Properties

Label 20.7.d.d
Level 20
Weight 7
Character orbit 20.d
Analytic conductor 4.601
Analytic rank 0
Dimension 12
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.6010816724\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} + ( 6 + \beta_{4} - \beta_{8} ) q^{4} + ( 37 + 3 \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{5} + ( -58 - 6 \beta_{4} + \beta_{5} ) q^{6} + ( 5 \beta_{2} - \beta_{3} + \beta_{7} ) q^{7} + ( -1 - 9 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{10} ) q^{8} + ( 162 + 6 \beta_{4} + 6 \beta_{8} - 3 \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} + ( 6 + \beta_{4} - \beta_{8} ) q^{4} + ( 37 + 3 \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{6} ) q^{5} + ( -58 - 6 \beta_{4} + \beta_{5} ) q^{6} + ( 5 \beta_{2} - \beta_{3} + \beta_{7} ) q^{7} + ( -1 - 9 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{10} ) q^{8} + ( 162 + 6 \beta_{4} + 6 \beta_{8} - 3 \beta_{11} ) q^{9} + ( 166 - 37 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 3 \beta_{6} + 5 \beta_{8} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{10} + ( -\beta_{1} + \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{6} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{11} + ( -1 + 54 \beta_{1} - 16 \beta_{2} + 5 \beta_{3} - \beta_{4} + 13 \beta_{6} + \beta_{7} - 11 \beta_{10} ) q^{12} + ( -1 + 16 \beta_{1} - \beta_{4} - 16 \beta_{6} + 18 \beta_{10} ) q^{13} + ( 42 + \beta_{1} - \beta_{3} - 26 \beta_{4} + 3 \beta_{5} + \beta_{6} - 10 \beta_{8} - 2 \beta_{9} - \beta_{10} + 12 \beta_{11} ) q^{14} + ( 8 - 91 \beta_{1} + 45 \beta_{2} - 34 \beta_{3} + 7 \beta_{4} - 12 \beta_{5} - \beta_{6} - 5 \beta_{7} - 20 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{15} + ( -1744 + 2 \beta_{1} - 2 \beta_{3} - 28 \beta_{4} - 12 \beta_{5} + 2 \beta_{6} - 16 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} ) q^{16} + ( 10 + 366 \beta_{1} - 90 \beta_{3} + 10 \beta_{4} - 6 \beta_{6} - 14 \beta_{10} ) q^{17} + ( 12 - 192 \beta_{1} - 60 \beta_{2} + 69 \beta_{3} + 12 \beta_{4} - 27 \beta_{6} - 12 \beta_{7} + 3 \beta_{10} ) q^{18} + ( -16 - 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 20 \beta_{5} - 3 \beta_{6} + 52 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 13 \beta_{11} ) q^{19} + ( -1127 - 178 \beta_{1} - 80 \beta_{2} + 21 \beta_{3} + 58 \beta_{4} + 20 \beta_{5} + 29 \beta_{6} - 15 \beta_{7} - 35 \beta_{8} + 5 \beta_{10} ) q^{20} + ( 4341 + 303 \beta_{4} + 56 \beta_{8} - 28 \beta_{11} ) q^{21} + ( 44 - 62 \beta_{1} - 168 \beta_{2} + 106 \beta_{3} + 44 \beta_{4} - 54 \beta_{6} + 20 \beta_{7} - 34 \beta_{10} ) q^{22} + ( 82 \beta_{1} + 101 \beta_{2} - 7 \beta_{3} - 9 \beta_{7} ) q^{23} + ( -1640 + 2 \beta_{1} - 2 \beta_{3} + 424 \beta_{4} + 28 \beta_{5} + 2 \beta_{6} + 124 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 8 \beta_{11} ) q^{24} + ( -4840 - 60 \beta_{1} + 20 \beta_{3} - 490 \beta_{4} - 20 \beta_{6} + 50 \beta_{8} + 50 \beta_{10} - 25 \beta_{11} ) q^{25} + ( 1020 + \beta_{1} - \beta_{3} - 632 \beta_{4} - 70 \beta_{5} + \beta_{6} - 70 \beta_{8} - 2 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{26} + ( 108 \beta_{1} - 126 \beta_{2} + 66 \beta_{3} + 30 \beta_{7} ) q^{27} + ( -87 + 126 \beta_{1} + 128 \beta_{2} - 433 \beta_{3} - 87 \beta_{4} + 199 \beta_{6} + 23 \beta_{7} - 25 \beta_{10} ) q^{28} + ( -1218 + 34 \beta_{4} - 220 \beta_{8} + 110 \beta_{11} ) q^{29} + ( 6974 - 59 \beta_{1} + 440 \beta_{2} + 263 \beta_{3} - 86 \beta_{4} + 55 \beta_{5} - 163 \beta_{6} + 20 \beta_{7} - 110 \beta_{8} + 10 \beta_{9} + 75 \beta_{10} - 60 \beta_{11} ) q^{30} + ( 16 + 8 \beta_{1} - 8 \beta_{3} - 40 \beta_{4} + 192 \beta_{5} + 8 \beta_{6} - 128 \beta_{8} - 16 \beta_{9} - 8 \beta_{10} - 56 \beta_{11} ) q^{31} + ( -120 + 1908 \beta_{1} + 688 \beta_{2} - 220 \beta_{3} - 120 \beta_{4} + 164 \beta_{6} - 8 \beta_{7} + 76 \beta_{10} ) q^{32} + ( -20 - 1942 \beta_{1} + 434 \beta_{3} - 20 \beta_{4} + 206 \beta_{6} - 166 \beta_{10} ) q^{33} + ( 21304 - 10 \beta_{1} + 10 \beta_{3} - 16 \beta_{4} + 36 \beta_{5} - 10 \beta_{6} + 396 \beta_{8} + 20 \beta_{9} + 10 \beta_{10} + 40 \beta_{11} ) q^{34} + ( 192 - 694 \beta_{1} - 335 \beta_{2} - 101 \beta_{3} + 123 \beta_{4} - 108 \beta_{5} + 11 \beta_{6} + 10 \beta_{7} - 500 \beta_{8} - 22 \beta_{9} - 11 \beta_{10} - 69 \beta_{11} ) q^{35} + ( -12006 - 24 \beta_{1} + 24 \beta_{3} + 231 \beta_{4} - 72 \beta_{5} - 24 \beta_{6} - 135 \beta_{8} + 48 \beta_{9} + 24 \beta_{10} - 96 \beta_{11} ) q^{36} + ( -21 - 1446 \beta_{1} + 354 \beta_{3} - 21 \beta_{4} + 30 \beta_{6} + 12 \beta_{10} ) q^{37} + ( 196 + 406 \beta_{1} + 968 \beta_{2} - 914 \beta_{3} + 196 \beta_{4} - 370 \beta_{6} - 4 \beta_{7} - 22 \beta_{10} ) q^{38} + ( -384 + 34 \beta_{1} - 34 \beta_{3} - 190 \beta_{4} - 8 \beta_{5} + 34 \beta_{6} + 1224 \beta_{8} - 68 \beta_{9} - 34 \beta_{10} + 194 \beta_{11} ) q^{39} + ( -2991 + 927 \beta_{1} - 1100 \beta_{2} + 218 \beta_{3} + 601 \beta_{4} - 36 \beta_{5} + 102 \beta_{6} + 55 \beta_{7} + 20 \beta_{8} - 4 \beta_{9} + 8 \beta_{10} - 248 \beta_{11} ) q^{40} + ( -16059 - 392 \beta_{4} - 166 \beta_{8} + 83 \beta_{11} ) q^{41} + ( 112 - 4621 \beta_{1} - 1548 \beta_{2} + 397 \beta_{3} + 112 \beta_{4} - 499 \beta_{6} - 112 \beta_{7} + 275 \beta_{10} ) q^{42} + ( 6111 \beta_{1} - 1063 \beta_{2} + 1806 \beta_{3} + 50 \beta_{7} ) q^{43} + ( -13152 - 24 \beta_{1} + 24 \beta_{3} + 1456 \beta_{4} - 160 \beta_{5} - 24 \beta_{6} + 48 \beta_{9} + 24 \beta_{10} + 416 \beta_{11} ) q^{44} + ( -5016 + 5151 \beta_{1} - 1317 \beta_{3} - 936 \beta_{4} + 117 \beta_{6} + 60 \beta_{8} - 390 \beta_{10} - 30 \beta_{11} ) q^{45} + ( -4366 - 9 \beta_{1} + 9 \beta_{3} - 578 \beta_{4} + 119 \beta_{5} - 9 \beta_{6} + 26 \beta_{8} + 18 \beta_{9} + 9 \beta_{10} - 108 \beta_{11} ) q^{46} + ( -5004 \beta_{1} + 301 \beta_{2} - 1397 \beta_{3} - 283 \beta_{7} ) q^{47} + ( -20 + 2064 \beta_{1} - 2720 \beta_{2} - 1972 \beta_{3} - 20 \beta_{4} - 148 \beta_{6} - 108 \beta_{7} + 188 \beta_{10} ) q^{48} + ( 13370 + 3322 \beta_{4} - 226 \beta_{8} + 113 \beta_{11} ) q^{49} + ( -3320 + 4590 \beta_{1} + 1600 \beta_{2} + 1085 \beta_{3} - 1380 \beta_{4} - 170 \beta_{5} + 315 \beta_{6} - 100 \beta_{7} - 250 \beta_{8} - 30 \beta_{9} - 515 \beta_{10} - 60 \beta_{11} ) q^{50} + ( 368 - 8 \beta_{1} + 8 \beta_{3} + 344 \beta_{4} - 640 \beta_{5} - 8 \beta_{6} - 800 \beta_{8} + 16 \beta_{9} + 8 \beta_{10} - 24 \beta_{11} ) q^{51} + ( -42 - 996 \beta_{1} + 5280 \beta_{2} + 1474 \beta_{3} - 42 \beta_{4} + 274 \beta_{6} - 22 \beta_{7} - 190 \beta_{10} ) q^{52} + ( 253 - 10880 \beta_{1} + 2992 \beta_{3} + 253 \beta_{4} - 1088 \beta_{6} + 582 \beta_{10} ) q^{53} + ( -9612 + 30 \beta_{1} - 30 \beta_{3} + 492 \beta_{4} - 186 \beta_{5} + 30 \beta_{6} + 84 \beta_{8} - 60 \beta_{9} - 30 \beta_{10} + 360 \beta_{11} ) q^{54} + ( 120 - 9885 \beta_{1} + 720 \beta_{2} - 2555 \beta_{3} - 85 \beta_{4} + 580 \beta_{5} - 45 \beta_{6} + 160 \beta_{7} - 740 \beta_{8} + 90 \beta_{9} + 45 \beta_{10} - 205 \beta_{11} ) q^{55} + ( 90216 + 110 \beta_{1} - 110 \beta_{3} + 1304 \beta_{4} + 356 \beta_{5} + 110 \beta_{6} + 356 \beta_{8} - 220 \beta_{9} - 110 \beta_{10} - 72 \beta_{11} ) q^{56} + ( -348 + 10302 \beta_{1} - 2634 \beta_{3} - 348 \beta_{4} + 234 \beta_{6} + 462 \beta_{10} ) q^{57} + ( -440 + 2318 \beta_{1} + 1184 \beta_{2} - 2784 \beta_{3} - 440 \beta_{4} + 736 \beta_{6} + 440 \beta_{7} + 144 \beta_{10} ) q^{58} + ( -240 - 141 \beta_{1} + 141 \beta_{3} - 205 \beta_{4} + 340 \beta_{5} - 141 \beta_{6} + 268 \beta_{8} + 282 \beta_{9} + 141 \beta_{10} + 35 \beta_{11} ) q^{59} + ( -76243 - 8554 \beta_{1} - 2400 \beta_{2} + 3819 \beta_{3} - 5027 \beta_{4} + 32 \beta_{5} - 109 \beta_{6} + 115 \beta_{7} - 1120 \beta_{8} + 48 \beta_{9} - 301 \beta_{10} + 416 \beta_{11} ) q^{60} + ( 15151 - 6347 \beta_{4} + 1364 \beta_{8} - 682 \beta_{11} ) q^{61} + ( -608 - 1920 \beta_{1} - 8192 \beta_{2} + 3168 \beta_{3} - 608 \beta_{4} + 1888 \beta_{6} + 96 \beta_{7} - 672 \beta_{10} ) q^{62} + ( 23238 \beta_{1} + 4071 \beta_{2} + 4803 \beta_{3} + 45 \beta_{7} ) q^{63} + ( -3232 + 112 \beta_{1} - 112 \beta_{3} - 6320 \beta_{4} + 640 \beta_{5} + 112 \beta_{6} + 1360 \beta_{8} - 224 \beta_{9} - 112 \beta_{10} - 576 \beta_{11} ) q^{64} + ( 60495 + 18990 \beta_{1} - 4730 \beta_{3} + 6690 \beta_{4} - 70 \beta_{6} + 1370 \beta_{8} + 820 \beta_{10} - 685 \beta_{11} ) q^{65} + ( -113952 + 20 \beta_{1} - 20 \beta_{3} + 8032 \beta_{4} + 704 \beta_{5} + 20 \beta_{6} - 1032 \beta_{8} - 40 \beta_{9} - 20 \beta_{10} - 80 \beta_{11} ) q^{66} + ( -8541 \beta_{1} + 3089 \beta_{2} - 2710 \beta_{3} + 790 \beta_{7} ) q^{67} + ( 780 - 19800 \beta_{1} - 896 \beta_{2} - 5004 \beta_{3} + 780 \beta_{4} - 748 \beta_{6} - 140 \beta_{7} - 812 \beta_{10} ) q^{68} + ( 83625 - 2571 \beta_{4} + 340 \beta_{8} - 170 \beta_{11} ) q^{69} + ( 44346 - 2156 \beta_{1} + 3000 \beta_{2} + 7112 \beta_{3} + 1566 \beta_{4} - 355 \beta_{5} + 188 \beta_{6} + 100 \beta_{7} - 420 \beta_{8} - 20 \beta_{9} + 1420 \beta_{10} + 120 \beta_{11} ) q^{70} + ( -144 - 106 \beta_{1} + 106 \beta_{3} + 166 \beta_{4} - 952 \beta_{5} - 106 \beta_{6} + 696 \beta_{8} + 212 \beta_{9} + 106 \beta_{10} + 310 \beta_{11} ) q^{71} + ( 1329 + 9921 \beta_{1} + 804 \beta_{2} + 5904 \beta_{3} + 1329 \beta_{4} - 3048 \beta_{6} + 207 \beta_{7} + 390 \beta_{10} ) q^{72} + ( -1088 - 33742 \beta_{1} + 7706 \beta_{3} - 1088 \beta_{4} + 2918 \beta_{6} - 742 \beta_{10} ) q^{73} + ( -84132 + 21 \beta_{1} - 21 \beta_{3} + 1320 \beta_{4} - 6 \beta_{5} + 21 \beta_{6} - 1422 \beta_{8} - 42 \beta_{9} - 21 \beta_{10} - 84 \beta_{11} ) q^{74} + ( -960 - 25905 \beta_{1} + 425 \beta_{2} - 6870 \beta_{3} - 490 \beta_{4} + 40 \beta_{5} + 70 \beta_{6} - 800 \beta_{7} + 3000 \beta_{8} - 140 \beta_{9} - 70 \beta_{10} + 470 \beta_{11} ) q^{75} + ( 207712 - 200 \beta_{1} + 200 \beta_{3} - 7216 \beta_{4} + 672 \beta_{5} - 200 \beta_{6} - 1216 \beta_{8} + 400 \beta_{9} + 200 \beta_{10} + 736 \beta_{11} ) q^{76} + ( 1932 + 24018 \beta_{1} - 5366 \beta_{3} + 1932 \beta_{4} - 2554 \beta_{6} - 1310 \beta_{10} ) q^{77} + ( -856 + 7804 \beta_{1} + 3408 \beta_{2} - 20180 \beta_{3} - 856 \beta_{4} + 2668 \beta_{6} - 1320 \beta_{7} - 956 \beta_{10} ) q^{78} + ( 1936 + 188 \beta_{1} - 188 \beta_{3} + 1068 \beta_{4} - 400 \beta_{5} + 188 \beta_{6} - 5232 \beta_{8} - 376 \beta_{9} - 188 \beta_{10} - 868 \beta_{11} ) q^{79} + ( -8908 - 682 \beta_{1} - 4160 \beta_{2} + 9694 \beta_{3} + 5392 \beta_{4} - 1140 \beta_{5} - 1894 \beta_{6} - 580 \beta_{7} + 1920 \beta_{8} - 220 \beta_{9} + 990 \beta_{10} + 440 \beta_{11} ) q^{80} + ( -189432 + 7380 \beta_{4} - 4158 \beta_{8} + 2079 \beta_{11} ) q^{81} + ( -332 + 16889 \beta_{1} + 2564 \beta_{2} - 1683 \beta_{3} - 332 \beta_{4} + 973 \beta_{6} + 332 \beta_{7} - 309 \beta_{10} ) q^{82} + ( 43331 \beta_{1} - 6687 \beta_{2} + 12282 \beta_{3} - 890 \beta_{7} ) q^{83} + ( 36816 - 224 \beta_{1} + 224 \beta_{3} + 1280 \beta_{4} - 2648 \beta_{5} - 224 \beta_{6} - 4336 \beta_{8} + 448 \beta_{9} + 224 \beta_{10} - 896 \beta_{11} ) q^{84} + ( -2550 + 11810 \beta_{1} - 2470 \beta_{3} - 4480 \beta_{4} - 1930 \beta_{6} - 4780 \beta_{8} + 1070 \beta_{10} + 2390 \beta_{11} ) q^{85} + ( -441426 + 50 \beta_{1} - 50 \beta_{3} - 846 \beta_{4} - 1163 \beta_{5} + 50 \beta_{6} + 6924 \beta_{8} - 100 \beta_{9} - 50 \beta_{10} + 600 \beta_{11} ) q^{86} + ( -13014 \beta_{1} - 18150 \beta_{2} + 1136 \beta_{3} - 592 \beta_{7} ) q^{87} + ( 528 + 20016 \beta_{1} + 5568 \beta_{2} - 18464 \beta_{3} + 528 \beta_{4} - 2464 \beta_{6} + 1008 \beta_{7} + 1408 \beta_{10} ) q^{88} + ( -365994 - 5858 \beta_{4} + 3724 \beta_{8} - 1862 \beta_{11} ) q^{89} + ( 298722 + 4716 \beta_{1} + 3930 \beta_{2} + 1959 \beta_{3} + 6708 \beta_{4} + 1287 \beta_{5} + 726 \beta_{6} - 120 \beta_{7} + 6555 \beta_{8} + 273 \beta_{9} - 966 \beta_{10} + 546 \beta_{11} ) q^{90} + ( -2160 + 362 \beta_{1} - 362 \beta_{3} - 1974 \beta_{4} + 3576 \beta_{5} + 362 \beta_{6} + 5416 \beta_{8} - 724 \beta_{9} - 362 \beta_{10} + 186 \beta_{11} ) q^{91} + ( 573 + 2078 \beta_{1} - 3744 \beta_{2} + 5139 \beta_{3} + 573 \beta_{4} + 107 \beta_{6} + 3 \beta_{7} - 1253 \beta_{10} ) q^{92} + ( 2560 - 104656 \beta_{1} + 27152 \beta_{3} + 2560 \beta_{4} - 3952 \beta_{6} - 1168 \beta_{10} ) q^{93} + ( 358986 - 283 \beta_{1} + 283 \beta_{3} + 3782 \beta_{4} + 867 \beta_{5} - 283 \beta_{6} - 3890 \beta_{8} + 566 \beta_{9} + 283 \beta_{10} - 3396 \beta_{11} ) q^{94} + ( -1320 - 43095 \beta_{1} - 4240 \beta_{2} - 9425 \beta_{3} - 175 \beta_{4} - 1940 \beta_{5} + 25 \beta_{6} + 1280 \beta_{7} + 4980 \beta_{8} - 50 \beta_{9} - 25 \beta_{10} + 1145 \beta_{11} ) q^{95} + ( 639200 - 88 \beta_{1} + 88 \beta_{3} + 5344 \beta_{4} - 3216 \beta_{5} - 88 \beta_{6} + 4464 \beta_{8} + 176 \beta_{9} + 88 \beta_{10} - 1376 \beta_{11} ) q^{96} + ( -3906 + 11622 \beta_{1} - 5282 \beta_{3} - 3906 \beta_{4} + 9506 \beta_{6} - 1694 \beta_{10} ) q^{97} + ( -452 - 12240 \beta_{1} - 11932 \beta_{2} - 6147 \beta_{3} - 452 \beta_{4} - 2531 \beta_{6} + 452 \beta_{7} + 3435 \beta_{10} ) q^{98} + ( 3024 + 357 \beta_{1} - 357 \beta_{3} + 1413 \beta_{4} + 396 \beta_{5} + 357 \beta_{6} - 8556 \beta_{8} - 714 \beta_{9} - 357 \beta_{10} - 1611 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 64q^{4} + 460q^{5} - 672q^{6} + 1956q^{9} + O(q^{10}) \) \( 12q + 64q^{4} + 460q^{5} - 672q^{6} + 1956q^{9} + 2000q^{10} + 512q^{14} - 20928q^{16} - 13760q^{20} + 51216q^{21} - 20928q^{24} - 55700q^{25} + 14496q^{26} - 16072q^{29} + 83520q^{30} + 257216q^{34} - 144960q^{36} - 36800q^{40} - 192136q^{41} - 165120q^{44} - 57180q^{45} - 49472q^{46} + 145796q^{49} - 36000q^{50} - 118656q^{54} + 1078208q^{56} - 902400q^{60} + 215384q^{61} - 6656q^{64} + 710400q^{65} - 1403520q^{66} + 1015824q^{69} + 530080q^{70} - 1020384q^{74} + 2515200q^{76} - 127040q^{80} - 2327652q^{81} + 424704q^{84} - 44800q^{85} - 5268832q^{86} - 4346152q^{89} + 3582000q^{90} + 4292992q^{94} + 7673088q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 16 x^{10} - 10 x^{8} - 1775 x^{6} - 1000 x^{4} + 160000 x^{2} + 1000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 9 \nu^{11} + 64 \nu^{9} - 970 \nu^{7} - 8775 \nu^{5} + 89000 \nu^{3} + 770000 \nu \)\()/50000\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{11} + 33 \nu^{9} + 260 \nu^{7} - 2900 \nu^{5} - 28375 \nu^{3} + 230000 \nu \)\()/12500\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{11} - 28 \nu^{9} + 350 \nu^{7} + 3125 \nu^{5} - 24500 \nu^{3} - 290000 \nu \)\()/6250\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{10} - 28 \nu^{8} + 350 \nu^{6} + 3125 \nu^{4} - 34500 \nu^{2} - 327500 \)\()/2500\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{10} - 16 \nu^{8} - 1050 \nu^{6} - 775 \nu^{4} + 87000 \nu^{2} - 20000 \)\()/1250\)
\(\beta_{6}\)\(=\)\((\)\( -61 \nu^{11} - 30 \nu^{10} - 576 \nu^{9} - 280 \nu^{8} + 5810 \nu^{7} + 3500 \nu^{6} + 69075 \nu^{5} + 31250 \nu^{4} - 453000 \nu^{3} - 345000 \nu^{2} - 7230000 \nu - 3250000 \)\()/50000\)
\(\beta_{7}\)\(=\)\((\)\( -103 \nu^{11} + 2512 \nu^{9} + 5590 \nu^{7} - 226775 \nu^{5} - 957000 \nu^{3} + 26490000 \nu \)\()/50000\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} - 56 \nu^{8} + 170 \nu^{6} + 5975 \nu^{4} - 5000 \nu^{2} - 586250 \)\()/1250\)
\(\beta_{9}\)\(=\)\((\)\( 121 \nu^{11} + 880 \nu^{10} + 1016 \nu^{9} - 1120 \nu^{8} - 12330 \nu^{7} - 76000 \nu^{6} - 107975 \nu^{5} - 314000 \nu^{4} + 1156000 \nu^{3} + 9020000 \nu^{2} + 10530000 \nu + 9550000 \)\()/50000\)
\(\beta_{10}\)\(=\)\((\)\( -27 \nu^{11} - 3 \nu^{10} - 232 \nu^{9} - 28 \nu^{8} + 2670 \nu^{7} + 350 \nu^{6} + 25125 \nu^{5} + 3125 \nu^{4} - 248000 \nu^{3} - 34500 \nu^{2} - 2520000 \nu - 325000 \)\()/5000\)
\(\beta_{11}\)\(=\)\((\)\( 61 \nu^{10} + 436 \nu^{8} - 6050 \nu^{6} - 49675 \nu^{4} + 547500 \nu^{2} + 4712500 \)\()/2500\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{7} - 42 \beta_{6} + 21 \beta_{4} + 62 \beta_{3} - 100 \beta_{2} - 151 \beta_{1} + 21\)\()/1680\)
\(\nu^{2}\)\(=\)\((\)\(-33 \beta_{11} + 8 \beta_{10} + 16 \beta_{9} + 130 \beta_{8} - 8 \beta_{6} - 76 \beta_{5} - 979 \beta_{4} + 8 \beta_{3} - 8 \beta_{1} - 9346\)\()/3360\)
\(\nu^{3}\)\(=\)\((\)\(-35 \beta_{10} + 10 \beta_{7} + 119 \beta_{6} - 42 \beta_{4} + 341 \beta_{3} - 60 \beta_{2} + 727 \beta_{1} - 42\)\()/560\)
\(\nu^{4}\)\(=\)\((\)\(333 \beta_{11} - 208 \beta_{10} - 416 \beta_{9} + 1870 \beta_{8} + 208 \beta_{6} - 124 \beta_{5} - 1321 \beta_{4} - 208 \beta_{3} + 208 \beta_{1} + 153746\)\()/3360\)
\(\nu^{5}\)\(=\)\((\)\(-1470 \beta_{10} + 295 \beta_{7} + 168 \beta_{6} + 651 \beta_{4} - 1298 \beta_{3} - 8420 \beta_{2} - 50531 \beta_{1} + 651\)\()/1680\)
\(\nu^{6}\)\(=\)\((\)\(507 \beta_{11} + 243 \beta_{10} + 486 \beta_{9} + 1980 \beta_{8} - 243 \beta_{6} - 2396 \beta_{5} + 1741 \beta_{4} + 243 \beta_{3} - 243 \beta_{1} + 69834\)\()/560\)
\(\nu^{7}\)\(=\)\((\)\(-9030 \beta_{10} - 2795 \beta_{7} - 10668 \beta_{6} + 9849 \beta_{4} + 133048 \beta_{3} + 58420 \beta_{2} + 31531 \beta_{1} + 9849\)\()/1680\)
\(\nu^{8}\)\(=\)\((\)\(4869 \beta_{11} - 2744 \beta_{10} - 5488 \beta_{9} + 19910 \beta_{8} + 2744 \beta_{6} - 5732 \beta_{5} - 29153 \beta_{4} - 2744 \beta_{3} + 2744 \beta_{1} - 2702822\)\()/480\)
\(\nu^{9}\)\(=\)\((\)\(-53865 \beta_{10} - 1360 \beta_{7} + 149681 \beta_{6} - 47908 \beta_{4} - 199641 \beta_{3} + 113860 \beta_{2} - 1048177 \beta_{1} - 47908\)\()/560\)
\(\nu^{10}\)\(=\)\((\)\(763167 \beta_{11} + 40708 \beta_{10} + 81416 \beta_{9} + 538130 \beta_{8} - 40708 \beta_{6} - 557876 \beta_{5} + 10205821 \beta_{4} + 40708 \beta_{3} - 40708 \beta_{1} + 126299254\)\()/3360\)
\(\nu^{11}\)\(=\)\((\)\(-219030 \beta_{10} - 709045 \beta_{7} - 4116168 \beta_{6} + 2167599 \beta_{4} + 1912298 \beta_{3} + 5993420 \beta_{2} - 22823719 \beta_{1} + 2167599\)\()/1680\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.826519 + 3.05235i
−0.826519 3.05235i
−0.954985 3.01463i
−0.954985 + 3.01463i
3.06674 0.771446i
3.06674 + 0.771446i
−3.06674 0.771446i
−3.06674 + 0.771446i
0.954985 3.01463i
0.954985 + 3.01463i
0.826519 + 3.05235i
0.826519 3.05235i
−7.42116 2.98770i 6.08848 46.1473 + 44.3444i 26.7958 122.094i −45.1836 18.1905i −422.345 −209.979 466.961i −691.930 −563.636 + 826.023i
19.2 −7.42116 + 2.98770i 6.08848 46.1473 44.3444i 26.7958 + 122.094i −45.1836 + 18.1905i −422.345 −209.979 + 466.961i −691.930 −563.636 826.023i
19.3 −5.94356 5.35482i 40.3729 6.65182 + 63.6534i −32.9271 + 120.585i −239.959 216.189i 450.705 301.317 413.947i 900.969 841.416 540.387i
19.4 −5.94356 + 5.35482i 40.3729 6.65182 63.6534i −32.9271 120.585i −239.959 + 216.189i 450.705 301.317 + 413.947i 900.969 841.416 + 540.387i
19.5 −3.68788 7.09927i −31.7642 −36.7991 + 52.3624i 121.131 + 30.8579i 117.142 + 225.502i 88.8045 507.445 + 68.1408i 279.961 −227.649 973.743i
19.6 −3.68788 + 7.09927i −31.7642 −36.7991 52.3624i 121.131 30.8579i 117.142 225.502i 88.8045 507.445 68.1408i 279.961 −227.649 + 973.743i
19.7 3.68788 7.09927i 31.7642 −36.7991 52.3624i 121.131 + 30.8579i 117.142 225.502i −88.8045 −507.445 + 68.1408i 279.961 665.785 746.143i
19.8 3.68788 + 7.09927i 31.7642 −36.7991 + 52.3624i 121.131 30.8579i 117.142 + 225.502i −88.8045 −507.445 68.1408i 279.961 665.785 + 746.143i
19.9 5.94356 5.35482i −40.3729 6.65182 63.6534i −32.9271 + 120.585i −239.959 + 216.189i −450.705 −301.317 413.947i 900.969 450.008 + 893.024i
19.10 5.94356 + 5.35482i −40.3729 6.65182 + 63.6534i −32.9271 120.585i −239.959 216.189i −450.705 −301.317 + 413.947i 900.969 450.008 893.024i
19.11 7.42116 2.98770i −6.08848 46.1473 44.3444i 26.7958 122.094i −45.1836 + 18.1905i 422.345 209.979 466.961i −691.930 −165.925 986.138i
19.12 7.42116 + 2.98770i −6.08848 46.1473 + 44.3444i 26.7958 + 122.094i −45.1836 18.1905i 422.345 209.979 + 466.961i −691.930 −165.925 + 986.138i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{6} - 2676 T_{3}^{4} + 1742400 T_{3}^{2} - 60963840 \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\).