Properties

Label 294.4.a.j.1.2
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +15.8995 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +15.8995 q^{5} +6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} -31.7990 q^{10} +57.3970 q^{11} -12.0000 q^{12} -5.69848 q^{13} -47.6985 q^{15} +16.0000 q^{16} +51.8995 q^{17} -18.0000 q^{18} -16.2010 q^{19} +63.5980 q^{20} -114.794 q^{22} -213.397 q^{23} +24.0000 q^{24} +127.794 q^{25} +11.3970 q^{26} -27.0000 q^{27} -218.191 q^{29} +95.3970 q^{30} +251.397 q^{31} -32.0000 q^{32} -172.191 q^{33} -103.799 q^{34} +36.0000 q^{36} +386.794 q^{37} +32.4020 q^{38} +17.0955 q^{39} -127.196 q^{40} +328.503 q^{41} -37.5879 q^{43} +229.588 q^{44} +143.095 q^{45} +426.794 q^{46} +254.995 q^{47} -48.0000 q^{48} -255.588 q^{50} -155.698 q^{51} -22.7939 q^{52} +211.588 q^{53} +54.0000 q^{54} +912.583 q^{55} +48.6030 q^{57} +436.382 q^{58} +412.201 q^{59} -190.794 q^{60} +836.693 q^{61} -502.794 q^{62} +64.0000 q^{64} -90.6030 q^{65} +344.382 q^{66} -165.588 q^{67} +207.598 q^{68} +640.191 q^{69} -465.015 q^{71} -72.0000 q^{72} -449.658 q^{73} -773.588 q^{74} -383.382 q^{75} -64.8040 q^{76} -34.1909 q^{78} -343.558 q^{79} +254.392 q^{80} +81.0000 q^{81} -657.005 q^{82} -1502.33 q^{83} +825.176 q^{85} +75.1758 q^{86} +654.573 q^{87} -459.176 q^{88} +341.085 q^{89} -286.191 q^{90} -853.588 q^{92} -754.191 q^{93} -509.990 q^{94} -257.588 q^{95} +96.0000 q^{96} +865.437 q^{97} +516.573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} - 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} - 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} - 6q^{3} + 8q^{4} + 12q^{5} + 12q^{6} - 16q^{8} + 18q^{9} - 24q^{10} - 4q^{11} - 24q^{12} + 48q^{13} - 36q^{15} + 32q^{16} + 84q^{17} - 36q^{18} - 72q^{19} + 48q^{20} + 8q^{22} - 308q^{23} + 48q^{24} + 18q^{25} - 96q^{26} - 54q^{27} - 80q^{29} + 72q^{30} + 384q^{31} - 64q^{32} + 12q^{33} - 168q^{34} + 72q^{36} + 536q^{37} + 144q^{38} - 144q^{39} - 96q^{40} + 756q^{41} + 400q^{43} - 16q^{44} + 108q^{45} + 616q^{46} + 312q^{47} - 96q^{48} - 36q^{50} - 252q^{51} + 192q^{52} - 52q^{53} + 108q^{54} + 1152q^{55} + 216q^{57} + 160q^{58} + 864q^{59} - 144q^{60} + 1416q^{61} - 768q^{62} + 128q^{64} - 300q^{65} - 24q^{66} + 144q^{67} + 336q^{68} + 924q^{69} - 1524q^{71} - 144q^{72} + 744q^{73} - 1072q^{74} - 54q^{75} - 288q^{76} + 288q^{78} + 976q^{79} + 192q^{80} + 162q^{81} - 1512q^{82} - 312q^{83} + 700q^{85} - 800q^{86} + 240q^{87} + 32q^{88} + 108q^{89} - 216q^{90} - 1232q^{92} - 1152q^{93} - 624q^{94} - 40q^{95} + 192q^{96} - 744q^{97} - 36q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 15.8995 1.42209 0.711047 0.703144i \(-0.248221\pi\)
0.711047 + 0.703144i \(0.248221\pi\)
\(6\) 6.00000 0.408248
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −31.7990 −1.00557
\(11\) 57.3970 1.57326 0.786629 0.617426i \(-0.211824\pi\)
0.786629 + 0.617426i \(0.211824\pi\)
\(12\) −12.0000 −0.288675
\(13\) −5.69848 −0.121575 −0.0607875 0.998151i \(-0.519361\pi\)
−0.0607875 + 0.998151i \(0.519361\pi\)
\(14\) 0 0
\(15\) −47.6985 −0.821046
\(16\) 16.0000 0.250000
\(17\) 51.8995 0.740440 0.370220 0.928944i \(-0.379282\pi\)
0.370220 + 0.928944i \(0.379282\pi\)
\(18\) −18.0000 −0.235702
\(19\) −16.2010 −0.195619 −0.0978096 0.995205i \(-0.531184\pi\)
−0.0978096 + 0.995205i \(0.531184\pi\)
\(20\) 63.5980 0.711047
\(21\) 0 0
\(22\) −114.794 −1.11246
\(23\) −213.397 −1.93462 −0.967312 0.253590i \(-0.918389\pi\)
−0.967312 + 0.253590i \(0.918389\pi\)
\(24\) 24.0000 0.204124
\(25\) 127.794 1.02235
\(26\) 11.3970 0.0859665
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −218.191 −1.39714 −0.698570 0.715542i \(-0.746180\pi\)
−0.698570 + 0.715542i \(0.746180\pi\)
\(30\) 95.3970 0.580567
\(31\) 251.397 1.45652 0.728262 0.685299i \(-0.240329\pi\)
0.728262 + 0.685299i \(0.240329\pi\)
\(32\) −32.0000 −0.176777
\(33\) −172.191 −0.908321
\(34\) −103.799 −0.523570
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 386.794 1.71861 0.859304 0.511464i \(-0.170897\pi\)
0.859304 + 0.511464i \(0.170897\pi\)
\(38\) 32.4020 0.138324
\(39\) 17.0955 0.0701914
\(40\) −127.196 −0.502786
\(41\) 328.503 1.25130 0.625652 0.780102i \(-0.284833\pi\)
0.625652 + 0.780102i \(0.284833\pi\)
\(42\) 0 0
\(43\) −37.5879 −0.133305 −0.0666523 0.997776i \(-0.521232\pi\)
−0.0666523 + 0.997776i \(0.521232\pi\)
\(44\) 229.588 0.786629
\(45\) 143.095 0.474031
\(46\) 426.794 1.36799
\(47\) 254.995 0.791379 0.395690 0.918384i \(-0.370506\pi\)
0.395690 + 0.918384i \(0.370506\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) −255.588 −0.722912
\(51\) −155.698 −0.427493
\(52\) −22.7939 −0.0607875
\(53\) 211.588 0.548374 0.274187 0.961676i \(-0.411591\pi\)
0.274187 + 0.961676i \(0.411591\pi\)
\(54\) 54.0000 0.136083
\(55\) 912.583 2.23732
\(56\) 0 0
\(57\) 48.6030 0.112941
\(58\) 436.382 0.987927
\(59\) 412.201 0.909559 0.454780 0.890604i \(-0.349718\pi\)
0.454780 + 0.890604i \(0.349718\pi\)
\(60\) −190.794 −0.410523
\(61\) 836.693 1.75619 0.878095 0.478486i \(-0.158814\pi\)
0.878095 + 0.478486i \(0.158814\pi\)
\(62\) −502.794 −1.02992
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −90.6030 −0.172891
\(66\) 344.382 0.642280
\(67\) −165.588 −0.301937 −0.150969 0.988539i \(-0.548239\pi\)
−0.150969 + 0.988539i \(0.548239\pi\)
\(68\) 207.598 0.370220
\(69\) 640.191 1.11696
\(70\) 0 0
\(71\) −465.015 −0.777284 −0.388642 0.921389i \(-0.627056\pi\)
−0.388642 + 0.921389i \(0.627056\pi\)
\(72\) −72.0000 −0.117851
\(73\) −449.658 −0.720938 −0.360469 0.932771i \(-0.617383\pi\)
−0.360469 + 0.932771i \(0.617383\pi\)
\(74\) −773.588 −1.21524
\(75\) −383.382 −0.590255
\(76\) −64.8040 −0.0978096
\(77\) 0 0
\(78\) −34.1909 −0.0496328
\(79\) −343.558 −0.489282 −0.244641 0.969614i \(-0.578670\pi\)
−0.244641 + 0.969614i \(0.578670\pi\)
\(80\) 254.392 0.355524
\(81\) 81.0000 0.111111
\(82\) −657.005 −0.884806
\(83\) −1502.33 −1.98677 −0.993387 0.114812i \(-0.963373\pi\)
−0.993387 + 0.114812i \(0.963373\pi\)
\(84\) 0 0
\(85\) 825.176 1.05298
\(86\) 75.1758 0.0942606
\(87\) 654.573 0.806639
\(88\) −459.176 −0.556231
\(89\) 341.085 0.406236 0.203118 0.979154i \(-0.434893\pi\)
0.203118 + 0.979154i \(0.434893\pi\)
\(90\) −286.191 −0.335191
\(91\) 0 0
\(92\) −853.588 −0.967312
\(93\) −754.191 −0.840924
\(94\) −509.990 −0.559590
\(95\) −257.588 −0.278189
\(96\) 96.0000 0.102062
\(97\) 865.437 0.905895 0.452947 0.891537i \(-0.350373\pi\)
0.452947 + 0.891537i \(0.350373\pi\)
\(98\) 0 0
\(99\) 516.573 0.524419
\(100\) 511.176 0.511176
\(101\) 243.256 0.239652 0.119826 0.992795i \(-0.461766\pi\)
0.119826 + 0.992795i \(0.461766\pi\)
\(102\) 311.397 0.302283
\(103\) 953.346 0.912000 0.456000 0.889980i \(-0.349282\pi\)
0.456000 + 0.889980i \(0.349282\pi\)
\(104\) 45.5879 0.0429833
\(105\) 0 0
\(106\) −423.176 −0.387759
\(107\) −1344.95 −1.21516 −0.607578 0.794260i \(-0.707859\pi\)
−0.607578 + 0.794260i \(0.707859\pi\)
\(108\) −108.000 −0.0962250
\(109\) 1734.35 1.52404 0.762022 0.647551i \(-0.224207\pi\)
0.762022 + 0.647551i \(0.224207\pi\)
\(110\) −1825.17 −1.58202
\(111\) −1160.38 −0.992239
\(112\) 0 0
\(113\) 1441.18 1.19977 0.599887 0.800085i \(-0.295212\pi\)
0.599887 + 0.800085i \(0.295212\pi\)
\(114\) −97.2061 −0.0798612
\(115\) −3392.90 −2.75122
\(116\) −872.764 −0.698570
\(117\) −51.2864 −0.0405250
\(118\) −824.402 −0.643156
\(119\) 0 0
\(120\) 381.588 0.290284
\(121\) 1963.41 1.47514
\(122\) −1673.39 −1.24181
\(123\) −985.508 −0.722441
\(124\) 1005.59 0.728262
\(125\) 44.4222 0.0317860
\(126\) 0 0
\(127\) 1184.70 0.827759 0.413880 0.910332i \(-0.364173\pi\)
0.413880 + 0.910332i \(0.364173\pi\)
\(128\) −128.000 −0.0883883
\(129\) 112.764 0.0769634
\(130\) 181.206 0.122252
\(131\) −297.588 −0.198476 −0.0992381 0.995064i \(-0.531641\pi\)
−0.0992381 + 0.995064i \(0.531641\pi\)
\(132\) −688.764 −0.454160
\(133\) 0 0
\(134\) 331.176 0.213502
\(135\) −429.286 −0.273682
\(136\) −415.196 −0.261785
\(137\) 620.985 0.387258 0.193629 0.981075i \(-0.437974\pi\)
0.193629 + 0.981075i \(0.437974\pi\)
\(138\) −1280.38 −0.789807
\(139\) 898.754 0.548426 0.274213 0.961669i \(-0.411583\pi\)
0.274213 + 0.961669i \(0.411583\pi\)
\(140\) 0 0
\(141\) −764.985 −0.456903
\(142\) 930.030 0.549623
\(143\) −327.076 −0.191269
\(144\) 144.000 0.0833333
\(145\) −3469.13 −1.98686
\(146\) 899.316 0.509780
\(147\) 0 0
\(148\) 1547.18 0.859304
\(149\) −3054.70 −1.67954 −0.839769 0.542945i \(-0.817309\pi\)
−0.839769 + 0.542945i \(0.817309\pi\)
\(150\) 766.764 0.417373
\(151\) −65.1455 −0.0351090 −0.0175545 0.999846i \(-0.505588\pi\)
−0.0175545 + 0.999846i \(0.505588\pi\)
\(152\) 129.608 0.0691619
\(153\) 467.095 0.246813
\(154\) 0 0
\(155\) 3997.08 2.07131
\(156\) 68.3818 0.0350957
\(157\) −1542.22 −0.783966 −0.391983 0.919973i \(-0.628211\pi\)
−0.391983 + 0.919973i \(0.628211\pi\)
\(158\) 687.115 0.345974
\(159\) −634.764 −0.316604
\(160\) −508.784 −0.251393
\(161\) 0 0
\(162\) −162.000 −0.0785674
\(163\) −2514.73 −1.20840 −0.604200 0.796833i \(-0.706507\pi\)
−0.604200 + 0.796833i \(0.706507\pi\)
\(164\) 1314.01 0.625652
\(165\) −2737.75 −1.29172
\(166\) 3004.66 1.40486
\(167\) −528.643 −0.244956 −0.122478 0.992471i \(-0.539084\pi\)
−0.122478 + 0.992471i \(0.539084\pi\)
\(168\) 0 0
\(169\) −2164.53 −0.985220
\(170\) −1650.35 −0.744566
\(171\) −145.809 −0.0652064
\(172\) −150.352 −0.0666523
\(173\) −96.8439 −0.0425602 −0.0212801 0.999774i \(-0.506774\pi\)
−0.0212801 + 0.999774i \(0.506774\pi\)
\(174\) −1309.15 −0.570380
\(175\) 0 0
\(176\) 918.352 0.393314
\(177\) −1236.60 −0.525134
\(178\) −682.171 −0.287252
\(179\) −534.542 −0.223204 −0.111602 0.993753i \(-0.535598\pi\)
−0.111602 + 0.993753i \(0.535598\pi\)
\(180\) 572.382 0.237016
\(181\) −2087.00 −0.857049 −0.428524 0.903530i \(-0.640966\pi\)
−0.428524 + 0.903530i \(0.640966\pi\)
\(182\) 0 0
\(183\) −2510.08 −1.01394
\(184\) 1707.18 0.683993
\(185\) 6149.83 2.44402
\(186\) 1508.38 0.594623
\(187\) 2978.87 1.16490
\(188\) 1019.98 0.395690
\(189\) 0 0
\(190\) 515.176 0.196709
\(191\) 3387.69 1.28337 0.641687 0.766966i \(-0.278235\pi\)
0.641687 + 0.766966i \(0.278235\pi\)
\(192\) −192.000 −0.0721688
\(193\) −1908.35 −0.711742 −0.355871 0.934535i \(-0.615816\pi\)
−0.355871 + 0.934535i \(0.615816\pi\)
\(194\) −1730.87 −0.640564
\(195\) 271.809 0.0998187
\(196\) 0 0
\(197\) −2061.88 −0.745699 −0.372850 0.927892i \(-0.621619\pi\)
−0.372850 + 0.927892i \(0.621619\pi\)
\(198\) −1033.15 −0.370820
\(199\) −3171.50 −1.12976 −0.564878 0.825174i \(-0.691077\pi\)
−0.564878 + 0.825174i \(0.691077\pi\)
\(200\) −1022.35 −0.361456
\(201\) 496.764 0.174323
\(202\) −486.512 −0.169460
\(203\) 0 0
\(204\) −622.794 −0.213747
\(205\) 5223.02 1.77947
\(206\) −1906.69 −0.644882
\(207\) −1920.57 −0.644875
\(208\) −91.1758 −0.0303938
\(209\) −929.889 −0.307760
\(210\) 0 0
\(211\) 1349.97 0.440454 0.220227 0.975449i \(-0.429320\pi\)
0.220227 + 0.975449i \(0.429320\pi\)
\(212\) 846.352 0.274187
\(213\) 1395.05 0.448765
\(214\) 2689.91 0.859245
\(215\) −597.628 −0.189572
\(216\) 216.000 0.0680414
\(217\) 0 0
\(218\) −3468.70 −1.07766
\(219\) 1348.97 0.416234
\(220\) 3650.33 1.11866
\(221\) −295.748 −0.0900190
\(222\) 2320.76 0.701619
\(223\) 1361.85 0.408951 0.204476 0.978872i \(-0.434451\pi\)
0.204476 + 0.978872i \(0.434451\pi\)
\(224\) 0 0
\(225\) 1150.15 0.340784
\(226\) −2882.35 −0.848368
\(227\) −1861.81 −0.544373 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(228\) 194.412 0.0564704
\(229\) 5358.78 1.54637 0.773184 0.634181i \(-0.218663\pi\)
0.773184 + 0.634181i \(0.218663\pi\)
\(230\) 6785.81 1.94540
\(231\) 0 0
\(232\) 1745.53 0.493963
\(233\) 5441.12 1.52987 0.764935 0.644107i \(-0.222771\pi\)
0.764935 + 0.644107i \(0.222771\pi\)
\(234\) 102.573 0.0286555
\(235\) 4054.29 1.12542
\(236\) 1648.80 0.454780
\(237\) 1030.67 0.282487
\(238\) 0 0
\(239\) −1157.28 −0.313213 −0.156607 0.987661i \(-0.550055\pi\)
−0.156607 + 0.987661i \(0.550055\pi\)
\(240\) −763.176 −0.205262
\(241\) 3969.38 1.06095 0.530477 0.847699i \(-0.322013\pi\)
0.530477 + 0.847699i \(0.322013\pi\)
\(242\) −3926.82 −1.04308
\(243\) −243.000 −0.0641500
\(244\) 3346.77 0.878095
\(245\) 0 0
\(246\) 1971.02 0.510843
\(247\) 92.3212 0.0237824
\(248\) −2011.18 −0.514959
\(249\) 4506.99 1.14706
\(250\) −88.8444 −0.0224761
\(251\) −5978.75 −1.50349 −0.751744 0.659455i \(-0.770787\pi\)
−0.751744 + 0.659455i \(0.770787\pi\)
\(252\) 0 0
\(253\) −12248.3 −3.04366
\(254\) −2369.41 −0.585314
\(255\) −2475.53 −0.607935
\(256\) 256.000 0.0625000
\(257\) −4650.15 −1.12867 −0.564335 0.825546i \(-0.690868\pi\)
−0.564335 + 0.825546i \(0.690868\pi\)
\(258\) −225.527 −0.0544214
\(259\) 0 0
\(260\) −362.412 −0.0864456
\(261\) −1963.72 −0.465713
\(262\) 595.176 0.140344
\(263\) 3695.37 0.866411 0.433205 0.901295i \(-0.357382\pi\)
0.433205 + 0.901295i \(0.357382\pi\)
\(264\) 1377.53 0.321140
\(265\) 3364.14 0.779840
\(266\) 0 0
\(267\) −1023.26 −0.234540
\(268\) −662.352 −0.150969
\(269\) 7157.69 1.62235 0.811175 0.584804i \(-0.198829\pi\)
0.811175 + 0.584804i \(0.198829\pi\)
\(270\) 858.573 0.193522
\(271\) −4038.37 −0.905216 −0.452608 0.891710i \(-0.649506\pi\)
−0.452608 + 0.891710i \(0.649506\pi\)
\(272\) 830.392 0.185110
\(273\) 0 0
\(274\) −1241.97 −0.273833
\(275\) 7334.98 1.60842
\(276\) 2560.76 0.558478
\(277\) −2754.82 −0.597550 −0.298775 0.954324i \(-0.596578\pi\)
−0.298775 + 0.954324i \(0.596578\pi\)
\(278\) −1797.51 −0.387796
\(279\) 2262.57 0.485508
\(280\) 0 0
\(281\) −772.742 −0.164050 −0.0820248 0.996630i \(-0.526139\pi\)
−0.0820248 + 0.996630i \(0.526139\pi\)
\(282\) 1529.97 0.323079
\(283\) 6745.49 1.41688 0.708441 0.705770i \(-0.249399\pi\)
0.708441 + 0.705770i \(0.249399\pi\)
\(284\) −1860.06 −0.388642
\(285\) 772.764 0.160613
\(286\) 654.152 0.135248
\(287\) 0 0
\(288\) −288.000 −0.0589256
\(289\) −2219.44 −0.451749
\(290\) 6938.25 1.40492
\(291\) −2596.31 −0.523019
\(292\) −1798.63 −0.360469
\(293\) −1922.69 −0.383362 −0.191681 0.981457i \(-0.561394\pi\)
−0.191681 + 0.981457i \(0.561394\pi\)
\(294\) 0 0
\(295\) 6553.79 1.29348
\(296\) −3094.35 −0.607620
\(297\) −1549.72 −0.302774
\(298\) 6109.41 1.18761
\(299\) 1216.04 0.235202
\(300\) −1533.53 −0.295127
\(301\) 0 0
\(302\) 130.291 0.0248258
\(303\) −729.768 −0.138363
\(304\) −259.216 −0.0489048
\(305\) 13303.0 2.49747
\(306\) −934.191 −0.174523
\(307\) −2016.68 −0.374913 −0.187456 0.982273i \(-0.560024\pi\)
−0.187456 + 0.982273i \(0.560024\pi\)
\(308\) 0 0
\(309\) −2860.04 −0.526544
\(310\) −7994.17 −1.46464
\(311\) −7149.99 −1.30366 −0.651831 0.758365i \(-0.725999\pi\)
−0.651831 + 0.758365i \(0.725999\pi\)
\(312\) −136.764 −0.0248164
\(313\) −8596.49 −1.55240 −0.776202 0.630484i \(-0.782856\pi\)
−0.776202 + 0.630484i \(0.782856\pi\)
\(314\) 3084.44 0.554347
\(315\) 0 0
\(316\) −1374.23 −0.244641
\(317\) −2853.24 −0.505532 −0.252766 0.967527i \(-0.581340\pi\)
−0.252766 + 0.967527i \(0.581340\pi\)
\(318\) 1269.53 0.223873
\(319\) −12523.5 −2.19806
\(320\) 1017.57 0.177762
\(321\) 4034.86 0.701570
\(322\) 0 0
\(323\) −840.824 −0.144844
\(324\) 324.000 0.0555556
\(325\) −728.232 −0.124292
\(326\) 5029.47 0.854467
\(327\) −5203.05 −0.879907
\(328\) −2628.02 −0.442403
\(329\) 0 0
\(330\) 5475.50 0.913382
\(331\) 1619.12 0.268866 0.134433 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(332\) −6009.33 −0.993387
\(333\) 3481.15 0.572870
\(334\) 1057.29 0.173210
\(335\) −2632.76 −0.429383
\(336\) 0 0
\(337\) −3278.67 −0.529972 −0.264986 0.964252i \(-0.585367\pi\)
−0.264986 + 0.964252i \(0.585367\pi\)
\(338\) 4329.05 0.696655
\(339\) −4323.53 −0.692690
\(340\) 3300.70 0.526488
\(341\) 14429.4 2.29149
\(342\) 291.618 0.0461079
\(343\) 0 0
\(344\) 300.703 0.0471303
\(345\) 10178.7 1.58842
\(346\) 193.688 0.0300946
\(347\) −2850.30 −0.440957 −0.220479 0.975392i \(-0.570762\pi\)
−0.220479 + 0.975392i \(0.570762\pi\)
\(348\) 2618.29 0.403319
\(349\) 4725.32 0.724758 0.362379 0.932031i \(-0.381965\pi\)
0.362379 + 0.932031i \(0.381965\pi\)
\(350\) 0 0
\(351\) 153.859 0.0233971
\(352\) −1836.70 −0.278115
\(353\) −6727.44 −1.01435 −0.507175 0.861843i \(-0.669310\pi\)
−0.507175 + 0.861843i \(0.669310\pi\)
\(354\) 2473.21 0.371326
\(355\) −7393.51 −1.10537
\(356\) 1364.34 0.203118
\(357\) 0 0
\(358\) 1069.08 0.157829
\(359\) −7331.89 −1.07789 −0.538945 0.842341i \(-0.681177\pi\)
−0.538945 + 0.842341i \(0.681177\pi\)
\(360\) −1144.76 −0.167595
\(361\) −6596.53 −0.961733
\(362\) 4174.01 0.606025
\(363\) −5890.24 −0.851673
\(364\) 0 0
\(365\) −7149.34 −1.02524
\(366\) 5020.16 0.716962
\(367\) −2774.43 −0.394616 −0.197308 0.980342i \(-0.563220\pi\)
−0.197308 + 0.980342i \(0.563220\pi\)
\(368\) −3414.35 −0.483656
\(369\) 2956.52 0.417101
\(370\) −12299.7 −1.72819
\(371\) 0 0
\(372\) −3016.76 −0.420462
\(373\) 2527.35 0.350834 0.175417 0.984494i \(-0.443873\pi\)
0.175417 + 0.984494i \(0.443873\pi\)
\(374\) −5957.75 −0.823711
\(375\) −133.267 −0.0183516
\(376\) −2039.96 −0.279795
\(377\) 1243.36 0.169857
\(378\) 0 0
\(379\) 3116.40 0.422371 0.211186 0.977446i \(-0.432268\pi\)
0.211186 + 0.977446i \(0.432268\pi\)
\(380\) −1030.35 −0.139095
\(381\) −3554.11 −0.477907
\(382\) −6775.38 −0.907483
\(383\) −1518.07 −0.202532 −0.101266 0.994859i \(-0.532289\pi\)
−0.101266 + 0.994859i \(0.532289\pi\)
\(384\) 384.000 0.0510310
\(385\) 0 0
\(386\) 3816.70 0.503277
\(387\) −338.291 −0.0444349
\(388\) 3461.75 0.452947
\(389\) −3246.77 −0.423182 −0.211591 0.977358i \(-0.567865\pi\)
−0.211591 + 0.977358i \(0.567865\pi\)
\(390\) −543.618 −0.0705825
\(391\) −11075.2 −1.43247
\(392\) 0 0
\(393\) 892.764 0.114590
\(394\) 4123.76 0.527289
\(395\) −5462.39 −0.695804
\(396\) 2066.29 0.262210
\(397\) 1830.62 0.231427 0.115713 0.993283i \(-0.463085\pi\)
0.115713 + 0.993283i \(0.463085\pi\)
\(398\) 6342.99 0.798858
\(399\) 0 0
\(400\) 2044.70 0.255588
\(401\) −3385.81 −0.421644 −0.210822 0.977524i \(-0.567614\pi\)
−0.210822 + 0.977524i \(0.567614\pi\)
\(402\) −993.527 −0.123265
\(403\) −1432.58 −0.177077
\(404\) 973.024 0.119826
\(405\) 1287.86 0.158010
\(406\) 0 0
\(407\) 22200.8 2.70382
\(408\) 1245.59 0.151142
\(409\) 9253.17 1.11868 0.559340 0.828938i \(-0.311055\pi\)
0.559340 + 0.828938i \(0.311055\pi\)
\(410\) −10446.0 −1.25828
\(411\) −1862.95 −0.223583
\(412\) 3813.39 0.456000
\(413\) 0 0
\(414\) 3841.15 0.455995
\(415\) −23886.3 −2.82538
\(416\) 182.352 0.0214916
\(417\) −2696.26 −0.316634
\(418\) 1859.78 0.217619
\(419\) 3547.52 0.413622 0.206811 0.978381i \(-0.433692\pi\)
0.206811 + 0.978381i \(0.433692\pi\)
\(420\) 0 0
\(421\) 7848.87 0.908624 0.454312 0.890843i \(-0.349885\pi\)
0.454312 + 0.890843i \(0.349885\pi\)
\(422\) −2699.94 −0.311448
\(423\) 2294.95 0.263793
\(424\) −1692.70 −0.193880
\(425\) 6632.44 0.756990
\(426\) −2790.09 −0.317325
\(427\) 0 0
\(428\) −5379.82 −0.607578
\(429\) 981.227 0.110429
\(430\) 1195.26 0.134047
\(431\) −4447.32 −0.497030 −0.248515 0.968628i \(-0.579943\pi\)
−0.248515 + 0.968628i \(0.579943\pi\)
\(432\) −432.000 −0.0481125
\(433\) −6994.82 −0.776327 −0.388164 0.921590i \(-0.626890\pi\)
−0.388164 + 0.921590i \(0.626890\pi\)
\(434\) 0 0
\(435\) 10407.4 1.14712
\(436\) 6937.41 0.762022
\(437\) 3457.25 0.378450
\(438\) −2697.95 −0.294322
\(439\) 636.182 0.0691647 0.0345823 0.999402i \(-0.488990\pi\)
0.0345823 + 0.999402i \(0.488990\pi\)
\(440\) −7300.66 −0.791012
\(441\) 0 0
\(442\) 591.497 0.0636530
\(443\) −4474.24 −0.479859 −0.239929 0.970790i \(-0.577124\pi\)
−0.239929 + 0.970790i \(0.577124\pi\)
\(444\) −4641.53 −0.496120
\(445\) 5423.08 0.577705
\(446\) −2723.70 −0.289172
\(447\) 9164.11 0.969681
\(448\) 0 0
\(449\) −2389.42 −0.251144 −0.125572 0.992085i \(-0.540077\pi\)
−0.125572 + 0.992085i \(0.540077\pi\)
\(450\) −2300.29 −0.240971
\(451\) 18855.0 1.96862
\(452\) 5764.70 0.599887
\(453\) 195.436 0.0202702
\(454\) 3723.62 0.384930
\(455\) 0 0
\(456\) −388.824 −0.0399306
\(457\) −4438.34 −0.454304 −0.227152 0.973859i \(-0.572941\pi\)
−0.227152 + 0.973859i \(0.572941\pi\)
\(458\) −10717.6 −1.09345
\(459\) −1401.29 −0.142498
\(460\) −13571.6 −1.37561
\(461\) −14079.8 −1.42248 −0.711240 0.702949i \(-0.751866\pi\)
−0.711240 + 0.702949i \(0.751866\pi\)
\(462\) 0 0
\(463\) 4687.50 0.470511 0.235255 0.971934i \(-0.424407\pi\)
0.235255 + 0.971934i \(0.424407\pi\)
\(464\) −3491.05 −0.349285
\(465\) −11991.3 −1.19587
\(466\) −10882.2 −1.08178
\(467\) 8447.26 0.837029 0.418514 0.908210i \(-0.362551\pi\)
0.418514 + 0.908210i \(0.362551\pi\)
\(468\) −205.145 −0.0202625
\(469\) 0 0
\(470\) −8108.58 −0.795789
\(471\) 4626.66 0.452623
\(472\) −3297.61 −0.321578
\(473\) −2157.43 −0.209723
\(474\) −2061.35 −0.199748
\(475\) −2070.39 −0.199992
\(476\) 0 0
\(477\) 1904.29 0.182791
\(478\) 2314.55 0.221475
\(479\) −4369.41 −0.416792 −0.208396 0.978045i \(-0.566824\pi\)
−0.208396 + 0.978045i \(0.566824\pi\)
\(480\) 1526.35 0.145142
\(481\) −2204.14 −0.208940
\(482\) −7938.75 −0.750208
\(483\) 0 0
\(484\) 7853.65 0.737570
\(485\) 13760.0 1.28827
\(486\) 486.000 0.0453609
\(487\) −14477.7 −1.34712 −0.673561 0.739132i \(-0.735236\pi\)
−0.673561 + 0.739132i \(0.735236\pi\)
\(488\) −6693.55 −0.620907
\(489\) 7544.20 0.697670
\(490\) 0 0
\(491\) 9306.12 0.855355 0.427677 0.903931i \(-0.359332\pi\)
0.427677 + 0.903931i \(0.359332\pi\)
\(492\) −3942.03 −0.361220
\(493\) −11324.0 −1.03450
\(494\) −184.642 −0.0168167
\(495\) 8213.25 0.745774
\(496\) 4022.35 0.364131
\(497\) 0 0
\(498\) −9013.99 −0.811097
\(499\) −12237.5 −1.09785 −0.548926 0.835871i \(-0.684963\pi\)
−0.548926 + 0.835871i \(0.684963\pi\)
\(500\) 177.689 0.0158930
\(501\) 1585.93 0.141425
\(502\) 11957.5 1.06313
\(503\) 5524.30 0.489694 0.244847 0.969562i \(-0.421262\pi\)
0.244847 + 0.969562i \(0.421262\pi\)
\(504\) 0 0
\(505\) 3867.65 0.340808
\(506\) 24496.7 2.15219
\(507\) 6493.58 0.568817
\(508\) 4738.81 0.413880
\(509\) −10079.6 −0.877743 −0.438871 0.898550i \(-0.644622\pi\)
−0.438871 + 0.898550i \(0.644622\pi\)
\(510\) 4951.05 0.429875
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 437.427 0.0376470
\(514\) 9300.30 0.798091
\(515\) 15157.7 1.29695
\(516\) 451.055 0.0384817
\(517\) 14635.9 1.24504
\(518\) 0 0
\(519\) 290.532 0.0245721
\(520\) 724.824 0.0611262
\(521\) 5706.61 0.479868 0.239934 0.970789i \(-0.422874\pi\)
0.239934 + 0.970789i \(0.422874\pi\)
\(522\) 3927.44 0.329309
\(523\) −10657.3 −0.891032 −0.445516 0.895274i \(-0.646980\pi\)
−0.445516 + 0.895274i \(0.646980\pi\)
\(524\) −1190.35 −0.0992381
\(525\) 0 0
\(526\) −7390.73 −0.612645
\(527\) 13047.4 1.07847
\(528\) −2755.05 −0.227080
\(529\) 33371.3 2.74277
\(530\) −6728.28 −0.551430
\(531\) 3709.81 0.303186
\(532\) 0 0
\(533\) −1871.97 −0.152127
\(534\) 2046.51 0.165845
\(535\) −21384.1 −1.72807
\(536\) 1324.70 0.106751
\(537\) 1603.63 0.128867
\(538\) −14315.4 −1.14717
\(539\) 0 0
\(540\) −1717.15 −0.136841
\(541\) −4010.48 −0.318714 −0.159357 0.987221i \(-0.550942\pi\)
−0.159357 + 0.987221i \(0.550942\pi\)
\(542\) 8076.74 0.640084
\(543\) 6261.01 0.494817
\(544\) −1660.78 −0.130892
\(545\) 27575.3 2.16733
\(546\) 0 0
\(547\) −17619.8 −1.37728 −0.688638 0.725105i \(-0.741791\pi\)
−0.688638 + 0.725105i \(0.741791\pi\)
\(548\) 2483.94 0.193629
\(549\) 7530.24 0.585397
\(550\) −14670.0 −1.13733
\(551\) 3534.91 0.273307
\(552\) −5121.53 −0.394903
\(553\) 0 0
\(554\) 5509.65 0.422532
\(555\) −18449.5 −1.41106
\(556\) 3595.01 0.274213
\(557\) −10337.7 −0.786395 −0.393198 0.919454i \(-0.628631\pi\)
−0.393198 + 0.919454i \(0.628631\pi\)
\(558\) −4525.15 −0.343306
\(559\) 214.194 0.0162065
\(560\) 0 0
\(561\) −8936.62 −0.672557
\(562\) 1545.48 0.116001
\(563\) −24023.7 −1.79836 −0.899180 0.437580i \(-0.855836\pi\)
−0.899180 + 0.437580i \(0.855836\pi\)
\(564\) −3059.94 −0.228452
\(565\) 22914.0 1.70619
\(566\) −13491.0 −1.00189
\(567\) 0 0
\(568\) 3720.12 0.274811
\(569\) −13179.2 −0.971001 −0.485501 0.874236i \(-0.661363\pi\)
−0.485501 + 0.874236i \(0.661363\pi\)
\(570\) −1545.53 −0.113570
\(571\) 7776.26 0.569924 0.284962 0.958539i \(-0.408019\pi\)
0.284962 + 0.958539i \(0.408019\pi\)
\(572\) −1308.30 −0.0956344
\(573\) −10163.1 −0.740956
\(574\) 0 0
\(575\) −27270.8 −1.97787
\(576\) 576.000 0.0416667
\(577\) 20167.0 1.45505 0.727525 0.686081i \(-0.240670\pi\)
0.727525 + 0.686081i \(0.240670\pi\)
\(578\) 4438.88 0.319435
\(579\) 5725.05 0.410924
\(580\) −13876.5 −0.993432
\(581\) 0 0
\(582\) 5192.62 0.369830
\(583\) 12144.5 0.862734
\(584\) 3597.26 0.254890
\(585\) −815.427 −0.0576304
\(586\) 3845.39 0.271078
\(587\) −8365.08 −0.588184 −0.294092 0.955777i \(-0.595017\pi\)
−0.294092 + 0.955777i \(0.595017\pi\)
\(588\) 0 0
\(589\) −4072.88 −0.284924
\(590\) −13107.6 −0.914628
\(591\) 6185.64 0.430530
\(592\) 6188.70 0.429652
\(593\) −27621.9 −1.91281 −0.956403 0.292050i \(-0.905663\pi\)
−0.956403 + 0.292050i \(0.905663\pi\)
\(594\) 3099.44 0.214093
\(595\) 0 0
\(596\) −12218.8 −0.839769
\(597\) 9514.49 0.652265
\(598\) −2432.08 −0.166313
\(599\) 538.318 0.0367197 0.0183598 0.999831i \(-0.494156\pi\)
0.0183598 + 0.999831i \(0.494156\pi\)
\(600\) 3067.05 0.208687
\(601\) 6958.64 0.472294 0.236147 0.971717i \(-0.424115\pi\)
0.236147 + 0.971717i \(0.424115\pi\)
\(602\) 0 0
\(603\) −1490.29 −0.100646
\(604\) −260.582 −0.0175545
\(605\) 31217.3 2.09779
\(606\) 1459.54 0.0978376
\(607\) −17297.6 −1.15665 −0.578326 0.815806i \(-0.696294\pi\)
−0.578326 + 0.815806i \(0.696294\pi\)
\(608\) 518.432 0.0345809
\(609\) 0 0
\(610\) −26606.0 −1.76598
\(611\) −1453.08 −0.0962120
\(612\) 1868.38 0.123407
\(613\) −839.158 −0.0552908 −0.0276454 0.999618i \(-0.508801\pi\)
−0.0276454 + 0.999618i \(0.508801\pi\)
\(614\) 4033.37 0.265103
\(615\) −15669.1 −1.02738
\(616\) 0 0
\(617\) −16040.0 −1.04659 −0.523295 0.852152i \(-0.675297\pi\)
−0.523295 + 0.852152i \(0.675297\pi\)
\(618\) 5720.08 0.372323
\(619\) −5429.28 −0.352538 −0.176269 0.984342i \(-0.556403\pi\)
−0.176269 + 0.984342i \(0.556403\pi\)
\(620\) 15988.3 1.03566
\(621\) 5761.72 0.372318
\(622\) 14300.0 0.921828
\(623\) 0 0
\(624\) 273.527 0.0175478
\(625\) −15268.0 −0.977149
\(626\) 17193.0 1.09772
\(627\) 2789.67 0.177685
\(628\) −6168.88 −0.391983
\(629\) 20074.4 1.27253
\(630\) 0 0
\(631\) −1807.86 −0.114057 −0.0570284 0.998373i \(-0.518163\pi\)
−0.0570284 + 0.998373i \(0.518163\pi\)
\(632\) 2748.46 0.172987
\(633\) −4049.91 −0.254296
\(634\) 5706.47 0.357465
\(635\) 18836.2 1.17715
\(636\) −2539.05 −0.158302
\(637\) 0 0
\(638\) 25047.0 1.55426
\(639\) −4185.14 −0.259095
\(640\) −2035.14 −0.125697
\(641\) −5904.56 −0.363832 −0.181916 0.983314i \(-0.558230\pi\)
−0.181916 + 0.983314i \(0.558230\pi\)
\(642\) −8069.73 −0.496085
\(643\) 8092.42 0.496320 0.248160 0.968719i \(-0.420174\pi\)
0.248160 + 0.968719i \(0.420174\pi\)
\(644\) 0 0
\(645\) 1792.88 0.109449
\(646\) 1681.65 0.102420
\(647\) 20192.7 1.22698 0.613490 0.789702i \(-0.289765\pi\)
0.613490 + 0.789702i \(0.289765\pi\)
\(648\) −648.000 −0.0392837
\(649\) 23659.1 1.43097
\(650\) 1456.46 0.0878880
\(651\) 0 0
\(652\) −10058.9 −0.604200
\(653\) −21180.8 −1.26933 −0.634664 0.772789i \(-0.718861\pi\)
−0.634664 + 0.772789i \(0.718861\pi\)
\(654\) 10406.1 0.622188
\(655\) −4731.50 −0.282252
\(656\) 5256.04 0.312826
\(657\) −4046.92 −0.240313
\(658\) 0 0
\(659\) 28411.3 1.67944 0.839718 0.543023i \(-0.182720\pi\)
0.839718 + 0.543023i \(0.182720\pi\)
\(660\) −10951.0 −0.645859
\(661\) 16704.9 0.982975 0.491488 0.870885i \(-0.336453\pi\)
0.491488 + 0.870885i \(0.336453\pi\)
\(662\) −3238.23 −0.190117
\(663\) 887.245 0.0519725
\(664\) 12018.7 0.702431
\(665\) 0 0
\(666\) −6962.29 −0.405080
\(667\) 46561.3 2.70294
\(668\) −2114.57 −0.122478
\(669\) −4085.55 −0.236108
\(670\) 5265.53 0.303620
\(671\) 48023.7 2.76294
\(672\) 0 0
\(673\) −9047.09 −0.518187 −0.259093 0.965852i \(-0.583424\pi\)
−0.259093 + 0.965852i \(0.583424\pi\)
\(674\) 6557.35 0.374747
\(675\) −3450.44 −0.196752
\(676\) −8658.11 −0.492610
\(677\) −7844.26 −0.445316 −0.222658 0.974897i \(-0.571473\pi\)
−0.222658 + 0.974897i \(0.571473\pi\)
\(678\) 8647.05 0.489806
\(679\) 0 0
\(680\) −6601.41 −0.372283
\(681\) 5585.43 0.314294
\(682\) −28858.8 −1.62033
\(683\) 25766.1 1.44350 0.721751 0.692153i \(-0.243338\pi\)
0.721751 + 0.692153i \(0.243338\pi\)
\(684\) −583.236 −0.0326032
\(685\) 9873.35 0.550717
\(686\) 0 0
\(687\) −16076.3 −0.892796
\(688\) −601.406 −0.0333261
\(689\) −1205.73 −0.0666686
\(690\) −20357.4 −1.12318
\(691\) 24674.1 1.35839 0.679195 0.733958i \(-0.262329\pi\)
0.679195 + 0.733958i \(0.262329\pi\)
\(692\) −387.376 −0.0212801
\(693\) 0 0
\(694\) 5700.60 0.311804
\(695\) 14289.7 0.779914
\(696\) −5236.58 −0.285190
\(697\) 17049.1 0.926515
\(698\) −9450.63 −0.512481
\(699\) −16323.4 −0.883271
\(700\) 0 0
\(701\) 29377.9 1.58286 0.791431 0.611258i \(-0.209336\pi\)
0.791431 + 0.611258i \(0.209336\pi\)
\(702\) −307.718 −0.0165443
\(703\) −6266.45 −0.336193
\(704\) 3673.41 0.196657
\(705\) −12162.9 −0.649759
\(706\) 13454.9 0.717253
\(707\) 0 0
\(708\) −4946.41 −0.262567
\(709\) 30594.3 1.62058 0.810291 0.586028i \(-0.199309\pi\)
0.810291 + 0.586028i \(0.199309\pi\)
\(710\) 14787.0 0.781615
\(711\) −3092.02 −0.163094
\(712\) −2728.68 −0.143626
\(713\) −53647.4 −2.81782
\(714\) 0 0
\(715\) −5200.34 −0.272002
\(716\) −2138.17 −0.111602
\(717\) 3471.83 0.180834
\(718\) 14663.8 0.762183
\(719\) 1946.94 0.100985 0.0504927 0.998724i \(-0.483921\pi\)
0.0504927 + 0.998724i \(0.483921\pi\)
\(720\) 2289.53 0.118508
\(721\) 0 0
\(722\) 13193.1 0.680048
\(723\) −11908.1 −0.612542
\(724\) −8348.02 −0.428524
\(725\) −27883.5 −1.42837
\(726\) 11780.5 0.602224
\(727\) −15750.6 −0.803518 −0.401759 0.915745i \(-0.631601\pi\)
−0.401759 + 0.915745i \(0.631601\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 14298.7 0.724956
\(731\) −1950.79 −0.0987040
\(732\) −10040.3 −0.506969
\(733\) −14349.6 −0.723074 −0.361537 0.932358i \(-0.617748\pi\)
−0.361537 + 0.932358i \(0.617748\pi\)
\(734\) 5548.86 0.279036
\(735\) 0 0
\(736\) 6828.70 0.341996
\(737\) −9504.24 −0.475025
\(738\) −5913.05 −0.294935
\(739\) −17758.1 −0.883953 −0.441976 0.897027i \(-0.645722\pi\)
−0.441976 + 0.897027i \(0.645722\pi\)
\(740\) 24599.3 1.22201
\(741\) −276.964 −0.0137308
\(742\) 0 0
\(743\) −29187.6 −1.44117 −0.720586 0.693366i \(-0.756127\pi\)
−0.720586 + 0.693366i \(0.756127\pi\)
\(744\) 6033.53 0.297312
\(745\) −48568.2 −2.38846
\(746\) −5054.69 −0.248077
\(747\) −13521.0 −0.662258
\(748\) 11915.5 0.582451
\(749\) 0 0
\(750\) 266.533 0.0129766
\(751\) 13781.9 0.669654 0.334827 0.942280i \(-0.391322\pi\)
0.334827 + 0.942280i \(0.391322\pi\)
\(752\) 4079.92 0.197845
\(753\) 17936.3 0.868039
\(754\) −2486.72 −0.120107
\(755\) −1035.78 −0.0499283
\(756\) 0 0
\(757\) 36952.7 1.77420 0.887099 0.461579i \(-0.152717\pi\)
0.887099 + 0.461579i \(0.152717\pi\)
\(758\) −6232.80 −0.298662
\(759\) 36745.0 1.75726
\(760\) 2060.70 0.0983547
\(761\) −28816.5 −1.37266 −0.686332 0.727288i \(-0.740780\pi\)
−0.686332 + 0.727288i \(0.740780\pi\)
\(762\) 7108.22 0.337931
\(763\) 0 0
\(764\) 13550.8 0.641687
\(765\) 7426.58 0.350992
\(766\) 3036.14 0.143212
\(767\) −2348.92 −0.110580
\(768\) −768.000 −0.0360844
\(769\) −25285.2 −1.18571 −0.592854 0.805310i \(-0.701999\pi\)
−0.592854 + 0.805310i \(0.701999\pi\)
\(770\) 0 0
\(771\) 13950.4 0.651638
\(772\) −7633.41 −0.355871
\(773\) 18418.2 0.856995 0.428497 0.903543i \(-0.359043\pi\)
0.428497 + 0.903543i \(0.359043\pi\)
\(774\) 676.582 0.0314202
\(775\) 32127.0 1.48908
\(776\) −6923.49 −0.320282
\(777\) 0 0
\(778\) 6493.55 0.299235
\(779\) −5322.07 −0.244779
\(780\) 1087.24 0.0499094
\(781\) −26690.5 −1.22287
\(782\) 22150.4 1.01291
\(783\) 5891.15 0.268880
\(784\) 0 0
\(785\) −24520.5 −1.11487
\(786\) −1785.53 −0.0810275
\(787\) 11075.8 0.501664 0.250832 0.968031i \(-0.419296\pi\)
0.250832 + 0.968031i \(0.419296\pi\)
\(788\) −8247.52 −0.372850
\(789\) −11086.1 −0.500223
\(790\) 10924.8 0.492008
\(791\) 0 0
\(792\) −4132.58 −0.185410
\(793\) −4767.88 −0.213509
\(794\) −3661.25 −0.163643
\(795\) −10092.4 −0.450241
\(796\) −12686.0 −0.564878
\(797\) 4838.83 0.215057 0.107528 0.994202i \(-0.465706\pi\)
0.107528 + 0.994202i \(0.465706\pi\)
\(798\) 0 0
\(799\) 13234.1 0.585969
\(800\) −4089.41 −0.180728
\(801\) 3069.77 0.135412
\(802\) 6771.62 0.298147
\(803\) −25809.0 −1.13422
\(804\) 1987.05 0.0871617
\(805\) 0 0
\(806\) 2865.16 0.125212
\(807\) −21473.1 −0.936664
\(808\) −1946.05 −0.0847299
\(809\) −31509.9 −1.36938 −0.684690 0.728834i \(-0.740062\pi\)
−0.684690 + 0.728834i \(0.740062\pi\)
\(810\) −2575.72 −0.111730
\(811\) −29463.3 −1.27570 −0.637851 0.770160i \(-0.720177\pi\)
−0.637851 + 0.770160i \(0.720177\pi\)
\(812\) 0 0
\(813\) 12115.1 0.522627
\(814\) −44401.6 −1.91189
\(815\) −39983.0 −1.71846
\(816\) −2491.18 −0.106873
\(817\) 608.962 0.0260770
\(818\) −18506.3 −0.791026
\(819\) 0 0
\(820\) 20892.1 0.889736
\(821\) 3502.22 0.148877 0.0744386 0.997226i \(-0.476284\pi\)
0.0744386 + 0.997226i \(0.476284\pi\)
\(822\) 3725.91 0.158097
\(823\) 39993.0 1.69389 0.846943 0.531684i \(-0.178440\pi\)
0.846943 + 0.531684i \(0.178440\pi\)
\(824\) −7626.77 −0.322441
\(825\) −22005.0 −0.928623
\(826\) 0 0
\(827\) −10733.6 −0.451322 −0.225661 0.974206i \(-0.572454\pi\)
−0.225661 + 0.974206i \(0.572454\pi\)
\(828\) −7682.29 −0.322437
\(829\) −14537.5 −0.609056 −0.304528 0.952503i \(-0.598499\pi\)
−0.304528 + 0.952503i \(0.598499\pi\)
\(830\) 47772.6 1.99785
\(831\) 8264.47 0.344996
\(832\) −364.703 −0.0151969
\(833\) 0 0
\(834\) 5392.52 0.223894
\(835\) −8405.16 −0.348351
\(836\) −3719.56 −0.153880
\(837\) −6787.72 −0.280308
\(838\) −7095.03 −0.292475
\(839\) 7353.57 0.302591 0.151295 0.988489i \(-0.451656\pi\)
0.151295 + 0.988489i \(0.451656\pi\)
\(840\) 0 0
\(841\) 23218.3 0.951998
\(842\) −15697.7 −0.642494
\(843\) 2318.23 0.0947141
\(844\) 5399.88 0.220227
\(845\) −34414.9 −1.40107
\(846\) −4589.91 −0.186530
\(847\) 0 0
\(848\) 3385.41 0.137094
\(849\) −20236.5 −0.818037
\(850\) −13264.9 −0.535273
\(851\) −82540.7 −3.32486
\(852\) 5580.18 0.224382
\(853\) 19293.6 0.774442 0.387221 0.921987i \(-0.373435\pi\)
0.387221 + 0.921987i \(0.373435\pi\)
\(854\) 0 0
\(855\) −2318.29 −0.0927297
\(856\) 10759.6 0.429622
\(857\) −14161.6 −0.564468 −0.282234 0.959346i \(-0.591076\pi\)
−0.282234 + 0.959346i \(0.591076\pi\)
\(858\) −1962.45 −0.0780852
\(859\) 8219.13 0.326465 0.163232 0.986588i \(-0.447808\pi\)
0.163232 + 0.986588i \(0.447808\pi\)
\(860\) −2390.51 −0.0947858
\(861\) 0 0
\(862\) 8894.65 0.351454
\(863\) 2574.95 0.101567 0.0507835 0.998710i \(-0.483828\pi\)
0.0507835 + 0.998710i \(0.483828\pi\)
\(864\) 864.000 0.0340207
\(865\) −1539.77 −0.0605246
\(866\) 13989.6 0.548946
\(867\) 6658.33 0.260817
\(868\) 0 0
\(869\) −19719.2 −0.769766
\(870\) −20814.8 −0.811134
\(871\) 943.600 0.0367080
\(872\) −13874.8 −0.538831
\(873\) 7788.93 0.301965
\(874\) −6914.49 −0.267604
\(875\) 0 0
\(876\) 5395.90 0.208117
\(877\) 30981.1 1.19288 0.596442 0.802656i \(-0.296581\pi\)
0.596442 + 0.802656i \(0.296581\pi\)
\(878\) −1272.36 −0.0489068
\(879\) 5768.08 0.221334
\(880\) 14601.3 0.559330
\(881\) −41781.8 −1.59780 −0.798902 0.601461i \(-0.794585\pi\)
−0.798902 + 0.601461i \(0.794585\pi\)
\(882\) 0 0
\(883\) −39289.6 −1.49740 −0.748699 0.662911i \(-0.769321\pi\)
−0.748699 + 0.662911i \(0.769321\pi\)
\(884\) −1182.99 −0.0450095
\(885\) −19661.4 −0.746790
\(886\) 8948.48 0.339312
\(887\) 5832.44 0.220783 0.110391 0.993888i \(-0.464790\pi\)
0.110391 + 0.993888i \(0.464790\pi\)
\(888\) 9283.05 0.350810
\(889\) 0 0
\(890\) −10846.2 −0.408499
\(891\) 4649.15 0.174806
\(892\) 5447.39 0.204476
\(893\) −4131.18 −0.154809
\(894\) −18328.2 −0.685668
\(895\) −8498.95 −0.317418
\(896\) 0 0
\(897\) −3648.12 −0.135794
\(898\) 4778.84 0.177586
\(899\) −54852.5 −2.03497
\(900\) 4600.58 0.170392
\(901\) 10981.3 0.406038
\(902\) −37710.1 −1.39203
\(903\) 0 0
\(904\) −11529.4 −0.424184
\(905\) −33182.3 −1.21880
\(906\) −390.873 −0.0143332
\(907\) 42387.9 1.55178 0.775892 0.630866i \(-0.217300\pi\)
0.775892 + 0.630866i \(0.217300\pi\)
\(908\) −7447.24 −0.272186
\(909\) 2189.30 0.0798841
\(910\) 0 0
\(911\) 2275.12 0.0827423 0.0413711 0.999144i \(-0.486827\pi\)
0.0413711 + 0.999144i \(0.486827\pi\)
\(912\) 777.648 0.0282352
\(913\) −86229.3 −3.12571
\(914\) 8876.68 0.321241
\(915\) −39909.0 −1.44191
\(916\) 21435.1 0.773184
\(917\) 0 0
\(918\) 2802.57 0.100761
\(919\) −31284.3 −1.12293 −0.561465 0.827500i \(-0.689762\pi\)
−0.561465 + 0.827500i \(0.689762\pi\)
\(920\) 27143.2 0.972702
\(921\) 6050.05 0.216456
\(922\) 28159.7 1.00585
\(923\) 2649.88 0.0944983
\(924\) 0 0
\(925\) 49429.9 1.75702
\(926\) −9374.99 −0.332701
\(927\) 8580.12 0.304000
\(928\) 6982.11 0.246982
\(929\) 32196.6 1.13707 0.568535 0.822659i \(-0.307511\pi\)
0.568535 + 0.822659i \(0.307511\pi\)
\(930\) 23982.5 0.845610
\(931\) 0 0
\(932\) 21764.5 0.764935
\(933\) 21450.0 0.752669
\(934\) −16894.5 −0.591869
\(935\) 47362.6 1.65660
\(936\) 410.291 0.0143278
\(937\) 22293.6 0.777269 0.388635 0.921392i \(-0.372947\pi\)
0.388635 + 0.921392i \(0.372947\pi\)
\(938\) 0 0
\(939\) 25789.5 0.896281
\(940\) 16217.2 0.562708
\(941\) −31809.7 −1.10198 −0.550991 0.834511i \(-0.685750\pi\)
−0.550991 + 0.834511i \(0.685750\pi\)
\(942\) −9253.32 −0.320053
\(943\) −70101.4 −2.42080
\(944\) 6595.22 0.227390
\(945\) 0 0
\(946\) 4314.86 0.148296
\(947\) 18982.6 0.651376 0.325688 0.945477i \(-0.394404\pi\)
0.325688 + 0.945477i \(0.394404\pi\)
\(948\) 4122.69 0.141243
\(949\) 2562.37 0.0876481
\(950\) 4140.78 0.141415
\(951\) 8559.71 0.291869
\(952\) 0 0
\(953\) 9254.58 0.314570 0.157285 0.987553i \(-0.449726\pi\)
0.157285 + 0.987553i \(0.449726\pi\)
\(954\) −3808.58 −0.129253
\(955\) 53862.5 1.82508
\(956\) −4629.10 −0.156607
\(957\) 37570.5 1.26905
\(958\) 8738.81 0.294716
\(959\) 0 0
\(960\) −3052.70 −0.102631
\(961\) 33409.4 1.12146
\(962\) 4408.28 0.147743
\(963\) −12104.6 −0.405052
\(964\) 15877.5 0.530477
\(965\) −30341.8 −1.01216
\(966\) 0 0
\(967\) 15317.5 0.509387 0.254694 0.967022i \(-0.418025\pi\)
0.254694 + 0.967022i \(0.418025\pi\)
\(968\) −15707.3 −0.521541
\(969\) 2522.47 0.0836259
\(970\) −27520.0 −0.910943
\(971\) 22432.6 0.741398 0.370699 0.928753i \(-0.379118\pi\)
0.370699 + 0.928753i \(0.379118\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) 28955.5 0.952559
\(975\) 2184.70 0.0717603
\(976\) 13387.1 0.439048
\(977\) −627.864 −0.0205600 −0.0102800 0.999947i \(-0.503272\pi\)
−0.0102800 + 0.999947i \(0.503272\pi\)
\(978\) −15088.4 −0.493327
\(979\) 19577.3 0.639114
\(980\) 0 0
\(981\) 15609.2 0.508015
\(982\) −18612.2 −0.604827
\(983\) 45032.6 1.46116 0.730579 0.682828i \(-0.239250\pi\)
0.730579 + 0.682828i \(0.239250\pi\)
\(984\) 7884.06 0.255421
\(985\) −32782.8 −1.06045
\(986\) 22648.0 0.731500
\(987\) 0 0
\(988\) 369.285 0.0118912
\(989\) 8021.14 0.257894
\(990\) −16426.5 −0.527342
\(991\) −31981.0 −1.02514 −0.512569 0.858646i \(-0.671306\pi\)
−0.512569 + 0.858646i \(0.671306\pi\)
\(992\) −8044.70 −0.257479
\(993\) −4857.35 −0.155230
\(994\) 0 0
\(995\) −50425.2 −1.60662
\(996\) 18028.0 0.573532
\(997\) 12378.8 0.393221 0.196611 0.980482i \(-0.437007\pi\)
0.196611 + 0.980482i \(0.437007\pi\)
\(998\) 24475.1 0.776298
\(999\) −10443.4 −0.330746
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 294.4.a.j.1.2 2
3.2 odd 2 882.4.a.bc.1.1 2
4.3 odd 2 2352.4.a.cd.1.2 2
7.2 even 3 294.4.e.o.67.1 4
7.3 odd 6 294.4.e.n.79.2 4
7.4 even 3 294.4.e.o.79.1 4
7.5 odd 6 294.4.e.n.67.2 4
7.6 odd 2 294.4.a.k.1.1 yes 2
21.2 odd 6 882.4.g.bd.361.2 4
21.5 even 6 882.4.g.y.361.1 4
21.11 odd 6 882.4.g.bd.667.2 4
21.17 even 6 882.4.g.y.667.1 4
21.20 even 2 882.4.a.bi.1.2 2
28.27 even 2 2352.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.2 2 1.1 even 1 trivial
294.4.a.k.1.1 yes 2 7.6 odd 2
294.4.e.n.67.2 4 7.5 odd 6
294.4.e.n.79.2 4 7.3 odd 6
294.4.e.o.67.1 4 7.2 even 3
294.4.e.o.79.1 4 7.4 even 3
882.4.a.bc.1.1 2 3.2 odd 2
882.4.a.bi.1.2 2 21.20 even 2
882.4.g.y.361.1 4 21.5 even 6
882.4.g.y.667.1 4 21.17 even 6
882.4.g.bd.361.2 4 21.2 odd 6
882.4.g.bd.667.2 4 21.11 odd 6
2352.4.a.bn.1.1 2 28.27 even 2
2352.4.a.cd.1.2 2 4.3 odd 2