Properties

Label 43.6.a.a.1.8
Level $43$
Weight $6$
Character 43.1
Self dual yes
Analytic conductor $6.897$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(10.9591\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} +O(q^{10})\) \(q+8.95911 q^{2} -25.1057 q^{3} +48.2657 q^{4} -61.2927 q^{5} -224.924 q^{6} -166.001 q^{7} +145.726 q^{8} +387.294 q^{9} -549.129 q^{10} +607.827 q^{11} -1211.74 q^{12} -1039.54 q^{13} -1487.22 q^{14} +1538.79 q^{15} -238.923 q^{16} +1439.76 q^{17} +3469.81 q^{18} -1332.62 q^{19} -2958.34 q^{20} +4167.57 q^{21} +5445.59 q^{22} -437.244 q^{23} -3658.56 q^{24} +631.800 q^{25} -9313.39 q^{26} -3622.60 q^{27} -8012.16 q^{28} -87.2656 q^{29} +13786.2 q^{30} +2654.45 q^{31} -6803.79 q^{32} -15259.9 q^{33} +12898.9 q^{34} +10174.7 q^{35} +18693.0 q^{36} -4671.70 q^{37} -11939.1 q^{38} +26098.4 q^{39} -8931.98 q^{40} -9012.91 q^{41} +37337.7 q^{42} -1849.00 q^{43} +29337.2 q^{44} -23738.3 q^{45} -3917.32 q^{46} -8623.49 q^{47} +5998.32 q^{48} +10749.4 q^{49} +5660.37 q^{50} -36146.0 q^{51} -50174.3 q^{52} -28358.3 q^{53} -32455.3 q^{54} -37255.4 q^{55} -24190.8 q^{56} +33456.3 q^{57} -781.822 q^{58} +48066.0 q^{59} +74271.0 q^{60} -39750.4 q^{61} +23781.5 q^{62} -64291.2 q^{63} -53310.4 q^{64} +63716.5 q^{65} -136715. q^{66} +19233.9 q^{67} +69490.8 q^{68} +10977.3 q^{69} +91156.0 q^{70} +17008.5 q^{71} +56439.0 q^{72} +22236.4 q^{73} -41854.3 q^{74} -15861.8 q^{75} -64319.9 q^{76} -100900. q^{77} +233819. q^{78} +36951.1 q^{79} +14644.3 q^{80} -3164.77 q^{81} -80747.7 q^{82} -117990. q^{83} +201151. q^{84} -88246.5 q^{85} -16565.4 q^{86} +2190.86 q^{87} +88576.5 q^{88} +41537.6 q^{89} -212674. q^{90} +172565. q^{91} -21103.9 q^{92} -66641.7 q^{93} -77258.8 q^{94} +81679.9 q^{95} +170814. q^{96} +32593.9 q^{97} +96304.8 q^{98} +235408. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} + O(q^{10}) \) \( 8 q - 12 q^{2} - 26 q^{3} + 122 q^{4} - 212 q^{5} - 69 q^{6} - 136 q^{7} - 666 q^{8} + 546 q^{9} - 617 q^{10} - 532 q^{11} - 4195 q^{12} - 2492 q^{13} - 4240 q^{14} - 1780 q^{15} + 1882 q^{16} - 2534 q^{17} - 3711 q^{18} - 1678 q^{19} - 2607 q^{20} - 2256 q^{21} + 11502 q^{22} - 2488 q^{23} + 19953 q^{24} + 4378 q^{25} + 4586 q^{26} - 8960 q^{27} + 18640 q^{28} - 4360 q^{29} + 25092 q^{30} + 5704 q^{31} - 18294 q^{32} - 12852 q^{33} + 30007 q^{34} + 5640 q^{35} + 67969 q^{36} - 3772 q^{37} - 6559 q^{38} + 11120 q^{39} + 14869 q^{40} - 10698 q^{41} + 78698 q^{42} - 14792 q^{43} - 356 q^{44} - 44912 q^{45} - 19389 q^{46} - 77864 q^{47} - 118727 q^{48} + 7188 q^{49} + 26877 q^{50} - 80246 q^{51} - 60736 q^{52} - 62352 q^{53} + 61026 q^{54} - 49552 q^{55} - 144528 q^{56} - 808 q^{57} + 52951 q^{58} - 26224 q^{59} + 101500 q^{60} - 82540 q^{61} - 9023 q^{62} - 61768 q^{63} + 153858 q^{64} - 5000 q^{65} - 48516 q^{66} + 27784 q^{67} + 40507 q^{68} - 93776 q^{69} + 185910 q^{70} - 9504 q^{71} - 186687 q^{72} + 14260 q^{73} - 15239 q^{74} + 167420 q^{75} + 1279 q^{76} - 218140 q^{77} + 264170 q^{78} + 160248 q^{79} - 1291 q^{80} + 161076 q^{81} - 47781 q^{82} - 77176 q^{83} + 16382 q^{84} + 141096 q^{85} + 22188 q^{86} + 268136 q^{87} + 129544 q^{88} - 265692 q^{89} + 48990 q^{90} + 401148 q^{91} + 190391 q^{92} - 123860 q^{93} + 248737 q^{94} + 135884 q^{95} + 950817 q^{96} + 144742 q^{97} - 292244 q^{98} + 239516 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.95911 1.58376 0.791881 0.610675i \(-0.209102\pi\)
0.791881 + 0.610675i \(0.209102\pi\)
\(3\) −25.1057 −1.61053 −0.805264 0.592916i \(-0.797977\pi\)
−0.805264 + 0.592916i \(0.797977\pi\)
\(4\) 48.2657 1.50830
\(5\) −61.2927 −1.09644 −0.548219 0.836335i \(-0.684694\pi\)
−0.548219 + 0.836335i \(0.684694\pi\)
\(6\) −224.924 −2.55070
\(7\) −166.001 −1.28046 −0.640230 0.768183i \(-0.721161\pi\)
−0.640230 + 0.768183i \(0.721161\pi\)
\(8\) 145.726 0.805033
\(9\) 387.294 1.59380
\(10\) −549.129 −1.73650
\(11\) 607.827 1.51460 0.757301 0.653066i \(-0.226518\pi\)
0.757301 + 0.653066i \(0.226518\pi\)
\(12\) −1211.74 −2.42917
\(13\) −1039.54 −1.70602 −0.853011 0.521894i \(-0.825226\pi\)
−0.853011 + 0.521894i \(0.825226\pi\)
\(14\) −1487.22 −2.02794
\(15\) 1538.79 1.76584
\(16\) −238.923 −0.233323
\(17\) 1439.76 1.20828 0.604138 0.796880i \(-0.293517\pi\)
0.604138 + 0.796880i \(0.293517\pi\)
\(18\) 3469.81 2.52421
\(19\) −1332.62 −0.846881 −0.423440 0.905924i \(-0.639178\pi\)
−0.423440 + 0.905924i \(0.639178\pi\)
\(20\) −2958.34 −1.65376
\(21\) 4167.57 2.06222
\(22\) 5445.59 2.39877
\(23\) −437.244 −0.172347 −0.0861737 0.996280i \(-0.527464\pi\)
−0.0861737 + 0.996280i \(0.527464\pi\)
\(24\) −3658.56 −1.29653
\(25\) 631.800 0.202176
\(26\) −9313.39 −2.70193
\(27\) −3622.60 −0.956336
\(28\) −8012.16 −1.93132
\(29\) −87.2656 −0.0192685 −0.00963425 0.999954i \(-0.503067\pi\)
−0.00963425 + 0.999954i \(0.503067\pi\)
\(30\) 13786.2 2.79668
\(31\) 2654.45 0.496101 0.248050 0.968747i \(-0.420210\pi\)
0.248050 + 0.968747i \(0.420210\pi\)
\(32\) −6803.79 −1.17456
\(33\) −15259.9 −2.43931
\(34\) 12898.9 1.91362
\(35\) 10174.7 1.40394
\(36\) 18693.0 2.40394
\(37\) −4671.70 −0.561010 −0.280505 0.959853i \(-0.590502\pi\)
−0.280505 + 0.959853i \(0.590502\pi\)
\(38\) −11939.1 −1.34126
\(39\) 26098.4 2.74760
\(40\) −8931.98 −0.882668
\(41\) −9012.91 −0.837347 −0.418674 0.908137i \(-0.637505\pi\)
−0.418674 + 0.908137i \(0.637505\pi\)
\(42\) 37337.7 3.26606
\(43\) −1849.00 −0.152499
\(44\) 29337.2 2.28448
\(45\) −23738.3 −1.74751
\(46\) −3917.32 −0.272957
\(47\) −8623.49 −0.569427 −0.284714 0.958613i \(-0.591899\pi\)
−0.284714 + 0.958613i \(0.591899\pi\)
\(48\) 5998.32 0.375774
\(49\) 10749.4 0.639577
\(50\) 5660.37 0.320199
\(51\) −36146.0 −1.94596
\(52\) −50174.3 −2.57320
\(53\) −28358.3 −1.38672 −0.693362 0.720590i \(-0.743871\pi\)
−0.693362 + 0.720590i \(0.743871\pi\)
\(54\) −32455.3 −1.51461
\(55\) −37255.4 −1.66067
\(56\) −24190.8 −1.03081
\(57\) 33456.3 1.36393
\(58\) −781.822 −0.0305167
\(59\) 48066.0 1.79766 0.898832 0.438294i \(-0.144417\pi\)
0.898832 + 0.438294i \(0.144417\pi\)
\(60\) 74271.0 2.66343
\(61\) −39750.4 −1.36778 −0.683891 0.729585i \(-0.739713\pi\)
−0.683891 + 0.729585i \(0.739713\pi\)
\(62\) 23781.5 0.785706
\(63\) −64291.2 −2.04080
\(64\) −53310.4 −1.62690
\(65\) 63716.5 1.87055
\(66\) −136715. −3.86329
\(67\) 19233.9 0.523456 0.261728 0.965142i \(-0.415708\pi\)
0.261728 + 0.965142i \(0.415708\pi\)
\(68\) 69490.8 1.82245
\(69\) 10977.3 0.277570
\(70\) 91156.0 2.22351
\(71\) 17008.5 0.400424 0.200212 0.979753i \(-0.435837\pi\)
0.200212 + 0.979753i \(0.435837\pi\)
\(72\) 56439.0 1.28306
\(73\) 22236.4 0.488380 0.244190 0.969727i \(-0.421478\pi\)
0.244190 + 0.969727i \(0.421478\pi\)
\(74\) −41854.3 −0.888506
\(75\) −15861.8 −0.325610
\(76\) −64319.9 −1.27735
\(77\) −100900. −1.93939
\(78\) 233819. 4.35154
\(79\) 36951.1 0.666131 0.333066 0.942904i \(-0.391917\pi\)
0.333066 + 0.942904i \(0.391917\pi\)
\(80\) 14644.3 0.255825
\(81\) −3164.77 −0.0535957
\(82\) −80747.7 −1.32616
\(83\) −117990. −1.87997 −0.939985 0.341216i \(-0.889161\pi\)
−0.939985 + 0.341216i \(0.889161\pi\)
\(84\) 201151. 3.11045
\(85\) −88246.5 −1.32480
\(86\) −16565.4 −0.241522
\(87\) 2190.86 0.0310325
\(88\) 88576.5 1.21930
\(89\) 41537.6 0.555861 0.277931 0.960601i \(-0.410351\pi\)
0.277931 + 0.960601i \(0.410351\pi\)
\(90\) −212674. −2.76763
\(91\) 172565. 2.18449
\(92\) −21103.9 −0.259952
\(93\) −66641.7 −0.798985
\(94\) −77258.8 −0.901838
\(95\) 81679.9 0.928552
\(96\) 170814. 1.89166
\(97\) 32593.9 0.351728 0.175864 0.984415i \(-0.443728\pi\)
0.175864 + 0.984415i \(0.443728\pi\)
\(98\) 96304.8 1.01294
\(99\) 235408. 2.41398
\(100\) 30494.3 0.304943
\(101\) −7934.83 −0.0773988 −0.0386994 0.999251i \(-0.512321\pi\)
−0.0386994 + 0.999251i \(0.512321\pi\)
\(102\) −323836. −3.08194
\(103\) 132558. 1.23116 0.615579 0.788075i \(-0.288922\pi\)
0.615579 + 0.788075i \(0.288922\pi\)
\(104\) −151489. −1.37340
\(105\) −255442. −2.26109
\(106\) −254065. −2.19624
\(107\) 120642. 1.01868 0.509342 0.860564i \(-0.329889\pi\)
0.509342 + 0.860564i \(0.329889\pi\)
\(108\) −174847. −1.44245
\(109\) 16769.1 0.135190 0.0675950 0.997713i \(-0.478467\pi\)
0.0675950 + 0.997713i \(0.478467\pi\)
\(110\) −333775. −2.63010
\(111\) 117286. 0.903522
\(112\) 39661.5 0.298761
\(113\) −18201.6 −0.134096 −0.0670478 0.997750i \(-0.521358\pi\)
−0.0670478 + 0.997750i \(0.521358\pi\)
\(114\) 299739. 2.16013
\(115\) 26799.9 0.188968
\(116\) −4211.94 −0.0290628
\(117\) −402609. −2.71906
\(118\) 430629. 2.84707
\(119\) −239001. −1.54715
\(120\) 224243. 1.42156
\(121\) 208403. 1.29402
\(122\) −356128. −2.16624
\(123\) 226275. 1.34857
\(124\) 128119. 0.748271
\(125\) 152815. 0.874764
\(126\) −575993. −3.23214
\(127\) −155225. −0.853987 −0.426994 0.904255i \(-0.640427\pi\)
−0.426994 + 0.904255i \(0.640427\pi\)
\(128\) −259892. −1.40207
\(129\) 46420.4 0.245603
\(130\) 570843. 2.96250
\(131\) 206749. 1.05261 0.526303 0.850297i \(-0.323578\pi\)
0.526303 + 0.850297i \(0.323578\pi\)
\(132\) −736530. −3.67922
\(133\) 221216. 1.08440
\(134\) 172319. 0.829030
\(135\) 222039. 1.04856
\(136\) 209810. 0.972702
\(137\) −219188. −0.997736 −0.498868 0.866678i \(-0.666251\pi\)
−0.498868 + 0.866678i \(0.666251\pi\)
\(138\) 98347.0 0.439606
\(139\) 83047.4 0.364577 0.182288 0.983245i \(-0.441650\pi\)
0.182288 + 0.983245i \(0.441650\pi\)
\(140\) 491087. 2.11757
\(141\) 216498. 0.917079
\(142\) 152381. 0.634176
\(143\) −631863. −2.58394
\(144\) −92533.5 −0.371871
\(145\) 5348.75 0.0211267
\(146\) 199219. 0.773478
\(147\) −269870. −1.03006
\(148\) −225483. −0.846173
\(149\) −505431. −1.86507 −0.932537 0.361074i \(-0.882410\pi\)
−0.932537 + 0.361074i \(0.882410\pi\)
\(150\) −142107. −0.515690
\(151\) −222212. −0.793095 −0.396548 0.918014i \(-0.629792\pi\)
−0.396548 + 0.918014i \(0.629792\pi\)
\(152\) −194198. −0.681767
\(153\) 557609. 1.92575
\(154\) −903974. −3.07153
\(155\) −162698. −0.543944
\(156\) 1.25966e6 4.14421
\(157\) 18138.1 0.0587275 0.0293638 0.999569i \(-0.490652\pi\)
0.0293638 + 0.999569i \(0.490652\pi\)
\(158\) 331049. 1.05499
\(159\) 711953. 2.23336
\(160\) 417023. 1.28783
\(161\) 72583.1 0.220684
\(162\) −28353.5 −0.0848828
\(163\) −428372. −1.26285 −0.631425 0.775437i \(-0.717530\pi\)
−0.631425 + 0.775437i \(0.717530\pi\)
\(164\) −435015. −1.26297
\(165\) 935321. 2.67455
\(166\) −1.05709e6 −2.97743
\(167\) 590110. 1.63735 0.818676 0.574256i \(-0.194709\pi\)
0.818676 + 0.574256i \(0.194709\pi\)
\(168\) 607325. 1.66015
\(169\) 709358. 1.91051
\(170\) −790611. −2.09817
\(171\) −516116. −1.34976
\(172\) −89243.3 −0.230014
\(173\) −194001. −0.492821 −0.246410 0.969166i \(-0.579251\pi\)
−0.246410 + 0.969166i \(0.579251\pi\)
\(174\) 19628.2 0.0491481
\(175\) −104880. −0.258878
\(176\) −145224. −0.353392
\(177\) −1.20673e6 −2.89519
\(178\) 372140. 0.880353
\(179\) −434140. −1.01274 −0.506369 0.862317i \(-0.669012\pi\)
−0.506369 + 0.862317i \(0.669012\pi\)
\(180\) −1.14575e6 −2.63577
\(181\) −626829. −1.42217 −0.711087 0.703104i \(-0.751797\pi\)
−0.711087 + 0.703104i \(0.751797\pi\)
\(182\) 1.54603e6 3.45972
\(183\) 997959. 2.20285
\(184\) −63718.1 −0.138745
\(185\) 286341. 0.615112
\(186\) −597050. −1.26540
\(187\) 875122. 1.83006
\(188\) −416219. −0.858870
\(189\) 601355. 1.22455
\(190\) 731780. 1.47061
\(191\) −109059. −0.216310 −0.108155 0.994134i \(-0.534494\pi\)
−0.108155 + 0.994134i \(0.534494\pi\)
\(192\) 1.33839e6 2.62017
\(193\) −31563.2 −0.0609941 −0.0304971 0.999535i \(-0.509709\pi\)
−0.0304971 + 0.999535i \(0.509709\pi\)
\(194\) 292012. 0.557053
\(195\) −1.59964e6 −3.01257
\(196\) 518826. 0.964676
\(197\) −276843. −0.508238 −0.254119 0.967173i \(-0.581786\pi\)
−0.254119 + 0.967173i \(0.581786\pi\)
\(198\) 2.10905e6 3.82316
\(199\) −777649. −1.39204 −0.696019 0.718024i \(-0.745047\pi\)
−0.696019 + 0.718024i \(0.745047\pi\)
\(200\) 92070.0 0.162758
\(201\) −482879. −0.843041
\(202\) −71089.0 −0.122581
\(203\) 14486.2 0.0246725
\(204\) −1.74461e6 −2.93511
\(205\) 552426. 0.918099
\(206\) 1.18761e6 1.94986
\(207\) −169342. −0.274688
\(208\) 248371. 0.398055
\(209\) −810002. −1.28269
\(210\) −2.28853e6 −3.58103
\(211\) −89486.6 −0.138373 −0.0691866 0.997604i \(-0.522040\pi\)
−0.0691866 + 0.997604i \(0.522040\pi\)
\(212\) −1.36873e6 −2.09160
\(213\) −427009. −0.644894
\(214\) 1.08085e6 1.61335
\(215\) 113330. 0.167205
\(216\) −527908. −0.769882
\(217\) −440641. −0.635237
\(218\) 150237. 0.214109
\(219\) −558260. −0.786550
\(220\) −1.79816e6 −2.50479
\(221\) −1.49669e6 −2.06135
\(222\) 1.05078e6 1.43096
\(223\) −952284. −1.28234 −0.641172 0.767398i \(-0.721551\pi\)
−0.641172 + 0.767398i \(0.721551\pi\)
\(224\) 1.12944e6 1.50398
\(225\) 244693. 0.322229
\(226\) −163071. −0.212376
\(227\) −465809. −0.599988 −0.299994 0.953941i \(-0.596985\pi\)
−0.299994 + 0.953941i \(0.596985\pi\)
\(228\) 1.61479e6 2.05721
\(229\) 477440. 0.601631 0.300815 0.953682i \(-0.402741\pi\)
0.300815 + 0.953682i \(0.402741\pi\)
\(230\) 240103. 0.299281
\(231\) 2.53316e6 3.12344
\(232\) −12716.9 −0.0155118
\(233\) 743589. 0.897312 0.448656 0.893705i \(-0.351903\pi\)
0.448656 + 0.893705i \(0.351903\pi\)
\(234\) −3.60702e6 −4.30635
\(235\) 528557. 0.624342
\(236\) 2.31994e6 2.71142
\(237\) −927682. −1.07282
\(238\) −2.14124e6 −2.45032
\(239\) 1.27115e6 1.43947 0.719734 0.694250i \(-0.244264\pi\)
0.719734 + 0.694250i \(0.244264\pi\)
\(240\) −367654. −0.412013
\(241\) −875960. −0.971497 −0.485749 0.874099i \(-0.661453\pi\)
−0.485749 + 0.874099i \(0.661453\pi\)
\(242\) 1.86710e6 2.04942
\(243\) 959745. 1.04265
\(244\) −1.91858e6 −2.06303
\(245\) −658858. −0.701256
\(246\) 2.02722e6 2.13582
\(247\) 1.38532e6 1.44480
\(248\) 386823. 0.399377
\(249\) 2.96222e6 3.02774
\(250\) 1.36909e6 1.38542
\(251\) −1.19674e6 −1.19899 −0.599494 0.800380i \(-0.704631\pi\)
−0.599494 + 0.800380i \(0.704631\pi\)
\(252\) −3.10306e6 −3.07815
\(253\) −265769. −0.261038
\(254\) −1.39068e6 −1.35251
\(255\) 2.21549e6 2.13363
\(256\) −622474. −0.593638
\(257\) −1.31630e6 −1.24315 −0.621573 0.783357i \(-0.713506\pi\)
−0.621573 + 0.783357i \(0.713506\pi\)
\(258\) 415885. 0.388977
\(259\) 775507. 0.718350
\(260\) 3.07532e6 2.82135
\(261\) −33797.4 −0.0307102
\(262\) 1.85229e6 1.66708
\(263\) 1.74612e6 1.55663 0.778314 0.627876i \(-0.216075\pi\)
0.778314 + 0.627876i \(0.216075\pi\)
\(264\) −2.22377e6 −1.96372
\(265\) 1.73816e6 1.52046
\(266\) 1.98190e6 1.71743
\(267\) −1.04283e6 −0.895231
\(268\) 928338. 0.789531
\(269\) −470733. −0.396637 −0.198319 0.980138i \(-0.563548\pi\)
−0.198319 + 0.980138i \(0.563548\pi\)
\(270\) 1.98927e6 1.66068
\(271\) 29459.5 0.0243670 0.0121835 0.999926i \(-0.496122\pi\)
0.0121835 + 0.999926i \(0.496122\pi\)
\(272\) −343991. −0.281919
\(273\) −4.33237e6 −3.51819
\(274\) −1.96373e6 −1.58018
\(275\) 384025. 0.306216
\(276\) 529828. 0.418661
\(277\) 1.01172e6 0.792244 0.396122 0.918198i \(-0.370356\pi\)
0.396122 + 0.918198i \(0.370356\pi\)
\(278\) 744031. 0.577403
\(279\) 1.02805e6 0.790687
\(280\) 1.48272e6 1.13022
\(281\) −74592.9 −0.0563549 −0.0281775 0.999603i \(-0.508970\pi\)
−0.0281775 + 0.999603i \(0.508970\pi\)
\(282\) 1.93963e6 1.45244
\(283\) −1.14789e6 −0.851993 −0.425996 0.904725i \(-0.640076\pi\)
−0.425996 + 0.904725i \(0.640076\pi\)
\(284\) 820927. 0.603961
\(285\) −2.05063e6 −1.49546
\(286\) −5.66093e6 −4.09235
\(287\) 1.49615e6 1.07219
\(288\) −2.63507e6 −1.87202
\(289\) 653038. 0.459932
\(290\) 47920.0 0.0334597
\(291\) −818290. −0.566467
\(292\) 1.07326e6 0.736625
\(293\) −1.30694e6 −0.889376 −0.444688 0.895686i \(-0.646685\pi\)
−0.444688 + 0.895686i \(0.646685\pi\)
\(294\) −2.41780e6 −1.63137
\(295\) −2.94610e6 −1.97103
\(296\) −680790. −0.451631
\(297\) −2.20191e6 −1.44847
\(298\) −4.52821e6 −2.95383
\(299\) 454535. 0.294028
\(300\) −765580. −0.491120
\(301\) 306936. 0.195268
\(302\) −1.99082e6 −1.25607
\(303\) 199209. 0.124653
\(304\) 318394. 0.197597
\(305\) 2.43641e6 1.49969
\(306\) 4.99568e6 3.04994
\(307\) −1.50999e6 −0.914380 −0.457190 0.889369i \(-0.651144\pi\)
−0.457190 + 0.889369i \(0.651144\pi\)
\(308\) −4.87001e6 −2.92518
\(309\) −3.32796e6 −1.98282
\(310\) −1.45763e6 −0.861478
\(311\) 1.69979e6 0.996539 0.498269 0.867022i \(-0.333969\pi\)
0.498269 + 0.867022i \(0.333969\pi\)
\(312\) 3.80323e6 2.21190
\(313\) 2.85190e6 1.64541 0.822703 0.568471i \(-0.192465\pi\)
0.822703 + 0.568471i \(0.192465\pi\)
\(314\) 162501. 0.0930104
\(315\) 3.94059e6 2.23761
\(316\) 1.78347e6 1.00473
\(317\) −158403. −0.0885351 −0.0442676 0.999020i \(-0.514095\pi\)
−0.0442676 + 0.999020i \(0.514095\pi\)
\(318\) 6.37847e6 3.53711
\(319\) −53042.4 −0.0291841
\(320\) 3.26754e6 1.78380
\(321\) −3.02880e6 −1.64062
\(322\) 650280. 0.349511
\(323\) −1.91865e6 −1.02327
\(324\) −152750. −0.0808386
\(325\) −656784. −0.344917
\(326\) −3.83783e6 −2.00006
\(327\) −421000. −0.217727
\(328\) −1.31342e6 −0.674092
\(329\) 1.43151e6 0.729129
\(330\) 8.37965e6 4.23585
\(331\) 1.38726e6 0.695968 0.347984 0.937501i \(-0.386866\pi\)
0.347984 + 0.937501i \(0.386866\pi\)
\(332\) −5.69488e6 −2.83557
\(333\) −1.80932e6 −0.894139
\(334\) 5.28687e6 2.59318
\(335\) −1.17890e6 −0.573937
\(336\) −995728. −0.481163
\(337\) 602990. 0.289225 0.144612 0.989488i \(-0.453807\pi\)
0.144612 + 0.989488i \(0.453807\pi\)
\(338\) 6.35522e6 3.02579
\(339\) 456964. 0.215965
\(340\) −4.25928e6 −1.99820
\(341\) 1.61345e6 0.751395
\(342\) −4.62394e6 −2.13770
\(343\) 1.00557e6 0.461507
\(344\) −269448. −0.122766
\(345\) −672829. −0.304339
\(346\) −1.73808e6 −0.780511
\(347\) 1.38847e6 0.619030 0.309515 0.950895i \(-0.399833\pi\)
0.309515 + 0.950895i \(0.399833\pi\)
\(348\) 105743. 0.0468064
\(349\) 1.90607e6 0.837676 0.418838 0.908061i \(-0.362437\pi\)
0.418838 + 0.908061i \(0.362437\pi\)
\(350\) −939628. −0.410002
\(351\) 3.76585e6 1.63153
\(352\) −4.13553e6 −1.77899
\(353\) −2.53782e6 −1.08399 −0.541993 0.840383i \(-0.682330\pi\)
−0.541993 + 0.840383i \(0.682330\pi\)
\(354\) −1.08112e7 −4.58529
\(355\) −1.04250e6 −0.439040
\(356\) 2.00484e6 0.838408
\(357\) 6.00028e6 2.49173
\(358\) −3.88951e6 −1.60394
\(359\) −589213. −0.241288 −0.120644 0.992696i \(-0.538496\pi\)
−0.120644 + 0.992696i \(0.538496\pi\)
\(360\) −3.45930e6 −1.40680
\(361\) −700223. −0.282793
\(362\) −5.61583e6 −2.25239
\(363\) −5.23209e6 −2.08405
\(364\) 8.32900e6 3.29488
\(365\) −1.36293e6 −0.535478
\(366\) 8.94083e6 3.48879
\(367\) 1.67673e6 0.649826 0.324913 0.945744i \(-0.394665\pi\)
0.324913 + 0.945744i \(0.394665\pi\)
\(368\) 104468. 0.0402127
\(369\) −3.49065e6 −1.33457
\(370\) 2.56536e6 0.974192
\(371\) 4.70750e6 1.77564
\(372\) −3.21651e6 −1.20511
\(373\) 3.12665e6 1.16361 0.581804 0.813329i \(-0.302347\pi\)
0.581804 + 0.813329i \(0.302347\pi\)
\(374\) 7.84032e6 2.89838
\(375\) −3.83652e6 −1.40883
\(376\) −1.25667e6 −0.458408
\(377\) 90716.4 0.0328725
\(378\) 5.38761e6 1.93940
\(379\) −1.98431e6 −0.709596 −0.354798 0.934943i \(-0.615450\pi\)
−0.354798 + 0.934943i \(0.615450\pi\)
\(380\) 3.94234e6 1.40054
\(381\) 3.89702e6 1.37537
\(382\) −977069. −0.342584
\(383\) 321813. 0.112100 0.0560502 0.998428i \(-0.482149\pi\)
0.0560502 + 0.998428i \(0.482149\pi\)
\(384\) 6.52477e6 2.25807
\(385\) 6.18444e6 2.12642
\(386\) −282778. −0.0966002
\(387\) −716107. −0.243053
\(388\) 1.57317e6 0.530512
\(389\) −1.34564e6 −0.450874 −0.225437 0.974258i \(-0.572381\pi\)
−0.225437 + 0.974258i \(0.572381\pi\)
\(390\) −1.43314e7 −4.77119
\(391\) −629525. −0.208243
\(392\) 1.56647e6 0.514880
\(393\) −5.19057e6 −1.69525
\(394\) −2.48026e6 −0.804929
\(395\) −2.26484e6 −0.730372
\(396\) 1.13621e7 3.64101
\(397\) 1.40299e6 0.446763 0.223382 0.974731i \(-0.428290\pi\)
0.223382 + 0.974731i \(0.428290\pi\)
\(398\) −6.96704e6 −2.20466
\(399\) −5.55378e6 −1.74645
\(400\) −150952. −0.0471724
\(401\) 3.44258e6 1.06911 0.534556 0.845133i \(-0.320479\pi\)
0.534556 + 0.845133i \(0.320479\pi\)
\(402\) −4.32617e6 −1.33518
\(403\) −2.75942e6 −0.846359
\(404\) −382980. −0.116741
\(405\) 193977. 0.0587643
\(406\) 129783. 0.0390754
\(407\) −2.83958e6 −0.849706
\(408\) −5.26743e6 −1.56656
\(409\) 6.44614e6 1.90542 0.952711 0.303877i \(-0.0982811\pi\)
0.952711 + 0.303877i \(0.0982811\pi\)
\(410\) 4.94925e6 1.45405
\(411\) 5.50286e6 1.60688
\(412\) 6.39802e6 1.85696
\(413\) −7.97902e6 −2.30184
\(414\) −1.51716e6 −0.435040
\(415\) 7.23194e6 2.06127
\(416\) 7.07283e6 2.00383
\(417\) −2.08496e6 −0.587161
\(418\) −7.25690e6 −2.03147
\(419\) −1.77865e6 −0.494943 −0.247472 0.968895i \(-0.579600\pi\)
−0.247472 + 0.968895i \(0.579600\pi\)
\(420\) −1.23291e7 −3.41042
\(421\) −3.37157e6 −0.927100 −0.463550 0.886071i \(-0.653425\pi\)
−0.463550 + 0.886071i \(0.653425\pi\)
\(422\) −801721. −0.219150
\(423\) −3.33983e6 −0.907555
\(424\) −4.13255e6 −1.11636
\(425\) 909638. 0.244285
\(426\) −3.82562e6 −1.02136
\(427\) 6.59861e6 1.75139
\(428\) 5.82288e6 1.53649
\(429\) 1.58633e7 4.16151
\(430\) 1.01534e6 0.264813
\(431\) 1.38644e6 0.359508 0.179754 0.983712i \(-0.442470\pi\)
0.179754 + 0.983712i \(0.442470\pi\)
\(432\) 865522. 0.223136
\(433\) 5.74536e6 1.47264 0.736322 0.676632i \(-0.236561\pi\)
0.736322 + 0.676632i \(0.236561\pi\)
\(434\) −3.94776e6 −1.00606
\(435\) −134284. −0.0340252
\(436\) 809375. 0.203908
\(437\) 582681. 0.145958
\(438\) −5.00152e6 −1.24571
\(439\) −5.14484e6 −1.27412 −0.637060 0.770814i \(-0.719850\pi\)
−0.637060 + 0.770814i \(0.719850\pi\)
\(440\) −5.42910e6 −1.33689
\(441\) 4.16317e6 1.01936
\(442\) −1.34090e7 −3.26468
\(443\) −5.65845e6 −1.36990 −0.684949 0.728591i \(-0.740176\pi\)
−0.684949 + 0.728591i \(0.740176\pi\)
\(444\) 5.66090e6 1.36279
\(445\) −2.54595e6 −0.609468
\(446\) −8.53162e6 −2.03093
\(447\) 1.26892e7 3.00376
\(448\) 8.84958e6 2.08318
\(449\) −956361. −0.223875 −0.111938 0.993715i \(-0.535706\pi\)
−0.111938 + 0.993715i \(0.535706\pi\)
\(450\) 2.19223e6 0.510334
\(451\) −5.47829e6 −1.26825
\(452\) −878515. −0.202257
\(453\) 5.57878e6 1.27730
\(454\) −4.17323e6 −0.950239
\(455\) −1.05770e7 −2.39516
\(456\) 4.87547e6 1.09800
\(457\) −5.50698e6 −1.23345 −0.616727 0.787177i \(-0.711542\pi\)
−0.616727 + 0.787177i \(0.711542\pi\)
\(458\) 4.27744e6 0.952840
\(459\) −5.21565e6 −1.15552
\(460\) 1.29352e6 0.285021
\(461\) 2.10202e6 0.460665 0.230333 0.973112i \(-0.426019\pi\)
0.230333 + 0.973112i \(0.426019\pi\)
\(462\) 2.26949e7 4.94678
\(463\) 2.36108e6 0.511868 0.255934 0.966694i \(-0.417617\pi\)
0.255934 + 0.966694i \(0.417617\pi\)
\(464\) 20849.8 0.00449579
\(465\) 4.08465e6 0.876037
\(466\) 6.66190e6 1.42113
\(467\) −8.56485e6 −1.81730 −0.908652 0.417554i \(-0.862887\pi\)
−0.908652 + 0.417554i \(0.862887\pi\)
\(468\) −1.94322e7 −4.10117
\(469\) −3.19285e6 −0.670264
\(470\) 4.73540e6 0.988809
\(471\) −455368. −0.0945823
\(472\) 7.00450e6 1.44718
\(473\) −1.12387e6 −0.230975
\(474\) −8.31121e6 −1.69910
\(475\) −841950. −0.171219
\(476\) −1.15356e7 −2.33357
\(477\) −1.09830e7 −2.21016
\(478\) 1.13884e7 2.27977
\(479\) 3.09570e6 0.616481 0.308241 0.951308i \(-0.400260\pi\)
0.308241 + 0.951308i \(0.400260\pi\)
\(480\) −1.04696e7 −2.07409
\(481\) 4.85643e6 0.957095
\(482\) −7.84782e6 −1.53862
\(483\) −1.82225e6 −0.355418
\(484\) 1.00587e7 1.95177
\(485\) −1.99777e6 −0.385648
\(486\) 8.59846e6 1.65132
\(487\) −4.69724e6 −0.897471 −0.448736 0.893665i \(-0.648125\pi\)
−0.448736 + 0.893665i \(0.648125\pi\)
\(488\) −5.79268e6 −1.10111
\(489\) 1.07546e7 2.03386
\(490\) −5.90279e6 −1.11062
\(491\) 4.40851e6 0.825255 0.412628 0.910900i \(-0.364611\pi\)
0.412628 + 0.910900i \(0.364611\pi\)
\(492\) 1.09213e7 2.03406
\(493\) −125641. −0.0232817
\(494\) 1.24112e7 2.28821
\(495\) −1.44288e7 −2.64677
\(496\) −634209. −0.115752
\(497\) −2.82343e6 −0.512726
\(498\) 2.65389e7 4.79523
\(499\) 3.52789e6 0.634255 0.317127 0.948383i \(-0.397282\pi\)
0.317127 + 0.948383i \(0.397282\pi\)
\(500\) 7.37573e6 1.31941
\(501\) −1.48151e7 −2.63700
\(502\) −1.07217e7 −1.89891
\(503\) 79054.9 0.0139319 0.00696593 0.999976i \(-0.497783\pi\)
0.00696593 + 0.999976i \(0.497783\pi\)
\(504\) −9.36894e6 −1.64291
\(505\) 486347. 0.0848629
\(506\) −2.38105e6 −0.413422
\(507\) −1.78089e7 −3.07693
\(508\) −7.49203e6 −1.28807
\(509\) 155909. 0.0266733 0.0133366 0.999911i \(-0.495755\pi\)
0.0133366 + 0.999911i \(0.495755\pi\)
\(510\) 1.98488e7 3.37916
\(511\) −3.69127e6 −0.625351
\(512\) 2.73974e6 0.461885
\(513\) 4.82754e6 0.809903
\(514\) −1.17929e7 −1.96885
\(515\) −8.12486e6 −1.34989
\(516\) 2.24051e6 0.370444
\(517\) −5.24159e6 −0.862456
\(518\) 6.94786e6 1.13770
\(519\) 4.87052e6 0.793702
\(520\) 9.28518e6 1.50585
\(521\) −4.24897e6 −0.685787 −0.342893 0.939374i \(-0.611407\pi\)
−0.342893 + 0.939374i \(0.611407\pi\)
\(522\) −302795. −0.0486377
\(523\) 1.30285e6 0.208276 0.104138 0.994563i \(-0.466792\pi\)
0.104138 + 0.994563i \(0.466792\pi\)
\(524\) 9.97890e6 1.58765
\(525\) 2.63307e6 0.416931
\(526\) 1.56437e7 2.46533
\(527\) 3.82176e6 0.599427
\(528\) 3.64594e6 0.569148
\(529\) −6.24516e6 −0.970296
\(530\) 1.55723e7 2.40804
\(531\) 1.86157e7 2.86512
\(532\) 1.06772e7 1.63560
\(533\) 9.36932e6 1.42853
\(534\) −9.34283e6 −1.41783
\(535\) −7.39449e6 −1.11692
\(536\) 2.80289e6 0.421399
\(537\) 1.08994e7 1.63104
\(538\) −4.21735e6 −0.628179
\(539\) 6.53376e6 0.968704
\(540\) 1.07169e7 1.58155
\(541\) 1.58581e6 0.232948 0.116474 0.993194i \(-0.462841\pi\)
0.116474 + 0.993194i \(0.462841\pi\)
\(542\) 263931. 0.0385915
\(543\) 1.57370e7 2.29045
\(544\) −9.79579e6 −1.41919
\(545\) −1.02783e6 −0.148227
\(546\) −3.88142e7 −5.57197
\(547\) −5.60569e6 −0.801052 −0.400526 0.916285i \(-0.631173\pi\)
−0.400526 + 0.916285i \(0.631173\pi\)
\(548\) −1.05793e7 −1.50489
\(549\) −1.53951e7 −2.17997
\(550\) 3.44053e6 0.484974
\(551\) 116292. 0.0163181
\(552\) 1.59968e6 0.223453
\(553\) −6.13393e6 −0.852954
\(554\) 9.06408e6 1.25473
\(555\) −7.18878e6 −0.990656
\(556\) 4.00834e6 0.549892
\(557\) −44346.9 −0.00605655 −0.00302828 0.999995i \(-0.500964\pi\)
−0.00302828 + 0.999995i \(0.500964\pi\)
\(558\) 9.21043e6 1.25226
\(559\) 1.92212e6 0.260166
\(560\) −2.43096e6 −0.327573
\(561\) −2.19705e7 −2.94736
\(562\) −668286. −0.0892528
\(563\) 4.30007e6 0.571748 0.285874 0.958267i \(-0.407716\pi\)
0.285874 + 0.958267i \(0.407716\pi\)
\(564\) 1.04495e7 1.38323
\(565\) 1.11563e6 0.147027
\(566\) −1.02841e7 −1.34935
\(567\) 525355. 0.0686271
\(568\) 2.47859e6 0.322354
\(569\) 6.54339e6 0.847271 0.423635 0.905833i \(-0.360754\pi\)
0.423635 + 0.905833i \(0.360754\pi\)
\(570\) −1.83718e7 −2.36845
\(571\) 8.65292e6 1.11064 0.555319 0.831637i \(-0.312596\pi\)
0.555319 + 0.831637i \(0.312596\pi\)
\(572\) −3.04973e7 −3.89737
\(573\) 2.73799e6 0.348374
\(574\) 1.34042e7 1.69809
\(575\) −276251. −0.0348445
\(576\) −2.06468e7 −2.59296
\(577\) 3.48407e6 0.435660 0.217830 0.975987i \(-0.430102\pi\)
0.217830 + 0.975987i \(0.430102\pi\)
\(578\) 5.85064e6 0.728423
\(579\) 792415. 0.0982328
\(580\) 258161. 0.0318655
\(581\) 1.95865e7 2.40723
\(582\) −7.33116e6 −0.897150
\(583\) −1.72369e7 −2.10033
\(584\) 3.24044e6 0.393162
\(585\) 2.46770e7 2.98128
\(586\) −1.17090e7 −1.40856
\(587\) 6.56208e6 0.786043 0.393021 0.919529i \(-0.371430\pi\)
0.393021 + 0.919529i \(0.371430\pi\)
\(588\) −1.30255e7 −1.55364
\(589\) −3.53737e6 −0.420138
\(590\) −2.63944e7 −3.12164
\(591\) 6.95032e6 0.818532
\(592\) 1.11618e6 0.130897
\(593\) 8.08785e6 0.944488 0.472244 0.881468i \(-0.343444\pi\)
0.472244 + 0.881468i \(0.343444\pi\)
\(594\) −1.97272e7 −2.29403
\(595\) 1.46490e7 1.69635
\(596\) −2.43950e7 −2.81310
\(597\) 1.95234e7 2.24192
\(598\) 4.07223e6 0.465671
\(599\) −9.16029e6 −1.04314 −0.521569 0.853209i \(-0.674653\pi\)
−0.521569 + 0.853209i \(0.674653\pi\)
\(600\) −2.31148e6 −0.262127
\(601\) −1.25891e7 −1.42171 −0.710853 0.703341i \(-0.751691\pi\)
−0.710853 + 0.703341i \(0.751691\pi\)
\(602\) 2.74988e6 0.309259
\(603\) 7.44917e6 0.834285
\(604\) −1.07252e7 −1.19623
\(605\) −1.27736e7 −1.41881
\(606\) 1.78474e6 0.197421
\(607\) −1.05315e7 −1.16017 −0.580083 0.814558i \(-0.696980\pi\)
−0.580083 + 0.814558i \(0.696980\pi\)
\(608\) 9.06686e6 0.994713
\(609\) −363685. −0.0397358
\(610\) 2.18281e7 2.37515
\(611\) 8.96450e6 0.971455
\(612\) 2.69134e7 2.90462
\(613\) 2.50815e6 0.269589 0.134794 0.990874i \(-0.456963\pi\)
0.134794 + 0.990874i \(0.456963\pi\)
\(614\) −1.35281e7 −1.44816
\(615\) −1.38690e7 −1.47863
\(616\) −1.47038e7 −1.56127
\(617\) 1.04940e7 1.10975 0.554876 0.831933i \(-0.312766\pi\)
0.554876 + 0.831933i \(0.312766\pi\)
\(618\) −2.98156e7 −3.14031
\(619\) −7.31973e6 −0.767836 −0.383918 0.923367i \(-0.625426\pi\)
−0.383918 + 0.923367i \(0.625426\pi\)
\(620\) −7.85276e6 −0.820433
\(621\) 1.58396e6 0.164822
\(622\) 1.52286e7 1.57828
\(623\) −6.89529e6 −0.711758
\(624\) −6.23552e6 −0.641078
\(625\) −1.13408e7 −1.16130
\(626\) 2.55505e7 2.60593
\(627\) 2.03356e7 2.06580
\(628\) 875446. 0.0885789
\(629\) −6.72610e6 −0.677855
\(630\) 3.53042e7 3.54384
\(631\) −6.30990e6 −0.630883 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(632\) 5.38476e6 0.536258
\(633\) 2.24662e6 0.222854
\(634\) −1.41915e6 −0.140219
\(635\) 9.51414e6 0.936344
\(636\) 3.43629e7 3.36858
\(637\) −1.11744e7 −1.09113
\(638\) −475213. −0.0462207
\(639\) 6.58729e6 0.638196
\(640\) 1.59295e7 1.53728
\(641\) 176304. 0.0169479 0.00847395 0.999964i \(-0.497303\pi\)
0.00847395 + 0.999964i \(0.497303\pi\)
\(642\) −2.71354e7 −2.59835
\(643\) −4.73829e6 −0.451954 −0.225977 0.974133i \(-0.572557\pi\)
−0.225977 + 0.974133i \(0.572557\pi\)
\(644\) 3.50327e6 0.332858
\(645\) −2.84523e6 −0.269289
\(646\) −1.71894e7 −1.62061
\(647\) −1.27894e7 −1.20113 −0.600563 0.799577i \(-0.705057\pi\)
−0.600563 + 0.799577i \(0.705057\pi\)
\(648\) −461191. −0.0431463
\(649\) 2.92158e7 2.72274
\(650\) −5.88420e6 −0.546266
\(651\) 1.10626e7 1.02307
\(652\) −2.06757e7 −1.90476
\(653\) 2.31597e6 0.212545 0.106272 0.994337i \(-0.466108\pi\)
0.106272 + 0.994337i \(0.466108\pi\)
\(654\) −3.77179e6 −0.344828
\(655\) −1.26722e7 −1.15412
\(656\) 2.15339e6 0.195373
\(657\) 8.61203e6 0.778381
\(658\) 1.28250e7 1.15477
\(659\) −4.32880e6 −0.388288 −0.194144 0.980973i \(-0.562193\pi\)
−0.194144 + 0.980973i \(0.562193\pi\)
\(660\) 4.51440e7 4.03404
\(661\) 1.34429e7 1.19671 0.598355 0.801231i \(-0.295821\pi\)
0.598355 + 0.801231i \(0.295821\pi\)
\(662\) 1.24287e7 1.10225
\(663\) 3.75754e7 3.31986
\(664\) −1.71943e7 −1.51344
\(665\) −1.35590e7 −1.18897
\(666\) −1.62099e7 −1.41610
\(667\) 38156.4 0.00332088
\(668\) 2.84821e7 2.46962
\(669\) 2.39077e7 2.06525
\(670\) −1.05619e7 −0.908980
\(671\) −2.41614e7 −2.07164
\(672\) −2.83552e7 −2.42220
\(673\) −1.30893e7 −1.11398 −0.556991 0.830518i \(-0.688044\pi\)
−0.556991 + 0.830518i \(0.688044\pi\)
\(674\) 5.40225e6 0.458063
\(675\) −2.28876e6 −0.193348
\(676\) 3.42377e7 2.88163
\(677\) 1.26746e6 0.106283 0.0531415 0.998587i \(-0.483077\pi\)
0.0531415 + 0.998587i \(0.483077\pi\)
\(678\) 4.09399e6 0.342037
\(679\) −5.41062e6 −0.450373
\(680\) −1.28599e7 −1.06651
\(681\) 1.16944e7 0.966298
\(682\) 1.44550e7 1.19003
\(683\) −1.77529e7 −1.45619 −0.728093 0.685479i \(-0.759593\pi\)
−0.728093 + 0.685479i \(0.759593\pi\)
\(684\) −2.49107e7 −2.03585
\(685\) 1.34346e7 1.09396
\(686\) 9.00905e6 0.730918
\(687\) −1.19864e7 −0.968944
\(688\) 441769. 0.0355815
\(689\) 2.94797e7 2.36578
\(690\) −6.02796e6 −0.482000
\(691\) 5.47133e6 0.435911 0.217955 0.975959i \(-0.430061\pi\)
0.217955 + 0.975959i \(0.430061\pi\)
\(692\) −9.36360e6 −0.743323
\(693\) −3.90780e7 −3.09100
\(694\) 1.24394e7 0.980396
\(695\) −5.09020e6 −0.399736
\(696\) 319266. 0.0249822
\(697\) −1.29764e7 −1.01175
\(698\) 1.70767e7 1.32668
\(699\) −1.86683e7 −1.44515
\(700\) −5.06209e6 −0.390467
\(701\) −1.16242e7 −0.893443 −0.446721 0.894673i \(-0.647408\pi\)
−0.446721 + 0.894673i \(0.647408\pi\)
\(702\) 3.37387e7 2.58396
\(703\) 6.22560e6 0.475108
\(704\) −3.24035e7 −2.46411
\(705\) −1.32698e7 −1.00552
\(706\) −2.27366e7 −1.71678
\(707\) 1.31719e6 0.0991060
\(708\) −5.82437e7 −4.36682
\(709\) −4.30447e6 −0.321591 −0.160795 0.986988i \(-0.551406\pi\)
−0.160795 + 0.986988i \(0.551406\pi\)
\(710\) −9.33985e6 −0.695335
\(711\) 1.43110e7 1.06168
\(712\) 6.05313e6 0.447487
\(713\) −1.16064e6 −0.0855017
\(714\) 5.37572e7 3.94631
\(715\) 3.87286e7 2.83313
\(716\) −2.09541e7 −1.52752
\(717\) −3.19130e7 −2.31830
\(718\) −5.27883e6 −0.382143
\(719\) 1.35107e7 0.974664 0.487332 0.873217i \(-0.337970\pi\)
0.487332 + 0.873217i \(0.337970\pi\)
\(720\) 5.67163e6 0.407734
\(721\) −2.20048e7 −1.57645
\(722\) −6.27338e6 −0.447877
\(723\) 2.19915e7 1.56462
\(724\) −3.02544e7 −2.14507
\(725\) −55134.4 −0.00389563
\(726\) −4.68749e7 −3.30064
\(727\) 2.62118e7 1.83934 0.919668 0.392697i \(-0.128458\pi\)
0.919668 + 0.392697i \(0.128458\pi\)
\(728\) 2.51474e7 1.75859
\(729\) −2.33260e7 −1.62563
\(730\) −1.22107e7 −0.848070
\(731\) −2.66211e6 −0.184260
\(732\) 4.81672e7 3.32257
\(733\) −2.19136e7 −1.50644 −0.753222 0.657766i \(-0.771501\pi\)
−0.753222 + 0.657766i \(0.771501\pi\)
\(734\) 1.50220e7 1.02917
\(735\) 1.65411e7 1.12939
\(736\) 2.97492e6 0.202433
\(737\) 1.16909e7 0.792827
\(738\) −3.12731e7 −2.11364
\(739\) −2.72080e6 −0.183267 −0.0916337 0.995793i \(-0.529209\pi\)
−0.0916337 + 0.995793i \(0.529209\pi\)
\(740\) 1.38205e7 0.927776
\(741\) −3.47793e7 −2.32689
\(742\) 4.21751e7 2.81220
\(743\) −5.79913e6 −0.385381 −0.192691 0.981260i \(-0.561721\pi\)
−0.192691 + 0.981260i \(0.561721\pi\)
\(744\) −9.71146e6 −0.643209
\(745\) 3.09792e7 2.04494
\(746\) 2.80120e7 1.84288
\(747\) −4.56969e7 −2.99630
\(748\) 4.22384e7 2.76028
\(749\) −2.00267e7 −1.30438
\(750\) −3.43718e7 −2.23126
\(751\) −2.22708e6 −0.144091 −0.0720455 0.997401i \(-0.522953\pi\)
−0.0720455 + 0.997401i \(0.522953\pi\)
\(752\) 2.06035e6 0.132861
\(753\) 3.00449e7 1.93100
\(754\) 812739. 0.0520622
\(755\) 1.36200e7 0.869579
\(756\) 2.90248e7 1.84699
\(757\) 1.68706e7 1.07002 0.535009 0.844846i \(-0.320308\pi\)
0.535009 + 0.844846i \(0.320308\pi\)
\(758\) −1.77776e7 −1.12383
\(759\) 6.67231e6 0.420409
\(760\) 1.19029e7 0.747515
\(761\) −6.73683e6 −0.421691 −0.210845 0.977519i \(-0.567622\pi\)
−0.210845 + 0.977519i \(0.567622\pi\)
\(762\) 3.49138e7 2.17826
\(763\) −2.78370e6 −0.173105
\(764\) −5.26380e6 −0.326261
\(765\) −3.41774e7 −2.11147
\(766\) 2.88316e6 0.177540
\(767\) −4.99668e7 −3.06685
\(768\) 1.56276e7 0.956071
\(769\) 2.99965e7 1.82917 0.914587 0.404390i \(-0.132516\pi\)
0.914587 + 0.404390i \(0.132516\pi\)
\(770\) 5.54071e7 3.36774
\(771\) 3.30466e7 2.00212
\(772\) −1.52342e6 −0.0919977
\(773\) 2.03964e7 1.22773 0.613867 0.789409i \(-0.289613\pi\)
0.613867 + 0.789409i \(0.289613\pi\)
\(774\) −6.41568e6 −0.384938
\(775\) 1.67708e6 0.100300
\(776\) 4.74979e6 0.283152
\(777\) −1.94696e7 −1.15692
\(778\) −1.20558e7 −0.714078
\(779\) 1.20108e7 0.709133
\(780\) −7.72080e7 −4.54387
\(781\) 1.03382e7 0.606482
\(782\) −5.63999e6 −0.329808
\(783\) 316128. 0.0184272
\(784\) −2.56827e6 −0.149228
\(785\) −1.11173e6 −0.0643911
\(786\) −4.65029e7 −2.68488
\(787\) −9.96285e6 −0.573386 −0.286693 0.958023i \(-0.592556\pi\)
−0.286693 + 0.958023i \(0.592556\pi\)
\(788\) −1.33620e7 −0.766578
\(789\) −4.38375e7 −2.50699
\(790\) −2.02909e7 −1.15674
\(791\) 3.02149e6 0.171704
\(792\) 3.43052e7 1.94333
\(793\) 4.13223e7 2.33346
\(794\) 1.25695e7 0.707567
\(795\) −4.36375e7 −2.44874
\(796\) −3.75338e7 −2.09961
\(797\) −4.33007e6 −0.241462 −0.120731 0.992685i \(-0.538524\pi\)
−0.120731 + 0.992685i \(0.538524\pi\)
\(798\) −4.97570e7 −2.76597
\(799\) −1.24157e7 −0.688026
\(800\) −4.29863e6 −0.237468
\(801\) 1.60873e7 0.885933
\(802\) 3.08425e7 1.69322
\(803\) 1.35159e7 0.739701
\(804\) −2.33065e7 −1.27156
\(805\) −4.44881e6 −0.241966
\(806\) −2.47219e7 −1.34043
\(807\) 1.18181e7 0.638796
\(808\) −1.15631e6 −0.0623085
\(809\) 4.65045e6 0.249818 0.124909 0.992168i \(-0.460136\pi\)
0.124909 + 0.992168i \(0.460136\pi\)
\(810\) 1.73787e6 0.0930687
\(811\) −2.23521e7 −1.19334 −0.596672 0.802486i \(-0.703510\pi\)
−0.596672 + 0.802486i \(0.703510\pi\)
\(812\) 699186. 0.0372137
\(813\) −739599. −0.0392437
\(814\) −2.54402e7 −1.34573
\(815\) 2.62561e7 1.38464
\(816\) 8.63611e6 0.454039
\(817\) 2.46401e6 0.129148
\(818\) 5.77517e7 3.01774
\(819\) 6.68336e7 3.48165
\(820\) 2.66633e7 1.38477
\(821\) 6.86450e6 0.355427 0.177714 0.984082i \(-0.443130\pi\)
0.177714 + 0.984082i \(0.443130\pi\)
\(822\) 4.93008e7 2.54492
\(823\) −3.78159e7 −1.94614 −0.973072 0.230502i \(-0.925963\pi\)
−0.973072 + 0.230502i \(0.925963\pi\)
\(824\) 1.93173e7 0.991123
\(825\) −9.64121e6 −0.493170
\(826\) −7.14849e7 −3.64556
\(827\) 3.11812e7 1.58536 0.792681 0.609637i \(-0.208685\pi\)
0.792681 + 0.609637i \(0.208685\pi\)
\(828\) −8.17342e6 −0.414313
\(829\) −7.56161e6 −0.382145 −0.191072 0.981576i \(-0.561196\pi\)
−0.191072 + 0.981576i \(0.561196\pi\)
\(830\) 6.47918e7 3.26456
\(831\) −2.53998e7 −1.27593
\(832\) 5.54185e7 2.77553
\(833\) 1.54765e7 0.772786
\(834\) −1.86794e7 −0.929924
\(835\) −3.61695e7 −1.79525
\(836\) −3.90954e7 −1.93468
\(837\) −9.61600e6 −0.474439
\(838\) −1.59351e7 −0.783873
\(839\) 1.20663e7 0.591792 0.295896 0.955220i \(-0.404382\pi\)
0.295896 + 0.955220i \(0.404382\pi\)
\(840\) −3.72246e7 −1.82025
\(841\) −2.05035e7 −0.999629
\(842\) −3.02062e7 −1.46831
\(843\) 1.87270e6 0.0907612
\(844\) −4.31914e6 −0.208709
\(845\) −4.34785e7 −2.09475
\(846\) −2.99219e7 −1.43735
\(847\) −3.45951e7 −1.65694
\(848\) 6.77544e6 0.323555
\(849\) 2.88186e7 1.37216
\(850\) 8.14955e6 0.386889
\(851\) 2.04267e6 0.0966886
\(852\) −2.06099e7 −0.972696
\(853\) 2.73307e7 1.28611 0.643055 0.765820i \(-0.277667\pi\)
0.643055 + 0.765820i \(0.277667\pi\)
\(854\) 5.91177e7 2.77378
\(855\) 3.16341e7 1.47993
\(856\) 1.75808e7 0.820074
\(857\) −3.10213e7 −1.44280 −0.721402 0.692516i \(-0.756502\pi\)
−0.721402 + 0.692516i \(0.756502\pi\)
\(858\) 1.42121e8 6.59085
\(859\) 2.98878e7 1.38201 0.691006 0.722849i \(-0.257168\pi\)
0.691006 + 0.722849i \(0.257168\pi\)
\(860\) 5.46997e6 0.252196
\(861\) −3.75619e7 −1.72679
\(862\) 1.24213e7 0.569375
\(863\) −2.37019e7 −1.08332 −0.541660 0.840598i \(-0.682204\pi\)
−0.541660 + 0.840598i \(0.682204\pi\)
\(864\) 2.46474e7 1.12328
\(865\) 1.18909e7 0.540347
\(866\) 5.14733e7 2.33232
\(867\) −1.63949e7 −0.740734
\(868\) −2.12679e7 −0.958131
\(869\) 2.24599e7 1.00892
\(870\) −1.20306e6 −0.0538878
\(871\) −1.99945e7 −0.893027
\(872\) 2.44371e6 0.108832
\(873\) 1.26234e7 0.560584
\(874\) 5.22030e6 0.231162
\(875\) −2.53675e7 −1.12010
\(876\) −2.69448e7 −1.18636
\(877\) 3.85645e7 1.69312 0.846562 0.532290i \(-0.178668\pi\)
0.846562 + 0.532290i \(0.178668\pi\)
\(878\) −4.60932e7 −2.01790
\(879\) 3.28115e7 1.43237
\(880\) 8.90117e6 0.387472
\(881\) −3.45951e7 −1.50167 −0.750836 0.660489i \(-0.770349\pi\)
−0.750836 + 0.660489i \(0.770349\pi\)
\(882\) 3.72983e7 1.61442
\(883\) −8.71550e6 −0.376175 −0.188088 0.982152i \(-0.560229\pi\)
−0.188088 + 0.982152i \(0.560229\pi\)
\(884\) −7.22388e7 −3.10914
\(885\) 7.39638e7 3.17439
\(886\) −5.06947e7 −2.16959
\(887\) 4.58909e6 0.195847 0.0979237 0.995194i \(-0.468780\pi\)
0.0979237 + 0.995194i \(0.468780\pi\)
\(888\) 1.70917e7 0.727365
\(889\) 2.57675e7 1.09350
\(890\) −2.28095e7 −0.965252
\(891\) −1.92363e6 −0.0811761
\(892\) −4.59627e7 −1.93416
\(893\) 1.14918e7 0.482237
\(894\) 1.13684e8 4.75723
\(895\) 2.66096e7 1.11040
\(896\) 4.31424e7 1.79529
\(897\) −1.14114e7 −0.473541
\(898\) −8.56815e6 −0.354565
\(899\) −231642. −0.00955912
\(900\) 1.18103e7 0.486019
\(901\) −4.08290e7 −1.67555
\(902\) −4.90807e7 −2.00860
\(903\) −7.70583e6 −0.314485
\(904\) −2.65246e6 −0.107951
\(905\) 3.84201e7 1.55933
\(906\) 4.99809e7 2.02294
\(907\) 1.10880e7 0.447544 0.223772 0.974642i \(-0.428163\pi\)
0.223772 + 0.974642i \(0.428163\pi\)
\(908\) −2.24826e7 −0.904965
\(909\) −3.07311e6 −0.123358
\(910\) −9.47606e7 −3.79336
\(911\) 3.54367e7 1.41468 0.707339 0.706875i \(-0.249896\pi\)
0.707339 + 0.706875i \(0.249896\pi\)
\(912\) −7.99348e6 −0.318236
\(913\) −7.17177e7 −2.84740
\(914\) −4.93377e7 −1.95350
\(915\) −6.11677e7 −2.41529
\(916\) 2.30440e7 0.907442
\(917\) −3.43206e7 −1.34782
\(918\) −4.67276e7 −1.83007
\(919\) 3.32990e7 1.30060 0.650298 0.759679i \(-0.274644\pi\)
0.650298 + 0.759679i \(0.274644\pi\)
\(920\) 3.90546e6 0.152126
\(921\) 3.79092e7 1.47264
\(922\) 1.88323e7 0.729584
\(923\) −1.76811e7 −0.683131
\(924\) 1.22265e8 4.71109
\(925\) −2.95158e6 −0.113423
\(926\) 2.11532e7 0.810677
\(927\) 5.13390e7 1.96222
\(928\) 593736. 0.0226320
\(929\) 3.43816e7 1.30703 0.653517 0.756912i \(-0.273293\pi\)
0.653517 + 0.756912i \(0.273293\pi\)
\(930\) 3.65949e7 1.38743
\(931\) −1.43248e7 −0.541645
\(932\) 3.58899e7 1.35342
\(933\) −4.26743e7 −1.60495
\(934\) −7.67335e7 −2.87818
\(935\) −5.36386e7 −2.00654
\(936\) −5.86708e7 −2.18893
\(937\) 3.81989e7 1.42135 0.710677 0.703518i \(-0.248389\pi\)
0.710677 + 0.703518i \(0.248389\pi\)
\(938\) −2.86051e7 −1.06154
\(939\) −7.15988e7 −2.64997
\(940\) 2.55112e7 0.941697
\(941\) 2.92661e7 1.07744 0.538718 0.842486i \(-0.318909\pi\)
0.538718 + 0.842486i \(0.318909\pi\)
\(942\) −4.07969e6 −0.149796
\(943\) 3.94085e6 0.144315
\(944\) −1.14841e7 −0.419437
\(945\) −3.68587e7 −1.34264
\(946\) −1.00689e7 −0.365809
\(947\) 4.49447e6 0.162856 0.0814280 0.996679i \(-0.474052\pi\)
0.0814280 + 0.996679i \(0.474052\pi\)
\(948\) −4.47753e7 −1.61814
\(949\) −2.31157e7 −0.833186
\(950\) −7.54312e6 −0.271170
\(951\) 3.97681e6 0.142588
\(952\) −3.48288e7 −1.24551
\(953\) −7.96508e6 −0.284091 −0.142046 0.989860i \(-0.545368\pi\)
−0.142046 + 0.989860i \(0.545368\pi\)
\(954\) −9.83978e7 −3.50038
\(955\) 6.68451e6 0.237171
\(956\) 6.13529e7 2.17115
\(957\) 1.33166e6 0.0470018
\(958\) 2.77347e7 0.976360
\(959\) 3.63855e7 1.27756
\(960\) −8.20337e7 −2.87286
\(961\) −2.15831e7 −0.753884
\(962\) 4.35094e7 1.51581
\(963\) 4.67240e7 1.62358
\(964\) −4.22788e7 −1.46531
\(965\) 1.93460e6 0.0668763
\(966\) −1.63257e7 −0.562897
\(967\) 3.91114e6 0.134505 0.0672523 0.997736i \(-0.478577\pi\)
0.0672523 + 0.997736i \(0.478577\pi\)
\(968\) 3.03698e7 1.04173
\(969\) 4.81689e7 1.64800
\(970\) −1.78982e7 −0.610774
\(971\) −2.98964e7 −1.01759 −0.508793 0.860889i \(-0.669908\pi\)
−0.508793 + 0.860889i \(0.669908\pi\)
\(972\) 4.63228e7 1.57264
\(973\) −1.37860e7 −0.466826
\(974\) −4.20831e7 −1.42138
\(975\) 1.64890e7 0.555498
\(976\) 9.49728e6 0.319135
\(977\) −4.36187e7 −1.46196 −0.730981 0.682398i \(-0.760937\pi\)
−0.730981 + 0.682398i \(0.760937\pi\)
\(978\) 9.63513e7 3.22115
\(979\) 2.52477e7 0.841909
\(980\) −3.18003e7 −1.05771
\(981\) 6.49459e6 0.215466
\(982\) 3.94964e7 1.30701
\(983\) −1.95377e7 −0.644895 −0.322448 0.946587i \(-0.604506\pi\)
−0.322448 + 0.946587i \(0.604506\pi\)
\(984\) 3.29743e7 1.08564
\(985\) 1.69684e7 0.557252
\(986\) −1.12563e6 −0.0368727
\(987\) −3.59390e7 −1.17428
\(988\) 6.68633e7 2.17919
\(989\) 808465. 0.0262827
\(990\) −1.29269e8 −4.19186
\(991\) 3.45462e7 1.11742 0.558709 0.829364i \(-0.311297\pi\)
0.558709 + 0.829364i \(0.311297\pi\)
\(992\) −1.80603e7 −0.582701
\(993\) −3.48282e7 −1.12088
\(994\) −2.52954e7 −0.812037
\(995\) 4.76642e7 1.52628
\(996\) 1.42974e8 4.56676
\(997\) 3.34457e7 1.06562 0.532810 0.846235i \(-0.321136\pi\)
0.532810 + 0.846235i \(0.321136\pi\)
\(998\) 3.16068e7 1.00451
\(999\) 1.69237e7 0.536514
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 43.6.a.a.1.8 8
3.2 odd 2 387.6.a.c.1.1 8
4.3 odd 2 688.6.a.e.1.7 8
5.4 even 2 1075.6.a.a.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.8 8 1.1 even 1 trivial
387.6.a.c.1.1 8 3.2 odd 2
688.6.a.e.1.7 8 4.3 odd 2
1075.6.a.a.1.1 8 5.4 even 2