Properties

Label 27.7.d.a
Level 27
Weight 7
Character orbit 27.d
Analytic conductor 6.211
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 27.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( \beta_{1} + \beta_{2} - 25 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( 29 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 15 \beta_{4} - \beta_{5} - \beta_{7} ) q^{5} \) \( + ( -18 + 9 \beta_{1} + 18 \beta_{2} - \beta_{3} - 17 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{7} \) \( + ( -35 + 23 \beta_{1} + 25 \beta_{2} + 2 \beta_{3} - 68 \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( \beta_{1} + \beta_{2} - 25 \beta_{4} + \beta_{5} ) q^{4} \) \( + ( 29 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 15 \beta_{4} - \beta_{5} - \beta_{7} ) q^{5} \) \( + ( -18 + 9 \beta_{1} + 18 \beta_{2} - \beta_{3} - 17 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{7} \) \( + ( -35 + 23 \beta_{1} + 25 \beta_{2} + 2 \beta_{3} - 68 \beta_{4} + 4 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{8} \) \( + ( -47 + 58 \beta_{1} - 56 \beta_{2} + 2 \beta_{3} + \beta_{6} + 4 \beta_{7} - 5 \beta_{8} ) q^{10} \) \( + ( -38 + 8 \beta_{1} + 5 \beta_{3} + 32 \beta_{4} - 5 \beta_{5} + 10 \beta_{6} + 4 \beta_{7} - 8 \beta_{8} + 4 \beta_{9} ) q^{11} \) \( + ( 8 - 168 \beta_{1} - 91 \beta_{2} + 130 \beta_{4} - 14 \beta_{5} - 16 \beta_{6} - 8 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} ) q^{13} \) \( + ( -1608 + 7 \beta_{1} + 50 \beta_{2} + 14 \beta_{3} - 799 \beta_{4} + 7 \beta_{5} - 10 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{14} \) \( + ( -280 + 197 \beta_{1} + 394 \beta_{2} + 12 \beta_{3} - 284 \beta_{4} + 12 \beta_{5} - \beta_{6} + 3 \beta_{7} - 5 \beta_{9} ) q^{16} \) \( + ( 1626 - 13 \beta_{1} - 27 \beta_{2} - 14 \beta_{3} + 3258 \beta_{4} - 28 \beta_{5} + 4 \beta_{6} + \beta_{7} - 3 \beta_{8} + 6 \beta_{9} ) q^{17} \) \( + ( 442 + 262 \beta_{1} - 289 \beta_{2} - 27 \beta_{3} + 7 \beta_{6} - 17 \beta_{7} + 10 \beta_{8} ) q^{19} \) \( + ( 4211 + 66 \beta_{1} - 49 \beta_{3} - 4185 \beta_{4} + 49 \beta_{5} - 41 \beta_{6} - 15 \beta_{7} + 30 \beta_{8} - 15 \beta_{9} ) q^{20} \) \( + ( -37 - 705 \beta_{1} - 313 \beta_{2} - 967 \beta_{4} + 79 \beta_{5} + 74 \beta_{6} + 37 \beta_{7} + 31 \beta_{8} - 31 \beta_{9} ) q^{22} \) \( + ( -7201 - 14 \beta_{1} - 187 \beta_{2} - 28 \beta_{3} - 3626 \beta_{4} - 14 \beta_{5} + 51 \beta_{7} + 14 \beta_{8} + 14 \beta_{9} ) q^{23} \) \( + ( 1807 + 207 \beta_{1} + 414 \beta_{2} - 49 \beta_{3} + 1820 \beta_{4} - 49 \beta_{5} + 37 \beta_{6} + 24 \beta_{7} + 50 \beta_{9} ) q^{25} \) \( + ( 7557 - 466 \beta_{1} - 458 \beta_{2} + 8 \beta_{3} + 15024 \beta_{4} + 16 \beta_{5} - 19 \beta_{6} + 26 \beta_{7} + 45 \beta_{8} - 90 \beta_{9} ) q^{26} \) \( + ( -1714 - 735 \beta_{1} + 876 \beta_{2} + 141 \beta_{3} - 87 \beta_{6} + 12 \beta_{7} + 75 \beta_{8} ) q^{28} \) \( + ( 5563 - 785 \beta_{1} + 168 \beta_{3} - 5582 \beta_{4} - 168 \beta_{5} + 6 \beta_{6} - 13 \beta_{7} + 26 \beta_{8} - 13 \beta_{9} ) q^{29} \) \( + ( -5 + 1148 \beta_{1} + 471 \beta_{2} + 5294 \beta_{4} - 206 \beta_{5} + 10 \beta_{6} + 5 \beta_{7} - 70 \beta_{8} + 70 \beta_{9} ) q^{31} \) \( + ( -29183 + 22 \beta_{1} - 225 \beta_{2} + 44 \beta_{3} - 14646 \beta_{4} + 22 \beta_{5} + 109 \beta_{7} - 39 \beta_{8} - 39 \beta_{9} ) q^{32} \) \( + ( 36 - 2417 \beta_{1} - 4834 \beta_{2} + 24 \beta_{3} - 56 \beta_{4} + 24 \beta_{5} + 31 \beta_{6} + 123 \beta_{7} - 61 \beta_{9} ) q^{34} \) \( + ( 11180 + 770 \beta_{1} + 947 \beta_{2} + 177 \beta_{3} + 22550 \beta_{4} + 354 \beta_{5} + 162 \beta_{6} + 67 \beta_{7} - 95 \beta_{8} + 190 \beta_{9} ) q^{35} \) \( + ( 1402 - 671 \beta_{1} + 374 \beta_{2} - 297 \beta_{3} + 253 \beta_{6} - 32 \beta_{7} - 221 \beta_{8} ) q^{37} \) \( + ( 24454 + 1661 \beta_{1} - 144 \beta_{3} - 24368 \beta_{4} + 144 \beta_{5} - 15 \beta_{6} + 71 \beta_{7} - 142 \beta_{8} + 71 \beta_{9} ) q^{38} \) \( + ( 240 + 7360 \beta_{1} + 3730 \beta_{2} - 8420 \beta_{4} + 100 \beta_{5} - 480 \beta_{6} - 240 \beta_{7} + 120 \beta_{8} - 120 \beta_{9} ) q^{40} \) \( + ( -30245 - 189 \beta_{1} + 1982 \beta_{2} - 378 \beta_{3} - 14756 \beta_{4} - 189 \beta_{5} - 733 \beta_{7} + 71 \beta_{8} + 71 \beta_{9} ) q^{41} \) \( + ( -17631 + 1658 \beta_{1} + 3316 \beta_{2} + 416 \beta_{3} - 17343 \beta_{4} + 416 \beta_{5} - 423 \beta_{6} - 711 \beta_{7} - 135 \beta_{9} ) q^{43} \) \( + ( 29357 + 1354 \beta_{1} + 815 \beta_{2} - 539 \beta_{3} + 58694 \beta_{4} - 1078 \beta_{5} - 590 \beta_{6} - 580 \beta_{7} + 10 \beta_{8} - 20 \beta_{9} ) q^{44} \) \( + ( 12507 - 1909 \beta_{1} + 1752 \beta_{2} - 157 \beta_{3} - 214 \beta_{6} + 224 \beta_{7} - 10 \beta_{8} ) q^{46} \) \( + ( 8704 + 1311 \beta_{1} - 461 \beta_{3} - 9417 \beta_{4} + 461 \beta_{5} + 812 \beta_{6} + 99 \beta_{7} - 198 \beta_{8} + 99 \beta_{9} ) q^{47} \) \( + ( -258 - 4958 \beta_{1} - 2057 \beta_{2} - 18713 \beta_{4} + 844 \beta_{5} + 516 \beta_{6} + 258 \beta_{7} - 309 \beta_{8} + 309 \beta_{9} ) q^{49} \) \( + ( -43989 + 586 \beta_{1} - 3569 \beta_{2} + 1172 \beta_{3} - 22330 \beta_{4} + 586 \beta_{5} + 671 \beta_{7} - 125 \beta_{8} - 125 \beta_{9} ) q^{50} \) \( + ( 35191 - 3914 \beta_{1} - 7828 \beta_{2} - 1179 \beta_{3} + 34931 \beta_{4} - 1179 \beta_{5} + 331 \beta_{6} + 591 \beta_{7} + 71 \beta_{9} ) q^{52} \) \( + ( -22606 + 261 \beta_{1} + 498 \beta_{2} + 237 \beta_{3} - 45130 \beta_{4} + 474 \beta_{5} + 735 \beta_{6} + 694 \beta_{7} - 41 \beta_{8} + 82 \beta_{9} ) q^{53} \) \( + ( -45882 + 7853 \beta_{1} - 5931 \beta_{2} + 1922 \beta_{3} - 109 \beta_{6} - 31 \beta_{7} + 140 \beta_{8} ) q^{55} \) \( + ( -20426 - 6742 \beta_{1} + 1022 \beta_{3} + 21710 \beta_{4} - 1022 \beta_{5} - 1502 \beta_{6} - 218 \beta_{7} + 436 \beta_{8} - 218 \beta_{9} ) q^{56} \) \( + ( -141 - 11192 \beta_{1} - 6680 \beta_{2} + 72088 \beta_{4} - 2168 \beta_{5} + 282 \beta_{6} + 141 \beta_{7} + 375 \beta_{8} - 375 \beta_{9} ) q^{58} \) \( + ( 135287 + 14 \beta_{1} + 6182 \beta_{2} + 28 \beta_{3} + 67281 \beta_{4} + 14 \beta_{5} + 725 \beta_{7} + 117 \beta_{8} + 117 \beta_{9} ) q^{59} \) \( + ( -3567 + 14267 \beta_{1} + 28534 \beta_{2} + 152 \beta_{3} - 3600 \beta_{4} + 152 \beta_{5} + 606 \beta_{6} + 639 \beta_{7} + 573 \beta_{9} ) q^{61} \) \( + ( -53323 - 13777 \beta_{1} - 12632 \beta_{2} + 1145 \beta_{3} - 107458 \beta_{4} + 2290 \beta_{5} + 446 \beta_{6} + 852 \beta_{7} + 406 \beta_{8} - 812 \beta_{9} ) q^{62} \) \( + ( 40785 - 2837 \beta_{1} + 203 \beta_{2} - 2634 \beta_{3} + 13 \beta_{6} - 812 \beta_{7} + 799 \beta_{8} ) q^{64} \) \( + ( -109529 + 1561 \beta_{1} + 36 \beta_{3} + 109660 \beta_{4} - 36 \beta_{5} - 546 \beta_{6} - 415 \beta_{7} + 830 \beta_{8} - 415 \beta_{9} ) q^{65} \) \( + ( -170 - 28849 \beta_{1} - 13842 \beta_{2} - 40672 \beta_{4} + 1165 \beta_{5} + 340 \beta_{6} + 170 \beta_{7} + 284 \beta_{8} - 284 \beta_{9} ) q^{67} \) \( + ( 222921 - 2476 \beta_{1} - 9555 \beta_{2} - 4952 \beta_{3} + 111676 \beta_{4} - 2476 \beta_{5} - 431 \beta_{7} + 77 \beta_{8} + 77 \beta_{9} ) q^{68} \) \( + ( -60283 + 2672 \beta_{1} + 5344 \beta_{2} + 4211 \beta_{3} - 60715 \beta_{4} + 4211 \beta_{5} + 432 \beta_{6} + 864 \beta_{7} ) q^{70} \) \( + ( -131444 + 13535 \beta_{1} + 12012 \beta_{2} - 1523 \beta_{3} - 263078 \beta_{4} - 3046 \beta_{5} - 1751 \beta_{6} - 1656 \beta_{7} + 95 \beta_{8} - 190 \beta_{9} ) q^{71} \) \( + ( 86144 + 10275 \beta_{1} - 12489 \beta_{2} - 2214 \beta_{3} + 240 \beta_{6} - 255 \beta_{7} + 15 \beta_{8} ) q^{73} \) \( + ( -43682 + 13226 \beta_{1} - 1278 \beta_{3} + 41938 \beta_{4} + 1278 \beta_{5} + 1884 \beta_{6} + 140 \beta_{7} - 280 \beta_{8} + 140 \beta_{9} ) q^{74} \) \( + ( 409 + 38706 \beta_{1} + 21109 \beta_{2} - 120552 \beta_{4} + 3512 \beta_{5} - 818 \beta_{6} - 409 \beta_{7} - 403 \beta_{8} + 403 \beta_{9} ) q^{76} \) \( + ( 285791 + 2217 \beta_{1} - 16285 \beta_{2} + 4434 \beta_{3} + 142877 \beta_{4} + 2217 \beta_{5} + 37 \beta_{7} - 50 \beta_{8} - 50 \beta_{9} ) q^{77} \) \( + ( 91467 - 6557 \beta_{1} - 13114 \beta_{2} - 5674 \beta_{3} + 91844 \beta_{4} - 5674 \beta_{5} - 1501 \beta_{6} - 1878 \beta_{7} - 1124 \beta_{9} ) q^{79} \) \( + ( -52730 + 22680 \beta_{1} + 23254 \beta_{2} + 574 \beta_{3} - 103580 \beta_{4} + 1148 \beta_{5} + 724 \beta_{6} - 216 \beta_{7} - 940 \beta_{8} + 1880 \beta_{9} ) q^{80} \) \( + ( -192454 - 14079 \beta_{1} + 21809 \beta_{2} + 7730 \beta_{3} + 780 \beta_{6} + 1608 \beta_{7} - 2388 \beta_{8} ) q^{82} \) \( + ( -195424 - 24401 \beta_{1} - 189 \beta_{3} + 196775 \beta_{4} + 189 \beta_{5} - 48 \beta_{6} + 1303 \beta_{7} - 2606 \beta_{8} + 1303 \beta_{9} ) q^{83} \) \( + ( 1061 + 11921 \beta_{1} + 2584 \beta_{2} + 242455 \beta_{4} - 6753 \beta_{5} - 2122 \beta_{6} - 1061 \beta_{7} - 365 \beta_{8} + 365 \beta_{9} ) q^{85} \) \( + ( -229327 + 2857 \beta_{1} + 66377 \beta_{2} + 5714 \beta_{3} - 113009 \beta_{4} + 2857 \beta_{5} - 3309 \beta_{7} - 25 \beta_{8} - 25 \beta_{9} ) q^{86} \) \( + ( -86408 - 29579 \beta_{1} - 59158 \beta_{2} - 2708 \beta_{3} - 84092 \beta_{4} - 2708 \beta_{5} - 3039 \beta_{6} - 5355 \beta_{7} - 723 \beta_{9} ) q^{88} \) \( + ( -167870 - 60017 \beta_{1} - 61698 \beta_{2} - 1681 \beta_{3} - 335294 \beta_{4} - 3362 \beta_{5} + 809 \beta_{6} + 586 \beta_{7} - 223 \beta_{8} + 446 \beta_{9} ) q^{89} \) \( + ( -19230 - 805 \beta_{1} - 811 \beta_{2} - 1616 \beta_{3} - 2521 \beta_{6} + 2993 \beta_{7} - 472 \beta_{8} ) q^{91} \) \( + ( 58937 + 20648 \beta_{1} + 1077 \beta_{3} - 61987 \beta_{4} - 1077 \beta_{5} + 3645 \beta_{6} + 595 \beta_{7} - 1190 \beta_{8} + 595 \beta_{9} ) q^{92} \) \( + ( -1460 + 54515 \beta_{1} + 28560 \beta_{2} - 145269 \beta_{4} + 2605 \beta_{5} + 2920 \beta_{6} + 1460 \beta_{7} - 2116 \beta_{8} + 2116 \beta_{9} ) q^{94} \) \( + ( 333504 - 686 \beta_{1} - 21956 \beta_{2} - 1372 \beta_{3} + 166040 \beta_{4} - 686 \beta_{5} + 1424 \beta_{7} - 1340 \beta_{8} - 1340 \beta_{9} ) q^{95} \) \( + ( 56412 - 13456 \beta_{1} - 26912 \beta_{2} + 8993 \beta_{3} + 53826 \beta_{4} + 8993 \beta_{5} + 3597 \beta_{6} + 6183 \beta_{7} + 1011 \beta_{9} ) q^{97} \) \( + ( 220807 + 64747 \beta_{1} + 66003 \beta_{2} + 1256 \beta_{3} + 442480 \beta_{4} + 2512 \beta_{5} + 1631 \beta_{6} + 1198 \beta_{7} - 433 \beta_{8} + 866 \beta_{9} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 127q^{4} \) \(\mathstrut +\mathstrut 219q^{5} \) \(\mathstrut -\mathstrut 121q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 127q^{4} \) \(\mathstrut +\mathstrut 219q^{5} \) \(\mathstrut -\mathstrut 121q^{7} \) \(\mathstrut -\mathstrut 132q^{10} \) \(\mathstrut -\mathstrut 483q^{11} \) \(\mathstrut -\mathstrut 841q^{13} \) \(\mathstrut -\mathstrut 12174q^{14} \) \(\mathstrut -\mathstrut 1985q^{16} \) \(\mathstrut +\mathstrut 6176q^{19} \) \(\mathstrut +\mathstrut 63186q^{20} \) \(\mathstrut +\mathstrut 3471q^{22} \) \(\mathstrut -\mathstrut 53565q^{23} \) \(\mathstrut +\mathstrut 8452q^{25} \) \(\mathstrut -\mathstrut 22660q^{28} \) \(\mathstrut +\mathstrut 80679q^{29} \) \(\mathstrut -\mathstrut 24601q^{31} \) \(\mathstrut -\mathstrut 218295q^{32} \) \(\mathstrut +\mathstrut 7425q^{34} \) \(\mathstrut +\mathstrut 12764q^{37} \) \(\mathstrut +\mathstrut 371877q^{38} \) \(\mathstrut +\mathstrut 54150q^{40} \) \(\mathstrut -\mathstrut 232251q^{41} \) \(\mathstrut -\mathstrut 93271q^{43} \) \(\mathstrut +\mathstrut 112512q^{46} \) \(\mathstrut +\mathstrut 142887q^{47} \) \(\mathstrut +\mathstrut 86238q^{49} \) \(\mathstrut -\mathstrut 318459q^{50} \) \(\mathstrut +\mathstrut 186920q^{52} \) \(\mathstrut -\mathstrut 419982q^{55} \) \(\mathstrut -\mathstrut 342546q^{56} \) \(\mathstrut -\mathstrut 380658q^{58} \) \(\mathstrut +\mathstrut 995061q^{59} \) \(\mathstrut -\mathstrut 59305q^{61} \) \(\mathstrut +\mathstrut 403066q^{64} \) \(\mathstrut -\mathstrut 1642029q^{65} \) \(\mathstrut +\mathstrut 158513q^{67} \) \(\mathstrut +\mathstrut 1693791q^{68} \) \(\mathstrut -\mathstrut 304788q^{70} \) \(\mathstrut +\mathstrut 933896q^{73} \) \(\mathstrut -\mathstrut 595182q^{74} \) \(\mathstrut +\mathstrut 666641q^{76} \) \(\mathstrut +\mathstrut 2198883q^{77} \) \(\mathstrut +\mathstrut 468707q^{79} \) \(\mathstrut -\mathstrut 2038470q^{82} \) \(\mathstrut -\mathstrut 3008337q^{83} \) \(\mathstrut -\mathstrut 1189944q^{85} \) \(\mathstrut -\mathstrut 1905549q^{86} \) \(\mathstrut -\mathstrut 349773q^{88} \) \(\mathstrut -\mathstrut 211778q^{91} \) \(\mathstrut +\mathstrut 973788q^{92} \) \(\mathstrut +\mathstrut 809124q^{94} \) \(\mathstrut +\mathstrut 2562954q^{95} \) \(\mathstrut +\mathstrut 336029q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(x^{9}\mathstrut +\mathstrut \) \(75\) \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut +\mathstrut \) \(4610\) \(x^{6}\mathstrut -\mathstrut \) \(2412\) \(x^{5}\mathstrut +\mathstrut \) \(66932\) \(x^{4}\mathstrut -\mathstrut \) \(174032\) \(x^{3}\mathstrut +\mathstrut \) \(870720\) \(x^{2}\mathstrut -\mathstrut \) \(1272832\) \(x\mathstrut +\mathstrut \) \(1982464\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(5928919\) \(\nu^{9}\mathstrut +\mathstrut \) \(4480309\) \(\nu^{8}\mathstrut -\mathstrut \) \(11157967\) \(\nu^{7}\mathstrut +\mathstrut \) \(1123354358\) \(\nu^{6}\mathstrut -\mathstrut \) \(339533226\) \(\nu^{5}\mathstrut +\mathstrut \) \(20243577060\) \(\nu^{4}\mathstrut -\mathstrut \) \(1662748272772\) \(\nu^{3}\mathstrut +\mathstrut \) \(303080290496\) \(\nu^{2}\mathstrut +\mathstrut \) \(24758157183040\) \(\nu\mathstrut +\mathstrut \) \(38990423194624\)\()/\)\(25215567395904\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(5928919\) \(\nu^{9}\mathstrut -\mathstrut \) \(4480309\) \(\nu^{8}\mathstrut +\mathstrut \) \(11157967\) \(\nu^{7}\mathstrut -\mathstrut \) \(1123354358\) \(\nu^{6}\mathstrut +\mathstrut \) \(339533226\) \(\nu^{5}\mathstrut -\mathstrut \) \(20243577060\) \(\nu^{4}\mathstrut +\mathstrut \) \(1662748272772\) \(\nu^{3}\mathstrut -\mathstrut \) \(303080290496\) \(\nu^{2}\mathstrut +\mathstrut \) \(13065193910816\) \(\nu\mathstrut -\mathstrut \) \(38990423194624\)\()/\)\(12607783697952\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(9684923\) \(\nu^{9}\mathstrut +\mathstrut \) \(688220647\) \(\nu^{8}\mathstrut -\mathstrut \) \(1713976261\) \(\nu^{7}\mathstrut +\mathstrut \) \(42631129082\) \(\nu^{6}\mathstrut -\mathstrut \) \(52155727758\) \(\nu^{5}\mathstrut +\mathstrut \) \(3109617595980\) \(\nu^{4}\mathstrut -\mathstrut \) \(3665101155628\) \(\nu^{3}\mathstrut +\mathstrut \) \(46556189231168\) \(\nu^{2}\mathstrut -\mathstrut \) \(82870605445664\) \(\nu\mathstrut +\mathstrut \) \(1178713975026976\)\()/\)\(12607783697952\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(3461507741\) \(\nu^{9}\mathstrut +\mathstrut \) \(4504997485\) \(\nu^{8}\mathstrut -\mathstrut \) \(258824546191\) \(\nu^{7}\mathstrut +\mathstrut \) \(4959213290\) \(\nu^{6}\mathstrut -\mathstrut \) \(15759840319002\) \(\nu^{5}\mathstrut +\mathstrut \) \(8289398823516\) \(\nu^{4}\mathstrut -\mathstrut \) \(228122766558052\) \(\nu^{3}\mathstrut +\mathstrut \) \(309769419173840\) \(\nu^{2}\mathstrut -\mathstrut \) \(2960661889116224\) \(\nu\mathstrut -\mathstrut \) \(112526238150656\)\()/\)\(4437939861679104\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(301534626821\) \(\nu^{9}\mathstrut +\mathstrut \) \(161056711573\) \(\nu^{8}\mathstrut -\mathstrut \) \(22437956373703\) \(\nu^{7}\mathstrut -\mathstrut \) \(13971656353030\) \(\nu^{6}\mathstrut -\mathstrut \) \(1384446245763690\) \(\nu^{5}\mathstrut -\mathstrut \) \(346140289804356\) \(\nu^{4}\mathstrut -\mathstrut \) \(19890741704909188\) \(\nu^{3}\mathstrut +\mathstrut \) \(24655545675519824\) \(\nu^{2}\mathstrut -\mathstrut \) \(247883787191899712\) \(\nu\mathstrut -\mathstrut \) \(9358563268702208\)\()/\)\(4437939861679104\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(228970339285\) \(\nu^{9}\mathstrut -\mathstrut \) \(1071624020857\) \(\nu^{8}\mathstrut -\mathstrut \) \(17892439228289\) \(\nu^{7}\mathstrut -\mathstrut \) \(92268001192520\) \(\nu^{6}\mathstrut -\mathstrut \) \(1185087940044102\) \(\nu^{5}\mathstrut -\mathstrut \) \(5233881275671536\) \(\nu^{4}\mathstrut -\mathstrut \) \(19899503053280492\) \(\nu^{3}\mathstrut -\mathstrut \) \(28091608583134376\) \(\nu^{2}\mathstrut -\mathstrut \) \(21510646254101248\) \(\nu\mathstrut -\mathstrut \) \(278089336893715072\)\()/\)\(554742482709888\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(680904382841\) \(\nu^{9}\mathstrut +\mathstrut \) \(1735301772337\) \(\nu^{8}\mathstrut -\mathstrut \) \(45444196783291\) \(\nu^{7}\mathstrut +\mathstrut \) \(81874683465962\) \(\nu^{6}\mathstrut -\mathstrut \) \(2664105725232018\) \(\nu^{5}\mathstrut +\mathstrut \) \(7056859532042268\) \(\nu^{4}\mathstrut -\mathstrut \) \(18842417040821140\) \(\nu^{3}\mathstrut +\mathstrut \) \(206189758524868592\) \(\nu^{2}\mathstrut -\mathstrut \) \(438995832667976192\) \(\nu\mathstrut +\mathstrut \) \(934989218885514496\)\()/\)\(1109484965419776\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(357064666657\) \(\nu^{9}\mathstrut +\mathstrut \) \(1468396024739\) \(\nu^{8}\mathstrut -\mathstrut \) \(24218221634837\) \(\nu^{7}\mathstrut +\mathstrut \) \(85319366526208\) \(\nu^{6}\mathstrut -\mathstrut \) \(1377579395474046\) \(\nu^{5}\mathstrut +\mathstrut \) \(6242802927799104\) \(\nu^{4}\mathstrut -\mathstrut \) \(9465337088444252\) \(\nu^{3}\mathstrut +\mathstrut \) \(143733596616975448\) \(\nu^{2}\mathstrut -\mathstrut \) \(280830061440619264\) \(\nu\mathstrut +\mathstrut \) \(761607762821734400\)\()/\)\(554742482709888\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(1183320135805\) \(\nu^{9}\mathstrut +\mathstrut \) \(2091348988481\) \(\nu^{8}\mathstrut -\mathstrut \) \(82073042998211\) \(\nu^{7}\mathstrut +\mathstrut \) \(80258730986950\) \(\nu^{6}\mathstrut -\mathstrut \) \(4933627466855202\) \(\nu^{5}\mathstrut +\mathstrut \) \(8188936742890116\) \(\nu^{4}\mathstrut -\mathstrut \) \(47715295575658388\) \(\nu^{3}\mathstrut +\mathstrut \) \(314020064100693472\) \(\nu^{2}\mathstrut -\mathstrut \) \(805271759552407552\) \(\nu\mathstrut +\mathstrut \) \(1468153012199075840\)\()/\)\(1109484965419776\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(89\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(89\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(153\) \(\beta_{2}\mathstrut -\mathstrut \) \(147\) \(\beta_{1}\mathstrut -\mathstrut \) \(107\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(5\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(204\) \(\beta_{5}\mathstrut +\mathstrut \) \(13276\) \(\beta_{4}\mathstrut -\mathstrut \) \(389\) \(\beta_{2}\mathstrut -\mathstrut \) \(574\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(217\) \(\beta_{9}\mathstrut +\mathstrut \) \(367\) \(\beta_{7}\mathstrut +\mathstrut \) \(75\) \(\beta_{6}\mathstrut -\mathstrut \) \(534\) \(\beta_{5}\mathstrut +\mathstrut \) \(23350\) \(\beta_{4}\mathstrut -\mathstrut \) \(534\) \(\beta_{3}\mathstrut -\mathstrut \) \(17770\) \(\beta_{2}\mathstrut -\mathstrut \) \(8885\) \(\beta_{1}\mathstrut +\mathstrut \) \(23642\)\()/9\)
\(\nu^{6}\)\(=\)\((\)\(89\) \(\beta_{8}\mathstrut -\mathstrut \) \(52\) \(\beta_{7}\mathstrut -\mathstrut \) \(37\) \(\beta_{6}\mathstrut -\mathstrut \) \(4230\) \(\beta_{3}\mathstrut -\mathstrut \) \(11123\) \(\beta_{2}\mathstrut +\mathstrut \) \(6893\) \(\beta_{1}\mathstrut +\mathstrut \) \(253495\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(4423\) \(\beta_{9}\mathstrut -\mathstrut \) \(4423\) \(\beta_{8}\mathstrut -\mathstrut \) \(1573\) \(\beta_{7}\mathstrut -\mathstrut \) \(3146\) \(\beta_{6}\mathstrut +\mathstrut \) \(14418\) \(\beta_{5}\mathstrut -\mathstrut \) \(755890\) \(\beta_{4}\mathstrut +\mathstrut \) \(180663\) \(\beta_{2}\mathstrut +\mathstrut \) \(346908\) \(\beta_{1}\mathstrut +\mathstrut \) \(1573\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(1997\) \(\beta_{9}\mathstrut -\mathstrut \) \(5061\) \(\beta_{7}\mathstrut -\mathstrut \) \(3529\) \(\beta_{6}\mathstrut +\mathstrut \) \(262018\) \(\beta_{5}\mathstrut -\mathstrut \) \(15267234\) \(\beta_{4}\mathstrut +\mathstrut \) \(262018\) \(\beta_{3}\mathstrut +\mathstrut \) \(1691622\) \(\beta_{2}\mathstrut +\mathstrut \) \(845811\) \(\beta_{1}\mathstrut -\mathstrut \) \(15268766\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(782853\) \(\beta_{8}\mathstrut -\mathstrut \) \(1072036\) \(\beta_{7}\mathstrut +\mathstrut \) \(289183\) \(\beta_{6}\mathstrut +\mathstrut \) \(3309126\) \(\beta_{3}\mathstrut +\mathstrut \) \(33892269\) \(\beta_{2}\mathstrut -\mathstrut \) \(30583143\) \(\beta_{1}\mathstrut -\mathstrut \) \(183654569\)\()/9\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−3.54866 6.14646i
−2.32209 4.02197i
1.07323 + 1.85889i
1.22025 + 2.11353i
4.07727 + 7.06203i
−3.54866 + 6.14646i
−2.32209 + 4.02197i
1.07323 1.85889i
1.22025 2.11353i
4.07727 7.06203i
−10.6460 6.14646i 0 43.5579 + 75.4444i 157.562 90.9682i 0 83.3541 144.373i 284.159i 0 −2236.53
8.2 −6.96626 4.02197i 0 0.352523 + 0.610587i −80.3236 + 46.3749i 0 60.0074 103.936i 509.141i 0 746.074
8.3 3.21969 + 1.85889i 0 −25.0891 43.4555i −136.563 + 78.8448i 0 −256.037 + 443.470i 424.489i 0 −586.255
8.4 3.66074 + 2.11353i 0 −23.0660 39.9514i 64.9866 37.5201i 0 181.066 313.616i 465.535i 0 317.199
8.5 12.2318 + 7.06203i 0 67.7447 + 117.337i 103.839 59.9512i 0 −128.891 + 223.245i 1009.72i 0 1693.51
17.1 −10.6460 + 6.14646i 0 43.5579 75.4444i 157.562 + 90.9682i 0 83.3541 + 144.373i 284.159i 0 −2236.53
17.2 −6.96626 + 4.02197i 0 0.352523 0.610587i −80.3236 46.3749i 0 60.0074 + 103.936i 509.141i 0 746.074
17.3 3.21969 1.85889i 0 −25.0891 + 43.4555i −136.563 78.8448i 0 −256.037 443.470i 424.489i 0 −586.255
17.4 3.66074 2.11353i 0 −23.0660 + 39.9514i 64.9866 + 37.5201i 0 181.066 + 313.616i 465.535i 0 317.199
17.5 12.2318 7.06203i 0 67.7447 117.337i 103.839 + 59.9512i 0 −128.891 223.245i 1009.72i 0 1693.51
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{7}^{\mathrm{new}}(27, [\chi])\).