Properties

Label 3920.2.a.bv.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\) \(q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +4.82843 q^{13} +0.414214 q^{15} -4.82843 q^{17} +2.82843 q^{19} -0.414214 q^{23} +1.00000 q^{25} +2.41421 q^{27} -1.00000 q^{29} -6.00000 q^{31} -0.343146 q^{33} -2.00000 q^{39} +7.82843 q^{41} -3.58579 q^{43} +2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -1.17157 q^{53} -0.828427 q^{55} -1.17157 q^{57} +4.48528 q^{59} -5.48528 q^{61} -4.82843 q^{65} -9.58579 q^{67} +0.171573 q^{69} -4.48528 q^{71} +0.828427 q^{73} -0.414214 q^{75} -14.8284 q^{79} +7.48528 q^{81} +13.7279 q^{83} +4.82843 q^{85} +0.414214 q^{87} +8.65685 q^{89} +2.48528 q^{93} -2.82843 q^{95} -11.6569 q^{97} -2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} - 4q^{11} + 4q^{13} - 2q^{15} - 4q^{17} + 2q^{23} + 2q^{25} + 2q^{27} - 2q^{29} - 12q^{31} - 12q^{33} - 4q^{39} + 10q^{41} - 10q^{43} + 4q^{47} + 4q^{51} - 8q^{53} + 4q^{55} - 8q^{57} - 8q^{59} + 6q^{61} - 4q^{65} - 22q^{67} + 6q^{69} + 8q^{71} - 4q^{73} + 2q^{75} - 24q^{79} - 2q^{81} + 2q^{83} + 4q^{85} - 2q^{87} + 6q^{89} - 12q^{93} - 12q^{97} - 16q^{99} + O(q^{100}) \)

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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0.414214 0.106949
\(16\) 0 0
\(17\) −4.82843 −1.17107 −0.585533 0.810649i \(-0.699115\pi\)
−0.585533 + 0.810649i \(0.699115\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.414214 −0.0863695 −0.0431847 0.999067i \(-0.513750\pi\)
−0.0431847 + 0.999067i \(0.513750\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −0.343146 −0.0597340
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 7.82843 1.22259 0.611297 0.791401i \(-0.290648\pi\)
0.611297 + 0.791401i \(0.290648\pi\)
\(42\) 0 0
\(43\) −3.58579 −0.546827 −0.273414 0.961897i \(-0.588153\pi\)
−0.273414 + 0.961897i \(0.588153\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −1.17157 −0.160928 −0.0804640 0.996758i \(-0.525640\pi\)
−0.0804640 + 0.996758i \(0.525640\pi\)
\(54\) 0 0
\(55\) −0.828427 −0.111705
\(56\) 0 0
\(57\) −1.17157 −0.155179
\(58\) 0 0
\(59\) 4.48528 0.583934 0.291967 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(60\) 0 0
\(61\) −5.48528 −0.702318 −0.351159 0.936316i \(-0.614212\pi\)
−0.351159 + 0.936316i \(0.614212\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.82843 −0.598893
\(66\) 0 0
\(67\) −9.58579 −1.17109 −0.585545 0.810640i \(-0.699119\pi\)
−0.585545 + 0.810640i \(0.699119\pi\)
\(68\) 0 0
\(69\) 0.171573 0.0206549
\(70\) 0 0
\(71\) −4.48528 −0.532305 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(72\) 0 0
\(73\) 0.828427 0.0969601 0.0484800 0.998824i \(-0.484562\pi\)
0.0484800 + 0.998824i \(0.484562\pi\)
\(74\) 0 0
\(75\) −0.414214 −0.0478293
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.8284 −1.66833 −0.834164 0.551516i \(-0.814049\pi\)
−0.834164 + 0.551516i \(0.814049\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) 13.7279 1.50684 0.753418 0.657542i \(-0.228404\pi\)
0.753418 + 0.657542i \(0.228404\pi\)
\(84\) 0 0
\(85\) 4.82843 0.523716
\(86\) 0 0
\(87\) 0.414214 0.0444084
\(88\) 0 0
\(89\) 8.65685 0.917625 0.458812 0.888533i \(-0.348275\pi\)
0.458812 + 0.888533i \(0.348275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.48528 0.257712
\(94\) 0 0
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −11.6569 −1.18357 −0.591787 0.806094i \(-0.701577\pi\)
−0.591787 + 0.806094i \(0.701577\pi\)
\(98\) 0 0
\(99\) −2.34315 −0.235495
\(100\) 0 0
\(101\) 10.3137 1.02625 0.513126 0.858313i \(-0.328487\pi\)
0.513126 + 0.858313i \(0.328487\pi\)
\(102\) 0 0
\(103\) −2.41421 −0.237880 −0.118940 0.992901i \(-0.537950\pi\)
−0.118940 + 0.992901i \(0.537950\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.2426 −1.08687 −0.543434 0.839452i \(-0.682876\pi\)
−0.543434 + 0.839452i \(0.682876\pi\)
\(108\) 0 0
\(109\) −13.4853 −1.29166 −0.645828 0.763483i \(-0.723488\pi\)
−0.645828 + 0.763483i \(0.723488\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.48528 −0.421940 −0.210970 0.977493i \(-0.567662\pi\)
−0.210970 + 0.977493i \(0.567662\pi\)
\(114\) 0 0
\(115\) 0.414214 0.0386256
\(116\) 0 0
\(117\) −13.6569 −1.26258
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −3.24264 −0.292379
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.31371 0.826458 0.413229 0.910627i \(-0.364401\pi\)
0.413229 + 0.910627i \(0.364401\pi\)
\(128\) 0 0
\(129\) 1.48528 0.130772
\(130\) 0 0
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −2.41421 −0.207782
\(136\) 0 0
\(137\) −9.65685 −0.825041 −0.412520 0.910948i \(-0.635351\pi\)
−0.412520 + 0.910948i \(0.635351\pi\)
\(138\) 0 0
\(139\) −16.1421 −1.36916 −0.684579 0.728939i \(-0.740014\pi\)
−0.684579 + 0.728939i \(0.740014\pi\)
\(140\) 0 0
\(141\) −0.828427 −0.0697661
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.17157 −0.177902 −0.0889511 0.996036i \(-0.528351\pi\)
−0.0889511 + 0.996036i \(0.528351\pi\)
\(150\) 0 0
\(151\) −11.6569 −0.948621 −0.474311 0.880358i \(-0.657303\pi\)
−0.474311 + 0.880358i \(0.657303\pi\)
\(152\) 0 0
\(153\) 13.6569 1.10409
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −17.3137 −1.38178 −0.690892 0.722958i \(-0.742782\pi\)
−0.690892 + 0.722958i \(0.742782\pi\)
\(158\) 0 0
\(159\) 0.485281 0.0384853
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.3431 −0.966790 −0.483395 0.875402i \(-0.660596\pi\)
−0.483395 + 0.875402i \(0.660596\pi\)
\(164\) 0 0
\(165\) 0.343146 0.0267139
\(166\) 0 0
\(167\) 22.4142 1.73446 0.867232 0.497904i \(-0.165897\pi\)
0.867232 + 0.497904i \(0.165897\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) −3.31371 −0.251937 −0.125968 0.992034i \(-0.540204\pi\)
−0.125968 + 0.992034i \(0.540204\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.85786 −0.139646
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −2.65685 −0.197482 −0.0987412 0.995113i \(-0.531482\pi\)
−0.0987412 + 0.995113i \(0.531482\pi\)
\(182\) 0 0
\(183\) 2.27208 0.167957
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.8284 0.928232 0.464116 0.885774i \(-0.346372\pi\)
0.464116 + 0.885774i \(0.346372\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −12.3431 −0.879413 −0.439706 0.898142i \(-0.644917\pi\)
−0.439706 + 0.898142i \(0.644917\pi\)
\(198\) 0 0
\(199\) −9.65685 −0.684556 −0.342278 0.939599i \(-0.611199\pi\)
−0.342278 + 0.939599i \(0.611199\pi\)
\(200\) 0 0
\(201\) 3.97056 0.280062
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −7.82843 −0.546761
\(206\) 0 0
\(207\) 1.17157 0.0814299
\(208\) 0 0
\(209\) 2.34315 0.162079
\(210\) 0 0
\(211\) −20.4853 −1.41026 −0.705132 0.709076i \(-0.749112\pi\)
−0.705132 + 0.709076i \(0.749112\pi\)
\(212\) 0 0
\(213\) 1.85786 0.127299
\(214\) 0 0
\(215\) 3.58579 0.244549
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.343146 −0.0231876
\(220\) 0 0
\(221\) −23.3137 −1.56825
\(222\) 0 0
\(223\) 0.343146 0.0229787 0.0114894 0.999934i \(-0.496343\pi\)
0.0114894 + 0.999934i \(0.496343\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) −6.97056 −0.462652 −0.231326 0.972876i \(-0.574306\pi\)
−0.231326 + 0.972876i \(0.574306\pi\)
\(228\) 0 0
\(229\) 11.6569 0.770307 0.385153 0.922853i \(-0.374149\pi\)
0.385153 + 0.922853i \(0.374149\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.8284 −1.10247 −0.551233 0.834351i \(-0.685843\pi\)
−0.551233 + 0.834351i \(0.685843\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 0 0
\(237\) 6.14214 0.398975
\(238\) 0 0
\(239\) 21.3137 1.37867 0.689335 0.724443i \(-0.257903\pi\)
0.689335 + 0.724443i \(0.257903\pi\)
\(240\) 0 0
\(241\) −27.6569 −1.78153 −0.890767 0.454460i \(-0.849832\pi\)
−0.890767 + 0.454460i \(0.849832\pi\)
\(242\) 0 0
\(243\) −10.3431 −0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.6569 0.868965
\(248\) 0 0
\(249\) −5.68629 −0.360354
\(250\) 0 0
\(251\) −9.31371 −0.587876 −0.293938 0.955824i \(-0.594966\pi\)
−0.293938 + 0.955824i \(0.594966\pi\)
\(252\) 0 0
\(253\) −0.343146 −0.0215734
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) −6.34315 −0.395675 −0.197837 0.980235i \(-0.563392\pi\)
−0.197837 + 0.980235i \(0.563392\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.82843 0.175075
\(262\) 0 0
\(263\) 29.0416 1.79078 0.895392 0.445279i \(-0.146896\pi\)
0.895392 + 0.445279i \(0.146896\pi\)
\(264\) 0 0
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) −3.58579 −0.219447
\(268\) 0 0
\(269\) −20.4558 −1.24721 −0.623607 0.781738i \(-0.714334\pi\)
−0.623607 + 0.781738i \(0.714334\pi\)
\(270\) 0 0
\(271\) 16.4853 1.00141 0.500705 0.865618i \(-0.333074\pi\)
0.500705 + 0.865618i \(0.333074\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.828427 0.0499560
\(276\) 0 0
\(277\) 16.1421 0.969887 0.484943 0.874546i \(-0.338840\pi\)
0.484943 + 0.874546i \(0.338840\pi\)
\(278\) 0 0
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) −30.2843 −1.80661 −0.903304 0.429001i \(-0.858866\pi\)
−0.903304 + 0.429001i \(0.858866\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 1.17157 0.0693980
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.31371 0.371395
\(290\) 0 0
\(291\) 4.82843 0.283047
\(292\) 0 0
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) −4.48528 −0.261143
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.27208 −0.245424
\(304\) 0 0
\(305\) 5.48528 0.314086
\(306\) 0 0
\(307\) −4.75736 −0.271517 −0.135758 0.990742i \(-0.543347\pi\)
−0.135758 + 0.990742i \(0.543347\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −13.1716 −0.746891 −0.373446 0.927652i \(-0.621824\pi\)
−0.373446 + 0.927652i \(0.621824\pi\)
\(312\) 0 0
\(313\) 6.34315 0.358536 0.179268 0.983800i \(-0.442627\pi\)
0.179268 + 0.983800i \(0.442627\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.7990 −0.775028 −0.387514 0.921864i \(-0.626666\pi\)
−0.387514 + 0.921864i \(0.626666\pi\)
\(318\) 0 0
\(319\) −0.828427 −0.0463830
\(320\) 0 0
\(321\) 4.65685 0.259920
\(322\) 0 0
\(323\) −13.6569 −0.759888
\(324\) 0 0
\(325\) 4.82843 0.267833
\(326\) 0 0
\(327\) 5.58579 0.308895
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.9706 −1.26258 −0.631288 0.775548i \(-0.717473\pi\)
−0.631288 + 0.775548i \(0.717473\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.58579 0.523727
\(336\) 0 0
\(337\) 9.17157 0.499607 0.249804 0.968296i \(-0.419634\pi\)
0.249804 + 0.968296i \(0.419634\pi\)
\(338\) 0 0
\(339\) 1.85786 0.100905
\(340\) 0 0
\(341\) −4.97056 −0.269171
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.171573 −0.00923717
\(346\) 0 0
\(347\) −7.92893 −0.425647 −0.212824 0.977091i \(-0.568266\pi\)
−0.212824 + 0.977091i \(0.568266\pi\)
\(348\) 0 0
\(349\) 15.3431 0.821300 0.410650 0.911793i \(-0.365302\pi\)
0.410650 + 0.911793i \(0.365302\pi\)
\(350\) 0 0
\(351\) 11.6569 0.622197
\(352\) 0 0
\(353\) 26.8284 1.42793 0.713967 0.700180i \(-0.246897\pi\)
0.713967 + 0.700180i \(0.246897\pi\)
\(354\) 0 0
\(355\) 4.48528 0.238054
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 4.27208 0.224226
\(364\) 0 0
\(365\) −0.828427 −0.0433619
\(366\) 0 0
\(367\) −2.75736 −0.143933 −0.0719665 0.997407i \(-0.522927\pi\)
−0.0719665 + 0.997407i \(0.522927\pi\)
\(368\) 0 0
\(369\) −22.1421 −1.15267
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −20.9706 −1.08581 −0.542907 0.839793i \(-0.682677\pi\)
−0.542907 + 0.839793i \(0.682677\pi\)
\(374\) 0 0
\(375\) 0.414214 0.0213899
\(376\) 0 0
\(377\) −4.82843 −0.248677
\(378\) 0 0
\(379\) −26.8284 −1.37808 −0.689042 0.724722i \(-0.741968\pi\)
−0.689042 + 0.724722i \(0.741968\pi\)
\(380\) 0 0
\(381\) −3.85786 −0.197644
\(382\) 0 0
\(383\) −2.89949 −0.148157 −0.0740786 0.997252i \(-0.523602\pi\)
−0.0740786 + 0.997252i \(0.523602\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.1421 0.515554
\(388\) 0 0
\(389\) 23.6569 1.19945 0.599725 0.800206i \(-0.295277\pi\)
0.599725 + 0.800206i \(0.295277\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) 14.8284 0.746099
\(396\) 0 0
\(397\) 16.6274 0.834506 0.417253 0.908790i \(-0.362993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.3137 1.51379 0.756897 0.653534i \(-0.226714\pi\)
0.756897 + 0.653534i \(0.226714\pi\)
\(402\) 0 0
\(403\) −28.9706 −1.44313
\(404\) 0 0
\(405\) −7.48528 −0.371947
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.7990 0.731763 0.365881 0.930661i \(-0.380768\pi\)
0.365881 + 0.930661i \(0.380768\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −13.7279 −0.673877
\(416\) 0 0
\(417\) 6.68629 0.327429
\(418\) 0 0
\(419\) −0.686292 −0.0335275 −0.0167638 0.999859i \(-0.505336\pi\)
−0.0167638 + 0.999859i \(0.505336\pi\)
\(420\) 0 0
\(421\) 13.4853 0.657232 0.328616 0.944464i \(-0.393418\pi\)
0.328616 + 0.944464i \(0.393418\pi\)
\(422\) 0 0
\(423\) −5.65685 −0.275046
\(424\) 0 0
\(425\) −4.82843 −0.234213
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.65685 −0.0799937
\(430\) 0 0
\(431\) −17.7990 −0.857347 −0.428674 0.903459i \(-0.641019\pi\)
−0.428674 + 0.903459i \(0.641019\pi\)
\(432\) 0 0
\(433\) −7.79899 −0.374796 −0.187398 0.982284i \(-0.560005\pi\)
−0.187398 + 0.982284i \(0.560005\pi\)
\(434\) 0 0
\(435\) −0.414214 −0.0198600
\(436\) 0 0
\(437\) −1.17157 −0.0560439
\(438\) 0 0
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −30.2132 −1.43547 −0.717736 0.696315i \(-0.754822\pi\)
−0.717736 + 0.696315i \(0.754822\pi\)
\(444\) 0 0
\(445\) −8.65685 −0.410374
\(446\) 0 0
\(447\) 0.899495 0.0425447
\(448\) 0 0
\(449\) 3.82843 0.180675 0.0903373 0.995911i \(-0.471205\pi\)
0.0903373 + 0.995911i \(0.471205\pi\)
\(450\) 0 0
\(451\) 6.48528 0.305380
\(452\) 0 0
\(453\) 4.82843 0.226859
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.2843 1.13597 0.567985 0.823039i \(-0.307723\pi\)
0.567985 + 0.823039i \(0.307723\pi\)
\(458\) 0 0
\(459\) −11.6569 −0.544095
\(460\) 0 0
\(461\) −41.3137 −1.92417 −0.962086 0.272748i \(-0.912068\pi\)
−0.962086 + 0.272748i \(0.912068\pi\)
\(462\) 0 0
\(463\) 37.0416 1.72147 0.860735 0.509053i \(-0.170004\pi\)
0.860735 + 0.509053i \(0.170004\pi\)
\(464\) 0 0
\(465\) −2.48528 −0.115252
\(466\) 0 0
\(467\) −3.10051 −0.143474 −0.0717371 0.997424i \(-0.522854\pi\)
−0.0717371 + 0.997424i \(0.522854\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 7.17157 0.330449
\(472\) 0 0
\(473\) −2.97056 −0.136587
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 0 0
\(477\) 3.31371 0.151724
\(478\) 0 0
\(479\) −35.6569 −1.62920 −0.814602 0.580021i \(-0.803044\pi\)
−0.814602 + 0.580021i \(0.803044\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.6569 0.529310
\(486\) 0 0
\(487\) −4.34315 −0.196807 −0.0984034 0.995147i \(-0.531374\pi\)
−0.0984034 + 0.995147i \(0.531374\pi\)
\(488\) 0 0
\(489\) 5.11270 0.231204
\(490\) 0 0
\(491\) −9.31371 −0.420322 −0.210161 0.977667i \(-0.567399\pi\)
−0.210161 + 0.977667i \(0.567399\pi\)
\(492\) 0 0
\(493\) 4.82843 0.217461
\(494\) 0 0
\(495\) 2.34315 0.105317
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.828427 0.0370855 0.0185427 0.999828i \(-0.494097\pi\)
0.0185427 + 0.999828i \(0.494097\pi\)
\(500\) 0 0
\(501\) −9.28427 −0.414791
\(502\) 0 0
\(503\) −15.8701 −0.707611 −0.353805 0.935319i \(-0.615113\pi\)
−0.353805 + 0.935319i \(0.615113\pi\)
\(504\) 0 0
\(505\) −10.3137 −0.458954
\(506\) 0 0
\(507\) −4.27208 −0.189730
\(508\) 0 0
\(509\) −13.3431 −0.591425 −0.295712 0.955277i \(-0.595557\pi\)
−0.295712 + 0.955277i \(0.595557\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.82843 0.301482
\(514\) 0 0
\(515\) 2.41421 0.106383
\(516\) 0 0
\(517\) 1.65685 0.0728684
\(518\) 0 0
\(519\) 1.37258 0.0602497
\(520\) 0 0
\(521\) −14.9706 −0.655872 −0.327936 0.944700i \(-0.606353\pi\)
−0.327936 + 0.944700i \(0.606353\pi\)
\(522\) 0 0
\(523\) 35.6569 1.55917 0.779583 0.626299i \(-0.215431\pi\)
0.779583 + 0.626299i \(0.215431\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.9706 1.26198
\(528\) 0 0
\(529\) −22.8284 −0.992540
\(530\) 0 0
\(531\) −12.6863 −0.550538
\(532\) 0 0
\(533\) 37.7990 1.63726
\(534\) 0 0
\(535\) 11.2426 0.486062
\(536\) 0 0
\(537\) −4.14214 −0.178746
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.34315 0.315706 0.157853 0.987463i \(-0.449543\pi\)
0.157853 + 0.987463i \(0.449543\pi\)
\(542\) 0 0
\(543\) 1.10051 0.0472272
\(544\) 0 0
\(545\) 13.4853 0.577646
\(546\) 0 0
\(547\) −24.8995 −1.06463 −0.532313 0.846548i \(-0.678677\pi\)
−0.532313 + 0.846548i \(0.678677\pi\)
\(548\) 0 0
\(549\) 15.5147 0.662152
\(550\) 0 0
\(551\) −2.82843 −0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.2843 0.944215 0.472107 0.881541i \(-0.343493\pi\)
0.472107 + 0.881541i \(0.343493\pi\)
\(558\) 0 0
\(559\) −17.3137 −0.732292
\(560\) 0 0
\(561\) 1.65685 0.0699524
\(562\) 0 0
\(563\) 41.7279 1.75862 0.879311 0.476248i \(-0.158003\pi\)
0.879311 + 0.476248i \(0.158003\pi\)
\(564\) 0 0
\(565\) 4.48528 0.188697
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.65685 0.320992 0.160496 0.987036i \(-0.448691\pi\)
0.160496 + 0.987036i \(0.448691\pi\)
\(570\) 0 0
\(571\) −9.17157 −0.383818 −0.191909 0.981413i \(-0.561468\pi\)
−0.191909 + 0.981413i \(0.561468\pi\)
\(572\) 0 0
\(573\) −5.31371 −0.221983
\(574\) 0 0
\(575\) −0.414214 −0.0172739
\(576\) 0 0
\(577\) 43.9411 1.82929 0.914646 0.404255i \(-0.132469\pi\)
0.914646 + 0.404255i \(0.132469\pi\)
\(578\) 0 0
\(579\) 0.828427 0.0344283
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.970563 −0.0401966
\(584\) 0 0
\(585\) 13.6569 0.564641
\(586\) 0 0
\(587\) −34.2843 −1.41506 −0.707532 0.706682i \(-0.750191\pi\)
−0.707532 + 0.706682i \(0.750191\pi\)
\(588\) 0 0
\(589\) −16.9706 −0.699260
\(590\) 0 0
\(591\) 5.11270 0.210308
\(592\) 0 0
\(593\) −4.20101 −0.172515 −0.0862574 0.996273i \(-0.527491\pi\)
−0.0862574 + 0.996273i \(0.527491\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −6.34315 −0.259174 −0.129587 0.991568i \(-0.541365\pi\)
−0.129587 + 0.991568i \(0.541365\pi\)
\(600\) 0 0
\(601\) −19.6569 −0.801820 −0.400910 0.916117i \(-0.631306\pi\)
−0.400910 + 0.916117i \(0.631306\pi\)
\(602\) 0 0
\(603\) 27.1127 1.10411
\(604\) 0 0
\(605\) 10.3137 0.419312
\(606\) 0 0
\(607\) 38.2132 1.55103 0.775513 0.631332i \(-0.217491\pi\)
0.775513 + 0.631332i \(0.217491\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.65685 0.390675
\(612\) 0 0
\(613\) 35.4558 1.43205 0.716024 0.698076i \(-0.245960\pi\)
0.716024 + 0.698076i \(0.245960\pi\)
\(614\) 0 0
\(615\) 3.24264 0.130756
\(616\) 0 0
\(617\) 11.3137 0.455473 0.227736 0.973723i \(-0.426868\pi\)
0.227736 + 0.973723i \(0.426868\pi\)
\(618\) 0 0
\(619\) 25.5147 1.02552 0.512762 0.858531i \(-0.328622\pi\)
0.512762 + 0.858531i \(0.328622\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.970563 −0.0387605
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 20.1421 0.801846 0.400923 0.916112i \(-0.368690\pi\)
0.400923 + 0.916112i \(0.368690\pi\)
\(632\) 0 0
\(633\) 8.48528 0.337260
\(634\) 0 0
\(635\) −9.31371 −0.369603
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 12.6863 0.501862
\(640\) 0 0
\(641\) 31.4853 1.24359 0.621797 0.783179i \(-0.286403\pi\)
0.621797 + 0.783179i \(0.286403\pi\)
\(642\) 0 0
\(643\) 26.2843 1.03655 0.518275 0.855214i \(-0.326574\pi\)
0.518275 + 0.855214i \(0.326574\pi\)
\(644\) 0 0
\(645\) −1.48528 −0.0584829
\(646\) 0 0
\(647\) −31.0416 −1.22037 −0.610186 0.792258i \(-0.708905\pi\)
−0.610186 + 0.792258i \(0.708905\pi\)
\(648\) 0 0
\(649\) 3.71573 0.145855
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.1716 −0.750242 −0.375121 0.926976i \(-0.622399\pi\)
−0.375121 + 0.926976i \(0.622399\pi\)
\(654\) 0 0
\(655\) 19.3137 0.754649
\(656\) 0 0
\(657\) −2.34315 −0.0914148
\(658\) 0 0
\(659\) −21.1716 −0.824727 −0.412364 0.911019i \(-0.635297\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(660\) 0 0
\(661\) −31.8284 −1.23798 −0.618991 0.785398i \(-0.712458\pi\)
−0.618991 + 0.785398i \(0.712458\pi\)
\(662\) 0 0
\(663\) 9.65685 0.375041
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.414214 0.0160384
\(668\) 0 0
\(669\) −0.142136 −0.00549528
\(670\) 0 0
\(671\) −4.54416 −0.175425
\(672\) 0 0
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) 0 0
\(675\) 2.41421 0.0929231
\(676\) 0 0
\(677\) 28.1421 1.08159 0.540795 0.841154i \(-0.318123\pi\)
0.540795 + 0.841154i \(0.318123\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 2.88730 0.110642
\(682\) 0 0
\(683\) 34.7574 1.32995 0.664977 0.746864i \(-0.268441\pi\)
0.664977 + 0.746864i \(0.268441\pi\)
\(684\) 0 0
\(685\) 9.65685 0.368969
\(686\) 0 0
\(687\) −4.82843 −0.184216
\(688\) 0 0
\(689\) −5.65685 −0.215509
\(690\) 0 0
\(691\) 0.828427 0.0315149 0.0157574 0.999876i \(-0.494984\pi\)
0.0157574 + 0.999876i \(0.494984\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.1421 0.612306
\(696\) 0 0
\(697\) −37.7990 −1.43174
\(698\) 0 0
\(699\) 6.97056 0.263651
\(700\) 0 0
\(701\) −3.20101 −0.120900 −0.0604502 0.998171i \(-0.519254\pi\)
−0.0604502 + 0.998171i \(0.519254\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0.828427 0.0312004
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 15.6863 0.589111 0.294556 0.955634i \(-0.404828\pi\)
0.294556 + 0.955634i \(0.404828\pi\)
\(710\) 0 0
\(711\) 41.9411 1.57292
\(712\) 0 0
\(713\) 2.48528 0.0930745
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −8.82843 −0.329704
\(718\) 0 0
\(719\) 21.1127 0.787371 0.393685 0.919245i \(-0.371200\pi\)
0.393685 + 0.919245i \(0.371200\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.4558 0.426047
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −37.5858 −1.39398 −0.696990 0.717081i \(-0.745478\pi\)
−0.696990 + 0.717081i \(0.745478\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 17.3137 0.640371
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.94113 −0.292515
\(738\) 0 0
\(739\) 21.1127 0.776643 0.388322 0.921524i \(-0.373055\pi\)
0.388322 + 0.921524i \(0.373055\pi\)
\(740\) 0 0
\(741\) −5.65685 −0.207810
\(742\) 0 0
\(743\) 16.0711 0.589590 0.294795 0.955560i \(-0.404749\pi\)
0.294795 + 0.955560i \(0.404749\pi\)
\(744\) 0 0
\(745\) 2.17157 0.0795603
\(746\) 0 0
\(747\) −38.8284 −1.42066
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 30.3431 1.10724 0.553619 0.832770i \(-0.313247\pi\)
0.553619 + 0.832770i \(0.313247\pi\)
\(752\) 0 0
\(753\) 3.85786 0.140588
\(754\) 0 0
\(755\) 11.6569 0.424236
\(756\) 0 0
\(757\) −31.4558 −1.14328 −0.571641 0.820504i \(-0.693693\pi\)
−0.571641 + 0.820504i \(0.693693\pi\)
\(758\) 0 0
\(759\) 0.142136 0.00515920
\(760\) 0 0
\(761\) −9.31371 −0.337622 −0.168811 0.985648i \(-0.553993\pi\)
−0.168811 + 0.985648i \(0.553993\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −13.6569 −0.493765
\(766\) 0 0
\(767\) 21.6569 0.781984
\(768\) 0 0
\(769\) 0.627417 0.0226252 0.0113126 0.999936i \(-0.496399\pi\)
0.0113126 + 0.999936i \(0.496399\pi\)
\(770\) 0 0
\(771\) 2.62742 0.0946241
\(772\) 0 0
\(773\) 37.1127 1.33485 0.667425 0.744677i \(-0.267396\pi\)
0.667425 + 0.744677i \(0.267396\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.1421 0.793324
\(780\) 0 0
\(781\) −3.71573 −0.132959
\(782\) 0 0
\(783\) −2.41421 −0.0862770
\(784\) 0 0
\(785\) 17.3137 0.617953
\(786\) 0 0
\(787\) 2.55635 0.0911240 0.0455620 0.998962i \(-0.485492\pi\)
0.0455620 + 0.998962i \(0.485492\pi\)
\(788\) 0 0
\(789\) −12.0294 −0.428259
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −26.4853 −0.940520
\(794\) 0 0
\(795\) −0.485281 −0.0172112
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −9.65685 −0.341635
\(800\) 0 0
\(801\) −24.4853 −0.865145
\(802\) 0 0
\(803\) 0.686292 0.0242187
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.47309 0.298267
\(808\) 0 0
\(809\) 35.6274 1.25259 0.626297 0.779585i \(-0.284570\pi\)
0.626297 + 0.779585i \(0.284570\pi\)
\(810\) 0 0
\(811\) −20.6274 −0.724327 −0.362163 0.932115i \(-0.617962\pi\)
−0.362163 + 0.932115i \(0.617962\pi\)
\(812\) 0 0
\(813\) −6.82843 −0.239483
\(814\) 0 0
\(815\) 12.3431 0.432362
\(816\) 0 0
\(817\) −10.1421 −0.354828
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 47.9411 1.67316 0.836578 0.547847i \(-0.184552\pi\)
0.836578 + 0.547847i \(0.184552\pi\)
\(822\) 0 0
\(823\) −2.07107 −0.0721929 −0.0360964 0.999348i \(-0.511492\pi\)
−0.0360964 + 0.999348i \(0.511492\pi\)
\(824\) 0 0
\(825\) −0.343146 −0.0119468
\(826\) 0 0
\(827\) 26.2132 0.911522 0.455761 0.890102i \(-0.349367\pi\)
0.455761 + 0.890102i \(0.349367\pi\)
\(828\) 0 0
\(829\) −29.3137 −1.01811 −0.509054 0.860735i \(-0.670005\pi\)
−0.509054 + 0.860735i \(0.670005\pi\)
\(830\) 0 0
\(831\) −6.68629 −0.231945
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −22.4142 −0.775676
\(836\) 0 0
\(837\) −14.4853 −0.500685
\(838\) 0 0
\(839\) 15.1716 0.523781 0.261890 0.965098i \(-0.415654\pi\)
0.261890 + 0.965098i \(0.415654\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 12.5442 0.432044
\(844\) 0 0
\(845\) −10.3137 −0.354802
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −5.79899 −0.199021
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.54416 −0.0871102 −0.0435551 0.999051i \(-0.513868\pi\)
−0.0435551 + 0.999051i \(0.513868\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −34.2843 −1.17113 −0.585564 0.810626i \(-0.699127\pi\)
−0.585564 + 0.810626i \(0.699127\pi\)
\(858\) 0 0
\(859\) 1.37258 0.0468319 0.0234160 0.999726i \(-0.492546\pi\)
0.0234160 + 0.999726i \(0.492546\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.5563 −0.495504 −0.247752 0.968823i \(-0.579692\pi\)
−0.247752 + 0.968823i \(0.579692\pi\)
\(864\) 0 0
\(865\) 3.31371 0.112669
\(866\) 0 0
\(867\) −2.61522 −0.0888177
\(868\) 0 0
\(869\) −12.2843 −0.416715
\(870\) 0 0
\(871\) −46.2843 −1.56828
\(872\) 0 0
\(873\) 32.9706 1.11588
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 25.1716 0.849984 0.424992 0.905197i \(-0.360277\pi\)
0.424992 + 0.905197i \(0.360277\pi\)
\(878\) 0 0
\(879\) −6.62742 −0.223537
\(880\) 0 0
\(881\) −1.82843 −0.0616013 −0.0308006 0.999526i \(-0.509806\pi\)
−0.0308006 + 0.999526i \(0.509806\pi\)
\(882\) 0 0
\(883\) −18.2843 −0.615315 −0.307657 0.951497i \(-0.599545\pi\)
−0.307657 + 0.951497i \(0.599545\pi\)
\(884\) 0 0
\(885\) 1.85786 0.0624514
\(886\) 0 0
\(887\) 29.9289 1.00492 0.502458 0.864602i \(-0.332429\pi\)
0.502458 + 0.864602i \(0.332429\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.20101 0.207742
\(892\) 0 0
\(893\) 5.65685 0.189299
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) 0 0
\(897\) 0.828427 0.0276604
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 5.65685 0.188457
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.65685 0.0883168
\(906\) 0 0
\(907\) 14.2132 0.471942 0.235971 0.971760i \(-0.424173\pi\)
0.235971 + 0.971760i \(0.424173\pi\)
\(908\) 0 0
\(909\) −29.1716 −0.967560
\(910\) 0 0
\(911\) 10.2010 0.337975 0.168987 0.985618i \(-0.445950\pi\)
0.168987 + 0.985618i \(0.445950\pi\)
\(912\) 0 0
\(913\) 11.3726 0.376378
\(914\) 0 0
\(915\) −2.27208 −0.0751126
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 43.1127 1.42216 0.711078 0.703113i \(-0.248207\pi\)
0.711078 + 0.703113i \(0.248207\pi\)
\(920\) 0 0
\(921\) 1.97056 0.0649323
\(922\) 0 0
\(923\) −21.6569 −0.712844
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.82843 0.224275
\(928\) 0 0
\(929\) −5.48528 −0.179966 −0.0899831 0.995943i \(-0.528681\pi\)
−0.0899831 + 0.995943i \(0.528681\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 5.45584 0.178616
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −34.6274 −1.13123 −0.565614 0.824670i \(-0.691361\pi\)
−0.565614 + 0.824670i \(0.691361\pi\)
\(938\) 0 0
\(939\) −2.62742 −0.0857425
\(940\) 0 0
\(941\) 46.2843 1.50882 0.754412 0.656401i \(-0.227922\pi\)
0.754412 + 0.656401i \(0.227922\pi\)
\(942\) 0 0
\(943\) −3.24264 −0.105595
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −33.1838 −1.07833 −0.539164 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 5.71573 0.185345
\(952\) 0 0
\(953\) −13.6569 −0.442389 −0.221194 0.975230i \(-0.570996\pi\)
−0.221194 + 0.975230i \(0.570996\pi\)
\(954\) 0 0
\(955\) −12.8284 −0.415118
\(956\) 0 0
\(957\) 0.343146 0.0110923
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 31.7990 1.02471
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −37.5269 −1.20678 −0.603392 0.797445i \(-0.706185\pi\)
−0.603392 + 0.797445i \(0.706185\pi\)
\(968\) 0 0
\(969\) 5.65685 0.181724
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) −1.31371 −0.0420293 −0.0210146 0.999779i \(-0.506690\pi\)
−0.0210146 + 0.999779i \(0.506690\pi\)
\(978\) 0 0
\(979\) 7.17157 0.229204
\(980\) 0 0
\(981\) 38.1421 1.21778
\(982\) 0 0
\(983\) 28.2132 0.899861 0.449931 0.893063i \(-0.351449\pi\)
0.449931 + 0.893063i \(0.351449\pi\)
\(984\) 0 0
\(985\) 12.3431 0.393285
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.48528 0.0472292
\(990\) 0 0
\(991\) 4.34315 0.137965 0.0689823 0.997618i \(-0.478025\pi\)
0.0689823 + 0.997618i \(0.478025\pi\)
\(992\) 0 0
\(993\) 9.51472 0.301940
\(994\) 0 0
\(995\) 9.65685 0.306143
\(996\) 0 0
\(997\) −33.4558 −1.05956 −0.529779 0.848136i \(-0.677725\pi\)
−0.529779 + 0.848136i \(0.677725\pi\)
\(998\) 0 0
\(999\) 0 0
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 3920.2.a.bv.1.1 2
4.3 odd 2 245.2.a.g.1.2 2
7.3 odd 6 560.2.q.k.401.1 4
7.5 odd 6 560.2.q.k.81.1 4
7.6 odd 2 3920.2.a.bq.1.2 2
12.11 even 2 2205.2.a.q.1.1 2
20.3 even 4 1225.2.b.h.99.1 4
20.7 even 4 1225.2.b.h.99.4 4
20.19 odd 2 1225.2.a.m.1.1 2
28.3 even 6 35.2.e.a.16.1 yes 4
28.11 odd 6 245.2.e.e.226.1 4
28.19 even 6 35.2.e.a.11.1 4
28.23 odd 6 245.2.e.e.116.1 4
28.27 even 2 245.2.a.h.1.2 2
84.47 odd 6 315.2.j.e.46.2 4
84.59 odd 6 315.2.j.e.226.2 4
84.83 odd 2 2205.2.a.n.1.1 2
140.3 odd 12 175.2.k.a.149.4 8
140.19 even 6 175.2.e.c.151.2 4
140.27 odd 4 1225.2.b.g.99.4 4
140.47 odd 12 175.2.k.a.74.4 8
140.59 even 6 175.2.e.c.51.2 4
140.83 odd 4 1225.2.b.g.99.1 4
140.87 odd 12 175.2.k.a.149.1 8
140.103 odd 12 175.2.k.a.74.1 8
140.139 even 2 1225.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.1 4 28.19 even 6
35.2.e.a.16.1 yes 4 28.3 even 6
175.2.e.c.51.2 4 140.59 even 6
175.2.e.c.151.2 4 140.19 even 6
175.2.k.a.74.1 8 140.103 odd 12
175.2.k.a.74.4 8 140.47 odd 12
175.2.k.a.149.1 8 140.87 odd 12
175.2.k.a.149.4 8 140.3 odd 12
245.2.a.g.1.2 2 4.3 odd 2
245.2.a.h.1.2 2 28.27 even 2
245.2.e.e.116.1 4 28.23 odd 6
245.2.e.e.226.1 4 28.11 odd 6
315.2.j.e.46.2 4 84.47 odd 6
315.2.j.e.226.2 4 84.59 odd 6
560.2.q.k.81.1 4 7.5 odd 6
560.2.q.k.401.1 4 7.3 odd 6
1225.2.a.k.1.1 2 140.139 even 2
1225.2.a.m.1.1 2 20.19 odd 2
1225.2.b.g.99.1 4 140.83 odd 4
1225.2.b.g.99.4 4 140.27 odd 4
1225.2.b.h.99.1 4 20.3 even 4
1225.2.b.h.99.4 4 20.7 even 4
2205.2.a.n.1.1 2 84.83 odd 2
2205.2.a.q.1.1 2 12.11 even 2
3920.2.a.bq.1.2 2 7.6 odd 2
3920.2.a.bv.1.1 2 1.1 even 1 trivial