Properties

Label 1323.2.a.bb.1.3
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.5642081874\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.874032\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.874032 q^{2} -1.23607 q^{4} +0.236068 q^{5} -2.82843 q^{8} +O(q^{10})\) \(q+0.874032 q^{2} -1.23607 q^{4} +0.236068 q^{5} -2.82843 q^{8} +0.206331 q^{10} +0.540182 q^{11} +0.874032 q^{13} -5.00000 q^{17} -4.03631 q^{19} -0.291796 q^{20} +0.472136 q^{22} +5.99070 q^{23} -4.94427 q^{25} +0.763932 q^{26} -8.61280 q^{29} -6.53089 q^{31} +5.65685 q^{32} -4.37016 q^{34} +8.70820 q^{37} -3.52786 q^{38} -0.667701 q^{40} -8.70820 q^{41} -2.23607 q^{43} -0.667701 q^{44} +5.23607 q^{46} -7.47214 q^{47} -4.32145 q^{50} -1.08036 q^{52} +3.16228 q^{53} +0.127520 q^{55} -7.52786 q^{58} -13.9443 q^{59} +0.540182 q^{61} -5.70820 q^{62} +4.94427 q^{64} +0.206331 q^{65} -6.76393 q^{67} +6.18034 q^{68} -6.73722 q^{71} +13.3956 q^{73} +7.61125 q^{74} +4.98915 q^{76} -2.52786 q^{79} -7.61125 q^{82} -4.23607 q^{83} -1.18034 q^{85} -1.95440 q^{86} -1.52786 q^{88} -11.7082 q^{89} -7.40492 q^{92} -6.53089 q^{94} -0.952843 q^{95} +5.11667 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 8q^{5} + O(q^{10}) \) \( 4q + 4q^{4} - 8q^{5} - 20q^{17} - 28q^{20} - 16q^{22} + 16q^{25} + 12q^{26} + 8q^{37} - 32q^{38} - 8q^{41} + 12q^{46} - 12q^{47} - 48q^{58} - 20q^{59} + 4q^{62} - 16q^{64} - 36q^{67} - 20q^{68} - 28q^{79} - 8q^{83} + 40q^{85} - 24q^{88} - 20q^{89} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.874032 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(3\) 0 0
\(4\) −1.23607 −0.618034
\(5\) 0.236068 0.105573 0.0527864 0.998606i \(-0.483190\pi\)
0.0527864 + 0.998606i \(0.483190\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 0.206331 0.0652476
\(11\) 0.540182 0.162871 0.0814354 0.996679i \(-0.474050\pi\)
0.0814354 + 0.996679i \(0.474050\pi\)
\(12\) 0 0
\(13\) 0.874032 0.242413 0.121206 0.992627i \(-0.461324\pi\)
0.121206 + 0.992627i \(0.461324\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) −4.03631 −0.925993 −0.462996 0.886360i \(-0.653226\pi\)
−0.462996 + 0.886360i \(0.653226\pi\)
\(20\) −0.291796 −0.0652476
\(21\) 0 0
\(22\) 0.472136 0.100660
\(23\) 5.99070 1.24915 0.624574 0.780966i \(-0.285273\pi\)
0.624574 + 0.780966i \(0.285273\pi\)
\(24\) 0 0
\(25\) −4.94427 −0.988854
\(26\) 0.763932 0.149819
\(27\) 0 0
\(28\) 0 0
\(29\) −8.61280 −1.59936 −0.799678 0.600428i \(-0.794997\pi\)
−0.799678 + 0.600428i \(0.794997\pi\)
\(30\) 0 0
\(31\) −6.53089 −1.17298 −0.586491 0.809956i \(-0.699491\pi\)
−0.586491 + 0.809956i \(0.699491\pi\)
\(32\) 5.65685 1.00000
\(33\) 0 0
\(34\) −4.37016 −0.749476
\(35\) 0 0
\(36\) 0 0
\(37\) 8.70820 1.43162 0.715810 0.698295i \(-0.246058\pi\)
0.715810 + 0.698295i \(0.246058\pi\)
\(38\) −3.52786 −0.572295
\(39\) 0 0
\(40\) −0.667701 −0.105573
\(41\) −8.70820 −1.35999 −0.679996 0.733215i \(-0.738019\pi\)
−0.679996 + 0.733215i \(0.738019\pi\)
\(42\) 0 0
\(43\) −2.23607 −0.340997 −0.170499 0.985358i \(-0.554538\pi\)
−0.170499 + 0.985358i \(0.554538\pi\)
\(44\) −0.667701 −0.100660
\(45\) 0 0
\(46\) 5.23607 0.772016
\(47\) −7.47214 −1.08992 −0.544962 0.838461i \(-0.683456\pi\)
−0.544962 + 0.838461i \(0.683456\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −4.32145 −0.611146
\(51\) 0 0
\(52\) −1.08036 −0.149819
\(53\) 3.16228 0.434372 0.217186 0.976130i \(-0.430312\pi\)
0.217186 + 0.976130i \(0.430312\pi\)
\(54\) 0 0
\(55\) 0.127520 0.0171947
\(56\) 0 0
\(57\) 0 0
\(58\) −7.52786 −0.988457
\(59\) −13.9443 −1.81539 −0.907695 0.419631i \(-0.862159\pi\)
−0.907695 + 0.419631i \(0.862159\pi\)
\(60\) 0 0
\(61\) 0.540182 0.0691632 0.0345816 0.999402i \(-0.488990\pi\)
0.0345816 + 0.999402i \(0.488990\pi\)
\(62\) −5.70820 −0.724943
\(63\) 0 0
\(64\) 4.94427 0.618034
\(65\) 0.206331 0.0255922
\(66\) 0 0
\(67\) −6.76393 −0.826346 −0.413173 0.910653i \(-0.635579\pi\)
−0.413173 + 0.910653i \(0.635579\pi\)
\(68\) 6.18034 0.749476
\(69\) 0 0
\(70\) 0 0
\(71\) −6.73722 −0.799561 −0.399780 0.916611i \(-0.630914\pi\)
−0.399780 + 0.916611i \(0.630914\pi\)
\(72\) 0 0
\(73\) 13.3956 1.56784 0.783920 0.620862i \(-0.213217\pi\)
0.783920 + 0.620862i \(0.213217\pi\)
\(74\) 7.61125 0.884790
\(75\) 0 0
\(76\) 4.98915 0.572295
\(77\) 0 0
\(78\) 0 0
\(79\) −2.52786 −0.284407 −0.142203 0.989837i \(-0.545419\pi\)
−0.142203 + 0.989837i \(0.545419\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.61125 −0.840522
\(83\) −4.23607 −0.464969 −0.232484 0.972600i \(-0.574685\pi\)
−0.232484 + 0.972600i \(0.574685\pi\)
\(84\) 0 0
\(85\) −1.18034 −0.128026
\(86\) −1.95440 −0.210748
\(87\) 0 0
\(88\) −1.52786 −0.162871
\(89\) −11.7082 −1.24107 −0.620534 0.784180i \(-0.713084\pi\)
−0.620534 + 0.784180i \(0.713084\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.40492 −0.772016
\(93\) 0 0
\(94\) −6.53089 −0.673609
\(95\) −0.952843 −0.0977597
\(96\) 0 0
\(97\) 5.11667 0.519519 0.259760 0.965673i \(-0.416357\pi\)
0.259760 + 0.965673i \(0.416357\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 6.11146 0.611146
\(101\) −4.76393 −0.474029 −0.237014 0.971506i \(-0.576169\pi\)
−0.237014 + 0.971506i \(0.576169\pi\)
\(102\) 0 0
\(103\) 16.8918 1.66439 0.832197 0.554480i \(-0.187083\pi\)
0.832197 + 0.554480i \(0.187083\pi\)
\(104\) −2.47214 −0.242413
\(105\) 0 0
\(106\) 2.76393 0.268457
\(107\) 9.89949 0.957020 0.478510 0.878082i \(-0.341177\pi\)
0.478510 + 0.878082i \(0.341177\pi\)
\(108\) 0 0
\(109\) 14.2361 1.36357 0.681784 0.731554i \(-0.261204\pi\)
0.681784 + 0.731554i \(0.261204\pi\)
\(110\) 0.111456 0.0106269
\(111\) 0 0
\(112\) 0 0
\(113\) 8.27895 0.778818 0.389409 0.921065i \(-0.372679\pi\)
0.389409 + 0.921065i \(0.372679\pi\)
\(114\) 0 0
\(115\) 1.41421 0.131876
\(116\) 10.6460 0.988457
\(117\) 0 0
\(118\) −12.1877 −1.12197
\(119\) 0 0
\(120\) 0 0
\(121\) −10.7082 −0.973473
\(122\) 0.472136 0.0427452
\(123\) 0 0
\(124\) 8.07262 0.724943
\(125\) −2.34752 −0.209969
\(126\) 0 0
\(127\) −3.47214 −0.308102 −0.154051 0.988063i \(-0.549232\pi\)
−0.154051 + 0.988063i \(0.549232\pi\)
\(128\) −6.99226 −0.618034
\(129\) 0 0
\(130\) 0.180340 0.0158169
\(131\) 6.94427 0.606724 0.303362 0.952875i \(-0.401891\pi\)
0.303362 + 0.952875i \(0.401891\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −5.91189 −0.510710
\(135\) 0 0
\(136\) 14.1421 1.21268
\(137\) −13.9358 −1.19062 −0.595308 0.803498i \(-0.702970\pi\)
−0.595308 + 0.803498i \(0.702970\pi\)
\(138\) 0 0
\(139\) 9.28050 0.787162 0.393581 0.919290i \(-0.371236\pi\)
0.393581 + 0.919290i \(0.371236\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.88854 −0.494156
\(143\) 0.472136 0.0394820
\(144\) 0 0
\(145\) −2.03321 −0.168849
\(146\) 11.7082 0.968978
\(147\) 0 0
\(148\) −10.7639 −0.884790
\(149\) 6.45207 0.528575 0.264287 0.964444i \(-0.414863\pi\)
0.264287 + 0.964444i \(0.414863\pi\)
\(150\) 0 0
\(151\) 10.4164 0.847675 0.423838 0.905738i \(-0.360683\pi\)
0.423838 + 0.905738i \(0.360683\pi\)
\(152\) 11.4164 0.925993
\(153\) 0 0
\(154\) 0 0
\(155\) −1.54173 −0.123835
\(156\) 0 0
\(157\) −17.5107 −1.39751 −0.698755 0.715361i \(-0.746262\pi\)
−0.698755 + 0.715361i \(0.746262\pi\)
\(158\) −2.20943 −0.175773
\(159\) 0 0
\(160\) 1.33540 0.105573
\(161\) 0 0
\(162\) 0 0
\(163\) 17.1803 1.34567 0.672834 0.739793i \(-0.265077\pi\)
0.672834 + 0.739793i \(0.265077\pi\)
\(164\) 10.7639 0.840522
\(165\) 0 0
\(166\) −3.70246 −0.287367
\(167\) 8.23607 0.637326 0.318663 0.947868i \(-0.396766\pi\)
0.318663 + 0.947868i \(0.396766\pi\)
\(168\) 0 0
\(169\) −12.2361 −0.941236
\(170\) −1.03165 −0.0791243
\(171\) 0 0
\(172\) 2.76393 0.210748
\(173\) −9.52786 −0.724390 −0.362195 0.932102i \(-0.617973\pi\)
−0.362195 + 0.932102i \(0.617973\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −10.2333 −0.767022
\(179\) −17.8446 −1.33377 −0.666884 0.745162i \(-0.732372\pi\)
−0.666884 + 0.745162i \(0.732372\pi\)
\(180\) 0 0
\(181\) −8.74032 −0.649663 −0.324831 0.945772i \(-0.605308\pi\)
−0.324831 + 0.945772i \(0.605308\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −16.9443 −1.24915
\(185\) 2.05573 0.151140
\(186\) 0 0
\(187\) −2.70091 −0.197510
\(188\) 9.23607 0.673609
\(189\) 0 0
\(190\) −0.832816 −0.0604188
\(191\) 4.24264 0.306987 0.153493 0.988150i \(-0.450948\pi\)
0.153493 + 0.988150i \(0.450948\pi\)
\(192\) 0 0
\(193\) −5.18034 −0.372889 −0.186445 0.982465i \(-0.559696\pi\)
−0.186445 + 0.982465i \(0.559696\pi\)
\(194\) 4.47214 0.321081
\(195\) 0 0
\(196\) 0 0
\(197\) 21.6746 1.54425 0.772125 0.635471i \(-0.219194\pi\)
0.772125 + 0.635471i \(0.219194\pi\)
\(198\) 0 0
\(199\) 26.1235 1.85185 0.925925 0.377709i \(-0.123288\pi\)
0.925925 + 0.377709i \(0.123288\pi\)
\(200\) 13.9845 0.988854
\(201\) 0 0
\(202\) −4.16383 −0.292966
\(203\) 0 0
\(204\) 0 0
\(205\) −2.05573 −0.143578
\(206\) 14.7639 1.02865
\(207\) 0 0
\(208\) 0 0
\(209\) −2.18034 −0.150817
\(210\) 0 0
\(211\) 9.41641 0.648252 0.324126 0.946014i \(-0.394930\pi\)
0.324126 + 0.946014i \(0.394930\pi\)
\(212\) −3.90879 −0.268457
\(213\) 0 0
\(214\) 8.65248 0.591471
\(215\) −0.527864 −0.0360000
\(216\) 0 0
\(217\) 0 0
\(218\) 12.4428 0.842731
\(219\) 0 0
\(220\) −0.157623 −0.0106269
\(221\) −4.37016 −0.293969
\(222\) 0 0
\(223\) 15.2712 1.02264 0.511318 0.859392i \(-0.329158\pi\)
0.511318 + 0.859392i \(0.329158\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.23607 0.481336
\(227\) 10.1803 0.675693 0.337846 0.941201i \(-0.390302\pi\)
0.337846 + 0.941201i \(0.390302\pi\)
\(228\) 0 0
\(229\) 23.2951 1.53938 0.769692 0.638415i \(-0.220410\pi\)
0.769692 + 0.638415i \(0.220410\pi\)
\(230\) 1.23607 0.0815039
\(231\) 0 0
\(232\) 24.3607 1.59936
\(233\) −15.6839 −1.02748 −0.513742 0.857945i \(-0.671741\pi\)
−0.513742 + 0.857945i \(0.671741\pi\)
\(234\) 0 0
\(235\) −1.76393 −0.115066
\(236\) 17.2361 1.12197
\(237\) 0 0
\(238\) 0 0
\(239\) 9.77198 0.632097 0.316048 0.948743i \(-0.397644\pi\)
0.316048 + 0.948743i \(0.397644\pi\)
\(240\) 0 0
\(241\) −2.62210 −0.168904 −0.0844520 0.996428i \(-0.526914\pi\)
−0.0844520 + 0.996428i \(0.526914\pi\)
\(242\) −9.35931 −0.601639
\(243\) 0 0
\(244\) −0.667701 −0.0427452
\(245\) 0 0
\(246\) 0 0
\(247\) −3.52786 −0.224473
\(248\) 18.4721 1.17298
\(249\) 0 0
\(250\) −2.05181 −0.129768
\(251\) 22.7082 1.43333 0.716665 0.697418i \(-0.245668\pi\)
0.716665 + 0.697418i \(0.245668\pi\)
\(252\) 0 0
\(253\) 3.23607 0.203450
\(254\) −3.03476 −0.190418
\(255\) 0 0
\(256\) −16.0000 −1.00000
\(257\) −21.4164 −1.33592 −0.667959 0.744198i \(-0.732832\pi\)
−0.667959 + 0.744198i \(0.732832\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.255039 −0.0158169
\(261\) 0 0
\(262\) 6.06952 0.374976
\(263\) −27.2526 −1.68047 −0.840234 0.542224i \(-0.817583\pi\)
−0.840234 + 0.542224i \(0.817583\pi\)
\(264\) 0 0
\(265\) 0.746512 0.0458579
\(266\) 0 0
\(267\) 0 0
\(268\) 8.36068 0.510710
\(269\) −7.00000 −0.426798 −0.213399 0.976965i \(-0.568453\pi\)
−0.213399 + 0.976965i \(0.568453\pi\)
\(270\) 0 0
\(271\) −19.7202 −1.19792 −0.598958 0.800781i \(-0.704418\pi\)
−0.598958 + 0.800781i \(0.704418\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −12.1803 −0.735841
\(275\) −2.67080 −0.161056
\(276\) 0 0
\(277\) −18.2361 −1.09570 −0.547850 0.836577i \(-0.684553\pi\)
−0.547850 + 0.836577i \(0.684553\pi\)
\(278\) 8.11146 0.486493
\(279\) 0 0
\(280\) 0 0
\(281\) 28.6181 1.70721 0.853607 0.520918i \(-0.174410\pi\)
0.853607 + 0.520918i \(0.174410\pi\)
\(282\) 0 0
\(283\) −12.1089 −0.719801 −0.359901 0.932991i \(-0.617189\pi\)
−0.359901 + 0.932991i \(0.617189\pi\)
\(284\) 8.32766 0.494156
\(285\) 0 0
\(286\) 0.412662 0.0244012
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) −1.77709 −0.104354
\(291\) 0 0
\(292\) −16.5579 −0.968978
\(293\) 1.47214 0.0860031 0.0430016 0.999075i \(-0.486308\pi\)
0.0430016 + 0.999075i \(0.486308\pi\)
\(294\) 0 0
\(295\) −3.29180 −0.191656
\(296\) −24.6305 −1.43162
\(297\) 0 0
\(298\) 5.63932 0.326677
\(299\) 5.23607 0.302810
\(300\) 0 0
\(301\) 0 0
\(302\) 9.10427 0.523892
\(303\) 0 0
\(304\) 0 0
\(305\) 0.127520 0.00730175
\(306\) 0 0
\(307\) −22.5486 −1.28692 −0.643458 0.765481i \(-0.722501\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.34752 −0.0765342
\(311\) 31.0689 1.76175 0.880877 0.473345i \(-0.156953\pi\)
0.880877 + 0.473345i \(0.156953\pi\)
\(312\) 0 0
\(313\) −16.3029 −0.921492 −0.460746 0.887532i \(-0.652418\pi\)
−0.460746 + 0.887532i \(0.652418\pi\)
\(314\) −15.3050 −0.863708
\(315\) 0 0
\(316\) 3.12461 0.175773
\(317\) −10.5672 −0.593513 −0.296756 0.954953i \(-0.595905\pi\)
−0.296756 + 0.954953i \(0.595905\pi\)
\(318\) 0 0
\(319\) −4.65248 −0.260489
\(320\) 1.16718 0.0652476
\(321\) 0 0
\(322\) 0 0
\(323\) 20.1815 1.12293
\(324\) 0 0
\(325\) −4.32145 −0.239711
\(326\) 15.0162 0.831669
\(327\) 0 0
\(328\) 24.6305 1.35999
\(329\) 0 0
\(330\) 0 0
\(331\) −8.41641 −0.462608 −0.231304 0.972882i \(-0.574299\pi\)
−0.231304 + 0.972882i \(0.574299\pi\)
\(332\) 5.23607 0.287367
\(333\) 0 0
\(334\) 7.19859 0.393889
\(335\) −1.59675 −0.0872396
\(336\) 0 0
\(337\) 8.23607 0.448647 0.224324 0.974515i \(-0.427983\pi\)
0.224324 + 0.974515i \(0.427983\pi\)
\(338\) −10.6947 −0.581716
\(339\) 0 0
\(340\) 1.45898 0.0791243
\(341\) −3.52786 −0.191045
\(342\) 0 0
\(343\) 0 0
\(344\) 6.32456 0.340997
\(345\) 0 0
\(346\) −8.32766 −0.447698
\(347\) −26.3299 −1.41346 −0.706731 0.707482i \(-0.749831\pi\)
−0.706731 + 0.707482i \(0.749831\pi\)
\(348\) 0 0
\(349\) −8.07262 −0.432117 −0.216059 0.976380i \(-0.569320\pi\)
−0.216059 + 0.976380i \(0.569320\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.05573 0.162871
\(353\) 10.1246 0.538879 0.269439 0.963017i \(-0.413162\pi\)
0.269439 + 0.963017i \(0.413162\pi\)
\(354\) 0 0
\(355\) −1.59044 −0.0844119
\(356\) 14.4721 0.767022
\(357\) 0 0
\(358\) −15.5967 −0.824314
\(359\) 23.6290 1.24709 0.623545 0.781788i \(-0.285692\pi\)
0.623545 + 0.781788i \(0.285692\pi\)
\(360\) 0 0
\(361\) −2.70820 −0.142537
\(362\) −7.63932 −0.401514
\(363\) 0 0
\(364\) 0 0
\(365\) 3.16228 0.165521
\(366\) 0 0
\(367\) −12.9830 −0.677705 −0.338853 0.940839i \(-0.610039\pi\)
−0.338853 + 0.940839i \(0.610039\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 1.79677 0.0934097
\(371\) 0 0
\(372\) 0 0
\(373\) 17.6525 0.914011 0.457005 0.889464i \(-0.348922\pi\)
0.457005 + 0.889464i \(0.348922\pi\)
\(374\) −2.36068 −0.122068
\(375\) 0 0
\(376\) 21.1344 1.08992
\(377\) −7.52786 −0.387705
\(378\) 0 0
\(379\) 4.88854 0.251108 0.125554 0.992087i \(-0.459929\pi\)
0.125554 + 0.992087i \(0.459929\pi\)
\(380\) 1.17778 0.0604188
\(381\) 0 0
\(382\) 3.70820 0.189728
\(383\) −11.1803 −0.571289 −0.285644 0.958336i \(-0.592208\pi\)
−0.285644 + 0.958336i \(0.592208\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.52778 −0.230458
\(387\) 0 0
\(388\) −6.32456 −0.321081
\(389\) −0.952843 −0.0483111 −0.0241555 0.999708i \(-0.507690\pi\)
−0.0241555 + 0.999708i \(0.507690\pi\)
\(390\) 0 0
\(391\) −29.9535 −1.51481
\(392\) 0 0
\(393\) 0 0
\(394\) 18.9443 0.954399
\(395\) −0.596748 −0.0300256
\(396\) 0 0
\(397\) −16.4791 −0.827062 −0.413531 0.910490i \(-0.635705\pi\)
−0.413531 + 0.910490i \(0.635705\pi\)
\(398\) 22.8328 1.14451
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0323 −1.49974 −0.749872 0.661583i \(-0.769885\pi\)
−0.749872 + 0.661583i \(0.769885\pi\)
\(402\) 0 0
\(403\) −5.70820 −0.284346
\(404\) 5.88854 0.292966
\(405\) 0 0
\(406\) 0 0
\(407\) 4.70401 0.233169
\(408\) 0 0
\(409\) −12.2364 −0.605053 −0.302527 0.953141i \(-0.597830\pi\)
−0.302527 + 0.953141i \(0.597830\pi\)
\(410\) −1.79677 −0.0887363
\(411\) 0 0
\(412\) −20.8794 −1.02865
\(413\) 0 0
\(414\) 0 0
\(415\) −1.00000 −0.0490881
\(416\) 4.94427 0.242413
\(417\) 0 0
\(418\) −1.90569 −0.0932102
\(419\) 15.6525 0.764673 0.382337 0.924023i \(-0.375119\pi\)
0.382337 + 0.924023i \(0.375119\pi\)
\(420\) 0 0
\(421\) 4.47214 0.217959 0.108979 0.994044i \(-0.465242\pi\)
0.108979 + 0.994044i \(0.465242\pi\)
\(422\) 8.23024 0.400642
\(423\) 0 0
\(424\) −8.94427 −0.434372
\(425\) 24.7214 1.19916
\(426\) 0 0
\(427\) 0 0
\(428\) −12.2364 −0.591471
\(429\) 0 0
\(430\) −0.461370 −0.0222492
\(431\) 11.5687 0.557247 0.278623 0.960400i \(-0.410122\pi\)
0.278623 + 0.960400i \(0.410122\pi\)
\(432\) 0 0
\(433\) 37.4860 1.80146 0.900730 0.434379i \(-0.143032\pi\)
0.900730 + 0.434379i \(0.143032\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −17.5967 −0.842731
\(437\) −24.1803 −1.15670
\(438\) 0 0
\(439\) −19.3863 −0.925259 −0.462629 0.886552i \(-0.653094\pi\)
−0.462629 + 0.886552i \(0.653094\pi\)
\(440\) −0.360680 −0.0171947
\(441\) 0 0
\(442\) −3.81966 −0.181683
\(443\) 25.8384 1.22762 0.613810 0.789454i \(-0.289636\pi\)
0.613810 + 0.789454i \(0.289636\pi\)
\(444\) 0 0
\(445\) −2.76393 −0.131023
\(446\) 13.3475 0.632024
\(447\) 0 0
\(448\) 0 0
\(449\) 6.27585 0.296176 0.148088 0.988974i \(-0.452688\pi\)
0.148088 + 0.988974i \(0.452688\pi\)
\(450\) 0 0
\(451\) −4.70401 −0.221503
\(452\) −10.2333 −0.481336
\(453\) 0 0
\(454\) 8.89794 0.417601
\(455\) 0 0
\(456\) 0 0
\(457\) −20.8328 −0.974518 −0.487259 0.873257i \(-0.662003\pi\)
−0.487259 + 0.873257i \(0.662003\pi\)
\(458\) 20.3607 0.951392
\(459\) 0 0
\(460\) −1.74806 −0.0815039
\(461\) −31.4721 −1.46580 −0.732902 0.680334i \(-0.761835\pi\)
−0.732902 + 0.680334i \(0.761835\pi\)
\(462\) 0 0
\(463\) 14.8197 0.688728 0.344364 0.938836i \(-0.388095\pi\)
0.344364 + 0.938836i \(0.388095\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −13.7082 −0.635020
\(467\) −11.5967 −0.536633 −0.268317 0.963331i \(-0.586467\pi\)
−0.268317 + 0.963331i \(0.586467\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.54173 −0.0711148
\(471\) 0 0
\(472\) 39.4404 1.81539
\(473\) −1.20788 −0.0555385
\(474\) 0 0
\(475\) 19.9566 0.915672
\(476\) 0 0
\(477\) 0 0
\(478\) 8.54102 0.390657
\(479\) −27.0689 −1.23681 −0.618404 0.785860i \(-0.712221\pi\)
−0.618404 + 0.785860i \(0.712221\pi\)
\(480\) 0 0
\(481\) 7.61125 0.347043
\(482\) −2.29180 −0.104388
\(483\) 0 0
\(484\) 13.2361 0.601639
\(485\) 1.20788 0.0548471
\(486\) 0 0
\(487\) 9.88854 0.448093 0.224046 0.974578i \(-0.428073\pi\)
0.224046 + 0.974578i \(0.428073\pi\)
\(488\) −1.52786 −0.0691632
\(489\) 0 0
\(490\) 0 0
\(491\) −39.6467 −1.78923 −0.894615 0.446838i \(-0.852550\pi\)
−0.894615 + 0.446838i \(0.852550\pi\)
\(492\) 0 0
\(493\) 43.0640 1.93951
\(494\) −3.08347 −0.138732
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.11146 0.0497556 0.0248778 0.999690i \(-0.492080\pi\)
0.0248778 + 0.999690i \(0.492080\pi\)
\(500\) 2.90170 0.129768
\(501\) 0 0
\(502\) 19.8477 0.885846
\(503\) 17.8328 0.795126 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(504\) 0 0
\(505\) −1.12461 −0.0500446
\(506\) 2.82843 0.125739
\(507\) 0 0
\(508\) 4.29180 0.190418
\(509\) −27.3607 −1.21274 −0.606370 0.795182i \(-0.707375\pi\)
−0.606370 + 0.795182i \(0.707375\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) −18.7186 −0.825643
\(515\) 3.98760 0.175715
\(516\) 0 0
\(517\) −4.03631 −0.177517
\(518\) 0 0
\(519\) 0 0
\(520\) −0.583592 −0.0255922
\(521\) 17.4721 0.765468 0.382734 0.923859i \(-0.374983\pi\)
0.382734 + 0.923859i \(0.374983\pi\)
\(522\) 0 0
\(523\) 36.8183 1.60995 0.804975 0.593309i \(-0.202179\pi\)
0.804975 + 0.593309i \(0.202179\pi\)
\(524\) −8.58359 −0.374976
\(525\) 0 0
\(526\) −23.8197 −1.03859
\(527\) 32.6544 1.42245
\(528\) 0 0
\(529\) 12.8885 0.560371
\(530\) 0.652476 0.0283417
\(531\) 0 0
\(532\) 0 0
\(533\) −7.61125 −0.329680
\(534\) 0 0
\(535\) 2.33695 0.101035
\(536\) 19.1313 0.826346
\(537\) 0 0
\(538\) −6.11822 −0.263775
\(539\) 0 0
\(540\) 0 0
\(541\) −34.3050 −1.47489 −0.737443 0.675410i \(-0.763967\pi\)
−0.737443 + 0.675410i \(0.763967\pi\)
\(542\) −17.2361 −0.740353
\(543\) 0 0
\(544\) −28.2843 −1.21268
\(545\) 3.36068 0.143956
\(546\) 0 0
\(547\) −22.1246 −0.945980 −0.472990 0.881068i \(-0.656825\pi\)
−0.472990 + 0.881068i \(0.656825\pi\)
\(548\) 17.2256 0.735841
\(549\) 0 0
\(550\) −2.33437 −0.0995378
\(551\) 34.7639 1.48099
\(552\) 0 0
\(553\) 0 0
\(554\) −15.9389 −0.677179
\(555\) 0 0
\(556\) −11.4713 −0.486493
\(557\) 6.32456 0.267980 0.133990 0.990983i \(-0.457221\pi\)
0.133990 + 0.990983i \(0.457221\pi\)
\(558\) 0 0
\(559\) −1.95440 −0.0826621
\(560\) 0 0
\(561\) 0 0
\(562\) 25.0132 1.05512
\(563\) −11.8197 −0.498139 −0.249070 0.968486i \(-0.580125\pi\)
−0.249070 + 0.968486i \(0.580125\pi\)
\(564\) 0 0
\(565\) 1.95440 0.0822220
\(566\) −10.5836 −0.444862
\(567\) 0 0
\(568\) 19.0557 0.799561
\(569\) −0.0788114 −0.00330395 −0.00165197 0.999999i \(-0.500526\pi\)
−0.00165197 + 0.999999i \(0.500526\pi\)
\(570\) 0 0
\(571\) 39.0689 1.63498 0.817491 0.575941i \(-0.195364\pi\)
0.817491 + 0.575941i \(0.195364\pi\)
\(572\) −0.583592 −0.0244012
\(573\) 0 0
\(574\) 0 0
\(575\) −29.6197 −1.23523
\(576\) 0 0
\(577\) −18.5610 −0.772705 −0.386352 0.922351i \(-0.626265\pi\)
−0.386352 + 0.922351i \(0.626265\pi\)
\(578\) 6.99226 0.290840
\(579\) 0 0
\(580\) 2.51318 0.104354
\(581\) 0 0
\(582\) 0 0
\(583\) 1.70820 0.0707466
\(584\) −37.8885 −1.56784
\(585\) 0 0
\(586\) 1.28669 0.0531528
\(587\) 9.52786 0.393257 0.196629 0.980478i \(-0.437001\pi\)
0.196629 + 0.980478i \(0.437001\pi\)
\(588\) 0 0
\(589\) 26.3607 1.08617
\(590\) −2.87714 −0.118450
\(591\) 0 0
\(592\) 0 0
\(593\) −2.52786 −0.103807 −0.0519035 0.998652i \(-0.516529\pi\)
−0.0519035 + 0.998652i \(0.516529\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.97520 −0.326677
\(597\) 0 0
\(598\) 4.57649 0.187147
\(599\) 2.54328 0.103916 0.0519579 0.998649i \(-0.483454\pi\)
0.0519579 + 0.998649i \(0.483454\pi\)
\(600\) 0 0
\(601\) −41.1397 −1.67812 −0.839062 0.544036i \(-0.816896\pi\)
−0.839062 + 0.544036i \(0.816896\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12.8754 −0.523892
\(605\) −2.52786 −0.102772
\(606\) 0 0
\(607\) 17.1769 0.697189 0.348594 0.937274i \(-0.386659\pi\)
0.348594 + 0.937274i \(0.386659\pi\)
\(608\) −22.8328 −0.925993
\(609\) 0 0
\(610\) 0.111456 0.00451273
\(611\) −6.53089 −0.264211
\(612\) 0 0
\(613\) 3.88854 0.157057 0.0785284 0.996912i \(-0.474978\pi\)
0.0785284 + 0.996912i \(0.474978\pi\)
\(614\) −19.7082 −0.795358
\(615\) 0 0
\(616\) 0 0
\(617\) 2.08191 0.0838147 0.0419074 0.999122i \(-0.486657\pi\)
0.0419074 + 0.999122i \(0.486657\pi\)
\(618\) 0 0
\(619\) 8.86784 0.356429 0.178214 0.983992i \(-0.442968\pi\)
0.178214 + 0.983992i \(0.442968\pi\)
\(620\) 1.90569 0.0765342
\(621\) 0 0
\(622\) 27.1552 1.08882
\(623\) 0 0
\(624\) 0 0
\(625\) 24.1672 0.966687
\(626\) −14.2492 −0.569514
\(627\) 0 0
\(628\) 21.6445 0.863708
\(629\) −43.5410 −1.73609
\(630\) 0 0
\(631\) −21.7639 −0.866408 −0.433204 0.901296i \(-0.642617\pi\)
−0.433204 + 0.901296i \(0.642617\pi\)
\(632\) 7.14988 0.284407
\(633\) 0 0
\(634\) −9.23607 −0.366811
\(635\) −0.819660 −0.0325272
\(636\) 0 0
\(637\) 0 0
\(638\) −4.06641 −0.160991
\(639\) 0 0
\(640\) −1.65065 −0.0652476
\(641\) 31.6529 1.25021 0.625107 0.780539i \(-0.285055\pi\)
0.625107 + 0.780539i \(0.285055\pi\)
\(642\) 0 0
\(643\) −8.56409 −0.337735 −0.168867 0.985639i \(-0.554011\pi\)
−0.168867 + 0.985639i \(0.554011\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 17.6393 0.694010
\(647\) −40.3607 −1.58674 −0.793371 0.608738i \(-0.791676\pi\)
−0.793371 + 0.608738i \(0.791676\pi\)
\(648\) 0 0
\(649\) −7.53244 −0.295674
\(650\) −3.77709 −0.148150
\(651\) 0 0
\(652\) −21.2361 −0.831669
\(653\) −22.6761 −0.887385 −0.443693 0.896179i \(-0.646332\pi\)
−0.443693 + 0.896179i \(0.646332\pi\)
\(654\) 0 0
\(655\) 1.63932 0.0640535
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −32.1931 −1.25406 −0.627032 0.778994i \(-0.715730\pi\)
−0.627032 + 0.778994i \(0.715730\pi\)
\(660\) 0 0
\(661\) 30.7487 1.19599 0.597994 0.801501i \(-0.295965\pi\)
0.597994 + 0.801501i \(0.295965\pi\)
\(662\) −7.35621 −0.285907
\(663\) 0 0
\(664\) 11.9814 0.464969
\(665\) 0 0
\(666\) 0 0
\(667\) −51.5967 −1.99783
\(668\) −10.1803 −0.393889
\(669\) 0 0
\(670\) −1.39561 −0.0539171
\(671\) 0.291796 0.0112647
\(672\) 0 0
\(673\) −36.7639 −1.41715 −0.708573 0.705638i \(-0.750661\pi\)
−0.708573 + 0.705638i \(0.750661\pi\)
\(674\) 7.19859 0.277279
\(675\) 0 0
\(676\) 15.1246 0.581716
\(677\) −47.8885 −1.84051 −0.920253 0.391324i \(-0.872017\pi\)
−0.920253 + 0.391324i \(0.872017\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.33851 0.128026
\(681\) 0 0
\(682\) −3.08347 −0.118072
\(683\) 14.8585 0.568546 0.284273 0.958743i \(-0.408248\pi\)
0.284273 + 0.958743i \(0.408248\pi\)
\(684\) 0 0
\(685\) −3.28980 −0.125697
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.76393 0.105297
\(690\) 0 0
\(691\) −27.6653 −1.05244 −0.526218 0.850349i \(-0.676391\pi\)
−0.526218 + 0.850349i \(0.676391\pi\)
\(692\) 11.7771 0.447698
\(693\) 0 0
\(694\) −23.0132 −0.873567
\(695\) 2.19083 0.0831029
\(696\) 0 0
\(697\) 43.5410 1.64923
\(698\) −7.05573 −0.267063
\(699\) 0 0
\(700\) 0 0
\(701\) −31.5741 −1.19254 −0.596268 0.802785i \(-0.703350\pi\)
−0.596268 + 0.802785i \(0.703350\pi\)
\(702\) 0 0
\(703\) −35.1490 −1.32567
\(704\) 2.67080 0.100660
\(705\) 0 0
\(706\) 8.84924 0.333045
\(707\) 0 0
\(708\) 0 0
\(709\) −12.5279 −0.470494 −0.235247 0.971936i \(-0.575590\pi\)
−0.235247 + 0.971936i \(0.575590\pi\)
\(710\) −1.39010 −0.0521694
\(711\) 0 0
\(712\) 33.1158 1.24107
\(713\) −39.1246 −1.46523
\(714\) 0 0
\(715\) 0.111456 0.00416822
\(716\) 22.0571 0.824314
\(717\) 0 0
\(718\) 20.6525 0.770744
\(719\) 26.4164 0.985166 0.492583 0.870266i \(-0.336053\pi\)
0.492583 + 0.870266i \(0.336053\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.36706 −0.0880927
\(723\) 0 0
\(724\) 10.8036 0.401514
\(725\) 42.5840 1.58153
\(726\) 0 0
\(727\) 19.6715 0.729574 0.364787 0.931091i \(-0.381142\pi\)
0.364787 + 0.931091i \(0.381142\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.76393 0.102298
\(731\) 11.1803 0.413520
\(732\) 0 0
\(733\) −6.32456 −0.233603 −0.116801 0.993155i \(-0.537264\pi\)
−0.116801 + 0.993155i \(0.537264\pi\)
\(734\) −11.3475 −0.418845
\(735\) 0 0
\(736\) 33.8885 1.24915
\(737\) −3.65375 −0.134588
\(738\) 0 0
\(739\) −48.3607 −1.77898 −0.889488 0.456958i \(-0.848939\pi\)
−0.889488 + 0.456958i \(0.848939\pi\)
\(740\) −2.54102 −0.0934097
\(741\) 0 0
\(742\) 0 0
\(743\) −3.78127 −0.138721 −0.0693607 0.997592i \(-0.522096\pi\)
−0.0693607 + 0.997592i \(0.522096\pi\)
\(744\) 0 0
\(745\) 1.52313 0.0558031
\(746\) 15.4288 0.564890
\(747\) 0 0
\(748\) 3.33851 0.122068
\(749\) 0 0
\(750\) 0 0
\(751\) 13.3475 0.487058 0.243529 0.969894i \(-0.421695\pi\)
0.243529 + 0.969894i \(0.421695\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −6.57959 −0.239615
\(755\) 2.45898 0.0894915
\(756\) 0 0
\(757\) 9.36068 0.340220 0.170110 0.985425i \(-0.445588\pi\)
0.170110 + 0.985425i \(0.445588\pi\)
\(758\) 4.27274 0.155193
\(759\) 0 0
\(760\) 2.69505 0.0977597
\(761\) 26.5279 0.961634 0.480817 0.876821i \(-0.340340\pi\)
0.480817 + 0.876821i \(0.340340\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.24419 −0.189728
\(765\) 0 0
\(766\) −9.77198 −0.353076
\(767\) −12.1877 −0.440074
\(768\) 0 0
\(769\) −47.6706 −1.71905 −0.859523 0.511097i \(-0.829239\pi\)
−0.859523 + 0.511097i \(0.829239\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.40325 0.230458
\(773\) −48.5967 −1.74790 −0.873952 0.486013i \(-0.838451\pi\)
−0.873952 + 0.486013i \(0.838451\pi\)
\(774\) 0 0
\(775\) 32.2905 1.15991
\(776\) −14.4721 −0.519519
\(777\) 0 0
\(778\) −0.832816 −0.0298579
\(779\) 35.1490 1.25934
\(780\) 0 0
\(781\) −3.63932 −0.130225
\(782\) −26.1803 −0.936207
\(783\) 0 0
\(784\) 0 0
\(785\) −4.13373 −0.147539
\(786\) 0 0
\(787\) 34.5300 1.23086 0.615431 0.788191i \(-0.288982\pi\)
0.615431 + 0.788191i \(0.288982\pi\)
\(788\) −26.7912 −0.954399
\(789\) 0 0
\(790\) −0.521577 −0.0185569
\(791\) 0 0
\(792\) 0 0
\(793\) 0.472136 0.0167660
\(794\) −14.4033 −0.511152
\(795\) 0 0
\(796\) −32.2905 −1.14451
\(797\) −42.0689 −1.49016 −0.745078 0.666977i \(-0.767588\pi\)
−0.745078 + 0.666977i \(0.767588\pi\)
\(798\) 0 0
\(799\) 37.3607 1.32173
\(800\) −27.9690 −0.988854
\(801\) 0 0
\(802\) −26.2492 −0.926892
\(803\) 7.23607 0.255355
\(804\) 0 0
\(805\) 0 0
\(806\) −4.98915 −0.175735
\(807\) 0 0
\(808\) 13.4744 0.474029
\(809\) −2.08191 −0.0731962 −0.0365981 0.999330i \(-0.511652\pi\)
−0.0365981 + 0.999330i \(0.511652\pi\)
\(810\) 0 0
\(811\) 28.4906 1.00044 0.500220 0.865898i \(-0.333252\pi\)
0.500220 + 0.865898i \(0.333252\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.11146 0.144106
\(815\) 4.05573 0.142066
\(816\) 0 0
\(817\) 9.02546 0.315761
\(818\) −10.6950 −0.373944
\(819\) 0 0
\(820\) 2.54102 0.0887363
\(821\) −32.0354 −1.11804 −0.559022 0.829153i \(-0.688823\pi\)
−0.559022 + 0.829153i \(0.688823\pi\)
\(822\) 0 0
\(823\) −36.4164 −1.26940 −0.634698 0.772760i \(-0.718876\pi\)
−0.634698 + 0.772760i \(0.718876\pi\)
\(824\) −47.7771 −1.66439
\(825\) 0 0
\(826\) 0 0
\(827\) −7.45363 −0.259188 −0.129594 0.991567i \(-0.541367\pi\)
−0.129594 + 0.991567i \(0.541367\pi\)
\(828\) 0 0
\(829\) −32.3206 −1.12254 −0.561270 0.827633i \(-0.689687\pi\)
−0.561270 + 0.827633i \(0.689687\pi\)
\(830\) −0.874032 −0.0303381
\(831\) 0 0
\(832\) 4.32145 0.149819
\(833\) 0 0
\(834\) 0 0
\(835\) 1.94427 0.0672843
\(836\) 2.69505 0.0932102
\(837\) 0 0
\(838\) 13.6808 0.472594
\(839\) 47.1803 1.62885 0.814423 0.580271i \(-0.197054\pi\)
0.814423 + 0.580271i \(0.197054\pi\)
\(840\) 0 0
\(841\) 45.1803 1.55794
\(842\) 3.90879 0.134706
\(843\) 0 0
\(844\) −11.6393 −0.400642
\(845\) −2.88854 −0.0993689
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 21.6073 0.741123
\(851\) 52.1683 1.78831
\(852\) 0 0
\(853\) 35.5316 1.21658 0.608289 0.793716i \(-0.291856\pi\)
0.608289 + 0.793716i \(0.291856\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −28.0000 −0.957020
\(857\) −19.1803 −0.655188 −0.327594 0.944819i \(-0.606238\pi\)
−0.327594 + 0.944819i \(0.606238\pi\)
\(858\) 0 0
\(859\) −3.31990 −0.113274 −0.0566368 0.998395i \(-0.518038\pi\)
−0.0566368 + 0.998395i \(0.518038\pi\)
\(860\) 0.652476 0.0222492
\(861\) 0 0
\(862\) 10.1115 0.344398
\(863\) 9.07417 0.308888 0.154444 0.988002i \(-0.450641\pi\)
0.154444 + 0.988002i \(0.450641\pi\)
\(864\) 0 0
\(865\) −2.24922 −0.0764759
\(866\) 32.7639 1.11336
\(867\) 0 0
\(868\) 0 0
\(869\) −1.36551 −0.0463216
\(870\) 0 0
\(871\) −5.91189 −0.200317
\(872\) −40.2657 −1.36357
\(873\) 0 0
\(874\) −21.1344 −0.714881
\(875\) 0 0
\(876\) 0 0
\(877\) 26.0132 0.878402 0.439201 0.898389i \(-0.355262\pi\)
0.439201 + 0.898389i \(0.355262\pi\)
\(878\) −16.9443 −0.571841
\(879\) 0 0
\(880\) 0 0
\(881\) 36.8328 1.24093 0.620465 0.784234i \(-0.286944\pi\)
0.620465 + 0.784234i \(0.286944\pi\)
\(882\) 0 0
\(883\) −18.8885 −0.635650 −0.317825 0.948149i \(-0.602952\pi\)
−0.317825 + 0.948149i \(0.602952\pi\)
\(884\) 5.40182 0.181683
\(885\) 0 0
\(886\) 22.5836 0.758711
\(887\) 46.1935 1.55103 0.775513 0.631332i \(-0.217491\pi\)
0.775513 + 0.631332i \(0.217491\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.41577 −0.0809766
\(891\) 0 0
\(892\) −18.8762 −0.632024
\(893\) 30.1599 1.00926
\(894\) 0 0
\(895\) −4.21254 −0.140810
\(896\) 0 0
\(897\) 0 0
\(898\) 5.48529 0.183047
\(899\) 56.2492 1.87602
\(900\) 0 0
\(901\) −15.8114 −0.526754
\(902\) −4.11146 −0.136897
\(903\) 0 0
\(904\) −23.4164 −0.778818
\(905\) −2.06331 −0.0685867
\(906\) 0 0
\(907\) 47.0000 1.56061 0.780305 0.625400i \(-0.215064\pi\)
0.780305 + 0.625400i \(0.215064\pi\)
\(908\) −12.5836 −0.417601
\(909\) 0 0
\(910\) 0 0
\(911\) −32.3206 −1.07083 −0.535414 0.844590i \(-0.679845\pi\)
−0.535414 + 0.844590i \(0.679845\pi\)
\(912\) 0 0
\(913\) −2.28825 −0.0757299
\(914\) −18.2085 −0.602285
\(915\) 0 0
\(916\) −28.7943 −0.951392
\(917\) 0 0
\(918\) 0 0
\(919\) −15.9443 −0.525953 −0.262976 0.964802i \(-0.584704\pi\)
−0.262976 + 0.964802i \(0.584704\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −27.5077 −0.905916
\(923\) −5.88854 −0.193824
\(924\) 0 0
\(925\) −43.0557 −1.41566
\(926\) 12.9529 0.425657
\(927\) 0 0
\(928\) −48.7214 −1.59936
\(929\) −35.8328 −1.17564 −0.587818 0.808993i \(-0.700013\pi\)
−0.587818 + 0.808993i \(0.700013\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 19.3863 0.635020
\(933\) 0 0
\(934\) −10.1359 −0.331658
\(935\) −0.637598 −0.0208517
\(936\) 0 0
\(937\) 14.6823 0.479650 0.239825 0.970816i \(-0.422910\pi\)
0.239825 + 0.970816i \(0.422910\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.18034 0.0711148
\(941\) −7.29180 −0.237706 −0.118853 0.992912i \(-0.537922\pi\)
−0.118853 + 0.992912i \(0.537922\pi\)
\(942\) 0 0
\(943\) −52.1683 −1.69883
\(944\) 0 0
\(945\) 0 0
\(946\) −1.05573 −0.0343247
\(947\) 35.1189 1.14121 0.570606 0.821224i \(-0.306709\pi\)
0.570606 + 0.821224i \(0.306709\pi\)
\(948\) 0 0
\(949\) 11.7082 0.380064
\(950\) 17.4427 0.565917
\(951\) 0 0
\(952\) 0 0
\(953\) 27.5378 0.892036 0.446018 0.895024i \(-0.352842\pi\)
0.446018 + 0.895024i \(0.352842\pi\)
\(954\) 0 0
\(955\) 1.00155 0.0324094
\(956\) −12.0788 −0.390657
\(957\) 0 0
\(958\) −23.6591 −0.764390
\(959\) 0 0
\(960\) 0 0
\(961\) 11.6525 0.375886
\(962\) 6.65248 0.214484
\(963\) 0 0
\(964\) 3.24109 0.104388
\(965\) −1.22291 −0.0393669
\(966\) 0 0
\(967\) 14.4721 0.465393 0.232696 0.972549i \(-0.425245\pi\)
0.232696 + 0.972549i \(0.425245\pi\)
\(968\) 30.2874 0.973473
\(969\) 0 0
\(970\) 1.05573 0.0338974
\(971\) 29.4721 0.945806 0.472903 0.881115i \(-0.343206\pi\)
0.472903 + 0.881115i \(0.343206\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 8.64290 0.276937
\(975\) 0 0
\(976\) 0 0
\(977\) −33.1158 −1.05947 −0.529734 0.848164i \(-0.677708\pi\)
−0.529734 + 0.848164i \(0.677708\pi\)
\(978\) 0 0
\(979\) −6.32456 −0.202134
\(980\) 0 0
\(981\) 0 0
\(982\) −34.6525 −1.10580
\(983\) 43.9443 1.40160 0.700802 0.713356i \(-0.252826\pi\)
0.700802 + 0.713356i \(0.252826\pi\)
\(984\) 0 0
\(985\) 5.11667 0.163031
\(986\) 37.6393 1.19868
\(987\) 0 0
\(988\) 4.36068 0.138732
\(989\) −13.3956 −0.425956
\(990\) 0 0
\(991\) −40.1246 −1.27460 −0.637300 0.770616i \(-0.719949\pi\)
−0.637300 + 0.770616i \(0.719949\pi\)
\(992\) −36.9443 −1.17298
\(993\) 0 0
\(994\) 0 0
\(995\) 6.16693 0.195505
\(996\) 0 0
\(997\) −17.6082 −0.557656 −0.278828 0.960341i \(-0.589946\pi\)
−0.278828 + 0.960341i \(0.589946\pi\)
\(998\) 0.971448 0.0307507
\(999\) 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 1323.2.a.bb.1.3 yes 4
3.2 odd 2 1323.2.a.be.1.2 yes 4
7.6 odd 2 1323.2.a.be.1.3 yes 4
21.20 even 2 inner 1323.2.a.bb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.bb.1.2 4 21.20 even 2 inner
1323.2.a.bb.1.3 yes 4 1.1 even 1 trivial
1323.2.a.be.1.2 yes 4 3.2 odd 2
1323.2.a.be.1.3 yes 4 7.6 odd 2