Properties

Label 8.13.d.b
Level 8
Weight 13
Character orbit 8.d
Analytic conductor 7.312
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31195053821\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + 28213427840 x^{3} + 1335549648384 x^{2} + 41051831642368 x + 824967906703360\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{3}\cdot 23 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -11 - \beta_{1} ) q^{2} + ( -66 + 3 \beta_{1} - \beta_{2} ) q^{3} + ( -244 + 12 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 48 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -13081 + 113 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{9} ) q^{6} + ( 323 \beta_{1} + 7 \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{7} + ( 2710 + 241 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} + ( 169275 - 763 \beta_{1} - 189 \beta_{2} + 11 \beta_{3} + 6 \beta_{4} - \beta_{6} - 4 \beta_{7} + 4 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -11 - \beta_{1} ) q^{2} + ( -66 + 3 \beta_{1} - \beta_{2} ) q^{3} + ( -244 + 12 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 48 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -13081 + 113 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} - \beta_{4} - \beta_{9} ) q^{6} + ( 323 \beta_{1} + 7 \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} ) q^{7} + ( 2710 + 241 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} + ( 169275 - 763 \beta_{1} - 189 \beta_{2} + 11 \beta_{3} + 6 \beta_{4} - \beta_{6} - 4 \beta_{7} + 4 \beta_{9} ) q^{9} + ( 187302 - 548 \beta_{1} - 92 \beta_{2} + 44 \beta_{3} + 9 \beta_{4} + 20 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} ) q^{10} + ( -459250 + 7249 \beta_{1} + 297 \beta_{2} + 242 \beta_{3} - 44 \beta_{4} - 22 \beta_{6} ) q^{11} + ( -542836 + 15674 \beta_{1} + 1218 \beta_{2} + 166 \beta_{3} - 38 \beta_{4} - 92 \beta_{5} - 8 \beta_{6} + 6 \beta_{7} + 14 \beta_{8} + 36 \beta_{9} ) q^{12} + ( 64 + 3760 \beta_{1} + 79 \beta_{2} - 216 \beta_{3} + 32 \beta_{4} + 17 \beta_{5} + 8 \beta_{6} - 16 \beta_{7} + 8 \beta_{8} - 104 \beta_{9} ) q^{13} + ( 1272852 - 4480 \beta_{1} - 2456 \beta_{2} + 104 \beta_{3} + 6 \beta_{4} - 280 \beta_{5} + 84 \beta_{6} - 28 \beta_{7} - 20 \beta_{8} - 60 \beta_{9} ) q^{14} + ( 896 + 10231 \beta_{1} - 53 \beta_{2} - 2256 \beta_{3} + 61 \beta_{4} - 32 \beta_{5} - 144 \beta_{6} + 32 \beta_{7} - 19 \beta_{8} + 205 \beta_{9} ) q^{15} + ( -3829556 + 7178 \beta_{1} - 1472 \beta_{2} + 540 \beta_{3} - 220 \beta_{4} + 660 \beta_{5} + 84 \beta_{6} + 70 \beta_{7} - 10 \beta_{8} + 52 \beta_{9} ) q^{16} + ( 4287290 - 127881 \beta_{1} - 8607 \beta_{2} + 825 \beta_{3} + 706 \beta_{4} - 75 \beta_{6} + 20 \beta_{7} - 20 \beta_{9} ) q^{17} + ( 822855 - 167931 \beta_{1} + 18304 \beta_{2} + 568 \beta_{3} - 440 \beta_{4} + 1184 \beta_{5} + 416 \beta_{6} + 48 \beta_{7} - 80 \beta_{8} - 184 \beta_{9} ) q^{18} + ( 3604718 - 56143 \beta_{1} - 13399 \beta_{2} + 3586 \beta_{3} + 372 \beta_{4} - 326 \beta_{6} + 128 \beta_{7} - 128 \beta_{9} ) q^{19} + ( 1761384 - 173228 \beta_{1} + 49216 \beta_{2} + 1184 \beta_{3} - 20 \beta_{4} - 1432 \beta_{5} + 736 \beta_{6} + 28 \beta_{7} - 244 \beta_{8} - 216 \beta_{9} ) q^{20} + ( 7616 + 361568 \beta_{1} + 5508 \beta_{2} - 19560 \beta_{3} + 1632 \beta_{4} - 100 \beta_{5} - 1096 \beta_{6} + 144 \beta_{7} - 200 \beta_{8} + 808 \beta_{9} ) q^{21} + ( -22702361 + 511313 \beta_{1} - 60852 \beta_{2} + 5797 \beta_{3} - 2849 \beta_{4} - 704 \beta_{5} + 704 \beta_{6} + 352 \beta_{7} + 352 \beta_{8} + 671 \beta_{9} ) q^{22} + ( 6528 - 436207 \beta_{1} - 10659 \beta_{2} - 18192 \beta_{3} - 741 \beta_{4} + 864 \beta_{5} - 464 \beta_{6} - 352 \beta_{7} + 139 \beta_{8} - 2325 \beta_{9} ) q^{23} + ( 16908372 + 440942 \beta_{1} - 158152 \beta_{2} + 6084 \beta_{3} + 3760 \beta_{4} - 2020 \beta_{5} + 2068 \beta_{6} - 814 \beta_{7} + 370 \beta_{8} - 420 \beta_{9} ) q^{24} + ( -70709183 - 395986 \beta_{1} + 41026 \beta_{2} + 33330 \beta_{3} + 516 \beta_{4} - 3030 \beta_{6} + 104 \beta_{7} - 104 \beta_{9} ) q^{25} + ( 15122586 + 103012 \beta_{1} + 275804 \beta_{2} + 17748 \beta_{3} - 10217 \beta_{4} - 7700 \beta_{5} - 710 \beta_{6} - 122 \beta_{7} + 250 \beta_{8} + 2626 \beta_{9} ) q^{26} + ( 138237804 + 1408380 \beta_{1} - 160512 \beta_{2} + 8250 \beta_{3} - 4572 \beta_{4} - 750 \beta_{6} - 1920 \beta_{7} + 1920 \beta_{9} ) q^{27} + ( -13215568 - 1056360 \beta_{1} + 462848 \beta_{2} + 18496 \beta_{3} + 11880 \beta_{4} + 14768 \beta_{5} + 832 \beta_{6} - 1144 \beta_{7} + 1704 \beta_{8} - 1744 \beta_{9} ) q^{28} + ( -7680 + 4916368 \beta_{1} + 107051 \beta_{2} + 19776 \beta_{3} + 16768 \beta_{4} + 85 \beta_{5} + 1088 \beta_{6} - 128 \beta_{7} + 2240 \beta_{8} + 1344 \beta_{9} ) q^{29} + ( 45609796 - 500736 \beta_{1} - 777528 \beta_{2} - 11320 \beta_{3} - 20562 \beta_{4} + 20296 \beta_{5} - 2940 \beta_{6} - 1708 \beta_{7} - 2500 \beta_{8} - 972 \beta_{9} ) q^{30} + ( -21376 - 7559524 \beta_{1} - 156340 \beta_{2} + 59088 \beta_{3} - 30668 \beta_{4} - 10720 \beta_{5} + 1680 \beta_{6} + 992 \beta_{7} - 380 \beta_{8} + 6564 \beta_{9} ) q^{31} + ( -265117592 + 4521676 \beta_{1} - 311712 \beta_{2} - 14072 \beta_{3} + 26600 \beta_{4} - 21096 \beta_{5} - 8904 \beta_{6} + 3668 \beta_{7} - 3148 \beta_{8} - 3112 \beta_{9} ) q^{32} + ( -47726580 - 12099879 \beta_{1} + 215567 \beta_{2} - 172425 \beta_{3} + 58014 \beta_{4} + 15675 \beta_{6} - 660 \beta_{7} + 660 \beta_{9} ) q^{33} + ( 445516274 - 5646074 \beta_{1} + 108672 \beta_{2} - 131096 \beta_{3} - 24168 \beta_{4} - 8480 \beta_{5} + 480 \beta_{6} + 1040 \beta_{7} + 1680 \beta_{8} - 12648 \beta_{9} ) q^{34} + ( -283796064 + 21135412 \beta_{1} + 549468 \beta_{2} - 116820 \beta_{3} - 104712 \beta_{4} + 10620 \beta_{6} + 11392 \beta_{7} - 11392 \beta_{9} ) q^{35} + ( -407402940 + 499540 \beta_{1} + 745939 \beta_{2} - 182035 \beta_{3} + 97456 \beta_{4} - 17440 \beta_{5} - 22464 \beta_{6} + 7504 \beta_{7} - 5360 \beta_{8} + 23008 \beta_{9} ) q^{36} + ( -153664 + 19079120 \beta_{1} + 447305 \beta_{2} + 395736 \beta_{3} + 62944 \beta_{4} + 1623 \beta_{5} + 21752 \beta_{6} - 2544 \beta_{7} - 14600 \beta_{8} - 32408 \beta_{9} ) q^{37} + ( 164449943 - 4575199 \beta_{1} - 864756 \beta_{2} - 91723 \beta_{3} - 64241 \beta_{4} - 49344 \beta_{5} - 1856 \beta_{6} + 4192 \beta_{7} + 8288 \beta_{8} - 15153 \beta_{9} ) q^{38} + ( -134144 - 45421239 \beta_{1} - 931691 \beta_{2} + 360576 \beta_{3} - 157565 \beta_{4} + 80640 \beta_{5} + 13952 \beta_{6} + 2816 \beta_{7} - 765 \beta_{8} + 18947 \beta_{9} ) q^{39} + ( 818121872 - 3801480 \beta_{1} + 1117632 \beta_{2} - 227824 \beta_{3} + 225008 \beta_{4} + 98032 \beta_{5} - 18576 \beta_{6} - 4536 \beta_{7} + 11144 \beta_{8} + 45424 \beta_{9} ) q^{40} + ( -1233871822 - 37314966 \beta_{1} + 2102502 \beta_{2} - 121418 \beta_{3} + 149068 \beta_{4} + 11038 \beta_{6} - 1096 \beta_{7} + 1096 \beta_{9} ) q^{41} + ( 1469134648 - 5944336 \beta_{1} - 4292912 \beta_{2} + 283376 \beta_{3} - 236972 \beta_{4} + 136848 \beta_{5} - 37448 \beta_{6} - 11384 \beta_{7} - 16392 \beta_{8} + 16920 \beta_{9} ) q^{42} + ( 2508180734 + 48826271 \beta_{1} - 2469645 \beta_{2} - 71500 \beta_{3} - 193848 \beta_{4} + 6500 \beta_{6} - 29440 \beta_{7} + 29440 \beta_{9} ) q^{43} + ( -2795254132 + 30459770 \beta_{1} - 1913758 \beta_{2} + 444422 \beta_{3} + 200090 \beta_{4} - 141020 \beta_{5} + 16632 \beta_{6} - 15290 \beta_{7} + 1870 \beta_{8} - 103004 \beta_{9} ) q^{44} + ( 126912 + 100901936 \beta_{1} + 2113987 \beta_{2} - 311592 \beta_{3} + 349792 \beta_{4} - 6307 \beta_{5} - 23048 \beta_{6} + 7184 \beta_{7} + 59512 \beta_{8} + 109800 \beta_{9} ) q^{45} + ( -1681345892 + 7445760 \beta_{1} + 5560568 \beta_{2} - 36360 \beta_{3} - 434206 \beta_{4} - 148360 \beta_{5} - 42148 \beta_{6} - 8948 \beta_{7} - 6940 \beta_{8} + 81452 \beta_{9} ) q^{46} + ( 166272 - 63919238 \beta_{1} - 1393774 \beta_{2} - 495888 \beta_{3} - 271106 \beta_{4} - 408224 \beta_{5} - 976 \beta_{6} - 19808 \beta_{7} + 7790 \beta_{8} - 130866 \beta_{9} ) q^{47} + ( 1330597000 - 15229156 \beta_{1} + 11378240 \beta_{2} + 1010920 \beta_{3} + 159864 \beta_{4} + 61240 \beta_{5} + 83960 \beta_{6} - 18620 \beta_{7} - 4636 \beta_{8} - 204296 \beta_{9} ) q^{48} + ( -3953924639 - 48476552 \beta_{1} - 493496 \beta_{2} + 1549064 \beta_{3} + 199824 \beta_{4} - 140824 \beta_{6} + 10144 \beta_{7} - 10144 \beta_{9} ) q^{49} + ( 2489173909 + 62694079 \beta_{1} - 6641920 \beta_{2} - 716976 \beta_{3} - 430160 \beta_{4} - 128576 \beta_{5} + 86976 \beta_{6} + 47648 \beta_{7} + 50976 \beta_{8} + 52144 \beta_{9} ) q^{50} + ( 3675394188 + 34437752 \beta_{1} - 6036364 \beta_{2} + 769406 \beta_{3} - 94836 \beta_{4} - 69946 \beta_{6} + 512 \beta_{7} - 512 \beta_{9} ) q^{51} + ( -3616909672 - 11368404 \beta_{1} - 10933312 \beta_{2} - 609440 \beta_{3} + 520340 \beta_{4} + 263320 \beta_{5} + 162080 \beta_{6} - 33180 \beta_{7} + 39284 \beta_{8} + 241624 \beta_{9} ) q^{52} + ( 1286208 + 55817808 \beta_{1} + 811829 \beta_{2} - 3327192 \beta_{3} + 282016 \beta_{4} - 4181 \beta_{5} - 177144 \beta_{6} + 16368 \beta_{7} - 146040 \beta_{8} - 31464 \beta_{9} ) q^{53} + ( -7401626754 - 117756654 \beta_{1} + 6653208 \beta_{2} + 2273082 \beta_{3} - 278130 \beta_{4} + 559680 \beta_{5} + 208320 \beta_{6} + 27360 \beta_{7} - 34080 \beta_{8} - 112050 \beta_{9} ) q^{54} + ( 1255936 - 83577945 \beta_{1} - 2094565 \beta_{2} - 3258816 \beta_{3} + 18381 \beta_{4} + 1457280 \beta_{5} - 169664 \beta_{6} + 12672 \beta_{7} - 13299 \beta_{8} + 75405 \beta_{9} ) q^{55} + ( 15445917408 - 27044592 \beta_{1} + 5950080 \beta_{2} - 1802272 \beta_{3} + 160032 \beta_{4} - 829920 \beta_{5} + 110368 \beta_{6} + 68976 \beta_{7} - 94480 \beta_{8} + 416032 \beta_{9} ) q^{56} + ( 8041262796 - 144294263 \beta_{1} + 1871071 \beta_{2} - 916025 \beta_{3} + 675390 \beta_{4} + 83275 \beta_{6} + 4780 \beta_{7} - 4780 \beta_{9} ) q^{57} + ( 19325448418 - 53664268 \beta_{1} - 5206900 \beta_{2} + 4106084 \beta_{3} + 259459 \beta_{4} - 702628 \beta_{5} + 210178 \beta_{6} - 45826 \beta_{7} - 24830 \beta_{8} - 142422 \beta_{9} ) q^{58} + ( -10945426594 - 39068077 \beta_{1} + 9074319 \beta_{2} + 1429824 \beta_{3} + 23680 \beta_{4} - 129984 \beta_{6} + 194176 \beta_{7} - 194176 \beta_{9} ) q^{59} + ( -34737076240 + 58717240 \beta_{1} + 17283072 \beta_{2} + 2324288 \beta_{3} - 1543224 \beta_{4} + 791792 \beta_{5} - 164288 \beta_{6} + 209000 \beta_{7} - 107768 \beta_{8} - 362896 \beta_{9} ) q^{60} + ( -851392 - 179649552 \beta_{1} - 3498413 \beta_{2} + 1982568 \beta_{3} - 628320 \beta_{4} + 72845 \beta_{5} + 190536 \beta_{6} - 84112 \beta_{7} + 148680 \beta_{8} - 440104 \beta_{9} ) q^{61} + ( -29920635504 + 74842752 \beta_{1} - 7489248 \beta_{2} - 7896544 \beta_{3} + 1417464 \beta_{4} + 568608 \beta_{5} + 181392 \beta_{6} - 18864 \beta_{7} + 74992 \beta_{8} - 258736 \beta_{9} ) q^{62} + ( -224640 + 500967737 \beta_{1} + 10518565 \beta_{2} + 969360 \beta_{3} + 1030867 \beta_{4} - 3725920 \beta_{5} - 98480 \beta_{6} + 126560 \beta_{7} - 39197 \beta_{8} + 846723 \beta_{9} ) q^{63} + ( 27619504880 + 185724936 \beta_{1} - 28503616 \beta_{2} + 3863472 \beta_{3} - 1903440 \beta_{4} + 431120 \beta_{5} - 362416 \beta_{6} - 41032 \beta_{7} + 274680 \beta_{8} - 304240 \beta_{9} ) q^{64} + ( 9649843728 + 540701266 \beta_{1} - 45840066 \beta_{2} - 3025330 \beta_{3} - 1832196 \beta_{4} + 275030 \beta_{6} - 96104 \beta_{7} + 96104 \beta_{9} ) q^{65} + ( 48632714636 - 68918476 \beta_{1} + 49619328 \beta_{2} - 12110824 \beta_{3} + 2287208 \beta_{4} + 702240 \beta_{5} - 438240 \beta_{6} - 245520 \beta_{7} - 266640 \beta_{8} - 204952 \beta_{9} ) q^{66} + ( -4985168722 - 392498463 \beta_{1} + 15407865 \beta_{2} - 2110350 \beta_{3} + 1680084 \beta_{4} + 191850 \beta_{6} - 456960 \beta_{7} + 456960 \beta_{9} ) q^{67} + ( -29679598024 - 370192408 \beta_{1} + 17127874 \beta_{2} - 4511618 \beta_{3} - 1285488 \beta_{4} - 1439584 \beta_{5} - 414016 \beta_{6} - 226448 \beta_{7} + 28336 \beta_{8} + 545440 \beta_{9} ) q^{68} + ( -5304256 - 868818784 \beta_{1} - 17018580 \beta_{2} + 13852968 \beta_{3} - 3436128 \beta_{4} - 90700 \beta_{5} + 686600 \beta_{6} - 23568 \beta_{7} + 286600 \beta_{8} + 121624 \beta_{9} ) q^{69} + ( -79913865248 + 491135072 \beta_{1} - 65864960 \beta_{2} + 19069792 \beta_{3} + 2994720 \beta_{4} - 3123328 \beta_{5} - 1433472 \beta_{6} - 261056 \beta_{7} + 103488 \beta_{8} + 1022112 \beta_{9} ) q^{70} + ( -6964608 + 583061179 \beta_{1} + 14647167 \beta_{2} + 17548176 \beta_{3} + 2971177 \beta_{4} + 6729120 \beta_{5} + 1115216 \beta_{6} - 244640 \beta_{7} + 174905 \beta_{8} - 1537575 \beta_{9} ) q^{71} + ( 107759549538 + 61065283 \beta_{1} - 121024380 \beta_{2} - 6204854 \beta_{3} - 2335500 \beta_{4} + 2649318 \beta_{5} - 559718 \beta_{6} - 120179 \beta_{7} - 627 \beta_{8} + 84710 \beta_{9} ) q^{72} + ( -5884699894 + 1282756125 \beta_{1} + 134716907 \beta_{2} + 2613875 \beta_{3} - 8017578 \beta_{4} - 237625 \beta_{6} + 12060 \beta_{7} - 12060 \beta_{9} ) q^{73} + ( 74265226230 - 153033668 \beta_{1} + 95973380 \beta_{2} + 21403852 \beta_{3} + 4843585 \beta_{4} + 2586100 \beta_{5} - 784298 \beta_{6} + 746218 \beta_{7} + 691350 \beta_{8} + 712334 \beta_{9} ) q^{74} + ( -25173117378 - 1170803881 \beta_{1} + 43343771 \beta_{2} - 14109700 \beta_{3} + 5237016 \beta_{4} + 1282700 \beta_{6} + 209024 \beta_{7} - 209024 \beta_{9} ) q^{75} + ( -69972883380 + 27169946 \beta_{1} + 3174658 \beta_{2} - 2863130 \beta_{3} - 2387782 \beta_{4} - 4021660 \beta_{5} + 78456 \beta_{6} - 624858 \beta_{7} + 321710 \beta_{8} - 606492 \beta_{9} ) q^{76} + ( 2033856 - 1729492512 \beta_{1} - 37979260 \beta_{2} - 3822984 \beta_{3} - 6300448 \beta_{4} - 403172 \beta_{5} - 759528 \beta_{6} + 505296 \beta_{7} - 1329768 \beta_{8} + 2207304 \beta_{9} ) q^{77} + ( -179565812164 + 483456896 \beta_{1} + 20717368 \beta_{2} - 44813000 \beta_{3} + 5964562 \beta_{4} - 989000 \beta_{5} + 168700 \beta_{6} + 751020 \beta_{7} - 312892 \beta_{8} - 825012 \beta_{9} ) q^{78} + ( -4264448 + 1635638870 \beta_{1} + 36048414 \beta_{2} + 10254528 \beta_{3} + 4457458 \beta_{4} - 7926400 \beta_{5} + 846272 \beta_{6} - 313216 \beta_{7} - 35150 \beta_{8} - 2227662 \beta_{9} ) q^{79} + ( 198198530400 - 1429761968 \beta_{1} - 73352832 \beta_{2} - 11852064 \beta_{3} - 3659808 \beta_{4} - 993120 \beta_{5} + 1048864 \beta_{6} + 147760 \beta_{7} - 1322576 \beta_{8} - 736352 \beta_{9} ) q^{80} + ( 25957437165 + 1234872201 \beta_{1} - 304495137 \beta_{2} - 13382457 \beta_{3} - 1203138 \beta_{4} + 1216587 \beta_{6} + 625836 \beta_{7} - 625836 \beta_{9} ) q^{81} + ( 165428141818 + 814746830 \beta_{1} + 49554688 \beta_{2} - 42564880 \beta_{3} + 2902544 \beta_{4} + 686400 \beta_{5} - 248000 \beta_{6} - 167840 \beta_{7} - 202912 \beta_{8} + 1162768 \beta_{9} ) q^{82} + ( -14624160514 - 1492621005 \beta_{1} - 2810673 \beta_{2} + 1060400 \beta_{3} + 7060192 \beta_{4} - 96400 \beta_{6} + 913280 \beta_{7} - 913280 \beta_{9} ) q^{83} + ( -314427300064 - 630488560 \beta_{1} + 17017088 \beta_{2} + 8690816 \beta_{3} - 8889616 \beta_{4} + 6170400 \beta_{5} + 230784 \beta_{6} + 1819824 \beta_{7} - 493200 \beta_{8} - 1546720 \beta_{9} ) q^{84} + ( 7265600 + 90335168 \beta_{1} + 537746 \beta_{2} - 19913400 \beta_{3} + 864160 \beta_{4} + 1087054 \beta_{5} - 627800 \beta_{6} - 280400 \beta_{7} + 2102440 \beta_{8} + 139640 \beta_{9} ) q^{85} + ( -226214802313 - 1883859935 \beta_{1} + 96593356 \beta_{2} + 65040373 \beta_{3} - 740657 \beta_{4} + 9157760 \beta_{5} + 2618240 \beta_{6} + 131520 \beta_{7} - 810560 \beta_{8} - 1074097 \beta_{9} ) q^{86} + ( 24343168 + 389503101 \beta_{1} + 660505 \beta_{2} - 60258672 \beta_{3} + 2560063 \beta_{4} + 3440544 \beta_{5} - 4256944 \beta_{6} + 1214048 \beta_{7} - 901041 \beta_{8} + 7597295 \beta_{9} ) q^{87} + ( 217018007316 + 2448941198 \beta_{1} + 303182264 \beta_{2} + 43645316 \beta_{3} - 6224592 \beta_{4} - 3631012 \beta_{5} + 1030612 \beta_{6} - 294734 \beta_{7} + 1884498 \beta_{8} - 1460580 \beta_{9} ) q^{88} + ( 91323136010 - 3152696603 \beta_{1} + 416293731 \beta_{2} + 25243691 \beta_{3} + 8394310 \beta_{4} - 2294881 \beta_{6} - 298116 \beta_{7} + 298116 \beta_{9} ) q^{89} + ( 398319828498 - 1240312556 \beta_{1} - 386274772 \beta_{2} + 74360772 \beta_{3} - 1271797 \beta_{4} - 13615748 \beta_{5} + 4988978 \beta_{6} - 2015474 \beta_{7} - 1693582 \beta_{8} - 4340070 \beta_{9} ) q^{90} + ( 36126681696 + 1813309132 \beta_{1} - 112186204 \beta_{2} + 64127316 \beta_{3} - 8461560 \beta_{4} - 5829756 \beta_{6} - 2293376 \beta_{7} + 2293376 \beta_{9} ) q^{91} + ( -264938071920 + 2310023688 \beta_{1} - 307742720 \beta_{2} - 1515840 \beta_{3} + 6585848 \beta_{4} + 13796240 \beta_{5} + 2944960 \beta_{6} - 529320 \beta_{7} + 273592 \beta_{8} + 7108112 \beta_{9} ) q^{92} + ( 6214144 + 1884616448 \beta_{1} + 39951248 \beta_{2} - 21648576 \beta_{3} + 8861696 \beta_{4} + 1085552 \beta_{5} + 1002048 \beta_{6} - 1778816 \beta_{7} - 1134272 \beta_{8} - 13585984 \beta_{9} ) q^{93} + ( -251725774760 + 856940672 \beta_{1} + 369088304 \beta_{2} - 39059920 \beta_{3} - 12279500 \beta_{4} - 7120 \beta_{5} + 1170904 \beta_{6} - 2841544 \beta_{7} + 2753320 \beta_{8} + 2867448 \beta_{9} ) q^{94} + ( 29283840 - 2857925673 \beta_{1} - 68089461 \beta_{2} - 77313600 \beta_{3} - 6702115 \beta_{4} + 10187136 \beta_{5} - 3512640 \beta_{6} - 147840 \beta_{7} + 1204125 \beta_{8} + 169245 \beta_{9} ) q^{95} + ( 283668065392 - 1949460792 \beta_{1} + 466679104 \beta_{2} - 18072016 \beta_{3} + 14632176 \beta_{4} - 13347440 \beta_{5} - 1716144 \beta_{6} + 1455608 \beta_{7} + 1834040 \beta_{8} + 13719312 \beta_{9} ) q^{96} + ( -147448328326 + 322170951 \beta_{1} - 951621615 \beta_{2} + 58859625 \beta_{3} + 10569762 \beta_{4} - 5350875 \beta_{6} - 2957100 \beta_{7} + 2957100 \beta_{9} ) q^{97} + ( 237815190229 + 3283451935 \beta_{1} - 243477504 \beta_{2} - 62522048 \beta_{3} - 23368000 \beta_{4} - 7590144 \beta_{5} + 3532544 \beta_{6} + 2172032 \beta_{7} + 2496640 \beta_{8} - 1079616 \beta_{9} ) q^{98} + ( -71728800198 + 6632125049 \beta_{1} + 74905677 \beta_{2} + 29189072 \beta_{3} - 33005280 \beta_{4} - 2653552 \beta_{6} + 3014528 \beta_{7} - 3014528 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 110q^{2} - 660q^{3} - 2444q^{4} - 130788q^{6} + 27160q^{8} + 1692798q^{9} + O(q^{10}) \) \( 10q - 110q^{2} - 660q^{3} - 2444q^{4} - 130788q^{6} + 27160q^{8} + 1692798q^{9} + 1873200q^{10} - 4591444q^{11} - 5427720q^{12} + 12728736q^{14} - 38294000q^{16} + 42876500q^{17} + 8229270q^{18} + 36062828q^{19} + 17615520q^{20} - 227004580q^{22} + 169102992q^{24} - 706946390q^{25} + 151295184q^{26} + 1382414040q^{27} - 132075840q^{28} + 456064800q^{30} - 2651204000q^{32} - 478018200q^{33} + 4454663012q^{34} - 2838470400q^{35} - 4074739428q^{36} + 1644178460q^{38} + 8180322240q^{40} - 12339248044q^{41} + 14692585920q^{42} + 25081495340q^{43} - 27950589832q^{44} - 16813594656q^{46} + 13310114400q^{48} - 39532486838q^{49} + 24888425650q^{50} + 36757299288q^{51} - 36172521120q^{52} - 74007907272q^{54} + 154450364544q^{56} + 80408630760q^{57} + 193270394640q^{58} - 109448026708q^{59} - 347360715840q^{60} - 299237961600q^{62} + 276213192256q^{64} + 96485235840q^{65} + 486280823688q^{66} - 49860896020q^{67} - 296812951960q^{68} - 799057954560q^{70} + 1077572984520q^{72} - 58835592940q^{73} + 742739480496q^{74} - 251792743380q^{75} - 699737494024q^{76} - 1795838526240q^{78} + 1981932232320q^{80} + 259515975474q^{81} + 1654109754980q^{82} - 146236977940q^{83} - 3144240693120q^{84} - 2261898070564q^{86} + 2170357811600q^{88} + 913341514388q^{89} + 3983485096080q^{90} + 361546645248q^{91} - 2649411172800q^{92} - 2517413216064q^{94} + 2836588548672q^{96} - 1474226441260q^{97} + 2377890492370q^{98} - 717160631484q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + 28213427840 x^{3} + 1335549648384 x^{2} + 41051831642368 x + 824967906703360\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{9} + \nu^{8} + 474 \nu^{7} + 4340 \nu^{6} + 736136 \nu^{5} + 33571824 \nu^{4} + 345001952 \nu^{3} + 30283439552 \nu^{2} + 1517250285696 \nu + 44108019403776 \)\()/ 549755813888 \)
\(\beta_{2}\)\(=\)\((\)\(-4493 \nu^{9} - 30093 \nu^{8} + 3196142 \nu^{7} - 128916068 \nu^{6} - 3415507432 \nu^{5} - 57254189872 \nu^{4} - 872932384352 \nu^{3} - 76346388882112 \nu^{2} - 7765245353563776 \nu - 145725105007139840\)\()/ 23639499997184 \)
\(\beta_{3}\)\(=\)\((\)\(-8707 \nu^{9} + 53757 \nu^{8} + 1815154 \nu^{7} - 101763804 \nu^{6} - 5862740632 \nu^{5} - 129969712080 \nu^{4} + 1042235363424 \nu^{3} - 153364515412288 \nu^{2} - 11188479510327680 \nu - 214530251328070656\)\()/ 23639499997184 \)
\(\beta_{4}\)\(=\)\((\)\(1615 \nu^{9} + 11215 \nu^{8} - 1231674 \nu^{7} + 48040268 \nu^{6} + 1229377784 \nu^{5} + 19124490000 \nu^{4} + 303242632736 \nu^{3} + 26513840492096 \nu^{2} + 3562413143783296 \nu + 51186634521883648\)\()/ 1477468749824 \)
\(\beta_{5}\)\(=\)\((\)\(-27321 \nu^{9} + 1186119 \nu^{8} - 6815370 \nu^{7} - 965949524 \nu^{6} + 20225317560 \nu^{5} - 647936292976 \nu^{4} + 15420159870752 \nu^{3} - 704336582180800 \nu^{2} - 21242122029100160 \nu + 644305990841488384\)\()/ 23639499997184 \)
\(\beta_{6}\)\(=\)\((\)\(136211 \nu^{9} + 361491 \nu^{8} - 76989650 \nu^{7} + 3208778268 \nu^{6} + 99804859160 \nu^{5} + 1795886598864 \nu^{4} + 12541834980768 \nu^{3} + 3832239415276864 \nu^{2} + 211929337247906176 \nu + 4199044269233464320\)\()/ 23639499997184 \)
\(\beta_{7}\)\(=\)\((\)\(38493 \nu^{9} - 13015459 \nu^{8} + 358357042 \nu^{7} - 2234490716 \nu^{6} - 180252307352 \nu^{5} - 795380527568 \nu^{4} - 192214906747808 \nu^{3} + 4323727691730624 \nu^{2} - 235260934463488384 \nu - 10401552655163827200\)\()/ 23639499997184 \)
\(\beta_{8}\)\(=\)\((\)\(26341 \nu^{9} - 1612827 \nu^{8} + 35745538 \nu^{7} - 95398332 \nu^{6} + 3422899368 \nu^{5} + 20630795696 \nu^{4} - 6505053142176 \nu^{3} + 681614306008256 \nu^{2} + 5123512746181248 \nu - 553898363154809856\)\()/ 2954937499648 \)
\(\beta_{9}\)\(=\)\((\)\(123429 \nu^{9} - 4683739 \nu^{8} + 61948546 \nu^{7} + 2245161796 \nu^{6} - 24203267416 \nu^{5} + 1104372863408 \nu^{4} - 29994699314336 \nu^{3} + 3078909051456704 \nu^{2} + 86730240289333888 \nu - 1405108855990273024\)\()/ 11819749998592 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + 6 \beta_{2} + 26 \beta_{1} + 256\)\()/512\)
\(\nu^{2}\)\(=\)\((\)\(8 \beta_{6} + 3 \beta_{4} + 40 \beta_{3} + 186 \beta_{2} + 390 \beta_{1} - 46656\)\()/512\)
\(\nu^{3}\)\(=\)\((\)\(32 \beta_{9} + 64 \beta_{8} - 96 \beta_{7} + 72 \beta_{6} - 128 \beta_{5} - 193 \beta_{4} + 2024 \beta_{3} + 2498 \beta_{2} + 66014 \beta_{1} - 583616\)\()/512\)
\(\nu^{4}\)\(=\)\((\)\(-4320 \beta_{9} + 1856 \beta_{8} - 1120 \beta_{7} + 552 \beta_{6} - 24192 \beta_{5} - 8449 \beta_{4} - 12984 \beta_{3} + 18594 \beta_{2} + 3979134 \beta_{1} - 126891712\)\()/512\)
\(\nu^{5}\)\(=\)\((\)\(-193888 \beta_{9} + 83008 \beta_{8} + 44832 \beta_{7} - 11416 \beta_{6} - 342144 \beta_{5} - 317977 \beta_{4} + 973192 \beta_{3} - 8415534 \beta_{2} - 6247474 \beta_{1} - 7937852096\)\()/512\)
\(\nu^{6}\)\(=\)\((\)\(-2658400 \beta_{9} + 65088 \beta_{8} + 2715680 \beta_{7} - 4321240 \beta_{6} - 3255424 \beta_{5} - 15187809 \beta_{4} + 92441928 \beta_{3} - 477634078 \beta_{2} + 2069551038 \beta_{1} + 2085743424\)\()/512\)
\(\nu^{7}\)\(=\)\((\)\(35934880 \beta_{9} - 57652160 \beta_{8} + 98783008 \beta_{7} - 216557976 \beta_{6} + 485832576 \beta_{5} - 53454153 \beta_{4} + 1358153864 \beta_{3} - 10219468110 \beta_{2} + 221797639662 \beta_{1} - 4460137581248\)\()/512\)
\(\nu^{8}\)\(=\)\((\)\(4710587296 \beta_{9} - 4144748992 \beta_{8} + 2120907808 \beta_{7} - 3951195992 \beta_{6} + 24607916928 \beta_{5} - 10898687281 \beta_{4} - 17279952952 \beta_{3} - 176195721854 \beta_{2} + 4161762999326 \beta_{1} - 451877316724928\)\()/512\)
\(\nu^{9}\)\(=\)\((\)\(266511889568 \beta_{9} - 114305216448 \beta_{8} - 23011923680 \beta_{7} - 151881717016 \beta_{6} + 867430313856 \beta_{5} - 921641522905 \beta_{4} - 3217809582328 \beta_{3} - 2934137801262 \beta_{2} - 39822522506034 \beta_{1} - 8697213876274880\)\()/512\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−25.4137 + 6.09761i
−25.4137 6.09761i
−11.5862 + 26.7344i
−11.5862 26.7344i
−10.5406 + 27.3936i
−10.5406 27.3936i
17.4695 + 29.8739i
17.4695 29.8739i
32.5710 + 17.8321i
32.5710 17.8321i
−62.8274 12.1952i 24.0930 3798.55 + 1532.39i 25133.2i −1513.70 293.819i 190438.i −219965. 142600.i −530861. −306505. + 1.57906e6i
3.2 −62.8274 + 12.1952i 24.0930 3798.55 1532.39i 25133.2i −1513.70 + 293.819i 190438.i −219965. + 142600.i −530861. −306505. 1.57906e6i
3.3 −35.1724 53.4687i 1272.20 −1621.80 + 3761.25i 19079.0i −44746.4 68023.0i 136156.i 258152. 45576.2i 1.08706e6 1.02013e6 671054.i
3.4 −35.1724 + 53.4687i 1272.20 −1621.80 3761.25i 19079.0i −44746.4 + 68023.0i 136156.i 258152. + 45576.2i 1.08706e6 1.02013e6 + 671054.i
3.5 −33.0812 54.7872i −875.418 −1907.27 + 3624.85i 2272.99i 28959.9 + 47961.7i 92079.6i 261690. 15420.3i 234916. 124531. 75193.1i
3.6 −33.0812 + 54.7872i −875.418 −1907.27 3624.85i 2272.99i 28959.9 47961.7i 92079.6i 261690. + 15420.3i 234916. 124531. + 75193.1i
3.7 22.9390 59.7478i 271.191 −3043.60 2741.11i 11036.2i 6220.87 16203.1i 58770.2i −233593. + 118970.i −457896. −659388. 253159.i
3.8 22.9390 + 59.7478i 271.191 −3043.60 + 2741.11i 11036.2i 6220.87 + 16203.1i 58770.2i −233593. 118970.i −457896. −659388. + 253159.i
3.9 53.1419 35.6642i −1022.07 1552.12 3790.53i 21249.1i −54314.6 + 36451.3i 149114.i −52703.6 256791.i 513182. 757833. + 1.12922e6i
3.10 53.1419 + 35.6642i −1022.07 1552.12 + 3790.53i 21249.1i −54314.6 36451.3i 149114.i −52703.6 + 256791.i 513182. 757833. 1.12922e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.13.d.b 10
3.b odd 2 1 72.13.b.b 10
4.b odd 2 1 32.13.d.b 10
8.b even 2 1 32.13.d.b 10
8.d odd 2 1 inner 8.13.d.b 10
24.f even 2 1 72.13.b.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.13.d.b 10 1.a even 1 1 trivial
8.13.d.b 10 8.d odd 2 1 inner
32.13.d.b 10 4.b odd 2 1
32.13.d.b 10 8.b even 2 1
72.13.b.b 10 3.b odd 2 1
72.13.b.b 10 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 330 T_{3}^{4} - 1697352 T_{3}^{3} - 685590480 T_{3}^{2} + 326191796496 T_{3} - \)\(74\!\cdots\!00\)\( \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 110 T + 7272 T^{2} + 347200 T^{3} + 22817792 T^{4} + 2004910080 T^{5} + 93461676032 T^{6} + 5825049395200 T^{7} + 499728034824192 T^{8} + 30962247438172160 T^{9} + 1152921504606846976 T^{10} \)
$3$ \( ( 1 + 330 T + 959853 T^{2} + 15911640 T^{3} + 444359828610 T^{4} - 176928653725380 T^{5} + 236151031676327010 T^{6} + 4493917109852538840 T^{7} + \)\(14\!\cdots\!13\)\( T^{8} + \)\(26\!\cdots\!30\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{2} \)
$5$ \( 1 - 867229930 T^{2} + 471532937915392845 T^{4} - \)\(19\!\cdots\!00\)\( T^{6} + \)\(62\!\cdots\!50\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(37\!\cdots\!50\)\( T^{12} - \)\(68\!\cdots\!00\)\( T^{14} + \)\(99\!\cdots\!25\)\( T^{16} - \)\(10\!\cdots\!50\)\( T^{18} + \)\(75\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 - 49440192586 T^{2} + \)\(16\!\cdots\!33\)\( T^{4} - \)\(36\!\cdots\!68\)\( T^{6} + \)\(68\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(13\!\cdots\!26\)\( T^{12} - \)\(13\!\cdots\!68\)\( T^{14} + \)\(11\!\cdots\!33\)\( T^{16} - \)\(66\!\cdots\!86\)\( T^{18} + \)\(25\!\cdots\!01\)\( T^{20} \)
$11$ \( ( 1 + 2295722 T + 9972896439693 T^{2} + 13526301537913731160 T^{3} + \)\(37\!\cdots\!22\)\( T^{4} + \)\(38\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!62\)\( T^{6} + \)\(13\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!73\)\( T^{8} + \)\(22\!\cdots\!82\)\( T^{9} + \)\(30\!\cdots\!01\)\( T^{10} )^{2} \)
$13$ \( 1 - 104531244325546 T^{2} + \)\(58\!\cdots\!13\)\( T^{4} - \)\(21\!\cdots\!88\)\( T^{6} + \)\(61\!\cdots\!26\)\( T^{8} - \)\(14\!\cdots\!12\)\( T^{10} + \)\(33\!\cdots\!86\)\( T^{12} - \)\(62\!\cdots\!48\)\( T^{14} + \)\(92\!\cdots\!53\)\( T^{16} - \)\(90\!\cdots\!86\)\( T^{18} + \)\(47\!\cdots\!01\)\( T^{20} \)
$17$ \( ( 1 - 21438250 T + 2412850471912173 T^{2} - \)\(42\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!10\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!10\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(47\!\cdots\!13\)\( T^{8} - \)\(24\!\cdots\!50\)\( T^{9} + \)\(67\!\cdots\!01\)\( T^{10} )^{2} \)
$19$ \( ( 1 - 18031414 T + 8471849079978669 T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(59\!\cdots\!60\)\( T^{5} + \)\(71\!\cdots\!26\)\( T^{6} - \)\(85\!\cdots\!80\)\( T^{7} + \)\(91\!\cdots\!89\)\( T^{8} - \)\(43\!\cdots\!74\)\( T^{9} + \)\(53\!\cdots\!01\)\( T^{10} )^{2} \)
$23$ \( 1 - 111670895161565386 T^{2} + \)\(67\!\cdots\!13\)\( T^{4} - \)\(27\!\cdots\!68\)\( T^{6} + \)\(87\!\cdots\!26\)\( T^{8} - \)\(21\!\cdots\!72\)\( T^{10} + \)\(41\!\cdots\!66\)\( T^{12} - \)\(64\!\cdots\!08\)\( T^{14} + \)\(74\!\cdots\!73\)\( T^{16} - \)\(59\!\cdots\!46\)\( T^{18} + \)\(25\!\cdots\!01\)\( T^{20} \)
$29$ \( 1 - 2227945431622225450 T^{2} + \)\(25\!\cdots\!05\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{8} - \)\(41\!\cdots\!52\)\( T^{10} + \)\(12\!\cdots\!50\)\( T^{12} - \)\(29\!\cdots\!80\)\( T^{14} + \)\(49\!\cdots\!05\)\( T^{16} - \)\(54\!\cdots\!50\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - 4692181618679558410 T^{2} + \)\(10\!\cdots\!45\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{10} + \)\(86\!\cdots\!10\)\( T^{12} - \)\(53\!\cdots\!20\)\( T^{14} + \)\(24\!\cdots\!45\)\( T^{16} - \)\(69\!\cdots\!10\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - 9412964374123010026 T^{2} + \)\(12\!\cdots\!53\)\( T^{4} - \)\(23\!\cdots\!28\)\( T^{6} + \)\(16\!\cdots\!06\)\( T^{8} + \)\(22\!\cdots\!88\)\( T^{10} + \)\(70\!\cdots\!66\)\( T^{12} - \)\(44\!\cdots\!88\)\( T^{14} + \)\(10\!\cdots\!93\)\( T^{16} - \)\(33\!\cdots\!66\)\( T^{18} + \)\(15\!\cdots\!01\)\( T^{20} \)
$41$ \( ( 1 + 6169624022 T + 92150338791973341453 T^{2} + \)\(36\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!42\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(76\!\cdots\!02\)\( T^{6} + \)\(18\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!73\)\( T^{8} + \)\(15\!\cdots\!62\)\( T^{9} + \)\(58\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( ( 1 - 12540747670 T + \)\(16\!\cdots\!37\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!42\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(41\!\cdots\!42\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!37\)\( T^{8} - \)\(31\!\cdots\!70\)\( T^{9} + \)\(10\!\cdots\!01\)\( T^{10} )^{2} \)
$47$ \( 1 - \)\(58\!\cdots\!66\)\( T^{2} + \)\(14\!\cdots\!13\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{6} + \)\(59\!\cdots\!26\)\( T^{8} + \)\(35\!\cdots\!88\)\( T^{10} + \)\(80\!\cdots\!06\)\( T^{12} - \)\(30\!\cdots\!48\)\( T^{14} + \)\(35\!\cdots\!33\)\( T^{16} - \)\(19\!\cdots\!86\)\( T^{18} + \)\(44\!\cdots\!01\)\( T^{20} \)
$53$ \( 1 - \)\(18\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!53\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{6} + \)\(69\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!12\)\( T^{10} + \)\(16\!\cdots\!86\)\( T^{12} - \)\(69\!\cdots\!88\)\( T^{14} + \)\(24\!\cdots\!73\)\( T^{16} - \)\(61\!\cdots\!66\)\( T^{18} + \)\(81\!\cdots\!01\)\( T^{20} \)
$59$ \( ( 1 + 54724013354 T + \)\(77\!\cdots\!53\)\( T^{2} + \)\(32\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!86\)\( T^{4} + \)\(80\!\cdots\!68\)\( T^{5} + \)\(45\!\cdots\!66\)\( T^{6} + \)\(10\!\cdots\!92\)\( T^{7} + \)\(43\!\cdots\!73\)\( T^{8} + \)\(54\!\cdots\!34\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \)
$61$ \( 1 - \)\(20\!\cdots\!90\)\( T^{2} + \)\(20\!\cdots\!25\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(55\!\cdots\!90\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{10} + \)\(39\!\cdots\!90\)\( T^{12} - \)\(64\!\cdots\!00\)\( T^{14} + \)\(72\!\cdots\!25\)\( T^{16} - \)\(51\!\cdots\!90\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$67$ \( ( 1 + 24930448010 T + \)\(26\!\cdots\!33\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!10\)\( T^{4} + \)\(25\!\cdots\!40\)\( T^{5} + \)\(28\!\cdots\!10\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!73\)\( T^{8} + \)\(11\!\cdots\!10\)\( T^{9} + \)\(36\!\cdots\!01\)\( T^{10} )^{2} \)
$71$ \( 1 - \)\(23\!\cdots\!50\)\( T^{2} + \)\(94\!\cdots\!05\)\( T^{4} - \)\(19\!\cdots\!80\)\( T^{6} + \)\(48\!\cdots\!50\)\( T^{8} - \)\(69\!\cdots\!52\)\( T^{10} + \)\(12\!\cdots\!50\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(18\!\cdots\!05\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{18} + \)\(14\!\cdots\!01\)\( T^{20} \)
$73$ \( ( 1 + 29417796470 T + \)\(32\!\cdots\!93\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(28\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!10\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(39\!\cdots\!73\)\( T^{8} + \)\(80\!\cdots\!70\)\( T^{9} + \)\(63\!\cdots\!01\)\( T^{10} )^{2} \)
$79$ \( 1 - \)\(34\!\cdots\!10\)\( T^{2} + \)\(58\!\cdots\!85\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{6} + \)\(55\!\cdots\!10\)\( T^{8} - \)\(36\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!10\)\( T^{12} - \)\(81\!\cdots\!60\)\( T^{14} + \)\(25\!\cdots\!85\)\( T^{16} - \)\(51\!\cdots\!10\)\( T^{18} + \)\(51\!\cdots\!01\)\( T^{20} \)
$83$ \( ( 1 + 73118488970 T + \)\(44\!\cdots\!13\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!10\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(90\!\cdots\!10\)\( T^{6} + \)\(19\!\cdots\!60\)\( T^{7} + \)\(53\!\cdots\!53\)\( T^{8} + \)\(95\!\cdots\!70\)\( T^{9} + \)\(13\!\cdots\!01\)\( T^{10} )^{2} \)
$89$ \( ( 1 - 456670757194 T + \)\(80\!\cdots\!49\)\( T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!26\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(84\!\cdots\!46\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!89\)\( T^{8} - \)\(16\!\cdots\!14\)\( T^{9} + \)\(91\!\cdots\!01\)\( T^{10} )^{2} \)
$97$ \( ( 1 + 737113220630 T + \)\(12\!\cdots\!77\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} - \)\(27\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} - \)\(19\!\cdots\!78\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!17\)\( T^{8} + \)\(17\!\cdots\!30\)\( T^{9} + \)\(16\!\cdots\!01\)\( T^{10} )^{2} \)
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