Properties

Label 441.2.c.b.440.4
Level $441$
Weight $2$
Character 441.440
Analytic conductor $3.521$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 440.4
Root \(0.923880 + 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 441.440
Dual form 441.2.c.b.440.6

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} -1.58579i q^{8} +O(q^{10})\) \(q-0.414214i q^{2} +1.82843 q^{4} +2.93015 q^{5} -1.58579i q^{8} -1.21371i q^{10} -4.82843i q^{11} +2.93015i q^{13} +3.00000 q^{16} -7.07401 q^{17} +5.86030i q^{19} +5.35757 q^{20} -2.00000 q^{22} +2.00000i q^{23} +3.58579 q^{25} +1.21371 q^{26} +0.828427i q^{29} -5.86030i q^{31} -4.41421i q^{32} +2.93015i q^{34} -5.41421 q^{37} +2.42742 q^{38} -4.64659i q^{40} -1.21371 q^{41} +4.48528 q^{43} -8.82843i q^{44} +0.828427 q^{46} +5.86030 q^{47} -1.48528i q^{50} +5.35757i q^{52} +7.07107i q^{53} -14.1480i q^{55} +0.343146 q^{58} +5.86030 q^{59} +1.21371i q^{61} -2.42742 q^{62} +4.17157 q^{64} +8.58579i q^{65} -8.48528 q^{67} -12.9343 q^{68} -0.828427i q^{71} +7.07401i q^{73} +2.24264i q^{74} +10.7151i q^{76} +1.65685 q^{79} +8.79045 q^{80} +0.502734i q^{82} -11.7206 q^{83} -20.7279 q^{85} -1.85786i q^{86} -7.65685 q^{88} -11.2179 q^{89} +3.65685i q^{92} -2.42742i q^{94} +17.1716i q^{95} -7.07401i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + O(q^{10}) \) \( 8q - 8q^{4} + 24q^{16} - 16q^{22} + 40q^{25} - 32q^{37} - 32q^{43} - 16q^{46} + 48q^{58} + 56q^{64} - 32q^{79} - 64q^{85} - 16q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) 0 0
\(4\) 1.82843 0.914214
\(5\) 2.93015 1.31040 0.655202 0.755454i \(-0.272584\pi\)
0.655202 + 0.755454i \(0.272584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 1.58579i − 0.560660i
\(9\) 0 0
\(10\) − 1.21371i − 0.383808i
\(11\) − 4.82843i − 1.45583i −0.685670 0.727913i \(-0.740491\pi\)
0.685670 0.727913i \(-0.259509\pi\)
\(12\) 0 0
\(13\) 2.93015i 0.812678i 0.913722 + 0.406339i \(0.133195\pi\)
−0.913722 + 0.406339i \(0.866805\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −7.07401 −1.71570 −0.857850 0.513900i \(-0.828200\pi\)
−0.857850 + 0.513900i \(0.828200\pi\)
\(18\) 0 0
\(19\) 5.86030i 1.34445i 0.740349 + 0.672223i \(0.234660\pi\)
−0.740349 + 0.672223i \(0.765340\pi\)
\(20\) 5.35757 1.19799
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 3.58579 0.717157
\(26\) 1.21371 0.238028
\(27\) 0 0
\(28\) 0 0
\(29\) 0.828427i 0.153835i 0.997037 + 0.0769175i \(0.0245078\pi\)
−0.997037 + 0.0769175i \(0.975492\pi\)
\(30\) 0 0
\(31\) − 5.86030i − 1.05254i −0.850317 0.526271i \(-0.823590\pi\)
0.850317 0.526271i \(-0.176410\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) 0 0
\(34\) 2.93015i 0.502517i
\(35\) 0 0
\(36\) 0 0
\(37\) −5.41421 −0.890091 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(38\) 2.42742 0.393779
\(39\) 0 0
\(40\) − 4.64659i − 0.734691i
\(41\) −1.21371 −0.189549 −0.0947747 0.995499i \(-0.530213\pi\)
−0.0947747 + 0.995499i \(0.530213\pi\)
\(42\) 0 0
\(43\) 4.48528 0.683999 0.341999 0.939700i \(-0.388896\pi\)
0.341999 + 0.939700i \(0.388896\pi\)
\(44\) − 8.82843i − 1.33094i
\(45\) 0 0
\(46\) 0.828427 0.122145
\(47\) 5.86030 0.854813 0.427406 0.904060i \(-0.359427\pi\)
0.427406 + 0.904060i \(0.359427\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 1.48528i − 0.210051i
\(51\) 0 0
\(52\) 5.35757i 0.742961i
\(53\) 7.07107i 0.971286i 0.874157 + 0.485643i \(0.161414\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) − 14.1480i − 1.90772i
\(56\) 0 0
\(57\) 0 0
\(58\) 0.343146 0.0450572
\(59\) 5.86030 0.762946 0.381473 0.924380i \(-0.375417\pi\)
0.381473 + 0.924380i \(0.375417\pi\)
\(60\) 0 0
\(61\) 1.21371i 0.155399i 0.996977 + 0.0776997i \(0.0247575\pi\)
−0.996977 + 0.0776997i \(0.975242\pi\)
\(62\) −2.42742 −0.308282
\(63\) 0 0
\(64\) 4.17157 0.521447
\(65\) 8.58579i 1.06494i
\(66\) 0 0
\(67\) −8.48528 −1.03664 −0.518321 0.855186i \(-0.673443\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) −12.9343 −1.56852
\(69\) 0 0
\(70\) 0 0
\(71\) − 0.828427i − 0.0983162i −0.998791 0.0491581i \(-0.984346\pi\)
0.998791 0.0491581i \(-0.0156538\pi\)
\(72\) 0 0
\(73\) 7.07401i 0.827950i 0.910288 + 0.413975i \(0.135860\pi\)
−0.910288 + 0.413975i \(0.864140\pi\)
\(74\) 2.24264i 0.260702i
\(75\) 0 0
\(76\) 10.7151i 1.22911i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.65685 0.186411 0.0932053 0.995647i \(-0.470289\pi\)
0.0932053 + 0.995647i \(0.470289\pi\)
\(80\) 8.79045 0.982803
\(81\) 0 0
\(82\) 0.502734i 0.0555177i
\(83\) −11.7206 −1.28650 −0.643252 0.765655i \(-0.722415\pi\)
−0.643252 + 0.765655i \(0.722415\pi\)
\(84\) 0 0
\(85\) −20.7279 −2.24826
\(86\) − 1.85786i − 0.200339i
\(87\) 0 0
\(88\) −7.65685 −0.816223
\(89\) −11.2179 −1.18909 −0.594546 0.804062i \(-0.702668\pi\)
−0.594546 + 0.804062i \(0.702668\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.65685i 0.381253i
\(93\) 0 0
\(94\) − 2.42742i − 0.250369i
\(95\) 17.1716i 1.76177i
\(96\) 0 0
\(97\) − 7.07401i − 0.718257i −0.933288 0.359128i \(-0.883074\pi\)
0.933288 0.359128i \(-0.116926\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 6.55635 0.655635
\(101\) −5.35757 −0.533098 −0.266549 0.963821i \(-0.585883\pi\)
−0.266549 + 0.963821i \(0.585883\pi\)
\(102\) 0 0
\(103\) − 14.1480i − 1.39405i −0.717049 0.697023i \(-0.754508\pi\)
0.717049 0.697023i \(-0.245492\pi\)
\(104\) 4.64659 0.455636
\(105\) 0 0
\(106\) 2.92893 0.284483
\(107\) 13.3137i 1.28708i 0.765410 + 0.643542i \(0.222536\pi\)
−0.765410 + 0.643542i \(0.777464\pi\)
\(108\) 0 0
\(109\) −5.41421 −0.518588 −0.259294 0.965798i \(-0.583490\pi\)
−0.259294 + 0.965798i \(0.583490\pi\)
\(110\) −5.86030 −0.558758
\(111\) 0 0
\(112\) 0 0
\(113\) 13.8995i 1.30755i 0.756687 + 0.653777i \(0.226817\pi\)
−0.756687 + 0.653777i \(0.773183\pi\)
\(114\) 0 0
\(115\) 5.86030i 0.546476i
\(116\) 1.51472i 0.140638i
\(117\) 0 0
\(118\) − 2.42742i − 0.223462i
\(119\) 0 0
\(120\) 0 0
\(121\) −12.3137 −1.11943
\(122\) 0.502734 0.0455154
\(123\) 0 0
\(124\) − 10.7151i − 0.962248i
\(125\) −4.14386 −0.370638
\(126\) 0 0
\(127\) 16.4853 1.46283 0.731416 0.681931i \(-0.238860\pi\)
0.731416 + 0.681931i \(0.238860\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) 0 0
\(130\) 3.55635 0.311912
\(131\) 8.28772 0.724101 0.362051 0.932158i \(-0.382077\pi\)
0.362051 + 0.932158i \(0.382077\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.51472i 0.303625i
\(135\) 0 0
\(136\) 11.2179i 0.961924i
\(137\) 4.82843i 0.412520i 0.978497 + 0.206260i \(0.0661293\pi\)
−0.978497 + 0.206260i \(0.933871\pi\)
\(138\) 0 0
\(139\) 8.28772i 0.702955i 0.936196 + 0.351478i \(0.114321\pi\)
−0.936196 + 0.351478i \(0.885679\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.343146 −0.0287962
\(143\) 14.1480 1.18312
\(144\) 0 0
\(145\) 2.42742i 0.201586i
\(146\) 2.93015 0.242501
\(147\) 0 0
\(148\) −9.89949 −0.813733
\(149\) − 4.24264i − 0.347571i −0.984784 0.173785i \(-0.944400\pi\)
0.984784 0.173785i \(-0.0555999\pi\)
\(150\) 0 0
\(151\) 4.48528 0.365007 0.182504 0.983205i \(-0.441580\pi\)
0.182504 + 0.983205i \(0.441580\pi\)
\(152\) 9.29319 0.753777
\(153\) 0 0
\(154\) 0 0
\(155\) − 17.1716i − 1.37925i
\(156\) 0 0
\(157\) − 22.9385i − 1.83069i −0.402671 0.915345i \(-0.631918\pi\)
0.402671 0.915345i \(-0.368082\pi\)
\(158\) − 0.686292i − 0.0545984i
\(159\) 0 0
\(160\) − 12.9343i − 1.02255i
\(161\) 0 0
\(162\) 0 0
\(163\) 9.17157 0.718373 0.359187 0.933266i \(-0.383054\pi\)
0.359187 + 0.933266i \(0.383054\pi\)
\(164\) −2.21918 −0.173289
\(165\) 0 0
\(166\) 4.85483i 0.376808i
\(167\) −14.1480 −1.09481 −0.547403 0.836869i \(-0.684384\pi\)
−0.547403 + 0.836869i \(0.684384\pi\)
\(168\) 0 0
\(169\) 4.41421 0.339555
\(170\) 8.58579i 0.658500i
\(171\) 0 0
\(172\) 8.20101 0.625321
\(173\) −1.21371 −0.0922765 −0.0461383 0.998935i \(-0.514691\pi\)
−0.0461383 + 0.998935i \(0.514691\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 14.4853i − 1.09187i
\(177\) 0 0
\(178\) 4.64659i 0.348277i
\(179\) − 26.4853i − 1.97960i −0.142454 0.989801i \(-0.545499\pi\)
0.142454 0.989801i \(-0.454501\pi\)
\(180\) 0 0
\(181\) 9.50143i 0.706236i 0.935579 + 0.353118i \(0.114879\pi\)
−0.935579 + 0.353118i \(0.885121\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3.17157 0.233811
\(185\) −15.8645 −1.16638
\(186\) 0 0
\(187\) 34.1563i 2.49776i
\(188\) 10.7151 0.781482
\(189\) 0 0
\(190\) 7.11270 0.516009
\(191\) − 15.1716i − 1.09778i −0.835896 0.548888i \(-0.815051\pi\)
0.835896 0.548888i \(-0.184949\pi\)
\(192\) 0 0
\(193\) −6.34315 −0.456590 −0.228295 0.973592i \(-0.573315\pi\)
−0.228295 + 0.973592i \(0.573315\pi\)
\(194\) −2.93015 −0.210373
\(195\) 0 0
\(196\) 0 0
\(197\) 7.75736i 0.552689i 0.961059 + 0.276344i \(0.0891231\pi\)
−0.961059 + 0.276344i \(0.910877\pi\)
\(198\) 0 0
\(199\) 16.5754i 1.17500i 0.809224 + 0.587501i \(0.199888\pi\)
−0.809224 + 0.587501i \(0.800112\pi\)
\(200\) − 5.68629i − 0.402082i
\(201\) 0 0
\(202\) 2.21918i 0.156141i
\(203\) 0 0
\(204\) 0 0
\(205\) −3.55635 −0.248386
\(206\) −5.86030 −0.408307
\(207\) 0 0
\(208\) 8.79045i 0.609508i
\(209\) 28.2960 1.95728
\(210\) 0 0
\(211\) −15.3137 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(212\) 12.9289i 0.887963i
\(213\) 0 0
\(214\) 5.51472 0.376978
\(215\) 13.1426 0.896315
\(216\) 0 0
\(217\) 0 0
\(218\) 2.24264i 0.151891i
\(219\) 0 0
\(220\) − 25.8686i − 1.74406i
\(221\) − 20.7279i − 1.39431i
\(222\) 0 0
\(223\) 3.43289i 0.229883i 0.993372 + 0.114942i \(0.0366681\pi\)
−0.993372 + 0.114942i \(0.963332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.75736 0.382974
\(227\) −14.1480 −0.939037 −0.469519 0.882923i \(-0.655573\pi\)
−0.469519 + 0.882923i \(0.655573\pi\)
\(228\) 0 0
\(229\) − 18.7946i − 1.24198i −0.783817 0.620992i \(-0.786730\pi\)
0.783817 0.620992i \(-0.213270\pi\)
\(230\) 2.42742 0.160059
\(231\) 0 0
\(232\) 1.31371 0.0862492
\(233\) − 5.17157i − 0.338801i −0.985547 0.169401i \(-0.945817\pi\)
0.985547 0.169401i \(-0.0541831\pi\)
\(234\) 0 0
\(235\) 17.1716 1.12015
\(236\) 10.7151 0.697496
\(237\) 0 0
\(238\) 0 0
\(239\) − 11.6569i − 0.754019i −0.926209 0.377010i \(-0.876952\pi\)
0.926209 0.377010i \(-0.123048\pi\)
\(240\) 0 0
\(241\) − 25.3659i − 1.63396i −0.576665 0.816980i \(-0.695646\pi\)
0.576665 0.816980i \(-0.304354\pi\)
\(242\) 5.10051i 0.327873i
\(243\) 0 0
\(244\) 2.21918i 0.142068i
\(245\) 0 0
\(246\) 0 0
\(247\) −17.1716 −1.09260
\(248\) −9.29319 −0.590118
\(249\) 0 0
\(250\) 1.71644i 0.108557i
\(251\) 10.7151 0.676333 0.338167 0.941086i \(-0.390193\pi\)
0.338167 + 0.941086i \(0.390193\pi\)
\(252\) 0 0
\(253\) 9.65685 0.607121
\(254\) − 6.82843i − 0.428454i
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 10.5069 0.655402 0.327701 0.944781i \(-0.393726\pi\)
0.327701 + 0.944781i \(0.393726\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 15.6985i 0.973579i
\(261\) 0 0
\(262\) − 3.43289i − 0.212084i
\(263\) − 4.14214i − 0.255415i −0.991812 0.127708i \(-0.959238\pi\)
0.991812 0.127708i \(-0.0407619\pi\)
\(264\) 0 0
\(265\) 20.7193i 1.27278i
\(266\) 0 0
\(267\) 0 0
\(268\) −15.5147 −0.947712
\(269\) −2.21918 −0.135306 −0.0676528 0.997709i \(-0.521551\pi\)
−0.0676528 + 0.997709i \(0.521551\pi\)
\(270\) 0 0
\(271\) − 5.86030i − 0.355988i −0.984032 0.177994i \(-0.943039\pi\)
0.984032 0.177994i \(-0.0569608\pi\)
\(272\) −21.2220 −1.28677
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) − 17.3137i − 1.04406i
\(276\) 0 0
\(277\) 12.9706 0.779326 0.389663 0.920958i \(-0.372592\pi\)
0.389663 + 0.920958i \(0.372592\pi\)
\(278\) 3.43289 0.205891
\(279\) 0 0
\(280\) 0 0
\(281\) 13.1716i 0.785750i 0.919592 + 0.392875i \(0.128520\pi\)
−0.919592 + 0.392875i \(0.871480\pi\)
\(282\) 0 0
\(283\) 9.29319i 0.552423i 0.961097 + 0.276211i \(0.0890790\pi\)
−0.961097 + 0.276211i \(0.910921\pi\)
\(284\) − 1.51472i − 0.0898820i
\(285\) 0 0
\(286\) − 5.86030i − 0.346527i
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0416 1.94363
\(290\) 1.00547 0.0590432
\(291\) 0 0
\(292\) 12.9343i 0.756923i
\(293\) 22.9385 1.34008 0.670040 0.742325i \(-0.266277\pi\)
0.670040 + 0.742325i \(0.266277\pi\)
\(294\) 0 0
\(295\) 17.1716 0.999768
\(296\) 8.58579i 0.499039i
\(297\) 0 0
\(298\) −1.75736 −0.101801
\(299\) −5.86030 −0.338910
\(300\) 0 0
\(301\) 0 0
\(302\) − 1.85786i − 0.106908i
\(303\) 0 0
\(304\) 17.5809i 1.00833i
\(305\) 3.55635i 0.203636i
\(306\) 0 0
\(307\) − 22.4357i − 1.28048i −0.768177 0.640238i \(-0.778836\pi\)
0.768177 0.640238i \(-0.221164\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −7.11270 −0.403974
\(311\) 25.8686 1.46688 0.733438 0.679757i \(-0.237915\pi\)
0.733438 + 0.679757i \(0.237915\pi\)
\(312\) 0 0
\(313\) − 5.35757i − 0.302828i −0.988470 0.151414i \(-0.951617\pi\)
0.988470 0.151414i \(-0.0483826\pi\)
\(314\) −9.50143 −0.536197
\(315\) 0 0
\(316\) 3.02944 0.170419
\(317\) 32.2426i 1.81093i 0.424424 + 0.905464i \(0.360477\pi\)
−0.424424 + 0.905464i \(0.639523\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 12.2233 0.683305
\(321\) 0 0
\(322\) 0 0
\(323\) − 41.4558i − 2.30666i
\(324\) 0 0
\(325\) 10.5069i 0.582818i
\(326\) − 3.79899i − 0.210407i
\(327\) 0 0
\(328\) 1.92468i 0.106273i
\(329\) 0 0
\(330\) 0 0
\(331\) 21.6569 1.19037 0.595184 0.803589i \(-0.297079\pi\)
0.595184 + 0.803589i \(0.297079\pi\)
\(332\) −21.4303 −1.17614
\(333\) 0 0
\(334\) 5.86030i 0.320661i
\(335\) −24.8632 −1.35842
\(336\) 0 0
\(337\) −27.0711 −1.47466 −0.737328 0.675535i \(-0.763913\pi\)
−0.737328 + 0.675535i \(0.763913\pi\)
\(338\) − 1.82843i − 0.0994533i
\(339\) 0 0
\(340\) −37.8995 −2.05539
\(341\) −28.2960 −1.53232
\(342\) 0 0
\(343\) 0 0
\(344\) − 7.11270i − 0.383491i
\(345\) 0 0
\(346\) 0.502734i 0.0270272i
\(347\) 14.9706i 0.803662i 0.915714 + 0.401831i \(0.131626\pi\)
−0.915714 + 0.401831i \(0.868374\pi\)
\(348\) 0 0
\(349\) 11.2179i 0.600479i 0.953864 + 0.300239i \(0.0970666\pi\)
−0.953864 + 0.300239i \(0.902933\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −21.3137 −1.13602
\(353\) 20.5111 1.09169 0.545847 0.837885i \(-0.316208\pi\)
0.545847 + 0.837885i \(0.316208\pi\)
\(354\) 0 0
\(355\) − 2.42742i − 0.128834i
\(356\) −20.5111 −1.08708
\(357\) 0 0
\(358\) −10.9706 −0.579812
\(359\) − 34.9706i − 1.84568i −0.385189 0.922838i \(-0.625864\pi\)
0.385189 0.922838i \(-0.374136\pi\)
\(360\) 0 0
\(361\) −15.3431 −0.807534
\(362\) 3.93562 0.206852
\(363\) 0 0
\(364\) 0 0
\(365\) 20.7279i 1.08495i
\(366\) 0 0
\(367\) 20.0083i 1.04443i 0.852815 + 0.522213i \(0.174893\pi\)
−0.852815 + 0.522213i \(0.825107\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 0 0
\(370\) 6.57128i 0.341624i
\(371\) 0 0
\(372\) 0 0
\(373\) 22.6274 1.17160 0.585802 0.810454i \(-0.300780\pi\)
0.585802 + 0.810454i \(0.300780\pi\)
\(374\) 14.1480 0.731577
\(375\) 0 0
\(376\) − 9.29319i − 0.479260i
\(377\) −2.42742 −0.125018
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 31.3970i 1.61063i
\(381\) 0 0
\(382\) −6.28427 −0.321531
\(383\) −23.4412 −1.19779 −0.598895 0.800828i \(-0.704393\pi\)
−0.598895 + 0.800828i \(0.704393\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.62742i 0.133732i
\(387\) 0 0
\(388\) − 12.9343i − 0.656640i
\(389\) 25.7990i 1.30806i 0.756468 + 0.654030i \(0.226923\pi\)
−0.756468 + 0.654030i \(0.773077\pi\)
\(390\) 0 0
\(391\) − 14.1480i − 0.715496i
\(392\) 0 0
\(393\) 0 0
\(394\) 3.21320 0.161879
\(395\) 4.85483 0.244273
\(396\) 0 0
\(397\) − 17.7891i − 0.892812i −0.894831 0.446406i \(-0.852704\pi\)
0.894831 0.446406i \(-0.147296\pi\)
\(398\) 6.86577 0.344150
\(399\) 0 0
\(400\) 10.7574 0.537868
\(401\) 5.17157i 0.258256i 0.991628 + 0.129128i \(0.0412178\pi\)
−0.991628 + 0.129128i \(0.958782\pi\)
\(402\) 0 0
\(403\) 17.1716 0.855377
\(404\) −9.79592 −0.487365
\(405\) 0 0
\(406\) 0 0
\(407\) 26.1421i 1.29582i
\(408\) 0 0
\(409\) 9.50143i 0.469815i 0.972018 + 0.234908i \(0.0754788\pi\)
−0.972018 + 0.234908i \(0.924521\pi\)
\(410\) 1.47309i 0.0727506i
\(411\) 0 0
\(412\) − 25.8686i − 1.27446i
\(413\) 0 0
\(414\) 0 0
\(415\) −34.3431 −1.68584
\(416\) 12.9343 0.634157
\(417\) 0 0
\(418\) − 11.7206i − 0.573274i
\(419\) 19.0029 0.928350 0.464175 0.885743i \(-0.346351\pi\)
0.464175 + 0.885743i \(0.346351\pi\)
\(420\) 0 0
\(421\) −0.686292 −0.0334478 −0.0167239 0.999860i \(-0.505324\pi\)
−0.0167239 + 0.999860i \(0.505324\pi\)
\(422\) 6.34315i 0.308780i
\(423\) 0 0
\(424\) 11.2132 0.544561
\(425\) −25.3659 −1.23043
\(426\) 0 0
\(427\) 0 0
\(428\) 24.3431i 1.17667i
\(429\) 0 0
\(430\) − 5.44382i − 0.262524i
\(431\) 35.4558i 1.70785i 0.520398 + 0.853924i \(0.325784\pi\)
−0.520398 + 0.853924i \(0.674216\pi\)
\(432\) 0 0
\(433\) − 1.21371i − 0.0583271i −0.999575 0.0291636i \(-0.990716\pi\)
0.999575 0.0291636i \(-0.00928436\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.89949 −0.474100
\(437\) −11.7206 −0.560673
\(438\) 0 0
\(439\) − 24.8632i − 1.18665i −0.804962 0.593327i \(-0.797814\pi\)
0.804962 0.593327i \(-0.202186\pi\)
\(440\) −22.4357 −1.06958
\(441\) 0 0
\(442\) −8.58579 −0.408384
\(443\) − 19.1716i − 0.910869i −0.890269 0.455434i \(-0.849484\pi\)
0.890269 0.455434i \(-0.150516\pi\)
\(444\) 0 0
\(445\) −32.8701 −1.55819
\(446\) 1.42195 0.0673312
\(447\) 0 0
\(448\) 0 0
\(449\) 15.7574i 0.743636i 0.928306 + 0.371818i \(0.121265\pi\)
−0.928306 + 0.371818i \(0.878735\pi\)
\(450\) 0 0
\(451\) 5.86030i 0.275951i
\(452\) 25.4142i 1.19538i
\(453\) 0 0
\(454\) 5.86030i 0.275038i
\(455\) 0 0
\(456\) 0 0
\(457\) −16.6274 −0.777798 −0.388899 0.921280i \(-0.627144\pi\)
−0.388899 + 0.921280i \(0.627144\pi\)
\(458\) −7.78498 −0.363768
\(459\) 0 0
\(460\) 10.7151i 0.499596i
\(461\) 10.5069 0.489355 0.244677 0.969605i \(-0.421318\pi\)
0.244677 + 0.969605i \(0.421318\pi\)
\(462\) 0 0
\(463\) 20.4853 0.952032 0.476016 0.879437i \(-0.342080\pi\)
0.476016 + 0.879437i \(0.342080\pi\)
\(464\) 2.48528i 0.115376i
\(465\) 0 0
\(466\) −2.14214 −0.0992325
\(467\) 14.1480 0.654692 0.327346 0.944904i \(-0.393846\pi\)
0.327346 + 0.944904i \(0.393846\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 7.11270i − 0.328084i
\(471\) 0 0
\(472\) − 9.29319i − 0.427754i
\(473\) − 21.6569i − 0.995783i
\(474\) 0 0
\(475\) 21.0138i 0.964179i
\(476\) 0 0
\(477\) 0 0
\(478\) −4.82843 −0.220847
\(479\) 42.4441 1.93932 0.969659 0.244460i \(-0.0786106\pi\)
0.969659 + 0.244460i \(0.0786106\pi\)
\(480\) 0 0
\(481\) − 15.8645i − 0.723357i
\(482\) −10.5069 −0.478576
\(483\) 0 0
\(484\) −22.5147 −1.02340
\(485\) − 20.7279i − 0.941206i
\(486\) 0 0
\(487\) −41.4558 −1.87854 −0.939272 0.343174i \(-0.888498\pi\)
−0.939272 + 0.343174i \(0.888498\pi\)
\(488\) 1.92468 0.0871263
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.3137i − 1.14239i −0.820814 0.571196i \(-0.806480\pi\)
0.820814 0.571196i \(-0.193520\pi\)
\(492\) 0 0
\(493\) − 5.86030i − 0.263935i
\(494\) 7.11270i 0.320015i
\(495\) 0 0
\(496\) − 17.5809i − 0.789406i
\(497\) 0 0
\(498\) 0 0
\(499\) 26.1421 1.17028 0.585141 0.810931i \(-0.301039\pi\)
0.585141 + 0.810931i \(0.301039\pi\)
\(500\) −7.57675 −0.338842
\(501\) 0 0
\(502\) − 4.43835i − 0.198093i
\(503\) −4.85483 −0.216466 −0.108233 0.994126i \(-0.534519\pi\)
−0.108233 + 0.994126i \(0.534519\pi\)
\(504\) 0 0
\(505\) −15.6985 −0.698573
\(506\) − 4.00000i − 0.177822i
\(507\) 0 0
\(508\) 30.1421 1.33734
\(509\) −26.0769 −1.15584 −0.577918 0.816095i \(-0.696135\pi\)
−0.577918 + 0.816095i \(0.696135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.7574i − 1.00574i
\(513\) 0 0
\(514\) − 4.35210i − 0.191963i
\(515\) − 41.4558i − 1.82676i
\(516\) 0 0
\(517\) − 28.2960i − 1.24446i
\(518\) 0 0
\(519\) 0 0
\(520\) 13.6152 0.597067
\(521\) −3.93562 −0.172423 −0.0862113 0.996277i \(-0.527476\pi\)
−0.0862113 + 0.996277i \(0.527476\pi\)
\(522\) 0 0
\(523\) 31.7289i 1.38741i 0.720260 + 0.693705i \(0.244023\pi\)
−0.720260 + 0.693705i \(0.755977\pi\)
\(524\) 15.1535 0.661983
\(525\) 0 0
\(526\) −1.71573 −0.0748093
\(527\) 41.4558i 1.80584i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 8.58221 0.372788
\(531\) 0 0
\(532\) 0 0
\(533\) − 3.55635i − 0.154043i
\(534\) 0 0
\(535\) 39.0112i 1.68660i
\(536\) 13.4558i 0.581204i
\(537\) 0 0
\(538\) 0.919213i 0.0396301i
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) −2.42742 −0.104266
\(543\) 0 0
\(544\) 31.2262i 1.33881i
\(545\) −15.8645 −0.679559
\(546\) 0 0
\(547\) 18.6274 0.796451 0.398225 0.917288i \(-0.369626\pi\)
0.398225 + 0.917288i \(0.369626\pi\)
\(548\) 8.82843i 0.377132i
\(549\) 0 0
\(550\) −7.17157 −0.305797
\(551\) −4.85483 −0.206823
\(552\) 0 0
\(553\) 0 0
\(554\) − 5.37258i − 0.228259i
\(555\) 0 0
\(556\) 15.1535i 0.642651i
\(557\) − 13.4142i − 0.568378i −0.958768 0.284189i \(-0.908276\pi\)
0.958768 0.284189i \(-0.0917244\pi\)
\(558\) 0 0
\(559\) 13.1426i 0.555871i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.45584 0.230141
\(563\) −14.1480 −0.596268 −0.298134 0.954524i \(-0.596364\pi\)
−0.298134 + 0.954524i \(0.596364\pi\)
\(564\) 0 0
\(565\) 40.7276i 1.71342i
\(566\) 3.84936 0.161801
\(567\) 0 0
\(568\) −1.31371 −0.0551220
\(569\) − 37.1127i − 1.55585i −0.628360 0.777923i \(-0.716274\pi\)
0.628360 0.777923i \(-0.283726\pi\)
\(570\) 0 0
\(571\) 11.3137 0.473464 0.236732 0.971575i \(-0.423924\pi\)
0.236732 + 0.971575i \(0.423924\pi\)
\(572\) 25.8686 1.08162
\(573\) 0 0
\(574\) 0 0
\(575\) 7.17157i 0.299075i
\(576\) 0 0
\(577\) − 29.5098i − 1.22851i −0.789109 0.614254i \(-0.789457\pi\)
0.789109 0.614254i \(-0.210543\pi\)
\(578\) − 13.6863i − 0.569275i
\(579\) 0 0
\(580\) 4.43835i 0.184293i
\(581\) 0 0
\(582\) 0 0
\(583\) 34.1421 1.41402
\(584\) 11.2179 0.464199
\(585\) 0 0
\(586\) − 9.50143i − 0.392500i
\(587\) −17.5809 −0.725642 −0.362821 0.931859i \(-0.618186\pi\)
−0.362821 + 0.931859i \(0.618186\pi\)
\(588\) 0 0
\(589\) 34.3431 1.41508
\(590\) − 7.11270i − 0.292825i
\(591\) 0 0
\(592\) −16.2426 −0.667568
\(593\) 23.6494 0.971166 0.485583 0.874190i \(-0.338607\pi\)
0.485583 + 0.874190i \(0.338607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 7.75736i − 0.317754i
\(597\) 0 0
\(598\) 2.42742i 0.0992645i
\(599\) − 10.4853i − 0.428417i −0.976788 0.214208i \(-0.931283\pi\)
0.976788 0.214208i \(-0.0687172\pi\)
\(600\) 0 0
\(601\) 40.5194i 1.65282i 0.563069 + 0.826410i \(0.309621\pi\)
−0.563069 + 0.826410i \(0.690379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.20101 0.333694
\(605\) −36.0810 −1.46690
\(606\) 0 0
\(607\) 3.43289i 0.139337i 0.997570 + 0.0696683i \(0.0221941\pi\)
−0.997570 + 0.0696683i \(0.977806\pi\)
\(608\) 25.8686 1.04911
\(609\) 0 0
\(610\) 1.47309 0.0596436
\(611\) 17.1716i 0.694687i
\(612\) 0 0
\(613\) 42.8701 1.73151 0.865753 0.500472i \(-0.166840\pi\)
0.865753 + 0.500472i \(0.166840\pi\)
\(614\) −9.29319 −0.375043
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.5147i − 0.544082i −0.962286 0.272041i \(-0.912301\pi\)
0.962286 0.272041i \(-0.0876986\pi\)
\(618\) 0 0
\(619\) − 44.8715i − 1.80354i −0.432219 0.901769i \(-0.642269\pi\)
0.432219 0.901769i \(-0.357731\pi\)
\(620\) − 31.3970i − 1.26093i
\(621\) 0 0
\(622\) − 10.7151i − 0.429638i
\(623\) 0 0
\(624\) 0 0
\(625\) −30.0711 −1.20284
\(626\) −2.21918 −0.0886962
\(627\) 0 0
\(628\) − 41.9413i − 1.67364i
\(629\) 38.3002 1.52713
\(630\) 0 0
\(631\) −26.8284 −1.06802 −0.534011 0.845477i \(-0.679316\pi\)
−0.534011 + 0.845477i \(0.679316\pi\)
\(632\) − 2.62742i − 0.104513i
\(633\) 0 0
\(634\) 13.3553 0.530408
\(635\) 48.3044 1.91690
\(636\) 0 0
\(637\) 0 0
\(638\) − 1.65685i − 0.0655955i
\(639\) 0 0
\(640\) − 30.9317i − 1.22268i
\(641\) 24.4853i 0.967110i 0.875314 + 0.483555i \(0.160655\pi\)
−0.875314 + 0.483555i \(0.839345\pi\)
\(642\) 0 0
\(643\) 5.86030i 0.231108i 0.993301 + 0.115554i \(0.0368643\pi\)
−0.993301 + 0.115554i \(0.963136\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −17.1716 −0.675606
\(647\) −10.7151 −0.421255 −0.210628 0.977566i \(-0.567551\pi\)
−0.210628 + 0.977566i \(0.567551\pi\)
\(648\) 0 0
\(649\) − 28.2960i − 1.11072i
\(650\) 4.35210 0.170703
\(651\) 0 0
\(652\) 16.7696 0.656746
\(653\) − 21.5147i − 0.841936i −0.907075 0.420968i \(-0.861690\pi\)
0.907075 0.420968i \(-0.138310\pi\)
\(654\) 0 0
\(655\) 24.2843 0.948865
\(656\) −3.64113 −0.142162
\(657\) 0 0
\(658\) 0 0
\(659\) 3.45584i 0.134621i 0.997732 + 0.0673103i \(0.0214417\pi\)
−0.997732 + 0.0673103i \(0.978558\pi\)
\(660\) 0 0
\(661\) 22.9385i 0.892203i 0.894982 + 0.446102i \(0.147188\pi\)
−0.894982 + 0.446102i \(0.852812\pi\)
\(662\) − 8.97056i − 0.348651i
\(663\) 0 0
\(664\) 18.5864i 0.721291i
\(665\) 0 0
\(666\) 0 0
\(667\) −1.65685 −0.0641537
\(668\) −25.8686 −1.00089
\(669\) 0 0
\(670\) 10.2987i 0.397872i
\(671\) 5.86030 0.226234
\(672\) 0 0
\(673\) −18.1005 −0.697723 −0.348862 0.937174i \(-0.613432\pi\)
−0.348862 + 0.937174i \(0.613432\pi\)
\(674\) 11.2132i 0.431916i
\(675\) 0 0
\(676\) 8.07107 0.310426
\(677\) 46.0852 1.77120 0.885599 0.464451i \(-0.153748\pi\)
0.885599 + 0.464451i \(0.153748\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 32.8701i 1.26051i
\(681\) 0 0
\(682\) 11.7206i 0.448805i
\(683\) − 34.7696i − 1.33042i −0.746656 0.665210i \(-0.768342\pi\)
0.746656 0.665210i \(-0.231658\pi\)
\(684\) 0 0
\(685\) 14.1480i 0.540568i
\(686\) 0 0
\(687\) 0 0
\(688\) 13.4558 0.512999
\(689\) −20.7193 −0.789342
\(690\) 0 0
\(691\) 8.28772i 0.315280i 0.987497 + 0.157640i \(0.0503885\pi\)
−0.987497 + 0.157640i \(0.949611\pi\)
\(692\) −2.21918 −0.0843605
\(693\) 0 0
\(694\) 6.20101 0.235387
\(695\) 24.2843i 0.921155i
\(696\) 0 0
\(697\) 8.58579 0.325210
\(698\) 4.64659 0.175876
\(699\) 0 0
\(700\) 0 0
\(701\) 32.7696i 1.23769i 0.785514 + 0.618844i \(0.212399\pi\)
−0.785514 + 0.618844i \(0.787601\pi\)
\(702\) 0 0
\(703\) − 31.7289i − 1.19668i
\(704\) − 20.1421i − 0.759135i
\(705\) 0 0
\(706\) − 8.49596i − 0.319750i
\(707\) 0 0
\(708\) 0 0
\(709\) −7.27208 −0.273109 −0.136554 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(710\) −1.00547 −0.0377346
\(711\) 0 0
\(712\) 17.7891i 0.666676i
\(713\) 11.7206 0.438940
\(714\) 0 0
\(715\) 41.4558 1.55036
\(716\) − 48.4264i − 1.80978i
\(717\) 0 0
\(718\) −14.4853 −0.540586
\(719\) −11.7206 −0.437105 −0.218552 0.975825i \(-0.570133\pi\)
−0.218552 + 0.975825i \(0.570133\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6.35534i 0.236521i
\(723\) 0 0
\(724\) 17.3727i 0.645650i
\(725\) 2.97056i 0.110324i
\(726\) 0 0
\(727\) − 21.0138i − 0.779358i −0.920951 0.389679i \(-0.872586\pi\)
0.920951 0.389679i \(-0.127414\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 8.58579 0.317774
\(731\) −31.7289 −1.17354
\(732\) 0 0
\(733\) 37.7975i 1.39608i 0.716058 + 0.698041i \(0.245945\pi\)
−0.716058 + 0.698041i \(0.754055\pi\)
\(734\) 8.28772 0.305905
\(735\) 0 0
\(736\) 8.82843 0.325420
\(737\) 40.9706i 1.50917i
\(738\) 0 0
\(739\) −13.1716 −0.484524 −0.242262 0.970211i \(-0.577889\pi\)
−0.242262 + 0.970211i \(0.577889\pi\)
\(740\) −29.0070 −1.06632
\(741\) 0 0
\(742\) 0 0
\(743\) − 32.4264i − 1.18961i −0.803870 0.594805i \(-0.797229\pi\)
0.803870 0.594805i \(-0.202771\pi\)
\(744\) 0 0
\(745\) − 12.4316i − 0.455458i
\(746\) − 9.37258i − 0.343155i
\(747\) 0 0
\(748\) 62.4524i 2.28349i
\(749\) 0 0
\(750\) 0 0
\(751\) 43.3137 1.58054 0.790270 0.612759i \(-0.209940\pi\)
0.790270 + 0.612759i \(0.209940\pi\)
\(752\) 17.5809 0.641110
\(753\) 0 0
\(754\) 1.00547i 0.0366170i
\(755\) 13.1426 0.478306
\(756\) 0 0
\(757\) −46.8701 −1.70352 −0.851761 0.523931i \(-0.824465\pi\)
−0.851761 + 0.523931i \(0.824465\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 27.2304 0.987752
\(761\) −17.0782 −0.619083 −0.309542 0.950886i \(-0.600176\pi\)
−0.309542 + 0.950886i \(0.600176\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 27.7401i − 1.00360i
\(765\) 0 0
\(766\) 9.70967i 0.350825i
\(767\) 17.1716i 0.620030i
\(768\) 0 0
\(769\) − 48.0961i − 1.73439i −0.497968 0.867195i \(-0.665920\pi\)
0.497968 0.867195i \(-0.334080\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −11.5980 −0.417420
\(773\) 24.6549 0.886776 0.443388 0.896330i \(-0.353776\pi\)
0.443388 + 0.896330i \(0.353776\pi\)
\(774\) 0 0
\(775\) − 21.0138i − 0.754838i
\(776\) −11.2179 −0.402698
\(777\) 0 0
\(778\) 10.6863 0.383122
\(779\) − 7.11270i − 0.254839i
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) −5.86030 −0.209564
\(783\) 0 0
\(784\) 0 0
\(785\) − 67.2132i − 2.39894i
\(786\) 0 0
\(787\) − 11.7206i − 0.417794i −0.977938 0.208897i \(-0.933013\pi\)
0.977938 0.208897i \(-0.0669874\pi\)
\(788\) 14.1838i 0.505276i
\(789\) 0 0
\(790\) − 2.01094i − 0.0715460i
\(791\) 0 0
\(792\) 0 0
\(793\) −3.55635 −0.126290
\(794\) −7.36851 −0.261498
\(795\) 0 0
\(796\) 30.3070i 1.07420i
\(797\) −51.2345 −1.81482 −0.907410 0.420247i \(-0.861944\pi\)
−0.907410 + 0.420247i \(0.861944\pi\)
\(798\) 0 0
\(799\) −41.4558 −1.46660
\(800\) − 15.8284i − 0.559619i
\(801\) 0 0
\(802\) 2.14214 0.0756414
\(803\) 34.1563 1.20535
\(804\) 0 0
\(805\) 0 0
\(806\) − 7.11270i − 0.250534i
\(807\) 0 0
\(808\) 8.49596i 0.298887i
\(809\) − 16.0416i − 0.563994i −0.959415 0.281997i \(-0.909003\pi\)
0.959415 0.281997i \(-0.0909968\pi\)
\(810\) 0 0
\(811\) − 8.28772i − 0.291021i −0.989357 0.145511i \(-0.953518\pi\)
0.989357 0.145511i \(-0.0464825\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 10.8284 0.379536
\(815\) 26.8741 0.941359
\(816\) 0 0
\(817\) 26.2851i 0.919599i
\(818\) 3.93562 0.137606
\(819\) 0 0
\(820\) −6.50253 −0.227078
\(821\) 23.7574i 0.829138i 0.910018 + 0.414569i \(0.136068\pi\)
−0.910018 + 0.414569i \(0.863932\pi\)
\(822\) 0 0
\(823\) −41.9411 −1.46198 −0.730988 0.682390i \(-0.760940\pi\)
−0.730988 + 0.682390i \(0.760940\pi\)
\(824\) −22.4357 −0.781586
\(825\) 0 0
\(826\) 0 0
\(827\) 39.6569i 1.37900i 0.724284 + 0.689502i \(0.242171\pi\)
−0.724284 + 0.689502i \(0.757829\pi\)
\(828\) 0 0
\(829\) 36.0810i 1.25315i 0.779363 + 0.626573i \(0.215543\pi\)
−0.779363 + 0.626573i \(0.784457\pi\)
\(830\) 14.2254i 0.493771i
\(831\) 0 0
\(832\) 12.2233i 0.423768i
\(833\) 0 0
\(834\) 0 0
\(835\) −41.4558 −1.43464
\(836\) 51.7373 1.78937
\(837\) 0 0
\(838\) − 7.87124i − 0.271907i
\(839\) −17.5809 −0.606960 −0.303480 0.952838i \(-0.598149\pi\)
−0.303480 + 0.952838i \(0.598149\pi\)
\(840\) 0 0
\(841\) 28.3137 0.976335
\(842\) 0.284271i 0.00979663i
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) 12.9343 0.444954
\(846\) 0 0
\(847\) 0 0
\(848\) 21.2132i 0.728464i
\(849\) 0 0
\(850\) 10.5069i 0.360384i
\(851\) − 10.8284i − 0.371194i
\(852\) 0 0
\(853\) − 16.0727i − 0.550319i −0.961399 0.275159i \(-0.911269\pi\)
0.961399 0.275159i \(-0.0887306\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 21.1127 0.721617
\(857\) −0.502734 −0.0171731 −0.00858654 0.999963i \(-0.502733\pi\)
−0.00858654 + 0.999963i \(0.502733\pi\)
\(858\) 0 0
\(859\) − 9.29319i − 0.317079i −0.987353 0.158540i \(-0.949321\pi\)
0.987353 0.158540i \(-0.0506786\pi\)
\(860\) 24.0302 0.819423
\(861\) 0 0
\(862\) 14.6863 0.500217
\(863\) − 10.6863i − 0.363766i −0.983320 0.181883i \(-0.941781\pi\)
0.983320 0.181883i \(-0.0582191\pi\)
\(864\) 0 0
\(865\) −3.55635 −0.120919
\(866\) −0.502734 −0.0170836
\(867\) 0 0
\(868\) 0 0
\(869\) − 8.00000i − 0.271381i
\(870\) 0 0
\(871\) − 24.8632i − 0.842456i
\(872\) 8.58579i 0.290751i
\(873\) 0 0
\(874\) 4.85483i 0.164217i
\(875\) 0 0
\(876\) 0 0
\(877\) −32.5269 −1.09836 −0.549178 0.835705i \(-0.685059\pi\)
−0.549178 + 0.835705i \(0.685059\pi\)
\(878\) −10.2987 −0.347563
\(879\) 0 0
\(880\) − 42.4441i − 1.43079i
\(881\) −23.6494 −0.796770 −0.398385 0.917218i \(-0.630429\pi\)
−0.398385 + 0.917218i \(0.630429\pi\)
\(882\) 0 0
\(883\) 13.4558 0.452825 0.226413 0.974031i \(-0.427300\pi\)
0.226413 + 0.974031i \(0.427300\pi\)
\(884\) − 37.8995i − 1.27470i
\(885\) 0 0
\(886\) −7.94113 −0.266787
\(887\) −50.7318 −1.70341 −0.851703 0.524024i \(-0.824430\pi\)
−0.851703 + 0.524024i \(0.824430\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 13.6152i 0.456383i
\(891\) 0 0
\(892\) 6.27678i 0.210162i
\(893\) 34.3431i 1.14925i
\(894\) 0 0
\(895\) − 77.6059i − 2.59408i
\(896\) 0 0
\(897\) 0 0
\(898\) 6.52691 0.217806
\(899\) 4.85483 0.161918
\(900\) 0 0
\(901\) − 50.0208i − 1.66643i
\(902\) 2.42742 0.0808241
\(903\) 0 0
\(904\) 22.0416 0.733094
\(905\) 27.8406i 0.925454i
\(906\) 0 0
\(907\) 28.7696 0.955277 0.477639 0.878556i \(-0.341493\pi\)
0.477639 + 0.878556i \(0.341493\pi\)
\(908\) −25.8686 −0.858481
\(909\) 0 0
\(910\) 0 0
\(911\) 42.4853i 1.40760i 0.710398 + 0.703800i \(0.248515\pi\)
−0.710398 + 0.703800i \(0.751485\pi\)
\(912\) 0 0
\(913\) 56.5921i 1.87292i
\(914\) 6.88730i 0.227812i
\(915\) 0 0
\(916\) − 34.3646i − 1.13544i
\(917\) 0 0
\(918\) 0 0
\(919\) −41.4558 −1.36750 −0.683751 0.729715i \(-0.739653\pi\)
−0.683751 + 0.729715i \(0.739653\pi\)
\(920\) 9.29319 0.306387
\(921\) 0 0
\(922\) − 4.35210i − 0.143329i
\(923\) 2.42742 0.0798994
\(924\) 0 0
\(925\) −19.4142 −0.638335
\(926\) − 8.48528i − 0.278844i
\(927\) 0 0
\(928\) 3.65685 0.120042
\(929\) −0.502734 −0.0164942 −0.00824709 0.999966i \(-0.502625\pi\)
−0.00824709 + 0.999966i \(0.502625\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 9.45584i − 0.309736i
\(933\) 0 0
\(934\) − 5.86030i − 0.191755i
\(935\) 100.083i 3.27307i
\(936\) 0 0
\(937\) 51.2345i 1.67376i 0.547387 + 0.836879i \(0.315622\pi\)
−0.547387 + 0.836879i \(0.684378\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 31.3970 1.02406
\(941\) 4.64659 0.151475 0.0757373 0.997128i \(-0.475869\pi\)
0.0757373 + 0.997128i \(0.475869\pi\)
\(942\) 0 0
\(943\) − 2.42742i − 0.0790476i
\(944\) 17.5809 0.572210
\(945\) 0 0
\(946\) −8.97056 −0.291658
\(947\) − 10.9706i − 0.356495i −0.983986 0.178248i \(-0.942957\pi\)
0.983986 0.178248i \(-0.0570428\pi\)
\(948\) 0 0
\(949\) −20.7279 −0.672857
\(950\) 8.70420 0.282401
\(951\) 0 0
\(952\) 0 0
\(953\) 10.3848i 0.336396i 0.985753 + 0.168198i \(0.0537948\pi\)
−0.985753 + 0.168198i \(0.946205\pi\)
\(954\) 0 0
\(955\) − 44.4550i − 1.43853i
\(956\) − 21.3137i − 0.689335i
\(957\) 0 0
\(958\) − 17.5809i − 0.568013i
\(959\) 0 0
\(960\) 0 0
\(961\) −3.34315 −0.107843
\(962\) −6.57128 −0.211866
\(963\) 0 0
\(964\) − 46.3797i − 1.49379i
\(965\) −18.5864 −0.598317
\(966\) 0 0
\(967\) −19.1127 −0.614623 −0.307311 0.951609i \(-0.599429\pi\)
−0.307311 + 0.951609i \(0.599429\pi\)
\(968\) 19.5269i 0.627619i
\(969\) 0 0
\(970\) −8.58579 −0.275673
\(971\) −4.85483 −0.155799 −0.0778995 0.996961i \(-0.524821\pi\)
−0.0778995 + 0.996961i \(0.524821\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 17.1716i 0.550213i
\(975\) 0 0
\(976\) 3.64113i 0.116550i
\(977\) 44.8284i 1.43419i 0.696976 + 0.717094i \(0.254528\pi\)
−0.696976 + 0.717094i \(0.745472\pi\)
\(978\) 0 0
\(979\) 54.1647i 1.73111i
\(980\) 0 0
\(981\) 0 0
\(982\) −10.4853 −0.334599
\(983\) 31.7289 1.01200 0.505998 0.862535i \(-0.331124\pi\)
0.505998 + 0.862535i \(0.331124\pi\)
\(984\) 0 0
\(985\) 22.7302i 0.724246i
\(986\) −2.42742 −0.0773047
\(987\) 0 0
\(988\) −31.3970 −0.998871
\(989\) 8.97056i 0.285247i
\(990\) 0 0
\(991\) −59.3970 −1.88681 −0.943403 0.331647i \(-0.892396\pi\)
−0.943403 + 0.331647i \(0.892396\pi\)
\(992\) −25.8686 −0.821330
\(993\) 0 0
\(994\) 0 0
\(995\) 48.5685i 1.53973i
\(996\) 0 0
\(997\) − 39.5139i − 1.25142i −0.780056 0.625709i \(-0.784810\pi\)
0.780056 0.625709i \(-0.215190\pi\)
\(998\) − 10.8284i − 0.342768i
\(999\) 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 441.2.c.b.440.4 yes 8
3.2 odd 2 inner 441.2.c.b.440.5 yes 8
4.3 odd 2 7056.2.k.g.881.6 8
7.2 even 3 441.2.p.c.80.3 16
7.3 odd 6 441.2.p.c.215.6 16
7.4 even 3 441.2.p.c.215.5 16
7.5 odd 6 441.2.p.c.80.4 16
7.6 odd 2 inner 441.2.c.b.440.3 8
12.11 even 2 7056.2.k.g.881.3 8
21.2 odd 6 441.2.p.c.80.6 16
21.5 even 6 441.2.p.c.80.5 16
21.11 odd 6 441.2.p.c.215.4 16
21.17 even 6 441.2.p.c.215.3 16
21.20 even 2 inner 441.2.c.b.440.6 yes 8
28.27 even 2 7056.2.k.g.881.4 8
84.83 odd 2 7056.2.k.g.881.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.c.b.440.3 8 7.6 odd 2 inner
441.2.c.b.440.4 yes 8 1.1 even 1 trivial
441.2.c.b.440.5 yes 8 3.2 odd 2 inner
441.2.c.b.440.6 yes 8 21.20 even 2 inner
441.2.p.c.80.3 16 7.2 even 3
441.2.p.c.80.4 16 7.5 odd 6
441.2.p.c.80.5 16 21.5 even 6
441.2.p.c.80.6 16 21.2 odd 6
441.2.p.c.215.3 16 21.17 even 6
441.2.p.c.215.4 16 21.11 odd 6
441.2.p.c.215.5 16 7.4 even 3
441.2.p.c.215.6 16 7.3 odd 6
7056.2.k.g.881.3 8 12.11 even 2
7056.2.k.g.881.4 8 28.27 even 2
7056.2.k.g.881.5 8 84.83 odd 2
7056.2.k.g.881.6 8 4.3 odd 2