Properties

Label 7.6.a.b.1.2
Level 7
Weight 6
Character 7.1
Self dual yes
Analytic conductor 1.123
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.12268673869\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 7.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.27492 q^{2} -25.6495 q^{3} +36.4743 q^{4} +28.7492 q^{5} -212.248 q^{6} +49.0000 q^{7} +37.0241 q^{8} +414.897 q^{9} +O(q^{10})\) \(q+8.27492 q^{2} -25.6495 q^{3} +36.4743 q^{4} +28.7492 q^{5} -212.248 q^{6} +49.0000 q^{7} +37.0241 q^{8} +414.897 q^{9} +237.897 q^{10} -270.090 q^{11} -935.547 q^{12} +300.640 q^{13} +405.471 q^{14} -737.402 q^{15} -860.805 q^{16} +613.106 q^{17} +3433.24 q^{18} -1700.95 q^{19} +1048.60 q^{20} -1256.83 q^{21} -2234.97 q^{22} +3188.15 q^{23} -949.650 q^{24} -2298.49 q^{25} +2487.77 q^{26} -4409.07 q^{27} +1787.24 q^{28} +4299.28 q^{29} -6101.94 q^{30} +2028.46 q^{31} -8307.86 q^{32} +6927.67 q^{33} +5073.40 q^{34} +1408.71 q^{35} +15133.1 q^{36} +5154.46 q^{37} -14075.2 q^{38} -7711.26 q^{39} +1064.41 q^{40} -7146.21 q^{41} -10400.1 q^{42} -19584.3 q^{43} -9851.32 q^{44} +11927.9 q^{45} +26381.7 q^{46} +19998.4 q^{47} +22079.2 q^{48} +2401.00 q^{49} -19019.8 q^{50} -15725.9 q^{51} +10965.6 q^{52} +3948.82 q^{53} -36484.7 q^{54} -7764.86 q^{55} +1814.18 q^{56} +43628.5 q^{57} +35576.2 q^{58} -29707.6 q^{59} -26896.2 q^{60} -50519.3 q^{61} +16785.3 q^{62} +20330.0 q^{63} -41201.1 q^{64} +8643.14 q^{65} +57325.9 q^{66} +5053.56 q^{67} +22362.6 q^{68} -81774.5 q^{69} +11657.0 q^{70} +32853.3 q^{71} +15361.2 q^{72} -11115.0 q^{73} +42652.7 q^{74} +58955.0 q^{75} -62040.8 q^{76} -13234.4 q^{77} -63810.0 q^{78} +81889.4 q^{79} -24747.4 q^{80} +12270.6 q^{81} -59134.3 q^{82} +118234. q^{83} -45841.8 q^{84} +17626.3 q^{85} -162058. q^{86} -110274. q^{87} -9999.83 q^{88} -41695.4 q^{89} +98702.8 q^{90} +14731.3 q^{91} +116286. q^{92} -52028.9 q^{93} +165485. q^{94} -48900.9 q^{95} +213092. q^{96} +43682.8 q^{97} +19868.1 q^{98} -112059. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 9q^{2} - 6q^{3} + 5q^{4} - 18q^{5} - 198q^{6} + 98q^{7} - 9q^{8} + 558q^{9} + O(q^{10}) \) \( 2q + 9q^{2} - 6q^{3} + 5q^{4} - 18q^{5} - 198q^{6} + 98q^{7} - 9q^{8} + 558q^{9} + 204q^{10} + 396q^{11} - 1554q^{12} - 350q^{13} + 441q^{14} - 1656q^{15} + 113q^{16} + 1800q^{17} + 3537q^{18} - 3266q^{19} + 2520q^{20} - 294q^{21} - 1752q^{22} + 2088q^{23} - 1854q^{24} - 3238q^{25} + 2016q^{26} - 6372q^{27} + 245q^{28} + 6696q^{29} - 6768q^{30} - 20q^{31} - 6129q^{32} + 20016q^{33} + 5934q^{34} - 882q^{35} + 10629q^{36} + 6232q^{37} - 15210q^{38} - 20496q^{39} + 3216q^{40} - 6048q^{41} - 9702q^{42} - 3020q^{43} - 30816q^{44} + 5238q^{45} + 25584q^{46} + 11700q^{47} + 41214q^{48} + 4802q^{49} - 19701q^{50} + 7596q^{51} + 31444q^{52} + 9468q^{53} - 37908q^{54} - 38904q^{55} - 441q^{56} + 12876q^{57} + 37314q^{58} - 43938q^{59} + 2016q^{60} - 64754q^{61} + 15300q^{62} + 27342q^{63} - 70783q^{64} + 39060q^{65} + 66816q^{66} + 24784q^{67} - 14994q^{68} - 103392q^{69} + 9996q^{70} + 97416q^{71} + 8775q^{72} + 17452q^{73} + 43434q^{74} + 40494q^{75} - 12782q^{76} + 19404q^{77} - 73080q^{78} + 51256q^{79} - 70272q^{80} - 61074q^{81} - 58338q^{82} + 117558q^{83} - 76146q^{84} - 37860q^{85} - 150048q^{86} - 63180q^{87} - 40656q^{88} + 84276q^{89} + 93852q^{90} - 17150q^{91} + 150912q^{92} - 92280q^{93} + 159468q^{94} + 24264q^{95} + 255906q^{96} + 20776q^{97} + 21609q^{98} - 16740q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.27492 1.46281 0.731406 0.681942i \(-0.238864\pi\)
0.731406 + 0.681942i \(0.238864\pi\)
\(3\) −25.6495 −1.64542 −0.822708 0.568464i \(-0.807538\pi\)
−0.822708 + 0.568464i \(0.807538\pi\)
\(4\) 36.4743 1.13982
\(5\) 28.7492 0.514281 0.257140 0.966374i \(-0.417220\pi\)
0.257140 + 0.966374i \(0.417220\pi\)
\(6\) −212.248 −2.40694
\(7\) 49.0000 0.377964
\(8\) 37.0241 0.204531
\(9\) 414.897 1.70740
\(10\) 237.897 0.752296
\(11\) −270.090 −0.673018 −0.336509 0.941680i \(-0.609246\pi\)
−0.336509 + 0.941680i \(0.609246\pi\)
\(12\) −935.547 −1.87548
\(13\) 300.640 0.493387 0.246694 0.969094i \(-0.420656\pi\)
0.246694 + 0.969094i \(0.420656\pi\)
\(14\) 405.471 0.552891
\(15\) −737.402 −0.846206
\(16\) −860.805 −0.840630
\(17\) 613.106 0.514533 0.257267 0.966340i \(-0.417178\pi\)
0.257267 + 0.966340i \(0.417178\pi\)
\(18\) 3433.24 2.49760
\(19\) −1700.95 −1.08095 −0.540477 0.841359i \(-0.681756\pi\)
−0.540477 + 0.841359i \(0.681756\pi\)
\(20\) 1048.60 0.586188
\(21\) −1256.83 −0.621909
\(22\) −2234.97 −0.984498
\(23\) 3188.15 1.25667 0.628333 0.777945i \(-0.283738\pi\)
0.628333 + 0.777945i \(0.283738\pi\)
\(24\) −949.650 −0.336539
\(25\) −2298.49 −0.735515
\(26\) 2487.77 0.721733
\(27\) −4409.07 −1.16396
\(28\) 1787.24 0.430812
\(29\) 4299.28 0.949294 0.474647 0.880176i \(-0.342576\pi\)
0.474647 + 0.880176i \(0.342576\pi\)
\(30\) −6101.94 −1.23784
\(31\) 2028.46 0.379106 0.189553 0.981870i \(-0.439296\pi\)
0.189553 + 0.981870i \(0.439296\pi\)
\(32\) −8307.86 −1.43421
\(33\) 6927.67 1.10739
\(34\) 5073.40 0.752666
\(35\) 1408.71 0.194380
\(36\) 15133.1 1.94612
\(37\) 5154.46 0.618983 0.309491 0.950902i \(-0.399841\pi\)
0.309491 + 0.950902i \(0.399841\pi\)
\(38\) −14075.2 −1.58123
\(39\) −7711.26 −0.811827
\(40\) 1064.41 0.105186
\(41\) −7146.21 −0.663921 −0.331960 0.943293i \(-0.607710\pi\)
−0.331960 + 0.943293i \(0.607710\pi\)
\(42\) −10400.1 −0.909736
\(43\) −19584.3 −1.61524 −0.807620 0.589703i \(-0.799245\pi\)
−0.807620 + 0.589703i \(0.799245\pi\)
\(44\) −9851.32 −0.767119
\(45\) 11927.9 0.878081
\(46\) 26381.7 1.83827
\(47\) 19998.4 1.32054 0.660268 0.751030i \(-0.270443\pi\)
0.660268 + 0.751030i \(0.270443\pi\)
\(48\) 22079.2 1.38319
\(49\) 2401.00 0.142857
\(50\) −19019.8 −1.07592
\(51\) −15725.9 −0.846621
\(52\) 10965.6 0.562373
\(53\) 3948.82 0.193098 0.0965489 0.995328i \(-0.469220\pi\)
0.0965489 + 0.995328i \(0.469220\pi\)
\(54\) −36484.7 −1.70265
\(55\) −7764.86 −0.346120
\(56\) 1814.18 0.0773055
\(57\) 43628.5 1.77862
\(58\) 35576.2 1.38864
\(59\) −29707.6 −1.11106 −0.555530 0.831497i \(-0.687484\pi\)
−0.555530 + 0.831497i \(0.687484\pi\)
\(60\) −26896.2 −0.964523
\(61\) −50519.3 −1.73833 −0.869165 0.494522i \(-0.835343\pi\)
−0.869165 + 0.494522i \(0.835343\pi\)
\(62\) 16785.3 0.554562
\(63\) 20330.0 0.645335
\(64\) −41201.1 −1.25736
\(65\) 8643.14 0.253740
\(66\) 57325.9 1.61991
\(67\) 5053.56 0.137534 0.0687671 0.997633i \(-0.478093\pi\)
0.0687671 + 0.997633i \(0.478093\pi\)
\(68\) 22362.6 0.586476
\(69\) −81774.5 −2.06774
\(70\) 11657.0 0.284341
\(71\) 32853.3 0.773453 0.386726 0.922195i \(-0.373606\pi\)
0.386726 + 0.922195i \(0.373606\pi\)
\(72\) 15361.2 0.349215
\(73\) −11115.0 −0.244119 −0.122059 0.992523i \(-0.538950\pi\)
−0.122059 + 0.992523i \(0.538950\pi\)
\(74\) 42652.7 0.905456
\(75\) 58955.0 1.21023
\(76\) −62040.8 −1.23209
\(77\) −13234.4 −0.254377
\(78\) −63810.0 −1.18755
\(79\) 81889.4 1.47625 0.738125 0.674664i \(-0.235712\pi\)
0.738125 + 0.674664i \(0.235712\pi\)
\(80\) −24747.4 −0.432320
\(81\) 12270.6 0.207803
\(82\) −59134.3 −0.971191
\(83\) 118234. 1.88385 0.941926 0.335819i \(-0.109013\pi\)
0.941926 + 0.335819i \(0.109013\pi\)
\(84\) −45841.8 −0.708865
\(85\) 17626.3 0.264615
\(86\) −162058. −2.36279
\(87\) −110274. −1.56198
\(88\) −9999.83 −0.137653
\(89\) −41695.4 −0.557972 −0.278986 0.960295i \(-0.589998\pi\)
−0.278986 + 0.960295i \(0.589998\pi\)
\(90\) 98702.8 1.28447
\(91\) 14731.3 0.186483
\(92\) 116286. 1.43237
\(93\) −52028.9 −0.623788
\(94\) 165485. 1.93170
\(95\) −48900.9 −0.555914
\(96\) 213092. 2.35988
\(97\) 43682.8 0.471391 0.235695 0.971827i \(-0.424263\pi\)
0.235695 + 0.971827i \(0.424263\pi\)
\(98\) 19868.1 0.208973
\(99\) −112059. −1.14911
\(100\) −83835.5 −0.838355
\(101\) 25648.1 0.250179 0.125090 0.992145i \(-0.460078\pi\)
0.125090 + 0.992145i \(0.460078\pi\)
\(102\) −130130. −1.23845
\(103\) −14320.0 −0.133000 −0.0664999 0.997786i \(-0.521183\pi\)
−0.0664999 + 0.997786i \(0.521183\pi\)
\(104\) 11130.9 0.100913
\(105\) −36132.7 −0.319836
\(106\) 32676.1 0.282466
\(107\) 17201.8 0.145249 0.0726247 0.997359i \(-0.476862\pi\)
0.0726247 + 0.997359i \(0.476862\pi\)
\(108\) −160818. −1.32670
\(109\) −86017.6 −0.693459 −0.346730 0.937965i \(-0.612708\pi\)
−0.346730 + 0.937965i \(0.612708\pi\)
\(110\) −64253.5 −0.506309
\(111\) −132209. −1.01848
\(112\) −42179.4 −0.317728
\(113\) 137568. 1.01349 0.506745 0.862096i \(-0.330848\pi\)
0.506745 + 0.862096i \(0.330848\pi\)
\(114\) 361022. 2.60179
\(115\) 91656.8 0.646279
\(116\) 156813. 1.08202
\(117\) 124734. 0.842407
\(118\) −245828. −1.62527
\(119\) 30042.2 0.194475
\(120\) −27301.6 −0.173075
\(121\) −88102.5 −0.547047
\(122\) −418043. −2.54285
\(123\) 183297. 1.09243
\(124\) 73986.4 0.432113
\(125\) −155921. −0.892542
\(126\) 168229. 0.944004
\(127\) −70567.1 −0.388233 −0.194117 0.980978i \(-0.562184\pi\)
−0.194117 + 0.980978i \(0.562184\pi\)
\(128\) −75084.2 −0.405064
\(129\) 502328. 2.65774
\(130\) 71521.3 0.371173
\(131\) −173712. −0.884408 −0.442204 0.896914i \(-0.645803\pi\)
−0.442204 + 0.896914i \(0.645803\pi\)
\(132\) 252682. 1.26223
\(133\) −83346.5 −0.408562
\(134\) 41817.8 0.201187
\(135\) −126757. −0.598602
\(136\) 22699.7 0.105238
\(137\) −1989.94 −0.00905813 −0.00452907 0.999990i \(-0.501442\pi\)
−0.00452907 + 0.999990i \(0.501442\pi\)
\(138\) −676678. −3.02471
\(139\) 366409. 1.60853 0.804264 0.594272i \(-0.202560\pi\)
0.804264 + 0.594272i \(0.202560\pi\)
\(140\) 51381.6 0.221558
\(141\) −512949. −2.17283
\(142\) 271859. 1.13142
\(143\) −81199.7 −0.332058
\(144\) −357145. −1.43529
\(145\) 123601. 0.488204
\(146\) −91975.4 −0.357100
\(147\) −61584.5 −0.235059
\(148\) 188005. 0.705529
\(149\) 140719. 0.519261 0.259631 0.965708i \(-0.416399\pi\)
0.259631 + 0.965708i \(0.416399\pi\)
\(150\) 487848. 1.77034
\(151\) 50064.6 0.178685 0.0893425 0.996001i \(-0.471523\pi\)
0.0893425 + 0.996001i \(0.471523\pi\)
\(152\) −62976.1 −0.221089
\(153\) 254376. 0.878512
\(154\) −109514. −0.372105
\(155\) 58316.4 0.194967
\(156\) −281262. −0.925337
\(157\) −89794.6 −0.290738 −0.145369 0.989378i \(-0.546437\pi\)
−0.145369 + 0.989378i \(0.546437\pi\)
\(158\) 677628. 2.15948
\(159\) −101285. −0.317726
\(160\) −238844. −0.737589
\(161\) 156219. 0.474975
\(162\) 101538. 0.303977
\(163\) −481230. −1.41868 −0.709339 0.704867i \(-0.751006\pi\)
−0.709339 + 0.704867i \(0.751006\pi\)
\(164\) −260653. −0.756750
\(165\) 199165. 0.569512
\(166\) 978376. 2.75572
\(167\) −86572.7 −0.240209 −0.120105 0.992761i \(-0.538323\pi\)
−0.120105 + 0.992761i \(0.538323\pi\)
\(168\) −46532.8 −0.127200
\(169\) −280909. −0.756569
\(170\) 145856. 0.387082
\(171\) −705718. −1.84562
\(172\) −714323. −1.84108
\(173\) −58137.4 −0.147686 −0.0738432 0.997270i \(-0.523526\pi\)
−0.0738432 + 0.997270i \(0.523526\pi\)
\(174\) −912511. −2.28489
\(175\) −112626. −0.277999
\(176\) 232495. 0.565759
\(177\) 761985. 1.82816
\(178\) −345026. −0.816209
\(179\) −209380. −0.488431 −0.244215 0.969721i \(-0.578530\pi\)
−0.244215 + 0.969721i \(0.578530\pi\)
\(180\) 435063. 1.00085
\(181\) 278996. 0.632996 0.316498 0.948593i \(-0.397493\pi\)
0.316498 + 0.948593i \(0.397493\pi\)
\(182\) 121901. 0.272789
\(183\) 1.29579e6 2.86028
\(184\) 118038. 0.257027
\(185\) 148186. 0.318331
\(186\) −430535. −0.912485
\(187\) −165594. −0.346290
\(188\) 729426. 1.50517
\(189\) −216045. −0.439935
\(190\) −404651. −0.813198
\(191\) −445132. −0.882888 −0.441444 0.897289i \(-0.645534\pi\)
−0.441444 + 0.897289i \(0.645534\pi\)
\(192\) 1.05679e6 2.06888
\(193\) −726811. −1.40452 −0.702260 0.711920i \(-0.747826\pi\)
−0.702260 + 0.711920i \(0.747826\pi\)
\(194\) 361471. 0.689556
\(195\) −221692. −0.417507
\(196\) 87574.7 0.162831
\(197\) −364897. −0.669892 −0.334946 0.942237i \(-0.608718\pi\)
−0.334946 + 0.942237i \(0.608718\pi\)
\(198\) −927282. −1.68093
\(199\) 289307. 0.517877 0.258938 0.965894i \(-0.416627\pi\)
0.258938 + 0.965894i \(0.416627\pi\)
\(200\) −85099.3 −0.150436
\(201\) −129621. −0.226301
\(202\) 212236. 0.365965
\(203\) 210665. 0.358799
\(204\) −573589. −0.964997
\(205\) −205448. −0.341442
\(206\) −118497. −0.194554
\(207\) 1.32276e6 2.14562
\(208\) −258792. −0.414756
\(209\) 459409. 0.727501
\(210\) −298995. −0.467860
\(211\) 750147. 1.15995 0.579976 0.814633i \(-0.303062\pi\)
0.579976 + 0.814633i \(0.303062\pi\)
\(212\) 144030. 0.220097
\(213\) −842672. −1.27265
\(214\) 142343. 0.212473
\(215\) −563033. −0.830687
\(216\) −163242. −0.238066
\(217\) 99394.3 0.143289
\(218\) −711788. −1.01440
\(219\) 285093. 0.401677
\(220\) −283217. −0.394515
\(221\) 184324. 0.253864
\(222\) −1.09402e6 −1.48985
\(223\) 534398. 0.719619 0.359810 0.933026i \(-0.382842\pi\)
0.359810 + 0.933026i \(0.382842\pi\)
\(224\) −407085. −0.542082
\(225\) −953635. −1.25582
\(226\) 1.13836e6 1.48255
\(227\) −410624. −0.528907 −0.264453 0.964398i \(-0.585192\pi\)
−0.264453 + 0.964398i \(0.585192\pi\)
\(228\) 1.59132e6 2.02731
\(229\) 1.03036e6 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(230\) 758452. 0.945385
\(231\) 339456. 0.418556
\(232\) 159177. 0.194160
\(233\) −119211. −0.143856 −0.0719278 0.997410i \(-0.522915\pi\)
−0.0719278 + 0.997410i \(0.522915\pi\)
\(234\) 1.03217e6 1.23228
\(235\) 574937. 0.679127
\(236\) −1.08356e6 −1.26641
\(237\) −2.10042e6 −2.42905
\(238\) 248597. 0.284481
\(239\) −254090. −0.287735 −0.143868 0.989597i \(-0.545954\pi\)
−0.143868 + 0.989597i \(0.545954\pi\)
\(240\) 634759. 0.711346
\(241\) 1.41251e6 1.56656 0.783282 0.621667i \(-0.213544\pi\)
0.783282 + 0.621667i \(0.213544\pi\)
\(242\) −729041. −0.800228
\(243\) 756671. 0.822037
\(244\) −1.84265e6 −1.98138
\(245\) 69026.8 0.0734687
\(246\) 1.51677e6 1.59801
\(247\) −511372. −0.533329
\(248\) 75101.7 0.0775391
\(249\) −3.03264e6 −3.09972
\(250\) −1.29023e6 −1.30562
\(251\) −1.67542e6 −1.67857 −0.839286 0.543690i \(-0.817027\pi\)
−0.839286 + 0.543690i \(0.817027\pi\)
\(252\) 741520. 0.735566
\(253\) −861087. −0.845758
\(254\) −583937. −0.567913
\(255\) −452106. −0.435401
\(256\) 697120. 0.664825
\(257\) 726996. 0.686593 0.343296 0.939227i \(-0.388456\pi\)
0.343296 + 0.939227i \(0.388456\pi\)
\(258\) 4.15672e6 3.88778
\(259\) 252568. 0.233953
\(260\) 315252. 0.289217
\(261\) 1.78376e6 1.62082
\(262\) −1.43746e6 −1.29372
\(263\) −225880. −0.201367 −0.100684 0.994918i \(-0.532103\pi\)
−0.100684 + 0.994918i \(0.532103\pi\)
\(264\) 256491. 0.226497
\(265\) 113525. 0.0993065
\(266\) −689685. −0.597650
\(267\) 1.06947e6 0.918097
\(268\) 184325. 0.156764
\(269\) 1.80527e6 1.52111 0.760557 0.649272i \(-0.224926\pi\)
0.760557 + 0.649272i \(0.224926\pi\)
\(270\) −1.04891e6 −0.875643
\(271\) −1.71380e6 −1.41754 −0.708771 0.705439i \(-0.750750\pi\)
−0.708771 + 0.705439i \(0.750750\pi\)
\(272\) −527765. −0.432532
\(273\) −377852. −0.306842
\(274\) −16466.6 −0.0132504
\(275\) 620797. 0.495015
\(276\) −2.98267e6 −2.35685
\(277\) 2.23055e6 1.74668 0.873338 0.487115i \(-0.161951\pi\)
0.873338 + 0.487115i \(0.161951\pi\)
\(278\) 3.03200e6 2.35298
\(279\) 841600. 0.647285
\(280\) 52156.2 0.0397567
\(281\) 1.67140e6 1.26274 0.631371 0.775481i \(-0.282493\pi\)
0.631371 + 0.775481i \(0.282493\pi\)
\(282\) −4.24461e6 −3.17845
\(283\) −396152. −0.294033 −0.147016 0.989134i \(-0.546967\pi\)
−0.147016 + 0.989134i \(0.546967\pi\)
\(284\) 1.19830e6 0.881597
\(285\) 1.25428e6 0.914710
\(286\) −671920. −0.485739
\(287\) −350164. −0.250938
\(288\) −3.44691e6 −2.44877
\(289\) −1.04396e6 −0.735256
\(290\) 1.02279e6 0.714150
\(291\) −1.12044e6 −0.775634
\(292\) −405410. −0.278251
\(293\) −929465. −0.632505 −0.316252 0.948675i \(-0.602425\pi\)
−0.316252 + 0.948675i \(0.602425\pi\)
\(294\) −509606. −0.343848
\(295\) −854068. −0.571397
\(296\) 190839. 0.126601
\(297\) 1.19085e6 0.783365
\(298\) 1.16443e6 0.759582
\(299\) 958485. 0.620022
\(300\) 2.15034e6 1.37944
\(301\) −959631. −0.610503
\(302\) 414280. 0.261383
\(303\) −657860. −0.411649
\(304\) 1.46418e6 0.908682
\(305\) −1.45239e6 −0.893990
\(306\) 2.10494e6 1.28510
\(307\) 1.83295e6 1.10995 0.554976 0.831866i \(-0.312727\pi\)
0.554976 + 0.831866i \(0.312727\pi\)
\(308\) −482715. −0.289944
\(309\) 367302. 0.218840
\(310\) 482563. 0.285200
\(311\) −2.29685e6 −1.34658 −0.673289 0.739379i \(-0.735119\pi\)
−0.673289 + 0.739379i \(0.735119\pi\)
\(312\) −285502. −0.166044
\(313\) −3.42470e6 −1.97589 −0.987943 0.154817i \(-0.950521\pi\)
−0.987943 + 0.154817i \(0.950521\pi\)
\(314\) −743043. −0.425295
\(315\) 584469. 0.331883
\(316\) 2.98685e6 1.68266
\(317\) 2.94305e6 1.64494 0.822470 0.568808i \(-0.192595\pi\)
0.822470 + 0.568808i \(0.192595\pi\)
\(318\) −838127. −0.464774
\(319\) −1.16119e6 −0.638891
\(320\) −1.18450e6 −0.646635
\(321\) −441217. −0.238996
\(322\) 1.29270e6 0.694799
\(323\) −1.04286e6 −0.556187
\(324\) 447560. 0.236858
\(325\) −691016. −0.362894
\(326\) −3.98214e6 −2.07526
\(327\) 2.20631e6 1.14103
\(328\) −264582. −0.135792
\(329\) 979921. 0.499116
\(330\) 1.64807e6 0.833089
\(331\) 966164. 0.484709 0.242354 0.970188i \(-0.422080\pi\)
0.242354 + 0.970188i \(0.422080\pi\)
\(332\) 4.31250e6 2.14725
\(333\) 2.13857e6 1.05685
\(334\) −716382. −0.351381
\(335\) 145286. 0.0707312
\(336\) 1.08188e6 0.522795
\(337\) 136417. 0.0654327 0.0327163 0.999465i \(-0.489584\pi\)
0.0327163 + 0.999465i \(0.489584\pi\)
\(338\) −2.32450e6 −1.10672
\(339\) −3.52854e6 −1.66761
\(340\) 642906. 0.301613
\(341\) −547865. −0.255145
\(342\) −5.83976e6 −2.69979
\(343\) 117649. 0.0539949
\(344\) −725091. −0.330367
\(345\) −2.35095e6 −1.06340
\(346\) −481082. −0.216038
\(347\) 355408. 0.158454 0.0792270 0.996857i \(-0.474755\pi\)
0.0792270 + 0.996857i \(0.474755\pi\)
\(348\) −4.02218e6 −1.78038
\(349\) −140128. −0.0615830 −0.0307915 0.999526i \(-0.509803\pi\)
−0.0307915 + 0.999526i \(0.509803\pi\)
\(350\) −931969. −0.406660
\(351\) −1.32554e6 −0.574283
\(352\) 2.24387e6 0.965252
\(353\) 3.48141e6 1.48703 0.743514 0.668721i \(-0.233158\pi\)
0.743514 + 0.668721i \(0.233158\pi\)
\(354\) 6.30536e6 2.67425
\(355\) 944507. 0.397772
\(356\) −1.52081e6 −0.635988
\(357\) −770568. −0.319993
\(358\) −1.73260e6 −0.714482
\(359\) 1.75285e6 0.717810 0.358905 0.933374i \(-0.383150\pi\)
0.358905 + 0.933374i \(0.383150\pi\)
\(360\) 441621. 0.179595
\(361\) 417127. 0.168461
\(362\) 2.30867e6 0.925955
\(363\) 2.25979e6 0.900121
\(364\) 537315. 0.212557
\(365\) −319546. −0.125546
\(366\) 1.07226e7 4.18405
\(367\) −1.76939e6 −0.685738 −0.342869 0.939383i \(-0.611399\pi\)
−0.342869 + 0.939383i \(0.611399\pi\)
\(368\) −2.74438e6 −1.05639
\(369\) −2.96494e6 −1.13357
\(370\) 1.22623e6 0.465658
\(371\) 193492. 0.0729841
\(372\) −1.89771e6 −0.711006
\(373\) −4.16212e6 −1.54897 −0.774485 0.632592i \(-0.781991\pi\)
−0.774485 + 0.632592i \(0.781991\pi\)
\(374\) −1.37027e6 −0.506557
\(375\) 3.99929e6 1.46860
\(376\) 740422. 0.270091
\(377\) 1.29253e6 0.468369
\(378\) −1.78775e6 −0.643543
\(379\) 618163. 0.221057 0.110529 0.993873i \(-0.464746\pi\)
0.110529 + 0.993873i \(0.464746\pi\)
\(380\) −1.78362e6 −0.633642
\(381\) 1.81001e6 0.638805
\(382\) −3.68343e6 −1.29150
\(383\) −4.11163e6 −1.43225 −0.716123 0.697974i \(-0.754085\pi\)
−0.716123 + 0.697974i \(0.754085\pi\)
\(384\) 1.92587e6 0.666498
\(385\) −380478. −0.130821
\(386\) −6.01430e6 −2.05455
\(387\) −8.12547e6 −2.75785
\(388\) 1.59330e6 0.537301
\(389\) 4.62076e6 1.54824 0.774122 0.633037i \(-0.218192\pi\)
0.774122 + 0.633037i \(0.218192\pi\)
\(390\) −1.83448e6 −0.610735
\(391\) 1.95468e6 0.646596
\(392\) 88894.8 0.0292187
\(393\) 4.45564e6 1.45522
\(394\) −3.01949e6 −0.979926
\(395\) 2.35425e6 0.759207
\(396\) −4.08728e6 −1.30978
\(397\) 5.07349e6 1.61559 0.807794 0.589465i \(-0.200661\pi\)
0.807794 + 0.589465i \(0.200661\pi\)
\(398\) 2.39399e6 0.757557
\(399\) 2.13780e6 0.672255
\(400\) 1.97855e6 0.618296
\(401\) −1.48056e6 −0.459795 −0.229898 0.973215i \(-0.573839\pi\)
−0.229898 + 0.973215i \(0.573839\pi\)
\(402\) −1.07261e6 −0.331036
\(403\) 609834. 0.187046
\(404\) 935495. 0.285160
\(405\) 352768. 0.106869
\(406\) 1.74323e6 0.524856
\(407\) −1.39217e6 −0.416586
\(408\) −582236. −0.173160
\(409\) −4.53379e6 −1.34015 −0.670075 0.742294i \(-0.733738\pi\)
−0.670075 + 0.742294i \(0.733738\pi\)
\(410\) −1.70006e6 −0.499465
\(411\) 51041.0 0.0149044
\(412\) −522313. −0.151596
\(413\) −1.45567e6 −0.419941
\(414\) 1.09457e7 3.13865
\(415\) 3.39913e6 0.968829
\(416\) −2.49767e6 −0.707623
\(417\) −9.39820e6 −2.64670
\(418\) 3.80157e6 1.06420
\(419\) 111026. 0.0308952 0.0154476 0.999881i \(-0.495083\pi\)
0.0154476 + 0.999881i \(0.495083\pi\)
\(420\) −1.31791e6 −0.364555
\(421\) −1.41151e6 −0.388132 −0.194066 0.980988i \(-0.562168\pi\)
−0.194066 + 0.980988i \(0.562168\pi\)
\(422\) 6.20740e6 1.69679
\(423\) 8.29727e6 2.25468
\(424\) 146201. 0.0394945
\(425\) −1.40922e6 −0.378447
\(426\) −6.97304e6 −1.86165
\(427\) −2.47544e6 −0.657027
\(428\) 627422. 0.165558
\(429\) 2.08273e6 0.546374
\(430\) −4.65905e6 −1.21514
\(431\) 1.07640e6 0.279113 0.139557 0.990214i \(-0.455432\pi\)
0.139557 + 0.990214i \(0.455432\pi\)
\(432\) 3.79535e6 0.978459
\(433\) −310172. −0.0795029 −0.0397515 0.999210i \(-0.512657\pi\)
−0.0397515 + 0.999210i \(0.512657\pi\)
\(434\) 822480. 0.209605
\(435\) −3.17030e6 −0.803298
\(436\) −3.13743e6 −0.790419
\(437\) −5.42288e6 −1.35840
\(438\) 2.35912e6 0.587578
\(439\) 5.67650e6 1.40579 0.702893 0.711296i \(-0.251891\pi\)
0.702893 + 0.711296i \(0.251891\pi\)
\(440\) −287487. −0.0707923
\(441\) 996168. 0.243914
\(442\) 1.52527e6 0.371356
\(443\) 4.05966e6 0.982834 0.491417 0.870924i \(-0.336479\pi\)
0.491417 + 0.870924i \(0.336479\pi\)
\(444\) −4.82223e6 −1.16089
\(445\) −1.19871e6 −0.286955
\(446\) 4.42210e6 1.05267
\(447\) −3.60936e6 −0.854401
\(448\) −2.01885e6 −0.475237
\(449\) −6.96544e6 −1.63054 −0.815272 0.579078i \(-0.803413\pi\)
−0.815272 + 0.579078i \(0.803413\pi\)
\(450\) −7.89125e6 −1.83702
\(451\) 1.93012e6 0.446830
\(452\) 5.01767e6 1.15520
\(453\) −1.28413e6 −0.294011
\(454\) −3.39788e6 −0.773692
\(455\) 423514. 0.0959045
\(456\) 1.61530e6 0.363783
\(457\) 1.79523e6 0.402096 0.201048 0.979581i \(-0.435565\pi\)
0.201048 + 0.979581i \(0.435565\pi\)
\(458\) 8.52616e6 1.89928
\(459\) −2.70323e6 −0.598896
\(460\) 3.34311e6 0.736642
\(461\) −2.11294e6 −0.463058 −0.231529 0.972828i \(-0.574373\pi\)
−0.231529 + 0.972828i \(0.574373\pi\)
\(462\) 2.80897e6 0.612268
\(463\) 1.26223e6 0.273643 0.136822 0.990596i \(-0.456311\pi\)
0.136822 + 0.990596i \(0.456311\pi\)
\(464\) −3.70084e6 −0.798005
\(465\) −1.49579e6 −0.320802
\(466\) −986462. −0.210434
\(467\) −3.58926e6 −0.761576 −0.380788 0.924662i \(-0.624347\pi\)
−0.380788 + 0.924662i \(0.624347\pi\)
\(468\) 4.54960e6 0.960192
\(469\) 247624. 0.0519830
\(470\) 4.75756e6 0.993435
\(471\) 2.30319e6 0.478384
\(472\) −1.09990e6 −0.227246
\(473\) 5.28952e6 1.08708
\(474\) −1.73808e7 −3.55324
\(475\) 3.90960e6 0.795058
\(476\) 1.09577e6 0.221667
\(477\) 1.63835e6 0.329694
\(478\) −2.10257e6 −0.420903
\(479\) −2.41693e6 −0.481311 −0.240655 0.970611i \(-0.577362\pi\)
−0.240655 + 0.970611i \(0.577362\pi\)
\(480\) 6.12623e6 1.21364
\(481\) 1.54963e6 0.305398
\(482\) 1.16884e7 2.29159
\(483\) −4.00695e6 −0.781531
\(484\) −3.21347e6 −0.623536
\(485\) 1.25584e6 0.242427
\(486\) 6.26139e6 1.20249
\(487\) −5.19403e6 −0.992388 −0.496194 0.868212i \(-0.665270\pi\)
−0.496194 + 0.868212i \(0.665270\pi\)
\(488\) −1.87043e6 −0.355543
\(489\) 1.23433e7 2.33432
\(490\) 571191. 0.107471
\(491\) 5.38961e6 1.00891 0.504456 0.863437i \(-0.331693\pi\)
0.504456 + 0.863437i \(0.331693\pi\)
\(492\) 6.68561e6 1.24517
\(493\) 2.63592e6 0.488443
\(494\) −4.23156e6 −0.780160
\(495\) −3.22162e6 −0.590964
\(496\) −1.74610e6 −0.318688
\(497\) 1.60981e6 0.292338
\(498\) −2.50949e7 −4.53431
\(499\) −3.29606e6 −0.592576 −0.296288 0.955099i \(-0.595749\pi\)
−0.296288 + 0.955099i \(0.595749\pi\)
\(500\) −5.68709e6 −1.01734
\(501\) 2.22055e6 0.395244
\(502\) −1.38640e7 −2.45544
\(503\) −1.06512e7 −1.87706 −0.938528 0.345204i \(-0.887810\pi\)
−0.938528 + 0.345204i \(0.887810\pi\)
\(504\) 752698. 0.131991
\(505\) 737361. 0.128662
\(506\) −7.12543e6 −1.23718
\(507\) 7.20517e6 1.24487
\(508\) −2.57388e6 −0.442516
\(509\) −2.74268e6 −0.469225 −0.234612 0.972089i \(-0.575382\pi\)
−0.234612 + 0.972089i \(0.575382\pi\)
\(510\) −3.74114e6 −0.636910
\(511\) −544633. −0.0922682
\(512\) 8.17130e6 1.37758
\(513\) 7.49961e6 1.25819
\(514\) 6.01583e6 1.00436
\(515\) −411689. −0.0683992
\(516\) 1.83220e7 3.02935
\(517\) −5.40136e6 −0.888744
\(518\) 2.08998e6 0.342230
\(519\) 1.49120e6 0.243006
\(520\) 320004. 0.0518976
\(521\) 4.97077e6 0.802286 0.401143 0.916015i \(-0.368613\pi\)
0.401143 + 0.916015i \(0.368613\pi\)
\(522\) 1.47605e7 2.37096
\(523\) 2.41579e6 0.386193 0.193096 0.981180i \(-0.438147\pi\)
0.193096 + 0.981180i \(0.438147\pi\)
\(524\) −6.33603e6 −1.00807
\(525\) 2.88880e6 0.457424
\(526\) −1.86914e6 −0.294563
\(527\) 1.24366e6 0.195063
\(528\) −5.96337e6 −0.930908
\(529\) 3.72798e6 0.579207
\(530\) 939412. 0.145267
\(531\) −1.23256e7 −1.89702
\(532\) −3.04000e6 −0.465688
\(533\) −2.14843e6 −0.327570
\(534\) 8.84974e6 1.34300
\(535\) 494537. 0.0746989
\(536\) 187103. 0.0281300
\(537\) 5.37050e6 0.803672
\(538\) 1.49385e7 2.22510
\(539\) −648485. −0.0961454
\(540\) −4.62337e6 −0.682299
\(541\) 472165. 0.0693587 0.0346794 0.999398i \(-0.488959\pi\)
0.0346794 + 0.999398i \(0.488959\pi\)
\(542\) −1.41815e7 −2.07360
\(543\) −7.15610e6 −1.04154
\(544\) −5.09360e6 −0.737951
\(545\) −2.47293e6 −0.356633
\(546\) −3.12669e6 −0.448852
\(547\) 7.63716e6 1.09135 0.545675 0.837997i \(-0.316273\pi\)
0.545675 + 0.837997i \(0.316273\pi\)
\(548\) −72581.6 −0.0103246
\(549\) −2.09603e7 −2.96802
\(550\) 5.13705e6 0.724114
\(551\) −7.31285e6 −1.02614
\(552\) −3.02763e6 −0.422917
\(553\) 4.01258e6 0.557970
\(554\) 1.84576e7 2.55506
\(555\) −3.80091e6 −0.523787
\(556\) 1.33645e7 1.83343
\(557\) −4.48807e6 −0.612946 −0.306473 0.951879i \(-0.599149\pi\)
−0.306473 + 0.951879i \(0.599149\pi\)
\(558\) 6.96417e6 0.946856
\(559\) −5.88782e6 −0.796938
\(560\) −1.21262e6 −0.163402
\(561\) 4.24740e6 0.569791
\(562\) 1.38307e7 1.84715
\(563\) −2.16500e6 −0.287864 −0.143932 0.989588i \(-0.545975\pi\)
−0.143932 + 0.989588i \(0.545975\pi\)
\(564\) −1.87094e7 −2.47664
\(565\) 3.95495e6 0.521219
\(566\) −3.27812e6 −0.430115
\(567\) 601258. 0.0785422
\(568\) 1.21637e6 0.158195
\(569\) −1.13325e7 −1.46739 −0.733696 0.679478i \(-0.762206\pi\)
−0.733696 + 0.679478i \(0.762206\pi\)
\(570\) 1.03791e7 1.33805
\(571\) −843773. −0.108302 −0.0541509 0.998533i \(-0.517245\pi\)
−0.0541509 + 0.998533i \(0.517245\pi\)
\(572\) −2.96170e6 −0.378487
\(573\) 1.14174e7 1.45272
\(574\) −2.89758e6 −0.367076
\(575\) −7.32792e6 −0.924296
\(576\) −1.70942e7 −2.14681
\(577\) −2.23784e6 −0.279827 −0.139914 0.990164i \(-0.544682\pi\)
−0.139914 + 0.990164i \(0.544682\pi\)
\(578\) −8.63866e6 −1.07554
\(579\) 1.86423e7 2.31102
\(580\) 4.50824e6 0.556464
\(581\) 5.79346e6 0.712029
\(582\) −9.27156e6 −1.13461
\(583\) −1.06653e6 −0.129958
\(584\) −411521. −0.0499299
\(585\) 3.58601e6 0.433234
\(586\) −7.69124e6 −0.925236
\(587\) 1.21190e7 1.45168 0.725839 0.687864i \(-0.241452\pi\)
0.725839 + 0.687864i \(0.241452\pi\)
\(588\) −2.24625e6 −0.267926
\(589\) −3.45030e6 −0.409797
\(590\) −7.06734e6 −0.835846
\(591\) 9.35942e6 1.10225
\(592\) −4.43698e6 −0.520335
\(593\) 8.00167e6 0.934424 0.467212 0.884145i \(-0.345258\pi\)
0.467212 + 0.884145i \(0.345258\pi\)
\(594\) 9.85415e6 1.14592
\(595\) 863689. 0.100015
\(596\) 5.13261e6 0.591865
\(597\) −7.42058e6 −0.852123
\(598\) 7.93138e6 0.906976
\(599\) 1.45899e7 1.66144 0.830719 0.556692i \(-0.187930\pi\)
0.830719 + 0.556692i \(0.187930\pi\)
\(600\) 2.18276e6 0.247529
\(601\) −8.67178e6 −0.979314 −0.489657 0.871915i \(-0.662878\pi\)
−0.489657 + 0.871915i \(0.662878\pi\)
\(602\) −7.94087e6 −0.893052
\(603\) 2.09671e6 0.234825
\(604\) 1.82607e6 0.203669
\(605\) −2.53287e6 −0.281336
\(606\) −5.44374e6 −0.602166
\(607\) −1.33059e7 −1.46580 −0.732898 0.680339i \(-0.761833\pi\)
−0.732898 + 0.680339i \(0.761833\pi\)
\(608\) 1.41312e7 1.55032
\(609\) −5.40344e6 −0.590374
\(610\) −1.20184e7 −1.30774
\(611\) 6.01231e6 0.651536
\(612\) 9.27817e6 1.00135
\(613\) 2.35101e6 0.252699 0.126350 0.991986i \(-0.459674\pi\)
0.126350 + 0.991986i \(0.459674\pi\)
\(614\) 1.51675e7 1.62365
\(615\) 5.26963e6 0.561814
\(616\) −489991. −0.0520280
\(617\) 9.63523e6 1.01894 0.509470 0.860488i \(-0.329841\pi\)
0.509470 + 0.860488i \(0.329841\pi\)
\(618\) 3.03939e6 0.320122
\(619\) −4.86148e6 −0.509967 −0.254983 0.966945i \(-0.582070\pi\)
−0.254983 + 0.966945i \(0.582070\pi\)
\(620\) 2.12705e6 0.222228
\(621\) −1.40568e7 −1.46271
\(622\) −1.90062e7 −1.96979
\(623\) −2.04307e6 −0.210894
\(624\) 6.63789e6 0.682446
\(625\) 2.70017e6 0.276498
\(626\) −2.83391e7 −2.89035
\(627\) −1.17836e7 −1.19704
\(628\) −3.27519e6 −0.331389
\(629\) 3.16023e6 0.318487
\(630\) 4.83644e6 0.485483
\(631\) −6.59770e6 −0.659659 −0.329829 0.944041i \(-0.606991\pi\)
−0.329829 + 0.944041i \(0.606991\pi\)
\(632\) 3.03188e6 0.301939
\(633\) −1.92409e7 −1.90860
\(634\) 2.43535e7 2.40624
\(635\) −2.02874e6 −0.199661
\(636\) −3.69430e6 −0.362151
\(637\) 721836. 0.0704839
\(638\) −9.60876e6 −0.934578
\(639\) 1.36308e7 1.32059
\(640\) −2.15861e6 −0.208317
\(641\) 1.44525e7 1.38930 0.694651 0.719347i \(-0.255559\pi\)
0.694651 + 0.719347i \(0.255559\pi\)
\(642\) −3.65104e6 −0.349606
\(643\) −1.54720e7 −1.47577 −0.737886 0.674926i \(-0.764176\pi\)
−0.737886 + 0.674926i \(0.764176\pi\)
\(644\) 5.69799e6 0.541386
\(645\) 1.44415e7 1.36683
\(646\) −8.62960e6 −0.813597
\(647\) 1.66647e7 1.56508 0.782540 0.622601i \(-0.213924\pi\)
0.782540 + 0.622601i \(0.213924\pi\)
\(648\) 454306. 0.0425022
\(649\) 8.02371e6 0.747762
\(650\) −5.71810e6 −0.530845
\(651\) −2.54941e6 −0.235770
\(652\) −1.75525e7 −1.61704
\(653\) −1.33451e7 −1.22472 −0.612361 0.790578i \(-0.709780\pi\)
−0.612361 + 0.790578i \(0.709780\pi\)
\(654\) 1.82570e7 1.66911
\(655\) −4.99409e6 −0.454834
\(656\) 6.15149e6 0.558111
\(657\) −4.61157e6 −0.416807
\(658\) 8.10877e6 0.730113
\(659\) −4.00667e6 −0.359393 −0.179697 0.983722i \(-0.557512\pi\)
−0.179697 + 0.983722i \(0.557512\pi\)
\(660\) 7.26438e6 0.649141
\(661\) 1.08005e7 0.961478 0.480739 0.876864i \(-0.340368\pi\)
0.480739 + 0.876864i \(0.340368\pi\)
\(662\) 7.99493e6 0.709038
\(663\) −4.72782e6 −0.417712
\(664\) 4.37750e6 0.385307
\(665\) −2.39614e6 −0.210116
\(666\) 1.76965e7 1.54597
\(667\) 1.37068e7 1.19294
\(668\) −3.15767e6 −0.273795
\(669\) −1.37070e7 −1.18407
\(670\) 1.20223e6 0.103466
\(671\) 1.36447e7 1.16993
\(672\) 1.04415e7 0.891951
\(673\) 1.09119e7 0.928676 0.464338 0.885658i \(-0.346292\pi\)
0.464338 + 0.885658i \(0.346292\pi\)
\(674\) 1.12884e6 0.0957158
\(675\) 1.01342e7 0.856110
\(676\) −1.02459e7 −0.862353
\(677\) −1.35765e7 −1.13846 −0.569229 0.822179i \(-0.692758\pi\)
−0.569229 + 0.822179i \(0.692758\pi\)
\(678\) −2.91984e7 −2.43941
\(679\) 2.14046e6 0.178169
\(680\) 652598. 0.0541219
\(681\) 1.05323e7 0.870272
\(682\) −4.53354e6 −0.373230
\(683\) −1.26726e7 −1.03948 −0.519738 0.854326i \(-0.673970\pi\)
−0.519738 + 0.854326i \(0.673970\pi\)
\(684\) −2.57406e7 −2.10367
\(685\) −57209.2 −0.00465843
\(686\) 973536. 0.0789845
\(687\) −2.64283e7 −2.13637
\(688\) 1.68583e7 1.35782
\(689\) 1.18717e6 0.0952720
\(690\) −1.94539e7 −1.55555
\(691\) 7.11964e6 0.567235 0.283617 0.958938i \(-0.408465\pi\)
0.283617 + 0.958938i \(0.408465\pi\)
\(692\) −2.12052e6 −0.168336
\(693\) −5.49091e6 −0.434322
\(694\) 2.94097e6 0.231789
\(695\) 1.05339e7 0.827235
\(696\) −4.08281e6 −0.319474
\(697\) −4.38139e6 −0.341609
\(698\) −1.15955e6 −0.0900844
\(699\) 3.05770e6 0.236702
\(700\) −4.10794e6 −0.316869
\(701\) −1.00155e7 −0.769803 −0.384902 0.922958i \(-0.625765\pi\)
−0.384902 + 0.922958i \(0.625765\pi\)
\(702\) −1.09687e7 −0.840068
\(703\) −8.76746e6 −0.669092
\(704\) 1.11280e7 0.846224
\(705\) −1.47469e7 −1.11745
\(706\) 2.88084e7 2.17524
\(707\) 1.25676e6 0.0945589
\(708\) 2.77928e7 2.08377
\(709\) −8.84454e6 −0.660784 −0.330392 0.943844i \(-0.607181\pi\)
−0.330392 + 0.943844i \(0.607181\pi\)
\(710\) 7.81571e6 0.581866
\(711\) 3.39757e7 2.52054
\(712\) −1.54373e6 −0.114123
\(713\) 6.46703e6 0.476410
\(714\) −6.37638e6 −0.468090
\(715\) −2.33442e6 −0.170771
\(716\) −7.63698e6 −0.556723
\(717\) 6.51728e6 0.473444
\(718\) 1.45047e7 1.05002
\(719\) 6.58086e6 0.474745 0.237373 0.971419i \(-0.423714\pi\)
0.237373 + 0.971419i \(0.423714\pi\)
\(720\) −1.02676e7 −0.738141
\(721\) −701682. −0.0502692
\(722\) 3.45169e6 0.246427
\(723\) −3.62301e7 −2.57765
\(724\) 1.01762e7 0.721502
\(725\) −9.88183e6 −0.698220
\(726\) 1.86995e7 1.31671
\(727\) 1.88401e7 1.32205 0.661023 0.750365i \(-0.270122\pi\)
0.661023 + 0.750365i \(0.270122\pi\)
\(728\) 545414. 0.0381415
\(729\) −2.23900e7 −1.56040
\(730\) −2.64422e6 −0.183650
\(731\) −1.20073e7 −0.831095
\(732\) 4.72631e7 3.26020
\(733\) −2.78330e6 −0.191337 −0.0956687 0.995413i \(-0.530499\pi\)
−0.0956687 + 0.995413i \(0.530499\pi\)
\(734\) −1.46416e7 −1.00311
\(735\) −1.77050e6 −0.120887
\(736\) −2.64867e7 −1.80233
\(737\) −1.36491e6 −0.0925629
\(738\) −2.45346e7 −1.65821
\(739\) −2.48970e7 −1.67701 −0.838505 0.544894i \(-0.816570\pi\)
−0.838505 + 0.544894i \(0.816570\pi\)
\(740\) 5.40499e6 0.362840
\(741\) 1.31164e7 0.877548
\(742\) 1.60113e6 0.106762
\(743\) −3.86085e6 −0.256573 −0.128286 0.991737i \(-0.540948\pi\)
−0.128286 + 0.991737i \(0.540948\pi\)
\(744\) −1.92632e6 −0.127584
\(745\) 4.04554e6 0.267046
\(746\) −3.44412e7 −2.26585
\(747\) 4.90549e7 3.21648
\(748\) −6.03991e6 −0.394708
\(749\) 842888. 0.0548991
\(750\) 3.30938e7 2.14829
\(751\) 6.72737e6 0.435257 0.217628 0.976032i \(-0.430168\pi\)
0.217628 + 0.976032i \(0.430168\pi\)
\(752\) −1.72147e7 −1.11008
\(753\) 4.29737e7 2.76195
\(754\) 1.06956e7 0.685136
\(755\) 1.43932e6 0.0918943
\(756\) −7.88007e6 −0.501447
\(757\) 2.17782e7 1.38128 0.690642 0.723197i \(-0.257328\pi\)
0.690642 + 0.723197i \(0.257328\pi\)
\(758\) 5.11525e6 0.323366
\(759\) 2.20865e7 1.39162
\(760\) −1.81051e6 −0.113702
\(761\) −2.57074e7 −1.60915 −0.804575 0.593851i \(-0.797607\pi\)
−0.804575 + 0.593851i \(0.797607\pi\)
\(762\) 1.49777e7 0.934453
\(763\) −4.21486e6 −0.262103
\(764\) −1.62359e7 −1.00633
\(765\) 7.31310e6 0.451802
\(766\) −3.40234e7 −2.09511
\(767\) −8.93127e6 −0.548182
\(768\) −1.78808e7 −1.09391
\(769\) −1.34375e7 −0.819413 −0.409706 0.912217i \(-0.634369\pi\)
−0.409706 + 0.912217i \(0.634369\pi\)
\(770\) −3.14842e6 −0.191367
\(771\) −1.86471e7 −1.12973
\(772\) −2.65099e7 −1.60090
\(773\) 3.05572e7 1.83935 0.919674 0.392682i \(-0.128453\pi\)
0.919674 + 0.392682i \(0.128453\pi\)
\(774\) −6.72376e7 −4.03422
\(775\) −4.66237e6 −0.278839
\(776\) 1.61731e6 0.0964140
\(777\) −6.47825e6 −0.384951
\(778\) 3.82364e7 2.26479
\(779\) 1.21553e7 0.717667
\(780\) −8.08606e6 −0.475883
\(781\) −8.87335e6 −0.520547
\(782\) 1.61748e7 0.945849
\(783\) −1.89558e7 −1.10494
\(784\) −2.06679e6 −0.120090
\(785\) −2.58152e6 −0.149521
\(786\) 3.68700e7 2.12871
\(787\) −2.07672e6 −0.119520 −0.0597602 0.998213i \(-0.519034\pi\)
−0.0597602 + 0.998213i \(0.519034\pi\)
\(788\) −1.33093e7 −0.763556
\(789\) 5.79372e6 0.331333
\(790\) 1.94812e7 1.11058
\(791\) 6.74081e6 0.383064
\(792\) −4.14890e6 −0.235028
\(793\) −1.51881e7 −0.857670
\(794\) 4.19827e7 2.36330
\(795\) −2.91187e6 −0.163401
\(796\) 1.05523e7 0.590286
\(797\) −5.98563e6 −0.333783 −0.166892 0.985975i \(-0.553373\pi\)
−0.166892 + 0.985975i \(0.553373\pi\)
\(798\) 1.76901e7 0.983383
\(799\) 1.22611e7 0.679460
\(800\) 1.90955e7 1.05489
\(801\) −1.72993e7 −0.952679
\(802\) −1.22515e7 −0.672594
\(803\) 3.00204e6 0.164296
\(804\) −4.72784e6 −0.257942
\(805\) 4.49118e6 0.244270
\(806\) 5.04633e6 0.273614
\(807\) −4.63043e7 −2.50286
\(808\) 949597. 0.0511695
\(809\) 1.96864e7 1.05754 0.528769 0.848766i \(-0.322654\pi\)
0.528769 + 0.848766i \(0.322654\pi\)
\(810\) 2.91913e6 0.156329
\(811\) 8.50101e6 0.453856 0.226928 0.973912i \(-0.427132\pi\)
0.226928 + 0.973912i \(0.427132\pi\)
\(812\) 7.68384e6 0.408967
\(813\) 4.39580e7 2.33245
\(814\) −1.15201e7 −0.609387
\(815\) −1.38350e7 −0.729599
\(816\) 1.35369e7 0.711695
\(817\) 3.33119e7 1.74600
\(818\) −3.75168e7 −1.96039
\(819\) 6.11199e6 0.318400
\(820\) −7.49355e6 −0.389182
\(821\) −1.36199e6 −0.0705204 −0.0352602 0.999378i \(-0.511226\pi\)
−0.0352602 + 0.999378i \(0.511226\pi\)
\(822\) 422360. 0.0218023
\(823\) −1.35934e6 −0.0699566 −0.0349783 0.999388i \(-0.511136\pi\)
−0.0349783 + 0.999388i \(0.511136\pi\)
\(824\) −530186. −0.0272026
\(825\) −1.59231e7 −0.814505
\(826\) −1.20456e7 −0.614295
\(827\) 1.00727e7 0.512132 0.256066 0.966659i \(-0.417574\pi\)
0.256066 + 0.966659i \(0.417574\pi\)
\(828\) 4.82465e7 2.44563
\(829\) −5.63984e6 −0.285023 −0.142512 0.989793i \(-0.545518\pi\)
−0.142512 + 0.989793i \(0.545518\pi\)
\(830\) 2.81275e7 1.41722
\(831\) −5.72125e7 −2.87401
\(832\) −1.23867e7 −0.620364
\(833\) 1.47207e6 0.0735048
\(834\) −7.77693e7 −3.87163
\(835\) −2.48889e6 −0.123535
\(836\) 1.67566e7 0.829220
\(837\) −8.94361e6 −0.441265
\(838\) 918733. 0.0451938
\(839\) −1.16351e7 −0.570642 −0.285321 0.958432i \(-0.592100\pi\)
−0.285321 + 0.958432i \(0.592100\pi\)
\(840\) −1.33778e6 −0.0654164
\(841\) −2.02735e6 −0.0988413
\(842\) −1.16801e7 −0.567764
\(843\) −4.28706e7 −2.07774
\(844\) 2.73610e7 1.32214
\(845\) −8.07590e6 −0.389089
\(846\) 6.86592e7 3.29817
\(847\) −4.31702e6 −0.206764
\(848\) −3.39916e6 −0.162324
\(849\) 1.01611e7 0.483806
\(850\) −1.16611e7 −0.553597
\(851\) 1.64332e7 0.777854
\(852\) −3.07358e7 −1.45059
\(853\) 2.85205e7 1.34210 0.671049 0.741413i \(-0.265844\pi\)
0.671049 + 0.741413i \(0.265844\pi\)
\(854\) −2.04841e7 −0.961108
\(855\) −2.02888e7 −0.949165
\(856\) 636880. 0.0297080
\(857\) −9.95725e6 −0.463113 −0.231557 0.972821i \(-0.574382\pi\)
−0.231557 + 0.972821i \(0.574382\pi\)
\(858\) 1.72344e7 0.799243
\(859\) −1.49322e7 −0.690463 −0.345232 0.938517i \(-0.612200\pi\)
−0.345232 + 0.938517i \(0.612200\pi\)
\(860\) −2.05362e7 −0.946834
\(861\) 8.98154e6 0.412898
\(862\) 8.90711e6 0.408290
\(863\) 3.84933e7 1.75937 0.879687 0.475553i \(-0.157752\pi\)
0.879687 + 0.475553i \(0.157752\pi\)
\(864\) 3.66300e7 1.66937
\(865\) −1.67140e6 −0.0759523
\(866\) −2.56665e6 −0.116298
\(867\) 2.67770e7 1.20980
\(868\) 3.62533e6 0.163323
\(869\) −2.21175e7 −0.993542
\(870\) −2.62339e7 −1.17507
\(871\) 1.51930e6 0.0678576
\(872\) −3.18472e6 −0.141834
\(873\) 1.81239e7 0.804850
\(874\) −4.48739e7 −1.98708
\(875\) −7.64011e6 −0.337349
\(876\) 1.03986e7 0.457839
\(877\) 9.40311e6 0.412831 0.206416 0.978464i \(-0.433820\pi\)
0.206416 + 0.978464i \(0.433820\pi\)
\(878\) 4.69726e7 2.05640
\(879\) 2.38403e7 1.04073
\(880\) 6.68403e6 0.290959
\(881\) 1.10395e6 0.0479194 0.0239597 0.999713i \(-0.492373\pi\)
0.0239597 + 0.999713i \(0.492373\pi\)
\(882\) 8.24321e6 0.356800
\(883\) 8.06579e6 0.348133 0.174067 0.984734i \(-0.444309\pi\)
0.174067 + 0.984734i \(0.444309\pi\)
\(884\) 6.72308e6 0.289359
\(885\) 2.19064e7 0.940185
\(886\) 3.35933e7 1.43770
\(887\) −1.49902e7 −0.639732 −0.319866 0.947463i \(-0.603638\pi\)
−0.319866 + 0.947463i \(0.603638\pi\)
\(888\) −4.89493e6 −0.208312
\(889\) −3.45779e6 −0.146738
\(890\) −9.91920e6 −0.419761
\(891\) −3.31415e6 −0.139855
\(892\) 1.94918e7 0.820237
\(893\) −3.40162e7 −1.42744
\(894\) −2.98672e7 −1.24983
\(895\) −6.01950e6 −0.251190
\(896\) −3.67912e6 −0.153100
\(897\) −2.45847e7 −1.02019
\(898\) −5.76384e7 −2.38518
\(899\) 8.72090e6 0.359883
\(900\) −3.47831e7 −1.43140
\(901\) 2.42104e6 0.0993553
\(902\) 1.59716e7 0.653629
\(903\) 2.46141e7 1.00453
\(904\) 5.09331e6 0.207290
\(905\) 8.02089e6 0.325538
\(906\) −1.06261e7 −0.430083
\(907\) 4.12622e6 0.166546 0.0832730 0.996527i \(-0.473463\pi\)
0.0832730 + 0.996527i \(0.473463\pi\)
\(908\) −1.49772e7 −0.602859
\(909\) 1.06413e7 0.427155
\(910\) 3.50454e6 0.140290
\(911\) 4.04272e7 1.61391 0.806953 0.590616i \(-0.201115\pi\)
0.806953 + 0.590616i \(0.201115\pi\)
\(912\) −3.75556e7 −1.49516
\(913\) −3.19338e7 −1.26787
\(914\) 1.48554e7 0.588191
\(915\) 3.72530e7 1.47099
\(916\) 3.75817e7 1.47992
\(917\) −8.51191e6 −0.334275
\(918\) −2.23690e7 −0.876072
\(919\) 2.18546e7 0.853600 0.426800 0.904346i \(-0.359641\pi\)
0.426800 + 0.904346i \(0.359641\pi\)
\(920\) 3.39351e6 0.132184
\(921\) −4.70142e7 −1.82633
\(922\) −1.74844e7 −0.677366
\(923\) 9.87702e6 0.381612
\(924\) 1.23814e7 0.477078
\(925\) −1.18474e7 −0.455271
\(926\) 1.04448e7 0.400289
\(927\) −5.94134e6 −0.227083
\(928\) −3.57178e7 −1.36149
\(929\) 1.06843e7 0.406169 0.203085 0.979161i \(-0.434903\pi\)
0.203085 + 0.979161i \(0.434903\pi\)
\(930\) −1.23775e7 −0.469274
\(931\) −4.08398e6 −0.154422
\(932\) −4.34813e6 −0.163970
\(933\) 5.89130e7 2.21568
\(934\) −2.97009e7 −1.11404
\(935\) −4.76068e6 −0.178090
\(936\) 4.61818e6 0.172298
\(937\) −3.99105e7 −1.48504 −0.742521 0.669823i \(-0.766370\pi\)
−0.742521 + 0.669823i \(0.766370\pi\)
\(938\) 2.04907e6 0.0760414
\(939\) 8.78419e7 3.25116
\(940\) 2.09704e7 0.774082
\(941\) 1.32350e6 0.0487248 0.0243624 0.999703i \(-0.492244\pi\)
0.0243624 + 0.999703i \(0.492244\pi\)
\(942\) 1.90587e7 0.699787
\(943\) −2.27832e7 −0.834326
\(944\) 2.55724e7 0.933990
\(945\) −6.21110e6 −0.226250
\(946\) 4.37703e7 1.59020
\(947\) −2.76322e7 −1.00124 −0.500622 0.865666i \(-0.666895\pi\)
−0.500622 + 0.865666i \(0.666895\pi\)
\(948\) −7.66113e7 −2.76868
\(949\) −3.34160e6 −0.120445
\(950\) 3.23517e7 1.16302
\(951\) −7.54879e7 −2.70661
\(952\) 1.11229e6 0.0397763
\(953\) −3.07901e7 −1.09819 −0.549096 0.835759i \(-0.685028\pi\)
−0.549096 + 0.835759i \(0.685028\pi\)
\(954\) 1.35572e7 0.482281
\(955\) −1.27972e7 −0.454053
\(956\) −9.26775e6 −0.327966
\(957\) 2.97840e7 1.05124
\(958\) −1.99999e7 −0.704068
\(959\) −97507.1 −0.00342365
\(960\) 3.03818e7 1.06398
\(961\) −2.45145e7 −0.856278
\(962\) 1.28231e7 0.446740
\(963\) 7.13697e6 0.247998
\(964\) 5.15202e7 1.78560
\(965\) −2.08952e7 −0.722318
\(966\) −3.31572e7 −1.14323
\(967\) 2.92557e6 0.100611 0.0503055 0.998734i \(-0.483981\pi\)
0.0503055 + 0.998734i \(0.483981\pi\)
\(968\) −3.26192e6 −0.111888
\(969\) 2.67489e7 0.915159
\(970\) 1.03920e7 0.354625
\(971\) 2.78109e6 0.0946601 0.0473301 0.998879i \(-0.484929\pi\)
0.0473301 + 0.998879i \(0.484929\pi\)
\(972\) 2.75990e7 0.936975
\(973\) 1.79540e7 0.607967
\(974\) −4.29801e7 −1.45168
\(975\) 1.77242e7 0.597111
\(976\) 4.34872e7 1.46129
\(977\) 7.48673e6 0.250932 0.125466 0.992098i \(-0.459957\pi\)
0.125466 + 0.992098i \(0.459957\pi\)
\(978\) 1.02140e8 3.41467
\(979\) 1.12615e7 0.375525
\(980\) 2.51770e6 0.0837411
\(981\) −3.56884e7 −1.18401
\(982\) 4.45985e7 1.47585
\(983\) 1.79815e7 0.593528 0.296764 0.954951i \(-0.404093\pi\)
0.296764 + 0.954951i \(0.404093\pi\)
\(984\) 6.78639e6 0.223435
\(985\) −1.04905e7 −0.344512
\(986\) 2.18120e7 0.714501
\(987\) −2.51345e7 −0.821253
\(988\) −1.86519e7 −0.607899
\(989\) −6.24378e7 −2.02982
\(990\) −2.66586e7 −0.864469
\(991\) 3.72778e7 1.20578 0.602888 0.797826i \(-0.294017\pi\)
0.602888 + 0.797826i \(0.294017\pi\)
\(992\) −1.68521e7 −0.543720
\(993\) −2.47816e7 −0.797548
\(994\) 1.33211e7 0.427635
\(995\) 8.31734e6 0.266334
\(996\) −1.10613e8 −3.53313
\(997\) 4.87422e7 1.55298 0.776492 0.630128i \(-0.216997\pi\)
0.776492 + 0.630128i \(0.216997\pi\)
\(998\) −2.72747e7 −0.866828
\(999\) −2.27264e7 −0.720471
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 7.6.a.b.1.2 2
3.2 odd 2 63.6.a.f.1.1 2
4.3 odd 2 112.6.a.h.1.2 2
5.2 odd 4 175.6.b.c.99.4 4
5.3 odd 4 175.6.b.c.99.1 4
5.4 even 2 175.6.a.c.1.1 2
7.2 even 3 49.6.c.e.18.1 4
7.3 odd 6 49.6.c.d.30.1 4
7.4 even 3 49.6.c.e.30.1 4
7.5 odd 6 49.6.c.d.18.1 4
7.6 odd 2 49.6.a.f.1.2 2
8.3 odd 2 448.6.a.u.1.1 2
8.5 even 2 448.6.a.w.1.2 2
11.10 odd 2 847.6.a.c.1.1 2
12.11 even 2 1008.6.a.bq.1.1 2
21.20 even 2 441.6.a.l.1.1 2
28.27 even 2 784.6.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.b.1.2 2 1.1 even 1 trivial
49.6.a.f.1.2 2 7.6 odd 2
49.6.c.d.18.1 4 7.5 odd 6
49.6.c.d.30.1 4 7.3 odd 6
49.6.c.e.18.1 4 7.2 even 3
49.6.c.e.30.1 4 7.4 even 3
63.6.a.f.1.1 2 3.2 odd 2
112.6.a.h.1.2 2 4.3 odd 2
175.6.a.c.1.1 2 5.4 even 2
175.6.b.c.99.1 4 5.3 odd 4
175.6.b.c.99.4 4 5.2 odd 4
441.6.a.l.1.1 2 21.20 even 2
448.6.a.u.1.1 2 8.3 odd 2
448.6.a.w.1.2 2 8.5 even 2
784.6.a.v.1.1 2 28.27 even 2
847.6.a.c.1.1 2 11.10 odd 2
1008.6.a.bq.1.1 2 12.11 even 2