Properties

Label 43.6.a.b.1.4
Level 43
Weight 6
Character 43.1
Self dual yes
Analytic conductor 6.897
Analytic rank 0
Dimension 10
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.50018\) of \(x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.50018 q^{2} +16.8892 q^{3} -25.7491 q^{4} +47.4635 q^{5} -42.2260 q^{6} +67.4603 q^{7} +144.383 q^{8} +42.2439 q^{9} +O(q^{10})\) \(q-2.50018 q^{2} +16.8892 q^{3} -25.7491 q^{4} +47.4635 q^{5} -42.2260 q^{6} +67.4603 q^{7} +144.383 q^{8} +42.2439 q^{9} -118.667 q^{10} +81.3987 q^{11} -434.881 q^{12} +1058.72 q^{13} -168.663 q^{14} +801.619 q^{15} +462.986 q^{16} +251.378 q^{17} -105.617 q^{18} +1612.95 q^{19} -1222.14 q^{20} +1139.35 q^{21} -203.512 q^{22} -32.7403 q^{23} +2438.51 q^{24} -872.217 q^{25} -2647.00 q^{26} -3390.60 q^{27} -1737.04 q^{28} -2583.75 q^{29} -2004.19 q^{30} -7206.51 q^{31} -5777.81 q^{32} +1374.76 q^{33} -628.491 q^{34} +3201.90 q^{35} -1087.74 q^{36} -6174.39 q^{37} -4032.66 q^{38} +17881.0 q^{39} +6852.93 q^{40} +15514.4 q^{41} -2848.58 q^{42} +1849.00 q^{43} -2095.94 q^{44} +2005.04 q^{45} +81.8569 q^{46} -1692.39 q^{47} +7819.45 q^{48} -12256.1 q^{49} +2180.70 q^{50} +4245.56 q^{51} -27261.2 q^{52} -25612.3 q^{53} +8477.13 q^{54} +3863.47 q^{55} +9740.14 q^{56} +27241.3 q^{57} +6459.83 q^{58} +24532.0 q^{59} -20641.0 q^{60} +8209.11 q^{61} +18017.6 q^{62} +2849.78 q^{63} -369.967 q^{64} +50250.7 q^{65} -3437.14 q^{66} -12302.3 q^{67} -6472.75 q^{68} -552.957 q^{69} -8005.34 q^{70} +18712.0 q^{71} +6099.31 q^{72} -12126.5 q^{73} +15437.1 q^{74} -14731.0 q^{75} -41531.9 q^{76} +5491.18 q^{77} -44705.7 q^{78} -52372.8 q^{79} +21974.9 q^{80} -67529.7 q^{81} -38788.9 q^{82} +28935.9 q^{83} -29337.2 q^{84} +11931.3 q^{85} -4622.84 q^{86} -43637.3 q^{87} +11752.6 q^{88} +117315. q^{89} -5012.97 q^{90} +71421.8 q^{91} +843.034 q^{92} -121712. q^{93} +4231.28 q^{94} +76556.1 q^{95} -97582.5 q^{96} -147143. q^{97} +30642.5 q^{98} +3438.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} + O(q^{10}) \) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} - 17q^{10} + 745q^{11} + 4627q^{12} + 1917q^{13} + 1936q^{14} + 1688q^{15} + 5354q^{16} + 4017q^{17} - 2725q^{18} - 2404q^{19} + 1311q^{20} - 228q^{21} - 5836q^{22} + 1733q^{23} - 10711q^{24} + 7120q^{25} - 1484q^{26} - 2324q^{27} - 15028q^{28} + 6996q^{29} - 48420q^{30} - 4899q^{31} - 7554q^{32} - 15734q^{33} - 27033q^{34} + 7084q^{35} + 4433q^{36} + 1466q^{37} + 13905q^{38} - 26542q^{39} - 93211q^{40} + 10297q^{41} - 37642q^{42} + 18490q^{43} - 36140q^{44} + 73822q^{45} + 17991q^{46} + 48592q^{47} + 83607q^{48} + 29458q^{49} + 983q^{50} + 92972q^{51} + 14232q^{52} + 127165q^{53} - 92002q^{54} + 106672q^{55} - 7780q^{56} + 34060q^{57} - 10305q^{58} + 99372q^{59} + 111372q^{60} + 17408q^{61} + 28265q^{62} + 2244q^{63} + 47202q^{64} + 54484q^{65} - 150292q^{66} - 2021q^{67} + 192151q^{68} + 1654q^{69} - 33194q^{70} + 11286q^{71} - 298365q^{72} + 49892q^{73} - 125431q^{74} - 44662q^{75} - 249803q^{76} + 98144q^{77} - 28494q^{78} - 91524q^{79} + 12251q^{80} - 26450q^{81} - 158909q^{82} - 105203q^{83} - 357682q^{84} - 87212q^{85} + 14792q^{86} + 181200q^{87} - 461824q^{88} - 62682q^{89} - 522670q^{90} - 295304q^{91} + 183783q^{92} - 238430q^{93} + 7259q^{94} - 305340q^{95} - 162399q^{96} + 108383q^{97} + 354656q^{98} - 270499q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50018 −0.441974 −0.220987 0.975277i \(-0.570928\pi\)
−0.220987 + 0.975277i \(0.570928\pi\)
\(3\) 16.8892 1.08344 0.541720 0.840559i \(-0.317773\pi\)
0.541720 + 0.840559i \(0.317773\pi\)
\(4\) −25.7491 −0.804659
\(5\) 47.4635 0.849053 0.424526 0.905416i \(-0.360441\pi\)
0.424526 + 0.905416i \(0.360441\pi\)
\(6\) −42.2260 −0.478853
\(7\) 67.4603 0.520359 0.260180 0.965560i \(-0.416218\pi\)
0.260180 + 0.965560i \(0.416218\pi\)
\(8\) 144.383 0.797612
\(9\) 42.2439 0.173843
\(10\) −118.667 −0.375259
\(11\) 81.3987 0.202832 0.101416 0.994844i \(-0.467663\pi\)
0.101416 + 0.994844i \(0.467663\pi\)
\(12\) −434.881 −0.871800
\(13\) 1058.72 1.73750 0.868749 0.495253i \(-0.164925\pi\)
0.868749 + 0.495253i \(0.164925\pi\)
\(14\) −168.663 −0.229985
\(15\) 801.619 0.919898
\(16\) 462.986 0.452135
\(17\) 251.378 0.210962 0.105481 0.994421i \(-0.466362\pi\)
0.105481 + 0.994421i \(0.466362\pi\)
\(18\) −105.617 −0.0768341
\(19\) 1612.95 1.02503 0.512515 0.858679i \(-0.328714\pi\)
0.512515 + 0.858679i \(0.328714\pi\)
\(20\) −1222.14 −0.683198
\(21\) 1139.35 0.563778
\(22\) −203.512 −0.0896463
\(23\) −32.7403 −0.0129052 −0.00645258 0.999979i \(-0.502054\pi\)
−0.00645258 + 0.999979i \(0.502054\pi\)
\(24\) 2438.51 0.864165
\(25\) −872.217 −0.279109
\(26\) −2647.00 −0.767929
\(27\) −3390.60 −0.895092
\(28\) −1737.04 −0.418712
\(29\) −2583.75 −0.570499 −0.285249 0.958453i \(-0.592076\pi\)
−0.285249 + 0.958453i \(0.592076\pi\)
\(30\) −2004.19 −0.406571
\(31\) −7206.51 −1.34685 −0.673427 0.739254i \(-0.735179\pi\)
−0.673427 + 0.739254i \(0.735179\pi\)
\(32\) −5777.81 −0.997444
\(33\) 1374.76 0.219756
\(34\) −628.491 −0.0932399
\(35\) 3201.90 0.441812
\(36\) −1087.74 −0.139884
\(37\) −6174.39 −0.741463 −0.370731 0.928740i \(-0.620893\pi\)
−0.370731 + 0.928740i \(0.620893\pi\)
\(38\) −4032.66 −0.453036
\(39\) 17881.0 1.88248
\(40\) 6852.93 0.677215
\(41\) 15514.4 1.44137 0.720685 0.693262i \(-0.243827\pi\)
0.720685 + 0.693262i \(0.243827\pi\)
\(42\) −2848.58 −0.249175
\(43\) 1849.00 0.152499
\(44\) −2095.94 −0.163210
\(45\) 2005.04 0.147602
\(46\) 81.8569 0.00570375
\(47\) −1692.39 −0.111752 −0.0558760 0.998438i \(-0.517795\pi\)
−0.0558760 + 0.998438i \(0.517795\pi\)
\(48\) 7819.45 0.489861
\(49\) −12256.1 −0.729226
\(50\) 2180.70 0.123359
\(51\) 4245.56 0.228565
\(52\) −27261.2 −1.39809
\(53\) −25612.3 −1.25245 −0.626223 0.779644i \(-0.715400\pi\)
−0.626223 + 0.779644i \(0.715400\pi\)
\(54\) 8477.13 0.395607
\(55\) 3863.47 0.172215
\(56\) 9740.14 0.415045
\(57\) 27241.3 1.11056
\(58\) 6459.83 0.252146
\(59\) 24532.0 0.917495 0.458748 0.888567i \(-0.348298\pi\)
0.458748 + 0.888567i \(0.348298\pi\)
\(60\) −20641.0 −0.740204
\(61\) 8209.11 0.282470 0.141235 0.989976i \(-0.454893\pi\)
0.141235 + 0.989976i \(0.454893\pi\)
\(62\) 18017.6 0.595275
\(63\) 2849.78 0.0904608
\(64\) −369.967 −0.0112905
\(65\) 50250.7 1.47523
\(66\) −3437.14 −0.0971264
\(67\) −12302.3 −0.334810 −0.167405 0.985888i \(-0.553539\pi\)
−0.167405 + 0.985888i \(0.553539\pi\)
\(68\) −6472.75 −0.169753
\(69\) −552.957 −0.0139820
\(70\) −8005.34 −0.195270
\(71\) 18712.0 0.440530 0.220265 0.975440i \(-0.429308\pi\)
0.220265 + 0.975440i \(0.429308\pi\)
\(72\) 6099.31 0.138659
\(73\) −12126.5 −0.266334 −0.133167 0.991094i \(-0.542515\pi\)
−0.133167 + 0.991094i \(0.542515\pi\)
\(74\) 15437.1 0.327707
\(75\) −14731.0 −0.302398
\(76\) −41531.9 −0.824799
\(77\) 5491.18 0.105545
\(78\) −44705.7 −0.832005
\(79\) −52372.8 −0.944144 −0.472072 0.881560i \(-0.656494\pi\)
−0.472072 + 0.881560i \(0.656494\pi\)
\(80\) 21974.9 0.383886
\(81\) −67529.7 −1.14362
\(82\) −38788.9 −0.637048
\(83\) 28935.9 0.461043 0.230521 0.973067i \(-0.425957\pi\)
0.230521 + 0.973067i \(0.425957\pi\)
\(84\) −29337.2 −0.453649
\(85\) 11931.3 0.179118
\(86\) −4622.84 −0.0674004
\(87\) −43637.3 −0.618101
\(88\) 11752.6 0.161781
\(89\) 117315. 1.56992 0.784962 0.619544i \(-0.212683\pi\)
0.784962 + 0.619544i \(0.212683\pi\)
\(90\) −5012.97 −0.0652362
\(91\) 71421.8 0.904123
\(92\) 843.034 0.0103843
\(93\) −121712. −1.45924
\(94\) 4231.28 0.0493915
\(95\) 76556.1 0.870304
\(96\) −97582.5 −1.08067
\(97\) −147143. −1.58785 −0.793925 0.608016i \(-0.791966\pi\)
−0.793925 + 0.608016i \(0.791966\pi\)
\(98\) 30642.5 0.322299
\(99\) 3438.60 0.0352609
\(100\) 22458.8 0.224588
\(101\) −118561. −1.15648 −0.578239 0.815867i \(-0.696260\pi\)
−0.578239 + 0.815867i \(0.696260\pi\)
\(102\) −10614.7 −0.101020
\(103\) 49153.5 0.456522 0.228261 0.973600i \(-0.426696\pi\)
0.228261 + 0.973600i \(0.426696\pi\)
\(104\) 152862. 1.38585
\(105\) 54077.4 0.478677
\(106\) 64035.4 0.553549
\(107\) −156632. −1.32258 −0.661290 0.750130i \(-0.729991\pi\)
−0.661290 + 0.750130i \(0.729991\pi\)
\(108\) 87304.9 0.720244
\(109\) 208919. 1.68427 0.842135 0.539267i \(-0.181299\pi\)
0.842135 + 0.539267i \(0.181299\pi\)
\(110\) −9659.37 −0.0761144
\(111\) −104280. −0.803331
\(112\) 31233.2 0.235273
\(113\) 33491.2 0.246737 0.123369 0.992361i \(-0.460630\pi\)
0.123369 + 0.992361i \(0.460630\pi\)
\(114\) −68108.3 −0.490838
\(115\) −1553.97 −0.0109572
\(116\) 66529.1 0.459057
\(117\) 44724.6 0.302052
\(118\) −61334.6 −0.405509
\(119\) 16958.0 0.109776
\(120\) 115740. 0.733722
\(121\) −154425. −0.958859
\(122\) −20524.3 −0.124844
\(123\) 262025. 1.56164
\(124\) 185561. 1.08376
\(125\) −189722. −1.08603
\(126\) −7124.98 −0.0399813
\(127\) 72430.2 0.398483 0.199242 0.979950i \(-0.436152\pi\)
0.199242 + 0.979950i \(0.436152\pi\)
\(128\) 185815. 1.00243
\(129\) 31228.1 0.165223
\(130\) −125636. −0.652012
\(131\) −73042.1 −0.371873 −0.185937 0.982562i \(-0.559532\pi\)
−0.185937 + 0.982562i \(0.559532\pi\)
\(132\) −35398.7 −0.176829
\(133\) 108810. 0.533383
\(134\) 30757.9 0.147977
\(135\) −160930. −0.759980
\(136\) 36294.8 0.168266
\(137\) 290889. 1.32412 0.662058 0.749453i \(-0.269683\pi\)
0.662058 + 0.749453i \(0.269683\pi\)
\(138\) 1382.49 0.00617967
\(139\) −375146. −1.64689 −0.823443 0.567399i \(-0.807950\pi\)
−0.823443 + 0.567399i \(0.807950\pi\)
\(140\) −82446.0 −0.355508
\(141\) −28583.0 −0.121077
\(142\) −46783.5 −0.194703
\(143\) 86178.7 0.352420
\(144\) 19558.3 0.0786005
\(145\) −122634. −0.484384
\(146\) 30318.3 0.117713
\(147\) −206995. −0.790073
\(148\) 158985. 0.596625
\(149\) 431136. 1.59092 0.795460 0.606006i \(-0.207229\pi\)
0.795460 + 0.606006i \(0.207229\pi\)
\(150\) 36830.2 0.133652
\(151\) 17252.4 0.0615755 0.0307878 0.999526i \(-0.490198\pi\)
0.0307878 + 0.999526i \(0.490198\pi\)
\(152\) 232883. 0.817576
\(153\) 10619.2 0.0366743
\(154\) −13729.0 −0.0466483
\(155\) −342046. −1.14355
\(156\) −460418. −1.51475
\(157\) −139497. −0.451663 −0.225832 0.974166i \(-0.572510\pi\)
−0.225832 + 0.974166i \(0.572510\pi\)
\(158\) 130942. 0.417287
\(159\) −432570. −1.35695
\(160\) −274235. −0.846883
\(161\) −2208.67 −0.00671532
\(162\) 168837. 0.505451
\(163\) −556197. −1.63968 −0.819841 0.572591i \(-0.805938\pi\)
−0.819841 + 0.572591i \(0.805938\pi\)
\(164\) −399482. −1.15981
\(165\) 65250.7 0.186584
\(166\) −72345.0 −0.203769
\(167\) −239206. −0.663713 −0.331857 0.943330i \(-0.607675\pi\)
−0.331857 + 0.943330i \(0.607675\pi\)
\(168\) 164503. 0.449676
\(169\) 749603. 2.01890
\(170\) −29830.4 −0.0791656
\(171\) 68137.1 0.178194
\(172\) −47610.1 −0.122709
\(173\) 79563.7 0.202116 0.101058 0.994881i \(-0.467777\pi\)
0.101058 + 0.994881i \(0.467777\pi\)
\(174\) 109101. 0.273185
\(175\) −58840.0 −0.145237
\(176\) 37686.5 0.0917073
\(177\) 414326. 0.994051
\(178\) −293309. −0.693865
\(179\) −95838.9 −0.223568 −0.111784 0.993733i \(-0.535656\pi\)
−0.111784 + 0.993733i \(0.535656\pi\)
\(180\) −51628.0 −0.118769
\(181\) −219413. −0.497813 −0.248906 0.968528i \(-0.580071\pi\)
−0.248906 + 0.968528i \(0.580071\pi\)
\(182\) −178568. −0.399599
\(183\) 138645. 0.306039
\(184\) −4727.16 −0.0102933
\(185\) −293058. −0.629541
\(186\) 304302. 0.644945
\(187\) 20461.8 0.0427898
\(188\) 43577.5 0.0899223
\(189\) −228731. −0.465769
\(190\) −191404. −0.384652
\(191\) 328729. 0.652011 0.326005 0.945368i \(-0.394297\pi\)
0.326005 + 0.945368i \(0.394297\pi\)
\(192\) −6248.43 −0.0122326
\(193\) 575743. 1.11259 0.556295 0.830985i \(-0.312222\pi\)
0.556295 + 0.830985i \(0.312222\pi\)
\(194\) 367884. 0.701788
\(195\) 848693. 1.59832
\(196\) 315584. 0.586779
\(197\) 962886. 1.76770 0.883852 0.467768i \(-0.154942\pi\)
0.883852 + 0.467768i \(0.154942\pi\)
\(198\) −8597.12 −0.0155844
\(199\) −760147. −1.36071 −0.680354 0.732884i \(-0.738174\pi\)
−0.680354 + 0.732884i \(0.738174\pi\)
\(200\) −125933. −0.222621
\(201\) −207775. −0.362747
\(202\) 296424. 0.511133
\(203\) −174300. −0.296864
\(204\) −109319. −0.183917
\(205\) 736368. 1.22380
\(206\) −122893. −0.201771
\(207\) −1383.08 −0.00224347
\(208\) 490175. 0.785584
\(209\) 131292. 0.207908
\(210\) −135203. −0.211563
\(211\) −979789. −1.51505 −0.757524 0.652807i \(-0.773591\pi\)
−0.757524 + 0.652807i \(0.773591\pi\)
\(212\) 659493. 1.00779
\(213\) 316031. 0.477288
\(214\) 391610. 0.584546
\(215\) 87760.0 0.129479
\(216\) −489546. −0.713936
\(217\) −486153. −0.700848
\(218\) −522336. −0.744404
\(219\) −204806. −0.288557
\(220\) −99480.7 −0.138574
\(221\) 266140. 0.366547
\(222\) 260720. 0.355051
\(223\) 668754. 0.900542 0.450271 0.892892i \(-0.351327\pi\)
0.450271 + 0.892892i \(0.351327\pi\)
\(224\) −389773. −0.519029
\(225\) −36845.8 −0.0485212
\(226\) −83734.2 −0.109052
\(227\) −1.40720e6 −1.81256 −0.906280 0.422677i \(-0.861090\pi\)
−0.906280 + 0.422677i \(0.861090\pi\)
\(228\) −701439. −0.893620
\(229\) 706746. 0.890583 0.445292 0.895386i \(-0.353100\pi\)
0.445292 + 0.895386i \(0.353100\pi\)
\(230\) 3885.21 0.00484278
\(231\) 92741.4 0.114352
\(232\) −373050. −0.455037
\(233\) 810873. 0.978505 0.489252 0.872142i \(-0.337270\pi\)
0.489252 + 0.872142i \(0.337270\pi\)
\(234\) −111820. −0.133499
\(235\) −80326.7 −0.0948834
\(236\) −631678. −0.738271
\(237\) −884533. −1.02292
\(238\) −42398.2 −0.0485182
\(239\) 81145.7 0.0918905 0.0459453 0.998944i \(-0.485370\pi\)
0.0459453 + 0.998944i \(0.485370\pi\)
\(240\) 371138. 0.415918
\(241\) 538808. 0.597574 0.298787 0.954320i \(-0.403418\pi\)
0.298787 + 0.954320i \(0.403418\pi\)
\(242\) 386091. 0.423791
\(243\) −316604. −0.343954
\(244\) −211377. −0.227292
\(245\) −581718. −0.619152
\(246\) −655111. −0.690204
\(247\) 1.70767e6 1.78099
\(248\) −1.04050e6 −1.07427
\(249\) 488703. 0.499512
\(250\) 474339. 0.479998
\(251\) −228910. −0.229340 −0.114670 0.993404i \(-0.536581\pi\)
−0.114670 + 0.993404i \(0.536581\pi\)
\(252\) −73379.3 −0.0727901
\(253\) −2665.02 −0.00261758
\(254\) −181089. −0.176119
\(255\) 201509. 0.194064
\(256\) −452733. −0.431759
\(257\) 1.42445e6 1.34528 0.672642 0.739968i \(-0.265160\pi\)
0.672642 + 0.739968i \(0.265160\pi\)
\(258\) −78075.9 −0.0730243
\(259\) −416526. −0.385827
\(260\) −1.29391e6 −1.18705
\(261\) −109147. −0.0991773
\(262\) 182618. 0.164358
\(263\) 60691.4 0.0541050 0.0270525 0.999634i \(-0.491388\pi\)
0.0270525 + 0.999634i \(0.491388\pi\)
\(264\) 198492. 0.175280
\(265\) −1.21565e6 −1.06339
\(266\) −272045. −0.235742
\(267\) 1.98135e6 1.70092
\(268\) 316772. 0.269408
\(269\) −1.85520e6 −1.56319 −0.781594 0.623788i \(-0.785593\pi\)
−0.781594 + 0.623788i \(0.785593\pi\)
\(270\) 402354. 0.335892
\(271\) −1.57226e6 −1.30047 −0.650236 0.759732i \(-0.725330\pi\)
−0.650236 + 0.759732i \(0.725330\pi\)
\(272\) 116385. 0.0953834
\(273\) 1.20625e6 0.979563
\(274\) −727276. −0.585225
\(275\) −70997.3 −0.0566122
\(276\) 14238.1 0.0112507
\(277\) 321513. 0.251767 0.125883 0.992045i \(-0.459823\pi\)
0.125883 + 0.992045i \(0.459823\pi\)
\(278\) 937934. 0.727881
\(279\) −304431. −0.234141
\(280\) 462301. 0.352395
\(281\) −501565. −0.378932 −0.189466 0.981887i \(-0.560676\pi\)
−0.189466 + 0.981887i \(0.560676\pi\)
\(282\) 71462.8 0.0535127
\(283\) 347650. 0.258034 0.129017 0.991642i \(-0.458818\pi\)
0.129017 + 0.991642i \(0.458818\pi\)
\(284\) −481818. −0.354476
\(285\) 1.29297e6 0.942922
\(286\) −215463. −0.155760
\(287\) 1.04661e6 0.750030
\(288\) −244077. −0.173399
\(289\) −1.35667e6 −0.955495
\(290\) 306606. 0.214085
\(291\) −2.48512e6 −1.72034
\(292\) 312245. 0.214308
\(293\) −1.60874e6 −1.09475 −0.547376 0.836886i \(-0.684373\pi\)
−0.547376 + 0.836886i \(0.684373\pi\)
\(294\) 517526. 0.349192
\(295\) 1.16438e6 0.779002
\(296\) −891478. −0.591400
\(297\) −275991. −0.181553
\(298\) −1.07792e6 −0.703145
\(299\) −34663.0 −0.0224227
\(300\) 379310. 0.243328
\(301\) 124734. 0.0793540
\(302\) −43134.2 −0.0272148
\(303\) −2.00239e6 −1.25298
\(304\) 746772. 0.463451
\(305\) 389633. 0.239832
\(306\) −26549.9 −0.0162091
\(307\) 2.50047e6 1.51417 0.757085 0.653316i \(-0.226623\pi\)
0.757085 + 0.653316i \(0.226623\pi\)
\(308\) −141393. −0.0849279
\(309\) 830162. 0.494614
\(310\) 855178. 0.505420
\(311\) 2.80353e6 1.64363 0.821816 0.569753i \(-0.192961\pi\)
0.821816 + 0.569753i \(0.192961\pi\)
\(312\) 2.58171e6 1.50149
\(313\) −102581. −0.0591841 −0.0295920 0.999562i \(-0.509421\pi\)
−0.0295920 + 0.999562i \(0.509421\pi\)
\(314\) 348767. 0.199623
\(315\) 135261. 0.0768060
\(316\) 1.34855e6 0.759714
\(317\) −263420. −0.147232 −0.0736158 0.997287i \(-0.523454\pi\)
−0.0736158 + 0.997287i \(0.523454\pi\)
\(318\) 1.08151e6 0.599737
\(319\) −210313. −0.115715
\(320\) −17559.9 −0.00958623
\(321\) −2.64539e6 −1.43294
\(322\) 5522.09 0.00296800
\(323\) 405459. 0.216243
\(324\) 1.73883e6 0.920225
\(325\) −923437. −0.484952
\(326\) 1.39059e6 0.724697
\(327\) 3.52847e6 1.82481
\(328\) 2.24002e6 1.14965
\(329\) −114169. −0.0581512
\(330\) −163139. −0.0824655
\(331\) −2.31706e6 −1.16243 −0.581216 0.813749i \(-0.697423\pi\)
−0.581216 + 0.813749i \(0.697423\pi\)
\(332\) −745072. −0.370982
\(333\) −260830. −0.128898
\(334\) 598058. 0.293344
\(335\) −583909. −0.284271
\(336\) 527502. 0.254904
\(337\) 1.40113e6 0.672055 0.336028 0.941852i \(-0.390916\pi\)
0.336028 + 0.941852i \(0.390916\pi\)
\(338\) −1.87414e6 −0.892301
\(339\) 565639. 0.267325
\(340\) −307219. −0.144129
\(341\) −586600. −0.273185
\(342\) −170355. −0.0787572
\(343\) −1.96061e6 −0.899819
\(344\) 266965. 0.121635
\(345\) −26245.3 −0.0118714
\(346\) −198924. −0.0893298
\(347\) −3.64231e6 −1.62388 −0.811940 0.583741i \(-0.801588\pi\)
−0.811940 + 0.583741i \(0.801588\pi\)
\(348\) 1.12362e6 0.497361
\(349\) 2.59872e6 1.14208 0.571039 0.820923i \(-0.306540\pi\)
0.571039 + 0.820923i \(0.306540\pi\)
\(350\) 147111. 0.0641910
\(351\) −3.58971e6 −1.55522
\(352\) −470306. −0.202313
\(353\) −4.35884e6 −1.86180 −0.930902 0.365269i \(-0.880977\pi\)
−0.930902 + 0.365269i \(0.880977\pi\)
\(354\) −1.03589e6 −0.439345
\(355\) 888139. 0.374033
\(356\) −3.02075e6 −1.26325
\(357\) 286407. 0.118936
\(358\) 239615. 0.0988112
\(359\) 3.30615e6 1.35390 0.676949 0.736029i \(-0.263302\pi\)
0.676949 + 0.736029i \(0.263302\pi\)
\(360\) 289494. 0.117729
\(361\) 125500. 0.0506845
\(362\) 548573. 0.220020
\(363\) −2.60811e6 −1.03887
\(364\) −1.83905e6 −0.727511
\(365\) −575564. −0.226132
\(366\) −346638. −0.135261
\(367\) −548156. −0.212441 −0.106221 0.994343i \(-0.533875\pi\)
−0.106221 + 0.994343i \(0.533875\pi\)
\(368\) −15158.3 −0.00583488
\(369\) 655389. 0.250572
\(370\) 732698. 0.278241
\(371\) −1.72781e6 −0.651722
\(372\) 3.13397e6 1.17419
\(373\) −549721. −0.204583 −0.102292 0.994754i \(-0.532617\pi\)
−0.102292 + 0.994754i \(0.532617\pi\)
\(374\) −51158.3 −0.0189120
\(375\) −3.20424e6 −1.17665
\(376\) −244353. −0.0891348
\(377\) −2.73547e6 −0.991240
\(378\) 571870. 0.205858
\(379\) −2.24483e6 −0.802760 −0.401380 0.915912i \(-0.631469\pi\)
−0.401380 + 0.915912i \(0.631469\pi\)
\(380\) −1.97125e6 −0.700298
\(381\) 1.22329e6 0.431733
\(382\) −821883. −0.288172
\(383\) 5.10313e6 1.77762 0.888812 0.458272i \(-0.151531\pi\)
0.888812 + 0.458272i \(0.151531\pi\)
\(384\) 3.13826e6 1.08608
\(385\) 260631. 0.0896135
\(386\) −1.43946e6 −0.491736
\(387\) 78108.9 0.0265108
\(388\) 3.78879e6 1.27768
\(389\) 4.70768e6 1.57737 0.788683 0.614800i \(-0.210763\pi\)
0.788683 + 0.614800i \(0.210763\pi\)
\(390\) −2.12189e6 −0.706416
\(391\) −8230.20 −0.00272250
\(392\) −1.76958e6 −0.581640
\(393\) −1.23362e6 −0.402902
\(394\) −2.40739e6 −0.781279
\(395\) −2.48580e6 −0.801628
\(396\) −88540.7 −0.0283730
\(397\) 3.81755e6 1.21565 0.607825 0.794071i \(-0.292042\pi\)
0.607825 + 0.794071i \(0.292042\pi\)
\(398\) 1.90051e6 0.601397
\(399\) 1.83771e6 0.577889
\(400\) −403824. −0.126195
\(401\) 1.31578e6 0.408622 0.204311 0.978906i \(-0.434505\pi\)
0.204311 + 0.978906i \(0.434505\pi\)
\(402\) 519476. 0.160325
\(403\) −7.62970e6 −2.34016
\(404\) 3.05283e6 0.930571
\(405\) −3.20520e6 −0.970995
\(406\) 435782. 0.131206
\(407\) −502587. −0.150392
\(408\) 612988. 0.182306
\(409\) 3.92071e6 1.15893 0.579464 0.814998i \(-0.303262\pi\)
0.579464 + 0.814998i \(0.303262\pi\)
\(410\) −1.84105e6 −0.540888
\(411\) 4.91287e6 1.43460
\(412\) −1.26566e6 −0.367344
\(413\) 1.65494e6 0.477427
\(414\) 3457.95 0.000991557 0
\(415\) 1.37340e6 0.391450
\(416\) −6.11711e6 −1.73306
\(417\) −6.33591e6 −1.78430
\(418\) −328253. −0.0918901
\(419\) 2.92842e6 0.814890 0.407445 0.913230i \(-0.366420\pi\)
0.407445 + 0.913230i \(0.366420\pi\)
\(420\) −1.39244e6 −0.385172
\(421\) 4.43337e6 1.21907 0.609535 0.792759i \(-0.291356\pi\)
0.609535 + 0.792759i \(0.291356\pi\)
\(422\) 2.44965e6 0.669612
\(423\) −71493.0 −0.0194273
\(424\) −3.69799e6 −0.998966
\(425\) −219256. −0.0588816
\(426\) −790135. −0.210949
\(427\) 553789. 0.146986
\(428\) 4.03314e6 1.06423
\(429\) 1.45549e6 0.381826
\(430\) −219416. −0.0572265
\(431\) 4.27188e6 1.10771 0.553854 0.832614i \(-0.313156\pi\)
0.553854 + 0.832614i \(0.313156\pi\)
\(432\) −1.56980e6 −0.404702
\(433\) −1.65833e6 −0.425061 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(434\) 1.21547e6 0.309757
\(435\) −2.07118e6 −0.524801
\(436\) −5.37947e6 −1.35526
\(437\) −52808.5 −0.0132282
\(438\) 512052. 0.127535
\(439\) −1.44252e6 −0.357240 −0.178620 0.983918i \(-0.557163\pi\)
−0.178620 + 0.983918i \(0.557163\pi\)
\(440\) 557820. 0.137361
\(441\) −517745. −0.126771
\(442\) −665398. −0.162004
\(443\) 5.80251e6 1.40477 0.702387 0.711795i \(-0.252118\pi\)
0.702387 + 0.711795i \(0.252118\pi\)
\(444\) 2.68512e6 0.646407
\(445\) 5.56818e6 1.33295
\(446\) −1.67201e6 −0.398016
\(447\) 7.28152e6 1.72367
\(448\) −24958.1 −0.00587511
\(449\) −1.86453e6 −0.436470 −0.218235 0.975896i \(-0.570030\pi\)
−0.218235 + 0.975896i \(0.570030\pi\)
\(450\) 92121.2 0.0214451
\(451\) 1.26285e6 0.292355
\(452\) −862368. −0.198539
\(453\) 291379. 0.0667134
\(454\) 3.51827e6 0.801105
\(455\) 3.38993e6 0.767648
\(456\) 3.93319e6 0.885795
\(457\) 3.61109e6 0.808812 0.404406 0.914580i \(-0.367478\pi\)
0.404406 + 0.914580i \(0.367478\pi\)
\(458\) −1.76699e6 −0.393615
\(459\) −852323. −0.188831
\(460\) 40013.3 0.00881679
\(461\) 5.82273e6 1.27607 0.638034 0.770008i \(-0.279748\pi\)
0.638034 + 0.770008i \(0.279748\pi\)
\(462\) −231871. −0.0505406
\(463\) 759569. 0.164670 0.0823351 0.996605i \(-0.473762\pi\)
0.0823351 + 0.996605i \(0.473762\pi\)
\(464\) −1.19624e6 −0.257942
\(465\) −5.77687e6 −1.23897
\(466\) −2.02733e6 −0.432474
\(467\) 3.30533e6 0.701330 0.350665 0.936501i \(-0.385956\pi\)
0.350665 + 0.936501i \(0.385956\pi\)
\(468\) −1.15162e6 −0.243049
\(469\) −829915. −0.174221
\(470\) 200831. 0.0419360
\(471\) −2.35598e6 −0.489350
\(472\) 3.54202e6 0.731805
\(473\) 150506. 0.0309315
\(474\) 2.21149e6 0.452106
\(475\) −1.40684e6 −0.286095
\(476\) −436654. −0.0883324
\(477\) −1.08196e6 −0.217729
\(478\) −202879. −0.0406132
\(479\) −7.63918e6 −1.52128 −0.760638 0.649176i \(-0.775114\pi\)
−0.760638 + 0.649176i \(0.775114\pi\)
\(480\) −4.63160e6 −0.917547
\(481\) −6.53697e6 −1.28829
\(482\) −1.34712e6 −0.264112
\(483\) −37302.7 −0.00727565
\(484\) 3.97631e6 0.771555
\(485\) −6.98391e6 −1.34817
\(486\) 791567. 0.152019
\(487\) −4.03922e6 −0.771747 −0.385873 0.922552i \(-0.626100\pi\)
−0.385873 + 0.922552i \(0.626100\pi\)
\(488\) 1.18526e6 0.225301
\(489\) −9.39371e6 −1.77650
\(490\) 1.45440e6 0.273649
\(491\) 7.62584e6 1.42753 0.713763 0.700387i \(-0.246989\pi\)
0.713763 + 0.700387i \(0.246989\pi\)
\(492\) −6.74691e6 −1.25659
\(493\) −649497. −0.120354
\(494\) −4.26948e6 −0.787149
\(495\) 163208. 0.0299383
\(496\) −3.33651e6 −0.608960
\(497\) 1.26232e6 0.229234
\(498\) −1.22185e6 −0.220772
\(499\) 9.55168e6 1.71723 0.858615 0.512621i \(-0.171325\pi\)
0.858615 + 0.512621i \(0.171325\pi\)
\(500\) 4.88516e6 0.873885
\(501\) −4.03999e6 −0.719094
\(502\) 572316. 0.101362
\(503\) −4.69119e6 −0.826730 −0.413365 0.910566i \(-0.635647\pi\)
−0.413365 + 0.910566i \(0.635647\pi\)
\(504\) 411461. 0.0721527
\(505\) −5.62731e6 −0.981911
\(506\) 6663.04 0.00115690
\(507\) 1.26602e7 2.18736
\(508\) −1.86501e6 −0.320643
\(509\) 2.37623e6 0.406531 0.203265 0.979124i \(-0.434845\pi\)
0.203265 + 0.979124i \(0.434845\pi\)
\(510\) −503810. −0.0857712
\(511\) −818054. −0.138589
\(512\) −4.81417e6 −0.811608
\(513\) −5.46886e6 −0.917495
\(514\) −3.56138e6 −0.594580
\(515\) 2.33300e6 0.387611
\(516\) −804094. −0.132948
\(517\) −137758. −0.0226668
\(518\) 1.04139e6 0.170526
\(519\) 1.34376e6 0.218980
\(520\) 7.25536e6 1.17666
\(521\) 6.31208e6 1.01878 0.509388 0.860537i \(-0.329872\pi\)
0.509388 + 0.860537i \(0.329872\pi\)
\(522\) 272888. 0.0438338
\(523\) −1.03668e7 −1.65725 −0.828627 0.559801i \(-0.810878\pi\)
−0.828627 + 0.559801i \(0.810878\pi\)
\(524\) 1.88077e6 0.299231
\(525\) −993758. −0.157356
\(526\) −151739. −0.0239130
\(527\) −1.81156e6 −0.284136
\(528\) 636493. 0.0993594
\(529\) −6.43527e6 −0.999833
\(530\) 3.03935e6 0.469992
\(531\) 1.03633e6 0.159500
\(532\) −2.80176e6 −0.429192
\(533\) 1.64255e7 2.50438
\(534\) −4.95374e6 −0.751762
\(535\) −7.43432e6 −1.12294
\(536\) −1.77624e6 −0.267049
\(537\) −1.61864e6 −0.242222
\(538\) 4.63835e6 0.690888
\(539\) −997631. −0.147910
\(540\) 4.14380e6 0.611525
\(541\) −5.71736e6 −0.839851 −0.419926 0.907559i \(-0.637944\pi\)
−0.419926 + 0.907559i \(0.637944\pi\)
\(542\) 3.93093e6 0.574775
\(543\) −3.70570e6 −0.539350
\(544\) −1.45242e6 −0.210423
\(545\) 9.91603e6 1.43003
\(546\) −3.01586e6 −0.432941
\(547\) 4.37335e6 0.624950 0.312475 0.949926i \(-0.398842\pi\)
0.312475 + 0.949926i \(0.398842\pi\)
\(548\) −7.49013e6 −1.06546
\(549\) 346785. 0.0491054
\(550\) 177506. 0.0250211
\(551\) −4.16744e6 −0.584778
\(552\) −79837.8 −0.0111522
\(553\) −3.53309e6 −0.491294
\(554\) −803841. −0.111274
\(555\) −4.94950e6 −0.682070
\(556\) 9.65967e6 1.32518
\(557\) −5.52441e6 −0.754481 −0.377240 0.926115i \(-0.623127\pi\)
−0.377240 + 0.926115i \(0.623127\pi\)
\(558\) 761133. 0.103484
\(559\) 1.95758e6 0.264966
\(560\) 1.48244e6 0.199759
\(561\) 345583. 0.0463602
\(562\) 1.25400e6 0.167478
\(563\) 1.38558e7 1.84230 0.921149 0.389211i \(-0.127252\pi\)
0.921149 + 0.389211i \(0.127252\pi\)
\(564\) 735987. 0.0974254
\(565\) 1.58961e6 0.209493
\(566\) −869189. −0.114044
\(567\) −4.55557e6 −0.595094
\(568\) 2.70171e6 0.351372
\(569\) 8.96028e6 1.16022 0.580111 0.814538i \(-0.303009\pi\)
0.580111 + 0.814538i \(0.303009\pi\)
\(570\) −3.23266e6 −0.416747
\(571\) 1.86281e6 0.239099 0.119550 0.992828i \(-0.461855\pi\)
0.119550 + 0.992828i \(0.461855\pi\)
\(572\) −2.21902e6 −0.283578
\(573\) 5.55196e6 0.706415
\(574\) −2.61671e6 −0.331494
\(575\) 28556.7 0.00360195
\(576\) −15628.8 −0.00196277
\(577\) −8.48683e6 −1.06122 −0.530610 0.847616i \(-0.678037\pi\)
−0.530610 + 0.847616i \(0.678037\pi\)
\(578\) 3.39191e6 0.422304
\(579\) 9.72382e6 1.20543
\(580\) 3.15770e6 0.389764
\(581\) 1.95202e6 0.239908
\(582\) 6.21325e6 0.760346
\(583\) −2.08481e6 −0.254036
\(584\) −1.75086e6 −0.212431
\(585\) 2.12279e6 0.256458
\(586\) 4.02214e6 0.483852
\(587\) −9.06080e6 −1.08535 −0.542677 0.839942i \(-0.682589\pi\)
−0.542677 + 0.839942i \(0.682589\pi\)
\(588\) 5.32994e6 0.635740
\(589\) −1.16237e7 −1.38056
\(590\) −2.91115e6 −0.344299
\(591\) 1.62623e7 1.91520
\(592\) −2.85866e6 −0.335241
\(593\) 1.56414e7 1.82658 0.913290 0.407310i \(-0.133533\pi\)
0.913290 + 0.407310i \(0.133533\pi\)
\(594\) 690027. 0.0802417
\(595\) 804887. 0.0932058
\(596\) −1.11013e7 −1.28015
\(597\) −1.28382e7 −1.47425
\(598\) 86663.8 0.00991025
\(599\) −1.40306e6 −0.159775 −0.0798877 0.996804i \(-0.525456\pi\)
−0.0798877 + 0.996804i \(0.525456\pi\)
\(600\) −2.12691e6 −0.241197
\(601\) −1.02325e6 −0.115557 −0.0577786 0.998329i \(-0.518402\pi\)
−0.0577786 + 0.998329i \(0.518402\pi\)
\(602\) −311858. −0.0350724
\(603\) −519696. −0.0582044
\(604\) −444235. −0.0495473
\(605\) −7.32956e6 −0.814122
\(606\) 5.00635e6 0.553783
\(607\) 6.79951e6 0.749042 0.374521 0.927219i \(-0.377807\pi\)
0.374521 + 0.927219i \(0.377807\pi\)
\(608\) −9.31931e6 −1.02241
\(609\) −2.94378e6 −0.321635
\(610\) −974154. −0.105999
\(611\) −1.79177e6 −0.194169
\(612\) −273434. −0.0295103
\(613\) 3.70194e6 0.397904 0.198952 0.980009i \(-0.436246\pi\)
0.198952 + 0.980009i \(0.436246\pi\)
\(614\) −6.25162e6 −0.669224
\(615\) 1.24366e7 1.32591
\(616\) 792834. 0.0841842
\(617\) 1.14193e7 1.20761 0.603806 0.797131i \(-0.293650\pi\)
0.603806 + 0.797131i \(0.293650\pi\)
\(618\) −2.07556e6 −0.218607
\(619\) −8.12725e6 −0.852545 −0.426272 0.904595i \(-0.640173\pi\)
−0.426272 + 0.904595i \(0.640173\pi\)
\(620\) 8.80737e6 0.920168
\(621\) 111010. 0.0115513
\(622\) −7.00934e6 −0.726443
\(623\) 7.91410e6 0.816924
\(624\) 8.27864e6 0.851133
\(625\) −6.27919e6 −0.642989
\(626\) 256470. 0.0261578
\(627\) 2.21741e6 0.225256
\(628\) 3.59191e6 0.363435
\(629\) −1.55210e6 −0.156421
\(630\) −338176. −0.0339463
\(631\) 8.36884e6 0.836743 0.418371 0.908276i \(-0.362601\pi\)
0.418371 + 0.908276i \(0.362601\pi\)
\(632\) −7.56176e6 −0.753061
\(633\) −1.65478e7 −1.64146
\(634\) 658599. 0.0650725
\(635\) 3.43779e6 0.338334
\(636\) 1.11383e7 1.09188
\(637\) −1.29758e7 −1.26703
\(638\) 525822. 0.0511431
\(639\) 790469. 0.0765831
\(640\) 8.81943e6 0.851120
\(641\) −1.04858e7 −1.00799 −0.503996 0.863706i \(-0.668137\pi\)
−0.503996 + 0.863706i \(0.668137\pi\)
\(642\) 6.61396e6 0.633321
\(643\) 1.22946e7 1.17270 0.586349 0.810058i \(-0.300565\pi\)
0.586349 + 0.810058i \(0.300565\pi\)
\(644\) 56871.3 0.00540354
\(645\) 1.48219e6 0.140283
\(646\) −1.01372e6 −0.0955736
\(647\) −1.39774e7 −1.31270 −0.656349 0.754457i \(-0.727900\pi\)
−0.656349 + 0.754457i \(0.727900\pi\)
\(648\) −9.75016e6 −0.912167
\(649\) 1.99688e6 0.186097
\(650\) 2.30876e6 0.214336
\(651\) −8.21072e6 −0.759327
\(652\) 1.43216e7 1.31939
\(653\) −5.00675e6 −0.459487 −0.229743 0.973251i \(-0.573789\pi\)
−0.229743 + 0.973251i \(0.573789\pi\)
\(654\) −8.82181e6 −0.806517
\(655\) −3.46683e6 −0.315740
\(656\) 7.18296e6 0.651694
\(657\) −512268. −0.0463003
\(658\) 285443. 0.0257013
\(659\) 5.50845e6 0.494101 0.247050 0.969003i \(-0.420539\pi\)
0.247050 + 0.969003i \(0.420539\pi\)
\(660\) −1.68015e6 −0.150137
\(661\) −6.69305e6 −0.595827 −0.297914 0.954593i \(-0.596291\pi\)
−0.297914 + 0.954593i \(0.596291\pi\)
\(662\) 5.79308e6 0.513765
\(663\) 4.49488e6 0.397131
\(664\) 4.17786e6 0.367733
\(665\) 5.16450e6 0.452870
\(666\) 652122. 0.0569697
\(667\) 84592.7 0.00736238
\(668\) 6.15933e6 0.534063
\(669\) 1.12947e7 0.975684
\(670\) 1.45988e6 0.125641
\(671\) 668211. 0.0572938
\(672\) −6.58294e6 −0.562337
\(673\) −7.54961e6 −0.642521 −0.321260 0.946991i \(-0.604106\pi\)
−0.321260 + 0.946991i \(0.604106\pi\)
\(674\) −3.50309e6 −0.297031
\(675\) 2.95734e6 0.249828
\(676\) −1.93016e7 −1.62453
\(677\) −1.51658e6 −0.127173 −0.0635863 0.997976i \(-0.520254\pi\)
−0.0635863 + 0.997976i \(0.520254\pi\)
\(678\) −1.41420e6 −0.118151
\(679\) −9.92629e6 −0.826252
\(680\) 1.72268e6 0.142867
\(681\) −2.37665e7 −1.96380
\(682\) 1.46661e6 0.120741
\(683\) −2.30637e7 −1.89181 −0.945904 0.324448i \(-0.894822\pi\)
−0.945904 + 0.324448i \(0.894822\pi\)
\(684\) −1.75447e6 −0.143386
\(685\) 1.38066e7 1.12424
\(686\) 4.90187e6 0.397696
\(687\) 1.19363e7 0.964894
\(688\) 856062. 0.0689499
\(689\) −2.71164e7 −2.17612
\(690\) 65618.0 0.00524687
\(691\) −1.55625e7 −1.23990 −0.619948 0.784643i \(-0.712846\pi\)
−0.619948 + 0.784643i \(0.712846\pi\)
\(692\) −2.04869e6 −0.162634
\(693\) 231969. 0.0183483
\(694\) 9.10645e6 0.717713
\(695\) −1.78058e7 −1.39829
\(696\) −6.30050e6 −0.493005
\(697\) 3.89998e6 0.304075
\(698\) −6.49727e6 −0.504769
\(699\) 1.36950e7 1.06015
\(700\) 1.51508e6 0.116866
\(701\) 1.48227e6 0.113929 0.0569643 0.998376i \(-0.481858\pi\)
0.0569643 + 0.998376i \(0.481858\pi\)
\(702\) 8.97494e6 0.687367
\(703\) −9.95896e6 −0.760021
\(704\) −30114.8 −0.00229007
\(705\) −1.35665e6 −0.102800
\(706\) 1.08979e7 0.822869
\(707\) −7.99814e6 −0.601784
\(708\) −1.06685e7 −0.799872
\(709\) 1.23095e7 0.919652 0.459826 0.888009i \(-0.347912\pi\)
0.459826 + 0.888009i \(0.347912\pi\)
\(710\) −2.22051e6 −0.165313
\(711\) −2.21243e6 −0.164133
\(712\) 1.69383e7 1.25219
\(713\) 235944. 0.0173814
\(714\) −716070. −0.0525666
\(715\) 4.09034e6 0.299223
\(716\) 2.46776e6 0.179896
\(717\) 1.37048e6 0.0995579
\(718\) −8.26598e6 −0.598388
\(719\) −2.07641e7 −1.49793 −0.748963 0.662612i \(-0.769448\pi\)
−0.748963 + 0.662612i \(0.769448\pi\)
\(720\) 928307. 0.0667360
\(721\) 3.31591e6 0.237555
\(722\) −313773. −0.0224012
\(723\) 9.10002e6 0.647435
\(724\) 5.64968e6 0.400569
\(725\) 2.25359e6 0.159232
\(726\) 6.52076e6 0.459152
\(727\) −7.15312e6 −0.501949 −0.250974 0.967994i \(-0.580751\pi\)
−0.250974 + 0.967994i \(0.580751\pi\)
\(728\) 1.03121e7 0.721140
\(729\) 1.10625e7 0.770968
\(730\) 1.43901e6 0.0999443
\(731\) 464798. 0.0321715
\(732\) −3.56998e6 −0.246257
\(733\) 1.65135e7 1.13521 0.567607 0.823299i \(-0.307869\pi\)
0.567607 + 0.823299i \(0.307869\pi\)
\(734\) 1.37049e6 0.0938936
\(735\) −9.82473e6 −0.670814
\(736\) 189168. 0.0128722
\(737\) −1.00139e6 −0.0679100
\(738\) −1.63859e6 −0.110746
\(739\) −7.25575e6 −0.488733 −0.244366 0.969683i \(-0.578580\pi\)
−0.244366 + 0.969683i \(0.578580\pi\)
\(740\) 7.54597e6 0.506566
\(741\) 2.88410e7 1.92959
\(742\) 4.31985e6 0.288044
\(743\) −1.59618e7 −1.06074 −0.530372 0.847765i \(-0.677948\pi\)
−0.530372 + 0.847765i \(0.677948\pi\)
\(744\) −1.75732e7 −1.16391
\(745\) 2.04632e7 1.35077
\(746\) 1.37440e6 0.0904205
\(747\) 1.22236e6 0.0801491
\(748\) −526874. −0.0344312
\(749\) −1.05665e7 −0.688217
\(750\) 8.01120e6 0.520049
\(751\) 2.56869e7 1.66192 0.830962 0.556329i \(-0.187791\pi\)
0.830962 + 0.556329i \(0.187791\pi\)
\(752\) −783553. −0.0505270
\(753\) −3.86610e6 −0.248476
\(754\) 6.83918e6 0.438102
\(755\) 818861. 0.0522809
\(756\) 5.88962e6 0.374785
\(757\) 7.97201e6 0.505624 0.252812 0.967515i \(-0.418645\pi\)
0.252812 + 0.967515i \(0.418645\pi\)
\(758\) 5.61249e6 0.354799
\(759\) −45010.0 −0.00283599
\(760\) 1.10534e7 0.694165
\(761\) 2.21020e7 1.38347 0.691736 0.722150i \(-0.256846\pi\)
0.691736 + 0.722150i \(0.256846\pi\)
\(762\) −3.05844e6 −0.190815
\(763\) 1.40937e7 0.876425
\(764\) −8.46448e6 −0.524646
\(765\) 504023. 0.0311385
\(766\) −1.27588e7 −0.785664
\(767\) 2.59727e7 1.59415
\(768\) −7.64628e6 −0.467786
\(769\) −2.64726e7 −1.61428 −0.807142 0.590357i \(-0.798987\pi\)
−0.807142 + 0.590357i \(0.798987\pi\)
\(770\) −651624. −0.0396068
\(771\) 2.40577e7 1.45753
\(772\) −1.48249e7 −0.895256
\(773\) 2.48033e7 1.49301 0.746503 0.665382i \(-0.231732\pi\)
0.746503 + 0.665382i \(0.231732\pi\)
\(774\) −195287. −0.0117171
\(775\) 6.28564e6 0.375920
\(776\) −2.12449e7 −1.26649
\(777\) −7.03477e6 −0.418021
\(778\) −1.17701e7 −0.697155
\(779\) 2.50239e7 1.47745
\(780\) −2.18531e7 −1.28610
\(781\) 1.52314e6 0.0893534
\(782\) 20577.0 0.00120328
\(783\) 8.76045e6 0.510649
\(784\) −5.67441e6 −0.329709
\(785\) −6.62100e6 −0.383486
\(786\) 3.08427e6 0.178072
\(787\) 8.09631e6 0.465962 0.232981 0.972481i \(-0.425152\pi\)
0.232981 + 0.972481i \(0.425152\pi\)
\(788\) −2.47934e7 −1.42240
\(789\) 1.02503e6 0.0586196
\(790\) 6.21495e6 0.354299
\(791\) 2.25933e6 0.128392
\(792\) 496476. 0.0281245
\(793\) 8.69118e6 0.490790
\(794\) −9.54457e6 −0.537286
\(795\) −2.05313e7 −1.15212
\(796\) 1.95731e7 1.09491
\(797\) 1.30377e7 0.727034 0.363517 0.931587i \(-0.381576\pi\)
0.363517 + 0.931587i \(0.381576\pi\)
\(798\) −4.59461e6 −0.255412
\(799\) −425429. −0.0235755
\(800\) 5.03951e6 0.278396
\(801\) 4.95584e6 0.272920
\(802\) −3.28968e6 −0.180600
\(803\) −987077. −0.0540210
\(804\) 5.35002e6 0.291887
\(805\) −104831. −0.00570166
\(806\) 1.90756e7 1.03429
\(807\) −3.13329e7 −1.69362
\(808\) −1.71182e7 −0.922422
\(809\) −3.29258e6 −0.176874 −0.0884371 0.996082i \(-0.528187\pi\)
−0.0884371 + 0.996082i \(0.528187\pi\)
\(810\) 8.01358e6 0.429155
\(811\) 8.58147e6 0.458152 0.229076 0.973409i \(-0.426430\pi\)
0.229076 + 0.973409i \(0.426430\pi\)
\(812\) 4.48807e6 0.238874
\(813\) −2.65541e7 −1.40898
\(814\) 1.25656e6 0.0664694
\(815\) −2.63991e7 −1.39218
\(816\) 1.96564e6 0.103342
\(817\) 2.98234e6 0.156315
\(818\) −9.80249e6 −0.512216
\(819\) 3.01713e6 0.157175
\(820\) −1.89608e7 −0.984741
\(821\) −7.34065e6 −0.380081 −0.190041 0.981776i \(-0.560862\pi\)
−0.190041 + 0.981776i \(0.560862\pi\)
\(822\) −1.22831e7 −0.634056
\(823\) 3.32707e7 1.71223 0.856116 0.516783i \(-0.172871\pi\)
0.856116 + 0.516783i \(0.172871\pi\)
\(824\) 7.09694e6 0.364127
\(825\) −1.19908e6 −0.0613359
\(826\) −4.13765e6 −0.211010
\(827\) −3.77399e7 −1.91883 −0.959417 0.281992i \(-0.909005\pi\)
−0.959417 + 0.281992i \(0.909005\pi\)
\(828\) 35613.0 0.00180523
\(829\) −2.81993e7 −1.42512 −0.712560 0.701611i \(-0.752465\pi\)
−0.712560 + 0.701611i \(0.752465\pi\)
\(830\) −3.43374e6 −0.173011
\(831\) 5.43008e6 0.272775
\(832\) −391693. −0.0196172
\(833\) −3.08092e6 −0.153839
\(834\) 1.58409e7 0.788616
\(835\) −1.13535e7 −0.563528
\(836\) −3.38064e6 −0.167295
\(837\) 2.44344e7 1.20556
\(838\) −7.32159e6 −0.360160
\(839\) −9.00386e6 −0.441594 −0.220797 0.975320i \(-0.570866\pi\)
−0.220797 + 0.975320i \(0.570866\pi\)
\(840\) 7.80788e6 0.381799
\(841\) −1.38354e7 −0.674531
\(842\) −1.10842e7 −0.538797
\(843\) −8.47102e6 −0.410551
\(844\) 2.52287e7 1.21910
\(845\) 3.55788e7 1.71415
\(846\) 178746. 0.00858637
\(847\) −1.04176e7 −0.498951
\(848\) −1.18581e7 −0.566275
\(849\) 5.87152e6 0.279564
\(850\) 548180. 0.0260241
\(851\) 202152. 0.00956870
\(852\) −8.13751e6 −0.384054
\(853\) 2.76660e7 1.30189 0.650944 0.759126i \(-0.274373\pi\)
0.650944 + 0.759126i \(0.274373\pi\)
\(854\) −1.38457e6 −0.0649638
\(855\) 3.23403e6 0.151296
\(856\) −2.26151e7 −1.05491
\(857\) −3.77780e7 −1.75706 −0.878531 0.477686i \(-0.841476\pi\)
−0.878531 + 0.477686i \(0.841476\pi\)
\(858\) −3.63898e6 −0.168757
\(859\) 2.02066e7 0.934352 0.467176 0.884164i \(-0.345271\pi\)
0.467176 + 0.884164i \(0.345271\pi\)
\(860\) −2.25974e6 −0.104187
\(861\) 1.76763e7 0.812613
\(862\) −1.06805e7 −0.489578
\(863\) −992399. −0.0453586 −0.0226793 0.999743i \(-0.507220\pi\)
−0.0226793 + 0.999743i \(0.507220\pi\)
\(864\) 1.95903e7 0.892804
\(865\) 3.77637e6 0.171607
\(866\) 4.14613e6 0.187866
\(867\) −2.29130e7 −1.03522
\(868\) 1.25180e7 0.563944
\(869\) −4.26308e6 −0.191502
\(870\) 5.17832e6 0.231948
\(871\) −1.30247e7 −0.581732
\(872\) 3.01644e7 1.34339
\(873\) −6.21588e6 −0.276037
\(874\) 132031. 0.00584651
\(875\) −1.27987e7 −0.565126
\(876\) 5.27356e6 0.232190
\(877\) 3.48750e7 1.53114 0.765572 0.643351i \(-0.222456\pi\)
0.765572 + 0.643351i \(0.222456\pi\)
\(878\) 3.60656e6 0.157891
\(879\) −2.71702e7 −1.18610
\(880\) 1.78873e6 0.0778643
\(881\) 2.68921e7 1.16731 0.583654 0.812002i \(-0.301622\pi\)
0.583654 + 0.812002i \(0.301622\pi\)
\(882\) 1.29446e6 0.0560295
\(883\) −2.40336e6 −0.103733 −0.0518666 0.998654i \(-0.516517\pi\)
−0.0518666 + 0.998654i \(0.516517\pi\)
\(884\) −6.85286e6 −0.294945
\(885\) 1.96653e7 0.844002
\(886\) −1.45073e7 −0.620874
\(887\) −2.56525e7 −1.09477 −0.547383 0.836882i \(-0.684376\pi\)
−0.547383 + 0.836882i \(0.684376\pi\)
\(888\) −1.50563e7 −0.640747
\(889\) 4.88616e6 0.207355
\(890\) −1.39215e7 −0.589128
\(891\) −5.49683e6 −0.231963
\(892\) −1.72198e7 −0.724629
\(893\) −2.72973e6 −0.114549
\(894\) −1.82051e7 −0.761816
\(895\) −4.54885e6 −0.189821
\(896\) 1.25351e7 0.521626
\(897\) −585429. −0.0242937
\(898\) 4.66167e6 0.192908
\(899\) 1.86198e7 0.768379
\(900\) 948746. 0.0390430
\(901\) −6.43837e6 −0.264219
\(902\) −3.15736e6 −0.129214
\(903\) 2.10665e6 0.0859753
\(904\) 4.83557e6 0.196801
\(905\) −1.04141e7 −0.422669
\(906\) −728501. −0.0294856
\(907\) −2.68686e7 −1.08450 −0.542248 0.840219i \(-0.682426\pi\)
−0.542248 + 0.840219i \(0.682426\pi\)
\(908\) 3.62342e7 1.45849
\(909\) −5.00847e6 −0.201046
\(910\) −8.47544e6 −0.339280
\(911\) 1.69356e7 0.676091 0.338045 0.941130i \(-0.390234\pi\)
0.338045 + 0.941130i \(0.390234\pi\)
\(912\) 1.26124e7 0.502122
\(913\) 2.35534e6 0.0935141
\(914\) −9.02838e6 −0.357474
\(915\) 6.58058e6 0.259843
\(916\) −1.81981e7 −0.716616
\(917\) −4.92744e6 −0.193508
\(918\) 2.13096e6 0.0834582
\(919\) −2.20166e7 −0.859927 −0.429964 0.902846i \(-0.641474\pi\)
−0.429964 + 0.902846i \(0.641474\pi\)
\(920\) −224367. −0.00873957
\(921\) 4.22308e7 1.64051
\(922\) −1.45579e7 −0.563989
\(923\) 1.98109e7 0.765420
\(924\) −2.38801e6 −0.0920144
\(925\) 5.38540e6 0.206949
\(926\) −1.89906e6 −0.0727799
\(927\) 2.07643e6 0.0793631
\(928\) 1.49284e7 0.569041
\(929\) −8.11263e6 −0.308406 −0.154203 0.988039i \(-0.549281\pi\)
−0.154203 + 0.988039i \(0.549281\pi\)
\(930\) 1.44432e7 0.547592
\(931\) −1.97685e7 −0.747478
\(932\) −2.08792e7 −0.787362
\(933\) 4.73493e7 1.78078
\(934\) −8.26392e6 −0.309970
\(935\) 971190. 0.0363308
\(936\) 6.45748e6 0.240920
\(937\) −1.80064e7 −0.670004 −0.335002 0.942217i \(-0.608737\pi\)
−0.335002 + 0.942217i \(0.608737\pi\)
\(938\) 2.07494e6 0.0770013
\(939\) −1.73250e6 −0.0641224
\(940\) 2.06834e6 0.0763488
\(941\) 2.90643e7 1.07000 0.535002 0.844851i \(-0.320311\pi\)
0.535002 + 0.844851i \(0.320311\pi\)
\(942\) 5.89039e6 0.216280
\(943\) −507947. −0.0186011
\(944\) 1.13580e7 0.414832
\(945\) −1.08564e7 −0.395463
\(946\) −376293. −0.0136709
\(947\) −1.97188e7 −0.714504 −0.357252 0.934008i \(-0.616286\pi\)
−0.357252 + 0.934008i \(0.616286\pi\)
\(948\) 2.27759e7 0.823105
\(949\) −1.28386e7 −0.462755
\(950\) 3.51736e6 0.126447
\(951\) −4.44895e6 −0.159517
\(952\) 2.44846e6 0.0875588
\(953\) −5.34568e6 −0.190665 −0.0953325 0.995445i \(-0.530391\pi\)
−0.0953325 + 0.995445i \(0.530391\pi\)
\(954\) 2.70510e6 0.0962306
\(955\) 1.56026e7 0.553592
\(956\) −2.08943e6 −0.0739405
\(957\) −3.55202e6 −0.125370
\(958\) 1.90993e7 0.672364
\(959\) 1.96235e7 0.689016
\(960\) −296572. −0.0103861
\(961\) 2.33046e7 0.814017
\(962\) 1.63436e7 0.569391
\(963\) −6.61676e6 −0.229922
\(964\) −1.38738e7 −0.480843
\(965\) 2.73268e7 0.944648
\(966\) 93263.4 0.00321565
\(967\) −605934. −0.0208381 −0.0104191 0.999946i \(-0.503317\pi\)
−0.0104191 + 0.999946i \(0.503317\pi\)
\(968\) −2.22964e7 −0.764798
\(969\) 6.84787e6 0.234286
\(970\) 1.74610e7 0.595855
\(971\) 2.31358e7 0.787475 0.393737 0.919223i \(-0.371182\pi\)
0.393737 + 0.919223i \(0.371182\pi\)
\(972\) 8.15226e6 0.276766
\(973\) −2.53075e7 −0.856972
\(974\) 1.00988e7 0.341092
\(975\) −1.55961e7 −0.525416
\(976\) 3.80071e6 0.127714
\(977\) 2.22967e6 0.0747315 0.0373658 0.999302i \(-0.488103\pi\)
0.0373658 + 0.999302i \(0.488103\pi\)
\(978\) 2.34860e7 0.785166
\(979\) 9.54928e6 0.318430
\(980\) 1.49787e7 0.498206
\(981\) 8.82555e6 0.292799
\(982\) −1.90660e7 −0.630929
\(983\) −2.21182e6 −0.0730071 −0.0365036 0.999334i \(-0.511622\pi\)
−0.0365036 + 0.999334i \(0.511622\pi\)
\(984\) 3.78321e7 1.24558
\(985\) 4.57019e7 1.50087
\(986\) 1.62386e6 0.0531932
\(987\) −1.92822e6 −0.0630033
\(988\) −4.39708e7 −1.43309
\(989\) −60536.9 −0.00196802
\(990\) −408049. −0.0132320
\(991\) −1.66837e7 −0.539645 −0.269823 0.962910i \(-0.586965\pi\)
−0.269823 + 0.962910i \(0.586965\pi\)
\(992\) 4.16379e7 1.34341
\(993\) −3.91332e7 −1.25943
\(994\) −3.15603e6 −0.101315
\(995\) −3.60792e7 −1.15531
\(996\) −1.25836e7 −0.401937
\(997\) 1.01549e7 0.323547 0.161774 0.986828i \(-0.448279\pi\)
0.161774 + 0.986828i \(0.448279\pi\)
\(998\) −2.38810e7 −0.758971
\(999\) 2.09349e7 0.663677
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.b.1.4 10
3.2 odd 2 387.6.a.e.1.7 10
4.3 odd 2 688.6.a.h.1.4 10
5.4 even 2 1075.6.a.b.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.4 10 1.1 even 1 trivial
387.6.a.e.1.7 10 3.2 odd 2
688.6.a.h.1.4 10 4.3 odd 2
1075.6.a.b.1.7 10 5.4 even 2