Properties

Label 4730.2.a.bb.1.5
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.580450\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.580450 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.580450 q^{6} +1.64698 q^{7} -1.00000 q^{8} -2.66308 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.580450 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.580450 q^{6} +1.64698 q^{7} -1.00000 q^{8} -2.66308 q^{9} +1.00000 q^{10} -1.00000 q^{11} -0.580450 q^{12} +2.95295 q^{13} -1.64698 q^{14} +0.580450 q^{15} +1.00000 q^{16} +1.81307 q^{17} +2.66308 q^{18} -3.80194 q^{19} -1.00000 q^{20} -0.955986 q^{21} +1.00000 q^{22} +0.952953 q^{23} +0.580450 q^{24} +1.00000 q^{25} -2.95295 q^{26} +3.28713 q^{27} +1.64698 q^{28} +3.68382 q^{29} -0.580450 q^{30} +3.41206 q^{31} -1.00000 q^{32} +0.580450 q^{33} -1.81307 q^{34} -1.64698 q^{35} -2.66308 q^{36} -2.19499 q^{37} +3.80194 q^{38} -1.71404 q^{39} +1.00000 q^{40} -5.45269 q^{41} +0.955986 q^{42} +1.00000 q^{43} -1.00000 q^{44} +2.66308 q^{45} -0.952953 q^{46} -11.1140 q^{47} -0.580450 q^{48} -4.28747 q^{49} -1.00000 q^{50} -1.05240 q^{51} +2.95295 q^{52} +8.91291 q^{53} -3.28713 q^{54} +1.00000 q^{55} -1.64698 q^{56} +2.20684 q^{57} -3.68382 q^{58} -3.10392 q^{59} +0.580450 q^{60} -9.87214 q^{61} -3.41206 q^{62} -4.38602 q^{63} +1.00000 q^{64} -2.95295 q^{65} -0.580450 q^{66} +6.18592 q^{67} +1.81307 q^{68} -0.553141 q^{69} +1.64698 q^{70} +12.8474 q^{71} +2.66308 q^{72} -0.229098 q^{73} +2.19499 q^{74} -0.580450 q^{75} -3.80194 q^{76} -1.64698 q^{77} +1.71404 q^{78} +15.2682 q^{79} -1.00000 q^{80} +6.08122 q^{81} +5.45269 q^{82} +2.72642 q^{83} -0.955986 q^{84} -1.81307 q^{85} -1.00000 q^{86} -2.13827 q^{87} +1.00000 q^{88} -5.52821 q^{89} -2.66308 q^{90} +4.86344 q^{91} +0.952953 q^{92} -1.98053 q^{93} +11.1140 q^{94} +3.80194 q^{95} +0.580450 q^{96} -3.05239 q^{97} +4.28747 q^{98} +2.66308 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 11q^{2} - q^{3} + 11q^{4} - 11q^{5} + q^{6} + 6q^{7} - 11q^{8} + 12q^{9} + O(q^{10}) \) \( 11q - 11q^{2} - q^{3} + 11q^{4} - 11q^{5} + q^{6} + 6q^{7} - 11q^{8} + 12q^{9} + 11q^{10} - 11q^{11} - q^{12} + 4q^{13} - 6q^{14} + q^{15} + 11q^{16} - 10q^{17} - 12q^{18} + 14q^{19} - 11q^{20} - 2q^{21} + 11q^{22} - 18q^{23} + q^{24} + 11q^{25} - 4q^{26} + 2q^{27} + 6q^{28} - 6q^{29} - q^{30} + 13q^{31} - 11q^{32} + q^{33} + 10q^{34} - 6q^{35} + 12q^{36} + q^{37} - 14q^{38} + 12q^{39} + 11q^{40} + 12q^{41} + 2q^{42} + 11q^{43} - 11q^{44} - 12q^{45} + 18q^{46} - 5q^{47} - q^{48} + 31q^{49} - 11q^{50} + q^{51} + 4q^{52} - 27q^{53} - 2q^{54} + 11q^{55} - 6q^{56} - 5q^{57} + 6q^{58} + 11q^{59} + q^{60} + 36q^{61} - 13q^{62} + 17q^{63} + 11q^{64} - 4q^{65} - q^{66} + 18q^{67} - 10q^{68} + 14q^{69} + 6q^{70} - 14q^{71} - 12q^{72} - 11q^{73} - q^{74} - q^{75} + 14q^{76} - 6q^{77} - 12q^{78} + 28q^{79} - 11q^{80} + 7q^{81} - 12q^{82} - 4q^{83} - 2q^{84} + 10q^{85} - 11q^{86} + 38q^{87} + 11q^{88} - 7q^{89} + 12q^{90} + 14q^{91} - 18q^{92} - 3q^{93} + 5q^{94} - 14q^{95} + q^{96} - q^{97} - 31q^{98} - 12q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.580450 −0.335123 −0.167561 0.985862i \(-0.553589\pi\)
−0.167561 + 0.985862i \(0.553589\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0.580450 0.236968
\(7\) 1.64698 0.622498 0.311249 0.950328i \(-0.399253\pi\)
0.311249 + 0.950328i \(0.399253\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.66308 −0.887693
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −0.580450 −0.167561
\(13\) 2.95295 0.819002 0.409501 0.912310i \(-0.365703\pi\)
0.409501 + 0.912310i \(0.365703\pi\)
\(14\) −1.64698 −0.440173
\(15\) 0.580450 0.149871
\(16\) 1.00000 0.250000
\(17\) 1.81307 0.439734 0.219867 0.975530i \(-0.429438\pi\)
0.219867 + 0.975530i \(0.429438\pi\)
\(18\) 2.66308 0.627694
\(19\) −3.80194 −0.872226 −0.436113 0.899892i \(-0.643645\pi\)
−0.436113 + 0.899892i \(0.643645\pi\)
\(20\) −1.00000 −0.223607
\(21\) −0.955986 −0.208613
\(22\) 1.00000 0.213201
\(23\) 0.952953 0.198704 0.0993522 0.995052i \(-0.468323\pi\)
0.0993522 + 0.995052i \(0.468323\pi\)
\(24\) 0.580450 0.118484
\(25\) 1.00000 0.200000
\(26\) −2.95295 −0.579122
\(27\) 3.28713 0.632609
\(28\) 1.64698 0.311249
\(29\) 3.68382 0.684068 0.342034 0.939688i \(-0.388884\pi\)
0.342034 + 0.939688i \(0.388884\pi\)
\(30\) −0.580450 −0.105975
\(31\) 3.41206 0.612824 0.306412 0.951899i \(-0.400872\pi\)
0.306412 + 0.951899i \(0.400872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.580450 0.101043
\(34\) −1.81307 −0.310939
\(35\) −1.64698 −0.278390
\(36\) −2.66308 −0.443846
\(37\) −2.19499 −0.360854 −0.180427 0.983588i \(-0.557748\pi\)
−0.180427 + 0.983588i \(0.557748\pi\)
\(38\) 3.80194 0.616757
\(39\) −1.71404 −0.274466
\(40\) 1.00000 0.158114
\(41\) −5.45269 −0.851568 −0.425784 0.904825i \(-0.640002\pi\)
−0.425784 + 0.904825i \(0.640002\pi\)
\(42\) 0.955986 0.147512
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 2.66308 0.396988
\(46\) −0.952953 −0.140505
\(47\) −11.1140 −1.62115 −0.810574 0.585636i \(-0.800845\pi\)
−0.810574 + 0.585636i \(0.800845\pi\)
\(48\) −0.580450 −0.0837807
\(49\) −4.28747 −0.612496
\(50\) −1.00000 −0.141421
\(51\) −1.05240 −0.147365
\(52\) 2.95295 0.409501
\(53\) 8.91291 1.22428 0.612141 0.790749i \(-0.290309\pi\)
0.612141 + 0.790749i \(0.290309\pi\)
\(54\) −3.28713 −0.447322
\(55\) 1.00000 0.134840
\(56\) −1.64698 −0.220086
\(57\) 2.20684 0.292303
\(58\) −3.68382 −0.483709
\(59\) −3.10392 −0.404096 −0.202048 0.979376i \(-0.564760\pi\)
−0.202048 + 0.979376i \(0.564760\pi\)
\(60\) 0.580450 0.0749357
\(61\) −9.87214 −1.26400 −0.631999 0.774969i \(-0.717765\pi\)
−0.631999 + 0.774969i \(0.717765\pi\)
\(62\) −3.41206 −0.433332
\(63\) −4.38602 −0.552587
\(64\) 1.00000 0.125000
\(65\) −2.95295 −0.366269
\(66\) −0.580450 −0.0714484
\(67\) 6.18592 0.755730 0.377865 0.925861i \(-0.376658\pi\)
0.377865 + 0.925861i \(0.376658\pi\)
\(68\) 1.81307 0.219867
\(69\) −0.553141 −0.0665904
\(70\) 1.64698 0.196851
\(71\) 12.8474 1.52471 0.762356 0.647158i \(-0.224043\pi\)
0.762356 + 0.647158i \(0.224043\pi\)
\(72\) 2.66308 0.313847
\(73\) −0.229098 −0.0268138 −0.0134069 0.999910i \(-0.504268\pi\)
−0.0134069 + 0.999910i \(0.504268\pi\)
\(74\) 2.19499 0.255162
\(75\) −0.580450 −0.0670246
\(76\) −3.80194 −0.436113
\(77\) −1.64698 −0.187690
\(78\) 1.71404 0.194077
\(79\) 15.2682 1.71781 0.858905 0.512134i \(-0.171145\pi\)
0.858905 + 0.512134i \(0.171145\pi\)
\(80\) −1.00000 −0.111803
\(81\) 6.08122 0.675691
\(82\) 5.45269 0.602149
\(83\) 2.72642 0.299263 0.149632 0.988742i \(-0.452191\pi\)
0.149632 + 0.988742i \(0.452191\pi\)
\(84\) −0.955986 −0.104307
\(85\) −1.81307 −0.196655
\(86\) −1.00000 −0.107833
\(87\) −2.13827 −0.229247
\(88\) 1.00000 0.106600
\(89\) −5.52821 −0.585989 −0.292995 0.956114i \(-0.594652\pi\)
−0.292995 + 0.956114i \(0.594652\pi\)
\(90\) −2.66308 −0.280713
\(91\) 4.86344 0.509827
\(92\) 0.952953 0.0993522
\(93\) −1.98053 −0.205371
\(94\) 11.1140 1.14632
\(95\) 3.80194 0.390071
\(96\) 0.580450 0.0592419
\(97\) −3.05239 −0.309923 −0.154962 0.987920i \(-0.549525\pi\)
−0.154962 + 0.987920i \(0.549525\pi\)
\(98\) 4.28747 0.433100
\(99\) 2.66308 0.267649
\(100\) 1.00000 0.100000
\(101\) 1.68377 0.167542 0.0837709 0.996485i \(-0.473304\pi\)
0.0837709 + 0.996485i \(0.473304\pi\)
\(102\) 1.05240 0.104203
\(103\) −1.85743 −0.183018 −0.0915091 0.995804i \(-0.529169\pi\)
−0.0915091 + 0.995804i \(0.529169\pi\)
\(104\) −2.95295 −0.289561
\(105\) 0.955986 0.0932947
\(106\) −8.91291 −0.865698
\(107\) −14.8768 −1.43819 −0.719097 0.694909i \(-0.755445\pi\)
−0.719097 + 0.694909i \(0.755445\pi\)
\(108\) 3.28713 0.316304
\(109\) 11.2283 1.07548 0.537738 0.843112i \(-0.319279\pi\)
0.537738 + 0.843112i \(0.319279\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 1.27408 0.120930
\(112\) 1.64698 0.155625
\(113\) 2.54599 0.239506 0.119753 0.992804i \(-0.461790\pi\)
0.119753 + 0.992804i \(0.461790\pi\)
\(114\) −2.20684 −0.206689
\(115\) −0.952953 −0.0888633
\(116\) 3.68382 0.342034
\(117\) −7.86395 −0.727022
\(118\) 3.10392 0.285739
\(119\) 2.98608 0.273734
\(120\) −0.580450 −0.0529876
\(121\) 1.00000 0.0909091
\(122\) 9.87214 0.893781
\(123\) 3.16501 0.285380
\(124\) 3.41206 0.306412
\(125\) −1.00000 −0.0894427
\(126\) 4.38602 0.390738
\(127\) −2.73781 −0.242941 −0.121471 0.992595i \(-0.538761\pi\)
−0.121471 + 0.992595i \(0.538761\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.580450 −0.0511057
\(130\) 2.95295 0.258991
\(131\) −20.4265 −1.78467 −0.892334 0.451375i \(-0.850934\pi\)
−0.892334 + 0.451375i \(0.850934\pi\)
\(132\) 0.580450 0.0505217
\(133\) −6.26171 −0.542959
\(134\) −6.18592 −0.534382
\(135\) −3.28713 −0.282911
\(136\) −1.81307 −0.155469
\(137\) −1.31421 −0.112281 −0.0561405 0.998423i \(-0.517879\pi\)
−0.0561405 + 0.998423i \(0.517879\pi\)
\(138\) 0.553141 0.0470865
\(139\) 19.4516 1.64986 0.824931 0.565234i \(-0.191214\pi\)
0.824931 + 0.565234i \(0.191214\pi\)
\(140\) −1.64698 −0.139195
\(141\) 6.45114 0.543284
\(142\) −12.8474 −1.07813
\(143\) −2.95295 −0.246938
\(144\) −2.66308 −0.221923
\(145\) −3.68382 −0.305925
\(146\) 0.229098 0.0189602
\(147\) 2.48866 0.205261
\(148\) −2.19499 −0.180427
\(149\) 0.630570 0.0516583 0.0258291 0.999666i \(-0.491777\pi\)
0.0258291 + 0.999666i \(0.491777\pi\)
\(150\) 0.580450 0.0473935
\(151\) 18.1908 1.48034 0.740172 0.672418i \(-0.234744\pi\)
0.740172 + 0.672418i \(0.234744\pi\)
\(152\) 3.80194 0.308378
\(153\) −4.82834 −0.390348
\(154\) 1.64698 0.132717
\(155\) −3.41206 −0.274063
\(156\) −1.71404 −0.137233
\(157\) 18.9796 1.51474 0.757368 0.652989i \(-0.226485\pi\)
0.757368 + 0.652989i \(0.226485\pi\)
\(158\) −15.2682 −1.21468
\(159\) −5.17349 −0.410285
\(160\) 1.00000 0.0790569
\(161\) 1.56949 0.123693
\(162\) −6.08122 −0.477786
\(163\) 7.78236 0.609562 0.304781 0.952422i \(-0.401417\pi\)
0.304781 + 0.952422i \(0.401417\pi\)
\(164\) −5.45269 −0.425784
\(165\) −0.580450 −0.0451879
\(166\) −2.72642 −0.211611
\(167\) −8.48582 −0.656652 −0.328326 0.944564i \(-0.606485\pi\)
−0.328326 + 0.944564i \(0.606485\pi\)
\(168\) 0.955986 0.0737560
\(169\) −4.28007 −0.329236
\(170\) 1.81307 0.139056
\(171\) 10.1249 0.774268
\(172\) 1.00000 0.0762493
\(173\) −4.62742 −0.351816 −0.175908 0.984407i \(-0.556286\pi\)
−0.175908 + 0.984407i \(0.556286\pi\)
\(174\) 2.13827 0.162102
\(175\) 1.64698 0.124500
\(176\) −1.00000 −0.0753778
\(177\) 1.80167 0.135422
\(178\) 5.52821 0.414357
\(179\) 11.4038 0.852356 0.426178 0.904639i \(-0.359860\pi\)
0.426178 + 0.904639i \(0.359860\pi\)
\(180\) 2.66308 0.198494
\(181\) 9.28441 0.690104 0.345052 0.938583i \(-0.387861\pi\)
0.345052 + 0.938583i \(0.387861\pi\)
\(182\) −4.86344 −0.360502
\(183\) 5.73028 0.423594
\(184\) −0.952953 −0.0702526
\(185\) 2.19499 0.161379
\(186\) 1.98053 0.145219
\(187\) −1.81307 −0.132585
\(188\) −11.1140 −0.810574
\(189\) 5.41383 0.393798
\(190\) −3.80194 −0.275822
\(191\) 3.13535 0.226866 0.113433 0.993546i \(-0.463815\pi\)
0.113433 + 0.993546i \(0.463815\pi\)
\(192\) −0.580450 −0.0418903
\(193\) 16.5144 1.18873 0.594365 0.804195i \(-0.297403\pi\)
0.594365 + 0.804195i \(0.297403\pi\)
\(194\) 3.05239 0.219149
\(195\) 1.71404 0.122745
\(196\) −4.28747 −0.306248
\(197\) −2.07563 −0.147882 −0.0739412 0.997263i \(-0.523558\pi\)
−0.0739412 + 0.997263i \(0.523558\pi\)
\(198\) −2.66308 −0.189257
\(199\) 4.55938 0.323206 0.161603 0.986856i \(-0.448334\pi\)
0.161603 + 0.986856i \(0.448334\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −3.59061 −0.253262
\(202\) −1.68377 −0.118470
\(203\) 6.06716 0.425831
\(204\) −1.05240 −0.0736824
\(205\) 5.45269 0.380833
\(206\) 1.85743 0.129413
\(207\) −2.53779 −0.176389
\(208\) 2.95295 0.204750
\(209\) 3.80194 0.262986
\(210\) −0.955986 −0.0659693
\(211\) −23.5716 −1.62274 −0.811368 0.584535i \(-0.801277\pi\)
−0.811368 + 0.584535i \(0.801277\pi\)
\(212\) 8.91291 0.612141
\(213\) −7.45729 −0.510965
\(214\) 14.8768 1.01696
\(215\) −1.00000 −0.0681994
\(216\) −3.28713 −0.223661
\(217\) 5.61957 0.381482
\(218\) −11.2283 −0.760477
\(219\) 0.132980 0.00898593
\(220\) 1.00000 0.0674200
\(221\) 5.35391 0.360143
\(222\) −1.27408 −0.0855107
\(223\) 7.34985 0.492183 0.246091 0.969247i \(-0.420854\pi\)
0.246091 + 0.969247i \(0.420854\pi\)
\(224\) −1.64698 −0.110043
\(225\) −2.66308 −0.177539
\(226\) −2.54599 −0.169356
\(227\) −23.1182 −1.53441 −0.767205 0.641402i \(-0.778353\pi\)
−0.767205 + 0.641402i \(0.778353\pi\)
\(228\) 2.20684 0.146151
\(229\) 28.5225 1.88482 0.942409 0.334462i \(-0.108555\pi\)
0.942409 + 0.334462i \(0.108555\pi\)
\(230\) 0.952953 0.0628359
\(231\) 0.955986 0.0628993
\(232\) −3.68382 −0.241855
\(233\) 15.4512 1.01224 0.506122 0.862462i \(-0.331079\pi\)
0.506122 + 0.862462i \(0.331079\pi\)
\(234\) 7.86395 0.514082
\(235\) 11.1140 0.725000
\(236\) −3.10392 −0.202048
\(237\) −8.86244 −0.575677
\(238\) −2.98608 −0.193559
\(239\) −5.45055 −0.352566 −0.176283 0.984339i \(-0.556407\pi\)
−0.176283 + 0.984339i \(0.556407\pi\)
\(240\) 0.580450 0.0374679
\(241\) −6.42694 −0.413996 −0.206998 0.978341i \(-0.566369\pi\)
−0.206998 + 0.978341i \(0.566369\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −13.3912 −0.859048
\(244\) −9.87214 −0.631999
\(245\) 4.28747 0.273916
\(246\) −3.16501 −0.201794
\(247\) −11.2270 −0.714355
\(248\) −3.41206 −0.216666
\(249\) −1.58255 −0.100290
\(250\) 1.00000 0.0632456
\(251\) 8.86310 0.559434 0.279717 0.960083i \(-0.409759\pi\)
0.279717 + 0.960083i \(0.409759\pi\)
\(252\) −4.38602 −0.276294
\(253\) −0.952953 −0.0599116
\(254\) 2.73781 0.171785
\(255\) 1.05240 0.0659035
\(256\) 1.00000 0.0625000
\(257\) 4.29084 0.267655 0.133828 0.991005i \(-0.457273\pi\)
0.133828 + 0.991005i \(0.457273\pi\)
\(258\) 0.580450 0.0361372
\(259\) −3.61509 −0.224631
\(260\) −2.95295 −0.183134
\(261\) −9.81030 −0.607242
\(262\) 20.4265 1.26195
\(263\) 23.3413 1.43929 0.719643 0.694344i \(-0.244306\pi\)
0.719643 + 0.694344i \(0.244306\pi\)
\(264\) −0.580450 −0.0357242
\(265\) −8.91291 −0.547515
\(266\) 6.26171 0.383930
\(267\) 3.20885 0.196378
\(268\) 6.18592 0.377865
\(269\) 9.64797 0.588247 0.294124 0.955767i \(-0.404972\pi\)
0.294124 + 0.955767i \(0.404972\pi\)
\(270\) 3.28713 0.200048
\(271\) 28.2847 1.71817 0.859086 0.511831i \(-0.171032\pi\)
0.859086 + 0.511831i \(0.171032\pi\)
\(272\) 1.81307 0.109933
\(273\) −2.82298 −0.170855
\(274\) 1.31421 0.0793946
\(275\) −1.00000 −0.0603023
\(276\) −0.553141 −0.0332952
\(277\) −9.51350 −0.571611 −0.285806 0.958288i \(-0.592261\pi\)
−0.285806 + 0.958288i \(0.592261\pi\)
\(278\) −19.4516 −1.16663
\(279\) −9.08657 −0.543999
\(280\) 1.64698 0.0984256
\(281\) 28.8476 1.72091 0.860453 0.509529i \(-0.170181\pi\)
0.860453 + 0.509529i \(0.170181\pi\)
\(282\) −6.45114 −0.384160
\(283\) −3.77444 −0.224367 −0.112184 0.993687i \(-0.535785\pi\)
−0.112184 + 0.993687i \(0.535785\pi\)
\(284\) 12.8474 0.762356
\(285\) −2.20684 −0.130722
\(286\) 2.95295 0.174612
\(287\) −8.98045 −0.530099
\(288\) 2.66308 0.156923
\(289\) −13.7128 −0.806634
\(290\) 3.68382 0.216321
\(291\) 1.77176 0.103862
\(292\) −0.229098 −0.0134069
\(293\) −20.0934 −1.17387 −0.586934 0.809635i \(-0.699665\pi\)
−0.586934 + 0.809635i \(0.699665\pi\)
\(294\) −2.48866 −0.145142
\(295\) 3.10392 0.180717
\(296\) 2.19499 0.127581
\(297\) −3.28713 −0.190739
\(298\) −0.630570 −0.0365279
\(299\) 2.81403 0.162739
\(300\) −0.580450 −0.0335123
\(301\) 1.64698 0.0949301
\(302\) −18.1908 −1.04676
\(303\) −0.977346 −0.0561471
\(304\) −3.80194 −0.218056
\(305\) 9.87214 0.565277
\(306\) 4.82834 0.276018
\(307\) 22.6639 1.29350 0.646749 0.762703i \(-0.276128\pi\)
0.646749 + 0.762703i \(0.276128\pi\)
\(308\) −1.64698 −0.0938451
\(309\) 1.07815 0.0613336
\(310\) 3.41206 0.193792
\(311\) 4.62256 0.262122 0.131061 0.991374i \(-0.458162\pi\)
0.131061 + 0.991374i \(0.458162\pi\)
\(312\) 1.71404 0.0970384
\(313\) 0.839632 0.0474588 0.0237294 0.999718i \(-0.492446\pi\)
0.0237294 + 0.999718i \(0.492446\pi\)
\(314\) −18.9796 −1.07108
\(315\) 4.38602 0.247125
\(316\) 15.2682 0.858905
\(317\) 27.1016 1.52218 0.761088 0.648649i \(-0.224665\pi\)
0.761088 + 0.648649i \(0.224665\pi\)
\(318\) 5.17349 0.290115
\(319\) −3.68382 −0.206254
\(320\) −1.00000 −0.0559017
\(321\) 8.63524 0.481972
\(322\) −1.56949 −0.0874643
\(323\) −6.89319 −0.383547
\(324\) 6.08122 0.337846
\(325\) 2.95295 0.163800
\(326\) −7.78236 −0.431025
\(327\) −6.51747 −0.360417
\(328\) 5.45269 0.301075
\(329\) −18.3045 −1.00916
\(330\) 0.580450 0.0319527
\(331\) 27.7639 1.52604 0.763020 0.646375i \(-0.223716\pi\)
0.763020 + 0.646375i \(0.223716\pi\)
\(332\) 2.72642 0.149632
\(333\) 5.84542 0.320327
\(334\) 8.48582 0.464323
\(335\) −6.18592 −0.337973
\(336\) −0.955986 −0.0521533
\(337\) 7.90859 0.430808 0.215404 0.976525i \(-0.430893\pi\)
0.215404 + 0.976525i \(0.430893\pi\)
\(338\) 4.28007 0.232805
\(339\) −1.47782 −0.0802640
\(340\) −1.81307 −0.0983275
\(341\) −3.41206 −0.184773
\(342\) −10.1249 −0.547490
\(343\) −18.5902 −1.00378
\(344\) −1.00000 −0.0539164
\(345\) 0.553141 0.0297801
\(346\) 4.62742 0.248772
\(347\) −5.46682 −0.293474 −0.146737 0.989176i \(-0.546877\pi\)
−0.146737 + 0.989176i \(0.546877\pi\)
\(348\) −2.13827 −0.114623
\(349\) 22.7588 1.21825 0.609126 0.793074i \(-0.291521\pi\)
0.609126 + 0.793074i \(0.291521\pi\)
\(350\) −1.64698 −0.0880346
\(351\) 9.70675 0.518108
\(352\) 1.00000 0.0533002
\(353\) −31.4006 −1.67129 −0.835644 0.549272i \(-0.814905\pi\)
−0.835644 + 0.549272i \(0.814905\pi\)
\(354\) −1.80167 −0.0957577
\(355\) −12.8474 −0.681872
\(356\) −5.52821 −0.292995
\(357\) −1.73327 −0.0917343
\(358\) −11.4038 −0.602707
\(359\) −16.0218 −0.845596 −0.422798 0.906224i \(-0.638952\pi\)
−0.422798 + 0.906224i \(0.638952\pi\)
\(360\) −2.66308 −0.140357
\(361\) −4.54522 −0.239222
\(362\) −9.28441 −0.487977
\(363\) −0.580450 −0.0304657
\(364\) 4.86344 0.254914
\(365\) 0.229098 0.0119915
\(366\) −5.73028 −0.299526
\(367\) −25.5328 −1.33280 −0.666401 0.745593i \(-0.732166\pi\)
−0.666401 + 0.745593i \(0.732166\pi\)
\(368\) 0.952953 0.0496761
\(369\) 14.5209 0.755930
\(370\) −2.19499 −0.114112
\(371\) 14.6793 0.762113
\(372\) −1.98053 −0.102686
\(373\) 12.2063 0.632018 0.316009 0.948756i \(-0.397657\pi\)
0.316009 + 0.948756i \(0.397657\pi\)
\(374\) 1.81307 0.0937515
\(375\) 0.580450 0.0299743
\(376\) 11.1140 0.573162
\(377\) 10.8781 0.560253
\(378\) −5.41383 −0.278457
\(379\) 24.9357 1.28086 0.640431 0.768015i \(-0.278756\pi\)
0.640431 + 0.768015i \(0.278756\pi\)
\(380\) 3.80194 0.195036
\(381\) 1.58916 0.0814152
\(382\) −3.13535 −0.160419
\(383\) 18.8612 0.963761 0.481881 0.876237i \(-0.339954\pi\)
0.481881 + 0.876237i \(0.339954\pi\)
\(384\) 0.580450 0.0296209
\(385\) 1.64698 0.0839377
\(386\) −16.5144 −0.840559
\(387\) −2.66308 −0.135372
\(388\) −3.05239 −0.154962
\(389\) 14.3498 0.727565 0.363782 0.931484i \(-0.381485\pi\)
0.363782 + 0.931484i \(0.381485\pi\)
\(390\) −1.71404 −0.0867938
\(391\) 1.72777 0.0873771
\(392\) 4.28747 0.216550
\(393\) 11.8565 0.598083
\(394\) 2.07563 0.104569
\(395\) −15.2682 −0.768228
\(396\) 2.66308 0.133825
\(397\) −24.8725 −1.24831 −0.624157 0.781299i \(-0.714558\pi\)
−0.624157 + 0.781299i \(0.714558\pi\)
\(398\) −4.55938 −0.228541
\(399\) 3.63461 0.181958
\(400\) 1.00000 0.0500000
\(401\) −2.38327 −0.119015 −0.0595075 0.998228i \(-0.518953\pi\)
−0.0595075 + 0.998228i \(0.518953\pi\)
\(402\) 3.59061 0.179084
\(403\) 10.0756 0.501904
\(404\) 1.68377 0.0837709
\(405\) −6.08122 −0.302178
\(406\) −6.06716 −0.301108
\(407\) 2.19499 0.108802
\(408\) 1.05240 0.0521013
\(409\) 32.5318 1.60859 0.804296 0.594228i \(-0.202542\pi\)
0.804296 + 0.594228i \(0.202542\pi\)
\(410\) −5.45269 −0.269289
\(411\) 0.762835 0.0376279
\(412\) −1.85743 −0.0915091
\(413\) −5.11209 −0.251549
\(414\) 2.53779 0.124726
\(415\) −2.72642 −0.133835
\(416\) −2.95295 −0.144780
\(417\) −11.2907 −0.552906
\(418\) −3.80194 −0.185959
\(419\) −6.83708 −0.334013 −0.167007 0.985956i \(-0.553410\pi\)
−0.167007 + 0.985956i \(0.553410\pi\)
\(420\) 0.955986 0.0466474
\(421\) 19.0503 0.928453 0.464227 0.885716i \(-0.346332\pi\)
0.464227 + 0.885716i \(0.346332\pi\)
\(422\) 23.5716 1.14745
\(423\) 29.5975 1.43908
\(424\) −8.91291 −0.432849
\(425\) 1.81307 0.0879467
\(426\) 7.45729 0.361307
\(427\) −16.2592 −0.786836
\(428\) −14.8768 −0.719097
\(429\) 1.71404 0.0827547
\(430\) 1.00000 0.0482243
\(431\) 16.5797 0.798615 0.399308 0.916817i \(-0.369251\pi\)
0.399308 + 0.916817i \(0.369251\pi\)
\(432\) 3.28713 0.158152
\(433\) 5.64945 0.271495 0.135748 0.990743i \(-0.456656\pi\)
0.135748 + 0.990743i \(0.456656\pi\)
\(434\) −5.61957 −0.269748
\(435\) 2.13827 0.102522
\(436\) 11.2283 0.537738
\(437\) −3.62307 −0.173315
\(438\) −0.132980 −0.00635401
\(439\) 40.1328 1.91543 0.957716 0.287716i \(-0.0928958\pi\)
0.957716 + 0.287716i \(0.0928958\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 11.4179 0.543708
\(442\) −5.35391 −0.254659
\(443\) −11.2633 −0.535137 −0.267569 0.963539i \(-0.586220\pi\)
−0.267569 + 0.963539i \(0.586220\pi\)
\(444\) 1.27408 0.0604652
\(445\) 5.52821 0.262062
\(446\) −7.34985 −0.348026
\(447\) −0.366014 −0.0173119
\(448\) 1.64698 0.0778123
\(449\) −10.8493 −0.512012 −0.256006 0.966675i \(-0.582407\pi\)
−0.256006 + 0.966675i \(0.582407\pi\)
\(450\) 2.66308 0.125539
\(451\) 5.45269 0.256757
\(452\) 2.54599 0.119753
\(453\) −10.5588 −0.496097
\(454\) 23.1182 1.08499
\(455\) −4.86344 −0.228002
\(456\) −2.20684 −0.103345
\(457\) −22.2951 −1.04292 −0.521461 0.853275i \(-0.674613\pi\)
−0.521461 + 0.853275i \(0.674613\pi\)
\(458\) −28.5225 −1.33277
\(459\) 5.95980 0.278179
\(460\) −0.952953 −0.0444317
\(461\) 39.4379 1.83681 0.918404 0.395644i \(-0.129479\pi\)
0.918404 + 0.395644i \(0.129479\pi\)
\(462\) −0.955986 −0.0444765
\(463\) 39.4578 1.83376 0.916879 0.399166i \(-0.130700\pi\)
0.916879 + 0.399166i \(0.130700\pi\)
\(464\) 3.68382 0.171017
\(465\) 1.98053 0.0918448
\(466\) −15.4512 −0.715765
\(467\) 33.2426 1.53829 0.769143 0.639077i \(-0.220683\pi\)
0.769143 + 0.639077i \(0.220683\pi\)
\(468\) −7.86395 −0.363511
\(469\) 10.1881 0.470441
\(470\) −11.1140 −0.512652
\(471\) −11.0167 −0.507622
\(472\) 3.10392 0.142870
\(473\) −1.00000 −0.0459800
\(474\) 8.86244 0.407065
\(475\) −3.80194 −0.174445
\(476\) 2.98608 0.136867
\(477\) −23.7358 −1.08679
\(478\) 5.45055 0.249302
\(479\) 36.6264 1.67350 0.836752 0.547582i \(-0.184451\pi\)
0.836752 + 0.547582i \(0.184451\pi\)
\(480\) −0.580450 −0.0264938
\(481\) −6.48170 −0.295540
\(482\) 6.42694 0.292739
\(483\) −0.911010 −0.0414524
\(484\) 1.00000 0.0454545
\(485\) 3.05239 0.138602
\(486\) 13.3912 0.607439
\(487\) 31.5638 1.43029 0.715145 0.698976i \(-0.246361\pi\)
0.715145 + 0.698976i \(0.246361\pi\)
\(488\) 9.87214 0.446891
\(489\) −4.51727 −0.204278
\(490\) −4.28747 −0.193688
\(491\) 7.65511 0.345470 0.172735 0.984968i \(-0.444740\pi\)
0.172735 + 0.984968i \(0.444740\pi\)
\(492\) 3.16501 0.142690
\(493\) 6.67902 0.300808
\(494\) 11.2270 0.505125
\(495\) −2.66308 −0.119696
\(496\) 3.41206 0.153206
\(497\) 21.1594 0.949130
\(498\) 1.58255 0.0709157
\(499\) 11.4695 0.513446 0.256723 0.966485i \(-0.417357\pi\)
0.256723 + 0.966485i \(0.417357\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 4.92559 0.220059
\(502\) −8.86310 −0.395579
\(503\) −18.5495 −0.827080 −0.413540 0.910486i \(-0.635708\pi\)
−0.413540 + 0.910486i \(0.635708\pi\)
\(504\) 4.38602 0.195369
\(505\) −1.68377 −0.0749269
\(506\) 0.952953 0.0423639
\(507\) 2.48436 0.110334
\(508\) −2.73781 −0.121471
\(509\) −9.83302 −0.435841 −0.217920 0.975967i \(-0.569927\pi\)
−0.217920 + 0.975967i \(0.569927\pi\)
\(510\) −1.05240 −0.0466008
\(511\) −0.377318 −0.0166916
\(512\) −1.00000 −0.0441942
\(513\) −12.4975 −0.551778
\(514\) −4.29084 −0.189261
\(515\) 1.85743 0.0818483
\(516\) −0.580450 −0.0255529
\(517\) 11.1140 0.488795
\(518\) 3.61509 0.158838
\(519\) 2.68598 0.117902
\(520\) 2.95295 0.129496
\(521\) −36.0224 −1.57817 −0.789084 0.614285i \(-0.789444\pi\)
−0.789084 + 0.614285i \(0.789444\pi\)
\(522\) 9.81030 0.429385
\(523\) −15.1116 −0.660783 −0.330392 0.943844i \(-0.607181\pi\)
−0.330392 + 0.943844i \(0.607181\pi\)
\(524\) −20.4265 −0.892334
\(525\) −0.955986 −0.0417227
\(526\) −23.3413 −1.01773
\(527\) 6.18629 0.269479
\(528\) 0.580450 0.0252608
\(529\) −22.0919 −0.960517
\(530\) 8.91291 0.387152
\(531\) 8.26599 0.358713
\(532\) −6.26171 −0.271480
\(533\) −16.1015 −0.697436
\(534\) −3.20885 −0.138860
\(535\) 14.8768 0.643180
\(536\) −6.18592 −0.267191
\(537\) −6.61930 −0.285644
\(538\) −9.64797 −0.415954
\(539\) 4.28747 0.184674
\(540\) −3.28713 −0.141456
\(541\) 38.2711 1.64540 0.822701 0.568474i \(-0.192466\pi\)
0.822701 + 0.568474i \(0.192466\pi\)
\(542\) −28.2847 −1.21493
\(543\) −5.38913 −0.231270
\(544\) −1.81307 −0.0777347
\(545\) −11.2283 −0.480968
\(546\) 2.82298 0.120813
\(547\) 6.64388 0.284072 0.142036 0.989862i \(-0.454635\pi\)
0.142036 + 0.989862i \(0.454635\pi\)
\(548\) −1.31421 −0.0561405
\(549\) 26.2903 1.12204
\(550\) 1.00000 0.0426401
\(551\) −14.0057 −0.596662
\(552\) 0.553141 0.0235433
\(553\) 25.1464 1.06933
\(554\) 9.51350 0.404190
\(555\) −1.27408 −0.0540817
\(556\) 19.4516 0.824931
\(557\) 11.8791 0.503334 0.251667 0.967814i \(-0.419021\pi\)
0.251667 + 0.967814i \(0.419021\pi\)
\(558\) 9.08657 0.384665
\(559\) 2.95295 0.124897
\(560\) −1.64698 −0.0695974
\(561\) 1.05240 0.0444322
\(562\) −28.8476 −1.21686
\(563\) 46.3455 1.95323 0.976616 0.214993i \(-0.0689728\pi\)
0.976616 + 0.214993i \(0.0689728\pi\)
\(564\) 6.45114 0.271642
\(565\) −2.54599 −0.107110
\(566\) 3.77444 0.158652
\(567\) 10.0156 0.420617
\(568\) −12.8474 −0.539067
\(569\) −10.6916 −0.448216 −0.224108 0.974564i \(-0.571947\pi\)
−0.224108 + 0.974564i \(0.571947\pi\)
\(570\) 2.20684 0.0924342
\(571\) −1.75530 −0.0734572 −0.0367286 0.999325i \(-0.511694\pi\)
−0.0367286 + 0.999325i \(0.511694\pi\)
\(572\) −2.95295 −0.123469
\(573\) −1.81992 −0.0760281
\(574\) 8.98045 0.374837
\(575\) 0.952953 0.0397409
\(576\) −2.66308 −0.110962
\(577\) −14.9153 −0.620932 −0.310466 0.950584i \(-0.600485\pi\)
−0.310466 + 0.950584i \(0.600485\pi\)
\(578\) 13.7128 0.570377
\(579\) −9.58576 −0.398371
\(580\) −3.68382 −0.152962
\(581\) 4.49035 0.186291
\(582\) −1.77176 −0.0734418
\(583\) −8.91291 −0.369135
\(584\) 0.229098 0.00948012
\(585\) 7.86395 0.325134
\(586\) 20.0934 0.830050
\(587\) 2.23127 0.0920945 0.0460472 0.998939i \(-0.485338\pi\)
0.0460472 + 0.998939i \(0.485338\pi\)
\(588\) 2.48866 0.102631
\(589\) −12.9724 −0.534520
\(590\) −3.10392 −0.127786
\(591\) 1.20480 0.0495587
\(592\) −2.19499 −0.0902135
\(593\) −9.25484 −0.380051 −0.190025 0.981779i \(-0.560857\pi\)
−0.190025 + 0.981779i \(0.560857\pi\)
\(594\) 3.28713 0.134873
\(595\) −2.98608 −0.122417
\(596\) 0.630570 0.0258291
\(597\) −2.64649 −0.108314
\(598\) −2.81403 −0.115074
\(599\) −16.4307 −0.671340 −0.335670 0.941980i \(-0.608963\pi\)
−0.335670 + 0.941980i \(0.608963\pi\)
\(600\) 0.580450 0.0236968
\(601\) −1.35830 −0.0554063 −0.0277032 0.999616i \(-0.508819\pi\)
−0.0277032 + 0.999616i \(0.508819\pi\)
\(602\) −1.64698 −0.0671257
\(603\) −16.4736 −0.670856
\(604\) 18.1908 0.740172
\(605\) −1.00000 −0.0406558
\(606\) 0.977346 0.0397020
\(607\) −2.07452 −0.0842020 −0.0421010 0.999113i \(-0.513405\pi\)
−0.0421010 + 0.999113i \(0.513405\pi\)
\(608\) 3.80194 0.154189
\(609\) −3.52168 −0.142706
\(610\) −9.87214 −0.399711
\(611\) −32.8192 −1.32772
\(612\) −4.82834 −0.195174
\(613\) −25.5758 −1.03300 −0.516498 0.856288i \(-0.672765\pi\)
−0.516498 + 0.856288i \(0.672765\pi\)
\(614\) −22.6639 −0.914641
\(615\) −3.16501 −0.127626
\(616\) 1.64698 0.0663585
\(617\) 1.68989 0.0680323 0.0340161 0.999421i \(-0.489170\pi\)
0.0340161 + 0.999421i \(0.489170\pi\)
\(618\) −1.07815 −0.0433694
\(619\) −12.4520 −0.500490 −0.250245 0.968183i \(-0.580511\pi\)
−0.250245 + 0.968183i \(0.580511\pi\)
\(620\) −3.41206 −0.137032
\(621\) 3.13248 0.125702
\(622\) −4.62256 −0.185348
\(623\) −9.10483 −0.364777
\(624\) −1.71404 −0.0686165
\(625\) 1.00000 0.0400000
\(626\) −0.839632 −0.0335585
\(627\) −2.20684 −0.0881326
\(628\) 18.9796 0.757368
\(629\) −3.97966 −0.158680
\(630\) −4.38602 −0.174743
\(631\) −30.6550 −1.22036 −0.610178 0.792265i \(-0.708902\pi\)
−0.610178 + 0.792265i \(0.708902\pi\)
\(632\) −15.2682 −0.607338
\(633\) 13.6821 0.543816
\(634\) −27.1016 −1.07634
\(635\) 2.73781 0.108647
\(636\) −5.17349 −0.205142
\(637\) −12.6607 −0.501635
\(638\) 3.68382 0.145844
\(639\) −34.2137 −1.35347
\(640\) 1.00000 0.0395285
\(641\) 2.25111 0.0889136 0.0444568 0.999011i \(-0.485844\pi\)
0.0444568 + 0.999011i \(0.485844\pi\)
\(642\) −8.63524 −0.340806
\(643\) 36.2268 1.42865 0.714323 0.699816i \(-0.246735\pi\)
0.714323 + 0.699816i \(0.246735\pi\)
\(644\) 1.56949 0.0618466
\(645\) 0.580450 0.0228552
\(646\) 6.89319 0.271209
\(647\) −5.58317 −0.219497 −0.109748 0.993959i \(-0.535005\pi\)
−0.109748 + 0.993959i \(0.535005\pi\)
\(648\) −6.08122 −0.238893
\(649\) 3.10392 0.121840
\(650\) −2.95295 −0.115824
\(651\) −3.26188 −0.127843
\(652\) 7.78236 0.304781
\(653\) 2.93736 0.114948 0.0574739 0.998347i \(-0.481695\pi\)
0.0574739 + 0.998347i \(0.481695\pi\)
\(654\) 6.51747 0.254853
\(655\) 20.4265 0.798128
\(656\) −5.45269 −0.212892
\(657\) 0.610105 0.0238024
\(658\) 18.3045 0.713585
\(659\) 19.2396 0.749469 0.374735 0.927132i \(-0.377734\pi\)
0.374735 + 0.927132i \(0.377734\pi\)
\(660\) −0.580450 −0.0225940
\(661\) 40.4661 1.57395 0.786974 0.616986i \(-0.211646\pi\)
0.786974 + 0.616986i \(0.211646\pi\)
\(662\) −27.7639 −1.07907
\(663\) −3.10767 −0.120692
\(664\) −2.72642 −0.105806
\(665\) 6.26171 0.242819
\(666\) −5.84542 −0.226506
\(667\) 3.51051 0.135927
\(668\) −8.48582 −0.328326
\(669\) −4.26622 −0.164942
\(670\) 6.18592 0.238983
\(671\) 9.87214 0.381110
\(672\) 0.955986 0.0368780
\(673\) 41.9506 1.61708 0.808539 0.588443i \(-0.200259\pi\)
0.808539 + 0.588443i \(0.200259\pi\)
\(674\) −7.90859 −0.304627
\(675\) 3.28713 0.126522
\(676\) −4.28007 −0.164618
\(677\) −17.7907 −0.683752 −0.341876 0.939745i \(-0.611062\pi\)
−0.341876 + 0.939745i \(0.611062\pi\)
\(678\) 1.47782 0.0567552
\(679\) −5.02722 −0.192927
\(680\) 1.81307 0.0695280
\(681\) 13.4190 0.514216
\(682\) 3.41206 0.130654
\(683\) −31.1953 −1.19365 −0.596827 0.802370i \(-0.703572\pi\)
−0.596827 + 0.802370i \(0.703572\pi\)
\(684\) 10.1249 0.387134
\(685\) 1.31421 0.0502136
\(686\) 18.5902 0.709777
\(687\) −16.5559 −0.631646
\(688\) 1.00000 0.0381246
\(689\) 26.3194 1.00269
\(690\) −0.553141 −0.0210577
\(691\) −5.32363 −0.202520 −0.101260 0.994860i \(-0.532287\pi\)
−0.101260 + 0.994860i \(0.532287\pi\)
\(692\) −4.62742 −0.175908
\(693\) 4.38602 0.166611
\(694\) 5.46682 0.207518
\(695\) −19.4516 −0.737841
\(696\) 2.13827 0.0810510
\(697\) −9.88611 −0.374463
\(698\) −22.7588 −0.861434
\(699\) −8.96866 −0.339226
\(700\) 1.64698 0.0622498
\(701\) −17.5252 −0.661917 −0.330958 0.943645i \(-0.607372\pi\)
−0.330958 + 0.943645i \(0.607372\pi\)
\(702\) −9.70675 −0.366358
\(703\) 8.34522 0.314746
\(704\) −1.00000 −0.0376889
\(705\) −6.45114 −0.242964
\(706\) 31.4006 1.18178
\(707\) 2.77313 0.104294
\(708\) 1.80167 0.0677109
\(709\) −34.6811 −1.30247 −0.651237 0.758874i \(-0.725750\pi\)
−0.651237 + 0.758874i \(0.725750\pi\)
\(710\) 12.8474 0.482156
\(711\) −40.6605 −1.52489
\(712\) 5.52821 0.207178
\(713\) 3.25153 0.121771
\(714\) 1.73327 0.0648660
\(715\) 2.95295 0.110434
\(716\) 11.4038 0.426178
\(717\) 3.16377 0.118153
\(718\) 16.0218 0.597927
\(719\) 4.09689 0.152788 0.0763942 0.997078i \(-0.475659\pi\)
0.0763942 + 0.997078i \(0.475659\pi\)
\(720\) 2.66308 0.0992471
\(721\) −3.05915 −0.113929
\(722\) 4.54522 0.169156
\(723\) 3.73052 0.138739
\(724\) 9.28441 0.345052
\(725\) 3.68382 0.136814
\(726\) 0.580450 0.0215425
\(727\) −42.1954 −1.56494 −0.782471 0.622687i \(-0.786041\pi\)
−0.782471 + 0.622687i \(0.786041\pi\)
\(728\) −4.86344 −0.180251
\(729\) −10.4707 −0.387804
\(730\) −0.229098 −0.00847928
\(731\) 1.81307 0.0670588
\(732\) 5.73028 0.211797
\(733\) 21.0209 0.776423 0.388212 0.921570i \(-0.373093\pi\)
0.388212 + 0.921570i \(0.373093\pi\)
\(734\) 25.5328 0.942434
\(735\) −2.48866 −0.0917957
\(736\) −0.952953 −0.0351263
\(737\) −6.18592 −0.227861
\(738\) −14.5209 −0.534524
\(739\) −14.0102 −0.515373 −0.257686 0.966229i \(-0.582960\pi\)
−0.257686 + 0.966229i \(0.582960\pi\)
\(740\) 2.19499 0.0806894
\(741\) 6.51669 0.239396
\(742\) −14.6793 −0.538895
\(743\) 7.10348 0.260601 0.130301 0.991475i \(-0.458406\pi\)
0.130301 + 0.991475i \(0.458406\pi\)
\(744\) 1.98053 0.0726097
\(745\) −0.630570 −0.0231023
\(746\) −12.2063 −0.446904
\(747\) −7.26067 −0.265654
\(748\) −1.81307 −0.0662924
\(749\) −24.5017 −0.895274
\(750\) −0.580450 −0.0211950
\(751\) 1.16751 0.0426030 0.0213015 0.999773i \(-0.493219\pi\)
0.0213015 + 0.999773i \(0.493219\pi\)
\(752\) −11.1140 −0.405287
\(753\) −5.14458 −0.187479
\(754\) −10.8781 −0.396159
\(755\) −18.1908 −0.662030
\(756\) 5.41383 0.196899
\(757\) −32.6233 −1.18571 −0.592857 0.805308i \(-0.702000\pi\)
−0.592857 + 0.805308i \(0.702000\pi\)
\(758\) −24.9357 −0.905707
\(759\) 0.553141 0.0200778
\(760\) −3.80194 −0.137911
\(761\) −22.3050 −0.808556 −0.404278 0.914636i \(-0.632477\pi\)
−0.404278 + 0.914636i \(0.632477\pi\)
\(762\) −1.58916 −0.0575692
\(763\) 18.4928 0.669483
\(764\) 3.13535 0.113433
\(765\) 4.82834 0.174569
\(766\) −18.8612 −0.681482
\(767\) −9.16574 −0.330956
\(768\) −0.580450 −0.0209452
\(769\) −42.7788 −1.54264 −0.771322 0.636445i \(-0.780404\pi\)
−0.771322 + 0.636445i \(0.780404\pi\)
\(770\) −1.64698 −0.0593529
\(771\) −2.49062 −0.0896973
\(772\) 16.5144 0.594365
\(773\) 41.5959 1.49610 0.748051 0.663642i \(-0.230990\pi\)
0.748051 + 0.663642i \(0.230990\pi\)
\(774\) 2.66308 0.0957224
\(775\) 3.41206 0.122565
\(776\) 3.05239 0.109574
\(777\) 2.09838 0.0752789
\(778\) −14.3498 −0.514466
\(779\) 20.7308 0.742759
\(780\) 1.71404 0.0613725
\(781\) −12.8474 −0.459718
\(782\) −1.72777 −0.0617849
\(783\) 12.1092 0.432748
\(784\) −4.28747 −0.153124
\(785\) −18.9796 −0.677410
\(786\) −11.8565 −0.422909
\(787\) −48.7051 −1.73615 −0.868074 0.496434i \(-0.834642\pi\)
−0.868074 + 0.496434i \(0.834642\pi\)
\(788\) −2.07563 −0.0739412
\(789\) −13.5484 −0.482337
\(790\) 15.2682 0.543219
\(791\) 4.19318 0.149092
\(792\) −2.66308 −0.0946284
\(793\) −29.1520 −1.03522
\(794\) 24.8725 0.882692
\(795\) 5.17349 0.183485
\(796\) 4.55938 0.161603
\(797\) −49.6197 −1.75762 −0.878811 0.477170i \(-0.841662\pi\)
−0.878811 + 0.477170i \(0.841662\pi\)
\(798\) −3.63461 −0.128664
\(799\) −20.1505 −0.712874
\(800\) −1.00000 −0.0353553
\(801\) 14.7221 0.520178
\(802\) 2.38327 0.0841564
\(803\) 0.229098 0.00808468
\(804\) −3.59061 −0.126631
\(805\) −1.56949 −0.0553173
\(806\) −10.0756 −0.354899
\(807\) −5.60016 −0.197135
\(808\) −1.68377 −0.0592350
\(809\) −25.6526 −0.901897 −0.450949 0.892550i \(-0.648914\pi\)
−0.450949 + 0.892550i \(0.648914\pi\)
\(810\) 6.08122 0.213672
\(811\) 12.0142 0.421876 0.210938 0.977499i \(-0.432348\pi\)
0.210938 + 0.977499i \(0.432348\pi\)
\(812\) 6.06716 0.212916
\(813\) −16.4178 −0.575799
\(814\) −2.19499 −0.0769343
\(815\) −7.78236 −0.272604
\(816\) −1.05240 −0.0368412
\(817\) −3.80194 −0.133013
\(818\) −32.5318 −1.13745
\(819\) −12.9517 −0.452570
\(820\) 5.45269 0.190416
\(821\) 36.3661 1.26918 0.634592 0.772847i \(-0.281168\pi\)
0.634592 + 0.772847i \(0.281168\pi\)
\(822\) −0.762835 −0.0266069
\(823\) 46.3560 1.61587 0.807934 0.589273i \(-0.200586\pi\)
0.807934 + 0.589273i \(0.200586\pi\)
\(824\) 1.85743 0.0647067
\(825\) 0.580450 0.0202087
\(826\) 5.11209 0.177872
\(827\) 51.0586 1.77548 0.887741 0.460344i \(-0.152274\pi\)
0.887741 + 0.460344i \(0.152274\pi\)
\(828\) −2.53779 −0.0881943
\(829\) 30.4558 1.05777 0.528886 0.848693i \(-0.322610\pi\)
0.528886 + 0.848693i \(0.322610\pi\)
\(830\) 2.72642 0.0946354
\(831\) 5.52211 0.191560
\(832\) 2.95295 0.102375
\(833\) −7.77348 −0.269335
\(834\) 11.2907 0.390964
\(835\) 8.48582 0.293664
\(836\) 3.80194 0.131493
\(837\) 11.2159 0.387678
\(838\) 6.83708 0.236183
\(839\) 8.16121 0.281756 0.140878 0.990027i \(-0.455007\pi\)
0.140878 + 0.990027i \(0.455007\pi\)
\(840\) −0.955986 −0.0329847
\(841\) −15.4295 −0.532051
\(842\) −19.0503 −0.656515
\(843\) −16.7446 −0.576715
\(844\) −23.5716 −0.811368
\(845\) 4.28007 0.147239
\(846\) −29.5975 −1.01758
\(847\) 1.64698 0.0565908
\(848\) 8.91291 0.306070
\(849\) 2.19087 0.0751906
\(850\) −1.81307 −0.0621877
\(851\) −2.09172 −0.0717033
\(852\) −7.45729 −0.255483
\(853\) 30.0565 1.02911 0.514557 0.857456i \(-0.327956\pi\)
0.514557 + 0.857456i \(0.327956\pi\)
\(854\) 16.2592 0.556377
\(855\) −10.1249 −0.346263
\(856\) 14.8768 0.508479
\(857\) −12.2194 −0.417407 −0.208703 0.977979i \(-0.566924\pi\)
−0.208703 + 0.977979i \(0.566924\pi\)
\(858\) −1.71404 −0.0585164
\(859\) −20.1048 −0.685966 −0.342983 0.939342i \(-0.611437\pi\)
−0.342983 + 0.939342i \(0.611437\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 5.21270 0.177648
\(862\) −16.5797 −0.564706
\(863\) 17.6080 0.599384 0.299692 0.954036i \(-0.403116\pi\)
0.299692 + 0.954036i \(0.403116\pi\)
\(864\) −3.28713 −0.111830
\(865\) 4.62742 0.157337
\(866\) −5.64945 −0.191976
\(867\) 7.95958 0.270322
\(868\) 5.61957 0.190741
\(869\) −15.2682 −0.517939
\(870\) −2.13827 −0.0724942
\(871\) 18.2667 0.618944
\(872\) −11.2283 −0.380238
\(873\) 8.12876 0.275117
\(874\) 3.62307 0.122552
\(875\) −1.64698 −0.0556779
\(876\) 0.132980 0.00449296
\(877\) 21.5061 0.726211 0.363105 0.931748i \(-0.381717\pi\)
0.363105 + 0.931748i \(0.381717\pi\)
\(878\) −40.1328 −1.35441
\(879\) 11.6632 0.393390
\(880\) 1.00000 0.0337100
\(881\) −10.0368 −0.338150 −0.169075 0.985603i \(-0.554078\pi\)
−0.169075 + 0.985603i \(0.554078\pi\)
\(882\) −11.4179 −0.384460
\(883\) −5.17892 −0.174284 −0.0871422 0.996196i \(-0.527773\pi\)
−0.0871422 + 0.996196i \(0.527773\pi\)
\(884\) 5.35391 0.180071
\(885\) −1.80167 −0.0605625
\(886\) 11.2633 0.378399
\(887\) 22.7889 0.765175 0.382588 0.923919i \(-0.375033\pi\)
0.382588 + 0.923919i \(0.375033\pi\)
\(888\) −1.27408 −0.0427553
\(889\) −4.50911 −0.151231
\(890\) −5.52821 −0.185306
\(891\) −6.08122 −0.203729
\(892\) 7.34985 0.246091
\(893\) 42.2549 1.41401
\(894\) 0.366014 0.0122413
\(895\) −11.4038 −0.381185
\(896\) −1.64698 −0.0550216
\(897\) −1.63340 −0.0545377
\(898\) 10.8493 0.362047
\(899\) 12.5694 0.419213
\(900\) −2.66308 −0.0887693
\(901\) 16.1597 0.538358
\(902\) −5.45269 −0.181555
\(903\) −0.955986 −0.0318132
\(904\) −2.54599 −0.0846782
\(905\) −9.28441 −0.308624
\(906\) 10.5588 0.350793
\(907\) 42.9039 1.42460 0.712300 0.701875i \(-0.247654\pi\)
0.712300 + 0.701875i \(0.247654\pi\)
\(908\) −23.1182 −0.767205
\(909\) −4.48402 −0.148726
\(910\) 4.86344 0.161222
\(911\) 24.6793 0.817661 0.408830 0.912610i \(-0.365937\pi\)
0.408830 + 0.912610i \(0.365937\pi\)
\(912\) 2.20684 0.0730757
\(913\) −2.72642 −0.0902313
\(914\) 22.2951 0.737457
\(915\) −5.73028 −0.189437
\(916\) 28.5225 0.942409
\(917\) −33.6419 −1.11095
\(918\) −5.95980 −0.196703
\(919\) 50.1392 1.65394 0.826969 0.562247i \(-0.190063\pi\)
0.826969 + 0.562247i \(0.190063\pi\)
\(920\) 0.952953 0.0314179
\(921\) −13.1553 −0.433480
\(922\) −39.4379 −1.29882
\(923\) 37.9379 1.24874
\(924\) 0.955986 0.0314496
\(925\) −2.19499 −0.0721708
\(926\) −39.4578 −1.29666
\(927\) 4.94649 0.162464
\(928\) −3.68382 −0.120927
\(929\) −18.0484 −0.592149 −0.296075 0.955165i \(-0.595678\pi\)
−0.296075 + 0.955165i \(0.595678\pi\)
\(930\) −1.98053 −0.0649440
\(931\) 16.3007 0.534235
\(932\) 15.4512 0.506122
\(933\) −2.68317 −0.0878429
\(934\) −33.2426 −1.08773
\(935\) 1.81307 0.0592937
\(936\) 7.86395 0.257041
\(937\) −37.8450 −1.23634 −0.618172 0.786043i \(-0.712126\pi\)
−0.618172 + 0.786043i \(0.712126\pi\)
\(938\) −10.1881 −0.332652
\(939\) −0.487364 −0.0159045
\(940\) 11.1140 0.362500
\(941\) 51.9990 1.69512 0.847559 0.530700i \(-0.178071\pi\)
0.847559 + 0.530700i \(0.178071\pi\)
\(942\) 11.0167 0.358943
\(943\) −5.19616 −0.169210
\(944\) −3.10392 −0.101024
\(945\) −5.41383 −0.176112
\(946\) 1.00000 0.0325128
\(947\) −37.0303 −1.20332 −0.601662 0.798751i \(-0.705495\pi\)
−0.601662 + 0.798751i \(0.705495\pi\)
\(948\) −8.86244 −0.287839
\(949\) −0.676514 −0.0219606
\(950\) 3.80194 0.123351
\(951\) −15.7311 −0.510116
\(952\) −2.98608 −0.0967794
\(953\) −21.6469 −0.701213 −0.350607 0.936523i \(-0.614025\pi\)
−0.350607 + 0.936523i \(0.614025\pi\)
\(954\) 23.7358 0.768474
\(955\) −3.13535 −0.101458
\(956\) −5.45055 −0.176283
\(957\) 2.13827 0.0691205
\(958\) −36.6264 −1.18335
\(959\) −2.16448 −0.0698947
\(960\) 0.580450 0.0187339
\(961\) −19.3579 −0.624447
\(962\) 6.48170 0.208978
\(963\) 39.6181 1.27668
\(964\) −6.42694 −0.206998
\(965\) −16.5144 −0.531616
\(966\) 0.911010 0.0293113
\(967\) −39.3479 −1.26534 −0.632671 0.774421i \(-0.718041\pi\)
−0.632671 + 0.774421i \(0.718041\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 4.00115 0.128535
\(970\) −3.05239 −0.0980064
\(971\) 33.1960 1.06531 0.532655 0.846332i \(-0.321194\pi\)
0.532655 + 0.846332i \(0.321194\pi\)
\(972\) −13.3912 −0.429524
\(973\) 32.0363 1.02704
\(974\) −31.5638 −1.01137
\(975\) −1.71404 −0.0548932
\(976\) −9.87214 −0.315999
\(977\) −49.9804 −1.59901 −0.799507 0.600657i \(-0.794906\pi\)
−0.799507 + 0.600657i \(0.794906\pi\)
\(978\) 4.51727 0.144446
\(979\) 5.52821 0.176682
\(980\) 4.28747 0.136958
\(981\) −29.9019 −0.954693
\(982\) −7.65511 −0.244284
\(983\) 55.7541 1.77828 0.889139 0.457636i \(-0.151304\pi\)
0.889139 + 0.457636i \(0.151304\pi\)
\(984\) −3.16501 −0.100897
\(985\) 2.07563 0.0661350
\(986\) −6.67902 −0.212703
\(987\) 10.6249 0.338193
\(988\) −11.2270 −0.357177
\(989\) 0.952953 0.0303021
\(990\) 2.66308 0.0846382
\(991\) −36.7684 −1.16799 −0.583994 0.811758i \(-0.698511\pi\)
−0.583994 + 0.811758i \(0.698511\pi\)
\(992\) −3.41206 −0.108333
\(993\) −16.1155 −0.511411
\(994\) −21.1594 −0.671136
\(995\) −4.55938 −0.144542
\(996\) −1.58255 −0.0501450
\(997\) −20.1652 −0.638637 −0.319319 0.947647i \(-0.603454\pi\)
−0.319319 + 0.947647i \(0.603454\pi\)
\(998\) −11.4695 −0.363061
\(999\) −7.21522 −0.228279
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 4730.2.a.bb.1.5 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bb.1.5 11 1.1 even 1 trivial