Properties

Label 1045.4.a.c.1.17
Level $1045$
Weight $4$
Character 1045.1
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + 42786958 x^{8} - 23777633 x^{7} - 102801526 x^{6} + 28436356 x^{5} + 122325928 x^{4} + 411232 x^{3} - 47350496 x^{2} - 4782848 x + 150528\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Root \(-3.59432\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} +O(q^{10})\) \(q+3.59432 q^{2} +4.02265 q^{3} +4.91912 q^{4} -5.00000 q^{5} +14.4587 q^{6} +10.5448 q^{7} -11.0737 q^{8} -10.8183 q^{9} -17.9716 q^{10} -11.0000 q^{11} +19.7879 q^{12} -33.8888 q^{13} +37.9012 q^{14} -20.1133 q^{15} -79.1552 q^{16} +70.8746 q^{17} -38.8842 q^{18} +19.0000 q^{19} -24.5956 q^{20} +42.4180 q^{21} -39.5375 q^{22} -83.8519 q^{23} -44.5455 q^{24} +25.0000 q^{25} -121.807 q^{26} -152.130 q^{27} +51.8710 q^{28} -121.086 q^{29} -72.2935 q^{30} -46.6505 q^{31} -195.920 q^{32} -44.2492 q^{33} +254.746 q^{34} -52.7238 q^{35} -53.2163 q^{36} +180.848 q^{37} +68.2920 q^{38} -136.323 q^{39} +55.3683 q^{40} -236.659 q^{41} +152.464 q^{42} -471.447 q^{43} -54.1103 q^{44} +54.0913 q^{45} -301.390 q^{46} +137.337 q^{47} -318.414 q^{48} -231.808 q^{49} +89.8579 q^{50} +285.104 q^{51} -166.703 q^{52} -491.854 q^{53} -546.803 q^{54} +55.0000 q^{55} -116.769 q^{56} +76.4304 q^{57} -435.223 q^{58} -294.905 q^{59} -98.9396 q^{60} -303.719 q^{61} -167.677 q^{62} -114.076 q^{63} -70.9559 q^{64} +169.444 q^{65} -159.046 q^{66} -466.369 q^{67} +348.641 q^{68} -337.307 q^{69} -189.506 q^{70} -266.496 q^{71} +119.798 q^{72} +1030.88 q^{73} +650.025 q^{74} +100.566 q^{75} +93.4633 q^{76} -115.992 q^{77} -489.988 q^{78} +1174.04 q^{79} +395.776 q^{80} -319.873 q^{81} -850.627 q^{82} -360.451 q^{83} +208.659 q^{84} -354.373 q^{85} -1694.53 q^{86} -487.089 q^{87} +121.810 q^{88} +988.640 q^{89} +194.421 q^{90} -357.350 q^{91} -412.478 q^{92} -187.659 q^{93} +493.633 q^{94} -95.0000 q^{95} -788.117 q^{96} +1294.58 q^{97} -833.191 q^{98} +119.001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + O(q^{10}) \) \( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.59432 1.27078 0.635392 0.772190i \(-0.280839\pi\)
0.635392 + 0.772190i \(0.280839\pi\)
\(3\) 4.02265 0.774160 0.387080 0.922046i \(-0.373484\pi\)
0.387080 + 0.922046i \(0.373484\pi\)
\(4\) 4.91912 0.614890
\(5\) −5.00000 −0.447214
\(6\) 14.4587 0.983790
\(7\) 10.5448 0.569364 0.284682 0.958622i \(-0.408112\pi\)
0.284682 + 0.958622i \(0.408112\pi\)
\(8\) −11.0737 −0.489391
\(9\) −10.8183 −0.400676
\(10\) −17.9716 −0.568312
\(11\) −11.0000 −0.301511
\(12\) 19.7879 0.476023
\(13\) −33.8888 −0.723005 −0.361503 0.932371i \(-0.617736\pi\)
−0.361503 + 0.932371i \(0.617736\pi\)
\(14\) 37.9012 0.723538
\(15\) −20.1133 −0.346215
\(16\) −79.1552 −1.23680
\(17\) 70.8746 1.01115 0.505577 0.862782i \(-0.331280\pi\)
0.505577 + 0.862782i \(0.331280\pi\)
\(18\) −38.8842 −0.509172
\(19\) 19.0000 0.229416
\(20\) −24.5956 −0.274987
\(21\) 42.4180 0.440779
\(22\) −39.5375 −0.383156
\(23\) −83.8519 −0.760189 −0.380094 0.924948i \(-0.624108\pi\)
−0.380094 + 0.924948i \(0.624108\pi\)
\(24\) −44.5455 −0.378867
\(25\) 25.0000 0.200000
\(26\) −121.807 −0.918783
\(27\) −152.130 −1.08435
\(28\) 51.8710 0.350096
\(29\) −121.086 −0.775351 −0.387676 0.921796i \(-0.626722\pi\)
−0.387676 + 0.921796i \(0.626722\pi\)
\(30\) −72.2935 −0.439964
\(31\) −46.6505 −0.270280 −0.135140 0.990827i \(-0.543148\pi\)
−0.135140 + 0.990827i \(0.543148\pi\)
\(32\) −195.920 −1.08231
\(33\) −44.2492 −0.233418
\(34\) 254.746 1.28496
\(35\) −52.7238 −0.254627
\(36\) −53.2163 −0.246372
\(37\) 180.848 0.803547 0.401773 0.915739i \(-0.368394\pi\)
0.401773 + 0.915739i \(0.368394\pi\)
\(38\) 68.2920 0.291538
\(39\) −136.323 −0.559722
\(40\) 55.3683 0.218862
\(41\) −236.659 −0.901461 −0.450730 0.892660i \(-0.648836\pi\)
−0.450730 + 0.892660i \(0.648836\pi\)
\(42\) 152.464 0.560134
\(43\) −471.447 −1.67198 −0.835988 0.548748i \(-0.815105\pi\)
−0.835988 + 0.548748i \(0.815105\pi\)
\(44\) −54.1103 −0.185396
\(45\) 54.0913 0.179188
\(46\) −301.390 −0.966035
\(47\) 137.337 0.426227 0.213114 0.977027i \(-0.431640\pi\)
0.213114 + 0.977027i \(0.431640\pi\)
\(48\) −318.414 −0.957482
\(49\) −231.808 −0.675825
\(50\) 89.8579 0.254157
\(51\) 285.104 0.782795
\(52\) −166.703 −0.444569
\(53\) −491.854 −1.27474 −0.637372 0.770556i \(-0.719978\pi\)
−0.637372 + 0.770556i \(0.719978\pi\)
\(54\) −546.803 −1.37797
\(55\) 55.0000 0.134840
\(56\) −116.769 −0.278642
\(57\) 76.4304 0.177605
\(58\) −435.223 −0.985303
\(59\) −294.905 −0.650736 −0.325368 0.945588i \(-0.605488\pi\)
−0.325368 + 0.945588i \(0.605488\pi\)
\(60\) −98.9396 −0.212884
\(61\) −303.719 −0.637495 −0.318747 0.947840i \(-0.603262\pi\)
−0.318747 + 0.947840i \(0.603262\pi\)
\(62\) −167.677 −0.343468
\(63\) −114.076 −0.228131
\(64\) −70.9559 −0.138586
\(65\) 169.444 0.323338
\(66\) −159.046 −0.296624
\(67\) −466.369 −0.850388 −0.425194 0.905102i \(-0.639794\pi\)
−0.425194 + 0.905102i \(0.639794\pi\)
\(68\) 348.641 0.621748
\(69\) −337.307 −0.588508
\(70\) −189.506 −0.323576
\(71\) −266.496 −0.445454 −0.222727 0.974881i \(-0.571496\pi\)
−0.222727 + 0.974881i \(0.571496\pi\)
\(72\) 119.798 0.196087
\(73\) 1030.88 1.65282 0.826409 0.563070i \(-0.190380\pi\)
0.826409 + 0.563070i \(0.190380\pi\)
\(74\) 650.025 1.02113
\(75\) 100.566 0.154832
\(76\) 93.4633 0.141065
\(77\) −115.992 −0.171670
\(78\) −489.988 −0.711285
\(79\) 1174.04 1.67202 0.836012 0.548710i \(-0.184881\pi\)
0.836012 + 0.548710i \(0.184881\pi\)
\(80\) 395.776 0.553114
\(81\) −319.873 −0.438783
\(82\) −850.627 −1.14556
\(83\) −360.451 −0.476682 −0.238341 0.971182i \(-0.576604\pi\)
−0.238341 + 0.971182i \(0.576604\pi\)
\(84\) 208.659 0.271031
\(85\) −354.373 −0.452202
\(86\) −1694.53 −2.12472
\(87\) −487.089 −0.600246
\(88\) 121.810 0.147557
\(89\) 988.640 1.17748 0.588739 0.808323i \(-0.299624\pi\)
0.588739 + 0.808323i \(0.299624\pi\)
\(90\) 194.421 0.227709
\(91\) −357.350 −0.411653
\(92\) −412.478 −0.467432
\(93\) −187.659 −0.209240
\(94\) 493.633 0.541643
\(95\) −95.0000 −0.102598
\(96\) −788.117 −0.837884
\(97\) 1294.58 1.35510 0.677550 0.735477i \(-0.263042\pi\)
0.677550 + 0.735477i \(0.263042\pi\)
\(98\) −833.191 −0.858827
\(99\) 119.001 0.120808
\(100\) 122.978 0.122978
\(101\) 920.017 0.906387 0.453194 0.891412i \(-0.350285\pi\)
0.453194 + 0.891412i \(0.350285\pi\)
\(102\) 1024.75 0.994763
\(103\) 20.3830 0.0194990 0.00974949 0.999952i \(-0.496897\pi\)
0.00974949 + 0.999952i \(0.496897\pi\)
\(104\) 375.274 0.353833
\(105\) −212.090 −0.197122
\(106\) −1767.88 −1.61992
\(107\) −851.740 −0.769540 −0.384770 0.923012i \(-0.625719\pi\)
−0.384770 + 0.923012i \(0.625719\pi\)
\(108\) −748.344 −0.666754
\(109\) −2012.22 −1.76821 −0.884107 0.467284i \(-0.845233\pi\)
−0.884107 + 0.467284i \(0.845233\pi\)
\(110\) 197.687 0.171352
\(111\) 727.489 0.622074
\(112\) −834.673 −0.704190
\(113\) 455.894 0.379530 0.189765 0.981830i \(-0.439227\pi\)
0.189765 + 0.981830i \(0.439227\pi\)
\(114\) 274.715 0.225697
\(115\) 419.260 0.339967
\(116\) −595.638 −0.476756
\(117\) 366.618 0.289691
\(118\) −1059.98 −0.826944
\(119\) 747.356 0.575715
\(120\) 222.728 0.169435
\(121\) 121.000 0.0909091
\(122\) −1091.66 −0.810118
\(123\) −951.996 −0.697875
\(124\) −229.480 −0.166193
\(125\) −125.000 −0.0894427
\(126\) −410.025 −0.289904
\(127\) −886.810 −0.619619 −0.309810 0.950799i \(-0.600265\pi\)
−0.309810 + 0.950799i \(0.600265\pi\)
\(128\) 1312.32 0.906201
\(129\) −1896.47 −1.29438
\(130\) 609.036 0.410892
\(131\) −66.4559 −0.0443227 −0.0221614 0.999754i \(-0.507055\pi\)
−0.0221614 + 0.999754i \(0.507055\pi\)
\(132\) −217.667 −0.143526
\(133\) 200.351 0.130621
\(134\) −1676.28 −1.08066
\(135\) 760.649 0.484935
\(136\) −784.842 −0.494850
\(137\) 2326.29 1.45072 0.725360 0.688370i \(-0.241673\pi\)
0.725360 + 0.688370i \(0.241673\pi\)
\(138\) −1212.39 −0.747866
\(139\) −1007.16 −0.614579 −0.307290 0.951616i \(-0.599422\pi\)
−0.307290 + 0.951616i \(0.599422\pi\)
\(140\) −259.355 −0.156568
\(141\) 552.460 0.329968
\(142\) −957.871 −0.566076
\(143\) 372.777 0.217994
\(144\) 856.321 0.495556
\(145\) 605.432 0.346748
\(146\) 3705.32 2.10037
\(147\) −932.483 −0.523197
\(148\) 889.613 0.494093
\(149\) 330.968 0.181973 0.0909864 0.995852i \(-0.470998\pi\)
0.0909864 + 0.995852i \(0.470998\pi\)
\(150\) 361.467 0.196758
\(151\) 1041.39 0.561242 0.280621 0.959819i \(-0.409460\pi\)
0.280621 + 0.959819i \(0.409460\pi\)
\(152\) −210.400 −0.112274
\(153\) −766.739 −0.405145
\(154\) −416.914 −0.218155
\(155\) 233.253 0.120873
\(156\) −670.589 −0.344167
\(157\) −1145.75 −0.582423 −0.291212 0.956659i \(-0.594058\pi\)
−0.291212 + 0.956659i \(0.594058\pi\)
\(158\) 4219.88 2.12478
\(159\) −1978.56 −0.986856
\(160\) 979.598 0.484025
\(161\) −884.199 −0.432824
\(162\) −1149.72 −0.557598
\(163\) −1722.98 −0.827939 −0.413969 0.910291i \(-0.635858\pi\)
−0.413969 + 0.910291i \(0.635858\pi\)
\(164\) −1164.15 −0.554299
\(165\) 221.246 0.104388
\(166\) −1295.57 −0.605759
\(167\) −781.510 −0.362126 −0.181063 0.983472i \(-0.557954\pi\)
−0.181063 + 0.983472i \(0.557954\pi\)
\(168\) −469.722 −0.215713
\(169\) −1048.55 −0.477263
\(170\) −1273.73 −0.574650
\(171\) −205.547 −0.0919214
\(172\) −2319.10 −1.02808
\(173\) 484.011 0.212709 0.106355 0.994328i \(-0.466082\pi\)
0.106355 + 0.994328i \(0.466082\pi\)
\(174\) −1750.75 −0.762782
\(175\) 263.619 0.113873
\(176\) 870.707 0.372909
\(177\) −1186.30 −0.503774
\(178\) 3553.49 1.49632
\(179\) −251.769 −0.105129 −0.0525646 0.998618i \(-0.516740\pi\)
−0.0525646 + 0.998618i \(0.516740\pi\)
\(180\) 266.081 0.110181
\(181\) 384.762 0.158006 0.0790031 0.996874i \(-0.474826\pi\)
0.0790031 + 0.996874i \(0.474826\pi\)
\(182\) −1284.43 −0.523122
\(183\) −1221.75 −0.493523
\(184\) 928.548 0.372030
\(185\) −904.240 −0.359357
\(186\) −674.506 −0.265899
\(187\) −779.621 −0.304874
\(188\) 675.578 0.262083
\(189\) −1604.17 −0.617388
\(190\) −341.460 −0.130380
\(191\) 863.324 0.327057 0.163529 0.986539i \(-0.447712\pi\)
0.163529 + 0.986539i \(0.447712\pi\)
\(192\) −285.431 −0.107287
\(193\) 3430.02 1.27927 0.639633 0.768680i \(-0.279086\pi\)
0.639633 + 0.768680i \(0.279086\pi\)
\(194\) 4653.13 1.72204
\(195\) 681.615 0.250315
\(196\) −1140.29 −0.415558
\(197\) −2567.10 −0.928417 −0.464209 0.885726i \(-0.653661\pi\)
−0.464209 + 0.885726i \(0.653661\pi\)
\(198\) 427.727 0.153521
\(199\) 5071.07 1.80642 0.903212 0.429195i \(-0.141203\pi\)
0.903212 + 0.429195i \(0.141203\pi\)
\(200\) −276.842 −0.0978783
\(201\) −1876.04 −0.658336
\(202\) 3306.83 1.15182
\(203\) −1276.83 −0.441457
\(204\) 1402.46 0.481333
\(205\) 1183.29 0.403146
\(206\) 73.2629 0.0247790
\(207\) 907.132 0.304589
\(208\) 2682.48 0.894213
\(209\) −209.000 −0.0691714
\(210\) −762.318 −0.250500
\(211\) −2862.49 −0.933944 −0.466972 0.884272i \(-0.654655\pi\)
−0.466972 + 0.884272i \(0.654655\pi\)
\(212\) −2419.49 −0.783827
\(213\) −1072.02 −0.344853
\(214\) −3061.42 −0.977919
\(215\) 2357.23 0.747730
\(216\) 1684.63 0.530670
\(217\) −491.919 −0.153888
\(218\) −7232.54 −2.24702
\(219\) 4146.89 1.27955
\(220\) 270.552 0.0829117
\(221\) −2401.86 −0.731070
\(222\) 2614.83 0.790521
\(223\) 1123.50 0.337376 0.168688 0.985669i \(-0.446047\pi\)
0.168688 + 0.985669i \(0.446047\pi\)
\(224\) −2065.93 −0.616230
\(225\) −270.456 −0.0801352
\(226\) 1638.63 0.482300
\(227\) 4297.36 1.25650 0.628250 0.778011i \(-0.283771\pi\)
0.628250 + 0.778011i \(0.283771\pi\)
\(228\) 375.970 0.109207
\(229\) 3859.37 1.11369 0.556843 0.830618i \(-0.312012\pi\)
0.556843 + 0.830618i \(0.312012\pi\)
\(230\) 1506.95 0.432024
\(231\) −466.598 −0.132900
\(232\) 1340.87 0.379450
\(233\) −1279.63 −0.359791 −0.179895 0.983686i \(-0.557576\pi\)
−0.179895 + 0.983686i \(0.557576\pi\)
\(234\) 1317.74 0.368134
\(235\) −686.686 −0.190615
\(236\) −1450.67 −0.400131
\(237\) 4722.76 1.29442
\(238\) 2686.24 0.731609
\(239\) −181.882 −0.0492259 −0.0246130 0.999697i \(-0.507835\pi\)
−0.0246130 + 0.999697i \(0.507835\pi\)
\(240\) 1592.07 0.428199
\(241\) 1712.24 0.457655 0.228827 0.973467i \(-0.426511\pi\)
0.228827 + 0.973467i \(0.426511\pi\)
\(242\) 434.912 0.115526
\(243\) 2820.77 0.744660
\(244\) −1494.03 −0.391989
\(245\) 1159.04 0.302238
\(246\) −3421.78 −0.886848
\(247\) −643.888 −0.165869
\(248\) 516.592 0.132273
\(249\) −1449.97 −0.369028
\(250\) −449.290 −0.113662
\(251\) −392.824 −0.0987842 −0.0493921 0.998779i \(-0.515728\pi\)
−0.0493921 + 0.998779i \(0.515728\pi\)
\(252\) −561.153 −0.140275
\(253\) 922.371 0.229205
\(254\) −3187.48 −0.787402
\(255\) −1425.52 −0.350077
\(256\) 5284.54 1.29017
\(257\) −1628.06 −0.395159 −0.197579 0.980287i \(-0.563308\pi\)
−0.197579 + 0.980287i \(0.563308\pi\)
\(258\) −6816.51 −1.64487
\(259\) 1907.00 0.457511
\(260\) 833.516 0.198817
\(261\) 1309.94 0.310665
\(262\) −238.864 −0.0563246
\(263\) −5121.43 −1.20076 −0.600382 0.799713i \(-0.704985\pi\)
−0.600382 + 0.799713i \(0.704985\pi\)
\(264\) 490.001 0.114233
\(265\) 2459.27 0.570083
\(266\) 720.124 0.165991
\(267\) 3976.96 0.911557
\(268\) −2294.12 −0.522895
\(269\) −940.184 −0.213100 −0.106550 0.994307i \(-0.533980\pi\)
−0.106550 + 0.994307i \(0.533980\pi\)
\(270\) 2734.01 0.616247
\(271\) 3839.71 0.860685 0.430342 0.902666i \(-0.358393\pi\)
0.430342 + 0.902666i \(0.358393\pi\)
\(272\) −5610.10 −1.25060
\(273\) −1437.49 −0.318686
\(274\) 8361.44 1.84355
\(275\) −275.000 −0.0603023
\(276\) −1659.26 −0.361867
\(277\) −5076.91 −1.10123 −0.550617 0.834758i \(-0.685608\pi\)
−0.550617 + 0.834758i \(0.685608\pi\)
\(278\) −3620.07 −0.780997
\(279\) 504.677 0.108295
\(280\) 583.846 0.124612
\(281\) −2830.73 −0.600951 −0.300476 0.953790i \(-0.597145\pi\)
−0.300476 + 0.953790i \(0.597145\pi\)
\(282\) 1985.72 0.419318
\(283\) −797.293 −0.167470 −0.0837352 0.996488i \(-0.526685\pi\)
−0.0837352 + 0.996488i \(0.526685\pi\)
\(284\) −1310.93 −0.273905
\(285\) −382.152 −0.0794272
\(286\) 1339.88 0.277024
\(287\) −2495.51 −0.513259
\(288\) 2119.51 0.433657
\(289\) 110.210 0.0224323
\(290\) 2176.11 0.440641
\(291\) 5207.65 1.04906
\(292\) 5071.04 1.01630
\(293\) −5690.33 −1.13458 −0.567291 0.823517i \(-0.692009\pi\)
−0.567291 + 0.823517i \(0.692009\pi\)
\(294\) −3351.64 −0.664869
\(295\) 1474.53 0.291018
\(296\) −2002.65 −0.393249
\(297\) 1673.43 0.326943
\(298\) 1189.60 0.231248
\(299\) 2841.64 0.549620
\(300\) 494.698 0.0952047
\(301\) −4971.30 −0.951963
\(302\) 3743.10 0.713216
\(303\) 3700.91 0.701689
\(304\) −1503.95 −0.283741
\(305\) 1518.59 0.285096
\(306\) −2755.91 −0.514852
\(307\) 3156.35 0.586784 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(308\) −570.581 −0.105558
\(309\) 81.9937 0.0150953
\(310\) 838.384 0.153603
\(311\) 2243.87 0.409127 0.204563 0.978853i \(-0.434423\pi\)
0.204563 + 0.978853i \(0.434423\pi\)
\(312\) 1509.60 0.273923
\(313\) −4984.70 −0.900165 −0.450082 0.892987i \(-0.648605\pi\)
−0.450082 + 0.892987i \(0.648605\pi\)
\(314\) −4118.17 −0.740134
\(315\) 570.380 0.102023
\(316\) 5775.25 1.02811
\(317\) −4774.12 −0.845872 −0.422936 0.906160i \(-0.639000\pi\)
−0.422936 + 0.906160i \(0.639000\pi\)
\(318\) −7111.57 −1.25408
\(319\) 1331.95 0.233777
\(320\) 354.779 0.0619774
\(321\) −3426.25 −0.595747
\(322\) −3178.09 −0.550025
\(323\) 1346.62 0.231975
\(324\) −1573.49 −0.269803
\(325\) −847.221 −0.144601
\(326\) −6192.93 −1.05213
\(327\) −8094.45 −1.36888
\(328\) 2620.68 0.441167
\(329\) 1448.19 0.242679
\(330\) 795.228 0.132654
\(331\) 3686.50 0.612169 0.306085 0.952004i \(-0.400981\pi\)
0.306085 + 0.952004i \(0.400981\pi\)
\(332\) −1773.10 −0.293107
\(333\) −1956.46 −0.321962
\(334\) −2808.99 −0.460183
\(335\) 2331.84 0.380305
\(336\) −3357.60 −0.545156
\(337\) −4919.26 −0.795160 −0.397580 0.917567i \(-0.630150\pi\)
−0.397580 + 0.917567i \(0.630150\pi\)
\(338\) −3768.81 −0.606498
\(339\) 1833.90 0.293817
\(340\) −1743.20 −0.278054
\(341\) 513.156 0.0814925
\(342\) −738.801 −0.116812
\(343\) −6061.22 −0.954154
\(344\) 5220.64 0.818251
\(345\) 1686.54 0.263189
\(346\) 1739.69 0.270307
\(347\) −2358.41 −0.364859 −0.182430 0.983219i \(-0.558396\pi\)
−0.182430 + 0.983219i \(0.558396\pi\)
\(348\) −2396.05 −0.369085
\(349\) −1502.56 −0.230459 −0.115229 0.993339i \(-0.536760\pi\)
−0.115229 + 0.993339i \(0.536760\pi\)
\(350\) 947.531 0.144708
\(351\) 5155.50 0.783989
\(352\) 2155.12 0.326330
\(353\) −6471.49 −0.975759 −0.487879 0.872911i \(-0.662229\pi\)
−0.487879 + 0.872911i \(0.662229\pi\)
\(354\) −4263.95 −0.640187
\(355\) 1332.48 0.199213
\(356\) 4863.24 0.724020
\(357\) 3006.36 0.445695
\(358\) −904.939 −0.133596
\(359\) 5482.69 0.806032 0.403016 0.915193i \(-0.367962\pi\)
0.403016 + 0.915193i \(0.367962\pi\)
\(360\) −598.989 −0.0876930
\(361\) 361.000 0.0526316
\(362\) 1382.96 0.200792
\(363\) 486.741 0.0703782
\(364\) −1757.85 −0.253121
\(365\) −5154.42 −0.739163
\(366\) −4391.37 −0.627161
\(367\) 1049.99 0.149343 0.0746714 0.997208i \(-0.476209\pi\)
0.0746714 + 0.997208i \(0.476209\pi\)
\(368\) 6637.32 0.940201
\(369\) 2560.23 0.361194
\(370\) −3250.13 −0.456665
\(371\) −5186.49 −0.725793
\(372\) −923.117 −0.128660
\(373\) 11106.9 1.54180 0.770902 0.636953i \(-0.219806\pi\)
0.770902 + 0.636953i \(0.219806\pi\)
\(374\) −2802.20 −0.387429
\(375\) −502.832 −0.0692430
\(376\) −1520.83 −0.208592
\(377\) 4103.48 0.560583
\(378\) −5765.91 −0.784567
\(379\) −1390.60 −0.188471 −0.0942353 0.995550i \(-0.530041\pi\)
−0.0942353 + 0.995550i \(0.530041\pi\)
\(380\) −467.316 −0.0630864
\(381\) −3567.33 −0.479685
\(382\) 3103.06 0.415619
\(383\) 1683.03 0.224540 0.112270 0.993678i \(-0.464188\pi\)
0.112270 + 0.993678i \(0.464188\pi\)
\(384\) 5279.01 0.701545
\(385\) 579.962 0.0767730
\(386\) 12328.6 1.62567
\(387\) 5100.23 0.669921
\(388\) 6368.20 0.833238
\(389\) −4021.88 −0.524210 −0.262105 0.965039i \(-0.584417\pi\)
−0.262105 + 0.965039i \(0.584417\pi\)
\(390\) 2449.94 0.318096
\(391\) −5942.97 −0.768668
\(392\) 2566.96 0.330743
\(393\) −267.329 −0.0343129
\(394\) −9226.97 −1.17982
\(395\) −5870.21 −0.747752
\(396\) 585.379 0.0742839
\(397\) 3946.62 0.498930 0.249465 0.968384i \(-0.419745\pi\)
0.249465 + 0.968384i \(0.419745\pi\)
\(398\) 18227.0 2.29557
\(399\) 805.941 0.101122
\(400\) −1978.88 −0.247360
\(401\) 4261.66 0.530716 0.265358 0.964150i \(-0.414510\pi\)
0.265358 + 0.964150i \(0.414510\pi\)
\(402\) −6743.08 −0.836603
\(403\) 1580.93 0.195414
\(404\) 4525.67 0.557328
\(405\) 1599.36 0.196230
\(406\) −4589.33 −0.560996
\(407\) −1989.33 −0.242279
\(408\) −3157.15 −0.383093
\(409\) 10343.7 1.25052 0.625260 0.780416i \(-0.284993\pi\)
0.625260 + 0.780416i \(0.284993\pi\)
\(410\) 4253.13 0.512311
\(411\) 9357.88 1.12309
\(412\) 100.266 0.0119897
\(413\) −3109.71 −0.370505
\(414\) 3260.52 0.387067
\(415\) 1802.25 0.213179
\(416\) 6639.49 0.782519
\(417\) −4051.47 −0.475783
\(418\) −751.212 −0.0879019
\(419\) −9746.66 −1.13641 −0.568205 0.822887i \(-0.692362\pi\)
−0.568205 + 0.822887i \(0.692362\pi\)
\(420\) −1043.29 −0.121209
\(421\) 4864.35 0.563121 0.281561 0.959543i \(-0.409148\pi\)
0.281561 + 0.959543i \(0.409148\pi\)
\(422\) −10288.7 −1.18684
\(423\) −1485.75 −0.170779
\(424\) 5446.63 0.623849
\(425\) 1771.87 0.202231
\(426\) −3853.18 −0.438233
\(427\) −3202.64 −0.362967
\(428\) −4189.81 −0.473182
\(429\) 1499.55 0.168763
\(430\) 8472.65 0.950203
\(431\) −12532.0 −1.40057 −0.700285 0.713863i \(-0.746944\pi\)
−0.700285 + 0.713863i \(0.746944\pi\)
\(432\) 12041.9 1.34112
\(433\) −12184.0 −1.35225 −0.676126 0.736786i \(-0.736342\pi\)
−0.676126 + 0.736786i \(0.736342\pi\)
\(434\) −1768.11 −0.195558
\(435\) 2435.44 0.268438
\(436\) −9898.33 −1.08726
\(437\) −1593.19 −0.174399
\(438\) 14905.2 1.62603
\(439\) 15031.0 1.63414 0.817072 0.576536i \(-0.195596\pi\)
0.817072 + 0.576536i \(0.195596\pi\)
\(440\) −609.051 −0.0659895
\(441\) 2507.76 0.270787
\(442\) −8633.04 −0.929031
\(443\) 6360.31 0.682138 0.341069 0.940038i \(-0.389211\pi\)
0.341069 + 0.940038i \(0.389211\pi\)
\(444\) 3578.61 0.382507
\(445\) −4943.20 −0.526585
\(446\) 4038.20 0.428732
\(447\) 1331.37 0.140876
\(448\) −748.213 −0.0789057
\(449\) 4578.96 0.481279 0.240640 0.970615i \(-0.422643\pi\)
0.240640 + 0.970615i \(0.422643\pi\)
\(450\) −972.106 −0.101834
\(451\) 2603.25 0.271801
\(452\) 2242.59 0.233369
\(453\) 4189.17 0.434491
\(454\) 15446.1 1.59674
\(455\) 1786.75 0.184097
\(456\) −846.365 −0.0869181
\(457\) −19236.5 −1.96903 −0.984513 0.175313i \(-0.943906\pi\)
−0.984513 + 0.175313i \(0.943906\pi\)
\(458\) 13871.8 1.41525
\(459\) −10782.1 −1.09644
\(460\) 2062.39 0.209042
\(461\) −465.907 −0.0470704 −0.0235352 0.999723i \(-0.507492\pi\)
−0.0235352 + 0.999723i \(0.507492\pi\)
\(462\) −1677.10 −0.168887
\(463\) −14416.8 −1.44710 −0.723551 0.690271i \(-0.757491\pi\)
−0.723551 + 0.690271i \(0.757491\pi\)
\(464\) 9584.62 0.958954
\(465\) 938.295 0.0935750
\(466\) −4599.39 −0.457216
\(467\) 5194.12 0.514679 0.257340 0.966321i \(-0.417154\pi\)
0.257340 + 0.966321i \(0.417154\pi\)
\(468\) 1803.44 0.178128
\(469\) −4917.75 −0.484180
\(470\) −2468.17 −0.242230
\(471\) −4608.94 −0.450889
\(472\) 3265.68 0.318464
\(473\) 5185.91 0.504120
\(474\) 16975.1 1.64492
\(475\) 475.000 0.0458831
\(476\) 3676.34 0.354001
\(477\) 5321.01 0.510759
\(478\) −653.743 −0.0625554
\(479\) 9212.89 0.878805 0.439403 0.898290i \(-0.355190\pi\)
0.439403 + 0.898290i \(0.355190\pi\)
\(480\) 3940.59 0.374713
\(481\) −6128.73 −0.580969
\(482\) 6154.32 0.581580
\(483\) −3556.83 −0.335075
\(484\) 595.213 0.0558991
\(485\) −6472.90 −0.606019
\(486\) 10138.7 0.946301
\(487\) −13265.2 −1.23430 −0.617152 0.786844i \(-0.711714\pi\)
−0.617152 + 0.786844i \(0.711714\pi\)
\(488\) 3363.28 0.311984
\(489\) −6930.94 −0.640957
\(490\) 4165.96 0.384079
\(491\) −10607.8 −0.974995 −0.487498 0.873124i \(-0.662090\pi\)
−0.487498 + 0.873124i \(0.662090\pi\)
\(492\) −4682.98 −0.429116
\(493\) −8581.95 −0.783999
\(494\) −2314.34 −0.210783
\(495\) −595.004 −0.0540271
\(496\) 3692.63 0.334283
\(497\) −2810.14 −0.253626
\(498\) −5211.64 −0.468955
\(499\) 15115.8 1.35607 0.678034 0.735030i \(-0.262832\pi\)
0.678034 + 0.735030i \(0.262832\pi\)
\(500\) −614.890 −0.0549974
\(501\) −3143.74 −0.280343
\(502\) −1411.94 −0.125533
\(503\) 8485.80 0.752213 0.376106 0.926577i \(-0.377263\pi\)
0.376106 + 0.926577i \(0.377263\pi\)
\(504\) 1263.24 0.111645
\(505\) −4600.08 −0.405349
\(506\) 3315.30 0.291270
\(507\) −4217.94 −0.369478
\(508\) −4362.32 −0.380998
\(509\) 9601.10 0.836073 0.418036 0.908430i \(-0.362718\pi\)
0.418036 + 0.908430i \(0.362718\pi\)
\(510\) −5123.77 −0.444871
\(511\) 10870.4 0.941055
\(512\) 8495.76 0.733326
\(513\) −2890.47 −0.248766
\(514\) −5851.78 −0.502161
\(515\) −101.915 −0.00872021
\(516\) −9328.95 −0.795899
\(517\) −1510.71 −0.128512
\(518\) 6854.37 0.581397
\(519\) 1947.01 0.164671
\(520\) −1876.37 −0.158239
\(521\) −3937.13 −0.331073 −0.165536 0.986204i \(-0.552936\pi\)
−0.165536 + 0.986204i \(0.552936\pi\)
\(522\) 4708.35 0.394787
\(523\) −3942.45 −0.329620 −0.164810 0.986325i \(-0.552701\pi\)
−0.164810 + 0.986325i \(0.552701\pi\)
\(524\) −326.905 −0.0272536
\(525\) 1060.45 0.0881558
\(526\) −18408.1 −1.52591
\(527\) −3306.34 −0.273295
\(528\) 3502.55 0.288692
\(529\) −5135.85 −0.422113
\(530\) 8839.41 0.724451
\(531\) 3190.36 0.260734
\(532\) 985.549 0.0803176
\(533\) 8020.09 0.651761
\(534\) 14294.4 1.15839
\(535\) 4258.70 0.344149
\(536\) 5164.41 0.416173
\(537\) −1012.78 −0.0813868
\(538\) −3379.32 −0.270804
\(539\) 2549.89 0.203769
\(540\) 3741.72 0.298182
\(541\) 6122.28 0.486539 0.243269 0.969959i \(-0.421780\pi\)
0.243269 + 0.969959i \(0.421780\pi\)
\(542\) 13801.1 1.09374
\(543\) 1547.76 0.122322
\(544\) −13885.7 −1.09439
\(545\) 10061.1 0.790770
\(546\) −5166.81 −0.404980
\(547\) 11279.7 0.881689 0.440844 0.897584i \(-0.354679\pi\)
0.440844 + 0.897584i \(0.354679\pi\)
\(548\) 11443.3 0.892033
\(549\) 3285.70 0.255429
\(550\) −988.437 −0.0766311
\(551\) −2300.64 −0.177878
\(552\) 3735.23 0.288011
\(553\) 12380.0 0.951991
\(554\) −18248.0 −1.39943
\(555\) −3637.45 −0.278200
\(556\) −4954.36 −0.377899
\(557\) −24386.0 −1.85506 −0.927529 0.373752i \(-0.878071\pi\)
−0.927529 + 0.373752i \(0.878071\pi\)
\(558\) 1813.97 0.137619
\(559\) 15976.8 1.20885
\(560\) 4173.37 0.314923
\(561\) −3136.14 −0.236022
\(562\) −10174.5 −0.763678
\(563\) −11983.0 −0.897022 −0.448511 0.893777i \(-0.648046\pi\)
−0.448511 + 0.893777i \(0.648046\pi\)
\(564\) 2717.62 0.202894
\(565\) −2279.47 −0.169731
\(566\) −2865.72 −0.212819
\(567\) −3372.98 −0.249827
\(568\) 2951.09 0.218001
\(569\) −8467.22 −0.623839 −0.311919 0.950109i \(-0.600972\pi\)
−0.311919 + 0.950109i \(0.600972\pi\)
\(570\) −1373.58 −0.100935
\(571\) −26265.3 −1.92499 −0.962493 0.271308i \(-0.912544\pi\)
−0.962493 + 0.271308i \(0.912544\pi\)
\(572\) 1833.74 0.134043
\(573\) 3472.86 0.253195
\(574\) −8969.66 −0.652241
\(575\) −2096.30 −0.152038
\(576\) 767.618 0.0555280
\(577\) 7838.60 0.565555 0.282777 0.959186i \(-0.408744\pi\)
0.282777 + 0.959186i \(0.408744\pi\)
\(578\) 396.130 0.0285066
\(579\) 13797.8 0.990357
\(580\) 2978.19 0.213212
\(581\) −3800.87 −0.271405
\(582\) 18717.9 1.33313
\(583\) 5410.40 0.384350
\(584\) −11415.7 −0.808875
\(585\) −1833.09 −0.129554
\(586\) −20452.9 −1.44181
\(587\) −21907.4 −1.54040 −0.770199 0.637804i \(-0.779843\pi\)
−0.770199 + 0.637804i \(0.779843\pi\)
\(588\) −4586.99 −0.321708
\(589\) −886.360 −0.0620065
\(590\) 5299.92 0.369820
\(591\) −10326.6 −0.718744
\(592\) −14315.1 −0.993827
\(593\) 19208.3 1.33017 0.665086 0.746767i \(-0.268395\pi\)
0.665086 + 0.746767i \(0.268395\pi\)
\(594\) 6014.83 0.415474
\(595\) −3736.78 −0.257467
\(596\) 1628.07 0.111893
\(597\) 20399.1 1.39846
\(598\) 10213.8 0.698448
\(599\) 29179.6 1.99040 0.995198 0.0978853i \(-0.0312078\pi\)
0.995198 + 0.0978853i \(0.0312078\pi\)
\(600\) −1113.64 −0.0757735
\(601\) −26008.0 −1.76521 −0.882603 0.470118i \(-0.844211\pi\)
−0.882603 + 0.470118i \(0.844211\pi\)
\(602\) −17868.4 −1.20974
\(603\) 5045.29 0.340730
\(604\) 5122.74 0.345102
\(605\) −605.000 −0.0406558
\(606\) 13302.2 0.891694
\(607\) −184.750 −0.0123538 −0.00617690 0.999981i \(-0.501966\pi\)
−0.00617690 + 0.999981i \(0.501966\pi\)
\(608\) −3722.47 −0.248300
\(609\) −5136.24 −0.341758
\(610\) 5458.31 0.362296
\(611\) −4654.20 −0.308165
\(612\) −3771.68 −0.249120
\(613\) 26.0343 0.00171536 0.000857681 1.00000i \(-0.499727\pi\)
0.000857681 1.00000i \(0.499727\pi\)
\(614\) 11344.9 0.745675
\(615\) 4759.98 0.312099
\(616\) 1284.46 0.0840137
\(617\) −8708.33 −0.568208 −0.284104 0.958794i \(-0.591696\pi\)
−0.284104 + 0.958794i \(0.591696\pi\)
\(618\) 294.711 0.0191829
\(619\) 9979.81 0.648017 0.324008 0.946054i \(-0.394969\pi\)
0.324008 + 0.946054i \(0.394969\pi\)
\(620\) 1147.40 0.0743236
\(621\) 12756.4 0.824309
\(622\) 8065.20 0.519911
\(623\) 10425.0 0.670414
\(624\) 10790.7 0.692264
\(625\) 625.000 0.0400000
\(626\) −17916.6 −1.14391
\(627\) −840.735 −0.0535498
\(628\) −5636.06 −0.358126
\(629\) 12817.5 0.812510
\(630\) 2050.13 0.129649
\(631\) −2728.74 −0.172154 −0.0860771 0.996288i \(-0.527433\pi\)
−0.0860771 + 0.996288i \(0.527433\pi\)
\(632\) −13000.9 −0.818275
\(633\) −11514.8 −0.723022
\(634\) −17159.7 −1.07492
\(635\) 4434.05 0.277102
\(636\) −9732.78 −0.606808
\(637\) 7855.70 0.488625
\(638\) 4787.45 0.297080
\(639\) 2883.02 0.178483
\(640\) −6561.60 −0.405266
\(641\) 9365.09 0.577065 0.288533 0.957470i \(-0.406833\pi\)
0.288533 + 0.957470i \(0.406833\pi\)
\(642\) −12315.0 −0.757066
\(643\) 8773.54 0.538094 0.269047 0.963127i \(-0.413291\pi\)
0.269047 + 0.963127i \(0.413291\pi\)
\(644\) −4349.48 −0.266139
\(645\) 9482.34 0.578863
\(646\) 4840.17 0.294789
\(647\) −15751.9 −0.957140 −0.478570 0.878049i \(-0.658845\pi\)
−0.478570 + 0.878049i \(0.658845\pi\)
\(648\) 3542.16 0.214736
\(649\) 3243.96 0.196204
\(650\) −3045.18 −0.183757
\(651\) −1978.82 −0.119134
\(652\) −8475.53 −0.509091
\(653\) −29366.4 −1.75987 −0.879935 0.475094i \(-0.842414\pi\)
−0.879935 + 0.475094i \(0.842414\pi\)
\(654\) −29094.0 −1.73955
\(655\) 332.280 0.0198217
\(656\) 18732.8 1.11493
\(657\) −11152.4 −0.662245
\(658\) 5205.25 0.308392
\(659\) −7085.46 −0.418832 −0.209416 0.977827i \(-0.567156\pi\)
−0.209416 + 0.977827i \(0.567156\pi\)
\(660\) 1088.34 0.0641870
\(661\) −6020.01 −0.354238 −0.177119 0.984189i \(-0.556678\pi\)
−0.177119 + 0.984189i \(0.556678\pi\)
\(662\) 13250.4 0.777935
\(663\) −9661.84 −0.565965
\(664\) 3991.51 0.233284
\(665\) −1001.75 −0.0584155
\(666\) −7032.14 −0.409144
\(667\) 10153.3 0.589413
\(668\) −3844.34 −0.222668
\(669\) 4519.44 0.261183
\(670\) 8381.38 0.483285
\(671\) 3340.90 0.192212
\(672\) −8310.51 −0.477061
\(673\) 9514.09 0.544935 0.272468 0.962165i \(-0.412160\pi\)
0.272468 + 0.962165i \(0.412160\pi\)
\(674\) −17681.4 −1.01048
\(675\) −3803.24 −0.216870
\(676\) −5157.93 −0.293464
\(677\) 25255.2 1.43373 0.716865 0.697212i \(-0.245576\pi\)
0.716865 + 0.697212i \(0.245576\pi\)
\(678\) 6591.63 0.373377
\(679\) 13651.1 0.771545
\(680\) 3924.21 0.221304
\(681\) 17286.8 0.972732
\(682\) 1844.45 0.103559
\(683\) −5594.79 −0.313439 −0.156719 0.987643i \(-0.550092\pi\)
−0.156719 + 0.987643i \(0.550092\pi\)
\(684\) −1011.11 −0.0565215
\(685\) −11631.5 −0.648782
\(686\) −21785.9 −1.21252
\(687\) 15524.9 0.862172
\(688\) 37317.5 2.06790
\(689\) 16668.4 0.921647
\(690\) 6061.95 0.334456
\(691\) 8966.72 0.493647 0.246824 0.969060i \(-0.420613\pi\)
0.246824 + 0.969060i \(0.420613\pi\)
\(692\) 2380.91 0.130793
\(693\) 1254.84 0.0687839
\(694\) −8476.88 −0.463657
\(695\) 5035.82 0.274848
\(696\) 5393.86 0.293755
\(697\) −16773.1 −0.911516
\(698\) −5400.68 −0.292863
\(699\) −5147.50 −0.278536
\(700\) 1296.77 0.0700192
\(701\) −9175.65 −0.494379 −0.247189 0.968967i \(-0.579507\pi\)
−0.247189 + 0.968967i \(0.579507\pi\)
\(702\) 18530.5 0.996280
\(703\) 3436.11 0.184346
\(704\) 780.514 0.0417851
\(705\) −2762.30 −0.147566
\(706\) −23260.6 −1.23998
\(707\) 9701.37 0.516064
\(708\) −5835.56 −0.309765
\(709\) −18524.6 −0.981247 −0.490624 0.871372i \(-0.663231\pi\)
−0.490624 + 0.871372i \(0.663231\pi\)
\(710\) 4789.36 0.253157
\(711\) −12701.1 −0.669940
\(712\) −10947.9 −0.576248
\(713\) 3911.74 0.205464
\(714\) 10805.8 0.566382
\(715\) −1863.89 −0.0974900
\(716\) −1238.48 −0.0646429
\(717\) −731.650 −0.0381087
\(718\) 19706.5 1.02429
\(719\) −376.118 −0.0195088 −0.00975441 0.999952i \(-0.503105\pi\)
−0.00975441 + 0.999952i \(0.503105\pi\)
\(720\) −4281.61 −0.221620
\(721\) 214.934 0.0111020
\(722\) 1297.55 0.0668833
\(723\) 6887.73 0.354298
\(724\) 1892.69 0.0971564
\(725\) −3027.16 −0.155070
\(726\) 1749.50 0.0894354
\(727\) 2761.55 0.140880 0.0704402 0.997516i \(-0.477560\pi\)
0.0704402 + 0.997516i \(0.477560\pi\)
\(728\) 3957.17 0.201460
\(729\) 19983.5 1.01527
\(730\) −18526.6 −0.939316
\(731\) −33413.6 −1.69063
\(732\) −6009.96 −0.303462
\(733\) −30604.4 −1.54215 −0.771077 0.636742i \(-0.780282\pi\)
−0.771077 + 0.636742i \(0.780282\pi\)
\(734\) 3773.98 0.189782
\(735\) 4662.41 0.233981
\(736\) 16428.2 0.822762
\(737\) 5130.05 0.256402
\(738\) 9202.30 0.458999
\(739\) −4943.77 −0.246088 −0.123044 0.992401i \(-0.539266\pi\)
−0.123044 + 0.992401i \(0.539266\pi\)
\(740\) −4448.07 −0.220965
\(741\) −2590.14 −0.128409
\(742\) −18641.9 −0.922326
\(743\) 40198.0 1.98482 0.992410 0.122974i \(-0.0392430\pi\)
0.992410 + 0.122974i \(0.0392430\pi\)
\(744\) 2078.07 0.102400
\(745\) −1654.84 −0.0813807
\(746\) 39921.7 1.95930
\(747\) 3899.45 0.190995
\(748\) −3835.05 −0.187464
\(749\) −8981.40 −0.438148
\(750\) −1807.34 −0.0879928
\(751\) −29166.1 −1.41716 −0.708579 0.705632i \(-0.750663\pi\)
−0.708579 + 0.705632i \(0.750663\pi\)
\(752\) −10871.0 −0.527158
\(753\) −1580.20 −0.0764748
\(754\) 14749.2 0.712379
\(755\) −5206.97 −0.250995
\(756\) −7891.12 −0.379626
\(757\) 21648.2 1.03939 0.519695 0.854352i \(-0.326046\pi\)
0.519695 + 0.854352i \(0.326046\pi\)
\(758\) −4998.26 −0.239505
\(759\) 3710.38 0.177442
\(760\) 1052.00 0.0502105
\(761\) 29779.1 1.41852 0.709259 0.704948i \(-0.249030\pi\)
0.709259 + 0.704948i \(0.249030\pi\)
\(762\) −12822.1 −0.609575
\(763\) −21218.4 −1.00676
\(764\) 4246.80 0.201104
\(765\) 3833.70 0.181186
\(766\) 6049.34 0.285342
\(767\) 9993.99 0.470485
\(768\) 21257.9 0.998799
\(769\) −7847.97 −0.368017 −0.184008 0.982925i \(-0.558907\pi\)
−0.184008 + 0.982925i \(0.558907\pi\)
\(770\) 2084.57 0.0975619
\(771\) −6549.13 −0.305916
\(772\) 16872.7 0.786608
\(773\) −4331.18 −0.201529 −0.100764 0.994910i \(-0.532129\pi\)
−0.100764 + 0.994910i \(0.532129\pi\)
\(774\) 18331.9 0.851324
\(775\) −1166.26 −0.0540560
\(776\) −14335.7 −0.663174
\(777\) 7671.21 0.354187
\(778\) −14455.9 −0.666157
\(779\) −4496.52 −0.206809
\(780\) 3352.95 0.153916
\(781\) 2931.46 0.134310
\(782\) −21360.9 −0.976810
\(783\) 18420.8 0.840750
\(784\) 18348.8 0.835860
\(785\) 5728.73 0.260468
\(786\) −960.866 −0.0436043
\(787\) −2772.13 −0.125560 −0.0627801 0.998027i \(-0.519997\pi\)
−0.0627801 + 0.998027i \(0.519997\pi\)
\(788\) −12627.9 −0.570874
\(789\) −20601.8 −0.929584
\(790\) −21099.4 −0.950231
\(791\) 4807.29 0.216091
\(792\) −1317.77 −0.0591226
\(793\) 10292.7 0.460912
\(794\) 14185.4 0.634032
\(795\) 9892.80 0.441335
\(796\) 24945.2 1.11075
\(797\) 27943.2 1.24190 0.620952 0.783848i \(-0.286746\pi\)
0.620952 + 0.783848i \(0.286746\pi\)
\(798\) 2896.81 0.128504
\(799\) 9733.72 0.430981
\(800\) −4897.99 −0.216463
\(801\) −10695.4 −0.471788
\(802\) 15317.8 0.674425
\(803\) −11339.7 −0.498344
\(804\) −9228.46 −0.404804
\(805\) 4421.00 0.193565
\(806\) 5682.37 0.248329
\(807\) −3782.03 −0.164974
\(808\) −10188.0 −0.443578
\(809\) 7822.12 0.339940 0.169970 0.985449i \(-0.445633\pi\)
0.169970 + 0.985449i \(0.445633\pi\)
\(810\) 5748.62 0.249365
\(811\) −26969.6 −1.16773 −0.583867 0.811850i \(-0.698461\pi\)
−0.583867 + 0.811850i \(0.698461\pi\)
\(812\) −6280.87 −0.271447
\(813\) 15445.8 0.666308
\(814\) −7150.28 −0.307883
\(815\) 8614.89 0.370265
\(816\) −22567.5 −0.968161
\(817\) −8957.49 −0.383578
\(818\) 37178.5 1.58914
\(819\) 3865.90 0.164940
\(820\) 5820.76 0.247890
\(821\) −17650.2 −0.750299 −0.375150 0.926964i \(-0.622409\pi\)
−0.375150 + 0.926964i \(0.622409\pi\)
\(822\) 33635.2 1.42720
\(823\) 39919.7 1.69078 0.845391 0.534148i \(-0.179368\pi\)
0.845391 + 0.534148i \(0.179368\pi\)
\(824\) −225.714 −0.00954263
\(825\) −1106.23 −0.0466836
\(826\) −11177.3 −0.470832
\(827\) −32658.1 −1.37320 −0.686598 0.727037i \(-0.740897\pi\)
−0.686598 + 0.727037i \(0.740897\pi\)
\(828\) 4462.29 0.187289
\(829\) −446.740 −0.0187164 −0.00935821 0.999956i \(-0.502979\pi\)
−0.00935821 + 0.999956i \(0.502979\pi\)
\(830\) 6477.87 0.270904
\(831\) −20422.7 −0.852532
\(832\) 2404.61 0.100198
\(833\) −16429.3 −0.683363
\(834\) −14562.3 −0.604617
\(835\) 3907.55 0.161948
\(836\) −1028.10 −0.0425328
\(837\) 7096.94 0.293078
\(838\) −35032.6 −1.44413
\(839\) 14975.6 0.616228 0.308114 0.951349i \(-0.400302\pi\)
0.308114 + 0.951349i \(0.400302\pi\)
\(840\) 2348.61 0.0964700
\(841\) −9727.08 −0.398831
\(842\) 17484.0 0.715605
\(843\) −11387.1 −0.465232
\(844\) −14080.9 −0.574273
\(845\) 5242.74 0.213439
\(846\) −5340.25 −0.217023
\(847\) 1275.92 0.0517604
\(848\) 38932.8 1.57660
\(849\) −3207.23 −0.129649
\(850\) 6368.65 0.256991
\(851\) −15164.5 −0.610847
\(852\) −5273.40 −0.212047
\(853\) 48986.0 1.96629 0.983146 0.182822i \(-0.0585233\pi\)
0.983146 + 0.182822i \(0.0585233\pi\)
\(854\) −11511.3 −0.461252
\(855\) 1027.73 0.0411085
\(856\) 9431.88 0.376606
\(857\) −19067.3 −0.760008 −0.380004 0.924985i \(-0.624077\pi\)
−0.380004 + 0.924985i \(0.624077\pi\)
\(858\) 5389.87 0.214461
\(859\) −19880.1 −0.789638 −0.394819 0.918759i \(-0.629193\pi\)
−0.394819 + 0.918759i \(0.629193\pi\)
\(860\) 11595.5 0.459772
\(861\) −10038.6 −0.397345
\(862\) −45044.0 −1.77982
\(863\) 6346.71 0.250341 0.125171 0.992135i \(-0.460052\pi\)
0.125171 + 0.992135i \(0.460052\pi\)
\(864\) 29805.2 1.17360
\(865\) −2420.06 −0.0951265
\(866\) −43793.1 −1.71842
\(867\) 443.337 0.0173662
\(868\) −2419.81 −0.0946241
\(869\) −12914.5 −0.504134
\(870\) 8753.76 0.341127
\(871\) 15804.7 0.614835
\(872\) 22282.6 0.865349
\(873\) −14005.1 −0.542956
\(874\) −5726.42 −0.221624
\(875\) −1318.10 −0.0509255
\(876\) 20399.0 0.786780
\(877\) 20298.2 0.781551 0.390776 0.920486i \(-0.372207\pi\)
0.390776 + 0.920486i \(0.372207\pi\)
\(878\) 54026.1 2.07664
\(879\) −22890.2 −0.878348
\(880\) −4353.54 −0.166770
\(881\) −40855.2 −1.56237 −0.781185 0.624299i \(-0.785385\pi\)
−0.781185 + 0.624299i \(0.785385\pi\)
\(882\) 9013.67 0.344111
\(883\) 16389.3 0.624623 0.312312 0.949980i \(-0.398897\pi\)
0.312312 + 0.949980i \(0.398897\pi\)
\(884\) −11815.0 −0.449527
\(885\) 5931.51 0.225294
\(886\) 22861.0 0.866850
\(887\) −30745.8 −1.16386 −0.581929 0.813240i \(-0.697702\pi\)
−0.581929 + 0.813240i \(0.697702\pi\)
\(888\) −8055.97 −0.304438
\(889\) −9351.20 −0.352789
\(890\) −17767.4 −0.669175
\(891\) 3518.60 0.132298
\(892\) 5526.61 0.207449
\(893\) 2609.41 0.0977833
\(894\) 4785.36 0.179023
\(895\) 1258.85 0.0470152
\(896\) 13838.1 0.515958
\(897\) 11431.0 0.425494
\(898\) 16458.2 0.611602
\(899\) 5648.75 0.209562
\(900\) −1330.41 −0.0492743
\(901\) −34860.0 −1.28896
\(902\) 9356.89 0.345400
\(903\) −19997.8 −0.736972
\(904\) −5048.41 −0.185739
\(905\) −1923.81 −0.0706625
\(906\) 15057.2 0.552144
\(907\) −40209.6 −1.47204 −0.736019 0.676961i \(-0.763297\pi\)
−0.736019 + 0.676961i \(0.763297\pi\)
\(908\) 21139.2 0.772609
\(909\) −9952.98 −0.363168
\(910\) 6422.14 0.233947
\(911\) −40716.8 −1.48080 −0.740400 0.672167i \(-0.765364\pi\)
−0.740400 + 0.672167i \(0.765364\pi\)
\(912\) −6049.87 −0.219661
\(913\) 3964.96 0.143725
\(914\) −69142.0 −2.50220
\(915\) 6108.77 0.220710
\(916\) 18984.7 0.684795
\(917\) −700.762 −0.0252358
\(918\) −38754.4 −1.39334
\(919\) −51957.8 −1.86500 −0.932498 0.361176i \(-0.882375\pi\)
−0.932498 + 0.361176i \(0.882375\pi\)
\(920\) −4642.74 −0.166377
\(921\) 12696.9 0.454265
\(922\) −1674.62 −0.0598162
\(923\) 9031.24 0.322066
\(924\) −2295.25 −0.0817188
\(925\) 4521.20 0.160709
\(926\) −51818.7 −1.83895
\(927\) −220.508 −0.00781278
\(928\) 23723.2 0.839173
\(929\) −38224.0 −1.34993 −0.674966 0.737848i \(-0.735842\pi\)
−0.674966 + 0.737848i \(0.735842\pi\)
\(930\) 3372.53 0.118914
\(931\) −4404.35 −0.155045
\(932\) −6294.65 −0.221232
\(933\) 9026.33 0.316730
\(934\) 18669.3 0.654046
\(935\) 3898.10 0.136344
\(936\) −4059.80 −0.141772
\(937\) 7129.07 0.248555 0.124278 0.992247i \(-0.460339\pi\)
0.124278 + 0.992247i \(0.460339\pi\)
\(938\) −17676.0 −0.615288
\(939\) −20051.7 −0.696872
\(940\) −3377.89 −0.117207
\(941\) 12566.0 0.435323 0.217661 0.976024i \(-0.430157\pi\)
0.217661 + 0.976024i \(0.430157\pi\)
\(942\) −16566.0 −0.572982
\(943\) 19844.3 0.685280
\(944\) 23343.3 0.804830
\(945\) 8020.87 0.276105
\(946\) 18639.8 0.640627
\(947\) 31374.5 1.07659 0.538297 0.842755i \(-0.319068\pi\)
0.538297 + 0.842755i \(0.319068\pi\)
\(948\) 23231.8 0.795923
\(949\) −34935.4 −1.19500
\(950\) 1707.30 0.0583075
\(951\) −19204.6 −0.654840
\(952\) −8275.97 −0.281750
\(953\) −26876.9 −0.913567 −0.456783 0.889578i \(-0.650999\pi\)
−0.456783 + 0.889578i \(0.650999\pi\)
\(954\) 19125.4 0.649064
\(955\) −4316.62 −0.146265
\(956\) −894.701 −0.0302685
\(957\) 5357.98 0.180981
\(958\) 33114.1 1.11677
\(959\) 24530.2 0.825988
\(960\) 1427.15 0.0479804
\(961\) −27614.7 −0.926949
\(962\) −22028.6 −0.738285
\(963\) 9214.34 0.308336
\(964\) 8422.69 0.281407
\(965\) −17150.1 −0.572105
\(966\) −12784.4 −0.425808
\(967\) 24688.0 0.821007 0.410504 0.911859i \(-0.365353\pi\)
0.410504 + 0.911859i \(0.365353\pi\)
\(968\) −1339.91 −0.0444901
\(969\) 5416.98 0.179586
\(970\) −23265.7 −0.770119
\(971\) 22539.0 0.744914 0.372457 0.928050i \(-0.378515\pi\)
0.372457 + 0.928050i \(0.378515\pi\)
\(972\) 13875.7 0.457884
\(973\) −10620.3 −0.349919
\(974\) −47679.5 −1.56853
\(975\) −3408.08 −0.111944
\(976\) 24040.9 0.788454
\(977\) 34992.1 1.14585 0.572925 0.819607i \(-0.305808\pi\)
0.572925 + 0.819607i \(0.305808\pi\)
\(978\) −24912.0 −0.814518
\(979\) −10875.0 −0.355023
\(980\) 5701.45 0.185843
\(981\) 21768.7 0.708481
\(982\) −38127.7 −1.23901
\(983\) −5752.54 −0.186651 −0.0933253 0.995636i \(-0.529750\pi\)
−0.0933253 + 0.995636i \(0.529750\pi\)
\(984\) 10542.1 0.341534
\(985\) 12835.5 0.415201
\(986\) −30846.3 −0.996293
\(987\) 5825.56 0.187872
\(988\) −3167.36 −0.101991
\(989\) 39531.7 1.27102
\(990\) −2138.63 −0.0686568
\(991\) −28229.1 −0.904871 −0.452436 0.891797i \(-0.649445\pi\)
−0.452436 + 0.891797i \(0.649445\pi\)
\(992\) 9139.76 0.292528
\(993\) 14829.5 0.473917
\(994\) −10100.5 −0.322303
\(995\) −25355.3 −0.807857
\(996\) −7132.56 −0.226912
\(997\) −50127.9 −1.59234 −0.796171 0.605071i \(-0.793145\pi\)
−0.796171 + 0.605071i \(0.793145\pi\)
\(998\) 54331.2 1.72327
\(999\) −27512.4 −0.871324
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 1045.4.a.c.1.17 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.4.a.c.1.17 20 1.1 even 1 trivial