Properties

Label 1078.2.a.t.1.1
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -2.41421 q^{6} +1.00000 q^{8} +2.82843 q^{9} -0.585786 q^{10} +1.00000 q^{11} -2.41421 q^{12} -3.82843 q^{13} +1.41421 q^{15} +1.00000 q^{16} +3.65685 q^{17} +2.82843 q^{18} -0.585786 q^{19} -0.585786 q^{20} +1.00000 q^{22} -6.24264 q^{23} -2.41421 q^{24} -4.65685 q^{25} -3.82843 q^{26} +0.414214 q^{27} +2.65685 q^{29} +1.41421 q^{30} -4.00000 q^{31} +1.00000 q^{32} -2.41421 q^{33} +3.65685 q^{34} +2.82843 q^{36} -9.41421 q^{37} -0.585786 q^{38} +9.24264 q^{39} -0.585786 q^{40} -5.41421 q^{41} -5.65685 q^{43} +1.00000 q^{44} -1.65685 q^{45} -6.24264 q^{46} -10.4853 q^{47} -2.41421 q^{48} -4.65685 q^{50} -8.82843 q^{51} -3.82843 q^{52} +7.89949 q^{53} +0.414214 q^{54} -0.585786 q^{55} +1.41421 q^{57} +2.65685 q^{58} -5.58579 q^{59} +1.41421 q^{60} +11.8284 q^{61} -4.00000 q^{62} +1.00000 q^{64} +2.24264 q^{65} -2.41421 q^{66} +2.75736 q^{67} +3.65685 q^{68} +15.0711 q^{69} -11.0711 q^{71} +2.82843 q^{72} -9.41421 q^{73} -9.41421 q^{74} +11.2426 q^{75} -0.585786 q^{76} +9.24264 q^{78} -13.2426 q^{79} -0.585786 q^{80} -9.48528 q^{81} -5.41421 q^{82} -12.1421 q^{83} -2.14214 q^{85} -5.65685 q^{86} -6.41421 q^{87} +1.00000 q^{88} +12.4853 q^{89} -1.65685 q^{90} -6.24264 q^{92} +9.65685 q^{93} -10.4853 q^{94} +0.343146 q^{95} -2.41421 q^{96} -3.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} - 4 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{16} - 4 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} - 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} - 2 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} - 16 q^{37} - 4 q^{38} + 10 q^{39} - 4 q^{40} - 8 q^{41} + 2 q^{44} + 8 q^{45} - 4 q^{46} - 4 q^{47} - 2 q^{48} + 2 q^{50} - 12 q^{51} - 2 q^{52} - 4 q^{53} - 2 q^{54} - 4 q^{55} - 6 q^{58} - 14 q^{59} + 18 q^{61} - 8 q^{62} + 2 q^{64} - 4 q^{65} - 2 q^{66} + 14 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 16 q^{74} + 14 q^{75} - 4 q^{76} + 10 q^{78} - 18 q^{79} - 4 q^{80} - 2 q^{81} - 8 q^{82} + 4 q^{83} + 24 q^{85} - 10 q^{87} + 2 q^{88} + 8 q^{89} + 8 q^{90} - 4 q^{92} + 8 q^{93} - 4 q^{94} + 12 q^{95} - 2 q^{96} - 2 q^{97} + O(q^{100}) \)

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) −2.41421 −0.985599
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 2.82843 0.942809
\(10\) −0.585786 −0.185242
\(11\) 1.00000 0.301511
\(12\) −2.41421 −0.696923
\(13\) −3.82843 −1.06181 −0.530907 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 1.00000 0.250000
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 2.82843 0.666667
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) −0.585786 −0.130986
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.24264 −1.30168 −0.650840 0.759215i \(-0.725583\pi\)
−0.650840 + 0.759215i \(0.725583\pi\)
\(24\) −2.41421 −0.492799
\(25\) −4.65685 −0.931371
\(26\) −3.82843 −0.750816
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 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
\(33\) −2.41421 −0.420261
\(34\) 3.65685 0.627145
\(35\) 0 0
\(36\) 2.82843 0.471405
\(37\) −9.41421 −1.54769 −0.773844 0.633377i \(-0.781668\pi\)
−0.773844 + 0.633377i \(0.781668\pi\)
\(38\) −0.585786 −0.0950271
\(39\) 9.24264 1.48001
\(40\) −0.585786 −0.0926210
\(41\) −5.41421 −0.845558 −0.422779 0.906233i \(-0.638945\pi\)
−0.422779 + 0.906233i \(0.638945\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.65685 −0.246989
\(46\) −6.24264 −0.920427
\(47\) −10.4853 −1.52944 −0.764718 0.644365i \(-0.777122\pi\)
−0.764718 + 0.644365i \(0.777122\pi\)
\(48\) −2.41421 −0.348462
\(49\) 0 0
\(50\) −4.65685 −0.658579
\(51\) −8.82843 −1.23623
\(52\) −3.82843 −0.530907
\(53\) 7.89949 1.08508 0.542540 0.840030i \(-0.317463\pi\)
0.542540 + 0.840030i \(0.317463\pi\)
\(54\) 0.414214 0.0563673
\(55\) −0.585786 −0.0789874
\(56\) 0 0
\(57\) 1.41421 0.187317
\(58\) 2.65685 0.348862
\(59\) −5.58579 −0.727207 −0.363604 0.931554i \(-0.618454\pi\)
−0.363604 + 0.931554i \(0.618454\pi\)
\(60\) 1.41421 0.182574
\(61\) 11.8284 1.51447 0.757237 0.653140i \(-0.226549\pi\)
0.757237 + 0.653140i \(0.226549\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.24264 0.278165
\(66\) −2.41421 −0.297169
\(67\) 2.75736 0.336865 0.168433 0.985713i \(-0.446129\pi\)
0.168433 + 0.985713i \(0.446129\pi\)
\(68\) 3.65685 0.443459
\(69\) 15.0711 1.81434
\(70\) 0 0
\(71\) −11.0711 −1.31389 −0.656947 0.753937i \(-0.728152\pi\)
−0.656947 + 0.753937i \(0.728152\pi\)
\(72\) 2.82843 0.333333
\(73\) −9.41421 −1.10185 −0.550925 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) −9.41421 −1.09438
\(75\) 11.2426 1.29819
\(76\) −0.585786 −0.0671943
\(77\) 0 0
\(78\) 9.24264 1.04652
\(79\) −13.2426 −1.48991 −0.744957 0.667113i \(-0.767530\pi\)
−0.744957 + 0.667113i \(0.767530\pi\)
\(80\) −0.585786 −0.0654929
\(81\) −9.48528 −1.05392
\(82\) −5.41421 −0.597900
\(83\) −12.1421 −1.33277 −0.666386 0.745607i \(-0.732160\pi\)
−0.666386 + 0.745607i \(0.732160\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) −5.65685 −0.609994
\(87\) −6.41421 −0.687676
\(88\) 1.00000 0.106600
\(89\) 12.4853 1.32344 0.661719 0.749752i \(-0.269827\pi\)
0.661719 + 0.749752i \(0.269827\pi\)
\(90\) −1.65685 −0.174648
\(91\) 0 0
\(92\) −6.24264 −0.650840
\(93\) 9.65685 1.00137
\(94\) −10.4853 −1.08147
\(95\) 0.343146 0.0352060
\(96\) −2.41421 −0.246400
\(97\) −3.82843 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) −4.65685 −0.465685
\(101\) 6.17157 0.614094 0.307047 0.951694i \(-0.400659\pi\)
0.307047 + 0.951694i \(0.400659\pi\)
\(102\) −8.82843 −0.874145
\(103\) 13.4142 1.32174 0.660871 0.750500i \(-0.270187\pi\)
0.660871 + 0.750500i \(0.270187\pi\)
\(104\) −3.82843 −0.375408
\(105\) 0 0
\(106\) 7.89949 0.767267
\(107\) −3.07107 −0.296891 −0.148446 0.988921i \(-0.547427\pi\)
−0.148446 + 0.988921i \(0.547427\pi\)
\(108\) 0.414214 0.0398577
\(109\) 16.4853 1.57900 0.789502 0.613748i \(-0.210339\pi\)
0.789502 + 0.613748i \(0.210339\pi\)
\(110\) −0.585786 −0.0558525
\(111\) 22.7279 2.15724
\(112\) 0 0
\(113\) −8.17157 −0.768717 −0.384358 0.923184i \(-0.625577\pi\)
−0.384358 + 0.923184i \(0.625577\pi\)
\(114\) 1.41421 0.132453
\(115\) 3.65685 0.341003
\(116\) 2.65685 0.246683
\(117\) −10.8284 −1.00109
\(118\) −5.58579 −0.514213
\(119\) 0 0
\(120\) 1.41421 0.129099
\(121\) 1.00000 0.0909091
\(122\) 11.8284 1.07090
\(123\) 13.0711 1.17858
\(124\) −4.00000 −0.359211
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 15.7279 1.39563 0.697814 0.716279i \(-0.254156\pi\)
0.697814 + 0.716279i \(0.254156\pi\)
\(128\) 1.00000 0.0883883
\(129\) 13.6569 1.20242
\(130\) 2.24264 0.196693
\(131\) −0.585786 −0.0511804 −0.0255902 0.999673i \(-0.508147\pi\)
−0.0255902 + 0.999673i \(0.508147\pi\)
\(132\) −2.41421 −0.210130
\(133\) 0 0
\(134\) 2.75736 0.238200
\(135\) −0.242641 −0.0208832
\(136\) 3.65685 0.313573
\(137\) −16.6569 −1.42309 −0.711546 0.702640i \(-0.752004\pi\)
−0.711546 + 0.702640i \(0.752004\pi\)
\(138\) 15.0711 1.28293
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 25.3137 2.13180
\(142\) −11.0711 −0.929063
\(143\) −3.82843 −0.320149
\(144\) 2.82843 0.235702
\(145\) −1.55635 −0.129248
\(146\) −9.41421 −0.779126
\(147\) 0 0
\(148\) −9.41421 −0.773844
\(149\) 17.6569 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(150\) 11.2426 0.917958
\(151\) −15.7279 −1.27992 −0.639960 0.768408i \(-0.721049\pi\)
−0.639960 + 0.768408i \(0.721049\pi\)
\(152\) −0.585786 −0.0475136
\(153\) 10.3431 0.836194
\(154\) 0 0
\(155\) 2.34315 0.188206
\(156\) 9.24264 0.740003
\(157\) 17.6569 1.40917 0.704585 0.709619i \(-0.251133\pi\)
0.704585 + 0.709619i \(0.251133\pi\)
\(158\) −13.2426 −1.05353
\(159\) −19.0711 −1.51243
\(160\) −0.585786 −0.0463105
\(161\) 0 0
\(162\) −9.48528 −0.745234
\(163\) 9.72792 0.761950 0.380975 0.924585i \(-0.375588\pi\)
0.380975 + 0.924585i \(0.375588\pi\)
\(164\) −5.41421 −0.422779
\(165\) 1.41421 0.110096
\(166\) −12.1421 −0.942412
\(167\) 13.7279 1.06230 0.531149 0.847278i \(-0.321760\pi\)
0.531149 + 0.847278i \(0.321760\pi\)
\(168\) 0 0
\(169\) 1.65685 0.127450
\(170\) −2.14214 −0.164294
\(171\) −1.65685 −0.126703
\(172\) −5.65685 −0.431331
\(173\) −9.82843 −0.747241 −0.373621 0.927582i \(-0.621884\pi\)
−0.373621 + 0.927582i \(0.621884\pi\)
\(174\) −6.41421 −0.486260
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 13.4853 1.01362
\(178\) 12.4853 0.935811
\(179\) 0.899495 0.0672314 0.0336157 0.999435i \(-0.489298\pi\)
0.0336157 + 0.999435i \(0.489298\pi\)
\(180\) −1.65685 −0.123495
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) 0 0
\(183\) −28.5563 −2.11095
\(184\) −6.24264 −0.460214
\(185\) 5.51472 0.405450
\(186\) 9.65685 0.708075
\(187\) 3.65685 0.267416
\(188\) −10.4853 −0.764718
\(189\) 0 0
\(190\) 0.343146 0.0248944
\(191\) −7.17157 −0.518917 −0.259458 0.965754i \(-0.583544\pi\)
−0.259458 + 0.965754i \(0.583544\pi\)
\(192\) −2.41421 −0.174231
\(193\) 21.8995 1.57636 0.788180 0.615445i \(-0.211024\pi\)
0.788180 + 0.615445i \(0.211024\pi\)
\(194\) −3.82843 −0.274865
\(195\) −5.41421 −0.387720
\(196\) 0 0
\(197\) −0.514719 −0.0366722 −0.0183361 0.999832i \(-0.505837\pi\)
−0.0183361 + 0.999832i \(0.505837\pi\)
\(198\) 2.82843 0.201008
\(199\) 0.100505 0.00712462 0.00356231 0.999994i \(-0.498866\pi\)
0.00356231 + 0.999994i \(0.498866\pi\)
\(200\) −4.65685 −0.329289
\(201\) −6.65685 −0.469538
\(202\) 6.17157 0.434230
\(203\) 0 0
\(204\) −8.82843 −0.618114
\(205\) 3.17157 0.221512
\(206\) 13.4142 0.934613
\(207\) −17.6569 −1.22724
\(208\) −3.82843 −0.265454
\(209\) −0.585786 −0.0405197
\(210\) 0 0
\(211\) 7.41421 0.510416 0.255208 0.966886i \(-0.417856\pi\)
0.255208 + 0.966886i \(0.417856\pi\)
\(212\) 7.89949 0.542540
\(213\) 26.7279 1.83137
\(214\) −3.07107 −0.209934
\(215\) 3.31371 0.225993
\(216\) 0.414214 0.0281837
\(217\) 0 0
\(218\) 16.4853 1.11652
\(219\) 22.7279 1.53581
\(220\) −0.585786 −0.0394937
\(221\) −14.0000 −0.941742
\(222\) 22.7279 1.52540
\(223\) 8.58579 0.574947 0.287473 0.957789i \(-0.407185\pi\)
0.287473 + 0.957789i \(0.407185\pi\)
\(224\) 0 0
\(225\) −13.1716 −0.878105
\(226\) −8.17157 −0.543565
\(227\) −28.8284 −1.91341 −0.956705 0.291059i \(-0.905992\pi\)
−0.956705 + 0.291059i \(0.905992\pi\)
\(228\) 1.41421 0.0936586
\(229\) 23.3137 1.54061 0.770307 0.637674i \(-0.220103\pi\)
0.770307 + 0.637674i \(0.220103\pi\)
\(230\) 3.65685 0.241126
\(231\) 0 0
\(232\) 2.65685 0.174431
\(233\) 1.41421 0.0926482 0.0463241 0.998926i \(-0.485249\pi\)
0.0463241 + 0.998926i \(0.485249\pi\)
\(234\) −10.8284 −0.707876
\(235\) 6.14214 0.400669
\(236\) −5.58579 −0.363604
\(237\) 31.9706 2.07671
\(238\) 0 0
\(239\) 20.2132 1.30748 0.653742 0.756718i \(-0.273198\pi\)
0.653742 + 0.756718i \(0.273198\pi\)
\(240\) 1.41421 0.0912871
\(241\) 12.2426 0.788618 0.394309 0.918978i \(-0.370984\pi\)
0.394309 + 0.918978i \(0.370984\pi\)
\(242\) 1.00000 0.0642824
\(243\) 21.6569 1.38929
\(244\) 11.8284 0.757237
\(245\) 0 0
\(246\) 13.0711 0.833381
\(247\) 2.24264 0.142696
\(248\) −4.00000 −0.254000
\(249\) 29.3137 1.85768
\(250\) 5.65685 0.357771
\(251\) −26.1421 −1.65008 −0.825038 0.565077i \(-0.808847\pi\)
−0.825038 + 0.565077i \(0.808847\pi\)
\(252\) 0 0
\(253\) −6.24264 −0.392471
\(254\) 15.7279 0.986858
\(255\) 5.17157 0.323856
\(256\) 1.00000 0.0625000
\(257\) −25.1421 −1.56832 −0.784162 0.620557i \(-0.786907\pi\)
−0.784162 + 0.620557i \(0.786907\pi\)
\(258\) 13.6569 0.850239
\(259\) 0 0
\(260\) 2.24264 0.139083
\(261\) 7.51472 0.465149
\(262\) −0.585786 −0.0361900
\(263\) −17.0416 −1.05083 −0.525416 0.850845i \(-0.676090\pi\)
−0.525416 + 0.850845i \(0.676090\pi\)
\(264\) −2.41421 −0.148585
\(265\) −4.62742 −0.284260
\(266\) 0 0
\(267\) −30.1421 −1.84467
\(268\) 2.75736 0.168433
\(269\) 2.34315 0.142864 0.0714321 0.997445i \(-0.477243\pi\)
0.0714321 + 0.997445i \(0.477243\pi\)
\(270\) −0.242641 −0.0147666
\(271\) 4.55635 0.276779 0.138389 0.990378i \(-0.455808\pi\)
0.138389 + 0.990378i \(0.455808\pi\)
\(272\) 3.65685 0.221729
\(273\) 0 0
\(274\) −16.6569 −1.00628
\(275\) −4.65685 −0.280819
\(276\) 15.0711 0.907172
\(277\) 1.82843 0.109860 0.0549298 0.998490i \(-0.482507\pi\)
0.0549298 + 0.998490i \(0.482507\pi\)
\(278\) 0 0
\(279\) −11.3137 −0.677334
\(280\) 0 0
\(281\) 8.72792 0.520664 0.260332 0.965519i \(-0.416168\pi\)
0.260332 + 0.965519i \(0.416168\pi\)
\(282\) 25.3137 1.50741
\(283\) −23.4142 −1.39183 −0.695915 0.718124i \(-0.745001\pi\)
−0.695915 + 0.718124i \(0.745001\pi\)
\(284\) −11.0711 −0.656947
\(285\) −0.828427 −0.0490718
\(286\) −3.82843 −0.226380
\(287\) 0 0
\(288\) 2.82843 0.166667
\(289\) −3.62742 −0.213377
\(290\) −1.55635 −0.0913920
\(291\) 9.24264 0.541813
\(292\) −9.41421 −0.550925
\(293\) −10.8284 −0.632603 −0.316302 0.948659i \(-0.602441\pi\)
−0.316302 + 0.948659i \(0.602441\pi\)
\(294\) 0 0
\(295\) 3.27208 0.190508
\(296\) −9.41421 −0.547190
\(297\) 0.414214 0.0240351
\(298\) 17.6569 1.02283
\(299\) 23.8995 1.38214
\(300\) 11.2426 0.649094
\(301\) 0 0
\(302\) −15.7279 −0.905040
\(303\) −14.8995 −0.855954
\(304\) −0.585786 −0.0335972
\(305\) −6.92893 −0.396750
\(306\) 10.3431 0.591278
\(307\) 9.89949 0.564994 0.282497 0.959268i \(-0.408837\pi\)
0.282497 + 0.959268i \(0.408837\pi\)
\(308\) 0 0
\(309\) −32.3848 −1.84231
\(310\) 2.34315 0.133082
\(311\) 16.7279 0.948553 0.474277 0.880376i \(-0.342710\pi\)
0.474277 + 0.880376i \(0.342710\pi\)
\(312\) 9.24264 0.523261
\(313\) −20.6569 −1.16759 −0.583797 0.811900i \(-0.698434\pi\)
−0.583797 + 0.811900i \(0.698434\pi\)
\(314\) 17.6569 0.996434
\(315\) 0 0
\(316\) −13.2426 −0.744957
\(317\) 8.68629 0.487871 0.243935 0.969791i \(-0.421562\pi\)
0.243935 + 0.969791i \(0.421562\pi\)
\(318\) −19.0711 −1.06945
\(319\) 2.65685 0.148755
\(320\) −0.585786 −0.0327465
\(321\) 7.41421 0.413821
\(322\) 0 0
\(323\) −2.14214 −0.119192
\(324\) −9.48528 −0.526960
\(325\) 17.8284 0.988943
\(326\) 9.72792 0.538780
\(327\) −39.7990 −2.20089
\(328\) −5.41421 −0.298950
\(329\) 0 0
\(330\) 1.41421 0.0778499
\(331\) −24.0711 −1.32307 −0.661533 0.749916i \(-0.730094\pi\)
−0.661533 + 0.749916i \(0.730094\pi\)
\(332\) −12.1421 −0.666386
\(333\) −26.6274 −1.45917
\(334\) 13.7279 0.751158
\(335\) −1.61522 −0.0882491
\(336\) 0 0
\(337\) −28.2426 −1.53847 −0.769237 0.638963i \(-0.779364\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(338\) 1.65685 0.0901210
\(339\) 19.7279 1.07147
\(340\) −2.14214 −0.116174
\(341\) −4.00000 −0.216612
\(342\) −1.65685 −0.0895924
\(343\) 0 0
\(344\) −5.65685 −0.304997
\(345\) −8.82843 −0.475307
\(346\) −9.82843 −0.528380
\(347\) 17.4142 0.934844 0.467422 0.884034i \(-0.345183\pi\)
0.467422 + 0.884034i \(0.345183\pi\)
\(348\) −6.41421 −0.343838
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −1.58579 −0.0846430
\(352\) 1.00000 0.0533002
\(353\) −2.68629 −0.142977 −0.0714884 0.997441i \(-0.522775\pi\)
−0.0714884 + 0.997441i \(0.522775\pi\)
\(354\) 13.4853 0.716735
\(355\) 6.48528 0.344203
\(356\) 12.4853 0.661719
\(357\) 0 0
\(358\) 0.899495 0.0475398
\(359\) 19.2426 1.01559 0.507794 0.861479i \(-0.330461\pi\)
0.507794 + 0.861479i \(0.330461\pi\)
\(360\) −1.65685 −0.0873239
\(361\) −18.6569 −0.981940
\(362\) −7.65685 −0.402435
\(363\) −2.41421 −0.126713
\(364\) 0 0
\(365\) 5.51472 0.288654
\(366\) −28.5563 −1.49266
\(367\) −10.7279 −0.559993 −0.279996 0.960001i \(-0.590333\pi\)
−0.279996 + 0.960001i \(0.590333\pi\)
\(368\) −6.24264 −0.325420
\(369\) −15.3137 −0.797200
\(370\) 5.51472 0.286697
\(371\) 0 0
\(372\) 9.65685 0.500685
\(373\) −19.9706 −1.03404 −0.517018 0.855974i \(-0.672958\pi\)
−0.517018 + 0.855974i \(0.672958\pi\)
\(374\) 3.65685 0.189091
\(375\) −13.6569 −0.705237
\(376\) −10.4853 −0.540737
\(377\) −10.1716 −0.523863
\(378\) 0 0
\(379\) 27.8701 1.43159 0.715794 0.698311i \(-0.246065\pi\)
0.715794 + 0.698311i \(0.246065\pi\)
\(380\) 0.343146 0.0176030
\(381\) −37.9706 −1.94529
\(382\) −7.17157 −0.366930
\(383\) −6.38478 −0.326247 −0.163123 0.986606i \(-0.552157\pi\)
−0.163123 + 0.986606i \(0.552157\pi\)
\(384\) −2.41421 −0.123200
\(385\) 0 0
\(386\) 21.8995 1.11465
\(387\) −16.0000 −0.813326
\(388\) −3.82843 −0.194359
\(389\) −22.7279 −1.15235 −0.576176 0.817326i \(-0.695456\pi\)
−0.576176 + 0.817326i \(0.695456\pi\)
\(390\) −5.41421 −0.274159
\(391\) −22.8284 −1.15448
\(392\) 0 0
\(393\) 1.41421 0.0713376
\(394\) −0.514719 −0.0259311
\(395\) 7.75736 0.390315
\(396\) 2.82843 0.142134
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0.100505 0.00503786
\(399\) 0 0
\(400\) −4.65685 −0.232843
\(401\) 18.3137 0.914543 0.457271 0.889327i \(-0.348827\pi\)
0.457271 + 0.889327i \(0.348827\pi\)
\(402\) −6.65685 −0.332014
\(403\) 15.3137 0.762830
\(404\) 6.17157 0.307047
\(405\) 5.55635 0.276097
\(406\) 0 0
\(407\) −9.41421 −0.466645
\(408\) −8.82843 −0.437072
\(409\) 2.72792 0.134887 0.0674435 0.997723i \(-0.478516\pi\)
0.0674435 + 0.997723i \(0.478516\pi\)
\(410\) 3.17157 0.156633
\(411\) 40.2132 1.98357
\(412\) 13.4142 0.660871
\(413\) 0 0
\(414\) −17.6569 −0.867787
\(415\) 7.11270 0.349149
\(416\) −3.82843 −0.187704
\(417\) 0 0
\(418\) −0.585786 −0.0286518
\(419\) −26.1421 −1.27713 −0.638563 0.769569i \(-0.720471\pi\)
−0.638563 + 0.769569i \(0.720471\pi\)
\(420\) 0 0
\(421\) 0.686292 0.0334478 0.0167239 0.999860i \(-0.494676\pi\)
0.0167239 + 0.999860i \(0.494676\pi\)
\(422\) 7.41421 0.360918
\(423\) −29.6569 −1.44197
\(424\) 7.89949 0.383633
\(425\) −17.0294 −0.826049
\(426\) 26.7279 1.29497
\(427\) 0 0
\(428\) −3.07107 −0.148446
\(429\) 9.24264 0.446239
\(430\) 3.31371 0.159801
\(431\) −17.5858 −0.847078 −0.423539 0.905878i \(-0.639212\pi\)
−0.423539 + 0.905878i \(0.639212\pi\)
\(432\) 0.414214 0.0199289
\(433\) 26.1421 1.25631 0.628155 0.778088i \(-0.283810\pi\)
0.628155 + 0.778088i \(0.283810\pi\)
\(434\) 0 0
\(435\) 3.75736 0.180152
\(436\) 16.4853 0.789502
\(437\) 3.65685 0.174931
\(438\) 22.7279 1.08598
\(439\) −27.3848 −1.30700 −0.653502 0.756925i \(-0.726701\pi\)
−0.653502 + 0.756925i \(0.726701\pi\)
\(440\) −0.585786 −0.0279263
\(441\) 0 0
\(442\) −14.0000 −0.665912
\(443\) 12.6274 0.599947 0.299973 0.953948i \(-0.403022\pi\)
0.299973 + 0.953948i \(0.403022\pi\)
\(444\) 22.7279 1.07862
\(445\) −7.31371 −0.346703
\(446\) 8.58579 0.406549
\(447\) −42.6274 −2.01621
\(448\) 0 0
\(449\) 22.3431 1.05444 0.527219 0.849729i \(-0.323235\pi\)
0.527219 + 0.849729i \(0.323235\pi\)
\(450\) −13.1716 −0.620914
\(451\) −5.41421 −0.254945
\(452\) −8.17157 −0.384358
\(453\) 37.9706 1.78401
\(454\) −28.8284 −1.35299
\(455\) 0 0
\(456\) 1.41421 0.0662266
\(457\) −11.6569 −0.545285 −0.272642 0.962115i \(-0.587898\pi\)
−0.272642 + 0.962115i \(0.587898\pi\)
\(458\) 23.3137 1.08938
\(459\) 1.51472 0.0707010
\(460\) 3.65685 0.170502
\(461\) −8.31371 −0.387208 −0.193604 0.981080i \(-0.562018\pi\)
−0.193604 + 0.981080i \(0.562018\pi\)
\(462\) 0 0
\(463\) 12.8284 0.596188 0.298094 0.954537i \(-0.403649\pi\)
0.298094 + 0.954537i \(0.403649\pi\)
\(464\) 2.65685 0.123341
\(465\) −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 \(0.788195\pi\)
\(468\) −10.8284 −0.500544
\(469\) 0 0
\(470\) 6.14214 0.283316
\(471\) −42.6274 −1.96417
\(472\) −5.58579 −0.257107
\(473\) −5.65685 −0.260102
\(474\) 31.9706 1.46846
\(475\) 2.72792 0.125166
\(476\) 0 0
\(477\) 22.3431 1.02302
\(478\) 20.2132 0.924530
\(479\) −11.9289 −0.545047 −0.272523 0.962149i \(-0.587858\pi\)
−0.272523 + 0.962149i \(0.587858\pi\)
\(480\) 1.41421 0.0645497
\(481\) 36.0416 1.64336
\(482\) 12.2426 0.557637
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 2.24264 0.101833
\(486\) 21.6569 0.982375
\(487\) −9.65685 −0.437594 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(488\) 11.8284 0.535448
\(489\) −23.4853 −1.06204
\(490\) 0 0
\(491\) 19.1716 0.865201 0.432600 0.901586i \(-0.357596\pi\)
0.432600 + 0.901586i \(0.357596\pi\)
\(492\) 13.0711 0.589289
\(493\) 9.71573 0.437574
\(494\) 2.24264 0.100901
\(495\) −1.65685 −0.0744701
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 29.3137 1.31358
\(499\) 22.1421 0.991218 0.495609 0.868546i \(-0.334945\pi\)
0.495609 + 0.868546i \(0.334945\pi\)
\(500\) 5.65685 0.252982
\(501\) −33.1421 −1.48068
\(502\) −26.1421 −1.16678
\(503\) 4.21320 0.187857 0.0939287 0.995579i \(-0.470057\pi\)
0.0939287 + 0.995579i \(0.470057\pi\)
\(504\) 0 0
\(505\) −3.61522 −0.160875
\(506\) −6.24264 −0.277519
\(507\) −4.00000 −0.177646
\(508\) 15.7279 0.697814
\(509\) 9.31371 0.412823 0.206411 0.978465i \(-0.433821\pi\)
0.206411 + 0.978465i \(0.433821\pi\)
\(510\) 5.17157 0.229001
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −0.242641 −0.0107128
\(514\) −25.1421 −1.10897
\(515\) −7.85786 −0.346259
\(516\) 13.6569 0.601209
\(517\) −10.4853 −0.461142
\(518\) 0 0
\(519\) 23.7279 1.04154
\(520\) 2.24264 0.0983463
\(521\) −28.2843 −1.23916 −0.619578 0.784935i \(-0.712696\pi\)
−0.619578 + 0.784935i \(0.712696\pi\)
\(522\) 7.51472 0.328910
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) −0.585786 −0.0255902
\(525\) 0 0
\(526\) −17.0416 −0.743050
\(527\) −14.6274 −0.637180
\(528\) −2.41421 −0.105065
\(529\) 15.9706 0.694372
\(530\) −4.62742 −0.201002
\(531\) −15.7990 −0.685618
\(532\) 0 0
\(533\) 20.7279 0.897826
\(534\) −30.1421 −1.30438
\(535\) 1.79899 0.0777771
\(536\) 2.75736 0.119100
\(537\) −2.17157 −0.0937103
\(538\) 2.34315 0.101020
\(539\) 0 0
\(540\) −0.242641 −0.0104416
\(541\) −12.8579 −0.552803 −0.276401 0.961042i \(-0.589142\pi\)
−0.276401 + 0.961042i \(0.589142\pi\)
\(542\) 4.55635 0.195712
\(543\) 18.4853 0.793279
\(544\) 3.65685 0.156786
\(545\) −9.65685 −0.413654
\(546\) 0 0
\(547\) 34.8701 1.49094 0.745468 0.666541i \(-0.232226\pi\)
0.745468 + 0.666541i \(0.232226\pi\)
\(548\) −16.6569 −0.711546
\(549\) 33.4558 1.42786
\(550\) −4.65685 −0.198569
\(551\) −1.55635 −0.0663027
\(552\) 15.0711 0.641467
\(553\) 0 0
\(554\) 1.82843 0.0776824
\(555\) −13.3137 −0.565135
\(556\) 0 0
\(557\) 7.51472 0.318409 0.159204 0.987246i \(-0.449107\pi\)
0.159204 + 0.987246i \(0.449107\pi\)
\(558\) −11.3137 −0.478947
\(559\) 21.6569 0.915987
\(560\) 0 0
\(561\) −8.82843 −0.372736
\(562\) 8.72792 0.368165
\(563\) 9.07107 0.382300 0.191150 0.981561i \(-0.438778\pi\)
0.191150 + 0.981561i \(0.438778\pi\)
\(564\) 25.3137 1.06590
\(565\) 4.78680 0.201382
\(566\) −23.4142 −0.984173
\(567\) 0 0
\(568\) −11.0711 −0.464532
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) −0.828427 −0.0346990
\(571\) 26.3848 1.10417 0.552084 0.833788i \(-0.313833\pi\)
0.552084 + 0.833788i \(0.313833\pi\)
\(572\) −3.82843 −0.160075
\(573\) 17.3137 0.723291
\(574\) 0 0
\(575\) 29.0711 1.21235
\(576\) 2.82843 0.117851
\(577\) −9.68629 −0.403246 −0.201623 0.979463i \(-0.564622\pi\)
−0.201623 + 0.979463i \(0.564622\pi\)
\(578\) −3.62742 −0.150881
\(579\) −52.8701 −2.19720
\(580\) −1.55635 −0.0646239
\(581\) 0 0
\(582\) 9.24264 0.383120
\(583\) 7.89949 0.327164
\(584\) −9.41421 −0.389563
\(585\) 6.34315 0.262257
\(586\) −10.8284 −0.447318
\(587\) −25.1005 −1.03601 −0.518004 0.855378i \(-0.673325\pi\)
−0.518004 + 0.855378i \(0.673325\pi\)
\(588\) 0 0
\(589\) 2.34315 0.0965476
\(590\) 3.27208 0.134709
\(591\) 1.24264 0.0511154
\(592\) −9.41421 −0.386922
\(593\) 23.6985 0.973180 0.486590 0.873630i \(-0.338241\pi\)
0.486590 + 0.873630i \(0.338241\pi\)
\(594\) 0.414214 0.0169954
\(595\) 0 0
\(596\) 17.6569 0.723253
\(597\) −0.242641 −0.00993062
\(598\) 23.8995 0.977323
\(599\) 42.6274 1.74171 0.870855 0.491541i \(-0.163566\pi\)
0.870855 + 0.491541i \(0.163566\pi\)
\(600\) 11.2426 0.458979
\(601\) 31.9411 1.30291 0.651453 0.758689i \(-0.274160\pi\)
0.651453 + 0.758689i \(0.274160\pi\)
\(602\) 0 0
\(603\) 7.79899 0.317599
\(604\) −15.7279 −0.639960
\(605\) −0.585786 −0.0238156
\(606\) −14.8995 −0.605251
\(607\) −42.9706 −1.74412 −0.872061 0.489398i \(-0.837217\pi\)
−0.872061 + 0.489398i \(0.837217\pi\)
\(608\) −0.585786 −0.0237568
\(609\) 0 0
\(610\) −6.92893 −0.280544
\(611\) 40.1421 1.62398
\(612\) 10.3431 0.418097
\(613\) 28.6274 1.15625 0.578125 0.815948i \(-0.303785\pi\)
0.578125 + 0.815948i \(0.303785\pi\)
\(614\) 9.89949 0.399511
\(615\) −7.65685 −0.308754
\(616\) 0 0
\(617\) −41.9706 −1.68967 −0.844836 0.535026i \(-0.820302\pi\)
−0.844836 + 0.535026i \(0.820302\pi\)
\(618\) −32.3848 −1.30271
\(619\) −21.9411 −0.881888 −0.440944 0.897535i \(-0.645356\pi\)
−0.440944 + 0.897535i \(0.645356\pi\)
\(620\) 2.34315 0.0941030
\(621\) −2.58579 −0.103764
\(622\) 16.7279 0.670729
\(623\) 0 0
\(624\) 9.24264 0.370002
\(625\) 19.9706 0.798823
\(626\) −20.6569 −0.825614
\(627\) 1.41421 0.0564782
\(628\) 17.6569 0.704585
\(629\) −34.4264 −1.37267
\(630\) 0 0
\(631\) 23.2721 0.926447 0.463223 0.886242i \(-0.346693\pi\)
0.463223 + 0.886242i \(0.346693\pi\)
\(632\) −13.2426 −0.526764
\(633\) −17.8995 −0.711441
\(634\) 8.68629 0.344977
\(635\) −9.21320 −0.365615
\(636\) −19.0711 −0.756217
\(637\) 0 0
\(638\) 2.65685 0.105186
\(639\) −31.3137 −1.23875
\(640\) −0.585786 −0.0231552
\(641\) 15.2843 0.603692 0.301846 0.953357i \(-0.402397\pi\)
0.301846 + 0.953357i \(0.402397\pi\)
\(642\) 7.41421 0.292616
\(643\) 1.58579 0.0625373 0.0312687 0.999511i \(-0.490045\pi\)
0.0312687 + 0.999511i \(0.490045\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) −2.14214 −0.0842812
\(647\) 30.1838 1.18665 0.593323 0.804964i \(-0.297816\pi\)
0.593323 + 0.804964i \(0.297816\pi\)
\(648\) −9.48528 −0.372617
\(649\) −5.58579 −0.219261
\(650\) 17.8284 0.699288
\(651\) 0 0
\(652\) 9.72792 0.380975
\(653\) −18.3848 −0.719452 −0.359726 0.933058i \(-0.617130\pi\)
−0.359726 + 0.933058i \(0.617130\pi\)
\(654\) −39.7990 −1.55626
\(655\) 0.343146 0.0134078
\(656\) −5.41421 −0.211390
\(657\) −26.6274 −1.03883
\(658\) 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
\(661\) 18.9706 0.737869 0.368935 0.929455i \(-0.379723\pi\)
0.368935 + 0.929455i \(0.379723\pi\)
\(662\) −24.0711 −0.935549
\(663\) 33.7990 1.31264
\(664\) −12.1421 −0.471206
\(665\) 0 0
\(666\) −26.6274 −1.03179
\(667\) −16.5858 −0.642204
\(668\) 13.7279 0.531149
\(669\) −20.7279 −0.801388
\(670\) −1.61522 −0.0624015
\(671\) 11.8284 0.456631
\(672\) 0 0
\(673\) 5.55635 0.214182 0.107091 0.994249i \(-0.465846\pi\)
0.107091 + 0.994249i \(0.465846\pi\)
\(674\) −28.2426 −1.08787
\(675\) −1.92893 −0.0742446
\(676\) 1.65685 0.0637252
\(677\) −35.3137 −1.35722 −0.678608 0.734501i \(-0.737416\pi\)
−0.678608 + 0.734501i \(0.737416\pi\)
\(678\) 19.7279 0.757646
\(679\) 0 0
\(680\) −2.14214 −0.0821472
\(681\) 69.5980 2.66700
\(682\) −4.00000 −0.153168
\(683\) 13.5858 0.519846 0.259923 0.965629i \(-0.416303\pi\)
0.259923 + 0.965629i \(0.416303\pi\)
\(684\) −1.65685 −0.0633514
\(685\) 9.75736 0.372810
\(686\) 0 0
\(687\) −56.2843 −2.14738
\(688\) −5.65685 −0.215666
\(689\) −30.2426 −1.15215
\(690\) −8.82843 −0.336092
\(691\) 28.0711 1.06787 0.533937 0.845524i \(-0.320712\pi\)
0.533937 + 0.845524i \(0.320712\pi\)
\(692\) −9.82843 −0.373621
\(693\) 0 0
\(694\) 17.4142 0.661035
\(695\) 0 0
\(696\) −6.41421 −0.243130
\(697\) −19.7990 −0.749940
\(698\) 0 0
\(699\) −3.41421 −0.129137
\(700\) 0 0
\(701\) −26.1127 −0.986263 −0.493132 0.869955i \(-0.664148\pi\)
−0.493132 + 0.869955i \(0.664148\pi\)
\(702\) −1.58579 −0.0598517
\(703\) 5.51472 0.207992
\(704\) 1.00000 0.0376889
\(705\) −14.8284 −0.558471
\(706\) −2.68629 −0.101100
\(707\) 0 0
\(708\) 13.4853 0.506808
\(709\) −12.0416 −0.452233 −0.226116 0.974100i \(-0.572603\pi\)
−0.226116 + 0.974100i \(0.572603\pi\)
\(710\) 6.48528 0.243388
\(711\) −37.4558 −1.40470
\(712\) 12.4853 0.467906
\(713\) 24.9706 0.935155
\(714\) 0 0
\(715\) 2.24264 0.0838700
\(716\) 0.899495 0.0336157
\(717\) −48.7990 −1.82243
\(718\) 19.2426 0.718129
\(719\) −1.51472 −0.0564895 −0.0282447 0.999601i \(-0.508992\pi\)
−0.0282447 + 0.999601i \(0.508992\pi\)
\(720\) −1.65685 −0.0617473
\(721\) 0 0
\(722\) −18.6569 −0.694336
\(723\) −29.5563 −1.09921
\(724\) −7.65685 −0.284565
\(725\) −12.3726 −0.459506
\(726\) −2.41421 −0.0895999
\(727\) 36.4264 1.35098 0.675490 0.737369i \(-0.263932\pi\)
0.675490 + 0.737369i \(0.263932\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 5.51472 0.204109
\(731\) −20.6863 −0.765110
\(732\) −28.5563 −1.05547
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) −10.7279 −0.395975
\(735\) 0 0
\(736\) −6.24264 −0.230107
\(737\) 2.75736 0.101569
\(738\) −15.3137 −0.563705
\(739\) −52.4264 −1.92854 −0.964268 0.264928i \(-0.914652\pi\)
−0.964268 + 0.264928i \(0.914652\pi\)
\(740\) 5.51472 0.202725
\(741\) −5.41421 −0.198896
\(742\) 0 0
\(743\) −13.3137 −0.488433 −0.244216 0.969721i \(-0.578531\pi\)
−0.244216 + 0.969721i \(0.578531\pi\)
\(744\) 9.65685 0.354037
\(745\) −10.3431 −0.378944
\(746\) −19.9706 −0.731174
\(747\) −34.3431 −1.25655
\(748\) 3.65685 0.133708
\(749\) 0 0
\(750\) −13.6569 −0.498678
\(751\) 45.6569 1.66604 0.833021 0.553241i \(-0.186609\pi\)
0.833021 + 0.553241i \(0.186609\pi\)
\(752\) −10.4853 −0.382359
\(753\) 63.1127 2.29995
\(754\) −10.1716 −0.370427
\(755\) 9.21320 0.335303
\(756\) 0 0
\(757\) 8.34315 0.303237 0.151618 0.988439i \(-0.451552\pi\)
0.151618 + 0.988439i \(0.451552\pi\)
\(758\) 27.8701 1.01229
\(759\) 15.0711 0.547045
\(760\) 0.343146 0.0124472
\(761\) 30.9706 1.12268 0.561341 0.827585i \(-0.310286\pi\)
0.561341 + 0.827585i \(0.310286\pi\)
\(762\) −37.9706 −1.37553
\(763\) 0 0
\(764\) −7.17157 −0.259458
\(765\) −6.05887 −0.219059
\(766\) −6.38478 −0.230691
\(767\) 21.3848 0.772160
\(768\) −2.41421 −0.0871154
\(769\) 22.9706 0.828340 0.414170 0.910200i \(-0.364072\pi\)
0.414170 + 0.910200i \(0.364072\pi\)
\(770\) 0 0
\(771\) 60.6985 2.18600
\(772\) 21.8995 0.788180
\(773\) −21.9411 −0.789167 −0.394584 0.918860i \(-0.629111\pi\)
−0.394584 + 0.918860i \(0.629111\pi\)
\(774\) −16.0000 −0.575108
\(775\) 18.6274 0.669117
\(776\) −3.82843 −0.137433
\(777\) 0 0
\(778\) −22.7279 −0.814835
\(779\) 3.17157 0.113633
\(780\) −5.41421 −0.193860
\(781\) −11.0711 −0.396154
\(782\) −22.8284 −0.816343
\(783\) 1.10051 0.0393288
\(784\) 0 0
\(785\) −10.3431 −0.369163
\(786\) 1.41421 0.0504433
\(787\) 13.5563 0.483232 0.241616 0.970372i \(-0.422323\pi\)
0.241616 + 0.970372i \(0.422323\pi\)
\(788\) −0.514719 −0.0183361
\(789\) 41.1421 1.46470
\(790\) 7.75736 0.275994
\(791\) 0 0
\(792\) 2.82843 0.100504
\(793\) −45.2843 −1.60809
\(794\) 22.0000 0.780751
\(795\) 11.1716 0.396215
\(796\) 0.100505 0.00356231
\(797\) −7.65685 −0.271220 −0.135610 0.990762i \(-0.543299\pi\)
−0.135610 + 0.990762i \(0.543299\pi\)
\(798\) 0 0
\(799\) −38.3431 −1.35648
\(800\) −4.65685 −0.164645
\(801\) 35.3137 1.24775
\(802\) 18.3137 0.646680
\(803\) −9.41421 −0.332220
\(804\) −6.65685 −0.234769
\(805\) 0 0
\(806\) 15.3137 0.539402
\(807\) −5.65685 −0.199131
\(808\) 6.17157 0.217115
\(809\) −46.6274 −1.63933 −0.819666 0.572841i \(-0.805841\pi\)
−0.819666 + 0.572841i \(0.805841\pi\)
\(810\) 5.55635 0.195230
\(811\) −24.2843 −0.852736 −0.426368 0.904550i \(-0.640207\pi\)
−0.426368 + 0.904550i \(0.640207\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) −9.41421 −0.329968
\(815\) −5.69848 −0.199609
\(816\) −8.82843 −0.309057
\(817\) 3.31371 0.115932
\(818\) 2.72792 0.0953796
\(819\) 0 0
\(820\) 3.17157 0.110756
\(821\) −29.4853 −1.02904 −0.514522 0.857477i \(-0.672031\pi\)
−0.514522 + 0.857477i \(0.672031\pi\)
\(822\) 40.2132 1.40260
\(823\) −43.4558 −1.51478 −0.757388 0.652965i \(-0.773525\pi\)
−0.757388 + 0.652965i \(0.773525\pi\)
\(824\) 13.4142 0.467306
\(825\) 11.2426 0.391419
\(826\) 0 0
\(827\) −21.3553 −0.742598 −0.371299 0.928513i \(-0.621087\pi\)
−0.371299 + 0.928513i \(0.621087\pi\)
\(828\) −17.6569 −0.613618
\(829\) 21.7574 0.755664 0.377832 0.925874i \(-0.376670\pi\)
0.377832 + 0.925874i \(0.376670\pi\)
\(830\) 7.11270 0.246885
\(831\) −4.41421 −0.153127
\(832\) −3.82843 −0.132727
\(833\) 0 0
\(834\) 0 0
\(835\) −8.04163 −0.278292
\(836\) −0.585786 −0.0202598
\(837\) −1.65685 −0.0572693
\(838\) −26.1421 −0.903065
\(839\) −18.4853 −0.638183 −0.319091 0.947724i \(-0.603378\pi\)
−0.319091 + 0.947724i \(0.603378\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) 0.686292 0.0236512
\(843\) −21.0711 −0.725726
\(844\) 7.41421 0.255208
\(845\) −0.970563 −0.0333884
\(846\) −29.6569 −1.01962
\(847\) 0 0
\(848\) 7.89949 0.271270
\(849\) 56.5269 1.94000
\(850\) −17.0294 −0.584105
\(851\) 58.7696 2.01459
\(852\) 26.7279 0.915684
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0.970563 0.0331925
\(856\) −3.07107 −0.104967
\(857\) −34.6274 −1.18285 −0.591425 0.806360i \(-0.701434\pi\)
−0.591425 + 0.806360i \(0.701434\pi\)
\(858\) 9.24264 0.315539
\(859\) −5.72792 −0.195434 −0.0977171 0.995214i \(-0.531154\pi\)
−0.0977171 + 0.995214i \(0.531154\pi\)
\(860\) 3.31371 0.112997
\(861\) 0 0
\(862\) −17.5858 −0.598974
\(863\) 35.2132 1.19867 0.599336 0.800498i \(-0.295431\pi\)
0.599336 + 0.800498i \(0.295431\pi\)
\(864\) 0.414214 0.0140918
\(865\) 5.75736 0.195756
\(866\) 26.1421 0.888346
\(867\) 8.75736 0.297416
\(868\) 0 0
\(869\) −13.2426 −0.449226
\(870\) 3.75736 0.127386
\(871\) −10.5563 −0.357688
\(872\) 16.4853 0.558262
\(873\) −10.8284 −0.366487
\(874\) 3.65685 0.123695
\(875\) 0 0
\(876\) 22.7279 0.767905
\(877\) 19.6274 0.662771 0.331385 0.943495i \(-0.392484\pi\)
0.331385 + 0.943495i \(0.392484\pi\)
\(878\) −27.3848 −0.924191
\(879\) 26.1421 0.881752
\(880\) −0.585786 −0.0197469
\(881\) 6.45584 0.217503 0.108751 0.994069i \(-0.465315\pi\)
0.108751 + 0.994069i \(0.465315\pi\)
\(882\) 0 0
\(883\) 15.7279 0.529287 0.264643 0.964346i \(-0.414746\pi\)
0.264643 + 0.964346i \(0.414746\pi\)
\(884\) −14.0000 −0.470871
\(885\) −7.89949 −0.265539
\(886\) 12.6274 0.424226
\(887\) −3.10051 −0.104105 −0.0520524 0.998644i \(-0.516576\pi\)
−0.0520524 + 0.998644i \(0.516576\pi\)
\(888\) 22.7279 0.762699
\(889\) 0 0
\(890\) −7.31371 −0.245156
\(891\) −9.48528 −0.317769
\(892\) 8.58579 0.287473
\(893\) 6.14214 0.205539
\(894\) −42.6274 −1.42567
\(895\) −0.526912 −0.0176127
\(896\) 0 0
\(897\) −57.6985 −1.92650
\(898\) 22.3431 0.745600
\(899\) −10.6274 −0.354444
\(900\) −13.1716 −0.439052
\(901\) 28.8873 0.962376
\(902\) −5.41421 −0.180274
\(903\) 0 0
\(904\) −8.17157 −0.271782
\(905\) 4.48528 0.149096
\(906\) 37.9706 1.26149
\(907\) 10.6863 0.354832 0.177416 0.984136i \(-0.443226\pi\)
0.177416 + 0.984136i \(0.443226\pi\)
\(908\) −28.8284 −0.956705
\(909\) 17.4558 0.578974
\(910\) 0 0
\(911\) −30.4853 −1.01002 −0.505011 0.863113i \(-0.668512\pi\)
−0.505011 + 0.863113i \(0.668512\pi\)
\(912\) 1.41421 0.0468293
\(913\) −12.1421 −0.401846
\(914\) −11.6569 −0.385574
\(915\) 16.7279 0.553008
\(916\) 23.3137 0.770307
\(917\) 0 0
\(918\) 1.51472 0.0499932
\(919\) 22.1421 0.730402 0.365201 0.930929i \(-0.381000\pi\)
0.365201 + 0.930929i \(0.381000\pi\)
\(920\) 3.65685 0.120563
\(921\) −23.8995 −0.787515
\(922\) −8.31371 −0.273797
\(923\) 42.3848 1.39511
\(924\) 0 0
\(925\) 43.8406 1.44147
\(926\) 12.8284 0.421568
\(927\) 37.9411 1.24615
\(928\) 2.65685 0.0872155
\(929\) −40.4558 −1.32731 −0.663657 0.748037i \(-0.730996\pi\)
−0.663657 + 0.748037i \(0.730996\pi\)
\(930\) −5.65685 −0.185496
\(931\) 0 0
\(932\) 1.41421 0.0463241
\(933\) −40.3848 −1.32214
\(934\) −34.0000 −1.11251
\(935\) −2.14214 −0.0700553
\(936\) −10.8284 −0.353938
\(937\) −19.4142 −0.634235 −0.317117 0.948386i \(-0.602715\pi\)
−0.317117 + 0.948386i \(0.602715\pi\)
\(938\) 0 0
\(939\) 49.8701 1.62745
\(940\) 6.14214 0.200334
\(941\) 24.6569 0.803790 0.401895 0.915686i \(-0.368352\pi\)
0.401895 + 0.915686i \(0.368352\pi\)
\(942\) −42.6274 −1.38888
\(943\) 33.7990 1.10065
\(944\) −5.58579 −0.181802
\(945\) 0 0
\(946\) −5.65685 −0.183920
\(947\) 34.8284 1.13177 0.565886 0.824484i \(-0.308534\pi\)
0.565886 + 0.824484i \(0.308534\pi\)
\(948\) 31.9706 1.03836
\(949\) 36.0416 1.16996
\(950\) 2.72792 0.0885055
\(951\) −20.9706 −0.680017
\(952\) 0 0
\(953\) 5.55635 0.179988 0.0899939 0.995942i \(-0.471315\pi\)
0.0899939 + 0.995942i \(0.471315\pi\)
\(954\) 22.3431 0.723386
\(955\) 4.20101 0.135941
\(956\) 20.2132 0.653742
\(957\) −6.41421 −0.207342
\(958\) −11.9289 −0.385406
\(959\) 0 0
\(960\) 1.41421 0.0456435
\(961\) −15.0000 −0.483871
\(962\) 36.0416 1.16203
\(963\) −8.68629 −0.279912
\(964\) 12.2426 0.394309
\(965\) −12.8284 −0.412962
\(966\) 0 0
\(967\) −36.2843 −1.16682 −0.583412 0.812177i \(-0.698283\pi\)
−0.583412 + 0.812177i \(0.698283\pi\)
\(968\) 1.00000 0.0321412
\(969\) 5.17157 0.166135
\(970\) 2.24264 0.0720069
\(971\) 22.2721 0.714745 0.357372 0.933962i \(-0.383673\pi\)
0.357372 + 0.933962i \(0.383673\pi\)
\(972\) 21.6569 0.694644
\(973\) 0 0
\(974\) −9.65685 −0.309426
\(975\) −43.0416 −1.37844
\(976\) 11.8284 0.378619
\(977\) 40.0000 1.27971 0.639857 0.768494i \(-0.278994\pi\)
0.639857 + 0.768494i \(0.278994\pi\)
\(978\) −23.4853 −0.750976
\(979\) 12.4853 0.399031
\(980\) 0 0
\(981\) 46.6274 1.48870
\(982\) 19.1716 0.611789
\(983\) 32.4264 1.03424 0.517121 0.855912i \(-0.327004\pi\)
0.517121 + 0.855912i \(0.327004\pi\)
\(984\) 13.0711 0.416690
\(985\) 0.301515 0.00960707
\(986\) 9.71573 0.309412
\(987\) 0 0
\(988\) 2.24264 0.0713479
\(989\) 35.3137 1.12291
\(990\) −1.65685 −0.0526583
\(991\) −1.61522 −0.0513093 −0.0256546 0.999671i \(-0.508167\pi\)
−0.0256546 + 0.999671i \(0.508167\pi\)
\(992\) −4.00000 −0.127000
\(993\) 58.1127 1.84415
\(994\) 0 0
\(995\) −0.0588745 −0.00186645
\(996\) 29.3137 0.928840
\(997\) 13.1716 0.417148 0.208574 0.978007i \(-0.433118\pi\)
0.208574 + 0.978007i \(0.433118\pi\)
\(998\) 22.1421 0.700897
\(999\) −3.89949 −0.123375
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.1.1 2
3.2 odd 2 9702.2.a.cx.1.1 2
4.3 odd 2 8624.2.a.cc.1.2 2
7.2 even 3 154.2.e.e.67.2 yes 4
7.3 odd 6 1078.2.e.m.177.1 4
7.4 even 3 154.2.e.e.23.2 4
7.5 odd 6 1078.2.e.m.67.1 4
7.6 odd 2 1078.2.a.x.1.2 2
21.2 odd 6 1386.2.k.t.991.2 4
21.11 odd 6 1386.2.k.t.793.2 4
21.20 even 2 9702.2.a.ch.1.2 2
28.11 odd 6 1232.2.q.f.177.1 4
28.23 odd 6 1232.2.q.f.529.1 4
28.27 even 2 8624.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 4 7.4 even 3
154.2.e.e.67.2 yes 4 7.2 even 3
1078.2.a.t.1.1 2 1.1 even 1 trivial
1078.2.a.x.1.2 2 7.6 odd 2
1078.2.e.m.67.1 4 7.5 odd 6
1078.2.e.m.177.1 4 7.3 odd 6
1232.2.q.f.177.1 4 28.11 odd 6
1232.2.q.f.529.1 4 28.23 odd 6
1386.2.k.t.793.2 4 21.11 odd 6
1386.2.k.t.991.2 4 21.2 odd 6
8624.2.a.bh.1.1 2 28.27 even 2
8624.2.a.cc.1.2 2 4.3 odd 2
9702.2.a.ch.1.2 2 21.20 even 2
9702.2.a.cx.1.1 2 3.2 odd 2