Properties

Label 7623.2.a.cp.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 6
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: \(6\)
Coefficient field: 6.6.7674048.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.70320\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} +1.00000 q^{7} -8.94020 q^{8} +O(q^{10})\) \(q-2.70320 q^{2} +5.30727 q^{4} +0.445072 q^{5} +1.00000 q^{7} -8.94020 q^{8} -1.20312 q^{10} +0.450933 q^{13} -2.70320 q^{14} +13.5526 q^{16} -4.83117 q^{17} -1.08644 q^{19} +2.36212 q^{20} -4.57222 q^{23} -4.80191 q^{25} -1.21896 q^{26} +5.30727 q^{28} +1.98431 q^{29} +8.25861 q^{31} -18.7549 q^{32} +13.0596 q^{34} +0.445072 q^{35} +7.31725 q^{37} +2.93685 q^{38} -3.97903 q^{40} -1.77073 q^{41} -11.4084 q^{43} +12.3596 q^{46} -1.02259 q^{47} +1.00000 q^{49} +12.9805 q^{50} +2.39322 q^{52} +3.57222 q^{53} -8.94020 q^{56} -5.36399 q^{58} +14.3996 q^{59} +4.92965 q^{61} -22.3246 q^{62} +23.5929 q^{64} +0.200698 q^{65} -6.18858 q^{67} -25.6403 q^{68} -1.20312 q^{70} +5.92165 q^{71} -1.65776 q^{73} -19.7800 q^{74} -5.76602 q^{76} -3.60833 q^{79} +6.03187 q^{80} +4.78663 q^{82} +10.8048 q^{83} -2.15022 q^{85} +30.8392 q^{86} +5.21170 q^{89} +0.450933 q^{91} -24.2660 q^{92} +2.76425 q^{94} -0.483543 q^{95} -5.30985 q^{97} -2.70320 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} + O(q^{10}) \) \( 6q - 4q^{2} + 4q^{4} + 4q^{5} + 6q^{7} - 12q^{8} - 8q^{10} + 4q^{13} - 4q^{14} + 8q^{16} - 22q^{17} + 6q^{19} - 2q^{20} - 2q^{23} + 4q^{25} - 6q^{26} + 4q^{28} - 12q^{29} - 2q^{31} - 8q^{32} + 24q^{34} + 4q^{35} + 14q^{37} + 22q^{38} + 18q^{40} - 26q^{41} - 4q^{43} + 12q^{46} + 16q^{47} + 6q^{49} + 4q^{50} + 12q^{52} - 4q^{53} - 12q^{56} - 2q^{58} + 4q^{59} - 8q^{61} - 20q^{62} + 26q^{64} - 24q^{65} + 6q^{67} - 12q^{68} - 8q^{70} - 22q^{71} + 14q^{73} - 44q^{74} - 30q^{76} - 28q^{79} + 4q^{80} - 4q^{82} - 22q^{83} - 24q^{85} + 30q^{86} + 4q^{91} - 10q^{92} - 38q^{94} + 24q^{95} - 4q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70320 −1.91145 −0.955724 0.294264i \(-0.904925\pi\)
−0.955724 + 0.294264i \(0.904925\pi\)
\(3\) 0 0
\(4\) 5.30727 2.65363
\(5\) 0.445072 0.199042 0.0995212 0.995035i \(-0.468269\pi\)
0.0995212 + 0.995035i \(0.468269\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −8.94020 −3.16084
\(9\) 0 0
\(10\) −1.20312 −0.380459
\(11\) 0 0
\(12\) 0 0
\(13\) 0.450933 0.125066 0.0625332 0.998043i \(-0.480082\pi\)
0.0625332 + 0.998043i \(0.480082\pi\)
\(14\) −2.70320 −0.722460
\(15\) 0 0
\(16\) 13.5526 3.38814
\(17\) −4.83117 −1.17173 −0.585866 0.810408i \(-0.699245\pi\)
−0.585866 + 0.810408i \(0.699245\pi\)
\(18\) 0 0
\(19\) −1.08644 −0.249246 −0.124623 0.992204i \(-0.539772\pi\)
−0.124623 + 0.992204i \(0.539772\pi\)
\(20\) 2.36212 0.528186
\(21\) 0 0
\(22\) 0 0
\(23\) −4.57222 −0.953373 −0.476687 0.879073i \(-0.658162\pi\)
−0.476687 + 0.879073i \(0.658162\pi\)
\(24\) 0 0
\(25\) −4.80191 −0.960382
\(26\) −1.21896 −0.239058
\(27\) 0 0
\(28\) 5.30727 1.00298
\(29\) 1.98431 0.368478 0.184239 0.982881i \(-0.441018\pi\)
0.184239 + 0.982881i \(0.441018\pi\)
\(30\) 0 0
\(31\) 8.25861 1.48329 0.741645 0.670793i \(-0.234046\pi\)
0.741645 + 0.670793i \(0.234046\pi\)
\(32\) −18.7549 −3.31542
\(33\) 0 0
\(34\) 13.0596 2.23970
\(35\) 0.445072 0.0752309
\(36\) 0 0
\(37\) 7.31725 1.20295 0.601474 0.798892i \(-0.294580\pi\)
0.601474 + 0.798892i \(0.294580\pi\)
\(38\) 2.93685 0.476421
\(39\) 0 0
\(40\) −3.97903 −0.629140
\(41\) −1.77073 −0.276542 −0.138271 0.990394i \(-0.544154\pi\)
−0.138271 + 0.990394i \(0.544154\pi\)
\(42\) 0 0
\(43\) −11.4084 −1.73977 −0.869884 0.493257i \(-0.835806\pi\)
−0.869884 + 0.493257i \(0.835806\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.3596 1.82232
\(47\) −1.02259 −0.149159 −0.0745797 0.997215i \(-0.523762\pi\)
−0.0745797 + 0.997215i \(0.523762\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 12.9805 1.83572
\(51\) 0 0
\(52\) 2.39322 0.331880
\(53\) 3.57222 0.490682 0.245341 0.969437i \(-0.421100\pi\)
0.245341 + 0.969437i \(0.421100\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.94020 −1.19468
\(57\) 0 0
\(58\) −5.36399 −0.704326
\(59\) 14.3996 1.87467 0.937333 0.348435i \(-0.113287\pi\)
0.937333 + 0.348435i \(0.113287\pi\)
\(60\) 0 0
\(61\) 4.92965 0.631177 0.315588 0.948896i \(-0.397798\pi\)
0.315588 + 0.948896i \(0.397798\pi\)
\(62\) −22.3246 −2.83523
\(63\) 0 0
\(64\) 23.5929 2.94911
\(65\) 0.200698 0.0248935
\(66\) 0 0
\(67\) −6.18858 −0.756055 −0.378028 0.925794i \(-0.623398\pi\)
−0.378028 + 0.925794i \(0.623398\pi\)
\(68\) −25.6403 −3.10935
\(69\) 0 0
\(70\) −1.20312 −0.143800
\(71\) 5.92165 0.702771 0.351385 0.936231i \(-0.385711\pi\)
0.351385 + 0.936231i \(0.385711\pi\)
\(72\) 0 0
\(73\) −1.65776 −0.194027 −0.0970133 0.995283i \(-0.530929\pi\)
−0.0970133 + 0.995283i \(0.530929\pi\)
\(74\) −19.7800 −2.29937
\(75\) 0 0
\(76\) −5.76602 −0.661407
\(77\) 0 0
\(78\) 0 0
\(79\) −3.60833 −0.405969 −0.202984 0.979182i \(-0.565064\pi\)
−0.202984 + 0.979182i \(0.565064\pi\)
\(80\) 6.03187 0.674384
\(81\) 0 0
\(82\) 4.78663 0.528595
\(83\) 10.8048 1.18598 0.592992 0.805208i \(-0.297947\pi\)
0.592992 + 0.805208i \(0.297947\pi\)
\(84\) 0 0
\(85\) −2.15022 −0.233224
\(86\) 30.8392 3.32548
\(87\) 0 0
\(88\) 0 0
\(89\) 5.21170 0.552439 0.276220 0.961095i \(-0.410918\pi\)
0.276220 + 0.961095i \(0.410918\pi\)
\(90\) 0 0
\(91\) 0.450933 0.0472706
\(92\) −24.2660 −2.52990
\(93\) 0 0
\(94\) 2.76425 0.285110
\(95\) −0.483543 −0.0496105
\(96\) 0 0
\(97\) −5.30985 −0.539133 −0.269567 0.962982i \(-0.586880\pi\)
−0.269567 + 0.962982i \(0.586880\pi\)
\(98\) −2.70320 −0.273064
\(99\) 0 0
\(100\) −25.4850 −2.54850
\(101\) −15.5604 −1.54832 −0.774158 0.632992i \(-0.781826\pi\)
−0.774158 + 0.632992i \(0.781826\pi\)
\(102\) 0 0
\(103\) −14.1713 −1.39634 −0.698172 0.715930i \(-0.746003\pi\)
−0.698172 + 0.715930i \(0.746003\pi\)
\(104\) −4.03143 −0.395314
\(105\) 0 0
\(106\) −9.65640 −0.937913
\(107\) 11.7551 1.13641 0.568205 0.822887i \(-0.307638\pi\)
0.568205 + 0.822887i \(0.307638\pi\)
\(108\) 0 0
\(109\) 15.7800 1.51145 0.755723 0.654891i \(-0.227286\pi\)
0.755723 + 0.654891i \(0.227286\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 13.5526 1.28060
\(113\) 5.92347 0.557233 0.278617 0.960402i \(-0.410124\pi\)
0.278617 + 0.960402i \(0.410124\pi\)
\(114\) 0 0
\(115\) −2.03497 −0.189762
\(116\) 10.5313 0.977805
\(117\) 0 0
\(118\) −38.9249 −3.58333
\(119\) −4.83117 −0.442873
\(120\) 0 0
\(121\) 0 0
\(122\) −13.3258 −1.20646
\(123\) 0 0
\(124\) 43.8306 3.93611
\(125\) −4.36256 −0.390199
\(126\) 0 0
\(127\) 10.1337 0.899222 0.449611 0.893224i \(-0.351563\pi\)
0.449611 + 0.893224i \(0.351563\pi\)
\(128\) −26.2666 −2.32166
\(129\) 0 0
\(130\) −0.542526 −0.0475826
\(131\) 6.83340 0.597037 0.298519 0.954404i \(-0.403508\pi\)
0.298519 + 0.954404i \(0.403508\pi\)
\(132\) 0 0
\(133\) −1.08644 −0.0942061
\(134\) 16.7289 1.44516
\(135\) 0 0
\(136\) 43.1916 3.70365
\(137\) −14.4673 −1.23602 −0.618012 0.786169i \(-0.712062\pi\)
−0.618012 + 0.786169i \(0.712062\pi\)
\(138\) 0 0
\(139\) −14.6957 −1.24647 −0.623235 0.782035i \(-0.714182\pi\)
−0.623235 + 0.782035i \(0.714182\pi\)
\(140\) 2.36212 0.199635
\(141\) 0 0
\(142\) −16.0074 −1.34331
\(143\) 0 0
\(144\) 0 0
\(145\) 0.883163 0.0733427
\(146\) 4.48126 0.370872
\(147\) 0 0
\(148\) 38.8346 3.19219
\(149\) −1.00140 −0.0820377 −0.0410189 0.999158i \(-0.513060\pi\)
−0.0410189 + 0.999158i \(0.513060\pi\)
\(150\) 0 0
\(151\) 1.37539 0.111927 0.0559636 0.998433i \(-0.482177\pi\)
0.0559636 + 0.998433i \(0.482177\pi\)
\(152\) 9.71297 0.787826
\(153\) 0 0
\(154\) 0 0
\(155\) 3.67568 0.295237
\(156\) 0 0
\(157\) 5.37668 0.429106 0.214553 0.976712i \(-0.431171\pi\)
0.214553 + 0.976712i \(0.431171\pi\)
\(158\) 9.75403 0.775989
\(159\) 0 0
\(160\) −8.34726 −0.659909
\(161\) −4.57222 −0.360341
\(162\) 0 0
\(163\) −9.42513 −0.738233 −0.369116 0.929383i \(-0.620340\pi\)
−0.369116 + 0.929383i \(0.620340\pi\)
\(164\) −9.39775 −0.733841
\(165\) 0 0
\(166\) −29.2076 −2.26695
\(167\) −20.0118 −1.54856 −0.774281 0.632842i \(-0.781888\pi\)
−0.774281 + 0.632842i \(0.781888\pi\)
\(168\) 0 0
\(169\) −12.7967 −0.984358
\(170\) 5.81247 0.445796
\(171\) 0 0
\(172\) −60.5476 −4.61671
\(173\) −10.8465 −0.824647 −0.412324 0.911037i \(-0.635283\pi\)
−0.412324 + 0.911037i \(0.635283\pi\)
\(174\) 0 0
\(175\) −4.80191 −0.362990
\(176\) 0 0
\(177\) 0 0
\(178\) −14.0883 −1.05596
\(179\) −16.8484 −1.25931 −0.629654 0.776876i \(-0.716803\pi\)
−0.629654 + 0.776876i \(0.716803\pi\)
\(180\) 0 0
\(181\) −20.3041 −1.50919 −0.754597 0.656188i \(-0.772168\pi\)
−0.754597 + 0.656188i \(0.772168\pi\)
\(182\) −1.21896 −0.0903554
\(183\) 0 0
\(184\) 40.8765 3.01346
\(185\) 3.25671 0.239438
\(186\) 0 0
\(187\) 0 0
\(188\) −5.42714 −0.395815
\(189\) 0 0
\(190\) 1.30711 0.0948279
\(191\) −2.54435 −0.184103 −0.0920513 0.995754i \(-0.529342\pi\)
−0.0920513 + 0.995754i \(0.529342\pi\)
\(192\) 0 0
\(193\) 17.5086 1.26030 0.630148 0.776475i \(-0.282994\pi\)
0.630148 + 0.776475i \(0.282994\pi\)
\(194\) 14.3536 1.03053
\(195\) 0 0
\(196\) 5.30727 0.379091
\(197\) 2.16558 0.154291 0.0771457 0.997020i \(-0.475419\pi\)
0.0771457 + 0.997020i \(0.475419\pi\)
\(198\) 0 0
\(199\) −14.5756 −1.03324 −0.516620 0.856215i \(-0.672810\pi\)
−0.516620 + 0.856215i \(0.672810\pi\)
\(200\) 42.9300 3.03561
\(201\) 0 0
\(202\) 42.0628 2.95953
\(203\) 1.98431 0.139272
\(204\) 0 0
\(205\) −0.788103 −0.0550435
\(206\) 38.3079 2.66904
\(207\) 0 0
\(208\) 6.11130 0.423743
\(209\) 0 0
\(210\) 0 0
\(211\) −3.63034 −0.249923 −0.124961 0.992162i \(-0.539881\pi\)
−0.124961 + 0.992162i \(0.539881\pi\)
\(212\) 18.9587 1.30209
\(213\) 0 0
\(214\) −31.7764 −2.17219
\(215\) −5.07757 −0.346287
\(216\) 0 0
\(217\) 8.25861 0.560631
\(218\) −42.6563 −2.88905
\(219\) 0 0
\(220\) 0 0
\(221\) −2.17854 −0.146544
\(222\) 0 0
\(223\) −4.84062 −0.324152 −0.162076 0.986778i \(-0.551819\pi\)
−0.162076 + 0.986778i \(0.551819\pi\)
\(224\) −18.7549 −1.25311
\(225\) 0 0
\(226\) −16.0123 −1.06512
\(227\) −22.7306 −1.50868 −0.754342 0.656482i \(-0.772044\pi\)
−0.754342 + 0.656482i \(0.772044\pi\)
\(228\) 0 0
\(229\) −22.6732 −1.49828 −0.749142 0.662409i \(-0.769534\pi\)
−0.749142 + 0.662409i \(0.769534\pi\)
\(230\) 5.50092 0.362720
\(231\) 0 0
\(232\) −17.7402 −1.16470
\(233\) 0.587282 0.0384741 0.0192371 0.999815i \(-0.493876\pi\)
0.0192371 + 0.999815i \(0.493876\pi\)
\(234\) 0 0
\(235\) −0.455124 −0.0296890
\(236\) 76.4225 4.97468
\(237\) 0 0
\(238\) 13.0596 0.846529
\(239\) 14.9721 0.968465 0.484233 0.874939i \(-0.339099\pi\)
0.484233 + 0.874939i \(0.339099\pi\)
\(240\) 0 0
\(241\) −18.2746 −1.17717 −0.588586 0.808435i \(-0.700315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 26.1630 1.67491
\(245\) 0.445072 0.0284346
\(246\) 0 0
\(247\) −0.489911 −0.0311723
\(248\) −73.8336 −4.68844
\(249\) 0 0
\(250\) 11.7929 0.745845
\(251\) −10.5649 −0.666850 −0.333425 0.942777i \(-0.608204\pi\)
−0.333425 + 0.942777i \(0.608204\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −27.3934 −1.71882
\(255\) 0 0
\(256\) 23.8178 1.48861
\(257\) 19.4680 1.21438 0.607189 0.794557i \(-0.292297\pi\)
0.607189 + 0.794557i \(0.292297\pi\)
\(258\) 0 0
\(259\) 7.31725 0.454672
\(260\) 1.06516 0.0660583
\(261\) 0 0
\(262\) −18.4720 −1.14121
\(263\) −21.1885 −1.30654 −0.653269 0.757126i \(-0.726603\pi\)
−0.653269 + 0.757126i \(0.726603\pi\)
\(264\) 0 0
\(265\) 1.58989 0.0976665
\(266\) 2.93685 0.180070
\(267\) 0 0
\(268\) −32.8444 −2.00629
\(269\) −23.1564 −1.41187 −0.705935 0.708276i \(-0.749473\pi\)
−0.705935 + 0.708276i \(0.749473\pi\)
\(270\) 0 0
\(271\) 26.8232 1.62939 0.814696 0.579888i \(-0.196904\pi\)
0.814696 + 0.579888i \(0.196904\pi\)
\(272\) −65.4748 −3.96999
\(273\) 0 0
\(274\) 39.1079 2.36260
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0508 −0.844230 −0.422115 0.906542i \(-0.638712\pi\)
−0.422115 + 0.906542i \(0.638712\pi\)
\(278\) 39.7253 2.38256
\(279\) 0 0
\(280\) −3.97903 −0.237793
\(281\) 1.74538 0.104120 0.0520602 0.998644i \(-0.483421\pi\)
0.0520602 + 0.998644i \(0.483421\pi\)
\(282\) 0 0
\(283\) −4.66531 −0.277324 −0.138662 0.990340i \(-0.544280\pi\)
−0.138662 + 0.990340i \(0.544280\pi\)
\(284\) 31.4278 1.86490
\(285\) 0 0
\(286\) 0 0
\(287\) −1.77073 −0.104523
\(288\) 0 0
\(289\) 6.34024 0.372955
\(290\) −2.38736 −0.140191
\(291\) 0 0
\(292\) −8.79820 −0.514876
\(293\) −14.5092 −0.847637 −0.423819 0.905747i \(-0.639311\pi\)
−0.423819 + 0.905747i \(0.639311\pi\)
\(294\) 0 0
\(295\) 6.40886 0.373138
\(296\) −65.4177 −3.80232
\(297\) 0 0
\(298\) 2.70698 0.156811
\(299\) −2.06176 −0.119235
\(300\) 0 0
\(301\) −11.4084 −0.657570
\(302\) −3.71794 −0.213943
\(303\) 0 0
\(304\) −14.7240 −0.844480
\(305\) 2.19405 0.125631
\(306\) 0 0
\(307\) 2.35679 0.134509 0.0672547 0.997736i \(-0.478576\pi\)
0.0672547 + 0.997736i \(0.478576\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9.93608 −0.564331
\(311\) 20.0050 1.13438 0.567190 0.823587i \(-0.308031\pi\)
0.567190 + 0.823587i \(0.308031\pi\)
\(312\) 0 0
\(313\) 18.7530 1.05998 0.529992 0.848003i \(-0.322195\pi\)
0.529992 + 0.848003i \(0.322195\pi\)
\(314\) −14.5342 −0.820214
\(315\) 0 0
\(316\) −19.1504 −1.07729
\(317\) −4.20203 −0.236009 −0.118005 0.993013i \(-0.537650\pi\)
−0.118005 + 0.993013i \(0.537650\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 10.5005 0.586999
\(321\) 0 0
\(322\) 12.3596 0.688774
\(323\) 5.24877 0.292049
\(324\) 0 0
\(325\) −2.16534 −0.120112
\(326\) 25.4780 1.41109
\(327\) 0 0
\(328\) 15.8307 0.874103
\(329\) −1.02259 −0.0563770
\(330\) 0 0
\(331\) −0.0682694 −0.00375242 −0.00187621 0.999998i \(-0.500597\pi\)
−0.00187621 + 0.999998i \(0.500597\pi\)
\(332\) 57.3441 3.14717
\(333\) 0 0
\(334\) 54.0959 2.96000
\(335\) −2.75436 −0.150487
\(336\) 0 0
\(337\) 21.7461 1.18459 0.592294 0.805722i \(-0.298223\pi\)
0.592294 + 0.805722i \(0.298223\pi\)
\(338\) 34.5919 1.88155
\(339\) 0 0
\(340\) −11.4118 −0.618892
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 101.994 5.49912
\(345\) 0 0
\(346\) 29.3203 1.57627
\(347\) −23.9351 −1.28490 −0.642452 0.766326i \(-0.722083\pi\)
−0.642452 + 0.766326i \(0.722083\pi\)
\(348\) 0 0
\(349\) −26.4151 −1.41397 −0.706983 0.707231i \(-0.749944\pi\)
−0.706983 + 0.707231i \(0.749944\pi\)
\(350\) 12.9805 0.693837
\(351\) 0 0
\(352\) 0 0
\(353\) 4.44182 0.236414 0.118207 0.992989i \(-0.462285\pi\)
0.118207 + 0.992989i \(0.462285\pi\)
\(354\) 0 0
\(355\) 2.63556 0.139881
\(356\) 27.6599 1.46597
\(357\) 0 0
\(358\) 45.5445 2.40710
\(359\) 14.5325 0.766997 0.383499 0.923541i \(-0.374719\pi\)
0.383499 + 0.923541i \(0.374719\pi\)
\(360\) 0 0
\(361\) −17.8197 −0.937876
\(362\) 54.8860 2.88475
\(363\) 0 0
\(364\) 2.39322 0.125439
\(365\) −0.737825 −0.0386195
\(366\) 0 0
\(367\) 4.98158 0.260037 0.130018 0.991512i \(-0.458496\pi\)
0.130018 + 0.991512i \(0.458496\pi\)
\(368\) −61.9653 −3.23016
\(369\) 0 0
\(370\) −8.80351 −0.457673
\(371\) 3.57222 0.185460
\(372\) 0 0
\(373\) −13.6638 −0.707485 −0.353743 0.935343i \(-0.615091\pi\)
−0.353743 + 0.935343i \(0.615091\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 9.14211 0.471469
\(377\) 0.894793 0.0460842
\(378\) 0 0
\(379\) −8.36232 −0.429544 −0.214772 0.976664i \(-0.568901\pi\)
−0.214772 + 0.976664i \(0.568901\pi\)
\(380\) −2.56629 −0.131648
\(381\) 0 0
\(382\) 6.87787 0.351902
\(383\) 1.98679 0.101520 0.0507601 0.998711i \(-0.483836\pi\)
0.0507601 + 0.998711i \(0.483836\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −47.3292 −2.40899
\(387\) 0 0
\(388\) −28.1808 −1.43066
\(389\) −14.8325 −0.752039 −0.376019 0.926612i \(-0.622707\pi\)
−0.376019 + 0.926612i \(0.622707\pi\)
\(390\) 0 0
\(391\) 22.0892 1.11710
\(392\) −8.94020 −0.451548
\(393\) 0 0
\(394\) −5.85400 −0.294920
\(395\) −1.60597 −0.0808050
\(396\) 0 0
\(397\) 11.4630 0.575314 0.287657 0.957734i \(-0.407124\pi\)
0.287657 + 0.957734i \(0.407124\pi\)
\(398\) 39.4008 1.97498
\(399\) 0 0
\(400\) −65.0782 −3.25391
\(401\) −4.31030 −0.215246 −0.107623 0.994192i \(-0.534324\pi\)
−0.107623 + 0.994192i \(0.534324\pi\)
\(402\) 0 0
\(403\) 3.72408 0.185510
\(404\) −82.5831 −4.10867
\(405\) 0 0
\(406\) −5.36399 −0.266210
\(407\) 0 0
\(408\) 0 0
\(409\) −9.31202 −0.460450 −0.230225 0.973137i \(-0.573946\pi\)
−0.230225 + 0.973137i \(0.573946\pi\)
\(410\) 2.13040 0.105213
\(411\) 0 0
\(412\) −75.2111 −3.70538
\(413\) 14.3996 0.708557
\(414\) 0 0
\(415\) 4.80893 0.236061
\(416\) −8.45719 −0.414648
\(417\) 0 0
\(418\) 0 0
\(419\) 26.3424 1.28691 0.643454 0.765485i \(-0.277501\pi\)
0.643454 + 0.765485i \(0.277501\pi\)
\(420\) 0 0
\(421\) 14.1792 0.691053 0.345526 0.938409i \(-0.387700\pi\)
0.345526 + 0.938409i \(0.387700\pi\)
\(422\) 9.81352 0.477715
\(423\) 0 0
\(424\) −31.9363 −1.55097
\(425\) 23.1989 1.12531
\(426\) 0 0
\(427\) 4.92965 0.238562
\(428\) 62.3876 3.01562
\(429\) 0 0
\(430\) 13.7257 0.661911
\(431\) −7.59531 −0.365853 −0.182927 0.983127i \(-0.558557\pi\)
−0.182927 + 0.983127i \(0.558557\pi\)
\(432\) 0 0
\(433\) 22.4008 1.07652 0.538258 0.842780i \(-0.319083\pi\)
0.538258 + 0.842780i \(0.319083\pi\)
\(434\) −22.3246 −1.07162
\(435\) 0 0
\(436\) 83.7485 4.01083
\(437\) 4.96743 0.237624
\(438\) 0 0
\(439\) 15.4051 0.735244 0.367622 0.929975i \(-0.380172\pi\)
0.367622 + 0.929975i \(0.380172\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.88901 0.280112
\(443\) −21.2099 −1.00771 −0.503857 0.863787i \(-0.668086\pi\)
−0.503857 + 0.863787i \(0.668086\pi\)
\(444\) 0 0
\(445\) 2.31958 0.109959
\(446\) 13.0851 0.619599
\(447\) 0 0
\(448\) 23.5929 1.11466
\(449\) 11.6316 0.548931 0.274466 0.961597i \(-0.411499\pi\)
0.274466 + 0.961597i \(0.411499\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 31.4375 1.47869
\(453\) 0 0
\(454\) 61.4453 2.88377
\(455\) 0.200698 0.00940886
\(456\) 0 0
\(457\) 31.5976 1.47807 0.739037 0.673665i \(-0.235281\pi\)
0.739037 + 0.673665i \(0.235281\pi\)
\(458\) 61.2900 2.86389
\(459\) 0 0
\(460\) −10.8001 −0.503558
\(461\) 8.86324 0.412802 0.206401 0.978467i \(-0.433825\pi\)
0.206401 + 0.978467i \(0.433825\pi\)
\(462\) 0 0
\(463\) −25.6132 −1.19035 −0.595173 0.803597i \(-0.702917\pi\)
−0.595173 + 0.803597i \(0.702917\pi\)
\(464\) 26.8925 1.24846
\(465\) 0 0
\(466\) −1.58754 −0.0735413
\(467\) −20.5402 −0.950486 −0.475243 0.879855i \(-0.657640\pi\)
−0.475243 + 0.879855i \(0.657640\pi\)
\(468\) 0 0
\(469\) −6.18858 −0.285762
\(470\) 1.23029 0.0567491
\(471\) 0 0
\(472\) −128.735 −5.92551
\(473\) 0 0
\(474\) 0 0
\(475\) 5.21698 0.239371
\(476\) −25.6403 −1.17522
\(477\) 0 0
\(478\) −40.4726 −1.85117
\(479\) 27.5487 1.25873 0.629367 0.777108i \(-0.283314\pi\)
0.629367 + 0.777108i \(0.283314\pi\)
\(480\) 0 0
\(481\) 3.29959 0.150448
\(482\) 49.3999 2.25010
\(483\) 0 0
\(484\) 0 0
\(485\) −2.36327 −0.107310
\(486\) 0 0
\(487\) −12.1013 −0.548364 −0.274182 0.961678i \(-0.588407\pi\)
−0.274182 + 0.961678i \(0.588407\pi\)
\(488\) −44.0720 −1.99505
\(489\) 0 0
\(490\) −1.20312 −0.0543513
\(491\) 39.3867 1.77750 0.888749 0.458394i \(-0.151575\pi\)
0.888749 + 0.458394i \(0.151575\pi\)
\(492\) 0 0
\(493\) −9.58656 −0.431757
\(494\) 1.32432 0.0595842
\(495\) 0 0
\(496\) 111.925 5.02560
\(497\) 5.92165 0.265622
\(498\) 0 0
\(499\) −34.7832 −1.55711 −0.778555 0.627576i \(-0.784047\pi\)
−0.778555 + 0.627576i \(0.784047\pi\)
\(500\) −23.1533 −1.03545
\(501\) 0 0
\(502\) 28.5590 1.27465
\(503\) −3.23224 −0.144119 −0.0720593 0.997400i \(-0.522957\pi\)
−0.0720593 + 0.997400i \(0.522957\pi\)
\(504\) 0 0
\(505\) −6.92550 −0.308181
\(506\) 0 0
\(507\) 0 0
\(508\) 53.7824 2.38621
\(509\) 13.2217 0.586042 0.293021 0.956106i \(-0.405339\pi\)
0.293021 + 0.956106i \(0.405339\pi\)
\(510\) 0 0
\(511\) −1.65776 −0.0733351
\(512\) −11.8512 −0.523752
\(513\) 0 0
\(514\) −52.6257 −2.32122
\(515\) −6.30727 −0.277931
\(516\) 0 0
\(517\) 0 0
\(518\) −19.7800 −0.869082
\(519\) 0 0
\(520\) −1.79428 −0.0786843
\(521\) −7.30239 −0.319924 −0.159962 0.987123i \(-0.551137\pi\)
−0.159962 + 0.987123i \(0.551137\pi\)
\(522\) 0 0
\(523\) 8.38007 0.366435 0.183217 0.983072i \(-0.441349\pi\)
0.183217 + 0.983072i \(0.441349\pi\)
\(524\) 36.2667 1.58432
\(525\) 0 0
\(526\) 57.2767 2.49738
\(527\) −39.8988 −1.73802
\(528\) 0 0
\(529\) −2.09483 −0.0910794
\(530\) −4.29780 −0.186684
\(531\) 0 0
\(532\) −5.76602 −0.249989
\(533\) −0.798481 −0.0345861
\(534\) 0 0
\(535\) 5.23188 0.226194
\(536\) 55.3271 2.38977
\(537\) 0 0
\(538\) 62.5963 2.69872
\(539\) 0 0
\(540\) 0 0
\(541\) −3.79768 −0.163275 −0.0816375 0.996662i \(-0.526015\pi\)
−0.0816375 + 0.996662i \(0.526015\pi\)
\(542\) −72.5083 −3.11450
\(543\) 0 0
\(544\) 90.6080 3.88478
\(545\) 7.02323 0.300842
\(546\) 0 0
\(547\) 9.08985 0.388654 0.194327 0.980937i \(-0.437748\pi\)
0.194327 + 0.980937i \(0.437748\pi\)
\(548\) −76.7818 −3.27996
\(549\) 0 0
\(550\) 0 0
\(551\) −2.15583 −0.0918416
\(552\) 0 0
\(553\) −3.60833 −0.153442
\(554\) 37.9820 1.61370
\(555\) 0 0
\(556\) −77.9939 −3.30768
\(557\) −18.2372 −0.772736 −0.386368 0.922345i \(-0.626270\pi\)
−0.386368 + 0.922345i \(0.626270\pi\)
\(558\) 0 0
\(559\) −5.14444 −0.217586
\(560\) 6.03187 0.254893
\(561\) 0 0
\(562\) −4.71809 −0.199021
\(563\) 15.5744 0.656384 0.328192 0.944611i \(-0.393561\pi\)
0.328192 + 0.944611i \(0.393561\pi\)
\(564\) 0 0
\(565\) 2.63637 0.110913
\(566\) 12.6112 0.530090
\(567\) 0 0
\(568\) −52.9407 −2.22134
\(569\) −24.4254 −1.02397 −0.511984 0.858995i \(-0.671089\pi\)
−0.511984 + 0.858995i \(0.671089\pi\)
\(570\) 0 0
\(571\) −14.9541 −0.625808 −0.312904 0.949785i \(-0.601302\pi\)
−0.312904 + 0.949785i \(0.601302\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4.78663 0.199790
\(575\) 21.9554 0.915603
\(576\) 0 0
\(577\) −21.4682 −0.893734 −0.446867 0.894600i \(-0.647460\pi\)
−0.446867 + 0.894600i \(0.647460\pi\)
\(578\) −17.1389 −0.712885
\(579\) 0 0
\(580\) 4.68718 0.194625
\(581\) 10.8048 0.448260
\(582\) 0 0
\(583\) 0 0
\(584\) 14.8207 0.613286
\(585\) 0 0
\(586\) 39.2212 1.62021
\(587\) 20.6760 0.853391 0.426695 0.904395i \(-0.359678\pi\)
0.426695 + 0.904395i \(0.359678\pi\)
\(588\) 0 0
\(589\) −8.97246 −0.369704
\(590\) −17.3244 −0.713234
\(591\) 0 0
\(592\) 99.1675 4.07576
\(593\) −3.30237 −0.135612 −0.0678060 0.997699i \(-0.521600\pi\)
−0.0678060 + 0.997699i \(0.521600\pi\)
\(594\) 0 0
\(595\) −2.15022 −0.0881505
\(596\) −5.31469 −0.217698
\(597\) 0 0
\(598\) 5.57335 0.227911
\(599\) −38.9212 −1.59028 −0.795140 0.606426i \(-0.792602\pi\)
−0.795140 + 0.606426i \(0.792602\pi\)
\(600\) 0 0
\(601\) 44.3783 1.81023 0.905115 0.425168i \(-0.139785\pi\)
0.905115 + 0.425168i \(0.139785\pi\)
\(602\) 30.8392 1.25691
\(603\) 0 0
\(604\) 7.29954 0.297014
\(605\) 0 0
\(606\) 0 0
\(607\) −18.9059 −0.767365 −0.383683 0.923465i \(-0.625344\pi\)
−0.383683 + 0.923465i \(0.625344\pi\)
\(608\) 20.3760 0.826355
\(609\) 0 0
\(610\) −5.93095 −0.240137
\(611\) −0.461118 −0.0186548
\(612\) 0 0
\(613\) −9.68116 −0.391018 −0.195509 0.980702i \(-0.562636\pi\)
−0.195509 + 0.980702i \(0.562636\pi\)
\(614\) −6.37088 −0.257108
\(615\) 0 0
\(616\) 0 0
\(617\) 26.2775 1.05789 0.528947 0.848655i \(-0.322587\pi\)
0.528947 + 0.848655i \(0.322587\pi\)
\(618\) 0 0
\(619\) −24.7954 −0.996610 −0.498305 0.867002i \(-0.666044\pi\)
−0.498305 + 0.867002i \(0.666044\pi\)
\(620\) 19.5078 0.783452
\(621\) 0 0
\(622\) −54.0774 −2.16831
\(623\) 5.21170 0.208802
\(624\) 0 0
\(625\) 22.0679 0.882716
\(626\) −50.6931 −2.02611
\(627\) 0 0
\(628\) 28.5355 1.13869
\(629\) −35.3509 −1.40953
\(630\) 0 0
\(631\) −11.8107 −0.470178 −0.235089 0.971974i \(-0.575538\pi\)
−0.235089 + 0.971974i \(0.575538\pi\)
\(632\) 32.2592 1.28320
\(633\) 0 0
\(634\) 11.3589 0.451120
\(635\) 4.51024 0.178983
\(636\) 0 0
\(637\) 0.450933 0.0178666
\(638\) 0 0
\(639\) 0 0
\(640\) −11.6905 −0.462108
\(641\) 22.5182 0.889416 0.444708 0.895676i \(-0.353307\pi\)
0.444708 + 0.895676i \(0.353307\pi\)
\(642\) 0 0
\(643\) 13.8901 0.547774 0.273887 0.961762i \(-0.411691\pi\)
0.273887 + 0.961762i \(0.411691\pi\)
\(644\) −24.2660 −0.956214
\(645\) 0 0
\(646\) −14.1885 −0.558237
\(647\) −50.8116 −1.99761 −0.998805 0.0488792i \(-0.984435\pi\)
−0.998805 + 0.0488792i \(0.984435\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 5.85334 0.229587
\(651\) 0 0
\(652\) −50.0217 −1.95900
\(653\) 4.81411 0.188391 0.0941953 0.995554i \(-0.469972\pi\)
0.0941953 + 0.995554i \(0.469972\pi\)
\(654\) 0 0
\(655\) 3.04136 0.118836
\(656\) −23.9979 −0.936962
\(657\) 0 0
\(658\) 2.76425 0.107762
\(659\) 28.0409 1.09232 0.546159 0.837681i \(-0.316089\pi\)
0.546159 + 0.837681i \(0.316089\pi\)
\(660\) 0 0
\(661\) −41.7390 −1.62346 −0.811730 0.584033i \(-0.801474\pi\)
−0.811730 + 0.584033i \(0.801474\pi\)
\(662\) 0.184545 0.00717256
\(663\) 0 0
\(664\) −96.5973 −3.74870
\(665\) −0.483543 −0.0187510
\(666\) 0 0
\(667\) −9.07271 −0.351297
\(668\) −106.208 −4.10932
\(669\) 0 0
\(670\) 7.44559 0.287648
\(671\) 0 0
\(672\) 0 0
\(673\) −28.7747 −1.10918 −0.554592 0.832123i \(-0.687126\pi\)
−0.554592 + 0.832123i \(0.687126\pi\)
\(674\) −58.7841 −2.26428
\(675\) 0 0
\(676\) −67.9153 −2.61213
\(677\) 12.2862 0.472195 0.236098 0.971729i \(-0.424131\pi\)
0.236098 + 0.971729i \(0.424131\pi\)
\(678\) 0 0
\(679\) −5.30985 −0.203773
\(680\) 19.2234 0.737184
\(681\) 0 0
\(682\) 0 0
\(683\) −30.5940 −1.17065 −0.585323 0.810800i \(-0.699032\pi\)
−0.585323 + 0.810800i \(0.699032\pi\)
\(684\) 0 0
\(685\) −6.43899 −0.246021
\(686\) −2.70320 −0.103209
\(687\) 0 0
\(688\) −154.613 −5.89458
\(689\) 1.61083 0.0613678
\(690\) 0 0
\(691\) −17.1121 −0.650976 −0.325488 0.945546i \(-0.605529\pi\)
−0.325488 + 0.945546i \(0.605529\pi\)
\(692\) −57.5655 −2.18831
\(693\) 0 0
\(694\) 64.7013 2.45603
\(695\) −6.54063 −0.248100
\(696\) 0 0
\(697\) 8.55471 0.324033
\(698\) 71.4051 2.70272
\(699\) 0 0
\(700\) −25.4850 −0.963244
\(701\) −13.5936 −0.513423 −0.256712 0.966488i \(-0.582639\pi\)
−0.256712 + 0.966488i \(0.582639\pi\)
\(702\) 0 0
\(703\) −7.94974 −0.299830
\(704\) 0 0
\(705\) 0 0
\(706\) −12.0071 −0.451893
\(707\) −15.5604 −0.585208
\(708\) 0 0
\(709\) 29.5995 1.11163 0.555816 0.831305i \(-0.312406\pi\)
0.555816 + 0.831305i \(0.312406\pi\)
\(710\) −7.12444 −0.267376
\(711\) 0 0
\(712\) −46.5936 −1.74617
\(713\) −37.7601 −1.41413
\(714\) 0 0
\(715\) 0 0
\(716\) −89.4190 −3.34174
\(717\) 0 0
\(718\) −39.2842 −1.46608
\(719\) −45.5407 −1.69838 −0.849190 0.528087i \(-0.822909\pi\)
−0.849190 + 0.528087i \(0.822909\pi\)
\(720\) 0 0
\(721\) −14.1713 −0.527768
\(722\) 48.1700 1.79270
\(723\) 0 0
\(724\) −107.759 −4.00485
\(725\) −9.52850 −0.353880
\(726\) 0 0
\(727\) −43.8796 −1.62740 −0.813702 0.581283i \(-0.802551\pi\)
−0.813702 + 0.581283i \(0.802551\pi\)
\(728\) −4.03143 −0.149415
\(729\) 0 0
\(730\) 1.99448 0.0738192
\(731\) 55.1161 2.03854
\(732\) 0 0
\(733\) 14.0878 0.520345 0.260172 0.965562i \(-0.416221\pi\)
0.260172 + 0.965562i \(0.416221\pi\)
\(734\) −13.4662 −0.497046
\(735\) 0 0
\(736\) 85.7513 3.16083
\(737\) 0 0
\(738\) 0 0
\(739\) −39.4975 −1.45294 −0.726470 0.687199i \(-0.758840\pi\)
−0.726470 + 0.687199i \(0.758840\pi\)
\(740\) 17.2842 0.635380
\(741\) 0 0
\(742\) −9.65640 −0.354498
\(743\) −18.6162 −0.682962 −0.341481 0.939889i \(-0.610928\pi\)
−0.341481 + 0.939889i \(0.610928\pi\)
\(744\) 0 0
\(745\) −0.445694 −0.0163290
\(746\) 36.9360 1.35232
\(747\) 0 0
\(748\) 0 0
\(749\) 11.7551 0.429523
\(750\) 0 0
\(751\) −15.8060 −0.576768 −0.288384 0.957515i \(-0.593118\pi\)
−0.288384 + 0.957515i \(0.593118\pi\)
\(752\) −13.8587 −0.505373
\(753\) 0 0
\(754\) −2.41880 −0.0880875
\(755\) 0.612146 0.0222783
\(756\) 0 0
\(757\) −3.61112 −0.131248 −0.0656241 0.997844i \(-0.520904\pi\)
−0.0656241 + 0.997844i \(0.520904\pi\)
\(758\) 22.6050 0.821051
\(759\) 0 0
\(760\) 4.32297 0.156811
\(761\) −39.5819 −1.43484 −0.717422 0.696639i \(-0.754678\pi\)
−0.717422 + 0.696639i \(0.754678\pi\)
\(762\) 0 0
\(763\) 15.7800 0.571273
\(764\) −13.5035 −0.488541
\(765\) 0 0
\(766\) −5.37068 −0.194051
\(767\) 6.49325 0.234458
\(768\) 0 0
\(769\) −10.5472 −0.380341 −0.190171 0.981751i \(-0.560904\pi\)
−0.190171 + 0.981751i \(0.560904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 92.9228 3.34437
\(773\) −41.5632 −1.49492 −0.747462 0.664305i \(-0.768728\pi\)
−0.747462 + 0.664305i \(0.768728\pi\)
\(774\) 0 0
\(775\) −39.6571 −1.42452
\(776\) 47.4711 1.70411
\(777\) 0 0
\(778\) 40.0952 1.43748
\(779\) 1.92379 0.0689269
\(780\) 0 0
\(781\) 0 0
\(782\) −59.7114 −2.13527
\(783\) 0 0
\(784\) 13.5526 0.484020
\(785\) 2.39301 0.0854103
\(786\) 0 0
\(787\) 11.5474 0.411620 0.205810 0.978592i \(-0.434017\pi\)
0.205810 + 0.978592i \(0.434017\pi\)
\(788\) 11.4933 0.409433
\(789\) 0 0
\(790\) 4.34125 0.154455
\(791\) 5.92347 0.210614
\(792\) 0 0
\(793\) 2.22294 0.0789390
\(794\) −30.9869 −1.09968
\(795\) 0 0
\(796\) −77.3569 −2.74184
\(797\) −42.3155 −1.49889 −0.749447 0.662065i \(-0.769680\pi\)
−0.749447 + 0.662065i \(0.769680\pi\)
\(798\) 0 0
\(799\) 4.94029 0.174775
\(800\) 90.0591 3.18407
\(801\) 0 0
\(802\) 11.6516 0.411431
\(803\) 0 0
\(804\) 0 0
\(805\) −2.03497 −0.0717232
\(806\) −10.0669 −0.354592
\(807\) 0 0
\(808\) 139.113 4.89397
\(809\) 17.9814 0.632194 0.316097 0.948727i \(-0.397628\pi\)
0.316097 + 0.948727i \(0.397628\pi\)
\(810\) 0 0
\(811\) 31.1200 1.09277 0.546386 0.837533i \(-0.316003\pi\)
0.546386 + 0.837533i \(0.316003\pi\)
\(812\) 10.5313 0.369576
\(813\) 0 0
\(814\) 0 0
\(815\) −4.19486 −0.146940
\(816\) 0 0
\(817\) 12.3945 0.433630
\(818\) 25.1722 0.880126
\(819\) 0 0
\(820\) −4.18268 −0.146065
\(821\) −48.8675 −1.70549 −0.852744 0.522329i \(-0.825063\pi\)
−0.852744 + 0.522329i \(0.825063\pi\)
\(822\) 0 0
\(823\) 29.5821 1.03117 0.515584 0.856839i \(-0.327575\pi\)
0.515584 + 0.856839i \(0.327575\pi\)
\(824\) 126.695 4.41361
\(825\) 0 0
\(826\) −38.9249 −1.35437
\(827\) −25.3924 −0.882981 −0.441490 0.897266i \(-0.645550\pi\)
−0.441490 + 0.897266i \(0.645550\pi\)
\(828\) 0 0
\(829\) 7.09580 0.246447 0.123224 0.992379i \(-0.460677\pi\)
0.123224 + 0.992379i \(0.460677\pi\)
\(830\) −12.9995 −0.451219
\(831\) 0 0
\(832\) 10.6388 0.368835
\(833\) −4.83117 −0.167390
\(834\) 0 0
\(835\) −8.90671 −0.308230
\(836\) 0 0
\(837\) 0 0
\(838\) −71.2086 −2.45986
\(839\) 23.7285 0.819197 0.409599 0.912266i \(-0.365669\pi\)
0.409599 + 0.912266i \(0.365669\pi\)
\(840\) 0 0
\(841\) −25.0625 −0.864224
\(842\) −38.3292 −1.32091
\(843\) 0 0
\(844\) −19.2672 −0.663204
\(845\) −5.69544 −0.195929
\(846\) 0 0
\(847\) 0 0
\(848\) 48.4127 1.66250
\(849\) 0 0
\(850\) −62.7111 −2.15097
\(851\) −33.4561 −1.14686
\(852\) 0 0
\(853\) −38.6542 −1.32350 −0.661748 0.749726i \(-0.730185\pi\)
−0.661748 + 0.749726i \(0.730185\pi\)
\(854\) −13.3258 −0.456000
\(855\) 0 0
\(856\) −105.093 −3.59201
\(857\) −34.0838 −1.16428 −0.582139 0.813089i \(-0.697784\pi\)
−0.582139 + 0.813089i \(0.697784\pi\)
\(858\) 0 0
\(859\) 56.7283 1.93555 0.967773 0.251825i \(-0.0810308\pi\)
0.967773 + 0.251825i \(0.0810308\pi\)
\(860\) −26.9480 −0.918920
\(861\) 0 0
\(862\) 20.5316 0.699310
\(863\) −24.3930 −0.830346 −0.415173 0.909742i \(-0.636279\pi\)
−0.415173 + 0.909742i \(0.636279\pi\)
\(864\) 0 0
\(865\) −4.82750 −0.164140
\(866\) −60.5539 −2.05770
\(867\) 0 0
\(868\) 43.8306 1.48771
\(869\) 0 0
\(870\) 0 0
\(871\) −2.79064 −0.0945571
\(872\) −141.076 −4.77744
\(873\) 0 0
\(874\) −13.4279 −0.454207
\(875\) −4.36256 −0.147481
\(876\) 0 0
\(877\) 30.4950 1.02974 0.514871 0.857267i \(-0.327840\pi\)
0.514871 + 0.857267i \(0.327840\pi\)
\(878\) −41.6429 −1.40538
\(879\) 0 0
\(880\) 0 0
\(881\) −2.10056 −0.0707697 −0.0353848 0.999374i \(-0.511266\pi\)
−0.0353848 + 0.999374i \(0.511266\pi\)
\(882\) 0 0
\(883\) 45.1955 1.52095 0.760475 0.649367i \(-0.224966\pi\)
0.760475 + 0.649367i \(0.224966\pi\)
\(884\) −11.5621 −0.388875
\(885\) 0 0
\(886\) 57.3346 1.92619
\(887\) −5.18758 −0.174182 −0.0870910 0.996200i \(-0.527757\pi\)
−0.0870910 + 0.996200i \(0.527757\pi\)
\(888\) 0 0
\(889\) 10.1337 0.339874
\(890\) −6.27029 −0.210181
\(891\) 0 0
\(892\) −25.6905 −0.860181
\(893\) 1.11098 0.0371774
\(894\) 0 0
\(895\) −7.49875 −0.250656
\(896\) −26.2666 −0.877504
\(897\) 0 0
\(898\) −31.4426 −1.04925
\(899\) 16.3877 0.546559
\(900\) 0 0
\(901\) −17.2580 −0.574947
\(902\) 0 0
\(903\) 0 0
\(904\) −52.9570 −1.76132
\(905\) −9.03681 −0.300394
\(906\) 0 0
\(907\) −38.6473 −1.28326 −0.641631 0.767013i \(-0.721742\pi\)
−0.641631 + 0.767013i \(0.721742\pi\)
\(908\) −120.637 −4.00350
\(909\) 0 0
\(910\) −0.542526 −0.0179846
\(911\) 18.3990 0.609585 0.304793 0.952419i \(-0.401413\pi\)
0.304793 + 0.952419i \(0.401413\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −85.4145 −2.82526
\(915\) 0 0
\(916\) −120.333 −3.97590
\(917\) 6.83340 0.225659
\(918\) 0 0
\(919\) 17.6533 0.582330 0.291165 0.956673i \(-0.405957\pi\)
0.291165 + 0.956673i \(0.405957\pi\)
\(920\) 18.1930 0.599806
\(921\) 0 0
\(922\) −23.9591 −0.789050
\(923\) 2.67027 0.0878930
\(924\) 0 0
\(925\) −35.1368 −1.15529
\(926\) 69.2375 2.27529
\(927\) 0 0
\(928\) −37.2155 −1.22166
\(929\) 13.4484 0.441226 0.220613 0.975361i \(-0.429194\pi\)
0.220613 + 0.975361i \(0.429194\pi\)
\(930\) 0 0
\(931\) −1.08644 −0.0356066
\(932\) 3.11687 0.102096
\(933\) 0 0
\(934\) 55.5241 1.81680
\(935\) 0 0
\(936\) 0 0
\(937\) −8.59580 −0.280813 −0.140406 0.990094i \(-0.544841\pi\)
−0.140406 + 0.990094i \(0.544841\pi\)
\(938\) 16.7289 0.546219
\(939\) 0 0
\(940\) −2.41547 −0.0787839
\(941\) 33.6975 1.09851 0.549254 0.835655i \(-0.314912\pi\)
0.549254 + 0.835655i \(0.314912\pi\)
\(942\) 0 0
\(943\) 8.09617 0.263647
\(944\) 195.151 6.35163
\(945\) 0 0
\(946\) 0 0
\(947\) 15.3289 0.498121 0.249061 0.968488i \(-0.419878\pi\)
0.249061 + 0.968488i \(0.419878\pi\)
\(948\) 0 0
\(949\) −0.747541 −0.0242662
\(950\) −14.1025 −0.457546
\(951\) 0 0
\(952\) 43.1916 1.39985
\(953\) −44.9376 −1.45567 −0.727835 0.685752i \(-0.759473\pi\)
−0.727835 + 0.685752i \(0.759473\pi\)
\(954\) 0 0
\(955\) −1.13242 −0.0366442
\(956\) 79.4610 2.56995
\(957\) 0 0
\(958\) −74.4697 −2.40601
\(959\) −14.4673 −0.467173
\(960\) 0 0
\(961\) 37.2046 1.20015
\(962\) −8.91944 −0.287574
\(963\) 0 0
\(964\) −96.9883 −3.12378
\(965\) 7.79259 0.250852
\(966\) 0 0
\(967\) 33.9453 1.09161 0.545804 0.837913i \(-0.316224\pi\)
0.545804 + 0.837913i \(0.316224\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.38837 0.205118
\(971\) 55.4600 1.77980 0.889899 0.456158i \(-0.150775\pi\)
0.889899 + 0.456158i \(0.150775\pi\)
\(972\) 0 0
\(973\) −14.6957 −0.471121
\(974\) 32.7123 1.04817
\(975\) 0 0
\(976\) 66.8094 2.13852
\(977\) 37.9269 1.21339 0.606694 0.794936i \(-0.292495\pi\)
0.606694 + 0.794936i \(0.292495\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.36212 0.0754551
\(981\) 0 0
\(982\) −106.470 −3.39760
\(983\) −31.5337 −1.00577 −0.502884 0.864354i \(-0.667728\pi\)
−0.502884 + 0.864354i \(0.667728\pi\)
\(984\) 0 0
\(985\) 0.963841 0.0307105
\(986\) 25.9144 0.825281
\(987\) 0 0
\(988\) −2.60009 −0.0827198
\(989\) 52.1618 1.65865
\(990\) 0 0
\(991\) 39.8173 1.26484 0.632419 0.774627i \(-0.282062\pi\)
0.632419 + 0.774627i \(0.282062\pi\)
\(992\) −154.889 −4.91773
\(993\) 0 0
\(994\) −16.0074 −0.507723
\(995\) −6.48721 −0.205659
\(996\) 0 0
\(997\) −18.7190 −0.592836 −0.296418 0.955058i \(-0.595792\pi\)
−0.296418 + 0.955058i \(0.595792\pi\)
\(998\) 94.0258 2.97634
\(999\) 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.cp.1.1 6
3.2 odd 2 847.2.a.n.1.6 yes 6
11.10 odd 2 7623.2.a.cs.1.6 6
21.20 even 2 5929.2.a.bm.1.6 6
33.2 even 10 847.2.f.z.323.6 24
33.5 odd 10 847.2.f.y.729.1 24
33.8 even 10 847.2.f.z.372.1 24
33.14 odd 10 847.2.f.y.372.6 24
33.17 even 10 847.2.f.z.729.6 24
33.20 odd 10 847.2.f.y.323.1 24
33.26 odd 10 847.2.f.y.148.6 24
33.29 even 10 847.2.f.z.148.1 24
33.32 even 2 847.2.a.m.1.1 6
231.230 odd 2 5929.2.a.bj.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.m.1.1 6 33.32 even 2
847.2.a.n.1.6 yes 6 3.2 odd 2
847.2.f.y.148.6 24 33.26 odd 10
847.2.f.y.323.1 24 33.20 odd 10
847.2.f.y.372.6 24 33.14 odd 10
847.2.f.y.729.1 24 33.5 odd 10
847.2.f.z.148.1 24 33.29 even 10
847.2.f.z.323.6 24 33.2 even 10
847.2.f.z.372.1 24 33.8 even 10
847.2.f.z.729.6 24 33.17 even 10
5929.2.a.bj.1.1 6 231.230 odd 2
5929.2.a.bm.1.6 6 21.20 even 2
7623.2.a.cp.1.1 6 1.1 even 1 trivial
7623.2.a.cs.1.6 6 11.10 odd 2