Properties

Label 8034.2.a.ba.1.1
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: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
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 1.1
Root \(4.07159\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.07159 q^{5} +1.00000 q^{6} +2.58120 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.07159 q^{5} +1.00000 q^{6} +2.58120 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.07159 q^{10} +3.75053 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.58120 q^{14} +4.07159 q^{15} +1.00000 q^{16} -4.96826 q^{17} -1.00000 q^{18} +3.27578 q^{19} -4.07159 q^{20} -2.58120 q^{21} -3.75053 q^{22} +6.21258 q^{23} +1.00000 q^{24} +11.5778 q^{25} +1.00000 q^{26} -1.00000 q^{27} +2.58120 q^{28} -3.00244 q^{29} -4.07159 q^{30} +0.515556 q^{31} -1.00000 q^{32} -3.75053 q^{33} +4.96826 q^{34} -10.5096 q^{35} +1.00000 q^{36} -0.169601 q^{37} -3.27578 q^{38} +1.00000 q^{39} +4.07159 q^{40} +9.14732 q^{41} +2.58120 q^{42} +5.16760 q^{43} +3.75053 q^{44} -4.07159 q^{45} -6.21258 q^{46} +9.19458 q^{47} -1.00000 q^{48} -0.337386 q^{49} -11.5778 q^{50} +4.96826 q^{51} -1.00000 q^{52} -1.36025 q^{53} +1.00000 q^{54} -15.2706 q^{55} -2.58120 q^{56} -3.27578 q^{57} +3.00244 q^{58} -7.18773 q^{59} +4.07159 q^{60} +8.72968 q^{61} -0.515556 q^{62} +2.58120 q^{63} +1.00000 q^{64} +4.07159 q^{65} +3.75053 q^{66} +5.57898 q^{67} -4.96826 q^{68} -6.21258 q^{69} +10.5096 q^{70} -8.81509 q^{71} -1.00000 q^{72} -6.75816 q^{73} +0.169601 q^{74} -11.5778 q^{75} +3.27578 q^{76} +9.68087 q^{77} -1.00000 q^{78} +11.5740 q^{79} -4.07159 q^{80} +1.00000 q^{81} -9.14732 q^{82} -12.6995 q^{83} -2.58120 q^{84} +20.2287 q^{85} -5.16760 q^{86} +3.00244 q^{87} -3.75053 q^{88} +1.52673 q^{89} +4.07159 q^{90} -2.58120 q^{91} +6.21258 q^{92} -0.515556 q^{93} -9.19458 q^{94} -13.3376 q^{95} +1.00000 q^{96} -1.76201 q^{97} +0.337386 q^{98} +3.75053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −4.07159 −1.82087 −0.910434 0.413653i \(-0.864253\pi\)
−0.910434 + 0.413653i \(0.864253\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.58120 0.975603 0.487802 0.872954i \(-0.337799\pi\)
0.487802 + 0.872954i \(0.337799\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.07159 1.28755
\(11\) 3.75053 1.13083 0.565413 0.824808i \(-0.308717\pi\)
0.565413 + 0.824808i \(0.308717\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −2.58120 −0.689856
\(15\) 4.07159 1.05128
\(16\) 1.00000 0.250000
\(17\) −4.96826 −1.20498 −0.602490 0.798127i \(-0.705825\pi\)
−0.602490 + 0.798127i \(0.705825\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.27578 0.751516 0.375758 0.926718i \(-0.377382\pi\)
0.375758 + 0.926718i \(0.377382\pi\)
\(20\) −4.07159 −0.910434
\(21\) −2.58120 −0.563265
\(22\) −3.75053 −0.799615
\(23\) 6.21258 1.29541 0.647706 0.761890i \(-0.275728\pi\)
0.647706 + 0.761890i \(0.275728\pi\)
\(24\) 1.00000 0.204124
\(25\) 11.5778 2.31556
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 2.58120 0.487802
\(29\) −3.00244 −0.557538 −0.278769 0.960358i \(-0.589926\pi\)
−0.278769 + 0.960358i \(0.589926\pi\)
\(30\) −4.07159 −0.743367
\(31\) 0.515556 0.0925966 0.0462983 0.998928i \(-0.485258\pi\)
0.0462983 + 0.998928i \(0.485258\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.75053 −0.652883
\(34\) 4.96826 0.852049
\(35\) −10.5096 −1.77645
\(36\) 1.00000 0.166667
\(37\) −0.169601 −0.0278822 −0.0139411 0.999903i \(-0.504438\pi\)
−0.0139411 + 0.999903i \(0.504438\pi\)
\(38\) −3.27578 −0.531402
\(39\) 1.00000 0.160128
\(40\) 4.07159 0.643774
\(41\) 9.14732 1.42857 0.714285 0.699854i \(-0.246752\pi\)
0.714285 + 0.699854i \(0.246752\pi\)
\(42\) 2.58120 0.398288
\(43\) 5.16760 0.788052 0.394026 0.919099i \(-0.371082\pi\)
0.394026 + 0.919099i \(0.371082\pi\)
\(44\) 3.75053 0.565413
\(45\) −4.07159 −0.606956
\(46\) −6.21258 −0.915995
\(47\) 9.19458 1.34117 0.670584 0.741834i \(-0.266044\pi\)
0.670584 + 0.741834i \(0.266044\pi\)
\(48\) −1.00000 −0.144338
\(49\) −0.337386 −0.0481980
\(50\) −11.5778 −1.63735
\(51\) 4.96826 0.695695
\(52\) −1.00000 −0.138675
\(53\) −1.36025 −0.186845 −0.0934223 0.995627i \(-0.529781\pi\)
−0.0934223 + 0.995627i \(0.529781\pi\)
\(54\) 1.00000 0.136083
\(55\) −15.2706 −2.05909
\(56\) −2.58120 −0.344928
\(57\) −3.27578 −0.433888
\(58\) 3.00244 0.394239
\(59\) −7.18773 −0.935763 −0.467881 0.883791i \(-0.654983\pi\)
−0.467881 + 0.883791i \(0.654983\pi\)
\(60\) 4.07159 0.525640
\(61\) 8.72968 1.11772 0.558860 0.829262i \(-0.311239\pi\)
0.558860 + 0.829262i \(0.311239\pi\)
\(62\) −0.515556 −0.0654757
\(63\) 2.58120 0.325201
\(64\) 1.00000 0.125000
\(65\) 4.07159 0.505018
\(66\) 3.75053 0.461658
\(67\) 5.57898 0.681581 0.340790 0.940139i \(-0.389305\pi\)
0.340790 + 0.940139i \(0.389305\pi\)
\(68\) −4.96826 −0.602490
\(69\) −6.21258 −0.747907
\(70\) 10.5096 1.25614
\(71\) −8.81509 −1.04616 −0.523079 0.852284i \(-0.675217\pi\)
−0.523079 + 0.852284i \(0.675217\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.75816 −0.790983 −0.395491 0.918470i \(-0.629426\pi\)
−0.395491 + 0.918470i \(0.629426\pi\)
\(74\) 0.169601 0.0197157
\(75\) −11.5778 −1.33689
\(76\) 3.27578 0.375758
\(77\) 9.68087 1.10324
\(78\) −1.00000 −0.113228
\(79\) 11.5740 1.30218 0.651089 0.759001i \(-0.274312\pi\)
0.651089 + 0.759001i \(0.274312\pi\)
\(80\) −4.07159 −0.455217
\(81\) 1.00000 0.111111
\(82\) −9.14732 −1.01015
\(83\) −12.6995 −1.39395 −0.696975 0.717096i \(-0.745471\pi\)
−0.696975 + 0.717096i \(0.745471\pi\)
\(84\) −2.58120 −0.281632
\(85\) 20.2287 2.19411
\(86\) −5.16760 −0.557237
\(87\) 3.00244 0.321895
\(88\) −3.75053 −0.399807
\(89\) 1.52673 0.161833 0.0809165 0.996721i \(-0.474215\pi\)
0.0809165 + 0.996721i \(0.474215\pi\)
\(90\) 4.07159 0.429183
\(91\) −2.58120 −0.270584
\(92\) 6.21258 0.647706
\(93\) −0.515556 −0.0534607
\(94\) −9.19458 −0.948349
\(95\) −13.3376 −1.36841
\(96\) 1.00000 0.102062
\(97\) −1.76201 −0.178905 −0.0894524 0.995991i \(-0.528512\pi\)
−0.0894524 + 0.995991i \(0.528512\pi\)
\(98\) 0.337386 0.0340811
\(99\) 3.75053 0.376942
\(100\) 11.5778 1.15778
\(101\) −9.92757 −0.987830 −0.493915 0.869510i \(-0.664435\pi\)
−0.493915 + 0.869510i \(0.664435\pi\)
\(102\) −4.96826 −0.491931
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 10.5096 1.02563
\(106\) 1.36025 0.132119
\(107\) −12.6827 −1.22608 −0.613041 0.790051i \(-0.710054\pi\)
−0.613041 + 0.790051i \(0.710054\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.63187 0.635218 0.317609 0.948222i \(-0.397120\pi\)
0.317609 + 0.948222i \(0.397120\pi\)
\(110\) 15.2706 1.45599
\(111\) 0.169601 0.0160978
\(112\) 2.58120 0.243901
\(113\) 16.1053 1.51506 0.757528 0.652802i \(-0.226407\pi\)
0.757528 + 0.652802i \(0.226407\pi\)
\(114\) 3.27578 0.306805
\(115\) −25.2951 −2.35878
\(116\) −3.00244 −0.278769
\(117\) −1.00000 −0.0924500
\(118\) 7.18773 0.661684
\(119\) −12.8241 −1.17558
\(120\) −4.07159 −0.371683
\(121\) 3.06644 0.278767
\(122\) −8.72968 −0.790348
\(123\) −9.14732 −0.824786
\(124\) 0.515556 0.0462983
\(125\) −26.7822 −2.39547
\(126\) −2.58120 −0.229952
\(127\) 14.4538 1.28257 0.641283 0.767305i \(-0.278403\pi\)
0.641283 + 0.767305i \(0.278403\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.16760 −0.454982
\(130\) −4.07159 −0.357102
\(131\) 7.73157 0.675511 0.337755 0.941234i \(-0.390332\pi\)
0.337755 + 0.941234i \(0.390332\pi\)
\(132\) −3.75053 −0.326441
\(133\) 8.45546 0.733181
\(134\) −5.57898 −0.481950
\(135\) 4.07159 0.350426
\(136\) 4.96826 0.426024
\(137\) 9.54400 0.815399 0.407699 0.913116i \(-0.366331\pi\)
0.407699 + 0.913116i \(0.366331\pi\)
\(138\) 6.21258 0.528850
\(139\) −8.60003 −0.729445 −0.364722 0.931116i \(-0.618836\pi\)
−0.364722 + 0.931116i \(0.618836\pi\)
\(140\) −10.5096 −0.888223
\(141\) −9.19458 −0.774323
\(142\) 8.81509 0.739746
\(143\) −3.75053 −0.313635
\(144\) 1.00000 0.0833333
\(145\) 12.2247 1.01520
\(146\) 6.75816 0.559309
\(147\) 0.337386 0.0278271
\(148\) −0.169601 −0.0139411
\(149\) 12.1260 0.993402 0.496701 0.867922i \(-0.334545\pi\)
0.496701 + 0.867922i \(0.334545\pi\)
\(150\) 11.5778 0.945325
\(151\) −9.15082 −0.744683 −0.372341 0.928096i \(-0.621445\pi\)
−0.372341 + 0.928096i \(0.621445\pi\)
\(152\) −3.27578 −0.265701
\(153\) −4.96826 −0.401660
\(154\) −9.68087 −0.780107
\(155\) −2.09913 −0.168606
\(156\) 1.00000 0.0800641
\(157\) 7.49283 0.597993 0.298997 0.954254i \(-0.403348\pi\)
0.298997 + 0.954254i \(0.403348\pi\)
\(158\) −11.5740 −0.920779
\(159\) 1.36025 0.107875
\(160\) 4.07159 0.321887
\(161\) 16.0359 1.26381
\(162\) −1.00000 −0.0785674
\(163\) −24.8572 −1.94696 −0.973482 0.228762i \(-0.926532\pi\)
−0.973482 + 0.228762i \(0.926532\pi\)
\(164\) 9.14732 0.714285
\(165\) 15.2706 1.18881
\(166\) 12.6995 0.985671
\(167\) −24.7241 −1.91321 −0.956606 0.291385i \(-0.905884\pi\)
−0.956606 + 0.291385i \(0.905884\pi\)
\(168\) 2.58120 0.199144
\(169\) 1.00000 0.0769231
\(170\) −20.2287 −1.55147
\(171\) 3.27578 0.250505
\(172\) 5.16760 0.394026
\(173\) −4.49671 −0.341878 −0.170939 0.985282i \(-0.554680\pi\)
−0.170939 + 0.985282i \(0.554680\pi\)
\(174\) −3.00244 −0.227614
\(175\) 29.8847 2.25907
\(176\) 3.75053 0.282707
\(177\) 7.18773 0.540263
\(178\) −1.52673 −0.114433
\(179\) −0.474184 −0.0354421 −0.0177211 0.999843i \(-0.505641\pi\)
−0.0177211 + 0.999843i \(0.505641\pi\)
\(180\) −4.07159 −0.303478
\(181\) −6.40924 −0.476395 −0.238198 0.971217i \(-0.576557\pi\)
−0.238198 + 0.971217i \(0.576557\pi\)
\(182\) 2.58120 0.191332
\(183\) −8.72968 −0.645316
\(184\) −6.21258 −0.457997
\(185\) 0.690546 0.0507699
\(186\) 0.515556 0.0378024
\(187\) −18.6336 −1.36262
\(188\) 9.19458 0.670584
\(189\) −2.58120 −0.187755
\(190\) 13.3376 0.967613
\(191\) −14.6130 −1.05736 −0.528681 0.848820i \(-0.677313\pi\)
−0.528681 + 0.848820i \(0.677313\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −9.13430 −0.657501 −0.328751 0.944417i \(-0.606628\pi\)
−0.328751 + 0.944417i \(0.606628\pi\)
\(194\) 1.76201 0.126505
\(195\) −4.07159 −0.291572
\(196\) −0.337386 −0.0240990
\(197\) −20.0325 −1.42726 −0.713629 0.700524i \(-0.752950\pi\)
−0.713629 + 0.700524i \(0.752950\pi\)
\(198\) −3.75053 −0.266538
\(199\) 1.01631 0.0720440 0.0360220 0.999351i \(-0.488531\pi\)
0.0360220 + 0.999351i \(0.488531\pi\)
\(200\) −11.5778 −0.818675
\(201\) −5.57898 −0.393511
\(202\) 9.92757 0.698501
\(203\) −7.74990 −0.543936
\(204\) 4.96826 0.347848
\(205\) −37.2441 −2.60124
\(206\) −1.00000 −0.0696733
\(207\) 6.21258 0.431804
\(208\) −1.00000 −0.0693375
\(209\) 12.2859 0.849834
\(210\) −10.5096 −0.725231
\(211\) 17.7043 1.21882 0.609408 0.792857i \(-0.291407\pi\)
0.609408 + 0.792857i \(0.291407\pi\)
\(212\) −1.36025 −0.0934223
\(213\) 8.81509 0.604000
\(214\) 12.6827 0.866970
\(215\) −21.0403 −1.43494
\(216\) 1.00000 0.0680414
\(217\) 1.33076 0.0903376
\(218\) −6.63187 −0.449167
\(219\) 6.75816 0.456674
\(220\) −15.2706 −1.02954
\(221\) 4.96826 0.334201
\(222\) −0.169601 −0.0113829
\(223\) 23.8931 1.60000 0.799999 0.600002i \(-0.204833\pi\)
0.799999 + 0.600002i \(0.204833\pi\)
\(224\) −2.58120 −0.172464
\(225\) 11.5778 0.771854
\(226\) −16.1053 −1.07131
\(227\) 11.5062 0.763692 0.381846 0.924226i \(-0.375288\pi\)
0.381846 + 0.924226i \(0.375288\pi\)
\(228\) −3.27578 −0.216944
\(229\) −5.01532 −0.331422 −0.165711 0.986174i \(-0.552992\pi\)
−0.165711 + 0.986174i \(0.552992\pi\)
\(230\) 25.2951 1.66791
\(231\) −9.68087 −0.636955
\(232\) 3.00244 0.197120
\(233\) 25.1837 1.64984 0.824921 0.565249i \(-0.191220\pi\)
0.824921 + 0.565249i \(0.191220\pi\)
\(234\) 1.00000 0.0653720
\(235\) −37.4365 −2.44209
\(236\) −7.18773 −0.467881
\(237\) −11.5740 −0.751813
\(238\) 12.8241 0.831262
\(239\) 8.78476 0.568239 0.284119 0.958789i \(-0.408299\pi\)
0.284119 + 0.958789i \(0.408299\pi\)
\(240\) 4.07159 0.262820
\(241\) 5.11482 0.329475 0.164737 0.986337i \(-0.447322\pi\)
0.164737 + 0.986337i \(0.447322\pi\)
\(242\) −3.06644 −0.197118
\(243\) −1.00000 −0.0641500
\(244\) 8.72968 0.558860
\(245\) 1.37370 0.0877622
\(246\) 9.14732 0.583212
\(247\) −3.27578 −0.208433
\(248\) −0.515556 −0.0327378
\(249\) 12.6995 0.804797
\(250\) 26.7822 1.69385
\(251\) −2.96547 −0.187179 −0.0935895 0.995611i \(-0.529834\pi\)
−0.0935895 + 0.995611i \(0.529834\pi\)
\(252\) 2.58120 0.162601
\(253\) 23.3004 1.46489
\(254\) −14.4538 −0.906911
\(255\) −20.2287 −1.26677
\(256\) 1.00000 0.0625000
\(257\) −19.7234 −1.23031 −0.615156 0.788406i \(-0.710907\pi\)
−0.615156 + 0.788406i \(0.710907\pi\)
\(258\) 5.16760 0.321721
\(259\) −0.437775 −0.0272020
\(260\) 4.07159 0.252509
\(261\) −3.00244 −0.185846
\(262\) −7.73157 −0.477658
\(263\) −27.6687 −1.70613 −0.853063 0.521809i \(-0.825258\pi\)
−0.853063 + 0.521809i \(0.825258\pi\)
\(264\) 3.75053 0.230829
\(265\) 5.53838 0.340220
\(266\) −8.45546 −0.518438
\(267\) −1.52673 −0.0934343
\(268\) 5.57898 0.340790
\(269\) −7.45176 −0.454342 −0.227171 0.973855i \(-0.572948\pi\)
−0.227171 + 0.973855i \(0.572948\pi\)
\(270\) −4.07159 −0.247789
\(271\) −1.58452 −0.0962527 −0.0481264 0.998841i \(-0.515325\pi\)
−0.0481264 + 0.998841i \(0.515325\pi\)
\(272\) −4.96826 −0.301245
\(273\) 2.58120 0.156222
\(274\) −9.54400 −0.576574
\(275\) 43.4229 2.61850
\(276\) −6.21258 −0.373953
\(277\) 1.30007 0.0781135 0.0390568 0.999237i \(-0.487565\pi\)
0.0390568 + 0.999237i \(0.487565\pi\)
\(278\) 8.60003 0.515795
\(279\) 0.515556 0.0308655
\(280\) 10.5096 0.628068
\(281\) 13.8619 0.826934 0.413467 0.910519i \(-0.364318\pi\)
0.413467 + 0.910519i \(0.364318\pi\)
\(282\) 9.19458 0.547529
\(283\) −23.0435 −1.36979 −0.684897 0.728639i \(-0.740153\pi\)
−0.684897 + 0.728639i \(0.740153\pi\)
\(284\) −8.81509 −0.523079
\(285\) 13.3376 0.790053
\(286\) 3.75053 0.221773
\(287\) 23.6111 1.39372
\(288\) −1.00000 −0.0589256
\(289\) 7.68357 0.451975
\(290\) −12.2247 −0.717858
\(291\) 1.76201 0.103291
\(292\) −6.75816 −0.395491
\(293\) −8.97969 −0.524599 −0.262300 0.964986i \(-0.584481\pi\)
−0.262300 + 0.964986i \(0.584481\pi\)
\(294\) −0.337386 −0.0196767
\(295\) 29.2655 1.70390
\(296\) 0.169601 0.00985786
\(297\) −3.75053 −0.217628
\(298\) −12.1260 −0.702441
\(299\) −6.21258 −0.359283
\(300\) −11.5778 −0.668446
\(301\) 13.3386 0.768826
\(302\) 9.15082 0.526570
\(303\) 9.92757 0.570324
\(304\) 3.27578 0.187879
\(305\) −35.5436 −2.03522
\(306\) 4.96826 0.284016
\(307\) 6.96717 0.397637 0.198819 0.980036i \(-0.436290\pi\)
0.198819 + 0.980036i \(0.436290\pi\)
\(308\) 9.68087 0.551619
\(309\) −1.00000 −0.0568880
\(310\) 2.09913 0.119223
\(311\) 12.9719 0.735570 0.367785 0.929911i \(-0.380116\pi\)
0.367785 + 0.929911i \(0.380116\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −5.37301 −0.303700 −0.151850 0.988404i \(-0.548523\pi\)
−0.151850 + 0.988404i \(0.548523\pi\)
\(314\) −7.49283 −0.422845
\(315\) −10.5096 −0.592149
\(316\) 11.5740 0.651089
\(317\) 6.85531 0.385032 0.192516 0.981294i \(-0.438335\pi\)
0.192516 + 0.981294i \(0.438335\pi\)
\(318\) −1.36025 −0.0762790
\(319\) −11.2607 −0.630479
\(320\) −4.07159 −0.227609
\(321\) 12.6827 0.707878
\(322\) −16.0359 −0.893648
\(323\) −16.2749 −0.905561
\(324\) 1.00000 0.0555556
\(325\) −11.5778 −0.642222
\(326\) 24.8572 1.37671
\(327\) −6.63187 −0.366743
\(328\) −9.14732 −0.505076
\(329\) 23.7331 1.30845
\(330\) −15.2706 −0.840618
\(331\) 28.8357 1.58495 0.792477 0.609902i \(-0.208791\pi\)
0.792477 + 0.609902i \(0.208791\pi\)
\(332\) −12.6995 −0.696975
\(333\) −0.169601 −0.00929408
\(334\) 24.7241 1.35284
\(335\) −22.7153 −1.24107
\(336\) −2.58120 −0.140816
\(337\) −30.7712 −1.67621 −0.838106 0.545507i \(-0.816337\pi\)
−0.838106 + 0.545507i \(0.816337\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −16.1053 −0.874718
\(340\) 20.2287 1.09705
\(341\) 1.93361 0.104711
\(342\) −3.27578 −0.177134
\(343\) −18.9393 −1.02263
\(344\) −5.16760 −0.278619
\(345\) 25.2951 1.36184
\(346\) 4.49671 0.241744
\(347\) 37.0016 1.98635 0.993175 0.116637i \(-0.0372115\pi\)
0.993175 + 0.116637i \(0.0372115\pi\)
\(348\) 3.00244 0.160947
\(349\) 19.4923 1.04340 0.521700 0.853129i \(-0.325298\pi\)
0.521700 + 0.853129i \(0.325298\pi\)
\(350\) −29.8847 −1.59740
\(351\) 1.00000 0.0533761
\(352\) −3.75053 −0.199904
\(353\) 19.9930 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(354\) −7.18773 −0.382023
\(355\) 35.8914 1.90492
\(356\) 1.52673 0.0809165
\(357\) 12.8241 0.678722
\(358\) 0.474184 0.0250614
\(359\) −16.0667 −0.847970 −0.423985 0.905669i \(-0.639369\pi\)
−0.423985 + 0.905669i \(0.639369\pi\)
\(360\) 4.07159 0.214591
\(361\) −8.26925 −0.435224
\(362\) 6.40924 0.336862
\(363\) −3.06644 −0.160946
\(364\) −2.58120 −0.135292
\(365\) 27.5164 1.44028
\(366\) 8.72968 0.456307
\(367\) 24.4433 1.27593 0.637964 0.770066i \(-0.279777\pi\)
0.637964 + 0.770066i \(0.279777\pi\)
\(368\) 6.21258 0.323853
\(369\) 9.14732 0.476190
\(370\) −0.690546 −0.0358998
\(371\) −3.51108 −0.182286
\(372\) −0.515556 −0.0267303
\(373\) 10.1846 0.527338 0.263669 0.964613i \(-0.415067\pi\)
0.263669 + 0.964613i \(0.415067\pi\)
\(374\) 18.6336 0.963519
\(375\) 26.7822 1.38302
\(376\) −9.19458 −0.474174
\(377\) 3.00244 0.154633
\(378\) 2.58120 0.132763
\(379\) 35.1348 1.80475 0.902376 0.430950i \(-0.141822\pi\)
0.902376 + 0.430950i \(0.141822\pi\)
\(380\) −13.3376 −0.684206
\(381\) −14.4538 −0.740490
\(382\) 14.6130 0.747668
\(383\) −4.80554 −0.245552 −0.122776 0.992434i \(-0.539180\pi\)
−0.122776 + 0.992434i \(0.539180\pi\)
\(384\) 1.00000 0.0510310
\(385\) −39.4165 −2.00885
\(386\) 9.13430 0.464924
\(387\) 5.16760 0.262684
\(388\) −1.76201 −0.0894524
\(389\) 11.0101 0.558236 0.279118 0.960257i \(-0.409958\pi\)
0.279118 + 0.960257i \(0.409958\pi\)
\(390\) 4.07159 0.206173
\(391\) −30.8657 −1.56094
\(392\) 0.337386 0.0170406
\(393\) −7.73157 −0.390006
\(394\) 20.0325 1.00922
\(395\) −47.1246 −2.37110
\(396\) 3.75053 0.188471
\(397\) 35.9964 1.80661 0.903303 0.429004i \(-0.141135\pi\)
0.903303 + 0.429004i \(0.141135\pi\)
\(398\) −1.01631 −0.0509428
\(399\) −8.45546 −0.423303
\(400\) 11.5778 0.578891
\(401\) 26.5560 1.32614 0.663071 0.748557i \(-0.269253\pi\)
0.663071 + 0.748557i \(0.269253\pi\)
\(402\) 5.57898 0.278254
\(403\) −0.515556 −0.0256817
\(404\) −9.92757 −0.493915
\(405\) −4.07159 −0.202319
\(406\) 7.74990 0.384621
\(407\) −0.636093 −0.0315300
\(408\) −4.96826 −0.245965
\(409\) 13.5214 0.668592 0.334296 0.942468i \(-0.391501\pi\)
0.334296 + 0.942468i \(0.391501\pi\)
\(410\) 37.2441 1.83935
\(411\) −9.54400 −0.470771
\(412\) 1.00000 0.0492665
\(413\) −18.5530 −0.912933
\(414\) −6.21258 −0.305332
\(415\) 51.7071 2.53820
\(416\) 1.00000 0.0490290
\(417\) 8.60003 0.421145
\(418\) −12.2859 −0.600923
\(419\) −3.26115 −0.159318 −0.0796588 0.996822i \(-0.525383\pi\)
−0.0796588 + 0.996822i \(0.525383\pi\)
\(420\) 10.5096 0.512816
\(421\) 10.8036 0.526536 0.263268 0.964723i \(-0.415200\pi\)
0.263268 + 0.964723i \(0.415200\pi\)
\(422\) −17.7043 −0.861834
\(423\) 9.19458 0.447056
\(424\) 1.36025 0.0660596
\(425\) −57.5216 −2.79021
\(426\) −8.81509 −0.427093
\(427\) 22.5331 1.09045
\(428\) −12.6827 −0.613041
\(429\) 3.75053 0.181077
\(430\) 21.0403 1.01466
\(431\) 9.42187 0.453836 0.226918 0.973914i \(-0.427135\pi\)
0.226918 + 0.973914i \(0.427135\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.6310 −0.703121 −0.351560 0.936165i \(-0.614349\pi\)
−0.351560 + 0.936165i \(0.614349\pi\)
\(434\) −1.33076 −0.0638783
\(435\) −12.2247 −0.586129
\(436\) 6.63187 0.317609
\(437\) 20.3511 0.973523
\(438\) −6.75816 −0.322917
\(439\) 0.578539 0.0276122 0.0138061 0.999905i \(-0.495605\pi\)
0.0138061 + 0.999905i \(0.495605\pi\)
\(440\) 15.2706 0.727997
\(441\) −0.337386 −0.0160660
\(442\) −4.96826 −0.236316
\(443\) −16.9554 −0.805575 −0.402787 0.915294i \(-0.631959\pi\)
−0.402787 + 0.915294i \(0.631959\pi\)
\(444\) 0.169601 0.00804891
\(445\) −6.21621 −0.294677
\(446\) −23.8931 −1.13137
\(447\) −12.1260 −0.573541
\(448\) 2.58120 0.121950
\(449\) 16.6918 0.787735 0.393867 0.919167i \(-0.371137\pi\)
0.393867 + 0.919167i \(0.371137\pi\)
\(450\) −11.5778 −0.545784
\(451\) 34.3072 1.61547
\(452\) 16.1053 0.757528
\(453\) 9.15082 0.429943
\(454\) −11.5062 −0.540012
\(455\) 10.5096 0.492697
\(456\) 3.27578 0.153403
\(457\) −0.498348 −0.0233117 −0.0116559 0.999932i \(-0.503710\pi\)
−0.0116559 + 0.999932i \(0.503710\pi\)
\(458\) 5.01532 0.234350
\(459\) 4.96826 0.231898
\(460\) −25.2951 −1.17939
\(461\) 2.29262 0.106778 0.0533889 0.998574i \(-0.482998\pi\)
0.0533889 + 0.998574i \(0.482998\pi\)
\(462\) 9.68087 0.450395
\(463\) 7.76040 0.360657 0.180328 0.983606i \(-0.442284\pi\)
0.180328 + 0.983606i \(0.442284\pi\)
\(464\) −3.00244 −0.139385
\(465\) 2.09913 0.0973449
\(466\) −25.1837 −1.16661
\(467\) −35.3243 −1.63461 −0.817307 0.576202i \(-0.804534\pi\)
−0.817307 + 0.576202i \(0.804534\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 14.4005 0.664953
\(470\) 37.4365 1.72682
\(471\) −7.49283 −0.345252
\(472\) 7.18773 0.330842
\(473\) 19.3812 0.891150
\(474\) 11.5740 0.531612
\(475\) 37.9264 1.74018
\(476\) −12.8241 −0.587791
\(477\) −1.36025 −0.0622816
\(478\) −8.78476 −0.401805
\(479\) 37.0768 1.69408 0.847042 0.531526i \(-0.178381\pi\)
0.847042 + 0.531526i \(0.178381\pi\)
\(480\) −4.07159 −0.185842
\(481\) 0.169601 0.00773314
\(482\) −5.11482 −0.232974
\(483\) −16.0359 −0.729660
\(484\) 3.06644 0.139384
\(485\) 7.17417 0.325762
\(486\) 1.00000 0.0453609
\(487\) 14.8710 0.673868 0.336934 0.941528i \(-0.390610\pi\)
0.336934 + 0.941528i \(0.390610\pi\)
\(488\) −8.72968 −0.395174
\(489\) 24.8572 1.12408
\(490\) −1.37370 −0.0620572
\(491\) 16.8665 0.761176 0.380588 0.924745i \(-0.375722\pi\)
0.380588 + 0.924745i \(0.375722\pi\)
\(492\) −9.14732 −0.412393
\(493\) 14.9169 0.671822
\(494\) 3.27578 0.147384
\(495\) −15.2706 −0.686362
\(496\) 0.515556 0.0231491
\(497\) −22.7536 −1.02064
\(498\) −12.6995 −0.569078
\(499\) 16.9371 0.758210 0.379105 0.925354i \(-0.376232\pi\)
0.379105 + 0.925354i \(0.376232\pi\)
\(500\) −26.7822 −1.19773
\(501\) 24.7241 1.10459
\(502\) 2.96547 0.132356
\(503\) 24.5868 1.09627 0.548135 0.836390i \(-0.315338\pi\)
0.548135 + 0.836390i \(0.315338\pi\)
\(504\) −2.58120 −0.114976
\(505\) 40.4209 1.79871
\(506\) −23.3004 −1.03583
\(507\) −1.00000 −0.0444116
\(508\) 14.4538 0.641283
\(509\) 27.8995 1.23662 0.618311 0.785934i \(-0.287817\pi\)
0.618311 + 0.785934i \(0.287817\pi\)
\(510\) 20.2287 0.895741
\(511\) −17.4442 −0.771685
\(512\) −1.00000 −0.0441942
\(513\) −3.27578 −0.144629
\(514\) 19.7234 0.869962
\(515\) −4.07159 −0.179416
\(516\) −5.16760 −0.227491
\(517\) 34.4845 1.51663
\(518\) 0.437775 0.0192347
\(519\) 4.49671 0.197383
\(520\) −4.07159 −0.178551
\(521\) 21.3868 0.936974 0.468487 0.883470i \(-0.344799\pi\)
0.468487 + 0.883470i \(0.344799\pi\)
\(522\) 3.00244 0.131413
\(523\) −19.9409 −0.871956 −0.435978 0.899957i \(-0.643597\pi\)
−0.435978 + 0.899957i \(0.643597\pi\)
\(524\) 7.73157 0.337755
\(525\) −29.8847 −1.30428
\(526\) 27.6687 1.20641
\(527\) −2.56141 −0.111577
\(528\) −3.75053 −0.163221
\(529\) 15.5961 0.678093
\(530\) −5.53838 −0.240572
\(531\) −7.18773 −0.311921
\(532\) 8.45546 0.366591
\(533\) −9.14732 −0.396214
\(534\) 1.52673 0.0660681
\(535\) 51.6386 2.23253
\(536\) −5.57898 −0.240975
\(537\) 0.474184 0.0204625
\(538\) 7.45176 0.321268
\(539\) −1.26537 −0.0545035
\(540\) 4.07159 0.175213
\(541\) −12.8726 −0.553437 −0.276719 0.960951i \(-0.589247\pi\)
−0.276719 + 0.960951i \(0.589247\pi\)
\(542\) 1.58452 0.0680610
\(543\) 6.40924 0.275047
\(544\) 4.96826 0.213012
\(545\) −27.0022 −1.15665
\(546\) −2.58120 −0.110465
\(547\) −3.66737 −0.156806 −0.0784028 0.996922i \(-0.524982\pi\)
−0.0784028 + 0.996922i \(0.524982\pi\)
\(548\) 9.54400 0.407699
\(549\) 8.72968 0.372573
\(550\) −43.4229 −1.85156
\(551\) −9.83533 −0.418999
\(552\) 6.21258 0.264425
\(553\) 29.8749 1.27041
\(554\) −1.30007 −0.0552346
\(555\) −0.690546 −0.0293120
\(556\) −8.60003 −0.364722
\(557\) 40.3235 1.70856 0.854280 0.519813i \(-0.173998\pi\)
0.854280 + 0.519813i \(0.173998\pi\)
\(558\) −0.515556 −0.0218252
\(559\) −5.16760 −0.218566
\(560\) −10.5096 −0.444111
\(561\) 18.6336 0.786710
\(562\) −13.8619 −0.584731
\(563\) 14.1289 0.595462 0.297731 0.954650i \(-0.403770\pi\)
0.297731 + 0.954650i \(0.403770\pi\)
\(564\) −9.19458 −0.387162
\(565\) −65.5740 −2.75872
\(566\) 23.0435 0.968591
\(567\) 2.58120 0.108400
\(568\) 8.81509 0.369873
\(569\) 36.8899 1.54651 0.773253 0.634097i \(-0.218628\pi\)
0.773253 + 0.634097i \(0.218628\pi\)
\(570\) −13.3376 −0.558652
\(571\) −18.4187 −0.770797 −0.385399 0.922750i \(-0.625936\pi\)
−0.385399 + 0.922750i \(0.625936\pi\)
\(572\) −3.75053 −0.156817
\(573\) 14.6130 0.610469
\(574\) −23.6111 −0.985508
\(575\) 71.9281 2.99961
\(576\) 1.00000 0.0416667
\(577\) 2.14056 0.0891128 0.0445564 0.999007i \(-0.485813\pi\)
0.0445564 + 0.999007i \(0.485813\pi\)
\(578\) −7.68357 −0.319595
\(579\) 9.13430 0.379608
\(580\) 12.2247 0.507602
\(581\) −32.7800 −1.35994
\(582\) −1.76201 −0.0730376
\(583\) −5.10165 −0.211289
\(584\) 6.75816 0.279655
\(585\) 4.07159 0.168339
\(586\) 8.97969 0.370948
\(587\) −20.2440 −0.835559 −0.417780 0.908548i \(-0.637192\pi\)
−0.417780 + 0.908548i \(0.637192\pi\)
\(588\) 0.337386 0.0139136
\(589\) 1.68885 0.0695878
\(590\) −29.2655 −1.20484
\(591\) 20.0325 0.824028
\(592\) −0.169601 −0.00697056
\(593\) 17.7664 0.729580 0.364790 0.931090i \(-0.381141\pi\)
0.364790 + 0.931090i \(0.381141\pi\)
\(594\) 3.75053 0.153886
\(595\) 52.2144 2.14058
\(596\) 12.1260 0.496701
\(597\) −1.01631 −0.0415946
\(598\) 6.21258 0.254051
\(599\) 13.4618 0.550035 0.275018 0.961439i \(-0.411316\pi\)
0.275018 + 0.961439i \(0.411316\pi\)
\(600\) 11.5778 0.472662
\(601\) −10.8488 −0.442531 −0.221266 0.975214i \(-0.571019\pi\)
−0.221266 + 0.975214i \(0.571019\pi\)
\(602\) −13.3386 −0.543642
\(603\) 5.57898 0.227194
\(604\) −9.15082 −0.372341
\(605\) −12.4853 −0.507599
\(606\) −9.92757 −0.403280
\(607\) 23.0492 0.935539 0.467770 0.883850i \(-0.345058\pi\)
0.467770 + 0.883850i \(0.345058\pi\)
\(608\) −3.27578 −0.132850
\(609\) 7.74990 0.314042
\(610\) 35.5436 1.43912
\(611\) −9.19458 −0.371973
\(612\) −4.96826 −0.200830
\(613\) −1.28093 −0.0517362 −0.0258681 0.999665i \(-0.508235\pi\)
−0.0258681 + 0.999665i \(0.508235\pi\)
\(614\) −6.96717 −0.281172
\(615\) 37.2441 1.50183
\(616\) −9.68087 −0.390053
\(617\) 17.0369 0.685881 0.342941 0.939357i \(-0.388577\pi\)
0.342941 + 0.939357i \(0.388577\pi\)
\(618\) 1.00000 0.0402259
\(619\) −1.79480 −0.0721390 −0.0360695 0.999349i \(-0.511484\pi\)
−0.0360695 + 0.999349i \(0.511484\pi\)
\(620\) −2.09913 −0.0843031
\(621\) −6.21258 −0.249302
\(622\) −12.9719 −0.520127
\(623\) 3.94080 0.157885
\(624\) 1.00000 0.0400320
\(625\) 51.1568 2.04627
\(626\) 5.37301 0.214749
\(627\) −12.2859 −0.490652
\(628\) 7.49283 0.298997
\(629\) 0.842622 0.0335975
\(630\) 10.5096 0.418712
\(631\) 39.1444 1.55831 0.779157 0.626829i \(-0.215648\pi\)
0.779157 + 0.626829i \(0.215648\pi\)
\(632\) −11.5740 −0.460390
\(633\) −17.7043 −0.703684
\(634\) −6.85531 −0.272259
\(635\) −58.8498 −2.33538
\(636\) 1.36025 0.0539374
\(637\) 0.337386 0.0133677
\(638\) 11.2607 0.445816
\(639\) −8.81509 −0.348720
\(640\) 4.07159 0.160944
\(641\) −3.41253 −0.134787 −0.0673934 0.997726i \(-0.521468\pi\)
−0.0673934 + 0.997726i \(0.521468\pi\)
\(642\) −12.6827 −0.500546
\(643\) −31.8411 −1.25569 −0.627845 0.778338i \(-0.716063\pi\)
−0.627845 + 0.778338i \(0.716063\pi\)
\(644\) 16.0359 0.631904
\(645\) 21.0403 0.828463
\(646\) 16.2749 0.640328
\(647\) −1.53527 −0.0603576 −0.0301788 0.999545i \(-0.509608\pi\)
−0.0301788 + 0.999545i \(0.509608\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −26.9578 −1.05818
\(650\) 11.5778 0.454119
\(651\) −1.33076 −0.0521564
\(652\) −24.8572 −0.973482
\(653\) 6.19853 0.242567 0.121284 0.992618i \(-0.461299\pi\)
0.121284 + 0.992618i \(0.461299\pi\)
\(654\) 6.63187 0.259327
\(655\) −31.4798 −1.23002
\(656\) 9.14732 0.357143
\(657\) −6.75816 −0.263661
\(658\) −23.7331 −0.925212
\(659\) −46.9403 −1.82853 −0.914267 0.405113i \(-0.867232\pi\)
−0.914267 + 0.405113i \(0.867232\pi\)
\(660\) 15.2706 0.594407
\(661\) −17.3549 −0.675028 −0.337514 0.941321i \(-0.609586\pi\)
−0.337514 + 0.941321i \(0.609586\pi\)
\(662\) −28.8357 −1.12073
\(663\) −4.96826 −0.192951
\(664\) 12.6995 0.492836
\(665\) −34.4271 −1.33503
\(666\) 0.169601 0.00657191
\(667\) −18.6529 −0.722242
\(668\) −24.7241 −0.956606
\(669\) −23.8931 −0.923759
\(670\) 22.7153 0.877568
\(671\) 32.7409 1.26395
\(672\) 2.58120 0.0995721
\(673\) −47.2284 −1.82052 −0.910261 0.414035i \(-0.864119\pi\)
−0.910261 + 0.414035i \(0.864119\pi\)
\(674\) 30.7712 1.18526
\(675\) −11.5778 −0.445630
\(676\) 1.00000 0.0384615
\(677\) −2.25869 −0.0868086 −0.0434043 0.999058i \(-0.513820\pi\)
−0.0434043 + 0.999058i \(0.513820\pi\)
\(678\) 16.1053 0.618519
\(679\) −4.54810 −0.174540
\(680\) −20.2287 −0.775735
\(681\) −11.5062 −0.440918
\(682\) −1.93361 −0.0740416
\(683\) −18.7360 −0.716915 −0.358457 0.933546i \(-0.616697\pi\)
−0.358457 + 0.933546i \(0.616697\pi\)
\(684\) 3.27578 0.125253
\(685\) −38.8592 −1.48473
\(686\) 18.9393 0.723105
\(687\) 5.01532 0.191346
\(688\) 5.16760 0.197013
\(689\) 1.36025 0.0518214
\(690\) −25.2951 −0.962966
\(691\) −15.3009 −0.582074 −0.291037 0.956712i \(-0.594000\pi\)
−0.291037 + 0.956712i \(0.594000\pi\)
\(692\) −4.49671 −0.170939
\(693\) 9.68087 0.367746
\(694\) −37.0016 −1.40456
\(695\) 35.0158 1.32822
\(696\) −3.00244 −0.113807
\(697\) −45.4462 −1.72140
\(698\) −19.4923 −0.737795
\(699\) −25.1837 −0.952536
\(700\) 29.8847 1.12954
\(701\) 29.4589 1.11265 0.556323 0.830966i \(-0.312212\pi\)
0.556323 + 0.830966i \(0.312212\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −0.555576 −0.0209540
\(704\) 3.75053 0.141353
\(705\) 37.4365 1.40994
\(706\) −19.9930 −0.752448
\(707\) −25.6251 −0.963730
\(708\) 7.18773 0.270131
\(709\) 7.78290 0.292293 0.146146 0.989263i \(-0.453313\pi\)
0.146146 + 0.989263i \(0.453313\pi\)
\(710\) −35.8914 −1.34698
\(711\) 11.5740 0.434059
\(712\) −1.52673 −0.0572166
\(713\) 3.20293 0.119951
\(714\) −12.8241 −0.479929
\(715\) 15.2706 0.571088
\(716\) −0.474184 −0.0177211
\(717\) −8.78476 −0.328073
\(718\) 16.0667 0.599605
\(719\) −0.207218 −0.00772794 −0.00386397 0.999993i \(-0.501230\pi\)
−0.00386397 + 0.999993i \(0.501230\pi\)
\(720\) −4.07159 −0.151739
\(721\) 2.58120 0.0961291
\(722\) 8.26925 0.307750
\(723\) −5.11482 −0.190222
\(724\) −6.40924 −0.238198
\(725\) −34.7617 −1.29102
\(726\) 3.06644 0.113806
\(727\) −30.8695 −1.14489 −0.572444 0.819944i \(-0.694005\pi\)
−0.572444 + 0.819944i \(0.694005\pi\)
\(728\) 2.58120 0.0956658
\(729\) 1.00000 0.0370370
\(730\) −27.5164 −1.01843
\(731\) −25.6740 −0.949587
\(732\) −8.72968 −0.322658
\(733\) −35.9740 −1.32873 −0.664366 0.747408i \(-0.731298\pi\)
−0.664366 + 0.747408i \(0.731298\pi\)
\(734\) −24.4433 −0.902218
\(735\) −1.37370 −0.0506695
\(736\) −6.21258 −0.228999
\(737\) 20.9241 0.770749
\(738\) −9.14732 −0.336717
\(739\) 25.9587 0.954907 0.477454 0.878657i \(-0.341560\pi\)
0.477454 + 0.878657i \(0.341560\pi\)
\(740\) 0.690546 0.0253850
\(741\) 3.27578 0.120339
\(742\) 3.51108 0.128896
\(743\) 36.4624 1.33768 0.668838 0.743408i \(-0.266792\pi\)
0.668838 + 0.743408i \(0.266792\pi\)
\(744\) 0.515556 0.0189012
\(745\) −49.3721 −1.80886
\(746\) −10.1846 −0.372884
\(747\) −12.6995 −0.464650
\(748\) −18.6336 −0.681311
\(749\) −32.7366 −1.19617
\(750\) −26.7822 −0.977946
\(751\) −50.8823 −1.85672 −0.928361 0.371679i \(-0.878782\pi\)
−0.928361 + 0.371679i \(0.878782\pi\)
\(752\) 9.19458 0.335292
\(753\) 2.96547 0.108068
\(754\) −3.00244 −0.109342
\(755\) 37.2583 1.35597
\(756\) −2.58120 −0.0938775
\(757\) 9.23185 0.335537 0.167769 0.985826i \(-0.446344\pi\)
0.167769 + 0.985826i \(0.446344\pi\)
\(758\) −35.1348 −1.27615
\(759\) −23.3004 −0.845752
\(760\) 13.3376 0.483807
\(761\) 38.0894 1.38074 0.690370 0.723456i \(-0.257448\pi\)
0.690370 + 0.723456i \(0.257448\pi\)
\(762\) 14.4538 0.523605
\(763\) 17.1182 0.619721
\(764\) −14.6130 −0.528681
\(765\) 20.2287 0.731370
\(766\) 4.80554 0.173631
\(767\) 7.18773 0.259534
\(768\) −1.00000 −0.0360844
\(769\) 45.4423 1.63869 0.819345 0.573301i \(-0.194337\pi\)
0.819345 + 0.573301i \(0.194337\pi\)
\(770\) 39.4165 1.42047
\(771\) 19.7234 0.710321
\(772\) −9.13430 −0.328751
\(773\) 8.18641 0.294445 0.147222 0.989103i \(-0.452967\pi\)
0.147222 + 0.989103i \(0.452967\pi\)
\(774\) −5.16760 −0.185746
\(775\) 5.96901 0.214413
\(776\) 1.76201 0.0632524
\(777\) 0.437775 0.0157051
\(778\) −11.0101 −0.394733
\(779\) 29.9646 1.07359
\(780\) −4.07159 −0.145786
\(781\) −33.0612 −1.18302
\(782\) 30.8657 1.10375
\(783\) 3.00244 0.107298
\(784\) −0.337386 −0.0120495
\(785\) −30.5077 −1.08887
\(786\) 7.73157 0.275776
\(787\) −2.63487 −0.0939231 −0.0469615 0.998897i \(-0.514954\pi\)
−0.0469615 + 0.998897i \(0.514954\pi\)
\(788\) −20.0325 −0.713629
\(789\) 27.6687 0.985032
\(790\) 47.1246 1.67662
\(791\) 41.5710 1.47809
\(792\) −3.75053 −0.133269
\(793\) −8.72968 −0.310000
\(794\) −35.9964 −1.27746
\(795\) −5.53838 −0.196426
\(796\) 1.01631 0.0360220
\(797\) −7.51714 −0.266271 −0.133135 0.991098i \(-0.542505\pi\)
−0.133135 + 0.991098i \(0.542505\pi\)
\(798\) 8.45546 0.299320
\(799\) −45.6810 −1.61608
\(800\) −11.5778 −0.409338
\(801\) 1.52673 0.0539443
\(802\) −26.5560 −0.937723
\(803\) −25.3466 −0.894464
\(804\) −5.57898 −0.196755
\(805\) −65.2917 −2.30123
\(806\) 0.515556 0.0181597
\(807\) 7.45176 0.262314
\(808\) 9.92757 0.349251
\(809\) 11.1468 0.391900 0.195950 0.980614i \(-0.437221\pi\)
0.195950 + 0.980614i \(0.437221\pi\)
\(810\) 4.07159 0.143061
\(811\) −9.67856 −0.339860 −0.169930 0.985456i \(-0.554354\pi\)
−0.169930 + 0.985456i \(0.554354\pi\)
\(812\) −7.74990 −0.271968
\(813\) 1.58452 0.0555715
\(814\) 0.636093 0.0222951
\(815\) 101.208 3.54517
\(816\) 4.96826 0.173924
\(817\) 16.9279 0.592234
\(818\) −13.5214 −0.472766
\(819\) −2.58120 −0.0901946
\(820\) −37.2441 −1.30062
\(821\) −3.72242 −0.129913 −0.0649567 0.997888i \(-0.520691\pi\)
−0.0649567 + 0.997888i \(0.520691\pi\)
\(822\) 9.54400 0.332885
\(823\) −20.5031 −0.714694 −0.357347 0.933972i \(-0.616319\pi\)
−0.357347 + 0.933972i \(0.616319\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −43.4229 −1.51179
\(826\) 18.5530 0.645541
\(827\) 3.08433 0.107253 0.0536263 0.998561i \(-0.482922\pi\)
0.0536263 + 0.998561i \(0.482922\pi\)
\(828\) 6.21258 0.215902
\(829\) −30.7010 −1.06629 −0.533144 0.846024i \(-0.678990\pi\)
−0.533144 + 0.846024i \(0.678990\pi\)
\(830\) −51.7071 −1.79478
\(831\) −1.30007 −0.0450989
\(832\) −1.00000 −0.0346688
\(833\) 1.67622 0.0580776
\(834\) −8.60003 −0.297795
\(835\) 100.666 3.48371
\(836\) 12.2859 0.424917
\(837\) −0.515556 −0.0178202
\(838\) 3.26115 0.112655
\(839\) −33.1287 −1.14373 −0.571864 0.820348i \(-0.693780\pi\)
−0.571864 + 0.820348i \(0.693780\pi\)
\(840\) −10.5096 −0.362615
\(841\) −19.9854 −0.689151
\(842\) −10.8036 −0.372317
\(843\) −13.8619 −0.477431
\(844\) 17.7043 0.609408
\(845\) −4.07159 −0.140067
\(846\) −9.19458 −0.316116
\(847\) 7.91511 0.271966
\(848\) −1.36025 −0.0467112
\(849\) 23.0435 0.790851
\(850\) 57.5216 1.97297
\(851\) −1.05366 −0.0361190
\(852\) 8.81509 0.302000
\(853\) 6.39008 0.218792 0.109396 0.993998i \(-0.465108\pi\)
0.109396 + 0.993998i \(0.465108\pi\)
\(854\) −22.5331 −0.771066
\(855\) −13.3376 −0.456137
\(856\) 12.6827 0.433485
\(857\) 20.9962 0.717216 0.358608 0.933488i \(-0.383251\pi\)
0.358608 + 0.933488i \(0.383251\pi\)
\(858\) −3.75053 −0.128041
\(859\) 5.21647 0.177984 0.0889919 0.996032i \(-0.471635\pi\)
0.0889919 + 0.996032i \(0.471635\pi\)
\(860\) −21.0403 −0.717470
\(861\) −23.6111 −0.804664
\(862\) −9.42187 −0.320910
\(863\) 28.5164 0.970710 0.485355 0.874317i \(-0.338690\pi\)
0.485355 + 0.874317i \(0.338690\pi\)
\(864\) 1.00000 0.0340207
\(865\) 18.3087 0.622515
\(866\) 14.6310 0.497181
\(867\) −7.68357 −0.260948
\(868\) 1.33076 0.0451688
\(869\) 43.4086 1.47254
\(870\) 12.2247 0.414455
\(871\) −5.57898 −0.189037
\(872\) −6.63187 −0.224583
\(873\) −1.76201 −0.0596349
\(874\) −20.3511 −0.688385
\(875\) −69.1302 −2.33703
\(876\) 6.75816 0.228337
\(877\) 43.7797 1.47834 0.739168 0.673521i \(-0.235219\pi\)
0.739168 + 0.673521i \(0.235219\pi\)
\(878\) −0.578539 −0.0195248
\(879\) 8.97969 0.302878
\(880\) −15.2706 −0.514771
\(881\) 48.0897 1.62018 0.810092 0.586303i \(-0.199417\pi\)
0.810092 + 0.586303i \(0.199417\pi\)
\(882\) 0.337386 0.0113604
\(883\) −35.3368 −1.18918 −0.594589 0.804030i \(-0.702685\pi\)
−0.594589 + 0.804030i \(0.702685\pi\)
\(884\) 4.96826 0.167101
\(885\) −29.2655 −0.983748
\(886\) 16.9554 0.569627
\(887\) −5.21276 −0.175027 −0.0875137 0.996163i \(-0.527892\pi\)
−0.0875137 + 0.996163i \(0.527892\pi\)
\(888\) −0.169601 −0.00569144
\(889\) 37.3082 1.25128
\(890\) 6.21621 0.208368
\(891\) 3.75053 0.125647
\(892\) 23.8931 0.799999
\(893\) 30.1194 1.00791
\(894\) 12.1260 0.405555
\(895\) 1.93068 0.0645355
\(896\) −2.58120 −0.0862320
\(897\) 6.21258 0.207432
\(898\) −16.6918 −0.557013
\(899\) −1.54792 −0.0516262
\(900\) 11.5778 0.385927
\(901\) 6.75807 0.225144
\(902\) −34.3072 −1.14231
\(903\) −13.3386 −0.443882
\(904\) −16.1053 −0.535653
\(905\) 26.0958 0.867453
\(906\) −9.15082 −0.304016
\(907\) 56.8488 1.88763 0.943816 0.330470i \(-0.107207\pi\)
0.943816 + 0.330470i \(0.107207\pi\)
\(908\) 11.5062 0.381846
\(909\) −9.92757 −0.329277
\(910\) −10.5096 −0.348390
\(911\) −8.13961 −0.269677 −0.134839 0.990868i \(-0.543052\pi\)
−0.134839 + 0.990868i \(0.543052\pi\)
\(912\) −3.27578 −0.108472
\(913\) −47.6297 −1.57631
\(914\) 0.498348 0.0164839
\(915\) 35.5436 1.17504
\(916\) −5.01532 −0.165711
\(917\) 19.9568 0.659030
\(918\) −4.96826 −0.163977
\(919\) −21.0280 −0.693649 −0.346824 0.937930i \(-0.612740\pi\)
−0.346824 + 0.937930i \(0.612740\pi\)
\(920\) 25.2951 0.833953
\(921\) −6.96717 −0.229576
\(922\) −2.29262 −0.0755034
\(923\) 8.81509 0.290152
\(924\) −9.68087 −0.318477
\(925\) −1.96361 −0.0645631
\(926\) −7.76040 −0.255023
\(927\) 1.00000 0.0328443
\(928\) 3.00244 0.0985598
\(929\) −52.0443 −1.70752 −0.853760 0.520667i \(-0.825683\pi\)
−0.853760 + 0.520667i \(0.825683\pi\)
\(930\) −2.09913 −0.0688332
\(931\) −1.10520 −0.0362215
\(932\) 25.1837 0.824921
\(933\) −12.9719 −0.424682
\(934\) 35.3243 1.15585
\(935\) 75.8682 2.48116
\(936\) 1.00000 0.0326860
\(937\) −10.5363 −0.344205 −0.172102 0.985079i \(-0.555056\pi\)
−0.172102 + 0.985079i \(0.555056\pi\)
\(938\) −14.4005 −0.470192
\(939\) 5.37301 0.175342
\(940\) −37.4365 −1.22105
\(941\) 6.00150 0.195643 0.0978217 0.995204i \(-0.468813\pi\)
0.0978217 + 0.995204i \(0.468813\pi\)
\(942\) 7.49283 0.244130
\(943\) 56.8284 1.85059
\(944\) −7.18773 −0.233941
\(945\) 10.5096 0.341877
\(946\) −19.3812 −0.630138
\(947\) −33.8206 −1.09902 −0.549511 0.835487i \(-0.685186\pi\)
−0.549511 + 0.835487i \(0.685186\pi\)
\(948\) −11.5740 −0.375907
\(949\) 6.75816 0.219379
\(950\) −37.9264 −1.23050
\(951\) −6.85531 −0.222299
\(952\) 12.8241 0.415631
\(953\) −6.31648 −0.204611 −0.102305 0.994753i \(-0.532622\pi\)
−0.102305 + 0.994753i \(0.532622\pi\)
\(954\) 1.36025 0.0440397
\(955\) 59.4983 1.92532
\(956\) 8.78476 0.284119
\(957\) 11.2607 0.364007
\(958\) −37.0768 −1.19790
\(959\) 24.6350 0.795506
\(960\) 4.07159 0.131410
\(961\) −30.7342 −0.991426
\(962\) −0.169601 −0.00546816
\(963\) −12.6827 −0.408694
\(964\) 5.11482 0.164737
\(965\) 37.1911 1.19722
\(966\) 16.0359 0.515948
\(967\) 58.9751 1.89651 0.948256 0.317506i \(-0.102845\pi\)
0.948256 + 0.317506i \(0.102845\pi\)
\(968\) −3.06644 −0.0985592
\(969\) 16.2749 0.522826
\(970\) −7.17417 −0.230349
\(971\) 47.2237 1.51548 0.757741 0.652556i \(-0.226303\pi\)
0.757741 + 0.652556i \(0.226303\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −22.1984 −0.711649
\(974\) −14.8710 −0.476497
\(975\) 11.5778 0.370787
\(976\) 8.72968 0.279430
\(977\) −47.3246 −1.51405 −0.757024 0.653387i \(-0.773348\pi\)
−0.757024 + 0.653387i \(0.773348\pi\)
\(978\) −24.8572 −0.794845
\(979\) 5.72604 0.183005
\(980\) 1.37370 0.0438811
\(981\) 6.63187 0.211739
\(982\) −16.8665 −0.538232
\(983\) 51.3562 1.63801 0.819004 0.573788i \(-0.194527\pi\)
0.819004 + 0.573788i \(0.194527\pi\)
\(984\) 9.14732 0.291606
\(985\) 81.5641 2.59885
\(986\) −14.9169 −0.475050
\(987\) −23.7331 −0.755433
\(988\) −3.27578 −0.104217
\(989\) 32.1041 1.02085
\(990\) 15.2706 0.485331
\(991\) 31.3541 0.995997 0.497998 0.867178i \(-0.334069\pi\)
0.497998 + 0.867178i \(0.334069\pi\)
\(992\) −0.515556 −0.0163689
\(993\) −28.8357 −0.915073
\(994\) 22.7536 0.721699
\(995\) −4.13798 −0.131183
\(996\) 12.6995 0.402399
\(997\) 49.2862 1.56091 0.780455 0.625212i \(-0.214987\pi\)
0.780455 + 0.625212i \(0.214987\pi\)
\(998\) −16.9371 −0.536135
\(999\) 0.169601 0.00536594
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.ba.1.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.1 14 1.1 even 1 trivial