# Properties

 Label 387.6.a.e Level 387 Weight 6 Character orbit 387.a Self dual yes Analytic conductor 62.069 Analytic rank 1 Dimension 10 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 387.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.0685382676$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 43) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 20 - \beta_{1} + \beta_{2} ) q^{4} + ( -14 - \beta_{7} ) q^{5} + ( 7 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} ) q^{7} + ( -32 + 20 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 20 - \beta_{1} + \beta_{2} ) q^{4} + ( -14 - \beta_{7} ) q^{5} + ( 7 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} ) q^{7} + ( -32 + 20 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{8} + ( 5 - 8 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{10} + ( -73 - 15 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} ) q^{11} + ( 191 + 11 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 9 \beta_{8} + 4 \beta_{9} ) q^{13} + ( -199 - 32 \beta_{1} + 2 \beta_{2} + 17 \beta_{3} - 6 \beta_{4} + \beta_{5} - 10 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{14} + ( 518 + 4 \beta_{1} + 19 \beta_{2} + 11 \beta_{3} - 12 \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} + 10 \beta_{8} - 5 \beta_{9} ) q^{16} + ( -383 - 56 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} - 7 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} + 4 \beta_{9} ) q^{17} + ( -249 - 23 \beta_{1} - 9 \beta_{2} + 19 \beta_{3} - 11 \beta_{4} + 20 \beta_{5} - 16 \beta_{6} - 15 \beta_{7} - 2 \beta_{8} + 7 \beta_{9} ) q^{19} + ( -96 - 158 \beta_{1} - 15 \beta_{2} + 21 \beta_{3} + 4 \beta_{4} + 36 \beta_{5} + 11 \beta_{6} - 16 \beta_{7} + 34 \beta_{8} + 4 \beta_{9} ) q^{20} + ( -561 + \beta_{1} - 31 \beta_{2} - 19 \beta_{3} + 6 \beta_{4} - 33 \beta_{5} + \beta_{6} + 29 \beta_{7} - 22 \beta_{8} + 5 \beta_{9} ) q^{22} + ( -183 + 21 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} - 17 \beta_{5} - 10 \beta_{6} + 9 \beta_{7} - 20 \beta_{8} + 4 \beta_{9} ) q^{23} + ( 707 - 3 \beta_{1} + 12 \beta_{2} + 37 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} + 12 \beta_{6} + 34 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{25} + ( 143 + 145 \beta_{1} - 25 \beta_{2} - 21 \beta_{3} + 26 \beta_{4} + 9 \beta_{5} - 17 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + 11 \beta_{9} ) q^{26} + ( -1422 - 112 \beta_{1} - 26 \beta_{2} - 88 \beta_{3} + 10 \beta_{4} - 34 \beta_{5} + 60 \beta_{6} + 38 \beta_{7} + 22 \beta_{8} - 2 \beta_{9} ) q^{28} + ( -608 - 25 \beta_{1} - 69 \beta_{2} - 52 \beta_{3} + 60 \beta_{4} + 20 \beta_{5} + 9 \beta_{6} + 42 \beta_{7} + 24 \beta_{8} - 6 \beta_{9} ) q^{29} + ( -423 - 52 \beta_{1} - 17 \beta_{2} - 20 \beta_{3} + 40 \beta_{4} + 27 \beta_{5} + 35 \beta_{6} + 34 \beta_{7} - 22 \beta_{8} - 36 \beta_{9} ) q^{31} + ( 668 + 396 \beta_{1} - 25 \beta_{2} - 39 \beta_{3} - 36 \beta_{4} - 73 \beta_{5} - 43 \beta_{6} + 31 \beta_{7} - 60 \beta_{8} + 11 \beta_{9} ) q^{32} + ( -2579 - 683 \beta_{1} - 53 \beta_{2} + 53 \beta_{3} + 14 \beta_{4} - 56 \beta_{5} - 45 \beta_{6} - 54 \beta_{7} + 38 \beta_{8} - 4 \beta_{9} ) q^{34} + ( -727 - 109 \beta_{1} - 6 \beta_{2} + 54 \beta_{3} - 32 \beta_{4} - 2 \beta_{5} - 67 \beta_{6} + 8 \beta_{7} - 103 \beta_{8} - 7 \beta_{9} ) q^{35} + ( 147 + 234 \beta_{1} - 54 \beta_{2} + 57 \beta_{3} + 33 \beta_{4} + 30 \beta_{5} + 11 \beta_{6} + 84 \beta_{7} - 84 \beta_{8} - 21 \beta_{9} ) q^{37} + ( -1172 - 482 \beta_{1} - 161 \beta_{2} - 113 \beta_{3} - 24 \beta_{4} + 136 \beta_{5} + 129 \beta_{6} - 18 \beta_{7} + 70 \beta_{8} ) q^{38} + ( -9148 - 100 \beta_{1} - 251 \beta_{2} - 149 \beta_{3} - 2 \beta_{4} + 176 \beta_{5} + 31 \beta_{6} - 8 \beta_{7} - 18 \beta_{8} + 66 \beta_{9} ) q^{40} + ( -1038 - 460 \beta_{1} + 120 \beta_{2} + 112 \beta_{3} + 40 \beta_{4} - 37 \beta_{5} - 161 \beta_{6} - 45 \beta_{7} - 155 \beta_{8} - 31 \beta_{9} ) q^{41} + 1849 q^{43} + ( 3660 - 1221 \beta_{1} + 122 \beta_{2} + 207 \beta_{3} - 128 \beta_{4} - 87 \beta_{5} - 49 \beta_{6} - 107 \beta_{7} + 132 \beta_{8} - 83 \beta_{9} ) q^{44} + ( 1803 - 501 \beta_{1} + 138 \beta_{2} + 14 \beta_{3} - 54 \beta_{4} - 175 \beta_{5} - 16 \beta_{6} - 43 \beta_{7} + 132 \beta_{8} - 97 \beta_{9} ) q^{46} + ( -4824 - 27 \beta_{1} + 100 \beta_{2} - 79 \beta_{3} - 25 \beta_{4} + 82 \beta_{5} + 72 \beta_{6} + 124 \beta_{7} + 139 \beta_{8} - 56 \beta_{9} ) q^{47} + ( 3104 - 603 \beta_{1} - 112 \beta_{2} - 108 \beta_{3} - 18 \beta_{4} - 74 \beta_{5} + 17 \beta_{6} - 134 \beta_{7} - 179 \beta_{8} - \beta_{9} ) q^{49} + ( -249 + 855 \beta_{1} + 164 \beta_{2} - 266 \beta_{3} - 4 \beta_{4} - 251 \beta_{5} - 36 \beta_{6} + 93 \beta_{7} - 24 \beta_{8} - 109 \beta_{9} ) q^{50} + ( 1516 - 917 \beta_{1} + 136 \beta_{2} + 25 \beta_{3} - 32 \beta_{4} + 95 \beta_{5} + 57 \beta_{6} - 205 \beta_{7} - 100 \beta_{8} - 37 \beta_{9} ) q^{52} + ( -12898 + 116 \beta_{1} + 109 \beta_{2} + 13 \beta_{3} - 223 \beta_{4} + 101 \beta_{5} - 52 \beta_{6} + 50 \beta_{7} + 236 \beta_{8} + 45 \beta_{9} ) q^{53} + ( 11087 - 2101 \beta_{1} + \beta_{2} + 175 \beta_{3} + 95 \beta_{4} - 96 \beta_{5} + 50 \beta_{6} + 94 \beta_{7} + 108 \beta_{8} + 71 \beta_{9} ) q^{55} + ( 970 - 1010 \beta_{1} + 74 \beta_{2} + 118 \beta_{3} + 296 \beta_{4} - 254 \beta_{5} - 212 \beta_{6} - 118 \beta_{7} + 58 \beta_{8} + 8 \beta_{9} ) q^{56} + ( -733 - 2872 \beta_{1} + 52 \beta_{2} + 345 \beta_{3} - 210 \beta_{4} + 426 \beta_{5} + 104 \beta_{6} - 318 \beta_{7} + 201 \beta_{8} + 237 \beta_{9} ) q^{58} + ( -9965 + 803 \beta_{1} - 32 \beta_{2} + 50 \beta_{3} + 108 \beta_{4} + 310 \beta_{5} + 293 \beta_{6} + 136 \beta_{7} - 49 \beta_{8} + 179 \beta_{9} ) q^{59} + ( 2015 - 2743 \beta_{1} - 37 \beta_{2} + 161 \beta_{3} - 443 \beta_{4} + 264 \beta_{5} - 30 \beta_{6} - 206 \beta_{7} - 80 \beta_{8} - 7 \beta_{9} ) q^{61} + ( -2923 - 925 \beta_{1} + 251 \beta_{2} + 327 \beta_{3} - 106 \beta_{4} + 390 \beta_{5} - 33 \beta_{6} - 208 \beta_{7} - 72 \beta_{8} + 104 \beta_{9} ) q^{62} + ( 4684 - 772 \beta_{1} + 295 \beta_{2} + 121 \beta_{3} + 144 \beta_{4} - 541 \beta_{5} - 295 \beta_{6} - 101 \beta_{7} + 164 \beta_{8} - 125 \beta_{9} ) q^{64} + ( -5651 + 213 \beta_{1} + 487 \beta_{2} - 127 \beta_{3} + 217 \beta_{4} - 296 \beta_{5} - 106 \beta_{6} - 248 \beta_{7} - 188 \beta_{8} + 57 \beta_{9} ) q^{65} + ( 463 - 1269 \beta_{1} - 163 \beta_{2} - 525 \beta_{3} + 541 \beta_{4} - 477 \beta_{5} - 171 \beta_{6} + 208 \beta_{7} - 297 \beta_{8} + 208 \beta_{9} ) q^{67} + ( -18106 - 2981 \beta_{1} - 1006 \beta_{2} - 451 \beta_{3} - 194 \beta_{4} - 44 \beta_{5} + 267 \beta_{6} + 152 \beta_{7} - 168 \beta_{8} + 36 \beta_{9} ) q^{68} + ( -3226 - 1398 \beta_{1} + 288 \beta_{2} - 52 \beta_{3} - 324 \beta_{4} - 86 \beta_{5} + 240 \beta_{6} - 142 \beta_{7} + 448 \beta_{8} - 322 \beta_{9} ) q^{70} + ( -998 - 440 \beta_{1} - 214 \beta_{2} + 156 \beta_{3} + 168 \beta_{4} - 482 \beta_{5} + 14 \beta_{6} - 110 \beta_{7} - 40 \beta_{8} + 64 \beta_{9} ) q^{71} + ( 4663 - 479 \beta_{1} + 863 \beta_{2} - 63 \beta_{3} - 207 \beta_{4} + 312 \beta_{5} + 174 \beta_{6} - 78 \beta_{7} + 108 \beta_{8} + 213 \beta_{9} ) q^{73} + ( 12458 - 1874 \beta_{1} + 838 \beta_{2} - 54 \beta_{3} - 316 \beta_{4} + 175 \beta_{5} + 210 \beta_{6} - 413 \beta_{7} + 426 \beta_{8} - 201 \beta_{9} ) q^{74} + ( -23918 - 3912 \beta_{1} - 707 \beta_{2} + 649 \beta_{3} + 582 \beta_{4} + 594 \beta_{5} - 77 \beta_{6} - 402 \beta_{7} - 16 \beta_{8} + 314 \beta_{9} ) q^{76} + ( -9711 + 197 \beta_{1} + 381 \beta_{2} - 113 \beta_{3} + 371 \beta_{4} + 572 \beta_{5} - 216 \beta_{6} + 510 \beta_{7} - 202 \beta_{8} + 201 \beta_{9} ) q^{77} + ( -8866 - 124 \beta_{1} + 553 \beta_{2} - 169 \beta_{3} + 809 \beta_{4} + 328 \beta_{5} - 319 \beta_{6} + 161 \beta_{7} - 81 \beta_{8} - 132 \beta_{9} ) q^{79} + ( 194 - 10246 \beta_{1} - 51 \beta_{2} + 887 \beta_{3} - 226 \beta_{4} + 676 \beta_{5} - 303 \beta_{6} - 1100 \beta_{7} + 72 \beta_{8} + 292 \beta_{9} ) q^{80} + ( -15733 + 1669 \beta_{1} - 120 \beta_{2} - 906 \beta_{3} - 438 \beta_{4} + 33 \beta_{5} + 1130 \beta_{6} + 461 \beta_{7} + 238 \beta_{8} - 347 \beta_{9} ) q^{82} + ( 11270 - 974 \beta_{1} - 57 \beta_{2} - 453 \beta_{3} + 483 \beta_{4} - 299 \beta_{5} + 636 \beta_{6} + 570 \beta_{7} + 28 \beta_{8} + 283 \beta_{9} ) q^{83} + ( -9281 + 4016 \beta_{1} - 44 \beta_{2} - 337 \beta_{3} + 175 \beta_{4} - 818 \beta_{5} - 57 \beta_{6} + 533 \beta_{7} + 20 \beta_{8} - 119 \beta_{9} ) q^{85} + ( -1849 + 1849 \beta_{1} ) q^{86} + ( -46676 + 5832 \beta_{1} - 990 \beta_{2} - 978 \beta_{3} - 262 \beta_{4} - 284 \beta_{5} - 218 \beta_{6} + 744 \beta_{7} - 902 \beta_{8} - 188 \beta_{9} ) q^{88} + ( 6109 + 147 \beta_{1} + 1099 \beta_{2} - 503 \beta_{3} + 237 \beta_{4} - 230 \beta_{5} + 50 \beta_{6} - 156 \beta_{7} - 480 \beta_{8} - 423 \beta_{9} ) q^{89} + ( -29467 - 1419 \beta_{1} + 345 \beta_{2} + 63 \beta_{3} - 525 \beta_{4} + 636 \beta_{5} + 224 \beta_{6} + 714 \beta_{7} + 762 \beta_{8} + 261 \beta_{9} ) q^{91} + ( -18714 + 3723 \beta_{1} - 343 \beta_{2} - 474 \beta_{3} + 166 \beta_{4} - 1031 \beta_{5} - 424 \beta_{6} + 1153 \beta_{7} - 884 \beta_{8} - 19 \beta_{9} ) q^{92} + ( 302 - 2196 \beta_{1} + 928 \beta_{2} + 990 \beta_{3} + 16 \beta_{4} + 143 \beta_{5} - 1244 \beta_{6} + 311 \beta_{7} - 636 \beta_{8} + 153 \beta_{9} ) q^{94} + ( 29615 + 3888 \beta_{1} + 1034 \beta_{2} + 81 \beta_{3} - 159 \beta_{4} - 420 \beta_{5} + 651 \beta_{6} + 602 \beta_{7} + 92 \beta_{8} - 837 \beta_{9} ) q^{95} + ( 10723 - 1265 \beta_{1} + 108 \beta_{2} + 700 \beta_{3} - 14 \beta_{4} - 1075 \beta_{5} - 266 \beta_{6} - 171 \beta_{7} - 1316 \beta_{8} - 434 \beta_{9} ) q^{97} + ( -35611 + 891 \beta_{1} - 990 \beta_{2} + 994 \beta_{3} + 108 \beta_{4} + 230 \beta_{5} + 134 \beta_{6} - 822 \beta_{7} + 336 \beta_{8} + 22 \beta_{9} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 8q^{2} + 202q^{4} - 138q^{5} + 60q^{7} - 294q^{8} + O(q^{10})$$ $$10q - 8q^{2} + 202q^{4} - 138q^{5} + 60q^{7} - 294q^{8} - 17q^{10} - 745q^{11} + 1917q^{13} - 1936q^{14} + 5354q^{16} - 4017q^{17} - 2404q^{19} - 1311q^{20} - 5836q^{22} - 1733q^{23} + 7120q^{25} + 1484q^{26} - 15028q^{28} - 6996q^{29} - 4899q^{31} + 7554q^{32} - 27033q^{34} - 7084q^{35} + 1466q^{37} - 13905q^{38} - 93211q^{40} - 10297q^{41} + 18490q^{43} + 36140q^{44} + 17991q^{46} - 48592q^{47} + 29458q^{49} - 983q^{50} + 14232q^{52} - 127165q^{53} + 106672q^{55} + 7780q^{56} - 10305q^{58} - 99372q^{59} + 17408q^{61} - 28265q^{62} + 47202q^{64} - 54484q^{65} - 2021q^{67} - 192151q^{68} - 33194q^{70} - 11286q^{71} + 49892q^{73} + 125431q^{74} - 249803q^{76} - 98144q^{77} - 91524q^{79} - 12251q^{80} - 158909q^{82} + 105203q^{83} - 87212q^{85} - 14792q^{86} - 461824q^{88} + 62682q^{89} - 295304q^{91} - 183783q^{92} + 7259q^{94} + 305340q^{95} + 108383q^{97} - 354656q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 51$$ $$\beta_{3}$$ $$=$$ $$($$$$-192531 \nu^{9} + 19231001 \nu^{8} - 40404153 \nu^{7} - 4078413669 \nu^{6} + 8269172671 \nu^{5} + 277010815095 \nu^{4} - 301260007075 \nu^{3} - 6223544561327 \nu^{2} + 3431882125456 \nu + 33312422351204$$$$)/ 197066179584$$ $$\beta_{4}$$ $$=$$ $$($$$$24107 \nu^{9} + 594245 \nu^{8} - 7832247 \nu^{7} - 116173337 \nu^{6} + 397671097 \nu^{5} + 6546797939 \nu^{4} + 20833059243 \nu^{3} - 64822510771 \nu^{2} - 762684015384 \nu + 167478232916$$$$)/ 24633272448$$ $$\beta_{5}$$ $$=$$ $$($$$$-144543 \nu^{9} - 820999 \nu^{8} + 35600547 \nu^{7} + 213763899 \nu^{6} - 2591602757 \nu^{5} - 15768164889 \nu^{4} + 54235511681 \nu^{3} + 287212616929 \nu^{2} - 418161182816 \nu - 912070606876$$$$)/ 24633272448$$ $$\beta_{6}$$ $$=$$ $$($$$$474917 \nu^{9} + 1144977 \nu^{8} - 116809809 \nu^{7} - 421392509 \nu^{6} + 8979038055 \nu^{5} + 39727817087 \nu^{4} - 220849424171 \nu^{3} - 981675569783 \nu^{2} + 1612360020176 \nu + 4700999183556$$$$)/ 21896242176$$ $$\beta_{7}$$ $$=$$ $$($$$$-6235841 \nu^{9} - 47565 \nu^{8} + 1552893885 \nu^{7} + 1540088777 \nu^{6} - 122844007131 \nu^{5} - 194780377283 \nu^{4} + 3248986969295 \nu^{3} + 4539319373339 \nu^{2} - 25027510353488 \nu - 19301122595508$$$$)/ 197066179584$$ $$\beta_{8}$$ $$=$$ $$($$$$-1603183 \nu^{9} - 3158223 \nu^{8} + 411977523 \nu^{7} + 1148308531 \nu^{6} - 33961389861 \nu^{5} - 106024108753 \nu^{4} + 953163113617 \nu^{3} + 2509398609001 \nu^{2} - 7514795536912 \nu - 12026382560220$$$$)/ 49266544896$$ $$\beta_{9}$$ $$=$$ $$($$$$-8421725 \nu^{9} + 11663207 \nu^{8} + 2044433577 \nu^{7} - 264140251 \nu^{6} - 158825858063 \nu^{5} - 139815344567 \nu^{4} + 4112939498419 \nu^{3} + 6054797547951 \nu^{2} - 25999649489104 \nu - 56109167945572$$$$)/ 197066179584$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 51$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - 2 \beta_{8} + 3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 84 \beta_{1} + 58$$ $$\nu^{4}$$ $$=$$ $$-9 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} - 4 \beta_{4} + 7 \beta_{3} + 125 \beta_{2} + 242 \beta_{1} + 4411$$ $$\nu^{5}$$ $$=$$ $$-152 \beta_{9} - 286 \beta_{8} + 430 \beta_{7} - 176 \beta_{6} - 226 \beta_{5} + 180 \beta_{4} - 122 \beta_{3} + 698 \beta_{2} + 8451 \beta_{1} + 13438$$ $$\nu^{6}$$ $$=$$ $$-1722 \beta_{9} - 22 \beta_{8} + 1924 \beta_{7} - 1166 \beta_{6} - 2292 \beta_{5} - 596 \beta_{4} + 1024 \beta_{3} + 14929 \beta_{2} + 39399 \beta_{1} + 450581$$ $$\nu^{7}$$ $$=$$ $$-21607 \beta_{9} - 34192 \beta_{8} + 55963 \beta_{7} - 24587 \beta_{6} - 41129 \beta_{5} + 13862 \beta_{4} - 13517 \beta_{3} + 103196 \beta_{2} + 921314 \beta_{1} + 2192128$$ $$\nu^{8}$$ $$=$$ $$-261787 \beta_{9} - 53612 \beta_{8} + 330037 \beta_{7} - 229441 \beta_{6} - 456771 \beta_{5} - 76448 \beta_{4} + 120455 \beta_{3} + 1813937 \beta_{2} + 5689340 \beta_{1} + 49716725$$ $$\nu^{9}$$ $$=$$ $$-3049570 \beta_{9} - 3990164 \beta_{8} + 7188442 \beta_{7} - 3369222 \beta_{6} - 6655062 \beta_{5} + 926440 \beta_{4} - 1450426 \beta_{3} + 14529034 \beta_{2} + 105558397 \beta_{1} + 318546212$$

## 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
 −9.70631 −8.57770 −6.91219 −4.38824 −1.48720 2.86024 3.50018 5.31531 9.86547 11.5305
−10.7063 0 82.6251 72.1865 0 −96.4803 −542.008 0 −772.851
1.2 −9.57770 0 59.7324 −28.1028 0 195.604 −265.613 0 269.160
1.3 −7.91219 0 30.6028 −79.5677 0 −172.354 11.0549 0 629.555
1.4 −5.38824 0 −2.96684 0.456695 0 166.517 188.410 0 −2.46078
1.5 −2.48720 0 −25.8138 −101.308 0 15.7005 143.795 0 251.974
1.6 1.86024 0 −28.5395 42.2365 0 202.971 −112.618 0 78.5700
1.7 2.50018 0 −25.7491 −47.4635 0 67.4603 −144.383 0 −118.667
1.8 4.31531 0 −13.3781 52.5837 0 −174.859 −195.821 0 226.915
1.9 8.86547 0 46.5966 37.8251 0 −124.747 129.406 0 335.337
1.10 10.5305 0 78.8905 −86.8464 0 −19.8137 493.778 0 −914.532
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 387.6.a.e 10
3.b odd 2 1 43.6.a.b 10
12.b even 2 1 688.6.a.h 10
15.d odd 2 1 1075.6.a.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b 10 3.b odd 2 1
387.6.a.e 10 1.a even 1 1 trivial
688.6.a.h 10 12.b even 2 1
1075.6.a.b 10 15.d odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(387))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T + 91 T^{2} + 570 T^{3} + 4178 T^{4} + 19212 T^{5} + 126300 T^{6} + 603512 T^{7} + 4655376 T^{8} + 24295776 T^{9} + 168985984 T^{10} + 777464832 T^{11} + 4767105024 T^{12} + 19775881216 T^{13} + 132435148800 T^{14} + 644647747584 T^{15} + 4486093340672 T^{16} + 19585050869760 T^{17} + 100055558127616 T^{18} + 281474976710656 T^{19} + 1125899906842624 T^{20}$$
$3$ 1
$5$ $$1 + 138 T + 21587 T^{2} + 2230608 T^{3} + 229459202 T^{4} + 19157670180 T^{5} + 1558167162738 T^{6} + 109829488059312 T^{7} + 7514260269046033 T^{8} + 457924168440376602 T^{9} + 27012319396517106654 T^{10} +$$$$14\!\cdots\!50$$$$T^{11} +$$$$73\!\cdots\!25$$$$T^{12} +$$$$33\!\cdots\!00$$$$T^{13} +$$$$14\!\cdots\!50$$$$T^{14} +$$$$57\!\cdots\!00$$$$T^{15} +$$$$21\!\cdots\!50$$$$T^{16} +$$$$64\!\cdots\!00$$$$T^{17} +$$$$19\!\cdots\!75$$$$T^{18} +$$$$39\!\cdots\!50$$$$T^{19} +$$$$88\!\cdots\!25$$$$T^{20}$$
$7$ $$1 - 60 T + 71106 T^{2} - 5672772 T^{3} + 2998880769 T^{4} - 247503266272 T^{5} + 92903828588736 T^{6} - 7232195163049824 T^{7} + 2203025939434531374 T^{8} -$$$$15\!\cdots\!56$$$$T^{9} +$$$$41\!\cdots\!04$$$$T^{10} -$$$$26\!\cdots\!92$$$$T^{11} +$$$$62\!\cdots\!26$$$$T^{12} -$$$$34\!\cdots\!32$$$$T^{13} +$$$$74\!\cdots\!36$$$$T^{14} -$$$$33\!\cdots\!04$$$$T^{15} +$$$$67\!\cdots\!81$$$$T^{16} -$$$$21\!\cdots\!96$$$$T^{17} +$$$$45\!\cdots\!06$$$$T^{18} -$$$$64\!\cdots\!20$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$
$11$ $$1 + 745 T + 1092200 T^{2} + 566790565 T^{3} + 495359905314 T^{4} + 194173124255379 T^{5} + 135383021483146674 T^{6} + 43098674076488688185 T^{7} +$$$$27\!\cdots\!81$$$$T^{8} +$$$$77\!\cdots\!58$$$$T^{9} +$$$$47\!\cdots\!16$$$$T^{10} +$$$$12\!\cdots\!58$$$$T^{11} +$$$$71\!\cdots\!81$$$$T^{12} +$$$$18\!\cdots\!35$$$$T^{13} +$$$$91\!\cdots\!74$$$$T^{14} +$$$$21\!\cdots\!29$$$$T^{15} +$$$$86\!\cdots\!14$$$$T^{16} +$$$$15\!\cdots\!15$$$$T^{17} +$$$$49\!\cdots\!00$$$$T^{18} +$$$$54\!\cdots\!95$$$$T^{19} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$1 - 1917 T + 3889346 T^{2} - 4848596699 T^{3} + 5963954609742 T^{4} - 5730569373162787 T^{5} + 5360926546177278684 T^{6} -$$$$42\!\cdots\!79$$$$T^{7} +$$$$32\!\cdots\!21$$$$T^{8} -$$$$21\!\cdots\!94$$$$T^{9} +$$$$14\!\cdots\!96$$$$T^{10} -$$$$81\!\cdots\!42$$$$T^{11} +$$$$44\!\cdots\!29$$$$T^{12} -$$$$21\!\cdots\!03$$$$T^{13} +$$$$10\!\cdots\!84$$$$T^{14} -$$$$40\!\cdots\!91$$$$T^{15} +$$$$15\!\cdots\!58$$$$T^{16} -$$$$47\!\cdots\!43$$$$T^{17} +$$$$14\!\cdots\!46$$$$T^{18} -$$$$25\!\cdots\!81$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20}$$
$17$ $$1 + 4017 T + 16731071 T^{2} + 43371481422 T^{3} + 109028323359989 T^{4} + 214433083727073801 T^{5} +$$$$40\!\cdots\!11$$$$T^{6} +$$$$64\!\cdots\!84$$$$T^{7} +$$$$98\!\cdots\!87$$$$T^{8} +$$$$13\!\cdots\!91$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!87$$$$T^{11} +$$$$19\!\cdots\!63$$$$T^{12} +$$$$18\!\cdots\!12$$$$T^{13} +$$$$16\!\cdots\!11$$$$T^{14} +$$$$12\!\cdots\!57$$$$T^{15} +$$$$89\!\cdots\!61$$$$T^{16} +$$$$50\!\cdots\!46$$$$T^{17} +$$$$27\!\cdots\!71$$$$T^{18} +$$$$94\!\cdots\!69$$$$T^{19} +$$$$33\!\cdots\!49$$$$T^{20}$$
$19$ $$1 + 2404 T + 15222557 T^{2} + 35388594664 T^{3} + 123780260921510 T^{4} + 260083719347815286 T^{5} +$$$$66\!\cdots\!54$$$$T^{6} +$$$$12\!\cdots\!42$$$$T^{7} +$$$$25\!\cdots\!37$$$$T^{8} +$$$$41\!\cdots\!16$$$$T^{9} +$$$$73\!\cdots\!66$$$$T^{10} +$$$$10\!\cdots\!84$$$$T^{11} +$$$$15\!\cdots\!37$$$$T^{12} +$$$$18\!\cdots\!58$$$$T^{13} +$$$$25\!\cdots\!54$$$$T^{14} +$$$$24\!\cdots\!14$$$$T^{15} +$$$$28\!\cdots\!10$$$$T^{16} +$$$$20\!\cdots\!36$$$$T^{17} +$$$$21\!\cdots\!57$$$$T^{18} +$$$$84\!\cdots\!96$$$$T^{19} +$$$$86\!\cdots\!01$$$$T^{20}$$
$23$ $$1 + 1733 T + 47557957 T^{2} + 72067471074 T^{3} + 1063199488621929 T^{4} + 1395811354288602239 T^{5} +$$$$14\!\cdots\!11$$$$T^{6} +$$$$16\!\cdots\!50$$$$T^{7} +$$$$14\!\cdots\!39$$$$T^{8} +$$$$14\!\cdots\!37$$$$T^{9} +$$$$10\!\cdots\!02$$$$T^{10} +$$$$93\!\cdots\!91$$$$T^{11} +$$$$61\!\cdots\!11$$$$T^{12} +$$$$45\!\cdots\!50$$$$T^{13} +$$$$25\!\cdots\!11$$$$T^{14} +$$$$15\!\cdots\!77$$$$T^{15} +$$$$75\!\cdots\!21$$$$T^{16} +$$$$32\!\cdots\!18$$$$T^{17} +$$$$14\!\cdots\!57$$$$T^{18} +$$$$32\!\cdots\!19$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20}$$
$29$ $$1 + 6996 T + 88174667 T^{2} + 354425388876 T^{3} + 2685110189553942 T^{4} + 3595496283776034360 T^{5} +$$$$29\!\cdots\!74$$$$T^{6} -$$$$15\!\cdots\!88$$$$T^{7} -$$$$13\!\cdots\!47$$$$T^{8} -$$$$63\!\cdots\!00$$$$T^{9} -$$$$82\!\cdots\!38$$$$T^{10} -$$$$13\!\cdots\!00$$$$T^{11} -$$$$58\!\cdots\!47$$$$T^{12} -$$$$13\!\cdots\!12$$$$T^{13} +$$$$52\!\cdots\!74$$$$T^{14} +$$$$13\!\cdots\!40$$$$T^{15} +$$$$19\!\cdots\!42$$$$T^{16} +$$$$54\!\cdots\!24$$$$T^{17} +$$$$27\!\cdots\!67$$$$T^{18} +$$$$44\!\cdots\!04$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$31$ $$1 + 4899 T + 169335305 T^{2} + 687234491898 T^{3} + 14314848605964873 T^{4} + 50115254632047551877 T^{5} +$$$$80\!\cdots\!39$$$$T^{6} +$$$$24\!\cdots\!98$$$$T^{7} +$$$$33\!\cdots\!11$$$$T^{8} +$$$$91\!\cdots\!15$$$$T^{9} +$$$$10\!\cdots\!46$$$$T^{10} +$$$$26\!\cdots\!65$$$$T^{11} +$$$$27\!\cdots\!11$$$$T^{12} +$$$$58\!\cdots\!98$$$$T^{13} +$$$$54\!\cdots\!39$$$$T^{14} +$$$$96\!\cdots\!27$$$$T^{15} +$$$$78\!\cdots\!73$$$$T^{16} +$$$$10\!\cdots\!98$$$$T^{17} +$$$$76\!\cdots\!05$$$$T^{18} +$$$$63\!\cdots\!49$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$37$ $$1 - 1466 T + 314595515 T^{2} - 1000159162152 T^{3} + 48977417173952218 T^{4} -$$$$27\!\cdots\!48$$$$T^{5} +$$$$50\!\cdots\!42$$$$T^{6} -$$$$41\!\cdots\!16$$$$T^{7} +$$$$40\!\cdots\!01$$$$T^{8} -$$$$41\!\cdots\!94$$$$T^{9} +$$$$29\!\cdots\!30$$$$T^{10} -$$$$28\!\cdots\!58$$$$T^{11} +$$$$19\!\cdots\!49$$$$T^{12} -$$$$13\!\cdots\!88$$$$T^{13} +$$$$11\!\cdots\!42$$$$T^{14} -$$$$43\!\cdots\!36$$$$T^{15} +$$$$54\!\cdots\!82$$$$T^{16} -$$$$77\!\cdots\!36$$$$T^{17} +$$$$16\!\cdots\!15$$$$T^{18} -$$$$54\!\cdots\!62$$$$T^{19} +$$$$25\!\cdots\!49$$$$T^{20}$$
$41$ $$1 + 10297 T + 512459663 T^{2} + 5996761727014 T^{3} + 151679237600092005 T^{4} +$$$$17\!\cdots\!49$$$$T^{5} +$$$$31\!\cdots\!39$$$$T^{6} +$$$$34\!\cdots\!68$$$$T^{7} +$$$$50\!\cdots\!19$$$$T^{8} +$$$$50\!\cdots\!39$$$$T^{9} +$$$$65\!\cdots\!96$$$$T^{10} +$$$$58\!\cdots\!39$$$$T^{11} +$$$$68\!\cdots\!19$$$$T^{12} +$$$$52\!\cdots\!68$$$$T^{13} +$$$$57\!\cdots\!39$$$$T^{14} +$$$$36\!\cdots\!49$$$$T^{15} +$$$$36\!\cdots\!05$$$$T^{16} +$$$$16\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!63$$$$T^{18} +$$$$38\!\cdots\!97$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20}$$
$43$ $$( 1 - 1849 T )^{10}$$
$47$ $$1 + 48592 T + 2515660075 T^{2} + 84810946501416 T^{3} + 2666904651640316270 T^{4} +$$$$68\!\cdots\!52$$$$T^{5} +$$$$16\!\cdots\!42$$$$T^{6} +$$$$33\!\cdots\!20$$$$T^{7} +$$$$64\!\cdots\!57$$$$T^{8} +$$$$11\!\cdots\!92$$$$T^{9} +$$$$17\!\cdots\!66$$$$T^{10} +$$$$25\!\cdots\!44$$$$T^{11} +$$$$34\!\cdots\!93$$$$T^{12} +$$$$40\!\cdots\!60$$$$T^{13} +$$$$45\!\cdots\!42$$$$T^{14} +$$$$43\!\cdots\!64$$$$T^{15} +$$$$38\!\cdots\!30$$$$T^{16} +$$$$28\!\cdots\!88$$$$T^{17} +$$$$19\!\cdots\!75$$$$T^{18} +$$$$85\!\cdots\!44$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$
$53$ $$1 + 127165 T + 9906770390 T^{2} + 550920294329671 T^{3} + 24581308628857110150 T^{4} +$$$$91\!\cdots\!35$$$$T^{5} +$$$$29\!\cdots\!20$$$$T^{6} +$$$$83\!\cdots\!31$$$$T^{7} +$$$$21\!\cdots\!05$$$$T^{8} +$$$$49\!\cdots\!78$$$$T^{9} +$$$$10\!\cdots\!76$$$$T^{10} +$$$$20\!\cdots\!54$$$$T^{11} +$$$$37\!\cdots\!45$$$$T^{12} +$$$$61\!\cdots\!67$$$$T^{13} +$$$$90\!\cdots\!20$$$$T^{14} +$$$$11\!\cdots\!55$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} +$$$$12\!\cdots\!47$$$$T^{17} +$$$$92\!\cdots\!90$$$$T^{18} +$$$$49\!\cdots\!45$$$$T^{19} +$$$$16\!\cdots\!49$$$$T^{20}$$
$59$ $$1 + 99372 T + 7563799766 T^{2} + 407898209580180 T^{3} + 19000827281933813957 T^{4} +$$$$74\!\cdots\!72$$$$T^{5} +$$$$26\!\cdots\!44$$$$T^{6} +$$$$87\!\cdots\!68$$$$T^{7} +$$$$26\!\cdots\!22$$$$T^{8} +$$$$76\!\cdots\!76$$$$T^{9} +$$$$20\!\cdots\!96$$$$T^{10} +$$$$54\!\cdots\!24$$$$T^{11} +$$$$13\!\cdots\!22$$$$T^{12} +$$$$31\!\cdots\!32$$$$T^{13} +$$$$70\!\cdots\!44$$$$T^{14} +$$$$13\!\cdots\!28$$$$T^{15} +$$$$25\!\cdots\!57$$$$T^{16} +$$$$38\!\cdots\!20$$$$T^{17} +$$$$51\!\cdots\!66$$$$T^{18} +$$$$48\!\cdots\!28$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 17408 T + 3107203610 T^{2} - 59824183442672 T^{3} + 5726081070022748105 T^{4} -$$$$10\!\cdots\!64$$$$T^{5} +$$$$76\!\cdots\!32$$$$T^{6} -$$$$13\!\cdots\!44$$$$T^{7} +$$$$84\!\cdots\!70$$$$T^{8} -$$$$13\!\cdots\!80$$$$T^{9} +$$$$77\!\cdots\!00$$$$T^{10} -$$$$11\!\cdots\!80$$$$T^{11} +$$$$60\!\cdots\!70$$$$T^{12} -$$$$78\!\cdots\!44$$$$T^{13} +$$$$38\!\cdots\!32$$$$T^{14} -$$$$44\!\cdots\!64$$$$T^{15} +$$$$20\!\cdots\!05$$$$T^{16} -$$$$18\!\cdots\!72$$$$T^{17} +$$$$80\!\cdots\!10$$$$T^{18} -$$$$38\!\cdots\!08$$$$T^{19} +$$$$18\!\cdots\!01$$$$T^{20}$$
$67$ $$1 + 2021 T + 3389673908 T^{2} - 95129046044495 T^{3} + 5951903574650911354 T^{4} -$$$$26\!\cdots\!45$$$$T^{5} +$$$$15\!\cdots\!46$$$$T^{6} -$$$$38\!\cdots\!75$$$$T^{7} +$$$$27\!\cdots\!93$$$$T^{8} -$$$$78\!\cdots\!74$$$$T^{9} +$$$$36\!\cdots\!28$$$$T^{10} -$$$$10\!\cdots\!18$$$$T^{11} +$$$$50\!\cdots\!57$$$$T^{12} -$$$$94\!\cdots\!25$$$$T^{13} +$$$$51\!\cdots\!46$$$$T^{14} -$$$$11\!\cdots\!15$$$$T^{15} +$$$$36\!\cdots\!46$$$$T^{16} -$$$$77\!\cdots\!85$$$$T^{17} +$$$$37\!\cdots\!08$$$$T^{18} +$$$$30\!\cdots\!47$$$$T^{19} +$$$$20\!\cdots\!49$$$$T^{20}$$
$71$ $$1 + 11286 T + 12540359910 T^{2} + 44404438984650 T^{3} + 71277357843109037437 T^{4} -$$$$33\!\cdots\!36$$$$T^{5} +$$$$24\!\cdots\!24$$$$T^{6} -$$$$30\!\cdots\!96$$$$T^{7} +$$$$61\!\cdots\!22$$$$T^{8} -$$$$10\!\cdots\!68$$$$T^{9} +$$$$12\!\cdots\!84$$$$T^{10} -$$$$18\!\cdots\!68$$$$T^{11} +$$$$20\!\cdots\!22$$$$T^{12} -$$$$17\!\cdots\!96$$$$T^{13} +$$$$26\!\cdots\!24$$$$T^{14} -$$$$63\!\cdots\!36$$$$T^{15} +$$$$24\!\cdots\!37$$$$T^{16} +$$$$27\!\cdots\!50$$$$T^{17} +$$$$14\!\cdots\!10$$$$T^{18} +$$$$22\!\cdots\!86$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$73$ $$1 - 49892 T + 9676771082 T^{2} - 435365481214596 T^{3} + 51788415419782211185 T^{4} -$$$$22\!\cdots\!88$$$$T^{5} +$$$$19\!\cdots\!88$$$$T^{6} -$$$$79\!\cdots\!24$$$$T^{7} +$$$$57\!\cdots\!14$$$$T^{8} -$$$$21\!\cdots\!32$$$$T^{9} +$$$$13\!\cdots\!60$$$$T^{10} -$$$$44\!\cdots\!76$$$$T^{11} +$$$$24\!\cdots\!86$$$$T^{12} -$$$$71\!\cdots\!68$$$$T^{13} +$$$$36\!\cdots\!88$$$$T^{14} -$$$$85\!\cdots\!84$$$$T^{15} +$$$$41\!\cdots\!65$$$$T^{16} -$$$$71\!\cdots\!72$$$$T^{17} +$$$$33\!\cdots\!82$$$$T^{18} -$$$$35\!\cdots\!56$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20}$$
$79$ $$1 + 91524 T + 16857154847 T^{2} + 842558519651240 T^{3} + 99200238774687370926 T^{4} +$$$$23\!\cdots\!36$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} +$$$$59\!\cdots\!56$$$$T^{7} +$$$$16\!\cdots\!17$$$$T^{8} +$$$$38\!\cdots\!04$$$$T^{9} +$$$$63\!\cdots\!18$$$$T^{10} +$$$$11\!\cdots\!96$$$$T^{11} +$$$$15\!\cdots\!17$$$$T^{12} +$$$$17\!\cdots\!44$$$$T^{13} +$$$$33\!\cdots\!90$$$$T^{14} +$$$$65\!\cdots\!64$$$$T^{15} +$$$$84\!\cdots\!26$$$$T^{16} +$$$$22\!\cdots\!60$$$$T^{17} +$$$$13\!\cdots\!47$$$$T^{18} +$$$$22\!\cdots\!76$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20}$$
$83$ $$1 - 105203 T + 28621889044 T^{2} - 2443253098090515 T^{3} +$$$$38\!\cdots\!90$$$$T^{4} -$$$$27\!\cdots\!61$$$$T^{5} +$$$$31\!\cdots\!66$$$$T^{6} -$$$$19\!\cdots\!11$$$$T^{7} +$$$$18\!\cdots\!57$$$$T^{8} -$$$$99\!\cdots\!22$$$$T^{9} +$$$$83\!\cdots\!32$$$$T^{10} -$$$$39\!\cdots\!46$$$$T^{11} +$$$$29\!\cdots\!93$$$$T^{12} -$$$$11\!\cdots\!77$$$$T^{13} +$$$$76\!\cdots\!66$$$$T^{14} -$$$$25\!\cdots\!23$$$$T^{15} +$$$$14\!\cdots\!10$$$$T^{16} -$$$$35\!\cdots\!05$$$$T^{17} +$$$$16\!\cdots\!44$$$$T^{18} -$$$$24\!\cdots\!29$$$$T^{19} +$$$$89\!\cdots\!49$$$$T^{20}$$
$89$ $$1 - 62682 T + 34400806898 T^{2} - 1591902274833594 T^{3} +$$$$57\!\cdots\!17$$$$T^{4} -$$$$20\!\cdots\!96$$$$T^{5} +$$$$63\!\cdots\!68$$$$T^{6} -$$$$16\!\cdots\!12$$$$T^{7} +$$$$50\!\cdots\!94$$$$T^{8} -$$$$10\!\cdots\!68$$$$T^{9} +$$$$31\!\cdots\!32$$$$T^{10} -$$$$60\!\cdots\!32$$$$T^{11} +$$$$15\!\cdots\!94$$$$T^{12} -$$$$29\!\cdots\!88$$$$T^{13} +$$$$61\!\cdots\!68$$$$T^{14} -$$$$10\!\cdots\!04$$$$T^{15} +$$$$17\!\cdots\!17$$$$T^{16} -$$$$26\!\cdots\!06$$$$T^{17} +$$$$32\!\cdots\!98$$$$T^{18} -$$$$33\!\cdots\!18$$$$T^{19} +$$$$29\!\cdots\!01$$$$T^{20}$$
$97$ $$1 - 108383 T + 55022053811 T^{2} - 4727913772355050 T^{3} +$$$$13\!\cdots\!17$$$$T^{4} -$$$$99\!\cdots\!31$$$$T^{5} +$$$$22\!\cdots\!35$$$$T^{6} -$$$$13\!\cdots\!96$$$$T^{7} +$$$$26\!\cdots\!83$$$$T^{8} -$$$$14\!\cdots\!33$$$$T^{9} +$$$$25\!\cdots\!28$$$$T^{10} -$$$$12\!\cdots\!81$$$$T^{11} +$$$$19\!\cdots\!67$$$$T^{12} -$$$$88\!\cdots\!28$$$$T^{13} +$$$$12\!\cdots\!35$$$$T^{14} -$$$$46\!\cdots\!67$$$$T^{15} +$$$$55\!\cdots\!33$$$$T^{16} -$$$$16\!\cdots\!50$$$$T^{17} +$$$$16\!\cdots\!11$$$$T^{18} -$$$$27\!\cdots\!31$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20}$$