Properties

Label 76.8.a.a.1.3
Level $76$
Weight $8$
Character 76.1
Self dual yes
Analytic conductor $23.741$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7412619368\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 5014 x^{3} + 113222 x^{2} - 625803 x + 567036\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(15.8518\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.09048 q^{3} +276.339 q^{5} -720.800 q^{7} -2121.54 q^{9} +O(q^{10})\) \(q+8.09048 q^{3} +276.339 q^{5} -720.800 q^{7} -2121.54 q^{9} -4974.98 q^{11} +8085.05 q^{13} +2235.72 q^{15} +1375.87 q^{17} -6859.00 q^{19} -5831.62 q^{21} -55033.5 q^{23} -1761.49 q^{25} -34858.2 q^{27} -117756. q^{29} -151040. q^{31} -40250.0 q^{33} -199185. q^{35} -76088.3 q^{37} +65411.9 q^{39} +420662. q^{41} -411546. q^{43} -586266. q^{45} -791764. q^{47} -303991. q^{49} +11131.5 q^{51} -800670. q^{53} -1.37478e6 q^{55} -55492.6 q^{57} +3.09461e6 q^{59} -2.19949e6 q^{61} +1.52921e6 q^{63} +2.23422e6 q^{65} -420260. q^{67} -445247. q^{69} +2.69010e6 q^{71} +598122. q^{73} -14251.3 q^{75} +3.58597e6 q^{77} +2.62187e6 q^{79} +4.35780e6 q^{81} -5.86850e6 q^{83} +380208. q^{85} -952705. q^{87} +6.26993e6 q^{89} -5.82770e6 q^{91} -1.22199e6 q^{93} -1.89541e6 q^{95} +1.21791e7 q^{97} +1.05546e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} + O(q^{10}) \) \( 5q - 14q^{3} - 280q^{5} + 414q^{7} + 3779q^{9} - 2662q^{11} - 602q^{13} - 20800q^{15} - 27366q^{17} - 34295q^{19} - 59964q^{21} - 67096q^{23} - 109115q^{25} - 178778q^{27} - 372398q^{29} - 271372q^{31} - 792700q^{33} - 608250q^{35} - 562630q^{37} - 963904q^{39} - 956714q^{41} - 827362q^{43} - 1165100q^{45} - 1812982q^{47} - 862031q^{49} - 2458254q^{51} + 486998q^{53} + 467930q^{55} + 96026q^{57} - 367182q^{59} + 1879732q^{61} - 1007274q^{63} + 1790920q^{65} - 1046394q^{67} + 7261712q^{69} - 4664572q^{71} + 4224942q^{73} + 8194850q^{75} + 8611110q^{77} + 9574024q^{79} + 11351813q^{81} + 11754804q^{83} + 18711750q^{85} + 3801472q^{87} + 2782542q^{89} + 7385214q^{91} + 29535004q^{93} + 1920520q^{95} + 1291574q^{97} - 9760310q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 8.09048 0.173002 0.0865008 0.996252i \(-0.472432\pi\)
0.0865008 + 0.996252i \(0.472432\pi\)
\(4\) 0 0
\(5\) 276.339 0.988662 0.494331 0.869274i \(-0.335413\pi\)
0.494331 + 0.869274i \(0.335413\pi\)
\(6\) 0 0
\(7\) −720.800 −0.794276 −0.397138 0.917759i \(-0.629997\pi\)
−0.397138 + 0.917759i \(0.629997\pi\)
\(8\) 0 0
\(9\) −2121.54 −0.970070
\(10\) 0 0
\(11\) −4974.98 −1.12698 −0.563491 0.826122i \(-0.690542\pi\)
−0.563491 + 0.826122i \(0.690542\pi\)
\(12\) 0 0
\(13\) 8085.05 1.02066 0.510330 0.859979i \(-0.329523\pi\)
0.510330 + 0.859979i \(0.329523\pi\)
\(14\) 0 0
\(15\) 2235.72 0.171040
\(16\) 0 0
\(17\) 1375.87 0.0679215 0.0339608 0.999423i \(-0.489188\pi\)
0.0339608 + 0.999423i \(0.489188\pi\)
\(18\) 0 0
\(19\) −6859.00 −0.229416
\(20\) 0 0
\(21\) −5831.62 −0.137411
\(22\) 0 0
\(23\) −55033.5 −0.943147 −0.471574 0.881827i \(-0.656314\pi\)
−0.471574 + 0.881827i \(0.656314\pi\)
\(24\) 0 0
\(25\) −1761.49 −0.0225471
\(26\) 0 0
\(27\) −34858.2 −0.340825
\(28\) 0 0
\(29\) −117756. −0.896585 −0.448292 0.893887i \(-0.647968\pi\)
−0.448292 + 0.893887i \(0.647968\pi\)
\(30\) 0 0
\(31\) −151040. −0.910598 −0.455299 0.890339i \(-0.650468\pi\)
−0.455299 + 0.890339i \(0.650468\pi\)
\(32\) 0 0
\(33\) −40250.0 −0.194970
\(34\) 0 0
\(35\) −199185. −0.785271
\(36\) 0 0
\(37\) −76088.3 −0.246952 −0.123476 0.992348i \(-0.539404\pi\)
−0.123476 + 0.992348i \(0.539404\pi\)
\(38\) 0 0
\(39\) 65411.9 0.176576
\(40\) 0 0
\(41\) 420662. 0.953212 0.476606 0.879117i \(-0.341867\pi\)
0.476606 + 0.879117i \(0.341867\pi\)
\(42\) 0 0
\(43\) −411546. −0.789366 −0.394683 0.918817i \(-0.629146\pi\)
−0.394683 + 0.918817i \(0.629146\pi\)
\(44\) 0 0
\(45\) −586266. −0.959072
\(46\) 0 0
\(47\) −791764. −1.11238 −0.556190 0.831055i \(-0.687737\pi\)
−0.556190 + 0.831055i \(0.687737\pi\)
\(48\) 0 0
\(49\) −303991. −0.369126
\(50\) 0 0
\(51\) 11131.5 0.0117505
\(52\) 0 0
\(53\) −800670. −0.738734 −0.369367 0.929284i \(-0.620425\pi\)
−0.369367 + 0.929284i \(0.620425\pi\)
\(54\) 0 0
\(55\) −1.37478e6 −1.11421
\(56\) 0 0
\(57\) −55492.6 −0.0396893
\(58\) 0 0
\(59\) 3.09461e6 1.96166 0.980831 0.194860i \(-0.0624254\pi\)
0.980831 + 0.194860i \(0.0624254\pi\)
\(60\) 0 0
\(61\) −2.19949e6 −1.24070 −0.620350 0.784325i \(-0.713010\pi\)
−0.620350 + 0.784325i \(0.713010\pi\)
\(62\) 0 0
\(63\) 1.52921e6 0.770504
\(64\) 0 0
\(65\) 2.23422e6 1.00909
\(66\) 0 0
\(67\) −420260. −0.170709 −0.0853544 0.996351i \(-0.527202\pi\)
−0.0853544 + 0.996351i \(0.527202\pi\)
\(68\) 0 0
\(69\) −445247. −0.163166
\(70\) 0 0
\(71\) 2.69010e6 0.891999 0.446000 0.895033i \(-0.352848\pi\)
0.446000 + 0.895033i \(0.352848\pi\)
\(72\) 0 0
\(73\) 598122. 0.179953 0.0899766 0.995944i \(-0.471321\pi\)
0.0899766 + 0.995944i \(0.471321\pi\)
\(74\) 0 0
\(75\) −14251.3 −0.00390069
\(76\) 0 0
\(77\) 3.58597e6 0.895136
\(78\) 0 0
\(79\) 2.62187e6 0.598296 0.299148 0.954207i \(-0.403298\pi\)
0.299148 + 0.954207i \(0.403298\pi\)
\(80\) 0 0
\(81\) 4.35780e6 0.911107
\(82\) 0 0
\(83\) −5.86850e6 −1.12656 −0.563280 0.826266i \(-0.690461\pi\)
−0.563280 + 0.826266i \(0.690461\pi\)
\(84\) 0 0
\(85\) 380208. 0.0671515
\(86\) 0 0
\(87\) −952705. −0.155110
\(88\) 0 0
\(89\) 6.26993e6 0.942752 0.471376 0.881932i \(-0.343757\pi\)
0.471376 + 0.881932i \(0.343757\pi\)
\(90\) 0 0
\(91\) −5.82770e6 −0.810685
\(92\) 0 0
\(93\) −1.22199e6 −0.157535
\(94\) 0 0
\(95\) −1.89541e6 −0.226815
\(96\) 0 0
\(97\) 1.21791e7 1.35492 0.677462 0.735558i \(-0.263080\pi\)
0.677462 + 0.735558i \(0.263080\pi\)
\(98\) 0 0
\(99\) 1.05546e7 1.09325
\(100\) 0 0
\(101\) 4.80256e6 0.463819 0.231909 0.972737i \(-0.425503\pi\)
0.231909 + 0.972737i \(0.425503\pi\)
\(102\) 0 0
\(103\) 4.18941e6 0.377766 0.188883 0.982000i \(-0.439513\pi\)
0.188883 + 0.982000i \(0.439513\pi\)
\(104\) 0 0
\(105\) −1.61151e6 −0.135853
\(106\) 0 0
\(107\) −1.54409e7 −1.21851 −0.609255 0.792974i \(-0.708531\pi\)
−0.609255 + 0.792974i \(0.708531\pi\)
\(108\) 0 0
\(109\) 2.21155e7 1.63570 0.817850 0.575432i \(-0.195166\pi\)
0.817850 + 0.575432i \(0.195166\pi\)
\(110\) 0 0
\(111\) −615591. −0.0427230
\(112\) 0 0
\(113\) −1.04919e7 −0.684036 −0.342018 0.939693i \(-0.611110\pi\)
−0.342018 + 0.939693i \(0.611110\pi\)
\(114\) 0 0
\(115\) −1.52079e7 −0.932454
\(116\) 0 0
\(117\) −1.71528e7 −0.990112
\(118\) 0 0
\(119\) −991730. −0.0539485
\(120\) 0 0
\(121\) 5.26330e6 0.270091
\(122\) 0 0
\(123\) 3.40336e6 0.164907
\(124\) 0 0
\(125\) −2.20758e7 −1.01095
\(126\) 0 0
\(127\) 1.77716e6 0.0769863 0.0384932 0.999259i \(-0.487744\pi\)
0.0384932 + 0.999259i \(0.487744\pi\)
\(128\) 0 0
\(129\) −3.32960e6 −0.136562
\(130\) 0 0
\(131\) −1.55541e7 −0.604498 −0.302249 0.953229i \(-0.597737\pi\)
−0.302249 + 0.953229i \(0.597737\pi\)
\(132\) 0 0
\(133\) 4.94396e6 0.182219
\(134\) 0 0
\(135\) −9.63269e6 −0.336961
\(136\) 0 0
\(137\) 4.72051e7 1.56844 0.784218 0.620485i \(-0.213064\pi\)
0.784218 + 0.620485i \(0.213064\pi\)
\(138\) 0 0
\(139\) −5.49244e7 −1.73466 −0.867329 0.497735i \(-0.834165\pi\)
−0.867329 + 0.497735i \(0.834165\pi\)
\(140\) 0 0
\(141\) −6.40575e6 −0.192443
\(142\) 0 0
\(143\) −4.02230e7 −1.15027
\(144\) 0 0
\(145\) −3.25407e7 −0.886419
\(146\) 0 0
\(147\) −2.45943e6 −0.0638593
\(148\) 0 0
\(149\) −1.28970e7 −0.319402 −0.159701 0.987165i \(-0.551053\pi\)
−0.159701 + 0.987165i \(0.551053\pi\)
\(150\) 0 0
\(151\) −1.86914e7 −0.441796 −0.220898 0.975297i \(-0.570899\pi\)
−0.220898 + 0.975297i \(0.570899\pi\)
\(152\) 0 0
\(153\) −2.91898e6 −0.0658887
\(154\) 0 0
\(155\) −4.17384e7 −0.900274
\(156\) 0 0
\(157\) 8.68724e7 1.79157 0.895784 0.444490i \(-0.146615\pi\)
0.895784 + 0.444490i \(0.146615\pi\)
\(158\) 0 0
\(159\) −6.47780e6 −0.127802
\(160\) 0 0
\(161\) 3.96681e7 0.749119
\(162\) 0 0
\(163\) 6.53832e7 1.18252 0.591262 0.806480i \(-0.298630\pi\)
0.591262 + 0.806480i \(0.298630\pi\)
\(164\) 0 0
\(165\) −1.11227e7 −0.192759
\(166\) 0 0
\(167\) −1.93667e7 −0.321772 −0.160886 0.986973i \(-0.551435\pi\)
−0.160886 + 0.986973i \(0.551435\pi\)
\(168\) 0 0
\(169\) 2.61950e6 0.0417461
\(170\) 0 0
\(171\) 1.45517e7 0.222549
\(172\) 0 0
\(173\) 1.41324e7 0.207518 0.103759 0.994602i \(-0.466913\pi\)
0.103759 + 0.994602i \(0.466913\pi\)
\(174\) 0 0
\(175\) 1.26968e6 0.0179086
\(176\) 0 0
\(177\) 2.50369e7 0.339370
\(178\) 0 0
\(179\) 613278. 0.00799229 0.00399615 0.999992i \(-0.498728\pi\)
0.00399615 + 0.999992i \(0.498728\pi\)
\(180\) 0 0
\(181\) −7.33331e7 −0.919232 −0.459616 0.888118i \(-0.652013\pi\)
−0.459616 + 0.888118i \(0.652013\pi\)
\(182\) 0 0
\(183\) −1.77949e7 −0.214643
\(184\) 0 0
\(185\) −2.10262e7 −0.244152
\(186\) 0 0
\(187\) −6.84495e6 −0.0765464
\(188\) 0 0
\(189\) 2.51258e7 0.270709
\(190\) 0 0
\(191\) 1.08428e7 0.112597 0.0562984 0.998414i \(-0.482070\pi\)
0.0562984 + 0.998414i \(0.482070\pi\)
\(192\) 0 0
\(193\) 1.57865e8 1.58065 0.790326 0.612687i \(-0.209911\pi\)
0.790326 + 0.612687i \(0.209911\pi\)
\(194\) 0 0
\(195\) 1.80759e7 0.174574
\(196\) 0 0
\(197\) 3.95231e7 0.368315 0.184157 0.982897i \(-0.441044\pi\)
0.184157 + 0.982897i \(0.441044\pi\)
\(198\) 0 0
\(199\) 6.52880e6 0.0587283 0.0293642 0.999569i \(-0.490652\pi\)
0.0293642 + 0.999569i \(0.490652\pi\)
\(200\) 0 0
\(201\) −3.40010e6 −0.0295329
\(202\) 0 0
\(203\) 8.48787e7 0.712136
\(204\) 0 0
\(205\) 1.16245e8 0.942405
\(206\) 0 0
\(207\) 1.16756e8 0.914919
\(208\) 0 0
\(209\) 3.41234e7 0.258548
\(210\) 0 0
\(211\) 1.25402e8 0.918999 0.459500 0.888178i \(-0.348029\pi\)
0.459500 + 0.888178i \(0.348029\pi\)
\(212\) 0 0
\(213\) 2.17642e7 0.154317
\(214\) 0 0
\(215\) −1.13726e8 −0.780416
\(216\) 0 0
\(217\) 1.08870e8 0.723266
\(218\) 0 0
\(219\) 4.83909e6 0.0311322
\(220\) 0 0
\(221\) 1.11240e7 0.0693248
\(222\) 0 0
\(223\) −2.73524e8 −1.65169 −0.825844 0.563899i \(-0.809301\pi\)
−0.825844 + 0.563899i \(0.809301\pi\)
\(224\) 0 0
\(225\) 3.73709e6 0.0218723
\(226\) 0 0
\(227\) −1.45717e8 −0.826835 −0.413418 0.910542i \(-0.635665\pi\)
−0.413418 + 0.910542i \(0.635665\pi\)
\(228\) 0 0
\(229\) 1.70169e8 0.936391 0.468196 0.883625i \(-0.344904\pi\)
0.468196 + 0.883625i \(0.344904\pi\)
\(230\) 0 0
\(231\) 2.90122e7 0.154860
\(232\) 0 0
\(233\) −2.20408e8 −1.14151 −0.570757 0.821119i \(-0.693350\pi\)
−0.570757 + 0.821119i \(0.693350\pi\)
\(234\) 0 0
\(235\) −2.18796e8 −1.09977
\(236\) 0 0
\(237\) 2.12122e7 0.103506
\(238\) 0 0
\(239\) 8.50266e7 0.402867 0.201434 0.979502i \(-0.435440\pi\)
0.201434 + 0.979502i \(0.435440\pi\)
\(240\) 0 0
\(241\) −1.65575e8 −0.761967 −0.380983 0.924582i \(-0.624414\pi\)
−0.380983 + 0.924582i \(0.624414\pi\)
\(242\) 0 0
\(243\) 1.11492e8 0.498448
\(244\) 0 0
\(245\) −8.40047e7 −0.364941
\(246\) 0 0
\(247\) −5.54554e7 −0.234155
\(248\) 0 0
\(249\) −4.74790e7 −0.194896
\(250\) 0 0
\(251\) −2.70250e8 −1.07872 −0.539359 0.842076i \(-0.681333\pi\)
−0.539359 + 0.842076i \(0.681333\pi\)
\(252\) 0 0
\(253\) 2.73791e8 1.06291
\(254\) 0 0
\(255\) 3.07607e6 0.0116173
\(256\) 0 0
\(257\) 5.67907e7 0.208695 0.104347 0.994541i \(-0.466725\pi\)
0.104347 + 0.994541i \(0.466725\pi\)
\(258\) 0 0
\(259\) 5.48444e7 0.196148
\(260\) 0 0
\(261\) 2.49825e8 0.869750
\(262\) 0 0
\(263\) −2.98253e6 −0.0101097 −0.00505487 0.999987i \(-0.501609\pi\)
−0.00505487 + 0.999987i \(0.501609\pi\)
\(264\) 0 0
\(265\) −2.21257e8 −0.730358
\(266\) 0 0
\(267\) 5.07267e7 0.163098
\(268\) 0 0
\(269\) −2.74367e8 −0.859407 −0.429703 0.902970i \(-0.641382\pi\)
−0.429703 + 0.902970i \(0.641382\pi\)
\(270\) 0 0
\(271\) −1.28592e8 −0.392483 −0.196241 0.980556i \(-0.562874\pi\)
−0.196241 + 0.980556i \(0.562874\pi\)
\(272\) 0 0
\(273\) −4.71489e7 −0.140250
\(274\) 0 0
\(275\) 8.76340e6 0.0254102
\(276\) 0 0
\(277\) −5.63559e8 −1.59316 −0.796582 0.604531i \(-0.793361\pi\)
−0.796582 + 0.604531i \(0.793361\pi\)
\(278\) 0 0
\(279\) 3.20439e8 0.883344
\(280\) 0 0
\(281\) −4.69126e8 −1.26130 −0.630648 0.776069i \(-0.717211\pi\)
−0.630648 + 0.776069i \(0.717211\pi\)
\(282\) 0 0
\(283\) 2.16530e8 0.567891 0.283946 0.958840i \(-0.408356\pi\)
0.283946 + 0.958840i \(0.408356\pi\)
\(284\) 0 0
\(285\) −1.53348e7 −0.0392393
\(286\) 0 0
\(287\) −3.03213e8 −0.757114
\(288\) 0 0
\(289\) −4.08446e8 −0.995387
\(290\) 0 0
\(291\) 9.85349e7 0.234404
\(292\) 0 0
\(293\) −3.53977e8 −0.822127 −0.411063 0.911607i \(-0.634843\pi\)
−0.411063 + 0.911607i \(0.634843\pi\)
\(294\) 0 0
\(295\) 8.55163e8 1.93942
\(296\) 0 0
\(297\) 1.73419e8 0.384104
\(298\) 0 0
\(299\) −4.44948e8 −0.962632
\(300\) 0 0
\(301\) 2.96642e8 0.626975
\(302\) 0 0
\(303\) 3.88550e7 0.0802414
\(304\) 0 0
\(305\) −6.07805e8 −1.22663
\(306\) 0 0
\(307\) −6.23436e8 −1.22972 −0.614862 0.788635i \(-0.710788\pi\)
−0.614862 + 0.788635i \(0.710788\pi\)
\(308\) 0 0
\(309\) 3.38943e7 0.0653540
\(310\) 0 0
\(311\) 5.17438e8 0.975432 0.487716 0.873002i \(-0.337830\pi\)
0.487716 + 0.873002i \(0.337830\pi\)
\(312\) 0 0
\(313\) −2.03327e8 −0.374791 −0.187395 0.982285i \(-0.560005\pi\)
−0.187395 + 0.982285i \(0.560005\pi\)
\(314\) 0 0
\(315\) 4.22581e8 0.761768
\(316\) 0 0
\(317\) −2.01407e6 −0.00355114 −0.00177557 0.999998i \(-0.500565\pi\)
−0.00177557 + 0.999998i \(0.500565\pi\)
\(318\) 0 0
\(319\) 5.85836e8 1.01044
\(320\) 0 0
\(321\) −1.24924e8 −0.210804
\(322\) 0 0
\(323\) −9.43712e6 −0.0155823
\(324\) 0 0
\(325\) −1.42418e7 −0.0230129
\(326\) 0 0
\(327\) 1.78925e8 0.282978
\(328\) 0 0
\(329\) 5.70703e8 0.883537
\(330\) 0 0
\(331\) 4.89819e8 0.742399 0.371200 0.928553i \(-0.378947\pi\)
0.371200 + 0.928553i \(0.378947\pi\)
\(332\) 0 0
\(333\) 1.61425e8 0.239560
\(334\) 0 0
\(335\) −1.16134e8 −0.168773
\(336\) 0 0
\(337\) 7.87197e8 1.12041 0.560207 0.828352i \(-0.310721\pi\)
0.560207 + 0.828352i \(0.310721\pi\)
\(338\) 0 0
\(339\) −8.48844e7 −0.118339
\(340\) 0 0
\(341\) 7.51423e8 1.02623
\(342\) 0 0
\(343\) 8.12726e8 1.08746
\(344\) 0 0
\(345\) −1.23039e8 −0.161316
\(346\) 0 0
\(347\) −2.92885e8 −0.376308 −0.188154 0.982140i \(-0.560250\pi\)
−0.188154 + 0.982140i \(0.560250\pi\)
\(348\) 0 0
\(349\) 1.32289e9 1.66584 0.832921 0.553392i \(-0.186667\pi\)
0.832921 + 0.553392i \(0.186667\pi\)
\(350\) 0 0
\(351\) −2.81830e8 −0.347866
\(352\) 0 0
\(353\) −1.07075e9 −1.29562 −0.647811 0.761801i \(-0.724315\pi\)
−0.647811 + 0.761801i \(0.724315\pi\)
\(354\) 0 0
\(355\) 7.43381e8 0.881886
\(356\) 0 0
\(357\) −8.02357e6 −0.00933316
\(358\) 0 0
\(359\) −5.11066e8 −0.582970 −0.291485 0.956575i \(-0.594149\pi\)
−0.291485 + 0.956575i \(0.594149\pi\)
\(360\) 0 0
\(361\) 4.70459e7 0.0526316
\(362\) 0 0
\(363\) 4.25826e7 0.0467261
\(364\) 0 0
\(365\) 1.65285e8 0.177913
\(366\) 0 0
\(367\) −3.57375e8 −0.377392 −0.188696 0.982036i \(-0.560426\pi\)
−0.188696 + 0.982036i \(0.560426\pi\)
\(368\) 0 0
\(369\) −8.92453e8 −0.924683
\(370\) 0 0
\(371\) 5.77123e8 0.586759
\(372\) 0 0
\(373\) −1.92494e8 −0.192060 −0.0960300 0.995378i \(-0.530614\pi\)
−0.0960300 + 0.995378i \(0.530614\pi\)
\(374\) 0 0
\(375\) −1.78604e8 −0.174897
\(376\) 0 0
\(377\) −9.52066e8 −0.915108
\(378\) 0 0
\(379\) 1.22141e9 1.15245 0.576226 0.817291i \(-0.304525\pi\)
0.576226 + 0.817291i \(0.304525\pi\)
\(380\) 0 0
\(381\) 1.43781e7 0.0133188
\(382\) 0 0
\(383\) 6.37917e7 0.0580188 0.0290094 0.999579i \(-0.490765\pi\)
0.0290094 + 0.999579i \(0.490765\pi\)
\(384\) 0 0
\(385\) 9.90944e8 0.884987
\(386\) 0 0
\(387\) 8.73112e8 0.765741
\(388\) 0 0
\(389\) −2.07082e9 −1.78369 −0.891845 0.452340i \(-0.850589\pi\)
−0.891845 + 0.452340i \(0.850589\pi\)
\(390\) 0 0
\(391\) −7.57192e7 −0.0640600
\(392\) 0 0
\(393\) −1.25840e8 −0.104579
\(394\) 0 0
\(395\) 7.24526e8 0.591513
\(396\) 0 0
\(397\) 1.48707e9 1.19280 0.596398 0.802689i \(-0.296598\pi\)
0.596398 + 0.802689i \(0.296598\pi\)
\(398\) 0 0
\(399\) 3.99991e7 0.0315242
\(400\) 0 0
\(401\) 4.94594e8 0.383039 0.191520 0.981489i \(-0.438658\pi\)
0.191520 + 0.981489i \(0.438658\pi\)
\(402\) 0 0
\(403\) −1.22117e9 −0.929411
\(404\) 0 0
\(405\) 1.20423e9 0.900777
\(406\) 0 0
\(407\) 3.78538e8 0.278310
\(408\) 0 0
\(409\) −1.77741e9 −1.28457 −0.642284 0.766467i \(-0.722013\pi\)
−0.642284 + 0.766467i \(0.722013\pi\)
\(410\) 0 0
\(411\) 3.81912e8 0.271342
\(412\) 0 0
\(413\) −2.23060e9 −1.55810
\(414\) 0 0
\(415\) −1.62170e9 −1.11379
\(416\) 0 0
\(417\) −4.44365e8 −0.300098
\(418\) 0 0
\(419\) −5.25911e8 −0.349271 −0.174636 0.984633i \(-0.555875\pi\)
−0.174636 + 0.984633i \(0.555875\pi\)
\(420\) 0 0
\(421\) −1.27157e9 −0.830527 −0.415264 0.909701i \(-0.636311\pi\)
−0.415264 + 0.909701i \(0.636311\pi\)
\(422\) 0 0
\(423\) 1.67976e9 1.07909
\(424\) 0 0
\(425\) −2.42359e6 −0.00153144
\(426\) 0 0
\(427\) 1.58539e9 0.985459
\(428\) 0 0
\(429\) −3.25423e8 −0.198998
\(430\) 0 0
\(431\) −2.07412e9 −1.24786 −0.623928 0.781482i \(-0.714464\pi\)
−0.623928 + 0.781482i \(0.714464\pi\)
\(432\) 0 0
\(433\) −2.00301e9 −1.18570 −0.592851 0.805312i \(-0.701998\pi\)
−0.592851 + 0.805312i \(0.701998\pi\)
\(434\) 0 0
\(435\) −2.63270e8 −0.153352
\(436\) 0 0
\(437\) 3.77475e8 0.216373
\(438\) 0 0
\(439\) 7.57162e8 0.427133 0.213566 0.976929i \(-0.431492\pi\)
0.213566 + 0.976929i \(0.431492\pi\)
\(440\) 0 0
\(441\) 6.44930e8 0.358078
\(442\) 0 0
\(443\) −4.44853e7 −0.0243110 −0.0121555 0.999926i \(-0.503869\pi\)
−0.0121555 + 0.999926i \(0.503869\pi\)
\(444\) 0 0
\(445\) 1.73263e9 0.932063
\(446\) 0 0
\(447\) −1.04343e8 −0.0552570
\(448\) 0 0
\(449\) −3.40918e9 −1.77741 −0.888706 0.458477i \(-0.848395\pi\)
−0.888706 + 0.458477i \(0.848395\pi\)
\(450\) 0 0
\(451\) −2.09279e9 −1.07425
\(452\) 0 0
\(453\) −1.51222e8 −0.0764314
\(454\) 0 0
\(455\) −1.61042e9 −0.801494
\(456\) 0 0
\(457\) 3.41561e9 1.67403 0.837013 0.547183i \(-0.184300\pi\)
0.837013 + 0.547183i \(0.184300\pi\)
\(458\) 0 0
\(459\) −4.79605e7 −0.0231494
\(460\) 0 0
\(461\) −1.71518e9 −0.815371 −0.407686 0.913122i \(-0.633664\pi\)
−0.407686 + 0.913122i \(0.633664\pi\)
\(462\) 0 0
\(463\) −2.34891e9 −1.09985 −0.549924 0.835215i \(-0.685343\pi\)
−0.549924 + 0.835215i \(0.685343\pi\)
\(464\) 0 0
\(465\) −3.37684e8 −0.155749
\(466\) 0 0
\(467\) −1.93983e9 −0.881364 −0.440682 0.897663i \(-0.645263\pi\)
−0.440682 + 0.897663i \(0.645263\pi\)
\(468\) 0 0
\(469\) 3.02923e8 0.135590
\(470\) 0 0
\(471\) 7.02840e8 0.309944
\(472\) 0 0
\(473\) 2.04743e9 0.889602
\(474\) 0 0
\(475\) 1.20821e7 0.00517266
\(476\) 0 0
\(477\) 1.69866e9 0.716624
\(478\) 0 0
\(479\) 4.99335e8 0.207595 0.103798 0.994598i \(-0.466901\pi\)
0.103798 + 0.994598i \(0.466901\pi\)
\(480\) 0 0
\(481\) −6.15177e8 −0.252053
\(482\) 0 0
\(483\) 3.20934e8 0.129599
\(484\) 0 0
\(485\) 3.36557e9 1.33956
\(486\) 0 0
\(487\) −2.33245e9 −0.915084 −0.457542 0.889188i \(-0.651270\pi\)
−0.457542 + 0.889188i \(0.651270\pi\)
\(488\) 0 0
\(489\) 5.28982e8 0.204578
\(490\) 0 0
\(491\) −1.75314e9 −0.668391 −0.334196 0.942504i \(-0.608465\pi\)
−0.334196 + 0.942504i \(0.608465\pi\)
\(492\) 0 0
\(493\) −1.62018e8 −0.0608974
\(494\) 0 0
\(495\) 2.91667e9 1.08086
\(496\) 0 0
\(497\) −1.93902e9 −0.708493
\(498\) 0 0
\(499\) 1.10432e9 0.397872 0.198936 0.980012i \(-0.436251\pi\)
0.198936 + 0.980012i \(0.436251\pi\)
\(500\) 0 0
\(501\) −1.56686e8 −0.0556671
\(502\) 0 0
\(503\) 2.39626e9 0.839548 0.419774 0.907629i \(-0.362109\pi\)
0.419774 + 0.907629i \(0.362109\pi\)
\(504\) 0 0
\(505\) 1.32714e9 0.458560
\(506\) 0 0
\(507\) 2.11930e7 0.00722213
\(508\) 0 0
\(509\) −1.57893e9 −0.530702 −0.265351 0.964152i \(-0.585488\pi\)
−0.265351 + 0.964152i \(0.585488\pi\)
\(510\) 0 0
\(511\) −4.31126e8 −0.142933
\(512\) 0 0
\(513\) 2.39092e8 0.0781907
\(514\) 0 0
\(515\) 1.15770e9 0.373483
\(516\) 0 0
\(517\) 3.93901e9 1.25363
\(518\) 0 0
\(519\) 1.14338e8 0.0359009
\(520\) 0 0
\(521\) 2.97045e9 0.920215 0.460108 0.887863i \(-0.347811\pi\)
0.460108 + 0.887863i \(0.347811\pi\)
\(522\) 0 0
\(523\) 8.32922e8 0.254594 0.127297 0.991865i \(-0.459370\pi\)
0.127297 + 0.991865i \(0.459370\pi\)
\(524\) 0 0
\(525\) 1.02724e7 0.00309822
\(526\) 0 0
\(527\) −2.07812e8 −0.0618492
\(528\) 0 0
\(529\) −3.76141e8 −0.110473
\(530\) 0 0
\(531\) −6.56536e9 −1.90295
\(532\) 0 0
\(533\) 3.40107e9 0.972905
\(534\) 0 0
\(535\) −4.26693e9 −1.20469
\(536\) 0 0
\(537\) 4.96171e6 0.00138268
\(538\) 0 0
\(539\) 1.51235e9 0.415998
\(540\) 0 0
\(541\) 8.68340e8 0.235776 0.117888 0.993027i \(-0.462388\pi\)
0.117888 + 0.993027i \(0.462388\pi\)
\(542\) 0 0
\(543\) −5.93300e8 −0.159028
\(544\) 0 0
\(545\) 6.11138e9 1.61715
\(546\) 0 0
\(547\) 6.78914e9 1.77362 0.886808 0.462139i \(-0.152918\pi\)
0.886808 + 0.462139i \(0.152918\pi\)
\(548\) 0 0
\(549\) 4.66631e9 1.20357
\(550\) 0 0
\(551\) 8.07691e8 0.205691
\(552\) 0 0
\(553\) −1.88984e9 −0.475212
\(554\) 0 0
\(555\) −1.70112e8 −0.0422386
\(556\) 0 0
\(557\) 7.20950e9 1.76771 0.883857 0.467757i \(-0.154938\pi\)
0.883857 + 0.467757i \(0.154938\pi\)
\(558\) 0 0
\(559\) −3.32737e9 −0.805674
\(560\) 0 0
\(561\) −5.53790e7 −0.0132426
\(562\) 0 0
\(563\) −7.85174e9 −1.85433 −0.927164 0.374655i \(-0.877761\pi\)
−0.927164 + 0.374655i \(0.877761\pi\)
\(564\) 0 0
\(565\) −2.89932e9 −0.676281
\(566\) 0 0
\(567\) −3.14110e9 −0.723671
\(568\) 0 0
\(569\) −4.52870e9 −1.03058 −0.515289 0.857017i \(-0.672315\pi\)
−0.515289 + 0.857017i \(0.672315\pi\)
\(570\) 0 0
\(571\) 7.34549e9 1.65118 0.825590 0.564271i \(-0.190843\pi\)
0.825590 + 0.564271i \(0.190843\pi\)
\(572\) 0 0
\(573\) 8.77237e7 0.0194794
\(574\) 0 0
\(575\) 9.69411e7 0.0212653
\(576\) 0 0
\(577\) 3.37956e8 0.0732393 0.0366197 0.999329i \(-0.488341\pi\)
0.0366197 + 0.999329i \(0.488341\pi\)
\(578\) 0 0
\(579\) 1.27721e9 0.273455
\(580\) 0 0
\(581\) 4.23002e9 0.894799
\(582\) 0 0
\(583\) 3.98332e9 0.832540
\(584\) 0 0
\(585\) −4.73999e9 −0.978886
\(586\) 0 0
\(587\) −5.74946e9 −1.17326 −0.586629 0.809856i \(-0.699545\pi\)
−0.586629 + 0.809856i \(0.699545\pi\)
\(588\) 0 0
\(589\) 1.03599e9 0.208906
\(590\) 0 0
\(591\) 3.19761e8 0.0637190
\(592\) 0 0
\(593\) 1.26534e9 0.249181 0.124590 0.992208i \(-0.460238\pi\)
0.124590 + 0.992208i \(0.460238\pi\)
\(594\) 0 0
\(595\) −2.74054e8 −0.0533368
\(596\) 0 0
\(597\) 5.28211e7 0.0101601
\(598\) 0 0
\(599\) 9.89338e9 1.88084 0.940418 0.340021i \(-0.110434\pi\)
0.940418 + 0.340021i \(0.110434\pi\)
\(600\) 0 0
\(601\) −7.60429e9 −1.42889 −0.714443 0.699693i \(-0.753320\pi\)
−0.714443 + 0.699693i \(0.753320\pi\)
\(602\) 0 0
\(603\) 8.91599e8 0.165600
\(604\) 0 0
\(605\) 1.45446e9 0.267028
\(606\) 0 0
\(607\) 4.96147e9 0.900430 0.450215 0.892920i \(-0.351347\pi\)
0.450215 + 0.892920i \(0.351347\pi\)
\(608\) 0 0
\(609\) 6.86710e8 0.123201
\(610\) 0 0
\(611\) −6.40145e9 −1.13536
\(612\) 0 0
\(613\) 5.04021e9 0.883765 0.441883 0.897073i \(-0.354311\pi\)
0.441883 + 0.897073i \(0.354311\pi\)
\(614\) 0 0
\(615\) 9.40482e8 0.163037
\(616\) 0 0
\(617\) −7.84954e9 −1.34538 −0.672692 0.739922i \(-0.734862\pi\)
−0.672692 + 0.739922i \(0.734862\pi\)
\(618\) 0 0
\(619\) −2.69511e9 −0.456729 −0.228365 0.973576i \(-0.573338\pi\)
−0.228365 + 0.973576i \(0.573338\pi\)
\(620\) 0 0
\(621\) 1.91837e9 0.321448
\(622\) 0 0
\(623\) −4.51936e9 −0.748805
\(624\) 0 0
\(625\) −5.96280e9 −0.976945
\(626\) 0 0
\(627\) 2.76075e8 0.0447291
\(628\) 0 0
\(629\) −1.04688e8 −0.0167733
\(630\) 0 0
\(631\) −5.44546e9 −0.862843 −0.431422 0.902150i \(-0.641988\pi\)
−0.431422 + 0.902150i \(0.641988\pi\)
\(632\) 0 0
\(633\) 1.01456e9 0.158988
\(634\) 0 0
\(635\) 4.91100e8 0.0761135
\(636\) 0 0
\(637\) −2.45778e9 −0.376752
\(638\) 0 0
\(639\) −5.70717e9 −0.865302
\(640\) 0 0
\(641\) −1.19511e10 −1.79228 −0.896139 0.443773i \(-0.853640\pi\)
−0.896139 + 0.443773i \(0.853640\pi\)
\(642\) 0 0
\(643\) −1.08936e10 −1.61597 −0.807987 0.589200i \(-0.799443\pi\)
−0.807987 + 0.589200i \(0.799443\pi\)
\(644\) 0 0
\(645\) −9.20101e8 −0.135013
\(646\) 0 0
\(647\) 1.48672e9 0.215807 0.107903 0.994161i \(-0.465586\pi\)
0.107903 + 0.994161i \(0.465586\pi\)
\(648\) 0 0
\(649\) −1.53956e10 −2.21076
\(650\) 0 0
\(651\) 8.80809e8 0.125126
\(652\) 0 0
\(653\) 8.66204e9 1.21737 0.608687 0.793411i \(-0.291697\pi\)
0.608687 + 0.793411i \(0.291697\pi\)
\(654\) 0 0
\(655\) −4.29821e9 −0.597644
\(656\) 0 0
\(657\) −1.26894e9 −0.174567
\(658\) 0 0
\(659\) −4.47037e9 −0.608477 −0.304239 0.952596i \(-0.598402\pi\)
−0.304239 + 0.952596i \(0.598402\pi\)
\(660\) 0 0
\(661\) 8.62221e9 1.16122 0.580608 0.814183i \(-0.302815\pi\)
0.580608 + 0.814183i \(0.302815\pi\)
\(662\) 0 0
\(663\) 8.99986e7 0.0119933
\(664\) 0 0
\(665\) 1.36621e9 0.180153
\(666\) 0 0
\(667\) 6.48054e9 0.845611
\(668\) 0 0
\(669\) −2.21294e9 −0.285744
\(670\) 0 0
\(671\) 1.09424e10 1.39825
\(672\) 0 0
\(673\) 2.08380e9 0.263513 0.131757 0.991282i \(-0.457938\pi\)
0.131757 + 0.991282i \(0.457938\pi\)
\(674\) 0 0
\(675\) 6.14025e7 0.00768463
\(676\) 0 0
\(677\) 7.68414e9 0.951777 0.475888 0.879506i \(-0.342127\pi\)
0.475888 + 0.879506i \(0.342127\pi\)
\(678\) 0 0
\(679\) −8.77871e9 −1.07618
\(680\) 0 0
\(681\) −1.17892e9 −0.143044
\(682\) 0 0
\(683\) 7.90692e9 0.949587 0.474794 0.880097i \(-0.342523\pi\)
0.474794 + 0.880097i \(0.342523\pi\)
\(684\) 0 0
\(685\) 1.30446e10 1.55065
\(686\) 0 0
\(687\) 1.37675e9 0.161997
\(688\) 0 0
\(689\) −6.47346e9 −0.753996
\(690\) 0 0
\(691\) 1.47299e10 1.69835 0.849175 0.528112i \(-0.177100\pi\)
0.849175 + 0.528112i \(0.177100\pi\)
\(692\) 0 0
\(693\) −7.60779e9 −0.868345
\(694\) 0 0
\(695\) −1.51778e10 −1.71499
\(696\) 0 0
\(697\) 5.78778e8 0.0647437
\(698\) 0 0
\(699\) −1.78320e9 −0.197484
\(700\) 0 0
\(701\) 1.76814e10 1.93867 0.969335 0.245744i \(-0.0790323\pi\)
0.969335 + 0.245744i \(0.0790323\pi\)
\(702\) 0 0
\(703\) 5.21889e8 0.0566546
\(704\) 0 0
\(705\) −1.77016e9 −0.190261
\(706\) 0 0
\(707\) −3.46169e9 −0.368400
\(708\) 0 0
\(709\) −3.52758e9 −0.371719 −0.185859 0.982576i \(-0.559507\pi\)
−0.185859 + 0.982576i \(0.559507\pi\)
\(710\) 0 0
\(711\) −5.56241e9 −0.580390
\(712\) 0 0
\(713\) 8.31227e9 0.858828
\(714\) 0 0
\(715\) −1.11152e10 −1.13722
\(716\) 0 0
\(717\) 6.87906e8 0.0696967
\(718\) 0 0
\(719\) −1.24573e10 −1.24990 −0.624949 0.780666i \(-0.714880\pi\)
−0.624949 + 0.780666i \(0.714880\pi\)
\(720\) 0 0
\(721\) −3.01972e9 −0.300050
\(722\) 0 0
\(723\) −1.33958e9 −0.131821
\(724\) 0 0
\(725\) 2.07427e8 0.0202154
\(726\) 0 0
\(727\) 1.43981e10 1.38975 0.694873 0.719132i \(-0.255460\pi\)
0.694873 + 0.719132i \(0.255460\pi\)
\(728\) 0 0
\(729\) −8.62848e9 −0.824875
\(730\) 0 0
\(731\) −5.66235e8 −0.0536150
\(732\) 0 0
\(733\) −1.94697e10 −1.82598 −0.912990 0.407983i \(-0.866232\pi\)
−0.912990 + 0.407983i \(0.866232\pi\)
\(734\) 0 0
\(735\) −6.79638e8 −0.0631353
\(736\) 0 0
\(737\) 2.09078e9 0.192386
\(738\) 0 0
\(739\) 1.90895e10 1.73996 0.869979 0.493089i \(-0.164132\pi\)
0.869979 + 0.493089i \(0.164132\pi\)
\(740\) 0 0
\(741\) −4.48660e8 −0.0405092
\(742\) 0 0
\(743\) 1.60614e10 1.43655 0.718277 0.695758i \(-0.244931\pi\)
0.718277 + 0.695758i \(0.244931\pi\)
\(744\) 0 0
\(745\) −3.56395e9 −0.315780
\(746\) 0 0
\(747\) 1.24503e10 1.09284
\(748\) 0 0
\(749\) 1.11298e10 0.967833
\(750\) 0 0
\(751\) 1.54515e10 1.33116 0.665579 0.746327i \(-0.268185\pi\)
0.665579 + 0.746327i \(0.268185\pi\)
\(752\) 0 0
\(753\) −2.18645e9 −0.186620
\(754\) 0 0
\(755\) −5.16516e9 −0.436787
\(756\) 0 0
\(757\) −1.26828e10 −1.06262 −0.531312 0.847176i \(-0.678301\pi\)
−0.531312 + 0.847176i \(0.678301\pi\)
\(758\) 0 0
\(759\) 2.21510e9 0.183885
\(760\) 0 0
\(761\) 1.77060e10 1.45638 0.728190 0.685375i \(-0.240362\pi\)
0.728190 + 0.685375i \(0.240362\pi\)
\(762\) 0 0
\(763\) −1.59408e10 −1.29920
\(764\) 0 0
\(765\) −8.06629e8 −0.0651416
\(766\) 0 0
\(767\) 2.50201e10 2.00219
\(768\) 0 0
\(769\) 1.88518e9 0.149489 0.0747446 0.997203i \(-0.476186\pi\)
0.0747446 + 0.997203i \(0.476186\pi\)
\(770\) 0 0
\(771\) 4.59464e8 0.0361045
\(772\) 0 0
\(773\) 6.99272e8 0.0544525 0.0272262 0.999629i \(-0.491333\pi\)
0.0272262 + 0.999629i \(0.491333\pi\)
\(774\) 0 0
\(775\) 2.66056e8 0.0205314
\(776\) 0 0
\(777\) 4.43718e8 0.0339338
\(778\) 0 0
\(779\) −2.88532e9 −0.218682
\(780\) 0 0
\(781\) −1.33832e10 −1.00527
\(782\) 0 0
\(783\) 4.10477e9 0.305579
\(784\) 0 0
\(785\) 2.40063e10 1.77126
\(786\) 0 0
\(787\) 2.19911e10 1.60818 0.804092 0.594505i \(-0.202652\pi\)
0.804092 + 0.594505i \(0.202652\pi\)
\(788\) 0 0
\(789\) −2.41301e7 −0.00174900
\(790\) 0 0
\(791\) 7.56255e9 0.543314
\(792\) 0 0
\(793\) −1.77830e10 −1.26633
\(794\) 0 0
\(795\) −1.79007e9 −0.126353
\(796\) 0 0
\(797\) 1.41708e10 0.991496 0.495748 0.868466i \(-0.334894\pi\)
0.495748 + 0.868466i \(0.334894\pi\)
\(798\) 0 0
\(799\) −1.08937e9 −0.0755546
\(800\) 0 0
\(801\) −1.33019e10 −0.914536
\(802\) 0 0
\(803\) −2.97565e9 −0.202804
\(804\) 0 0
\(805\) 1.09619e10 0.740626
\(806\) 0 0
\(807\) −2.21976e9 −0.148679
\(808\) 0 0
\(809\) −2.00198e10 −1.32935 −0.664677 0.747131i \(-0.731431\pi\)
−0.664677 + 0.747131i \(0.731431\pi\)
\(810\) 0 0
\(811\) −6.19672e9 −0.407933 −0.203967 0.978978i \(-0.565383\pi\)
−0.203967 + 0.978978i \(0.565383\pi\)
\(812\) 0 0
\(813\) −1.04037e9 −0.0679001
\(814\) 0 0
\(815\) 1.80680e10 1.16912
\(816\) 0 0
\(817\) 2.82279e9 0.181093
\(818\) 0 0
\(819\) 1.23637e10 0.786422
\(820\) 0 0
\(821\) −1.36180e10 −0.858839 −0.429419 0.903105i \(-0.641282\pi\)
−0.429419 + 0.903105i \(0.641282\pi\)
\(822\) 0 0
\(823\) −9.96103e9 −0.622880 −0.311440 0.950266i \(-0.600811\pi\)
−0.311440 + 0.950266i \(0.600811\pi\)
\(824\) 0 0
\(825\) 7.09001e7 0.00439601
\(826\) 0 0
\(827\) 2.53194e10 1.55663 0.778313 0.627876i \(-0.216076\pi\)
0.778313 + 0.627876i \(0.216076\pi\)
\(828\) 0 0
\(829\) −2.10063e10 −1.28059 −0.640294 0.768130i \(-0.721187\pi\)
−0.640294 + 0.768130i \(0.721187\pi\)
\(830\) 0 0
\(831\) −4.55946e9 −0.275620
\(832\) 0 0
\(833\) −4.18253e8 −0.0250716
\(834\) 0 0
\(835\) −5.35179e9 −0.318124
\(836\) 0 0
\(837\) 5.26499e9 0.310355
\(838\) 0 0
\(839\) −1.76564e10 −1.03213 −0.516066 0.856549i \(-0.672604\pi\)
−0.516066 + 0.856549i \(0.672604\pi\)
\(840\) 0 0
\(841\) −3.38332e9 −0.196136
\(842\) 0 0
\(843\) −3.79545e9 −0.218206
\(844\) 0 0
\(845\) 7.23872e8 0.0412727
\(846\) 0 0
\(847\) −3.79379e9 −0.214526
\(848\) 0 0
\(849\) 1.75183e9 0.0982461
\(850\) 0 0
\(851\) 4.18740e9 0.232912
\(852\) 0 0
\(853\) −1.73063e9 −0.0954734 −0.0477367 0.998860i \(-0.515201\pi\)
−0.0477367 + 0.998860i \(0.515201\pi\)
\(854\) 0 0
\(855\) 4.02120e9 0.220026
\(856\) 0 0
\(857\) −1.07792e8 −0.00584995 −0.00292498 0.999996i \(-0.500931\pi\)
−0.00292498 + 0.999996i \(0.500931\pi\)
\(858\) 0 0
\(859\) −1.21663e10 −0.654911 −0.327455 0.944867i \(-0.606191\pi\)
−0.327455 + 0.944867i \(0.606191\pi\)
\(860\) 0 0
\(861\) −2.45314e9 −0.130982
\(862\) 0 0
\(863\) −2.32114e10 −1.22932 −0.614659 0.788793i \(-0.710707\pi\)
−0.614659 + 0.788793i \(0.710707\pi\)
\(864\) 0 0
\(865\) 3.90535e9 0.205165
\(866\) 0 0
\(867\) −3.30452e9 −0.172203
\(868\) 0 0
\(869\) −1.30438e10 −0.674270
\(870\) 0 0
\(871\) −3.39782e9 −0.174236
\(872\) 0 0
\(873\) −2.58385e10 −1.31437
\(874\) 0 0
\(875\) 1.59122e10 0.802976
\(876\) 0 0
\(877\) −2.30902e10 −1.15593 −0.577963 0.816063i \(-0.696152\pi\)
−0.577963 + 0.816063i \(0.696152\pi\)
\(878\) 0 0
\(879\) −2.86385e9 −0.142229
\(880\) 0 0
\(881\) 6.17232e9 0.304112 0.152056 0.988372i \(-0.451411\pi\)
0.152056 + 0.988372i \(0.451411\pi\)
\(882\) 0 0
\(883\) −1.18206e10 −0.577801 −0.288901 0.957359i \(-0.593290\pi\)
−0.288901 + 0.957359i \(0.593290\pi\)
\(884\) 0 0
\(885\) 6.91868e9 0.335523
\(886\) 0 0
\(887\) 4.11165e10 1.97826 0.989129 0.147049i \(-0.0469775\pi\)
0.989129 + 0.147049i \(0.0469775\pi\)
\(888\) 0 0
\(889\) −1.28098e9 −0.0611484
\(890\) 0 0
\(891\) −2.16800e10 −1.02680
\(892\) 0 0
\(893\) 5.43071e9 0.255197
\(894\) 0 0
\(895\) 1.69473e8 0.00790168
\(896\) 0 0
\(897\) −3.59985e9 −0.166537
\(898\) 0 0
\(899\) 1.77859e10 0.816428
\(900\) 0 0
\(901\) −1.10162e9 −0.0501759
\(902\) 0 0
\(903\) 2.39998e9 0.108468
\(904\) 0 0
\(905\) −2.02648e10 −0.908810
\(906\) 0 0
\(907\) −1.36671e10 −0.608204 −0.304102 0.952639i \(-0.598356\pi\)
−0.304102 + 0.952639i \(0.598356\pi\)
\(908\) 0 0
\(909\) −1.01889e10 −0.449937
\(910\) 0 0
\(911\) −3.65765e10 −1.60283 −0.801417 0.598106i \(-0.795920\pi\)
−0.801417 + 0.598106i \(0.795920\pi\)
\(912\) 0 0
\(913\) 2.91957e10 1.26961
\(914\) 0 0
\(915\) −4.91744e9 −0.212210
\(916\) 0 0
\(917\) 1.12114e10 0.480138
\(918\) 0 0
\(919\) −3.71530e9 −0.157903 −0.0789514 0.996878i \(-0.525157\pi\)
−0.0789514 + 0.996878i \(0.525157\pi\)
\(920\) 0 0
\(921\) −5.04390e9 −0.212744
\(922\) 0 0
\(923\) 2.17496e10 0.910427
\(924\) 0 0
\(925\) 1.34029e8 0.00556805
\(926\) 0 0
\(927\) −8.88801e9 −0.366459
\(928\) 0 0
\(929\) −3.28521e10 −1.34434 −0.672168 0.740398i \(-0.734637\pi\)
−0.672168 + 0.740398i \(0.734637\pi\)
\(930\) 0 0
\(931\) 2.08507e9 0.0846832
\(932\) 0 0
\(933\) 4.18632e9 0.168751
\(934\) 0 0
\(935\) −1.89153e9 −0.0756786
\(936\) 0 0
\(937\) 2.29737e10 0.912310 0.456155 0.889900i \(-0.349226\pi\)
0.456155 + 0.889900i \(0.349226\pi\)
\(938\) 0 0
\(939\) −1.64501e9 −0.0648394
\(940\) 0 0
\(941\) 1.00811e10 0.394407 0.197204 0.980363i \(-0.436814\pi\)
0.197204 + 0.980363i \(0.436814\pi\)
\(942\) 0 0
\(943\) −2.31505e10 −0.899020
\(944\) 0 0
\(945\) 6.94324e9 0.267640
\(946\) 0 0
\(947\) −1.75260e10 −0.670589 −0.335295 0.942113i \(-0.608836\pi\)
−0.335295 + 0.942113i \(0.608836\pi\)
\(948\) 0 0
\(949\) 4.83584e9 0.183671
\(950\) 0 0
\(951\) −1.62948e7 −0.000614353 0
\(952\) 0 0
\(953\) −2.66888e10 −0.998859 −0.499429 0.866355i \(-0.666457\pi\)
−0.499429 + 0.866355i \(0.666457\pi\)
\(954\) 0 0
\(955\) 2.99630e9 0.111320
\(956\) 0 0
\(957\) 4.73969e9 0.174807
\(958\) 0 0
\(959\) −3.40254e10 −1.24577
\(960\) 0 0
\(961\) −4.69946e9 −0.170811
\(962\) 0 0
\(963\) 3.27585e10 1.18204
\(964\) 0 0
\(965\) 4.36244e10 1.56273
\(966\) 0 0
\(967\) 2.40155e10 0.854080 0.427040 0.904233i \(-0.359556\pi\)
0.427040 + 0.904233i \(0.359556\pi\)
\(968\) 0 0
\(969\) −7.63508e7 −0.00269576
\(970\) 0 0
\(971\) −3.24355e10 −1.13698 −0.568491 0.822689i \(-0.692473\pi\)
−0.568491 + 0.822689i \(0.692473\pi\)
\(972\) 0 0
\(973\) 3.95895e10 1.37780
\(974\) 0 0
\(975\) −1.15223e8 −0.00398127
\(976\) 0 0
\(977\) −1.53419e10 −0.526317 −0.263158 0.964753i \(-0.584764\pi\)
−0.263158 + 0.964753i \(0.584764\pi\)
\(978\) 0 0
\(979\) −3.11928e10 −1.06247
\(980\) 0 0
\(981\) −4.69189e10 −1.58674
\(982\) 0 0
\(983\) −6.48494e9 −0.217755 −0.108878 0.994055i \(-0.534726\pi\)
−0.108878 + 0.994055i \(0.534726\pi\)
\(984\) 0 0
\(985\) 1.09218e10 0.364139
\(986\) 0 0
\(987\) 4.61726e9 0.152853
\(988\) 0 0
\(989\) 2.26488e10 0.744489
\(990\) 0 0
\(991\) 3.82059e10 1.24702 0.623510 0.781816i \(-0.285706\pi\)
0.623510 + 0.781816i \(0.285706\pi\)
\(992\) 0 0
\(993\) 3.96287e9 0.128436
\(994\) 0 0
\(995\) 1.80416e9 0.0580625
\(996\) 0 0
\(997\) 4.90627e10 1.56790 0.783950 0.620824i \(-0.213202\pi\)
0.783950 + 0.620824i \(0.213202\pi\)
\(998\) 0 0
\(999\) 2.65230e9 0.0841673
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 76.8.a.a.1.3 5
4.3 odd 2 304.8.a.g.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.a.a.1.3 5 1.1 even 1 trivial
304.8.a.g.1.3 5 4.3 odd 2