Properties

Label 15.8.b.a
Level 15
Weight 8
Character orbit 15.b
Analytic conductor 4.686
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 15.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.68577538226\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{12}\cdot 5^{3} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( -83 - \beta_{4} ) q^{4} + ( -55 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{5} + ( 61 + \beta_{4} - \beta_{7} ) q^{6} + ( -5 \beta_{1} - 11 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( 99 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{8} -729 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + \beta_{2} q^{3} + ( -83 - \beta_{4} ) q^{4} + ( -55 - 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{5} + ( 61 + \beta_{4} - \beta_{7} ) q^{6} + ( -5 \beta_{1} - 11 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{7} + ( 99 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{8} -729 q^{9} + ( -837 + 80 \beta_{1} + 23 \beta_{2} - \beta_{3} + 5 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{10} + ( 1352 - \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{11} + ( -81 \beta_{1} - 75 \beta_{2} + 9 \beta_{3} - 9 \beta_{5} ) q^{12} + ( -505 \beta_{1} + 125 \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + 7 \beta_{6} ) q^{13} + ( -1694 - 2 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} - 18 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 36 \beta_{7} ) q^{14} + ( -2176 - 90 \beta_{1} - 45 \beta_{2} - 10 \beta_{4} + 9 \beta_{6} - 8 \beta_{7} ) q^{15} + ( 10783 + 6 \beta_{1} + 18 \beta_{2} + 42 \beta_{3} + 61 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} - 24 \beta_{7} ) q^{16} + ( 125 \beta_{1} - 45 \beta_{2} - 9 \beta_{3} + 30 \beta_{5} + 21 \beta_{6} ) q^{17} + 729 \beta_{1} q^{18} + ( -3888 + 12 \beta_{1} + 36 \beta_{2} + 84 \beta_{3} - 112 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} + 96 \beta_{7} ) q^{19} + ( 10926 + 85 \beta_{1} - 921 \beta_{2} + 27 \beta_{3} + 210 \beta_{4} - 55 \beta_{5} - 4 \beta_{6} - 72 \beta_{7} ) q^{20} + ( 8028 - 9 \beta_{1} - 27 \beta_{2} - 63 \beta_{3} + 144 \beta_{4} - 18 \beta_{5} + 9 \beta_{6} ) q^{21} + ( 620 \beta_{1} + 320 \beta_{2} - 88 \beta_{3} + 50 \beta_{5} - 38 \beta_{6} ) q^{22} + ( 92 \beta_{1} + 1032 \beta_{2} + 84 \beta_{3} - 72 \beta_{5} + 12 \beta_{6} ) q^{23} + ( -13687 - 18 \beta_{1} - 54 \beta_{2} - 126 \beta_{3} - 181 \beta_{4} - 36 \beta_{5} + 18 \beta_{6} + 64 \beta_{7} ) q^{24} + ( 15973 - 2695 \beta_{1} - 1081 \beta_{2} + 47 \beta_{3} - 120 \beta_{4} + 50 \beta_{5} - 17 \beta_{6} - 96 \beta_{7} ) q^{25} + ( -98926 - 10 \beta_{1} - 30 \beta_{2} - 70 \beta_{3} - 570 \beta_{4} - 20 \beta_{5} + 10 \beta_{6} - 144 \beta_{7} ) q^{26} -729 \beta_{2} q^{27} + ( -330 \beta_{1} + 6162 \beta_{2} - 246 \beta_{3} + 126 \beta_{5} - 120 \beta_{6} ) q^{28} + ( 29990 - 13 \beta_{1} - 39 \beta_{2} - 91 \beta_{3} + 336 \beta_{4} - 26 \beta_{5} + 13 \beta_{6} ) q^{29} + ( -21366 + 4815 \beta_{1} - 1309 \beta_{2} - 117 \beta_{3} - 360 \beta_{4} - 45 \beta_{5} - 36 \beta_{6} + 27 \beta_{7} ) q^{30} + ( 28884 + 30 \beta_{1} + 90 \beta_{2} + 210 \beta_{3} + 536 \beta_{4} + 60 \beta_{5} - 30 \beta_{6} + 96 \beta_{7} ) q^{31} + ( -5383 \beta_{1} - 5325 \beta_{2} + 255 \beta_{3} - 353 \beta_{5} - 98 \beta_{6} ) q^{32} + ( -1863 \beta_{1} + 1575 \beta_{2} + 171 \beta_{3} - 90 \beta_{5} + 81 \beta_{6} ) q^{33} + ( 23186 + 18 \beta_{1} + 54 \beta_{2} + 126 \beta_{3} - 82 \beta_{4} + 36 \beta_{5} - 18 \beta_{6} - 324 \beta_{7} ) q^{34} + ( 73916 + 8405 \beta_{1} - 4293 \beta_{2} - 349 \beta_{3} + 660 \beta_{4} + 370 \beta_{5} + 91 \beta_{6} + 288 \beta_{7} ) q^{35} + ( 60507 + 729 \beta_{4} ) q^{36} + ( 16287 \beta_{1} - 7299 \beta_{2} + 333 \beta_{3} - 78 \beta_{5} + 255 \beta_{6} ) q^{37} + ( 5184 \beta_{1} + 17088 \beta_{2} - 288 \beta_{3} + 424 \beta_{5} + 136 \beta_{6} ) q^{38} + ( -57060 + 63 \beta_{1} + 189 \beta_{2} + 441 \beta_{3} + 396 \beta_{4} + 126 \beta_{5} - 63 \beta_{6} - 432 \beta_{7} ) q^{39} + ( -143521 - 21500 \beta_{1} - 12668 \beta_{2} + 196 \beta_{3} - 835 \beta_{4} - 230 \beta_{5} + 74 \beta_{6} + 792 \beta_{7} ) q^{40} + ( 63286 + 10 \beta_{1} + 30 \beta_{2} + 70 \beta_{3} - 1824 \beta_{4} + 20 \beta_{5} - 10 \beta_{6} + 1152 \beta_{7} ) q^{41} + ( -26568 \beta_{1} + 612 \beta_{2} - 36 \beta_{3} + 198 \beta_{5} + 162 \beta_{6} ) q^{42} + ( -15128 \beta_{1} + 11740 \beta_{2} + 952 \beta_{3} - 608 \beta_{5} + 344 \beta_{6} ) q^{43} + ( 322218 + 48 \beta_{1} + 144 \beta_{2} + 336 \beta_{3} + 1470 \beta_{4} + 96 \beta_{5} - 48 \beta_{6} - 1008 \beta_{7} ) q^{44} + ( 40095 + 3645 \beta_{1} - 2187 \beta_{2} + 729 \beta_{3} ) q^{45} + ( 83804 - 168 \beta_{1} - 504 \beta_{2} - 1176 \beta_{3} - 484 \beta_{4} - 336 \beta_{5} + 168 \beta_{6} - 84 \beta_{7} ) q^{46} + ( -27218 \beta_{1} + 750 \beta_{2} - 1302 \beta_{3} + 1180 \beta_{5} - 122 \beta_{6} ) q^{47} + ( 25839 \beta_{1} + 8261 \beta_{2} - 711 \beta_{3} + 225 \beta_{5} - 486 \beta_{6} ) q^{48} + ( -406841 - 294 \beta_{1} - 882 \beta_{2} - 2058 \beta_{3} + 2352 \beta_{4} - 588 \beta_{5} + 294 \beta_{6} - 1344 \beta_{7} ) q^{49} + ( -639628 + 10495 \beta_{1} - 19764 \beta_{2} + 1268 \beta_{3} - 3180 \beta_{4} - 950 \beta_{5} - 338 \beta_{6} + 756 \beta_{7} ) q^{50} + ( 32584 + 189 \beta_{1} + 567 \beta_{2} + 1323 \beta_{3} - 2180 \beta_{4} + 378 \beta_{5} - 189 \beta_{6} + 128 \beta_{7} ) q^{51} + ( 136662 \beta_{1} - 4686 \beta_{2} - 726 \beta_{3} + 54 \beta_{5} - 672 \beta_{6} ) q^{52} + ( 15365 \beta_{1} + 16563 \beta_{2} - 129 \beta_{3} - 906 \beta_{5} - 1035 \beta_{6} ) q^{53} + ( -44469 - 729 \beta_{4} + 729 \beta_{7} ) q^{54} + ( 442212 - 26945 \beta_{1} - 7763 \beta_{2} - 2339 \beta_{3} + 1920 \beta_{4} - 410 \beta_{5} - 133 \beta_{6} - 1344 \beta_{7} ) q^{55} + ( 86306 + 236 \beta_{1} + 708 \beta_{2} + 1652 \beta_{3} + 9510 \beta_{4} + 472 \beta_{5} - 236 \beta_{6} - 3312 \beta_{7} ) q^{56} + ( -60588 \beta_{1} + 572 \beta_{2} - 612 \beta_{3} - 360 \beta_{5} - 972 \beta_{6} ) q^{57} + ( -75202 \beta_{1} - 268 \beta_{2} + 332 \beta_{3} + 158 \beta_{5} + 490 \beta_{6} ) q^{58} + ( 136624 + 217 \beta_{1} + 651 \beta_{2} + 1519 \beta_{3} - 7428 \beta_{4} + 434 \beta_{5} - 217 \beta_{6} + 1152 \beta_{7} ) q^{59} + ( 662637 + 61470 \beta_{1} + 5562 \beta_{2} - 1494 \beta_{3} + 3495 \beta_{4} + 1170 \beta_{5} + 252 \beta_{6} + 456 \beta_{7} ) q^{60} + ( -801366 - 384 \beta_{1} - 1152 \beta_{2} - 2688 \beta_{3} - 6688 \beta_{4} - 768 \beta_{5} + 384 \beta_{6} + 1536 \beta_{7} ) q^{61} + ( -125292 \beta_{1} + 7440 \beta_{2} + 2592 \beta_{3} - 908 \beta_{5} + 1684 \beta_{6} ) q^{62} + ( 3645 \beta_{1} + 8019 \beta_{2} - 729 \beta_{3} + 1458 \beta_{5} + 729 \beta_{6} ) q^{63} + ( -74295 + 258 \beta_{1} + 774 \beta_{2} + 1806 \beta_{3} - 5765 \beta_{4} + 516 \beta_{5} - 258 \beta_{6} + 6744 \beta_{7} ) q^{64} + ( -317528 + 44365 \beta_{1} + 40743 \beta_{2} + 19 \beta_{3} + 720 \beta_{4} + 1130 \beta_{5} + 167 \beta_{6} - 5184 \beta_{7} ) q^{65} + ( -294822 - 342 \beta_{1} - 1026 \beta_{2} - 2394 \beta_{3} - 4410 \beta_{4} - 684 \beta_{5} + 342 \beta_{6} - 324 \beta_{7} ) q^{66} + ( 89306 \beta_{1} - 24670 \beta_{2} + 4334 \beta_{3} - 3148 \beta_{5} + 1186 \beta_{6} ) q^{67} + ( 40406 \beta_{1} - 71694 \beta_{2} + 1482 \beta_{3} + 646 \beta_{5} + 2128 \beta_{6} ) q^{68} + ( -744884 + 108 \beta_{1} + 324 \beta_{2} + 756 \beta_{3} + 5152 \beta_{4} + 216 \beta_{5} - 108 \beta_{6} + 896 \beta_{7} ) q^{69} + ( 1507548 - 202360 \beta_{1} + 53216 \beta_{2} + 4088 \beta_{3} + 7980 \beta_{4} + 3110 \beta_{5} + 1198 \beta_{6} - 636 \beta_{7} ) q^{70} + ( -175704 - 258 \beta_{1} - 774 \beta_{2} - 1806 \beta_{3} + 9000 \beta_{4} - 516 \beta_{5} + 258 \beta_{6} + 4032 \beta_{7} ) q^{71} + ( -72171 \beta_{1} - 6561 \beta_{2} + 2187 \beta_{3} - 729 \beta_{5} + 1458 \beta_{6} ) q^{72} + ( 82578 \beta_{1} - 59922 \beta_{2} - 5130 \beta_{3} + 2940 \beta_{5} - 2190 \beta_{6} ) q^{73} + ( 2994330 - 666 \beta_{1} - 1998 \beta_{2} - 4662 \beta_{3} - 498 \beta_{4} - 1332 \beta_{5} + 666 \beta_{6} + 8568 \beta_{7} ) q^{74} + ( 972612 + 62595 \beta_{1} + 10246 \beta_{2} + 873 \beta_{3} + 720 \beta_{4} - 2250 \beta_{5} - 873 \beta_{6} - 3024 \beta_{7} ) q^{75} + ( 1631288 + 2112 \beta_{1} + 6336 \beta_{2} + 14784 \beta_{3} + 10792 \beta_{4} + 4224 \beta_{5} - 2112 \beta_{6} - 10176 \beta_{7} ) q^{76} + ( 65670 \beta_{1} - 654 \beta_{2} - 3054 \beta_{3} - 444 \beta_{5} - 3498 \beta_{6} ) q^{77} + ( 38232 \beta_{1} - 100728 \beta_{2} + 7056 \beta_{3} - 6246 \beta_{5} + 810 \beta_{6} ) q^{78} + ( -1799316 + 414 \beta_{1} + 1242 \beta_{2} + 2898 \beta_{3} + 5240 \beta_{4} + 828 \beta_{5} - 414 \beta_{6} - 3168 \beta_{7} ) q^{79} + ( -3902126 + 178525 \beta_{1} + 43935 \beta_{2} - 8165 \beta_{3} - 12810 \beta_{4} - 545 \beta_{5} - 206 \beta_{6} + 6192 \beta_{7} ) q^{80} + 531441 q^{81} + ( 54082 \beta_{1} + 237784 \beta_{2} - 11864 \beta_{3} + 10660 \beta_{5} - 1204 \beta_{6} ) q^{82} + ( -58768 \beta_{1} - 117420 \beta_{2} - 1200 \beta_{3} + 720 \beta_{5} - 480 \beta_{6} ) q^{83} + ( -4543722 - 1080 \beta_{1} - 3240 \beta_{2} - 7560 \beta_{3} - 9342 \beta_{4} - 2160 \beta_{5} + 1080 \beta_{6} - 3024 \beta_{7} ) q^{84} + ( -178608 - 88945 \beta_{1} + 102545 \beta_{2} + 5465 \beta_{3} - 8080 \beta_{4} - 6850 \beta_{5} - 2903 \beta_{6} - 3264 \beta_{7} ) q^{85} + ( -2462252 - 1904 \beta_{1} - 5712 \beta_{2} - 13328 \beta_{3} - 24732 \beta_{4} - 3808 \beta_{5} + 1904 \beta_{6} - 3492 \beta_{7} ) q^{86} + ( 15633 \beta_{1} + 28953 \beta_{2} - 2205 \beta_{3} + 3258 \beta_{5} + 1053 \beta_{6} ) q^{87} + ( -341258 \beta_{1} - 177998 \beta_{2} + 1690 \beta_{3} - 4958 \beta_{5} - 3268 \beta_{6} ) q^{88} + ( 5732154 - 312 \beta_{1} - 936 \beta_{2} - 2184 \beta_{3} - 672 \beta_{4} - 624 \beta_{5} + 312 \beta_{6} - 16128 \beta_{7} ) q^{89} + ( 610173 - 58320 \beta_{1} - 16767 \beta_{2} + 729 \beta_{3} - 3645 \beta_{4} - 3645 \beta_{5} + 1458 \beta_{6} + 4374 \beta_{7} ) q^{90} + ( 3010728 - 1578 \beta_{1} - 4734 \beta_{2} - 11046 \beta_{3} - 15384 \beta_{4} - 3156 \beta_{5} + 1578 \beta_{6} + 20928 \beta_{7} ) q^{91} + ( 49076 \beta_{1} + 155772 \beta_{2} + 1068 \beta_{3} - 3020 \beta_{5} - 1952 \beta_{6} ) q^{92} + ( 7938 \beta_{1} + 25874 \beta_{2} - 7578 \beta_{3} + 5148 \beta_{5} - 2430 \beta_{6} ) q^{93} + ( -5720400 + 2604 \beta_{1} + 7812 \beta_{2} + 18228 \beta_{3} - 2504 \beta_{4} + 5208 \beta_{5} - 2604 \beta_{6} - 16212 \beta_{7} ) q^{94} + ( -3238208 - 282660 \beta_{1} + 165876 \beta_{2} + 12148 \beta_{3} + 45120 \beta_{4} + 440 \beta_{5} + 4412 \beta_{6} + 2016 \beta_{7} ) q^{95} + ( 4196823 - 882 \beta_{1} - 2646 \beta_{2} - 6174 \beta_{3} + 30309 \beta_{4} - 1764 \beta_{5} + 882 \beta_{6} - 3480 \beta_{7} ) q^{96} + ( -30032 \beta_{1} - 163304 \beta_{2} + 688 \beta_{3} - 1424 \beta_{5} - 736 \beta_{6} ) q^{97} + ( 306545 \beta_{1} - 219576 \beta_{2} - 168 \beta_{3} - 1932 \beta_{5} - 2100 \beta_{6} ) q^{98} + ( -985608 + 729 \beta_{1} + 2187 \beta_{2} + 5103 \beta_{3} + 8748 \beta_{4} + 1458 \beta_{5} - 729 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 666q^{4} - 444q^{5} + 486q^{6} - 5832q^{9} + O(q^{10}) \) \( 8q - 666q^{4} - 444q^{5} + 486q^{6} - 5832q^{9} - 6686q^{10} + 10752q^{11} - 13524q^{14} - 17496q^{15} + 86530q^{16} - 30464q^{19} + 87444q^{20} + 64152q^{21} - 110322q^{24} + 127616q^{25} - 793524q^{26} + 240072q^{29} - 172044q^{30} + 233728q^{31} + 184748q^{34} + 593520q^{35} + 485514q^{36} - 454896q^{39} - 1147102q^{40} + 507648q^{41} + 2578572q^{44} + 323676q^{45} + 662408q^{46} - 3267160q^{49} - 5117736q^{50} + 264384q^{51} - 354294q^{54} + 3525696q^{55} + 705660q^{56} + 1091424q^{59} + 5307606q^{60} - 6433520q^{61} - 568594q^{64} - 2555592q^{65} - 2382372q^{66} - 5940864q^{69} + 12097800q^{70} - 1381824q^{71} + 23961276q^{74} + 7768224q^{75} + 13115664q^{76} - 14380160q^{79} - 31251876q^{80} + 4251528q^{81} - 36423756q^{84} - 1452008q^{85} - 19837608q^{86} + 45778896q^{89} + 4874094q^{90} + 24075648q^{91} - 45728896q^{94} - 25774656q^{95} + 33586002q^{96} - 7838208q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 162 x^{6} + 7361 x^{4} + 87300 x^{2} + 160000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{7} - 486 \nu^{5} - 22883 \nu^{3} - 286700 \nu \)\()/20000\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{7} + 1458 \nu^{5} + 62649 \nu^{3} + 494100 \nu \)\()/20000\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{7} + 160 \nu^{6} + 86 \nu^{5} + 21920 \nu^{4} - 29517 \nu^{3} + 609760 \nu^{2} - 1453300 \nu - 996000 \)\()/20000\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 242 \nu^{4} + 13441 \nu^{2} + 81400 \)\()/500\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{7} + 20 \nu^{6} + 1408 \nu^{5} + 2740 \nu^{4} + 60849 \nu^{3} + 76220 \nu^{2} + 791350 \nu - 124500 \)\()/2500\)
\(\beta_{6}\)\(=\)\((\)\( 189 \nu^{7} - 160 \nu^{6} + 27018 \nu^{5} - 21920 \nu^{4} + 932029 \nu^{3} - 609760 \nu^{2} + 3684100 \nu + 996000 \)\()/20000\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 152 \nu^{4} + 6151 \nu^{2} + 45300 \)\()/50\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{5} - 3 \beta_{3} - 5 \beta_{2} + 54 \beta_{1}\)\()/270\)
\(\nu^{2}\)\(=\)\((\)\(14 \beta_{7} + 3 \beta_{6} - 6 \beta_{5} - 20 \beta_{4} - 21 \beta_{3} - 9 \beta_{2} - 3 \beta_{1} - 10922\)\()/270\)
\(\nu^{3}\)\(=\)\((\)\(-183 \beta_{5} + 183 \beta_{3} - 595 \beta_{2} - 5994 \beta_{1}\)\()/270\)
\(\nu^{4}\)\(=\)\((\)\(-428 \beta_{7} - 81 \beta_{6} + 162 \beta_{5} + 1040 \beta_{4} + 567 \beta_{3} + 243 \beta_{2} + 81 \beta_{1} + 258794\)\()/90\)
\(\nu^{5}\)\(=\)\((\)\(-1350 \beta_{6} + 13623 \beta_{5} - 14973 \beta_{3} + 103745 \beta_{2} + 538164 \beta_{1}\)\()/270\)
\(\nu^{6}\)\(=\)\((\)\(122554 \beta_{7} + 18483 \beta_{6} - 36966 \beta_{5} - 351220 \beta_{4} - 129381 \beta_{3} - 55449 \beta_{2} - 18483 \beta_{1} - 63059842\)\()/270\)
\(\nu^{7}\)\(=\)\((\)\(218700 \beta_{6} - 1097763 \beta_{5} + 1316463 \beta_{3} - 11790395 \beta_{2} - 48422934 \beta_{1}\)\()/270\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
9.60626i
1.49423i
3.83609i
7.26440i
7.26440i
3.83609i
1.49423i
9.60626i
21.0486i 27.0000i −315.042 −238.301 + 146.073i 568.311 923.886i 3936.97i −729.000 3074.63 + 5015.90i
4.2 17.7812i 27.0000i −188.170 57.4967 273.531i −480.092 1233.23i 1069.90i −729.000 −4863.70 1022.36i
4.3 8.75511i 27.0000i 51.3480 231.605 156.474i 236.388 536.160i 1570.21i −729.000 −1369.95 2027.73i
4.4 3.02250i 27.0000i 118.865 −272.800 60.8703i −81.6074 1505.28i 746.147i −729.000 −183.980 + 824.537i
4.5 3.02250i 27.0000i 118.865 −272.800 + 60.8703i −81.6074 1505.28i 746.147i −729.000 −183.980 824.537i
4.6 8.75511i 27.0000i 51.3480 231.605 + 156.474i 236.388 536.160i 1570.21i −729.000 −1369.95 + 2027.73i
4.7 17.7812i 27.0000i −188.170 57.4967 + 273.531i −480.092 1233.23i 1069.90i −729.000 −4863.70 + 1022.36i
4.8 21.0486i 27.0000i −315.042 −238.301 146.073i 568.311 923.886i 3936.97i −729.000 3074.63 5015.90i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{8}^{\mathrm{new}}(15, [\chi])\).