 \beta_{4} + 595120 \beta_{5} - 15664 \beta_{6} + 595120 \beta_{7} - 275408 \beta_{8} + 15664 \beta_{9} ) q^{66} + ( -26836615 + 3520839 \beta_{1} + 26836615 \beta_{3} + 45732 \beta_{4} + 133127 \beta_{5} + 106498 \beta_{6} + 133127 \beta_{7} - 45732 \beta_{8} - 106498 \beta_{9} ) q^{67} + ( 3144119 \beta_{1} - 3144119 \beta_{2} + 7892559 \beta_{3} + 488529 \beta_{7} - 138033 \beta_{8} - 36448 \beta_{9} ) q^{68} + ( 134567397 - 5336292 \beta_{2} - 49728 \beta_{4} - 146436 \beta_{5} - 56583 \beta_{6} ) q^{69} + ( 37265529 + 3414712 \beta_{1} + 702660 \beta_{2} + 63096173 \beta_{3} + 68257 \beta_{4} - 374115 \beta_{5} - 28910 \beta_{6} - 853825 \beta_{7} - 13867 \beta_{8} - 18326 \beta_{9} ) q^{70} + ( -118841568 + 618504 \beta_{2} + 247788 \beta_{4} - 577176 \beta_{5} + 2478 \beta_{6} ) q^{71} + ( -3874644 \beta_{1} + 3874644 \beta_{2} - 192184524 \beta_{3} - 1384836 \beta_{7} + 116436 \beta_{8} - 18816 \beta_{9} ) q^{72} + ( -173198403 + 2667688 \beta_{1} + 173198403 \beta_{3} - 81536 \beta_{4} + 681424 \beta_{5} - 22698 \beta_{6} + 681424 \beta_{7} + 81536 \beta_{8} + 22698 \beta_{9} ) q^{73} + ( 120976554 - 6502975 \beta_{1} - 120976554 \beta_{3} + 135790 \beta_{4} + 77382 \beta_{5} - 69132 \beta_{6} + 77382 \beta_{7} - 135790 \beta_{8} + 69132 \beta_{9} ) q^{74} + ( -11219744 \beta_{1} + 11219744 \beta_{2} - 38065836 \beta_{3} + 496520 \beta_{7} + 151564 \beta_{8} - 13718 \beta_{9} ) q^{75} + ( 107515751 + 5990733 \beta_{2} - 142569 \beta_{4} + 429807 \beta_{5} + 87696 \beta_{6} ) q^{76} + ( 190811013 - 2401520 \beta_{1} + 3572614 \beta_{2} - 983785 \beta_{3} - 261140 \beta_{4} - 606342 \beta_{5} + 9180 \beta_{6} - 510240 \beta_{7} + 81568 \beta_{8} + 70363 \beta_{9} ) q^{77} + ( 57522990 - 13203372 \beta_{2} - 454370 \beta_{4} - 1104922 \beta_{5} + 131740 \beta_{6} ) q^{78} + ( 11509413 \beta_{1} - 11509413 \beta_{2} - 100369637 \beta_{3} + 1186521 \beta_{7} - 190092 \beta_{8} - 7728 \beta_{9} ) q^{79} + ( -252792184 + 3541920 \beta_{1} + 252792184 \beta_{3} - 400120 \beta_{4} - 527880 \beta_{5} + 90032 \beta_{6} - 527880 \beta_{7} + 400120 \beta_{8} - 90032 \beta_{9} ) q^{80} + ( 109718097 + 1207626 \beta_{1} - 109718097 \beta_{3} - 243152 \beta_{4} + 386246 \beta_{5} - 179333 \beta_{6} + 386246 \beta_{7} + 243152 \beta_{8} + 179333 \beta_{9} ) q^{81} + ( -5363238 \beta_{1} + 5363238 \beta_{2} - 271025884 \beta_{3} - 1325932 \beta_{7} + 212268 \beta_{8} + 130136 \beta_{9} ) q^{82} + ( -11965020 - 6939344 \beta_{2} + 779044 \beta_{4} + 977208 \beta_{5} + 48406 \beta_{6} ) q^{83} + ( -118515222 + 18467071 \beta_{1} - 6083497 \beta_{2} + 331413999 \beta_{3} + 201194 \beta_{4} + 1853866 \beta_{5} - 26656 \beta_{6} + 2793441 \beta_{7} - 526897 \beta_{8} + 43904 \beta_{9} ) q^{84} + ( -117971535 - 6540962 \beta_{2} - 501212 \beta_{4} + 1989570 \beta_{5} - 110432 \beta_{6} ) q^{85} + ( -6522284 \beta_{1} + 6522284 \beta_{2} - 137616464 \beta_{3} + 1318944 \beta_{7} + 686976 \beta_{8} - 142800 \beta_{9} ) q^{86} + ( -70609326 + 8113446 \beta_{1} + 70609326 \beta_{3} + 114898 \beta_{4} - 1964398 \beta_{5} + 41041 \beta_{6} - 1964398 \beta_{7} - 114898 \beta_{8} - 41041 \beta_{9} ) q^{87} + ( 185530650 - 11927534 \beta_{1} - 185530650 \beta_{3} - 430550 \beta_{4} - 403278 \beta_{5} + 139216 \beta_{6} - 403278 \beta_{7} + 430550 \beta_{8} - 139216 \beta_{9} ) q^{88} + ( -5558276 \beta_{1} + 5558276 \beta_{2} + 330210049 \beta_{3} + 654420 \beta_{7} - 985432 \beta_{8} - 144296 \beta_{9} ) q^{89} + ( 195712764 + 5212662 \beta_{2} - 125468 \beta_{4} - 2520700 \beta_{5} - 96920 \beta_{6} ) q^{90} + ( 109658962 - 10990700 \beta_{1} + 14868854 \beta_{2} - 81478180 \beta_{3} + 502250 \beta_{4} + 412678 \beta_{5} - 76881 \beta_{6} + 162092 \beta_{7} - 89768 \beta_{8} - 77812 \beta_{9} ) q^{91} + ( -135172605 + 10684041 \beta_{2} + 1387667 \beta_{4} + 2452731 \beta_{5} + 28336 \beta_{6} ) q^{92} + ( 1356734 \beta_{1} - 1356734 \beta_{2} - 85029801 \beta_{3} - 1890534 \beta_{7} - 591932 \beta_{8} + 215992 \beta_{9} ) q^{93} + ( -48863379 - 13086278 \beta_{1} + 48863379 \beta_{3} + 863485 \beta_{4} - 322695 \beta_{5} - 291278 \beta_{6} - 322695 \beta_{7} - 863485 \beta_{8} + 291278 \beta_{9} ) q^{94} + ( -241561379 + 9505677 \beta_{1} + 241561379 \beta_{3} + 684952 \beta_{4} - 4148415 \beta_{5} + 326914 \beta_{6} - 4148415 \beta_{7} - 684952 \beta_{8} - 326914 \beta_{9} ) q^{95} + ( 13074456 \beta_{1} - 13074456 \beta_{2} - 283520904 \beta_{3} - 1174520 \beta_{7} + 439096 \beta_{8} - 39392 \beta_{9} ) q^{96} + ( 90081768 - 6851474 \beta_{2} - 1237712 \beta_{4} - 1405558 \beta_{5} - 123697 \beta_{6} ) q^{97} + ( -185733352 - 8106357 \beta_{1} - 13252771 \beta_{2} - 93269792 \beta_{3} - 781704 \beta_{4} - 3896424 \beta_{5} + 306656 \beta_{6} - 1487556 \beta_{7} + 1446564 \beta_{8} - 330232 \beta_{9} ) q^{98} + ( -200734674 + 25930614 \beta_{2} + 219734 \beta_{4} + 1130938 \beta_{5} - 309295 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 18q^{2} + 161q^{3} - 940q^{4} + 1533q^{5} - 8708q^{6} - 1036q^{7} + 34272q^{8} - 35734q^{9} + O(q^{10}) \) \( 10q - 18q^{2} + 161q^{3} - 940q^{4} + 1533q^{5} - 8708q^{6} - 1036q^{7} + 34272q^{8} - 35734q^{9} + 4298q^{10} + 42213q^{11} + 135604q^{12} - 319676q^{13} - 39522q^{14} + 151394q^{15} + 322064q^{16} + 324681q^{17} - 1012868q^{18} - 16121q^{19} - 350616q^{20} - 1557857q^{21} - 62692q^{22} + 2638863q^{23} + 8449728q^{24} - 1304092q^{25} + 4179252q^{26} - 18331558q^{27} - 22156316q^{28} + 15292500q^{29} + 20557942q^{30} + 19179237q^{31} - 6263520q^{32} + 1689359q^{33} - 62909700q^{34} - 43746759q^{35} + 71476528q^{36} + 39566985q^{37} + 67365270q^{38} - 44299486q^{39} + 5721744q^{40} - 53436852q^{41} - 183129856q^{42} + 101835992q^{43} + 99704916q^{44} + 85098230q^{45} - 14489202q^{46} + 32509659q^{47} - 185141600q^{48} - 49024598q^{49} + 3328464q^{50} + 44168403q^{51} + 103893272q^{52} - 25714707q^{53} + 51200926q^{54} - 144695222q^{55} + 115352832q^{56} - 121710346q^{57} - 46645516q^{58} + 46776513q^{59} + 132391756q^{60} - 113075039q^{61} + 467465628q^{62} + 318071530q^{63} - 192008960q^{64} - 338113566q^{65} - 836682602q^{66} - 126707879q^{67} + 32262636q^{68} + 1323616182q^{69} + 697712470q^{70} - 1188736032q^{71} - 950557728q^{72} - 859257651q^{73} + 591757530q^{74} - 169061732q^{75} + 1101475592q^{76} + 1911891891q^{77} + 519432424q^{78} - 527065417q^{79} - 1257352656q^{80} + 551662715q^{81} - 1341703076q^{82} - 144863208q^{83} + 486452204q^{84} - 1197360222q^{85} - 678648216q^{86} - 340781350q^{87} + 903700608q^{88} + 1661554797q^{89} + 1967758744q^{90} + 726641384q^{91} - 1301840952q^{92} - 423057489q^{93} - 272580882q^{94} - 1197123495q^{95} - 1441922272q^{96} + 869770188q^{97} - 2404833858q^{98} - 1900777180q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(174976544647 \nu^{9} + 543280294119 \nu^{8} - 2080323777608 \nu^{7} + 306747495137243 \nu^{6} - 527825025841115 \nu^{5} + 79251473775272906 \nu^{4} - 8455568604050729964 \nu^{3} + 8633477355188858864 \nu^{2} + 6390233594342311200 \nu - 728499442105846008000\)\()/ \)\(49\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(439915122044593 \nu^{9} - 359425911506973 \nu^{8} + 189413411414469730 \nu^{7} + 25877873507020493 \nu^{6} + 64699967753173288309 \nu^{5} + 10142716689341838644 \nu^{4} + 7135341367123388280552 \nu^{3} + 9561128363649180087824 \nu^{2} + 619444498139966557897152 \nu + 457911692363235156681600\)\()/ \)\(45\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(9108600924689 \nu^{9} + 744234087958953 \nu^{8} - 2849814149431096 \nu^{7} + 263155948009015381 \nu^{6} - 723062074883078005 \nu^{5} + 108565778922995107222 \nu^{4} - 451992977525138313948 \nu^{3} + 11826911844414978154768 \nu^{2} + 8753915285372555954400 \nu + 1498083923068056865537920\)\()/ \)\(23\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-44987322634457 \nu^{9} - 60129676436289 \nu^{8} + 230247989821048 \nu^{7} - 51400958387300173 \nu^{6} + 58419104028555565 \nu^{5} - 8771467558811176486 \nu^{4} + 1145664200820139449324 \nu^{3} - 955543953106873546384 \nu^{2} - 707264155426816927200 \nu + 261687922576606706402880\)\()/ \)\(69\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-10228081391315 \nu^{9} + 1952450458321161 \nu^{8} - 7476304878007352 \nu^{7} + 897435986095250741 \nu^{6} - 1896907038176377685 \nu^{5} + 284815366892851129814 \nu^{4} - 297921422945101274472 \nu^{3} + 31027145658538503980816 \nu^{2} + 22965336024722535832800 \nu + 1379362333342473784891584\)\()/ \)\(69\!\cdots\!72\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-24593260288252157 \nu^{9} + 42692492469767637 \nu^{8} - 13545670542188346350 \nu^{7} + 9874684454720909303 \nu^{6} - 5152118620797417711701 \nu^{5} + 2305464026693842938944 \nu^{4} - 830194281200809808698368 \nu^{3} - 538575932743903852034896 \nu^{2} - 81614206561156272103139328 \nu - 60375780584089979592902400\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(2067492150283470103 \nu^{9} - 1639595498403371523 \nu^{8} + 883784357122680443350 \nu^{7} + 146171516473387214363 \nu^{6} + 300834508490249091273979 \nu^{5} + 53897632103545345431524 \nu^{4} + 32599014210284667922809672 \nu^{3} + 57713122284011650146364784 \nu^{2} + 2809343727409862629926750912 \nu + 2076654230999456261004489600\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(1042602787806930287 \nu^{9} - 1349683788498539907 \nu^{8} + 446731469300652376670 \nu^{7} + 69678488713722111187 \nu^{6} + 152247759899824475413931 \nu^{5} + 26156328254249575048396 \nu^{4} + 16592811419754117068838168 \nu^{3} + 40035573671756716045423216 \nu^{2} + 1433445544800945125082504768 \nu + 1059612788993006122246454400\)\()/ \)\(53\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} - \beta_{7} - 687 \beta_{3} - \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-27 \beta_{5} + \beta_{4} - 507 \beta_{2} - 375\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(14 \beta_{9} - 169 \beta_{8} + 155 \beta_{7} - 14 \beta_{6} + 155 \beta_{5} + 169 \beta_{4} + 87639 \beta_{3} - 134 \beta_{1} - 87639\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(28 \beta_{9} - 583 \beta_{8} - 11457 \beta_{7} + 99345 \beta_{3} + 142643 \beta_{2} - 142643 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(12040 \beta_{6} - 89029 \beta_{5} - 107929 \beta_{4} + 71655 \beta_{2} + 49481343\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-9450 \beta_{9} + 115386 \beta_{8} + 1949460 \beta_{7} + 9450 \beta_{6} + 1949460 \beta_{5} - 115386 \beta_{4} - 13417998 \beta_{3} + 21095143 \beta_{1} + 13417998\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-4129692 \beta_{9} + 33731905 \beta_{8} - 25531489 \beta_{7} - 14657685447 \beta_{3} - 22648273 \beta_{2} + 22648273 \beta_{1}\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-8200416 \beta_{6} - 1242725823 \beta_{5} + 81699181 \beta_{4} - 12756581151 \beta_{2} - 8505431979\)\()/4\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−8.71912 15.1020i
−5.11725 8.86334i
−0.371984 0.644295i
5.89912 + 10.2176i
8.80924 + 15.2580i
−8.71912 + 15.1020i
−5.11725 + 8.86334i
−0.371984 + 0.644295i
5.89912 10.2176i
8.80924 15.2580i
−19.4382 + 33.6680i 113.728 + 196.982i −499.691 865.489i −162.760 + 281.909i −8842.67 5234.95 3598.46i 18947.7 −16026.5 + 27758.7i −6327.54 10959.6i
2.2 −12.2345 + 21.1908i −79.7348 138.105i −43.3662 75.1124i 1014.15 1756.56i 3902.06 −4235.51 4734.35i −10405.9 −2873.78 + 4977.54i 24815.2 + 42981.2i
2.3 −2.74397 + 4.75269i −1.70307 2.94981i 240.941 + 417.323i −828.924 + 1435.74i 18.6927 2822.68 + 5690.88i −5454.36 9835.70 17035.9i −4549.08 7879.24i
2.4 9.79824 16.9710i 104.977 + 181.826i 63.9892 + 110.832i 983.791 1703.98i 4114.37 −5768.52 + 2660.41i 12541.3 −12198.9 + 21129.2i −19278.8 33391.9i
2.5 15.6185 27.0520i −56.7670 98.3234i −231.874 401.617i −239.755 + 415.269i −3546.46 1428.40 6189.77i 1507.26 3396.51 5882.92i 7489.23 + 12971.7i
4.1 −19.4382 33.6680i 113.728 196.982i −499.691 + 865.489i −162.760 281.909i −8842.67 5234.95 + 3598.46i 18947.7 −16026.5 27758.7i −6327.54 + 10959.6i
4.2 −12.2345 21.1908i −79.7348 + 138.105i −43.3662 + 75.1124i 1014.15 + 1756.56i 3902.06 −4235.51 + 4734.35i −10405.9 −2873.78 4977.54i 24815.2 42981.2i
4.3 −2.74397 4.75269i −1.70307 + 2.94981i 240.941 417.323i −828.924 1435.74i 18.6927 2822.68 5690.88i −5454.36 9835.70 + 17035.9i −4549.08 + 7879.24i
4.4 9.79824 + 16.9710i 104.977 181.826i 63.9892 110.832i 983.791 + 1703.98i 4114.37 −5768.52 2660.41i 12541.3 −12198.9 21129.2i −19278.8 + 33391.9i
4.5 15.6185 + 27.0520i −56.7670 + 98.3234i −231.874 + 401.617i −239.755 415.269i −3546.46 1428.40 + 6189.77i 1507.26 3396.51 + 5882.92i 7489.23 12971.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.10.c.a 10
3.b odd 2 1 63.10.e.b 10
4.b odd 2 1 112.10.i.c 10
7.b odd 2 1 49.10.c.g 10
7.c even 3 1 inner 7.10.c.a 10
7.c even 3 1 49.10.a.e 5
7.d odd 6 1 49.10.a.f 5
7.d odd 6 1 49.10.c.g 10
21.h odd 6 1 63.10.e.b 10
28.g odd 6 1 112.10.i.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.c.a 10 1.a even 1 1 trivial
7.10.c.a 10 7.c even 3 1 inner
49.10.a.e 5 7.c even 3 1
49.10.a.f 5 7.d odd 6 1
49.10.c.g 10 7.b odd 2 1
49.10.c.g 10 7.d odd 6 1
63.10.e.b 10 3.b odd 2 1
63.10.e.b 10 21.h odd 6 1
112.10.i.c 10 4.b odd 2 1
112.10.i.c 10 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 18 T - 648 T^{2} - 18888 T^{3} - 94672 T^{4} + 4437696 T^{5} + 145005568 T^{6} + 202951680 T^{7} - 31516762112 T^{8} + 65792360448 T^{9} + 3292007038976 T^{10} + 33685688549376 T^{11} - 8261930087088128 T^{12} + 27239713383383040 T^{13} + 9964706756766466048 T^{14} + \)\(15\!\cdots\!72\)\( T^{15} - \)\(17\!\cdots\!48\)\( T^{16} - \)\(17\!\cdots\!04\)\( T^{17} - \)\(30\!\cdots\!08\)\( T^{18} + \)\(43\!\cdots\!36\)\( T^{19} + \)\(12\!\cdots\!24\)\( T^{20} \)
$3$ \( 1 - 161 T - 18380 T^{2} + 8348151 T^{3} - 844491330 T^{4} - 17310184731 T^{5} + 23710706388306 T^{6} - 4343339165895009 T^{7} + 332958830166204561 T^{8} + 60409189510371073386 T^{9} - \)\(16\!\cdots\!64\)\( T^{10} + \)\(11\!\cdots\!38\)\( T^{11} + \)\(12\!\cdots\!29\)\( T^{12} - \)\(33\!\cdots\!83\)\( T^{13} + \)\(35\!\cdots\!26\)\( T^{14} - \)\(51\!\cdots\!33\)\( T^{15} - \)\(49\!\cdots\!70\)\( T^{16} + \)\(95\!\cdots\!77\)\( T^{17} - \)\(41\!\cdots\!80\)\( T^{18} - \)\(71\!\cdots\!83\)\( T^{19} + \)\(87\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - 1533 T - 3055722 T^{2} + 7403950197 T^{3} - 320742802594 T^{4} - 6624362577933555 T^{5} + 7817200764353129800 T^{6} - \)\(26\!\cdots\!75\)\( T^{7} + \)\(32\!\cdots\!25\)\( T^{8} + \)\(44\!\cdots\!50\)\( T^{9} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(86\!\cdots\!50\)\( T^{11} + \)\(12\!\cdots\!25\)\( T^{12} - \)\(19\!\cdots\!75\)\( T^{13} + \)\(11\!\cdots\!00\)\( T^{14} - \)\(18\!\cdots\!75\)\( T^{15} - \)\(17\!\cdots\!50\)\( T^{16} + \)\(80\!\cdots\!25\)\( T^{17} - \)\(64\!\cdots\!50\)\( T^{18} - \)\(63\!\cdots\!25\)\( T^{19} + \)\(80\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + 1036 T + 25048947 T^{2} + 247958108048 T^{3} + 1787726185230650 T^{4} + 135641181930574152 T^{5} + \)\(72\!\cdots\!50\)\( T^{6} + \)\(40\!\cdots\!52\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} + \)\(27\!\cdots\!36\)\( T^{9} + \)\(10\!\cdots\!07\)\( T^{10} \)
$11$ \( 1 - 42213 T - 6326783460 T^{2} + 327121353142323 T^{3} + 17750681164843604918 T^{4} - \)\(93\!\cdots\!51\)\( T^{5} - \)\(48\!\cdots\!18\)\( T^{6} + \)\(92\!\cdots\!83\)\( T^{7} + \)\(18\!\cdots\!73\)\( T^{8} - \)\(42\!\cdots\!58\)\( T^{9} - \)\(56\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!78\)\( T^{11} + \)\(10\!\cdots\!13\)\( T^{12} + \)\(12\!\cdots\!93\)\( T^{13} - \)\(14\!\cdots\!98\)\( T^{14} - \)\(68\!\cdots\!01\)\( T^{15} + \)\(30\!\cdots\!38\)\( T^{16} + \)\(13\!\cdots\!13\)\( T^{17} - \)\(60\!\cdots\!60\)\( T^{18} - \)\(95\!\cdots\!43\)\( T^{19} + \)\(53\!\cdots\!01\)\( T^{20} \)
$13$ \( ( 1 + 159838 T + 52194785337 T^{2} + 6011850787968872 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} + \)\(92\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!18\)\( T^{6} + \)\(67\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!29\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{9} + \)\(13\!\cdots\!93\)\( T^{10} )^{2} \)
$17$ \( 1 - 324681 T - 328834212334 T^{2} + 62765598850452981 T^{3} + \)\(66\!\cdots\!42\)\( T^{4} - \)\(30\!\cdots\!55\)\( T^{5} - \)\(12\!\cdots\!16\)\( T^{6} + \)\(59\!\cdots\!33\)\( T^{7} + \)\(18\!\cdots\!29\)\( T^{8} - \)\(67\!\cdots\!18\)\( T^{9} - \)\(22\!\cdots\!44\)\( T^{10} - \)\(80\!\cdots\!46\)\( T^{11} + \)\(26\!\cdots\!61\)\( T^{12} + \)\(98\!\cdots\!09\)\( T^{13} - \)\(25\!\cdots\!96\)\( T^{14} - \)\(71\!\cdots\!35\)\( T^{15} + \)\(18\!\cdots\!18\)\( T^{16} + \)\(20\!\cdots\!53\)\( T^{17} - \)\(12\!\cdots\!74\)\( T^{18} - \)\(15\!\cdots\!77\)\( T^{19} + \)\(55\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 + 16121 T - 434040959924 T^{2} - 420368046459486167 T^{3} + \)\(18\!\cdots\!38\)\( T^{4} + \)\(22\!\cdots\!55\)\( T^{5} + \)\(44\!\cdots\!02\)\( T^{6} - \)\(88\!\cdots\!79\)\( T^{7} - \)\(41\!\cdots\!79\)\( T^{8} + \)\(95\!\cdots\!02\)\( T^{9} + \)\(20\!\cdots\!04\)\( T^{10} + \)\(30\!\cdots\!58\)\( T^{11} - \)\(43\!\cdots\!39\)\( T^{12} - \)\(29\!\cdots\!81\)\( T^{13} + \)\(47\!\cdots\!62\)\( T^{14} + \)\(77\!\cdots\!45\)\( T^{15} + \)\(20\!\cdots\!98\)\( T^{16} - \)\(15\!\cdots\!53\)\( T^{17} - \)\(51\!\cdots\!64\)\( T^{18} + \)\(61\!\cdots\!99\)\( T^{19} + \)\(12\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - 2638863 T - 480527621872 T^{2} - 311022444212974755 T^{3} + \)\(13\!\cdots\!02\)\( T^{4} - \)\(34\!\cdots\!73\)\( T^{5} - \)\(90\!\cdots\!82\)\( T^{6} - \)\(28\!\cdots\!95\)\( T^{7} + \)\(11\!\cdots\!53\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{9} + \)\(39\!\cdots\!12\)\( T^{10} + \)\(23\!\cdots\!50\)\( T^{11} + \)\(36\!\cdots\!57\)\( T^{12} - \)\(16\!\cdots\!65\)\( T^{13} - \)\(95\!\cdots\!02\)\( T^{14} - \)\(66\!\cdots\!39\)\( T^{15} + \)\(47\!\cdots\!18\)\( T^{16} - \)\(19\!\cdots\!85\)\( T^{17} - \)\(53\!\cdots\!12\)\( T^{18} - \)\(52\!\cdots\!49\)\( T^{19} + \)\(35\!\cdots\!49\)\( T^{20} \)
$29$ \( ( 1 - 7646250 T + 88492818721737 T^{2} - \)\(45\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!74\)\( T^{4} - \)\(98\!\cdots\!56\)\( T^{5} + \)\(40\!\cdots\!06\)\( T^{6} - \)\(95\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!33\)\( T^{8} - \)\(33\!\cdots\!50\)\( T^{9} + \)\(64\!\cdots\!49\)\( T^{10} )^{2} \)
$31$ \( 1 - 19179237 T + 120960263219896 T^{2} - \)\(28\!\cdots\!01\)\( T^{3} + \)\(22\!\cdots\!62\)\( T^{4} - \)\(27\!\cdots\!71\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} - \)\(34\!\cdots\!25\)\( T^{7} + \)\(39\!\cdots\!25\)\( T^{8} - \)\(24\!\cdots\!10\)\( T^{9} + \)\(10\!\cdots\!20\)\( T^{10} - \)\(64\!\cdots\!10\)\( T^{11} + \)\(27\!\cdots\!25\)\( T^{12} - \)\(64\!\cdots\!75\)\( T^{13} + \)\(53\!\cdots\!50\)\( T^{14} - \)\(35\!\cdots\!21\)\( T^{15} + \)\(78\!\cdots\!02\)\( T^{16} - \)\(25\!\cdots\!91\)\( T^{17} + \)\(28\!\cdots\!56\)\( T^{18} - \)\(12\!\cdots\!47\)\( T^{19} + \)\(16\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - 39566985 T + 448975929019726 T^{2} - \)\(17\!\cdots\!23\)\( T^{3} + \)\(74\!\cdots\!22\)\( T^{4} - \)\(13\!\cdots\!03\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} + \)\(18\!\cdots\!29\)\( T^{7} + \)\(18\!\cdots\!53\)\( T^{8} - \)\(52\!\cdots\!46\)\( T^{9} - \)\(22\!\cdots\!36\)\( T^{10} - \)\(68\!\cdots\!42\)\( T^{11} + \)\(30\!\cdots\!37\)\( T^{12} + \)\(40\!\cdots\!57\)\( T^{13} + \)\(30\!\cdots\!36\)\( T^{14} - \)\(50\!\cdots\!71\)\( T^{15} + \)\(35\!\cdots\!58\)\( T^{16} - \)\(11\!\cdots\!19\)\( T^{17} + \)\(36\!\cdots\!06\)\( T^{18} - \)\(41\!\cdots\!45\)\( T^{19} + \)\(13\!\cdots\!49\)\( T^{20} \)
$41$ \( ( 1 + 26718426 T + 918529945217061 T^{2} + \)\(23\!\cdots\!36\)\( T^{3} + \)\(56\!\cdots\!46\)\( T^{4} + \)\(93\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!06\)\( T^{6} + \)\(24\!\cdots\!56\)\( T^{7} + \)\(32\!\cdots\!41\)\( T^{8} + \)\(30\!\cdots\!66\)\( T^{9} + \)\(37\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( ( 1 - 50917996 T + 2693125333085463 T^{2} - \)\(86\!\cdots\!48\)\( T^{3} + \)\(25\!\cdots\!62\)\( T^{4} - \)\(60\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!66\)\( T^{6} - \)\(21\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} - \)\(32\!\cdots\!96\)\( T^{9} + \)\(32\!\cdots\!43\)\( T^{10} )^{2} \)
$47$ \( 1 - 32509659 T - 3410631284393464 T^{2} + \)\(10\!\cdots\!49\)\( T^{3} + \)\(68\!\cdots\!10\)\( T^{4} - \)\(15\!\cdots\!25\)\( T^{5} - \)\(11\!\cdots\!42\)\( T^{6} + \)\(13\!\cdots\!77\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} - \)\(49\!\cdots\!58\)\( T^{9} - \)\(21\!\cdots\!56\)\( T^{10} - \)\(55\!\cdots\!86\)\( T^{11} + \)\(21\!\cdots\!29\)\( T^{12} + \)\(18\!\cdots\!51\)\( T^{13} - \)\(17\!\cdots\!82\)\( T^{14} - \)\(27\!\cdots\!75\)\( T^{15} + \)\(13\!\cdots\!90\)\( T^{16} + \)\(22\!\cdots\!27\)\( T^{17} - \)\(83\!\cdots\!24\)\( T^{18} - \)\(89\!\cdots\!73\)\( T^{19} + \)\(30\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 + 25714707 T - 5546541912716410 T^{2} - \)\(63\!\cdots\!55\)\( T^{3} - \)\(57\!\cdots\!02\)\( T^{4} + \)\(20\!\cdots\!13\)\( T^{5} + \)\(13\!\cdots\!12\)\( T^{6} + \)\(36\!\cdots\!85\)\( T^{7} - \)\(87\!\cdots\!67\)\( T^{8} - \)\(15\!\cdots\!94\)\( T^{9} - \)\(13\!\cdots\!68\)\( T^{10} - \)\(52\!\cdots\!02\)\( T^{11} - \)\(95\!\cdots\!63\)\( T^{12} + \)\(13\!\cdots\!45\)\( T^{13} + \)\(16\!\cdots\!52\)\( T^{14} + \)\(82\!\cdots\!09\)\( T^{15} - \)\(74\!\cdots\!38\)\( T^{16} - \)\(27\!\cdots\!35\)\( T^{17} - \)\(77\!\cdots\!10\)\( T^{18} + \)\(11\!\cdots\!71\)\( T^{19} + \)\(15\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 - 46776513 T - 27103348513629988 T^{2} + \)\(28\!\cdots\!83\)\( T^{3} + \)\(37\!\cdots\!62\)\( T^{4} - \)\(54\!\cdots\!63\)\( T^{5} - \)\(15\!\cdots\!10\)\( T^{6} + \)\(60\!\cdots\!99\)\( T^{7} - \)\(16\!\cdots\!03\)\( T^{8} - \)\(22\!\cdots\!46\)\( T^{9} + \)\(32\!\cdots\!56\)\( T^{10} - \)\(19\!\cdots\!94\)\( T^{11} - \)\(12\!\cdots\!63\)\( T^{12} + \)\(39\!\cdots\!81\)\( T^{13} - \)\(87\!\cdots\!10\)\( T^{14} - \)\(26\!\cdots\!37\)\( T^{15} + \)\(16\!\cdots\!82\)\( T^{16} + \)\(10\!\cdots\!57\)\( T^{17} - \)\(85\!\cdots\!28\)\( T^{18} - \)\(12\!\cdots\!67\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 + 113075039 T - 40080640716766298 T^{2} - \)\(29\!\cdots\!31\)\( T^{3} + \)\(11\!\cdots\!98\)\( T^{4} + \)\(48\!\cdots\!45\)\( T^{5} - \)\(22\!\cdots\!44\)\( T^{6} - \)\(47\!\cdots\!63\)\( T^{7} + \)\(35\!\cdots\!01\)\( T^{8} + \)\(19\!\cdots\!06\)\( T^{9} - \)\(45\!\cdots\!48\)\( T^{10} + \)\(23\!\cdots\!46\)\( T^{11} + \)\(47\!\cdots\!81\)\( T^{12} - \)\(76\!\cdots\!23\)\( T^{13} - \)\(42\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!45\)\( T^{15} + \)\(28\!\cdots\!18\)\( T^{16} - \)\(88\!\cdots\!11\)\( T^{17} - \)\(14\!\cdots\!58\)\( T^{18} + \)\(46\!\cdots\!79\)\( T^{19} + \)\(47\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 + 126707879 T - 43788049157476460 T^{2} - \)\(33\!\cdots\!93\)\( T^{3} + \)\(41\!\cdots\!94\)\( T^{4} - \)\(50\!\cdots\!91\)\( T^{5} - \)\(15\!\cdots\!78\)\( T^{6} - \)\(40\!\cdots\!93\)\( T^{7} + \)\(55\!\cdots\!05\)\( T^{8} + \)\(14\!\cdots\!02\)\( T^{9} - \)\(51\!\cdots\!60\)\( T^{10} + \)\(39\!\cdots\!94\)\( T^{11} + \)\(40\!\cdots\!45\)\( T^{12} - \)\(81\!\cdots\!39\)\( T^{13} - \)\(85\!\cdots\!18\)\( T^{14} - \)\(74\!\cdots\!37\)\( T^{15} + \)\(16\!\cdots\!26\)\( T^{16} - \)\(36\!\cdots\!59\)\( T^{17} - \)\(13\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!93\)\( T^{19} + \)\(22\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 + 594368016 T + 309193099286863459 T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(31\!\cdots\!50\)\( T^{4} + \)\(69\!\cdots\!04\)\( T^{5} + \)\(14\!\cdots\!50\)\( T^{6} + \)\(21\!\cdots\!92\)\( T^{7} + \)\(29\!\cdots\!69\)\( T^{8} + \)\(26\!\cdots\!36\)\( T^{9} + \)\(20\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( 1 + 859257651 T + 199070264658270994 T^{2} + \)\(11\!\cdots\!21\)\( T^{3} + \)\(18\!\cdots\!02\)\( T^{4} + \)\(85\!\cdots\!25\)\( T^{5} + \)\(51\!\cdots\!68\)\( T^{6} + \)\(92\!\cdots\!93\)\( T^{7} + \)\(17\!\cdots\!81\)\( T^{8} + \)\(29\!\cdots\!82\)\( T^{9} + \)\(16\!\cdots\!12\)\( T^{10} + \)\(17\!\cdots\!66\)\( T^{11} + \)\(59\!\cdots\!89\)\( T^{12} + \)\(18\!\cdots\!21\)\( T^{13} + \)\(62\!\cdots\!48\)\( T^{14} + \)\(60\!\cdots\!25\)\( T^{15} + \)\(77\!\cdots\!18\)\( T^{16} + \)\(27\!\cdots\!57\)\( T^{17} + \)\(28\!\cdots\!74\)\( T^{18} + \)\(72\!\cdots\!23\)\( T^{19} + \)\(50\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 + 527065417 T - 105527969267521112 T^{2} + \)\(26\!\cdots\!45\)\( T^{3} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(24\!\cdots\!85\)\( T^{5} + \)\(26\!\cdots\!82\)\( T^{6} + \)\(34\!\cdots\!09\)\( T^{7} - \)\(57\!\cdots\!11\)\( T^{8} + \)\(65\!\cdots\!50\)\( T^{9} + \)\(17\!\cdots\!24\)\( T^{10} + \)\(78\!\cdots\!50\)\( T^{11} - \)\(83\!\cdots\!71\)\( T^{12} + \)\(59\!\cdots\!31\)\( T^{13} + \)\(54\!\cdots\!22\)\( T^{14} - \)\(60\!\cdots\!15\)\( T^{15} + \)\(19\!\cdots\!98\)\( T^{16} + \)\(94\!\cdots\!55\)\( T^{17} - \)\(44\!\cdots\!92\)\( T^{18} + \)\(26\!\cdots\!43\)\( T^{19} + \)\(61\!\cdots\!01\)\( T^{20} \)
$83$ \( ( 1 + 72431604 T + 478832888086868191 T^{2} + \)\(61\!\cdots\!88\)\( T^{3} + \)\(12\!\cdots\!02\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!06\)\( T^{6} + \)\(21\!\cdots\!92\)\( T^{7} + \)\(31\!\cdots\!57\)\( T^{8} + \)\(88\!\cdots\!24\)\( T^{9} + \)\(22\!\cdots\!43\)\( T^{10} )^{2} \)
$89$ \( 1 - 1661554797 T + 593701432891643778 T^{2} + \)\(11\!\cdots\!53\)\( T^{3} + \)\(34\!\cdots\!50\)\( T^{4} - \)\(23\!\cdots\!83\)\( T^{5} - \)\(69\!\cdots\!72\)\( T^{6} + \)\(66\!\cdots\!81\)\( T^{7} - \)\(20\!\cdots\!67\)\( T^{8} + \)\(28\!\cdots\!94\)\( T^{9} - \)\(25\!\cdots\!60\)\( T^{10} + \)\(10\!\cdots\!46\)\( T^{11} - \)\(25\!\cdots\!27\)\( T^{12} + \)\(28\!\cdots\!49\)\( T^{13} - \)\(10\!\cdots\!92\)\( T^{14} - \)\(12\!\cdots\!67\)\( T^{15} + \)\(63\!\cdots\!50\)\( T^{16} + \)\(72\!\cdots\!57\)\( T^{17} + \)\(13\!\cdots\!38\)\( T^{18} - \)\(13\!\cdots\!33\)\( T^{19} + \)\(27\!\cdots\!01\)\( T^{20} \)
$97$ \( ( 1 - 434885094 T + 2848708413464846333 T^{2} - \)\(88\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!86\)\( T^{4} - \)\(83\!\cdots\!40\)\( T^{5} + \)\(27\!\cdots\!62\)\( T^{6} - \)\(50\!\cdots\!36\)\( T^{7} + \)\(12\!\cdots\!29\)\( T^{8} - \)\(14\!\cdots\!74\)\( T^{9} + \)\(25\!\cdots\!57\)\( T^{10} )^{2} \)
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