Properties

Label 1856.4.a.z.1.2
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.69859\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.11424 q^{3} -16.3850 q^{5} +5.74609 q^{7} +23.6124 q^{9} +O(q^{10})\) \(q-7.11424 q^{3} -16.3850 q^{5} +5.74609 q^{7} +23.6124 q^{9} -23.3693 q^{11} +18.0318 q^{13} +116.567 q^{15} -24.1314 q^{17} +12.0191 q^{19} -40.8790 q^{21} -144.669 q^{23} +143.469 q^{25} +24.1004 q^{27} -29.0000 q^{29} +6.27664 q^{31} +166.255 q^{33} -94.1498 q^{35} -28.6658 q^{37} -128.282 q^{39} -436.340 q^{41} -495.080 q^{43} -386.889 q^{45} +351.351 q^{47} -309.982 q^{49} +171.676 q^{51} +58.0933 q^{53} +382.906 q^{55} -85.5068 q^{57} -485.535 q^{59} -607.926 q^{61} +135.679 q^{63} -295.451 q^{65} +296.974 q^{67} +1029.21 q^{69} -330.178 q^{71} -662.978 q^{73} -1020.67 q^{75} -134.282 q^{77} +145.394 q^{79} -808.990 q^{81} -1077.01 q^{83} +395.393 q^{85} +206.313 q^{87} -851.923 q^{89} +103.612 q^{91} -44.6535 q^{93} -196.933 q^{95} -227.768 q^{97} -551.804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 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
\(3\) −7.11424 −1.36914 −0.684568 0.728949i \(-0.740009\pi\)
−0.684568 + 0.728949i \(0.740009\pi\)
\(4\) 0 0
\(5\) −16.3850 −1.46552 −0.732760 0.680487i \(-0.761768\pi\)
−0.732760 + 0.680487i \(0.761768\pi\)
\(6\) 0 0
\(7\) 5.74609 0.310260 0.155130 0.987894i \(-0.450420\pi\)
0.155130 + 0.987894i \(0.450420\pi\)
\(8\) 0 0
\(9\) 23.6124 0.874532
\(10\) 0 0
\(11\) −23.3693 −0.640554 −0.320277 0.947324i \(-0.603776\pi\)
−0.320277 + 0.947324i \(0.603776\pi\)
\(12\) 0 0
\(13\) 18.0318 0.384701 0.192351 0.981326i \(-0.438389\pi\)
0.192351 + 0.981326i \(0.438389\pi\)
\(14\) 0 0
\(15\) 116.567 2.00650
\(16\) 0 0
\(17\) −24.1314 −0.344278 −0.172139 0.985073i \(-0.555068\pi\)
−0.172139 + 0.985073i \(0.555068\pi\)
\(18\) 0 0
\(19\) 12.0191 0.145125 0.0725624 0.997364i \(-0.476882\pi\)
0.0725624 + 0.997364i \(0.476882\pi\)
\(20\) 0 0
\(21\) −40.8790 −0.424788
\(22\) 0 0
\(23\) −144.669 −1.31155 −0.655776 0.754956i \(-0.727658\pi\)
−0.655776 + 0.754956i \(0.727658\pi\)
\(24\) 0 0
\(25\) 143.469 1.14775
\(26\) 0 0
\(27\) 24.1004 0.171782
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 6.27664 0.0363651 0.0181826 0.999835i \(-0.494212\pi\)
0.0181826 + 0.999835i \(0.494212\pi\)
\(32\) 0 0
\(33\) 166.255 0.877006
\(34\) 0 0
\(35\) −94.1498 −0.454692
\(36\) 0 0
\(37\) −28.6658 −0.127368 −0.0636842 0.997970i \(-0.520285\pi\)
−0.0636842 + 0.997970i \(0.520285\pi\)
\(38\) 0 0
\(39\) −128.282 −0.526708
\(40\) 0 0
\(41\) −436.340 −1.66207 −0.831035 0.556220i \(-0.812251\pi\)
−0.831035 + 0.556220i \(0.812251\pi\)
\(42\) 0 0
\(43\) −495.080 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(44\) 0 0
\(45\) −386.889 −1.28165
\(46\) 0 0
\(47\) 351.351 1.09042 0.545211 0.838299i \(-0.316450\pi\)
0.545211 + 0.838299i \(0.316450\pi\)
\(48\) 0 0
\(49\) −309.982 −0.903739
\(50\) 0 0
\(51\) 171.676 0.471363
\(52\) 0 0
\(53\) 58.0933 0.150561 0.0752804 0.997162i \(-0.476015\pi\)
0.0752804 + 0.997162i \(0.476015\pi\)
\(54\) 0 0
\(55\) 382.906 0.938746
\(56\) 0 0
\(57\) −85.5068 −0.198696
\(58\) 0 0
\(59\) −485.535 −1.07138 −0.535689 0.844415i \(-0.679948\pi\)
−0.535689 + 0.844415i \(0.679948\pi\)
\(60\) 0 0
\(61\) −607.926 −1.27602 −0.638008 0.770030i \(-0.720241\pi\)
−0.638008 + 0.770030i \(0.720241\pi\)
\(62\) 0 0
\(63\) 135.679 0.271332
\(64\) 0 0
\(65\) −295.451 −0.563788
\(66\) 0 0
\(67\) 296.974 0.541510 0.270755 0.962648i \(-0.412727\pi\)
0.270755 + 0.962648i \(0.412727\pi\)
\(68\) 0 0
\(69\) 1029.21 1.79569
\(70\) 0 0
\(71\) −330.178 −0.551901 −0.275950 0.961172i \(-0.588993\pi\)
−0.275950 + 0.961172i \(0.588993\pi\)
\(72\) 0 0
\(73\) −662.978 −1.06296 −0.531478 0.847072i \(-0.678363\pi\)
−0.531478 + 0.847072i \(0.678363\pi\)
\(74\) 0 0
\(75\) −1020.67 −1.57143
\(76\) 0 0
\(77\) −134.282 −0.198738
\(78\) 0 0
\(79\) 145.394 0.207065 0.103532 0.994626i \(-0.466985\pi\)
0.103532 + 0.994626i \(0.466985\pi\)
\(80\) 0 0
\(81\) −808.990 −1.10973
\(82\) 0 0
\(83\) −1077.01 −1.42431 −0.712154 0.702023i \(-0.752280\pi\)
−0.712154 + 0.702023i \(0.752280\pi\)
\(84\) 0 0
\(85\) 395.393 0.504546
\(86\) 0 0
\(87\) 206.313 0.254242
\(88\) 0 0
\(89\) −851.923 −1.01465 −0.507324 0.861755i \(-0.669365\pi\)
−0.507324 + 0.861755i \(0.669365\pi\)
\(90\) 0 0
\(91\) 103.612 0.119357
\(92\) 0 0
\(93\) −44.6535 −0.0497888
\(94\) 0 0
\(95\) −196.933 −0.212684
\(96\) 0 0
\(97\) −227.768 −0.238416 −0.119208 0.992869i \(-0.538036\pi\)
−0.119208 + 0.992869i \(0.538036\pi\)
\(98\) 0 0
\(99\) −551.804 −0.560186
\(100\) 0 0
\(101\) 77.3495 0.0762036 0.0381018 0.999274i \(-0.487869\pi\)
0.0381018 + 0.999274i \(0.487869\pi\)
\(102\) 0 0
\(103\) −751.327 −0.718742 −0.359371 0.933195i \(-0.617009\pi\)
−0.359371 + 0.933195i \(0.617009\pi\)
\(104\) 0 0
\(105\) 669.804 0.622535
\(106\) 0 0
\(107\) 761.498 0.688007 0.344004 0.938968i \(-0.388217\pi\)
0.344004 + 0.938968i \(0.388217\pi\)
\(108\) 0 0
\(109\) 661.001 0.580848 0.290424 0.956898i \(-0.406204\pi\)
0.290424 + 0.956898i \(0.406204\pi\)
\(110\) 0 0
\(111\) 203.935 0.174385
\(112\) 0 0
\(113\) −2266.45 −1.88681 −0.943404 0.331646i \(-0.892396\pi\)
−0.943404 + 0.331646i \(0.892396\pi\)
\(114\) 0 0
\(115\) 2370.41 1.92211
\(116\) 0 0
\(117\) 425.773 0.336434
\(118\) 0 0
\(119\) −138.661 −0.106815
\(120\) 0 0
\(121\) −784.877 −0.589690
\(122\) 0 0
\(123\) 3104.23 2.27560
\(124\) 0 0
\(125\) −302.612 −0.216532
\(126\) 0 0
\(127\) 420.522 0.293821 0.146911 0.989150i \(-0.453067\pi\)
0.146911 + 0.989150i \(0.453067\pi\)
\(128\) 0 0
\(129\) 3522.12 2.40392
\(130\) 0 0
\(131\) 208.443 0.139021 0.0695103 0.997581i \(-0.477856\pi\)
0.0695103 + 0.997581i \(0.477856\pi\)
\(132\) 0 0
\(133\) 69.0629 0.0450264
\(134\) 0 0
\(135\) −394.885 −0.251750
\(136\) 0 0
\(137\) −188.285 −0.117418 −0.0587089 0.998275i \(-0.518698\pi\)
−0.0587089 + 0.998275i \(0.518698\pi\)
\(138\) 0 0
\(139\) 1460.15 0.890996 0.445498 0.895283i \(-0.353027\pi\)
0.445498 + 0.895283i \(0.353027\pi\)
\(140\) 0 0
\(141\) −2499.59 −1.49293
\(142\) 0 0
\(143\) −421.390 −0.246422
\(144\) 0 0
\(145\) 475.166 0.272140
\(146\) 0 0
\(147\) 2205.29 1.23734
\(148\) 0 0
\(149\) −665.203 −0.365742 −0.182871 0.983137i \(-0.558539\pi\)
−0.182871 + 0.983137i \(0.558539\pi\)
\(150\) 0 0
\(151\) 3248.63 1.75079 0.875396 0.483406i \(-0.160601\pi\)
0.875396 + 0.483406i \(0.160601\pi\)
\(152\) 0 0
\(153\) −569.799 −0.301082
\(154\) 0 0
\(155\) −102.843 −0.0532938
\(156\) 0 0
\(157\) −1907.56 −0.969681 −0.484840 0.874603i \(-0.661122\pi\)
−0.484840 + 0.874603i \(0.661122\pi\)
\(158\) 0 0
\(159\) −413.289 −0.206138
\(160\) 0 0
\(161\) −831.284 −0.406921
\(162\) 0 0
\(163\) 2961.93 1.42329 0.711645 0.702539i \(-0.247950\pi\)
0.711645 + 0.702539i \(0.247950\pi\)
\(164\) 0 0
\(165\) −2724.08 −1.28527
\(166\) 0 0
\(167\) 2952.52 1.36810 0.684051 0.729434i \(-0.260217\pi\)
0.684051 + 0.729434i \(0.260217\pi\)
\(168\) 0 0
\(169\) −1871.85 −0.852005
\(170\) 0 0
\(171\) 283.800 0.126916
\(172\) 0 0
\(173\) −3500.71 −1.53846 −0.769232 0.638969i \(-0.779361\pi\)
−0.769232 + 0.638969i \(0.779361\pi\)
\(174\) 0 0
\(175\) 824.385 0.356101
\(176\) 0 0
\(177\) 3454.21 1.46686
\(178\) 0 0
\(179\) −4562.48 −1.90511 −0.952557 0.304359i \(-0.901558\pi\)
−0.952557 + 0.304359i \(0.901558\pi\)
\(180\) 0 0
\(181\) −326.513 −0.134086 −0.0670430 0.997750i \(-0.521356\pi\)
−0.0670430 + 0.997750i \(0.521356\pi\)
\(182\) 0 0
\(183\) 4324.93 1.74704
\(184\) 0 0
\(185\) 469.690 0.186661
\(186\) 0 0
\(187\) 563.933 0.220529
\(188\) 0 0
\(189\) 138.483 0.0532970
\(190\) 0 0
\(191\) 2623.16 0.993744 0.496872 0.867824i \(-0.334482\pi\)
0.496872 + 0.867824i \(0.334482\pi\)
\(192\) 0 0
\(193\) 894.480 0.333607 0.166803 0.985990i \(-0.446656\pi\)
0.166803 + 0.985990i \(0.446656\pi\)
\(194\) 0 0
\(195\) 2101.91 0.771902
\(196\) 0 0
\(197\) −2182.51 −0.789326 −0.394663 0.918826i \(-0.629139\pi\)
−0.394663 + 0.918826i \(0.629139\pi\)
\(198\) 0 0
\(199\) −635.874 −0.226512 −0.113256 0.993566i \(-0.536128\pi\)
−0.113256 + 0.993566i \(0.536128\pi\)
\(200\) 0 0
\(201\) −2112.74 −0.741400
\(202\) 0 0
\(203\) −166.637 −0.0576138
\(204\) 0 0
\(205\) 7149.44 2.43580
\(206\) 0 0
\(207\) −3415.99 −1.14699
\(208\) 0 0
\(209\) −280.878 −0.0929604
\(210\) 0 0
\(211\) 745.334 0.243180 0.121590 0.992580i \(-0.461201\pi\)
0.121590 + 0.992580i \(0.461201\pi\)
\(212\) 0 0
\(213\) 2348.97 0.755627
\(214\) 0 0
\(215\) 8111.90 2.57315
\(216\) 0 0
\(217\) 36.0661 0.0112826
\(218\) 0 0
\(219\) 4716.58 1.45533
\(220\) 0 0
\(221\) −435.132 −0.132444
\(222\) 0 0
\(223\) 159.305 0.0478380 0.0239190 0.999714i \(-0.492386\pi\)
0.0239190 + 0.999714i \(0.492386\pi\)
\(224\) 0 0
\(225\) 3387.64 1.00375
\(226\) 0 0
\(227\) 3699.17 1.08160 0.540798 0.841153i \(-0.318122\pi\)
0.540798 + 0.841153i \(0.318122\pi\)
\(228\) 0 0
\(229\) 6713.39 1.93726 0.968632 0.248499i \(-0.0799372\pi\)
0.968632 + 0.248499i \(0.0799372\pi\)
\(230\) 0 0
\(231\) 955.313 0.272100
\(232\) 0 0
\(233\) 1250.99 0.351740 0.175870 0.984413i \(-0.443726\pi\)
0.175870 + 0.984413i \(0.443726\pi\)
\(234\) 0 0
\(235\) −5756.89 −1.59803
\(236\) 0 0
\(237\) −1034.37 −0.283499
\(238\) 0 0
\(239\) −3073.10 −0.831726 −0.415863 0.909427i \(-0.636520\pi\)
−0.415863 + 0.909427i \(0.636520\pi\)
\(240\) 0 0
\(241\) 4745.39 1.26837 0.634185 0.773181i \(-0.281336\pi\)
0.634185 + 0.773181i \(0.281336\pi\)
\(242\) 0 0
\(243\) 5104.64 1.34758
\(244\) 0 0
\(245\) 5079.07 1.32445
\(246\) 0 0
\(247\) 216.726 0.0558298
\(248\) 0 0
\(249\) 7662.13 1.95007
\(250\) 0 0
\(251\) −1919.33 −0.482658 −0.241329 0.970443i \(-0.577583\pi\)
−0.241329 + 0.970443i \(0.577583\pi\)
\(252\) 0 0
\(253\) 3380.82 0.840120
\(254\) 0 0
\(255\) −2812.92 −0.690792
\(256\) 0 0
\(257\) 4680.45 1.13602 0.568012 0.823020i \(-0.307713\pi\)
0.568012 + 0.823020i \(0.307713\pi\)
\(258\) 0 0
\(259\) −164.716 −0.0395173
\(260\) 0 0
\(261\) −684.759 −0.162397
\(262\) 0 0
\(263\) −4404.71 −1.03272 −0.516361 0.856371i \(-0.672714\pi\)
−0.516361 + 0.856371i \(0.672714\pi\)
\(264\) 0 0
\(265\) −951.859 −0.220650
\(266\) 0 0
\(267\) 6060.78 1.38919
\(268\) 0 0
\(269\) −8244.86 −1.86877 −0.934383 0.356271i \(-0.884048\pi\)
−0.934383 + 0.356271i \(0.884048\pi\)
\(270\) 0 0
\(271\) −1050.02 −0.235366 −0.117683 0.993051i \(-0.537547\pi\)
−0.117683 + 0.993051i \(0.537547\pi\)
\(272\) 0 0
\(273\) −737.122 −0.163416
\(274\) 0 0
\(275\) −3352.76 −0.735197
\(276\) 0 0
\(277\) −3630.09 −0.787405 −0.393703 0.919238i \(-0.628806\pi\)
−0.393703 + 0.919238i \(0.628806\pi\)
\(278\) 0 0
\(279\) 148.206 0.0318025
\(280\) 0 0
\(281\) 6718.58 1.42632 0.713162 0.700999i \(-0.247262\pi\)
0.713162 + 0.700999i \(0.247262\pi\)
\(282\) 0 0
\(283\) 1096.86 0.230394 0.115197 0.993343i \(-0.463250\pi\)
0.115197 + 0.993343i \(0.463250\pi\)
\(284\) 0 0
\(285\) 1401.03 0.291193
\(286\) 0 0
\(287\) −2507.25 −0.515673
\(288\) 0 0
\(289\) −4330.68 −0.881473
\(290\) 0 0
\(291\) 1620.40 0.326424
\(292\) 0 0
\(293\) 1991.65 0.397111 0.198556 0.980090i \(-0.436375\pi\)
0.198556 + 0.980090i \(0.436375\pi\)
\(294\) 0 0
\(295\) 7955.50 1.57013
\(296\) 0 0
\(297\) −563.208 −0.110036
\(298\) 0 0
\(299\) −2608.65 −0.504556
\(300\) 0 0
\(301\) −2844.78 −0.544751
\(302\) 0 0
\(303\) −550.283 −0.104333
\(304\) 0 0
\(305\) 9960.88 1.87003
\(306\) 0 0
\(307\) 3847.15 0.715207 0.357604 0.933873i \(-0.383594\pi\)
0.357604 + 0.933873i \(0.383594\pi\)
\(308\) 0 0
\(309\) 5345.12 0.984055
\(310\) 0 0
\(311\) 8281.96 1.51005 0.755027 0.655693i \(-0.227624\pi\)
0.755027 + 0.655693i \(0.227624\pi\)
\(312\) 0 0
\(313\) −8930.42 −1.61271 −0.806354 0.591434i \(-0.798562\pi\)
−0.806354 + 0.591434i \(0.798562\pi\)
\(314\) 0 0
\(315\) −2223.10 −0.397643
\(316\) 0 0
\(317\) 2716.45 0.481297 0.240649 0.970612i \(-0.422640\pi\)
0.240649 + 0.970612i \(0.422640\pi\)
\(318\) 0 0
\(319\) 677.709 0.118948
\(320\) 0 0
\(321\) −5417.48 −0.941975
\(322\) 0 0
\(323\) −290.038 −0.0499633
\(324\) 0 0
\(325\) 2587.00 0.441541
\(326\) 0 0
\(327\) −4702.52 −0.795260
\(328\) 0 0
\(329\) 2018.89 0.338314
\(330\) 0 0
\(331\) −6445.84 −1.07038 −0.535189 0.844732i \(-0.679760\pi\)
−0.535189 + 0.844732i \(0.679760\pi\)
\(332\) 0 0
\(333\) −676.868 −0.111388
\(334\) 0 0
\(335\) −4865.92 −0.793593
\(336\) 0 0
\(337\) 6898.90 1.11515 0.557577 0.830125i \(-0.311731\pi\)
0.557577 + 0.830125i \(0.311731\pi\)
\(338\) 0 0
\(339\) 16124.0 2.58330
\(340\) 0 0
\(341\) −146.681 −0.0232938
\(342\) 0 0
\(343\) −3752.10 −0.590653
\(344\) 0 0
\(345\) −16863.7 −2.63162
\(346\) 0 0
\(347\) −4463.85 −0.690583 −0.345291 0.938496i \(-0.612220\pi\)
−0.345291 + 0.938496i \(0.612220\pi\)
\(348\) 0 0
\(349\) 9789.78 1.50153 0.750766 0.660568i \(-0.229685\pi\)
0.750766 + 0.660568i \(0.229685\pi\)
\(350\) 0 0
\(351\) 434.572 0.0660848
\(352\) 0 0
\(353\) −3601.80 −0.543072 −0.271536 0.962428i \(-0.587532\pi\)
−0.271536 + 0.962428i \(0.587532\pi\)
\(354\) 0 0
\(355\) 5409.98 0.808822
\(356\) 0 0
\(357\) 986.468 0.146245
\(358\) 0 0
\(359\) −11426.7 −1.67988 −0.839940 0.542679i \(-0.817410\pi\)
−0.839940 + 0.542679i \(0.817410\pi\)
\(360\) 0 0
\(361\) −6714.54 −0.978939
\(362\) 0 0
\(363\) 5583.80 0.807366
\(364\) 0 0
\(365\) 10862.9 1.55778
\(366\) 0 0
\(367\) 6545.41 0.930975 0.465487 0.885055i \(-0.345879\pi\)
0.465487 + 0.885055i \(0.345879\pi\)
\(368\) 0 0
\(369\) −10303.0 −1.45353
\(370\) 0 0
\(371\) 333.809 0.0467129
\(372\) 0 0
\(373\) −12488.0 −1.73352 −0.866761 0.498724i \(-0.833802\pi\)
−0.866761 + 0.498724i \(0.833802\pi\)
\(374\) 0 0
\(375\) 2152.85 0.296461
\(376\) 0 0
\(377\) −522.922 −0.0714373
\(378\) 0 0
\(379\) −10431.7 −1.41383 −0.706915 0.707298i \(-0.749914\pi\)
−0.706915 + 0.707298i \(0.749914\pi\)
\(380\) 0 0
\(381\) −2991.69 −0.402281
\(382\) 0 0
\(383\) −9500.35 −1.26748 −0.633741 0.773545i \(-0.718481\pi\)
−0.633741 + 0.773545i \(0.718481\pi\)
\(384\) 0 0
\(385\) 2200.21 0.291255
\(386\) 0 0
\(387\) −11690.0 −1.53550
\(388\) 0 0
\(389\) −12956.1 −1.68869 −0.844345 0.535800i \(-0.820010\pi\)
−0.844345 + 0.535800i \(0.820010\pi\)
\(390\) 0 0
\(391\) 3491.08 0.451538
\(392\) 0 0
\(393\) −1482.91 −0.190338
\(394\) 0 0
\(395\) −2382.28 −0.303457
\(396\) 0 0
\(397\) 167.727 0.0212040 0.0106020 0.999944i \(-0.496625\pi\)
0.0106020 + 0.999944i \(0.496625\pi\)
\(398\) 0 0
\(399\) −491.330 −0.0616473
\(400\) 0 0
\(401\) 13225.9 1.64705 0.823526 0.567279i \(-0.192004\pi\)
0.823526 + 0.567279i \(0.192004\pi\)
\(402\) 0 0
\(403\) 113.179 0.0139897
\(404\) 0 0
\(405\) 13255.3 1.62633
\(406\) 0 0
\(407\) 669.899 0.0815864
\(408\) 0 0
\(409\) −674.413 −0.0815344 −0.0407672 0.999169i \(-0.512980\pi\)
−0.0407672 + 0.999169i \(0.512980\pi\)
\(410\) 0 0
\(411\) 1339.50 0.160761
\(412\) 0 0
\(413\) −2789.93 −0.332405
\(414\) 0 0
\(415\) 17646.9 2.08735
\(416\) 0 0
\(417\) −10387.9 −1.21989
\(418\) 0 0
\(419\) 12355.6 1.44059 0.720296 0.693667i \(-0.244006\pi\)
0.720296 + 0.693667i \(0.244006\pi\)
\(420\) 0 0
\(421\) 8788.18 1.01736 0.508682 0.860955i \(-0.330133\pi\)
0.508682 + 0.860955i \(0.330133\pi\)
\(422\) 0 0
\(423\) 8296.23 0.953609
\(424\) 0 0
\(425\) −3462.10 −0.395145
\(426\) 0 0
\(427\) −3493.20 −0.395896
\(428\) 0 0
\(429\) 2997.87 0.337385
\(430\) 0 0
\(431\) 15171.7 1.69559 0.847793 0.530328i \(-0.177931\pi\)
0.847793 + 0.530328i \(0.177931\pi\)
\(432\) 0 0
\(433\) 3884.76 0.431154 0.215577 0.976487i \(-0.430837\pi\)
0.215577 + 0.976487i \(0.430837\pi\)
\(434\) 0 0
\(435\) −3380.44 −0.372597
\(436\) 0 0
\(437\) −1738.80 −0.190339
\(438\) 0 0
\(439\) 14998.9 1.63066 0.815329 0.578998i \(-0.196556\pi\)
0.815329 + 0.578998i \(0.196556\pi\)
\(440\) 0 0
\(441\) −7319.42 −0.790349
\(442\) 0 0
\(443\) −4945.82 −0.530435 −0.265218 0.964189i \(-0.585444\pi\)
−0.265218 + 0.964189i \(0.585444\pi\)
\(444\) 0 0
\(445\) 13958.8 1.48699
\(446\) 0 0
\(447\) 4732.41 0.500750
\(448\) 0 0
\(449\) 13041.0 1.37070 0.685351 0.728213i \(-0.259649\pi\)
0.685351 + 0.728213i \(0.259649\pi\)
\(450\) 0 0
\(451\) 10196.9 1.06465
\(452\) 0 0
\(453\) −23111.5 −2.39707
\(454\) 0 0
\(455\) −1697.69 −0.174921
\(456\) 0 0
\(457\) 3833.64 0.392408 0.196204 0.980563i \(-0.437139\pi\)
0.196204 + 0.980563i \(0.437139\pi\)
\(458\) 0 0
\(459\) −581.575 −0.0591407
\(460\) 0 0
\(461\) 2806.70 0.283560 0.141780 0.989898i \(-0.454717\pi\)
0.141780 + 0.989898i \(0.454717\pi\)
\(462\) 0 0
\(463\) −4240.08 −0.425600 −0.212800 0.977096i \(-0.568258\pi\)
−0.212800 + 0.977096i \(0.568258\pi\)
\(464\) 0 0
\(465\) 731.649 0.0729665
\(466\) 0 0
\(467\) −7538.50 −0.746981 −0.373491 0.927634i \(-0.621839\pi\)
−0.373491 + 0.927634i \(0.621839\pi\)
\(468\) 0 0
\(469\) 1706.44 0.168009
\(470\) 0 0
\(471\) 13570.8 1.32762
\(472\) 0 0
\(473\) 11569.7 1.12468
\(474\) 0 0
\(475\) 1724.37 0.166567
\(476\) 0 0
\(477\) 1371.72 0.131670
\(478\) 0 0
\(479\) −14810.8 −1.41279 −0.706393 0.707820i \(-0.749679\pi\)
−0.706393 + 0.707820i \(0.749679\pi\)
\(480\) 0 0
\(481\) −516.896 −0.0489988
\(482\) 0 0
\(483\) 5913.95 0.557131
\(484\) 0 0
\(485\) 3731.98 0.349403
\(486\) 0 0
\(487\) 2951.06 0.274590 0.137295 0.990530i \(-0.456159\pi\)
0.137295 + 0.990530i \(0.456159\pi\)
\(488\) 0 0
\(489\) −21071.9 −1.94868
\(490\) 0 0
\(491\) −19549.3 −1.79683 −0.898417 0.439143i \(-0.855282\pi\)
−0.898417 + 0.439143i \(0.855282\pi\)
\(492\) 0 0
\(493\) 699.810 0.0639308
\(494\) 0 0
\(495\) 9041.32 0.820964
\(496\) 0 0
\(497\) −1897.23 −0.171233
\(498\) 0 0
\(499\) 5713.30 0.512550 0.256275 0.966604i \(-0.417505\pi\)
0.256275 + 0.966604i \(0.417505\pi\)
\(500\) 0 0
\(501\) −21004.9 −1.87312
\(502\) 0 0
\(503\) 7379.92 0.654184 0.327092 0.944993i \(-0.393931\pi\)
0.327092 + 0.944993i \(0.393931\pi\)
\(504\) 0 0
\(505\) −1267.37 −0.111678
\(506\) 0 0
\(507\) 13316.8 1.16651
\(508\) 0 0
\(509\) −4642.94 −0.404312 −0.202156 0.979353i \(-0.564795\pi\)
−0.202156 + 0.979353i \(0.564795\pi\)
\(510\) 0 0
\(511\) −3809.53 −0.329792
\(512\) 0 0
\(513\) 289.665 0.0249299
\(514\) 0 0
\(515\) 12310.5 1.05333
\(516\) 0 0
\(517\) −8210.81 −0.698474
\(518\) 0 0
\(519\) 24904.9 2.10637
\(520\) 0 0
\(521\) −11184.4 −0.940494 −0.470247 0.882535i \(-0.655835\pi\)
−0.470247 + 0.882535i \(0.655835\pi\)
\(522\) 0 0
\(523\) −10752.9 −0.899025 −0.449512 0.893274i \(-0.648402\pi\)
−0.449512 + 0.893274i \(0.648402\pi\)
\(524\) 0 0
\(525\) −5864.87 −0.487550
\(526\) 0 0
\(527\) −151.464 −0.0125197
\(528\) 0 0
\(529\) 8762.26 0.720166
\(530\) 0 0
\(531\) −11464.6 −0.936955
\(532\) 0 0
\(533\) −7867.99 −0.639401
\(534\) 0 0
\(535\) −12477.2 −1.00829
\(536\) 0 0
\(537\) 32458.5 2.60836
\(538\) 0 0
\(539\) 7244.06 0.578894
\(540\) 0 0
\(541\) 20175.9 1.60338 0.801691 0.597738i \(-0.203934\pi\)
0.801691 + 0.597738i \(0.203934\pi\)
\(542\) 0 0
\(543\) 2322.89 0.183582
\(544\) 0 0
\(545\) −10830.5 −0.851245
\(546\) 0 0
\(547\) 9678.22 0.756510 0.378255 0.925702i \(-0.376524\pi\)
0.378255 + 0.925702i \(0.376524\pi\)
\(548\) 0 0
\(549\) −14354.6 −1.11592
\(550\) 0 0
\(551\) −348.554 −0.0269490
\(552\) 0 0
\(553\) 835.447 0.0642438
\(554\) 0 0
\(555\) −3341.49 −0.255564
\(556\) 0 0
\(557\) −19870.3 −1.51155 −0.755775 0.654832i \(-0.772740\pi\)
−0.755775 + 0.654832i \(0.772740\pi\)
\(558\) 0 0
\(559\) −8927.18 −0.675455
\(560\) 0 0
\(561\) −4011.95 −0.301934
\(562\) 0 0
\(563\) −9424.02 −0.705462 −0.352731 0.935725i \(-0.614747\pi\)
−0.352731 + 0.935725i \(0.614747\pi\)
\(564\) 0 0
\(565\) 37135.8 2.76516
\(566\) 0 0
\(567\) −4648.53 −0.344303
\(568\) 0 0
\(569\) −14765.7 −1.08789 −0.543945 0.839121i \(-0.683070\pi\)
−0.543945 + 0.839121i \(0.683070\pi\)
\(570\) 0 0
\(571\) 10061.1 0.737380 0.368690 0.929552i \(-0.379806\pi\)
0.368690 + 0.929552i \(0.379806\pi\)
\(572\) 0 0
\(573\) −18661.8 −1.36057
\(574\) 0 0
\(575\) −20755.6 −1.50533
\(576\) 0 0
\(577\) −26307.2 −1.89806 −0.949032 0.315179i \(-0.897935\pi\)
−0.949032 + 0.315179i \(0.897935\pi\)
\(578\) 0 0
\(579\) −6363.54 −0.456753
\(580\) 0 0
\(581\) −6188.62 −0.441905
\(582\) 0 0
\(583\) −1357.60 −0.0964424
\(584\) 0 0
\(585\) −6976.30 −0.493051
\(586\) 0 0
\(587\) −2053.70 −0.144404 −0.0722021 0.997390i \(-0.523003\pi\)
−0.0722021 + 0.997390i \(0.523003\pi\)
\(588\) 0 0
\(589\) 75.4397 0.00527748
\(590\) 0 0
\(591\) 15526.9 1.08069
\(592\) 0 0
\(593\) 12767.4 0.884140 0.442070 0.896981i \(-0.354244\pi\)
0.442070 + 0.896981i \(0.354244\pi\)
\(594\) 0 0
\(595\) 2271.96 0.156540
\(596\) 0 0
\(597\) 4523.76 0.310126
\(598\) 0 0
\(599\) −22284.1 −1.52004 −0.760021 0.649899i \(-0.774811\pi\)
−0.760021 + 0.649899i \(0.774811\pi\)
\(600\) 0 0
\(601\) 1145.05 0.0777166 0.0388583 0.999245i \(-0.487628\pi\)
0.0388583 + 0.999245i \(0.487628\pi\)
\(602\) 0 0
\(603\) 7012.26 0.473568
\(604\) 0 0
\(605\) 12860.2 0.864203
\(606\) 0 0
\(607\) 5726.01 0.382886 0.191443 0.981504i \(-0.438683\pi\)
0.191443 + 0.981504i \(0.438683\pi\)
\(608\) 0 0
\(609\) 1185.49 0.0788811
\(610\) 0 0
\(611\) 6335.49 0.419487
\(612\) 0 0
\(613\) −17973.2 −1.18423 −0.592113 0.805855i \(-0.701706\pi\)
−0.592113 + 0.805855i \(0.701706\pi\)
\(614\) 0 0
\(615\) −50862.8 −3.33494
\(616\) 0 0
\(617\) 28779.6 1.87783 0.938916 0.344146i \(-0.111832\pi\)
0.938916 + 0.344146i \(0.111832\pi\)
\(618\) 0 0
\(619\) 20981.6 1.36239 0.681197 0.732100i \(-0.261460\pi\)
0.681197 + 0.732100i \(0.261460\pi\)
\(620\) 0 0
\(621\) −3486.59 −0.225301
\(622\) 0 0
\(623\) −4895.22 −0.314804
\(624\) 0 0
\(625\) −12975.3 −0.830419
\(626\) 0 0
\(627\) 1998.23 0.127275
\(628\) 0 0
\(629\) 691.746 0.0438501
\(630\) 0 0
\(631\) 618.760 0.0390372 0.0195186 0.999809i \(-0.493787\pi\)
0.0195186 + 0.999809i \(0.493787\pi\)
\(632\) 0 0
\(633\) −5302.49 −0.332946
\(634\) 0 0
\(635\) −6890.26 −0.430601
\(636\) 0 0
\(637\) −5589.54 −0.347670
\(638\) 0 0
\(639\) −7796.29 −0.482655
\(640\) 0 0
\(641\) −21528.5 −1.32656 −0.663280 0.748371i \(-0.730836\pi\)
−0.663280 + 0.748371i \(0.730836\pi\)
\(642\) 0 0
\(643\) −8142.79 −0.499410 −0.249705 0.968322i \(-0.580334\pi\)
−0.249705 + 0.968322i \(0.580334\pi\)
\(644\) 0 0
\(645\) −57710.0 −3.52299
\(646\) 0 0
\(647\) 12779.6 0.776538 0.388269 0.921546i \(-0.373073\pi\)
0.388269 + 0.921546i \(0.373073\pi\)
\(648\) 0 0
\(649\) 11346.6 0.686276
\(650\) 0 0
\(651\) −256.583 −0.0154474
\(652\) 0 0
\(653\) 19463.5 1.16641 0.583205 0.812325i \(-0.301799\pi\)
0.583205 + 0.812325i \(0.301799\pi\)
\(654\) 0 0
\(655\) −3415.33 −0.203738
\(656\) 0 0
\(657\) −15654.5 −0.929589
\(658\) 0 0
\(659\) 23009.3 1.36011 0.680056 0.733160i \(-0.261955\pi\)
0.680056 + 0.733160i \(0.261955\pi\)
\(660\) 0 0
\(661\) −4135.64 −0.243355 −0.121678 0.992570i \(-0.538827\pi\)
−0.121678 + 0.992570i \(0.538827\pi\)
\(662\) 0 0
\(663\) 3095.63 0.181334
\(664\) 0 0
\(665\) −1131.60 −0.0659871
\(666\) 0 0
\(667\) 4195.42 0.243549
\(668\) 0 0
\(669\) −1133.34 −0.0654967
\(670\) 0 0
\(671\) 14206.8 0.817357
\(672\) 0 0
\(673\) 19479.7 1.11573 0.557865 0.829931i \(-0.311621\pi\)
0.557865 + 0.829931i \(0.311621\pi\)
\(674\) 0 0
\(675\) 3457.65 0.197163
\(676\) 0 0
\(677\) −5334.44 −0.302835 −0.151417 0.988470i \(-0.548384\pi\)
−0.151417 + 0.988470i \(0.548384\pi\)
\(678\) 0 0
\(679\) −1308.78 −0.0739708
\(680\) 0 0
\(681\) −26316.7 −1.48085
\(682\) 0 0
\(683\) −8454.59 −0.473654 −0.236827 0.971552i \(-0.576107\pi\)
−0.236827 + 0.971552i \(0.576107\pi\)
\(684\) 0 0
\(685\) 3085.05 0.172078
\(686\) 0 0
\(687\) −47760.7 −2.65238
\(688\) 0 0
\(689\) 1047.53 0.0579210
\(690\) 0 0
\(691\) −10176.2 −0.560231 −0.280115 0.959966i \(-0.590373\pi\)
−0.280115 + 0.959966i \(0.590373\pi\)
\(692\) 0 0
\(693\) −3170.71 −0.173803
\(694\) 0 0
\(695\) −23924.6 −1.30577
\(696\) 0 0
\(697\) 10529.5 0.572214
\(698\) 0 0
\(699\) −8899.87 −0.481579
\(700\) 0 0
\(701\) 20411.6 1.09976 0.549882 0.835243i \(-0.314673\pi\)
0.549882 + 0.835243i \(0.314673\pi\)
\(702\) 0 0
\(703\) −344.538 −0.0184843
\(704\) 0 0
\(705\) 40955.9 2.18793
\(706\) 0 0
\(707\) 444.457 0.0236429
\(708\) 0 0
\(709\) −8169.75 −0.432752 −0.216376 0.976310i \(-0.569424\pi\)
−0.216376 + 0.976310i \(0.569424\pi\)
\(710\) 0 0
\(711\) 3433.10 0.181085
\(712\) 0 0
\(713\) −908.039 −0.0476947
\(714\) 0 0
\(715\) 6904.48 0.361137
\(716\) 0 0
\(717\) 21862.8 1.13875
\(718\) 0 0
\(719\) 2017.01 0.104620 0.0523101 0.998631i \(-0.483342\pi\)
0.0523101 + 0.998631i \(0.483342\pi\)
\(720\) 0 0
\(721\) −4317.19 −0.222997
\(722\) 0 0
\(723\) −33759.8 −1.73657
\(724\) 0 0
\(725\) −4160.60 −0.213132
\(726\) 0 0
\(727\) 9836.46 0.501807 0.250904 0.968012i \(-0.419272\pi\)
0.250904 + 0.968012i \(0.419272\pi\)
\(728\) 0 0
\(729\) −14472.9 −0.735298
\(730\) 0 0
\(731\) 11947.0 0.604480
\(732\) 0 0
\(733\) −14843.6 −0.747969 −0.373984 0.927435i \(-0.622009\pi\)
−0.373984 + 0.927435i \(0.622009\pi\)
\(734\) 0 0
\(735\) −36133.7 −1.81335
\(736\) 0 0
\(737\) −6940.06 −0.346866
\(738\) 0 0
\(739\) −2834.15 −0.141077 −0.0705385 0.997509i \(-0.522472\pi\)
−0.0705385 + 0.997509i \(0.522472\pi\)
\(740\) 0 0
\(741\) −1541.84 −0.0764385
\(742\) 0 0
\(743\) −22076.2 −1.09004 −0.545019 0.838424i \(-0.683477\pi\)
−0.545019 + 0.838424i \(0.683477\pi\)
\(744\) 0 0
\(745\) 10899.4 0.536002
\(746\) 0 0
\(747\) −25430.9 −1.24560
\(748\) 0 0
\(749\) 4375.63 0.213461
\(750\) 0 0
\(751\) 24518.8 1.19135 0.595675 0.803225i \(-0.296885\pi\)
0.595675 + 0.803225i \(0.296885\pi\)
\(752\) 0 0
\(753\) 13654.6 0.660824
\(754\) 0 0
\(755\) −53228.8 −2.56582
\(756\) 0 0
\(757\) 19834.9 0.952326 0.476163 0.879357i \(-0.342027\pi\)
0.476163 + 0.879357i \(0.342027\pi\)
\(758\) 0 0
\(759\) −24052.0 −1.15024
\(760\) 0 0
\(761\) 17719.2 0.844048 0.422024 0.906585i \(-0.361320\pi\)
0.422024 + 0.906585i \(0.361320\pi\)
\(762\) 0 0
\(763\) 3798.17 0.180214
\(764\) 0 0
\(765\) 9336.17 0.441242
\(766\) 0 0
\(767\) −8755.07 −0.412161
\(768\) 0 0
\(769\) −17318.8 −0.812134 −0.406067 0.913843i \(-0.633100\pi\)
−0.406067 + 0.913843i \(0.633100\pi\)
\(770\) 0 0
\(771\) −33297.8 −1.55537
\(772\) 0 0
\(773\) −6232.89 −0.290015 −0.145007 0.989431i \(-0.546321\pi\)
−0.145007 + 0.989431i \(0.546321\pi\)
\(774\) 0 0
\(775\) 900.502 0.0417381
\(776\) 0 0
\(777\) 1171.83 0.0541045
\(778\) 0 0
\(779\) −5244.42 −0.241208
\(780\) 0 0
\(781\) 7716.02 0.353522
\(782\) 0 0
\(783\) −698.910 −0.0318991
\(784\) 0 0
\(785\) 31255.4 1.42109
\(786\) 0 0
\(787\) 18625.8 0.843634 0.421817 0.906681i \(-0.361393\pi\)
0.421817 + 0.906681i \(0.361393\pi\)
\(788\) 0 0
\(789\) 31336.1 1.41394
\(790\) 0 0
\(791\) −13023.2 −0.585400
\(792\) 0 0
\(793\) −10962.0 −0.490885
\(794\) 0 0
\(795\) 6771.75 0.302100
\(796\) 0 0
\(797\) 39096.7 1.73761 0.868805 0.495155i \(-0.164889\pi\)
0.868805 + 0.495155i \(0.164889\pi\)
\(798\) 0 0
\(799\) −8478.59 −0.375408
\(800\) 0 0
\(801\) −20115.9 −0.887342
\(802\) 0 0
\(803\) 15493.3 0.680881
\(804\) 0 0
\(805\) 13620.6 0.596352
\(806\) 0 0
\(807\) 58655.9 2.55859
\(808\) 0 0
\(809\) −13821.7 −0.600672 −0.300336 0.953833i \(-0.597099\pi\)
−0.300336 + 0.953833i \(0.597099\pi\)
\(810\) 0 0
\(811\) 7106.62 0.307703 0.153852 0.988094i \(-0.450832\pi\)
0.153852 + 0.988094i \(0.450832\pi\)
\(812\) 0 0
\(813\) 7470.10 0.322248
\(814\) 0 0
\(815\) −48531.3 −2.08586
\(816\) 0 0
\(817\) −5950.42 −0.254809
\(818\) 0 0
\(819\) 2446.53 0.104382
\(820\) 0 0
\(821\) 22099.3 0.939427 0.469714 0.882819i \(-0.344357\pi\)
0.469714 + 0.882819i \(0.344357\pi\)
\(822\) 0 0
\(823\) −11652.0 −0.493517 −0.246758 0.969077i \(-0.579365\pi\)
−0.246758 + 0.969077i \(0.579365\pi\)
\(824\) 0 0
\(825\) 23852.3 1.00658
\(826\) 0 0
\(827\) 14728.5 0.619298 0.309649 0.950851i \(-0.399788\pi\)
0.309649 + 0.950851i \(0.399788\pi\)
\(828\) 0 0
\(829\) 11640.5 0.487684 0.243842 0.969815i \(-0.421592\pi\)
0.243842 + 0.969815i \(0.421592\pi\)
\(830\) 0 0
\(831\) 25825.4 1.07806
\(832\) 0 0
\(833\) 7480.31 0.311137
\(834\) 0 0
\(835\) −48377.1 −2.00498
\(836\) 0 0
\(837\) 151.269 0.00624687
\(838\) 0 0
\(839\) 17270.0 0.710638 0.355319 0.934745i \(-0.384372\pi\)
0.355319 + 0.934745i \(0.384372\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −47797.6 −1.95283
\(844\) 0 0
\(845\) 30670.4 1.24863
\(846\) 0 0
\(847\) −4509.98 −0.182957
\(848\) 0 0
\(849\) −7803.32 −0.315441
\(850\) 0 0
\(851\) 4147.07 0.167050
\(852\) 0 0
\(853\) −12968.3 −0.520547 −0.260274 0.965535i \(-0.583813\pi\)
−0.260274 + 0.965535i \(0.583813\pi\)
\(854\) 0 0
\(855\) −4650.06 −0.185999
\(856\) 0 0
\(857\) −9597.62 −0.382554 −0.191277 0.981536i \(-0.561263\pi\)
−0.191277 + 0.981536i \(0.561263\pi\)
\(858\) 0 0
\(859\) −18055.5 −0.717164 −0.358582 0.933498i \(-0.616740\pi\)
−0.358582 + 0.933498i \(0.616740\pi\)
\(860\) 0 0
\(861\) 17837.2 0.706027
\(862\) 0 0
\(863\) −1068.84 −0.0421594 −0.0210797 0.999778i \(-0.506710\pi\)
−0.0210797 + 0.999778i \(0.506710\pi\)
\(864\) 0 0
\(865\) 57359.3 2.25465
\(866\) 0 0
\(867\) 30809.5 1.20686
\(868\) 0 0
\(869\) −3397.75 −0.132636
\(870\) 0 0
\(871\) 5354.97 0.208319
\(872\) 0 0
\(873\) −5378.15 −0.208502
\(874\) 0 0
\(875\) −1738.84 −0.0671810
\(876\) 0 0
\(877\) −50968.4 −1.96246 −0.981231 0.192837i \(-0.938231\pi\)
−0.981231 + 0.192837i \(0.938231\pi\)
\(878\) 0 0
\(879\) −14169.1 −0.543699
\(880\) 0 0
\(881\) −11100.3 −0.424495 −0.212247 0.977216i \(-0.568078\pi\)
−0.212247 + 0.977216i \(0.568078\pi\)
\(882\) 0 0
\(883\) −24959.0 −0.951232 −0.475616 0.879653i \(-0.657775\pi\)
−0.475616 + 0.879653i \(0.657775\pi\)
\(884\) 0 0
\(885\) −56597.3 −2.14972
\(886\) 0 0
\(887\) −22444.1 −0.849604 −0.424802 0.905286i \(-0.639656\pi\)
−0.424802 + 0.905286i \(0.639656\pi\)
\(888\) 0 0
\(889\) 2416.36 0.0911609
\(890\) 0 0
\(891\) 18905.5 0.710839
\(892\) 0 0
\(893\) 4222.93 0.158247
\(894\) 0 0
\(895\) 74756.3 2.79199
\(896\) 0 0
\(897\) 18558.6 0.690805
\(898\) 0 0
\(899\) −182.023 −0.00675283
\(900\) 0 0
\(901\) −1401.87 −0.0518347
\(902\) 0 0
\(903\) 20238.4 0.745838
\(904\) 0 0
\(905\) 5349.93 0.196506
\(906\) 0 0
\(907\) −18929.6 −0.692994 −0.346497 0.938051i \(-0.612629\pi\)
−0.346497 + 0.938051i \(0.612629\pi\)
\(908\) 0 0
\(909\) 1826.41 0.0666425
\(910\) 0 0
\(911\) −32210.8 −1.17145 −0.585726 0.810509i \(-0.699190\pi\)
−0.585726 + 0.810509i \(0.699190\pi\)
\(912\) 0 0
\(913\) 25169.0 0.912347
\(914\) 0 0
\(915\) −70864.1 −2.56032
\(916\) 0 0
\(917\) 1197.73 0.0431325
\(918\) 0 0
\(919\) −13885.5 −0.498411 −0.249206 0.968451i \(-0.580169\pi\)
−0.249206 + 0.968451i \(0.580169\pi\)
\(920\) 0 0
\(921\) −27369.6 −0.979216
\(922\) 0 0
\(923\) −5953.70 −0.212317
\(924\) 0 0
\(925\) −4112.65 −0.146187
\(926\) 0 0
\(927\) −17740.6 −0.628563
\(928\) 0 0
\(929\) 46561.6 1.64439 0.822194 0.569207i \(-0.192750\pi\)
0.822194 + 0.569207i \(0.192750\pi\)
\(930\) 0 0
\(931\) −3725.71 −0.131155
\(932\) 0 0
\(933\) −58919.9 −2.06747
\(934\) 0 0
\(935\) −9240.05 −0.323189
\(936\) 0 0
\(937\) −3046.54 −0.106218 −0.0531088 0.998589i \(-0.516913\pi\)
−0.0531088 + 0.998589i \(0.516913\pi\)
\(938\) 0 0
\(939\) 63533.2 2.20801
\(940\) 0 0
\(941\) −24897.4 −0.862521 −0.431261 0.902227i \(-0.641931\pi\)
−0.431261 + 0.902227i \(0.641931\pi\)
\(942\) 0 0
\(943\) 63125.1 2.17989
\(944\) 0 0
\(945\) −2269.04 −0.0781079
\(946\) 0 0
\(947\) −2787.52 −0.0956519 −0.0478259 0.998856i \(-0.515229\pi\)
−0.0478259 + 0.998856i \(0.515229\pi\)
\(948\) 0 0
\(949\) −11954.7 −0.408920
\(950\) 0 0
\(951\) −19325.5 −0.658962
\(952\) 0 0
\(953\) 45400.9 1.54321 0.771605 0.636102i \(-0.219454\pi\)
0.771605 + 0.636102i \(0.219454\pi\)
\(954\) 0 0
\(955\) −42980.5 −1.45635
\(956\) 0 0
\(957\) −4821.38 −0.162856
\(958\) 0 0
\(959\) −1081.90 −0.0364300
\(960\) 0 0
\(961\) −29751.6 −0.998678
\(962\) 0 0
\(963\) 17980.8 0.601685
\(964\) 0 0
\(965\) −14656.1 −0.488907
\(966\) 0 0
\(967\) −43256.9 −1.43852 −0.719259 0.694742i \(-0.755519\pi\)
−0.719259 + 0.694742i \(0.755519\pi\)
\(968\) 0 0
\(969\) 2063.40 0.0684065
\(970\) 0 0
\(971\) −35734.1 −1.18101 −0.590505 0.807034i \(-0.701072\pi\)
−0.590505 + 0.807034i \(0.701072\pi\)
\(972\) 0 0
\(973\) 8390.16 0.276440
\(974\) 0 0
\(975\) −18404.5 −0.604530
\(976\) 0 0
\(977\) −15675.2 −0.513299 −0.256650 0.966505i \(-0.582619\pi\)
−0.256650 + 0.966505i \(0.582619\pi\)
\(978\) 0 0
\(979\) 19908.8 0.649937
\(980\) 0 0
\(981\) 15607.8 0.507970
\(982\) 0 0
\(983\) 17324.1 0.562109 0.281055 0.959692i \(-0.409316\pi\)
0.281055 + 0.959692i \(0.409316\pi\)
\(984\) 0 0
\(985\) 35760.4 1.15677
\(986\) 0 0
\(987\) −14362.9 −0.463197
\(988\) 0 0
\(989\) 71623.0 2.30281
\(990\) 0 0
\(991\) 22853.9 0.732572 0.366286 0.930502i \(-0.380629\pi\)
0.366286 + 0.930502i \(0.380629\pi\)
\(992\) 0 0
\(993\) 45857.2 1.46549
\(994\) 0 0
\(995\) 10418.8 0.331958
\(996\) 0 0
\(997\) 20703.7 0.657667 0.328834 0.944388i \(-0.393344\pi\)
0.328834 + 0.944388i \(0.393344\pi\)
\(998\) 0 0
\(999\) −690.856 −0.0218796
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 1856.4.a.z.1.2 5
4.3 odd 2 1856.4.a.ba.1.4 5
8.3 odd 2 464.4.a.m.1.2 5
8.5 even 2 232.4.a.d.1.4 5
24.5 odd 2 2088.4.a.f.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.d.1.4 5 8.5 even 2
464.4.a.m.1.2 5 8.3 odd 2
1856.4.a.z.1.2 5 1.1 even 1 trivial
1856.4.a.ba.1.4 5 4.3 odd 2
2088.4.a.f.1.1 5 24.5 odd 2