Properties

Label 4235.2.a.y.1.2
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.173513.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 3 x^{2} + 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.52979\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +O(q^{10})\) \(q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +1.65369 q^{10} -0.838578 q^{12} +5.60081 q^{13} -1.65369 q^{14} +1.14142 q^{15} -4.92960 q^{16} -4.57876 q^{17} +2.80656 q^{18} +1.88149 q^{19} -0.734678 q^{20} -1.14142 q^{21} +2.08099 q^{23} -2.38836 q^{24} +1.00000 q^{25} -9.26199 q^{26} +5.36144 q^{27} +0.734678 q^{28} -5.51585 q^{29} -1.88755 q^{30} -5.46081 q^{31} +3.96713 q^{32} +7.57184 q^{34} -1.00000 q^{35} -1.24686 q^{36} -9.34694 q^{37} -3.11139 q^{38} -6.39289 q^{39} -2.09245 q^{40} +11.0136 q^{41} +1.88755 q^{42} -8.43344 q^{43} +1.69716 q^{45} -3.44131 q^{46} +1.60936 q^{47} +5.62676 q^{48} +1.00000 q^{49} -1.65369 q^{50} +5.22630 q^{51} +4.11479 q^{52} +6.21331 q^{53} -8.86613 q^{54} +2.09245 q^{56} -2.14757 q^{57} +9.12149 q^{58} +13.5862 q^{59} +0.838578 q^{60} -11.1145 q^{61} +9.03046 q^{62} -1.69716 q^{63} +3.29883 q^{64} -5.60081 q^{65} -5.60924 q^{67} -3.36392 q^{68} -2.37529 q^{69} +1.65369 q^{70} +0.696594 q^{71} -3.55121 q^{72} +13.8873 q^{73} +15.4569 q^{74} -1.14142 q^{75} +1.38229 q^{76} +10.5718 q^{78} -8.47388 q^{79} +4.92960 q^{80} -1.02820 q^{81} -18.2131 q^{82} -9.50230 q^{83} -0.838578 q^{84} +4.57876 q^{85} +13.9463 q^{86} +6.29592 q^{87} +1.49863 q^{89} -2.80656 q^{90} +5.60081 q^{91} +1.52886 q^{92} +6.23309 q^{93} -2.66137 q^{94} -1.88149 q^{95} -4.52817 q^{96} +5.93697 q^{97} -1.65369 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + O(q^{10}) \) \( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.65369 −1.16933 −0.584666 0.811274i \(-0.698775\pi\)
−0.584666 + 0.811274i \(0.698775\pi\)
\(3\) −1.14142 −0.659000 −0.329500 0.944156i \(-0.606880\pi\)
−0.329500 + 0.944156i \(0.606880\pi\)
\(4\) 0.734678 0.367339
\(5\) −1.00000 −0.447214
\(6\) 1.88755 0.770591
\(7\) 1.00000 0.377964
\(8\) 2.09245 0.739791
\(9\) −1.69716 −0.565718
\(10\) 1.65369 0.522941
\(11\) 0 0
\(12\) −0.838578 −0.242076
\(13\) 5.60081 1.55339 0.776693 0.629879i \(-0.216896\pi\)
0.776693 + 0.629879i \(0.216896\pi\)
\(14\) −1.65369 −0.441966
\(15\) 1.14142 0.294714
\(16\) −4.92960 −1.23240
\(17\) −4.57876 −1.11051 −0.555257 0.831679i \(-0.687380\pi\)
−0.555257 + 0.831679i \(0.687380\pi\)
\(18\) 2.80656 0.661513
\(19\) 1.88149 0.431642 0.215821 0.976433i \(-0.430757\pi\)
0.215821 + 0.976433i \(0.430757\pi\)
\(20\) −0.734678 −0.164279
\(21\) −1.14142 −0.249079
\(22\) 0 0
\(23\) 2.08099 0.433917 0.216958 0.976181i \(-0.430386\pi\)
0.216958 + 0.976181i \(0.430386\pi\)
\(24\) −2.38836 −0.487523
\(25\) 1.00000 0.200000
\(26\) −9.26199 −1.81642
\(27\) 5.36144 1.03181
\(28\) 0.734678 0.138841
\(29\) −5.51585 −1.02427 −0.512134 0.858905i \(-0.671145\pi\)
−0.512134 + 0.858905i \(0.671145\pi\)
\(30\) −1.88755 −0.344619
\(31\) −5.46081 −0.980790 −0.490395 0.871500i \(-0.663148\pi\)
−0.490395 + 0.871500i \(0.663148\pi\)
\(32\) 3.96713 0.701296
\(33\) 0 0
\(34\) 7.57184 1.29856
\(35\) −1.00000 −0.169031
\(36\) −1.24686 −0.207810
\(37\) −9.34694 −1.53663 −0.768314 0.640073i \(-0.778904\pi\)
−0.768314 + 0.640073i \(0.778904\pi\)
\(38\) −3.11139 −0.504734
\(39\) −6.39289 −1.02368
\(40\) −2.09245 −0.330845
\(41\) 11.0136 1.72004 0.860020 0.510261i \(-0.170451\pi\)
0.860020 + 0.510261i \(0.170451\pi\)
\(42\) 1.88755 0.291256
\(43\) −8.43344 −1.28609 −0.643044 0.765829i \(-0.722329\pi\)
−0.643044 + 0.765829i \(0.722329\pi\)
\(44\) 0 0
\(45\) 1.69716 0.252997
\(46\) −3.44131 −0.507393
\(47\) 1.60936 0.234749 0.117375 0.993088i \(-0.462552\pi\)
0.117375 + 0.993088i \(0.462552\pi\)
\(48\) 5.62676 0.812153
\(49\) 1.00000 0.142857
\(50\) −1.65369 −0.233867
\(51\) 5.22630 0.731829
\(52\) 4.11479 0.570619
\(53\) 6.21331 0.853463 0.426732 0.904378i \(-0.359665\pi\)
0.426732 + 0.904378i \(0.359665\pi\)
\(54\) −8.86613 −1.20653
\(55\) 0 0
\(56\) 2.09245 0.279615
\(57\) −2.14757 −0.284453
\(58\) 9.12149 1.19771
\(59\) 13.5862 1.76877 0.884386 0.466756i \(-0.154577\pi\)
0.884386 + 0.466756i \(0.154577\pi\)
\(60\) 0.838578 0.108260
\(61\) −11.1145 −1.42307 −0.711533 0.702653i \(-0.751999\pi\)
−0.711533 + 0.702653i \(0.751999\pi\)
\(62\) 9.03046 1.14687
\(63\) −1.69716 −0.213821
\(64\) 3.29883 0.412353
\(65\) −5.60081 −0.694695
\(66\) 0 0
\(67\) −5.60924 −0.685277 −0.342639 0.939467i \(-0.611321\pi\)
−0.342639 + 0.939467i \(0.611321\pi\)
\(68\) −3.36392 −0.407935
\(69\) −2.37529 −0.285951
\(70\) 1.65369 0.197653
\(71\) 0.696594 0.0826705 0.0413353 0.999145i \(-0.486839\pi\)
0.0413353 + 0.999145i \(0.486839\pi\)
\(72\) −3.55121 −0.418514
\(73\) 13.8873 1.62539 0.812694 0.582690i \(-0.198000\pi\)
0.812694 + 0.582690i \(0.198000\pi\)
\(74\) 15.4569 1.79683
\(75\) −1.14142 −0.131800
\(76\) 1.38229 0.158559
\(77\) 0 0
\(78\) 10.5718 1.19702
\(79\) −8.47388 −0.953386 −0.476693 0.879070i \(-0.658165\pi\)
−0.476693 + 0.879070i \(0.658165\pi\)
\(80\) 4.92960 0.551146
\(81\) −1.02820 −0.114244
\(82\) −18.2131 −2.01130
\(83\) −9.50230 −1.04301 −0.521506 0.853247i \(-0.674630\pi\)
−0.521506 + 0.853247i \(0.674630\pi\)
\(84\) −0.838578 −0.0914963
\(85\) 4.57876 0.496637
\(86\) 13.9463 1.50386
\(87\) 6.29592 0.674993
\(88\) 0 0
\(89\) 1.49863 0.158854 0.0794272 0.996841i \(-0.474691\pi\)
0.0794272 + 0.996841i \(0.474691\pi\)
\(90\) −2.80656 −0.295838
\(91\) 5.60081 0.587125
\(92\) 1.52886 0.159395
\(93\) 6.23309 0.646341
\(94\) −2.66137 −0.274500
\(95\) −1.88149 −0.193036
\(96\) −4.52817 −0.462154
\(97\) 5.93697 0.602808 0.301404 0.953497i \(-0.402545\pi\)
0.301404 + 0.953497i \(0.402545\pi\)
\(98\) −1.65369 −0.167048
\(99\) 0 0
\(100\) 0.734678 0.0734678
\(101\) −3.15826 −0.314259 −0.157129 0.987578i \(-0.550224\pi\)
−0.157129 + 0.987578i \(0.550224\pi\)
\(102\) −8.64266 −0.855751
\(103\) 1.24686 0.122857 0.0614285 0.998111i \(-0.480434\pi\)
0.0614285 + 0.998111i \(0.480434\pi\)
\(104\) 11.7194 1.14918
\(105\) 1.14142 0.111391
\(106\) −10.2749 −0.997983
\(107\) 10.2684 0.992687 0.496343 0.868126i \(-0.334676\pi\)
0.496343 + 0.868126i \(0.334676\pi\)
\(108\) 3.93893 0.379024
\(109\) −5.67907 −0.543956 −0.271978 0.962303i \(-0.587678\pi\)
−0.271978 + 0.962303i \(0.587678\pi\)
\(110\) 0 0
\(111\) 10.6688 1.01264
\(112\) −4.92960 −0.465804
\(113\) 17.9797 1.69139 0.845694 0.533669i \(-0.179187\pi\)
0.845694 + 0.533669i \(0.179187\pi\)
\(114\) 3.55141 0.332620
\(115\) −2.08099 −0.194053
\(116\) −4.05238 −0.376254
\(117\) −9.50545 −0.878779
\(118\) −22.4673 −2.06828
\(119\) −4.57876 −0.419734
\(120\) 2.38836 0.218027
\(121\) 0 0
\(122\) 18.3799 1.66404
\(123\) −12.5712 −1.13351
\(124\) −4.01194 −0.360282
\(125\) −1.00000 −0.0894427
\(126\) 2.80656 0.250028
\(127\) −12.0909 −1.07289 −0.536447 0.843934i \(-0.680234\pi\)
−0.536447 + 0.843934i \(0.680234\pi\)
\(128\) −13.3895 −1.18347
\(129\) 9.62612 0.847533
\(130\) 9.26199 0.812330
\(131\) −1.04441 −0.0912502 −0.0456251 0.998959i \(-0.514528\pi\)
−0.0456251 + 0.998959i \(0.514528\pi\)
\(132\) 0 0
\(133\) 1.88149 0.163146
\(134\) 9.27592 0.801317
\(135\) −5.36144 −0.461439
\(136\) −9.58081 −0.821548
\(137\) −1.43820 −0.122874 −0.0614368 0.998111i \(-0.519568\pi\)
−0.0614368 + 0.998111i \(0.519568\pi\)
\(138\) 3.92798 0.334372
\(139\) 21.6489 1.83623 0.918117 0.396310i \(-0.129709\pi\)
0.918117 + 0.396310i \(0.129709\pi\)
\(140\) −0.734678 −0.0620916
\(141\) −1.83696 −0.154700
\(142\) −1.15195 −0.0966694
\(143\) 0 0
\(144\) 8.36630 0.697192
\(145\) 5.51585 0.458067
\(146\) −22.9653 −1.90062
\(147\) −1.14142 −0.0941429
\(148\) −6.86699 −0.564463
\(149\) 12.6540 1.03666 0.518329 0.855181i \(-0.326554\pi\)
0.518329 + 0.855181i \(0.326554\pi\)
\(150\) 1.88755 0.154118
\(151\) −2.48546 −0.202264 −0.101132 0.994873i \(-0.532246\pi\)
−0.101132 + 0.994873i \(0.532246\pi\)
\(152\) 3.93691 0.319325
\(153\) 7.77087 0.628238
\(154\) 0 0
\(155\) 5.46081 0.438623
\(156\) −4.69672 −0.376038
\(157\) −14.2773 −1.13945 −0.569727 0.821834i \(-0.692951\pi\)
−0.569727 + 0.821834i \(0.692951\pi\)
\(158\) 14.0131 1.11483
\(159\) −7.09201 −0.562433
\(160\) −3.96713 −0.313629
\(161\) 2.08099 0.164005
\(162\) 1.70032 0.133589
\(163\) −9.75758 −0.764272 −0.382136 0.924106i \(-0.624811\pi\)
−0.382136 + 0.924106i \(0.624811\pi\)
\(164\) 8.09147 0.631837
\(165\) 0 0
\(166\) 15.7138 1.21963
\(167\) −11.3890 −0.881308 −0.440654 0.897677i \(-0.645253\pi\)
−0.440654 + 0.897677i \(0.645253\pi\)
\(168\) −2.38836 −0.184266
\(169\) 18.3691 1.41301
\(170\) −7.57184 −0.580733
\(171\) −3.19317 −0.244188
\(172\) −6.19586 −0.472430
\(173\) −10.3544 −0.787227 −0.393613 0.919276i \(-0.628775\pi\)
−0.393613 + 0.919276i \(0.628775\pi\)
\(174\) −10.4115 −0.789292
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −15.5076 −1.16562
\(178\) −2.47826 −0.185754
\(179\) 18.8195 1.40664 0.703318 0.710875i \(-0.251701\pi\)
0.703318 + 0.710875i \(0.251701\pi\)
\(180\) 1.24686 0.0929356
\(181\) −7.75369 −0.576327 −0.288164 0.957581i \(-0.593045\pi\)
−0.288164 + 0.957581i \(0.593045\pi\)
\(182\) −9.26199 −0.686544
\(183\) 12.6863 0.937801
\(184\) 4.35436 0.321008
\(185\) 9.34694 0.687201
\(186\) −10.3076 −0.755788
\(187\) 0 0
\(188\) 1.18236 0.0862325
\(189\) 5.36144 0.389987
\(190\) 3.11139 0.225724
\(191\) 15.9410 1.15345 0.576723 0.816939i \(-0.304331\pi\)
0.576723 + 0.816939i \(0.304331\pi\)
\(192\) −3.76535 −0.271741
\(193\) −8.52230 −0.613448 −0.306724 0.951798i \(-0.599233\pi\)
−0.306724 + 0.951798i \(0.599233\pi\)
\(194\) −9.81789 −0.704883
\(195\) 6.39289 0.457804
\(196\) 0.734678 0.0524770
\(197\) 19.0556 1.35765 0.678826 0.734299i \(-0.262489\pi\)
0.678826 + 0.734299i \(0.262489\pi\)
\(198\) 0 0
\(199\) −12.7118 −0.901118 −0.450559 0.892747i \(-0.648775\pi\)
−0.450559 + 0.892747i \(0.648775\pi\)
\(200\) 2.09245 0.147958
\(201\) 6.40251 0.451598
\(202\) 5.22277 0.367473
\(203\) −5.51585 −0.387137
\(204\) 3.83965 0.268829
\(205\) −11.0136 −0.769225
\(206\) −2.06192 −0.143661
\(207\) −3.53177 −0.245475
\(208\) −27.6098 −1.91439
\(209\) 0 0
\(210\) −1.88755 −0.130254
\(211\) 28.4506 1.95862 0.979311 0.202363i \(-0.0648620\pi\)
0.979311 + 0.202363i \(0.0648620\pi\)
\(212\) 4.56478 0.313510
\(213\) −0.795108 −0.0544799
\(214\) −16.9808 −1.16078
\(215\) 8.43344 0.575156
\(216\) 11.2185 0.763323
\(217\) −5.46081 −0.370704
\(218\) 9.39140 0.636066
\(219\) −15.8513 −1.07113
\(220\) 0 0
\(221\) −25.6448 −1.72506
\(222\) −17.6429 −1.18411
\(223\) −23.9632 −1.60470 −0.802348 0.596856i \(-0.796416\pi\)
−0.802348 + 0.596856i \(0.796416\pi\)
\(224\) 3.96713 0.265065
\(225\) −1.69716 −0.113144
\(226\) −29.7328 −1.97779
\(227\) −20.6007 −1.36732 −0.683658 0.729803i \(-0.739612\pi\)
−0.683658 + 0.729803i \(0.739612\pi\)
\(228\) −1.57777 −0.104490
\(229\) 0.903360 0.0596957 0.0298479 0.999554i \(-0.490498\pi\)
0.0298479 + 0.999554i \(0.490498\pi\)
\(230\) 3.44131 0.226913
\(231\) 0 0
\(232\) −11.5416 −0.757745
\(233\) −23.2459 −1.52289 −0.761444 0.648231i \(-0.775509\pi\)
−0.761444 + 0.648231i \(0.775509\pi\)
\(234\) 15.7190 1.02759
\(235\) −1.60936 −0.104983
\(236\) 9.98148 0.649739
\(237\) 9.67228 0.628282
\(238\) 7.57184 0.490809
\(239\) −8.21214 −0.531199 −0.265600 0.964083i \(-0.585570\pi\)
−0.265600 + 0.964083i \(0.585570\pi\)
\(240\) −5.62676 −0.363206
\(241\) 8.33040 0.536608 0.268304 0.963334i \(-0.413537\pi\)
0.268304 + 0.963334i \(0.413537\pi\)
\(242\) 0 0
\(243\) −14.9107 −0.956522
\(244\) −8.16557 −0.522747
\(245\) −1.00000 −0.0638877
\(246\) 20.7888 1.32545
\(247\) 10.5378 0.670507
\(248\) −11.4264 −0.725580
\(249\) 10.8461 0.687346
\(250\) 1.65369 0.104588
\(251\) −9.77541 −0.617018 −0.308509 0.951221i \(-0.599830\pi\)
−0.308509 + 0.951221i \(0.599830\pi\)
\(252\) −1.24686 −0.0785450
\(253\) 0 0
\(254\) 19.9946 1.25457
\(255\) −5.22630 −0.327284
\(256\) 15.5443 0.971521
\(257\) 4.10056 0.255786 0.127893 0.991788i \(-0.459179\pi\)
0.127893 + 0.991788i \(0.459179\pi\)
\(258\) −15.9186 −0.991047
\(259\) −9.34694 −0.580791
\(260\) −4.11479 −0.255189
\(261\) 9.36126 0.579448
\(262\) 1.72712 0.106702
\(263\) −25.6453 −1.58136 −0.790678 0.612232i \(-0.790272\pi\)
−0.790678 + 0.612232i \(0.790272\pi\)
\(264\) 0 0
\(265\) −6.21331 −0.381680
\(266\) −3.11139 −0.190771
\(267\) −1.71057 −0.104685
\(268\) −4.12098 −0.251729
\(269\) 13.7027 0.835470 0.417735 0.908569i \(-0.362824\pi\)
0.417735 + 0.908569i \(0.362824\pi\)
\(270\) 8.86613 0.539576
\(271\) 8.60468 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(272\) 22.5715 1.36860
\(273\) −6.39289 −0.386915
\(274\) 2.37833 0.143680
\(275\) 0 0
\(276\) −1.74507 −0.105041
\(277\) −28.8920 −1.73595 −0.867975 0.496607i \(-0.834579\pi\)
−0.867975 + 0.496607i \(0.834579\pi\)
\(278\) −35.8005 −2.14717
\(279\) 9.26784 0.554851
\(280\) −2.09245 −0.125048
\(281\) −7.98397 −0.476284 −0.238142 0.971230i \(-0.576538\pi\)
−0.238142 + 0.971230i \(0.576538\pi\)
\(282\) 3.03775 0.180895
\(283\) 7.39498 0.439586 0.219793 0.975547i \(-0.429462\pi\)
0.219793 + 0.975547i \(0.429462\pi\)
\(284\) 0.511772 0.0303681
\(285\) 2.14757 0.127211
\(286\) 0 0
\(287\) 11.0136 0.650114
\(288\) −6.73283 −0.396736
\(289\) 3.96507 0.233239
\(290\) −9.12149 −0.535632
\(291\) −6.77659 −0.397251
\(292\) 10.2027 0.597068
\(293\) −13.7881 −0.805508 −0.402754 0.915308i \(-0.631947\pi\)
−0.402754 + 0.915308i \(0.631947\pi\)
\(294\) 1.88755 0.110084
\(295\) −13.5862 −0.791019
\(296\) −19.5580 −1.13678
\(297\) 0 0
\(298\) −20.9258 −1.21220
\(299\) 11.6552 0.674040
\(300\) −0.838578 −0.0484153
\(301\) −8.43344 −0.486096
\(302\) 4.11017 0.236514
\(303\) 3.60491 0.207097
\(304\) −9.27498 −0.531957
\(305\) 11.1145 0.636414
\(306\) −12.8506 −0.734619
\(307\) 5.89759 0.336593 0.168297 0.985736i \(-0.446173\pi\)
0.168297 + 0.985736i \(0.446173\pi\)
\(308\) 0 0
\(309\) −1.42320 −0.0809628
\(310\) −9.03046 −0.512896
\(311\) −26.0053 −1.47462 −0.737311 0.675553i \(-0.763905\pi\)
−0.737311 + 0.675553i \(0.763905\pi\)
\(312\) −13.3768 −0.757311
\(313\) −12.1886 −0.688938 −0.344469 0.938798i \(-0.611941\pi\)
−0.344469 + 0.938798i \(0.611941\pi\)
\(314\) 23.6102 1.33240
\(315\) 1.69716 0.0956239
\(316\) −6.22557 −0.350216
\(317\) 13.5894 0.763258 0.381629 0.924316i \(-0.375363\pi\)
0.381629 + 0.924316i \(0.375363\pi\)
\(318\) 11.7280 0.657671
\(319\) 0 0
\(320\) −3.29883 −0.184410
\(321\) −11.7206 −0.654181
\(322\) −3.44131 −0.191777
\(323\) −8.61488 −0.479345
\(324\) −0.755394 −0.0419663
\(325\) 5.60081 0.310677
\(326\) 16.1360 0.893689
\(327\) 6.48222 0.358468
\(328\) 23.0454 1.27247
\(329\) 1.60936 0.0887268
\(330\) 0 0
\(331\) 18.3402 1.00807 0.504034 0.863684i \(-0.331849\pi\)
0.504034 + 0.863684i \(0.331849\pi\)
\(332\) −6.98113 −0.383139
\(333\) 15.8632 0.869299
\(334\) 18.8338 1.03054
\(335\) 5.60924 0.306465
\(336\) 5.62676 0.306965
\(337\) −25.2973 −1.37803 −0.689015 0.724747i \(-0.741957\pi\)
−0.689015 + 0.724747i \(0.741957\pi\)
\(338\) −30.3767 −1.65228
\(339\) −20.5224 −1.11462
\(340\) 3.36392 0.182434
\(341\) 0 0
\(342\) 5.28051 0.285537
\(343\) 1.00000 0.0539949
\(344\) −17.6465 −0.951437
\(345\) 2.37529 0.127881
\(346\) 17.1228 0.920530
\(347\) −14.4944 −0.778098 −0.389049 0.921217i \(-0.627196\pi\)
−0.389049 + 0.921217i \(0.627196\pi\)
\(348\) 4.62547 0.247951
\(349\) −2.97717 −0.159364 −0.0796822 0.996820i \(-0.525391\pi\)
−0.0796822 + 0.996820i \(0.525391\pi\)
\(350\) −1.65369 −0.0883932
\(351\) 30.0284 1.60280
\(352\) 0 0
\(353\) 14.0782 0.749304 0.374652 0.927165i \(-0.377762\pi\)
0.374652 + 0.927165i \(0.377762\pi\)
\(354\) 25.6447 1.36300
\(355\) −0.696594 −0.0369714
\(356\) 1.10101 0.0583534
\(357\) 5.22630 0.276605
\(358\) −31.1216 −1.64483
\(359\) −15.3313 −0.809154 −0.404577 0.914504i \(-0.632581\pi\)
−0.404577 + 0.914504i \(0.632581\pi\)
\(360\) 3.55121 0.187165
\(361\) −15.4600 −0.813685
\(362\) 12.8222 0.673918
\(363\) 0 0
\(364\) 4.11479 0.215674
\(365\) −13.8873 −0.726896
\(366\) −20.9792 −1.09660
\(367\) 13.7180 0.716072 0.358036 0.933708i \(-0.383447\pi\)
0.358036 + 0.933708i \(0.383447\pi\)
\(368\) −10.2585 −0.534759
\(369\) −18.6918 −0.973058
\(370\) −15.4569 −0.803567
\(371\) 6.21331 0.322579
\(372\) 4.57931 0.237426
\(373\) −12.6631 −0.655670 −0.327835 0.944735i \(-0.606319\pi\)
−0.327835 + 0.944735i \(0.606319\pi\)
\(374\) 0 0
\(375\) 1.14142 0.0589428
\(376\) 3.36750 0.173665
\(377\) −30.8933 −1.59108
\(378\) −8.86613 −0.456025
\(379\) 12.0373 0.618314 0.309157 0.951011i \(-0.399953\pi\)
0.309157 + 0.951011i \(0.399953\pi\)
\(380\) −1.38229 −0.0709098
\(381\) 13.8008 0.707038
\(382\) −26.3613 −1.34876
\(383\) −9.96182 −0.509025 −0.254513 0.967069i \(-0.581915\pi\)
−0.254513 + 0.967069i \(0.581915\pi\)
\(384\) 15.2830 0.779910
\(385\) 0 0
\(386\) 14.0932 0.717325
\(387\) 14.3129 0.727564
\(388\) 4.36176 0.221435
\(389\) 22.1252 1.12179 0.560895 0.827887i \(-0.310457\pi\)
0.560895 + 0.827887i \(0.310457\pi\)
\(390\) −10.5718 −0.535326
\(391\) −9.52837 −0.481870
\(392\) 2.09245 0.105684
\(393\) 1.19211 0.0601339
\(394\) −31.5119 −1.58755
\(395\) 8.47388 0.426367
\(396\) 0 0
\(397\) −26.4968 −1.32984 −0.664918 0.746916i \(-0.731533\pi\)
−0.664918 + 0.746916i \(0.731533\pi\)
\(398\) 21.0214 1.05371
\(399\) −2.14757 −0.107513
\(400\) −4.92960 −0.246480
\(401\) −2.98212 −0.148920 −0.0744600 0.997224i \(-0.523723\pi\)
−0.0744600 + 0.997224i \(0.523723\pi\)
\(402\) −10.5877 −0.528068
\(403\) −30.5850 −1.52355
\(404\) −2.32031 −0.115439
\(405\) 1.02820 0.0510915
\(406\) 9.12149 0.452692
\(407\) 0 0
\(408\) 10.9358 0.541400
\(409\) −1.71895 −0.0849966 −0.0424983 0.999097i \(-0.513532\pi\)
−0.0424983 + 0.999097i \(0.513532\pi\)
\(410\) 18.2131 0.899480
\(411\) 1.64159 0.0809738
\(412\) 0.916042 0.0451302
\(413\) 13.5862 0.668533
\(414\) 5.84043 0.287042
\(415\) 9.50230 0.466449
\(416\) 22.2191 1.08938
\(417\) −24.7105 −1.21008
\(418\) 0 0
\(419\) −24.6056 −1.20206 −0.601031 0.799226i \(-0.705243\pi\)
−0.601031 + 0.799226i \(0.705243\pi\)
\(420\) 0.838578 0.0409184
\(421\) −29.8732 −1.45593 −0.727965 0.685615i \(-0.759533\pi\)
−0.727965 + 0.685615i \(0.759533\pi\)
\(422\) −47.0484 −2.29028
\(423\) −2.73133 −0.132802
\(424\) 13.0010 0.631385
\(425\) −4.57876 −0.222103
\(426\) 1.31486 0.0637052
\(427\) −11.1145 −0.537868
\(428\) 7.54399 0.364652
\(429\) 0 0
\(430\) −13.9463 −0.672549
\(431\) 15.5900 0.750943 0.375472 0.926834i \(-0.377481\pi\)
0.375472 + 0.926834i \(0.377481\pi\)
\(432\) −26.4298 −1.27160
\(433\) 10.6888 0.513671 0.256835 0.966455i \(-0.417320\pi\)
0.256835 + 0.966455i \(0.417320\pi\)
\(434\) 9.03046 0.433476
\(435\) −6.29592 −0.301866
\(436\) −4.17229 −0.199816
\(437\) 3.91536 0.187297
\(438\) 26.2131 1.25251
\(439\) −18.1537 −0.866427 −0.433214 0.901291i \(-0.642620\pi\)
−0.433214 + 0.901291i \(0.642620\pi\)
\(440\) 0 0
\(441\) −1.69716 −0.0808169
\(442\) 42.4084 2.01716
\(443\) −9.93668 −0.472106 −0.236053 0.971740i \(-0.575854\pi\)
−0.236053 + 0.971740i \(0.575854\pi\)
\(444\) 7.83814 0.371982
\(445\) −1.49863 −0.0710418
\(446\) 39.6277 1.87642
\(447\) −14.4436 −0.683158
\(448\) 3.29883 0.155855
\(449\) −18.0920 −0.853813 −0.426906 0.904296i \(-0.640397\pi\)
−0.426906 + 0.904296i \(0.640397\pi\)
\(450\) 2.80656 0.132303
\(451\) 0 0
\(452\) 13.2093 0.621312
\(453\) 2.83696 0.133292
\(454\) 34.0671 1.59885
\(455\) −5.60081 −0.262570
\(456\) −4.49367 −0.210436
\(457\) 15.9901 0.747987 0.373993 0.927431i \(-0.377988\pi\)
0.373993 + 0.927431i \(0.377988\pi\)
\(458\) −1.49387 −0.0698041
\(459\) −24.5487 −1.14584
\(460\) −1.52886 −0.0712834
\(461\) −27.9612 −1.30228 −0.651140 0.758957i \(-0.725709\pi\)
−0.651140 + 0.758957i \(0.725709\pi\)
\(462\) 0 0
\(463\) −30.8389 −1.43320 −0.716602 0.697482i \(-0.754304\pi\)
−0.716602 + 0.697482i \(0.754304\pi\)
\(464\) 27.1910 1.26231
\(465\) −6.23309 −0.289053
\(466\) 38.4414 1.78076
\(467\) −8.41735 −0.389509 −0.194754 0.980852i \(-0.562391\pi\)
−0.194754 + 0.980852i \(0.562391\pi\)
\(468\) −6.98344 −0.322810
\(469\) −5.60924 −0.259011
\(470\) 2.66137 0.122760
\(471\) 16.2964 0.750900
\(472\) 28.4284 1.30852
\(473\) 0 0
\(474\) −15.9949 −0.734671
\(475\) 1.88149 0.0863285
\(476\) −3.36392 −0.154185
\(477\) −10.5449 −0.482820
\(478\) 13.5803 0.621149
\(479\) −29.6865 −1.35641 −0.678205 0.734873i \(-0.737242\pi\)
−0.678205 + 0.734873i \(0.737242\pi\)
\(480\) 4.52817 0.206682
\(481\) −52.3505 −2.38698
\(482\) −13.7759 −0.627474
\(483\) −2.37529 −0.108079
\(484\) 0 0
\(485\) −5.93697 −0.269584
\(486\) 24.6576 1.11849
\(487\) 37.8313 1.71430 0.857150 0.515067i \(-0.172233\pi\)
0.857150 + 0.515067i \(0.172233\pi\)
\(488\) −23.2565 −1.05277
\(489\) 11.1375 0.503656
\(490\) 1.65369 0.0747059
\(491\) −18.6985 −0.843850 −0.421925 0.906631i \(-0.638645\pi\)
−0.421925 + 0.906631i \(0.638645\pi\)
\(492\) −9.23578 −0.416381
\(493\) 25.2558 1.13746
\(494\) −17.4263 −0.784046
\(495\) 0 0
\(496\) 26.9196 1.20873
\(497\) 0.696594 0.0312465
\(498\) −17.9361 −0.803736
\(499\) 22.8758 1.02406 0.512030 0.858967i \(-0.328894\pi\)
0.512030 + 0.858967i \(0.328894\pi\)
\(500\) −0.734678 −0.0328558
\(501\) 12.9997 0.580782
\(502\) 16.1655 0.721499
\(503\) 11.6087 0.517607 0.258804 0.965930i \(-0.416672\pi\)
0.258804 + 0.965930i \(0.416672\pi\)
\(504\) −3.55121 −0.158183
\(505\) 3.15826 0.140541
\(506\) 0 0
\(507\) −20.9669 −0.931173
\(508\) −8.88292 −0.394116
\(509\) −13.8562 −0.614163 −0.307082 0.951683i \(-0.599353\pi\)
−0.307082 + 0.951683i \(0.599353\pi\)
\(510\) 8.64266 0.382704
\(511\) 13.8873 0.614339
\(512\) 1.07349 0.0474422
\(513\) 10.0875 0.445373
\(514\) −6.78105 −0.299099
\(515\) −1.24686 −0.0549433
\(516\) 7.07210 0.311332
\(517\) 0 0
\(518\) 15.4569 0.679138
\(519\) 11.8187 0.518783
\(520\) −11.7194 −0.513930
\(521\) −25.2028 −1.10415 −0.552077 0.833793i \(-0.686165\pi\)
−0.552077 + 0.833793i \(0.686165\pi\)
\(522\) −15.4806 −0.677567
\(523\) 28.9229 1.26471 0.632355 0.774679i \(-0.282089\pi\)
0.632355 + 0.774679i \(0.282089\pi\)
\(524\) −0.767303 −0.0335198
\(525\) −1.14142 −0.0498157
\(526\) 42.4093 1.84913
\(527\) 25.0038 1.08918
\(528\) 0 0
\(529\) −18.6695 −0.811716
\(530\) 10.2749 0.446311
\(531\) −23.0579 −1.00063
\(532\) 1.38229 0.0599297
\(533\) 61.6852 2.67188
\(534\) 2.82874 0.122412
\(535\) −10.2684 −0.443943
\(536\) −11.7370 −0.506962
\(537\) −21.4810 −0.926974
\(538\) −22.6600 −0.976943
\(539\) 0 0
\(540\) −3.93893 −0.169505
\(541\) −13.9092 −0.598003 −0.299001 0.954253i \(-0.596653\pi\)
−0.299001 + 0.954253i \(0.596653\pi\)
\(542\) −14.2294 −0.611207
\(543\) 8.85023 0.379800
\(544\) −18.1645 −0.778798
\(545\) 5.67907 0.243265
\(546\) 10.5718 0.452433
\(547\) −12.4603 −0.532766 −0.266383 0.963867i \(-0.585829\pi\)
−0.266383 + 0.963867i \(0.585829\pi\)
\(548\) −1.05661 −0.0451363
\(549\) 18.8630 0.805054
\(550\) 0 0
\(551\) −10.3780 −0.442118
\(552\) −4.97017 −0.211544
\(553\) −8.47388 −0.360346
\(554\) 47.7783 2.02990
\(555\) −10.6688 −0.452866
\(556\) 15.9050 0.674520
\(557\) −37.5067 −1.58921 −0.794605 0.607127i \(-0.792322\pi\)
−0.794605 + 0.607127i \(0.792322\pi\)
\(558\) −15.3261 −0.648806
\(559\) −47.2341 −1.99779
\(560\) 4.92960 0.208314
\(561\) 0 0
\(562\) 13.2030 0.556934
\(563\) 24.5527 1.03477 0.517387 0.855752i \(-0.326905\pi\)
0.517387 + 0.855752i \(0.326905\pi\)
\(564\) −1.34957 −0.0568272
\(565\) −17.9797 −0.756411
\(566\) −12.2290 −0.514022
\(567\) −1.02820 −0.0431802
\(568\) 1.45759 0.0611590
\(569\) 39.6609 1.66267 0.831336 0.555771i \(-0.187577\pi\)
0.831336 + 0.555771i \(0.187577\pi\)
\(570\) −3.55141 −0.148752
\(571\) −31.3014 −1.30992 −0.654961 0.755663i \(-0.727315\pi\)
−0.654961 + 0.755663i \(0.727315\pi\)
\(572\) 0 0
\(573\) −18.1954 −0.760122
\(574\) −18.2131 −0.760199
\(575\) 2.08099 0.0867834
\(576\) −5.59862 −0.233276
\(577\) −33.8941 −1.41103 −0.705516 0.708694i \(-0.749285\pi\)
−0.705516 + 0.708694i \(0.749285\pi\)
\(578\) −6.55698 −0.272734
\(579\) 9.72754 0.404263
\(580\) 4.05238 0.168266
\(581\) −9.50230 −0.394222
\(582\) 11.2064 0.464518
\(583\) 0 0
\(584\) 29.0585 1.20245
\(585\) 9.50545 0.393002
\(586\) 22.8011 0.941907
\(587\) −32.3331 −1.33453 −0.667264 0.744821i \(-0.732535\pi\)
−0.667264 + 0.744821i \(0.732535\pi\)
\(588\) −0.838578 −0.0345824
\(589\) −10.2744 −0.423351
\(590\) 22.4673 0.924964
\(591\) −21.7504 −0.894693
\(592\) 46.0767 1.89374
\(593\) −25.6741 −1.05431 −0.527153 0.849770i \(-0.676741\pi\)
−0.527153 + 0.849770i \(0.676741\pi\)
\(594\) 0 0
\(595\) 4.57876 0.187711
\(596\) 9.29663 0.380805
\(597\) 14.5096 0.593837
\(598\) −19.2741 −0.788177
\(599\) −21.7518 −0.888756 −0.444378 0.895839i \(-0.646575\pi\)
−0.444378 + 0.895839i \(0.646575\pi\)
\(600\) −2.38836 −0.0975046
\(601\) 15.6850 0.639806 0.319903 0.947450i \(-0.396350\pi\)
0.319903 + 0.947450i \(0.396350\pi\)
\(602\) 13.9463 0.568407
\(603\) 9.51975 0.387674
\(604\) −1.82601 −0.0742994
\(605\) 0 0
\(606\) −5.96139 −0.242165
\(607\) −22.9458 −0.931343 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(608\) 7.46409 0.302709
\(609\) 6.29592 0.255123
\(610\) −18.3799 −0.744180
\(611\) 9.01372 0.364656
\(612\) 5.70909 0.230776
\(613\) −40.0317 −1.61687 −0.808433 0.588588i \(-0.799684\pi\)
−0.808433 + 0.588588i \(0.799684\pi\)
\(614\) −9.75276 −0.393589
\(615\) 12.5712 0.506920
\(616\) 0 0
\(617\) 14.7371 0.593293 0.296647 0.954987i \(-0.404132\pi\)
0.296647 + 0.954987i \(0.404132\pi\)
\(618\) 2.35352 0.0946725
\(619\) 25.4079 1.02123 0.510616 0.859809i \(-0.329417\pi\)
0.510616 + 0.859809i \(0.329417\pi\)
\(620\) 4.01194 0.161123
\(621\) 11.1571 0.447719
\(622\) 43.0045 1.72432
\(623\) 1.49863 0.0600413
\(624\) 31.5144 1.26159
\(625\) 1.00000 0.0400000
\(626\) 20.1560 0.805597
\(627\) 0 0
\(628\) −10.4892 −0.418566
\(629\) 42.7974 1.70645
\(630\) −2.80656 −0.111816
\(631\) −26.1912 −1.04265 −0.521327 0.853357i \(-0.674563\pi\)
−0.521327 + 0.853357i \(0.674563\pi\)
\(632\) −17.7311 −0.705307
\(633\) −32.4742 −1.29073
\(634\) −22.4726 −0.892503
\(635\) 12.0909 0.479813
\(636\) −5.21034 −0.206603
\(637\) 5.60081 0.221912
\(638\) 0 0
\(639\) −1.18223 −0.0467683
\(640\) 13.3895 0.529266
\(641\) 8.22192 0.324746 0.162373 0.986729i \(-0.448085\pi\)
0.162373 + 0.986729i \(0.448085\pi\)
\(642\) 19.3822 0.764955
\(643\) 31.8519 1.25612 0.628059 0.778166i \(-0.283850\pi\)
0.628059 + 0.778166i \(0.283850\pi\)
\(644\) 1.52886 0.0602455
\(645\) −9.62612 −0.379028
\(646\) 14.2463 0.560513
\(647\) −22.5883 −0.888036 −0.444018 0.896018i \(-0.646447\pi\)
−0.444018 + 0.896018i \(0.646447\pi\)
\(648\) −2.15145 −0.0845168
\(649\) 0 0
\(650\) −9.26199 −0.363285
\(651\) 6.23309 0.244294
\(652\) −7.16867 −0.280747
\(653\) −0.763209 −0.0298667 −0.0149333 0.999888i \(-0.504754\pi\)
−0.0149333 + 0.999888i \(0.504754\pi\)
\(654\) −10.7196 −0.419168
\(655\) 1.04441 0.0408083
\(656\) −54.2928 −2.11978
\(657\) −23.5690 −0.919512
\(658\) −2.66137 −0.103751
\(659\) 42.2863 1.64724 0.823619 0.567143i \(-0.191951\pi\)
0.823619 + 0.567143i \(0.191951\pi\)
\(660\) 0 0
\(661\) −17.2730 −0.671843 −0.335921 0.941890i \(-0.609048\pi\)
−0.335921 + 0.941890i \(0.609048\pi\)
\(662\) −30.3289 −1.17877
\(663\) 29.2715 1.13681
\(664\) −19.8830 −0.771612
\(665\) −1.88149 −0.0729609
\(666\) −26.2328 −1.01650
\(667\) −11.4784 −0.444447
\(668\) −8.36725 −0.323739
\(669\) 27.3522 1.05750
\(670\) −9.27592 −0.358360
\(671\) 0 0
\(672\) −4.52817 −0.174678
\(673\) 39.3343 1.51623 0.758113 0.652124i \(-0.226122\pi\)
0.758113 + 0.652124i \(0.226122\pi\)
\(674\) 41.8338 1.61138
\(675\) 5.36144 0.206362
\(676\) 13.4954 0.519053
\(677\) −4.86816 −0.187098 −0.0935492 0.995615i \(-0.529821\pi\)
−0.0935492 + 0.995615i \(0.529821\pi\)
\(678\) 33.9376 1.30337
\(679\) 5.93697 0.227840
\(680\) 9.58081 0.367407
\(681\) 23.5141 0.901062
\(682\) 0 0
\(683\) 28.9705 1.10853 0.554263 0.832341i \(-0.313000\pi\)
0.554263 + 0.832341i \(0.313000\pi\)
\(684\) −2.34595 −0.0896998
\(685\) 1.43820 0.0549507
\(686\) −1.65369 −0.0631380
\(687\) −1.03112 −0.0393395
\(688\) 41.5735 1.58498
\(689\) 34.7996 1.32576
\(690\) −3.92798 −0.149536
\(691\) −15.4179 −0.586526 −0.293263 0.956032i \(-0.594741\pi\)
−0.293263 + 0.956032i \(0.594741\pi\)
\(692\) −7.60711 −0.289179
\(693\) 0 0
\(694\) 23.9691 0.909855
\(695\) −21.6489 −0.821189
\(696\) 13.1739 0.499354
\(697\) −50.4288 −1.91013
\(698\) 4.92331 0.186350
\(699\) 26.5334 1.00358
\(700\) 0.734678 0.0277682
\(701\) −6.78227 −0.256163 −0.128081 0.991764i \(-0.540882\pi\)
−0.128081 + 0.991764i \(0.540882\pi\)
\(702\) −49.6576 −1.87420
\(703\) −17.5861 −0.663274
\(704\) 0 0
\(705\) 1.83696 0.0691838
\(706\) −23.2808 −0.876186
\(707\) −3.15826 −0.118779
\(708\) −11.3931 −0.428178
\(709\) −21.1656 −0.794891 −0.397446 0.917626i \(-0.630103\pi\)
−0.397446 + 0.917626i \(0.630103\pi\)
\(710\) 1.15195 0.0432319
\(711\) 14.3815 0.539348
\(712\) 3.13580 0.117519
\(713\) −11.3639 −0.425581
\(714\) −8.64266 −0.323443
\(715\) 0 0
\(716\) 13.8263 0.516712
\(717\) 9.37352 0.350061
\(718\) 25.3531 0.946170
\(719\) 32.6186 1.21647 0.608234 0.793758i \(-0.291878\pi\)
0.608234 + 0.793758i \(0.291878\pi\)
\(720\) −8.36630 −0.311794
\(721\) 1.24686 0.0464356
\(722\) 25.5660 0.951468
\(723\) −9.50851 −0.353625
\(724\) −5.69646 −0.211707
\(725\) −5.51585 −0.204854
\(726\) 0 0
\(727\) 22.4245 0.831678 0.415839 0.909438i \(-0.363488\pi\)
0.415839 + 0.909438i \(0.363488\pi\)
\(728\) 11.7194 0.434350
\(729\) 20.1040 0.744593
\(730\) 22.9653 0.849983
\(731\) 38.6147 1.42822
\(732\) 9.32037 0.344491
\(733\) −28.7607 −1.06230 −0.531150 0.847278i \(-0.678240\pi\)
−0.531150 + 0.847278i \(0.678240\pi\)
\(734\) −22.6852 −0.837326
\(735\) 1.14142 0.0421020
\(736\) 8.25556 0.304304
\(737\) 0 0
\(738\) 30.9104 1.13783
\(739\) 2.12241 0.0780742 0.0390371 0.999238i \(-0.487571\pi\)
0.0390371 + 0.999238i \(0.487571\pi\)
\(740\) 6.86699 0.252436
\(741\) −12.0281 −0.441865
\(742\) −10.2749 −0.377202
\(743\) 38.6935 1.41953 0.709763 0.704440i \(-0.248802\pi\)
0.709763 + 0.704440i \(0.248802\pi\)
\(744\) 13.0424 0.478158
\(745\) −12.6540 −0.463607
\(746\) 20.9408 0.766696
\(747\) 16.1269 0.590052
\(748\) 0 0
\(749\) 10.2684 0.375200
\(750\) −1.88755 −0.0689237
\(751\) 2.01833 0.0736498 0.0368249 0.999322i \(-0.488276\pi\)
0.0368249 + 0.999322i \(0.488276\pi\)
\(752\) −7.93350 −0.289305
\(753\) 11.1579 0.406615
\(754\) 51.0878 1.86051
\(755\) 2.48546 0.0904551
\(756\) 3.93893 0.143257
\(757\) 39.9006 1.45021 0.725107 0.688637i \(-0.241790\pi\)
0.725107 + 0.688637i \(0.241790\pi\)
\(758\) −19.9059 −0.723015
\(759\) 0 0
\(760\) −3.93691 −0.142807
\(761\) −28.9628 −1.04990 −0.524950 0.851133i \(-0.675916\pi\)
−0.524950 + 0.851133i \(0.675916\pi\)
\(762\) −22.8222 −0.826762
\(763\) −5.67907 −0.205596
\(764\) 11.7115 0.423706
\(765\) −7.77087 −0.280956
\(766\) 16.4737 0.595220
\(767\) 76.0937 2.74759
\(768\) −17.7427 −0.640233
\(769\) −5.26499 −0.189860 −0.0949302 0.995484i \(-0.530263\pi\)
−0.0949302 + 0.995484i \(0.530263\pi\)
\(770\) 0 0
\(771\) −4.68047 −0.168563
\(772\) −6.26114 −0.225344
\(773\) −0.286828 −0.0103165 −0.00515825 0.999987i \(-0.501642\pi\)
−0.00515825 + 0.999987i \(0.501642\pi\)
\(774\) −23.6690 −0.850764
\(775\) −5.46081 −0.196158
\(776\) 12.4228 0.445952
\(777\) 10.6688 0.382741
\(778\) −36.5881 −1.31175
\(779\) 20.7220 0.742442
\(780\) 4.69672 0.168169
\(781\) 0 0
\(782\) 15.7569 0.563467
\(783\) −29.5729 −1.05685
\(784\) −4.92960 −0.176057
\(785\) 14.2773 0.509579
\(786\) −1.97137 −0.0703166
\(787\) −27.4622 −0.978923 −0.489462 0.872025i \(-0.662807\pi\)
−0.489462 + 0.872025i \(0.662807\pi\)
\(788\) 13.9997 0.498718
\(789\) 29.2721 1.04211
\(790\) −14.0131 −0.498565
\(791\) 17.9797 0.639284
\(792\) 0 0
\(793\) −62.2502 −2.21057
\(794\) 43.8174 1.55502
\(795\) 7.09201 0.251528
\(796\) −9.33911 −0.331016
\(797\) 22.6304 0.801609 0.400805 0.916164i \(-0.368731\pi\)
0.400805 + 0.916164i \(0.368731\pi\)
\(798\) 3.55141 0.125718
\(799\) −7.36887 −0.260692
\(800\) 3.96713 0.140259
\(801\) −2.54341 −0.0898668
\(802\) 4.93149 0.174137
\(803\) 0 0
\(804\) 4.70378 0.165890
\(805\) −2.08099 −0.0733453
\(806\) 50.5779 1.78153
\(807\) −15.6406 −0.550575
\(808\) −6.60849 −0.232486
\(809\) 37.2584 1.30994 0.654968 0.755656i \(-0.272682\pi\)
0.654968 + 0.755656i \(0.272682\pi\)
\(810\) −1.70032 −0.0597430
\(811\) 0.190250 0.00668059 0.00334030 0.999994i \(-0.498937\pi\)
0.00334030 + 0.999994i \(0.498937\pi\)
\(812\) −4.05238 −0.142211
\(813\) −9.82158 −0.344458
\(814\) 0 0
\(815\) 9.75758 0.341793
\(816\) −25.7636 −0.901906
\(817\) −15.8674 −0.555130
\(818\) 2.84260 0.0993893
\(819\) −9.50545 −0.332147
\(820\) −8.09147 −0.282566
\(821\) 30.0724 1.04953 0.524767 0.851246i \(-0.324152\pi\)
0.524767 + 0.851246i \(0.324152\pi\)
\(822\) −2.71468 −0.0946853
\(823\) 20.6546 0.719975 0.359987 0.932957i \(-0.382781\pi\)
0.359987 + 0.932957i \(0.382781\pi\)
\(824\) 2.60899 0.0908885
\(825\) 0 0
\(826\) −22.4673 −0.781738
\(827\) −25.3871 −0.882795 −0.441397 0.897312i \(-0.645517\pi\)
−0.441397 + 0.897312i \(0.645517\pi\)
\(828\) −2.59471 −0.0901724
\(829\) 6.53298 0.226900 0.113450 0.993544i \(-0.463810\pi\)
0.113450 + 0.993544i \(0.463810\pi\)
\(830\) −15.7138 −0.545435
\(831\) 32.9779 1.14399
\(832\) 18.4761 0.640544
\(833\) −4.57876 −0.158645
\(834\) 40.8634 1.41498
\(835\) 11.3890 0.394133
\(836\) 0 0
\(837\) −29.2778 −1.01199
\(838\) 40.6899 1.40561
\(839\) −13.3320 −0.460271 −0.230136 0.973159i \(-0.573917\pi\)
−0.230136 + 0.973159i \(0.573917\pi\)
\(840\) 2.38836 0.0824064
\(841\) 1.42465 0.0491257
\(842\) 49.4008 1.70247
\(843\) 9.11308 0.313871
\(844\) 20.9020 0.719478
\(845\) −18.3691 −0.631916
\(846\) 4.51677 0.155290
\(847\) 0 0
\(848\) −30.6291 −1.05181
\(849\) −8.44080 −0.289687
\(850\) 7.57184 0.259712
\(851\) −19.4509 −0.666769
\(852\) −0.584148 −0.0200126
\(853\) −45.3221 −1.55180 −0.775899 0.630857i \(-0.782704\pi\)
−0.775899 + 0.630857i \(0.782704\pi\)
\(854\) 18.3799 0.628947
\(855\) 3.19317 0.109204
\(856\) 21.4861 0.734381
\(857\) −26.0895 −0.891199 −0.445599 0.895232i \(-0.647009\pi\)
−0.445599 + 0.895232i \(0.647009\pi\)
\(858\) 0 0
\(859\) 40.0079 1.36505 0.682527 0.730861i \(-0.260881\pi\)
0.682527 + 0.730861i \(0.260881\pi\)
\(860\) 6.19586 0.211277
\(861\) −12.5712 −0.428425
\(862\) −25.7809 −0.878103
\(863\) 58.6608 1.99684 0.998418 0.0562309i \(-0.0179083\pi\)
0.998418 + 0.0562309i \(0.0179083\pi\)
\(864\) 21.2695 0.723603
\(865\) 10.3544 0.352059
\(866\) −17.6759 −0.600652
\(867\) −4.52582 −0.153705
\(868\) −4.01194 −0.136174
\(869\) 0 0
\(870\) 10.4115 0.352982
\(871\) −31.4163 −1.06450
\(872\) −11.8832 −0.402414
\(873\) −10.0760 −0.341020
\(874\) −6.47477 −0.219012
\(875\) −1.00000 −0.0338062
\(876\) −11.6456 −0.393468
\(877\) −33.3513 −1.12619 −0.563096 0.826392i \(-0.690390\pi\)
−0.563096 + 0.826392i \(0.690390\pi\)
\(878\) 30.0205 1.01314
\(879\) 15.7380 0.530830
\(880\) 0 0
\(881\) 21.4875 0.723933 0.361967 0.932191i \(-0.382105\pi\)
0.361967 + 0.932191i \(0.382105\pi\)
\(882\) 2.80656 0.0945019
\(883\) −43.7638 −1.47277 −0.736385 0.676563i \(-0.763469\pi\)
−0.736385 + 0.676563i \(0.763469\pi\)
\(884\) −18.8407 −0.633680
\(885\) 15.5076 0.521282
\(886\) 16.4321 0.552049
\(887\) 4.54756 0.152692 0.0763460 0.997081i \(-0.475675\pi\)
0.0763460 + 0.997081i \(0.475675\pi\)
\(888\) 22.3239 0.749141
\(889\) −12.0909 −0.405516
\(890\) 2.47826 0.0830715
\(891\) 0 0
\(892\) −17.6053 −0.589467
\(893\) 3.02799 0.101328
\(894\) 23.8851 0.798839
\(895\) −18.8195 −0.629067
\(896\) −13.3895 −0.447311
\(897\) −13.3036 −0.444193
\(898\) 29.9184 0.998391
\(899\) 30.1210 1.00459
\(900\) −1.24686 −0.0415621
\(901\) −28.4493 −0.947782
\(902\) 0 0
\(903\) 9.62612 0.320337
\(904\) 37.6215 1.25127
\(905\) 7.75369 0.257741
\(906\) −4.69144 −0.155863
\(907\) −17.4034 −0.577869 −0.288935 0.957349i \(-0.593301\pi\)
−0.288935 + 0.957349i \(0.593301\pi\)
\(908\) −15.1349 −0.502268
\(909\) 5.36006 0.177782
\(910\) 9.26199 0.307032
\(911\) 4.17760 0.138410 0.0692050 0.997602i \(-0.477954\pi\)
0.0692050 + 0.997602i \(0.477954\pi\)
\(912\) 10.5867 0.350560
\(913\) 0 0
\(914\) −26.4427 −0.874645
\(915\) −12.6863 −0.419397
\(916\) 0.663678 0.0219286
\(917\) −1.04441 −0.0344894
\(918\) 40.5959 1.33987
\(919\) −18.4017 −0.607018 −0.303509 0.952829i \(-0.598158\pi\)
−0.303509 + 0.952829i \(0.598158\pi\)
\(920\) −4.35436 −0.143559
\(921\) −6.73164 −0.221815
\(922\) 46.2390 1.52280
\(923\) 3.90149 0.128419
\(924\) 0 0
\(925\) −9.34694 −0.307326
\(926\) 50.9978 1.67589
\(927\) −2.11612 −0.0695025
\(928\) −21.8821 −0.718315
\(929\) −52.1959 −1.71249 −0.856245 0.516569i \(-0.827209\pi\)
−0.856245 + 0.516569i \(0.827209\pi\)
\(930\) 10.3076 0.337999
\(931\) 1.88149 0.0616632
\(932\) −17.0782 −0.559416
\(933\) 29.6830 0.971777
\(934\) 13.9197 0.455465
\(935\) 0 0
\(936\) −19.8896 −0.650113
\(937\) 12.3471 0.403363 0.201681 0.979451i \(-0.435359\pi\)
0.201681 + 0.979451i \(0.435359\pi\)
\(938\) 9.27592 0.302869
\(939\) 13.9123 0.454010
\(940\) −1.18236 −0.0385643
\(941\) −13.8509 −0.451525 −0.225763 0.974182i \(-0.572487\pi\)
−0.225763 + 0.974182i \(0.572487\pi\)
\(942\) −26.9492 −0.878052
\(943\) 22.9193 0.746354
\(944\) −66.9746 −2.17984
\(945\) −5.36144 −0.174408
\(946\) 0 0
\(947\) −21.7800 −0.707756 −0.353878 0.935292i \(-0.615137\pi\)
−0.353878 + 0.935292i \(0.615137\pi\)
\(948\) 7.10601 0.230792
\(949\) 77.7803 2.52486
\(950\) −3.11139 −0.100947
\(951\) −15.5113 −0.502988
\(952\) −9.58081 −0.310516
\(953\) −7.56150 −0.244941 −0.122471 0.992472i \(-0.539082\pi\)
−0.122471 + 0.992472i \(0.539082\pi\)
\(954\) 17.4380 0.564577
\(955\) −15.9410 −0.515837
\(956\) −6.03328 −0.195130
\(957\) 0 0
\(958\) 49.0921 1.58609
\(959\) −1.43820 −0.0464419
\(960\) 3.76535 0.121526
\(961\) −1.17956 −0.0380503
\(962\) 86.5713 2.79117
\(963\) −17.4271 −0.561581
\(964\) 6.12016 0.197117
\(965\) 8.52230 0.274343
\(966\) 3.92798 0.126381
\(967\) 24.0488 0.773357 0.386679 0.922215i \(-0.373622\pi\)
0.386679 + 0.922215i \(0.373622\pi\)
\(968\) 0 0
\(969\) 9.83321 0.315888
\(970\) 9.81789 0.315233
\(971\) 16.0049 0.513622 0.256811 0.966462i \(-0.417328\pi\)
0.256811 + 0.966462i \(0.417328\pi\)
\(972\) −10.9546 −0.351368
\(973\) 21.6489 0.694031
\(974\) −62.5611 −2.00459
\(975\) −6.39289 −0.204736
\(976\) 54.7901 1.75379
\(977\) 16.4515 0.526330 0.263165 0.964751i \(-0.415234\pi\)
0.263165 + 0.964751i \(0.415234\pi\)
\(978\) −18.4180 −0.588941
\(979\) 0 0
\(980\) −0.734678 −0.0234684
\(981\) 9.63827 0.307726
\(982\) 30.9214 0.986741
\(983\) 30.2550 0.964984 0.482492 0.875900i \(-0.339732\pi\)
0.482492 + 0.875900i \(0.339732\pi\)
\(984\) −26.3045 −0.838558
\(985\) −19.0556 −0.607160
\(986\) −41.7651 −1.33007
\(987\) −1.83696 −0.0584710
\(988\) 7.74192 0.246303
\(989\) −17.5499 −0.558055
\(990\) 0 0
\(991\) −13.0778 −0.415428 −0.207714 0.978190i \(-0.566602\pi\)
−0.207714 + 0.978190i \(0.566602\pi\)
\(992\) −21.6637 −0.687824
\(993\) −20.9339 −0.664317
\(994\) −1.15195 −0.0365376
\(995\) 12.7118 0.402992
\(996\) 7.96841 0.252489
\(997\) −31.8380 −1.00832 −0.504160 0.863610i \(-0.668198\pi\)
−0.504160 + 0.863610i \(0.668198\pi\)
\(998\) −37.8294 −1.19747
\(999\) −50.1131 −1.58551
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 4235.2.a.y.1.2 5
11.10 odd 2 4235.2.a.be.1.4 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.y.1.2 5 1.1 even 1 trivial
4235.2.a.be.1.4 yes 5 11.10 odd 2