Properties

Label 12.13.d.a
Level 12
Weight 13
Character orbit 12.d
Analytic conductor 10.968
Analytic rank 0
Dimension 12
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 13 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(10.9679258073\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{25} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -8 + \beta_{2} ) q^{2} \) \( + ( \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -386 + 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -867 - \beta_{1} + 17 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{5} \) \( + ( -4010 - 7 \beta_{1} - 7 \beta_{2} - \beta_{3} - \beta_{7} ) q^{6} \) \( + ( 181 + 42 \beta_{1} - 318 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} \) \( + ( -53671 + 169 \beta_{1} - 499 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{8} \) \( -177147 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -8 + \beta_{2} ) q^{2} \) \( + ( \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -386 + 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -867 - \beta_{1} + 17 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{5} \) \( + ( -4010 - 7 \beta_{1} - 7 \beta_{2} - \beta_{3} - \beta_{7} ) q^{6} \) \( + ( 181 + 42 \beta_{1} - 318 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} \) \( + ( -53671 + 169 \beta_{1} - 499 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{8} \) \( -177147 q^{9} \) \( + ( 77420 + 314 \beta_{1} - 703 \beta_{2} - 18 \beta_{3} + 26 \beta_{4} - 9 \beta_{5} + 11 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} - \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{10} \) \( + ( 5020 + 54 \beta_{1} - 9978 \beta_{2} + 52 \beta_{3} - 32 \beta_{4} + 14 \beta_{5} + 4 \beta_{6} - 16 \beta_{7} + 4 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{11} \) \( + ( -249403 - 295 \beta_{1} - 4530 \beta_{2} + 10 \beta_{3} + 34 \beta_{4} - \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 19 \beta_{8} + 8 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} ) q^{12} \) \( + ( 171825 - 221 \beta_{1} + 5268 \beta_{2} + 156 \beta_{3} - 24 \beta_{4} - 117 \beta_{6} - 13 \beta_{7} - 63 \beta_{8} + 25 \beta_{9} - 14 \beta_{10} + 2 \beta_{11} ) q^{13} \) \( + ( 1281500 + 478 \beta_{1} + 2277 \beta_{2} + 294 \beta_{3} - 7 \beta_{5} - 177 \beta_{6} - 55 \beta_{7} - 44 \beta_{8} + \beta_{9} - 38 \beta_{10} + 17 \beta_{11} ) q^{14} \) \( + ( -6528 - 1089 \beta_{1} + 11999 \beta_{2} - 168 \beta_{3} + 8 \beta_{4} - 59 \beta_{5} - 18 \beta_{6} - 38 \beta_{7} - 14 \beta_{8} - 14 \beta_{9} + 2 \beta_{10} + 17 \beta_{11} ) q^{15} \) \( + ( -5092570 + 4994 \beta_{1} - 64478 \beta_{2} + 786 \beta_{3} - 56 \beta_{4} + 84 \beta_{5} - 188 \beta_{6} - 176 \beta_{7} - 90 \beta_{8} - 58 \beta_{9} + 82 \beta_{10} - 2 \beta_{11} ) q^{16} \) \( + ( -924982 + 4088 \beta_{1} - 162774 \beta_{2} - 12 \beta_{3} + 808 \beta_{4} + 24 \beta_{5} + 452 \beta_{6} + 226 \beta_{7} - 194 \beta_{8} + 54 \beta_{9} + 76 \beta_{10} + 4 \beta_{11} ) q^{17} \) \( + ( 1417176 - 177147 \beta_{2} ) q^{18} \) \( + ( -74090 + 24184 \beta_{1} + 170872 \beta_{2} + 886 \beta_{3} + 524 \beta_{4} + 258 \beta_{5} + 104 \beta_{6} - 918 \beta_{7} + 192 \beta_{8} + 102 \beta_{9} + 160 \beta_{10} + 48 \beta_{11} ) q^{19} \) \( + ( -31475556 + 8036 \beta_{1} + 19072 \beta_{2} + 1802 \beta_{3} - 536 \beta_{4} - 792 \beta_{5} - 384 \beta_{6} - 424 \beta_{7} - 112 \beta_{8} + 240 \beta_{9} - 328 \beta_{10} + 56 \beta_{11} ) q^{20} \) \( + ( -6234304 + 808 \beta_{1} - 40645 \beta_{2} + 10 \beta_{3} + 472 \beta_{4} + 272 \beta_{5} - 306 \beta_{6} + 687 \beta_{7} + 389 \beta_{8} + 173 \beta_{9} - 422 \beta_{10} + 58 \beta_{11} ) q^{21} \) \( + ( 40206448 + 60000 \beta_{1} + 71762 \beta_{2} + 10680 \beta_{3} - 832 \beta_{4} - 370 \beta_{5} + 34 \beta_{6} - 782 \beta_{7} - 360 \beta_{8} + 254 \beta_{9} + 204 \beta_{10} - 226 \beta_{11} ) q^{22} \) \( + ( -7630 - 94380 \beta_{1} - 80124 \beta_{2} - 19254 \beta_{3} - 1396 \beta_{4} + 590 \beta_{5} + 2800 \beta_{6} - 1690 \beta_{7} - 80 \beta_{8} + 250 \beta_{9} - 80 \beta_{10} - 40 \beta_{11} ) q^{23} \) \( + ( -27541143 - 48531 \beta_{1} - 307103 \beta_{2} + 4185 \beta_{3} + 1801 \beta_{4} + 194 \beta_{5} + 2043 \beta_{6} + 1112 \beta_{7} + 392 \beta_{8} - 823 \beta_{9} - 137 \beta_{10} + 10 \beta_{11} ) q^{24} \) \( + ( 87429145 - 32834 \beta_{1} + 1580064 \beta_{2} - 15240 \beta_{3} - 5952 \beta_{4} + 688 \beta_{5} + 2542 \beta_{6} + 5830 \beta_{7} + 882 \beta_{8} - 526 \beta_{9} + 2084 \beta_{10} - 12 \beta_{11} ) q^{25} \) \( + ( 20186632 + 33156 \beta_{1} + 218136 \beta_{2} - 7156 \beta_{3} + 2020 \beta_{4} - 506 \beta_{5} - 2898 \beta_{6} + 1454 \beta_{7} + 612 \beta_{8} - 2410 \beta_{9} - 1092 \beta_{10} - 310 \beta_{11} ) q^{26} \) \( + ( -177147 \beta_{1} - 177147 \beta_{2} ) q^{27} \) \( + ( -16104308 - 42228 \beta_{1} + 1270648 \beta_{2} - 5200 \beta_{3} + 2736 \beta_{4} - 2092 \beta_{5} - 23956 \beta_{6} + 876 \beta_{7} + 420 \beta_{8} - 1824 \beta_{9} - 156 \beta_{10} - 896 \beta_{11} ) q^{28} \) \( + ( 164636393 - 91941 \beta_{1} + 3591835 \beta_{2} + 10830 \beta_{3} - 9824 \beta_{4} + 4176 \beta_{5} + 14657 \beta_{6} + 17006 \beta_{7} + 5642 \beta_{8} + 1194 \beta_{9} - 1516 \beta_{10} + 644 \beta_{11} ) q^{29} \) \( + ( -48060280 + 80736 \beta_{1} - 88997 \beta_{2} - 10748 \beta_{3} + 352 \beta_{4} - 4027 \beta_{5} - 21501 \beta_{6} + 939 \beta_{7} + 356 \beta_{8} + 1517 \beta_{9} - 1262 \beta_{10} - 35 \beta_{11} ) q^{30} \) \( + ( -555575 - 511762 \beta_{1} + 576134 \beta_{2} - 15755 \beta_{3} + 3270 \beta_{4} + 7999 \beta_{5} + 8008 \beta_{6} - 13325 \beta_{7} + 3240 \beta_{8} + 2909 \beta_{9} + 1832 \beta_{10} - 492 \beta_{11} ) q^{31} \) \( + ( 146901704 - 77360 \beta_{1} - 4947792 \beta_{2} + 66912 \beta_{3} - 18228 \beta_{4} - 5264 \beta_{5} + 28908 \beta_{6} - 9456 \beta_{7} - 3476 \beta_{8} + 536 \beta_{9} - 1976 \beta_{10} - 1884 \beta_{11} ) q^{32} \) \( + ( 28263026 + 94078 \beta_{1} - 2745850 \beta_{2} - 66524 \beta_{3} + 13624 \beta_{4} + 1016 \beta_{5} + 3042 \beta_{6} + 11820 \beta_{7} + 620 \beta_{8} - 1324 \beta_{9} + 5512 \beta_{10} - 128 \beta_{11} ) q^{33} \) \( + ( -649742344 + 946316 \beta_{1} - 2276684 \beta_{2} + 204068 \beta_{3} + 1164 \beta_{4} - 10094 \beta_{5} + 28330 \beta_{6} - 3958 \beta_{7} + 2892 \beta_{8} - 5630 \beta_{9} - 1676 \beta_{10} + 574 \beta_{11} ) q^{34} \) \( + ( -4471912 + 625434 \beta_{1} + 9524970 \beta_{2} - 381264 \beta_{3} - 3112 \beta_{4} + 170 \beta_{5} + 47620 \beta_{6} - 51772 \beta_{7} - 1772 \beta_{8} + 4516 \beta_{9} + 1108 \beta_{10} + 3434 \beta_{11} ) q^{35} \) \( + ( 68378742 - 354294 \beta_{1} + 1417176 \beta_{2} + 177147 \beta_{3} ) q^{36} \) \( + ( -74635715 + 290303 \beta_{1} - 10488294 \beta_{2} - 102048 \beta_{3} + 64632 \beta_{4} + 13920 \beta_{5} - 3309 \beta_{6} + 58909 \beta_{7} + 399 \beta_{8} + 9815 \beta_{9} - 4498 \beta_{10} + 2494 \beta_{11} ) q^{37} \) \( + ( -687630208 + 3590296 \beta_{1} - 1230814 \beta_{2} - 84000 \beta_{3} - 45056 \beta_{4} - 11146 \beta_{5} - 31174 \beta_{6} - 43886 \beta_{7} - 32904 \beta_{8} + 7078 \beta_{9} - 4260 \beta_{10} - 2042 \beta_{11} ) q^{38} \) \( + ( -693675 + 31685 \beta_{1} + 1369565 \beta_{2} + 361737 \beta_{3} + 35382 \beta_{4} + 16584 \beta_{5} - 33642 \beta_{6} - 5295 \beta_{7} + 11994 \beta_{8} - 885 \beta_{9} + 7914 \beta_{10} - 123 \beta_{11} ) q^{39} \) \( + ( 345927422 - 3702786 \beta_{1} - 31301962 \beta_{2} - 110410 \beta_{3} + 60178 \beta_{4} - 10468 \beta_{5} - 204546 \beta_{6} + 7296 \beta_{7} - 5556 \beta_{8} - 2130 \beta_{9} - 8302 \beta_{10} + 5200 \beta_{11} ) q^{40} \) \( + ( 478529910 - 418844 \beta_{1} + 10271902 \beta_{2} + 464876 \beta_{3} - 43752 \beta_{4} - 9144 \beta_{5} + 83848 \beta_{6} - 35662 \beta_{7} - 59266 \beta_{8} + 13254 \beta_{9} + 1580 \beta_{10} - 364 \beta_{11} ) q^{41} \) \( + ( -111893860 + 1449322 \beta_{1} - 6590305 \beta_{2} + 63358 \beta_{3} + 22378 \beta_{4} + 25535 \beta_{5} - 137709 \beta_{6} + 9891 \beta_{7} + 11018 \beta_{8} - 16441 \beta_{9} + 18838 \beta_{10} - 743 \beta_{11} ) q^{42} \) \( + ( 11439526 - 3357372 \beta_{1} - 26111964 \beta_{2} + 560854 \beta_{3} - 55460 \beta_{4} - 31546 \beta_{5} - 83280 \beta_{6} + 20114 \beta_{7} - 2040 \beta_{8} - 24434 \beta_{9} + 6120 \beta_{10} + 11220 \beta_{11} ) q^{43} \) \( + ( 391277780 - 7388796 \beta_{1} + 41613272 \beta_{2} - 218248 \beta_{3} - 45288 \beta_{4} - 4772 \beta_{5} + 222028 \beta_{6} - 71676 \beta_{7} - 40700 \beta_{8} - 7648 \beta_{9} - 10676 \beta_{10} + 7176 \beta_{11} ) q^{44} \) \( + ( 153586449 + 177147 \beta_{1} - 3011499 \beta_{2} - 354294 \beta_{3} - 177147 \beta_{6} ) q^{45} \) \( + ( 349818952 + 9038036 \beta_{1} + 715746 \beta_{2} + 213620 \beta_{3} + 323840 \beta_{4} + 43250 \beta_{5} + 115870 \beta_{6} + 122554 \beta_{7} - 4440 \beta_{8} + 16130 \beta_{9} - 11340 \beta_{10} - 3550 \beta_{11} ) q^{46} \) \( + ( 12155962 + 7176864 \beta_{1} - 16995888 \beta_{2} - 1497550 \beta_{3} - 175172 \beta_{4} + 27258 \beta_{5} + 197640 \beta_{6} + 44358 \beta_{7} - 30600 \beta_{8} + 12042 \beta_{9} - 33480 \beta_{10} - 19620 \beta_{11} ) q^{47} \) \( + ( -636919454 - 4635042 \beta_{1} - 29370274 \beta_{2} + 263102 \beta_{3} + 38732 \beta_{4} - 12260 \beta_{5} + 46800 \beta_{6} + 51744 \beta_{7} + 24478 \beta_{8} + 8818 \beta_{9} + 45782 \beta_{10} - 4594 \beta_{11} ) q^{48} \) \( + ( -5053078869 - 527470 \beta_{1} + 25815276 \beta_{2} - 324192 \beta_{3} - 185808 \beta_{4} - 33184 \beta_{5} - 215350 \beta_{6} - 173098 \beta_{7} + 88818 \beta_{8} - 46526 \beta_{9} - 8828 \beta_{10} - 6780 \beta_{11} ) q^{49} \) \( + ( 5675106744 + 24944120 \beta_{1} + 100582583 \beta_{2} - 1303832 \beta_{3} - 460872 \beta_{4} - 13964 \beta_{5} + 480164 \beta_{6} - 98268 \beta_{7} + 106936 \beta_{8} + 56660 \beta_{9} - 25976 \beta_{10} + 17772 \beta_{11} ) q^{50} \) \( + ( -21632250 - 2414778 \beta_{1} + 40819526 \beta_{2} + 1321446 \beta_{3} + 253676 \beta_{4} - 79262 \beta_{5} - 225144 \beta_{6} + 143770 \beta_{7} + 352 \beta_{8} + 4726 \beta_{9} + 320 \beta_{10} + 128 \beta_{11} ) q^{51} \) \( + ( -2615531588 - 33469116 \beta_{1} + 15744144 \beta_{2} - 750346 \beta_{3} + 221072 \beta_{4} + 103568 \beta_{5} + 208384 \beta_{6} + 105456 \beta_{7} + 124320 \beta_{8} + 38496 \beta_{9} + 75184 \beta_{10} - 16976 \beta_{11} ) q^{52} \) \( + ( 3549046841 - 7253 \beta_{1} - 18137005 \beta_{2} + 1565118 \beta_{3} - 31040 \beta_{4} - 100560 \beta_{5} + 457697 \beta_{6} - 528522 \beta_{7} + 77058 \beta_{8} - 73470 \beta_{9} - 41916 \beta_{10} - 15852 \beta_{11} ) q^{53} \) \( + ( 710359470 + 1240029 \beta_{1} + 1240029 \beta_{2} + 177147 \beta_{3} + 177147 \beta_{7} ) q^{54} \) \( + ( 43572356 + 7033908 \beta_{1} - 79604556 \beta_{2} + 3845540 \beta_{3} + 45032 \beta_{4} - 31664 \beta_{5} - 476088 \beta_{6} + 616516 \beta_{7} + 12840 \beta_{8} + 35084 \beta_{9} - 33624 \beta_{10} - 63276 \beta_{11} ) q^{55} \) \( + ( -3189385900 - 58847356 \beta_{1} - 21624620 \beta_{2} - 2161484 \beta_{3} - 775548 \beta_{4} + 234024 \beta_{5} + 495404 \beta_{6} + 226528 \beta_{7} + 116272 \beta_{8} + 48084 \beta_{9} - 66452 \beta_{10} - 5960 \beta_{11} ) q^{56} \) \( + ( -4777750548 + 4485672 \beta_{1} - 160700814 \beta_{2} - 1307772 \beta_{3} + 688392 \beta_{4} + 1080 \beta_{5} - 43740 \beta_{6} - 5670 \beta_{7} + 139590 \beta_{8} - 45090 \beta_{9} + 540 \beta_{10} - 2700 \beta_{11} ) q^{57} \) \( + ( 13239994580 + 63528086 \beta_{1} + 194771195 \beta_{2} - 3079806 \beta_{3} - 97546 \beta_{4} + 136385 \beta_{5} - 648947 \beta_{6} - 111907 \beta_{7} + 209814 \beta_{8} - 117383 \beta_{9} + 255914 \beta_{10} + 33351 \beta_{11} ) q^{58} \) \( + ( -94600780 + 7537320 \beta_{1} + 197026440 \beta_{2} - 2670524 \beta_{3} + 507608 \beta_{4} - 379632 \beta_{5} - 60552 \beta_{6} + 513348 \beta_{7} - 192888 \beta_{8} - 108180 \beta_{9} - 116280 \beta_{10} + 18468 \beta_{11} ) q^{59} \) \( + ( -1653218450 - 31209418 \beta_{1} - 52454172 \beta_{2} - 393964 \beta_{3} - 581372 \beta_{4} + 145562 \beta_{5} - 944766 \beta_{6} + 71926 \beta_{7} + 72374 \beta_{8} - 13648 \beta_{9} - 131006 \beta_{10} + 27820 \beta_{11} ) q^{60} \) \( + ( 11324794797 - 6308081 \beta_{1} + 239771142 \beta_{2} + 746856 \beta_{3} - 1364040 \beta_{4} - 214912 \beta_{5} - 904933 \beta_{6} - 781335 \beta_{7} - 385677 \beta_{8} - 51333 \beta_{9} + 144438 \beta_{10} - 34586 \beta_{11} ) q^{61} \) \( + ( -2288319508 + 65429014 \beta_{1} - 3867515 \beta_{2} + 752910 \beta_{3} + 22400 \beta_{4} + 184721 \beta_{5} + 1104631 \beta_{6} + 313785 \beta_{7} - 474316 \beta_{8} + 32089 \beta_{9} - 8758 \beta_{10} - 28311 \beta_{11} ) q^{62} \) \( + ( -32063607 - 7440174 \beta_{1} + 56332746 \beta_{2} - 177147 \beta_{3} + 354294 \beta_{4} + 177147 \beta_{5} + 177147 \beta_{7} - 177147 \beta_{9} ) q^{63} \) \( + ( 15564428328 - 87037240 \beta_{1} + 175007240 \beta_{2} + 2915048 \beta_{3} + 2062216 \beta_{4} - 34896 \beta_{5} + 1598200 \beta_{6} + 178080 \beta_{7} - 250272 \beta_{8} - 15480 \beta_{9} + 96536 \beta_{10} + 65616 \beta_{11} ) q^{64} \) \( + ( -27247118440 + 4907364 \beta_{1} - 210771974 \beta_{2} - 189340 \beta_{3} + 916552 \beta_{4} + 56472 \beta_{5} - 1547952 \beta_{6} - 172530 \beta_{7} - 225822 \beta_{8} + 219066 \beta_{9} - 350124 \beta_{10} + 31692 \beta_{11} ) q^{65} \) \( + ( -11375673880 + 39192364 \beta_{1} + 4891694 \beta_{2} + 4141828 \beta_{3} - 1142804 \beta_{4} - 68734 \beta_{5} + 1331226 \beta_{6} - 220230 \beta_{7} + 241964 \beta_{8} + 143090 \beta_{9} - 122540 \beta_{10} + 37006 \beta_{11} ) q^{66} \) \( + ( 470191748 + 5808144 \beta_{1} - 932235792 \beta_{2} + 2243732 \beta_{3} - 3498760 \beta_{4} + 371632 \beta_{5} + 399960 \beta_{6} - 1019948 \beta_{7} + 150120 \beta_{8} - 72292 \beta_{9} + 153000 \beta_{10} + 79380 \beta_{11} ) q^{67} \) \( + ( 6757988956 - 122605404 \beta_{1} - 649046192 \beta_{2} + 82614 \beta_{3} + 106800 \beta_{4} - 760528 \beta_{5} - 1545984 \beta_{6} - 1248176 \beta_{7} - 395040 \beta_{8} + 161824 \beta_{9} + 360336 \beta_{10} + 51856 \beta_{11} ) q^{68} \) \( + ( 18041446528 + 8181368 \beta_{1} - 353259194 \beta_{2} + 1675508 \beta_{3} + 1645904 \beta_{4} - 12032 \beta_{5} + 697284 \beta_{6} - 60906 \beta_{7} - 423278 \beta_{8} + 140482 \beta_{9} - 30460 \beta_{10} + 8228 \beta_{11} ) q^{69} \) \( + ( -37771341608 + 220366508 \beta_{1} - 66192198 \beta_{2} - 5295652 \beta_{3} + 5253056 \beta_{4} - 219022 \beta_{5} - 3068834 \beta_{6} - 337502 \beta_{7} - 512280 \beta_{8} + 696578 \beta_{9} - 580044 \beta_{10} - 70558 \beta_{11} ) q^{70} \) \( + ( -178691010 + 4433640 \beta_{1} + 358770360 \beta_{2} + 8776262 \beta_{3} + 1687860 \beta_{4} + 245438 \beta_{5} - 757256 \beta_{6} - 1623838 \beta_{7} + 468904 \beta_{8} + 291790 \beta_{9} + 392296 \beta_{10} + 119540 \beta_{11} ) q^{71} \) \( + ( 9507656637 - 29937843 \beta_{1} + 88396353 \beta_{2} - 2302911 \beta_{3} - 1240029 \beta_{4} - 354294 \beta_{5} - 177147 \beta_{6} - 354294 \beta_{8} - 177147 \beta_{9} + 177147 \beta_{10} ) q^{72} \) \( + ( -67551716792 - 27828466 \beta_{1} + 1001161860 \beta_{2} + 9427968 \beta_{3} - 3431088 \beta_{4} + 502688 \beta_{5} + 4314806 \beta_{6} + 2251130 \beta_{7} + 159870 \beta_{8} + 273838 \beta_{9} - 57764 \beta_{10} + 81756 \beta_{11} ) q^{73} \) \( + ( -41723568096 + 186483432 \beta_{1} - 163010498 \beta_{2} + 15320248 \beta_{3} - 1098040 \beta_{4} + 400988 \beta_{5} - 2044596 \beta_{6} - 16532 \beta_{7} + 1131144 \beta_{8} - 922820 \beta_{9} + 290136 \beta_{10} + 26980 \beta_{11} ) q^{74} \) \( + ( -547284294 + 70071153 \beta_{1} + 1160872065 \beta_{2} - 15650478 \beta_{3} + 2641116 \beta_{4} - 330600 \beta_{5} + 1601100 \beta_{6} - 1327854 \beta_{7} - 99804 \beta_{8} + 722022 \beta_{9} - 91644 \beta_{10} - 37662 \beta_{11} ) q^{75} \) \( + ( 12261458332 - 139926548 \beta_{1} - 675507320 \beta_{2} - 1966136 \beta_{3} + 4535336 \beta_{4} - 6476 \beta_{5} + 1696548 \beta_{6} - 2557204 \beta_{7} - 22068 \beta_{8} - 391328 \beta_{9} - 886012 \beta_{10} - 101000 \beta_{11} ) q^{76} \) \( + ( 115878354372 + 35586868 \beta_{1} - 1100116436 \beta_{2} - 21855960 \beta_{3} + 5814576 \beta_{4} + 719472 \beta_{5} + 932028 \beta_{6} + 4978576 \beta_{7} + 571456 \beta_{8} - 155568 \beta_{9} + 1294432 \beta_{10} + 39760 \beta_{11} ) q^{77} \) \( + ( -5985854716 + 18848638 \beta_{1} - 11974484 \beta_{2} - 46214 \beta_{3} - 6818592 \beta_{4} - 430062 \beta_{5} + 1214622 \beta_{6} - 1330016 \beta_{7} - 1269816 \beta_{8} - 181662 \beta_{9} + 342708 \beta_{10} - 100830 \beta_{11} ) q^{78} \) \( + ( 618209685 - 170505466 \beta_{1} - 1400622178 \beta_{2} - 32874879 \beta_{3} - 7765090 \beta_{4} - 872029 \beta_{5} + 4122136 \beta_{6} - 2174809 \beta_{7} - 771288 \beta_{8} + 83017 \beta_{9} - 372376 \beta_{10} + 212724 \beta_{11} ) q^{79} \) \( + ( -35003118668 - 140669124 \beta_{1} + 256466172 \beta_{2} + 31189404 \beta_{3} - 6188528 \beta_{4} + 322904 \beta_{5} - 9647464 \beta_{6} + 4839648 \beta_{7} + 1064084 \beta_{8} - 295436 \beta_{9} - 779812 \beta_{10} - 269532 \beta_{11} ) q^{80} \) \( + 31381059609 q^{81} \) \( + ( 38112105896 + 18720212 \beta_{1} + 570328912 \beta_{2} - 13195780 \beta_{3} + 5323092 \beta_{4} - 2629378 \beta_{5} + 3671078 \beta_{6} + 16582 \beta_{7} - 950892 \beta_{8} - 1451506 \beta_{9} - 409300 \beta_{10} - 55246 \beta_{11} ) q^{82} \) \( + ( -1129769936 - 225136974 \beta_{1} + 2019226722 \beta_{2} + 25920280 \beta_{3} + 12302776 \beta_{4} + 4160486 \beta_{5} - 2121620 \beta_{6} + 1064996 \beta_{7} + 1477660 \beta_{8} - 617516 \beta_{9} + 679900 \beta_{10} - 457810 \beta_{11} ) q^{83} \) \( + ( 2387222000 - 19647728 \beta_{1} - 118294960 \beta_{2} + 8450836 \beta_{3} - 11148248 \beta_{4} + 362024 \beta_{5} + 4680576 \beta_{6} - 2444520 \beta_{7} + 47888 \beta_{8} + 394352 \beta_{9} - 249224 \beta_{10} - 350984 \beta_{11} ) q^{84} \) \( + ( 117690472366 - 109240734 \beta_{1} + 4192136766 \beta_{2} + 12150204 \beta_{3} - 16755456 \beta_{4} + 1594624 \beta_{5} - 1018082 \beta_{6} + 4967360 \beta_{7} + 2792256 \beta_{8} + 607552 \beta_{9} - 1716608 \beta_{10} + 290944 \beta_{11} ) q^{85} \) \( + ( 104446365536 - 175898696 \beta_{1} + 181576398 \beta_{2} + 21263568 \beta_{3} - 9329280 \beta_{4} - 3479062 \beta_{5} - 13309722 \beta_{6} + 1534526 \beta_{7} + 245896 \beta_{8} + 509626 \beta_{9} - 8348 \beta_{10} - 162598 \beta_{11} ) q^{86} \) \( + ( -1385683086 + 119790423 \beta_{1} + 2880089543 \beta_{2} - 32766126 \beta_{3} + 8412452 \beta_{4} + 1007311 \beta_{5} + 3007350 \beta_{6} - 1895384 \beta_{7} - 203270 \beta_{8} - 942476 \beta_{9} + 17930 \beta_{10} + 230165 \beta_{11} ) q^{87} \) \( + ( 3916822964 + 352144708 \beta_{1} + 448414420 \beta_{2} - 40355020 \beta_{3} + 17808084 \beta_{4} - 2546520 \beta_{5} - 3440196 \beta_{6} + 8053984 \beta_{7} - 117888 \beta_{8} + 143156 \beta_{9} + 284492 \beta_{10} - 326552 \beta_{11} ) q^{88} \) \( + ( -117285674214 + 82323400 \beta_{1} - 2508270116 \beta_{2} - 54071368 \beta_{3} + 10813968 \beta_{4} + 302064 \beta_{5} - 4502480 \beta_{6} + 4478948 \beta_{7} + 2443292 \beta_{8} - 1411764 \beta_{9} + 2525528 \beta_{10} - 129400 \beta_{11} ) q^{89} \) \( + ( -13714720740 - 55624158 \beta_{1} + 124534341 \beta_{2} + 3188646 \beta_{3} - 4605822 \beta_{4} + 1594323 \beta_{5} - 1948617 \beta_{6} + 885735 \beta_{7} + 1062882 \beta_{8} + 177147 \beta_{9} - 1062882 \beta_{10} - 177147 \beta_{11} ) q^{90} \) \( + ( 3434051850 - 79712588 \beta_{1} - 6924645644 \beta_{2} + 3204234 \beta_{3} - 26161196 \beta_{4} + 697414 \beta_{5} + 3667688 \beta_{6} - 1504730 \beta_{7} + 84048 \beta_{8} + 1781546 \beta_{9} - 281744 \beta_{10} - 506664 \beta_{11} ) q^{91} \) \( + ( -288409809840 + 124476976 \beta_{1} - 183193920 \beta_{2} + 9592272 \beta_{3} + 516112 \beta_{4} + 591312 \beta_{5} + 7592000 \beta_{6} - 7975056 \beta_{7} + 1080384 \beta_{8} - 2748288 \beta_{9} + 2088336 \beta_{10} + 110928 \beta_{11} ) q^{92} \) \( + ( 89408457316 + 64647572 \beta_{1} - 2642607461 \beta_{2} + 7860170 \beta_{3} + 12437000 \beta_{4} + 46048 \beta_{5} + 11643786 \beta_{6} - 56325 \beta_{7} + 1088185 \beta_{8} - 282767 \beta_{9} - 140254 \beta_{10} - 7534 \beta_{11} ) q^{93} \) \( + ( 70530828936 - 80042716 \beta_{1} + 137506582 \beta_{2} + 12385748 \beta_{3} + 31094400 \beta_{4} + 7067406 \beta_{5} + 24093666 \beta_{6} - 1649778 \beta_{7} + 4805592 \beta_{8} - 764418 \beta_{9} + 389964 \beta_{10} + 154974 \beta_{11} ) q^{94} \) \( + ( -1157840352 + 231812904 \beta_{1} + 2550345000 \beta_{2} + 22143136 \beta_{3} + 7105728 \beta_{4} - 8351360 \beta_{5} - 7124800 \beta_{6} + 1183648 \beta_{7} - 1043392 \beta_{8} + 2242016 \beta_{9} - 248512 \beta_{10} + 670624 \beta_{11} ) q^{95} \) \( + ( 28808942864 + 143890216 \beta_{1} - 618062328 \beta_{2} + 34638328 \beta_{3} - 14314996 \beta_{4} - 1449248 \beta_{5} + 8889084 \beta_{6} + 1521104 \beta_{7} - 1385132 \beta_{8} + 1057504 \beta_{9} - 809440 \beta_{10} + 773612 \beta_{11} ) q^{96} \) \( + ( -341534272370 - 194250580 \beta_{1} + 6787211796 \beta_{2} + 77350488 \beta_{3} - 32959920 \beta_{4} - 3528688 \beta_{5} + 11087396 \beta_{6} - 12956272 \beta_{7} - 6729312 \beta_{8} - 674288 \beta_{9} + 2248288 \beta_{10} - 553488 \beta_{11} ) q^{97} \) \( + ( 143925253912 - 785640656 \beta_{1} - 4844606887 \beta_{2} - 41535088 \beta_{3} - 2784912 \beta_{4} + 3849416 \beta_{5} - 5174552 \beta_{6} + 1262760 \beta_{7} - 2462224 \beta_{8} + 4709128 \beta_{9} - 319408 \beta_{10} - 219720 \beta_{11} ) q^{98} \) \( + ( -889277940 - 9565938 \beta_{1} + 1767572766 \beta_{2} - 9211644 \beta_{3} + 5668704 \beta_{4} - 2480058 \beta_{5} - 708588 \beta_{6} + 2834352 \beta_{7} - 708588 \beta_{8} + 1417176 \beta_{9} - 708588 \beta_{10} - 354294 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 90q^{2} \) \(\mathstrut -\mathstrut 4692q^{4} \) \(\mathstrut -\mathstrut 10296q^{5} \) \(\mathstrut -\mathstrut 48114q^{6} \) \(\mathstrut -\mathstrut 648000q^{8} \) \(\mathstrut -\mathstrut 2125764q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 90q^{2} \) \(\mathstrut -\mathstrut 4692q^{4} \) \(\mathstrut -\mathstrut 10296q^{5} \) \(\mathstrut -\mathstrut 48114q^{6} \) \(\mathstrut -\mathstrut 648000q^{8} \) \(\mathstrut -\mathstrut 2125764q^{9} \) \(\mathstrut +\mathstrut 923028q^{10} \) \(\mathstrut -\mathstrut 3018060q^{12} \) \(\mathstrut +\mathstrut 2094840q^{13} \) \(\mathstrut +\mathstrut 15389208q^{14} \) \(\mathstrut -\mathstrut 61526928q^{16} \) \(\mathstrut -\mathstrut 12097800q^{17} \) \(\mathstrut +\mathstrut 15943230q^{18} \) \(\mathstrut -\mathstrut 377644248q^{20} \) \(\mathstrut -\mathstrut 75057840q^{21} \) \(\mathstrut +\mathstrut 482545560q^{22} \) \(\mathstrut -\mathstrut 332039088q^{24} \) \(\mathstrut +\mathstrut 1058748132q^{25} \) \(\mathstrut +\mathstrut 243358236q^{26} \) \(\mathstrut -\mathstrut 185369520q^{28} \) \(\mathstrut +\mathstrut 1997608680q^{29} \) \(\mathstrut -\mathstrut 577761660q^{30} \) \(\mathstrut +\mathstrut 1733536800q^{32} \) \(\mathstrut +\mathstrut 322101360q^{33} \) \(\mathstrut -\mathstrut 7816269348q^{34} \) \(\mathstrut +\mathstrut 831173724q^{36} \) \(\mathstrut -\mathstrut 960170280q^{37} \) \(\mathstrut -\mathstrut 8280525240q^{38} \) \(\mathstrut +\mathstrut 3985807104q^{40} \) \(\mathstrut +\mathstrut 5806392696q^{41} \) \(\mathstrut -\mathstrut 1390844520q^{42} \) \(\mathstrut +\mathstrut 4989496464q^{44} \) \(\mathstrut +\mathstrut 1823905512q^{45} \) \(\mathstrut +\mathstrut 4149450240q^{46} \) \(\mathstrut -\mathstrut 7791843600q^{48} \) \(\mathstrut -\mathstrut 60479071668q^{49} \) \(\mathstrut +\mathstrut 68552901522q^{50} \) \(\mathstrut -\mathstrut 31090133640q^{52} \) \(\mathstrut +\mathstrut 42482511720q^{53} \) \(\mathstrut +\mathstrut 8523250758q^{54} \) \(\mathstrut -\mathstrut 38053468224q^{56} \) \(\mathstrut -\mathstrut 58319941680q^{57} \) \(\mathstrut +\mathstrut 159666562500q^{58} \) \(\mathstrut -\mathstrut 19968517224q^{60} \) \(\mathstrut +\mathstrut 137368568088q^{61} \) \(\mathstrut -\mathstrut 27876030840q^{62} \) \(\mathstrut +\mathstrut 188355529344q^{64} \) \(\mathstrut -\mathstrut 328250713392q^{65} \) \(\mathstrut -\mathstrut 136719325224q^{66} \) \(\mathstrut +\mathstrut 77938316280q^{68} \) \(\mathstrut +\mathstrut 214339017024q^{69} \) \(\mathstrut -\mathstrut 454939318704q^{70} \) \(\mathstrut +\mathstrut 114791256000q^{72} \) \(\mathstrut -\mathstrut 804477880680q^{73} \) \(\mathstrut -\mathstrut 502785766548q^{74} \) \(\mathstrut +\mathstrut 143972453808q^{76} \) \(\mathstrut +\mathstrut 1383727360320q^{77} \) \(\mathstrut -\mathstrut 72052158420q^{78} \) \(\mathstrut -\mathstrut 417712547808q^{80} \) \(\mathstrut +\mathstrut 376572715308q^{81} \) \(\mathstrut +\mathstrut 460673773020q^{82} \) \(\mathstrut +\mathstrut 28008331632q^{84} \) \(\mathstrut +\mathstrut 1437981718224q^{85} \) \(\mathstrut +\mathstrut 1255416205464q^{86} \) \(\mathstrut +\mathstrut 47622991680q^{88} \) \(\mathstrut -\mathstrut 1422946205928q^{89} \) \(\mathstrut -\mathstrut 163511641116q^{90} \) \(\mathstrut -\mathstrut 3462722444160q^{92} \) \(\mathstrut +\mathstrut 1056734080560q^{93} \) \(\mathstrut +\mathstrut 847910842896q^{94} \) \(\mathstrut +\mathstrut 341032101984q^{96} \) \(\mathstrut -\mathstrut 4056673857000q^{97} \) \(\mathstrut +\mathstrut 1702751294790q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut +\mathstrut \) \(1570\) \(x^{10}\mathstrut -\mathstrut \) \(4077\) \(x^{9}\mathstrut +\mathstrut \) \(1884069\) \(x^{8}\mathstrut -\mathstrut \) \(3551868\) \(x^{7}\mathstrut +\mathstrut \) \(881574992\) \(x^{6}\mathstrut +\mathstrut \) \(2167220880\) \(x^{5}\mathstrut +\mathstrut \) \(304613421264\) \(x^{4}\mathstrut +\mathstrut \) \(27792635520\) \(x^{3}\mathstrut +\mathstrut \) \(5806366144000\) \(x^{2}\mathstrut +\mathstrut \) \(1120796928000\) \(x\mathstrut +\mathstrut \) \(104882177440000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(15\!\cdots\!17\) \(\nu^{11}\mathstrut -\mathstrut \) \(71\!\cdots\!89\) \(\nu^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!70\) \(\nu^{9}\mathstrut -\mathstrut \) \(11\!\cdots\!91\) \(\nu^{8}\mathstrut -\mathstrut \) \(25\!\cdots\!93\) \(\nu^{7}\mathstrut -\mathstrut \) \(13\!\cdots\!04\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!44\) \(\nu^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(64\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(22\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(22\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(23\!\cdots\!61\) \(\nu^{11}\mathstrut -\mathstrut \) \(78\!\cdots\!63\) \(\nu^{10}\mathstrut +\mathstrut \) \(37\!\cdots\!10\) \(\nu^{9}\mathstrut -\mathstrut \) \(10\!\cdots\!97\) \(\nu^{8}\mathstrut +\mathstrut \) \(45\!\cdots\!69\) \(\nu^{7}\mathstrut -\mathstrut \) \(96\!\cdots\!68\) \(\nu^{6}\mathstrut +\mathstrut \) \(21\!\cdots\!52\) \(\nu^{5}\mathstrut +\mathstrut \) \(46\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(72\!\cdots\!04\) \(\nu^{3}\mathstrut -\mathstrut \) \(24\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(91\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(49\!\cdots\!67\) \(\nu^{11}\mathstrut +\mathstrut \) \(94\!\cdots\!55\) \(\nu^{10}\mathstrut +\mathstrut \) \(76\!\cdots\!42\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!21\) \(\nu^{8}\mathstrut +\mathstrut \) \(89\!\cdots\!91\) \(\nu^{7}\mathstrut +\mathstrut \) \(18\!\cdots\!28\) \(\nu^{6}\mathstrut +\mathstrut \) \(40\!\cdots\!16\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!92\) \(\nu^{4}\mathstrut +\mathstrut \) \(16\!\cdots\!28\) \(\nu^{3}\mathstrut +\mathstrut \) \(32\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(61\!\cdots\!40\) \(\nu\mathstrut +\mathstrut \) \(44\!\cdots\!00\)\()/\)\(17\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(28\!\cdots\!41\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!43\) \(\nu^{10}\mathstrut -\mathstrut \) \(42\!\cdots\!30\) \(\nu^{9}\mathstrut +\mathstrut \) \(29\!\cdots\!57\) \(\nu^{8}\mathstrut -\mathstrut \) \(50\!\cdots\!69\) \(\nu^{7}\mathstrut +\mathstrut \) \(31\!\cdots\!68\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(44\!\cdots\!60\) \(\nu^{4}\mathstrut -\mathstrut \) \(65\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(41\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(51\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(54\!\cdots\!00\)\()/\)\(17\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(24\!\cdots\!44\) \(\nu^{11}\mathstrut +\mathstrut \) \(11\!\cdots\!27\) \(\nu^{10}\mathstrut -\mathstrut \) \(43\!\cdots\!15\) \(\nu^{9}\mathstrut +\mathstrut \) \(18\!\cdots\!88\) \(\nu^{8}\mathstrut -\mathstrut \) \(54\!\cdots\!51\) \(\nu^{7}\mathstrut +\mathstrut \) \(21\!\cdots\!97\) \(\nu^{6}\mathstrut -\mathstrut \) \(30\!\cdots\!58\) \(\nu^{5}\mathstrut +\mathstrut \) \(90\!\cdots\!20\) \(\nu^{4}\mathstrut -\mathstrut \) \(73\!\cdots\!16\) \(\nu^{3}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(83\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(19\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(38\!\cdots\!71\) \(\nu^{11}\mathstrut -\mathstrut \) \(23\!\cdots\!68\) \(\nu^{10}\mathstrut +\mathstrut \) \(60\!\cdots\!85\) \(\nu^{9}\mathstrut -\mathstrut \) \(33\!\cdots\!67\) \(\nu^{8}\mathstrut +\mathstrut \) \(72\!\cdots\!34\) \(\nu^{7}\mathstrut -\mathstrut \) \(35\!\cdots\!73\) \(\nu^{6}\mathstrut +\mathstrut \) \(33\!\cdots\!22\) \(\nu^{5}\mathstrut -\mathstrut \) \(67\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!44\) \(\nu^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(45\!\cdots\!91\) \(\nu^{11}\mathstrut -\mathstrut \) \(16\!\cdots\!53\) \(\nu^{10}\mathstrut +\mathstrut \) \(70\!\cdots\!10\) \(\nu^{9}\mathstrut -\mathstrut \) \(23\!\cdots\!07\) \(\nu^{8}\mathstrut +\mathstrut \) \(83\!\cdots\!39\) \(\nu^{7}\mathstrut -\mathstrut \) \(22\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(37\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(63\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!24\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(34\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(29\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(12\!\cdots\!66\) \(\nu^{11}\mathstrut +\mathstrut \) \(53\!\cdots\!03\) \(\nu^{10}\mathstrut -\mathstrut \) \(19\!\cdots\!85\) \(\nu^{9}\mathstrut +\mathstrut \) \(72\!\cdots\!82\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!39\) \(\nu^{7}\mathstrut +\mathstrut \) \(71\!\cdots\!83\) \(\nu^{6}\mathstrut -\mathstrut \) \(10\!\cdots\!62\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!20\) \(\nu^{4}\mathstrut -\mathstrut \) \(32\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(66\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(76\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(51\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(29\!\cdots\!31\) \(\nu^{11}\mathstrut -\mathstrut \) \(42\!\cdots\!73\) \(\nu^{10}\mathstrut +\mathstrut \) \(46\!\cdots\!10\) \(\nu^{9}\mathstrut -\mathstrut \) \(63\!\cdots\!87\) \(\nu^{8}\mathstrut +\mathstrut \) \(55\!\cdots\!99\) \(\nu^{7}\mathstrut -\mathstrut \) \(35\!\cdots\!28\) \(\nu^{6}\mathstrut +\mathstrut \) \(26\!\cdots\!92\) \(\nu^{5}\mathstrut +\mathstrut \) \(89\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(91\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(22\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(18\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(49\!\cdots\!09\) \(\nu^{11}\mathstrut -\mathstrut \) \(10\!\cdots\!22\) \(\nu^{10}\mathstrut +\mathstrut \) \(76\!\cdots\!65\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!93\) \(\nu^{8}\mathstrut +\mathstrut \) \(91\!\cdots\!36\) \(\nu^{7}\mathstrut -\mathstrut \) \(83\!\cdots\!17\) \(\nu^{6}\mathstrut +\mathstrut \) \(42\!\cdots\!38\) \(\nu^{5}\mathstrut +\mathstrut \) \(15\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(14\!\cdots\!76\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(38\!\cdots\!00\)\()/\)\(85\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(32\!\cdots\!73\) \(\nu^{11}\mathstrut -\mathstrut \) \(27\!\cdots\!44\) \(\nu^{10}\mathstrut +\mathstrut \) \(53\!\cdots\!35\) \(\nu^{9}\mathstrut -\mathstrut \) \(41\!\cdots\!21\) \(\nu^{8}\mathstrut +\mathstrut \) \(64\!\cdots\!62\) \(\nu^{7}\mathstrut -\mathstrut \) \(45\!\cdots\!39\) \(\nu^{6}\mathstrut +\mathstrut \) \(30\!\cdots\!66\) \(\nu^{5}\mathstrut -\mathstrut \) \(91\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!72\) \(\nu^{3}\mathstrut -\mathstrut \) \(55\!\cdots\!40\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(94\!\cdots\!00\)\()/\)\(42\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(20\) \(\beta_{10}\mathstrut +\mathstrut \) \(22\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\) \(\beta_{8}\mathstrut -\mathstrut \) \(90\) \(\beta_{7}\mathstrut +\mathstrut \) \(144\) \(\beta_{6}\mathstrut +\mathstrut \) \(58\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(661\) \(\beta_{3}\mathstrut -\mathstrut \) \(71621\) \(\beta_{2}\mathstrut +\mathstrut \) \(1538\) \(\beta_{1}\mathstrut +\mathstrut \) \(129832\)\()/373248\)
\(\nu^{2}\)\(=\)\((\)\(80\) \(\beta_{11}\mathstrut -\mathstrut \) \(286\) \(\beta_{10}\mathstrut -\mathstrut \) \(401\) \(\beta_{9}\mathstrut +\mathstrut \) \(355\) \(\beta_{8}\mathstrut +\mathstrut \) \(567\) \(\beta_{7}\mathstrut +\mathstrut \) \(4572\) \(\beta_{6}\mathstrut -\mathstrut \) \(527\) \(\beta_{5}\mathstrut -\mathstrut \) \(712\) \(\beta_{4}\mathstrut -\mathstrut \) \(24259\) \(\beta_{3}\mathstrut +\mathstrut \) \(96304\) \(\beta_{2}\mathstrut -\mathstrut \) \(403252\) \(\beta_{1}\mathstrut -\mathstrut \) \(97635602\)\()/373248\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(4028\) \(\beta_{11}\mathstrut -\mathstrut \) \(14930\) \(\beta_{10}\mathstrut -\mathstrut \) \(7351\) \(\beta_{9}\mathstrut -\mathstrut \) \(23551\) \(\beta_{8}\mathstrut -\mathstrut \) \(167631\) \(\beta_{7}\mathstrut -\mathstrut \) \(42696\) \(\beta_{6}\mathstrut -\mathstrut \) \(32281\) \(\beta_{5}\mathstrut -\mathstrut \) \(270725\) \(\beta_{4}\mathstrut +\mathstrut \) \(578044\) \(\beta_{3}\mathstrut +\mathstrut \) \(51537707\) \(\beta_{2}\mathstrut -\mathstrut \) \(1492058\) \(\beta_{1}\mathstrut -\mathstrut \) \(84683362\)\()/373248\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(3660\) \(\beta_{11}\mathstrut +\mathstrut \) \(147396\) \(\beta_{10}\mathstrut +\mathstrut \) \(9726\) \(\beta_{9}\mathstrut +\mathstrut \) \(570354\) \(\beta_{8}\mathstrut +\mathstrut \) \(431998\) \(\beta_{7}\mathstrut -\mathstrut \) \(114804\) \(\beta_{6}\mathstrut +\mathstrut \) \(247218\) \(\beta_{5}\mathstrut -\mathstrut \) \(291453\) \(\beta_{4}\mathstrut +\mathstrut \) \(7565419\) \(\beta_{3}\mathstrut -\mathstrut \) \(74313893\) \(\beta_{2}\mathstrut +\mathstrut \) \(113021770\) \(\beta_{1}\mathstrut -\mathstrut \) \(27007880872\)\()/124416\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(4995128\) \(\beta_{11}\mathstrut -\mathstrut \) \(7412198\) \(\beta_{10}\mathstrut -\mathstrut \) \(5144365\) \(\beta_{9}\mathstrut +\mathstrut \) \(1707263\) \(\beta_{8}\mathstrut +\mathstrut \) \(198042267\) \(\beta_{7}\mathstrut -\mathstrut \) \(24452316\) \(\beta_{6}\mathstrut -\mathstrut \) \(9624403\) \(\beta_{5}\mathstrut +\mathstrut \) \(296965726\) \(\beta_{4}\mathstrut -\mathstrut \) \(393758789\) \(\beta_{3}\mathstrut +\mathstrut \) \(22344896342\) \(\beta_{2}\mathstrut -\mathstrut \) \(1022876480\) \(\beta_{1}\mathstrut -\mathstrut \) \(184700568850\)\()/373248\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(71870492\) \(\beta_{11}\mathstrut -\mathstrut \) \(89038526\) \(\beta_{10}\mathstrut +\mathstrut \) \(315768383\) \(\beta_{9}\mathstrut -\mathstrut \) \(2296611649\) \(\beta_{8}\mathstrut -\mathstrut \) \(3441470793\) \(\beta_{7}\mathstrut -\mathstrut \) \(3433883472\) \(\beta_{6}\mathstrut -\mathstrut \) \(755107759\) \(\beta_{5}\mathstrut -\mathstrut \) \(349646105\) \(\beta_{4}\mathstrut +\mathstrut \) \(6758350690\) \(\beta_{3}\mathstrut -\mathstrut \) \(143575862953\) \(\beta_{2}\mathstrut +\mathstrut \) \(430415398\) \(\beta_{1}\mathstrut +\mathstrut \) \(146255720944826\)\()/373248\)
\(\nu^{7}\)\(=\)\((\)\(9032756612\) \(\beta_{11}\mathstrut +\mathstrut \) \(24633390032\) \(\beta_{10}\mathstrut +\mathstrut \) \(6792482104\) \(\beta_{9}\mathstrut +\mathstrut \) \(39768341236\) \(\beta_{8}\mathstrut -\mathstrut \) \(14313673560\) \(\beta_{7}\mathstrut +\mathstrut \) \(31971234852\) \(\beta_{6}\mathstrut +\mathstrut \) \(41614844920\) \(\beta_{5}\mathstrut -\mathstrut \) \(71672245393\) \(\beta_{4}\mathstrut -\mathstrut \) \(115986906163\) \(\beta_{3}\mathstrut -\mathstrut \) \(62652383312873\) \(\beta_{2}\mathstrut +\mathstrut \) \(3216307486010\) \(\beta_{1}\mathstrut -\mathstrut \) \(378740060495780\)\()/373248\)
\(\nu^{8}\)\(=\)\((\)\(6806105008\) \(\beta_{11}\mathstrut -\mathstrut \) \(130871998130\) \(\beta_{10}\mathstrut -\mathstrut \) \(154570173319\) \(\beta_{9}\mathstrut +\mathstrut \) \(185237328869\) \(\beta_{8}\mathstrut +\mathstrut \) \(924146285937\) \(\beta_{7}\mathstrut +\mathstrut \) \(1201681762884\) \(\beta_{6}\mathstrut +\mathstrut \) \(25865178407\) \(\beta_{5}\mathstrut +\mathstrut \) \(1004323707844\) \(\beta_{4}\mathstrut -\mathstrut \) \(9564718113857\) \(\beta_{3}\mathstrut +\mathstrut \) \(264041859366172\) \(\beta_{2}\mathstrut -\mathstrut \) \(98530952298564\) \(\beta_{1}\mathstrut -\mathstrut \) \(23054187130452334\)\()/124416\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(3890527022236\) \(\beta_{11}\mathstrut -\mathstrut \) \(17094744642106\) \(\beta_{10}\mathstrut +\mathstrut \) \(675981456253\) \(\beta_{9}\mathstrut -\mathstrut \) \(52514458837499\) \(\beta_{8}\mathstrut -\mathstrut \) \(184147797318267\) \(\beta_{7}\mathstrut -\mathstrut \) \(27057908537784\) \(\beta_{6}\mathstrut -\mathstrut \) \(35735893066541\) \(\beta_{5}\mathstrut -\mathstrut \) \(195967183014841\) \(\beta_{4}\mathstrut +\mathstrut \) \(680641178199020\) \(\beta_{3}\mathstrut +\mathstrut \) \(34611580977286087\) \(\beta_{2}\mathstrut -\mathstrut \) \(1111537917597346\) \(\beta_{1}\mathstrut +\mathstrut \) \(1281938520726663574\)\()/373248\)
\(\nu^{10}\)\(=\)\((\)\(103575226611500\) \(\beta_{11}\mathstrut +\mathstrut \) \(539346476025884\) \(\beta_{10}\mathstrut +\mathstrut \) \(255009375397138\) \(\beta_{9}\mathstrut +\mathstrut \) \(1907367326055166\) \(\beta_{8}\mathstrut +\mathstrut \) \(1275052164187026\) \(\beta_{7}\mathstrut -\mathstrut \) \(217826666440812\) \(\beta_{6}\mathstrut +\mathstrut \) \(911376903535294\) \(\beta_{5}\mathstrut -\mathstrut \) \(2309715108434395\) \(\beta_{4}\mathstrut +\mathstrut \) \(18761777053329149\) \(\beta_{3}\mathstrut -\mathstrut \) \(988572404602276595\) \(\beta_{2}\mathstrut +\mathstrut \) \(294991881221699558\) \(\beta_{1}\mathstrut -\mathstrut \) \(65781971876789877272\)\()/373248\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(5161662809172248\) \(\beta_{11}\mathstrut -\mathstrut \) \(9834346103673518\) \(\beta_{10}\mathstrut -\mathstrut \) \(7244821572875593\) \(\beta_{9}\mathstrut +\mathstrut \) \(9170421841361459\) \(\beta_{8}\mathstrut +\mathstrut \) \(193096328321140191\) \(\beta_{7}\mathstrut +\mathstrut \) \(18501212696116500\) \(\beta_{6}\mathstrut -\mathstrut \) \(5544291838927543\) \(\beta_{5}\mathstrut +\mathstrut \) \(269412390923297494\) \(\beta_{4}\mathstrut -\mathstrut \) \(760421748113454161\) \(\beta_{3}\mathstrut +\mathstrut \) \(29001804793532807918\) \(\beta_{2}\mathstrut -\mathstrut \) \(4010514877261495904\) \(\beta_{1}\mathstrut -\mathstrut \) \(893140877498609938474\)\()/373248\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
14.6572 + 25.3870i
14.6572 25.3870i
2.19768 3.80649i
2.19768 + 3.80649i
13.5291 + 23.4332i
13.5291 23.4332i
−16.4016 + 28.4085i
−16.4016 28.4085i
−2.17114 3.76053i
−2.17114 + 3.76053i
−10.3112 17.8596i
−10.3112 + 17.8596i
−62.3881 14.2733i 420.888i 3688.55 + 1780.96i −21622.0 −6007.45 + 26258.4i 155240.i −204701. 163759.i −177147. 1.34896e6 + 308617.i
7.2 −62.3881 + 14.2733i 420.888i 3688.55 1780.96i −21622.0 −6007.45 26258.4i 155240.i −204701. + 163759.i −177147. 1.34896e6 308617.i
7.3 −46.3006 44.1844i 420.888i 191.485 + 4091.52i 3884.37 18596.7 19487.4i 57100.7i 171915. 197900.i −177147. −179848. 171628.i
7.4 −46.3006 + 44.1844i 420.888i 191.485 4091.52i 3884.37 18596.7 + 19487.4i 57100.7i 171915. + 197900.i −177147. −179848. + 171628.i
7.5 −35.2919 53.3899i 420.888i −1604.96 + 3768.46i 8409.28 −22471.2 + 14854.0i 192052.i 257840. 47307.4i −177147. −296780. 448971.i
7.6 −35.2919 + 53.3899i 420.888i −1604.96 3768.46i 8409.28 −22471.2 14854.0i 192052.i 257840. + 47307.4i −177147. −296780. + 448971.i
7.7 18.1780 61.3642i 420.888i −3435.12 2230.95i −204.488 25827.5 + 7650.89i 121613.i −199344. + 170239.i −177147. −3717.17 + 12548.2i
7.8 18.1780 + 61.3642i 420.888i −3435.12 + 2230.95i −204.488 25827.5 7650.89i 121613.i −199344. 170239.i −177147. −3717.17 12548.2i
7.9 29.4812 56.8054i 420.888i −2357.72 3349.39i 28943.6 −23908.8 12408.3i 157943.i −259772. + 35187.3i −177147. 853291. 1.64415e6i
7.10 29.4812 + 56.8054i 420.888i −2357.72 + 3349.39i 28943.6 −23908.8 + 12408.3i 157943.i −259772. 35187.3i −177147. 853291. + 1.64415e6i
7.11 51.3214 38.2376i 420.888i 1171.77 3924.81i −24558.7 −16093.8 21600.6i 96476.2i −89938.5 246233.i −177147. −1.26039e6 + 939066.i
7.12 51.3214 + 38.2376i 420.888i 1171.77 + 3924.81i −24558.7 −16093.8 + 21600.6i 96476.2i −89938.5 + 246233.i −177147. −1.26039e6 939066.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{13}^{\mathrm{new}}(12, [\chi])\).