Properties

Label 230.6.a.f.1.4
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - x^{4} - 774 x^{3} - 197 x^{2} + 66287 x + 154128\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(10.8366\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +10.8366 q^{3} +16.0000 q^{4} +25.0000 q^{5} -43.3465 q^{6} +46.1561 q^{7} -64.0000 q^{8} -125.567 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +10.8366 q^{3} +16.0000 q^{4} +25.0000 q^{5} -43.3465 q^{6} +46.1561 q^{7} -64.0000 q^{8} -125.567 q^{9} -100.000 q^{10} +8.88634 q^{11} +173.386 q^{12} +229.106 q^{13} -184.624 q^{14} +270.916 q^{15} +256.000 q^{16} +561.534 q^{17} +502.269 q^{18} +1957.16 q^{19} +400.000 q^{20} +500.176 q^{21} -35.5454 q^{22} -529.000 q^{23} -693.545 q^{24} +625.000 q^{25} -916.423 q^{26} -3994.03 q^{27} +738.497 q^{28} +576.364 q^{29} -1083.66 q^{30} +7333.04 q^{31} -1024.00 q^{32} +96.2981 q^{33} -2246.14 q^{34} +1153.90 q^{35} -2009.08 q^{36} +2330.56 q^{37} -7828.64 q^{38} +2482.74 q^{39} -1600.00 q^{40} +89.3538 q^{41} -2000.71 q^{42} -1692.41 q^{43} +142.181 q^{44} -3139.18 q^{45} +2116.00 q^{46} -2617.45 q^{47} +2774.18 q^{48} -14676.6 q^{49} -2500.00 q^{50} +6085.14 q^{51} +3665.69 q^{52} +19388.3 q^{53} +15976.1 q^{54} +222.159 q^{55} -2953.99 q^{56} +21209.0 q^{57} -2305.45 q^{58} +37460.4 q^{59} +4334.65 q^{60} -17456.6 q^{61} -29332.2 q^{62} -5795.69 q^{63} +4096.00 q^{64} +5727.65 q^{65} -385.192 q^{66} +25612.4 q^{67} +8984.55 q^{68} -5732.58 q^{69} -4615.61 q^{70} -6079.31 q^{71} +8036.31 q^{72} +67969.4 q^{73} -9322.23 q^{74} +6772.90 q^{75} +31314.5 q^{76} +410.159 q^{77} -9930.95 q^{78} +70755.0 q^{79} +6400.00 q^{80} -12769.0 q^{81} -357.415 q^{82} +54086.0 q^{83} +8002.82 q^{84} +14038.4 q^{85} +6769.65 q^{86} +6245.84 q^{87} -568.726 q^{88} +67687.8 q^{89} +12556.7 q^{90} +10574.6 q^{91} -8464.00 q^{92} +79465.5 q^{93} +10469.8 q^{94} +48929.0 q^{95} -11096.7 q^{96} +103840. q^{97} +58706.5 q^{98} -1115.83 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9} + O(q^{10}) \) \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9} - 500 q^{10} + 251 q^{11} + 16 q^{12} + 1743 q^{13} - 408 q^{14} + 25 q^{15} + 1280 q^{16} + 1944 q^{17} - 1336 q^{18} - 845 q^{19} + 2000 q^{20} + 4682 q^{21} - 1004 q^{22} - 2645 q^{23} - 64 q^{24} + 3125 q^{25} - 6972 q^{26} + 2428 q^{27} + 1632 q^{28} - 4021 q^{29} - 100 q^{30} - 15752 q^{31} - 5120 q^{32} + 2931 q^{33} - 7776 q^{34} + 2550 q^{35} + 5344 q^{36} - 3455 q^{37} + 3380 q^{38} - 16708 q^{39} - 8000 q^{40} - 11898 q^{41} - 18728 q^{42} + 6968 q^{43} + 4016 q^{44} + 8350 q^{45} + 10580 q^{46} + 13412 q^{47} + 256 q^{48} + 91041 q^{49} - 12500 q^{50} - 2115 q^{51} + 27888 q^{52} + 53029 q^{53} - 9712 q^{54} + 6275 q^{55} - 6528 q^{56} - 21730 q^{57} + 16084 q^{58} - 31223 q^{59} + 400 q^{60} + 71477 q^{61} + 63008 q^{62} + 262199 q^{63} + 20480 q^{64} + 43575 q^{65} - 11724 q^{66} + 76003 q^{67} + 31104 q^{68} - 529 q^{69} - 10200 q^{70} + 54418 q^{71} - 21376 q^{72} + 69418 q^{73} + 13820 q^{74} + 625 q^{75} - 13520 q^{76} + 283598 q^{77} + 66832 q^{78} + 105024 q^{79} + 32000 q^{80} + 102913 q^{81} + 47592 q^{82} + 89399 q^{83} + 74912 q^{84} + 48600 q^{85} - 27872 q^{86} + 276726 q^{87} - 16064 q^{88} + 96240 q^{89} - 33400 q^{90} + 59261 q^{91} - 42320 q^{92} + 84434 q^{93} - 53648 q^{94} - 21125 q^{95} - 1024 q^{96} + 216087 q^{97} - 364164 q^{98} + 386925 q^{99} + O(q^{100}) \)

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
\(3\) 10.8366 0.695171 0.347585 0.937648i \(-0.387002\pi\)
0.347585 + 0.937648i \(0.387002\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) −43.3465 −0.491560
\(7\) 46.1561 0.356028 0.178014 0.984028i \(-0.443033\pi\)
0.178014 + 0.984028i \(0.443033\pi\)
\(8\) −64.0000 −0.353553
\(9\) −125.567 −0.516738
\(10\) −100.000 −0.316228
\(11\) 8.88634 0.0221432 0.0110716 0.999939i \(-0.496476\pi\)
0.0110716 + 0.999939i \(0.496476\pi\)
\(12\) 173.386 0.347585
\(13\) 229.106 0.375991 0.187996 0.982170i \(-0.439801\pi\)
0.187996 + 0.982170i \(0.439801\pi\)
\(14\) −184.624 −0.251750
\(15\) 270.916 0.310890
\(16\) 256.000 0.250000
\(17\) 561.534 0.471253 0.235626 0.971844i \(-0.424286\pi\)
0.235626 + 0.971844i \(0.424286\pi\)
\(18\) 502.269 0.365389
\(19\) 1957.16 1.24378 0.621888 0.783106i \(-0.286366\pi\)
0.621888 + 0.783106i \(0.286366\pi\)
\(20\) 400.000 0.223607
\(21\) 500.176 0.247500
\(22\) −35.5454 −0.0156576
\(23\) −529.000 −0.208514
\(24\) −693.545 −0.245780
\(25\) 625.000 0.200000
\(26\) −916.423 −0.265866
\(27\) −3994.03 −1.05439
\(28\) 738.497 0.178014
\(29\) 576.364 0.127263 0.0636314 0.997973i \(-0.479732\pi\)
0.0636314 + 0.997973i \(0.479732\pi\)
\(30\) −1083.66 −0.219832
\(31\) 7333.04 1.37050 0.685251 0.728307i \(-0.259692\pi\)
0.685251 + 0.728307i \(0.259692\pi\)
\(32\) −1024.00 −0.176777
\(33\) 96.2981 0.0153933
\(34\) −2246.14 −0.333226
\(35\) 1153.90 0.159220
\(36\) −2009.08 −0.258369
\(37\) 2330.56 0.279869 0.139935 0.990161i \(-0.455311\pi\)
0.139935 + 0.990161i \(0.455311\pi\)
\(38\) −7828.64 −0.879482
\(39\) 2482.74 0.261378
\(40\) −1600.00 −0.158114
\(41\) 89.3538 0.00830144 0.00415072 0.999991i \(-0.498679\pi\)
0.00415072 + 0.999991i \(0.498679\pi\)
\(42\) −2000.71 −0.175009
\(43\) −1692.41 −0.139584 −0.0697919 0.997562i \(-0.522234\pi\)
−0.0697919 + 0.997562i \(0.522234\pi\)
\(44\) 142.181 0.0110716
\(45\) −3139.18 −0.231092
\(46\) 2116.00 0.147442
\(47\) −2617.45 −0.172836 −0.0864179 0.996259i \(-0.527542\pi\)
−0.0864179 + 0.996259i \(0.527542\pi\)
\(48\) 2774.18 0.173793
\(49\) −14676.6 −0.873244
\(50\) −2500.00 −0.141421
\(51\) 6085.14 0.327601
\(52\) 3665.69 0.187996
\(53\) 19388.3 0.948092 0.474046 0.880500i \(-0.342793\pi\)
0.474046 + 0.880500i \(0.342793\pi\)
\(54\) 15976.1 0.745567
\(55\) 222.159 0.00990276
\(56\) −2953.99 −0.125875
\(57\) 21209.0 0.864636
\(58\) −2305.45 −0.0899884
\(59\) 37460.4 1.40101 0.700507 0.713646i \(-0.252957\pi\)
0.700507 + 0.713646i \(0.252957\pi\)
\(60\) 4334.65 0.155445
\(61\) −17456.6 −0.600670 −0.300335 0.953834i \(-0.597098\pi\)
−0.300335 + 0.953834i \(0.597098\pi\)
\(62\) −29332.2 −0.969091
\(63\) −5795.69 −0.183973
\(64\) 4096.00 0.125000
\(65\) 5727.65 0.168148
\(66\) −385.192 −0.0108847
\(67\) 25612.4 0.697050 0.348525 0.937300i \(-0.386683\pi\)
0.348525 + 0.937300i \(0.386683\pi\)
\(68\) 8984.55 0.235626
\(69\) −5732.58 −0.144953
\(70\) −4615.61 −0.112586
\(71\) −6079.31 −0.143123 −0.0715614 0.997436i \(-0.522798\pi\)
−0.0715614 + 0.997436i \(0.522798\pi\)
\(72\) 8036.31 0.182694
\(73\) 67969.4 1.49282 0.746408 0.665488i \(-0.231777\pi\)
0.746408 + 0.665488i \(0.231777\pi\)
\(74\) −9322.23 −0.197898
\(75\) 6772.90 0.139034
\(76\) 31314.5 0.621888
\(77\) 410.159 0.00788361
\(78\) −9930.95 −0.184822
\(79\) 70755.0 1.27553 0.637763 0.770233i \(-0.279860\pi\)
0.637763 + 0.770233i \(0.279860\pi\)
\(80\) 6400.00 0.111803
\(81\) −12769.0 −0.216244
\(82\) −357.415 −0.00587000
\(83\) 54086.0 0.861766 0.430883 0.902408i \(-0.358202\pi\)
0.430883 + 0.902408i \(0.358202\pi\)
\(84\) 8002.82 0.123750
\(85\) 14038.4 0.210751
\(86\) 6769.65 0.0987007
\(87\) 6245.84 0.0884694
\(88\) −568.726 −0.00782882
\(89\) 67687.8 0.905807 0.452903 0.891560i \(-0.350388\pi\)
0.452903 + 0.891560i \(0.350388\pi\)
\(90\) 12556.7 0.163407
\(91\) 10574.6 0.133863
\(92\) −8464.00 −0.104257
\(93\) 79465.5 0.952733
\(94\) 10469.8 0.122213
\(95\) 48929.0 0.556233
\(96\) −11096.7 −0.122890
\(97\) 103840. 1.12056 0.560281 0.828303i \(-0.310693\pi\)
0.560281 + 0.828303i \(0.310693\pi\)
\(98\) 58706.5 0.617477
\(99\) −1115.83 −0.0114423
\(100\) 10000.0 0.100000
\(101\) −90122.4 −0.879082 −0.439541 0.898223i \(-0.644859\pi\)
−0.439541 + 0.898223i \(0.644859\pi\)
\(102\) −24340.6 −0.231649
\(103\) −86411.2 −0.802559 −0.401280 0.915956i \(-0.631434\pi\)
−0.401280 + 0.915956i \(0.631434\pi\)
\(104\) −14662.8 −0.132933
\(105\) 12504.4 0.110685
\(106\) −77553.3 −0.670403
\(107\) 135629. 1.14523 0.572617 0.819823i \(-0.305928\pi\)
0.572617 + 0.819823i \(0.305928\pi\)
\(108\) −63904.5 −0.527196
\(109\) 180022. 1.45131 0.725654 0.688060i \(-0.241537\pi\)
0.725654 + 0.688060i \(0.241537\pi\)
\(110\) −888.634 −0.00700231
\(111\) 25255.4 0.194557
\(112\) 11816.0 0.0890069
\(113\) −159472. −1.17486 −0.587432 0.809274i \(-0.699861\pi\)
−0.587432 + 0.809274i \(0.699861\pi\)
\(114\) −84836.1 −0.611390
\(115\) −13225.0 −0.0932505
\(116\) 9221.82 0.0636314
\(117\) −28768.2 −0.194289
\(118\) −149842. −0.990666
\(119\) 25918.2 0.167779
\(120\) −17338.6 −0.109916
\(121\) −160972. −0.999510
\(122\) 69826.5 0.424738
\(123\) 968.294 0.00577091
\(124\) 117329. 0.685251
\(125\) 15625.0 0.0894427
\(126\) 23182.8 0.130089
\(127\) 63419.2 0.348909 0.174454 0.984665i \(-0.444184\pi\)
0.174454 + 0.984665i \(0.444184\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −18340.1 −0.0970346
\(130\) −22910.6 −0.118899
\(131\) −189912. −0.966881 −0.483441 0.875377i \(-0.660613\pi\)
−0.483441 + 0.875377i \(0.660613\pi\)
\(132\) 1540.77 0.00769667
\(133\) 90334.8 0.442819
\(134\) −102450. −0.492889
\(135\) −99850.8 −0.471538
\(136\) −35938.2 −0.166613
\(137\) 75422.5 0.343320 0.171660 0.985156i \(-0.445087\pi\)
0.171660 + 0.985156i \(0.445087\pi\)
\(138\) 22930.3 0.102497
\(139\) 47836.5 0.210002 0.105001 0.994472i \(-0.466515\pi\)
0.105001 + 0.994472i \(0.466515\pi\)
\(140\) 18462.4 0.0796102
\(141\) −28364.4 −0.120150
\(142\) 24317.3 0.101203
\(143\) 2035.91 0.00832567
\(144\) −32145.2 −0.129184
\(145\) 14409.1 0.0569137
\(146\) −271878. −1.05558
\(147\) −159045. −0.607054
\(148\) 37288.9 0.139935
\(149\) −415547. −1.53340 −0.766699 0.642007i \(-0.778102\pi\)
−0.766699 + 0.642007i \(0.778102\pi\)
\(150\) −27091.6 −0.0983120
\(151\) −246.134 −0.000878474 0 −0.000439237 1.00000i \(-0.500140\pi\)
−0.000439237 1.00000i \(0.500140\pi\)
\(152\) −125258. −0.439741
\(153\) −70510.3 −0.243514
\(154\) −1640.63 −0.00557455
\(155\) 183326. 0.612907
\(156\) 39723.8 0.130689
\(157\) 49843.7 0.161384 0.0806922 0.996739i \(-0.474287\pi\)
0.0806922 + 0.996739i \(0.474287\pi\)
\(158\) −283020. −0.901933
\(159\) 210104. 0.659086
\(160\) −25600.0 −0.0790569
\(161\) −24416.6 −0.0742369
\(162\) 51076.0 0.152908
\(163\) −428050. −1.26190 −0.630951 0.775823i \(-0.717335\pi\)
−0.630951 + 0.775823i \(0.717335\pi\)
\(164\) 1429.66 0.00415072
\(165\) 2407.45 0.00688411
\(166\) −216344. −0.609361
\(167\) 387299. 1.07462 0.537310 0.843385i \(-0.319440\pi\)
0.537310 + 0.843385i \(0.319440\pi\)
\(168\) −32011.3 −0.0875044
\(169\) −318804. −0.858631
\(170\) −56153.4 −0.149023
\(171\) −245755. −0.642706
\(172\) −27078.6 −0.0697919
\(173\) −168861. −0.428959 −0.214479 0.976729i \(-0.568805\pi\)
−0.214479 + 0.976729i \(0.568805\pi\)
\(174\) −24983.4 −0.0625573
\(175\) 28847.5 0.0712055
\(176\) 2274.90 0.00553581
\(177\) 405945. 0.973944
\(178\) −270751. −0.640502
\(179\) −507780. −1.18452 −0.592260 0.805747i \(-0.701764\pi\)
−0.592260 + 0.805747i \(0.701764\pi\)
\(180\) −50226.9 −0.115546
\(181\) 22218.5 0.0504102 0.0252051 0.999682i \(-0.491976\pi\)
0.0252051 + 0.999682i \(0.491976\pi\)
\(182\) −42298.5 −0.0946556
\(183\) −189171. −0.417568
\(184\) 33856.0 0.0737210
\(185\) 58263.9 0.125161
\(186\) −317862. −0.673684
\(187\) 4989.99 0.0104351
\(188\) −41879.2 −0.0864179
\(189\) −184349. −0.375392
\(190\) −195716. −0.393316
\(191\) −543158. −1.07732 −0.538658 0.842525i \(-0.681068\pi\)
−0.538658 + 0.842525i \(0.681068\pi\)
\(192\) 44386.9 0.0868963
\(193\) 647088. 1.25046 0.625231 0.780440i \(-0.285005\pi\)
0.625231 + 0.780440i \(0.285005\pi\)
\(194\) −415360. −0.792357
\(195\) 62068.4 0.116892
\(196\) −234826. −0.436622
\(197\) −37986.4 −0.0697368 −0.0348684 0.999392i \(-0.511101\pi\)
−0.0348684 + 0.999392i \(0.511101\pi\)
\(198\) 4463.34 0.00809089
\(199\) 552686. 0.989340 0.494670 0.869081i \(-0.335289\pi\)
0.494670 + 0.869081i \(0.335289\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 277553. 0.484568
\(202\) 360489. 0.621605
\(203\) 26602.7 0.0453091
\(204\) 97362.3 0.163801
\(205\) 2233.84 0.00371252
\(206\) 345645. 0.567495
\(207\) 66425.1 0.107747
\(208\) 58651.1 0.0939978
\(209\) 17392.0 0.0275412
\(210\) −50017.6 −0.0782663
\(211\) −49704.3 −0.0768578 −0.0384289 0.999261i \(-0.512235\pi\)
−0.0384289 + 0.999261i \(0.512235\pi\)
\(212\) 310213. 0.474046
\(213\) −65879.3 −0.0994947
\(214\) −542518. −0.809803
\(215\) −42310.3 −0.0624238
\(216\) 255618. 0.372784
\(217\) 338464. 0.487937
\(218\) −720088. −1.02623
\(219\) 736560. 1.03776
\(220\) 3554.54 0.00495138
\(221\) 128651. 0.177187
\(222\) −101022. −0.137573
\(223\) 49037.1 0.0660333 0.0330166 0.999455i \(-0.489489\pi\)
0.0330166 + 0.999455i \(0.489489\pi\)
\(224\) −47263.8 −0.0629374
\(225\) −78479.6 −0.103348
\(226\) 637887. 0.830754
\(227\) 228515. 0.294340 0.147170 0.989111i \(-0.452984\pi\)
0.147170 + 0.989111i \(0.452984\pi\)
\(228\) 339344. 0.432318
\(229\) −202746. −0.255484 −0.127742 0.991807i \(-0.540773\pi\)
−0.127742 + 0.991807i \(0.540773\pi\)
\(230\) 52900.0 0.0659380
\(231\) 4444.74 0.00548045
\(232\) −36887.3 −0.0449942
\(233\) −327542. −0.395255 −0.197628 0.980277i \(-0.563324\pi\)
−0.197628 + 0.980277i \(0.563324\pi\)
\(234\) 115073. 0.137383
\(235\) −65436.3 −0.0772946
\(236\) 599366. 0.700507
\(237\) 766746. 0.886708
\(238\) −103673. −0.118638
\(239\) −544856. −0.617002 −0.308501 0.951224i \(-0.599827\pi\)
−0.308501 + 0.951224i \(0.599827\pi\)
\(240\) 69354.5 0.0777224
\(241\) 41076.8 0.0455569 0.0227784 0.999741i \(-0.492749\pi\)
0.0227784 + 0.999741i \(0.492749\pi\)
\(242\) 643888. 0.706760
\(243\) 832176. 0.904065
\(244\) −279306. −0.300335
\(245\) −366915. −0.390527
\(246\) −3873.18 −0.00408065
\(247\) 448396. 0.467649
\(248\) −469315. −0.484546
\(249\) 586110. 0.599074
\(250\) −62500.0 −0.0632456
\(251\) 325565. 0.326177 0.163088 0.986611i \(-0.447854\pi\)
0.163088 + 0.986611i \(0.447854\pi\)
\(252\) −92731.1 −0.0919865
\(253\) −4700.87 −0.00461719
\(254\) −253677. −0.246716
\(255\) 152129. 0.146508
\(256\) 65536.0 0.0625000
\(257\) 164107. 0.154986 0.0774932 0.996993i \(-0.475308\pi\)
0.0774932 + 0.996993i \(0.475308\pi\)
\(258\) 73360.3 0.0686138
\(259\) 107569. 0.0996413
\(260\) 91642.3 0.0840742
\(261\) −72372.4 −0.0657615
\(262\) 759646. 0.683688
\(263\) 414875. 0.369852 0.184926 0.982752i \(-0.440795\pi\)
0.184926 + 0.982752i \(0.440795\pi\)
\(264\) −6163.08 −0.00544236
\(265\) 484708. 0.424000
\(266\) −361339. −0.313120
\(267\) 733508. 0.629690
\(268\) 409799. 0.348525
\(269\) −1.03395e6 −0.871199 −0.435600 0.900141i \(-0.643464\pi\)
−0.435600 + 0.900141i \(0.643464\pi\)
\(270\) 399403. 0.333428
\(271\) −905248. −0.748763 −0.374381 0.927275i \(-0.622145\pi\)
−0.374381 + 0.927275i \(0.622145\pi\)
\(272\) 143753. 0.117813
\(273\) 114593. 0.0930578
\(274\) −301690. −0.242764
\(275\) 5553.96 0.00442865
\(276\) −91721.3 −0.0724765
\(277\) −736609. −0.576817 −0.288408 0.957507i \(-0.593126\pi\)
−0.288408 + 0.957507i \(0.593126\pi\)
\(278\) −191346. −0.148494
\(279\) −920790. −0.708190
\(280\) −73849.7 −0.0562929
\(281\) −1.52991e6 −1.15584 −0.577921 0.816093i \(-0.696136\pi\)
−0.577921 + 0.816093i \(0.696136\pi\)
\(282\) 113457. 0.0849592
\(283\) 136848. 0.101572 0.0507860 0.998710i \(-0.483827\pi\)
0.0507860 + 0.998710i \(0.483827\pi\)
\(284\) −97269.0 −0.0715614
\(285\) 530226. 0.386677
\(286\) −8143.65 −0.00588713
\(287\) 4124.22 0.00295554
\(288\) 128581. 0.0913472
\(289\) −1.10454e6 −0.777921
\(290\) −57636.4 −0.0402440
\(291\) 1.12528e6 0.778981
\(292\) 1.08751e6 0.746408
\(293\) −509931. −0.347010 −0.173505 0.984833i \(-0.555509\pi\)
−0.173505 + 0.984833i \(0.555509\pi\)
\(294\) 636181. 0.429252
\(295\) 936510. 0.626552
\(296\) −149156. −0.0989488
\(297\) −35492.3 −0.0233477
\(298\) 1.66219e6 1.08428
\(299\) −121197. −0.0783996
\(300\) 108366. 0.0695171
\(301\) −78115.1 −0.0496957
\(302\) 984.535 0.000621175 0
\(303\) −976623. −0.611112
\(304\) 501033. 0.310944
\(305\) −436416. −0.268628
\(306\) 282041. 0.172191
\(307\) 60837.0 0.0368402 0.0184201 0.999830i \(-0.494136\pi\)
0.0184201 + 0.999830i \(0.494136\pi\)
\(308\) 6562.54 0.00394180
\(309\) −936407. −0.557916
\(310\) −733304. −0.433391
\(311\) −2.46113e6 −1.44289 −0.721447 0.692470i \(-0.756522\pi\)
−0.721447 + 0.692470i \(0.756522\pi\)
\(312\) −158895. −0.0924111
\(313\) 1.07570e6 0.620628 0.310314 0.950634i \(-0.399566\pi\)
0.310314 + 0.950634i \(0.399566\pi\)
\(314\) −199375. −0.114116
\(315\) −144892. −0.0822752
\(316\) 1.13208e6 0.637763
\(317\) 183297. 0.102449 0.0512244 0.998687i \(-0.483688\pi\)
0.0512244 + 0.998687i \(0.483688\pi\)
\(318\) −840417. −0.466044
\(319\) 5121.76 0.00281801
\(320\) 102400. 0.0559017
\(321\) 1.46977e6 0.796133
\(322\) 97666.2 0.0524934
\(323\) 1.09901e6 0.586133
\(324\) −204304. −0.108122
\(325\) 143191. 0.0751982
\(326\) 1.71220e6 0.892299
\(327\) 1.95083e6 1.00891
\(328\) −5718.64 −0.00293500
\(329\) −120811. −0.0615343
\(330\) −9629.81 −0.00486780
\(331\) −620020. −0.311054 −0.155527 0.987832i \(-0.549708\pi\)
−0.155527 + 0.987832i \(0.549708\pi\)
\(332\) 865375. 0.430883
\(333\) −292642. −0.144619
\(334\) −1.54920e6 −0.759871
\(335\) 640311. 0.311730
\(336\) 128045. 0.0618750
\(337\) 3.89502e6 1.86825 0.934125 0.356947i \(-0.116182\pi\)
0.934125 + 0.356947i \(0.116182\pi\)
\(338\) 1.27521e6 0.607144
\(339\) −1.72814e6 −0.816731
\(340\) 224614. 0.105375
\(341\) 65163.9 0.0303474
\(342\) 983021. 0.454462
\(343\) −1.45316e6 −0.666927
\(344\) 108314. 0.0493503
\(345\) −143315. −0.0648250
\(346\) 675446. 0.303319
\(347\) −2.39495e6 −1.06776 −0.533879 0.845561i \(-0.679266\pi\)
−0.533879 + 0.845561i \(0.679266\pi\)
\(348\) 99933.5 0.0442347
\(349\) 3.14342e6 1.38146 0.690731 0.723112i \(-0.257289\pi\)
0.690731 + 0.723112i \(0.257289\pi\)
\(350\) −115390. −0.0503499
\(351\) −915055. −0.396442
\(352\) −9099.61 −0.00391441
\(353\) −2.35756e6 −1.00699 −0.503497 0.863997i \(-0.667953\pi\)
−0.503497 + 0.863997i \(0.667953\pi\)
\(354\) −1.62378e6 −0.688682
\(355\) −151983. −0.0640064
\(356\) 1.08300e6 0.452903
\(357\) 280866. 0.116635
\(358\) 2.03112e6 0.837583
\(359\) −3.97376e6 −1.62729 −0.813647 0.581360i \(-0.802521\pi\)
−0.813647 + 0.581360i \(0.802521\pi\)
\(360\) 200908. 0.0817034
\(361\) 1.35437e6 0.546978
\(362\) −88874.0 −0.0356454
\(363\) −1.74440e6 −0.694830
\(364\) 169194. 0.0669316
\(365\) 1.69924e6 0.667608
\(366\) 756685. 0.295265
\(367\) 1.44453e6 0.559837 0.279919 0.960024i \(-0.409692\pi\)
0.279919 + 0.960024i \(0.409692\pi\)
\(368\) −135424. −0.0521286
\(369\) −11219.9 −0.00428967
\(370\) −233056. −0.0885025
\(371\) 894889. 0.337547
\(372\) 1.27145e6 0.476366
\(373\) −1.45357e6 −0.540957 −0.270478 0.962726i \(-0.587182\pi\)
−0.270478 + 0.962726i \(0.587182\pi\)
\(374\) −19959.9 −0.00737871
\(375\) 169322. 0.0621779
\(376\) 167517. 0.0611067
\(377\) 132048. 0.0478497
\(378\) 737395. 0.265443
\(379\) −5.11661e6 −1.82972 −0.914860 0.403771i \(-0.867699\pi\)
−0.914860 + 0.403771i \(0.867699\pi\)
\(380\) 782864. 0.278117
\(381\) 687251. 0.242551
\(382\) 2.17263e6 0.761777
\(383\) −1.70140e6 −0.592664 −0.296332 0.955085i \(-0.595763\pi\)
−0.296332 + 0.955085i \(0.595763\pi\)
\(384\) −177547. −0.0614450
\(385\) 10254.0 0.00352566
\(386\) −2.58835e6 −0.884210
\(387\) 212512. 0.0721283
\(388\) 1.66144e6 0.560281
\(389\) −2.48418e6 −0.832357 −0.416179 0.909283i \(-0.636631\pi\)
−0.416179 + 0.909283i \(0.636631\pi\)
\(390\) −248274. −0.0826550
\(391\) −297052. −0.0982630
\(392\) 939304. 0.308739
\(393\) −2.05800e6 −0.672147
\(394\) 151945. 0.0493114
\(395\) 1.76887e6 0.570433
\(396\) −17853.3 −0.00572113
\(397\) 2.81139e6 0.895251 0.447626 0.894221i \(-0.352270\pi\)
0.447626 + 0.894221i \(0.352270\pi\)
\(398\) −2.21074e6 −0.699569
\(399\) 978925. 0.307834
\(400\) 160000. 0.0500000
\(401\) 584078. 0.181389 0.0906943 0.995879i \(-0.471091\pi\)
0.0906943 + 0.995879i \(0.471091\pi\)
\(402\) −1.11021e6 −0.342642
\(403\) 1.68004e6 0.515297
\(404\) −1.44196e6 −0.439541
\(405\) −319225. −0.0967073
\(406\) −106411. −0.0320384
\(407\) 20710.1 0.00619722
\(408\) −389449. −0.115824
\(409\) 1.06930e6 0.316076 0.158038 0.987433i \(-0.449483\pi\)
0.158038 + 0.987433i \(0.449483\pi\)
\(410\) −8935.38 −0.00262514
\(411\) 817326. 0.238666
\(412\) −1.38258e6 −0.401280
\(413\) 1.72902e6 0.498800
\(414\) −265700. −0.0761888
\(415\) 1.35215e6 0.385393
\(416\) −234604. −0.0664665
\(417\) 518387. 0.145987
\(418\) −69567.9 −0.0194746
\(419\) −1.73494e6 −0.482780 −0.241390 0.970428i \(-0.577603\pi\)
−0.241390 + 0.970428i \(0.577603\pi\)
\(420\) 200071. 0.0553427
\(421\) −4.58077e6 −1.25960 −0.629801 0.776757i \(-0.716864\pi\)
−0.629801 + 0.776757i \(0.716864\pi\)
\(422\) 198817. 0.0543467
\(423\) 328666. 0.0893108
\(424\) −1.24085e6 −0.335201
\(425\) 350959. 0.0942506
\(426\) 263517. 0.0703534
\(427\) −805729. −0.213855
\(428\) 2.17007e6 0.572617
\(429\) 22062.4 0.00578776
\(430\) 169241. 0.0441403
\(431\) −2.49767e6 −0.647652 −0.323826 0.946117i \(-0.604969\pi\)
−0.323826 + 0.946117i \(0.604969\pi\)
\(432\) −1.02247e6 −0.263598
\(433\) 3.20773e6 0.822202 0.411101 0.911590i \(-0.365144\pi\)
0.411101 + 0.911590i \(0.365144\pi\)
\(434\) −1.35386e6 −0.345023
\(435\) 156146. 0.0395647
\(436\) 2.88035e6 0.725654
\(437\) −1.03534e6 −0.259345
\(438\) −2.94624e6 −0.733809
\(439\) 540078. 0.133750 0.0668752 0.997761i \(-0.478697\pi\)
0.0668752 + 0.997761i \(0.478697\pi\)
\(440\) −14218.1 −0.00350115
\(441\) 1.84290e6 0.451238
\(442\) −514603. −0.125290
\(443\) −5.25495e6 −1.27221 −0.636106 0.771601i \(-0.719456\pi\)
−0.636106 + 0.771601i \(0.719456\pi\)
\(444\) 404086. 0.0972785
\(445\) 1.69220e6 0.405089
\(446\) −196149. −0.0466926
\(447\) −4.50314e6 −1.06597
\(448\) 189055. 0.0445034
\(449\) 4.84754e6 1.13476 0.567382 0.823455i \(-0.307956\pi\)
0.567382 + 0.823455i \(0.307956\pi\)
\(450\) 313918. 0.0730778
\(451\) 794.028 0.000183821 0
\(452\) −2.55155e6 −0.587432
\(453\) −2667.26 −0.000610689 0
\(454\) −914059. −0.208130
\(455\) 264366. 0.0598655
\(456\) −1.35738e6 −0.305695
\(457\) 3.76038e6 0.842250 0.421125 0.907003i \(-0.361635\pi\)
0.421125 + 0.907003i \(0.361635\pi\)
\(458\) 810986. 0.180655
\(459\) −2.24278e6 −0.496885
\(460\) −211600. −0.0466252
\(461\) 759278. 0.166398 0.0831991 0.996533i \(-0.473486\pi\)
0.0831991 + 0.996533i \(0.473486\pi\)
\(462\) −17779.0 −0.00387526
\(463\) −5.74798e6 −1.24613 −0.623064 0.782171i \(-0.714112\pi\)
−0.623064 + 0.782171i \(0.714112\pi\)
\(464\) 147549. 0.0318157
\(465\) 1.98664e6 0.426075
\(466\) 1.31017e6 0.279488
\(467\) −4.81148e6 −1.02091 −0.510454 0.859905i \(-0.670523\pi\)
−0.510454 + 0.859905i \(0.670523\pi\)
\(468\) −460291. −0.0971445
\(469\) 1.18217e6 0.248169
\(470\) 261745. 0.0546555
\(471\) 540138. 0.112190
\(472\) −2.39747e6 −0.495333
\(473\) −15039.4 −0.00309084
\(474\) −3.06698e6 −0.626997
\(475\) 1.22322e6 0.248755
\(476\) 414691. 0.0838895
\(477\) −2.43454e6 −0.489915
\(478\) 2.17942e6 0.436287
\(479\) 2.53290e6 0.504405 0.252202 0.967675i \(-0.418845\pi\)
0.252202 + 0.967675i \(0.418845\pi\)
\(480\) −277418. −0.0549581
\(481\) 533944. 0.105228
\(482\) −164307. −0.0322136
\(483\) −264593. −0.0516073
\(484\) −2.57555e6 −0.499755
\(485\) 2.59600e6 0.501130
\(486\) −3.32871e6 −0.639270
\(487\) 7.92170e6 1.51355 0.756774 0.653677i \(-0.226774\pi\)
0.756774 + 0.653677i \(0.226774\pi\)
\(488\) 1.11722e6 0.212369
\(489\) −4.63862e6 −0.877237
\(490\) 1.46766e6 0.276144
\(491\) −6.16346e6 −1.15378 −0.576888 0.816824i \(-0.695733\pi\)
−0.576888 + 0.816824i \(0.695733\pi\)
\(492\) 15492.7 0.00288546
\(493\) 323648. 0.0599730
\(494\) −1.79359e6 −0.330678
\(495\) −27895.8 −0.00511713
\(496\) 1.87726e6 0.342626
\(497\) −280597. −0.0509557
\(498\) −2.34444e6 −0.423609
\(499\) −3.52751e6 −0.634186 −0.317093 0.948394i \(-0.602707\pi\)
−0.317093 + 0.948394i \(0.602707\pi\)
\(500\) 250000. 0.0447214
\(501\) 4.19702e6 0.747044
\(502\) −1.30226e6 −0.230642
\(503\) −9.79646e6 −1.72643 −0.863216 0.504835i \(-0.831553\pi\)
−0.863216 + 0.504835i \(0.831553\pi\)
\(504\) 370924. 0.0650443
\(505\) −2.25306e6 −0.393137
\(506\) 18803.5 0.00326484
\(507\) −3.45476e6 −0.596895
\(508\) 1.01471e6 0.174454
\(509\) 5.69781e6 0.974795 0.487398 0.873180i \(-0.337946\pi\)
0.487398 + 0.873180i \(0.337946\pi\)
\(510\) −608514. −0.103597
\(511\) 3.13720e6 0.531484
\(512\) −262144. −0.0441942
\(513\) −7.81695e6 −1.31143
\(514\) −656427. −0.109592
\(515\) −2.16028e6 −0.358915
\(516\) −293441. −0.0485173
\(517\) −23259.6 −0.00382715
\(518\) −430277. −0.0704570
\(519\) −1.82989e6 −0.298199
\(520\) −366569. −0.0594494
\(521\) −226080. −0.0364895 −0.0182447 0.999834i \(-0.505808\pi\)
−0.0182447 + 0.999834i \(0.505808\pi\)
\(522\) 289490. 0.0465004
\(523\) −8.28092e6 −1.32381 −0.661903 0.749589i \(-0.730251\pi\)
−0.661903 + 0.749589i \(0.730251\pi\)
\(524\) −3.03858e6 −0.483441
\(525\) 312610. 0.0495000
\(526\) −1.65950e6 −0.261525
\(527\) 4.11775e6 0.645853
\(528\) 24652.3 0.00384833
\(529\) 279841. 0.0434783
\(530\) −1.93883e6 −0.299813
\(531\) −4.70380e6 −0.723957
\(532\) 1.44536e6 0.221409
\(533\) 20471.5 0.00312127
\(534\) −2.93403e6 −0.445258
\(535\) 3.39074e6 0.512165
\(536\) −1.63920e6 −0.246444
\(537\) −5.50262e6 −0.823444
\(538\) 4.13579e6 0.616031
\(539\) −130421. −0.0193365
\(540\) −1.59761e6 −0.235769
\(541\) 1.32891e7 1.95210 0.976049 0.217551i \(-0.0698069\pi\)
0.976049 + 0.217551i \(0.0698069\pi\)
\(542\) 3.62099e6 0.529455
\(543\) 240774. 0.0350437
\(544\) −575011. −0.0833065
\(545\) 4.50055e6 0.649044
\(546\) −458373. −0.0658018
\(547\) −8.07986e6 −1.15461 −0.577306 0.816528i \(-0.695896\pi\)
−0.577306 + 0.816528i \(0.695896\pi\)
\(548\) 1.20676e6 0.171660
\(549\) 2.19198e6 0.310389
\(550\) −22215.9 −0.00313153
\(551\) 1.12804e6 0.158286
\(552\) 366885. 0.0512487
\(553\) 3.26577e6 0.454123
\(554\) 2.94644e6 0.407871
\(555\) 631385. 0.0870085
\(556\) 765384. 0.105001
\(557\) 49436.4 0.00675163 0.00337582 0.999994i \(-0.498925\pi\)
0.00337582 + 0.999994i \(0.498925\pi\)
\(558\) 3.68316e6 0.500766
\(559\) −387742. −0.0524823
\(560\) 295399. 0.0398051
\(561\) 54074.7 0.00725415
\(562\) 6.11962e6 0.817304
\(563\) −1.23559e6 −0.164286 −0.0821432 0.996621i \(-0.526176\pi\)
−0.0821432 + 0.996621i \(0.526176\pi\)
\(564\) −453830. −0.0600752
\(565\) −3.98679e6 −0.525415
\(566\) −547394. −0.0718222
\(567\) −589367. −0.0769889
\(568\) 389076. 0.0506015
\(569\) 8.02461e6 1.03907 0.519533 0.854450i \(-0.326106\pi\)
0.519533 + 0.854450i \(0.326106\pi\)
\(570\) −2.12090e6 −0.273422
\(571\) −8.09912e6 −1.03956 −0.519778 0.854302i \(-0.673985\pi\)
−0.519778 + 0.854302i \(0.673985\pi\)
\(572\) 32574.6 0.00416283
\(573\) −5.88601e6 −0.748918
\(574\) −16496.9 −0.00208988
\(575\) −330625. −0.0417029
\(576\) −514324. −0.0645922
\(577\) 2.77223e6 0.346648 0.173324 0.984865i \(-0.444549\pi\)
0.173324 + 0.984865i \(0.444549\pi\)
\(578\) 4.41815e6 0.550073
\(579\) 7.01226e6 0.869284
\(580\) 230545. 0.0284568
\(581\) 2.49639e6 0.306812
\(582\) −4.50111e6 −0.550823
\(583\) 172291. 0.0209938
\(584\) −4.35004e6 −0.527790
\(585\) −719205. −0.0868886
\(586\) 2.03972e6 0.245373
\(587\) 1.19837e7 1.43548 0.717739 0.696312i \(-0.245177\pi\)
0.717739 + 0.696312i \(0.245177\pi\)
\(588\) −2.54472e6 −0.303527
\(589\) 1.43519e7 1.70460
\(590\) −3.74604e6 −0.443039
\(591\) −411644. −0.0484790
\(592\) 596623. 0.0699674
\(593\) 2.37354e6 0.277179 0.138590 0.990350i \(-0.455743\pi\)
0.138590 + 0.990350i \(0.455743\pi\)
\(594\) 141969. 0.0165093
\(595\) 647955. 0.0750331
\(596\) −6.64876e6 −0.766699
\(597\) 5.98926e6 0.687760
\(598\) 484788. 0.0554369
\(599\) 7.06059e6 0.804033 0.402017 0.915632i \(-0.368309\pi\)
0.402017 + 0.915632i \(0.368309\pi\)
\(600\) −433465. −0.0491560
\(601\) 3.04137e6 0.343466 0.171733 0.985144i \(-0.445063\pi\)
0.171733 + 0.985144i \(0.445063\pi\)
\(602\) 312460. 0.0351402
\(603\) −3.21608e6 −0.360192
\(604\) −3938.14 −0.000439237 0
\(605\) −4.02430e6 −0.446994
\(606\) 3.90649e6 0.432121
\(607\) 1.15515e6 0.127252 0.0636262 0.997974i \(-0.479733\pi\)
0.0636262 + 0.997974i \(0.479733\pi\)
\(608\) −2.00413e6 −0.219871
\(609\) 288284. 0.0314975
\(610\) 1.74566e6 0.189949
\(611\) −599673. −0.0649848
\(612\) −1.12817e6 −0.121757
\(613\) 3.83998e6 0.412742 0.206371 0.978474i \(-0.433835\pi\)
0.206371 + 0.978474i \(0.433835\pi\)
\(614\) −243348. −0.0260499
\(615\) 24207.4 0.00258083
\(616\) −26250.1 −0.00278728
\(617\) 2.47164e6 0.261380 0.130690 0.991423i \(-0.458281\pi\)
0.130690 + 0.991423i \(0.458281\pi\)
\(618\) 3.74563e6 0.394506
\(619\) 1.47166e7 1.54376 0.771879 0.635769i \(-0.219317\pi\)
0.771879 + 0.635769i \(0.219317\pi\)
\(620\) 2.93322e6 0.306454
\(621\) 2.11284e6 0.219856
\(622\) 9.84453e6 1.02028
\(623\) 3.12420e6 0.322492
\(624\) 635581. 0.0653445
\(625\) 390625. 0.0400000
\(626\) −4.30281e6 −0.438850
\(627\) 188471. 0.0191459
\(628\) 797500. 0.0806922
\(629\) 1.30869e6 0.131889
\(630\) 579569. 0.0581774
\(631\) −6.71247e6 −0.671134 −0.335567 0.942016i \(-0.608928\pi\)
−0.335567 + 0.942016i \(0.608928\pi\)
\(632\) −4.52832e6 −0.450967
\(633\) −538628. −0.0534293
\(634\) −733188. −0.0724423
\(635\) 1.58548e6 0.156037
\(636\) 3.36167e6 0.329543
\(637\) −3.36250e6 −0.328332
\(638\) −20487.1 −0.00199264
\(639\) 763363. 0.0739570
\(640\) −409600. −0.0395285
\(641\) 1.45715e7 1.40075 0.700373 0.713777i \(-0.253017\pi\)
0.700373 + 0.713777i \(0.253017\pi\)
\(642\) −5.87907e6 −0.562951
\(643\) 3.97709e6 0.379348 0.189674 0.981847i \(-0.439257\pi\)
0.189674 + 0.981847i \(0.439257\pi\)
\(644\) −390665. −0.0371184
\(645\) −458502. −0.0433952
\(646\) −4.39605e6 −0.414459
\(647\) 1.12008e7 1.05194 0.525968 0.850504i \(-0.323703\pi\)
0.525968 + 0.850504i \(0.323703\pi\)
\(648\) 817216. 0.0764538
\(649\) 332886. 0.0310230
\(650\) −572765. −0.0531732
\(651\) 3.66781e6 0.339199
\(652\) −6.84880e6 −0.630951
\(653\) −5.84459e6 −0.536378 −0.268189 0.963366i \(-0.586425\pi\)
−0.268189 + 0.963366i \(0.586425\pi\)
\(654\) −7.80333e6 −0.713404
\(655\) −4.74779e6 −0.432402
\(656\) 22874.6 0.00207536
\(657\) −8.53474e6 −0.771395
\(658\) 483245. 0.0435113
\(659\) −8.73888e6 −0.783867 −0.391933 0.919994i \(-0.628194\pi\)
−0.391933 + 0.919994i \(0.628194\pi\)
\(660\) 38519.2 0.00344205
\(661\) −326006. −0.0290217 −0.0145108 0.999895i \(-0.504619\pi\)
−0.0145108 + 0.999895i \(0.504619\pi\)
\(662\) 2.48008e6 0.219948
\(663\) 1.39414e6 0.123175
\(664\) −3.46150e6 −0.304680
\(665\) 2.25837e6 0.198034
\(666\) 1.17057e6 0.102261
\(667\) −304896. −0.0265361
\(668\) 6.19678e6 0.537310
\(669\) 531398. 0.0459044
\(670\) −2.56124e6 −0.220426
\(671\) −155126. −0.0133008
\(672\) −512181. −0.0437522
\(673\) 1.23201e7 1.04852 0.524260 0.851558i \(-0.324342\pi\)
0.524260 + 0.851558i \(0.324342\pi\)
\(674\) −1.55801e7 −1.32105
\(675\) −2.49627e6 −0.210878
\(676\) −5.10086e6 −0.429315
\(677\) 1.46730e7 1.23040 0.615202 0.788369i \(-0.289074\pi\)
0.615202 + 0.788369i \(0.289074\pi\)
\(678\) 6.91255e6 0.577516
\(679\) 4.79285e6 0.398951
\(680\) −898455. −0.0745116
\(681\) 2.47633e6 0.204617
\(682\) −260656. −0.0214588
\(683\) −4.14801e6 −0.340242 −0.170121 0.985423i \(-0.554416\pi\)
−0.170121 + 0.985423i \(0.554416\pi\)
\(684\) −3.93208e6 −0.321353
\(685\) 1.88556e6 0.153538
\(686\) 5.81264e6 0.471588
\(687\) −2.19709e6 −0.177605
\(688\) −433258. −0.0348960
\(689\) 4.44198e6 0.356474
\(690\) 573258. 0.0458382
\(691\) 1.42489e7 1.13524 0.567618 0.823292i \(-0.307865\pi\)
0.567618 + 0.823292i \(0.307865\pi\)
\(692\) −2.70178e6 −0.214479
\(693\) −51502.5 −0.00407376
\(694\) 9.57980e6 0.755019
\(695\) 1.19591e6 0.0939156
\(696\) −399734. −0.0312786
\(697\) 50175.2 0.00391208
\(698\) −1.25737e7 −0.976842
\(699\) −3.54946e6 −0.274770
\(700\) 461561. 0.0356028
\(701\) −1.37111e7 −1.05385 −0.526924 0.849913i \(-0.676655\pi\)
−0.526924 + 0.849913i \(0.676655\pi\)
\(702\) 3.66022e6 0.280327
\(703\) 4.56127e6 0.348095
\(704\) 36398.5 0.00276791
\(705\) −709109. −0.0537329
\(706\) 9.43025e6 0.712052
\(707\) −4.15969e6 −0.312977
\(708\) 6.49512e6 0.486972
\(709\) 1.45280e7 1.08540 0.542701 0.839926i \(-0.317402\pi\)
0.542701 + 0.839926i \(0.317402\pi\)
\(710\) 607931. 0.0452594
\(711\) −8.88451e6 −0.659113
\(712\) −4.33202e6 −0.320251
\(713\) −3.87918e6 −0.285769
\(714\) −1.12346e6 −0.0824734
\(715\) 50897.8 0.00372335
\(716\) −8.12447e6 −0.592260
\(717\) −5.90440e6 −0.428922
\(718\) 1.58951e7 1.15067
\(719\) −1.31122e6 −0.0945918 −0.0472959 0.998881i \(-0.515060\pi\)
−0.0472959 + 0.998881i \(0.515060\pi\)
\(720\) −803631. −0.0577730
\(721\) −3.98840e6 −0.285733
\(722\) −5.41749e6 −0.386772
\(723\) 445134. 0.0316698
\(724\) 355496. 0.0252051
\(725\) 360227. 0.0254526
\(726\) 6.97758e6 0.491319
\(727\) 1.77409e7 1.24491 0.622456 0.782655i \(-0.286135\pi\)
0.622456 + 0.782655i \(0.286135\pi\)
\(728\) −676776. −0.0473278
\(729\) 1.21209e7 0.844723
\(730\) −6.79694e6 −0.472070
\(731\) −950348. −0.0657793
\(732\) −3.02674e6 −0.208784
\(733\) 1.04193e7 0.716272 0.358136 0.933669i \(-0.383412\pi\)
0.358136 + 0.933669i \(0.383412\pi\)
\(734\) −5.77813e6 −0.395865
\(735\) −3.97613e6 −0.271483
\(736\) 541696. 0.0368605
\(737\) 227601. 0.0154349
\(738\) 44879.7 0.00303325
\(739\) 1.11987e7 0.754318 0.377159 0.926148i \(-0.376901\pi\)
0.377159 + 0.926148i \(0.376901\pi\)
\(740\) 932223. 0.0625807
\(741\) 4.85911e6 0.325096
\(742\) −3.57956e6 −0.238682
\(743\) −1.18961e7 −0.790557 −0.395279 0.918561i \(-0.629352\pi\)
−0.395279 + 0.918561i \(0.629352\pi\)
\(744\) −5.08579e6 −0.336842
\(745\) −1.03887e7 −0.685756
\(746\) 5.81426e6 0.382514
\(747\) −6.79143e6 −0.445307
\(748\) 79839.8 0.00521753
\(749\) 6.26012e6 0.407735
\(750\) −677290. −0.0439664
\(751\) −1.35790e7 −0.878552 −0.439276 0.898352i \(-0.644765\pi\)
−0.439276 + 0.898352i \(0.644765\pi\)
\(752\) −670067. −0.0432090
\(753\) 3.52803e6 0.226748
\(754\) −528193. −0.0338349
\(755\) −6153.35 −0.000392866 0
\(756\) −2.94958e6 −0.187696
\(757\) −5.81377e6 −0.368738 −0.184369 0.982857i \(-0.559024\pi\)
−0.184369 + 0.982857i \(0.559024\pi\)
\(758\) 2.04665e7 1.29381
\(759\) −50941.7 −0.00320973
\(760\) −3.13145e6 −0.196658
\(761\) −1.50951e7 −0.944874 −0.472437 0.881365i \(-0.656626\pi\)
−0.472437 + 0.881365i \(0.656626\pi\)
\(762\) −2.74900e6 −0.171509
\(763\) 8.30911e6 0.516705
\(764\) −8.69053e6 −0.538658
\(765\) −1.76276e6 −0.108903
\(766\) 6.80559e6 0.419077
\(767\) 8.58240e6 0.526769
\(768\) 710190. 0.0434482
\(769\) −8.65959e6 −0.528058 −0.264029 0.964515i \(-0.585051\pi\)
−0.264029 + 0.964515i \(0.585051\pi\)
\(770\) −41015.9 −0.00249302
\(771\) 1.77836e6 0.107742
\(772\) 1.03534e7 0.625231
\(773\) 2.22807e7 1.34116 0.670579 0.741838i \(-0.266046\pi\)
0.670579 + 0.741838i \(0.266046\pi\)
\(774\) −850047. −0.0510024
\(775\) 4.58315e6 0.274100
\(776\) −6.64577e6 −0.396178
\(777\) 1.16569e6 0.0692677
\(778\) 9.93673e6 0.588565
\(779\) 174880. 0.0103251
\(780\) 993095. 0.0584459
\(781\) −54022.9 −0.00316920
\(782\) 1.18821e6 0.0694824
\(783\) −2.30201e6 −0.134185
\(784\) −3.75721e6 −0.218311
\(785\) 1.24609e6 0.0721733
\(786\) 8.23201e6 0.475280
\(787\) −3.09584e6 −0.178173 −0.0890865 0.996024i \(-0.528395\pi\)
−0.0890865 + 0.996024i \(0.528395\pi\)
\(788\) −607782. −0.0348684
\(789\) 4.49585e6 0.257110
\(790\) −7.07550e6 −0.403357
\(791\) −7.36058e6 −0.418284
\(792\) 71413.4 0.00404545
\(793\) −3.99942e6 −0.225847
\(794\) −1.12456e7 −0.633038
\(795\) 5.25261e6 0.294752
\(796\) 8.84297e6 0.494670
\(797\) −1.55315e7 −0.866100 −0.433050 0.901370i \(-0.642563\pi\)
−0.433050 + 0.901370i \(0.642563\pi\)
\(798\) −3.91570e6 −0.217672
\(799\) −1.46979e6 −0.0814494
\(800\) −640000. −0.0353553
\(801\) −8.49938e6 −0.468065
\(802\) −2.33631e6 −0.128261
\(803\) 604000. 0.0330558
\(804\) 4.44084e6 0.242284
\(805\) −610414. −0.0331997
\(806\) −6.72017e6 −0.364370
\(807\) −1.12045e7 −0.605632
\(808\) 5.76783e6 0.310802
\(809\) 1.34979e7 0.725097 0.362549 0.931965i \(-0.381907\pi\)
0.362549 + 0.931965i \(0.381907\pi\)
\(810\) 1.27690e6 0.0683824
\(811\) −2.88614e6 −0.154087 −0.0770434 0.997028i \(-0.524548\pi\)
−0.0770434 + 0.997028i \(0.524548\pi\)
\(812\) 425643. 0.0226545
\(813\) −9.80984e6 −0.520518
\(814\) −82840.5 −0.00438209
\(815\) −1.07012e7 −0.564339
\(816\) 1.55780e6 0.0819003
\(817\) −3.31232e6 −0.173611
\(818\) −4.27721e6 −0.223500
\(819\) −1.32783e6 −0.0691722
\(820\) 35741.5 0.00185626
\(821\) −1.86299e7 −0.964614 −0.482307 0.876002i \(-0.660201\pi\)
−0.482307 + 0.876002i \(0.660201\pi\)
\(822\) −3.26931e6 −0.168763
\(823\) 1.26283e7 0.649899 0.324949 0.945731i \(-0.394653\pi\)
0.324949 + 0.945731i \(0.394653\pi\)
\(824\) 5.53032e6 0.283748
\(825\) 60186.3 0.00307867
\(826\) −6.91610e6 −0.352705
\(827\) −1.71877e7 −0.873882 −0.436941 0.899490i \(-0.643938\pi\)
−0.436941 + 0.899490i \(0.643938\pi\)
\(828\) 1.06280e6 0.0538736
\(829\) 5.98252e6 0.302342 0.151171 0.988508i \(-0.451696\pi\)
0.151171 + 0.988508i \(0.451696\pi\)
\(830\) −5.40860e6 −0.272514
\(831\) −7.98237e6 −0.400986
\(832\) 938417. 0.0469989
\(833\) −8.24142e6 −0.411519
\(834\) −2.07355e6 −0.103228
\(835\) 9.68247e6 0.480585
\(836\) 278272. 0.0137706
\(837\) −2.92884e7 −1.44505
\(838\) 6.93975e6 0.341377
\(839\) 1.51819e7 0.744597 0.372299 0.928113i \(-0.378570\pi\)
0.372299 + 0.928113i \(0.378570\pi\)
\(840\) −800282. −0.0391332
\(841\) −2.01790e7 −0.983804
\(842\) 1.83231e7 0.890673
\(843\) −1.65790e7 −0.803508
\(844\) −795269. −0.0384289
\(845\) −7.97009e6 −0.383991
\(846\) −1.31466e6 −0.0631523
\(847\) −7.42984e6 −0.355853
\(848\) 4.96341e6 0.237023
\(849\) 1.48298e6 0.0706098
\(850\) −1.40384e6 −0.0666452
\(851\) −1.23286e6 −0.0583568
\(852\) −1.05407e6 −0.0497474
\(853\) 1.77368e7 0.834648 0.417324 0.908758i \(-0.362968\pi\)
0.417324 + 0.908758i \(0.362968\pi\)
\(854\) 3.22292e6 0.151218
\(855\) −6.14388e6 −0.287427
\(856\) −8.68028e6 −0.404902
\(857\) −1.92352e7 −0.894633 −0.447316 0.894376i \(-0.647620\pi\)
−0.447316 + 0.894376i \(0.647620\pi\)
\(858\) −88249.8 −0.00409256
\(859\) −3.42966e7 −1.58587 −0.792935 0.609306i \(-0.791448\pi\)
−0.792935 + 0.609306i \(0.791448\pi\)
\(860\) −676965. −0.0312119
\(861\) 44692.7 0.00205460
\(862\) 9.99067e6 0.457959
\(863\) 1.92707e7 0.880784 0.440392 0.897806i \(-0.354839\pi\)
0.440392 + 0.897806i \(0.354839\pi\)
\(864\) 4.08989e6 0.186392
\(865\) −4.22154e6 −0.191836
\(866\) −1.28309e7 −0.581385
\(867\) −1.19695e7 −0.540788
\(868\) 5.41543e6 0.243968
\(869\) 628753. 0.0282443
\(870\) −624584. −0.0279765
\(871\) 5.86796e6 0.262085
\(872\) −1.15214e7 −0.513115
\(873\) −1.30389e7 −0.579037
\(874\) 4.14135e6 0.183385
\(875\) 721188. 0.0318441
\(876\) 1.17850e7 0.518881
\(877\) −3.64202e7 −1.59898 −0.799490 0.600680i \(-0.794897\pi\)
−0.799490 + 0.600680i \(0.794897\pi\)
\(878\) −2.16031e6 −0.0945759
\(879\) −5.52593e6 −0.241231
\(880\) 56872.6 0.00247569
\(881\) −3.02637e7 −1.31366 −0.656828 0.754040i \(-0.728102\pi\)
−0.656828 + 0.754040i \(0.728102\pi\)
\(882\) −7.37161e6 −0.319074
\(883\) −3.95818e7 −1.70842 −0.854208 0.519932i \(-0.825957\pi\)
−0.854208 + 0.519932i \(0.825957\pi\)
\(884\) 2.05841e6 0.0885935
\(885\) 1.01486e7 0.435561
\(886\) 2.10198e7 0.899590
\(887\) −1.51763e7 −0.647673 −0.323837 0.946113i \(-0.604973\pi\)
−0.323837 + 0.946113i \(0.604973\pi\)
\(888\) −1.61635e6 −0.0687863
\(889\) 2.92718e6 0.124221
\(890\) −6.76878e6 −0.286441
\(891\) −113470. −0.00478835
\(892\) 784594. 0.0330166
\(893\) −5.12277e6 −0.214969
\(894\) 1.80125e7 0.753757
\(895\) −1.26945e7 −0.529734
\(896\) −756221. −0.0314687
\(897\) −1.31337e6 −0.0545011
\(898\) −1.93902e7 −0.802399
\(899\) 4.22650e6 0.174414
\(900\) −1.25567e6 −0.0516738
\(901\) 1.08872e7 0.446791
\(902\) −3176.11 −0.000129981 0
\(903\) −846505. −0.0345470
\(904\) 1.02062e7 0.415377
\(905\) 555463. 0.0225441
\(906\) 10669.1 0.000431823 0
\(907\) −3.82649e7 −1.54448 −0.772240 0.635331i \(-0.780864\pi\)
−0.772240 + 0.635331i \(0.780864\pi\)
\(908\) 3.65624e6 0.147170
\(909\) 1.13164e7 0.454255
\(910\) −1.05746e6 −0.0423313
\(911\) 1.77305e7 0.707822 0.353911 0.935279i \(-0.384852\pi\)
0.353911 + 0.935279i \(0.384852\pi\)
\(912\) 5.42951e6 0.216159
\(913\) 480626. 0.0190823
\(914\) −1.50415e7 −0.595560
\(915\) −4.72928e6 −0.186742
\(916\) −3.24394e6 −0.127742
\(917\) −8.76557e6 −0.344236
\(918\) 8.97114e6 0.351351
\(919\) 1.45775e7 0.569371 0.284686 0.958621i \(-0.408111\pi\)
0.284686 + 0.958621i \(0.408111\pi\)
\(920\) 846400. 0.0329690
\(921\) 659268. 0.0256102
\(922\) −3.03711e6 −0.117661
\(923\) −1.39281e6 −0.0538129
\(924\) 71115.8 0.00274023
\(925\) 1.45660e6 0.0559739
\(926\) 2.29919e7 0.881146
\(927\) 1.08504e7 0.414713
\(928\) −590196. −0.0224971
\(929\) 1.28138e7 0.487123 0.243561 0.969885i \(-0.421684\pi\)
0.243561 + 0.969885i \(0.421684\pi\)
\(930\) −7.94655e6 −0.301281
\(931\) −2.87245e7 −1.08612
\(932\) −5.24068e6 −0.197628
\(933\) −2.66704e7 −1.00306
\(934\) 1.92459e7 0.721891
\(935\) 124750. 0.00466670
\(936\) 1.84116e6 0.0686915
\(937\) 1.33046e7 0.495055 0.247528 0.968881i \(-0.420382\pi\)
0.247528 + 0.968881i \(0.420382\pi\)
\(938\) −4.72868e6 −0.175482
\(939\) 1.16570e7 0.431442
\(940\) −1.04698e6 −0.0386473
\(941\) 9.32459e6 0.343286 0.171643 0.985159i \(-0.445093\pi\)
0.171643 + 0.985159i \(0.445093\pi\)
\(942\) −2.16055e6 −0.0793301
\(943\) −47268.1 −0.00173097
\(944\) 9.58986e6 0.350253
\(945\) −4.60872e6 −0.167881
\(946\) 60157.4 0.00218555
\(947\) 2.89898e7 1.05044 0.525220 0.850967i \(-0.323983\pi\)
0.525220 + 0.850967i \(0.323983\pi\)
\(948\) 1.22679e7 0.443354
\(949\) 1.55722e7 0.561286
\(950\) −4.89290e6 −0.175896
\(951\) 1.98632e6 0.0712194
\(952\) −1.65877e6 −0.0593188
\(953\) −2.60957e7 −0.930757 −0.465379 0.885112i \(-0.654082\pi\)
−0.465379 + 0.885112i \(0.654082\pi\)
\(954\) 9.73816e6 0.346422
\(955\) −1.35789e7 −0.481790
\(956\) −8.71769e6 −0.308501
\(957\) 55502.7 0.00195900
\(958\) −1.01316e7 −0.356668
\(959\) 3.48121e6 0.122232
\(960\) 1.10967e6 0.0388612
\(961\) 2.51443e7 0.878277
\(962\) −2.13578e6 −0.0744078
\(963\) −1.70306e7 −0.591786
\(964\) 657228. 0.0227784
\(965\) 1.61772e7 0.559223
\(966\) 1.05837e6 0.0364919
\(967\) −3.75886e7 −1.29268 −0.646338 0.763051i \(-0.723700\pi\)
−0.646338 + 0.763051i \(0.723700\pi\)
\(968\) 1.03022e7 0.353380
\(969\) 1.19096e7 0.407462
\(970\) −1.03840e7 −0.354353
\(971\) −2.83881e7 −0.966248 −0.483124 0.875552i \(-0.660498\pi\)
−0.483124 + 0.875552i \(0.660498\pi\)
\(972\) 1.33148e7 0.452032
\(973\) 2.20795e6 0.0747664
\(974\) −3.16868e7 −1.07024
\(975\) 1.55171e6 0.0522756
\(976\) −4.46890e6 −0.150167
\(977\) −1.11673e7 −0.374293 −0.187146 0.982332i \(-0.559924\pi\)
−0.187146 + 0.982332i \(0.559924\pi\)
\(978\) 1.85545e7 0.620300
\(979\) 601497. 0.0200575
\(980\) −5.87065e6 −0.195263
\(981\) −2.26049e7 −0.749945
\(982\) 2.46539e7 0.815842
\(983\) 4.82265e7 1.59185 0.795924 0.605397i \(-0.206986\pi\)
0.795924 + 0.605397i \(0.206986\pi\)
\(984\) −61970.8 −0.00204033
\(985\) −949659. −0.0311872
\(986\) −1.29459e6 −0.0424073
\(987\) −1.30919e6 −0.0427769
\(988\) 7.17434e6 0.233824
\(989\) 895286. 0.0291052
\(990\) 111583. 0.00361836
\(991\) 2.90191e6 0.0938643 0.0469322 0.998898i \(-0.485056\pi\)
0.0469322 + 0.998898i \(0.485056\pi\)
\(992\) −7.50903e6 −0.242273
\(993\) −6.71893e6 −0.216236
\(994\) 1.12239e6 0.0360311
\(995\) 1.38171e7 0.442446
\(996\) 9.37776e6 0.299537
\(997\) −3.01833e7 −0.961675 −0.480837 0.876810i \(-0.659667\pi\)
−0.480837 + 0.876810i \(0.659667\pi\)
\(998\) 1.41100e7 0.448437
\(999\) −9.30832e6 −0.295092
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 230.6.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.f.1.4 5 1.1 even 1 trivial