Properties

Label 43.6.a.a
Level $43$
Weight $6$
Character orbit 43.a
Self dual yes
Analytic conductor $6.897$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -2 - \beta_{1} + \beta_{4} - \beta_{7} ) q^{3} + ( 15 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{4} + ( -28 - \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{5} + ( -5 - 10 \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{6} + ( -10 - 13 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -82 + 7 \beta_{1} + 5 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} - 6 \beta_{7} ) q^{8} + ( 62 - 8 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} - 15 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -2 - \beta_{1} + \beta_{4} - \beta_{7} ) q^{3} + ( 15 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{4} + ( -28 - \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{5} + ( -5 - 10 \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{6} + ( -10 - 13 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -82 + 7 \beta_{1} + 5 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} - 6 \beta_{7} ) q^{8} + ( 62 - 8 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} - 15 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} + ( -59 - 35 \beta_{1} - 13 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} ) q^{10} + ( -74 + 28 \beta_{1} + 19 \beta_{2} - 9 \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{11} + ( -499 + 10 \beta_{1} + 19 \beta_{3} + 52 \beta_{4} - 3 \beta_{5} + 7 \beta_{6} - 20 \beta_{7} ) q^{12} + ( -292 - 4 \beta_{1} + \beta_{2} + 13 \beta_{3} + 24 \beta_{4} + 2 \beta_{5} - 19 \beta_{6} - \beta_{7} ) q^{13} + ( -534 + 24 \beta_{1} - 8 \beta_{2} - 24 \beta_{3} + 36 \beta_{4} - 18 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{14} + ( -297 + 90 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 54 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 33 \beta_{7} ) q^{15} + ( 226 - 53 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} - 88 \beta_{4} - 29 \beta_{5} - 36 \beta_{6} + 22 \beta_{7} ) q^{16} + ( -419 + 109 \beta_{1} - 17 \beta_{2} - 12 \beta_{3} - 67 \beta_{4} - 8 \beta_{5} + 9 \beta_{6} + 32 \beta_{7} ) q^{17} + ( -513 + 182 \beta_{1} + 71 \beta_{2} + 40 \beta_{3} + 50 \beta_{4} + 12 \beta_{5} + 63 \beta_{6} - 10 \beta_{7} ) q^{18} + ( -284 - 17 \beta_{1} - 105 \beta_{2} - 62 \beta_{3} - 9 \beta_{4} + 64 \beta_{5} + 9 \beta_{6} + 38 \beta_{7} ) q^{19} + ( -275 - 33 \beta_{1} + 5 \beta_{2} - 40 \beta_{3} + 92 \beta_{4} - 16 \beta_{5} - 15 \beta_{6} - 6 \beta_{7} ) q^{20} + ( -351 + 189 \beta_{1} + 65 \beta_{2} + 87 \beta_{3} - 4 \beta_{4} + 24 \beta_{5} + 30 \beta_{6} + 23 \beta_{7} ) q^{21} + ( 1542 - 84 \beta_{1} + 107 \beta_{2} + 37 \beta_{3} - 8 \beta_{4} + 57 \beta_{5} + 22 \beta_{6} - 76 \beta_{7} ) q^{22} + ( -329 - 2 \beta_{1} - 21 \beta_{2} + 112 \beta_{3} - 125 \beta_{4} - 70 \beta_{5} - 26 \beta_{6} + 20 \beta_{7} ) q^{23} + ( 2531 - 488 \beta_{1} - 190 \beta_{2} - 103 \beta_{3} - 268 \beta_{4} + 25 \beta_{5} - 109 \beta_{6} + 90 \beta_{7} ) q^{24} + ( 750 - 41 \beta_{1} + 158 \beta_{2} - 21 \beta_{3} + 221 \beta_{4} + 34 \beta_{5} - 26 \beta_{6} - 115 \beta_{7} ) q^{25} + ( 830 - 394 \beta_{1} - 55 \beta_{2} - 65 \beta_{3} + 168 \beta_{4} - 29 \beta_{5} - 54 \beta_{6} - 44 \beta_{7} ) q^{26} + ( -1216 + 125 \beta_{1} - 98 \beta_{2} - 74 \beta_{3} + 150 \beta_{4} + 6 \beta_{5} + 93 \beta_{6} + 28 \beta_{7} ) q^{27} + ( 2668 - 678 \beta_{1} - 102 \beta_{2} + 50 \beta_{3} - 56 \beta_{4} - 30 \beta_{5} - 108 \beta_{6} - 40 \beta_{7} ) q^{28} + ( -492 + 103 \beta_{1} + 52 \beta_{2} + 54 \beta_{3} + 19 \beta_{4} - 122 \beta_{5} + 12 \beta_{6} - 139 \beta_{7} ) q^{29} + ( 3014 + 135 \beta_{1} + 65 \beta_{2} - 19 \beta_{3} - 76 \beta_{4} + 137 \beta_{5} + 176 \beta_{6} - 40 \beta_{7} ) q^{30} + ( 631 + 137 \beta_{1} + 155 \beta_{2} - 140 \beta_{3} - 83 \beta_{4} - 202 \beta_{5} + 39 \beta_{6} + 56 \beta_{7} ) q^{31} + ( -1964 + 241 \beta_{1} + 177 \beta_{2} + 247 \beta_{3} + 692 \beta_{4} - 49 \beta_{5} + 108 \beta_{6} - 204 \beta_{7} ) q^{32} + ( -1582 - 61 \beta_{1} - 161 \beta_{2} - 34 \beta_{3} + 164 \beta_{4} - 118 \beta_{5} - 146 \beta_{6} + 133 \beta_{7} ) q^{33} + ( 3749 - 92 \beta_{1} + 61 \beta_{2} + 128 \beta_{3} - 82 \beta_{4} + 172 \beta_{5} + 193 \beta_{6} + 6 \beta_{7} ) q^{34} + ( 323 + 539 \beta_{1} + 13 \beta_{2} - 323 \beta_{3} - 60 \beta_{4} + 56 \beta_{5} + 72 \beta_{6} - 29 \beta_{7} ) q^{35} + ( 7982 - 33 \beta_{1} - 5 \beta_{2} - 120 \beta_{3} - 1070 \beta_{4} - 91 \beta_{5} - 265 \beta_{6} + 467 \beta_{7} ) q^{36} + ( -176 - 422 \beta_{1} - 307 \beta_{2} + 114 \beta_{3} + 217 \beta_{4} + 280 \beta_{5} + 98 \beta_{6} - 398 \beta_{7} ) q^{37} + ( -739 - 223 \beta_{1} - 375 \beta_{2} + 378 \beta_{3} + 122 \beta_{4} + 142 \beta_{5} - 61 \beta_{6} + 74 \beta_{7} ) q^{38} + ( 778 + 1023 \beta_{1} + 445 \beta_{2} + 242 \beta_{3} - 418 \beta_{4} - 2 \beta_{5} - 30 \beta_{6} + 477 \beta_{7} ) q^{39} + ( 2189 - 149 \beta_{1} + 201 \beta_{2} + 84 \beta_{3} + 336 \beta_{4} - 104 \beta_{5} - 247 \beta_{6} + 110 \beta_{7} ) q^{40} + ( -1208 - 295 \beta_{1} - 274 \beta_{2} - 173 \beta_{3} - \beta_{4} + 138 \beta_{5} - 214 \beta_{6} - 245 \beta_{7} ) q^{41} + ( 8790 + 576 \beta_{1} - 56 \beta_{2} - 642 \beta_{3} - 1140 \beta_{4} + 150 \beta_{5} + 22 \beta_{6} + 336 \beta_{7} ) q^{42} -1849 q^{43} + ( -1275 + 1667 \beta_{1} + 73 \beta_{2} - 253 \beta_{3} - 538 \beta_{4} - 82 \beta_{5} + 260 \beta_{6} + 279 \beta_{7} ) q^{44} + ( -5010 - 232 \beta_{1} + 49 \beta_{2} + 404 \beta_{3} + 546 \beta_{4} - 26 \beta_{5} - 437 \beta_{6} - 137 \beta_{7} ) q^{45} + ( -2729 + 407 \beta_{1} + 120 \beta_{2} - 465 \beta_{3} + 350 \beta_{4} - 277 \beta_{5} + 441 \beta_{6} + 230 \beta_{7} ) q^{46} + ( -9863 + 439 \beta_{1} + 8 \beta_{2} + 193 \beta_{3} - 25 \beta_{4} - 300 \beta_{5} + 108 \beta_{6} - 99 \beta_{7} ) q^{47} + ( -15453 + 3208 \beta_{1} + 510 \beta_{2} + 349 \beta_{3} + 1916 \beta_{4} - 171 \beta_{5} + 811 \beta_{6} - 470 \beta_{7} ) q^{48} + ( 1696 - 1383 \beta_{1} - 61 \beta_{2} + 205 \beta_{3} - 254 \beta_{4} + 516 \beta_{5} + 162 \beta_{6} - 741 \beta_{7} ) q^{49} + ( 3741 - 263 \beta_{1} + 362 \beta_{2} - 189 \beta_{3} - 210 \beta_{4} - 37 \beta_{5} - 525 \beta_{6} - 278 \beta_{7} ) q^{50} + ( -9866 - 398 \beta_{1} - 80 \beta_{2} - 54 \beta_{3} + 269 \beta_{4} - 482 \beta_{5} - 21 \beta_{6} + 419 \beta_{7} ) q^{51} + ( -6723 - 1067 \beta_{1} - 485 \beta_{2} + 97 \beta_{3} + 670 \beta_{4} - 500 \beta_{5} + 4 \beta_{6} - 481 \beta_{7} ) q^{52} + ( -6787 - 1303 \beta_{1} - 48 \beta_{2} - 100 \beta_{3} + 568 \beta_{4} + 406 \beta_{5} - 423 \beta_{6} - 348 \beta_{7} ) q^{53} + ( 8084 - 2755 \beta_{1} - 1229 \beta_{2} + 465 \beta_{3} - 1512 \beta_{4} + 237 \beta_{5} - 906 \beta_{6} + 552 \beta_{7} ) q^{54} + ( -5326 - 2591 \beta_{1} - 1005 \beta_{2} + 516 \beta_{3} - 308 \beta_{4} + 322 \beta_{5} + 244 \beta_{6} - 99 \beta_{7} ) q^{55} + ( -17948 + 1818 \beta_{1} + 426 \beta_{2} + 698 \beta_{3} + 1608 \beta_{4} - 414 \beta_{5} + 220 \beta_{6} - 248 \beta_{7} ) q^{56} + ( -112 - 17 \beta_{1} + 342 \beta_{2} - 1038 \beta_{3} - 84 \beta_{4} + 350 \beta_{5} + 617 \beta_{6} - 910 \beta_{7} ) q^{57} + ( 6943 - 1424 \beta_{1} + 422 \beta_{2} - 283 \beta_{3} - 662 \beta_{4} - 353 \beta_{5} - 31 \beta_{6} + 744 \beta_{7} ) q^{58} + ( -3531 + 815 \beta_{1} + 1163 \beta_{2} - 445 \beta_{3} + 226 \beta_{4} + 728 \beta_{5} + 388 \beta_{6} + 179 \beta_{7} ) q^{59} + ( 11054 + 2135 \beta_{1} + 275 \beta_{2} - 729 \beta_{3} - 1132 \beta_{4} + 423 \beta_{5} + 98 \beta_{6} - 40 \beta_{7} ) q^{60} + ( -8856 - 1315 \beta_{1} + 1067 \beta_{2} - 296 \beta_{3} + 1356 \beta_{4} - 646 \beta_{5} + 204 \beta_{6} + 161 \beta_{7} ) q^{61} + ( 339 - 746 \beta_{1} + 689 \beta_{2} + 1106 \beta_{3} + 422 \beta_{4} + 482 \beta_{5} + 151 \beta_{6} - 1330 \beta_{7} ) q^{62} + ( -7702 - 877 \beta_{1} - 97 \beta_{2} + 484 \beta_{3} - 894 \beta_{4} - 1414 \beta_{5} - 1076 \beta_{6} + 1639 \beta_{7} ) q^{63} + ( 19272 - 3883 \beta_{1} - 1823 \beta_{2} - 1393 \beta_{3} - 1860 \beta_{4} + 395 \beta_{5} - 1500 \beta_{6} + 1328 \beta_{7} ) q^{64} + ( -2522 + 211 \beta_{1} - 1247 \beta_{2} - 796 \beta_{3} - 2214 \beta_{4} + 1154 \beta_{5} + 1134 \beta_{6} - 355 \beta_{7} ) q^{65} + ( -2634 - 4458 \beta_{1} - 1444 \beta_{2} + 880 \beta_{3} + 2476 \beta_{4} - 6 \beta_{5} - 776 \beta_{6} - 1066 \beta_{7} ) q^{66} + ( 3110 + 50 \beta_{1} + 1093 \beta_{2} + 77 \beta_{3} - 948 \beta_{4} - 874 \beta_{5} + 159 \beta_{6} + 1283 \beta_{7} ) q^{67} + ( 2796 + 3405 \beta_{1} + 429 \beta_{2} - 976 \beta_{3} - 1054 \beta_{4} + 153 \beta_{5} - 119 \beta_{6} + 259 \beta_{7} ) q^{68} + ( -12360 + 1856 \beta_{1} + 963 \beta_{2} + 1728 \beta_{3} + 142 \beta_{4} - 578 \beta_{5} + 449 \beta_{6} + 1377 \beta_{7} ) q^{69} + ( 23722 - 286 \beta_{1} + 894 \beta_{2} + 1712 \beta_{3} - 844 \beta_{4} + 1196 \beta_{5} + 194 \beta_{6} - 244 \beta_{7} ) q^{70} + ( -2506 - 838 \beta_{1} - 1682 \beta_{2} + 816 \beta_{3} - 1340 \beta_{4} + 1644 \beta_{5} + 612 \beta_{6} + 1198 \beta_{7} ) q^{71} + ( -24007 + 8416 \beta_{1} + 1064 \beta_{2} + 359 \beta_{3} + 5912 \beta_{4} + 661 \beta_{5} + 2231 \beta_{6} - 3924 \beta_{7} ) q^{72} + ( 780 + 1677 \beta_{1} - 65 \beta_{2} - 280 \beta_{3} - 908 \beta_{4} - 714 \beta_{5} - 32 \beta_{6} - 463 \beta_{7} ) q^{73} + ( -4821 + 174 \beta_{1} - 746 \beta_{2} - 1433 \beta_{3} - 2774 \beta_{4} - 1473 \beta_{5} - 693 \beta_{6} + 4022 \beta_{7} ) q^{74} + ( 21510 - 1282 \beta_{1} - 787 \beta_{2} - 276 \beta_{3} - 134 \beta_{4} + 472 \beta_{5} - 1295 \beta_{6} - 243 \beta_{7} ) q^{75} + ( 9 + 1007 \beta_{1} + 645 \beta_{2} - 58 \beta_{3} - 304 \beta_{4} - 3112 \beta_{5} - 961 \beta_{6} + 1040 \beta_{7} ) q^{76} + ( -28556 + 1371 \beta_{1} - 907 \beta_{2} - 1714 \beta_{3} + 544 \beta_{4} - 1842 \beta_{5} - 328 \beta_{6} + 935 \beta_{7} ) q^{77} + ( 29672 + 6678 \beta_{1} + 1148 \beta_{2} - 1386 \beta_{3} - 800 \beta_{4} + 1936 \beta_{5} + 1698 \beta_{6} - 2326 \beta_{7} ) q^{78} + ( 19609 + 1226 \beta_{1} + 60 \beta_{2} - 639 \beta_{3} + 1839 \beta_{4} + 1676 \beta_{5} + 195 \beta_{6} + 275 \beta_{7} ) q^{79} + ( 437 + 843 \beta_{1} - 671 \beta_{2} + 1140 \beta_{3} + 232 \beta_{4} + 512 \beta_{5} - 383 \beta_{6} - 2314 \beta_{7} ) q^{80} + ( 20406 - 1448 \beta_{1} + 249 \beta_{2} - 467 \beta_{3} - 1522 \beta_{4} - 1210 \beta_{5} - 701 \beta_{6} + 493 \beta_{7} ) q^{81} + ( -3185 - 2105 \beta_{1} + 1014 \beta_{2} + 1005 \beta_{3} + 3514 \beta_{4} - 575 \beta_{5} + 541 \beta_{6} + 402 \beta_{7} ) q^{82} + ( -7731 - 1607 \beta_{1} - 502 \beta_{2} + 884 \beta_{3} + 384 \beta_{4} - 2078 \beta_{5} - 1885 \beta_{6} - 546 \beta_{7} ) q^{83} + ( 530 + 9052 \beta_{1} + 2280 \beta_{2} + 1138 \beta_{3} + 3888 \beta_{4} + 1858 \beta_{5} + 3314 \beta_{6} - 3924 \beta_{7} ) q^{84} + ( 23712 - 4392 \beta_{1} + 151 \beta_{2} + 1274 \beta_{3} + 3908 \beta_{4} + 666 \beta_{5} - 2305 \beta_{6} - 4121 \beta_{7} ) q^{85} + ( 3698 - 1849 \beta_{1} ) q^{86} + ( 33093 - 2474 \beta_{1} - 1040 \beta_{2} - 639 \beta_{3} - 1871 \beta_{4} - 80 \beta_{5} - 327 \beta_{6} + 1337 \beta_{7} ) q^{87} + ( 10698 + 3934 \beta_{1} - 2452 \beta_{2} + 444 \beta_{3} - 3908 \beta_{4} + 898 \beta_{5} + 640 \beta_{6} + 2618 \beta_{7} ) q^{88} + ( -35082 + 4015 \beta_{1} + 1847 \beta_{2} + 1528 \beta_{3} - 430 \beta_{4} - 1342 \beta_{5} + 1516 \beta_{6} + 1507 \beta_{7} ) q^{89} + ( 10914 - 7473 \beta_{1} - 963 \beta_{2} - 2121 \beta_{3} + 3652 \beta_{4} - 1291 \beta_{5} - 1150 \beta_{6} - 338 \beta_{7} ) q^{90} + ( 48192 + 2775 \beta_{1} - 3 \beta_{2} - 2070 \beta_{3} - 1056 \beta_{4} + 894 \beta_{5} + 2052 \beta_{6} - 2601 \beta_{7} ) q^{91} + ( 27760 - 8902 \beta_{1} - 1838 \beta_{2} - 571 \beta_{3} - 814 \beta_{4} + 3880 \beta_{5} - 1837 \beta_{6} - 1795 \beta_{7} ) q^{92} + ( -17226 + 3476 \beta_{1} - 1948 \beta_{2} - 1312 \beta_{3} + 1783 \beta_{4} - 638 \beta_{5} + 25 \beta_{6} - 829 \beta_{7} ) q^{93} + ( 34811 - 11390 \beta_{1} - 482 \beta_{2} - 773 \beta_{3} - 2710 \beta_{4} - 437 \beta_{5} - 309 \beta_{6} + 1914 \beta_{7} ) q^{94} + ( 20120 - 270 \beta_{1} + 3451 \beta_{2} + 2330 \beta_{3} + 2375 \beta_{4} + 2092 \beta_{5} - 1150 \beta_{6} - 550 \beta_{7} ) q^{95} + ( 117271 - 11172 \beta_{1} - 966 \beta_{2} - 247 \beta_{3} - 15020 \beta_{4} + 1109 \beta_{5} - 4989 \beta_{6} + 5918 \beta_{7} ) q^{96} + ( 22025 - 1322 \beta_{1} + 1891 \beta_{2} + 1116 \beta_{3} + 4143 \beta_{4} + 2578 \beta_{5} - 760 \beta_{6} - 2298 \beta_{7} ) q^{97} + ( -42998 + 8381 \beta_{1} + 3356 \beta_{2} - 3104 \beta_{3} - 2328 \beta_{4} - 2820 \beta_{5} + 1888 \beta_{6} + 4712 \beta_{7} ) q^{98} + ( 23821 + 7573 \beta_{1} + 202 \beta_{2} + 1668 \beta_{3} - 3870 \beta_{4} + 1126 \beta_{5} + 2105 \beta_{6} + 1884 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 8125 \nu^{7} + 510320 \nu^{6} - 2147429 \nu^{5} - 76756386 \nu^{4} + 85006462 \nu^{3} + 2758106176 \nu^{2} - 206135304 \nu - 20471935488 \)\()/ 547265856 \)
\(\beta_{3}\)\(=\)\((\)\( 12949 \nu^{7} - 681148 \nu^{6} + 2601727 \nu^{5} + 80176782 \nu^{4} - 378607946 \nu^{3} - 1837549088 \nu^{2} + 7674423864 \nu + 1709840448 \)\()/ 547265856 \)
\(\beta_{4}\)\(=\)\((\)\( -17485 \nu^{7} + 269956 \nu^{6} + 1611305 \nu^{5} - 37491046 \nu^{4} + 6237658 \nu^{3} + 1274965584 \nu^{2} - 1052986520 \nu - 6710235008 \)\()/ 364843904 \)
\(\beta_{5}\)\(=\)\((\)\( 38033 \nu^{7} - 126518 \nu^{6} - 5886547 \nu^{5} + 8711358 \nu^{4} + 252968474 \nu^{3} + 152252096 \nu^{2} - 3214801896 \nu - 7712512416 \)\()/ 273632928 \)
\(\beta_{6}\)\(=\)\((\)\( -83131 \nu^{7} + 19780 \nu^{6} + 13177295 \nu^{5} + 14631678 \nu^{4} - 470063386 \nu^{3} - 1263726976 \nu^{2} + 1067316504 \nu + 17642049216 \)\()/ 547265856 \)
\(\beta_{7}\)\(=\)\((\)\( -128521 \nu^{7} + 1062904 \nu^{6} + 16607009 \nu^{5} - 129895854 \nu^{4} - 487223974 \nu^{3} + 4067658416 \nu^{2} + 2176112520 \nu - 28238112000 \)\()/ 547265856 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{1} + 43\)
\(\nu^{3}\)\(=\)\(8 \beta_{6} + 7 \beta_{5} - 4 \beta_{4} - \beta_{3} + 5 \beta_{2} + 71 \beta_{1} + 56\)
\(\nu^{4}\)\(=\)\(94 \beta_{7} + 28 \beta_{6} + 99 \beta_{5} - 264 \beta_{4} - 23 \beta_{3} + 31 \beta_{2} + 307 \beta_{1} + 3114\)
\(\nu^{5}\)\(=\)\(48 \beta_{7} + 1092 \beta_{6} + 869 \beta_{5} - 924 \beta_{4} - 71 \beta_{3} + 927 \beta_{2} + 6567 \beta_{1} + 9672\)
\(\nu^{6}\)\(=\)\(8760 \beta_{7} + 5444 \beta_{6} + 10339 \beta_{5} - 29780 \beta_{4} - 3425 \beta_{3} + 6801 \beta_{2} + 40661 \beta_{1} + 285312\)
\(\nu^{7}\)\(=\)\(11036 \beta_{7} + 127500 \beta_{6} + 102849 \beta_{5} - 146996 \beta_{4} - 10463 \beta_{3} + 125743 \beta_{2} + 685627 \beta_{1} + 1391004\)

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
−9.16809
−6.09504
−5.06235
−2.08717
2.58275
5.65705
7.21373
10.9591
−11.1681 −28.1764 92.7262 −10.1483 314.677 135.849 −678.196 550.912 113.337
1.2 −8.09504 11.1803 33.5297 −63.3756 −90.5052 223.489 −12.3830 −118.000 513.028
1.3 −7.06235 −9.19190 17.8767 73.4416 64.9164 −4.24720 99.7434 −158.509 −518.670
1.4 −4.08717 25.6605 −15.2950 −61.4284 −104.879 −184.774 193.303 415.460 251.069
1.5 0.582753 3.05838 −31.6604 27.7074 1.78228 −103.690 −37.0983 −233.646 16.1466
1.6 3.65705 7.84314 −18.6260 −107.102 28.6827 −25.5214 −185.142 −181.485 −391.677
1.7 5.21373 −11.2683 −4.81697 −9.80186 −58.7500 −11.1041 −191.954 −116.025 −51.1043
1.8 8.95911 −25.1057 48.2657 −61.2927 −224.924 −166.001 145.726 387.294 −549.129
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(43\) \(1\)

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 43.6.a.a 8
3.b odd 2 1 387.6.a.c 8
4.b odd 2 1 688.6.a.e 8
5.b even 2 1 1075.6.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 1.a even 1 1 trivial
387.6.a.c 8 3.b odd 2 1
688.6.a.e 8 4.b odd 2 1
1075.6.a.a 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 259776 - 439936 T - 39464 T^{2} + 49104 T^{3} + 3358 T^{4} - 1502 T^{5} - 117 T^{6} + 12 T^{7} + T^{8} \)
$3$ \( 504223128 - 156321960 T - 14906374 T^{2} + 3627880 T^{3} + 184435 T^{4} - 22242 T^{5} - 907 T^{6} + 26 T^{7} + T^{8} \)
$5$ \( 5172924974752 + 1078880170848 T + 54379881442 T^{2} - 883062964 T^{3} - 81381385 T^{4} - 1016128 T^{5} + 7783 T^{6} + 212 T^{7} + T^{8} \)
$7$ \( 116222354316288 + 43459429678848 T + 4329955769184 T^{2} + 129844227552 T^{3} + 573109808 T^{4} - 9763880 T^{5} - 61574 T^{6} + 136 T^{7} + T^{8} \)
$11$ \( 16753808850459134976 + 624343314702054528 T + 8339795208994704 T^{2} + 45000841976988 T^{3} + 46616679553 T^{4} - 322346400 T^{5} - 558018 T^{6} + 532 T^{7} + T^{8} \)
$13$ \( -\)\(20\!\cdots\!44\)\( + 7478047501760359440 T + 6802553866384368 T^{2} - 370621947684948 T^{3} - 1445788688279 T^{4} - 1375580104 T^{5} + 1195110 T^{6} + 2492 T^{7} + T^{8} \)
$17$ \( \)\(37\!\cdots\!33\)\( - \)\(29\!\cdots\!02\)\( T + 196078714064136693 T^{2} + 2317785169226190 T^{3} - 1750346026308 T^{4} - 6313795694 T^{5} - 1270643 T^{6} + 2534 T^{7} + T^{8} \)
$19$ \( \)\(20\!\cdots\!68\)\( - \)\(57\!\cdots\!56\)\( T - 30881750045973655680 T^{2} + 103160205820683760 T^{3} + 57520540020045 T^{4} - 25395335782 T^{5} - 15002885 T^{6} + 1678 T^{7} + T^{8} \)
$23$ \( -\)\(18\!\cdots\!83\)\( - \)\(85\!\cdots\!36\)\( T - \)\(82\!\cdots\!05\)\( T^{2} + 532680614614099516 T^{3} + 296571412629528 T^{4} - 70979412460 T^{5} - 31683297 T^{6} + 2488 T^{7} + T^{8} \)
$29$ \( -\)\(39\!\cdots\!36\)\( - \)\(43\!\cdots\!72\)\( T + \)\(23\!\cdots\!22\)\( T^{2} + 764932230807720084 T^{3} - 102909589543017 T^{4} - 237071828860 T^{5} - 40668269 T^{6} + 4360 T^{7} + T^{8} \)
$31$ \( \)\(10\!\cdots\!13\)\( - \)\(41\!\cdots\!72\)\( T + \)\(37\!\cdots\!35\)\( T^{2} - 11705033314802358652 T^{3} + 55447831561440 T^{4} + 606806231860 T^{5} - 74495573 T^{6} - 5704 T^{7} + T^{8} \)
$37$ \( -\)\(10\!\cdots\!04\)\( - \)\(55\!\cdots\!12\)\( T - \)\(13\!\cdots\!08\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + 33804397527288373 T^{4} - 2110832852496 T^{5} - 363678631 T^{6} + 3772 T^{7} + T^{8} \)
$41$ \( \)\(17\!\cdots\!57\)\( - \)\(14\!\cdots\!18\)\( T - \)\(29\!\cdots\!19\)\( T^{2} + 97139036739429657510 T^{3} + 13684977957931252 T^{4} - 1850906058934 T^{5} - 208766819 T^{6} + 10698 T^{7} + T^{8} \)
$43$ \( ( 1849 + T )^{8} \)
$47$ \( -\)\(19\!\cdots\!04\)\( - \)\(10\!\cdots\!72\)\( T - \)\(20\!\cdots\!32\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + 184514805755443205 T^{4} + 34288673842832 T^{5} + 2375107969 T^{6} + 77864 T^{7} + T^{8} \)
$53$ \( \)\(55\!\cdots\!84\)\( + \)\(73\!\cdots\!52\)\( T + \)\(13\!\cdots\!56\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} - 391239772753902151 T^{4} - 17547197551260 T^{5} + 759433570 T^{6} + 62352 T^{7} + T^{8} \)
$59$ \( -\)\(68\!\cdots\!68\)\( - \)\(10\!\cdots\!48\)\( T + \)\(57\!\cdots\!08\)\( T^{2} + \)\(58\!\cdots\!44\)\( T^{3} + 1572083239240996464 T^{4} - 85817413362304 T^{5} - 2944879380 T^{6} + 26224 T^{7} + T^{8} \)
$61$ \( \)\(23\!\cdots\!84\)\( + \)\(63\!\cdots\!84\)\( T + \)\(33\!\cdots\!20\)\( T^{2} - \)\(40\!\cdots\!76\)\( T^{3} - 4757919938906569384 T^{4} - 146433759288344 T^{5} + 310045522 T^{6} + 82540 T^{7} + T^{8} \)
$67$ \( \)\(69\!\cdots\!48\)\( + \)\(69\!\cdots\!68\)\( T - \)\(61\!\cdots\!84\)\( T^{2} - \)\(44\!\cdots\!08\)\( T^{3} + 2253346285790674161 T^{4} + 71143970372344 T^{5} - 2814867770 T^{6} - 27784 T^{7} + T^{8} \)
$71$ \( -\)\(15\!\cdots\!64\)\( + \)\(58\!\cdots\!60\)\( T - \)\(54\!\cdots\!80\)\( T^{2} - \)\(72\!\cdots\!20\)\( T^{3} + 15044739621952512640 T^{4} + 3736008685216 T^{5} - 7671997080 T^{6} + 9504 T^{7} + T^{8} \)
$73$ \( \)\(80\!\cdots\!92\)\( - \)\(13\!\cdots\!44\)\( T - \)\(28\!\cdots\!56\)\( T^{2} - \)\(95\!\cdots\!68\)\( T^{3} + 1545293860286742712 T^{4} + 23615008093864 T^{5} - 2401149454 T^{6} - 14260 T^{7} + T^{8} \)
$79$ \( \)\(19\!\cdots\!12\)\( - \)\(47\!\cdots\!36\)\( T + \)\(22\!\cdots\!36\)\( T^{2} + \)\(29\!\cdots\!88\)\( T^{3} - 38441230790897773763 T^{4} + 637841681315296 T^{5} + 3511120093 T^{6} - 160248 T^{7} + T^{8} \)
$83$ \( -\)\(19\!\cdots\!08\)\( - \)\(56\!\cdots\!20\)\( T - \)\(34\!\cdots\!60\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + 37880169769568642313 T^{4} - 551728279600100 T^{5} - 11424361174 T^{6} + 77176 T^{7} + T^{8} \)
$89$ \( -\)\(13\!\cdots\!64\)\( - \)\(12\!\cdots\!52\)\( T + \)\(83\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!28\)\( T^{3} - 72968680454010964328 T^{4} - 877164197088320 T^{5} + 15614540094 T^{6} + 265692 T^{7} + T^{8} \)
$97$ \( \)\(38\!\cdots\!17\)\( + \)\(16\!\cdots\!26\)\( T - \)\(55\!\cdots\!79\)\( T^{2} - \)\(36\!\cdots\!58\)\( T^{3} + 81215980328465108268 T^{4} + 1748586412389886 T^{5} - 15555161907 T^{6} - 144742 T^{7} + T^{8} \)
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