Properties

Label 12.11.d.a
Level 12
Weight 11
Character orbit 12.d
Analytic conductor 7.624
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(7.62428703208\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{18} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 2 - \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( -65 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 319 + 37 \beta_{2} - \beta_{5} ) q^{5} \) \( + ( 243 + 2 \beta_{1} + 3 \beta_{4} ) q^{6} \) \( + ( -49 - 16 \beta_{1} - 242 \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 2027 + 35 \beta_{1} + 61 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{8} \) \( -19683 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 - \beta_{2} ) q^{2} \) \( + \beta_{1} q^{3} \) \( + ( -65 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( 319 + 37 \beta_{2} - \beta_{5} ) q^{5} \) \( + ( 243 + 2 \beta_{1} + 3 \beta_{4} ) q^{6} \) \( + ( -49 - 16 \beta_{1} - 242 \beta_{2} - 3 \beta_{3} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{7} \) \( + ( 2027 + 35 \beta_{1} + 61 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 4 \beta_{5} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{8} \) \( -19683 q^{9} \) \( + ( -36986 - 62 \beta_{1} - 369 \beta_{2} + 55 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} + 8 \beta_{8} - \beta_{9} ) q^{10} \) \( + ( -18 - 68 \beta_{1} - 190 \beta_{2} + 68 \beta_{3} - 26 \beta_{4} - 12 \beta_{5} + 10 \beta_{6} - 26 \beta_{7} + 14 \beta_{8} + 12 \beta_{9} ) q^{11} \) \( + ( 30978 - 65 \beta_{1} - 171 \beta_{2} + 9 \beta_{4} + 12 \beta_{5} + 6 \beta_{6} - 18 \beta_{7} + 3 \beta_{8} + 15 \beta_{9} ) q^{12} \) \( + ( -29100 + 118 \beta_{1} - 2962 \beta_{2} + 114 \beta_{3} + 24 \beta_{4} - 76 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} - 38 \beta_{9} ) q^{13} \) \( + ( -249738 - 6 \beta_{1} - 571 \beta_{2} - 233 \beta_{3} - 55 \beta_{4} - 19 \beta_{5} - 26 \beta_{6} - 20 \beta_{7} - 40 \beta_{8} - \beta_{9} ) q^{14} \) \( + ( 333 + 178 \beta_{1} + 1881 \beta_{2} - 216 \beta_{3} - 129 \beta_{4} + 12 \beta_{5} + 33 \beta_{6} + 9 \beta_{7} - 51 \beta_{8} - 12 \beta_{9} ) q^{15} \) \( + ( 358722 + 30 \beta_{1} - 1204 \beta_{2} + 140 \beta_{3} + 48 \beta_{4} - 170 \beta_{5} - 128 \beta_{6} - 28 \beta_{7} - 26 \beta_{8} - 40 \beta_{9} ) q^{16} \) \( + ( 225856 - 268 \beta_{1} - 3242 \beta_{2} - 436 \beta_{3} + 736 \beta_{4} + 110 \beta_{5} + 170 \beta_{6} + 60 \beta_{7} + 100 \beta_{8} + 124 \beta_{9} ) q^{17} \) \( + ( -39366 + 19683 \beta_{2} ) q^{18} \) \( + ( -3870 - 3836 \beta_{1} - 20444 \beta_{2} + 422 \beta_{3} - 768 \beta_{4} - 30 \beta_{5} + 112 \beta_{6} + 290 \beta_{7} + 98 \beta_{8} + 30 \beta_{9} ) q^{19} \) \( + ( 126322 + 7764 \beta_{1} + 35452 \beta_{2} - 630 \beta_{3} - 72 \beta_{4} + 600 \beta_{5} - 224 \beta_{6} + 224 \beta_{8} - 56 \beta_{9} ) q^{20} \) \( + ( 356301 + 126 \beta_{1} + 2313 \beta_{2} + 810 \beta_{3} + 792 \beta_{4} + 237 \beta_{5} + 159 \beta_{6} + 90 \beta_{7} - 42 \beta_{8} + 114 \beta_{9} ) q^{21} \) \( + ( -214152 - 6988 \beta_{1} + 1262 \beta_{2} - 134 \beta_{3} + 90 \beta_{4} + 1246 \beta_{5} - 636 \beta_{6} + 200 \beta_{7} - 240 \beta_{8} - 406 \beta_{9} ) q^{22} \) \( + ( -1170 + 9176 \beta_{1} - 8860 \beta_{2} + 834 \beta_{3} - 3288 \beta_{4} - 10 \beta_{5} + 664 \beta_{6} + 622 \beta_{7} + 206 \beta_{8} + 10 \beta_{9} ) q^{23} \) \( + ( -692829 + 1679 \beta_{1} - 33201 \beta_{2} - 513 \beta_{3} - 15 \beta_{4} + 870 \beta_{5} - 240 \beta_{6} + 126 \beta_{7} - 579 \beta_{8} - 141 \beta_{9} ) q^{24} \) \( + ( 2306263 + 892 \beta_{1} + 2852 \beta_{2} + 3636 \beta_{3} + 4752 \beta_{4} + 304 \beta_{5} + 926 \beta_{6} + 500 \beta_{7} - 148 \beta_{8} + 452 \beta_{9} ) q^{25} \) \( + ( 2967736 - 22876 \beta_{1} + 40424 \beta_{2} - 1346 \beta_{3} + 170 \beta_{4} - 1318 \beta_{5} - 716 \beta_{6} - 1048 \beta_{7} + 1232 \beta_{8} + 590 \beta_{9} ) q^{26} \) \( -19683 \beta_{1} q^{27} \) \( + ( -4109240 + 49748 \beta_{1} + 229908 \beta_{2} - 1288 \beta_{3} - 1020 \beta_{4} - 2184 \beta_{5} - 600 \beta_{6} - 568 \beta_{7} - 1100 \beta_{8} - 420 \beta_{9} ) q^{28} \) \( + ( -4708331 - 2628 \beta_{1} - 192269 \beta_{2} - 14028 \beta_{3} + 1488 \beta_{4} - 2291 \beta_{5} + 382 \beta_{6} - 204 \beta_{7} + 1580 \beta_{8} - 572 \beta_{9} ) q^{29} \) \( + ( 1998900 - 34378 \beta_{1} + 13329 \beta_{2} + 3483 \beta_{3} - 261 \beta_{4} - 3111 \beta_{5} - 786 \beta_{6} + 252 \beta_{7} - 1608 \beta_{8} - 669 \beta_{9} ) q^{30} \) \( + ( -16889 - 22728 \beta_{1} - 75830 \beta_{2} - 8807 \beta_{3} - 4980 \beta_{4} - 701 \beta_{5} + 1556 \beta_{6} - 1945 \beta_{7} - 2377 \beta_{8} + 701 \beta_{9} ) q^{31} \) \( + ( -8454736 + 147416 \beta_{1} - 414252 \beta_{2} - 2924 \beta_{3} - 796 \beta_{4} - 1948 \beta_{5} - 480 \beta_{6} - 2368 \beta_{7} + 2976 \beta_{8} + 1276 \beta_{9} ) q^{32} \) \( + ( 2244906 + 10296 \beta_{1} - 116658 \beta_{2} + 18360 \beta_{3} - 1872 \beta_{4} - 1902 \beta_{5} - 1140 \beta_{6} + 72 \beta_{7} - 2568 \beta_{8} - 2472 \beta_{9} ) q^{33} \) \( + ( 3718256 - 222164 \beta_{1} - 168620 \beta_{2} + 314 \beta_{3} + 702 \beta_{4} + 5774 \beta_{5} + 3004 \beta_{6} - 2952 \beta_{7} - 2512 \beta_{8} + 874 \beta_{9} ) q^{34} \) \( + ( 14778 + 89552 \beta_{1} + 41474 \beta_{2} + 30528 \beta_{3} + 11598 \beta_{4} - 2008 \beta_{5} - 2174 \beta_{6} - 4910 \beta_{7} + 7130 \beta_{8} + 2008 \beta_{9} ) q^{35} \) \( + ( 1279395 + 39366 \beta_{1} + 39366 \beta_{2} + 19683 \beta_{3} ) q^{36} \) \( + ( -19259546 - 646 \beta_{1} + 924424 \beta_{2} + 20766 \beta_{3} - 20760 \beta_{4} - 98 \beta_{5} - 3907 \beta_{6} - 1202 \beta_{7} - 3006 \beta_{8} + 1526 \beta_{9} ) q^{37} \) \( + ( -21871008 - 300780 \beta_{1} + 25718 \beta_{2} - 11822 \beta_{3} - 10246 \beta_{4} - 5082 \beta_{5} + 4148 \beta_{6} + 6184 \beta_{7} + 5200 \beta_{8} - 2046 \beta_{9} ) q^{38} \) \( + ( 81000 - 52006 \beta_{1} + 451251 \beta_{2} - 30699 \beta_{3} + 8883 \beta_{4} + 3915 \beta_{5} - 1971 \beta_{6} + 2916 \beta_{7} - 6696 \beta_{8} - 3915 \beta_{9} ) q^{39} \) \( + ( 33987690 + 502650 \beta_{1} - 164858 \beta_{2} + 12934 \beta_{3} + 22290 \beta_{4} + 9672 \beta_{5} + 7040 \beta_{6} + 7044 \beta_{7} - 826 \beta_{8} - 3834 \beta_{9} ) q^{40} \) \( + ( 27602980 - 33100 \beta_{1} - 23558 \beta_{2} - 75220 \beta_{3} - 33152 \beta_{4} + 10458 \beta_{5} - 4502 \beta_{6} - 4164 \beta_{7} + 7652 \beta_{8} + 2652 \beta_{9} ) q^{41} \) \( + ( -1647198 - 247230 \beta_{1} - 304767 \beta_{2} + 6615 \beta_{3} + 5661 \beta_{4} + 7125 \beta_{5} + 3738 \beta_{6} + 180 \beta_{7} - 696 \beta_{8} + 7455 \beta_{9} ) q^{42} \) \( + ( -304714 - 18580 \beta_{1} - 1484936 \beta_{2} - 13002 \beta_{3} + 53292 \beta_{4} + 674 \beta_{5} - 15708 \beta_{6} + 10214 \beta_{7} - 3082 \beta_{8} - 674 \beta_{9} ) q^{43} \) \( + ( 53315304 + 898268 \beta_{1} + 108868 \beta_{2} - 38576 \beta_{3} - 23404 \beta_{4} - 4448 \beta_{5} + 15000 \beta_{6} + 11960 \beta_{7} + 1932 \beta_{8} + 9292 \beta_{9} ) q^{44} \) \( + ( -6278877 - 728271 \beta_{2} + 19683 \beta_{5} ) q^{45} \) \( + ( -6949356 - 1094364 \beta_{1} + 259626 \beta_{2} + 24846 \beta_{3} + 29850 \beta_{4} - 17030 \beta_{5} + 396 \beta_{6} + 23384 \beta_{7} + 4272 \beta_{8} - 15394 \beta_{9} ) q^{46} \) \( + ( 723690 + 239296 \beta_{1} + 3716376 \beta_{2} - 47750 \beta_{3} + 38756 \beta_{4} + 18350 \beta_{5} - 7140 \beta_{6} + 8154 \beta_{7} - 7350 \beta_{8} - 18350 \beta_{9} ) q^{47} \) \( + ( 2830158 + 331978 \beta_{1} + 606240 \beta_{2} - 49248 \beta_{3} + 2988 \beta_{4} - 15546 \beta_{5} + 6240 \beta_{6} + 6876 \beta_{7} - 9030 \beta_{8} + 10092 \beta_{9} ) q^{48} \) \( + ( 66013641 - 12292 \beta_{1} + 3257008 \beta_{2} + 25812 \beta_{3} - 68688 \beta_{4} - 23020 \beta_{5} - 12482 \beta_{6} - 4940 \beta_{7} - 5204 \beta_{8} + 3940 \beta_{9} ) q^{49} \) \( + ( 1720206 - 1594984 \beta_{1} - 1943663 \beta_{2} + 45076 \beta_{3} + 25820 \beta_{4} + 27132 \beta_{5} + 13944 \beta_{6} - 15376 \beta_{7} + 4320 \beta_{8} + 37492 \beta_{9} ) q^{50} \) \( + ( 855234 + 197506 \beta_{1} + 4236372 \beta_{2} + 47574 \beta_{3} + 3696 \beta_{4} + 7986 \beta_{5} + 4992 \beta_{6} - 15678 \beta_{7} + 13890 \beta_{8} - 7986 \beta_{9} ) q^{51} \) \( + ( -96630786 + 1403356 \beta_{1} - 3510100 \beta_{2} + 2862 \beta_{3} - 72720 \beta_{4} + 49200 \beta_{5} - 9920 \beta_{6} - 14336 \beta_{7} + 6336 \beta_{8} - 26352 \beta_{9} ) q^{52} \) \( + ( -153091355 + 106924 \beta_{1} - 4162157 \beta_{2} + 147652 \beta_{3} + 22736 \beta_{4} + 11517 \beta_{5} - 3786 \beta_{6} + 3204 \beta_{7} - 20388 \beta_{8} - 28268 \beta_{9} ) q^{53} \) \( + ( -4782969 - 39366 \beta_{1} - 59049 \beta_{4} ) q^{54} \) \( + ( -2447456 - 291944 \beta_{1} - 12596116 \beta_{2} + 195828 \beta_{3} - 103188 \beta_{4} - 66164 \beta_{5} + 19380 \beta_{6} - 40496 \beta_{7} + 32416 \beta_{8} + 66164 \beta_{9} ) q^{55} \) \( + ( -62483436 + 1271140 \beta_{1} + 3768996 \beta_{2} + 250084 \beta_{3} + 130268 \beta_{4} - 59544 \beta_{5} - 30400 \beta_{6} - 29880 \beta_{7} + 1132 \beta_{8} + 2580 \beta_{9} ) q^{56} \) \( + ( 83305530 - 1404 \beta_{1} - 5577282 \beta_{2} - 166212 \beta_{3} + 56160 \beta_{4} - 10602 \beta_{5} + 9522 \beta_{6} + 108 \beta_{7} + 18612 \beta_{8} - 17748 \beta_{9} ) q^{57} \) \( + ( 186099490 - 678626 \beta_{1} + 5246205 \beta_{2} - 171623 \beta_{3} - 54669 \beta_{4} - 27013 \beta_{5} - 14938 \beta_{6} - 60852 \beta_{7} - 5640 \beta_{8} - 66479 \beta_{9} ) q^{58} \) \( + ( 72720 - 1346452 \beta_{1} + 599916 \beta_{2} - 235804 \beta_{3} - 133748 \beta_{4} + 2140 \beta_{5} + 36852 \beta_{6} - 11520 \beta_{7} - 58416 \beta_{8} - 2140 \beta_{9} ) q^{59} \) \( + ( -171287604 + 243634 \beta_{1} - 2421378 \beta_{2} + 67608 \beta_{3} - 113322 \beta_{4} - 86160 \beta_{5} - 19212 \beta_{6} - 37404 \beta_{7} + 36186 \beta_{8} + 35130 \beta_{9} ) q^{60} \) \( + ( -345526710 - 256478 \beta_{1} + 16226348 \beta_{2} - 210282 \beta_{3} + 205896 \beta_{4} - 120414 \beta_{5} + 63825 \beta_{6} + 19526 \beta_{7} + 49546 \beta_{8} + 107982 \beta_{9} ) q^{61} \) \( + ( -80660946 - 1214934 \beta_{1} + 125333 \beta_{2} - 16889 \beta_{3} - 95311 \beta_{4} - 79043 \beta_{5} - 84282 \beta_{6} - 6740 \beta_{7} - 113064 \beta_{8} - 41105 \beta_{9} ) q^{62} \) \( + ( 964467 + 314928 \beta_{1} + 4763286 \beta_{2} + 59049 \beta_{3} + 19683 \beta_{5} + 19683 \beta_{7} + 19683 \beta_{8} - 19683 \beta_{9} ) q^{63} \) \( + ( -86324648 + 1124120 \beta_{1} + 7899176 \beta_{2} - 476504 \beta_{3} + 439800 \beta_{4} + 125856 \beta_{5} - 31552 \beta_{6} - 12176 \beta_{7} - 5528 \beta_{8} - 73944 \beta_{9} ) q^{64} \) \( + ( 670623290 + 397228 \beta_{1} - 10200562 \beta_{2} + 708532 \beta_{3} + 438464 \beta_{4} - 28834 \beta_{5} + 59910 \beta_{6} + 47844 \beta_{7} - 71556 \beta_{8} - 55292 \beta_{9} ) q^{65} \) \( + ( 124114500 - 176724 \beta_{1} - 1976598 \beta_{2} - 70470 \beta_{3} + 51390 \beta_{4} - 65778 \beta_{5} - 35076 \beta_{6} + 5112 \beta_{7} + 77232 \beta_{8} + 82410 \beta_{9} ) q^{66} \) \( + ( -4055696 + 2444836 \beta_{1} - 20661028 \beta_{2} - 25260 \beta_{3} - 575172 \beta_{4} - 65684 \beta_{5} + 105156 \beta_{6} + 88864 \beta_{7} - 22736 \beta_{8} + 65684 \beta_{9} ) q^{67} \) \( + ( 307632982 - 3088564 \beta_{1} - 2404164 \beta_{2} - 176922 \beta_{3} - 634416 \beta_{4} + 76176 \beta_{5} - 73024 \beta_{6} + 53248 \beta_{7} - 169664 \beta_{8} + 5936 \beta_{9} ) q^{68} \) \( + ( -174272814 + 113580 \beta_{1} - 21734766 \beta_{2} - 306396 \beta_{3} + 13104 \beta_{4} + 58026 \beta_{5} - 11946 \beta_{6} - 11196 \beta_{7} + 20892 \beta_{8} - 94476 \beta_{9} ) q^{69} \) \( + ( 59647116 + 3636612 \beta_{1} - 793758 \beta_{2} - 142602 \beta_{3} + 385434 \beta_{4} + 503026 \beta_{5} - 11940 \beta_{6} - 5896 \beta_{7} + 127344 \beta_{8} + 15494 \beta_{9} ) q^{70} \) \( + ( 5511294 - 3014712 \beta_{1} + 27343688 \beta_{2} + 256174 \beta_{3} + 34220 \beta_{4} + 117706 \beta_{5} - 7532 \beta_{6} + 113614 \beta_{7} + 93470 \beta_{8} - 117706 \beta_{9} ) q^{71} \) \( + ( -39897441 - 688905 \beta_{1} - 1200663 \beta_{2} + 59049 \beta_{3} + 177147 \beta_{4} + 78732 \beta_{5} + 39366 \beta_{7} + 59049 \beta_{8} + 59049 \beta_{9} ) q^{72} \) \( + ( 314108314 - 501052 \beta_{1} + 24827296 \beta_{2} - 168084 \beta_{3} + 61776 \beta_{4} + 246044 \beta_{5} + 58402 \beta_{6} + 15244 \beta_{7} + 55828 \beta_{8} + 217564 \beta_{9} ) q^{73} \) \( + ( -979126100 + 7524760 \beta_{1} + 15756930 \beta_{2} + 812276 \beta_{3} + 28348 \beta_{4} - 56420 \beta_{5} - 27496 \beta_{6} + 156592 \beta_{7} + 17440 \beta_{8} + 20692 \beta_{9} ) q^{74} \) \( + ( 6192936 + 1876827 \beta_{1} + 31026726 \beta_{2} + 46818 \beta_{3} + 5238 \beta_{4} + 128574 \beta_{5} + 9450 \beta_{6} + 85536 \beta_{7} + 43848 \beta_{8} - 128574 \beta_{9} ) q^{75} \) \( + ( 280068216 - 8699948 \beta_{1} + 24784300 \beta_{2} + 303632 \beta_{3} - 950628 \beta_{4} - 145600 \beta_{5} + 50120 \beta_{6} + 1832 \beta_{7} + 133156 \beta_{8} - 145148 \beta_{9} ) q^{76} \) \( + ( -461473284 - 91616 \beta_{1} - 18311268 \beta_{2} - 270464 \beta_{3} - 115552 \beta_{4} + 470132 \beta_{5} - 18064 \beta_{6} - 15456 \beta_{7} + 25696 \beta_{8} - 3072 \beta_{9} ) q^{77} \) \( + ( 451199322 + 3323800 \beta_{1} + 208818 \beta_{2} + 410022 \beta_{3} - 276168 \beta_{4} - 368766 \beta_{5} + 57564 \beta_{6} - 104328 \beta_{7} - 82512 \beta_{8} + 82998 \beta_{9} ) q^{78} \) \( + ( -11952181 - 4778648 \beta_{1} - 59780526 \beta_{2} + 55717 \beta_{3} + 551628 \beta_{4} - 184249 \beta_{5} - 181708 \beta_{6} - 9045 \beta_{7} - 32133 \beta_{8} + 184249 \beta_{9} ) q^{79} \) \( + ( 1666097788 - 12066364 \beta_{1} - 27779352 \beta_{2} + 71784 \beta_{3} + 1597728 \beta_{4} + 24532 \beta_{5} + 137472 \beta_{6} + 3640 \beta_{7} + 47412 \beta_{8} + 181968 \beta_{9} ) q^{80} \) \( + 387420489 q^{81} \) \( + ( 76573784 + 13662868 \beta_{1} - 30774008 \beta_{2} - 621850 \beta_{3} - 310878 \beta_{4} + 38866 \beta_{5} + 25988 \beta_{6} + 206856 \beta_{7} - 293936 \beta_{8} - 498250 \beta_{9} ) q^{82} \) \( + ( 1391382 + 9348052 \beta_{1} + 8903942 \beta_{2} - 1132216 \beta_{3} + 753562 \beta_{4} + 44688 \beta_{5} - 127658 \beta_{6} - 198242 \beta_{7} - 271882 \beta_{8} - 44688 \beta_{9} ) q^{83} \) \( + ( -1026284220 - 4395528 \beta_{1} + 338976 \beta_{2} - 288036 \beta_{3} - 629064 \beta_{4} + 191064 \beta_{5} + 68640 \beta_{6} + 92160 \beta_{7} - 156192 \beta_{8} + 67272 \beta_{9} ) q^{84} \) \( + ( -314693654 + 505344 \beta_{1} + 69595062 \beta_{2} + 1307136 \beta_{3} - 1886208 \beta_{4} - 1371646 \beta_{5} - 416000 \beta_{6} - 142848 \beta_{7} - 260608 \beta_{8} - 206336 \beta_{9} ) q^{85} \) \( + ( -1520053296 + 14677164 \beta_{1} - 7200454 \beta_{2} - 1905218 \beta_{3} - 113146 \beta_{4} - 210934 \beta_{5} + 499948 \beta_{6} - 332264 \beta_{7} + 340784 \beta_{8} + 458510 \beta_{9} ) q^{86} \) \( + ( 2170863 - 5040302 \beta_{1} + 11970225 \beta_{2} - 602694 \beta_{3} + 415923 \beta_{4} - 45438 \beta_{5} - 9651 \beta_{6} - 422757 \beta_{7} - 162033 \beta_{8} + 45438 \beta_{9} ) q^{87} \) \( + ( -1247724724 - 19574532 \beta_{1} - 51394180 \beta_{2} + 869948 \beta_{3} + 2671236 \beta_{4} - 391784 \beta_{5} + 95552 \beta_{6} + 78712 \beta_{7} - 149964 \beta_{8} - 220660 \beta_{9} ) q^{88} \) \( + ( -1368162170 - 481048 \beta_{1} - 1743788 \beta_{2} - 2599144 \beta_{3} - 2008256 \beta_{4} - 862380 \beta_{5} - 394412 \beta_{6} - 239496 \beta_{7} + 169160 \beta_{8} - 287880 \beta_{9} ) q^{89} \) \( + ( 727995438 + 1220346 \beta_{1} + 7263027 \beta_{2} - 1082565 \beta_{3} + 59049 \beta_{4} + 216513 \beta_{5} + 118098 \beta_{6} + 236196 \beta_{7} - 157464 \beta_{8} + 19683 \beta_{9} ) q^{90} \) \( + ( -2650346 + 3909344 \beta_{1} - 14516124 \beta_{2} + 1740074 \beta_{3} + 1794456 \beta_{4} - 195794 \beta_{5} - 368456 \beta_{6} - 516426 \beta_{7} + 386070 \beta_{8} + 195794 \beta_{9} ) q^{91} \) \( + ( 1263474048 - 17176784 \beta_{1} + 14046720 \beta_{2} + 959728 \beta_{3} - 3327424 \beta_{4} - 565872 \beta_{5} + 369696 \beta_{6} - 331488 \beta_{7} + 838608 \beta_{8} + 304576 \beta_{9} ) q^{92} \) \( + ( 390431493 + 522918 \beta_{1} - 23057559 \beta_{2} + 1640898 \beta_{3} + 410904 \beta_{4} + 906981 \beta_{5} + 53715 \beta_{6} + 69138 \beta_{7} - 169122 \beta_{8} + 11370 \beta_{9} ) q^{93} \) \( + ( 3849337620 + 15063532 \beta_{1} + 5046790 \beta_{2} + 3322370 \beta_{3} + 496734 \beta_{4} - 473802 \beta_{5} + 261780 \beta_{6} - 194328 \beta_{7} + 68304 \beta_{8} + 240306 \beta_{9} ) q^{94} \) \( + ( 12799296 + 1012584 \beta_{1} + 62079328 \beta_{2} + 1769696 \beta_{3} + 146656 \beta_{4} + 64544 \beta_{5} + 49952 \beta_{6} - 281920 \beta_{7} + 458560 \beta_{8} - 64544 \beta_{9} ) q^{95} \) \( + ( -2786505480 - 7449328 \beta_{1} - 6839676 \beta_{2} + 1272132 \beta_{3} + 882084 \beta_{4} - 677508 \beta_{5} + 15648 \beta_{6} - 284976 \beta_{7} - 187656 \beta_{8} + 41820 \beta_{9} ) q^{96} \) \( + ( -1892607098 + 467744 \beta_{1} - 61299212 \beta_{2} + 1333440 \beta_{3} - 378144 \beta_{4} + 1947644 \beta_{5} - 105296 \beta_{6} - 7904 \beta_{7} - 178976 \beta_{8} - 66752 \beta_{9} ) q^{97} \) \( + ( -3182148142 + 23608720 \beta_{1} - 77540825 \beta_{2} + 3198392 \beta_{3} - 129944 \beta_{4} - 449112 \beta_{5} - 234096 \beta_{6} + 172576 \beta_{7} + 156864 \beta_{8} - 168712 \beta_{9} ) q^{98} \) \( + ( 354294 + 1338444 \beta_{1} + 3739770 \beta_{2} - 1338444 \beta_{3} + 511758 \beta_{4} + 236196 \beta_{5} - 196830 \beta_{6} + 511758 \beta_{7} - 275562 \beta_{8} - 236196 \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 22q^{2} \) \(\mathstrut -\mathstrut 644q^{4} \) \(\mathstrut +\mathstrut 3116q^{5} \) \(\mathstrut +\mathstrut 2430q^{6} \) \(\mathstrut +\mathstrut 20176q^{8} \) \(\mathstrut -\mathstrut 196830q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 22q^{2} \) \(\mathstrut -\mathstrut 644q^{4} \) \(\mathstrut +\mathstrut 3116q^{5} \) \(\mathstrut +\mathstrut 2430q^{6} \) \(\mathstrut +\mathstrut 20176q^{8} \) \(\mathstrut -\mathstrut 196830q^{9} \) \(\mathstrut -\mathstrut 369196q^{10} \) \(\mathstrut +\mathstrut 310068q^{12} \) \(\mathstrut -\mathstrut 285100q^{13} \) \(\mathstrut -\mathstrut 2495544q^{14} \) \(\mathstrut +\mathstrut 3590128q^{16} \) \(\mathstrut +\mathstrut 2264420q^{17} \) \(\mathstrut -\mathstrut 433026q^{18} \) \(\mathstrut +\mathstrut 1194248q^{20} \) \(\mathstrut +\mathstrut 3555576q^{21} \) \(\mathstrut -\mathstrut 2139528q^{22} \) \(\mathstrut -\mathstrut 6858432q^{24} \) \(\mathstrut +\mathstrut 23043438q^{25} \) \(\mathstrut +\mathstrut 29599340q^{26} \) \(\mathstrut -\mathstrut 41542224q^{28} \) \(\mathstrut -\mathstrut 46672708q^{29} \) \(\mathstrut +\mathstrut 19963908q^{30} \) \(\mathstrut -\mathstrut 83717408q^{32} \) \(\mathstrut +\mathstrut 22665096q^{33} \) \(\mathstrut +\mathstrut 37514588q^{34} \) \(\mathstrut +\mathstrut 12675852q^{36} \) \(\mathstrut -\mathstrut 194467900q^{37} \) \(\mathstrut -\mathstrut 218769048q^{38} \) \(\mathstrut +\mathstrut 340155488q^{40} \) \(\mathstrut +\mathstrut 276227780q^{41} \) \(\mathstrut -\mathstrut 15919416q^{42} \) \(\mathstrut +\mathstrut 532887504q^{44} \) \(\mathstrut -\mathstrut 61332228q^{45} \) \(\mathstrut -\mathstrut 70057824q^{46} \) \(\mathstrut +\mathstrut 27126576q^{48} \) \(\mathstrut +\mathstrut 653625226q^{49} \) \(\mathstrut +\mathstrut 20815602q^{50} \) \(\mathstrut -\mathstrut 959132296q^{52} \) \(\mathstrut -\mathstrut 1522721956q^{53} \) \(\mathstrut -\mathstrut 47829690q^{54} \) \(\mathstrut -\mathstrut 632703744q^{56} \) \(\mathstrut +\mathstrut 844537752q^{57} \) \(\mathstrut +\mathstrut 1851304388q^{58} \) \(\mathstrut -\mathstrut 1708229736q^{60} \) \(\mathstrut -\mathstrut 3488124604q^{61} \) \(\mathstrut -\mathstrut 806085192q^{62} \) \(\mathstrut -\mathstrut 877634432q^{64} \) \(\mathstrut +\mathstrut 6725245912q^{65} \) \(\mathstrut +\mathstrut 1244885112q^{66} \) \(\mathstrut +\mathstrut 3081993176q^{68} \) \(\mathstrut -\mathstrut 1698239520q^{69} \) \(\mathstrut +\mathstrut 598086768q^{70} \) \(\mathstrut -\mathstrut 397124208q^{72} \) \(\mathstrut +\mathstrut 3090518708q^{73} \) \(\mathstrut -\mathstrut 9824720260q^{74} \) \(\mathstrut +\mathstrut 2750616432q^{76} \) \(\mathstrut -\mathstrut 4577505312q^{77} \) \(\mathstrut +\mathstrut 4510566972q^{78} \) \(\mathstrut +\mathstrut 16715013152q^{80} \) \(\mathstrut +\mathstrut 3874204890q^{81} \) \(\mathstrut +\mathstrut 830592764q^{82} \) \(\mathstrut -\mathstrut 10263359664q^{84} \) \(\mathstrut -\mathstrut 3285444680q^{85} \) \(\mathstrut -\mathstrut 15186172488q^{86} \) \(\mathstrut -\mathstrut 12375555840q^{88} \) \(\mathstrut -\mathstrut 13670065996q^{89} \) \(\mathstrut +\mathstrut 7266884868q^{90} \) \(\mathstrut +\mathstrut 12601016256q^{92} \) \(\mathstrut +\mathstrut 3947087880q^{93} \) \(\mathstrut +\mathstrut 38474881584q^{94} \) \(\mathstrut -\mathstrut 27853204320q^{96} \) \(\mathstrut -\mathstrut 18805077484q^{97} \) \(\mathstrut -\mathstrut 31671844202q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut +\mathstrut \) \(198\) \(x^{8}\mathstrut -\mathstrut \) \(1066\) \(x^{7}\mathstrut +\mathstrut \) \(30608\) \(x^{6}\mathstrut -\mathstrut \) \(147714\) \(x^{5}\mathstrut +\mathstrut \) \(2092829\) \(x^{4}\mathstrut -\mathstrut \) \(9587938\) \(x^{3}\mathstrut +\mathstrut \) \(101463432\) \(x^{2}\mathstrut -\mathstrut \) \(344403128\) \(x\mathstrut +\mathstrut \) \(1648684816\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(4777429206717243\) \(\nu^{9}\mathstrut -\mathstrut \) \(10617846828144012\) \(\nu^{8}\mathstrut +\mathstrut \) \(866840028580050498\) \(\nu^{7}\mathstrut -\mathstrut \) \(4135098999535025202\) \(\nu^{6}\mathstrut +\mathstrut \) \(133217088577972139196\) \(\nu^{5}\mathstrut -\mathstrut \) \(520588536343518703086\) \(\nu^{4}\mathstrut +\mathstrut \) \(7421508231889413592827\) \(\nu^{3}\mathstrut -\mathstrut \) \(9972499972996597707108\) \(\nu^{2}\mathstrut +\mathstrut \) \(335690363418459138322488\) \(\nu\mathstrut -\mathstrut \) \(319611005063159694582672\)\()/\)\(94\!\cdots\!08\)
\(\beta_{2}\)\(=\)\((\)\(244514118165802049\) \(\nu^{9}\mathstrut +\mathstrut \) \(4759286524274815484\) \(\nu^{8}\mathstrut +\mathstrut \) \(62309791757744002046\) \(\nu^{7}\mathstrut +\mathstrut \) \(664185951330364058330\) \(\nu^{6}\mathstrut +\mathstrut \) \(5391190466481343383452\) \(\nu^{5}\mathstrut +\mathstrut \) \(81450882246208754247238\) \(\nu^{4}\mathstrut +\mathstrut \) \(290635608588960074612913\) \(\nu^{3}\mathstrut +\mathstrut \) \(3590167781275253920297724\) \(\nu^{2}\mathstrut +\mathstrut \) \(2077866539674224928688160\) \(\nu\mathstrut +\mathstrut \) \(138176154449247025883471192\)\()/\)\(85\!\cdots\!72\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(6956003609714575371\) \(\nu^{9}\mathstrut +\mathstrut \) \(61094477451775937552\) \(\nu^{8}\mathstrut -\mathstrut \) \(867568192030995250890\) \(\nu^{7}\mathstrut +\mathstrut \) \(17399371203657263226522\) \(\nu^{6}\mathstrut -\mathstrut \) \(161065151537227499757796\) \(\nu^{5}\mathstrut +\mathstrut \) \(2057279143818991878258574\) \(\nu^{4}\mathstrut -\mathstrut \) \(7881913751830270469003155\) \(\nu^{3}\mathstrut +\mathstrut \) \(119946400642994070806778488\) \(\nu^{2}\mathstrut -\mathstrut \) \(472197871339060434705012064\) \(\nu\mathstrut +\mathstrut \) \(3562184677354643152346156020\)\()/\)\(42\!\cdots\!36\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(4074401261735809353\) \(\nu^{9}\mathstrut +\mathstrut \) \(2043647962356372540\) \(\nu^{8}\mathstrut -\mathstrut \) \(662445338394291108078\) \(\nu^{7}\mathstrut +\mathstrut \) \(4021307961432448536294\) \(\nu^{6}\mathstrut -\mathstrut \) \(92231550063765008493660\) \(\nu^{5}\mathstrut +\mathstrut \) \(461018153155023600411690\) \(\nu^{4}\mathstrut -\mathstrut \) \(4650356582597071798686153\) \(\nu^{3}\mathstrut +\mathstrut \) \(32501892010779144078818412\) \(\nu^{2}\mathstrut -\mathstrut \) \(200862412340721645600310560\) \(\nu\mathstrut +\mathstrut \) \(799325383998374735364623280\)\()/\)\(94\!\cdots\!08\)
\(\beta_{5}\)\(=\)\((\)\(40946879772659454085\) \(\nu^{9}\mathstrut -\mathstrut \) \(140390274454448131732\) \(\nu^{8}\mathstrut +\mathstrut \) \(6137478113104649769142\) \(\nu^{7}\mathstrut -\mathstrut \) \(62266727766819475748990\) \(\nu^{6}\mathstrut +\mathstrut \) \(940173598313698671124364\) \(\nu^{5}\mathstrut -\mathstrut \) \(7297565567994471237856994\) \(\nu^{4}\mathstrut +\mathstrut \) \(47432635866637246745233525\) \(\nu^{3}\mathstrut -\mathstrut \) \(463566037968121253114083732\) \(\nu^{2}\mathstrut +\mathstrut \) \(2310091957847957517733893280\) \(\nu\mathstrut -\mathstrut \) \(12798241643837765335247478288\)\()/\)\(85\!\cdots\!72\)
\(\beta_{6}\)\(=\)\((\)\(40040567016769138229\) \(\nu^{9}\mathstrut +\mathstrut \) \(427161687787190661692\) \(\nu^{8}\mathstrut +\mathstrut \) \(8443283742927685413934\) \(\nu^{7}\mathstrut +\mathstrut \) \(42748973619824852650578\) \(\nu^{6}\mathstrut +\mathstrut \) \(869620934184691905258164\) \(\nu^{5}\mathstrut +\mathstrut \) \(5476710502601004804923566\) \(\nu^{4}\mathstrut +\mathstrut \) \(44867912595724383218728245\) \(\nu^{3}\mathstrut +\mathstrut \) \(189955483629097698382984484\) \(\nu^{2}\mathstrut +\mathstrut \) \(1008201159213673133314361368\) \(\nu\mathstrut +\mathstrut \) \(9328228357856593825858481080\)\()/\)\(42\!\cdots\!36\)
\(\beta_{7}\)\(=\)\((\)\(53958474794391512307\) \(\nu^{9}\mathstrut +\mathstrut \) \(449020066197220706832\) \(\nu^{8}\mathstrut +\mathstrut \) \(10913793378645562939958\) \(\nu^{7}\mathstrut +\mathstrut \) \(35397998674387297358206\) \(\nu^{6}\mathstrut +\mathstrut \) \(1196834826639495118486344\) \(\nu^{5}\mathstrut +\mathstrut \) \(4654520791073348634739386\) \(\nu^{4}\mathstrut +\mathstrut \) \(61421330382795854754272307\) \(\nu^{3}\mathstrut +\mathstrut \) \(122242577853610486327425852\) \(\nu^{2}\mathstrut +\mathstrut \) \(1599969854882240469364611340\) \(\nu\mathstrut +\mathstrut \) \(8027818694497427725620649220\)\()/\)\(42\!\cdots\!36\)
\(\beta_{8}\)\(=\)\((\)\(54900553566834144609\) \(\nu^{9}\mathstrut -\mathstrut \) \(206758686747266430884\) \(\nu^{8}\mathstrut +\mathstrut \) \(8034975102562038780138\) \(\nu^{7}\mathstrut -\mathstrut \) \(87521298290337040914174\) \(\nu^{6}\mathstrut +\mathstrut \) \(1242244064941865331524176\) \(\nu^{5}\mathstrut -\mathstrut \) \(10207533804115811223296962\) \(\nu^{4}\mathstrut +\mathstrut \) \(61350948513513740325556393\) \(\nu^{3}\mathstrut -\mathstrut \) \(642895758724408004714681504\) \(\nu^{2}\mathstrut +\mathstrut \) \(3063784151383619291861511676\) \(\nu\mathstrut -\mathstrut \) \(17717548849782258618425143948\)\()/\)\(42\!\cdots\!36\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(178045243840232938623\) \(\nu^{9}\mathstrut -\mathstrut \) \(36415201309600845492\) \(\nu^{8}\mathstrut -\mathstrut \) \(29256442348958154057106\) \(\nu^{7}\mathstrut +\mathstrut \) \(152929966958268613377370\) \(\nu^{6}\mathstrut -\mathstrut \) \(4002728165471237713272660\) \(\nu^{5}\mathstrut +\mathstrut \) \(17095941284939570978393958\) \(\nu^{4}\mathstrut -\mathstrut \) \(201418898124871689874699023\) \(\nu^{3}\mathstrut +\mathstrut \) \(1266245414440285888599400188\) \(\nu^{2}\mathstrut -\mathstrut \) \(8537636403937481754889384112\) \(\nu\mathstrut +\mathstrut \) \(29386959163296740160283748288\)\()/\)\(85\!\cdots\!72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(68\) \(\beta_{8}\mathstrut -\mathstrut \) \(60\) \(\beta_{7}\mathstrut +\mathstrut \) \(62\) \(\beta_{6}\mathstrut -\mathstrut \) \(155\) \(\beta_{5}\mathstrut -\mathstrut \) \(231\) \(\beta_{4}\mathstrut -\mathstrut \) \(549\) \(\beta_{3}\mathstrut +\mathstrut \) \(4785\) \(\beta_{2}\mathstrut +\mathstrut \) \(78\) \(\beta_{1}\mathstrut +\mathstrut \) \(25734\)\()/124416\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(51\) \(\beta_{9}\mathstrut +\mathstrut \) \(168\) \(\beta_{8}\mathstrut -\mathstrut \) \(72\) \(\beta_{7}\mathstrut -\mathstrut \) \(60\) \(\beta_{6}\mathstrut -\mathstrut \) \(237\) \(\beta_{5}\mathstrut -\mathstrut \) \(117\) \(\beta_{4}\mathstrut +\mathstrut \) \(909\) \(\beta_{3}\mathstrut +\mathstrut \) \(9099\) \(\beta_{2}\mathstrut +\mathstrut \) \(20542\) \(\beta_{1}\mathstrut -\mathstrut \) \(1623726\)\()/41472\)
\(\nu^{3}\)\(=\)\((\)\(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(1282\) \(\beta_{8}\mathstrut +\mathstrut \) \(1194\) \(\beta_{7}\mathstrut +\mathstrut \) \(1747\) \(\beta_{6}\mathstrut +\mathstrut \) \(15248\) \(\beta_{5}\mathstrut +\mathstrut \) \(10554\) \(\beta_{4}\mathstrut +\mathstrut \) \(15048\) \(\beta_{3}\mathstrut -\mathstrut \) \(627378\) \(\beta_{2}\mathstrut +\mathstrut \) \(6114\) \(\beta_{1}\mathstrut +\mathstrut \) \(12431796\)\()/62208\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(11613\) \(\beta_{9}\mathstrut -\mathstrut \) \(41964\) \(\beta_{8}\mathstrut +\mathstrut \) \(6444\) \(\beta_{7}\mathstrut -\mathstrut \) \(6150\) \(\beta_{6}\mathstrut -\mathstrut \) \(42099\) \(\beta_{5}\mathstrut -\mathstrut \) \(74007\) \(\beta_{4}\mathstrut -\mathstrut \) \(129645\) \(\beta_{3}\mathstrut -\mathstrut \) \(1514511\) \(\beta_{2}\mathstrut -\mathstrut \) \(2003674\) \(\beta_{1}\mathstrut -\mathstrut \) \(160624026\)\()/41472\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(489877\) \(\beta_{9}\mathstrut +\mathstrut \) \(1319800\) \(\beta_{8}\mathstrut -\mathstrut \) \(131928\) \(\beta_{7}\mathstrut -\mathstrut \) \(1196308\) \(\beta_{6}\mathstrut -\mathstrut \) \(1555787\) \(\beta_{5}\mathstrut +\mathstrut \) \(1616349\) \(\beta_{4}\mathstrut +\mathstrut \) \(3650283\) \(\beta_{3}\mathstrut +\mathstrut \) \(205381149\) \(\beta_{2}\mathstrut +\mathstrut \) \(33557874\) \(\beta_{1}\mathstrut -\mathstrut \) \(2663160738\)\()/124416\)
\(\nu^{6}\)\(=\)\((\)\(181262\) \(\beta_{9}\mathstrut +\mathstrut \) \(91354\) \(\beta_{8}\mathstrut +\mathstrut \) \(79758\) \(\beta_{7}\mathstrut +\mathstrut \) \(205193\) \(\beta_{6}\mathstrut +\mathstrut \) \(748018\) \(\beta_{5}\mathstrut +\mathstrut \) \(878784\) \(\beta_{4}\mathstrut -\mathstrut \) \(230538\) \(\beta_{3}\mathstrut -\mathstrut \) \(17977236\) \(\beta_{2}\mathstrut -\mathstrut \) \(305958\) \(\beta_{1}\mathstrut +\mathstrut \) \(2094460440\)\()/2304\)
\(\nu^{7}\)\(=\)\((\)\(26615909\) \(\beta_{9}\mathstrut -\mathstrut \) \(200051060\) \(\beta_{8}\mathstrut -\mathstrut \) \(10789260\) \(\beta_{7}\mathstrut +\mathstrut \) \(37213190\) \(\beta_{6}\mathstrut -\mathstrut \) \(325888373\) \(\beta_{5}\mathstrut -\mathstrut \) \(803013009\) \(\beta_{4}\mathstrut -\mathstrut \) \(791950347\) \(\beta_{3}\mathstrut -\mathstrut \) \(8114780793\) \(\beta_{2}\mathstrut -\mathstrut \) \(5734603254\) \(\beta_{1}\mathstrut -\mathstrut \) \(458252865366\)\()/124416\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(326062383\) \(\beta_{9}\mathstrut +\mathstrut \) \(719028600\) \(\beta_{8}\mathstrut -\mathstrut \) \(356038488\) \(\beta_{7}\mathstrut -\mathstrut \) \(509101332\) \(\beta_{6}\mathstrut -\mathstrut \) \(930499761\) \(\beta_{5}\mathstrut -\mathstrut \) \(63683721\) \(\beta_{4}\mathstrut +\mathstrut \) \(3493536849\) \(\beta_{3}\mathstrut +\mathstrut \) \(103033122903\) \(\beta_{2}\mathstrut +\mathstrut \) \(31696916134\) \(\beta_{1}\mathstrut -\mathstrut \) \(2463595346214\)\()/41472\)
\(\nu^{9}\)\(=\)\((\)\(5095241728\) \(\beta_{9}\mathstrut +\mathstrut \) \(2327484254\) \(\beta_{8}\mathstrut +\mathstrut \) \(4576200522\) \(\beta_{7}\mathstrut +\mathstrut \) \(10316143171\) \(\beta_{6}\mathstrut +\mathstrut \) \(39770992352\) \(\beta_{5}\mathstrut +\mathstrut \) \(49897932522\) \(\beta_{4}\mathstrut +\mathstrut \) \(4780169784\) \(\beta_{3}\mathstrut -\mathstrut \) \(1616508888450\) \(\beta_{2}\mathstrut -\mathstrut \) \(1116850398\) \(\beta_{1}\mathstrut +\mathstrut \) \(72045577607316\)\()/62208\)

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
4.69893 8.13879i
4.69893 + 8.13879i
3.24812 + 5.62591i
3.24812 5.62591i
−4.11365 + 7.12504i
−4.11365 7.12504i
−6.13017 10.6178i
−6.13017 + 10.6178i
3.29676 5.71016i
3.29676 + 5.71016i
−30.5317 9.58205i 140.296i 840.369 + 585.112i 2598.88 −1344.32 + 4283.48i 4231.42i −20051.3 25916.9i −19683.0 −79348.1 24902.6i
7.2 −30.5317 + 9.58205i 140.296i 840.369 585.112i 2598.88 −1344.32 4283.48i 4231.42i −20051.3 + 25916.9i −19683.0 −79348.1 + 24902.6i
7.3 −6.13922 31.4056i 140.296i −948.620 + 385.612i 5105.39 4406.08 861.309i 14850.7i 17934.1 + 27424.6i −19683.0 −31343.1 160338.i
7.4 −6.13922 + 31.4056i 140.296i −948.620 385.612i 5105.39 4406.08 + 861.309i 14850.7i 17934.1 27424.6i −19683.0 −31343.1 + 160338.i
7.5 −4.31497 31.7077i 140.296i −986.762 + 273.636i −1853.03 −4448.47 + 605.373i 2434.59i 12934.2 + 30107.3i −19683.0 7995.78 + 58755.5i
7.6 −4.31497 + 31.7077i 140.296i −986.762 273.636i −1853.03 −4448.47 605.373i 2434.59i 12934.2 30107.3i −19683.0 7995.78 58755.5i
7.7 20.5505 24.5291i 140.296i −179.355 1008.17i −4874.35 3441.34 + 2883.15i 22628.8i −28415.4 16319.0i −19683.0 −100170. + 119563.i
7.8 20.5505 + 24.5291i 140.296i −179.355 + 1008.17i −4874.35 3441.34 2883.15i 22628.8i −28415.4 + 16319.0i −19683.0 −100170. 119563.i
7.9 31.4354 5.98464i 140.296i 952.368 376.259i 581.119 −839.621 4410.26i 18141.8i 27686.3 17527.4i −19683.0 18267.7 3477.79i
7.10 31.4354 + 5.98464i 140.296i 952.368 + 376.259i 581.119 −839.621 + 4410.26i 18141.8i 27686.3 + 17527.4i −19683.0 18267.7 + 3477.79i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.10
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 \(S_{11}^{\mathrm{new}}(12, [\chi])\).