Properties

Label 2009.2.a.n.1.3
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.633117.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.460315\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.460315 q^{2} -0.539685 q^{3} -1.78811 q^{4} -4.10136 q^{5} +0.248425 q^{6} +1.74372 q^{8} -2.70874 q^{9} +O(q^{10})\) \(q-0.460315 q^{2} -0.539685 q^{3} -1.78811 q^{4} -4.10136 q^{5} +0.248425 q^{6} +1.74372 q^{8} -2.70874 q^{9} +1.88791 q^{10} +2.76768 q^{11} +0.965017 q^{12} +4.05697 q^{13} +2.21344 q^{15} +2.77356 q^{16} -5.22447 q^{17} +1.24687 q^{18} +0.109209 q^{19} +7.33368 q^{20} -1.27400 q^{22} +6.08681 q^{23} -0.941062 q^{24} +11.8211 q^{25} -1.86748 q^{26} +3.08092 q^{27} +2.25431 q^{29} -1.01888 q^{30} +1.18073 q^{31} -4.76415 q^{32} -1.49368 q^{33} +2.40490 q^{34} +4.84353 q^{36} -8.95429 q^{37} -0.0502704 q^{38} -2.18949 q^{39} -7.15163 q^{40} +1.00000 q^{41} -7.93974 q^{43} -4.94891 q^{44} +11.1095 q^{45} -2.80185 q^{46} +10.5714 q^{47} -1.49685 q^{48} -5.44143 q^{50} +2.81957 q^{51} -7.25431 q^{52} +6.23100 q^{53} -1.41819 q^{54} -11.3512 q^{55} -0.0589384 q^{57} -1.03769 q^{58} +9.43006 q^{59} -3.95788 q^{60} -6.89732 q^{61} -0.543506 q^{62} -3.35411 q^{64} -16.6391 q^{65} +0.687561 q^{66} -1.35699 q^{67} +9.34193 q^{68} -3.28496 q^{69} -7.90672 q^{71} -4.72329 q^{72} -9.88791 q^{73} +4.12179 q^{74} -6.37969 q^{75} -0.195277 q^{76} +1.00785 q^{78} -12.5668 q^{79} -11.3754 q^{80} +6.46349 q^{81} -0.460315 q^{82} +14.9633 q^{83} +21.4274 q^{85} +3.65478 q^{86} -1.21662 q^{87} +4.82606 q^{88} +0.852931 q^{89} -5.11387 q^{90} -10.8839 q^{92} -0.637221 q^{93} -4.86616 q^{94} -0.447904 q^{95} +2.57115 q^{96} -9.92806 q^{97} -7.49692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.460315 −0.325492 −0.162746 0.986668i \(-0.552035\pi\)
−0.162746 + 0.986668i \(0.552035\pi\)
\(3\) −0.539685 −0.311588 −0.155794 0.987790i \(-0.549794\pi\)
−0.155794 + 0.987790i \(0.549794\pi\)
\(4\) −1.78811 −0.894055
\(5\) −4.10136 −1.83418 −0.917091 0.398678i \(-0.869469\pi\)
−0.917091 + 0.398678i \(0.869469\pi\)
\(6\) 0.248425 0.101419
\(7\) 0 0
\(8\) 1.74372 0.616499
\(9\) −2.70874 −0.902913
\(10\) 1.88791 0.597011
\(11\) 2.76768 0.834486 0.417243 0.908795i \(-0.362996\pi\)
0.417243 + 0.908795i \(0.362996\pi\)
\(12\) 0.965017 0.278576
\(13\) 4.05697 1.12520 0.562600 0.826729i \(-0.309801\pi\)
0.562600 + 0.826729i \(0.309801\pi\)
\(14\) 0 0
\(15\) 2.21344 0.571508
\(16\) 2.77356 0.693390
\(17\) −5.22447 −1.26712 −0.633560 0.773694i \(-0.718407\pi\)
−0.633560 + 0.773694i \(0.718407\pi\)
\(18\) 1.24687 0.293891
\(19\) 0.109209 0.0250542 0.0125271 0.999922i \(-0.496012\pi\)
0.0125271 + 0.999922i \(0.496012\pi\)
\(20\) 7.33368 1.63986
\(21\) 0 0
\(22\) −1.27400 −0.271618
\(23\) 6.08681 1.26919 0.634593 0.772846i \(-0.281168\pi\)
0.634593 + 0.772846i \(0.281168\pi\)
\(24\) −0.941062 −0.192093
\(25\) 11.8211 2.36422
\(26\) −1.86748 −0.366243
\(27\) 3.08092 0.592924
\(28\) 0 0
\(29\) 2.25431 0.418614 0.209307 0.977850i \(-0.432879\pi\)
0.209307 + 0.977850i \(0.432879\pi\)
\(30\) −1.01888 −0.186021
\(31\) 1.18073 0.212065 0.106032 0.994363i \(-0.466185\pi\)
0.106032 + 0.994363i \(0.466185\pi\)
\(32\) −4.76415 −0.842192
\(33\) −1.49368 −0.260016
\(34\) 2.40490 0.412437
\(35\) 0 0
\(36\) 4.84353 0.807254
\(37\) −8.95429 −1.47208 −0.736038 0.676940i \(-0.763305\pi\)
−0.736038 + 0.676940i \(0.763305\pi\)
\(38\) −0.0502704 −0.00815494
\(39\) −2.18949 −0.350598
\(40\) −7.15163 −1.13077
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −7.93974 −1.21080 −0.605399 0.795922i \(-0.706987\pi\)
−0.605399 + 0.795922i \(0.706987\pi\)
\(44\) −4.94891 −0.746077
\(45\) 11.1095 1.65611
\(46\) −2.80185 −0.413110
\(47\) 10.5714 1.54199 0.770997 0.636839i \(-0.219758\pi\)
0.770997 + 0.636839i \(0.219758\pi\)
\(48\) −1.49685 −0.216052
\(49\) 0 0
\(50\) −5.44143 −0.769535
\(51\) 2.81957 0.394819
\(52\) −7.25431 −1.00599
\(53\) 6.23100 0.855893 0.427947 0.903804i \(-0.359237\pi\)
0.427947 + 0.903804i \(0.359237\pi\)
\(54\) −1.41819 −0.192992
\(55\) −11.3512 −1.53060
\(56\) 0 0
\(57\) −0.0589384 −0.00780658
\(58\) −1.03769 −0.136255
\(59\) 9.43006 1.22769 0.613845 0.789427i \(-0.289622\pi\)
0.613845 + 0.789427i \(0.289622\pi\)
\(60\) −3.95788 −0.510960
\(61\) −6.89732 −0.883111 −0.441556 0.897234i \(-0.645573\pi\)
−0.441556 + 0.897234i \(0.645573\pi\)
\(62\) −0.543506 −0.0690253
\(63\) 0 0
\(64\) −3.35411 −0.419264
\(65\) −16.6391 −2.06382
\(66\) 0.687561 0.0846329
\(67\) −1.35699 −0.165782 −0.0828912 0.996559i \(-0.526415\pi\)
−0.0828912 + 0.996559i \(0.526415\pi\)
\(68\) 9.34193 1.13288
\(69\) −3.28496 −0.395463
\(70\) 0 0
\(71\) −7.90672 −0.938356 −0.469178 0.883104i \(-0.655450\pi\)
−0.469178 + 0.883104i \(0.655450\pi\)
\(72\) −4.72329 −0.556645
\(73\) −9.88791 −1.15729 −0.578646 0.815579i \(-0.696419\pi\)
−0.578646 + 0.815579i \(0.696419\pi\)
\(74\) 4.12179 0.479148
\(75\) −6.37969 −0.736663
\(76\) −0.195277 −0.0223999
\(77\) 0 0
\(78\) 1.00785 0.114117
\(79\) −12.5668 −1.41388 −0.706939 0.707275i \(-0.749924\pi\)
−0.706939 + 0.707275i \(0.749924\pi\)
\(80\) −11.3754 −1.27180
\(81\) 6.46349 0.718165
\(82\) −0.460315 −0.0508332
\(83\) 14.9633 1.64244 0.821220 0.570611i \(-0.193294\pi\)
0.821220 + 0.570611i \(0.193294\pi\)
\(84\) 0 0
\(85\) 21.4274 2.32413
\(86\) 3.65478 0.394105
\(87\) −1.21662 −0.130435
\(88\) 4.82606 0.514460
\(89\) 0.852931 0.0904105 0.0452053 0.998978i \(-0.485606\pi\)
0.0452053 + 0.998978i \(0.485606\pi\)
\(90\) −5.11387 −0.539049
\(91\) 0 0
\(92\) −10.8839 −1.13472
\(93\) −0.637221 −0.0660768
\(94\) −4.86616 −0.501906
\(95\) −0.447904 −0.0459540
\(96\) 2.57115 0.262416
\(97\) −9.92806 −1.00804 −0.504021 0.863691i \(-0.668147\pi\)
−0.504021 + 0.863691i \(0.668147\pi\)
\(98\) 0 0
\(99\) −7.49692 −0.753469
\(100\) −21.1375 −2.11375
\(101\) −11.2908 −1.12348 −0.561740 0.827314i \(-0.689868\pi\)
−0.561740 + 0.827314i \(0.689868\pi\)
\(102\) −1.29789 −0.128510
\(103\) 10.8690 1.07096 0.535479 0.844549i \(-0.320131\pi\)
0.535479 + 0.844549i \(0.320131\pi\)
\(104\) 7.07423 0.693685
\(105\) 0 0
\(106\) −2.86822 −0.278586
\(107\) 1.99215 0.192588 0.0962941 0.995353i \(-0.469301\pi\)
0.0962941 + 0.995353i \(0.469301\pi\)
\(108\) −5.50903 −0.530107
\(109\) −17.8607 −1.71074 −0.855371 0.518016i \(-0.826671\pi\)
−0.855371 + 0.518016i \(0.826671\pi\)
\(110\) 5.22514 0.498197
\(111\) 4.83250 0.458680
\(112\) 0 0
\(113\) −16.2536 −1.52901 −0.764506 0.644616i \(-0.777017\pi\)
−0.764506 + 0.644616i \(0.777017\pi\)
\(114\) 0.0271302 0.00254098
\(115\) −24.9642 −2.32792
\(116\) −4.03095 −0.374264
\(117\) −10.9893 −1.01596
\(118\) −4.34079 −0.399602
\(119\) 0 0
\(120\) 3.85963 0.352334
\(121\) −3.33996 −0.303633
\(122\) 3.17494 0.287445
\(123\) −0.539685 −0.0486618
\(124\) −2.11127 −0.189598
\(125\) −27.9759 −2.50224
\(126\) 0 0
\(127\) 17.3566 1.54015 0.770073 0.637955i \(-0.220220\pi\)
0.770073 + 0.637955i \(0.220220\pi\)
\(128\) 11.0723 0.978658
\(129\) 4.28496 0.377270
\(130\) 7.65921 0.671757
\(131\) 0.803568 0.0702080 0.0351040 0.999384i \(-0.488824\pi\)
0.0351040 + 0.999384i \(0.488824\pi\)
\(132\) 2.67086 0.232468
\(133\) 0 0
\(134\) 0.624641 0.0539608
\(135\) −12.6360 −1.08753
\(136\) −9.11002 −0.781178
\(137\) −4.63907 −0.396343 −0.198171 0.980167i \(-0.563500\pi\)
−0.198171 + 0.980167i \(0.563500\pi\)
\(138\) 1.51212 0.128720
\(139\) 0.0641748 0.00544323 0.00272162 0.999996i \(-0.499134\pi\)
0.00272162 + 0.999996i \(0.499134\pi\)
\(140\) 0 0
\(141\) −5.70522 −0.480466
\(142\) 3.63958 0.305427
\(143\) 11.2284 0.938964
\(144\) −7.51285 −0.626071
\(145\) −9.24572 −0.767815
\(146\) 4.55155 0.376689
\(147\) 0 0
\(148\) 16.0113 1.31612
\(149\) 14.2436 1.16688 0.583440 0.812156i \(-0.301706\pi\)
0.583440 + 0.812156i \(0.301706\pi\)
\(150\) 2.93666 0.239778
\(151\) 12.8350 1.04450 0.522249 0.852793i \(-0.325093\pi\)
0.522249 + 0.852793i \(0.325093\pi\)
\(152\) 0.190430 0.0154459
\(153\) 14.1517 1.14410
\(154\) 0 0
\(155\) −4.84258 −0.388966
\(156\) 3.91504 0.313454
\(157\) −14.3051 −1.14167 −0.570835 0.821064i \(-0.693381\pi\)
−0.570835 + 0.821064i \(0.693381\pi\)
\(158\) 5.78469 0.460205
\(159\) −3.36278 −0.266686
\(160\) 19.5395 1.54473
\(161\) 0 0
\(162\) −2.97524 −0.233757
\(163\) 5.60958 0.439376 0.219688 0.975570i \(-0.429496\pi\)
0.219688 + 0.975570i \(0.429496\pi\)
\(164\) −1.78811 −0.139628
\(165\) 6.12610 0.476916
\(166\) −6.88785 −0.534600
\(167\) −4.49350 −0.347717 −0.173859 0.984771i \(-0.555624\pi\)
−0.173859 + 0.984771i \(0.555624\pi\)
\(168\) 0 0
\(169\) 3.45899 0.266076
\(170\) −9.86335 −0.756484
\(171\) −0.295818 −0.0226218
\(172\) 14.1971 1.08252
\(173\) −14.2512 −1.08350 −0.541749 0.840541i \(-0.682238\pi\)
−0.541749 + 0.840541i \(0.682238\pi\)
\(174\) 0.560026 0.0424555
\(175\) 0 0
\(176\) 7.67632 0.578625
\(177\) −5.08927 −0.382533
\(178\) −0.392617 −0.0294279
\(179\) 5.78841 0.432646 0.216323 0.976322i \(-0.430594\pi\)
0.216323 + 0.976322i \(0.430594\pi\)
\(180\) −19.8650 −1.48065
\(181\) 14.6662 1.09013 0.545065 0.838394i \(-0.316505\pi\)
0.545065 + 0.838394i \(0.316505\pi\)
\(182\) 0 0
\(183\) 3.72238 0.275166
\(184\) 10.6137 0.782452
\(185\) 36.7247 2.70006
\(186\) 0.293322 0.0215074
\(187\) −14.4596 −1.05739
\(188\) −18.9028 −1.37863
\(189\) 0 0
\(190\) 0.206177 0.0149576
\(191\) 8.41984 0.609238 0.304619 0.952474i \(-0.401471\pi\)
0.304619 + 0.952474i \(0.401471\pi\)
\(192\) 1.81016 0.130637
\(193\) −8.76374 −0.630828 −0.315414 0.948954i \(-0.602143\pi\)
−0.315414 + 0.948954i \(0.602143\pi\)
\(194\) 4.57003 0.328109
\(195\) 8.97986 0.643061
\(196\) 0 0
\(197\) −18.9953 −1.35336 −0.676680 0.736277i \(-0.736582\pi\)
−0.676680 + 0.736277i \(0.736582\pi\)
\(198\) 3.45094 0.245248
\(199\) 6.81886 0.483376 0.241688 0.970354i \(-0.422299\pi\)
0.241688 + 0.970354i \(0.422299\pi\)
\(200\) 20.6128 1.45754
\(201\) 0.732347 0.0516558
\(202\) 5.19734 0.365684
\(203\) 0 0
\(204\) −5.04170 −0.352990
\(205\) −4.10136 −0.286451
\(206\) −5.00317 −0.348588
\(207\) −16.4876 −1.14597
\(208\) 11.2522 0.780203
\(209\) 0.302255 0.0209074
\(210\) 0 0
\(211\) 5.02073 0.345641 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(212\) −11.1417 −0.765216
\(213\) 4.26714 0.292380
\(214\) −0.917014 −0.0626858
\(215\) 32.5637 2.22083
\(216\) 5.37228 0.365537
\(217\) 0 0
\(218\) 8.22152 0.556832
\(219\) 5.33636 0.360598
\(220\) 20.2973 1.36844
\(221\) −21.1955 −1.42576
\(222\) −2.22447 −0.149297
\(223\) −6.96231 −0.466231 −0.233115 0.972449i \(-0.574892\pi\)
−0.233115 + 0.972449i \(0.574892\pi\)
\(224\) 0 0
\(225\) −32.0203 −2.13469
\(226\) 7.48178 0.497681
\(227\) −10.1094 −0.670987 −0.335493 0.942043i \(-0.608903\pi\)
−0.335493 + 0.942043i \(0.608903\pi\)
\(228\) 0.105388 0.00697952
\(229\) 2.32356 0.153545 0.0767725 0.997049i \(-0.475538\pi\)
0.0767725 + 0.997049i \(0.475538\pi\)
\(230\) 11.4914 0.757718
\(231\) 0 0
\(232\) 3.93089 0.258075
\(233\) 17.5655 1.15075 0.575377 0.817888i \(-0.304855\pi\)
0.575377 + 0.817888i \(0.304855\pi\)
\(234\) 5.05852 0.330686
\(235\) −43.3570 −2.82830
\(236\) −16.8620 −1.09762
\(237\) 6.78213 0.440547
\(238\) 0 0
\(239\) −23.0063 −1.48815 −0.744077 0.668093i \(-0.767111\pi\)
−0.744077 + 0.668093i \(0.767111\pi\)
\(240\) 6.13912 0.396278
\(241\) 12.6808 0.816839 0.408420 0.912794i \(-0.366080\pi\)
0.408420 + 0.912794i \(0.366080\pi\)
\(242\) 1.53743 0.0988298
\(243\) −12.7310 −0.816695
\(244\) 12.3332 0.789550
\(245\) 0 0
\(246\) 0.248425 0.0158390
\(247\) 0.443057 0.0281910
\(248\) 2.05886 0.130738
\(249\) −8.07550 −0.511764
\(250\) 12.8777 0.814457
\(251\) 18.5629 1.17168 0.585840 0.810427i \(-0.300765\pi\)
0.585840 + 0.810427i \(0.300765\pi\)
\(252\) 0 0
\(253\) 16.8463 1.05912
\(254\) −7.98948 −0.501305
\(255\) −11.5641 −0.724170
\(256\) 1.61150 0.100719
\(257\) −18.1358 −1.13128 −0.565639 0.824653i \(-0.691370\pi\)
−0.565639 + 0.824653i \(0.691370\pi\)
\(258\) −1.97243 −0.122798
\(259\) 0 0
\(260\) 29.7525 1.84517
\(261\) −6.10633 −0.377972
\(262\) −0.369894 −0.0228521
\(263\) 4.24808 0.261948 0.130974 0.991386i \(-0.458190\pi\)
0.130974 + 0.991386i \(0.458190\pi\)
\(264\) −2.60456 −0.160299
\(265\) −25.5555 −1.56986
\(266\) 0 0
\(267\) −0.460315 −0.0281708
\(268\) 2.42644 0.148219
\(269\) 1.31427 0.0801326 0.0400663 0.999197i \(-0.487243\pi\)
0.0400663 + 0.999197i \(0.487243\pi\)
\(270\) 5.81652 0.353982
\(271\) 7.91618 0.480874 0.240437 0.970665i \(-0.422709\pi\)
0.240437 + 0.970665i \(0.422709\pi\)
\(272\) −14.4904 −0.878608
\(273\) 0 0
\(274\) 2.13543 0.129006
\(275\) 32.7171 1.97291
\(276\) 5.87387 0.353566
\(277\) 0.0684353 0.00411188 0.00205594 0.999998i \(-0.499346\pi\)
0.00205594 + 0.999998i \(0.499346\pi\)
\(278\) −0.0295406 −0.00177173
\(279\) −3.19828 −0.191476
\(280\) 0 0
\(281\) 8.33050 0.496956 0.248478 0.968638i \(-0.420070\pi\)
0.248478 + 0.968638i \(0.420070\pi\)
\(282\) 2.62619 0.156388
\(283\) −3.09424 −0.183934 −0.0919668 0.995762i \(-0.529315\pi\)
−0.0919668 + 0.995762i \(0.529315\pi\)
\(284\) 14.1381 0.838942
\(285\) 0.241727 0.0143187
\(286\) −5.16859 −0.305625
\(287\) 0 0
\(288\) 12.9049 0.760426
\(289\) 10.2951 0.605593
\(290\) 4.25594 0.249917
\(291\) 5.35803 0.314093
\(292\) 17.6807 1.03468
\(293\) −4.47354 −0.261347 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(294\) 0 0
\(295\) −38.6760 −2.25181
\(296\) −15.6138 −0.907533
\(297\) 8.52700 0.494787
\(298\) −6.55653 −0.379809
\(299\) 24.6940 1.42809
\(300\) 11.4076 0.658617
\(301\) 0 0
\(302\) −5.90813 −0.339975
\(303\) 6.09350 0.350063
\(304\) 0.302897 0.0173723
\(305\) 28.2884 1.61979
\(306\) −6.51425 −0.372395
\(307\) −25.8316 −1.47429 −0.737143 0.675737i \(-0.763825\pi\)
−0.737143 + 0.675737i \(0.763825\pi\)
\(308\) 0 0
\(309\) −5.86586 −0.333697
\(310\) 2.22911 0.126605
\(311\) −13.6802 −0.775735 −0.387867 0.921715i \(-0.626788\pi\)
−0.387867 + 0.921715i \(0.626788\pi\)
\(312\) −3.81786 −0.216144
\(313\) −24.0537 −1.35959 −0.679797 0.733400i \(-0.737932\pi\)
−0.679797 + 0.733400i \(0.737932\pi\)
\(314\) 6.58484 0.371604
\(315\) 0 0
\(316\) 22.4709 1.26408
\(317\) 3.80766 0.213859 0.106930 0.994267i \(-0.465898\pi\)
0.106930 + 0.994267i \(0.465898\pi\)
\(318\) 1.54794 0.0868039
\(319\) 6.23920 0.349328
\(320\) 13.7564 0.769006
\(321\) −1.07513 −0.0600081
\(322\) 0 0
\(323\) −0.570558 −0.0317467
\(324\) −11.5574 −0.642080
\(325\) 47.9579 2.66023
\(326\) −2.58217 −0.143013
\(327\) 9.63914 0.533046
\(328\) 1.74372 0.0962810
\(329\) 0 0
\(330\) −2.81993 −0.155232
\(331\) −21.1336 −1.16161 −0.580805 0.814043i \(-0.697262\pi\)
−0.580805 + 0.814043i \(0.697262\pi\)
\(332\) −26.7561 −1.46843
\(333\) 24.2548 1.32916
\(334\) 2.06842 0.113179
\(335\) 5.56549 0.304075
\(336\) 0 0
\(337\) −20.7801 −1.13197 −0.565983 0.824417i \(-0.691503\pi\)
−0.565983 + 0.824417i \(0.691503\pi\)
\(338\) −1.59222 −0.0866055
\(339\) 8.77185 0.476421
\(340\) −38.3146 −2.07790
\(341\) 3.26787 0.176965
\(342\) 0.136169 0.00736320
\(343\) 0 0
\(344\) −13.8447 −0.746456
\(345\) 13.4728 0.725351
\(346\) 6.56003 0.352669
\(347\) 21.8355 1.17219 0.586096 0.810242i \(-0.300664\pi\)
0.586096 + 0.810242i \(0.300664\pi\)
\(348\) 2.17545 0.116616
\(349\) 6.23178 0.333579 0.166790 0.985992i \(-0.446660\pi\)
0.166790 + 0.985992i \(0.446660\pi\)
\(350\) 0 0
\(351\) 12.4992 0.667158
\(352\) −13.1856 −0.702797
\(353\) −28.5933 −1.52187 −0.760933 0.648830i \(-0.775258\pi\)
−0.760933 + 0.648830i \(0.775258\pi\)
\(354\) 2.34266 0.124511
\(355\) 32.4283 1.72112
\(356\) −1.52514 −0.0808320
\(357\) 0 0
\(358\) −2.66449 −0.140823
\(359\) 6.89008 0.363644 0.181822 0.983331i \(-0.441800\pi\)
0.181822 + 0.983331i \(0.441800\pi\)
\(360\) 19.3719 1.02099
\(361\) −18.9881 −0.999372
\(362\) −6.75107 −0.354828
\(363\) 1.80253 0.0946081
\(364\) 0 0
\(365\) 40.5539 2.12269
\(366\) −1.71347 −0.0895644
\(367\) −27.9358 −1.45824 −0.729119 0.684387i \(-0.760070\pi\)
−0.729119 + 0.684387i \(0.760070\pi\)
\(368\) 16.8821 0.880042
\(369\) −2.70874 −0.141011
\(370\) −16.9049 −0.878845
\(371\) 0 0
\(372\) 1.13942 0.0590763
\(373\) 6.20063 0.321057 0.160528 0.987031i \(-0.448680\pi\)
0.160528 + 0.987031i \(0.448680\pi\)
\(374\) 6.65599 0.344173
\(375\) 15.0982 0.779666
\(376\) 18.4335 0.950637
\(377\) 9.14565 0.471025
\(378\) 0 0
\(379\) −18.4006 −0.945174 −0.472587 0.881284i \(-0.656680\pi\)
−0.472587 + 0.881284i \(0.656680\pi\)
\(380\) 0.800902 0.0410854
\(381\) −9.36709 −0.479890
\(382\) −3.87577 −0.198302
\(383\) 34.2322 1.74918 0.874591 0.484861i \(-0.161130\pi\)
0.874591 + 0.484861i \(0.161130\pi\)
\(384\) −5.97554 −0.304938
\(385\) 0 0
\(386\) 4.03408 0.205329
\(387\) 21.5067 1.09325
\(388\) 17.7525 0.901245
\(389\) −23.0399 −1.16817 −0.584085 0.811693i \(-0.698546\pi\)
−0.584085 + 0.811693i \(0.698546\pi\)
\(390\) −4.13356 −0.209311
\(391\) −31.8003 −1.60821
\(392\) 0 0
\(393\) −0.433674 −0.0218760
\(394\) 8.74382 0.440507
\(395\) 51.5410 2.59331
\(396\) 13.4053 0.673643
\(397\) −1.35123 −0.0678165 −0.0339082 0.999425i \(-0.510795\pi\)
−0.0339082 + 0.999425i \(0.510795\pi\)
\(398\) −3.13882 −0.157335
\(399\) 0 0
\(400\) 32.7866 1.63933
\(401\) 1.54256 0.0770319 0.0385160 0.999258i \(-0.487737\pi\)
0.0385160 + 0.999258i \(0.487737\pi\)
\(402\) −0.337110 −0.0168135
\(403\) 4.79017 0.238615
\(404\) 20.1893 1.00445
\(405\) −26.5091 −1.31725
\(406\) 0 0
\(407\) −24.7826 −1.22843
\(408\) 4.91655 0.243405
\(409\) −4.34571 −0.214882 −0.107441 0.994211i \(-0.534266\pi\)
−0.107441 + 0.994211i \(0.534266\pi\)
\(410\) 1.88791 0.0932374
\(411\) 2.50364 0.123495
\(412\) −19.4350 −0.957495
\(413\) 0 0
\(414\) 7.58947 0.373002
\(415\) −61.3700 −3.01254
\(416\) −19.3280 −0.947634
\(417\) −0.0346342 −0.00169604
\(418\) −0.139132 −0.00680518
\(419\) −27.2835 −1.33288 −0.666442 0.745557i \(-0.732184\pi\)
−0.666442 + 0.745557i \(0.732184\pi\)
\(420\) 0 0
\(421\) −16.6168 −0.809853 −0.404927 0.914349i \(-0.632703\pi\)
−0.404927 + 0.914349i \(0.632703\pi\)
\(422\) −2.31111 −0.112503
\(423\) −28.6351 −1.39229
\(424\) 10.8651 0.527657
\(425\) −61.7591 −2.99576
\(426\) −1.96423 −0.0951672
\(427\) 0 0
\(428\) −3.56218 −0.172184
\(429\) −6.05979 −0.292570
\(430\) −14.9895 −0.722860
\(431\) −9.36507 −0.451099 −0.225550 0.974232i \(-0.572418\pi\)
−0.225550 + 0.974232i \(0.572418\pi\)
\(432\) 8.54513 0.411128
\(433\) 3.87136 0.186046 0.0930228 0.995664i \(-0.470347\pi\)
0.0930228 + 0.995664i \(0.470347\pi\)
\(434\) 0 0
\(435\) 4.98978 0.239242
\(436\) 31.9368 1.52950
\(437\) 0.664733 0.0317985
\(438\) −2.45641 −0.117372
\(439\) −29.6011 −1.41279 −0.706393 0.707820i \(-0.749679\pi\)
−0.706393 + 0.707820i \(0.749679\pi\)
\(440\) −19.7934 −0.943613
\(441\) 0 0
\(442\) 9.75660 0.464074
\(443\) −0.570558 −0.0271080 −0.0135540 0.999908i \(-0.504315\pi\)
−0.0135540 + 0.999908i \(0.504315\pi\)
\(444\) −8.64104 −0.410086
\(445\) −3.49817 −0.165829
\(446\) 3.20485 0.151754
\(447\) −7.68705 −0.363585
\(448\) 0 0
\(449\) 18.8280 0.888551 0.444275 0.895890i \(-0.353461\pi\)
0.444275 + 0.895890i \(0.353461\pi\)
\(450\) 14.7394 0.694823
\(451\) 2.76768 0.130325
\(452\) 29.0633 1.36702
\(453\) −6.92686 −0.325452
\(454\) 4.65352 0.218401
\(455\) 0 0
\(456\) −0.102772 −0.00481275
\(457\) 22.2732 1.04190 0.520949 0.853588i \(-0.325578\pi\)
0.520949 + 0.853588i \(0.325578\pi\)
\(458\) −1.06957 −0.0499776
\(459\) −16.0962 −0.751306
\(460\) 44.6387 2.08129
\(461\) −4.47354 −0.208354 −0.104177 0.994559i \(-0.533221\pi\)
−0.104177 + 0.994559i \(0.533221\pi\)
\(462\) 0 0
\(463\) −3.62240 −0.168347 −0.0841736 0.996451i \(-0.526825\pi\)
−0.0841736 + 0.996451i \(0.526825\pi\)
\(464\) 6.25246 0.290263
\(465\) 2.61347 0.121197
\(466\) −8.08565 −0.374561
\(467\) 1.54346 0.0714226 0.0357113 0.999362i \(-0.488630\pi\)
0.0357113 + 0.999362i \(0.488630\pi\)
\(468\) 19.6500 0.908323
\(469\) 0 0
\(470\) 19.9578 0.920587
\(471\) 7.72025 0.355730
\(472\) 16.4434 0.756869
\(473\) −21.9746 −1.01039
\(474\) −3.12191 −0.143394
\(475\) 1.29097 0.0592338
\(476\) 0 0
\(477\) −16.8781 −0.772797
\(478\) 10.5901 0.484382
\(479\) 14.5913 0.666693 0.333347 0.942804i \(-0.391822\pi\)
0.333347 + 0.942804i \(0.391822\pi\)
\(480\) −10.5452 −0.481320
\(481\) −36.3273 −1.65638
\(482\) −5.83713 −0.265874
\(483\) 0 0
\(484\) 5.97221 0.271464
\(485\) 40.7185 1.84893
\(486\) 5.86027 0.265827
\(487\) −1.52440 −0.0690771 −0.0345385 0.999403i \(-0.510996\pi\)
−0.0345385 + 0.999403i \(0.510996\pi\)
\(488\) −12.0270 −0.544437
\(489\) −3.02741 −0.136904
\(490\) 0 0
\(491\) −34.8827 −1.57424 −0.787118 0.616802i \(-0.788428\pi\)
−0.787118 + 0.616802i \(0.788428\pi\)
\(492\) 0.965017 0.0435063
\(493\) −11.7776 −0.530435
\(494\) −0.203945 −0.00917594
\(495\) 30.7475 1.38200
\(496\) 3.27482 0.147044
\(497\) 0 0
\(498\) 3.71727 0.166575
\(499\) 14.7766 0.661490 0.330745 0.943720i \(-0.392700\pi\)
0.330745 + 0.943720i \(0.392700\pi\)
\(500\) 50.0239 2.23714
\(501\) 2.42507 0.108344
\(502\) −8.54478 −0.381372
\(503\) −18.9590 −0.845338 −0.422669 0.906284i \(-0.638907\pi\)
−0.422669 + 0.906284i \(0.638907\pi\)
\(504\) 0 0
\(505\) 46.3078 2.06067
\(506\) −7.75461 −0.344734
\(507\) −1.86677 −0.0829060
\(508\) −31.0355 −1.37698
\(509\) 17.0790 0.757011 0.378506 0.925599i \(-0.376438\pi\)
0.378506 + 0.925599i \(0.376438\pi\)
\(510\) 5.32311 0.235711
\(511\) 0 0
\(512\) −22.8863 −1.01144
\(513\) 0.336464 0.0148552
\(514\) 8.34815 0.368221
\(515\) −44.5778 −1.96433
\(516\) −7.66198 −0.337300
\(517\) 29.2582 1.28677
\(518\) 0 0
\(519\) 7.69116 0.337604
\(520\) −29.0139 −1.27234
\(521\) −25.3095 −1.10883 −0.554415 0.832240i \(-0.687058\pi\)
−0.554415 + 0.832240i \(0.687058\pi\)
\(522\) 2.81083 0.123027
\(523\) 4.43499 0.193928 0.0969642 0.995288i \(-0.469087\pi\)
0.0969642 + 0.995288i \(0.469087\pi\)
\(524\) −1.43687 −0.0627699
\(525\) 0 0
\(526\) −1.95545 −0.0852617
\(527\) −6.16867 −0.268712
\(528\) −4.14280 −0.180292
\(529\) 14.0492 0.610835
\(530\) 11.7636 0.510978
\(531\) −25.5436 −1.10850
\(532\) 0 0
\(533\) 4.05697 0.175727
\(534\) 0.211889 0.00916935
\(535\) −8.17051 −0.353242
\(536\) −2.36621 −0.102205
\(537\) −3.12392 −0.134807
\(538\) −0.604979 −0.0260825
\(539\) 0 0
\(540\) 22.5945 0.972313
\(541\) −5.57149 −0.239537 −0.119769 0.992802i \(-0.538215\pi\)
−0.119769 + 0.992802i \(0.538215\pi\)
\(542\) −3.64393 −0.156520
\(543\) −7.91514 −0.339671
\(544\) 24.8902 1.06716
\(545\) 73.2529 3.13781
\(546\) 0 0
\(547\) 38.0432 1.62661 0.813304 0.581839i \(-0.197666\pi\)
0.813304 + 0.581839i \(0.197666\pi\)
\(548\) 8.29517 0.354352
\(549\) 18.6830 0.797373
\(550\) −15.0601 −0.642167
\(551\) 0.246190 0.0104881
\(552\) −5.72806 −0.243802
\(553\) 0 0
\(554\) −0.0315018 −0.00133838
\(555\) −19.8198 −0.841304
\(556\) −0.114752 −0.00486655
\(557\) −19.0835 −0.808594 −0.404297 0.914628i \(-0.632484\pi\)
−0.404297 + 0.914628i \(0.632484\pi\)
\(558\) 1.47222 0.0623239
\(559\) −32.2113 −1.36239
\(560\) 0 0
\(561\) 7.80366 0.329471
\(562\) −3.83465 −0.161755
\(563\) 19.1867 0.808621 0.404311 0.914622i \(-0.367511\pi\)
0.404311 + 0.914622i \(0.367511\pi\)
\(564\) 10.2016 0.429563
\(565\) 66.6619 2.80449
\(566\) 1.42432 0.0598688
\(567\) 0 0
\(568\) −13.7871 −0.578495
\(569\) 29.3295 1.22956 0.614778 0.788700i \(-0.289246\pi\)
0.614778 + 0.788700i \(0.289246\pi\)
\(570\) −0.111271 −0.00466061
\(571\) −21.5864 −0.903362 −0.451681 0.892180i \(-0.649175\pi\)
−0.451681 + 0.892180i \(0.649175\pi\)
\(572\) −20.0776 −0.839486
\(573\) −4.54406 −0.189831
\(574\) 0 0
\(575\) 71.9529 3.00064
\(576\) 9.08541 0.378559
\(577\) −5.28198 −0.219892 −0.109946 0.993938i \(-0.535068\pi\)
−0.109946 + 0.993938i \(0.535068\pi\)
\(578\) −4.73898 −0.197115
\(579\) 4.72966 0.196558
\(580\) 16.5324 0.686469
\(581\) 0 0
\(582\) −2.46638 −0.102235
\(583\) 17.2454 0.714231
\(584\) −17.2418 −0.713470
\(585\) 45.0709 1.86345
\(586\) 2.05924 0.0850662
\(587\) 41.8869 1.72886 0.864428 0.502757i \(-0.167681\pi\)
0.864428 + 0.502757i \(0.167681\pi\)
\(588\) 0 0
\(589\) 0.128946 0.00531312
\(590\) 17.8031 0.732944
\(591\) 10.2515 0.421690
\(592\) −24.8353 −1.02072
\(593\) −25.8605 −1.06196 −0.530982 0.847383i \(-0.678177\pi\)
−0.530982 + 0.847383i \(0.678177\pi\)
\(594\) −3.92510 −0.161049
\(595\) 0 0
\(596\) −25.4691 −1.04325
\(597\) −3.68004 −0.150614
\(598\) −11.3670 −0.464831
\(599\) −39.4470 −1.61176 −0.805881 0.592078i \(-0.798308\pi\)
−0.805881 + 0.592078i \(0.798308\pi\)
\(600\) −11.1244 −0.454152
\(601\) −40.0435 −1.63341 −0.816704 0.577056i \(-0.804201\pi\)
−0.816704 + 0.577056i \(0.804201\pi\)
\(602\) 0 0
\(603\) 3.67573 0.149687
\(604\) −22.9504 −0.933838
\(605\) 13.6984 0.556918
\(606\) −2.80493 −0.113942
\(607\) 10.7580 0.436653 0.218327 0.975876i \(-0.429940\pi\)
0.218327 + 0.975876i \(0.429940\pi\)
\(608\) −0.520288 −0.0211004
\(609\) 0 0
\(610\) −13.0215 −0.527227
\(611\) 42.8877 1.73505
\(612\) −25.3049 −1.02289
\(613\) 16.7220 0.675393 0.337697 0.941255i \(-0.390352\pi\)
0.337697 + 0.941255i \(0.390352\pi\)
\(614\) 11.8906 0.479867
\(615\) 2.21344 0.0892546
\(616\) 0 0
\(617\) −42.3381 −1.70447 −0.852234 0.523160i \(-0.824753\pi\)
−0.852234 + 0.523160i \(0.824753\pi\)
\(618\) 2.70014 0.108616
\(619\) −32.0830 −1.28952 −0.644762 0.764384i \(-0.723043\pi\)
−0.644762 + 0.764384i \(0.723043\pi\)
\(620\) 8.65907 0.347757
\(621\) 18.7530 0.752531
\(622\) 6.29721 0.252495
\(623\) 0 0
\(624\) −6.07267 −0.243101
\(625\) 55.6333 2.22533
\(626\) 11.0723 0.442536
\(627\) −0.163123 −0.00651449
\(628\) 25.5791 1.02072
\(629\) 46.7814 1.86530
\(630\) 0 0
\(631\) −44.9458 −1.78927 −0.894633 0.446802i \(-0.852563\pi\)
−0.894633 + 0.446802i \(0.852563\pi\)
\(632\) −21.9130 −0.871654
\(633\) −2.70961 −0.107698
\(634\) −1.75272 −0.0696094
\(635\) −71.1855 −2.82491
\(636\) 6.01302 0.238432
\(637\) 0 0
\(638\) −2.87199 −0.113703
\(639\) 21.4173 0.847254
\(640\) −45.4113 −1.79504
\(641\) 10.4353 0.412171 0.206085 0.978534i \(-0.433928\pi\)
0.206085 + 0.978534i \(0.433928\pi\)
\(642\) 0.494899 0.0195321
\(643\) 33.1360 1.30676 0.653378 0.757032i \(-0.273351\pi\)
0.653378 + 0.757032i \(0.273351\pi\)
\(644\) 0 0
\(645\) −17.5742 −0.691981
\(646\) 0.262636 0.0103333
\(647\) 29.4094 1.15620 0.578101 0.815965i \(-0.303794\pi\)
0.578101 + 0.815965i \(0.303794\pi\)
\(648\) 11.2705 0.442748
\(649\) 26.0994 1.02449
\(650\) −22.0757 −0.865881
\(651\) 0 0
\(652\) −10.0306 −0.392827
\(653\) −41.0121 −1.60493 −0.802464 0.596701i \(-0.796478\pi\)
−0.802464 + 0.596701i \(0.796478\pi\)
\(654\) −4.43704 −0.173502
\(655\) −3.29572 −0.128774
\(656\) 2.77356 0.108289
\(657\) 26.7838 1.04493
\(658\) 0 0
\(659\) 41.6194 1.62126 0.810631 0.585558i \(-0.199124\pi\)
0.810631 + 0.585558i \(0.199124\pi\)
\(660\) −10.9541 −0.426389
\(661\) 43.0712 1.67527 0.837637 0.546227i \(-0.183936\pi\)
0.837637 + 0.546227i \(0.183936\pi\)
\(662\) 9.72812 0.378094
\(663\) 11.4389 0.444250
\(664\) 26.0919 1.01256
\(665\) 0 0
\(666\) −11.1649 −0.432629
\(667\) 13.7215 0.531300
\(668\) 8.03487 0.310878
\(669\) 3.75746 0.145272
\(670\) −2.56188 −0.0989739
\(671\) −19.0896 −0.736944
\(672\) 0 0
\(673\) 44.0679 1.69869 0.849346 0.527837i \(-0.176997\pi\)
0.849346 + 0.527837i \(0.176997\pi\)
\(674\) 9.56540 0.368445
\(675\) 36.4200 1.40181
\(676\) −6.18506 −0.237887
\(677\) −5.02017 −0.192941 −0.0964704 0.995336i \(-0.530755\pi\)
−0.0964704 + 0.995336i \(0.530755\pi\)
\(678\) −4.03781 −0.155071
\(679\) 0 0
\(680\) 37.3635 1.43282
\(681\) 5.45592 0.209071
\(682\) −1.50425 −0.0576007
\(683\) 24.9476 0.954595 0.477297 0.878742i \(-0.341616\pi\)
0.477297 + 0.878742i \(0.341616\pi\)
\(684\) 0.528956 0.0202251
\(685\) 19.0265 0.726965
\(686\) 0 0
\(687\) −1.25399 −0.0478427
\(688\) −22.0213 −0.839556
\(689\) 25.2790 0.963052
\(690\) −6.20172 −0.236096
\(691\) −42.8508 −1.63012 −0.815061 0.579375i \(-0.803297\pi\)
−0.815061 + 0.579375i \(0.803297\pi\)
\(692\) 25.4827 0.968706
\(693\) 0 0
\(694\) −10.0512 −0.381538
\(695\) −0.263204 −0.00998389
\(696\) −2.12144 −0.0804131
\(697\) −5.22447 −0.197891
\(698\) −2.86858 −0.108577
\(699\) −9.47984 −0.358560
\(700\) 0 0
\(701\) −46.3548 −1.75079 −0.875397 0.483404i \(-0.839400\pi\)
−0.875397 + 0.483404i \(0.839400\pi\)
\(702\) −5.75357 −0.217154
\(703\) −0.977887 −0.0368817
\(704\) −9.28310 −0.349870
\(705\) 23.3991 0.881262
\(706\) 13.1619 0.495355
\(707\) 0 0
\(708\) 9.10017 0.342005
\(709\) 10.9521 0.411316 0.205658 0.978624i \(-0.434067\pi\)
0.205658 + 0.978624i \(0.434067\pi\)
\(710\) −14.9272 −0.560208
\(711\) 34.0402 1.27661
\(712\) 1.48728 0.0557380
\(713\) 7.18686 0.269150
\(714\) 0 0
\(715\) −46.0516 −1.72223
\(716\) −10.3503 −0.386809
\(717\) 12.4162 0.463691
\(718\) −3.17160 −0.118363
\(719\) −7.62700 −0.284439 −0.142220 0.989835i \(-0.545424\pi\)
−0.142220 + 0.989835i \(0.545424\pi\)
\(720\) 30.8129 1.14833
\(721\) 0 0
\(722\) 8.74049 0.325287
\(723\) −6.84362 −0.254517
\(724\) −26.2248 −0.974637
\(725\) 26.6484 0.989698
\(726\) −0.829729 −0.0307941
\(727\) 37.2537 1.38166 0.690832 0.723015i \(-0.257244\pi\)
0.690832 + 0.723015i \(0.257244\pi\)
\(728\) 0 0
\(729\) −12.5197 −0.463693
\(730\) −18.6675 −0.690916
\(731\) 41.4809 1.53423
\(732\) −6.65603 −0.246014
\(733\) −7.99745 −0.295393 −0.147696 0.989033i \(-0.547186\pi\)
−0.147696 + 0.989033i \(0.547186\pi\)
\(734\) 12.8593 0.474644
\(735\) 0 0
\(736\) −28.9985 −1.06890
\(737\) −3.75571 −0.138343
\(738\) 1.24687 0.0458980
\(739\) 19.7183 0.725350 0.362675 0.931916i \(-0.381864\pi\)
0.362675 + 0.931916i \(0.381864\pi\)
\(740\) −65.6679 −2.41400
\(741\) −0.239111 −0.00878397
\(742\) 0 0
\(743\) 31.1345 1.14221 0.571106 0.820876i \(-0.306514\pi\)
0.571106 + 0.820876i \(0.306514\pi\)
\(744\) −1.11114 −0.0407363
\(745\) −58.4180 −2.14027
\(746\) −2.85424 −0.104501
\(747\) −40.5318 −1.48298
\(748\) 25.8555 0.945369
\(749\) 0 0
\(750\) −6.94990 −0.253775
\(751\) −40.7231 −1.48601 −0.743004 0.669287i \(-0.766600\pi\)
−0.743004 + 0.669287i \(0.766600\pi\)
\(752\) 29.3203 1.06920
\(753\) −10.0181 −0.365081
\(754\) −4.20988 −0.153315
\(755\) −52.6409 −1.91580
\(756\) 0 0
\(757\) 34.9086 1.26878 0.634388 0.773015i \(-0.281252\pi\)
0.634388 + 0.773015i \(0.281252\pi\)
\(758\) 8.47005 0.307646
\(759\) −9.09171 −0.330008
\(760\) −0.781021 −0.0283306
\(761\) −4.76771 −0.172830 −0.0864148 0.996259i \(-0.527541\pi\)
−0.0864148 + 0.996259i \(0.527541\pi\)
\(762\) 4.31181 0.156200
\(763\) 0 0
\(764\) −15.0556 −0.544693
\(765\) −58.0413 −2.09849
\(766\) −15.7576 −0.569344
\(767\) 38.2574 1.38140
\(768\) −0.869704 −0.0313827
\(769\) −51.9550 −1.87354 −0.936772 0.349941i \(-0.886202\pi\)
−0.936772 + 0.349941i \(0.886202\pi\)
\(770\) 0 0
\(771\) 9.78760 0.352492
\(772\) 15.6705 0.563995
\(773\) −19.6577 −0.707038 −0.353519 0.935427i \(-0.615015\pi\)
−0.353519 + 0.935427i \(0.615015\pi\)
\(774\) −9.89984 −0.355842
\(775\) 13.9575 0.501369
\(776\) −17.3118 −0.621457
\(777\) 0 0
\(778\) 10.6056 0.380229
\(779\) 0.109209 0.00391281
\(780\) −16.0570 −0.574932
\(781\) −21.8833 −0.783045
\(782\) 14.6382 0.523459
\(783\) 6.94535 0.248207
\(784\) 0 0
\(785\) 58.6703 2.09403
\(786\) 0.199626 0.00712044
\(787\) −16.1094 −0.574238 −0.287119 0.957895i \(-0.592698\pi\)
−0.287119 + 0.957895i \(0.592698\pi\)
\(788\) 33.9657 1.20998
\(789\) −2.29263 −0.0816196
\(790\) −23.7251 −0.844100
\(791\) 0 0
\(792\) −13.0725 −0.464513
\(793\) −27.9822 −0.993677
\(794\) 0.621992 0.0220737
\(795\) 13.7920 0.489150
\(796\) −12.1929 −0.432165
\(797\) −20.3109 −0.719447 −0.359724 0.933059i \(-0.617129\pi\)
−0.359724 + 0.933059i \(0.617129\pi\)
\(798\) 0 0
\(799\) −55.2298 −1.95389
\(800\) −56.3177 −1.99113
\(801\) −2.31037 −0.0816329
\(802\) −0.710064 −0.0250732
\(803\) −27.3666 −0.965745
\(804\) −1.30952 −0.0461831
\(805\) 0 0
\(806\) −2.20499 −0.0776673
\(807\) −0.709294 −0.0249683
\(808\) −19.6881 −0.692625
\(809\) 3.62656 0.127503 0.0637515 0.997966i \(-0.479693\pi\)
0.0637515 + 0.997966i \(0.479693\pi\)
\(810\) 12.2025 0.428753
\(811\) −10.2870 −0.361225 −0.180613 0.983554i \(-0.557808\pi\)
−0.180613 + 0.983554i \(0.557808\pi\)
\(812\) 0 0
\(813\) −4.27225 −0.149834
\(814\) 11.4078 0.399843
\(815\) −23.0069 −0.805896
\(816\) 7.82025 0.273763
\(817\) −0.867089 −0.0303356
\(818\) 2.00039 0.0699422
\(819\) 0 0
\(820\) 7.33368 0.256103
\(821\) 6.42929 0.224384 0.112192 0.993687i \(-0.464213\pi\)
0.112192 + 0.993687i \(0.464213\pi\)
\(822\) −1.15246 −0.0401967
\(823\) 4.04441 0.140979 0.0704896 0.997513i \(-0.477544\pi\)
0.0704896 + 0.997513i \(0.477544\pi\)
\(824\) 18.9526 0.660244
\(825\) −17.6569 −0.614735
\(826\) 0 0
\(827\) 4.70225 0.163513 0.0817566 0.996652i \(-0.473947\pi\)
0.0817566 + 0.996652i \(0.473947\pi\)
\(828\) 29.4816 1.02456
\(829\) 6.74357 0.234214 0.117107 0.993119i \(-0.462638\pi\)
0.117107 + 0.993119i \(0.462638\pi\)
\(830\) 28.2495 0.980555
\(831\) −0.0369335 −0.00128121
\(832\) −13.6075 −0.471756
\(833\) 0 0
\(834\) 0.0159426 0.000552048 0
\(835\) 18.4294 0.637777
\(836\) −0.540465 −0.0186924
\(837\) 3.63773 0.125738
\(838\) 12.5590 0.433842
\(839\) −50.7412 −1.75178 −0.875890 0.482511i \(-0.839725\pi\)
−0.875890 + 0.482511i \(0.839725\pi\)
\(840\) 0 0
\(841\) −23.9181 −0.824762
\(842\) 7.64895 0.263600
\(843\) −4.49585 −0.154845
\(844\) −8.97762 −0.309022
\(845\) −14.1866 −0.488032
\(846\) 13.1812 0.453177
\(847\) 0 0
\(848\) 17.2820 0.593468
\(849\) 1.66992 0.0573114
\(850\) 28.4286 0.975093
\(851\) −54.5030 −1.86834
\(852\) −7.63013 −0.261404
\(853\) −9.55166 −0.327042 −0.163521 0.986540i \(-0.552285\pi\)
−0.163521 + 0.986540i \(0.552285\pi\)
\(854\) 0 0
\(855\) 1.21326 0.0414925
\(856\) 3.47375 0.118730
\(857\) 25.8962 0.884598 0.442299 0.896868i \(-0.354163\pi\)
0.442299 + 0.896868i \(0.354163\pi\)
\(858\) 2.78941 0.0952289
\(859\) 41.3258 1.41002 0.705010 0.709198i \(-0.250943\pi\)
0.705010 + 0.709198i \(0.250943\pi\)
\(860\) −58.2275 −1.98554
\(861\) 0 0
\(862\) 4.31088 0.146829
\(863\) −1.76472 −0.0600719 −0.0300360 0.999549i \(-0.509562\pi\)
−0.0300360 + 0.999549i \(0.509562\pi\)
\(864\) −14.6780 −0.499356
\(865\) 58.4492 1.98733
\(866\) −1.78204 −0.0605563
\(867\) −5.55611 −0.188695
\(868\) 0 0
\(869\) −34.7809 −1.17986
\(870\) −2.29687 −0.0778711
\(871\) −5.50526 −0.186539
\(872\) −31.1440 −1.05467
\(873\) 26.8925 0.910175
\(874\) −0.305986 −0.0103501
\(875\) 0 0
\(876\) −9.54201 −0.322395
\(877\) −37.4713 −1.26532 −0.632658 0.774432i \(-0.718036\pi\)
−0.632658 + 0.774432i \(0.718036\pi\)
\(878\) 13.6258 0.459850
\(879\) 2.41430 0.0814325
\(880\) −31.4833 −1.06130
\(881\) 14.9319 0.503068 0.251534 0.967849i \(-0.419065\pi\)
0.251534 + 0.967849i \(0.419065\pi\)
\(882\) 0 0
\(883\) 35.3025 1.18802 0.594012 0.804456i \(-0.297543\pi\)
0.594012 + 0.804456i \(0.297543\pi\)
\(884\) 37.8999 1.27471
\(885\) 20.8729 0.701635
\(886\) 0.262636 0.00882343
\(887\) 6.34513 0.213049 0.106524 0.994310i \(-0.466028\pi\)
0.106524 + 0.994310i \(0.466028\pi\)
\(888\) 8.42654 0.282776
\(889\) 0 0
\(890\) 1.61026 0.0539761
\(891\) 17.8889 0.599299
\(892\) 12.4494 0.416836
\(893\) 1.15449 0.0386334
\(894\) 3.53846 0.118344
\(895\) −23.7403 −0.793551
\(896\) 0 0
\(897\) −13.3270 −0.444975
\(898\) −8.66682 −0.289216
\(899\) 2.66172 0.0887734
\(900\) 57.2559 1.90853
\(901\) −32.5537 −1.08452
\(902\) −1.27400 −0.0424196
\(903\) 0 0
\(904\) −28.3418 −0.942635
\(905\) −60.1513 −1.99950
\(906\) 3.18853 0.105932
\(907\) −21.6718 −0.719599 −0.359800 0.933030i \(-0.617155\pi\)
−0.359800 + 0.933030i \(0.617155\pi\)
\(908\) 18.0768 0.599899
\(909\) 30.5840 1.01441
\(910\) 0 0
\(911\) −5.13876 −0.170255 −0.0851274 0.996370i \(-0.527130\pi\)
−0.0851274 + 0.996370i \(0.527130\pi\)
\(912\) −0.163469 −0.00541301
\(913\) 41.4137 1.37059
\(914\) −10.2527 −0.339129
\(915\) −15.2668 −0.504705
\(916\) −4.15478 −0.137278
\(917\) 0 0
\(918\) 7.40931 0.244544
\(919\) −34.5055 −1.13823 −0.569116 0.822257i \(-0.692715\pi\)
−0.569116 + 0.822257i \(0.692715\pi\)
\(920\) −43.5306 −1.43516
\(921\) 13.9409 0.459369
\(922\) 2.05924 0.0678173
\(923\) −32.0773 −1.05584
\(924\) 0 0
\(925\) −105.850 −3.48032
\(926\) 1.66744 0.0547956
\(927\) −29.4414 −0.966982
\(928\) −10.7399 −0.352553
\(929\) 5.42071 0.177848 0.0889238 0.996038i \(-0.471657\pi\)
0.0889238 + 0.996038i \(0.471657\pi\)
\(930\) −1.20302 −0.0394485
\(931\) 0 0
\(932\) −31.4090 −1.02884
\(933\) 7.38302 0.241709
\(934\) −0.710475 −0.0232475
\(935\) 59.3042 1.93945
\(936\) −19.1622 −0.626337
\(937\) −46.9692 −1.53442 −0.767208 0.641398i \(-0.778355\pi\)
−0.767208 + 0.641398i \(0.778355\pi\)
\(938\) 0 0
\(939\) 12.9814 0.423633
\(940\) 77.5271 2.52865
\(941\) −3.58221 −0.116777 −0.0583884 0.998294i \(-0.518596\pi\)
−0.0583884 + 0.998294i \(0.518596\pi\)
\(942\) −3.55374 −0.115787
\(943\) 6.08681 0.198214
\(944\) 26.1548 0.851267
\(945\) 0 0
\(946\) 10.1152 0.328875
\(947\) −35.5839 −1.15632 −0.578161 0.815923i \(-0.696229\pi\)
−0.578161 + 0.815923i \(0.696229\pi\)
\(948\) −12.1272 −0.393873
\(949\) −40.1150 −1.30219
\(950\) −0.594253 −0.0192801
\(951\) −2.05494 −0.0666359
\(952\) 0 0
\(953\) −43.5164 −1.40963 −0.704817 0.709389i \(-0.748971\pi\)
−0.704817 + 0.709389i \(0.748971\pi\)
\(954\) 7.76926 0.251539
\(955\) −34.5328 −1.11745
\(956\) 41.1378 1.33049
\(957\) −3.36720 −0.108846
\(958\) −6.71659 −0.217003
\(959\) 0 0
\(960\) −7.42413 −0.239613
\(961\) −29.6059 −0.955028
\(962\) 16.7220 0.539138
\(963\) −5.39621 −0.173890
\(964\) −22.6746 −0.730299
\(965\) 35.9432 1.15705
\(966\) 0 0
\(967\) −22.2431 −0.715289 −0.357645 0.933858i \(-0.616420\pi\)
−0.357645 + 0.933858i \(0.616420\pi\)
\(968\) −5.82396 −0.187189
\(969\) 0.307922 0.00989188
\(970\) −18.7433 −0.601812
\(971\) −10.0764 −0.323366 −0.161683 0.986843i \(-0.551692\pi\)
−0.161683 + 0.986843i \(0.551692\pi\)
\(972\) 22.7645 0.730171
\(973\) 0 0
\(974\) 0.701702 0.0224840
\(975\) −25.8822 −0.828893
\(976\) −19.1301 −0.612341
\(977\) 60.5914 1.93849 0.969245 0.246097i \(-0.0791481\pi\)
0.969245 + 0.246097i \(0.0791481\pi\)
\(978\) 1.39356 0.0445612
\(979\) 2.36064 0.0754463
\(980\) 0 0
\(981\) 48.3799 1.54465
\(982\) 16.0570 0.512401
\(983\) −41.9563 −1.33820 −0.669100 0.743173i \(-0.733320\pi\)
−0.669100 + 0.743173i \(0.733320\pi\)
\(984\) −0.941062 −0.0299999
\(985\) 77.9066 2.48231
\(986\) 5.42138 0.172652
\(987\) 0 0
\(988\) −0.792234 −0.0252043
\(989\) −48.3276 −1.53673
\(990\) −14.1535 −0.449829
\(991\) 40.0186 1.27123 0.635616 0.772005i \(-0.280746\pi\)
0.635616 + 0.772005i \(0.280746\pi\)
\(992\) −5.62517 −0.178599
\(993\) 11.4055 0.361943
\(994\) 0 0
\(995\) −27.9666 −0.886599
\(996\) 14.4399 0.457545
\(997\) −6.59339 −0.208815 −0.104407 0.994535i \(-0.533295\pi\)
−0.104407 + 0.994535i \(0.533295\pi\)
\(998\) −6.80187 −0.215310
\(999\) −27.5875 −0.872829
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 2009.2.a.n.1.3 5
7.6 odd 2 287.2.a.e.1.3 5
21.20 even 2 2583.2.a.r.1.3 5
28.27 even 2 4592.2.a.bb.1.3 5
35.34 odd 2 7175.2.a.n.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.a.e.1.3 5 7.6 odd 2
2009.2.a.n.1.3 5 1.1 even 1 trivial
2583.2.a.r.1.3 5 21.20 even 2
4592.2.a.bb.1.3 5 28.27 even 2
7175.2.a.n.1.3 5 35.34 odd 2