Properties

Label 4008.2.a.k.1.9
Level $4008$
Weight $2$
Character 4008.1
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-2.34947\) of defining polynomial
Character \(\chi\) \(=\) 4008.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.34947 q^{5} +3.54581 q^{7} +1.00000 q^{9} -5.22121 q^{11} +5.27416 q^{13} -3.34947 q^{15} +6.86903 q^{17} +5.25090 q^{19} -3.54581 q^{21} +2.32838 q^{23} +6.21896 q^{25} -1.00000 q^{27} -0.922527 q^{29} -10.9921 q^{31} +5.22121 q^{33} +11.8766 q^{35} -10.2033 q^{37} -5.27416 q^{39} +2.58506 q^{41} +4.38272 q^{43} +3.34947 q^{45} +6.30219 q^{47} +5.57274 q^{49} -6.86903 q^{51} +10.7064 q^{53} -17.4883 q^{55} -5.25090 q^{57} +10.3875 q^{59} +10.7958 q^{61} +3.54581 q^{63} +17.6657 q^{65} -8.19936 q^{67} -2.32838 q^{69} -3.27323 q^{71} -9.70252 q^{73} -6.21896 q^{75} -18.5134 q^{77} -14.2308 q^{79} +1.00000 q^{81} -12.2173 q^{83} +23.0076 q^{85} +0.922527 q^{87} +1.83867 q^{89} +18.7011 q^{91} +10.9921 q^{93} +17.5878 q^{95} -10.0691 q^{97} -5.22121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} + O(q^{10}) \) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} - q^{11} + 10q^{13} - 10q^{15} + 17q^{17} + 2q^{19} + q^{21} - 3q^{23} + 21q^{25} - 11q^{27} + 17q^{29} - 15q^{31} + q^{33} + 11q^{35} + 4q^{37} - 10q^{39} + 16q^{41} + 10q^{43} + 10q^{45} - 16q^{47} + 22q^{49} - 17q^{51} + 42q^{53} - 5q^{55} - 2q^{57} - 2q^{59} + 12q^{61} - q^{63} + 10q^{65} - q^{67} + 3q^{69} - 9q^{71} + 24q^{73} - 21q^{75} + 22q^{77} - 30q^{79} + 11q^{81} + 16q^{83} + 25q^{85} - 17q^{87} + 37q^{89} + q^{91} + 15q^{93} + 5q^{95} + 4q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) 3.34947 1.49793 0.748965 0.662610i \(-0.230551\pi\)
0.748965 + 0.662610i \(0.230551\pi\)
\(6\) 0 0
\(7\) 3.54581 1.34019 0.670094 0.742276i \(-0.266254\pi\)
0.670094 + 0.742276i \(0.266254\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.22121 −1.57425 −0.787127 0.616791i \(-0.788433\pi\)
−0.787127 + 0.616791i \(0.788433\pi\)
\(12\) 0 0
\(13\) 5.27416 1.46279 0.731394 0.681955i \(-0.238870\pi\)
0.731394 + 0.681955i \(0.238870\pi\)
\(14\) 0 0
\(15\) −3.34947 −0.864830
\(16\) 0 0
\(17\) 6.86903 1.66598 0.832992 0.553285i \(-0.186626\pi\)
0.832992 + 0.553285i \(0.186626\pi\)
\(18\) 0 0
\(19\) 5.25090 1.20464 0.602320 0.798255i \(-0.294243\pi\)
0.602320 + 0.798255i \(0.294243\pi\)
\(20\) 0 0
\(21\) −3.54581 −0.773758
\(22\) 0 0
\(23\) 2.32838 0.485500 0.242750 0.970089i \(-0.421950\pi\)
0.242750 + 0.970089i \(0.421950\pi\)
\(24\) 0 0
\(25\) 6.21896 1.24379
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.922527 −0.171309 −0.0856545 0.996325i \(-0.527298\pi\)
−0.0856545 + 0.996325i \(0.527298\pi\)
\(30\) 0 0
\(31\) −10.9921 −1.97425 −0.987123 0.159963i \(-0.948863\pi\)
−0.987123 + 0.159963i \(0.948863\pi\)
\(32\) 0 0
\(33\) 5.22121 0.908896
\(34\) 0 0
\(35\) 11.8766 2.00751
\(36\) 0 0
\(37\) −10.2033 −1.67742 −0.838708 0.544581i \(-0.816689\pi\)
−0.838708 + 0.544581i \(0.816689\pi\)
\(38\) 0 0
\(39\) −5.27416 −0.844541
\(40\) 0 0
\(41\) 2.58506 0.403719 0.201859 0.979415i \(-0.435302\pi\)
0.201859 + 0.979415i \(0.435302\pi\)
\(42\) 0 0
\(43\) 4.38272 0.668359 0.334179 0.942509i \(-0.391541\pi\)
0.334179 + 0.942509i \(0.391541\pi\)
\(44\) 0 0
\(45\) 3.34947 0.499310
\(46\) 0 0
\(47\) 6.30219 0.919270 0.459635 0.888108i \(-0.347980\pi\)
0.459635 + 0.888108i \(0.347980\pi\)
\(48\) 0 0
\(49\) 5.57274 0.796106
\(50\) 0 0
\(51\) −6.86903 −0.961857
\(52\) 0 0
\(53\) 10.7064 1.47064 0.735319 0.677721i \(-0.237032\pi\)
0.735319 + 0.677721i \(0.237032\pi\)
\(54\) 0 0
\(55\) −17.4883 −2.35812
\(56\) 0 0
\(57\) −5.25090 −0.695499
\(58\) 0 0
\(59\) 10.3875 1.35234 0.676169 0.736747i \(-0.263639\pi\)
0.676169 + 0.736747i \(0.263639\pi\)
\(60\) 0 0
\(61\) 10.7958 1.38226 0.691129 0.722732i \(-0.257114\pi\)
0.691129 + 0.722732i \(0.257114\pi\)
\(62\) 0 0
\(63\) 3.54581 0.446730
\(64\) 0 0
\(65\) 17.6657 2.19115
\(66\) 0 0
\(67\) −8.19936 −1.00171 −0.500856 0.865531i \(-0.666981\pi\)
−0.500856 + 0.865531i \(0.666981\pi\)
\(68\) 0 0
\(69\) −2.32838 −0.280304
\(70\) 0 0
\(71\) −3.27323 −0.388460 −0.194230 0.980956i \(-0.562221\pi\)
−0.194230 + 0.980956i \(0.562221\pi\)
\(72\) 0 0
\(73\) −9.70252 −1.13559 −0.567797 0.823169i \(-0.692204\pi\)
−0.567797 + 0.823169i \(0.692204\pi\)
\(74\) 0 0
\(75\) −6.21896 −0.718104
\(76\) 0 0
\(77\) −18.5134 −2.10980
\(78\) 0 0
\(79\) −14.2308 −1.60109 −0.800543 0.599275i \(-0.795456\pi\)
−0.800543 + 0.599275i \(0.795456\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2173 −1.34103 −0.670513 0.741898i \(-0.733926\pi\)
−0.670513 + 0.741898i \(0.733926\pi\)
\(84\) 0 0
\(85\) 23.0076 2.49553
\(86\) 0 0
\(87\) 0.922527 0.0989053
\(88\) 0 0
\(89\) 1.83867 0.194899 0.0974494 0.995240i \(-0.468932\pi\)
0.0974494 + 0.995240i \(0.468932\pi\)
\(90\) 0 0
\(91\) 18.7011 1.96041
\(92\) 0 0
\(93\) 10.9921 1.13983
\(94\) 0 0
\(95\) 17.5878 1.80447
\(96\) 0 0
\(97\) −10.0691 −1.02237 −0.511183 0.859472i \(-0.670792\pi\)
−0.511183 + 0.859472i \(0.670792\pi\)
\(98\) 0 0
\(99\) −5.22121 −0.524752
\(100\) 0 0
\(101\) 3.62392 0.360593 0.180297 0.983612i \(-0.442294\pi\)
0.180297 + 0.983612i \(0.442294\pi\)
\(102\) 0 0
\(103\) 1.34478 0.132505 0.0662527 0.997803i \(-0.478896\pi\)
0.0662527 + 0.997803i \(0.478896\pi\)
\(104\) 0 0
\(105\) −11.8766 −1.15904
\(106\) 0 0
\(107\) 5.14056 0.496957 0.248479 0.968637i \(-0.420069\pi\)
0.248479 + 0.968637i \(0.420069\pi\)
\(108\) 0 0
\(109\) −8.05520 −0.771549 −0.385774 0.922593i \(-0.626066\pi\)
−0.385774 + 0.922593i \(0.626066\pi\)
\(110\) 0 0
\(111\) 10.2033 0.968457
\(112\) 0 0
\(113\) −18.0162 −1.69482 −0.847412 0.530936i \(-0.821841\pi\)
−0.847412 + 0.530936i \(0.821841\pi\)
\(114\) 0 0
\(115\) 7.79884 0.727245
\(116\) 0 0
\(117\) 5.27416 0.487596
\(118\) 0 0
\(119\) 24.3563 2.23273
\(120\) 0 0
\(121\) 16.2611 1.47828
\(122\) 0 0
\(123\) −2.58506 −0.233087
\(124\) 0 0
\(125\) 4.08289 0.365185
\(126\) 0 0
\(127\) −6.40394 −0.568258 −0.284129 0.958786i \(-0.591704\pi\)
−0.284129 + 0.958786i \(0.591704\pi\)
\(128\) 0 0
\(129\) −4.38272 −0.385877
\(130\) 0 0
\(131\) 17.2197 1.50449 0.752246 0.658882i \(-0.228970\pi\)
0.752246 + 0.658882i \(0.228970\pi\)
\(132\) 0 0
\(133\) 18.6187 1.61445
\(134\) 0 0
\(135\) −3.34947 −0.288277
\(136\) 0 0
\(137\) −16.8321 −1.43806 −0.719032 0.694977i \(-0.755415\pi\)
−0.719032 + 0.694977i \(0.755415\pi\)
\(138\) 0 0
\(139\) −9.08472 −0.770556 −0.385278 0.922801i \(-0.625894\pi\)
−0.385278 + 0.922801i \(0.625894\pi\)
\(140\) 0 0
\(141\) −6.30219 −0.530741
\(142\) 0 0
\(143\) −27.5375 −2.30280
\(144\) 0 0
\(145\) −3.08998 −0.256609
\(146\) 0 0
\(147\) −5.57274 −0.459632
\(148\) 0 0
\(149\) −10.5718 −0.866077 −0.433038 0.901376i \(-0.642559\pi\)
−0.433038 + 0.901376i \(0.642559\pi\)
\(150\) 0 0
\(151\) 9.42466 0.766968 0.383484 0.923548i \(-0.374724\pi\)
0.383484 + 0.923548i \(0.374724\pi\)
\(152\) 0 0
\(153\) 6.86903 0.555328
\(154\) 0 0
\(155\) −36.8179 −2.95728
\(156\) 0 0
\(157\) −11.0413 −0.881195 −0.440597 0.897705i \(-0.645233\pi\)
−0.440597 + 0.897705i \(0.645233\pi\)
\(158\) 0 0
\(159\) −10.7064 −0.849073
\(160\) 0 0
\(161\) 8.25598 0.650662
\(162\) 0 0
\(163\) 0.204447 0.0160135 0.00800676 0.999968i \(-0.497451\pi\)
0.00800676 + 0.999968i \(0.497451\pi\)
\(164\) 0 0
\(165\) 17.4883 1.36146
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 14.8168 1.13975
\(170\) 0 0
\(171\) 5.25090 0.401547
\(172\) 0 0
\(173\) −5.27958 −0.401399 −0.200700 0.979653i \(-0.564322\pi\)
−0.200700 + 0.979653i \(0.564322\pi\)
\(174\) 0 0
\(175\) 22.0512 1.66692
\(176\) 0 0
\(177\) −10.3875 −0.780773
\(178\) 0 0
\(179\) 19.1223 1.42926 0.714632 0.699501i \(-0.246594\pi\)
0.714632 + 0.699501i \(0.246594\pi\)
\(180\) 0 0
\(181\) 24.5626 1.82572 0.912861 0.408270i \(-0.133868\pi\)
0.912861 + 0.408270i \(0.133868\pi\)
\(182\) 0 0
\(183\) −10.7958 −0.798046
\(184\) 0 0
\(185\) −34.1758 −2.51265
\(186\) 0 0
\(187\) −35.8647 −2.62268
\(188\) 0 0
\(189\) −3.54581 −0.257919
\(190\) 0 0
\(191\) 11.5678 0.837016 0.418508 0.908213i \(-0.362553\pi\)
0.418508 + 0.908213i \(0.362553\pi\)
\(192\) 0 0
\(193\) −4.06662 −0.292722 −0.146361 0.989231i \(-0.546756\pi\)
−0.146361 + 0.989231i \(0.546756\pi\)
\(194\) 0 0
\(195\) −17.6657 −1.26506
\(196\) 0 0
\(197\) 24.7058 1.76021 0.880106 0.474777i \(-0.157471\pi\)
0.880106 + 0.474777i \(0.157471\pi\)
\(198\) 0 0
\(199\) −6.95811 −0.493247 −0.246624 0.969111i \(-0.579321\pi\)
−0.246624 + 0.969111i \(0.579321\pi\)
\(200\) 0 0
\(201\) 8.19936 0.578338
\(202\) 0 0
\(203\) −3.27110 −0.229586
\(204\) 0 0
\(205\) 8.65859 0.604742
\(206\) 0 0
\(207\) 2.32838 0.161833
\(208\) 0 0
\(209\) −27.4161 −1.89641
\(210\) 0 0
\(211\) −12.0883 −0.832196 −0.416098 0.909320i \(-0.636603\pi\)
−0.416098 + 0.909320i \(0.636603\pi\)
\(212\) 0 0
\(213\) 3.27323 0.224278
\(214\) 0 0
\(215\) 14.6798 1.00115
\(216\) 0 0
\(217\) −38.9760 −2.64586
\(218\) 0 0
\(219\) 9.70252 0.655635
\(220\) 0 0
\(221\) 36.2284 2.43698
\(222\) 0 0
\(223\) 6.74178 0.451463 0.225732 0.974190i \(-0.427523\pi\)
0.225732 + 0.974190i \(0.427523\pi\)
\(224\) 0 0
\(225\) 6.21896 0.414598
\(226\) 0 0
\(227\) −16.6461 −1.10484 −0.552419 0.833566i \(-0.686295\pi\)
−0.552419 + 0.833566i \(0.686295\pi\)
\(228\) 0 0
\(229\) 4.34626 0.287209 0.143604 0.989635i \(-0.454131\pi\)
0.143604 + 0.989635i \(0.454131\pi\)
\(230\) 0 0
\(231\) 18.5134 1.21809
\(232\) 0 0
\(233\) 15.4375 1.01135 0.505674 0.862725i \(-0.331244\pi\)
0.505674 + 0.862725i \(0.331244\pi\)
\(234\) 0 0
\(235\) 21.1090 1.37700
\(236\) 0 0
\(237\) 14.2308 0.924388
\(238\) 0 0
\(239\) 3.91114 0.252990 0.126495 0.991967i \(-0.459627\pi\)
0.126495 + 0.991967i \(0.459627\pi\)
\(240\) 0 0
\(241\) 7.89534 0.508584 0.254292 0.967128i \(-0.418158\pi\)
0.254292 + 0.967128i \(0.418158\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 18.6657 1.19251
\(246\) 0 0
\(247\) 27.6941 1.76213
\(248\) 0 0
\(249\) 12.2173 0.774241
\(250\) 0 0
\(251\) −9.57349 −0.604273 −0.302137 0.953265i \(-0.597700\pi\)
−0.302137 + 0.953265i \(0.597700\pi\)
\(252\) 0 0
\(253\) −12.1570 −0.764301
\(254\) 0 0
\(255\) −23.0076 −1.44079
\(256\) 0 0
\(257\) −22.3318 −1.39302 −0.696511 0.717546i \(-0.745265\pi\)
−0.696511 + 0.717546i \(0.745265\pi\)
\(258\) 0 0
\(259\) −36.1790 −2.24805
\(260\) 0 0
\(261\) −0.922527 −0.0571030
\(262\) 0 0
\(263\) 19.8026 1.22108 0.610539 0.791986i \(-0.290953\pi\)
0.610539 + 0.791986i \(0.290953\pi\)
\(264\) 0 0
\(265\) 35.8608 2.20291
\(266\) 0 0
\(267\) −1.83867 −0.112525
\(268\) 0 0
\(269\) 13.4173 0.818068 0.409034 0.912519i \(-0.365866\pi\)
0.409034 + 0.912519i \(0.365866\pi\)
\(270\) 0 0
\(271\) −25.1007 −1.52476 −0.762379 0.647130i \(-0.775969\pi\)
−0.762379 + 0.647130i \(0.775969\pi\)
\(272\) 0 0
\(273\) −18.7011 −1.13184
\(274\) 0 0
\(275\) −32.4705 −1.95805
\(276\) 0 0
\(277\) −3.87307 −0.232710 −0.116355 0.993208i \(-0.537121\pi\)
−0.116355 + 0.993208i \(0.537121\pi\)
\(278\) 0 0
\(279\) −10.9921 −0.658082
\(280\) 0 0
\(281\) −3.69139 −0.220210 −0.110105 0.993920i \(-0.535119\pi\)
−0.110105 + 0.993920i \(0.535119\pi\)
\(282\) 0 0
\(283\) 7.75352 0.460899 0.230449 0.973084i \(-0.425980\pi\)
0.230449 + 0.973084i \(0.425980\pi\)
\(284\) 0 0
\(285\) −17.5878 −1.04181
\(286\) 0 0
\(287\) 9.16613 0.541059
\(288\) 0 0
\(289\) 30.1836 1.77550
\(290\) 0 0
\(291\) 10.0691 0.590263
\(292\) 0 0
\(293\) −1.01242 −0.0591460 −0.0295730 0.999563i \(-0.509415\pi\)
−0.0295730 + 0.999563i \(0.509415\pi\)
\(294\) 0 0
\(295\) 34.7927 2.02571
\(296\) 0 0
\(297\) 5.22121 0.302965
\(298\) 0 0
\(299\) 12.2802 0.710184
\(300\) 0 0
\(301\) 15.5403 0.895727
\(302\) 0 0
\(303\) −3.62392 −0.208189
\(304\) 0 0
\(305\) 36.1601 2.07052
\(306\) 0 0
\(307\) 25.9773 1.48260 0.741301 0.671173i \(-0.234209\pi\)
0.741301 + 0.671173i \(0.234209\pi\)
\(308\) 0 0
\(309\) −1.34478 −0.0765020
\(310\) 0 0
\(311\) −10.2568 −0.581608 −0.290804 0.956783i \(-0.593923\pi\)
−0.290804 + 0.956783i \(0.593923\pi\)
\(312\) 0 0
\(313\) 13.2574 0.749351 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(314\) 0 0
\(315\) 11.8766 0.669169
\(316\) 0 0
\(317\) 21.0416 1.18181 0.590907 0.806739i \(-0.298770\pi\)
0.590907 + 0.806739i \(0.298770\pi\)
\(318\) 0 0
\(319\) 4.81671 0.269684
\(320\) 0 0
\(321\) −5.14056 −0.286918
\(322\) 0 0
\(323\) 36.0686 2.00691
\(324\) 0 0
\(325\) 32.7998 1.81941
\(326\) 0 0
\(327\) 8.05520 0.445454
\(328\) 0 0
\(329\) 22.3464 1.23199
\(330\) 0 0
\(331\) 11.4570 0.629735 0.314868 0.949136i \(-0.398040\pi\)
0.314868 + 0.949136i \(0.398040\pi\)
\(332\) 0 0
\(333\) −10.2033 −0.559139
\(334\) 0 0
\(335\) −27.4635 −1.50049
\(336\) 0 0
\(337\) −4.81048 −0.262044 −0.131022 0.991379i \(-0.541826\pi\)
−0.131022 + 0.991379i \(0.541826\pi\)
\(338\) 0 0
\(339\) 18.0162 0.978507
\(340\) 0 0
\(341\) 57.3923 3.10797
\(342\) 0 0
\(343\) −5.06078 −0.273256
\(344\) 0 0
\(345\) −7.79884 −0.419875
\(346\) 0 0
\(347\) 10.5296 0.565258 0.282629 0.959229i \(-0.408793\pi\)
0.282629 + 0.959229i \(0.408793\pi\)
\(348\) 0 0
\(349\) −8.11284 −0.434271 −0.217135 0.976142i \(-0.569671\pi\)
−0.217135 + 0.976142i \(0.569671\pi\)
\(350\) 0 0
\(351\) −5.27416 −0.281514
\(352\) 0 0
\(353\) 16.7942 0.893867 0.446933 0.894567i \(-0.352516\pi\)
0.446933 + 0.894567i \(0.352516\pi\)
\(354\) 0 0
\(355\) −10.9636 −0.581886
\(356\) 0 0
\(357\) −24.3563 −1.28907
\(358\) 0 0
\(359\) 5.05875 0.266991 0.133495 0.991049i \(-0.457380\pi\)
0.133495 + 0.991049i \(0.457380\pi\)
\(360\) 0 0
\(361\) 8.57200 0.451158
\(362\) 0 0
\(363\) −16.2611 −0.853484
\(364\) 0 0
\(365\) −32.4983 −1.70104
\(366\) 0 0
\(367\) 18.3665 0.958723 0.479362 0.877617i \(-0.340868\pi\)
0.479362 + 0.877617i \(0.340868\pi\)
\(368\) 0 0
\(369\) 2.58506 0.134573
\(370\) 0 0
\(371\) 37.9628 1.97093
\(372\) 0 0
\(373\) −12.5025 −0.647355 −0.323678 0.946167i \(-0.604919\pi\)
−0.323678 + 0.946167i \(0.604919\pi\)
\(374\) 0 0
\(375\) −4.08289 −0.210839
\(376\) 0 0
\(377\) −4.86555 −0.250589
\(378\) 0 0
\(379\) −20.8449 −1.07073 −0.535365 0.844621i \(-0.679826\pi\)
−0.535365 + 0.844621i \(0.679826\pi\)
\(380\) 0 0
\(381\) 6.40394 0.328084
\(382\) 0 0
\(383\) −31.4740 −1.60825 −0.804123 0.594463i \(-0.797365\pi\)
−0.804123 + 0.594463i \(0.797365\pi\)
\(384\) 0 0
\(385\) −62.0101 −3.16033
\(386\) 0 0
\(387\) 4.38272 0.222786
\(388\) 0 0
\(389\) 8.57981 0.435014 0.217507 0.976059i \(-0.430208\pi\)
0.217507 + 0.976059i \(0.430208\pi\)
\(390\) 0 0
\(391\) 15.9937 0.808836
\(392\) 0 0
\(393\) −17.2197 −0.868619
\(394\) 0 0
\(395\) −47.6656 −2.39831
\(396\) 0 0
\(397\) 20.3543 1.02155 0.510776 0.859714i \(-0.329358\pi\)
0.510776 + 0.859714i \(0.329358\pi\)
\(398\) 0 0
\(399\) −18.6187 −0.932100
\(400\) 0 0
\(401\) 16.1277 0.805378 0.402689 0.915337i \(-0.368076\pi\)
0.402689 + 0.915337i \(0.368076\pi\)
\(402\) 0 0
\(403\) −57.9743 −2.88790
\(404\) 0 0
\(405\) 3.34947 0.166437
\(406\) 0 0
\(407\) 53.2737 2.64068
\(408\) 0 0
\(409\) −18.1245 −0.896199 −0.448099 0.893984i \(-0.647899\pi\)
−0.448099 + 0.893984i \(0.647899\pi\)
\(410\) 0 0
\(411\) 16.8321 0.830267
\(412\) 0 0
\(413\) 36.8321 1.81239
\(414\) 0 0
\(415\) −40.9216 −2.00876
\(416\) 0 0
\(417\) 9.08472 0.444881
\(418\) 0 0
\(419\) −23.7981 −1.16261 −0.581307 0.813684i \(-0.697459\pi\)
−0.581307 + 0.813684i \(0.697459\pi\)
\(420\) 0 0
\(421\) −14.4408 −0.703800 −0.351900 0.936038i \(-0.614464\pi\)
−0.351900 + 0.936038i \(0.614464\pi\)
\(422\) 0 0
\(423\) 6.30219 0.306423
\(424\) 0 0
\(425\) 42.7183 2.07214
\(426\) 0 0
\(427\) 38.2797 1.85249
\(428\) 0 0
\(429\) 27.5375 1.32952
\(430\) 0 0
\(431\) −12.8595 −0.619420 −0.309710 0.950831i \(-0.600232\pi\)
−0.309710 + 0.950831i \(0.600232\pi\)
\(432\) 0 0
\(433\) 16.3060 0.783618 0.391809 0.920047i \(-0.371849\pi\)
0.391809 + 0.920047i \(0.371849\pi\)
\(434\) 0 0
\(435\) 3.08998 0.148153
\(436\) 0 0
\(437\) 12.2261 0.584853
\(438\) 0 0
\(439\) 10.2780 0.490544 0.245272 0.969454i \(-0.421123\pi\)
0.245272 + 0.969454i \(0.421123\pi\)
\(440\) 0 0
\(441\) 5.57274 0.265369
\(442\) 0 0
\(443\) −32.8390 −1.56023 −0.780113 0.625638i \(-0.784839\pi\)
−0.780113 + 0.625638i \(0.784839\pi\)
\(444\) 0 0
\(445\) 6.15858 0.291945
\(446\) 0 0
\(447\) 10.5718 0.500030
\(448\) 0 0
\(449\) 37.3707 1.76363 0.881817 0.471592i \(-0.156321\pi\)
0.881817 + 0.471592i \(0.156321\pi\)
\(450\) 0 0
\(451\) −13.4972 −0.635556
\(452\) 0 0
\(453\) −9.42466 −0.442809
\(454\) 0 0
\(455\) 62.6390 2.93656
\(456\) 0 0
\(457\) 32.2479 1.50849 0.754246 0.656592i \(-0.228003\pi\)
0.754246 + 0.656592i \(0.228003\pi\)
\(458\) 0 0
\(459\) −6.86903 −0.320619
\(460\) 0 0
\(461\) −27.8346 −1.29639 −0.648193 0.761476i \(-0.724475\pi\)
−0.648193 + 0.761476i \(0.724475\pi\)
\(462\) 0 0
\(463\) 6.04994 0.281164 0.140582 0.990069i \(-0.455103\pi\)
0.140582 + 0.990069i \(0.455103\pi\)
\(464\) 0 0
\(465\) 36.8179 1.70739
\(466\) 0 0
\(467\) 17.8938 0.828028 0.414014 0.910271i \(-0.364126\pi\)
0.414014 + 0.910271i \(0.364126\pi\)
\(468\) 0 0
\(469\) −29.0733 −1.34248
\(470\) 0 0
\(471\) 11.0413 0.508758
\(472\) 0 0
\(473\) −22.8831 −1.05217
\(474\) 0 0
\(475\) 32.6552 1.49832
\(476\) 0 0
\(477\) 10.7064 0.490213
\(478\) 0 0
\(479\) −14.3503 −0.655683 −0.327841 0.944733i \(-0.606321\pi\)
−0.327841 + 0.944733i \(0.606321\pi\)
\(480\) 0 0
\(481\) −53.8140 −2.45371
\(482\) 0 0
\(483\) −8.25598 −0.375660
\(484\) 0 0
\(485\) −33.7263 −1.53143
\(486\) 0 0
\(487\) −16.1796 −0.733170 −0.366585 0.930385i \(-0.619473\pi\)
−0.366585 + 0.930385i \(0.619473\pi\)
\(488\) 0 0
\(489\) −0.204447 −0.00924541
\(490\) 0 0
\(491\) 33.6911 1.52046 0.760228 0.649656i \(-0.225087\pi\)
0.760228 + 0.649656i \(0.225087\pi\)
\(492\) 0 0
\(493\) −6.33687 −0.285398
\(494\) 0 0
\(495\) −17.4883 −0.786041
\(496\) 0 0
\(497\) −11.6062 −0.520610
\(498\) 0 0
\(499\) −24.5927 −1.10092 −0.550461 0.834861i \(-0.685548\pi\)
−0.550461 + 0.834861i \(0.685548\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 8.11857 0.361989 0.180995 0.983484i \(-0.442068\pi\)
0.180995 + 0.983484i \(0.442068\pi\)
\(504\) 0 0
\(505\) 12.1382 0.540143
\(506\) 0 0
\(507\) −14.8168 −0.658035
\(508\) 0 0
\(509\) 5.74908 0.254823 0.127412 0.991850i \(-0.459333\pi\)
0.127412 + 0.991850i \(0.459333\pi\)
\(510\) 0 0
\(511\) −34.4032 −1.52191
\(512\) 0 0
\(513\) −5.25090 −0.231833
\(514\) 0 0
\(515\) 4.50431 0.198484
\(516\) 0 0
\(517\) −32.9051 −1.44716
\(518\) 0 0
\(519\) 5.27958 0.231748
\(520\) 0 0
\(521\) −32.3615 −1.41778 −0.708891 0.705318i \(-0.750804\pi\)
−0.708891 + 0.705318i \(0.750804\pi\)
\(522\) 0 0
\(523\) 36.4106 1.59213 0.796063 0.605214i \(-0.206912\pi\)
0.796063 + 0.605214i \(0.206912\pi\)
\(524\) 0 0
\(525\) −22.0512 −0.962395
\(526\) 0 0
\(527\) −75.5053 −3.28906
\(528\) 0 0
\(529\) −17.5787 −0.764289
\(530\) 0 0
\(531\) 10.3875 0.450779
\(532\) 0 0
\(533\) 13.6340 0.590555
\(534\) 0 0
\(535\) 17.2182 0.744407
\(536\) 0 0
\(537\) −19.1223 −0.825186
\(538\) 0 0
\(539\) −29.0965 −1.25327
\(540\) 0 0
\(541\) −17.8083 −0.765637 −0.382818 0.923824i \(-0.625047\pi\)
−0.382818 + 0.923824i \(0.625047\pi\)
\(542\) 0 0
\(543\) −24.5626 −1.05408
\(544\) 0 0
\(545\) −26.9807 −1.15573
\(546\) 0 0
\(547\) −11.2379 −0.480500 −0.240250 0.970711i \(-0.577229\pi\)
−0.240250 + 0.970711i \(0.577229\pi\)
\(548\) 0 0
\(549\) 10.7958 0.460752
\(550\) 0 0
\(551\) −4.84410 −0.206366
\(552\) 0 0
\(553\) −50.4595 −2.14576
\(554\) 0 0
\(555\) 34.1758 1.45068
\(556\) 0 0
\(557\) −6.44916 −0.273260 −0.136630 0.990622i \(-0.543627\pi\)
−0.136630 + 0.990622i \(0.543627\pi\)
\(558\) 0 0
\(559\) 23.1152 0.977668
\(560\) 0 0
\(561\) 35.8647 1.51421
\(562\) 0 0
\(563\) 13.4201 0.565590 0.282795 0.959180i \(-0.408738\pi\)
0.282795 + 0.959180i \(0.408738\pi\)
\(564\) 0 0
\(565\) −60.3449 −2.53873
\(566\) 0 0
\(567\) 3.54581 0.148910
\(568\) 0 0
\(569\) −13.4681 −0.564613 −0.282306 0.959324i \(-0.591099\pi\)
−0.282306 + 0.959324i \(0.591099\pi\)
\(570\) 0 0
\(571\) 19.5925 0.819919 0.409959 0.912104i \(-0.365543\pi\)
0.409959 + 0.912104i \(0.365543\pi\)
\(572\) 0 0
\(573\) −11.5678 −0.483251
\(574\) 0 0
\(575\) 14.4801 0.603862
\(576\) 0 0
\(577\) −21.3611 −0.889274 −0.444637 0.895711i \(-0.646667\pi\)
−0.444637 + 0.895711i \(0.646667\pi\)
\(578\) 0 0
\(579\) 4.06662 0.169003
\(580\) 0 0
\(581\) −43.3203 −1.79723
\(582\) 0 0
\(583\) −55.9004 −2.31516
\(584\) 0 0
\(585\) 17.6657 0.730385
\(586\) 0 0
\(587\) −28.4585 −1.17461 −0.587305 0.809366i \(-0.699811\pi\)
−0.587305 + 0.809366i \(0.699811\pi\)
\(588\) 0 0
\(589\) −57.7187 −2.37826
\(590\) 0 0
\(591\) −24.7058 −1.01626
\(592\) 0 0
\(593\) −14.7257 −0.604711 −0.302355 0.953195i \(-0.597773\pi\)
−0.302355 + 0.953195i \(0.597773\pi\)
\(594\) 0 0
\(595\) 81.5806 3.34448
\(596\) 0 0
\(597\) 6.95811 0.284777
\(598\) 0 0
\(599\) 24.5093 1.00142 0.500712 0.865614i \(-0.333072\pi\)
0.500712 + 0.865614i \(0.333072\pi\)
\(600\) 0 0
\(601\) −34.6537 −1.41355 −0.706777 0.707436i \(-0.749852\pi\)
−0.706777 + 0.707436i \(0.749852\pi\)
\(602\) 0 0
\(603\) −8.19936 −0.333904
\(604\) 0 0
\(605\) 54.4659 2.21436
\(606\) 0 0
\(607\) −9.43186 −0.382827 −0.191414 0.981509i \(-0.561307\pi\)
−0.191414 + 0.981509i \(0.561307\pi\)
\(608\) 0 0
\(609\) 3.27110 0.132552
\(610\) 0 0
\(611\) 33.2388 1.34470
\(612\) 0 0
\(613\) 36.6729 1.48121 0.740603 0.671943i \(-0.234540\pi\)
0.740603 + 0.671943i \(0.234540\pi\)
\(614\) 0 0
\(615\) −8.65859 −0.349148
\(616\) 0 0
\(617\) −8.27551 −0.333159 −0.166580 0.986028i \(-0.553272\pi\)
−0.166580 + 0.986028i \(0.553272\pi\)
\(618\) 0 0
\(619\) 26.8480 1.07911 0.539556 0.841950i \(-0.318592\pi\)
0.539556 + 0.841950i \(0.318592\pi\)
\(620\) 0 0
\(621\) −2.32838 −0.0934346
\(622\) 0 0
\(623\) 6.51957 0.261201
\(624\) 0 0
\(625\) −17.4193 −0.696772
\(626\) 0 0
\(627\) 27.4161 1.09489
\(628\) 0 0
\(629\) −70.0870 −2.79455
\(630\) 0 0
\(631\) −8.42971 −0.335581 −0.167791 0.985823i \(-0.553663\pi\)
−0.167791 + 0.985823i \(0.553663\pi\)
\(632\) 0 0
\(633\) 12.0883 0.480468
\(634\) 0 0
\(635\) −21.4498 −0.851210
\(636\) 0 0
\(637\) 29.3915 1.16453
\(638\) 0 0
\(639\) −3.27323 −0.129487
\(640\) 0 0
\(641\) 43.5628 1.72063 0.860313 0.509766i \(-0.170268\pi\)
0.860313 + 0.509766i \(0.170268\pi\)
\(642\) 0 0
\(643\) −10.3839 −0.409500 −0.204750 0.978814i \(-0.565638\pi\)
−0.204750 + 0.978814i \(0.565638\pi\)
\(644\) 0 0
\(645\) −14.6798 −0.578017
\(646\) 0 0
\(647\) 13.4049 0.527003 0.263501 0.964659i \(-0.415123\pi\)
0.263501 + 0.964659i \(0.415123\pi\)
\(648\) 0 0
\(649\) −54.2354 −2.12892
\(650\) 0 0
\(651\) 38.9760 1.52759
\(652\) 0 0
\(653\) 24.6733 0.965539 0.482770 0.875747i \(-0.339631\pi\)
0.482770 + 0.875747i \(0.339631\pi\)
\(654\) 0 0
\(655\) 57.6769 2.25362
\(656\) 0 0
\(657\) −9.70252 −0.378531
\(658\) 0 0
\(659\) 33.9385 1.32206 0.661028 0.750362i \(-0.270121\pi\)
0.661028 + 0.750362i \(0.270121\pi\)
\(660\) 0 0
\(661\) −20.3567 −0.791783 −0.395891 0.918297i \(-0.629564\pi\)
−0.395891 + 0.918297i \(0.629564\pi\)
\(662\) 0 0
\(663\) −36.2284 −1.40699
\(664\) 0 0
\(665\) 62.3628 2.41833
\(666\) 0 0
\(667\) −2.14799 −0.0831706
\(668\) 0 0
\(669\) −6.74178 −0.260652
\(670\) 0 0
\(671\) −56.3670 −2.17602
\(672\) 0 0
\(673\) −23.5647 −0.908352 −0.454176 0.890912i \(-0.650066\pi\)
−0.454176 + 0.890912i \(0.650066\pi\)
\(674\) 0 0
\(675\) −6.21896 −0.239368
\(676\) 0 0
\(677\) 20.6458 0.793482 0.396741 0.917931i \(-0.370141\pi\)
0.396741 + 0.917931i \(0.370141\pi\)
\(678\) 0 0
\(679\) −35.7032 −1.37016
\(680\) 0 0
\(681\) 16.6461 0.637879
\(682\) 0 0
\(683\) −15.2872 −0.584950 −0.292475 0.956273i \(-0.594479\pi\)
−0.292475 + 0.956273i \(0.594479\pi\)
\(684\) 0 0
\(685\) −56.3787 −2.15412
\(686\) 0 0
\(687\) −4.34626 −0.165820
\(688\) 0 0
\(689\) 56.4673 2.15123
\(690\) 0 0
\(691\) 1.05755 0.0402310 0.0201155 0.999798i \(-0.493597\pi\)
0.0201155 + 0.999798i \(0.493597\pi\)
\(692\) 0 0
\(693\) −18.5134 −0.703266
\(694\) 0 0
\(695\) −30.4290 −1.15424
\(696\) 0 0
\(697\) 17.7569 0.672589
\(698\) 0 0
\(699\) −15.4375 −0.583902
\(700\) 0 0
\(701\) −27.2205 −1.02810 −0.514051 0.857759i \(-0.671856\pi\)
−0.514051 + 0.857759i \(0.671856\pi\)
\(702\) 0 0
\(703\) −53.5767 −2.02068
\(704\) 0 0
\(705\) −21.1090 −0.795012
\(706\) 0 0
\(707\) 12.8497 0.483263
\(708\) 0 0
\(709\) 3.84618 0.144446 0.0722231 0.997388i \(-0.476991\pi\)
0.0722231 + 0.997388i \(0.476991\pi\)
\(710\) 0 0
\(711\) −14.2308 −0.533695
\(712\) 0 0
\(713\) −25.5938 −0.958497
\(714\) 0 0
\(715\) −92.2361 −3.44943
\(716\) 0 0
\(717\) −3.91114 −0.146064
\(718\) 0 0
\(719\) 31.7863 1.18543 0.592715 0.805412i \(-0.298056\pi\)
0.592715 + 0.805412i \(0.298056\pi\)
\(720\) 0 0
\(721\) 4.76834 0.177582
\(722\) 0 0
\(723\) −7.89534 −0.293631
\(724\) 0 0
\(725\) −5.73716 −0.213073
\(726\) 0 0
\(727\) 11.8047 0.437813 0.218907 0.975746i \(-0.429751\pi\)
0.218907 + 0.975746i \(0.429751\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.1050 1.11348
\(732\) 0 0
\(733\) −15.2574 −0.563546 −0.281773 0.959481i \(-0.590923\pi\)
−0.281773 + 0.959481i \(0.590923\pi\)
\(734\) 0 0
\(735\) −18.6657 −0.688496
\(736\) 0 0
\(737\) 42.8106 1.57695
\(738\) 0 0
\(739\) −39.0262 −1.43560 −0.717802 0.696248i \(-0.754852\pi\)
−0.717802 + 0.696248i \(0.754852\pi\)
\(740\) 0 0
\(741\) −27.6941 −1.01737
\(742\) 0 0
\(743\) −9.20570 −0.337724 −0.168862 0.985640i \(-0.554009\pi\)
−0.168862 + 0.985640i \(0.554009\pi\)
\(744\) 0 0
\(745\) −35.4100 −1.29732
\(746\) 0 0
\(747\) −12.2173 −0.447008
\(748\) 0 0
\(749\) 18.2274 0.666016
\(750\) 0 0
\(751\) −38.3585 −1.39972 −0.699861 0.714279i \(-0.746755\pi\)
−0.699861 + 0.714279i \(0.746755\pi\)
\(752\) 0 0
\(753\) 9.57349 0.348877
\(754\) 0 0
\(755\) 31.5676 1.14886
\(756\) 0 0
\(757\) 37.1329 1.34962 0.674810 0.737992i \(-0.264226\pi\)
0.674810 + 0.737992i \(0.264226\pi\)
\(758\) 0 0
\(759\) 12.1570 0.441269
\(760\) 0 0
\(761\) −25.9072 −0.939137 −0.469568 0.882896i \(-0.655590\pi\)
−0.469568 + 0.882896i \(0.655590\pi\)
\(762\) 0 0
\(763\) −28.5622 −1.03402
\(764\) 0 0
\(765\) 23.0076 0.831843
\(766\) 0 0
\(767\) 54.7854 1.97818
\(768\) 0 0
\(769\) −4.98095 −0.179618 −0.0898089 0.995959i \(-0.528626\pi\)
−0.0898089 + 0.995959i \(0.528626\pi\)
\(770\) 0 0
\(771\) 22.3318 0.804262
\(772\) 0 0
\(773\) 3.71962 0.133785 0.0668927 0.997760i \(-0.478691\pi\)
0.0668927 + 0.997760i \(0.478691\pi\)
\(774\) 0 0
\(775\) −68.3597 −2.45555
\(776\) 0 0
\(777\) 36.1790 1.29791
\(778\) 0 0
\(779\) 13.5739 0.486336
\(780\) 0 0
\(781\) 17.0902 0.611536
\(782\) 0 0
\(783\) 0.922527 0.0329684
\(784\) 0 0
\(785\) −36.9827 −1.31997
\(786\) 0 0
\(787\) −1.36356 −0.0486056 −0.0243028 0.999705i \(-0.507737\pi\)
−0.0243028 + 0.999705i \(0.507737\pi\)
\(788\) 0 0
\(789\) −19.8026 −0.704990
\(790\) 0 0
\(791\) −63.8821 −2.27138
\(792\) 0 0
\(793\) 56.9386 2.02195
\(794\) 0 0
\(795\) −35.8608 −1.27185
\(796\) 0 0
\(797\) −35.5615 −1.25965 −0.629826 0.776736i \(-0.716874\pi\)
−0.629826 + 0.776736i \(0.716874\pi\)
\(798\) 0 0
\(799\) 43.2900 1.53149
\(800\) 0 0
\(801\) 1.83867 0.0649663
\(802\) 0 0
\(803\) 50.6589 1.78771
\(804\) 0 0
\(805\) 27.6532 0.974646
\(806\) 0 0
\(807\) −13.4173 −0.472312
\(808\) 0 0
\(809\) −27.8127 −0.977842 −0.488921 0.872328i \(-0.662609\pi\)
−0.488921 + 0.872328i \(0.662609\pi\)
\(810\) 0 0
\(811\) −54.8403 −1.92570 −0.962851 0.270033i \(-0.912965\pi\)
−0.962851 + 0.270033i \(0.912965\pi\)
\(812\) 0 0
\(813\) 25.1007 0.880320
\(814\) 0 0
\(815\) 0.684789 0.0239871
\(816\) 0 0
\(817\) 23.0133 0.805132
\(818\) 0 0
\(819\) 18.7011 0.653471
\(820\) 0 0
\(821\) −20.8424 −0.727404 −0.363702 0.931515i \(-0.618487\pi\)
−0.363702 + 0.931515i \(0.618487\pi\)
\(822\) 0 0
\(823\) 14.4050 0.502125 0.251063 0.967971i \(-0.419220\pi\)
0.251063 + 0.967971i \(0.419220\pi\)
\(824\) 0 0
\(825\) 32.4705 1.13048
\(826\) 0 0
\(827\) −18.2464 −0.634490 −0.317245 0.948344i \(-0.602758\pi\)
−0.317245 + 0.948344i \(0.602758\pi\)
\(828\) 0 0
\(829\) −18.2942 −0.635385 −0.317693 0.948194i \(-0.602908\pi\)
−0.317693 + 0.948194i \(0.602908\pi\)
\(830\) 0 0
\(831\) 3.87307 0.134355
\(832\) 0 0
\(833\) 38.2793 1.32630
\(834\) 0 0
\(835\) 3.34947 0.115913
\(836\) 0 0
\(837\) 10.9921 0.379944
\(838\) 0 0
\(839\) −1.08068 −0.0373092 −0.0186546 0.999826i \(-0.505938\pi\)
−0.0186546 + 0.999826i \(0.505938\pi\)
\(840\) 0 0
\(841\) −28.1489 −0.970653
\(842\) 0 0
\(843\) 3.69139 0.127138
\(844\) 0 0
\(845\) 49.6283 1.70727
\(846\) 0 0
\(847\) 57.6585 1.98117
\(848\) 0 0
\(849\) −7.75352 −0.266100
\(850\) 0 0
\(851\) −23.7572 −0.814386
\(852\) 0 0
\(853\) −25.1917 −0.862546 −0.431273 0.902221i \(-0.641935\pi\)
−0.431273 + 0.902221i \(0.641935\pi\)
\(854\) 0 0
\(855\) 17.5878 0.601489
\(856\) 0 0
\(857\) −40.0093 −1.36669 −0.683345 0.730095i \(-0.739476\pi\)
−0.683345 + 0.730095i \(0.739476\pi\)
\(858\) 0 0
\(859\) −55.1265 −1.88089 −0.940445 0.339945i \(-0.889591\pi\)
−0.940445 + 0.339945i \(0.889591\pi\)
\(860\) 0 0
\(861\) −9.16613 −0.312381
\(862\) 0 0
\(863\) −40.1875 −1.36800 −0.684000 0.729482i \(-0.739761\pi\)
−0.684000 + 0.729482i \(0.739761\pi\)
\(864\) 0 0
\(865\) −17.6838 −0.601268
\(866\) 0 0
\(867\) −30.1836 −1.02509
\(868\) 0 0
\(869\) 74.3019 2.52052
\(870\) 0 0
\(871\) −43.2447 −1.46529
\(872\) 0 0
\(873\) −10.0691 −0.340789
\(874\) 0 0
\(875\) 14.4771 0.489416
\(876\) 0 0
\(877\) 26.9315 0.909412 0.454706 0.890642i \(-0.349744\pi\)
0.454706 + 0.890642i \(0.349744\pi\)
\(878\) 0 0
\(879\) 1.01242 0.0341480
\(880\) 0 0
\(881\) 43.4956 1.46540 0.732701 0.680550i \(-0.238259\pi\)
0.732701 + 0.680550i \(0.238259\pi\)
\(882\) 0 0
\(883\) 54.9488 1.84917 0.924587 0.380970i \(-0.124410\pi\)
0.924587 + 0.380970i \(0.124410\pi\)
\(884\) 0 0
\(885\) −34.7927 −1.16954
\(886\) 0 0
\(887\) −45.4217 −1.52511 −0.762556 0.646923i \(-0.776056\pi\)
−0.762556 + 0.646923i \(0.776056\pi\)
\(888\) 0 0
\(889\) −22.7071 −0.761572
\(890\) 0 0
\(891\) −5.22121 −0.174917
\(892\) 0 0
\(893\) 33.0922 1.10739
\(894\) 0 0
\(895\) 64.0495 2.14094
\(896\) 0 0
\(897\) −12.2802 −0.410025
\(898\) 0 0
\(899\) 10.1405 0.338206
\(900\) 0 0
\(901\) 73.5426 2.45006
\(902\) 0 0
\(903\) −15.5403 −0.517148
\(904\) 0 0
\(905\) 82.2717 2.73480
\(906\) 0 0
\(907\) 3.43231 0.113968 0.0569840 0.998375i \(-0.481852\pi\)
0.0569840 + 0.998375i \(0.481852\pi\)
\(908\) 0 0
\(909\) 3.62392 0.120198
\(910\) 0 0
\(911\) −16.2715 −0.539098 −0.269549 0.962987i \(-0.586875\pi\)
−0.269549 + 0.962987i \(0.586875\pi\)
\(912\) 0 0
\(913\) 63.7892 2.11112
\(914\) 0 0
\(915\) −36.1601 −1.19542
\(916\) 0 0
\(917\) 61.0577 2.01630
\(918\) 0 0
\(919\) 47.6271 1.57107 0.785537 0.618815i \(-0.212387\pi\)
0.785537 + 0.618815i \(0.212387\pi\)
\(920\) 0 0
\(921\) −25.9773 −0.855980
\(922\) 0 0
\(923\) −17.2635 −0.568236
\(924\) 0 0
\(925\) −63.4541 −2.08636
\(926\) 0 0
\(927\) 1.34478 0.0441685
\(928\) 0 0
\(929\) 10.6196 0.348417 0.174208 0.984709i \(-0.444263\pi\)
0.174208 + 0.984709i \(0.444263\pi\)
\(930\) 0 0
\(931\) 29.2619 0.959021
\(932\) 0 0
\(933\) 10.2568 0.335792
\(934\) 0 0
\(935\) −120.128 −3.92860
\(936\) 0 0
\(937\) −32.3052 −1.05537 −0.527683 0.849441i \(-0.676939\pi\)
−0.527683 + 0.849441i \(0.676939\pi\)
\(938\) 0 0
\(939\) −13.2574 −0.432638
\(940\) 0 0
\(941\) −34.9102 −1.13804 −0.569020 0.822324i \(-0.692677\pi\)
−0.569020 + 0.822324i \(0.692677\pi\)
\(942\) 0 0
\(943\) 6.01900 0.196006
\(944\) 0 0
\(945\) −11.8766 −0.386345
\(946\) 0 0
\(947\) 1.60729 0.0522299 0.0261149 0.999659i \(-0.491686\pi\)
0.0261149 + 0.999659i \(0.491686\pi\)
\(948\) 0 0
\(949\) −51.1726 −1.66113
\(950\) 0 0
\(951\) −21.0416 −0.682321
\(952\) 0 0
\(953\) −17.1682 −0.556133 −0.278067 0.960562i \(-0.589694\pi\)
−0.278067 + 0.960562i \(0.589694\pi\)
\(954\) 0 0
\(955\) 38.7460 1.25379
\(956\) 0 0
\(957\) −4.81671 −0.155702
\(958\) 0 0
\(959\) −59.6834 −1.92728
\(960\) 0 0
\(961\) 89.8271 2.89765
\(962\) 0 0
\(963\) 5.14056 0.165652
\(964\) 0 0
\(965\) −13.6210 −0.438476
\(966\) 0 0
\(967\) −19.4971 −0.626984 −0.313492 0.949591i \(-0.601499\pi\)
−0.313492 + 0.949591i \(0.601499\pi\)
\(968\) 0 0
\(969\) −36.0686 −1.15869
\(970\) 0 0
\(971\) −20.8748 −0.669903 −0.334952 0.942235i \(-0.608720\pi\)
−0.334952 + 0.942235i \(0.608720\pi\)
\(972\) 0 0
\(973\) −32.2126 −1.03269
\(974\) 0 0
\(975\) −32.7998 −1.05043
\(976\) 0 0
\(977\) −52.7385 −1.68725 −0.843627 0.536930i \(-0.819584\pi\)
−0.843627 + 0.536930i \(0.819584\pi\)
\(978\) 0 0
\(979\) −9.60009 −0.306820
\(980\) 0 0
\(981\) −8.05520 −0.257183
\(982\) 0 0
\(983\) −17.3260 −0.552613 −0.276307 0.961070i \(-0.589110\pi\)
−0.276307 + 0.961070i \(0.589110\pi\)
\(984\) 0 0
\(985\) 82.7512 2.63667
\(986\) 0 0
\(987\) −22.3464 −0.711293
\(988\) 0 0
\(989\) 10.2046 0.324488
\(990\) 0 0
\(991\) 46.9567 1.49163 0.745814 0.666154i \(-0.232061\pi\)
0.745814 + 0.666154i \(0.232061\pi\)
\(992\) 0 0
\(993\) −11.4570 −0.363578
\(994\) 0 0
\(995\) −23.3060 −0.738850
\(996\) 0 0
\(997\) −31.3277 −0.992159 −0.496080 0.868277i \(-0.665228\pi\)
−0.496080 + 0.868277i \(0.665228\pi\)
\(998\) 0 0
\(999\) 10.2033 0.322819
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 4008.2.a.k.1.9 11
4.3 odd 2 8016.2.a.be.1.9 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.k.1.9 11 1.1 even 1 trivial
8016.2.a.be.1.9 11 4.3 odd 2