Properties

Label 570.6.a.k.1.4
Level $570$
Weight $6$
Character 570.1
Self dual yes
Analytic conductor $91.419$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.4187772934\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 26355 x^{2} - 7203 x + 128450070\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(142.069\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +154.089 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +154.089 q^{7} -64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} -170.915 q^{11} +144.000 q^{12} +428.866 q^{13} -616.358 q^{14} -225.000 q^{15} +256.000 q^{16} +643.732 q^{17} -324.000 q^{18} -361.000 q^{19} -400.000 q^{20} +1386.80 q^{21} +683.661 q^{22} +603.612 q^{23} -576.000 q^{24} +625.000 q^{25} -1715.46 q^{26} +729.000 q^{27} +2465.43 q^{28} -2796.64 q^{29} +900.000 q^{30} +8519.91 q^{31} -1024.00 q^{32} -1538.24 q^{33} -2574.93 q^{34} -3852.24 q^{35} +1296.00 q^{36} +7657.76 q^{37} +1444.00 q^{38} +3859.79 q^{39} +1600.00 q^{40} -10006.3 q^{41} -5547.22 q^{42} -18239.6 q^{43} -2734.64 q^{44} -2025.00 q^{45} -2414.45 q^{46} +22320.0 q^{47} +2304.00 q^{48} +6936.55 q^{49} -2500.00 q^{50} +5793.59 q^{51} +6861.86 q^{52} +18742.6 q^{53} -2916.00 q^{54} +4272.88 q^{55} -9861.72 q^{56} -3249.00 q^{57} +11186.6 q^{58} -40593.3 q^{59} -3600.00 q^{60} +4700.35 q^{61} -34079.6 q^{62} +12481.2 q^{63} +4096.00 q^{64} -10721.6 q^{65} +6152.95 q^{66} +43815.5 q^{67} +10299.7 q^{68} +5432.51 q^{69} +15408.9 q^{70} -44371.6 q^{71} -5184.00 q^{72} +40128.8 q^{73} -30631.0 q^{74} +5625.00 q^{75} -5776.00 q^{76} -26336.2 q^{77} -15439.2 q^{78} +65555.6 q^{79} -6400.00 q^{80} +6561.00 q^{81} +40025.1 q^{82} +43398.6 q^{83} +22188.9 q^{84} -16093.3 q^{85} +72958.4 q^{86} -25169.7 q^{87} +10938.6 q^{88} -58336.8 q^{89} +8100.00 q^{90} +66083.7 q^{91} +9657.79 q^{92} +76679.2 q^{93} -89279.8 q^{94} +9025.00 q^{95} -9216.00 q^{96} -183324. q^{97} -27746.2 q^{98} -13844.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + 400q^{10} - 472q^{11} + 576q^{12} - 482q^{13} - 104q^{14} - 900q^{15} + 1024q^{16} + 1816q^{17} - 1296q^{18} - 1444q^{19} - 1600q^{20} + 234q^{21} + 1888q^{22} - 418q^{23} - 2304q^{24} + 2500q^{25} + 1928q^{26} + 2916q^{27} + 416q^{28} - 10396q^{29} + 3600q^{30} + 528q^{31} - 4096q^{32} - 4248q^{33} - 7264q^{34} - 650q^{35} + 5184q^{36} + 5774q^{37} + 5776q^{38} - 4338q^{39} + 6400q^{40} + 9620q^{41} - 936q^{42} + 21098q^{43} - 7552q^{44} - 8100q^{45} + 1672q^{46} - 18858q^{47} + 9216q^{48} + 2148q^{49} - 10000q^{50} + 16344q^{51} - 7712q^{52} + 822q^{53} - 11664q^{54} + 11800q^{55} - 1664q^{56} - 12996q^{57} + 41584q^{58} + 28672q^{59} - 14400q^{60} + 77748q^{61} - 2112q^{62} + 2106q^{63} + 16384q^{64} + 12050q^{65} + 16992q^{66} + 82400q^{67} + 29056q^{68} - 3762q^{69} + 2600q^{70} + 928q^{71} - 20736q^{72} + 121100q^{73} - 23096q^{74} + 22500q^{75} - 23104q^{76} - 120220q^{77} + 17352q^{78} + 144284q^{79} - 25600q^{80} + 26244q^{81} - 38480q^{82} - 6082q^{83} + 3744q^{84} - 45400q^{85} - 84392q^{86} - 93564q^{87} + 30208q^{88} + 43260q^{89} + 32400q^{90} + 135148q^{91} - 6688q^{92} + 4752q^{93} + 75432q^{94} + 36100q^{95} - 36864q^{96} - 6862q^{97} - 8592q^{98} - 38232q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 9.00000 0.577350
\(4\) 16.0000 0.500000
\(5\) −25.0000 −0.447214
\(6\) −36.0000 −0.408248
\(7\) 154.089 1.18858 0.594289 0.804252i \(-0.297434\pi\)
0.594289 + 0.804252i \(0.297434\pi\)
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 100.000 0.316228
\(11\) −170.915 −0.425892 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(12\) 144.000 0.288675
\(13\) 428.866 0.703823 0.351911 0.936033i \(-0.385532\pi\)
0.351911 + 0.936033i \(0.385532\pi\)
\(14\) −616.358 −0.840452
\(15\) −225.000 −0.258199
\(16\) 256.000 0.250000
\(17\) 643.732 0.540235 0.270118 0.962827i \(-0.412937\pi\)
0.270118 + 0.962827i \(0.412937\pi\)
\(18\) −324.000 −0.235702
\(19\) −361.000 −0.229416
\(20\) −400.000 −0.223607
\(21\) 1386.80 0.686226
\(22\) 683.661 0.301151
\(23\) 603.612 0.237924 0.118962 0.992899i \(-0.462043\pi\)
0.118962 + 0.992899i \(0.462043\pi\)
\(24\) −576.000 −0.204124
\(25\) 625.000 0.200000
\(26\) −1715.46 −0.497678
\(27\) 729.000 0.192450
\(28\) 2465.43 0.594289
\(29\) −2796.64 −0.617506 −0.308753 0.951142i \(-0.599912\pi\)
−0.308753 + 0.951142i \(0.599912\pi\)
\(30\) 900.000 0.182574
\(31\) 8519.91 1.59232 0.796161 0.605085i \(-0.206861\pi\)
0.796161 + 0.605085i \(0.206861\pi\)
\(32\) −1024.00 −0.176777
\(33\) −1538.24 −0.245889
\(34\) −2574.93 −0.382004
\(35\) −3852.24 −0.531548
\(36\) 1296.00 0.166667
\(37\) 7657.76 0.919597 0.459798 0.888023i \(-0.347922\pi\)
0.459798 + 0.888023i \(0.347922\pi\)
\(38\) 1444.00 0.162221
\(39\) 3859.79 0.406352
\(40\) 1600.00 0.158114
\(41\) −10006.3 −0.929636 −0.464818 0.885406i \(-0.653880\pi\)
−0.464818 + 0.885406i \(0.653880\pi\)
\(42\) −5547.22 −0.485235
\(43\) −18239.6 −1.50433 −0.752167 0.658972i \(-0.770991\pi\)
−0.752167 + 0.658972i \(0.770991\pi\)
\(44\) −2734.64 −0.212946
\(45\) −2025.00 −0.149071
\(46\) −2414.45 −0.168238
\(47\) 22320.0 1.47383 0.736917 0.675983i \(-0.236281\pi\)
0.736917 + 0.675983i \(0.236281\pi\)
\(48\) 2304.00 0.144338
\(49\) 6936.55 0.412718
\(50\) −2500.00 −0.141421
\(51\) 5793.59 0.311905
\(52\) 6861.86 0.351911
\(53\) 18742.6 0.916517 0.458259 0.888819i \(-0.348473\pi\)
0.458259 + 0.888819i \(0.348473\pi\)
\(54\) −2916.00 −0.136083
\(55\) 4272.88 0.190465
\(56\) −9861.72 −0.420226
\(57\) −3249.00 −0.132453
\(58\) 11186.6 0.436643
\(59\) −40593.3 −1.51818 −0.759092 0.650983i \(-0.774357\pi\)
−0.759092 + 0.650983i \(0.774357\pi\)
\(60\) −3600.00 −0.129099
\(61\) 4700.35 0.161736 0.0808678 0.996725i \(-0.474231\pi\)
0.0808678 + 0.996725i \(0.474231\pi\)
\(62\) −34079.6 −1.12594
\(63\) 12481.2 0.396193
\(64\) 4096.00 0.125000
\(65\) −10721.6 −0.314759
\(66\) 6152.95 0.173870
\(67\) 43815.5 1.19245 0.596226 0.802817i \(-0.296666\pi\)
0.596226 + 0.802817i \(0.296666\pi\)
\(68\) 10299.7 0.270118
\(69\) 5432.51 0.137365
\(70\) 15408.9 0.375861
\(71\) −44371.6 −1.04462 −0.522312 0.852755i \(-0.674930\pi\)
−0.522312 + 0.852755i \(0.674930\pi\)
\(72\) −5184.00 −0.117851
\(73\) 40128.8 0.881352 0.440676 0.897666i \(-0.354739\pi\)
0.440676 + 0.897666i \(0.354739\pi\)
\(74\) −30631.0 −0.650253
\(75\) 5625.00 0.115470
\(76\) −5776.00 −0.114708
\(77\) −26336.2 −0.506205
\(78\) −15439.2 −0.287334
\(79\) 65555.6 1.18180 0.590898 0.806747i \(-0.298774\pi\)
0.590898 + 0.806747i \(0.298774\pi\)
\(80\) −6400.00 −0.111803
\(81\) 6561.00 0.111111
\(82\) 40025.1 0.657352
\(83\) 43398.6 0.691481 0.345741 0.938330i \(-0.387628\pi\)
0.345741 + 0.938330i \(0.387628\pi\)
\(84\) 22188.9 0.343113
\(85\) −16093.3 −0.241600
\(86\) 72958.4 1.06373
\(87\) −25169.7 −0.356517
\(88\) 10938.6 0.150575
\(89\) −58336.8 −0.780670 −0.390335 0.920673i \(-0.627641\pi\)
−0.390335 + 0.920673i \(0.627641\pi\)
\(90\) 8100.00 0.105409
\(91\) 66083.7 0.836548
\(92\) 9657.79 0.118962
\(93\) 76679.2 0.919327
\(94\) −89279.8 −1.04216
\(95\) 9025.00 0.102598
\(96\) −9216.00 −0.102062
\(97\) −183324. −1.97829 −0.989146 0.146939i \(-0.953058\pi\)
−0.989146 + 0.146939i \(0.953058\pi\)
\(98\) −27746.2 −0.291836
\(99\) −13844.1 −0.141964
\(100\) 10000.0 0.100000
\(101\) −3464.89 −0.0337976 −0.0168988 0.999857i \(-0.505379\pi\)
−0.0168988 + 0.999857i \(0.505379\pi\)
\(102\) −23174.4 −0.220550
\(103\) 18567.0 0.172444 0.0862222 0.996276i \(-0.472521\pi\)
0.0862222 + 0.996276i \(0.472521\pi\)
\(104\) −27447.4 −0.248839
\(105\) −34670.1 −0.306890
\(106\) −74970.5 −0.648075
\(107\) 52951.8 0.447118 0.223559 0.974690i \(-0.428233\pi\)
0.223559 + 0.974690i \(0.428233\pi\)
\(108\) 11664.0 0.0962250
\(109\) 113888. 0.918142 0.459071 0.888400i \(-0.348182\pi\)
0.459071 + 0.888400i \(0.348182\pi\)
\(110\) −17091.5 −0.134679
\(111\) 68919.8 0.530929
\(112\) 39446.9 0.297145
\(113\) 73805.0 0.543738 0.271869 0.962334i \(-0.412358\pi\)
0.271869 + 0.962334i \(0.412358\pi\)
\(114\) 12996.0 0.0936586
\(115\) −15090.3 −0.106403
\(116\) −44746.2 −0.308753
\(117\) 34738.1 0.234608
\(118\) 162373. 1.07352
\(119\) 99192.3 0.642112
\(120\) 14400.0 0.0912871
\(121\) −131839. −0.818616
\(122\) −18801.4 −0.114364
\(123\) −90056.5 −0.536726
\(124\) 136319. 0.796161
\(125\) −15625.0 −0.0894427
\(126\) −49925.0 −0.280151
\(127\) −34162.6 −0.187950 −0.0939748 0.995575i \(-0.529957\pi\)
−0.0939748 + 0.995575i \(0.529957\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −164157. −0.868528
\(130\) 42886.6 0.222568
\(131\) 45985.4 0.234122 0.117061 0.993125i \(-0.462653\pi\)
0.117061 + 0.993125i \(0.462653\pi\)
\(132\) −24611.8 −0.122944
\(133\) −55626.3 −0.272679
\(134\) −175262. −0.843190
\(135\) −18225.0 −0.0860663
\(136\) −41198.8 −0.191002
\(137\) −40735.7 −0.185427 −0.0927136 0.995693i \(-0.529554\pi\)
−0.0927136 + 0.995693i \(0.529554\pi\)
\(138\) −21730.0 −0.0971320
\(139\) −101466. −0.445435 −0.222718 0.974883i \(-0.571493\pi\)
−0.222718 + 0.974883i \(0.571493\pi\)
\(140\) −61635.8 −0.265774
\(141\) 200880. 0.850919
\(142\) 177487. 0.738660
\(143\) −73299.7 −0.299752
\(144\) 20736.0 0.0833333
\(145\) 69916.0 0.276157
\(146\) −160515. −0.623210
\(147\) 62428.9 0.238283
\(148\) 122524. 0.459798
\(149\) 385179. 1.42134 0.710668 0.703528i \(-0.248393\pi\)
0.710668 + 0.703528i \(0.248393\pi\)
\(150\) −22500.0 −0.0816497
\(151\) 124234. 0.443403 0.221701 0.975115i \(-0.428839\pi\)
0.221701 + 0.975115i \(0.428839\pi\)
\(152\) 23104.0 0.0811107
\(153\) 52142.3 0.180078
\(154\) 105345. 0.357941
\(155\) −212998. −0.712108
\(156\) 61756.7 0.203176
\(157\) 180803. 0.585406 0.292703 0.956203i \(-0.405445\pi\)
0.292703 + 0.956203i \(0.405445\pi\)
\(158\) −262222. −0.835655
\(159\) 168684. 0.529151
\(160\) 25600.0 0.0790569
\(161\) 93010.2 0.282791
\(162\) −26244.0 −0.0785674
\(163\) 259451. 0.764869 0.382434 0.923983i \(-0.375086\pi\)
0.382434 + 0.923983i \(0.375086\pi\)
\(164\) −160100. −0.464818
\(165\) 38455.9 0.109965
\(166\) −173594. −0.488951
\(167\) −153015. −0.424563 −0.212282 0.977209i \(-0.568089\pi\)
−0.212282 + 0.977209i \(0.568089\pi\)
\(168\) −88755.5 −0.242617
\(169\) −187367. −0.504634
\(170\) 64373.2 0.170837
\(171\) −29241.0 −0.0764719
\(172\) −291834. −0.752167
\(173\) 711467. 1.80734 0.903670 0.428230i \(-0.140863\pi\)
0.903670 + 0.428230i \(0.140863\pi\)
\(174\) 100679. 0.252096
\(175\) 96305.9 0.237716
\(176\) −43754.3 −0.106473
\(177\) −365340. −0.876524
\(178\) 233347. 0.552017
\(179\) 25929.2 0.0604863 0.0302431 0.999543i \(-0.490372\pi\)
0.0302431 + 0.999543i \(0.490372\pi\)
\(180\) −32400.0 −0.0745356
\(181\) 782725. 1.77588 0.887939 0.459962i \(-0.152137\pi\)
0.887939 + 0.459962i \(0.152137\pi\)
\(182\) −264335. −0.591529
\(183\) 42303.1 0.0933781
\(184\) −38631.1 −0.0841188
\(185\) −191444. −0.411256
\(186\) −306717. −0.650062
\(187\) −110024. −0.230082
\(188\) 357119. 0.736917
\(189\) 112331. 0.228742
\(190\) −36100.0 −0.0725476
\(191\) 348184. 0.690599 0.345299 0.938493i \(-0.387777\pi\)
0.345299 + 0.938493i \(0.387777\pi\)
\(192\) 36864.0 0.0721688
\(193\) 880820. 1.70214 0.851068 0.525056i \(-0.175956\pi\)
0.851068 + 0.525056i \(0.175956\pi\)
\(194\) 733296. 1.39886
\(195\) −96494.8 −0.181726
\(196\) 110985. 0.206359
\(197\) 509269. 0.934935 0.467467 0.884010i \(-0.345167\pi\)
0.467467 + 0.884010i \(0.345167\pi\)
\(198\) 55376.5 0.100384
\(199\) 477521. 0.854791 0.427396 0.904065i \(-0.359431\pi\)
0.427396 + 0.904065i \(0.359431\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 394339. 0.688462
\(202\) 13859.6 0.0238985
\(203\) −430932. −0.733954
\(204\) 92697.4 0.155952
\(205\) 250157. 0.415746
\(206\) −74268.1 −0.121937
\(207\) 48892.5 0.0793080
\(208\) 109790. 0.175956
\(209\) 61700.4 0.0977062
\(210\) 138680. 0.217004
\(211\) 420820. 0.650713 0.325357 0.945591i \(-0.394516\pi\)
0.325357 + 0.945591i \(0.394516\pi\)
\(212\) 299882. 0.458259
\(213\) −399345. −0.603114
\(214\) −211807. −0.316160
\(215\) 455990. 0.672759
\(216\) −46656.0 −0.0680414
\(217\) 1.31283e6 1.89260
\(218\) −455550. −0.649224
\(219\) 361160. 0.508849
\(220\) 68366.1 0.0952323
\(221\) 276075. 0.380230
\(222\) −275679. −0.375424
\(223\) −563018. −0.758158 −0.379079 0.925364i \(-0.623759\pi\)
−0.379079 + 0.925364i \(0.623759\pi\)
\(224\) −157788. −0.210113
\(225\) 50625.0 0.0666667
\(226\) −295220. −0.384481
\(227\) 876104. 1.12847 0.564236 0.825613i \(-0.309171\pi\)
0.564236 + 0.825613i \(0.309171\pi\)
\(228\) −51984.0 −0.0662266
\(229\) 1.20221e6 1.51493 0.757465 0.652875i \(-0.226438\pi\)
0.757465 + 0.652875i \(0.226438\pi\)
\(230\) 60361.2 0.0752381
\(231\) −237026. −0.292258
\(232\) 178985. 0.218321
\(233\) 151909. 0.183313 0.0916565 0.995791i \(-0.470784\pi\)
0.0916565 + 0.995791i \(0.470784\pi\)
\(234\) −138953. −0.165893
\(235\) −557999. −0.659119
\(236\) −649493. −0.759092
\(237\) 590001. 0.682310
\(238\) −396769. −0.454041
\(239\) 537847. 0.609065 0.304533 0.952502i \(-0.401500\pi\)
0.304533 + 0.952502i \(0.401500\pi\)
\(240\) −57600.0 −0.0645497
\(241\) 853488. 0.946574 0.473287 0.880908i \(-0.343067\pi\)
0.473287 + 0.880908i \(0.343067\pi\)
\(242\) 527356. 0.578849
\(243\) 59049.0 0.0641500
\(244\) 75205.6 0.0808678
\(245\) −173414. −0.184573
\(246\) 360226. 0.379522
\(247\) −154821. −0.161468
\(248\) −545274. −0.562971
\(249\) 390587. 0.399227
\(250\) 62500.0 0.0632456
\(251\) 25664.5 0.0257127 0.0128564 0.999917i \(-0.495908\pi\)
0.0128564 + 0.999917i \(0.495908\pi\)
\(252\) 199700. 0.198096
\(253\) −103166. −0.101330
\(254\) 136650. 0.132900
\(255\) −144840. −0.139488
\(256\) 65536.0 0.0625000
\(257\) 927552. 0.876003 0.438001 0.898974i \(-0.355686\pi\)
0.438001 + 0.898974i \(0.355686\pi\)
\(258\) 656626. 0.614142
\(259\) 1.17998e6 1.09301
\(260\) −171546. −0.157380
\(261\) −226528. −0.205835
\(262\) −183942. −0.165549
\(263\) −163358. −0.145630 −0.0728152 0.997345i \(-0.523198\pi\)
−0.0728152 + 0.997345i \(0.523198\pi\)
\(264\) 98447.2 0.0869348
\(265\) −468565. −0.409879
\(266\) 222505. 0.192813
\(267\) −525031. −0.450720
\(268\) 701048. 0.596226
\(269\) 795370. 0.670175 0.335088 0.942187i \(-0.391234\pi\)
0.335088 + 0.942187i \(0.391234\pi\)
\(270\) 72900.0 0.0608581
\(271\) −1.09810e6 −0.908274 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(272\) 164795. 0.135059
\(273\) 594753. 0.482981
\(274\) 162943. 0.131117
\(275\) −106822. −0.0851783
\(276\) 86920.1 0.0686827
\(277\) 2.11726e6 1.65796 0.828981 0.559277i \(-0.188921\pi\)
0.828981 + 0.559277i \(0.188921\pi\)
\(278\) 405865. 0.314970
\(279\) 690112. 0.530774
\(280\) 246543. 0.187931
\(281\) −1.17403e6 −0.886983 −0.443491 0.896279i \(-0.646260\pi\)
−0.443491 + 0.896279i \(0.646260\pi\)
\(282\) −803518. −0.601690
\(283\) −1.75925e6 −1.30576 −0.652879 0.757462i \(-0.726439\pi\)
−0.652879 + 0.757462i \(0.726439\pi\)
\(284\) −709946. −0.522312
\(285\) 81225.0 0.0592349
\(286\) 293199. 0.211957
\(287\) −1.54186e6 −1.10494
\(288\) −82944.0 −0.0589256
\(289\) −1.00547e6 −0.708146
\(290\) −279664. −0.195273
\(291\) −1.64992e6 −1.14217
\(292\) 642062. 0.440676
\(293\) −2.56876e6 −1.74805 −0.874027 0.485878i \(-0.838500\pi\)
−0.874027 + 0.485878i \(0.838500\pi\)
\(294\) −249716. −0.168491
\(295\) 1.01483e6 0.678952
\(296\) −490097. −0.325127
\(297\) −124597. −0.0819629
\(298\) −1.54072e6 −1.00504
\(299\) 258869. 0.167456
\(300\) 90000.0 0.0577350
\(301\) −2.81053e6 −1.78802
\(302\) −496936. −0.313533
\(303\) −31184.0 −0.0195130
\(304\) −92416.0 −0.0573539
\(305\) −117509. −0.0723303
\(306\) −208569. −0.127335
\(307\) 968764. 0.586641 0.293320 0.956014i \(-0.405240\pi\)
0.293320 + 0.956014i \(0.405240\pi\)
\(308\) −421380. −0.253103
\(309\) 167103. 0.0995608
\(310\) 851991. 0.503536
\(311\) −1.11053e6 −0.651075 −0.325538 0.945529i \(-0.605545\pi\)
−0.325538 + 0.945529i \(0.605545\pi\)
\(312\) −247027. −0.143667
\(313\) 735913. 0.424586 0.212293 0.977206i \(-0.431907\pi\)
0.212293 + 0.977206i \(0.431907\pi\)
\(314\) −723213. −0.413944
\(315\) −312031. −0.177183
\(316\) 1.04889e6 0.590898
\(317\) −2.95111e6 −1.64944 −0.824720 0.565541i \(-0.808667\pi\)
−0.824720 + 0.565541i \(0.808667\pi\)
\(318\) −674734. −0.374167
\(319\) 477988. 0.262991
\(320\) −102400. −0.0559017
\(321\) 476567. 0.258143
\(322\) −372041. −0.199964
\(323\) −232387. −0.123938
\(324\) 104976. 0.0555556
\(325\) 268041. 0.140765
\(326\) −1.03781e6 −0.540844
\(327\) 1.02499e6 0.530089
\(328\) 640402. 0.328676
\(329\) 3.43927e6 1.75177
\(330\) −153824. −0.0777568
\(331\) 857181. 0.430034 0.215017 0.976610i \(-0.431019\pi\)
0.215017 + 0.976610i \(0.431019\pi\)
\(332\) 694377. 0.345741
\(333\) 620279. 0.306532
\(334\) 612060. 0.300212
\(335\) −1.09539e6 −0.533280
\(336\) 355022. 0.171556
\(337\) 271384. 0.130169 0.0650847 0.997880i \(-0.479268\pi\)
0.0650847 + 0.997880i \(0.479268\pi\)
\(338\) 749468. 0.356830
\(339\) 664245. 0.313927
\(340\) −257493. −0.120800
\(341\) −1.45618e6 −0.678156
\(342\) 116964. 0.0540738
\(343\) −1.52093e6 −0.698031
\(344\) 1.16734e6 0.531863
\(345\) −135813. −0.0614317
\(346\) −2.84587e6 −1.27798
\(347\) −1.21101e6 −0.539915 −0.269958 0.962872i \(-0.587010\pi\)
−0.269958 + 0.962872i \(0.587010\pi\)
\(348\) −402716. −0.178259
\(349\) 606148. 0.266388 0.133194 0.991090i \(-0.457477\pi\)
0.133194 + 0.991090i \(0.457477\pi\)
\(350\) −385224. −0.168090
\(351\) 312643. 0.135451
\(352\) 175017. 0.0752877
\(353\) −3.13708e6 −1.33995 −0.669976 0.742383i \(-0.733696\pi\)
−0.669976 + 0.742383i \(0.733696\pi\)
\(354\) 1.46136e6 0.619796
\(355\) 1.10929e6 0.467170
\(356\) −933389. −0.390335
\(357\) 892731. 0.370723
\(358\) −103717. −0.0427703
\(359\) −3.72604e6 −1.52585 −0.762925 0.646487i \(-0.776237\pi\)
−0.762925 + 0.646487i \(0.776237\pi\)
\(360\) 129600. 0.0527046
\(361\) 130321. 0.0526316
\(362\) −3.13090e6 −1.25573
\(363\) −1.18655e6 −0.472628
\(364\) 1.05734e6 0.418274
\(365\) −1.00322e6 −0.394153
\(366\) −169213. −0.0660283
\(367\) 97146.7 0.0376498 0.0188249 0.999823i \(-0.494007\pi\)
0.0188249 + 0.999823i \(0.494007\pi\)
\(368\) 154525. 0.0594810
\(369\) −810509. −0.309879
\(370\) 765776. 0.290802
\(371\) 2.88804e6 1.08935
\(372\) 1.22687e6 0.459663
\(373\) −2.54000e6 −0.945283 −0.472641 0.881255i \(-0.656699\pi\)
−0.472641 + 0.881255i \(0.656699\pi\)
\(374\) 440094. 0.162692
\(375\) −140625. −0.0516398
\(376\) −1.42848e6 −0.521079
\(377\) −1.19938e6 −0.434615
\(378\) −449325. −0.161745
\(379\) 5877.82 0.00210193 0.00105097 0.999999i \(-0.499665\pi\)
0.00105097 + 0.999999i \(0.499665\pi\)
\(380\) 144400. 0.0512989
\(381\) −307463. −0.108513
\(382\) −1.39274e6 −0.488327
\(383\) −368963. −0.128524 −0.0642622 0.997933i \(-0.520469\pi\)
−0.0642622 + 0.997933i \(0.520469\pi\)
\(384\) −147456. −0.0510310
\(385\) 658406. 0.226382
\(386\) −3.52328e6 −1.20359
\(387\) −1.47741e6 −0.501445
\(388\) −2.93319e6 −0.989146
\(389\) 3.59863e6 1.20576 0.602882 0.797830i \(-0.294019\pi\)
0.602882 + 0.797830i \(0.294019\pi\)
\(390\) 385979. 0.128500
\(391\) 388564. 0.128535
\(392\) −443939. −0.145918
\(393\) 413869. 0.135170
\(394\) −2.03707e6 −0.661099
\(395\) −1.63889e6 −0.528515
\(396\) −221506. −0.0709819
\(397\) 4.55438e6 1.45029 0.725143 0.688599i \(-0.241774\pi\)
0.725143 + 0.688599i \(0.241774\pi\)
\(398\) −1.91008e6 −0.604429
\(399\) −500637. −0.157431
\(400\) 160000. 0.0500000
\(401\) 2.74479e6 0.852411 0.426205 0.904626i \(-0.359850\pi\)
0.426205 + 0.904626i \(0.359850\pi\)
\(402\) −1.57736e6 −0.486816
\(403\) 3.65390e6 1.12071
\(404\) −55438.2 −0.0168988
\(405\) −164025. −0.0496904
\(406\) 1.72373e6 0.518984
\(407\) −1.30883e6 −0.391648
\(408\) −370790. −0.110275
\(409\) 3.10446e6 0.917651 0.458826 0.888526i \(-0.348270\pi\)
0.458826 + 0.888526i \(0.348270\pi\)
\(410\) −1.00063e6 −0.293977
\(411\) −366621. −0.107056
\(412\) 297072. 0.0862222
\(413\) −6.25500e6 −1.80448
\(414\) −195570. −0.0560792
\(415\) −1.08496e6 −0.309240
\(416\) −439159. −0.124419
\(417\) −913197. −0.257172
\(418\) −246802. −0.0690887
\(419\) 1.95553e6 0.544163 0.272081 0.962274i \(-0.412288\pi\)
0.272081 + 0.962274i \(0.412288\pi\)
\(420\) −554722. −0.153445
\(421\) 935570. 0.257259 0.128630 0.991693i \(-0.458942\pi\)
0.128630 + 0.991693i \(0.458942\pi\)
\(422\) −1.68328e6 −0.460124
\(423\) 1.80792e6 0.491278
\(424\) −1.19953e6 −0.324038
\(425\) 402333. 0.108047
\(426\) 1.59738e6 0.426466
\(427\) 724274. 0.192235
\(428\) 847230. 0.223559
\(429\) −659698. −0.173062
\(430\) −1.82396e6 −0.475712
\(431\) −695955. −0.180463 −0.0902314 0.995921i \(-0.528761\pi\)
−0.0902314 + 0.995921i \(0.528761\pi\)
\(432\) 186624. 0.0481125
\(433\) −6.73236e6 −1.72563 −0.862815 0.505520i \(-0.831301\pi\)
−0.862815 + 0.505520i \(0.831301\pi\)
\(434\) −5.25131e6 −1.33827
\(435\) 629244. 0.159439
\(436\) 1.82220e6 0.459071
\(437\) −217904. −0.0545835
\(438\) −1.44464e6 −0.359811
\(439\) −601243. −0.148898 −0.0744490 0.997225i \(-0.523720\pi\)
−0.0744490 + 0.997225i \(0.523720\pi\)
\(440\) −273464. −0.0673394
\(441\) 561860. 0.137573
\(442\) −1.10430e6 −0.268863
\(443\) −3.75205e6 −0.908363 −0.454182 0.890909i \(-0.650068\pi\)
−0.454182 + 0.890909i \(0.650068\pi\)
\(444\) 1.10272e6 0.265465
\(445\) 1.45842e6 0.349126
\(446\) 2.25207e6 0.536099
\(447\) 3.46661e6 0.820609
\(448\) 631150. 0.148572
\(449\) −924053. −0.216312 −0.108156 0.994134i \(-0.534495\pi\)
−0.108156 + 0.994134i \(0.534495\pi\)
\(450\) −202500. −0.0471405
\(451\) 1.71023e6 0.395924
\(452\) 1.18088e6 0.271869
\(453\) 1.11811e6 0.255999
\(454\) −3.50441e6 −0.797950
\(455\) −1.65209e6 −0.374116
\(456\) 207936. 0.0468293
\(457\) −5.65144e6 −1.26581 −0.632906 0.774229i \(-0.718138\pi\)
−0.632906 + 0.774229i \(0.718138\pi\)
\(458\) −4.80885e6 −1.07122
\(459\) 469281. 0.103968
\(460\) −241445. −0.0532014
\(461\) 5.34397e6 1.17115 0.585574 0.810619i \(-0.300869\pi\)
0.585574 + 0.810619i \(0.300869\pi\)
\(462\) 948104. 0.206657
\(463\) −2.67530e6 −0.579988 −0.289994 0.957028i \(-0.593653\pi\)
−0.289994 + 0.957028i \(0.593653\pi\)
\(464\) −715939. −0.154377
\(465\) −1.91698e6 −0.411136
\(466\) −607635. −0.129622
\(467\) 1.81414e6 0.384927 0.192463 0.981304i \(-0.438352\pi\)
0.192463 + 0.981304i \(0.438352\pi\)
\(468\) 555810. 0.117304
\(469\) 6.75150e6 1.41732
\(470\) 2.23200e6 0.466067
\(471\) 1.62723e6 0.337984
\(472\) 2.59797e6 0.536759
\(473\) 3.11743e6 0.640683
\(474\) −2.36000e6 −0.482466
\(475\) −225625. −0.0458831
\(476\) 1.58708e6 0.321056
\(477\) 1.51815e6 0.305506
\(478\) −2.15139e6 −0.430674
\(479\) 1.34799e6 0.268440 0.134220 0.990952i \(-0.457147\pi\)
0.134220 + 0.990952i \(0.457147\pi\)
\(480\) 230400. 0.0456435
\(481\) 3.28415e6 0.647233
\(482\) −3.41395e6 −0.669329
\(483\) 837092. 0.163270
\(484\) −2.10942e6 −0.409308
\(485\) 4.58310e6 0.884719
\(486\) −236196. −0.0453609
\(487\) 1.95283e6 0.373114 0.186557 0.982444i \(-0.440267\pi\)
0.186557 + 0.982444i \(0.440267\pi\)
\(488\) −300822. −0.0571822
\(489\) 2.33506e6 0.441597
\(490\) 693655. 0.130513
\(491\) −9.95354e6 −1.86326 −0.931631 0.363406i \(-0.881614\pi\)
−0.931631 + 0.363406i \(0.881614\pi\)
\(492\) −1.44090e6 −0.268363
\(493\) −1.80029e6 −0.333599
\(494\) 619282. 0.114175
\(495\) 346103. 0.0634882
\(496\) 2.18110e6 0.398080
\(497\) −6.83720e6 −1.24162
\(498\) −1.56235e6 −0.282296
\(499\) −6.49751e6 −1.16814 −0.584071 0.811703i \(-0.698541\pi\)
−0.584071 + 0.811703i \(0.698541\pi\)
\(500\) −250000. −0.0447214
\(501\) −1.37713e6 −0.245122
\(502\) −102658. −0.0181816
\(503\) 3.07853e6 0.542530 0.271265 0.962505i \(-0.412558\pi\)
0.271265 + 0.962505i \(0.412558\pi\)
\(504\) −798800. −0.140075
\(505\) 86622.2 0.0151147
\(506\) 412666. 0.0716510
\(507\) −1.68630e6 −0.291350
\(508\) −546601. −0.0939748
\(509\) 966859. 0.165413 0.0827063 0.996574i \(-0.473644\pi\)
0.0827063 + 0.996574i \(0.473644\pi\)
\(510\) 579359. 0.0986330
\(511\) 6.18343e6 1.04756
\(512\) −262144. −0.0441942
\(513\) −263169. −0.0441511
\(514\) −3.71021e6 −0.619428
\(515\) −464175. −0.0771195
\(516\) −2.62650e6 −0.434264
\(517\) −3.81482e6 −0.627694
\(518\) −4.71992e6 −0.772877
\(519\) 6.40321e6 1.04347
\(520\) 686186. 0.111284
\(521\) −9.40205e6 −1.51750 −0.758750 0.651382i \(-0.774189\pi\)
−0.758750 + 0.651382i \(0.774189\pi\)
\(522\) 906111. 0.145548
\(523\) −7.04712e6 −1.12657 −0.563284 0.826263i \(-0.690462\pi\)
−0.563284 + 0.826263i \(0.690462\pi\)
\(524\) 735766. 0.117061
\(525\) 866753. 0.137245
\(526\) 653434. 0.102976
\(527\) 5.48454e6 0.860228
\(528\) −393789. −0.0614722
\(529\) −6.07200e6 −0.943392
\(530\) 1.87426e6 0.289828
\(531\) −3.28806e6 −0.506061
\(532\) −890020. −0.136339
\(533\) −4.29135e6 −0.654299
\(534\) 2.10012e6 0.318707
\(535\) −1.32380e6 −0.199957
\(536\) −2.80419e6 −0.421595
\(537\) 233363. 0.0349218
\(538\) −3.18148e6 −0.473886
\(539\) −1.18556e6 −0.175773
\(540\) −291600. −0.0430331
\(541\) −6.07424e6 −0.892276 −0.446138 0.894964i \(-0.647201\pi\)
−0.446138 + 0.894964i \(0.647201\pi\)
\(542\) 4.39238e6 0.642247
\(543\) 7.04453e6 1.02530
\(544\) −659182. −0.0955010
\(545\) −2.84719e6 −0.410606
\(546\) −2.37901e6 −0.341519
\(547\) 1.34725e7 1.92522 0.962609 0.270895i \(-0.0873195\pi\)
0.962609 + 0.270895i \(0.0873195\pi\)
\(548\) −651771. −0.0927136
\(549\) 380728. 0.0539119
\(550\) 427288. 0.0602302
\(551\) 1.00959e6 0.141666
\(552\) −347680. −0.0485660
\(553\) 1.01014e7 1.40466
\(554\) −8.46904e6 −1.17236
\(555\) −1.72300e6 −0.237439
\(556\) −1.62346e6 −0.222718
\(557\) −8.30144e6 −1.13374 −0.566872 0.823806i \(-0.691847\pi\)
−0.566872 + 0.823806i \(0.691847\pi\)
\(558\) −2.76045e6 −0.375314
\(559\) −7.82235e6 −1.05878
\(560\) −986172. −0.132887
\(561\) −990212. −0.132838
\(562\) 4.69614e6 0.627191
\(563\) 5.95933e6 0.792367 0.396184 0.918171i \(-0.370334\pi\)
0.396184 + 0.918171i \(0.370334\pi\)
\(564\) 3.21407e6 0.425459
\(565\) −1.84512e6 −0.243167
\(566\) 7.03702e6 0.923310
\(567\) 1.01098e6 0.132064
\(568\) 2.83979e6 0.369330
\(569\) −492041. −0.0637119 −0.0318559 0.999492i \(-0.510142\pi\)
−0.0318559 + 0.999492i \(0.510142\pi\)
\(570\) −324900. −0.0418854
\(571\) 5.28641e6 0.678533 0.339266 0.940690i \(-0.389821\pi\)
0.339266 + 0.940690i \(0.389821\pi\)
\(572\) −1.17280e6 −0.149876
\(573\) 3.13366e6 0.398717
\(574\) 6.16745e6 0.781314
\(575\) 377257. 0.0475848
\(576\) 331776. 0.0416667
\(577\) 7.71097e6 0.964206 0.482103 0.876115i \(-0.339873\pi\)
0.482103 + 0.876115i \(0.339873\pi\)
\(578\) 4.02186e6 0.500735
\(579\) 7.92738e6 0.982728
\(580\) 1.11866e6 0.138079
\(581\) 6.68726e6 0.821879
\(582\) 6.59967e6 0.807634
\(583\) −3.20340e6 −0.390337
\(584\) −2.56825e6 −0.311605
\(585\) −868454. −0.104920
\(586\) 1.02750e7 1.23606
\(587\) 5.18100e6 0.620609 0.310305 0.950637i \(-0.399569\pi\)
0.310305 + 0.950637i \(0.399569\pi\)
\(588\) 998863. 0.119141
\(589\) −3.07569e6 −0.365304
\(590\) −4.05933e6 −0.480092
\(591\) 4.58342e6 0.539785
\(592\) 1.96039e6 0.229899
\(593\) −1.28200e7 −1.49710 −0.748548 0.663080i \(-0.769249\pi\)
−0.748548 + 0.663080i \(0.769249\pi\)
\(594\) 498389. 0.0579565
\(595\) −2.47981e6 −0.287161
\(596\) 6.16286e6 0.710668
\(597\) 4.29769e6 0.493514
\(598\) −1.03547e6 −0.118409
\(599\) −601719. −0.0685214 −0.0342607 0.999413i \(-0.510908\pi\)
−0.0342607 + 0.999413i \(0.510908\pi\)
\(600\) −360000. −0.0408248
\(601\) 1.06391e7 1.20149 0.600745 0.799440i \(-0.294871\pi\)
0.600745 + 0.799440i \(0.294871\pi\)
\(602\) 1.12421e7 1.26432
\(603\) 3.54905e6 0.397484
\(604\) 1.98775e6 0.221701
\(605\) 3.29597e6 0.366096
\(606\) 124736. 0.0137978
\(607\) 1.09517e6 0.120645 0.0603223 0.998179i \(-0.480787\pi\)
0.0603223 + 0.998179i \(0.480787\pi\)
\(608\) 369664. 0.0405554
\(609\) −3.87839e6 −0.423749
\(610\) 470035. 0.0511453
\(611\) 9.57227e6 1.03732
\(612\) 834277. 0.0900392
\(613\) 5.09667e6 0.547816 0.273908 0.961756i \(-0.411684\pi\)
0.273908 + 0.961756i \(0.411684\pi\)
\(614\) −3.87506e6 −0.414818
\(615\) 2.25141e6 0.240031
\(616\) 1.68552e6 0.178971
\(617\) −1.74045e7 −1.84055 −0.920276 0.391270i \(-0.872036\pi\)
−0.920276 + 0.391270i \(0.872036\pi\)
\(618\) −668413. −0.0704001
\(619\) 1.44084e7 1.51143 0.755716 0.654900i \(-0.227289\pi\)
0.755716 + 0.654900i \(0.227289\pi\)
\(620\) −3.40796e6 −0.356054
\(621\) 440033. 0.0457885
\(622\) 4.44214e6 0.460380
\(623\) −8.98908e6 −0.927887
\(624\) 988107. 0.101588
\(625\) 390625. 0.0400000
\(626\) −2.94365e6 −0.300228
\(627\) 555304. 0.0564107
\(628\) 2.89285e6 0.292703
\(629\) 4.92954e6 0.496798
\(630\) 1.24812e6 0.125287
\(631\) 2.95859e6 0.295809 0.147905 0.989002i \(-0.452747\pi\)
0.147905 + 0.989002i \(0.452747\pi\)
\(632\) −4.19556e6 −0.417828
\(633\) 3.78738e6 0.375690
\(634\) 1.18044e7 1.16633
\(635\) 854064. 0.0840536
\(636\) 2.69894e6 0.264576
\(637\) 2.97485e6 0.290480
\(638\) −1.91195e6 −0.185963
\(639\) −3.59410e6 −0.348208
\(640\) 409600. 0.0395285
\(641\) −9.85922e6 −0.947758 −0.473879 0.880590i \(-0.657147\pi\)
−0.473879 + 0.880590i \(0.657147\pi\)
\(642\) −1.90627e6 −0.182535
\(643\) −1.41271e7 −1.34749 −0.673743 0.738966i \(-0.735315\pi\)
−0.673743 + 0.738966i \(0.735315\pi\)
\(644\) 1.48816e6 0.141396
\(645\) 4.10391e6 0.388418
\(646\) 929549. 0.0876377
\(647\) 1.46105e7 1.37216 0.686081 0.727525i \(-0.259330\pi\)
0.686081 + 0.727525i \(0.259330\pi\)
\(648\) −419904. −0.0392837
\(649\) 6.93801e6 0.646582
\(650\) −1.07216e6 −0.0995356
\(651\) 1.18154e7 1.09269
\(652\) 4.15122e6 0.382434
\(653\) 9.07832e6 0.833149 0.416574 0.909102i \(-0.363231\pi\)
0.416574 + 0.909102i \(0.363231\pi\)
\(654\) −4.09995e6 −0.374830
\(655\) −1.14963e6 −0.104702
\(656\) −2.56161e6 −0.232409
\(657\) 3.25044e6 0.293784
\(658\) −1.37571e7 −1.23869
\(659\) 1.00017e7 0.897143 0.448571 0.893747i \(-0.351933\pi\)
0.448571 + 0.893747i \(0.351933\pi\)
\(660\) 615295. 0.0549824
\(661\) 4.82308e6 0.429359 0.214680 0.976685i \(-0.431129\pi\)
0.214680 + 0.976685i \(0.431129\pi\)
\(662\) −3.42873e6 −0.304080
\(663\) 2.48467e6 0.219526
\(664\) −2.77751e6 −0.244475
\(665\) 1.39066e6 0.121946
\(666\) −2.48111e6 −0.216751
\(667\) −1.68808e6 −0.146919
\(668\) −2.44824e6 −0.212282
\(669\) −5.06716e6 −0.437723
\(670\) 4.38155e6 0.377086
\(671\) −803361. −0.0688818
\(672\) −1.42009e6 −0.121309
\(673\) −1.94846e7 −1.65827 −0.829133 0.559051i \(-0.811165\pi\)
−0.829133 + 0.559051i \(0.811165\pi\)
\(674\) −1.08553e6 −0.0920437
\(675\) 455625. 0.0384900
\(676\) −2.99787e6 −0.252317
\(677\) −9.45369e6 −0.792738 −0.396369 0.918091i \(-0.629730\pi\)
−0.396369 + 0.918091i \(0.629730\pi\)
\(678\) −2.65698e6 −0.221980
\(679\) −2.82483e7 −2.35135
\(680\) 1.02997e6 0.0854187
\(681\) 7.88493e6 0.651524
\(682\) 5.82473e6 0.479529
\(683\) −1.69988e7 −1.39434 −0.697168 0.716907i \(-0.745557\pi\)
−0.697168 + 0.716907i \(0.745557\pi\)
\(684\) −467856. −0.0382360
\(685\) 1.01839e6 0.0829256
\(686\) 6.08373e6 0.493582
\(687\) 1.08199e7 0.874646
\(688\) −4.66934e6 −0.376084
\(689\) 8.03807e6 0.645065
\(690\) 543251. 0.0434388
\(691\) −1.70617e7 −1.35934 −0.679668 0.733520i \(-0.737876\pi\)
−0.679668 + 0.733520i \(0.737876\pi\)
\(692\) 1.13835e7 0.903670
\(693\) −2.13323e6 −0.168735
\(694\) 4.84406e6 0.381778
\(695\) 2.53666e6 0.199205
\(696\) 1.61086e6 0.126048
\(697\) −6.44136e6 −0.502222
\(698\) −2.42459e6 −0.188365
\(699\) 1.36718e6 0.105836
\(700\) 1.54089e6 0.118858
\(701\) −4.49531e6 −0.345513 −0.172756 0.984965i \(-0.555267\pi\)
−0.172756 + 0.984965i \(0.555267\pi\)
\(702\) −1.25057e6 −0.0957781
\(703\) −2.76445e6 −0.210970
\(704\) −700069. −0.0532364
\(705\) −5.02199e6 −0.380542
\(706\) 1.25483e7 0.947489
\(707\) −533903. −0.0401711
\(708\) −5.84544e6 −0.438262
\(709\) −1.89043e6 −0.141236 −0.0706179 0.997503i \(-0.522497\pi\)
−0.0706179 + 0.997503i \(0.522497\pi\)
\(710\) −4.43716e6 −0.330339
\(711\) 5.31001e6 0.393932
\(712\) 3.73356e6 0.276009
\(713\) 5.14272e6 0.378851
\(714\) −3.57092e6 −0.262141
\(715\) 1.83249e6 0.134053
\(716\) 414868. 0.0302431
\(717\) 4.84062e6 0.351644
\(718\) 1.49042e7 1.07894
\(719\) 4.36868e6 0.315157 0.157579 0.987506i \(-0.449631\pi\)
0.157579 + 0.987506i \(0.449631\pi\)
\(720\) −518400. −0.0372678
\(721\) 2.86098e6 0.204964
\(722\) −521284. −0.0372161
\(723\) 7.68139e6 0.546505
\(724\) 1.25236e7 0.887939
\(725\) −1.74790e6 −0.123501
\(726\) 4.74620e6 0.334199
\(727\) −2.31384e7 −1.62367 −0.811834 0.583889i \(-0.801531\pi\)
−0.811834 + 0.583889i \(0.801531\pi\)
\(728\) −4.22936e6 −0.295764
\(729\) 531441. 0.0370370
\(730\) 4.01288e6 0.278708
\(731\) −1.17414e7 −0.812694
\(732\) 676850. 0.0466890
\(733\) 3.71737e6 0.255550 0.127775 0.991803i \(-0.459216\pi\)
0.127775 + 0.991803i \(0.459216\pi\)
\(734\) −388587. −0.0266224
\(735\) −1.56072e6 −0.106563
\(736\) −618098. −0.0420594
\(737\) −7.48873e6 −0.507855
\(738\) 3.24203e6 0.219117
\(739\) −567967. −0.0382571 −0.0191286 0.999817i \(-0.506089\pi\)
−0.0191286 + 0.999817i \(0.506089\pi\)
\(740\) −3.06310e6 −0.205628
\(741\) −1.39339e6 −0.0932236
\(742\) −1.15522e7 −0.770288
\(743\) −1.58076e7 −1.05049 −0.525246 0.850950i \(-0.676027\pi\)
−0.525246 + 0.850950i \(0.676027\pi\)
\(744\) −4.90747e6 −0.325031
\(745\) −9.62947e6 −0.635641
\(746\) 1.01600e7 0.668416
\(747\) 3.51529e6 0.230494
\(748\) −1.76038e6 −0.115041
\(749\) 8.15932e6 0.531434
\(750\) 562500. 0.0365148
\(751\) −6.58077e6 −0.425772 −0.212886 0.977077i \(-0.568286\pi\)
−0.212886 + 0.977077i \(0.568286\pi\)
\(752\) 5.71391e6 0.368459
\(753\) 230980. 0.0148452
\(754\) 4.79753e6 0.307319
\(755\) −3.10585e6 −0.198296
\(756\) 1.79730e6 0.114371
\(757\) −2.37466e7 −1.50613 −0.753064 0.657947i \(-0.771425\pi\)
−0.753064 + 0.657947i \(0.771425\pi\)
\(758\) −23511.3 −0.00148629
\(759\) −928498. −0.0585028
\(760\) −577600. −0.0362738
\(761\) −1.14055e7 −0.713922 −0.356961 0.934119i \(-0.616187\pi\)
−0.356961 + 0.934119i \(0.616187\pi\)
\(762\) 1.22985e6 0.0767301
\(763\) 1.75489e7 1.09128
\(764\) 5.57095e6 0.345299
\(765\) −1.30356e6 −0.0805335
\(766\) 1.47585e6 0.0908805
\(767\) −1.74091e7 −1.06853
\(768\) 589824. 0.0360844
\(769\) 3.12375e6 0.190485 0.0952424 0.995454i \(-0.469637\pi\)
0.0952424 + 0.995454i \(0.469637\pi\)
\(770\) −2.63362e6 −0.160076
\(771\) 8.34797e6 0.505761
\(772\) 1.40931e7 0.851068
\(773\) 2.00014e7 1.20396 0.601979 0.798512i \(-0.294379\pi\)
0.601979 + 0.798512i \(0.294379\pi\)
\(774\) 5.90963e6 0.354575
\(775\) 5.32494e6 0.318464
\(776\) 1.17327e7 0.699431
\(777\) 1.06198e7 0.631051
\(778\) −1.43945e7 −0.852604
\(779\) 3.61227e6 0.213273
\(780\) −1.54392e6 −0.0908631
\(781\) 7.58379e6 0.444896
\(782\) −1.55426e6 −0.0908879
\(783\) −2.03875e6 −0.118839
\(784\) 1.77576e6 0.103179
\(785\) −4.52008e6 −0.261801
\(786\) −1.65547e6 −0.0955798
\(787\) −1.76725e7 −1.01710 −0.508548 0.861034i \(-0.669817\pi\)
−0.508548 + 0.861034i \(0.669817\pi\)
\(788\) 8.14830e6 0.467467
\(789\) −1.47023e6 −0.0840798
\(790\) 6.55556e6 0.373716
\(791\) 1.13726e7 0.646275
\(792\) 886025. 0.0501918
\(793\) 2.01582e6 0.113833
\(794\) −1.82175e7 −1.02551
\(795\) −4.21709e6 −0.236644
\(796\) 7.64034e6 0.427396
\(797\) 1.82957e7 1.02024 0.510121 0.860102i \(-0.329601\pi\)
0.510121 + 0.860102i \(0.329601\pi\)
\(798\) 2.00255e6 0.111321
\(799\) 1.43681e7 0.796217
\(800\) −640000. −0.0353553
\(801\) −4.72528e6 −0.260223
\(802\) −1.09792e7 −0.602745
\(803\) −6.85863e6 −0.375361
\(804\) 6.30943e6 0.344231
\(805\) −2.32525e6 −0.126468
\(806\) −1.46156e7 −0.792463
\(807\) 7.15833e6 0.386926
\(808\) 221753. 0.0119493
\(809\) 1.58557e7 0.851751 0.425876 0.904782i \(-0.359966\pi\)
0.425876 + 0.904782i \(0.359966\pi\)
\(810\) 656100. 0.0351364
\(811\) 1.41608e7 0.756027 0.378013 0.925800i \(-0.376607\pi\)
0.378013 + 0.925800i \(0.376607\pi\)
\(812\) −6.89492e6 −0.366977
\(813\) −9.88286e6 −0.524393
\(814\) 5.23531e6 0.276937
\(815\) −6.48628e6 −0.342060
\(816\) 1.48316e6 0.0779762
\(817\) 6.58450e6 0.345118
\(818\) −1.24178e7 −0.648878
\(819\) 5.35278e6 0.278849
\(820\) 4.00251e6 0.207873
\(821\) 2.71518e7 1.40585 0.702927 0.711262i \(-0.251876\pi\)
0.702927 + 0.711262i \(0.251876\pi\)
\(822\) 1.46648e6 0.0757004
\(823\) 1.45900e6 0.0750856 0.0375428 0.999295i \(-0.488047\pi\)
0.0375428 + 0.999295i \(0.488047\pi\)
\(824\) −1.18829e6 −0.0609683
\(825\) −961398. −0.0491777
\(826\) 2.50200e7 1.27596
\(827\) 2.52723e7 1.28494 0.642468 0.766312i \(-0.277910\pi\)
0.642468 + 0.766312i \(0.277910\pi\)
\(828\) 782281. 0.0396540
\(829\) −8.40734e6 −0.424886 −0.212443 0.977173i \(-0.568142\pi\)
−0.212443 + 0.977173i \(0.568142\pi\)
\(830\) 4.33986e6 0.218666
\(831\) 1.90553e7 0.957225
\(832\) 1.75664e6 0.0879778
\(833\) 4.46528e6 0.222965
\(834\) 3.65279e6 0.181848
\(835\) 3.82537e6 0.189871
\(836\) 987206. 0.0488531
\(837\) 6.21101e6 0.306442
\(838\) −7.82211e6 −0.384781
\(839\) −1.41890e6 −0.0695899 −0.0347950 0.999394i \(-0.511078\pi\)
−0.0347950 + 0.999394i \(0.511078\pi\)
\(840\) 2.21889e6 0.108502
\(841\) −1.26900e7 −0.618686
\(842\) −3.74228e6 −0.181910
\(843\) −1.05663e7 −0.512100
\(844\) 6.73311e6 0.325357
\(845\) 4.68417e6 0.225679
\(846\) −7.23167e6 −0.347386
\(847\) −2.03150e7 −0.972989
\(848\) 4.79811e6 0.229129
\(849\) −1.58333e7 −0.753879
\(850\) −1.60933e6 −0.0764008
\(851\) 4.62231e6 0.218794
\(852\) −6.38952e6 −0.301557
\(853\) −4.20130e7 −1.97702 −0.988510 0.151154i \(-0.951701\pi\)
−0.988510 + 0.151154i \(0.951701\pi\)
\(854\) −2.89710e6 −0.135931
\(855\) 731025. 0.0341993
\(856\) −3.38892e6 −0.158080
\(857\) 1.54482e7 0.718500 0.359250 0.933241i \(-0.383032\pi\)
0.359250 + 0.933241i \(0.383032\pi\)
\(858\) 2.63879e6 0.122373
\(859\) 3.81032e7 1.76189 0.880945 0.473219i \(-0.156908\pi\)
0.880945 + 0.473219i \(0.156908\pi\)
\(860\) 7.29584e6 0.336379
\(861\) −1.38768e7 −0.637940
\(862\) 2.78382e6 0.127606
\(863\) 2.13308e6 0.0974947 0.0487473 0.998811i \(-0.484477\pi\)
0.0487473 + 0.998811i \(0.484477\pi\)
\(864\) −746496. −0.0340207
\(865\) −1.77867e7 −0.808267
\(866\) 2.69294e7 1.22020
\(867\) −9.04919e6 −0.408848
\(868\) 2.10052e7 0.946299
\(869\) −1.12045e7 −0.503317
\(870\) −2.51697e6 −0.112741
\(871\) 1.87910e7 0.839274
\(872\) −7.28880e6 −0.324612
\(873\) −1.48493e7 −0.659430
\(874\) 871615. 0.0385964
\(875\) −2.40765e6 −0.106310
\(876\) 5.77855e6 0.254425
\(877\) −1.74743e7 −0.767188 −0.383594 0.923502i \(-0.625314\pi\)
−0.383594 + 0.923502i \(0.625314\pi\)
\(878\) 2.40497e6 0.105287
\(879\) −2.31188e7 −1.00924
\(880\) 1.09386e6 0.0476161
\(881\) −2.60458e7 −1.13057 −0.565286 0.824895i \(-0.691234\pi\)
−0.565286 + 0.824895i \(0.691234\pi\)
\(882\) −2.24744e6 −0.0972785
\(883\) −1.29495e7 −0.558920 −0.279460 0.960157i \(-0.590155\pi\)
−0.279460 + 0.960157i \(0.590155\pi\)
\(884\) 4.41720e6 0.190115
\(885\) 9.13349e6 0.391993
\(886\) 1.50082e7 0.642310
\(887\) 2.78281e7 1.18761 0.593807 0.804608i \(-0.297624\pi\)
0.593807 + 0.804608i \(0.297624\pi\)
\(888\) −4.41087e6 −0.187712
\(889\) −5.26409e6 −0.223393
\(890\) −5.83368e6 −0.246870
\(891\) −1.12137e6 −0.0473213
\(892\) −9.00828e6 −0.379079
\(893\) −8.05750e6 −0.338121
\(894\) −1.38664e7 −0.580258
\(895\) −648231. −0.0270503
\(896\) −2.52460e6 −0.105056
\(897\) 2.32982e6 0.0966809
\(898\) 3.69621e6 0.152956
\(899\) −2.38271e7 −0.983268
\(900\) 810000. 0.0333333
\(901\) 1.20652e7 0.495135
\(902\) −6.84090e6 −0.279961
\(903\) −2.52948e7 −1.03231
\(904\) −4.72352e6 −0.192240
\(905\) −1.95681e7 −0.794197
\(906\) −4.47243e6 −0.181018
\(907\) 2.15417e7 0.869483 0.434741 0.900555i \(-0.356840\pi\)
0.434741 + 0.900555i \(0.356840\pi\)
\(908\) 1.40177e7 0.564236
\(909\) −280656. −0.0112659
\(910\) 6.60837e6 0.264540
\(911\) 3.11633e7 1.24408 0.622039 0.782986i \(-0.286304\pi\)
0.622039 + 0.782986i \(0.286304\pi\)
\(912\) −831744. −0.0331133
\(913\) −7.41748e6 −0.294496
\(914\) 2.26058e7 0.895064
\(915\) −1.05758e6 −0.0417599
\(916\) 1.92354e7 0.757465
\(917\) 7.08586e6 0.278272
\(918\) −1.87712e6 −0.0735167
\(919\) 7.83154e6 0.305885 0.152942 0.988235i \(-0.451125\pi\)
0.152942 + 0.988235i \(0.451125\pi\)
\(920\) 965779. 0.0376191
\(921\) 8.71888e6 0.338697
\(922\) −2.13759e7 −0.828127
\(923\) −1.90295e7 −0.735229
\(924\) −3.79242e6 −0.146129
\(925\) 4.78610e6 0.183919
\(926\) 1.07012e7 0.410114
\(927\) 1.50393e6 0.0574815
\(928\) 2.86376e6 0.109161
\(929\) 4.52091e7 1.71865 0.859324 0.511431i \(-0.170885\pi\)
0.859324 + 0.511431i \(0.170885\pi\)
\(930\) 7.66792e6 0.290717
\(931\) −2.50409e6 −0.0946839
\(932\) 2.43054e6 0.0916565
\(933\) −9.99481e6 −0.375898
\(934\) −7.25655e6 −0.272184
\(935\) 2.75059e6 0.102896
\(936\) −2.22324e6 −0.0829463
\(937\) −1.43912e7 −0.535487 −0.267743 0.963490i \(-0.586278\pi\)
−0.267743 + 0.963490i \(0.586278\pi\)
\(938\) −2.70060e7 −1.00220
\(939\) 6.62322e6 0.245135
\(940\) −8.92798e6 −0.329559
\(941\) 2.42598e7 0.893128 0.446564 0.894752i \(-0.352647\pi\)
0.446564 + 0.894752i \(0.352647\pi\)
\(942\) −6.50892e6 −0.238991
\(943\) −6.03991e6 −0.221183
\(944\) −1.03919e7 −0.379546
\(945\) −2.80828e6 −0.102297
\(946\) −1.24697e7 −0.453032
\(947\) 7.95718e6 0.288326 0.144163 0.989554i \(-0.453951\pi\)
0.144163 + 0.989554i \(0.453951\pi\)
\(948\) 9.44001e6 0.341155
\(949\) 1.72099e7 0.620316
\(950\) 902500. 0.0324443
\(951\) −2.65600e7 −0.952305
\(952\) −6.34831e6 −0.227021
\(953\) −3.82773e7 −1.36524 −0.682621 0.730773i \(-0.739160\pi\)
−0.682621 + 0.730773i \(0.739160\pi\)
\(954\) −6.07261e6 −0.216025
\(955\) −8.70461e6 −0.308845
\(956\) 8.60555e6 0.304533
\(957\) 4.30189e6 0.151838
\(958\) −5.39196e6 −0.189816
\(959\) −6.27694e6 −0.220395
\(960\) −921600. −0.0322749
\(961\) 4.39597e7 1.53549
\(962\) −1.31366e7 −0.457663
\(963\) 4.28910e6 0.149039
\(964\) 1.36558e7 0.473287
\(965\) −2.20205e7 −0.761218
\(966\) −3.34837e6 −0.115449
\(967\) 2.02758e7 0.697287 0.348643 0.937255i \(-0.386642\pi\)
0.348643 + 0.937255i \(0.386642\pi\)
\(968\) 8.43770e6 0.289425
\(969\) −2.09149e6 −0.0715559
\(970\) −1.83324e7 −0.625591
\(971\) 2.62269e7 0.892687 0.446343 0.894862i \(-0.352726\pi\)
0.446343 + 0.894862i \(0.352726\pi\)
\(972\) 944784. 0.0320750
\(973\) −1.56349e7 −0.529435
\(974\) −7.81131e6 −0.263831
\(975\) 2.41237e6 0.0812704
\(976\) 1.20329e6 0.0404339
\(977\) −4.22437e6 −0.141588 −0.0707938 0.997491i \(-0.522553\pi\)
−0.0707938 + 0.997491i \(0.522553\pi\)
\(978\) −9.34025e6 −0.312256
\(979\) 9.97065e6 0.332481
\(980\) −2.77462e6 −0.0922865
\(981\) 9.22489e6 0.306047
\(982\) 3.98142e7 1.31753
\(983\) −1.69659e7 −0.560005 −0.280003 0.959999i \(-0.590335\pi\)
−0.280003 + 0.959999i \(0.590335\pi\)
\(984\) 5.76362e6 0.189761
\(985\) −1.27317e7 −0.418116
\(986\) 7.20114e6 0.235890
\(987\) 3.09534e7 1.01138
\(988\) −2.47713e6 −0.0807340
\(989\) −1.10096e7 −0.357917
\(990\) −1.38441e6 −0.0448929
\(991\) −3.59220e7 −1.16192 −0.580960 0.813932i \(-0.697323\pi\)
−0.580960 + 0.813932i \(0.697323\pi\)
\(992\) −8.72439e6 −0.281485
\(993\) 7.71463e6 0.248280
\(994\) 2.73488e7 0.877955
\(995\) −1.19380e7 −0.382274
\(996\) 6.24940e6 0.199613
\(997\) −1.10205e7 −0.351128 −0.175564 0.984468i \(-0.556175\pi\)
−0.175564 + 0.984468i \(0.556175\pi\)
\(998\) 2.59900e7 0.826001
\(999\) 5.58251e6 0.176976
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 570.6.a.k.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.6.a.k.1.4 4 1.1 even 1 trivial