# Properties

 Label 387.8.a.b Level $387$ Weight $8$ Character orbit 387.a Self dual yes Analytic conductor $120.893$ Analytic rank $0$ Dimension $11$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$387 = 3^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 387.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$120.893004862$$ Analytic rank: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}\cdot 7$$ 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 $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta_{1} ) q^{2} + ( 54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 68 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -4 + 12 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{7} + ( 338 + 19 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{8} +O(q^{10})$$ $$q + ( 2 + \beta_{1} ) q^{2} + ( 54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 68 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{5} + ( -4 + 12 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{7} + ( 338 + 19 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{8} + ( -162 + 194 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{10} + ( -94 - 145 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} - \beta_{9} - \beta_{10} ) q^{11} + ( -1641 + 93 \beta_{1} + 20 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} + 16 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + \beta_{10} ) q^{13} + ( 2042 - 43 \beta_{1} + 11 \beta_{2} + 36 \beta_{3} + 8 \beta_{4} + 17 \beta_{5} - 22 \beta_{6} + 6 \beta_{7} - 14 \beta_{8} + 7 \beta_{9} - 6 \beta_{10} ) q^{14} + ( -3271 + 623 \beta_{1} + 29 \beta_{2} + 29 \beta_{3} - 29 \beta_{4} - 15 \beta_{5} - 44 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} + 17 \beta_{9} + 8 \beta_{10} ) q^{16} + ( 5691 + 333 \beta_{1} + 71 \beta_{2} - 33 \beta_{3} + 47 \beta_{4} - 21 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 13 \beta_{8} - 15 \beta_{9} + 6 \beta_{10} ) q^{17} + ( -5047 + 279 \beta_{1} + 33 \beta_{2} - 7 \beta_{3} - 46 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} + 39 \beta_{7} - 4 \beta_{8} + 54 \beta_{9} + 2 \beta_{10} ) q^{19} + ( 25298 + 979 \beta_{1} + 196 \beta_{2} + 45 \beta_{3} - 51 \beta_{4} - 16 \beta_{5} - 58 \beta_{6} - 30 \beta_{7} + 16 \beta_{8} + 4 \beta_{9} - 27 \beta_{10} ) q^{20} + ( -25986 - 1424 \beta_{1} - 189 \beta_{2} - 25 \beta_{3} + 32 \beta_{4} + 101 \beta_{5} - \beta_{6} + 67 \beta_{7} + 44 \beta_{8} + 20 \beta_{10} ) q^{22} + ( 12158 + 1748 \beta_{1} + 4 \beta_{2} + 87 \beta_{3} - 135 \beta_{4} - 81 \beta_{5} + 57 \beta_{6} + 34 \beta_{7} - 53 \beta_{8} - 99 \beta_{9} + 19 \beta_{10} ) q^{23} + ( 423 - 1502 \beta_{1} + 193 \beta_{2} + 85 \beta_{3} - 147 \beta_{4} - 57 \beta_{5} - 20 \beta_{6} + 15 \beta_{7} + 106 \beta_{8} - 3 \beta_{9} - 85 \beta_{10} ) q^{25} + ( 12058 - 770 \beta_{1} - 349 \beta_{2} + 51 \beta_{3} - 284 \beta_{4} + 45 \beta_{5} + 219 \beta_{6} + 391 \beta_{7} + 180 \beta_{8} - 104 \beta_{9} + 28 \beta_{10} ) q^{26} + ( -1800 + 1826 \beta_{1} + 2 \beta_{2} + 282 \beta_{3} - 432 \beta_{4} - 82 \beta_{5} + 74 \beta_{6} + 150 \beta_{7} + 8 \beta_{8} + 20 \beta_{9} - 16 \beta_{10} ) q^{28} + ( 27822 + 1742 \beta_{1} - 154 \beta_{2} + 18 \beta_{3} + 66 \beta_{4} + 194 \beta_{5} - 211 \beta_{6} - 133 \beta_{7} + 69 \beta_{8} + 127 \beta_{9} + 51 \beta_{10} ) q^{29} + ( -19050 - 33 \beta_{1} + 295 \beta_{2} - 199 \beta_{3} - 112 \beta_{4} - 31 \beta_{5} + 180 \beta_{6} + 283 \beta_{7} - 98 \beta_{8} - 84 \beta_{9} - 26 \beta_{10} ) q^{31} + ( 58517 - 591 \beta_{1} + 943 \beta_{2} + 345 \beta_{3} + 175 \beta_{4} - 223 \beta_{5} - 424 \beta_{6} - 510 \beta_{7} + 67 \beta_{8} + 25 \beta_{9} - 160 \beta_{10} ) q^{32} + ( 67440 + 8202 \beta_{1} - 164 \beta_{2} - 89 \beta_{3} - 649 \beta_{4} + 4 \beta_{5} + 152 \beta_{6} + 70 \beta_{7} + 214 \beta_{8} + 272 \beta_{9} + 153 \beta_{10} ) q^{34} + ( 51984 + 364 \beta_{1} + 775 \beta_{2} - 66 \beta_{3} - 683 \beta_{4} - 534 \beta_{5} + 504 \beta_{6} + 220 \beta_{7} + 134 \beta_{8} - 385 \beta_{9} - 146 \beta_{10} ) q^{35} + ( -26532 + 96 \beta_{1} + 698 \beta_{2} - 663 \beta_{3} + 576 \beta_{4} - 579 \beta_{5} + 102 \beta_{6} - 1065 \beta_{7} - 660 \beta_{8} - 486 \beta_{9} - 111 \beta_{10} ) q^{37} + ( 33538 - 689 \beta_{1} + 1654 \beta_{2} - 111 \beta_{3} + 155 \beta_{4} - 238 \beta_{5} - 310 \beta_{6} - 120 \beta_{7} - 214 \beta_{8} + 516 \beta_{9} - 187 \beta_{10} ) q^{38} + ( 234168 + 23585 \beta_{1} + 944 \beta_{2} + 707 \beta_{3} - 943 \beta_{4} - 228 \beta_{5} - 780 \beta_{6} - 420 \beta_{7} + 106 \beta_{8} + 594 \beta_{9} + 79 \beta_{10} ) q^{40} + ( 120903 + 9334 \beta_{1} + 2539 \beta_{2} - 1377 \beta_{3} + 395 \beta_{4} + 69 \beta_{5} - 594 \beta_{6} - 489 \beta_{7} + 132 \beta_{8} + 633 \beta_{9} - 141 \beta_{10} ) q^{41} + 79507 q^{43} + ( -278759 - 28328 \beta_{1} - 1484 \beta_{2} - 603 \beta_{3} + 729 \beta_{4} + 585 \beta_{5} + 1338 \beta_{6} - 288 \beta_{7} + 29 \beta_{8} - 857 \beta_{9} - 44 \beta_{10} ) q^{44} + ( 322265 + 18745 \beta_{1} + 3253 \beta_{2} + 1376 \beta_{3} - 290 \beta_{4} + 333 \beta_{5} - 292 \beta_{6} - 108 \beta_{7} - 769 \beta_{8} - 687 \beta_{9} - 171 \beta_{10} ) q^{46} + ( -48052 + 11048 \beta_{1} + 611 \beta_{2} + 1209 \beta_{3} - 819 \beta_{4} - 81 \beta_{5} + 30 \beta_{6} - 259 \beta_{7} + 10 \beta_{8} - 839 \beta_{9} - 203 \beta_{10} ) q^{47} + ( 234325 - 18354 \beta_{1} - 1241 \beta_{2} - 360 \beta_{3} - 1623 \beta_{4} + 1844 \beta_{5} - 278 \beta_{6} + 604 \beta_{7} + 1316 \beta_{8} + 1047 \beta_{9} - 166 \beta_{10} ) q^{49} + ( -282810 + 29207 \beta_{1} + 1475 \beta_{2} + 2148 \beta_{3} + 603 \beta_{4} + 625 \beta_{5} - 2231 \beta_{6} + 331 \beta_{7} - 1292 \beta_{8} + 1246 \beta_{9} + 663 \beta_{10} ) q^{50} + ( 96069 - 21722 \beta_{1} - 554 \beta_{2} - 1075 \beta_{3} + 2121 \beta_{4} + 2305 \beta_{5} + 434 \beta_{6} - 104 \beta_{7} - 771 \beta_{8} - 1321 \beta_{9} - 316 \beta_{10} ) q^{52} + ( 195623 + 30275 \beta_{1} + 77 \beta_{2} - 1194 \beta_{3} + 791 \beta_{4} + 1130 \beta_{5} + 1361 \beta_{6} + 1413 \beta_{7} + 955 \beta_{8} + 817 \beta_{9} + 405 \beta_{10} ) q^{53} + ( -160351 - 40400 \beta_{1} - 912 \beta_{2} - 2061 \beta_{3} - 239 \beta_{4} - 1211 \beta_{5} + 2748 \beta_{6} - 1061 \beta_{7} - 680 \beta_{8} - 1846 \beta_{9} - 246 \beta_{10} ) q^{55} + ( 36198 + 31838 \beta_{1} + 7838 \beta_{2} - 1866 \beta_{3} + 230 \beta_{4} - 1970 \beta_{5} + 364 \beta_{6} - 1192 \beta_{7} - 702 \beta_{8} - 1254 \beta_{9} - 68 \beta_{10} ) q^{56} + ( 394443 + 19291 \beta_{1} + 1080 \beta_{2} + 270 \beta_{3} + 80 \beta_{4} - 1266 \beta_{5} - 366 \beta_{6} + 282 \beta_{7} + 1995 \beta_{8} - 312 \beta_{9} + 787 \beta_{10} ) q^{58} + ( 532654 - 5798 \beta_{1} - 1593 \beta_{2} - 2214 \beta_{3} - 159 \beta_{4} + 1992 \beta_{5} + 252 \beta_{6} - 3180 \beta_{7} + 1142 \beta_{8} + 67 \beta_{9} - 2308 \beta_{10} ) q^{59} + ( -410657 + 29014 \beta_{1} + 1770 \beta_{2} - 3693 \beta_{3} - 1919 \beta_{4} + 755 \beta_{5} - 724 \beta_{6} + 3455 \beta_{7} + 288 \beta_{8} + 592 \beta_{9} + 2462 \beta_{10} ) q^{61} + ( -89476 - 7398 \beta_{1} - 2384 \beta_{2} - 951 \beta_{3} - 2345 \beta_{4} + 1652 \beta_{5} + 2702 \beta_{6} + 3656 \beta_{7} - 158 \beta_{8} - 234 \beta_{9} - 1019 \beta_{10} ) q^{62} + ( 413609 + 82303 \beta_{1} + 465 \beta_{2} + 6079 \beta_{3} - 289 \beta_{4} - 2073 \beta_{5} - 1714 \beta_{6} - 736 \beta_{7} - 221 \beta_{8} + 3149 \beta_{9} + 1932 \beta_{10} ) q^{64} + ( 201273 + 4304 \beta_{1} + 280 \beta_{2} + 1683 \beta_{3} + 9523 \beta_{4} + 2467 \beta_{5} + 268 \beta_{6} + 2115 \beta_{7} - 3208 \beta_{8} - 1744 \beta_{9} - 642 \beta_{10} ) q^{65} + ( -623972 + 4139 \beta_{1} - 428 \beta_{2} - 3506 \beta_{3} + 655 \beta_{4} - 2496 \beta_{5} + 3896 \beta_{6} - 5066 \beta_{7} + 1658 \beta_{8} - 3551 \beta_{9} + 1133 \beta_{10} ) q^{67} + ( 844050 + 52208 \beta_{1} + 12327 \beta_{2} + 1773 \beta_{3} + 3943 \beta_{4} + 2820 \beta_{5} - 1752 \beta_{6} - 1364 \beta_{7} - 750 \beta_{8} + 2662 \beta_{9} + 1629 \beta_{10} ) q^{68} + ( 52991 + 144756 \beta_{1} + 9968 \beta_{2} + 3428 \beta_{3} + 4745 \beta_{4} + 4886 \beta_{5} - 3003 \beta_{6} + 1419 \beta_{7} - 6769 \beta_{8} + 2423 \beta_{9} + 2708 \beta_{10} ) q^{70} + ( 977207 - 20634 \beta_{1} + 1368 \beta_{2} - 5124 \beta_{3} - 1847 \beta_{4} - 3720 \beta_{5} + 1401 \beta_{6} + 5401 \beta_{7} + 2967 \beta_{8} - 3125 \beta_{9} - 450 \beta_{10} ) q^{71} + ( -431280 + 130632 \beta_{1} + 3046 \beta_{2} - 6159 \beta_{3} - 5554 \beta_{4} + 365 \beta_{5} - 3109 \beta_{6} + 476 \beta_{7} + 4245 \beta_{8} + 1885 \beta_{9} + 448 \beta_{10} ) q^{73} + ( -94203 - 17604 \beta_{1} - 15527 \beta_{2} + 4206 \beta_{3} - 752 \beta_{4} - 5591 \beta_{5} + 6076 \beta_{6} + 120 \beta_{7} - 375 \beta_{8} + 2401 \beta_{9} + 1513 \beta_{10} ) q^{74} + ( 504126 + 136291 \beta_{1} + 580 \beta_{2} + 6199 \beta_{3} - 1151 \beta_{4} - 4390 \beta_{5} - 3020 \beta_{6} - 3348 \beta_{7} - 160 \beta_{8} + 2408 \beta_{9} - 823 \beta_{10} ) q^{76} + ( 1563901 - 76358 \beta_{1} - 19592 \beta_{2} - 1563 \beta_{3} - 2753 \beta_{4} + 1097 \beta_{5} - 3910 \beta_{6} - 561 \beta_{7} + 4018 \beta_{8} + 4648 \beta_{9} - 5494 \beta_{10} ) q^{77} + ( -1422246 + 76915 \beta_{1} - 1928 \beta_{2} - 4229 \beta_{3} + 10952 \beta_{4} + 2359 \beta_{5} + 3949 \beta_{6} + 4546 \beta_{7} + 5513 \beta_{8} - 6520 \beta_{9} + 184 \beta_{10} ) q^{79} + ( 1362356 + 296359 \beta_{1} + 26644 \beta_{2} + 10269 \beta_{3} + 7095 \beta_{4} - 6764 \beta_{5} - 6884 \beta_{6} + 1028 \beta_{7} - 4174 \beta_{8} + 6934 \beta_{9} + 7365 \beta_{10} ) q^{80} + ( 1770172 + 306005 \beta_{1} + 4721 \beta_{2} + 1656 \beta_{3} - 4139 \beta_{4} - 6049 \beta_{5} + 859 \beta_{6} + 11537 \beta_{7} + 5578 \beta_{8} + 14206 \beta_{9} + 8079 \beta_{10} ) q^{82} + ( 1018726 - 33503 \beta_{1} - 3843 \beta_{2} + 4578 \beta_{3} - 12288 \beta_{4} - 1760 \beta_{5} + 406 \beta_{6} - 3494 \beta_{7} - 7478 \beta_{8} - 5702 \beta_{9} - 2383 \beta_{10} ) q^{83} + ( -1162954 + 183661 \beta_{1} + 14309 \beta_{2} - 1101 \beta_{3} + 6859 \beta_{4} + 1075 \beta_{5} - 7770 \beta_{6} - 5919 \beta_{7} - 4684 \beta_{8} - 1250 \beta_{9} + 109 \beta_{10} ) q^{85} + ( 159014 + 79507 \beta_{1} ) q^{86} + ( -2150865 - 293494 \beta_{1} - 35190 \beta_{2} - 15010 \beta_{3} + 10019 \beta_{4} - 4050 \beta_{5} + 18057 \beta_{6} - 6089 \beta_{7} - 7089 \beta_{8} - 14621 \beta_{9} - 3342 \beta_{10} ) q^{88} + ( 1267095 - 130482 \beta_{1} - 21278 \beta_{2} - 15147 \beta_{3} + 5923 \beta_{4} - 4871 \beta_{5} - 302 \beta_{6} - 18869 \beta_{7} + 3378 \beta_{8} - 5494 \beta_{9} - 8368 \beta_{10} ) q^{89} + ( -1839667 + 237088 \beta_{1} - 30714 \beta_{2} + 21447 \beta_{3} - 3921 \beta_{4} + 2473 \beta_{5} + 2656 \beta_{6} + 19615 \beta_{7} - 8700 \beta_{8} + 5816 \beta_{9} - 1614 \beta_{10} ) q^{91} + ( 2267979 + 436339 \beta_{1} + 12098 \beta_{2} + 19398 \beta_{3} - 16292 \beta_{4} + 3299 \beta_{5} - 8554 \beta_{6} + 13292 \beta_{7} + 8313 \beta_{8} + 7663 \beta_{9} - 4225 \beta_{10} ) q^{92} + ( 1818888 + 93596 \beta_{1} + 16393 \beta_{2} + 20320 \beta_{3} - 5507 \beta_{4} + 2055 \beta_{5} - 6841 \beta_{6} + 4049 \beta_{7} - 4916 \beta_{8} - 5842 \beta_{9} + 1693 \beta_{10} ) q^{94} + ( 1089010 + 129714 \beta_{1} - 8660 \beta_{2} + 10953 \beta_{3} - 1402 \beta_{4} - 3025 \beta_{5} - 5572 \beta_{6} - 4583 \beta_{7} + 2098 \beta_{8} + 218 \beta_{9} + 2705 \beta_{10} ) q^{95} + ( -992439 - 22346 \beta_{1} - 39602 \beta_{2} + 5411 \beta_{3} + 1776 \beta_{4} + 11903 \beta_{5} + 4606 \beta_{6} - 10783 \beta_{7} + 7790 \beta_{8} + 4774 \beta_{9} + 10525 \beta_{10} ) q^{97} + ( -2745065 + 203855 \beta_{1} - 12396 \beta_{2} - 6414 \beta_{3} + 20093 \beta_{4} + 6918 \beta_{5} + 1203 \beta_{6} + 18857 \beta_{7} + 3605 \beta_{8} - 5259 \beta_{9} + 3254 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q + 24q^{2} + 602q^{4} + 752q^{5} - 12q^{7} + 3810q^{8} + O(q^{10})$$ $$11q + 24q^{2} + 602q^{4} + 752q^{5} - 12q^{7} + 3810q^{8} - 1333q^{10} - 1333q^{11} - 17967q^{13} + 22352q^{14} - 34406q^{16} + 63095q^{17} - 54524q^{19} + 280995q^{20} - 289358q^{22} + 138139q^{23} + 3455q^{25} + 132946q^{26} - 12704q^{28} + 308658q^{29} - 209523q^{31} + 644934q^{32} + 762435q^{34} + 578892q^{35} - 298472q^{37} + 369707q^{38} + 2633173q^{40} + 1346735q^{41} + 874577q^{43} - 3134292q^{44} + 3588111q^{46} - 499284q^{47} + 2544563q^{49} - 3049745q^{50} + 983088q^{52} + 2210495q^{53} - 1855072q^{55} + 469976q^{56} + 4397067q^{58} + 5824216q^{59} - 4453034q^{61} - 1002789q^{62} + 4757538q^{64} + 2162872q^{65} - 6859513q^{67} + 9397005q^{68} + 845078q^{70} + 10726554q^{71} - 4456898q^{73} - 1046637q^{74} + 5861267q^{76} + 17019816q^{77} - 15541320q^{79} + 15680911q^{80} + 20233655q^{82} + 11146767q^{83} - 12471976q^{85} + 1908168q^{86} - 24463544q^{88} + 13531356q^{89} - 19746448q^{91} + 26023161q^{92} + 20288857q^{94} + 12291624q^{95} - 10999901q^{97} - 29909168q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 178$$ $$\beta_{3}$$ $$=$$ $$($$$$-1500599134629001 \nu^{10} - 39183746302943970 \nu^{9} + 664832435218620337 \nu^{8} + 24515803722851066612 \nu^{7} + 67209956153990217778 \nu^{6} - 3472321812967918657468 \nu^{5} - 52006805018854165868720 \nu^{4} - 86962422456018334411072 \nu^{3} + 5816603974274884482789248 \nu^{2} + 20251203606917738939518208 \nu - 152547096099166884427779072$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$2537638357898367 \nu^{10} + 3943244548083470 \nu^{9} - 1628799915307686999 \nu^{8} - 804903210724535564 \nu^{7} + 242092152331931307554 \nu^{6} - 453344375740036975004 \nu^{5} + 10430567452590387855440 \nu^{4} + 29643268429800372462784 \nu^{3} - 2974328985066183090142336 \nu^{2} + 6432997678553769789887744 \nu + 68998149783161691615417344$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$44082674675011 \nu^{10} + 868207465743318 \nu^{9} - 38489149606439611 \nu^{8} - 735871367912450396 \nu^{7} + 11771785592905208858 \nu^{6} + 203453192555128423636 \nu^{5} - 1573313378247863242480 \nu^{4} - 20534728206647225735744 \nu^{3} + 91159995442397982641536 \nu^{2} + 548442611133774445781248 \nu - 1859002308800284282174464$$$$)/$$$$24\!\cdots\!52$$ $$\beta_{6}$$ $$=$$ $$($$$$-6970668805319197 \nu^{10} - 58848026278326730 \nu^{9} + 5406184685771909989 \nu^{8} + 43770032756855928324 \nu^{7} - 1298843231045805189414 \nu^{6} - 9761906883295197975276 \nu^{5} + 89208792813294927465360 \nu^{4} + 658304812827375950986176 \nu^{3} + 3478856203926691302541696 \nu^{2} + 2670286972075641915863296 \nu - 251762078824321604521729024$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-9798468099969729 \nu^{10} - 135598692615217010 \nu^{9} + 8717368078423901673 \nu^{8} + 103828813201334493428 \nu^{7} - 2790356918408797448798 \nu^{6} - 25434543944531325289372 \nu^{5} + 404662938221299312555600 \nu^{4} + 2350790628670385956897472 \nu^{3} - 24842951078069978264709248 \nu^{2} - 67138267300992344115170048 \nu + 416612947090106170508833792$$$$)/$$$$27\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$3266643518058507 \nu^{10} + 44072850657173290 \nu^{9} - 2655475128449888139 \nu^{8} - 33154890484697645824 \nu^{7} + 721865558146059121354 \nu^{6} + 7872425702454351826796 \nu^{5} - 79521382930120278299360 \nu^{4} - 688305088482367009769536 \nu^{3} + 3237789793856720396293504 \nu^{2} + 16812743664771525034465024 \nu - 43714760928708310030678016$$$$)/$$$$34\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$-1847875868774081 \nu^{10} - 13025999644003570 \nu^{9} + 1634408559024771305 \nu^{8} + 10005406636137431924 \nu^{7} - 500526086076007585886 \nu^{6} - 2404703196261121554844 \nu^{5} + 63692959965362967409744 \nu^{4} + 203856762786007024154816 \nu^{3} - 3035229804891562332200064 \nu^{2} - 4200110409497461229513472 \nu + 36527953719388860818871296$$$$)/$$$$18\!\cdots\!32$$ $$\beta_{10}$$ $$=$$ $$($$$$24395717495229789 \nu^{10} + 242974029137269690 \nu^{9} - 20678531372967524613 \nu^{8} - 182317709313163411828 \nu^{7} + 6052126047627667082278 \nu^{6} + 42670881479478950332172 \nu^{5} - 754689293610768064087760 \nu^{4} - 3553201609758346553394112 \nu^{3} + 37619968029519414788702848 \nu^{2} + 70841626666134288569022208 \nu - 516049764394654432562250752$$$$)/$$$$13\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 178$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{10} + 2 \beta_{9} + 3 \beta_{7} - \beta_{6} - \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - \beta_{2} + 269 \beta_{1} - 226$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{10} + \beta_{9} - 5 \beta_{8} - 34 \beta_{7} - 36 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 397 \beta_{2} - 385 \beta_{1} + 47753$$ $$\nu^{5}$$ $$=$$ $$864 \beta_{10} + 959 \beta_{9} + 117 \beta_{8} + 1246 \beta_{7} - 536 \beta_{6} - 625 \beta_{5} - 1743 \beta_{4} + 1711 \beta_{3} - 507 \beta_{2} + 84147 \beta_{1} - 87349$$ $$\nu^{6}$$ $$=$$ $$-3156 \beta_{10} + 2141 \beta_{9} - 4525 \beta_{8} - 20528 \beta_{7} - 21122 \beta_{6} - 3593 \beta_{5} + 2527 \beta_{4} + 3327 \beta_{3} + 148665 \beta_{2} - 106265 \beta_{1} + 14944937$$ $$\nu^{7}$$ $$=$$ $$328892 \beta_{10} + 385245 \beta_{9} + 79803 \beta_{8} + 463972 \beta_{7} - 220158 \beta_{6} - 312245 \beta_{5} - 691945 \beta_{4} + 735667 \beta_{3} - 160303 \beta_{2} + 28561327 \beta_{1} - 25795951$$ $$\nu^{8}$$ $$=$$ $$-887636 \beta_{10} + 1548065 \beta_{9} - 2513553 \beta_{8} - 9235496 \beta_{7} - 9622802 \beta_{6} - 1382717 \beta_{5} + 1533875 \beta_{4} + 1698147 \beta_{3} + 55890457 \beta_{2} - 20175697 \beta_{1} + 5071030165$$ $$\nu^{9}$$ $$=$$ $$122206260 \beta_{10} + 148649593 \beta_{9} + 40920183 \beta_{8} + 170420984 \beta_{7} - 85222882 \beta_{6} - 138977653 \beta_{5} - 273133389 \beta_{4} + 292040643 \beta_{3} - 38747431 \beta_{2} + 10162237047 \beta_{1} - 6107186547$$ $$\nu^{10}$$ $$=$$ $$-190355732 \beta_{10} + 824440657 \beta_{9} - 1178473041 \beta_{8} - 3759830416 \beta_{7} - 4034912506 \beta_{6} - 499014901 \beta_{5} + 781352723 \beta_{4} + 802180395 \beta_{3} + 21161681769 \beta_{2} + 221717471 \beta_{1} + 1802621769685$$

## 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
 −19.3827 −17.2185 −14.3182 −10.3097 −5.14919 3.52766 6.31419 11.4671 13.1261 14.1572 19.7860
−17.3827 0 174.160 230.747 0 −1110.68 −802.384 0 −4011.02
1.2 −15.2185 0 103.603 537.911 0 1471.62 371.288 0 −8186.21
1.3 −12.3182 0 23.7379 −330.186 0 −467.397 1284.32 0 4067.29
1.4 −8.30971 0 −58.9487 187.284 0 −1501.21 1553.49 0 −1556.27
1.5 −3.14919 0 −118.083 39.2891 0 434.472 774.961 0 −123.729
1.6 5.52766 0 −97.4450 −304.619 0 1421.71 −1246.18 0 −1683.83
1.7 8.31419 0 −58.8742 −148.086 0 122.189 −1553.71 0 −1231.21
1.8 13.4671 0 53.3630 −131.138 0 −712.280 −1005.14 0 −1766.05
1.9 15.1261 0 100.799 −12.0619 0 −1248.65 −411.441 0 −182.450
1.10 16.1572 0 133.057 272.935 0 1101.65 81.7025 0 4409.88
1.11 21.7860 0 346.631 409.924 0 476.571 4763.10 0 8930.61
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.8.a.b 11
3.b odd 2 1 43.8.a.a 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.a.a 11 3.b odd 2 1
387.8.a.b 11 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{11} - \cdots$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(387))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$281015823360 + 26122731136 T - 25840429824 T^{2} - 34580064 T^{3} + 584457696 T^{4} - 13609592 T^{5} - 5061216 T^{6} + 169726 T^{7} + 18498 T^{8} - 717 T^{9} - 24 T^{10} + T^{11}$$
$3$ $$T^{11}$$
$5$ $$24\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T -$$$$48\!\cdots\!00$$$$T^{2} - 28565547925139138300 T^{3} + 459116654974581840 T^{4} + 1028996267258884 T^{5} - 15408511756684 T^{6} - 2680774591 T^{7} + 188879052 T^{8} - 148663 T^{9} - 752 T^{10} + T^{11}$$
$7$ $$40\!\cdots\!60$$$$-$$$$36\!\cdots\!96$$$$T -$$$$13\!\cdots\!20$$$$T^{2} +$$$$33\!\cdots\!56$$$$T^{3} -$$$$35\!\cdots\!44$$$$T^{4} - 10243901756898828448 T^{5} + 603608136785648 T^{6} + 11909995219748 T^{7} - 211679696 T^{8} - 5801696 T^{9} + 12 T^{10} + T^{11}$$
$11$ $$21\!\cdots\!12$$$$-$$$$65\!\cdots\!84$$$$T -$$$$42\!\cdots\!28$$$$T^{2} +$$$$17\!\cdots\!24$$$$T^{3} +$$$$13\!\cdots\!43$$$$T^{4} -$$$$53\!\cdots\!73$$$$T^{5} + 3087077307972469775 T^{6} + 4175185169015655 T^{7} - 129871555135 T^{8} - 113220383 T^{9} + 1333 T^{10} + T^{11}$$
$13$ $$43\!\cdots\!00$$$$-$$$$53\!\cdots\!60$$$$T -$$$$87\!\cdots\!04$$$$T^{2} -$$$$31\!\cdots\!28$$$$T^{3} -$$$$28\!\cdots\!15$$$$T^{4} +$$$$48\!\cdots\!47$$$$T^{5} +$$$$95\!\cdots\!69$$$$T^{6} + 5679855731653215 T^{7} - 7530949156773 T^{8} - 305242879 T^{9} + 17967 T^{10} + T^{11}$$
$17$ $$-$$$$77\!\cdots\!91$$$$-$$$$27\!\cdots\!55$$$$T +$$$$67\!\cdots\!26$$$$T^{2} +$$$$19\!\cdots\!40$$$$T^{3} -$$$$70\!\cdots\!65$$$$T^{4} +$$$$59\!\cdots\!89$$$$T^{5} -$$$$51\!\cdots\!43$$$$T^{6} - 1716664395658149193 T^{7} + 75891027680628 T^{8} + 81714758 T^{9} - 63095 T^{10} + T^{11}$$
$19$ $$19\!\cdots\!00$$$$+$$$$22\!\cdots\!00$$$$T +$$$$27\!\cdots\!20$$$$T^{2} -$$$$86\!\cdots\!04$$$$T^{3} -$$$$83\!\cdots\!64$$$$T^{4} +$$$$69\!\cdots\!88$$$$T^{5} +$$$$77\!\cdots\!14$$$$T^{6} + 143984052477338765 T^{7} - 129983407666036 T^{8} - 1963257117 T^{9} + 54524 T^{10} + T^{11}$$
$23$ $$90\!\cdots\!15$$$$+$$$$22\!\cdots\!79$$$$T -$$$$11\!\cdots\!14$$$$T^{2} -$$$$52\!\cdots\!18$$$$T^{3} +$$$$63\!\cdots\!67$$$$T^{4} +$$$$85\!\cdots\!99$$$$T^{5} -$$$$65\!\cdots\!75$$$$T^{6} + 3163769113048920469 T^{7} + 1850372016393690 T^{8} - 9940477486 T^{9} - 138139 T^{10} + T^{11}$$
$29$ $$19\!\cdots\!00$$$$+$$$$37\!\cdots\!20$$$$T -$$$$23\!\cdots\!76$$$$T^{2} -$$$$20\!\cdots\!32$$$$T^{3} +$$$$89\!\cdots\!36$$$$T^{4} +$$$$33\!\cdots\!28$$$$T^{5} -$$$$14\!\cdots\!50$$$$T^{6} - 74903092533507315195 T^{7} + 11129264495493282 T^{8} - 19042918263 T^{9} - 308658 T^{10} + T^{11}$$
$31$ $$21\!\cdots\!45$$$$-$$$$40\!\cdots\!13$$$$T +$$$$12\!\cdots\!10$$$$T^{2} +$$$$58\!\cdots\!30$$$$T^{3} -$$$$18\!\cdots\!47$$$$T^{4} -$$$$14\!\cdots\!49$$$$T^{5} +$$$$31\!\cdots\!79$$$$T^{6} +$$$$15\!\cdots\!93$$$$T^{7} - 16437622252566070 T^{8} - 76915247122 T^{9} + 209523 T^{10} + T^{11}$$
$37$ $$13\!\cdots\!00$$$$+$$$$38\!\cdots\!00$$$$T +$$$$29\!\cdots\!60$$$$T^{2} +$$$$55\!\cdots\!76$$$$T^{3} -$$$$71\!\cdots\!44$$$$T^{4} -$$$$13\!\cdots\!64$$$$T^{5} +$$$$60\!\cdots\!84$$$$T^{6} +$$$$16\!\cdots\!97$$$$T^{7} - 222373841681655188 T^{8} - 678107201451 T^{9} + 298472 T^{10} + T^{11}$$
$41$ $$-$$$$14\!\cdots\!75$$$$+$$$$50\!\cdots\!05$$$$T -$$$$39\!\cdots\!38$$$$T^{2} -$$$$93\!\cdots\!04$$$$T^{3} +$$$$79\!\cdots\!07$$$$T^{4} +$$$$42\!\cdots\!13$$$$T^{5} -$$$$53\!\cdots\!95$$$$T^{6} +$$$$71\!\cdots\!27$$$$T^{7} + 1449544017958490252 T^{8} - 736254337058 T^{9} - 1346735 T^{10} + T^{11}$$
$43$ $$( -79507 + T )^{11}$$
$47$ $$61\!\cdots\!00$$$$+$$$$86\!\cdots\!92$$$$T +$$$$32\!\cdots\!20$$$$T^{2} -$$$$69\!\cdots\!72$$$$T^{3} -$$$$64\!\cdots\!20$$$$T^{4} -$$$$57\!\cdots\!76$$$$T^{5} +$$$$33\!\cdots\!44$$$$T^{6} +$$$$53\!\cdots\!73$$$$T^{7} - 666099867725593932 T^{8} - 1329581302839 T^{9} + 499284 T^{10} + T^{11}$$
$53$ $$15\!\cdots\!00$$$$+$$$$13\!\cdots\!60$$$$T -$$$$10\!\cdots\!64$$$$T^{2} -$$$$93\!\cdots\!48$$$$T^{3} +$$$$12\!\cdots\!51$$$$T^{4} -$$$$28\!\cdots\!41$$$$T^{5} -$$$$45\!\cdots\!57$$$$T^{6} +$$$$19\!\cdots\!43$$$$T^{7} + 5915591359309490873 T^{8} - 2481662094443 T^{9} - 2210495 T^{10} + T^{11}$$
$59$ $$47\!\cdots\!00$$$$-$$$$24\!\cdots\!80$$$$T +$$$$42\!\cdots\!28$$$$T^{2} -$$$$21\!\cdots\!04$$$$T^{3} -$$$$25\!\cdots\!76$$$$T^{4} +$$$$34\!\cdots\!96$$$$T^{5} -$$$$64\!\cdots\!68$$$$T^{6} -$$$$96\!\cdots\!64$$$$T^{7} + 50551309640986651344 T^{8} + 1718681997224 T^{9} - 5824216 T^{10} + T^{11}$$
$61$ $$46\!\cdots\!80$$$$+$$$$81\!\cdots\!72$$$$T +$$$$14\!\cdots\!40$$$$T^{2} +$$$$89\!\cdots\!96$$$$T^{3} +$$$$24\!\cdots\!60$$$$T^{4} +$$$$30\!\cdots\!20$$$$T^{5} +$$$$13\!\cdots\!28$$$$T^{6} -$$$$45\!\cdots\!56$$$$T^{7} - 55856950104763679848 T^{8} - 7073916965796 T^{9} + 4453034 T^{10} + T^{11}$$
$67$ $$-$$$$63\!\cdots\!64$$$$-$$$$23\!\cdots\!40$$$$T -$$$$31\!\cdots\!76$$$$T^{2} -$$$$16\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!19$$$$T^{4} +$$$$49\!\cdots\!55$$$$T^{5} +$$$$14\!\cdots\!71$$$$T^{6} -$$$$28\!\cdots\!77$$$$T^{7} -$$$$19\!\cdots\!23$$$$T^{8} - 12368696822019 T^{9} + 6859513 T^{10} + T^{11}$$
$71$ $$-$$$$11\!\cdots\!56$$$$+$$$$54\!\cdots\!36$$$$T +$$$$35\!\cdots\!28$$$$T^{2} -$$$$93\!\cdots\!16$$$$T^{3} -$$$$17\!\cdots\!92$$$$T^{4} +$$$$56\!\cdots\!52$$$$T^{5} -$$$$94\!\cdots\!80$$$$T^{6} -$$$$77\!\cdots\!08$$$$T^{7} +$$$$25\!\cdots\!44$$$$T^{8} + 8611552441100 T^{9} - 10726554 T^{10} + T^{11}$$
$73$ $$-$$$$53\!\cdots\!28$$$$-$$$$29\!\cdots\!76$$$$T +$$$$16\!\cdots\!72$$$$T^{2} +$$$$61\!\cdots\!80$$$$T^{3} -$$$$14\!\cdots\!24$$$$T^{4} -$$$$42\!\cdots\!40$$$$T^{5} +$$$$27\!\cdots\!92$$$$T^{6} +$$$$74\!\cdots\!92$$$$T^{7} -$$$$19\!\cdots\!36$$$$T^{8} - 47269704355236 T^{9} + 4456898 T^{10} + T^{11}$$
$79$ $$65\!\cdots\!60$$$$+$$$$13\!\cdots\!88$$$$T -$$$$45\!\cdots\!92$$$$T^{2} -$$$$68\!\cdots\!64$$$$T^{3} -$$$$16\!\cdots\!04$$$$T^{4} +$$$$10\!\cdots\!24$$$$T^{5} +$$$$15\!\cdots\!92$$$$T^{6} -$$$$47\!\cdots\!07$$$$T^{7} -$$$$11\!\cdots\!16$$$$T^{8} - 9338192974147 T^{9} + 15541320 T^{10} + T^{11}$$
$83$ $$62\!\cdots\!80$$$$+$$$$19\!\cdots\!88$$$$T -$$$$10\!\cdots\!48$$$$T^{2} -$$$$75\!\cdots\!12$$$$T^{3} +$$$$38\!\cdots\!83$$$$T^{4} +$$$$36\!\cdots\!67$$$$T^{5} -$$$$32\!\cdots\!41$$$$T^{6} +$$$$69\!\cdots\!95$$$$T^{7} +$$$$10\!\cdots\!21$$$$T^{8} - 67946821803891 T^{9} - 11146767 T^{10} + T^{11}$$
$89$ $$-$$$$11\!\cdots\!00$$$$+$$$$17\!\cdots\!68$$$$T -$$$$55\!\cdots\!32$$$$T^{2} -$$$$11\!\cdots\!16$$$$T^{3} +$$$$58\!\cdots\!76$$$$T^{4} +$$$$66\!\cdots\!48$$$$T^{5} -$$$$20\!\cdots\!28$$$$T^{6} +$$$$81\!\cdots\!64$$$$T^{7} +$$$$28\!\cdots\!56$$$$T^{8} - 175477347945756 T^{9} - 13531356 T^{10} + T^{11}$$
$97$ $$-$$$$28\!\cdots\!07$$$$+$$$$25\!\cdots\!57$$$$T +$$$$10\!\cdots\!42$$$$T^{2} -$$$$11\!\cdots\!48$$$$T^{3} -$$$$65\!\cdots\!69$$$$T^{4} -$$$$19\!\cdots\!15$$$$T^{5} +$$$$10\!\cdots\!77$$$$T^{6} +$$$$64\!\cdots\!55$$$$T^{7} -$$$$58\!\cdots\!00$$$$T^{8} - 465260564352906 T^{9} + 10999901 T^{10} + T^{11}$$