Properties

Label 29.8.a.b
Level $29$
Weight $8$
Character orbit 29.a
Self dual yes
Analytic conductor $9.059$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.05916573904\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + 2140673280 x^{2} - 1918315520 x - 9372051456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{2} ) q^{3} + ( 92 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 18 - 4 \beta_{1} + \beta_{2} - \beta_{7} ) q^{5} + ( 36 - 26 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 104 - 22 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{7} + ( -460 - 110 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 4 \beta_{8} - 5 \beta_{9} ) q^{8} + ( 1098 - 31 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{2} ) q^{3} + ( 92 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 18 - 4 \beta_{1} + \beta_{2} - \beta_{7} ) q^{5} + ( 36 - 26 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{6} + ( 104 - 22 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{7} + ( -460 - 110 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 4 \beta_{8} - 5 \beta_{9} ) q^{8} + ( 1098 - 31 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} - \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} ) q^{9} + ( 847 + 9 \beta_{1} - 48 \beta_{2} + 10 \beta_{3} - 16 \beta_{5} + 11 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} ) q^{10} + ( 743 + 9 \beta_{1} - 33 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - 12 \beta_{9} ) q^{11} + ( 4964 + 66 \beta_{1} + 43 \beta_{2} + 11 \beta_{3} + 10 \beta_{4} + 23 \beta_{5} - 11 \beta_{6} - 23 \beta_{7} + 8 \beta_{8} + 9 \beta_{9} ) q^{12} + ( 2079 + 87 \beta_{1} + 3 \beta_{2} + 11 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} + 14 \beta_{6} - 2 \beta_{7} - 23 \beta_{8} + 4 \beta_{9} ) q^{13} + ( 5112 - 80 \beta_{1} - 60 \beta_{2} + 12 \beta_{3} - 8 \beta_{5} - 38 \beta_{6} + 22 \beta_{7} - 24 \beta_{8} - 14 \beta_{9} ) q^{14} + ( 4341 + 635 \beta_{1} + 130 \beta_{2} - 14 \beta_{3} - 3 \beta_{4} + \beta_{5} + 38 \beta_{6} + 5 \beta_{7} + 25 \beta_{8} + 3 \beta_{9} ) q^{15} + ( 12186 + 1108 \beta_{1} + 79 \beta_{2} + 53 \beta_{3} - 54 \beta_{4} + 14 \beta_{5} + 20 \beta_{6} + 14 \beta_{7} + 22 \beta_{8} + 2 \beta_{9} ) q^{16} + ( -1141 + 269 \beta_{1} + 63 \beta_{2} - 70 \beta_{3} + 3 \beta_{4} - 25 \beta_{5} + 26 \beta_{6} + 23 \beta_{7} - \beta_{8} - 35 \beta_{9} ) q^{17} + ( 6621 - 652 \beta_{1} - 88 \beta_{2} - 32 \beta_{3} + 104 \beta_{4} - 104 \beta_{5} - 61 \beta_{6} + 54 \beta_{7} + 40 \beta_{8} + 82 \beta_{9} ) q^{18} + ( 7543 - 397 \beta_{1} + 74 \beta_{2} - 61 \beta_{3} + 90 \beta_{4} - 35 \beta_{5} - 110 \beta_{6} - 5 \beta_{7} - 15 \beta_{8} + 40 \beta_{9} ) q^{19} + ( -4344 - 86 \beta_{1} + 157 \beta_{2} - 27 \beta_{3} - 36 \beta_{4} + 186 \beta_{5} + 138 \beta_{6} - 234 \beta_{7} - 16 \beta_{8} - 18 \beta_{9} ) q^{20} + ( 5203 + 1027 \beta_{1} + 291 \beta_{2} - 112 \beta_{3} - 121 \beta_{4} + 65 \beta_{5} - 66 \beta_{6} + 37 \beta_{7} + \beta_{8} - 163 \beta_{9} ) q^{21} + ( -3732 + 154 \beta_{1} - 414 \beta_{2} + 141 \beta_{3} - 183 \beta_{4} + 35 \beta_{5} + 152 \beta_{6} - 2 \beta_{7} - 77 \beta_{8} - 66 \beta_{9} ) q^{22} + ( 6170 - 904 \beta_{1} - 65 \beta_{2} + 35 \beta_{3} + 135 \beta_{4} + 12 \beta_{5} - 36 \beta_{6} - 146 \beta_{7} + 24 \beta_{8} + 135 \beta_{9} ) q^{23} + ( -20490 - 4564 \beta_{1} - 927 \beta_{2} + 27 \beta_{3} + 70 \beta_{4} - 216 \beta_{5} - 146 \beta_{6} + 244 \beta_{7} - 158 \beta_{8} - 28 \beta_{9} ) q^{24} + ( 26009 + 1684 \beta_{1} - 66 \beta_{2} + 343 \beta_{3} - 57 \beta_{4} - 64 \beta_{5} + 32 \beta_{6} + 175 \beta_{7} + 32 \beta_{8} + 123 \beta_{9} ) q^{25} + ( -20175 - 2397 \beta_{1} - 756 \beta_{2} + 42 \beta_{3} - 66 \beta_{4} - 118 \beta_{5} + 41 \beta_{6} + 48 \beta_{7} + 134 \beta_{8} + 12 \beta_{9} ) q^{26} + ( -2967 + 1777 \beta_{1} + 771 \beta_{2} + 91 \beta_{3} + 88 \beta_{4} - 17 \beta_{5} + 310 \beta_{6} + 145 \beta_{7} + 267 \beta_{8} + 154 \beta_{9} ) q^{27} + ( -2464 - 4172 \beta_{1} + 762 \beta_{2} - 38 \beta_{3} + 96 \beta_{4} + 564 \beta_{5} - 484 \beta_{6} - 444 \beta_{7} - 80 \beta_{8} - 132 \beta_{9} ) q^{28} -24389 q^{29} + ( -128390 - 5586 \beta_{1} - 114 \beta_{2} - 575 \beta_{3} + 269 \beta_{4} - 425 \beta_{5} + 364 \beta_{6} + 280 \beta_{7} + 243 \beta_{8} - 104 \beta_{9} ) q^{30} + ( 20106 + 3236 \beta_{1} + 8 \beta_{2} + 277 \beta_{3} + 165 \beta_{4} + 48 \beta_{5} - 492 \beta_{6} - 266 \beta_{7} - 68 \beta_{8} + 5 \beta_{9} ) q^{31} + ( -175740 - 11190 \beta_{1} - 965 \beta_{2} - 821 \beta_{3} + 12 \beta_{4} - 117 \beta_{5} - 223 \beta_{6} + 57 \beta_{7} - 540 \beta_{8} - 329 \beta_{9} ) q^{32} + ( -106700 + 6066 \beta_{1} + 650 \beta_{2} - 101 \beta_{3} - 863 \beta_{4} + 226 \beta_{5} + 588 \beta_{6} + 403 \beta_{7} - 814 \beta_{8} - 399 \beta_{9} ) q^{33} + ( -50706 + 9412 \beta_{1} + 1416 \beta_{2} + 122 \beta_{3} - 204 \beta_{4} - 12 \beta_{5} + 604 \beta_{6} - 194 \beta_{7} + 598 \beta_{8} + 252 \beta_{9} ) q^{34} + ( 10724 + 13618 \beta_{1} - 2177 \beta_{2} + 439 \beta_{3} - 339 \beta_{4} - 798 \beta_{5} - 8 \beta_{6} - 88 \beta_{7} + 358 \beta_{8} + 153 \beta_{9} ) q^{35} + ( -2956 + 6844 \beta_{1} + 4962 \beta_{2} - 118 \beta_{3} + 360 \beta_{4} + 872 \beta_{5} - 308 \beta_{6} - 1040 \beta_{7} + 320 \beta_{8} + 864 \beta_{9} ) q^{36} + ( -36780 + 1506 \beta_{1} + 660 \beta_{2} + 220 \beta_{3} + 102 \beta_{4} + 870 \beta_{5} + 204 \beta_{6} - 128 \beta_{7} - 1074 \beta_{8} - 190 \beta_{9} ) q^{37} + ( 76450 - 192 \beta_{1} + 2780 \beta_{2} - 440 \beta_{3} + 630 \beta_{4} + 142 \beta_{5} - 1286 \beta_{6} - 256 \beta_{7} + 1144 \beta_{8} + 1630 \beta_{9} ) q^{38} + ( 39562 + 11010 \beta_{1} + 490 \beta_{2} - 477 \beta_{3} + 425 \beta_{4} + 114 \beta_{5} - 616 \beta_{6} - 582 \beta_{7} + 494 \beta_{8} - 979 \beta_{9} ) q^{39} + ( -86392 + 10494 \beta_{1} - 5961 \beta_{2} + 1715 \beta_{3} - 12 \beta_{4} - 2855 \beta_{5} + 1231 \beta_{6} + 2859 \beta_{7} + 288 \beta_{8} - 919 \beta_{9} ) q^{40} + ( 93297 - 3745 \beta_{1} - 2795 \beta_{2} - 116 \beta_{3} + 399 \beta_{4} - 83 \beta_{5} - 474 \beta_{6} + 39 \beta_{7} + 397 \beta_{8} + 173 \beta_{9} ) q^{41} + ( -204982 - 13230 \beta_{1} + 2016 \beta_{2} - 878 \beta_{3} - 1256 \beta_{4} + 1336 \beta_{5} - 344 \beta_{6} + 1358 \beta_{7} - 2322 \beta_{8} - 1912 \beta_{9} ) q^{42} + ( 144017 - 837 \beta_{1} - 2851 \beta_{2} + 525 \beta_{3} + 612 \beta_{4} + 1053 \beta_{5} + 570 \beta_{6} - 1007 \beta_{7} - 63 \beta_{8} - 390 \beta_{9} ) q^{43} + ( -133036 - 5130 \beta_{1} - 3435 \beta_{2} + 1837 \beta_{3} - 1170 \beta_{4} + 521 \beta_{5} + 2211 \beta_{6} + 167 \beta_{7} - 1424 \beta_{8} - 921 \beta_{9} ) q^{44} + ( 423925 + 12953 \beta_{1} + 182 \beta_{2} + 1011 \beta_{3} + 952 \beta_{4} - 1077 \beta_{5} - 854 \beta_{6} - 545 \beta_{7} + 1483 \beta_{8} + 2654 \beta_{9} ) q^{45} + ( 175608 + 1104 \beta_{1} - 4528 \beta_{2} + 898 \beta_{3} + 1494 \beta_{4} - 2654 \beta_{5} - 90 \beta_{6} + 1046 \beta_{7} + 2098 \beta_{8} + 1938 \beta_{9} ) q^{46} + ( -28465 - 24441 \beta_{1} - 4680 \beta_{2} - 2368 \beta_{3} - 729 \beta_{4} + 181 \beta_{5} + 1630 \beta_{6} + 95 \beta_{7} + 469 \beta_{8} - 1067 \beta_{9} ) q^{47} + ( 317944 + 36562 \beta_{1} + 9113 \beta_{2} + 1845 \beta_{3} - 3096 \beta_{4} + 3135 \beta_{5} - 767 \beta_{6} - 3651 \beta_{7} - 1088 \beta_{8} + 715 \beta_{9} ) q^{48} + ( 472067 - 18438 \beta_{1} + 1246 \beta_{2} - 1932 \beta_{3} + 1134 \beta_{4} - 322 \beta_{5} - 2828 \beta_{6} + 602 \beta_{7} + 686 \beta_{8} + 658 \beta_{9} ) q^{49} + ( -366739 - 73934 \beta_{1} + 2540 \beta_{2} - 4588 \beta_{3} + 702 \beta_{4} + 3370 \beta_{5} - 2057 \beta_{6} - 2926 \beta_{7} - 2644 \beta_{8} - 1712 \beta_{9} ) q^{50} + ( 145806 - 22136 \beta_{1} - 5843 \beta_{2} - 1635 \beta_{3} + 313 \beta_{4} - 2868 \beta_{5} - 100 \beta_{6} + 78 \beta_{7} - 376 \beta_{8} + 1457 \beta_{9} ) q^{51} + ( 256088 + 20110 \beta_{1} + 1151 \beta_{2} + 839 \beta_{3} + 228 \beta_{4} + 354 \beta_{5} - 854 \beta_{6} - 2530 \beta_{7} + 880 \beta_{8} - 1554 \beta_{9} ) q^{52} + ( 394145 - 13215 \beta_{1} + 2215 \beta_{2} + 1319 \beta_{3} - 2856 \beta_{4} + 63 \beta_{5} + 3666 \beta_{6} + 2474 \beta_{7} + 431 \beta_{8} + 406 \beta_{9} ) q^{53} + ( -314258 - 20194 \beta_{1} + 8538 \beta_{2} - 3051 \beta_{3} + 3509 \beta_{4} - 1641 \beta_{5} + 1118 \beta_{6} - 14 \beta_{7} + 1643 \beta_{8} - 546 \beta_{9} ) q^{54} + ( 398324 - 14384 \beta_{1} - 5742 \beta_{2} + 885 \beta_{3} - 2967 \beta_{4} + 1260 \beta_{5} + 1412 \beta_{6} + 992 \beta_{7} - 3128 \beta_{8} - 1391 \beta_{9} ) q^{55} + ( 207840 - 18452 \beta_{1} + 2574 \beta_{2} + 3334 \beta_{3} + 264 \beta_{4} - 2774 \beta_{5} + 390 \beta_{6} + 8270 \beta_{7} + 480 \beta_{8} + 602 \beta_{9} ) q^{56} + ( 205941 - 33321 \beta_{1} + 14162 \beta_{2} - 3031 \beta_{3} + 2820 \beta_{4} - 883 \beta_{5} + 126 \beta_{6} + 3073 \beta_{7} + 757 \beta_{8} - 1354 \beta_{9} ) q^{57} + 24389 \beta_{1} q^{58} + ( 671536 - 67410 \beta_{1} - 2745 \beta_{2} + 2129 \beta_{3} + 5259 \beta_{4} - 546 \beta_{5} - 3352 \beta_{6} - 696 \beta_{7} + 1562 \beta_{8} - 1313 \beta_{9} ) q^{59} + ( 743612 + 156906 \beta_{1} + 10423 \beta_{2} + 10079 \beta_{3} + 1118 \beta_{4} - 619 \beta_{5} + 1599 \beta_{6} - 6493 \beta_{7} + 4864 \beta_{8} + 4395 \beta_{9} ) q^{60} + ( 190509 + 28643 \beta_{1} + 14179 \beta_{2} - 5004 \beta_{3} - 4941 \beta_{4} + 4505 \beta_{5} + 2934 \beta_{6} + 397 \beta_{7} - 399 \beta_{8} - 1983 \beta_{9} ) q^{61} + ( -803212 - 70830 \beta_{1} - 11154 \beta_{2} - 5165 \beta_{3} - 171 \beta_{4} + 643 \beta_{5} - 2926 \beta_{6} + 2520 \beta_{7} - 1567 \beta_{8} + 830 \beta_{9} ) q^{62} + ( 364896 + 40296 \beta_{1} - 12598 \beta_{2} + 6266 \beta_{3} - 5950 \beta_{4} + 416 \beta_{5} - 2120 \beta_{6} - 1512 \beta_{7} - 7816 \beta_{8} + 674 \beta_{9} ) q^{63} + ( 841570 + 185828 \beta_{1} + 3739 \beta_{2} + 9001 \beta_{3} + 1506 \beta_{4} + 266 \beta_{5} - 2528 \beta_{6} - 4958 \beta_{7} + 2446 \beta_{8} + 5694 \beta_{9} ) q^{64} + ( 467637 - 4275 \beta_{1} + 1062 \beta_{2} + 4138 \beta_{3} + 1623 \beta_{4} - 1237 \beta_{5} - 5486 \beta_{6} - 4448 \beta_{7} - 349 \beta_{8} - 1479 \beta_{9} ) q^{65} + ( -1241485 + 101339 \beta_{1} - 13348 \beta_{2} - 1360 \beta_{3} - 7830 \beta_{4} + 3782 \beta_{5} + 2337 \beta_{6} - 562 \beta_{7} - 3416 \beta_{8} - 6708 \beta_{9} ) q^{66} + ( -271404 + 8908 \beta_{1} - 2838 \beta_{2} - 10930 \beta_{3} - 1638 \beta_{4} + 380 \beta_{5} + 1744 \beta_{6} + 1952 \beta_{7} + 2036 \beta_{8} + 4690 \beta_{9} ) q^{67} + ( -1769488 - 32640 \beta_{1} - 30996 \beta_{2} - 356 \beta_{3} + 4740 \beta_{4} - 6310 \beta_{5} + 3702 \beta_{6} + 2494 \beta_{7} + 128 \beta_{8} + 406 \beta_{9} ) q^{68} + ( 108369 + 56365 \beta_{1} + 30485 \beta_{2} + 1614 \beta_{3} + 8519 \beta_{4} - 1473 \beta_{5} + 3546 \beta_{6} + 729 \beta_{7} + 8967 \beta_{8} + 33 \beta_{9} ) q^{69} + ( -3041296 - 38500 \beta_{1} - 39476 \beta_{2} - 15880 \beta_{3} - 2508 \beta_{4} + 3388 \beta_{5} + 6490 \beta_{6} - 10346 \beta_{7} - 6020 \beta_{8} - 2310 \beta_{9} ) q^{70} + ( 339980 - 50800 \beta_{1} + 1690 \beta_{2} + 11616 \beta_{3} + 336 \beta_{4} + 7888 \beta_{5} + 4944 \beta_{6} - 2224 \beta_{7} - 2192 \beta_{8} + 1712 \beta_{9} ) q^{71} + ( -2241952 - 7932 \beta_{1} - 25154 \beta_{2} - 8578 \beta_{3} + 2608 \beta_{4} - 11170 \beta_{5} - 878 \beta_{6} + 19930 \beta_{7} + 2648 \beta_{8} - 5682 \beta_{9} ) q^{72} + ( -250958 - 23632 \beta_{1} - 8706 \beta_{2} - 2624 \beta_{3} - 1920 \beta_{4} + 1856 \beta_{5} - 7800 \beta_{6} - 7442 \beta_{7} - 7336 \beta_{8} - 6536 \beta_{9} ) q^{73} + ( -422826 + 17820 \beta_{1} - 9480 \beta_{2} + 5840 \beta_{3} - 3648 \beta_{4} - 3456 \beta_{5} - 2534 \beta_{6} + 8340 \beta_{7} + 5960 \beta_{8} + 1028 \beta_{9} ) q^{74} + ( 108704 - 14802 \beta_{1} + 7720 \beta_{2} - 1016 \beta_{3} + 1326 \beta_{4} + 7762 \beta_{5} - 5612 \beta_{6} - 6408 \beta_{7} + 2562 \beta_{8} - 3174 \beta_{9} ) q^{75} + ( -823616 - 88784 \beta_{1} + 9828 \beta_{2} - 16836 \beta_{3} + 12864 \beta_{4} - 3164 \beta_{5} - 15780 \beta_{6} + 8876 \beta_{7} + 968 \beta_{8} + 5468 \beta_{9} ) q^{76} + ( -398955 - 62579 \beta_{1} + 29785 \beta_{2} + 2724 \beta_{3} + 549 \beta_{4} - 11593 \beta_{5} + 2954 \beta_{6} + 1027 \beta_{7} + 5439 \beta_{8} + 11351 \beta_{9} ) q^{77} + ( -2481090 - 50562 \beta_{1} - 4650 \beta_{2} - 3661 \beta_{3} - 4947 \beta_{4} - 2269 \beta_{5} + 9620 \beta_{6} + 10824 \beta_{7} - 1191 \beta_{8} - 2034 \beta_{9} ) q^{78} + ( 416355 + 57923 \beta_{1} + 18354 \beta_{2} + 4376 \beta_{3} - 5229 \beta_{4} - 3567 \beta_{5} + 11534 \beta_{6} + 10567 \beta_{7} - 1799 \beta_{8} + 1553 \beta_{9} ) q^{79} + ( -1773890 + 34056 \beta_{1} + 75587 \beta_{2} - 11075 \beta_{3} - 13074 \beta_{4} + 29524 \beta_{5} + 3442 \beta_{6} - 35544 \beta_{7} - 5022 \beta_{8} - 5064 \beta_{9} ) q^{80} + ( 511406 - 78389 \beta_{1} - 24692 \beta_{2} + 19159 \beta_{3} + 10126 \beta_{4} - 9287 \beta_{5} + 1414 \beta_{6} - 4739 \beta_{7} + 4833 \beta_{8} + 7552 \beta_{9} ) q^{81} + ( 702016 - 44994 \beta_{1} + 17352 \beta_{2} + 1246 \beta_{3} + 1536 \beta_{4} + 5128 \beta_{5} - 6194 \beta_{6} - 6562 \beta_{7} - 3174 \beta_{8} + 3164 \beta_{9} ) q^{82} + ( 351980 - 101410 \beta_{1} + 10165 \beta_{2} + 579 \beta_{3} - 1059 \beta_{4} - 11658 \beta_{5} - 5392 \beta_{6} + 3924 \beta_{7} + 8090 \beta_{8} - 2159 \beta_{9} ) q^{83} + ( 2286760 + 71508 \beta_{1} + 33374 \beta_{2} + 34878 \beta_{3} - 7852 \beta_{4} + 20010 \beta_{5} - 7322 \beta_{6} - 946 \beta_{7} - 14768 \beta_{8} + 7014 \beta_{9} ) q^{84} + ( 187717 - 138971 \beta_{1} + 50593 \beta_{2} - 11494 \beta_{3} + 579 \beta_{4} - 1441 \beta_{5} - 3422 \beta_{6} + 13545 \beta_{7} - 4545 \beta_{8} - 9147 \beta_{9} ) q^{85} + ( -40680 - 119754 \beta_{1} - 35462 \beta_{2} + 20445 \beta_{3} - 4203 \beta_{4} - 15041 \beta_{5} + 14416 \beta_{6} + 13362 \beta_{7} + 5091 \beta_{8} - 2330 \beta_{9} ) q^{86} + ( -195112 - 24389 \beta_{2} ) q^{87} + ( 1509546 + 52804 \beta_{1} + 327 \beta_{2} + 12013 \beta_{3} + 1866 \beta_{4} - 2456 \beta_{5} + 8754 \beta_{6} - 5268 \beta_{7} + 4382 \beta_{8} - 11188 \beta_{9} ) q^{88} + ( -26229 - 11123 \beta_{1} + 15981 \beta_{2} + 24766 \beta_{3} - 9993 \beta_{4} + 647 \beta_{5} - 926 \beta_{6} + 305 \beta_{7} - 23945 \beta_{8} - 21031 \beta_{9} ) q^{89} + ( -2885908 - 544446 \beta_{1} - 37252 \beta_{2} - 42076 \beta_{3} + 29146 \beta_{4} - 11578 \beta_{5} - 19356 \beta_{6} - 8184 \beta_{7} + 5956 \beta_{8} + 17014 \beta_{9} ) q^{90} + ( 2716388 + 16710 \beta_{1} - 87015 \beta_{2} - 13879 \beta_{3} + 387 \beta_{4} + 4262 \beta_{5} + 3304 \beta_{6} - 4104 \beta_{7} + 21778 \beta_{8} + 10135 \beta_{9} ) q^{91} + ( -1076520 - 23848 \beta_{1} + 66480 \beta_{2} - 30864 \beta_{3} + 3348 \beta_{4} + 12970 \beta_{5} + 574 \beta_{6} - 26786 \beta_{7} + 6336 \beta_{8} + 1894 \beta_{9} ) q^{92} + ( 105326 + 241230 \beta_{1} + 38357 \beta_{2} + 10826 \beta_{3} - 10484 \beta_{4} + 24662 \beta_{5} + 7284 \beta_{6} - 3219 \beta_{7} - 13706 \beta_{8} - 7016 \beta_{9} ) q^{93} + ( 5551372 + 467274 \beta_{1} + 19882 \beta_{2} + 51355 \beta_{3} - 11769 \beta_{4} - 9795 \beta_{5} + 25266 \beta_{6} - 2512 \beta_{7} + 2209 \beta_{8} - 5032 \beta_{9} ) q^{94} + ( -876421 + 69961 \beta_{1} - 50142 \beta_{2} - 23773 \beta_{3} + 19416 \beta_{4} - 17201 \beta_{5} - 7778 \beta_{6} - 1421 \beta_{7} + 7451 \beta_{8} + 18138 \beta_{9} ) q^{95} + ( -5311894 - 295416 \beta_{1} - 167943 \beta_{2} - 42009 \beta_{3} - 6278 \beta_{4} - 23232 \beta_{5} + 8130 \beta_{6} + 49364 \beta_{7} - 15978 \beta_{8} - 31060 \beta_{9} ) q^{96} + ( 494479 + 109349 \beta_{1} - 2577 \beta_{2} - 50144 \beta_{3} + 10617 \beta_{4} - 3753 \beta_{5} - 614 \beta_{6} + 117 \beta_{7} + 24367 \beta_{8} + 12059 \beta_{9} ) q^{97} + ( 4015592 - 356497 \beta_{1} + 113344 \beta_{2} - 10024 \beta_{3} + 15204 \beta_{4} + 17444 \beta_{5} - 43876 \beta_{6} - 5572 \beta_{7} - 1568 \beta_{8} + 20104 \beta_{9} ) q^{98} + ( -1429911 + 227833 \beta_{1} - 148246 \beta_{2} - 26205 \beta_{3} - 17832 \beta_{4} + 8535 \beta_{5} - 21322 \beta_{6} - 5399 \beta_{7} - 7837 \beta_{8} - 7974 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 80q^{3} + 922q^{4} + 180q^{5} + 358q^{6} + 1040q^{7} - 4620q^{8} + 10986q^{9} + O(q^{10}) \) \( 10q + 80q^{3} + 922q^{4} + 180q^{5} + 358q^{6} + 1040q^{7} - 4620q^{8} + 10986q^{9} + 8496q^{10} + 7384q^{11} + 49720q^{12} + 20820q^{13} + 50976q^{14} + 43516q^{15} + 122082q^{16} - 11620q^{17} + 66060q^{18} + 75068q^{19} - 42914q^{20} + 51480q^{21} - 36950q^{22} + 62040q^{23} - 205942q^{24} + 261022q^{25} - 201528q^{26} - 28060q^{27} - 24980q^{28} - 243890q^{29} - 1284894q^{30} + 200600q^{31} - 1761460q^{32} - 1068000q^{33} - 503932q^{34} + 107528q^{35} - 26300q^{36} - 367740q^{37} + 766880q^{38} + 392692q^{39} - 865000q^{40} + 932764q^{41} - 2058060q^{42} + 1443560q^{43} - 1325912q^{44} + 4245684q^{45} + 1760460q^{46} - 286960q^{47} + 3187120q^{48} + 4713194q^{49} - 3682652q^{50} + 1451016q^{51} + 2560210q^{52} + 3953220q^{53} - 3147534q^{54} + 3981316q^{55} + 2082464q^{56} + 2050640q^{57} + 6712320q^{59} + 7476756q^{60} + 1905196q^{61} - 8048490q^{62} + 3643800q^{63} + 8445458q^{64} + 4667544q^{65} - 12425580q^{66} - 2718200q^{67} - 17699740q^{68} + 1109064q^{69} - 30441624q^{70} + 3447736q^{71} - 22466840q^{72} - 2554460q^{73} - 4214584q^{74} + 1088084q^{75} - 8294848q^{76} - 3967800q^{77} - 24809970q^{78} + 4187744q^{79} - 17715290q^{80} + 5161402q^{81} + 7020500q^{82} + 3498720q^{83} + 22947224q^{84} + 1817072q^{85} - 361638q^{86} - 1951120q^{87} + 15118470q^{88} - 303268q^{89} - 28959160q^{90} + 27215080q^{91} - 10783380q^{92} + 1097360q^{93} + 55641726q^{94} - 8810536q^{95} - 53327238q^{96} + 4908620q^{97} + 40120080q^{98} - 14408716q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + 2140673280 x^{2} - 1918315520 x - 9372051456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-14005209807 \nu^{9} + 59170210660 \nu^{8} + 14265030073043 \nu^{7} - 19758503934520 \nu^{6} - 4799744739677380 \nu^{5} - 4484184535987040 \nu^{4} + 553721440219482272 \nu^{3} + 1113537869125544960 \nu^{2} - 13671818781375387136 \nu - 10999486194010383360\)\()/ 259990674547368960 \)
\(\beta_{3}\)\(=\)\((\)\(14005209807 \nu^{9} - 59170210660 \nu^{8} - 14265030073043 \nu^{7} + 19758503934520 \nu^{6} + 4799744739677380 \nu^{5} + 4484184535987040 \nu^{4} - 553721440219482272 \nu^{3} - 853547194578176000 \nu^{2} + 13151837432280649216 \nu - 46198462206410787840\)\()/ 259990674547368960 \)
\(\beta_{4}\)\(=\)\((\)\(-10384500596 \nu^{9} + 30939298131 \nu^{8} + 10038072254884 \nu^{7} - 2197921748399 \nu^{6} - 3022432375049340 \nu^{5} - 8988731024606340 \nu^{4} + 248428981035941776 \nu^{3} + 2044275900677672704 \nu^{2} + 5597485689483884672 \nu - 53768627900084404992\)\()/ 129995337273684480 \)
\(\beta_{5}\)\(=\)\((\)\(-29512730997 \nu^{9} + 180010810744 \nu^{8} + 27002918567993 \nu^{7} - 47739245380996 \nu^{6} - 7989438820016980 \nu^{5} - 19059537068353040 \nu^{4} + 714186836150737472 \nu^{3} + 3993166508334928256 \nu^{2} + 285923858056296704 \nu - 60819265834069582848\)\()/ 129995337273684480 \)
\(\beta_{6}\)\(=\)\((\)\(-11916632683 \nu^{9} - 174800952576 \nu^{8} + 14992620801047 \nu^{7} + 168334959164684 \nu^{6} - 5780375488193340 \nu^{5} - 53096897357294640 \nu^{4} + 717937240969453568 \nu^{3} + 5173932072820822016 \nu^{2} - 17314653838431820544 \nu - 41584411397991909888\)\()/ 43331779091228160 \)
\(\beta_{7}\)\(=\)\((\)\(-51681392253 \nu^{9} + 157018466129 \nu^{8} + 49336581999097 \nu^{7} + 7370824996639 \nu^{6} - 15432004992829520 \nu^{5} - 44049762052803100 \nu^{4} + 1601042868034054288 \nu^{3} + 6816647931116584576 \nu^{2} - 25943190360687645824 \nu - 71378897385067438848\)\()/ 129995337273684480 \)
\(\beta_{8}\)\(=\)\((\)\(151314526971 \nu^{9} + 647264453194 \nu^{8} - 164885198551999 \nu^{7} - 935728438657246 \nu^{6} + 58490652470742860 \nu^{5} + 380044109429268280 \nu^{4} - 7027806690032748736 \nu^{3} - 42758193075244562944 \nu^{2} + 176190842444414160128 \nu + 369112706533483924992\)\()/ 259990674547368960 \)
\(\beta_{9}\)\(=\)\((\)\(-2454351473 \nu^{9} - 5753235209 \nu^{8} + 2590223575437 \nu^{7} + 11490332531601 \nu^{6} - 890885912097160 \nu^{5} - 5233883508657860 \nu^{4} + 102989865726840048 \nu^{3} + 611201810967715584 \nu^{2} - 2439432847487560064 \nu - 5300606801502880512\)\()/ 3333213776248320 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 2 \beta_{1} + 220\)
\(\nu^{3}\)\(=\)\(5 \beta_{9} + 4 \beta_{8} - \beta_{7} - \beta_{6} - 3 \beta_{5} + 5 \beta_{3} + 5 \beta_{2} + 366 \beta_{1} + 460\)
\(\nu^{4}\)\(=\)\(2 \beta_{9} + 22 \beta_{8} + 14 \beta_{7} + 20 \beta_{6} + 14 \beta_{5} - 54 \beta_{4} + 437 \beta_{3} + 463 \beta_{2} + 1876 \beta_{1} + 80282\)
\(\nu^{5}\)\(=\)\(2889 \beta_{9} + 2588 \beta_{8} - 569 \beta_{7} - 289 \beta_{6} - 1419 \beta_{5} - 12 \beta_{4} + 3381 \beta_{3} + 3525 \beta_{2} + 149430 \beta_{1} + 411260\)
\(\nu^{6}\)\(=\)\(6974 \beta_{9} + 16526 \beta_{8} + 4002 \beta_{7} + 10272 \beta_{6} + 9226 \beta_{5} - 33054 \beta_{4} + 190377 \beta_{3} + 201755 \beta_{2} + 1189860 \beta_{1} + 32692322\)
\(\nu^{7}\)\(=\)\(1360817 \beta_{9} + 1284308 \beta_{8} - 308305 \beta_{7} - 60289 \beta_{6} - 559619 \beta_{5} - 63540 \beta_{4} + 1913513 \beta_{3} + 2224097 \beta_{2} + 64077830 \beta_{1} + 259537956\)
\(\nu^{8}\)\(=\)\(6265666 \beta_{9} + 9944574 \beta_{8} - 491682 \beta_{7} + 4193788 \beta_{6} + 4829694 \beta_{5} - 16077006 \beta_{4} + 83942829 \beta_{3} + 91444191 \beta_{2} + 667945372 \beta_{1} + 13994765650\)
\(\nu^{9}\)\(=\)\(609645753 \beta_{9} + 591000076 \beta_{8} - 170764441 \beta_{7} - 5077777 \beta_{6} - 199397579 \beta_{5} - 64607340 \beta_{4} + 1013643517 \beta_{3} + 1269390653 \beta_{2} + 28251194358 \beta_{1} + 145602030764\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
22.0686
21.4083
13.0110
4.34965
3.85204
−1.69492
−9.91427
−14.7228
−19.0438
−19.3138
−22.0686 73.8979 359.023 376.792 −1630.82 647.379 −5098.36 3273.90 −8315.26
1.2 −21.4083 −17.3673 330.317 −555.983 371.805 −1113.12 −4331.27 −1885.38 11902.7
1.3 −13.0110 25.6161 41.2859 124.561 −333.291 −962.222 1128.24 −1530.82 −1620.67
1.4 −4.34965 −40.5558 −109.081 −341.807 176.403 956.613 1031.22 −542.229 1486.74
1.5 −3.85204 −70.5970 −113.162 −69.8627 271.943 −1359.73 928.965 2796.94 269.114
1.6 1.69492 56.8671 −125.127 194.958 96.3850 1221.62 −429.029 1046.87 330.438
1.7 9.91427 −83.3983 −29.7073 326.241 −826.834 1247.39 −1563.55 4768.28 3234.44
1.8 14.7228 83.8824 88.7619 246.177 1234.99 −1219.72 −577.696 4849.26 3624.43
1.9 19.0438 0.836957 234.668 287.010 15.9389 −124.374 2031.37 −2186.30 5465.77
1.10 19.3138 50.8179 245.021 −408.086 981.485 1746.17 2260.12 395.462 −7881.68
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.8.a.b 10
3.b odd 2 1 261.8.a.f 10
4.b odd 2 1 464.8.a.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.8.a.b 10 1.a even 1 1 trivial
261.8.a.f 10 3.b odd 2 1
464.8.a.g 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -9372051456 + 1918315520 T + 2140673280 T^{2} + 87078400 T^{3} - 54569376 T^{4} - 870160 T^{5} + 405148 T^{6} + 1540 T^{7} - 1101 T^{8} + T^{10} \)
$3$ \( 1592673248644596 - 1892361691916460 T - 18022100824683 T^{2} + 6475913448300 T^{3} - 37887674064 T^{4} - 4823822020 T^{5} + 49434610 T^{6} + 1121620 T^{7} - 13228 T^{8} - 80 T^{9} + T^{10} \)
$5$ \( \)\(11\!\cdots\!00\)\( - \)\(56\!\cdots\!00\)\( T - \)\(19\!\cdots\!75\)\( T^{2} + 1814179104029773400 T^{3} - 1752170074932580 T^{4} - 29112800930304 T^{5} + 73606803758 T^{6} + 140825384 T^{7} - 504936 T^{8} - 180 T^{9} + T^{10} \)
$7$ \( -\)\(36\!\cdots\!64\)\( - \)\(24\!\cdots\!40\)\( T + \)\(49\!\cdots\!36\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} - 12706194596175179008 T^{4} - 10030974026129280 T^{5} + 13020047545680 T^{6} + 5334338080 T^{7} - 5933512 T^{8} - 1040 T^{9} + T^{10} \)
$11$ \( -\)\(55\!\cdots\!76\)\( - \)\(13\!\cdots\!68\)\( T - \)\(29\!\cdots\!07\)\( T^{2} + \)\(68\!\cdots\!04\)\( T^{3} + \)\(28\!\cdots\!36\)\( T^{4} - 10403109067466730828 T^{5} + 637508886110282 T^{6} + 549299326796 T^{7} - 62780820 T^{8} - 7384 T^{9} + T^{10} \)
$13$ \( -\)\(23\!\cdots\!44\)\( - \)\(10\!\cdots\!00\)\( T - \)\(12\!\cdots\!63\)\( T^{2} - \)\(51\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!92\)\( T^{4} - 7935986665128914680 T^{5} - 25048329283210570 T^{6} + 4094828348880 T^{7} - 71678432 T^{8} - 20820 T^{9} + T^{10} \)
$17$ \( -\)\(19\!\cdots\!24\)\( - \)\(45\!\cdots\!60\)\( T - \)\(50\!\cdots\!28\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} - \)\(24\!\cdots\!64\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + 1139263887492096368 T^{6} - 10088828734560 T^{7} - 1825885764 T^{8} + 11620 T^{9} + T^{10} \)
$19$ \( -\)\(67\!\cdots\!96\)\( - \)\(13\!\cdots\!32\)\( T - \)\(15\!\cdots\!84\)\( T^{2} + \)\(42\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!92\)\( T^{4} - \)\(23\!\cdots\!76\)\( T^{5} - 1502248924863570624 T^{6} + 276570552127808 T^{7} - 2385078892 T^{8} - 75068 T^{9} + T^{10} \)
$23$ \( -\)\(91\!\cdots\!16\)\( + \)\(97\!\cdots\!80\)\( T + \)\(73\!\cdots\!84\)\( T^{2} + \)\(25\!\cdots\!40\)\( T^{3} - \)\(12\!\cdots\!40\)\( T^{4} - \)\(81\!\cdots\!40\)\( T^{5} + 74535866981962874768 T^{6} + 536534348044000 T^{7} - 14645922680 T^{8} - 62040 T^{9} + T^{10} \)
$29$ \( ( 24389 + T )^{10} \)
$31$ \( \)\(53\!\cdots\!56\)\( - \)\(15\!\cdots\!88\)\( T - \)\(52\!\cdots\!31\)\( T^{2} + \)\(39\!\cdots\!24\)\( T^{3} - \)\(91\!\cdots\!08\)\( T^{4} - \)\(37\!\cdots\!68\)\( T^{5} + \)\(15\!\cdots\!22\)\( T^{6} + 14702262584823132 T^{7} - 69755231996 T^{8} - 200600 T^{9} + T^{10} \)
$37$ \( \)\(44\!\cdots\!36\)\( + \)\(39\!\cdots\!00\)\( T - \)\(12\!\cdots\!28\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(69\!\cdots\!16\)\( T^{4} + \)\(31\!\cdots\!80\)\( T^{5} + \)\(50\!\cdots\!60\)\( T^{6} - 207475463785770560 T^{7} - 466256646428 T^{8} + 367740 T^{9} + T^{10} \)
$41$ \( \)\(10\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( T - \)\(50\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} - \)\(17\!\cdots\!36\)\( T^{6} + 73331872070800608 T^{7} + 143796333500 T^{8} - 932764 T^{9} + T^{10} \)
$43$ \( -\)\(15\!\cdots\!56\)\( + \)\(78\!\cdots\!60\)\( T - \)\(42\!\cdots\!79\)\( T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!04\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{5} - \)\(41\!\cdots\!94\)\( T^{6} + 1073388107936261660 T^{7} - 115150628052 T^{8} - 1443560 T^{9} + T^{10} \)
$47$ \( -\)\(51\!\cdots\!76\)\( - \)\(15\!\cdots\!80\)\( T + \)\(20\!\cdots\!17\)\( T^{2} - \)\(44\!\cdots\!80\)\( T^{3} - \)\(13\!\cdots\!48\)\( T^{4} + \)\(34\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!86\)\( T^{6} - 601584561145470980 T^{7} - 3066926527396 T^{8} + 286960 T^{9} + T^{10} \)
$53$ \( -\)\(81\!\cdots\!24\)\( + \)\(53\!\cdots\!80\)\( T - \)\(30\!\cdots\!51\)\( T^{2} - \)\(37\!\cdots\!80\)\( T^{3} + \)\(68\!\cdots\!68\)\( T^{4} + \)\(22\!\cdots\!00\)\( T^{5} - \)\(13\!\cdots\!46\)\( T^{6} + 8829399525944243280 T^{7} + 2407450249216 T^{8} - 3953220 T^{9} + T^{10} \)
$59$ \( -\)\(22\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T - \)\(14\!\cdots\!00\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!00\)\( T^{4} - \)\(90\!\cdots\!40\)\( T^{5} - \)\(80\!\cdots\!36\)\( T^{6} + 47127124972706556640 T^{7} + 5130385782584 T^{8} - 6712320 T^{9} + T^{10} \)
$61$ \( \)\(11\!\cdots\!56\)\( - \)\(12\!\cdots\!92\)\( T + \)\(82\!\cdots\!12\)\( T^{2} + \)\(20\!\cdots\!92\)\( T^{3} - \)\(14\!\cdots\!40\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{5} + \)\(75\!\cdots\!96\)\( T^{6} + 29070763557509433312 T^{7} - 15700726489508 T^{8} - 1905196 T^{9} + T^{10} \)
$67$ \( -\)\(70\!\cdots\!84\)\( + \)\(16\!\cdots\!40\)\( T - \)\(15\!\cdots\!28\)\( T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(77\!\cdots\!52\)\( T^{4} + \)\(38\!\cdots\!40\)\( T^{5} + \)\(93\!\cdots\!52\)\( T^{6} - 64406784372799020800 T^{7} - 21307032469264 T^{8} + 2718200 T^{9} + T^{10} \)
$71$ \( -\)\(20\!\cdots\!84\)\( - \)\(42\!\cdots\!16\)\( T + \)\(24\!\cdots\!44\)\( T^{2} + \)\(52\!\cdots\!04\)\( T^{3} - \)\(10\!\cdots\!68\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!48\)\( T^{7} - 39737660574432 T^{8} - 3447736 T^{9} + T^{10} \)
$73$ \( -\)\(10\!\cdots\!76\)\( + \)\(20\!\cdots\!60\)\( T - \)\(71\!\cdots\!00\)\( T^{2} + \)\(95\!\cdots\!40\)\( T^{3} - \)\(45\!\cdots\!96\)\( T^{4} - \)\(46\!\cdots\!80\)\( T^{5} + \)\(83\!\cdots\!88\)\( T^{6} - 48734761889687031360 T^{7} - 48524828659836 T^{8} + 2554460 T^{9} + T^{10} \)
$79$ \( -\)\(93\!\cdots\!56\)\( - \)\(53\!\cdots\!60\)\( T - \)\(25\!\cdots\!99\)\( T^{2} + \)\(74\!\cdots\!08\)\( T^{3} + \)\(26\!\cdots\!88\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{5} + \)\(51\!\cdots\!98\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} - 50238327411172 T^{8} - 4187744 T^{9} + T^{10} \)
$83$ \( -\)\(42\!\cdots\!16\)\( + \)\(18\!\cdots\!80\)\( T - \)\(19\!\cdots\!96\)\( T^{2} + \)\(63\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!52\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!84\)\( T^{6} + \)\(42\!\cdots\!20\)\( T^{7} - 79446576451976 T^{8} - 3498720 T^{9} + T^{10} \)
$89$ \( \)\(71\!\cdots\!84\)\( + \)\(40\!\cdots\!96\)\( T + \)\(58\!\cdots\!08\)\( T^{2} + \)\(76\!\cdots\!24\)\( T^{3} - \)\(85\!\cdots\!72\)\( T^{4} + \)\(77\!\cdots\!64\)\( T^{5} + \)\(26\!\cdots\!88\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} - 288697132792484 T^{8} + 303268 T^{9} + T^{10} \)
$97$ \( \)\(33\!\cdots\!76\)\( + \)\(34\!\cdots\!20\)\( T + \)\(97\!\cdots\!68\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} - \)\(43\!\cdots\!36\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(68\!\cdots\!36\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} - 451993668721668 T^{8} - 4908620 T^{9} + T^{10} \)
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