Properties

Label 570.6.a.k.1.3
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 1.3
Root \(-80.8568\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +94.6269 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +94.6269 q^{7} -64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} -613.114 q^{11} +144.000 q^{12} -983.389 q^{13} -378.507 q^{14} -225.000 q^{15} +256.000 q^{16} -1418.72 q^{17} -324.000 q^{18} -361.000 q^{19} -400.000 q^{20} +851.642 q^{21} +2452.46 q^{22} +517.954 q^{23} -576.000 q^{24} +625.000 q^{25} +3933.55 q^{26} +729.000 q^{27} +1514.03 q^{28} +3308.73 q^{29} +900.000 q^{30} +1286.62 q^{31} -1024.00 q^{32} -5518.03 q^{33} +5674.90 q^{34} -2365.67 q^{35} +1296.00 q^{36} +5560.25 q^{37} +1444.00 q^{38} -8850.50 q^{39} +1600.00 q^{40} +8443.04 q^{41} -3406.57 q^{42} +23850.1 q^{43} -9809.83 q^{44} -2025.00 q^{45} -2071.82 q^{46} -14363.8 q^{47} +2304.00 q^{48} -7852.76 q^{49} -2500.00 q^{50} -12768.5 q^{51} -15734.2 q^{52} -16913.3 q^{53} -2916.00 q^{54} +15327.9 q^{55} -6056.12 q^{56} -3249.00 q^{57} -13234.9 q^{58} +14310.4 q^{59} -3600.00 q^{60} +42281.9 q^{61} -5146.48 q^{62} +7664.78 q^{63} +4096.00 q^{64} +24584.7 q^{65} +22072.1 q^{66} +20705.8 q^{67} -22699.6 q^{68} +4661.59 q^{69} +9462.69 q^{70} +37.7768 q^{71} -5184.00 q^{72} +30975.2 q^{73} -22241.0 q^{74} +5625.00 q^{75} -5776.00 q^{76} -58017.1 q^{77} +35402.0 q^{78} -64612.5 q^{79} -6400.00 q^{80} +6561.00 q^{81} -33772.2 q^{82} +100570. q^{83} +13626.3 q^{84} +35468.1 q^{85} -95400.6 q^{86} +29778.5 q^{87} +39239.3 q^{88} +48895.8 q^{89} +8100.00 q^{90} -93055.0 q^{91} +8287.27 q^{92} +11579.6 q^{93} +57455.4 q^{94} +9025.00 q^{95} -9216.00 q^{96} +92291.9 q^{97} +31411.0 q^{98} -49662.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + O(q^{10}) \) \( 4q - 16q^{2} + 36q^{3} + 64q^{4} - 100q^{5} - 144q^{6} + 26q^{7} - 256q^{8} + 324q^{9} + 400q^{10} - 472q^{11} + 576q^{12} - 482q^{13} - 104q^{14} - 900q^{15} + 1024q^{16} + 1816q^{17} - 1296q^{18} - 1444q^{19} - 1600q^{20} + 234q^{21} + 1888q^{22} - 418q^{23} - 2304q^{24} + 2500q^{25} + 1928q^{26} + 2916q^{27} + 416q^{28} - 10396q^{29} + 3600q^{30} + 528q^{31} - 4096q^{32} - 4248q^{33} - 7264q^{34} - 650q^{35} + 5184q^{36} + 5774q^{37} + 5776q^{38} - 4338q^{39} + 6400q^{40} + 9620q^{41} - 936q^{42} + 21098q^{43} - 7552q^{44} - 8100q^{45} + 1672q^{46} - 18858q^{47} + 9216q^{48} + 2148q^{49} - 10000q^{50} + 16344q^{51} - 7712q^{52} + 822q^{53} - 11664q^{54} + 11800q^{55} - 1664q^{56} - 12996q^{57} + 41584q^{58} + 28672q^{59} - 14400q^{60} + 77748q^{61} - 2112q^{62} + 2106q^{63} + 16384q^{64} + 12050q^{65} + 16992q^{66} + 82400q^{67} + 29056q^{68} - 3762q^{69} + 2600q^{70} + 928q^{71} - 20736q^{72} + 121100q^{73} - 23096q^{74} + 22500q^{75} - 23104q^{76} - 120220q^{77} + 17352q^{78} + 144284q^{79} - 25600q^{80} + 26244q^{81} - 38480q^{82} - 6082q^{83} + 3744q^{84} - 45400q^{85} - 84392q^{86} - 93564q^{87} + 30208q^{88} + 43260q^{89} + 32400q^{90} + 135148q^{91} - 6688q^{92} + 4752q^{93} + 75432q^{94} + 36100q^{95} - 36864q^{96} - 6862q^{97} - 8592q^{98} - 38232q^{99} + O(q^{100}) \)

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
\(7\) 94.6269 0.729910 0.364955 0.931025i \(-0.381084\pi\)
0.364955 + 0.931025i \(0.381084\pi\)
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 100.000 0.316228
\(11\) −613.114 −1.52778 −0.763888 0.645349i \(-0.776712\pi\)
−0.763888 + 0.645349i \(0.776712\pi\)
\(12\) 144.000 0.288675
\(13\) −983.389 −1.61386 −0.806932 0.590645i \(-0.798873\pi\)
−0.806932 + 0.590645i \(0.798873\pi\)
\(14\) −378.507 −0.516124
\(15\) −225.000 −0.258199
\(16\) 256.000 0.250000
\(17\) −1418.72 −1.19063 −0.595313 0.803494i \(-0.702972\pi\)
−0.595313 + 0.803494i \(0.702972\pi\)
\(18\) −324.000 −0.235702
\(19\) −361.000 −0.229416
\(20\) −400.000 −0.223607
\(21\) 851.642 0.421414
\(22\) 2452.46 1.08030
\(23\) 517.954 0.204161 0.102080 0.994776i \(-0.467450\pi\)
0.102080 + 0.994776i \(0.467450\pi\)
\(24\) −576.000 −0.204124
\(25\) 625.000 0.200000
\(26\) 3933.55 1.14117
\(27\) 729.000 0.192450
\(28\) 1514.03 0.364955
\(29\) 3308.73 0.730577 0.365288 0.930894i \(-0.380970\pi\)
0.365288 + 0.930894i \(0.380970\pi\)
\(30\) 900.000 0.182574
\(31\) 1286.62 0.240462 0.120231 0.992746i \(-0.461637\pi\)
0.120231 + 0.992746i \(0.461637\pi\)
\(32\) −1024.00 −0.176777
\(33\) −5518.03 −0.882062
\(34\) 5674.90 0.841900
\(35\) −2365.67 −0.326426
\(36\) 1296.00 0.166667
\(37\) 5560.25 0.667713 0.333856 0.942624i \(-0.391650\pi\)
0.333856 + 0.942624i \(0.391650\pi\)
\(38\) 1444.00 0.162221
\(39\) −8850.50 −0.931764
\(40\) 1600.00 0.158114
\(41\) 8443.04 0.784403 0.392202 0.919879i \(-0.371714\pi\)
0.392202 + 0.919879i \(0.371714\pi\)
\(42\) −3406.57 −0.297985
\(43\) 23850.1 1.96707 0.983535 0.180716i \(-0.0578416\pi\)
0.983535 + 0.180716i \(0.0578416\pi\)
\(44\) −9809.83 −0.763888
\(45\) −2025.00 −0.149071
\(46\) −2071.82 −0.144363
\(47\) −14363.8 −0.948475 −0.474238 0.880397i \(-0.657276\pi\)
−0.474238 + 0.880397i \(0.657276\pi\)
\(48\) 2304.00 0.144338
\(49\) −7852.76 −0.467231
\(50\) −2500.00 −0.141421
\(51\) −12768.5 −0.687409
\(52\) −15734.2 −0.806932
\(53\) −16913.3 −0.827063 −0.413531 0.910490i \(-0.635705\pi\)
−0.413531 + 0.910490i \(0.635705\pi\)
\(54\) −2916.00 −0.136083
\(55\) 15327.9 0.683242
\(56\) −6056.12 −0.258062
\(57\) −3249.00 −0.132453
\(58\) −13234.9 −0.516596
\(59\) 14310.4 0.535209 0.267604 0.963529i \(-0.413768\pi\)
0.267604 + 0.963529i \(0.413768\pi\)
\(60\) −3600.00 −0.129099
\(61\) 42281.9 1.45489 0.727445 0.686166i \(-0.240708\pi\)
0.727445 + 0.686166i \(0.240708\pi\)
\(62\) −5146.48 −0.170032
\(63\) 7664.78 0.243303
\(64\) 4096.00 0.125000
\(65\) 24584.7 0.721742
\(66\) 22072.1 0.623712
\(67\) 20705.8 0.563514 0.281757 0.959486i \(-0.409083\pi\)
0.281757 + 0.959486i \(0.409083\pi\)
\(68\) −22699.6 −0.595313
\(69\) 4661.59 0.117872
\(70\) 9462.69 0.230818
\(71\) 37.7768 0.000889362 0 0.000444681 1.00000i \(-0.499858\pi\)
0.000444681 1.00000i \(0.499858\pi\)
\(72\) −5184.00 −0.117851
\(73\) 30975.2 0.680310 0.340155 0.940369i \(-0.389520\pi\)
0.340155 + 0.940369i \(0.389520\pi\)
\(74\) −22241.0 −0.472144
\(75\) 5625.00 0.115470
\(76\) −5776.00 −0.114708
\(77\) −58017.1 −1.11514
\(78\) 35402.0 0.658857
\(79\) −64612.5 −1.16479 −0.582396 0.812905i \(-0.697885\pi\)
−0.582396 + 0.812905i \(0.697885\pi\)
\(80\) −6400.00 −0.111803
\(81\) 6561.00 0.111111
\(82\) −33772.2 −0.554657
\(83\) 100570. 1.60241 0.801204 0.598391i \(-0.204193\pi\)
0.801204 + 0.598391i \(0.204193\pi\)
\(84\) 13626.3 0.210707
\(85\) 35468.1 0.532464
\(86\) −95400.6 −1.39093
\(87\) 29778.5 0.421799
\(88\) 39239.3 0.540150
\(89\) 48895.8 0.654329 0.327165 0.944967i \(-0.393907\pi\)
0.327165 + 0.944967i \(0.393907\pi\)
\(90\) 8100.00 0.105409
\(91\) −93055.0 −1.17798
\(92\) 8287.27 0.102080
\(93\) 11579.6 0.138831
\(94\) 57455.4 0.670673
\(95\) 9025.00 0.102598
\(96\) −9216.00 −0.102062
\(97\) 92291.9 0.995943 0.497971 0.867193i \(-0.334078\pi\)
0.497971 + 0.867193i \(0.334078\pi\)
\(98\) 31411.0 0.330382
\(99\) −49662.3 −0.509259
\(100\) 10000.0 0.100000
\(101\) −16865.5 −0.164511 −0.0822556 0.996611i \(-0.526212\pi\)
−0.0822556 + 0.996611i \(0.526212\pi\)
\(102\) 51074.1 0.486071
\(103\) 80870.7 0.751101 0.375550 0.926802i \(-0.377454\pi\)
0.375550 + 0.926802i \(0.377454\pi\)
\(104\) 62936.9 0.570587
\(105\) −21291.0 −0.188462
\(106\) 67653.2 0.584822
\(107\) −200516. −1.69313 −0.846566 0.532284i \(-0.821334\pi\)
−0.846566 + 0.532284i \(0.821334\pi\)
\(108\) 11664.0 0.0962250
\(109\) −115645. −0.932310 −0.466155 0.884703i \(-0.654361\pi\)
−0.466155 + 0.884703i \(0.654361\pi\)
\(110\) −61311.4 −0.483125
\(111\) 50042.2 0.385504
\(112\) 24224.5 0.182478
\(113\) −24053.7 −0.177209 −0.0886046 0.996067i \(-0.528241\pi\)
−0.0886046 + 0.996067i \(0.528241\pi\)
\(114\) 12996.0 0.0936586
\(115\) −12948.9 −0.0913034
\(116\) 52939.6 0.365288
\(117\) −79654.5 −0.537954
\(118\) −57241.8 −0.378450
\(119\) −134249. −0.869050
\(120\) 14400.0 0.0912871
\(121\) 214858. 1.33410
\(122\) −169128. −1.02876
\(123\) 75987.4 0.452875
\(124\) 20585.9 0.120231
\(125\) −15625.0 −0.0894427
\(126\) −30659.1 −0.172041
\(127\) 15990.8 0.0879756 0.0439878 0.999032i \(-0.485994\pi\)
0.0439878 + 0.999032i \(0.485994\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 214651. 1.13569
\(130\) −98338.9 −0.510348
\(131\) −17110.3 −0.0871121 −0.0435560 0.999051i \(-0.513869\pi\)
−0.0435560 + 0.999051i \(0.513869\pi\)
\(132\) −88288.5 −0.441031
\(133\) −34160.3 −0.167453
\(134\) −82823.1 −0.398465
\(135\) −18225.0 −0.0860663
\(136\) 90798.3 0.420950
\(137\) −123786. −0.563471 −0.281735 0.959492i \(-0.590910\pi\)
−0.281735 + 0.959492i \(0.590910\pi\)
\(138\) −18646.4 −0.0833482
\(139\) −86142.9 −0.378166 −0.189083 0.981961i \(-0.560552\pi\)
−0.189083 + 0.981961i \(0.560552\pi\)
\(140\) −37850.7 −0.163213
\(141\) −129275. −0.547602
\(142\) −151.107 −0.000628874 0
\(143\) 602930. 2.46562
\(144\) 20736.0 0.0833333
\(145\) −82718.1 −0.326724
\(146\) −123901. −0.481052
\(147\) −70674.8 −0.269756
\(148\) 88964.0 0.333856
\(149\) −15526.0 −0.0572919 −0.0286459 0.999590i \(-0.509120\pi\)
−0.0286459 + 0.999590i \(0.509120\pi\)
\(150\) −22500.0 −0.0816497
\(151\) 2995.97 0.0106929 0.00534645 0.999986i \(-0.498298\pi\)
0.00534645 + 0.999986i \(0.498298\pi\)
\(152\) 23104.0 0.0811107
\(153\) −114917. −0.396876
\(154\) 232068. 0.788523
\(155\) −32165.5 −0.107538
\(156\) −141608. −0.465882
\(157\) 365967. 1.18493 0.592466 0.805596i \(-0.298155\pi\)
0.592466 + 0.805596i \(0.298155\pi\)
\(158\) 258450. 0.823633
\(159\) −152220. −0.477505
\(160\) 25600.0 0.0790569
\(161\) 49012.4 0.149019
\(162\) −26244.0 −0.0785674
\(163\) −436835. −1.28780 −0.643901 0.765109i \(-0.722685\pi\)
−0.643901 + 0.765109i \(0.722685\pi\)
\(164\) 135089. 0.392202
\(165\) 137951. 0.394470
\(166\) −402280. −1.13307
\(167\) 313346. 0.869428 0.434714 0.900569i \(-0.356849\pi\)
0.434714 + 0.900569i \(0.356849\pi\)
\(168\) −54505.1 −0.148992
\(169\) 595760. 1.60456
\(170\) −141872. −0.376509
\(171\) −29241.0 −0.0764719
\(172\) 381602. 0.983535
\(173\) 86867.9 0.220670 0.110335 0.993894i \(-0.464808\pi\)
0.110335 + 0.993894i \(0.464808\pi\)
\(174\) −119114. −0.298257
\(175\) 59141.8 0.145982
\(176\) −156957. −0.381944
\(177\) 128794. 0.309003
\(178\) −195583. −0.462680
\(179\) 577947. 1.34820 0.674101 0.738639i \(-0.264531\pi\)
0.674101 + 0.738639i \(0.264531\pi\)
\(180\) −32400.0 −0.0745356
\(181\) 570827. 1.29512 0.647558 0.762017i \(-0.275791\pi\)
0.647558 + 0.762017i \(0.275791\pi\)
\(182\) 372220. 0.832954
\(183\) 380537. 0.839981
\(184\) −33149.1 −0.0721817
\(185\) −139006. −0.298610
\(186\) −46318.3 −0.0981681
\(187\) 869840. 1.81901
\(188\) −229821. −0.474238
\(189\) 68983.0 0.140471
\(190\) −36100.0 −0.0725476
\(191\) 210943. 0.418390 0.209195 0.977874i \(-0.432916\pi\)
0.209195 + 0.977874i \(0.432916\pi\)
\(192\) 36864.0 0.0721688
\(193\) −418042. −0.807842 −0.403921 0.914794i \(-0.632353\pi\)
−0.403921 + 0.914794i \(0.632353\pi\)
\(194\) −369168. −0.704238
\(195\) 221262. 0.416698
\(196\) −125644. −0.233616
\(197\) 921835. 1.69234 0.846170 0.532913i \(-0.178903\pi\)
0.846170 + 0.532913i \(0.178903\pi\)
\(198\) 198649. 0.360100
\(199\) 720466. 1.28968 0.644838 0.764319i \(-0.276925\pi\)
0.644838 + 0.764319i \(0.276925\pi\)
\(200\) −40000.0 −0.0707107
\(201\) 186352. 0.325345
\(202\) 67462.0 0.116327
\(203\) 313094. 0.533255
\(204\) −204296. −0.343704
\(205\) −211076. −0.350796
\(206\) −323483. −0.531108
\(207\) 41954.3 0.0680535
\(208\) −251747. −0.403466
\(209\) 221334. 0.350496
\(210\) 85164.2 0.133263
\(211\) 561945. 0.868936 0.434468 0.900687i \(-0.356936\pi\)
0.434468 + 0.900687i \(0.356936\pi\)
\(212\) −270613. −0.413531
\(213\) 339.991 0.000513474 0
\(214\) 802066. 1.19722
\(215\) −596254. −0.879701
\(216\) −46656.0 −0.0680414
\(217\) 121749. 0.175515
\(218\) 462580. 0.659242
\(219\) 278777. 0.392777
\(220\) 245246. 0.341621
\(221\) 1.39516e6 1.92151
\(222\) −200169. −0.272593
\(223\) 121273. 0.163305 0.0816527 0.996661i \(-0.473980\pi\)
0.0816527 + 0.996661i \(0.473980\pi\)
\(224\) −96897.9 −0.129031
\(225\) 50625.0 0.0666667
\(226\) 96214.9 0.125306
\(227\) −678530. −0.873985 −0.436993 0.899465i \(-0.643956\pi\)
−0.436993 + 0.899465i \(0.643956\pi\)
\(228\) −51984.0 −0.0662266
\(229\) 212100. 0.267271 0.133635 0.991031i \(-0.457335\pi\)
0.133635 + 0.991031i \(0.457335\pi\)
\(230\) 51795.4 0.0645613
\(231\) −522154. −0.643826
\(232\) −211758. −0.258298
\(233\) −719902. −0.868728 −0.434364 0.900738i \(-0.643027\pi\)
−0.434364 + 0.900738i \(0.643027\pi\)
\(234\) 318618. 0.380391
\(235\) 359096. 0.424171
\(236\) 228967. 0.267604
\(237\) −581512. −0.672493
\(238\) 536998. 0.614511
\(239\) 239724. 0.271466 0.135733 0.990745i \(-0.456661\pi\)
0.135733 + 0.990745i \(0.456661\pi\)
\(240\) −57600.0 −0.0645497
\(241\) 1.13435e6 1.25807 0.629036 0.777376i \(-0.283450\pi\)
0.629036 + 0.777376i \(0.283450\pi\)
\(242\) −859433. −0.943351
\(243\) 59049.0 0.0641500
\(244\) 676511. 0.727445
\(245\) 196319. 0.208952
\(246\) −303950. −0.320231
\(247\) 355003. 0.370246
\(248\) −82343.6 −0.0850160
\(249\) 905130. 0.925151
\(250\) 62500.0 0.0632456
\(251\) −1.52013e6 −1.52299 −0.761493 0.648173i \(-0.775533\pi\)
−0.761493 + 0.648173i \(0.775533\pi\)
\(252\) 122636. 0.121652
\(253\) −317565. −0.311912
\(254\) −63963.4 −0.0622081
\(255\) 319213. 0.307418
\(256\) 65536.0 0.0625000
\(257\) 642217. 0.606526 0.303263 0.952907i \(-0.401924\pi\)
0.303263 + 0.952907i \(0.401924\pi\)
\(258\) −858605. −0.803053
\(259\) 526149. 0.487370
\(260\) 393355. 0.360871
\(261\) 268007. 0.243526
\(262\) 68441.0 0.0615975
\(263\) −229857. −0.204912 −0.102456 0.994738i \(-0.532670\pi\)
−0.102456 + 0.994738i \(0.532670\pi\)
\(264\) 353154. 0.311856
\(265\) 422832. 0.369874
\(266\) 136641. 0.118407
\(267\) 440062. 0.377777
\(268\) 331293. 0.281757
\(269\) 783628. 0.660282 0.330141 0.943932i \(-0.392904\pi\)
0.330141 + 0.943932i \(0.392904\pi\)
\(270\) 72900.0 0.0608581
\(271\) 966852. 0.799718 0.399859 0.916577i \(-0.369059\pi\)
0.399859 + 0.916577i \(0.369059\pi\)
\(272\) −363193. −0.297657
\(273\) −837495. −0.680104
\(274\) 495146. 0.398434
\(275\) −383196. −0.305555
\(276\) 74585.4 0.0589361
\(277\) 121779. 0.0953617 0.0476808 0.998863i \(-0.484817\pi\)
0.0476808 + 0.998863i \(0.484817\pi\)
\(278\) 344572. 0.267404
\(279\) 104216. 0.0801539
\(280\) 151403. 0.115409
\(281\) −1.47565e6 −1.11486 −0.557428 0.830226i \(-0.688212\pi\)
−0.557428 + 0.830226i \(0.688212\pi\)
\(282\) 517098. 0.387213
\(283\) −744205. −0.552365 −0.276183 0.961105i \(-0.589069\pi\)
−0.276183 + 0.961105i \(0.589069\pi\)
\(284\) 604.428 0.000444681 0
\(285\) 81225.0 0.0592349
\(286\) −2.41172e6 −1.74346
\(287\) 798939. 0.572544
\(288\) −82944.0 −0.0589256
\(289\) 592921. 0.417592
\(290\) 330873. 0.231029
\(291\) 830627. 0.575008
\(292\) 495603. 0.340155
\(293\) 122599. 0.0834289 0.0417145 0.999130i \(-0.486718\pi\)
0.0417145 + 0.999130i \(0.486718\pi\)
\(294\) 282699. 0.190746
\(295\) −357761. −0.239353
\(296\) −355856. −0.236072
\(297\) −446960. −0.294021
\(298\) 62103.9 0.0405115
\(299\) −509351. −0.329487
\(300\) 90000.0 0.0577350
\(301\) 2.25686e6 1.43578
\(302\) −11983.9 −0.00756102
\(303\) −151789. −0.0949806
\(304\) −92416.0 −0.0573539
\(305\) −1.05705e6 −0.650646
\(306\) 459667. 0.280633
\(307\) −2.36136e6 −1.42994 −0.714968 0.699157i \(-0.753559\pi\)
−0.714968 + 0.699157i \(0.753559\pi\)
\(308\) −928273. −0.557570
\(309\) 727836. 0.433648
\(310\) 128662. 0.0760407
\(311\) −2.31762e6 −1.35875 −0.679377 0.733789i \(-0.737750\pi\)
−0.679377 + 0.733789i \(0.737750\pi\)
\(312\) 566432. 0.329428
\(313\) −558773. −0.322385 −0.161192 0.986923i \(-0.551534\pi\)
−0.161192 + 0.986923i \(0.551534\pi\)
\(314\) −1.46387e6 −0.837873
\(315\) −191619. −0.108809
\(316\) −1.03380e6 −0.582396
\(317\) −1.36549e6 −0.763201 −0.381601 0.924327i \(-0.624627\pi\)
−0.381601 + 0.924327i \(0.624627\pi\)
\(318\) 608879. 0.337647
\(319\) −2.02863e6 −1.11616
\(320\) −102400. −0.0559017
\(321\) −1.80465e6 −0.977530
\(322\) −196050. −0.105372
\(323\) 512159. 0.273148
\(324\) 104976. 0.0555556
\(325\) −614618. −0.322773
\(326\) 1.74734e6 0.910613
\(327\) −1.04080e6 −0.538269
\(328\) −540355. −0.277328
\(329\) −1.35921e6 −0.692302
\(330\) −551803. −0.278933
\(331\) −2.09707e6 −1.05207 −0.526033 0.850464i \(-0.676321\pi\)
−0.526033 + 0.850464i \(0.676321\pi\)
\(332\) 1.60912e6 0.801204
\(333\) 450380. 0.222571
\(334\) −1.25339e6 −0.614778
\(335\) −517645. −0.252011
\(336\) 218020. 0.105353
\(337\) 1.99343e6 0.956152 0.478076 0.878319i \(-0.341334\pi\)
0.478076 + 0.878319i \(0.341334\pi\)
\(338\) −2.38304e6 −1.13459
\(339\) −216484. −0.102312
\(340\) 567490. 0.266232
\(341\) −788845. −0.367372
\(342\) 116964. 0.0540738
\(343\) −2.33348e6 −1.07095
\(344\) −1.52641e6 −0.695464
\(345\) −116540. −0.0527140
\(346\) −347472. −0.156038
\(347\) 2.46044e6 1.09695 0.548477 0.836166i \(-0.315208\pi\)
0.548477 + 0.836166i \(0.315208\pi\)
\(348\) 476456. 0.210899
\(349\) 4.02725e6 1.76989 0.884943 0.465699i \(-0.154197\pi\)
0.884943 + 0.465699i \(0.154197\pi\)
\(350\) −236567. −0.103225
\(351\) −716890. −0.310588
\(352\) 627829. 0.270075
\(353\) −624915. −0.266922 −0.133461 0.991054i \(-0.542609\pi\)
−0.133461 + 0.991054i \(0.542609\pi\)
\(354\) −515176. −0.218498
\(355\) −944.419 −0.000397735 0
\(356\) 782332. 0.327165
\(357\) −1.20824e6 −0.501746
\(358\) −2.31179e6 −0.953323
\(359\) 3.80986e6 1.56017 0.780087 0.625671i \(-0.215175\pi\)
0.780087 + 0.625671i \(0.215175\pi\)
\(360\) 129600. 0.0527046
\(361\) 130321. 0.0526316
\(362\) −2.28331e6 −0.915785
\(363\) 1.93372e6 0.770243
\(364\) −1.48888e6 −0.588988
\(365\) −774380. −0.304244
\(366\) −1.52215e6 −0.593956
\(367\) 1.24404e6 0.482137 0.241068 0.970508i \(-0.422502\pi\)
0.241068 + 0.970508i \(0.422502\pi\)
\(368\) 132596. 0.0510402
\(369\) 683886. 0.261468
\(370\) 556025. 0.211149
\(371\) −1.60045e6 −0.603682
\(372\) 185273. 0.0694153
\(373\) 3.30542e6 1.23014 0.615069 0.788473i \(-0.289128\pi\)
0.615069 + 0.788473i \(0.289128\pi\)
\(374\) −3.47936e6 −1.28624
\(375\) −140625. −0.0516398
\(376\) 919286. 0.335337
\(377\) −3.25376e6 −1.17905
\(378\) −275932. −0.0993282
\(379\) 166417. 0.0595114 0.0297557 0.999557i \(-0.490527\pi\)
0.0297557 + 0.999557i \(0.490527\pi\)
\(380\) 144400. 0.0512989
\(381\) 143918. 0.0507927
\(382\) −843770. −0.295846
\(383\) −3.24313e6 −1.12971 −0.564855 0.825190i \(-0.691068\pi\)
−0.564855 + 0.825190i \(0.691068\pi\)
\(384\) −147456. −0.0510310
\(385\) 1.45043e6 0.498705
\(386\) 1.67217e6 0.571231
\(387\) 1.93186e6 0.655690
\(388\) 1.47667e6 0.497971
\(389\) 2.36343e6 0.791898 0.395949 0.918272i \(-0.370416\pi\)
0.395949 + 0.918272i \(0.370416\pi\)
\(390\) −885050. −0.294650
\(391\) −734834. −0.243079
\(392\) 502576. 0.165191
\(393\) −153992. −0.0502942
\(394\) −3.68734e6 −1.19667
\(395\) 1.61531e6 0.520911
\(396\) −794596. −0.254629
\(397\) −4.68658e6 −1.49238 −0.746190 0.665733i \(-0.768119\pi\)
−0.746190 + 0.665733i \(0.768119\pi\)
\(398\) −2.88186e6 −0.911939
\(399\) −307443. −0.0966789
\(400\) 160000. 0.0500000
\(401\) −1.51865e6 −0.471626 −0.235813 0.971799i \(-0.575775\pi\)
−0.235813 + 0.971799i \(0.575775\pi\)
\(402\) −745408. −0.230054
\(403\) −1.26525e6 −0.388072
\(404\) −269848. −0.0822556
\(405\) −164025. −0.0496904
\(406\) −1.25238e6 −0.377068
\(407\) −3.40907e6 −1.02012
\(408\) 817185. 0.243036
\(409\) 5.38381e6 1.59141 0.795704 0.605686i \(-0.207101\pi\)
0.795704 + 0.605686i \(0.207101\pi\)
\(410\) 844304. 0.248050
\(411\) −1.11408e6 −0.325320
\(412\) 1.29393e6 0.375550
\(413\) 1.35415e6 0.390654
\(414\) −167817. −0.0481211
\(415\) −2.51425e6 −0.716619
\(416\) 1.00699e6 0.285293
\(417\) −775287. −0.218334
\(418\) −885337. −0.247838
\(419\) −3.99972e6 −1.11300 −0.556499 0.830849i \(-0.687856\pi\)
−0.556499 + 0.830849i \(0.687856\pi\)
\(420\) −340657. −0.0942310
\(421\) 4.80526e6 1.32133 0.660666 0.750680i \(-0.270274\pi\)
0.660666 + 0.750680i \(0.270274\pi\)
\(422\) −2.24778e6 −0.614431
\(423\) −1.16347e6 −0.316158
\(424\) 1.08245e6 0.292411
\(425\) −886702. −0.238125
\(426\) −1359.96 −0.000363081 0
\(427\) 4.00100e6 1.06194
\(428\) −3.20826e6 −0.846566
\(429\) 5.42637e6 1.42353
\(430\) 2.38501e6 0.622042
\(431\) −2.77677e6 −0.720023 −0.360012 0.932948i \(-0.617227\pi\)
−0.360012 + 0.932948i \(0.617227\pi\)
\(432\) 186624. 0.0481125
\(433\) −1.83502e6 −0.470349 −0.235174 0.971953i \(-0.575566\pi\)
−0.235174 + 0.971953i \(0.575566\pi\)
\(434\) −486995. −0.124108
\(435\) −744463. −0.188634
\(436\) −1.85032e6 −0.466155
\(437\) −186982. −0.0468377
\(438\) −1.11511e6 −0.277735
\(439\) −1.23249e6 −0.305227 −0.152614 0.988286i \(-0.548769\pi\)
−0.152614 + 0.988286i \(0.548769\pi\)
\(440\) −980983. −0.241563
\(441\) −636073. −0.155744
\(442\) −5.58063e6 −1.35871
\(443\) −1.44347e6 −0.349461 −0.174730 0.984616i \(-0.555905\pi\)
−0.174730 + 0.984616i \(0.555905\pi\)
\(444\) 800676. 0.192752
\(445\) −1.22239e6 −0.292625
\(446\) −485090. −0.115474
\(447\) −139734. −0.0330775
\(448\) 387592. 0.0912388
\(449\) 6.78670e6 1.58870 0.794352 0.607458i \(-0.207811\pi\)
0.794352 + 0.607458i \(0.207811\pi\)
\(450\) −202500. −0.0471405
\(451\) −5.17655e6 −1.19839
\(452\) −384860. −0.0886046
\(453\) 26963.8 0.00617355
\(454\) 2.71412e6 0.618001
\(455\) 2.32637e6 0.526806
\(456\) 207936. 0.0468293
\(457\) 5.70240e6 1.27723 0.638613 0.769528i \(-0.279509\pi\)
0.638613 + 0.769528i \(0.279509\pi\)
\(458\) −848399. −0.188989
\(459\) −1.03425e6 −0.229136
\(460\) −207182. −0.0456517
\(461\) −3.10351e6 −0.680143 −0.340072 0.940400i \(-0.610451\pi\)
−0.340072 + 0.940400i \(0.610451\pi\)
\(462\) 2.08862e6 0.455254
\(463\) −2.39485e6 −0.519189 −0.259594 0.965718i \(-0.583589\pi\)
−0.259594 + 0.965718i \(0.583589\pi\)
\(464\) 847034. 0.182644
\(465\) −289489. −0.0620869
\(466\) 2.87961e6 0.614283
\(467\) 7.82454e6 1.66022 0.830112 0.557597i \(-0.188276\pi\)
0.830112 + 0.557597i \(0.188276\pi\)
\(468\) −1.27447e6 −0.268977
\(469\) 1.95932e6 0.411315
\(470\) −1.43638e6 −0.299934
\(471\) 3.29371e6 0.684120
\(472\) −915868. −0.189225
\(473\) −1.46229e7 −3.00524
\(474\) 2.32605e6 0.475524
\(475\) −225625. −0.0458831
\(476\) −2.14799e6 −0.434525
\(477\) −1.36998e6 −0.275688
\(478\) −958894. −0.191956
\(479\) 317612. 0.0632496 0.0316248 0.999500i \(-0.489932\pi\)
0.0316248 + 0.999500i \(0.489932\pi\)
\(480\) 230400. 0.0456435
\(481\) −5.46788e6 −1.07760
\(482\) −4.53741e6 −0.889592
\(483\) 441112. 0.0860361
\(484\) 3.43773e6 0.667050
\(485\) −2.30730e6 −0.445399
\(486\) −236196. −0.0453609
\(487\) 2.12267e6 0.405565 0.202783 0.979224i \(-0.435002\pi\)
0.202783 + 0.979224i \(0.435002\pi\)
\(488\) −2.70604e6 −0.514381
\(489\) −3.93152e6 −0.743513
\(490\) −785276. −0.147752
\(491\) 7.61911e6 1.42627 0.713133 0.701029i \(-0.247276\pi\)
0.713133 + 0.701029i \(0.247276\pi\)
\(492\) 1.21580e6 0.226438
\(493\) −4.69417e6 −0.869844
\(494\) −1.42001e6 −0.261803
\(495\) 1.24156e6 0.227747
\(496\) 329375. 0.0601154
\(497\) 3574.70 0.000649154 0
\(498\) −3.62052e6 −0.654180
\(499\) −1.11962e6 −0.201289 −0.100644 0.994922i \(-0.532090\pi\)
−0.100644 + 0.994922i \(0.532090\pi\)
\(500\) −250000. −0.0447214
\(501\) 2.82012e6 0.501964
\(502\) 6.08051e6 1.07691
\(503\) −9.34148e6 −1.64625 −0.823125 0.567860i \(-0.807771\pi\)
−0.823125 + 0.567860i \(0.807771\pi\)
\(504\) −490546. −0.0860207
\(505\) 421637. 0.0735717
\(506\) 1.27026e6 0.220555
\(507\) 5.36184e6 0.926390
\(508\) 255853. 0.0439878
\(509\) 3.60261e6 0.616343 0.308172 0.951331i \(-0.400283\pi\)
0.308172 + 0.951331i \(0.400283\pi\)
\(510\) −1.27685e6 −0.217378
\(511\) 2.93109e6 0.496565
\(512\) −262144. −0.0441942
\(513\) −263169. −0.0441511
\(514\) −2.56887e6 −0.428878
\(515\) −2.02177e6 −0.335902
\(516\) 3.43442e6 0.567844
\(517\) 8.80668e6 1.44906
\(518\) −2.10459e6 −0.344623
\(519\) 781811. 0.127404
\(520\) −1.57342e6 −0.255174
\(521\) −7.78592e6 −1.25665 −0.628327 0.777950i \(-0.716260\pi\)
−0.628327 + 0.777950i \(0.716260\pi\)
\(522\) −1.07203e6 −0.172199
\(523\) 1.13511e7 1.81461 0.907304 0.420475i \(-0.138137\pi\)
0.907304 + 0.420475i \(0.138137\pi\)
\(524\) −273764. −0.0435560
\(525\) 532276. 0.0842828
\(526\) 919428. 0.144895
\(527\) −1.82536e6 −0.286300
\(528\) −1.41262e6 −0.220516
\(529\) −6.16807e6 −0.958318
\(530\) −1.69133e6 −0.261540
\(531\) 1.15915e6 0.178403
\(532\) −546565. −0.0837264
\(533\) −8.30279e6 −1.26592
\(534\) −1.76025e6 −0.267129
\(535\) 5.01291e6 0.757191
\(536\) −1.32517e6 −0.199232
\(537\) 5.20152e6 0.778385
\(538\) −3.13451e6 −0.466890
\(539\) 4.81464e6 0.713825
\(540\) −291600. −0.0430331
\(541\) 6.97868e6 1.02513 0.512566 0.858648i \(-0.328695\pi\)
0.512566 + 0.858648i \(0.328695\pi\)
\(542\) −3.86741e6 −0.565486
\(543\) 5.13745e6 0.747735
\(544\) 1.45277e6 0.210475
\(545\) 2.89112e6 0.416942
\(546\) 3.34998e6 0.480906
\(547\) −6.40187e6 −0.914826 −0.457413 0.889254i \(-0.651224\pi\)
−0.457413 + 0.889254i \(0.651224\pi\)
\(548\) −1.98058e6 −0.281735
\(549\) 3.42483e6 0.484963
\(550\) 1.53279e6 0.216060
\(551\) −1.19445e6 −0.167606
\(552\) −298342. −0.0416741
\(553\) −6.11407e6 −0.850194
\(554\) −487117. −0.0674309
\(555\) −1.25106e6 −0.172403
\(556\) −1.37829e6 −0.189083
\(557\) −1.37919e7 −1.88358 −0.941791 0.336199i \(-0.890859\pi\)
−0.941791 + 0.336199i \(0.890859\pi\)
\(558\) −416865. −0.0566774
\(559\) −2.34540e7 −3.17458
\(560\) −605612. −0.0816064
\(561\) 7.82856e6 1.05021
\(562\) 5.90261e6 0.788322
\(563\) 1.05556e6 0.140350 0.0701749 0.997535i \(-0.477644\pi\)
0.0701749 + 0.997535i \(0.477644\pi\)
\(564\) −2.06839e6 −0.273801
\(565\) 601343. 0.0792503
\(566\) 2.97682e6 0.390581
\(567\) 620847. 0.0811011
\(568\) −2417.71 −0.000314437 0
\(569\) −1.24916e6 −0.161747 −0.0808736 0.996724i \(-0.525771\pi\)
−0.0808736 + 0.996724i \(0.525771\pi\)
\(570\) −324900. −0.0418854
\(571\) 1.16283e7 1.49255 0.746273 0.665640i \(-0.231841\pi\)
0.746273 + 0.665640i \(0.231841\pi\)
\(572\) 9.64687e6 1.23281
\(573\) 1.89848e6 0.241557
\(574\) −3.19575e6 −0.404850
\(575\) 323722. 0.0408321
\(576\) 331776. 0.0416667
\(577\) −6.27747e6 −0.784956 −0.392478 0.919761i \(-0.628382\pi\)
−0.392478 + 0.919761i \(0.628382\pi\)
\(578\) −2.37168e6 −0.295282
\(579\) −3.76238e6 −0.466408
\(580\) −1.32349e6 −0.163362
\(581\) 9.51662e6 1.16961
\(582\) −3.32251e6 −0.406592
\(583\) 1.03698e7 1.26357
\(584\) −1.98241e6 −0.240526
\(585\) 1.99136e6 0.240581
\(586\) −490395. −0.0589932
\(587\) 7.88932e6 0.945027 0.472513 0.881323i \(-0.343347\pi\)
0.472513 + 0.881323i \(0.343347\pi\)
\(588\) −1.13080e6 −0.134878
\(589\) −464470. −0.0551657
\(590\) 1.43104e6 0.169248
\(591\) 8.29651e6 0.977073
\(592\) 1.42342e6 0.166928
\(593\) 8.29668e6 0.968874 0.484437 0.874826i \(-0.339024\pi\)
0.484437 + 0.874826i \(0.339024\pi\)
\(594\) 1.78784e6 0.207904
\(595\) 3.35623e6 0.388651
\(596\) −248415. −0.0286459
\(597\) 6.48419e6 0.744595
\(598\) 2.03740e6 0.232983
\(599\) 1.16906e7 1.33128 0.665638 0.746274i \(-0.268159\pi\)
0.665638 + 0.746274i \(0.268159\pi\)
\(600\) −360000. −0.0408248
\(601\) −8.58241e6 −0.969221 −0.484611 0.874730i \(-0.661039\pi\)
−0.484611 + 0.874730i \(0.661039\pi\)
\(602\) −9.02746e6 −1.01525
\(603\) 1.67717e6 0.187838
\(604\) 47935.6 0.00534645
\(605\) −5.37146e6 −0.596628
\(606\) 607158. 0.0671615
\(607\) −5.59550e6 −0.616406 −0.308203 0.951321i \(-0.599728\pi\)
−0.308203 + 0.951321i \(0.599728\pi\)
\(608\) 369664. 0.0405554
\(609\) 2.81785e6 0.307875
\(610\) 4.22819e6 0.460076
\(611\) 1.41252e7 1.53071
\(612\) −1.83867e6 −0.198438
\(613\) −1.22426e7 −1.31589 −0.657947 0.753064i \(-0.728575\pi\)
−0.657947 + 0.753064i \(0.728575\pi\)
\(614\) 9.44545e6 1.01112
\(615\) −1.89968e6 −0.202532
\(616\) 3.71309e6 0.394261
\(617\) 6.16362e6 0.651813 0.325907 0.945402i \(-0.394331\pi\)
0.325907 + 0.945402i \(0.394331\pi\)
\(618\) −2.91135e6 −0.306636
\(619\) 1.03768e6 0.108852 0.0544260 0.998518i \(-0.482667\pi\)
0.0544260 + 0.998518i \(0.482667\pi\)
\(620\) −514648. −0.0537689
\(621\) 377589. 0.0392907
\(622\) 9.27047e6 0.960784
\(623\) 4.62685e6 0.477601
\(624\) −2.26573e6 −0.232941
\(625\) 390625. 0.0400000
\(626\) 2.23509e6 0.227961
\(627\) 1.99201e6 0.202359
\(628\) 5.85548e6 0.592466
\(629\) −7.88846e6 −0.794997
\(630\) 766478. 0.0769393
\(631\) 1.02819e7 1.02802 0.514008 0.857785i \(-0.328160\pi\)
0.514008 + 0.857785i \(0.328160\pi\)
\(632\) 4.13520e6 0.411816
\(633\) 5.05751e6 0.501680
\(634\) 5.46194e6 0.539665
\(635\) −399771. −0.0393439
\(636\) −2.43551e6 −0.238753
\(637\) 7.72231e6 0.754048
\(638\) 8.11451e6 0.789243
\(639\) 3059.92 0.000296454 0
\(640\) 409600. 0.0395285
\(641\) 4.63984e6 0.446023 0.223012 0.974816i \(-0.428411\pi\)
0.223012 + 0.974816i \(0.428411\pi\)
\(642\) 7.21859e6 0.691218
\(643\) −1.41813e7 −1.35266 −0.676332 0.736597i \(-0.736431\pi\)
−0.676332 + 0.736597i \(0.736431\pi\)
\(644\) 784198. 0.0745094
\(645\) −5.36628e6 −0.507895
\(646\) −2.04864e6 −0.193145
\(647\) −7.60555e6 −0.714282 −0.357141 0.934050i \(-0.616248\pi\)
−0.357141 + 0.934050i \(0.616248\pi\)
\(648\) −419904. −0.0392837
\(649\) −8.77394e6 −0.817679
\(650\) 2.45847e6 0.228235
\(651\) 1.09574e6 0.101334
\(652\) −6.98937e6 −0.643901
\(653\) −1.93820e7 −1.77875 −0.889376 0.457177i \(-0.848861\pi\)
−0.889376 + 0.457177i \(0.848861\pi\)
\(654\) 4.16322e6 0.380614
\(655\) 427756. 0.0389577
\(656\) 2.16142e6 0.196101
\(657\) 2.50899e6 0.226770
\(658\) 5.43682e6 0.489531
\(659\) −1.52426e6 −0.136725 −0.0683623 0.997661i \(-0.521777\pi\)
−0.0683623 + 0.997661i \(0.521777\pi\)
\(660\) 2.20721e6 0.197235
\(661\) 4.95206e6 0.440841 0.220421 0.975405i \(-0.429257\pi\)
0.220421 + 0.975405i \(0.429257\pi\)
\(662\) 8.38828e6 0.743923
\(663\) 1.25564e7 1.10938
\(664\) −6.43648e6 −0.566537
\(665\) 854007. 0.0748872
\(666\) −1.80152e6 −0.157381
\(667\) 1.71377e6 0.149155
\(668\) 5.01354e6 0.434714
\(669\) 1.09145e6 0.0942844
\(670\) 2.07058e6 0.178199
\(671\) −2.59236e7 −2.22275
\(672\) −872081. −0.0744961
\(673\) −1.19068e7 −1.01335 −0.506674 0.862138i \(-0.669125\pi\)
−0.506674 + 0.862138i \(0.669125\pi\)
\(674\) −7.97373e6 −0.676101
\(675\) 455625. 0.0384900
\(676\) 9.53216e6 0.802278
\(677\) −1.09393e7 −0.917316 −0.458658 0.888613i \(-0.651670\pi\)
−0.458658 + 0.888613i \(0.651670\pi\)
\(678\) 865934. 0.0723453
\(679\) 8.73330e6 0.726949
\(680\) −2.26996e6 −0.188255
\(681\) −6.10677e6 −0.504596
\(682\) 3.15538e6 0.259771
\(683\) 2.25582e7 1.85035 0.925173 0.379546i \(-0.123920\pi\)
0.925173 + 0.379546i \(0.123920\pi\)
\(684\) −467856. −0.0382360
\(685\) 3.09466e6 0.251992
\(686\) 9.33390e6 0.757274
\(687\) 1.90890e6 0.154309
\(688\) 6.10564e6 0.491768
\(689\) 1.66323e7 1.33477
\(690\) 466159. 0.0372745
\(691\) 1.92705e7 1.53531 0.767657 0.640861i \(-0.221423\pi\)
0.767657 + 0.640861i \(0.221423\pi\)
\(692\) 1.38989e6 0.110335
\(693\) −4.69938e6 −0.371713
\(694\) −9.84175e6 −0.775664
\(695\) 2.15357e6 0.169121
\(696\) −1.90583e6 −0.149128
\(697\) −1.19783e7 −0.933931
\(698\) −1.61090e7 −1.25150
\(699\) −6.47912e6 −0.501560
\(700\) 946269. 0.0729910
\(701\) 3.93645e6 0.302559 0.151280 0.988491i \(-0.451661\pi\)
0.151280 + 0.988491i \(0.451661\pi\)
\(702\) 2.86756e6 0.219619
\(703\) −2.00725e6 −0.153184
\(704\) −2.51132e6 −0.190972
\(705\) 3.23186e6 0.244895
\(706\) 2.49966e6 0.188742
\(707\) −1.59593e6 −0.120078
\(708\) 2.06070e6 0.154501
\(709\) 1.86732e7 1.39509 0.697546 0.716540i \(-0.254275\pi\)
0.697546 + 0.716540i \(0.254275\pi\)
\(710\) 3777.68 0.000281241 0
\(711\) −5.23361e6 −0.388264
\(712\) −3.12933e6 −0.231340
\(713\) 666410. 0.0490928
\(714\) 4.83298e6 0.354788
\(715\) −1.50732e7 −1.10266
\(716\) 9.24715e6 0.674101
\(717\) 2.15751e6 0.156731
\(718\) −1.52395e7 −1.10321
\(719\) −1.00226e7 −0.723034 −0.361517 0.932366i \(-0.617741\pi\)
−0.361517 + 0.932366i \(0.617741\pi\)
\(720\) −518400. −0.0372678
\(721\) 7.65254e6 0.548236
\(722\) −521284. −0.0372161
\(723\) 1.02092e7 0.726348
\(724\) 9.13324e6 0.647558
\(725\) 2.06795e6 0.146115
\(726\) −7.73490e6 −0.544644
\(727\) −1.12364e7 −0.788479 −0.394239 0.919008i \(-0.628992\pi\)
−0.394239 + 0.919008i \(0.628992\pi\)
\(728\) 5.95552e6 0.416477
\(729\) 531441. 0.0370370
\(730\) 3.09752e6 0.215133
\(731\) −3.38368e7 −2.34205
\(732\) 6.08860e6 0.419990
\(733\) −1.41728e7 −0.974309 −0.487155 0.873316i \(-0.661965\pi\)
−0.487155 + 0.873316i \(0.661965\pi\)
\(734\) −4.97617e6 −0.340922
\(735\) 1.76687e6 0.120639
\(736\) −530385. −0.0360908
\(737\) −1.26950e7 −0.860923
\(738\) −2.73555e6 −0.184886
\(739\) 1.54330e7 1.03953 0.519767 0.854308i \(-0.326019\pi\)
0.519767 + 0.854308i \(0.326019\pi\)
\(740\) −2.22410e6 −0.149305
\(741\) 3.19503e6 0.213761
\(742\) 6.40181e6 0.426867
\(743\) −2.07933e6 −0.138182 −0.0690908 0.997610i \(-0.522010\pi\)
−0.0690908 + 0.997610i \(0.522010\pi\)
\(744\) −741093. −0.0490840
\(745\) 388149. 0.0256217
\(746\) −1.32217e7 −0.869840
\(747\) 8.14617e6 0.534136
\(748\) 1.39174e7 0.909506
\(749\) −1.89742e7 −1.23583
\(750\) 562500. 0.0365148
\(751\) −1.90164e7 −1.23035 −0.615175 0.788391i \(-0.710915\pi\)
−0.615175 + 0.788391i \(0.710915\pi\)
\(752\) −3.67714e6 −0.237119
\(753\) −1.36812e7 −0.879297
\(754\) 1.30151e7 0.833715
\(755\) −74899.3 −0.00478201
\(756\) 1.10373e6 0.0702356
\(757\) −1.37514e6 −0.0872182 −0.0436091 0.999049i \(-0.513886\pi\)
−0.0436091 + 0.999049i \(0.513886\pi\)
\(758\) −665669. −0.0420809
\(759\) −2.85809e6 −0.180082
\(760\) −577600. −0.0362738
\(761\) 1.05023e7 0.657390 0.328695 0.944436i \(-0.393391\pi\)
0.328695 + 0.944436i \(0.393391\pi\)
\(762\) −575670. −0.0359159
\(763\) −1.09431e7 −0.680502
\(764\) 3.37508e6 0.209195
\(765\) 2.87292e6 0.177488
\(766\) 1.29725e7 0.798826
\(767\) −1.40727e7 −0.863753
\(768\) 589824. 0.0360844
\(769\) −4.99693e6 −0.304711 −0.152355 0.988326i \(-0.548686\pi\)
−0.152355 + 0.988326i \(0.548686\pi\)
\(770\) −5.80171e6 −0.352638
\(771\) 5.77996e6 0.350178
\(772\) −6.68867e6 −0.403921
\(773\) −3.43234e6 −0.206606 −0.103303 0.994650i \(-0.532941\pi\)
−0.103303 + 0.994650i \(0.532941\pi\)
\(774\) −7.72745e6 −0.463643
\(775\) 804137. 0.0480923
\(776\) −5.90668e6 −0.352119
\(777\) 4.73534e6 0.281383
\(778\) −9.45374e6 −0.559957
\(779\) −3.04794e6 −0.179954
\(780\) 3.54020e6 0.208349
\(781\) −23161.5 −0.00135875
\(782\) 2.93934e6 0.171883
\(783\) 2.41206e6 0.140600
\(784\) −2.01031e6 −0.116808
\(785\) −9.14918e6 −0.529917
\(786\) 615969. 0.0355633
\(787\) 1.76794e7 1.01749 0.508746 0.860917i \(-0.330109\pi\)
0.508746 + 0.860917i \(0.330109\pi\)
\(788\) 1.47494e7 0.846170
\(789\) −2.06871e6 −0.118306
\(790\) −6.46125e6 −0.368340
\(791\) −2.27613e6 −0.129347
\(792\) 3.17838e6 0.180050
\(793\) −4.15796e7 −2.34799
\(794\) 1.87463e7 1.05527
\(795\) 3.80549e6 0.213547
\(796\) 1.15275e7 0.644838
\(797\) 2.92233e7 1.62961 0.814804 0.579737i \(-0.196845\pi\)
0.814804 + 0.579737i \(0.196845\pi\)
\(798\) 1.22977e6 0.0683623
\(799\) 2.03783e7 1.12928
\(800\) −640000. −0.0353553
\(801\) 3.96056e6 0.218110
\(802\) 6.07461e6 0.333490
\(803\) −1.89913e7 −1.03936
\(804\) 2.98163e6 0.162672
\(805\) −1.22531e6 −0.0666433
\(806\) 5.06099e6 0.274409
\(807\) 7.05265e6 0.381214
\(808\) 1.07939e6 0.0581635
\(809\) 2.33358e7 1.25358 0.626789 0.779189i \(-0.284369\pi\)
0.626789 + 0.779189i \(0.284369\pi\)
\(810\) 656100. 0.0351364
\(811\) 2.56264e7 1.36815 0.684077 0.729409i \(-0.260205\pi\)
0.684077 + 0.729409i \(0.260205\pi\)
\(812\) 5.00951e6 0.266628
\(813\) 8.70167e6 0.461717
\(814\) 1.36363e7 0.721331
\(815\) 1.09209e7 0.575922
\(816\) −3.26874e6 −0.171852
\(817\) −8.60990e6 −0.451277
\(818\) −2.15352e7 −1.12530
\(819\) −7.53745e6 −0.392658
\(820\) −3.37722e6 −0.175398
\(821\) 3.39526e7 1.75799 0.878993 0.476834i \(-0.158216\pi\)
0.878993 + 0.476834i \(0.158216\pi\)
\(822\) 4.45631e6 0.230036
\(823\) −3.55156e6 −0.182776 −0.0913882 0.995815i \(-0.529130\pi\)
−0.0913882 + 0.995815i \(0.529130\pi\)
\(824\) −5.17572e6 −0.265554
\(825\) −3.44877e6 −0.176412
\(826\) −5.41661e6 −0.276234
\(827\) −6.74117e6 −0.342745 −0.171373 0.985206i \(-0.554820\pi\)
−0.171373 + 0.985206i \(0.554820\pi\)
\(828\) 671269. 0.0340268
\(829\) 1.21627e7 0.614674 0.307337 0.951601i \(-0.400562\pi\)
0.307337 + 0.951601i \(0.400562\pi\)
\(830\) 1.00570e7 0.506726
\(831\) 1.09601e6 0.0550571
\(832\) −4.02796e6 −0.201733
\(833\) 1.11409e7 0.556298
\(834\) 3.10115e6 0.154386
\(835\) −7.83366e6 −0.388820
\(836\) 3.54135e6 0.175248
\(837\) 937946. 0.0462769
\(838\) 1.59989e7 0.787008
\(839\) 2.04276e7 1.00187 0.500935 0.865485i \(-0.332989\pi\)
0.500935 + 0.865485i \(0.332989\pi\)
\(840\) 1.36263e6 0.0666314
\(841\) −9.56349e6 −0.466258
\(842\) −1.92211e7 −0.934323
\(843\) −1.32809e7 −0.643662
\(844\) 8.99112e6 0.434468
\(845\) −1.48940e7 −0.717579
\(846\) 4.65389e6 0.223558
\(847\) 2.03314e7 0.973773
\(848\) −4.32980e6 −0.206766
\(849\) −6.69784e6 −0.318908
\(850\) 3.54681e6 0.168380
\(851\) 2.87995e6 0.136321
\(852\) 5439.85 0.000256737 0
\(853\) 3.21514e7 1.51296 0.756479 0.654018i \(-0.226918\pi\)
0.756479 + 0.654018i \(0.226918\pi\)
\(854\) −1.60040e7 −0.750904
\(855\) 731025. 0.0341993
\(856\) 1.28331e7 0.598612
\(857\) −2.63220e7 −1.22424 −0.612121 0.790764i \(-0.709683\pi\)
−0.612121 + 0.790764i \(0.709683\pi\)
\(858\) −2.17055e7 −1.00659
\(859\) 1.21188e6 0.0560374 0.0280187 0.999607i \(-0.491080\pi\)
0.0280187 + 0.999607i \(0.491080\pi\)
\(860\) −9.54006e6 −0.439850
\(861\) 7.19045e6 0.330558
\(862\) 1.11071e7 0.509133
\(863\) −5.47409e6 −0.250199 −0.125099 0.992144i \(-0.539925\pi\)
−0.125099 + 0.992144i \(0.539925\pi\)
\(864\) −746496. −0.0340207
\(865\) −2.17170e6 −0.0986868
\(866\) 7.34007e6 0.332587
\(867\) 5.33628e6 0.241097
\(868\) 1.94798e6 0.0877577
\(869\) 3.96148e7 1.77954
\(870\) 2.97785e6 0.133384
\(871\) −2.03618e7 −0.909435
\(872\) 7.40127e6 0.329621
\(873\) 7.47565e6 0.331981
\(874\) 747926. 0.0331192
\(875\) −1.47854e6 −0.0652851
\(876\) 4.46043e6 0.196389
\(877\) −2.98986e7 −1.31266 −0.656330 0.754474i \(-0.727892\pi\)
−0.656330 + 0.754474i \(0.727892\pi\)
\(878\) 4.92998e6 0.215828
\(879\) 1.10339e6 0.0481677
\(880\) 3.92393e6 0.170811
\(881\) 892818. 0.0387546 0.0193773 0.999812i \(-0.493832\pi\)
0.0193773 + 0.999812i \(0.493832\pi\)
\(882\) 2.54429e6 0.110127
\(883\) −9.71539e6 −0.419332 −0.209666 0.977773i \(-0.567238\pi\)
−0.209666 + 0.977773i \(0.567238\pi\)
\(884\) 2.23225e7 0.960754
\(885\) −3.21985e6 −0.138190
\(886\) 5.77388e6 0.247106
\(887\) 3.82580e7 1.63273 0.816364 0.577538i \(-0.195987\pi\)
0.816364 + 0.577538i \(0.195987\pi\)
\(888\) −3.20270e6 −0.136296
\(889\) 1.51316e6 0.0642142
\(890\) 4.88958e6 0.206917
\(891\) −4.02264e6 −0.169753
\(892\) 1.94036e6 0.0816527
\(893\) 5.18535e6 0.217595
\(894\) 558935. 0.0233893
\(895\) −1.44487e7 −0.602934
\(896\) −1.55037e6 −0.0645155
\(897\) −4.58415e6 −0.190230
\(898\) −2.71468e7 −1.12338
\(899\) 4.25707e6 0.175676
\(900\) 810000. 0.0333333
\(901\) 2.39953e7 0.984723
\(902\) 2.07062e7 0.847392
\(903\) 2.03118e7 0.828951
\(904\) 1.53944e6 0.0626529
\(905\) −1.42707e7 −0.579193
\(906\) −107855. −0.00436536
\(907\) −3.97217e7 −1.60328 −0.801641 0.597806i \(-0.796039\pi\)
−0.801641 + 0.597806i \(0.796039\pi\)
\(908\) −1.08565e7 −0.436993
\(909\) −1.36611e6 −0.0548371
\(910\) −9.30550e6 −0.372508
\(911\) 9.37882e6 0.374414 0.187207 0.982320i \(-0.440056\pi\)
0.187207 + 0.982320i \(0.440056\pi\)
\(912\) −831744. −0.0331133
\(913\) −6.16609e7 −2.44812
\(914\) −2.28096e7 −0.903135
\(915\) −9.51343e6 −0.375651
\(916\) 3.39360e6 0.133635
\(917\) −1.61909e6 −0.0635840
\(918\) 4.13700e6 0.162024
\(919\) −3.04741e7 −1.19026 −0.595131 0.803629i \(-0.702900\pi\)
−0.595131 + 0.803629i \(0.702900\pi\)
\(920\) 828727. 0.0322806
\(921\) −2.12523e7 −0.825574
\(922\) 1.24140e7 0.480934
\(923\) −37149.2 −0.00143531
\(924\) −8.35446e6 −0.321913
\(925\) 3.47515e6 0.133543
\(926\) 9.57939e6 0.367122
\(927\) 6.55053e6 0.250367
\(928\) −3.38813e6 −0.129149
\(929\) −2.66683e7 −1.01381 −0.506905 0.862002i \(-0.669211\pi\)
−0.506905 + 0.862002i \(0.669211\pi\)
\(930\) 1.15796e6 0.0439021
\(931\) 2.83485e6 0.107190
\(932\) −1.15184e7 −0.434364
\(933\) −2.08586e7 −0.784477
\(934\) −3.12982e7 −1.17396
\(935\) −2.17460e7 −0.813487
\(936\) 5.09789e6 0.190196
\(937\) −1.31362e7 −0.488788 −0.244394 0.969676i \(-0.578589\pi\)
−0.244394 + 0.969676i \(0.578589\pi\)
\(938\) −7.83729e6 −0.290843
\(939\) −5.02896e6 −0.186129
\(940\) 5.74554e6 0.212086
\(941\) −2.01910e6 −0.0743333 −0.0371666 0.999309i \(-0.511833\pi\)
−0.0371666 + 0.999309i \(0.511833\pi\)
\(942\) −1.31748e7 −0.483746
\(943\) 4.37311e6 0.160144
\(944\) 3.66347e6 0.133802
\(945\) −1.72457e6 −0.0628207
\(946\) 5.84915e7 2.12503
\(947\) 3.06998e7 1.11240 0.556200 0.831048i \(-0.312259\pi\)
0.556200 + 0.831048i \(0.312259\pi\)
\(948\) −9.30419e6 −0.336247
\(949\) −3.04607e7 −1.09793
\(950\) 902500. 0.0324443
\(951\) −1.22894e7 −0.440634
\(952\) 8.59196e6 0.307256
\(953\) 4.06107e7 1.44847 0.724233 0.689556i \(-0.242194\pi\)
0.724233 + 0.689556i \(0.242194\pi\)
\(954\) 5.47991e6 0.194941
\(955\) −5.27356e6 −0.187109
\(956\) 3.83558e6 0.135733
\(957\) −1.82576e7 −0.644414
\(958\) −1.27045e6 −0.0447242
\(959\) −1.17135e7 −0.411283
\(960\) −921600. −0.0322749
\(961\) −2.69738e7 −0.942178
\(962\) 2.18715e7 0.761976
\(963\) −1.62418e7 −0.564377
\(964\) 1.81496e7 0.629036
\(965\) 1.04511e7 0.361278
\(966\) −1.76445e6 −0.0608367
\(967\) −2.25532e7 −0.775608 −0.387804 0.921742i \(-0.626766\pi\)
−0.387804 + 0.921742i \(0.626766\pi\)
\(968\) −1.37509e7 −0.471676
\(969\) 4.60943e6 0.157702
\(970\) 9.22919e6 0.314945
\(971\) 4.45749e7 1.51720 0.758598 0.651559i \(-0.225885\pi\)
0.758598 + 0.651559i \(0.225885\pi\)
\(972\) 944784. 0.0320750
\(973\) −8.15144e6 −0.276027
\(974\) −8.49069e6 −0.286778
\(975\) −5.53156e6 −0.186353
\(976\) 1.08242e7 0.363722
\(977\) 4.21455e7 1.41259 0.706293 0.707920i \(-0.250366\pi\)
0.706293 + 0.707920i \(0.250366\pi\)
\(978\) 1.57261e7 0.525743
\(979\) −2.99787e7 −0.999668
\(980\) 3.14110e6 0.104476
\(981\) −9.36724e6 −0.310770
\(982\) −3.04764e7 −1.00852
\(983\) −7.47617e6 −0.246772 −0.123386 0.992359i \(-0.539375\pi\)
−0.123386 + 0.992359i \(0.539375\pi\)
\(984\) −4.86319e6 −0.160116
\(985\) −2.30459e7 −0.756837
\(986\) 1.87767e7 0.615073
\(987\) −1.22328e7 −0.399701
\(988\) 5.68005e6 0.185123
\(989\) 1.23533e7 0.401598
\(990\) −4.96623e6 −0.161042
\(991\) 5.19925e7 1.68173 0.840865 0.541245i \(-0.182047\pi\)
0.840865 + 0.541245i \(0.182047\pi\)
\(992\) −1.31750e6 −0.0425080
\(993\) −1.88736e7 −0.607411
\(994\) −14298.8 −0.000459022 0
\(995\) −1.80116e7 −0.576761
\(996\) 1.44821e7 0.462575
\(997\) 5.14642e7 1.63971 0.819856 0.572570i \(-0.194054\pi\)
0.819856 + 0.572570i \(0.194054\pi\)
\(998\) 4.47848e6 0.142333
\(999\) 4.05342e6 0.128501
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.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.6.a.k.1.3 4 1.1 even 1 trivial