Properties

Label 1520.2.d.f.609.3
Level $1520$
Weight $2$
Character 1520.609
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Defining polynomial: \(x^{4} + 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.3
Root \(0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.f.609.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.874032i q^{3} +2.23607 q^{5} -2.82843i q^{7} +2.23607 q^{9} +O(q^{10})\) \(q+0.874032i q^{3} +2.23607 q^{5} -2.82843i q^{7} +2.23607 q^{9} +0.763932 q^{11} +5.45052i q^{13} +1.95440i q^{15} -7.40492i q^{17} -1.00000 q^{19} +2.47214 q^{21} -1.08036i q^{23} +5.00000 q^{25} +4.57649i q^{27} +4.47214 q^{29} +4.00000 q^{31} +0.667701i q^{33} -6.32456i q^{35} -2.62210i q^{37} -4.76393 q^{39} -6.00000 q^{41} +8.48528i q^{43} +5.00000 q^{45} -8.48528i q^{47} -1.00000 q^{49} +6.47214 q^{51} -2.62210i q^{53} +1.70820 q^{55} -0.874032i q^{57} -1.52786 q^{59} -11.7082 q^{61} -6.32456i q^{63} +12.1877i q^{65} -11.1074i q^{67} +0.944272 q^{69} +10.4721 q^{71} -5.24419i q^{73} +4.37016i q^{75} -2.16073i q^{77} +15.4164 q^{79} +2.70820 q^{81} +13.7295i q^{83} -16.5579i q^{85} +3.90879i q^{87} -2.94427 q^{89} +15.4164 q^{91} +3.49613i q^{93} -2.23607 q^{95} +13.9358i q^{97} +1.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 12q^{11} - 4q^{19} - 8q^{21} + 20q^{25} + 16q^{31} - 28q^{39} - 24q^{41} + 20q^{45} - 4q^{49} + 8q^{51} - 20q^{55} - 24q^{59} - 20q^{61} - 32q^{69} + 24q^{71} + 8q^{79} - 16q^{81} + 24q^{89} + 8q^{91} - 20q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.874032i 0.504623i 0.967646 + 0.252311i \(0.0811907\pi\)
−0.967646 + 0.252311i \(0.918809\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 2.23607 0.745356
\(10\) 0 0
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) 5.45052i 1.51170i 0.654743 + 0.755852i \(0.272777\pi\)
−0.654743 + 0.755852i \(0.727223\pi\)
\(14\) 0 0
\(15\) 1.95440i 0.504623i
\(16\) 0 0
\(17\) − 7.40492i − 1.79596i −0.440040 0.897978i \(-0.645036\pi\)
0.440040 0.897978i \(-0.354964\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.47214 0.539464
\(22\) 0 0
\(23\) − 1.08036i − 0.225271i −0.993636 0.112636i \(-0.964071\pi\)
0.993636 0.112636i \(-0.0359293\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 4.57649i 0.880746i
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0.667701i 0.116232i
\(34\) 0 0
\(35\) − 6.32456i − 1.06904i
\(36\) 0 0
\(37\) − 2.62210i − 0.431070i −0.976496 0.215535i \(-0.930850\pi\)
0.976496 0.215535i \(-0.0691495\pi\)
\(38\) 0 0
\(39\) −4.76393 −0.762840
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 5.00000 0.745356
\(46\) 0 0
\(47\) − 8.48528i − 1.23771i −0.785507 0.618853i \(-0.787598\pi\)
0.785507 0.618853i \(-0.212402\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 6.47214 0.906280
\(52\) 0 0
\(53\) − 2.62210i − 0.360173i −0.983651 0.180086i \(-0.942362\pi\)
0.983651 0.180086i \(-0.0576377\pi\)
\(54\) 0 0
\(55\) 1.70820 0.230334
\(56\) 0 0
\(57\) − 0.874032i − 0.115768i
\(58\) 0 0
\(59\) −1.52786 −0.198911 −0.0994555 0.995042i \(-0.531710\pi\)
−0.0994555 + 0.995042i \(0.531710\pi\)
\(60\) 0 0
\(61\) −11.7082 −1.49908 −0.749541 0.661958i \(-0.769726\pi\)
−0.749541 + 0.661958i \(0.769726\pi\)
\(62\) 0 0
\(63\) − 6.32456i − 0.796819i
\(64\) 0 0
\(65\) 12.1877i 1.51170i
\(66\) 0 0
\(67\) − 11.1074i − 1.35698i −0.734609 0.678491i \(-0.762634\pi\)
0.734609 0.678491i \(-0.237366\pi\)
\(68\) 0 0
\(69\) 0.944272 0.113677
\(70\) 0 0
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 0 0
\(73\) − 5.24419i − 0.613786i −0.951744 0.306893i \(-0.900711\pi\)
0.951744 0.306893i \(-0.0992894\pi\)
\(74\) 0 0
\(75\) 4.37016i 0.504623i
\(76\) 0 0
\(77\) − 2.16073i − 0.246238i
\(78\) 0 0
\(79\) 15.4164 1.73448 0.867241 0.497889i \(-0.165891\pi\)
0.867241 + 0.497889i \(0.165891\pi\)
\(80\) 0 0
\(81\) 2.70820 0.300912
\(82\) 0 0
\(83\) 13.7295i 1.50701i 0.657445 + 0.753503i \(0.271637\pi\)
−0.657445 + 0.753503i \(0.728363\pi\)
\(84\) 0 0
\(85\) − 16.5579i − 1.79596i
\(86\) 0 0
\(87\) 3.90879i 0.419066i
\(88\) 0 0
\(89\) −2.94427 −0.312092 −0.156046 0.987750i \(-0.549875\pi\)
−0.156046 + 0.987750i \(0.549875\pi\)
\(90\) 0 0
\(91\) 15.4164 1.61608
\(92\) 0 0
\(93\) 3.49613i 0.362532i
\(94\) 0 0
\(95\) −2.23607 −0.229416
\(96\) 0 0
\(97\) 13.9358i 1.41497i 0.706730 + 0.707483i \(0.250170\pi\)
−0.706730 + 0.707483i \(0.749830\pi\)
\(98\) 0 0
\(99\) 1.70820 0.171681
\(100\) 0 0
\(101\) 5.23607 0.521008 0.260504 0.965473i \(-0.416111\pi\)
0.260504 + 0.965473i \(0.416111\pi\)
\(102\) 0 0
\(103\) − 5.86319i − 0.577717i −0.957372 0.288858i \(-0.906724\pi\)
0.957372 0.288858i \(-0.0932757\pi\)
\(104\) 0 0
\(105\) 5.52786 0.539464
\(106\) 0 0
\(107\) 13.2681i 1.28268i 0.767258 + 0.641338i \(0.221620\pi\)
−0.767258 + 0.641338i \(0.778380\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 2.29180 0.217528
\(112\) 0 0
\(113\) 16.3516i 1.53823i 0.639113 + 0.769113i \(0.279302\pi\)
−0.639113 + 0.769113i \(0.720698\pi\)
\(114\) 0 0
\(115\) − 2.41577i − 0.225271i
\(116\) 0 0
\(117\) 12.1877i 1.12676i
\(118\) 0 0
\(119\) −20.9443 −1.91996
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0 0
\(123\) − 5.24419i − 0.472853i
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 19.5927i 1.73857i 0.494314 + 0.869284i \(0.335419\pi\)
−0.494314 + 0.869284i \(0.664581\pi\)
\(128\) 0 0
\(129\) −7.41641 −0.652978
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 2.82843i 0.245256i
\(134\) 0 0
\(135\) 10.2333i 0.880746i
\(136\) 0 0
\(137\) − 7.40492i − 0.632645i −0.948652 0.316322i \(-0.897552\pi\)
0.948652 0.316322i \(-0.102448\pi\)
\(138\) 0 0
\(139\) 2.29180 0.194388 0.0971938 0.995265i \(-0.469013\pi\)
0.0971938 + 0.995265i \(0.469013\pi\)
\(140\) 0 0
\(141\) 7.41641 0.624574
\(142\) 0 0
\(143\) 4.16383i 0.348197i
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) − 0.874032i − 0.0720889i
\(148\) 0 0
\(149\) 12.6525 1.03653 0.518266 0.855220i \(-0.326578\pi\)
0.518266 + 0.855220i \(0.326578\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) − 16.5579i − 1.33863i
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) − 0.412662i − 0.0329340i −0.999864 0.0164670i \(-0.994758\pi\)
0.999864 0.0164670i \(-0.00524185\pi\)
\(158\) 0 0
\(159\) 2.29180 0.181751
\(160\) 0 0
\(161\) −3.05573 −0.240825
\(162\) 0 0
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 1.49302i 0.116232i
\(166\) 0 0
\(167\) − 17.4319i − 1.34892i −0.738310 0.674462i \(-0.764376\pi\)
0.738310 0.674462i \(-0.235624\pi\)
\(168\) 0 0
\(169\) −16.7082 −1.28525
\(170\) 0 0
\(171\) −2.23607 −0.170996
\(172\) 0 0
\(173\) − 8.94665i − 0.680201i −0.940389 0.340101i \(-0.889539\pi\)
0.940389 0.340101i \(-0.110461\pi\)
\(174\) 0 0
\(175\) − 14.1421i − 1.06904i
\(176\) 0 0
\(177\) − 1.33540i − 0.100375i
\(178\) 0 0
\(179\) −22.4721 −1.67965 −0.839823 0.542860i \(-0.817341\pi\)
−0.839823 + 0.542860i \(0.817341\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) − 10.2333i − 0.756471i
\(184\) 0 0
\(185\) − 5.86319i − 0.431070i
\(186\) 0 0
\(187\) − 5.65685i − 0.413670i
\(188\) 0 0
\(189\) 12.9443 0.941557
\(190\) 0 0
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) 0 0
\(193\) − 13.9358i − 1.00312i −0.865123 0.501561i \(-0.832759\pi\)
0.865123 0.501561i \(-0.167241\pi\)
\(194\) 0 0
\(195\) −10.6525 −0.762840
\(196\) 0 0
\(197\) 12.6491i 0.901212i 0.892723 + 0.450606i \(0.148792\pi\)
−0.892723 + 0.450606i \(0.851208\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) 0 0
\(203\) − 12.6491i − 0.887794i
\(204\) 0 0
\(205\) −13.4164 −0.937043
\(206\) 0 0
\(207\) − 2.41577i − 0.167907i
\(208\) 0 0
\(209\) −0.763932 −0.0528423
\(210\) 0 0
\(211\) −15.4164 −1.06131 −0.530655 0.847588i \(-0.678054\pi\)
−0.530655 + 0.847588i \(0.678054\pi\)
\(212\) 0 0
\(213\) 9.15298i 0.627152i
\(214\) 0 0
\(215\) 18.9737i 1.29399i
\(216\) 0 0
\(217\) − 11.3137i − 0.768025i
\(218\) 0 0
\(219\) 4.58359 0.309730
\(220\) 0 0
\(221\) 40.3607 2.71495
\(222\) 0 0
\(223\) 18.7673i 1.25675i 0.777909 + 0.628377i \(0.216280\pi\)
−0.777909 + 0.628377i \(0.783720\pi\)
\(224\) 0 0
\(225\) 11.1803 0.745356
\(226\) 0 0
\(227\) − 5.86319i − 0.389153i −0.980887 0.194577i \(-0.937667\pi\)
0.980887 0.194577i \(-0.0623333\pi\)
\(228\) 0 0
\(229\) −7.70820 −0.509372 −0.254686 0.967024i \(-0.581972\pi\)
−0.254686 + 0.967024i \(0.581972\pi\)
\(230\) 0 0
\(231\) 1.88854 0.124257
\(232\) 0 0
\(233\) 12.6491i 0.828671i 0.910124 + 0.414335i \(0.135986\pi\)
−0.910124 + 0.414335i \(0.864014\pi\)
\(234\) 0 0
\(235\) − 18.9737i − 1.23771i
\(236\) 0 0
\(237\) 13.4744i 0.875259i
\(238\) 0 0
\(239\) −29.8885 −1.93333 −0.966665 0.256046i \(-0.917580\pi\)
−0.966665 + 0.256046i \(0.917580\pi\)
\(240\) 0 0
\(241\) −28.8328 −1.85728 −0.928642 0.370976i \(-0.879023\pi\)
−0.928642 + 0.370976i \(0.879023\pi\)
\(242\) 0 0
\(243\) 16.0965i 1.03259i
\(244\) 0 0
\(245\) −2.23607 −0.142857
\(246\) 0 0
\(247\) − 5.45052i − 0.346808i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 5.88854 0.371682 0.185841 0.982580i \(-0.440499\pi\)
0.185841 + 0.982580i \(0.440499\pi\)
\(252\) 0 0
\(253\) − 0.825324i − 0.0518877i
\(254\) 0 0
\(255\) 14.4721 0.906280
\(256\) 0 0
\(257\) − 17.4319i − 1.08737i −0.839288 0.543687i \(-0.817028\pi\)
0.839288 0.543687i \(-0.182972\pi\)
\(258\) 0 0
\(259\) −7.41641 −0.460833
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 15.8902i 0.979832i 0.871770 + 0.489916i \(0.162973\pi\)
−0.871770 + 0.489916i \(0.837027\pi\)
\(264\) 0 0
\(265\) − 5.86319i − 0.360173i
\(266\) 0 0
\(267\) − 2.57339i − 0.157489i
\(268\) 0 0
\(269\) −14.9443 −0.911168 −0.455584 0.890193i \(-0.650570\pi\)
−0.455584 + 0.890193i \(0.650570\pi\)
\(270\) 0 0
\(271\) −21.1246 −1.28323 −0.641614 0.767027i \(-0.721735\pi\)
−0.641614 + 0.767027i \(0.721735\pi\)
\(272\) 0 0
\(273\) 13.4744i 0.815510i
\(274\) 0 0
\(275\) 3.81966 0.230334
\(276\) 0 0
\(277\) − 22.2148i − 1.33476i −0.744719 0.667378i \(-0.767417\pi\)
0.744719 0.667378i \(-0.232583\pi\)
\(278\) 0 0
\(279\) 8.94427 0.535480
\(280\) 0 0
\(281\) −25.4164 −1.51622 −0.758108 0.652129i \(-0.773876\pi\)
−0.758108 + 0.652129i \(0.773876\pi\)
\(282\) 0 0
\(283\) − 2.41577i − 0.143602i −0.997419 0.0718012i \(-0.977125\pi\)
0.997419 0.0718012i \(-0.0228747\pi\)
\(284\) 0 0
\(285\) − 1.95440i − 0.115768i
\(286\) 0 0
\(287\) 16.9706i 1.00174i
\(288\) 0 0
\(289\) −37.8328 −2.22546
\(290\) 0 0
\(291\) −12.1803 −0.714024
\(292\) 0 0
\(293\) − 2.62210i − 0.153184i −0.997062 0.0765922i \(-0.975596\pi\)
0.997062 0.0765922i \(-0.0244040\pi\)
\(294\) 0 0
\(295\) −3.41641 −0.198911
\(296\) 0 0
\(297\) 3.49613i 0.202866i
\(298\) 0 0
\(299\) 5.88854 0.340543
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 0 0
\(303\) 4.57649i 0.262913i
\(304\) 0 0
\(305\) −26.1803 −1.49908
\(306\) 0 0
\(307\) 5.45052i 0.311078i 0.987830 + 0.155539i \(0.0497114\pi\)
−0.987830 + 0.155539i \(0.950289\pi\)
\(308\) 0 0
\(309\) 5.12461 0.291529
\(310\) 0 0
\(311\) 9.70820 0.550502 0.275251 0.961372i \(-0.411239\pi\)
0.275251 + 0.961372i \(0.411239\pi\)
\(312\) 0 0
\(313\) 10.4884i 0.592839i 0.955058 + 0.296419i \(0.0957926\pi\)
−0.955058 + 0.296419i \(0.904207\pi\)
\(314\) 0 0
\(315\) − 14.1421i − 0.796819i
\(316\) 0 0
\(317\) − 8.94665i − 0.502494i −0.967923 0.251247i \(-0.919159\pi\)
0.967923 0.251247i \(-0.0808406\pi\)
\(318\) 0 0
\(319\) 3.41641 0.191282
\(320\) 0 0
\(321\) −11.5967 −0.647267
\(322\) 0 0
\(323\) 7.40492i 0.412021i
\(324\) 0 0
\(325\) 27.2526i 1.51170i
\(326\) 0 0
\(327\) − 1.74806i − 0.0966682i
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −7.41641 −0.407643 −0.203821 0.979008i \(-0.565336\pi\)
−0.203821 + 0.979008i \(0.565336\pi\)
\(332\) 0 0
\(333\) − 5.86319i − 0.321301i
\(334\) 0 0
\(335\) − 24.8369i − 1.35698i
\(336\) 0 0
\(337\) 16.7642i 0.913206i 0.889671 + 0.456603i \(0.150934\pi\)
−0.889671 + 0.456603i \(0.849066\pi\)
\(338\) 0 0
\(339\) −14.2918 −0.776224
\(340\) 0 0
\(341\) 3.05573 0.165477
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 2.11146 0.113677
\(346\) 0 0
\(347\) 9.40802i 0.505049i 0.967590 + 0.252525i \(0.0812608\pi\)
−0.967590 + 0.252525i \(0.918739\pi\)
\(348\) 0 0
\(349\) −25.4164 −1.36051 −0.680255 0.732976i \(-0.738131\pi\)
−0.680255 + 0.732976i \(0.738131\pi\)
\(350\) 0 0
\(351\) −24.9443 −1.33143
\(352\) 0 0
\(353\) − 9.56564i − 0.509128i −0.967056 0.254564i \(-0.918068\pi\)
0.967056 0.254564i \(-0.0819319\pi\)
\(354\) 0 0
\(355\) 23.4164 1.24281
\(356\) 0 0
\(357\) − 18.3060i − 0.968854i
\(358\) 0 0
\(359\) 29.1246 1.53714 0.768569 0.639767i \(-0.220969\pi\)
0.768569 + 0.639767i \(0.220969\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 9.10427i − 0.477850i
\(364\) 0 0
\(365\) − 11.7264i − 0.613786i
\(366\) 0 0
\(367\) − 8.48528i − 0.442928i −0.975169 0.221464i \(-0.928916\pi\)
0.975169 0.221464i \(-0.0710835\pi\)
\(368\) 0 0
\(369\) −13.4164 −0.698430
\(370\) 0 0
\(371\) −7.41641 −0.385041
\(372\) 0 0
\(373\) − 13.1105i − 0.678835i −0.940636 0.339417i \(-0.889770\pi\)
0.940636 0.339417i \(-0.110230\pi\)
\(374\) 0 0
\(375\) 9.77198i 0.504623i
\(376\) 0 0
\(377\) 24.3755i 1.25540i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −17.1246 −0.877320
\(382\) 0 0
\(383\) − 36.4056i − 1.86024i −0.367257 0.930120i \(-0.619703\pi\)
0.367257 0.930120i \(-0.380297\pi\)
\(384\) 0 0
\(385\) − 4.83153i − 0.246238i
\(386\) 0 0
\(387\) 18.9737i 0.964486i
\(388\) 0 0
\(389\) 14.9443 0.757705 0.378852 0.925457i \(-0.376319\pi\)
0.378852 + 0.925457i \(0.376319\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 34.4721 1.73448
\(396\) 0 0
\(397\) 11.7264i 0.588530i 0.955724 + 0.294265i \(0.0950748\pi\)
−0.955724 + 0.294265i \(0.904925\pi\)
\(398\) 0 0
\(399\) −2.47214 −0.123762
\(400\) 0 0
\(401\) −4.47214 −0.223328 −0.111664 0.993746i \(-0.535618\pi\)
−0.111664 + 0.993746i \(0.535618\pi\)
\(402\) 0 0
\(403\) 21.8021i 1.08604i
\(404\) 0 0
\(405\) 6.05573 0.300912
\(406\) 0 0
\(407\) − 2.00310i − 0.0992901i
\(408\) 0 0
\(409\) −17.4164 −0.861186 −0.430593 0.902546i \(-0.641696\pi\)
−0.430593 + 0.902546i \(0.641696\pi\)
\(410\) 0 0
\(411\) 6.47214 0.319247
\(412\) 0 0
\(413\) 4.32145i 0.212645i
\(414\) 0 0
\(415\) 30.7000i 1.50701i
\(416\) 0 0
\(417\) 2.00310i 0.0980924i
\(418\) 0 0
\(419\) −20.9443 −1.02319 −0.511597 0.859225i \(-0.670946\pi\)
−0.511597 + 0.859225i \(0.670946\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) − 18.9737i − 0.922531i
\(424\) 0 0
\(425\) − 37.0246i − 1.79596i
\(426\) 0 0
\(427\) 33.1158i 1.60259i
\(428\) 0 0
\(429\) −3.63932 −0.175708
\(430\) 0 0
\(431\) −1.52786 −0.0735946 −0.0367973 0.999323i \(-0.511716\pi\)
−0.0367973 + 0.999323i \(0.511716\pi\)
\(432\) 0 0
\(433\) 28.4906i 1.36917i 0.728933 + 0.684585i \(0.240017\pi\)
−0.728933 + 0.684585i \(0.759983\pi\)
\(434\) 0 0
\(435\) 8.74032i 0.419066i
\(436\) 0 0
\(437\) 1.08036i 0.0516808i
\(438\) 0 0
\(439\) −18.8328 −0.898841 −0.449421 0.893320i \(-0.648370\pi\)
−0.449421 + 0.893320i \(0.648370\pi\)
\(440\) 0 0
\(441\) −2.23607 −0.106479
\(442\) 0 0
\(443\) − 13.7295i − 0.652307i −0.945317 0.326153i \(-0.894247\pi\)
0.945317 0.326153i \(-0.105753\pi\)
\(444\) 0 0
\(445\) −6.58359 −0.312092
\(446\) 0 0
\(447\) 11.0587i 0.523057i
\(448\) 0 0
\(449\) 16.4721 0.777368 0.388684 0.921371i \(-0.372930\pi\)
0.388684 + 0.921371i \(0.372930\pi\)
\(450\) 0 0
\(451\) −4.58359 −0.215833
\(452\) 0 0
\(453\) 6.99226i 0.328525i
\(454\) 0 0
\(455\) 34.4721 1.61608
\(456\) 0 0
\(457\) 37.9473i 1.77510i 0.460710 + 0.887551i \(0.347595\pi\)
−0.460710 + 0.887551i \(0.652405\pi\)
\(458\) 0 0
\(459\) 33.8885 1.58178
\(460\) 0 0
\(461\) 23.8885 1.11260 0.556300 0.830981i \(-0.312220\pi\)
0.556300 + 0.830981i \(0.312220\pi\)
\(462\) 0 0
\(463\) 14.5548i 0.676419i 0.941071 + 0.338209i \(0.109821\pi\)
−0.941071 + 0.338209i \(0.890179\pi\)
\(464\) 0 0
\(465\) 7.81758i 0.362532i
\(466\) 0 0
\(467\) − 30.7000i − 1.42063i −0.703885 0.710314i \(-0.748553\pi\)
0.703885 0.710314i \(-0.251447\pi\)
\(468\) 0 0
\(469\) −31.4164 −1.45067
\(470\) 0 0
\(471\) 0.360680 0.0166192
\(472\) 0 0
\(473\) 6.48218i 0.298051i
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) − 5.86319i − 0.268457i
\(478\) 0 0
\(479\) 11.2361 0.513389 0.256695 0.966493i \(-0.417367\pi\)
0.256695 + 0.966493i \(0.417367\pi\)
\(480\) 0 0
\(481\) 14.2918 0.651650
\(482\) 0 0
\(483\) − 2.67080i − 0.121526i
\(484\) 0 0
\(485\) 31.1614i 1.41497i
\(486\) 0 0
\(487\) 10.6947i 0.484624i 0.970198 + 0.242312i \(0.0779057\pi\)
−0.970198 + 0.242312i \(0.922094\pi\)
\(488\) 0 0
\(489\) 7.41641 0.335382
\(490\) 0 0
\(491\) −5.88854 −0.265746 −0.132873 0.991133i \(-0.542420\pi\)
−0.132873 + 0.991133i \(0.542420\pi\)
\(492\) 0 0
\(493\) − 33.1158i − 1.49146i
\(494\) 0 0
\(495\) 3.81966 0.171681
\(496\) 0 0
\(497\) − 29.6197i − 1.32862i
\(498\) 0 0
\(499\) −17.1246 −0.766603 −0.383301 0.923623i \(-0.625213\pi\)
−0.383301 + 0.923623i \(0.625213\pi\)
\(500\) 0 0
\(501\) 15.2361 0.680697
\(502\) 0 0
\(503\) 20.2117i 0.901193i 0.892728 + 0.450597i \(0.148789\pi\)
−0.892728 + 0.450597i \(0.851211\pi\)
\(504\) 0 0
\(505\) 11.7082 0.521008
\(506\) 0 0
\(507\) − 14.6035i − 0.648564i
\(508\) 0 0
\(509\) 10.5836 0.469109 0.234555 0.972103i \(-0.424637\pi\)
0.234555 + 0.972103i \(0.424637\pi\)
\(510\) 0 0
\(511\) −14.8328 −0.656165
\(512\) 0 0
\(513\) − 4.57649i − 0.202057i
\(514\) 0 0
\(515\) − 13.1105i − 0.577717i
\(516\) 0 0
\(517\) − 6.48218i − 0.285086i
\(518\) 0 0
\(519\) 7.81966 0.343245
\(520\) 0 0
\(521\) 0.111456 0.00488298 0.00244149 0.999997i \(-0.499223\pi\)
0.00244149 + 0.999997i \(0.499223\pi\)
\(522\) 0 0
\(523\) − 24.8369i − 1.08604i −0.839720 0.543020i \(-0.817281\pi\)
0.839720 0.543020i \(-0.182719\pi\)
\(524\) 0 0
\(525\) 12.3607 0.539464
\(526\) 0 0
\(527\) − 29.6197i − 1.29025i
\(528\) 0 0
\(529\) 21.8328 0.949253
\(530\) 0 0
\(531\) −3.41641 −0.148259
\(532\) 0 0
\(533\) − 32.7031i − 1.41653i
\(534\) 0 0
\(535\) 29.6684i 1.28268i
\(536\) 0 0
\(537\) − 19.6414i − 0.847588i
\(538\) 0 0
\(539\) −0.763932 −0.0329049
\(540\) 0 0
\(541\) 11.7082 0.503375 0.251688 0.967809i \(-0.419014\pi\)
0.251688 + 0.967809i \(0.419014\pi\)
\(542\) 0 0
\(543\) 5.24419i 0.225050i
\(544\) 0 0
\(545\) −4.47214 −0.191565
\(546\) 0 0
\(547\) 8.27895i 0.353982i 0.984212 + 0.176991i \(0.0566364\pi\)
−0.984212 + 0.176991i \(0.943364\pi\)
\(548\) 0 0
\(549\) −26.1803 −1.11735
\(550\) 0 0
\(551\) −4.47214 −0.190519
\(552\) 0 0
\(553\) − 43.6042i − 1.85424i
\(554\) 0 0
\(555\) 5.12461 0.217528
\(556\) 0 0
\(557\) 20.0540i 0.849716i 0.905260 + 0.424858i \(0.139676\pi\)
−0.905260 + 0.424858i \(0.860324\pi\)
\(558\) 0 0
\(559\) −46.2492 −1.95613
\(560\) 0 0
\(561\) 4.94427 0.208747
\(562\) 0 0
\(563\) 30.0810i 1.26776i 0.773429 + 0.633882i \(0.218540\pi\)
−0.773429 + 0.633882i \(0.781460\pi\)
\(564\) 0 0
\(565\) 36.5632i 1.53823i
\(566\) 0 0
\(567\) − 7.65996i − 0.321688i
\(568\) 0 0
\(569\) 38.9443 1.63263 0.816314 0.577608i \(-0.196014\pi\)
0.816314 + 0.577608i \(0.196014\pi\)
\(570\) 0 0
\(571\) −25.1246 −1.05143 −0.525716 0.850660i \(-0.676203\pi\)
−0.525716 + 0.850660i \(0.676203\pi\)
\(572\) 0 0
\(573\) 2.67080i 0.111574i
\(574\) 0 0
\(575\) − 5.40182i − 0.225271i
\(576\) 0 0
\(577\) − 5.65685i − 0.235498i −0.993043 0.117749i \(-0.962432\pi\)
0.993043 0.117749i \(-0.0375678\pi\)
\(578\) 0 0
\(579\) 12.1803 0.506198
\(580\) 0 0
\(581\) 38.8328 1.61106
\(582\) 0 0
\(583\) − 2.00310i − 0.0829601i
\(584\) 0 0
\(585\) 27.2526i 1.12676i
\(586\) 0 0
\(587\) − 5.40182i − 0.222957i −0.993767 0.111478i \(-0.964441\pi\)
0.993767 0.111478i \(-0.0355586\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −11.0557 −0.454772
\(592\) 0 0
\(593\) 2.16073i 0.0887304i 0.999015 + 0.0443652i \(0.0141265\pi\)
−0.999015 + 0.0443652i \(0.985873\pi\)
\(594\) 0 0
\(595\) −46.8328 −1.91996
\(596\) 0 0
\(597\) − 13.9845i − 0.572348i
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 16.8328 0.686625 0.343312 0.939221i \(-0.388451\pi\)
0.343312 + 0.939221i \(0.388451\pi\)
\(602\) 0 0
\(603\) − 24.8369i − 1.01143i
\(604\) 0 0
\(605\) −23.2918 −0.946946
\(606\) 0 0
\(607\) − 22.4211i − 0.910044i −0.890480 0.455022i \(-0.849631\pi\)
0.890480 0.455022i \(-0.150369\pi\)
\(608\) 0 0
\(609\) 11.0557 0.448001
\(610\) 0 0
\(611\) 46.2492 1.87104
\(612\) 0 0
\(613\) 39.1853i 1.58268i 0.611376 + 0.791340i \(0.290616\pi\)
−0.611376 + 0.791340i \(0.709384\pi\)
\(614\) 0 0
\(615\) − 11.7264i − 0.472853i
\(616\) 0 0
\(617\) 3.08347i 0.124136i 0.998072 + 0.0620678i \(0.0197695\pi\)
−0.998072 + 0.0620678i \(0.980230\pi\)
\(618\) 0 0
\(619\) 41.1246 1.65294 0.826469 0.562982i \(-0.190346\pi\)
0.826469 + 0.562982i \(0.190346\pi\)
\(620\) 0 0
\(621\) 4.94427 0.198407
\(622\) 0 0
\(623\) 8.32766i 0.333641i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) − 0.667701i − 0.0266654i
\(628\) 0 0
\(629\) −19.4164 −0.774183
\(630\) 0 0
\(631\) 1.70820 0.0680025 0.0340013 0.999422i \(-0.489175\pi\)
0.0340013 + 0.999422i \(0.489175\pi\)
\(632\) 0 0
\(633\) − 13.4744i − 0.535561i
\(634\) 0 0
\(635\) 43.8105i 1.73857i
\(636\) 0 0
\(637\) − 5.45052i − 0.215958i
\(638\) 0 0
\(639\) 23.4164 0.926339
\(640\) 0 0
\(641\) −12.1115 −0.478374 −0.239187 0.970974i \(-0.576881\pi\)
−0.239187 + 0.970974i \(0.576881\pi\)
\(642\) 0 0
\(643\) − 30.7000i − 1.21069i −0.795963 0.605346i \(-0.793035\pi\)
0.795963 0.605346i \(-0.206965\pi\)
\(644\) 0 0
\(645\) −16.5836 −0.652978
\(646\) 0 0
\(647\) 9.40802i 0.369867i 0.982751 + 0.184934i \(0.0592071\pi\)
−0.982751 + 0.184934i \(0.940793\pi\)
\(648\) 0 0
\(649\) −1.16718 −0.0458160
\(650\) 0 0
\(651\) 9.88854 0.387563
\(652\) 0 0
\(653\) 22.2148i 0.869331i 0.900592 + 0.434665i \(0.143133\pi\)
−0.900592 + 0.434665i \(0.856867\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 11.7264i − 0.457489i
\(658\) 0 0
\(659\) −44.9443 −1.75078 −0.875390 0.483417i \(-0.839395\pi\)
−0.875390 + 0.483417i \(0.839395\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 0 0
\(663\) 35.2765i 1.37003i
\(664\) 0 0
\(665\) 6.32456i 0.245256i
\(666\) 0 0
\(667\) − 4.83153i − 0.187078i
\(668\) 0 0
\(669\) −16.4033 −0.634186
\(670\) 0 0
\(671\) −8.94427 −0.345290
\(672\) 0 0
\(673\) 4.62520i 0.178288i 0.996019 + 0.0891442i \(0.0284132\pi\)
−0.996019 + 0.0891442i \(0.971587\pi\)
\(674\) 0 0
\(675\) 22.8825i 0.880746i
\(676\) 0 0
\(677\) 42.7302i 1.64225i 0.570746 + 0.821127i \(0.306654\pi\)
−0.570746 + 0.821127i \(0.693346\pi\)
\(678\) 0 0
\(679\) 39.4164 1.51266
\(680\) 0 0
\(681\) 5.12461 0.196376
\(682\) 0 0
\(683\) 19.5927i 0.749692i 0.927087 + 0.374846i \(0.122304\pi\)
−0.927087 + 0.374846i \(0.877696\pi\)
\(684\) 0 0
\(685\) − 16.5579i − 0.632645i
\(686\) 0 0
\(687\) − 6.73722i − 0.257041i
\(688\) 0 0
\(689\) 14.2918 0.544474
\(690\) 0 0
\(691\) −25.1246 −0.955785 −0.477893 0.878418i \(-0.658599\pi\)
−0.477893 + 0.878418i \(0.658599\pi\)
\(692\) 0 0
\(693\) − 4.83153i − 0.183535i
\(694\) 0 0
\(695\) 5.12461 0.194388
\(696\) 0 0
\(697\) 44.4295i 1.68289i
\(698\) 0 0
\(699\) −11.0557 −0.418166
\(700\) 0 0
\(701\) 0.875388 0.0330630 0.0165315 0.999863i \(-0.494738\pi\)
0.0165315 + 0.999863i \(0.494738\pi\)
\(702\) 0 0
\(703\) 2.62210i 0.0988942i
\(704\) 0 0
\(705\) 16.5836 0.624574
\(706\) 0 0
\(707\) − 14.8098i − 0.556981i
\(708\) 0 0
\(709\) −33.4164 −1.25498 −0.627490 0.778625i \(-0.715918\pi\)
−0.627490 + 0.778625i \(0.715918\pi\)
\(710\) 0 0
\(711\) 34.4721 1.29281
\(712\) 0 0
\(713\) − 4.32145i − 0.161840i
\(714\) 0 0
\(715\) 9.31061i 0.348197i
\(716\) 0 0
\(717\) − 26.1235i − 0.975602i
\(718\) 0 0
\(719\) 27.5967 1.02919 0.514593 0.857435i \(-0.327943\pi\)
0.514593 + 0.857435i \(0.327943\pi\)
\(720\) 0 0
\(721\) −16.5836 −0.617605
\(722\) 0 0
\(723\) − 25.2008i − 0.937228i
\(724\) 0 0
\(725\) 22.3607 0.830455
\(726\) 0 0
\(727\) − 8.48528i − 0.314702i −0.987543 0.157351i \(-0.949705\pi\)
0.987543 0.157351i \(-0.0502953\pi\)
\(728\) 0 0
\(729\) −5.94427 −0.220158
\(730\) 0 0
\(731\) 62.8328 2.32396
\(732\) 0 0
\(733\) 33.9411i 1.25364i 0.779162 + 0.626822i \(0.215645\pi\)
−0.779162 + 0.626822i \(0.784355\pi\)
\(734\) 0 0
\(735\) − 1.95440i − 0.0720889i
\(736\) 0 0
\(737\) − 8.48528i − 0.312559i
\(738\) 0 0
\(739\) −18.8328 −0.692776 −0.346388 0.938091i \(-0.612592\pi\)
−0.346388 + 0.938091i \(0.612592\pi\)
\(740\) 0 0
\(741\) 4.76393 0.175007
\(742\) 0 0
\(743\) − 2.62210i − 0.0961954i −0.998843 0.0480977i \(-0.984684\pi\)
0.998843 0.0480977i \(-0.0153159\pi\)
\(744\) 0 0
\(745\) 28.2918 1.03653
\(746\) 0 0
\(747\) 30.7000i 1.12326i
\(748\) 0 0
\(749\) 37.5279 1.37124
\(750\) 0 0
\(751\) −20.5836 −0.751106 −0.375553 0.926801i \(-0.622547\pi\)
−0.375553 + 0.926801i \(0.622547\pi\)
\(752\) 0 0
\(753\) 5.14678i 0.187559i
\(754\) 0 0
\(755\) 17.8885 0.651031
\(756\) 0 0
\(757\) − 38.7727i − 1.40922i −0.709597 0.704608i \(-0.751123\pi\)
0.709597 0.704608i \(-0.248877\pi\)
\(758\) 0 0
\(759\) 0.721360 0.0261837
\(760\) 0 0
\(761\) −42.5410 −1.54211 −0.771055 0.636768i \(-0.780271\pi\)
−0.771055 + 0.636768i \(0.780271\pi\)
\(762\) 0 0
\(763\) 5.65685i 0.204792i
\(764\) 0 0
\(765\) − 37.0246i − 1.33863i
\(766\) 0 0
\(767\) − 8.32766i − 0.300694i
\(768\) 0 0
\(769\) 38.5410 1.38982 0.694912 0.719094i \(-0.255443\pi\)
0.694912 + 0.719094i \(0.255443\pi\)
\(770\) 0 0
\(771\) 15.2361 0.548714
\(772\) 0 0
\(773\) − 5.86319i − 0.210884i −0.994425 0.105442i \(-0.966374\pi\)
0.994425 0.105442i \(-0.0336258\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) 0 0
\(777\) − 6.48218i − 0.232547i
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 20.4667i 0.731420i
\(784\) 0 0
\(785\) − 0.922740i − 0.0329340i
\(786\) 0 0
\(787\) − 24.8369i − 0.885338i −0.896685 0.442669i \(-0.854032\pi\)
0.896685 0.442669i \(-0.145968\pi\)
\(788\) 0 0
\(789\) −13.8885 −0.494445
\(790\) 0 0
\(791\) 46.2492 1.64443
\(792\) 0 0
\(793\) − 63.8158i − 2.26617i
\(794\) 0 0
\(795\) 5.12461 0.181751
\(796\) 0 0
\(797\) − 52.2958i − 1.85241i −0.377018 0.926206i \(-0.623050\pi\)
0.377018 0.926206i \(-0.376950\pi\)
\(798\) 0 0
\(799\) −62.8328 −2.22287
\(800\) 0 0
\(801\) −6.58359 −0.232620
\(802\) 0 0
\(803\) − 4.00621i − 0.141376i
\(804\) 0 0
\(805\) −6.83282 −0.240825
\(806\) 0 0
\(807\) − 13.0618i − 0.459796i
\(808\) 0 0
\(809\) 7.52786 0.264666 0.132333 0.991205i \(-0.457753\pi\)
0.132333 + 0.991205i \(0.457753\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) − 18.4636i − 0.647546i
\(814\) 0 0
\(815\) − 18.9737i − 0.664619i
\(816\) 0 0
\(817\) − 8.48528i − 0.296862i
\(818\) 0 0
\(819\) 34.4721 1.20455
\(820\) 0 0
\(821\) 38.9443 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(822\) 0 0
\(823\) − 35.9442i − 1.25294i −0.779447 0.626469i \(-0.784500\pi\)
0.779447 0.626469i \(-0.215500\pi\)
\(824\) 0 0
\(825\) 3.33851i 0.116232i
\(826\) 0 0
\(827\) 7.86629i 0.273538i 0.990603 + 0.136769i \(0.0436717\pi\)
−0.990603 + 0.136769i \(0.956328\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 19.4164 0.673548
\(832\) 0 0
\(833\) 7.40492i 0.256565i
\(834\) 0 0
\(835\) − 38.9790i − 1.34892i
\(836\) 0 0
\(837\) 18.3060i 0.632747i
\(838\) 0 0
\(839\) 49.3050 1.70220 0.851098 0.525007i \(-0.175937\pi\)
0.851098 + 0.525007i \(0.175937\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) − 22.2148i − 0.765117i
\(844\) 0 0
\(845\) −37.3607 −1.28525
\(846\) 0 0
\(847\) 29.4621i 1.01233i
\(848\) 0 0
\(849\) 2.11146 0.0724650
\(850\) 0 0
\(851\) −2.83282 −0.0971077
\(852\) 0 0
\(853\) 27.4589i 0.940176i 0.882619 + 0.470088i \(0.155778\pi\)
−0.882619 + 0.470088i \(0.844222\pi\)
\(854\) 0 0
\(855\) −5.00000 −0.170996
\(856\) 0 0
\(857\) 31.3190i 1.06984i 0.844903 + 0.534919i \(0.179658\pi\)
−0.844903 + 0.534919i \(0.820342\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) −14.8328 −0.505501
\(862\) 0 0
\(863\) 7.86629i 0.267772i 0.990997 + 0.133886i \(0.0427455\pi\)
−0.990997 + 0.133886i \(0.957254\pi\)
\(864\) 0 0
\(865\) − 20.0053i − 0.680201i
\(866\) 0 0
\(867\) − 33.0671i − 1.12302i
\(868\) 0 0
\(869\) 11.7771 0.399510
\(870\) 0 0
\(871\) 60.5410 2.05135
\(872\) 0 0
\(873\) 31.1614i 1.05465i
\(874\) 0 0
\(875\) − 31.6228i − 1.06904i
\(876\) 0 0
\(877\) 13.9358i 0.470579i 0.971925 + 0.235289i \(0.0756038\pi\)
−0.971925 + 0.235289i \(0.924396\pi\)
\(878\) 0 0
\(879\) 2.29180 0.0773004
\(880\) 0 0
\(881\) 0.652476 0.0219825 0.0109912 0.999940i \(-0.496501\pi\)
0.0109912 + 0.999940i \(0.496501\pi\)
\(882\) 0 0
\(883\) − 52.9148i − 1.78072i −0.455253 0.890362i \(-0.650451\pi\)
0.455253 0.890362i \(-0.349549\pi\)
\(884\) 0 0
\(885\) − 2.98605i − 0.100375i
\(886\) 0 0
\(887\) 49.0547i 1.64710i 0.567247 + 0.823548i \(0.308009\pi\)
−0.567247 + 0.823548i \(0.691991\pi\)
\(888\) 0 0
\(889\) 55.4164 1.85861
\(890\) 0 0
\(891\) 2.06888 0.0693102
\(892\) 0 0
\(893\) 8.48528i 0.283949i
\(894\) 0 0
\(895\) −50.2492 −1.67965
\(896\) 0 0
\(897\) 5.14678i 0.171846i
\(898\) 0 0
\(899\) 17.8885 0.596616
\(900\) 0 0
\(901\) −19.4164 −0.646854
\(902\) 0 0
\(903\) 20.9768i 0.698063i
\(904\) 0 0
\(905\) 13.4164 0.445976
\(906\) 0 0
\(907\) 38.5663i 1.28057i 0.768136 + 0.640287i \(0.221185\pi\)
−0.768136 + 0.640287i \(0.778815\pi\)
\(908\) 0 0
\(909\) 11.7082 0.388337
\(910\) 0 0
\(911\) 46.4721 1.53969 0.769845 0.638231i \(-0.220333\pi\)
0.769845 + 0.638231i \(0.220333\pi\)
\(912\) 0 0
\(913\) 10.4884i 0.347115i
\(914\) 0 0
\(915\) − 22.8825i − 0.756471i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 26.8328 0.885133 0.442566 0.896736i \(-0.354068\pi\)
0.442566 + 0.896736i \(0.354068\pi\)
\(920\) 0 0
\(921\) −4.76393 −0.156977
\(922\) 0 0
\(923\) 57.0786i 1.87877i
\(924\) 0 0
\(925\) − 13.1105i − 0.431070i
\(926\) 0 0
\(927\) − 13.1105i − 0.430605i
\(928\) 0 0
\(929\) −7.52786 −0.246981 −0.123491 0.992346i \(-0.539409\pi\)
−0.123491 + 0.992346i \(0.539409\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 8.48528i 0.277796i
\(934\) 0 0
\(935\) − 12.6491i − 0.413670i
\(936\) 0 0
\(937\) − 43.1915i − 1.41101i −0.708707 0.705503i \(-0.750721\pi\)
0.708707 0.705503i \(-0.249279\pi\)
\(938\) 0 0
\(939\) −9.16718 −0.299160
\(940\) 0 0
\(941\) −38.9443 −1.26955 −0.634773 0.772698i \(-0.718907\pi\)
−0.634773 + 0.772698i \(0.718907\pi\)
\(942\) 0 0
\(943\) 6.48218i 0.211089i
\(944\) 0 0
\(945\) 28.9443 0.941557
\(946\) 0 0
\(947\) 47.6706i 1.54909i 0.632521 + 0.774543i \(0.282020\pi\)
−0.632521 + 0.774543i \(0.717980\pi\)
\(948\) 0 0
\(949\) 28.5836 0.927863
\(950\) 0 0
\(951\) 7.81966 0.253570
\(952\) 0 0
\(953\) 36.5632i 1.18440i 0.805791 + 0.592199i \(0.201740\pi\)
−0.805791 + 0.592199i \(0.798260\pi\)
\(954\) 0 0
\(955\) 6.83282 0.221105
\(956\) 0 0
\(957\) 2.98605i 0.0965253i
\(958\) 0 0
\(959\) −20.9443 −0.676326
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 29.6684i 0.956050i
\(964\) 0 0
\(965\) − 31.1614i − 1.00312i
\(966\) 0 0
\(967\) 2.41577i 0.0776858i 0.999245 + 0.0388429i \(0.0123672\pi\)
−0.999245 + 0.0388429i \(0.987633\pi\)
\(968\) 0 0
\(969\) −6.47214 −0.207915
\(970\) 0 0
\(971\) 40.3607 1.29524 0.647618 0.761965i \(-0.275765\pi\)
0.647618 + 0.761965i \(0.275765\pi\)
\(972\) 0 0
\(973\) − 6.48218i − 0.207809i
\(974\) 0 0
\(975\) −23.8197 −0.762840
\(976\) 0 0
\(977\) 24.9945i 0.799644i 0.916593 + 0.399822i \(0.130928\pi\)
−0.916593 + 0.399822i \(0.869072\pi\)
\(978\) 0 0
\(979\) −2.24922 −0.0718855
\(980\) 0 0
\(981\) −4.47214 −0.142784
\(982\) 0 0
\(983\) − 30.2387i − 0.964464i −0.876044 0.482232i \(-0.839826\pi\)
0.876044 0.482232i \(-0.160174\pi\)
\(984\) 0 0
\(985\) 28.2843i 0.901212i
\(986\) 0 0
\(987\) − 20.9768i − 0.667698i
\(988\) 0 0
\(989\) 9.16718 0.291500
\(990\) 0 0
\(991\) −38.8328 −1.23357 −0.616783 0.787134i \(-0.711564\pi\)
−0.616783 + 0.787134i \(0.711564\pi\)
\(992\) 0 0
\(993\) − 6.48218i − 0.205706i
\(994\) 0 0
\(995\) −35.7771 −1.13421
\(996\) 0 0
\(997\) − 11.7264i − 0.371378i −0.982609 0.185689i \(-0.940548\pi\)
0.982609 0.185689i \(-0.0594517\pi\)
\(998\) 0 0
\(999\) 12.0000 0.379663
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 1520.2.d.f.609.3 4
4.3 odd 2 380.2.c.a.229.2 4
5.2 odd 4 7600.2.a.ce.1.3 4
5.3 odd 4 7600.2.a.ce.1.2 4
5.4 even 2 inner 1520.2.d.f.609.2 4
12.11 even 2 3420.2.f.a.1369.2 4
20.3 even 4 1900.2.a.j.1.3 4
20.7 even 4 1900.2.a.j.1.2 4
20.19 odd 2 380.2.c.a.229.3 yes 4
60.59 even 2 3420.2.f.a.1369.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 4 4.3 odd 2
380.2.c.a.229.3 yes 4 20.19 odd 2
1520.2.d.f.609.2 4 5.4 even 2 inner
1520.2.d.f.609.3 4 1.1 even 1 trivial
1900.2.a.j.1.2 4 20.7 even 4
1900.2.a.j.1.3 4 20.3 even 4
3420.2.f.a.1369.1 4 60.59 even 2
3420.2.f.a.1369.2 4 12.11 even 2
7600.2.a.ce.1.2 4 5.3 odd 4
7600.2.a.ce.1.3 4 5.2 odd 4