Properties

Label 1400.2.a.r.1.2
Level $1400$
Weight $2$
Character 1400.1
Self dual yes
Analytic conductor $11.179$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 1400.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.37228 q^{3} +1.00000 q^{7} +8.37228 q^{9} +O(q^{10})\) \(q+3.37228 q^{3} +1.00000 q^{7} +8.37228 q^{9} +0.627719 q^{11} +1.37228 q^{13} -5.37228 q^{17} +6.74456 q^{19} +3.37228 q^{21} -6.74456 q^{23} +18.1168 q^{27} +1.37228 q^{29} -8.00000 q^{31} +2.11684 q^{33} +2.00000 q^{37} +4.62772 q^{39} -4.74456 q^{41} -2.74456 q^{43} -10.1168 q^{47} +1.00000 q^{49} -18.1168 q^{51} +0.744563 q^{53} +22.7446 q^{57} +8.00000 q^{59} +8.74456 q^{61} +8.37228 q^{63} +4.00000 q^{67} -22.7446 q^{69} +8.00000 q^{71} +6.00000 q^{73} +0.627719 q^{77} -2.11684 q^{79} +35.9783 q^{81} -13.4891 q^{83} +4.62772 q^{87} +3.25544 q^{89} +1.37228 q^{91} -26.9783 q^{93} -18.8614 q^{97} +5.25544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 2q^{7} + 11q^{9} + O(q^{10}) \) \( 2q + q^{3} + 2q^{7} + 11q^{9} + 7q^{11} - 3q^{13} - 5q^{17} + 2q^{19} + q^{21} - 2q^{23} + 19q^{27} - 3q^{29} - 16q^{31} - 13q^{33} + 4q^{37} + 15q^{39} + 2q^{41} + 6q^{43} - 3q^{47} + 2q^{49} - 19q^{51} - 10q^{53} + 34q^{57} + 16q^{59} + 6q^{61} + 11q^{63} + 8q^{67} - 34q^{69} + 16q^{71} + 12q^{73} + 7q^{77} + 13q^{79} + 26q^{81} - 4q^{83} + 15q^{87} + 18q^{89} - 3q^{91} - 8q^{93} - 9q^{97} + 22q^{99} + O(q^{100}) \)

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
\(3\) 3.37228 1.94699 0.973494 0.228714i \(-0.0734519\pi\)
0.973494 + 0.228714i \(0.0734519\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 8.37228 2.79076
\(10\) 0 0
\(11\) 0.627719 0.189264 0.0946322 0.995512i \(-0.469833\pi\)
0.0946322 + 0.995512i \(0.469833\pi\)
\(12\) 0 0
\(13\) 1.37228 0.380602 0.190301 0.981726i \(-0.439054\pi\)
0.190301 + 0.981726i \(0.439054\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) 6.74456 1.54731 0.773654 0.633608i \(-0.218427\pi\)
0.773654 + 0.633608i \(0.218427\pi\)
\(20\) 0 0
\(21\) 3.37228 0.735892
\(22\) 0 0
\(23\) −6.74456 −1.40634 −0.703169 0.711022i \(-0.748232\pi\)
−0.703169 + 0.711022i \(0.748232\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 18.1168 3.48659
\(28\) 0 0
\(29\) 1.37228 0.254826 0.127413 0.991850i \(-0.459333\pi\)
0.127413 + 0.991850i \(0.459333\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.11684 0.368495
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 4.62772 0.741028
\(40\) 0 0
\(41\) −4.74456 −0.740976 −0.370488 0.928837i \(-0.620810\pi\)
−0.370488 + 0.928837i \(0.620810\pi\)
\(42\) 0 0
\(43\) −2.74456 −0.418542 −0.209271 0.977858i \(-0.567109\pi\)
−0.209271 + 0.977858i \(0.567109\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.1168 −1.47569 −0.737847 0.674968i \(-0.764157\pi\)
−0.737847 + 0.674968i \(0.764157\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −18.1168 −2.53687
\(52\) 0 0
\(53\) 0.744563 0.102274 0.0511368 0.998692i \(-0.483716\pi\)
0.0511368 + 0.998692i \(0.483716\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 22.7446 3.01259
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 8.74456 1.11963 0.559813 0.828619i \(-0.310873\pi\)
0.559813 + 0.828619i \(0.310873\pi\)
\(62\) 0 0
\(63\) 8.37228 1.05481
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −22.7446 −2.73812
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.627719 0.0715352
\(78\) 0 0
\(79\) −2.11684 −0.238164 −0.119082 0.992884i \(-0.537995\pi\)
−0.119082 + 0.992884i \(0.537995\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) −13.4891 −1.48062 −0.740312 0.672264i \(-0.765322\pi\)
−0.740312 + 0.672264i \(0.765322\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.62772 0.496144
\(88\) 0 0
\(89\) 3.25544 0.345076 0.172538 0.985003i \(-0.444803\pi\)
0.172538 + 0.985003i \(0.444803\pi\)
\(90\) 0 0
\(91\) 1.37228 0.143854
\(92\) 0 0
\(93\) −26.9783 −2.79751
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.8614 −1.91509 −0.957543 0.288291i \(-0.906913\pi\)
−0.957543 + 0.288291i \(0.906913\pi\)
\(98\) 0 0
\(99\) 5.25544 0.528191
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 11.3723 1.12054 0.560272 0.828309i \(-0.310697\pi\)
0.560272 + 0.828309i \(0.310697\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.74456 0.265327 0.132663 0.991161i \(-0.457647\pi\)
0.132663 + 0.991161i \(0.457647\pi\)
\(108\) 0 0
\(109\) −5.37228 −0.514571 −0.257286 0.966335i \(-0.582828\pi\)
−0.257286 + 0.966335i \(0.582828\pi\)
\(110\) 0 0
\(111\) 6.74456 0.640166
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11.4891 1.06217
\(118\) 0 0
\(119\) −5.37228 −0.492476
\(120\) 0 0
\(121\) −10.6060 −0.964179
\(122\) 0 0
\(123\) −16.0000 −1.44267
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −9.25544 −0.814896
\(130\) 0 0
\(131\) 6.74456 0.589275 0.294638 0.955609i \(-0.404801\pi\)
0.294638 + 0.955609i \(0.404801\pi\)
\(132\) 0 0
\(133\) 6.74456 0.584828
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.25544 −0.278131 −0.139065 0.990283i \(-0.544410\pi\)
−0.139065 + 0.990283i \(0.544410\pi\)
\(138\) 0 0
\(139\) 6.74456 0.572066 0.286033 0.958220i \(-0.407663\pi\)
0.286033 + 0.958220i \(0.407663\pi\)
\(140\) 0 0
\(141\) −34.1168 −2.87316
\(142\) 0 0
\(143\) 0.861407 0.0720344
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.37228 0.278141
\(148\) 0 0
\(149\) −7.48913 −0.613533 −0.306767 0.951785i \(-0.599247\pi\)
−0.306767 + 0.951785i \(0.599247\pi\)
\(150\) 0 0
\(151\) −2.11684 −0.172266 −0.0861332 0.996284i \(-0.527451\pi\)
−0.0861332 + 0.996284i \(0.527451\pi\)
\(152\) 0 0
\(153\) −44.9783 −3.63628
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.48913 −0.597697 −0.298849 0.954301i \(-0.596603\pi\)
−0.298849 + 0.954301i \(0.596603\pi\)
\(158\) 0 0
\(159\) 2.51087 0.199125
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 0 0
\(163\) 5.25544 0.411638 0.205819 0.978590i \(-0.434014\pi\)
0.205819 + 0.978590i \(0.434014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3723 0.880014 0.440007 0.897994i \(-0.354976\pi\)
0.440007 + 0.897994i \(0.354976\pi\)
\(168\) 0 0
\(169\) −11.1168 −0.855142
\(170\) 0 0
\(171\) 56.4674 4.31817
\(172\) 0 0
\(173\) −5.37228 −0.408447 −0.204223 0.978924i \(-0.565467\pi\)
−0.204223 + 0.978924i \(0.565467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 26.9783 2.02781
\(178\) 0 0
\(179\) 22.9783 1.71748 0.858738 0.512416i \(-0.171249\pi\)
0.858738 + 0.512416i \(0.171249\pi\)
\(180\) 0 0
\(181\) −18.2337 −1.35530 −0.677650 0.735385i \(-0.737001\pi\)
−0.677650 + 0.735385i \(0.737001\pi\)
\(182\) 0 0
\(183\) 29.4891 2.17990
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.37228 −0.246606
\(188\) 0 0
\(189\) 18.1168 1.31781
\(190\) 0 0
\(191\) −24.8614 −1.79891 −0.899454 0.437015i \(-0.856036\pi\)
−0.899454 + 0.437015i \(0.856036\pi\)
\(192\) 0 0
\(193\) 4.74456 0.341521 0.170761 0.985313i \(-0.445378\pi\)
0.170761 + 0.985313i \(0.445378\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.2337 −1.86907 −0.934536 0.355867i \(-0.884185\pi\)
−0.934536 + 0.355867i \(0.884185\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 13.4891 0.951450
\(202\) 0 0
\(203\) 1.37228 0.0963153
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −56.4674 −3.92475
\(208\) 0 0
\(209\) 4.23369 0.292850
\(210\) 0 0
\(211\) −8.62772 −0.593957 −0.296978 0.954884i \(-0.595979\pi\)
−0.296978 + 0.954884i \(0.595979\pi\)
\(212\) 0 0
\(213\) 26.9783 1.84852
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 20.2337 1.36727
\(220\) 0 0
\(221\) −7.37228 −0.495913
\(222\) 0 0
\(223\) 11.3723 0.761544 0.380772 0.924669i \(-0.375658\pi\)
0.380772 + 0.924669i \(0.375658\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.86141 0.588152 0.294076 0.955782i \(-0.404988\pi\)
0.294076 + 0.955782i \(0.404988\pi\)
\(228\) 0 0
\(229\) 22.2337 1.46924 0.734622 0.678477i \(-0.237360\pi\)
0.734622 + 0.678477i \(0.237360\pi\)
\(230\) 0 0
\(231\) 2.11684 0.139278
\(232\) 0 0
\(233\) 12.7446 0.834924 0.417462 0.908694i \(-0.362920\pi\)
0.417462 + 0.908694i \(0.362920\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −7.13859 −0.463701
\(238\) 0 0
\(239\) 19.3723 1.25309 0.626544 0.779386i \(-0.284469\pi\)
0.626544 + 0.779386i \(0.284469\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 66.9783 4.29666
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.25544 0.588909
\(248\) 0 0
\(249\) −45.4891 −2.88276
\(250\) 0 0
\(251\) −6.74456 −0.425713 −0.212857 0.977083i \(-0.568277\pi\)
−0.212857 + 0.977083i \(0.568277\pi\)
\(252\) 0 0
\(253\) −4.23369 −0.266170
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.510875 0.0318675 0.0159337 0.999873i \(-0.494928\pi\)
0.0159337 + 0.999873i \(0.494928\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 11.4891 0.711159
\(262\) 0 0
\(263\) 12.2337 0.754362 0.377181 0.926140i \(-0.376894\pi\)
0.377181 + 0.926140i \(0.376894\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 10.9783 0.671858
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −13.4891 −0.819406 −0.409703 0.912219i \(-0.634368\pi\)
−0.409703 + 0.912219i \(0.634368\pi\)
\(272\) 0 0
\(273\) 4.62772 0.280082
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.9783 1.26046 0.630230 0.776408i \(-0.282960\pi\)
0.630230 + 0.776408i \(0.282960\pi\)
\(278\) 0 0
\(279\) −66.9783 −4.00988
\(280\) 0 0
\(281\) −21.6060 −1.28890 −0.644452 0.764645i \(-0.722914\pi\)
−0.644452 + 0.764645i \(0.722914\pi\)
\(282\) 0 0
\(283\) 26.1168 1.55249 0.776243 0.630434i \(-0.217123\pi\)
0.776243 + 0.630434i \(0.217123\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.74456 −0.280063
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) −63.6060 −3.72865
\(292\) 0 0
\(293\) −7.88316 −0.460539 −0.230269 0.973127i \(-0.573961\pi\)
−0.230269 + 0.973127i \(0.573961\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 11.3723 0.659887
\(298\) 0 0
\(299\) −9.25544 −0.535256
\(300\) 0 0
\(301\) −2.74456 −0.158194
\(302\) 0 0
\(303\) −20.2337 −1.16240
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −13.8832 −0.792354 −0.396177 0.918174i \(-0.629663\pi\)
−0.396177 + 0.918174i \(0.629663\pi\)
\(308\) 0 0
\(309\) 38.3505 2.18169
\(310\) 0 0
\(311\) −1.25544 −0.0711893 −0.0355947 0.999366i \(-0.511333\pi\)
−0.0355947 + 0.999366i \(0.511333\pi\)
\(312\) 0 0
\(313\) −20.1168 −1.13707 −0.568536 0.822659i \(-0.692490\pi\)
−0.568536 + 0.822659i \(0.692490\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 0.861407 0.0482295
\(320\) 0 0
\(321\) 9.25544 0.516588
\(322\) 0 0
\(323\) −36.2337 −2.01610
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −18.1168 −1.00186
\(328\) 0 0
\(329\) −10.1168 −0.557760
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 16.7446 0.917596
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.4891 −0.843746 −0.421873 0.906655i \(-0.638627\pi\)
−0.421873 + 0.906655i \(0.638627\pi\)
\(338\) 0 0
\(339\) −6.74456 −0.366314
\(340\) 0 0
\(341\) −5.02175 −0.271943
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.2554 0.711589 0.355795 0.934564i \(-0.384210\pi\)
0.355795 + 0.934564i \(0.384210\pi\)
\(348\) 0 0
\(349\) −3.48913 −0.186769 −0.0933843 0.995630i \(-0.529769\pi\)
−0.0933843 + 0.995630i \(0.529769\pi\)
\(350\) 0 0
\(351\) 24.8614 1.32700
\(352\) 0 0
\(353\) −26.8614 −1.42969 −0.714844 0.699284i \(-0.753502\pi\)
−0.714844 + 0.699284i \(0.753502\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −18.1168 −0.958845
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 0 0
\(363\) −35.7663 −1.87724
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.86141 0.462562 0.231281 0.972887i \(-0.425708\pi\)
0.231281 + 0.972887i \(0.425708\pi\)
\(368\) 0 0
\(369\) −39.7228 −2.06789
\(370\) 0 0
\(371\) 0.744563 0.0386558
\(372\) 0 0
\(373\) 19.2554 0.997009 0.498504 0.866887i \(-0.333883\pi\)
0.498504 + 0.866887i \(0.333883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.88316 0.0969875
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −26.9783 −1.38214
\(382\) 0 0
\(383\) −5.48913 −0.280481 −0.140241 0.990117i \(-0.544788\pi\)
−0.140241 + 0.990117i \(0.544788\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −22.9783 −1.16805
\(388\) 0 0
\(389\) −10.8614 −0.550695 −0.275348 0.961345i \(-0.588793\pi\)
−0.275348 + 0.961345i \(0.588793\pi\)
\(390\) 0 0
\(391\) 36.2337 1.83242
\(392\) 0 0
\(393\) 22.7446 1.14731
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −37.3723 −1.87566 −0.937831 0.347094i \(-0.887169\pi\)
−0.937831 + 0.347094i \(0.887169\pi\)
\(398\) 0 0
\(399\) 22.7446 1.13865
\(400\) 0 0
\(401\) 1.60597 0.0801983 0.0400991 0.999196i \(-0.487233\pi\)
0.0400991 + 0.999196i \(0.487233\pi\)
\(402\) 0 0
\(403\) −10.9783 −0.546866
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.25544 0.0622297
\(408\) 0 0
\(409\) −11.4891 −0.568101 −0.284050 0.958809i \(-0.591678\pi\)
−0.284050 + 0.958809i \(0.591678\pi\)
\(410\) 0 0
\(411\) −10.9783 −0.541517
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 22.7446 1.11381
\(418\) 0 0
\(419\) −37.4891 −1.83146 −0.915732 0.401790i \(-0.868388\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(420\) 0 0
\(421\) 21.6060 1.05301 0.526505 0.850172i \(-0.323502\pi\)
0.526505 + 0.850172i \(0.323502\pi\)
\(422\) 0 0
\(423\) −84.7011 −4.11831
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.74456 0.423179
\(428\) 0 0
\(429\) 2.90491 0.140250
\(430\) 0 0
\(431\) 12.6277 0.608256 0.304128 0.952631i \(-0.401635\pi\)
0.304128 + 0.952631i \(0.401635\pi\)
\(432\) 0 0
\(433\) 16.9783 0.815923 0.407961 0.912999i \(-0.366240\pi\)
0.407961 + 0.912999i \(0.366240\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −45.4891 −2.17604
\(438\) 0 0
\(439\) 28.2337 1.34752 0.673760 0.738950i \(-0.264678\pi\)
0.673760 + 0.738950i \(0.264678\pi\)
\(440\) 0 0
\(441\) 8.37228 0.398680
\(442\) 0 0
\(443\) 5.25544 0.249693 0.124847 0.992176i \(-0.460156\pi\)
0.124847 + 0.992176i \(0.460156\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −25.2554 −1.19454
\(448\) 0 0
\(449\) −0.116844 −0.00551421 −0.00275710 0.999996i \(-0.500878\pi\)
−0.00275710 + 0.999996i \(0.500878\pi\)
\(450\) 0 0
\(451\) −2.97825 −0.140240
\(452\) 0 0
\(453\) −7.13859 −0.335400
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16.7446 −0.783278 −0.391639 0.920119i \(-0.628092\pi\)
−0.391639 + 0.920119i \(0.628092\pi\)
\(458\) 0 0
\(459\) −97.3288 −4.54292
\(460\) 0 0
\(461\) −12.7446 −0.593573 −0.296787 0.954944i \(-0.595915\pi\)
−0.296787 + 0.954944i \(0.595915\pi\)
\(462\) 0 0
\(463\) 29.4891 1.37048 0.685238 0.728319i \(-0.259698\pi\)
0.685238 + 0.728319i \(0.259698\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.6060 1.46255 0.731275 0.682083i \(-0.238926\pi\)
0.731275 + 0.682083i \(0.238926\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −25.2554 −1.16371
\(472\) 0 0
\(473\) −1.72281 −0.0792150
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.23369 0.285421
\(478\) 0 0
\(479\) −12.2337 −0.558971 −0.279486 0.960150i \(-0.590164\pi\)
−0.279486 + 0.960150i \(0.590164\pi\)
\(480\) 0 0
\(481\) 2.74456 0.125141
\(482\) 0 0
\(483\) −22.7446 −1.03491
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.23369 0.191847 0.0959234 0.995389i \(-0.469420\pi\)
0.0959234 + 0.995389i \(0.469420\pi\)
\(488\) 0 0
\(489\) 17.7228 0.801453
\(490\) 0 0
\(491\) 17.8832 0.807056 0.403528 0.914967i \(-0.367784\pi\)
0.403528 + 0.914967i \(0.367784\pi\)
\(492\) 0 0
\(493\) −7.37228 −0.332031
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 3.13859 0.140503 0.0702514 0.997529i \(-0.477620\pi\)
0.0702514 + 0.997529i \(0.477620\pi\)
\(500\) 0 0
\(501\) 38.3505 1.71338
\(502\) 0 0
\(503\) 12.6277 0.563042 0.281521 0.959555i \(-0.409161\pi\)
0.281521 + 0.959555i \(0.409161\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −37.4891 −1.66495
\(508\) 0 0
\(509\) 4.97825 0.220657 0.110329 0.993895i \(-0.464810\pi\)
0.110329 + 0.993895i \(0.464810\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 122.190 5.39483
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.35053 −0.279296
\(518\) 0 0
\(519\) −18.1168 −0.795241
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −32.4674 −1.41970 −0.709850 0.704353i \(-0.751237\pi\)
−0.709850 + 0.704353i \(0.751237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.9783 1.87216
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) 66.9783 2.90661
\(532\) 0 0
\(533\) −6.51087 −0.282017
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 77.4891 3.34390
\(538\) 0 0
\(539\) 0.627719 0.0270378
\(540\) 0 0
\(541\) 20.3505 0.874938 0.437469 0.899234i \(-0.355875\pi\)
0.437469 + 0.899234i \(0.355875\pi\)
\(542\) 0 0
\(543\) −61.4891 −2.63875
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −14.9783 −0.640424 −0.320212 0.947346i \(-0.603754\pi\)
−0.320212 + 0.947346i \(0.603754\pi\)
\(548\) 0 0
\(549\) 73.2119 3.12461
\(550\) 0 0
\(551\) 9.25544 0.394295
\(552\) 0 0
\(553\) −2.11684 −0.0900174
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.2337 1.62001 0.810007 0.586421i \(-0.199463\pi\)
0.810007 + 0.586421i \(0.199463\pi\)
\(558\) 0 0
\(559\) −3.76631 −0.159298
\(560\) 0 0
\(561\) −11.3723 −0.480138
\(562\) 0 0
\(563\) 5.48913 0.231339 0.115670 0.993288i \(-0.463099\pi\)
0.115670 + 0.993288i \(0.463099\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 35.9783 1.51094
\(568\) 0 0
\(569\) 20.9783 0.879454 0.439727 0.898131i \(-0.355075\pi\)
0.439727 + 0.898131i \(0.355075\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −83.8397 −3.50245
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.6060 −0.732946 −0.366473 0.930429i \(-0.619435\pi\)
−0.366473 + 0.930429i \(0.619435\pi\)
\(578\) 0 0
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) −13.4891 −0.559623
\(582\) 0 0
\(583\) 0.467376 0.0193567
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.9783 −0.453121 −0.226560 0.973997i \(-0.572748\pi\)
−0.226560 + 0.973997i \(0.572748\pi\)
\(588\) 0 0
\(589\) −53.9565 −2.22324
\(590\) 0 0
\(591\) −88.4674 −3.63906
\(592\) 0 0
\(593\) 25.3723 1.04191 0.520957 0.853583i \(-0.325575\pi\)
0.520957 + 0.853583i \(0.325575\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −53.9565 −2.20829
\(598\) 0 0
\(599\) 7.60597 0.310771 0.155386 0.987854i \(-0.450338\pi\)
0.155386 + 0.987854i \(0.450338\pi\)
\(600\) 0 0
\(601\) 39.4891 1.61080 0.805398 0.592735i \(-0.201952\pi\)
0.805398 + 0.592735i \(0.201952\pi\)
\(602\) 0 0
\(603\) 33.4891 1.36378
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.6060 0.633427 0.316713 0.948521i \(-0.397421\pi\)
0.316713 + 0.948521i \(0.397421\pi\)
\(608\) 0 0
\(609\) 4.62772 0.187525
\(610\) 0 0
\(611\) −13.8832 −0.561652
\(612\) 0 0
\(613\) 31.4891 1.27183 0.635917 0.771758i \(-0.280622\pi\)
0.635917 + 0.771758i \(0.280622\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −1.25544 −0.0504603 −0.0252301 0.999682i \(-0.508032\pi\)
−0.0252301 + 0.999682i \(0.508032\pi\)
\(620\) 0 0
\(621\) −122.190 −4.90332
\(622\) 0 0
\(623\) 3.25544 0.130426
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 14.2772 0.570176
\(628\) 0 0
\(629\) −10.7446 −0.428414
\(630\) 0 0
\(631\) −3.37228 −0.134248 −0.0671242 0.997745i \(-0.521382\pi\)
−0.0671242 + 0.997745i \(0.521382\pi\)
\(632\) 0 0
\(633\) −29.0951 −1.15643
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.37228 0.0543718
\(638\) 0 0
\(639\) 66.9783 2.64962
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) −12.6277 −0.497989 −0.248994 0.968505i \(-0.580100\pi\)
−0.248994 + 0.968505i \(0.580100\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) 5.02175 0.197121
\(650\) 0 0
\(651\) −26.9783 −1.05736
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 50.2337 1.95980
\(658\) 0 0
\(659\) 6.11684 0.238278 0.119139 0.992878i \(-0.461987\pi\)
0.119139 + 0.992878i \(0.461987\pi\)
\(660\) 0 0
\(661\) 3.25544 0.126622 0.0633109 0.997994i \(-0.479834\pi\)
0.0633109 + 0.997994i \(0.479834\pi\)
\(662\) 0 0
\(663\) −24.8614 −0.965537
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.25544 −0.358372
\(668\) 0 0
\(669\) 38.3505 1.48272
\(670\) 0 0
\(671\) 5.48913 0.211905
\(672\) 0 0
\(673\) 31.7228 1.22282 0.611412 0.791312i \(-0.290602\pi\)
0.611412 + 0.791312i \(0.290602\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.3505 1.39706 0.698532 0.715579i \(-0.253837\pi\)
0.698532 + 0.715579i \(0.253837\pi\)
\(678\) 0 0
\(679\) −18.8614 −0.723834
\(680\) 0 0
\(681\) 29.8832 1.14513
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 74.9783 2.86060
\(688\) 0 0
\(689\) 1.02175 0.0389256
\(690\) 0 0
\(691\) 13.4891 0.513151 0.256575 0.966524i \(-0.417406\pi\)
0.256575 + 0.966524i \(0.417406\pi\)
\(692\) 0 0
\(693\) 5.25544 0.199638
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 25.4891 0.965469
\(698\) 0 0
\(699\) 42.9783 1.62559
\(700\) 0 0
\(701\) 30.8614 1.16562 0.582810 0.812609i \(-0.301953\pi\)
0.582810 + 0.812609i \(0.301953\pi\)
\(702\) 0 0
\(703\) 13.4891 0.508752
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −33.6060 −1.26210 −0.631049 0.775743i \(-0.717375\pi\)
−0.631049 + 0.775743i \(0.717375\pi\)
\(710\) 0 0
\(711\) −17.7228 −0.664657
\(712\) 0 0
\(713\) 53.9565 2.02069
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 65.3288 2.43975
\(718\) 0 0
\(719\) 14.7446 0.549879 0.274940 0.961461i \(-0.411342\pi\)
0.274940 + 0.961461i \(0.411342\pi\)
\(720\) 0 0
\(721\) 11.3723 0.423526
\(722\) 0 0
\(723\) 87.6793 3.26083
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 117.935 4.36795
\(730\) 0 0
\(731\) 14.7446 0.545347
\(732\) 0 0
\(733\) 38.8614 1.43538 0.717689 0.696363i \(-0.245200\pi\)
0.717689 + 0.696363i \(0.245200\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.51087 0.0924893
\(738\) 0 0
\(739\) 19.6060 0.721217 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(740\) 0 0
\(741\) 31.2119 1.14660
\(742\) 0 0
\(743\) 29.4891 1.08185 0.540926 0.841070i \(-0.318074\pi\)
0.540926 + 0.841070i \(0.318074\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −112.935 −4.13207
\(748\) 0 0
\(749\) 2.74456 0.100284
\(750\) 0 0
\(751\) 48.8614 1.78298 0.891489 0.453042i \(-0.149661\pi\)
0.891489 + 0.453042i \(0.149661\pi\)
\(752\) 0 0
\(753\) −22.7446 −0.828858
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.2337 1.38963 0.694814 0.719190i \(-0.255487\pi\)
0.694814 + 0.719190i \(0.255487\pi\)
\(758\) 0 0
\(759\) −14.2772 −0.518229
\(760\) 0 0
\(761\) −40.9783 −1.48546 −0.742730 0.669591i \(-0.766469\pi\)
−0.742730 + 0.669591i \(0.766469\pi\)
\(762\) 0 0
\(763\) −5.37228 −0.194490
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.9783 0.396402
\(768\) 0 0
\(769\) −19.4891 −0.702796 −0.351398 0.936226i \(-0.614294\pi\)
−0.351398 + 0.936226i \(0.614294\pi\)
\(770\) 0 0
\(771\) 1.72281 0.0620456
\(772\) 0 0
\(773\) 1.37228 0.0493575 0.0246788 0.999695i \(-0.492144\pi\)
0.0246788 + 0.999695i \(0.492144\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.74456 0.241960
\(778\) 0 0
\(779\) −32.0000 −1.14652
\(780\) 0 0
\(781\) 5.02175 0.179692
\(782\) 0 0
\(783\) 24.8614 0.888474
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.8832 0.494881 0.247441 0.968903i \(-0.420411\pi\)
0.247441 + 0.968903i \(0.420411\pi\)
\(788\) 0 0
\(789\) 41.2554 1.46873
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.3723 −0.757045 −0.378523 0.925592i \(-0.623568\pi\)
−0.378523 + 0.925592i \(0.623568\pi\)
\(798\) 0 0
\(799\) 54.3505 1.92278
\(800\) 0 0
\(801\) 27.2554 0.963024
\(802\) 0 0
\(803\) 3.76631 0.132910
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20.2337 −0.712260
\(808\) 0 0
\(809\) 14.6277 0.514283 0.257142 0.966374i \(-0.417219\pi\)
0.257142 + 0.966374i \(0.417219\pi\)
\(810\) 0 0
\(811\) 1.25544 0.0440844 0.0220422 0.999757i \(-0.492983\pi\)
0.0220422 + 0.999757i \(0.492983\pi\)
\(812\) 0 0
\(813\) −45.4891 −1.59537
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −18.5109 −0.647614
\(818\) 0 0
\(819\) 11.4891 0.401463
\(820\) 0 0
\(821\) −7.88316 −0.275124 −0.137562 0.990493i \(-0.543927\pi\)
−0.137562 + 0.990493i \(0.543927\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −13.2554 −0.460937 −0.230468 0.973080i \(-0.574026\pi\)
−0.230468 + 0.973080i \(0.574026\pi\)
\(828\) 0 0
\(829\) 22.2337 0.772208 0.386104 0.922455i \(-0.373821\pi\)
0.386104 + 0.922455i \(0.373821\pi\)
\(830\) 0 0
\(831\) 70.7446 2.45410
\(832\) 0 0
\(833\) −5.37228 −0.186139
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −144.935 −5.00968
\(838\) 0 0
\(839\) −22.7446 −0.785230 −0.392615 0.919703i \(-0.628429\pi\)
−0.392615 + 0.919703i \(0.628429\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) −72.8614 −2.50948
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.6060 −0.364425
\(848\) 0 0
\(849\) 88.0733 3.02267
\(850\) 0 0
\(851\) −13.4891 −0.462401
\(852\) 0 0
\(853\) 16.5109 0.565322 0.282661 0.959220i \(-0.408783\pi\)
0.282661 + 0.959220i \(0.408783\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −24.4674 −0.834816 −0.417408 0.908719i \(-0.637061\pi\)
−0.417408 + 0.908719i \(0.637061\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) −2.51087 −0.0854712 −0.0427356 0.999086i \(-0.513607\pi\)
−0.0427356 + 0.999086i \(0.513607\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 40.0000 1.35847
\(868\) 0 0
\(869\) −1.32878 −0.0450759
\(870\) 0 0
\(871\) 5.48913 0.185992
\(872\) 0 0
\(873\) −157.913 −5.34455
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 52.9783 1.78895 0.894474 0.447120i \(-0.147550\pi\)
0.894474 + 0.447120i \(0.147550\pi\)
\(878\) 0 0
\(879\) −26.5842 −0.896663
\(880\) 0 0
\(881\) −36.7446 −1.23796 −0.618978 0.785408i \(-0.712453\pi\)
−0.618978 + 0.785408i \(0.712453\pi\)
\(882\) 0 0
\(883\) 33.4891 1.12700 0.563499 0.826116i \(-0.309455\pi\)
0.563499 + 0.826116i \(0.309455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 58.9783 1.98030 0.990148 0.140025i \(-0.0447184\pi\)
0.990148 + 0.140025i \(0.0447184\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 22.5842 0.756600
\(892\) 0 0
\(893\) −68.2337 −2.28335
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −31.2119 −1.04214
\(898\) 0 0
\(899\) −10.9783 −0.366145
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) −9.25544 −0.308002
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 45.7228 1.51820 0.759101 0.650973i \(-0.225639\pi\)
0.759101 + 0.650973i \(0.225639\pi\)
\(908\) 0 0
\(909\) −50.2337 −1.66615
\(910\) 0 0
\(911\) 45.9565 1.52261 0.761303 0.648396i \(-0.224560\pi\)
0.761303 + 0.648396i \(0.224560\pi\)
\(912\) 0 0
\(913\) −8.46738 −0.280229
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.74456 0.222725
\(918\) 0 0
\(919\) 19.3723 0.639033 0.319516 0.947581i \(-0.396480\pi\)
0.319516 + 0.947581i \(0.396480\pi\)
\(920\) 0 0
\(921\) −46.8179 −1.54270
\(922\) 0 0
\(923\) 10.9783 0.361354
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 95.2119 3.12717
\(928\) 0 0
\(929\) −42.2337 −1.38564 −0.692821 0.721109i \(-0.743632\pi\)
−0.692821 + 0.721109i \(0.743632\pi\)
\(930\) 0 0
\(931\) 6.74456 0.221044
\(932\) 0 0
\(933\) −4.23369 −0.138605
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0951 1.14651 0.573253 0.819378i \(-0.305681\pi\)
0.573253 + 0.819378i \(0.305681\pi\)
\(938\) 0 0
\(939\) −67.8397 −2.21386
\(940\) 0 0
\(941\) −2.23369 −0.0728161 −0.0364081 0.999337i \(-0.511592\pi\)
−0.0364081 + 0.999337i \(0.511592\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 8.23369 0.267277
\(950\) 0 0
\(951\) −47.2119 −1.53095
\(952\) 0 0
\(953\) −48.7446 −1.57899 −0.789496 0.613756i \(-0.789658\pi\)
−0.789496 + 0.613756i \(0.789658\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.90491 0.0939023
\(958\) 0 0
\(959\) −3.25544 −0.105124
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 22.9783 0.740464
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −36.2337 −1.16520 −0.582598 0.812760i \(-0.697964\pi\)
−0.582598 + 0.812760i \(0.697964\pi\)
\(968\) 0 0
\(969\) −122.190 −3.92531
\(970\) 0 0
\(971\) 10.5109 0.337310 0.168655 0.985675i \(-0.446058\pi\)
0.168655 + 0.985675i \(0.446058\pi\)
\(972\) 0 0
\(973\) 6.74456 0.216221
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.02175 −0.224646 −0.112323 0.993672i \(-0.535829\pi\)
−0.112323 + 0.993672i \(0.535829\pi\)
\(978\) 0 0
\(979\) 2.04350 0.0653105
\(980\) 0 0
\(981\) −44.9783 −1.43605
\(982\) 0 0
\(983\) 4.62772 0.147601 0.0738007 0.997273i \(-0.476487\pi\)
0.0738007 + 0.997273i \(0.476487\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −34.1168 −1.08595
\(988\) 0 0
\(989\) 18.5109 0.588612
\(990\) 0 0
\(991\) 53.9565 1.71398 0.856992 0.515329i \(-0.172330\pi\)
0.856992 + 0.515329i \(0.172330\pi\)
\(992\) 0 0
\(993\) 40.4674 1.28419
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 24.1168 0.763788 0.381894 0.924206i \(-0.375272\pi\)
0.381894 + 0.924206i \(0.375272\pi\)
\(998\) 0 0
\(999\) 36.2337 1.14638
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 1400.2.a.r.1.2 2
4.3 odd 2 2800.2.a.bk.1.1 2
5.2 odd 4 1400.2.g.i.449.1 4
5.3 odd 4 1400.2.g.i.449.4 4
5.4 even 2 280.2.a.c.1.1 2
7.6 odd 2 9800.2.a.bu.1.1 2
15.14 odd 2 2520.2.a.x.1.2 2
20.3 even 4 2800.2.g.r.449.1 4
20.7 even 4 2800.2.g.r.449.4 4
20.19 odd 2 560.2.a.h.1.2 2
35.4 even 6 1960.2.q.t.961.2 4
35.9 even 6 1960.2.q.t.361.2 4
35.19 odd 6 1960.2.q.r.361.1 4
35.24 odd 6 1960.2.q.r.961.1 4
35.34 odd 2 1960.2.a.s.1.2 2
40.19 odd 2 2240.2.a.bg.1.1 2
40.29 even 2 2240.2.a.bk.1.2 2
60.59 even 2 5040.2.a.by.1.1 2
140.139 even 2 3920.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.1 2 5.4 even 2
560.2.a.h.1.2 2 20.19 odd 2
1400.2.a.r.1.2 2 1.1 even 1 trivial
1400.2.g.i.449.1 4 5.2 odd 4
1400.2.g.i.449.4 4 5.3 odd 4
1960.2.a.s.1.2 2 35.34 odd 2
1960.2.q.r.361.1 4 35.19 odd 6
1960.2.q.r.961.1 4 35.24 odd 6
1960.2.q.t.361.2 4 35.9 even 6
1960.2.q.t.961.2 4 35.4 even 6
2240.2.a.bg.1.1 2 40.19 odd 2
2240.2.a.bk.1.2 2 40.29 even 2
2520.2.a.x.1.2 2 15.14 odd 2
2800.2.a.bk.1.1 2 4.3 odd 2
2800.2.g.r.449.1 4 20.3 even 4
2800.2.g.r.449.4 4 20.7 even 4
3920.2.a.bt.1.1 2 140.139 even 2
5040.2.a.by.1.1 2 60.59 even 2
9800.2.a.bu.1.1 2 7.6 odd 2