Properties

Label 8016.2.a.w.1.3
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.674271\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.68509 q^{5} +2.23045 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.68509 q^{5} +2.23045 q^{7} +1.00000 q^{9} +3.03364 q^{11} -4.55392 q^{13} -1.68509 q^{15} -2.98814 q^{17} +2.14913 q^{19} +2.23045 q^{21} -1.24231 q^{23} -2.16046 q^{25} +1.00000 q^{27} -6.48215 q^{29} -0.905758 q^{31} +3.03364 q^{33} -3.75852 q^{35} +5.19406 q^{37} -4.55392 q^{39} +0.662614 q^{41} +10.7062 q^{43} -1.68509 q^{45} -6.58538 q^{47} -2.02508 q^{49} -2.98814 q^{51} -8.56261 q^{53} -5.11196 q^{55} +2.14913 q^{57} +0.576526 q^{59} -9.76943 q^{61} +2.23045 q^{63} +7.67378 q^{65} -4.66641 q^{67} -1.24231 q^{69} -2.31288 q^{71} +13.5937 q^{73} -2.16046 q^{75} +6.76638 q^{77} -6.77552 q^{79} +1.00000 q^{81} -11.3095 q^{83} +5.03530 q^{85} -6.48215 q^{87} -1.55582 q^{89} -10.1573 q^{91} -0.905758 q^{93} -3.62148 q^{95} -12.8855 q^{97} +3.03364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.68509 −0.753597 −0.376799 0.926295i \(-0.622975\pi\)
−0.376799 + 0.926295i \(0.622975\pi\)
\(6\) 0 0
\(7\) 2.23045 0.843032 0.421516 0.906821i \(-0.361498\pi\)
0.421516 + 0.906821i \(0.361498\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.03364 0.914676 0.457338 0.889293i \(-0.348803\pi\)
0.457338 + 0.889293i \(0.348803\pi\)
\(12\) 0 0
\(13\) −4.55392 −1.26303 −0.631514 0.775364i \(-0.717566\pi\)
−0.631514 + 0.775364i \(0.717566\pi\)
\(14\) 0 0
\(15\) −1.68509 −0.435089
\(16\) 0 0
\(17\) −2.98814 −0.724731 −0.362366 0.932036i \(-0.618031\pi\)
−0.362366 + 0.932036i \(0.618031\pi\)
\(18\) 0 0
\(19\) 2.14913 0.493043 0.246522 0.969137i \(-0.420712\pi\)
0.246522 + 0.969137i \(0.420712\pi\)
\(20\) 0 0
\(21\) 2.23045 0.486725
\(22\) 0 0
\(23\) −1.24231 −0.259039 −0.129520 0.991577i \(-0.541344\pi\)
−0.129520 + 0.991577i \(0.541344\pi\)
\(24\) 0 0
\(25\) −2.16046 −0.432091
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.48215 −1.20371 −0.601853 0.798607i \(-0.705571\pi\)
−0.601853 + 0.798607i \(0.705571\pi\)
\(30\) 0 0
\(31\) −0.905758 −0.162679 −0.0813394 0.996686i \(-0.525920\pi\)
−0.0813394 + 0.996686i \(0.525920\pi\)
\(32\) 0 0
\(33\) 3.03364 0.528088
\(34\) 0 0
\(35\) −3.75852 −0.635306
\(36\) 0 0
\(37\) 5.19406 0.853897 0.426949 0.904276i \(-0.359588\pi\)
0.426949 + 0.904276i \(0.359588\pi\)
\(38\) 0 0
\(39\) −4.55392 −0.729210
\(40\) 0 0
\(41\) 0.662614 0.103483 0.0517415 0.998661i \(-0.483523\pi\)
0.0517415 + 0.998661i \(0.483523\pi\)
\(42\) 0 0
\(43\) 10.7062 1.63268 0.816338 0.577574i \(-0.196000\pi\)
0.816338 + 0.577574i \(0.196000\pi\)
\(44\) 0 0
\(45\) −1.68509 −0.251199
\(46\) 0 0
\(47\) −6.58538 −0.960576 −0.480288 0.877111i \(-0.659468\pi\)
−0.480288 + 0.877111i \(0.659468\pi\)
\(48\) 0 0
\(49\) −2.02508 −0.289297
\(50\) 0 0
\(51\) −2.98814 −0.418424
\(52\) 0 0
\(53\) −8.56261 −1.17616 −0.588082 0.808801i \(-0.700117\pi\)
−0.588082 + 0.808801i \(0.700117\pi\)
\(54\) 0 0
\(55\) −5.11196 −0.689297
\(56\) 0 0
\(57\) 2.14913 0.284659
\(58\) 0 0
\(59\) 0.576526 0.0750573 0.0375286 0.999296i \(-0.488051\pi\)
0.0375286 + 0.999296i \(0.488051\pi\)
\(60\) 0 0
\(61\) −9.76943 −1.25085 −0.625424 0.780285i \(-0.715074\pi\)
−0.625424 + 0.780285i \(0.715074\pi\)
\(62\) 0 0
\(63\) 2.23045 0.281011
\(64\) 0 0
\(65\) 7.67378 0.951815
\(66\) 0 0
\(67\) −4.66641 −0.570093 −0.285046 0.958514i \(-0.592009\pi\)
−0.285046 + 0.958514i \(0.592009\pi\)
\(68\) 0 0
\(69\) −1.24231 −0.149557
\(70\) 0 0
\(71\) −2.31288 −0.274489 −0.137244 0.990537i \(-0.543825\pi\)
−0.137244 + 0.990537i \(0.543825\pi\)
\(72\) 0 0
\(73\) 13.5937 1.59102 0.795511 0.605939i \(-0.207203\pi\)
0.795511 + 0.605939i \(0.207203\pi\)
\(74\) 0 0
\(75\) −2.16046 −0.249468
\(76\) 0 0
\(77\) 6.76638 0.771101
\(78\) 0 0
\(79\) −6.77552 −0.762305 −0.381153 0.924512i \(-0.624473\pi\)
−0.381153 + 0.924512i \(0.624473\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.3095 −1.24138 −0.620692 0.784054i \(-0.713148\pi\)
−0.620692 + 0.784054i \(0.713148\pi\)
\(84\) 0 0
\(85\) 5.03530 0.546155
\(86\) 0 0
\(87\) −6.48215 −0.694960
\(88\) 0 0
\(89\) −1.55582 −0.164916 −0.0824582 0.996595i \(-0.526277\pi\)
−0.0824582 + 0.996595i \(0.526277\pi\)
\(90\) 0 0
\(91\) −10.1573 −1.06477
\(92\) 0 0
\(93\) −0.905758 −0.0939227
\(94\) 0 0
\(95\) −3.62148 −0.371556
\(96\) 0 0
\(97\) −12.8855 −1.30833 −0.654164 0.756352i \(-0.726980\pi\)
−0.654164 + 0.756352i \(0.726980\pi\)
\(98\) 0 0
\(99\) 3.03364 0.304892
\(100\) 0 0
\(101\) 3.21883 0.320285 0.160143 0.987094i \(-0.448805\pi\)
0.160143 + 0.987094i \(0.448805\pi\)
\(102\) 0 0
\(103\) 12.7950 1.26073 0.630365 0.776299i \(-0.282905\pi\)
0.630365 + 0.776299i \(0.282905\pi\)
\(104\) 0 0
\(105\) −3.75852 −0.366794
\(106\) 0 0
\(107\) 5.79714 0.560431 0.280216 0.959937i \(-0.409594\pi\)
0.280216 + 0.959937i \(0.409594\pi\)
\(108\) 0 0
\(109\) −11.0690 −1.06021 −0.530107 0.847931i \(-0.677848\pi\)
−0.530107 + 0.847931i \(0.677848\pi\)
\(110\) 0 0
\(111\) 5.19406 0.492998
\(112\) 0 0
\(113\) −12.1323 −1.14131 −0.570654 0.821191i \(-0.693310\pi\)
−0.570654 + 0.821191i \(0.693310\pi\)
\(114\) 0 0
\(115\) 2.09341 0.195211
\(116\) 0 0
\(117\) −4.55392 −0.421010
\(118\) 0 0
\(119\) −6.66491 −0.610972
\(120\) 0 0
\(121\) −1.79705 −0.163368
\(122\) 0 0
\(123\) 0.662614 0.0597459
\(124\) 0 0
\(125\) 12.0660 1.07922
\(126\) 0 0
\(127\) −16.7289 −1.48445 −0.742226 0.670150i \(-0.766230\pi\)
−0.742226 + 0.670150i \(0.766230\pi\)
\(128\) 0 0
\(129\) 10.7062 0.942626
\(130\) 0 0
\(131\) −9.52434 −0.832145 −0.416073 0.909331i \(-0.636594\pi\)
−0.416073 + 0.909331i \(0.636594\pi\)
\(132\) 0 0
\(133\) 4.79352 0.415651
\(134\) 0 0
\(135\) −1.68509 −0.145030
\(136\) 0 0
\(137\) 11.1055 0.948810 0.474405 0.880307i \(-0.342663\pi\)
0.474405 + 0.880307i \(0.342663\pi\)
\(138\) 0 0
\(139\) 3.18986 0.270561 0.135280 0.990807i \(-0.456807\pi\)
0.135280 + 0.990807i \(0.456807\pi\)
\(140\) 0 0
\(141\) −6.58538 −0.554589
\(142\) 0 0
\(143\) −13.8149 −1.15526
\(144\) 0 0
\(145\) 10.9230 0.907109
\(146\) 0 0
\(147\) −2.02508 −0.167026
\(148\) 0 0
\(149\) 8.87024 0.726678 0.363339 0.931657i \(-0.381637\pi\)
0.363339 + 0.931657i \(0.381637\pi\)
\(150\) 0 0
\(151\) 1.14248 0.0929738 0.0464869 0.998919i \(-0.485197\pi\)
0.0464869 + 0.998919i \(0.485197\pi\)
\(152\) 0 0
\(153\) −2.98814 −0.241577
\(154\) 0 0
\(155\) 1.52629 0.122594
\(156\) 0 0
\(157\) 9.53736 0.761164 0.380582 0.924747i \(-0.375724\pi\)
0.380582 + 0.924747i \(0.375724\pi\)
\(158\) 0 0
\(159\) −8.56261 −0.679059
\(160\) 0 0
\(161\) −2.77091 −0.218379
\(162\) 0 0
\(163\) −20.9850 −1.64367 −0.821836 0.569724i \(-0.807050\pi\)
−0.821836 + 0.569724i \(0.807050\pi\)
\(164\) 0 0
\(165\) −5.11196 −0.397966
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 7.73814 0.595242
\(170\) 0 0
\(171\) 2.14913 0.164348
\(172\) 0 0
\(173\) −9.53795 −0.725157 −0.362578 0.931953i \(-0.618103\pi\)
−0.362578 + 0.931953i \(0.618103\pi\)
\(174\) 0 0
\(175\) −4.81880 −0.364267
\(176\) 0 0
\(177\) 0.576526 0.0433343
\(178\) 0 0
\(179\) 2.12282 0.158667 0.0793336 0.996848i \(-0.474721\pi\)
0.0793336 + 0.996848i \(0.474721\pi\)
\(180\) 0 0
\(181\) −8.06276 −0.599300 −0.299650 0.954049i \(-0.596870\pi\)
−0.299650 + 0.954049i \(0.596870\pi\)
\(182\) 0 0
\(183\) −9.76943 −0.722177
\(184\) 0 0
\(185\) −8.75247 −0.643495
\(186\) 0 0
\(187\) −9.06494 −0.662894
\(188\) 0 0
\(189\) 2.23045 0.162242
\(190\) 0 0
\(191\) 2.87699 0.208172 0.104086 0.994568i \(-0.466808\pi\)
0.104086 + 0.994568i \(0.466808\pi\)
\(192\) 0 0
\(193\) −26.0937 −1.87826 −0.939132 0.343557i \(-0.888368\pi\)
−0.939132 + 0.343557i \(0.888368\pi\)
\(194\) 0 0
\(195\) 7.67378 0.549531
\(196\) 0 0
\(197\) 24.0013 1.71003 0.855013 0.518607i \(-0.173549\pi\)
0.855013 + 0.518607i \(0.173549\pi\)
\(198\) 0 0
\(199\) 17.2330 1.22162 0.610809 0.791778i \(-0.290844\pi\)
0.610809 + 0.791778i \(0.290844\pi\)
\(200\) 0 0
\(201\) −4.66641 −0.329143
\(202\) 0 0
\(203\) −14.4581 −1.01476
\(204\) 0 0
\(205\) −1.11657 −0.0779845
\(206\) 0 0
\(207\) −1.24231 −0.0863465
\(208\) 0 0
\(209\) 6.51967 0.450975
\(210\) 0 0
\(211\) 28.7011 1.97587 0.987934 0.154876i \(-0.0494979\pi\)
0.987934 + 0.154876i \(0.0494979\pi\)
\(212\) 0 0
\(213\) −2.31288 −0.158476
\(214\) 0 0
\(215\) −18.0409 −1.23038
\(216\) 0 0
\(217\) −2.02025 −0.137143
\(218\) 0 0
\(219\) 13.5937 0.918577
\(220\) 0 0
\(221\) 13.6078 0.915356
\(222\) 0 0
\(223\) −13.2997 −0.890614 −0.445307 0.895378i \(-0.646905\pi\)
−0.445307 + 0.895378i \(0.646905\pi\)
\(224\) 0 0
\(225\) −2.16046 −0.144030
\(226\) 0 0
\(227\) −5.09741 −0.338327 −0.169164 0.985588i \(-0.554107\pi\)
−0.169164 + 0.985588i \(0.554107\pi\)
\(228\) 0 0
\(229\) 26.5563 1.75489 0.877444 0.479679i \(-0.159247\pi\)
0.877444 + 0.479679i \(0.159247\pi\)
\(230\) 0 0
\(231\) 6.76638 0.445195
\(232\) 0 0
\(233\) 28.9849 1.89886 0.949432 0.313972i \(-0.101660\pi\)
0.949432 + 0.313972i \(0.101660\pi\)
\(234\) 0 0
\(235\) 11.0970 0.723887
\(236\) 0 0
\(237\) −6.77552 −0.440117
\(238\) 0 0
\(239\) −15.1035 −0.976965 −0.488483 0.872574i \(-0.662449\pi\)
−0.488483 + 0.872574i \(0.662449\pi\)
\(240\) 0 0
\(241\) 19.7378 1.27142 0.635711 0.771927i \(-0.280707\pi\)
0.635711 + 0.771927i \(0.280707\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.41245 0.218013
\(246\) 0 0
\(247\) −9.78694 −0.622728
\(248\) 0 0
\(249\) −11.3095 −0.716714
\(250\) 0 0
\(251\) −11.0371 −0.696655 −0.348328 0.937373i \(-0.613250\pi\)
−0.348328 + 0.937373i \(0.613250\pi\)
\(252\) 0 0
\(253\) −3.76872 −0.236937
\(254\) 0 0
\(255\) 5.03530 0.315323
\(256\) 0 0
\(257\) −21.9098 −1.36669 −0.683347 0.730094i \(-0.739476\pi\)
−0.683347 + 0.730094i \(0.739476\pi\)
\(258\) 0 0
\(259\) 11.5851 0.719863
\(260\) 0 0
\(261\) −6.48215 −0.401235
\(262\) 0 0
\(263\) −20.7484 −1.27940 −0.639702 0.768623i \(-0.720942\pi\)
−0.639702 + 0.768623i \(0.720942\pi\)
\(264\) 0 0
\(265\) 14.4288 0.886354
\(266\) 0 0
\(267\) −1.55582 −0.0952145
\(268\) 0 0
\(269\) −9.59849 −0.585230 −0.292615 0.956230i \(-0.594525\pi\)
−0.292615 + 0.956230i \(0.594525\pi\)
\(270\) 0 0
\(271\) −8.78291 −0.533524 −0.266762 0.963762i \(-0.585954\pi\)
−0.266762 + 0.963762i \(0.585954\pi\)
\(272\) 0 0
\(273\) −10.1573 −0.614747
\(274\) 0 0
\(275\) −6.55404 −0.395224
\(276\) 0 0
\(277\) −10.9897 −0.660304 −0.330152 0.943928i \(-0.607100\pi\)
−0.330152 + 0.943928i \(0.607100\pi\)
\(278\) 0 0
\(279\) −0.905758 −0.0542263
\(280\) 0 0
\(281\) 1.74882 0.104326 0.0521629 0.998639i \(-0.483388\pi\)
0.0521629 + 0.998639i \(0.483388\pi\)
\(282\) 0 0
\(283\) −5.33783 −0.317301 −0.158650 0.987335i \(-0.550714\pi\)
−0.158650 + 0.987335i \(0.550714\pi\)
\(284\) 0 0
\(285\) −3.62148 −0.214518
\(286\) 0 0
\(287\) 1.47793 0.0872395
\(288\) 0 0
\(289\) −8.07100 −0.474765
\(290\) 0 0
\(291\) −12.8855 −0.755364
\(292\) 0 0
\(293\) −23.7683 −1.38856 −0.694280 0.719705i \(-0.744277\pi\)
−0.694280 + 0.719705i \(0.744277\pi\)
\(294\) 0 0
\(295\) −0.971500 −0.0565629
\(296\) 0 0
\(297\) 3.03364 0.176029
\(298\) 0 0
\(299\) 5.65737 0.327174
\(300\) 0 0
\(301\) 23.8796 1.37640
\(302\) 0 0
\(303\) 3.21883 0.184917
\(304\) 0 0
\(305\) 16.4624 0.942635
\(306\) 0 0
\(307\) 25.4316 1.45146 0.725728 0.687981i \(-0.241503\pi\)
0.725728 + 0.687981i \(0.241503\pi\)
\(308\) 0 0
\(309\) 12.7950 0.727883
\(310\) 0 0
\(311\) −34.0914 −1.93315 −0.966573 0.256392i \(-0.917466\pi\)
−0.966573 + 0.256392i \(0.917466\pi\)
\(312\) 0 0
\(313\) 5.72030 0.323330 0.161665 0.986846i \(-0.448314\pi\)
0.161665 + 0.986846i \(0.448314\pi\)
\(314\) 0 0
\(315\) −3.75852 −0.211769
\(316\) 0 0
\(317\) 1.84667 0.103719 0.0518596 0.998654i \(-0.483485\pi\)
0.0518596 + 0.998654i \(0.483485\pi\)
\(318\) 0 0
\(319\) −19.6645 −1.10100
\(320\) 0 0
\(321\) 5.79714 0.323565
\(322\) 0 0
\(323\) −6.42190 −0.357324
\(324\) 0 0
\(325\) 9.83854 0.545744
\(326\) 0 0
\(327\) −11.0690 −0.612115
\(328\) 0 0
\(329\) −14.6884 −0.809796
\(330\) 0 0
\(331\) 28.9536 1.59143 0.795716 0.605670i \(-0.207095\pi\)
0.795716 + 0.605670i \(0.207095\pi\)
\(332\) 0 0
\(333\) 5.19406 0.284632
\(334\) 0 0
\(335\) 7.86334 0.429620
\(336\) 0 0
\(337\) 2.43852 0.132835 0.0664173 0.997792i \(-0.478843\pi\)
0.0664173 + 0.997792i \(0.478843\pi\)
\(338\) 0 0
\(339\) −12.1323 −0.658934
\(340\) 0 0
\(341\) −2.74774 −0.148798
\(342\) 0 0
\(343\) −20.1300 −1.08692
\(344\) 0 0
\(345\) 2.09341 0.112705
\(346\) 0 0
\(347\) −4.29284 −0.230452 −0.115226 0.993339i \(-0.536759\pi\)
−0.115226 + 0.993339i \(0.536759\pi\)
\(348\) 0 0
\(349\) −22.7374 −1.21710 −0.608551 0.793515i \(-0.708249\pi\)
−0.608551 + 0.793515i \(0.708249\pi\)
\(350\) 0 0
\(351\) −4.55392 −0.243070
\(352\) 0 0
\(353\) −6.60400 −0.351496 −0.175748 0.984435i \(-0.556234\pi\)
−0.175748 + 0.984435i \(0.556234\pi\)
\(354\) 0 0
\(355\) 3.89743 0.206854
\(356\) 0 0
\(357\) −6.66491 −0.352745
\(358\) 0 0
\(359\) −34.3313 −1.81194 −0.905969 0.423343i \(-0.860856\pi\)
−0.905969 + 0.423343i \(0.860856\pi\)
\(360\) 0 0
\(361\) −14.3813 −0.756908
\(362\) 0 0
\(363\) −1.79705 −0.0943206
\(364\) 0 0
\(365\) −22.9067 −1.19899
\(366\) 0 0
\(367\) −12.3087 −0.642510 −0.321255 0.946993i \(-0.604105\pi\)
−0.321255 + 0.946993i \(0.604105\pi\)
\(368\) 0 0
\(369\) 0.662614 0.0344943
\(370\) 0 0
\(371\) −19.0985 −0.991544
\(372\) 0 0
\(373\) 10.6805 0.553013 0.276507 0.961012i \(-0.410823\pi\)
0.276507 + 0.961012i \(0.410823\pi\)
\(374\) 0 0
\(375\) 12.0660 0.623088
\(376\) 0 0
\(377\) 29.5192 1.52031
\(378\) 0 0
\(379\) 24.4104 1.25388 0.626938 0.779069i \(-0.284308\pi\)
0.626938 + 0.779069i \(0.284308\pi\)
\(380\) 0 0
\(381\) −16.7289 −0.857049
\(382\) 0 0
\(383\) 5.92490 0.302748 0.151374 0.988477i \(-0.451630\pi\)
0.151374 + 0.988477i \(0.451630\pi\)
\(384\) 0 0
\(385\) −11.4020 −0.581099
\(386\) 0 0
\(387\) 10.7062 0.544226
\(388\) 0 0
\(389\) −21.7011 −1.10029 −0.550146 0.835069i \(-0.685428\pi\)
−0.550146 + 0.835069i \(0.685428\pi\)
\(390\) 0 0
\(391\) 3.71220 0.187734
\(392\) 0 0
\(393\) −9.52434 −0.480439
\(394\) 0 0
\(395\) 11.4174 0.574471
\(396\) 0 0
\(397\) 20.8145 1.04465 0.522326 0.852746i \(-0.325064\pi\)
0.522326 + 0.852746i \(0.325064\pi\)
\(398\) 0 0
\(399\) 4.79352 0.239976
\(400\) 0 0
\(401\) −13.4584 −0.672080 −0.336040 0.941848i \(-0.609088\pi\)
−0.336040 + 0.941848i \(0.609088\pi\)
\(402\) 0 0
\(403\) 4.12474 0.205468
\(404\) 0 0
\(405\) −1.68509 −0.0837330
\(406\) 0 0
\(407\) 15.7569 0.781039
\(408\) 0 0
\(409\) −0.684734 −0.0338579 −0.0169290 0.999857i \(-0.505389\pi\)
−0.0169290 + 0.999857i \(0.505389\pi\)
\(410\) 0 0
\(411\) 11.1055 0.547796
\(412\) 0 0
\(413\) 1.28591 0.0632757
\(414\) 0 0
\(415\) 19.0577 0.935504
\(416\) 0 0
\(417\) 3.18986 0.156208
\(418\) 0 0
\(419\) −23.5683 −1.15139 −0.575694 0.817665i \(-0.695268\pi\)
−0.575694 + 0.817665i \(0.695268\pi\)
\(420\) 0 0
\(421\) −8.78562 −0.428185 −0.214092 0.976813i \(-0.568679\pi\)
−0.214092 + 0.976813i \(0.568679\pi\)
\(422\) 0 0
\(423\) −6.58538 −0.320192
\(424\) 0 0
\(425\) 6.45575 0.313150
\(426\) 0 0
\(427\) −21.7903 −1.05450
\(428\) 0 0
\(429\) −13.8149 −0.666991
\(430\) 0 0
\(431\) 15.9236 0.767014 0.383507 0.923538i \(-0.374716\pi\)
0.383507 + 0.923538i \(0.374716\pi\)
\(432\) 0 0
\(433\) 33.9313 1.63063 0.815316 0.579016i \(-0.196563\pi\)
0.815316 + 0.579016i \(0.196563\pi\)
\(434\) 0 0
\(435\) 10.9230 0.523720
\(436\) 0 0
\(437\) −2.66988 −0.127718
\(438\) 0 0
\(439\) −6.67326 −0.318497 −0.159249 0.987239i \(-0.550907\pi\)
−0.159249 + 0.987239i \(0.550907\pi\)
\(440\) 0 0
\(441\) −2.02508 −0.0964324
\(442\) 0 0
\(443\) 2.91953 0.138711 0.0693555 0.997592i \(-0.477906\pi\)
0.0693555 + 0.997592i \(0.477906\pi\)
\(444\) 0 0
\(445\) 2.62170 0.124281
\(446\) 0 0
\(447\) 8.87024 0.419548
\(448\) 0 0
\(449\) −33.6453 −1.58782 −0.793910 0.608035i \(-0.791958\pi\)
−0.793910 + 0.608035i \(0.791958\pi\)
\(450\) 0 0
\(451\) 2.01013 0.0946534
\(452\) 0 0
\(453\) 1.14248 0.0536784
\(454\) 0 0
\(455\) 17.1160 0.802410
\(456\) 0 0
\(457\) −11.8562 −0.554608 −0.277304 0.960782i \(-0.589441\pi\)
−0.277304 + 0.960782i \(0.589441\pi\)
\(458\) 0 0
\(459\) −2.98814 −0.139475
\(460\) 0 0
\(461\) −30.1143 −1.40256 −0.701282 0.712884i \(-0.747389\pi\)
−0.701282 + 0.712884i \(0.747389\pi\)
\(462\) 0 0
\(463\) 30.5085 1.41785 0.708924 0.705284i \(-0.249181\pi\)
0.708924 + 0.705284i \(0.249181\pi\)
\(464\) 0 0
\(465\) 1.52629 0.0707799
\(466\) 0 0
\(467\) −2.58448 −0.119596 −0.0597978 0.998211i \(-0.519046\pi\)
−0.0597978 + 0.998211i \(0.519046\pi\)
\(468\) 0 0
\(469\) −10.4082 −0.480606
\(470\) 0 0
\(471\) 9.53736 0.439458
\(472\) 0 0
\(473\) 32.4787 1.49337
\(474\) 0 0
\(475\) −4.64309 −0.213040
\(476\) 0 0
\(477\) −8.56261 −0.392055
\(478\) 0 0
\(479\) −25.6801 −1.17336 −0.586678 0.809820i \(-0.699564\pi\)
−0.586678 + 0.809820i \(0.699564\pi\)
\(480\) 0 0
\(481\) −23.6533 −1.07850
\(482\) 0 0
\(483\) −2.77091 −0.126081
\(484\) 0 0
\(485\) 21.7134 0.985953
\(486\) 0 0
\(487\) −1.86501 −0.0845119 −0.0422559 0.999107i \(-0.513454\pi\)
−0.0422559 + 0.999107i \(0.513454\pi\)
\(488\) 0 0
\(489\) −20.9850 −0.948974
\(490\) 0 0
\(491\) −26.7924 −1.20913 −0.604563 0.796557i \(-0.706652\pi\)
−0.604563 + 0.796557i \(0.706652\pi\)
\(492\) 0 0
\(493\) 19.3696 0.872363
\(494\) 0 0
\(495\) −5.11196 −0.229766
\(496\) 0 0
\(497\) −5.15878 −0.231403
\(498\) 0 0
\(499\) 20.9253 0.936746 0.468373 0.883531i \(-0.344840\pi\)
0.468373 + 0.883531i \(0.344840\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 4.03993 0.180132 0.0900658 0.995936i \(-0.471292\pi\)
0.0900658 + 0.995936i \(0.471292\pi\)
\(504\) 0 0
\(505\) −5.42403 −0.241366
\(506\) 0 0
\(507\) 7.73814 0.343663
\(508\) 0 0
\(509\) −6.65081 −0.294792 −0.147396 0.989078i \(-0.547089\pi\)
−0.147396 + 0.989078i \(0.547089\pi\)
\(510\) 0 0
\(511\) 30.3201 1.34128
\(512\) 0 0
\(513\) 2.14913 0.0948862
\(514\) 0 0
\(515\) −21.5608 −0.950082
\(516\) 0 0
\(517\) −19.9776 −0.878616
\(518\) 0 0
\(519\) −9.53795 −0.418669
\(520\) 0 0
\(521\) −23.7666 −1.04124 −0.520618 0.853790i \(-0.674298\pi\)
−0.520618 + 0.853790i \(0.674298\pi\)
\(522\) 0 0
\(523\) 6.42494 0.280943 0.140471 0.990085i \(-0.455138\pi\)
0.140471 + 0.990085i \(0.455138\pi\)
\(524\) 0 0
\(525\) −4.81880 −0.210310
\(526\) 0 0
\(527\) 2.70653 0.117898
\(528\) 0 0
\(529\) −21.4567 −0.932899
\(530\) 0 0
\(531\) 0.576526 0.0250191
\(532\) 0 0
\(533\) −3.01749 −0.130702
\(534\) 0 0
\(535\) −9.76874 −0.422339
\(536\) 0 0
\(537\) 2.12282 0.0916066
\(538\) 0 0
\(539\) −6.14336 −0.264613
\(540\) 0 0
\(541\) 6.56151 0.282101 0.141051 0.990002i \(-0.454952\pi\)
0.141051 + 0.990002i \(0.454952\pi\)
\(542\) 0 0
\(543\) −8.06276 −0.346006
\(544\) 0 0
\(545\) 18.6522 0.798975
\(546\) 0 0
\(547\) −12.7097 −0.543429 −0.271714 0.962378i \(-0.587591\pi\)
−0.271714 + 0.962378i \(0.587591\pi\)
\(548\) 0 0
\(549\) −9.76943 −0.416949
\(550\) 0 0
\(551\) −13.9310 −0.593479
\(552\) 0 0
\(553\) −15.1125 −0.642648
\(554\) 0 0
\(555\) −8.75247 −0.371522
\(556\) 0 0
\(557\) 35.7402 1.51436 0.757180 0.653207i \(-0.226577\pi\)
0.757180 + 0.653207i \(0.226577\pi\)
\(558\) 0 0
\(559\) −48.7550 −2.06212
\(560\) 0 0
\(561\) −9.06494 −0.382722
\(562\) 0 0
\(563\) −41.4699 −1.74775 −0.873873 0.486154i \(-0.838399\pi\)
−0.873873 + 0.486154i \(0.838399\pi\)
\(564\) 0 0
\(565\) 20.4440 0.860086
\(566\) 0 0
\(567\) 2.23045 0.0936702
\(568\) 0 0
\(569\) −17.6005 −0.737852 −0.368926 0.929459i \(-0.620274\pi\)
−0.368926 + 0.929459i \(0.620274\pi\)
\(570\) 0 0
\(571\) −3.72137 −0.155734 −0.0778672 0.996964i \(-0.524811\pi\)
−0.0778672 + 0.996964i \(0.524811\pi\)
\(572\) 0 0
\(573\) 2.87699 0.120188
\(574\) 0 0
\(575\) 2.68396 0.111929
\(576\) 0 0
\(577\) −24.7882 −1.03194 −0.515972 0.856605i \(-0.672569\pi\)
−0.515972 + 0.856605i \(0.672569\pi\)
\(578\) 0 0
\(579\) −26.0937 −1.08442
\(580\) 0 0
\(581\) −25.2254 −1.04653
\(582\) 0 0
\(583\) −25.9758 −1.07581
\(584\) 0 0
\(585\) 7.67378 0.317272
\(586\) 0 0
\(587\) 10.7714 0.444582 0.222291 0.974980i \(-0.428647\pi\)
0.222291 + 0.974980i \(0.428647\pi\)
\(588\) 0 0
\(589\) −1.94659 −0.0802077
\(590\) 0 0
\(591\) 24.0013 0.987283
\(592\) 0 0
\(593\) 3.45137 0.141731 0.0708653 0.997486i \(-0.477424\pi\)
0.0708653 + 0.997486i \(0.477424\pi\)
\(594\) 0 0
\(595\) 11.2310 0.460426
\(596\) 0 0
\(597\) 17.2330 0.705301
\(598\) 0 0
\(599\) −18.8839 −0.771574 −0.385787 0.922588i \(-0.626070\pi\)
−0.385787 + 0.922588i \(0.626070\pi\)
\(600\) 0 0
\(601\) −37.9861 −1.54949 −0.774743 0.632276i \(-0.782121\pi\)
−0.774743 + 0.632276i \(0.782121\pi\)
\(602\) 0 0
\(603\) −4.66641 −0.190031
\(604\) 0 0
\(605\) 3.02820 0.123114
\(606\) 0 0
\(607\) −30.1849 −1.22517 −0.612583 0.790406i \(-0.709870\pi\)
−0.612583 + 0.790406i \(0.709870\pi\)
\(608\) 0 0
\(609\) −14.4581 −0.585873
\(610\) 0 0
\(611\) 29.9892 1.21324
\(612\) 0 0
\(613\) −5.96309 −0.240847 −0.120423 0.992723i \(-0.538425\pi\)
−0.120423 + 0.992723i \(0.538425\pi\)
\(614\) 0 0
\(615\) −1.11657 −0.0450244
\(616\) 0 0
\(617\) −16.0868 −0.647630 −0.323815 0.946120i \(-0.604966\pi\)
−0.323815 + 0.946120i \(0.604966\pi\)
\(618\) 0 0
\(619\) 17.6051 0.707610 0.353805 0.935319i \(-0.384888\pi\)
0.353805 + 0.935319i \(0.384888\pi\)
\(620\) 0 0
\(621\) −1.24231 −0.0498522
\(622\) 0 0
\(623\) −3.47018 −0.139030
\(624\) 0 0
\(625\) −9.53014 −0.381206
\(626\) 0 0
\(627\) 6.51967 0.260370
\(628\) 0 0
\(629\) −15.5206 −0.618846
\(630\) 0 0
\(631\) 20.7751 0.827044 0.413522 0.910494i \(-0.364298\pi\)
0.413522 + 0.910494i \(0.364298\pi\)
\(632\) 0 0
\(633\) 28.7011 1.14077
\(634\) 0 0
\(635\) 28.1898 1.11868
\(636\) 0 0
\(637\) 9.22204 0.365391
\(638\) 0 0
\(639\) −2.31288 −0.0914962
\(640\) 0 0
\(641\) 35.2600 1.39269 0.696343 0.717709i \(-0.254809\pi\)
0.696343 + 0.717709i \(0.254809\pi\)
\(642\) 0 0
\(643\) 7.93781 0.313036 0.156518 0.987675i \(-0.449973\pi\)
0.156518 + 0.987675i \(0.449973\pi\)
\(644\) 0 0
\(645\) −18.0409 −0.710360
\(646\) 0 0
\(647\) 13.5098 0.531124 0.265562 0.964094i \(-0.414443\pi\)
0.265562 + 0.964094i \(0.414443\pi\)
\(648\) 0 0
\(649\) 1.74897 0.0686531
\(650\) 0 0
\(651\) −2.02025 −0.0791798
\(652\) 0 0
\(653\) −18.8369 −0.737146 −0.368573 0.929599i \(-0.620154\pi\)
−0.368573 + 0.929599i \(0.620154\pi\)
\(654\) 0 0
\(655\) 16.0494 0.627102
\(656\) 0 0
\(657\) 13.5937 0.530341
\(658\) 0 0
\(659\) 4.71786 0.183782 0.0918909 0.995769i \(-0.470709\pi\)
0.0918909 + 0.995769i \(0.470709\pi\)
\(660\) 0 0
\(661\) 20.3849 0.792882 0.396441 0.918060i \(-0.370245\pi\)
0.396441 + 0.918060i \(0.370245\pi\)
\(662\) 0 0
\(663\) 13.6078 0.528481
\(664\) 0 0
\(665\) −8.07754 −0.313234
\(666\) 0 0
\(667\) 8.05284 0.311807
\(668\) 0 0
\(669\) −13.2997 −0.514196
\(670\) 0 0
\(671\) −29.6369 −1.14412
\(672\) 0 0
\(673\) −20.2303 −0.779822 −0.389911 0.920853i \(-0.627494\pi\)
−0.389911 + 0.920853i \(0.627494\pi\)
\(674\) 0 0
\(675\) −2.16046 −0.0831560
\(676\) 0 0
\(677\) −33.1478 −1.27397 −0.636987 0.770875i \(-0.719819\pi\)
−0.636987 + 0.770875i \(0.719819\pi\)
\(678\) 0 0
\(679\) −28.7406 −1.10296
\(680\) 0 0
\(681\) −5.09741 −0.195333
\(682\) 0 0
\(683\) 16.5351 0.632698 0.316349 0.948643i \(-0.397543\pi\)
0.316349 + 0.948643i \(0.397543\pi\)
\(684\) 0 0
\(685\) −18.7139 −0.715020
\(686\) 0 0
\(687\) 26.5563 1.01319
\(688\) 0 0
\(689\) 38.9934 1.48553
\(690\) 0 0
\(691\) 32.9967 1.25525 0.627627 0.778514i \(-0.284026\pi\)
0.627627 + 0.778514i \(0.284026\pi\)
\(692\) 0 0
\(693\) 6.76638 0.257034
\(694\) 0 0
\(695\) −5.37522 −0.203894
\(696\) 0 0
\(697\) −1.97999 −0.0749973
\(698\) 0 0
\(699\) 28.9849 1.09631
\(700\) 0 0
\(701\) 36.1210 1.36427 0.682136 0.731225i \(-0.261051\pi\)
0.682136 + 0.731225i \(0.261051\pi\)
\(702\) 0 0
\(703\) 11.1627 0.421008
\(704\) 0 0
\(705\) 11.0970 0.417936
\(706\) 0 0
\(707\) 7.17944 0.270011
\(708\) 0 0
\(709\) −5.15320 −0.193533 −0.0967663 0.995307i \(-0.530850\pi\)
−0.0967663 + 0.995307i \(0.530850\pi\)
\(710\) 0 0
\(711\) −6.77552 −0.254102
\(712\) 0 0
\(713\) 1.12523 0.0421402
\(714\) 0 0
\(715\) 23.2794 0.870602
\(716\) 0 0
\(717\) −15.1035 −0.564051
\(718\) 0 0
\(719\) 38.8596 1.44922 0.724609 0.689160i \(-0.242020\pi\)
0.724609 + 0.689160i \(0.242020\pi\)
\(720\) 0 0
\(721\) 28.5387 1.06284
\(722\) 0 0
\(723\) 19.7378 0.734055
\(724\) 0 0
\(725\) 14.0044 0.520111
\(726\) 0 0
\(727\) −7.08653 −0.262825 −0.131412 0.991328i \(-0.541951\pi\)
−0.131412 + 0.991328i \(0.541951\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −31.9916 −1.18325
\(732\) 0 0
\(733\) −9.95448 −0.367677 −0.183839 0.982956i \(-0.558852\pi\)
−0.183839 + 0.982956i \(0.558852\pi\)
\(734\) 0 0
\(735\) 3.41245 0.125870
\(736\) 0 0
\(737\) −14.1562 −0.521450
\(738\) 0 0
\(739\) 41.9772 1.54416 0.772078 0.635528i \(-0.219218\pi\)
0.772078 + 0.635528i \(0.219218\pi\)
\(740\) 0 0
\(741\) −9.78694 −0.359532
\(742\) 0 0
\(743\) 8.11387 0.297669 0.148834 0.988862i \(-0.452448\pi\)
0.148834 + 0.988862i \(0.452448\pi\)
\(744\) 0 0
\(745\) −14.9472 −0.547623
\(746\) 0 0
\(747\) −11.3095 −0.413795
\(748\) 0 0
\(749\) 12.9303 0.472461
\(750\) 0 0
\(751\) 35.3736 1.29080 0.645400 0.763845i \(-0.276691\pi\)
0.645400 + 0.763845i \(0.276691\pi\)
\(752\) 0 0
\(753\) −11.0371 −0.402214
\(754\) 0 0
\(755\) −1.92519 −0.0700648
\(756\) 0 0
\(757\) 7.15401 0.260017 0.130008 0.991513i \(-0.458500\pi\)
0.130008 + 0.991513i \(0.458500\pi\)
\(758\) 0 0
\(759\) −3.76872 −0.136796
\(760\) 0 0
\(761\) 2.69995 0.0978730 0.0489365 0.998802i \(-0.484417\pi\)
0.0489365 + 0.998802i \(0.484417\pi\)
\(762\) 0 0
\(763\) −24.6888 −0.893795
\(764\) 0 0
\(765\) 5.03530 0.182052
\(766\) 0 0
\(767\) −2.62545 −0.0947995
\(768\) 0 0
\(769\) 39.3940 1.42058 0.710291 0.703908i \(-0.248563\pi\)
0.710291 + 0.703908i \(0.248563\pi\)
\(770\) 0 0
\(771\) −21.9098 −0.789061
\(772\) 0 0
\(773\) 15.7509 0.566522 0.283261 0.959043i \(-0.408584\pi\)
0.283261 + 0.959043i \(0.408584\pi\)
\(774\) 0 0
\(775\) 1.95685 0.0702921
\(776\) 0 0
\(777\) 11.5851 0.415613
\(778\) 0 0
\(779\) 1.42404 0.0510216
\(780\) 0 0
\(781\) −7.01645 −0.251068
\(782\) 0 0
\(783\) −6.48215 −0.231653
\(784\) 0 0
\(785\) −16.0713 −0.573611
\(786\) 0 0
\(787\) 26.1812 0.933261 0.466630 0.884452i \(-0.345468\pi\)
0.466630 + 0.884452i \(0.345468\pi\)
\(788\) 0 0
\(789\) −20.7484 −0.738664
\(790\) 0 0
\(791\) −27.0604 −0.962158
\(792\) 0 0
\(793\) 44.4892 1.57986
\(794\) 0 0
\(795\) 14.4288 0.511737
\(796\) 0 0
\(797\) 14.3201 0.507242 0.253621 0.967304i \(-0.418378\pi\)
0.253621 + 0.967304i \(0.418378\pi\)
\(798\) 0 0
\(799\) 19.6780 0.696159
\(800\) 0 0
\(801\) −1.55582 −0.0549721
\(802\) 0 0
\(803\) 41.2383 1.45527
\(804\) 0 0
\(805\) 4.66925 0.164569
\(806\) 0 0
\(807\) −9.59849 −0.337883
\(808\) 0 0
\(809\) −30.4050 −1.06898 −0.534492 0.845174i \(-0.679497\pi\)
−0.534492 + 0.845174i \(0.679497\pi\)
\(810\) 0 0
\(811\) −18.4602 −0.648224 −0.324112 0.946019i \(-0.605065\pi\)
−0.324112 + 0.946019i \(0.605065\pi\)
\(812\) 0 0
\(813\) −8.78291 −0.308030
\(814\) 0 0
\(815\) 35.3617 1.23867
\(816\) 0 0
\(817\) 23.0089 0.804980
\(818\) 0 0
\(819\) −10.1573 −0.354925
\(820\) 0 0
\(821\) 19.7093 0.687859 0.343930 0.938995i \(-0.388242\pi\)
0.343930 + 0.938995i \(0.388242\pi\)
\(822\) 0 0
\(823\) −2.37531 −0.0827982 −0.0413991 0.999143i \(-0.513182\pi\)
−0.0413991 + 0.999143i \(0.513182\pi\)
\(824\) 0 0
\(825\) −6.55404 −0.228182
\(826\) 0 0
\(827\) −1.85683 −0.0645683 −0.0322841 0.999479i \(-0.510278\pi\)
−0.0322841 + 0.999479i \(0.510278\pi\)
\(828\) 0 0
\(829\) −2.41323 −0.0838148 −0.0419074 0.999121i \(-0.513343\pi\)
−0.0419074 + 0.999121i \(0.513343\pi\)
\(830\) 0 0
\(831\) −10.9897 −0.381227
\(832\) 0 0
\(833\) 6.05123 0.209663
\(834\) 0 0
\(835\) −1.68509 −0.0583151
\(836\) 0 0
\(837\) −0.905758 −0.0313076
\(838\) 0 0
\(839\) 9.76856 0.337248 0.168624 0.985680i \(-0.446068\pi\)
0.168624 + 0.985680i \(0.446068\pi\)
\(840\) 0 0
\(841\) 13.0183 0.448907
\(842\) 0 0
\(843\) 1.74882 0.0602326
\(844\) 0 0
\(845\) −13.0395 −0.448572
\(846\) 0 0
\(847\) −4.00823 −0.137725
\(848\) 0 0
\(849\) −5.33783 −0.183194
\(850\) 0 0
\(851\) −6.45263 −0.221193
\(852\) 0 0
\(853\) 39.4709 1.35146 0.675729 0.737150i \(-0.263829\pi\)
0.675729 + 0.737150i \(0.263829\pi\)
\(854\) 0 0
\(855\) −3.62148 −0.123852
\(856\) 0 0
\(857\) −7.76665 −0.265304 −0.132652 0.991163i \(-0.542349\pi\)
−0.132652 + 0.991163i \(0.542349\pi\)
\(858\) 0 0
\(859\) −12.0800 −0.412163 −0.206081 0.978535i \(-0.566071\pi\)
−0.206081 + 0.978535i \(0.566071\pi\)
\(860\) 0 0
\(861\) 1.47793 0.0503677
\(862\) 0 0
\(863\) 23.3890 0.796170 0.398085 0.917349i \(-0.369675\pi\)
0.398085 + 0.917349i \(0.369675\pi\)
\(864\) 0 0
\(865\) 16.0723 0.546476
\(866\) 0 0
\(867\) −8.07100 −0.274106
\(868\) 0 0
\(869\) −20.5545 −0.697262
\(870\) 0 0
\(871\) 21.2504 0.720043
\(872\) 0 0
\(873\) −12.8855 −0.436110
\(874\) 0 0
\(875\) 26.9127 0.909817
\(876\) 0 0
\(877\) 43.6990 1.47561 0.737805 0.675014i \(-0.235862\pi\)
0.737805 + 0.675014i \(0.235862\pi\)
\(878\) 0 0
\(879\) −23.7683 −0.801685
\(880\) 0 0
\(881\) 36.3074 1.22323 0.611614 0.791156i \(-0.290520\pi\)
0.611614 + 0.791156i \(0.290520\pi\)
\(882\) 0 0
\(883\) 28.2760 0.951563 0.475782 0.879563i \(-0.342165\pi\)
0.475782 + 0.879563i \(0.342165\pi\)
\(884\) 0 0
\(885\) −0.971500 −0.0326566
\(886\) 0 0
\(887\) −19.2679 −0.646952 −0.323476 0.946236i \(-0.604851\pi\)
−0.323476 + 0.946236i \(0.604851\pi\)
\(888\) 0 0
\(889\) −37.3131 −1.25144
\(890\) 0 0
\(891\) 3.03364 0.101631
\(892\) 0 0
\(893\) −14.1528 −0.473605
\(894\) 0 0
\(895\) −3.57716 −0.119571
\(896\) 0 0
\(897\) 5.65737 0.188894
\(898\) 0 0
\(899\) 5.87126 0.195817
\(900\) 0 0
\(901\) 25.5863 0.852403
\(902\) 0 0
\(903\) 23.8796 0.794664
\(904\) 0 0
\(905\) 13.5865 0.451631
\(906\) 0 0
\(907\) 28.4693 0.945306 0.472653 0.881249i \(-0.343296\pi\)
0.472653 + 0.881249i \(0.343296\pi\)
\(908\) 0 0
\(909\) 3.21883 0.106762
\(910\) 0 0
\(911\) 15.3135 0.507360 0.253680 0.967288i \(-0.418359\pi\)
0.253680 + 0.967288i \(0.418359\pi\)
\(912\) 0 0
\(913\) −34.3091 −1.13546
\(914\) 0 0
\(915\) 16.4624 0.544231
\(916\) 0 0
\(917\) −21.2436 −0.701525
\(918\) 0 0
\(919\) −15.0291 −0.495765 −0.247883 0.968790i \(-0.579735\pi\)
−0.247883 + 0.968790i \(0.579735\pi\)
\(920\) 0 0
\(921\) 25.4316 0.837999
\(922\) 0 0
\(923\) 10.5327 0.346687
\(924\) 0 0
\(925\) −11.2215 −0.368962
\(926\) 0 0
\(927\) 12.7950 0.420243
\(928\) 0 0
\(929\) −15.5114 −0.508912 −0.254456 0.967084i \(-0.581896\pi\)
−0.254456 + 0.967084i \(0.581896\pi\)
\(930\) 0 0
\(931\) −4.35215 −0.142636
\(932\) 0 0
\(933\) −34.0914 −1.11610
\(934\) 0 0
\(935\) 15.2753 0.499555
\(936\) 0 0
\(937\) 29.5850 0.966500 0.483250 0.875482i \(-0.339456\pi\)
0.483250 + 0.875482i \(0.339456\pi\)
\(938\) 0 0
\(939\) 5.72030 0.186675
\(940\) 0 0
\(941\) −14.1685 −0.461881 −0.230940 0.972968i \(-0.574180\pi\)
−0.230940 + 0.972968i \(0.574180\pi\)
\(942\) 0 0
\(943\) −0.823172 −0.0268062
\(944\) 0 0
\(945\) −3.75852 −0.122265
\(946\) 0 0
\(947\) 27.9383 0.907874 0.453937 0.891034i \(-0.350019\pi\)
0.453937 + 0.891034i \(0.350019\pi\)
\(948\) 0 0
\(949\) −61.9045 −2.00951
\(950\) 0 0
\(951\) 1.84667 0.0598823
\(952\) 0 0
\(953\) 14.8870 0.482238 0.241119 0.970496i \(-0.422486\pi\)
0.241119 + 0.970496i \(0.422486\pi\)
\(954\) 0 0
\(955\) −4.84800 −0.156877
\(956\) 0 0
\(957\) −19.6645 −0.635663
\(958\) 0 0
\(959\) 24.7704 0.799877
\(960\) 0 0
\(961\) −30.1796 −0.973536
\(962\) 0 0
\(963\) 5.79714 0.186810
\(964\) 0 0
\(965\) 43.9703 1.41545
\(966\) 0 0
\(967\) 21.6077 0.694857 0.347429 0.937706i \(-0.387055\pi\)
0.347429 + 0.937706i \(0.387055\pi\)
\(968\) 0 0
\(969\) −6.42190 −0.206301
\(970\) 0 0
\(971\) −0.916978 −0.0294272 −0.0147136 0.999892i \(-0.504684\pi\)
−0.0147136 + 0.999892i \(0.504684\pi\)
\(972\) 0 0
\(973\) 7.11484 0.228091
\(974\) 0 0
\(975\) 9.83854 0.315085
\(976\) 0 0
\(977\) 26.3437 0.842810 0.421405 0.906873i \(-0.361537\pi\)
0.421405 + 0.906873i \(0.361537\pi\)
\(978\) 0 0
\(979\) −4.71979 −0.150845
\(980\) 0 0
\(981\) −11.0690 −0.353405
\(982\) 0 0
\(983\) 1.16941 0.0372984 0.0186492 0.999826i \(-0.494063\pi\)
0.0186492 + 0.999826i \(0.494063\pi\)
\(984\) 0 0
\(985\) −40.4445 −1.28867
\(986\) 0 0
\(987\) −14.6884 −0.467536
\(988\) 0 0
\(989\) −13.3004 −0.422928
\(990\) 0 0
\(991\) 47.2036 1.49947 0.749737 0.661736i \(-0.230180\pi\)
0.749737 + 0.661736i \(0.230180\pi\)
\(992\) 0 0
\(993\) 28.9536 0.918814
\(994\) 0 0
\(995\) −29.0393 −0.920607
\(996\) 0 0
\(997\) −12.0724 −0.382337 −0.191168 0.981557i \(-0.561228\pi\)
−0.191168 + 0.981557i \(0.561228\pi\)
\(998\) 0 0
\(999\) 5.19406 0.164333
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 8016.2.a.w.1.3 7
4.3 odd 2 4008.2.a.g.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.3 7 4.3 odd 2
8016.2.a.w.1.3 7 1.1 even 1 trivial