Properties

Label 43.6.a.a.1.2
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 1.2
Root \(-6.09504\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.09504 q^{2} +11.1803 q^{3} +33.5297 q^{4} -63.3756 q^{5} -90.5052 q^{6} +223.489 q^{7} -12.3830 q^{8} -118.000 q^{9} +O(q^{10})\) \(q-8.09504 q^{2} +11.1803 q^{3} +33.5297 q^{4} -63.3756 q^{5} -90.5052 q^{6} +223.489 q^{7} -12.3830 q^{8} -118.000 q^{9} +513.028 q^{10} -631.897 q^{11} +374.873 q^{12} +28.5724 q^{13} -1809.15 q^{14} -708.560 q^{15} -972.709 q^{16} -1743.07 q^{17} +955.217 q^{18} -2027.92 q^{19} -2124.97 q^{20} +2498.68 q^{21} +5115.23 q^{22} +2980.86 q^{23} -138.447 q^{24} +891.469 q^{25} -231.295 q^{26} -4036.10 q^{27} +7493.51 q^{28} +766.139 q^{29} +5735.83 q^{30} -8355.33 q^{31} +8270.38 q^{32} -7064.81 q^{33} +14110.3 q^{34} -14163.7 q^{35} -3956.51 q^{36} +14892.6 q^{37} +16416.1 q^{38} +319.449 q^{39} +784.783 q^{40} -5342.20 q^{41} -20226.9 q^{42} -1849.00 q^{43} -21187.3 q^{44} +7478.34 q^{45} -24130.2 q^{46} -6282.09 q^{47} -10875.2 q^{48} +33140.1 q^{49} -7216.48 q^{50} -19488.1 q^{51} +958.024 q^{52} -915.172 q^{53} +32672.4 q^{54} +40046.8 q^{55} -2767.47 q^{56} -22672.8 q^{57} -6201.92 q^{58} -14644.5 q^{59} -23757.8 q^{60} -21324.9 q^{61} +67636.7 q^{62} -26371.7 q^{63} -35822.4 q^{64} -1810.79 q^{65} +57189.9 q^{66} -12868.9 q^{67} -58444.8 q^{68} +33327.0 q^{69} +114656. q^{70} +56454.6 q^{71} +1461.20 q^{72} -25591.3 q^{73} -120556. q^{74} +9966.92 q^{75} -67995.4 q^{76} -141222. q^{77} -2585.95 q^{78} +5795.13 q^{79} +61646.1 q^{80} -16450.9 q^{81} +43245.4 q^{82} -7857.24 q^{83} +83779.9 q^{84} +110468. q^{85} +14967.7 q^{86} +8565.68 q^{87} +7824.80 q^{88} -7560.11 q^{89} -60537.5 q^{90} +6385.60 q^{91} +99947.5 q^{92} -93415.3 q^{93} +50853.8 q^{94} +128520. q^{95} +92465.6 q^{96} +111712. q^{97} -268271. q^{98} +74563.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{2} - 26q^{3} + 122q^{4} - 212q^{5} - 69q^{6} - 136q^{7} - 666q^{8} + 546q^{9} + O(q^{10}) \) \( 8q - 12q^{2} - 26q^{3} + 122q^{4} - 212q^{5} - 69q^{6} - 136q^{7} - 666q^{8} + 546q^{9} - 617q^{10} - 532q^{11} - 4195q^{12} - 2492q^{13} - 4240q^{14} - 1780q^{15} + 1882q^{16} - 2534q^{17} - 3711q^{18} - 1678q^{19} - 2607q^{20} - 2256q^{21} + 11502q^{22} - 2488q^{23} + 19953q^{24} + 4378q^{25} + 4586q^{26} - 8960q^{27} + 18640q^{28} - 4360q^{29} + 25092q^{30} + 5704q^{31} - 18294q^{32} - 12852q^{33} + 30007q^{34} + 5640q^{35} + 67969q^{36} - 3772q^{37} - 6559q^{38} + 11120q^{39} + 14869q^{40} - 10698q^{41} + 78698q^{42} - 14792q^{43} - 356q^{44} - 44912q^{45} - 19389q^{46} - 77864q^{47} - 118727q^{48} + 7188q^{49} + 26877q^{50} - 80246q^{51} - 60736q^{52} - 62352q^{53} + 61026q^{54} - 49552q^{55} - 144528q^{56} - 808q^{57} + 52951q^{58} - 26224q^{59} + 101500q^{60} - 82540q^{61} - 9023q^{62} - 61768q^{63} + 153858q^{64} - 5000q^{65} - 48516q^{66} + 27784q^{67} + 40507q^{68} - 93776q^{69} + 185910q^{70} - 9504q^{71} - 186687q^{72} + 14260q^{73} - 15239q^{74} + 167420q^{75} + 1279q^{76} - 218140q^{77} + 264170q^{78} + 160248q^{79} - 1291q^{80} + 161076q^{81} - 47781q^{82} - 77176q^{83} + 16382q^{84} + 141096q^{85} + 22188q^{86} + 268136q^{87} + 129544q^{88} - 265692q^{89} + 48990q^{90} + 401148q^{91} + 190391q^{92} - 123860q^{93} + 248737q^{94} + 135884q^{95} + 950817q^{96} + 144742q^{97} - 292244q^{98} + 239516q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.09504 −1.43101 −0.715507 0.698605i \(-0.753804\pi\)
−0.715507 + 0.698605i \(0.753804\pi\)
\(3\) 11.1803 0.717218 0.358609 0.933488i \(-0.383251\pi\)
0.358609 + 0.933488i \(0.383251\pi\)
\(4\) 33.5297 1.04780
\(5\) −63.3756 −1.13370 −0.566849 0.823822i \(-0.691838\pi\)
−0.566849 + 0.823822i \(0.691838\pi\)
\(6\) −90.5052 −1.02635
\(7\) 223.489 1.72389 0.861946 0.507000i \(-0.169245\pi\)
0.861946 + 0.507000i \(0.169245\pi\)
\(8\) −12.3830 −0.0684073
\(9\) −118.000 −0.485598
\(10\) 513.028 1.62234
\(11\) −631.897 −1.57458 −0.787289 0.616584i \(-0.788516\pi\)
−0.787289 + 0.616584i \(0.788516\pi\)
\(12\) 374.873 0.751504
\(13\) 28.5724 0.0468909 0.0234454 0.999725i \(-0.492536\pi\)
0.0234454 + 0.999725i \(0.492536\pi\)
\(14\) −1809.15 −2.46692
\(15\) −708.560 −0.813109
\(16\) −972.709 −0.949911
\(17\) −1743.07 −1.46283 −0.731414 0.681933i \(-0.761139\pi\)
−0.731414 + 0.681933i \(0.761139\pi\)
\(18\) 955.217 0.694897
\(19\) −2027.92 −1.28874 −0.644371 0.764713i \(-0.722881\pi\)
−0.644371 + 0.764713i \(0.722881\pi\)
\(20\) −2124.97 −1.18789
\(21\) 2498.68 1.23641
\(22\) 5115.23 2.25324
\(23\) 2980.86 1.17496 0.587479 0.809239i \(-0.300120\pi\)
0.587479 + 0.809239i \(0.300120\pi\)
\(24\) −138.447 −0.0490630
\(25\) 891.469 0.285270
\(26\) −231.295 −0.0671015
\(27\) −4036.10 −1.06550
\(28\) 7493.51 1.80630
\(29\) 766.139 0.169166 0.0845829 0.996416i \(-0.473044\pi\)
0.0845829 + 0.996416i \(0.473044\pi\)
\(30\) 5735.83 1.16357
\(31\) −8355.33 −1.56156 −0.780781 0.624805i \(-0.785178\pi\)
−0.780781 + 0.624805i \(0.785178\pi\)
\(32\) 8270.38 1.42774
\(33\) −7064.81 −1.12932
\(34\) 14110.3 2.09333
\(35\) −14163.7 −1.95437
\(36\) −3956.51 −0.508811
\(37\) 14892.6 1.78841 0.894204 0.447661i \(-0.147743\pi\)
0.894204 + 0.447661i \(0.147743\pi\)
\(38\) 16416.1 1.84421
\(39\) 319.449 0.0336310
\(40\) 784.783 0.0775532
\(41\) −5342.20 −0.496319 −0.248159 0.968719i \(-0.579826\pi\)
−0.248159 + 0.968719i \(0.579826\pi\)
\(42\) −20226.9 −1.76932
\(43\) −1849.00 −0.152499
\(44\) −21187.3 −1.64985
\(45\) 7478.34 0.550521
\(46\) −24130.2 −1.68138
\(47\) −6282.09 −0.414820 −0.207410 0.978254i \(-0.566503\pi\)
−0.207410 + 0.978254i \(0.566503\pi\)
\(48\) −10875.2 −0.681294
\(49\) 33140.1 1.97181
\(50\) −7216.48 −0.408226
\(51\) −19488.1 −1.04917
\(52\) 958.024 0.0491324
\(53\) −915.172 −0.0447521 −0.0223760 0.999750i \(-0.507123\pi\)
−0.0223760 + 0.999750i \(0.507123\pi\)
\(54\) 32672.4 1.52474
\(55\) 40046.8 1.78510
\(56\) −2767.47 −0.117927
\(57\) −22672.8 −0.924310
\(58\) −6201.92 −0.242079
\(59\) −14644.5 −0.547704 −0.273852 0.961772i \(-0.588298\pi\)
−0.273852 + 0.961772i \(0.588298\pi\)
\(60\) −23757.8 −0.851978
\(61\) −21324.9 −0.733775 −0.366887 0.930265i \(-0.619577\pi\)
−0.366887 + 0.930265i \(0.619577\pi\)
\(62\) 67636.7 2.23462
\(63\) −26371.7 −0.837118
\(64\) −35822.4 −1.09321
\(65\) −1810.79 −0.0531601
\(66\) 57189.9 1.61607
\(67\) −12868.9 −0.350232 −0.175116 0.984548i \(-0.556030\pi\)
−0.175116 + 0.984548i \(0.556030\pi\)
\(68\) −58444.8 −1.53276
\(69\) 33327.0 0.842702
\(70\) 114656. 2.79674
\(71\) 56454.6 1.32909 0.664544 0.747249i \(-0.268626\pi\)
0.664544 + 0.747249i \(0.268626\pi\)
\(72\) 1461.20 0.0332184
\(73\) −25591.3 −0.562064 −0.281032 0.959698i \(-0.590677\pi\)
−0.281032 + 0.959698i \(0.590677\pi\)
\(74\) −120556. −2.55924
\(75\) 9966.92 0.204601
\(76\) −67995.4 −1.35035
\(77\) −141222. −2.71440
\(78\) −2585.95 −0.0481265
\(79\) 5795.13 0.104471 0.0522355 0.998635i \(-0.483365\pi\)
0.0522355 + 0.998635i \(0.483365\pi\)
\(80\) 61646.1 1.07691
\(81\) −16450.9 −0.278597
\(82\) 43245.4 0.710240
\(83\) −7857.24 −0.125192 −0.0625958 0.998039i \(-0.519938\pi\)
−0.0625958 + 0.998039i \(0.519938\pi\)
\(84\) 83779.9 1.29551
\(85\) 110468. 1.65841
\(86\) 14967.7 0.218228
\(87\) 8565.68 0.121329
\(88\) 7824.80 0.107713
\(89\) −7560.11 −0.101170 −0.0505852 0.998720i \(-0.516109\pi\)
−0.0505852 + 0.998720i \(0.516109\pi\)
\(90\) −60537.5 −0.787803
\(91\) 6385.60 0.0808348
\(92\) 99947.5 1.23113
\(93\) −93415.3 −1.11998
\(94\) 50853.8 0.593613
\(95\) 128520. 1.46104
\(96\) 92465.6 1.02400
\(97\) 111712. 1.20551 0.602754 0.797927i \(-0.294070\pi\)
0.602754 + 0.797927i \(0.294070\pi\)
\(98\) −268271. −2.82168
\(99\) 74563.9 0.764611
\(100\) 29890.7 0.298907
\(101\) 10492.3 0.102345 0.0511724 0.998690i \(-0.483704\pi\)
0.0511724 + 0.998690i \(0.483704\pi\)
\(102\) 157757. 1.50137
\(103\) −11286.6 −0.104826 −0.0524130 0.998625i \(-0.516691\pi\)
−0.0524130 + 0.998625i \(0.516691\pi\)
\(104\) −353.813 −0.00320768
\(105\) −158355. −1.40171
\(106\) 7408.36 0.0640409
\(107\) 39045.4 0.329693 0.164847 0.986319i \(-0.447287\pi\)
0.164847 + 0.986319i \(0.447287\pi\)
\(108\) −135329. −1.11643
\(109\) −14622.8 −0.117887 −0.0589433 0.998261i \(-0.518773\pi\)
−0.0589433 + 0.998261i \(0.518773\pi\)
\(110\) −324181. −2.55450
\(111\) 166504. 1.28268
\(112\) −217389. −1.63755
\(113\) −859.121 −0.00632934 −0.00316467 0.999995i \(-0.501007\pi\)
−0.00316467 + 0.999995i \(0.501007\pi\)
\(114\) 183537. 1.32270
\(115\) −188914. −1.33205
\(116\) 25688.4 0.177252
\(117\) −3371.55 −0.0227701
\(118\) 118548. 0.783772
\(119\) −389557. −2.52176
\(120\) 8774.13 0.0556226
\(121\) 238242. 1.47930
\(122\) 172626. 1.05004
\(123\) −59727.6 −0.355969
\(124\) −280152. −1.63621
\(125\) 141551. 0.810288
\(126\) 213480. 1.19793
\(127\) 193396. 1.06399 0.531997 0.846746i \(-0.321442\pi\)
0.531997 + 0.846746i \(0.321442\pi\)
\(128\) 25331.5 0.136658
\(129\) −20672.4 −0.109375
\(130\) 14658.4 0.0760728
\(131\) −269599. −1.37259 −0.686293 0.727325i \(-0.740763\pi\)
−0.686293 + 0.727325i \(0.740763\pi\)
\(132\) −236881. −1.18330
\(133\) −453216. −2.22165
\(134\) 104175. 0.501187
\(135\) 255790. 1.20795
\(136\) 21584.6 0.100068
\(137\) 148987. 0.678183 0.339091 0.940753i \(-0.389880\pi\)
0.339091 + 0.940753i \(0.389880\pi\)
\(138\) −269784. −1.20592
\(139\) −16355.6 −0.0718008 −0.0359004 0.999355i \(-0.511430\pi\)
−0.0359004 + 0.999355i \(0.511430\pi\)
\(140\) −474906. −2.04780
\(141\) −70235.8 −0.297516
\(142\) −457003. −1.90194
\(143\) −18054.8 −0.0738333
\(144\) 114780. 0.461275
\(145\) −48554.5 −0.191783
\(146\) 207163. 0.804322
\(147\) 370518. 1.41422
\(148\) 499345. 1.87390
\(149\) −150536. −0.555489 −0.277744 0.960655i \(-0.589587\pi\)
−0.277744 + 0.960655i \(0.589587\pi\)
\(150\) −80682.6 −0.292787
\(151\) 428141. 1.52807 0.764037 0.645172i \(-0.223214\pi\)
0.764037 + 0.645172i \(0.223214\pi\)
\(152\) 25111.8 0.0881594
\(153\) 205683. 0.710346
\(154\) 1.14320e6 3.88435
\(155\) 529524. 1.77034
\(156\) 10711.0 0.0352387
\(157\) −434190. −1.40582 −0.702912 0.711277i \(-0.748117\pi\)
−0.702912 + 0.711277i \(0.748117\pi\)
\(158\) −46911.8 −0.149499
\(159\) −10231.9 −0.0320970
\(160\) −524140. −1.61863
\(161\) 666189. 2.02550
\(162\) 133171. 0.398677
\(163\) −571830. −1.68577 −0.842884 0.538095i \(-0.819144\pi\)
−0.842884 + 0.538095i \(0.819144\pi\)
\(164\) −179122. −0.520044
\(165\) 447737. 1.28030
\(166\) 63604.7 0.179151
\(167\) 605744. 1.68073 0.840365 0.542020i \(-0.182340\pi\)
0.840365 + 0.542020i \(0.182340\pi\)
\(168\) −30941.2 −0.0845793
\(169\) −370477. −0.997801
\(170\) −894246. −2.37320
\(171\) 239295. 0.625810
\(172\) −61996.4 −0.159789
\(173\) −85238.8 −0.216532 −0.108266 0.994122i \(-0.534530\pi\)
−0.108266 + 0.994122i \(0.534530\pi\)
\(174\) −69339.6 −0.173623
\(175\) 199233. 0.491775
\(176\) 614652. 1.49571
\(177\) −163731. −0.392823
\(178\) 61199.4 0.144776
\(179\) 77941.8 0.181818 0.0909092 0.995859i \(-0.471023\pi\)
0.0909092 + 0.995859i \(0.471023\pi\)
\(180\) 250746. 0.576838
\(181\) −347661. −0.788787 −0.394393 0.918942i \(-0.629045\pi\)
−0.394393 + 0.918942i \(0.629045\pi\)
\(182\) −51691.7 −0.115676
\(183\) −238420. −0.526277
\(184\) −36912.2 −0.0803757
\(185\) −943828. −2.02751
\(186\) 756201. 1.60271
\(187\) 1.10144e6 2.30334
\(188\) −210637. −0.434650
\(189\) −902023. −1.83680
\(190\) −1.04038e6 −2.09078
\(191\) −192163. −0.381142 −0.190571 0.981673i \(-0.561034\pi\)
−0.190571 + 0.981673i \(0.561034\pi\)
\(192\) −400506. −0.784072
\(193\) −271724. −0.525091 −0.262546 0.964920i \(-0.584562\pi\)
−0.262546 + 0.964920i \(0.584562\pi\)
\(194\) −904313. −1.72510
\(195\) −20245.3 −0.0381274
\(196\) 1.11118e6 2.06606
\(197\) 76788.1 0.140971 0.0704853 0.997513i \(-0.477545\pi\)
0.0704853 + 0.997513i \(0.477545\pi\)
\(198\) −603598. −1.09417
\(199\) −694776. −1.24369 −0.621845 0.783140i \(-0.713617\pi\)
−0.621845 + 0.783140i \(0.713617\pi\)
\(200\) −11039.1 −0.0195146
\(201\) −143879. −0.251193
\(202\) −84935.3 −0.146457
\(203\) 171223. 0.291624
\(204\) −653432. −1.09932
\(205\) 338565. 0.562675
\(206\) 91365.3 0.150008
\(207\) −351743. −0.570557
\(208\) −27792.6 −0.0445422
\(209\) 1.28143e6 2.02923
\(210\) 1.28189e6 2.00587
\(211\) −433533. −0.670372 −0.335186 0.942152i \(-0.608799\pi\)
−0.335186 + 0.942152i \(0.608799\pi\)
\(212\) −30685.5 −0.0468914
\(213\) 631181. 0.953246
\(214\) −316074. −0.471796
\(215\) 117182. 0.172887
\(216\) 49979.2 0.0728878
\(217\) −1.86732e6 −2.69196
\(218\) 118372. 0.168698
\(219\) −286119. −0.403123
\(220\) 1.34276e6 1.87043
\(221\) −49803.8 −0.0685933
\(222\) −1.34786e6 −1.83553
\(223\) 1.04204e6 1.40320 0.701602 0.712569i \(-0.252469\pi\)
0.701602 + 0.712569i \(0.252469\pi\)
\(224\) 1.84834e6 2.46128
\(225\) −105194. −0.138526
\(226\) 6954.62 0.00905737
\(227\) 924970. 1.19141 0.595707 0.803202i \(-0.296872\pi\)
0.595707 + 0.803202i \(0.296872\pi\)
\(228\) −760211. −0.968495
\(229\) −137140. −0.172812 −0.0864062 0.996260i \(-0.527538\pi\)
−0.0864062 + 0.996260i \(0.527538\pi\)
\(230\) 1.52927e6 1.90618
\(231\) −1.57890e6 −1.94682
\(232\) −9487.13 −0.0115722
\(233\) 428602. 0.517208 0.258604 0.965983i \(-0.416738\pi\)
0.258604 + 0.965983i \(0.416738\pi\)
\(234\) 27292.8 0.0325843
\(235\) 398131. 0.470280
\(236\) −491027. −0.573886
\(237\) 64791.5 0.0749285
\(238\) 3.15348e6 3.60868
\(239\) −1.17134e6 −1.32645 −0.663223 0.748422i \(-0.730812\pi\)
−0.663223 + 0.748422i \(0.730812\pi\)
\(240\) 689223. 0.772381
\(241\) 119252. 0.132258 0.0661291 0.997811i \(-0.478935\pi\)
0.0661291 + 0.997811i \(0.478935\pi\)
\(242\) −1.92858e6 −2.11690
\(243\) 796846. 0.865683
\(244\) −715018. −0.768852
\(245\) −2.10028e6 −2.23543
\(246\) 483497. 0.509397
\(247\) −57942.4 −0.0604302
\(248\) 103464. 0.106822
\(249\) −87846.6 −0.0897897
\(250\) −1.14586e6 −1.15953
\(251\) −141272. −0.141538 −0.0707690 0.997493i \(-0.522545\pi\)
−0.0707690 + 0.997493i \(0.522545\pi\)
\(252\) −884235. −0.877135
\(253\) −1.88360e6 −1.85006
\(254\) −1.56555e6 −1.52259
\(255\) 1.23507e6 1.18944
\(256\) 941257. 0.897652
\(257\) −196749. −0.185815 −0.0929074 0.995675i \(-0.529616\pi\)
−0.0929074 + 0.995675i \(0.529616\pi\)
\(258\) 167344. 0.156517
\(259\) 3.32833e6 3.08302
\(260\) −60715.4 −0.0557013
\(261\) −90404.5 −0.0821465
\(262\) 2.18241e6 1.96419
\(263\) −1.96879e6 −1.75513 −0.877565 0.479457i \(-0.840834\pi\)
−0.877565 + 0.479457i \(0.840834\pi\)
\(264\) 87483.9 0.0772535
\(265\) 57999.6 0.0507353
\(266\) 3.66880e6 3.17922
\(267\) −84524.6 −0.0725612
\(268\) −431492. −0.366974
\(269\) −531007. −0.447424 −0.223712 0.974655i \(-0.571818\pi\)
−0.223712 + 0.974655i \(0.571818\pi\)
\(270\) −2.07063e6 −1.72860
\(271\) −1.68769e6 −1.39595 −0.697975 0.716122i \(-0.745916\pi\)
−0.697975 + 0.716122i \(0.745916\pi\)
\(272\) 1.69550e6 1.38956
\(273\) 71393.2 0.0579762
\(274\) −1.20606e6 −0.970489
\(275\) −563316. −0.449180
\(276\) 1.11745e6 0.882986
\(277\) 13630.1 0.0106733 0.00533665 0.999986i \(-0.498301\pi\)
0.00533665 + 0.999986i \(0.498301\pi\)
\(278\) 132399. 0.102748
\(279\) 985930. 0.758291
\(280\) 175390. 0.133693
\(281\) 1.12070e6 0.846692 0.423346 0.905968i \(-0.360855\pi\)
0.423346 + 0.905968i \(0.360855\pi\)
\(282\) 568562. 0.425750
\(283\) −1.52947e6 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(284\) 1.89291e6 1.39262
\(285\) 1.43690e6 1.04789
\(286\) 146154. 0.105657
\(287\) −1.19392e6 −0.855600
\(288\) −975907. −0.693309
\(289\) 1.61845e6 1.13987
\(290\) 393051. 0.274444
\(291\) 1.24898e6 0.864613
\(292\) −858070. −0.588932
\(293\) −1.58484e6 −1.07849 −0.539246 0.842148i \(-0.681291\pi\)
−0.539246 + 0.842148i \(0.681291\pi\)
\(294\) −2.99936e6 −2.02376
\(295\) 928107. 0.620930
\(296\) −184416. −0.122340
\(297\) 2.55040e6 1.67771
\(298\) 1.21860e6 0.794912
\(299\) 85170.4 0.0550948
\(300\) 334188. 0.214382
\(301\) −413230. −0.262891
\(302\) −3.46582e6 −2.18670
\(303\) 117307. 0.0734035
\(304\) 1.97257e6 1.22419
\(305\) 1.35148e6 0.831879
\(306\) −1.66501e6 −1.01652
\(307\) 695751. 0.421316 0.210658 0.977560i \(-0.432439\pi\)
0.210658 + 0.977560i \(0.432439\pi\)
\(308\) −4.73512e6 −2.84416
\(309\) −126188. −0.0751831
\(310\) −4.28652e6 −2.53338
\(311\) −988763. −0.579684 −0.289842 0.957075i \(-0.593603\pi\)
−0.289842 + 0.957075i \(0.593603\pi\)
\(312\) −3955.75 −0.00230061
\(313\) −3.07551e6 −1.77442 −0.887208 0.461369i \(-0.847358\pi\)
−0.887208 + 0.461369i \(0.847358\pi\)
\(314\) 3.51479e6 2.01176
\(315\) 1.67132e6 0.949039
\(316\) 194309. 0.109465
\(317\) 904527. 0.505561 0.252780 0.967524i \(-0.418655\pi\)
0.252780 + 0.967524i \(0.418655\pi\)
\(318\) 82827.9 0.0459313
\(319\) −484120. −0.266365
\(320\) 2.27027e6 1.23937
\(321\) 436540. 0.236462
\(322\) −5.39283e6 −2.89852
\(323\) 3.53481e6 1.88521
\(324\) −551594. −0.291915
\(325\) 25471.4 0.0133766
\(326\) 4.62899e6 2.41236
\(327\) −163488. −0.0845505
\(328\) 66152.7 0.0339518
\(329\) −1.40398e6 −0.715105
\(330\) −3.62445e6 −1.83213
\(331\) −2.09309e6 −1.05007 −0.525034 0.851081i \(-0.675947\pi\)
−0.525034 + 0.851081i \(0.675947\pi\)
\(332\) −263451. −0.131176
\(333\) −1.75733e6 −0.868446
\(334\) −4.90353e6 −2.40515
\(335\) 815577. 0.397057
\(336\) −2.43049e6 −1.17448
\(337\) −647484. −0.310566 −0.155283 0.987870i \(-0.549629\pi\)
−0.155283 + 0.987870i \(0.549629\pi\)
\(338\) 2.99902e6 1.42787
\(339\) −9605.25 −0.00453952
\(340\) 3.70397e6 1.73768
\(341\) 5.27970e6 2.45880
\(342\) −1.93710e6 −0.895544
\(343\) 3.65027e6 1.67529
\(344\) 22896.2 0.0104320
\(345\) −2.11212e6 −0.955369
\(346\) 690012. 0.309860
\(347\) −462018. −0.205985 −0.102992 0.994682i \(-0.532842\pi\)
−0.102992 + 0.994682i \(0.532842\pi\)
\(348\) 287205. 0.127129
\(349\) 2.85075e6 1.25284 0.626420 0.779486i \(-0.284520\pi\)
0.626420 + 0.779486i \(0.284520\pi\)
\(350\) −1.61280e6 −0.703737
\(351\) −115321. −0.0499621
\(352\) −5.22602e6 −2.24810
\(353\) −809444. −0.345740 −0.172870 0.984945i \(-0.555304\pi\)
−0.172870 + 0.984945i \(0.555304\pi\)
\(354\) 1.32541e6 0.562136
\(355\) −3.57785e6 −1.50678
\(356\) −253488. −0.106007
\(357\) −4.35538e6 −1.80865
\(358\) −630942. −0.260185
\(359\) 3.46118e6 1.41739 0.708694 0.705516i \(-0.249285\pi\)
0.708694 + 0.705516i \(0.249285\pi\)
\(360\) −92604.6 −0.0376596
\(361\) 1.63635e6 0.660856
\(362\) 2.81433e6 1.12877
\(363\) 2.66363e6 1.06098
\(364\) 214107. 0.0846990
\(365\) 1.62187e6 0.637210
\(366\) 1.93002e6 0.753110
\(367\) 2.13602e6 0.827827 0.413914 0.910316i \(-0.364162\pi\)
0.413914 + 0.910316i \(0.364162\pi\)
\(368\) −2.89951e6 −1.11611
\(369\) 630381. 0.241011
\(370\) 7.64033e6 2.90140
\(371\) −204531. −0.0771478
\(372\) −3.13219e6 −1.17352
\(373\) 2.37193e6 0.882734 0.441367 0.897327i \(-0.354494\pi\)
0.441367 + 0.897327i \(0.354494\pi\)
\(374\) −8.91622e6 −3.29611
\(375\) 1.58259e6 0.581153
\(376\) 77791.4 0.0283767
\(377\) 21890.4 0.00793233
\(378\) 7.30191e6 2.62849
\(379\) −5.47562e6 −1.95810 −0.979051 0.203616i \(-0.934731\pi\)
−0.979051 + 0.203616i \(0.934731\pi\)
\(380\) 4.30925e6 1.53089
\(381\) 2.16224e6 0.763116
\(382\) 1.55557e6 0.545419
\(383\) 3.16966e6 1.10412 0.552059 0.833805i \(-0.313842\pi\)
0.552059 + 0.833805i \(0.313842\pi\)
\(384\) 283214. 0.0980138
\(385\) 8.95001e6 3.07731
\(386\) 2.19962e6 0.751413
\(387\) 218182. 0.0740529
\(388\) 3.74567e6 1.26314
\(389\) −311235. −0.104283 −0.0521416 0.998640i \(-0.516605\pi\)
−0.0521416 + 0.998640i \(0.516605\pi\)
\(390\) 163886. 0.0545609
\(391\) −5.19587e6 −1.71876
\(392\) −410376. −0.134886
\(393\) −3.01420e6 −0.984444
\(394\) −621603. −0.201731
\(395\) −367270. −0.118438
\(396\) 2.50011e6 0.801162
\(397\) −496314. −0.158045 −0.0790224 0.996873i \(-0.525180\pi\)
−0.0790224 + 0.996873i \(0.525180\pi\)
\(398\) 5.62424e6 1.77974
\(399\) −5.06711e6 −1.59341
\(400\) −867140. −0.270981
\(401\) −5.63670e6 −1.75051 −0.875253 0.483665i \(-0.839305\pi\)
−0.875253 + 0.483665i \(0.839305\pi\)
\(402\) 1.16471e6 0.359461
\(403\) −238732. −0.0732230
\(404\) 351802. 0.107237
\(405\) 1.04259e6 0.315845
\(406\) −1.38606e6 −0.417318
\(407\) −9.41059e6 −2.81599
\(408\) 241323. 0.0717707
\(409\) −720515. −0.212978 −0.106489 0.994314i \(-0.533961\pi\)
−0.106489 + 0.994314i \(0.533961\pi\)
\(410\) −2.74070e6 −0.805197
\(411\) 1.66572e6 0.486405
\(412\) −378435. −0.109837
\(413\) −3.27289e6 −0.944182
\(414\) 2.84737e6 0.816476
\(415\) 497958. 0.141929
\(416\) 236305. 0.0669482
\(417\) −182861. −0.0514969
\(418\) −1.03733e7 −2.90385
\(419\) −3.94151e6 −1.09680 −0.548400 0.836216i \(-0.684763\pi\)
−0.548400 + 0.836216i \(0.684763\pi\)
\(420\) −5.30960e6 −1.46872
\(421\) 4.39506e6 1.20854 0.604268 0.796781i \(-0.293466\pi\)
0.604268 + 0.796781i \(0.293466\pi\)
\(422\) 3.50947e6 0.959312
\(423\) 741288. 0.201436
\(424\) 11332.6 0.00306137
\(425\) −1.55390e6 −0.417301
\(426\) −5.10944e6 −1.36411
\(427\) −4.76588e6 −1.26495
\(428\) 1.30918e6 0.345454
\(429\) −201859. −0.0529546
\(430\) −948589. −0.247404
\(431\) 1.24913e6 0.323903 0.161951 0.986799i \(-0.448221\pi\)
0.161951 + 0.986799i \(0.448221\pi\)
\(432\) 3.92595e6 1.01213
\(433\) 971902. 0.249117 0.124558 0.992212i \(-0.460249\pi\)
0.124558 + 0.992212i \(0.460249\pi\)
\(434\) 1.51160e7 3.85224
\(435\) −542855. −0.137550
\(436\) −490299. −0.123522
\(437\) −6.04494e6 −1.51422
\(438\) 2.31615e6 0.576874
\(439\) −1.32453e6 −0.328021 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(440\) −495902. −0.122114
\(441\) −3.91054e6 −0.957504
\(442\) 403164. 0.0981581
\(443\) 4.76446e6 1.15346 0.576732 0.816933i \(-0.304328\pi\)
0.576732 + 0.816933i \(0.304328\pi\)
\(444\) 5.58284e6 1.34399
\(445\) 479127. 0.114697
\(446\) −8.43533e6 −2.00801
\(447\) −1.68304e6 −0.398407
\(448\) −8.00589e6 −1.88458
\(449\) −5.40007e6 −1.26411 −0.632053 0.774926i \(-0.717787\pi\)
−0.632053 + 0.774926i \(0.717787\pi\)
\(450\) 851546. 0.198233
\(451\) 3.37572e6 0.781493
\(452\) −28806.1 −0.00663190
\(453\) 4.78676e6 1.09596
\(454\) −7.48767e6 −1.70493
\(455\) −404692. −0.0916422
\(456\) 280758. 0.0632295
\(457\) 2.12752e6 0.476523 0.238261 0.971201i \(-0.423423\pi\)
0.238261 + 0.971201i \(0.423423\pi\)
\(458\) 1.11015e6 0.247297
\(459\) 7.03522e6 1.55864
\(460\) −6.33423e6 −1.39572
\(461\) 2.80905e6 0.615612 0.307806 0.951449i \(-0.400405\pi\)
0.307806 + 0.951449i \(0.400405\pi\)
\(462\) 1.27813e7 2.78593
\(463\) −2.85342e6 −0.618605 −0.309302 0.950964i \(-0.600096\pi\)
−0.309302 + 0.950964i \(0.600096\pi\)
\(464\) −745230. −0.160692
\(465\) 5.92025e6 1.26972
\(466\) −3.46955e6 −0.740132
\(467\) −559463. −0.118708 −0.0593539 0.998237i \(-0.518904\pi\)
−0.0593539 + 0.998237i \(0.518904\pi\)
\(468\) −113047. −0.0238586
\(469\) −2.87606e6 −0.603762
\(470\) −3.22289e6 −0.672978
\(471\) −4.85439e6 −1.00828
\(472\) 181344. 0.0374669
\(473\) 1.16838e6 0.240121
\(474\) −524490. −0.107224
\(475\) −1.80782e6 −0.367640
\(476\) −1.30617e7 −2.64231
\(477\) 107991. 0.0217315
\(478\) 9.48208e6 1.89816
\(479\) 5.46747e6 1.08880 0.544399 0.838826i \(-0.316758\pi\)
0.544399 + 0.838826i \(0.316758\pi\)
\(480\) −5.86006e6 −1.16091
\(481\) 425517. 0.0838600
\(482\) −965349. −0.189263
\(483\) 7.44821e6 1.45273
\(484\) 7.98819e6 1.55001
\(485\) −7.07981e6 −1.36668
\(486\) −6.45050e6 −1.23881
\(487\) 9.34759e6 1.78598 0.892991 0.450074i \(-0.148603\pi\)
0.892991 + 0.450074i \(0.148603\pi\)
\(488\) 264067. 0.0501955
\(489\) −6.39325e6 −1.20906
\(490\) 1.70018e7 3.19893
\(491\) −1.04964e7 −1.96488 −0.982438 0.186589i \(-0.940257\pi\)
−0.982438 + 0.186589i \(0.940257\pi\)
\(492\) −2.00265e6 −0.372986
\(493\) −1.33544e6 −0.247460
\(494\) 469046. 0.0864766
\(495\) −4.72554e6 −0.866838
\(496\) 8.12730e6 1.48335
\(497\) 1.26170e7 2.29120
\(498\) 711122. 0.128490
\(499\) 5.74842e6 1.03347 0.516734 0.856146i \(-0.327148\pi\)
0.516734 + 0.856146i \(0.327148\pi\)
\(500\) 4.74618e6 0.849022
\(501\) 6.77242e6 1.20545
\(502\) 1.14361e6 0.202543
\(503\) 3.10341e6 0.546915 0.273457 0.961884i \(-0.411833\pi\)
0.273457 + 0.961884i \(0.411833\pi\)
\(504\) 326562. 0.0572650
\(505\) −664953. −0.116028
\(506\) 1.52478e7 2.64747
\(507\) −4.14205e6 −0.715642
\(508\) 6.48453e6 1.11486
\(509\) −7.92714e6 −1.35620 −0.678098 0.734972i \(-0.737195\pi\)
−0.678098 + 0.734972i \(0.737195\pi\)
\(510\) −9.99797e6 −1.70210
\(511\) −5.71937e6 −0.968938
\(512\) −8.43012e6 −1.42121
\(513\) 8.18488e6 1.37315
\(514\) 1.59269e6 0.265904
\(515\) 715293. 0.118841
\(516\) −693141. −0.114603
\(517\) 3.96963e6 0.653166
\(518\) −2.69429e7 −4.41185
\(519\) −952998. −0.155301
\(520\) 22423.1 0.00363654
\(521\) −4.37136e6 −0.705541 −0.352771 0.935710i \(-0.614760\pi\)
−0.352771 + 0.935710i \(0.614760\pi\)
\(522\) 731828. 0.117553
\(523\) 1.11680e6 0.178534 0.0892671 0.996008i \(-0.471548\pi\)
0.0892671 + 0.996008i \(0.471548\pi\)
\(524\) −9.03957e6 −1.43820
\(525\) 2.22749e6 0.352710
\(526\) 1.59374e7 2.51162
\(527\) 1.45640e7 2.28430
\(528\) 6.87201e6 1.07275
\(529\) 2.44920e6 0.380527
\(530\) −469509. −0.0726030
\(531\) 1.72806e6 0.265964
\(532\) −1.51962e7 −2.32786
\(533\) −152640. −0.0232728
\(534\) 684230. 0.103836
\(535\) −2.47453e6 −0.373773
\(536\) 159357. 0.0239584
\(537\) 871415. 0.130403
\(538\) 4.29852e6 0.640271
\(539\) −2.09411e7 −3.10476
\(540\) 8.57658e6 1.26570
\(541\) −6212.12 −0.000912528 0 −0.000456264 1.00000i \(-0.500145\pi\)
−0.000456264 1.00000i \(0.500145\pi\)
\(542\) 1.36619e7 1.99763
\(543\) −3.88696e6 −0.565732
\(544\) −1.44159e7 −2.08855
\(545\) 926730. 0.133648
\(546\) −577931. −0.0829648
\(547\) −9.82839e6 −1.40448 −0.702238 0.711942i \(-0.747816\pi\)
−0.702238 + 0.711942i \(0.747816\pi\)
\(548\) 4.99549e6 0.710602
\(549\) 2.51635e6 0.356319
\(550\) 4.56007e6 0.642783
\(551\) −1.55367e6 −0.218011
\(552\) −412690. −0.0576470
\(553\) 1.29515e6 0.180097
\(554\) −110336. −0.0152737
\(555\) −1.05523e7 −1.45417
\(556\) −548398. −0.0752332
\(557\) −6.27045e6 −0.856369 −0.428184 0.903691i \(-0.640847\pi\)
−0.428184 + 0.903691i \(0.640847\pi\)
\(558\) −7.98115e6 −1.08513
\(559\) −52830.4 −0.00715079
\(560\) 1.37772e7 1.85648
\(561\) 1.23145e7 1.65200
\(562\) −9.07215e6 −1.21163
\(563\) −6.71129e6 −0.892350 −0.446175 0.894946i \(-0.647214\pi\)
−0.446175 + 0.894946i \(0.647214\pi\)
\(564\) −2.35499e6 −0.311739
\(565\) 54447.3 0.00717555
\(566\) 1.23812e7 1.62450
\(567\) −3.67659e6 −0.480272
\(568\) −699080. −0.0909193
\(569\) −7.01697e6 −0.908592 −0.454296 0.890851i \(-0.650109\pi\)
−0.454296 + 0.890851i \(0.650109\pi\)
\(570\) −1.16318e7 −1.49954
\(571\) 1.61627e6 0.207455 0.103727 0.994606i \(-0.466923\pi\)
0.103727 + 0.994606i \(0.466923\pi\)
\(572\) −605372. −0.0773628
\(573\) −2.14845e6 −0.273362
\(574\) 9.66484e6 1.22438
\(575\) 2.65735e6 0.335181
\(576\) 4.22705e6 0.530861
\(577\) 1.08712e7 1.35937 0.679686 0.733503i \(-0.262116\pi\)
0.679686 + 0.733503i \(0.262116\pi\)
\(578\) −1.31014e7 −1.63117
\(579\) −3.03796e6 −0.376605
\(580\) −1.62802e6 −0.200951
\(581\) −1.75600e6 −0.215817
\(582\) −1.01105e7 −1.23727
\(583\) 578294. 0.0704656
\(584\) 316898. 0.0384493
\(585\) 213674. 0.0258144
\(586\) 1.28294e7 1.54334
\(587\) 8.65320e6 1.03653 0.518264 0.855221i \(-0.326578\pi\)
0.518264 + 0.855221i \(0.326578\pi\)
\(588\) 1.24233e7 1.48182
\(589\) 1.69439e7 2.01245
\(590\) −7.51306e6 −0.888560
\(591\) 858517. 0.101107
\(592\) −1.44862e7 −1.69883
\(593\) 5.13574e6 0.599745 0.299872 0.953979i \(-0.403056\pi\)
0.299872 + 0.953979i \(0.403056\pi\)
\(594\) −2.06456e7 −2.40083
\(595\) 2.46884e7 2.85891
\(596\) −5.04743e6 −0.582043
\(597\) −7.76782e6 −0.891997
\(598\) −689458. −0.0788415
\(599\) 7.56853e6 0.861875 0.430937 0.902382i \(-0.358183\pi\)
0.430937 + 0.902382i \(0.358183\pi\)
\(600\) −123421. −0.0139962
\(601\) 2.56891e6 0.290110 0.145055 0.989424i \(-0.453664\pi\)
0.145055 + 0.989424i \(0.453664\pi\)
\(602\) 3.34512e6 0.376201
\(603\) 1.51854e6 0.170072
\(604\) 1.43554e7 1.60112
\(605\) −1.50987e7 −1.67707
\(606\) −949604. −0.105042
\(607\) 2.62172e6 0.288812 0.144406 0.989519i \(-0.453873\pi\)
0.144406 + 0.989519i \(0.453873\pi\)
\(608\) −1.67716e7 −1.83999
\(609\) 1.91433e6 0.209158
\(610\) −1.09403e7 −1.19043
\(611\) −179494. −0.0194513
\(612\) 6.89650e6 0.744303
\(613\) 537167. 0.0577376 0.0288688 0.999583i \(-0.490810\pi\)
0.0288688 + 0.999583i \(0.490810\pi\)
\(614\) −5.63213e6 −0.602909
\(615\) 3.78527e6 0.403561
\(616\) 1.74875e6 0.185685
\(617\) 1.15522e7 1.22166 0.610830 0.791762i \(-0.290836\pi\)
0.610830 + 0.791762i \(0.290836\pi\)
\(618\) 1.02149e6 0.107588
\(619\) 2.20752e6 0.231568 0.115784 0.993274i \(-0.463062\pi\)
0.115784 + 0.993274i \(0.463062\pi\)
\(620\) 1.77548e7 1.85497
\(621\) −1.20311e7 −1.25192
\(622\) 8.00408e6 0.829536
\(623\) −1.68960e6 −0.174407
\(624\) −310731. −0.0319465
\(625\) −1.17567e7 −1.20389
\(626\) 2.48963e7 2.53922
\(627\) 1.43268e7 1.45540
\(628\) −1.45583e7 −1.47303
\(629\) −2.59589e7 −2.61613
\(630\) −1.35294e7 −1.35809
\(631\) −1.43943e7 −1.43919 −0.719594 0.694395i \(-0.755672\pi\)
−0.719594 + 0.694395i \(0.755672\pi\)
\(632\) −71761.3 −0.00714657
\(633\) −4.84704e6 −0.480803
\(634\) −7.32219e6 −0.723465
\(635\) −1.22566e7 −1.20625
\(636\) −343074. −0.0336314
\(637\) 946893. 0.0924597
\(638\) 3.91897e6 0.381172
\(639\) −6.66166e6 −0.645402
\(640\) −1.60540e6 −0.154929
\(641\) 4.90224e6 0.471248 0.235624 0.971844i \(-0.424287\pi\)
0.235624 + 0.971844i \(0.424287\pi\)
\(642\) −3.53381e6 −0.338381
\(643\) −1.80253e7 −1.71931 −0.859656 0.510874i \(-0.829322\pi\)
−0.859656 + 0.510874i \(0.829322\pi\)
\(644\) 2.23371e7 2.12233
\(645\) 1.31013e6 0.123998
\(646\) −2.86144e7 −2.69776
\(647\) −1.24242e7 −1.16683 −0.583415 0.812174i \(-0.698284\pi\)
−0.583415 + 0.812174i \(0.698284\pi\)
\(648\) 203712. 0.0190581
\(649\) 9.25383e6 0.862402
\(650\) −206192. −0.0191421
\(651\) −2.08772e7 −1.93073
\(652\) −1.91733e7 −1.76635
\(653\) −1.28553e7 −1.17977 −0.589887 0.807486i \(-0.700828\pi\)
−0.589887 + 0.807486i \(0.700828\pi\)
\(654\) 1.32344e6 0.120993
\(655\) 1.70860e7 1.55610
\(656\) 5.19641e6 0.471459
\(657\) 3.01978e6 0.272937
\(658\) 1.13652e7 1.02333
\(659\) −5.18218e6 −0.464835 −0.232417 0.972616i \(-0.574664\pi\)
−0.232417 + 0.972616i \(0.574664\pi\)
\(660\) 1.50125e7 1.34151
\(661\) 6.47130e6 0.576087 0.288043 0.957617i \(-0.406995\pi\)
0.288043 + 0.957617i \(0.406995\pi\)
\(662\) 1.69436e7 1.50266
\(663\) −556823. −0.0491964
\(664\) 97296.6 0.00856402
\(665\) 2.87229e7 2.51868
\(666\) 1.42257e7 1.24276
\(667\) 2.28375e6 0.198763
\(668\) 2.03104e7 1.76108
\(669\) 1.16503e7 1.00640
\(670\) −6.60213e6 −0.568194
\(671\) 1.34751e7 1.15539
\(672\) 2.06650e7 1.76527
\(673\) 6.21171e6 0.528657 0.264328 0.964433i \(-0.414850\pi\)
0.264328 + 0.964433i \(0.414850\pi\)
\(674\) 5.24141e6 0.444425
\(675\) −3.59806e6 −0.303955
\(676\) −1.24220e7 −1.04550
\(677\) 1.85078e7 1.55197 0.775984 0.630752i \(-0.217253\pi\)
0.775984 + 0.630752i \(0.217253\pi\)
\(678\) 77754.9 0.00649611
\(679\) 2.49663e7 2.07817
\(680\) −1.36793e6 −0.113447
\(681\) 1.03415e7 0.854504
\(682\) −4.27394e7 −3.51858
\(683\) −4.91590e6 −0.403228 −0.201614 0.979465i \(-0.564619\pi\)
−0.201614 + 0.979465i \(0.564619\pi\)
\(684\) 8.02348e6 0.655726
\(685\) −9.44214e6 −0.768854
\(686\) −2.95491e7 −2.39736
\(687\) −1.53327e6 −0.123944
\(688\) 1.79854e6 0.144860
\(689\) −26148.7 −0.00209846
\(690\) 1.70977e7 1.36715
\(691\) −1.07645e7 −0.857630 −0.428815 0.903392i \(-0.641069\pi\)
−0.428815 + 0.903392i \(0.641069\pi\)
\(692\) −2.85803e6 −0.226883
\(693\) 1.66642e7 1.31811
\(694\) 3.74005e6 0.294767
\(695\) 1.03655e6 0.0814004
\(696\) −106069. −0.00829977
\(697\) 9.31185e6 0.726029
\(698\) −2.30769e7 −1.79283
\(699\) 4.79192e6 0.370951
\(700\) 6.68023e6 0.515284
\(701\) 9.02819e6 0.693914 0.346957 0.937881i \(-0.387215\pi\)
0.346957 + 0.937881i \(0.387215\pi\)
\(702\) 933529. 0.0714966
\(703\) −3.02010e7 −2.30480
\(704\) 2.26360e7 1.72135
\(705\) 4.45124e6 0.337294
\(706\) 6.55248e6 0.494760
\(707\) 2.34490e6 0.176431
\(708\) −5.48984e6 −0.411601
\(709\) 4.91059e6 0.366875 0.183438 0.983031i \(-0.441277\pi\)
0.183438 + 0.983031i \(0.441277\pi\)
\(710\) 2.89628e7 2.15623
\(711\) −683827. −0.0507308
\(712\) 93617.2 0.00692079
\(713\) −2.49061e7 −1.83477
\(714\) 3.52570e7 2.58821
\(715\) 1.14423e6 0.0837047
\(716\) 2.61336e6 0.190510
\(717\) −1.30960e7 −0.951352
\(718\) −2.80184e7 −2.02830
\(719\) 1.44751e7 1.04424 0.522119 0.852873i \(-0.325142\pi\)
0.522119 + 0.852873i \(0.325142\pi\)
\(720\) −7.27425e6 −0.522946
\(721\) −2.52242e6 −0.180709
\(722\) −1.32463e7 −0.945695
\(723\) 1.33328e6 0.0948580
\(724\) −1.16570e7 −0.826493
\(725\) 682989. 0.0482579
\(726\) −2.15622e7 −1.51828
\(727\) 1.55189e7 1.08899 0.544495 0.838764i \(-0.316721\pi\)
0.544495 + 0.838764i \(0.316721\pi\)
\(728\) −79073.2 −0.00552969
\(729\) 1.29066e7 0.899481
\(730\) −1.31291e7 −0.911857
\(731\) 3.22294e6 0.223079
\(732\) −7.99414e6 −0.551435
\(733\) 1.03870e6 0.0714050 0.0357025 0.999362i \(-0.488633\pi\)
0.0357025 + 0.999362i \(0.488633\pi\)
\(734\) −1.72912e7 −1.18463
\(735\) −2.34818e7 −1.60329
\(736\) 2.46529e7 1.67754
\(737\) 8.13184e6 0.551467
\(738\) −5.10296e6 −0.344891
\(739\) 1.33918e6 0.0902046 0.0451023 0.998982i \(-0.485639\pi\)
0.0451023 + 0.998982i \(0.485639\pi\)
\(740\) −3.16463e7 −2.12443
\(741\) −647816. −0.0433417
\(742\) 1.65568e6 0.110400
\(743\) 1.56518e7 1.04014 0.520070 0.854124i \(-0.325906\pi\)
0.520070 + 0.854124i \(0.325906\pi\)
\(744\) 1.15677e6 0.0766149
\(745\) 9.54032e6 0.629756
\(746\) −1.92009e7 −1.26321
\(747\) 927157. 0.0607927
\(748\) 3.69310e7 2.41345
\(749\) 8.72620e6 0.568356
\(750\) −1.28111e7 −0.831639
\(751\) 1.92274e7 1.24400 0.622002 0.783016i \(-0.286320\pi\)
0.622002 + 0.783016i \(0.286320\pi\)
\(752\) 6.11065e6 0.394042
\(753\) −1.57947e6 −0.101514
\(754\) −177204. −0.0113513
\(755\) −2.71337e7 −1.73237
\(756\) −3.02446e7 −1.92461
\(757\) −2.73627e7 −1.73548 −0.867739 0.497021i \(-0.834427\pi\)
−0.867739 + 0.497021i \(0.834427\pi\)
\(758\) 4.43254e7 2.80207
\(759\) −2.10592e7 −1.32690
\(760\) −1.59147e6 −0.0999461
\(761\) 5.86301e6 0.366994 0.183497 0.983020i \(-0.441258\pi\)
0.183497 + 0.983020i \(0.441258\pi\)
\(762\) −1.75034e7 −1.09203
\(763\) −3.26803e6 −0.203224
\(764\) −6.44317e6 −0.399361
\(765\) −1.30353e7 −0.805318
\(766\) −2.56585e7 −1.58001
\(767\) −418430. −0.0256823
\(768\) 1.05236e7 0.643813
\(769\) −1.36162e7 −0.830308 −0.415154 0.909751i \(-0.636272\pi\)
−0.415154 + 0.909751i \(0.636272\pi\)
\(770\) −7.24507e7 −4.40368
\(771\) −2.19972e6 −0.133270
\(772\) −9.11083e6 −0.550192
\(773\) −9.44787e6 −0.568703 −0.284351 0.958720i \(-0.591778\pi\)
−0.284351 + 0.958720i \(0.591778\pi\)
\(774\) −1.76620e6 −0.105971
\(775\) −7.44851e6 −0.445467
\(776\) −1.38333e6 −0.0824655
\(777\) 3.72118e7 2.21120
\(778\) 2.51946e6 0.149231
\(779\) 1.08335e7 0.639627
\(780\) −678818. −0.0399500
\(781\) −3.56735e7 −2.09275
\(782\) 4.20608e7 2.45958
\(783\) −3.09221e6 −0.180246
\(784\) −3.22357e7 −1.87304
\(785\) 2.75171e7 1.59378
\(786\) 2.44001e7 1.40875
\(787\) 2.97678e7 1.71320 0.856602 0.515977i \(-0.172571\pi\)
0.856602 + 0.515977i \(0.172571\pi\)
\(788\) 2.57468e6 0.147709
\(789\) −2.20117e7 −1.25881
\(790\) 2.97307e6 0.169487
\(791\) −192004. −0.0109111
\(792\) −923328. −0.0523050
\(793\) −609304. −0.0344073
\(794\) 4.01768e6 0.226164
\(795\) 648455. 0.0363883
\(796\) −2.32956e7 −1.30314
\(797\) −1.80924e7 −1.00891 −0.504453 0.863439i \(-0.668306\pi\)
−0.504453 + 0.863439i \(0.668306\pi\)
\(798\) 4.10184e7 2.28019
\(799\) 1.09501e7 0.606810
\(800\) 7.37279e6 0.407293
\(801\) 892095. 0.0491281
\(802\) 4.56293e7 2.50500
\(803\) 1.61711e7 0.885013
\(804\) −4.82422e6 −0.263201
\(805\) −4.22201e7 −2.29631
\(806\) 1.93254e6 0.104783
\(807\) −5.93683e6 −0.320901
\(808\) −129926. −0.00700113
\(809\) −1.57182e7 −0.844366 −0.422183 0.906511i \(-0.638736\pi\)
−0.422183 + 0.906511i \(0.638736\pi\)
\(810\) −8.43977e6 −0.451979
\(811\) −1.57562e6 −0.0841202 −0.0420601 0.999115i \(-0.513392\pi\)
−0.0420601 + 0.999115i \(0.513392\pi\)
\(812\) 5.74106e6 0.305564
\(813\) −1.88690e7 −1.00120
\(814\) 7.61791e7 4.02972
\(815\) 3.62401e7 1.91115
\(816\) 1.89563e7 0.996617
\(817\) 3.74962e6 0.196531
\(818\) 5.83260e6 0.304775
\(819\) −753503. −0.0392532
\(820\) 1.13520e7 0.589573
\(821\) 2.61911e7 1.35611 0.678056 0.735010i \(-0.262823\pi\)
0.678056 + 0.735010i \(0.262823\pi\)
\(822\) −1.34841e7 −0.696053
\(823\) −3.66910e7 −1.88825 −0.944126 0.329585i \(-0.893091\pi\)
−0.944126 + 0.329585i \(0.893091\pi\)
\(824\) 139762. 0.00717086
\(825\) −6.29806e6 −0.322160
\(826\) 2.64942e7 1.35114
\(827\) 1.38402e7 0.703685 0.351843 0.936059i \(-0.385555\pi\)
0.351843 + 0.936059i \(0.385555\pi\)
\(828\) −1.17938e7 −0.597832
\(829\) −2.83219e7 −1.43132 −0.715660 0.698449i \(-0.753874\pi\)
−0.715660 + 0.698449i \(0.753874\pi\)
\(830\) −4.03099e6 −0.203103
\(831\) 152389. 0.00765509
\(832\) −1.02353e6 −0.0512617
\(833\) −5.77657e7 −2.88441
\(834\) 1.48027e6 0.0736928
\(835\) −3.83894e7 −1.90544
\(836\) 4.29661e7 2.12623
\(837\) 3.37229e7 1.66384
\(838\) 3.19067e7 1.56954
\(839\) 2.80231e7 1.37439 0.687197 0.726471i \(-0.258841\pi\)
0.687197 + 0.726471i \(0.258841\pi\)
\(840\) 1.96092e6 0.0958873
\(841\) −1.99242e7 −0.971383
\(842\) −3.55782e7 −1.72943
\(843\) 1.25298e7 0.607263
\(844\) −1.45362e7 −0.702418
\(845\) 2.34792e7 1.13120
\(846\) −6.00076e6 −0.288257
\(847\) 5.32444e7 2.55015
\(848\) 890197. 0.0425105
\(849\) −1.71000e7 −0.814193
\(850\) 1.25789e7 0.597164
\(851\) 4.43928e7 2.10130
\(852\) 2.11633e7 0.998815
\(853\) −2.38300e7 −1.12138 −0.560689 0.828027i \(-0.689463\pi\)
−0.560689 + 0.828027i \(0.689463\pi\)
\(854\) 3.85800e7 1.81016
\(855\) −1.51654e7 −0.709479
\(856\) −483501. −0.0225534
\(857\) −3.83129e7 −1.78194 −0.890971 0.454061i \(-0.849975\pi\)
−0.890971 + 0.454061i \(0.849975\pi\)
\(858\) 1.63405e6 0.0757789
\(859\) 2.68650e7 1.24224 0.621118 0.783717i \(-0.286679\pi\)
0.621118 + 0.783717i \(0.286679\pi\)
\(860\) 3.92906e6 0.181152
\(861\) −1.33484e7 −0.613652
\(862\) −1.01118e7 −0.463510
\(863\) −1.63558e7 −0.747558 −0.373779 0.927518i \(-0.621938\pi\)
−0.373779 + 0.927518i \(0.621938\pi\)
\(864\) −3.33801e7 −1.52126
\(865\) 5.40206e6 0.245482
\(866\) −7.86758e6 −0.356489
\(867\) 1.80948e7 0.817535
\(868\) −6.26107e7 −2.82065
\(869\) −3.66192e6 −0.164498
\(870\) 4.39444e6 0.196836
\(871\) −367696. −0.0164227
\(872\) 181075. 0.00806431
\(873\) −1.31820e7 −0.585392
\(874\) 4.89341e7 2.16687
\(875\) 3.16351e7 1.39685
\(876\) −9.59350e6 −0.422393
\(877\) −2.30604e7 −1.01243 −0.506217 0.862406i \(-0.668957\pi\)
−0.506217 + 0.862406i \(0.668957\pi\)
\(878\) 1.07221e7 0.469402
\(879\) −1.77191e7 −0.773515
\(880\) −3.89539e7 −1.69568
\(881\) −3.15095e7 −1.36774 −0.683868 0.729606i \(-0.739704\pi\)
−0.683868 + 0.729606i \(0.739704\pi\)
\(882\) 3.16560e7 1.37020
\(883\) 2.86348e7 1.23593 0.617963 0.786208i \(-0.287958\pi\)
0.617963 + 0.786208i \(0.287958\pi\)
\(884\) −1.66991e6 −0.0718723
\(885\) 1.03765e7 0.445343
\(886\) −3.85685e7 −1.65063
\(887\) −2.45670e7 −1.04844 −0.524220 0.851583i \(-0.675643\pi\)
−0.524220 + 0.851583i \(0.675643\pi\)
\(888\) −2.06183e6 −0.0877446
\(889\) 4.32219e7 1.83421
\(890\) −3.87855e6 −0.164132
\(891\) 1.03953e7 0.438673
\(892\) 3.49392e7 1.47028
\(893\) 1.27396e7 0.534596
\(894\) 1.36243e7 0.570126
\(895\) −4.93961e6 −0.206127
\(896\) 5.66130e6 0.235584
\(897\) 952233. 0.0395150
\(898\) 4.37138e7 1.80895
\(899\) −6.40134e6 −0.264163
\(900\) −3.52711e6 −0.145149
\(901\) 1.59521e6 0.0654646
\(902\) −2.73266e7 −1.11833
\(903\) −4.62005e6 −0.188550
\(904\) 10638.5 0.000432973 0
\(905\) 2.20332e7 0.894245
\(906\) −3.87490e7 −1.56834
\(907\) 1.90832e7 0.770254 0.385127 0.922864i \(-0.374158\pi\)
0.385127 + 0.922864i \(0.374158\pi\)
\(908\) 3.10140e7 1.24837
\(909\) −1.23809e6 −0.0496984
\(910\) 3.27600e6 0.131141
\(911\) −3.04616e7 −1.21606 −0.608032 0.793912i \(-0.708041\pi\)
−0.608032 + 0.793912i \(0.708041\pi\)
\(912\) 2.20540e7 0.878012
\(913\) 4.96497e6 0.197124
\(914\) −1.72224e7 −0.681911
\(915\) 1.51100e7 0.596639
\(916\) −4.59826e6 −0.181073
\(917\) −6.02522e7 −2.36619
\(918\) −5.69504e7 −2.23044
\(919\) 2.73866e7 1.06967 0.534834 0.844957i \(-0.320374\pi\)
0.534834 + 0.844957i \(0.320374\pi\)
\(920\) 2.33933e6 0.0911218
\(921\) 7.77872e6 0.302175
\(922\) −2.27394e7 −0.880950
\(923\) 1.61304e6 0.0623221
\(924\) −5.29402e7 −2.03989
\(925\) 1.32763e7 0.510179
\(926\) 2.30986e7 0.885232
\(927\) 1.33182e6 0.0509033
\(928\) 6.33626e6 0.241525
\(929\) −2.65451e7 −1.00913 −0.504563 0.863375i \(-0.668346\pi\)
−0.504563 + 0.863375i \(0.668346\pi\)
\(930\) −4.79247e7 −1.81699
\(931\) −6.72054e7 −2.54115
\(932\) 1.43709e7 0.541932
\(933\) −1.10547e7 −0.415760
\(934\) 4.52888e6 0.169873
\(935\) −6.98046e7 −2.61129
\(936\) 41750.0 0.00155764
\(937\) −2.34660e7 −0.873151 −0.436576 0.899668i \(-0.643809\pi\)
−0.436576 + 0.899668i \(0.643809\pi\)
\(938\) 2.32818e7 0.863993
\(939\) −3.43852e7 −1.27264
\(940\) 1.33492e7 0.492761
\(941\) −2.18303e7 −0.803686 −0.401843 0.915709i \(-0.631630\pi\)
−0.401843 + 0.915709i \(0.631630\pi\)
\(942\) 3.92965e7 1.44287
\(943\) −1.59244e7 −0.583154
\(944\) 1.42449e7 0.520270
\(945\) 5.71662e7 2.08238
\(946\) −9.45806e6 −0.343617
\(947\) 1.87197e7 0.678305 0.339152 0.940731i \(-0.389860\pi\)
0.339152 + 0.940731i \(0.389860\pi\)
\(948\) 2.17244e6 0.0785103
\(949\) −731206. −0.0263557
\(950\) 1.46344e7 0.526098
\(951\) 1.01129e7 0.362598
\(952\) 4.82390e6 0.172507
\(953\) 5.37413e7 1.91680 0.958399 0.285432i \(-0.0921372\pi\)
0.958399 + 0.285432i \(0.0921372\pi\)
\(954\) −874188. −0.0310981
\(955\) 1.21785e7 0.432099
\(956\) −3.92748e7 −1.38986
\(957\) −5.41262e6 −0.191042
\(958\) −4.42594e7 −1.55809
\(959\) 3.32969e7 1.16911
\(960\) 2.53823e7 0.888901
\(961\) 4.11823e7 1.43847
\(962\) −3.44458e6 −0.120005
\(963\) −4.60736e6 −0.160098
\(964\) 3.99848e6 0.138581
\(965\) 1.72207e7 0.595295
\(966\) −6.02936e7 −2.07887
\(967\) 1.78136e7 0.612611 0.306306 0.951933i \(-0.400907\pi\)
0.306306 + 0.951933i \(0.400907\pi\)
\(968\) −2.95016e6 −0.101195
\(969\) 3.95203e7 1.35211
\(970\) 5.73114e7 1.95574
\(971\) 3.06231e7 1.04232 0.521160 0.853459i \(-0.325500\pi\)
0.521160 + 0.853459i \(0.325500\pi\)
\(972\) 2.67180e7 0.907065
\(973\) −3.65529e6 −0.123777
\(974\) −7.56691e7 −2.55577
\(975\) 284779. 0.00959392
\(976\) 2.07429e7 0.697021
\(977\) −1.93935e7 −0.650011 −0.325005 0.945712i \(-0.605366\pi\)
−0.325005 + 0.945712i \(0.605366\pi\)
\(978\) 5.17536e7 1.73019
\(979\) 4.77721e6 0.159301
\(980\) −7.04217e7 −2.34229
\(981\) 1.72550e6 0.0572455
\(982\) 8.49685e7 2.81177
\(983\) 3.41089e7 1.12586 0.562929 0.826505i \(-0.309675\pi\)
0.562929 + 0.826505i \(0.309675\pi\)
\(984\) 739609. 0.0243509
\(985\) −4.86650e6 −0.159818
\(986\) 1.08104e7 0.354120
\(987\) −1.56969e7 −0.512886
\(988\) −1.94279e6 −0.0633190
\(989\) −5.51162e6 −0.179179
\(990\) 3.82534e7 1.24046
\(991\) −1.15321e7 −0.373013 −0.186507 0.982454i \(-0.559717\pi\)
−0.186507 + 0.982454i \(0.559717\pi\)
\(992\) −6.91017e7 −2.22951
\(993\) −2.34014e7 −0.753128
\(994\) −1.02135e8 −3.27875
\(995\) 4.40319e7 1.40997
\(996\) −2.94547e6 −0.0940819
\(997\) 3.45745e7 1.10159 0.550793 0.834642i \(-0.314325\pi\)
0.550793 + 0.834642i \(0.314325\pi\)
\(998\) −4.65337e7 −1.47891
\(999\) −6.01081e7 −1.90554
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.1.2 8
3.2 odd 2 387.6.a.c.1.7 8
4.3 odd 2 688.6.a.e.1.2 8
5.4 even 2 1075.6.a.a.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.2 8 1.1 even 1 trivial
387.6.a.c.1.7 8 3.2 odd 2
688.6.a.e.1.2 8 4.3 odd 2
1075.6.a.a.1.7 8 5.4 even 2