Properties

Label 8034.2.a.z.1.4
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.05100\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.05100 q^{5} +1.00000 q^{6} +5.03739 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.05100 q^{5} +1.00000 q^{6} +5.03739 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.05100 q^{10} +3.25636 q^{11} -1.00000 q^{12} +1.00000 q^{13} -5.03739 q^{14} +2.05100 q^{15} +1.00000 q^{16} +0.479851 q^{17} -1.00000 q^{18} +5.58203 q^{19} -2.05100 q^{20} -5.03739 q^{21} -3.25636 q^{22} -6.13582 q^{23} +1.00000 q^{24} -0.793401 q^{25} -1.00000 q^{26} -1.00000 q^{27} +5.03739 q^{28} +3.21614 q^{29} -2.05100 q^{30} +8.30084 q^{31} -1.00000 q^{32} -3.25636 q^{33} -0.479851 q^{34} -10.3317 q^{35} +1.00000 q^{36} -3.88622 q^{37} -5.58203 q^{38} -1.00000 q^{39} +2.05100 q^{40} +5.27024 q^{41} +5.03739 q^{42} +3.12359 q^{43} +3.25636 q^{44} -2.05100 q^{45} +6.13582 q^{46} +3.55585 q^{47} -1.00000 q^{48} +18.3753 q^{49} +0.793401 q^{50} -0.479851 q^{51} +1.00000 q^{52} -6.76178 q^{53} +1.00000 q^{54} -6.67880 q^{55} -5.03739 q^{56} -5.58203 q^{57} -3.21614 q^{58} +13.1023 q^{59} +2.05100 q^{60} +6.84529 q^{61} -8.30084 q^{62} +5.03739 q^{63} +1.00000 q^{64} -2.05100 q^{65} +3.25636 q^{66} -3.51638 q^{67} +0.479851 q^{68} +6.13582 q^{69} +10.3317 q^{70} +0.616815 q^{71} -1.00000 q^{72} +13.2386 q^{73} +3.88622 q^{74} +0.793401 q^{75} +5.58203 q^{76} +16.4036 q^{77} +1.00000 q^{78} +2.05078 q^{79} -2.05100 q^{80} +1.00000 q^{81} -5.27024 q^{82} -2.18812 q^{83} -5.03739 q^{84} -0.984174 q^{85} -3.12359 q^{86} -3.21614 q^{87} -3.25636 q^{88} -3.35651 q^{89} +2.05100 q^{90} +5.03739 q^{91} -6.13582 q^{92} -8.30084 q^{93} -3.55585 q^{94} -11.4487 q^{95} +1.00000 q^{96} -1.04552 q^{97} -18.3753 q^{98} +3.25636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.05100 −0.917235 −0.458617 0.888634i \(-0.651655\pi\)
−0.458617 + 0.888634i \(0.651655\pi\)
\(6\) 1.00000 0.408248
\(7\) 5.03739 1.90396 0.951978 0.306167i \(-0.0990467\pi\)
0.951978 + 0.306167i \(0.0990467\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.05100 0.648583
\(11\) 3.25636 0.981831 0.490915 0.871207i \(-0.336662\pi\)
0.490915 + 0.871207i \(0.336662\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −5.03739 −1.34630
\(15\) 2.05100 0.529566
\(16\) 1.00000 0.250000
\(17\) 0.479851 0.116381 0.0581905 0.998305i \(-0.481467\pi\)
0.0581905 + 0.998305i \(0.481467\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.58203 1.28061 0.640303 0.768123i \(-0.278809\pi\)
0.640303 + 0.768123i \(0.278809\pi\)
\(20\) −2.05100 −0.458617
\(21\) −5.03739 −1.09925
\(22\) −3.25636 −0.694259
\(23\) −6.13582 −1.27941 −0.639703 0.768622i \(-0.720943\pi\)
−0.639703 + 0.768622i \(0.720943\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.793401 −0.158680
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 5.03739 0.951978
\(29\) 3.21614 0.597222 0.298611 0.954375i \(-0.403477\pi\)
0.298611 + 0.954375i \(0.403477\pi\)
\(30\) −2.05100 −0.374460
\(31\) 8.30084 1.49087 0.745437 0.666576i \(-0.232241\pi\)
0.745437 + 0.666576i \(0.232241\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.25636 −0.566860
\(34\) −0.479851 −0.0822938
\(35\) −10.3317 −1.74637
\(36\) 1.00000 0.166667
\(37\) −3.88622 −0.638891 −0.319445 0.947605i \(-0.603497\pi\)
−0.319445 + 0.947605i \(0.603497\pi\)
\(38\) −5.58203 −0.905525
\(39\) −1.00000 −0.160128
\(40\) 2.05100 0.324291
\(41\) 5.27024 0.823073 0.411536 0.911393i \(-0.364992\pi\)
0.411536 + 0.911393i \(0.364992\pi\)
\(42\) 5.03739 0.777287
\(43\) 3.12359 0.476343 0.238171 0.971223i \(-0.423452\pi\)
0.238171 + 0.971223i \(0.423452\pi\)
\(44\) 3.25636 0.490915
\(45\) −2.05100 −0.305745
\(46\) 6.13582 0.904677
\(47\) 3.55585 0.518674 0.259337 0.965787i \(-0.416496\pi\)
0.259337 + 0.965787i \(0.416496\pi\)
\(48\) −1.00000 −0.144338
\(49\) 18.3753 2.62505
\(50\) 0.793401 0.112204
\(51\) −0.479851 −0.0671926
\(52\) 1.00000 0.138675
\(53\) −6.76178 −0.928802 −0.464401 0.885625i \(-0.653730\pi\)
−0.464401 + 0.885625i \(0.653730\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.67880 −0.900570
\(56\) −5.03739 −0.673150
\(57\) −5.58203 −0.739358
\(58\) −3.21614 −0.422300
\(59\) 13.1023 1.70577 0.852887 0.522096i \(-0.174850\pi\)
0.852887 + 0.522096i \(0.174850\pi\)
\(60\) 2.05100 0.264783
\(61\) 6.84529 0.876450 0.438225 0.898865i \(-0.355607\pi\)
0.438225 + 0.898865i \(0.355607\pi\)
\(62\) −8.30084 −1.05421
\(63\) 5.03739 0.634652
\(64\) 1.00000 0.125000
\(65\) −2.05100 −0.254395
\(66\) 3.25636 0.400831
\(67\) −3.51638 −0.429594 −0.214797 0.976659i \(-0.568909\pi\)
−0.214797 + 0.976659i \(0.568909\pi\)
\(68\) 0.479851 0.0581905
\(69\) 6.13582 0.738666
\(70\) 10.3317 1.23487
\(71\) 0.616815 0.0732025 0.0366013 0.999330i \(-0.488347\pi\)
0.0366013 + 0.999330i \(0.488347\pi\)
\(72\) −1.00000 −0.117851
\(73\) 13.2386 1.54946 0.774731 0.632292i \(-0.217885\pi\)
0.774731 + 0.632292i \(0.217885\pi\)
\(74\) 3.88622 0.451764
\(75\) 0.793401 0.0916140
\(76\) 5.58203 0.640303
\(77\) 16.4036 1.86936
\(78\) 1.00000 0.113228
\(79\) 2.05078 0.230730 0.115365 0.993323i \(-0.463196\pi\)
0.115365 + 0.993323i \(0.463196\pi\)
\(80\) −2.05100 −0.229309
\(81\) 1.00000 0.111111
\(82\) −5.27024 −0.582000
\(83\) −2.18812 −0.240178 −0.120089 0.992763i \(-0.538318\pi\)
−0.120089 + 0.992763i \(0.538318\pi\)
\(84\) −5.03739 −0.549625
\(85\) −0.984174 −0.106749
\(86\) −3.12359 −0.336825
\(87\) −3.21614 −0.344806
\(88\) −3.25636 −0.347130
\(89\) −3.35651 −0.355789 −0.177895 0.984050i \(-0.556929\pi\)
−0.177895 + 0.984050i \(0.556929\pi\)
\(90\) 2.05100 0.216194
\(91\) 5.03739 0.528062
\(92\) −6.13582 −0.639703
\(93\) −8.30084 −0.860757
\(94\) −3.55585 −0.366758
\(95\) −11.4487 −1.17462
\(96\) 1.00000 0.102062
\(97\) −1.04552 −0.106156 −0.0530780 0.998590i \(-0.516903\pi\)
−0.0530780 + 0.998590i \(0.516903\pi\)
\(98\) −18.3753 −1.85619
\(99\) 3.25636 0.327277
\(100\) −0.793401 −0.0793401
\(101\) 6.17717 0.614651 0.307326 0.951604i \(-0.400566\pi\)
0.307326 + 0.951604i \(0.400566\pi\)
\(102\) 0.479851 0.0475123
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 10.3317 1.00827
\(106\) 6.76178 0.656762
\(107\) −18.0835 −1.74819 −0.874097 0.485752i \(-0.838546\pi\)
−0.874097 + 0.485752i \(0.838546\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.59970 −0.153224 −0.0766120 0.997061i \(-0.524410\pi\)
−0.0766120 + 0.997061i \(0.524410\pi\)
\(110\) 6.67880 0.636799
\(111\) 3.88622 0.368864
\(112\) 5.03739 0.475989
\(113\) 2.73914 0.257676 0.128838 0.991666i \(-0.458875\pi\)
0.128838 + 0.991666i \(0.458875\pi\)
\(114\) 5.58203 0.522805
\(115\) 12.5846 1.17352
\(116\) 3.21614 0.298611
\(117\) 1.00000 0.0924500
\(118\) −13.1023 −1.20616
\(119\) 2.41720 0.221584
\(120\) −2.05100 −0.187230
\(121\) −0.396088 −0.0360080
\(122\) −6.84529 −0.619744
\(123\) −5.27024 −0.475201
\(124\) 8.30084 0.745437
\(125\) 11.8823 1.06278
\(126\) −5.03739 −0.448767
\(127\) −11.9402 −1.05952 −0.529762 0.848146i \(-0.677719\pi\)
−0.529762 + 0.848146i \(0.677719\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −3.12359 −0.275017
\(130\) 2.05100 0.179885
\(131\) 3.36895 0.294347 0.147173 0.989111i \(-0.452982\pi\)
0.147173 + 0.989111i \(0.452982\pi\)
\(132\) −3.25636 −0.283430
\(133\) 28.1189 2.43822
\(134\) 3.51638 0.303769
\(135\) 2.05100 0.176522
\(136\) −0.479851 −0.0411469
\(137\) −1.91274 −0.163416 −0.0817082 0.996656i \(-0.526038\pi\)
−0.0817082 + 0.996656i \(0.526038\pi\)
\(138\) −6.13582 −0.522315
\(139\) −20.1841 −1.71199 −0.855996 0.516983i \(-0.827055\pi\)
−0.855996 + 0.516983i \(0.827055\pi\)
\(140\) −10.3317 −0.873187
\(141\) −3.55585 −0.299457
\(142\) −0.616815 −0.0517620
\(143\) 3.25636 0.272311
\(144\) 1.00000 0.0833333
\(145\) −6.59630 −0.547793
\(146\) −13.2386 −1.09563
\(147\) −18.3753 −1.51557
\(148\) −3.88622 −0.319445
\(149\) −0.778189 −0.0637517 −0.0318759 0.999492i \(-0.510148\pi\)
−0.0318759 + 0.999492i \(0.510148\pi\)
\(150\) −0.793401 −0.0647809
\(151\) 20.3436 1.65554 0.827771 0.561066i \(-0.189608\pi\)
0.827771 + 0.561066i \(0.189608\pi\)
\(152\) −5.58203 −0.452763
\(153\) 0.479851 0.0387937
\(154\) −16.4036 −1.32184
\(155\) −17.0250 −1.36748
\(156\) −1.00000 −0.0800641
\(157\) −21.3836 −1.70660 −0.853300 0.521420i \(-0.825403\pi\)
−0.853300 + 0.521420i \(0.825403\pi\)
\(158\) −2.05078 −0.163151
\(159\) 6.76178 0.536244
\(160\) 2.05100 0.162146
\(161\) −30.9085 −2.43593
\(162\) −1.00000 −0.0785674
\(163\) −14.8949 −1.16666 −0.583328 0.812236i \(-0.698250\pi\)
−0.583328 + 0.812236i \(0.698250\pi\)
\(164\) 5.27024 0.411536
\(165\) 6.67880 0.519944
\(166\) 2.18812 0.169831
\(167\) 6.70885 0.519146 0.259573 0.965723i \(-0.416418\pi\)
0.259573 + 0.965723i \(0.416418\pi\)
\(168\) 5.03739 0.388643
\(169\) 1.00000 0.0769231
\(170\) 0.984174 0.0754827
\(171\) 5.58203 0.426869
\(172\) 3.12359 0.238171
\(173\) −14.8394 −1.12822 −0.564110 0.825699i \(-0.690781\pi\)
−0.564110 + 0.825699i \(0.690781\pi\)
\(174\) 3.21614 0.243815
\(175\) −3.99667 −0.302120
\(176\) 3.25636 0.245458
\(177\) −13.1023 −0.984829
\(178\) 3.35651 0.251581
\(179\) 8.75510 0.654387 0.327193 0.944957i \(-0.393897\pi\)
0.327193 + 0.944957i \(0.393897\pi\)
\(180\) −2.05100 −0.152872
\(181\) 26.8429 1.99522 0.997610 0.0691034i \(-0.0220138\pi\)
0.997610 + 0.0691034i \(0.0220138\pi\)
\(182\) −5.03739 −0.373396
\(183\) −6.84529 −0.506019
\(184\) 6.13582 0.452338
\(185\) 7.97064 0.586013
\(186\) 8.30084 0.608647
\(187\) 1.56257 0.114266
\(188\) 3.55585 0.259337
\(189\) −5.03739 −0.366416
\(190\) 11.4487 0.830579
\(191\) −18.2739 −1.32225 −0.661125 0.750275i \(-0.729921\pi\)
−0.661125 + 0.750275i \(0.729921\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.22242 0.663844 0.331922 0.943307i \(-0.392303\pi\)
0.331922 + 0.943307i \(0.392303\pi\)
\(194\) 1.04552 0.0750637
\(195\) 2.05100 0.146875
\(196\) 18.3753 1.31252
\(197\) 21.2145 1.51147 0.755734 0.654878i \(-0.227280\pi\)
0.755734 + 0.654878i \(0.227280\pi\)
\(198\) −3.25636 −0.231420
\(199\) −21.4058 −1.51742 −0.758708 0.651431i \(-0.774169\pi\)
−0.758708 + 0.651431i \(0.774169\pi\)
\(200\) 0.793401 0.0561019
\(201\) 3.51638 0.248026
\(202\) −6.17717 −0.434624
\(203\) 16.2009 1.13708
\(204\) −0.479851 −0.0335963
\(205\) −10.8093 −0.754951
\(206\) 1.00000 0.0696733
\(207\) −6.13582 −0.426469
\(208\) 1.00000 0.0693375
\(209\) 18.1771 1.25734
\(210\) −10.3317 −0.712954
\(211\) 3.59640 0.247586 0.123793 0.992308i \(-0.460494\pi\)
0.123793 + 0.992308i \(0.460494\pi\)
\(212\) −6.76178 −0.464401
\(213\) −0.616815 −0.0422635
\(214\) 18.0835 1.23616
\(215\) −6.40648 −0.436918
\(216\) 1.00000 0.0680414
\(217\) 41.8146 2.83856
\(218\) 1.59970 0.108346
\(219\) −13.2386 −0.894582
\(220\) −6.67880 −0.450285
\(221\) 0.479851 0.0322783
\(222\) −3.88622 −0.260826
\(223\) −27.5119 −1.84233 −0.921167 0.389168i \(-0.872763\pi\)
−0.921167 + 0.389168i \(0.872763\pi\)
\(224\) −5.03739 −0.336575
\(225\) −0.793401 −0.0528934
\(226\) −2.73914 −0.182205
\(227\) 0.778045 0.0516407 0.0258203 0.999667i \(-0.491780\pi\)
0.0258203 + 0.999667i \(0.491780\pi\)
\(228\) −5.58203 −0.369679
\(229\) 8.61245 0.569126 0.284563 0.958657i \(-0.408151\pi\)
0.284563 + 0.958657i \(0.408151\pi\)
\(230\) −12.5846 −0.829801
\(231\) −16.4036 −1.07928
\(232\) −3.21614 −0.211150
\(233\) −18.5994 −1.21849 −0.609243 0.792984i \(-0.708527\pi\)
−0.609243 + 0.792984i \(0.708527\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −7.29305 −0.475746
\(236\) 13.1023 0.852887
\(237\) −2.05078 −0.133212
\(238\) −2.41720 −0.156684
\(239\) 6.01961 0.389376 0.194688 0.980865i \(-0.437631\pi\)
0.194688 + 0.980865i \(0.437631\pi\)
\(240\) 2.05100 0.132391
\(241\) 8.74673 0.563426 0.281713 0.959499i \(-0.409097\pi\)
0.281713 + 0.959499i \(0.409097\pi\)
\(242\) 0.396088 0.0254615
\(243\) −1.00000 −0.0641500
\(244\) 6.84529 0.438225
\(245\) −37.6878 −2.40778
\(246\) 5.27024 0.336018
\(247\) 5.58203 0.355176
\(248\) −8.30084 −0.527104
\(249\) 2.18812 0.138667
\(250\) −11.8823 −0.751500
\(251\) −2.31076 −0.145854 −0.0729268 0.997337i \(-0.523234\pi\)
−0.0729268 + 0.997337i \(0.523234\pi\)
\(252\) 5.03739 0.317326
\(253\) −19.9805 −1.25616
\(254\) 11.9402 0.749197
\(255\) 0.984174 0.0616314
\(256\) 1.00000 0.0625000
\(257\) −30.3838 −1.89529 −0.947645 0.319326i \(-0.896544\pi\)
−0.947645 + 0.319326i \(0.896544\pi\)
\(258\) 3.12359 0.194466
\(259\) −19.5764 −1.21642
\(260\) −2.05100 −0.127198
\(261\) 3.21614 0.199074
\(262\) −3.36895 −0.208135
\(263\) 15.0290 0.926730 0.463365 0.886168i \(-0.346642\pi\)
0.463365 + 0.886168i \(0.346642\pi\)
\(264\) 3.25636 0.200415
\(265\) 13.8684 0.851929
\(266\) −28.1189 −1.72408
\(267\) 3.35651 0.205415
\(268\) −3.51638 −0.214797
\(269\) −19.3049 −1.17704 −0.588519 0.808483i \(-0.700289\pi\)
−0.588519 + 0.808483i \(0.700289\pi\)
\(270\) −2.05100 −0.124820
\(271\) 5.46064 0.331710 0.165855 0.986150i \(-0.446962\pi\)
0.165855 + 0.986150i \(0.446962\pi\)
\(272\) 0.479851 0.0290952
\(273\) −5.03739 −0.304877
\(274\) 1.91274 0.115553
\(275\) −2.58360 −0.155797
\(276\) 6.13582 0.369333
\(277\) −18.3298 −1.10133 −0.550665 0.834726i \(-0.685626\pi\)
−0.550665 + 0.834726i \(0.685626\pi\)
\(278\) 20.1841 1.21056
\(279\) 8.30084 0.496958
\(280\) 10.3317 0.617437
\(281\) 7.86816 0.469375 0.234688 0.972071i \(-0.424593\pi\)
0.234688 + 0.972071i \(0.424593\pi\)
\(282\) 3.55585 0.211748
\(283\) 14.4497 0.858943 0.429472 0.903080i \(-0.358700\pi\)
0.429472 + 0.903080i \(0.358700\pi\)
\(284\) 0.616815 0.0366013
\(285\) 11.4487 0.678165
\(286\) −3.25636 −0.192553
\(287\) 26.5483 1.56709
\(288\) −1.00000 −0.0589256
\(289\) −16.7697 −0.986455
\(290\) 6.59630 0.387348
\(291\) 1.04552 0.0612892
\(292\) 13.2386 0.774731
\(293\) 7.67257 0.448236 0.224118 0.974562i \(-0.428050\pi\)
0.224118 + 0.974562i \(0.428050\pi\)
\(294\) 18.3753 1.07167
\(295\) −26.8728 −1.56460
\(296\) 3.88622 0.225882
\(297\) −3.25636 −0.188953
\(298\) 0.778189 0.0450793
\(299\) −6.13582 −0.354843
\(300\) 0.793401 0.0458070
\(301\) 15.7347 0.906936
\(302\) −20.3436 −1.17065
\(303\) −6.17717 −0.354869
\(304\) 5.58203 0.320151
\(305\) −14.0397 −0.803910
\(306\) −0.479851 −0.0274313
\(307\) 8.31383 0.474495 0.237248 0.971449i \(-0.423755\pi\)
0.237248 + 0.971449i \(0.423755\pi\)
\(308\) 16.4036 0.934681
\(309\) 1.00000 0.0568880
\(310\) 17.0250 0.966956
\(311\) −18.2570 −1.03526 −0.517630 0.855605i \(-0.673186\pi\)
−0.517630 + 0.855605i \(0.673186\pi\)
\(312\) 1.00000 0.0566139
\(313\) 28.5611 1.61437 0.807186 0.590298i \(-0.200990\pi\)
0.807186 + 0.590298i \(0.200990\pi\)
\(314\) 21.3836 1.20675
\(315\) −10.3317 −0.582125
\(316\) 2.05078 0.115365
\(317\) 18.8153 1.05677 0.528387 0.849004i \(-0.322797\pi\)
0.528387 + 0.849004i \(0.322797\pi\)
\(318\) −6.76178 −0.379182
\(319\) 10.4729 0.586371
\(320\) −2.05100 −0.114654
\(321\) 18.0835 1.00932
\(322\) 30.9085 1.72246
\(323\) 2.67854 0.149038
\(324\) 1.00000 0.0555556
\(325\) −0.793401 −0.0440100
\(326\) 14.8949 0.824951
\(327\) 1.59970 0.0884639
\(328\) −5.27024 −0.291000
\(329\) 17.9122 0.987533
\(330\) −6.67880 −0.367656
\(331\) −23.2918 −1.28023 −0.640115 0.768279i \(-0.721113\pi\)
−0.640115 + 0.768279i \(0.721113\pi\)
\(332\) −2.18812 −0.120089
\(333\) −3.88622 −0.212964
\(334\) −6.70885 −0.367092
\(335\) 7.21210 0.394039
\(336\) −5.03739 −0.274812
\(337\) 31.2626 1.70298 0.851490 0.524371i \(-0.175699\pi\)
0.851490 + 0.524371i \(0.175699\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −2.73914 −0.148769
\(340\) −0.984174 −0.0533743
\(341\) 27.0305 1.46379
\(342\) −5.58203 −0.301842
\(343\) 57.3020 3.09401
\(344\) −3.12359 −0.168413
\(345\) −12.5846 −0.677530
\(346\) 14.8394 0.797772
\(347\) −26.8471 −1.44123 −0.720613 0.693338i \(-0.756139\pi\)
−0.720613 + 0.693338i \(0.756139\pi\)
\(348\) −3.21614 −0.172403
\(349\) 8.81086 0.471635 0.235817 0.971797i \(-0.424223\pi\)
0.235817 + 0.971797i \(0.424223\pi\)
\(350\) 3.99667 0.213631
\(351\) −1.00000 −0.0533761
\(352\) −3.25636 −0.173565
\(353\) −18.6641 −0.993392 −0.496696 0.867925i \(-0.665454\pi\)
−0.496696 + 0.867925i \(0.665454\pi\)
\(354\) 13.1023 0.696379
\(355\) −1.26509 −0.0671439
\(356\) −3.35651 −0.177895
\(357\) −2.41720 −0.127932
\(358\) −8.75510 −0.462721
\(359\) 34.4613 1.81880 0.909398 0.415926i \(-0.136543\pi\)
0.909398 + 0.415926i \(0.136543\pi\)
\(360\) 2.05100 0.108097
\(361\) 12.1591 0.639951
\(362\) −26.8429 −1.41083
\(363\) 0.396088 0.0207892
\(364\) 5.03739 0.264031
\(365\) −27.1524 −1.42122
\(366\) 6.84529 0.357809
\(367\) 34.7488 1.81387 0.906936 0.421268i \(-0.138415\pi\)
0.906936 + 0.421268i \(0.138415\pi\)
\(368\) −6.13582 −0.319852
\(369\) 5.27024 0.274358
\(370\) −7.97064 −0.414374
\(371\) −34.0617 −1.76840
\(372\) −8.30084 −0.430378
\(373\) 16.4025 0.849291 0.424645 0.905360i \(-0.360399\pi\)
0.424645 + 0.905360i \(0.360399\pi\)
\(374\) −1.56257 −0.0807986
\(375\) −11.8823 −0.613597
\(376\) −3.55585 −0.183379
\(377\) 3.21614 0.165640
\(378\) 5.03739 0.259096
\(379\) −21.1040 −1.08404 −0.542019 0.840366i \(-0.682340\pi\)
−0.542019 + 0.840366i \(0.682340\pi\)
\(380\) −11.4487 −0.587308
\(381\) 11.9402 0.611717
\(382\) 18.2739 0.934973
\(383\) 26.8489 1.37191 0.685957 0.727642i \(-0.259384\pi\)
0.685957 + 0.727642i \(0.259384\pi\)
\(384\) 1.00000 0.0510310
\(385\) −33.6437 −1.71464
\(386\) −9.22242 −0.469409
\(387\) 3.12359 0.158781
\(388\) −1.04552 −0.0530780
\(389\) 0.453851 0.0230111 0.0115056 0.999934i \(-0.496338\pi\)
0.0115056 + 0.999934i \(0.496338\pi\)
\(390\) −2.05100 −0.103856
\(391\) −2.94428 −0.148899
\(392\) −18.3753 −0.928094
\(393\) −3.36895 −0.169941
\(394\) −21.2145 −1.06877
\(395\) −4.20614 −0.211634
\(396\) 3.25636 0.163638
\(397\) −10.5105 −0.527507 −0.263754 0.964590i \(-0.584961\pi\)
−0.263754 + 0.964590i \(0.584961\pi\)
\(398\) 21.4058 1.07298
\(399\) −28.1189 −1.40770
\(400\) −0.793401 −0.0396700
\(401\) 19.8785 0.992685 0.496342 0.868127i \(-0.334676\pi\)
0.496342 + 0.868127i \(0.334676\pi\)
\(402\) −3.51638 −0.175381
\(403\) 8.30084 0.413494
\(404\) 6.17717 0.307326
\(405\) −2.05100 −0.101915
\(406\) −16.2009 −0.804040
\(407\) −12.6550 −0.627283
\(408\) 0.479851 0.0237562
\(409\) −11.4709 −0.567198 −0.283599 0.958943i \(-0.591528\pi\)
−0.283599 + 0.958943i \(0.591528\pi\)
\(410\) 10.8093 0.533831
\(411\) 1.91274 0.0943485
\(412\) −1.00000 −0.0492665
\(413\) 66.0014 3.24772
\(414\) 6.13582 0.301559
\(415\) 4.48784 0.220300
\(416\) −1.00000 −0.0490290
\(417\) 20.1841 0.988419
\(418\) −18.1771 −0.889072
\(419\) −34.7936 −1.69978 −0.849889 0.526961i \(-0.823331\pi\)
−0.849889 + 0.526961i \(0.823331\pi\)
\(420\) 10.3317 0.504135
\(421\) −6.18218 −0.301301 −0.150650 0.988587i \(-0.548137\pi\)
−0.150650 + 0.988587i \(0.548137\pi\)
\(422\) −3.59640 −0.175070
\(423\) 3.55585 0.172891
\(424\) 6.76178 0.328381
\(425\) −0.380714 −0.0184674
\(426\) 0.616815 0.0298848
\(427\) 34.4824 1.66872
\(428\) −18.0835 −0.874097
\(429\) −3.25636 −0.157219
\(430\) 6.40648 0.308948
\(431\) 2.84511 0.137044 0.0685220 0.997650i \(-0.478172\pi\)
0.0685220 + 0.997650i \(0.478172\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.12180 −0.246138 −0.123069 0.992398i \(-0.539274\pi\)
−0.123069 + 0.992398i \(0.539274\pi\)
\(434\) −41.8146 −2.00716
\(435\) 6.59630 0.316268
\(436\) −1.59970 −0.0766120
\(437\) −34.2503 −1.63842
\(438\) 13.2386 0.632565
\(439\) −8.67259 −0.413920 −0.206960 0.978349i \(-0.566357\pi\)
−0.206960 + 0.978349i \(0.566357\pi\)
\(440\) 6.67880 0.318399
\(441\) 18.3753 0.875015
\(442\) −0.479851 −0.0228242
\(443\) −5.13670 −0.244052 −0.122026 0.992527i \(-0.538939\pi\)
−0.122026 + 0.992527i \(0.538939\pi\)
\(444\) 3.88622 0.184432
\(445\) 6.88420 0.326342
\(446\) 27.5119 1.30273
\(447\) 0.778189 0.0368071
\(448\) 5.03739 0.237994
\(449\) 0.593133 0.0279917 0.0139958 0.999902i \(-0.495545\pi\)
0.0139958 + 0.999902i \(0.495545\pi\)
\(450\) 0.793401 0.0374013
\(451\) 17.1618 0.808118
\(452\) 2.73914 0.128838
\(453\) −20.3436 −0.955828
\(454\) −0.778045 −0.0365155
\(455\) −10.3317 −0.484357
\(456\) 5.58203 0.261403
\(457\) −21.1234 −0.988113 −0.494056 0.869430i \(-0.664486\pi\)
−0.494056 + 0.869430i \(0.664486\pi\)
\(458\) −8.61245 −0.402433
\(459\) −0.479851 −0.0223975
\(460\) 12.5846 0.586758
\(461\) 21.7463 1.01283 0.506413 0.862291i \(-0.330971\pi\)
0.506413 + 0.862291i \(0.330971\pi\)
\(462\) 16.4036 0.763164
\(463\) 31.9048 1.48274 0.741371 0.671096i \(-0.234176\pi\)
0.741371 + 0.671096i \(0.234176\pi\)
\(464\) 3.21614 0.149305
\(465\) 17.0250 0.789516
\(466\) 18.5994 0.861599
\(467\) 0.266545 0.0123342 0.00616711 0.999981i \(-0.498037\pi\)
0.00616711 + 0.999981i \(0.498037\pi\)
\(468\) 1.00000 0.0462250
\(469\) −17.7134 −0.817929
\(470\) 7.29305 0.336403
\(471\) 21.3836 0.985306
\(472\) −13.1023 −0.603082
\(473\) 10.1715 0.467688
\(474\) 2.05078 0.0941952
\(475\) −4.42879 −0.203207
\(476\) 2.41720 0.110792
\(477\) −6.76178 −0.309601
\(478\) −6.01961 −0.275331
\(479\) −21.6179 −0.987748 −0.493874 0.869533i \(-0.664420\pi\)
−0.493874 + 0.869533i \(0.664420\pi\)
\(480\) −2.05100 −0.0936149
\(481\) −3.88622 −0.177196
\(482\) −8.74673 −0.398403
\(483\) 30.9085 1.40639
\(484\) −0.396088 −0.0180040
\(485\) 2.14435 0.0973701
\(486\) 1.00000 0.0453609
\(487\) −15.2920 −0.692946 −0.346473 0.938060i \(-0.612621\pi\)
−0.346473 + 0.938060i \(0.612621\pi\)
\(488\) −6.84529 −0.309872
\(489\) 14.8949 0.673570
\(490\) 37.6878 1.70256
\(491\) 29.3696 1.32543 0.662715 0.748871i \(-0.269404\pi\)
0.662715 + 0.748871i \(0.269404\pi\)
\(492\) −5.27024 −0.237601
\(493\) 1.54327 0.0695053
\(494\) −5.58203 −0.251147
\(495\) −6.67880 −0.300190
\(496\) 8.30084 0.372719
\(497\) 3.10714 0.139374
\(498\) −2.18812 −0.0980522
\(499\) 19.4220 0.869450 0.434725 0.900563i \(-0.356846\pi\)
0.434725 + 0.900563i \(0.356846\pi\)
\(500\) 11.8823 0.531391
\(501\) −6.70885 −0.299729
\(502\) 2.31076 0.103134
\(503\) 12.2856 0.547789 0.273895 0.961760i \(-0.411688\pi\)
0.273895 + 0.961760i \(0.411688\pi\)
\(504\) −5.03739 −0.224383
\(505\) −12.6694 −0.563780
\(506\) 19.9805 0.888240
\(507\) −1.00000 −0.0444116
\(508\) −11.9402 −0.529762
\(509\) −3.08185 −0.136600 −0.0683002 0.997665i \(-0.521758\pi\)
−0.0683002 + 0.997665i \(0.521758\pi\)
\(510\) −0.984174 −0.0435800
\(511\) 66.6880 2.95010
\(512\) −1.00000 −0.0441942
\(513\) −5.58203 −0.246453
\(514\) 30.3838 1.34017
\(515\) 2.05100 0.0903778
\(516\) −3.12359 −0.137508
\(517\) 11.5792 0.509250
\(518\) 19.5764 0.860139
\(519\) 14.8394 0.651378
\(520\) 2.05100 0.0899423
\(521\) 13.3801 0.586191 0.293096 0.956083i \(-0.405315\pi\)
0.293096 + 0.956083i \(0.405315\pi\)
\(522\) −3.21614 −0.140767
\(523\) −42.4084 −1.85439 −0.927194 0.374581i \(-0.877787\pi\)
−0.927194 + 0.374581i \(0.877787\pi\)
\(524\) 3.36895 0.147173
\(525\) 3.99667 0.174429
\(526\) −15.0290 −0.655297
\(527\) 3.98316 0.173509
\(528\) −3.25636 −0.141715
\(529\) 14.6483 0.636881
\(530\) −13.8684 −0.602405
\(531\) 13.1023 0.568591
\(532\) 28.1189 1.21911
\(533\) 5.27024 0.228279
\(534\) −3.35651 −0.145250
\(535\) 37.0891 1.60350
\(536\) 3.51638 0.151885
\(537\) −8.75510 −0.377810
\(538\) 19.3049 0.832292
\(539\) 59.8368 2.57735
\(540\) 2.05100 0.0882610
\(541\) 8.14551 0.350203 0.175101 0.984550i \(-0.443975\pi\)
0.175101 + 0.984550i \(0.443975\pi\)
\(542\) −5.46064 −0.234555
\(543\) −26.8429 −1.15194
\(544\) −0.479851 −0.0205734
\(545\) 3.28099 0.140542
\(546\) 5.03739 0.215580
\(547\) −10.4160 −0.445354 −0.222677 0.974892i \(-0.571480\pi\)
−0.222677 + 0.974892i \(0.571480\pi\)
\(548\) −1.91274 −0.0817082
\(549\) 6.84529 0.292150
\(550\) 2.58360 0.110165
\(551\) 17.9526 0.764806
\(552\) −6.13582 −0.261158
\(553\) 10.3306 0.439300
\(554\) 18.3298 0.778758
\(555\) −7.97064 −0.338335
\(556\) −20.1841 −0.855996
\(557\) −11.9603 −0.506774 −0.253387 0.967365i \(-0.581545\pi\)
−0.253387 + 0.967365i \(0.581545\pi\)
\(558\) −8.30084 −0.351402
\(559\) 3.12359 0.132114
\(560\) −10.3317 −0.436594
\(561\) −1.56257 −0.0659718
\(562\) −7.86816 −0.331898
\(563\) 35.2621 1.48612 0.743060 0.669224i \(-0.233373\pi\)
0.743060 + 0.669224i \(0.233373\pi\)
\(564\) −3.55585 −0.149728
\(565\) −5.61797 −0.236350
\(566\) −14.4497 −0.607365
\(567\) 5.03739 0.211551
\(568\) −0.616815 −0.0258810
\(569\) −37.9481 −1.59087 −0.795433 0.606042i \(-0.792756\pi\)
−0.795433 + 0.606042i \(0.792756\pi\)
\(570\) −11.4487 −0.479535
\(571\) −13.8437 −0.579342 −0.289671 0.957126i \(-0.593546\pi\)
−0.289671 + 0.957126i \(0.593546\pi\)
\(572\) 3.25636 0.136155
\(573\) 18.2739 0.763402
\(574\) −26.5483 −1.10810
\(575\) 4.86816 0.203016
\(576\) 1.00000 0.0416667
\(577\) 45.6191 1.89915 0.949574 0.313543i \(-0.101516\pi\)
0.949574 + 0.313543i \(0.101516\pi\)
\(578\) 16.7697 0.697529
\(579\) −9.22242 −0.383271
\(580\) −6.59630 −0.273896
\(581\) −11.0224 −0.457288
\(582\) −1.04552 −0.0433380
\(583\) −22.0188 −0.911926
\(584\) −13.2386 −0.547817
\(585\) −2.05100 −0.0847984
\(586\) −7.67257 −0.316951
\(587\) −1.92423 −0.0794217 −0.0397108 0.999211i \(-0.512644\pi\)
−0.0397108 + 0.999211i \(0.512644\pi\)
\(588\) −18.3753 −0.757785
\(589\) 46.3355 1.90922
\(590\) 26.8728 1.10634
\(591\) −21.2145 −0.872647
\(592\) −3.88622 −0.159723
\(593\) 20.6977 0.849951 0.424975 0.905205i \(-0.360283\pi\)
0.424975 + 0.905205i \(0.360283\pi\)
\(594\) 3.25636 0.133610
\(595\) −4.95767 −0.203245
\(596\) −0.778189 −0.0318759
\(597\) 21.4058 0.876081
\(598\) 6.13582 0.250912
\(599\) 16.4988 0.674124 0.337062 0.941482i \(-0.390567\pi\)
0.337062 + 0.941482i \(0.390567\pi\)
\(600\) −0.793401 −0.0323905
\(601\) 8.49981 0.346715 0.173357 0.984859i \(-0.444538\pi\)
0.173357 + 0.984859i \(0.444538\pi\)
\(602\) −15.7347 −0.641300
\(603\) −3.51638 −0.143198
\(604\) 20.3436 0.827771
\(605\) 0.812376 0.0330278
\(606\) 6.17717 0.250930
\(607\) 26.9659 1.09451 0.547256 0.836965i \(-0.315673\pi\)
0.547256 + 0.836965i \(0.315673\pi\)
\(608\) −5.58203 −0.226381
\(609\) −16.2009 −0.656496
\(610\) 14.0397 0.568451
\(611\) 3.55585 0.143854
\(612\) 0.479851 0.0193968
\(613\) −22.8824 −0.924213 −0.462107 0.886824i \(-0.652906\pi\)
−0.462107 + 0.886824i \(0.652906\pi\)
\(614\) −8.31383 −0.335519
\(615\) 10.8093 0.435871
\(616\) −16.4036 −0.660919
\(617\) 40.8773 1.64566 0.822829 0.568289i \(-0.192394\pi\)
0.822829 + 0.568289i \(0.192394\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −15.2553 −0.613161 −0.306580 0.951845i \(-0.599185\pi\)
−0.306580 + 0.951845i \(0.599185\pi\)
\(620\) −17.0250 −0.683741
\(621\) 6.13582 0.246222
\(622\) 18.2570 0.732039
\(623\) −16.9081 −0.677407
\(624\) −1.00000 −0.0400320
\(625\) −20.4035 −0.816140
\(626\) −28.5611 −1.14153
\(627\) −18.1771 −0.725925
\(628\) −21.3836 −0.853300
\(629\) −1.86481 −0.0743547
\(630\) 10.3317 0.411624
\(631\) −31.9766 −1.27297 −0.636485 0.771289i \(-0.719612\pi\)
−0.636485 + 0.771289i \(0.719612\pi\)
\(632\) −2.05078 −0.0815755
\(633\) −3.59640 −0.142944
\(634\) −18.8153 −0.747252
\(635\) 24.4894 0.971833
\(636\) 6.76178 0.268122
\(637\) 18.3753 0.728057
\(638\) −10.4729 −0.414627
\(639\) 0.616815 0.0244008
\(640\) 2.05100 0.0810729
\(641\) 30.5631 1.20717 0.603584 0.797299i \(-0.293739\pi\)
0.603584 + 0.797299i \(0.293739\pi\)
\(642\) −18.0835 −0.713697
\(643\) −34.1633 −1.34727 −0.673634 0.739066i \(-0.735267\pi\)
−0.673634 + 0.739066i \(0.735267\pi\)
\(644\) −30.9085 −1.21797
\(645\) 6.40648 0.252255
\(646\) −2.67854 −0.105386
\(647\) 35.6220 1.40045 0.700223 0.713924i \(-0.253084\pi\)
0.700223 + 0.713924i \(0.253084\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 42.6659 1.67478
\(650\) 0.793401 0.0311197
\(651\) −41.8146 −1.63884
\(652\) −14.8949 −0.583328
\(653\) −21.1952 −0.829432 −0.414716 0.909951i \(-0.636119\pi\)
−0.414716 + 0.909951i \(0.636119\pi\)
\(654\) −1.59970 −0.0625534
\(655\) −6.90972 −0.269985
\(656\) 5.27024 0.205768
\(657\) 13.2386 0.516487
\(658\) −17.9122 −0.698291
\(659\) 0.843335 0.0328516 0.0164258 0.999865i \(-0.494771\pi\)
0.0164258 + 0.999865i \(0.494771\pi\)
\(660\) 6.67880 0.259972
\(661\) −0.0808243 −0.00314370 −0.00157185 0.999999i \(-0.500500\pi\)
−0.00157185 + 0.999999i \(0.500500\pi\)
\(662\) 23.2918 0.905260
\(663\) −0.479851 −0.0186359
\(664\) 2.18812 0.0849157
\(665\) −57.6718 −2.23642
\(666\) 3.88622 0.150588
\(667\) −19.7336 −0.764089
\(668\) 6.70885 0.259573
\(669\) 27.5119 1.06367
\(670\) −7.21210 −0.278628
\(671\) 22.2908 0.860526
\(672\) 5.03739 0.194322
\(673\) 11.5783 0.446310 0.223155 0.974783i \(-0.428364\pi\)
0.223155 + 0.974783i \(0.428364\pi\)
\(674\) −31.2626 −1.20419
\(675\) 0.793401 0.0305380
\(676\) 1.00000 0.0384615
\(677\) 8.07324 0.310280 0.155140 0.987893i \(-0.450417\pi\)
0.155140 + 0.987893i \(0.450417\pi\)
\(678\) 2.73914 0.105196
\(679\) −5.26668 −0.202116
\(680\) 0.984174 0.0377414
\(681\) −0.778045 −0.0298147
\(682\) −27.0305 −1.03505
\(683\) −46.8207 −1.79154 −0.895772 0.444513i \(-0.853377\pi\)
−0.895772 + 0.444513i \(0.853377\pi\)
\(684\) 5.58203 0.213434
\(685\) 3.92303 0.149891
\(686\) −57.3020 −2.18780
\(687\) −8.61245 −0.328585
\(688\) 3.12359 0.119086
\(689\) −6.76178 −0.257603
\(690\) 12.5846 0.479086
\(691\) 0.322636 0.0122737 0.00613683 0.999981i \(-0.498047\pi\)
0.00613683 + 0.999981i \(0.498047\pi\)
\(692\) −14.8394 −0.564110
\(693\) 16.4036 0.623121
\(694\) 26.8471 1.01910
\(695\) 41.3975 1.57030
\(696\) 3.21614 0.121907
\(697\) 2.52893 0.0957900
\(698\) −8.81086 −0.333496
\(699\) 18.5994 0.703493
\(700\) −3.99667 −0.151060
\(701\) −1.11915 −0.0422698 −0.0211349 0.999777i \(-0.506728\pi\)
−0.0211349 + 0.999777i \(0.506728\pi\)
\(702\) 1.00000 0.0377426
\(703\) −21.6930 −0.818167
\(704\) 3.25636 0.122729
\(705\) 7.29305 0.274672
\(706\) 18.6641 0.702434
\(707\) 31.1168 1.17027
\(708\) −13.1023 −0.492415
\(709\) −31.9614 −1.20034 −0.600168 0.799874i \(-0.704900\pi\)
−0.600168 + 0.799874i \(0.704900\pi\)
\(710\) 1.26509 0.0474779
\(711\) 2.05078 0.0769101
\(712\) 3.35651 0.125791
\(713\) −50.9324 −1.90743
\(714\) 2.41720 0.0904614
\(715\) −6.67880 −0.249773
\(716\) 8.75510 0.327193
\(717\) −6.01961 −0.224806
\(718\) −34.4613 −1.28608
\(719\) −11.8588 −0.442260 −0.221130 0.975244i \(-0.570974\pi\)
−0.221130 + 0.975244i \(0.570974\pi\)
\(720\) −2.05100 −0.0764362
\(721\) −5.03739 −0.187602
\(722\) −12.1591 −0.452514
\(723\) −8.74673 −0.325294
\(724\) 26.8429 0.997610
\(725\) −2.55169 −0.0947673
\(726\) −0.396088 −0.0147002
\(727\) 6.51833 0.241752 0.120876 0.992668i \(-0.461430\pi\)
0.120876 + 0.992668i \(0.461430\pi\)
\(728\) −5.03739 −0.186698
\(729\) 1.00000 0.0370370
\(730\) 27.1524 1.00495
\(731\) 1.49886 0.0554373
\(732\) −6.84529 −0.253009
\(733\) 25.8385 0.954368 0.477184 0.878803i \(-0.341658\pi\)
0.477184 + 0.878803i \(0.341658\pi\)
\(734\) −34.7488 −1.28260
\(735\) 37.6878 1.39013
\(736\) 6.13582 0.226169
\(737\) −11.4506 −0.421789
\(738\) −5.27024 −0.194000
\(739\) 18.6136 0.684711 0.342356 0.939570i \(-0.388775\pi\)
0.342356 + 0.939570i \(0.388775\pi\)
\(740\) 7.97064 0.293006
\(741\) −5.58203 −0.205061
\(742\) 34.0617 1.25045
\(743\) −17.1841 −0.630423 −0.315212 0.949021i \(-0.602076\pi\)
−0.315212 + 0.949021i \(0.602076\pi\)
\(744\) 8.30084 0.304323
\(745\) 1.59607 0.0584753
\(746\) −16.4025 −0.600539
\(747\) −2.18812 −0.0800593
\(748\) 1.56257 0.0571332
\(749\) −91.0934 −3.32848
\(750\) 11.8823 0.433879
\(751\) 40.1392 1.46470 0.732350 0.680929i \(-0.238424\pi\)
0.732350 + 0.680929i \(0.238424\pi\)
\(752\) 3.55585 0.129669
\(753\) 2.31076 0.0842086
\(754\) −3.21614 −0.117125
\(755\) −41.7248 −1.51852
\(756\) −5.03739 −0.183208
\(757\) 50.7558 1.84475 0.922375 0.386295i \(-0.126245\pi\)
0.922375 + 0.386295i \(0.126245\pi\)
\(758\) 21.1040 0.766530
\(759\) 19.9805 0.725245
\(760\) 11.4487 0.415290
\(761\) 34.2737 1.24242 0.621210 0.783644i \(-0.286641\pi\)
0.621210 + 0.783644i \(0.286641\pi\)
\(762\) −11.9402 −0.432549
\(763\) −8.05834 −0.291732
\(764\) −18.2739 −0.661125
\(765\) −0.984174 −0.0355829
\(766\) −26.8489 −0.970090
\(767\) 13.1023 0.473097
\(768\) −1.00000 −0.0360844
\(769\) −50.1599 −1.80881 −0.904406 0.426672i \(-0.859686\pi\)
−0.904406 + 0.426672i \(0.859686\pi\)
\(770\) 33.6437 1.21244
\(771\) 30.3838 1.09425
\(772\) 9.22242 0.331922
\(773\) 35.1265 1.26341 0.631705 0.775208i \(-0.282355\pi\)
0.631705 + 0.775208i \(0.282355\pi\)
\(774\) −3.12359 −0.112275
\(775\) −6.58589 −0.236572
\(776\) 1.04552 0.0375318
\(777\) 19.5764 0.702300
\(778\) −0.453851 −0.0162713
\(779\) 29.4186 1.05403
\(780\) 2.05100 0.0734376
\(781\) 2.00858 0.0718725
\(782\) 2.94428 0.105287
\(783\) −3.21614 −0.114935
\(784\) 18.3753 0.656261
\(785\) 43.8579 1.56535
\(786\) 3.36895 0.120167
\(787\) 7.10009 0.253091 0.126545 0.991961i \(-0.459611\pi\)
0.126545 + 0.991961i \(0.459611\pi\)
\(788\) 21.2145 0.755734
\(789\) −15.0290 −0.535048
\(790\) 4.20614 0.149648
\(791\) 13.7981 0.490604
\(792\) −3.25636 −0.115710
\(793\) 6.84529 0.243083
\(794\) 10.5105 0.373004
\(795\) −13.8684 −0.491862
\(796\) −21.4058 −0.758708
\(797\) 43.3628 1.53599 0.767995 0.640455i \(-0.221254\pi\)
0.767995 + 0.640455i \(0.221254\pi\)
\(798\) 28.1189 0.995398
\(799\) 1.70628 0.0603638
\(800\) 0.793401 0.0280510
\(801\) −3.35651 −0.118596
\(802\) −19.8785 −0.701934
\(803\) 43.1097 1.52131
\(804\) 3.51638 0.124013
\(805\) 63.3934 2.23432
\(806\) −8.30084 −0.292384
\(807\) 19.3049 0.679563
\(808\) −6.17717 −0.217312
\(809\) −4.30489 −0.151352 −0.0756760 0.997132i \(-0.524111\pi\)
−0.0756760 + 0.997132i \(0.524111\pi\)
\(810\) 2.05100 0.0720648
\(811\) 12.6095 0.442779 0.221390 0.975185i \(-0.428941\pi\)
0.221390 + 0.975185i \(0.428941\pi\)
\(812\) 16.2009 0.568542
\(813\) −5.46064 −0.191513
\(814\) 12.6550 0.443556
\(815\) 30.5494 1.07010
\(816\) −0.479851 −0.0167981
\(817\) 17.4360 0.610008
\(818\) 11.4709 0.401069
\(819\) 5.03739 0.176021
\(820\) −10.8093 −0.377476
\(821\) −28.7016 −1.00169 −0.500847 0.865536i \(-0.666978\pi\)
−0.500847 + 0.865536i \(0.666978\pi\)
\(822\) −1.91274 −0.0667145
\(823\) 1.62898 0.0567828 0.0283914 0.999597i \(-0.490962\pi\)
0.0283914 + 0.999597i \(0.490962\pi\)
\(824\) 1.00000 0.0348367
\(825\) 2.58360 0.0899495
\(826\) −66.0014 −2.29648
\(827\) 16.5940 0.577031 0.288516 0.957475i \(-0.406838\pi\)
0.288516 + 0.957475i \(0.406838\pi\)
\(828\) −6.13582 −0.213234
\(829\) −18.9337 −0.657593 −0.328797 0.944401i \(-0.606643\pi\)
−0.328797 + 0.944401i \(0.606643\pi\)
\(830\) −4.48784 −0.155775
\(831\) 18.3298 0.635853
\(832\) 1.00000 0.0346688
\(833\) 8.81742 0.305505
\(834\) −20.1841 −0.698917
\(835\) −13.7598 −0.476179
\(836\) 18.1771 0.628669
\(837\) −8.30084 −0.286919
\(838\) 34.7936 1.20192
\(839\) −2.03863 −0.0703812 −0.0351906 0.999381i \(-0.511204\pi\)
−0.0351906 + 0.999381i \(0.511204\pi\)
\(840\) −10.3317 −0.356477
\(841\) −18.6565 −0.643326
\(842\) 6.18218 0.213052
\(843\) −7.86816 −0.270994
\(844\) 3.59640 0.123793
\(845\) −2.05100 −0.0705565
\(846\) −3.55585 −0.122253
\(847\) −1.99525 −0.0685576
\(848\) −6.76178 −0.232200
\(849\) −14.4497 −0.495911
\(850\) 0.380714 0.0130584
\(851\) 23.8451 0.817401
\(852\) −0.616815 −0.0211317
\(853\) 0.868769 0.0297461 0.0148730 0.999889i \(-0.495266\pi\)
0.0148730 + 0.999889i \(0.495266\pi\)
\(854\) −34.4824 −1.17996
\(855\) −11.4487 −0.391539
\(856\) 18.0835 0.618080
\(857\) −7.49315 −0.255961 −0.127981 0.991777i \(-0.540850\pi\)
−0.127981 + 0.991777i \(0.540850\pi\)
\(858\) 3.25636 0.111170
\(859\) 35.6824 1.21747 0.608734 0.793374i \(-0.291678\pi\)
0.608734 + 0.793374i \(0.291678\pi\)
\(860\) −6.40648 −0.218459
\(861\) −26.5483 −0.904762
\(862\) −2.84511 −0.0969048
\(863\) 17.3230 0.589682 0.294841 0.955546i \(-0.404733\pi\)
0.294841 + 0.955546i \(0.404733\pi\)
\(864\) 1.00000 0.0340207
\(865\) 30.4357 1.03484
\(866\) 5.12180 0.174046
\(867\) 16.7697 0.569530
\(868\) 41.8146 1.41928
\(869\) 6.67807 0.226538
\(870\) −6.59630 −0.223635
\(871\) −3.51638 −0.119148
\(872\) 1.59970 0.0541728
\(873\) −1.04552 −0.0353854
\(874\) 34.2503 1.15853
\(875\) 59.8556 2.02349
\(876\) −13.2386 −0.447291
\(877\) −31.0758 −1.04936 −0.524678 0.851301i \(-0.675814\pi\)
−0.524678 + 0.851301i \(0.675814\pi\)
\(878\) 8.67259 0.292686
\(879\) −7.67257 −0.258789
\(880\) −6.67880 −0.225142
\(881\) 16.4398 0.553870 0.276935 0.960889i \(-0.410681\pi\)
0.276935 + 0.960889i \(0.410681\pi\)
\(882\) −18.3753 −0.618729
\(883\) −1.87158 −0.0629836 −0.0314918 0.999504i \(-0.510026\pi\)
−0.0314918 + 0.999504i \(0.510026\pi\)
\(884\) 0.479851 0.0161391
\(885\) 26.8728 0.903320
\(886\) 5.13670 0.172571
\(887\) 41.3963 1.38995 0.694975 0.719034i \(-0.255415\pi\)
0.694975 + 0.719034i \(0.255415\pi\)
\(888\) −3.88622 −0.130413
\(889\) −60.1477 −2.01729
\(890\) −6.88420 −0.230759
\(891\) 3.25636 0.109092
\(892\) −27.5119 −0.921167
\(893\) 19.8489 0.664217
\(894\) −0.778189 −0.0260265
\(895\) −17.9567 −0.600227
\(896\) −5.03739 −0.168287
\(897\) 6.13582 0.204869
\(898\) −0.593133 −0.0197931
\(899\) 26.6966 0.890382
\(900\) −0.793401 −0.0264467
\(901\) −3.24465 −0.108095
\(902\) −17.1618 −0.571426
\(903\) −15.7347 −0.523620
\(904\) −2.73914 −0.0911023
\(905\) −55.0548 −1.83008
\(906\) 20.3436 0.675872
\(907\) 21.5957 0.717074 0.358537 0.933516i \(-0.383276\pi\)
0.358537 + 0.933516i \(0.383276\pi\)
\(908\) 0.778045 0.0258203
\(909\) 6.17717 0.204884
\(910\) 10.3317 0.342492
\(911\) −54.0446 −1.79058 −0.895289 0.445486i \(-0.853031\pi\)
−0.895289 + 0.445486i \(0.853031\pi\)
\(912\) −5.58203 −0.184840
\(913\) −7.12533 −0.235814
\(914\) 21.1234 0.698701
\(915\) 14.0397 0.464138
\(916\) 8.61245 0.284563
\(917\) 16.9707 0.560423
\(918\) 0.479851 0.0158374
\(919\) −29.4124 −0.970227 −0.485113 0.874451i \(-0.661222\pi\)
−0.485113 + 0.874451i \(0.661222\pi\)
\(920\) −12.5846 −0.414901
\(921\) −8.31383 −0.273950
\(922\) −21.7463 −0.716176
\(923\) 0.616815 0.0203027
\(924\) −16.4036 −0.539638
\(925\) 3.08333 0.101379
\(926\) −31.9048 −1.04846
\(927\) −1.00000 −0.0328443
\(928\) −3.21614 −0.105575
\(929\) −7.67002 −0.251645 −0.125823 0.992053i \(-0.540157\pi\)
−0.125823 + 0.992053i \(0.540157\pi\)
\(930\) −17.0250 −0.558272
\(931\) 102.572 3.36165
\(932\) −18.5994 −0.609243
\(933\) 18.2570 0.597708
\(934\) −0.266545 −0.00872162
\(935\) −3.20483 −0.104809
\(936\) −1.00000 −0.0326860
\(937\) −42.3946 −1.38497 −0.692485 0.721432i \(-0.743484\pi\)
−0.692485 + 0.721432i \(0.743484\pi\)
\(938\) 17.7134 0.578363
\(939\) −28.5611 −0.932058
\(940\) −7.29305 −0.237873
\(941\) −32.7077 −1.06624 −0.533120 0.846040i \(-0.678980\pi\)
−0.533120 + 0.846040i \(0.678980\pi\)
\(942\) −21.3836 −0.696717
\(943\) −32.3372 −1.05304
\(944\) 13.1023 0.426444
\(945\) 10.3317 0.336090
\(946\) −10.1715 −0.330706
\(947\) 8.41560 0.273470 0.136735 0.990608i \(-0.456339\pi\)
0.136735 + 0.990608i \(0.456339\pi\)
\(948\) −2.05078 −0.0666061
\(949\) 13.2386 0.429743
\(950\) 4.42879 0.143689
\(951\) −18.8153 −0.610129
\(952\) −2.41720 −0.0783418
\(953\) −6.22137 −0.201530 −0.100765 0.994910i \(-0.532129\pi\)
−0.100765 + 0.994910i \(0.532129\pi\)
\(954\) 6.76178 0.218921
\(955\) 37.4797 1.21281
\(956\) 6.01961 0.194688
\(957\) −10.4729 −0.338541
\(958\) 21.6179 0.698443
\(959\) −9.63522 −0.311138
\(960\) 2.05100 0.0661957
\(961\) 37.9039 1.22271
\(962\) 3.88622 0.125297
\(963\) −18.0835 −0.582731
\(964\) 8.74673 0.281713
\(965\) −18.9152 −0.608901
\(966\) −30.9085 −0.994465
\(967\) 8.10148 0.260526 0.130263 0.991479i \(-0.458418\pi\)
0.130263 + 0.991479i \(0.458418\pi\)
\(968\) 0.396088 0.0127308
\(969\) −2.67854 −0.0860472
\(970\) −2.14435 −0.0688510
\(971\) −52.7015 −1.69127 −0.845636 0.533760i \(-0.820779\pi\)
−0.845636 + 0.533760i \(0.820779\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −101.675 −3.25955
\(974\) 15.2920 0.489987
\(975\) 0.793401 0.0254092
\(976\) 6.84529 0.219112
\(977\) 43.3117 1.38567 0.692833 0.721098i \(-0.256362\pi\)
0.692833 + 0.721098i \(0.256362\pi\)
\(978\) −14.8949 −0.476286
\(979\) −10.9300 −0.349325
\(980\) −37.6878 −1.20389
\(981\) −1.59970 −0.0510746
\(982\) −29.3696 −0.937221
\(983\) −23.0411 −0.734896 −0.367448 0.930044i \(-0.619768\pi\)
−0.367448 + 0.930044i \(0.619768\pi\)
\(984\) 5.27024 0.168009
\(985\) −43.5109 −1.38637
\(986\) −1.54327 −0.0491476
\(987\) −17.9122 −0.570152
\(988\) 5.58203 0.177588
\(989\) −19.1658 −0.609436
\(990\) 6.67880 0.212266
\(991\) 55.5151 1.76350 0.881748 0.471721i \(-0.156367\pi\)
0.881748 + 0.471721i \(0.156367\pi\)
\(992\) −8.30084 −0.263552
\(993\) 23.2918 0.739142
\(994\) −3.10714 −0.0985525
\(995\) 43.9033 1.39183
\(996\) 2.18812 0.0693334
\(997\) −51.9836 −1.64634 −0.823168 0.567797i \(-0.807796\pi\)
−0.823168 + 0.567797i \(0.807796\pi\)
\(998\) −19.4220 −0.614794
\(999\) 3.88622 0.122955
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 8034.2.a.z.1.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.4 14 1.1 even 1 trivial