Properties

Label 7623.2.a.cn.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
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: \(1\)
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 1.1
Root \(-0.326909\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} +O(q^{10})\) \(q-2.05896 q^{2} +2.23931 q^{4} +3.05896 q^{5} +1.00000 q^{7} -0.492737 q^{8} -6.29827 q^{10} -3.32691 q^{13} -2.05896 q^{14} -3.46410 q^{16} +5.37651 q^{17} -8.29827 q^{19} +6.84997 q^{20} +1.74141 q^{23} +4.35723 q^{25} +6.84997 q^{26} +2.23931 q^{28} -8.16277 q^{29} -5.44999 q^{31} +8.11792 q^{32} -11.0700 q^{34} +3.05896 q^{35} +4.53242 q^{37} +17.0858 q^{38} -1.50726 q^{40} -6.84997 q^{41} +1.84997 q^{43} -3.58550 q^{46} +7.28416 q^{47} +1.00000 q^{49} -8.97136 q^{50} -7.44999 q^{52} +0.985474 q^{53} -0.492737 q^{56} +16.8068 q^{58} -4.52306 q^{59} +12.0845 q^{61} +11.2213 q^{62} -9.78626 q^{64} -10.1769 q^{65} +0.170993 q^{67} +12.0397 q^{68} -6.29827 q^{70} +10.0017 q^{71} -12.7910 q^{73} -9.33207 q^{74} -18.5824 q^{76} -14.6794 q^{79} -10.5965 q^{80} +14.1038 q^{82} -1.62993 q^{83} +16.4465 q^{85} -3.80901 q^{86} -13.9872 q^{89} -3.32691 q^{91} +3.89957 q^{92} -14.9978 q^{94} -25.3841 q^{95} +3.44483 q^{97} -2.05896 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} - 10q^{10} - 10q^{13} + 2q^{14} + 6q^{17} - 18q^{19} + 2q^{23} - 8q^{25} + 4q^{28} + 6q^{29} + 12q^{32} - 2q^{34} + 2q^{35} - 4q^{37} - 8q^{40} - 20q^{43} - 16q^{46} + 6q^{47} + 4q^{49} - 24q^{50} - 8q^{52} + 24q^{58} + 6q^{59} + 10q^{61} - 16q^{64} - 10q^{65} + 4q^{67} + 28q^{68} - 10q^{70} + 6q^{71} - 34q^{73} - 36q^{74} - 36q^{76} - 24q^{79} - 12q^{80} + 28q^{82} + 6q^{83} + 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} + 6q^{94} - 18q^{95} - 10q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.05896 −1.45590 −0.727952 0.685628i \(-0.759528\pi\)
−0.727952 + 0.685628i \(0.759528\pi\)
\(3\) 0 0
\(4\) 2.23931 1.11966
\(5\) 3.05896 1.36801 0.684004 0.729478i \(-0.260237\pi\)
0.684004 + 0.729478i \(0.260237\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −0.492737 −0.174209
\(9\) 0 0
\(10\) −6.29827 −1.99169
\(11\) 0 0
\(12\) 0 0
\(13\) −3.32691 −0.922718 −0.461359 0.887213i \(-0.652638\pi\)
−0.461359 + 0.887213i \(0.652638\pi\)
\(14\) −2.05896 −0.550280
\(15\) 0 0
\(16\) −3.46410 −0.866025
\(17\) 5.37651 1.30399 0.651997 0.758221i \(-0.273931\pi\)
0.651997 + 0.758221i \(0.273931\pi\)
\(18\) 0 0
\(19\) −8.29827 −1.90375 −0.951877 0.306480i \(-0.900849\pi\)
−0.951877 + 0.306480i \(0.900849\pi\)
\(20\) 6.84997 1.53170
\(21\) 0 0
\(22\) 0 0
\(23\) 1.74141 0.363110 0.181555 0.983381i \(-0.441887\pi\)
0.181555 + 0.983381i \(0.441887\pi\)
\(24\) 0 0
\(25\) 4.35723 0.871446
\(26\) 6.84997 1.34339
\(27\) 0 0
\(28\) 2.23931 0.423191
\(29\) −8.16277 −1.51579 −0.757894 0.652378i \(-0.773772\pi\)
−0.757894 + 0.652378i \(0.773772\pi\)
\(30\) 0 0
\(31\) −5.44999 −0.978847 −0.489424 0.872046i \(-0.662793\pi\)
−0.489424 + 0.872046i \(0.662793\pi\)
\(32\) 8.11792 1.43506
\(33\) 0 0
\(34\) −11.0700 −1.89849
\(35\) 3.05896 0.517059
\(36\) 0 0
\(37\) 4.53242 0.745126 0.372563 0.928007i \(-0.378479\pi\)
0.372563 + 0.928007i \(0.378479\pi\)
\(38\) 17.0858 2.77168
\(39\) 0 0
\(40\) −1.50726 −0.238319
\(41\) −6.84997 −1.06979 −0.534893 0.844920i \(-0.679648\pi\)
−0.534893 + 0.844920i \(0.679648\pi\)
\(42\) 0 0
\(43\) 1.84997 0.282118 0.141059 0.990001i \(-0.454949\pi\)
0.141059 + 0.990001i \(0.454949\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.58550 −0.528653
\(47\) 7.28416 1.06250 0.531252 0.847214i \(-0.321722\pi\)
0.531252 + 0.847214i \(0.321722\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.97136 −1.26874
\(51\) 0 0
\(52\) −7.44999 −1.03313
\(53\) 0.985474 0.135365 0.0676827 0.997707i \(-0.478439\pi\)
0.0676827 + 0.997707i \(0.478439\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.492737 −0.0658448
\(57\) 0 0
\(58\) 16.8068 2.20684
\(59\) −4.52306 −0.588852 −0.294426 0.955674i \(-0.595129\pi\)
−0.294426 + 0.955674i \(0.595129\pi\)
\(60\) 0 0
\(61\) 12.0845 1.54727 0.773633 0.633634i \(-0.218437\pi\)
0.773633 + 0.633634i \(0.218437\pi\)
\(62\) 11.2213 1.42511
\(63\) 0 0
\(64\) −9.78626 −1.22328
\(65\) −10.1769 −1.26229
\(66\) 0 0
\(67\) 0.170993 0.0208901 0.0104451 0.999945i \(-0.496675\pi\)
0.0104451 + 0.999945i \(0.496675\pi\)
\(68\) 12.0397 1.46003
\(69\) 0 0
\(70\) −6.29827 −0.752788
\(71\) 10.0017 1.18698 0.593491 0.804841i \(-0.297749\pi\)
0.593491 + 0.804841i \(0.297749\pi\)
\(72\) 0 0
\(73\) −12.7910 −1.49707 −0.748537 0.663093i \(-0.769243\pi\)
−0.748537 + 0.663093i \(0.769243\pi\)
\(74\) −9.33207 −1.08483
\(75\) 0 0
\(76\) −18.5824 −2.13155
\(77\) 0 0
\(78\) 0 0
\(79\) −14.6794 −1.65156 −0.825780 0.563992i \(-0.809265\pi\)
−0.825780 + 0.563992i \(0.809265\pi\)
\(80\) −10.5965 −1.18473
\(81\) 0 0
\(82\) 14.1038 1.55751
\(83\) −1.62993 −0.178908 −0.0894540 0.995991i \(-0.528512\pi\)
−0.0894540 + 0.995991i \(0.528512\pi\)
\(84\) 0 0
\(85\) 16.4465 1.78388
\(86\) −3.80901 −0.410736
\(87\) 0 0
\(88\) 0 0
\(89\) −13.9872 −1.48264 −0.741318 0.671154i \(-0.765799\pi\)
−0.741318 + 0.671154i \(0.765799\pi\)
\(90\) 0 0
\(91\) −3.32691 −0.348755
\(92\) 3.89957 0.406558
\(93\) 0 0
\(94\) −14.9978 −1.54690
\(95\) −25.3841 −2.60435
\(96\) 0 0
\(97\) 3.44483 0.349769 0.174885 0.984589i \(-0.444045\pi\)
0.174885 + 0.984589i \(0.444045\pi\)
\(98\) −2.05896 −0.207986
\(99\) 0 0
\(100\) 9.75721 0.975721
\(101\) 9.35554 0.930911 0.465456 0.885071i \(-0.345890\pi\)
0.465456 + 0.885071i \(0.345890\pi\)
\(102\) 0 0
\(103\) 12.6952 1.25089 0.625447 0.780267i \(-0.284917\pi\)
0.625447 + 0.780267i \(0.284917\pi\)
\(104\) 1.63929 0.160746
\(105\) 0 0
\(106\) −2.02905 −0.197079
\(107\) 11.7589 1.13678 0.568388 0.822761i \(-0.307567\pi\)
0.568388 + 0.822761i \(0.307567\pi\)
\(108\) 0 0
\(109\) −15.9038 −1.52330 −0.761652 0.647986i \(-0.775611\pi\)
−0.761652 + 0.647986i \(0.775611\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.46410 −0.327327
\(113\) −7.30763 −0.687444 −0.343722 0.939071i \(-0.611688\pi\)
−0.343722 + 0.939071i \(0.611688\pi\)
\(114\) 0 0
\(115\) 5.32691 0.496737
\(116\) −18.2790 −1.69716
\(117\) 0 0
\(118\) 9.31280 0.857313
\(119\) 5.37651 0.492864
\(120\) 0 0
\(121\) 0 0
\(122\) −24.8816 −2.25267
\(123\) 0 0
\(124\) −12.2042 −1.09597
\(125\) −1.96620 −0.175862
\(126\) 0 0
\(127\) 4.32860 0.384101 0.192050 0.981385i \(-0.438486\pi\)
0.192050 + 0.981385i \(0.438486\pi\)
\(128\) 3.91368 0.345923
\(129\) 0 0
\(130\) 20.9538 1.83777
\(131\) 8.71236 0.761202 0.380601 0.924739i \(-0.375717\pi\)
0.380601 + 0.924739i \(0.375717\pi\)
\(132\) 0 0
\(133\) −8.29827 −0.719552
\(134\) −0.352068 −0.0304140
\(135\) 0 0
\(136\) −2.64920 −0.227167
\(137\) 0.234149 0.0200047 0.0100024 0.999950i \(-0.496816\pi\)
0.0100024 + 0.999950i \(0.496816\pi\)
\(138\) 0 0
\(139\) −19.3200 −1.63870 −0.819349 0.573296i \(-0.805665\pi\)
−0.819349 + 0.573296i \(0.805665\pi\)
\(140\) 6.84997 0.578928
\(141\) 0 0
\(142\) −20.5931 −1.72813
\(143\) 0 0
\(144\) 0 0
\(145\) −24.9696 −2.07361
\(146\) 26.3362 2.17960
\(147\) 0 0
\(148\) 10.1495 0.834285
\(149\) −5.20090 −0.426074 −0.213037 0.977044i \(-0.568336\pi\)
−0.213037 + 0.977044i \(0.568336\pi\)
\(150\) 0 0
\(151\) −6.35682 −0.517310 −0.258655 0.965970i \(-0.583279\pi\)
−0.258655 + 0.965970i \(0.583279\pi\)
\(152\) 4.08887 0.331651
\(153\) 0 0
\(154\) 0 0
\(155\) −16.6713 −1.33907
\(156\) 0 0
\(157\) −10.0141 −0.799213 −0.399606 0.916687i \(-0.630853\pi\)
−0.399606 + 0.916687i \(0.630853\pi\)
\(158\) 30.2243 2.40451
\(159\) 0 0
\(160\) 24.8324 1.96317
\(161\) 1.74141 0.137242
\(162\) 0 0
\(163\) −8.50337 −0.666035 −0.333018 0.942921i \(-0.608067\pi\)
−0.333018 + 0.942921i \(0.608067\pi\)
\(164\) −15.3392 −1.19779
\(165\) 0 0
\(166\) 3.35596 0.260473
\(167\) 2.93756 0.227316 0.113658 0.993520i \(-0.463743\pi\)
0.113658 + 0.993520i \(0.463743\pi\)
\(168\) 0 0
\(169\) −1.93168 −0.148591
\(170\) −33.8627 −2.59715
\(171\) 0 0
\(172\) 4.14266 0.315875
\(173\) −20.6841 −1.57259 −0.786293 0.617854i \(-0.788002\pi\)
−0.786293 + 0.617854i \(0.788002\pi\)
\(174\) 0 0
\(175\) 4.35723 0.329376
\(176\) 0 0
\(177\) 0 0
\(178\) 28.7990 2.15858
\(179\) −6.14697 −0.459446 −0.229723 0.973256i \(-0.573782\pi\)
−0.229723 + 0.973256i \(0.573782\pi\)
\(180\) 0 0
\(181\) −19.2458 −1.43053 −0.715263 0.698856i \(-0.753693\pi\)
−0.715263 + 0.698856i \(0.753693\pi\)
\(182\) 6.84997 0.507754
\(183\) 0 0
\(184\) −0.858058 −0.0632569
\(185\) 13.8645 1.01934
\(186\) 0 0
\(187\) 0 0
\(188\) 16.3115 1.18964
\(189\) 0 0
\(190\) 52.2648 3.79169
\(191\) −25.5978 −1.85219 −0.926097 0.377287i \(-0.876857\pi\)
−0.926097 + 0.377287i \(0.876857\pi\)
\(192\) 0 0
\(193\) −11.1962 −0.805917 −0.402958 0.915218i \(-0.632018\pi\)
−0.402958 + 0.915218i \(0.632018\pi\)
\(194\) −7.09276 −0.509230
\(195\) 0 0
\(196\) 2.23931 0.159951
\(197\) −17.4687 −1.24459 −0.622297 0.782781i \(-0.713800\pi\)
−0.622297 + 0.782781i \(0.713800\pi\)
\(198\) 0 0
\(199\) −6.71963 −0.476342 −0.238171 0.971223i \(-0.576548\pi\)
−0.238171 + 0.971223i \(0.576548\pi\)
\(200\) −2.14697 −0.151814
\(201\) 0 0
\(202\) −19.2627 −1.35532
\(203\) −8.16277 −0.572914
\(204\) 0 0
\(205\) −20.9538 −1.46348
\(206\) −26.1389 −1.82118
\(207\) 0 0
\(208\) 11.5247 0.799098
\(209\) 0 0
\(210\) 0 0
\(211\) 20.2293 1.39264 0.696321 0.717730i \(-0.254819\pi\)
0.696321 + 0.717730i \(0.254819\pi\)
\(212\) 2.20679 0.151563
\(213\) 0 0
\(214\) −24.2111 −1.65504
\(215\) 5.65898 0.385939
\(216\) 0 0
\(217\) −5.44999 −0.369970
\(218\) 32.7452 2.21779
\(219\) 0 0
\(220\) 0 0
\(221\) −17.8871 −1.20322
\(222\) 0 0
\(223\) 11.8705 0.794909 0.397454 0.917622i \(-0.369894\pi\)
0.397454 + 0.917622i \(0.369894\pi\)
\(224\) 8.11792 0.542401
\(225\) 0 0
\(226\) 15.0461 1.00085
\(227\) 13.9521 0.926033 0.463016 0.886350i \(-0.346767\pi\)
0.463016 + 0.886350i \(0.346767\pi\)
\(228\) 0 0
\(229\) 24.8324 1.64097 0.820485 0.571668i \(-0.193703\pi\)
0.820485 + 0.571668i \(0.193703\pi\)
\(230\) −10.9679 −0.723201
\(231\) 0 0
\(232\) 4.02210 0.264064
\(233\) −11.8265 −0.774780 −0.387390 0.921916i \(-0.626623\pi\)
−0.387390 + 0.921916i \(0.626623\pi\)
\(234\) 0 0
\(235\) 22.2820 1.45351
\(236\) −10.1286 −0.659313
\(237\) 0 0
\(238\) −11.0700 −0.717562
\(239\) −8.86522 −0.573443 −0.286722 0.958014i \(-0.592565\pi\)
−0.286722 + 0.958014i \(0.592565\pi\)
\(240\) 0 0
\(241\) −11.3922 −0.733834 −0.366917 0.930254i \(-0.619587\pi\)
−0.366917 + 0.930254i \(0.619587\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 27.0611 1.73241
\(245\) 3.05896 0.195430
\(246\) 0 0
\(247\) 27.6076 1.75663
\(248\) 2.68541 0.170524
\(249\) 0 0
\(250\) 4.04833 0.256039
\(251\) −9.71866 −0.613436 −0.306718 0.951800i \(-0.599231\pi\)
−0.306718 + 0.951800i \(0.599231\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.91241 −0.559214
\(255\) 0 0
\(256\) 11.5144 0.719651
\(257\) 20.0456 1.25041 0.625204 0.780461i \(-0.285016\pi\)
0.625204 + 0.780461i \(0.285016\pi\)
\(258\) 0 0
\(259\) 4.53242 0.281631
\(260\) −22.7892 −1.41333
\(261\) 0 0
\(262\) −17.9384 −1.10824
\(263\) 12.9345 0.797576 0.398788 0.917043i \(-0.369431\pi\)
0.398788 + 0.917043i \(0.369431\pi\)
\(264\) 0 0
\(265\) 3.01453 0.185181
\(266\) 17.0858 1.04760
\(267\) 0 0
\(268\) 0.382907 0.0233898
\(269\) −4.44958 −0.271295 −0.135648 0.990757i \(-0.543312\pi\)
−0.135648 + 0.990757i \(0.543312\pi\)
\(270\) 0 0
\(271\) −9.59696 −0.582974 −0.291487 0.956575i \(-0.594150\pi\)
−0.291487 + 0.956575i \(0.594150\pi\)
\(272\) −18.6248 −1.12929
\(273\) 0 0
\(274\) −0.482104 −0.0291249
\(275\) 0 0
\(276\) 0 0
\(277\) 2.80037 0.168258 0.0841290 0.996455i \(-0.473189\pi\)
0.0841290 + 0.996455i \(0.473189\pi\)
\(278\) 39.7790 2.38579
\(279\) 0 0
\(280\) −1.50726 −0.0900762
\(281\) 15.4721 0.922988 0.461494 0.887143i \(-0.347314\pi\)
0.461494 + 0.887143i \(0.347314\pi\)
\(282\) 0 0
\(283\) 25.9076 1.54004 0.770022 0.638017i \(-0.220245\pi\)
0.770022 + 0.638017i \(0.220245\pi\)
\(284\) 22.3969 1.32901
\(285\) 0 0
\(286\) 0 0
\(287\) −6.84997 −0.404341
\(288\) 0 0
\(289\) 11.9068 0.700401
\(290\) 51.4113 3.01898
\(291\) 0 0
\(292\) −28.6431 −1.67621
\(293\) 23.3799 1.36587 0.682934 0.730480i \(-0.260704\pi\)
0.682934 + 0.730480i \(0.260704\pi\)
\(294\) 0 0
\(295\) −13.8359 −0.805555
\(296\) −2.23329 −0.129808
\(297\) 0 0
\(298\) 10.7084 0.620323
\(299\) −5.79352 −0.335048
\(300\) 0 0
\(301\) 1.84997 0.106630
\(302\) 13.0884 0.753154
\(303\) 0 0
\(304\) 28.7461 1.64870
\(305\) 36.9661 2.11667
\(306\) 0 0
\(307\) 10.0742 0.574965 0.287483 0.957786i \(-0.407182\pi\)
0.287483 + 0.957786i \(0.407182\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 34.3255 1.94956
\(311\) −8.26014 −0.468390 −0.234195 0.972190i \(-0.575245\pi\)
−0.234195 + 0.972190i \(0.575245\pi\)
\(312\) 0 0
\(313\) −7.39189 −0.417814 −0.208907 0.977935i \(-0.566991\pi\)
−0.208907 + 0.977935i \(0.566991\pi\)
\(314\) 20.6186 1.16358
\(315\) 0 0
\(316\) −32.8718 −1.84918
\(317\) −31.1600 −1.75012 −0.875060 0.484014i \(-0.839178\pi\)
−0.875060 + 0.484014i \(0.839178\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −29.9358 −1.67346
\(321\) 0 0
\(322\) −3.58550 −0.199812
\(323\) −44.6157 −2.48249
\(324\) 0 0
\(325\) −14.4961 −0.804100
\(326\) 17.5081 0.969684
\(327\) 0 0
\(328\) 3.37523 0.186366
\(329\) 7.28416 0.401589
\(330\) 0 0
\(331\) −20.0110 −1.09991 −0.549953 0.835195i \(-0.685355\pi\)
−0.549953 + 0.835195i \(0.685355\pi\)
\(332\) −3.64992 −0.200316
\(333\) 0 0
\(334\) −6.04833 −0.330950
\(335\) 0.523061 0.0285779
\(336\) 0 0
\(337\) 20.4973 1.11656 0.558278 0.829654i \(-0.311462\pi\)
0.558278 + 0.829654i \(0.311462\pi\)
\(338\) 3.97725 0.216334
\(339\) 0 0
\(340\) 36.8289 1.99733
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −0.911549 −0.0491474
\(345\) 0 0
\(346\) 42.5878 2.28953
\(347\) 33.7959 1.81426 0.907131 0.420849i \(-0.138268\pi\)
0.907131 + 0.420849i \(0.138268\pi\)
\(348\) 0 0
\(349\) −33.0030 −1.76661 −0.883304 0.468801i \(-0.844686\pi\)
−0.883304 + 0.468801i \(0.844686\pi\)
\(350\) −8.97136 −0.479540
\(351\) 0 0
\(352\) 0 0
\(353\) −9.49315 −0.505270 −0.252635 0.967562i \(-0.581297\pi\)
−0.252635 + 0.967562i \(0.581297\pi\)
\(354\) 0 0
\(355\) 30.5948 1.62380
\(356\) −31.3216 −1.66004
\(357\) 0 0
\(358\) 12.6564 0.668910
\(359\) −13.0490 −0.688702 −0.344351 0.938841i \(-0.611901\pi\)
−0.344351 + 0.938841i \(0.611901\pi\)
\(360\) 0 0
\(361\) 49.8613 2.62428
\(362\) 39.6262 2.08271
\(363\) 0 0
\(364\) −7.44999 −0.390486
\(365\) −39.1272 −2.04801
\(366\) 0 0
\(367\) −1.93419 −0.100964 −0.0504819 0.998725i \(-0.516076\pi\)
−0.0504819 + 0.998725i \(0.516076\pi\)
\(368\) −6.03243 −0.314462
\(369\) 0 0
\(370\) −28.5464 −1.48406
\(371\) 0.985474 0.0511633
\(372\) 0 0
\(373\) −23.8816 −1.23654 −0.618270 0.785966i \(-0.712166\pi\)
−0.618270 + 0.785966i \(0.712166\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −3.58918 −0.185098
\(377\) 27.1568 1.39865
\(378\) 0 0
\(379\) −38.0262 −1.95327 −0.976637 0.214895i \(-0.931059\pi\)
−0.976637 + 0.214895i \(0.931059\pi\)
\(380\) −56.8429 −2.91598
\(381\) 0 0
\(382\) 52.7049 2.69662
\(383\) −31.5678 −1.61304 −0.806519 0.591208i \(-0.798651\pi\)
−0.806519 + 0.591208i \(0.798651\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.0524 1.17334
\(387\) 0 0
\(388\) 7.71405 0.391621
\(389\) −10.1308 −0.513652 −0.256826 0.966458i \(-0.582677\pi\)
−0.256826 + 0.966458i \(0.582677\pi\)
\(390\) 0 0
\(391\) 9.36271 0.473493
\(392\) −0.492737 −0.0248870
\(393\) 0 0
\(394\) 35.9674 1.81201
\(395\) −44.9037 −2.25935
\(396\) 0 0
\(397\) −7.51188 −0.377010 −0.188505 0.982072i \(-0.560364\pi\)
−0.188505 + 0.982072i \(0.560364\pi\)
\(398\) 13.8354 0.693508
\(399\) 0 0
\(400\) −15.0939 −0.754695
\(401\) −19.2645 −0.962022 −0.481011 0.876715i \(-0.659730\pi\)
−0.481011 + 0.876715i \(0.659730\pi\)
\(402\) 0 0
\(403\) 18.1316 0.903201
\(404\) 20.9500 1.04230
\(405\) 0 0
\(406\) 16.8068 0.834108
\(407\) 0 0
\(408\) 0 0
\(409\) −13.0605 −0.645801 −0.322900 0.946433i \(-0.604658\pi\)
−0.322900 + 0.946433i \(0.604658\pi\)
\(410\) 43.1430 2.13068
\(411\) 0 0
\(412\) 28.4285 1.40057
\(413\) −4.52306 −0.222565
\(414\) 0 0
\(415\) −4.98589 −0.244748
\(416\) −27.0076 −1.32416
\(417\) 0 0
\(418\) 0 0
\(419\) 30.7589 1.50267 0.751335 0.659921i \(-0.229410\pi\)
0.751335 + 0.659921i \(0.229410\pi\)
\(420\) 0 0
\(421\) 16.0657 0.782993 0.391497 0.920180i \(-0.371957\pi\)
0.391497 + 0.920180i \(0.371957\pi\)
\(422\) −41.6513 −2.02755
\(423\) 0 0
\(424\) −0.485580 −0.0235818
\(425\) 23.4267 1.13636
\(426\) 0 0
\(427\) 12.0845 0.584812
\(428\) 26.3319 1.27280
\(429\) 0 0
\(430\) −11.6516 −0.561891
\(431\) 13.0752 0.629809 0.314904 0.949123i \(-0.398028\pi\)
0.314904 + 0.949123i \(0.398028\pi\)
\(432\) 0 0
\(433\) 12.9757 0.623572 0.311786 0.950152i \(-0.399073\pi\)
0.311786 + 0.950152i \(0.399073\pi\)
\(434\) 11.2213 0.538640
\(435\) 0 0
\(436\) −35.6135 −1.70558
\(437\) −14.4507 −0.691271
\(438\) 0 0
\(439\) 15.9709 0.762252 0.381126 0.924523i \(-0.375536\pi\)
0.381126 + 0.924523i \(0.375536\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.8289 1.75177
\(443\) 21.2695 1.01055 0.505273 0.862959i \(-0.331392\pi\)
0.505273 + 0.862959i \(0.331392\pi\)
\(444\) 0 0
\(445\) −42.7862 −2.02826
\(446\) −24.4409 −1.15731
\(447\) 0 0
\(448\) −9.78626 −0.462357
\(449\) 6.63021 0.312899 0.156449 0.987686i \(-0.449995\pi\)
0.156449 + 0.987686i \(0.449995\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −16.3641 −0.769702
\(453\) 0 0
\(454\) −28.7268 −1.34821
\(455\) −10.1769 −0.477099
\(456\) 0 0
\(457\) −17.1191 −0.800796 −0.400398 0.916341i \(-0.631128\pi\)
−0.400398 + 0.916341i \(0.631128\pi\)
\(458\) −51.1289 −2.38910
\(459\) 0 0
\(460\) 11.9286 0.556175
\(461\) 19.1173 0.890380 0.445190 0.895436i \(-0.353136\pi\)
0.445190 + 0.895436i \(0.353136\pi\)
\(462\) 0 0
\(463\) 16.0828 0.747433 0.373717 0.927543i \(-0.378083\pi\)
0.373717 + 0.927543i \(0.378083\pi\)
\(464\) 28.2767 1.31271
\(465\) 0 0
\(466\) 24.3503 1.12800
\(467\) 13.4920 0.624336 0.312168 0.950027i \(-0.398945\pi\)
0.312168 + 0.950027i \(0.398945\pi\)
\(468\) 0 0
\(469\) 0.170993 0.00789573
\(470\) −45.8776 −2.11618
\(471\) 0 0
\(472\) 2.22868 0.102583
\(473\) 0 0
\(474\) 0 0
\(475\) −36.1575 −1.65902
\(476\) 12.0397 0.551838
\(477\) 0 0
\(478\) 18.2531 0.834878
\(479\) 25.0471 1.14443 0.572215 0.820103i \(-0.306084\pi\)
0.572215 + 0.820103i \(0.306084\pi\)
\(480\) 0 0
\(481\) −15.0790 −0.687541
\(482\) 23.4560 1.06839
\(483\) 0 0
\(484\) 0 0
\(485\) 10.5376 0.478487
\(486\) 0 0
\(487\) −30.7888 −1.39517 −0.697587 0.716500i \(-0.745743\pi\)
−0.697587 + 0.716500i \(0.745743\pi\)
\(488\) −5.95450 −0.269548
\(489\) 0 0
\(490\) −6.29827 −0.284527
\(491\) 39.1200 1.76546 0.882731 0.469879i \(-0.155702\pi\)
0.882731 + 0.469879i \(0.155702\pi\)
\(492\) 0 0
\(493\) −43.8872 −1.97658
\(494\) −56.8429 −2.55748
\(495\) 0 0
\(496\) 18.8793 0.847707
\(497\) 10.0017 0.448637
\(498\) 0 0
\(499\) 13.5522 0.606682 0.303341 0.952882i \(-0.401898\pi\)
0.303341 + 0.952882i \(0.401898\pi\)
\(500\) −4.40294 −0.196905
\(501\) 0 0
\(502\) 20.0103 0.893105
\(503\) −18.1214 −0.807995 −0.403997 0.914760i \(-0.632380\pi\)
−0.403997 + 0.914760i \(0.632380\pi\)
\(504\) 0 0
\(505\) 28.6182 1.27349
\(506\) 0 0
\(507\) 0 0
\(508\) 9.69309 0.430061
\(509\) 40.4192 1.79155 0.895774 0.444510i \(-0.146622\pi\)
0.895774 + 0.444510i \(0.146622\pi\)
\(510\) 0 0
\(511\) −12.7910 −0.565841
\(512\) −31.5351 −1.39367
\(513\) 0 0
\(514\) −41.2730 −1.82047
\(515\) 38.8341 1.71123
\(516\) 0 0
\(517\) 0 0
\(518\) −9.33207 −0.410028
\(519\) 0 0
\(520\) 5.01453 0.219902
\(521\) 7.74795 0.339444 0.169722 0.985492i \(-0.445713\pi\)
0.169722 + 0.985492i \(0.445713\pi\)
\(522\) 0 0
\(523\) −21.6047 −0.944706 −0.472353 0.881409i \(-0.656595\pi\)
−0.472353 + 0.881409i \(0.656595\pi\)
\(524\) 19.5097 0.852286
\(525\) 0 0
\(526\) −26.6316 −1.16119
\(527\) −29.3019 −1.27641
\(528\) 0 0
\(529\) −19.9675 −0.868151
\(530\) −6.20679 −0.269606
\(531\) 0 0
\(532\) −18.5824 −0.805651
\(533\) 22.7892 0.987111
\(534\) 0 0
\(535\) 35.9700 1.55512
\(536\) −0.0842547 −0.00363925
\(537\) 0 0
\(538\) 9.16150 0.394980
\(539\) 0 0
\(540\) 0 0
\(541\) −35.2507 −1.51555 −0.757773 0.652519i \(-0.773712\pi\)
−0.757773 + 0.652519i \(0.773712\pi\)
\(542\) 19.7598 0.848754
\(543\) 0 0
\(544\) 43.6460 1.87131
\(545\) −48.6490 −2.08389
\(546\) 0 0
\(547\) −5.44775 −0.232929 −0.116465 0.993195i \(-0.537156\pi\)
−0.116465 + 0.993195i \(0.537156\pi\)
\(548\) 0.524333 0.0223984
\(549\) 0 0
\(550\) 0 0
\(551\) 67.7369 2.88569
\(552\) 0 0
\(553\) −14.6794 −0.624231
\(554\) −5.76585 −0.244968
\(555\) 0 0
\(556\) −43.2634 −1.83478
\(557\) 8.73429 0.370084 0.185042 0.982731i \(-0.440758\pi\)
0.185042 + 0.982731i \(0.440758\pi\)
\(558\) 0 0
\(559\) −6.15468 −0.260315
\(560\) −10.5965 −0.447786
\(561\) 0 0
\(562\) −31.8564 −1.34378
\(563\) −15.2200 −0.641448 −0.320724 0.947173i \(-0.603926\pi\)
−0.320724 + 0.947173i \(0.603926\pi\)
\(564\) 0 0
\(565\) −22.3538 −0.940430
\(566\) −53.3426 −2.24216
\(567\) 0 0
\(568\) −4.92820 −0.206783
\(569\) −0.0166300 −0.000697167 0 −0.000348583 1.00000i \(-0.500111\pi\)
−0.000348583 1.00000i \(0.500111\pi\)
\(570\) 0 0
\(571\) −6.08632 −0.254705 −0.127352 0.991858i \(-0.540648\pi\)
−0.127352 + 0.991858i \(0.540648\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 14.1038 0.588682
\(575\) 7.58774 0.316430
\(576\) 0 0
\(577\) −36.6359 −1.52517 −0.762585 0.646888i \(-0.776070\pi\)
−0.762585 + 0.646888i \(0.776070\pi\)
\(578\) −24.5157 −1.01972
\(579\) 0 0
\(580\) −55.9147 −2.32173
\(581\) −1.62993 −0.0676209
\(582\) 0 0
\(583\) 0 0
\(584\) 6.30261 0.260804
\(585\) 0 0
\(586\) −48.1382 −1.98857
\(587\) −5.52836 −0.228180 −0.114090 0.993470i \(-0.536395\pi\)
−0.114090 + 0.993470i \(0.536395\pi\)
\(588\) 0 0
\(589\) 45.2255 1.86349
\(590\) 28.4875 1.17281
\(591\) 0 0
\(592\) −15.7008 −0.645298
\(593\) −33.1440 −1.36106 −0.680531 0.732719i \(-0.738251\pi\)
−0.680531 + 0.732719i \(0.738251\pi\)
\(594\) 0 0
\(595\) 16.4465 0.674241
\(596\) −11.6464 −0.477057
\(597\) 0 0
\(598\) 11.9286 0.487798
\(599\) −18.8290 −0.769332 −0.384666 0.923056i \(-0.625683\pi\)
−0.384666 + 0.923056i \(0.625683\pi\)
\(600\) 0 0
\(601\) 7.88491 0.321632 0.160816 0.986984i \(-0.448587\pi\)
0.160816 + 0.986984i \(0.448587\pi\)
\(602\) −3.80901 −0.155244
\(603\) 0 0
\(604\) −14.2349 −0.579210
\(605\) 0 0
\(606\) 0 0
\(607\) −1.34563 −0.0546175 −0.0273087 0.999627i \(-0.508694\pi\)
−0.0273087 + 0.999627i \(0.508694\pi\)
\(608\) −67.3647 −2.73200
\(609\) 0 0
\(610\) −76.1117 −3.08167
\(611\) −24.2337 −0.980392
\(612\) 0 0
\(613\) −21.8217 −0.881372 −0.440686 0.897661i \(-0.645265\pi\)
−0.440686 + 0.897661i \(0.645265\pi\)
\(614\) −20.7424 −0.837094
\(615\) 0 0
\(616\) 0 0
\(617\) −30.1973 −1.21570 −0.607849 0.794053i \(-0.707967\pi\)
−0.607849 + 0.794053i \(0.707967\pi\)
\(618\) 0 0
\(619\) −39.2011 −1.57562 −0.787812 0.615915i \(-0.788786\pi\)
−0.787812 + 0.615915i \(0.788786\pi\)
\(620\) −37.3323 −1.49930
\(621\) 0 0
\(622\) 17.0073 0.681930
\(623\) −13.9872 −0.560384
\(624\) 0 0
\(625\) −27.8007 −1.11203
\(626\) 15.2196 0.608298
\(627\) 0 0
\(628\) −22.4247 −0.894844
\(629\) 24.3686 0.971640
\(630\) 0 0
\(631\) 5.44272 0.216671 0.108336 0.994114i \(-0.465448\pi\)
0.108336 + 0.994114i \(0.465448\pi\)
\(632\) 7.23308 0.287717
\(633\) 0 0
\(634\) 64.1572 2.54801
\(635\) 13.2410 0.525453
\(636\) 0 0
\(637\) −3.32691 −0.131817
\(638\) 0 0
\(639\) 0 0
\(640\) 11.9718 0.473226
\(641\) −45.6066 −1.80135 −0.900677 0.434489i \(-0.856929\pi\)
−0.900677 + 0.434489i \(0.856929\pi\)
\(642\) 0 0
\(643\) 21.2599 0.838407 0.419204 0.907892i \(-0.362309\pi\)
0.419204 + 0.907892i \(0.362309\pi\)
\(644\) 3.89957 0.153664
\(645\) 0 0
\(646\) 91.8620 3.61426
\(647\) 18.5756 0.730283 0.365141 0.930952i \(-0.381021\pi\)
0.365141 + 0.930952i \(0.381021\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 29.8469 1.17069
\(651\) 0 0
\(652\) −19.0417 −0.745731
\(653\) 32.1882 1.25962 0.629812 0.776748i \(-0.283132\pi\)
0.629812 + 0.776748i \(0.283132\pi\)
\(654\) 0 0
\(655\) 26.6508 1.04133
\(656\) 23.7290 0.926461
\(657\) 0 0
\(658\) −14.9978 −0.584675
\(659\) 40.6203 1.58234 0.791172 0.611594i \(-0.209471\pi\)
0.791172 + 0.611594i \(0.209471\pi\)
\(660\) 0 0
\(661\) 7.69278 0.299215 0.149607 0.988746i \(-0.452199\pi\)
0.149607 + 0.988746i \(0.452199\pi\)
\(662\) 41.2019 1.60136
\(663\) 0 0
\(664\) 0.803127 0.0311674
\(665\) −25.3841 −0.984352
\(666\) 0 0
\(667\) −14.2147 −0.550397
\(668\) 6.57813 0.254515
\(669\) 0 0
\(670\) −1.07696 −0.0416066
\(671\) 0 0
\(672\) 0 0
\(673\) −7.24740 −0.279367 −0.139683 0.990196i \(-0.544608\pi\)
−0.139683 + 0.990196i \(0.544608\pi\)
\(674\) −42.2030 −1.62560
\(675\) 0 0
\(676\) −4.32564 −0.166371
\(677\) 6.52296 0.250698 0.125349 0.992113i \(-0.459995\pi\)
0.125349 + 0.992113i \(0.459995\pi\)
\(678\) 0 0
\(679\) 3.44483 0.132200
\(680\) −8.10381 −0.310767
\(681\) 0 0
\(682\) 0 0
\(683\) −2.12170 −0.0811846 −0.0405923 0.999176i \(-0.512924\pi\)
−0.0405923 + 0.999176i \(0.512924\pi\)
\(684\) 0 0
\(685\) 0.716253 0.0273666
\(686\) −2.05896 −0.0786114
\(687\) 0 0
\(688\) −6.40848 −0.244321
\(689\) −3.27858 −0.124904
\(690\) 0 0
\(691\) −5.66834 −0.215634 −0.107817 0.994171i \(-0.534386\pi\)
−0.107817 + 0.994171i \(0.534386\pi\)
\(692\) −46.3183 −1.76076
\(693\) 0 0
\(694\) −69.5845 −2.64139
\(695\) −59.0990 −2.24175
\(696\) 0 0
\(697\) −36.8289 −1.39499
\(698\) 67.9518 2.57201
\(699\) 0 0
\(700\) 9.75721 0.368788
\(701\) 8.22087 0.310498 0.155249 0.987875i \(-0.450382\pi\)
0.155249 + 0.987875i \(0.450382\pi\)
\(702\) 0 0
\(703\) −37.6113 −1.41854
\(704\) 0 0
\(705\) 0 0
\(706\) 19.5460 0.735624
\(707\) 9.35554 0.351851
\(708\) 0 0
\(709\) −6.38116 −0.239649 −0.119825 0.992795i \(-0.538233\pi\)
−0.119825 + 0.992795i \(0.538233\pi\)
\(710\) −62.9934 −2.36410
\(711\) 0 0
\(712\) 6.89199 0.258288
\(713\) −9.49068 −0.355429
\(714\) 0 0
\(715\) 0 0
\(716\) −13.7650 −0.514422
\(717\) 0 0
\(718\) 26.8675 1.00268
\(719\) −22.6376 −0.844241 −0.422121 0.906540i \(-0.638714\pi\)
−0.422121 + 0.906540i \(0.638714\pi\)
\(720\) 0 0
\(721\) 12.6952 0.472794
\(722\) −102.662 −3.82070
\(723\) 0 0
\(724\) −43.0973 −1.60170
\(725\) −35.5671 −1.32093
\(726\) 0 0
\(727\) 24.6482 0.914153 0.457076 0.889427i \(-0.348897\pi\)
0.457076 + 0.889427i \(0.348897\pi\)
\(728\) 1.63929 0.0607562
\(729\) 0 0
\(730\) 80.5613 2.98171
\(731\) 9.94637 0.367880
\(732\) 0 0
\(733\) 42.4229 1.56693 0.783464 0.621438i \(-0.213451\pi\)
0.783464 + 0.621438i \(0.213451\pi\)
\(734\) 3.98241 0.146994
\(735\) 0 0
\(736\) 14.1366 0.521084
\(737\) 0 0
\(738\) 0 0
\(739\) −17.2443 −0.634343 −0.317172 0.948368i \(-0.602733\pi\)
−0.317172 + 0.948368i \(0.602733\pi\)
\(740\) 31.0470 1.14131
\(741\) 0 0
\(742\) −2.02905 −0.0744888
\(743\) 31.2944 1.14808 0.574040 0.818827i \(-0.305375\pi\)
0.574040 + 0.818827i \(0.305375\pi\)
\(744\) 0 0
\(745\) −15.9093 −0.582873
\(746\) 49.1712 1.80028
\(747\) 0 0
\(748\) 0 0
\(749\) 11.7589 0.429661
\(750\) 0 0
\(751\) −31.4340 −1.14704 −0.573521 0.819191i \(-0.694423\pi\)
−0.573521 + 0.819191i \(0.694423\pi\)
\(752\) −25.2331 −0.920156
\(753\) 0 0
\(754\) −55.9147 −2.03629
\(755\) −19.4452 −0.707685
\(756\) 0 0
\(757\) 25.3641 0.921873 0.460937 0.887433i \(-0.347514\pi\)
0.460937 + 0.887433i \(0.347514\pi\)
\(758\) 78.2944 2.84378
\(759\) 0 0
\(760\) 12.5077 0.453701
\(761\) 10.0619 0.364744 0.182372 0.983230i \(-0.441622\pi\)
0.182372 + 0.983230i \(0.441622\pi\)
\(762\) 0 0
\(763\) −15.9038 −0.575755
\(764\) −57.3215 −2.07382
\(765\) 0 0
\(766\) 64.9968 2.34843
\(767\) 15.0478 0.543345
\(768\) 0 0
\(769\) 0.408482 0.0147303 0.00736513 0.999973i \(-0.497656\pi\)
0.00736513 + 0.999973i \(0.497656\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25.0717 −0.902350
\(773\) −11.9310 −0.429129 −0.214565 0.976710i \(-0.568833\pi\)
−0.214565 + 0.976710i \(0.568833\pi\)
\(774\) 0 0
\(775\) −23.7469 −0.853013
\(776\) −1.69739 −0.0609329
\(777\) 0 0
\(778\) 20.8589 0.747827
\(779\) 56.8429 2.03661
\(780\) 0 0
\(781\) 0 0
\(782\) −19.2774 −0.689360
\(783\) 0 0
\(784\) −3.46410 −0.123718
\(785\) −30.6328 −1.09333
\(786\) 0 0
\(787\) −20.3396 −0.725030 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(788\) −39.1179 −1.39352
\(789\) 0 0
\(790\) 92.4548 3.28939
\(791\) −7.30763 −0.259830
\(792\) 0 0
\(793\) −40.2041 −1.42769
\(794\) 15.4666 0.548891
\(795\) 0 0
\(796\) −15.0474 −0.533339
\(797\) −6.82763 −0.241847 −0.120924 0.992662i \(-0.538586\pi\)
−0.120924 + 0.992662i \(0.538586\pi\)
\(798\) 0 0
\(799\) 39.1634 1.38550
\(800\) 35.3717 1.25058
\(801\) 0 0
\(802\) 39.6648 1.40061
\(803\) 0 0
\(804\) 0 0
\(805\) 5.32691 0.187749
\(806\) −37.3323 −1.31497
\(807\) 0 0
\(808\) −4.60982 −0.162173
\(809\) 15.3890 0.541047 0.270524 0.962713i \(-0.412803\pi\)
0.270524 + 0.962713i \(0.412803\pi\)
\(810\) 0 0
\(811\) −9.42694 −0.331025 −0.165512 0.986208i \(-0.552928\pi\)
−0.165512 + 0.986208i \(0.552928\pi\)
\(812\) −18.2790 −0.641467
\(813\) 0 0
\(814\) 0 0
\(815\) −26.0115 −0.911142
\(816\) 0 0
\(817\) −15.3516 −0.537083
\(818\) 26.8911 0.940224
\(819\) 0 0
\(820\) −46.9221 −1.63859
\(821\) 32.3301 1.12833 0.564165 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(822\) 0 0
\(823\) −18.3223 −0.638676 −0.319338 0.947641i \(-0.603461\pi\)
−0.319338 + 0.947641i \(0.603461\pi\)
\(824\) −6.25539 −0.217917
\(825\) 0 0
\(826\) 9.31280 0.324034
\(827\) −6.37527 −0.221690 −0.110845 0.993838i \(-0.535356\pi\)
−0.110845 + 0.993838i \(0.535356\pi\)
\(828\) 0 0
\(829\) 0.949545 0.0329791 0.0164895 0.999864i \(-0.494751\pi\)
0.0164895 + 0.999864i \(0.494751\pi\)
\(830\) 10.2657 0.356329
\(831\) 0 0
\(832\) 32.5580 1.12875
\(833\) 5.37651 0.186285
\(834\) 0 0
\(835\) 8.98589 0.310970
\(836\) 0 0
\(837\) 0 0
\(838\) −63.3313 −2.18774
\(839\) −14.3937 −0.496925 −0.248462 0.968642i \(-0.579925\pi\)
−0.248462 + 0.968642i \(0.579925\pi\)
\(840\) 0 0
\(841\) 37.6308 1.29761
\(842\) −33.0786 −1.13996
\(843\) 0 0
\(844\) 45.2998 1.55928
\(845\) −5.90893 −0.203273
\(846\) 0 0
\(847\) 0 0
\(848\) −3.41378 −0.117230
\(849\) 0 0
\(850\) −48.2346 −1.65443
\(851\) 7.89281 0.270562
\(852\) 0 0
\(853\) 39.9349 1.36735 0.683673 0.729788i \(-0.260381\pi\)
0.683673 + 0.729788i \(0.260381\pi\)
\(854\) −24.8816 −0.851430
\(855\) 0 0
\(856\) −5.79405 −0.198036
\(857\) −52.2332 −1.78425 −0.892127 0.451786i \(-0.850787\pi\)
−0.892127 + 0.451786i \(0.850787\pi\)
\(858\) 0 0
\(859\) −37.1209 −1.26655 −0.633274 0.773928i \(-0.718289\pi\)
−0.633274 + 0.773928i \(0.718289\pi\)
\(860\) 12.6722 0.432120
\(861\) 0 0
\(862\) −26.9213 −0.916941
\(863\) 34.0072 1.15762 0.578809 0.815463i \(-0.303518\pi\)
0.578809 + 0.815463i \(0.303518\pi\)
\(864\) 0 0
\(865\) −63.2719 −2.15131
\(866\) −26.7164 −0.907861
\(867\) 0 0
\(868\) −12.2042 −0.414239
\(869\) 0 0
\(870\) 0 0
\(871\) −0.568878 −0.0192757
\(872\) 7.83638 0.265373
\(873\) 0 0
\(874\) 29.7534 1.00642
\(875\) −1.96620 −0.0664697
\(876\) 0 0
\(877\) −45.3869 −1.53261 −0.766303 0.642479i \(-0.777906\pi\)
−0.766303 + 0.642479i \(0.777906\pi\)
\(878\) −32.8835 −1.10977
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0836 −0.609251 −0.304626 0.952472i \(-0.598531\pi\)
−0.304626 + 0.952472i \(0.598531\pi\)
\(882\) 0 0
\(883\) 14.8564 0.499958 0.249979 0.968251i \(-0.419576\pi\)
0.249979 + 0.968251i \(0.419576\pi\)
\(884\) −40.0549 −1.34719
\(885\) 0 0
\(886\) −43.7931 −1.47126
\(887\) −29.3108 −0.984162 −0.492081 0.870549i \(-0.663764\pi\)
−0.492081 + 0.870549i \(0.663764\pi\)
\(888\) 0 0
\(889\) 4.32860 0.145176
\(890\) 88.0950 2.95295
\(891\) 0 0
\(892\) 26.5818 0.890025
\(893\) −60.4460 −2.02275
\(894\) 0 0
\(895\) −18.8033 −0.628526
\(896\) 3.91368 0.130747
\(897\) 0 0
\(898\) −13.6513 −0.455551
\(899\) 44.4870 1.48373
\(900\) 0 0
\(901\) 5.29841 0.176516
\(902\) 0 0
\(903\) 0 0
\(904\) 3.60074 0.119759
\(905\) −58.8720 −1.95697
\(906\) 0 0
\(907\) 0.784709 0.0260558 0.0130279 0.999915i \(-0.495853\pi\)
0.0130279 + 0.999915i \(0.495853\pi\)
\(908\) 31.2431 1.03684
\(909\) 0 0
\(910\) 20.9538 0.694611
\(911\) −49.3824 −1.63611 −0.818056 0.575139i \(-0.804948\pi\)
−0.818056 + 0.575139i \(0.804948\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 35.2474 1.16588
\(915\) 0 0
\(916\) 55.6075 1.83732
\(917\) 8.71236 0.287707
\(918\) 0 0
\(919\) −1.20242 −0.0396641 −0.0198320 0.999803i \(-0.506313\pi\)
−0.0198320 + 0.999803i \(0.506313\pi\)
\(920\) −2.62477 −0.0865360
\(921\) 0 0
\(922\) −39.3617 −1.29631
\(923\) −33.2747 −1.09525
\(924\) 0 0
\(925\) 19.7488 0.649337
\(926\) −33.1139 −1.08819
\(927\) 0 0
\(928\) −66.2647 −2.17525
\(929\) 16.9944 0.557569 0.278785 0.960354i \(-0.410068\pi\)
0.278785 + 0.960354i \(0.410068\pi\)
\(930\) 0 0
\(931\) −8.29827 −0.271965
\(932\) −26.4832 −0.867487
\(933\) 0 0
\(934\) −27.7795 −0.908973
\(935\) 0 0
\(936\) 0 0
\(937\) 28.4196 0.928427 0.464213 0.885723i \(-0.346337\pi\)
0.464213 + 0.885723i \(0.346337\pi\)
\(938\) −0.352068 −0.0114954
\(939\) 0 0
\(940\) 49.8963 1.62744
\(941\) −27.9160 −0.910034 −0.455017 0.890483i \(-0.650367\pi\)
−0.455017 + 0.890483i \(0.650367\pi\)
\(942\) 0 0
\(943\) −11.9286 −0.388449
\(944\) 15.6683 0.509961
\(945\) 0 0
\(946\) 0 0
\(947\) −34.5418 −1.12246 −0.561229 0.827660i \(-0.689671\pi\)
−0.561229 + 0.827660i \(0.689671\pi\)
\(948\) 0 0
\(949\) 42.5545 1.38138
\(950\) 74.4468 2.41537
\(951\) 0 0
\(952\) −2.64920 −0.0858612
\(953\) −13.8398 −0.448314 −0.224157 0.974553i \(-0.571963\pi\)
−0.224157 + 0.974553i \(0.571963\pi\)
\(954\) 0 0
\(955\) −78.3027 −2.53382
\(956\) −19.8520 −0.642060
\(957\) 0 0
\(958\) −51.5709 −1.66618
\(959\) 0.234149 0.00756107
\(960\) 0 0
\(961\) −1.29759 −0.0418577
\(962\) 31.0470 1.00099
\(963\) 0 0
\(964\) −25.5106 −0.821642
\(965\) −34.2486 −1.10250
\(966\) 0 0
\(967\) 7.67468 0.246801 0.123401 0.992357i \(-0.460620\pi\)
0.123401 + 0.992357i \(0.460620\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −21.6965 −0.696631
\(971\) −6.99587 −0.224508 −0.112254 0.993680i \(-0.535807\pi\)
−0.112254 + 0.993680i \(0.535807\pi\)
\(972\) 0 0
\(973\) −19.3200 −0.619369
\(974\) 63.3929 2.03124
\(975\) 0 0
\(976\) −41.8621 −1.33997
\(977\) 11.0312 0.352918 0.176459 0.984308i \(-0.443536\pi\)
0.176459 + 0.984308i \(0.443536\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.84997 0.218814
\(981\) 0 0
\(982\) −80.5465 −2.57034
\(983\) −13.4873 −0.430178 −0.215089 0.976594i \(-0.569004\pi\)
−0.215089 + 0.976594i \(0.569004\pi\)
\(984\) 0 0
\(985\) −53.4361 −1.70262
\(986\) 90.3619 2.87771
\(987\) 0 0
\(988\) 61.8221 1.96682
\(989\) 3.22156 0.102440
\(990\) 0 0
\(991\) 0.759034 0.0241115 0.0120558 0.999927i \(-0.496162\pi\)
0.0120558 + 0.999927i \(0.496162\pi\)
\(992\) −44.2426 −1.40470
\(993\) 0 0
\(994\) −20.5931 −0.653173
\(995\) −20.5551 −0.651640
\(996\) 0 0
\(997\) −47.7793 −1.51319 −0.756593 0.653886i \(-0.773138\pi\)
−0.756593 + 0.653886i \(0.773138\pi\)
\(998\) −27.9035 −0.883271
\(999\) 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.cn.1.1 4
3.2 odd 2 2541.2.a.bl.1.4 4
11.10 odd 2 7623.2.a.cg.1.4 4
33.32 even 2 2541.2.a.bp.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bl.1.4 4 3.2 odd 2
2541.2.a.bp.1.1 yes 4 33.32 even 2
7623.2.a.cg.1.4 4 11.10 odd 2
7623.2.a.cn.1.1 4 1.1 even 1 trivial