Properties

Label 547.2.a.b.1.16
Level $547$
Weight $2$
Character 547.1
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36781699056\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-1.87675\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.87675 q^{2} -1.13919 q^{3} +1.52220 q^{4} -1.30620 q^{5} -2.13798 q^{6} -1.71403 q^{7} -0.896704 q^{8} -1.70224 q^{9} +O(q^{10})\) \(q+1.87675 q^{2} -1.13919 q^{3} +1.52220 q^{4} -1.30620 q^{5} -2.13798 q^{6} -1.71403 q^{7} -0.896704 q^{8} -1.70224 q^{9} -2.45142 q^{10} -5.30835 q^{11} -1.73408 q^{12} +2.18823 q^{13} -3.21682 q^{14} +1.48801 q^{15} -4.72730 q^{16} +0.392924 q^{17} -3.19469 q^{18} +0.498929 q^{19} -1.98831 q^{20} +1.95261 q^{21} -9.96246 q^{22} +8.33591 q^{23} +1.02152 q^{24} -3.29383 q^{25} +4.10676 q^{26} +5.35675 q^{27} -2.60911 q^{28} -4.50948 q^{29} +2.79264 q^{30} -2.96386 q^{31} -7.07857 q^{32} +6.04722 q^{33} +0.737421 q^{34} +2.23887 q^{35} -2.59116 q^{36} +3.06843 q^{37} +0.936368 q^{38} -2.49281 q^{39} +1.17128 q^{40} -10.0885 q^{41} +3.66457 q^{42} +9.93284 q^{43} -8.08039 q^{44} +2.22348 q^{45} +15.6444 q^{46} +0.714605 q^{47} +5.38530 q^{48} -4.06209 q^{49} -6.18172 q^{50} -0.447615 q^{51} +3.33093 q^{52} -10.3693 q^{53} +10.0533 q^{54} +6.93378 q^{55} +1.53698 q^{56} -0.568376 q^{57} -8.46318 q^{58} -6.58293 q^{59} +2.26506 q^{60} -0.889139 q^{61} -5.56243 q^{62} +2.91770 q^{63} -3.83013 q^{64} -2.85827 q^{65} +11.3491 q^{66} +4.67997 q^{67} +0.598110 q^{68} -9.49619 q^{69} +4.20181 q^{70} +11.4239 q^{71} +1.52641 q^{72} +5.35047 q^{73} +5.75869 q^{74} +3.75231 q^{75} +0.759473 q^{76} +9.09868 q^{77} -4.67838 q^{78} -7.79987 q^{79} +6.17481 q^{80} -0.995634 q^{81} -18.9336 q^{82} -17.9975 q^{83} +2.97227 q^{84} -0.513238 q^{85} +18.6415 q^{86} +5.13716 q^{87} +4.76002 q^{88} +13.3683 q^{89} +4.17292 q^{90} -3.75069 q^{91} +12.6890 q^{92} +3.37640 q^{93} +1.34114 q^{94} -0.651703 q^{95} +8.06385 q^{96} -13.2660 q^{97} -7.62355 q^{98} +9.03610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.87675 1.32707 0.663533 0.748147i \(-0.269056\pi\)
0.663533 + 0.748147i \(0.269056\pi\)
\(3\) −1.13919 −0.657712 −0.328856 0.944380i \(-0.606663\pi\)
−0.328856 + 0.944380i \(0.606663\pi\)
\(4\) 1.52220 0.761102
\(5\) −1.30620 −0.584152 −0.292076 0.956395i \(-0.594346\pi\)
−0.292076 + 0.956395i \(0.594346\pi\)
\(6\) −2.13798 −0.872827
\(7\) −1.71403 −0.647843 −0.323922 0.946084i \(-0.605001\pi\)
−0.323922 + 0.946084i \(0.605001\pi\)
\(8\) −0.896704 −0.317033
\(9\) −1.70224 −0.567415
\(10\) −2.45142 −0.775207
\(11\) −5.30835 −1.60053 −0.800263 0.599649i \(-0.795307\pi\)
−0.800263 + 0.599649i \(0.795307\pi\)
\(12\) −1.73408 −0.500586
\(13\) 2.18823 0.606905 0.303452 0.952847i \(-0.401861\pi\)
0.303452 + 0.952847i \(0.401861\pi\)
\(14\) −3.21682 −0.859730
\(15\) 1.48801 0.384204
\(16\) −4.72730 −1.18183
\(17\) 0.392924 0.0952980 0.0476490 0.998864i \(-0.484827\pi\)
0.0476490 + 0.998864i \(0.484827\pi\)
\(18\) −3.19469 −0.752996
\(19\) 0.498929 0.114462 0.0572311 0.998361i \(-0.481773\pi\)
0.0572311 + 0.998361i \(0.481773\pi\)
\(20\) −1.98831 −0.444599
\(21\) 1.95261 0.426094
\(22\) −9.96246 −2.12400
\(23\) 8.33591 1.73816 0.869079 0.494674i \(-0.164713\pi\)
0.869079 + 0.494674i \(0.164713\pi\)
\(24\) 1.02152 0.208516
\(25\) −3.29383 −0.658767
\(26\) 4.10676 0.805402
\(27\) 5.35675 1.03091
\(28\) −2.60911 −0.493075
\(29\) −4.50948 −0.837389 −0.418695 0.908127i \(-0.637512\pi\)
−0.418695 + 0.908127i \(0.637512\pi\)
\(30\) 2.79264 0.509863
\(31\) −2.96386 −0.532324 −0.266162 0.963928i \(-0.585756\pi\)
−0.266162 + 0.963928i \(0.585756\pi\)
\(32\) −7.07857 −1.25133
\(33\) 6.04722 1.05269
\(34\) 0.737421 0.126467
\(35\) 2.23887 0.378439
\(36\) −2.59116 −0.431861
\(37\) 3.06843 0.504447 0.252223 0.967669i \(-0.418838\pi\)
0.252223 + 0.967669i \(0.418838\pi\)
\(38\) 0.936368 0.151899
\(39\) −2.49281 −0.399169
\(40\) 1.17128 0.185195
\(41\) −10.0885 −1.57556 −0.787780 0.615957i \(-0.788770\pi\)
−0.787780 + 0.615957i \(0.788770\pi\)
\(42\) 3.66457 0.565455
\(43\) 9.93284 1.51474 0.757372 0.652984i \(-0.226483\pi\)
0.757372 + 0.652984i \(0.226483\pi\)
\(44\) −8.08039 −1.21816
\(45\) 2.22348 0.331456
\(46\) 15.6444 2.30665
\(47\) 0.714605 0.104236 0.0521179 0.998641i \(-0.483403\pi\)
0.0521179 + 0.998641i \(0.483403\pi\)
\(48\) 5.38530 0.777301
\(49\) −4.06209 −0.580299
\(50\) −6.18172 −0.874227
\(51\) −0.447615 −0.0626787
\(52\) 3.33093 0.461916
\(53\) −10.3693 −1.42433 −0.712165 0.702012i \(-0.752285\pi\)
−0.712165 + 0.702012i \(0.752285\pi\)
\(54\) 10.0533 1.36808
\(55\) 6.93378 0.934950
\(56\) 1.53698 0.205388
\(57\) −0.568376 −0.0752832
\(58\) −8.46318 −1.11127
\(59\) −6.58293 −0.857024 −0.428512 0.903536i \(-0.640962\pi\)
−0.428512 + 0.903536i \(0.640962\pi\)
\(60\) 2.26506 0.292418
\(61\) −0.889139 −0.113843 −0.0569213 0.998379i \(-0.518128\pi\)
−0.0569213 + 0.998379i \(0.518128\pi\)
\(62\) −5.56243 −0.706429
\(63\) 2.91770 0.367596
\(64\) −3.83013 −0.478767
\(65\) −2.85827 −0.354524
\(66\) 11.3491 1.39698
\(67\) 4.67997 0.571750 0.285875 0.958267i \(-0.407716\pi\)
0.285875 + 0.958267i \(0.407716\pi\)
\(68\) 0.598110 0.0725315
\(69\) −9.49619 −1.14321
\(70\) 4.20181 0.502213
\(71\) 11.4239 1.35577 0.677886 0.735167i \(-0.262896\pi\)
0.677886 + 0.735167i \(0.262896\pi\)
\(72\) 1.52641 0.179889
\(73\) 5.35047 0.626225 0.313113 0.949716i \(-0.398628\pi\)
0.313113 + 0.949716i \(0.398628\pi\)
\(74\) 5.75869 0.669434
\(75\) 3.75231 0.433279
\(76\) 0.759473 0.0871175
\(77\) 9.09868 1.03689
\(78\) −4.67838 −0.529723
\(79\) −7.79987 −0.877554 −0.438777 0.898596i \(-0.644588\pi\)
−0.438777 + 0.898596i \(0.644588\pi\)
\(80\) 6.17481 0.690365
\(81\) −0.995634 −0.110626
\(82\) −18.9336 −2.09087
\(83\) −17.9975 −1.97548 −0.987740 0.156107i \(-0.950105\pi\)
−0.987740 + 0.156107i \(0.950105\pi\)
\(84\) 2.97227 0.324301
\(85\) −0.513238 −0.0556685
\(86\) 18.6415 2.01016
\(87\) 5.13716 0.550761
\(88\) 4.76002 0.507420
\(89\) 13.3683 1.41704 0.708519 0.705691i \(-0.249363\pi\)
0.708519 + 0.705691i \(0.249363\pi\)
\(90\) 4.17292 0.439864
\(91\) −3.75069 −0.393179
\(92\) 12.6890 1.32292
\(93\) 3.37640 0.350116
\(94\) 1.34114 0.138328
\(95\) −0.651703 −0.0668633
\(96\) 8.06385 0.823013
\(97\) −13.2660 −1.34696 −0.673480 0.739206i \(-0.735201\pi\)
−0.673480 + 0.739206i \(0.735201\pi\)
\(98\) −7.62355 −0.770095
\(99\) 9.03610 0.908162
\(100\) −5.01389 −0.501389
\(101\) −5.06820 −0.504305 −0.252152 0.967688i \(-0.581138\pi\)
−0.252152 + 0.967688i \(0.581138\pi\)
\(102\) −0.840064 −0.0831787
\(103\) −1.26789 −0.124929 −0.0624644 0.998047i \(-0.519896\pi\)
−0.0624644 + 0.998047i \(0.519896\pi\)
\(104\) −1.96219 −0.192409
\(105\) −2.55050 −0.248904
\(106\) −19.4606 −1.89018
\(107\) 11.5228 1.11395 0.556974 0.830530i \(-0.311962\pi\)
0.556974 + 0.830530i \(0.311962\pi\)
\(108\) 8.15407 0.784626
\(109\) 1.84704 0.176914 0.0884570 0.996080i \(-0.471806\pi\)
0.0884570 + 0.996080i \(0.471806\pi\)
\(110\) 13.0130 1.24074
\(111\) −3.49553 −0.331781
\(112\) 8.10275 0.765638
\(113\) −10.8186 −1.01773 −0.508865 0.860846i \(-0.669935\pi\)
−0.508865 + 0.860846i \(0.669935\pi\)
\(114\) −1.06670 −0.0999057
\(115\) −10.8884 −1.01535
\(116\) −6.86435 −0.637339
\(117\) −3.72489 −0.344367
\(118\) −12.3545 −1.13733
\(119\) −0.673484 −0.0617382
\(120\) −1.33431 −0.121805
\(121\) 17.1786 1.56169
\(122\) −1.66869 −0.151077
\(123\) 11.4927 1.03626
\(124\) −4.51159 −0.405153
\(125\) 10.8334 0.968971
\(126\) 5.47580 0.487823
\(127\) 16.9759 1.50637 0.753185 0.657809i \(-0.228517\pi\)
0.753185 + 0.657809i \(0.228517\pi\)
\(128\) 6.96893 0.615972
\(129\) −11.3154 −0.996266
\(130\) −5.36426 −0.470477
\(131\) 0.273878 0.0239289 0.0119644 0.999928i \(-0.496192\pi\)
0.0119644 + 0.999928i \(0.496192\pi\)
\(132\) 9.20511 0.801202
\(133\) −0.855181 −0.0741536
\(134\) 8.78316 0.758749
\(135\) −6.99701 −0.602206
\(136\) −0.352337 −0.0302126
\(137\) −5.63213 −0.481185 −0.240593 0.970626i \(-0.577342\pi\)
−0.240593 + 0.970626i \(0.577342\pi\)
\(138\) −17.8220 −1.51711
\(139\) −22.7017 −1.92553 −0.962767 0.270333i \(-0.912866\pi\)
−0.962767 + 0.270333i \(0.912866\pi\)
\(140\) 3.40802 0.288030
\(141\) −0.814072 −0.0685572
\(142\) 21.4399 1.79920
\(143\) −11.6159 −0.971367
\(144\) 8.04702 0.670585
\(145\) 5.89029 0.489162
\(146\) 10.0415 0.831042
\(147\) 4.62750 0.381670
\(148\) 4.67078 0.383936
\(149\) −3.23862 −0.265318 −0.132659 0.991162i \(-0.542352\pi\)
−0.132659 + 0.991162i \(0.542352\pi\)
\(150\) 7.04216 0.574990
\(151\) 18.3287 1.49157 0.745783 0.666189i \(-0.232076\pi\)
0.745783 + 0.666189i \(0.232076\pi\)
\(152\) −0.447392 −0.0362883
\(153\) −0.668852 −0.0540735
\(154\) 17.0760 1.37602
\(155\) 3.87140 0.310958
\(156\) −3.79456 −0.303808
\(157\) −2.10134 −0.167705 −0.0838527 0.996478i \(-0.526723\pi\)
−0.0838527 + 0.996478i \(0.526723\pi\)
\(158\) −14.6384 −1.16457
\(159\) 11.8126 0.936799
\(160\) 9.24605 0.730965
\(161\) −14.2880 −1.12605
\(162\) −1.86856 −0.146808
\(163\) −23.0706 −1.80703 −0.903514 0.428558i \(-0.859022\pi\)
−0.903514 + 0.428558i \(0.859022\pi\)
\(164\) −15.3568 −1.19916
\(165\) −7.89890 −0.614928
\(166\) −33.7768 −2.62159
\(167\) −11.3441 −0.877832 −0.438916 0.898528i \(-0.644637\pi\)
−0.438916 + 0.898528i \(0.644637\pi\)
\(168\) −1.75091 −0.135086
\(169\) −8.21167 −0.631667
\(170\) −0.963222 −0.0738757
\(171\) −0.849299 −0.0649476
\(172\) 15.1198 1.15287
\(173\) −17.2109 −1.30852 −0.654259 0.756271i \(-0.727019\pi\)
−0.654259 + 0.756271i \(0.727019\pi\)
\(174\) 9.64118 0.730896
\(175\) 5.64574 0.426778
\(176\) 25.0942 1.89154
\(177\) 7.49921 0.563675
\(178\) 25.0890 1.88050
\(179\) −11.9899 −0.896168 −0.448084 0.893991i \(-0.647893\pi\)
−0.448084 + 0.893991i \(0.647893\pi\)
\(180\) 3.38458 0.252272
\(181\) 1.09690 0.0815319 0.0407660 0.999169i \(-0.487020\pi\)
0.0407660 + 0.999169i \(0.487020\pi\)
\(182\) −7.03912 −0.521774
\(183\) 1.01290 0.0748756
\(184\) −7.47485 −0.551053
\(185\) −4.00799 −0.294673
\(186\) 6.33667 0.464627
\(187\) −2.08578 −0.152527
\(188\) 1.08778 0.0793342
\(189\) −9.18165 −0.667867
\(190\) −1.22309 −0.0887320
\(191\) 1.44468 0.104533 0.0522667 0.998633i \(-0.483355\pi\)
0.0522667 + 0.998633i \(0.483355\pi\)
\(192\) 4.36325 0.314891
\(193\) 1.41359 0.101753 0.0508763 0.998705i \(-0.483799\pi\)
0.0508763 + 0.998705i \(0.483799\pi\)
\(194\) −24.8970 −1.78750
\(195\) 3.25611 0.233175
\(196\) −6.18334 −0.441667
\(197\) −5.14237 −0.366379 −0.183189 0.983078i \(-0.558642\pi\)
−0.183189 + 0.983078i \(0.558642\pi\)
\(198\) 16.9585 1.20519
\(199\) −16.7502 −1.18739 −0.593695 0.804690i \(-0.702332\pi\)
−0.593695 + 0.804690i \(0.702332\pi\)
\(200\) 2.95360 0.208851
\(201\) −5.33139 −0.376047
\(202\) −9.51176 −0.669245
\(203\) 7.72939 0.542497
\(204\) −0.681362 −0.0477049
\(205\) 13.1776 0.920366
\(206\) −2.37951 −0.165789
\(207\) −14.1897 −0.986256
\(208\) −10.3444 −0.717255
\(209\) −2.64849 −0.183200
\(210\) −4.78667 −0.330311
\(211\) −11.7475 −0.808734 −0.404367 0.914597i \(-0.632508\pi\)
−0.404367 + 0.914597i \(0.632508\pi\)
\(212\) −15.7842 −1.08406
\(213\) −13.0141 −0.891708
\(214\) 21.6254 1.47828
\(215\) −12.9743 −0.884840
\(216\) −4.80343 −0.326832
\(217\) 5.08014 0.344863
\(218\) 3.46643 0.234777
\(219\) −6.09521 −0.411876
\(220\) 10.5546 0.711593
\(221\) 0.859806 0.0578368
\(222\) −6.56025 −0.440295
\(223\) 26.9958 1.80778 0.903888 0.427770i \(-0.140701\pi\)
0.903888 + 0.427770i \(0.140701\pi\)
\(224\) 12.1329 0.810664
\(225\) 5.60691 0.373794
\(226\) −20.3039 −1.35059
\(227\) −6.16279 −0.409039 −0.204519 0.978863i \(-0.565563\pi\)
−0.204519 + 0.978863i \(0.565563\pi\)
\(228\) −0.865184 −0.0572982
\(229\) 5.04203 0.333187 0.166593 0.986026i \(-0.446723\pi\)
0.166593 + 0.986026i \(0.446723\pi\)
\(230\) −20.4348 −1.34743
\(231\) −10.3651 −0.681975
\(232\) 4.04367 0.265480
\(233\) −8.21008 −0.537860 −0.268930 0.963160i \(-0.586670\pi\)
−0.268930 + 0.963160i \(0.586670\pi\)
\(234\) −6.99071 −0.456997
\(235\) −0.933419 −0.0608896
\(236\) −10.0206 −0.652283
\(237\) 8.88554 0.577178
\(238\) −1.26396 −0.0819306
\(239\) 25.4229 1.64447 0.822235 0.569148i \(-0.192727\pi\)
0.822235 + 0.569148i \(0.192727\pi\)
\(240\) −7.03429 −0.454062
\(241\) 26.7739 1.72466 0.862328 0.506350i \(-0.169006\pi\)
0.862328 + 0.506350i \(0.169006\pi\)
\(242\) 32.2399 2.07246
\(243\) −14.9360 −0.958148
\(244\) −1.35345 −0.0866458
\(245\) 5.30592 0.338983
\(246\) 21.5690 1.37519
\(247\) 1.09177 0.0694677
\(248\) 2.65770 0.168764
\(249\) 20.5026 1.29930
\(250\) 20.3317 1.28589
\(251\) 23.7572 1.49954 0.749770 0.661699i \(-0.230164\pi\)
0.749770 + 0.661699i \(0.230164\pi\)
\(252\) 4.44134 0.279778
\(253\) −44.2499 −2.78197
\(254\) 31.8596 1.99905
\(255\) 0.584676 0.0366138
\(256\) 20.7392 1.29620
\(257\) −6.92184 −0.431773 −0.215886 0.976418i \(-0.569264\pi\)
−0.215886 + 0.976418i \(0.569264\pi\)
\(258\) −21.2362 −1.32211
\(259\) −5.25939 −0.326802
\(260\) −4.35087 −0.269829
\(261\) 7.67623 0.475147
\(262\) 0.514002 0.0317551
\(263\) −9.02135 −0.556281 −0.278140 0.960540i \(-0.589718\pi\)
−0.278140 + 0.960540i \(0.589718\pi\)
\(264\) −5.42257 −0.333736
\(265\) 13.5444 0.832025
\(266\) −1.60496 −0.0984066
\(267\) −15.2291 −0.932004
\(268\) 7.12388 0.435160
\(269\) 7.43191 0.453131 0.226566 0.973996i \(-0.427250\pi\)
0.226566 + 0.973996i \(0.427250\pi\)
\(270\) −13.1317 −0.799167
\(271\) 24.3754 1.48070 0.740351 0.672220i \(-0.234659\pi\)
0.740351 + 0.672220i \(0.234659\pi\)
\(272\) −1.85747 −0.112626
\(273\) 4.27275 0.258599
\(274\) −10.5701 −0.638564
\(275\) 17.4848 1.05437
\(276\) −14.4551 −0.870098
\(277\) 15.8453 0.952052 0.476026 0.879431i \(-0.342077\pi\)
0.476026 + 0.879431i \(0.342077\pi\)
\(278\) −42.6055 −2.55531
\(279\) 5.04520 0.302049
\(280\) −2.00761 −0.119977
\(281\) 5.55329 0.331281 0.165641 0.986186i \(-0.447031\pi\)
0.165641 + 0.986186i \(0.447031\pi\)
\(282\) −1.52781 −0.0909799
\(283\) −21.5322 −1.27996 −0.639978 0.768394i \(-0.721056\pi\)
−0.639978 + 0.768394i \(0.721056\pi\)
\(284\) 17.3896 1.03188
\(285\) 0.742414 0.0439768
\(286\) −21.8001 −1.28907
\(287\) 17.2920 1.02072
\(288\) 12.0495 0.710021
\(289\) −16.8456 −0.990918
\(290\) 11.0546 0.649150
\(291\) 15.1125 0.885912
\(292\) 8.14451 0.476621
\(293\) −23.9442 −1.39884 −0.699418 0.714713i \(-0.746557\pi\)
−0.699418 + 0.714713i \(0.746557\pi\)
\(294\) 8.68468 0.506501
\(295\) 8.59863 0.500632
\(296\) −2.75148 −0.159926
\(297\) −28.4355 −1.65000
\(298\) −6.07809 −0.352094
\(299\) 18.2408 1.05490
\(300\) 5.71178 0.329770
\(301\) −17.0252 −0.981316
\(302\) 34.3984 1.97941
\(303\) 5.77365 0.331687
\(304\) −2.35859 −0.135274
\(305\) 1.16140 0.0665013
\(306\) −1.25527 −0.0717591
\(307\) 1.31760 0.0751996 0.0375998 0.999293i \(-0.488029\pi\)
0.0375998 + 0.999293i \(0.488029\pi\)
\(308\) 13.8500 0.789180
\(309\) 1.44437 0.0821672
\(310\) 7.26566 0.412662
\(311\) −21.9974 −1.24736 −0.623679 0.781680i \(-0.714363\pi\)
−0.623679 + 0.781680i \(0.714363\pi\)
\(312\) 2.23531 0.126550
\(313\) −0.891821 −0.0504087 −0.0252044 0.999682i \(-0.508024\pi\)
−0.0252044 + 0.999682i \(0.508024\pi\)
\(314\) −3.94370 −0.222556
\(315\) −3.81111 −0.214732
\(316\) −11.8730 −0.667908
\(317\) −24.3281 −1.36640 −0.683202 0.730229i \(-0.739413\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(318\) 22.1693 1.24319
\(319\) 23.9379 1.34026
\(320\) 5.00293 0.279672
\(321\) −13.1266 −0.732657
\(322\) −26.8151 −1.49435
\(323\) 0.196041 0.0109080
\(324\) −1.51556 −0.0841977
\(325\) −7.20765 −0.399809
\(326\) −43.2978 −2.39804
\(327\) −2.10413 −0.116359
\(328\) 9.04641 0.499504
\(329\) −1.22486 −0.0675285
\(330\) −14.8243 −0.816050
\(331\) 27.7538 1.52549 0.762745 0.646700i \(-0.223851\pi\)
0.762745 + 0.646700i \(0.223851\pi\)
\(332\) −27.3958 −1.50354
\(333\) −5.22322 −0.286231
\(334\) −21.2901 −1.16494
\(335\) −6.11300 −0.333989
\(336\) −9.23058 −0.503569
\(337\) 6.10740 0.332691 0.166346 0.986068i \(-0.446803\pi\)
0.166346 + 0.986068i \(0.446803\pi\)
\(338\) −15.4113 −0.838263
\(339\) 12.3245 0.669374
\(340\) −0.781253 −0.0423694
\(341\) 15.7332 0.851999
\(342\) −1.59393 −0.0861896
\(343\) 18.9608 1.02379
\(344\) −8.90682 −0.480224
\(345\) 12.4040 0.667806
\(346\) −32.3005 −1.73649
\(347\) 27.7441 1.48938 0.744691 0.667409i \(-0.232597\pi\)
0.744691 + 0.667409i \(0.232597\pi\)
\(348\) 7.81981 0.419186
\(349\) −14.3589 −0.768616 −0.384308 0.923205i \(-0.625560\pi\)
−0.384308 + 0.923205i \(0.625560\pi\)
\(350\) 10.5957 0.566362
\(351\) 11.7218 0.625663
\(352\) 37.5755 2.00278
\(353\) 1.89459 0.100839 0.0504194 0.998728i \(-0.483944\pi\)
0.0504194 + 0.998728i \(0.483944\pi\)
\(354\) 14.0742 0.748034
\(355\) −14.9220 −0.791977
\(356\) 20.3493 1.07851
\(357\) 0.767227 0.0406060
\(358\) −22.5021 −1.18927
\(359\) 10.0371 0.529740 0.264870 0.964284i \(-0.414671\pi\)
0.264870 + 0.964284i \(0.414671\pi\)
\(360\) −1.99380 −0.105083
\(361\) −18.7511 −0.986898
\(362\) 2.05861 0.108198
\(363\) −19.5697 −1.02714
\(364\) −5.70931 −0.299249
\(365\) −6.98880 −0.365810
\(366\) 1.90096 0.0993649
\(367\) 5.02023 0.262054 0.131027 0.991379i \(-0.458173\pi\)
0.131027 + 0.991379i \(0.458173\pi\)
\(368\) −39.4064 −2.05420
\(369\) 17.1731 0.893996
\(370\) −7.52201 −0.391051
\(371\) 17.7733 0.922742
\(372\) 5.13957 0.266474
\(373\) −3.12699 −0.161910 −0.0809548 0.996718i \(-0.525797\pi\)
−0.0809548 + 0.996718i \(0.525797\pi\)
\(374\) −3.91449 −0.202413
\(375\) −12.3413 −0.637304
\(376\) −0.640790 −0.0330462
\(377\) −9.86776 −0.508215
\(378\) −17.2317 −0.886302
\(379\) 16.8372 0.864867 0.432433 0.901666i \(-0.357655\pi\)
0.432433 + 0.901666i \(0.357655\pi\)
\(380\) −0.992025 −0.0508898
\(381\) −19.3388 −0.990757
\(382\) 2.71131 0.138723
\(383\) −3.31954 −0.169621 −0.0848104 0.996397i \(-0.527028\pi\)
−0.0848104 + 0.996397i \(0.527028\pi\)
\(384\) −7.93894 −0.405132
\(385\) −11.8847 −0.605701
\(386\) 2.65297 0.135032
\(387\) −16.9081 −0.859488
\(388\) −20.1936 −1.02517
\(389\) −13.8035 −0.699867 −0.349933 0.936775i \(-0.613796\pi\)
−0.349933 + 0.936775i \(0.613796\pi\)
\(390\) 6.11092 0.309438
\(391\) 3.27538 0.165643
\(392\) 3.64250 0.183974
\(393\) −0.312000 −0.0157383
\(394\) −9.65096 −0.486208
\(395\) 10.1882 0.512625
\(396\) 13.7548 0.691204
\(397\) 13.6460 0.684875 0.342438 0.939541i \(-0.388747\pi\)
0.342438 + 0.939541i \(0.388747\pi\)
\(398\) −31.4360 −1.57574
\(399\) 0.974214 0.0487717
\(400\) 15.5710 0.778548
\(401\) −4.21927 −0.210700 −0.105350 0.994435i \(-0.533596\pi\)
−0.105350 + 0.994435i \(0.533596\pi\)
\(402\) −10.0057 −0.499039
\(403\) −6.48558 −0.323070
\(404\) −7.71484 −0.383828
\(405\) 1.30050 0.0646224
\(406\) 14.5062 0.719929
\(407\) −16.2883 −0.807381
\(408\) 0.401379 0.0198712
\(409\) 3.32926 0.164621 0.0823105 0.996607i \(-0.473770\pi\)
0.0823105 + 0.996607i \(0.473770\pi\)
\(410\) 24.7312 1.22139
\(411\) 6.41607 0.316481
\(412\) −1.92999 −0.0950836
\(413\) 11.2833 0.555217
\(414\) −26.6307 −1.30883
\(415\) 23.5084 1.15398
\(416\) −15.4895 −0.759436
\(417\) 25.8616 1.26645
\(418\) −4.97056 −0.243118
\(419\) −17.2091 −0.840718 −0.420359 0.907358i \(-0.638096\pi\)
−0.420359 + 0.907358i \(0.638096\pi\)
\(420\) −3.88239 −0.189441
\(421\) 5.90690 0.287884 0.143942 0.989586i \(-0.454022\pi\)
0.143942 + 0.989586i \(0.454022\pi\)
\(422\) −22.0472 −1.07324
\(423\) −1.21643 −0.0591450
\(424\) 9.29818 0.451559
\(425\) −1.29423 −0.0627792
\(426\) −24.4242 −1.18336
\(427\) 1.52401 0.0737521
\(428\) 17.5400 0.847828
\(429\) 13.2327 0.638880
\(430\) −24.3496 −1.17424
\(431\) −27.0646 −1.30366 −0.651829 0.758366i \(-0.725998\pi\)
−0.651829 + 0.758366i \(0.725998\pi\)
\(432\) −25.3230 −1.21835
\(433\) −5.65104 −0.271571 −0.135786 0.990738i \(-0.543356\pi\)
−0.135786 + 0.990738i \(0.543356\pi\)
\(434\) 9.53418 0.457655
\(435\) −6.71017 −0.321728
\(436\) 2.81157 0.134650
\(437\) 4.15903 0.198953
\(438\) −11.4392 −0.546586
\(439\) −26.7713 −1.27772 −0.638861 0.769322i \(-0.720594\pi\)
−0.638861 + 0.769322i \(0.720594\pi\)
\(440\) −6.21755 −0.296410
\(441\) 6.91468 0.329270
\(442\) 1.61364 0.0767532
\(443\) 34.1995 1.62487 0.812435 0.583052i \(-0.198142\pi\)
0.812435 + 0.583052i \(0.198142\pi\)
\(444\) −5.32091 −0.252519
\(445\) −17.4617 −0.827765
\(446\) 50.6646 2.39904
\(447\) 3.68940 0.174503
\(448\) 6.56497 0.310166
\(449\) −7.11797 −0.335918 −0.167959 0.985794i \(-0.553718\pi\)
−0.167959 + 0.985794i \(0.553718\pi\)
\(450\) 10.5228 0.496049
\(451\) 53.5533 2.52173
\(452\) −16.4682 −0.774597
\(453\) −20.8799 −0.981021
\(454\) −11.5660 −0.542821
\(455\) 4.89916 0.229676
\(456\) 0.509665 0.0238673
\(457\) 25.8336 1.20845 0.604223 0.796815i \(-0.293484\pi\)
0.604223 + 0.796815i \(0.293484\pi\)
\(458\) 9.46265 0.442161
\(459\) 2.10480 0.0982435
\(460\) −16.5744 −0.772783
\(461\) −38.8134 −1.80772 −0.903860 0.427829i \(-0.859279\pi\)
−0.903860 + 0.427829i \(0.859279\pi\)
\(462\) −19.4528 −0.905026
\(463\) 22.8908 1.06383 0.531913 0.846799i \(-0.321473\pi\)
0.531913 + 0.846799i \(0.321473\pi\)
\(464\) 21.3177 0.989648
\(465\) −4.41026 −0.204521
\(466\) −15.4083 −0.713775
\(467\) −12.1901 −0.564089 −0.282045 0.959401i \(-0.591013\pi\)
−0.282045 + 0.959401i \(0.591013\pi\)
\(468\) −5.67005 −0.262098
\(469\) −8.02163 −0.370404
\(470\) −1.75180 −0.0808044
\(471\) 2.39383 0.110302
\(472\) 5.90294 0.271705
\(473\) −52.7270 −2.42439
\(474\) 16.6760 0.765953
\(475\) −1.64339 −0.0754039
\(476\) −1.02518 −0.0469891
\(477\) 17.6510 0.808186
\(478\) 47.7125 2.18232
\(479\) −35.1228 −1.60480 −0.802400 0.596786i \(-0.796444\pi\)
−0.802400 + 0.596786i \(0.796444\pi\)
\(480\) −10.5330 −0.480764
\(481\) 6.71442 0.306151
\(482\) 50.2479 2.28873
\(483\) 16.2768 0.740619
\(484\) 26.1493 1.18860
\(485\) 17.3281 0.786828
\(486\) −28.0313 −1.27152
\(487\) 7.59211 0.344031 0.172016 0.985094i \(-0.444972\pi\)
0.172016 + 0.985094i \(0.444972\pi\)
\(488\) 0.797295 0.0360918
\(489\) 26.2818 1.18850
\(490\) 9.95790 0.449852
\(491\) 18.2927 0.825537 0.412768 0.910836i \(-0.364562\pi\)
0.412768 + 0.910836i \(0.364562\pi\)
\(492\) 17.4943 0.788704
\(493\) −1.77188 −0.0798015
\(494\) 2.04898 0.0921881
\(495\) −11.8030 −0.530504
\(496\) 14.0110 0.629114
\(497\) −19.5810 −0.878328
\(498\) 38.4783 1.72425
\(499\) 5.76535 0.258093 0.129046 0.991639i \(-0.458808\pi\)
0.129046 + 0.991639i \(0.458808\pi\)
\(500\) 16.4907 0.737486
\(501\) 12.9231 0.577361
\(502\) 44.5864 1.98999
\(503\) −21.1805 −0.944394 −0.472197 0.881493i \(-0.656539\pi\)
−0.472197 + 0.881493i \(0.656539\pi\)
\(504\) −2.61631 −0.116540
\(505\) 6.62010 0.294590
\(506\) −83.0462 −3.69185
\(507\) 9.35466 0.415455
\(508\) 25.8408 1.14650
\(509\) 18.1907 0.806289 0.403145 0.915136i \(-0.367917\pi\)
0.403145 + 0.915136i \(0.367917\pi\)
\(510\) 1.09729 0.0485890
\(511\) −9.17088 −0.405696
\(512\) 24.9846 1.10417
\(513\) 2.67264 0.118000
\(514\) −12.9906 −0.572991
\(515\) 1.65612 0.0729773
\(516\) −17.2244 −0.758260
\(517\) −3.79337 −0.166832
\(518\) −9.87058 −0.433688
\(519\) 19.6065 0.860628
\(520\) 2.56302 0.112396
\(521\) 21.4553 0.939976 0.469988 0.882673i \(-0.344258\pi\)
0.469988 + 0.882673i \(0.344258\pi\)
\(522\) 14.4064 0.630551
\(523\) 11.6184 0.508038 0.254019 0.967199i \(-0.418247\pi\)
0.254019 + 0.967199i \(0.418247\pi\)
\(524\) 0.416899 0.0182123
\(525\) −6.43157 −0.280697
\(526\) −16.9309 −0.738221
\(527\) −1.16457 −0.0507294
\(528\) −28.5870 −1.24409
\(529\) 46.4874 2.02119
\(530\) 25.4195 1.10415
\(531\) 11.2057 0.486288
\(532\) −1.30176 −0.0564385
\(533\) −22.0759 −0.956214
\(534\) −28.5812 −1.23683
\(535\) −15.0511 −0.650714
\(536\) −4.19655 −0.181264
\(537\) 13.6588 0.589421
\(538\) 13.9479 0.601335
\(539\) 21.5630 0.928785
\(540\) −10.6509 −0.458341
\(541\) 15.8443 0.681198 0.340599 0.940209i \(-0.389370\pi\)
0.340599 + 0.940209i \(0.389370\pi\)
\(542\) 45.7467 1.96499
\(543\) −1.24958 −0.0536245
\(544\) −2.78134 −0.119249
\(545\) −2.41260 −0.103345
\(546\) 8.01890 0.343177
\(547\) −1.00000 −0.0427569
\(548\) −8.57325 −0.366231
\(549\) 1.51353 0.0645959
\(550\) 32.8147 1.39922
\(551\) −2.24991 −0.0958495
\(552\) 8.51528 0.362434
\(553\) 13.3692 0.568517
\(554\) 29.7377 1.26343
\(555\) 4.56587 0.193810
\(556\) −34.5566 −1.46553
\(557\) 11.0799 0.469471 0.234735 0.972059i \(-0.424578\pi\)
0.234735 + 0.972059i \(0.424578\pi\)
\(558\) 9.46861 0.400838
\(559\) 21.7353 0.919305
\(560\) −10.5838 −0.447248
\(561\) 2.37610 0.100319
\(562\) 10.4221 0.439632
\(563\) 41.1603 1.73470 0.867351 0.497697i \(-0.165821\pi\)
0.867351 + 0.497697i \(0.165821\pi\)
\(564\) −1.23918 −0.0521791
\(565\) 14.1313 0.594509
\(566\) −40.4106 −1.69858
\(567\) 1.70655 0.0716683
\(568\) −10.2439 −0.429825
\(569\) −9.52628 −0.399363 −0.199681 0.979861i \(-0.563991\pi\)
−0.199681 + 0.979861i \(0.563991\pi\)
\(570\) 1.39333 0.0583601
\(571\) 4.76393 0.199364 0.0996822 0.995019i \(-0.468217\pi\)
0.0996822 + 0.995019i \(0.468217\pi\)
\(572\) −17.6817 −0.739310
\(573\) −1.64577 −0.0687530
\(574\) 32.4529 1.35456
\(575\) −27.4571 −1.14504
\(576\) 6.51982 0.271659
\(577\) 33.8345 1.40855 0.704274 0.709929i \(-0.251273\pi\)
0.704274 + 0.709929i \(0.251273\pi\)
\(578\) −31.6151 −1.31501
\(579\) −1.61035 −0.0669240
\(580\) 8.96623 0.372303
\(581\) 30.8483 1.27980
\(582\) 28.3625 1.17566
\(583\) 55.0437 2.27968
\(584\) −4.79779 −0.198534
\(585\) 4.86547 0.201162
\(586\) −44.9374 −1.85635
\(587\) −12.2101 −0.503964 −0.251982 0.967732i \(-0.581082\pi\)
−0.251982 + 0.967732i \(0.581082\pi\)
\(588\) 7.04400 0.290490
\(589\) −1.47875 −0.0609310
\(590\) 16.1375 0.664371
\(591\) 5.85814 0.240972
\(592\) −14.5054 −0.596168
\(593\) 32.7830 1.34624 0.673118 0.739535i \(-0.264954\pi\)
0.673118 + 0.739535i \(0.264954\pi\)
\(594\) −53.3664 −2.18965
\(595\) 0.879707 0.0360645
\(596\) −4.92984 −0.201934
\(597\) 19.0817 0.780961
\(598\) 34.2336 1.39992
\(599\) −26.1846 −1.06987 −0.534936 0.844892i \(-0.679664\pi\)
−0.534936 + 0.844892i \(0.679664\pi\)
\(600\) −3.36471 −0.137364
\(601\) −14.1011 −0.575196 −0.287598 0.957751i \(-0.592857\pi\)
−0.287598 + 0.957751i \(0.592857\pi\)
\(602\) −31.9521 −1.30227
\(603\) −7.96646 −0.324419
\(604\) 27.9000 1.13523
\(605\) −22.4387 −0.912262
\(606\) 10.8357 0.440171
\(607\) 16.3651 0.664239 0.332120 0.943237i \(-0.392236\pi\)
0.332120 + 0.943237i \(0.392236\pi\)
\(608\) −3.53171 −0.143230
\(609\) −8.80525 −0.356807
\(610\) 2.17965 0.0882516
\(611\) 1.56372 0.0632612
\(612\) −1.01813 −0.0411555
\(613\) −13.3441 −0.538961 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(614\) 2.47282 0.0997948
\(615\) −15.0118 −0.605336
\(616\) −8.15882 −0.328728
\(617\) 34.5802 1.39215 0.696074 0.717971i \(-0.254929\pi\)
0.696074 + 0.717971i \(0.254929\pi\)
\(618\) 2.71072 0.109041
\(619\) −20.0240 −0.804832 −0.402416 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(620\) 5.89306 0.236671
\(621\) 44.6534 1.79188
\(622\) −41.2837 −1.65533
\(623\) −22.9137 −0.918019
\(624\) 11.7843 0.471748
\(625\) 2.31852 0.0927408
\(626\) −1.67373 −0.0668957
\(627\) 3.01714 0.120493
\(628\) −3.19868 −0.127641
\(629\) 1.20566 0.0480728
\(630\) −7.15251 −0.284963
\(631\) −1.64737 −0.0655808 −0.0327904 0.999462i \(-0.510439\pi\)
−0.0327904 + 0.999462i \(0.510439\pi\)
\(632\) 6.99418 0.278213
\(633\) 13.3827 0.531914
\(634\) −45.6579 −1.81331
\(635\) −22.1740 −0.879948
\(636\) 17.9812 0.713000
\(637\) −8.88878 −0.352186
\(638\) 44.9255 1.77862
\(639\) −19.4463 −0.769285
\(640\) −9.10283 −0.359821
\(641\) 30.9733 1.22337 0.611686 0.791100i \(-0.290491\pi\)
0.611686 + 0.791100i \(0.290491\pi\)
\(642\) −24.6354 −0.972283
\(643\) 24.8226 0.978907 0.489454 0.872029i \(-0.337196\pi\)
0.489454 + 0.872029i \(0.337196\pi\)
\(644\) −21.7493 −0.857042
\(645\) 14.7802 0.581970
\(646\) 0.367921 0.0144757
\(647\) 36.5122 1.43544 0.717721 0.696331i \(-0.245185\pi\)
0.717721 + 0.696331i \(0.245185\pi\)
\(648\) 0.892790 0.0350721
\(649\) 34.9445 1.37169
\(650\) −13.5270 −0.530572
\(651\) −5.78725 −0.226820
\(652\) −35.1182 −1.37533
\(653\) −26.5164 −1.03767 −0.518834 0.854875i \(-0.673634\pi\)
−0.518834 + 0.854875i \(0.673634\pi\)
\(654\) −3.94893 −0.154415
\(655\) −0.357740 −0.0139781
\(656\) 47.6914 1.86204
\(657\) −9.10781 −0.355329
\(658\) −2.29875 −0.0896147
\(659\) −25.7938 −1.00478 −0.502391 0.864641i \(-0.667546\pi\)
−0.502391 + 0.864641i \(0.667546\pi\)
\(660\) −12.0237 −0.468023
\(661\) −41.9540 −1.63182 −0.815910 0.578179i \(-0.803764\pi\)
−0.815910 + 0.578179i \(0.803764\pi\)
\(662\) 52.0871 2.02442
\(663\) −0.979483 −0.0380400
\(664\) 16.1384 0.626292
\(665\) 1.11704 0.0433169
\(666\) −9.80269 −0.379847
\(667\) −37.5906 −1.45551
\(668\) −17.2680 −0.668120
\(669\) −30.7534 −1.18900
\(670\) −11.4726 −0.443225
\(671\) 4.71986 0.182208
\(672\) −13.8217 −0.533183
\(673\) −3.20648 −0.123601 −0.0618003 0.998089i \(-0.519684\pi\)
−0.0618003 + 0.998089i \(0.519684\pi\)
\(674\) 11.4621 0.441503
\(675\) −17.6443 −0.679128
\(676\) −12.4998 −0.480763
\(677\) 4.84844 0.186341 0.0931704 0.995650i \(-0.470300\pi\)
0.0931704 + 0.995650i \(0.470300\pi\)
\(678\) 23.1300 0.888303
\(679\) 22.7384 0.872618
\(680\) 0.460223 0.0176487
\(681\) 7.02059 0.269030
\(682\) 29.5273 1.13066
\(683\) −32.3263 −1.23693 −0.618466 0.785812i \(-0.712245\pi\)
−0.618466 + 0.785812i \(0.712245\pi\)
\(684\) −1.29281 −0.0494317
\(685\) 7.35670 0.281085
\(686\) 35.5847 1.35863
\(687\) −5.74384 −0.219141
\(688\) −46.9555 −1.79016
\(689\) −22.6903 −0.864432
\(690\) 23.2792 0.886223
\(691\) −9.59431 −0.364985 −0.182492 0.983207i \(-0.558416\pi\)
−0.182492 + 0.983207i \(0.558416\pi\)
\(692\) −26.1985 −0.995916
\(693\) −15.4882 −0.588347
\(694\) 52.0689 1.97651
\(695\) 29.6530 1.12480
\(696\) −4.60651 −0.174609
\(697\) −3.96401 −0.150148
\(698\) −26.9482 −1.02000
\(699\) 9.35285 0.353757
\(700\) 8.59397 0.324821
\(701\) −48.0675 −1.81549 −0.907743 0.419527i \(-0.862196\pi\)
−0.907743 + 0.419527i \(0.862196\pi\)
\(702\) 21.9989 0.830295
\(703\) 1.53093 0.0577401
\(704\) 20.3317 0.766279
\(705\) 1.06334 0.0400478
\(706\) 3.55568 0.133820
\(707\) 8.68706 0.326710
\(708\) 11.4153 0.429014
\(709\) 12.6771 0.476099 0.238049 0.971253i \(-0.423492\pi\)
0.238049 + 0.971253i \(0.423492\pi\)
\(710\) −28.0049 −1.05100
\(711\) 13.2773 0.497937
\(712\) −11.9874 −0.449248
\(713\) −24.7064 −0.925263
\(714\) 1.43990 0.0538867
\(715\) 15.1727 0.567426
\(716\) −18.2511 −0.682076
\(717\) −28.9615 −1.08159
\(718\) 18.8372 0.702999
\(719\) 14.9369 0.557053 0.278526 0.960429i \(-0.410154\pi\)
0.278526 + 0.960429i \(0.410154\pi\)
\(720\) −10.5110 −0.391723
\(721\) 2.17320 0.0809343
\(722\) −35.1911 −1.30968
\(723\) −30.5005 −1.13433
\(724\) 1.66971 0.0620541
\(725\) 14.8535 0.551644
\(726\) −36.7274 −1.36308
\(727\) −16.9039 −0.626932 −0.313466 0.949599i \(-0.601490\pi\)
−0.313466 + 0.949599i \(0.601490\pi\)
\(728\) 3.36326 0.124651
\(729\) 20.0019 0.740811
\(730\) −13.1163 −0.485454
\(731\) 3.90285 0.144352
\(732\) 1.54184 0.0569880
\(733\) 17.0731 0.630609 0.315305 0.948991i \(-0.397893\pi\)
0.315305 + 0.948991i \(0.397893\pi\)
\(734\) 9.42174 0.347763
\(735\) −6.04445 −0.222953
\(736\) −59.0063 −2.17500
\(737\) −24.8429 −0.915101
\(738\) 32.2297 1.18639
\(739\) −0.313121 −0.0115183 −0.00575917 0.999983i \(-0.501833\pi\)
−0.00575917 + 0.999983i \(0.501833\pi\)
\(740\) −6.10098 −0.224277
\(741\) −1.24373 −0.0456897
\(742\) 33.3561 1.22454
\(743\) −19.0657 −0.699451 −0.349726 0.936852i \(-0.613725\pi\)
−0.349726 + 0.936852i \(0.613725\pi\)
\(744\) −3.02763 −0.110998
\(745\) 4.23029 0.154986
\(746\) −5.86860 −0.214865
\(747\) 30.6361 1.12092
\(748\) −3.17498 −0.116089
\(749\) −19.7504 −0.721663
\(750\) −23.1617 −0.845744
\(751\) −14.1901 −0.517803 −0.258901 0.965904i \(-0.583360\pi\)
−0.258901 + 0.965904i \(0.583360\pi\)
\(752\) −3.37816 −0.123189
\(753\) −27.0640 −0.986266
\(754\) −18.5194 −0.674435
\(755\) −23.9410 −0.871301
\(756\) −13.9763 −0.508315
\(757\) −32.0979 −1.16662 −0.583309 0.812250i \(-0.698242\pi\)
−0.583309 + 0.812250i \(0.698242\pi\)
\(758\) 31.5992 1.14773
\(759\) 50.4091 1.82973
\(760\) 0.584385 0.0211979
\(761\) 27.3107 0.990013 0.495006 0.868889i \(-0.335166\pi\)
0.495006 + 0.868889i \(0.335166\pi\)
\(762\) −36.2942 −1.31480
\(763\) −3.16588 −0.114613
\(764\) 2.19910 0.0795607
\(765\) 0.873656 0.0315871
\(766\) −6.22997 −0.225098
\(767\) −14.4049 −0.520132
\(768\) −23.6259 −0.852528
\(769\) 15.5549 0.560923 0.280461 0.959865i \(-0.409513\pi\)
0.280461 + 0.959865i \(0.409513\pi\)
\(770\) −22.3047 −0.803805
\(771\) 7.88530 0.283982
\(772\) 2.15178 0.0774442
\(773\) −49.6932 −1.78734 −0.893670 0.448724i \(-0.851878\pi\)
−0.893670 + 0.448724i \(0.851878\pi\)
\(774\) −31.7324 −1.14060
\(775\) 9.76245 0.350678
\(776\) 11.8957 0.427030
\(777\) 5.99145 0.214942
\(778\) −25.9058 −0.928769
\(779\) −5.03345 −0.180342
\(780\) 4.95647 0.177470
\(781\) −60.6423 −2.16995
\(782\) 6.14708 0.219819
\(783\) −24.1562 −0.863271
\(784\) 19.2028 0.685813
\(785\) 2.74478 0.0979654
\(786\) −0.585546 −0.0208858
\(787\) −14.9449 −0.532729 −0.266365 0.963872i \(-0.585823\pi\)
−0.266365 + 0.963872i \(0.585823\pi\)
\(788\) −7.82774 −0.278852
\(789\) 10.2770 0.365873
\(790\) 19.1208 0.680286
\(791\) 18.5435 0.659330
\(792\) −8.10271 −0.287917
\(793\) −1.94564 −0.0690916
\(794\) 25.6103 0.908874
\(795\) −15.4296 −0.547233
\(796\) −25.4972 −0.903726
\(797\) −13.1154 −0.464573 −0.232286 0.972647i \(-0.574621\pi\)
−0.232286 + 0.972647i \(0.574621\pi\)
\(798\) 1.82836 0.0647233
\(799\) 0.280785 0.00993347
\(800\) 23.3157 0.824333
\(801\) −22.7561 −0.804049
\(802\) −7.91853 −0.279613
\(803\) −28.4022 −1.00229
\(804\) −8.11546 −0.286210
\(805\) 18.6630 0.657786
\(806\) −12.1718 −0.428735
\(807\) −8.46636 −0.298030
\(808\) 4.54468 0.159881
\(809\) −19.4909 −0.685262 −0.342631 0.939470i \(-0.611318\pi\)
−0.342631 + 0.939470i \(0.611318\pi\)
\(810\) 2.44072 0.0857581
\(811\) 18.5367 0.650912 0.325456 0.945557i \(-0.394482\pi\)
0.325456 + 0.945557i \(0.394482\pi\)
\(812\) 11.7657 0.412896
\(813\) −27.7683 −0.973876
\(814\) −30.5691 −1.07145
\(815\) 30.1349 1.05558
\(816\) 2.11601 0.0740753
\(817\) 4.95579 0.173381
\(818\) 6.24819 0.218463
\(819\) 6.38459 0.223095
\(820\) 20.0590 0.700492
\(821\) −12.1526 −0.424130 −0.212065 0.977256i \(-0.568019\pi\)
−0.212065 + 0.977256i \(0.568019\pi\)
\(822\) 12.0414 0.419992
\(823\) 15.5837 0.543214 0.271607 0.962408i \(-0.412445\pi\)
0.271607 + 0.962408i \(0.412445\pi\)
\(824\) 1.13692 0.0396065
\(825\) −19.9185 −0.693475
\(826\) 21.1761 0.736809
\(827\) 14.2481 0.495456 0.247728 0.968830i \(-0.420316\pi\)
0.247728 + 0.968830i \(0.420316\pi\)
\(828\) −21.5997 −0.750641
\(829\) 52.8914 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(830\) 44.1194 1.53141
\(831\) −18.0508 −0.626176
\(832\) −8.38120 −0.290566
\(833\) −1.59609 −0.0553014
\(834\) 48.5358 1.68066
\(835\) 14.8177 0.512787
\(836\) −4.03154 −0.139434
\(837\) −15.8766 −0.548777
\(838\) −32.2972 −1.11569
\(839\) −25.6169 −0.884395 −0.442198 0.896918i \(-0.645801\pi\)
−0.442198 + 0.896918i \(0.645801\pi\)
\(840\) 2.28705 0.0789107
\(841\) −8.66460 −0.298779
\(842\) 11.0858 0.382041
\(843\) −6.32625 −0.217888
\(844\) −17.8821 −0.615529
\(845\) 10.7261 0.368989
\(846\) −2.28294 −0.0784892
\(847\) −29.4446 −1.01173
\(848\) 49.0187 1.68331
\(849\) 24.5293 0.841842
\(850\) −2.42894 −0.0833121
\(851\) 25.5782 0.876808
\(852\) −19.8101 −0.678681
\(853\) 23.7883 0.814498 0.407249 0.913317i \(-0.366488\pi\)
0.407249 + 0.913317i \(0.366488\pi\)
\(854\) 2.86020 0.0978739
\(855\) 1.10936 0.0379392
\(856\) −10.3325 −0.353158
\(857\) −1.27557 −0.0435727 −0.0217863 0.999763i \(-0.506935\pi\)
−0.0217863 + 0.999763i \(0.506935\pi\)
\(858\) 24.8345 0.847835
\(859\) −38.6456 −1.31857 −0.659285 0.751893i \(-0.729141\pi\)
−0.659285 + 0.751893i \(0.729141\pi\)
\(860\) −19.7495 −0.673454
\(861\) −19.6989 −0.671337
\(862\) −50.7937 −1.73004
\(863\) 13.3344 0.453908 0.226954 0.973905i \(-0.427123\pi\)
0.226954 + 0.973905i \(0.427123\pi\)
\(864\) −37.9182 −1.29000
\(865\) 22.4809 0.764373
\(866\) −10.6056 −0.360393
\(867\) 19.1904 0.651739
\(868\) 7.73302 0.262476
\(869\) 41.4044 1.40455
\(870\) −12.5933 −0.426954
\(871\) 10.2408 0.346998
\(872\) −1.65625 −0.0560876
\(873\) 22.5820 0.764284
\(874\) 7.80547 0.264024
\(875\) −18.5688 −0.627741
\(876\) −9.27815 −0.313480
\(877\) −5.39699 −0.182244 −0.0911218 0.995840i \(-0.529045\pi\)
−0.0911218 + 0.995840i \(0.529045\pi\)
\(878\) −50.2431 −1.69562
\(879\) 27.2770 0.920031
\(880\) −32.7781 −1.10495
\(881\) −5.04786 −0.170067 −0.0850333 0.996378i \(-0.527100\pi\)
−0.0850333 + 0.996378i \(0.527100\pi\)
\(882\) 12.9771 0.436963
\(883\) −36.4145 −1.22545 −0.612723 0.790298i \(-0.709926\pi\)
−0.612723 + 0.790298i \(0.709926\pi\)
\(884\) 1.30880 0.0440197
\(885\) −9.79549 −0.329272
\(886\) 64.1841 2.15631
\(887\) −24.9346 −0.837224 −0.418612 0.908165i \(-0.637483\pi\)
−0.418612 + 0.908165i \(0.637483\pi\)
\(888\) 3.13446 0.105185
\(889\) −29.0973 −0.975891
\(890\) −32.7714 −1.09850
\(891\) 5.28517 0.177060
\(892\) 41.0932 1.37590
\(893\) 0.356538 0.0119311
\(894\) 6.92410 0.231577
\(895\) 15.6613 0.523498
\(896\) −11.9450 −0.399053
\(897\) −20.7798 −0.693818
\(898\) −13.3587 −0.445785
\(899\) 13.3654 0.445763
\(900\) 8.53486 0.284495
\(901\) −4.07434 −0.135736
\(902\) 100.506 3.34649
\(903\) 19.3950 0.645424
\(904\) 9.70111 0.322654
\(905\) −1.43277 −0.0476270
\(906\) −39.1863 −1.30188
\(907\) −27.1951 −0.903000 −0.451500 0.892271i \(-0.649111\pi\)
−0.451500 + 0.892271i \(0.649111\pi\)
\(908\) −9.38102 −0.311320
\(909\) 8.62731 0.286150
\(910\) 9.19451 0.304795
\(911\) −47.9588 −1.58895 −0.794473 0.607299i \(-0.792253\pi\)
−0.794473 + 0.607299i \(0.792253\pi\)
\(912\) 2.68688 0.0889716
\(913\) 95.5369 3.16181
\(914\) 48.4834 1.60369
\(915\) −1.32305 −0.0437387
\(916\) 7.67500 0.253589
\(917\) −0.469436 −0.0155021
\(918\) 3.95018 0.130376
\(919\) −20.3674 −0.671857 −0.335928 0.941887i \(-0.609050\pi\)
−0.335928 + 0.941887i \(0.609050\pi\)
\(920\) 9.76366 0.321898
\(921\) −1.50100 −0.0494597
\(922\) −72.8432 −2.39896
\(923\) 24.9982 0.822825
\(924\) −15.7778 −0.519053
\(925\) −10.1069 −0.332313
\(926\) 42.9604 1.41177
\(927\) 2.15826 0.0708864
\(928\) 31.9207 1.04785
\(929\) 3.43001 0.112535 0.0562675 0.998416i \(-0.482080\pi\)
0.0562675 + 0.998416i \(0.482080\pi\)
\(930\) −8.27697 −0.271413
\(931\) −2.02670 −0.0664224
\(932\) −12.4974 −0.409367
\(933\) 25.0593 0.820403
\(934\) −22.8778 −0.748583
\(935\) 2.72445 0.0890989
\(936\) 3.34013 0.109176
\(937\) −55.8441 −1.82435 −0.912173 0.409805i \(-0.865597\pi\)
−0.912173 + 0.409805i \(0.865597\pi\)
\(938\) −15.0546 −0.491551
\(939\) 1.01595 0.0331544
\(940\) −1.42086 −0.0463432
\(941\) −35.2757 −1.14996 −0.574978 0.818169i \(-0.694989\pi\)
−0.574978 + 0.818169i \(0.694989\pi\)
\(942\) 4.49263 0.146378
\(943\) −84.0969 −2.73857
\(944\) 31.1195 1.01285
\(945\) 11.9931 0.390135
\(946\) −98.9555 −3.21732
\(947\) −45.2546 −1.47058 −0.735289 0.677754i \(-0.762954\pi\)
−0.735289 + 0.677754i \(0.762954\pi\)
\(948\) 13.5256 0.439291
\(949\) 11.7080 0.380059
\(950\) −3.08424 −0.100066
\(951\) 27.7144 0.898701
\(952\) 0.603916 0.0195730
\(953\) 39.0690 1.26557 0.632784 0.774328i \(-0.281912\pi\)
0.632784 + 0.774328i \(0.281912\pi\)
\(954\) 33.1267 1.07252
\(955\) −1.88705 −0.0610634
\(956\) 38.6988 1.25161
\(957\) −27.2698 −0.881508
\(958\) −65.9168 −2.12968
\(959\) 9.65365 0.311733
\(960\) −5.69929 −0.183944
\(961\) −22.2156 −0.716631
\(962\) 12.6013 0.406283
\(963\) −19.6146 −0.632070
\(964\) 40.7553 1.31264
\(965\) −1.84644 −0.0594390
\(966\) 30.5475 0.982850
\(967\) 11.8771 0.381942 0.190971 0.981596i \(-0.438836\pi\)
0.190971 + 0.981596i \(0.438836\pi\)
\(968\) −15.4041 −0.495106
\(969\) −0.223328 −0.00717434
\(970\) 32.5206 1.04417
\(971\) −43.3430 −1.39094 −0.695471 0.718554i \(-0.744804\pi\)
−0.695471 + 0.718554i \(0.744804\pi\)
\(972\) −22.7357 −0.729248
\(973\) 38.9115 1.24744
\(974\) 14.2485 0.456552
\(975\) 8.21089 0.262959
\(976\) 4.20323 0.134542
\(977\) 30.9084 0.988846 0.494423 0.869221i \(-0.335379\pi\)
0.494423 + 0.869221i \(0.335379\pi\)
\(978\) 49.3245 1.57722
\(979\) −70.9637 −2.26801
\(980\) 8.07669 0.258001
\(981\) −3.14411 −0.100384
\(982\) 34.3308 1.09554
\(983\) 35.0128 1.11673 0.558367 0.829594i \(-0.311428\pi\)
0.558367 + 0.829594i \(0.311428\pi\)
\(984\) −10.3056 −0.328530
\(985\) 6.71698 0.214021
\(986\) −3.32539 −0.105902
\(987\) 1.39535 0.0444143
\(988\) 1.66190 0.0528720
\(989\) 82.7992 2.63286
\(990\) −22.1513 −0.704014
\(991\) −22.7329 −0.722135 −0.361067 0.932540i \(-0.617588\pi\)
−0.361067 + 0.932540i \(0.617588\pi\)
\(992\) 20.9799 0.666112
\(993\) −31.6169 −1.00333
\(994\) −36.7487 −1.16560
\(995\) 21.8792 0.693616
\(996\) 31.2091 0.988898
\(997\) −10.6754 −0.338095 −0.169047 0.985608i \(-0.554069\pi\)
−0.169047 + 0.985608i \(0.554069\pi\)
\(998\) 10.8201 0.342506
\(999\) 16.4368 0.520038
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 547.2.a.b.1.16 18
3.2 odd 2 4923.2.a.l.1.3 18
4.3 odd 2 8752.2.a.s.1.11 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.16 18 1.1 even 1 trivial
4923.2.a.l.1.3 18 3.2 odd 2
8752.2.a.s.1.11 18 4.3 odd 2