Properties

Label 320.3.b.c.191.1
Level $320$
Weight $3$
Character 320.191
Analytic conductor $8.719$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 320.191
Dual form 320.3.b.c.191.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.80423i q^{3} -2.23607 q^{5} -8.50651i q^{7} -5.47214 q^{9} +O(q^{10})\) \(q-3.80423i q^{3} -2.23607 q^{5} -8.50651i q^{7} -5.47214 q^{9} -1.79611i q^{11} -0.472136 q^{13} +8.50651i q^{15} -23.8885 q^{17} +9.40456i q^{19} -32.3607 q^{21} -16.1150i q^{23} +5.00000 q^{25} -13.4208i q^{27} -6.94427 q^{29} +47.4468i q^{31} -6.83282 q^{33} +19.0211i q^{35} -26.3607 q^{37} +1.79611i q^{39} -41.4164 q^{41} +2.00811i q^{43} +12.2361 q^{45} -35.3481i q^{47} -23.3607 q^{49} +90.8774i q^{51} +21.6393 q^{53} +4.01623i q^{55} +35.7771 q^{57} -73.8644i q^{59} +26.1378 q^{61} +46.5488i q^{63} +1.05573 q^{65} -88.8693i q^{67} -61.3050 q^{69} -39.4144i q^{71} +137.554 q^{73} -19.0211i q^{75} -15.2786 q^{77} -113.703i q^{79} -100.305 q^{81} -21.2412i q^{83} +53.4164 q^{85} +26.4176i q^{87} +67.4427 q^{89} +4.01623i q^{91} +180.498 q^{93} -21.0292i q^{95} -39.1672 q^{97} +9.82857i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{9} + 16 q^{13} - 24 q^{17} - 40 q^{21} + 20 q^{25} + 8 q^{29} + 80 q^{33} - 16 q^{37} - 112 q^{41} + 40 q^{45} - 4 q^{49} + 176 q^{53} - 128 q^{61} + 40 q^{65} - 120 q^{69} + 264 q^{73} - 240 q^{77} - 276 q^{81} + 160 q^{85} - 88 q^{89} + 400 q^{93} - 264 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 3.80423i − 1.26808i −0.773302 0.634038i \(-0.781396\pi\)
0.773302 0.634038i \(-0.218604\pi\)
\(4\) 0 0
\(5\) −2.23607 −0.447214
\(6\) 0 0
\(7\) − 8.50651i − 1.21522i −0.794237 0.607608i \(-0.792129\pi\)
0.794237 0.607608i \(-0.207871\pi\)
\(8\) 0 0
\(9\) −5.47214 −0.608015
\(10\) 0 0
\(11\) − 1.79611i − 0.163283i −0.996662 0.0816415i \(-0.973984\pi\)
0.996662 0.0816415i \(-0.0260162\pi\)
\(12\) 0 0
\(13\) −0.472136 −0.0363182 −0.0181591 0.999835i \(-0.505781\pi\)
−0.0181591 + 0.999835i \(0.505781\pi\)
\(14\) 0 0
\(15\) 8.50651i 0.567101i
\(16\) 0 0
\(17\) −23.8885 −1.40521 −0.702604 0.711581i \(-0.747980\pi\)
−0.702604 + 0.711581i \(0.747980\pi\)
\(18\) 0 0
\(19\) 9.40456i 0.494977i 0.968891 + 0.247489i \(0.0796053\pi\)
−0.968891 + 0.247489i \(0.920395\pi\)
\(20\) 0 0
\(21\) −32.3607 −1.54098
\(22\) 0 0
\(23\) − 16.1150i − 0.700650i −0.936628 0.350325i \(-0.886071\pi\)
0.936628 0.350325i \(-0.113929\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) − 13.4208i − 0.497066i
\(28\) 0 0
\(29\) −6.94427 −0.239458 −0.119729 0.992807i \(-0.538203\pi\)
−0.119729 + 0.992807i \(0.538203\pi\)
\(30\) 0 0
\(31\) 47.4468i 1.53054i 0.643708 + 0.765271i \(0.277395\pi\)
−0.643708 + 0.765271i \(0.722605\pi\)
\(32\) 0 0
\(33\) −6.83282 −0.207055
\(34\) 0 0
\(35\) 19.0211i 0.543461i
\(36\) 0 0
\(37\) −26.3607 −0.712451 −0.356225 0.934400i \(-0.615936\pi\)
−0.356225 + 0.934400i \(0.615936\pi\)
\(38\) 0 0
\(39\) 1.79611i 0.0460542i
\(40\) 0 0
\(41\) −41.4164 −1.01016 −0.505078 0.863074i \(-0.668536\pi\)
−0.505078 + 0.863074i \(0.668536\pi\)
\(42\) 0 0
\(43\) 2.00811i 0.0467003i 0.999727 + 0.0233502i \(0.00743326\pi\)
−0.999727 + 0.0233502i \(0.992567\pi\)
\(44\) 0 0
\(45\) 12.2361 0.271913
\(46\) 0 0
\(47\) − 35.3481i − 0.752087i −0.926602 0.376044i \(-0.877284\pi\)
0.926602 0.376044i \(-0.122716\pi\)
\(48\) 0 0
\(49\) −23.3607 −0.476749
\(50\) 0 0
\(51\) 90.8774i 1.78191i
\(52\) 0 0
\(53\) 21.6393 0.408289 0.204145 0.978941i \(-0.434559\pi\)
0.204145 + 0.978941i \(0.434559\pi\)
\(54\) 0 0
\(55\) 4.01623i 0.0730223i
\(56\) 0 0
\(57\) 35.7771 0.627668
\(58\) 0 0
\(59\) − 73.8644i − 1.25194i −0.779848 0.625970i \(-0.784703\pi\)
0.779848 0.625970i \(-0.215297\pi\)
\(60\) 0 0
\(61\) 26.1378 0.428488 0.214244 0.976780i \(-0.431271\pi\)
0.214244 + 0.976780i \(0.431271\pi\)
\(62\) 0 0
\(63\) 46.5488i 0.738869i
\(64\) 0 0
\(65\) 1.05573 0.0162420
\(66\) 0 0
\(67\) − 88.8693i − 1.32641i −0.748439 0.663204i \(-0.769196\pi\)
0.748439 0.663204i \(-0.230804\pi\)
\(68\) 0 0
\(69\) −61.3050 −0.888478
\(70\) 0 0
\(71\) − 39.4144i − 0.555132i −0.960707 0.277566i \(-0.910472\pi\)
0.960707 0.277566i \(-0.0895277\pi\)
\(72\) 0 0
\(73\) 137.554 1.88430 0.942152 0.335186i \(-0.108799\pi\)
0.942152 + 0.335186i \(0.108799\pi\)
\(74\) 0 0
\(75\) − 19.0211i − 0.253615i
\(76\) 0 0
\(77\) −15.2786 −0.198424
\(78\) 0 0
\(79\) − 113.703i − 1.43928i −0.694350 0.719638i \(-0.744308\pi\)
0.694350 0.719638i \(-0.255692\pi\)
\(80\) 0 0
\(81\) −100.305 −1.23833
\(82\) 0 0
\(83\) − 21.2412i − 0.255919i −0.991779 0.127959i \(-0.959157\pi\)
0.991779 0.127959i \(-0.0408427\pi\)
\(84\) 0 0
\(85\) 53.4164 0.628428
\(86\) 0 0
\(87\) 26.4176i 0.303650i
\(88\) 0 0
\(89\) 67.4427 0.757783 0.378892 0.925441i \(-0.376305\pi\)
0.378892 + 0.925441i \(0.376305\pi\)
\(90\) 0 0
\(91\) 4.01623i 0.0441344i
\(92\) 0 0
\(93\) 180.498 1.94084
\(94\) 0 0
\(95\) − 21.0292i − 0.221360i
\(96\) 0 0
\(97\) −39.1672 −0.403785 −0.201893 0.979408i \(-0.564709\pi\)
−0.201893 + 0.979408i \(0.564709\pi\)
\(98\) 0 0
\(99\) 9.82857i 0.0992785i
\(100\) 0 0
\(101\) −99.8885 −0.988995 −0.494498 0.869179i \(-0.664648\pi\)
−0.494498 + 0.869179i \(0.664648\pi\)
\(102\) 0 0
\(103\) − 35.7721i − 0.347302i −0.984807 0.173651i \(-0.944444\pi\)
0.984807 0.173651i \(-0.0555565\pi\)
\(104\) 0 0
\(105\) 72.3607 0.689149
\(106\) 0 0
\(107\) 121.099i 1.13177i 0.824485 + 0.565884i \(0.191465\pi\)
−0.824485 + 0.565884i \(0.808535\pi\)
\(108\) 0 0
\(109\) 197.469 1.81164 0.905821 0.423660i \(-0.139255\pi\)
0.905821 + 0.423660i \(0.139255\pi\)
\(110\) 0 0
\(111\) 100.282i 0.903441i
\(112\) 0 0
\(113\) 81.2786 0.719280 0.359640 0.933091i \(-0.382900\pi\)
0.359640 + 0.933091i \(0.382900\pi\)
\(114\) 0 0
\(115\) 36.0341i 0.313340i
\(116\) 0 0
\(117\) 2.58359 0.0220820
\(118\) 0 0
\(119\) 203.208i 1.70763i
\(120\) 0 0
\(121\) 117.774 0.973339
\(122\) 0 0
\(123\) 157.557i 1.28095i
\(124\) 0 0
\(125\) −11.1803 −0.0894427
\(126\) 0 0
\(127\) − 1.84616i − 0.0145367i −0.999974 0.00726834i \(-0.997686\pi\)
0.999974 0.00726834i \(-0.00231361\pi\)
\(128\) 0 0
\(129\) 7.63932 0.0592195
\(130\) 0 0
\(131\) − 225.609i − 1.72221i −0.508428 0.861105i \(-0.669773\pi\)
0.508428 0.861105i \(-0.330227\pi\)
\(132\) 0 0
\(133\) 80.0000 0.601504
\(134\) 0 0
\(135\) 30.0098i 0.222295i
\(136\) 0 0
\(137\) −52.8328 −0.385641 −0.192820 0.981234i \(-0.561764\pi\)
−0.192820 + 0.981234i \(0.561764\pi\)
\(138\) 0 0
\(139\) 125.852i 0.905407i 0.891661 + 0.452703i \(0.149540\pi\)
−0.891661 + 0.452703i \(0.850460\pi\)
\(140\) 0 0
\(141\) −134.472 −0.953703
\(142\) 0 0
\(143\) 0.848009i 0.00593013i
\(144\) 0 0
\(145\) 15.5279 0.107089
\(146\) 0 0
\(147\) 88.8693i 0.604553i
\(148\) 0 0
\(149\) −132.971 −0.892420 −0.446210 0.894928i \(-0.647227\pi\)
−0.446210 + 0.894928i \(0.647227\pi\)
\(150\) 0 0
\(151\) − 151.221i − 1.00146i −0.865603 0.500732i \(-0.833064\pi\)
0.865603 0.500732i \(-0.166936\pi\)
\(152\) 0 0
\(153\) 130.721 0.854388
\(154\) 0 0
\(155\) − 106.094i − 0.684480i
\(156\) 0 0
\(157\) 36.7477 0.234062 0.117031 0.993128i \(-0.462662\pi\)
0.117031 + 0.993128i \(0.462662\pi\)
\(158\) 0 0
\(159\) − 82.3209i − 0.517741i
\(160\) 0 0
\(161\) −137.082 −0.851441
\(162\) 0 0
\(163\) − 302.854i − 1.85800i −0.370079 0.929000i \(-0.620669\pi\)
0.370079 0.929000i \(-0.379331\pi\)
\(164\) 0 0
\(165\) 15.2786 0.0925978
\(166\) 0 0
\(167\) 99.3839i 0.595113i 0.954704 + 0.297557i \(0.0961717\pi\)
−0.954704 + 0.297557i \(0.903828\pi\)
\(168\) 0 0
\(169\) −168.777 −0.998681
\(170\) 0 0
\(171\) − 51.4631i − 0.300954i
\(172\) 0 0
\(173\) 181.639 1.04994 0.524969 0.851121i \(-0.324077\pi\)
0.524969 + 0.851121i \(0.324077\pi\)
\(174\) 0 0
\(175\) − 42.5325i − 0.243043i
\(176\) 0 0
\(177\) −280.997 −1.58755
\(178\) 0 0
\(179\) 260.907i 1.45758i 0.684735 + 0.728792i \(0.259918\pi\)
−0.684735 + 0.728792i \(0.740082\pi\)
\(180\) 0 0
\(181\) −157.777 −0.871697 −0.435848 0.900020i \(-0.643552\pi\)
−0.435848 + 0.900020i \(0.643552\pi\)
\(182\) 0 0
\(183\) − 99.4340i − 0.543355i
\(184\) 0 0
\(185\) 58.9443 0.318618
\(186\) 0 0
\(187\) 42.9065i 0.229447i
\(188\) 0 0
\(189\) −114.164 −0.604043
\(190\) 0 0
\(191\) − 324.095i − 1.69683i −0.529328 0.848417i \(-0.677556\pi\)
0.529328 0.848417i \(-0.322444\pi\)
\(192\) 0 0
\(193\) 181.777 0.941850 0.470925 0.882173i \(-0.343920\pi\)
0.470925 + 0.882173i \(0.343920\pi\)
\(194\) 0 0
\(195\) − 4.01623i − 0.0205960i
\(196\) 0 0
\(197\) −140.525 −0.713324 −0.356662 0.934234i \(-0.616085\pi\)
−0.356662 + 0.934234i \(0.616085\pi\)
\(198\) 0 0
\(199\) 168.234i 0.845397i 0.906270 + 0.422698i \(0.138917\pi\)
−0.906270 + 0.422698i \(0.861083\pi\)
\(200\) 0 0
\(201\) −338.079 −1.68198
\(202\) 0 0
\(203\) 59.0715i 0.290993i
\(204\) 0 0
\(205\) 92.6099 0.451756
\(206\) 0 0
\(207\) 88.1833i 0.426006i
\(208\) 0 0
\(209\) 16.8916 0.0808213
\(210\) 0 0
\(211\) − 93.9455i − 0.445240i −0.974905 0.222620i \(-0.928539\pi\)
0.974905 0.222620i \(-0.0714608\pi\)
\(212\) 0 0
\(213\) −149.941 −0.703949
\(214\) 0 0
\(215\) − 4.49028i − 0.0208850i
\(216\) 0 0
\(217\) 403.607 1.85994
\(218\) 0 0
\(219\) − 523.287i − 2.38944i
\(220\) 0 0
\(221\) 11.2786 0.0510346
\(222\) 0 0
\(223\) 214.035i 0.959797i 0.877324 + 0.479899i \(0.159327\pi\)
−0.877324 + 0.479899i \(0.840673\pi\)
\(224\) 0 0
\(225\) −27.3607 −0.121603
\(226\) 0 0
\(227\) 41.4225i 0.182478i 0.995829 + 0.0912389i \(0.0290827\pi\)
−0.995829 + 0.0912389i \(0.970917\pi\)
\(228\) 0 0
\(229\) −73.2786 −0.319994 −0.159997 0.987117i \(-0.551148\pi\)
−0.159997 + 0.987117i \(0.551148\pi\)
\(230\) 0 0
\(231\) 58.1234i 0.251616i
\(232\) 0 0
\(233\) −307.050 −1.31781 −0.658905 0.752227i \(-0.728980\pi\)
−0.658905 + 0.752227i \(0.728980\pi\)
\(234\) 0 0
\(235\) 79.0407i 0.336344i
\(236\) 0 0
\(237\) −432.551 −1.82511
\(238\) 0 0
\(239\) − 42.9065i − 0.179525i −0.995963 0.0897625i \(-0.971389\pi\)
0.995963 0.0897625i \(-0.0286108\pi\)
\(240\) 0 0
\(241\) −135.082 −0.560506 −0.280253 0.959926i \(-0.590418\pi\)
−0.280253 + 0.959926i \(0.590418\pi\)
\(242\) 0 0
\(243\) 260.796i 1.07323i
\(244\) 0 0
\(245\) 52.2361 0.213208
\(246\) 0 0
\(247\) − 4.44023i − 0.0179767i
\(248\) 0 0
\(249\) −80.8065 −0.324524
\(250\) 0 0
\(251\) 221.169i 0.881152i 0.897715 + 0.440576i \(0.145226\pi\)
−0.897715 + 0.440576i \(0.854774\pi\)
\(252\) 0 0
\(253\) −28.9443 −0.114404
\(254\) 0 0
\(255\) − 203.208i − 0.796894i
\(256\) 0 0
\(257\) −257.056 −1.00022 −0.500108 0.865963i \(-0.666707\pi\)
−0.500108 + 0.865963i \(0.666707\pi\)
\(258\) 0 0
\(259\) 224.237i 0.865781i
\(260\) 0 0
\(261\) 38.0000 0.145594
\(262\) 0 0
\(263\) − 164.168i − 0.624212i −0.950047 0.312106i \(-0.898966\pi\)
0.950047 0.312106i \(-0.101034\pi\)
\(264\) 0 0
\(265\) −48.3870 −0.182592
\(266\) 0 0
\(267\) − 256.567i − 0.960926i
\(268\) 0 0
\(269\) 35.4752 0.131878 0.0659391 0.997824i \(-0.478996\pi\)
0.0659391 + 0.997824i \(0.478996\pi\)
\(270\) 0 0
\(271\) 298.950i 1.10314i 0.834130 + 0.551568i \(0.185970\pi\)
−0.834130 + 0.551568i \(0.814030\pi\)
\(272\) 0 0
\(273\) 15.2786 0.0559657
\(274\) 0 0
\(275\) − 8.98056i − 0.0326566i
\(276\) 0 0
\(277\) 457.246 1.65071 0.825354 0.564616i \(-0.190976\pi\)
0.825354 + 0.564616i \(0.190976\pi\)
\(278\) 0 0
\(279\) − 259.635i − 0.930593i
\(280\) 0 0
\(281\) −5.63932 −0.0200688 −0.0100344 0.999950i \(-0.503194\pi\)
−0.0100344 + 0.999950i \(0.503194\pi\)
\(282\) 0 0
\(283\) 169.918i 0.600418i 0.953874 + 0.300209i \(0.0970563\pi\)
−0.953874 + 0.300209i \(0.902944\pi\)
\(284\) 0 0
\(285\) −80.0000 −0.280702
\(286\) 0 0
\(287\) 352.309i 1.22756i
\(288\) 0 0
\(289\) 281.663 0.974611
\(290\) 0 0
\(291\) 149.001i 0.512030i
\(292\) 0 0
\(293\) 26.8591 0.0916694 0.0458347 0.998949i \(-0.485405\pi\)
0.0458347 + 0.998949i \(0.485405\pi\)
\(294\) 0 0
\(295\) 165.166i 0.559884i
\(296\) 0 0
\(297\) −24.1052 −0.0811624
\(298\) 0 0
\(299\) 7.60845i 0.0254463i
\(300\) 0 0
\(301\) 17.0820 0.0567510
\(302\) 0 0
\(303\) 379.999i 1.25412i
\(304\) 0 0
\(305\) −58.4458 −0.191626
\(306\) 0 0
\(307\) 118.031i 0.384466i 0.981349 + 0.192233i \(0.0615730\pi\)
−0.981349 + 0.192233i \(0.938427\pi\)
\(308\) 0 0
\(309\) −136.085 −0.440405
\(310\) 0 0
\(311\) 121.835i 0.391753i 0.980629 + 0.195877i \(0.0627552\pi\)
−0.980629 + 0.195877i \(0.937245\pi\)
\(312\) 0 0
\(313\) 219.548 0.701431 0.350716 0.936482i \(-0.385938\pi\)
0.350716 + 0.936482i \(0.385938\pi\)
\(314\) 0 0
\(315\) − 104.086i − 0.330432i
\(316\) 0 0
\(317\) −366.859 −1.15728 −0.578642 0.815582i \(-0.696417\pi\)
−0.578642 + 0.815582i \(0.696417\pi\)
\(318\) 0 0
\(319\) 12.4727i 0.0390993i
\(320\) 0 0
\(321\) 460.689 1.43517
\(322\) 0 0
\(323\) − 224.661i − 0.695546i
\(324\) 0 0
\(325\) −2.36068 −0.00726363
\(326\) 0 0
\(327\) − 751.217i − 2.29730i
\(328\) 0 0
\(329\) −300.689 −0.913948
\(330\) 0 0
\(331\) 162.846i 0.491981i 0.969272 + 0.245990i \(0.0791132\pi\)
−0.969272 + 0.245990i \(0.920887\pi\)
\(332\) 0 0
\(333\) 144.249 0.433181
\(334\) 0 0
\(335\) 198.718i 0.593187i
\(336\) 0 0
\(337\) 17.1084 0.0507666 0.0253833 0.999678i \(-0.491919\pi\)
0.0253833 + 0.999678i \(0.491919\pi\)
\(338\) 0 0
\(339\) − 309.202i − 0.912101i
\(340\) 0 0
\(341\) 85.2198 0.249911
\(342\) 0 0
\(343\) − 218.101i − 0.635863i
\(344\) 0 0
\(345\) 137.082 0.397339
\(346\) 0 0
\(347\) − 167.498i − 0.482703i −0.970438 0.241351i \(-0.922409\pi\)
0.970438 0.241351i \(-0.0775906\pi\)
\(348\) 0 0
\(349\) 483.495 1.38537 0.692687 0.721239i \(-0.256427\pi\)
0.692687 + 0.721239i \(0.256427\pi\)
\(350\) 0 0
\(351\) 6.33644i 0.0180525i
\(352\) 0 0
\(353\) 307.994 0.872504 0.436252 0.899825i \(-0.356306\pi\)
0.436252 + 0.899825i \(0.356306\pi\)
\(354\) 0 0
\(355\) 88.1332i 0.248263i
\(356\) 0 0
\(357\) 773.050 2.16540
\(358\) 0 0
\(359\) − 23.2494i − 0.0647615i −0.999476 0.0323807i \(-0.989691\pi\)
0.999476 0.0323807i \(-0.0103089\pi\)
\(360\) 0 0
\(361\) 272.554 0.754998
\(362\) 0 0
\(363\) − 448.039i − 1.23427i
\(364\) 0 0
\(365\) −307.580 −0.842686
\(366\) 0 0
\(367\) − 517.325i − 1.40960i −0.709404 0.704802i \(-0.751036\pi\)
0.709404 0.704802i \(-0.248964\pi\)
\(368\) 0 0
\(369\) 226.636 0.614190
\(370\) 0 0
\(371\) − 184.075i − 0.496159i
\(372\) 0 0
\(373\) 88.3545 0.236875 0.118438 0.992961i \(-0.462211\pi\)
0.118438 + 0.992961i \(0.462211\pi\)
\(374\) 0 0
\(375\) 42.5325i 0.113420i
\(376\) 0 0
\(377\) 3.27864 0.00869666
\(378\) 0 0
\(379\) − 19.3332i − 0.0510112i −0.999675 0.0255056i \(-0.991880\pi\)
0.999675 0.0255056i \(-0.00811956\pi\)
\(380\) 0 0
\(381\) −7.02321 −0.0184336
\(382\) 0 0
\(383\) − 431.612i − 1.12692i −0.826142 0.563462i \(-0.809469\pi\)
0.826142 0.563462i \(-0.190531\pi\)
\(384\) 0 0
\(385\) 34.1641 0.0887379
\(386\) 0 0
\(387\) − 10.9887i − 0.0283945i
\(388\) 0 0
\(389\) 296.354 0.761837 0.380918 0.924609i \(-0.375608\pi\)
0.380918 + 0.924609i \(0.375608\pi\)
\(390\) 0 0
\(391\) 384.963i 0.984560i
\(392\) 0 0
\(393\) −858.269 −2.18389
\(394\) 0 0
\(395\) 254.247i 0.643664i
\(396\) 0 0
\(397\) 86.1904 0.217104 0.108552 0.994091i \(-0.465379\pi\)
0.108552 + 0.994091i \(0.465379\pi\)
\(398\) 0 0
\(399\) − 304.338i − 0.762752i
\(400\) 0 0
\(401\) 442.997 1.10473 0.552365 0.833602i \(-0.313725\pi\)
0.552365 + 0.833602i \(0.313725\pi\)
\(402\) 0 0
\(403\) − 22.4014i − 0.0555865i
\(404\) 0 0
\(405\) 224.289 0.553799
\(406\) 0 0
\(407\) 47.3467i 0.116331i
\(408\) 0 0
\(409\) 63.4102 0.155037 0.0775186 0.996991i \(-0.475300\pi\)
0.0775186 + 0.996991i \(0.475300\pi\)
\(410\) 0 0
\(411\) 200.988i 0.489022i
\(412\) 0 0
\(413\) −628.328 −1.52138
\(414\) 0 0
\(415\) 47.4969i 0.114450i
\(416\) 0 0
\(417\) 478.768 1.14812
\(418\) 0 0
\(419\) 435.678i 1.03980i 0.854226 + 0.519902i \(0.174032\pi\)
−0.854226 + 0.519902i \(0.825968\pi\)
\(420\) 0 0
\(421\) 582.912 1.38459 0.692294 0.721615i \(-0.256600\pi\)
0.692294 + 0.721615i \(0.256600\pi\)
\(422\) 0 0
\(423\) 193.430i 0.457280i
\(424\) 0 0
\(425\) −119.443 −0.281042
\(426\) 0 0
\(427\) − 222.341i − 0.520705i
\(428\) 0 0
\(429\) 3.22602 0.00751986
\(430\) 0 0
\(431\) 375.882i 0.872117i 0.899918 + 0.436058i \(0.143626\pi\)
−0.899918 + 0.436058i \(0.856374\pi\)
\(432\) 0 0
\(433\) −368.164 −0.850263 −0.425132 0.905131i \(-0.639772\pi\)
−0.425132 + 0.905131i \(0.639772\pi\)
\(434\) 0 0
\(435\) − 59.0715i − 0.135797i
\(436\) 0 0
\(437\) 151.554 0.346806
\(438\) 0 0
\(439\) 483.549i 1.10148i 0.834677 + 0.550739i \(0.185654\pi\)
−0.834677 + 0.550739i \(0.814346\pi\)
\(440\) 0 0
\(441\) 127.833 0.289870
\(442\) 0 0
\(443\) 279.181i 0.630205i 0.949058 + 0.315102i \(0.102039\pi\)
−0.949058 + 0.315102i \(0.897961\pi\)
\(444\) 0 0
\(445\) −150.807 −0.338891
\(446\) 0 0
\(447\) 505.850i 1.13166i
\(448\) 0 0
\(449\) 756.079 1.68392 0.841959 0.539542i \(-0.181403\pi\)
0.841959 + 0.539542i \(0.181403\pi\)
\(450\) 0 0
\(451\) 74.3885i 0.164941i
\(452\) 0 0
\(453\) −575.279 −1.26993
\(454\) 0 0
\(455\) − 8.98056i − 0.0197375i
\(456\) 0 0
\(457\) 285.672 0.625103 0.312551 0.949901i \(-0.398816\pi\)
0.312551 + 0.949901i \(0.398816\pi\)
\(458\) 0 0
\(459\) 320.603i 0.698482i
\(460\) 0 0
\(461\) 99.1146 0.214999 0.107500 0.994205i \(-0.465716\pi\)
0.107500 + 0.994205i \(0.465716\pi\)
\(462\) 0 0
\(463\) − 630.603i − 1.36199i −0.732286 0.680997i \(-0.761547\pi\)
0.732286 0.680997i \(-0.238453\pi\)
\(464\) 0 0
\(465\) −403.607 −0.867972
\(466\) 0 0
\(467\) − 496.010i − 1.06212i −0.847334 0.531060i \(-0.821794\pi\)
0.847334 0.531060i \(-0.178206\pi\)
\(468\) 0 0
\(469\) −755.967 −1.61187
\(470\) 0 0
\(471\) − 139.796i − 0.296808i
\(472\) 0 0
\(473\) 3.60680 0.00762537
\(474\) 0 0
\(475\) 47.0228i 0.0989954i
\(476\) 0 0
\(477\) −118.413 −0.248246
\(478\) 0 0
\(479\) 579.090i 1.20896i 0.796621 + 0.604478i \(0.206618\pi\)
−0.796621 + 0.604478i \(0.793382\pi\)
\(480\) 0 0
\(481\) 12.4458 0.0258749
\(482\) 0 0
\(483\) 521.491i 1.07969i
\(484\) 0 0
\(485\) 87.5805 0.180578
\(486\) 0 0
\(487\) 626.363i 1.28617i 0.765796 + 0.643084i \(0.222345\pi\)
−0.765796 + 0.643084i \(0.777655\pi\)
\(488\) 0 0
\(489\) −1152.13 −2.35608
\(490\) 0 0
\(491\) − 22.3013i − 0.0454201i −0.999742 0.0227100i \(-0.992771\pi\)
0.999742 0.0227100i \(-0.00722945\pi\)
\(492\) 0 0
\(493\) 165.889 0.336488
\(494\) 0 0
\(495\) − 21.9773i − 0.0443987i
\(496\) 0 0
\(497\) −335.279 −0.674605
\(498\) 0 0
\(499\) − 627.362i − 1.25724i −0.777714 0.628619i \(-0.783621\pi\)
0.777714 0.628619i \(-0.216379\pi\)
\(500\) 0 0
\(501\) 378.079 0.754649
\(502\) 0 0
\(503\) 780.853i 1.55239i 0.630492 + 0.776196i \(0.282853\pi\)
−0.630492 + 0.776196i \(0.717147\pi\)
\(504\) 0 0
\(505\) 223.358 0.442292
\(506\) 0 0
\(507\) 642.066i 1.26640i
\(508\) 0 0
\(509\) 288.950 0.567683 0.283841 0.958871i \(-0.408391\pi\)
0.283841 + 0.958871i \(0.408391\pi\)
\(510\) 0 0
\(511\) − 1170.11i − 2.28984i
\(512\) 0 0
\(513\) 126.217 0.246036
\(514\) 0 0
\(515\) 79.9888i 0.155318i
\(516\) 0 0
\(517\) −63.4891 −0.122803
\(518\) 0 0
\(519\) − 690.997i − 1.33140i
\(520\) 0 0
\(521\) −602.984 −1.15736 −0.578680 0.815555i \(-0.696432\pi\)
−0.578680 + 0.815555i \(0.696432\pi\)
\(522\) 0 0
\(523\) − 367.962i − 0.703560i −0.936083 0.351780i \(-0.885577\pi\)
0.936083 0.351780i \(-0.114423\pi\)
\(524\) 0 0
\(525\) −161.803 −0.308197
\(526\) 0 0
\(527\) − 1133.44i − 2.15073i
\(528\) 0 0
\(529\) 269.308 0.509089
\(530\) 0 0
\(531\) 404.196i 0.761198i
\(532\) 0 0
\(533\) 19.5542 0.0366870
\(534\) 0 0
\(535\) − 270.786i − 0.506142i
\(536\) 0 0
\(537\) 992.551 1.84833
\(538\) 0 0
\(539\) 41.9584i 0.0778449i
\(540\) 0 0
\(541\) 616.885 1.14027 0.570134 0.821551i \(-0.306891\pi\)
0.570134 + 0.821551i \(0.306891\pi\)
\(542\) 0 0
\(543\) 600.220i 1.10538i
\(544\) 0 0
\(545\) −441.554 −0.810191
\(546\) 0 0
\(547\) 97.8499i 0.178885i 0.995992 + 0.0894423i \(0.0285085\pi\)
−0.995992 + 0.0894423i \(0.971492\pi\)
\(548\) 0 0
\(549\) −143.029 −0.260527
\(550\) 0 0
\(551\) − 65.3078i − 0.118526i
\(552\) 0 0
\(553\) −967.214 −1.74903
\(554\) 0 0
\(555\) − 224.237i − 0.404031i
\(556\) 0 0
\(557\) −896.302 −1.60916 −0.804580 0.593845i \(-0.797609\pi\)
−0.804580 + 0.593845i \(0.797609\pi\)
\(558\) 0 0
\(559\) − 0.948103i − 0.00169607i
\(560\) 0 0
\(561\) 163.226 0.290955
\(562\) 0 0
\(563\) − 771.186i − 1.36978i −0.728647 0.684890i \(-0.759850\pi\)
0.728647 0.684890i \(-0.240150\pi\)
\(564\) 0 0
\(565\) −181.745 −0.321672
\(566\) 0 0
\(567\) 853.245i 1.50484i
\(568\) 0 0
\(569\) 8.74767 0.0153738 0.00768688 0.999970i \(-0.497553\pi\)
0.00768688 + 0.999970i \(0.497553\pi\)
\(570\) 0 0
\(571\) − 511.138i − 0.895164i −0.894243 0.447582i \(-0.852285\pi\)
0.894243 0.447582i \(-0.147715\pi\)
\(572\) 0 0
\(573\) −1232.93 −2.15171
\(574\) 0 0
\(575\) − 80.5748i − 0.140130i
\(576\) 0 0
\(577\) −713.712 −1.23694 −0.618468 0.785810i \(-0.712246\pi\)
−0.618468 + 0.785810i \(0.712246\pi\)
\(578\) 0 0
\(579\) − 691.521i − 1.19434i
\(580\) 0 0
\(581\) −180.689 −0.310996
\(582\) 0 0
\(583\) − 38.8666i − 0.0666666i
\(584\) 0 0
\(585\) −5.77709 −0.00987536
\(586\) 0 0
\(587\) 422.169i 0.719198i 0.933107 + 0.359599i \(0.117086\pi\)
−0.933107 + 0.359599i \(0.882914\pi\)
\(588\) 0 0
\(589\) −446.217 −0.757584
\(590\) 0 0
\(591\) 534.588i 0.904548i
\(592\) 0 0
\(593\) −308.663 −0.520510 −0.260255 0.965540i \(-0.583807\pi\)
−0.260255 + 0.965540i \(0.583807\pi\)
\(594\) 0 0
\(595\) − 454.387i − 0.763676i
\(596\) 0 0
\(597\) 640.000 1.07203
\(598\) 0 0
\(599\) − 462.196i − 0.771612i −0.922580 0.385806i \(-0.873923\pi\)
0.922580 0.385806i \(-0.126077\pi\)
\(600\) 0 0
\(601\) 355.358 0.591277 0.295639 0.955300i \(-0.404468\pi\)
0.295639 + 0.955300i \(0.404468\pi\)
\(602\) 0 0
\(603\) 486.305i 0.806476i
\(604\) 0 0
\(605\) −263.351 −0.435290
\(606\) 0 0
\(607\) 630.403i 1.03856i 0.854605 + 0.519278i \(0.173799\pi\)
−0.854605 + 0.519278i \(0.826201\pi\)
\(608\) 0 0
\(609\) 224.721 0.369001
\(610\) 0 0
\(611\) 16.6891i 0.0273144i
\(612\) 0 0
\(613\) −812.525 −1.32549 −0.662745 0.748846i \(-0.730608\pi\)
−0.662745 + 0.748846i \(0.730608\pi\)
\(614\) 0 0
\(615\) − 352.309i − 0.572860i
\(616\) 0 0
\(617\) −437.935 −0.709781 −0.354891 0.934908i \(-0.615482\pi\)
−0.354891 + 0.934908i \(0.615482\pi\)
\(618\) 0 0
\(619\) 770.250i 1.24435i 0.782880 + 0.622173i \(0.213750\pi\)
−0.782880 + 0.622173i \(0.786250\pi\)
\(620\) 0 0
\(621\) −216.276 −0.348270
\(622\) 0 0
\(623\) − 573.702i − 0.920870i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) − 64.2597i − 0.102487i
\(628\) 0 0
\(629\) 629.718 1.00114
\(630\) 0 0
\(631\) − 875.496i − 1.38747i −0.720228 0.693737i \(-0.755963\pi\)
0.720228 0.693737i \(-0.244037\pi\)
\(632\) 0 0
\(633\) −357.390 −0.564597
\(634\) 0 0
\(635\) 4.12814i 0.00650100i
\(636\) 0 0
\(637\) 11.0294 0.0173146
\(638\) 0 0
\(639\) 215.681i 0.337529i
\(640\) 0 0
\(641\) 842.571 1.31446 0.657232 0.753689i \(-0.271727\pi\)
0.657232 + 0.753689i \(0.271727\pi\)
\(642\) 0 0
\(643\) 1153.20i 1.79348i 0.442563 + 0.896738i \(0.354069\pi\)
−0.442563 + 0.896738i \(0.645931\pi\)
\(644\) 0 0
\(645\) −17.0820 −0.0264838
\(646\) 0 0
\(647\) − 355.751i − 0.549847i −0.961466 0.274924i \(-0.911347\pi\)
0.961466 0.274924i \(-0.0886526\pi\)
\(648\) 0 0
\(649\) −132.669 −0.204420
\(650\) 0 0
\(651\) − 1535.41i − 2.35854i
\(652\) 0 0
\(653\) 557.915 0.854387 0.427194 0.904160i \(-0.359502\pi\)
0.427194 + 0.904160i \(0.359502\pi\)
\(654\) 0 0
\(655\) 504.478i 0.770195i
\(656\) 0 0
\(657\) −752.715 −1.14569
\(658\) 0 0
\(659\) − 284.157i − 0.431194i −0.976482 0.215597i \(-0.930830\pi\)
0.976482 0.215597i \(-0.0691697\pi\)
\(660\) 0 0
\(661\) −716.735 −1.08432 −0.542160 0.840275i \(-0.682393\pi\)
−0.542160 + 0.840275i \(0.682393\pi\)
\(662\) 0 0
\(663\) − 42.9065i − 0.0647157i
\(664\) 0 0
\(665\) −178.885 −0.269001
\(666\) 0 0
\(667\) 111.907i 0.167776i
\(668\) 0 0
\(669\) 814.237 1.21710
\(670\) 0 0
\(671\) − 46.9464i − 0.0699648i
\(672\) 0 0
\(673\) −695.378 −1.03325 −0.516625 0.856212i \(-0.672812\pi\)
−0.516625 + 0.856212i \(0.672812\pi\)
\(674\) 0 0
\(675\) − 67.1040i − 0.0994133i
\(676\) 0 0
\(677\) 820.237 1.21158 0.605788 0.795626i \(-0.292858\pi\)
0.605788 + 0.795626i \(0.292858\pi\)
\(678\) 0 0
\(679\) 333.176i 0.490686i
\(680\) 0 0
\(681\) 157.580 0.231396
\(682\) 0 0
\(683\) − 335.508i − 0.491227i −0.969368 0.245613i \(-0.921011\pi\)
0.969368 0.245613i \(-0.0789894\pi\)
\(684\) 0 0
\(685\) 118.138 0.172464
\(686\) 0 0
\(687\) 278.769i 0.405777i
\(688\) 0 0
\(689\) −10.2167 −0.0148283
\(690\) 0 0
\(691\) − 336.568i − 0.487074i −0.969892 0.243537i \(-0.921692\pi\)
0.969892 0.243537i \(-0.0783077\pi\)
\(692\) 0 0
\(693\) 83.6068 0.120645
\(694\) 0 0
\(695\) − 281.413i − 0.404910i
\(696\) 0 0
\(697\) 989.378 1.41948
\(698\) 0 0
\(699\) 1168.09i 1.67108i
\(700\) 0 0
\(701\) 429.364 0.612502 0.306251 0.951951i \(-0.400925\pi\)
0.306251 + 0.951951i \(0.400925\pi\)
\(702\) 0 0
\(703\) − 247.911i − 0.352647i
\(704\) 0 0
\(705\) 300.689 0.426509
\(706\) 0 0
\(707\) 849.703i 1.20184i
\(708\) 0 0
\(709\) −1224.60 −1.72722 −0.863609 0.504162i \(-0.831801\pi\)
−0.863609 + 0.504162i \(0.831801\pi\)
\(710\) 0 0
\(711\) 622.197i 0.875101i
\(712\) 0 0
\(713\) 764.604 1.07238
\(714\) 0 0
\(715\) − 1.89621i − 0.00265204i
\(716\) 0 0
\(717\) −163.226 −0.227651
\(718\) 0 0
\(719\) 496.022i 0.689877i 0.938625 + 0.344939i \(0.112100\pi\)
−0.938625 + 0.344939i \(0.887900\pi\)
\(720\) 0 0
\(721\) −304.296 −0.422047
\(722\) 0 0
\(723\) 513.883i 0.710764i
\(724\) 0 0
\(725\) −34.7214 −0.0478915
\(726\) 0 0
\(727\) − 152.843i − 0.210238i −0.994460 0.105119i \(-0.966478\pi\)
0.994460 0.105119i \(-0.0335224\pi\)
\(728\) 0 0
\(729\) 89.3808 0.122607
\(730\) 0 0
\(731\) − 47.9709i − 0.0656237i
\(732\) 0 0
\(733\) 761.286 1.03859 0.519295 0.854595i \(-0.326195\pi\)
0.519295 + 0.854595i \(0.326195\pi\)
\(734\) 0 0
\(735\) − 198.718i − 0.270364i
\(736\) 0 0
\(737\) −159.619 −0.216580
\(738\) 0 0
\(739\) 183.975i 0.248951i 0.992223 + 0.124476i \(0.0397249\pi\)
−0.992223 + 0.124476i \(0.960275\pi\)
\(740\) 0 0
\(741\) −16.8916 −0.0227957
\(742\) 0 0
\(743\) 495.247i 0.666551i 0.942830 + 0.333275i \(0.108154\pi\)
−0.942830 + 0.333275i \(0.891846\pi\)
\(744\) 0 0
\(745\) 297.331 0.399102
\(746\) 0 0
\(747\) 116.235i 0.155602i
\(748\) 0 0
\(749\) 1030.13 1.37534
\(750\) 0 0
\(751\) 800.059i 1.06533i 0.846328 + 0.532663i \(0.178809\pi\)
−0.846328 + 0.532663i \(0.821191\pi\)
\(752\) 0 0
\(753\) 841.378 1.11737
\(754\) 0 0
\(755\) 338.140i 0.447868i
\(756\) 0 0
\(757\) 276.367 0.365082 0.182541 0.983198i \(-0.441568\pi\)
0.182541 + 0.983198i \(0.441568\pi\)
\(758\) 0 0
\(759\) 110.111i 0.145073i
\(760\) 0 0
\(761\) 891.207 1.17110 0.585550 0.810636i \(-0.300879\pi\)
0.585550 + 0.810636i \(0.300879\pi\)
\(762\) 0 0
\(763\) − 1679.77i − 2.20154i
\(764\) 0 0
\(765\) −292.302 −0.382094
\(766\) 0 0
\(767\) 34.8740i 0.0454681i
\(768\) 0 0
\(769\) −835.430 −1.08639 −0.543193 0.839608i \(-0.682785\pi\)
−0.543193 + 0.839608i \(0.682785\pi\)
\(770\) 0 0
\(771\) 977.898i 1.26835i
\(772\) 0 0
\(773\) −213.522 −0.276225 −0.138112 0.990417i \(-0.544103\pi\)
−0.138112 + 0.990417i \(0.544103\pi\)
\(774\) 0 0
\(775\) 237.234i 0.306109i
\(776\) 0 0
\(777\) 853.050 1.09788
\(778\) 0 0
\(779\) − 389.503i − 0.500004i
\(780\) 0 0
\(781\) −70.7926 −0.0906436
\(782\) 0 0
\(783\) 93.1976i 0.119026i
\(784\) 0 0
\(785\) −82.1703 −0.104676
\(786\) 0 0
\(787\) 370.182i 0.470371i 0.971951 + 0.235185i \(0.0755697\pi\)
−0.971951 + 0.235185i \(0.924430\pi\)
\(788\) 0 0
\(789\) −624.531 −0.791547
\(790\) 0 0
\(791\) − 691.397i − 0.874080i
\(792\) 0 0
\(793\) −12.3406 −0.0155619
\(794\) 0 0
\(795\) 184.075i 0.231541i
\(796\) 0 0
\(797\) −274.426 −0.344323 −0.172162 0.985069i \(-0.555075\pi\)
−0.172162 + 0.985069i \(0.555075\pi\)
\(798\) 0 0
\(799\) 844.414i 1.05684i
\(800\) 0 0
\(801\) −369.056 −0.460744
\(802\) 0 0
\(803\) − 247.063i − 0.307675i
\(804\) 0 0
\(805\) 306.525 0.380776
\(806\) 0 0
\(807\) − 134.956i − 0.167232i
\(808\) 0 0
\(809\) 665.214 0.822266 0.411133 0.911575i \(-0.365133\pi\)
0.411133 + 0.911575i \(0.365133\pi\)
\(810\) 0 0
\(811\) 360.665i 0.444717i 0.974965 + 0.222358i \(0.0713755\pi\)
−0.974965 + 0.222358i \(0.928624\pi\)
\(812\) 0 0
\(813\) 1137.27 1.39886
\(814\) 0 0
\(815\) 677.202i 0.830923i
\(816\) 0 0
\(817\) −18.8854 −0.0231156
\(818\) 0 0
\(819\) − 21.9773i − 0.0268344i
\(820\) 0 0
\(821\) 666.899 0.812301 0.406151 0.913806i \(-0.366871\pi\)
0.406151 + 0.913806i \(0.366871\pi\)
\(822\) 0 0
\(823\) − 122.433i − 0.148764i −0.997230 0.0743822i \(-0.976302\pi\)
0.997230 0.0743822i \(-0.0236985\pi\)
\(824\) 0 0
\(825\) −34.1641 −0.0414110
\(826\) 0 0
\(827\) − 1532.98i − 1.85366i −0.375477 0.926832i \(-0.622521\pi\)
0.375477 0.926832i \(-0.377479\pi\)
\(828\) 0 0
\(829\) 195.475 0.235796 0.117898 0.993026i \(-0.462384\pi\)
0.117898 + 0.993026i \(0.462384\pi\)
\(830\) 0 0
\(831\) − 1739.47i − 2.09322i
\(832\) 0 0
\(833\) 558.053 0.669931
\(834\) 0 0
\(835\) − 222.229i − 0.266143i
\(836\) 0 0
\(837\) 636.774 0.760781
\(838\) 0 0
\(839\) − 1325.97i − 1.58041i −0.612840 0.790207i \(-0.709973\pi\)
0.612840 0.790207i \(-0.290027\pi\)
\(840\) 0 0
\(841\) −792.777 −0.942660
\(842\) 0 0
\(843\) 21.4532i 0.0254487i
\(844\) 0 0
\(845\) 377.397 0.446624
\(846\) 0 0
\(847\) − 1001.85i − 1.18282i
\(848\) 0 0
\(849\) 646.407 0.761375
\(850\) 0 0
\(851\) 424.801i 0.499179i
\(852\) 0 0
\(853\) −1055.28 −1.23714 −0.618570 0.785730i \(-0.712288\pi\)
−0.618570 + 0.785730i \(0.712288\pi\)
\(854\) 0 0
\(855\) 115.075i 0.134591i
\(856\) 0 0
\(857\) 155.378 0.181304 0.0906521 0.995883i \(-0.471105\pi\)
0.0906521 + 0.995883i \(0.471105\pi\)
\(858\) 0 0
\(859\) − 226.033i − 0.263136i −0.991307 0.131568i \(-0.957999\pi\)
0.991307 0.131568i \(-0.0420011\pi\)
\(860\) 0 0
\(861\) 1340.26 1.55664
\(862\) 0 0
\(863\) − 930.702i − 1.07845i −0.842162 0.539225i \(-0.818717\pi\)
0.842162 0.539225i \(-0.181283\pi\)
\(864\) 0 0
\(865\) −406.158 −0.469547
\(866\) 0 0
\(867\) − 1071.51i − 1.23588i
\(868\) 0 0
\(869\) −204.223 −0.235009
\(870\) 0 0
\(871\) 41.9584i 0.0481727i
\(872\) 0 0
\(873\) 214.328 0.245508
\(874\) 0 0
\(875\) 95.1057i 0.108692i
\(876\) 0 0
\(877\) 33.5217 0.0382231 0.0191115 0.999817i \(-0.493916\pi\)
0.0191115 + 0.999817i \(0.493916\pi\)
\(878\) 0 0
\(879\) − 102.178i − 0.116244i
\(880\) 0 0
\(881\) −933.850 −1.05999 −0.529994 0.848001i \(-0.677806\pi\)
−0.529994 + 0.848001i \(0.677806\pi\)
\(882\) 0 0
\(883\) 542.308i 0.614166i 0.951683 + 0.307083i \(0.0993529\pi\)
−0.951683 + 0.307083i \(0.900647\pi\)
\(884\) 0 0
\(885\) 628.328 0.709975
\(886\) 0 0
\(887\) 714.720i 0.805773i 0.915250 + 0.402886i \(0.131993\pi\)
−0.915250 + 0.402886i \(0.868007\pi\)
\(888\) 0 0
\(889\) −15.7044 −0.0176652
\(890\) 0 0
\(891\) 180.159i 0.202199i
\(892\) 0 0
\(893\) 332.433 0.372266
\(894\) 0 0
\(895\) − 583.407i − 0.651851i
\(896\) 0 0
\(897\) 28.9443 0.0322679
\(898\) 0 0
\(899\) − 329.484i − 0.366500i
\(900\) 0 0
\(901\) −516.932 −0.573731
\(902\) 0 0
\(903\) − 64.9839i − 0.0719645i
\(904\) 0 0
\(905\) 352.800 0.389835
\(906\) 0 0
\(907\) − 347.233i − 0.382837i −0.981509 0.191418i \(-0.938691\pi\)
0.981509 0.191418i \(-0.0613087\pi\)
\(908\) 0 0
\(909\) 546.604 0.601324
\(910\) 0 0
\(911\) 1427.54i 1.56701i 0.621386 + 0.783504i \(0.286570\pi\)
−0.621386 + 0.783504i \(0.713430\pi\)
\(912\) 0 0
\(913\) −38.1517 −0.0417871
\(914\) 0 0
\(915\) 222.341i 0.242996i
\(916\) 0 0
\(917\) −1919.15 −2.09286
\(918\) 0 0
\(919\) − 569.162i − 0.619327i −0.950846 0.309664i \(-0.899784\pi\)
0.950846 0.309664i \(-0.100216\pi\)
\(920\) 0 0
\(921\) 449.017 0.487532
\(922\) 0 0
\(923\) 18.6089i 0.0201614i
\(924\) 0 0
\(925\) −131.803 −0.142490
\(926\) 0 0
\(927\) 195.750i 0.211165i
\(928\) 0 0
\(929\) 1535.96 1.65335 0.826675 0.562680i \(-0.190230\pi\)
0.826675 + 0.562680i \(0.190230\pi\)
\(930\) 0 0
\(931\) − 219.697i − 0.235980i
\(932\) 0 0
\(933\) 463.489 0.496773
\(934\) 0 0
\(935\) − 95.9418i − 0.102612i
\(936\) 0 0
\(937\) 338.721 0.361496 0.180748 0.983529i \(-0.442148\pi\)
0.180748 + 0.983529i \(0.442148\pi\)
\(938\) 0 0
\(939\) − 835.210i − 0.889468i
\(940\) 0 0
\(941\) −1439.77 −1.53004 −0.765022 0.644004i \(-0.777272\pi\)
−0.765022 + 0.644004i \(0.777272\pi\)
\(942\) 0 0
\(943\) 667.424i 0.707766i
\(944\) 0 0
\(945\) 255.279 0.270136
\(946\) 0 0
\(947\) − 656.135i − 0.692856i −0.938077 0.346428i \(-0.887394\pi\)
0.938077 0.346428i \(-0.112606\pi\)
\(948\) 0 0
\(949\) −64.9443 −0.0684344
\(950\) 0 0
\(951\) 1395.62i 1.46752i
\(952\) 0 0
\(953\) 436.675 0.458211 0.229105 0.973402i \(-0.426420\pi\)
0.229105 + 0.973402i \(0.426420\pi\)
\(954\) 0 0
\(955\) 724.699i 0.758847i
\(956\) 0 0
\(957\) 47.4489 0.0495809
\(958\) 0 0
\(959\) 449.423i 0.468637i
\(960\) 0 0
\(961\) −1290.20 −1.34256
\(962\) 0 0
\(963\) − 662.671i − 0.688132i
\(964\) 0 0
\(965\) −406.466 −0.421208
\(966\) 0 0
\(967\) − 903.436i − 0.934267i −0.884187 0.467133i \(-0.845287\pi\)
0.884187 0.467133i \(-0.154713\pi\)
\(968\) 0 0
\(969\) −854.663 −0.882005
\(970\) 0 0
\(971\) 1866.89i 1.92265i 0.275420 + 0.961324i \(0.411183\pi\)
−0.275420 + 0.961324i \(0.588817\pi\)
\(972\) 0 0
\(973\) 1070.56 1.10026
\(974\) 0 0
\(975\) 8.98056i 0.00921083i
\(976\) 0 0
\(977\) −1073.95 −1.09923 −0.549615 0.835418i \(-0.685226\pi\)
−0.549615 + 0.835418i \(0.685226\pi\)
\(978\) 0 0
\(979\) − 121.135i − 0.123733i
\(980\) 0 0
\(981\) −1080.58 −1.10151
\(982\) 0 0
\(983\) 534.114i 0.543351i 0.962389 + 0.271675i \(0.0875777\pi\)
−0.962389 + 0.271675i \(0.912422\pi\)
\(984\) 0 0
\(985\) 314.223 0.319008
\(986\) 0 0
\(987\) 1143.89i 1.15895i
\(988\) 0 0
\(989\) 32.3607 0.0327206
\(990\) 0 0
\(991\) 520.419i 0.525146i 0.964912 + 0.262573i \(0.0845710\pi\)
−0.964912 + 0.262573i \(0.915429\pi\)
\(992\) 0 0
\(993\) 619.502 0.623869
\(994\) 0 0
\(995\) − 376.183i − 0.378073i
\(996\) 0 0
\(997\) −457.680 −0.459057 −0.229528 0.973302i \(-0.573718\pi\)
−0.229528 + 0.973302i \(0.573718\pi\)
\(998\) 0 0
\(999\) 353.781i 0.354135i
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 320.3.b.c.191.1 4
3.2 odd 2 2880.3.e.e.2431.3 4
4.3 odd 2 inner 320.3.b.c.191.4 4
5.2 odd 4 1600.3.h.n.1599.1 8
5.3 odd 4 1600.3.h.n.1599.7 8
5.4 even 2 1600.3.b.s.1151.4 4
8.3 odd 2 20.3.b.a.11.1 4
8.5 even 2 20.3.b.a.11.2 yes 4
12.11 even 2 2880.3.e.e.2431.4 4
16.3 odd 4 1280.3.g.e.1151.2 8
16.5 even 4 1280.3.g.e.1151.1 8
16.11 odd 4 1280.3.g.e.1151.7 8
16.13 even 4 1280.3.g.e.1151.8 8
20.3 even 4 1600.3.h.n.1599.2 8
20.7 even 4 1600.3.h.n.1599.8 8
20.19 odd 2 1600.3.b.s.1151.1 4
24.5 odd 2 180.3.c.a.91.3 4
24.11 even 2 180.3.c.a.91.4 4
40.3 even 4 100.3.d.b.99.4 8
40.13 odd 4 100.3.d.b.99.6 8
40.19 odd 2 100.3.b.f.51.4 4
40.27 even 4 100.3.d.b.99.5 8
40.29 even 2 100.3.b.f.51.3 4
40.37 odd 4 100.3.d.b.99.3 8
120.29 odd 2 900.3.c.k.451.2 4
120.53 even 4 900.3.f.e.199.3 8
120.59 even 2 900.3.c.k.451.1 4
120.77 even 4 900.3.f.e.199.6 8
120.83 odd 4 900.3.f.e.199.5 8
120.107 odd 4 900.3.f.e.199.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.1 4 8.3 odd 2
20.3.b.a.11.2 yes 4 8.5 even 2
100.3.b.f.51.3 4 40.29 even 2
100.3.b.f.51.4 4 40.19 odd 2
100.3.d.b.99.3 8 40.37 odd 4
100.3.d.b.99.4 8 40.3 even 4
100.3.d.b.99.5 8 40.27 even 4
100.3.d.b.99.6 8 40.13 odd 4
180.3.c.a.91.3 4 24.5 odd 2
180.3.c.a.91.4 4 24.11 even 2
320.3.b.c.191.1 4 1.1 even 1 trivial
320.3.b.c.191.4 4 4.3 odd 2 inner
900.3.c.k.451.1 4 120.59 even 2
900.3.c.k.451.2 4 120.29 odd 2
900.3.f.e.199.3 8 120.53 even 4
900.3.f.e.199.4 8 120.107 odd 4
900.3.f.e.199.5 8 120.83 odd 4
900.3.f.e.199.6 8 120.77 even 4
1280.3.g.e.1151.1 8 16.5 even 4
1280.3.g.e.1151.2 8 16.3 odd 4
1280.3.g.e.1151.7 8 16.11 odd 4
1280.3.g.e.1151.8 8 16.13 even 4
1600.3.b.s.1151.1 4 20.19 odd 2
1600.3.b.s.1151.4 4 5.4 even 2
1600.3.h.n.1599.1 8 5.2 odd 4
1600.3.h.n.1599.2 8 20.3 even 4
1600.3.h.n.1599.7 8 5.3 odd 4
1600.3.h.n.1599.8 8 20.7 even 4
2880.3.e.e.2431.3 4 3.2 odd 2
2880.3.e.e.2431.4 4 12.11 even 2