Properties

Label 7623.2.a.ci.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.477260\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} -1.00000 q^{7} +1.52274 q^{8} +O(q^{10})\) \(q-1.77222 q^{2} +1.14077 q^{4} -0.589926 q^{5} -1.00000 q^{7} +1.52274 q^{8} +1.04548 q^{10} -1.54445 q^{13} +1.77222 q^{14} -4.98018 q^{16} -7.14511 q^{17} -6.19059 q^{19} -0.672972 q^{20} -3.76314 q^{23} -4.65199 q^{25} +2.73710 q^{26} -1.14077 q^{28} -0.607298 q^{29} -6.87147 q^{31} +5.78051 q^{32} +12.6627 q^{34} +0.589926 q^{35} -7.70693 q^{37} +10.9711 q^{38} -0.898304 q^{40} -7.48287 q^{41} -10.9537 q^{43} +6.66913 q^{46} +10.2072 q^{47} +1.00000 q^{49} +8.24436 q^{50} -1.76186 q^{52} -7.04469 q^{53} -1.52274 q^{56} +1.07627 q^{58} +3.35386 q^{59} -3.37044 q^{61} +12.1778 q^{62} -0.283993 q^{64} +0.911108 q^{65} -2.19138 q^{67} -8.15095 q^{68} -1.04548 q^{70} +13.9430 q^{71} -2.58993 q^{73} +13.6584 q^{74} -7.06206 q^{76} -4.66351 q^{79} +2.93794 q^{80} +13.2613 q^{82} +3.50970 q^{83} +4.21508 q^{85} +19.4125 q^{86} -8.15095 q^{89} +1.54445 q^{91} -4.29289 q^{92} -18.0894 q^{94} +3.65199 q^{95} -4.79710 q^{97} -1.77222 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + O(q^{10}) \) \( 4q - q^{2} + 3q^{4} + 4q^{5} - 4q^{7} + 9q^{8} + 10q^{10} + 6q^{13} + q^{14} - 3q^{16} - 8q^{17} - 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} - 3q^{28} - 18q^{31} + 2q^{32} + 18q^{34} - 4q^{35} - 2q^{37} + 8q^{38} + 6q^{40} - 10q^{41} - 4q^{43} + 11q^{46} - 4q^{47} + 4q^{49} + 9q^{50} + 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} + 14q^{61} - 11q^{64} + 28q^{65} - 28q^{67} + 16q^{68} - 10q^{70} + 18q^{71} - 4q^{73} + 41q^{74} - 4q^{76} - 20q^{79} + 36q^{80} - 24q^{82} - 6q^{83} - 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} - 16q^{94} - 16q^{95} + 32q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.77222 −1.25315 −0.626575 0.779361i \(-0.715544\pi\)
−0.626575 + 0.779361i \(0.715544\pi\)
\(3\) 0 0
\(4\) 1.14077 0.570387
\(5\) −0.589926 −0.263823 −0.131911 0.991262i \(-0.542111\pi\)
−0.131911 + 0.991262i \(0.542111\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.52274 0.538370
\(9\) 0 0
\(10\) 1.04548 0.330610
\(11\) 0 0
\(12\) 0 0
\(13\) −1.54445 −0.428352 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(14\) 1.77222 0.473646
\(15\) 0 0
\(16\) −4.98018 −1.24505
\(17\) −7.14511 −1.73294 −0.866472 0.499226i \(-0.833618\pi\)
−0.866472 + 0.499226i \(0.833618\pi\)
\(18\) 0 0
\(19\) −6.19059 −1.42022 −0.710109 0.704092i \(-0.751354\pi\)
−0.710109 + 0.704092i \(0.751354\pi\)
\(20\) −0.672972 −0.150481
\(21\) 0 0
\(22\) 0 0
\(23\) −3.76314 −0.784669 −0.392335 0.919823i \(-0.628332\pi\)
−0.392335 + 0.919823i \(0.628332\pi\)
\(24\) 0 0
\(25\) −4.65199 −0.930398
\(26\) 2.73710 0.536790
\(27\) 0 0
\(28\) −1.14077 −0.215586
\(29\) −0.607298 −0.112772 −0.0563862 0.998409i \(-0.517958\pi\)
−0.0563862 + 0.998409i \(0.517958\pi\)
\(30\) 0 0
\(31\) −6.87147 −1.23415 −0.617077 0.786903i \(-0.711683\pi\)
−0.617077 + 0.786903i \(0.711683\pi\)
\(32\) 5.78051 1.02186
\(33\) 0 0
\(34\) 12.6627 2.17164
\(35\) 0.589926 0.0997157
\(36\) 0 0
\(37\) −7.70693 −1.26701 −0.633505 0.773738i \(-0.718385\pi\)
−0.633505 + 0.773738i \(0.718385\pi\)
\(38\) 10.9711 1.77975
\(39\) 0 0
\(40\) −0.898304 −0.142034
\(41\) −7.48287 −1.16863 −0.584314 0.811528i \(-0.698636\pi\)
−0.584314 + 0.811528i \(0.698636\pi\)
\(42\) 0 0
\(43\) −10.9537 −1.67043 −0.835214 0.549925i \(-0.814656\pi\)
−0.835214 + 0.549925i \(0.814656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.66913 0.983309
\(47\) 10.2072 1.48887 0.744434 0.667696i \(-0.232719\pi\)
0.744434 + 0.667696i \(0.232719\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.24436 1.16593
\(51\) 0 0
\(52\) −1.76186 −0.244327
\(53\) −7.04469 −0.967663 −0.483831 0.875161i \(-0.660755\pi\)
−0.483831 + 0.875161i \(0.660755\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.52274 −0.203485
\(57\) 0 0
\(58\) 1.07627 0.141321
\(59\) 3.35386 0.436635 0.218318 0.975878i \(-0.429943\pi\)
0.218318 + 0.975878i \(0.429943\pi\)
\(60\) 0 0
\(61\) −3.37044 −0.431541 −0.215770 0.976444i \(-0.569226\pi\)
−0.215770 + 0.976444i \(0.569226\pi\)
\(62\) 12.1778 1.54658
\(63\) 0 0
\(64\) −0.283993 −0.0354992
\(65\) 0.911108 0.113009
\(66\) 0 0
\(67\) −2.19138 −0.267719 −0.133860 0.991000i \(-0.542737\pi\)
−0.133860 + 0.991000i \(0.542737\pi\)
\(68\) −8.15095 −0.988448
\(69\) 0 0
\(70\) −1.04548 −0.124959
\(71\) 13.9430 1.65473 0.827364 0.561665i \(-0.189839\pi\)
0.827364 + 0.561665i \(0.189839\pi\)
\(72\) 0 0
\(73\) −2.58993 −0.303128 −0.151564 0.988447i \(-0.548431\pi\)
−0.151564 + 0.988447i \(0.548431\pi\)
\(74\) 13.6584 1.58776
\(75\) 0 0
\(76\) −7.06206 −0.810074
\(77\) 0 0
\(78\) 0 0
\(79\) −4.66351 −0.524686 −0.262343 0.964975i \(-0.584495\pi\)
−0.262343 + 0.964975i \(0.584495\pi\)
\(80\) 2.93794 0.328471
\(81\) 0 0
\(82\) 13.2613 1.46447
\(83\) 3.50970 0.385240 0.192620 0.981273i \(-0.438302\pi\)
0.192620 + 0.981273i \(0.438302\pi\)
\(84\) 0 0
\(85\) 4.21508 0.457190
\(86\) 19.4125 2.09330
\(87\) 0 0
\(88\) 0 0
\(89\) −8.15095 −0.863999 −0.432000 0.901874i \(-0.642192\pi\)
−0.432000 + 0.901874i \(0.642192\pi\)
\(90\) 0 0
\(91\) 1.54445 0.161902
\(92\) −4.29289 −0.447565
\(93\) 0 0
\(94\) −18.0894 −1.86578
\(95\) 3.65199 0.374686
\(96\) 0 0
\(97\) −4.79710 −0.487071 −0.243536 0.969892i \(-0.578307\pi\)
−0.243536 + 0.969892i \(0.578307\pi\)
\(98\) −1.77222 −0.179022
\(99\) 0 0
\(100\) −5.30687 −0.530687
\(101\) 1.42666 0.141958 0.0709788 0.997478i \(-0.477388\pi\)
0.0709788 + 0.997478i \(0.477388\pi\)
\(102\) 0 0
\(103\) 17.6172 1.73588 0.867939 0.496670i \(-0.165444\pi\)
0.867939 + 0.496670i \(0.165444\pi\)
\(104\) −2.35179 −0.230612
\(105\) 0 0
\(106\) 12.4848 1.21263
\(107\) 7.20005 0.696055 0.348027 0.937484i \(-0.386852\pi\)
0.348027 + 0.937484i \(0.386852\pi\)
\(108\) 0 0
\(109\) −8.88678 −0.851199 −0.425599 0.904912i \(-0.639937\pi\)
−0.425599 + 0.904912i \(0.639937\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.98018 0.470583
\(113\) 7.80862 0.734573 0.367287 0.930108i \(-0.380287\pi\)
0.367287 + 0.930108i \(0.380287\pi\)
\(114\) 0 0
\(115\) 2.21997 0.207014
\(116\) −0.692789 −0.0643239
\(117\) 0 0
\(118\) −5.94378 −0.547170
\(119\) 7.14511 0.654991
\(120\) 0 0
\(121\) 0 0
\(122\) 5.97317 0.540785
\(123\) 0 0
\(124\) −7.83880 −0.703945
\(125\) 5.69396 0.509283
\(126\) 0 0
\(127\) 8.25186 0.732234 0.366117 0.930569i \(-0.380687\pi\)
0.366117 + 0.930569i \(0.380687\pi\)
\(128\) −11.0577 −0.977374
\(129\) 0 0
\(130\) −1.61469 −0.141617
\(131\) −12.1162 −1.05860 −0.529299 0.848435i \(-0.677545\pi\)
−0.529299 + 0.848435i \(0.677545\pi\)
\(132\) 0 0
\(133\) 6.19059 0.536792
\(134\) 3.88361 0.335493
\(135\) 0 0
\(136\) −10.8801 −0.932964
\(137\) 17.4548 1.49126 0.745631 0.666360i \(-0.232148\pi\)
0.745631 + 0.666360i \(0.232148\pi\)
\(138\) 0 0
\(139\) 14.7624 1.25213 0.626063 0.779772i \(-0.284665\pi\)
0.626063 + 0.779772i \(0.284665\pi\)
\(140\) 0.672972 0.0568765
\(141\) 0 0
\(142\) −24.7101 −2.07362
\(143\) 0 0
\(144\) 0 0
\(145\) 0.358261 0.0297519
\(146\) 4.58993 0.379865
\(147\) 0 0
\(148\) −8.79186 −0.722686
\(149\) −12.5258 −1.02615 −0.513077 0.858343i \(-0.671494\pi\)
−0.513077 + 0.858343i \(0.671494\pi\)
\(150\) 0 0
\(151\) 0.328572 0.0267388 0.0133694 0.999911i \(-0.495744\pi\)
0.0133694 + 0.999911i \(0.495744\pi\)
\(152\) −9.42666 −0.764603
\(153\) 0 0
\(154\) 0 0
\(155\) 4.05366 0.325598
\(156\) 0 0
\(157\) −2.68089 −0.213958 −0.106979 0.994261i \(-0.534118\pi\)
−0.106979 + 0.994261i \(0.534118\pi\)
\(158\) 8.26479 0.657511
\(159\) 0 0
\(160\) −3.41007 −0.269590
\(161\) 3.76314 0.296577
\(162\) 0 0
\(163\) −0.891668 −0.0698408 −0.0349204 0.999390i \(-0.511118\pi\)
−0.0349204 + 0.999390i \(0.511118\pi\)
\(164\) −8.53627 −0.666570
\(165\) 0 0
\(166\) −6.21997 −0.482764
\(167\) −11.5610 −0.894619 −0.447310 0.894379i \(-0.647618\pi\)
−0.447310 + 0.894379i \(0.647618\pi\)
\(168\) 0 0
\(169\) −10.6147 −0.816514
\(170\) −7.47007 −0.572928
\(171\) 0 0
\(172\) −12.4957 −0.952790
\(173\) −8.98926 −0.683441 −0.341721 0.939802i \(-0.611010\pi\)
−0.341721 + 0.939802i \(0.611010\pi\)
\(174\) 0 0
\(175\) 4.65199 0.351657
\(176\) 0 0
\(177\) 0 0
\(178\) 14.4453 1.08272
\(179\) 6.14590 0.459366 0.229683 0.973265i \(-0.426231\pi\)
0.229683 + 0.973265i \(0.426231\pi\)
\(180\) 0 0
\(181\) 14.9662 1.11243 0.556215 0.831039i \(-0.312253\pi\)
0.556215 + 0.831039i \(0.312253\pi\)
\(182\) −2.73710 −0.202888
\(183\) 0 0
\(184\) −5.73029 −0.422442
\(185\) 4.54651 0.334266
\(186\) 0 0
\(187\) 0 0
\(188\) 11.6441 0.849231
\(189\) 0 0
\(190\) −6.47214 −0.469538
\(191\) −9.87351 −0.714422 −0.357211 0.934024i \(-0.616272\pi\)
−0.357211 + 0.934024i \(0.616272\pi\)
\(192\) 0 0
\(193\) −6.18113 −0.444927 −0.222464 0.974941i \(-0.571410\pi\)
−0.222464 + 0.974941i \(0.571410\pi\)
\(194\) 8.50152 0.610374
\(195\) 0 0
\(196\) 1.14077 0.0814839
\(197\) 19.0720 1.35882 0.679412 0.733757i \(-0.262235\pi\)
0.679412 + 0.733757i \(0.262235\pi\)
\(198\) 0 0
\(199\) −11.8479 −0.839878 −0.419939 0.907552i \(-0.637948\pi\)
−0.419939 + 0.907552i \(0.637948\pi\)
\(200\) −7.08377 −0.500898
\(201\) 0 0
\(202\) −2.52835 −0.177894
\(203\) 0.607298 0.0426239
\(204\) 0 0
\(205\) 4.41434 0.308311
\(206\) −31.2217 −2.17532
\(207\) 0 0
\(208\) 7.69162 0.533318
\(209\) 0 0
\(210\) 0 0
\(211\) −20.9810 −1.44439 −0.722197 0.691687i \(-0.756868\pi\)
−0.722197 + 0.691687i \(0.756868\pi\)
\(212\) −8.03640 −0.551942
\(213\) 0 0
\(214\) −12.7601 −0.872262
\(215\) 6.46189 0.440697
\(216\) 0 0
\(217\) 6.87147 0.466466
\(218\) 15.7493 1.06668
\(219\) 0 0
\(220\) 0 0
\(221\) 11.0352 0.742310
\(222\) 0 0
\(223\) 3.38860 0.226918 0.113459 0.993543i \(-0.463807\pi\)
0.113459 + 0.993543i \(0.463807\pi\)
\(224\) −5.78051 −0.386227
\(225\) 0 0
\(226\) −13.8386 −0.920531
\(227\) −19.3973 −1.28744 −0.643721 0.765260i \(-0.722610\pi\)
−0.643721 + 0.765260i \(0.722610\pi\)
\(228\) 0 0
\(229\) −9.86405 −0.651835 −0.325917 0.945398i \(-0.605673\pi\)
−0.325917 + 0.945398i \(0.605673\pi\)
\(230\) −3.93429 −0.259419
\(231\) 0 0
\(232\) −0.924756 −0.0607132
\(233\) −7.12619 −0.466852 −0.233426 0.972375i \(-0.574994\pi\)
−0.233426 + 0.972375i \(0.574994\pi\)
\(234\) 0 0
\(235\) −6.02147 −0.392798
\(236\) 3.82599 0.249051
\(237\) 0 0
\(238\) −12.6627 −0.820802
\(239\) −22.2894 −1.44178 −0.720892 0.693048i \(-0.756267\pi\)
−0.720892 + 0.693048i \(0.756267\pi\)
\(240\) 0 0
\(241\) 14.8753 0.958199 0.479100 0.877761i \(-0.340963\pi\)
0.479100 + 0.877761i \(0.340963\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −3.84491 −0.246145
\(245\) −0.589926 −0.0376890
\(246\) 0 0
\(247\) 9.56103 0.608354
\(248\) −10.4635 −0.664431
\(249\) 0 0
\(250\) −10.0910 −0.638208
\(251\) 25.9233 1.63626 0.818132 0.575031i \(-0.195010\pi\)
0.818132 + 0.575031i \(0.195010\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −14.6241 −0.917600
\(255\) 0 0
\(256\) 20.1647 1.26030
\(257\) −0.871474 −0.0543610 −0.0271805 0.999631i \(-0.508653\pi\)
−0.0271805 + 0.999631i \(0.508653\pi\)
\(258\) 0 0
\(259\) 7.70693 0.478885
\(260\) 1.03937 0.0644589
\(261\) 0 0
\(262\) 21.4726 1.32658
\(263\) −26.8873 −1.65794 −0.828970 0.559293i \(-0.811073\pi\)
−0.828970 + 0.559293i \(0.811073\pi\)
\(264\) 0 0
\(265\) 4.15584 0.255291
\(266\) −10.9711 −0.672681
\(267\) 0 0
\(268\) −2.49987 −0.152704
\(269\) 0.992574 0.0605183 0.0302592 0.999542i \(-0.490367\pi\)
0.0302592 + 0.999542i \(0.490367\pi\)
\(270\) 0 0
\(271\) 26.8650 1.63193 0.815966 0.578099i \(-0.196205\pi\)
0.815966 + 0.578099i \(0.196205\pi\)
\(272\) 35.5839 2.15759
\(273\) 0 0
\(274\) −30.9337 −1.86878
\(275\) 0 0
\(276\) 0 0
\(277\) −27.6302 −1.66014 −0.830069 0.557660i \(-0.811699\pi\)
−0.830069 + 0.557660i \(0.811699\pi\)
\(278\) −26.1622 −1.56910
\(279\) 0 0
\(280\) 0.898304 0.0536839
\(281\) −23.8418 −1.42228 −0.711141 0.703050i \(-0.751821\pi\)
−0.711141 + 0.703050i \(0.751821\pi\)
\(282\) 0 0
\(283\) 8.74735 0.519976 0.259988 0.965612i \(-0.416281\pi\)
0.259988 + 0.965612i \(0.416281\pi\)
\(284\) 15.9058 0.943836
\(285\) 0 0
\(286\) 0 0
\(287\) 7.48287 0.441700
\(288\) 0 0
\(289\) 34.0526 2.00309
\(290\) −0.634918 −0.0372836
\(291\) 0 0
\(292\) −2.95452 −0.172900
\(293\) 19.6252 1.14651 0.573257 0.819376i \(-0.305680\pi\)
0.573257 + 0.819376i \(0.305680\pi\)
\(294\) 0 0
\(295\) −1.97853 −0.115194
\(296\) −11.7356 −0.682120
\(297\) 0 0
\(298\) 22.1985 1.28592
\(299\) 5.81197 0.336115
\(300\) 0 0
\(301\) 10.9537 0.631362
\(302\) −0.582303 −0.0335078
\(303\) 0 0
\(304\) 30.8303 1.76824
\(305\) 1.98831 0.113850
\(306\) 0 0
\(307\) −17.0835 −0.975009 −0.487504 0.873121i \(-0.662093\pi\)
−0.487504 + 0.873121i \(0.662093\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −7.18399 −0.408023
\(311\) 7.23351 0.410175 0.205087 0.978744i \(-0.434252\pi\)
0.205087 + 0.978744i \(0.434252\pi\)
\(312\) 0 0
\(313\) 13.7784 0.778804 0.389402 0.921068i \(-0.372682\pi\)
0.389402 + 0.921068i \(0.372682\pi\)
\(314\) 4.75113 0.268122
\(315\) 0 0
\(316\) −5.32002 −0.299274
\(317\) 6.95629 0.390704 0.195352 0.980733i \(-0.437415\pi\)
0.195352 + 0.980733i \(0.437415\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.167535 0.00936550
\(321\) 0 0
\(322\) −6.66913 −0.371656
\(323\) 44.2324 2.46116
\(324\) 0 0
\(325\) 7.18474 0.398538
\(326\) 1.58023 0.0875211
\(327\) 0 0
\(328\) −11.3945 −0.629154
\(329\) −10.2072 −0.562739
\(330\) 0 0
\(331\) −29.6698 −1.63080 −0.815401 0.578896i \(-0.803484\pi\)
−0.815401 + 0.578896i \(0.803484\pi\)
\(332\) 4.00378 0.219736
\(333\) 0 0
\(334\) 20.4887 1.12109
\(335\) 1.29275 0.0706305
\(336\) 0 0
\(337\) 19.5714 1.06612 0.533062 0.846076i \(-0.321041\pi\)
0.533062 + 0.846076i \(0.321041\pi\)
\(338\) 18.8116 1.02322
\(339\) 0 0
\(340\) 4.80846 0.260775
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −16.6797 −0.899308
\(345\) 0 0
\(346\) 15.9310 0.856455
\(347\) 6.49184 0.348500 0.174250 0.984701i \(-0.444250\pi\)
0.174250 + 0.984701i \(0.444250\pi\)
\(348\) 0 0
\(349\) 7.87965 0.421788 0.210894 0.977509i \(-0.432362\pi\)
0.210894 + 0.977509i \(0.432362\pi\)
\(350\) −8.24436 −0.440680
\(351\) 0 0
\(352\) 0 0
\(353\) 5.34594 0.284536 0.142268 0.989828i \(-0.454561\pi\)
0.142268 + 0.989828i \(0.454561\pi\)
\(354\) 0 0
\(355\) −8.22533 −0.436555
\(356\) −9.29840 −0.492814
\(357\) 0 0
\(358\) −10.8919 −0.575655
\(359\) 17.8515 0.942169 0.471084 0.882088i \(-0.343863\pi\)
0.471084 + 0.882088i \(0.343863\pi\)
\(360\) 0 0
\(361\) 19.3234 1.01702
\(362\) −26.5235 −1.39404
\(363\) 0 0
\(364\) 1.76186 0.0923467
\(365\) 1.52786 0.0799721
\(366\) 0 0
\(367\) −29.9357 −1.56263 −0.781316 0.624135i \(-0.785451\pi\)
−0.781316 + 0.624135i \(0.785451\pi\)
\(368\) 18.7411 0.976949
\(369\) 0 0
\(370\) −8.05744 −0.418886
\(371\) 7.04469 0.365742
\(372\) 0 0
\(373\) 23.4915 1.21635 0.608173 0.793805i \(-0.291903\pi\)
0.608173 + 0.793805i \(0.291903\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 15.5429 0.801562
\(377\) 0.937938 0.0483063
\(378\) 0 0
\(379\) −31.4532 −1.61564 −0.807821 0.589428i \(-0.799353\pi\)
−0.807821 + 0.589428i \(0.799353\pi\)
\(380\) 4.16609 0.213716
\(381\) 0 0
\(382\) 17.4981 0.895278
\(383\) 5.62749 0.287551 0.143776 0.989610i \(-0.454076\pi\)
0.143776 + 0.989610i \(0.454076\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.9543 0.557561
\(387\) 0 0
\(388\) −5.47240 −0.277819
\(389\) −26.1072 −1.32369 −0.661845 0.749641i \(-0.730226\pi\)
−0.661845 + 0.749641i \(0.730226\pi\)
\(390\) 0 0
\(391\) 26.8881 1.35979
\(392\) 1.52274 0.0769100
\(393\) 0 0
\(394\) −33.7998 −1.70281
\(395\) 2.75113 0.138424
\(396\) 0 0
\(397\) 10.3828 0.521096 0.260548 0.965461i \(-0.416097\pi\)
0.260548 + 0.965461i \(0.416097\pi\)
\(398\) 20.9972 1.05249
\(399\) 0 0
\(400\) 23.1677 1.15839
\(401\) −3.68876 −0.184208 −0.0921040 0.995749i \(-0.529359\pi\)
−0.0921040 + 0.995749i \(0.529359\pi\)
\(402\) 0 0
\(403\) 10.6126 0.528652
\(404\) 1.62749 0.0809708
\(405\) 0 0
\(406\) −1.07627 −0.0534142
\(407\) 0 0
\(408\) 0 0
\(409\) 1.08022 0.0534136 0.0267068 0.999643i \(-0.491498\pi\)
0.0267068 + 0.999643i \(0.491498\pi\)
\(410\) −7.82319 −0.386360
\(411\) 0 0
\(412\) 20.0973 0.990123
\(413\) −3.35386 −0.165033
\(414\) 0 0
\(415\) −2.07046 −0.101635
\(416\) −8.92769 −0.437716
\(417\) 0 0
\(418\) 0 0
\(419\) −10.0766 −0.492273 −0.246136 0.969235i \(-0.579161\pi\)
−0.246136 + 0.969235i \(0.579161\pi\)
\(420\) 0 0
\(421\) 11.9757 0.583658 0.291829 0.956470i \(-0.405736\pi\)
0.291829 + 0.956470i \(0.405736\pi\)
\(422\) 37.1831 1.81004
\(423\) 0 0
\(424\) −10.7272 −0.520960
\(425\) 33.2390 1.61233
\(426\) 0 0
\(427\) 3.37044 0.163107
\(428\) 8.21363 0.397021
\(429\) 0 0
\(430\) −11.4519 −0.552260
\(431\) 0.548547 0.0264226 0.0132113 0.999913i \(-0.495795\pi\)
0.0132113 + 0.999913i \(0.495795\pi\)
\(432\) 0 0
\(433\) 19.5879 0.941332 0.470666 0.882311i \(-0.344014\pi\)
0.470666 + 0.882311i \(0.344014\pi\)
\(434\) −12.1778 −0.584552
\(435\) 0 0
\(436\) −10.1378 −0.485513
\(437\) 23.2961 1.11440
\(438\) 0 0
\(439\) 37.4069 1.78533 0.892667 0.450716i \(-0.148831\pi\)
0.892667 + 0.450716i \(0.148831\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −19.5569 −0.930226
\(443\) −21.4478 −1.01902 −0.509508 0.860466i \(-0.670173\pi\)
−0.509508 + 0.860466i \(0.670173\pi\)
\(444\) 0 0
\(445\) 4.80846 0.227943
\(446\) −6.00536 −0.284362
\(447\) 0 0
\(448\) 0.283993 0.0134174
\(449\) −6.39904 −0.301989 −0.150995 0.988535i \(-0.548248\pi\)
−0.150995 + 0.988535i \(0.548248\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 8.90787 0.418991
\(453\) 0 0
\(454\) 34.3763 1.61336
\(455\) −0.911108 −0.0427134
\(456\) 0 0
\(457\) −36.9430 −1.72812 −0.864060 0.503389i \(-0.832086\pi\)
−0.864060 + 0.503389i \(0.832086\pi\)
\(458\) 17.4813 0.816847
\(459\) 0 0
\(460\) 2.53249 0.118078
\(461\) −7.39825 −0.344571 −0.172285 0.985047i \(-0.555115\pi\)
−0.172285 + 0.985047i \(0.555115\pi\)
\(462\) 0 0
\(463\) −14.3717 −0.667910 −0.333955 0.942589i \(-0.608383\pi\)
−0.333955 + 0.942589i \(0.608383\pi\)
\(464\) 3.02445 0.140407
\(465\) 0 0
\(466\) 12.6292 0.585036
\(467\) −36.1626 −1.67341 −0.836704 0.547656i \(-0.815520\pi\)
−0.836704 + 0.547656i \(0.815520\pi\)
\(468\) 0 0
\(469\) 2.19138 0.101188
\(470\) 10.6714 0.492235
\(471\) 0 0
\(472\) 5.10705 0.235071
\(473\) 0 0
\(474\) 0 0
\(475\) 28.7985 1.32137
\(476\) 8.15095 0.373598
\(477\) 0 0
\(478\) 39.5018 1.80677
\(479\) −16.0252 −0.732212 −0.366106 0.930573i \(-0.619309\pi\)
−0.366106 + 0.930573i \(0.619309\pi\)
\(480\) 0 0
\(481\) 11.9029 0.542727
\(482\) −26.3623 −1.20077
\(483\) 0 0
\(484\) 0 0
\(485\) 2.82993 0.128501
\(486\) 0 0
\(487\) 1.55059 0.0702641 0.0351320 0.999383i \(-0.488815\pi\)
0.0351320 + 0.999383i \(0.488815\pi\)
\(488\) −5.13230 −0.232328
\(489\) 0 0
\(490\) 1.04548 0.0472300
\(491\) 0.0331635 0.00149665 0.000748324 1.00000i \(-0.499762\pi\)
0.000748324 1.00000i \(0.499762\pi\)
\(492\) 0 0
\(493\) 4.33921 0.195428
\(494\) −16.9443 −0.762359
\(495\) 0 0
\(496\) 34.2212 1.53658
\(497\) −13.9430 −0.625429
\(498\) 0 0
\(499\) −27.0314 −1.21009 −0.605046 0.796190i \(-0.706845\pi\)
−0.605046 + 0.796190i \(0.706845\pi\)
\(500\) 6.49552 0.290488
\(501\) 0 0
\(502\) −45.9418 −2.05048
\(503\) 29.3341 1.30794 0.653972 0.756519i \(-0.273101\pi\)
0.653972 + 0.756519i \(0.273101\pi\)
\(504\) 0 0
\(505\) −0.841621 −0.0374516
\(506\) 0 0
\(507\) 0 0
\(508\) 9.41351 0.417657
\(509\) 16.6329 0.737239 0.368620 0.929580i \(-0.379831\pi\)
0.368620 + 0.929580i \(0.379831\pi\)
\(510\) 0 0
\(511\) 2.58993 0.114572
\(512\) −13.6210 −0.601967
\(513\) 0 0
\(514\) 1.54445 0.0681226
\(515\) −10.3929 −0.457964
\(516\) 0 0
\(517\) 0 0
\(518\) −13.6584 −0.600115
\(519\) 0 0
\(520\) 1.38738 0.0608407
\(521\) −35.8923 −1.57247 −0.786236 0.617927i \(-0.787973\pi\)
−0.786236 + 0.617927i \(0.787973\pi\)
\(522\) 0 0
\(523\) 7.27873 0.318276 0.159138 0.987256i \(-0.449128\pi\)
0.159138 + 0.987256i \(0.449128\pi\)
\(524\) −13.8219 −0.603811
\(525\) 0 0
\(526\) 47.6502 2.07765
\(527\) 49.0974 2.13872
\(528\) 0 0
\(529\) −8.83876 −0.384294
\(530\) −7.36508 −0.319919
\(531\) 0 0
\(532\) 7.06206 0.306179
\(533\) 11.5569 0.500585
\(534\) 0 0
\(535\) −4.24749 −0.183635
\(536\) −3.33690 −0.144132
\(537\) 0 0
\(538\) −1.75906 −0.0758386
\(539\) 0 0
\(540\) 0 0
\(541\) −10.4709 −0.450177 −0.225089 0.974338i \(-0.572267\pi\)
−0.225089 + 0.974338i \(0.572267\pi\)
\(542\) −47.6108 −2.04506
\(543\) 0 0
\(544\) −41.3024 −1.77083
\(545\) 5.24254 0.224566
\(546\) 0 0
\(547\) 8.16609 0.349157 0.174578 0.984643i \(-0.444144\pi\)
0.174578 + 0.984643i \(0.444144\pi\)
\(548\) 19.9119 0.850596
\(549\) 0 0
\(550\) 0 0
\(551\) 3.75953 0.160161
\(552\) 0 0
\(553\) 4.66351 0.198313
\(554\) 48.9669 2.08040
\(555\) 0 0
\(556\) 16.8405 0.714197
\(557\) −11.8765 −0.503225 −0.251612 0.967828i \(-0.580961\pi\)
−0.251612 + 0.967828i \(0.580961\pi\)
\(558\) 0 0
\(559\) 16.9174 0.715532
\(560\) −2.93794 −0.124151
\(561\) 0 0
\(562\) 42.2530 1.78233
\(563\) −4.83929 −0.203952 −0.101976 0.994787i \(-0.532516\pi\)
−0.101976 + 0.994787i \(0.532516\pi\)
\(564\) 0 0
\(565\) −4.60651 −0.193797
\(566\) −15.5023 −0.651608
\(567\) 0 0
\(568\) 21.2316 0.890856
\(569\) −6.52993 −0.273749 −0.136875 0.990588i \(-0.543706\pi\)
−0.136875 + 0.990588i \(0.543706\pi\)
\(570\) 0 0
\(571\) −33.8691 −1.41738 −0.708689 0.705521i \(-0.750713\pi\)
−0.708689 + 0.705521i \(0.750713\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −13.2613 −0.553517
\(575\) 17.5061 0.730054
\(576\) 0 0
\(577\) −37.3609 −1.55536 −0.777678 0.628662i \(-0.783603\pi\)
−0.777678 + 0.628662i \(0.783603\pi\)
\(578\) −60.3487 −2.51018
\(579\) 0 0
\(580\) 0.408694 0.0169701
\(581\) −3.50970 −0.145607
\(582\) 0 0
\(583\) 0 0
\(584\) −3.94378 −0.163195
\(585\) 0 0
\(586\) −34.7802 −1.43675
\(587\) 2.07974 0.0858399 0.0429199 0.999079i \(-0.486334\pi\)
0.0429199 + 0.999079i \(0.486334\pi\)
\(588\) 0 0
\(589\) 42.5385 1.75277
\(590\) 3.50639 0.144356
\(591\) 0 0
\(592\) 38.3819 1.57749
\(593\) −6.54983 −0.268969 −0.134485 0.990916i \(-0.542938\pi\)
−0.134485 + 0.990916i \(0.542938\pi\)
\(594\) 0 0
\(595\) −4.21508 −0.172802
\(596\) −14.2891 −0.585304
\(597\) 0 0
\(598\) −10.3001 −0.421203
\(599\) 4.98187 0.203554 0.101777 0.994807i \(-0.467547\pi\)
0.101777 + 0.994807i \(0.467547\pi\)
\(600\) 0 0
\(601\) −32.4713 −1.32453 −0.662266 0.749269i \(-0.730405\pi\)
−0.662266 + 0.749269i \(0.730405\pi\)
\(602\) −19.4125 −0.791192
\(603\) 0 0
\(604\) 0.374827 0.0152515
\(605\) 0 0
\(606\) 0 0
\(607\) 3.13388 0.127200 0.0636002 0.997975i \(-0.479742\pi\)
0.0636002 + 0.997975i \(0.479742\pi\)
\(608\) −35.7848 −1.45126
\(609\) 0 0
\(610\) −3.52373 −0.142672
\(611\) −15.7644 −0.637760
\(612\) 0 0
\(613\) 46.8484 1.89219 0.946094 0.323892i \(-0.104991\pi\)
0.946094 + 0.323892i \(0.104991\pi\)
\(614\) 30.2758 1.22183
\(615\) 0 0
\(616\) 0 0
\(617\) −26.5924 −1.07057 −0.535286 0.844671i \(-0.679796\pi\)
−0.535286 + 0.844671i \(0.679796\pi\)
\(618\) 0 0
\(619\) −32.9765 −1.32544 −0.662718 0.748869i \(-0.730597\pi\)
−0.662718 + 0.748869i \(0.730597\pi\)
\(620\) 4.62431 0.185717
\(621\) 0 0
\(622\) −12.8194 −0.514011
\(623\) 8.15095 0.326561
\(624\) 0 0
\(625\) 19.9009 0.796037
\(626\) −24.4185 −0.975959
\(627\) 0 0
\(628\) −3.05828 −0.122039
\(629\) 55.0668 2.19566
\(630\) 0 0
\(631\) 11.1721 0.444754 0.222377 0.974961i \(-0.428618\pi\)
0.222377 + 0.974961i \(0.428618\pi\)
\(632\) −7.10132 −0.282475
\(633\) 0 0
\(634\) −12.3281 −0.489611
\(635\) −4.86798 −0.193180
\(636\) 0 0
\(637\) −1.54445 −0.0611932
\(638\) 0 0
\(639\) 0 0
\(640\) 6.52324 0.257854
\(641\) 41.9153 1.65556 0.827778 0.561056i \(-0.189605\pi\)
0.827778 + 0.561056i \(0.189605\pi\)
\(642\) 0 0
\(643\) 29.0873 1.14709 0.573546 0.819173i \(-0.305567\pi\)
0.573546 + 0.819173i \(0.305567\pi\)
\(644\) 4.29289 0.169164
\(645\) 0 0
\(646\) −78.3897 −3.08420
\(647\) −1.89804 −0.0746196 −0.0373098 0.999304i \(-0.511879\pi\)
−0.0373098 + 0.999304i \(0.511879\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −12.7330 −0.499428
\(651\) 0 0
\(652\) −1.01719 −0.0398363
\(653\) −20.2519 −0.792517 −0.396258 0.918139i \(-0.629692\pi\)
−0.396258 + 0.918139i \(0.629692\pi\)
\(654\) 0 0
\(655\) 7.14766 0.279282
\(656\) 37.2661 1.45500
\(657\) 0 0
\(658\) 18.0894 0.705197
\(659\) 11.5620 0.450392 0.225196 0.974314i \(-0.427698\pi\)
0.225196 + 0.974314i \(0.427698\pi\)
\(660\) 0 0
\(661\) −32.6894 −1.27147 −0.635735 0.771907i \(-0.719303\pi\)
−0.635735 + 0.771907i \(0.719303\pi\)
\(662\) 52.5816 2.04364
\(663\) 0 0
\(664\) 5.34436 0.207402
\(665\) −3.65199 −0.141618
\(666\) 0 0
\(667\) 2.28535 0.0884890
\(668\) −13.1885 −0.510279
\(669\) 0 0
\(670\) −2.29104 −0.0885107
\(671\) 0 0
\(672\) 0 0
\(673\) −20.1249 −0.775758 −0.387879 0.921710i \(-0.626792\pi\)
−0.387879 + 0.921710i \(0.626792\pi\)
\(674\) −34.6850 −1.33601
\(675\) 0 0
\(676\) −12.1090 −0.465729
\(677\) −23.5746 −0.906045 −0.453022 0.891499i \(-0.649654\pi\)
−0.453022 + 0.891499i \(0.649654\pi\)
\(678\) 0 0
\(679\) 4.79710 0.184096
\(680\) 6.41848 0.246137
\(681\) 0 0
\(682\) 0 0
\(683\) −11.5812 −0.443143 −0.221572 0.975144i \(-0.571119\pi\)
−0.221572 + 0.975144i \(0.571119\pi\)
\(684\) 0 0
\(685\) −10.2970 −0.393429
\(686\) 1.77222 0.0676638
\(687\) 0 0
\(688\) 54.5516 2.07976
\(689\) 10.8801 0.414500
\(690\) 0 0
\(691\) 43.8164 1.66685 0.833426 0.552630i \(-0.186376\pi\)
0.833426 + 0.552630i \(0.186376\pi\)
\(692\) −10.2547 −0.389826
\(693\) 0 0
\(694\) −11.5050 −0.436723
\(695\) −8.70869 −0.330340
\(696\) 0 0
\(697\) 53.4659 2.02517
\(698\) −13.9645 −0.528564
\(699\) 0 0
\(700\) 5.30687 0.200581
\(701\) 22.8165 0.861769 0.430884 0.902407i \(-0.358202\pi\)
0.430884 + 0.902407i \(0.358202\pi\)
\(702\) 0 0
\(703\) 47.7104 1.79943
\(704\) 0 0
\(705\) 0 0
\(706\) −9.47420 −0.356566
\(707\) −1.42666 −0.0536549
\(708\) 0 0
\(709\) 39.9746 1.50128 0.750638 0.660713i \(-0.229746\pi\)
0.750638 + 0.660713i \(0.229746\pi\)
\(710\) 14.5771 0.547070
\(711\) 0 0
\(712\) −12.4118 −0.465151
\(713\) 25.8583 0.968402
\(714\) 0 0
\(715\) 0 0
\(716\) 7.01108 0.262016
\(717\) 0 0
\(718\) −31.6369 −1.18068
\(719\) 26.2200 0.977840 0.488920 0.872329i \(-0.337391\pi\)
0.488920 + 0.872329i \(0.337391\pi\)
\(720\) 0 0
\(721\) −17.6172 −0.656100
\(722\) −34.2453 −1.27448
\(723\) 0 0
\(724\) 17.0731 0.634515
\(725\) 2.82514 0.104923
\(726\) 0 0
\(727\) −22.2366 −0.824708 −0.412354 0.911024i \(-0.635293\pi\)
−0.412354 + 0.911024i \(0.635293\pi\)
\(728\) 2.35179 0.0871631
\(729\) 0 0
\(730\) −2.70772 −0.100217
\(731\) 78.2656 2.89476
\(732\) 0 0
\(733\) −21.1284 −0.780395 −0.390197 0.920731i \(-0.627593\pi\)
−0.390197 + 0.920731i \(0.627593\pi\)
\(734\) 53.0528 1.95821
\(735\) 0 0
\(736\) −21.7529 −0.801822
\(737\) 0 0
\(738\) 0 0
\(739\) 37.5036 1.37959 0.689795 0.724004i \(-0.257701\pi\)
0.689795 + 0.724004i \(0.257701\pi\)
\(740\) 5.18655 0.190661
\(741\) 0 0
\(742\) −12.4848 −0.458330
\(743\) 36.1464 1.32608 0.663041 0.748583i \(-0.269265\pi\)
0.663041 + 0.748583i \(0.269265\pi\)
\(744\) 0 0
\(745\) 7.38929 0.270723
\(746\) −41.6322 −1.52426
\(747\) 0 0
\(748\) 0 0
\(749\) −7.20005 −0.263084
\(750\) 0 0
\(751\) −43.5533 −1.58928 −0.794641 0.607080i \(-0.792341\pi\)
−0.794641 + 0.607080i \(0.792341\pi\)
\(752\) −50.8336 −1.85371
\(753\) 0 0
\(754\) −1.66224 −0.0605351
\(755\) −0.193833 −0.00705431
\(756\) 0 0
\(757\) −44.4442 −1.61535 −0.807675 0.589628i \(-0.799274\pi\)
−0.807675 + 0.589628i \(0.799274\pi\)
\(758\) 55.7421 2.02464
\(759\) 0 0
\(760\) 5.56103 0.201720
\(761\) 15.9613 0.578598 0.289299 0.957239i \(-0.406578\pi\)
0.289299 + 0.957239i \(0.406578\pi\)
\(762\) 0 0
\(763\) 8.88678 0.321723
\(764\) −11.2634 −0.407497
\(765\) 0 0
\(766\) −9.97317 −0.360345
\(767\) −5.17985 −0.187034
\(768\) 0 0
\(769\) 5.19899 0.187480 0.0937402 0.995597i \(-0.470118\pi\)
0.0937402 + 0.995597i \(0.470118\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.05127 −0.253781
\(773\) 12.1614 0.437416 0.218708 0.975790i \(-0.429816\pi\)
0.218708 + 0.975790i \(0.429816\pi\)
\(774\) 0 0
\(775\) 31.9660 1.14825
\(776\) −7.30473 −0.262224
\(777\) 0 0
\(778\) 46.2678 1.65878
\(779\) 46.3234 1.65971
\(780\) 0 0
\(781\) 0 0
\(782\) −47.6516 −1.70402
\(783\) 0 0
\(784\) −4.98018 −0.177864
\(785\) 1.58152 0.0564470
\(786\) 0 0
\(787\) −39.6430 −1.41312 −0.706559 0.707654i \(-0.749754\pi\)
−0.706559 + 0.707654i \(0.749754\pi\)
\(788\) 21.7569 0.775056
\(789\) 0 0
\(790\) −4.87561 −0.173466
\(791\) −7.80862 −0.277643
\(792\) 0 0
\(793\) 5.20546 0.184851
\(794\) −18.4006 −0.653011
\(795\) 0 0
\(796\) −13.5158 −0.479055
\(797\) 35.1118 1.24372 0.621862 0.783127i \(-0.286376\pi\)
0.621862 + 0.783127i \(0.286376\pi\)
\(798\) 0 0
\(799\) −72.9313 −2.58012
\(800\) −26.8909 −0.950736
\(801\) 0 0
\(802\) 6.53731 0.230840
\(803\) 0 0
\(804\) 0 0
\(805\) −2.21997 −0.0782438
\(806\) −18.8079 −0.662481
\(807\) 0 0
\(808\) 2.17243 0.0764257
\(809\) 41.7873 1.46916 0.734581 0.678521i \(-0.237379\pi\)
0.734581 + 0.678521i \(0.237379\pi\)
\(810\) 0 0
\(811\) −9.89879 −0.347594 −0.173797 0.984782i \(-0.555604\pi\)
−0.173797 + 0.984782i \(0.555604\pi\)
\(812\) 0.692789 0.0243121
\(813\) 0 0
\(814\) 0 0
\(815\) 0.526018 0.0184256
\(816\) 0 0
\(817\) 67.8100 2.37237
\(818\) −1.91440 −0.0669353
\(819\) 0 0
\(820\) 5.03576 0.175856
\(821\) −16.1936 −0.565161 −0.282580 0.959244i \(-0.591190\pi\)
−0.282580 + 0.959244i \(0.591190\pi\)
\(822\) 0 0
\(823\) −3.86642 −0.134775 −0.0673874 0.997727i \(-0.521466\pi\)
−0.0673874 + 0.997727i \(0.521466\pi\)
\(824\) 26.8265 0.934545
\(825\) 0 0
\(826\) 5.94378 0.206811
\(827\) 46.8179 1.62802 0.814009 0.580852i \(-0.197280\pi\)
0.814009 + 0.580852i \(0.197280\pi\)
\(828\) 0 0
\(829\) −41.5599 −1.44344 −0.721718 0.692187i \(-0.756647\pi\)
−0.721718 + 0.692187i \(0.756647\pi\)
\(830\) 3.66932 0.127364
\(831\) 0 0
\(832\) 0.438613 0.0152062
\(833\) −7.14511 −0.247563
\(834\) 0 0
\(835\) 6.82015 0.236021
\(836\) 0 0
\(837\) 0 0
\(838\) 17.8579 0.616892
\(839\) −5.14119 −0.177494 −0.0887469 0.996054i \(-0.528286\pi\)
−0.0887469 + 0.996054i \(0.528286\pi\)
\(840\) 0 0
\(841\) −28.6312 −0.987282
\(842\) −21.2236 −0.731412
\(843\) 0 0
\(844\) −23.9346 −0.823864
\(845\) 6.26188 0.215415
\(846\) 0 0
\(847\) 0 0
\(848\) 35.0838 1.20478
\(849\) 0 0
\(850\) −58.9068 −2.02049
\(851\) 29.0023 0.994184
\(852\) 0 0
\(853\) 6.69649 0.229283 0.114642 0.993407i \(-0.463428\pi\)
0.114642 + 0.993407i \(0.463428\pi\)
\(854\) −5.97317 −0.204398
\(855\) 0 0
\(856\) 10.9638 0.374735
\(857\) 36.1064 1.23337 0.616686 0.787209i \(-0.288475\pi\)
0.616686 + 0.787209i \(0.288475\pi\)
\(858\) 0 0
\(859\) −37.1034 −1.26595 −0.632976 0.774171i \(-0.718167\pi\)
−0.632976 + 0.774171i \(0.718167\pi\)
\(860\) 7.37155 0.251368
\(861\) 0 0
\(862\) −0.972147 −0.0331115
\(863\) −38.3724 −1.30621 −0.653105 0.757267i \(-0.726534\pi\)
−0.653105 + 0.757267i \(0.726534\pi\)
\(864\) 0 0
\(865\) 5.30300 0.180307
\(866\) −34.7140 −1.17963
\(867\) 0 0
\(868\) 7.83880 0.266066
\(869\) 0 0
\(870\) 0 0
\(871\) 3.38446 0.114678
\(872\) −13.5323 −0.458260
\(873\) 0 0
\(874\) −41.2858 −1.39651
\(875\) −5.69396 −0.192491
\(876\) 0 0
\(877\) −0.811156 −0.0273908 −0.0136954 0.999906i \(-0.504360\pi\)
−0.0136954 + 0.999906i \(0.504360\pi\)
\(878\) −66.2934 −2.23729
\(879\) 0 0
\(880\) 0 0
\(881\) 32.5076 1.09521 0.547605 0.836737i \(-0.315540\pi\)
0.547605 + 0.836737i \(0.315540\pi\)
\(882\) 0 0
\(883\) 48.0995 1.61868 0.809338 0.587343i \(-0.199826\pi\)
0.809338 + 0.587343i \(0.199826\pi\)
\(884\) 12.5887 0.423404
\(885\) 0 0
\(886\) 38.0103 1.27698
\(887\) −11.3847 −0.382260 −0.191130 0.981565i \(-0.561215\pi\)
−0.191130 + 0.981565i \(0.561215\pi\)
\(888\) 0 0
\(889\) −8.25186 −0.276758
\(890\) −8.52166 −0.285647
\(891\) 0 0
\(892\) 3.86563 0.129431
\(893\) −63.1884 −2.11452
\(894\) 0 0
\(895\) −3.62562 −0.121191
\(896\) 11.0577 0.369413
\(897\) 0 0
\(898\) 11.3405 0.378438
\(899\) 4.17303 0.139178
\(900\) 0 0
\(901\) 50.3351 1.67690
\(902\) 0 0
\(903\) 0 0
\(904\) 11.8905 0.395472
\(905\) −8.82895 −0.293484
\(906\) 0 0
\(907\) 13.4259 0.445799 0.222899 0.974841i \(-0.428448\pi\)
0.222899 + 0.974841i \(0.428448\pi\)
\(908\) −22.1279 −0.734340
\(909\) 0 0
\(910\) 1.61469 0.0535264
\(911\) −3.39303 −0.112416 −0.0562080 0.998419i \(-0.517901\pi\)
−0.0562080 + 0.998419i \(0.517901\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 65.4712 2.16560
\(915\) 0 0
\(916\) −11.2526 −0.371798
\(917\) 12.1162 0.400112
\(918\) 0 0
\(919\) −44.5668 −1.47012 −0.735062 0.678000i \(-0.762847\pi\)
−0.735062 + 0.678000i \(0.762847\pi\)
\(920\) 3.38044 0.111450
\(921\) 0 0
\(922\) 13.1113 0.431799
\(923\) −21.5342 −0.708807
\(924\) 0 0
\(925\) 35.8525 1.17882
\(926\) 25.4699 0.836992
\(927\) 0 0
\(928\) −3.51049 −0.115238
\(929\) 7.15205 0.234651 0.117325 0.993094i \(-0.462568\pi\)
0.117325 + 0.993094i \(0.462568\pi\)
\(930\) 0 0
\(931\) −6.19059 −0.202888
\(932\) −8.12937 −0.266287
\(933\) 0 0
\(934\) 64.0883 2.09703
\(935\) 0 0
\(936\) 0 0
\(937\) −1.30811 −0.0427342 −0.0213671 0.999772i \(-0.506802\pi\)
−0.0213671 + 0.999772i \(0.506802\pi\)
\(938\) −3.88361 −0.126804
\(939\) 0 0
\(940\) −6.86914 −0.224047
\(941\) 22.1225 0.721174 0.360587 0.932726i \(-0.382576\pi\)
0.360587 + 0.932726i \(0.382576\pi\)
\(942\) 0 0
\(943\) 28.1591 0.916987
\(944\) −16.7028 −0.543631
\(945\) 0 0
\(946\) 0 0
\(947\) 17.6842 0.574658 0.287329 0.957832i \(-0.407233\pi\)
0.287329 + 0.957832i \(0.407233\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −51.0374 −1.65587
\(951\) 0 0
\(952\) 10.8801 0.352627
\(953\) 23.7741 0.770118 0.385059 0.922892i \(-0.374181\pi\)
0.385059 + 0.922892i \(0.374181\pi\)
\(954\) 0 0
\(955\) 5.82464 0.188481
\(956\) −25.4272 −0.822374
\(957\) 0 0
\(958\) 28.4003 0.917573
\(959\) −17.4548 −0.563644
\(960\) 0 0
\(961\) 16.2172 0.523134
\(962\) −21.0946 −0.680119
\(963\) 0 0
\(964\) 16.9693 0.546544
\(965\) 3.64641 0.117382
\(966\) 0 0
\(967\) 21.1829 0.681195 0.340597 0.940209i \(-0.389371\pi\)
0.340597 + 0.940209i \(0.389371\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −5.01527 −0.161031
\(971\) 47.4360 1.52229 0.761147 0.648579i \(-0.224637\pi\)
0.761147 + 0.648579i \(0.224637\pi\)
\(972\) 0 0
\(973\) −14.7624 −0.473259
\(974\) −2.74800 −0.0880515
\(975\) 0 0
\(976\) 16.7854 0.537288
\(977\) 13.1860 0.421858 0.210929 0.977501i \(-0.432351\pi\)
0.210929 + 0.977501i \(0.432351\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.672972 −0.0214973
\(981\) 0 0
\(982\) −0.0587731 −0.00187553
\(983\) 4.07740 0.130049 0.0650244 0.997884i \(-0.479287\pi\)
0.0650244 + 0.997884i \(0.479287\pi\)
\(984\) 0 0
\(985\) −11.2511 −0.358489
\(986\) −7.69004 −0.244901
\(987\) 0 0
\(988\) 10.9070 0.346997
\(989\) 41.2204 1.31073
\(990\) 0 0
\(991\) −5.49154 −0.174445 −0.0872223 0.996189i \(-0.527799\pi\)
−0.0872223 + 0.996189i \(0.527799\pi\)
\(992\) −39.7206 −1.26113
\(993\) 0 0
\(994\) 24.7101 0.783757
\(995\) 6.98940 0.221579
\(996\) 0 0
\(997\) 22.9725 0.727548 0.363774 0.931487i \(-0.381488\pi\)
0.363774 + 0.931487i \(0.381488\pi\)
\(998\) 47.9057 1.51643
\(999\) 0 0
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 7623.2.a.ci.1.1 4
3.2 odd 2 2541.2.a.bn.1.4 4
11.5 even 5 693.2.m.f.190.2 8
11.9 even 5 693.2.m.f.631.2 8
11.10 odd 2 7623.2.a.cl.1.4 4
33.5 odd 10 231.2.j.f.190.1 yes 8
33.20 odd 10 231.2.j.f.169.1 8
33.32 even 2 2541.2.a.bm.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.f.169.1 8 33.20 odd 10
231.2.j.f.190.1 yes 8 33.5 odd 10
693.2.m.f.190.2 8 11.5 even 5
693.2.m.f.631.2 8 11.9 even 5
2541.2.a.bm.1.1 4 33.32 even 2
2541.2.a.bn.1.4 4 3.2 odd 2
7623.2.a.ci.1.1 4 1.1 even 1 trivial
7623.2.a.cl.1.4 4 11.10 odd 2