Properties

Label 7623.2.a.cm.1.2
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: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.43091\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} +O(q^{10})\) \(q+0.301143 q^{2} -1.90931 q^{4} -1.30114 q^{5} -1.00000 q^{7} -1.17726 q^{8} -0.391830 q^{10} -3.38772 q^{13} -0.301143 q^{14} +3.46410 q^{16} -0.478405 q^{17} +1.60817 q^{19} +2.48429 q^{20} -3.47841 q^{23} -3.30703 q^{25} -1.02019 q^{26} +1.90931 q^{28} -0.942507 q^{29} -1.17726 q^{31} +3.39771 q^{32} -0.144068 q^{34} +1.30114 q^{35} -3.55479 q^{37} +0.484289 q^{38} +1.53179 q^{40} -3.94839 q^{41} -2.87024 q^{43} -1.04750 q^{46} -12.1113 q^{47} +1.00000 q^{49} -0.995888 q^{50} +6.46821 q^{52} -5.28273 q^{53} +1.17726 q^{56} -0.283829 q^{58} +9.69345 q^{59} +3.00588 q^{61} -0.354524 q^{62} -5.90501 q^{64} +4.40791 q^{65} +7.74683 q^{67} +0.913425 q^{68} +0.391830 q^{70} +8.46568 q^{71} -13.3709 q^{73} -1.07050 q^{74} -3.07050 q^{76} -8.86612 q^{79} -4.50729 q^{80} -1.18903 q^{82} +11.4700 q^{83} +0.622473 q^{85} -0.864351 q^{86} -7.54068 q^{89} +3.38772 q^{91} +6.64136 q^{92} -3.64725 q^{94} -2.09246 q^{95} -1.71364 q^{97} +0.301143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 4q^{4} - 6q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 2q^{2} + 4q^{4} - 6q^{5} - 4q^{7} - 14q^{10} + 2q^{13} - 2q^{14} + 2q^{17} - 6q^{19} - 8q^{20} - 10q^{23} - 4q^{28} + 14q^{29} + 12q^{32} - 2q^{34} + 6q^{35} - 12q^{37} - 16q^{38} - 8q^{40} + 16q^{41} - 20q^{43} - 8q^{46} - 2q^{47} + 4q^{49} + 24q^{50} + 40q^{52} + 16q^{53} - 8q^{58} - 2q^{59} - 2q^{61} + 8q^{62} - 16q^{64} - 2q^{65} - 20q^{67} + 20q^{68} + 14q^{70} + 10q^{71} + 10q^{73} - 20q^{74} - 28q^{76} - 16q^{79} - 12q^{80} + 28q^{82} + 18q^{83} - 26q^{86} + 14q^{89} - 2q^{91} + 8q^{92} + 18q^{94} + 22q^{95} + 38q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.301143 0.212940 0.106470 0.994316i \(-0.466045\pi\)
0.106470 + 0.994316i \(0.466045\pi\)
\(3\) 0 0
\(4\) −1.90931 −0.954656
\(5\) −1.30114 −0.581889 −0.290944 0.956740i \(-0.593969\pi\)
−0.290944 + 0.956740i \(0.593969\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.17726 −0.416225
\(9\) 0 0
\(10\) −0.391830 −0.123908
\(11\) 0 0
\(12\) 0 0
\(13\) −3.38772 −0.939584 −0.469792 0.882777i \(-0.655671\pi\)
−0.469792 + 0.882777i \(0.655671\pi\)
\(14\) −0.301143 −0.0804838
\(15\) 0 0
\(16\) 3.46410 0.866025
\(17\) −0.478405 −0.116030 −0.0580151 0.998316i \(-0.518477\pi\)
−0.0580151 + 0.998316i \(0.518477\pi\)
\(18\) 0 0
\(19\) 1.60817 0.368939 0.184470 0.982838i \(-0.440943\pi\)
0.184470 + 0.982838i \(0.440943\pi\)
\(20\) 2.48429 0.555504
\(21\) 0 0
\(22\) 0 0
\(23\) −3.47841 −0.725298 −0.362649 0.931926i \(-0.618128\pi\)
−0.362649 + 0.931926i \(0.618128\pi\)
\(24\) 0 0
\(25\) −3.30703 −0.661405
\(26\) −1.02019 −0.200075
\(27\) 0 0
\(28\) 1.90931 0.360826
\(29\) −0.942507 −0.175019 −0.0875095 0.996164i \(-0.527891\pi\)
−0.0875095 + 0.996164i \(0.527891\pi\)
\(30\) 0 0
\(31\) −1.17726 −0.211443 −0.105721 0.994396i \(-0.533715\pi\)
−0.105721 + 0.994396i \(0.533715\pi\)
\(32\) 3.39771 0.600637
\(33\) 0 0
\(34\) −0.144068 −0.0247075
\(35\) 1.30114 0.219933
\(36\) 0 0
\(37\) −3.55479 −0.584404 −0.292202 0.956357i \(-0.594388\pi\)
−0.292202 + 0.956357i \(0.594388\pi\)
\(38\) 0.484289 0.0785621
\(39\) 0 0
\(40\) 1.53179 0.242197
\(41\) −3.94839 −0.616635 −0.308318 0.951284i \(-0.599766\pi\)
−0.308318 + 0.951284i \(0.599766\pi\)
\(42\) 0 0
\(43\) −2.87024 −0.437707 −0.218853 0.975758i \(-0.570232\pi\)
−0.218853 + 0.975758i \(0.570232\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.04750 −0.154445
\(47\) −12.1113 −1.76662 −0.883311 0.468788i \(-0.844691\pi\)
−0.883311 + 0.468788i \(0.844691\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −0.995888 −0.140840
\(51\) 0 0
\(52\) 6.46821 0.896980
\(53\) −5.28273 −0.725638 −0.362819 0.931860i \(-0.618186\pi\)
−0.362819 + 0.931860i \(0.618186\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.17726 0.157318
\(57\) 0 0
\(58\) −0.283829 −0.0372686
\(59\) 9.69345 1.26198 0.630990 0.775791i \(-0.282649\pi\)
0.630990 + 0.775791i \(0.282649\pi\)
\(60\) 0 0
\(61\) 3.00588 0.384864 0.192432 0.981310i \(-0.438363\pi\)
0.192432 + 0.981310i \(0.438363\pi\)
\(62\) −0.354524 −0.0450246
\(63\) 0 0
\(64\) −5.90501 −0.738126
\(65\) 4.40791 0.546733
\(66\) 0 0
\(67\) 7.74683 0.946426 0.473213 0.880948i \(-0.343094\pi\)
0.473213 + 0.880948i \(0.343094\pi\)
\(68\) 0.913425 0.110769
\(69\) 0 0
\(70\) 0.391830 0.0468326
\(71\) 8.46568 1.00469 0.502346 0.864667i \(-0.332471\pi\)
0.502346 + 0.864667i \(0.332471\pi\)
\(72\) 0 0
\(73\) −13.3709 −1.56494 −0.782472 0.622686i \(-0.786041\pi\)
−0.782472 + 0.622686i \(0.786041\pi\)
\(74\) −1.07050 −0.124443
\(75\) 0 0
\(76\) −3.07050 −0.352210
\(77\) 0 0
\(78\) 0 0
\(79\) −8.86612 −0.997517 −0.498758 0.866741i \(-0.666210\pi\)
−0.498758 + 0.866741i \(0.666210\pi\)
\(80\) −4.50729 −0.503931
\(81\) 0 0
\(82\) −1.18903 −0.131306
\(83\) 11.4700 1.25899 0.629497 0.777003i \(-0.283261\pi\)
0.629497 + 0.777003i \(0.283261\pi\)
\(84\) 0 0
\(85\) 0.622473 0.0675167
\(86\) −0.864351 −0.0932054
\(87\) 0 0
\(88\) 0 0
\(89\) −7.54068 −0.799311 −0.399655 0.916666i \(-0.630870\pi\)
−0.399655 + 0.916666i \(0.630870\pi\)
\(90\) 0 0
\(91\) 3.38772 0.355129
\(92\) 6.64136 0.692410
\(93\) 0 0
\(94\) −3.64725 −0.376185
\(95\) −2.09246 −0.214682
\(96\) 0 0
\(97\) −1.71364 −0.173993 −0.0869967 0.996209i \(-0.527727\pi\)
−0.0869967 + 0.996209i \(0.527727\pi\)
\(98\) 0.301143 0.0304200
\(99\) 0 0
\(100\) 6.31415 0.631415
\(101\) −9.53637 −0.948905 −0.474452 0.880281i \(-0.657354\pi\)
−0.474452 + 0.880281i \(0.657354\pi\)
\(102\) 0 0
\(103\) −1.22634 −0.120834 −0.0604172 0.998173i \(-0.519243\pi\)
−0.0604172 + 0.998173i \(0.519243\pi\)
\(104\) 3.98823 0.391078
\(105\) 0 0
\(106\) −1.59086 −0.154518
\(107\) 9.16296 0.885817 0.442908 0.896567i \(-0.353947\pi\)
0.442908 + 0.896567i \(0.353947\pi\)
\(108\) 0 0
\(109\) −16.2268 −1.55425 −0.777123 0.629348i \(-0.783322\pi\)
−0.777123 + 0.629348i \(0.783322\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.46410 −0.327327
\(113\) −17.4473 −1.64130 −0.820650 0.571431i \(-0.806389\pi\)
−0.820650 + 0.571431i \(0.806389\pi\)
\(114\) 0 0
\(115\) 4.52590 0.422043
\(116\) 1.79954 0.167083
\(117\) 0 0
\(118\) 2.91911 0.268726
\(119\) 0.478405 0.0438553
\(120\) 0 0
\(121\) 0 0
\(122\) 0.905201 0.0819530
\(123\) 0 0
\(124\) 2.24776 0.201855
\(125\) 10.8086 0.966753
\(126\) 0 0
\(127\) −2.60248 −0.230933 −0.115466 0.993311i \(-0.536836\pi\)
−0.115466 + 0.993311i \(0.536836\pi\)
\(128\) −8.57368 −0.757813
\(129\) 0 0
\(130\) 1.32741 0.116422
\(131\) 10.2853 0.898628 0.449314 0.893374i \(-0.351669\pi\)
0.449314 + 0.893374i \(0.351669\pi\)
\(132\) 0 0
\(133\) −1.60817 −0.139446
\(134\) 2.33290 0.201532
\(135\) 0 0
\(136\) 0.563208 0.0482947
\(137\) 19.9098 1.70101 0.850504 0.525969i \(-0.176297\pi\)
0.850504 + 0.525969i \(0.176297\pi\)
\(138\) 0 0
\(139\) −4.64089 −0.393635 −0.196818 0.980440i \(-0.563061\pi\)
−0.196818 + 0.980440i \(0.563061\pi\)
\(140\) −2.48429 −0.209961
\(141\) 0 0
\(142\) 2.54938 0.213939
\(143\) 0 0
\(144\) 0 0
\(145\) 1.22634 0.101842
\(146\) −4.02655 −0.333239
\(147\) 0 0
\(148\) 6.78720 0.557905
\(149\) 23.4807 1.92361 0.961805 0.273736i \(-0.0882593\pi\)
0.961805 + 0.273736i \(0.0882593\pi\)
\(150\) 0 0
\(151\) −13.3893 −1.08961 −0.544803 0.838564i \(-0.683395\pi\)
−0.544803 + 0.838564i \(0.683395\pi\)
\(152\) −1.89324 −0.153562
\(153\) 0 0
\(154\) 0 0
\(155\) 1.53179 0.123036
\(156\) 0 0
\(157\) 12.6414 1.00889 0.504445 0.863444i \(-0.331697\pi\)
0.504445 + 0.863444i \(0.331697\pi\)
\(158\) −2.66997 −0.212411
\(159\) 0 0
\(160\) −4.42091 −0.349504
\(161\) 3.47841 0.274137
\(162\) 0 0
\(163\) −0.854355 −0.0669183 −0.0334591 0.999440i \(-0.510652\pi\)
−0.0334591 + 0.999440i \(0.510652\pi\)
\(164\) 7.53871 0.588675
\(165\) 0 0
\(166\) 3.45411 0.268091
\(167\) 8.16727 0.632002 0.316001 0.948759i \(-0.397660\pi\)
0.316001 + 0.948759i \(0.397660\pi\)
\(168\) 0 0
\(169\) −1.52337 −0.117182
\(170\) 0.187453 0.0143770
\(171\) 0 0
\(172\) 5.48018 0.417860
\(173\) 20.9111 1.58984 0.794920 0.606714i \(-0.207513\pi\)
0.794920 + 0.606714i \(0.207513\pi\)
\(174\) 0 0
\(175\) 3.30703 0.249988
\(176\) 0 0
\(177\) 0 0
\(178\) −2.27082 −0.170205
\(179\) −8.41230 −0.628765 −0.314382 0.949296i \(-0.601797\pi\)
−0.314382 + 0.949296i \(0.601797\pi\)
\(180\) 0 0
\(181\) −20.6995 −1.53858 −0.769292 0.638898i \(-0.779391\pi\)
−0.769292 + 0.638898i \(0.779391\pi\)
\(182\) 1.02019 0.0756213
\(183\) 0 0
\(184\) 4.09499 0.301887
\(185\) 4.62529 0.340058
\(186\) 0 0
\(187\) 0 0
\(188\) 23.1244 1.68652
\(189\) 0 0
\(190\) −0.630129 −0.0457144
\(191\) 14.6830 1.06242 0.531211 0.847239i \(-0.321737\pi\)
0.531211 + 0.847239i \(0.321737\pi\)
\(192\) 0 0
\(193\) 25.2657 1.81866 0.909332 0.416071i \(-0.136593\pi\)
0.909332 + 0.416071i \(0.136593\pi\)
\(194\) −0.516049 −0.0370502
\(195\) 0 0
\(196\) −1.90931 −0.136379
\(197\) −10.3936 −0.740513 −0.370257 0.928929i \(-0.620730\pi\)
−0.370257 + 0.928929i \(0.620730\pi\)
\(198\) 0 0
\(199\) 11.1704 0.791850 0.395925 0.918283i \(-0.370424\pi\)
0.395925 + 0.918283i \(0.370424\pi\)
\(200\) 3.89324 0.275293
\(201\) 0 0
\(202\) −2.87181 −0.202060
\(203\) 0.942507 0.0661510
\(204\) 0 0
\(205\) 5.13742 0.358813
\(206\) −0.369302 −0.0257305
\(207\) 0 0
\(208\) −11.7354 −0.813704
\(209\) 0 0
\(210\) 0 0
\(211\) −11.1918 −0.770478 −0.385239 0.922817i \(-0.625881\pi\)
−0.385239 + 0.922817i \(0.625881\pi\)
\(212\) 10.0864 0.692735
\(213\) 0 0
\(214\) 2.75936 0.188626
\(215\) 3.73459 0.254697
\(216\) 0 0
\(217\) 1.17726 0.0799178
\(218\) −4.88659 −0.330962
\(219\) 0 0
\(220\) 0 0
\(221\) 1.62070 0.109020
\(222\) 0 0
\(223\) −20.4514 −1.36953 −0.684763 0.728766i \(-0.740094\pi\)
−0.684763 + 0.728766i \(0.740094\pi\)
\(224\) −3.39771 −0.227019
\(225\) 0 0
\(226\) −5.25412 −0.349499
\(227\) 28.0554 1.86210 0.931052 0.364886i \(-0.118892\pi\)
0.931052 + 0.364886i \(0.118892\pi\)
\(228\) 0 0
\(229\) 9.82178 0.649041 0.324521 0.945879i \(-0.394797\pi\)
0.324521 + 0.945879i \(0.394797\pi\)
\(230\) 1.36294 0.0898698
\(231\) 0 0
\(232\) 1.10958 0.0728473
\(233\) −3.37294 −0.220969 −0.110484 0.993878i \(-0.535240\pi\)
−0.110484 + 0.993878i \(0.535240\pi\)
\(234\) 0 0
\(235\) 15.7586 1.02798
\(236\) −18.5078 −1.20476
\(237\) 0 0
\(238\) 0.144068 0.00933856
\(239\) 25.1604 1.62749 0.813746 0.581220i \(-0.197425\pi\)
0.813746 + 0.581220i \(0.197425\pi\)
\(240\) 0 0
\(241\) 12.9077 0.831460 0.415730 0.909488i \(-0.363526\pi\)
0.415730 + 0.909488i \(0.363526\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −5.73917 −0.367413
\(245\) −1.30114 −0.0831270
\(246\) 0 0
\(247\) −5.44803 −0.346650
\(248\) 1.38595 0.0880077
\(249\) 0 0
\(250\) 3.25494 0.205861
\(251\) 10.8704 0.686135 0.343068 0.939311i \(-0.388534\pi\)
0.343068 + 0.939311i \(0.388534\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −0.783719 −0.0491749
\(255\) 0 0
\(256\) 9.22811 0.576757
\(257\) −10.1400 −0.632513 −0.316257 0.948674i \(-0.602426\pi\)
−0.316257 + 0.948674i \(0.602426\pi\)
\(258\) 0 0
\(259\) 3.55479 0.220884
\(260\) −8.41607 −0.521943
\(261\) 0 0
\(262\) 3.09733 0.191354
\(263\) 20.7929 1.28214 0.641072 0.767480i \(-0.278490\pi\)
0.641072 + 0.767480i \(0.278490\pi\)
\(264\) 0 0
\(265\) 6.87358 0.422241
\(266\) −0.484289 −0.0296937
\(267\) 0 0
\(268\) −14.7911 −0.903512
\(269\) 1.24235 0.0757476 0.0378738 0.999283i \(-0.487942\pi\)
0.0378738 + 0.999283i \(0.487942\pi\)
\(270\) 0 0
\(271\) −22.3996 −1.36068 −0.680340 0.732897i \(-0.738168\pi\)
−0.680340 + 0.732897i \(0.738168\pi\)
\(272\) −1.65724 −0.100485
\(273\) 0 0
\(274\) 5.99569 0.362213
\(275\) 0 0
\(276\) 0 0
\(277\) 12.7509 0.766130 0.383065 0.923721i \(-0.374869\pi\)
0.383065 + 0.923721i \(0.374869\pi\)
\(278\) −1.39757 −0.0838207
\(279\) 0 0
\(280\) −1.53179 −0.0915417
\(281\) −1.56957 −0.0936325 −0.0468163 0.998904i \(-0.514908\pi\)
−0.0468163 + 0.998904i \(0.514908\pi\)
\(282\) 0 0
\(283\) 14.3841 0.855045 0.427522 0.904005i \(-0.359387\pi\)
0.427522 + 0.904005i \(0.359387\pi\)
\(284\) −16.1636 −0.959135
\(285\) 0 0
\(286\) 0 0
\(287\) 3.94839 0.233066
\(288\) 0 0
\(289\) −16.7711 −0.986537
\(290\) 0.369302 0.0216862
\(291\) 0 0
\(292\) 25.5292 1.49398
\(293\) 17.3984 1.01642 0.508212 0.861232i \(-0.330307\pi\)
0.508212 + 0.861232i \(0.330307\pi\)
\(294\) 0 0
\(295\) −12.6126 −0.734332
\(296\) 4.18492 0.243243
\(297\) 0 0
\(298\) 7.07103 0.409614
\(299\) 11.7839 0.681478
\(300\) 0 0
\(301\) 2.87024 0.165438
\(302\) −4.03209 −0.232021
\(303\) 0 0
\(304\) 5.57086 0.319511
\(305\) −3.91108 −0.223948
\(306\) 0 0
\(307\) 11.3404 0.647232 0.323616 0.946189i \(-0.395101\pi\)
0.323616 + 0.946189i \(0.395101\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.461287 0.0261993
\(311\) −13.7568 −0.780078 −0.390039 0.920798i \(-0.627538\pi\)
−0.390039 + 0.920798i \(0.627538\pi\)
\(312\) 0 0
\(313\) 6.41359 0.362518 0.181259 0.983435i \(-0.441983\pi\)
0.181259 + 0.983435i \(0.441983\pi\)
\(314\) 3.80686 0.214833
\(315\) 0 0
\(316\) 16.9282 0.952286
\(317\) −10.6661 −0.599070 −0.299535 0.954085i \(-0.596832\pi\)
−0.299535 + 0.954085i \(0.596832\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.68326 0.429507
\(321\) 0 0
\(322\) 1.04750 0.0583747
\(323\) −0.769357 −0.0428081
\(324\) 0 0
\(325\) 11.2033 0.621446
\(326\) −0.257283 −0.0142496
\(327\) 0 0
\(328\) 4.64829 0.256659
\(329\) 12.1113 0.667720
\(330\) 0 0
\(331\) 9.96427 0.547686 0.273843 0.961774i \(-0.411705\pi\)
0.273843 + 0.961774i \(0.411705\pi\)
\(332\) −21.8998 −1.20191
\(333\) 0 0
\(334\) 2.45951 0.134579
\(335\) −10.0797 −0.550715
\(336\) 0 0
\(337\) 19.0653 1.03855 0.519276 0.854607i \(-0.326202\pi\)
0.519276 + 0.854607i \(0.326202\pi\)
\(338\) −0.458751 −0.0249528
\(339\) 0 0
\(340\) −1.18850 −0.0644553
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 3.37902 0.182184
\(345\) 0 0
\(346\) 6.29723 0.338541
\(347\) 0.262065 0.0140684 0.00703420 0.999975i \(-0.497761\pi\)
0.00703420 + 0.999975i \(0.497761\pi\)
\(348\) 0 0
\(349\) 4.96728 0.265892 0.132946 0.991123i \(-0.457556\pi\)
0.132946 + 0.991123i \(0.457556\pi\)
\(350\) 0.995888 0.0532324
\(351\) 0 0
\(352\) 0 0
\(353\) 19.8104 1.05440 0.527201 0.849741i \(-0.323242\pi\)
0.527201 + 0.849741i \(0.323242\pi\)
\(354\) 0 0
\(355\) −11.0151 −0.584619
\(356\) 14.3975 0.763067
\(357\) 0 0
\(358\) −2.53330 −0.133889
\(359\) −19.8539 −1.04785 −0.523924 0.851765i \(-0.675532\pi\)
−0.523924 + 0.851765i \(0.675532\pi\)
\(360\) 0 0
\(361\) −16.4138 −0.863884
\(362\) −6.23352 −0.327626
\(363\) 0 0
\(364\) −6.46821 −0.339027
\(365\) 17.3974 0.910623
\(366\) 0 0
\(367\) 30.9087 1.61342 0.806710 0.590947i \(-0.201246\pi\)
0.806710 + 0.590947i \(0.201246\pi\)
\(368\) −12.0495 −0.628126
\(369\) 0 0
\(370\) 1.39287 0.0724120
\(371\) 5.28273 0.274266
\(372\) 0 0
\(373\) 37.1879 1.92552 0.962759 0.270361i \(-0.0871431\pi\)
0.962759 + 0.270361i \(0.0871431\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 14.2582 0.735312
\(377\) 3.19295 0.164445
\(378\) 0 0
\(379\) −19.9729 −1.02594 −0.512969 0.858407i \(-0.671454\pi\)
−0.512969 + 0.858407i \(0.671454\pi\)
\(380\) 3.99516 0.204947
\(381\) 0 0
\(382\) 4.42168 0.226233
\(383\) −20.3331 −1.03897 −0.519486 0.854479i \(-0.673877\pi\)
−0.519486 + 0.854479i \(0.673877\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.60859 0.387267
\(387\) 0 0
\(388\) 3.27187 0.166104
\(389\) −10.7754 −0.546336 −0.273168 0.961966i \(-0.588072\pi\)
−0.273168 + 0.961966i \(0.588072\pi\)
\(390\) 0 0
\(391\) 1.66409 0.0841565
\(392\) −1.17726 −0.0594607
\(393\) 0 0
\(394\) −3.12996 −0.157685
\(395\) 11.5361 0.580444
\(396\) 0 0
\(397\) −0.247762 −0.0124348 −0.00621740 0.999981i \(-0.501979\pi\)
−0.00621740 + 0.999981i \(0.501979\pi\)
\(398\) 3.36389 0.168617
\(399\) 0 0
\(400\) −11.4559 −0.572794
\(401\) 24.8441 1.24065 0.620327 0.784343i \(-0.287000\pi\)
0.620327 + 0.784343i \(0.287000\pi\)
\(402\) 0 0
\(403\) 3.98823 0.198668
\(404\) 18.2079 0.905878
\(405\) 0 0
\(406\) 0.283829 0.0140862
\(407\) 0 0
\(408\) 0 0
\(409\) 18.5332 0.916409 0.458204 0.888847i \(-0.348493\pi\)
0.458204 + 0.888847i \(0.348493\pi\)
\(410\) 1.54710 0.0764057
\(411\) 0 0
\(412\) 2.34146 0.115355
\(413\) −9.69345 −0.476983
\(414\) 0 0
\(415\) −14.9241 −0.732595
\(416\) −11.5105 −0.564349
\(417\) 0 0
\(418\) 0 0
\(419\) 19.5238 0.953802 0.476901 0.878957i \(-0.341760\pi\)
0.476901 + 0.878957i \(0.341760\pi\)
\(420\) 0 0
\(421\) −8.78687 −0.428246 −0.214123 0.976807i \(-0.568689\pi\)
−0.214123 + 0.976807i \(0.568689\pi\)
\(422\) −3.37035 −0.164066
\(423\) 0 0
\(424\) 6.21915 0.302029
\(425\) 1.58210 0.0767430
\(426\) 0 0
\(427\) −3.00588 −0.145465
\(428\) −17.4950 −0.845651
\(429\) 0 0
\(430\) 1.12464 0.0542352
\(431\) −18.1532 −0.874407 −0.437203 0.899363i \(-0.644031\pi\)
−0.437203 + 0.899363i \(0.644031\pi\)
\(432\) 0 0
\(433\) 21.3423 1.02564 0.512822 0.858495i \(-0.328600\pi\)
0.512822 + 0.858495i \(0.328600\pi\)
\(434\) 0.354524 0.0170177
\(435\) 0 0
\(436\) 30.9821 1.48377
\(437\) −5.59387 −0.267591
\(438\) 0 0
\(439\) −7.59086 −0.362292 −0.181146 0.983456i \(-0.557981\pi\)
−0.181146 + 0.983456i \(0.557981\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.488063 0.0232148
\(443\) 37.8911 1.80026 0.900130 0.435621i \(-0.143471\pi\)
0.900130 + 0.435621i \(0.143471\pi\)
\(444\) 0 0
\(445\) 9.81150 0.465110
\(446\) −6.15879 −0.291627
\(447\) 0 0
\(448\) 5.90501 0.278985
\(449\) −15.2640 −0.720354 −0.360177 0.932884i \(-0.617284\pi\)
−0.360177 + 0.932884i \(0.617284\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 33.3123 1.56688
\(453\) 0 0
\(454\) 8.44870 0.396517
\(455\) −4.40791 −0.206646
\(456\) 0 0
\(457\) −16.7762 −0.784757 −0.392379 0.919804i \(-0.628348\pi\)
−0.392379 + 0.919804i \(0.628348\pi\)
\(458\) 2.95776 0.138207
\(459\) 0 0
\(460\) −8.64136 −0.402906
\(461\) −36.0385 −1.67848 −0.839239 0.543763i \(-0.816999\pi\)
−0.839239 + 0.543763i \(0.816999\pi\)
\(462\) 0 0
\(463\) −25.7197 −1.19530 −0.597648 0.801758i \(-0.703898\pi\)
−0.597648 + 0.801758i \(0.703898\pi\)
\(464\) −3.26494 −0.151571
\(465\) 0 0
\(466\) −1.01574 −0.0470531
\(467\) −27.6780 −1.28079 −0.640394 0.768047i \(-0.721229\pi\)
−0.640394 + 0.768047i \(0.721229\pi\)
\(468\) 0 0
\(469\) −7.74683 −0.357715
\(470\) 4.74559 0.218898
\(471\) 0 0
\(472\) −11.4117 −0.525267
\(473\) 0 0
\(474\) 0 0
\(475\) −5.31826 −0.244019
\(476\) −0.913425 −0.0418668
\(477\) 0 0
\(478\) 7.57688 0.346559
\(479\) 13.8413 0.632427 0.316214 0.948688i \(-0.397588\pi\)
0.316214 + 0.948688i \(0.397588\pi\)
\(480\) 0 0
\(481\) 12.0426 0.549096
\(482\) 3.88707 0.177051
\(483\) 0 0
\(484\) 0 0
\(485\) 2.22968 0.101245
\(486\) 0 0
\(487\) −10.9228 −0.494959 −0.247480 0.968893i \(-0.579602\pi\)
−0.247480 + 0.968893i \(0.579602\pi\)
\(488\) −3.53871 −0.160190
\(489\) 0 0
\(490\) −0.391830 −0.0177011
\(491\) −26.6839 −1.20423 −0.602114 0.798410i \(-0.705675\pi\)
−0.602114 + 0.798410i \(0.705675\pi\)
\(492\) 0 0
\(493\) 0.450900 0.0203075
\(494\) −1.64063 −0.0738156
\(495\) 0 0
\(496\) −4.07816 −0.183115
\(497\) −8.46568 −0.379738
\(498\) 0 0
\(499\) −36.1374 −1.61773 −0.808867 0.587992i \(-0.799919\pi\)
−0.808867 + 0.587992i \(0.799919\pi\)
\(500\) −20.6371 −0.922917
\(501\) 0 0
\(502\) 3.27355 0.146106
\(503\) 27.2718 1.21599 0.607994 0.793942i \(-0.291974\pi\)
0.607994 + 0.793942i \(0.291974\pi\)
\(504\) 0 0
\(505\) 12.4082 0.552157
\(506\) 0 0
\(507\) 0 0
\(508\) 4.96895 0.220462
\(509\) −35.5186 −1.57434 −0.787168 0.616739i \(-0.788453\pi\)
−0.787168 + 0.616739i \(0.788453\pi\)
\(510\) 0 0
\(511\) 13.3709 0.591493
\(512\) 19.9263 0.880628
\(513\) 0 0
\(514\) −3.05358 −0.134687
\(515\) 1.59564 0.0703122
\(516\) 0 0
\(517\) 0 0
\(518\) 1.07050 0.0470351
\(519\) 0 0
\(520\) −5.18926 −0.227564
\(521\) −19.1607 −0.839446 −0.419723 0.907652i \(-0.637873\pi\)
−0.419723 + 0.907652i \(0.637873\pi\)
\(522\) 0 0
\(523\) −4.44268 −0.194265 −0.0971323 0.995271i \(-0.530967\pi\)
−0.0971323 + 0.995271i \(0.530967\pi\)
\(524\) −19.6378 −0.857881
\(525\) 0 0
\(526\) 6.26163 0.273020
\(527\) 0.563208 0.0245337
\(528\) 0 0
\(529\) −10.9007 −0.473943
\(530\) 2.06993 0.0899121
\(531\) 0 0
\(532\) 3.07050 0.133123
\(533\) 13.3760 0.579380
\(534\) 0 0
\(535\) −11.9223 −0.515447
\(536\) −9.12005 −0.393926
\(537\) 0 0
\(538\) 0.374126 0.0161297
\(539\) 0 0
\(540\) 0 0
\(541\) 23.7902 1.02282 0.511411 0.859336i \(-0.329123\pi\)
0.511411 + 0.859336i \(0.329123\pi\)
\(542\) −6.74549 −0.289744
\(543\) 0 0
\(544\) −1.62548 −0.0696920
\(545\) 21.1134 0.904399
\(546\) 0 0
\(547\) 22.0421 0.942452 0.471226 0.882013i \(-0.343812\pi\)
0.471226 + 0.882013i \(0.343812\pi\)
\(548\) −38.0140 −1.62388
\(549\) 0 0
\(550\) 0 0
\(551\) −1.51571 −0.0645715
\(552\) 0 0
\(553\) 8.86612 0.377026
\(554\) 3.83986 0.163140
\(555\) 0 0
\(556\) 8.86091 0.375786
\(557\) 29.7068 1.25872 0.629359 0.777115i \(-0.283318\pi\)
0.629359 + 0.777115i \(0.283318\pi\)
\(558\) 0 0
\(559\) 9.72355 0.411262
\(560\) 4.50729 0.190468
\(561\) 0 0
\(562\) −0.472664 −0.0199381
\(563\) 7.06070 0.297573 0.148786 0.988869i \(-0.452463\pi\)
0.148786 + 0.988869i \(0.452463\pi\)
\(564\) 0 0
\(565\) 22.7014 0.955054
\(566\) 4.33166 0.182073
\(567\) 0 0
\(568\) −9.96632 −0.418178
\(569\) 22.6262 0.948538 0.474269 0.880380i \(-0.342712\pi\)
0.474269 + 0.880380i \(0.342712\pi\)
\(570\) 0 0
\(571\) 18.4644 0.772713 0.386356 0.922350i \(-0.373733\pi\)
0.386356 + 0.922350i \(0.373733\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.18903 0.0496292
\(575\) 11.5032 0.479716
\(576\) 0 0
\(577\) 2.19908 0.0915488 0.0457744 0.998952i \(-0.485424\pi\)
0.0457744 + 0.998952i \(0.485424\pi\)
\(578\) −5.05051 −0.210073
\(579\) 0 0
\(580\) −2.34146 −0.0972238
\(581\) −11.4700 −0.475855
\(582\) 0 0
\(583\) 0 0
\(584\) 15.7410 0.651368
\(585\) 0 0
\(586\) 5.23940 0.216438
\(587\) 19.1979 0.792383 0.396192 0.918168i \(-0.370332\pi\)
0.396192 + 0.918168i \(0.370332\pi\)
\(588\) 0 0
\(589\) −1.89324 −0.0780095
\(590\) −3.79818 −0.156369
\(591\) 0 0
\(592\) −12.3141 −0.506108
\(593\) −19.1328 −0.785691 −0.392846 0.919604i \(-0.628509\pi\)
−0.392846 + 0.919604i \(0.628509\pi\)
\(594\) 0 0
\(595\) −0.622473 −0.0255189
\(596\) −44.8319 −1.83639
\(597\) 0 0
\(598\) 3.54863 0.145114
\(599\) 4.42406 0.180762 0.0903812 0.995907i \(-0.471191\pi\)
0.0903812 + 0.995907i \(0.471191\pi\)
\(600\) 0 0
\(601\) 39.9277 1.62869 0.814343 0.580384i \(-0.197098\pi\)
0.814343 + 0.580384i \(0.197098\pi\)
\(602\) 0.864351 0.0352283
\(603\) 0 0
\(604\) 25.5644 1.04020
\(605\) 0 0
\(606\) 0 0
\(607\) 22.7804 0.924626 0.462313 0.886717i \(-0.347020\pi\)
0.462313 + 0.886717i \(0.347020\pi\)
\(608\) 5.46410 0.221599
\(609\) 0 0
\(610\) −1.17780 −0.0476876
\(611\) 41.0298 1.65989
\(612\) 0 0
\(613\) −40.4017 −1.63181 −0.815904 0.578187i \(-0.803760\pi\)
−0.815904 + 0.578187i \(0.803760\pi\)
\(614\) 3.41509 0.137822
\(615\) 0 0
\(616\) 0 0
\(617\) −21.2137 −0.854033 −0.427017 0.904244i \(-0.640435\pi\)
−0.427017 + 0.904244i \(0.640435\pi\)
\(618\) 0 0
\(619\) 13.0445 0.524302 0.262151 0.965027i \(-0.415568\pi\)
0.262151 + 0.965027i \(0.415568\pi\)
\(620\) −2.92466 −0.117457
\(621\) 0 0
\(622\) −4.14277 −0.166110
\(623\) 7.54068 0.302111
\(624\) 0 0
\(625\) 2.47156 0.0988625
\(626\) 1.93141 0.0771946
\(627\) 0 0
\(628\) −24.1363 −0.963144
\(629\) 1.70063 0.0678085
\(630\) 0 0
\(631\) 37.0202 1.47375 0.736875 0.676029i \(-0.236301\pi\)
0.736875 + 0.676029i \(0.236301\pi\)
\(632\) 10.4378 0.415191
\(633\) 0 0
\(634\) −3.21203 −0.127566
\(635\) 3.38620 0.134377
\(636\) 0 0
\(637\) −3.38772 −0.134226
\(638\) 0 0
\(639\) 0 0
\(640\) 11.1556 0.440963
\(641\) −21.3679 −0.843980 −0.421990 0.906600i \(-0.638668\pi\)
−0.421990 + 0.906600i \(0.638668\pi\)
\(642\) 0 0
\(643\) 18.1845 0.717128 0.358564 0.933505i \(-0.383267\pi\)
0.358564 + 0.933505i \(0.383267\pi\)
\(644\) −6.64136 −0.261706
\(645\) 0 0
\(646\) −0.231686 −0.00911558
\(647\) 9.43913 0.371091 0.185545 0.982636i \(-0.440595\pi\)
0.185545 + 0.982636i \(0.440595\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 3.37379 0.132331
\(651\) 0 0
\(652\) 1.63123 0.0638839
\(653\) 21.9975 0.860827 0.430414 0.902632i \(-0.358368\pi\)
0.430414 + 0.902632i \(0.358368\pi\)
\(654\) 0 0
\(655\) −13.3826 −0.522901
\(656\) −13.6776 −0.534022
\(657\) 0 0
\(658\) 3.64725 0.142185
\(659\) 46.9739 1.82984 0.914922 0.403631i \(-0.132252\pi\)
0.914922 + 0.403631i \(0.132252\pi\)
\(660\) 0 0
\(661\) −11.4368 −0.444840 −0.222420 0.974951i \(-0.571396\pi\)
−0.222420 + 0.974951i \(0.571396\pi\)
\(662\) 3.00067 0.116624
\(663\) 0 0
\(664\) −13.5032 −0.524025
\(665\) 2.09246 0.0811421
\(666\) 0 0
\(667\) 3.27842 0.126941
\(668\) −15.5939 −0.603345
\(669\) 0 0
\(670\) −3.03544 −0.117269
\(671\) 0 0
\(672\) 0 0
\(673\) 4.68715 0.180676 0.0903381 0.995911i \(-0.471205\pi\)
0.0903381 + 0.995911i \(0.471205\pi\)
\(674\) 5.74138 0.221150
\(675\) 0 0
\(676\) 2.90858 0.111869
\(677\) −8.71493 −0.334942 −0.167471 0.985877i \(-0.553560\pi\)
−0.167471 + 0.985877i \(0.553560\pi\)
\(678\) 0 0
\(679\) 1.71364 0.0657633
\(680\) −0.732814 −0.0281021
\(681\) 0 0
\(682\) 0 0
\(683\) 39.2999 1.50377 0.751884 0.659295i \(-0.229145\pi\)
0.751884 + 0.659295i \(0.229145\pi\)
\(684\) 0 0
\(685\) −25.9055 −0.989798
\(686\) −0.301143 −0.0114977
\(687\) 0 0
\(688\) −9.94279 −0.379065
\(689\) 17.8964 0.681798
\(690\) 0 0
\(691\) −21.4190 −0.814819 −0.407409 0.913246i \(-0.633568\pi\)
−0.407409 + 0.913246i \(0.633568\pi\)
\(692\) −39.9258 −1.51775
\(693\) 0 0
\(694\) 0.0789191 0.00299573
\(695\) 6.03846 0.229052
\(696\) 0 0
\(697\) 1.88893 0.0715483
\(698\) 1.49586 0.0566192
\(699\) 0 0
\(700\) −6.31415 −0.238652
\(701\) 24.6197 0.929875 0.464937 0.885344i \(-0.346077\pi\)
0.464937 + 0.885344i \(0.346077\pi\)
\(702\) 0 0
\(703\) −5.71670 −0.215610
\(704\) 0 0
\(705\) 0 0
\(706\) 5.96576 0.224524
\(707\) 9.53637 0.358652
\(708\) 0 0
\(709\) −10.9882 −0.412672 −0.206336 0.978481i \(-0.566154\pi\)
−0.206336 + 0.978481i \(0.566154\pi\)
\(710\) −3.31711 −0.124489
\(711\) 0 0
\(712\) 8.87736 0.332693
\(713\) 4.09499 0.153359
\(714\) 0 0
\(715\) 0 0
\(716\) 16.0617 0.600254
\(717\) 0 0
\(718\) −5.97885 −0.223129
\(719\) −33.4441 −1.24725 −0.623627 0.781722i \(-0.714342\pi\)
−0.623627 + 0.781722i \(0.714342\pi\)
\(720\) 0 0
\(721\) 1.22634 0.0456711
\(722\) −4.94290 −0.183956
\(723\) 0 0
\(724\) 39.5219 1.46882
\(725\) 3.11690 0.115759
\(726\) 0 0
\(727\) 44.9100 1.66562 0.832810 0.553560i \(-0.186731\pi\)
0.832810 + 0.553560i \(0.186731\pi\)
\(728\) −3.98823 −0.147814
\(729\) 0 0
\(730\) 5.23911 0.193908
\(731\) 1.37314 0.0507872
\(732\) 0 0
\(733\) −23.4850 −0.867437 −0.433718 0.901048i \(-0.642799\pi\)
−0.433718 + 0.901048i \(0.642799\pi\)
\(734\) 9.30793 0.343562
\(735\) 0 0
\(736\) −11.8186 −0.435640
\(737\) 0 0
\(738\) 0 0
\(739\) −17.2692 −0.635259 −0.317630 0.948215i \(-0.602887\pi\)
−0.317630 + 0.948215i \(0.602887\pi\)
\(740\) −8.83112 −0.324639
\(741\) 0 0
\(742\) 1.59086 0.0584022
\(743\) −7.15344 −0.262434 −0.131217 0.991354i \(-0.541888\pi\)
−0.131217 + 0.991354i \(0.541888\pi\)
\(744\) 0 0
\(745\) −30.5517 −1.11933
\(746\) 11.1989 0.410020
\(747\) 0 0
\(748\) 0 0
\(749\) −9.16296 −0.334807
\(750\) 0 0
\(751\) −52.3091 −1.90879 −0.954394 0.298551i \(-0.903497\pi\)
−0.954394 + 0.298551i \(0.903497\pi\)
\(752\) −41.9549 −1.52994
\(753\) 0 0
\(754\) 0.961534 0.0350170
\(755\) 17.4214 0.634029
\(756\) 0 0
\(757\) −11.0149 −0.400344 −0.200172 0.979761i \(-0.564150\pi\)
−0.200172 + 0.979761i \(0.564150\pi\)
\(758\) −6.01469 −0.218464
\(759\) 0 0
\(760\) 2.46337 0.0893559
\(761\) −11.2716 −0.408596 −0.204298 0.978909i \(-0.565491\pi\)
−0.204298 + 0.978909i \(0.565491\pi\)
\(762\) 0 0
\(763\) 16.2268 0.587450
\(764\) −28.0344 −1.01425
\(765\) 0 0
\(766\) −6.12317 −0.221239
\(767\) −32.8387 −1.18574
\(768\) 0 0
\(769\) 1.37733 0.0496678 0.0248339 0.999692i \(-0.492094\pi\)
0.0248339 + 0.999692i \(0.492094\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −48.2401 −1.73620
\(773\) −1.30100 −0.0467937 −0.0233969 0.999726i \(-0.507448\pi\)
−0.0233969 + 0.999726i \(0.507448\pi\)
\(774\) 0 0
\(775\) 3.89324 0.139849
\(776\) 2.01740 0.0724204
\(777\) 0 0
\(778\) −3.24495 −0.116337
\(779\) −6.34968 −0.227501
\(780\) 0 0
\(781\) 0 0
\(782\) 0.501128 0.0179203
\(783\) 0 0
\(784\) 3.46410 0.123718
\(785\) −16.4482 −0.587062
\(786\) 0 0
\(787\) 50.2646 1.79174 0.895871 0.444315i \(-0.146553\pi\)
0.895871 + 0.444315i \(0.146553\pi\)
\(788\) 19.8446 0.706936
\(789\) 0 0
\(790\) 3.47401 0.123600
\(791\) 17.4473 0.620353
\(792\) 0 0
\(793\) −10.1831 −0.361612
\(794\) −0.0746117 −0.00264787
\(795\) 0 0
\(796\) −21.3278 −0.755945
\(797\) 7.51222 0.266096 0.133048 0.991110i \(-0.457523\pi\)
0.133048 + 0.991110i \(0.457523\pi\)
\(798\) 0 0
\(799\) 5.79413 0.204982
\(800\) −11.2363 −0.397264
\(801\) 0 0
\(802\) 7.48162 0.264185
\(803\) 0 0
\(804\) 0 0
\(805\) −4.52590 −0.159517
\(806\) 1.20103 0.0423044
\(807\) 0 0
\(808\) 11.2268 0.394958
\(809\) 6.79104 0.238760 0.119380 0.992849i \(-0.461909\pi\)
0.119380 + 0.992849i \(0.461909\pi\)
\(810\) 0 0
\(811\) −32.0227 −1.12447 −0.562235 0.826978i \(-0.690058\pi\)
−0.562235 + 0.826978i \(0.690058\pi\)
\(812\) −1.79954 −0.0631515
\(813\) 0 0
\(814\) 0 0
\(815\) 1.11164 0.0389390
\(816\) 0 0
\(817\) −4.61583 −0.161487
\(818\) 5.58115 0.195140
\(819\) 0 0
\(820\) −9.80894 −0.342543
\(821\) −35.3145 −1.23249 −0.616243 0.787556i \(-0.711346\pi\)
−0.616243 + 0.787556i \(0.711346\pi\)
\(822\) 0 0
\(823\) −2.00171 −0.0697753 −0.0348877 0.999391i \(-0.511107\pi\)
−0.0348877 + 0.999391i \(0.511107\pi\)
\(824\) 1.44372 0.0502943
\(825\) 0 0
\(826\) −2.91911 −0.101569
\(827\) 8.42435 0.292943 0.146472 0.989215i \(-0.453208\pi\)
0.146472 + 0.989215i \(0.453208\pi\)
\(828\) 0 0
\(829\) 14.1931 0.492945 0.246473 0.969150i \(-0.420728\pi\)
0.246473 + 0.969150i \(0.420728\pi\)
\(830\) −4.49429 −0.155999
\(831\) 0 0
\(832\) 20.0045 0.693531
\(833\) −0.478405 −0.0165758
\(834\) 0 0
\(835\) −10.6268 −0.367755
\(836\) 0 0
\(837\) 0 0
\(838\) 5.87947 0.203103
\(839\) −8.44234 −0.291462 −0.145731 0.989324i \(-0.546553\pi\)
−0.145731 + 0.989324i \(0.546553\pi\)
\(840\) 0 0
\(841\) −28.1117 −0.969368
\(842\) −2.64610 −0.0911907
\(843\) 0 0
\(844\) 21.3687 0.735542
\(845\) 1.98212 0.0681869
\(846\) 0 0
\(847\) 0 0
\(848\) −18.2999 −0.628421
\(849\) 0 0
\(850\) 0.476438 0.0163417
\(851\) 12.3650 0.423867
\(852\) 0 0
\(853\) −57.3123 −1.96234 −0.981168 0.193158i \(-0.938127\pi\)
−0.981168 + 0.193158i \(0.938127\pi\)
\(854\) −0.905201 −0.0309753
\(855\) 0 0
\(856\) −10.7872 −0.368699
\(857\) 13.5681 0.463479 0.231739 0.972778i \(-0.425558\pi\)
0.231739 + 0.972778i \(0.425558\pi\)
\(858\) 0 0
\(859\) −15.4832 −0.528280 −0.264140 0.964484i \(-0.585088\pi\)
−0.264140 + 0.964484i \(0.585088\pi\)
\(860\) −7.13049 −0.243148
\(861\) 0 0
\(862\) −5.46670 −0.186196
\(863\) 21.5553 0.733752 0.366876 0.930270i \(-0.380427\pi\)
0.366876 + 0.930270i \(0.380427\pi\)
\(864\) 0 0
\(865\) −27.2083 −0.925110
\(866\) 6.42708 0.218401
\(867\) 0 0
\(868\) −2.24776 −0.0762940
\(869\) 0 0
\(870\) 0 0
\(871\) −26.2441 −0.889246
\(872\) 19.1032 0.646916
\(873\) 0 0
\(874\) −1.68455 −0.0569809
\(875\) −10.8086 −0.365398
\(876\) 0 0
\(877\) 18.6275 0.629007 0.314504 0.949256i \(-0.398162\pi\)
0.314504 + 0.949256i \(0.398162\pi\)
\(878\) −2.28593 −0.0771465
\(879\) 0 0
\(880\) 0 0
\(881\) −13.7010 −0.461600 −0.230800 0.973001i \(-0.574134\pi\)
−0.230800 + 0.973001i \(0.574134\pi\)
\(882\) 0 0
\(883\) −29.4327 −0.990489 −0.495244 0.868754i \(-0.664921\pi\)
−0.495244 + 0.868754i \(0.664921\pi\)
\(884\) −3.09443 −0.104077
\(885\) 0 0
\(886\) 11.4106 0.383348
\(887\) 14.4530 0.485283 0.242641 0.970116i \(-0.421986\pi\)
0.242641 + 0.970116i \(0.421986\pi\)
\(888\) 0 0
\(889\) 2.60248 0.0872844
\(890\) 2.95467 0.0990406
\(891\) 0 0
\(892\) 39.0481 1.30743
\(893\) −19.4771 −0.651777
\(894\) 0 0
\(895\) 10.9456 0.365871
\(896\) 8.57368 0.286427
\(897\) 0 0
\(898\) −4.59666 −0.153392
\(899\) 1.10958 0.0370065
\(900\) 0 0
\(901\) 2.52728 0.0841960
\(902\) 0 0
\(903\) 0 0
\(904\) 20.5400 0.683150
\(905\) 26.9330 0.895285
\(906\) 0 0
\(907\) 33.5011 1.11239 0.556193 0.831053i \(-0.312261\pi\)
0.556193 + 0.831053i \(0.312261\pi\)
\(908\) −53.5666 −1.77767
\(909\) 0 0
\(910\) −1.32741 −0.0440032
\(911\) 21.2100 0.702718 0.351359 0.936241i \(-0.385720\pi\)
0.351359 + 0.936241i \(0.385720\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −5.05203 −0.167106
\(915\) 0 0
\(916\) −18.7528 −0.619612
\(917\) −10.2853 −0.339649
\(918\) 0 0
\(919\) −46.8324 −1.54486 −0.772430 0.635100i \(-0.780959\pi\)
−0.772430 + 0.635100i \(0.780959\pi\)
\(920\) −5.32817 −0.175665
\(921\) 0 0
\(922\) −10.8527 −0.357415
\(923\) −28.6793 −0.943992
\(924\) 0 0
\(925\) 11.7558 0.386528
\(926\) −7.74531 −0.254527
\(927\) 0 0
\(928\) −3.20237 −0.105123
\(929\) 10.9210 0.358307 0.179154 0.983821i \(-0.442664\pi\)
0.179154 + 0.983821i \(0.442664\pi\)
\(930\) 0 0
\(931\) 1.60817 0.0527056
\(932\) 6.44000 0.210949
\(933\) 0 0
\(934\) −8.33505 −0.272731
\(935\) 0 0
\(936\) 0 0
\(937\) −7.85086 −0.256476 −0.128238 0.991743i \(-0.540932\pi\)
−0.128238 + 0.991743i \(0.540932\pi\)
\(938\) −2.33290 −0.0761720
\(939\) 0 0
\(940\) −30.0881 −0.981365
\(941\) −27.6122 −0.900132 −0.450066 0.892995i \(-0.648599\pi\)
−0.450066 + 0.892995i \(0.648599\pi\)
\(942\) 0 0
\(943\) 13.7341 0.447244
\(944\) 33.5791 1.09291
\(945\) 0 0
\(946\) 0 0
\(947\) −56.9267 −1.84987 −0.924934 0.380127i \(-0.875880\pi\)
−0.924934 + 0.380127i \(0.875880\pi\)
\(948\) 0 0
\(949\) 45.2968 1.47040
\(950\) −1.60156 −0.0519614
\(951\) 0 0
\(952\) −0.563208 −0.0182537
\(953\) −25.2881 −0.819163 −0.409581 0.912274i \(-0.634325\pi\)
−0.409581 + 0.912274i \(0.634325\pi\)
\(954\) 0 0
\(955\) −19.1047 −0.618212
\(956\) −48.0391 −1.55370
\(957\) 0 0
\(958\) 4.16822 0.134669
\(959\) −19.9098 −0.642921
\(960\) 0 0
\(961\) −29.6141 −0.955292
\(962\) 3.62655 0.116925
\(963\) 0 0
\(964\) −24.6449 −0.793759
\(965\) −32.8743 −1.05826
\(966\) 0 0
\(967\) 19.2122 0.617824 0.308912 0.951091i \(-0.400035\pi\)
0.308912 + 0.951091i \(0.400035\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.671454 0.0215591
\(971\) −50.9532 −1.63517 −0.817583 0.575811i \(-0.804686\pi\)
−0.817583 + 0.575811i \(0.804686\pi\)
\(972\) 0 0
\(973\) 4.64089 0.148780
\(974\) −3.28932 −0.105397
\(975\) 0 0
\(976\) 10.4127 0.333302
\(977\) 54.7278 1.75090 0.875449 0.483310i \(-0.160565\pi\)
0.875449 + 0.483310i \(0.160565\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.48429 0.0793577
\(981\) 0 0
\(982\) −8.03568 −0.256429
\(983\) −6.20739 −0.197985 −0.0989924 0.995088i \(-0.531562\pi\)
−0.0989924 + 0.995088i \(0.531562\pi\)
\(984\) 0 0
\(985\) 13.5236 0.430897
\(986\) 0.135785 0.00432429
\(987\) 0 0
\(988\) 10.4020 0.330931
\(989\) 9.98384 0.317468
\(990\) 0 0
\(991\) −4.64753 −0.147634 −0.0738168 0.997272i \(-0.523518\pi\)
−0.0738168 + 0.997272i \(0.523518\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) −2.54938 −0.0808614
\(995\) −14.5343 −0.460769
\(996\) 0 0
\(997\) 34.0235 1.07754 0.538768 0.842454i \(-0.318890\pi\)
0.538768 + 0.842454i \(0.318890\pi\)
\(998\) −10.8825 −0.344480
\(999\) 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.cm.1.2 4
3.2 odd 2 2541.2.a.bk.1.3 4
11.10 odd 2 7623.2.a.cf.1.3 4
33.32 even 2 2541.2.a.bo.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bk.1.3 4 3.2 odd 2
2541.2.a.bo.1.2 yes 4 33.32 even 2
7623.2.a.cf.1.3 4 11.10 odd 2
7623.2.a.cm.1.2 4 1.1 even 1 trivial