Properties

Label 8.12.b.a
Level 8
Weight 12
Character orbit 8.b
Analytic conductor 6.147
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 8.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.14674544448\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{2}\cdot 7 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 2 + \beta_{1} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 43 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{5} + ( -2431 + 2 \beta_{1} + \beta_{2} + \beta_{7} ) q^{6} + ( -3385 + 115 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{7} + ( -20862 + 67 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{8} + ( -47267 + 166 \beta_{1} - 11 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 6 \beta_{7} + 2 \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( 2 + \beta_{1} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( 43 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{5} + ( -2431 + 2 \beta_{1} + \beta_{2} + \beta_{7} ) q^{6} + ( -3385 + 115 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{7} + \beta_{8} ) q^{7} + ( -20862 + 67 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{8} + ( -47267 + 166 \beta_{1} - 11 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 6 \beta_{7} + 2 \beta_{9} ) q^{9} + ( 2956 + 20 \beta_{1} + 5 \beta_{2} - 125 \beta_{3} + 16 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} ) q^{10} + ( -252 + 1161 \beta_{1} + 96 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 16 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} ) q^{11} + ( -47070 - 2414 \beta_{1} + 12 \beta_{2} + 200 \beta_{3} + 98 \beta_{4} - 12 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - 10 \beta_{8} + 6 \beta_{9} ) q^{12} + ( -1032 + 4941 \beta_{1} + 272 \beta_{2} + 106 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 32 \beta_{7} - 8 \beta_{8} + 8 \beta_{9} ) q^{13} + ( 227516 - 3984 \beta_{1} + 62 \beta_{2} + 1806 \beta_{3} - 184 \beta_{4} + 10 \beta_{5} - 20 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} - 12 \beta_{9} ) q^{14} + ( 336863 + 12587 \beta_{1} - 1050 \beta_{2} + 104 \beta_{3} - 49 \beta_{4} + 8 \beta_{5} + 24 \beta_{6} - 41 \beta_{7} - 7 \beta_{8} - 16 \beta_{9} ) q^{15} + ( 379004 - 20558 \beta_{1} + 56 \beta_{2} - 3882 \beta_{3} - 242 \beta_{4} - 74 \beta_{5} - 40 \beta_{6} - 16 \beta_{7} - 34 \beta_{8} - 2 \beta_{9} ) q^{16} + ( 266276 - 9270 \beta_{1} - 2013 \beta_{2} + 557 \beta_{3} - 59 \beta_{4} + 137 \beta_{5} + 55 \beta_{6} - 22 \beta_{7} - 32 \beta_{8} - 18 \beta_{9} ) q^{17} + ( 239926 - 45617 \beta_{1} + 268 \beta_{2} - 10708 \beta_{3} + 464 \beta_{4} + 68 \beta_{5} - 72 \beta_{6} + 32 \beta_{7} - 88 \beta_{8} + 8 \beta_{9} ) q^{18} + ( 3284 - 18603 \beta_{1} + 2400 \beta_{2} + 177 \beta_{3} - 190 \beta_{4} - 172 \beta_{5} + 106 \beta_{6} - 176 \beta_{7} - 44 \beta_{8} + 44 \beta_{9} ) q^{19} + ( 833868 - 1492 \beta_{1} - 80 \beta_{2} + 19816 \beta_{3} - 180 \beta_{4} - 352 \beta_{5} - 136 \beta_{6} - 176 \beta_{7} + 148 \beta_{8} - 76 \beta_{9} ) q^{20} + ( -36760 + 179572 \beta_{1} + 4400 \beta_{2} + 1112 \beta_{3} - 232 \beta_{4} + 664 \beta_{5} + 100 \beta_{6} + 416 \beta_{7} + 104 \beta_{8} - 104 \beta_{9} ) q^{21} + ( -2411891 - 4614 \beta_{1} + 821 \beta_{2} + 36792 \beta_{3} + 1408 \beta_{4} + 488 \beta_{5} - 80 \beta_{6} + 93 \beta_{7} - 16 \beta_{8} + 176 \beta_{9} ) q^{22} + ( -4595147 + 301433 \beta_{1} - 3470 \beta_{2} - 3608 \beta_{3} + 341 \beta_{4} - 1144 \beta_{5} + 152 \beta_{6} + 733 \beta_{7} - 13 \beta_{8} + 240 \beta_{9} ) q^{23} + ( 1061412 - 35178 \beta_{1} - 868 \beta_{2} - 68690 \beta_{3} + 2346 \beta_{4} - 1402 \beta_{5} - 80 \beta_{6} + 352 \beta_{7} + 354 \beta_{8} - 94 \beta_{9} ) q^{24} + ( -4057151 - 468924 \beta_{1} + 3070 \beta_{2} + 5602 \beta_{3} - 462 \beta_{4} + 1914 \beta_{5} - 58 \beta_{6} - 508 \beta_{7} + 544 \beta_{8} + 12 \beta_{9} ) q^{25} + ( -10212828 + 20892 \beta_{1} + 4471 \beta_{2} - 63231 \beta_{3} - 4432 \beta_{4} + 2603 \beta_{5} + 170 \beta_{6} - 182 \beta_{7} + 514 \beta_{8} + 170 \beta_{9} ) q^{26} + ( 195956 - 974890 \beta_{1} - 11680 \beta_{2} - 16878 \beta_{3} + 2642 \beta_{4} - 3596 \beta_{5} - 326 \beta_{6} + 464 \beta_{7} + 116 \beta_{8} - 116 \beta_{9} ) q^{27} + ( -9596088 + 222288 \beta_{1} - 4536 \beta_{2} + 87920 \beta_{3} - 3744 \beta_{4} - 4336 \beta_{5} + 608 \beta_{6} + 1728 \beta_{7} - 832 \beta_{8} + 320 \beta_{9} ) q^{28} + ( -216512 + 1113873 \beta_{1} - 16256 \beta_{2} + 26274 \beta_{3} + 1231 \beta_{4} + 4800 \beta_{5} - 608 \beta_{6} - 1792 \beta_{7} - 448 \beta_{8} + 448 \beta_{9} ) q^{29} + ( 26093852 + 287344 \beta_{1} + 9710 \beta_{2} + 102718 \beta_{3} + 648 \beta_{4} + 8602 \beta_{5} + 716 \beta_{6} - 144 \beta_{7} - 348 \beta_{8} - 1004 \beta_{9} ) q^{30} + ( 43802764 + 1953596 \beta_{1} + 26848 \beta_{2} - 25768 \beta_{3} - 1484 \beta_{4} - 8520 \beta_{5} - 1496 \beta_{6} - 4372 \beta_{7} + 196 \beta_{8} - 1392 \beta_{9} ) q^{31} + ( 34315832 + 358148 \beta_{1} - 18656 \beta_{2} - 50148 \beta_{3} - 3332 \beta_{4} - 10116 \beta_{5} + 1552 \beta_{6} - 3424 \beta_{7} - 1540 \beta_{8} + 700 \beta_{9} ) q^{32} + ( -4664270 - 4085498 \beta_{1} + 68069 \beta_{2} + 30859 \beta_{3} + 5843 \beta_{4} + 11343 \beta_{5} - 1231 \beta_{6} + 4966 \beta_{7} - 3904 \beta_{8} + 354 \beta_{9} ) q^{33} + ( -18746068 + 261506 \beta_{1} - 12588 \beta_{2} + 97972 \beta_{3} + 7856 \beta_{4} + 17500 \beta_{5} + 1288 \beta_{6} + 224 \beta_{7} - 1384 \beta_{8} - 1736 \beta_{9} ) q^{34} + ( 988600 - 4882312 \beta_{1} - 61760 \beta_{2} - 18464 \beta_{3} - 19348 \beta_{4} - 18120 \beta_{5} - 1700 \beta_{6} + 1760 \beta_{7} + 440 \beta_{8} - 440 \beta_{9} ) q^{35} + ( -43240267 + 357758 \beta_{1} - 33409 \beta_{2} - 311328 \beta_{3} + 15296 \beta_{4} - 18656 \beta_{5} + 1216 \beta_{6} - 9856 \beta_{7} + 1664 \beta_{8} - 128 \beta_{9} ) q^{36} + ( -1014680 + 4998159 \beta_{1} - 10704 \beta_{2} - 43890 \beta_{3} + 1405 \beta_{4} + 21400 \beta_{5} - 1180 \beta_{6} + 416 \beta_{7} + 104 \beta_{8} - 104 \beta_{9} ) q^{37} + ( 37435817 + 11186 \beta_{1} - 26127 \beta_{2} - 310888 \beta_{3} - 29824 \beta_{4} + 23176 \beta_{5} + 1136 \beta_{6} - 839 \beta_{7} + 3504 \beta_{8} + 2416 \beta_{9} ) q^{38} + ( -44571851 + 7145401 \beta_{1} - 17414 \beta_{2} - 92480 \beta_{3} + 5661 \beta_{4} - 29248 \beta_{5} + 832 \beta_{6} + 9693 \beta_{7} - 93 \beta_{8} + 3200 \beta_{9} ) q^{39} + ( 151947256 + 547444 \beta_{1} - 6200 \beta_{2} + 642948 \beta_{3} - 24308 \beta_{4} - 28204 \beta_{5} - 992 \beta_{6} + 19008 \beta_{7} + 1436 \beta_{8} - 1252 \beta_{9} ) q^{40} + ( -64291914 - 5985652 \beta_{1} - 41318 \beta_{2} + 99078 \beta_{3} - 17418 \beta_{4} + 31822 \beta_{5} + 882 \beta_{6} - 17460 \beta_{7} + 14688 \beta_{8} - 924 \beta_{9} ) q^{41} + ( -369645488 + 176080 \beta_{1} + 170276 \beta_{2} + 989276 \beta_{3} + 31552 \beta_{4} + 23412 \beta_{5} - 2024 \beta_{6} + 2040 \beta_{7} + 952 \beta_{8} + 6168 \beta_{9} ) q^{42} + ( 1220456 - 6308091 \beta_{1} + 87232 \beta_{2} - 269059 \beta_{3} + 99812 \beta_{4} - 29592 \beta_{5} + 5044 \beta_{6} - 10848 \beta_{7} - 2712 \beta_{8} + 2712 \beta_{9} ) q^{43} + ( -412594726 - 1739638 \beta_{1} + 22684 \beta_{2} - 1315224 \beta_{3} + 12058 \beta_{4} - 31772 \beta_{5} - 6132 \beta_{6} + 35304 \beta_{7} + 3422 \beta_{8} - 2770 \beta_{9} ) q^{44} + ( -2114680 + 10977401 \beta_{1} + 266480 \beta_{2} + 727842 \beta_{3} - 14461 \beta_{4} + 37240 \beta_{5} + 6580 \beta_{6} + 20000 \beta_{7} + 5000 \beta_{8} - 5000 \beta_{9} ) q^{45} + ( 603578548 - 5005360 \beta_{1} + 305786 \beta_{2} - 1278742 \beta_{3} + 61144 \beta_{4} + 17246 \beta_{5} - 8380 \beta_{6} + 4048 \beta_{7} - 11636 \beta_{8} + 92 \beta_{9} ) q^{46} + ( -57393782 + 7235794 \beta_{1} - 341428 \beta_{2} - 25976 \beta_{3} - 2254 \beta_{4} - 13656 \beta_{5} + 10232 \beta_{6} + 10714 \beta_{7} - 2506 \beta_{8} + 2736 \beta_{9} ) q^{47} + ( 935962648 - 79276 \beta_{1} - 38192 \beta_{2} + 965596 \beta_{3} + 55852 \beta_{4} - 16804 \beta_{5} - 13264 \beta_{6} - 63776 \beta_{7} + 12172 \beta_{8} - 4660 \beta_{9} ) q^{48} + ( 114547289 - 384784 \beta_{1} - 398648 \beta_{2} + 70328 \beta_{3} + 9208 \beta_{4} + 13656 \beta_{5} + 11944 \beta_{6} + 18416 \beta_{7} - 26624 \beta_{8} - 2736 \beta_{9} ) q^{49} + ( -961771734 - 3213043 \beta_{1} - 480760 \beta_{2} + 858824 \beta_{3} - 103456 \beta_{4} - 12584 \beta_{5} - 11312 \beta_{6} - 8512 \beta_{7} + 3696 \beta_{8} - 4432 \beta_{9} ) q^{50} + ( -2052116 + 10280854 \beta_{1} + 349216 \beta_{2} + 754554 \beta_{3} - 338018 \beta_{4} + 30956 \beta_{5} + 11126 \beta_{6} + 2992 \beta_{7} + 748 \beta_{8} - 748 \beta_{9} ) q^{51} + ( -1157688060 - 9053020 \beta_{1} - 14320 \beta_{2} + 537464 \beta_{3} - 150780 \beta_{4} + 26464 \beta_{5} - 7512 \beta_{6} - 77968 \beta_{7} - 23908 \beta_{8} + 5884 \beta_{9} ) q^{52} + ( 2582232 - 15615861 \beta_{1} + 45648 \beta_{2} - 2729690 \beta_{3} + 2889 \beta_{4} - 61400 \beta_{5} + 7868 \beta_{6} - 50336 \beta_{7} - 12584 \beta_{8} + 12584 \beta_{9} ) q^{53} + ( 2005805898 - 2071308 \beta_{1} - 963246 \beta_{2} + 529496 \beta_{3} + 116608 \beta_{4} - 69432 \beta_{5} - 7184 \beta_{6} - 1958 \beta_{7} + 1840 \beta_{8} - 12048 \beta_{9} ) q^{54} + ( 568575571 - 26619281 \beta_{1} - 19850 \beta_{2} + 415968 \beta_{3} - 57973 \beta_{4} + 122976 \beta_{5} - 7392 \beta_{6} - 96277 \beta_{7} + 3541 \beta_{8} - 30912 \beta_{9} ) q^{55} + ( 1871677552 - 11276504 \beta_{1} + 195312 \beta_{2} - 2316568 \beta_{3} + 142552 \beta_{4} + 111816 \beta_{5} + 14592 \beta_{6} + 118272 \beta_{7} - 42824 \beta_{8} + 21688 \beta_{9} ) q^{56} + ( -85805878 + 36825230 \beta_{1} + 208193 \beta_{2} - 531601 \beta_{3} + 33687 \beta_{4} - 165149 \beta_{5} - 5731 \beta_{6} + 40302 \beta_{7} - 3264 \beta_{8} + 12346 \beta_{9} ) q^{57} + ( -2284591180 + 3142060 \beta_{1} + 1213675 \beta_{2} - 4376051 \beta_{3} - 118032 \beta_{4} - 140497 \beta_{5} + 8610 \beta_{6} - 350 \beta_{7} - 9446 \beta_{8} - 32350 \beta_{9} ) q^{58} + ( -8461184 + 42222853 \beta_{1} - 483968 \beta_{2} - 940155 \beta_{3} + 703296 \beta_{4} + 202240 \beta_{5} - 31296 \beta_{6} + 84480 \beta_{7} + 21120 \beta_{8} - 21120 \beta_{9} ) q^{59} + ( -4372983288 + 29627856 \beta_{1} + 253960 \beta_{2} + 5819632 \beta_{3} + 133216 \beta_{4} + 249232 \beta_{5} + 26976 \beta_{6} + 83136 \beta_{7} + 33472 \beta_{8} + 6976 \beta_{9} ) q^{60} + ( 16295416 - 72162927 \beta_{1} - 1242608 \beta_{2} + 7484402 \beta_{3} + 70579 \beta_{4} - 300280 \beta_{5} - 30196 \beta_{6} - 43040 \beta_{7} - 10760 \beta_{8} + 10760 \beta_{9} ) q^{61} + ( 4070838192 + 38370368 \beta_{1} + 1877720 \beta_{2} + 7064344 \beta_{3} - 435808 \beta_{4} - 275768 \beta_{5} + 46192 \beta_{6} - 25408 \beta_{7} + 87632 \beta_{8} + 17680 \beta_{9} ) q^{62} + ( -1430334359 - 91562819 \beta_{1} + 1960906 \beta_{2} + 584824 \beta_{3} + 116313 \beta_{4} + 249944 \beta_{5} - 29432 \beta_{6} + 139953 \beta_{7} + 19263 \beta_{8} + 53072 \beta_{9} ) q^{63} + ( 5519192752 + 29542376 \beta_{1} + 226560 \beta_{2} - 6447464 \beta_{3} - 333288 \beta_{4} + 392408 \beta_{5} + 43296 \beta_{6} - 42176 \beta_{7} + 9112 \beta_{8} - 7912 \beta_{9} ) q^{64} + ( -27488980 + 94595020 \beta_{1} + 1712570 \beta_{2} - 1420570 \beta_{3} - 46250 \beta_{4} - 425810 \beta_{5} - 63470 \beta_{6} - 101620 \beta_{7} + 125920 \beta_{8} + 8100 \beta_{9} ) q^{65} + ( -8315022176 + 6223044 \beta_{1} - 3684404 \beta_{2} - 10252564 \beta_{3} + 665296 \beta_{4} - 432508 \beta_{5} + 65336 \beta_{6} + 68128 \beta_{7} + 13864 \beta_{8} + 93320 \beta_{9} ) q^{66} + ( -16267180 + 83669949 \beta_{1} - 973600 \beta_{2} + 3143833 \beta_{3} - 878590 \beta_{4} + 352852 \beta_{5} - 36950 \beta_{6} - 129712 \beta_{7} - 32428 \beta_{8} + 32428 \beta_{9} ) q^{67} + ( -8326532250 - 11545564 \beta_{1} + 290130 \beta_{2} + 8041248 \beta_{3} + 539200 \beta_{4} + 478688 \beta_{5} + 51520 \beta_{6} + 66176 \beta_{7} + 60800 \beta_{8} - 25472 \beta_{9} ) q^{68} + ( 22037272 - 131532356 \beta_{1} - 1072688 \beta_{2} - 23509752 \beta_{3} - 32264 \beta_{4} - 375832 \beta_{5} - 52132 \beta_{6} + 339040 \beta_{7} + 84760 \beta_{8} - 84760 \beta_{9} ) q^{69} + ( 10016571312 - 10850400 \beta_{1} - 5008800 \beta_{2} + 5455760 \beta_{3} + 9472 \beta_{4} - 397008 \beta_{5} + 32416 \beta_{6} + 52496 \beta_{7} - 186848 \beta_{8} - 864 \beta_{9} ) q^{70} + ( -1153643697 - 117774885 \beta_{1} + 1959894 \beta_{2} + 903960 \beta_{3} + 134991 \beta_{4} + 343416 \beta_{5} - 31128 \beta_{6} + 135879 \beta_{7} - 39831 \beta_{8} + 32016 \beta_{9} ) q^{71} + ( 11296915518 - 55686435 \beta_{1} + 85950 \beta_{2} + 803621 \beta_{3} - 808733 \beta_{4} + 451233 \beta_{5} - 34304 \beta_{6} - 349184 \beta_{7} + 201823 \beta_{8} - 103329 \beta_{9} ) q^{72} + ( 3217528776 + 66203286 \beta_{1} - 1225667 \beta_{2} - 595661 \beta_{3} + 93819 \beta_{4} - 263721 \beta_{5} + 10793 \beta_{6} + 17654 \beta_{7} - 278528 \beta_{8} - 86958 \beta_{9} ) q^{73} + ( -10213380980 + 10305460 \beta_{1} + 5212701 \beta_{2} + 166411 \beta_{3} + 160144 \beta_{4} - 262951 \beta_{5} - 2738 \beta_{6} - 102610 \beta_{7} - 37866 \beta_{8} - 35506 \beta_{9} ) q^{74} + ( -15054920 + 73409749 \beta_{1} + 1574080 \beta_{2} - 106947 \beta_{3} + 571756 \beta_{4} + 217400 \beta_{5} + 73820 \beta_{6} - 315680 \beta_{7} - 78920 \beta_{8} + 78920 \beta_{9} ) q^{75} + ( -9761081326 + 48893698 \beta_{1} - 125460 \beta_{2} - 3697336 \beta_{3} - 935854 \beta_{4} + 149652 \beta_{5} - 78692 \beta_{6} - 431928 \beta_{7} - 252474 \beta_{8} - 25770 \beta_{9} ) q^{76} + ( 1666408 + 38355204 \beta_{1} + 1004848 \beta_{2} + 47958904 \beta_{3} - 209144 \beta_{4} - 101480 \beta_{5} + 60132 \beta_{6} - 256608 \beta_{7} - 64152 \beta_{8} + 64152 \beta_{9} ) q^{77} + ( 14451020212 - 56442032 \beta_{1} + 7206010 \beta_{2} - 18013462 \beta_{3} + 685784 \beta_{4} - 113314 \beta_{5} - 113340 \beta_{6} + 52688 \beta_{7} - 116596 \beta_{8} + 38492 \beta_{9} ) q^{78} + ( 5721090582 + 45497870 \beta_{1} - 6088596 \beta_{2} + 1220512 \beta_{3} - 442202 \beta_{4} + 193056 \beta_{5} + 100448 \beta_{6} - 561978 \beta_{7} - 98694 \beta_{8} - 220224 \beta_{9} ) q^{79} + ( 12809337424 + 135217688 \beta_{1} + 376160 \beta_{2} + 16560776 \beta_{3} + 1496296 \beta_{4} - 426616 \beta_{5} - 37728 \beta_{6} + 877632 \beta_{7} - 316376 \beta_{8} + 158632 \beta_{9} ) q^{80} + ( -2312093605 - 19731850 \beta_{1} - 6363859 \beta_{2} + 1806275 \beta_{3} - 414261 \beta_{4} + 494119 \beta_{5} + 182873 \beta_{6} - 210474 \beta_{7} + 273216 \beta_{8} + 20914 \beta_{9} ) q^{81} + ( -12312528924 - 57929510 \beta_{1} - 6725224 \beta_{2} + 32720856 \beta_{3} - 2787936 \beta_{4} + 502536 \beta_{5} - 260496 \beta_{6} - 249792 \beta_{7} + 56400 \beta_{8} - 222960 \beta_{9} ) q^{82} + ( 20078112 - 106008263 \beta_{1} + 3376512 \beta_{2} - 5040743 \beta_{3} + 818384 \beta_{4} - 419680 \beta_{5} + 66448 \beta_{6} + 860288 \beta_{7} + 215072 \beta_{8} - 215072 \beta_{9} ) q^{83} + ( -19938649552 - 341457488 \beta_{1} + 941120 \beta_{2} - 41470944 \beta_{3} - 498128 \beta_{4} - 1058816 \beta_{5} - 215328 \beta_{6} + 894784 \beta_{7} + 171088 \beta_{8} + 58576 \beta_{9} ) q^{84} + ( -63920840 + 225581814 \beta_{1} + 7059600 \beta_{2} - 84241092 \beta_{3} + 1966 \beta_{4} + 973000 \beta_{5} + 285740 \beta_{6} - 850720 \beta_{7} - 212680 \beta_{8} + 212680 \beta_{9} ) q^{85} + ( 13015261421 - 7166566 \beta_{1} - 6112387 \beta_{2} - 33821776 \beta_{3} + 65280 \beta_{4} + 1204880 \beta_{5} - 150048 \beta_{6} - 271731 \beta_{7} + 828512 \beta_{8} - 118816 \beta_{9} ) q^{86} + ( -7474307855 + 458162149 \beta_{1} - 8762758 \beta_{2} - 4149032 \beta_{3} - 51391 \beta_{4} - 1404360 \beta_{5} + 278696 \beta_{6} + 473977 \beta_{7} + 266039 \beta_{8} + 246672 \beta_{9} ) q^{87} + ( 17015026452 - 434915202 \beta_{1} - 1340756 \beta_{2} + 25004086 \beta_{3} + 3631170 \beta_{4} - 1806162 \beta_{5} - 186640 \beta_{6} - 935968 \beta_{7} - 264038 \beta_{8} + 181722 \beta_{9} ) q^{88} + ( -10738746248 - 442959210 \beta_{1} + 3292029 \beta_{2} + 3733747 \beta_{3} + 660923 \beta_{4} + 1443991 \beta_{5} + 86633 \beta_{6} + 1133302 \beta_{7} + 23936 \beta_{8} + 385746 \beta_{9} ) q^{89} + ( -22845470316 + 12857580 \beta_{1} + 10418475 \beta_{2} + 48830765 \beta_{3} + 1322224 \beta_{4} + 1521039 \beta_{5} - 94238 \beta_{6} + 752962 \beta_{7} + 93274 \beta_{8} + 356322 \beta_{9} ) q^{90} + ( 85050920 - 423745560 \beta_{1} + 2849344 \beta_{2} + 4825376 \beta_{3} - 4794940 \beta_{4} - 1792344 \beta_{5} + 126100 \beta_{6} + 493728 \beta_{7} + 123432 \beta_{8} - 123432 \beta_{9} ) q^{91} + ( -14245349800 + 628517872 \beta_{1} - 3472808 \beta_{2} - 35868592 \beta_{3} + 1327136 \beta_{4} - 2374864 \beta_{5} + 178976 \beta_{6} - 1188800 \beta_{7} + 513600 \beta_{8} + 250816 \beta_{9} ) q^{92} + ( -133840320 + 809857328 \beta_{1} + 5536128 \beta_{2} + 149129696 \beta_{3} + 796080 \beta_{4} + 2765760 \beta_{5} - 17760 \beta_{6} + 1452288 \beta_{7} + 363072 \beta_{8} - 363072 \beta_{9} ) q^{93} + ( 14543774440 - 71763168 \beta_{1} + 6760756 \beta_{2} - 10948588 \beta_{3} + 1946800 \beta_{4} + 2620796 \beta_{5} - 48376 \beta_{6} + 83872 \beta_{7} - 414056 \beta_{8} - 242632 \beta_{9} ) q^{94} + ( 6883763639 + 638615331 \beta_{1} + 10832190 \beta_{2} - 9799648 \beta_{3} + 61903 \beta_{4} - 2994016 \beta_{5} - 400928 \beta_{6} - 331153 \beta_{7} + 354769 \beta_{8} + 7872 \beta_{9} ) q^{95} + ( 10770606896 + 921848744 \beta_{1} - 237632 \beta_{2} + 23132440 \beta_{3} - 5903272 \beta_{4} - 2293800 \beta_{5} - 53344 \beta_{6} + 462912 \beta_{7} + 987992 \beta_{8} - 545064 \beta_{9} ) q^{96} + ( 6261753092 - 814494854 \beta_{1} + 6482635 \beta_{2} + 8644101 \beta_{3} + 322397 \beta_{4} + 2803329 \beta_{5} - 163521 \beta_{6} - 121702 \beta_{7} - 720032 \beta_{8} - 280578 \beta_{9} ) q^{97} + ( -647303662 + 118290729 \beta_{1} - 51232 \beta_{2} - 21080864 \beta_{3} + 5542016 \beta_{4} + 3474592 \beta_{5} + 630976 \beta_{6} + 382208 \beta_{7} - 412096 \beta_{8} - 117440 \beta_{9} ) q^{98} + ( 181894176 - 905730717 \beta_{1} - 13258368 \beta_{2} - 23125053 \beta_{3} + 9910224 \beta_{4} - 3673440 \beta_{5} - 115056 \beta_{6} - 2828160 \beta_{7} - 707040 \beta_{8} + 707040 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 22q^{2} + 436q^{4} - 24308q^{6} - 33616q^{7} - 208472q^{8} - 472394q^{9} + O(q^{10}) \) \( 10q + 22q^{2} + 436q^{4} - 24308q^{6} - 33616q^{7} - 208472q^{8} - 472394q^{9} + 29864q^{10} - 476216q^{12} + 2264272q^{14} + 3391792q^{15} + 3757072q^{16} + 2639732q^{17} + 2329130q^{18} + 8296432q^{20} - 24201892q^{22} - 45357808q^{23} + 10668944q^{24} - 41502654q^{25} - 101931080q^{26} - 95722144q^{28} + 261362768q^{30} + 442035392q^{31} + 343908512q^{32} - 54732216q^{33} - 187103796q^{34} - 431195700q^{36} + 375100844q^{38} - 431459312q^{39} + 1519138784q^{40} - 654907964q^{41} - 3697737056q^{42} - 4127012952q^{44} + 6028867440q^{46} - 560226528q^{47} + 9357606048q^{48} + 1143661722q^{49} - 9626584226q^{50} - 11595427248q^{52} + 20050494008q^{54} + 5632783984q^{55} + 18698733760q^{56} - 783862360q^{57} - 22828679464q^{58} - 43681325472q^{60} + 40774631744q^{62} - 14483624624q^{63} + 55266728512q^{64} - 80291360q^{65} - 83128448712q^{66} - 83303223768q^{68} + 100120831168q^{70} - 11769460176q^{71} + 112862051672q^{72} + 32304890372q^{73} - 102103996184q^{74} - 97501928568q^{76} + 144445560944q^{78} + 57291670496q^{79} + 128322038976q^{80} - 23172996094q^{81} - 123309929604q^{82} - 199990865984q^{84} + 130201908124q^{86} - 73844404272q^{87} + 169216119632q^{88} - 108276011804q^{89} - 228506434984q^{90} - 141139403232q^{92} + 145336130016q^{94} + 70146835376q^{95} + 109508068928q^{96} + 60995282900q^{97} - 6194726778q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - 48 x^{8} + 8836 x^{7} - 92554 x^{6} - 2200342 x^{5} - 88894588 x^{4} + 2040335460 x^{3} + 7439251985 x^{2} - 747501940689 x + 34431558027300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} + 4 \nu - 39 \)
\(\beta_{3}\)\(=\)\((\)\(73 \nu^{9} + 4718 \nu^{8} + 80354 \nu^{7} - 2051134 \nu^{6} - 24450588 \nu^{5} - 408604458 \nu^{4} - 11443621426 \nu^{3} - 308359737162 \nu^{2} + 4232086908963 \nu + 403377232281156\)\()/ 360777252864 \)
\(\beta_{4}\)\(=\)\((\)\(-41 \nu^{9} - 13102 \nu^{8} - 155170 \nu^{7} + 2037758 \nu^{6} + 23134236 \nu^{5} - 525426582 \nu^{4} + 3615136370 \nu^{3} + 1106797363338 \nu^{2} - 14788734861891 \nu - 643828474284996\)\()/ 45097156608 \)
\(\beta_{5}\)\(=\)\((\)\(823 \nu^{9} - 6510 \nu^{8} - 121570 \nu^{7} + 10009406 \nu^{6} - 66436068 \nu^{5} - 1472015318 \nu^{4} - 66325309646 \nu^{3} + 1733232903242 \nu^{2} + 1489646040669 \nu - 1005622921756740\)\()/ 120259084288 \)
\(\beta_{6}\)\(=\)\((\)\(817 \nu^{9} + 36510 \nu^{8} - 446926 \nu^{7} + 8893202 \nu^{6} + 459904388 \nu^{5} - 8773528250 \nu^{4} - 183165333250 \nu^{3} - 416681889754 \nu^{2} + 71508096851483 \nu - 1436607317152028\)\()/ 60129542144 \)
\(\beta_{7}\)\(=\)\((\)\(19 \nu^{9} + 346 \nu^{8} - 10218 \nu^{7} - 77130 \nu^{6} - 1064180 \nu^{5} - 20948910 \nu^{4} - 1831046278 \nu^{3} + 13205710066 \nu^{2} + 1802560181073 \nu - 6502213267284\)\()/ 704643072 \)
\(\beta_{8}\)\(=\)\((\)\(-2899 \nu^{9} - 9690 \nu^{8} + 835178 \nu^{7} + 607178 \nu^{6} - 145501964 \nu^{5} - 2057896146 \nu^{4} + 322600568326 \nu^{3} - 5329960471922 \nu^{2} - 133161865489425 \nu + 1779580494171732\)\()/ 45097156608 \)
\(\beta_{9}\)\(=\)\((\)\(12521 \nu^{9} - 116434 \nu^{8} - 5184222 \nu^{7} + 31910658 \nu^{6} + 698838116 \nu^{5} - 1303068522 \nu^{4} + 67991356558 \nu^{3} + 39316578282614 \nu^{2} + 346711472029443 \nu - 11565474023541564\)\()/ 180388626432 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1} + 39\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9} + \beta_{8} - \beta_{5} - 3 \beta_{4} - 5 \beta_{3} - 4 \beta_{2} + 67 \beta_{1} - 21104\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{9} - 21 \beta_{8} - 8 \beta_{7} - 20 \beta_{6} - 33 \beta_{5} - 109 \beta_{4} - 1921 \beta_{3} + 32 \beta_{2} - 10539 \beta_{1} + 273442\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(95 \beta_{9} - 145 \beta_{8} - 408 \beta_{7} + 244 \beta_{6} - 1177 \beta_{5} - 129 \beta_{4} - 1441 \beta_{3} - 2402 \beta_{2} + 70791 \beta_{1} + 3711000\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-1037 \beta_{9} + 1587 \beta_{8} - 128 \beta_{7} + 1392 \beta_{6} + 31845 \beta_{5} - 19209 \beta_{4} - 379863 \beta_{3} + 28357 \beta_{2} + 1500043 \beta_{1} + 320843183\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(32187 \beta_{9} - 84261 \beta_{8} - 485424 \beta_{7} + 12840 \beta_{6} + 339441 \beta_{5} - 265529 \beta_{4} + 13340845 \beta_{3} + 1309856 \beta_{2} + 331410209 \beta_{1} - 14601281944\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-251631 \beta_{9} + 2142913 \beta_{8} + 4932632 \beta_{7} + 1984220 \beta_{6} + 2358769 \beta_{5} - 26529503 \beta_{4} - 222424471 \beta_{3} + 164103968 \beta_{2} - 7471547601 \beta_{1} - 111706686750\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(28743150 \beta_{9} - 3619858 \beta_{8} - 27438368 \beta_{7} - 3161440 \beta_{6} + 33384786 \beta_{5} - 59845738 \beta_{4} + 1344906586 \beta_{3} - 1015777294 \beta_{2} - 11867736769 \beta_{1} - 461581578530\)\()/2\)

Character Values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−23.4302 2.98090i
−23.4302 + 2.98090i
−13.4162 18.9166i
−13.4162 + 18.9166i
7.36621 21.0240i
7.36621 + 21.0240i
9.13304 20.2317i
9.13304 + 20.2317i
20.8472 5.89062i
20.8472 + 5.89062i
−44.8604 5.96180i 614.710i 1976.91 + 534.897i 4397.69i −3664.78 + 27576.1i −41556.6 −85496.2 35781.7i −200721. −26218.2 + 197282.i
5.2 −44.8604 + 5.96180i 614.710i 1976.91 534.897i 4397.69i −3664.78 27576.1i −41556.6 −85496.2 + 35781.7i −200721. −26218.2 197282.i
5.3 −24.8325 37.8331i 102.287i −814.694 + 1878.98i 2319.82i 3869.83 2540.04i 38165.0 91318.7 15837.4i 166684. 87766.1 57606.9i
5.4 −24.8325 + 37.8331i 102.287i −814.694 1878.98i 2319.82i 3869.83 + 2540.04i 38165.0 91318.7 + 15837.4i 166684. 87766.1 + 57606.9i
5.5 16.7324 42.0479i 284.352i −1488.05 1407.13i 10467.0i 11956.4 + 4757.90i −70503.2 −84065.4 + 39024.9i 96290.8 −440117. 175139.i
5.6 16.7324 + 42.0479i 284.352i −1488.05 + 1407.13i 10467.0i 11956.4 4757.90i −70503.2 −84065.4 39024.9i 96290.8 −440117. + 175139.i
5.7 20.2661 40.4634i 712.065i −1226.57 1640.07i 7122.63i −28812.6 14430.8i 46911.8 −91220.5 + 16393.5i −329890. 288206. + 144348.i
5.8 20.2661 + 40.4634i 712.065i −1226.57 + 1640.07i 7122.63i −28812.6 + 14430.8i 46911.8 −91220.5 16393.5i −329890. 288206. 144348.i
5.9 43.6944 11.7812i 381.717i 1770.40 1029.55i 8937.55i 4497.10 + 16678.9i 10175.0 65227.5 65843.1i 31439.0 105295. + 390521.i
5.10 43.6944 + 11.7812i 381.717i 1770.40 + 1029.55i 8937.55i 4497.10 16678.9i 10175.0 65227.5 + 65843.1i 31439.0 105295. 390521.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{12}^{\mathrm{new}}(8, [\chi])\).