Properties

Label 1155.2.a.u.1.2
Level 1155
Weight 2
Character 1155.1
Self dual yes
Analytic conductor 9.223
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Defining polynomial: \(x^{4} - 7 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.546295\) of defining polynomial
Character \(\chi\) \(=\) 1155.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} -0.546295 q^{6} -1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.546295 q^{2} +1.00000 q^{3} -1.70156 q^{4} -1.00000 q^{5} -0.546295 q^{6} -1.00000 q^{7} +2.02214 q^{8} +1.00000 q^{9} +0.546295 q^{10} -1.00000 q^{11} -1.70156 q^{12} +4.56844 q^{13} +0.546295 q^{14} -1.00000 q^{15} +2.29844 q^{16} +2.70156 q^{17} -0.546295 q^{18} -8.02362 q^{19} +1.70156 q^{20} -1.00000 q^{21} +0.546295 q^{22} -3.52790 q^{23} +2.02214 q^{24} +1.00000 q^{25} -2.49571 q^{26} +1.00000 q^{27} +1.70156 q^{28} +0.133124 q^{29} +0.546295 q^{30} +2.33897 q^{31} -5.29991 q^{32} -1.00000 q^{33} -1.47585 q^{34} +1.00000 q^{35} -1.70156 q^{36} +6.75362 q^{37} +4.38326 q^{38} +4.56844 q^{39} -2.02214 q^{40} +10.9831 q^{41} +0.546295 q^{42} +9.45518 q^{43} +1.70156 q^{44} -1.00000 q^{45} +1.92728 q^{46} -0.649507 q^{47} +2.29844 q^{48} +1.00000 q^{49} -0.546295 q^{50} +2.70156 q^{51} -7.77348 q^{52} +11.8384 q^{53} -0.546295 q^{54} +1.00000 q^{55} -2.02214 q^{56} -8.02362 q^{57} -0.0727248 q^{58} -0.959466 q^{59} +1.70156 q^{60} +13.1162 q^{61} -1.27777 q^{62} -1.00000 q^{63} -1.70156 q^{64} -4.56844 q^{65} +0.546295 q^{66} +6.41464 q^{67} -4.59688 q^{68} -3.52790 q^{69} -0.546295 q^{70} -2.56844 q^{71} +2.02214 q^{72} +12.5400 q^{73} -3.68947 q^{74} +1.00000 q^{75} +13.6527 q^{76} +1.00000 q^{77} -2.49571 q^{78} +2.56844 q^{79} -2.29844 q^{80} +1.00000 q^{81} -6.00000 q^{82} -7.75737 q^{83} +1.70156 q^{84} -2.70156 q^{85} -5.16531 q^{86} +0.133124 q^{87} -2.02214 q^{88} +13.3741 q^{89} +0.546295 q^{90} -4.56844 q^{91} +6.00295 q^{92} +2.33897 q^{93} +0.354822 q^{94} +8.02362 q^{95} -5.29991 q^{96} -1.78581 q^{97} -0.546295 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 6q^{4} - 4q^{5} - 4q^{7} + 4q^{9} - 4q^{11} + 6q^{12} + 8q^{13} - 4q^{15} + 22q^{16} - 2q^{17} + 10q^{19} - 6q^{20} - 4q^{21} - 2q^{23} + 4q^{25} + 20q^{26} + 4q^{27} - 6q^{28} - 2q^{29} + 24q^{31} - 4q^{33} + 4q^{35} + 6q^{36} + 8q^{37} + 16q^{38} + 8q^{39} + 6q^{43} - 6q^{44} - 4q^{45} - 12q^{46} + 4q^{47} + 22q^{48} + 4q^{49} - 2q^{51} + 12q^{52} + 14q^{53} + 4q^{55} + 10q^{57} - 20q^{58} - 2q^{59} - 6q^{60} + 6q^{61} + 8q^{62} - 4q^{63} + 6q^{64} - 8q^{65} - 8q^{67} - 44q^{68} - 2q^{69} + 4q^{73} - 36q^{74} + 4q^{75} + 56q^{76} + 4q^{77} + 20q^{78} - 22q^{80} + 4q^{81} - 24q^{82} + 6q^{83} - 6q^{84} + 2q^{85} - 36q^{86} - 2q^{87} + 18q^{89} - 8q^{91} - 44q^{92} + 24q^{93} - 36q^{94} - 10q^{95} - 6q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.546295 −0.386289 −0.193144 0.981170i \(-0.561869\pi\)
−0.193144 + 0.981170i \(0.561869\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.70156 −0.850781
\(5\) −1.00000 −0.447214
\(6\) −0.546295 −0.223024
\(7\) −1.00000 −0.377964
\(8\) 2.02214 0.714936
\(9\) 1.00000 0.333333
\(10\) 0.546295 0.172754
\(11\) −1.00000 −0.301511
\(12\) −1.70156 −0.491199
\(13\) 4.56844 1.26706 0.633528 0.773719i \(-0.281606\pi\)
0.633528 + 0.773719i \(0.281606\pi\)
\(14\) 0.546295 0.146003
\(15\) −1.00000 −0.258199
\(16\) 2.29844 0.574609
\(17\) 2.70156 0.655225 0.327613 0.944812i \(-0.393756\pi\)
0.327613 + 0.944812i \(0.393756\pi\)
\(18\) −0.546295 −0.128763
\(19\) −8.02362 −1.84074 −0.920372 0.391044i \(-0.872114\pi\)
−0.920372 + 0.391044i \(0.872114\pi\)
\(20\) 1.70156 0.380481
\(21\) −1.00000 −0.218218
\(22\) 0.546295 0.116470
\(23\) −3.52790 −0.735619 −0.367809 0.929901i \(-0.619892\pi\)
−0.367809 + 0.929901i \(0.619892\pi\)
\(24\) 2.02214 0.412768
\(25\) 1.00000 0.200000
\(26\) −2.49571 −0.489450
\(27\) 1.00000 0.192450
\(28\) 1.70156 0.321565
\(29\) 0.133124 0.0247205 0.0123602 0.999924i \(-0.496066\pi\)
0.0123602 + 0.999924i \(0.496066\pi\)
\(30\) 0.546295 0.0997393
\(31\) 2.33897 0.420092 0.210046 0.977692i \(-0.432639\pi\)
0.210046 + 0.977692i \(0.432639\pi\)
\(32\) −5.29991 −0.936901
\(33\) −1.00000 −0.174078
\(34\) −1.47585 −0.253106
\(35\) 1.00000 0.169031
\(36\) −1.70156 −0.283594
\(37\) 6.75362 1.11029 0.555144 0.831754i \(-0.312663\pi\)
0.555144 + 0.831754i \(0.312663\pi\)
\(38\) 4.38326 0.711059
\(39\) 4.56844 0.731536
\(40\) −2.02214 −0.319729
\(41\) 10.9831 1.71527 0.857635 0.514259i \(-0.171933\pi\)
0.857635 + 0.514259i \(0.171933\pi\)
\(42\) 0.546295 0.0842951
\(43\) 9.45518 1.44190 0.720951 0.692986i \(-0.243705\pi\)
0.720951 + 0.692986i \(0.243705\pi\)
\(44\) 1.70156 0.256520
\(45\) −1.00000 −0.149071
\(46\) 1.92728 0.284161
\(47\) −0.649507 −0.0947403 −0.0473702 0.998877i \(-0.515084\pi\)
−0.0473702 + 0.998877i \(0.515084\pi\)
\(48\) 2.29844 0.331751
\(49\) 1.00000 0.142857
\(50\) −0.546295 −0.0772577
\(51\) 2.70156 0.378294
\(52\) −7.77348 −1.07799
\(53\) 11.8384 1.62613 0.813067 0.582170i \(-0.197796\pi\)
0.813067 + 0.582170i \(0.197796\pi\)
\(54\) −0.546295 −0.0743413
\(55\) 1.00000 0.134840
\(56\) −2.02214 −0.270220
\(57\) −8.02362 −1.06275
\(58\) −0.0727248 −0.00954923
\(59\) −0.959466 −0.124912 −0.0624559 0.998048i \(-0.519893\pi\)
−0.0624559 + 0.998048i \(0.519893\pi\)
\(60\) 1.70156 0.219671
\(61\) 13.1162 1.67936 0.839679 0.543083i \(-0.182743\pi\)
0.839679 + 0.543083i \(0.182743\pi\)
\(62\) −1.27777 −0.162277
\(63\) −1.00000 −0.125988
\(64\) −1.70156 −0.212695
\(65\) −4.56844 −0.566645
\(66\) 0.546295 0.0672442
\(67\) 6.41464 0.783674 0.391837 0.920035i \(-0.371840\pi\)
0.391837 + 0.920035i \(0.371840\pi\)
\(68\) −4.59688 −0.557453
\(69\) −3.52790 −0.424710
\(70\) −0.546295 −0.0652947
\(71\) −2.56844 −0.304818 −0.152409 0.988318i \(-0.548703\pi\)
−0.152409 + 0.988318i \(0.548703\pi\)
\(72\) 2.02214 0.238312
\(73\) 12.5400 1.46770 0.733848 0.679314i \(-0.237722\pi\)
0.733848 + 0.679314i \(0.237722\pi\)
\(74\) −3.68947 −0.428892
\(75\) 1.00000 0.115470
\(76\) 13.6527 1.56607
\(77\) 1.00000 0.113961
\(78\) −2.49571 −0.282584
\(79\) 2.56844 0.288972 0.144486 0.989507i \(-0.453847\pi\)
0.144486 + 0.989507i \(0.453847\pi\)
\(80\) −2.29844 −0.256973
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −7.75737 −0.851482 −0.425741 0.904845i \(-0.639987\pi\)
−0.425741 + 0.904845i \(0.639987\pi\)
\(84\) 1.70156 0.185656
\(85\) −2.70156 −0.293026
\(86\) −5.16531 −0.556990
\(87\) 0.133124 0.0142724
\(88\) −2.02214 −0.215561
\(89\) 13.3741 1.41765 0.708826 0.705383i \(-0.249225\pi\)
0.708826 + 0.705383i \(0.249225\pi\)
\(90\) 0.546295 0.0575845
\(91\) −4.56844 −0.478902
\(92\) 6.00295 0.625851
\(93\) 2.33897 0.242540
\(94\) 0.354822 0.0365971
\(95\) 8.02362 0.823206
\(96\) −5.29991 −0.540920
\(97\) −1.78581 −0.181321 −0.0906606 0.995882i \(-0.528898\pi\)
−0.0906606 + 0.995882i \(0.528898\pi\)
\(98\) −0.546295 −0.0551841
\(99\) −1.00000 −0.100504
\(100\) −1.70156 −0.170156
\(101\) −11.6326 −1.15749 −0.578743 0.815510i \(-0.696457\pi\)
−0.578743 + 0.815510i \(0.696457\pi\)
\(102\) −1.47585 −0.146131
\(103\) −9.18893 −0.905412 −0.452706 0.891660i \(-0.649541\pi\)
−0.452706 + 0.891660i \(0.649541\pi\)
\(104\) 9.23804 0.905864
\(105\) 1.00000 0.0975900
\(106\) −6.46728 −0.628157
\(107\) −13.0199 −1.25868 −0.629339 0.777131i \(-0.716674\pi\)
−0.629339 + 0.777131i \(0.716674\pi\)
\(108\) −1.70156 −0.163733
\(109\) 0.109506 0.0104888 0.00524439 0.999986i \(-0.498331\pi\)
0.00524439 + 0.999986i \(0.498331\pi\)
\(110\) −0.546295 −0.0520872
\(111\) 6.75362 0.641025
\(112\) −2.29844 −0.217182
\(113\) −3.83844 −0.361090 −0.180545 0.983567i \(-0.557786\pi\)
−0.180545 + 0.983567i \(0.557786\pi\)
\(114\) 4.38326 0.410530
\(115\) 3.52790 0.328979
\(116\) −0.226518 −0.0210317
\(117\) 4.56844 0.422352
\(118\) 0.524151 0.0482520
\(119\) −2.70156 −0.247652
\(120\) −2.02214 −0.184596
\(121\) 1.00000 0.0909091
\(122\) −7.16531 −0.648717
\(123\) 10.9831 0.990311
\(124\) −3.97991 −0.357406
\(125\) −1.00000 −0.0894427
\(126\) 0.546295 0.0486678
\(127\) −5.45518 −0.484069 −0.242034 0.970268i \(-0.577815\pi\)
−0.242034 + 0.970268i \(0.577815\pi\)
\(128\) 11.5294 1.01906
\(129\) 9.45518 0.832482
\(130\) 2.49571 0.218889
\(131\) −21.1084 −1.84425 −0.922126 0.386889i \(-0.873550\pi\)
−0.922126 + 0.386889i \(0.873550\pi\)
\(132\) 1.70156 0.148102
\(133\) 8.02362 0.695736
\(134\) −3.50429 −0.302724
\(135\) −1.00000 −0.0860663
\(136\) 5.46295 0.468444
\(137\) −3.40312 −0.290749 −0.145374 0.989377i \(-0.546439\pi\)
−0.145374 + 0.989377i \(0.546439\pi\)
\(138\) 1.92728 0.164061
\(139\) 21.2410 1.80164 0.900818 0.434196i \(-0.142967\pi\)
0.900818 + 0.434196i \(0.142967\pi\)
\(140\) −1.70156 −0.143808
\(141\) −0.649507 −0.0546984
\(142\) 1.40312 0.117748
\(143\) −4.56844 −0.382032
\(144\) 2.29844 0.191536
\(145\) −0.133124 −0.0110553
\(146\) −6.85054 −0.566954
\(147\) 1.00000 0.0824786
\(148\) −11.4917 −0.944612
\(149\) −11.5072 −0.942709 −0.471355 0.881944i \(-0.656235\pi\)
−0.471355 + 0.881944i \(0.656235\pi\)
\(150\) −0.546295 −0.0446048
\(151\) 7.58830 0.617527 0.308764 0.951139i \(-0.400085\pi\)
0.308764 + 0.951139i \(0.400085\pi\)
\(152\) −16.2249 −1.31601
\(153\) 2.70156 0.218408
\(154\) −0.546295 −0.0440217
\(155\) −2.33897 −0.187871
\(156\) −7.77348 −0.622377
\(157\) −12.8583 −1.02620 −0.513102 0.858328i \(-0.671504\pi\)
−0.513102 + 0.858328i \(0.671504\pi\)
\(158\) −1.40312 −0.111627
\(159\) 11.8384 0.938849
\(160\) 5.29991 0.418995
\(161\) 3.52790 0.278038
\(162\) −0.546295 −0.0429210
\(163\) 22.6525 1.77428 0.887139 0.461503i \(-0.152690\pi\)
0.887139 + 0.461503i \(0.152690\pi\)
\(164\) −18.6884 −1.45932
\(165\) 1.00000 0.0778499
\(166\) 4.23781 0.328918
\(167\) −3.21795 −0.249012 −0.124506 0.992219i \(-0.539735\pi\)
−0.124506 + 0.992219i \(0.539735\pi\)
\(168\) −2.02214 −0.156012
\(169\) 7.87063 0.605433
\(170\) 1.47585 0.113192
\(171\) −8.02362 −0.613581
\(172\) −16.0886 −1.22674
\(173\) −18.2094 −1.38443 −0.692216 0.721690i \(-0.743366\pi\)
−0.692216 + 0.721690i \(0.743366\pi\)
\(174\) −0.0727248 −0.00551325
\(175\) −1.00000 −0.0755929
\(176\) −2.29844 −0.173251
\(177\) −0.959466 −0.0721179
\(178\) −7.30621 −0.547623
\(179\) −4.75362 −0.355302 −0.177651 0.984094i \(-0.556850\pi\)
−0.177651 + 0.984094i \(0.556850\pi\)
\(180\) 1.70156 0.126827
\(181\) 0.347316 0.0258158 0.0129079 0.999917i \(-0.495891\pi\)
0.0129079 + 0.999917i \(0.495891\pi\)
\(182\) 2.49571 0.184995
\(183\) 13.1162 0.969578
\(184\) −7.13393 −0.525920
\(185\) −6.75362 −0.496536
\(186\) −1.27777 −0.0936905
\(187\) −2.70156 −0.197558
\(188\) 1.10518 0.0806033
\(189\) −1.00000 −0.0727393
\(190\) −4.38326 −0.317995
\(191\) 11.8379 0.856558 0.428279 0.903647i \(-0.359120\pi\)
0.428279 + 0.903647i \(0.359120\pi\)
\(192\) −1.70156 −0.122800
\(193\) −4.14840 −0.298608 −0.149304 0.988791i \(-0.547703\pi\)
−0.149304 + 0.988791i \(0.547703\pi\)
\(194\) 0.975577 0.0700424
\(195\) −4.56844 −0.327153
\(196\) −1.70156 −0.121540
\(197\) 19.3663 1.37979 0.689897 0.723907i \(-0.257656\pi\)
0.689897 + 0.723907i \(0.257656\pi\)
\(198\) 0.546295 0.0388235
\(199\) 13.2622 0.940135 0.470067 0.882631i \(-0.344230\pi\)
0.470067 + 0.882631i \(0.344230\pi\)
\(200\) 2.02214 0.142987
\(201\) 6.41464 0.452454
\(202\) 6.35482 0.447124
\(203\) −0.133124 −0.00934345
\(204\) −4.59688 −0.321846
\(205\) −10.9831 −0.767092
\(206\) 5.01986 0.349751
\(207\) −3.52790 −0.245206
\(208\) 10.5003 0.728063
\(209\) 8.02362 0.555005
\(210\) −0.546295 −0.0376979
\(211\) 2.54000 0.174861 0.0874304 0.996171i \(-0.472134\pi\)
0.0874304 + 0.996171i \(0.472134\pi\)
\(212\) −20.1438 −1.38348
\(213\) −2.56844 −0.175986
\(214\) 7.11268 0.486213
\(215\) −9.45518 −0.644838
\(216\) 2.02214 0.137589
\(217\) −2.33897 −0.158780
\(218\) −0.0598226 −0.00405170
\(219\) 12.5400 0.847375
\(220\) −1.70156 −0.114719
\(221\) 12.3419 0.830207
\(222\) −3.68947 −0.247621
\(223\) 20.5110 1.37352 0.686759 0.726886i \(-0.259033\pi\)
0.686759 + 0.726886i \(0.259033\pi\)
\(224\) 5.29991 0.354115
\(225\) 1.00000 0.0666667
\(226\) 2.09692 0.139485
\(227\) 19.6119 1.30169 0.650844 0.759211i \(-0.274415\pi\)
0.650844 + 0.759211i \(0.274415\pi\)
\(228\) 13.6527 0.904171
\(229\) 20.4230 1.34959 0.674795 0.738006i \(-0.264232\pi\)
0.674795 + 0.738006i \(0.264232\pi\)
\(230\) −1.92728 −0.127081
\(231\) 1.00000 0.0657952
\(232\) 0.269195 0.0176735
\(233\) −17.3304 −1.13535 −0.567676 0.823252i \(-0.692157\pi\)
−0.567676 + 0.823252i \(0.692157\pi\)
\(234\) −2.49571 −0.163150
\(235\) 0.649507 0.0423692
\(236\) 1.63259 0.106273
\(237\) 2.56844 0.166838
\(238\) 1.47585 0.0956651
\(239\) −11.6404 −0.752952 −0.376476 0.926426i \(-0.622864\pi\)
−0.376476 + 0.926426i \(0.622864\pi\)
\(240\) −2.29844 −0.148364
\(241\) −22.5429 −1.45212 −0.726059 0.687632i \(-0.758650\pi\)
−0.726059 + 0.687632i \(0.758650\pi\)
\(242\) −0.546295 −0.0351172
\(243\) 1.00000 0.0641500
\(244\) −22.3180 −1.42877
\(245\) −1.00000 −0.0638877
\(246\) −6.00000 −0.382546
\(247\) −36.6554 −2.33233
\(248\) 4.72974 0.300339
\(249\) −7.75737 −0.491603
\(250\) 0.546295 0.0345507
\(251\) 11.4474 0.722554 0.361277 0.932458i \(-0.382341\pi\)
0.361277 + 0.932458i \(0.382341\pi\)
\(252\) 1.70156 0.107188
\(253\) 3.52790 0.221797
\(254\) 2.98014 0.186990
\(255\) −2.70156 −0.169178
\(256\) −2.89531 −0.180957
\(257\) −28.6157 −1.78500 −0.892498 0.451051i \(-0.851049\pi\)
−0.892498 + 0.451051i \(0.851049\pi\)
\(258\) −5.16531 −0.321578
\(259\) −6.75362 −0.419649
\(260\) 7.77348 0.482091
\(261\) 0.133124 0.00824015
\(262\) 11.5314 0.712414
\(263\) −8.67794 −0.535105 −0.267552 0.963543i \(-0.586215\pi\)
−0.267552 + 0.963543i \(0.586215\pi\)
\(264\) −2.02214 −0.124454
\(265\) −11.8384 −0.727230
\(266\) −4.38326 −0.268755
\(267\) 13.3741 0.818482
\(268\) −10.9149 −0.666735
\(269\) 29.8911 1.82249 0.911245 0.411864i \(-0.135122\pi\)
0.911245 + 0.411864i \(0.135122\pi\)
\(270\) 0.546295 0.0332464
\(271\) 20.9678 1.27370 0.636852 0.770986i \(-0.280236\pi\)
0.636852 + 0.770986i \(0.280236\pi\)
\(272\) 6.20937 0.376499
\(273\) −4.56844 −0.276494
\(274\) 1.85911 0.112313
\(275\) −1.00000 −0.0603023
\(276\) 6.00295 0.361335
\(277\) −13.7137 −0.823974 −0.411987 0.911190i \(-0.635165\pi\)
−0.411987 + 0.911190i \(0.635165\pi\)
\(278\) −11.6038 −0.695952
\(279\) 2.33897 0.140031
\(280\) 2.02214 0.120846
\(281\) 4.91036 0.292927 0.146464 0.989216i \(-0.453211\pi\)
0.146464 + 0.989216i \(0.453211\pi\)
\(282\) 0.354822 0.0211294
\(283\) −25.4788 −1.51456 −0.757279 0.653092i \(-0.773472\pi\)
−0.757279 + 0.653092i \(0.773472\pi\)
\(284\) 4.37036 0.259333
\(285\) 8.02362 0.475278
\(286\) 2.49571 0.147575
\(287\) −10.9831 −0.648311
\(288\) −5.29991 −0.312300
\(289\) −9.70156 −0.570680
\(290\) 0.0727248 0.00427055
\(291\) −1.78581 −0.104686
\(292\) −21.3376 −1.24869
\(293\) −2.16907 −0.126718 −0.0633591 0.997991i \(-0.520181\pi\)
−0.0633591 + 0.997991i \(0.520181\pi\)
\(294\) −0.546295 −0.0318606
\(295\) 0.959466 0.0558622
\(296\) 13.6568 0.793784
\(297\) −1.00000 −0.0580259
\(298\) 6.28634 0.364158
\(299\) −16.1170 −0.932071
\(300\) −1.70156 −0.0982397
\(301\) −9.45518 −0.544987
\(302\) −4.14545 −0.238544
\(303\) −11.6326 −0.668275
\(304\) −18.4418 −1.05771
\(305\) −13.1162 −0.751032
\(306\) −1.47585 −0.0843687
\(307\) 12.3548 0.705127 0.352563 0.935788i \(-0.385310\pi\)
0.352563 + 0.935788i \(0.385310\pi\)
\(308\) −1.70156 −0.0969555
\(309\) −9.18893 −0.522740
\(310\) 1.27777 0.0725724
\(311\) 22.6073 1.28194 0.640972 0.767564i \(-0.278531\pi\)
0.640972 + 0.767564i \(0.278531\pi\)
\(312\) 9.23804 0.523001
\(313\) −34.2614 −1.93657 −0.968285 0.249848i \(-0.919620\pi\)
−0.968285 + 0.249848i \(0.919620\pi\)
\(314\) 7.02442 0.396411
\(315\) 1.00000 0.0563436
\(316\) −4.37036 −0.245852
\(317\) 8.28929 0.465573 0.232786 0.972528i \(-0.425216\pi\)
0.232786 + 0.972528i \(0.425216\pi\)
\(318\) −6.46728 −0.362667
\(319\) −0.133124 −0.00745350
\(320\) 1.70156 0.0951202
\(321\) −13.0199 −0.726698
\(322\) −1.92728 −0.107403
\(323\) −21.6763 −1.20610
\(324\) −1.70156 −0.0945312
\(325\) 4.56844 0.253411
\(326\) −12.3749 −0.685383
\(327\) 0.109506 0.00605570
\(328\) 22.2094 1.22631
\(329\) 0.649507 0.0358085
\(330\) −0.546295 −0.0300725
\(331\) 21.4267 1.17772 0.588860 0.808235i \(-0.299577\pi\)
0.588860 + 0.808235i \(0.299577\pi\)
\(332\) 13.1996 0.724425
\(333\) 6.75362 0.370096
\(334\) 1.75795 0.0961906
\(335\) −6.41464 −0.350469
\(336\) −2.29844 −0.125390
\(337\) −20.2585 −1.10355 −0.551775 0.833993i \(-0.686049\pi\)
−0.551775 + 0.833993i \(0.686049\pi\)
\(338\) −4.29968 −0.233872
\(339\) −3.83844 −0.208475
\(340\) 4.59688 0.249301
\(341\) −2.33897 −0.126662
\(342\) 4.38326 0.237020
\(343\) −1.00000 −0.0539949
\(344\) 19.1197 1.03087
\(345\) 3.52790 0.189936
\(346\) 9.94768 0.534791
\(347\) −23.9431 −1.28533 −0.642667 0.766146i \(-0.722172\pi\)
−0.642667 + 0.766146i \(0.722172\pi\)
\(348\) −0.226518 −0.0121427
\(349\) −25.5751 −1.36901 −0.684503 0.729010i \(-0.739981\pi\)
−0.684503 + 0.729010i \(0.739981\pi\)
\(350\) 0.546295 0.0292007
\(351\) 4.56844 0.243845
\(352\) 5.29991 0.282486
\(353\) 0.330628 0.0175976 0.00879878 0.999961i \(-0.497199\pi\)
0.00879878 + 0.999961i \(0.497199\pi\)
\(354\) 0.524151 0.0278583
\(355\) 2.56844 0.136319
\(356\) −22.7569 −1.20611
\(357\) −2.70156 −0.142982
\(358\) 2.59688 0.137249
\(359\) −23.6248 −1.24687 −0.623435 0.781875i \(-0.714263\pi\)
−0.623435 + 0.781875i \(0.714263\pi\)
\(360\) −2.02214 −0.106576
\(361\) 45.3784 2.38834
\(362\) −0.189737 −0.00997235
\(363\) 1.00000 0.0524864
\(364\) 7.77348 0.407441
\(365\) −12.5400 −0.656374
\(366\) −7.16531 −0.374537
\(367\) −22.3730 −1.16786 −0.583932 0.811803i \(-0.698486\pi\)
−0.583932 + 0.811803i \(0.698486\pi\)
\(368\) −8.10867 −0.422694
\(369\) 10.9831 0.571756
\(370\) 3.68947 0.191806
\(371\) −11.8384 −0.614621
\(372\) −3.97991 −0.206349
\(373\) −11.9922 −0.620934 −0.310467 0.950584i \(-0.600485\pi\)
−0.310467 + 0.950584i \(0.600485\pi\)
\(374\) 1.47585 0.0763143
\(375\) −1.00000 −0.0516398
\(376\) −1.31340 −0.0677333
\(377\) 0.608168 0.0313222
\(378\) 0.546295 0.0280984
\(379\) 18.2088 0.935323 0.467662 0.883908i \(-0.345097\pi\)
0.467662 + 0.883908i \(0.345097\pi\)
\(380\) −13.6527 −0.700368
\(381\) −5.45518 −0.279477
\(382\) −6.46696 −0.330879
\(383\) 22.3494 1.14200 0.571001 0.820949i \(-0.306555\pi\)
0.571001 + 0.820949i \(0.306555\pi\)
\(384\) 11.5294 0.588356
\(385\) −1.00000 −0.0509647
\(386\) 2.26625 0.115349
\(387\) 9.45518 0.480634
\(388\) 3.03866 0.154265
\(389\) 25.4863 1.29221 0.646103 0.763250i \(-0.276397\pi\)
0.646103 + 0.763250i \(0.276397\pi\)
\(390\) 2.49571 0.126375
\(391\) −9.53085 −0.481996
\(392\) 2.02214 0.102134
\(393\) −21.1084 −1.06478
\(394\) −10.5797 −0.532999
\(395\) −2.56844 −0.129232
\(396\) 1.70156 0.0855067
\(397\) 10.3548 0.519694 0.259847 0.965650i \(-0.416328\pi\)
0.259847 + 0.965650i \(0.416328\pi\)
\(398\) −7.24509 −0.363163
\(399\) 8.02362 0.401683
\(400\) 2.29844 0.114922
\(401\) 0.0939709 0.00469268 0.00234634 0.999997i \(-0.499253\pi\)
0.00234634 + 0.999997i \(0.499253\pi\)
\(402\) −3.50429 −0.174778
\(403\) 10.6855 0.532280
\(404\) 19.7936 0.984767
\(405\) −1.00000 −0.0496904
\(406\) 0.0727248 0.00360927
\(407\) −6.75362 −0.334764
\(408\) 5.46295 0.270456
\(409\) 2.90741 0.143762 0.0718811 0.997413i \(-0.477100\pi\)
0.0718811 + 0.997413i \(0.477100\pi\)
\(410\) 6.00000 0.296319
\(411\) −3.40312 −0.167864
\(412\) 15.6355 0.770308
\(413\) 0.959466 0.0472122
\(414\) 1.92728 0.0947204
\(415\) 7.75737 0.380794
\(416\) −24.2123 −1.18711
\(417\) 21.2410 1.04018
\(418\) −4.38326 −0.214392
\(419\) 16.2660 0.794645 0.397323 0.917679i \(-0.369939\pi\)
0.397323 + 0.917679i \(0.369939\pi\)
\(420\) −1.70156 −0.0830277
\(421\) −16.1047 −0.784894 −0.392447 0.919775i \(-0.628371\pi\)
−0.392447 + 0.919775i \(0.628371\pi\)
\(422\) −1.38759 −0.0675468
\(423\) −0.649507 −0.0315801
\(424\) 23.9390 1.16258
\(425\) 2.70156 0.131045
\(426\) 1.40312 0.0679816
\(427\) −13.1162 −0.634738
\(428\) 22.1541 1.07086
\(429\) −4.56844 −0.220566
\(430\) 5.16531 0.249094
\(431\) −19.3221 −0.930711 −0.465355 0.885124i \(-0.654073\pi\)
−0.465355 + 0.885124i \(0.654073\pi\)
\(432\) 2.29844 0.110584
\(433\) 37.0462 1.78033 0.890163 0.455643i \(-0.150591\pi\)
0.890163 + 0.455643i \(0.150591\pi\)
\(434\) 1.27777 0.0613348
\(435\) −0.133124 −0.00638279
\(436\) −0.186331 −0.00892366
\(437\) 28.3066 1.35409
\(438\) −6.85054 −0.327331
\(439\) 14.4750 0.690856 0.345428 0.938445i \(-0.387734\pi\)
0.345428 + 0.938445i \(0.387734\pi\)
\(440\) 2.02214 0.0964019
\(441\) 1.00000 0.0476190
\(442\) −6.74233 −0.320700
\(443\) −26.4146 −1.25500 −0.627499 0.778618i \(-0.715921\pi\)
−0.627499 + 0.778618i \(0.715921\pi\)
\(444\) −11.4917 −0.545372
\(445\) −13.3741 −0.633994
\(446\) −11.2050 −0.530574
\(447\) −11.5072 −0.544274
\(448\) 1.70156 0.0803913
\(449\) −13.7526 −0.649023 −0.324511 0.945882i \(-0.605200\pi\)
−0.324511 + 0.945882i \(0.605200\pi\)
\(450\) −0.546295 −0.0257526
\(451\) −10.9831 −0.517173
\(452\) 6.53134 0.307208
\(453\) 7.58830 0.356530
\(454\) −10.7139 −0.502828
\(455\) 4.56844 0.214172
\(456\) −16.2249 −0.759801
\(457\) −9.98473 −0.467066 −0.233533 0.972349i \(-0.575029\pi\)
−0.233533 + 0.972349i \(0.575029\pi\)
\(458\) −11.1570 −0.521331
\(459\) 2.70156 0.126098
\(460\) −6.00295 −0.279889
\(461\) 3.31911 0.154586 0.0772931 0.997008i \(-0.475372\pi\)
0.0772931 + 0.997008i \(0.475372\pi\)
\(462\) −0.546295 −0.0254159
\(463\) −20.1726 −0.937500 −0.468750 0.883331i \(-0.655295\pi\)
−0.468750 + 0.883331i \(0.655295\pi\)
\(464\) 0.305977 0.0142046
\(465\) −2.33897 −0.108467
\(466\) 9.46751 0.438574
\(467\) 3.69241 0.170864 0.0854322 0.996344i \(-0.472773\pi\)
0.0854322 + 0.996344i \(0.472773\pi\)
\(468\) −7.77348 −0.359329
\(469\) −6.41464 −0.296201
\(470\) −0.354822 −0.0163667
\(471\) −12.8583 −0.592479
\(472\) −1.94018 −0.0893039
\(473\) −9.45518 −0.434750
\(474\) −1.40312 −0.0644476
\(475\) −8.02362 −0.368149
\(476\) 4.59688 0.210697
\(477\) 11.8384 0.542045
\(478\) 6.35907 0.290857
\(479\) 41.0516 1.87569 0.937847 0.347049i \(-0.112816\pi\)
0.937847 + 0.347049i \(0.112816\pi\)
\(480\) 5.29991 0.241907
\(481\) 30.8535 1.40680
\(482\) 12.3151 0.560937
\(483\) 3.52790 0.160525
\(484\) −1.70156 −0.0773437
\(485\) 1.78581 0.0810893
\(486\) −0.546295 −0.0247804
\(487\) −8.08402 −0.366322 −0.183161 0.983083i \(-0.558633\pi\)
−0.183161 + 0.983083i \(0.558633\pi\)
\(488\) 26.5229 1.20063
\(489\) 22.6525 1.02438
\(490\) 0.546295 0.0246791
\(491\) −3.47348 −0.156756 −0.0783779 0.996924i \(-0.524974\pi\)
−0.0783779 + 0.996924i \(0.524974\pi\)
\(492\) −18.6884 −0.842538
\(493\) 0.359642 0.0161975
\(494\) 20.0247 0.900952
\(495\) 1.00000 0.0449467
\(496\) 5.37598 0.241389
\(497\) 2.56844 0.115210
\(498\) 4.23781 0.189901
\(499\) −29.0719 −1.30144 −0.650719 0.759319i \(-0.725532\pi\)
−0.650719 + 0.759319i \(0.725532\pi\)
\(500\) 1.70156 0.0760962
\(501\) −3.21795 −0.143767
\(502\) −6.25366 −0.279115
\(503\) 30.6109 1.36487 0.682435 0.730946i \(-0.260921\pi\)
0.682435 + 0.730946i \(0.260921\pi\)
\(504\) −2.02214 −0.0900734
\(505\) 11.6326 0.517643
\(506\) −1.92728 −0.0856778
\(507\) 7.87063 0.349547
\(508\) 9.28233 0.411837
\(509\) −0.696166 −0.0308570 −0.0154285 0.999881i \(-0.504911\pi\)
−0.0154285 + 0.999881i \(0.504911\pi\)
\(510\) 1.47585 0.0653517
\(511\) −12.5400 −0.554737
\(512\) −21.4771 −0.949161
\(513\) −8.02362 −0.354251
\(514\) 15.6326 0.689524
\(515\) 9.18893 0.404913
\(516\) −16.0886 −0.708260
\(517\) 0.649507 0.0285653
\(518\) 3.68947 0.162106
\(519\) −18.2094 −0.799303
\(520\) −9.23804 −0.405115
\(521\) 14.2142 0.622735 0.311368 0.950290i \(-0.399213\pi\)
0.311368 + 0.950290i \(0.399213\pi\)
\(522\) −0.0727248 −0.00318308
\(523\) 5.04830 0.220747 0.110373 0.993890i \(-0.464795\pi\)
0.110373 + 0.993890i \(0.464795\pi\)
\(524\) 35.9173 1.56906
\(525\) −1.00000 −0.0436436
\(526\) 4.74071 0.206705
\(527\) 6.31888 0.275255
\(528\) −2.29844 −0.100027
\(529\) −10.5539 −0.458865
\(530\) 6.46728 0.280921
\(531\) −0.959466 −0.0416373
\(532\) −13.6527 −0.591919
\(533\) 50.1755 2.17334
\(534\) −7.30621 −0.316170
\(535\) 13.0199 0.562898
\(536\) 12.9713 0.560276
\(537\) −4.75362 −0.205134
\(538\) −16.3293 −0.704008
\(539\) −1.00000 −0.0430730
\(540\) 1.70156 0.0732236
\(541\) 36.1154 1.55272 0.776361 0.630288i \(-0.217063\pi\)
0.776361 + 0.630288i \(0.217063\pi\)
\(542\) −11.4546 −0.492017
\(543\) 0.347316 0.0149048
\(544\) −14.3180 −0.613881
\(545\) −0.109506 −0.00469073
\(546\) 2.49571 0.106807
\(547\) 11.4213 0.488341 0.244171 0.969732i \(-0.421484\pi\)
0.244171 + 0.969732i \(0.421484\pi\)
\(548\) 5.79063 0.247363
\(549\) 13.1162 0.559786
\(550\) 0.546295 0.0232941
\(551\) −1.06813 −0.0455040
\(552\) −7.13393 −0.303640
\(553\) −2.56844 −0.109221
\(554\) 7.49170 0.318292
\(555\) −6.75362 −0.286675
\(556\) −36.1429 −1.53280
\(557\) 3.52733 0.149458 0.0747288 0.997204i \(-0.476191\pi\)
0.0747288 + 0.997204i \(0.476191\pi\)
\(558\) −1.27777 −0.0540922
\(559\) 43.1954 1.82697
\(560\) 2.29844 0.0971267
\(561\) −2.70156 −0.114060
\(562\) −2.68250 −0.113155
\(563\) −10.5400 −0.444208 −0.222104 0.975023i \(-0.571292\pi\)
−0.222104 + 0.975023i \(0.571292\pi\)
\(564\) 1.10518 0.0465363
\(565\) 3.83844 0.161484
\(566\) 13.9189 0.585056
\(567\) −1.00000 −0.0419961
\(568\) −5.19375 −0.217925
\(569\) −15.6173 −0.654712 −0.327356 0.944901i \(-0.606158\pi\)
−0.327356 + 0.944901i \(0.606158\pi\)
\(570\) −4.38326 −0.183595
\(571\) −9.94737 −0.416284 −0.208142 0.978099i \(-0.566742\pi\)
−0.208142 + 0.978099i \(0.566742\pi\)
\(572\) 7.77348 0.325026
\(573\) 11.8379 0.494534
\(574\) 6.00000 0.250435
\(575\) −3.52790 −0.147124
\(576\) −1.70156 −0.0708984
\(577\) 29.1030 1.21158 0.605788 0.795626i \(-0.292858\pi\)
0.605788 + 0.795626i \(0.292858\pi\)
\(578\) 5.29991 0.220447
\(579\) −4.14840 −0.172402
\(580\) 0.226518 0.00940566
\(581\) 7.75737 0.321830
\(582\) 0.975577 0.0404390
\(583\) −11.8384 −0.490298
\(584\) 25.3577 1.04931
\(585\) −4.56844 −0.188882
\(586\) 1.18495 0.0489498
\(587\) 12.7536 0.526398 0.263199 0.964742i \(-0.415222\pi\)
0.263199 + 0.964742i \(0.415222\pi\)
\(588\) −1.70156 −0.0701712
\(589\) −18.7670 −0.773282
\(590\) −0.524151 −0.0215790
\(591\) 19.3663 0.796625
\(592\) 15.5228 0.637982
\(593\) −3.91893 −0.160931 −0.0804656 0.996757i \(-0.525641\pi\)
−0.0804656 + 0.996757i \(0.525641\pi\)
\(594\) 0.546295 0.0224147
\(595\) 2.70156 0.110753
\(596\) 19.5803 0.802039
\(597\) 13.2622 0.542787
\(598\) 8.80464 0.360048
\(599\) −11.9420 −0.487936 −0.243968 0.969783i \(-0.578449\pi\)
−0.243968 + 0.969783i \(0.578449\pi\)
\(600\) 2.02214 0.0825537
\(601\) 16.6878 0.680710 0.340355 0.940297i \(-0.389453\pi\)
0.340355 + 0.940297i \(0.389453\pi\)
\(602\) 5.16531 0.210522
\(603\) 6.41464 0.261225
\(604\) −12.9120 −0.525381
\(605\) −1.00000 −0.0406558
\(606\) 6.35482 0.258147
\(607\) 4.72518 0.191789 0.0958946 0.995391i \(-0.469429\pi\)
0.0958946 + 0.995391i \(0.469429\pi\)
\(608\) 42.5245 1.72459
\(609\) −0.133124 −0.00539445
\(610\) 7.16531 0.290115
\(611\) −2.96723 −0.120041
\(612\) −4.59688 −0.185818
\(613\) −6.84938 −0.276644 −0.138322 0.990387i \(-0.544171\pi\)
−0.138322 + 0.990387i \(0.544171\pi\)
\(614\) −6.74937 −0.272383
\(615\) −10.9831 −0.442881
\(616\) 2.02214 0.0814745
\(617\) 21.0725 0.848347 0.424173 0.905581i \(-0.360565\pi\)
0.424173 + 0.905581i \(0.360565\pi\)
\(618\) 5.01986 0.201929
\(619\) −29.2209 −1.17449 −0.587243 0.809410i \(-0.699787\pi\)
−0.587243 + 0.809410i \(0.699787\pi\)
\(620\) 3.97991 0.159837
\(621\) −3.52790 −0.141570
\(622\) −12.3503 −0.495200
\(623\) −13.3741 −0.535822
\(624\) 10.5003 0.420347
\(625\) 1.00000 0.0400000
\(626\) 18.7168 0.748075
\(627\) 8.02362 0.320432
\(628\) 21.8792 0.873075
\(629\) 18.2453 0.727488
\(630\) −0.546295 −0.0217649
\(631\) −23.8201 −0.948265 −0.474132 0.880454i \(-0.657238\pi\)
−0.474132 + 0.880454i \(0.657238\pi\)
\(632\) 5.19375 0.206596
\(633\) 2.54000 0.100956
\(634\) −4.52839 −0.179846
\(635\) 5.45518 0.216482
\(636\) −20.1438 −0.798755
\(637\) 4.56844 0.181008
\(638\) 0.0727248 0.00287920
\(639\) −2.56844 −0.101606
\(640\) −11.5294 −0.455739
\(641\) −30.9179 −1.22118 −0.610591 0.791946i \(-0.709068\pi\)
−0.610591 + 0.791946i \(0.709068\pi\)
\(642\) 7.11268 0.280715
\(643\) 12.4224 0.489892 0.244946 0.969537i \(-0.421230\pi\)
0.244946 + 0.969537i \(0.421230\pi\)
\(644\) −6.00295 −0.236549
\(645\) −9.45518 −0.372297
\(646\) 11.8416 0.465903
\(647\) 31.6924 1.24596 0.622979 0.782239i \(-0.285922\pi\)
0.622979 + 0.782239i \(0.285922\pi\)
\(648\) 2.02214 0.0794373
\(649\) 0.959466 0.0376623
\(650\) −2.49571 −0.0978899
\(651\) −2.33897 −0.0916715
\(652\) −38.5446 −1.50952
\(653\) 25.4037 0.994124 0.497062 0.867715i \(-0.334412\pi\)
0.497062 + 0.867715i \(0.334412\pi\)
\(654\) −0.0598226 −0.00233925
\(655\) 21.1084 0.824775
\(656\) 25.2439 0.985610
\(657\) 12.5400 0.489232
\(658\) −0.354822 −0.0138324
\(659\) −17.1004 −0.666135 −0.333068 0.942903i \(-0.608084\pi\)
−0.333068 + 0.942903i \(0.608084\pi\)
\(660\) −1.70156 −0.0662332
\(661\) 17.8675 0.694963 0.347482 0.937687i \(-0.387037\pi\)
0.347482 + 0.937687i \(0.387037\pi\)
\(662\) −11.7053 −0.454940
\(663\) 12.3419 0.479320
\(664\) −15.6865 −0.608755
\(665\) −8.02362 −0.311143
\(666\) −3.68947 −0.142964
\(667\) −0.469648 −0.0181848
\(668\) 5.47553 0.211855
\(669\) 20.5110 0.793001
\(670\) 3.50429 0.135382
\(671\) −13.1162 −0.506346
\(672\) 5.29991 0.204449
\(673\) −30.7973 −1.18715 −0.593575 0.804779i \(-0.702284\pi\)
−0.593575 + 0.804779i \(0.702284\pi\)
\(674\) 11.0671 0.426289
\(675\) 1.00000 0.0384900
\(676\) −13.3924 −0.515091
\(677\) 1.54800 0.0594944 0.0297472 0.999557i \(-0.490530\pi\)
0.0297472 + 0.999557i \(0.490530\pi\)
\(678\) 2.09692 0.0805317
\(679\) 1.78581 0.0685330
\(680\) −5.46295 −0.209494
\(681\) 19.6119 0.751530
\(682\) 1.27777 0.0489283
\(683\) −28.2767 −1.08198 −0.540989 0.841030i \(-0.681950\pi\)
−0.540989 + 0.841030i \(0.681950\pi\)
\(684\) 13.6527 0.522023
\(685\) 3.40312 0.130027
\(686\) 0.546295 0.0208576
\(687\) 20.4230 0.779186
\(688\) 21.7321 0.828530
\(689\) 54.0832 2.06041
\(690\) −1.92728 −0.0733701
\(691\) −11.4061 −0.433907 −0.216954 0.976182i \(-0.569612\pi\)
−0.216954 + 0.976182i \(0.569612\pi\)
\(692\) 30.9844 1.17785
\(693\) 1.00000 0.0379869
\(694\) 13.0800 0.496510
\(695\) −21.2410 −0.805717
\(696\) 0.269195 0.0102038
\(697\) 29.6715 1.12389
\(698\) 13.9716 0.528831
\(699\) −17.3304 −0.655496
\(700\) 1.70156 0.0643130
\(701\) −4.13312 −0.156106 −0.0780530 0.996949i \(-0.524870\pi\)
−0.0780530 + 0.996949i \(0.524870\pi\)
\(702\) −2.49571 −0.0941946
\(703\) −54.1884 −2.04376
\(704\) 1.70156 0.0641300
\(705\) 0.649507 0.0244619
\(706\) −0.180620 −0.00679774
\(707\) 11.6326 0.437489
\(708\) 1.63259 0.0613565
\(709\) 38.5867 1.44915 0.724576 0.689195i \(-0.242036\pi\)
0.724576 + 0.689195i \(0.242036\pi\)
\(710\) −1.40312 −0.0526583
\(711\) 2.56844 0.0963240
\(712\) 27.0444 1.01353
\(713\) −8.25167 −0.309027
\(714\) 1.47585 0.0552323
\(715\) 4.56844 0.170850
\(716\) 8.08857 0.302284
\(717\) −11.6404 −0.434717
\(718\) 12.9061 0.481652
\(719\) −13.4920 −0.503165 −0.251583 0.967836i \(-0.580951\pi\)
−0.251583 + 0.967836i \(0.580951\pi\)
\(720\) −2.29844 −0.0856577
\(721\) 9.18893 0.342214
\(722\) −24.7900 −0.922588
\(723\) −22.5429 −0.838381
\(724\) −0.590980 −0.0219636
\(725\) 0.133124 0.00494409
\(726\) −0.546295 −0.0202749
\(727\) −15.0923 −0.559743 −0.279872 0.960037i \(-0.590292\pi\)
−0.279872 + 0.960037i \(0.590292\pi\)
\(728\) −9.23804 −0.342385
\(729\) 1.00000 0.0370370
\(730\) 6.85054 0.253550
\(731\) 25.5438 0.944770
\(732\) −22.3180 −0.824899
\(733\) 5.07134 0.187314 0.0936572 0.995605i \(-0.470144\pi\)
0.0936572 + 0.995605i \(0.470144\pi\)
\(734\) 12.2223 0.451132
\(735\) −1.00000 −0.0368856
\(736\) 18.6976 0.689202
\(737\) −6.41464 −0.236286
\(738\) −6.00000 −0.220863
\(739\) −22.7021 −0.835112 −0.417556 0.908651i \(-0.637113\pi\)
−0.417556 + 0.908651i \(0.637113\pi\)
\(740\) 11.4917 0.422443
\(741\) −36.6554 −1.34657
\(742\) 6.46728 0.237421
\(743\) 29.2727 1.07391 0.536955 0.843611i \(-0.319574\pi\)
0.536955 + 0.843611i \(0.319574\pi\)
\(744\) 4.72974 0.173401
\(745\) 11.5072 0.421592
\(746\) 6.55129 0.239860
\(747\) −7.75737 −0.283827
\(748\) 4.59688 0.168078
\(749\) 13.0199 0.475735
\(750\) 0.546295 0.0199479
\(751\) 7.65211 0.279229 0.139615 0.990206i \(-0.455414\pi\)
0.139615 + 0.990206i \(0.455414\pi\)
\(752\) −1.49285 −0.0544387
\(753\) 11.4474 0.417167
\(754\) −0.332239 −0.0120994
\(755\) −7.58830 −0.276167
\(756\) 1.70156 0.0618852
\(757\) −20.9576 −0.761717 −0.380858 0.924633i \(-0.624371\pi\)
−0.380858 + 0.924633i \(0.624371\pi\)
\(758\) −9.94737 −0.361305
\(759\) 3.52790 0.128055
\(760\) 16.2249 0.588539
\(761\) −8.16509 −0.295984 −0.147992 0.988989i \(-0.547281\pi\)
−0.147992 + 0.988989i \(0.547281\pi\)
\(762\) 2.98014 0.107959
\(763\) −0.109506 −0.00396439
\(764\) −20.1429 −0.728743
\(765\) −2.70156 −0.0976752
\(766\) −12.2094 −0.441143
\(767\) −4.38326 −0.158270
\(768\) −2.89531 −0.104476
\(769\) −46.3583 −1.67172 −0.835862 0.548939i \(-0.815032\pi\)
−0.835862 + 0.548939i \(0.815032\pi\)
\(770\) 0.546295 0.0196871
\(771\) −28.6157 −1.03057
\(772\) 7.05876 0.254050
\(773\) −24.2169 −0.871021 −0.435510 0.900184i \(-0.643432\pi\)
−0.435510 + 0.900184i \(0.643432\pi\)
\(774\) −5.16531 −0.185663
\(775\) 2.33897 0.0840184
\(776\) −3.61116 −0.129633
\(777\) −6.75362 −0.242285
\(778\) −13.9230 −0.499165
\(779\) −88.1241 −3.15737
\(780\) 7.77348 0.278335
\(781\) 2.56844 0.0919060
\(782\) 5.20665 0.186190
\(783\) 0.133124 0.00475745
\(784\) 2.29844 0.0820871
\(785\) 12.8583 0.458933
\(786\) 11.5314 0.411312
\(787\) 47.1916 1.68220 0.841100 0.540880i \(-0.181909\pi\)
0.841100 + 0.540880i \(0.181909\pi\)
\(788\) −32.9530 −1.17390
\(789\) −8.67794 −0.308943
\(790\) 1.40312 0.0499209
\(791\) 3.83844 0.136479
\(792\) −2.02214 −0.0718537
\(793\) 59.9206 2.12784
\(794\) −5.65678 −0.200752
\(795\) −11.8384 −0.419866
\(796\) −22.5665 −0.799849
\(797\) −6.97474 −0.247058 −0.123529 0.992341i \(-0.539421\pi\)
−0.123529 + 0.992341i \(0.539421\pi\)
\(798\) −4.38326 −0.155166
\(799\) −1.75468 −0.0620763
\(800\) −5.29991 −0.187380
\(801\) 13.3741 0.472551
\(802\) −0.0513358 −0.00181273
\(803\) −12.5400 −0.442527
\(804\) −10.9149 −0.384939
\(805\) −3.52790 −0.124342
\(806\) −5.83740 −0.205614
\(807\) 29.8911 1.05222
\(808\) −23.5228 −0.827528
\(809\) −39.1428 −1.37619 −0.688093 0.725622i \(-0.741552\pi\)
−0.688093 + 0.725622i \(0.741552\pi\)
\(810\) 0.546295 0.0191948
\(811\) −22.1116 −0.776444 −0.388222 0.921566i \(-0.626911\pi\)
−0.388222 + 0.921566i \(0.626911\pi\)
\(812\) 0.226518 0.00794923
\(813\) 20.9678 0.735373
\(814\) 3.68947 0.129316
\(815\) −22.6525 −0.793481
\(816\) 6.20937 0.217372
\(817\) −75.8647 −2.65417
\(818\) −1.58830 −0.0555337
\(819\) −4.56844 −0.159634
\(820\) 18.6884 0.652627
\(821\) −49.1868 −1.71663 −0.858316 0.513122i \(-0.828489\pi\)
−0.858316 + 0.513122i \(0.828489\pi\)
\(822\) 1.85911 0.0648439
\(823\) −4.04853 −0.141123 −0.0705615 0.997507i \(-0.522479\pi\)
−0.0705615 + 0.997507i \(0.522479\pi\)
\(824\) −18.5813 −0.647312
\(825\) −1.00000 −0.0348155
\(826\) −0.524151 −0.0182375
\(827\) 20.5217 0.713610 0.356805 0.934179i \(-0.383866\pi\)
0.356805 + 0.934179i \(0.383866\pi\)
\(828\) 6.00295 0.208617
\(829\) 22.7665 0.790714 0.395357 0.918528i \(-0.370621\pi\)
0.395357 + 0.918528i \(0.370621\pi\)
\(830\) −4.23781 −0.147097
\(831\) −13.7137 −0.475722
\(832\) −7.77348 −0.269497
\(833\) 2.70156 0.0936036
\(834\) −11.6038 −0.401808
\(835\) 3.21795 0.111362
\(836\) −13.6527 −0.472188
\(837\) 2.33897 0.0808467
\(838\) −8.88602 −0.306963
\(839\) 9.06473 0.312949 0.156475 0.987682i \(-0.449987\pi\)
0.156475 + 0.987682i \(0.449987\pi\)
\(840\) 2.02214 0.0697706
\(841\) −28.9823 −0.999389
\(842\) 8.79790 0.303196
\(843\) 4.91036 0.169122
\(844\) −4.32197 −0.148768
\(845\) −7.87063 −0.270758
\(846\) 0.354822 0.0121990
\(847\) −1.00000 −0.0343604
\(848\) 27.2099 0.934392
\(849\) −25.4788 −0.874430
\(850\) −1.47585 −0.0506212
\(851\) −23.8261 −0.816749
\(852\) 4.37036 0.149726
\(853\) −13.0854 −0.448035 −0.224018 0.974585i \(-0.571917\pi\)
−0.224018 + 0.974585i \(0.571917\pi\)
\(854\) 7.16531 0.245192
\(855\) 8.02362 0.274402
\(856\) −26.3280 −0.899874
\(857\) 38.4589 1.31373 0.656866 0.754007i \(-0.271882\pi\)
0.656866 + 0.754007i \(0.271882\pi\)
\(858\) 2.49571 0.0852023
\(859\) −0.435576 −0.0148617 −0.00743083 0.999972i \(-0.502365\pi\)
−0.00743083 + 0.999972i \(0.502365\pi\)
\(860\) 16.0886 0.548616
\(861\) −10.9831 −0.374302
\(862\) 10.5555 0.359523
\(863\) 38.7362 1.31860 0.659298 0.751882i \(-0.270854\pi\)
0.659298 + 0.751882i \(0.270854\pi\)
\(864\) −5.29991 −0.180307
\(865\) 18.2094 0.619137
\(866\) −20.2381 −0.687719
\(867\) −9.70156 −0.329482
\(868\) 3.97991 0.135087
\(869\) −2.56844 −0.0871283
\(870\) 0.0727248 0.00246560
\(871\) 29.3049 0.992959
\(872\) 0.221437 0.00749881
\(873\) −1.78581 −0.0604404
\(874\) −15.4637 −0.523068
\(875\) 1.00000 0.0338062
\(876\) −21.3376 −0.720930
\(877\) −34.8044 −1.17526 −0.587630 0.809130i \(-0.699939\pi\)
−0.587630 + 0.809130i \(0.699939\pi\)
\(878\) −7.90764 −0.266870
\(879\) −2.16907 −0.0731608
\(880\) 2.29844 0.0774803
\(881\) 28.8583 0.972261 0.486130 0.873886i \(-0.338408\pi\)
0.486130 + 0.873886i \(0.338408\pi\)
\(882\) −0.546295 −0.0183947
\(883\) −27.5988 −0.928772 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(884\) −21.0005 −0.706325
\(885\) 0.959466 0.0322521
\(886\) 14.4302 0.484791
\(887\) 34.0939 1.14476 0.572380 0.819988i \(-0.306020\pi\)
0.572380 + 0.819988i \(0.306020\pi\)
\(888\) 13.6568 0.458292
\(889\) 5.45518 0.182961
\(890\) 7.30621 0.244905
\(891\) −1.00000 −0.0335013
\(892\) −34.9007 −1.16856
\(893\) 5.21140 0.174393
\(894\) 6.28634 0.210247
\(895\) 4.75362 0.158896
\(896\) −11.5294 −0.385169
\(897\) −16.1170 −0.538131
\(898\) 7.51295 0.250710
\(899\) 0.311373 0.0103849
\(900\) −1.70156 −0.0567187
\(901\) 31.9823 1.06548
\(902\) 6.00000 0.199778
\(903\) −9.45518 −0.314649
\(904\) −7.76188 −0.258156
\(905\) −0.347316 −0.0115452
\(906\) −4.14545 −0.137723
\(907\) −26.7979 −0.889810 −0.444905 0.895578i \(-0.646763\pi\)
−0.444905 + 0.895578i \(0.646763\pi\)
\(908\) −33.3709 −1.10745
\(909\) −11.6326 −0.385829
\(910\) −2.49571 −0.0827321
\(911\) −27.2083 −0.901451 −0.450726 0.892663i \(-0.648835\pi\)
−0.450726 + 0.892663i \(0.648835\pi\)
\(912\) −18.4418 −0.610669
\(913\) 7.75737 0.256731
\(914\) 5.45460 0.180422
\(915\) −13.1162 −0.433608
\(916\) −34.7510 −1.14820
\(917\) 21.1084 0.697062
\(918\) −1.47585 −0.0487103
\(919\) 1.61134 0.0531533 0.0265767 0.999647i \(-0.491539\pi\)
0.0265767 + 0.999647i \(0.491539\pi\)
\(920\) 7.13393 0.235199
\(921\) 12.3548 0.407105
\(922\) −1.81321 −0.0597149
\(923\) −11.7338 −0.386221
\(924\) −1.70156 −0.0559773
\(925\) 6.75362 0.222058
\(926\) 11.0202 0.362146
\(927\) −9.18893 −0.301804
\(928\) −0.705544 −0.0231606
\(929\) 53.7396 1.76314 0.881570 0.472053i \(-0.156487\pi\)
0.881570 + 0.472053i \(0.156487\pi\)
\(930\) 1.27777 0.0418997
\(931\) −8.02362 −0.262963
\(932\) 29.4888 0.965936
\(933\) 22.6073 0.740131
\(934\) −2.01715 −0.0660030
\(935\) 2.70156 0.0883505
\(936\) 9.23804 0.301955
\(937\) −26.0601 −0.851348 −0.425674 0.904877i \(-0.639963\pi\)
−0.425674 + 0.904877i \(0.639963\pi\)
\(938\) 3.50429 0.114419
\(939\) −34.2614 −1.11808
\(940\) −1.10518 −0.0360469
\(941\) −57.6176 −1.87828 −0.939139 0.343537i \(-0.888375\pi\)
−0.939139 + 0.343537i \(0.888375\pi\)
\(942\) 7.02442 0.228868
\(943\) −38.7473 −1.26178
\(944\) −2.20527 −0.0717755
\(945\) 1.00000 0.0325300
\(946\) 5.16531 0.167939
\(947\) 58.3245 1.89529 0.947646 0.319323i \(-0.103455\pi\)
0.947646 + 0.319323i \(0.103455\pi\)
\(948\) −4.37036 −0.141943
\(949\) 57.2882 1.85965
\(950\) 4.38326 0.142212
\(951\) 8.28929 0.268799
\(952\) −5.46295 −0.177055
\(953\) 2.27670 0.0737496 0.0368748 0.999320i \(-0.488260\pi\)
0.0368748 + 0.999320i \(0.488260\pi\)
\(954\) −6.46728 −0.209386
\(955\) −11.8379 −0.383064
\(956\) 19.8068 0.640597
\(957\) −0.133124 −0.00430328
\(958\) −22.4263 −0.724559
\(959\) 3.40312 0.109893
\(960\) 1.70156 0.0549177
\(961\) −25.5292 −0.823523
\(962\) −16.8551 −0.543430
\(963\) −13.0199 −0.419559
\(964\) 38.3582 1.23544
\(965\) 4.14840 0.133542
\(966\) −1.92728 −0.0620091
\(967\) 51.7348 1.66368 0.831840 0.555016i \(-0.187288\pi\)
0.831840 + 0.555016i \(0.187288\pi\)
\(968\) 2.02214 0.0649942
\(969\) −21.6763 −0.696343
\(970\) −0.975577 −0.0313239
\(971\) −21.6846 −0.695893 −0.347947 0.937514i \(-0.613121\pi\)
−0.347947 + 0.937514i \(0.613121\pi\)
\(972\) −1.70156 −0.0545776
\(973\) −21.2410 −0.680955
\(974\) 4.41626 0.141506
\(975\) 4.56844 0.146307
\(976\) 30.1468 0.964975
\(977\) −15.7235 −0.503041 −0.251520 0.967852i \(-0.580931\pi\)
−0.251520 + 0.967852i \(0.580931\pi\)
\(978\) −12.3749 −0.395706
\(979\) −13.3741 −0.427438
\(980\) 1.70156 0.0543544
\(981\) 0.109506 0.00349626
\(982\) 1.89754 0.0605530
\(983\) 26.0134 0.829699 0.414849 0.909890i \(-0.363834\pi\)
0.414849 + 0.909890i \(0.363834\pi\)
\(984\) 22.2094 0.708009
\(985\) −19.3663 −0.617063
\(986\) −0.196471 −0.00625690
\(987\) 0.649507 0.0206740
\(988\) 62.3714 1.98430
\(989\) −33.3570 −1.06069
\(990\) −0.546295 −0.0173624
\(991\) 39.8932 1.26725 0.633624 0.773641i \(-0.281567\pi\)
0.633624 + 0.773641i \(0.281567\pi\)
\(992\) −12.3963 −0.393584
\(993\) 21.4267 0.679957
\(994\) −1.40312 −0.0445044
\(995\) −13.2622 −0.420441
\(996\) 13.1996 0.418247
\(997\) 10.9914 0.348102 0.174051 0.984737i \(-0.444314\pi\)
0.174051 + 0.984737i \(0.444314\pi\)
\(998\) 15.8818 0.502731
\(999\) 6.75362 0.213675
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 1155.2.a.u.1.2 4
3.2 odd 2 3465.2.a.bl.1.3 4
5.4 even 2 5775.2.a.bz.1.3 4
7.6 odd 2 8085.2.a.bn.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.2 4 1.1 even 1 trivial
3465.2.a.bl.1.3 4 3.2 odd 2
5775.2.a.bz.1.3 4 5.4 even 2
8085.2.a.bn.1.2 4 7.6 odd 2