Properties

Label 1078.2.a.t.1.2
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -3.41421 q^{5} +0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.414214 q^{3} +1.00000 q^{4} -3.41421 q^{5} +0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} -3.41421 q^{10} +1.00000 q^{11} +0.414214 q^{12} +1.82843 q^{13} -1.41421 q^{15} +1.00000 q^{16} -7.65685 q^{17} -2.82843 q^{18} -3.41421 q^{19} -3.41421 q^{20} +1.00000 q^{22} +2.24264 q^{23} +0.414214 q^{24} +6.65685 q^{25} +1.82843 q^{26} -2.41421 q^{27} -8.65685 q^{29} -1.41421 q^{30} -4.00000 q^{31} +1.00000 q^{32} +0.414214 q^{33} -7.65685 q^{34} -2.82843 q^{36} -6.58579 q^{37} -3.41421 q^{38} +0.757359 q^{39} -3.41421 q^{40} -2.58579 q^{41} +5.65685 q^{43} +1.00000 q^{44} +9.65685 q^{45} +2.24264 q^{46} +6.48528 q^{47} +0.414214 q^{48} +6.65685 q^{50} -3.17157 q^{51} +1.82843 q^{52} -11.8995 q^{53} -2.41421 q^{54} -3.41421 q^{55} -1.41421 q^{57} -8.65685 q^{58} -8.41421 q^{59} -1.41421 q^{60} +6.17157 q^{61} -4.00000 q^{62} +1.00000 q^{64} -6.24264 q^{65} +0.414214 q^{66} +11.2426 q^{67} -7.65685 q^{68} +0.928932 q^{69} +3.07107 q^{71} -2.82843 q^{72} -6.58579 q^{73} -6.58579 q^{74} +2.75736 q^{75} -3.41421 q^{76} +0.757359 q^{78} -4.75736 q^{79} -3.41421 q^{80} +7.48528 q^{81} -2.58579 q^{82} +16.1421 q^{83} +26.1421 q^{85} +5.65685 q^{86} -3.58579 q^{87} +1.00000 q^{88} -4.48528 q^{89} +9.65685 q^{90} +2.24264 q^{92} -1.65685 q^{93} +6.48528 q^{94} +11.6569 q^{95} +0.414214 q^{96} +1.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} - 4 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{16} - 4 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} - 16 q^{37} - 4 q^{38} + 10 q^{39} - 4 q^{40} - 8 q^{41} + 2 q^{44} + 8 q^{45} - 4 q^{46} - 4 q^{47} - 2 q^{48} + 2 q^{50} - 12 q^{51} - 2 q^{52} - 4 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{58} - 14 q^{59} + 18 q^{61} - 8 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} + 14 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 16 q^{74} + 14 q^{75} - 4 q^{76} + 10 q^{78} - 18 q^{79} - 4 q^{80} - 2 q^{81} - 8 q^{82} + 4 q^{83} + 24 q^{85} - 10 q^{87} + 2 q^{88} + 8 q^{89} + 8 q^{90} - 4 q^{92} + 8 q^{93} - 4 q^{94} + 12 q^{95} - 2 q^{96} - 2 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.414214 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0.414214 0.169102
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.82843 −0.942809
\(10\) −3.41421 −1.07967
\(11\) 1.00000 0.301511
\(12\) 0.414214 0.119573
\(13\) 1.82843 0.507114 0.253557 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 1.00000 0.250000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) −2.82843 −0.666667
\(19\) −3.41421 −0.783274 −0.391637 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(20\) −3.41421 −0.763441
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.24264 0.467623 0.233811 0.972282i \(-0.424880\pi\)
0.233811 + 0.972282i \(0.424880\pi\)
\(24\) 0.414214 0.0845510
\(25\) 6.65685 1.33137
\(26\) 1.82843 0.358584
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) −1.41421 −0.258199
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.414214 0.0721053
\(34\) −7.65685 −1.31314
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) −6.58579 −1.08270 −0.541348 0.840798i \(-0.682086\pi\)
−0.541348 + 0.840798i \(0.682086\pi\)
\(38\) −3.41421 −0.553859
\(39\) 0.757359 0.121275
\(40\) −3.41421 −0.539835
\(41\) −2.58579 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 1.00000 0.150756
\(45\) 9.65685 1.43956
\(46\) 2.24264 0.330659
\(47\) 6.48528 0.945976 0.472988 0.881069i \(-0.343175\pi\)
0.472988 + 0.881069i \(0.343175\pi\)
\(48\) 0.414214 0.0597866
\(49\) 0 0
\(50\) 6.65685 0.941421
\(51\) −3.17157 −0.444109
\(52\) 1.82843 0.253557
\(53\) −11.8995 −1.63452 −0.817261 0.576268i \(-0.804508\pi\)
−0.817261 + 0.576268i \(0.804508\pi\)
\(54\) −2.41421 −0.328533
\(55\) −3.41421 −0.460372
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) −8.65685 −1.13670
\(59\) −8.41421 −1.09544 −0.547719 0.836663i \(-0.684504\pi\)
−0.547719 + 0.836663i \(0.684504\pi\)
\(60\) −1.41421 −0.182574
\(61\) 6.17157 0.790189 0.395094 0.918640i \(-0.370712\pi\)
0.395094 + 0.918640i \(0.370712\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.24264 −0.774304
\(66\) 0.414214 0.0509862
\(67\) 11.2426 1.37351 0.686754 0.726890i \(-0.259035\pi\)
0.686754 + 0.726890i \(0.259035\pi\)
\(68\) −7.65685 −0.928530
\(69\) 0.928932 0.111830
\(70\) 0 0
\(71\) 3.07107 0.364469 0.182234 0.983255i \(-0.441667\pi\)
0.182234 + 0.983255i \(0.441667\pi\)
\(72\) −2.82843 −0.333333
\(73\) −6.58579 −0.770808 −0.385404 0.922748i \(-0.625938\pi\)
−0.385404 + 0.922748i \(0.625938\pi\)
\(74\) −6.58579 −0.765582
\(75\) 2.75736 0.318392
\(76\) −3.41421 −0.391637
\(77\) 0 0
\(78\) 0.757359 0.0857541
\(79\) −4.75736 −0.535245 −0.267622 0.963524i \(-0.586238\pi\)
−0.267622 + 0.963524i \(0.586238\pi\)
\(80\) −3.41421 −0.381721
\(81\) 7.48528 0.831698
\(82\) −2.58579 −0.285552
\(83\) 16.1421 1.77183 0.885915 0.463848i \(-0.153532\pi\)
0.885915 + 0.463848i \(0.153532\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 5.65685 0.609994
\(87\) −3.58579 −0.384437
\(88\) 1.00000 0.106600
\(89\) −4.48528 −0.475439 −0.237719 0.971334i \(-0.576400\pi\)
−0.237719 + 0.971334i \(0.576400\pi\)
\(90\) 9.65685 1.01792
\(91\) 0 0
\(92\) 2.24264 0.233811
\(93\) −1.65685 −0.171808
\(94\) 6.48528 0.668906
\(95\) 11.6569 1.19597
\(96\) 0.414214 0.0422755
\(97\) 1.82843 0.185649 0.0928243 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
\(100\) 6.65685 0.665685
\(101\) 11.8284 1.17697 0.588486 0.808507i \(-0.299724\pi\)
0.588486 + 0.808507i \(0.299724\pi\)
\(102\) −3.17157 −0.314033
\(103\) 10.5858 1.04305 0.521524 0.853236i \(-0.325364\pi\)
0.521524 + 0.853236i \(0.325364\pi\)
\(104\) 1.82843 0.179292
\(105\) 0 0
\(106\) −11.8995 −1.15578
\(107\) 11.0711 1.07028 0.535140 0.844763i \(-0.320259\pi\)
0.535140 + 0.844763i \(0.320259\pi\)
\(108\) −2.41421 −0.232308
\(109\) −0.485281 −0.0464815 −0.0232408 0.999730i \(-0.507398\pi\)
−0.0232408 + 0.999730i \(0.507398\pi\)
\(110\) −3.41421 −0.325532
\(111\) −2.72792 −0.258923
\(112\) 0 0
\(113\) −13.8284 −1.30087 −0.650434 0.759562i \(-0.725413\pi\)
−0.650434 + 0.759562i \(0.725413\pi\)
\(114\) −1.41421 −0.132453
\(115\) −7.65685 −0.714005
\(116\) −8.65685 −0.803769
\(117\) −5.17157 −0.478112
\(118\) −8.41421 −0.774591
\(119\) 0 0
\(120\) −1.41421 −0.129099
\(121\) 1.00000 0.0909091
\(122\) 6.17157 0.558748
\(123\) −1.07107 −0.0965749
\(124\) −4.00000 −0.359211
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −9.72792 −0.863213 −0.431607 0.902062i \(-0.642053\pi\)
−0.431607 + 0.902062i \(0.642053\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.34315 0.206302
\(130\) −6.24264 −0.547516
\(131\) −3.41421 −0.298301 −0.149151 0.988814i \(-0.547654\pi\)
−0.149151 + 0.988814i \(0.547654\pi\)
\(132\) 0.414214 0.0360527
\(133\) 0 0
\(134\) 11.2426 0.971216
\(135\) 8.24264 0.709414
\(136\) −7.65685 −0.656570
\(137\) −5.34315 −0.456496 −0.228248 0.973603i \(-0.573300\pi\)
−0.228248 + 0.973603i \(0.573300\pi\)
\(138\) 0.928932 0.0790760
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 2.68629 0.226227
\(142\) 3.07107 0.257718
\(143\) 1.82843 0.152901
\(144\) −2.82843 −0.235702
\(145\) 29.5563 2.45452
\(146\) −6.58579 −0.545044
\(147\) 0 0
\(148\) −6.58579 −0.541348
\(149\) 6.34315 0.519651 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(150\) 2.75736 0.225137
\(151\) 9.72792 0.791647 0.395824 0.918327i \(-0.370459\pi\)
0.395824 + 0.918327i \(0.370459\pi\)
\(152\) −3.41421 −0.276929
\(153\) 21.6569 1.75085
\(154\) 0 0
\(155\) 13.6569 1.09694
\(156\) 0.757359 0.0606373
\(157\) 6.34315 0.506238 0.253119 0.967435i \(-0.418544\pi\)
0.253119 + 0.967435i \(0.418544\pi\)
\(158\) −4.75736 −0.378475
\(159\) −4.92893 −0.390890
\(160\) −3.41421 −0.269917
\(161\) 0 0
\(162\) 7.48528 0.588099
\(163\) −15.7279 −1.23191 −0.615953 0.787783i \(-0.711229\pi\)
−0.615953 + 0.787783i \(0.711229\pi\)
\(164\) −2.58579 −0.201916
\(165\) −1.41421 −0.110096
\(166\) 16.1421 1.25287
\(167\) −11.7279 −0.907534 −0.453767 0.891120i \(-0.649920\pi\)
−0.453767 + 0.891120i \(0.649920\pi\)
\(168\) 0 0
\(169\) −9.65685 −0.742835
\(170\) 26.1421 2.00501
\(171\) 9.65685 0.738478
\(172\) 5.65685 0.431331
\(173\) −4.17157 −0.317159 −0.158579 0.987346i \(-0.550691\pi\)
−0.158579 + 0.987346i \(0.550691\pi\)
\(174\) −3.58579 −0.271838
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −3.48528 −0.261970
\(178\) −4.48528 −0.336186
\(179\) −18.8995 −1.41261 −0.706307 0.707905i \(-0.749640\pi\)
−0.706307 + 0.707905i \(0.749640\pi\)
\(180\) 9.65685 0.719779
\(181\) 3.65685 0.271812 0.135906 0.990722i \(-0.456606\pi\)
0.135906 + 0.990722i \(0.456606\pi\)
\(182\) 0 0
\(183\) 2.55635 0.188971
\(184\) 2.24264 0.165330
\(185\) 22.4853 1.65315
\(186\) −1.65685 −0.121486
\(187\) −7.65685 −0.559925
\(188\) 6.48528 0.472988
\(189\) 0 0
\(190\) 11.6569 0.845677
\(191\) −12.8284 −0.928232 −0.464116 0.885774i \(-0.653628\pi\)
−0.464116 + 0.885774i \(0.653628\pi\)
\(192\) 0.414214 0.0298933
\(193\) 2.10051 0.151198 0.0755988 0.997138i \(-0.475913\pi\)
0.0755988 + 0.997138i \(0.475913\pi\)
\(194\) 1.82843 0.131273
\(195\) −2.58579 −0.185172
\(196\) 0 0
\(197\) −17.4853 −1.24577 −0.622887 0.782312i \(-0.714041\pi\)
−0.622887 + 0.782312i \(0.714041\pi\)
\(198\) −2.82843 −0.201008
\(199\) 19.8995 1.41064 0.705319 0.708890i \(-0.250804\pi\)
0.705319 + 0.708890i \(0.250804\pi\)
\(200\) 6.65685 0.470711
\(201\) 4.65685 0.328469
\(202\) 11.8284 0.832245
\(203\) 0 0
\(204\) −3.17157 −0.222055
\(205\) 8.82843 0.616604
\(206\) 10.5858 0.737547
\(207\) −6.34315 −0.440879
\(208\) 1.82843 0.126779
\(209\) −3.41421 −0.236166
\(210\) 0 0
\(211\) 4.58579 0.315699 0.157849 0.987463i \(-0.449544\pi\)
0.157849 + 0.987463i \(0.449544\pi\)
\(212\) −11.8995 −0.817261
\(213\) 1.27208 0.0871613
\(214\) 11.0711 0.756803
\(215\) −19.3137 −1.31718
\(216\) −2.41421 −0.164266
\(217\) 0 0
\(218\) −0.485281 −0.0328674
\(219\) −2.72792 −0.184336
\(220\) −3.41421 −0.230186
\(221\) −14.0000 −0.941742
\(222\) −2.72792 −0.183086
\(223\) 11.4142 0.764352 0.382176 0.924089i \(-0.375175\pi\)
0.382176 + 0.924089i \(0.375175\pi\)
\(224\) 0 0
\(225\) −18.8284 −1.25523
\(226\) −13.8284 −0.919853
\(227\) −23.1716 −1.53795 −0.768976 0.639278i \(-0.779233\pi\)
−0.768976 + 0.639278i \(0.779233\pi\)
\(228\) −1.41421 −0.0936586
\(229\) 0.686292 0.0453514 0.0226757 0.999743i \(-0.492781\pi\)
0.0226757 + 0.999743i \(0.492781\pi\)
\(230\) −7.65685 −0.504878
\(231\) 0 0
\(232\) −8.65685 −0.568350
\(233\) −1.41421 −0.0926482 −0.0463241 0.998926i \(-0.514751\pi\)
−0.0463241 + 0.998926i \(0.514751\pi\)
\(234\) −5.17157 −0.338076
\(235\) −22.1421 −1.44439
\(236\) −8.41421 −0.547719
\(237\) −1.97056 −0.128002
\(238\) 0 0
\(239\) −22.2132 −1.43685 −0.718426 0.695603i \(-0.755137\pi\)
−0.718426 + 0.695603i \(0.755137\pi\)
\(240\) −1.41421 −0.0912871
\(241\) 3.75736 0.242033 0.121016 0.992651i \(-0.461385\pi\)
0.121016 + 0.992651i \(0.461385\pi\)
\(242\) 1.00000 0.0642824
\(243\) 10.3431 0.663513
\(244\) 6.17157 0.395094
\(245\) 0 0
\(246\) −1.07107 −0.0682888
\(247\) −6.24264 −0.397210
\(248\) −4.00000 −0.254000
\(249\) 6.68629 0.423727
\(250\) −5.65685 −0.357771
\(251\) 2.14214 0.135210 0.0676052 0.997712i \(-0.478464\pi\)
0.0676052 + 0.997712i \(0.478464\pi\)
\(252\) 0 0
\(253\) 2.24264 0.140994
\(254\) −9.72792 −0.610384
\(255\) 10.8284 0.678102
\(256\) 1.00000 0.0625000
\(257\) 3.14214 0.196001 0.0980005 0.995186i \(-0.468755\pi\)
0.0980005 + 0.995186i \(0.468755\pi\)
\(258\) 2.34315 0.145878
\(259\) 0 0
\(260\) −6.24264 −0.387152
\(261\) 24.4853 1.51560
\(262\) −3.41421 −0.210931
\(263\) 31.0416 1.91411 0.957054 0.289908i \(-0.0936248\pi\)
0.957054 + 0.289908i \(0.0936248\pi\)
\(264\) 0.414214 0.0254931
\(265\) 40.6274 2.49572
\(266\) 0 0
\(267\) −1.85786 −0.113699
\(268\) 11.2426 0.686754
\(269\) 13.6569 0.832673 0.416337 0.909211i \(-0.363314\pi\)
0.416337 + 0.909211i \(0.363314\pi\)
\(270\) 8.24264 0.501631
\(271\) −26.5563 −1.61318 −0.806592 0.591109i \(-0.798690\pi\)
−0.806592 + 0.591109i \(0.798690\pi\)
\(272\) −7.65685 −0.464265
\(273\) 0 0
\(274\) −5.34315 −0.322791
\(275\) 6.65685 0.401423
\(276\) 0.928932 0.0559151
\(277\) −3.82843 −0.230028 −0.115014 0.993364i \(-0.536691\pi\)
−0.115014 + 0.993364i \(0.536691\pi\)
\(278\) 0 0
\(279\) 11.3137 0.677334
\(280\) 0 0
\(281\) −16.7279 −0.997904 −0.498952 0.866630i \(-0.666282\pi\)
−0.498952 + 0.866630i \(0.666282\pi\)
\(282\) 2.68629 0.159966
\(283\) −20.5858 −1.22370 −0.611849 0.790975i \(-0.709574\pi\)
−0.611849 + 0.790975i \(0.709574\pi\)
\(284\) 3.07107 0.182234
\(285\) 4.82843 0.286011
\(286\) 1.82843 0.108117
\(287\) 0 0
\(288\) −2.82843 −0.166667
\(289\) 41.6274 2.44867
\(290\) 29.5563 1.73561
\(291\) 0.757359 0.0443972
\(292\) −6.58579 −0.385404
\(293\) −5.17157 −0.302127 −0.151063 0.988524i \(-0.548270\pi\)
−0.151063 + 0.988524i \(0.548270\pi\)
\(294\) 0 0
\(295\) 28.7279 1.67260
\(296\) −6.58579 −0.382791
\(297\) −2.41421 −0.140087
\(298\) 6.34315 0.367449
\(299\) 4.10051 0.237138
\(300\) 2.75736 0.159196
\(301\) 0 0
\(302\) 9.72792 0.559779
\(303\) 4.89949 0.281469
\(304\) −3.41421 −0.195819
\(305\) −21.0711 −1.20653
\(306\) 21.6569 1.23804
\(307\) −9.89949 −0.564994 −0.282497 0.959268i \(-0.591163\pi\)
−0.282497 + 0.959268i \(0.591163\pi\)
\(308\) 0 0
\(309\) 4.38478 0.249441
\(310\) 13.6569 0.775657
\(311\) −8.72792 −0.494915 −0.247458 0.968899i \(-0.579595\pi\)
−0.247458 + 0.968899i \(0.579595\pi\)
\(312\) 0.757359 0.0428770
\(313\) −9.34315 −0.528106 −0.264053 0.964508i \(-0.585059\pi\)
−0.264053 + 0.964508i \(0.585059\pi\)
\(314\) 6.34315 0.357964
\(315\) 0 0
\(316\) −4.75736 −0.267622
\(317\) 31.3137 1.75875 0.879377 0.476127i \(-0.157960\pi\)
0.879377 + 0.476127i \(0.157960\pi\)
\(318\) −4.92893 −0.276401
\(319\) −8.65685 −0.484691
\(320\) −3.41421 −0.190860
\(321\) 4.58579 0.255954
\(322\) 0 0
\(323\) 26.1421 1.45459
\(324\) 7.48528 0.415849
\(325\) 12.1716 0.675157
\(326\) −15.7279 −0.871089
\(327\) −0.201010 −0.0111159
\(328\) −2.58579 −0.142776
\(329\) 0 0
\(330\) −1.41421 −0.0778499
\(331\) −9.92893 −0.545743 −0.272872 0.962050i \(-0.587973\pi\)
−0.272872 + 0.962050i \(0.587973\pi\)
\(332\) 16.1421 0.885915
\(333\) 18.6274 1.02078
\(334\) −11.7279 −0.641723
\(335\) −38.3848 −2.09718
\(336\) 0 0
\(337\) −19.7574 −1.07625 −0.538126 0.842864i \(-0.680868\pi\)
−0.538126 + 0.842864i \(0.680868\pi\)
\(338\) −9.65685 −0.525264
\(339\) −5.72792 −0.311098
\(340\) 26.1421 1.41776
\(341\) −4.00000 −0.216612
\(342\) 9.65685 0.522183
\(343\) 0 0
\(344\) 5.65685 0.304997
\(345\) −3.17157 −0.170752
\(346\) −4.17157 −0.224265
\(347\) 14.5858 0.783006 0.391503 0.920177i \(-0.371955\pi\)
0.391503 + 0.920177i \(0.371955\pi\)
\(348\) −3.58579 −0.192218
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −4.41421 −0.235613
\(352\) 1.00000 0.0533002
\(353\) −25.3137 −1.34731 −0.673656 0.739045i \(-0.735277\pi\)
−0.673656 + 0.739045i \(0.735277\pi\)
\(354\) −3.48528 −0.185241
\(355\) −10.4853 −0.556501
\(356\) −4.48528 −0.237719
\(357\) 0 0
\(358\) −18.8995 −0.998869
\(359\) 10.7574 0.567752 0.283876 0.958861i \(-0.408380\pi\)
0.283876 + 0.958861i \(0.408380\pi\)
\(360\) 9.65685 0.508961
\(361\) −7.34315 −0.386481
\(362\) 3.65685 0.192200
\(363\) 0.414214 0.0217406
\(364\) 0 0
\(365\) 22.4853 1.17693
\(366\) 2.55635 0.133623
\(367\) 14.7279 0.768791 0.384396 0.923168i \(-0.374410\pi\)
0.384396 + 0.923168i \(0.374410\pi\)
\(368\) 2.24264 0.116906
\(369\) 7.31371 0.380736
\(370\) 22.4853 1.16895
\(371\) 0 0
\(372\) −1.65685 −0.0859039
\(373\) 13.9706 0.723368 0.361684 0.932301i \(-0.382202\pi\)
0.361684 + 0.932301i \(0.382202\pi\)
\(374\) −7.65685 −0.395927
\(375\) −2.34315 −0.121000
\(376\) 6.48528 0.334453
\(377\) −15.8284 −0.815205
\(378\) 0 0
\(379\) −25.8701 −1.32886 −0.664428 0.747352i \(-0.731325\pi\)
−0.664428 + 0.747352i \(0.731325\pi\)
\(380\) 11.6569 0.597984
\(381\) −4.02944 −0.206434
\(382\) −12.8284 −0.656359
\(383\) 30.3848 1.55259 0.776295 0.630370i \(-0.217097\pi\)
0.776295 + 0.630370i \(0.217097\pi\)
\(384\) 0.414214 0.0211377
\(385\) 0 0
\(386\) 2.10051 0.106913
\(387\) −16.0000 −0.813326
\(388\) 1.82843 0.0928243
\(389\) 2.72792 0.138311 0.0691556 0.997606i \(-0.477970\pi\)
0.0691556 + 0.997606i \(0.477970\pi\)
\(390\) −2.58579 −0.130936
\(391\) −17.1716 −0.868404
\(392\) 0 0
\(393\) −1.41421 −0.0713376
\(394\) −17.4853 −0.880896
\(395\) 16.2426 0.817256
\(396\) −2.82843 −0.142134
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 19.8995 0.997472
\(399\) 0 0
\(400\) 6.65685 0.332843
\(401\) −4.31371 −0.215416 −0.107708 0.994183i \(-0.534351\pi\)
−0.107708 + 0.994183i \(0.534351\pi\)
\(402\) 4.65685 0.232263
\(403\) −7.31371 −0.364322
\(404\) 11.8284 0.588486
\(405\) −25.5563 −1.26991
\(406\) 0 0
\(407\) −6.58579 −0.326445
\(408\) −3.17157 −0.157016
\(409\) −22.7279 −1.12382 −0.561912 0.827197i \(-0.689934\pi\)
−0.561912 + 0.827197i \(0.689934\pi\)
\(410\) 8.82843 0.436005
\(411\) −2.21320 −0.109169
\(412\) 10.5858 0.521524
\(413\) 0 0
\(414\) −6.34315 −0.311749
\(415\) −55.1127 −2.70538
\(416\) 1.82843 0.0896460
\(417\) 0 0
\(418\) −3.41421 −0.166995
\(419\) 2.14214 0.104650 0.0523251 0.998630i \(-0.483337\pi\)
0.0523251 + 0.998630i \(0.483337\pi\)
\(420\) 0 0
\(421\) 23.3137 1.13624 0.568120 0.822946i \(-0.307671\pi\)
0.568120 + 0.822946i \(0.307671\pi\)
\(422\) 4.58579 0.223233
\(423\) −18.3431 −0.891874
\(424\) −11.8995 −0.577891
\(425\) −50.9706 −2.47244
\(426\) 1.27208 0.0616324
\(427\) 0 0
\(428\) 11.0711 0.535140
\(429\) 0.757359 0.0365657
\(430\) −19.3137 −0.931390
\(431\) −20.4142 −0.983318 −0.491659 0.870788i \(-0.663609\pi\)
−0.491659 + 0.870788i \(0.663609\pi\)
\(432\) −2.41421 −0.116154
\(433\) −2.14214 −0.102944 −0.0514722 0.998674i \(-0.516391\pi\)
−0.0514722 + 0.998674i \(0.516391\pi\)
\(434\) 0 0
\(435\) 12.2426 0.586990
\(436\) −0.485281 −0.0232408
\(437\) −7.65685 −0.366277
\(438\) −2.72792 −0.130345
\(439\) 9.38478 0.447911 0.223955 0.974599i \(-0.428103\pi\)
0.223955 + 0.974599i \(0.428103\pi\)
\(440\) −3.41421 −0.162766
\(441\) 0 0
\(442\) −14.0000 −0.665912
\(443\) −32.6274 −1.55018 −0.775088 0.631854i \(-0.782294\pi\)
−0.775088 + 0.631854i \(0.782294\pi\)
\(444\) −2.72792 −0.129461
\(445\) 15.3137 0.725939
\(446\) 11.4142 0.540479
\(447\) 2.62742 0.124273
\(448\) 0 0
\(449\) 33.6569 1.58837 0.794183 0.607679i \(-0.207899\pi\)
0.794183 + 0.607679i \(0.207899\pi\)
\(450\) −18.8284 −0.887581
\(451\) −2.58579 −0.121760
\(452\) −13.8284 −0.650434
\(453\) 4.02944 0.189319
\(454\) −23.1716 −1.08750
\(455\) 0 0
\(456\) −1.41421 −0.0662266
\(457\) −0.343146 −0.0160517 −0.00802584 0.999968i \(-0.502555\pi\)
−0.00802584 + 0.999968i \(0.502555\pi\)
\(458\) 0.686292 0.0320683
\(459\) 18.4853 0.862819
\(460\) −7.65685 −0.357003
\(461\) 14.3137 0.666656 0.333328 0.942811i \(-0.391828\pi\)
0.333328 + 0.942811i \(0.391828\pi\)
\(462\) 0 0
\(463\) 7.17157 0.333291 0.166646 0.986017i \(-0.446706\pi\)
0.166646 + 0.986017i \(0.446706\pi\)
\(464\) −8.65685 −0.401884
\(465\) 5.65685 0.262330
\(466\) −1.41421 −0.0655122
\(467\) −34.0000 −1.57333 −0.786666 0.617379i \(-0.788195\pi\)
−0.786666 + 0.617379i \(0.788195\pi\)
\(468\) −5.17157 −0.239056
\(469\) 0 0
\(470\) −22.1421 −1.02134
\(471\) 2.62742 0.121065
\(472\) −8.41421 −0.387296
\(473\) 5.65685 0.260102
\(474\) −1.97056 −0.0905109
\(475\) −22.7279 −1.04283
\(476\) 0 0
\(477\) 33.6569 1.54104
\(478\) −22.2132 −1.01601
\(479\) −26.0711 −1.19122 −0.595609 0.803275i \(-0.703089\pi\)
−0.595609 + 0.803275i \(0.703089\pi\)
\(480\) −1.41421 −0.0645497
\(481\) −12.0416 −0.549051
\(482\) 3.75736 0.171143
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −6.24264 −0.283464
\(486\) 10.3431 0.469175
\(487\) 1.65685 0.0750792 0.0375396 0.999295i \(-0.488048\pi\)
0.0375396 + 0.999295i \(0.488048\pi\)
\(488\) 6.17157 0.279374
\(489\) −6.51472 −0.294606
\(490\) 0 0
\(491\) 24.8284 1.12049 0.560246 0.828327i \(-0.310707\pi\)
0.560246 + 0.828327i \(0.310707\pi\)
\(492\) −1.07107 −0.0482875
\(493\) 66.2843 2.98529
\(494\) −6.24264 −0.280870
\(495\) 9.65685 0.434043
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 6.68629 0.299620
\(499\) −6.14214 −0.274960 −0.137480 0.990505i \(-0.543900\pi\)
−0.137480 + 0.990505i \(0.543900\pi\)
\(500\) −5.65685 −0.252982
\(501\) −4.85786 −0.217033
\(502\) 2.14214 0.0956082
\(503\) −38.2132 −1.70384 −0.851921 0.523670i \(-0.824563\pi\)
−0.851921 + 0.523670i \(0.824563\pi\)
\(504\) 0 0
\(505\) −40.3848 −1.79710
\(506\) 2.24264 0.0996975
\(507\) −4.00000 −0.177646
\(508\) −9.72792 −0.431607
\(509\) −13.3137 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(510\) 10.8284 0.479491
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 8.24264 0.363921
\(514\) 3.14214 0.138594
\(515\) −36.1421 −1.59261
\(516\) 2.34315 0.103151
\(517\) 6.48528 0.285222
\(518\) 0 0
\(519\) −1.72792 −0.0758474
\(520\) −6.24264 −0.273758
\(521\) 28.2843 1.23916 0.619578 0.784935i \(-0.287304\pi\)
0.619578 + 0.784935i \(0.287304\pi\)
\(522\) 24.4853 1.07169
\(523\) 12.7279 0.556553 0.278277 0.960501i \(-0.410237\pi\)
0.278277 + 0.960501i \(0.410237\pi\)
\(524\) −3.41421 −0.149151
\(525\) 0 0
\(526\) 31.0416 1.35348
\(527\) 30.6274 1.33415
\(528\) 0.414214 0.0180263
\(529\) −17.9706 −0.781329
\(530\) 40.6274 1.76474
\(531\) 23.7990 1.03279
\(532\) 0 0
\(533\) −4.72792 −0.204789
\(534\) −1.85786 −0.0803977
\(535\) −37.7990 −1.63419
\(536\) 11.2426 0.485608
\(537\) −7.82843 −0.337822
\(538\) 13.6569 0.588789
\(539\) 0 0
\(540\) 8.24264 0.354707
\(541\) −41.1421 −1.76884 −0.884419 0.466693i \(-0.845445\pi\)
−0.884419 + 0.466693i \(0.845445\pi\)
\(542\) −26.5563 −1.14069
\(543\) 1.51472 0.0650028
\(544\) −7.65685 −0.328285
\(545\) 1.65685 0.0709718
\(546\) 0 0
\(547\) −18.8701 −0.806825 −0.403413 0.915018i \(-0.632176\pi\)
−0.403413 + 0.915018i \(0.632176\pi\)
\(548\) −5.34315 −0.228248
\(549\) −17.4558 −0.744997
\(550\) 6.65685 0.283849
\(551\) 29.5563 1.25914
\(552\) 0.928932 0.0395380
\(553\) 0 0
\(554\) −3.82843 −0.162654
\(555\) 9.31371 0.395345
\(556\) 0 0
\(557\) 24.4853 1.03747 0.518737 0.854934i \(-0.326402\pi\)
0.518737 + 0.854934i \(0.326402\pi\)
\(558\) 11.3137 0.478947
\(559\) 10.3431 0.437468
\(560\) 0 0
\(561\) −3.17157 −0.133904
\(562\) −16.7279 −0.705625
\(563\) −5.07107 −0.213720 −0.106860 0.994274i \(-0.534080\pi\)
−0.106860 + 0.994274i \(0.534080\pi\)
\(564\) 2.68629 0.113113
\(565\) 47.2132 1.98627
\(566\) −20.5858 −0.865285
\(567\) 0 0
\(568\) 3.07107 0.128859
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 4.82843 0.202241
\(571\) −10.3848 −0.434589 −0.217295 0.976106i \(-0.569723\pi\)
−0.217295 + 0.976106i \(0.569723\pi\)
\(572\) 1.82843 0.0764504
\(573\) −5.31371 −0.221983
\(574\) 0 0
\(575\) 14.9289 0.622580
\(576\) −2.82843 −0.117851
\(577\) −32.3137 −1.34524 −0.672619 0.739989i \(-0.734831\pi\)
−0.672619 + 0.739989i \(0.734831\pi\)
\(578\) 41.6274 1.73147
\(579\) 0.870058 0.0361584
\(580\) 29.5563 1.22726
\(581\) 0 0
\(582\) 0.757359 0.0313936
\(583\) −11.8995 −0.492827
\(584\) −6.58579 −0.272522
\(585\) 17.6569 0.730021
\(586\) −5.17157 −0.213636
\(587\) −44.8995 −1.85320 −0.926600 0.376048i \(-0.877283\pi\)
−0.926600 + 0.376048i \(0.877283\pi\)
\(588\) 0 0
\(589\) 13.6569 0.562721
\(590\) 28.7279 1.18271
\(591\) −7.24264 −0.297922
\(592\) −6.58579 −0.270674
\(593\) −35.6985 −1.46596 −0.732981 0.680250i \(-0.761871\pi\)
−0.732981 + 0.680250i \(0.761871\pi\)
\(594\) −2.41421 −0.0990564
\(595\) 0 0
\(596\) 6.34315 0.259825
\(597\) 8.24264 0.337349
\(598\) 4.10051 0.167682
\(599\) −2.62742 −0.107353 −0.0536767 0.998558i \(-0.517094\pi\)
−0.0536767 + 0.998558i \(0.517094\pi\)
\(600\) 2.75736 0.112569
\(601\) −35.9411 −1.46607 −0.733035 0.680191i \(-0.761897\pi\)
−0.733035 + 0.680191i \(0.761897\pi\)
\(602\) 0 0
\(603\) −31.7990 −1.29495
\(604\) 9.72792 0.395824
\(605\) −3.41421 −0.138808
\(606\) 4.89949 0.199028
\(607\) −9.02944 −0.366494 −0.183247 0.983067i \(-0.558661\pi\)
−0.183247 + 0.983067i \(0.558661\pi\)
\(608\) −3.41421 −0.138465
\(609\) 0 0
\(610\) −21.0711 −0.853143
\(611\) 11.8579 0.479718
\(612\) 21.6569 0.875426
\(613\) −16.6274 −0.671575 −0.335788 0.941938i \(-0.609002\pi\)
−0.335788 + 0.941938i \(0.609002\pi\)
\(614\) −9.89949 −0.399511
\(615\) 3.65685 0.147459
\(616\) 0 0
\(617\) −8.02944 −0.323253 −0.161626 0.986852i \(-0.551674\pi\)
−0.161626 + 0.986852i \(0.551674\pi\)
\(618\) 4.38478 0.176382
\(619\) 45.9411 1.84653 0.923265 0.384164i \(-0.125510\pi\)
0.923265 + 0.384164i \(0.125510\pi\)
\(620\) 13.6569 0.548472
\(621\) −5.41421 −0.217265
\(622\) −8.72792 −0.349958
\(623\) 0 0
\(624\) 0.757359 0.0303186
\(625\) −13.9706 −0.558823
\(626\) −9.34315 −0.373427
\(627\) −1.41421 −0.0564782
\(628\) 6.34315 0.253119
\(629\) 50.4264 2.01063
\(630\) 0 0
\(631\) 48.7279 1.93983 0.969914 0.243448i \(-0.0782785\pi\)
0.969914 + 0.243448i \(0.0782785\pi\)
\(632\) −4.75736 −0.189238
\(633\) 1.89949 0.0754981
\(634\) 31.3137 1.24363
\(635\) 33.2132 1.31803
\(636\) −4.92893 −0.195445
\(637\) 0 0
\(638\) −8.65685 −0.342728
\(639\) −8.68629 −0.343624
\(640\) −3.41421 −0.134959
\(641\) −41.2843 −1.63063 −0.815315 0.579017i \(-0.803436\pi\)
−0.815315 + 0.579017i \(0.803436\pi\)
\(642\) 4.58579 0.180987
\(643\) 4.41421 0.174080 0.0870398 0.996205i \(-0.472259\pi\)
0.0870398 + 0.996205i \(0.472259\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 26.1421 1.02855
\(647\) −46.1838 −1.81567 −0.907836 0.419326i \(-0.862266\pi\)
−0.907836 + 0.419326i \(0.862266\pi\)
\(648\) 7.48528 0.294050
\(649\) −8.41421 −0.330287
\(650\) 12.1716 0.477408
\(651\) 0 0
\(652\) −15.7279 −0.615953
\(653\) 18.3848 0.719452 0.359726 0.933058i \(-0.382870\pi\)
0.359726 + 0.933058i \(0.382870\pi\)
\(654\) −0.201010 −0.00786012
\(655\) 11.6569 0.455471
\(656\) −2.58579 −0.100958
\(657\) 18.6274 0.726725
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −1.41421 −0.0550482
\(661\) −14.9706 −0.582287 −0.291144 0.956679i \(-0.594036\pi\)
−0.291144 + 0.956679i \(0.594036\pi\)
\(662\) −9.92893 −0.385899
\(663\) −5.79899 −0.225214
\(664\) 16.1421 0.626436
\(665\) 0 0
\(666\) 18.6274 0.721798
\(667\) −19.4142 −0.751721
\(668\) −11.7279 −0.453767
\(669\) 4.72792 0.182792
\(670\) −38.3848 −1.48293
\(671\) 6.17157 0.238251
\(672\) 0 0
\(673\) −25.5563 −0.985125 −0.492562 0.870277i \(-0.663940\pi\)
−0.492562 + 0.870277i \(0.663940\pi\)
\(674\) −19.7574 −0.761025
\(675\) −16.0711 −0.618576
\(676\) −9.65685 −0.371417
\(677\) −12.6863 −0.487574 −0.243787 0.969829i \(-0.578390\pi\)
−0.243787 + 0.969829i \(0.578390\pi\)
\(678\) −5.72792 −0.219980
\(679\) 0 0
\(680\) 26.1421 1.00251
\(681\) −9.59798 −0.367795
\(682\) −4.00000 −0.153168
\(683\) 16.4142 0.628072 0.314036 0.949411i \(-0.398319\pi\)
0.314036 + 0.949411i \(0.398319\pi\)
\(684\) 9.65685 0.369239
\(685\) 18.2426 0.697015
\(686\) 0 0
\(687\) 0.284271 0.0108456
\(688\) 5.65685 0.215666
\(689\) −21.7574 −0.828889
\(690\) −3.17157 −0.120740
\(691\) 13.9289 0.529882 0.264941 0.964265i \(-0.414648\pi\)
0.264941 + 0.964265i \(0.414648\pi\)
\(692\) −4.17157 −0.158579
\(693\) 0 0
\(694\) 14.5858 0.553669
\(695\) 0 0
\(696\) −3.58579 −0.135919
\(697\) 19.7990 0.749940
\(698\) 0 0
\(699\) −0.585786 −0.0221565
\(700\) 0 0
\(701\) 36.1127 1.36396 0.681979 0.731372i \(-0.261120\pi\)
0.681979 + 0.731372i \(0.261120\pi\)
\(702\) −4.41421 −0.166604
\(703\) 22.4853 0.848048
\(704\) 1.00000 0.0376889
\(705\) −9.17157 −0.345421
\(706\) −25.3137 −0.952694
\(707\) 0 0
\(708\) −3.48528 −0.130985
\(709\) 36.0416 1.35357 0.676786 0.736180i \(-0.263372\pi\)
0.676786 + 0.736180i \(0.263372\pi\)
\(710\) −10.4853 −0.393506
\(711\) 13.4558 0.504634
\(712\) −4.48528 −0.168093
\(713\) −8.97056 −0.335950
\(714\) 0 0
\(715\) −6.24264 −0.233462
\(716\) −18.8995 −0.706307
\(717\) −9.20101 −0.343618
\(718\) 10.7574 0.401461
\(719\) −18.4853 −0.689385 −0.344692 0.938716i \(-0.612017\pi\)
−0.344692 + 0.938716i \(0.612017\pi\)
\(720\) 9.65685 0.359890
\(721\) 0 0
\(722\) −7.34315 −0.273284
\(723\) 1.55635 0.0578812
\(724\) 3.65685 0.135906
\(725\) −57.6274 −2.14023
\(726\) 0.414214 0.0153729
\(727\) −48.4264 −1.79604 −0.898018 0.439959i \(-0.854993\pi\)
−0.898018 + 0.439959i \(0.854993\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 22.4853 0.832218
\(731\) −43.3137 −1.60202
\(732\) 2.55635 0.0944854
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 14.7279 0.543618
\(735\) 0 0
\(736\) 2.24264 0.0826648
\(737\) 11.2426 0.414128
\(738\) 7.31371 0.269221
\(739\) 32.4264 1.19282 0.596412 0.802678i \(-0.296592\pi\)
0.596412 + 0.802678i \(0.296592\pi\)
\(740\) 22.4853 0.826575
\(741\) −2.58579 −0.0949912
\(742\) 0 0
\(743\) 9.31371 0.341687 0.170843 0.985298i \(-0.445351\pi\)
0.170843 + 0.985298i \(0.445351\pi\)
\(744\) −1.65685 −0.0607432
\(745\) −21.6569 −0.793446
\(746\) 13.9706 0.511499
\(747\) −45.6569 −1.67050
\(748\) −7.65685 −0.279962
\(749\) 0 0
\(750\) −2.34315 −0.0855596
\(751\) 34.3431 1.25320 0.626600 0.779341i \(-0.284446\pi\)
0.626600 + 0.779341i \(0.284446\pi\)
\(752\) 6.48528 0.236494
\(753\) 0.887302 0.0323351
\(754\) −15.8284 −0.576437
\(755\) −33.2132 −1.20875
\(756\) 0 0
\(757\) 19.6569 0.714441 0.357220 0.934020i \(-0.383725\pi\)
0.357220 + 0.934020i \(0.383725\pi\)
\(758\) −25.8701 −0.939643
\(759\) 0.928932 0.0337181
\(760\) 11.6569 0.422839
\(761\) −2.97056 −0.107683 −0.0538414 0.998549i \(-0.517147\pi\)
−0.0538414 + 0.998549i \(0.517147\pi\)
\(762\) −4.02944 −0.145971
\(763\) 0 0
\(764\) −12.8284 −0.464116
\(765\) −73.9411 −2.67335
\(766\) 30.3848 1.09785
\(767\) −15.3848 −0.555512
\(768\) 0.414214 0.0149466
\(769\) −10.9706 −0.395609 −0.197804 0.980242i \(-0.563381\pi\)
−0.197804 + 0.980242i \(0.563381\pi\)
\(770\) 0 0
\(771\) 1.30152 0.0468729
\(772\) 2.10051 0.0755988
\(773\) 45.9411 1.65239 0.826194 0.563386i \(-0.190502\pi\)
0.826194 + 0.563386i \(0.190502\pi\)
\(774\) −16.0000 −0.575108
\(775\) −26.6274 −0.956485
\(776\) 1.82843 0.0656367
\(777\) 0 0
\(778\) 2.72792 0.0978007
\(779\) 8.82843 0.316311
\(780\) −2.58579 −0.0925860
\(781\) 3.07107 0.109891
\(782\) −17.1716 −0.614054
\(783\) 20.8995 0.746887
\(784\) 0 0
\(785\) −21.6569 −0.772966
\(786\) −1.41421 −0.0504433
\(787\) −17.5563 −0.625816 −0.312908 0.949783i \(-0.601303\pi\)
−0.312908 + 0.949783i \(0.601303\pi\)
\(788\) −17.4853 −0.622887
\(789\) 12.8579 0.457752
\(790\) 16.2426 0.577887
\(791\) 0 0
\(792\) −2.82843 −0.100504
\(793\) 11.2843 0.400716
\(794\) 22.0000 0.780751
\(795\) 16.8284 0.596843
\(796\) 19.8995 0.705319
\(797\) 3.65685 0.129532 0.0647662 0.997900i \(-0.479370\pi\)
0.0647662 + 0.997900i \(0.479370\pi\)
\(798\) 0 0
\(799\) −49.6569 −1.75673
\(800\) 6.65685 0.235355
\(801\) 12.6863 0.448248
\(802\) −4.31371 −0.152322
\(803\) −6.58579 −0.232407
\(804\) 4.65685 0.164235
\(805\) 0 0
\(806\) −7.31371 −0.257614
\(807\) 5.65685 0.199131
\(808\) 11.8284 0.416123
\(809\) −1.37258 −0.0482574 −0.0241287 0.999709i \(-0.507681\pi\)
−0.0241287 + 0.999709i \(0.507681\pi\)
\(810\) −25.5563 −0.897959
\(811\) 32.2843 1.13365 0.566827 0.823837i \(-0.308171\pi\)
0.566827 + 0.823837i \(0.308171\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) −6.58579 −0.230832
\(815\) 53.6985 1.88098
\(816\) −3.17157 −0.111027
\(817\) −19.3137 −0.675701
\(818\) −22.7279 −0.794663
\(819\) 0 0
\(820\) 8.82843 0.308302
\(821\) −12.5147 −0.436767 −0.218383 0.975863i \(-0.570078\pi\)
−0.218383 + 0.975863i \(0.570078\pi\)
\(822\) −2.21320 −0.0771943
\(823\) 7.45584 0.259894 0.129947 0.991521i \(-0.458519\pi\)
0.129947 + 0.991521i \(0.458519\pi\)
\(824\) 10.5858 0.368773
\(825\) 2.75736 0.0959989
\(826\) 0 0
\(827\) 49.3553 1.71625 0.858127 0.513438i \(-0.171628\pi\)
0.858127 + 0.513438i \(0.171628\pi\)
\(828\) −6.34315 −0.220440
\(829\) 30.2426 1.05037 0.525185 0.850988i \(-0.323996\pi\)
0.525185 + 0.850988i \(0.323996\pi\)
\(830\) −55.1127 −1.91299
\(831\) −1.58579 −0.0550103
\(832\) 1.82843 0.0633893
\(833\) 0 0
\(834\) 0 0
\(835\) 40.0416 1.38570
\(836\) −3.41421 −0.118083
\(837\) 9.65685 0.333790
\(838\) 2.14214 0.0739988
\(839\) −1.51472 −0.0522939 −0.0261469 0.999658i \(-0.508324\pi\)
−0.0261469 + 0.999658i \(0.508324\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) 23.3137 0.803443
\(843\) −6.92893 −0.238645
\(844\) 4.58579 0.157849
\(845\) 32.9706 1.13422
\(846\) −18.3431 −0.630650
\(847\) 0 0
\(848\) −11.8995 −0.408630
\(849\) −8.52691 −0.292643
\(850\) −50.9706 −1.74828
\(851\) −14.7696 −0.506294
\(852\) 1.27208 0.0435807
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −32.9706 −1.12757
\(856\) 11.0711 0.378401
\(857\) 10.6274 0.363026 0.181513 0.983389i \(-0.441901\pi\)
0.181513 + 0.983389i \(0.441901\pi\)
\(858\) 0.757359 0.0258558
\(859\) 19.7279 0.673108 0.336554 0.941664i \(-0.390739\pi\)
0.336554 + 0.941664i \(0.390739\pi\)
\(860\) −19.3137 −0.658592
\(861\) 0 0
\(862\) −20.4142 −0.695311
\(863\) −7.21320 −0.245540 −0.122770 0.992435i \(-0.539178\pi\)
−0.122770 + 0.992435i \(0.539178\pi\)
\(864\) −2.41421 −0.0821332
\(865\) 14.2426 0.484264
\(866\) −2.14214 −0.0727927
\(867\) 17.2426 0.585591
\(868\) 0 0
\(869\) −4.75736 −0.161382
\(870\) 12.2426 0.415064
\(871\) 20.5563 0.696525
\(872\) −0.485281 −0.0164337
\(873\) −5.17157 −0.175031
\(874\) −7.65685 −0.258997
\(875\) 0 0
\(876\) −2.72792 −0.0921679
\(877\) −25.6274 −0.865376 −0.432688 0.901544i \(-0.642435\pi\)
−0.432688 + 0.901544i \(0.642435\pi\)
\(878\) 9.38478 0.316721
\(879\) −2.14214 −0.0722524
\(880\) −3.41421 −0.115093
\(881\) −44.4558 −1.49776 −0.748878 0.662708i \(-0.769407\pi\)
−0.748878 + 0.662708i \(0.769407\pi\)
\(882\) 0 0
\(883\) −9.72792 −0.327371 −0.163685 0.986513i \(-0.552338\pi\)
−0.163685 + 0.986513i \(0.552338\pi\)
\(884\) −14.0000 −0.470871
\(885\) 11.8995 0.399997
\(886\) −32.6274 −1.09614
\(887\) −22.8995 −0.768890 −0.384445 0.923148i \(-0.625607\pi\)
−0.384445 + 0.923148i \(0.625607\pi\)
\(888\) −2.72792 −0.0915431
\(889\) 0 0
\(890\) 15.3137 0.513317
\(891\) 7.48528 0.250766
\(892\) 11.4142 0.382176
\(893\) −22.1421 −0.740958
\(894\) 2.62742 0.0878740
\(895\) 64.5269 2.15690
\(896\) 0 0
\(897\) 1.69848 0.0567108
\(898\) 33.6569 1.12314
\(899\) 34.6274 1.15489
\(900\) −18.8284 −0.627614
\(901\) 91.1127 3.03540
\(902\) −2.58579 −0.0860973
\(903\) 0 0
\(904\) −13.8284 −0.459927
\(905\) −12.4853 −0.415025
\(906\) 4.02944 0.133869
\(907\) 33.3137 1.10616 0.553082 0.833127i \(-0.313452\pi\)
0.553082 + 0.833127i \(0.313452\pi\)
\(908\) −23.1716 −0.768976
\(909\) −33.4558 −1.10966
\(910\) 0 0
\(911\) −13.5147 −0.447763 −0.223881 0.974616i \(-0.571873\pi\)
−0.223881 + 0.974616i \(0.571873\pi\)
\(912\) −1.41421 −0.0468293
\(913\) 16.1421 0.534227
\(914\) −0.343146 −0.0113503
\(915\) −8.72792 −0.288536
\(916\) 0.686292 0.0226757
\(917\) 0 0
\(918\) 18.4853 0.610105
\(919\) −6.14214 −0.202610 −0.101305 0.994855i \(-0.532302\pi\)
−0.101305 + 0.994855i \(0.532302\pi\)
\(920\) −7.65685 −0.252439
\(921\) −4.10051 −0.135116
\(922\) 14.3137 0.471397
\(923\) 5.61522 0.184827
\(924\) 0 0
\(925\) −43.8406 −1.44147
\(926\) 7.17157 0.235673
\(927\) −29.9411 −0.983396
\(928\) −8.65685 −0.284175
\(929\) 10.4558 0.343045 0.171523 0.985180i \(-0.445131\pi\)
0.171523 + 0.985180i \(0.445131\pi\)
\(930\) 5.65685 0.185496
\(931\) 0 0
\(932\) −1.41421 −0.0463241
\(933\) −3.61522 −0.118357
\(934\) −34.0000 −1.11251
\(935\) 26.1421 0.854939
\(936\) −5.17157 −0.169038
\(937\) −16.5858 −0.541834 −0.270917 0.962603i \(-0.587327\pi\)
−0.270917 + 0.962603i \(0.587327\pi\)
\(938\) 0 0
\(939\) −3.87006 −0.126295
\(940\) −22.1421 −0.722197
\(941\) 13.3431 0.434974 0.217487 0.976063i \(-0.430214\pi\)
0.217487 + 0.976063i \(0.430214\pi\)
\(942\) 2.62742 0.0856059
\(943\) −5.79899 −0.188841
\(944\) −8.41421 −0.273859
\(945\) 0 0
\(946\) 5.65685 0.183920
\(947\) 29.1716 0.947949 0.473974 0.880539i \(-0.342819\pi\)
0.473974 + 0.880539i \(0.342819\pi\)
\(948\) −1.97056 −0.0640009
\(949\) −12.0416 −0.390888
\(950\) −22.7279 −0.737391
\(951\) 12.9706 0.420599
\(952\) 0 0
\(953\) −25.5563 −0.827851 −0.413926 0.910311i \(-0.635843\pi\)
−0.413926 + 0.910311i \(0.635843\pi\)
\(954\) 33.6569 1.08968
\(955\) 43.7990 1.41730
\(956\) −22.2132 −0.718426
\(957\) −3.58579 −0.115912
\(958\) −26.0711 −0.842318
\(959\) 0 0
\(960\) −1.41421 −0.0456435
\(961\) −15.0000 −0.483871
\(962\) −12.0416 −0.388238
\(963\) −31.3137 −1.00907
\(964\) 3.75736 0.121016
\(965\) −7.17157 −0.230861
\(966\) 0 0
\(967\) 20.2843 0.652298 0.326149 0.945318i \(-0.394249\pi\)
0.326149 + 0.945318i \(0.394249\pi\)
\(968\) 1.00000 0.0321412
\(969\) 10.8284 0.347859
\(970\) −6.24264 −0.200439
\(971\) 47.7279 1.53166 0.765831 0.643042i \(-0.222328\pi\)
0.765831 + 0.643042i \(0.222328\pi\)
\(972\) 10.3431 0.331757
\(973\) 0 0
\(974\) 1.65685 0.0530890
\(975\) 5.04163 0.161461
\(976\) 6.17157 0.197547
\(977\) 40.0000 1.27971 0.639857 0.768494i \(-0.278994\pi\)
0.639857 + 0.768494i \(0.278994\pi\)
\(978\) −6.51472 −0.208318
\(979\) −4.48528 −0.143350
\(980\) 0 0
\(981\) 1.37258 0.0438232
\(982\) 24.8284 0.792307
\(983\) −52.4264 −1.67214 −0.836071 0.548621i \(-0.815153\pi\)
−0.836071 + 0.548621i \(0.815153\pi\)
\(984\) −1.07107 −0.0341444
\(985\) 59.6985 1.90215
\(986\) 66.2843 2.11092
\(987\) 0 0
\(988\) −6.24264 −0.198605
\(989\) 12.6863 0.403401
\(990\) 9.65685 0.306915
\(991\) −38.3848 −1.21933 −0.609666 0.792658i \(-0.708697\pi\)
−0.609666 + 0.792658i \(0.708697\pi\)
\(992\) −4.00000 −0.127000
\(993\) −4.11270 −0.130513
\(994\) 0 0
\(995\) −67.9411 −2.15388
\(996\) 6.68629 0.211863
\(997\) 18.8284 0.596302 0.298151 0.954519i \(-0.403630\pi\)
0.298151 + 0.954519i \(0.403630\pi\)
\(998\) −6.14214 −0.194426
\(999\) 15.8995 0.503038
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 1078.2.a.t.1.2 2
3.2 odd 2 9702.2.a.cx.1.2 2
4.3 odd 2 8624.2.a.cc.1.1 2
7.2 even 3 154.2.e.e.67.1 yes 4
7.3 odd 6 1078.2.e.m.177.2 4
7.4 even 3 154.2.e.e.23.1 4
7.5 odd 6 1078.2.e.m.67.2 4
7.6 odd 2 1078.2.a.x.1.1 2
21.2 odd 6 1386.2.k.t.991.1 4
21.11 odd 6 1386.2.k.t.793.1 4
21.20 even 2 9702.2.a.ch.1.1 2
28.11 odd 6 1232.2.q.f.177.2 4
28.23 odd 6 1232.2.q.f.529.2 4
28.27 even 2 8624.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 4 7.4 even 3
154.2.e.e.67.1 yes 4 7.2 even 3
1078.2.a.t.1.2 2 1.1 even 1 trivial
1078.2.a.x.1.1 2 7.6 odd 2
1078.2.e.m.67.2 4 7.5 odd 6
1078.2.e.m.177.2 4 7.3 odd 6
1232.2.q.f.177.2 4 28.11 odd 6
1232.2.q.f.529.2 4 28.23 odd 6
1386.2.k.t.793.1 4 21.11 odd 6
1386.2.k.t.991.1 4 21.2 odd 6
8624.2.a.bh.1.2 2 28.27 even 2
8624.2.a.cc.1.1 2 4.3 odd 2
9702.2.a.ch.1.1 2 21.20 even 2
9702.2.a.cx.1.2 2 3.2 odd 2