Properties

Label 8004.2.a.e.1.5
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 13 x^{7} + 32 x^{6} + 40 x^{5} - 79 x^{4} - 39 x^{3} + 58 x^{2} + 9 x - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.761439\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.461945 q^{5} +0.732054 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.461945 q^{5} +0.732054 q^{7} +1.00000 q^{9} +0.317021 q^{11} +1.73156 q^{13} -0.461945 q^{15} -1.85694 q^{17} -0.400818 q^{19} -0.732054 q^{21} -1.00000 q^{23} -4.78661 q^{25} -1.00000 q^{27} +1.00000 q^{29} -2.76393 q^{31} -0.317021 q^{33} +0.338169 q^{35} +6.86987 q^{37} -1.73156 q^{39} +2.88726 q^{41} +7.54434 q^{43} +0.461945 q^{45} -11.3820 q^{47} -6.46410 q^{49} +1.85694 q^{51} -10.8018 q^{53} +0.146446 q^{55} +0.400818 q^{57} +3.88177 q^{59} -4.01686 q^{61} +0.732054 q^{63} +0.799886 q^{65} -6.53778 q^{67} +1.00000 q^{69} -0.411185 q^{71} -13.9499 q^{73} +4.78661 q^{75} +0.232077 q^{77} +11.1974 q^{79} +1.00000 q^{81} +6.15780 q^{83} -0.857805 q^{85} -1.00000 q^{87} -4.04181 q^{89} +1.26760 q^{91} +2.76393 q^{93} -0.185156 q^{95} +10.3181 q^{97} +0.317021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) 0.461945 0.206588 0.103294 0.994651i \(-0.467062\pi\)
0.103294 + 0.994651i \(0.467062\pi\)
\(6\) 0 0
\(7\) 0.732054 0.276691 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.317021 0.0955854 0.0477927 0.998857i \(-0.484781\pi\)
0.0477927 + 0.998857i \(0.484781\pi\)
\(12\) 0 0
\(13\) 1.73156 0.480249 0.240124 0.970742i \(-0.422812\pi\)
0.240124 + 0.970742i \(0.422812\pi\)
\(14\) 0 0
\(15\) −0.461945 −0.119274
\(16\) 0 0
\(17\) −1.85694 −0.450375 −0.225187 0.974315i \(-0.572299\pi\)
−0.225187 + 0.974315i \(0.572299\pi\)
\(18\) 0 0
\(19\) −0.400818 −0.0919540 −0.0459770 0.998942i \(-0.514640\pi\)
−0.0459770 + 0.998942i \(0.514640\pi\)
\(20\) 0 0
\(21\) −0.732054 −0.159747
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.78661 −0.957321
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −2.76393 −0.496417 −0.248208 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) 0 0
\(33\) −0.317021 −0.0551863
\(34\) 0 0
\(35\) 0.338169 0.0571610
\(36\) 0 0
\(37\) 6.86987 1.12940 0.564700 0.825296i \(-0.308992\pi\)
0.564700 + 0.825296i \(0.308992\pi\)
\(38\) 0 0
\(39\) −1.73156 −0.277272
\(40\) 0 0
\(41\) 2.88726 0.450915 0.225457 0.974253i \(-0.427612\pi\)
0.225457 + 0.974253i \(0.427612\pi\)
\(42\) 0 0
\(43\) 7.54434 1.15050 0.575250 0.817977i \(-0.304905\pi\)
0.575250 + 0.817977i \(0.304905\pi\)
\(44\) 0 0
\(45\) 0.461945 0.0688627
\(46\) 0 0
\(47\) −11.3820 −1.66024 −0.830118 0.557588i \(-0.811727\pi\)
−0.830118 + 0.557588i \(0.811727\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923442
\(50\) 0 0
\(51\) 1.85694 0.260024
\(52\) 0 0
\(53\) −10.8018 −1.48374 −0.741872 0.670541i \(-0.766062\pi\)
−0.741872 + 0.670541i \(0.766062\pi\)
\(54\) 0 0
\(55\) 0.146446 0.0197468
\(56\) 0 0
\(57\) 0.400818 0.0530896
\(58\) 0 0
\(59\) 3.88177 0.505364 0.252682 0.967549i \(-0.418687\pi\)
0.252682 + 0.967549i \(0.418687\pi\)
\(60\) 0 0
\(61\) −4.01686 −0.514306 −0.257153 0.966371i \(-0.582784\pi\)
−0.257153 + 0.966371i \(0.582784\pi\)
\(62\) 0 0
\(63\) 0.732054 0.0922302
\(64\) 0 0
\(65\) 0.799886 0.0992137
\(66\) 0 0
\(67\) −6.53778 −0.798716 −0.399358 0.916795i \(-0.630767\pi\)
−0.399358 + 0.916795i \(0.630767\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.411185 −0.0487987 −0.0243994 0.999702i \(-0.507767\pi\)
−0.0243994 + 0.999702i \(0.507767\pi\)
\(72\) 0 0
\(73\) −13.9499 −1.63271 −0.816354 0.577552i \(-0.804008\pi\)
−0.816354 + 0.577552i \(0.804008\pi\)
\(74\) 0 0
\(75\) 4.78661 0.552710
\(76\) 0 0
\(77\) 0.232077 0.0264476
\(78\) 0 0
\(79\) 11.1974 1.25980 0.629902 0.776675i \(-0.283095\pi\)
0.629902 + 0.776675i \(0.283095\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.15780 0.675906 0.337953 0.941163i \(-0.390265\pi\)
0.337953 + 0.941163i \(0.390265\pi\)
\(84\) 0 0
\(85\) −0.857805 −0.0930420
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −4.04181 −0.428431 −0.214216 0.976786i \(-0.568720\pi\)
−0.214216 + 0.976786i \(0.568720\pi\)
\(90\) 0 0
\(91\) 1.26760 0.132880
\(92\) 0 0
\(93\) 2.76393 0.286606
\(94\) 0 0
\(95\) −0.185156 −0.0189966
\(96\) 0 0
\(97\) 10.3181 1.04765 0.523823 0.851827i \(-0.324505\pi\)
0.523823 + 0.851827i \(0.324505\pi\)
\(98\) 0 0
\(99\) 0.317021 0.0318618
\(100\) 0 0
\(101\) −11.5404 −1.14832 −0.574159 0.818744i \(-0.694671\pi\)
−0.574159 + 0.818744i \(0.694671\pi\)
\(102\) 0 0
\(103\) −0.595012 −0.0586283 −0.0293142 0.999570i \(-0.509332\pi\)
−0.0293142 + 0.999570i \(0.509332\pi\)
\(104\) 0 0
\(105\) −0.338169 −0.0330019
\(106\) 0 0
\(107\) 4.20690 0.406696 0.203348 0.979107i \(-0.434818\pi\)
0.203348 + 0.979107i \(0.434818\pi\)
\(108\) 0 0
\(109\) 1.09440 0.104824 0.0524122 0.998626i \(-0.483309\pi\)
0.0524122 + 0.998626i \(0.483309\pi\)
\(110\) 0 0
\(111\) −6.86987 −0.652059
\(112\) 0 0
\(113\) −2.54021 −0.238963 −0.119481 0.992836i \(-0.538123\pi\)
−0.119481 + 0.992836i \(0.538123\pi\)
\(114\) 0 0
\(115\) −0.461945 −0.0430766
\(116\) 0 0
\(117\) 1.73156 0.160083
\(118\) 0 0
\(119\) −1.35938 −0.124614
\(120\) 0 0
\(121\) −10.8995 −0.990863
\(122\) 0 0
\(123\) −2.88726 −0.260336
\(124\) 0 0
\(125\) −4.52087 −0.404359
\(126\) 0 0
\(127\) −3.35997 −0.298149 −0.149075 0.988826i \(-0.547630\pi\)
−0.149075 + 0.988826i \(0.547630\pi\)
\(128\) 0 0
\(129\) −7.54434 −0.664242
\(130\) 0 0
\(131\) 4.65661 0.406850 0.203425 0.979091i \(-0.434793\pi\)
0.203425 + 0.979091i \(0.434793\pi\)
\(132\) 0 0
\(133\) −0.293421 −0.0254428
\(134\) 0 0
\(135\) −0.461945 −0.0397579
\(136\) 0 0
\(137\) −13.5798 −1.16020 −0.580102 0.814544i \(-0.696987\pi\)
−0.580102 + 0.814544i \(0.696987\pi\)
\(138\) 0 0
\(139\) −11.8371 −1.00401 −0.502005 0.864865i \(-0.667404\pi\)
−0.502005 + 0.864865i \(0.667404\pi\)
\(140\) 0 0
\(141\) 11.3820 0.958538
\(142\) 0 0
\(143\) 0.548941 0.0459048
\(144\) 0 0
\(145\) 0.461945 0.0383624
\(146\) 0 0
\(147\) 6.46410 0.533150
\(148\) 0 0
\(149\) 2.83537 0.232283 0.116141 0.993233i \(-0.462947\pi\)
0.116141 + 0.993233i \(0.462947\pi\)
\(150\) 0 0
\(151\) −7.46767 −0.607711 −0.303855 0.952718i \(-0.598274\pi\)
−0.303855 + 0.952718i \(0.598274\pi\)
\(152\) 0 0
\(153\) −1.85694 −0.150125
\(154\) 0 0
\(155\) −1.27678 −0.102554
\(156\) 0 0
\(157\) 15.3488 1.22497 0.612483 0.790484i \(-0.290171\pi\)
0.612483 + 0.790484i \(0.290171\pi\)
\(158\) 0 0
\(159\) 10.8018 0.856641
\(160\) 0 0
\(161\) −0.732054 −0.0576940
\(162\) 0 0
\(163\) −0.837150 −0.0655706 −0.0327853 0.999462i \(-0.510438\pi\)
−0.0327853 + 0.999462i \(0.510438\pi\)
\(164\) 0 0
\(165\) −0.146446 −0.0114008
\(166\) 0 0
\(167\) 18.4774 1.42982 0.714912 0.699214i \(-0.246467\pi\)
0.714912 + 0.699214i \(0.246467\pi\)
\(168\) 0 0
\(169\) −10.0017 −0.769361
\(170\) 0 0
\(171\) −0.400818 −0.0306513
\(172\) 0 0
\(173\) −11.9553 −0.908943 −0.454471 0.890761i \(-0.650172\pi\)
−0.454471 + 0.890761i \(0.650172\pi\)
\(174\) 0 0
\(175\) −3.50406 −0.264882
\(176\) 0 0
\(177\) −3.88177 −0.291772
\(178\) 0 0
\(179\) −5.15766 −0.385502 −0.192751 0.981248i \(-0.561741\pi\)
−0.192751 + 0.981248i \(0.561741\pi\)
\(180\) 0 0
\(181\) 26.3614 1.95942 0.979712 0.200409i \(-0.0642273\pi\)
0.979712 + 0.200409i \(0.0642273\pi\)
\(182\) 0 0
\(183\) 4.01686 0.296935
\(184\) 0 0
\(185\) 3.17350 0.233321
\(186\) 0 0
\(187\) −0.588689 −0.0430492
\(188\) 0 0
\(189\) −0.732054 −0.0532491
\(190\) 0 0
\(191\) −12.6094 −0.912385 −0.456192 0.889881i \(-0.650787\pi\)
−0.456192 + 0.889881i \(0.650787\pi\)
\(192\) 0 0
\(193\) 5.78062 0.416098 0.208049 0.978118i \(-0.433289\pi\)
0.208049 + 0.978118i \(0.433289\pi\)
\(194\) 0 0
\(195\) −0.799886 −0.0572810
\(196\) 0 0
\(197\) 9.44739 0.673099 0.336549 0.941666i \(-0.390740\pi\)
0.336549 + 0.941666i \(0.390740\pi\)
\(198\) 0 0
\(199\) 14.8323 1.05143 0.525717 0.850659i \(-0.323797\pi\)
0.525717 + 0.850659i \(0.323797\pi\)
\(200\) 0 0
\(201\) 6.53778 0.461139
\(202\) 0 0
\(203\) 0.732054 0.0513801
\(204\) 0 0
\(205\) 1.33376 0.0931536
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −0.127068 −0.00878946
\(210\) 0 0
\(211\) −0.306888 −0.0211270 −0.0105635 0.999944i \(-0.503363\pi\)
−0.0105635 + 0.999944i \(0.503363\pi\)
\(212\) 0 0
\(213\) 0.411185 0.0281739
\(214\) 0 0
\(215\) 3.48507 0.237680
\(216\) 0 0
\(217\) −2.02335 −0.137354
\(218\) 0 0
\(219\) 13.9499 0.942645
\(220\) 0 0
\(221\) −3.21541 −0.216292
\(222\) 0 0
\(223\) 6.86424 0.459664 0.229832 0.973230i \(-0.426182\pi\)
0.229832 + 0.973230i \(0.426182\pi\)
\(224\) 0 0
\(225\) −4.78661 −0.319107
\(226\) 0 0
\(227\) 11.9875 0.795640 0.397820 0.917463i \(-0.369767\pi\)
0.397820 + 0.917463i \(0.369767\pi\)
\(228\) 0 0
\(229\) −8.07143 −0.533375 −0.266687 0.963783i \(-0.585929\pi\)
−0.266687 + 0.963783i \(0.585929\pi\)
\(230\) 0 0
\(231\) −0.232077 −0.0152695
\(232\) 0 0
\(233\) 19.4991 1.27743 0.638715 0.769444i \(-0.279466\pi\)
0.638715 + 0.769444i \(0.279466\pi\)
\(234\) 0 0
\(235\) −5.25786 −0.342985
\(236\) 0 0
\(237\) −11.1974 −0.727348
\(238\) 0 0
\(239\) −26.4626 −1.71173 −0.855863 0.517203i \(-0.826973\pi\)
−0.855863 + 0.517203i \(0.826973\pi\)
\(240\) 0 0
\(241\) 4.62601 0.297988 0.148994 0.988838i \(-0.452397\pi\)
0.148994 + 0.988838i \(0.452397\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.98606 −0.190772
\(246\) 0 0
\(247\) −0.694041 −0.0441608
\(248\) 0 0
\(249\) −6.15780 −0.390235
\(250\) 0 0
\(251\) −24.5425 −1.54911 −0.774554 0.632507i \(-0.782026\pi\)
−0.774554 + 0.632507i \(0.782026\pi\)
\(252\) 0 0
\(253\) −0.317021 −0.0199309
\(254\) 0 0
\(255\) 0.857805 0.0537178
\(256\) 0 0
\(257\) 4.88448 0.304685 0.152343 0.988328i \(-0.451318\pi\)
0.152343 + 0.988328i \(0.451318\pi\)
\(258\) 0 0
\(259\) 5.02912 0.312494
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −2.37549 −0.146479 −0.0732394 0.997314i \(-0.523334\pi\)
−0.0732394 + 0.997314i \(0.523334\pi\)
\(264\) 0 0
\(265\) −4.98985 −0.306524
\(266\) 0 0
\(267\) 4.04181 0.247355
\(268\) 0 0
\(269\) −12.0953 −0.737462 −0.368731 0.929536i \(-0.620208\pi\)
−0.368731 + 0.929536i \(0.620208\pi\)
\(270\) 0 0
\(271\) 8.76866 0.532658 0.266329 0.963882i \(-0.414189\pi\)
0.266329 + 0.963882i \(0.414189\pi\)
\(272\) 0 0
\(273\) −1.26760 −0.0767185
\(274\) 0 0
\(275\) −1.51745 −0.0915060
\(276\) 0 0
\(277\) −31.7407 −1.90712 −0.953558 0.301209i \(-0.902610\pi\)
−0.953558 + 0.301209i \(0.902610\pi\)
\(278\) 0 0
\(279\) −2.76393 −0.165472
\(280\) 0 0
\(281\) −1.92770 −0.114997 −0.0574985 0.998346i \(-0.518312\pi\)
−0.0574985 + 0.998346i \(0.518312\pi\)
\(282\) 0 0
\(283\) −3.84260 −0.228419 −0.114209 0.993457i \(-0.536433\pi\)
−0.114209 + 0.993457i \(0.536433\pi\)
\(284\) 0 0
\(285\) 0.185156 0.0109677
\(286\) 0 0
\(287\) 2.11363 0.124764
\(288\) 0 0
\(289\) −13.5518 −0.797163
\(290\) 0 0
\(291\) −10.3181 −0.604858
\(292\) 0 0
\(293\) −19.6858 −1.15005 −0.575027 0.818134i \(-0.695008\pi\)
−0.575027 + 0.818134i \(0.695008\pi\)
\(294\) 0 0
\(295\) 1.79317 0.104402
\(296\) 0 0
\(297\) −0.317021 −0.0183954
\(298\) 0 0
\(299\) −1.73156 −0.100139
\(300\) 0 0
\(301\) 5.52287 0.318333
\(302\) 0 0
\(303\) 11.5404 0.662981
\(304\) 0 0
\(305\) −1.85557 −0.106250
\(306\) 0 0
\(307\) −16.2630 −0.928177 −0.464089 0.885789i \(-0.653618\pi\)
−0.464089 + 0.885789i \(0.653618\pi\)
\(308\) 0 0
\(309\) 0.595012 0.0338491
\(310\) 0 0
\(311\) −11.8702 −0.673098 −0.336549 0.941666i \(-0.609260\pi\)
−0.336549 + 0.941666i \(0.609260\pi\)
\(312\) 0 0
\(313\) 12.8326 0.725339 0.362669 0.931918i \(-0.381865\pi\)
0.362669 + 0.931918i \(0.381865\pi\)
\(314\) 0 0
\(315\) 0.338169 0.0190537
\(316\) 0 0
\(317\) −8.37675 −0.470485 −0.235243 0.971937i \(-0.575588\pi\)
−0.235243 + 0.971937i \(0.575588\pi\)
\(318\) 0 0
\(319\) 0.317021 0.0177498
\(320\) 0 0
\(321\) −4.20690 −0.234806
\(322\) 0 0
\(323\) 0.744296 0.0414137
\(324\) 0 0
\(325\) −8.28830 −0.459752
\(326\) 0 0
\(327\) −1.09440 −0.0605203
\(328\) 0 0
\(329\) −8.33224 −0.459372
\(330\) 0 0
\(331\) 31.7411 1.74465 0.872323 0.488930i \(-0.162613\pi\)
0.872323 + 0.488930i \(0.162613\pi\)
\(332\) 0 0
\(333\) 6.86987 0.376467
\(334\) 0 0
\(335\) −3.02009 −0.165005
\(336\) 0 0
\(337\) 0.506781 0.0276061 0.0138031 0.999905i \(-0.495606\pi\)
0.0138031 + 0.999905i \(0.495606\pi\)
\(338\) 0 0
\(339\) 2.54021 0.137965
\(340\) 0 0
\(341\) −0.876224 −0.0474502
\(342\) 0 0
\(343\) −9.85645 −0.532198
\(344\) 0 0
\(345\) 0.461945 0.0248703
\(346\) 0 0
\(347\) −34.6598 −1.86064 −0.930319 0.366751i \(-0.880470\pi\)
−0.930319 + 0.366751i \(0.880470\pi\)
\(348\) 0 0
\(349\) −7.55271 −0.404288 −0.202144 0.979356i \(-0.564791\pi\)
−0.202144 + 0.979356i \(0.564791\pi\)
\(350\) 0 0
\(351\) −1.73156 −0.0924239
\(352\) 0 0
\(353\) −11.3957 −0.606530 −0.303265 0.952906i \(-0.598077\pi\)
−0.303265 + 0.952906i \(0.598077\pi\)
\(354\) 0 0
\(355\) −0.189945 −0.0100812
\(356\) 0 0
\(357\) 1.35938 0.0719461
\(358\) 0 0
\(359\) −19.8148 −1.04579 −0.522894 0.852398i \(-0.675147\pi\)
−0.522894 + 0.852398i \(0.675147\pi\)
\(360\) 0 0
\(361\) −18.8393 −0.991544
\(362\) 0 0
\(363\) 10.8995 0.572075
\(364\) 0 0
\(365\) −6.44407 −0.337298
\(366\) 0 0
\(367\) −6.49275 −0.338919 −0.169459 0.985537i \(-0.554202\pi\)
−0.169459 + 0.985537i \(0.554202\pi\)
\(368\) 0 0
\(369\) 2.88726 0.150305
\(370\) 0 0
\(371\) −7.90752 −0.410538
\(372\) 0 0
\(373\) 3.33591 0.172727 0.0863634 0.996264i \(-0.472475\pi\)
0.0863634 + 0.996264i \(0.472475\pi\)
\(374\) 0 0
\(375\) 4.52087 0.233457
\(376\) 0 0
\(377\) 1.73156 0.0891800
\(378\) 0 0
\(379\) 21.1538 1.08660 0.543298 0.839540i \(-0.317176\pi\)
0.543298 + 0.839540i \(0.317176\pi\)
\(380\) 0 0
\(381\) 3.35997 0.172137
\(382\) 0 0
\(383\) −0.256376 −0.0131002 −0.00655010 0.999979i \(-0.502085\pi\)
−0.00655010 + 0.999979i \(0.502085\pi\)
\(384\) 0 0
\(385\) 0.107207 0.00546375
\(386\) 0 0
\(387\) 7.54434 0.383500
\(388\) 0 0
\(389\) −30.2812 −1.53532 −0.767659 0.640859i \(-0.778578\pi\)
−0.767659 + 0.640859i \(0.778578\pi\)
\(390\) 0 0
\(391\) 1.85694 0.0939096
\(392\) 0 0
\(393\) −4.65661 −0.234895
\(394\) 0 0
\(395\) 5.17257 0.260260
\(396\) 0 0
\(397\) −21.3664 −1.07235 −0.536174 0.844107i \(-0.680131\pi\)
−0.536174 + 0.844107i \(0.680131\pi\)
\(398\) 0 0
\(399\) 0.293421 0.0146894
\(400\) 0 0
\(401\) 0.637327 0.0318266 0.0159133 0.999873i \(-0.494934\pi\)
0.0159133 + 0.999873i \(0.494934\pi\)
\(402\) 0 0
\(403\) −4.78592 −0.238404
\(404\) 0 0
\(405\) 0.461945 0.0229542
\(406\) 0 0
\(407\) 2.17789 0.107954
\(408\) 0 0
\(409\) −25.4021 −1.25605 −0.628025 0.778193i \(-0.716137\pi\)
−0.628025 + 0.778193i \(0.716137\pi\)
\(410\) 0 0
\(411\) 13.5798 0.669844
\(412\) 0 0
\(413\) 2.84167 0.139829
\(414\) 0 0
\(415\) 2.84456 0.139634
\(416\) 0 0
\(417\) 11.8371 0.579665
\(418\) 0 0
\(419\) 10.0239 0.489698 0.244849 0.969561i \(-0.421262\pi\)
0.244849 + 0.969561i \(0.421262\pi\)
\(420\) 0 0
\(421\) −17.9276 −0.873736 −0.436868 0.899526i \(-0.643912\pi\)
−0.436868 + 0.899526i \(0.643912\pi\)
\(422\) 0 0
\(423\) −11.3820 −0.553412
\(424\) 0 0
\(425\) 8.88845 0.431153
\(426\) 0 0
\(427\) −2.94056 −0.142304
\(428\) 0 0
\(429\) −0.548941 −0.0265031
\(430\) 0 0
\(431\) −13.7837 −0.663935 −0.331968 0.943291i \(-0.607713\pi\)
−0.331968 + 0.943291i \(0.607713\pi\)
\(432\) 0 0
\(433\) −0.149916 −0.00720452 −0.00360226 0.999994i \(-0.501147\pi\)
−0.00360226 + 0.999994i \(0.501147\pi\)
\(434\) 0 0
\(435\) −0.461945 −0.0221486
\(436\) 0 0
\(437\) 0.400818 0.0191737
\(438\) 0 0
\(439\) 23.7877 1.13533 0.567663 0.823261i \(-0.307848\pi\)
0.567663 + 0.823261i \(0.307848\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) 0 0
\(443\) 15.0166 0.713462 0.356731 0.934207i \(-0.383891\pi\)
0.356731 + 0.934207i \(0.383891\pi\)
\(444\) 0 0
\(445\) −1.86710 −0.0885088
\(446\) 0 0
\(447\) −2.83537 −0.134108
\(448\) 0 0
\(449\) −1.20526 −0.0568798 −0.0284399 0.999596i \(-0.509054\pi\)
−0.0284399 + 0.999596i \(0.509054\pi\)
\(450\) 0 0
\(451\) 0.915323 0.0431009
\(452\) 0 0
\(453\) 7.46767 0.350862
\(454\) 0 0
\(455\) 0.585560 0.0274515
\(456\) 0 0
\(457\) −34.3509 −1.60687 −0.803433 0.595395i \(-0.796996\pi\)
−0.803433 + 0.595395i \(0.796996\pi\)
\(458\) 0 0
\(459\) 1.85694 0.0866746
\(460\) 0 0
\(461\) 23.6738 1.10260 0.551301 0.834307i \(-0.314132\pi\)
0.551301 + 0.834307i \(0.314132\pi\)
\(462\) 0 0
\(463\) −7.41151 −0.344442 −0.172221 0.985058i \(-0.555094\pi\)
−0.172221 + 0.985058i \(0.555094\pi\)
\(464\) 0 0
\(465\) 1.27678 0.0592095
\(466\) 0 0
\(467\) 26.8498 1.24246 0.621230 0.783628i \(-0.286633\pi\)
0.621230 + 0.783628i \(0.286633\pi\)
\(468\) 0 0
\(469\) −4.78601 −0.220997
\(470\) 0 0
\(471\) −15.3488 −0.707234
\(472\) 0 0
\(473\) 2.39171 0.109971
\(474\) 0 0
\(475\) 1.91856 0.0880295
\(476\) 0 0
\(477\) −10.8018 −0.494582
\(478\) 0 0
\(479\) −19.7862 −0.904054 −0.452027 0.892004i \(-0.649299\pi\)
−0.452027 + 0.892004i \(0.649299\pi\)
\(480\) 0 0
\(481\) 11.8956 0.542393
\(482\) 0 0
\(483\) 0.732054 0.0333096
\(484\) 0 0
\(485\) 4.76640 0.216431
\(486\) 0 0
\(487\) 14.0323 0.635864 0.317932 0.948114i \(-0.397012\pi\)
0.317932 + 0.948114i \(0.397012\pi\)
\(488\) 0 0
\(489\) 0.837150 0.0378572
\(490\) 0 0
\(491\) 15.7827 0.712263 0.356132 0.934436i \(-0.384095\pi\)
0.356132 + 0.934436i \(0.384095\pi\)
\(492\) 0 0
\(493\) −1.85694 −0.0836325
\(494\) 0 0
\(495\) 0.146446 0.00658227
\(496\) 0 0
\(497\) −0.301010 −0.0135021
\(498\) 0 0
\(499\) 33.7879 1.51255 0.756277 0.654251i \(-0.227016\pi\)
0.756277 + 0.654251i \(0.227016\pi\)
\(500\) 0 0
\(501\) −18.4774 −0.825510
\(502\) 0 0
\(503\) 9.28079 0.413810 0.206905 0.978361i \(-0.433661\pi\)
0.206905 + 0.978361i \(0.433661\pi\)
\(504\) 0 0
\(505\) −5.33105 −0.237229
\(506\) 0 0
\(507\) 10.0017 0.444191
\(508\) 0 0
\(509\) −33.4886 −1.48436 −0.742178 0.670203i \(-0.766207\pi\)
−0.742178 + 0.670203i \(0.766207\pi\)
\(510\) 0 0
\(511\) −10.2121 −0.451755
\(512\) 0 0
\(513\) 0.400818 0.0176965
\(514\) 0 0
\(515\) −0.274863 −0.0121119
\(516\) 0 0
\(517\) −3.60833 −0.158694
\(518\) 0 0
\(519\) 11.9553 0.524778
\(520\) 0 0
\(521\) −5.16840 −0.226432 −0.113216 0.993570i \(-0.536115\pi\)
−0.113216 + 0.993570i \(0.536115\pi\)
\(522\) 0 0
\(523\) −41.5421 −1.81651 −0.908255 0.418418i \(-0.862585\pi\)
−0.908255 + 0.418418i \(0.862585\pi\)
\(524\) 0 0
\(525\) 3.50406 0.152930
\(526\) 0 0
\(527\) 5.13246 0.223574
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.88177 0.168455
\(532\) 0 0
\(533\) 4.99947 0.216551
\(534\) 0 0
\(535\) 1.94336 0.0840186
\(536\) 0 0
\(537\) 5.15766 0.222570
\(538\) 0 0
\(539\) −2.04925 −0.0882676
\(540\) 0 0
\(541\) −32.0593 −1.37834 −0.689169 0.724601i \(-0.742024\pi\)
−0.689169 + 0.724601i \(0.742024\pi\)
\(542\) 0 0
\(543\) −26.3614 −1.13127
\(544\) 0 0
\(545\) 0.505552 0.0216555
\(546\) 0 0
\(547\) −7.66351 −0.327668 −0.163834 0.986488i \(-0.552386\pi\)
−0.163834 + 0.986488i \(0.552386\pi\)
\(548\) 0 0
\(549\) −4.01686 −0.171435
\(550\) 0 0
\(551\) −0.400818 −0.0170754
\(552\) 0 0
\(553\) 8.19709 0.348576
\(554\) 0 0
\(555\) −3.17350 −0.134708
\(556\) 0 0
\(557\) −12.8923 −0.546266 −0.273133 0.961976i \(-0.588060\pi\)
−0.273133 + 0.961976i \(0.588060\pi\)
\(558\) 0 0
\(559\) 13.0635 0.552527
\(560\) 0 0
\(561\) 0.588689 0.0248545
\(562\) 0 0
\(563\) 4.73204 0.199432 0.0997158 0.995016i \(-0.468207\pi\)
0.0997158 + 0.995016i \(0.468207\pi\)
\(564\) 0 0
\(565\) −1.17344 −0.0493668
\(566\) 0 0
\(567\) 0.732054 0.0307434
\(568\) 0 0
\(569\) 5.82884 0.244358 0.122179 0.992508i \(-0.461012\pi\)
0.122179 + 0.992508i \(0.461012\pi\)
\(570\) 0 0
\(571\) −42.1737 −1.76492 −0.882458 0.470392i \(-0.844113\pi\)
−0.882458 + 0.470392i \(0.844113\pi\)
\(572\) 0 0
\(573\) 12.6094 0.526766
\(574\) 0 0
\(575\) 4.78661 0.199615
\(576\) 0 0
\(577\) −26.8790 −1.11899 −0.559495 0.828834i \(-0.689005\pi\)
−0.559495 + 0.828834i \(0.689005\pi\)
\(578\) 0 0
\(579\) −5.78062 −0.240234
\(580\) 0 0
\(581\) 4.50784 0.187017
\(582\) 0 0
\(583\) −3.42441 −0.141824
\(584\) 0 0
\(585\) 0.799886 0.0330712
\(586\) 0 0
\(587\) −7.54458 −0.311398 −0.155699 0.987805i \(-0.549763\pi\)
−0.155699 + 0.987805i \(0.549763\pi\)
\(588\) 0 0
\(589\) 1.10783 0.0456475
\(590\) 0 0
\(591\) −9.44739 −0.388614
\(592\) 0 0
\(593\) −8.59461 −0.352938 −0.176469 0.984306i \(-0.556468\pi\)
−0.176469 + 0.984306i \(0.556468\pi\)
\(594\) 0 0
\(595\) −0.627960 −0.0257438
\(596\) 0 0
\(597\) −14.8323 −0.607046
\(598\) 0 0
\(599\) 32.5984 1.33193 0.665967 0.745982i \(-0.268019\pi\)
0.665967 + 0.745982i \(0.268019\pi\)
\(600\) 0 0
\(601\) 40.4526 1.65010 0.825049 0.565061i \(-0.191148\pi\)
0.825049 + 0.565061i \(0.191148\pi\)
\(602\) 0 0
\(603\) −6.53778 −0.266239
\(604\) 0 0
\(605\) −5.03497 −0.204701
\(606\) 0 0
\(607\) 12.1015 0.491185 0.245592 0.969373i \(-0.421018\pi\)
0.245592 + 0.969373i \(0.421018\pi\)
\(608\) 0 0
\(609\) −0.732054 −0.0296643
\(610\) 0 0
\(611\) −19.7086 −0.797326
\(612\) 0 0
\(613\) −20.4357 −0.825391 −0.412695 0.910869i \(-0.635413\pi\)
−0.412695 + 0.910869i \(0.635413\pi\)
\(614\) 0 0
\(615\) −1.33376 −0.0537823
\(616\) 0 0
\(617\) −5.24679 −0.211228 −0.105614 0.994407i \(-0.533681\pi\)
−0.105614 + 0.994407i \(0.533681\pi\)
\(618\) 0 0
\(619\) 20.6009 0.828021 0.414011 0.910272i \(-0.364128\pi\)
0.414011 + 0.910272i \(0.364128\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −2.95883 −0.118543
\(624\) 0 0
\(625\) 21.8446 0.873786
\(626\) 0 0
\(627\) 0.127068 0.00507460
\(628\) 0 0
\(629\) −12.7570 −0.508653
\(630\) 0 0
\(631\) −34.3896 −1.36903 −0.684515 0.728999i \(-0.739986\pi\)
−0.684515 + 0.728999i \(0.739986\pi\)
\(632\) 0 0
\(633\) 0.306888 0.0121977
\(634\) 0 0
\(635\) −1.55212 −0.0615941
\(636\) 0 0
\(637\) −11.1930 −0.443482
\(638\) 0 0
\(639\) −0.411185 −0.0162662
\(640\) 0 0
\(641\) −0.0313102 −0.00123668 −0.000618339 1.00000i \(-0.500197\pi\)
−0.000618339 1.00000i \(0.500197\pi\)
\(642\) 0 0
\(643\) 19.9158 0.785404 0.392702 0.919666i \(-0.371540\pi\)
0.392702 + 0.919666i \(0.371540\pi\)
\(644\) 0 0
\(645\) −3.48507 −0.137224
\(646\) 0 0
\(647\) 10.9631 0.431003 0.215501 0.976504i \(-0.430861\pi\)
0.215501 + 0.976504i \(0.430861\pi\)
\(648\) 0 0
\(649\) 1.23060 0.0483054
\(650\) 0 0
\(651\) 2.02335 0.0793013
\(652\) 0 0
\(653\) −40.5105 −1.58530 −0.792650 0.609678i \(-0.791299\pi\)
−0.792650 + 0.609678i \(0.791299\pi\)
\(654\) 0 0
\(655\) 2.15110 0.0840503
\(656\) 0 0
\(657\) −13.9499 −0.544236
\(658\) 0 0
\(659\) −30.5209 −1.18893 −0.594463 0.804123i \(-0.702635\pi\)
−0.594463 + 0.804123i \(0.702635\pi\)
\(660\) 0 0
\(661\) 29.5037 1.14756 0.573780 0.819009i \(-0.305476\pi\)
0.573780 + 0.819009i \(0.305476\pi\)
\(662\) 0 0
\(663\) 3.21541 0.124876
\(664\) 0 0
\(665\) −0.135544 −0.00525618
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −6.86424 −0.265387
\(670\) 0 0
\(671\) −1.27343 −0.0491602
\(672\) 0 0
\(673\) 11.1124 0.428352 0.214176 0.976795i \(-0.431293\pi\)
0.214176 + 0.976795i \(0.431293\pi\)
\(674\) 0 0
\(675\) 4.78661 0.184237
\(676\) 0 0
\(677\) −18.8437 −0.724224 −0.362112 0.932135i \(-0.617944\pi\)
−0.362112 + 0.932135i \(0.617944\pi\)
\(678\) 0 0
\(679\) 7.55342 0.289874
\(680\) 0 0
\(681\) −11.9875 −0.459363
\(682\) 0 0
\(683\) −36.7086 −1.40461 −0.702307 0.711874i \(-0.747847\pi\)
−0.702307 + 0.711874i \(0.747847\pi\)
\(684\) 0 0
\(685\) −6.27314 −0.239684
\(686\) 0 0
\(687\) 8.07143 0.307944
\(688\) 0 0
\(689\) −18.7040 −0.712567
\(690\) 0 0
\(691\) 10.0074 0.380698 0.190349 0.981717i \(-0.439038\pi\)
0.190349 + 0.981717i \(0.439038\pi\)
\(692\) 0 0
\(693\) 0.232077 0.00881586
\(694\) 0 0
\(695\) −5.46809 −0.207416
\(696\) 0 0
\(697\) −5.36148 −0.203081
\(698\) 0 0
\(699\) −19.4991 −0.737524
\(700\) 0 0
\(701\) −16.3727 −0.618387 −0.309194 0.950999i \(-0.600059\pi\)
−0.309194 + 0.950999i \(0.600059\pi\)
\(702\) 0 0
\(703\) −2.75357 −0.103853
\(704\) 0 0
\(705\) 5.25786 0.198022
\(706\) 0 0
\(707\) −8.44824 −0.317729
\(708\) 0 0
\(709\) 32.6058 1.22454 0.612269 0.790649i \(-0.290257\pi\)
0.612269 + 0.790649i \(0.290257\pi\)
\(710\) 0 0
\(711\) 11.1974 0.419934
\(712\) 0 0
\(713\) 2.76393 0.103510
\(714\) 0 0
\(715\) 0.253581 0.00948338
\(716\) 0 0
\(717\) 26.4626 0.988265
\(718\) 0 0
\(719\) −28.9583 −1.07996 −0.539981 0.841677i \(-0.681569\pi\)
−0.539981 + 0.841677i \(0.681569\pi\)
\(720\) 0 0
\(721\) −0.435581 −0.0162219
\(722\) 0 0
\(723\) −4.62601 −0.172043
\(724\) 0 0
\(725\) −4.78661 −0.177770
\(726\) 0 0
\(727\) 9.81774 0.364120 0.182060 0.983287i \(-0.441724\pi\)
0.182060 + 0.983287i \(0.441724\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.0094 −0.518156
\(732\) 0 0
\(733\) −16.1524 −0.596602 −0.298301 0.954472i \(-0.596420\pi\)
−0.298301 + 0.954472i \(0.596420\pi\)
\(734\) 0 0
\(735\) 2.98606 0.110142
\(736\) 0 0
\(737\) −2.07261 −0.0763456
\(738\) 0 0
\(739\) −20.0100 −0.736080 −0.368040 0.929810i \(-0.619971\pi\)
−0.368040 + 0.929810i \(0.619971\pi\)
\(740\) 0 0
\(741\) 0.694041 0.0254962
\(742\) 0 0
\(743\) 39.3919 1.44515 0.722574 0.691293i \(-0.242959\pi\)
0.722574 + 0.691293i \(0.242959\pi\)
\(744\) 0 0
\(745\) 1.30978 0.0479868
\(746\) 0 0
\(747\) 6.15780 0.225302
\(748\) 0 0
\(749\) 3.07968 0.112529
\(750\) 0 0
\(751\) 34.4945 1.25872 0.629361 0.777113i \(-0.283317\pi\)
0.629361 + 0.777113i \(0.283317\pi\)
\(752\) 0 0
\(753\) 24.5425 0.894378
\(754\) 0 0
\(755\) −3.44966 −0.125546
\(756\) 0 0
\(757\) 19.9383 0.724668 0.362334 0.932048i \(-0.381980\pi\)
0.362334 + 0.932048i \(0.381980\pi\)
\(758\) 0 0
\(759\) 0.317021 0.0115071
\(760\) 0 0
\(761\) −33.1351 −1.20115 −0.600573 0.799570i \(-0.705061\pi\)
−0.600573 + 0.799570i \(0.705061\pi\)
\(762\) 0 0
\(763\) 0.801159 0.0290039
\(764\) 0 0
\(765\) −0.857805 −0.0310140
\(766\) 0 0
\(767\) 6.72153 0.242700
\(768\) 0 0
\(769\) 27.4214 0.988842 0.494421 0.869223i \(-0.335380\pi\)
0.494421 + 0.869223i \(0.335380\pi\)
\(770\) 0 0
\(771\) −4.88448 −0.175910
\(772\) 0 0
\(773\) −5.99277 −0.215545 −0.107773 0.994176i \(-0.534372\pi\)
−0.107773 + 0.994176i \(0.534372\pi\)
\(774\) 0 0
\(775\) 13.2299 0.475231
\(776\) 0 0
\(777\) −5.02912 −0.180419
\(778\) 0 0
\(779\) −1.15727 −0.0414634
\(780\) 0 0
\(781\) −0.130354 −0.00466444
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 7.09029 0.253063
\(786\) 0 0
\(787\) −2.17329 −0.0774694 −0.0387347 0.999250i \(-0.512333\pi\)
−0.0387347 + 0.999250i \(0.512333\pi\)
\(788\) 0 0
\(789\) 2.37549 0.0845696
\(790\) 0 0
\(791\) −1.85957 −0.0661187
\(792\) 0 0
\(793\) −6.95544 −0.246995
\(794\) 0 0
\(795\) 4.98985 0.176972
\(796\) 0 0
\(797\) −11.7622 −0.416638 −0.208319 0.978061i \(-0.566799\pi\)
−0.208319 + 0.978061i \(0.566799\pi\)
\(798\) 0 0
\(799\) 21.1357 0.747728
\(800\) 0 0
\(801\) −4.04181 −0.142810
\(802\) 0 0
\(803\) −4.42240 −0.156063
\(804\) 0 0
\(805\) −0.338169 −0.0119189
\(806\) 0 0
\(807\) 12.0953 0.425774
\(808\) 0 0
\(809\) 37.8832 1.33190 0.665950 0.745996i \(-0.268026\pi\)
0.665950 + 0.745996i \(0.268026\pi\)
\(810\) 0 0
\(811\) 13.4144 0.471044 0.235522 0.971869i \(-0.424320\pi\)
0.235522 + 0.971869i \(0.424320\pi\)
\(812\) 0 0
\(813\) −8.76866 −0.307530
\(814\) 0 0
\(815\) −0.386717 −0.0135461
\(816\) 0 0
\(817\) −3.02391 −0.105793
\(818\) 0 0
\(819\) 1.26760 0.0442934
\(820\) 0 0
\(821\) −27.6086 −0.963548 −0.481774 0.876296i \(-0.660007\pi\)
−0.481774 + 0.876296i \(0.660007\pi\)
\(822\) 0 0
\(823\) 13.5881 0.473650 0.236825 0.971552i \(-0.423893\pi\)
0.236825 + 0.971552i \(0.423893\pi\)
\(824\) 0 0
\(825\) 1.51745 0.0528310
\(826\) 0 0
\(827\) −47.2467 −1.64293 −0.821465 0.570259i \(-0.806843\pi\)
−0.821465 + 0.570259i \(0.806843\pi\)
\(828\) 0 0
\(829\) 22.2463 0.772646 0.386323 0.922364i \(-0.373745\pi\)
0.386323 + 0.922364i \(0.373745\pi\)
\(830\) 0 0
\(831\) 31.7407 1.10107
\(832\) 0 0
\(833\) 12.0035 0.415895
\(834\) 0 0
\(835\) 8.53555 0.295385
\(836\) 0 0
\(837\) 2.76393 0.0955355
\(838\) 0 0
\(839\) −7.31344 −0.252488 −0.126244 0.991999i \(-0.540292\pi\)
−0.126244 + 0.991999i \(0.540292\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 1.92770 0.0663935
\(844\) 0 0
\(845\) −4.62023 −0.158941
\(846\) 0 0
\(847\) −7.97902 −0.274163
\(848\) 0 0
\(849\) 3.84260 0.131878
\(850\) 0 0
\(851\) −6.86987 −0.235496
\(852\) 0 0
\(853\) −2.07973 −0.0712084 −0.0356042 0.999366i \(-0.511336\pi\)
−0.0356042 + 0.999366i \(0.511336\pi\)
\(854\) 0 0
\(855\) −0.185156 −0.00633220
\(856\) 0 0
\(857\) 15.1501 0.517519 0.258759 0.965942i \(-0.416686\pi\)
0.258759 + 0.965942i \(0.416686\pi\)
\(858\) 0 0
\(859\) 3.02894 0.103346 0.0516731 0.998664i \(-0.483545\pi\)
0.0516731 + 0.998664i \(0.483545\pi\)
\(860\) 0 0
\(861\) −2.11363 −0.0720324
\(862\) 0 0
\(863\) 8.84761 0.301176 0.150588 0.988597i \(-0.451883\pi\)
0.150588 + 0.988597i \(0.451883\pi\)
\(864\) 0 0
\(865\) −5.52268 −0.187777
\(866\) 0 0
\(867\) 13.5518 0.460242
\(868\) 0 0
\(869\) 3.54980 0.120419
\(870\) 0 0
\(871\) −11.3206 −0.383583
\(872\) 0 0
\(873\) 10.3181 0.349215
\(874\) 0 0
\(875\) −3.30953 −0.111882
\(876\) 0 0
\(877\) 28.8391 0.973826 0.486913 0.873450i \(-0.338123\pi\)
0.486913 + 0.873450i \(0.338123\pi\)
\(878\) 0 0
\(879\) 19.6858 0.663984
\(880\) 0 0
\(881\) 10.2832 0.346449 0.173224 0.984882i \(-0.444581\pi\)
0.173224 + 0.984882i \(0.444581\pi\)
\(882\) 0 0
\(883\) 18.4793 0.621877 0.310939 0.950430i \(-0.399357\pi\)
0.310939 + 0.950430i \(0.399357\pi\)
\(884\) 0 0
\(885\) −1.79317 −0.0602766
\(886\) 0 0
\(887\) −7.63853 −0.256477 −0.128238 0.991743i \(-0.540932\pi\)
−0.128238 + 0.991743i \(0.540932\pi\)
\(888\) 0 0
\(889\) −2.45968 −0.0824951
\(890\) 0 0
\(891\) 0.317021 0.0106206
\(892\) 0 0
\(893\) 4.56211 0.152665
\(894\) 0 0
\(895\) −2.38256 −0.0796401
\(896\) 0 0
\(897\) 1.73156 0.0578152
\(898\) 0 0
\(899\) −2.76393 −0.0921823
\(900\) 0 0
\(901\) 20.0584 0.668241
\(902\) 0 0
\(903\) −5.52287 −0.183789
\(904\) 0 0
\(905\) 12.1775 0.404794
\(906\) 0 0
\(907\) 10.2717 0.341066 0.170533 0.985352i \(-0.445451\pi\)
0.170533 + 0.985352i \(0.445451\pi\)
\(908\) 0 0
\(909\) −11.5404 −0.382773
\(910\) 0 0
\(911\) −35.0741 −1.16206 −0.581028 0.813883i \(-0.697349\pi\)
−0.581028 + 0.813883i \(0.697349\pi\)
\(912\) 0 0
\(913\) 1.95215 0.0646068
\(914\) 0 0
\(915\) 1.85557 0.0613432
\(916\) 0 0
\(917\) 3.40889 0.112572
\(918\) 0 0
\(919\) 9.48065 0.312738 0.156369 0.987699i \(-0.450021\pi\)
0.156369 + 0.987699i \(0.450021\pi\)
\(920\) 0 0
\(921\) 16.2630 0.535883
\(922\) 0 0
\(923\) −0.711992 −0.0234355
\(924\) 0 0
\(925\) −32.8834 −1.08120
\(926\) 0 0
\(927\) −0.595012 −0.0195428
\(928\) 0 0
\(929\) −27.7195 −0.909446 −0.454723 0.890633i \(-0.650262\pi\)
−0.454723 + 0.890633i \(0.650262\pi\)
\(930\) 0 0
\(931\) 2.59093 0.0849142
\(932\) 0 0
\(933\) 11.8702 0.388613
\(934\) 0 0
\(935\) −0.271942 −0.00889346
\(936\) 0 0
\(937\) −4.67875 −0.152848 −0.0764241 0.997075i \(-0.524350\pi\)
−0.0764241 + 0.997075i \(0.524350\pi\)
\(938\) 0 0
\(939\) −12.8326 −0.418774
\(940\) 0 0
\(941\) 10.2302 0.333494 0.166747 0.986000i \(-0.446674\pi\)
0.166747 + 0.986000i \(0.446674\pi\)
\(942\) 0 0
\(943\) −2.88726 −0.0940222
\(944\) 0 0
\(945\) −0.338169 −0.0110006
\(946\) 0 0
\(947\) 5.94865 0.193305 0.0966526 0.995318i \(-0.469186\pi\)
0.0966526 + 0.995318i \(0.469186\pi\)
\(948\) 0 0
\(949\) −24.1551 −0.784106
\(950\) 0 0
\(951\) 8.37675 0.271635
\(952\) 0 0
\(953\) −38.7092 −1.25391 −0.626957 0.779054i \(-0.715700\pi\)
−0.626957 + 0.779054i \(0.715700\pi\)
\(954\) 0 0
\(955\) −5.82485 −0.188488
\(956\) 0 0
\(957\) −0.317021 −0.0102478
\(958\) 0 0
\(959\) −9.94118 −0.321017
\(960\) 0 0
\(961\) −23.3607 −0.753570
\(962\) 0 0
\(963\) 4.20690 0.135565
\(964\) 0 0
\(965\) 2.67033 0.0859609
\(966\) 0 0
\(967\) 16.2411 0.522279 0.261140 0.965301i \(-0.415902\pi\)
0.261140 + 0.965301i \(0.415902\pi\)
\(968\) 0 0
\(969\) −0.744296 −0.0239102
\(970\) 0 0
\(971\) −35.6266 −1.14331 −0.571656 0.820494i \(-0.693699\pi\)
−0.571656 + 0.820494i \(0.693699\pi\)
\(972\) 0 0
\(973\) −8.66540 −0.277800
\(974\) 0 0
\(975\) 8.28830 0.265438
\(976\) 0 0
\(977\) −40.0107 −1.28005 −0.640027 0.768352i \(-0.721077\pi\)
−0.640027 + 0.768352i \(0.721077\pi\)
\(978\) 0 0
\(979\) −1.28134 −0.0409518
\(980\) 0 0
\(981\) 1.09440 0.0349414
\(982\) 0 0
\(983\) 40.1699 1.28122 0.640611 0.767865i \(-0.278681\pi\)
0.640611 + 0.767865i \(0.278681\pi\)
\(984\) 0 0
\(985\) 4.36418 0.139054
\(986\) 0 0
\(987\) 8.33224 0.265218
\(988\) 0 0
\(989\) −7.54434 −0.239896
\(990\) 0 0
\(991\) 24.3207 0.772571 0.386286 0.922379i \(-0.373758\pi\)
0.386286 + 0.922379i \(0.373758\pi\)
\(992\) 0 0
\(993\) −31.7411 −1.00727
\(994\) 0 0
\(995\) 6.85171 0.217214
\(996\) 0 0
\(997\) −39.3032 −1.24475 −0.622373 0.782721i \(-0.713831\pi\)
−0.622373 + 0.782721i \(0.713831\pi\)
\(998\) 0 0
\(999\) −6.86987 −0.217353
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 8004.2.a.e.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.5 9 1.1 even 1 trivial