Properties

Label 1014.2.a.m.1.1
Level $1014$
Weight $2$
Character 1014.1
Self dual yes
Analytic conductor $8.097$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 1014.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} +0.356896 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} +0.356896 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.692021 q^{10} +2.93900 q^{11} +1.00000 q^{12} -0.356896 q^{14} -0.692021 q^{15} +1.00000 q^{16} +6.71379 q^{17} -1.00000 q^{18} -7.20775 q^{19} -0.692021 q^{20} +0.356896 q^{21} -2.93900 q^{22} +2.39612 q^{23} -1.00000 q^{24} -4.52111 q^{25} +1.00000 q^{27} +0.356896 q^{28} +7.82908 q^{29} +0.692021 q^{30} -2.76271 q^{31} -1.00000 q^{32} +2.93900 q^{33} -6.71379 q^{34} -0.246980 q^{35} +1.00000 q^{36} +10.0978 q^{37} +7.20775 q^{38} +0.692021 q^{40} +4.89008 q^{41} -0.356896 q^{42} +6.59179 q^{43} +2.93900 q^{44} -0.692021 q^{45} -2.39612 q^{46} +4.98792 q^{47} +1.00000 q^{48} -6.87263 q^{49} +4.52111 q^{50} +6.71379 q^{51} -8.88769 q^{53} -1.00000 q^{54} -2.03385 q^{55} -0.356896 q^{56} -7.20775 q^{57} -7.82908 q^{58} +1.64310 q^{59} -0.692021 q^{60} -6.49396 q^{61} +2.76271 q^{62} +0.356896 q^{63} +1.00000 q^{64} -2.93900 q^{66} +13.5254 q^{67} +6.71379 q^{68} +2.39612 q^{69} +0.246980 q^{70} +6.81163 q^{71} -1.00000 q^{72} +3.18598 q^{73} -10.0978 q^{74} -4.52111 q^{75} -7.20775 q^{76} +1.04892 q^{77} +15.0465 q^{79} -0.692021 q^{80} +1.00000 q^{81} -4.89008 q^{82} -14.8267 q^{83} +0.356896 q^{84} -4.64609 q^{85} -6.59179 q^{86} +7.82908 q^{87} -2.93900 q^{88} +0.396125 q^{89} +0.692021 q^{90} +2.39612 q^{92} -2.76271 q^{93} -4.98792 q^{94} +4.98792 q^{95} -1.00000 q^{96} -0.417895 q^{97} +6.87263 q^{98} +2.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} - q^{11} + 3 q^{12} + 3 q^{14} + 3 q^{15} + 3 q^{16} + 12 q^{17} - 3 q^{18} - 4 q^{19} + 3 q^{20} - 3 q^{21} + q^{22} + 16 q^{23} - 3 q^{24} + 2 q^{25} + 3 q^{27} - 3 q^{28} + 13 q^{29} - 3 q^{30} + 9 q^{31} - 3 q^{32} - q^{33} - 12 q^{34} + 4 q^{35} + 3 q^{36} + 12 q^{37} + 4 q^{38} - 3 q^{40} + 14 q^{41} + 3 q^{42} - 8 q^{43} - q^{44} + 3 q^{45} - 16 q^{46} - 4 q^{47} + 3 q^{48} - 4 q^{49} - 2 q^{50} + 12 q^{51} + 15 q^{53} - 3 q^{54} - 22 q^{55} + 3 q^{56} - 4 q^{57} - 13 q^{58} + 9 q^{59} + 3 q^{60} - 10 q^{61} - 9 q^{62} - 3 q^{63} + 3 q^{64} + q^{66} + 6 q^{67} + 12 q^{68} + 16 q^{69} - 4 q^{70} - 6 q^{71} - 3 q^{72} - 5 q^{73} - 12 q^{74} + 2 q^{75} - 4 q^{76} - 6 q^{77} - 5 q^{79} + 3 q^{80} + 3 q^{81} - 14 q^{82} + 7 q^{83} - 3 q^{84} + 26 q^{85} + 8 q^{86} + 13 q^{87} + q^{88} + 10 q^{89} - 3 q^{90} + 16 q^{92} + 9 q^{93} + 4 q^{94} - 4 q^{95} - 3 q^{96} - 7 q^{97} + 4 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −0.692021 −0.309481 −0.154741 0.987955i \(-0.549454\pi\)
−0.154741 + 0.987955i \(0.549454\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.356896 0.134894 0.0674470 0.997723i \(-0.478515\pi\)
0.0674470 + 0.997723i \(0.478515\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.692021 0.218836
\(11\) 2.93900 0.886142 0.443071 0.896486i \(-0.353889\pi\)
0.443071 + 0.896486i \(0.353889\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −0.356896 −0.0953844
\(15\) −0.692021 −0.178679
\(16\) 1.00000 0.250000
\(17\) 6.71379 1.62833 0.814167 0.580631i \(-0.197194\pi\)
0.814167 + 0.580631i \(0.197194\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.20775 −1.65357 −0.826786 0.562517i \(-0.809833\pi\)
−0.826786 + 0.562517i \(0.809833\pi\)
\(20\) −0.692021 −0.154741
\(21\) 0.356896 0.0778811
\(22\) −2.93900 −0.626597
\(23\) 2.39612 0.499627 0.249813 0.968294i \(-0.419631\pi\)
0.249813 + 0.968294i \(0.419631\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.52111 −0.904221
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.356896 0.0674470
\(29\) 7.82908 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(30\) 0.692021 0.126345
\(31\) −2.76271 −0.496197 −0.248099 0.968735i \(-0.579806\pi\)
−0.248099 + 0.968735i \(0.579806\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.93900 0.511614
\(34\) −6.71379 −1.15141
\(35\) −0.246980 −0.0417472
\(36\) 1.00000 0.166667
\(37\) 10.0978 1.66007 0.830037 0.557708i \(-0.188319\pi\)
0.830037 + 0.557708i \(0.188319\pi\)
\(38\) 7.20775 1.16925
\(39\) 0 0
\(40\) 0.692021 0.109418
\(41\) 4.89008 0.763703 0.381851 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(42\) −0.356896 −0.0550702
\(43\) 6.59179 1.00524 0.502620 0.864508i \(-0.332370\pi\)
0.502620 + 0.864508i \(0.332370\pi\)
\(44\) 2.93900 0.443071
\(45\) −0.692021 −0.103160
\(46\) −2.39612 −0.353289
\(47\) 4.98792 0.727563 0.363781 0.931484i \(-0.381486\pi\)
0.363781 + 0.931484i \(0.381486\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.87263 −0.981804
\(50\) 4.52111 0.639381
\(51\) 6.71379 0.940119
\(52\) 0 0
\(53\) −8.88769 −1.22082 −0.610409 0.792086i \(-0.708995\pi\)
−0.610409 + 0.792086i \(0.708995\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.03385 −0.274245
\(56\) −0.356896 −0.0476922
\(57\) −7.20775 −0.954690
\(58\) −7.82908 −1.02801
\(59\) 1.64310 0.213914 0.106957 0.994264i \(-0.465889\pi\)
0.106957 + 0.994264i \(0.465889\pi\)
\(60\) −0.692021 −0.0893396
\(61\) −6.49396 −0.831466 −0.415733 0.909487i \(-0.636475\pi\)
−0.415733 + 0.909487i \(0.636475\pi\)
\(62\) 2.76271 0.350864
\(63\) 0.356896 0.0449647
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.93900 −0.361766
\(67\) 13.5254 1.65239 0.826196 0.563382i \(-0.190500\pi\)
0.826196 + 0.563382i \(0.190500\pi\)
\(68\) 6.71379 0.814167
\(69\) 2.39612 0.288459
\(70\) 0.246980 0.0295197
\(71\) 6.81163 0.808391 0.404196 0.914673i \(-0.367551\pi\)
0.404196 + 0.914673i \(0.367551\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.18598 0.372891 0.186445 0.982465i \(-0.440303\pi\)
0.186445 + 0.982465i \(0.440303\pi\)
\(74\) −10.0978 −1.17385
\(75\) −4.52111 −0.522052
\(76\) −7.20775 −0.826786
\(77\) 1.04892 0.119535
\(78\) 0 0
\(79\) 15.0465 1.69287 0.846433 0.532495i \(-0.178745\pi\)
0.846433 + 0.532495i \(0.178745\pi\)
\(80\) −0.692021 −0.0773704
\(81\) 1.00000 0.111111
\(82\) −4.89008 −0.540019
\(83\) −14.8267 −1.62744 −0.813720 0.581256i \(-0.802561\pi\)
−0.813720 + 0.581256i \(0.802561\pi\)
\(84\) 0.356896 0.0389405
\(85\) −4.64609 −0.503939
\(86\) −6.59179 −0.710811
\(87\) 7.82908 0.839366
\(88\) −2.93900 −0.313299
\(89\) 0.396125 0.0419891 0.0209946 0.999780i \(-0.493317\pi\)
0.0209946 + 0.999780i \(0.493317\pi\)
\(90\) 0.692021 0.0729455
\(91\) 0 0
\(92\) 2.39612 0.249813
\(93\) −2.76271 −0.286480
\(94\) −4.98792 −0.514465
\(95\) 4.98792 0.511750
\(96\) −1.00000 −0.102062
\(97\) −0.417895 −0.0424308 −0.0212154 0.999775i \(-0.506754\pi\)
−0.0212154 + 0.999775i \(0.506754\pi\)
\(98\) 6.87263 0.694240
\(99\) 2.93900 0.295381
\(100\) −4.52111 −0.452111
\(101\) 10.0151 0.996536 0.498268 0.867023i \(-0.333970\pi\)
0.498268 + 0.867023i \(0.333970\pi\)
\(102\) −6.71379 −0.664764
\(103\) −9.62565 −0.948443 −0.474222 0.880406i \(-0.657270\pi\)
−0.474222 + 0.880406i \(0.657270\pi\)
\(104\) 0 0
\(105\) −0.246980 −0.0241027
\(106\) 8.88769 0.863249
\(107\) −6.63102 −0.641045 −0.320523 0.947241i \(-0.603859\pi\)
−0.320523 + 0.947241i \(0.603859\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.9879 1.24402 0.622008 0.783011i \(-0.286317\pi\)
0.622008 + 0.783011i \(0.286317\pi\)
\(110\) 2.03385 0.193920
\(111\) 10.0978 0.958444
\(112\) 0.356896 0.0337235
\(113\) 0.792249 0.0745285 0.0372643 0.999305i \(-0.488136\pi\)
0.0372643 + 0.999305i \(0.488136\pi\)
\(114\) 7.20775 0.675068
\(115\) −1.65817 −0.154625
\(116\) 7.82908 0.726912
\(117\) 0 0
\(118\) −1.64310 −0.151260
\(119\) 2.39612 0.219652
\(120\) 0.692021 0.0631726
\(121\) −2.36227 −0.214752
\(122\) 6.49396 0.587935
\(123\) 4.89008 0.440924
\(124\) −2.76271 −0.248099
\(125\) 6.58881 0.589321
\(126\) −0.356896 −0.0317948
\(127\) −18.2174 −1.61654 −0.808268 0.588815i \(-0.799595\pi\)
−0.808268 + 0.588815i \(0.799595\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.59179 0.580375
\(130\) 0 0
\(131\) 2.73556 0.239007 0.119504 0.992834i \(-0.461870\pi\)
0.119504 + 0.992834i \(0.461870\pi\)
\(132\) 2.93900 0.255807
\(133\) −2.57242 −0.223057
\(134\) −13.5254 −1.16842
\(135\) −0.692021 −0.0595597
\(136\) −6.71379 −0.575703
\(137\) −7.64742 −0.653363 −0.326681 0.945135i \(-0.605930\pi\)
−0.326681 + 0.945135i \(0.605930\pi\)
\(138\) −2.39612 −0.203972
\(139\) 3.38404 0.287031 0.143515 0.989648i \(-0.454159\pi\)
0.143515 + 0.989648i \(0.454159\pi\)
\(140\) −0.246980 −0.0208736
\(141\) 4.98792 0.420059
\(142\) −6.81163 −0.571619
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.41789 −0.449932
\(146\) −3.18598 −0.263674
\(147\) −6.87263 −0.566845
\(148\) 10.0978 0.830037
\(149\) 20.8170 1.70540 0.852698 0.522405i \(-0.174965\pi\)
0.852698 + 0.522405i \(0.174965\pi\)
\(150\) 4.52111 0.369147
\(151\) 0.895461 0.0728715 0.0364358 0.999336i \(-0.488400\pi\)
0.0364358 + 0.999336i \(0.488400\pi\)
\(152\) 7.20775 0.584626
\(153\) 6.71379 0.542778
\(154\) −1.04892 −0.0845242
\(155\) 1.91185 0.153564
\(156\) 0 0
\(157\) 8.59179 0.685700 0.342850 0.939390i \(-0.388608\pi\)
0.342850 + 0.939390i \(0.388608\pi\)
\(158\) −15.0465 −1.19704
\(159\) −8.88769 −0.704840
\(160\) 0.692021 0.0547091
\(161\) 0.855167 0.0673966
\(162\) −1.00000 −0.0785674
\(163\) −1.72587 −0.135181 −0.0675904 0.997713i \(-0.521531\pi\)
−0.0675904 + 0.997713i \(0.521531\pi\)
\(164\) 4.89008 0.381851
\(165\) −2.03385 −0.158335
\(166\) 14.8267 1.15077
\(167\) −21.1400 −1.63587 −0.817933 0.575314i \(-0.804880\pi\)
−0.817933 + 0.575314i \(0.804880\pi\)
\(168\) −0.356896 −0.0275351
\(169\) 0 0
\(170\) 4.64609 0.356339
\(171\) −7.20775 −0.551190
\(172\) 6.59179 0.502620
\(173\) 9.35450 0.711210 0.355605 0.934636i \(-0.384275\pi\)
0.355605 + 0.934636i \(0.384275\pi\)
\(174\) −7.82908 −0.593521
\(175\) −1.61356 −0.121974
\(176\) 2.93900 0.221536
\(177\) 1.64310 0.123503
\(178\) −0.396125 −0.0296908
\(179\) 3.17523 0.237328 0.118664 0.992934i \(-0.462139\pi\)
0.118664 + 0.992934i \(0.462139\pi\)
\(180\) −0.692021 −0.0515802
\(181\) −19.7995 −1.47169 −0.735844 0.677151i \(-0.763214\pi\)
−0.735844 + 0.677151i \(0.763214\pi\)
\(182\) 0 0
\(183\) −6.49396 −0.480047
\(184\) −2.39612 −0.176645
\(185\) −6.98792 −0.513762
\(186\) 2.76271 0.202572
\(187\) 19.7318 1.44294
\(188\) 4.98792 0.363781
\(189\) 0.356896 0.0259604
\(190\) −4.98792 −0.361862
\(191\) 15.2620 1.10432 0.552161 0.833737i \(-0.313803\pi\)
0.552161 + 0.833737i \(0.313803\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.76809 −0.343214 −0.171607 0.985165i \(-0.554896\pi\)
−0.171607 + 0.985165i \(0.554896\pi\)
\(194\) 0.417895 0.0300031
\(195\) 0 0
\(196\) −6.87263 −0.490902
\(197\) −12.2349 −0.871700 −0.435850 0.900019i \(-0.643552\pi\)
−0.435850 + 0.900019i \(0.643552\pi\)
\(198\) −2.93900 −0.208866
\(199\) −11.8485 −0.839915 −0.419958 0.907544i \(-0.637955\pi\)
−0.419958 + 0.907544i \(0.637955\pi\)
\(200\) 4.52111 0.319690
\(201\) 13.5254 0.954009
\(202\) −10.0151 −0.704658
\(203\) 2.79417 0.196112
\(204\) 6.71379 0.470059
\(205\) −3.38404 −0.236352
\(206\) 9.62565 0.670651
\(207\) 2.39612 0.166542
\(208\) 0 0
\(209\) −21.1836 −1.46530
\(210\) 0.246980 0.0170432
\(211\) −17.2620 −1.18837 −0.594184 0.804329i \(-0.702525\pi\)
−0.594184 + 0.804329i \(0.702525\pi\)
\(212\) −8.88769 −0.610409
\(213\) 6.81163 0.466725
\(214\) 6.63102 0.453287
\(215\) −4.56166 −0.311103
\(216\) −1.00000 −0.0680414
\(217\) −0.985999 −0.0669340
\(218\) −12.9879 −0.879653
\(219\) 3.18598 0.215289
\(220\) −2.03385 −0.137122
\(221\) 0 0
\(222\) −10.0978 −0.677722
\(223\) −6.76809 −0.453225 −0.226612 0.973985i \(-0.572765\pi\)
−0.226612 + 0.973985i \(0.572765\pi\)
\(224\) −0.356896 −0.0238461
\(225\) −4.52111 −0.301407
\(226\) −0.792249 −0.0526996
\(227\) −23.6799 −1.57169 −0.785846 0.618422i \(-0.787772\pi\)
−0.785846 + 0.618422i \(0.787772\pi\)
\(228\) −7.20775 −0.477345
\(229\) 8.29829 0.548366 0.274183 0.961677i \(-0.411593\pi\)
0.274183 + 0.961677i \(0.411593\pi\)
\(230\) 1.65817 0.109336
\(231\) 1.04892 0.0690137
\(232\) −7.82908 −0.514005
\(233\) −23.9651 −1.57000 −0.785002 0.619493i \(-0.787338\pi\)
−0.785002 + 0.619493i \(0.787338\pi\)
\(234\) 0 0
\(235\) −3.45175 −0.225167
\(236\) 1.64310 0.106957
\(237\) 15.0465 0.977377
\(238\) −2.39612 −0.155318
\(239\) −12.6160 −0.816058 −0.408029 0.912969i \(-0.633784\pi\)
−0.408029 + 0.912969i \(0.633784\pi\)
\(240\) −0.692021 −0.0446698
\(241\) −26.3937 −1.70017 −0.850085 0.526646i \(-0.823449\pi\)
−0.850085 + 0.526646i \(0.823449\pi\)
\(242\) 2.36227 0.151853
\(243\) 1.00000 0.0641500
\(244\) −6.49396 −0.415733
\(245\) 4.75600 0.303850
\(246\) −4.89008 −0.311780
\(247\) 0 0
\(248\) 2.76271 0.175432
\(249\) −14.8267 −0.939603
\(250\) −6.58881 −0.416713
\(251\) −30.0344 −1.89576 −0.947879 0.318632i \(-0.896777\pi\)
−0.947879 + 0.318632i \(0.896777\pi\)
\(252\) 0.356896 0.0224823
\(253\) 7.04221 0.442740
\(254\) 18.2174 1.14306
\(255\) −4.64609 −0.290949
\(256\) 1.00000 0.0625000
\(257\) 11.2620 0.702507 0.351254 0.936280i \(-0.385756\pi\)
0.351254 + 0.936280i \(0.385756\pi\)
\(258\) −6.59179 −0.410387
\(259\) 3.60388 0.223934
\(260\) 0 0
\(261\) 7.82908 0.484608
\(262\) −2.73556 −0.169004
\(263\) −5.54958 −0.342202 −0.171101 0.985254i \(-0.554732\pi\)
−0.171101 + 0.985254i \(0.554732\pi\)
\(264\) −2.93900 −0.180883
\(265\) 6.15047 0.377821
\(266\) 2.57242 0.157725
\(267\) 0.396125 0.0242424
\(268\) 13.5254 0.826196
\(269\) 16.6872 1.01744 0.508719 0.860932i \(-0.330119\pi\)
0.508719 + 0.860932i \(0.330119\pi\)
\(270\) 0.692021 0.0421151
\(271\) 6.61356 0.401745 0.200873 0.979617i \(-0.435622\pi\)
0.200873 + 0.979617i \(0.435622\pi\)
\(272\) 6.71379 0.407083
\(273\) 0 0
\(274\) 7.64742 0.461997
\(275\) −13.2875 −0.801269
\(276\) 2.39612 0.144230
\(277\) −21.7995 −1.30981 −0.654904 0.755712i \(-0.727291\pi\)
−0.654904 + 0.755712i \(0.727291\pi\)
\(278\) −3.38404 −0.202961
\(279\) −2.76271 −0.165399
\(280\) 0.246980 0.0147599
\(281\) −20.5918 −1.22840 −0.614202 0.789149i \(-0.710522\pi\)
−0.614202 + 0.789149i \(0.710522\pi\)
\(282\) −4.98792 −0.297026
\(283\) −13.0121 −0.773488 −0.386744 0.922187i \(-0.626400\pi\)
−0.386744 + 0.922187i \(0.626400\pi\)
\(284\) 6.81163 0.404196
\(285\) 4.98792 0.295459
\(286\) 0 0
\(287\) 1.74525 0.103019
\(288\) −1.00000 −0.0589256
\(289\) 28.0750 1.65147
\(290\) 5.41789 0.318150
\(291\) −0.417895 −0.0244974
\(292\) 3.18598 0.186445
\(293\) 14.9390 0.872746 0.436373 0.899766i \(-0.356263\pi\)
0.436373 + 0.899766i \(0.356263\pi\)
\(294\) 6.87263 0.400820
\(295\) −1.13706 −0.0662024
\(296\) −10.0978 −0.586925
\(297\) 2.93900 0.170538
\(298\) −20.8170 −1.20590
\(299\) 0 0
\(300\) −4.52111 −0.261026
\(301\) 2.35258 0.135601
\(302\) −0.895461 −0.0515280
\(303\) 10.0151 0.575350
\(304\) −7.20775 −0.413393
\(305\) 4.49396 0.257323
\(306\) −6.71379 −0.383802
\(307\) −26.0301 −1.48562 −0.742809 0.669503i \(-0.766507\pi\)
−0.742809 + 0.669503i \(0.766507\pi\)
\(308\) 1.04892 0.0597676
\(309\) −9.62565 −0.547584
\(310\) −1.91185 −0.108586
\(311\) −4.81163 −0.272842 −0.136421 0.990651i \(-0.543560\pi\)
−0.136421 + 0.990651i \(0.543560\pi\)
\(312\) 0 0
\(313\) −26.0411 −1.47193 −0.735966 0.677018i \(-0.763272\pi\)
−0.735966 + 0.677018i \(0.763272\pi\)
\(314\) −8.59179 −0.484863
\(315\) −0.246980 −0.0139157
\(316\) 15.0465 0.846433
\(317\) 11.5211 0.647090 0.323545 0.946213i \(-0.395125\pi\)
0.323545 + 0.946213i \(0.395125\pi\)
\(318\) 8.88769 0.498397
\(319\) 23.0097 1.28830
\(320\) −0.692021 −0.0386852
\(321\) −6.63102 −0.370108
\(322\) −0.855167 −0.0476566
\(323\) −48.3913 −2.69257
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.72587 0.0955873
\(327\) 12.9879 0.718234
\(328\) −4.89008 −0.270010
\(329\) 1.78017 0.0981438
\(330\) 2.03385 0.111960
\(331\) −3.43834 −0.188988 −0.0944940 0.995525i \(-0.530123\pi\)
−0.0944940 + 0.995525i \(0.530123\pi\)
\(332\) −14.8267 −0.813720
\(333\) 10.0978 0.553358
\(334\) 21.1400 1.15673
\(335\) −9.35988 −0.511385
\(336\) 0.356896 0.0194703
\(337\) 8.20105 0.446739 0.223370 0.974734i \(-0.428294\pi\)
0.223370 + 0.974734i \(0.428294\pi\)
\(338\) 0 0
\(339\) 0.792249 0.0430291
\(340\) −4.64609 −0.251970
\(341\) −8.11960 −0.439701
\(342\) 7.20775 0.389751
\(343\) −4.95108 −0.267333
\(344\) −6.59179 −0.355406
\(345\) −1.65817 −0.0892729
\(346\) −9.35450 −0.502901
\(347\) 7.86294 0.422105 0.211052 0.977475i \(-0.432311\pi\)
0.211052 + 0.977475i \(0.432311\pi\)
\(348\) 7.82908 0.419683
\(349\) 18.7245 1.00230 0.501151 0.865360i \(-0.332910\pi\)
0.501151 + 0.865360i \(0.332910\pi\)
\(350\) 1.61356 0.0862486
\(351\) 0 0
\(352\) −2.93900 −0.156649
\(353\) 31.5448 1.67896 0.839480 0.543391i \(-0.182860\pi\)
0.839480 + 0.543391i \(0.182860\pi\)
\(354\) −1.64310 −0.0873300
\(355\) −4.71379 −0.250182
\(356\) 0.396125 0.0209946
\(357\) 2.39612 0.126816
\(358\) −3.17523 −0.167816
\(359\) −2.39612 −0.126463 −0.0632313 0.997999i \(-0.520141\pi\)
−0.0632313 + 0.997999i \(0.520141\pi\)
\(360\) 0.692021 0.0364727
\(361\) 32.9517 1.73430
\(362\) 19.7995 1.04064
\(363\) −2.36227 −0.123987
\(364\) 0 0
\(365\) −2.20477 −0.115403
\(366\) 6.49396 0.339445
\(367\) −0.00431187 −0.000225078 0 −0.000112539 1.00000i \(-0.500036\pi\)
−0.000112539 1.00000i \(0.500036\pi\)
\(368\) 2.39612 0.124907
\(369\) 4.89008 0.254568
\(370\) 6.98792 0.363285
\(371\) −3.17198 −0.164681
\(372\) −2.76271 −0.143240
\(373\) −32.3129 −1.67310 −0.836549 0.547892i \(-0.815430\pi\)
−0.836549 + 0.547892i \(0.815430\pi\)
\(374\) −19.7318 −1.02031
\(375\) 6.58881 0.340245
\(376\) −4.98792 −0.257232
\(377\) 0 0
\(378\) −0.356896 −0.0183567
\(379\) −19.7560 −1.01480 −0.507399 0.861711i \(-0.669393\pi\)
−0.507399 + 0.861711i \(0.669393\pi\)
\(380\) 4.98792 0.255875
\(381\) −18.2174 −0.933308
\(382\) −15.2620 −0.780874
\(383\) −28.8116 −1.47221 −0.736103 0.676870i \(-0.763336\pi\)
−0.736103 + 0.676870i \(0.763336\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.725873 −0.0369939
\(386\) 4.76809 0.242689
\(387\) 6.59179 0.335080
\(388\) −0.417895 −0.0212154
\(389\) 34.7821 1.76352 0.881761 0.471697i \(-0.156358\pi\)
0.881761 + 0.471697i \(0.156358\pi\)
\(390\) 0 0
\(391\) 16.0871 0.813559
\(392\) 6.87263 0.347120
\(393\) 2.73556 0.137991
\(394\) 12.2349 0.616385
\(395\) −10.4125 −0.523911
\(396\) 2.93900 0.147690
\(397\) 5.15346 0.258645 0.129322 0.991603i \(-0.458720\pi\)
0.129322 + 0.991603i \(0.458720\pi\)
\(398\) 11.8485 0.593910
\(399\) −2.57242 −0.128782
\(400\) −4.52111 −0.226055
\(401\) −13.3250 −0.665417 −0.332708 0.943030i \(-0.607962\pi\)
−0.332708 + 0.943030i \(0.607962\pi\)
\(402\) −13.5254 −0.674587
\(403\) 0 0
\(404\) 10.0151 0.498268
\(405\) −0.692021 −0.0343868
\(406\) −2.79417 −0.138672
\(407\) 29.6775 1.47106
\(408\) −6.71379 −0.332382
\(409\) 24.0237 1.18789 0.593947 0.804504i \(-0.297569\pi\)
0.593947 + 0.804504i \(0.297569\pi\)
\(410\) 3.38404 0.167126
\(411\) −7.64742 −0.377219
\(412\) −9.62565 −0.474222
\(413\) 0.586417 0.0288557
\(414\) −2.39612 −0.117763
\(415\) 10.2604 0.503663
\(416\) 0 0
\(417\) 3.38404 0.165717
\(418\) 21.1836 1.03612
\(419\) 13.8049 0.674415 0.337207 0.941430i \(-0.390518\pi\)
0.337207 + 0.941430i \(0.390518\pi\)
\(420\) −0.246980 −0.0120514
\(421\) −7.72587 −0.376536 −0.188268 0.982118i \(-0.560287\pi\)
−0.188268 + 0.982118i \(0.560287\pi\)
\(422\) 17.2620 0.840303
\(423\) 4.98792 0.242521
\(424\) 8.88769 0.431624
\(425\) −30.3538 −1.47237
\(426\) −6.81163 −0.330024
\(427\) −2.31767 −0.112160
\(428\) −6.63102 −0.320523
\(429\) 0 0
\(430\) 4.56166 0.219983
\(431\) −0.640120 −0.0308335 −0.0154168 0.999881i \(-0.504907\pi\)
−0.0154168 + 0.999881i \(0.504907\pi\)
\(432\) 1.00000 0.0481125
\(433\) 21.2760 1.02246 0.511231 0.859443i \(-0.329190\pi\)
0.511231 + 0.859443i \(0.329190\pi\)
\(434\) 0.985999 0.0473295
\(435\) −5.41789 −0.259768
\(436\) 12.9879 0.622008
\(437\) −17.2707 −0.826168
\(438\) −3.18598 −0.152232
\(439\) 12.7181 0.607002 0.303501 0.952831i \(-0.401844\pi\)
0.303501 + 0.952831i \(0.401844\pi\)
\(440\) 2.03385 0.0969601
\(441\) −6.87263 −0.327268
\(442\) 0 0
\(443\) 22.5972 1.07362 0.536812 0.843702i \(-0.319628\pi\)
0.536812 + 0.843702i \(0.319628\pi\)
\(444\) 10.0978 0.479222
\(445\) −0.274127 −0.0129949
\(446\) 6.76809 0.320478
\(447\) 20.8170 0.984610
\(448\) 0.356896 0.0168617
\(449\) −11.6474 −0.549676 −0.274838 0.961491i \(-0.588624\pi\)
−0.274838 + 0.961491i \(0.588624\pi\)
\(450\) 4.52111 0.213127
\(451\) 14.3720 0.676749
\(452\) 0.792249 0.0372643
\(453\) 0.895461 0.0420724
\(454\) 23.6799 1.11135
\(455\) 0 0
\(456\) 7.20775 0.337534
\(457\) 21.1890 0.991178 0.495589 0.868557i \(-0.334952\pi\)
0.495589 + 0.868557i \(0.334952\pi\)
\(458\) −8.29829 −0.387754
\(459\) 6.71379 0.313373
\(460\) −1.65817 −0.0773126
\(461\) −24.0694 −1.12102 −0.560511 0.828147i \(-0.689395\pi\)
−0.560511 + 0.828147i \(0.689395\pi\)
\(462\) −1.04892 −0.0488001
\(463\) 18.1715 0.844502 0.422251 0.906479i \(-0.361240\pi\)
0.422251 + 0.906479i \(0.361240\pi\)
\(464\) 7.82908 0.363456
\(465\) 1.91185 0.0886601
\(466\) 23.9651 1.11016
\(467\) −2.93123 −0.135641 −0.0678206 0.997698i \(-0.521605\pi\)
−0.0678206 + 0.997698i \(0.521605\pi\)
\(468\) 0 0
\(469\) 4.82717 0.222898
\(470\) 3.45175 0.159217
\(471\) 8.59179 0.395889
\(472\) −1.64310 −0.0756300
\(473\) 19.3733 0.890785
\(474\) −15.0465 −0.691110
\(475\) 32.5870 1.49519
\(476\) 2.39612 0.109826
\(477\) −8.88769 −0.406939
\(478\) 12.6160 0.577040
\(479\) 30.7090 1.40313 0.701565 0.712605i \(-0.252485\pi\)
0.701565 + 0.712605i \(0.252485\pi\)
\(480\) 0.692021 0.0315863
\(481\) 0 0
\(482\) 26.3937 1.20220
\(483\) 0.855167 0.0389114
\(484\) −2.36227 −0.107376
\(485\) 0.289192 0.0131315
\(486\) −1.00000 −0.0453609
\(487\) −24.1497 −1.09433 −0.547164 0.837025i \(-0.684293\pi\)
−0.547164 + 0.837025i \(0.684293\pi\)
\(488\) 6.49396 0.293968
\(489\) −1.72587 −0.0780467
\(490\) −4.75600 −0.214854
\(491\) 14.5972 0.658761 0.329381 0.944197i \(-0.393160\pi\)
0.329381 + 0.944197i \(0.393160\pi\)
\(492\) 4.89008 0.220462
\(493\) 52.5628 2.36731
\(494\) 0 0
\(495\) −2.03385 −0.0914148
\(496\) −2.76271 −0.124049
\(497\) 2.43104 0.109047
\(498\) 14.8267 0.664400
\(499\) −6.85517 −0.306879 −0.153440 0.988158i \(-0.549035\pi\)
−0.153440 + 0.988158i \(0.549035\pi\)
\(500\) 6.58881 0.294661
\(501\) −21.1400 −0.944468
\(502\) 30.0344 1.34050
\(503\) −26.7332 −1.19197 −0.595987 0.802994i \(-0.703239\pi\)
−0.595987 + 0.802994i \(0.703239\pi\)
\(504\) −0.356896 −0.0158974
\(505\) −6.93064 −0.308409
\(506\) −7.04221 −0.313065
\(507\) 0 0
\(508\) −18.2174 −0.808268
\(509\) 20.6595 0.915716 0.457858 0.889025i \(-0.348617\pi\)
0.457858 + 0.889025i \(0.348617\pi\)
\(510\) 4.64609 0.205732
\(511\) 1.13706 0.0503007
\(512\) −1.00000 −0.0441942
\(513\) −7.20775 −0.318230
\(514\) −11.2620 −0.496748
\(515\) 6.66115 0.293525
\(516\) 6.59179 0.290188
\(517\) 14.6595 0.644724
\(518\) −3.60388 −0.158345
\(519\) 9.35450 0.410617
\(520\) 0 0
\(521\) −15.0965 −0.661390 −0.330695 0.943738i \(-0.607283\pi\)
−0.330695 + 0.943738i \(0.607283\pi\)
\(522\) −7.82908 −0.342670
\(523\) 0.0349168 0.00152680 0.000763402 1.00000i \(-0.499757\pi\)
0.000763402 1.00000i \(0.499757\pi\)
\(524\) 2.73556 0.119504
\(525\) −1.61356 −0.0704217
\(526\) 5.54958 0.241973
\(527\) −18.5483 −0.807975
\(528\) 2.93900 0.127904
\(529\) −17.2586 −0.750373
\(530\) −6.15047 −0.267159
\(531\) 1.64310 0.0713046
\(532\) −2.57242 −0.111528
\(533\) 0 0
\(534\) −0.396125 −0.0171420
\(535\) 4.58881 0.198392
\(536\) −13.5254 −0.584209
\(537\) 3.17523 0.137021
\(538\) −16.6872 −0.719438
\(539\) −20.1987 −0.870018
\(540\) −0.692021 −0.0297799
\(541\) −13.0858 −0.562600 −0.281300 0.959620i \(-0.590766\pi\)
−0.281300 + 0.959620i \(0.590766\pi\)
\(542\) −6.61356 −0.284077
\(543\) −19.7995 −0.849680
\(544\) −6.71379 −0.287851
\(545\) −8.98792 −0.385000
\(546\) 0 0
\(547\) −5.97584 −0.255508 −0.127754 0.991806i \(-0.540777\pi\)
−0.127754 + 0.991806i \(0.540777\pi\)
\(548\) −7.64742 −0.326681
\(549\) −6.49396 −0.277155
\(550\) 13.2875 0.566582
\(551\) −56.4301 −2.40400
\(552\) −2.39612 −0.101986
\(553\) 5.37004 0.228357
\(554\) 21.7995 0.926174
\(555\) −6.98792 −0.296621
\(556\) 3.38404 0.143515
\(557\) 10.4397 0.442343 0.221171 0.975235i \(-0.429012\pi\)
0.221171 + 0.975235i \(0.429012\pi\)
\(558\) 2.76271 0.116955
\(559\) 0 0
\(560\) −0.246980 −0.0104368
\(561\) 19.7318 0.833079
\(562\) 20.5918 0.868612
\(563\) −5.66056 −0.238564 −0.119282 0.992860i \(-0.538059\pi\)
−0.119282 + 0.992860i \(0.538059\pi\)
\(564\) 4.98792 0.210029
\(565\) −0.548253 −0.0230652
\(566\) 13.0121 0.546939
\(567\) 0.356896 0.0149882
\(568\) −6.81163 −0.285809
\(569\) 20.6568 0.865980 0.432990 0.901399i \(-0.357459\pi\)
0.432990 + 0.901399i \(0.357459\pi\)
\(570\) −4.98792 −0.208921
\(571\) 44.3672 1.85671 0.928354 0.371697i \(-0.121224\pi\)
0.928354 + 0.371697i \(0.121224\pi\)
\(572\) 0 0
\(573\) 15.2620 0.637581
\(574\) −1.74525 −0.0728454
\(575\) −10.8331 −0.451773
\(576\) 1.00000 0.0416667
\(577\) −29.4426 −1.22571 −0.612857 0.790194i \(-0.709980\pi\)
−0.612857 + 0.790194i \(0.709980\pi\)
\(578\) −28.0750 −1.16777
\(579\) −4.76809 −0.198155
\(580\) −5.41789 −0.224966
\(581\) −5.29159 −0.219532
\(582\) 0.417895 0.0173223
\(583\) −26.1209 −1.08182
\(584\) −3.18598 −0.131837
\(585\) 0 0
\(586\) −14.9390 −0.617124
\(587\) 19.5636 0.807475 0.403738 0.914875i \(-0.367711\pi\)
0.403738 + 0.914875i \(0.367711\pi\)
\(588\) −6.87263 −0.283422
\(589\) 19.9129 0.820498
\(590\) 1.13706 0.0468122
\(591\) −12.2349 −0.503276
\(592\) 10.0978 0.415018
\(593\) 10.8793 0.446761 0.223380 0.974731i \(-0.428291\pi\)
0.223380 + 0.974731i \(0.428291\pi\)
\(594\) −2.93900 −0.120589
\(595\) −1.65817 −0.0679783
\(596\) 20.8170 0.852698
\(597\) −11.8485 −0.484925
\(598\) 0 0
\(599\) 16.0543 0.655961 0.327980 0.944685i \(-0.393632\pi\)
0.327980 + 0.944685i \(0.393632\pi\)
\(600\) 4.52111 0.184573
\(601\) 12.8955 0.526017 0.263008 0.964794i \(-0.415285\pi\)
0.263008 + 0.964794i \(0.415285\pi\)
\(602\) −2.35258 −0.0958842
\(603\) 13.5254 0.550798
\(604\) 0.895461 0.0364358
\(605\) 1.63474 0.0664618
\(606\) −10.0151 −0.406834
\(607\) 44.4741 1.80515 0.902574 0.430534i \(-0.141675\pi\)
0.902574 + 0.430534i \(0.141675\pi\)
\(608\) 7.20775 0.292313
\(609\) 2.79417 0.113225
\(610\) −4.49396 −0.181955
\(611\) 0 0
\(612\) 6.71379 0.271389
\(613\) −42.0253 −1.69739 −0.848694 0.528884i \(-0.822610\pi\)
−0.848694 + 0.528884i \(0.822610\pi\)
\(614\) 26.0301 1.05049
\(615\) −3.38404 −0.136458
\(616\) −1.04892 −0.0422621
\(617\) 16.6655 0.670926 0.335463 0.942053i \(-0.391107\pi\)
0.335463 + 0.942053i \(0.391107\pi\)
\(618\) 9.62565 0.387200
\(619\) −39.7512 −1.59774 −0.798868 0.601506i \(-0.794568\pi\)
−0.798868 + 0.601506i \(0.794568\pi\)
\(620\) 1.91185 0.0767819
\(621\) 2.39612 0.0961532
\(622\) 4.81163 0.192929
\(623\) 0.141375 0.00566408
\(624\) 0 0
\(625\) 18.0459 0.721837
\(626\) 26.0411 1.04081
\(627\) −21.1836 −0.845991
\(628\) 8.59179 0.342850
\(629\) 67.7948 2.70315
\(630\) 0.246980 0.00983990
\(631\) −14.5767 −0.580290 −0.290145 0.956983i \(-0.593704\pi\)
−0.290145 + 0.956983i \(0.593704\pi\)
\(632\) −15.0465 −0.598519
\(633\) −17.2620 −0.686105
\(634\) −11.5211 −0.457562
\(635\) 12.6069 0.500288
\(636\) −8.88769 −0.352420
\(637\) 0 0
\(638\) −23.0097 −0.910962
\(639\) 6.81163 0.269464
\(640\) 0.692021 0.0273546
\(641\) 38.4349 1.51809 0.759043 0.651040i \(-0.225667\pi\)
0.759043 + 0.651040i \(0.225667\pi\)
\(642\) 6.63102 0.261706
\(643\) 1.04221 0.0411009 0.0205504 0.999789i \(-0.493458\pi\)
0.0205504 + 0.999789i \(0.493458\pi\)
\(644\) 0.855167 0.0336983
\(645\) −4.56166 −0.179615
\(646\) 48.3913 1.90393
\(647\) −28.1608 −1.10711 −0.553557 0.832811i \(-0.686730\pi\)
−0.553557 + 0.832811i \(0.686730\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.82908 0.189558
\(650\) 0 0
\(651\) −0.985999 −0.0386444
\(652\) −1.72587 −0.0675904
\(653\) 10.6203 0.415603 0.207802 0.978171i \(-0.433369\pi\)
0.207802 + 0.978171i \(0.433369\pi\)
\(654\) −12.9879 −0.507868
\(655\) −1.89307 −0.0739683
\(656\) 4.89008 0.190926
\(657\) 3.18598 0.124297
\(658\) −1.78017 −0.0693982
\(659\) −40.3629 −1.57231 −0.786157 0.618027i \(-0.787932\pi\)
−0.786157 + 0.618027i \(0.787932\pi\)
\(660\) −2.03385 −0.0791676
\(661\) 31.9168 1.24142 0.620709 0.784041i \(-0.286845\pi\)
0.620709 + 0.784041i \(0.286845\pi\)
\(662\) 3.43834 0.133635
\(663\) 0 0
\(664\) 14.8267 0.575387
\(665\) 1.78017 0.0690319
\(666\) −10.0978 −0.391283
\(667\) 18.7595 0.726369
\(668\) −21.1400 −0.817933
\(669\) −6.76809 −0.261669
\(670\) 9.35988 0.361604
\(671\) −19.0858 −0.736797
\(672\) −0.356896 −0.0137676
\(673\) 3.82802 0.147559 0.0737797 0.997275i \(-0.476494\pi\)
0.0737797 + 0.997275i \(0.476494\pi\)
\(674\) −8.20105 −0.315892
\(675\) −4.52111 −0.174017
\(676\) 0 0
\(677\) 1.78927 0.0687670 0.0343835 0.999409i \(-0.489053\pi\)
0.0343835 + 0.999409i \(0.489053\pi\)
\(678\) −0.792249 −0.0304261
\(679\) −0.149145 −0.00572366
\(680\) 4.64609 0.178169
\(681\) −23.6799 −0.907417
\(682\) 8.11960 0.310916
\(683\) −0.759725 −0.0290701 −0.0145350 0.999894i \(-0.504627\pi\)
−0.0145350 + 0.999894i \(0.504627\pi\)
\(684\) −7.20775 −0.275595
\(685\) 5.29218 0.202204
\(686\) 4.95108 0.189033
\(687\) 8.29829 0.316600
\(688\) 6.59179 0.251310
\(689\) 0 0
\(690\) 1.65817 0.0631254
\(691\) −4.65950 −0.177256 −0.0886278 0.996065i \(-0.528248\pi\)
−0.0886278 + 0.996065i \(0.528248\pi\)
\(692\) 9.35450 0.355605
\(693\) 1.04892 0.0398451
\(694\) −7.86294 −0.298473
\(695\) −2.34183 −0.0888307
\(696\) −7.82908 −0.296761
\(697\) 32.8310 1.24356
\(698\) −18.7245 −0.708734
\(699\) −23.9651 −0.906443
\(700\) −1.61356 −0.0609870
\(701\) −24.3284 −0.918872 −0.459436 0.888211i \(-0.651948\pi\)
−0.459436 + 0.888211i \(0.651948\pi\)
\(702\) 0 0
\(703\) −72.7827 −2.74505
\(704\) 2.93900 0.110768
\(705\) −3.45175 −0.130000
\(706\) −31.5448 −1.18720
\(707\) 3.57434 0.134427
\(708\) 1.64310 0.0617516
\(709\) 29.3927 1.10386 0.551932 0.833889i \(-0.313891\pi\)
0.551932 + 0.833889i \(0.313891\pi\)
\(710\) 4.71379 0.176905
\(711\) 15.0465 0.564289
\(712\) −0.396125 −0.0148454
\(713\) −6.61979 −0.247913
\(714\) −2.39612 −0.0896727
\(715\) 0 0
\(716\) 3.17523 0.118664
\(717\) −12.6160 −0.471152
\(718\) 2.39612 0.0894226
\(719\) −51.4878 −1.92017 −0.960086 0.279704i \(-0.909764\pi\)
−0.960086 + 0.279704i \(0.909764\pi\)
\(720\) −0.692021 −0.0257901
\(721\) −3.43535 −0.127939
\(722\) −32.9517 −1.22633
\(723\) −26.3937 −0.981593
\(724\) −19.7995 −0.735844
\(725\) −35.3961 −1.31458
\(726\) 2.36227 0.0876722
\(727\) −3.67324 −0.136233 −0.0681164 0.997677i \(-0.521699\pi\)
−0.0681164 + 0.997677i \(0.521699\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.20477 0.0816021
\(731\) 44.2559 1.63686
\(732\) −6.49396 −0.240024
\(733\) −18.3612 −0.678187 −0.339093 0.940753i \(-0.610120\pi\)
−0.339093 + 0.940753i \(0.610120\pi\)
\(734\) 0.00431187 0.000159154 0
\(735\) 4.75600 0.175428
\(736\) −2.39612 −0.0883223
\(737\) 39.7512 1.46425
\(738\) −4.89008 −0.180006
\(739\) −1.68233 −0.0618856 −0.0309428 0.999521i \(-0.509851\pi\)
−0.0309428 + 0.999521i \(0.509851\pi\)
\(740\) −6.98792 −0.256881
\(741\) 0 0
\(742\) 3.17198 0.116447
\(743\) −48.2935 −1.77172 −0.885858 0.463956i \(-0.846430\pi\)
−0.885858 + 0.463956i \(0.846430\pi\)
\(744\) 2.76271 0.101286
\(745\) −14.4058 −0.527788
\(746\) 32.3129 1.18306
\(747\) −14.8267 −0.542480
\(748\) 19.7318 0.721468
\(749\) −2.36658 −0.0864731
\(750\) −6.58881 −0.240589
\(751\) 11.7011 0.426980 0.213490 0.976945i \(-0.431517\pi\)
0.213490 + 0.976945i \(0.431517\pi\)
\(752\) 4.98792 0.181891
\(753\) −30.0344 −1.09452
\(754\) 0 0
\(755\) −0.619678 −0.0225524
\(756\) 0.356896 0.0129802
\(757\) 25.1836 0.915313 0.457657 0.889129i \(-0.348689\pi\)
0.457657 + 0.889129i \(0.348689\pi\)
\(758\) 19.7560 0.717570
\(759\) 7.04221 0.255616
\(760\) −4.98792 −0.180931
\(761\) −22.4155 −0.812561 −0.406281 0.913748i \(-0.633174\pi\)
−0.406281 + 0.913748i \(0.633174\pi\)
\(762\) 18.2174 0.659948
\(763\) 4.63533 0.167810
\(764\) 15.2620 0.552161
\(765\) −4.64609 −0.167980
\(766\) 28.8116 1.04101
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 0.132751 0.00478714 0.00239357 0.999997i \(-0.499238\pi\)
0.00239357 + 0.999997i \(0.499238\pi\)
\(770\) 0.725873 0.0261587
\(771\) 11.2620 0.405593
\(772\) −4.76809 −0.171607
\(773\) 48.0694 1.72893 0.864467 0.502689i \(-0.167656\pi\)
0.864467 + 0.502689i \(0.167656\pi\)
\(774\) −6.59179 −0.236937
\(775\) 12.4905 0.448672
\(776\) 0.417895 0.0150015
\(777\) 3.60388 0.129288
\(778\) −34.7821 −1.24700
\(779\) −35.2465 −1.26284
\(780\) 0 0
\(781\) 20.0194 0.716350
\(782\) −16.0871 −0.575273
\(783\) 7.82908 0.279789
\(784\) −6.87263 −0.245451
\(785\) −5.94571 −0.212211
\(786\) −2.73556 −0.0975743
\(787\) −6.20908 −0.221330 −0.110665 0.993858i \(-0.535298\pi\)
−0.110665 + 0.993858i \(0.535298\pi\)
\(788\) −12.2349 −0.435850
\(789\) −5.54958 −0.197570
\(790\) 10.4125 0.370461
\(791\) 0.282750 0.0100534
\(792\) −2.93900 −0.104433
\(793\) 0 0
\(794\) −5.15346 −0.182889
\(795\) 6.15047 0.218135
\(796\) −11.8485 −0.419958
\(797\) 0.327830 0.0116123 0.00580616 0.999983i \(-0.498152\pi\)
0.00580616 + 0.999983i \(0.498152\pi\)
\(798\) 2.57242 0.0910626
\(799\) 33.4878 1.18471
\(800\) 4.52111 0.159845
\(801\) 0.396125 0.0139964
\(802\) 13.3250 0.470521
\(803\) 9.36360 0.330434
\(804\) 13.5254 0.477005
\(805\) −0.591794 −0.0208580
\(806\) 0 0
\(807\) 16.6872 0.587419
\(808\) −10.0151 −0.352329
\(809\) −37.4383 −1.31626 −0.658131 0.752904i \(-0.728653\pi\)
−0.658131 + 0.752904i \(0.728653\pi\)
\(810\) 0.692021 0.0243152
\(811\) −17.1448 −0.602037 −0.301018 0.953618i \(-0.597327\pi\)
−0.301018 + 0.953618i \(0.597327\pi\)
\(812\) 2.79417 0.0980561
\(813\) 6.61356 0.231948
\(814\) −29.6775 −1.04020
\(815\) 1.19434 0.0418360
\(816\) 6.71379 0.235030
\(817\) −47.5120 −1.66223
\(818\) −24.0237 −0.839969
\(819\) 0 0
\(820\) −3.38404 −0.118176
\(821\) 17.7885 0.620824 0.310412 0.950602i \(-0.399533\pi\)
0.310412 + 0.950602i \(0.399533\pi\)
\(822\) 7.64742 0.266734
\(823\) 12.2301 0.426315 0.213157 0.977018i \(-0.431625\pi\)
0.213157 + 0.977018i \(0.431625\pi\)
\(824\) 9.62565 0.335325
\(825\) −13.2875 −0.462613
\(826\) −0.586417 −0.0204041
\(827\) −20.5623 −0.715020 −0.357510 0.933909i \(-0.616374\pi\)
−0.357510 + 0.933909i \(0.616374\pi\)
\(828\) 2.39612 0.0832711
\(829\) −25.4470 −0.883809 −0.441905 0.897062i \(-0.645697\pi\)
−0.441905 + 0.897062i \(0.645697\pi\)
\(830\) −10.2604 −0.356143
\(831\) −21.7995 −0.756218
\(832\) 0 0
\(833\) −46.1414 −1.59870
\(834\) −3.38404 −0.117180
\(835\) 14.6294 0.506270
\(836\) −21.1836 −0.732650
\(837\) −2.76271 −0.0954932
\(838\) −13.8049 −0.476883
\(839\) −5.76676 −0.199091 −0.0995453 0.995033i \(-0.531739\pi\)
−0.0995453 + 0.995033i \(0.531739\pi\)
\(840\) 0.246980 0.00852161
\(841\) 32.2946 1.11361
\(842\) 7.72587 0.266251
\(843\) −20.5918 −0.709219
\(844\) −17.2620 −0.594184
\(845\) 0 0
\(846\) −4.98792 −0.171488
\(847\) −0.843085 −0.0289688
\(848\) −8.88769 −0.305205
\(849\) −13.0121 −0.446573
\(850\) 30.3538 1.04113
\(851\) 24.1957 0.829417
\(852\) 6.81163 0.233362
\(853\) 21.1728 0.724944 0.362472 0.931995i \(-0.381933\pi\)
0.362472 + 0.931995i \(0.381933\pi\)
\(854\) 2.31767 0.0793089
\(855\) 4.98792 0.170583
\(856\) 6.63102 0.226644
\(857\) 12.0086 0.410207 0.205103 0.978740i \(-0.434247\pi\)
0.205103 + 0.978740i \(0.434247\pi\)
\(858\) 0 0
\(859\) −1.66296 −0.0567393 −0.0283697 0.999598i \(-0.509032\pi\)
−0.0283697 + 0.999598i \(0.509032\pi\)
\(860\) −4.56166 −0.155551
\(861\) 1.74525 0.0594780
\(862\) 0.640120 0.0218026
\(863\) 23.8323 0.811262 0.405631 0.914037i \(-0.367052\pi\)
0.405631 + 0.914037i \(0.367052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.47352 −0.220106
\(866\) −21.2760 −0.722989
\(867\) 28.0750 0.953477
\(868\) −0.985999 −0.0334670
\(869\) 44.2218 1.50012
\(870\) 5.41789 0.183684
\(871\) 0 0
\(872\) −12.9879 −0.439826
\(873\) −0.417895 −0.0141436
\(874\) 17.2707 0.584189
\(875\) 2.35152 0.0794959
\(876\) 3.18598 0.107644
\(877\) 42.3177 1.42897 0.714483 0.699653i \(-0.246662\pi\)
0.714483 + 0.699653i \(0.246662\pi\)
\(878\) −12.7181 −0.429215
\(879\) 14.9390 0.503880
\(880\) −2.03385 −0.0685611
\(881\) 22.2741 0.750434 0.375217 0.926937i \(-0.377568\pi\)
0.375217 + 0.926937i \(0.377568\pi\)
\(882\) 6.87263 0.231413
\(883\) −8.54229 −0.287471 −0.143735 0.989616i \(-0.545911\pi\)
−0.143735 + 0.989616i \(0.545911\pi\)
\(884\) 0 0
\(885\) −1.13706 −0.0382220
\(886\) −22.5972 −0.759167
\(887\) −18.9142 −0.635078 −0.317539 0.948245i \(-0.602856\pi\)
−0.317539 + 0.948245i \(0.602856\pi\)
\(888\) −10.0978 −0.338861
\(889\) −6.50173 −0.218061
\(890\) 0.274127 0.00918875
\(891\) 2.93900 0.0984602
\(892\) −6.76809 −0.226612
\(893\) −35.9517 −1.20308
\(894\) −20.8170 −0.696225
\(895\) −2.19733 −0.0734485
\(896\) −0.356896 −0.0119231
\(897\) 0 0
\(898\) 11.6474 0.388679
\(899\) −21.6295 −0.721384
\(900\) −4.52111 −0.150704
\(901\) −59.6701 −1.98790
\(902\) −14.3720 −0.478534
\(903\) 2.35258 0.0782891
\(904\) −0.792249 −0.0263498
\(905\) 13.7017 0.455460
\(906\) −0.895461 −0.0297497
\(907\) 13.9517 0.463258 0.231629 0.972804i \(-0.425595\pi\)
0.231629 + 0.972804i \(0.425595\pi\)
\(908\) −23.6799 −0.785846
\(909\) 10.0151 0.332179
\(910\) 0 0
\(911\) 45.0422 1.49232 0.746158 0.665769i \(-0.231897\pi\)
0.746158 + 0.665769i \(0.231897\pi\)
\(912\) −7.20775 −0.238672
\(913\) −43.5757 −1.44214
\(914\) −21.1890 −0.700869
\(915\) 4.49396 0.148566
\(916\) 8.29829 0.274183
\(917\) 0.976311 0.0322406
\(918\) −6.71379 −0.221588
\(919\) 39.9976 1.31940 0.659700 0.751529i \(-0.270683\pi\)
0.659700 + 0.751529i \(0.270683\pi\)
\(920\) 1.65817 0.0546682
\(921\) −26.0301 −0.857722
\(922\) 24.0694 0.792682
\(923\) 0 0
\(924\) 1.04892 0.0345068
\(925\) −45.6534 −1.50107
\(926\) −18.1715 −0.597153
\(927\) −9.62565 −0.316148
\(928\) −7.82908 −0.257002
\(929\) −34.6848 −1.13797 −0.568986 0.822347i \(-0.692664\pi\)
−0.568986 + 0.822347i \(0.692664\pi\)
\(930\) −1.91185 −0.0626922
\(931\) 49.5362 1.62348
\(932\) −23.9651 −0.785002
\(933\) −4.81163 −0.157526
\(934\) 2.93123 0.0959128
\(935\) −13.6549 −0.446562
\(936\) 0 0
\(937\) −19.1260 −0.624821 −0.312410 0.949947i \(-0.601136\pi\)
−0.312410 + 0.949947i \(0.601136\pi\)
\(938\) −4.82717 −0.157613
\(939\) −26.0411 −0.849821
\(940\) −3.45175 −0.112584
\(941\) 22.5972 0.736647 0.368323 0.929698i \(-0.379932\pi\)
0.368323 + 0.929698i \(0.379932\pi\)
\(942\) −8.59179 −0.279936
\(943\) 11.7172 0.381566
\(944\) 1.64310 0.0534785
\(945\) −0.246980 −0.00803425
\(946\) −19.3733 −0.629880
\(947\) 27.4359 0.891548 0.445774 0.895145i \(-0.352928\pi\)
0.445774 + 0.895145i \(0.352928\pi\)
\(948\) 15.0465 0.488688
\(949\) 0 0
\(950\) −32.5870 −1.05726
\(951\) 11.5211 0.373597
\(952\) −2.39612 −0.0776588
\(953\) −1.84787 −0.0598584 −0.0299292 0.999552i \(-0.509528\pi\)
−0.0299292 + 0.999552i \(0.509528\pi\)
\(954\) 8.88769 0.287750
\(955\) −10.5617 −0.341767
\(956\) −12.6160 −0.408029
\(957\) 23.0097 0.743798
\(958\) −30.7090 −0.992163
\(959\) −2.72933 −0.0881347
\(960\) −0.692021 −0.0223349
\(961\) −23.3674 −0.753788
\(962\) 0 0
\(963\) −6.63102 −0.213682
\(964\) −26.3937 −0.850085
\(965\) 3.29962 0.106218
\(966\) −0.855167 −0.0275145
\(967\) 8.88471 0.285713 0.142856 0.989743i \(-0.454371\pi\)
0.142856 + 0.989743i \(0.454371\pi\)
\(968\) 2.36227 0.0759263
\(969\) −48.3913 −1.55455
\(970\) −0.289192 −0.00928540
\(971\) 35.0863 1.12597 0.562987 0.826466i \(-0.309652\pi\)
0.562987 + 0.826466i \(0.309652\pi\)
\(972\) 1.00000 0.0320750
\(973\) 1.20775 0.0387187
\(974\) 24.1497 0.773807
\(975\) 0 0
\(976\) −6.49396 −0.207867
\(977\) 8.33704 0.266726 0.133363 0.991067i \(-0.457422\pi\)
0.133363 + 0.991067i \(0.457422\pi\)
\(978\) 1.72587 0.0551873
\(979\) 1.16421 0.0372083
\(980\) 4.75600 0.151925
\(981\) 12.9879 0.414672
\(982\) −14.5972 −0.465814
\(983\) 55.6051 1.77353 0.886763 0.462224i \(-0.152949\pi\)
0.886763 + 0.462224i \(0.152949\pi\)
\(984\) −4.89008 −0.155890
\(985\) 8.46681 0.269775
\(986\) −52.5628 −1.67394
\(987\) 1.78017 0.0566634
\(988\) 0 0
\(989\) 15.7948 0.502244
\(990\) 2.03385 0.0646401
\(991\) −43.5967 −1.38489 −0.692447 0.721468i \(-0.743468\pi\)
−0.692447 + 0.721468i \(0.743468\pi\)
\(992\) 2.76271 0.0877161
\(993\) −3.43834 −0.109112
\(994\) −2.43104 −0.0771079
\(995\) 8.19939 0.259938
\(996\) −14.8267 −0.469802
\(997\) 22.4590 0.711285 0.355643 0.934622i \(-0.384262\pi\)
0.355643 + 0.934622i \(0.384262\pi\)
\(998\) 6.85517 0.216997
\(999\) 10.0978 0.319481
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 1014.2.a.m.1.1 3
3.2 odd 2 3042.2.a.be.1.3 3
4.3 odd 2 8112.2.a.ce.1.1 3
13.2 odd 12 1014.2.i.g.823.3 12
13.3 even 3 1014.2.e.m.529.1 6
13.4 even 6 1014.2.e.k.991.3 6
13.5 odd 4 1014.2.b.g.337.6 6
13.6 odd 12 1014.2.i.g.361.6 12
13.7 odd 12 1014.2.i.g.361.1 12
13.8 odd 4 1014.2.b.g.337.1 6
13.9 even 3 1014.2.e.m.991.1 6
13.10 even 6 1014.2.e.k.529.3 6
13.11 odd 12 1014.2.i.g.823.4 12
13.12 even 2 1014.2.a.o.1.3 yes 3
39.5 even 4 3042.2.b.r.1351.1 6
39.8 even 4 3042.2.b.r.1351.6 6
39.38 odd 2 3042.2.a.bd.1.1 3
52.51 odd 2 8112.2.a.bz.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1014.2.a.m.1.1 3 1.1 even 1 trivial
1014.2.a.o.1.3 yes 3 13.12 even 2
1014.2.b.g.337.1 6 13.8 odd 4
1014.2.b.g.337.6 6 13.5 odd 4
1014.2.e.k.529.3 6 13.10 even 6
1014.2.e.k.991.3 6 13.4 even 6
1014.2.e.m.529.1 6 13.3 even 3
1014.2.e.m.991.1 6 13.9 even 3
1014.2.i.g.361.1 12 13.7 odd 12
1014.2.i.g.361.6 12 13.6 odd 12
1014.2.i.g.823.3 12 13.2 odd 12
1014.2.i.g.823.4 12 13.11 odd 12
3042.2.a.bd.1.1 3 39.38 odd 2
3042.2.a.be.1.3 3 3.2 odd 2
3042.2.b.r.1351.1 6 39.5 even 4
3042.2.b.r.1351.6 6 39.8 even 4
8112.2.a.bz.1.3 3 52.51 odd 2
8112.2.a.ce.1.1 3 4.3 odd 2