Properties

Label 126.9.n.b
Level $126$
Weight $9$
Character orbit 126.n
Analytic conductor $51.330$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3297048677\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + 6901058684 x^{5} + 370996492174 x^{4} + 1285719886320 x^{3} + 55526550982200 x^{2} - 240706289358000 x + 3600954063202500\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{10}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 128 \beta_{1} q^{4} + ( -93 + 93 \beta_{1} + 15 \beta_{2} - 8 \beta_{3} + \beta_{4} ) q^{5} + ( 39 + 295 \beta_{1} - 9 \beta_{2} + 63 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{7} -128 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + 128 \beta_{1} q^{4} + ( -93 + 93 \beta_{1} + 15 \beta_{2} - 8 \beta_{3} + \beta_{4} ) q^{5} + ( 39 + 295 \beta_{1} - 9 \beta_{2} + 63 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{7} -128 \beta_{2} q^{8} + ( 1966 + 984 \beta_{1} - 95 \beta_{2} - 95 \beta_{3} + 6 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 6 \beta_{10} ) q^{10} + ( 1 + 1715 \beta_{1} - 619 \beta_{2} + 616 \beta_{3} + 3 \beta_{4} - \beta_{5} - 13 \beta_{6} + 3 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{11} + ( 1461 + 2917 \beta_{1} - 845 \beta_{2} + 1705 \beta_{3} - 5 \beta_{4} - 14 \beta_{5} + 8 \beta_{6} + 15 \beta_{7} + 15 \beta_{8} + 10 \beta_{10} ) q^{13} + ( -1287 + 6913 \beta_{1} - 276 \beta_{2} + 23 \beta_{3} + 15 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 7 \beta_{7} + 28 \beta_{8} + 2 \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{14} + ( -16384 - 16384 \beta_{1} ) q^{16} + ( -19247 - 9629 \beta_{1} + 1160 \beta_{2} + 1221 \beta_{3} - 31 \beta_{4} + 27 \beta_{5} - 10 \beta_{6} + \beta_{7} - 40 \beta_{8} - 10 \beta_{9} - 7 \beta_{10} + 20 \beta_{11} ) q^{17} + ( 22539 - 22558 \beta_{1} - 1105 \beta_{2} + 539 \beta_{3} + 52 \beta_{4} + 68 \beta_{5} + 81 \beta_{6} - 20 \beta_{7} + 7 \beta_{8} - 12 \beta_{9} + 13 \beta_{10} + 6 \beta_{11} ) q^{19} + ( -11904 - 23808 \beta_{1} - 896 \beta_{2} + 1920 \beta_{3} + 128 \beta_{8} ) q^{20} + ( -78437 - \beta_{1} - 1759 \beta_{2} - 34 \beta_{3} + 84 \beta_{4} - 200 \beta_{5} + 14 \beta_{7} + 21 \beta_{8} - 15 \beta_{9} - 41 \beta_{10} + 15 \beta_{11} ) q^{22} + ( -26435 - 26418 \beta_{1} + 112 \beta_{2} - 3217 \beta_{3} + 109 \beta_{4} + 340 \beta_{5} - 235 \beta_{6} - 88 \beta_{7} + 167 \beta_{8} - 105 \beta_{10} - 20 \beta_{11} ) q^{23} + ( 6 - 139790 \beta_{1} - 1888 \beta_{2} + 1952 \beta_{3} - 136 \beta_{4} - 6 \beta_{5} + 40 \beta_{6} + 90 \beta_{7} + 118 \beta_{8} + 60 \beta_{9} + 6 \beta_{10} ) q^{25} + ( -108836 + 108814 \beta_{1} - 2980 \beta_{2} + 1318 \beta_{3} + 362 \beta_{4} - 30 \beta_{5} - 4 \beta_{6} - 56 \beta_{7} + 30 \beta_{8} + 8 \beta_{9} + 26 \beta_{10} - 4 \beta_{11} ) q^{26} + ( -37760 - 32768 \beta_{1} - 6912 \beta_{2} - 1152 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} + 128 \beta_{6} - 128 \beta_{7} - 256 \beta_{8} ) q^{28} + ( 362799 - 17 \beta_{1} + 12131 \beta_{2} - 257 \beta_{3} + 487 \beta_{4} - 964 \beta_{5} - 61 \beta_{7} + 335 \beta_{8} + 44 \beta_{9} + 200 \beta_{10} - 44 \beta_{11} ) q^{29} + ( 502360 + 251213 \beta_{1} + 987 \beta_{2} + 1068 \beta_{3} - 81 \beta_{4} - 52 \beta_{5} + 125 \beta_{6} - 66 \beta_{7} - 147 \beta_{8} - 198 \beta_{10} ) q^{31} + ( 16384 \beta_{2} - 16384 \beta_{3} ) q^{32} + ( 152234 + 304466 \beta_{1} + 10003 \beta_{2} - 19430 \beta_{3} - 2 \beta_{4} - 1284 \beta_{5} + 2560 \beta_{6} + 6 \beta_{7} + 576 \beta_{8} + 4 \beta_{10} ) q^{34} + ( 955588 + 1033001 \beta_{1} + 85342 \beta_{2} - 34990 \beta_{3} - 245 \beta_{4} + 1640 \beta_{5} + 335 \beta_{6} - 285 \beta_{7} + 154 \beta_{8} - 50 \beta_{9} + 115 \beta_{10} - 60 \beta_{11} ) q^{35} + ( 22635 + 22708 \beta_{1} - 348 \beta_{2} + 76740 \beta_{3} - 395 \beta_{4} - 499 \beta_{5} + 984 \beta_{6} - 412 \beta_{7} - 1009 \beta_{8} - 485 \beta_{10} - 120 \beta_{11} ) q^{37} + ( -141720 - 70722 \beta_{1} + 22429 \beta_{2} + 22427 \beta_{3} - 223 \beta_{4} - 1158 \beta_{5} + 843 \beta_{6} - 201 \beta_{7} - 349 \beta_{8} + 75 \beta_{9} - 528 \beta_{10} - 150 \beta_{11} ) q^{38} + ( -125952 + 125696 \beta_{1} + 24320 \beta_{2} - 12160 \beta_{3} + 256 \beta_{4} - 256 \beta_{5} - 512 \beta_{7} + 256 \beta_{8} + 256 \beta_{10} ) q^{40} + ( -262957 - 525157 \beta_{1} - 38559 \beta_{2} + 79055 \beta_{3} + 627 \beta_{4} - 1386 \beta_{5} + 5020 \beta_{6} - 1621 \beta_{7} + 2327 \beta_{8} + 130 \beta_{9} - 1124 \beta_{10} + 130 \beta_{11} ) q^{41} + ( -1080524 + 198 \beta_{1} + 20566 \beta_{2} - 2108 \beta_{3} + 3538 \beta_{4} + 610 \beta_{5} - 282 \beta_{7} + 2192 \beta_{8} + 480 \beta_{9} + 648 \beta_{10} - 480 \beta_{11} ) q^{43} + ( -219776 - 219648 \beta_{1} + 384 \beta_{2} - 79232 \beta_{3} + 768 \beta_{4} - 1280 \beta_{5} + 1664 \beta_{6} - 256 \beta_{7} + 1152 \beta_{8} - 384 \beta_{10} + 256 \beta_{11} ) q^{44} + ( -144 - 390624 \beta_{1} + 28519 \beta_{2} - 26921 \beta_{3} - 1055 \beta_{4} + 144 \beta_{5} - 2305 \beta_{6} - 975 \beta_{7} + 1487 \beta_{8} - 255 \beta_{9} - 144 \beta_{10} ) q^{46} + ( -1021629 + 1021449 \beta_{1} - 385161 \beta_{2} + 191695 \beta_{3} + 1951 \beta_{4} - 3051 \beta_{5} - 2871 \beta_{6} - 360 \beta_{7} + 180 \beta_{8} + 180 \beta_{10} ) q^{47} + ( 732272 + 2075101 \beta_{1} - 290369 \beta_{2} + 328135 \beta_{3} - 2129 \beta_{4} + 1156 \beta_{5} - 5576 \beta_{6} - 153 \beta_{7} - 3755 \beta_{8} + 414 \beta_{9} + 26 \beta_{10} + 102 \beta_{11} ) q^{49} + ( -248556 - 268 \beta_{1} + 140882 \beta_{2} - 584 \beta_{3} + 1336 \beta_{4} + 6208 \beta_{5} - 368 \beta_{7} + 1220 \beta_{8} + 100 \beta_{9} + 1372 \beta_{10} - 100 \beta_{11} ) q^{50} + ( -373376 - 186368 \beta_{1} - 110080 \beta_{2} - 108160 \beta_{3} - 1920 \beta_{4} + 2944 \beta_{5} - 512 \beta_{6} - 640 \beta_{7} - 2560 \beta_{8} - 1920 \beta_{10} ) q^{52} + ( -159 - 2756433 \beta_{1} - 311804 \beta_{2} + 311284 \beta_{3} + 958 \beta_{4} + 159 \beta_{5} - 276 \beta_{6} - 915 \beta_{7} - 481 \beta_{8} - 120 \beta_{9} - 159 \beta_{10} ) q^{53} + ( -1224636 - 2446569 \beta_{1} + 123519 \beta_{2} - 238800 \beta_{3} + 2343 \beta_{4} + 3841 \beta_{5} + 970 \beta_{6} - 6309 \beta_{7} + 9318 \beta_{8} + 360 \beta_{9} - 4326 \beta_{10} + 360 \beta_{11} ) q^{55} + ( -884608 - 1049216 \beta_{1} + 32384 \beta_{2} - 35328 \beta_{3} - 3584 \beta_{4} - 1280 \beta_{5} + 256 \beta_{6} + 1536 \beta_{7} - 1664 \beta_{8} + 128 \beta_{9} + 896 \beta_{10} - 384 \beta_{11} ) q^{56} + ( 1529300 + 1529638 \beta_{1} - 5344 \beta_{2} + 367268 \beta_{3} - 4286 \beta_{4} + 7322 \beta_{5} - 6352 \beta_{6} - 632 \beta_{7} - 9586 \beta_{8} - 970 \beta_{10} + 720 \beta_{11} ) q^{58} + ( -3463018 - 1731817 \beta_{1} + 356317 \beta_{2} + 365883 \beta_{3} - 8066 \beta_{4} - 3978 \beta_{5} + 2065 \beta_{6} + 116 \beta_{7} - 8450 \beta_{8} - 500 \beta_{9} - 152 \beta_{10} + 1000 \beta_{11} ) q^{59} + ( -4745039 + 4744168 \beta_{1} + 1158320 \beta_{2} - 573868 \beta_{3} - 9701 \beta_{4} - 4447 \beta_{5} - 3588 \beta_{6} - 1706 \beta_{7} + 847 \beta_{8} - 24 \beta_{9} + 859 \beta_{10} + 12 \beta_{11} ) q^{61} + ( 138061 + 276085 \beta_{1} - 248971 \beta_{2} + 501416 \beta_{3} - 162 \beta_{4} - 324 \beta_{5} - 250 \beta_{6} + 736 \beta_{7} + 3849 \beta_{8} + 125 \beta_{9} + 449 \beta_{10} + 125 \beta_{11} ) q^{62} + 2097152 q^{64} + ( 7755100 + 7756923 \beta_{1} + 2483 \beta_{2} - 1419894 \beta_{3} + 3896 \beta_{4} - 7445 \beta_{5} + 16970 \beta_{6} - 7702 \beta_{7} + 2323 \beta_{8} - 9525 \beta_{10} - 410 \beta_{11} ) q^{65} + ( -934 + 6293595 \beta_{1} - 12413 \beta_{2} + 7471 \beta_{3} + 4590 \beta_{4} + 934 \beta_{5} + 17069 \beta_{6} - 2450 \beta_{7} - 1788 \beta_{8} + 2220 \beta_{9} - 934 \beta_{10} ) q^{67} + ( 1231232 - 1229824 \beta_{1} - 304768 \beta_{2} + 148480 \beta_{3} + 5120 \beta_{4} - 1152 \beta_{5} - 1280 \beta_{6} - 1024 \beta_{7} + 1152 \beta_{8} + 2560 \beta_{9} - 128 \beta_{10} - 1280 \beta_{11} ) q^{68} + ( 10864934 + 6389778 \beta_{1} - 1032365 \beta_{2} + 957415 \beta_{3} - 4417 \beta_{4} - 8248 \beta_{5} - 5865 \beta_{6} - 541 \beta_{7} - 3717 \beta_{8} + 225 \beta_{9} - 192 \beta_{10} - 2040 \beta_{11} ) q^{70} + ( -3783594 + 4376 \beta_{1} + 458940 \beta_{2} + 9490 \beta_{3} - 25616 \beta_{4} + 38854 \beta_{5} + 2116 \beta_{7} - 15982 \beta_{8} + 2260 \beta_{9} - 10724 \beta_{10} - 2260 \beta_{11} ) q^{71} + ( 1050914 + 527697 \beta_{1} - 695272 \beta_{2} - 690214 \beta_{3} - 10998 \beta_{4} + 5296 \beta_{5} + 112 \beta_{6} - 2500 \beta_{7} - 11518 \beta_{8} + 1980 \beta_{9} - 5520 \beta_{10} - 3960 \beta_{11} ) q^{73} + ( 1176 + 9803856 \beta_{1} - 22953 \beta_{2} + 25897 \beta_{3} - 6160 \beta_{4} - 1176 \beta_{5} - 16224 \beta_{6} + 6744 \beta_{7} + 2632 \beta_{8} + 864 \beta_{9} + 1176 \beta_{10} ) q^{74} + ( 2885888 + 5771648 \beta_{1} + 72448 \beta_{2} - 141440 \beta_{3} - 896 \beta_{4} + 7808 \beta_{5} - 20736 \beta_{6} + 4224 \beta_{7} + 5760 \beta_{8} + 768 \beta_{9} + 2560 \beta_{10} + 768 \beta_{11} ) q^{76} + ( 7184787 - 2384668 \beta_{1} + 1826009 \beta_{2} - 871788 \beta_{3} - 19908 \beta_{4} - 34646 \beta_{5} + 6984 \beta_{6} - 4091 \beta_{7} + 4452 \beta_{8} + 460 \beta_{9} + 10198 \beta_{10} + 20 \beta_{11} ) q^{77} + ( 9964371 + 9969272 \beta_{1} - 18108 \beta_{2} + 1504869 \beta_{3} - 12727 \beta_{4} + 15130 \beta_{5} + 8895 \beta_{6} - 19124 \beta_{7} - 40157 \beta_{8} - 24025 \beta_{10} + 480 \beta_{11} ) q^{79} + ( 3047424 + 1523712 \beta_{1} - 131072 \beta_{2} - 114688 \beta_{3} - 16384 \beta_{4} - 16384 \beta_{8} ) q^{80} + ( -5185704 + 5185550 \beta_{1} + 545036 \beta_{2} - 260374 \beta_{3} - 26774 \beta_{4} + 11206 \beta_{5} + 14000 \beta_{6} - 8228 \beta_{7} + 5434 \beta_{8} + 5280 \beta_{9} + 2794 \beta_{10} - 2640 \beta_{11} ) q^{82} + ( 10581456 + 21161688 \beta_{1} + 105046 \beta_{2} - 255726 \beta_{3} - 1044 \beta_{4} + 31440 \beta_{5} - 66696 \beta_{6} + 2772 \beta_{7} - 46174 \beta_{8} - 180 \beta_{9} + 1908 \beta_{10} - 180 \beta_{11} ) q^{83} + ( -5378658 + 5169 \beta_{1} - 446434 \beta_{2} + 36458 \beta_{3} - 81325 \beta_{4} - 34624 \beta_{5} + 1929 \beta_{7} - 43556 \beta_{8} + 3240 \beta_{9} - 10956 \beta_{10} - 3240 \beta_{11} ) q^{85} + ( 2530490 + 2532786 \beta_{1} - 14572 \beta_{2} - 1072474 \beta_{3} - 14014 \beta_{4} + 72920 \beta_{5} - 59702 \beta_{6} - 10922 \beta_{7} - 34916 \beta_{8} - 13218 \beta_{10} - 1738 \beta_{11} ) q^{86} + ( -1792 - 10039808 \beta_{1} + 229504 \beta_{2} - 225152 \beta_{3} - 2688 \beta_{4} + 1792 \beta_{5} + 30848 \beta_{6} - 7040 \beta_{7} + 8064 \beta_{8} + 1920 \beta_{9} - 1792 \beta_{10} ) q^{88} + ( -17962089 + 17948739 \beta_{1} - 823920 \beta_{2} + 413160 \beta_{3} + 11526 \beta_{4} - 7758 \beta_{5} + 5016 \beta_{6} - 24972 \beta_{7} + 12198 \beta_{8} - 1152 \beta_{9} + 12774 \beta_{10} + 576 \beta_{11} ) q^{89} + ( -33955903 - 23415957 \beta_{1} + 1073179 \beta_{2} + 1409449 \beta_{3} - 18339 \beta_{4} + 41486 \beta_{5} - 30896 \beta_{6} - 6407 \beta_{7} + 12611 \beta_{8} + 7458 \beta_{9} + 4102 \beta_{10} + 978 \beta_{11} ) q^{91} + ( 3381504 - 4736 \beta_{1} + 397440 \beta_{2} + 14336 \beta_{3} - 21376 \beta_{4} - 41344 \beta_{5} - 2176 \beta_{7} - 7424 \beta_{8} - 2560 \beta_{9} + 11264 \beta_{10} + 2560 \beta_{11} ) q^{92} + ( -49270964 - 24636582 \beta_{1} - 1034513 \beta_{2} - 1005767 \beta_{3} - 20133 \beta_{4} + 10626 \beta_{5} - 2871 \beta_{6} - 671 \beta_{7} - 23675 \beta_{8} - 2871 \beta_{9} - 4884 \beta_{10} + 5742 \beta_{11} ) q^{94} + ( 2755 - 29325235 \beta_{1} - 1395215 \beta_{2} + 1376805 \beta_{3} + 3260 \beta_{4} - 2755 \beta_{5} - 97675 \beta_{6} + 23415 \beta_{7} - 11525 \beta_{8} + 9640 \beta_{9} + 2755 \beta_{10} ) q^{95} + ( 16632853 + 33277629 \beta_{1} + 753991 \beta_{2} - 1590911 \beta_{3} + 10273 \beta_{4} + 12494 \beta_{5} + 12804 \beta_{6} - 27519 \beta_{7} - 77979 \beta_{8} + 1650 \beta_{9} - 18896 \beta_{10} + 1650 \beta_{11} ) q^{97} + ( -36929688 + 4830098 \beta_{1} - 2081424 \beta_{2} + 761799 \beta_{3} - 33286 \beta_{4} + 43282 \beta_{5} + 14136 \beta_{6} + 17536 \beta_{7} - 14206 \beta_{8} - 5060 \beta_{9} + 13690 \beta_{10} + 3980 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 768 q^{4} - 1674 q^{5} - 1308 q^{7} + O(q^{10}) \) \( 12 q - 768 q^{4} - 1674 q^{5} - 1308 q^{7} + 17664 q^{10} - 10302 q^{11} - 56832 q^{14} - 98304 q^{16} - 173178 q^{17} + 405978 q^{19} - 941568 q^{22} - 158934 q^{23} + 838668 q^{25} - 1958400 q^{26} - 255744 q^{28} + 4355256 q^{29} + 4520250 q^{31} + 5270790 q^{35} + 134214 q^{37} - 1278720 q^{38} - 2260992 q^{40} - 12961896 q^{43} - 1318656 q^{44} + 2345472 q^{46} - 18385002 q^{47} - 3659172 q^{49} - 2970624 q^{50} - 3369984 q^{52} + 16540506 q^{53} - 4325376 q^{56} + 9176064 q^{58} - 31163922 q^{59} - 85390158 q^{61} + 25165824 q^{64} + 46506264 q^{65} - 37750362 q^{67} + 22166784 q^{68} + 92031744 q^{70} - 45506424 q^{71} + 9414786 q^{73} - 58837248 q^{74} + 100614066 q^{77} + 59730294 q^{79} + 27426816 q^{80} - 93259776 q^{82} - 64652220 q^{85} + 15144960 q^{86} + 60260352 q^{88} - 323014482 q^{89} - 266861424 q^{91} + 40687104 q^{92} - 443440128 q^{94} + 175918350 q^{95} - 472166400 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + 6901058684 x^{5} + 370996492174 x^{4} + 1285719886320 x^{3} + 55526550982200 x^{2} - 240706289358000 x + 3600954063202500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(24\!\cdots\!14\)\( \nu^{11} + \)\(17\!\cdots\!42\)\( \nu^{10} + \)\(40\!\cdots\!04\)\( \nu^{9} + \)\(10\!\cdots\!77\)\( \nu^{8} + \)\(62\!\cdots\!18\)\( \nu^{7} + \)\(11\!\cdots\!79\)\( \nu^{6} + \)\(28\!\cdots\!40\)\( \nu^{5} + \)\(34\!\cdots\!51\)\( \nu^{4} + \)\(83\!\cdots\!16\)\( \nu^{3} + \)\(11\!\cdots\!10\)\( \nu^{2} + \)\(97\!\cdots\!00\)\( \nu - \)\(26\!\cdots\!50\)\(\)\()/ \)\(12\!\cdots\!50\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-202090052997757213384 \nu^{11} - 1989997430335076733412 \nu^{10} - 255656756837032256753104 \nu^{9} - 8227695102102134864192572 \nu^{8} - 492429624969135883257007488 \nu^{7} - 4647099311970982545404004644 \nu^{6} - 177379614919651411062138004000 \nu^{5} - 210613223065982349309540476456 \nu^{4} - 78770721438988807079781895174936 \nu^{3} + 216994393067807114987978096647200 \nu^{2} - 2348955365028536232136922784228000 \nu + 150034933393435109178188851023054000\)\()/ \)\(14\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(41\!\cdots\!32\)\( \nu^{11} - \)\(21\!\cdots\!04\)\( \nu^{10} + \)\(64\!\cdots\!52\)\( \nu^{9} + \)\(13\!\cdots\!76\)\( \nu^{8} + \)\(73\!\cdots\!84\)\( \nu^{7} + \)\(12\!\cdots\!52\)\( \nu^{6} + \)\(34\!\cdots\!20\)\( \nu^{5} + \)\(53\!\cdots\!88\)\( \nu^{4} + \)\(95\!\cdots\!08\)\( \nu^{3} + \)\(12\!\cdots\!80\)\( \nu^{2} + \)\(26\!\cdots\!00\)\( \nu + \)\(11\!\cdots\!00\)\(\)\()/ \)\(22\!\cdots\!75\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(43\!\cdots\!71\)\( \nu^{11} - \)\(19\!\cdots\!37\)\( \nu^{10} + \)\(10\!\cdots\!81\)\( \nu^{9} - \)\(19\!\cdots\!22\)\( \nu^{8} + \)\(87\!\cdots\!27\)\( \nu^{7} - \)\(22\!\cdots\!44\)\( \nu^{6} + \)\(45\!\cdots\!10\)\( \nu^{5} - \)\(91\!\cdots\!61\)\( \nu^{4} + \)\(12\!\cdots\!24\)\( \nu^{3} - \)\(29\!\cdots\!60\)\( \nu^{2} + \)\(21\!\cdots\!00\)\( \nu - \)\(13\!\cdots\!00\)\(\)\()/ \)\(22\!\cdots\!75\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(22\!\cdots\!98\)\( \nu^{11} + \)\(10\!\cdots\!11\)\( \nu^{10} - \)\(49\!\cdots\!88\)\( \nu^{9} + \)\(89\!\cdots\!41\)\( \nu^{8} - \)\(32\!\cdots\!86\)\( \nu^{7} + \)\(10\!\cdots\!57\)\( \nu^{6} - \)\(13\!\cdots\!50\)\( \nu^{5} + \)\(24\!\cdots\!18\)\( \nu^{4} - \)\(40\!\cdots\!42\)\( \nu^{3} + \)\(62\!\cdots\!00\)\( \nu^{2} - \)\(41\!\cdots\!00\)\( \nu - \)\(52\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!50\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(14\!\cdots\!01\)\( \nu^{11} + \)\(35\!\cdots\!38\)\( \nu^{10} + \)\(15\!\cdots\!91\)\( \nu^{9} + \)\(11\!\cdots\!28\)\( \nu^{8} + \)\(30\!\cdots\!17\)\( \nu^{7} + \)\(10\!\cdots\!56\)\( \nu^{6} + \)\(10\!\cdots\!20\)\( \nu^{5} + \)\(36\!\cdots\!84\)\( \nu^{4} + \)\(27\!\cdots\!34\)\( \nu^{3} + \)\(85\!\cdots\!30\)\( \nu^{2} - \)\(39\!\cdots\!00\)\( \nu + \)\(27\!\cdots\!00\)\(\)\()/ \)\(49\!\cdots\!50\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(13\!\cdots\!92\)\( \nu^{11} + \)\(86\!\cdots\!36\)\( \nu^{10} + \)\(13\!\cdots\!42\)\( \nu^{9} + \)\(20\!\cdots\!91\)\( \nu^{8} + \)\(26\!\cdots\!84\)\( \nu^{7} + \)\(19\!\cdots\!57\)\( \nu^{6} + \)\(41\!\cdots\!80\)\( \nu^{5} + \)\(67\!\cdots\!03\)\( \nu^{4} - \)\(23\!\cdots\!12\)\( \nu^{3} + \)\(16\!\cdots\!70\)\( \nu^{2} - \)\(95\!\cdots\!00\)\( \nu + \)\(93\!\cdots\!00\)\(\)\()/ \)\(44\!\cdots\!50\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(70\!\cdots\!12\)\( \nu^{11} + \)\(42\!\cdots\!91\)\( \nu^{10} + \)\(84\!\cdots\!22\)\( \nu^{9} + \)\(33\!\cdots\!71\)\( \nu^{8} + \)\(12\!\cdots\!84\)\( \nu^{7} + \)\(27\!\cdots\!17\)\( \nu^{6} + \)\(24\!\cdots\!50\)\( \nu^{5} + \)\(94\!\cdots\!08\)\( \nu^{4} + \)\(56\!\cdots\!98\)\( \nu^{3} + \)\(27\!\cdots\!00\)\( \nu^{2} - \)\(17\!\cdots\!00\)\( \nu + \)\(18\!\cdots\!50\)\(\)\()/ \)\(22\!\cdots\!75\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(60\!\cdots\!17\)\( \nu^{11} - \)\(95\!\cdots\!14\)\( \nu^{10} + \)\(45\!\cdots\!17\)\( \nu^{9} - \)\(18\!\cdots\!59\)\( \nu^{8} + \)\(38\!\cdots\!09\)\( \nu^{7} - \)\(13\!\cdots\!93\)\( \nu^{6} + \)\(35\!\cdots\!80\)\( \nu^{5} - \)\(55\!\cdots\!47\)\( \nu^{4} + \)\(10\!\cdots\!38\)\( \nu^{3} - \)\(15\!\cdots\!80\)\( \nu^{2} + \)\(26\!\cdots\!00\)\( \nu - \)\(17\!\cdots\!00\)\(\)\()/ \)\(14\!\cdots\!50\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(28\!\cdots\!94\)\( \nu^{11} - \)\(69\!\cdots\!43\)\( \nu^{10} + \)\(53\!\cdots\!34\)\( \nu^{9} - \)\(16\!\cdots\!58\)\( \nu^{8} + \)\(50\!\cdots\!78\)\( \nu^{7} - \)\(33\!\cdots\!16\)\( \nu^{6} + \)\(20\!\cdots\!90\)\( \nu^{5} - \)\(11\!\cdots\!79\)\( \nu^{4} + \)\(61\!\cdots\!86\)\( \nu^{3} - \)\(46\!\cdots\!90\)\( \nu^{2} + \)\(38\!\cdots\!00\)\( \nu - \)\(31\!\cdots\!50\)\(\)\()/ \)\(44\!\cdots\!50\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(13\!\cdots\!67\)\( \nu^{11} + \)\(16\!\cdots\!03\)\( \nu^{10} - \)\(18\!\cdots\!05\)\( \nu^{9} - \)\(25\!\cdots\!52\)\( \nu^{8} - \)\(19\!\cdots\!87\)\( \nu^{7} - \)\(96\!\cdots\!54\)\( \nu^{6} - \)\(33\!\cdots\!86\)\( \nu^{5} - \)\(27\!\cdots\!93\)\( \nu^{4} - \)\(56\!\cdots\!12\)\( \nu^{3} - \)\(14\!\cdots\!24\)\( \nu^{2} + \)\(29\!\cdots\!40\)\( \nu - \)\(23\!\cdots\!50\)\(\)\()/ \)\(29\!\cdots\!50\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{11} - 16 \beta_{10} - \beta_{8} - 13 \beta_{7} + 25 \beta_{6} - 9 \beta_{5} + 4 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} + 346 \beta_{1} + 343\)\()/1008\)
\(\nu^{2}\)\(=\)\((\)\(-18 \beta_{10} - 49 \beta_{9} - 221 \beta_{8} - 139 \beta_{7} + 161 \beta_{6} + 18 \beta_{5} + 275 \beta_{4} - 22062 \beta_{3} + 21872 \beta_{2} + 594420 \beta_{1} - 18\)\()/1008\)
\(\nu^{3}\)\(=\)\((\)\(1719 \beta_{11} + 12399 \beta_{10} - 1719 \beta_{9} + 10197 \beta_{8} - 2670 \beta_{7} + 14840 \beta_{5} + 12384 \beta_{4} - 3138 \beta_{3} + 942552 \beta_{2} - 4389 \beta_{1} - 7463217\)\()/1008\)
\(\nu^{4}\)\(=\)\((\)\(38339 \beta_{11} + 180779 \beta_{10} + 422426 \beta_{8} + 152291 \beta_{7} - 338667 \beta_{6} + 157888 \beta_{5} + 168481 \beta_{4} + 20004934 \beta_{3} + 158630 \beta_{2} - 313790723 \beta_{1} - 313762235\)\()/504\)
\(\nu^{5}\)\(=\)\((\)\(1740162 \beta_{10} + 1425607 \beta_{9} + 4226891 \beta_{8} + 10126417 \beta_{7} - 19778119 \beta_{6} - 1740162 \beta_{5} - 9447377 \beta_{4} + 897854151 \beta_{3} - 893312705 \beta_{2} - 8764846812 \beta_{1} + 1740162\)\()/504\)
\(\nu^{6}\)\(=\)\((\)\(-8880085 \beta_{11} - 41399221 \beta_{10} + 8880085 \beta_{9} - 60656063 \beta_{8} + 8129784 \beta_{7} - 56912680 \beta_{5} - 96922774 \beta_{4} + 35516410 \beta_{3} - 5065718862 \beta_{2} + 17009869 \beta_{1} + 64028213221\)\()/72\)
\(\nu^{7}\)\(=\)\((\)\(-2441634696 \beta_{11} - 15870609651 \beta_{10} - 25238396103 \beta_{8} - 13184814660 \beta_{7} + 31611606560 \beta_{6} - 15740996909 \beta_{5} - 8590505565 \beta_{4} - 1521779220600 \beta_{3} - 8834665860 \beta_{2} + 16285944462429 \beta_{1} + 16283258667438\)\()/504\)
\(\nu^{8}\)\(=\)\((\)\(-25493071377 \beta_{10} - 25897143496 \beta_{9} - 101211458282 \beta_{8} - 153362500381 \beta_{7} + 307354544994 \beta_{6} + 25493071377 \beta_{5} + 177690672413 \beta_{4} - 15335513385057 \beta_{3} + 15234705998894 \beta_{2} + 179909518274172 \beta_{1} - 25493071377\)\()/126\)
\(\nu^{9}\)\(=\)\((\)\(4181874760289 \beta_{11} + 21831088563341 \beta_{10} - 4181874760289 \beta_{9} + 28694475501481 \beta_{8} - 4412303450763 \beta_{7} + 30689756347068 \beta_{5} + 44152040650673 \beta_{4} - 15687993839666 \beta_{3} + 2583615563299296 \beta_{2} - 8594178211052 \beta_{1} - 28463072943642074\)\()/504\)
\(\nu^{10}\)\(=\)\((\)\(87304514607604 \beta_{11} + 528704325307279 \beta_{10} + 934094766761179 \beta_{8} + 440424363167344 \beta_{7} - 1060919765017952 \beta_{6} + 532215439710673 \beta_{5} + 334627440170687 \beta_{4} + 52207229765269760 \beta_{3} + 335602887703018 \beta_{2} - 600624307838982445 \beta_{1} - 600536027876842510\)\()/252\)
\(\nu^{11}\)\(=\)\((\)\(7408129914931449 \beta_{10} + 7135763060222049 \beta_{9} + 26797101213372018 \beta_{8} + 44176412634879294 \beta_{7} - 88533032853576553 \beta_{6} - 7408129914931449 \beta_{5} - 49021490958166365 \beta_{4} + 4371379426685956410 \beta_{3} - 4344309958617874992 \beta_{2} - 48807279280252259418 \beta_{1} + 7408129914931449\)\()/504\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
20.6104 35.6982i
−10.5605 + 18.2913i
−9.54988 + 16.5409i
−12.5688 + 21.7698i
3.92504 6.79836i
9.14374 15.8374i
20.6104 + 35.6982i
−10.5605 18.2913i
−9.54988 16.5409i
−12.5688 21.7698i
3.92504 + 6.79836i
9.14374 + 15.8374i
−5.65685 + 9.79796i 0 −64.0000 110.851i −928.109 535.844i 0 2212.86 + 931.703i 1448.15 0 10500.4 6062.38i
19.2 −5.65685 + 9.79796i 0 −64.0000 110.851i −372.872 215.278i 0 −1545.60 + 1837.37i 1448.15 0 4218.56 2435.59i
19.3 −5.65685 + 9.79796i 0 −64.0000 110.851i 492.158 + 284.147i 0 −1740.97 1653.43i 1448.15 0 −5568.13 + 3214.76i
19.4 5.65685 9.79796i 0 −64.0000 110.851i −753.641 435.115i 0 −1836.02 + 1547.20i −1448.15 0 −8526.48 + 4922.77i
19.5 5.65685 9.79796i 0 −64.0000 110.851i −32.9005 18.9951i 0 1399.84 1950.71i −1448.15 0 −372.227 + 214.905i
19.6 5.65685 9.79796i 0 −64.0000 110.851i 758.365 + 437.842i 0 855.887 2243.27i −1448.15 0 8579.92 4953.62i
73.1 −5.65685 9.79796i 0 −64.0000 + 110.851i −928.109 + 535.844i 0 2212.86 931.703i 1448.15 0 10500.4 + 6062.38i
73.2 −5.65685 9.79796i 0 −64.0000 + 110.851i −372.872 + 215.278i 0 −1545.60 1837.37i 1448.15 0 4218.56 + 2435.59i
73.3 −5.65685 9.79796i 0 −64.0000 + 110.851i 492.158 284.147i 0 −1740.97 + 1653.43i 1448.15 0 −5568.13 3214.76i
73.4 5.65685 + 9.79796i 0 −64.0000 + 110.851i −753.641 + 435.115i 0 −1836.02 1547.20i −1448.15 0 −8526.48 4922.77i
73.5 5.65685 + 9.79796i 0 −64.0000 + 110.851i −32.9005 + 18.9951i 0 1399.84 + 1950.71i −1448.15 0 −372.227 214.905i
73.6 5.65685 + 9.79796i 0 −64.0000 + 110.851i 758.365 437.842i 0 855.887 + 2243.27i −1448.15 0 8579.92 + 4953.62i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.6
Significant digits:
Format:

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 126.9.n.b 12
3.b odd 2 1 14.9.d.a 12
7.d odd 6 1 inner 126.9.n.b 12
12.b even 2 1 112.9.s.c 12
21.c even 2 1 98.9.d.b 12
21.g even 6 1 14.9.d.a 12
21.g even 6 1 98.9.b.c 12
21.h odd 6 1 98.9.b.c 12
21.h odd 6 1 98.9.d.b 12
84.j odd 6 1 112.9.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.d.a 12 3.b odd 2 1
14.9.d.a 12 21.g even 6 1
98.9.b.c 12 21.g even 6 1
98.9.b.c 12 21.h odd 6 1
98.9.d.b 12 21.c even 2 1
98.9.d.b 12 21.h odd 6 1
112.9.s.c 12 12.b even 2 1
112.9.s.c 12 84.j odd 6 1
126.9.n.b 12 1.a even 1 1 trivial
126.9.n.b 12 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(16\!\cdots\!50\)\( T_{5}^{7} + \)\(51\!\cdots\!25\)\( T_{5}^{6} - \)\(47\!\cdots\!50\)\( T_{5}^{5} - \)\(13\!\cdots\!50\)\( T_{5}^{4} + \)\(96\!\cdots\!50\)\( T_{5}^{3} + \)\(46\!\cdots\!25\)\( T_{5}^{2} + \)\(27\!\cdots\!50\)\( T_{5} + \)\(57\!\cdots\!25\)\( \)">\(T_{5}^{12} + \cdots\) acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16384 + 128 T^{2} + T^{4} )^{3} \)
$3$ \( T^{12} \)
$5$ \( \)\(57\!\cdots\!25\)\( + \)\(27\!\cdots\!50\)\( T + \)\(46\!\cdots\!25\)\( T^{2} + \)\(96\!\cdots\!50\)\( T^{3} - \)\(13\!\cdots\!50\)\( T^{4} - \)\(47\!\cdots\!50\)\( T^{5} + 516425178513746025 T^{6} + 1612081221903450 T^{7} + 18486646794 T^{8} - 1881848862 T^{9} - 190071 T^{10} + 1674 T^{11} + T^{12} \)
$7$ \( \)\(36\!\cdots\!01\)\( + \)\(83\!\cdots\!08\)\( T + \)\(29\!\cdots\!18\)\( T^{2} - \)\(42\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} + 16422695158604759052 T^{6} - 17664438572780328 T^{7} + 56377019557215 T^{8} - 2234556212 T^{9} + 2685018 T^{10} + 1308 T^{11} + T^{12} \)
$11$ \( \)\(48\!\cdots\!81\)\( + \)\(66\!\cdots\!38\)\( T + \)\(10\!\cdots\!61\)\( T^{2} - \)\(23\!\cdots\!18\)\( T^{3} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(38\!\cdots\!02\)\( T^{5} + \)\(51\!\cdots\!01\)\( T^{6} - \)\(18\!\cdots\!58\)\( T^{7} + 326160370076618934 T^{8} - 6525798653934 T^{9} + 740755377 T^{10} + 10302 T^{11} + T^{12} \)
$13$ \( \)\(35\!\cdots\!24\)\( + \)\(39\!\cdots\!40\)\( T^{2} + \)\(25\!\cdots\!12\)\( T^{4} + \)\(64\!\cdots\!16\)\( T^{6} + 7906434877372039776 T^{8} + 4612551504 T^{10} + T^{12} \)
$17$ \( \)\(30\!\cdots\!01\)\( + \)\(20\!\cdots\!18\)\( T + \)\(51\!\cdots\!97\)\( T^{2} + \)\(42\!\cdots\!02\)\( T^{3} - \)\(36\!\cdots\!14\)\( T^{4} - \)\(69\!\cdots\!22\)\( T^{5} + \)\(28\!\cdots\!53\)\( T^{6} + \)\(98\!\cdots\!18\)\( T^{7} + \)\(25\!\cdots\!46\)\( T^{8} - 4491697539933198 T^{9} - 15940009863 T^{10} + 173178 T^{11} + T^{12} \)
$19$ \( \)\(94\!\cdots\!25\)\( - \)\(32\!\cdots\!50\)\( T + \)\(46\!\cdots\!25\)\( T^{2} - \)\(34\!\cdots\!50\)\( T^{3} + \)\(14\!\cdots\!46\)\( T^{4} - \)\(31\!\cdots\!14\)\( T^{5} + \)\(36\!\cdots\!41\)\( T^{6} - \)\(15\!\cdots\!66\)\( T^{7} - \)\(74\!\cdots\!82\)\( T^{8} + 5967426272557302 T^{9} + 40240488069 T^{10} - 405978 T^{11} + T^{12} \)
$23$ \( \)\(25\!\cdots\!41\)\( - \)\(93\!\cdots\!54\)\( T + \)\(60\!\cdots\!61\)\( T^{2} - \)\(67\!\cdots\!62\)\( T^{3} + \)\(53\!\cdots\!86\)\( T^{4} - \)\(58\!\cdots\!42\)\( T^{5} + \)\(30\!\cdots\!49\)\( T^{6} + \)\(11\!\cdots\!94\)\( T^{7} + \)\(60\!\cdots\!10\)\( T^{8} + 66695318786854170 T^{9} + 257114823633 T^{10} + 158934 T^{11} + T^{12} \)
$29$ \( ( -\)\(79\!\cdots\!12\)\( + \)\(14\!\cdots\!08\)\( T - \)\(74\!\cdots\!56\)\( T^{2} + 940925167313519808 T^{3} + 819765973332 T^{4} - 2177628 T^{5} + T^{6} )^{2} \)
$31$ \( \)\(95\!\cdots\!29\)\( - \)\(52\!\cdots\!14\)\( T + \)\(12\!\cdots\!85\)\( T^{2} - \)\(16\!\cdots\!82\)\( T^{3} + \)\(12\!\cdots\!22\)\( T^{4} - \)\(64\!\cdots\!18\)\( T^{5} + \)\(22\!\cdots\!77\)\( T^{6} - \)\(56\!\cdots\!82\)\( T^{7} + \)\(97\!\cdots\!78\)\( T^{8} - 11797369442909678250 T^{9} + 9420779822373 T^{10} - 4520250 T^{11} + T^{12} \)
$37$ \( \)\(20\!\cdots\!21\)\( - \)\(67\!\cdots\!46\)\( T + \)\(17\!\cdots\!41\)\( T^{2} - \)\(22\!\cdots\!98\)\( T^{3} + \)\(24\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!66\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} - \)\(45\!\cdots\!34\)\( T^{7} + \)\(38\!\cdots\!14\)\( T^{8} - 8596405628770429042 T^{9} + 7075094642901 T^{10} - 134214 T^{11} + T^{12} \)
$41$ \( \)\(50\!\cdots\!96\)\( + \)\(29\!\cdots\!88\)\( T^{2} + \)\(27\!\cdots\!12\)\( T^{4} + \)\(96\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!48\)\( T^{8} + 63126434226384 T^{10} + T^{12} \)
$43$ \( ( -\)\(32\!\cdots\!64\)\( + \)\(76\!\cdots\!12\)\( T + \)\(98\!\cdots\!12\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} - 19549380996084 T^{4} + 6480948 T^{5} + T^{6} )^{2} \)
$47$ \( \)\(85\!\cdots\!49\)\( + \)\(10\!\cdots\!06\)\( T + \)\(28\!\cdots\!69\)\( T^{2} - \)\(12\!\cdots\!86\)\( T^{3} - \)\(56\!\cdots\!22\)\( T^{4} + \)\(19\!\cdots\!94\)\( T^{5} + \)\(12\!\cdots\!49\)\( T^{6} + \)\(81\!\cdots\!38\)\( T^{7} - \)\(33\!\cdots\!58\)\( T^{8} - \)\(32\!\cdots\!58\)\( T^{9} + 95082902343189 T^{10} + 18385002 T^{11} + T^{12} \)
$53$ \( \)\(10\!\cdots\!41\)\( - \)\(10\!\cdots\!06\)\( T + \)\(56\!\cdots\!73\)\( T^{2} + \)\(35\!\cdots\!46\)\( T^{3} + \)\(19\!\cdots\!26\)\( T^{4} - \)\(89\!\cdots\!78\)\( T^{5} + \)\(31\!\cdots\!45\)\( T^{6} - \)\(11\!\cdots\!82\)\( T^{7} + \)\(59\!\cdots\!86\)\( T^{8} - \)\(12\!\cdots\!18\)\( T^{9} + 204949252399797 T^{10} - 16540506 T^{11} + T^{12} \)
$59$ \( \)\(28\!\cdots\!01\)\( + \)\(15\!\cdots\!82\)\( T + \)\(26\!\cdots\!69\)\( T^{2} - \)\(46\!\cdots\!98\)\( T^{3} - \)\(55\!\cdots\!90\)\( T^{4} - \)\(51\!\cdots\!74\)\( T^{5} + \)\(12\!\cdots\!21\)\( T^{6} + \)\(11\!\cdots\!22\)\( T^{7} - \)\(29\!\cdots\!70\)\( T^{8} - \)\(63\!\cdots\!34\)\( T^{9} + 120060743343381 T^{10} + 31163922 T^{11} + T^{12} \)
$61$ \( \)\(13\!\cdots\!69\)\( + \)\(11\!\cdots\!10\)\( T - \)\(20\!\cdots\!11\)\( T^{2} - \)\(18\!\cdots\!90\)\( T^{3} + \)\(26\!\cdots\!50\)\( T^{4} + \)\(40\!\cdots\!74\)\( T^{5} + \)\(10\!\cdots\!65\)\( T^{6} - \)\(96\!\cdots\!22\)\( T^{7} - \)\(42\!\cdots\!38\)\( T^{8} + \)\(28\!\cdots\!34\)\( T^{9} + 2767349780555961 T^{10} + 85390158 T^{11} + T^{12} \)
$67$ \( \)\(23\!\cdots\!89\)\( - \)\(14\!\cdots\!54\)\( T + \)\(79\!\cdots\!13\)\( T^{2} - \)\(11\!\cdots\!50\)\( T^{3} + \)\(26\!\cdots\!86\)\( T^{4} + \)\(16\!\cdots\!14\)\( T^{5} + \)\(62\!\cdots\!73\)\( T^{6} + \)\(24\!\cdots\!94\)\( T^{7} + \)\(36\!\cdots\!30\)\( T^{8} + \)\(11\!\cdots\!02\)\( T^{9} + 1574633884496769 T^{10} + 37750362 T^{11} + T^{12} \)
$71$ \( ( -\)\(21\!\cdots\!64\)\( - \)\(79\!\cdots\!68\)\( T + \)\(15\!\cdots\!48\)\( T^{2} - \)\(20\!\cdots\!72\)\( T^{3} - 2426256761646132 T^{4} + 22753212 T^{5} + T^{6} )^{2} \)
$73$ \( \)\(26\!\cdots\!49\)\( - \)\(10\!\cdots\!18\)\( T + \)\(17\!\cdots\!69\)\( T^{2} - \)\(14\!\cdots\!14\)\( T^{3} + \)\(60\!\cdots\!06\)\( T^{4} - \)\(71\!\cdots\!66\)\( T^{5} - \)\(11\!\cdots\!95\)\( T^{6} + \)\(20\!\cdots\!02\)\( T^{7} + \)\(40\!\cdots\!74\)\( T^{8} + \)\(18\!\cdots\!86\)\( T^{9} - 1980813382799319 T^{10} - 9414786 T^{11} + T^{12} \)
$79$ \( \)\(92\!\cdots\!25\)\( + \)\(14\!\cdots\!50\)\( T + \)\(25\!\cdots\!25\)\( T^{2} - \)\(41\!\cdots\!50\)\( T^{3} + \)\(48\!\cdots\!50\)\( T^{4} - \)\(24\!\cdots\!50\)\( T^{5} + \)\(96\!\cdots\!25\)\( T^{6} - \)\(22\!\cdots\!90\)\( T^{7} + \)\(43\!\cdots\!14\)\( T^{8} - \)\(48\!\cdots\!38\)\( T^{9} + 7328718281480049 T^{10} - 59730294 T^{11} + T^{12} \)
$83$ \( \)\(17\!\cdots\!84\)\( + \)\(39\!\cdots\!56\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{4} + \)\(37\!\cdots\!52\)\( T^{6} + \)\(31\!\cdots\!64\)\( T^{8} + 9881353210358592 T^{10} + T^{12} \)
$89$ \( \)\(13\!\cdots\!21\)\( + \)\(58\!\cdots\!90\)\( T - \)\(39\!\cdots\!03\)\( T^{2} - \)\(21\!\cdots\!70\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} + \)\(58\!\cdots\!62\)\( T^{5} - \)\(19\!\cdots\!35\)\( T^{6} - \)\(38\!\cdots\!90\)\( T^{7} + \)\(31\!\cdots\!98\)\( T^{8} + \)\(22\!\cdots\!22\)\( T^{9} + 41683197526157529 T^{10} + 323014482 T^{11} + T^{12} \)
$97$ \( \)\(52\!\cdots\!16\)\( + \)\(19\!\cdots\!52\)\( T^{2} + \)\(25\!\cdots\!56\)\( T^{4} + \)\(13\!\cdots\!68\)\( T^{6} + \)\(32\!\cdots\!76\)\( T^{8} + 32297903455130832 T^{10} + T^{12} \)
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