Properties

Label 12.9.d.a
Level 12
Weight 9
Character orbit 12.d
Analytic conductor 4.889
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.88854332073\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( -6 + \beta_{2} - \beta_{3} ) q^{4} + ( -37 + 17 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{5} + ( 142 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{6} + ( -7 - 20 \beta_{1} + \beta_{2} + 10 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{7} + ( -1621 + 4 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{8} -2187 q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + ( -\beta_{1} - \beta_{3} ) q^{3} + ( -6 + \beta_{2} - \beta_{3} ) q^{4} + ( -37 + 17 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{5} + ( 142 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{6} + ( -7 - 20 \beta_{1} + \beta_{2} + 10 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{7} + ( -1621 + 4 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{8} -2187 q^{9} + ( 4570 - 66 \beta_{1} + 28 \beta_{2} + 24 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{10} + ( 78 + 268 \beta_{1} - 34 \beta_{2} - 48 \beta_{3} - 22 \beta_{4} - 22 \beta_{5} - 8 \beta_{6} - 12 \beta_{7} ) q^{11} + ( -1377 + 140 \beta_{1} + 5 \beta_{3} - 36 \beta_{4} + 18 \beta_{5} + 9 \beta_{6} + 9 \beta_{7} ) q^{12} + ( -364 - 38 \beta_{1} + 66 \beta_{2} - 42 \beta_{3} + 54 \beta_{4} - 12 \beta_{5} + 40 \beta_{6} - 40 \beta_{7} ) q^{13} + ( 6620 - 172 \beta_{1} - 20 \beta_{2} - 348 \beta_{3} + 24 \beta_{4} - 56 \beta_{5} + 16 \beta_{6} + 60 \beta_{7} ) q^{14} + ( -19 + 50 \beta_{1} - 91 \beta_{2} + 104 \beta_{3} - 73 \beta_{4} - 9 \beta_{5} + 4 \beta_{6} - 18 \beta_{7} ) q^{15} + ( 11906 - 1352 \beta_{1} + 14 \beta_{2} + 708 \beta_{3} + 24 \beta_{4} - 108 \beta_{5} + 26 \beta_{6} + 58 \beta_{7} ) q^{16} + ( -24032 + 846 \beta_{1} - 162 \beta_{2} - 142 \beta_{3} + 306 \beta_{4} - 164 \beta_{5} + 48 \beta_{6} - 48 \beta_{7} ) q^{17} + ( -2187 - 2187 \beta_{1} ) q^{18} + ( -474 - 1532 \beta_{1} - 426 \beta_{2} + 280 \beta_{3} - 318 \beta_{4} + 66 \beta_{5} - 24 \beta_{6} - 108 \beta_{7} ) q^{19} + ( 42884 + 5024 \beta_{1} + 86 \beta_{2} + 1282 \beta_{3} + 160 \beta_{4} + 112 \beta_{5} - 72 \beta_{6} + 120 \beta_{7} ) q^{20} + ( 14625 - 2727 \beta_{1} + 369 \beta_{2} + 135 \beta_{3} - 153 \beta_{4} + 18 \beta_{5} + 72 \beta_{6} - 72 \beta_{7} ) q^{21} + ( -69040 - 1196 \beta_{1} + 112 \beta_{2} - 2892 \beta_{3} - 552 \beta_{4} + 240 \beta_{5} + 36 \beta_{6} + 72 \beta_{7} ) q^{22} + ( 1906 + 5952 \beta_{1} + 770 \beta_{2} - 1412 \beta_{3} + 710 \beta_{4} - 314 \beta_{5} - 200 \beta_{6} + 60 \beta_{7} ) q^{23} + ( 27163 - 860 \beta_{1} + 343 \beta_{2} + 1840 \beta_{3} - 188 \beta_{4} + 630 \beta_{5} - 13 \beta_{6} - 117 \beta_{7} ) q^{24} + ( -70257 + 7084 \beta_{1} + 252 \beta_{2} - 204 \beta_{3} - 780 \beta_{4} + 984 \beta_{5} - 176 \beta_{6} + 176 \beta_{7} ) q^{25} + ( 3650 - 3926 \beta_{1} + 24 \beta_{2} - 8576 \beta_{3} + 1280 \beta_{4} - 16 \beta_{5} - 424 \beta_{6} - 216 \beta_{7} ) q^{26} + ( 2187 \beta_{1} + 2187 \beta_{3} ) q^{27} + ( -74140 + 12688 \beta_{1} - 776 \beta_{2} + 12164 \beta_{3} - 816 \beta_{4} - 936 \beta_{5} - 212 \beta_{6} - 532 \beta_{7} ) q^{28} + ( 256625 - 3961 \beta_{1} - 1433 \beta_{2} + 905 \beta_{3} - 439 \beta_{4} - 466 \beta_{5} - 624 \beta_{6} + 624 \beta_{7} ) q^{29} + ( 6552 - 2306 \beta_{1} + 72 \beta_{2} - 4466 \beta_{3} - 828 \beta_{4} + 936 \beta_{5} + 342 \beta_{6} - 180 \beta_{7} ) q^{30} + ( -9887 - 41564 \beta_{1} + 1769 \beta_{2} - 1934 \beta_{3} + 987 \beta_{4} + 2523 \beta_{5} + 700 \beta_{6} + 782 \beta_{7} ) q^{31} + ( -449432 + 19872 \beta_{1} - 848 \beta_{2} + 13224 \beta_{3} + 1504 \beta_{4} - 656 \beta_{5} + 632 \beta_{6} - 552 \beta_{7} ) q^{32} + ( -121302 - 23490 \beta_{1} - 450 \beta_{2} + 1890 \beta_{3} - 414 \beta_{4} - 1476 \beta_{5} - 288 \beta_{6} + 288 \beta_{7} ) q^{33} + ( 191458 - 30630 \beta_{1} - 1400 \beta_{2} - 15616 \beta_{3} + 1536 \beta_{4} - 5808 \beta_{5} + 456 \beta_{6} - 1224 \beta_{7} ) q^{34} + ( 6706 + 29832 \beta_{1} + 866 \beta_{2} + 2500 \beta_{3} - 202 \beta_{4} - 1994 \beta_{5} + 1576 \beta_{6} + 1068 \beta_{7} ) q^{35} + ( 13122 - 2187 \beta_{2} + 2187 \beta_{3} ) q^{36} + ( 947822 + 54164 \beta_{1} - 948 \beta_{2} - 3684 \beta_{3} + 828 \beta_{4} + 2856 \beta_{5} - 40 \beta_{6} + 40 \beta_{7} ) q^{37} + ( 460448 - 12676 \beta_{1} - 1600 \beta_{2} - 23716 \beta_{3} - 4216 \beta_{4} + 3888 \beta_{5} + 1564 \beta_{6} - 216 \beta_{7} ) q^{38} + ( 17802 + 67078 \beta_{1} - 342 \beta_{2} - 4526 \beta_{3} - 666 \beta_{4} - 4698 \beta_{5} + 720 \beta_{6} + 324 \beta_{7} ) q^{39} + ( 225818 + 48696 \beta_{1} + 6090 \beta_{2} + 21128 \beta_{3} + 3384 \beta_{4} - 3756 \beta_{5} + 1594 \beta_{6} + 554 \beta_{7} ) q^{40} + ( -1130396 - 87678 \beta_{1} + 3666 \beta_{2} + 5246 \beta_{3} - 1218 \beta_{4} - 4028 \beta_{5} + 816 \beta_{6} - 816 \beta_{7} ) q^{41} + ( -655776 + 13860 \beta_{1} - 324 \beta_{2} - 8064 \beta_{3} + 2304 \beta_{4} + 5976 \beta_{5} - 1764 \beta_{6} + 612 \beta_{7} ) q^{42} + ( -28294 - 119060 \beta_{1} - 374 \beta_{2} - 4712 \beta_{3} + 1278 \beta_{4} + 7806 \beta_{5} - 2824 \beta_{6} - 1652 \beta_{7} ) q^{43} + ( 281916 - 61264 \beta_{1} - 880 \beta_{2} + 20708 \beta_{3} - 8976 \beta_{4} + 10248 \beta_{5} - 3580 \beta_{6} + 132 \beta_{7} ) q^{44} + ( 80919 - 37179 \beta_{1} - 2187 \beta_{2} + 2187 \beta_{3} + 2187 \beta_{4} - 4374 \beta_{5} ) q^{45} + ( -1705976 + 12736 \beta_{1} + 3752 \beta_{2} + 14432 \beta_{3} + 4608 \beta_{4} - 9840 \beta_{5} - 3576 \beta_{6} + 3960 \beta_{7} ) q^{46} + ( 3214 + 35336 \beta_{1} - 12738 \beta_{2} + 21964 \beta_{3} - 7014 \beta_{4} - 1126 \beta_{5} - 5208 \beta_{6} - 5724 \beta_{7} ) q^{47} + ( 1371942 + 13160 \beta_{1} + 4338 \beta_{2} - 18700 \beta_{3} + 5832 \beta_{4} + 6300 \beta_{5} + 702 \beta_{6} + 1854 \beta_{7} ) q^{48} + ( -2278151 + 331928 \beta_{1} - 216 \beta_{2} - 24312 \beta_{3} + 5064 \beta_{4} + 19248 \beta_{5} + 1616 \beta_{6} - 1616 \beta_{7} ) q^{49} + ( 1800819 - 66013 \beta_{1} + 12624 \beta_{2} + 40192 \beta_{3} - 5632 \beta_{4} + 13088 \beta_{5} - 304 \beta_{6} + 3120 \beta_{7} ) q^{50} + ( 34598 + 147878 \beta_{1} + 6230 \beta_{2} + 12986 \beta_{3} + 7490 \beta_{4} - 6462 \beta_{5} - 4760 \beta_{6} - 1260 \beta_{7} ) q^{51} + ( 2331444 - 12736 \beta_{1} - 18406 \beta_{2} - 43306 \beta_{3} - 10176 \beta_{4} - 23712 \beta_{5} - 6992 \beta_{6} + 5168 \beta_{7} ) q^{52} + ( 1061001 - 124097 \beta_{1} + 21599 \beta_{2} + 2513 \beta_{3} - 2159 \beta_{4} - 354 \beta_{5} + 6480 \beta_{6} - 6480 \beta_{7} ) q^{53} + ( -310554 - 2187 \beta_{1} + 4374 \beta_{2} - 2187 \beta_{3} + 4374 \beta_{4} + 2187 \beta_{6} ) q^{54} + ( -44696 - 261784 \beta_{1} - 12568 \beta_{2} - 87016 \beta_{3} - 11304 \beta_{4} + 8664 \beta_{5} + 2752 \beta_{6} - 1264 \beta_{7} ) q^{55} + ( -5696852 - 126832 \beta_{1} + 22716 \beta_{2} - 123680 \beta_{3} + 11280 \beta_{4} + 13016 \beta_{5} + 12876 \beta_{6} + 3372 \beta_{7} ) q^{56} + ( -348246 - 201690 \beta_{1} - 5130 \beta_{2} + 9882 \beta_{3} + 11610 \beta_{4} - 21492 \beta_{5} + 2160 \beta_{6} - 2160 \beta_{7} ) q^{57} + ( -1054358 + 319590 \beta_{1} - 9212 \beta_{2} + 131328 \beta_{3} - 19968 \beta_{4} - 9432 \beta_{5} + 8228 \beta_{6} + 1756 \beta_{7} ) q^{58} + ( 53720 + 278548 \beta_{1} - 7464 \beta_{2} + 57572 \beta_{3} - 11352 \beta_{4} - 17240 \beta_{5} + 9984 \beta_{6} + 3888 \beta_{7} ) q^{59} + ( 3550874 - 19960 \beta_{1} - 616 \beta_{2} - 35242 \beta_{3} - 9112 \beta_{4} + 18828 \beta_{5} - 5690 \beta_{6} - 2682 \beta_{7} ) q^{60} + ( 1670450 - 19880 \beta_{1} - 17040 \beta_{2} + 2136 \beta_{3} + 10104 \beta_{4} - 12240 \beta_{5} - 2312 \beta_{6} + 2312 \beta_{7} ) q^{61} + ( 10052972 + 78428 \beta_{1} - 33924 \beta_{2} + 206380 \beta_{3} + 23560 \beta_{4} - 9208 \beta_{5} - 5336 \beta_{6} - 7908 \beta_{7} ) q^{62} + ( 15309 + 43740 \beta_{1} - 2187 \beta_{2} - 21870 \beta_{3} - 6561 \beta_{4} - 6561 \beta_{5} + 8748 \beta_{6} + 4374 \beta_{7} ) q^{63} + ( 58552 - 534112 \beta_{1} + 23640 \beta_{2} - 165952 \beta_{3} + 40608 \beta_{4} - 23952 \beta_{5} + 15096 \beta_{6} - 10056 \beta_{7} ) q^{64} + ( 866522 - 122634 \beta_{1} - 45402 \beta_{2} + 19594 \beta_{3} + 6090 \beta_{4} - 25684 \beta_{5} - 13104 \beta_{6} + 13104 \beta_{7} ) q^{65} + ( -6440148 - 51228 \beta_{1} - 25272 \beta_{2} + 87552 \beta_{3} - 9216 \beta_{4} + 720 \beta_{5} + 2952 \beta_{6} + 1656 \beta_{7} ) q^{66} + ( -97832 - 160900 \beta_{1} + 35288 \beta_{2} + 237644 \beta_{3} + 26664 \beta_{4} + 28968 \beta_{5} + 1408 \beta_{6} + 8624 \beta_{7} ) q^{67} + ( -14563212 + 157376 \beta_{1} - 68694 \beta_{2} - 255450 \beta_{3} - 29504 \beta_{4} - 23520 \beta_{5} - 17648 \beta_{6} - 6768 \beta_{7} ) q^{68} + ( -1026630 + 141426 \beta_{1} + 23778 \beta_{2} + 1998 \beta_{3} - 42354 \beta_{4} + 40356 \beta_{5} - 6192 \beta_{6} + 6192 \beta_{7} ) q^{69} + ( -7562600 + 95320 \beta_{1} + 35768 \beta_{2} + 183032 \beta_{3} + 16080 \beta_{4} + 9744 \beta_{5} - 4176 \beta_{6} - 21960 \beta_{7} ) q^{70} + ( 8858 - 311056 \beta_{1} + 91946 \beta_{2} - 323236 \beta_{3} + 75038 \beta_{4} + 12318 \beta_{5} - 6344 \beta_{6} + 16908 \beta_{7} ) q^{71} + ( 3545127 - 8748 \beta_{1} - 6561 \beta_{2} - 26244 \beta_{3} - 8748 \beta_{4} - 13122 \beta_{5} + 10935 \beta_{6} - 6561 \beta_{7} ) q^{72} + ( 11859434 - 71560 \beta_{1} + 69480 \beta_{2} + 21480 \beta_{3} - 86232 \beta_{4} + 64752 \beta_{5} - 5584 \beta_{6} + 5584 \beta_{7} ) q^{73} + ( 15183410 + 847938 \beta_{1} + 49008 \beta_{2} - 65152 \beta_{3} - 1280 \beta_{4} - 21152 \beta_{5} + 3952 \beta_{6} - 3312 \beta_{7} ) q^{74} + ( -92628 - 237891 \beta_{1} - 40788 \beta_{2} + 123045 \beta_{3} - 32364 \beta_{4} + 17172 \beta_{5} + 1152 \beta_{6} - 8424 \beta_{7} ) q^{75} + ( 18205140 + 385936 \beta_{1} - 15024 \beta_{2} - 51284 \beta_{3} - 66480 \beta_{4} + 77400 \beta_{5} - 21204 \beta_{6} - 20052 \beta_{7} ) q^{76} + ( -6883316 + 947756 \beta_{1} - 66452 \beta_{2} - 39276 \beta_{3} - 7276 \beta_{4} + 46552 \beta_{5} - 24576 \beta_{6} + 24576 \beta_{7} ) q^{77} + ( -17547820 + 33578 \beta_{1} + 55892 \beta_{2} - 5302 \beta_{3} - 19924 \beta_{4} + 11664 \beta_{5} - 9314 \beta_{6} - 11016 \beta_{7} ) q^{78} + ( 196493 + 389700 \beta_{1} + 7285 \beta_{2} - 395910 \beta_{3} + 4095 \beta_{4} - 48897 \beta_{5} + 2828 \beta_{6} + 3190 \beta_{7} ) q^{79} + ( -20436612 + 307856 \beta_{1} + 76324 \beta_{2} + 95288 \beta_{3} + 112080 \beta_{4} - 19240 \beta_{5} - 5684 \beta_{6} - 8052 \beta_{7} ) q^{80} + 4782969 q^{81} + ( -23815886 - 1034502 \beta_{1} - 69320 \beta_{2} - 40192 \beta_{3} + 26112 \beta_{4} + 55344 \beta_{5} - 17928 \beta_{6} + 4872 \beta_{7} ) q^{82} + ( -58482 + 268564 \beta_{1} - 3106 \beta_{2} + 507480 \beta_{3} + 4682 \beta_{4} + 17738 \beta_{5} - 12776 \beta_{6} - 7788 \beta_{7} ) q^{83} + ( 24765408 - 682848 \beta_{1} + 3708 \beta_{2} + 116892 \beta_{3} - 7776 \beta_{4} - 58896 \beta_{5} + 6264 \beta_{6} + 17208 \beta_{7} ) q^{84} + ( -25742030 - 2113794 \beta_{1} - 45282 \beta_{2} + 120546 \beta_{3} + 77538 \beta_{4} - 198084 \beta_{5} + 10752 \beta_{6} - 10752 \beta_{7} ) q^{85} + ( 30025664 - 143788 \beta_{1} - 120416 \beta_{2} - 203916 \beta_{3} - 7464 \beta_{4} - 28976 \beta_{5} + 8212 \beta_{6} + 40920 \beta_{7} ) q^{86} + ( -266053 - 1280950 \beta_{1} + 34979 \beta_{2} - 202300 \beta_{3} + 35537 \beta_{4} + 75537 \beta_{5} - 14996 \beta_{6} - 558 \beta_{7} ) q^{87} + ( 19571516 + 288848 \beta_{1} + 5164 \beta_{2} + 188288 \beta_{3} - 51504 \beta_{4} + 133752 \beta_{5} + 32924 \beta_{6} + 52348 \beta_{7} ) q^{88} + ( 23695798 + 269412 \beta_{1} + 88836 \beta_{2} - 89188 \beta_{3} + 98076 \beta_{4} - 8888 \beta_{5} + 62304 \beta_{6} - 62304 \beta_{7} ) q^{89} + ( -9994590 + 144342 \beta_{1} - 61236 \beta_{2} - 52488 \beta_{5} + 8748 \beta_{6} - 8748 \beta_{7} ) q^{90} + ( 656866 + 3737400 \beta_{1} - 189838 \beta_{2} + 1063572 \beta_{3} - 152874 \beta_{4} - 193194 \beta_{5} + 9400 \beta_{6} - 36964 \beta_{7} ) q^{91} + ( -48979280 - 1160128 \beta_{1} - 3088 \beta_{2} + 898688 \beta_{3} + 16960 \beta_{4} - 113568 \beta_{5} + 9264 \beta_{6} + 37296 \beta_{7} ) q^{92} + ( -6443055 + 1575585 \beta_{1} + 27225 \beta_{2} - 152577 \beta_{3} + 87903 \beta_{4} + 64674 \beta_{5} + 38376 \beta_{6} - 38376 \beta_{7} ) q^{93} + ( -4659448 - 597576 \beta_{1} + 21736 \beta_{2} - 1256680 \beta_{3} - 120048 \beta_{4} + 64368 \beta_{5} + 63744 \beta_{6} + 59400 \beta_{7} ) q^{94} + ( -313664 - 2415368 \beta_{1} - 252224 \beta_{2} - 1229960 \beta_{3} - 212672 \beta_{4} + 35136 \beta_{5} + 29696 \beta_{6} - 39552 \beta_{7} ) q^{95} + ( 31029088 + 1463488 \beta_{1} + 2488 \beta_{2} + 369880 \beta_{3} + 47488 \beta_{4} - 89568 \beta_{5} - 21616 \beta_{6} - 9360 \beta_{7} ) q^{96} + ( -15640954 - 686092 \beta_{1} - 78156 \beta_{2} + 15180 \beta_{3} + 123468 \beta_{4} - 138648 \beta_{5} + 15104 \beta_{6} - 15104 \beta_{7} ) q^{97} + ( 85881265 - 3052399 \beta_{1} + 328224 \beta_{2} - 754432 \beta_{3} + 51712 \beta_{4} - 69824 \beta_{5} - 5600 \beta_{6} - 20256 \beta_{7} ) q^{98} + ( -170586 - 586116 \beta_{1} + 74358 \beta_{2} + 104976 \beta_{3} + 48114 \beta_{4} + 48114 \beta_{5} + 17496 \beta_{6} + 26244 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{2} - 52q^{4} - 336q^{5} + 1134q^{6} - 12960q^{8} - 17496q^{9} + O(q^{10}) \) \( 8q + 6q^{2} - 52q^{4} - 336q^{5} + 1134q^{6} - 12960q^{8} - 17496q^{9} + 36628q^{10} - 11340q^{12} - 2864q^{13} + 52728q^{14} + 99440q^{16} - 193200q^{17} - 13122q^{18} + 335592q^{20} + 121824q^{21} - 556968q^{22} + 221616q^{24} - 579048q^{25} + 21564q^{26} - 594672q^{28} + 2063472q^{29} + 46980q^{30} - 3602784q^{32} - 920160q^{33} + 1568476q^{34} + 113724q^{36} + 7470352q^{37} + 3659400q^{38} + 1749184q^{40} - 8865456q^{41} - 5288328q^{42} + 2395920q^{44} + 734832q^{45} - 13649856q^{46} + 10916208q^{48} - 18923896q^{49} + 14581842q^{50} + 18592888q^{52} + 8706672q^{53} - 2480058q^{54} - 45565632q^{56} - 2325024q^{57} - 8816444q^{58} + 28348056q^{60} + 13457296q^{61} + 80783976q^{62} + 1268864q^{64} + 7293408q^{65} - 51205608q^{66} - 117288264q^{68} - 8636544q^{69} - 60373104q^{70} + 28343520q^{72} + 94738960q^{73} + 119548428q^{74} + 144621360q^{76} - 56971392q^{77} - 140630580q^{78} - 163857888q^{80} + 38263752q^{81} - 188383460q^{82} + 199712304q^{84} - 201200416q^{85} + 240327384q^{86} + 156323520q^{88} + 188992272q^{89} - 80105436q^{90} - 387657984q^{92} - 54802656q^{93} - 38749872q^{94} + 246092256q^{96} - 123291632q^{97} + 691081830q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 40 x^{6} - 395 x^{5} + 403 x^{4} + 8998 x^{3} + 74584 x^{2} + 217224 x + 269328\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-13071742 \nu^{7} - 40510221 \nu^{6} + 365720344 \nu^{5} + 4477637544 \nu^{4} + 21313950753 \nu^{3} + 10344779632 \nu^{2} - 18901658816 \nu - 864197730939\)\()/ 48999153875 \)
\(\beta_{2}\)\(=\)\((\)\(-442676109 \nu^{7} + 14876179533 \nu^{6} - 20899272462 \nu^{5} - 244299281437 \nu^{4} - 3890047326769 \nu^{3} - 319558226336 \nu^{2} + 19966153994968 \nu + 257458199902472\)\()/ 783986462000 \)
\(\beta_{3}\)\(=\)\((\)\(16258001 \nu^{7} - 126478257 \nu^{6} - 1086101402 \nu^{5} - 4794860127 \nu^{4} + 44454963621 \nu^{3} + 311217070624 \nu^{2} + 1515996537928 \nu + 2529327015672\)\()/ 22399613200 \)
\(\beta_{4}\)\(=\)\((\)\(909414791 \nu^{7} + 1589452833 \nu^{6} + 7394147338 \nu^{5} - 139320190137 \nu^{4} - 1333799641269 \nu^{3} - 15999239006336 \nu^{2} - 53245255804232 \nu - 157888370611128\)\()/ 391993231000 \)
\(\beta_{5}\)\(=\)\((\)\(1924762227 \nu^{7} + 1444346301 \nu^{6} + 32579772786 \nu^{5} - 555685145789 \nu^{4} - 1808597634593 \nu^{3} - 28806873474592 \nu^{2} - 97146730558504 \nu - 343266156881416\)\()/ 783986462000 \)
\(\beta_{6}\)\(=\)\((\)\(-2088295459 \nu^{7} + 1743279843 \nu^{6} + 40266101998 \nu^{5} + 817766792173 \nu^{4} + 814888424641 \nu^{3} - 2033961609696 \nu^{2} - 83293581219672 \nu - 153540313829368\)\()/ 156797292400 \)
\(\beta_{7}\)\(=\)\((\)\(-844419621 \nu^{7} + 4157547477 \nu^{6} + 19453004322 \nu^{5} + 248245173547 \nu^{4} - 610375035161 \nu^{3} - 3612345119584 \nu^{2} - 37207566428008 \nu - 74886881534632\)\()/ 55999033000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} - 5 \beta_{6} - 12 \beta_{5} + 8 \beta_{4} - 7 \beta_{2} + 54 \beta_{1} + 335\)\()/864\)
\(\nu^{2}\)\(=\)\((\)\(12 \beta_{7} - 11 \beta_{6} - 12 \beta_{5} - 22 \beta_{4} - 16 \beta_{3} - 55 \beta_{2} - 358 \beta_{1} + 4715\)\()/432\)
\(\nu^{3}\)\(=\)\((\)\(207 \beta_{7} - 175 \beta_{6} - 18 \beta_{5} + 64 \beta_{4} + 914 \beta_{3} - 413 \beta_{2} + 788 \beta_{1} + 169669\)\()/864\)
\(\nu^{4}\)\(=\)\((\)\(1395 \beta_{7} - 1505 \beta_{6} - 3942 \beta_{5} + 1940 \beta_{4} + 374 \beta_{3} - 4999 \beta_{2} - 11524 \beta_{1} + 843359\)\()/864\)
\(\nu^{5}\)\(=\)\((\)\(6099 \beta_{7} - 6978 \beta_{6} - 7368 \beta_{5} - 2886 \beta_{4} - 5956 \beta_{3} - 22716 \beta_{2} - 57400 \beta_{1} + 4054608\)\()/432\)
\(\nu^{6}\)\(=\)\((\)\(114921 \beta_{7} - 109501 \beta_{6} - 117126 \beta_{5} + 41656 \beta_{4} + 242822 \beta_{3} - 286079 \beta_{2} - 478564 \beta_{1} + 70038967\)\()/864\)
\(\nu^{7}\)\(=\)\((\)\(815151 \beta_{7} - 862161 \beta_{6} - 1430598 \beta_{5} + 431916 \beta_{4} + 507302 \beta_{3} - 2847231 \beta_{2} - 8274772 \beta_{1} + 525219687\)\()/864\)

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
−1.97054 1.25304i
−1.97054 + 1.25304i
8.23534 + 0.0522875i
8.23534 0.0522875i
−1.11858 4.64627i
−1.11858 + 4.64627i
−3.64622 4.31154i
−3.64622 + 4.31154i
−15.8645 2.07809i 46.7654i 247.363 + 65.9356i −374.901 97.1827 741.908i 4472.52i −3787.26 1560.08i −2187.00 5947.61 + 779.079i
7.2 −15.8645 + 2.07809i 46.7654i 247.363 65.9356i −374.901 97.1827 + 741.908i 4472.52i −3787.26 + 1560.08i −2187.00 5947.61 779.079i
7.3 −2.26258 15.8392i 46.7654i −245.761 + 71.6749i −538.046 740.727 105.810i 3350.97i 1691.33 + 3730.50i −2187.00 1217.37 + 8522.23i
7.4 −2.26258 + 15.8392i 46.7654i −245.761 71.6749i −538.046 740.727 + 105.810i 3350.97i 1691.33 3730.50i −2187.00 1217.37 8522.23i
7.5 7.47945 14.1442i 46.7654i −144.116 211.581i −159.249 −661.458 349.779i 707.133i −4070.55 + 455.883i −2187.00 −1191.10 + 2252.45i
7.6 7.47945 + 14.1442i 46.7654i −144.116 + 211.581i −159.249 −661.458 + 349.779i 707.133i −4070.55 455.883i −2187.00 −1191.10 2252.45i
7.7 13.6476 8.35123i 46.7654i 116.514 227.948i 904.196 390.548 + 638.235i 888.085i −313.513 4083.98i −2187.00 12340.1 7551.15i
7.8 13.6476 + 8.35123i 46.7654i 116.514 + 227.948i 904.196 390.548 638.235i 888.085i −313.513 + 4083.98i −2187.00 12340.1 + 7551.15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
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 \(S_{9}^{\mathrm{new}}(12, [\chi])\).