Properties

Label 8004.2.a.e.1.9
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 1.9
Root \(-0.553378\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.29187 q^{5} -1.02028 q^{7} +1.00000 q^{9} -1.88720 q^{11} -3.64262 q^{13} -3.29187 q^{15} -1.28578 q^{17} +5.72913 q^{19} +1.02028 q^{21} -1.00000 q^{23} +5.83643 q^{25} -1.00000 q^{27} +1.00000 q^{29} -7.90835 q^{31} +1.88720 q^{33} -3.35864 q^{35} -4.37633 q^{37} +3.64262 q^{39} -4.79282 q^{41} +9.68133 q^{43} +3.29187 q^{45} +6.33639 q^{47} -5.95902 q^{49} +1.28578 q^{51} -3.39752 q^{53} -6.21243 q^{55} -5.72913 q^{57} +4.11544 q^{59} -0.134655 q^{61} -1.02028 q^{63} -11.9911 q^{65} +12.1554 q^{67} +1.00000 q^{69} -1.67588 q^{71} -0.127917 q^{73} -5.83643 q^{75} +1.92548 q^{77} -0.342687 q^{79} +1.00000 q^{81} +0.0974823 q^{83} -4.23263 q^{85} -1.00000 q^{87} -14.4602 q^{89} +3.71650 q^{91} +7.90835 q^{93} +18.8596 q^{95} -18.0506 q^{97} -1.88720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 3q^{5} + 7q^{7} + 9q^{9} - 2q^{11} - 7q^{13} + 3q^{15} - q^{19} - 7q^{21} - 9q^{23} + 2q^{25} - 9q^{27} + 9q^{29} + 8q^{31} + 2q^{33} - 5q^{35} - 8q^{37} + 7q^{39} - 19q^{41} - 3q^{43} - 3q^{45} - 3q^{47} - 18q^{49} - 17q^{53} + 9q^{55} + q^{57} - 10q^{59} + q^{61} + 7q^{63} - 16q^{65} + 12q^{67} + 9q^{69} - 7q^{71} + 13q^{73} - 2q^{75} - 15q^{77} - 10q^{79} + 9q^{81} + 9q^{83} - 6q^{85} - 9q^{87} - 5q^{89} - 18q^{91} - 8q^{93} + 31q^{95} - 7q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) 3.29187 1.47217 0.736085 0.676889i \(-0.236672\pi\)
0.736085 + 0.676889i \(0.236672\pi\)
\(6\) 0 0
\(7\) −1.02028 −0.385630 −0.192815 0.981235i \(-0.561762\pi\)
−0.192815 + 0.981235i \(0.561762\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.88720 −0.569013 −0.284506 0.958674i \(-0.591830\pi\)
−0.284506 + 0.958674i \(0.591830\pi\)
\(12\) 0 0
\(13\) −3.64262 −1.01028 −0.505141 0.863037i \(-0.668560\pi\)
−0.505141 + 0.863037i \(0.668560\pi\)
\(14\) 0 0
\(15\) −3.29187 −0.849958
\(16\) 0 0
\(17\) −1.28578 −0.311848 −0.155924 0.987769i \(-0.549835\pi\)
−0.155924 + 0.987769i \(0.549835\pi\)
\(18\) 0 0
\(19\) 5.72913 1.31435 0.657177 0.753737i \(-0.271751\pi\)
0.657177 + 0.753737i \(0.271751\pi\)
\(20\) 0 0
\(21\) 1.02028 0.222644
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 5.83643 1.16729
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −7.90835 −1.42038 −0.710191 0.704009i \(-0.751391\pi\)
−0.710191 + 0.704009i \(0.751391\pi\)
\(32\) 0 0
\(33\) 1.88720 0.328520
\(34\) 0 0
\(35\) −3.35864 −0.567714
\(36\) 0 0
\(37\) −4.37633 −0.719464 −0.359732 0.933056i \(-0.617132\pi\)
−0.359732 + 0.933056i \(0.617132\pi\)
\(38\) 0 0
\(39\) 3.64262 0.583287
\(40\) 0 0
\(41\) −4.79282 −0.748512 −0.374256 0.927325i \(-0.622102\pi\)
−0.374256 + 0.927325i \(0.622102\pi\)
\(42\) 0 0
\(43\) 9.68133 1.47639 0.738195 0.674588i \(-0.235678\pi\)
0.738195 + 0.674588i \(0.235678\pi\)
\(44\) 0 0
\(45\) 3.29187 0.490723
\(46\) 0 0
\(47\) 6.33639 0.924257 0.462129 0.886813i \(-0.347086\pi\)
0.462129 + 0.886813i \(0.347086\pi\)
\(48\) 0 0
\(49\) −5.95902 −0.851289
\(50\) 0 0
\(51\) 1.28578 0.180045
\(52\) 0 0
\(53\) −3.39752 −0.466685 −0.233343 0.972395i \(-0.574966\pi\)
−0.233343 + 0.972395i \(0.574966\pi\)
\(54\) 0 0
\(55\) −6.21243 −0.837683
\(56\) 0 0
\(57\) −5.72913 −0.758842
\(58\) 0 0
\(59\) 4.11544 0.535784 0.267892 0.963449i \(-0.413673\pi\)
0.267892 + 0.963449i \(0.413673\pi\)
\(60\) 0 0
\(61\) −0.134655 −0.0172408 −0.00862042 0.999963i \(-0.502744\pi\)
−0.00862042 + 0.999963i \(0.502744\pi\)
\(62\) 0 0
\(63\) −1.02028 −0.128543
\(64\) 0 0
\(65\) −11.9911 −1.48731
\(66\) 0 0
\(67\) 12.1554 1.48502 0.742510 0.669835i \(-0.233635\pi\)
0.742510 + 0.669835i \(0.233635\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.67588 −0.198890 −0.0994450 0.995043i \(-0.531707\pi\)
−0.0994450 + 0.995043i \(0.531707\pi\)
\(72\) 0 0
\(73\) −0.127917 −0.0149715 −0.00748577 0.999972i \(-0.502383\pi\)
−0.00748577 + 0.999972i \(0.502383\pi\)
\(74\) 0 0
\(75\) −5.83643 −0.673933
\(76\) 0 0
\(77\) 1.92548 0.219429
\(78\) 0 0
\(79\) −0.342687 −0.0385553 −0.0192777 0.999814i \(-0.506137\pi\)
−0.0192777 + 0.999814i \(0.506137\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.0974823 0.0107001 0.00535004 0.999986i \(-0.498297\pi\)
0.00535004 + 0.999986i \(0.498297\pi\)
\(84\) 0 0
\(85\) −4.23263 −0.459093
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −14.4602 −1.53278 −0.766390 0.642375i \(-0.777949\pi\)
−0.766390 + 0.642375i \(0.777949\pi\)
\(90\) 0 0
\(91\) 3.71650 0.389596
\(92\) 0 0
\(93\) 7.90835 0.820058
\(94\) 0 0
\(95\) 18.8596 1.93495
\(96\) 0 0
\(97\) −18.0506 −1.83276 −0.916382 0.400305i \(-0.868904\pi\)
−0.916382 + 0.400305i \(0.868904\pi\)
\(98\) 0 0
\(99\) −1.88720 −0.189671
\(100\) 0 0
\(101\) −0.307106 −0.0305582 −0.0152791 0.999883i \(-0.504864\pi\)
−0.0152791 + 0.999883i \(0.504864\pi\)
\(102\) 0 0
\(103\) −9.96209 −0.981594 −0.490797 0.871274i \(-0.663294\pi\)
−0.490797 + 0.871274i \(0.663294\pi\)
\(104\) 0 0
\(105\) 3.35864 0.327770
\(106\) 0 0
\(107\) −4.95885 −0.479390 −0.239695 0.970848i \(-0.577047\pi\)
−0.239695 + 0.970848i \(0.577047\pi\)
\(108\) 0 0
\(109\) −11.9944 −1.14886 −0.574428 0.818555i \(-0.694776\pi\)
−0.574428 + 0.818555i \(0.694776\pi\)
\(110\) 0 0
\(111\) 4.37633 0.415383
\(112\) 0 0
\(113\) 1.20091 0.112972 0.0564862 0.998403i \(-0.482010\pi\)
0.0564862 + 0.998403i \(0.482010\pi\)
\(114\) 0 0
\(115\) −3.29187 −0.306969
\(116\) 0 0
\(117\) −3.64262 −0.336761
\(118\) 0 0
\(119\) 1.31186 0.120258
\(120\) 0 0
\(121\) −7.43847 −0.676225
\(122\) 0 0
\(123\) 4.79282 0.432154
\(124\) 0 0
\(125\) 2.75342 0.246273
\(126\) 0 0
\(127\) −2.85660 −0.253483 −0.126741 0.991936i \(-0.540452\pi\)
−0.126741 + 0.991936i \(0.540452\pi\)
\(128\) 0 0
\(129\) −9.68133 −0.852394
\(130\) 0 0
\(131\) 12.5796 1.09908 0.549542 0.835466i \(-0.314802\pi\)
0.549542 + 0.835466i \(0.314802\pi\)
\(132\) 0 0
\(133\) −5.84533 −0.506855
\(134\) 0 0
\(135\) −3.29187 −0.283319
\(136\) 0 0
\(137\) −1.28529 −0.109810 −0.0549048 0.998492i \(-0.517486\pi\)
−0.0549048 + 0.998492i \(0.517486\pi\)
\(138\) 0 0
\(139\) −5.27614 −0.447516 −0.223758 0.974645i \(-0.571833\pi\)
−0.223758 + 0.974645i \(0.571833\pi\)
\(140\) 0 0
\(141\) −6.33639 −0.533620
\(142\) 0 0
\(143\) 6.87436 0.574863
\(144\) 0 0
\(145\) 3.29187 0.273375
\(146\) 0 0
\(147\) 5.95902 0.491492
\(148\) 0 0
\(149\) −15.0902 −1.23624 −0.618119 0.786084i \(-0.712105\pi\)
−0.618119 + 0.786084i \(0.712105\pi\)
\(150\) 0 0
\(151\) −0.0808502 −0.00657950 −0.00328975 0.999995i \(-0.501047\pi\)
−0.00328975 + 0.999995i \(0.501047\pi\)
\(152\) 0 0
\(153\) −1.28578 −0.103949
\(154\) 0 0
\(155\) −26.0333 −2.09104
\(156\) 0 0
\(157\) 13.2238 1.05538 0.527688 0.849438i \(-0.323059\pi\)
0.527688 + 0.849438i \(0.323059\pi\)
\(158\) 0 0
\(159\) 3.39752 0.269441
\(160\) 0 0
\(161\) 1.02028 0.0804095
\(162\) 0 0
\(163\) −9.32641 −0.730501 −0.365250 0.930909i \(-0.619017\pi\)
−0.365250 + 0.930909i \(0.619017\pi\)
\(164\) 0 0
\(165\) 6.21243 0.483637
\(166\) 0 0
\(167\) −2.65640 −0.205559 −0.102779 0.994704i \(-0.532774\pi\)
−0.102779 + 0.994704i \(0.532774\pi\)
\(168\) 0 0
\(169\) 0.268711 0.0206701
\(170\) 0 0
\(171\) 5.72913 0.438118
\(172\) 0 0
\(173\) −20.8900 −1.58824 −0.794120 0.607761i \(-0.792068\pi\)
−0.794120 + 0.607761i \(0.792068\pi\)
\(174\) 0 0
\(175\) −5.95480 −0.450141
\(176\) 0 0
\(177\) −4.11544 −0.309335
\(178\) 0 0
\(179\) 7.03363 0.525718 0.262859 0.964834i \(-0.415335\pi\)
0.262859 + 0.964834i \(0.415335\pi\)
\(180\) 0 0
\(181\) 10.4995 0.780420 0.390210 0.920726i \(-0.372402\pi\)
0.390210 + 0.920726i \(0.372402\pi\)
\(182\) 0 0
\(183\) 0.134655 0.00995400
\(184\) 0 0
\(185\) −14.4063 −1.05917
\(186\) 0 0
\(187\) 2.42653 0.177445
\(188\) 0 0
\(189\) 1.02028 0.0742146
\(190\) 0 0
\(191\) 12.9567 0.937513 0.468757 0.883327i \(-0.344702\pi\)
0.468757 + 0.883327i \(0.344702\pi\)
\(192\) 0 0
\(193\) −27.1033 −1.95094 −0.975470 0.220134i \(-0.929351\pi\)
−0.975470 + 0.220134i \(0.929351\pi\)
\(194\) 0 0
\(195\) 11.9911 0.858697
\(196\) 0 0
\(197\) 14.7303 1.04949 0.524744 0.851260i \(-0.324161\pi\)
0.524744 + 0.851260i \(0.324161\pi\)
\(198\) 0 0
\(199\) 1.88160 0.133383 0.0666915 0.997774i \(-0.478756\pi\)
0.0666915 + 0.997774i \(0.478756\pi\)
\(200\) 0 0
\(201\) −12.1554 −0.857376
\(202\) 0 0
\(203\) −1.02028 −0.0716098
\(204\) 0 0
\(205\) −15.7773 −1.10194
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −10.8120 −0.747883
\(210\) 0 0
\(211\) −22.5394 −1.55167 −0.775837 0.630933i \(-0.782672\pi\)
−0.775837 + 0.630933i \(0.782672\pi\)
\(212\) 0 0
\(213\) 1.67588 0.114829
\(214\) 0 0
\(215\) 31.8697 2.17350
\(216\) 0 0
\(217\) 8.06875 0.547743
\(218\) 0 0
\(219\) 0.127917 0.00864382
\(220\) 0 0
\(221\) 4.68362 0.315054
\(222\) 0 0
\(223\) 0.0899811 0.00602558 0.00301279 0.999995i \(-0.499041\pi\)
0.00301279 + 0.999995i \(0.499041\pi\)
\(224\) 0 0
\(225\) 5.83643 0.389095
\(226\) 0 0
\(227\) −24.3526 −1.61634 −0.808168 0.588951i \(-0.799541\pi\)
−0.808168 + 0.588951i \(0.799541\pi\)
\(228\) 0 0
\(229\) 13.3336 0.881112 0.440556 0.897725i \(-0.354781\pi\)
0.440556 + 0.897725i \(0.354781\pi\)
\(230\) 0 0
\(231\) −1.92548 −0.126687
\(232\) 0 0
\(233\) −6.28577 −0.411794 −0.205897 0.978574i \(-0.566011\pi\)
−0.205897 + 0.978574i \(0.566011\pi\)
\(234\) 0 0
\(235\) 20.8586 1.36066
\(236\) 0 0
\(237\) 0.342687 0.0222599
\(238\) 0 0
\(239\) −13.2321 −0.855916 −0.427958 0.903799i \(-0.640767\pi\)
−0.427958 + 0.903799i \(0.640767\pi\)
\(240\) 0 0
\(241\) 19.0873 1.22952 0.614760 0.788714i \(-0.289253\pi\)
0.614760 + 0.788714i \(0.289253\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −19.6164 −1.25324
\(246\) 0 0
\(247\) −20.8691 −1.32787
\(248\) 0 0
\(249\) −0.0974823 −0.00617769
\(250\) 0 0
\(251\) 8.09378 0.510875 0.255438 0.966826i \(-0.417780\pi\)
0.255438 + 0.966826i \(0.417780\pi\)
\(252\) 0 0
\(253\) 1.88720 0.118647
\(254\) 0 0
\(255\) 4.23263 0.265058
\(256\) 0 0
\(257\) 11.6298 0.725446 0.362723 0.931897i \(-0.381847\pi\)
0.362723 + 0.931897i \(0.381847\pi\)
\(258\) 0 0
\(259\) 4.46509 0.277447
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −7.51527 −0.463411 −0.231706 0.972786i \(-0.574431\pi\)
−0.231706 + 0.972786i \(0.574431\pi\)
\(264\) 0 0
\(265\) −11.1842 −0.687040
\(266\) 0 0
\(267\) 14.4602 0.884951
\(268\) 0 0
\(269\) 1.14454 0.0697836 0.0348918 0.999391i \(-0.488891\pi\)
0.0348918 + 0.999391i \(0.488891\pi\)
\(270\) 0 0
\(271\) −12.5549 −0.762654 −0.381327 0.924440i \(-0.624533\pi\)
−0.381327 + 0.924440i \(0.624533\pi\)
\(272\) 0 0
\(273\) −3.71650 −0.224933
\(274\) 0 0
\(275\) −11.0145 −0.664200
\(276\) 0 0
\(277\) −2.89067 −0.173683 −0.0868417 0.996222i \(-0.527677\pi\)
−0.0868417 + 0.996222i \(0.527677\pi\)
\(278\) 0 0
\(279\) −7.90835 −0.473461
\(280\) 0 0
\(281\) 11.1332 0.664150 0.332075 0.943253i \(-0.392251\pi\)
0.332075 + 0.943253i \(0.392251\pi\)
\(282\) 0 0
\(283\) −0.153847 −0.00914527 −0.00457264 0.999990i \(-0.501456\pi\)
−0.00457264 + 0.999990i \(0.501456\pi\)
\(284\) 0 0
\(285\) −18.8596 −1.11715
\(286\) 0 0
\(287\) 4.89002 0.288649
\(288\) 0 0
\(289\) −15.3468 −0.902751
\(290\) 0 0
\(291\) 18.0506 1.05815
\(292\) 0 0
\(293\) 32.7574 1.91371 0.956855 0.290567i \(-0.0938440\pi\)
0.956855 + 0.290567i \(0.0938440\pi\)
\(294\) 0 0
\(295\) 13.5475 0.788766
\(296\) 0 0
\(297\) 1.88720 0.109507
\(298\) 0 0
\(299\) 3.64262 0.210658
\(300\) 0 0
\(301\) −9.87769 −0.569341
\(302\) 0 0
\(303\) 0.307106 0.0176428
\(304\) 0 0
\(305\) −0.443268 −0.0253815
\(306\) 0 0
\(307\) 10.3644 0.591527 0.295764 0.955261i \(-0.404426\pi\)
0.295764 + 0.955261i \(0.404426\pi\)
\(308\) 0 0
\(309\) 9.96209 0.566724
\(310\) 0 0
\(311\) −0.192809 −0.0109332 −0.00546661 0.999985i \(-0.501740\pi\)
−0.00546661 + 0.999985i \(0.501740\pi\)
\(312\) 0 0
\(313\) −18.5661 −1.04942 −0.524709 0.851282i \(-0.675826\pi\)
−0.524709 + 0.851282i \(0.675826\pi\)
\(314\) 0 0
\(315\) −3.35864 −0.189238
\(316\) 0 0
\(317\) 16.1487 0.907004 0.453502 0.891255i \(-0.350175\pi\)
0.453502 + 0.891255i \(0.350175\pi\)
\(318\) 0 0
\(319\) −1.88720 −0.105663
\(320\) 0 0
\(321\) 4.95885 0.276776
\(322\) 0 0
\(323\) −7.36642 −0.409878
\(324\) 0 0
\(325\) −21.2599 −1.17929
\(326\) 0 0
\(327\) 11.9944 0.663293
\(328\) 0 0
\(329\) −6.46491 −0.356422
\(330\) 0 0
\(331\) −18.9714 −1.04277 −0.521383 0.853323i \(-0.674584\pi\)
−0.521383 + 0.853323i \(0.674584\pi\)
\(332\) 0 0
\(333\) −4.37633 −0.239821
\(334\) 0 0
\(335\) 40.0141 2.18620
\(336\) 0 0
\(337\) 19.9061 1.08436 0.542178 0.840264i \(-0.317600\pi\)
0.542178 + 0.840264i \(0.317600\pi\)
\(338\) 0 0
\(339\) −1.20091 −0.0652247
\(340\) 0 0
\(341\) 14.9247 0.808215
\(342\) 0 0
\(343\) 13.2219 0.713913
\(344\) 0 0
\(345\) 3.29187 0.177228
\(346\) 0 0
\(347\) 0.225778 0.0121204 0.00606020 0.999982i \(-0.498071\pi\)
0.00606020 + 0.999982i \(0.498071\pi\)
\(348\) 0 0
\(349\) 11.5780 0.619756 0.309878 0.950776i \(-0.399712\pi\)
0.309878 + 0.950776i \(0.399712\pi\)
\(350\) 0 0
\(351\) 3.64262 0.194429
\(352\) 0 0
\(353\) −30.4567 −1.62105 −0.810523 0.585706i \(-0.800817\pi\)
−0.810523 + 0.585706i \(0.800817\pi\)
\(354\) 0 0
\(355\) −5.51677 −0.292800
\(356\) 0 0
\(357\) −1.31186 −0.0694310
\(358\) 0 0
\(359\) 12.9315 0.682499 0.341249 0.939973i \(-0.389150\pi\)
0.341249 + 0.939973i \(0.389150\pi\)
\(360\) 0 0
\(361\) 13.8230 0.727525
\(362\) 0 0
\(363\) 7.43847 0.390419
\(364\) 0 0
\(365\) −0.421086 −0.0220407
\(366\) 0 0
\(367\) 20.6009 1.07536 0.537678 0.843150i \(-0.319302\pi\)
0.537678 + 0.843150i \(0.319302\pi\)
\(368\) 0 0
\(369\) −4.79282 −0.249504
\(370\) 0 0
\(371\) 3.46643 0.179968
\(372\) 0 0
\(373\) −31.6195 −1.63720 −0.818599 0.574365i \(-0.805249\pi\)
−0.818599 + 0.574365i \(0.805249\pi\)
\(374\) 0 0
\(375\) −2.75342 −0.142186
\(376\) 0 0
\(377\) −3.64262 −0.187605
\(378\) 0 0
\(379\) −33.8695 −1.73976 −0.869878 0.493266i \(-0.835803\pi\)
−0.869878 + 0.493266i \(0.835803\pi\)
\(380\) 0 0
\(381\) 2.85660 0.146348
\(382\) 0 0
\(383\) 24.3113 1.24225 0.621125 0.783712i \(-0.286676\pi\)
0.621125 + 0.783712i \(0.286676\pi\)
\(384\) 0 0
\(385\) 6.33843 0.323036
\(386\) 0 0
\(387\) 9.68133 0.492130
\(388\) 0 0
\(389\) 30.3825 1.54046 0.770228 0.637769i \(-0.220143\pi\)
0.770228 + 0.637769i \(0.220143\pi\)
\(390\) 0 0
\(391\) 1.28578 0.0650248
\(392\) 0 0
\(393\) −12.5796 −0.634556
\(394\) 0 0
\(395\) −1.12808 −0.0567600
\(396\) 0 0
\(397\) 28.9430 1.45261 0.726304 0.687373i \(-0.241236\pi\)
0.726304 + 0.687373i \(0.241236\pi\)
\(398\) 0 0
\(399\) 5.84533 0.292633
\(400\) 0 0
\(401\) −32.0873 −1.60236 −0.801182 0.598421i \(-0.795795\pi\)
−0.801182 + 0.598421i \(0.795795\pi\)
\(402\) 0 0
\(403\) 28.8072 1.43499
\(404\) 0 0
\(405\) 3.29187 0.163574
\(406\) 0 0
\(407\) 8.25902 0.409384
\(408\) 0 0
\(409\) 16.8190 0.831646 0.415823 0.909445i \(-0.363494\pi\)
0.415823 + 0.909445i \(0.363494\pi\)
\(410\) 0 0
\(411\) 1.28529 0.0633987
\(412\) 0 0
\(413\) −4.19891 −0.206615
\(414\) 0 0
\(415\) 0.320899 0.0157523
\(416\) 0 0
\(417\) 5.27614 0.258374
\(418\) 0 0
\(419\) −23.5339 −1.14971 −0.574853 0.818257i \(-0.694941\pi\)
−0.574853 + 0.818257i \(0.694941\pi\)
\(420\) 0 0
\(421\) −4.05228 −0.197496 −0.0987481 0.995112i \(-0.531484\pi\)
−0.0987481 + 0.995112i \(0.531484\pi\)
\(422\) 0 0
\(423\) 6.33639 0.308086
\(424\) 0 0
\(425\) −7.50437 −0.364016
\(426\) 0 0
\(427\) 0.137386 0.00664859
\(428\) 0 0
\(429\) −6.87436 −0.331897
\(430\) 0 0
\(431\) 4.17887 0.201289 0.100644 0.994922i \(-0.467910\pi\)
0.100644 + 0.994922i \(0.467910\pi\)
\(432\) 0 0
\(433\) 20.8989 1.00434 0.502168 0.864770i \(-0.332536\pi\)
0.502168 + 0.864770i \(0.332536\pi\)
\(434\) 0 0
\(435\) −3.29187 −0.157833
\(436\) 0 0
\(437\) −5.72913 −0.274062
\(438\) 0 0
\(439\) −35.9387 −1.71526 −0.857631 0.514266i \(-0.828064\pi\)
−0.857631 + 0.514266i \(0.828064\pi\)
\(440\) 0 0
\(441\) −5.95902 −0.283763
\(442\) 0 0
\(443\) −0.911071 −0.0432863 −0.0216432 0.999766i \(-0.506890\pi\)
−0.0216432 + 0.999766i \(0.506890\pi\)
\(444\) 0 0
\(445\) −47.6012 −2.25651
\(446\) 0 0
\(447\) 15.0902 0.713743
\(448\) 0 0
\(449\) −20.5377 −0.969235 −0.484617 0.874726i \(-0.661041\pi\)
−0.484617 + 0.874726i \(0.661041\pi\)
\(450\) 0 0
\(451\) 9.04501 0.425913
\(452\) 0 0
\(453\) 0.0808502 0.00379867
\(454\) 0 0
\(455\) 12.2343 0.573551
\(456\) 0 0
\(457\) 38.8892 1.81916 0.909581 0.415527i \(-0.136403\pi\)
0.909581 + 0.415527i \(0.136403\pi\)
\(458\) 0 0
\(459\) 1.28578 0.0600152
\(460\) 0 0
\(461\) −36.7804 −1.71304 −0.856518 0.516118i \(-0.827377\pi\)
−0.856518 + 0.516118i \(0.827377\pi\)
\(462\) 0 0
\(463\) −7.42298 −0.344975 −0.172488 0.985012i \(-0.555180\pi\)
−0.172488 + 0.985012i \(0.555180\pi\)
\(464\) 0 0
\(465\) 26.0333 1.20727
\(466\) 0 0
\(467\) −27.8689 −1.28962 −0.644808 0.764344i \(-0.723063\pi\)
−0.644808 + 0.764344i \(0.723063\pi\)
\(468\) 0 0
\(469\) −12.4019 −0.572669
\(470\) 0 0
\(471\) −13.2238 −0.609322
\(472\) 0 0
\(473\) −18.2706 −0.840084
\(474\) 0 0
\(475\) 33.4377 1.53423
\(476\) 0 0
\(477\) −3.39752 −0.155562
\(478\) 0 0
\(479\) −10.8615 −0.496277 −0.248138 0.968725i \(-0.579819\pi\)
−0.248138 + 0.968725i \(0.579819\pi\)
\(480\) 0 0
\(481\) 15.9413 0.726862
\(482\) 0 0
\(483\) −1.02028 −0.0464244
\(484\) 0 0
\(485\) −59.4204 −2.69814
\(486\) 0 0
\(487\) 15.1469 0.686374 0.343187 0.939267i \(-0.388494\pi\)
0.343187 + 0.939267i \(0.388494\pi\)
\(488\) 0 0
\(489\) 9.32641 0.421755
\(490\) 0 0
\(491\) −20.1888 −0.911107 −0.455553 0.890208i \(-0.650559\pi\)
−0.455553 + 0.890208i \(0.650559\pi\)
\(492\) 0 0
\(493\) −1.28578 −0.0579087
\(494\) 0 0
\(495\) −6.21243 −0.279228
\(496\) 0 0
\(497\) 1.70987 0.0766980
\(498\) 0 0
\(499\) −2.12975 −0.0953406 −0.0476703 0.998863i \(-0.515180\pi\)
−0.0476703 + 0.998863i \(0.515180\pi\)
\(500\) 0 0
\(501\) 2.65640 0.118679
\(502\) 0 0
\(503\) −33.5893 −1.49767 −0.748835 0.662756i \(-0.769387\pi\)
−0.748835 + 0.662756i \(0.769387\pi\)
\(504\) 0 0
\(505\) −1.01096 −0.0449869
\(506\) 0 0
\(507\) −0.268711 −0.0119339
\(508\) 0 0
\(509\) −29.0969 −1.28970 −0.644848 0.764311i \(-0.723079\pi\)
−0.644848 + 0.764311i \(0.723079\pi\)
\(510\) 0 0
\(511\) 0.130511 0.00577348
\(512\) 0 0
\(513\) −5.72913 −0.252947
\(514\) 0 0
\(515\) −32.7939 −1.44507
\(516\) 0 0
\(517\) −11.9580 −0.525914
\(518\) 0 0
\(519\) 20.8900 0.916971
\(520\) 0 0
\(521\) −39.7968 −1.74353 −0.871764 0.489925i \(-0.837024\pi\)
−0.871764 + 0.489925i \(0.837024\pi\)
\(522\) 0 0
\(523\) 10.3933 0.454469 0.227235 0.973840i \(-0.427032\pi\)
0.227235 + 0.973840i \(0.427032\pi\)
\(524\) 0 0
\(525\) 5.95480 0.259889
\(526\) 0 0
\(527\) 10.1684 0.442943
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.11544 0.178595
\(532\) 0 0
\(533\) 17.4584 0.756208
\(534\) 0 0
\(535\) −16.3239 −0.705743
\(536\) 0 0
\(537\) −7.03363 −0.303523
\(538\) 0 0
\(539\) 11.2459 0.484394
\(540\) 0 0
\(541\) −0.986445 −0.0424106 −0.0212053 0.999775i \(-0.506750\pi\)
−0.0212053 + 0.999775i \(0.506750\pi\)
\(542\) 0 0
\(543\) −10.4995 −0.450576
\(544\) 0 0
\(545\) −39.4841 −1.69131
\(546\) 0 0
\(547\) −11.0767 −0.473607 −0.236804 0.971558i \(-0.576100\pi\)
−0.236804 + 0.971558i \(0.576100\pi\)
\(548\) 0 0
\(549\) −0.134655 −0.00574695
\(550\) 0 0
\(551\) 5.72913 0.244069
\(552\) 0 0
\(553\) 0.349638 0.0148681
\(554\) 0 0
\(555\) 14.4063 0.611515
\(556\) 0 0
\(557\) 1.81205 0.0767789 0.0383895 0.999263i \(-0.487777\pi\)
0.0383895 + 0.999263i \(0.487777\pi\)
\(558\) 0 0
\(559\) −35.2655 −1.49157
\(560\) 0 0
\(561\) −2.42653 −0.102448
\(562\) 0 0
\(563\) 2.19519 0.0925163 0.0462581 0.998930i \(-0.485270\pi\)
0.0462581 + 0.998930i \(0.485270\pi\)
\(564\) 0 0
\(565\) 3.95326 0.166315
\(566\) 0 0
\(567\) −1.02028 −0.0428478
\(568\) 0 0
\(569\) −20.4169 −0.855921 −0.427961 0.903797i \(-0.640768\pi\)
−0.427961 + 0.903797i \(0.640768\pi\)
\(570\) 0 0
\(571\) −23.1651 −0.969429 −0.484714 0.874672i \(-0.661076\pi\)
−0.484714 + 0.874672i \(0.661076\pi\)
\(572\) 0 0
\(573\) −12.9567 −0.541273
\(574\) 0 0
\(575\) −5.83643 −0.243396
\(576\) 0 0
\(577\) −29.6341 −1.23368 −0.616842 0.787087i \(-0.711588\pi\)
−0.616842 + 0.787087i \(0.711588\pi\)
\(578\) 0 0
\(579\) 27.1033 1.12638
\(580\) 0 0
\(581\) −0.0994595 −0.00412627
\(582\) 0 0
\(583\) 6.41181 0.265550
\(584\) 0 0
\(585\) −11.9911 −0.495769
\(586\) 0 0
\(587\) 21.0958 0.870717 0.435359 0.900257i \(-0.356622\pi\)
0.435359 + 0.900257i \(0.356622\pi\)
\(588\) 0 0
\(589\) −45.3080 −1.86688
\(590\) 0 0
\(591\) −14.7303 −0.605922
\(592\) 0 0
\(593\) −24.5799 −1.00938 −0.504688 0.863302i \(-0.668392\pi\)
−0.504688 + 0.863302i \(0.668392\pi\)
\(594\) 0 0
\(595\) 4.31848 0.177040
\(596\) 0 0
\(597\) −1.88160 −0.0770087
\(598\) 0 0
\(599\) 33.2567 1.35883 0.679415 0.733754i \(-0.262234\pi\)
0.679415 + 0.733754i \(0.262234\pi\)
\(600\) 0 0
\(601\) 6.74958 0.275321 0.137660 0.990479i \(-0.456042\pi\)
0.137660 + 0.990479i \(0.456042\pi\)
\(602\) 0 0
\(603\) 12.1554 0.495006
\(604\) 0 0
\(605\) −24.4865 −0.995518
\(606\) 0 0
\(607\) −13.7679 −0.558821 −0.279410 0.960172i \(-0.590139\pi\)
−0.279410 + 0.960172i \(0.590139\pi\)
\(608\) 0 0
\(609\) 1.02028 0.0413439
\(610\) 0 0
\(611\) −23.0811 −0.933761
\(612\) 0 0
\(613\) −38.6304 −1.56027 −0.780134 0.625612i \(-0.784849\pi\)
−0.780134 + 0.625612i \(0.784849\pi\)
\(614\) 0 0
\(615\) 15.7773 0.636204
\(616\) 0 0
\(617\) 15.2750 0.614949 0.307475 0.951556i \(-0.400516\pi\)
0.307475 + 0.951556i \(0.400516\pi\)
\(618\) 0 0
\(619\) −21.8995 −0.880213 −0.440107 0.897945i \(-0.645059\pi\)
−0.440107 + 0.897945i \(0.645059\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 14.7535 0.591087
\(624\) 0 0
\(625\) −20.1182 −0.804730
\(626\) 0 0
\(627\) 10.8120 0.431791
\(628\) 0 0
\(629\) 5.62701 0.224364
\(630\) 0 0
\(631\) 35.2070 1.40157 0.700784 0.713374i \(-0.252834\pi\)
0.700784 + 0.713374i \(0.252834\pi\)
\(632\) 0 0
\(633\) 22.5394 0.895859
\(634\) 0 0
\(635\) −9.40358 −0.373170
\(636\) 0 0
\(637\) 21.7065 0.860042
\(638\) 0 0
\(639\) −1.67588 −0.0662966
\(640\) 0 0
\(641\) −38.4422 −1.51838 −0.759188 0.650872i \(-0.774403\pi\)
−0.759188 + 0.650872i \(0.774403\pi\)
\(642\) 0 0
\(643\) 11.4506 0.451569 0.225785 0.974177i \(-0.427505\pi\)
0.225785 + 0.974177i \(0.427505\pi\)
\(644\) 0 0
\(645\) −31.8697 −1.25487
\(646\) 0 0
\(647\) −29.5280 −1.16087 −0.580433 0.814308i \(-0.697117\pi\)
−0.580433 + 0.814308i \(0.697117\pi\)
\(648\) 0 0
\(649\) −7.76666 −0.304868
\(650\) 0 0
\(651\) −8.06875 −0.316239
\(652\) 0 0
\(653\) −23.5186 −0.920353 −0.460176 0.887828i \(-0.652214\pi\)
−0.460176 + 0.887828i \(0.652214\pi\)
\(654\) 0 0
\(655\) 41.4104 1.61804
\(656\) 0 0
\(657\) −0.127917 −0.00499051
\(658\) 0 0
\(659\) −20.8705 −0.812999 −0.406499 0.913651i \(-0.633251\pi\)
−0.406499 + 0.913651i \(0.633251\pi\)
\(660\) 0 0
\(661\) −36.2819 −1.41120 −0.705602 0.708608i \(-0.749323\pi\)
−0.705602 + 0.708608i \(0.749323\pi\)
\(662\) 0 0
\(663\) −4.68362 −0.181897
\(664\) 0 0
\(665\) −19.2421 −0.746176
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −0.0899811 −0.00347887
\(670\) 0 0
\(671\) 0.254122 0.00981025
\(672\) 0 0
\(673\) 27.2490 1.05037 0.525185 0.850988i \(-0.323996\pi\)
0.525185 + 0.850988i \(0.323996\pi\)
\(674\) 0 0
\(675\) −5.83643 −0.224644
\(676\) 0 0
\(677\) 35.3520 1.35869 0.679344 0.733820i \(-0.262264\pi\)
0.679344 + 0.733820i \(0.262264\pi\)
\(678\) 0 0
\(679\) 18.4167 0.706769
\(680\) 0 0
\(681\) 24.3526 0.933193
\(682\) 0 0
\(683\) 29.2266 1.11832 0.559162 0.829058i \(-0.311123\pi\)
0.559162 + 0.829058i \(0.311123\pi\)
\(684\) 0 0
\(685\) −4.23101 −0.161659
\(686\) 0 0
\(687\) −13.3336 −0.508710
\(688\) 0 0
\(689\) 12.3759 0.471484
\(690\) 0 0
\(691\) 50.6850 1.92815 0.964074 0.265635i \(-0.0855816\pi\)
0.964074 + 0.265635i \(0.0855816\pi\)
\(692\) 0 0
\(693\) 1.92548 0.0731428
\(694\) 0 0
\(695\) −17.3684 −0.658820
\(696\) 0 0
\(697\) 6.16251 0.233422
\(698\) 0 0
\(699\) 6.28577 0.237750
\(700\) 0 0
\(701\) −17.6090 −0.665082 −0.332541 0.943089i \(-0.607906\pi\)
−0.332541 + 0.943089i \(0.607906\pi\)
\(702\) 0 0
\(703\) −25.0726 −0.945631
\(704\) 0 0
\(705\) −20.8586 −0.785580
\(706\) 0 0
\(707\) 0.313335 0.0117842
\(708\) 0 0
\(709\) 16.9287 0.635771 0.317885 0.948129i \(-0.397027\pi\)
0.317885 + 0.948129i \(0.397027\pi\)
\(710\) 0 0
\(711\) −0.342687 −0.0128518
\(712\) 0 0
\(713\) 7.90835 0.296170
\(714\) 0 0
\(715\) 22.6295 0.846297
\(716\) 0 0
\(717\) 13.2321 0.494163
\(718\) 0 0
\(719\) 0.578253 0.0215652 0.0107826 0.999942i \(-0.496568\pi\)
0.0107826 + 0.999942i \(0.496568\pi\)
\(720\) 0 0
\(721\) 10.1641 0.378533
\(722\) 0 0
\(723\) −19.0873 −0.709864
\(724\) 0 0
\(725\) 5.83643 0.216760
\(726\) 0 0
\(727\) 33.2629 1.23365 0.616827 0.787099i \(-0.288418\pi\)
0.616827 + 0.787099i \(0.288418\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.4481 −0.460409
\(732\) 0 0
\(733\) 37.4181 1.38207 0.691035 0.722821i \(-0.257155\pi\)
0.691035 + 0.722821i \(0.257155\pi\)
\(734\) 0 0
\(735\) 19.6164 0.723560
\(736\) 0 0
\(737\) −22.9397 −0.844995
\(738\) 0 0
\(739\) 15.7570 0.579631 0.289815 0.957083i \(-0.406406\pi\)
0.289815 + 0.957083i \(0.406406\pi\)
\(740\) 0 0
\(741\) 20.8691 0.766645
\(742\) 0 0
\(743\) 6.49549 0.238296 0.119148 0.992876i \(-0.461984\pi\)
0.119148 + 0.992876i \(0.461984\pi\)
\(744\) 0 0
\(745\) −49.6751 −1.81995
\(746\) 0 0
\(747\) 0.0974823 0.00356669
\(748\) 0 0
\(749\) 5.05942 0.184867
\(750\) 0 0
\(751\) 16.7238 0.610260 0.305130 0.952311i \(-0.401300\pi\)
0.305130 + 0.952311i \(0.401300\pi\)
\(752\) 0 0
\(753\) −8.09378 −0.294954
\(754\) 0 0
\(755\) −0.266149 −0.00968614
\(756\) 0 0
\(757\) 13.2545 0.481742 0.240871 0.970557i \(-0.422567\pi\)
0.240871 + 0.970557i \(0.422567\pi\)
\(758\) 0 0
\(759\) −1.88720 −0.0685011
\(760\) 0 0
\(761\) −35.7001 −1.29413 −0.647064 0.762436i \(-0.724003\pi\)
−0.647064 + 0.762436i \(0.724003\pi\)
\(762\) 0 0
\(763\) 12.2377 0.443034
\(764\) 0 0
\(765\) −4.23263 −0.153031
\(766\) 0 0
\(767\) −14.9910 −0.541293
\(768\) 0 0
\(769\) 31.0686 1.12036 0.560182 0.828370i \(-0.310731\pi\)
0.560182 + 0.828370i \(0.310731\pi\)
\(770\) 0 0
\(771\) −11.6298 −0.418837
\(772\) 0 0
\(773\) −21.8747 −0.786779 −0.393390 0.919372i \(-0.628698\pi\)
−0.393390 + 0.919372i \(0.628698\pi\)
\(774\) 0 0
\(775\) −46.1565 −1.65799
\(776\) 0 0
\(777\) −4.46509 −0.160184
\(778\) 0 0
\(779\) −27.4587 −0.983809
\(780\) 0 0
\(781\) 3.16272 0.113171
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 43.5312 1.55369
\(786\) 0 0
\(787\) −6.97490 −0.248628 −0.124314 0.992243i \(-0.539673\pi\)
−0.124314 + 0.992243i \(0.539673\pi\)
\(788\) 0 0
\(789\) 7.51527 0.267551
\(790\) 0 0
\(791\) −1.22527 −0.0435656
\(792\) 0 0
\(793\) 0.490498 0.0174181
\(794\) 0 0
\(795\) 11.1842 0.396663
\(796\) 0 0
\(797\) 12.1684 0.431025 0.215513 0.976501i \(-0.430858\pi\)
0.215513 + 0.976501i \(0.430858\pi\)
\(798\) 0 0
\(799\) −8.14722 −0.288228
\(800\) 0 0
\(801\) −14.4602 −0.510927
\(802\) 0 0
\(803\) 0.241405 0.00851900
\(804\) 0 0
\(805\) 3.35864 0.118376
\(806\) 0 0
\(807\) −1.14454 −0.0402896
\(808\) 0 0
\(809\) 29.8416 1.04918 0.524588 0.851356i \(-0.324220\pi\)
0.524588 + 0.851356i \(0.324220\pi\)
\(810\) 0 0
\(811\) −13.5969 −0.477451 −0.238725 0.971087i \(-0.576730\pi\)
−0.238725 + 0.971087i \(0.576730\pi\)
\(812\) 0 0
\(813\) 12.5549 0.440319
\(814\) 0 0
\(815\) −30.7014 −1.07542
\(816\) 0 0
\(817\) 55.4657 1.94050
\(818\) 0 0
\(819\) 3.71650 0.129865
\(820\) 0 0
\(821\) −4.64965 −0.162274 −0.0811369 0.996703i \(-0.525855\pi\)
−0.0811369 + 0.996703i \(0.525855\pi\)
\(822\) 0 0
\(823\) 20.5466 0.716210 0.358105 0.933681i \(-0.383423\pi\)
0.358105 + 0.933681i \(0.383423\pi\)
\(824\) 0 0
\(825\) 11.0145 0.383476
\(826\) 0 0
\(827\) −3.31525 −0.115282 −0.0576412 0.998337i \(-0.518358\pi\)
−0.0576412 + 0.998337i \(0.518358\pi\)
\(828\) 0 0
\(829\) 9.08377 0.315492 0.157746 0.987480i \(-0.449577\pi\)
0.157746 + 0.987480i \(0.449577\pi\)
\(830\) 0 0
\(831\) 2.89067 0.100276
\(832\) 0 0
\(833\) 7.66201 0.265473
\(834\) 0 0
\(835\) −8.74455 −0.302617
\(836\) 0 0
\(837\) 7.90835 0.273353
\(838\) 0 0
\(839\) −36.9667 −1.27623 −0.638116 0.769940i \(-0.720286\pi\)
−0.638116 + 0.769940i \(0.720286\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −11.1332 −0.383447
\(844\) 0 0
\(845\) 0.884563 0.0304299
\(846\) 0 0
\(847\) 7.58934 0.260773
\(848\) 0 0
\(849\) 0.153847 0.00528002
\(850\) 0 0
\(851\) 4.37633 0.150019
\(852\) 0 0
\(853\) 4.56076 0.156158 0.0780788 0.996947i \(-0.475121\pi\)
0.0780788 + 0.996947i \(0.475121\pi\)
\(854\) 0 0
\(855\) 18.8596 0.644984
\(856\) 0 0
\(857\) −30.9348 −1.05671 −0.528356 0.849023i \(-0.677191\pi\)
−0.528356 + 0.849023i \(0.677191\pi\)
\(858\) 0 0
\(859\) 39.4985 1.34767 0.673835 0.738882i \(-0.264646\pi\)
0.673835 + 0.738882i \(0.264646\pi\)
\(860\) 0 0
\(861\) −4.89002 −0.166652
\(862\) 0 0
\(863\) −5.03237 −0.171304 −0.0856520 0.996325i \(-0.527297\pi\)
−0.0856520 + 0.996325i \(0.527297\pi\)
\(864\) 0 0
\(865\) −68.7674 −2.33816
\(866\) 0 0
\(867\) 15.3468 0.521203
\(868\) 0 0
\(869\) 0.646719 0.0219385
\(870\) 0 0
\(871\) −44.2776 −1.50029
\(872\) 0 0
\(873\) −18.0506 −0.610921
\(874\) 0 0
\(875\) −2.80926 −0.0949704
\(876\) 0 0
\(877\) −1.80048 −0.0607979 −0.0303989 0.999538i \(-0.509678\pi\)
−0.0303989 + 0.999538i \(0.509678\pi\)
\(878\) 0 0
\(879\) −32.7574 −1.10488
\(880\) 0 0
\(881\) −17.4943 −0.589397 −0.294698 0.955590i \(-0.595219\pi\)
−0.294698 + 0.955590i \(0.595219\pi\)
\(882\) 0 0
\(883\) 12.9385 0.435415 0.217708 0.976014i \(-0.430142\pi\)
0.217708 + 0.976014i \(0.430142\pi\)
\(884\) 0 0
\(885\) −13.5475 −0.455394
\(886\) 0 0
\(887\) −16.9871 −0.570372 −0.285186 0.958472i \(-0.592055\pi\)
−0.285186 + 0.958472i \(0.592055\pi\)
\(888\) 0 0
\(889\) 2.91454 0.0977506
\(890\) 0 0
\(891\) −1.88720 −0.0632236
\(892\) 0 0
\(893\) 36.3020 1.21480
\(894\) 0 0
\(895\) 23.1538 0.773947
\(896\) 0 0
\(897\) −3.64262 −0.121624
\(898\) 0 0
\(899\) −7.90835 −0.263758
\(900\) 0 0
\(901\) 4.36847 0.145535
\(902\) 0 0
\(903\) 9.87769 0.328709
\(904\) 0 0
\(905\) 34.5629 1.14891
\(906\) 0 0
\(907\) −53.7530 −1.78484 −0.892419 0.451207i \(-0.850994\pi\)
−0.892419 + 0.451207i \(0.850994\pi\)
\(908\) 0 0
\(909\) −0.307106 −0.0101861
\(910\) 0 0
\(911\) −0.961864 −0.0318680 −0.0159340 0.999873i \(-0.505072\pi\)
−0.0159340 + 0.999873i \(0.505072\pi\)
\(912\) 0 0
\(913\) −0.183969 −0.00608848
\(914\) 0 0
\(915\) 0.443268 0.0146540
\(916\) 0 0
\(917\) −12.8347 −0.423840
\(918\) 0 0
\(919\) −25.5693 −0.843454 −0.421727 0.906723i \(-0.638576\pi\)
−0.421727 + 0.906723i \(0.638576\pi\)
\(920\) 0 0
\(921\) −10.3644 −0.341518
\(922\) 0 0
\(923\) 6.10459 0.200935
\(924\) 0 0
\(925\) −25.5421 −0.839821
\(926\) 0 0
\(927\) −9.96209 −0.327198
\(928\) 0 0
\(929\) −20.9124 −0.686113 −0.343057 0.939315i \(-0.611462\pi\)
−0.343057 + 0.939315i \(0.611462\pi\)
\(930\) 0 0
\(931\) −34.1400 −1.11889
\(932\) 0 0
\(933\) 0.192809 0.00631229
\(934\) 0 0
\(935\) 7.98783 0.261230
\(936\) 0 0
\(937\) 49.7499 1.62526 0.812629 0.582782i \(-0.198036\pi\)
0.812629 + 0.582782i \(0.198036\pi\)
\(938\) 0 0
\(939\) 18.5661 0.605882
\(940\) 0 0
\(941\) 22.8626 0.745300 0.372650 0.927972i \(-0.378449\pi\)
0.372650 + 0.927972i \(0.378449\pi\)
\(942\) 0 0
\(943\) 4.79282 0.156076
\(944\) 0 0
\(945\) 3.35864 0.109257
\(946\) 0 0
\(947\) −44.1781 −1.43559 −0.717797 0.696253i \(-0.754849\pi\)
−0.717797 + 0.696253i \(0.754849\pi\)
\(948\) 0 0
\(949\) 0.465953 0.0151255
\(950\) 0 0
\(951\) −16.1487 −0.523659
\(952\) 0 0
\(953\) 43.7614 1.41757 0.708786 0.705424i \(-0.249243\pi\)
0.708786 + 0.705424i \(0.249243\pi\)
\(954\) 0 0
\(955\) 42.6518 1.38018
\(956\) 0 0
\(957\) 1.88720 0.0610045
\(958\) 0 0
\(959\) 1.31136 0.0423460
\(960\) 0 0
\(961\) 31.5421 1.01749
\(962\) 0 0
\(963\) −4.95885 −0.159797
\(964\) 0 0
\(965\) −89.2207 −2.87212
\(966\) 0 0
\(967\) −12.6578 −0.407048 −0.203524 0.979070i \(-0.565240\pi\)
−0.203524 + 0.979070i \(0.565240\pi\)
\(968\) 0 0
\(969\) 7.36642 0.236643
\(970\) 0 0
\(971\) 25.3998 0.815118 0.407559 0.913179i \(-0.366380\pi\)
0.407559 + 0.913179i \(0.366380\pi\)
\(972\) 0 0
\(973\) 5.38315 0.172576
\(974\) 0 0
\(975\) 21.2599 0.680862
\(976\) 0 0
\(977\) 30.7723 0.984494 0.492247 0.870455i \(-0.336176\pi\)
0.492247 + 0.870455i \(0.336176\pi\)
\(978\) 0 0
\(979\) 27.2893 0.872171
\(980\) 0 0
\(981\) −11.9944 −0.382952
\(982\) 0 0
\(983\) 14.3927 0.459057 0.229528 0.973302i \(-0.426282\pi\)
0.229528 + 0.973302i \(0.426282\pi\)
\(984\) 0 0
\(985\) 48.4902 1.54502
\(986\) 0 0
\(987\) 6.46491 0.205780
\(988\) 0 0
\(989\) −9.68133 −0.307849
\(990\) 0 0
\(991\) −19.9935 −0.635116 −0.317558 0.948239i \(-0.602863\pi\)
−0.317558 + 0.948239i \(0.602863\pi\)
\(992\) 0 0
\(993\) 18.9714 0.602041
\(994\) 0 0
\(995\) 6.19398 0.196362
\(996\) 0 0
\(997\) 8.84129 0.280006 0.140003 0.990151i \(-0.455289\pi\)
0.140003 + 0.990151i \(0.455289\pi\)
\(998\) 0 0
\(999\) 4.37633 0.138461
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.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.e.1.9 9 1.1 even 1 trivial