Properties

Label 272.4.a.h.1.1
Level $272$
Weight $4$
Character 272.1
Self dual yes
Analytic conductor $16.049$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0485195216\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.58966\) of defining polynomial
Character \(\chi\) \(=\) 272.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +O(q^{10})\) \(q-8.47535 q^{3} +0.885690 q^{5} -3.81828 q^{7} +44.8316 q^{9} +52.3720 q^{11} -8.06025 q^{13} -7.50653 q^{15} -17.0000 q^{17} +66.5154 q^{19} +32.3613 q^{21} -180.226 q^{23} -124.216 q^{25} -151.129 q^{27} -41.2800 q^{29} +34.9114 q^{31} -443.871 q^{33} -3.38182 q^{35} +130.368 q^{37} +68.3134 q^{39} -17.9081 q^{41} -277.620 q^{43} +39.7069 q^{45} -463.789 q^{47} -328.421 q^{49} +144.081 q^{51} -329.944 q^{53} +46.3853 q^{55} -563.741 q^{57} -678.656 q^{59} +340.280 q^{61} -171.180 q^{63} -7.13888 q^{65} -15.3925 q^{67} +1527.48 q^{69} +670.203 q^{71} +193.480 q^{73} +1052.77 q^{75} -199.971 q^{77} -1080.15 q^{79} +70.4207 q^{81} +865.668 q^{83} -15.0567 q^{85} +349.863 q^{87} +1129.46 q^{89} +30.7763 q^{91} -295.886 q^{93} +58.9120 q^{95} -379.412 q^{97} +2347.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} - 8 q^{5} - 22 q^{7} + 59 q^{9} + 28 q^{11} + 30 q^{13} - 108 q^{15} - 51 q^{17} - 80 q^{19} - 192 q^{21} - 142 q^{23} - 223 q^{25} + 20 q^{27} - 456 q^{29} - 230 q^{31} - 332 q^{33} + 332 q^{35} + 356 q^{37} - 268 q^{39} - 294 q^{41} - 556 q^{43} - 384 q^{45} - 640 q^{47} - 269 q^{49} + 68 q^{51} + 302 q^{53} - 76 q^{55} - 720 q^{57} - 636 q^{59} - 84 q^{61} - 1122 q^{63} + 408 q^{65} - 1008 q^{67} + 576 q^{69} + 402 q^{71} + 838 q^{73} + 1548 q^{75} - 504 q^{77} + 594 q^{79} - 505 q^{81} + 2396 q^{83} + 136 q^{85} - 1428 q^{87} - 170 q^{89} + 1016 q^{91} + 632 q^{93} + 472 q^{95} - 270 q^{97} + 2920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −8.47535 −1.63108 −0.815541 0.578699i \(-0.803561\pi\)
−0.815541 + 0.578699i \(0.803561\pi\)
\(4\) 0 0
\(5\) 0.885690 0.0792185 0.0396092 0.999215i \(-0.487389\pi\)
0.0396092 + 0.999215i \(0.487389\pi\)
\(6\) 0 0
\(7\) −3.81828 −0.206168 −0.103084 0.994673i \(-0.532871\pi\)
−0.103084 + 0.994673i \(0.532871\pi\)
\(8\) 0 0
\(9\) 44.8316 1.66043
\(10\) 0 0
\(11\) 52.3720 1.43552 0.717761 0.696289i \(-0.245167\pi\)
0.717761 + 0.696289i \(0.245167\pi\)
\(12\) 0 0
\(13\) −8.06025 −0.171962 −0.0859811 0.996297i \(-0.527402\pi\)
−0.0859811 + 0.996297i \(0.527402\pi\)
\(14\) 0 0
\(15\) −7.50653 −0.129212
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 66.5154 0.803141 0.401570 0.915828i \(-0.368465\pi\)
0.401570 + 0.915828i \(0.368465\pi\)
\(20\) 0 0
\(21\) 32.3613 0.336277
\(22\) 0 0
\(23\) −180.226 −1.63390 −0.816948 0.576711i \(-0.804336\pi\)
−0.816948 + 0.576711i \(0.804336\pi\)
\(24\) 0 0
\(25\) −124.216 −0.993724
\(26\) 0 0
\(27\) −151.129 −1.07722
\(28\) 0 0
\(29\) −41.2800 −0.264328 −0.132164 0.991228i \(-0.542193\pi\)
−0.132164 + 0.991228i \(0.542193\pi\)
\(30\) 0 0
\(31\) 34.9114 0.202267 0.101133 0.994873i \(-0.467753\pi\)
0.101133 + 0.994873i \(0.467753\pi\)
\(32\) 0 0
\(33\) −443.871 −2.34146
\(34\) 0 0
\(35\) −3.38182 −0.0163323
\(36\) 0 0
\(37\) 130.368 0.579255 0.289627 0.957139i \(-0.406469\pi\)
0.289627 + 0.957139i \(0.406469\pi\)
\(38\) 0 0
\(39\) 68.3134 0.280485
\(40\) 0 0
\(41\) −17.9081 −0.0682142 −0.0341071 0.999418i \(-0.510859\pi\)
−0.0341071 + 0.999418i \(0.510859\pi\)
\(42\) 0 0
\(43\) −277.620 −0.984573 −0.492287 0.870433i \(-0.663839\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(44\) 0 0
\(45\) 39.7069 0.131537
\(46\) 0 0
\(47\) −463.789 −1.43937 −0.719687 0.694299i \(-0.755715\pi\)
−0.719687 + 0.694299i \(0.755715\pi\)
\(48\) 0 0
\(49\) −328.421 −0.957495
\(50\) 0 0
\(51\) 144.081 0.395596
\(52\) 0 0
\(53\) −329.944 −0.855118 −0.427559 0.903987i \(-0.640626\pi\)
−0.427559 + 0.903987i \(0.640626\pi\)
\(54\) 0 0
\(55\) 46.3853 0.113720
\(56\) 0 0
\(57\) −563.741 −1.30999
\(58\) 0 0
\(59\) −678.656 −1.49752 −0.748759 0.662843i \(-0.769350\pi\)
−0.748759 + 0.662843i \(0.769350\pi\)
\(60\) 0 0
\(61\) 340.280 0.714237 0.357118 0.934059i \(-0.383759\pi\)
0.357118 + 0.934059i \(0.383759\pi\)
\(62\) 0 0
\(63\) −171.180 −0.342328
\(64\) 0 0
\(65\) −7.13888 −0.0136226
\(66\) 0 0
\(67\) −15.3925 −0.0280671 −0.0140336 0.999902i \(-0.504467\pi\)
−0.0140336 + 0.999902i \(0.504467\pi\)
\(68\) 0 0
\(69\) 1527.48 2.66502
\(70\) 0 0
\(71\) 670.203 1.12026 0.560130 0.828405i \(-0.310751\pi\)
0.560130 + 0.828405i \(0.310751\pi\)
\(72\) 0 0
\(73\) 193.480 0.310207 0.155103 0.987898i \(-0.450429\pi\)
0.155103 + 0.987898i \(0.450429\pi\)
\(74\) 0 0
\(75\) 1052.77 1.62085
\(76\) 0 0
\(77\) −199.971 −0.295959
\(78\) 0 0
\(79\) −1080.15 −1.53831 −0.769156 0.639061i \(-0.779323\pi\)
−0.769156 + 0.639061i \(0.779323\pi\)
\(80\) 0 0
\(81\) 70.4207 0.0965990
\(82\) 0 0
\(83\) 865.668 1.14481 0.572406 0.819970i \(-0.306010\pi\)
0.572406 + 0.819970i \(0.306010\pi\)
\(84\) 0 0
\(85\) −15.0567 −0.0192133
\(86\) 0 0
\(87\) 349.863 0.431141
\(88\) 0 0
\(89\) 1129.46 1.34520 0.672599 0.740008i \(-0.265178\pi\)
0.672599 + 0.740008i \(0.265178\pi\)
\(90\) 0 0
\(91\) 30.7763 0.0354531
\(92\) 0 0
\(93\) −295.886 −0.329914
\(94\) 0 0
\(95\) 58.9120 0.0636236
\(96\) 0 0
\(97\) −379.412 −0.397149 −0.198574 0.980086i \(-0.563631\pi\)
−0.198574 + 0.980086i \(0.563631\pi\)
\(98\) 0 0
\(99\) 2347.92 2.38359
\(100\) 0 0
\(101\) 131.732 0.129780 0.0648902 0.997892i \(-0.479330\pi\)
0.0648902 + 0.997892i \(0.479330\pi\)
\(102\) 0 0
\(103\) −195.988 −0.187488 −0.0937442 0.995596i \(-0.529884\pi\)
−0.0937442 + 0.995596i \(0.529884\pi\)
\(104\) 0 0
\(105\) 28.6621 0.0266394
\(106\) 0 0
\(107\) 485.147 0.438326 0.219163 0.975688i \(-0.429667\pi\)
0.219163 + 0.975688i \(0.429667\pi\)
\(108\) 0 0
\(109\) −1255.12 −1.10292 −0.551460 0.834201i \(-0.685929\pi\)
−0.551460 + 0.834201i \(0.685929\pi\)
\(110\) 0 0
\(111\) −1104.92 −0.944812
\(112\) 0 0
\(113\) −1013.35 −0.843612 −0.421806 0.906686i \(-0.638604\pi\)
−0.421806 + 0.906686i \(0.638604\pi\)
\(114\) 0 0
\(115\) −159.624 −0.129435
\(116\) 0 0
\(117\) −361.354 −0.285531
\(118\) 0 0
\(119\) 64.9108 0.0500031
\(120\) 0 0
\(121\) 1411.83 1.06073
\(122\) 0 0
\(123\) 151.778 0.111263
\(124\) 0 0
\(125\) −220.728 −0.157940
\(126\) 0 0
\(127\) −1927.72 −1.34691 −0.673456 0.739227i \(-0.735191\pi\)
−0.673456 + 0.739227i \(0.735191\pi\)
\(128\) 0 0
\(129\) 2352.93 1.60592
\(130\) 0 0
\(131\) 406.738 0.271274 0.135637 0.990759i \(-0.456692\pi\)
0.135637 + 0.990759i \(0.456692\pi\)
\(132\) 0 0
\(133\) −253.975 −0.165582
\(134\) 0 0
\(135\) −133.854 −0.0853355
\(136\) 0 0
\(137\) −130.552 −0.0814149 −0.0407074 0.999171i \(-0.512961\pi\)
−0.0407074 + 0.999171i \(0.512961\pi\)
\(138\) 0 0
\(139\) −2073.54 −1.26529 −0.632644 0.774443i \(-0.718030\pi\)
−0.632644 + 0.774443i \(0.718030\pi\)
\(140\) 0 0
\(141\) 3930.78 2.34774
\(142\) 0 0
\(143\) −422.131 −0.246856
\(144\) 0 0
\(145\) −36.5613 −0.0209397
\(146\) 0 0
\(147\) 2783.48 1.56175
\(148\) 0 0
\(149\) −1852.73 −1.01867 −0.509334 0.860569i \(-0.670108\pi\)
−0.509334 + 0.860569i \(0.670108\pi\)
\(150\) 0 0
\(151\) −2050.86 −1.10527 −0.552637 0.833422i \(-0.686378\pi\)
−0.552637 + 0.833422i \(0.686378\pi\)
\(152\) 0 0
\(153\) −762.138 −0.402714
\(154\) 0 0
\(155\) 30.9207 0.0160233
\(156\) 0 0
\(157\) −262.991 −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(158\) 0 0
\(159\) 2796.39 1.39477
\(160\) 0 0
\(161\) 688.152 0.336857
\(162\) 0 0
\(163\) 1444.98 0.694354 0.347177 0.937800i \(-0.387140\pi\)
0.347177 + 0.937800i \(0.387140\pi\)
\(164\) 0 0
\(165\) −393.132 −0.185487
\(166\) 0 0
\(167\) 501.565 0.232409 0.116204 0.993225i \(-0.462927\pi\)
0.116204 + 0.993225i \(0.462927\pi\)
\(168\) 0 0
\(169\) −2132.03 −0.970429
\(170\) 0 0
\(171\) 2981.99 1.33356
\(172\) 0 0
\(173\) −2590.14 −1.13829 −0.569146 0.822237i \(-0.692726\pi\)
−0.569146 + 0.822237i \(0.692726\pi\)
\(174\) 0 0
\(175\) 474.290 0.204874
\(176\) 0 0
\(177\) 5751.85 2.44257
\(178\) 0 0
\(179\) −2165.65 −0.904294 −0.452147 0.891943i \(-0.649342\pi\)
−0.452147 + 0.891943i \(0.649342\pi\)
\(180\) 0 0
\(181\) −1925.56 −0.790750 −0.395375 0.918520i \(-0.629385\pi\)
−0.395375 + 0.918520i \(0.629385\pi\)
\(182\) 0 0
\(183\) −2884.00 −1.16498
\(184\) 0 0
\(185\) 115.466 0.0458877
\(186\) 0 0
\(187\) −890.324 −0.348165
\(188\) 0 0
\(189\) 577.055 0.222088
\(190\) 0 0
\(191\) 2783.52 1.05449 0.527247 0.849712i \(-0.323224\pi\)
0.527247 + 0.849712i \(0.323224\pi\)
\(192\) 0 0
\(193\) 2258.27 0.842246 0.421123 0.907004i \(-0.361636\pi\)
0.421123 + 0.907004i \(0.361636\pi\)
\(194\) 0 0
\(195\) 60.5045 0.0222196
\(196\) 0 0
\(197\) −1270.70 −0.459560 −0.229780 0.973243i \(-0.573801\pi\)
−0.229780 + 0.973243i \(0.573801\pi\)
\(198\) 0 0
\(199\) 4794.36 1.70786 0.853928 0.520392i \(-0.174214\pi\)
0.853928 + 0.520392i \(0.174214\pi\)
\(200\) 0 0
\(201\) 130.457 0.0457798
\(202\) 0 0
\(203\) 157.619 0.0544960
\(204\) 0 0
\(205\) −15.8611 −0.00540383
\(206\) 0 0
\(207\) −8079.80 −2.71297
\(208\) 0 0
\(209\) 3483.54 1.15293
\(210\) 0 0
\(211\) 2807.00 0.915837 0.457918 0.888994i \(-0.348595\pi\)
0.457918 + 0.888994i \(0.348595\pi\)
\(212\) 0 0
\(213\) −5680.21 −1.82724
\(214\) 0 0
\(215\) −245.885 −0.0779964
\(216\) 0 0
\(217\) −133.302 −0.0417009
\(218\) 0 0
\(219\) −1639.81 −0.505973
\(220\) 0 0
\(221\) 137.024 0.0417070
\(222\) 0 0
\(223\) −4684.30 −1.40665 −0.703327 0.710866i \(-0.748303\pi\)
−0.703327 + 0.710866i \(0.748303\pi\)
\(224\) 0 0
\(225\) −5568.79 −1.65001
\(226\) 0 0
\(227\) 1395.72 0.408095 0.204047 0.978961i \(-0.434590\pi\)
0.204047 + 0.978961i \(0.434590\pi\)
\(228\) 0 0
\(229\) 894.638 0.258163 0.129082 0.991634i \(-0.458797\pi\)
0.129082 + 0.991634i \(0.458797\pi\)
\(230\) 0 0
\(231\) 1694.83 0.482733
\(232\) 0 0
\(233\) 1196.13 0.336313 0.168156 0.985760i \(-0.446219\pi\)
0.168156 + 0.985760i \(0.446219\pi\)
\(234\) 0 0
\(235\) −410.773 −0.114025
\(236\) 0 0
\(237\) 9154.67 2.50911
\(238\) 0 0
\(239\) −4948.82 −1.33938 −0.669691 0.742639i \(-0.733574\pi\)
−0.669691 + 0.742639i \(0.733574\pi\)
\(240\) 0 0
\(241\) −6702.73 −1.79154 −0.895770 0.444518i \(-0.853375\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(242\) 0 0
\(243\) 3483.65 0.919656
\(244\) 0 0
\(245\) −290.879 −0.0758513
\(246\) 0 0
\(247\) −536.130 −0.138110
\(248\) 0 0
\(249\) −7336.85 −1.86728
\(250\) 0 0
\(251\) 4756.08 1.19602 0.598010 0.801489i \(-0.295958\pi\)
0.598010 + 0.801489i \(0.295958\pi\)
\(252\) 0 0
\(253\) −9438.77 −2.34550
\(254\) 0 0
\(255\) 127.611 0.0313385
\(256\) 0 0
\(257\) 2892.84 0.702143 0.351071 0.936349i \(-0.385817\pi\)
0.351071 + 0.936349i \(0.385817\pi\)
\(258\) 0 0
\(259\) −497.784 −0.119424
\(260\) 0 0
\(261\) −1850.65 −0.438898
\(262\) 0 0
\(263\) −5415.48 −1.26971 −0.634853 0.772633i \(-0.718939\pi\)
−0.634853 + 0.772633i \(0.718939\pi\)
\(264\) 0 0
\(265\) −292.228 −0.0677412
\(266\) 0 0
\(267\) −9572.58 −2.19413
\(268\) 0 0
\(269\) 5787.00 1.31167 0.655835 0.754904i \(-0.272317\pi\)
0.655835 + 0.754904i \(0.272317\pi\)
\(270\) 0 0
\(271\) −5465.13 −1.22503 −0.612515 0.790459i \(-0.709842\pi\)
−0.612515 + 0.790459i \(0.709842\pi\)
\(272\) 0 0
\(273\) −260.840 −0.0578270
\(274\) 0 0
\(275\) −6505.42 −1.42651
\(276\) 0 0
\(277\) −1207.65 −0.261952 −0.130976 0.991386i \(-0.541811\pi\)
−0.130976 + 0.991386i \(0.541811\pi\)
\(278\) 0 0
\(279\) 1565.13 0.335850
\(280\) 0 0
\(281\) −1197.18 −0.254155 −0.127077 0.991893i \(-0.540560\pi\)
−0.127077 + 0.991893i \(0.540560\pi\)
\(282\) 0 0
\(283\) −3164.73 −0.664748 −0.332374 0.943148i \(-0.607850\pi\)
−0.332374 + 0.943148i \(0.607850\pi\)
\(284\) 0 0
\(285\) −499.300 −0.103775
\(286\) 0 0
\(287\) 68.3784 0.0140636
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) 3215.65 0.647782
\(292\) 0 0
\(293\) 7456.21 1.48668 0.743339 0.668915i \(-0.233241\pi\)
0.743339 + 0.668915i \(0.233241\pi\)
\(294\) 0 0
\(295\) −601.079 −0.118631
\(296\) 0 0
\(297\) −7914.94 −1.54637
\(298\) 0 0
\(299\) 1452.66 0.280969
\(300\) 0 0
\(301\) 1060.03 0.202988
\(302\) 0 0
\(303\) −1116.47 −0.211683
\(304\) 0 0
\(305\) 301.383 0.0565808
\(306\) 0 0
\(307\) 6535.48 1.21498 0.607491 0.794327i \(-0.292176\pi\)
0.607491 + 0.794327i \(0.292176\pi\)
\(308\) 0 0
\(309\) 1661.07 0.305809
\(310\) 0 0
\(311\) 8935.89 1.62928 0.814642 0.579963i \(-0.196933\pi\)
0.814642 + 0.579963i \(0.196933\pi\)
\(312\) 0 0
\(313\) −2628.71 −0.474707 −0.237353 0.971423i \(-0.576280\pi\)
−0.237353 + 0.971423i \(0.576280\pi\)
\(314\) 0 0
\(315\) −151.612 −0.0271187
\(316\) 0 0
\(317\) 4268.54 0.756293 0.378147 0.925746i \(-0.376562\pi\)
0.378147 + 0.925746i \(0.376562\pi\)
\(318\) 0 0
\(319\) −2161.92 −0.379449
\(320\) 0 0
\(321\) −4111.79 −0.714946
\(322\) 0 0
\(323\) −1130.76 −0.194790
\(324\) 0 0
\(325\) 1001.21 0.170883
\(326\) 0 0
\(327\) 10637.5 1.79895
\(328\) 0 0
\(329\) 1770.88 0.296753
\(330\) 0 0
\(331\) −992.298 −0.164778 −0.0823892 0.996600i \(-0.526255\pi\)
−0.0823892 + 0.996600i \(0.526255\pi\)
\(332\) 0 0
\(333\) 5844.63 0.961812
\(334\) 0 0
\(335\) −13.6330 −0.00222344
\(336\) 0 0
\(337\) 8042.26 1.29997 0.649985 0.759947i \(-0.274775\pi\)
0.649985 + 0.759947i \(0.274775\pi\)
\(338\) 0 0
\(339\) 8588.52 1.37600
\(340\) 0 0
\(341\) 1828.38 0.290359
\(342\) 0 0
\(343\) 2563.68 0.403573
\(344\) 0 0
\(345\) 1352.87 0.211119
\(346\) 0 0
\(347\) 7414.16 1.14701 0.573506 0.819202i \(-0.305583\pi\)
0.573506 + 0.819202i \(0.305583\pi\)
\(348\) 0 0
\(349\) −859.194 −0.131781 −0.0658905 0.997827i \(-0.520989\pi\)
−0.0658905 + 0.997827i \(0.520989\pi\)
\(350\) 0 0
\(351\) 1218.14 0.185241
\(352\) 0 0
\(353\) 569.084 0.0858053 0.0429027 0.999079i \(-0.486339\pi\)
0.0429027 + 0.999079i \(0.486339\pi\)
\(354\) 0 0
\(355\) 593.592 0.0887453
\(356\) 0 0
\(357\) −550.142 −0.0815592
\(358\) 0 0
\(359\) 5005.21 0.735835 0.367918 0.929858i \(-0.380071\pi\)
0.367918 + 0.929858i \(0.380071\pi\)
\(360\) 0 0
\(361\) −2434.71 −0.354965
\(362\) 0 0
\(363\) −11965.7 −1.73013
\(364\) 0 0
\(365\) 171.363 0.0245741
\(366\) 0 0
\(367\) 10975.3 1.56105 0.780523 0.625127i \(-0.214953\pi\)
0.780523 + 0.625127i \(0.214953\pi\)
\(368\) 0 0
\(369\) −802.851 −0.113265
\(370\) 0 0
\(371\) 1259.82 0.176298
\(372\) 0 0
\(373\) −3211.72 −0.445835 −0.222918 0.974837i \(-0.571558\pi\)
−0.222918 + 0.974837i \(0.571558\pi\)
\(374\) 0 0
\(375\) 1870.74 0.257613
\(376\) 0 0
\(377\) 332.727 0.0454544
\(378\) 0 0
\(379\) −8051.48 −1.09123 −0.545616 0.838035i \(-0.683704\pi\)
−0.545616 + 0.838035i \(0.683704\pi\)
\(380\) 0 0
\(381\) 16338.1 2.19692
\(382\) 0 0
\(383\) 2584.16 0.344763 0.172382 0.985030i \(-0.444854\pi\)
0.172382 + 0.985030i \(0.444854\pi\)
\(384\) 0 0
\(385\) −177.112 −0.0234454
\(386\) 0 0
\(387\) −12446.2 −1.63482
\(388\) 0 0
\(389\) −5174.31 −0.674417 −0.337208 0.941430i \(-0.609483\pi\)
−0.337208 + 0.941430i \(0.609483\pi\)
\(390\) 0 0
\(391\) 3063.83 0.396278
\(392\) 0 0
\(393\) −3447.25 −0.442470
\(394\) 0 0
\(395\) −956.680 −0.121863
\(396\) 0 0
\(397\) −5149.36 −0.650980 −0.325490 0.945545i \(-0.605529\pi\)
−0.325490 + 0.945545i \(0.605529\pi\)
\(398\) 0 0
\(399\) 2152.53 0.270078
\(400\) 0 0
\(401\) 8700.49 1.08350 0.541748 0.840541i \(-0.317763\pi\)
0.541748 + 0.840541i \(0.317763\pi\)
\(402\) 0 0
\(403\) −281.394 −0.0347823
\(404\) 0 0
\(405\) 62.3709 0.00765243
\(406\) 0 0
\(407\) 6827.65 0.831533
\(408\) 0 0
\(409\) 12346.0 1.49260 0.746299 0.665611i \(-0.231829\pi\)
0.746299 + 0.665611i \(0.231829\pi\)
\(410\) 0 0
\(411\) 1106.48 0.132794
\(412\) 0 0
\(413\) 2591.30 0.308740
\(414\) 0 0
\(415\) 766.713 0.0906903
\(416\) 0 0
\(417\) 17574.0 2.06379
\(418\) 0 0
\(419\) 5763.33 0.671974 0.335987 0.941867i \(-0.390930\pi\)
0.335987 + 0.941867i \(0.390930\pi\)
\(420\) 0 0
\(421\) −1876.12 −0.217188 −0.108594 0.994086i \(-0.534635\pi\)
−0.108594 + 0.994086i \(0.534635\pi\)
\(422\) 0 0
\(423\) −20792.4 −2.38998
\(424\) 0 0
\(425\) 2111.66 0.241014
\(426\) 0 0
\(427\) −1299.29 −0.147253
\(428\) 0 0
\(429\) 3577.71 0.402642
\(430\) 0 0
\(431\) −83.9299 −0.00937996 −0.00468998 0.999989i \(-0.501493\pi\)
−0.00468998 + 0.999989i \(0.501493\pi\)
\(432\) 0 0
\(433\) −15345.0 −1.70308 −0.851539 0.524291i \(-0.824331\pi\)
−0.851539 + 0.524291i \(0.824331\pi\)
\(434\) 0 0
\(435\) 309.870 0.0341543
\(436\) 0 0
\(437\) −11987.8 −1.31225
\(438\) 0 0
\(439\) −3064.74 −0.333194 −0.166597 0.986025i \(-0.553278\pi\)
−0.166597 + 0.986025i \(0.553278\pi\)
\(440\) 0 0
\(441\) −14723.6 −1.58985
\(442\) 0 0
\(443\) 1792.97 0.192295 0.0961474 0.995367i \(-0.469348\pi\)
0.0961474 + 0.995367i \(0.469348\pi\)
\(444\) 0 0
\(445\) 1000.35 0.106564
\(446\) 0 0
\(447\) 15702.6 1.66153
\(448\) 0 0
\(449\) 2499.19 0.262681 0.131341 0.991337i \(-0.458072\pi\)
0.131341 + 0.991337i \(0.458072\pi\)
\(450\) 0 0
\(451\) −937.885 −0.0979231
\(452\) 0 0
\(453\) 17381.7 1.80279
\(454\) 0 0
\(455\) 27.2583 0.00280854
\(456\) 0 0
\(457\) 14784.4 1.51331 0.756656 0.653813i \(-0.226832\pi\)
0.756656 + 0.653813i \(0.226832\pi\)
\(458\) 0 0
\(459\) 2569.20 0.261263
\(460\) 0 0
\(461\) −17746.9 −1.79297 −0.896483 0.443078i \(-0.853887\pi\)
−0.896483 + 0.443078i \(0.853887\pi\)
\(462\) 0 0
\(463\) −18486.4 −1.85559 −0.927793 0.373096i \(-0.878296\pi\)
−0.927793 + 0.373096i \(0.878296\pi\)
\(464\) 0 0
\(465\) −262.064 −0.0261353
\(466\) 0 0
\(467\) −7406.57 −0.733908 −0.366954 0.930239i \(-0.619599\pi\)
−0.366954 + 0.930239i \(0.619599\pi\)
\(468\) 0 0
\(469\) 58.7731 0.00578655
\(470\) 0 0
\(471\) 2228.94 0.218055
\(472\) 0 0
\(473\) −14539.5 −1.41338
\(474\) 0 0
\(475\) −8262.24 −0.798101
\(476\) 0 0
\(477\) −14791.9 −1.41986
\(478\) 0 0
\(479\) 18550.9 1.76955 0.884775 0.466019i \(-0.154312\pi\)
0.884775 + 0.466019i \(0.154312\pi\)
\(480\) 0 0
\(481\) −1050.80 −0.0996100
\(482\) 0 0
\(483\) −5832.34 −0.549442
\(484\) 0 0
\(485\) −336.041 −0.0314615
\(486\) 0 0
\(487\) −10203.4 −0.949406 −0.474703 0.880146i \(-0.657444\pi\)
−0.474703 + 0.880146i \(0.657444\pi\)
\(488\) 0 0
\(489\) −12246.7 −1.13255
\(490\) 0 0
\(491\) 1247.46 0.114658 0.0573290 0.998355i \(-0.481742\pi\)
0.0573290 + 0.998355i \(0.481742\pi\)
\(492\) 0 0
\(493\) 701.760 0.0641089
\(494\) 0 0
\(495\) 2079.53 0.188824
\(496\) 0 0
\(497\) −2559.03 −0.230962
\(498\) 0 0
\(499\) −70.0303 −0.00628254 −0.00314127 0.999995i \(-0.501000\pi\)
−0.00314127 + 0.999995i \(0.501000\pi\)
\(500\) 0 0
\(501\) −4250.94 −0.379078
\(502\) 0 0
\(503\) −1444.29 −0.128028 −0.0640138 0.997949i \(-0.520390\pi\)
−0.0640138 + 0.997949i \(0.520390\pi\)
\(504\) 0 0
\(505\) 116.674 0.0102810
\(506\) 0 0
\(507\) 18069.7 1.58285
\(508\) 0 0
\(509\) 14272.8 1.24289 0.621445 0.783458i \(-0.286546\pi\)
0.621445 + 0.783458i \(0.286546\pi\)
\(510\) 0 0
\(511\) −738.761 −0.0639547
\(512\) 0 0
\(513\) −10052.4 −0.865157
\(514\) 0 0
\(515\) −173.585 −0.0148525
\(516\) 0 0
\(517\) −24289.6 −2.06625
\(518\) 0 0
\(519\) 21952.3 1.85665
\(520\) 0 0
\(521\) 14874.0 1.25075 0.625376 0.780324i \(-0.284946\pi\)
0.625376 + 0.780324i \(0.284946\pi\)
\(522\) 0 0
\(523\) 8142.90 0.680811 0.340406 0.940279i \(-0.389436\pi\)
0.340406 + 0.940279i \(0.389436\pi\)
\(524\) 0 0
\(525\) −4019.78 −0.334167
\(526\) 0 0
\(527\) −593.494 −0.0490569
\(528\) 0 0
\(529\) 20314.2 1.66962
\(530\) 0 0
\(531\) −30425.3 −2.48652
\(532\) 0 0
\(533\) 144.344 0.0117303
\(534\) 0 0
\(535\) 429.689 0.0347235
\(536\) 0 0
\(537\) 18354.7 1.47498
\(538\) 0 0
\(539\) −17200.0 −1.37451
\(540\) 0 0
\(541\) 3179.67 0.252689 0.126344 0.991986i \(-0.459676\pi\)
0.126344 + 0.991986i \(0.459676\pi\)
\(542\) 0 0
\(543\) 16319.8 1.28978
\(544\) 0 0
\(545\) −1111.64 −0.0873716
\(546\) 0 0
\(547\) −2107.07 −0.164702 −0.0823509 0.996603i \(-0.526243\pi\)
−0.0823509 + 0.996603i \(0.526243\pi\)
\(548\) 0 0
\(549\) 15255.3 1.18594
\(550\) 0 0
\(551\) −2745.76 −0.212292
\(552\) 0 0
\(553\) 4124.33 0.317151
\(554\) 0 0
\(555\) −978.614 −0.0748466
\(556\) 0 0
\(557\) 467.382 0.0355540 0.0177770 0.999842i \(-0.494341\pi\)
0.0177770 + 0.999842i \(0.494341\pi\)
\(558\) 0 0
\(559\) 2237.69 0.169309
\(560\) 0 0
\(561\) 7545.81 0.567887
\(562\) 0 0
\(563\) −14612.6 −1.09387 −0.546935 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(564\) 0 0
\(565\) −897.515 −0.0668297
\(566\) 0 0
\(567\) −268.886 −0.0199156
\(568\) 0 0
\(569\) 11602.3 0.854821 0.427410 0.904058i \(-0.359426\pi\)
0.427410 + 0.904058i \(0.359426\pi\)
\(570\) 0 0
\(571\) 10534.9 0.772104 0.386052 0.922477i \(-0.373839\pi\)
0.386052 + 0.922477i \(0.373839\pi\)
\(572\) 0 0
\(573\) −23591.3 −1.71997
\(574\) 0 0
\(575\) 22386.8 1.62364
\(576\) 0 0
\(577\) 14404.7 1.03930 0.519650 0.854379i \(-0.326062\pi\)
0.519650 + 0.854379i \(0.326062\pi\)
\(578\) 0 0
\(579\) −19139.6 −1.37377
\(580\) 0 0
\(581\) −3305.37 −0.236024
\(582\) 0 0
\(583\) −17279.8 −1.22754
\(584\) 0 0
\(585\) −320.047 −0.0226194
\(586\) 0 0
\(587\) 11004.9 0.773799 0.386900 0.922122i \(-0.373546\pi\)
0.386900 + 0.922122i \(0.373546\pi\)
\(588\) 0 0
\(589\) 2322.14 0.162449
\(590\) 0 0
\(591\) 10769.6 0.749581
\(592\) 0 0
\(593\) 1853.59 0.128361 0.0641804 0.997938i \(-0.479557\pi\)
0.0641804 + 0.997938i \(0.479557\pi\)
\(594\) 0 0
\(595\) 57.4909 0.00396117
\(596\) 0 0
\(597\) −40633.9 −2.78565
\(598\) 0 0
\(599\) −19074.7 −1.30112 −0.650559 0.759456i \(-0.725465\pi\)
−0.650559 + 0.759456i \(0.725465\pi\)
\(600\) 0 0
\(601\) −27776.0 −1.88520 −0.942600 0.333923i \(-0.891627\pi\)
−0.942600 + 0.333923i \(0.891627\pi\)
\(602\) 0 0
\(603\) −690.073 −0.0466035
\(604\) 0 0
\(605\) 1250.44 0.0840291
\(606\) 0 0
\(607\) −18728.3 −1.25232 −0.626159 0.779695i \(-0.715374\pi\)
−0.626159 + 0.779695i \(0.715374\pi\)
\(608\) 0 0
\(609\) −1335.88 −0.0888874
\(610\) 0 0
\(611\) 3738.25 0.247518
\(612\) 0 0
\(613\) −24405.3 −1.60802 −0.804012 0.594613i \(-0.797305\pi\)
−0.804012 + 0.594613i \(0.797305\pi\)
\(614\) 0 0
\(615\) 134.428 0.00881409
\(616\) 0 0
\(617\) −22516.4 −1.46917 −0.734584 0.678518i \(-0.762623\pi\)
−0.734584 + 0.678518i \(0.762623\pi\)
\(618\) 0 0
\(619\) 5146.53 0.334179 0.167089 0.985942i \(-0.446563\pi\)
0.167089 + 0.985942i \(0.446563\pi\)
\(620\) 0 0
\(621\) 27237.4 1.76006
\(622\) 0 0
\(623\) −4312.60 −0.277337
\(624\) 0 0
\(625\) 15331.4 0.981213
\(626\) 0 0
\(627\) −29524.3 −1.88052
\(628\) 0 0
\(629\) −2216.26 −0.140490
\(630\) 0 0
\(631\) 3858.77 0.243447 0.121724 0.992564i \(-0.461158\pi\)
0.121724 + 0.992564i \(0.461158\pi\)
\(632\) 0 0
\(633\) −23790.3 −1.49381
\(634\) 0 0
\(635\) −1707.36 −0.106700
\(636\) 0 0
\(637\) 2647.15 0.164653
\(638\) 0 0
\(639\) 30046.3 1.86011
\(640\) 0 0
\(641\) 18689.3 1.15161 0.575805 0.817587i \(-0.304689\pi\)
0.575805 + 0.817587i \(0.304689\pi\)
\(642\) 0 0
\(643\) −26473.5 −1.62366 −0.811831 0.583893i \(-0.801529\pi\)
−0.811831 + 0.583893i \(0.801529\pi\)
\(644\) 0 0
\(645\) 2083.96 0.127219
\(646\) 0 0
\(647\) −14397.7 −0.874855 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(648\) 0 0
\(649\) −35542.6 −2.14972
\(650\) 0 0
\(651\) 1129.78 0.0680177
\(652\) 0 0
\(653\) 20939.5 1.25486 0.627431 0.778672i \(-0.284107\pi\)
0.627431 + 0.778672i \(0.284107\pi\)
\(654\) 0 0
\(655\) 360.244 0.0214899
\(656\) 0 0
\(657\) 8674.02 0.515077
\(658\) 0 0
\(659\) −4031.76 −0.238323 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(660\) 0 0
\(661\) 6691.52 0.393752 0.196876 0.980428i \(-0.436920\pi\)
0.196876 + 0.980428i \(0.436920\pi\)
\(662\) 0 0
\(663\) −1161.33 −0.0680275
\(664\) 0 0
\(665\) −224.943 −0.0131171
\(666\) 0 0
\(667\) 7439.71 0.431884
\(668\) 0 0
\(669\) 39701.1 2.29437
\(670\) 0 0
\(671\) 17821.2 1.02530
\(672\) 0 0
\(673\) 10319.2 0.591048 0.295524 0.955335i \(-0.404506\pi\)
0.295524 + 0.955335i \(0.404506\pi\)
\(674\) 0 0
\(675\) 18772.6 1.07046
\(676\) 0 0
\(677\) −19813.3 −1.12480 −0.562398 0.826866i \(-0.690121\pi\)
−0.562398 + 0.826866i \(0.690121\pi\)
\(678\) 0 0
\(679\) 1448.70 0.0818793
\(680\) 0 0
\(681\) −11829.3 −0.665636
\(682\) 0 0
\(683\) −5924.61 −0.331916 −0.165958 0.986133i \(-0.553072\pi\)
−0.165958 + 0.986133i \(0.553072\pi\)
\(684\) 0 0
\(685\) −115.629 −0.00644957
\(686\) 0 0
\(687\) −7582.38 −0.421085
\(688\) 0 0
\(689\) 2659.43 0.147048
\(690\) 0 0
\(691\) −1973.16 −0.108629 −0.0543143 0.998524i \(-0.517297\pi\)
−0.0543143 + 0.998524i \(0.517297\pi\)
\(692\) 0 0
\(693\) −8965.03 −0.491419
\(694\) 0 0
\(695\) −1836.51 −0.100234
\(696\) 0 0
\(697\) 304.439 0.0165444
\(698\) 0 0
\(699\) −10137.6 −0.548554
\(700\) 0 0
\(701\) −12840.1 −0.691815 −0.345907 0.938269i \(-0.612429\pi\)
−0.345907 + 0.938269i \(0.612429\pi\)
\(702\) 0 0
\(703\) 8671.50 0.465223
\(704\) 0 0
\(705\) 3481.45 0.185984
\(706\) 0 0
\(707\) −502.990 −0.0267566
\(708\) 0 0
\(709\) −27749.7 −1.46990 −0.734952 0.678119i \(-0.762796\pi\)
−0.734952 + 0.678119i \(0.762796\pi\)
\(710\) 0 0
\(711\) −48425.0 −2.55426
\(712\) 0 0
\(713\) −6291.92 −0.330483
\(714\) 0 0
\(715\) −373.877 −0.0195555
\(716\) 0 0
\(717\) 41943.0 2.18464
\(718\) 0 0
\(719\) −16888.3 −0.875979 −0.437989 0.898980i \(-0.644309\pi\)
−0.437989 + 0.898980i \(0.644309\pi\)
\(720\) 0 0
\(721\) 748.339 0.0386541
\(722\) 0 0
\(723\) 56808.0 2.92215
\(724\) 0 0
\(725\) 5127.62 0.262669
\(726\) 0 0
\(727\) −2135.25 −0.108930 −0.0544649 0.998516i \(-0.517345\pi\)
−0.0544649 + 0.998516i \(0.517345\pi\)
\(728\) 0 0
\(729\) −31426.5 −1.59663
\(730\) 0 0
\(731\) 4719.54 0.238794
\(732\) 0 0
\(733\) 4795.27 0.241633 0.120817 0.992675i \(-0.461449\pi\)
0.120817 + 0.992675i \(0.461449\pi\)
\(734\) 0 0
\(735\) 2465.30 0.123720
\(736\) 0 0
\(737\) −806.138 −0.0402910
\(738\) 0 0
\(739\) 32747.6 1.63010 0.815048 0.579393i \(-0.196710\pi\)
0.815048 + 0.579393i \(0.196710\pi\)
\(740\) 0 0
\(741\) 4543.89 0.225269
\(742\) 0 0
\(743\) −12299.4 −0.607298 −0.303649 0.952784i \(-0.598205\pi\)
−0.303649 + 0.952784i \(0.598205\pi\)
\(744\) 0 0
\(745\) −1640.94 −0.0806974
\(746\) 0 0
\(747\) 38809.3 1.90088
\(748\) 0 0
\(749\) −1852.43 −0.0903688
\(750\) 0 0
\(751\) −30102.6 −1.46266 −0.731332 0.682021i \(-0.761101\pi\)
−0.731332 + 0.682021i \(0.761101\pi\)
\(752\) 0 0
\(753\) −40309.4 −1.95081
\(754\) 0 0
\(755\) −1816.42 −0.0875581
\(756\) 0 0
\(757\) 38826.3 1.86416 0.932078 0.362257i \(-0.117994\pi\)
0.932078 + 0.362257i \(0.117994\pi\)
\(758\) 0 0
\(759\) 79996.9 3.82570
\(760\) 0 0
\(761\) 19981.6 0.951815 0.475907 0.879495i \(-0.342120\pi\)
0.475907 + 0.879495i \(0.342120\pi\)
\(762\) 0 0
\(763\) 4792.39 0.227387
\(764\) 0 0
\(765\) −675.017 −0.0319024
\(766\) 0 0
\(767\) 5470.14 0.257517
\(768\) 0 0
\(769\) −22407.7 −1.05077 −0.525384 0.850865i \(-0.676078\pi\)
−0.525384 + 0.850865i \(0.676078\pi\)
\(770\) 0 0
\(771\) −24517.9 −1.14525
\(772\) 0 0
\(773\) −6902.77 −0.321184 −0.160592 0.987021i \(-0.551340\pi\)
−0.160592 + 0.987021i \(0.551340\pi\)
\(774\) 0 0
\(775\) −4336.54 −0.200997
\(776\) 0 0
\(777\) 4218.89 0.194790
\(778\) 0 0
\(779\) −1191.17 −0.0547856
\(780\) 0 0
\(781\) 35099.9 1.60816
\(782\) 0 0
\(783\) 6238.62 0.284738
\(784\) 0 0
\(785\) −232.928 −0.0105905
\(786\) 0 0
\(787\) 22185.9 1.00488 0.502442 0.864611i \(-0.332435\pi\)
0.502442 + 0.864611i \(0.332435\pi\)
\(788\) 0 0
\(789\) 45898.1 2.07099
\(790\) 0 0
\(791\) 3869.27 0.173926
\(792\) 0 0
\(793\) −2742.74 −0.122822
\(794\) 0 0
\(795\) 2476.73 0.110491
\(796\) 0 0
\(797\) −16291.1 −0.724040 −0.362020 0.932170i \(-0.617913\pi\)
−0.362020 + 0.932170i \(0.617913\pi\)
\(798\) 0 0
\(799\) 7884.41 0.349100
\(800\) 0 0
\(801\) 50635.5 2.23361
\(802\) 0 0
\(803\) 10132.9 0.445309
\(804\) 0 0
\(805\) 609.489 0.0266853
\(806\) 0 0
\(807\) −49046.8 −2.13944
\(808\) 0 0
\(809\) 17696.8 0.769082 0.384541 0.923108i \(-0.374360\pi\)
0.384541 + 0.923108i \(0.374360\pi\)
\(810\) 0 0
\(811\) 3095.34 0.134022 0.0670111 0.997752i \(-0.478654\pi\)
0.0670111 + 0.997752i \(0.478654\pi\)
\(812\) 0 0
\(813\) 46318.9 1.99812
\(814\) 0 0
\(815\) 1279.81 0.0550057
\(816\) 0 0
\(817\) −18466.0 −0.790751
\(818\) 0 0
\(819\) 1379.75 0.0588674
\(820\) 0 0
\(821\) 12323.5 0.523864 0.261932 0.965086i \(-0.415640\pi\)
0.261932 + 0.965086i \(0.415640\pi\)
\(822\) 0 0
\(823\) 34436.5 1.45854 0.729271 0.684225i \(-0.239860\pi\)
0.729271 + 0.684225i \(0.239860\pi\)
\(824\) 0 0
\(825\) 55135.7 2.32676
\(826\) 0 0
\(827\) −18761.6 −0.788880 −0.394440 0.918922i \(-0.629061\pi\)
−0.394440 + 0.918922i \(0.629061\pi\)
\(828\) 0 0
\(829\) 22423.8 0.939457 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(830\) 0 0
\(831\) 10235.3 0.427265
\(832\) 0 0
\(833\) 5583.15 0.232227
\(834\) 0 0
\(835\) 444.231 0.0184111
\(836\) 0 0
\(837\) −5276.13 −0.217885
\(838\) 0 0
\(839\) −9128.63 −0.375632 −0.187816 0.982204i \(-0.560141\pi\)
−0.187816 + 0.982204i \(0.560141\pi\)
\(840\) 0 0
\(841\) −22685.0 −0.930131
\(842\) 0 0
\(843\) 10146.5 0.414547
\(844\) 0 0
\(845\) −1888.32 −0.0768759
\(846\) 0 0
\(847\) −5390.75 −0.218688
\(848\) 0 0
\(849\) 26822.2 1.08426
\(850\) 0 0
\(851\) −23495.7 −0.946442
\(852\) 0 0
\(853\) 27204.8 1.09200 0.545999 0.837786i \(-0.316150\pi\)
0.545999 + 0.837786i \(0.316150\pi\)
\(854\) 0 0
\(855\) 2641.12 0.105643
\(856\) 0 0
\(857\) −38060.0 −1.51704 −0.758520 0.651649i \(-0.774077\pi\)
−0.758520 + 0.651649i \(0.774077\pi\)
\(858\) 0 0
\(859\) 33326.2 1.32372 0.661860 0.749627i \(-0.269767\pi\)
0.661860 + 0.749627i \(0.269767\pi\)
\(860\) 0 0
\(861\) −579.531 −0.0229389
\(862\) 0 0
\(863\) 41724.2 1.64578 0.822890 0.568201i \(-0.192360\pi\)
0.822890 + 0.568201i \(0.192360\pi\)
\(864\) 0 0
\(865\) −2294.06 −0.0901737
\(866\) 0 0
\(867\) −2449.38 −0.0959460
\(868\) 0 0
\(869\) −56569.7 −2.20828
\(870\) 0 0
\(871\) 124.068 0.00482649
\(872\) 0 0
\(873\) −17009.6 −0.659438
\(874\) 0 0
\(875\) 842.801 0.0325621
\(876\) 0 0
\(877\) −49337.3 −1.89966 −0.949830 0.312767i \(-0.898744\pi\)
−0.949830 + 0.312767i \(0.898744\pi\)
\(878\) 0 0
\(879\) −63194.0 −2.42489
\(880\) 0 0
\(881\) 8845.46 0.338265 0.169132 0.985593i \(-0.445903\pi\)
0.169132 + 0.985593i \(0.445903\pi\)
\(882\) 0 0
\(883\) −14724.2 −0.561165 −0.280582 0.959830i \(-0.590528\pi\)
−0.280582 + 0.959830i \(0.590528\pi\)
\(884\) 0 0
\(885\) 5094.36 0.193497
\(886\) 0 0
\(887\) −3864.38 −0.146283 −0.0731415 0.997322i \(-0.523302\pi\)
−0.0731415 + 0.997322i \(0.523302\pi\)
\(888\) 0 0
\(889\) 7360.60 0.277690
\(890\) 0 0
\(891\) 3688.07 0.138670
\(892\) 0 0
\(893\) −30849.1 −1.15602
\(894\) 0 0
\(895\) −1918.10 −0.0716368
\(896\) 0 0
\(897\) −12311.8 −0.458283
\(898\) 0 0
\(899\) −1441.14 −0.0534648
\(900\) 0 0
\(901\) 5609.04 0.207397
\(902\) 0 0
\(903\) −8984.15 −0.331089
\(904\) 0 0
\(905\) −1705.45 −0.0626421
\(906\) 0 0
\(907\) 743.409 0.0272155 0.0136078 0.999907i \(-0.495668\pi\)
0.0136078 + 0.999907i \(0.495668\pi\)
\(908\) 0 0
\(909\) 5905.76 0.215491
\(910\) 0 0
\(911\) −16291.0 −0.592475 −0.296238 0.955114i \(-0.595732\pi\)
−0.296238 + 0.955114i \(0.595732\pi\)
\(912\) 0 0
\(913\) 45336.8 1.64340
\(914\) 0 0
\(915\) −2554.33 −0.0922879
\(916\) 0 0
\(917\) −1553.04 −0.0559280
\(918\) 0 0
\(919\) 6188.99 0.222150 0.111075 0.993812i \(-0.464571\pi\)
0.111075 + 0.993812i \(0.464571\pi\)
\(920\) 0 0
\(921\) −55390.5 −1.98174
\(922\) 0 0
\(923\) −5402.00 −0.192643
\(924\) 0 0
\(925\) −16193.8 −0.575620
\(926\) 0 0
\(927\) −8786.47 −0.311311
\(928\) 0 0
\(929\) −31661.7 −1.11818 −0.559089 0.829108i \(-0.688849\pi\)
−0.559089 + 0.829108i \(0.688849\pi\)
\(930\) 0 0
\(931\) −21845.0 −0.769003
\(932\) 0 0
\(933\) −75734.8 −2.65750
\(934\) 0 0
\(935\) −788.551 −0.0275811
\(936\) 0 0
\(937\) 35010.5 1.22064 0.610322 0.792153i \(-0.291040\pi\)
0.610322 + 0.792153i \(0.291040\pi\)
\(938\) 0 0
\(939\) 22279.2 0.774286
\(940\) 0 0
\(941\) −45625.8 −1.58061 −0.790307 0.612711i \(-0.790079\pi\)
−0.790307 + 0.612711i \(0.790079\pi\)
\(942\) 0 0
\(943\) 3227.51 0.111455
\(944\) 0 0
\(945\) 511.092 0.0175934
\(946\) 0 0
\(947\) 21508.4 0.738044 0.369022 0.929421i \(-0.379693\pi\)
0.369022 + 0.929421i \(0.379693\pi\)
\(948\) 0 0
\(949\) −1559.50 −0.0533439
\(950\) 0 0
\(951\) −36177.4 −1.23358
\(952\) 0 0
\(953\) 35686.7 1.21302 0.606509 0.795076i \(-0.292569\pi\)
0.606509 + 0.795076i \(0.292569\pi\)
\(954\) 0 0
\(955\) 2465.34 0.0835355
\(956\) 0 0
\(957\) 18323.0 0.618912
\(958\) 0 0
\(959\) 498.486 0.0167851
\(960\) 0 0
\(961\) −28572.2 −0.959088
\(962\) 0 0
\(963\) 21749.9 0.727810
\(964\) 0 0
\(965\) 2000.12 0.0667215
\(966\) 0 0
\(967\) 3731.33 0.124086 0.0620432 0.998073i \(-0.480238\pi\)
0.0620432 + 0.998073i \(0.480238\pi\)
\(968\) 0 0
\(969\) 9583.60 0.317719
\(970\) 0 0
\(971\) −17645.1 −0.583171 −0.291585 0.956545i \(-0.594183\pi\)
−0.291585 + 0.956545i \(0.594183\pi\)
\(972\) 0 0
\(973\) 7917.35 0.260862
\(974\) 0 0
\(975\) −8485.59 −0.278725
\(976\) 0 0
\(977\) 24941.2 0.816723 0.408362 0.912820i \(-0.366100\pi\)
0.408362 + 0.912820i \(0.366100\pi\)
\(978\) 0 0
\(979\) 59152.1 1.93106
\(980\) 0 0
\(981\) −56268.9 −1.83132
\(982\) 0 0
\(983\) 22506.2 0.730252 0.365126 0.930958i \(-0.381026\pi\)
0.365126 + 0.930958i \(0.381026\pi\)
\(984\) 0 0
\(985\) −1125.44 −0.0364057
\(986\) 0 0
\(987\) −15008.8 −0.484029
\(988\) 0 0
\(989\) 50034.2 1.60869
\(990\) 0 0
\(991\) −32694.1 −1.04799 −0.523997 0.851720i \(-0.675560\pi\)
−0.523997 + 0.851720i \(0.675560\pi\)
\(992\) 0 0
\(993\) 8410.08 0.268767
\(994\) 0 0
\(995\) 4246.32 0.135294
\(996\) 0 0
\(997\) 18248.8 0.579686 0.289843 0.957074i \(-0.406397\pi\)
0.289843 + 0.957074i \(0.406397\pi\)
\(998\) 0 0
\(999\) −19702.5 −0.623983
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 272.4.a.h.1.1 3
3.2 odd 2 2448.4.a.bi.1.2 3
4.3 odd 2 17.4.a.b.1.1 3
8.3 odd 2 1088.4.a.v.1.1 3
8.5 even 2 1088.4.a.x.1.3 3
12.11 even 2 153.4.a.g.1.3 3
20.3 even 4 425.4.b.f.324.6 6
20.7 even 4 425.4.b.f.324.1 6
20.19 odd 2 425.4.a.g.1.3 3
28.27 even 2 833.4.a.d.1.1 3
44.43 even 2 2057.4.a.e.1.3 3
68.47 odd 4 289.4.b.b.288.6 6
68.55 odd 4 289.4.b.b.288.5 6
68.67 odd 2 289.4.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 4.3 odd 2
153.4.a.g.1.3 3 12.11 even 2
272.4.a.h.1.1 3 1.1 even 1 trivial
289.4.a.b.1.1 3 68.67 odd 2
289.4.b.b.288.5 6 68.55 odd 4
289.4.b.b.288.6 6 68.47 odd 4
425.4.a.g.1.3 3 20.19 odd 2
425.4.b.f.324.1 6 20.7 even 4
425.4.b.f.324.6 6 20.3 even 4
833.4.a.d.1.1 3 28.27 even 2
1088.4.a.v.1.1 3 8.3 odd 2
1088.4.a.x.1.3 3 8.5 even 2
2057.4.a.e.1.3 3 44.43 even 2
2448.4.a.bi.1.2 3 3.2 odd 2