Properties

Label 5775.2.a.bz.1.3
Level 5775
Weight 2
Character 5775.1
Self dual yes
Analytic conductor 46.114
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
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: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.546295\) of defining polynomial
Character \(\chi\) \(=\) 5775.1

$q$-expansion

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

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 \(-0.438131\pi\)
0.193144 + 0.981170i \(0.438131\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.70156 −0.850781
\(5\) 0 0
\(6\) −0.546295 −0.223024
\(7\) 1.00000 0.377964
\(8\) −2.02214 −0.714936
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 1.70156 0.491199
\(13\) −4.56844 −1.26706 −0.633528 0.773719i \(-0.718394\pi\)
−0.633528 + 0.773719i \(0.718394\pi\)
\(14\) 0.546295 0.146003
\(15\) 0 0
\(16\) 2.29844 0.574609
\(17\) −2.70156 −0.655225 −0.327613 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) 0.546295 0.128763
\(19\) −8.02362 −1.84074 −0.920372 0.391044i \(-0.872114\pi\)
−0.920372 + 0.391044i \(0.872114\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −0.546295 −0.116470
\(23\) 3.52790 0.735619 0.367809 0.929901i \(-0.380108\pi\)
0.367809 + 0.929901i \(0.380108\pi\)
\(24\) 2.02214 0.412768
\(25\) 0 0
\(26\) −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 0
\(31\) 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
\(35\) 0 0
\(36\) −1.70156 −0.283594
\(37\) −6.75362 −1.11029 −0.555144 0.831754i \(-0.687337\pi\)
−0.555144 + 0.831754i \(0.687337\pi\)
\(38\) −4.38326 −0.711059
\(39\) 4.56844 0.731536
\(40\) 0 0
\(41\) 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 \(-0.756295\pi\)
−0.720951 + 0.692986i \(0.756295\pi\)
\(44\) 1.70156 0.256520
\(45\) 0 0
\(46\) 1.92728 0.284161
\(47\) 0.649507 0.0947403 0.0473702 0.998877i \(-0.484916\pi\)
0.0473702 + 0.998877i \(0.484916\pi\)
\(48\) −2.29844 −0.331751
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.70156 0.378294
\(52\) 7.77348 1.07799
\(53\) −11.8384 −1.62613 −0.813067 0.582170i \(-0.802204\pi\)
−0.813067 + 0.582170i \(0.802204\pi\)
\(54\) −0.546295 −0.0743413
\(55\) 0 0
\(56\) −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 \(0.519893\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) 0.546295 0.0672442
\(67\) −6.41464 −0.783674 −0.391837 0.920035i \(-0.628160\pi\)
−0.391837 + 0.920035i \(0.628160\pi\)
\(68\) 4.59688 0.557453
\(69\) −3.52790 −0.424710
\(70\) 0 0
\(71\) −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 \(-0.762278\pi\)
−0.733848 + 0.679314i \(0.762278\pi\)
\(74\) −3.68947 −0.428892
\(75\) 0 0
\(76\) 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 \(0.453847\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 7.75737 0.851482 0.425741 0.904845i \(-0.360013\pi\)
0.425741 + 0.904845i \(0.360013\pi\)
\(84\) 1.70156 0.185656
\(85\) 0 0
\(86\) −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 0
\(91\) −4.56844 −0.478902
\(92\) −6.00295 −0.625851
\(93\) −2.33897 −0.242540
\(94\) 0.354822 0.0365971
\(95\) 0 0
\(96\) −5.29991 −0.540920
\(97\) 1.78581 0.181321 0.0906606 0.995882i \(-0.471102\pi\)
0.0906606 + 0.995882i \(0.471102\pi\)
\(98\) 0.546295 0.0551841
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −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 \(-0.350459\pi\)
0.452706 + 0.891660i \(0.350459\pi\)
\(104\) 9.23804 0.905864
\(105\) 0 0
\(106\) −6.46728 −0.628157
\(107\) 13.0199 1.25868 0.629339 0.777131i \(-0.283326\pi\)
0.629339 + 0.777131i \(0.283326\pi\)
\(108\) 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 0
\(111\) 6.75362 0.641025
\(112\) 2.29844 0.217182
\(113\) 3.83844 0.361090 0.180545 0.983567i \(-0.442214\pi\)
0.180545 + 0.983567i \(0.442214\pi\)
\(114\) 4.38326 0.410530
\(115\) 0 0
\(116\) −0.226518 −0.0210317
\(117\) −4.56844 −0.422352
\(118\) −0.524151 −0.0482520
\(119\) −2.70156 −0.247652
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 7.16531 0.648717
\(123\) −10.9831 −0.990311
\(124\) −3.97991 −0.357406
\(125\) 0 0
\(126\) 0.546295 0.0486678
\(127\) 5.45518 0.484069 0.242034 0.970268i \(-0.422185\pi\)
0.242034 + 0.970268i \(0.422185\pi\)
\(128\) −11.5294 −1.01906
\(129\) 9.45518 0.832482
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) 5.46295 0.468444
\(137\) 3.40312 0.290749 0.145374 0.989377i \(-0.453561\pi\)
0.145374 + 0.989377i \(0.453561\pi\)
\(138\) −1.92728 −0.164061
\(139\) 21.2410 1.80164 0.900818 0.434196i \(-0.142967\pi\)
0.900818 + 0.434196i \(0.142967\pi\)
\(140\) 0 0
\(141\) −0.649507 −0.0546984
\(142\) −1.40312 −0.117748
\(143\) 4.56844 0.382032
\(144\) 2.29844 0.191536
\(145\) 0 0
\(146\) −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 0
\(151\) 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
\(155\) 0 0
\(156\) −7.77348 −0.622377
\(157\) 12.8583 1.02620 0.513102 0.858328i \(-0.328496\pi\)
0.513102 + 0.858328i \(0.328496\pi\)
\(158\) 1.40312 0.111627
\(159\) 11.8384 0.938849
\(160\) 0 0
\(161\) 3.52790 0.278038
\(162\) 0.546295 0.0429210
\(163\) −22.6525 −1.77428 −0.887139 0.461503i \(-0.847310\pi\)
−0.887139 + 0.461503i \(0.847310\pi\)
\(164\) −18.6884 −1.45932
\(165\) 0 0
\(166\) 4.23781 0.328918
\(167\) 3.21795 0.249012 0.124506 0.992219i \(-0.460265\pi\)
0.124506 + 0.992219i \(0.460265\pi\)
\(168\) 2.02214 0.156012
\(169\) 7.87063 0.605433
\(170\) 0 0
\(171\) −8.02362 −0.613581
\(172\) 16.0886 1.22674
\(173\) 18.2094 1.38443 0.692216 0.721690i \(-0.256634\pi\)
0.692216 + 0.721690i \(0.256634\pi\)
\(174\) −0.0727248 −0.00551325
\(175\) 0 0
\(176\) −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 \(0.556850\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −1.27777 −0.0936905
\(187\) 2.70156 0.197558
\(188\) −1.10518 −0.0806033
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 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 \(-0.452297\pi\)
0.149304 + 0.988791i \(0.452297\pi\)
\(194\) 0.975577 0.0700424
\(195\) 0 0
\(196\) −1.70156 −0.121540
\(197\) −19.3663 −1.37979 −0.689897 0.723907i \(-0.742344\pi\)
−0.689897 + 0.723907i \(0.742344\pi\)
\(198\) −0.546295 −0.0388235
\(199\) 13.2622 0.940135 0.470067 0.882631i \(-0.344230\pi\)
0.470067 + 0.882631i \(0.344230\pi\)
\(200\) 0 0
\(201\) 6.41464 0.452454
\(202\) −6.35482 −0.447124
\(203\) 0.133124 0.00934345
\(204\) −4.59688 −0.321846
\(205\) 0 0
\(206\) 5.01986 0.349751
\(207\) 3.52790 0.245206
\(208\) −10.5003 −0.728063
\(209\) 8.02362 0.555005
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) 2.02214 0.137589
\(217\) 2.33897 0.158780
\(218\) 0.0598226 0.00405170
\(219\) 12.5400 0.847375
\(220\) 0 0
\(221\) 12.3419 0.830207
\(222\) 3.68947 0.247621
\(223\) −20.5110 −1.37352 −0.686759 0.726886i \(-0.740967\pi\)
−0.686759 + 0.726886i \(0.740967\pi\)
\(224\) 5.29991 0.354115
\(225\) 0 0
\(226\) 2.09692 0.139485
\(227\) −19.6119 −1.30169 −0.650844 0.759211i \(-0.725585\pi\)
−0.650844 + 0.759211i \(0.725585\pi\)
\(228\) −13.6527 −0.904171
\(229\) 20.4230 1.34959 0.674795 0.738006i \(-0.264232\pi\)
0.674795 + 0.738006i \(0.264232\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −0.269195 −0.0176735
\(233\) 17.3304 1.13535 0.567676 0.823252i \(-0.307843\pi\)
0.567676 + 0.823252i \(0.307843\pi\)
\(234\) −2.49571 −0.163150
\(235\) 0 0
\(236\) 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 \(0.622864\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 36.6554 2.33233
\(248\) −4.72974 −0.300339
\(249\) −7.75737 −0.491603
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) −2.89531 −0.180957
\(257\) 28.6157 1.78500 0.892498 0.451051i \(-0.148951\pi\)
0.892498 + 0.451051i \(0.148951\pi\)
\(258\) 5.16531 0.321578
\(259\) −6.75362 −0.419649
\(260\) 0 0
\(261\) 0.133124 0.00824015
\(262\) −11.5314 −0.712414
\(263\) 8.67794 0.535105 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(264\) −2.02214 −0.124454
\(265\) 0 0
\(266\) −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 0
\(271\) 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
\(275\) 0 0
\(276\) 6.00295 0.361335
\(277\) 13.7137 0.823974 0.411987 0.911190i \(-0.364835\pi\)
0.411987 + 0.911190i \(0.364835\pi\)
\(278\) 11.6038 0.695952
\(279\) 2.33897 0.140031
\(280\) 0 0
\(281\) 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 \(-0.226528\pi\)
0.757279 + 0.653092i \(0.226528\pi\)
\(284\) 4.37036 0.259333
\(285\) 0 0
\(286\) 2.49571 0.147575
\(287\) 10.9831 0.648311
\(288\) 5.29991 0.312300
\(289\) −9.70156 −0.570680
\(290\) 0 0
\(291\) −1.78581 −0.104686
\(292\) 21.3376 1.24869
\(293\) 2.16907 0.126718 0.0633591 0.997991i \(-0.479819\pi\)
0.0633591 + 0.997991i \(0.479819\pi\)
\(294\) −0.546295 −0.0318606
\(295\) 0 0
\(296\) 13.6568 0.793784
\(297\) 1.00000 0.0580259
\(298\) −6.28634 −0.364158
\(299\) −16.1170 −0.932071
\(300\) 0 0
\(301\) −9.45518 −0.544987
\(302\) 4.14545 0.238544
\(303\) 11.6326 0.668275
\(304\) −18.4418 −1.05771
\(305\) 0 0
\(306\) −1.47585 −0.0843687
\(307\) −12.3548 −0.705127 −0.352563 0.935788i \(-0.614690\pi\)
−0.352563 + 0.935788i \(0.614690\pi\)
\(308\) 1.70156 0.0969555
\(309\) −9.18893 −0.522740
\(310\) 0 0
\(311\) 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 \(-0.0803805\pi\)
0.968285 + 0.249848i \(0.0803805\pi\)
\(314\) 7.02442 0.396411
\(315\) 0 0
\(316\) −4.37036 −0.245852
\(317\) −8.28929 −0.465573 −0.232786 0.972528i \(-0.574784\pi\)
−0.232786 + 0.972528i \(0.574784\pi\)
\(318\) 6.46728 0.362667
\(319\) −0.133124 −0.00745350
\(320\) 0 0
\(321\) −13.0199 −0.726698
\(322\) 1.92728 0.107403
\(323\) 21.6763 1.20610
\(324\) −1.70156 −0.0945312
\(325\) 0 0
\(326\) −12.3749 −0.685383
\(327\) −0.109506 −0.00605570
\(328\) −22.2094 −1.22631
\(329\) 0.649507 0.0358085
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) −2.29844 −0.125390
\(337\) 20.2585 1.10355 0.551775 0.833993i \(-0.313951\pi\)
0.551775 + 0.833993i \(0.313951\pi\)
\(338\) 4.29968 0.233872
\(339\) −3.83844 −0.208475
\(340\) 0 0
\(341\) −2.33897 −0.126662
\(342\) −4.38326 −0.237020
\(343\) 1.00000 0.0539949
\(344\) 19.1197 1.03087
\(345\) 0 0
\(346\) 9.94768 0.534791
\(347\) 23.9431 1.28533 0.642667 0.766146i \(-0.277828\pi\)
0.642667 + 0.766146i \(0.277828\pi\)
\(348\) 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 0
\(351\) 4.56844 0.243845
\(352\) −5.29991 −0.282486
\(353\) −0.330628 −0.0175976 −0.00879878 0.999961i \(-0.502801\pi\)
−0.00879878 + 0.999961i \(0.502801\pi\)
\(354\) 0.524151 0.0278583
\(355\) 0 0
\(356\) −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 \(0.714263\pi\)
\(360\) 0 0
\(361\) 45.3784 2.38834
\(362\) 0.189737 0.00997235
\(363\) −1.00000 −0.0524864
\(364\) 7.77348 0.407441
\(365\) 0 0
\(366\) −7.16531 −0.374537
\(367\) 22.3730 1.16786 0.583932 0.811803i \(-0.301514\pi\)
0.583932 + 0.811803i \(0.301514\pi\)
\(368\) 8.10867 0.422694
\(369\) 10.9831 0.571756
\(370\) 0 0
\(371\) −11.8384 −0.614621
\(372\) 3.97991 0.206349
\(373\) 11.9922 0.620934 0.310467 0.950584i \(-0.399515\pi\)
0.310467 + 0.950584i \(0.399515\pi\)
\(374\) 1.47585 0.0763143
\(375\) 0 0
\(376\) −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 \(0.345097\pi\)
\(380\) 0 0
\(381\) −5.45518 −0.279477
\(382\) 6.46696 0.330879
\(383\) −22.3494 −1.14200 −0.571001 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(384\) 11.5294 0.588356
\(385\) 0 0
\(386\) 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 \(0.276397\pi\)
\(390\) 0 0
\(391\) −9.53085 −0.481996
\(392\) −2.02214 −0.102134
\(393\) 21.1084 1.06478
\(394\) −10.5797 −0.532999
\(395\) 0 0
\(396\) 1.70156 0.0855067
\(397\) −10.3548 −0.519694 −0.259847 0.965650i \(-0.583672\pi\)
−0.259847 + 0.965650i \(0.583672\pi\)
\(398\) 7.24509 0.363163
\(399\) 8.02362 0.401683
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 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 \(0.477100\pi\)
\(410\) 0 0
\(411\) −3.40312 −0.167864
\(412\) −15.6355 −0.770308
\(413\) −0.959466 −0.0472122
\(414\) 1.92728 0.0947204
\(415\) 0 0
\(416\) −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 \(0.369939\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) 1.40312 0.0679816
\(427\) 13.1162 0.634738
\(428\) −22.1541 −1.07086
\(429\) −4.56844 −0.220566
\(430\) 0 0
\(431\) −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 \(-0.849409\pi\)
−0.890163 + 0.455643i \(0.849409\pi\)
\(434\) 1.27777 0.0613348
\(435\) 0 0
\(436\) −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 \(0.387734\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 6.74233 0.320700
\(443\) 26.4146 1.25500 0.627499 0.778618i \(-0.284079\pi\)
0.627499 + 0.778618i \(0.284079\pi\)
\(444\) −11.4917 −0.545372
\(445\) 0 0
\(446\) −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 0
\(451\) −10.9831 −0.517173
\(452\) −6.53134 −0.307208
\(453\) −7.58830 −0.356530
\(454\) −10.7139 −0.502828
\(455\) 0 0
\(456\) −16.2249 −0.759801
\(457\) 9.98473 0.467066 0.233533 0.972349i \(-0.424971\pi\)
0.233533 + 0.972349i \(0.424971\pi\)
\(458\) 11.1570 0.521331
\(459\) 2.70156 0.126098
\(460\) 0 0
\(461\) 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 \(-0.344705\pi\)
0.468750 + 0.883331i \(0.344705\pi\)
\(464\) 0.305977 0.0142046
\(465\) 0 0
\(466\) 9.46751 0.438574
\(467\) −3.69241 −0.170864 −0.0854322 0.996344i \(-0.527227\pi\)
−0.0854322 + 0.996344i \(0.527227\pi\)
\(468\) 7.77348 0.359329
\(469\) −6.41464 −0.296201
\(470\) 0 0
\(471\) −12.8583 −0.592479
\(472\) 1.94018 0.0893039
\(473\) 9.45518 0.434750
\(474\) −1.40312 −0.0644476
\(475\) 0 0
\(476\) 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 \(0.112816\pi\)
\(480\) 0 0
\(481\) 30.8535 1.40680
\(482\) −12.3151 −0.560937
\(483\) −3.52790 −0.160525
\(484\) −1.70156 −0.0773437
\(485\) 0 0
\(486\) −0.546295 −0.0247804
\(487\) 8.08402 0.366322 0.183161 0.983083i \(-0.441367\pi\)
0.183161 + 0.983083i \(0.441367\pi\)
\(488\) −26.5229 −1.20063
\(489\) 22.6525 1.02438
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.725532\pi\)
\(500\) 0 0
\(501\) −3.21795 −0.143767
\(502\) 6.25366 0.279115
\(503\) −30.6109 −1.36487 −0.682435 0.730946i \(-0.739079\pi\)
−0.682435 + 0.730946i \(0.739079\pi\)
\(504\) −2.02214 −0.0900734
\(505\) 0 0
\(506\) −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 \(0.504911\pi\)
\(510\) 0 0
\(511\) −12.5400 −0.554737
\(512\) 21.4771 0.949161
\(513\) 8.02362 0.354251
\(514\) 15.6326 0.689524
\(515\) 0 0
\(516\) −16.0886 −0.708260
\(517\) −0.649507 −0.0285653
\(518\) −3.68947 −0.162106
\(519\) −18.2094 −0.799303
\(520\) 0 0
\(521\) 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 \(-0.535205\pi\)
−0.110373 + 0.993890i \(0.535205\pi\)
\(524\) 35.9173 1.56906
\(525\) 0 0
\(526\) 4.74071 0.206705
\(527\) −6.31888 −0.275255
\(528\) 2.29844 0.100027
\(529\) −10.5539 −0.458865
\(530\) 0 0
\(531\) −0.959466 −0.0416373
\(532\) 13.6527 0.591919
\(533\) −50.1755 −2.17334
\(534\) −7.30621 −0.316170
\(535\) 0 0
\(536\) 12.9713 0.560276
\(537\) 4.75362 0.205134
\(538\) 16.3293 0.704008
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 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 0
\(546\) 2.49571 0.106807
\(547\) −11.4213 −0.488341 −0.244171 0.969732i \(-0.578516\pi\)
−0.244171 + 0.969732i \(0.578516\pi\)
\(548\) −5.79063 −0.247363
\(549\) 13.1162 0.559786
\(550\) 0 0
\(551\) −1.06813 −0.0455040
\(552\) 7.13393 0.303640
\(553\) 2.56844 0.109221
\(554\) 7.49170 0.318292
\(555\) 0 0
\(556\) −36.1429 −1.53280
\(557\) −3.52733 −0.149458 −0.0747288 0.997204i \(-0.523809\pi\)
−0.0747288 + 0.997204i \(0.523809\pi\)
\(558\) 1.27777 0.0540922
\(559\) 43.1954 1.82697
\(560\) 0 0
\(561\) −2.70156 −0.114060
\(562\) 2.68250 0.113155
\(563\) 10.5400 0.444208 0.222104 0.975023i \(-0.428708\pi\)
0.222104 + 0.975023i \(0.428708\pi\)
\(564\) 1.10518 0.0465363
\(565\) 0 0
\(566\) 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 \(0.606158\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) −1.70156 −0.0708984
\(577\) −29.1030 −1.21158 −0.605788 0.795626i \(-0.707142\pi\)
−0.605788 + 0.795626i \(0.707142\pi\)
\(578\) −5.29991 −0.220447
\(579\) −4.14840 −0.172402
\(580\) 0 0
\(581\) 7.75737 0.321830
\(582\) −0.975577 −0.0404390
\(583\) 11.8384 0.490298
\(584\) 25.3577 1.04931
\(585\) 0 0
\(586\) 1.18495 0.0489498
\(587\) −12.7536 −0.526398 −0.263199 0.964742i \(-0.584778\pi\)
−0.263199 + 0.964742i \(0.584778\pi\)
\(588\) 1.70156 0.0701712
\(589\) −18.7670 −0.773282
\(590\) 0 0
\(591\) 19.3663 0.796625
\(592\) −15.5228 −0.637982
\(593\) 3.91893 0.160931 0.0804656 0.996757i \(-0.474359\pi\)
0.0804656 + 0.996757i \(0.474359\pi\)
\(594\) 0.546295 0.0224147
\(595\) 0 0
\(596\) 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 \(0.578449\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 6.35482 0.258147
\(607\) −4.72518 −0.191789 −0.0958946 0.995391i \(-0.530571\pi\)
−0.0958946 + 0.995391i \(0.530571\pi\)
\(608\) −42.5245 −1.72459
\(609\) −0.133124 −0.00539445
\(610\) 0 0
\(611\) −2.96723 −0.120041
\(612\) 4.59688 0.185818
\(613\) 6.84938 0.276644 0.138322 0.990387i \(-0.455829\pi\)
0.138322 + 0.990387i \(0.455829\pi\)
\(614\) −6.74937 −0.272383
\(615\) 0 0
\(616\) 2.02214 0.0814745
\(617\) −21.0725 −0.848347 −0.424173 0.905581i \(-0.639435\pi\)
−0.424173 + 0.905581i \(0.639435\pi\)
\(618\) −5.01986 −0.201929
\(619\) −29.2209 −1.17449 −0.587243 0.809410i \(-0.699787\pi\)
−0.587243 + 0.809410i \(0.699787\pi\)
\(620\) 0 0
\(621\) −3.52790 −0.141570
\(622\) 12.3503 0.495200
\(623\) 13.3741 0.535822
\(624\) 10.5003 0.420347
\(625\) 0 0
\(626\) 18.7168 0.748075
\(627\) −8.02362 −0.320432
\(628\) −21.8792 −0.873075
\(629\) 18.2453 0.727488
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) −20.1438 −0.798755
\(637\) −4.56844 −0.181008
\(638\) −0.0727248 −0.00287920
\(639\) −2.56844 −0.101606
\(640\) 0 0
\(641\) −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 \(-0.578770\pi\)
−0.244946 + 0.969537i \(0.578770\pi\)
\(644\) −6.00295 −0.236549
\(645\) 0 0
\(646\) 11.8416 0.465903
\(647\) −31.6924 −1.24596 −0.622979 0.782239i \(-0.714078\pi\)
−0.622979 + 0.782239i \(0.714078\pi\)
\(648\) −2.02214 −0.0794373
\(649\) 0.959466 0.0376623
\(650\) 0 0
\(651\) −2.33897 −0.0916715
\(652\) 38.5446 1.50952
\(653\) −25.4037 −0.994124 −0.497062 0.867715i \(-0.665588\pi\)
−0.497062 + 0.867715i \(0.665588\pi\)
\(654\) −0.0598226 −0.00233925
\(655\) 0 0
\(656\) 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 \(0.608084\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −3.68947 −0.142964
\(667\) 0.469648 0.0181848
\(668\) −5.47553 −0.211855
\(669\) 20.5110 0.793001
\(670\) 0 0
\(671\) −13.1162 −0.506346
\(672\) −5.29991 −0.204449
\(673\) 30.7973 1.18715 0.593575 0.804779i \(-0.297716\pi\)
0.593575 + 0.804779i \(0.297716\pi\)
\(674\) 11.0671 0.426289
\(675\) 0 0
\(676\) −13.3924 −0.515091
\(677\) −1.54800 −0.0594944 −0.0297472 0.999557i \(-0.509470\pi\)
−0.0297472 + 0.999557i \(0.509470\pi\)
\(678\) −2.09692 −0.0805317
\(679\) 1.78581 0.0685330
\(680\) 0 0
\(681\) 19.6119 0.751530
\(682\) −1.27777 −0.0489283
\(683\) 28.2767 1.08198 0.540989 0.841030i \(-0.318050\pi\)
0.540989 + 0.841030i \(0.318050\pi\)
\(684\) 13.6527 0.522023
\(685\) 0 0
\(686\) 0.546295 0.0208576
\(687\) −20.4230 −0.779186
\(688\) −21.7321 −0.828530
\(689\) 54.0832 2.06041
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 0.269195 0.0102038
\(697\) −29.6715 −1.12389
\(698\) −13.9716 −0.528831
\(699\) −17.3304 −0.655496
\(700\) 0 0
\(701\) −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 0
\(706\) −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 \(0.242036\pi\)
\(710\) 0 0
\(711\) 2.56844 0.0963240
\(712\) −27.0444 −1.01353
\(713\) 8.25167 0.309027
\(714\) 1.47585 0.0552323
\(715\) 0 0
\(716\) 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 \(0.580951\pi\)
\(720\) 0 0
\(721\) 9.18893 0.342214
\(722\) 24.7900 0.922588
\(723\) 22.5429 0.838381
\(724\) −0.590980 −0.0219636
\(725\) 0 0
\(726\) −0.546295 −0.0202749
\(727\) 15.0923 0.559743 0.279872 0.960037i \(-0.409708\pi\)
0.279872 + 0.960037i \(0.409708\pi\)
\(728\) 9.23804 0.342385
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 25.5438 0.944770
\(732\) 22.3180 0.824899
\(733\) −5.07134 −0.187314 −0.0936572 0.995605i \(-0.529856\pi\)
−0.0936572 + 0.995605i \(0.529856\pi\)
\(734\) 12.2223 0.451132
\(735\) 0 0
\(736\) 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 \(0.637113\pi\)
\(740\) 0 0
\(741\) −36.6554 −1.34657
\(742\) −6.46728 −0.237421
\(743\) −29.2727 −1.07391 −0.536955 0.843611i \(-0.680426\pi\)
−0.536955 + 0.843611i \(0.680426\pi\)
\(744\) 4.72974 0.173401
\(745\) 0 0
\(746\) 6.55129 0.239860
\(747\) 7.75737 0.283827
\(748\) −4.59688 −0.168078
\(749\) 13.0199 0.475735
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 1.70156 0.0618852
\(757\) 20.9576 0.761717 0.380858 0.924633i \(-0.375629\pi\)
0.380858 + 0.924633i \(0.375629\pi\)
\(758\) 9.94737 0.361305
\(759\) 3.52790 0.128055
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) −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 0
\(771\) −28.6157 −1.03057
\(772\) −7.05876 −0.254050
\(773\) 24.2169 0.871021 0.435510 0.900184i \(-0.356568\pi\)
0.435510 + 0.900184i \(0.356568\pi\)
\(774\) −5.16531 −0.185663
\(775\) 0 0
\(776\) −3.61116 −0.129633
\(777\) 6.75362 0.242285
\(778\) 13.9230 0.499165
\(779\) −88.1241 −3.15737
\(780\) 0 0
\(781\) 2.56844 0.0919060
\(782\) −5.20665 −0.186190
\(783\) −0.133124 −0.00475745
\(784\) 2.29844 0.0820871
\(785\) 0 0
\(786\) 11.5314 0.411312
\(787\) −47.1916 −1.68220 −0.841100 0.540880i \(-0.818091\pi\)
−0.841100 + 0.540880i \(0.818091\pi\)
\(788\) 32.9530 1.17390
\(789\) −8.67794 −0.308943
\(790\) 0 0
\(791\) 3.83844 0.136479
\(792\) 2.02214 0.0718537
\(793\) −59.9206 −2.12784
\(794\) −5.65678 −0.200752
\(795\) 0 0
\(796\) −22.5665 −0.799849
\(797\) 6.97474 0.247058 0.123529 0.992341i \(-0.460579\pi\)
0.123529 + 0.992341i \(0.460579\pi\)
\(798\) 4.38326 0.155166
\(799\) −1.75468 −0.0620763
\(800\) 0 0
\(801\) 13.3741 0.472551
\(802\) 0.0513358 0.00181273
\(803\) 12.5400 0.442527
\(804\) −10.9149 −0.384939
\(805\) 0 0
\(806\) −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 0
\(811\) −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
\(815\) 0 0
\(816\) 6.20937 0.217372
\(817\) 75.8647 2.65417
\(818\) 1.58830 0.0555337
\(819\) −4.56844 −0.159634
\(820\) 0 0
\(821\) −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 \(-0.477521\pi\)
0.0705615 + 0.997507i \(0.477521\pi\)
\(824\) −18.5813 −0.647312
\(825\) 0 0
\(826\) −0.524151 −0.0182375
\(827\) −20.5217 −0.713610 −0.356805 0.934179i \(-0.616134\pi\)
−0.356805 + 0.934179i \(0.616134\pi\)
\(828\) −6.00295 −0.208617
\(829\) 22.7665 0.790714 0.395357 0.918528i \(-0.370621\pi\)
0.395357 + 0.918528i \(0.370621\pi\)
\(830\) 0 0
\(831\) −13.7137 −0.475722
\(832\) 7.77348 0.269497
\(833\) −2.70156 −0.0936036
\(834\) −11.6038 −0.401808
\(835\) 0 0
\(836\) −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 \(0.449987\pi\)
\(840\) 0 0
\(841\) −28.9823 −0.999389
\(842\) −8.79790 −0.303196
\(843\) −4.91036 −0.169122
\(844\) −4.32197 −0.148768
\(845\) 0 0
\(846\) 0.354822 0.0121990
\(847\) 1.00000 0.0343604
\(848\) −27.2099 −0.934392
\(849\) −25.4788 −0.874430
\(850\) 0 0
\(851\) −23.8261 −0.816749
\(852\) −4.37036 −0.149726
\(853\) 13.0854 0.448035 0.224018 0.974585i \(-0.428083\pi\)
0.224018 + 0.974585i \(0.428083\pi\)
\(854\) 7.16531 0.245192
\(855\) 0 0
\(856\) −26.3280 −0.899874
\(857\) −38.4589 −1.31373 −0.656866 0.754007i \(-0.728118\pi\)
−0.656866 + 0.754007i \(0.728118\pi\)
\(858\) −2.49571 −0.0852023
\(859\) −0.435576 −0.0148617 −0.00743083 0.999972i \(-0.502365\pi\)
−0.00743083 + 0.999972i \(0.502365\pi\)
\(860\) 0 0
\(861\) −10.9831 −0.374302
\(862\) −10.5555 −0.359523
\(863\) −38.7362 −1.31860 −0.659298 0.751882i \(-0.729146\pi\)
−0.659298 + 0.751882i \(0.729146\pi\)
\(864\) −5.29991 −0.180307
\(865\) 0 0
\(866\) −20.2381 −0.687719
\(867\) 9.70156 0.329482
\(868\) −3.97991 −0.135087
\(869\) −2.56844 −0.0871283
\(870\) 0 0
\(871\) 29.3049 0.992959
\(872\) −0.221437 −0.00749881
\(873\) 1.78581 0.0604404
\(874\) −15.4637 −0.523068
\(875\) 0 0
\(876\) −21.3376 −0.720930
\(877\) 34.8044 1.17526 0.587630 0.809130i \(-0.300061\pi\)
0.587630 + 0.809130i \(0.300061\pi\)
\(878\) 7.90764 0.266870
\(879\) −2.16907 −0.0731608
\(880\) 0 0
\(881\) 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 \(-0.346275\pi\)
0.464386 + 0.885633i \(0.346275\pi\)
\(884\) −21.0005 −0.706325
\(885\) 0 0
\(886\) 14.4302 0.484791
\(887\) −34.0939 −1.14476 −0.572380 0.819988i \(-0.693980\pi\)
−0.572380 + 0.819988i \(0.693980\pi\)
\(888\) −13.6568 −0.458292
\(889\) 5.45518 0.182961
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 34.9007 1.16856
\(893\) −5.21140 −0.174393
\(894\) 6.28634 0.210247
\(895\) 0 0
\(896\) −11.5294 −0.385169
\(897\) 16.1170 0.538131
\(898\) −7.51295 −0.250710
\(899\) 0.311373 0.0103849
\(900\) 0 0
\(901\) 31.9823 1.06548
\(902\) −6.00000 −0.199778
\(903\) 9.45518 0.314649
\(904\) −7.76188 −0.258156
\(905\) 0 0
\(906\) −4.14545 −0.137723
\(907\) 26.7979 0.889810 0.444905 0.895578i \(-0.353237\pi\)
0.444905 + 0.895578i \(0.353237\pi\)
\(908\) 33.3709 1.10745
\(909\) −11.6326 −0.385829
\(910\) 0 0
\(911\) −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
\(915\) 0 0
\(916\) −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 \(0.491539\pi\)
\(920\) 0 0
\(921\) 12.3548 0.407105
\(922\) 1.81321 0.0597149
\(923\) 11.7338 0.386221
\(924\) −1.70156 −0.0559773
\(925\) 0 0
\(926\) 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 \(0.156487\pi\)
\(930\) 0 0
\(931\) −8.02362 −0.262963
\(932\) −29.4888 −0.965936
\(933\) −22.6073 −0.740131
\(934\) −2.01715 −0.0660030
\(935\) 0 0
\(936\) 9.23804 0.301955
\(937\) 26.0601 0.851348 0.425674 0.904877i \(-0.360037\pi\)
0.425674 + 0.904877i \(0.360037\pi\)
\(938\) −3.50429 −0.114419
\(939\) −34.2614 −1.11808
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 5.16531 0.167939
\(947\) −58.3245 −1.89529 −0.947646 0.319323i \(-0.896545\pi\)
−0.947646 + 0.319323i \(0.896545\pi\)
\(948\) 4.37036 0.141943
\(949\) 57.2882 1.85965
\(950\) 0 0
\(951\) 8.28929 0.268799
\(952\) 5.46295 0.177055
\(953\) −2.27670 −0.0737496 −0.0368748 0.999320i \(-0.511740\pi\)
−0.0368748 + 0.999320i \(0.511740\pi\)
\(954\) −6.46728 −0.209386
\(955\) 0 0
\(956\) 19.8068 0.640597
\(957\) 0.133124 0.00430328
\(958\) 22.4263 0.724559
\(959\) 3.40312 0.109893
\(960\) 0 0
\(961\) −25.5292 −0.823523
\(962\) 16.8551 0.543430
\(963\) 13.0199 0.419559
\(964\) 38.3582 1.23544
\(965\) 0 0
\(966\) −1.92728 −0.0620091
\(967\) −51.7348 −1.66368 −0.831840 0.555016i \(-0.812712\pi\)
−0.831840 + 0.555016i \(0.812712\pi\)
\(968\) −2.02214 −0.0649942
\(969\) −21.6763 −0.696343
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) 30.1468 0.964975
\(977\) 15.7235 0.503041 0.251520 0.967852i \(-0.419069\pi\)
0.251520 + 0.967852i \(0.419069\pi\)
\(978\) 12.3749 0.395706
\(979\) −13.3741 −0.427438
\(980\) 0 0
\(981\) 0.109506 0.00349626
\(982\) −1.89754 −0.0605530
\(983\) −26.0134 −0.829699 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(984\) 22.2094 0.708009
\(985\) 0 0
\(986\) −0.196471 −0.00625690
\(987\) −0.649507 −0.0206740
\(988\) −62.3714 −1.98430
\(989\) −33.3570 −1.06069
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) 13.1996 0.418247
\(997\) −10.9914 −0.348102 −0.174051 0.984737i \(-0.555686\pi\)
−0.174051 + 0.984737i \(0.555686\pi\)
\(998\) −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 5775.2.a.bz.1.3 4
5.4 even 2 1155.2.a.u.1.2 4
15.14 odd 2 3465.2.a.bl.1.3 4
35.34 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 5.4 even 2
3465.2.a.bl.1.3 4 15.14 odd 2
5775.2.a.bz.1.3 4 1.1 even 1 trivial
8085.2.a.bn.1.2 4 35.34 odd 2