Properties

Label 2151.2.a.k.1.6
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.25296\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.25296 q^{2} -0.430099 q^{4} -1.67805 q^{5} -0.437804 q^{7} +3.04481 q^{8} +O(q^{10})\) \(q-1.25296 q^{2} -0.430099 q^{4} -1.67805 q^{5} -0.437804 q^{7} +3.04481 q^{8} +2.10252 q^{10} +3.01794 q^{11} +3.79768 q^{13} +0.548550 q^{14} -2.95482 q^{16} +4.87279 q^{17} -7.63286 q^{19} +0.721726 q^{20} -3.78135 q^{22} +0.261247 q^{23} -2.18416 q^{25} -4.75833 q^{26} +0.188299 q^{28} +0.689712 q^{29} +8.79905 q^{31} -2.38736 q^{32} -6.10540 q^{34} +0.734656 q^{35} -6.63032 q^{37} +9.56364 q^{38} -5.10934 q^{40} -0.687470 q^{41} -7.99902 q^{43} -1.29801 q^{44} -0.327332 q^{46} +0.785372 q^{47} -6.80833 q^{49} +2.73665 q^{50} -1.63338 q^{52} +12.0733 q^{53} -5.06424 q^{55} -1.33303 q^{56} -0.864179 q^{58} -11.3147 q^{59} -1.44871 q^{61} -11.0248 q^{62} +8.90089 q^{64} -6.37269 q^{65} +6.28276 q^{67} -2.09578 q^{68} -0.920493 q^{70} +11.4266 q^{71} +9.10142 q^{73} +8.30751 q^{74} +3.28288 q^{76} -1.32127 q^{77} +5.17440 q^{79} +4.95832 q^{80} +0.861370 q^{82} -1.59126 q^{83} -8.17678 q^{85} +10.0224 q^{86} +9.18904 q^{88} +16.4673 q^{89} -1.66264 q^{91} -0.112362 q^{92} -0.984037 q^{94} +12.8083 q^{95} +2.60081 q^{97} +8.53054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25296 −0.885974 −0.442987 0.896528i \(-0.646081\pi\)
−0.442987 + 0.896528i \(0.646081\pi\)
\(3\) 0 0
\(4\) −0.430099 −0.215049
\(5\) −1.67805 −0.750446 −0.375223 0.926935i \(-0.622434\pi\)
−0.375223 + 0.926935i \(0.622434\pi\)
\(6\) 0 0
\(7\) −0.437804 −0.165474 −0.0827372 0.996571i \(-0.526366\pi\)
−0.0827372 + 0.996571i \(0.526366\pi\)
\(8\) 3.04481 1.07650
\(9\) 0 0
\(10\) 2.10252 0.664876
\(11\) 3.01794 0.909942 0.454971 0.890506i \(-0.349650\pi\)
0.454971 + 0.890506i \(0.349650\pi\)
\(12\) 0 0
\(13\) 3.79768 1.05329 0.526643 0.850086i \(-0.323450\pi\)
0.526643 + 0.850086i \(0.323450\pi\)
\(14\) 0.548550 0.146606
\(15\) 0 0
\(16\) −2.95482 −0.738704
\(17\) 4.87279 1.18183 0.590913 0.806735i \(-0.298768\pi\)
0.590913 + 0.806735i \(0.298768\pi\)
\(18\) 0 0
\(19\) −7.63286 −1.75110 −0.875549 0.483129i \(-0.839500\pi\)
−0.875549 + 0.483129i \(0.839500\pi\)
\(20\) 0.721726 0.161383
\(21\) 0 0
\(22\) −3.78135 −0.806186
\(23\) 0.261247 0.0544738 0.0272369 0.999629i \(-0.491329\pi\)
0.0272369 + 0.999629i \(0.491329\pi\)
\(24\) 0 0
\(25\) −2.18416 −0.436831
\(26\) −4.75833 −0.933185
\(27\) 0 0
\(28\) 0.188299 0.0355852
\(29\) 0.689712 0.128076 0.0640381 0.997947i \(-0.479602\pi\)
0.0640381 + 0.997947i \(0.479602\pi\)
\(30\) 0 0
\(31\) 8.79905 1.58036 0.790178 0.612877i \(-0.209988\pi\)
0.790178 + 0.612877i \(0.209988\pi\)
\(32\) −2.38736 −0.422030
\(33\) 0 0
\(34\) −6.10540 −1.04707
\(35\) 0.734656 0.124180
\(36\) 0 0
\(37\) −6.63032 −1.09002 −0.545009 0.838430i \(-0.683474\pi\)
−0.545009 + 0.838430i \(0.683474\pi\)
\(38\) 9.56364 1.55143
\(39\) 0 0
\(40\) −5.10934 −0.807857
\(41\) −0.687470 −0.107365 −0.0536824 0.998558i \(-0.517096\pi\)
−0.0536824 + 0.998558i \(0.517096\pi\)
\(42\) 0 0
\(43\) −7.99902 −1.21984 −0.609920 0.792463i \(-0.708798\pi\)
−0.609920 + 0.792463i \(0.708798\pi\)
\(44\) −1.29801 −0.195683
\(45\) 0 0
\(46\) −0.327332 −0.0482624
\(47\) 0.785372 0.114558 0.0572791 0.998358i \(-0.481757\pi\)
0.0572791 + 0.998358i \(0.481757\pi\)
\(48\) 0 0
\(49\) −6.80833 −0.972618
\(50\) 2.73665 0.387021
\(51\) 0 0
\(52\) −1.63338 −0.226509
\(53\) 12.0733 1.65840 0.829199 0.558954i \(-0.188797\pi\)
0.829199 + 0.558954i \(0.188797\pi\)
\(54\) 0 0
\(55\) −5.06424 −0.682862
\(56\) −1.33303 −0.178134
\(57\) 0 0
\(58\) −0.864179 −0.113472
\(59\) −11.3147 −1.47305 −0.736525 0.676411i \(-0.763534\pi\)
−0.736525 + 0.676411i \(0.763534\pi\)
\(60\) 0 0
\(61\) −1.44871 −0.185488 −0.0927439 0.995690i \(-0.529564\pi\)
−0.0927439 + 0.995690i \(0.529564\pi\)
\(62\) −11.0248 −1.40016
\(63\) 0 0
\(64\) 8.90089 1.11261
\(65\) −6.37269 −0.790434
\(66\) 0 0
\(67\) 6.28276 0.767561 0.383780 0.923424i \(-0.374622\pi\)
0.383780 + 0.923424i \(0.374622\pi\)
\(68\) −2.09578 −0.254151
\(69\) 0 0
\(70\) −0.920493 −0.110020
\(71\) 11.4266 1.35609 0.678044 0.735022i \(-0.262828\pi\)
0.678044 + 0.735022i \(0.262828\pi\)
\(72\) 0 0
\(73\) 9.10142 1.06524 0.532620 0.846354i \(-0.321207\pi\)
0.532620 + 0.846354i \(0.321207\pi\)
\(74\) 8.30751 0.965728
\(75\) 0 0
\(76\) 3.28288 0.376573
\(77\) −1.32127 −0.150572
\(78\) 0 0
\(79\) 5.17440 0.582165 0.291083 0.956698i \(-0.405985\pi\)
0.291083 + 0.956698i \(0.405985\pi\)
\(80\) 4.95832 0.554357
\(81\) 0 0
\(82\) 0.861370 0.0951224
\(83\) −1.59126 −0.174663 −0.0873317 0.996179i \(-0.527834\pi\)
−0.0873317 + 0.996179i \(0.527834\pi\)
\(84\) 0 0
\(85\) −8.17678 −0.886896
\(86\) 10.0224 1.08075
\(87\) 0 0
\(88\) 9.18904 0.979555
\(89\) 16.4673 1.74553 0.872766 0.488138i \(-0.162324\pi\)
0.872766 + 0.488138i \(0.162324\pi\)
\(90\) 0 0
\(91\) −1.66264 −0.174292
\(92\) −0.112362 −0.0117146
\(93\) 0 0
\(94\) −0.984037 −0.101496
\(95\) 12.8083 1.31410
\(96\) 0 0
\(97\) 2.60081 0.264073 0.132036 0.991245i \(-0.457848\pi\)
0.132036 + 0.991245i \(0.457848\pi\)
\(98\) 8.53054 0.861715
\(99\) 0 0
\(100\) 0.939403 0.0939403
\(101\) 6.29708 0.626583 0.313292 0.949657i \(-0.398568\pi\)
0.313292 + 0.949657i \(0.398568\pi\)
\(102\) 0 0
\(103\) 2.23873 0.220589 0.110294 0.993899i \(-0.464821\pi\)
0.110294 + 0.993899i \(0.464821\pi\)
\(104\) 11.5632 1.13387
\(105\) 0 0
\(106\) −15.1273 −1.46930
\(107\) −11.8071 −1.14143 −0.570716 0.821148i \(-0.693334\pi\)
−0.570716 + 0.821148i \(0.693334\pi\)
\(108\) 0 0
\(109\) −8.93712 −0.856020 −0.428010 0.903774i \(-0.640785\pi\)
−0.428010 + 0.903774i \(0.640785\pi\)
\(110\) 6.34528 0.604999
\(111\) 0 0
\(112\) 1.29363 0.122237
\(113\) 18.4672 1.73725 0.868625 0.495470i \(-0.165004\pi\)
0.868625 + 0.495470i \(0.165004\pi\)
\(114\) 0 0
\(115\) −0.438385 −0.0408796
\(116\) −0.296644 −0.0275427
\(117\) 0 0
\(118\) 14.1768 1.30508
\(119\) −2.13333 −0.195562
\(120\) 0 0
\(121\) −1.89205 −0.172005
\(122\) 1.81517 0.164337
\(123\) 0 0
\(124\) −3.78446 −0.339855
\(125\) 12.0554 1.07826
\(126\) 0 0
\(127\) 18.2247 1.61718 0.808592 0.588369i \(-0.200230\pi\)
0.808592 + 0.588369i \(0.200230\pi\)
\(128\) −6.37772 −0.563716
\(129\) 0 0
\(130\) 7.98470 0.700305
\(131\) −0.446703 −0.0390286 −0.0195143 0.999810i \(-0.506212\pi\)
−0.0195143 + 0.999810i \(0.506212\pi\)
\(132\) 0 0
\(133\) 3.34170 0.289762
\(134\) −7.87202 −0.680039
\(135\) 0 0
\(136\) 14.8367 1.27224
\(137\) 8.14824 0.696152 0.348076 0.937466i \(-0.386835\pi\)
0.348076 + 0.937466i \(0.386835\pi\)
\(138\) 0 0
\(139\) −7.92325 −0.672041 −0.336021 0.941855i \(-0.609081\pi\)
−0.336021 + 0.941855i \(0.609081\pi\)
\(140\) −0.315975 −0.0267048
\(141\) 0 0
\(142\) −14.3170 −1.20146
\(143\) 11.4612 0.958430
\(144\) 0 0
\(145\) −1.15737 −0.0961143
\(146\) −11.4037 −0.943776
\(147\) 0 0
\(148\) 2.85169 0.234408
\(149\) −5.04368 −0.413194 −0.206597 0.978426i \(-0.566239\pi\)
−0.206597 + 0.978426i \(0.566239\pi\)
\(150\) 0 0
\(151\) −14.2381 −1.15868 −0.579340 0.815086i \(-0.696689\pi\)
−0.579340 + 0.815086i \(0.696689\pi\)
\(152\) −23.2406 −1.88506
\(153\) 0 0
\(154\) 1.65549 0.133403
\(155\) −14.7652 −1.18597
\(156\) 0 0
\(157\) 18.0010 1.43664 0.718318 0.695715i \(-0.244912\pi\)
0.718318 + 0.695715i \(0.244912\pi\)
\(158\) −6.48330 −0.515783
\(159\) 0 0
\(160\) 4.00610 0.316710
\(161\) −0.114375 −0.00901402
\(162\) 0 0
\(163\) 10.7570 0.842556 0.421278 0.906932i \(-0.361582\pi\)
0.421278 + 0.906932i \(0.361582\pi\)
\(164\) 0.295680 0.0230887
\(165\) 0 0
\(166\) 1.99378 0.154747
\(167\) 3.56318 0.275727 0.137863 0.990451i \(-0.455976\pi\)
0.137863 + 0.990451i \(0.455976\pi\)
\(168\) 0 0
\(169\) 1.42236 0.109412
\(170\) 10.2451 0.785767
\(171\) 0 0
\(172\) 3.44037 0.262326
\(173\) −10.6387 −0.808845 −0.404423 0.914572i \(-0.632528\pi\)
−0.404423 + 0.914572i \(0.632528\pi\)
\(174\) 0 0
\(175\) 0.956233 0.0722844
\(176\) −8.91745 −0.672178
\(177\) 0 0
\(178\) −20.6328 −1.54650
\(179\) 2.79518 0.208922 0.104461 0.994529i \(-0.466688\pi\)
0.104461 + 0.994529i \(0.466688\pi\)
\(180\) 0 0
\(181\) 2.45006 0.182112 0.0910559 0.995846i \(-0.470976\pi\)
0.0910559 + 0.995846i \(0.470976\pi\)
\(182\) 2.08322 0.154418
\(183\) 0 0
\(184\) 0.795448 0.0586412
\(185\) 11.1260 0.817999
\(186\) 0 0
\(187\) 14.7058 1.07539
\(188\) −0.337788 −0.0246357
\(189\) 0 0
\(190\) −16.0483 −1.16426
\(191\) −17.0298 −1.23223 −0.616116 0.787656i \(-0.711295\pi\)
−0.616116 + 0.787656i \(0.711295\pi\)
\(192\) 0 0
\(193\) −5.17425 −0.372451 −0.186225 0.982507i \(-0.559625\pi\)
−0.186225 + 0.982507i \(0.559625\pi\)
\(194\) −3.25871 −0.233961
\(195\) 0 0
\(196\) 2.92825 0.209161
\(197\) 20.2790 1.44482 0.722411 0.691464i \(-0.243034\pi\)
0.722411 + 0.691464i \(0.243034\pi\)
\(198\) 0 0
\(199\) 18.8206 1.33415 0.667077 0.744988i \(-0.267545\pi\)
0.667077 + 0.744988i \(0.267545\pi\)
\(200\) −6.65034 −0.470250
\(201\) 0 0
\(202\) −7.88998 −0.555137
\(203\) −0.301959 −0.0211933
\(204\) 0 0
\(205\) 1.15361 0.0805714
\(206\) −2.80503 −0.195436
\(207\) 0 0
\(208\) −11.2214 −0.778067
\(209\) −23.0355 −1.59340
\(210\) 0 0
\(211\) 27.8687 1.91856 0.959280 0.282458i \(-0.0911497\pi\)
0.959280 + 0.282458i \(0.0911497\pi\)
\(212\) −5.19272 −0.356638
\(213\) 0 0
\(214\) 14.7937 1.01128
\(215\) 13.4227 0.915423
\(216\) 0 0
\(217\) −3.85226 −0.261509
\(218\) 11.1978 0.758412
\(219\) 0 0
\(220\) 2.17813 0.146849
\(221\) 18.5053 1.24480
\(222\) 0 0
\(223\) −12.8425 −0.859997 −0.429999 0.902830i \(-0.641486\pi\)
−0.429999 + 0.902830i \(0.641486\pi\)
\(224\) 1.04520 0.0698351
\(225\) 0 0
\(226\) −23.1386 −1.53916
\(227\) 3.74994 0.248892 0.124446 0.992226i \(-0.460285\pi\)
0.124446 + 0.992226i \(0.460285\pi\)
\(228\) 0 0
\(229\) −4.60000 −0.303977 −0.151988 0.988382i \(-0.548568\pi\)
−0.151988 + 0.988382i \(0.548568\pi\)
\(230\) 0.549278 0.0362183
\(231\) 0 0
\(232\) 2.10004 0.137874
\(233\) 28.5275 1.86890 0.934449 0.356097i \(-0.115893\pi\)
0.934449 + 0.356097i \(0.115893\pi\)
\(234\) 0 0
\(235\) −1.31789 −0.0859698
\(236\) 4.86644 0.316778
\(237\) 0 0
\(238\) 2.67297 0.173263
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 13.0672 0.841732 0.420866 0.907123i \(-0.361726\pi\)
0.420866 + 0.907123i \(0.361726\pi\)
\(242\) 2.37066 0.152392
\(243\) 0 0
\(244\) 0.623087 0.0398890
\(245\) 11.4247 0.729897
\(246\) 0 0
\(247\) −28.9871 −1.84441
\(248\) 26.7914 1.70126
\(249\) 0 0
\(250\) −15.1048 −0.955314
\(251\) −13.0772 −0.825426 −0.412713 0.910861i \(-0.635419\pi\)
−0.412713 + 0.910861i \(0.635419\pi\)
\(252\) 0 0
\(253\) 0.788428 0.0495680
\(254\) −22.8348 −1.43278
\(255\) 0 0
\(256\) −9.81078 −0.613174
\(257\) −22.4358 −1.39951 −0.699753 0.714385i \(-0.746706\pi\)
−0.699753 + 0.714385i \(0.746706\pi\)
\(258\) 0 0
\(259\) 2.90278 0.180370
\(260\) 2.74088 0.169982
\(261\) 0 0
\(262\) 0.559699 0.0345783
\(263\) −6.21793 −0.383414 −0.191707 0.981452i \(-0.561402\pi\)
−0.191707 + 0.981452i \(0.561402\pi\)
\(264\) 0 0
\(265\) −20.2596 −1.24454
\(266\) −4.18700 −0.256722
\(267\) 0 0
\(268\) −2.70221 −0.165064
\(269\) −18.5021 −1.12809 −0.564045 0.825744i \(-0.690756\pi\)
−0.564045 + 0.825744i \(0.690756\pi\)
\(270\) 0 0
\(271\) 24.5430 1.49088 0.745442 0.666571i \(-0.232239\pi\)
0.745442 + 0.666571i \(0.232239\pi\)
\(272\) −14.3982 −0.873020
\(273\) 0 0
\(274\) −10.2094 −0.616772
\(275\) −6.59165 −0.397491
\(276\) 0 0
\(277\) 1.77904 0.106892 0.0534462 0.998571i \(-0.482979\pi\)
0.0534462 + 0.998571i \(0.482979\pi\)
\(278\) 9.92749 0.595411
\(279\) 0 0
\(280\) 2.23689 0.133680
\(281\) 12.6890 0.756963 0.378482 0.925609i \(-0.376446\pi\)
0.378482 + 0.925609i \(0.376446\pi\)
\(282\) 0 0
\(283\) 24.2666 1.44250 0.721250 0.692675i \(-0.243568\pi\)
0.721250 + 0.692675i \(0.243568\pi\)
\(284\) −4.91456 −0.291626
\(285\) 0 0
\(286\) −14.3603 −0.849144
\(287\) 0.300977 0.0177661
\(288\) 0 0
\(289\) 6.74410 0.396712
\(290\) 1.45013 0.0851548
\(291\) 0 0
\(292\) −3.91451 −0.229079
\(293\) −15.3635 −0.897545 −0.448772 0.893646i \(-0.648139\pi\)
−0.448772 + 0.893646i \(0.648139\pi\)
\(294\) 0 0
\(295\) 18.9866 1.10544
\(296\) −20.1881 −1.17341
\(297\) 0 0
\(298\) 6.31951 0.366080
\(299\) 0.992133 0.0573765
\(300\) 0 0
\(301\) 3.50201 0.201852
\(302\) 17.8397 1.02656
\(303\) 0 0
\(304\) 22.5537 1.29354
\(305\) 2.43100 0.139198
\(306\) 0 0
\(307\) −5.88479 −0.335863 −0.167931 0.985799i \(-0.553709\pi\)
−0.167931 + 0.985799i \(0.553709\pi\)
\(308\) 0.568275 0.0323805
\(309\) 0 0
\(310\) 18.5002 1.05074
\(311\) 19.4787 1.10453 0.552267 0.833668i \(-0.313763\pi\)
0.552267 + 0.833668i \(0.313763\pi\)
\(312\) 0 0
\(313\) 16.4195 0.928087 0.464044 0.885812i \(-0.346398\pi\)
0.464044 + 0.885812i \(0.346398\pi\)
\(314\) −22.5545 −1.27282
\(315\) 0 0
\(316\) −2.22550 −0.125194
\(317\) 32.9447 1.85036 0.925178 0.379533i \(-0.123915\pi\)
0.925178 + 0.379533i \(0.123915\pi\)
\(318\) 0 0
\(319\) 2.08151 0.116542
\(320\) −14.9361 −0.834955
\(321\) 0 0
\(322\) 0.143307 0.00798619
\(323\) −37.1933 −2.06949
\(324\) 0 0
\(325\) −8.29472 −0.460108
\(326\) −13.4781 −0.746483
\(327\) 0 0
\(328\) −2.09321 −0.115578
\(329\) −0.343839 −0.0189565
\(330\) 0 0
\(331\) 29.5028 1.62162 0.810810 0.585310i \(-0.199027\pi\)
0.810810 + 0.585310i \(0.199027\pi\)
\(332\) 0.684399 0.0375613
\(333\) 0 0
\(334\) −4.46451 −0.244287
\(335\) −10.5428 −0.576013
\(336\) 0 0
\(337\) −15.6639 −0.853268 −0.426634 0.904424i \(-0.640301\pi\)
−0.426634 + 0.904424i \(0.640301\pi\)
\(338\) −1.78216 −0.0969367
\(339\) 0 0
\(340\) 3.51682 0.190726
\(341\) 26.5550 1.43803
\(342\) 0 0
\(343\) 6.04534 0.326418
\(344\) −24.3555 −1.31316
\(345\) 0 0
\(346\) 13.3298 0.716616
\(347\) 23.0843 1.23923 0.619616 0.784905i \(-0.287288\pi\)
0.619616 + 0.784905i \(0.287288\pi\)
\(348\) 0 0
\(349\) −8.93132 −0.478083 −0.239041 0.971009i \(-0.576833\pi\)
−0.239041 + 0.971009i \(0.576833\pi\)
\(350\) −1.19812 −0.0640421
\(351\) 0 0
\(352\) −7.20490 −0.384023
\(353\) −28.7892 −1.53230 −0.766148 0.642664i \(-0.777829\pi\)
−0.766148 + 0.642664i \(0.777829\pi\)
\(354\) 0 0
\(355\) −19.1744 −1.01767
\(356\) −7.08258 −0.375376
\(357\) 0 0
\(358\) −3.50225 −0.185099
\(359\) 32.7177 1.72677 0.863387 0.504543i \(-0.168339\pi\)
0.863387 + 0.504543i \(0.168339\pi\)
\(360\) 0 0
\(361\) 39.2605 2.06634
\(362\) −3.06982 −0.161346
\(363\) 0 0
\(364\) 0.715099 0.0374814
\(365\) −15.2726 −0.799405
\(366\) 0 0
\(367\) −36.0851 −1.88363 −0.941814 0.336136i \(-0.890880\pi\)
−0.941814 + 0.336136i \(0.890880\pi\)
\(368\) −0.771938 −0.0402400
\(369\) 0 0
\(370\) −13.9404 −0.724727
\(371\) −5.28575 −0.274422
\(372\) 0 0
\(373\) −34.8068 −1.80223 −0.901114 0.433582i \(-0.857249\pi\)
−0.901114 + 0.433582i \(0.857249\pi\)
\(374\) −18.4257 −0.952771
\(375\) 0 0
\(376\) 2.39131 0.123322
\(377\) 2.61930 0.134901
\(378\) 0 0
\(379\) 14.3593 0.737588 0.368794 0.929511i \(-0.379771\pi\)
0.368794 + 0.929511i \(0.379771\pi\)
\(380\) −5.50884 −0.282597
\(381\) 0 0
\(382\) 21.3376 1.09173
\(383\) −8.25786 −0.421957 −0.210978 0.977491i \(-0.567665\pi\)
−0.210978 + 0.977491i \(0.567665\pi\)
\(384\) 0 0
\(385\) 2.21715 0.112996
\(386\) 6.48311 0.329982
\(387\) 0 0
\(388\) −1.11861 −0.0567886
\(389\) 28.9801 1.46935 0.734676 0.678418i \(-0.237334\pi\)
0.734676 + 0.678418i \(0.237334\pi\)
\(390\) 0 0
\(391\) 1.27300 0.0643786
\(392\) −20.7301 −1.04703
\(393\) 0 0
\(394\) −25.4088 −1.28007
\(395\) −8.68288 −0.436883
\(396\) 0 0
\(397\) 3.66863 0.184123 0.0920615 0.995753i \(-0.470654\pi\)
0.0920615 + 0.995753i \(0.470654\pi\)
\(398\) −23.5814 −1.18203
\(399\) 0 0
\(400\) 6.45378 0.322689
\(401\) −5.67684 −0.283488 −0.141744 0.989903i \(-0.545271\pi\)
−0.141744 + 0.989903i \(0.545271\pi\)
\(402\) 0 0
\(403\) 33.4160 1.66457
\(404\) −2.70837 −0.134746
\(405\) 0 0
\(406\) 0.378341 0.0187768
\(407\) −20.0099 −0.991854
\(408\) 0 0
\(409\) 16.3901 0.810440 0.405220 0.914219i \(-0.367195\pi\)
0.405220 + 0.914219i \(0.367195\pi\)
\(410\) −1.44542 −0.0713842
\(411\) 0 0
\(412\) −0.962876 −0.0474375
\(413\) 4.95363 0.243752
\(414\) 0 0
\(415\) 2.67021 0.131075
\(416\) −9.06643 −0.444518
\(417\) 0 0
\(418\) 28.8625 1.41171
\(419\) 33.3282 1.62819 0.814094 0.580733i \(-0.197234\pi\)
0.814094 + 0.580733i \(0.197234\pi\)
\(420\) 0 0
\(421\) 26.0005 1.26719 0.633594 0.773666i \(-0.281579\pi\)
0.633594 + 0.773666i \(0.281579\pi\)
\(422\) −34.9183 −1.69979
\(423\) 0 0
\(424\) 36.7610 1.78527
\(425\) −10.6429 −0.516258
\(426\) 0 0
\(427\) 0.634249 0.0306935
\(428\) 5.07820 0.245464
\(429\) 0 0
\(430\) −16.8181 −0.811042
\(431\) −14.5215 −0.699478 −0.349739 0.936847i \(-0.613730\pi\)
−0.349739 + 0.936847i \(0.613730\pi\)
\(432\) 0 0
\(433\) 8.16667 0.392465 0.196232 0.980557i \(-0.437129\pi\)
0.196232 + 0.980557i \(0.437129\pi\)
\(434\) 4.82672 0.231690
\(435\) 0 0
\(436\) 3.84384 0.184087
\(437\) −1.99406 −0.0953890
\(438\) 0 0
\(439\) −13.2750 −0.633581 −0.316791 0.948496i \(-0.602605\pi\)
−0.316791 + 0.948496i \(0.602605\pi\)
\(440\) −15.4197 −0.735103
\(441\) 0 0
\(442\) −23.1863 −1.10286
\(443\) −7.07100 −0.335953 −0.167977 0.985791i \(-0.553723\pi\)
−0.167977 + 0.985791i \(0.553723\pi\)
\(444\) 0 0
\(445\) −27.6330 −1.30993
\(446\) 16.0911 0.761936
\(447\) 0 0
\(448\) −3.89685 −0.184109
\(449\) −13.1410 −0.620163 −0.310081 0.950710i \(-0.600356\pi\)
−0.310081 + 0.950710i \(0.600356\pi\)
\(450\) 0 0
\(451\) −2.07474 −0.0976957
\(452\) −7.94273 −0.373595
\(453\) 0 0
\(454\) −4.69852 −0.220512
\(455\) 2.78999 0.130797
\(456\) 0 0
\(457\) 30.8808 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(458\) 5.76361 0.269316
\(459\) 0 0
\(460\) 0.188549 0.00879115
\(461\) −4.10612 −0.191241 −0.0956205 0.995418i \(-0.530484\pi\)
−0.0956205 + 0.995418i \(0.530484\pi\)
\(462\) 0 0
\(463\) −33.2340 −1.54452 −0.772259 0.635308i \(-0.780873\pi\)
−0.772259 + 0.635308i \(0.780873\pi\)
\(464\) −2.03797 −0.0946105
\(465\) 0 0
\(466\) −35.7437 −1.65580
\(467\) 24.3933 1.12879 0.564393 0.825506i \(-0.309110\pi\)
0.564393 + 0.825506i \(0.309110\pi\)
\(468\) 0 0
\(469\) −2.75062 −0.127012
\(470\) 1.65126 0.0761670
\(471\) 0 0
\(472\) −34.4511 −1.58574
\(473\) −24.1406 −1.10998
\(474\) 0 0
\(475\) 16.6714 0.764934
\(476\) 0.917542 0.0420555
\(477\) 0 0
\(478\) −1.25296 −0.0573089
\(479\) 24.1619 1.10399 0.551994 0.833848i \(-0.313867\pi\)
0.551994 + 0.833848i \(0.313867\pi\)
\(480\) 0 0
\(481\) −25.1798 −1.14810
\(482\) −16.3726 −0.745753
\(483\) 0 0
\(484\) 0.813770 0.0369896
\(485\) −4.36429 −0.198172
\(486\) 0 0
\(487\) −10.8428 −0.491334 −0.245667 0.969354i \(-0.579007\pi\)
−0.245667 + 0.969354i \(0.579007\pi\)
\(488\) −4.41103 −0.199678
\(489\) 0 0
\(490\) −14.3147 −0.646670
\(491\) −8.33251 −0.376041 −0.188020 0.982165i \(-0.560207\pi\)
−0.188020 + 0.982165i \(0.560207\pi\)
\(492\) 0 0
\(493\) 3.36082 0.151364
\(494\) 36.3196 1.63410
\(495\) 0 0
\(496\) −25.9996 −1.16742
\(497\) −5.00261 −0.224398
\(498\) 0 0
\(499\) −5.29529 −0.237050 −0.118525 0.992951i \(-0.537817\pi\)
−0.118525 + 0.992951i \(0.537817\pi\)
\(500\) −5.18500 −0.231880
\(501\) 0 0
\(502\) 16.3852 0.731306
\(503\) −13.6978 −0.610756 −0.305378 0.952231i \(-0.598783\pi\)
−0.305378 + 0.952231i \(0.598783\pi\)
\(504\) 0 0
\(505\) −10.5668 −0.470217
\(506\) −0.987866 −0.0439160
\(507\) 0 0
\(508\) −7.83844 −0.347775
\(509\) −19.4548 −0.862319 −0.431159 0.902276i \(-0.641895\pi\)
−0.431159 + 0.902276i \(0.641895\pi\)
\(510\) 0 0
\(511\) −3.98464 −0.176270
\(512\) 25.0479 1.10697
\(513\) 0 0
\(514\) 28.1111 1.23993
\(515\) −3.75670 −0.165540
\(516\) 0 0
\(517\) 2.37020 0.104241
\(518\) −3.63706 −0.159803
\(519\) 0 0
\(520\) −19.4036 −0.850905
\(521\) −10.3167 −0.451983 −0.225992 0.974129i \(-0.572562\pi\)
−0.225992 + 0.974129i \(0.572562\pi\)
\(522\) 0 0
\(523\) 1.78034 0.0778489 0.0389245 0.999242i \(-0.487607\pi\)
0.0389245 + 0.999242i \(0.487607\pi\)
\(524\) 0.192126 0.00839308
\(525\) 0 0
\(526\) 7.79079 0.339695
\(527\) 42.8760 1.86771
\(528\) 0 0
\(529\) −22.9317 −0.997033
\(530\) 25.3844 1.10263
\(531\) 0 0
\(532\) −1.43726 −0.0623132
\(533\) −2.61079 −0.113086
\(534\) 0 0
\(535\) 19.8128 0.856582
\(536\) 19.1298 0.826281
\(537\) 0 0
\(538\) 23.1823 0.999459
\(539\) −20.5471 −0.885027
\(540\) 0 0
\(541\) 7.99971 0.343934 0.171967 0.985103i \(-0.444988\pi\)
0.171967 + 0.985103i \(0.444988\pi\)
\(542\) −30.7514 −1.32088
\(543\) 0 0
\(544\) −11.6331 −0.498765
\(545\) 14.9969 0.642397
\(546\) 0 0
\(547\) −43.8106 −1.87321 −0.936603 0.350392i \(-0.886048\pi\)
−0.936603 + 0.350392i \(0.886048\pi\)
\(548\) −3.50455 −0.149707
\(549\) 0 0
\(550\) 8.25905 0.352167
\(551\) −5.26447 −0.224274
\(552\) 0 0
\(553\) −2.26537 −0.0963334
\(554\) −2.22906 −0.0947039
\(555\) 0 0
\(556\) 3.40778 0.144522
\(557\) −43.3503 −1.83681 −0.918405 0.395641i \(-0.870522\pi\)
−0.918405 + 0.395641i \(0.870522\pi\)
\(558\) 0 0
\(559\) −30.3777 −1.28484
\(560\) −2.17078 −0.0917320
\(561\) 0 0
\(562\) −15.8988 −0.670650
\(563\) 0.411863 0.0173579 0.00867897 0.999962i \(-0.497237\pi\)
0.00867897 + 0.999962i \(0.497237\pi\)
\(564\) 0 0
\(565\) −30.9889 −1.30371
\(566\) −30.4050 −1.27802
\(567\) 0 0
\(568\) 34.7918 1.45983
\(569\) −17.9961 −0.754436 −0.377218 0.926124i \(-0.623119\pi\)
−0.377218 + 0.926124i \(0.623119\pi\)
\(570\) 0 0
\(571\) 20.8344 0.871891 0.435946 0.899973i \(-0.356414\pi\)
0.435946 + 0.899973i \(0.356414\pi\)
\(572\) −4.92943 −0.206110
\(573\) 0 0
\(574\) −0.377111 −0.0157403
\(575\) −0.570605 −0.0237959
\(576\) 0 0
\(577\) 28.4066 1.18258 0.591291 0.806458i \(-0.298618\pi\)
0.591291 + 0.806458i \(0.298618\pi\)
\(578\) −8.45007 −0.351477
\(579\) 0 0
\(580\) 0.497783 0.0206693
\(581\) 0.696660 0.0289023
\(582\) 0 0
\(583\) 36.4365 1.50905
\(584\) 27.7121 1.14673
\(585\) 0 0
\(586\) 19.2498 0.795202
\(587\) −8.95210 −0.369493 −0.184746 0.982786i \(-0.559146\pi\)
−0.184746 + 0.982786i \(0.559146\pi\)
\(588\) 0 0
\(589\) −67.1619 −2.76736
\(590\) −23.7894 −0.979395
\(591\) 0 0
\(592\) 19.5914 0.805201
\(593\) 18.0335 0.740547 0.370273 0.928923i \(-0.379264\pi\)
0.370273 + 0.928923i \(0.379264\pi\)
\(594\) 0 0
\(595\) 3.57983 0.146759
\(596\) 2.16928 0.0888572
\(597\) 0 0
\(598\) −1.24310 −0.0508341
\(599\) 14.5505 0.594517 0.297259 0.954797i \(-0.403928\pi\)
0.297259 + 0.954797i \(0.403928\pi\)
\(600\) 0 0
\(601\) 21.6209 0.881936 0.440968 0.897523i \(-0.354635\pi\)
0.440968 + 0.897523i \(0.354635\pi\)
\(602\) −4.38786 −0.178836
\(603\) 0 0
\(604\) 6.12379 0.249173
\(605\) 3.17496 0.129080
\(606\) 0 0
\(607\) −24.8442 −1.00840 −0.504198 0.863588i \(-0.668212\pi\)
−0.504198 + 0.863588i \(0.668212\pi\)
\(608\) 18.2224 0.739015
\(609\) 0 0
\(610\) −3.04593 −0.123326
\(611\) 2.98259 0.120663
\(612\) 0 0
\(613\) −33.5479 −1.35499 −0.677494 0.735528i \(-0.736934\pi\)
−0.677494 + 0.735528i \(0.736934\pi\)
\(614\) 7.37339 0.297566
\(615\) 0 0
\(616\) −4.02300 −0.162091
\(617\) −2.01405 −0.0810826 −0.0405413 0.999178i \(-0.512908\pi\)
−0.0405413 + 0.999178i \(0.512908\pi\)
\(618\) 0 0
\(619\) −16.5501 −0.665205 −0.332603 0.943067i \(-0.607927\pi\)
−0.332603 + 0.943067i \(0.607927\pi\)
\(620\) 6.35051 0.255043
\(621\) 0 0
\(622\) −24.4059 −0.978588
\(623\) −7.20946 −0.288841
\(624\) 0 0
\(625\) −9.30868 −0.372347
\(626\) −20.5730 −0.822262
\(627\) 0 0
\(628\) −7.74221 −0.308948
\(629\) −32.3082 −1.28821
\(630\) 0 0
\(631\) −12.6654 −0.504204 −0.252102 0.967701i \(-0.581122\pi\)
−0.252102 + 0.967701i \(0.581122\pi\)
\(632\) 15.7551 0.626702
\(633\) 0 0
\(634\) −41.2782 −1.63937
\(635\) −30.5820 −1.21361
\(636\) 0 0
\(637\) −25.8558 −1.02445
\(638\) −2.60804 −0.103253
\(639\) 0 0
\(640\) 10.7021 0.423038
\(641\) 16.4831 0.651043 0.325522 0.945535i \(-0.394460\pi\)
0.325522 + 0.945535i \(0.394460\pi\)
\(642\) 0 0
\(643\) 4.60496 0.181602 0.0908009 0.995869i \(-0.471057\pi\)
0.0908009 + 0.995869i \(0.471057\pi\)
\(644\) 0.0491926 0.00193846
\(645\) 0 0
\(646\) 46.6017 1.83352
\(647\) −34.5212 −1.35717 −0.678584 0.734523i \(-0.737406\pi\)
−0.678584 + 0.734523i \(0.737406\pi\)
\(648\) 0 0
\(649\) −34.1471 −1.34039
\(650\) 10.3929 0.407644
\(651\) 0 0
\(652\) −4.62659 −0.181191
\(653\) 21.9138 0.857554 0.428777 0.903410i \(-0.358945\pi\)
0.428777 + 0.903410i \(0.358945\pi\)
\(654\) 0 0
\(655\) 0.749588 0.0292888
\(656\) 2.03135 0.0793108
\(657\) 0 0
\(658\) 0.430816 0.0167949
\(659\) 2.42455 0.0944471 0.0472235 0.998884i \(-0.484963\pi\)
0.0472235 + 0.998884i \(0.484963\pi\)
\(660\) 0 0
\(661\) 22.1526 0.861638 0.430819 0.902438i \(-0.358225\pi\)
0.430819 + 0.902438i \(0.358225\pi\)
\(662\) −36.9657 −1.43671
\(663\) 0 0
\(664\) −4.84508 −0.188026
\(665\) −5.60753 −0.217451
\(666\) 0 0
\(667\) 0.180185 0.00697680
\(668\) −1.53252 −0.0592949
\(669\) 0 0
\(670\) 13.2096 0.510332
\(671\) −4.37210 −0.168783
\(672\) 0 0
\(673\) 23.3640 0.900615 0.450307 0.892874i \(-0.351314\pi\)
0.450307 + 0.892874i \(0.351314\pi\)
\(674\) 19.6262 0.755974
\(675\) 0 0
\(676\) −0.611757 −0.0235291
\(677\) −11.8456 −0.455264 −0.227632 0.973747i \(-0.573098\pi\)
−0.227632 + 0.973747i \(0.573098\pi\)
\(678\) 0 0
\(679\) −1.13865 −0.0436973
\(680\) −24.8967 −0.954746
\(681\) 0 0
\(682\) −33.2723 −1.27406
\(683\) −5.41333 −0.207135 −0.103568 0.994622i \(-0.533026\pi\)
−0.103568 + 0.994622i \(0.533026\pi\)
\(684\) 0 0
\(685\) −13.6731 −0.522424
\(686\) −7.57456 −0.289198
\(687\) 0 0
\(688\) 23.6357 0.901101
\(689\) 45.8506 1.74677
\(690\) 0 0
\(691\) −31.7264 −1.20693 −0.603464 0.797390i \(-0.706213\pi\)
−0.603464 + 0.797390i \(0.706213\pi\)
\(692\) 4.57569 0.173942
\(693\) 0 0
\(694\) −28.9237 −1.09793
\(695\) 13.2956 0.504331
\(696\) 0 0
\(697\) −3.34990 −0.126886
\(698\) 11.1906 0.423569
\(699\) 0 0
\(700\) −0.411275 −0.0155447
\(701\) 12.1452 0.458717 0.229359 0.973342i \(-0.426337\pi\)
0.229359 + 0.973342i \(0.426337\pi\)
\(702\) 0 0
\(703\) 50.6083 1.90873
\(704\) 26.8623 1.01241
\(705\) 0 0
\(706\) 36.0717 1.35758
\(707\) −2.75689 −0.103684
\(708\) 0 0
\(709\) −38.9674 −1.46345 −0.731726 0.681599i \(-0.761285\pi\)
−0.731726 + 0.681599i \(0.761285\pi\)
\(710\) 24.0247 0.901629
\(711\) 0 0
\(712\) 50.1399 1.87907
\(713\) 2.29873 0.0860881
\(714\) 0 0
\(715\) −19.2324 −0.719250
\(716\) −1.20221 −0.0449285
\(717\) 0 0
\(718\) −40.9938 −1.52988
\(719\) 16.5847 0.618506 0.309253 0.950980i \(-0.399921\pi\)
0.309253 + 0.950980i \(0.399921\pi\)
\(720\) 0 0
\(721\) −0.980126 −0.0365018
\(722\) −49.1918 −1.83073
\(723\) 0 0
\(724\) −1.05377 −0.0391630
\(725\) −1.50644 −0.0559477
\(726\) 0 0
\(727\) 11.7059 0.434146 0.217073 0.976155i \(-0.430349\pi\)
0.217073 + 0.976155i \(0.430349\pi\)
\(728\) −5.06242 −0.187626
\(729\) 0 0
\(730\) 19.1359 0.708253
\(731\) −38.9776 −1.44164
\(732\) 0 0
\(733\) −18.9548 −0.700112 −0.350056 0.936729i \(-0.613837\pi\)
−0.350056 + 0.936729i \(0.613837\pi\)
\(734\) 45.2131 1.66885
\(735\) 0 0
\(736\) −0.623691 −0.0229896
\(737\) 18.9610 0.698436
\(738\) 0 0
\(739\) −0.435521 −0.0160209 −0.00801045 0.999968i \(-0.502550\pi\)
−0.00801045 + 0.999968i \(0.502550\pi\)
\(740\) −4.78528 −0.175910
\(741\) 0 0
\(742\) 6.62282 0.243131
\(743\) −11.5604 −0.424110 −0.212055 0.977258i \(-0.568016\pi\)
−0.212055 + 0.977258i \(0.568016\pi\)
\(744\) 0 0
\(745\) 8.46354 0.310080
\(746\) 43.6114 1.59673
\(747\) 0 0
\(748\) −6.32494 −0.231263
\(749\) 5.16918 0.188878
\(750\) 0 0
\(751\) −18.9573 −0.691762 −0.345881 0.938278i \(-0.612420\pi\)
−0.345881 + 0.938278i \(0.612420\pi\)
\(752\) −2.32063 −0.0846247
\(753\) 0 0
\(754\) −3.28187 −0.119519
\(755\) 23.8922 0.869526
\(756\) 0 0
\(757\) −36.1482 −1.31383 −0.656914 0.753965i \(-0.728139\pi\)
−0.656914 + 0.753965i \(0.728139\pi\)
\(758\) −17.9916 −0.653484
\(759\) 0 0
\(760\) 38.9988 1.41464
\(761\) 33.4183 1.21141 0.605707 0.795688i \(-0.292890\pi\)
0.605707 + 0.795688i \(0.292890\pi\)
\(762\) 0 0
\(763\) 3.91271 0.141650
\(764\) 7.32449 0.264991
\(765\) 0 0
\(766\) 10.3467 0.373843
\(767\) −42.9696 −1.55154
\(768\) 0 0
\(769\) 46.7975 1.68756 0.843781 0.536688i \(-0.180325\pi\)
0.843781 + 0.536688i \(0.180325\pi\)
\(770\) −2.77799 −0.100112
\(771\) 0 0
\(772\) 2.22544 0.0800953
\(773\) 49.1281 1.76702 0.883508 0.468416i \(-0.155175\pi\)
0.883508 + 0.468416i \(0.155175\pi\)
\(774\) 0 0
\(775\) −19.2185 −0.690349
\(776\) 7.91898 0.284275
\(777\) 0 0
\(778\) −36.3109 −1.30181
\(779\) 5.24736 0.188006
\(780\) 0 0
\(781\) 34.4847 1.23396
\(782\) −1.59502 −0.0570378
\(783\) 0 0
\(784\) 20.1174 0.718477
\(785\) −30.2065 −1.07812
\(786\) 0 0
\(787\) −25.2734 −0.900901 −0.450450 0.892801i \(-0.648737\pi\)
−0.450450 + 0.892801i \(0.648737\pi\)
\(788\) −8.72199 −0.310708
\(789\) 0 0
\(790\) 10.8793 0.387067
\(791\) −8.08503 −0.287470
\(792\) 0 0
\(793\) −5.50172 −0.195372
\(794\) −4.59663 −0.163128
\(795\) 0 0
\(796\) −8.09471 −0.286909
\(797\) −16.3302 −0.578447 −0.289223 0.957262i \(-0.593397\pi\)
−0.289223 + 0.957262i \(0.593397\pi\)
\(798\) 0 0
\(799\) 3.82695 0.135388
\(800\) 5.21437 0.184356
\(801\) 0 0
\(802\) 7.11284 0.251163
\(803\) 27.4675 0.969308
\(804\) 0 0
\(805\) 0.191927 0.00676454
\(806\) −41.8688 −1.47476
\(807\) 0 0
\(808\) 19.1734 0.674519
\(809\) −6.61012 −0.232400 −0.116200 0.993226i \(-0.537071\pi\)
−0.116200 + 0.993226i \(0.537071\pi\)
\(810\) 0 0
\(811\) 8.25580 0.289900 0.144950 0.989439i \(-0.453698\pi\)
0.144950 + 0.989439i \(0.453698\pi\)
\(812\) 0.129872 0.00455762
\(813\) 0 0
\(814\) 25.0715 0.878757
\(815\) −18.0508 −0.632292
\(816\) 0 0
\(817\) 61.0554 2.13606
\(818\) −20.5361 −0.718029
\(819\) 0 0
\(820\) −0.496165 −0.0173268
\(821\) 32.1939 1.12358 0.561788 0.827282i \(-0.310114\pi\)
0.561788 + 0.827282i \(0.310114\pi\)
\(822\) 0 0
\(823\) −34.2382 −1.19347 −0.596735 0.802439i \(-0.703536\pi\)
−0.596735 + 0.802439i \(0.703536\pi\)
\(824\) 6.81651 0.237464
\(825\) 0 0
\(826\) −6.20668 −0.215958
\(827\) 35.3643 1.22974 0.614869 0.788629i \(-0.289209\pi\)
0.614869 + 0.788629i \(0.289209\pi\)
\(828\) 0 0
\(829\) 6.39310 0.222041 0.111021 0.993818i \(-0.464588\pi\)
0.111021 + 0.993818i \(0.464588\pi\)
\(830\) −3.34566 −0.116129
\(831\) 0 0
\(832\) 33.8027 1.17190
\(833\) −33.1756 −1.14947
\(834\) 0 0
\(835\) −5.97918 −0.206918
\(836\) 9.90754 0.342659
\(837\) 0 0
\(838\) −41.7588 −1.44253
\(839\) 47.5595 1.64194 0.820968 0.570974i \(-0.193434\pi\)
0.820968 + 0.570974i \(0.193434\pi\)
\(840\) 0 0
\(841\) −28.5243 −0.983596
\(842\) −32.5775 −1.12270
\(843\) 0 0
\(844\) −11.9863 −0.412585
\(845\) −2.38679 −0.0821081
\(846\) 0 0
\(847\) 0.828349 0.0284624
\(848\) −35.6744 −1.22507
\(849\) 0 0
\(850\) 13.3351 0.457392
\(851\) −1.73215 −0.0593775
\(852\) 0 0
\(853\) 27.9031 0.955384 0.477692 0.878527i \(-0.341474\pi\)
0.477692 + 0.878527i \(0.341474\pi\)
\(854\) −0.794687 −0.0271936
\(855\) 0 0
\(856\) −35.9502 −1.22875
\(857\) 41.0114 1.40092 0.700462 0.713690i \(-0.252977\pi\)
0.700462 + 0.713690i \(0.252977\pi\)
\(858\) 0 0
\(859\) 17.8846 0.610215 0.305107 0.952318i \(-0.401308\pi\)
0.305107 + 0.952318i \(0.401308\pi\)
\(860\) −5.77311 −0.196861
\(861\) 0 0
\(862\) 18.1949 0.619719
\(863\) 14.1941 0.483174 0.241587 0.970379i \(-0.422332\pi\)
0.241587 + 0.970379i \(0.422332\pi\)
\(864\) 0 0
\(865\) 17.8522 0.606995
\(866\) −10.2325 −0.347714
\(867\) 0 0
\(868\) 1.65685 0.0562373
\(869\) 15.6160 0.529737
\(870\) 0 0
\(871\) 23.8599 0.808461
\(872\) −27.2118 −0.921508
\(873\) 0 0
\(874\) 2.49848 0.0845122
\(875\) −5.27789 −0.178425
\(876\) 0 0
\(877\) 16.4208 0.554490 0.277245 0.960799i \(-0.410579\pi\)
0.277245 + 0.960799i \(0.410579\pi\)
\(878\) 16.6330 0.561337
\(879\) 0 0
\(880\) 14.9639 0.504433
\(881\) 13.1795 0.444029 0.222014 0.975043i \(-0.428737\pi\)
0.222014 + 0.975043i \(0.428737\pi\)
\(882\) 0 0
\(883\) −57.2316 −1.92600 −0.962999 0.269506i \(-0.913140\pi\)
−0.962999 + 0.269506i \(0.913140\pi\)
\(884\) −7.95911 −0.267694
\(885\) 0 0
\(886\) 8.85966 0.297646
\(887\) −38.0535 −1.27771 −0.638857 0.769326i \(-0.720592\pi\)
−0.638857 + 0.769326i \(0.720592\pi\)
\(888\) 0 0
\(889\) −7.97887 −0.267603
\(890\) 34.6229 1.16056
\(891\) 0 0
\(892\) 5.52354 0.184942
\(893\) −5.99463 −0.200603
\(894\) 0 0
\(895\) −4.69045 −0.156785
\(896\) 2.79219 0.0932806
\(897\) 0 0
\(898\) 16.4651 0.549448
\(899\) 6.06881 0.202406
\(900\) 0 0
\(901\) 58.8308 1.95994
\(902\) 2.59956 0.0865559
\(903\) 0 0
\(904\) 56.2292 1.87015
\(905\) −4.11132 −0.136665
\(906\) 0 0
\(907\) −23.2951 −0.773500 −0.386750 0.922185i \(-0.626402\pi\)
−0.386750 + 0.922185i \(0.626402\pi\)
\(908\) −1.61285 −0.0535242
\(909\) 0 0
\(910\) −3.49574 −0.115883
\(911\) −29.0411 −0.962175 −0.481088 0.876673i \(-0.659758\pi\)
−0.481088 + 0.876673i \(0.659758\pi\)
\(912\) 0 0
\(913\) −4.80232 −0.158934
\(914\) −38.6924 −1.27983
\(915\) 0 0
\(916\) 1.97846 0.0653701
\(917\) 0.195568 0.00645823
\(918\) 0 0
\(919\) −37.6246 −1.24112 −0.620561 0.784158i \(-0.713095\pi\)
−0.620561 + 0.784158i \(0.713095\pi\)
\(920\) −1.33480 −0.0440070
\(921\) 0 0
\(922\) 5.14479 0.169435
\(923\) 43.3945 1.42835
\(924\) 0 0
\(925\) 14.4817 0.476154
\(926\) 41.6408 1.36840
\(927\) 0 0
\(928\) −1.64659 −0.0540520
\(929\) 53.0432 1.74029 0.870145 0.492796i \(-0.164025\pi\)
0.870145 + 0.492796i \(0.164025\pi\)
\(930\) 0 0
\(931\) 51.9670 1.70315
\(932\) −12.2696 −0.401906
\(933\) 0 0
\(934\) −30.5637 −1.00008
\(935\) −24.6770 −0.807024
\(936\) 0 0
\(937\) −12.0063 −0.392228 −0.196114 0.980581i \(-0.562832\pi\)
−0.196114 + 0.980581i \(0.562832\pi\)
\(938\) 3.44640 0.112529
\(939\) 0 0
\(940\) 0.566824 0.0184878
\(941\) −30.1289 −0.982173 −0.491086 0.871111i \(-0.663400\pi\)
−0.491086 + 0.871111i \(0.663400\pi\)
\(942\) 0 0
\(943\) −0.179600 −0.00584857
\(944\) 33.4329 1.08815
\(945\) 0 0
\(946\) 30.2471 0.983417
\(947\) 16.5223 0.536902 0.268451 0.963293i \(-0.413488\pi\)
0.268451 + 0.963293i \(0.413488\pi\)
\(948\) 0 0
\(949\) 34.5643 1.12200
\(950\) −20.8885 −0.677712
\(951\) 0 0
\(952\) −6.49558 −0.210523
\(953\) 51.7178 1.67531 0.837653 0.546204i \(-0.183927\pi\)
0.837653 + 0.546204i \(0.183927\pi\)
\(954\) 0 0
\(955\) 28.5768 0.924723
\(956\) −0.430099 −0.0139104
\(957\) 0 0
\(958\) −30.2739 −0.978104
\(959\) −3.56734 −0.115195
\(960\) 0 0
\(961\) 46.4233 1.49753
\(962\) 31.5492 1.01719
\(963\) 0 0
\(964\) −5.62019 −0.181014
\(965\) 8.68263 0.279504
\(966\) 0 0
\(967\) −31.8965 −1.02572 −0.512862 0.858471i \(-0.671415\pi\)
−0.512862 + 0.858471i \(0.671415\pi\)
\(968\) −5.76094 −0.185164
\(969\) 0 0
\(970\) 5.46826 0.175575
\(971\) 52.5319 1.68583 0.842915 0.538047i \(-0.180838\pi\)
0.842915 + 0.538047i \(0.180838\pi\)
\(972\) 0 0
\(973\) 3.46883 0.111206
\(974\) 13.5856 0.435310
\(975\) 0 0
\(976\) 4.28066 0.137021
\(977\) 52.8123 1.68961 0.844807 0.535071i \(-0.179715\pi\)
0.844807 + 0.535071i \(0.179715\pi\)
\(978\) 0 0
\(979\) 49.6974 1.58833
\(980\) −4.91375 −0.156964
\(981\) 0 0
\(982\) 10.4403 0.333163
\(983\) −9.10617 −0.290442 −0.145221 0.989399i \(-0.546389\pi\)
−0.145221 + 0.989399i \(0.546389\pi\)
\(984\) 0 0
\(985\) −34.0292 −1.08426
\(986\) −4.21097 −0.134104
\(987\) 0 0
\(988\) 12.4673 0.396639
\(989\) −2.08972 −0.0664493
\(990\) 0 0
\(991\) 15.1055 0.479843 0.239921 0.970792i \(-0.422878\pi\)
0.239921 + 0.970792i \(0.422878\pi\)
\(992\) −21.0065 −0.666957
\(993\) 0 0
\(994\) 6.26806 0.198811
\(995\) −31.5818 −1.00121
\(996\) 0 0
\(997\) 7.86287 0.249019 0.124510 0.992218i \(-0.460264\pi\)
0.124510 + 0.992218i \(0.460264\pi\)
\(998\) 6.63477 0.210020
\(999\) 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 2151.2.a.k.1.6 yes 20
3.2 odd 2 2151.2.a.j.1.15 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.15 20 3.2 odd 2
2151.2.a.k.1.6 yes 20 1.1 even 1 trivial