Properties

Label 3330.2.a.bj.1.3
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.339102\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.84556 q^{7} -1.00000 q^{8} +1.00000 q^{10} -0.832640 q^{11} +3.21973 q^{13} -1.84556 q^{14} +1.00000 q^{16} +7.95030 q^{17} +4.20681 q^{19} -1.00000 q^{20} +0.832640 q^{22} +6.57614 q^{23} +1.00000 q^{25} -3.21973 q^{26} +1.84556 q^{28} -3.03945 q^{29} -4.73057 q^{31} -1.00000 q^{32} -7.95030 q^{34} -1.84556 q^{35} -1.00000 q^{37} -4.20681 q^{38} +1.00000 q^{40} -2.73057 q^{41} -5.03945 q^{43} -0.832640 q^{44} -6.57614 q^{46} -9.11766 q^{47} -3.59390 q^{49} -1.00000 q^{50} +3.21973 q^{52} -3.95030 q^{53} +0.832640 q^{55} -1.84556 q^{56} +3.03945 q^{58} +3.69113 q^{59} +5.40878 q^{61} +4.73057 q^{62} +1.00000 q^{64} -3.21973 q^{65} +11.7959 q^{67} +7.95030 q^{68} +1.84556 q^{70} +13.7700 q^{71} -3.52860 q^{73} +1.00000 q^{74} +4.20681 q^{76} -1.53669 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.73057 q^{82} +3.55445 q^{83} -7.95030 q^{85} +5.03945 q^{86} +0.832640 q^{88} +6.55029 q^{89} +5.94222 q^{91} +6.57614 q^{92} +9.11766 q^{94} -4.20681 q^{95} -10.4918 q^{97} +3.59390 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{5} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{5} + 4q^{7} - 4q^{8} + 4q^{10} - 2q^{11} - 3q^{13} - 4q^{14} + 4q^{16} - 6q^{17} + 3q^{19} - 4q^{20} + 2q^{22} + q^{23} + 4q^{25} + 3q^{26} + 4q^{28} + 3q^{29} + 3q^{31} - 4q^{32} + 6q^{34} - 4q^{35} - 4q^{37} - 3q^{38} + 4q^{40} + 11q^{41} - 5q^{43} - 2q^{44} - q^{46} + 14q^{49} - 4q^{50} - 3q^{52} + 22q^{53} + 2q^{55} - 4q^{56} - 3q^{58} + 8q^{59} - 5q^{61} - 3q^{62} + 4q^{64} + 3q^{65} + 6q^{67} - 6q^{68} + 4q^{70} + 18q^{71} - 5q^{73} + 4q^{74} + 3q^{76} + 4q^{77} + 16q^{79} - 4q^{80} - 11q^{82} + q^{83} + 6q^{85} + 5q^{86} + 2q^{88} + 5q^{89} - 13q^{91} + q^{92} - 3q^{95} + 7q^{97} - 14q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.84556 0.697557 0.348779 0.937205i \(-0.386596\pi\)
0.348779 + 0.937205i \(0.386596\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −0.832640 −0.251050 −0.125525 0.992090i \(-0.540062\pi\)
−0.125525 + 0.992090i \(0.540062\pi\)
\(12\) 0 0
\(13\) 3.21973 0.892992 0.446496 0.894786i \(-0.352672\pi\)
0.446496 + 0.894786i \(0.352672\pi\)
\(14\) −1.84556 −0.493248
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.95030 1.92823 0.964116 0.265482i \(-0.0855311\pi\)
0.964116 + 0.265482i \(0.0855311\pi\)
\(18\) 0 0
\(19\) 4.20681 0.965108 0.482554 0.875866i \(-0.339709\pi\)
0.482554 + 0.875866i \(0.339709\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0.832640 0.177519
\(23\) 6.57614 1.37122 0.685610 0.727969i \(-0.259536\pi\)
0.685610 + 0.727969i \(0.259536\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.21973 −0.631441
\(27\) 0 0
\(28\) 1.84556 0.348779
\(29\) −3.03945 −0.564411 −0.282206 0.959354i \(-0.591066\pi\)
−0.282206 + 0.959354i \(0.591066\pi\)
\(30\) 0 0
\(31\) −4.73057 −0.849636 −0.424818 0.905279i \(-0.639662\pi\)
−0.424818 + 0.905279i \(0.639662\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.95030 −1.36347
\(35\) −1.84556 −0.311957
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −4.20681 −0.682434
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.73057 −0.426444 −0.213222 0.977004i \(-0.568396\pi\)
−0.213222 + 0.977004i \(0.568396\pi\)
\(42\) 0 0
\(43\) −5.03945 −0.768508 −0.384254 0.923227i \(-0.625541\pi\)
−0.384254 + 0.923227i \(0.625541\pi\)
\(44\) −0.832640 −0.125525
\(45\) 0 0
\(46\) −6.57614 −0.969598
\(47\) −9.11766 −1.32995 −0.664974 0.746867i \(-0.731557\pi\)
−0.664974 + 0.746867i \(0.731557\pi\)
\(48\) 0 0
\(49\) −3.59390 −0.513414
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 3.21973 0.446496
\(53\) −3.95030 −0.542616 −0.271308 0.962493i \(-0.587456\pi\)
−0.271308 + 0.962493i \(0.587456\pi\)
\(54\) 0 0
\(55\) 0.832640 0.112273
\(56\) −1.84556 −0.246624
\(57\) 0 0
\(58\) 3.03945 0.399099
\(59\) 3.69113 0.480544 0.240272 0.970706i \(-0.422763\pi\)
0.240272 + 0.970706i \(0.422763\pi\)
\(60\) 0 0
\(61\) 5.40878 0.692523 0.346261 0.938138i \(-0.387451\pi\)
0.346261 + 0.938138i \(0.387451\pi\)
\(62\) 4.73057 0.600783
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.21973 −0.399358
\(66\) 0 0
\(67\) 11.7959 1.44109 0.720547 0.693406i \(-0.243891\pi\)
0.720547 + 0.693406i \(0.243891\pi\)
\(68\) 7.95030 0.964116
\(69\) 0 0
\(70\) 1.84556 0.220587
\(71\) 13.7700 1.63420 0.817100 0.576495i \(-0.195580\pi\)
0.817100 + 0.576495i \(0.195580\pi\)
\(72\) 0 0
\(73\) −3.52860 −0.412992 −0.206496 0.978447i \(-0.566206\pi\)
−0.206496 + 0.978447i \(0.566206\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 4.20681 0.482554
\(77\) −1.53669 −0.175122
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.73057 0.301541
\(83\) 3.55445 0.390151 0.195076 0.980788i \(-0.437505\pi\)
0.195076 + 0.980788i \(0.437505\pi\)
\(84\) 0 0
\(85\) −7.95030 −0.862331
\(86\) 5.03945 0.543417
\(87\) 0 0
\(88\) 0.832640 0.0887597
\(89\) 6.55029 0.694329 0.347165 0.937804i \(-0.387144\pi\)
0.347165 + 0.937804i \(0.387144\pi\)
\(90\) 0 0
\(91\) 5.94222 0.622913
\(92\) 6.57614 0.685610
\(93\) 0 0
\(94\) 9.11766 0.940415
\(95\) −4.20681 −0.431609
\(96\) 0 0
\(97\) −10.4918 −1.06528 −0.532642 0.846341i \(-0.678801\pi\)
−0.532642 + 0.846341i \(0.678801\pi\)
\(98\) 3.59390 0.363038
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 3.63067 0.361265 0.180633 0.983551i \(-0.442186\pi\)
0.180633 + 0.983551i \(0.442186\pi\)
\(102\) 0 0
\(103\) −9.11766 −0.898390 −0.449195 0.893434i \(-0.648289\pi\)
−0.449195 + 0.893434i \(0.648289\pi\)
\(104\) −3.21973 −0.315720
\(105\) 0 0
\(106\) 3.95030 0.383687
\(107\) −8.91086 −0.861445 −0.430723 0.902484i \(-0.641741\pi\)
−0.430723 + 0.902484i \(0.641741\pi\)
\(108\) 0 0
\(109\) 11.2979 1.08215 0.541073 0.840975i \(-0.318018\pi\)
0.541073 + 0.840975i \(0.318018\pi\)
\(110\) −0.832640 −0.0793891
\(111\) 0 0
\(112\) 1.84556 0.174389
\(113\) 11.0394 1.03850 0.519252 0.854621i \(-0.326211\pi\)
0.519252 + 0.854621i \(0.326211\pi\)
\(114\) 0 0
\(115\) −6.57614 −0.613228
\(116\) −3.03945 −0.282206
\(117\) 0 0
\(118\) −3.69113 −0.339796
\(119\) 14.6728 1.34505
\(120\) 0 0
\(121\) −10.3067 −0.936974
\(122\) −5.40878 −0.489688
\(123\) 0 0
\(124\) −4.73057 −0.424818
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.136678 −0.0121282 −0.00606410 0.999982i \(-0.501930\pi\)
−0.00606410 + 0.999982i \(0.501930\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.21973 0.282389
\(131\) −2.07889 −0.181634 −0.0908169 0.995868i \(-0.528948\pi\)
−0.0908169 + 0.995868i \(0.528948\pi\)
\(132\) 0 0
\(133\) 7.76393 0.673218
\(134\) −11.7959 −1.01901
\(135\) 0 0
\(136\) −7.95030 −0.681733
\(137\) −3.06597 −0.261943 −0.130972 0.991386i \(-0.541810\pi\)
−0.130972 + 0.991386i \(0.541810\pi\)
\(138\) 0 0
\(139\) 22.1917 1.88228 0.941139 0.338021i \(-0.109757\pi\)
0.941139 + 0.338021i \(0.109757\pi\)
\(140\) −1.84556 −0.155979
\(141\) 0 0
\(142\) −13.7700 −1.15555
\(143\) −2.68088 −0.224186
\(144\) 0 0
\(145\) 3.03945 0.252412
\(146\) 3.52860 0.292029
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 14.4999 1.18788 0.593940 0.804510i \(-0.297572\pi\)
0.593940 + 0.804510i \(0.297572\pi\)
\(150\) 0 0
\(151\) 8.24142 0.670677 0.335339 0.942098i \(-0.391149\pi\)
0.335339 + 0.942098i \(0.391149\pi\)
\(152\) −4.20681 −0.341217
\(153\) 0 0
\(154\) 1.53669 0.123830
\(155\) 4.73057 0.379969
\(156\) 0 0
\(157\) −6.11283 −0.487857 −0.243928 0.969793i \(-0.578436\pi\)
−0.243928 + 0.969793i \(0.578436\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 12.1367 0.956504
\(162\) 0 0
\(163\) −21.4373 −1.67910 −0.839549 0.543283i \(-0.817181\pi\)
−0.839549 + 0.543283i \(0.817181\pi\)
\(164\) −2.73057 −0.213222
\(165\) 0 0
\(166\) −3.55445 −0.275879
\(167\) 22.5815 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(168\) 0 0
\(169\) −2.63334 −0.202565
\(170\) 7.95030 0.609760
\(171\) 0 0
\(172\) −5.03945 −0.384254
\(173\) 6.48916 0.493361 0.246681 0.969097i \(-0.420660\pi\)
0.246681 + 0.969097i \(0.420660\pi\)
\(174\) 0 0
\(175\) 1.84556 0.139511
\(176\) −0.832640 −0.0627626
\(177\) 0 0
\(178\) −6.55029 −0.490965
\(179\) −16.2095 −1.21155 −0.605777 0.795635i \(-0.707138\pi\)
−0.605777 + 0.795635i \(0.707138\pi\)
\(180\) 0 0
\(181\) 5.38225 0.400060 0.200030 0.979790i \(-0.435896\pi\)
0.200030 + 0.979790i \(0.435896\pi\)
\(182\) −5.94222 −0.440466
\(183\) 0 0
\(184\) −6.57614 −0.484799
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −6.61974 −0.484083
\(188\) −9.11766 −0.664974
\(189\) 0 0
\(190\) 4.20681 0.305194
\(191\) 10.5680 0.764677 0.382339 0.924022i \(-0.375119\pi\)
0.382339 + 0.924022i \(0.375119\pi\)
\(192\) 0 0
\(193\) 3.11766 0.224414 0.112207 0.993685i \(-0.464208\pi\)
0.112207 + 0.993685i \(0.464208\pi\)
\(194\) 10.4918 0.753269
\(195\) 0 0
\(196\) −3.59390 −0.256707
\(197\) −0.576137 −0.0410481 −0.0205240 0.999789i \(-0.506533\pi\)
−0.0205240 + 0.999789i \(0.506533\pi\)
\(198\) 0 0
\(199\) 6.64359 0.470952 0.235476 0.971880i \(-0.424335\pi\)
0.235476 + 0.971880i \(0.424335\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −3.63067 −0.255453
\(203\) −5.60949 −0.393709
\(204\) 0 0
\(205\) 2.73057 0.190712
\(206\) 9.11766 0.635258
\(207\) 0 0
\(208\) 3.21973 0.223248
\(209\) −3.50276 −0.242291
\(210\) 0 0
\(211\) −8.83531 −0.608248 −0.304124 0.952632i \(-0.598364\pi\)
−0.304124 + 0.952632i \(0.598364\pi\)
\(212\) −3.95030 −0.271308
\(213\) 0 0
\(214\) 8.91086 0.609134
\(215\) 5.03945 0.343687
\(216\) 0 0
\(217\) −8.73057 −0.592670
\(218\) −11.2979 −0.765193
\(219\) 0 0
\(220\) 0.832640 0.0561366
\(221\) 25.5978 1.72190
\(222\) 0 0
\(223\) 17.8828 1.19752 0.598762 0.800927i \(-0.295660\pi\)
0.598762 + 0.800927i \(0.295660\pi\)
\(224\) −1.84556 −0.123312
\(225\) 0 0
\(226\) −11.0394 −0.734333
\(227\) −25.1959 −1.67231 −0.836155 0.548494i \(-0.815201\pi\)
−0.836155 + 0.548494i \(0.815201\pi\)
\(228\) 0 0
\(229\) −5.48699 −0.362591 −0.181295 0.983429i \(-0.558029\pi\)
−0.181295 + 0.983429i \(0.558029\pi\)
\(230\) 6.57614 0.433618
\(231\) 0 0
\(232\) 3.03945 0.199549
\(233\) 28.5788 1.87226 0.936130 0.351654i \(-0.114381\pi\)
0.936130 + 0.351654i \(0.114381\pi\)
\(234\) 0 0
\(235\) 9.11766 0.594771
\(236\) 3.69113 0.240272
\(237\) 0 0
\(238\) −14.6728 −0.951096
\(239\) −1.77811 −0.115016 −0.0575081 0.998345i \(-0.518316\pi\)
−0.0575081 + 0.998345i \(0.518316\pi\)
\(240\) 0 0
\(241\) 22.2612 1.43397 0.716984 0.697090i \(-0.245522\pi\)
0.716984 + 0.697090i \(0.245522\pi\)
\(242\) 10.3067 0.662540
\(243\) 0 0
\(244\) 5.40878 0.346261
\(245\) 3.59390 0.229606
\(246\) 0 0
\(247\) 13.5448 0.861834
\(248\) 4.73057 0.300392
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 28.6748 1.80994 0.904968 0.425479i \(-0.139894\pi\)
0.904968 + 0.425479i \(0.139894\pi\)
\(252\) 0 0
\(253\) −5.47556 −0.344245
\(254\) 0.136678 0.00857593
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.19388 −0.573499 −0.286749 0.958006i \(-0.592575\pi\)
−0.286749 + 0.958006i \(0.592575\pi\)
\(258\) 0 0
\(259\) −1.84556 −0.114678
\(260\) −3.21973 −0.199679
\(261\) 0 0
\(262\) 2.07889 0.128434
\(263\) 8.02652 0.494937 0.247468 0.968896i \(-0.420401\pi\)
0.247468 + 0.968896i \(0.420401\pi\)
\(264\) 0 0
\(265\) 3.95030 0.242665
\(266\) −7.76393 −0.476037
\(267\) 0 0
\(268\) 11.7959 0.720547
\(269\) −21.3849 −1.30386 −0.651931 0.758278i \(-0.726041\pi\)
−0.651931 + 0.758278i \(0.726041\pi\)
\(270\) 0 0
\(271\) 2.64359 0.160587 0.0802934 0.996771i \(-0.474414\pi\)
0.0802934 + 0.996771i \(0.474414\pi\)
\(272\) 7.95030 0.482058
\(273\) 0 0
\(274\) 3.06597 0.185222
\(275\) −0.832640 −0.0502101
\(276\) 0 0
\(277\) −14.4456 −0.867949 −0.433975 0.900925i \(-0.642889\pi\)
−0.433975 + 0.900925i \(0.642889\pi\)
\(278\) −22.1917 −1.33097
\(279\) 0 0
\(280\) 1.84556 0.110294
\(281\) 11.2197 0.669313 0.334656 0.942340i \(-0.391380\pi\)
0.334656 + 0.942340i \(0.391380\pi\)
\(282\) 0 0
\(283\) 3.86332 0.229651 0.114825 0.993386i \(-0.463369\pi\)
0.114825 + 0.993386i \(0.463369\pi\)
\(284\) 13.7700 0.817100
\(285\) 0 0
\(286\) 2.68088 0.158524
\(287\) −5.03945 −0.297469
\(288\) 0 0
\(289\) 46.2073 2.71808
\(290\) −3.03945 −0.178482
\(291\) 0 0
\(292\) −3.52860 −0.206496
\(293\) 16.1544 0.943752 0.471876 0.881665i \(-0.343577\pi\)
0.471876 + 0.881665i \(0.343577\pi\)
\(294\) 0 0
\(295\) −3.69113 −0.214906
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −14.4999 −0.839958
\(299\) 21.1734 1.22449
\(300\) 0 0
\(301\) −9.30062 −0.536079
\(302\) −8.24142 −0.474240
\(303\) 0 0
\(304\) 4.20681 0.241277
\(305\) −5.40878 −0.309706
\(306\) 0 0
\(307\) 6.28303 0.358591 0.179296 0.983795i \(-0.442618\pi\)
0.179296 + 0.983795i \(0.442618\pi\)
\(308\) −1.53669 −0.0875611
\(309\) 0 0
\(310\) −4.73057 −0.268679
\(311\) −20.6312 −1.16989 −0.584943 0.811074i \(-0.698883\pi\)
−0.584943 + 0.811074i \(0.698883\pi\)
\(312\) 0 0
\(313\) −7.32180 −0.413852 −0.206926 0.978357i \(-0.566346\pi\)
−0.206926 + 0.978357i \(0.566346\pi\)
\(314\) 6.11283 0.344967
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −19.1442 −1.07524 −0.537622 0.843186i \(-0.680677\pi\)
−0.537622 + 0.843186i \(0.680677\pi\)
\(318\) 0 0
\(319\) 2.53077 0.141696
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −12.1367 −0.676351
\(323\) 33.4454 1.86095
\(324\) 0 0
\(325\) 3.21973 0.178598
\(326\) 21.4373 1.18730
\(327\) 0 0
\(328\) 2.73057 0.150771
\(329\) −16.8272 −0.927715
\(330\) 0 0
\(331\) 0.908184 0.0499183 0.0249591 0.999688i \(-0.492054\pi\)
0.0249591 + 0.999688i \(0.492054\pi\)
\(332\) 3.55445 0.195076
\(333\) 0 0
\(334\) −22.5815 −1.23560
\(335\) −11.7959 −0.644477
\(336\) 0 0
\(337\) −8.11083 −0.441825 −0.220913 0.975294i \(-0.570904\pi\)
−0.220913 + 0.975294i \(0.570904\pi\)
\(338\) 2.63334 0.143235
\(339\) 0 0
\(340\) −7.95030 −0.431166
\(341\) 3.93887 0.213302
\(342\) 0 0
\(343\) −19.5517 −1.05569
\(344\) 5.03945 0.271709
\(345\) 0 0
\(346\) −6.48916 −0.348859
\(347\) −9.35641 −0.502278 −0.251139 0.967951i \(-0.580805\pi\)
−0.251139 + 0.967951i \(0.580805\pi\)
\(348\) 0 0
\(349\) −35.4665 −1.89848 −0.949239 0.314556i \(-0.898144\pi\)
−0.949239 + 0.314556i \(0.898144\pi\)
\(350\) −1.84556 −0.0986495
\(351\) 0 0
\(352\) 0.832640 0.0443799
\(353\) 36.0923 1.92100 0.960500 0.278279i \(-0.0897639\pi\)
0.960500 + 0.278279i \(0.0897639\pi\)
\(354\) 0 0
\(355\) −13.7700 −0.730837
\(356\) 6.55029 0.347165
\(357\) 0 0
\(358\) 16.2095 0.856698
\(359\) −2.07889 −0.109720 −0.0548599 0.998494i \(-0.517471\pi\)
−0.0548599 + 0.998494i \(0.517471\pi\)
\(360\) 0 0
\(361\) −1.30278 −0.0685674
\(362\) −5.38225 −0.282885
\(363\) 0 0
\(364\) 5.94222 0.311457
\(365\) 3.52860 0.184696
\(366\) 0 0
\(367\) 34.2645 1.78859 0.894297 0.447474i \(-0.147676\pi\)
0.894297 + 0.447474i \(0.147676\pi\)
\(368\) 6.57614 0.342805
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −7.29053 −0.378506
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 6.61974 0.342299
\(375\) 0 0
\(376\) 9.11766 0.470208
\(377\) −9.78620 −0.504015
\(378\) 0 0
\(379\) 29.8217 1.53184 0.765919 0.642937i \(-0.222284\pi\)
0.765919 + 0.642937i \(0.222284\pi\)
\(380\) −4.20681 −0.215805
\(381\) 0 0
\(382\) −10.5680 −0.540708
\(383\) 6.78027 0.346456 0.173228 0.984882i \(-0.444580\pi\)
0.173228 + 0.984882i \(0.444580\pi\)
\(384\) 0 0
\(385\) 1.53669 0.0783170
\(386\) −3.11766 −0.158685
\(387\) 0 0
\(388\) −10.4918 −0.532642
\(389\) −35.1959 −1.78450 −0.892251 0.451540i \(-0.850875\pi\)
−0.892251 + 0.451540i \(0.850875\pi\)
\(390\) 0 0
\(391\) 52.2823 2.64403
\(392\) 3.59390 0.181519
\(393\) 0 0
\(394\) 0.576137 0.0290254
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −3.56054 −0.178698 −0.0893492 0.996000i \(-0.528479\pi\)
−0.0893492 + 0.996000i \(0.528479\pi\)
\(398\) −6.64359 −0.333013
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −1.44971 −0.0723950 −0.0361975 0.999345i \(-0.511525\pi\)
−0.0361975 + 0.999345i \(0.511525\pi\)
\(402\) 0 0
\(403\) −15.2312 −0.758718
\(404\) 3.63067 0.180633
\(405\) 0 0
\(406\) 5.60949 0.278394
\(407\) 0.832640 0.0412724
\(408\) 0 0
\(409\) 25.1781 1.24498 0.622489 0.782629i \(-0.286122\pi\)
0.622489 + 0.782629i \(0.286122\pi\)
\(410\) −2.73057 −0.134853
\(411\) 0 0
\(412\) −9.11766 −0.449195
\(413\) 6.81221 0.335207
\(414\) 0 0
\(415\) −3.55445 −0.174481
\(416\) −3.21973 −0.157860
\(417\) 0 0
\(418\) 3.50276 0.171325
\(419\) −31.6938 −1.54834 −0.774172 0.632976i \(-0.781833\pi\)
−0.774172 + 0.632976i \(0.781833\pi\)
\(420\) 0 0
\(421\) −5.14351 −0.250679 −0.125340 0.992114i \(-0.540002\pi\)
−0.125340 + 0.992114i \(0.540002\pi\)
\(422\) 8.83531 0.430096
\(423\) 0 0
\(424\) 3.95030 0.191844
\(425\) 7.95030 0.385646
\(426\) 0 0
\(427\) 9.98224 0.483075
\(428\) −8.91086 −0.430723
\(429\) 0 0
\(430\) −5.03945 −0.243024
\(431\) −2.72448 −0.131234 −0.0656168 0.997845i \(-0.520902\pi\)
−0.0656168 + 0.997845i \(0.520902\pi\)
\(432\) 0 0
\(433\) −8.68088 −0.417176 −0.208588 0.978004i \(-0.566887\pi\)
−0.208588 + 0.978004i \(0.566887\pi\)
\(434\) 8.73057 0.419081
\(435\) 0 0
\(436\) 11.2979 0.541073
\(437\) 27.6645 1.32337
\(438\) 0 0
\(439\) 7.64752 0.364996 0.182498 0.983206i \(-0.441582\pi\)
0.182498 + 0.983206i \(0.441582\pi\)
\(440\) −0.832640 −0.0396946
\(441\) 0 0
\(442\) −25.5978 −1.21756
\(443\) −16.7917 −0.797798 −0.398899 0.916995i \(-0.630608\pi\)
−0.398899 + 0.916995i \(0.630608\pi\)
\(444\) 0 0
\(445\) −6.55029 −0.310514
\(446\) −17.8828 −0.846777
\(447\) 0 0
\(448\) 1.84556 0.0871947
\(449\) −26.1047 −1.23196 −0.615979 0.787762i \(-0.711240\pi\)
−0.615979 + 0.787762i \(0.711240\pi\)
\(450\) 0 0
\(451\) 2.27359 0.107059
\(452\) 11.0394 0.519252
\(453\) 0 0
\(454\) 25.1959 1.18250
\(455\) −5.94222 −0.278575
\(456\) 0 0
\(457\) −17.6646 −0.826315 −0.413158 0.910660i \(-0.635574\pi\)
−0.413158 + 0.910660i \(0.635574\pi\)
\(458\) 5.48699 0.256390
\(459\) 0 0
\(460\) −6.57614 −0.306614
\(461\) 21.2231 0.988457 0.494229 0.869332i \(-0.335451\pi\)
0.494229 + 0.869332i \(0.335451\pi\)
\(462\) 0 0
\(463\) 32.5530 1.51286 0.756432 0.654072i \(-0.226941\pi\)
0.756432 + 0.654072i \(0.226941\pi\)
\(464\) −3.03945 −0.141103
\(465\) 0 0
\(466\) −28.5788 −1.32389
\(467\) −15.1170 −0.699531 −0.349765 0.936837i \(-0.613739\pi\)
−0.349765 + 0.936837i \(0.613739\pi\)
\(468\) 0 0
\(469\) 21.7700 1.00525
\(470\) −9.11766 −0.420566
\(471\) 0 0
\(472\) −3.69113 −0.169898
\(473\) 4.19605 0.192934
\(474\) 0 0
\(475\) 4.20681 0.193022
\(476\) 14.6728 0.672526
\(477\) 0 0
\(478\) 1.77811 0.0813288
\(479\) −3.44971 −0.157621 −0.0788106 0.996890i \(-0.525112\pi\)
−0.0788106 + 0.996890i \(0.525112\pi\)
\(480\) 0 0
\(481\) −3.21973 −0.146807
\(482\) −22.2612 −1.01397
\(483\) 0 0
\(484\) −10.3067 −0.468487
\(485\) 10.4918 0.476409
\(486\) 0 0
\(487\) −15.6619 −0.709707 −0.354853 0.934922i \(-0.615469\pi\)
−0.354853 + 0.934922i \(0.615469\pi\)
\(488\) −5.40878 −0.244844
\(489\) 0 0
\(490\) −3.59390 −0.162356
\(491\) −7.89793 −0.356429 −0.178214 0.983992i \(-0.557032\pi\)
−0.178214 + 0.983992i \(0.557032\pi\)
\(492\) 0 0
\(493\) −24.1645 −1.08832
\(494\) −13.5448 −0.609408
\(495\) 0 0
\(496\) −4.73057 −0.212409
\(497\) 25.4134 1.13995
\(498\) 0 0
\(499\) 5.87209 0.262871 0.131435 0.991325i \(-0.458041\pi\)
0.131435 + 0.991325i \(0.458041\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −28.6748 −1.27982
\(503\) −3.33056 −0.148502 −0.0742512 0.997240i \(-0.523657\pi\)
−0.0742512 + 0.997240i \(0.523657\pi\)
\(504\) 0 0
\(505\) −3.63067 −0.161563
\(506\) 5.47556 0.243418
\(507\) 0 0
\(508\) −0.136678 −0.00606410
\(509\) 30.9508 1.37187 0.685935 0.727663i \(-0.259393\pi\)
0.685935 + 0.727663i \(0.259393\pi\)
\(510\) 0 0
\(511\) −6.51226 −0.288085
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 9.19388 0.405525
\(515\) 9.11766 0.401772
\(516\) 0 0
\(517\) 7.59173 0.333884
\(518\) 1.84556 0.0810894
\(519\) 0 0
\(520\) 3.21973 0.141194
\(521\) −7.45306 −0.326524 −0.163262 0.986583i \(-0.552202\pi\)
−0.163262 + 0.986583i \(0.552202\pi\)
\(522\) 0 0
\(523\) 31.5917 1.38141 0.690705 0.723137i \(-0.257300\pi\)
0.690705 + 0.723137i \(0.257300\pi\)
\(524\) −2.07889 −0.0908169
\(525\) 0 0
\(526\) −8.02652 −0.349973
\(527\) −37.6095 −1.63830
\(528\) 0 0
\(529\) 20.2456 0.880242
\(530\) −3.95030 −0.171590
\(531\) 0 0
\(532\) 7.76393 0.336609
\(533\) −8.79171 −0.380811
\(534\) 0 0
\(535\) 8.91086 0.385250
\(536\) −11.7959 −0.509504
\(537\) 0 0
\(538\) 21.3849 0.921970
\(539\) 2.99242 0.128893
\(540\) 0 0
\(541\) 36.0816 1.55127 0.775634 0.631183i \(-0.217430\pi\)
0.775634 + 0.631183i \(0.217430\pi\)
\(542\) −2.64359 −0.113552
\(543\) 0 0
\(544\) −7.95030 −0.340866
\(545\) −11.2979 −0.483951
\(546\) 0 0
\(547\) −38.7420 −1.65649 −0.828244 0.560367i \(-0.810660\pi\)
−0.828244 + 0.560367i \(0.810660\pi\)
\(548\) −3.06597 −0.130972
\(549\) 0 0
\(550\) 0.832640 0.0355039
\(551\) −12.7864 −0.544717
\(552\) 0 0
\(553\) 7.38225 0.313925
\(554\) 14.4456 0.613733
\(555\) 0 0
\(556\) 22.1917 0.941139
\(557\) 23.3618 0.989869 0.494935 0.868930i \(-0.335192\pi\)
0.494935 + 0.868930i \(0.335192\pi\)
\(558\) 0 0
\(559\) −16.2257 −0.686272
\(560\) −1.84556 −0.0779893
\(561\) 0 0
\(562\) −11.2197 −0.473276
\(563\) −39.0351 −1.64513 −0.822567 0.568668i \(-0.807459\pi\)
−0.822567 + 0.568668i \(0.807459\pi\)
\(564\) 0 0
\(565\) −11.0394 −0.464433
\(566\) −3.86332 −0.162388
\(567\) 0 0
\(568\) −13.7700 −0.577777
\(569\) −35.4809 −1.48744 −0.743718 0.668493i \(-0.766940\pi\)
−0.743718 + 0.668493i \(0.766940\pi\)
\(570\) 0 0
\(571\) 37.0093 1.54879 0.774395 0.632702i \(-0.218054\pi\)
0.774395 + 0.632702i \(0.218054\pi\)
\(572\) −2.68088 −0.112093
\(573\) 0 0
\(574\) 5.03945 0.210342
\(575\) 6.57614 0.274244
\(576\) 0 0
\(577\) −15.7354 −0.655074 −0.327537 0.944838i \(-0.606219\pi\)
−0.327537 + 0.944838i \(0.606219\pi\)
\(578\) −46.2073 −1.92197
\(579\) 0 0
\(580\) 3.03945 0.126206
\(581\) 6.55996 0.272153
\(582\) 0 0
\(583\) 3.28918 0.136224
\(584\) 3.52860 0.146015
\(585\) 0 0
\(586\) −16.1544 −0.667334
\(587\) −37.3006 −1.53956 −0.769781 0.638309i \(-0.779634\pi\)
−0.769781 + 0.638309i \(0.779634\pi\)
\(588\) 0 0
\(589\) −19.9006 −0.819990
\(590\) 3.69113 0.151961
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 17.9829 0.738470 0.369235 0.929336i \(-0.379620\pi\)
0.369235 + 0.929336i \(0.379620\pi\)
\(594\) 0 0
\(595\) −14.6728 −0.601526
\(596\) 14.4999 0.593940
\(597\) 0 0
\(598\) −21.1734 −0.865844
\(599\) 32.9998 1.34834 0.674168 0.738578i \(-0.264502\pi\)
0.674168 + 0.738578i \(0.264502\pi\)
\(600\) 0 0
\(601\) 33.0334 1.34746 0.673729 0.738978i \(-0.264691\pi\)
0.673729 + 0.738978i \(0.264691\pi\)
\(602\) 9.30062 0.379065
\(603\) 0 0
\(604\) 8.24142 0.335339
\(605\) 10.3067 0.419027
\(606\) 0 0
\(607\) 27.6102 1.12066 0.560331 0.828269i \(-0.310674\pi\)
0.560331 + 0.828269i \(0.310674\pi\)
\(608\) −4.20681 −0.170609
\(609\) 0 0
\(610\) 5.40878 0.218995
\(611\) −29.3564 −1.18763
\(612\) 0 0
\(613\) 1.62999 0.0658348 0.0329174 0.999458i \(-0.489520\pi\)
0.0329174 + 0.999458i \(0.489520\pi\)
\(614\) −6.28303 −0.253562
\(615\) 0 0
\(616\) 1.53669 0.0619150
\(617\) −28.6143 −1.15197 −0.575985 0.817460i \(-0.695381\pi\)
−0.575985 + 0.817460i \(0.695381\pi\)
\(618\) 0 0
\(619\) 19.6830 0.791128 0.395564 0.918438i \(-0.370549\pi\)
0.395564 + 0.918438i \(0.370549\pi\)
\(620\) 4.73057 0.189984
\(621\) 0 0
\(622\) 20.6312 0.827235
\(623\) 12.0890 0.484335
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 7.32180 0.292638
\(627\) 0 0
\(628\) −6.11283 −0.243928
\(629\) −7.95030 −0.316999
\(630\) 0 0
\(631\) −5.76193 −0.229379 −0.114689 0.993401i \(-0.536587\pi\)
−0.114689 + 0.993401i \(0.536587\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) 19.1442 0.760313
\(635\) 0.136678 0.00542390
\(636\) 0 0
\(637\) −11.5714 −0.458474
\(638\) −2.53077 −0.100194
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −20.8967 −0.825369 −0.412685 0.910874i \(-0.635409\pi\)
−0.412685 + 0.910874i \(0.635409\pi\)
\(642\) 0 0
\(643\) 30.6197 1.20752 0.603762 0.797164i \(-0.293668\pi\)
0.603762 + 0.797164i \(0.293668\pi\)
\(644\) 12.1367 0.478252
\(645\) 0 0
\(646\) −33.4454 −1.31589
\(647\) 10.6278 0.417823 0.208912 0.977935i \(-0.433008\pi\)
0.208912 + 0.977935i \(0.433008\pi\)
\(648\) 0 0
\(649\) −3.07338 −0.120641
\(650\) −3.21973 −0.126288
\(651\) 0 0
\(652\) −21.4373 −0.839549
\(653\) 5.90061 0.230909 0.115454 0.993313i \(-0.463168\pi\)
0.115454 + 0.993313i \(0.463168\pi\)
\(654\) 0 0
\(655\) 2.07889 0.0812291
\(656\) −2.73057 −0.106611
\(657\) 0 0
\(658\) 16.8272 0.655994
\(659\) −19.8305 −0.772486 −0.386243 0.922397i \(-0.626227\pi\)
−0.386243 + 0.922397i \(0.626227\pi\)
\(660\) 0 0
\(661\) −9.32379 −0.362653 −0.181327 0.983423i \(-0.558039\pi\)
−0.181327 + 0.983423i \(0.558039\pi\)
\(662\) −0.908184 −0.0352976
\(663\) 0 0
\(664\) −3.55445 −0.137939
\(665\) −7.76393 −0.301072
\(666\) 0 0
\(667\) −19.9878 −0.773931
\(668\) 22.5815 0.873704
\(669\) 0 0
\(670\) 11.7959 0.455714
\(671\) −4.50357 −0.173858
\(672\) 0 0
\(673\) 33.0113 1.27249 0.636245 0.771487i \(-0.280487\pi\)
0.636245 + 0.771487i \(0.280487\pi\)
\(674\) 8.11083 0.312418
\(675\) 0 0
\(676\) −2.63334 −0.101282
\(677\) −26.1245 −1.00405 −0.502023 0.864854i \(-0.667411\pi\)
−0.502023 + 0.864854i \(0.667411\pi\)
\(678\) 0 0
\(679\) −19.3633 −0.743097
\(680\) 7.95030 0.304880
\(681\) 0 0
\(682\) −3.93887 −0.150827
\(683\) 39.3957 1.50743 0.753717 0.657199i \(-0.228259\pi\)
0.753717 + 0.657199i \(0.228259\pi\)
\(684\) 0 0
\(685\) 3.06597 0.117145
\(686\) 19.5517 0.746488
\(687\) 0 0
\(688\) −5.03945 −0.192127
\(689\) −12.7189 −0.484552
\(690\) 0 0
\(691\) −9.45306 −0.359611 −0.179806 0.983702i \(-0.557547\pi\)
−0.179806 + 0.983702i \(0.557547\pi\)
\(692\) 6.48916 0.246681
\(693\) 0 0
\(694\) 9.35641 0.355164
\(695\) −22.1917 −0.841780
\(696\) 0 0
\(697\) −21.7089 −0.822283
\(698\) 35.4665 1.34243
\(699\) 0 0
\(700\) 1.84556 0.0697557
\(701\) −37.2842 −1.40821 −0.704103 0.710098i \(-0.748651\pi\)
−0.704103 + 0.710098i \(0.748651\pi\)
\(702\) 0 0
\(703\) −4.20681 −0.158663
\(704\) −0.832640 −0.0313813
\(705\) 0 0
\(706\) −36.0923 −1.35835
\(707\) 6.70063 0.252003
\(708\) 0 0
\(709\) −19.8110 −0.744016 −0.372008 0.928230i \(-0.621331\pi\)
−0.372008 + 0.928230i \(0.621331\pi\)
\(710\) 13.7700 0.516780
\(711\) 0 0
\(712\) −6.55029 −0.245483
\(713\) −31.1089 −1.16504
\(714\) 0 0
\(715\) 2.68088 0.100259
\(716\) −16.2095 −0.605777
\(717\) 0 0
\(718\) 2.07889 0.0775836
\(719\) 19.2136 0.716548 0.358274 0.933617i \(-0.383365\pi\)
0.358274 + 0.933617i \(0.383365\pi\)
\(720\) 0 0
\(721\) −16.8272 −0.626679
\(722\) 1.30278 0.0484845
\(723\) 0 0
\(724\) 5.38225 0.200030
\(725\) −3.03945 −0.112882
\(726\) 0 0
\(727\) −45.0183 −1.66964 −0.834818 0.550527i \(-0.814427\pi\)
−0.834818 + 0.550527i \(0.814427\pi\)
\(728\) −5.94222 −0.220233
\(729\) 0 0
\(730\) −3.52860 −0.130599
\(731\) −40.0651 −1.48186
\(732\) 0 0
\(733\) −20.7262 −0.765541 −0.382771 0.923843i \(-0.625030\pi\)
−0.382771 + 0.923843i \(0.625030\pi\)
\(734\) −34.2645 −1.26473
\(735\) 0 0
\(736\) −6.57614 −0.242400
\(737\) −9.82171 −0.361787
\(738\) 0 0
\(739\) 24.2176 0.890858 0.445429 0.895317i \(-0.353051\pi\)
0.445429 + 0.895317i \(0.353051\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 7.29053 0.267644
\(743\) −13.9530 −0.511885 −0.255943 0.966692i \(-0.582386\pi\)
−0.255943 + 0.966692i \(0.582386\pi\)
\(744\) 0 0
\(745\) −14.4999 −0.531236
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) −6.61974 −0.242042
\(749\) −16.4456 −0.600907
\(750\) 0 0
\(751\) 41.6176 1.51865 0.759324 0.650713i \(-0.225530\pi\)
0.759324 + 0.650713i \(0.225530\pi\)
\(752\) −9.11766 −0.332487
\(753\) 0 0
\(754\) 9.78620 0.356392
\(755\) −8.24142 −0.299936
\(756\) 0 0
\(757\) −45.4034 −1.65021 −0.825107 0.564977i \(-0.808885\pi\)
−0.825107 + 0.564977i \(0.808885\pi\)
\(758\) −29.8217 −1.08317
\(759\) 0 0
\(760\) 4.20681 0.152597
\(761\) −18.9347 −0.686383 −0.343191 0.939266i \(-0.611508\pi\)
−0.343191 + 0.939266i \(0.611508\pi\)
\(762\) 0 0
\(763\) 20.8511 0.754860
\(764\) 10.5680 0.382339
\(765\) 0 0
\(766\) −6.78027 −0.244981
\(767\) 11.8844 0.429122
\(768\) 0 0
\(769\) 31.0884 1.12108 0.560538 0.828129i \(-0.310594\pi\)
0.560538 + 0.828129i \(0.310594\pi\)
\(770\) −1.53669 −0.0553785
\(771\) 0 0
\(772\) 3.11766 0.112207
\(773\) −34.3542 −1.23564 −0.617818 0.786321i \(-0.711983\pi\)
−0.617818 + 0.786321i \(0.711983\pi\)
\(774\) 0 0
\(775\) −4.73057 −0.169927
\(776\) 10.4918 0.376635
\(777\) 0 0
\(778\) 35.1959 1.26183
\(779\) −11.4870 −0.411564
\(780\) 0 0
\(781\) −11.4655 −0.410267
\(782\) −52.2823 −1.86961
\(783\) 0 0
\(784\) −3.59390 −0.128353
\(785\) 6.11283 0.218176
\(786\) 0 0
\(787\) 36.0053 1.28345 0.641726 0.766934i \(-0.278219\pi\)
0.641726 + 0.766934i \(0.278219\pi\)
\(788\) −0.576137 −0.0205240
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 20.3740 0.724416
\(792\) 0 0
\(793\) 17.4148 0.618418
\(794\) 3.56054 0.126359
\(795\) 0 0
\(796\) 6.64359 0.235476
\(797\) −43.2107 −1.53060 −0.765300 0.643674i \(-0.777409\pi\)
−0.765300 + 0.643674i \(0.777409\pi\)
\(798\) 0 0
\(799\) −72.4882 −2.56445
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 1.44971 0.0511910
\(803\) 2.93806 0.103682
\(804\) 0 0
\(805\) −12.1367 −0.427762
\(806\) 15.2312 0.536495
\(807\) 0 0
\(808\) −3.63067 −0.127727
\(809\) 32.2209 1.13283 0.566414 0.824121i \(-0.308331\pi\)
0.566414 + 0.824121i \(0.308331\pi\)
\(810\) 0 0
\(811\) 8.87892 0.311781 0.155890 0.987774i \(-0.450175\pi\)
0.155890 + 0.987774i \(0.450175\pi\)
\(812\) −5.60949 −0.196855
\(813\) 0 0
\(814\) −0.832640 −0.0291840
\(815\) 21.4373 0.750916
\(816\) 0 0
\(817\) −21.2000 −0.741693
\(818\) −25.1781 −0.880332
\(819\) 0 0
\(820\) 2.73057 0.0953558
\(821\) −2.31155 −0.0806735 −0.0403368 0.999186i \(-0.512843\pi\)
−0.0403368 + 0.999186i \(0.512843\pi\)
\(822\) 0 0
\(823\) 13.6280 0.475042 0.237521 0.971382i \(-0.423665\pi\)
0.237521 + 0.971382i \(0.423665\pi\)
\(824\) 9.11766 0.317629
\(825\) 0 0
\(826\) −6.81221 −0.237027
\(827\) 17.8298 0.620003 0.310001 0.950736i \(-0.399670\pi\)
0.310001 + 0.950736i \(0.399670\pi\)
\(828\) 0 0
\(829\) −29.7388 −1.03287 −0.516435 0.856326i \(-0.672741\pi\)
−0.516435 + 0.856326i \(0.672741\pi\)
\(830\) 3.55445 0.123377
\(831\) 0 0
\(832\) 3.21973 0.111624
\(833\) −28.5726 −0.989980
\(834\) 0 0
\(835\) −22.5815 −0.781464
\(836\) −3.50276 −0.121145
\(837\) 0 0
\(838\) 31.6938 1.09484
\(839\) 10.4556 0.360969 0.180484 0.983578i \(-0.442234\pi\)
0.180484 + 0.983578i \(0.442234\pi\)
\(840\) 0 0
\(841\) −19.7618 −0.681440
\(842\) 5.14351 0.177257
\(843\) 0 0
\(844\) −8.83531 −0.304124
\(845\) 2.63334 0.0905897
\(846\) 0 0
\(847\) −19.0217 −0.653593
\(848\) −3.95030 −0.135654
\(849\) 0 0
\(850\) −7.95030 −0.272693
\(851\) −6.57614 −0.225427
\(852\) 0 0
\(853\) −15.6333 −0.535275 −0.267638 0.963520i \(-0.586243\pi\)
−0.267638 + 0.963520i \(0.586243\pi\)
\(854\) −9.98224 −0.341585
\(855\) 0 0
\(856\) 8.91086 0.304567
\(857\) 14.4945 0.495123 0.247561 0.968872i \(-0.420371\pi\)
0.247561 + 0.968872i \(0.420371\pi\)
\(858\) 0 0
\(859\) 3.38493 0.115492 0.0577461 0.998331i \(-0.481609\pi\)
0.0577461 + 0.998331i \(0.481609\pi\)
\(860\) 5.03945 0.171844
\(861\) 0 0
\(862\) 2.72448 0.0927962
\(863\) 56.3924 1.91962 0.959810 0.280649i \(-0.0905498\pi\)
0.959810 + 0.280649i \(0.0905498\pi\)
\(864\) 0 0
\(865\) −6.48916 −0.220638
\(866\) 8.68088 0.294988
\(867\) 0 0
\(868\) −8.73057 −0.296335
\(869\) −3.33056 −0.112982
\(870\) 0 0
\(871\) 37.9795 1.28689
\(872\) −11.2979 −0.382597
\(873\) 0 0
\(874\) −27.6645 −0.935767
\(875\) −1.84556 −0.0623914
\(876\) 0 0
\(877\) −32.1755 −1.08649 −0.543245 0.839574i \(-0.682805\pi\)
−0.543245 + 0.839574i \(0.682805\pi\)
\(878\) −7.64752 −0.258091
\(879\) 0 0
\(880\) 0.832640 0.0280683
\(881\) −14.6097 −0.492212 −0.246106 0.969243i \(-0.579151\pi\)
−0.246106 + 0.969243i \(0.579151\pi\)
\(882\) 0 0
\(883\) 18.5203 0.623259 0.311630 0.950204i \(-0.399125\pi\)
0.311630 + 0.950204i \(0.399125\pi\)
\(884\) 25.5978 0.860948
\(885\) 0 0
\(886\) 16.7917 0.564128
\(887\) 28.5611 0.958986 0.479493 0.877546i \(-0.340821\pi\)
0.479493 + 0.877546i \(0.340821\pi\)
\(888\) 0 0
\(889\) −0.252248 −0.00846012
\(890\) 6.55029 0.219566
\(891\) 0 0
\(892\) 17.8828 0.598762
\(893\) −38.3562 −1.28354
\(894\) 0 0
\(895\) 16.2095 0.541823
\(896\) −1.84556 −0.0616559
\(897\) 0 0
\(898\) 26.1047 0.871126
\(899\) 14.3783 0.479544
\(900\) 0 0
\(901\) −31.4061 −1.04629
\(902\) −2.27359 −0.0757021
\(903\) 0 0
\(904\) −11.0394 −0.367167
\(905\) −5.38225 −0.178912
\(906\) 0 0
\(907\) −25.8428 −0.858097 −0.429048 0.903282i \(-0.641151\pi\)
−0.429048 + 0.903282i \(0.641151\pi\)
\(908\) −25.1959 −0.836155
\(909\) 0 0
\(910\) 5.94222 0.196983
\(911\) −24.9998 −0.828281 −0.414141 0.910213i \(-0.635918\pi\)
−0.414141 + 0.910213i \(0.635918\pi\)
\(912\) 0 0
\(913\) −2.95958 −0.0979477
\(914\) 17.6646 0.584293
\(915\) 0 0
\(916\) −5.48699 −0.181295
\(917\) −3.83673 −0.126700
\(918\) 0 0
\(919\) −34.3931 −1.13452 −0.567262 0.823537i \(-0.691997\pi\)
−0.567262 + 0.823537i \(0.691997\pi\)
\(920\) 6.57614 0.216809
\(921\) 0 0
\(922\) −21.2231 −0.698945
\(923\) 44.3357 1.45933
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −32.5530 −1.06976
\(927\) 0 0
\(928\) 3.03945 0.0997747
\(929\) −55.6310 −1.82519 −0.912597 0.408860i \(-0.865926\pi\)
−0.912597 + 0.408860i \(0.865926\pi\)
\(930\) 0 0
\(931\) −15.1188 −0.495499
\(932\) 28.5788 0.936130
\(933\) 0 0
\(934\) 15.1170 0.494643
\(935\) 6.61974 0.216489
\(936\) 0 0
\(937\) −53.9576 −1.76272 −0.881360 0.472446i \(-0.843371\pi\)
−0.881360 + 0.472446i \(0.843371\pi\)
\(938\) −21.7700 −0.710816
\(939\) 0 0
\(940\) 9.11766 0.297385
\(941\) 50.1092 1.63351 0.816756 0.576983i \(-0.195770\pi\)
0.816756 + 0.576983i \(0.195770\pi\)
\(942\) 0 0
\(943\) −17.9566 −0.584748
\(944\) 3.69113 0.120136
\(945\) 0 0
\(946\) −4.19605 −0.136425
\(947\) −39.3795 −1.27966 −0.639831 0.768516i \(-0.720996\pi\)
−0.639831 + 0.768516i \(0.720996\pi\)
\(948\) 0 0
\(949\) −11.3611 −0.368798
\(950\) −4.20681 −0.136487
\(951\) 0 0
\(952\) −14.6728 −0.475548
\(953\) 20.5096 0.664371 0.332185 0.943214i \(-0.392214\pi\)
0.332185 + 0.943214i \(0.392214\pi\)
\(954\) 0 0
\(955\) −10.5680 −0.341974
\(956\) −1.77811 −0.0575081
\(957\) 0 0
\(958\) 3.44971 0.111455
\(959\) −5.65844 −0.182721
\(960\) 0 0
\(961\) −8.62168 −0.278119
\(962\) 3.21973 0.103808
\(963\) 0 0
\(964\) 22.2612 0.716984
\(965\) −3.11766 −0.100361
\(966\) 0 0
\(967\) 18.6265 0.598988 0.299494 0.954098i \(-0.403182\pi\)
0.299494 + 0.954098i \(0.403182\pi\)
\(968\) 10.3067 0.331270
\(969\) 0 0
\(970\) −10.4918 −0.336872
\(971\) 3.56656 0.114456 0.0572282 0.998361i \(-0.481774\pi\)
0.0572282 + 0.998361i \(0.481774\pi\)
\(972\) 0 0
\(973\) 40.9562 1.31300
\(974\) 15.6619 0.501838
\(975\) 0 0
\(976\) 5.40878 0.173131
\(977\) 43.6998 1.39808 0.699041 0.715082i \(-0.253611\pi\)
0.699041 + 0.715082i \(0.253611\pi\)
\(978\) 0 0
\(979\) −5.45404 −0.174312
\(980\) 3.59390 0.114803
\(981\) 0 0
\(982\) 7.89793 0.252033
\(983\) −53.1096 −1.69393 −0.846966 0.531647i \(-0.821573\pi\)
−0.846966 + 0.531647i \(0.821573\pi\)
\(984\) 0 0
\(985\) 0.576137 0.0183572
\(986\) 24.1645 0.769555
\(987\) 0 0
\(988\) 13.5448 0.430917
\(989\) −33.1401 −1.05379
\(990\) 0 0
\(991\) −9.14283 −0.290432 −0.145216 0.989400i \(-0.546388\pi\)
−0.145216 + 0.989400i \(0.546388\pi\)
\(992\) 4.73057 0.150196
\(993\) 0 0
\(994\) −25.4134 −0.806066
\(995\) −6.64359 −0.210616
\(996\) 0 0
\(997\) −15.6280 −0.494944 −0.247472 0.968895i \(-0.579600\pi\)
−0.247472 + 0.968895i \(0.579600\pi\)
\(998\) −5.87209 −0.185878
\(999\) 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 3330.2.a.bj.1.3 4
3.2 odd 2 1110.2.a.s.1.3 4
12.11 even 2 8880.2.a.cg.1.2 4
15.14 odd 2 5550.2.a.cj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.3 4 3.2 odd 2
3330.2.a.bj.1.3 4 1.1 even 1 trivial
5550.2.a.cj.1.2 4 15.14 odd 2
8880.2.a.cg.1.2 4 12.11 even 2