Properties

Label 722.2.a.l.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.87939 q^{6} +5.06418 q^{7} +1.00000 q^{8} +0.532089 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.87939 q^{6} +5.06418 q^{7} +1.00000 q^{8} +0.532089 q^{9} +2.00000 q^{10} -1.41147 q^{11} -1.87939 q^{12} -1.30541 q^{13} +5.06418 q^{14} -3.75877 q^{15} +1.00000 q^{16} +2.38919 q^{17} +0.532089 q^{18} +2.00000 q^{20} -9.51754 q^{21} -1.41147 q^{22} -3.06418 q^{23} -1.87939 q^{24} -1.00000 q^{25} -1.30541 q^{26} +4.63816 q^{27} +5.06418 q^{28} +8.45336 q^{29} -3.75877 q^{30} +0.369585 q^{31} +1.00000 q^{32} +2.65270 q^{33} +2.38919 q^{34} +10.1284 q^{35} +0.532089 q^{36} +4.82295 q^{37} +2.45336 q^{39} +2.00000 q^{40} -1.53209 q^{41} -9.51754 q^{42} -0.758770 q^{43} -1.41147 q^{44} +1.06418 q^{45} -3.06418 q^{46} -10.2121 q^{47} -1.87939 q^{48} +18.6459 q^{49} -1.00000 q^{50} -4.49020 q^{51} -1.30541 q^{52} +1.67499 q^{53} +4.63816 q^{54} -2.82295 q^{55} +5.06418 q^{56} +8.45336 q^{58} +0.716881 q^{59} -3.75877 q^{60} +9.75877 q^{61} +0.369585 q^{62} +2.69459 q^{63} +1.00000 q^{64} -2.61081 q^{65} +2.65270 q^{66} -1.40373 q^{67} +2.38919 q^{68} +5.75877 q^{69} +10.1284 q^{70} -6.36959 q^{71} +0.532089 q^{72} -4.55943 q^{73} +4.82295 q^{74} +1.87939 q^{75} -7.14796 q^{77} +2.45336 q^{78} -2.24123 q^{79} +2.00000 q^{80} -10.3131 q^{81} -1.53209 q^{82} -3.98545 q^{83} -9.51754 q^{84} +4.77837 q^{85} -0.758770 q^{86} -15.8871 q^{87} -1.41147 q^{88} -10.6459 q^{89} +1.06418 q^{90} -6.61081 q^{91} -3.06418 q^{92} -0.694593 q^{93} -10.2121 q^{94} -1.87939 q^{96} -1.53209 q^{97} +18.6459 q^{98} -0.751030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} + 3 q^{8} - 3 q^{9} + 6 q^{10} + 6 q^{11} - 6 q^{13} + 6 q^{14} + 3 q^{16} + 3 q^{17} - 3 q^{18} + 6 q^{20} - 6 q^{21} + 6 q^{22} - 3 q^{25} - 6 q^{26} - 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} + 3 q^{32} + 9 q^{33} + 3 q^{34} + 12 q^{35} - 3 q^{36} - 6 q^{37} - 6 q^{39} + 6 q^{40} - 6 q^{42} + 9 q^{43} + 6 q^{44} - 6 q^{45} - 6 q^{47} + 15 q^{49} - 3 q^{50} - 12 q^{51} - 6 q^{52} - 3 q^{54} + 12 q^{55} + 6 q^{56} + 12 q^{58} - 6 q^{59} + 18 q^{61} - 6 q^{62} + 6 q^{63} + 3 q^{64} - 12 q^{65} + 9 q^{66} - 18 q^{67} + 3 q^{68} + 6 q^{69} + 12 q^{70} - 12 q^{71} - 3 q^{72} + 12 q^{73} - 6 q^{74} - 6 q^{77} - 6 q^{78} - 18 q^{79} + 6 q^{80} - 9 q^{81} + 6 q^{83} - 6 q^{84} + 6 q^{85} + 9 q^{86} - 18 q^{87} + 6 q^{88} + 9 q^{89} - 6 q^{90} - 24 q^{91} - 6 q^{94} + 15 q^{98} - 15 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.87939 −1.08506 −0.542532 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.87939 −0.767256
\(7\) 5.06418 1.91408 0.957040 0.289957i \(-0.0936410\pi\)
0.957040 + 0.289957i \(0.0936410\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.532089 0.177363
\(10\) 2.00000 0.632456
\(11\) −1.41147 −0.425575 −0.212788 0.977098i \(-0.568254\pi\)
−0.212788 + 0.977098i \(0.568254\pi\)
\(12\) −1.87939 −0.542532
\(13\) −1.30541 −0.362055 −0.181027 0.983478i \(-0.557942\pi\)
−0.181027 + 0.983478i \(0.557942\pi\)
\(14\) 5.06418 1.35346
\(15\) −3.75877 −0.970510
\(16\) 1.00000 0.250000
\(17\) 2.38919 0.579463 0.289731 0.957108i \(-0.406434\pi\)
0.289731 + 0.957108i \(0.406434\pi\)
\(18\) 0.532089 0.125415
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) −9.51754 −2.07690
\(22\) −1.41147 −0.300927
\(23\) −3.06418 −0.638925 −0.319463 0.947599i \(-0.603502\pi\)
−0.319463 + 0.947599i \(0.603502\pi\)
\(24\) −1.87939 −0.383628
\(25\) −1.00000 −0.200000
\(26\) −1.30541 −0.256011
\(27\) 4.63816 0.892613
\(28\) 5.06418 0.957040
\(29\) 8.45336 1.56975 0.784875 0.619654i \(-0.212727\pi\)
0.784875 + 0.619654i \(0.212727\pi\)
\(30\) −3.75877 −0.686254
\(31\) 0.369585 0.0663794 0.0331897 0.999449i \(-0.489433\pi\)
0.0331897 + 0.999449i \(0.489433\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.65270 0.461776
\(34\) 2.38919 0.409742
\(35\) 10.1284 1.71200
\(36\) 0.532089 0.0886815
\(37\) 4.82295 0.792888 0.396444 0.918059i \(-0.370244\pi\)
0.396444 + 0.918059i \(0.370244\pi\)
\(38\) 0 0
\(39\) 2.45336 0.392853
\(40\) 2.00000 0.316228
\(41\) −1.53209 −0.239272 −0.119636 0.992818i \(-0.538173\pi\)
−0.119636 + 0.992818i \(0.538173\pi\)
\(42\) −9.51754 −1.46859
\(43\) −0.758770 −0.115711 −0.0578557 0.998325i \(-0.518426\pi\)
−0.0578557 + 0.998325i \(0.518426\pi\)
\(44\) −1.41147 −0.212788
\(45\) 1.06418 0.158638
\(46\) −3.06418 −0.451788
\(47\) −10.2121 −1.48959 −0.744796 0.667292i \(-0.767453\pi\)
−0.744796 + 0.667292i \(0.767453\pi\)
\(48\) −1.87939 −0.271266
\(49\) 18.6459 2.66370
\(50\) −1.00000 −0.141421
\(51\) −4.49020 −0.628754
\(52\) −1.30541 −0.181027
\(53\) 1.67499 0.230078 0.115039 0.993361i \(-0.463301\pi\)
0.115039 + 0.993361i \(0.463301\pi\)
\(54\) 4.63816 0.631173
\(55\) −2.82295 −0.380646
\(56\) 5.06418 0.676729
\(57\) 0 0
\(58\) 8.45336 1.10998
\(59\) 0.716881 0.0933300 0.0466650 0.998911i \(-0.485141\pi\)
0.0466650 + 0.998911i \(0.485141\pi\)
\(60\) −3.75877 −0.485255
\(61\) 9.75877 1.24948 0.624741 0.780832i \(-0.285204\pi\)
0.624741 + 0.780832i \(0.285204\pi\)
\(62\) 0.369585 0.0469373
\(63\) 2.69459 0.339487
\(64\) 1.00000 0.125000
\(65\) −2.61081 −0.323832
\(66\) 2.65270 0.326525
\(67\) −1.40373 −0.171493 −0.0857467 0.996317i \(-0.527328\pi\)
−0.0857467 + 0.996317i \(0.527328\pi\)
\(68\) 2.38919 0.289731
\(69\) 5.75877 0.693274
\(70\) 10.1284 1.21057
\(71\) −6.36959 −0.755931 −0.377965 0.925820i \(-0.623376\pi\)
−0.377965 + 0.925820i \(0.623376\pi\)
\(72\) 0.532089 0.0627073
\(73\) −4.55943 −0.533641 −0.266820 0.963746i \(-0.585973\pi\)
−0.266820 + 0.963746i \(0.585973\pi\)
\(74\) 4.82295 0.560656
\(75\) 1.87939 0.217013
\(76\) 0 0
\(77\) −7.14796 −0.814585
\(78\) 2.45336 0.277789
\(79\) −2.24123 −0.252158 −0.126079 0.992020i \(-0.540239\pi\)
−0.126079 + 0.992020i \(0.540239\pi\)
\(80\) 2.00000 0.223607
\(81\) −10.3131 −1.14591
\(82\) −1.53209 −0.169191
\(83\) −3.98545 −0.437460 −0.218730 0.975785i \(-0.570191\pi\)
−0.218730 + 0.975785i \(0.570191\pi\)
\(84\) −9.51754 −1.03845
\(85\) 4.77837 0.518287
\(86\) −0.758770 −0.0818203
\(87\) −15.8871 −1.70328
\(88\) −1.41147 −0.150464
\(89\) −10.6459 −1.12846 −0.564231 0.825617i \(-0.690827\pi\)
−0.564231 + 0.825617i \(0.690827\pi\)
\(90\) 1.06418 0.112174
\(91\) −6.61081 −0.693002
\(92\) −3.06418 −0.319463
\(93\) −0.694593 −0.0720259
\(94\) −10.2121 −1.05330
\(95\) 0 0
\(96\) −1.87939 −0.191814
\(97\) −1.53209 −0.155560 −0.0777800 0.996971i \(-0.524783\pi\)
−0.0777800 + 0.996971i \(0.524783\pi\)
\(98\) 18.6459 1.88352
\(99\) −0.751030 −0.0754813
\(100\) −1.00000 −0.100000
\(101\) 8.82295 0.877916 0.438958 0.898508i \(-0.355348\pi\)
0.438958 + 0.898508i \(0.355348\pi\)
\(102\) −4.49020 −0.444596
\(103\) −7.14796 −0.704309 −0.352155 0.935942i \(-0.614551\pi\)
−0.352155 + 0.935942i \(0.614551\pi\)
\(104\) −1.30541 −0.128006
\(105\) −19.0351 −1.85763
\(106\) 1.67499 0.162690
\(107\) −9.36959 −0.905792 −0.452896 0.891563i \(-0.649609\pi\)
−0.452896 + 0.891563i \(0.649609\pi\)
\(108\) 4.63816 0.446307
\(109\) 11.0642 1.05976 0.529878 0.848074i \(-0.322238\pi\)
0.529878 + 0.848074i \(0.322238\pi\)
\(110\) −2.82295 −0.269158
\(111\) −9.06418 −0.860334
\(112\) 5.06418 0.478520
\(113\) −13.2986 −1.25103 −0.625514 0.780213i \(-0.715110\pi\)
−0.625514 + 0.780213i \(0.715110\pi\)
\(114\) 0 0
\(115\) −6.12836 −0.571472
\(116\) 8.45336 0.784875
\(117\) −0.694593 −0.0642151
\(118\) 0.716881 0.0659943
\(119\) 12.0993 1.10914
\(120\) −3.75877 −0.343127
\(121\) −9.00774 −0.818886
\(122\) 9.75877 0.883518
\(123\) 2.87939 0.259625
\(124\) 0.369585 0.0331897
\(125\) −12.0000 −1.07331
\(126\) 2.69459 0.240053
\(127\) −5.71419 −0.507053 −0.253526 0.967328i \(-0.581590\pi\)
−0.253526 + 0.967328i \(0.581590\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.42602 0.125554
\(130\) −2.61081 −0.228984
\(131\) −9.87939 −0.863166 −0.431583 0.902073i \(-0.642045\pi\)
−0.431583 + 0.902073i \(0.642045\pi\)
\(132\) 2.65270 0.230888
\(133\) 0 0
\(134\) −1.40373 −0.121264
\(135\) 9.27631 0.798378
\(136\) 2.38919 0.204871
\(137\) 5.39693 0.461091 0.230545 0.973062i \(-0.425949\pi\)
0.230545 + 0.973062i \(0.425949\pi\)
\(138\) 5.75877 0.490219
\(139\) −2.33956 −0.198439 −0.0992193 0.995066i \(-0.531635\pi\)
−0.0992193 + 0.995066i \(0.531635\pi\)
\(140\) 10.1284 0.856002
\(141\) 19.1925 1.61630
\(142\) −6.36959 −0.534524
\(143\) 1.84255 0.154082
\(144\) 0.532089 0.0443407
\(145\) 16.9067 1.40403
\(146\) −4.55943 −0.377341
\(147\) −35.0428 −2.89028
\(148\) 4.82295 0.396444
\(149\) −14.3696 −1.17720 −0.588601 0.808424i \(-0.700321\pi\)
−0.588601 + 0.808424i \(0.700321\pi\)
\(150\) 1.87939 0.153451
\(151\) −20.8384 −1.69581 −0.847904 0.530150i \(-0.822135\pi\)
−0.847904 + 0.530150i \(0.822135\pi\)
\(152\) 0 0
\(153\) 1.27126 0.102775
\(154\) −7.14796 −0.575999
\(155\) 0.739170 0.0593716
\(156\) 2.45336 0.196426
\(157\) 6.36959 0.508348 0.254174 0.967158i \(-0.418196\pi\)
0.254174 + 0.967158i \(0.418196\pi\)
\(158\) −2.24123 −0.178303
\(159\) −3.14796 −0.249649
\(160\) 2.00000 0.158114
\(161\) −15.5175 −1.22295
\(162\) −10.3131 −0.810277
\(163\) 4.73143 0.370594 0.185297 0.982683i \(-0.440675\pi\)
0.185297 + 0.982683i \(0.440675\pi\)
\(164\) −1.53209 −0.119636
\(165\) 5.30541 0.413025
\(166\) −3.98545 −0.309331
\(167\) 17.2763 1.33688 0.668441 0.743766i \(-0.266962\pi\)
0.668441 + 0.743766i \(0.266962\pi\)
\(168\) −9.51754 −0.734294
\(169\) −11.2959 −0.868916
\(170\) 4.77837 0.366484
\(171\) 0 0
\(172\) −0.758770 −0.0578557
\(173\) 18.8229 1.43108 0.715541 0.698571i \(-0.246180\pi\)
0.715541 + 0.698571i \(0.246180\pi\)
\(174\) −15.8871 −1.20440
\(175\) −5.06418 −0.382816
\(176\) −1.41147 −0.106394
\(177\) −1.34730 −0.101269
\(178\) −10.6459 −0.797944
\(179\) −13.8280 −1.03355 −0.516777 0.856120i \(-0.672868\pi\)
−0.516777 + 0.856120i \(0.672868\pi\)
\(180\) 1.06418 0.0793191
\(181\) 12.2567 0.911034 0.455517 0.890227i \(-0.349454\pi\)
0.455517 + 0.890227i \(0.349454\pi\)
\(182\) −6.61081 −0.490026
\(183\) −18.3405 −1.35577
\(184\) −3.06418 −0.225894
\(185\) 9.64590 0.709180
\(186\) −0.694593 −0.0509300
\(187\) −3.37227 −0.246605
\(188\) −10.2121 −0.744796
\(189\) 23.4884 1.70853
\(190\) 0 0
\(191\) −20.0993 −1.45433 −0.727166 0.686462i \(-0.759163\pi\)
−0.727166 + 0.686462i \(0.759163\pi\)
\(192\) −1.87939 −0.135633
\(193\) 16.6851 1.20102 0.600510 0.799617i \(-0.294964\pi\)
0.600510 + 0.799617i \(0.294964\pi\)
\(194\) −1.53209 −0.109998
\(195\) 4.90673 0.351378
\(196\) 18.6459 1.33185
\(197\) 12.9905 0.925535 0.462768 0.886480i \(-0.346856\pi\)
0.462768 + 0.886480i \(0.346856\pi\)
\(198\) −0.751030 −0.0533734
\(199\) 17.1925 1.21875 0.609373 0.792884i \(-0.291421\pi\)
0.609373 + 0.792884i \(0.291421\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.63816 0.186081
\(202\) 8.82295 0.620780
\(203\) 42.8093 3.00463
\(204\) −4.49020 −0.314377
\(205\) −3.06418 −0.214011
\(206\) −7.14796 −0.498022
\(207\) −1.63041 −0.113322
\(208\) −1.30541 −0.0905137
\(209\) 0 0
\(210\) −19.0351 −1.31355
\(211\) −16.1088 −1.10897 −0.554486 0.832193i \(-0.687085\pi\)
−0.554486 + 0.832193i \(0.687085\pi\)
\(212\) 1.67499 0.115039
\(213\) 11.9709 0.820233
\(214\) −9.36959 −0.640492
\(215\) −1.51754 −0.103495
\(216\) 4.63816 0.315587
\(217\) 1.87164 0.127056
\(218\) 11.0642 0.749361
\(219\) 8.56893 0.579034
\(220\) −2.82295 −0.190323
\(221\) −3.11886 −0.209797
\(222\) −9.06418 −0.608348
\(223\) 4.08378 0.273470 0.136735 0.990608i \(-0.456339\pi\)
0.136735 + 0.990608i \(0.456339\pi\)
\(224\) 5.06418 0.338365
\(225\) −0.532089 −0.0354726
\(226\) −13.2986 −0.884610
\(227\) 13.6604 0.906676 0.453338 0.891339i \(-0.350233\pi\)
0.453338 + 0.891339i \(0.350233\pi\)
\(228\) 0 0
\(229\) −5.22163 −0.345055 −0.172527 0.985005i \(-0.555193\pi\)
−0.172527 + 0.985005i \(0.555193\pi\)
\(230\) −6.12836 −0.404092
\(231\) 13.4338 0.883877
\(232\) 8.45336 0.554990
\(233\) −27.0428 −1.77163 −0.885817 0.464035i \(-0.846401\pi\)
−0.885817 + 0.464035i \(0.846401\pi\)
\(234\) −0.694593 −0.0454069
\(235\) −20.4243 −1.33233
\(236\) 0.716881 0.0466650
\(237\) 4.21213 0.273607
\(238\) 12.0993 0.784279
\(239\) −0.285807 −0.0184873 −0.00924366 0.999957i \(-0.502942\pi\)
−0.00924366 + 0.999957i \(0.502942\pi\)
\(240\) −3.75877 −0.242628
\(241\) 3.10101 0.199754 0.0998769 0.995000i \(-0.468155\pi\)
0.0998769 + 0.995000i \(0.468155\pi\)
\(242\) −9.00774 −0.579040
\(243\) 5.46791 0.350767
\(244\) 9.75877 0.624741
\(245\) 37.2918 2.38249
\(246\) 2.87939 0.183583
\(247\) 0 0
\(248\) 0.369585 0.0234687
\(249\) 7.49020 0.474672
\(250\) −12.0000 −0.758947
\(251\) −12.6578 −0.798950 −0.399475 0.916744i \(-0.630808\pi\)
−0.399475 + 0.916744i \(0.630808\pi\)
\(252\) 2.69459 0.169743
\(253\) 4.32501 0.271911
\(254\) −5.71419 −0.358540
\(255\) −8.98040 −0.562374
\(256\) 1.00000 0.0625000
\(257\) −30.3928 −1.89585 −0.947926 0.318492i \(-0.896824\pi\)
−0.947926 + 0.318492i \(0.896824\pi\)
\(258\) 1.42602 0.0887803
\(259\) 24.4243 1.51765
\(260\) −2.61081 −0.161916
\(261\) 4.49794 0.278416
\(262\) −9.87939 −0.610350
\(263\) 21.9026 1.35057 0.675286 0.737556i \(-0.264020\pi\)
0.675286 + 0.737556i \(0.264020\pi\)
\(264\) 2.65270 0.163263
\(265\) 3.34998 0.205788
\(266\) 0 0
\(267\) 20.0077 1.22445
\(268\) −1.40373 −0.0857467
\(269\) −1.43376 −0.0874181 −0.0437090 0.999044i \(-0.513917\pi\)
−0.0437090 + 0.999044i \(0.513917\pi\)
\(270\) 9.27631 0.564538
\(271\) 16.1729 0.982436 0.491218 0.871037i \(-0.336552\pi\)
0.491218 + 0.871037i \(0.336552\pi\)
\(272\) 2.38919 0.144866
\(273\) 12.4243 0.751951
\(274\) 5.39693 0.326040
\(275\) 1.41147 0.0851151
\(276\) 5.75877 0.346637
\(277\) 11.1088 0.667460 0.333730 0.942669i \(-0.391693\pi\)
0.333730 + 0.942669i \(0.391693\pi\)
\(278\) −2.33956 −0.140317
\(279\) 0.196652 0.0117733
\(280\) 10.1284 0.605285
\(281\) 4.24897 0.253472 0.126736 0.991936i \(-0.459550\pi\)
0.126736 + 0.991936i \(0.459550\pi\)
\(282\) 19.1925 1.14290
\(283\) −5.45605 −0.324329 −0.162164 0.986764i \(-0.551847\pi\)
−0.162164 + 0.986764i \(0.551847\pi\)
\(284\) −6.36959 −0.377965
\(285\) 0 0
\(286\) 1.84255 0.108952
\(287\) −7.75877 −0.457986
\(288\) 0.532089 0.0313536
\(289\) −11.2918 −0.664223
\(290\) 16.9067 0.992797
\(291\) 2.87939 0.168793
\(292\) −4.55943 −0.266820
\(293\) 17.8135 1.04067 0.520337 0.853961i \(-0.325807\pi\)
0.520337 + 0.853961i \(0.325807\pi\)
\(294\) −35.0428 −2.04374
\(295\) 1.43376 0.0834769
\(296\) 4.82295 0.280328
\(297\) −6.54664 −0.379874
\(298\) −14.3696 −0.832408
\(299\) 4.00000 0.231326
\(300\) 1.87939 0.108506
\(301\) −3.84255 −0.221481
\(302\) −20.8384 −1.19912
\(303\) −16.5817 −0.952595
\(304\) 0 0
\(305\) 19.5175 1.11757
\(306\) 1.27126 0.0726730
\(307\) −28.6587 −1.63564 −0.817819 0.575476i \(-0.804817\pi\)
−0.817819 + 0.575476i \(0.804817\pi\)
\(308\) −7.14796 −0.407293
\(309\) 13.4338 0.764220
\(310\) 0.739170 0.0419820
\(311\) 15.8135 0.896699 0.448349 0.893858i \(-0.352012\pi\)
0.448349 + 0.893858i \(0.352012\pi\)
\(312\) 2.45336 0.138894
\(313\) 13.1402 0.742729 0.371364 0.928487i \(-0.378890\pi\)
0.371364 + 0.928487i \(0.378890\pi\)
\(314\) 6.36959 0.359456
\(315\) 5.38919 0.303646
\(316\) −2.24123 −0.126079
\(317\) −21.4047 −1.20221 −0.601103 0.799172i \(-0.705272\pi\)
−0.601103 + 0.799172i \(0.705272\pi\)
\(318\) −3.14796 −0.176529
\(319\) −11.9317 −0.668047
\(320\) 2.00000 0.111803
\(321\) 17.6091 0.982842
\(322\) −15.5175 −0.864759
\(323\) 0 0
\(324\) −10.3131 −0.572953
\(325\) 1.30541 0.0724110
\(326\) 4.73143 0.262050
\(327\) −20.7939 −1.14990
\(328\) −1.53209 −0.0845955
\(329\) −51.7161 −2.85120
\(330\) 5.30541 0.292053
\(331\) −25.3979 −1.39599 −0.697996 0.716101i \(-0.745925\pi\)
−0.697996 + 0.716101i \(0.745925\pi\)
\(332\) −3.98545 −0.218730
\(333\) 2.56624 0.140629
\(334\) 17.2763 0.945318
\(335\) −2.80747 −0.153388
\(336\) −9.51754 −0.519224
\(337\) 26.3773 1.43686 0.718432 0.695597i \(-0.244860\pi\)
0.718432 + 0.695597i \(0.244860\pi\)
\(338\) −11.2959 −0.614417
\(339\) 24.9932 1.35744
\(340\) 4.77837 0.259144
\(341\) −0.521660 −0.0282495
\(342\) 0 0
\(343\) 58.9769 3.18445
\(344\) −0.758770 −0.0409102
\(345\) 11.5175 0.620084
\(346\) 18.8229 1.01193
\(347\) −25.6313 −1.37596 −0.687981 0.725728i \(-0.741503\pi\)
−0.687981 + 0.725728i \(0.741503\pi\)
\(348\) −15.8871 −0.851639
\(349\) −5.84255 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(350\) −5.06418 −0.270692
\(351\) −6.05468 −0.323175
\(352\) −1.41147 −0.0752318
\(353\) −22.4097 −1.19275 −0.596375 0.802706i \(-0.703393\pi\)
−0.596375 + 0.802706i \(0.703393\pi\)
\(354\) −1.34730 −0.0716080
\(355\) −12.7392 −0.676125
\(356\) −10.6459 −0.564231
\(357\) −22.7392 −1.20348
\(358\) −13.8280 −0.730833
\(359\) 9.61680 0.507555 0.253778 0.967263i \(-0.418327\pi\)
0.253778 + 0.967263i \(0.418327\pi\)
\(360\) 1.06418 0.0560871
\(361\) 0 0
\(362\) 12.2567 0.644198
\(363\) 16.9290 0.888543
\(364\) −6.61081 −0.346501
\(365\) −9.11886 −0.477303
\(366\) −18.3405 −0.958673
\(367\) 23.3601 1.21939 0.609693 0.792637i \(-0.291293\pi\)
0.609693 + 0.792637i \(0.291293\pi\)
\(368\) −3.06418 −0.159731
\(369\) −0.815207 −0.0424380
\(370\) 9.64590 0.501466
\(371\) 8.48246 0.440387
\(372\) −0.694593 −0.0360130
\(373\) −25.9418 −1.34322 −0.671608 0.740907i \(-0.734396\pi\)
−0.671608 + 0.740907i \(0.734396\pi\)
\(374\) −3.37227 −0.174376
\(375\) 22.5526 1.16461
\(376\) −10.2121 −0.526651
\(377\) −11.0351 −0.568336
\(378\) 23.4884 1.20812
\(379\) 9.47834 0.486870 0.243435 0.969917i \(-0.421726\pi\)
0.243435 + 0.969917i \(0.421726\pi\)
\(380\) 0 0
\(381\) 10.7392 0.550184
\(382\) −20.0993 −1.02837
\(383\) −13.4884 −0.689227 −0.344614 0.938745i \(-0.611990\pi\)
−0.344614 + 0.938745i \(0.611990\pi\)
\(384\) −1.87939 −0.0959070
\(385\) −14.2959 −0.728587
\(386\) 16.6851 0.849249
\(387\) −0.403733 −0.0205229
\(388\) −1.53209 −0.0777800
\(389\) −25.1480 −1.27505 −0.637526 0.770429i \(-0.720042\pi\)
−0.637526 + 0.770429i \(0.720042\pi\)
\(390\) 4.90673 0.248462
\(391\) −7.32089 −0.370233
\(392\) 18.6459 0.941760
\(393\) 18.5672 0.936590
\(394\) 12.9905 0.654452
\(395\) −4.48246 −0.225537
\(396\) −0.751030 −0.0377407
\(397\) 6.56624 0.329550 0.164775 0.986331i \(-0.447310\pi\)
0.164775 + 0.986331i \(0.447310\pi\)
\(398\) 17.1925 0.861784
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −29.5107 −1.47370 −0.736848 0.676059i \(-0.763687\pi\)
−0.736848 + 0.676059i \(0.763687\pi\)
\(402\) 2.63816 0.131579
\(403\) −0.482459 −0.0240330
\(404\) 8.82295 0.438958
\(405\) −20.6263 −1.02493
\(406\) 42.8093 2.12459
\(407\) −6.80747 −0.337434
\(408\) −4.49020 −0.222298
\(409\) 21.1310 1.04486 0.522431 0.852681i \(-0.325025\pi\)
0.522431 + 0.852681i \(0.325025\pi\)
\(410\) −3.06418 −0.151329
\(411\) −10.1429 −0.500313
\(412\) −7.14796 −0.352155
\(413\) 3.63041 0.178641
\(414\) −1.63041 −0.0801305
\(415\) −7.97090 −0.391276
\(416\) −1.30541 −0.0640029
\(417\) 4.39693 0.215318
\(418\) 0 0
\(419\) −27.8830 −1.36217 −0.681087 0.732202i \(-0.738492\pi\)
−0.681087 + 0.732202i \(0.738492\pi\)
\(420\) −19.0351 −0.928817
\(421\) −0.0445774 −0.00217257 −0.00108629 0.999999i \(-0.500346\pi\)
−0.00108629 + 0.999999i \(0.500346\pi\)
\(422\) −16.1088 −0.784162
\(423\) −5.43376 −0.264199
\(424\) 1.67499 0.0813448
\(425\) −2.38919 −0.115893
\(426\) 11.9709 0.579992
\(427\) 49.4201 2.39161
\(428\) −9.36959 −0.452896
\(429\) −3.46286 −0.167188
\(430\) −1.51754 −0.0731823
\(431\) −11.7879 −0.567802 −0.283901 0.958854i \(-0.591629\pi\)
−0.283901 + 0.958854i \(0.591629\pi\)
\(432\) 4.63816 0.223153
\(433\) 27.6459 1.32858 0.664289 0.747476i \(-0.268735\pi\)
0.664289 + 0.747476i \(0.268735\pi\)
\(434\) 1.87164 0.0898418
\(435\) −31.7743 −1.52346
\(436\) 11.0642 0.529878
\(437\) 0 0
\(438\) 8.56893 0.409439
\(439\) 37.4492 1.78735 0.893677 0.448710i \(-0.148116\pi\)
0.893677 + 0.448710i \(0.148116\pi\)
\(440\) −2.82295 −0.134579
\(441\) 9.92127 0.472442
\(442\) −3.11886 −0.148349
\(443\) 20.9881 0.997177 0.498588 0.866839i \(-0.333852\pi\)
0.498588 + 0.866839i \(0.333852\pi\)
\(444\) −9.06418 −0.430167
\(445\) −21.2918 −1.00933
\(446\) 4.08378 0.193372
\(447\) 27.0060 1.27734
\(448\) 5.06418 0.239260
\(449\) −21.8949 −1.03328 −0.516641 0.856202i \(-0.672818\pi\)
−0.516641 + 0.856202i \(0.672818\pi\)
\(450\) −0.532089 −0.0250829
\(451\) 2.16250 0.101828
\(452\) −13.2986 −0.625514
\(453\) 39.1634 1.84006
\(454\) 13.6604 0.641116
\(455\) −13.2216 −0.619840
\(456\) 0 0
\(457\) −13.0496 −0.610436 −0.305218 0.952283i \(-0.598729\pi\)
−0.305218 + 0.952283i \(0.598729\pi\)
\(458\) −5.22163 −0.243991
\(459\) 11.0814 0.517236
\(460\) −6.12836 −0.285736
\(461\) 4.40879 0.205338 0.102669 0.994716i \(-0.467262\pi\)
0.102669 + 0.994716i \(0.467262\pi\)
\(462\) 13.4338 0.624995
\(463\) 26.6655 1.23925 0.619625 0.784898i \(-0.287285\pi\)
0.619625 + 0.784898i \(0.287285\pi\)
\(464\) 8.45336 0.392438
\(465\) −1.38919 −0.0644219
\(466\) −27.0428 −1.25273
\(467\) 30.1138 1.39350 0.696750 0.717314i \(-0.254629\pi\)
0.696750 + 0.717314i \(0.254629\pi\)
\(468\) −0.694593 −0.0321076
\(469\) −7.10876 −0.328252
\(470\) −20.4243 −0.942101
\(471\) −11.9709 −0.551590
\(472\) 0.716881 0.0329971
\(473\) 1.07098 0.0492439
\(474\) 4.21213 0.193470
\(475\) 0 0
\(476\) 12.0993 0.554569
\(477\) 0.891245 0.0408073
\(478\) −0.285807 −0.0130725
\(479\) 8.16756 0.373185 0.186593 0.982437i \(-0.440256\pi\)
0.186593 + 0.982437i \(0.440256\pi\)
\(480\) −3.75877 −0.171564
\(481\) −6.29591 −0.287069
\(482\) 3.10101 0.141247
\(483\) 29.1634 1.32698
\(484\) −9.00774 −0.409443
\(485\) −3.06418 −0.139137
\(486\) 5.46791 0.248029
\(487\) 26.5871 1.20478 0.602388 0.798203i \(-0.294216\pi\)
0.602388 + 0.798203i \(0.294216\pi\)
\(488\) 9.75877 0.441759
\(489\) −8.89218 −0.402118
\(490\) 37.2918 1.68467
\(491\) 38.8289 1.75233 0.876163 0.482016i \(-0.160095\pi\)
0.876163 + 0.482016i \(0.160095\pi\)
\(492\) 2.87939 0.129813
\(493\) 20.1967 0.909611
\(494\) 0 0
\(495\) −1.50206 −0.0675125
\(496\) 0.369585 0.0165949
\(497\) −32.2567 −1.44691
\(498\) 7.49020 0.335644
\(499\) 20.6587 0.924810 0.462405 0.886669i \(-0.346987\pi\)
0.462405 + 0.886669i \(0.346987\pi\)
\(500\) −12.0000 −0.536656
\(501\) −32.4688 −1.45060
\(502\) −12.6578 −0.564943
\(503\) 0.0736733 0.00328493 0.00164246 0.999999i \(-0.499477\pi\)
0.00164246 + 0.999999i \(0.499477\pi\)
\(504\) 2.69459 0.120027
\(505\) 17.6459 0.785232
\(506\) 4.32501 0.192270
\(507\) 21.2294 0.942829
\(508\) −5.71419 −0.253526
\(509\) 15.0196 0.665732 0.332866 0.942974i \(-0.391984\pi\)
0.332866 + 0.942974i \(0.391984\pi\)
\(510\) −8.98040 −0.397659
\(511\) −23.0898 −1.02143
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −30.3928 −1.34057
\(515\) −14.2959 −0.629953
\(516\) 1.42602 0.0627771
\(517\) 14.4142 0.633934
\(518\) 24.4243 1.07314
\(519\) −35.3756 −1.55282
\(520\) −2.61081 −0.114492
\(521\) 45.1712 1.97899 0.989493 0.144583i \(-0.0461842\pi\)
0.989493 + 0.144583i \(0.0461842\pi\)
\(522\) 4.49794 0.196870
\(523\) 0.167556 0.00732672 0.00366336 0.999993i \(-0.498834\pi\)
0.00366336 + 0.999993i \(0.498834\pi\)
\(524\) −9.87939 −0.431583
\(525\) 9.51754 0.415380
\(526\) 21.9026 0.954999
\(527\) 0.883007 0.0384644
\(528\) 2.65270 0.115444
\(529\) −13.6108 −0.591775
\(530\) 3.34998 0.145514
\(531\) 0.381445 0.0165533
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 20.0077 0.865820
\(535\) −18.7392 −0.810165
\(536\) −1.40373 −0.0606320
\(537\) 25.9881 1.12147
\(538\) −1.43376 −0.0618139
\(539\) −26.3182 −1.13361
\(540\) 9.27631 0.399189
\(541\) −0.980400 −0.0421507 −0.0210753 0.999778i \(-0.506709\pi\)
−0.0210753 + 0.999778i \(0.506709\pi\)
\(542\) 16.1729 0.694687
\(543\) −23.0351 −0.988530
\(544\) 2.38919 0.102435
\(545\) 22.1284 0.947875
\(546\) 12.4243 0.531710
\(547\) 28.4047 1.21450 0.607248 0.794512i \(-0.292273\pi\)
0.607248 + 0.794512i \(0.292273\pi\)
\(548\) 5.39693 0.230545
\(549\) 5.19253 0.221612
\(550\) 1.41147 0.0601855
\(551\) 0 0
\(552\) 5.75877 0.245110
\(553\) −11.3500 −0.482650
\(554\) 11.1088 0.471966
\(555\) −18.1284 −0.769506
\(556\) −2.33956 −0.0992193
\(557\) 35.6323 1.50979 0.754894 0.655847i \(-0.227688\pi\)
0.754894 + 0.655847i \(0.227688\pi\)
\(558\) 0.196652 0.00832495
\(559\) 0.990505 0.0418939
\(560\) 10.1284 0.428001
\(561\) 6.33780 0.267582
\(562\) 4.24897 0.179232
\(563\) 8.62773 0.363615 0.181808 0.983334i \(-0.441805\pi\)
0.181808 + 0.983334i \(0.441805\pi\)
\(564\) 19.1925 0.808151
\(565\) −26.5972 −1.11895
\(566\) −5.45605 −0.229335
\(567\) −52.2276 −2.19335
\(568\) −6.36959 −0.267262
\(569\) 22.3310 0.936164 0.468082 0.883685i \(-0.344945\pi\)
0.468082 + 0.883685i \(0.344945\pi\)
\(570\) 0 0
\(571\) −9.56448 −0.400261 −0.200131 0.979769i \(-0.564137\pi\)
−0.200131 + 0.979769i \(0.564137\pi\)
\(572\) 1.84255 0.0770408
\(573\) 37.7743 1.57804
\(574\) −7.75877 −0.323845
\(575\) 3.06418 0.127785
\(576\) 0.532089 0.0221704
\(577\) 22.4757 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(578\) −11.2918 −0.469677
\(579\) −31.3577 −1.30318
\(580\) 16.9067 0.702014
\(581\) −20.1830 −0.837334
\(582\) 2.87939 0.119354
\(583\) −2.36421 −0.0979155
\(584\) −4.55943 −0.188671
\(585\) −1.38919 −0.0574357
\(586\) 17.8135 0.735867
\(587\) 4.14796 0.171204 0.0856022 0.996329i \(-0.472719\pi\)
0.0856022 + 0.996329i \(0.472719\pi\)
\(588\) −35.0428 −1.44514
\(589\) 0 0
\(590\) 1.43376 0.0590271
\(591\) −24.4142 −1.00426
\(592\) 4.82295 0.198222
\(593\) 10.4175 0.427794 0.213897 0.976856i \(-0.431384\pi\)
0.213897 + 0.976856i \(0.431384\pi\)
\(594\) −6.54664 −0.268612
\(595\) 24.1985 0.992043
\(596\) −14.3696 −0.588601
\(597\) −32.3114 −1.32242
\(598\) 4.00000 0.163572
\(599\) −39.6560 −1.62030 −0.810150 0.586222i \(-0.800614\pi\)
−0.810150 + 0.586222i \(0.800614\pi\)
\(600\) 1.87939 0.0767256
\(601\) 23.8648 0.973467 0.486734 0.873551i \(-0.338188\pi\)
0.486734 + 0.873551i \(0.338188\pi\)
\(602\) −3.84255 −0.156611
\(603\) −0.746911 −0.0304166
\(604\) −20.8384 −0.847904
\(605\) −18.0155 −0.732433
\(606\) −16.5817 −0.673586
\(607\) −29.9317 −1.21489 −0.607445 0.794362i \(-0.707806\pi\)
−0.607445 + 0.794362i \(0.707806\pi\)
\(608\) 0 0
\(609\) −80.4552 −3.26021
\(610\) 19.5175 0.790242
\(611\) 13.3310 0.539314
\(612\) 1.27126 0.0513876
\(613\) −35.5776 −1.43697 −0.718483 0.695545i \(-0.755163\pi\)
−0.718483 + 0.695545i \(0.755163\pi\)
\(614\) −28.6587 −1.15657
\(615\) 5.75877 0.232216
\(616\) −7.14796 −0.287999
\(617\) 12.0324 0.484406 0.242203 0.970226i \(-0.422130\pi\)
0.242203 + 0.970226i \(0.422130\pi\)
\(618\) 13.4338 0.540385
\(619\) −17.1129 −0.687824 −0.343912 0.939002i \(-0.611752\pi\)
−0.343912 + 0.939002i \(0.611752\pi\)
\(620\) 0.739170 0.0296858
\(621\) −14.2121 −0.570313
\(622\) 15.8135 0.634062
\(623\) −53.9127 −2.15997
\(624\) 2.45336 0.0982131
\(625\) −19.0000 −0.760000
\(626\) 13.1402 0.525189
\(627\) 0 0
\(628\) 6.36959 0.254174
\(629\) 11.5229 0.459449
\(630\) 5.38919 0.214710
\(631\) 18.3405 0.730123 0.365062 0.930983i \(-0.381048\pi\)
0.365062 + 0.930983i \(0.381048\pi\)
\(632\) −2.24123 −0.0891513
\(633\) 30.2746 1.20331
\(634\) −21.4047 −0.850088
\(635\) −11.4284 −0.453522
\(636\) −3.14796 −0.124825
\(637\) −24.3405 −0.964405
\(638\) −11.9317 −0.472381
\(639\) −3.38919 −0.134074
\(640\) 2.00000 0.0790569
\(641\) −31.1780 −1.23146 −0.615728 0.787959i \(-0.711138\pi\)
−0.615728 + 0.787959i \(0.711138\pi\)
\(642\) 17.6091 0.694974
\(643\) −31.9495 −1.25997 −0.629984 0.776608i \(-0.716938\pi\)
−0.629984 + 0.776608i \(0.716938\pi\)
\(644\) −15.5175 −0.611477
\(645\) 2.85204 0.112299
\(646\) 0 0
\(647\) 2.99588 0.117780 0.0588901 0.998264i \(-0.481244\pi\)
0.0588901 + 0.998264i \(0.481244\pi\)
\(648\) −10.3131 −0.405139
\(649\) −1.01186 −0.0397190
\(650\) 1.30541 0.0512023
\(651\) −3.51754 −0.137863
\(652\) 4.73143 0.185297
\(653\) 0.935822 0.0366216 0.0183108 0.999832i \(-0.494171\pi\)
0.0183108 + 0.999832i \(0.494171\pi\)
\(654\) −20.7939 −0.813104
\(655\) −19.7588 −0.772039
\(656\) −1.53209 −0.0598180
\(657\) −2.42602 −0.0946481
\(658\) −51.7161 −2.01610
\(659\) −12.2371 −0.476690 −0.238345 0.971181i \(-0.576605\pi\)
−0.238345 + 0.971181i \(0.576605\pi\)
\(660\) 5.30541 0.206513
\(661\) −11.9554 −0.465012 −0.232506 0.972595i \(-0.574693\pi\)
−0.232506 + 0.972595i \(0.574693\pi\)
\(662\) −25.3979 −0.987116
\(663\) 5.86154 0.227643
\(664\) −3.98545 −0.154666
\(665\) 0 0
\(666\) 2.56624 0.0994397
\(667\) −25.9026 −1.00295
\(668\) 17.2763 0.668441
\(669\) −7.67499 −0.296732
\(670\) −2.80747 −0.108462
\(671\) −13.7743 −0.531749
\(672\) −9.51754 −0.367147
\(673\) 44.8634 1.72936 0.864679 0.502325i \(-0.167522\pi\)
0.864679 + 0.502325i \(0.167522\pi\)
\(674\) 26.3773 1.01602
\(675\) −4.63816 −0.178523
\(676\) −11.2959 −0.434458
\(677\) 0.945927 0.0363549 0.0181775 0.999835i \(-0.494214\pi\)
0.0181775 + 0.999835i \(0.494214\pi\)
\(678\) 24.9932 0.959858
\(679\) −7.75877 −0.297754
\(680\) 4.77837 0.183242
\(681\) −25.6732 −0.983801
\(682\) −0.521660 −0.0199754
\(683\) −5.92221 −0.226607 −0.113303 0.993560i \(-0.536143\pi\)
−0.113303 + 0.993560i \(0.536143\pi\)
\(684\) 0 0
\(685\) 10.7939 0.412412
\(686\) 58.9769 2.25175
\(687\) 9.81345 0.374407
\(688\) −0.758770 −0.0289279
\(689\) −2.18655 −0.0833008
\(690\) 11.5175 0.438465
\(691\) 0.206148 0.00784222 0.00392111 0.999992i \(-0.498752\pi\)
0.00392111 + 0.999992i \(0.498752\pi\)
\(692\) 18.8229 0.715541
\(693\) −3.80335 −0.144477
\(694\) −25.6313 −0.972953
\(695\) −4.67911 −0.177489
\(696\) −15.8871 −0.602200
\(697\) −3.66044 −0.138649
\(698\) −5.84255 −0.221144
\(699\) 50.8239 1.92234
\(700\) −5.06418 −0.191408
\(701\) 4.36959 0.165037 0.0825185 0.996590i \(-0.473704\pi\)
0.0825185 + 0.996590i \(0.473704\pi\)
\(702\) −6.05468 −0.228519
\(703\) 0 0
\(704\) −1.41147 −0.0531969
\(705\) 38.3851 1.44567
\(706\) −22.4097 −0.843401
\(707\) 44.6810 1.68040
\(708\) −1.34730 −0.0506345
\(709\) 8.75465 0.328788 0.164394 0.986395i \(-0.447433\pi\)
0.164394 + 0.986395i \(0.447433\pi\)
\(710\) −12.7392 −0.478093
\(711\) −1.19253 −0.0447235
\(712\) −10.6459 −0.398972
\(713\) −1.13247 −0.0424115
\(714\) −22.7392 −0.850992
\(715\) 3.68510 0.137815
\(716\) −13.8280 −0.516777
\(717\) 0.537141 0.0200599
\(718\) 9.61680 0.358896
\(719\) −27.6067 −1.02956 −0.514778 0.857324i \(-0.672126\pi\)
−0.514778 + 0.857324i \(0.672126\pi\)
\(720\) 1.06418 0.0396596
\(721\) −36.1985 −1.34810
\(722\) 0 0
\(723\) −5.82800 −0.216746
\(724\) 12.2567 0.455517
\(725\) −8.45336 −0.313950
\(726\) 16.9290 0.628295
\(727\) 28.6364 1.06207 0.531033 0.847351i \(-0.321804\pi\)
0.531033 + 0.847351i \(0.321804\pi\)
\(728\) −6.61081 −0.245013
\(729\) 20.6631 0.765301
\(730\) −9.11886 −0.337504
\(731\) −1.81284 −0.0670504
\(732\) −18.3405 −0.677884
\(733\) 36.9614 1.36520 0.682600 0.730792i \(-0.260849\pi\)
0.682600 + 0.730792i \(0.260849\pi\)
\(734\) 23.3601 0.862237
\(735\) −70.0856 −2.58515
\(736\) −3.06418 −0.112947
\(737\) 1.98133 0.0729833
\(738\) −0.815207 −0.0300082
\(739\) 46.0265 1.69311 0.846556 0.532299i \(-0.178672\pi\)
0.846556 + 0.532299i \(0.178672\pi\)
\(740\) 9.64590 0.354590
\(741\) 0 0
\(742\) 8.48246 0.311401
\(743\) 26.4107 0.968913 0.484456 0.874815i \(-0.339017\pi\)
0.484456 + 0.874815i \(0.339017\pi\)
\(744\) −0.694593 −0.0254650
\(745\) −28.7392 −1.05292
\(746\) −25.9418 −0.949797
\(747\) −2.12061 −0.0775892
\(748\) −3.37227 −0.123303
\(749\) −47.4492 −1.73376
\(750\) 22.5526 0.823505
\(751\) −27.1771 −0.991705 −0.495852 0.868407i \(-0.665144\pi\)
−0.495852 + 0.868407i \(0.665144\pi\)
\(752\) −10.2121 −0.372398
\(753\) 23.7888 0.866912
\(754\) −11.0351 −0.401874
\(755\) −41.6769 −1.51678
\(756\) 23.4884 0.854266
\(757\) 19.4047 0.705275 0.352637 0.935760i \(-0.385285\pi\)
0.352637 + 0.935760i \(0.385285\pi\)
\(758\) 9.47834 0.344269
\(759\) −8.12836 −0.295041
\(760\) 0 0
\(761\) 40.2645 1.45959 0.729793 0.683669i \(-0.239617\pi\)
0.729793 + 0.683669i \(0.239617\pi\)
\(762\) 10.7392 0.389039
\(763\) 56.0310 2.02846
\(764\) −20.0993 −0.727166
\(765\) 2.54252 0.0919249
\(766\) −13.4884 −0.487357
\(767\) −0.935822 −0.0337906
\(768\) −1.87939 −0.0678165
\(769\) −38.9418 −1.40428 −0.702139 0.712040i \(-0.747771\pi\)
−0.702139 + 0.712040i \(0.747771\pi\)
\(770\) −14.2959 −0.515189
\(771\) 57.1198 2.05712
\(772\) 16.6851 0.600510
\(773\) −6.76289 −0.243244 −0.121622 0.992576i \(-0.538810\pi\)
−0.121622 + 0.992576i \(0.538810\pi\)
\(774\) −0.403733 −0.0145119
\(775\) −0.369585 −0.0132759
\(776\) −1.53209 −0.0549988
\(777\) −45.9026 −1.64675
\(778\) −25.1480 −0.901598
\(779\) 0 0
\(780\) 4.90673 0.175689
\(781\) 8.99050 0.321706
\(782\) −7.32089 −0.261794
\(783\) 39.2080 1.40118
\(784\) 18.6459 0.665925
\(785\) 12.7392 0.454680
\(786\) 18.5672 0.662269
\(787\) 5.80478 0.206918 0.103459 0.994634i \(-0.467009\pi\)
0.103459 + 0.994634i \(0.467009\pi\)
\(788\) 12.9905 0.462768
\(789\) −41.1634 −1.46546
\(790\) −4.48246 −0.159479
\(791\) −67.3465 −2.39456
\(792\) −0.751030 −0.0266867
\(793\) −12.7392 −0.452381
\(794\) 6.56624 0.233027
\(795\) −6.29591 −0.223293
\(796\) 17.1925 0.609373
\(797\) −39.2181 −1.38918 −0.694589 0.719407i \(-0.744414\pi\)
−0.694589 + 0.719407i \(0.744414\pi\)
\(798\) 0 0
\(799\) −24.3987 −0.863163
\(800\) −1.00000 −0.0353553
\(801\) −5.66456 −0.200148
\(802\) −29.5107 −1.04206
\(803\) 6.43552 0.227104
\(804\) 2.63816 0.0930406
\(805\) −31.0351 −1.09384
\(806\) −0.482459 −0.0169939
\(807\) 2.69459 0.0948542
\(808\) 8.82295 0.310390
\(809\) 7.00269 0.246201 0.123101 0.992394i \(-0.460716\pi\)
0.123101 + 0.992394i \(0.460716\pi\)
\(810\) −20.6263 −0.724734
\(811\) 43.6851 1.53399 0.766996 0.641652i \(-0.221751\pi\)
0.766996 + 0.641652i \(0.221751\pi\)
\(812\) 42.8093 1.50231
\(813\) −30.3952 −1.06601
\(814\) −6.80747 −0.238602
\(815\) 9.46286 0.331469
\(816\) −4.49020 −0.157188
\(817\) 0 0
\(818\) 21.1310 0.738830
\(819\) −3.51754 −0.122913
\(820\) −3.06418 −0.107006
\(821\) 12.3851 0.432242 0.216121 0.976367i \(-0.430659\pi\)
0.216121 + 0.976367i \(0.430659\pi\)
\(822\) −10.1429 −0.353774
\(823\) −12.8283 −0.447167 −0.223584 0.974685i \(-0.571776\pi\)
−0.223584 + 0.974685i \(0.571776\pi\)
\(824\) −7.14796 −0.249011
\(825\) −2.65270 −0.0923553
\(826\) 3.63041 0.126318
\(827\) 25.0966 0.872693 0.436347 0.899779i \(-0.356272\pi\)
0.436347 + 0.899779i \(0.356272\pi\)
\(828\) −1.63041 −0.0566608
\(829\) 28.3269 0.983833 0.491917 0.870642i \(-0.336297\pi\)
0.491917 + 0.870642i \(0.336297\pi\)
\(830\) −7.97090 −0.276674
\(831\) −20.8776 −0.724237
\(832\) −1.30541 −0.0452569
\(833\) 44.5485 1.54351
\(834\) 4.39693 0.152253
\(835\) 34.5526 1.19574
\(836\) 0 0
\(837\) 1.71419 0.0592512
\(838\) −27.8830 −0.963203
\(839\) −30.3304 −1.04712 −0.523561 0.851988i \(-0.675397\pi\)
−0.523561 + 0.851988i \(0.675397\pi\)
\(840\) −19.0351 −0.656773
\(841\) 42.4593 1.46412
\(842\) −0.0445774 −0.00153624
\(843\) −7.98545 −0.275034
\(844\) −16.1088 −0.554486
\(845\) −22.5918 −0.777182
\(846\) −5.43376 −0.186817
\(847\) −45.6168 −1.56741
\(848\) 1.67499 0.0575195
\(849\) 10.2540 0.351917
\(850\) −2.38919 −0.0819484
\(851\) −14.7784 −0.506596
\(852\) 11.9709 0.410116
\(853\) −43.6323 −1.49394 −0.746970 0.664857i \(-0.768492\pi\)
−0.746970 + 0.664857i \(0.768492\pi\)
\(854\) 49.4201 1.69112
\(855\) 0 0
\(856\) −9.36959 −0.320246
\(857\) −31.5098 −1.07635 −0.538177 0.842832i \(-0.680887\pi\)
−0.538177 + 0.842832i \(0.680887\pi\)
\(858\) −3.46286 −0.118220
\(859\) −18.6979 −0.637964 −0.318982 0.947761i \(-0.603341\pi\)
−0.318982 + 0.947761i \(0.603341\pi\)
\(860\) −1.51754 −0.0517477
\(861\) 14.5817 0.496944
\(862\) −11.7879 −0.401496
\(863\) 10.6263 0.361723 0.180862 0.983509i \(-0.442111\pi\)
0.180862 + 0.983509i \(0.442111\pi\)
\(864\) 4.63816 0.157793
\(865\) 37.6459 1.28000
\(866\) 27.6459 0.939446
\(867\) 21.2216 0.720724
\(868\) 1.87164 0.0635278
\(869\) 3.16344 0.107312
\(870\) −31.7743 −1.07725
\(871\) 1.83244 0.0620900
\(872\) 11.0642 0.374680
\(873\) −0.815207 −0.0275906
\(874\) 0 0
\(875\) −60.7701 −2.05441
\(876\) 8.56893 0.289517
\(877\) 44.8985 1.51611 0.758057 0.652188i \(-0.226149\pi\)
0.758057 + 0.652188i \(0.226149\pi\)
\(878\) 37.4492 1.26385
\(879\) −33.4783 −1.12920
\(880\) −2.82295 −0.0951616
\(881\) −18.0164 −0.606988 −0.303494 0.952833i \(-0.598153\pi\)
−0.303494 + 0.952833i \(0.598153\pi\)
\(882\) 9.92127 0.334067
\(883\) −46.5981 −1.56815 −0.784076 0.620665i \(-0.786863\pi\)
−0.784076 + 0.620665i \(0.786863\pi\)
\(884\) −3.11886 −0.104899
\(885\) −2.69459 −0.0905777
\(886\) 20.9881 0.705110
\(887\) 7.07966 0.237712 0.118856 0.992912i \(-0.462077\pi\)
0.118856 + 0.992912i \(0.462077\pi\)
\(888\) −9.06418 −0.304174
\(889\) −28.9377 −0.970539
\(890\) −21.2918 −0.713703
\(891\) 14.5567 0.487669
\(892\) 4.08378 0.136735
\(893\) 0 0
\(894\) 27.0060 0.903215
\(895\) −27.6560 −0.924438
\(896\) 5.06418 0.169182
\(897\) −7.51754 −0.251003
\(898\) −21.8949 −0.730641
\(899\) 3.12424 0.104199
\(900\) −0.532089 −0.0177363
\(901\) 4.00187 0.133322
\(902\) 2.16250 0.0720035
\(903\) 7.22163 0.240321
\(904\) −13.2986 −0.442305
\(905\) 24.5134 0.814854
\(906\) 39.1634 1.30112
\(907\) 15.1156 0.501904 0.250952 0.968000i \(-0.419256\pi\)
0.250952 + 0.968000i \(0.419256\pi\)
\(908\) 13.6604 0.453338
\(909\) 4.69459 0.155710
\(910\) −13.2216 −0.438293
\(911\) −12.8366 −0.425294 −0.212647 0.977129i \(-0.568208\pi\)
−0.212647 + 0.977129i \(0.568208\pi\)
\(912\) 0 0
\(913\) 5.62536 0.186172
\(914\) −13.0496 −0.431643
\(915\) −36.6810 −1.21264
\(916\) −5.22163 −0.172527
\(917\) −50.0310 −1.65217
\(918\) 11.0814 0.365741
\(919\) −20.6791 −0.682141 −0.341070 0.940038i \(-0.610789\pi\)
−0.341070 + 0.940038i \(0.610789\pi\)
\(920\) −6.12836 −0.202046
\(921\) 53.8607 1.77477
\(922\) 4.40879 0.145196
\(923\) 8.31490 0.273688
\(924\) 13.4338 0.441938
\(925\) −4.82295 −0.158578
\(926\) 26.6655 0.876283
\(927\) −3.80335 −0.124918
\(928\) 8.45336 0.277495
\(929\) −18.2499 −0.598760 −0.299380 0.954134i \(-0.596780\pi\)
−0.299380 + 0.954134i \(0.596780\pi\)
\(930\) −1.38919 −0.0455532
\(931\) 0 0
\(932\) −27.0428 −0.885817
\(933\) −29.7196 −0.972975
\(934\) 30.1138 0.985354
\(935\) −6.74455 −0.220570
\(936\) −0.694593 −0.0227035
\(937\) −4.74691 −0.155075 −0.0775374 0.996989i \(-0.524706\pi\)
−0.0775374 + 0.996989i \(0.524706\pi\)
\(938\) −7.10876 −0.232109
\(939\) −24.6955 −0.805908
\(940\) −20.4243 −0.666166
\(941\) 47.0215 1.53286 0.766428 0.642330i \(-0.222032\pi\)
0.766428 + 0.642330i \(0.222032\pi\)
\(942\) −11.9709 −0.390033
\(943\) 4.69459 0.152877
\(944\) 0.716881 0.0233325
\(945\) 46.9769 1.52816
\(946\) 1.07098 0.0348207
\(947\) 36.9959 1.20220 0.601102 0.799172i \(-0.294728\pi\)
0.601102 + 0.799172i \(0.294728\pi\)
\(948\) 4.21213 0.136804
\(949\) 5.95191 0.193207
\(950\) 0 0
\(951\) 40.2276 1.30447
\(952\) 12.0993 0.392139
\(953\) −37.7093 −1.22152 −0.610761 0.791815i \(-0.709136\pi\)
−0.610761 + 0.791815i \(0.709136\pi\)
\(954\) 0.891245 0.0288551
\(955\) −40.1985 −1.30079
\(956\) −0.285807 −0.00924366
\(957\) 22.4243 0.724874
\(958\) 8.16756 0.263882
\(959\) 27.3310 0.882564
\(960\) −3.75877 −0.121314
\(961\) −30.8634 −0.995594
\(962\) −6.29591 −0.202988
\(963\) −4.98545 −0.160654
\(964\) 3.10101 0.0998769
\(965\) 33.3702 1.07422
\(966\) 29.1634 0.938318
\(967\) 40.5134 1.30282 0.651412 0.758724i \(-0.274177\pi\)
0.651412 + 0.758724i \(0.274177\pi\)
\(968\) −9.00774 −0.289520
\(969\) 0 0
\(970\) −3.06418 −0.0983848
\(971\) −14.0779 −0.451781 −0.225891 0.974153i \(-0.572529\pi\)
−0.225891 + 0.974153i \(0.572529\pi\)
\(972\) 5.46791 0.175383
\(973\) −11.8479 −0.379827
\(974\) 26.5871 0.851905
\(975\) −2.45336 −0.0785705
\(976\) 9.75877 0.312371
\(977\) −35.2057 −1.12633 −0.563164 0.826345i \(-0.690416\pi\)
−0.563164 + 0.826345i \(0.690416\pi\)
\(978\) −8.89218 −0.284341
\(979\) 15.0264 0.480246
\(980\) 37.2918 1.19124
\(981\) 5.88713 0.187961
\(982\) 38.8289 1.23908
\(983\) 22.4397 0.715717 0.357858 0.933776i \(-0.383507\pi\)
0.357858 + 0.933776i \(0.383507\pi\)
\(984\) 2.87939 0.0917914
\(985\) 25.9810 0.827824
\(986\) 20.1967 0.643192
\(987\) 97.1944 3.09373
\(988\) 0 0
\(989\) 2.32501 0.0739309
\(990\) −1.50206 −0.0477386
\(991\) 27.1034 0.860967 0.430484 0.902598i \(-0.358343\pi\)
0.430484 + 0.902598i \(0.358343\pi\)
\(992\) 0.369585 0.0117343
\(993\) 47.7324 1.51474
\(994\) −32.2567 −1.02312
\(995\) 34.3851 1.09008
\(996\) 7.49020 0.237336
\(997\) 27.9472 0.885096 0.442548 0.896745i \(-0.354075\pi\)
0.442548 + 0.896745i \(0.354075\pi\)
\(998\) 20.6587 0.653939
\(999\) 22.3696 0.707742
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 722.2.a.l.1.1 3
3.2 odd 2 6498.2.a.bl.1.3 3
4.3 odd 2 5776.2.a.bn.1.3 3
19.2 odd 18 722.2.e.k.99.1 6
19.3 odd 18 722.2.e.l.389.1 6
19.4 even 9 722.2.e.m.415.1 6
19.5 even 9 722.2.e.m.595.1 6
19.6 even 9 722.2.e.b.245.1 6
19.7 even 3 722.2.c.k.429.3 6
19.8 odd 6 722.2.c.l.653.1 6
19.9 even 9 38.2.e.a.5.1 6
19.10 odd 18 722.2.e.k.423.1 6
19.11 even 3 722.2.c.k.653.3 6
19.12 odd 6 722.2.c.l.429.1 6
19.13 odd 18 722.2.e.l.245.1 6
19.14 odd 18 722.2.e.a.595.1 6
19.15 odd 18 722.2.e.a.415.1 6
19.16 even 9 722.2.e.b.389.1 6
19.17 even 9 38.2.e.a.23.1 yes 6
19.18 odd 2 722.2.a.k.1.3 3
57.17 odd 18 342.2.u.c.289.1 6
57.47 odd 18 342.2.u.c.271.1 6
57.56 even 2 6498.2.a.bq.1.3 3
76.47 odd 18 304.2.u.c.81.1 6
76.55 odd 18 304.2.u.c.289.1 6
76.75 even 2 5776.2.a.bo.1.1 3
95.9 even 18 950.2.l.d.651.1 6
95.17 odd 36 950.2.u.b.99.2 12
95.28 odd 36 950.2.u.b.499.2 12
95.47 odd 36 950.2.u.b.499.1 12
95.74 even 18 950.2.l.d.251.1 6
95.93 odd 36 950.2.u.b.99.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.e.a.5.1 6 19.9 even 9
38.2.e.a.23.1 yes 6 19.17 even 9
304.2.u.c.81.1 6 76.47 odd 18
304.2.u.c.289.1 6 76.55 odd 18
342.2.u.c.271.1 6 57.47 odd 18
342.2.u.c.289.1 6 57.17 odd 18
722.2.a.k.1.3 3 19.18 odd 2
722.2.a.l.1.1 3 1.1 even 1 trivial
722.2.c.k.429.3 6 19.7 even 3
722.2.c.k.653.3 6 19.11 even 3
722.2.c.l.429.1 6 19.12 odd 6
722.2.c.l.653.1 6 19.8 odd 6
722.2.e.a.415.1 6 19.15 odd 18
722.2.e.a.595.1 6 19.14 odd 18
722.2.e.b.245.1 6 19.6 even 9
722.2.e.b.389.1 6 19.16 even 9
722.2.e.k.99.1 6 19.2 odd 18
722.2.e.k.423.1 6 19.10 odd 18
722.2.e.l.245.1 6 19.13 odd 18
722.2.e.l.389.1 6 19.3 odd 18
722.2.e.m.415.1 6 19.4 even 9
722.2.e.m.595.1 6 19.5 even 9
950.2.l.d.251.1 6 95.74 even 18
950.2.l.d.651.1 6 95.9 even 18
950.2.u.b.99.1 12 95.93 odd 36
950.2.u.b.99.2 12 95.17 odd 36
950.2.u.b.499.1 12 95.47 odd 36
950.2.u.b.499.2 12 95.28 odd 36
5776.2.a.bn.1.3 3 4.3 odd 2
5776.2.a.bo.1.1 3 76.75 even 2
6498.2.a.bl.1.3 3 3.2 odd 2
6498.2.a.bq.1.3 3 57.56 even 2