Properties

Label 8034.2.a.bb.1.8
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.290663\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.290663 q^{5} -1.00000 q^{6} -3.15490 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.290663 q^{5} -1.00000 q^{6} -3.15490 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.290663 q^{10} +5.57708 q^{11} +1.00000 q^{12} -1.00000 q^{13} +3.15490 q^{14} +0.290663 q^{15} +1.00000 q^{16} -2.13717 q^{17} -1.00000 q^{18} -5.05516 q^{19} +0.290663 q^{20} -3.15490 q^{21} -5.57708 q^{22} +1.96903 q^{23} -1.00000 q^{24} -4.91552 q^{25} +1.00000 q^{26} +1.00000 q^{27} -3.15490 q^{28} -0.444744 q^{29} -0.290663 q^{30} +2.60815 q^{31} -1.00000 q^{32} +5.57708 q^{33} +2.13717 q^{34} -0.917011 q^{35} +1.00000 q^{36} -7.55167 q^{37} +5.05516 q^{38} -1.00000 q^{39} -0.290663 q^{40} +8.43427 q^{41} +3.15490 q^{42} -1.60263 q^{43} +5.57708 q^{44} +0.290663 q^{45} -1.96903 q^{46} +9.45661 q^{47} +1.00000 q^{48} +2.95338 q^{49} +4.91552 q^{50} -2.13717 q^{51} -1.00000 q^{52} +13.2019 q^{53} -1.00000 q^{54} +1.62105 q^{55} +3.15490 q^{56} -5.05516 q^{57} +0.444744 q^{58} -10.9582 q^{59} +0.290663 q^{60} -0.462620 q^{61} -2.60815 q^{62} -3.15490 q^{63} +1.00000 q^{64} -0.290663 q^{65} -5.57708 q^{66} +13.6224 q^{67} -2.13717 q^{68} +1.96903 q^{69} +0.917011 q^{70} -8.17349 q^{71} -1.00000 q^{72} -15.1964 q^{73} +7.55167 q^{74} -4.91552 q^{75} -5.05516 q^{76} -17.5951 q^{77} +1.00000 q^{78} -14.5749 q^{79} +0.290663 q^{80} +1.00000 q^{81} -8.43427 q^{82} -12.2435 q^{83} -3.15490 q^{84} -0.621194 q^{85} +1.60263 q^{86} -0.444744 q^{87} -5.57708 q^{88} -7.96168 q^{89} -0.290663 q^{90} +3.15490 q^{91} +1.96903 q^{92} +2.60815 q^{93} -9.45661 q^{94} -1.46935 q^{95} -1.00000 q^{96} +17.9490 q^{97} -2.95338 q^{98} +5.57708 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) 0.290663 0.129988 0.0649942 0.997886i \(-0.479297\pi\)
0.0649942 + 0.997886i \(0.479297\pi\)
\(6\) −1.00000 −0.408248
\(7\) −3.15490 −1.19244 −0.596220 0.802821i \(-0.703331\pi\)
−0.596220 + 0.802821i \(0.703331\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.290663 −0.0919156
\(11\) 5.57708 1.68155 0.840776 0.541383i \(-0.182099\pi\)
0.840776 + 0.541383i \(0.182099\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 3.15490 0.843182
\(15\) 0.290663 0.0750488
\(16\) 1.00000 0.250000
\(17\) −2.13717 −0.518339 −0.259169 0.965832i \(-0.583449\pi\)
−0.259169 + 0.965832i \(0.583449\pi\)
\(18\) −1.00000 −0.235702
\(19\) −5.05516 −1.15973 −0.579867 0.814711i \(-0.696895\pi\)
−0.579867 + 0.814711i \(0.696895\pi\)
\(20\) 0.290663 0.0649942
\(21\) −3.15490 −0.688455
\(22\) −5.57708 −1.18904
\(23\) 1.96903 0.410571 0.205286 0.978702i \(-0.434188\pi\)
0.205286 + 0.978702i \(0.434188\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.91552 −0.983103
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −3.15490 −0.596220
\(29\) −0.444744 −0.0825870 −0.0412935 0.999147i \(-0.513148\pi\)
−0.0412935 + 0.999147i \(0.513148\pi\)
\(30\) −0.290663 −0.0530675
\(31\) 2.60815 0.468438 0.234219 0.972184i \(-0.424747\pi\)
0.234219 + 0.972184i \(0.424747\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.57708 0.970845
\(34\) 2.13717 0.366521
\(35\) −0.917011 −0.155003
\(36\) 1.00000 0.166667
\(37\) −7.55167 −1.24149 −0.620743 0.784014i \(-0.713169\pi\)
−0.620743 + 0.784014i \(0.713169\pi\)
\(38\) 5.05516 0.820055
\(39\) −1.00000 −0.160128
\(40\) −0.290663 −0.0459578
\(41\) 8.43427 1.31721 0.658606 0.752488i \(-0.271147\pi\)
0.658606 + 0.752488i \(0.271147\pi\)
\(42\) 3.15490 0.486811
\(43\) −1.60263 −0.244399 −0.122199 0.992506i \(-0.538995\pi\)
−0.122199 + 0.992506i \(0.538995\pi\)
\(44\) 5.57708 0.840776
\(45\) 0.290663 0.0433294
\(46\) −1.96903 −0.290318
\(47\) 9.45661 1.37939 0.689694 0.724101i \(-0.257745\pi\)
0.689694 + 0.724101i \(0.257745\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.95338 0.421911
\(50\) 4.91552 0.695159
\(51\) −2.13717 −0.299263
\(52\) −1.00000 −0.138675
\(53\) 13.2019 1.81342 0.906708 0.421758i \(-0.138587\pi\)
0.906708 + 0.421758i \(0.138587\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.62105 0.218582
\(56\) 3.15490 0.421591
\(57\) −5.05516 −0.669572
\(58\) 0.444744 0.0583978
\(59\) −10.9582 −1.42663 −0.713317 0.700841i \(-0.752808\pi\)
−0.713317 + 0.700841i \(0.752808\pi\)
\(60\) 0.290663 0.0375244
\(61\) −0.462620 −0.0592324 −0.0296162 0.999561i \(-0.509429\pi\)
−0.0296162 + 0.999561i \(0.509429\pi\)
\(62\) −2.60815 −0.331236
\(63\) −3.15490 −0.397480
\(64\) 1.00000 0.125000
\(65\) −0.290663 −0.0360523
\(66\) −5.57708 −0.686491
\(67\) 13.6224 1.66424 0.832118 0.554599i \(-0.187128\pi\)
0.832118 + 0.554599i \(0.187128\pi\)
\(68\) −2.13717 −0.259169
\(69\) 1.96903 0.237043
\(70\) 0.917011 0.109604
\(71\) −8.17349 −0.970014 −0.485007 0.874510i \(-0.661183\pi\)
−0.485007 + 0.874510i \(0.661183\pi\)
\(72\) −1.00000 −0.117851
\(73\) −15.1964 −1.77860 −0.889302 0.457320i \(-0.848809\pi\)
−0.889302 + 0.457320i \(0.848809\pi\)
\(74\) 7.55167 0.877863
\(75\) −4.91552 −0.567595
\(76\) −5.05516 −0.579867
\(77\) −17.5951 −2.00515
\(78\) 1.00000 0.113228
\(79\) −14.5749 −1.63980 −0.819901 0.572505i \(-0.805972\pi\)
−0.819901 + 0.572505i \(0.805972\pi\)
\(80\) 0.290663 0.0324971
\(81\) 1.00000 0.111111
\(82\) −8.43427 −0.931409
\(83\) −12.2435 −1.34389 −0.671947 0.740599i \(-0.734542\pi\)
−0.671947 + 0.740599i \(0.734542\pi\)
\(84\) −3.15490 −0.344228
\(85\) −0.621194 −0.0673780
\(86\) 1.60263 0.172816
\(87\) −0.444744 −0.0476816
\(88\) −5.57708 −0.594519
\(89\) −7.96168 −0.843937 −0.421968 0.906611i \(-0.638661\pi\)
−0.421968 + 0.906611i \(0.638661\pi\)
\(90\) −0.290663 −0.0306385
\(91\) 3.15490 0.330723
\(92\) 1.96903 0.205286
\(93\) 2.60815 0.270453
\(94\) −9.45661 −0.975374
\(95\) −1.46935 −0.150752
\(96\) −1.00000 −0.102062
\(97\) 17.9490 1.82244 0.911222 0.411916i \(-0.135140\pi\)
0.911222 + 0.411916i \(0.135140\pi\)
\(98\) −2.95338 −0.298336
\(99\) 5.57708 0.560518
\(100\) −4.91552 −0.491552
\(101\) −5.29560 −0.526932 −0.263466 0.964669i \(-0.584866\pi\)
−0.263466 + 0.964669i \(0.584866\pi\)
\(102\) 2.13717 0.211611
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −0.917011 −0.0894911
\(106\) −13.2019 −1.28228
\(107\) 1.03117 0.0996867 0.0498433 0.998757i \(-0.484128\pi\)
0.0498433 + 0.998757i \(0.484128\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.95257 −0.953284 −0.476642 0.879098i \(-0.658146\pi\)
−0.476642 + 0.879098i \(0.658146\pi\)
\(110\) −1.62105 −0.154561
\(111\) −7.55167 −0.716772
\(112\) −3.15490 −0.298110
\(113\) −8.67226 −0.815818 −0.407909 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(114\) 5.05516 0.473459
\(115\) 0.572324 0.0533695
\(116\) −0.444744 −0.0412935
\(117\) −1.00000 −0.0924500
\(118\) 10.9582 1.00878
\(119\) 6.74254 0.618088
\(120\) −0.290663 −0.0265338
\(121\) 20.1038 1.82762
\(122\) 0.462620 0.0418836
\(123\) 8.43427 0.760492
\(124\) 2.60815 0.234219
\(125\) −2.88207 −0.257780
\(126\) 3.15490 0.281061
\(127\) −1.72805 −0.153340 −0.0766699 0.997057i \(-0.524429\pi\)
−0.0766699 + 0.997057i \(0.524429\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.60263 −0.141104
\(130\) 0.290663 0.0254928
\(131\) −13.9485 −1.21869 −0.609343 0.792907i \(-0.708567\pi\)
−0.609343 + 0.792907i \(0.708567\pi\)
\(132\) 5.57708 0.485422
\(133\) 15.9485 1.38291
\(134\) −13.6224 −1.17679
\(135\) 0.290663 0.0250163
\(136\) 2.13717 0.183260
\(137\) −7.85127 −0.670779 −0.335390 0.942079i \(-0.608868\pi\)
−0.335390 + 0.942079i \(0.608868\pi\)
\(138\) −1.96903 −0.167615
\(139\) 15.4944 1.31422 0.657108 0.753796i \(-0.271779\pi\)
0.657108 + 0.753796i \(0.271779\pi\)
\(140\) −0.917011 −0.0775016
\(141\) 9.45661 0.796390
\(142\) 8.17349 0.685904
\(143\) −5.57708 −0.466379
\(144\) 1.00000 0.0833333
\(145\) −0.129271 −0.0107353
\(146\) 15.1964 1.25766
\(147\) 2.95338 0.243591
\(148\) −7.55167 −0.620743
\(149\) −17.7998 −1.45822 −0.729109 0.684398i \(-0.760065\pi\)
−0.729109 + 0.684398i \(0.760065\pi\)
\(150\) 4.91552 0.401350
\(151\) −6.49777 −0.528781 −0.264391 0.964416i \(-0.585171\pi\)
−0.264391 + 0.964416i \(0.585171\pi\)
\(152\) 5.05516 0.410028
\(153\) −2.13717 −0.172780
\(154\) 17.5951 1.41785
\(155\) 0.758093 0.0608915
\(156\) −1.00000 −0.0800641
\(157\) −11.9055 −0.950165 −0.475082 0.879941i \(-0.657582\pi\)
−0.475082 + 0.879941i \(0.657582\pi\)
\(158\) 14.5749 1.15952
\(159\) 13.2019 1.04698
\(160\) −0.290663 −0.0229789
\(161\) −6.21209 −0.489581
\(162\) −1.00000 −0.0785674
\(163\) 3.87572 0.303570 0.151785 0.988414i \(-0.451498\pi\)
0.151785 + 0.988414i \(0.451498\pi\)
\(164\) 8.43427 0.658606
\(165\) 1.62105 0.126199
\(166\) 12.2435 0.950277
\(167\) 4.93497 0.381880 0.190940 0.981602i \(-0.438846\pi\)
0.190940 + 0.981602i \(0.438846\pi\)
\(168\) 3.15490 0.243406
\(169\) 1.00000 0.0769231
\(170\) 0.621194 0.0476434
\(171\) −5.05516 −0.386578
\(172\) −1.60263 −0.122199
\(173\) 24.0000 1.82469 0.912345 0.409422i \(-0.134270\pi\)
0.912345 + 0.409422i \(0.134270\pi\)
\(174\) 0.444744 0.0337160
\(175\) 15.5079 1.17229
\(176\) 5.57708 0.420388
\(177\) −10.9582 −0.823668
\(178\) 7.96168 0.596753
\(179\) 18.1673 1.35789 0.678946 0.734189i \(-0.262437\pi\)
0.678946 + 0.734189i \(0.262437\pi\)
\(180\) 0.290663 0.0216647
\(181\) −7.40216 −0.550198 −0.275099 0.961416i \(-0.588711\pi\)
−0.275099 + 0.961416i \(0.588711\pi\)
\(182\) −3.15490 −0.233857
\(183\) −0.462620 −0.0341978
\(184\) −1.96903 −0.145159
\(185\) −2.19499 −0.161379
\(186\) −2.60815 −0.191239
\(187\) −11.9191 −0.871614
\(188\) 9.45661 0.689694
\(189\) −3.15490 −0.229485
\(190\) 1.46935 0.106598
\(191\) 1.61795 0.117071 0.0585354 0.998285i \(-0.481357\pi\)
0.0585354 + 0.998285i \(0.481357\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.89904 −0.208677 −0.104339 0.994542i \(-0.533273\pi\)
−0.104339 + 0.994542i \(0.533273\pi\)
\(194\) −17.9490 −1.28866
\(195\) −0.290663 −0.0208148
\(196\) 2.95338 0.210956
\(197\) −5.32513 −0.379400 −0.189700 0.981842i \(-0.560751\pi\)
−0.189700 + 0.981842i \(0.560751\pi\)
\(198\) −5.57708 −0.396346
\(199\) −15.2964 −1.08433 −0.542167 0.840271i \(-0.682396\pi\)
−0.542167 + 0.840271i \(0.682396\pi\)
\(200\) 4.91552 0.347579
\(201\) 13.6224 0.960847
\(202\) 5.29560 0.372597
\(203\) 1.40312 0.0984799
\(204\) −2.13717 −0.149632
\(205\) 2.45153 0.171222
\(206\) −1.00000 −0.0696733
\(207\) 1.96903 0.136857
\(208\) −1.00000 −0.0693375
\(209\) −28.1930 −1.95015
\(210\) 0.917011 0.0632798
\(211\) −5.18412 −0.356890 −0.178445 0.983950i \(-0.557107\pi\)
−0.178445 + 0.983950i \(0.557107\pi\)
\(212\) 13.2019 0.906708
\(213\) −8.17349 −0.560038
\(214\) −1.03117 −0.0704891
\(215\) −0.465824 −0.0317690
\(216\) −1.00000 −0.0680414
\(217\) −8.22846 −0.558584
\(218\) 9.95257 0.674073
\(219\) −15.1964 −1.02688
\(220\) 1.62105 0.109291
\(221\) 2.13717 0.143761
\(222\) 7.55167 0.506835
\(223\) 18.0617 1.20950 0.604749 0.796416i \(-0.293273\pi\)
0.604749 + 0.796416i \(0.293273\pi\)
\(224\) 3.15490 0.210795
\(225\) −4.91552 −0.327701
\(226\) 8.67226 0.576870
\(227\) 8.33048 0.552913 0.276457 0.961026i \(-0.410840\pi\)
0.276457 + 0.961026i \(0.410840\pi\)
\(228\) −5.05516 −0.334786
\(229\) −1.04971 −0.0693666 −0.0346833 0.999398i \(-0.511042\pi\)
−0.0346833 + 0.999398i \(0.511042\pi\)
\(230\) −0.572324 −0.0377379
\(231\) −17.5951 −1.15767
\(232\) 0.444744 0.0291989
\(233\) −17.9285 −1.17454 −0.587268 0.809392i \(-0.699797\pi\)
−0.587268 + 0.809392i \(0.699797\pi\)
\(234\) 1.00000 0.0653720
\(235\) 2.74868 0.179304
\(236\) −10.9582 −0.713317
\(237\) −14.5749 −0.946740
\(238\) −6.74254 −0.437054
\(239\) −28.4472 −1.84010 −0.920049 0.391803i \(-0.871851\pi\)
−0.920049 + 0.391803i \(0.871851\pi\)
\(240\) 0.290663 0.0187622
\(241\) −23.7447 −1.52953 −0.764766 0.644309i \(-0.777145\pi\)
−0.764766 + 0.644309i \(0.777145\pi\)
\(242\) −20.1038 −1.29232
\(243\) 1.00000 0.0641500
\(244\) −0.462620 −0.0296162
\(245\) 0.858438 0.0548436
\(246\) −8.43427 −0.537749
\(247\) 5.05516 0.321652
\(248\) −2.60815 −0.165618
\(249\) −12.2435 −0.775898
\(250\) 2.88207 0.182278
\(251\) 2.69531 0.170127 0.0850634 0.996376i \(-0.472891\pi\)
0.0850634 + 0.996376i \(0.472891\pi\)
\(252\) −3.15490 −0.198740
\(253\) 10.9814 0.690397
\(254\) 1.72805 0.108428
\(255\) −0.621194 −0.0389007
\(256\) 1.00000 0.0625000
\(257\) 29.5475 1.84312 0.921562 0.388230i \(-0.126913\pi\)
0.921562 + 0.388230i \(0.126913\pi\)
\(258\) 1.60263 0.0997753
\(259\) 23.8247 1.48040
\(260\) −0.290663 −0.0180261
\(261\) −0.444744 −0.0275290
\(262\) 13.9485 0.861740
\(263\) −8.69529 −0.536175 −0.268087 0.963395i \(-0.586392\pi\)
−0.268087 + 0.963395i \(0.586392\pi\)
\(264\) −5.57708 −0.343245
\(265\) 3.83729 0.235723
\(266\) −15.9485 −0.977866
\(267\) −7.96168 −0.487247
\(268\) 13.6224 0.832118
\(269\) 2.38908 0.145665 0.0728323 0.997344i \(-0.476796\pi\)
0.0728323 + 0.997344i \(0.476796\pi\)
\(270\) −0.290663 −0.0176892
\(271\) −27.3531 −1.66158 −0.830790 0.556586i \(-0.812111\pi\)
−0.830790 + 0.556586i \(0.812111\pi\)
\(272\) −2.13717 −0.129585
\(273\) 3.15490 0.190943
\(274\) 7.85127 0.474313
\(275\) −27.4142 −1.65314
\(276\) 1.96903 0.118522
\(277\) 7.20989 0.433200 0.216600 0.976260i \(-0.430503\pi\)
0.216600 + 0.976260i \(0.430503\pi\)
\(278\) −15.4944 −0.929292
\(279\) 2.60815 0.156146
\(280\) 0.917011 0.0548019
\(281\) 9.84747 0.587451 0.293725 0.955890i \(-0.405105\pi\)
0.293725 + 0.955890i \(0.405105\pi\)
\(282\) −9.45661 −0.563133
\(283\) −11.7273 −0.697116 −0.348558 0.937287i \(-0.613329\pi\)
−0.348558 + 0.937287i \(0.613329\pi\)
\(284\) −8.17349 −0.485007
\(285\) −1.46935 −0.0870366
\(286\) 5.57708 0.329780
\(287\) −26.6092 −1.57069
\(288\) −1.00000 −0.0589256
\(289\) −12.4325 −0.731325
\(290\) 0.129271 0.00759103
\(291\) 17.9490 1.05219
\(292\) −15.1964 −0.889302
\(293\) −23.3098 −1.36177 −0.680887 0.732388i \(-0.738406\pi\)
−0.680887 + 0.732388i \(0.738406\pi\)
\(294\) −2.95338 −0.172245
\(295\) −3.18514 −0.185446
\(296\) 7.55167 0.438932
\(297\) 5.57708 0.323615
\(298\) 17.7998 1.03112
\(299\) −1.96903 −0.113872
\(300\) −4.91552 −0.283797
\(301\) 5.05613 0.291430
\(302\) 6.49777 0.373905
\(303\) −5.29560 −0.304224
\(304\) −5.05516 −0.289933
\(305\) −0.134466 −0.00769952
\(306\) 2.13717 0.122174
\(307\) 14.8883 0.849718 0.424859 0.905260i \(-0.360324\pi\)
0.424859 + 0.905260i \(0.360324\pi\)
\(308\) −17.5951 −1.00257
\(309\) 1.00000 0.0568880
\(310\) −0.758093 −0.0430568
\(311\) 32.4727 1.84136 0.920679 0.390321i \(-0.127636\pi\)
0.920679 + 0.390321i \(0.127636\pi\)
\(312\) 1.00000 0.0566139
\(313\) 21.6485 1.22365 0.611824 0.790994i \(-0.290436\pi\)
0.611824 + 0.790994i \(0.290436\pi\)
\(314\) 11.9055 0.671868
\(315\) −0.917011 −0.0516677
\(316\) −14.5749 −0.819901
\(317\) −11.4084 −0.640762 −0.320381 0.947289i \(-0.603811\pi\)
−0.320381 + 0.947289i \(0.603811\pi\)
\(318\) −13.2019 −0.740324
\(319\) −2.48037 −0.138874
\(320\) 0.290663 0.0162485
\(321\) 1.03117 0.0575541
\(322\) 6.21209 0.346186
\(323\) 10.8037 0.601135
\(324\) 1.00000 0.0555556
\(325\) 4.91552 0.272664
\(326\) −3.87572 −0.214656
\(327\) −9.95257 −0.550379
\(328\) −8.43427 −0.465704
\(329\) −29.8346 −1.64484
\(330\) −1.62105 −0.0892358
\(331\) −15.1788 −0.834300 −0.417150 0.908838i \(-0.636971\pi\)
−0.417150 + 0.908838i \(0.636971\pi\)
\(332\) −12.2435 −0.671947
\(333\) −7.55167 −0.413829
\(334\) −4.93497 −0.270030
\(335\) 3.95951 0.216331
\(336\) −3.15490 −0.172114
\(337\) −2.69714 −0.146923 −0.0734614 0.997298i \(-0.523405\pi\)
−0.0734614 + 0.997298i \(0.523405\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −8.67226 −0.471013
\(340\) −0.621194 −0.0336890
\(341\) 14.5459 0.787703
\(342\) 5.05516 0.273352
\(343\) 12.7667 0.689336
\(344\) 1.60263 0.0864079
\(345\) 0.572324 0.0308129
\(346\) −24.0000 −1.29025
\(347\) 10.1270 0.543647 0.271824 0.962347i \(-0.412373\pi\)
0.271824 + 0.962347i \(0.412373\pi\)
\(348\) −0.444744 −0.0238408
\(349\) −17.6111 −0.942698 −0.471349 0.881947i \(-0.656233\pi\)
−0.471349 + 0.881947i \(0.656233\pi\)
\(350\) −15.5079 −0.828935
\(351\) −1.00000 −0.0533761
\(352\) −5.57708 −0.297259
\(353\) −7.45823 −0.396961 −0.198481 0.980105i \(-0.563601\pi\)
−0.198481 + 0.980105i \(0.563601\pi\)
\(354\) 10.9582 0.582421
\(355\) −2.37573 −0.126091
\(356\) −7.96168 −0.421968
\(357\) 6.74254 0.356853
\(358\) −18.1673 −0.960174
\(359\) −0.109489 −0.00577861 −0.00288931 0.999996i \(-0.500920\pi\)
−0.00288931 + 0.999996i \(0.500920\pi\)
\(360\) −0.290663 −0.0153193
\(361\) 6.55465 0.344982
\(362\) 7.40216 0.389049
\(363\) 20.1038 1.05518
\(364\) 3.15490 0.165362
\(365\) −4.41703 −0.231198
\(366\) 0.462620 0.0241815
\(367\) −23.9306 −1.24916 −0.624582 0.780959i \(-0.714731\pi\)
−0.624582 + 0.780959i \(0.714731\pi\)
\(368\) 1.96903 0.102643
\(369\) 8.43427 0.439070
\(370\) 2.19499 0.114112
\(371\) −41.6506 −2.16239
\(372\) 2.60815 0.135226
\(373\) 2.83392 0.146735 0.0733673 0.997305i \(-0.476625\pi\)
0.0733673 + 0.997305i \(0.476625\pi\)
\(374\) 11.9191 0.616324
\(375\) −2.88207 −0.148830
\(376\) −9.45661 −0.487687
\(377\) 0.444744 0.0229055
\(378\) 3.15490 0.162270
\(379\) 2.43968 0.125318 0.0626590 0.998035i \(-0.480042\pi\)
0.0626590 + 0.998035i \(0.480042\pi\)
\(380\) −1.46935 −0.0753759
\(381\) −1.72805 −0.0885308
\(382\) −1.61795 −0.0827815
\(383\) 7.68470 0.392670 0.196335 0.980537i \(-0.437096\pi\)
0.196335 + 0.980537i \(0.437096\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −5.11424 −0.260646
\(386\) 2.89904 0.147557
\(387\) −1.60263 −0.0814662
\(388\) 17.9490 0.911222
\(389\) −15.0757 −0.764370 −0.382185 0.924086i \(-0.624828\pi\)
−0.382185 + 0.924086i \(0.624828\pi\)
\(390\) 0.290663 0.0147183
\(391\) −4.20814 −0.212815
\(392\) −2.95338 −0.149168
\(393\) −13.9485 −0.703608
\(394\) 5.32513 0.268276
\(395\) −4.23638 −0.213155
\(396\) 5.57708 0.280259
\(397\) −19.8678 −0.997138 −0.498569 0.866850i \(-0.666141\pi\)
−0.498569 + 0.866850i \(0.666141\pi\)
\(398\) 15.2964 0.766740
\(399\) 15.9485 0.798424
\(400\) −4.91552 −0.245776
\(401\) 8.07833 0.403412 0.201706 0.979446i \(-0.435351\pi\)
0.201706 + 0.979446i \(0.435351\pi\)
\(402\) −13.6224 −0.679421
\(403\) −2.60815 −0.129921
\(404\) −5.29560 −0.263466
\(405\) 0.290663 0.0144431
\(406\) −1.40312 −0.0696358
\(407\) −42.1162 −2.08762
\(408\) 2.13717 0.105805
\(409\) −1.67295 −0.0827219 −0.0413609 0.999144i \(-0.513169\pi\)
−0.0413609 + 0.999144i \(0.513169\pi\)
\(410\) −2.45153 −0.121072
\(411\) −7.85127 −0.387275
\(412\) 1.00000 0.0492665
\(413\) 34.5720 1.70118
\(414\) −1.96903 −0.0967725
\(415\) −3.55872 −0.174691
\(416\) 1.00000 0.0490290
\(417\) 15.4944 0.758764
\(418\) 28.1930 1.37897
\(419\) −29.6169 −1.44688 −0.723439 0.690388i \(-0.757440\pi\)
−0.723439 + 0.690388i \(0.757440\pi\)
\(420\) −0.917011 −0.0447456
\(421\) 21.6503 1.05517 0.527586 0.849502i \(-0.323097\pi\)
0.527586 + 0.849502i \(0.323097\pi\)
\(422\) 5.18412 0.252359
\(423\) 9.45661 0.459796
\(424\) −13.2019 −0.641140
\(425\) 10.5053 0.509580
\(426\) 8.17349 0.396007
\(427\) 1.45952 0.0706310
\(428\) 1.03117 0.0498433
\(429\) −5.57708 −0.269264
\(430\) 0.465824 0.0224640
\(431\) −12.8563 −0.619267 −0.309633 0.950856i \(-0.600206\pi\)
−0.309633 + 0.950856i \(0.600206\pi\)
\(432\) 1.00000 0.0481125
\(433\) 5.64923 0.271485 0.135742 0.990744i \(-0.456658\pi\)
0.135742 + 0.990744i \(0.456658\pi\)
\(434\) 8.22846 0.394978
\(435\) −0.129271 −0.00619805
\(436\) −9.95257 −0.476642
\(437\) −9.95376 −0.476153
\(438\) 15.1964 0.726112
\(439\) −3.24402 −0.154829 −0.0774144 0.996999i \(-0.524666\pi\)
−0.0774144 + 0.996999i \(0.524666\pi\)
\(440\) −1.62105 −0.0772805
\(441\) 2.95338 0.140637
\(442\) −2.13717 −0.101655
\(443\) 3.93720 0.187062 0.0935309 0.995616i \(-0.470185\pi\)
0.0935309 + 0.995616i \(0.470185\pi\)
\(444\) −7.55167 −0.358386
\(445\) −2.31416 −0.109702
\(446\) −18.0617 −0.855245
\(447\) −17.7998 −0.841902
\(448\) −3.15490 −0.149055
\(449\) −32.6440 −1.54056 −0.770282 0.637703i \(-0.779885\pi\)
−0.770282 + 0.637703i \(0.779885\pi\)
\(450\) 4.91552 0.231720
\(451\) 47.0386 2.21496
\(452\) −8.67226 −0.407909
\(453\) −6.49777 −0.305292
\(454\) −8.33048 −0.390969
\(455\) 0.917011 0.0429902
\(456\) 5.05516 0.236730
\(457\) 30.0476 1.40557 0.702784 0.711403i \(-0.251940\pi\)
0.702784 + 0.711403i \(0.251940\pi\)
\(458\) 1.04971 0.0490496
\(459\) −2.13717 −0.0997543
\(460\) 0.572324 0.0266847
\(461\) −28.1535 −1.31124 −0.655621 0.755090i \(-0.727593\pi\)
−0.655621 + 0.755090i \(0.727593\pi\)
\(462\) 17.5951 0.818599
\(463\) −28.2939 −1.31493 −0.657464 0.753486i \(-0.728371\pi\)
−0.657464 + 0.753486i \(0.728371\pi\)
\(464\) −0.444744 −0.0206467
\(465\) 0.758093 0.0351557
\(466\) 17.9285 0.830523
\(467\) −17.8245 −0.824821 −0.412410 0.910998i \(-0.635313\pi\)
−0.412410 + 0.910998i \(0.635313\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −42.9771 −1.98450
\(470\) −2.74868 −0.126787
\(471\) −11.9055 −0.548578
\(472\) 10.9582 0.504392
\(473\) −8.93798 −0.410969
\(474\) 14.5749 0.669446
\(475\) 24.8487 1.14014
\(476\) 6.74254 0.309044
\(477\) 13.2019 0.604472
\(478\) 28.4472 1.30115
\(479\) −3.47708 −0.158872 −0.0794359 0.996840i \(-0.525312\pi\)
−0.0794359 + 0.996840i \(0.525312\pi\)
\(480\) −0.290663 −0.0132669
\(481\) 7.55167 0.344326
\(482\) 23.7447 1.08154
\(483\) −6.21209 −0.282660
\(484\) 20.1038 0.913809
\(485\) 5.21710 0.236896
\(486\) −1.00000 −0.0453609
\(487\) −11.5461 −0.523205 −0.261602 0.965176i \(-0.584251\pi\)
−0.261602 + 0.965176i \(0.584251\pi\)
\(488\) 0.462620 0.0209418
\(489\) 3.87572 0.175266
\(490\) −0.858438 −0.0387803
\(491\) −7.19621 −0.324761 −0.162380 0.986728i \(-0.551917\pi\)
−0.162380 + 0.986728i \(0.551917\pi\)
\(492\) 8.43427 0.380246
\(493\) 0.950492 0.0428080
\(494\) −5.05516 −0.227442
\(495\) 1.62105 0.0728607
\(496\) 2.60815 0.117110
\(497\) 25.7865 1.15668
\(498\) 12.2435 0.548643
\(499\) −6.96624 −0.311852 −0.155926 0.987769i \(-0.549836\pi\)
−0.155926 + 0.987769i \(0.549836\pi\)
\(500\) −2.88207 −0.128890
\(501\) 4.93497 0.220478
\(502\) −2.69531 −0.120298
\(503\) −14.9610 −0.667078 −0.333539 0.942736i \(-0.608243\pi\)
−0.333539 + 0.942736i \(0.608243\pi\)
\(504\) 3.15490 0.140530
\(505\) −1.53923 −0.0684950
\(506\) −10.9814 −0.488184
\(507\) 1.00000 0.0444116
\(508\) −1.72805 −0.0766699
\(509\) 4.23418 0.187677 0.0938384 0.995587i \(-0.470086\pi\)
0.0938384 + 0.995587i \(0.470086\pi\)
\(510\) 0.621194 0.0275070
\(511\) 47.9431 2.12088
\(512\) −1.00000 −0.0441942
\(513\) −5.05516 −0.223191
\(514\) −29.5475 −1.30329
\(515\) 0.290663 0.0128081
\(516\) −1.60263 −0.0705518
\(517\) 52.7402 2.31951
\(518\) −23.8247 −1.04680
\(519\) 24.0000 1.05349
\(520\) 0.290663 0.0127464
\(521\) −30.4410 −1.33364 −0.666822 0.745217i \(-0.732346\pi\)
−0.666822 + 0.745217i \(0.732346\pi\)
\(522\) 0.444744 0.0194659
\(523\) 4.86408 0.212691 0.106346 0.994329i \(-0.466085\pi\)
0.106346 + 0.994329i \(0.466085\pi\)
\(524\) −13.9485 −0.609343
\(525\) 15.5079 0.676822
\(526\) 8.69529 0.379133
\(527\) −5.57405 −0.242810
\(528\) 5.57708 0.242711
\(529\) −19.1229 −0.831431
\(530\) −3.83729 −0.166681
\(531\) −10.9582 −0.475545
\(532\) 15.9485 0.691456
\(533\) −8.43427 −0.365329
\(534\) 7.96168 0.344536
\(535\) 0.299722 0.0129581
\(536\) −13.6224 −0.588396
\(537\) 18.1673 0.783979
\(538\) −2.38908 −0.103000
\(539\) 16.4712 0.709466
\(540\) 0.290663 0.0125081
\(541\) −11.0402 −0.474656 −0.237328 0.971430i \(-0.576272\pi\)
−0.237328 + 0.971430i \(0.576272\pi\)
\(542\) 27.3531 1.17491
\(543\) −7.40216 −0.317657
\(544\) 2.13717 0.0916302
\(545\) −2.89284 −0.123916
\(546\) −3.15490 −0.135017
\(547\) 14.7243 0.629564 0.314782 0.949164i \(-0.398069\pi\)
0.314782 + 0.949164i \(0.398069\pi\)
\(548\) −7.85127 −0.335390
\(549\) −0.462620 −0.0197441
\(550\) 27.4142 1.16895
\(551\) 2.24825 0.0957789
\(552\) −1.96903 −0.0838075
\(553\) 45.9823 1.95536
\(554\) −7.20989 −0.306319
\(555\) −2.19499 −0.0931721
\(556\) 15.4944 0.657108
\(557\) −20.3026 −0.860251 −0.430125 0.902769i \(-0.641531\pi\)
−0.430125 + 0.902769i \(0.641531\pi\)
\(558\) −2.60815 −0.110412
\(559\) 1.60263 0.0677840
\(560\) −0.917011 −0.0387508
\(561\) −11.9191 −0.503227
\(562\) −9.84747 −0.415390
\(563\) 5.06971 0.213663 0.106831 0.994277i \(-0.465929\pi\)
0.106831 + 0.994277i \(0.465929\pi\)
\(564\) 9.45661 0.398195
\(565\) −2.52070 −0.106047
\(566\) 11.7273 0.492936
\(567\) −3.15490 −0.132493
\(568\) 8.17349 0.342952
\(569\) 12.9588 0.543262 0.271631 0.962401i \(-0.412437\pi\)
0.271631 + 0.962401i \(0.412437\pi\)
\(570\) 1.46935 0.0615442
\(571\) 3.92266 0.164158 0.0820791 0.996626i \(-0.473844\pi\)
0.0820791 + 0.996626i \(0.473844\pi\)
\(572\) −5.57708 −0.233189
\(573\) 1.61795 0.0675908
\(574\) 26.6092 1.11065
\(575\) −9.67880 −0.403634
\(576\) 1.00000 0.0416667
\(577\) −34.3669 −1.43071 −0.715356 0.698760i \(-0.753735\pi\)
−0.715356 + 0.698760i \(0.753735\pi\)
\(578\) 12.4325 0.517125
\(579\) −2.89904 −0.120480
\(580\) −0.129271 −0.00536767
\(581\) 38.6269 1.60251
\(582\) −17.9490 −0.744009
\(583\) 73.6279 3.04936
\(584\) 15.1964 0.628831
\(585\) −0.290663 −0.0120174
\(586\) 23.3098 0.962920
\(587\) −33.4509 −1.38066 −0.690332 0.723492i \(-0.742536\pi\)
−0.690332 + 0.723492i \(0.742536\pi\)
\(588\) 2.95338 0.121795
\(589\) −13.1846 −0.543263
\(590\) 3.18514 0.131130
\(591\) −5.32513 −0.219047
\(592\) −7.55167 −0.310372
\(593\) 16.7073 0.686085 0.343043 0.939320i \(-0.388542\pi\)
0.343043 + 0.939320i \(0.388542\pi\)
\(594\) −5.57708 −0.228830
\(595\) 1.95981 0.0803442
\(596\) −17.7998 −0.729109
\(597\) −15.2964 −0.626041
\(598\) 1.96903 0.0805196
\(599\) 0.588732 0.0240549 0.0120275 0.999928i \(-0.496171\pi\)
0.0120275 + 0.999928i \(0.496171\pi\)
\(600\) 4.91552 0.200675
\(601\) 13.9648 0.569636 0.284818 0.958582i \(-0.408067\pi\)
0.284818 + 0.958582i \(0.408067\pi\)
\(602\) −5.05613 −0.206072
\(603\) 13.6224 0.554745
\(604\) −6.49777 −0.264391
\(605\) 5.84343 0.237569
\(606\) 5.29560 0.215119
\(607\) 19.5716 0.794388 0.397194 0.917735i \(-0.369984\pi\)
0.397194 + 0.917735i \(0.369984\pi\)
\(608\) 5.05516 0.205014
\(609\) 1.40312 0.0568574
\(610\) 0.134466 0.00544438
\(611\) −9.45661 −0.382573
\(612\) −2.13717 −0.0863898
\(613\) 19.0031 0.767529 0.383764 0.923431i \(-0.374627\pi\)
0.383764 + 0.923431i \(0.374627\pi\)
\(614\) −14.8883 −0.600841
\(615\) 2.45153 0.0988551
\(616\) 17.5951 0.708927
\(617\) 6.22176 0.250479 0.125239 0.992127i \(-0.460030\pi\)
0.125239 + 0.992127i \(0.460030\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 25.4935 1.02467 0.512335 0.858786i \(-0.328781\pi\)
0.512335 + 0.858786i \(0.328781\pi\)
\(620\) 0.758093 0.0304457
\(621\) 1.96903 0.0790145
\(622\) −32.4727 −1.30204
\(623\) 25.1183 1.00634
\(624\) −1.00000 −0.0400320
\(625\) 23.7399 0.949595
\(626\) −21.6485 −0.865250
\(627\) −28.1930 −1.12592
\(628\) −11.9055 −0.475082
\(629\) 16.1392 0.643511
\(630\) 0.917011 0.0365346
\(631\) 11.1486 0.443819 0.221910 0.975067i \(-0.428771\pi\)
0.221910 + 0.975067i \(0.428771\pi\)
\(632\) 14.5749 0.579758
\(633\) −5.18412 −0.206050
\(634\) 11.4084 0.453087
\(635\) −0.502280 −0.0199324
\(636\) 13.2019 0.523488
\(637\) −2.95338 −0.117017
\(638\) 2.48037 0.0981990
\(639\) −8.17349 −0.323338
\(640\) −0.290663 −0.0114895
\(641\) 10.9535 0.432636 0.216318 0.976323i \(-0.430595\pi\)
0.216318 + 0.976323i \(0.430595\pi\)
\(642\) −1.03117 −0.0406969
\(643\) −41.6162 −1.64118 −0.820592 0.571514i \(-0.806356\pi\)
−0.820592 + 0.571514i \(0.806356\pi\)
\(644\) −6.21209 −0.244791
\(645\) −0.465824 −0.0183418
\(646\) −10.8037 −0.425067
\(647\) −32.9026 −1.29354 −0.646768 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −61.1147 −2.39896
\(650\) −4.91552 −0.192802
\(651\) −8.22846 −0.322499
\(652\) 3.87572 0.151785
\(653\) 10.7035 0.418861 0.209430 0.977824i \(-0.432839\pi\)
0.209430 + 0.977824i \(0.432839\pi\)
\(654\) 9.95257 0.389176
\(655\) −4.05431 −0.158415
\(656\) 8.43427 0.329303
\(657\) −15.1964 −0.592868
\(658\) 29.8346 1.16307
\(659\) −8.40362 −0.327359 −0.163679 0.986514i \(-0.552336\pi\)
−0.163679 + 0.986514i \(0.552336\pi\)
\(660\) 1.62105 0.0630993
\(661\) 14.7129 0.572267 0.286133 0.958190i \(-0.407630\pi\)
0.286133 + 0.958190i \(0.407630\pi\)
\(662\) 15.1788 0.589939
\(663\) 2.13717 0.0830006
\(664\) 12.2435 0.475138
\(665\) 4.63564 0.179762
\(666\) 7.55167 0.292621
\(667\) −0.875715 −0.0339078
\(668\) 4.93497 0.190940
\(669\) 18.0617 0.698304
\(670\) −3.95951 −0.152969
\(671\) −2.58007 −0.0996024
\(672\) 3.15490 0.121703
\(673\) 25.7263 0.991676 0.495838 0.868415i \(-0.334861\pi\)
0.495838 + 0.868415i \(0.334861\pi\)
\(674\) 2.69714 0.103890
\(675\) −4.91552 −0.189198
\(676\) 1.00000 0.0384615
\(677\) 39.5284 1.51920 0.759601 0.650390i \(-0.225394\pi\)
0.759601 + 0.650390i \(0.225394\pi\)
\(678\) 8.67226 0.333056
\(679\) −56.6272 −2.17315
\(680\) 0.621194 0.0238217
\(681\) 8.33048 0.319225
\(682\) −14.5459 −0.556990
\(683\) −43.2833 −1.65619 −0.828095 0.560588i \(-0.810575\pi\)
−0.828095 + 0.560588i \(0.810575\pi\)
\(684\) −5.05516 −0.193289
\(685\) −2.28207 −0.0871935
\(686\) −12.7667 −0.487434
\(687\) −1.04971 −0.0400488
\(688\) −1.60263 −0.0610996
\(689\) −13.2019 −0.502951
\(690\) −0.572324 −0.0217880
\(691\) 18.0548 0.686838 0.343419 0.939182i \(-0.388415\pi\)
0.343419 + 0.939182i \(0.388415\pi\)
\(692\) 24.0000 0.912345
\(693\) −17.5951 −0.668383
\(694\) −10.1270 −0.384417
\(695\) 4.50364 0.170833
\(696\) 0.444744 0.0168580
\(697\) −18.0254 −0.682762
\(698\) 17.6111 0.666588
\(699\) −17.9285 −0.678119
\(700\) 15.5079 0.586145
\(701\) 12.7773 0.482591 0.241296 0.970452i \(-0.422428\pi\)
0.241296 + 0.970452i \(0.422428\pi\)
\(702\) 1.00000 0.0377426
\(703\) 38.1749 1.43979
\(704\) 5.57708 0.210194
\(705\) 2.74868 0.103521
\(706\) 7.45823 0.280694
\(707\) 16.7071 0.628334
\(708\) −10.9582 −0.411834
\(709\) −24.2744 −0.911644 −0.455822 0.890071i \(-0.650655\pi\)
−0.455822 + 0.890071i \(0.650655\pi\)
\(710\) 2.37573 0.0891595
\(711\) −14.5749 −0.546601
\(712\) 7.96168 0.298377
\(713\) 5.13553 0.192327
\(714\) −6.74254 −0.252333
\(715\) −1.62105 −0.0606238
\(716\) 18.1673 0.678946
\(717\) −28.4472 −1.06238
\(718\) 0.109489 0.00408609
\(719\) −20.4853 −0.763973 −0.381987 0.924168i \(-0.624760\pi\)
−0.381987 + 0.924168i \(0.624760\pi\)
\(720\) 0.290663 0.0108324
\(721\) −3.15490 −0.117495
\(722\) −6.55465 −0.243939
\(723\) −23.7447 −0.883075
\(724\) −7.40216 −0.275099
\(725\) 2.18615 0.0811915
\(726\) −20.1038 −0.746122
\(727\) −18.8760 −0.700073 −0.350037 0.936736i \(-0.613831\pi\)
−0.350037 + 0.936736i \(0.613831\pi\)
\(728\) −3.15490 −0.116928
\(729\) 1.00000 0.0370370
\(730\) 4.41703 0.163482
\(731\) 3.42508 0.126681
\(732\) −0.462620 −0.0170989
\(733\) 16.0005 0.590991 0.295496 0.955344i \(-0.404515\pi\)
0.295496 + 0.955344i \(0.404515\pi\)
\(734\) 23.9306 0.883293
\(735\) 0.858438 0.0316639
\(736\) −1.96903 −0.0725794
\(737\) 75.9729 2.79850
\(738\) −8.43427 −0.310470
\(739\) 27.2275 1.00158 0.500789 0.865569i \(-0.333043\pi\)
0.500789 + 0.865569i \(0.333043\pi\)
\(740\) −2.19499 −0.0806894
\(741\) 5.05516 0.185706
\(742\) 41.6506 1.52904
\(743\) 0.764203 0.0280359 0.0140180 0.999902i \(-0.495538\pi\)
0.0140180 + 0.999902i \(0.495538\pi\)
\(744\) −2.60815 −0.0956195
\(745\) −5.17374 −0.189551
\(746\) −2.83392 −0.103757
\(747\) −12.2435 −0.447965
\(748\) −11.9191 −0.435807
\(749\) −3.25323 −0.118870
\(750\) 2.88207 0.105238
\(751\) −17.1952 −0.627462 −0.313731 0.949512i \(-0.601579\pi\)
−0.313731 + 0.949512i \(0.601579\pi\)
\(752\) 9.45661 0.344847
\(753\) 2.69531 0.0982227
\(754\) −0.444744 −0.0161966
\(755\) −1.88866 −0.0687354
\(756\) −3.15490 −0.114743
\(757\) 42.9856 1.56234 0.781169 0.624320i \(-0.214624\pi\)
0.781169 + 0.624320i \(0.214624\pi\)
\(758\) −2.43968 −0.0886133
\(759\) 10.9814 0.398601
\(760\) 1.46935 0.0532988
\(761\) 9.75380 0.353575 0.176788 0.984249i \(-0.443429\pi\)
0.176788 + 0.984249i \(0.443429\pi\)
\(762\) 1.72805 0.0626007
\(763\) 31.3994 1.13673
\(764\) 1.61795 0.0585354
\(765\) −0.621194 −0.0224593
\(766\) −7.68470 −0.277660
\(767\) 10.9582 0.395677
\(768\) 1.00000 0.0360844
\(769\) 23.2481 0.838349 0.419174 0.907906i \(-0.362320\pi\)
0.419174 + 0.907906i \(0.362320\pi\)
\(770\) 5.11424 0.184305
\(771\) 29.5475 1.06413
\(772\) −2.89904 −0.104339
\(773\) 40.8722 1.47007 0.735036 0.678028i \(-0.237165\pi\)
0.735036 + 0.678028i \(0.237165\pi\)
\(774\) 1.60263 0.0576053
\(775\) −12.8204 −0.460523
\(776\) −17.9490 −0.644331
\(777\) 23.8247 0.854708
\(778\) 15.0757 0.540491
\(779\) −42.6366 −1.52761
\(780\) −0.290663 −0.0104074
\(781\) −45.5842 −1.63113
\(782\) 4.20814 0.150483
\(783\) −0.444744 −0.0158939
\(784\) 2.95338 0.105478
\(785\) −3.46050 −0.123510
\(786\) 13.9485 0.497526
\(787\) −21.5873 −0.769503 −0.384751 0.923020i \(-0.625713\pi\)
−0.384751 + 0.923020i \(0.625713\pi\)
\(788\) −5.32513 −0.189700
\(789\) −8.69529 −0.309561
\(790\) 4.23638 0.150723
\(791\) 27.3601 0.972813
\(792\) −5.57708 −0.198173
\(793\) 0.462620 0.0164281
\(794\) 19.8678 0.705083
\(795\) 3.83729 0.136095
\(796\) −15.2964 −0.542167
\(797\) 21.6888 0.768256 0.384128 0.923280i \(-0.374502\pi\)
0.384128 + 0.923280i \(0.374502\pi\)
\(798\) −15.9485 −0.564571
\(799\) −20.2103 −0.714990
\(800\) 4.91552 0.173790
\(801\) −7.96168 −0.281312
\(802\) −8.07833 −0.285256
\(803\) −84.7515 −2.99082
\(804\) 13.6224 0.480423
\(805\) −1.80562 −0.0636398
\(806\) 2.60815 0.0918683
\(807\) 2.38908 0.0840994
\(808\) 5.29560 0.186299
\(809\) −9.87247 −0.347098 −0.173549 0.984825i \(-0.555523\pi\)
−0.173549 + 0.984825i \(0.555523\pi\)
\(810\) −0.290663 −0.0102128
\(811\) 14.8157 0.520248 0.260124 0.965575i \(-0.416236\pi\)
0.260124 + 0.965575i \(0.416236\pi\)
\(812\) 1.40312 0.0492400
\(813\) −27.3531 −0.959314
\(814\) 42.1162 1.47617
\(815\) 1.12653 0.0394605
\(816\) −2.13717 −0.0748158
\(817\) 8.10154 0.283437
\(818\) 1.67295 0.0584932
\(819\) 3.15490 0.110241
\(820\) 2.45153 0.0856110
\(821\) −23.7220 −0.827903 −0.413951 0.910299i \(-0.635852\pi\)
−0.413951 + 0.910299i \(0.635852\pi\)
\(822\) 7.85127 0.273845
\(823\) −21.7641 −0.758649 −0.379325 0.925264i \(-0.623844\pi\)
−0.379325 + 0.925264i \(0.623844\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −27.4142 −0.954440
\(826\) −34.5720 −1.20291
\(827\) −21.9738 −0.764105 −0.382053 0.924141i \(-0.624783\pi\)
−0.382053 + 0.924141i \(0.624783\pi\)
\(828\) 1.96903 0.0684285
\(829\) −50.6407 −1.75883 −0.879413 0.476060i \(-0.842064\pi\)
−0.879413 + 0.476060i \(0.842064\pi\)
\(830\) 3.55872 0.123525
\(831\) 7.20989 0.250108
\(832\) −1.00000 −0.0346688
\(833\) −6.31186 −0.218693
\(834\) −15.4944 −0.536527
\(835\) 1.43441 0.0496399
\(836\) −28.1930 −0.975076
\(837\) 2.60815 0.0901509
\(838\) 29.6169 1.02310
\(839\) −40.9488 −1.41371 −0.706855 0.707358i \(-0.749887\pi\)
−0.706855 + 0.707358i \(0.749887\pi\)
\(840\) 0.917011 0.0316399
\(841\) −28.8022 −0.993179
\(842\) −21.6503 −0.746119
\(843\) 9.84747 0.339165
\(844\) −5.18412 −0.178445
\(845\) 0.290663 0.00999910
\(846\) −9.45661 −0.325125
\(847\) −63.4255 −2.17932
\(848\) 13.2019 0.453354
\(849\) −11.7273 −0.402480
\(850\) −10.5053 −0.360328
\(851\) −14.8695 −0.509718
\(852\) −8.17349 −0.280019
\(853\) −39.2025 −1.34227 −0.671134 0.741336i \(-0.734193\pi\)
−0.671134 + 0.741336i \(0.734193\pi\)
\(854\) −1.45952 −0.0499437
\(855\) −1.46935 −0.0502506
\(856\) −1.03117 −0.0352446
\(857\) 31.1171 1.06294 0.531470 0.847077i \(-0.321640\pi\)
0.531470 + 0.847077i \(0.321640\pi\)
\(858\) 5.57708 0.190398
\(859\) 3.08785 0.105356 0.0526780 0.998612i \(-0.483224\pi\)
0.0526780 + 0.998612i \(0.483224\pi\)
\(860\) −0.465824 −0.0158845
\(861\) −26.6092 −0.906841
\(862\) 12.8563 0.437888
\(863\) −18.3232 −0.623729 −0.311864 0.950127i \(-0.600953\pi\)
−0.311864 + 0.950127i \(0.600953\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.97592 0.237188
\(866\) −5.64923 −0.191969
\(867\) −12.4325 −0.422231
\(868\) −8.22846 −0.279292
\(869\) −81.2853 −2.75741
\(870\) 0.129271 0.00438269
\(871\) −13.6224 −0.461576
\(872\) 9.95257 0.337037
\(873\) 17.9490 0.607481
\(874\) 9.95376 0.336691
\(875\) 9.09264 0.307387
\(876\) −15.1964 −0.513439
\(877\) −49.1191 −1.65863 −0.829317 0.558778i \(-0.811270\pi\)
−0.829317 + 0.558778i \(0.811270\pi\)
\(878\) 3.24402 0.109480
\(879\) −23.3098 −0.786221
\(880\) 1.62105 0.0546456
\(881\) 39.1803 1.32002 0.660009 0.751258i \(-0.270552\pi\)
0.660009 + 0.751258i \(0.270552\pi\)
\(882\) −2.95338 −0.0994455
\(883\) −4.99667 −0.168151 −0.0840756 0.996459i \(-0.526794\pi\)
−0.0840756 + 0.996459i \(0.526794\pi\)
\(884\) 2.13717 0.0718807
\(885\) −3.18514 −0.107067
\(886\) −3.93720 −0.132273
\(887\) −51.7082 −1.73619 −0.868096 0.496396i \(-0.834656\pi\)
−0.868096 + 0.496396i \(0.834656\pi\)
\(888\) 7.55167 0.253417
\(889\) 5.45183 0.182848
\(890\) 2.31416 0.0775710
\(891\) 5.57708 0.186839
\(892\) 18.0617 0.604749
\(893\) −47.8047 −1.59972
\(894\) 17.7998 0.595315
\(895\) 5.28057 0.176510
\(896\) 3.15490 0.105398
\(897\) −1.96903 −0.0657440
\(898\) 32.6440 1.08934
\(899\) −1.15996 −0.0386869
\(900\) −4.91552 −0.163851
\(901\) −28.2146 −0.939964
\(902\) −47.0386 −1.56621
\(903\) 5.05613 0.168257
\(904\) 8.67226 0.288435
\(905\) −2.15153 −0.0715194
\(906\) 6.49777 0.215874
\(907\) 45.2516 1.50255 0.751277 0.659987i \(-0.229438\pi\)
0.751277 + 0.659987i \(0.229438\pi\)
\(908\) 8.33048 0.276457
\(909\) −5.29560 −0.175644
\(910\) −0.917011 −0.0303986
\(911\) −51.2211 −1.69703 −0.848516 0.529170i \(-0.822503\pi\)
−0.848516 + 0.529170i \(0.822503\pi\)
\(912\) −5.05516 −0.167393
\(913\) −68.2827 −2.25983
\(914\) −30.0476 −0.993887
\(915\) −0.134466 −0.00444532
\(916\) −1.04971 −0.0346833
\(917\) 44.0061 1.45321
\(918\) 2.13717 0.0705370
\(919\) 49.3019 1.62632 0.813160 0.582040i \(-0.197745\pi\)
0.813160 + 0.582040i \(0.197745\pi\)
\(920\) −0.572324 −0.0188690
\(921\) 14.8883 0.490585
\(922\) 28.1535 0.927188
\(923\) 8.17349 0.269034
\(924\) −17.5951 −0.578837
\(925\) 37.1203 1.22051
\(926\) 28.2939 0.929794
\(927\) 1.00000 0.0328443
\(928\) 0.444744 0.0145995
\(929\) −37.4227 −1.22780 −0.613899 0.789385i \(-0.710400\pi\)
−0.613899 + 0.789385i \(0.710400\pi\)
\(930\) −0.758093 −0.0248588
\(931\) −14.9298 −0.489305
\(932\) −17.9285 −0.587268
\(933\) 32.4727 1.06311
\(934\) 17.8245 0.583236
\(935\) −3.46445 −0.113300
\(936\) 1.00000 0.0326860
\(937\) 4.00375 0.130797 0.0653984 0.997859i \(-0.479168\pi\)
0.0653984 + 0.997859i \(0.479168\pi\)
\(938\) 42.9771 1.40325
\(939\) 21.6485 0.706474
\(940\) 2.74868 0.0896522
\(941\) 27.1284 0.884362 0.442181 0.896926i \(-0.354205\pi\)
0.442181 + 0.896926i \(0.354205\pi\)
\(942\) 11.9055 0.387903
\(943\) 16.6073 0.540809
\(944\) −10.9582 −0.356659
\(945\) −0.917011 −0.0298304
\(946\) 8.93798 0.290599
\(947\) −1.79919 −0.0584658 −0.0292329 0.999573i \(-0.509306\pi\)
−0.0292329 + 0.999573i \(0.509306\pi\)
\(948\) −14.5749 −0.473370
\(949\) 15.1964 0.493296
\(950\) −24.8487 −0.806199
\(951\) −11.4084 −0.369944
\(952\) −6.74254 −0.218527
\(953\) 41.8700 1.35630 0.678151 0.734923i \(-0.262782\pi\)
0.678151 + 0.734923i \(0.262782\pi\)
\(954\) −13.2019 −0.427426
\(955\) 0.470278 0.0152178
\(956\) −28.4472 −0.920049
\(957\) −2.48037 −0.0801791
\(958\) 3.47708 0.112339
\(959\) 24.7700 0.799864
\(960\) 0.290663 0.00938110
\(961\) −24.1975 −0.780566
\(962\) −7.55167 −0.243476
\(963\) 1.03117 0.0332289
\(964\) −23.7447 −0.764766
\(965\) −0.842643 −0.0271256
\(966\) 6.21209 0.199871
\(967\) 0.719652 0.0231424 0.0115712 0.999933i \(-0.496317\pi\)
0.0115712 + 0.999933i \(0.496317\pi\)
\(968\) −20.1038 −0.646161
\(969\) 10.8037 0.347065
\(970\) −5.21710 −0.167511
\(971\) 19.1042 0.613083 0.306541 0.951857i \(-0.400828\pi\)
0.306541 + 0.951857i \(0.400828\pi\)
\(972\) 1.00000 0.0320750
\(973\) −48.8832 −1.56712
\(974\) 11.5461 0.369962
\(975\) 4.91552 0.157422
\(976\) −0.462620 −0.0148081
\(977\) 11.2186 0.358915 0.179458 0.983766i \(-0.442566\pi\)
0.179458 + 0.983766i \(0.442566\pi\)
\(978\) −3.87572 −0.123932
\(979\) −44.4029 −1.41912
\(980\) 0.858438 0.0274218
\(981\) −9.95257 −0.317761
\(982\) 7.19621 0.229640
\(983\) −3.27103 −0.104330 −0.0521648 0.998638i \(-0.516612\pi\)
−0.0521648 + 0.998638i \(0.516612\pi\)
\(984\) −8.43427 −0.268875
\(985\) −1.54782 −0.0493175
\(986\) −0.950492 −0.0302698
\(987\) −29.8346 −0.949647
\(988\) 5.05516 0.160826
\(989\) −3.15562 −0.100343
\(990\) −1.62105 −0.0515203
\(991\) −23.3751 −0.742534 −0.371267 0.928526i \(-0.621076\pi\)
−0.371267 + 0.928526i \(0.621076\pi\)
\(992\) −2.60815 −0.0828089
\(993\) −15.1788 −0.481683
\(994\) −25.7865 −0.817899
\(995\) −4.44610 −0.140951
\(996\) −12.2435 −0.387949
\(997\) 48.6147 1.53964 0.769822 0.638259i \(-0.220345\pi\)
0.769822 + 0.638259i \(0.220345\pi\)
\(998\) 6.96624 0.220512
\(999\) −7.55167 −0.238924
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.bb.1.8 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.8 14 1.1 even 1 trivial