Properties

Label 1216.4.a.i.1.2
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{93}) \)
Defining polynomial: \(x^{2} - x - 23\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.32183\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +21.2873 q^{5} -31.6437 q^{7} -26.0000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +21.2873 q^{5} -31.6437 q^{7} -26.0000 q^{9} +0.712698 q^{11} +38.9310 q^{13} -21.2873 q^{15} -54.1492 q^{17} +19.0000 q^{19} +31.6437 q^{21} +61.7817 q^{23} +328.149 q^{25} +53.0000 q^{27} +225.229 q^{29} -131.586 q^{31} -0.712698 q^{33} -673.608 q^{35} -94.1381 q^{37} -38.9310 q^{39} -108.873 q^{41} -205.287 q^{43} -553.470 q^{45} -523.448 q^{47} +658.321 q^{49} +54.1492 q^{51} -560.240 q^{53} +15.1714 q^{55} -19.0000 q^{57} -498.425 q^{59} -179.011 q^{61} +822.735 q^{63} +828.735 q^{65} +985.022 q^{67} -61.7817 q^{69} -904.713 q^{71} -724.149 q^{73} -328.149 q^{75} -22.5524 q^{77} -1168.20 q^{79} +649.000 q^{81} +1131.93 q^{83} -1152.69 q^{85} -225.229 q^{87} +208.370 q^{89} -1231.92 q^{91} +131.586 q^{93} +404.459 q^{95} -683.470 q^{97} -18.5302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} - 44q^{7} - 52q^{9} + 40q^{11} + 20q^{13} - 4q^{15} + 46q^{17} + 38q^{19} + 44q^{21} + 220q^{23} + 502q^{25} + 106q^{27} + 84q^{29} + 84q^{31} - 40q^{33} - 460q^{35} - 304q^{37} - 20q^{39} + 168q^{41} - 372q^{43} - 104q^{45} - 584q^{47} + 468q^{49} - 46q^{51} - 484q^{53} - 664q^{55} - 38q^{57} - 1074q^{59} - 88q^{61} + 1144q^{63} + 1156q^{65} + 1430q^{67} - 220q^{69} - 1848q^{71} - 1294q^{73} - 502q^{75} - 508q^{77} - 832q^{79} + 1298q^{81} + 528q^{83} - 2884q^{85} - 84q^{87} + 1844q^{89} - 998q^{91} - 84q^{93} + 76q^{95} - 364q^{97} - 1040q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.192450 −0.0962250 0.995360i \(-0.530677\pi\)
−0.0962250 + 0.995360i \(0.530677\pi\)
\(4\) 0 0
\(5\) 21.2873 1.90399 0.951997 0.306107i \(-0.0990266\pi\)
0.951997 + 0.306107i \(0.0990266\pi\)
\(6\) 0 0
\(7\) −31.6437 −1.70860 −0.854298 0.519783i \(-0.826013\pi\)
−0.854298 + 0.519783i \(0.826013\pi\)
\(8\) 0 0
\(9\) −26.0000 −0.962963
\(10\) 0 0
\(11\) 0.712698 0.0195352 0.00976758 0.999952i \(-0.496891\pi\)
0.00976758 + 0.999952i \(0.496891\pi\)
\(12\) 0 0
\(13\) 38.9310 0.830577 0.415289 0.909690i \(-0.363681\pi\)
0.415289 + 0.909690i \(0.363681\pi\)
\(14\) 0 0
\(15\) −21.2873 −0.366424
\(16\) 0 0
\(17\) −54.1492 −0.772536 −0.386268 0.922387i \(-0.626236\pi\)
−0.386268 + 0.922387i \(0.626236\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 31.6437 0.328820
\(22\) 0 0
\(23\) 61.7817 0.560104 0.280052 0.959985i \(-0.409648\pi\)
0.280052 + 0.959985i \(0.409648\pi\)
\(24\) 0 0
\(25\) 328.149 2.62519
\(26\) 0 0
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) 225.229 1.44221 0.721104 0.692827i \(-0.243635\pi\)
0.721104 + 0.692827i \(0.243635\pi\)
\(30\) 0 0
\(31\) −131.586 −0.762371 −0.381185 0.924499i \(-0.624484\pi\)
−0.381185 + 0.924499i \(0.624484\pi\)
\(32\) 0 0
\(33\) −0.712698 −0.00375954
\(34\) 0 0
\(35\) −673.608 −3.25316
\(36\) 0 0
\(37\) −94.1381 −0.418276 −0.209138 0.977886i \(-0.567066\pi\)
−0.209138 + 0.977886i \(0.567066\pi\)
\(38\) 0 0
\(39\) −38.9310 −0.159845
\(40\) 0 0
\(41\) −108.873 −0.414710 −0.207355 0.978266i \(-0.566486\pi\)
−0.207355 + 0.978266i \(0.566486\pi\)
\(42\) 0 0
\(43\) −205.287 −0.728047 −0.364023 0.931390i \(-0.618597\pi\)
−0.364023 + 0.931390i \(0.618597\pi\)
\(44\) 0 0
\(45\) −553.470 −1.83348
\(46\) 0 0
\(47\) −523.448 −1.62453 −0.812263 0.583292i \(-0.801764\pi\)
−0.812263 + 0.583292i \(0.801764\pi\)
\(48\) 0 0
\(49\) 658.321 1.91930
\(50\) 0 0
\(51\) 54.1492 0.148675
\(52\) 0 0
\(53\) −560.240 −1.45198 −0.725990 0.687705i \(-0.758618\pi\)
−0.725990 + 0.687705i \(0.758618\pi\)
\(54\) 0 0
\(55\) 15.1714 0.0371948
\(56\) 0 0
\(57\) −19.0000 −0.0441511
\(58\) 0 0
\(59\) −498.425 −1.09982 −0.549911 0.835223i \(-0.685338\pi\)
−0.549911 + 0.835223i \(0.685338\pi\)
\(60\) 0 0
\(61\) −179.011 −0.375738 −0.187869 0.982194i \(-0.560158\pi\)
−0.187869 + 0.982194i \(0.560158\pi\)
\(62\) 0 0
\(63\) 822.735 1.64532
\(64\) 0 0
\(65\) 828.735 1.58141
\(66\) 0 0
\(67\) 985.022 1.79611 0.898057 0.439879i \(-0.144979\pi\)
0.898057 + 0.439879i \(0.144979\pi\)
\(68\) 0 0
\(69\) −61.7817 −0.107792
\(70\) 0 0
\(71\) −904.713 −1.51225 −0.756124 0.654428i \(-0.772909\pi\)
−0.756124 + 0.654428i \(0.772909\pi\)
\(72\) 0 0
\(73\) −724.149 −1.16103 −0.580515 0.814249i \(-0.697149\pi\)
−0.580515 + 0.814249i \(0.697149\pi\)
\(74\) 0 0
\(75\) −328.149 −0.505219
\(76\) 0 0
\(77\) −22.5524 −0.0333777
\(78\) 0 0
\(79\) −1168.20 −1.66371 −0.831856 0.554991i \(-0.812722\pi\)
−0.831856 + 0.554991i \(0.812722\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 1131.93 1.49693 0.748466 0.663174i \(-0.230791\pi\)
0.748466 + 0.663174i \(0.230791\pi\)
\(84\) 0 0
\(85\) −1152.69 −1.47090
\(86\) 0 0
\(87\) −225.229 −0.277553
\(88\) 0 0
\(89\) 208.370 0.248170 0.124085 0.992272i \(-0.460400\pi\)
0.124085 + 0.992272i \(0.460400\pi\)
\(90\) 0 0
\(91\) −1231.92 −1.41912
\(92\) 0 0
\(93\) 131.586 0.146718
\(94\) 0 0
\(95\) 404.459 0.436806
\(96\) 0 0
\(97\) −683.470 −0.715421 −0.357711 0.933833i \(-0.616443\pi\)
−0.357711 + 0.933833i \(0.616443\pi\)
\(98\) 0 0
\(99\) −18.5302 −0.0188116
\(100\) 0 0
\(101\) −360.806 −0.355461 −0.177731 0.984079i \(-0.556876\pi\)
−0.177731 + 0.984079i \(0.556876\pi\)
\(102\) 0 0
\(103\) 155.514 0.148770 0.0743848 0.997230i \(-0.476301\pi\)
0.0743848 + 0.997230i \(0.476301\pi\)
\(104\) 0 0
\(105\) 673.608 0.626071
\(106\) 0 0
\(107\) −1443.09 −1.30382 −0.651910 0.758297i \(-0.726032\pi\)
−0.651910 + 0.758297i \(0.726032\pi\)
\(108\) 0 0
\(109\) −980.953 −0.862003 −0.431001 0.902351i \(-0.641840\pi\)
−0.431001 + 0.902351i \(0.641840\pi\)
\(110\) 0 0
\(111\) 94.1381 0.0804972
\(112\) 0 0
\(113\) −842.917 −0.701726 −0.350863 0.936427i \(-0.614112\pi\)
−0.350863 + 0.936427i \(0.614112\pi\)
\(114\) 0 0
\(115\) 1315.17 1.06643
\(116\) 0 0
\(117\) −1012.20 −0.799815
\(118\) 0 0
\(119\) 1713.48 1.31995
\(120\) 0 0
\(121\) −1330.49 −0.999618
\(122\) 0 0
\(123\) 108.873 0.0798110
\(124\) 0 0
\(125\) 4324.50 3.09436
\(126\) 0 0
\(127\) 489.310 0.341883 0.170942 0.985281i \(-0.445319\pi\)
0.170942 + 0.985281i \(0.445319\pi\)
\(128\) 0 0
\(129\) 205.287 0.140113
\(130\) 0 0
\(131\) 927.310 0.618469 0.309234 0.950986i \(-0.399927\pi\)
0.309234 + 0.950986i \(0.399927\pi\)
\(132\) 0 0
\(133\) −601.229 −0.391979
\(134\) 0 0
\(135\) 1128.23 0.719276
\(136\) 0 0
\(137\) 1897.02 1.18302 0.591509 0.806298i \(-0.298532\pi\)
0.591509 + 0.806298i \(0.298532\pi\)
\(138\) 0 0
\(139\) 590.713 0.360458 0.180229 0.983625i \(-0.442316\pi\)
0.180229 + 0.983625i \(0.442316\pi\)
\(140\) 0 0
\(141\) 523.448 0.312640
\(142\) 0 0
\(143\) 27.7460 0.0162255
\(144\) 0 0
\(145\) 4794.53 2.74596
\(146\) 0 0
\(147\) −658.321 −0.369370
\(148\) 0 0
\(149\) −824.641 −0.453404 −0.226702 0.973964i \(-0.572794\pi\)
−0.226702 + 0.973964i \(0.572794\pi\)
\(150\) 0 0
\(151\) −1424.71 −0.767822 −0.383911 0.923370i \(-0.625423\pi\)
−0.383911 + 0.923370i \(0.625423\pi\)
\(152\) 0 0
\(153\) 1407.88 0.743924
\(154\) 0 0
\(155\) −2801.10 −1.45155
\(156\) 0 0
\(157\) −2499.51 −1.27059 −0.635295 0.772270i \(-0.719121\pi\)
−0.635295 + 0.772270i \(0.719121\pi\)
\(158\) 0 0
\(159\) 560.240 0.279434
\(160\) 0 0
\(161\) −1955.00 −0.956991
\(162\) 0 0
\(163\) −1102.62 −0.529841 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(164\) 0 0
\(165\) −15.1714 −0.00715815
\(166\) 0 0
\(167\) −161.613 −0.0748860 −0.0374430 0.999299i \(-0.511921\pi\)
−0.0374430 + 0.999299i \(0.511921\pi\)
\(168\) 0 0
\(169\) −681.381 −0.310142
\(170\) 0 0
\(171\) −494.000 −0.220919
\(172\) 0 0
\(173\) 1829.59 0.804051 0.402026 0.915628i \(-0.368306\pi\)
0.402026 + 0.915628i \(0.368306\pi\)
\(174\) 0 0
\(175\) −10383.8 −4.48540
\(176\) 0 0
\(177\) 498.425 0.211661
\(178\) 0 0
\(179\) −1048.00 −0.437604 −0.218802 0.975769i \(-0.570215\pi\)
−0.218802 + 0.975769i \(0.570215\pi\)
\(180\) 0 0
\(181\) −897.354 −0.368507 −0.184254 0.982879i \(-0.558987\pi\)
−0.184254 + 0.982879i \(0.558987\pi\)
\(182\) 0 0
\(183\) 179.011 0.0723108
\(184\) 0 0
\(185\) −2003.95 −0.796395
\(186\) 0 0
\(187\) −38.5921 −0.0150916
\(188\) 0 0
\(189\) −1677.11 −0.645461
\(190\) 0 0
\(191\) −4113.98 −1.55852 −0.779259 0.626702i \(-0.784404\pi\)
−0.779259 + 0.626702i \(0.784404\pi\)
\(192\) 0 0
\(193\) 3511.87 1.30979 0.654897 0.755718i \(-0.272712\pi\)
0.654897 + 0.755718i \(0.272712\pi\)
\(194\) 0 0
\(195\) −828.735 −0.304343
\(196\) 0 0
\(197\) 2488.28 0.899910 0.449955 0.893051i \(-0.351440\pi\)
0.449955 + 0.893051i \(0.351440\pi\)
\(198\) 0 0
\(199\) 386.556 0.137700 0.0688499 0.997627i \(-0.478067\pi\)
0.0688499 + 0.997627i \(0.478067\pi\)
\(200\) 0 0
\(201\) −985.022 −0.345662
\(202\) 0 0
\(203\) −7127.08 −2.46415
\(204\) 0 0
\(205\) −2317.61 −0.789605
\(206\) 0 0
\(207\) −1606.33 −0.539359
\(208\) 0 0
\(209\) 13.5413 0.00448167
\(210\) 0 0
\(211\) −4851.06 −1.58275 −0.791376 0.611330i \(-0.790635\pi\)
−0.791376 + 0.611330i \(0.790635\pi\)
\(212\) 0 0
\(213\) 904.713 0.291032
\(214\) 0 0
\(215\) −4370.01 −1.38620
\(216\) 0 0
\(217\) 4163.85 1.30258
\(218\) 0 0
\(219\) 724.149 0.223440
\(220\) 0 0
\(221\) −2108.08 −0.641651
\(222\) 0 0
\(223\) 802.187 0.240890 0.120445 0.992720i \(-0.461568\pi\)
0.120445 + 0.992720i \(0.461568\pi\)
\(224\) 0 0
\(225\) −8531.88 −2.52796
\(226\) 0 0
\(227\) 4889.70 1.42969 0.714847 0.699281i \(-0.246496\pi\)
0.714847 + 0.699281i \(0.246496\pi\)
\(228\) 0 0
\(229\) −569.132 −0.164233 −0.0821163 0.996623i \(-0.526168\pi\)
−0.0821163 + 0.996623i \(0.526168\pi\)
\(230\) 0 0
\(231\) 22.5524 0.00642354
\(232\) 0 0
\(233\) −3835.88 −1.07853 −0.539264 0.842137i \(-0.681297\pi\)
−0.539264 + 0.842137i \(0.681297\pi\)
\(234\) 0 0
\(235\) −11142.8 −3.09309
\(236\) 0 0
\(237\) 1168.20 0.320182
\(238\) 0 0
\(239\) −527.821 −0.142853 −0.0714266 0.997446i \(-0.522755\pi\)
−0.0714266 + 0.997446i \(0.522755\pi\)
\(240\) 0 0
\(241\) −5523.03 −1.47622 −0.738111 0.674679i \(-0.764282\pi\)
−0.738111 + 0.674679i \(0.764282\pi\)
\(242\) 0 0
\(243\) −2080.00 −0.549103
\(244\) 0 0
\(245\) 14013.9 3.65434
\(246\) 0 0
\(247\) 739.688 0.190547
\(248\) 0 0
\(249\) −1131.93 −0.288085
\(250\) 0 0
\(251\) 7573.11 1.90442 0.952212 0.305438i \(-0.0988030\pi\)
0.952212 + 0.305438i \(0.0988030\pi\)
\(252\) 0 0
\(253\) 44.0318 0.0109417
\(254\) 0 0
\(255\) 1152.69 0.283076
\(256\) 0 0
\(257\) −6349.60 −1.54116 −0.770578 0.637346i \(-0.780032\pi\)
−0.770578 + 0.637346i \(0.780032\pi\)
\(258\) 0 0
\(259\) 2978.87 0.714665
\(260\) 0 0
\(261\) −5855.96 −1.38879
\(262\) 0 0
\(263\) 3915.06 0.917921 0.458960 0.888457i \(-0.348222\pi\)
0.458960 + 0.888457i \(0.348222\pi\)
\(264\) 0 0
\(265\) −11926.0 −2.76456
\(266\) 0 0
\(267\) −208.370 −0.0477604
\(268\) 0 0
\(269\) 7171.13 1.62540 0.812699 0.582684i \(-0.197998\pi\)
0.812699 + 0.582684i \(0.197998\pi\)
\(270\) 0 0
\(271\) 1100.04 0.246577 0.123289 0.992371i \(-0.460656\pi\)
0.123289 + 0.992371i \(0.460656\pi\)
\(272\) 0 0
\(273\) 1231.92 0.273110
\(274\) 0 0
\(275\) 233.871 0.0512836
\(276\) 0 0
\(277\) −1361.51 −0.295326 −0.147663 0.989038i \(-0.547175\pi\)
−0.147663 + 0.989038i \(0.547175\pi\)
\(278\) 0 0
\(279\) 3421.23 0.734135
\(280\) 0 0
\(281\) 4438.51 0.942275 0.471138 0.882060i \(-0.343843\pi\)
0.471138 + 0.882060i \(0.343843\pi\)
\(282\) 0 0
\(283\) −9190.64 −1.93048 −0.965241 0.261360i \(-0.915829\pi\)
−0.965241 + 0.261360i \(0.915829\pi\)
\(284\) 0 0
\(285\) −404.459 −0.0840634
\(286\) 0 0
\(287\) 3445.14 0.708572
\(288\) 0 0
\(289\) −1980.86 −0.403188
\(290\) 0 0
\(291\) 683.470 0.137683
\(292\) 0 0
\(293\) −3316.50 −0.661269 −0.330635 0.943759i \(-0.607263\pi\)
−0.330635 + 0.943759i \(0.607263\pi\)
\(294\) 0 0
\(295\) −10610.1 −2.09405
\(296\) 0 0
\(297\) 37.7730 0.00737984
\(298\) 0 0
\(299\) 2405.22 0.465209
\(300\) 0 0
\(301\) 6496.04 1.24394
\(302\) 0 0
\(303\) 360.806 0.0684085
\(304\) 0 0
\(305\) −3810.66 −0.715403
\(306\) 0 0
\(307\) 1574.75 0.292755 0.146378 0.989229i \(-0.453239\pi\)
0.146378 + 0.989229i \(0.453239\pi\)
\(308\) 0 0
\(309\) −155.514 −0.0286307
\(310\) 0 0
\(311\) −4614.64 −0.841389 −0.420695 0.907202i \(-0.638214\pi\)
−0.420695 + 0.907202i \(0.638214\pi\)
\(312\) 0 0
\(313\) 4738.42 0.855690 0.427845 0.903852i \(-0.359273\pi\)
0.427845 + 0.903852i \(0.359273\pi\)
\(314\) 0 0
\(315\) 17513.8 3.13267
\(316\) 0 0
\(317\) −1039.66 −0.184206 −0.0921028 0.995750i \(-0.529359\pi\)
−0.0921028 + 0.995750i \(0.529359\pi\)
\(318\) 0 0
\(319\) 160.521 0.0281738
\(320\) 0 0
\(321\) 1443.09 0.250920
\(322\) 0 0
\(323\) −1028.83 −0.177232
\(324\) 0 0
\(325\) 12775.2 2.18043
\(326\) 0 0
\(327\) 980.953 0.165892
\(328\) 0 0
\(329\) 16563.8 2.77566
\(330\) 0 0
\(331\) 4634.52 0.769597 0.384798 0.923001i \(-0.374271\pi\)
0.384798 + 0.923001i \(0.374271\pi\)
\(332\) 0 0
\(333\) 2447.59 0.402784
\(334\) 0 0
\(335\) 20968.5 3.41979
\(336\) 0 0
\(337\) 5219.30 0.843660 0.421830 0.906675i \(-0.361388\pi\)
0.421830 + 0.906675i \(0.361388\pi\)
\(338\) 0 0
\(339\) 842.917 0.135047
\(340\) 0 0
\(341\) −93.7809 −0.0148930
\(342\) 0 0
\(343\) −9977.90 −1.57072
\(344\) 0 0
\(345\) −1315.17 −0.205235
\(346\) 0 0
\(347\) 7019.77 1.08600 0.542999 0.839734i \(-0.317289\pi\)
0.542999 + 0.839734i \(0.317289\pi\)
\(348\) 0 0
\(349\) −6552.97 −1.00508 −0.502539 0.864554i \(-0.667601\pi\)
−0.502539 + 0.864554i \(0.667601\pi\)
\(350\) 0 0
\(351\) 2063.34 0.313769
\(352\) 0 0
\(353\) −8837.11 −1.33244 −0.666221 0.745754i \(-0.732089\pi\)
−0.666221 + 0.745754i \(0.732089\pi\)
\(354\) 0 0
\(355\) −19258.9 −2.87931
\(356\) 0 0
\(357\) −1713.48 −0.254025
\(358\) 0 0
\(359\) −2521.14 −0.370643 −0.185321 0.982678i \(-0.559333\pi\)
−0.185321 + 0.982678i \(0.559333\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 1330.49 0.192377
\(364\) 0 0
\(365\) −15415.2 −2.21060
\(366\) 0 0
\(367\) 5652.41 0.803960 0.401980 0.915648i \(-0.368322\pi\)
0.401980 + 0.915648i \(0.368322\pi\)
\(368\) 0 0
\(369\) 2830.70 0.399350
\(370\) 0 0
\(371\) 17728.1 2.48085
\(372\) 0 0
\(373\) −6864.49 −0.952895 −0.476447 0.879203i \(-0.658076\pi\)
−0.476447 + 0.879203i \(0.658076\pi\)
\(374\) 0 0
\(375\) −4324.50 −0.595510
\(376\) 0 0
\(377\) 8768.39 1.19787
\(378\) 0 0
\(379\) 6393.09 0.866467 0.433234 0.901282i \(-0.357373\pi\)
0.433234 + 0.901282i \(0.357373\pi\)
\(380\) 0 0
\(381\) −489.310 −0.0657955
\(382\) 0 0
\(383\) −7948.16 −1.06040 −0.530198 0.847874i \(-0.677882\pi\)
−0.530198 + 0.847874i \(0.677882\pi\)
\(384\) 0 0
\(385\) −480.079 −0.0635509
\(386\) 0 0
\(387\) 5337.47 0.701082
\(388\) 0 0
\(389\) 8896.48 1.15956 0.579781 0.814773i \(-0.303138\pi\)
0.579781 + 0.814773i \(0.303138\pi\)
\(390\) 0 0
\(391\) −3345.43 −0.432700
\(392\) 0 0
\(393\) −927.310 −0.119024
\(394\) 0 0
\(395\) −24867.9 −3.16770
\(396\) 0 0
\(397\) −2865.66 −0.362276 −0.181138 0.983458i \(-0.557978\pi\)
−0.181138 + 0.983458i \(0.557978\pi\)
\(398\) 0 0
\(399\) 601.229 0.0754364
\(400\) 0 0
\(401\) 10635.1 1.32442 0.662208 0.749320i \(-0.269619\pi\)
0.662208 + 0.749320i \(0.269619\pi\)
\(402\) 0 0
\(403\) −5122.76 −0.633208
\(404\) 0 0
\(405\) 13815.5 1.69505
\(406\) 0 0
\(407\) −67.0921 −0.00817108
\(408\) 0 0
\(409\) 9261.74 1.11972 0.559858 0.828589i \(-0.310856\pi\)
0.559858 + 0.828589i \(0.310856\pi\)
\(410\) 0 0
\(411\) −1897.02 −0.227672
\(412\) 0 0
\(413\) 15772.0 1.87915
\(414\) 0 0
\(415\) 24095.7 2.85015
\(416\) 0 0
\(417\) −590.713 −0.0693701
\(418\) 0 0
\(419\) 3241.64 0.377959 0.188979 0.981981i \(-0.439482\pi\)
0.188979 + 0.981981i \(0.439482\pi\)
\(420\) 0 0
\(421\) 3075.85 0.356075 0.178038 0.984024i \(-0.443025\pi\)
0.178038 + 0.984024i \(0.443025\pi\)
\(422\) 0 0
\(423\) 13609.6 1.56436
\(424\) 0 0
\(425\) −17769.0 −2.02806
\(426\) 0 0
\(427\) 5664.57 0.641985
\(428\) 0 0
\(429\) −27.7460 −0.00312259
\(430\) 0 0
\(431\) 4730.26 0.528651 0.264326 0.964433i \(-0.414851\pi\)
0.264326 + 0.964433i \(0.414851\pi\)
\(432\) 0 0
\(433\) −8846.65 −0.981854 −0.490927 0.871201i \(-0.663342\pi\)
−0.490927 + 0.871201i \(0.663342\pi\)
\(434\) 0 0
\(435\) −4794.53 −0.528460
\(436\) 0 0
\(437\) 1173.85 0.128497
\(438\) 0 0
\(439\) 14485.9 1.57489 0.787443 0.616387i \(-0.211404\pi\)
0.787443 + 0.616387i \(0.211404\pi\)
\(440\) 0 0
\(441\) −17116.3 −1.84822
\(442\) 0 0
\(443\) 1271.64 0.136383 0.0681914 0.997672i \(-0.478277\pi\)
0.0681914 + 0.997672i \(0.478277\pi\)
\(444\) 0 0
\(445\) 4435.63 0.472515
\(446\) 0 0
\(447\) 824.641 0.0872577
\(448\) 0 0
\(449\) 4706.71 0.494706 0.247353 0.968925i \(-0.420439\pi\)
0.247353 + 0.968925i \(0.420439\pi\)
\(450\) 0 0
\(451\) −77.5936 −0.00810142
\(452\) 0 0
\(453\) 1424.71 0.147767
\(454\) 0 0
\(455\) −26224.2 −2.70200
\(456\) 0 0
\(457\) 4850.91 0.496534 0.248267 0.968692i \(-0.420139\pi\)
0.248267 + 0.968692i \(0.420139\pi\)
\(458\) 0 0
\(459\) −2869.91 −0.291843
\(460\) 0 0
\(461\) −17447.0 −1.76267 −0.881333 0.472495i \(-0.843353\pi\)
−0.881333 + 0.472495i \(0.843353\pi\)
\(462\) 0 0
\(463\) 5892.21 0.591435 0.295717 0.955275i \(-0.404441\pi\)
0.295717 + 0.955275i \(0.404441\pi\)
\(464\) 0 0
\(465\) 2801.10 0.279351
\(466\) 0 0
\(467\) 11880.9 1.17727 0.588634 0.808400i \(-0.299666\pi\)
0.588634 + 0.808400i \(0.299666\pi\)
\(468\) 0 0
\(469\) −31169.7 −3.06883
\(470\) 0 0
\(471\) 2499.51 0.244525
\(472\) 0 0
\(473\) −146.308 −0.0142225
\(474\) 0 0
\(475\) 6234.83 0.602261
\(476\) 0 0
\(477\) 14566.3 1.39820
\(478\) 0 0
\(479\) −9454.03 −0.901807 −0.450904 0.892573i \(-0.648898\pi\)
−0.450904 + 0.892573i \(0.648898\pi\)
\(480\) 0 0
\(481\) −3664.89 −0.347410
\(482\) 0 0
\(483\) 1955.00 0.184173
\(484\) 0 0
\(485\) −14549.2 −1.36216
\(486\) 0 0
\(487\) −17015.1 −1.58322 −0.791608 0.611029i \(-0.790756\pi\)
−0.791608 + 0.611029i \(0.790756\pi\)
\(488\) 0 0
\(489\) 1102.62 0.101968
\(490\) 0 0
\(491\) −4075.78 −0.374618 −0.187309 0.982301i \(-0.559977\pi\)
−0.187309 + 0.982301i \(0.559977\pi\)
\(492\) 0 0
\(493\) −12196.0 −1.11416
\(494\) 0 0
\(495\) −394.457 −0.0358172
\(496\) 0 0
\(497\) 28628.4 2.58382
\(498\) 0 0
\(499\) 6632.38 0.595002 0.297501 0.954721i \(-0.403847\pi\)
0.297501 + 0.954721i \(0.403847\pi\)
\(500\) 0 0
\(501\) 161.613 0.0144118
\(502\) 0 0
\(503\) 9052.55 0.802452 0.401226 0.915979i \(-0.368584\pi\)
0.401226 + 0.915979i \(0.368584\pi\)
\(504\) 0 0
\(505\) −7680.59 −0.676796
\(506\) 0 0
\(507\) 681.381 0.0596868
\(508\) 0 0
\(509\) −8676.14 −0.755527 −0.377763 0.925902i \(-0.623307\pi\)
−0.377763 + 0.925902i \(0.623307\pi\)
\(510\) 0 0
\(511\) 22914.7 1.98373
\(512\) 0 0
\(513\) 1007.00 0.0866669
\(514\) 0 0
\(515\) 3310.48 0.283257
\(516\) 0 0
\(517\) −373.060 −0.0317353
\(518\) 0 0
\(519\) −1829.59 −0.154740
\(520\) 0 0
\(521\) −2967.88 −0.249568 −0.124784 0.992184i \(-0.539824\pi\)
−0.124784 + 0.992184i \(0.539824\pi\)
\(522\) 0 0
\(523\) −9815.59 −0.820661 −0.410330 0.911937i \(-0.634587\pi\)
−0.410330 + 0.911937i \(0.634587\pi\)
\(524\) 0 0
\(525\) 10383.8 0.863215
\(526\) 0 0
\(527\) 7125.26 0.588959
\(528\) 0 0
\(529\) −8350.02 −0.686284
\(530\) 0 0
\(531\) 12959.1 1.05909
\(532\) 0 0
\(533\) −4238.53 −0.344449
\(534\) 0 0
\(535\) −30719.5 −2.48246
\(536\) 0 0
\(537\) 1048.00 0.0842170
\(538\) 0 0
\(539\) 469.184 0.0374939
\(540\) 0 0
\(541\) −4090.55 −0.325077 −0.162538 0.986702i \(-0.551968\pi\)
−0.162538 + 0.986702i \(0.551968\pi\)
\(542\) 0 0
\(543\) 897.354 0.0709192
\(544\) 0 0
\(545\) −20881.8 −1.64125
\(546\) 0 0
\(547\) −2001.12 −0.156420 −0.0782100 0.996937i \(-0.524920\pi\)
−0.0782100 + 0.996937i \(0.524920\pi\)
\(548\) 0 0
\(549\) 4654.29 0.361822
\(550\) 0 0
\(551\) 4279.36 0.330865
\(552\) 0 0
\(553\) 36966.3 2.84261
\(554\) 0 0
\(555\) 2003.95 0.153266
\(556\) 0 0
\(557\) 6727.24 0.511746 0.255873 0.966710i \(-0.417637\pi\)
0.255873 + 0.966710i \(0.417637\pi\)
\(558\) 0 0
\(559\) −7992.03 −0.604699
\(560\) 0 0
\(561\) 38.5921 0.00290438
\(562\) 0 0
\(563\) 2255.21 0.168820 0.0844100 0.996431i \(-0.473099\pi\)
0.0844100 + 0.996431i \(0.473099\pi\)
\(564\) 0 0
\(565\) −17943.4 −1.33608
\(566\) 0 0
\(567\) −20536.7 −1.52110
\(568\) 0 0
\(569\) 12206.7 0.899349 0.449675 0.893192i \(-0.351540\pi\)
0.449675 + 0.893192i \(0.351540\pi\)
\(570\) 0 0
\(571\) 19976.5 1.46408 0.732042 0.681259i \(-0.238567\pi\)
0.732042 + 0.681259i \(0.238567\pi\)
\(572\) 0 0
\(573\) 4113.98 0.299937
\(574\) 0 0
\(575\) 20273.6 1.47038
\(576\) 0 0
\(577\) 7025.87 0.506916 0.253458 0.967346i \(-0.418432\pi\)
0.253458 + 0.967346i \(0.418432\pi\)
\(578\) 0 0
\(579\) −3511.87 −0.252070
\(580\) 0 0
\(581\) −35818.4 −2.55765
\(582\) 0 0
\(583\) −399.283 −0.0283647
\(584\) 0 0
\(585\) −21547.1 −1.52284
\(586\) 0 0
\(587\) −15464.1 −1.08735 −0.543674 0.839297i \(-0.682967\pi\)
−0.543674 + 0.839297i \(0.682967\pi\)
\(588\) 0 0
\(589\) −2500.13 −0.174900
\(590\) 0 0
\(591\) −2488.28 −0.173188
\(592\) 0 0
\(593\) 15847.9 1.09746 0.548731 0.835999i \(-0.315111\pi\)
0.548731 + 0.835999i \(0.315111\pi\)
\(594\) 0 0
\(595\) 36475.3 2.51318
\(596\) 0 0
\(597\) −386.556 −0.0265003
\(598\) 0 0
\(599\) 19135.1 1.30524 0.652621 0.757684i \(-0.273669\pi\)
0.652621 + 0.757684i \(0.273669\pi\)
\(600\) 0 0
\(601\) −2997.59 −0.203451 −0.101726 0.994812i \(-0.532436\pi\)
−0.101726 + 0.994812i \(0.532436\pi\)
\(602\) 0 0
\(603\) −25610.6 −1.72959
\(604\) 0 0
\(605\) −28322.6 −1.90327
\(606\) 0 0
\(607\) 18978.6 1.26906 0.634528 0.772900i \(-0.281195\pi\)
0.634528 + 0.772900i \(0.281195\pi\)
\(608\) 0 0
\(609\) 7127.08 0.474226
\(610\) 0 0
\(611\) −20378.3 −1.34929
\(612\) 0 0
\(613\) 11248.7 0.741156 0.370578 0.928801i \(-0.379160\pi\)
0.370578 + 0.928801i \(0.379160\pi\)
\(614\) 0 0
\(615\) 2317.61 0.151960
\(616\) 0 0
\(617\) −4803.53 −0.313425 −0.156712 0.987644i \(-0.550090\pi\)
−0.156712 + 0.987644i \(0.550090\pi\)
\(618\) 0 0
\(619\) −11306.6 −0.734168 −0.367084 0.930188i \(-0.619644\pi\)
−0.367084 + 0.930188i \(0.619644\pi\)
\(620\) 0 0
\(621\) 3274.43 0.211592
\(622\) 0 0
\(623\) −6593.58 −0.424023
\(624\) 0 0
\(625\) 51038.3 3.26645
\(626\) 0 0
\(627\) −13.5413 −0.000862498 0
\(628\) 0 0
\(629\) 5097.50 0.323133
\(630\) 0 0
\(631\) −28148.8 −1.77589 −0.887945 0.459950i \(-0.847867\pi\)
−0.887945 + 0.459950i \(0.847867\pi\)
\(632\) 0 0
\(633\) 4851.06 0.304601
\(634\) 0 0
\(635\) 10416.1 0.650944
\(636\) 0 0
\(637\) 25629.0 1.59413
\(638\) 0 0
\(639\) 23522.5 1.45624
\(640\) 0 0
\(641\) −28222.5 −1.73904 −0.869519 0.493900i \(-0.835571\pi\)
−0.869519 + 0.493900i \(0.835571\pi\)
\(642\) 0 0
\(643\) 11441.5 0.701723 0.350861 0.936427i \(-0.385889\pi\)
0.350861 + 0.936427i \(0.385889\pi\)
\(644\) 0 0
\(645\) 4370.01 0.266774
\(646\) 0 0
\(647\) 26346.6 1.60091 0.800457 0.599390i \(-0.204590\pi\)
0.800457 + 0.599390i \(0.204590\pi\)
\(648\) 0 0
\(649\) −355.227 −0.0214852
\(650\) 0 0
\(651\) −4163.85 −0.250682
\(652\) 0 0
\(653\) −29759.4 −1.78342 −0.891710 0.452607i \(-0.850494\pi\)
−0.891710 + 0.452607i \(0.850494\pi\)
\(654\) 0 0
\(655\) 19739.9 1.17756
\(656\) 0 0
\(657\) 18827.9 1.11803
\(658\) 0 0
\(659\) 15959.4 0.943386 0.471693 0.881763i \(-0.343643\pi\)
0.471693 + 0.881763i \(0.343643\pi\)
\(660\) 0 0
\(661\) −3165.29 −0.186256 −0.0931282 0.995654i \(-0.529687\pi\)
−0.0931282 + 0.995654i \(0.529687\pi\)
\(662\) 0 0
\(663\) 2108.08 0.123486
\(664\) 0 0
\(665\) −12798.6 −0.746326
\(666\) 0 0
\(667\) 13915.1 0.807786
\(668\) 0 0
\(669\) −802.187 −0.0463593
\(670\) 0 0
\(671\) −127.581 −0.00734010
\(672\) 0 0
\(673\) 29152.9 1.66978 0.834890 0.550416i \(-0.185531\pi\)
0.834890 + 0.550416i \(0.185531\pi\)
\(674\) 0 0
\(675\) 17391.9 0.991726
\(676\) 0 0
\(677\) 15830.4 0.898687 0.449343 0.893359i \(-0.351658\pi\)
0.449343 + 0.893359i \(0.351658\pi\)
\(678\) 0 0
\(679\) 21627.5 1.22237
\(680\) 0 0
\(681\) −4889.70 −0.275145
\(682\) 0 0
\(683\) 9998.31 0.560139 0.280069 0.959980i \(-0.409643\pi\)
0.280069 + 0.959980i \(0.409643\pi\)
\(684\) 0 0
\(685\) 40382.5 2.25246
\(686\) 0 0
\(687\) 569.132 0.0316066
\(688\) 0 0
\(689\) −21810.7 −1.20598
\(690\) 0 0
\(691\) 7331.53 0.403625 0.201812 0.979424i \(-0.435317\pi\)
0.201812 + 0.979424i \(0.435317\pi\)
\(692\) 0 0
\(693\) 586.362 0.0321415
\(694\) 0 0
\(695\) 12574.7 0.686309
\(696\) 0 0
\(697\) 5895.39 0.320378
\(698\) 0 0
\(699\) 3835.88 0.207563
\(700\) 0 0
\(701\) −9928.34 −0.534933 −0.267467 0.963567i \(-0.586187\pi\)
−0.267467 + 0.963567i \(0.586187\pi\)
\(702\) 0 0
\(703\) −1788.62 −0.0959591
\(704\) 0 0
\(705\) 11142.8 0.595265
\(706\) 0 0
\(707\) 11417.2 0.607340
\(708\) 0 0
\(709\) −1613.88 −0.0854874 −0.0427437 0.999086i \(-0.513610\pi\)
−0.0427437 + 0.999086i \(0.513610\pi\)
\(710\) 0 0
\(711\) 30373.3 1.60209
\(712\) 0 0
\(713\) −8129.60 −0.427007
\(714\) 0 0
\(715\) 590.638 0.0308932
\(716\) 0 0
\(717\) 527.821 0.0274921
\(718\) 0 0
\(719\) −26007.1 −1.34896 −0.674480 0.738293i \(-0.735632\pi\)
−0.674480 + 0.738293i \(0.735632\pi\)
\(720\) 0 0
\(721\) −4921.04 −0.254187
\(722\) 0 0
\(723\) 5523.03 0.284099
\(724\) 0 0
\(725\) 73908.8 3.78608
\(726\) 0 0
\(727\) 18096.2 0.923180 0.461590 0.887093i \(-0.347279\pi\)
0.461590 + 0.887093i \(0.347279\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 0 0
\(731\) 11116.1 0.562442
\(732\) 0 0
\(733\) −25273.4 −1.27353 −0.636764 0.771059i \(-0.719727\pi\)
−0.636764 + 0.771059i \(0.719727\pi\)
\(734\) 0 0
\(735\) −14013.9 −0.703278
\(736\) 0 0
\(737\) 702.024 0.0350874
\(738\) 0 0
\(739\) −15373.0 −0.765229 −0.382615 0.923908i \(-0.624976\pi\)
−0.382615 + 0.923908i \(0.624976\pi\)
\(740\) 0 0
\(741\) −739.688 −0.0366709
\(742\) 0 0
\(743\) −25357.0 −1.25203 −0.626016 0.779810i \(-0.715315\pi\)
−0.626016 + 0.779810i \(0.715315\pi\)
\(744\) 0 0
\(745\) −17554.4 −0.863279
\(746\) 0 0
\(747\) −29430.1 −1.44149
\(748\) 0 0
\(749\) 45664.6 2.22770
\(750\) 0 0
\(751\) −2019.97 −0.0981486 −0.0490743 0.998795i \(-0.515627\pi\)
−0.0490743 + 0.998795i \(0.515627\pi\)
\(752\) 0 0
\(753\) −7573.11 −0.366506
\(754\) 0 0
\(755\) −30328.2 −1.46193
\(756\) 0 0
\(757\) −38997.9 −1.87239 −0.936197 0.351477i \(-0.885680\pi\)
−0.936197 + 0.351477i \(0.885680\pi\)
\(758\) 0 0
\(759\) −44.0318 −0.00210573
\(760\) 0 0
\(761\) −4691.83 −0.223493 −0.111747 0.993737i \(-0.535645\pi\)
−0.111747 + 0.993737i \(0.535645\pi\)
\(762\) 0 0
\(763\) 31040.9 1.47281
\(764\) 0 0
\(765\) 29970.0 1.41643
\(766\) 0 0
\(767\) −19404.2 −0.913487
\(768\) 0 0
\(769\) −18338.5 −0.859952 −0.429976 0.902840i \(-0.641478\pi\)
−0.429976 + 0.902840i \(0.641478\pi\)
\(770\) 0 0
\(771\) 6349.60 0.296596
\(772\) 0 0
\(773\) 8104.70 0.377110 0.188555 0.982063i \(-0.439620\pi\)
0.188555 + 0.982063i \(0.439620\pi\)
\(774\) 0 0
\(775\) −43179.7 −2.00137
\(776\) 0 0
\(777\) −2978.87 −0.137537
\(778\) 0 0
\(779\) −2068.59 −0.0951410
\(780\) 0 0
\(781\) −644.787 −0.0295420
\(782\) 0 0
\(783\) 11937.2 0.544827
\(784\) 0 0
\(785\) −53207.8 −2.41920
\(786\) 0 0
\(787\) 31649.8 1.43354 0.716769 0.697311i \(-0.245620\pi\)
0.716769 + 0.697311i \(0.245620\pi\)
\(788\) 0 0
\(789\) −3915.06 −0.176654
\(790\) 0 0
\(791\) 26673.0 1.19897
\(792\) 0 0
\(793\) −6969.07 −0.312079
\(794\) 0 0
\(795\) 11926.0 0.532040
\(796\) 0 0
\(797\) −1846.41 −0.0820615 −0.0410308 0.999158i \(-0.513064\pi\)
−0.0410308 + 0.999158i \(0.513064\pi\)
\(798\) 0 0
\(799\) 28344.3 1.25500
\(800\) 0 0
\(801\) −5417.62 −0.238979
\(802\) 0 0
\(803\) −516.100 −0.0226809
\(804\) 0 0
\(805\) −41616.7 −1.82211
\(806\) 0 0
\(807\) −7171.13 −0.312808
\(808\) 0 0
\(809\) −1277.18 −0.0555047 −0.0277523 0.999615i \(-0.508835\pi\)
−0.0277523 + 0.999615i \(0.508835\pi\)
\(810\) 0 0
\(811\) −7935.98 −0.343613 −0.171806 0.985131i \(-0.554960\pi\)
−0.171806 + 0.985131i \(0.554960\pi\)
\(812\) 0 0
\(813\) −1100.04 −0.0474538
\(814\) 0 0
\(815\) −23471.9 −1.00882
\(816\) 0 0
\(817\) −3900.46 −0.167025
\(818\) 0 0
\(819\) 32029.9 1.36656
\(820\) 0 0
\(821\) 19972.8 0.849034 0.424517 0.905420i \(-0.360444\pi\)
0.424517 + 0.905420i \(0.360444\pi\)
\(822\) 0 0
\(823\) 10986.0 0.465307 0.232654 0.972560i \(-0.425259\pi\)
0.232654 + 0.972560i \(0.425259\pi\)
\(824\) 0 0
\(825\) −233.871 −0.00986953
\(826\) 0 0
\(827\) −9581.30 −0.402871 −0.201436 0.979502i \(-0.564561\pi\)
−0.201436 + 0.979502i \(0.564561\pi\)
\(828\) 0 0
\(829\) 23501.8 0.984621 0.492311 0.870420i \(-0.336152\pi\)
0.492311 + 0.870420i \(0.336152\pi\)
\(830\) 0 0
\(831\) 1361.51 0.0568354
\(832\) 0 0
\(833\) −35647.5 −1.48273
\(834\) 0 0
\(835\) −3440.30 −0.142583
\(836\) 0 0
\(837\) −6974.04 −0.288003
\(838\) 0 0
\(839\) −14939.3 −0.614733 −0.307367 0.951591i \(-0.599448\pi\)
−0.307367 + 0.951591i \(0.599448\pi\)
\(840\) 0 0
\(841\) 26339.3 1.07997
\(842\) 0 0
\(843\) −4438.51 −0.181341
\(844\) 0 0
\(845\) −14504.8 −0.590508
\(846\) 0 0
\(847\) 42101.6 1.70794
\(848\) 0 0
\(849\) 9190.64 0.371522
\(850\) 0 0
\(851\) −5816.02 −0.234278
\(852\) 0 0
\(853\) 14952.6 0.600196 0.300098 0.953908i \(-0.402981\pi\)
0.300098 + 0.953908i \(0.402981\pi\)
\(854\) 0 0
\(855\) −10515.9 −0.420628
\(856\) 0 0
\(857\) 9152.49 0.364811 0.182406 0.983223i \(-0.441612\pi\)
0.182406 + 0.983223i \(0.441612\pi\)
\(858\) 0 0
\(859\) −37436.7 −1.48699 −0.743494 0.668743i \(-0.766833\pi\)
−0.743494 + 0.668743i \(0.766833\pi\)
\(860\) 0 0
\(861\) −3445.14 −0.136365
\(862\) 0 0
\(863\) −29723.0 −1.17240 −0.586200 0.810166i \(-0.699377\pi\)
−0.586200 + 0.810166i \(0.699377\pi\)
\(864\) 0 0
\(865\) 38946.9 1.53091
\(866\) 0 0
\(867\) 1980.86 0.0775936
\(868\) 0 0
\(869\) −832.578 −0.0325009
\(870\) 0 0
\(871\) 38347.9 1.49181
\(872\) 0 0
\(873\) 17770.2 0.688924
\(874\) 0 0
\(875\) −136843. −5.28701
\(876\) 0 0
\(877\) −42664.1 −1.64272 −0.821360 0.570411i \(-0.806784\pi\)
−0.821360 + 0.570411i \(0.806784\pi\)
\(878\) 0 0
\(879\) 3316.50 0.127261
\(880\) 0 0
\(881\) −42198.0 −1.61372 −0.806859 0.590743i \(-0.798835\pi\)
−0.806859 + 0.590743i \(0.798835\pi\)
\(882\) 0 0
\(883\) 41031.1 1.56377 0.781884 0.623424i \(-0.214259\pi\)
0.781884 + 0.623424i \(0.214259\pi\)
\(884\) 0 0
\(885\) 10610.1 0.403001
\(886\) 0 0
\(887\) −42729.3 −1.61749 −0.808743 0.588162i \(-0.799852\pi\)
−0.808743 + 0.588162i \(0.799852\pi\)
\(888\) 0 0
\(889\) −15483.5 −0.584141
\(890\) 0 0
\(891\) 462.541 0.0173914
\(892\) 0 0
\(893\) −9945.50 −0.372692
\(894\) 0 0
\(895\) −22309.1 −0.833196
\(896\) 0 0
\(897\) −2405.22 −0.0895296
\(898\) 0 0
\(899\) −29637.0 −1.09950
\(900\) 0 0
\(901\) 30336.6 1.12171
\(902\) 0 0
\(903\) −6496.04 −0.239396
\(904\) 0 0
\(905\) −19102.2 −0.701635
\(906\) 0 0
\(907\) 39315.7 1.43931 0.719656 0.694330i \(-0.244299\pi\)
0.719656 + 0.694330i \(0.244299\pi\)
\(908\) 0 0
\(909\) 9380.97 0.342296
\(910\) 0 0
\(911\) 5025.05 0.182752 0.0913761 0.995816i \(-0.470873\pi\)
0.0913761 + 0.995816i \(0.470873\pi\)
\(912\) 0 0
\(913\) 806.724 0.0292428
\(914\) 0 0
\(915\) 3810.66 0.137679
\(916\) 0 0
\(917\) −29343.5 −1.05671
\(918\) 0 0
\(919\) −14259.7 −0.511843 −0.255922 0.966698i \(-0.582379\pi\)
−0.255922 + 0.966698i \(0.582379\pi\)
\(920\) 0 0
\(921\) −1574.75 −0.0563408
\(922\) 0 0
\(923\) −35221.3 −1.25604
\(924\) 0 0
\(925\) −30891.3 −1.09806
\(926\) 0 0
\(927\) −4043.37 −0.143260
\(928\) 0 0
\(929\) 27668.1 0.977137 0.488568 0.872526i \(-0.337519\pi\)
0.488568 + 0.872526i \(0.337519\pi\)
\(930\) 0 0
\(931\) 12508.1 0.440318
\(932\) 0 0
\(933\) 4614.64 0.161925
\(934\) 0 0
\(935\) −821.521 −0.0287343
\(936\) 0 0
\(937\) −25122.4 −0.875893 −0.437947 0.899001i \(-0.644294\pi\)
−0.437947 + 0.899001i \(0.644294\pi\)
\(938\) 0 0
\(939\) −4738.42 −0.164678
\(940\) 0 0
\(941\) 19335.4 0.669836 0.334918 0.942247i \(-0.391291\pi\)
0.334918 + 0.942247i \(0.391291\pi\)
\(942\) 0 0
\(943\) −6726.36 −0.232281
\(944\) 0 0
\(945\) −35701.2 −1.22895
\(946\) 0 0
\(947\) −6752.27 −0.231700 −0.115850 0.993267i \(-0.536959\pi\)
−0.115850 + 0.993267i \(0.536959\pi\)
\(948\) 0 0
\(949\) −28191.8 −0.964326
\(950\) 0 0
\(951\) 1039.66 0.0354504
\(952\) 0 0
\(953\) 42620.6 1.44870 0.724352 0.689430i \(-0.242139\pi\)
0.724352 + 0.689430i \(0.242139\pi\)
\(954\) 0 0
\(955\) −87575.5 −2.96741
\(956\) 0 0
\(957\) −160.521 −0.00542204
\(958\) 0 0
\(959\) −60028.7 −2.02130
\(960\) 0 0
\(961\) −12476.2 −0.418791
\(962\) 0 0
\(963\) 37520.3 1.25553
\(964\) 0 0
\(965\) 74758.3 2.49384
\(966\) 0 0
\(967\) 17814.9 0.592441 0.296220 0.955120i \(-0.404274\pi\)
0.296220 + 0.955120i \(0.404274\pi\)
\(968\) 0 0
\(969\) 1028.83 0.0341083
\(970\) 0 0
\(971\) −49040.2 −1.62078 −0.810388 0.585893i \(-0.800744\pi\)
−0.810388 + 0.585893i \(0.800744\pi\)
\(972\) 0 0
\(973\) −18692.3 −0.615876
\(974\) 0 0
\(975\) −12775.2 −0.419623
\(976\) 0 0
\(977\) −4853.73 −0.158940 −0.0794702 0.996837i \(-0.525323\pi\)
−0.0794702 + 0.996837i \(0.525323\pi\)
\(978\) 0 0
\(979\) 148.505 0.00484805
\(980\) 0 0
\(981\) 25504.8 0.830076
\(982\) 0 0
\(983\) −37770.7 −1.22553 −0.612767 0.790264i \(-0.709944\pi\)
−0.612767 + 0.790264i \(0.709944\pi\)
\(984\) 0 0
\(985\) 52968.7 1.71342
\(986\) 0 0
\(987\) −16563.8 −0.534176
\(988\) 0 0
\(989\) −12683.0 −0.407782
\(990\) 0 0
\(991\) 46048.7 1.47607 0.738036 0.674762i \(-0.235754\pi\)
0.738036 + 0.674762i \(0.235754\pi\)
\(992\) 0 0
\(993\) −4634.52 −0.148109
\(994\) 0 0
\(995\) 8228.74 0.262180
\(996\) 0 0
\(997\) 43033.7 1.36699 0.683495 0.729955i \(-0.260459\pi\)
0.683495 + 0.729955i \(0.260459\pi\)
\(998\) 0 0
\(999\) −4989.32 −0.158013
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 1216.4.a.i.1.2 2
4.3 odd 2 1216.4.a.n.1.2 2
8.3 odd 2 608.4.a.c.1.1 2
8.5 even 2 608.4.a.d.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.c.1.1 2 8.3 odd 2
608.4.a.d.1.1 yes 2 8.5 even 2
1216.4.a.i.1.2 2 1.1 even 1 trivial
1216.4.a.n.1.2 2 4.3 odd 2