Properties

Label 12.8.b.b
Level 12
Weight 8
Character orbit 12.b
Analytic conductor 3.749
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(3.74862030581\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{5} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -41 + \beta_{2} - \beta_{4} ) q^{4} \) \( + ( 5 \beta_{1} + \beta_{7} ) q^{5} \) \( + ( -189 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{6} \) \( + ( 4 \beta_{1} + 8 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{7} \) \( + ( -40 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{8} \) \( + ( 261 + 51 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -41 + \beta_{2} - \beta_{4} ) q^{4} \) \( + ( 5 \beta_{1} + \beta_{7} ) q^{5} \) \( + ( -189 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{6} \) \( + ( 4 \beta_{1} + 8 \beta_{2} + \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{7} \) \( + ( -40 \beta_{1} - 4 \beta_{2} + 4 \beta_{5} + 4 \beta_{7} ) q^{8} \) \( + ( 261 + 51 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{9} \) \( + ( -610 + 2 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} ) q^{10} \) \( + ( 127 \beta_{1} + 31 \beta_{2} - 46 \beta_{5} + 5 \beta_{6} ) q^{11} \) \( + ( -1521 - 208 \beta_{1} - 59 \beta_{2} - 9 \beta_{4} + 12 \beta_{5} - 24 \beta_{6} - 12 \beta_{7} ) q^{12} \) \( + ( 6806 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - 2 \beta_{6} ) q^{13} \) \( + ( -44 \beta_{1} - 108 \beta_{2} + 42 \beta_{5} + 22 \beta_{6} - 16 \beta_{7} ) q^{14} \) \( + ( -20 \beta_{1} - 14 \beta_{2} + 21 \beta_{3} + 84 \beta_{4} - 30 \beta_{5} - 10 \beta_{6} ) q^{15} \) \( + ( -1340 - 32 \beta_{1} - 132 \beta_{2} - 32 \beta_{3} + 36 \beta_{4} - 32 \beta_{6} ) q^{16} \) \( + ( -382 \beta_{1} - 28 \beta_{2} - 14 \beta_{5} + 14 \beta_{6} - 12 \beta_{7} ) q^{17} \) \( + ( -6282 + 93 \beta_{1} - 198 \beta_{2} + 24 \beta_{3} - 42 \beta_{4} - 48 \beta_{5} + 96 \beta_{6} + 48 \beta_{7} ) q^{18} \) \( + ( 49 \beta_{1} + 267 \beta_{2} - 30 \beta_{3} - 120 \beta_{4} + 49 \beta_{6} ) q^{19} \) \( + ( -160 \beta_{1} + 600 \beta_{2} + 120 \beta_{5} - 240 \beta_{6} + 8 \beta_{7} ) q^{20} \) \( + ( -16722 + 1737 \beta_{1} + 168 \beta_{2} + 36 \beta_{3} - 144 \beta_{4} + 66 \beta_{5} - 102 \beta_{6} + 51 \beta_{7} ) q^{21} \) \( + ( 17538 + 328 \beta_{1} + 982 \beta_{2} + 2 \beta_{4} + 328 \beta_{6} ) q^{22} \) \( + ( 298 \beta_{1} - 514 \beta_{2} + 208 \beta_{5} + 102 \beta_{6} ) q^{23} \) \( + ( 4644 - 1672 \beta_{1} - 136 \beta_{2} + 96 \beta_{3} + 132 \beta_{4} - 348 \beta_{5} - 32 \beta_{6} + 36 \beta_{7} ) q^{24} \) \( + ( -24435 + 130 \beta_{1} - 130 \beta_{2} - 130 \beta_{3} + 520 \beta_{4} + 130 \beta_{6} ) q^{25} \) \( + ( 6694 \beta_{1} - 160 \beta_{2} - 32 \beta_{5} + 64 \beta_{6} + 32 \beta_{7} ) q^{26} \) \( + ( -3726 \beta_{1} - 144 \beta_{2} - 54 \beta_{3} - 216 \beta_{4} + 702 \beta_{5} - 495 \beta_{6} ) q^{27} \) \( + ( 25266 - 512 \beta_{1} - 1122 \beta_{2} + 128 \beta_{3} - 414 \beta_{4} - 512 \beta_{6} ) q^{28} \) \( + ( -12817 \beta_{1} - 1128 \beta_{2} - 564 \beta_{5} + 564 \beta_{6} + 31 \beta_{7} ) q^{29} \) \( + ( -5490 - 920 \beta_{1} - 550 \beta_{2} + 270 \beta_{4} - 600 \beta_{5} + 960 \beta_{6} - 336 \beta_{7} ) q^{30} \) \( + ( -530 \beta_{1} - 1346 \beta_{2} - 61 \beta_{3} - 244 \beta_{4} - 530 \beta_{6} ) q^{31} \) \( + ( 288 \beta_{1} + 2960 \beta_{2} - 272 \beta_{5} - 896 \beta_{6} - 144 \beta_{7} ) q^{32} \) \( + ( 22896 + 16347 \beta_{1} + 1383 \beta_{2} - 183 \beta_{3} + 732 \beta_{4} + 783 \beta_{5} - 600 \beta_{6} - 369 \beta_{7} ) q^{33} \) \( + ( 45176 - 248 \beta_{2} + 96 \beta_{3} + 248 \beta_{4} ) q^{34} \) \( + ( 10840 \beta_{1} - 1990 \beta_{2} - 1430 \beta_{5} + 1140 \beta_{6} ) q^{35} \) \( + ( 129483 - 7968 \beta_{1} - 2331 \beta_{2} - 384 \beta_{3} - 813 \beta_{4} - 168 \beta_{5} + 336 \beta_{6} + 168 \beta_{7} ) q^{36} \) \( + ( -2722 - 526 \beta_{1} + 526 \beta_{2} + 526 \beta_{3} - 2104 \beta_{4} - 526 \beta_{6} ) q^{37} \) \( + ( 1996 \beta_{1} + 1212 \beta_{2} + 1782 \beta_{5} - 998 \beta_{6} + 480 \beta_{7} ) q^{38} \) \( + ( -8846 \beta_{1} + 6584 \beta_{2} - 18 \beta_{3} - 72 \beta_{4} + 234 \beta_{5} - 408 \beta_{6} ) q^{39} \) \( + ( -378520 + 960 \beta_{1} + 536 \beta_{2} - 64 \beta_{3} + 2344 \beta_{4} + 960 \beta_{6} ) q^{40} \) \( + ( -17110 \beta_{1} - 1600 \beta_{2} - 800 \beta_{5} + 800 \beta_{6} + 258 \beta_{7} ) q^{41} \) \( + ( -209574 - 18738 \beta_{1} - 1722 \beta_{2} - 408 \beta_{3} - 1158 \beta_{4} - 576 \beta_{5} + 1152 \beta_{6} + 576 \beta_{7} ) q^{42} \) \( + ( 2137 \beta_{1} + 2675 \beta_{2} + 934 \beta_{3} + 3736 \beta_{4} + 2137 \beta_{6} ) q^{43} \) \( + ( 18848 \beta_{1} - 3928 \beta_{2} + 5896 \beta_{5} - 656 \beta_{6} - 8 \beta_{7} ) q^{44} \) \( + ( 249120 + 36825 \beta_{1} + 2610 \beta_{2} + 30 \beta_{3} - 120 \beta_{4} + 1290 \beta_{5} - 1320 \beta_{6} + 1437 \beta_{7} ) q^{45} \) \( + ( 62004 - 2480 \beta_{1} - 5188 \beta_{2} - 2252 \beta_{4} - 2480 \beta_{6} ) q^{46} \) \( + ( 7848 \beta_{1} + 3508 \beta_{2} - 3700 \beta_{5} + 64 \beta_{6} ) q^{47} \) \( + ( 229860 + 1280 \beta_{1} - 1172 \beta_{2} - 288 \beta_{3} + 2628 \beta_{4} - 3216 \beta_{5} + 5472 \beta_{6} - 528 \beta_{7} ) q^{48} \) \( + ( 119755 - 154 \beta_{1} + 154 \beta_{2} + 154 \beta_{3} - 616 \beta_{4} - 154 \beta_{6} ) q^{49} \) \( + ( -17155 \beta_{1} + 10400 \beta_{2} + 2080 \beta_{5} - 4160 \beta_{6} - 2080 \beta_{7} ) q^{50} \) \( + ( 2410 \beta_{1} + 700 \beta_{2} - 420 \beta_{3} - 1680 \beta_{4} - 858 \beta_{5} - 286 \beta_{6} ) q^{51} \) \( + ( -185590 - 256 \beta_{1} + 6406 \beta_{2} - 256 \beta_{3} - 7174 \beta_{4} - 256 \beta_{6} ) q^{52} \) \( + ( -30511 \beta_{1} - 1664 \beta_{2} - 832 \beta_{5} + 832 \beta_{6} - 2275 \beta_{7} ) q^{53} \) \( + ( -526419 - 702 \beta_{1} - 8091 \beta_{2} + 4077 \beta_{4} - 1809 \beta_{5} - 4419 \beta_{6} + 864 \beta_{7} ) q^{54} \) \( + ( -5110 \beta_{1} - 10970 \beta_{2} - 1090 \beta_{3} - 4360 \beta_{4} - 5110 \beta_{6} ) q^{55} \) \( + ( 16464 \beta_{1} - 5240 \beta_{2} - 9352 \beta_{5} + 4864 \beta_{6} + 1656 \beta_{7} ) q^{56} \) \( + ( -474822 - 26253 \beta_{1} - 477 \beta_{2} + 441 \beta_{3} - 1764 \beta_{4} - 459 \beta_{5} + 18 \beta_{6} - 3051 \beta_{7} ) q^{57} \) \( + ( 1506146 - 8962 \beta_{2} - 248 \beta_{3} + 8962 \beta_{4} ) q^{58} \) \( + ( 5085 \beta_{1} - 18745 \beta_{2} + 8920 \beta_{5} + 3275 \beta_{6} ) q^{59} \) \( + ( 1281960 - 4640 \beta_{1} + 688 \beta_{2} + 2688 \beta_{3} - 4248 \beta_{4} + 4920 \beta_{5} - 880 \beta_{6} - 1080 \beta_{7} ) q^{60} \) \( + ( -378490 + 2866 \beta_{1} - 2866 \beta_{2} - 2866 \beta_{3} + 11464 \beta_{4} + 2866 \beta_{6} ) q^{61} \) \( + ( 1540 \beta_{1} + 10020 \beta_{2} - 7710 \beta_{5} - 770 \beta_{6} + 976 \beta_{7} ) q^{62} \) \( + ( -31248 \beta_{1} - 23940 \beta_{2} + 1539 \beta_{3} + 6156 \beta_{4} + 8424 \beta_{5} - 5940 \beta_{6} ) q^{63} \) \( + ( -1134224 + 9344 \beta_{1} + 17808 \beta_{2} + 1152 \beta_{3} + 10224 \beta_{4} + 9344 \beta_{6} ) q^{64} \) \( + ( 57730 \beta_{1} + 1720 \beta_{2} + 860 \beta_{5} - 860 \beta_{6} + 7590 \beta_{7} ) q^{65} \) \( + ( -1892142 + 33144 \beta_{1} + 26430 \beta_{2} + 2952 \beta_{3} - 11790 \beta_{4} + 2928 \beta_{5} - 5856 \beta_{6} - 2928 \beta_{7} ) q^{66} \) \( + ( 7379 \beta_{1} + 38041 \beta_{2} - 3976 \beta_{3} - 15904 \beta_{4} + 7379 \beta_{6} ) q^{67} \) \( + ( 39552 \beta_{1} - 6304 \beta_{2} - 2336 \beta_{5} + 2880 \beta_{6} - 992 \beta_{7} ) q^{68} \) \( + ( -1018656 - 92982 \beta_{1} - 9618 \beta_{2} - 294 \beta_{3} + 1176 \beta_{4} - 4662 \beta_{5} + 4956 \beta_{6} + 2790 \beta_{7} ) q^{69} \) \( + ( 1661700 + 2320 \beta_{1} + 24620 \beta_{2} - 17660 \beta_{4} + 2320 \beta_{6} ) q^{70} \) \( + ( -140798 \beta_{1} + 43026 \beta_{2} + 9324 \beta_{5} - 17450 \beta_{6} ) q^{71} \) \( + ( 1323144 + 145848 \beta_{1} + 16452 \beta_{2} - 1344 \beta_{3} + 840 \beta_{4} + 1716 \beta_{5} - 15168 \beta_{6} + 3252 \beta_{7} ) q^{72} \) \( + ( 460346 - 1312 \beta_{1} + 1312 \beta_{2} + 1312 \beta_{3} - 5248 \beta_{4} - 1312 \beta_{6} ) q^{73} \) \( + ( -32178 \beta_{1} - 42080 \beta_{2} - 8416 \beta_{5} + 16832 \beta_{6} + 8416 \beta_{7} ) q^{74} \) \( + ( 157035 \beta_{1} - 10005 \beta_{2} + 1170 \beta_{3} + 4680 \beta_{4} - 15210 \beta_{5} + 26520 \beta_{6} ) q^{75} \) \( + ( -2300274 - 6272 \beta_{1} - 22110 \beta_{2} - 3840 \beta_{3} + 3294 \beta_{4} - 6272 \beta_{6} ) q^{76} \) \( + ( 282426 \beta_{1} + 29344 \beta_{2} + 14672 \beta_{5} - 14672 \beta_{6} - 11006 \beta_{7} ) q^{77} \) \( + ( -1566702 + 13516 \beta_{1} - 7142 \beta_{2} + 23442 \beta_{4} + 17106 \beta_{5} + 6374 \beta_{6} + 288 \beta_{7} ) q^{78} \) \( + ( -8706 \beta_{1} - 51250 \beta_{2} + 6283 \beta_{3} + 25132 \beta_{4} - 8706 \beta_{6} ) q^{79} \) \( + ( -373440 \beta_{1} + 2720 \beta_{2} + 8800 \beta_{5} - 3840 \beta_{6} - 9376 \beta_{7} ) q^{80} \) \( + ( 2082105 - 304452 \beta_{1} - 23328 \beta_{2} + 3132 \beta_{3} - 12528 \beta_{4} - 13230 \beta_{5} + 10098 \beta_{6} + 594 \beta_{7} ) q^{81} \) \( + ( 2005820 - 12284 \beta_{2} - 2064 \beta_{3} + 12284 \beta_{4} ) q^{82} \) \( + ( 115741 \beta_{1} - 14039 \beta_{2} - 19150 \beta_{5} + 11063 \beta_{6} ) q^{83} \) \( + ( 2367810 - 190176 \beta_{1} + 2454 \beta_{2} - 4608 \beta_{3} + 10098 \beta_{4} + 10344 \beta_{5} - 16848 \beta_{6} + 4632 \beta_{7} ) q^{84} \) \( + ( 1504000 - 3240 \beta_{1} + 3240 \beta_{2} + 3240 \beta_{3} - 12960 \beta_{4} - 3240 \beta_{6} ) q^{85} \) \( + ( -47492 \beta_{1} - 81684 \beta_{2} + 10446 \beta_{5} + 23746 \beta_{6} - 14944 \beta_{7} ) q^{86} \) \( + ( 86800 \beta_{1} + 20998 \beta_{2} - 6117 \beta_{3} - 24468 \beta_{4} - 49998 \beta_{5} - 16666 \beta_{6} ) q^{87} \) \( + ( -3706344 - 41920 \beta_{1} - 90520 \beta_{2} + 64 \beta_{3} - 35240 \beta_{4} - 41920 \beta_{6} ) q^{88} \) \( + ( 266108 \beta_{1} + 24212 \beta_{2} + 12106 \beta_{5} - 12106 \beta_{6} - 2466 \beta_{7} ) q^{89} \) \( + ( -4364730 + 247440 \beta_{1} + 21114 \beta_{2} - 11496 \beta_{3} - 23514 \beta_{4} - 480 \beta_{5} + 960 \beta_{6} + 480 \beta_{7} ) q^{90} \) \( + ( 23508 \beta_{1} + 39892 \beta_{2} + 7658 \beta_{3} + 30632 \beta_{4} + 23508 \beta_{6} ) q^{91} \) \( + ( 54336 \beta_{1} + 20752 \beta_{2} - 35632 \beta_{5} + 4960 \beta_{6} + 9008 \beta_{7} ) q^{92} \) \( + ( 2672550 - 149715 \beta_{1} - 17970 \beta_{2} - 4770 \beta_{3} + 19080 \beta_{4} - 6600 \beta_{5} + 11370 \beta_{6} - 537 \beta_{7} ) q^{93} \) \( + ( 1027176 + 29088 \beta_{1} + 81016 \beta_{2} + 6248 \beta_{4} + 29088 \beta_{6} ) q^{94} \) \( + ( -347170 \beta_{1} + 125610 \beta_{2} + 12480 \beta_{5} - 46030 \beta_{6} ) q^{95} \) \( + ( 5659632 + 254624 \beta_{1} + 28448 \beta_{2} + 4224 \beta_{3} - 20880 \beta_{4} + 37488 \beta_{5} - 12032 \beta_{6} - 10512 \beta_{7} ) q^{96} \) \( + ( -1985902 - 5866 \beta_{1} + 5866 \beta_{2} + 5866 \beta_{3} - 23464 \beta_{4} - 5866 \beta_{6} ) q^{97} \) \( + ( 111131 \beta_{1} - 12320 \beta_{2} - 2464 \beta_{5} + 4928 \beta_{6} + 2464 \beta_{7} ) q^{98} \) \( + ( 49779 \beta_{1} + 18621 \beta_{2} + 3294 \beta_{3} + 13176 \beta_{4} + 57780 \beta_{5} + 63729 \beta_{6} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 328q^{4} \) \(\mathstrut -\mathstrut 1512q^{6} \) \(\mathstrut +\mathstrut 2088q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 328q^{4} \) \(\mathstrut -\mathstrut 1512q^{6} \) \(\mathstrut +\mathstrut 2088q^{9} \) \(\mathstrut -\mathstrut 4880q^{10} \) \(\mathstrut -\mathstrut 12168q^{12} \) \(\mathstrut +\mathstrut 54448q^{13} \) \(\mathstrut -\mathstrut 10720q^{16} \) \(\mathstrut -\mathstrut 50256q^{18} \) \(\mathstrut -\mathstrut 133776q^{21} \) \(\mathstrut +\mathstrut 140304q^{22} \) \(\mathstrut +\mathstrut 37152q^{24} \) \(\mathstrut -\mathstrut 195480q^{25} \) \(\mathstrut +\mathstrut 202128q^{28} \) \(\mathstrut -\mathstrut 43920q^{30} \) \(\mathstrut +\mathstrut 183168q^{33} \) \(\mathstrut +\mathstrut 361408q^{34} \) \(\mathstrut +\mathstrut 1035864q^{36} \) \(\mathstrut -\mathstrut 21776q^{37} \) \(\mathstrut -\mathstrut 3028160q^{40} \) \(\mathstrut -\mathstrut 1676592q^{42} \) \(\mathstrut +\mathstrut 1992960q^{45} \) \(\mathstrut +\mathstrut 496032q^{46} \) \(\mathstrut +\mathstrut 1838880q^{48} \) \(\mathstrut +\mathstrut 958040q^{49} \) \(\mathstrut -\mathstrut 1484720q^{52} \) \(\mathstrut -\mathstrut 4211352q^{54} \) \(\mathstrut -\mathstrut 3798576q^{57} \) \(\mathstrut +\mathstrut 12049168q^{58} \) \(\mathstrut +\mathstrut 10255680q^{60} \) \(\mathstrut -\mathstrut 3027920q^{61} \) \(\mathstrut -\mathstrut 9073792q^{64} \) \(\mathstrut -\mathstrut 15137136q^{66} \) \(\mathstrut -\mathstrut 8149248q^{69} \) \(\mathstrut +\mathstrut 13293600q^{70} \) \(\mathstrut +\mathstrut 10585152q^{72} \) \(\mathstrut +\mathstrut 3682768q^{73} \) \(\mathstrut -\mathstrut 18402192q^{76} \) \(\mathstrut -\mathstrut 12533616q^{78} \) \(\mathstrut +\mathstrut 16656840q^{81} \) \(\mathstrut +\mathstrut 16046560q^{82} \) \(\mathstrut +\mathstrut 18942480q^{84} \) \(\mathstrut +\mathstrut 12032000q^{85} \) \(\mathstrut -\mathstrut 29650752q^{88} \) \(\mathstrut -\mathstrut 34917840q^{90} \) \(\mathstrut +\mathstrut 21380400q^{93} \) \(\mathstrut +\mathstrut 8217408q^{94} \) \(\mathstrut +\mathstrut 45277056q^{96} \) \(\mathstrut -\mathstrut 15887216q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut +\mathstrut \) \(41\) \(x^{6}\mathstrut +\mathstrut \) \(1008\) \(x^{4}\mathstrut +\mathstrut \) \(41984\) \(x^{2}\mathstrut +\mathstrut \) \(1048576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 41 \nu^{5} + 1008 \nu^{3} + 41984 \nu \)\()/16384\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 32 \nu^{6} - 87 \nu^{5} - 288 \nu^{4} - 144 \nu^{3} - 23040 \nu^{2} + 97280 \nu - 827392 \)\()/24576\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 9 \nu^{4} + 1328 \nu^{2} - 4864 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 64 \nu^{6} - 87 \nu^{5} + 3648 \nu^{4} - 144 \nu^{3} + 73728 \nu^{2} + 97280 \nu + 2195456 \)\()/24576\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{7} - 32 \nu^{6} - 57 \nu^{5} - 288 \nu^{4} - 11376 \nu^{3} - 23040 \nu^{2} - 203776 \nu - 827392 \)\()/24576\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 64 \nu^{6} + 133 \nu^{5} - 576 \nu^{4} - 720 \nu^{3} - 46080 \nu^{2} - 236544 \nu - 1654784 \)\()/16384\)
\(\beta_{7}\)\(=\)\((\)\( -19 \nu^{7} - 267 \nu^{5} + 18224 \nu^{3} - 68608 \nu \)\()/16384\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(164\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(29\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(21\) \(\beta_{1}\)\()/32\)
\(\nu^{4}\)\(=\)\((\)\(73\) \(\beta_{6}\mathstrut +\mathstrut \) \(128\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(91\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut -\mathstrut \) \(1340\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(144\) \(\beta_{7}\mathstrut +\mathstrut \) \(475\) \(\beta_{6}\mathstrut +\mathstrut \) \(693\) \(\beta_{5}\mathstrut -\mathstrut \) \(2118\) \(\beta_{2}\mathstrut +\mathstrut \) \(7955\) \(\beta_{1}\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(1985\) \(\beta_{6}\mathstrut -\mathstrut \) \(1152\) \(\beta_{4}\mathstrut -\mathstrut \) \(639\) \(\beta_{3}\mathstrut -\mathstrut \) \(4803\) \(\beta_{2}\mathstrut -\mathstrut \) \(1985\) \(\beta_{1}\mathstrut -\mathstrut \) \(283556\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(10224\) \(\beta_{7}\mathstrut +\mathstrut \) \(9405\) \(\beta_{6}\mathstrut -\mathstrut \) \(41165\) \(\beta_{5}\mathstrut +\mathstrut \) \(12950\) \(\beta_{2}\mathstrut -\mathstrut \) \(116923\) \(\beta_{1}\)\()/32\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
4.51977 3.40172i
4.51977 + 3.40172i
1.14965 5.53880i
1.14965 + 5.53880i
−1.14965 5.53880i
−1.14965 + 5.53880i
−4.51977 3.40172i
−4.51977 + 3.40172i
−9.03954 6.80344i 46.2722 + 6.77368i 35.4264 + 123.000i 426.666i −372.195 376.041i 780.788i 516.584 1352.88i 2095.23 + 626.867i −2902.79 + 3856.86i
11.2 −9.03954 + 6.80344i 46.2722 6.77368i 35.4264 123.000i 426.666i −372.195 + 376.041i 780.788i 516.584 + 1352.88i 2095.23 626.867i −2902.79 3856.86i
11.3 −2.29930 11.0776i −17.5181 + 43.3603i −117.426 + 50.9415i 151.910i 520.608 + 94.3598i 893.278i 834.308 + 1183.67i −1573.23 1519.18i 1682.79 349.286i
11.4 −2.29930 + 11.0776i −17.5181 43.3603i −117.426 50.9415i 151.910i 520.608 94.3598i 893.278i 834.308 1183.67i −1573.23 + 1519.18i 1682.79 + 349.286i
11.5 2.29930 11.0776i 17.5181 43.3603i −117.426 50.9415i 151.910i −440.049 293.757i 893.278i −834.308 + 1183.67i −1573.23 1519.18i 1682.79 + 349.286i
11.6 2.29930 + 11.0776i 17.5181 + 43.3603i −117.426 + 50.9415i 151.910i −440.049 + 293.757i 893.278i −834.308 1183.67i −1573.23 + 1519.18i 1682.79 349.286i
11.7 9.03954 6.80344i −46.2722 6.77368i 35.4264 123.000i 426.666i −464.364 + 253.579i 780.788i −516.584 1352.88i 2095.23 + 626.867i −2902.79 3856.86i
11.8 9.03954 + 6.80344i −46.2722 + 6.77368i 35.4264 + 123.000i 426.666i −464.364 253.579i 780.788i −516.584 + 1352.88i 2095.23 626.867i −2902.79 + 3856.86i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{4} \) \(\mathstrut +\mathstrut 205120 T_{5}^{2} \) \(\mathstrut +\mathstrut 4200928000 \) acting on \(S_{8}^{\mathrm{new}}(12, [\chi])\).