Properties

Label 177.6.a.c.1.1
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + 21512960 x + 49172480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(12.3715\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.3715 q^{2} +9.00000 q^{3} +75.5670 q^{4} +33.2592 q^{5} -93.3431 q^{6} -56.9512 q^{7} -451.853 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-10.3715 q^{2} +9.00000 q^{3} +75.5670 q^{4} +33.2592 q^{5} -93.3431 q^{6} -56.9512 q^{7} -451.853 q^{8} +81.0000 q^{9} -344.946 q^{10} +432.466 q^{11} +680.103 q^{12} +178.063 q^{13} +590.667 q^{14} +299.333 q^{15} +2268.23 q^{16} +1658.66 q^{17} -840.088 q^{18} -168.835 q^{19} +2513.30 q^{20} -512.561 q^{21} -4485.30 q^{22} +143.318 q^{23} -4066.68 q^{24} -2018.83 q^{25} -1846.77 q^{26} +729.000 q^{27} -4303.63 q^{28} +275.265 q^{29} -3104.51 q^{30} -2273.75 q^{31} -9065.52 q^{32} +3892.20 q^{33} -17202.7 q^{34} -1894.15 q^{35} +6120.93 q^{36} -1120.19 q^{37} +1751.06 q^{38} +1602.57 q^{39} -15028.3 q^{40} -2114.42 q^{41} +5316.00 q^{42} +3840.06 q^{43} +32680.2 q^{44} +2693.99 q^{45} -1486.42 q^{46} +14809.7 q^{47} +20414.1 q^{48} -13563.6 q^{49} +20938.2 q^{50} +14927.9 q^{51} +13455.7 q^{52} +9651.99 q^{53} -7560.79 q^{54} +14383.5 q^{55} +25733.6 q^{56} -1519.51 q^{57} -2854.90 q^{58} -3481.00 q^{59} +22619.7 q^{60} +29254.1 q^{61} +23582.1 q^{62} -4613.05 q^{63} +21439.3 q^{64} +5922.23 q^{65} -40367.7 q^{66} +15092.5 q^{67} +125340. q^{68} +1289.86 q^{69} +19645.1 q^{70} +13271.8 q^{71} -36600.1 q^{72} +42497.1 q^{73} +11618.0 q^{74} -18169.4 q^{75} -12758.3 q^{76} -24629.5 q^{77} -16620.9 q^{78} -55585.7 q^{79} +75439.4 q^{80} +6561.00 q^{81} +21929.6 q^{82} +81043.5 q^{83} -38732.7 q^{84} +55165.7 q^{85} -39827.0 q^{86} +2477.39 q^{87} -195411. q^{88} +107408. q^{89} -27940.6 q^{90} -10140.9 q^{91} +10830.1 q^{92} -20463.7 q^{93} -153598. q^{94} -5615.31 q^{95} -81589.7 q^{96} -92436.4 q^{97} +140674. q^{98} +35029.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.3715 −1.83343 −0.916715 0.399541i \(-0.869170\pi\)
−0.916715 + 0.399541i \(0.869170\pi\)
\(3\) 9.00000 0.577350
\(4\) 75.5670 2.36147
\(5\) 33.2592 0.594958 0.297479 0.954728i \(-0.403854\pi\)
0.297479 + 0.954728i \(0.403854\pi\)
\(6\) −93.3431 −1.05853
\(7\) −56.9512 −0.439297 −0.219648 0.975579i \(-0.570491\pi\)
−0.219648 + 0.975579i \(0.570491\pi\)
\(8\) −451.853 −2.49616
\(9\) 81.0000 0.333333
\(10\) −344.946 −1.09081
\(11\) 432.466 1.07763 0.538816 0.842423i \(-0.318872\pi\)
0.538816 + 0.842423i \(0.318872\pi\)
\(12\) 680.103 1.36339
\(13\) 178.063 0.292224 0.146112 0.989268i \(-0.453324\pi\)
0.146112 + 0.989268i \(0.453324\pi\)
\(14\) 590.667 0.805420
\(15\) 299.333 0.343499
\(16\) 2268.23 2.21507
\(17\) 1658.66 1.39199 0.695993 0.718048i \(-0.254964\pi\)
0.695993 + 0.718048i \(0.254964\pi\)
\(18\) −840.088 −0.611144
\(19\) −168.835 −0.107295 −0.0536473 0.998560i \(-0.517085\pi\)
−0.0536473 + 0.998560i \(0.517085\pi\)
\(20\) 2513.30 1.40498
\(21\) −512.561 −0.253628
\(22\) −4485.30 −1.97576
\(23\) 143.318 0.0564914 0.0282457 0.999601i \(-0.491008\pi\)
0.0282457 + 0.999601i \(0.491008\pi\)
\(24\) −4066.68 −1.44116
\(25\) −2018.83 −0.646025
\(26\) −1846.77 −0.535772
\(27\) 729.000 0.192450
\(28\) −4303.63 −1.03739
\(29\) 275.265 0.0607794 0.0303897 0.999538i \(-0.490325\pi\)
0.0303897 + 0.999538i \(0.490325\pi\)
\(30\) −3104.51 −0.629782
\(31\) −2273.75 −0.424950 −0.212475 0.977166i \(-0.568152\pi\)
−0.212475 + 0.977166i \(0.568152\pi\)
\(32\) −9065.52 −1.56501
\(33\) 3892.20 0.622171
\(34\) −17202.7 −2.55211
\(35\) −1894.15 −0.261363
\(36\) 6120.93 0.787156
\(37\) −1120.19 −0.134520 −0.0672601 0.997735i \(-0.521426\pi\)
−0.0672601 + 0.997735i \(0.521426\pi\)
\(38\) 1751.06 0.196717
\(39\) 1602.57 0.168715
\(40\) −15028.3 −1.48511
\(41\) −2114.42 −0.196441 −0.0982203 0.995165i \(-0.531315\pi\)
−0.0982203 + 0.995165i \(0.531315\pi\)
\(42\) 5316.00 0.465009
\(43\) 3840.06 0.316714 0.158357 0.987382i \(-0.449380\pi\)
0.158357 + 0.987382i \(0.449380\pi\)
\(44\) 32680.2 2.54479
\(45\) 2693.99 0.198319
\(46\) −1486.42 −0.103573
\(47\) 14809.7 0.977917 0.488959 0.872307i \(-0.337377\pi\)
0.488959 + 0.872307i \(0.337377\pi\)
\(48\) 20414.1 1.27887
\(49\) −13563.6 −0.807019
\(50\) 20938.2 1.18444
\(51\) 14927.9 0.803664
\(52\) 13455.7 0.690077
\(53\) 9651.99 0.471984 0.235992 0.971755i \(-0.424166\pi\)
0.235992 + 0.971755i \(0.424166\pi\)
\(54\) −7560.79 −0.352844
\(55\) 14383.5 0.641146
\(56\) 25733.6 1.09655
\(57\) −1519.51 −0.0619466
\(58\) −2854.90 −0.111435
\(59\) −3481.00 −0.130189
\(60\) 22619.7 0.811163
\(61\) 29254.1 1.00661 0.503307 0.864108i \(-0.332117\pi\)
0.503307 + 0.864108i \(0.332117\pi\)
\(62\) 23582.1 0.779117
\(63\) −4613.05 −0.146432
\(64\) 21439.3 0.654275
\(65\) 5922.23 0.173861
\(66\) −40367.7 −1.14071
\(67\) 15092.5 0.410747 0.205374 0.978684i \(-0.434159\pi\)
0.205374 + 0.978684i \(0.434159\pi\)
\(68\) 125340. 3.28713
\(69\) 1289.86 0.0326153
\(70\) 19645.1 0.479191
\(71\) 13271.8 0.312453 0.156226 0.987721i \(-0.450067\pi\)
0.156226 + 0.987721i \(0.450067\pi\)
\(72\) −36600.1 −0.832053
\(73\) 42497.1 0.933366 0.466683 0.884425i \(-0.345449\pi\)
0.466683 + 0.884425i \(0.345449\pi\)
\(74\) 11618.0 0.246634
\(75\) −18169.4 −0.372983
\(76\) −12758.3 −0.253373
\(77\) −24629.5 −0.473400
\(78\) −16620.9 −0.309328
\(79\) −55585.7 −1.00206 −0.501032 0.865429i \(-0.667046\pi\)
−0.501032 + 0.865429i \(0.667046\pi\)
\(80\) 75439.4 1.31787
\(81\) 6561.00 0.111111
\(82\) 21929.6 0.360160
\(83\) 81043.5 1.29129 0.645643 0.763639i \(-0.276589\pi\)
0.645643 + 0.763639i \(0.276589\pi\)
\(84\) −38732.7 −0.598935
\(85\) 55165.7 0.828174
\(86\) −39827.0 −0.580673
\(87\) 2477.39 0.0350910
\(88\) −195411. −2.68994
\(89\) 107408. 1.43734 0.718672 0.695349i \(-0.244750\pi\)
0.718672 + 0.695349i \(0.244750\pi\)
\(90\) −27940.6 −0.363605
\(91\) −10140.9 −0.128373
\(92\) 10830.1 0.133403
\(93\) −20463.7 −0.245345
\(94\) −153598. −1.79294
\(95\) −5615.31 −0.0638358
\(96\) −81589.7 −0.903560
\(97\) −92436.4 −0.997502 −0.498751 0.866745i \(-0.666208\pi\)
−0.498751 + 0.866745i \(0.666208\pi\)
\(98\) 140674. 1.47961
\(99\) 35029.8 0.359211
\(100\) −152557. −1.52557
\(101\) −134769. −1.31458 −0.657291 0.753637i \(-0.728298\pi\)
−0.657291 + 0.753637i \(0.728298\pi\)
\(102\) −154824. −1.47346
\(103\) 166146. 1.54311 0.771553 0.636165i \(-0.219480\pi\)
0.771553 + 0.636165i \(0.219480\pi\)
\(104\) −80458.3 −0.729437
\(105\) −17047.3 −0.150898
\(106\) −100105. −0.865350
\(107\) 120010. 1.01335 0.506674 0.862138i \(-0.330875\pi\)
0.506674 + 0.862138i \(0.330875\pi\)
\(108\) 55088.3 0.454465
\(109\) 188031. 1.51587 0.757937 0.652328i \(-0.226207\pi\)
0.757937 + 0.652328i \(0.226207\pi\)
\(110\) −149177. −1.17550
\(111\) −10081.7 −0.0776653
\(112\) −129178. −0.973071
\(113\) 114073. 0.840401 0.420201 0.907431i \(-0.361960\pi\)
0.420201 + 0.907431i \(0.361960\pi\)
\(114\) 15759.6 0.113575
\(115\) 4766.65 0.0336100
\(116\) 20801.0 0.143529
\(117\) 14423.1 0.0974079
\(118\) 36103.0 0.238692
\(119\) −94462.7 −0.611495
\(120\) −135254. −0.857429
\(121\) 25976.1 0.161291
\(122\) −303408. −1.84556
\(123\) −19029.8 −0.113415
\(124\) −171820. −1.00351
\(125\) −171079. −0.979316
\(126\) 47844.0 0.268473
\(127\) 305300. 1.67964 0.839822 0.542861i \(-0.182659\pi\)
0.839822 + 0.542861i \(0.182659\pi\)
\(128\) 67740.1 0.365444
\(129\) 34560.6 0.182855
\(130\) −61422.1 −0.318762
\(131\) −152752. −0.777695 −0.388848 0.921302i \(-0.627127\pi\)
−0.388848 + 0.921302i \(0.627127\pi\)
\(132\) 294122. 1.46924
\(133\) 9615.35 0.0471342
\(134\) −156531. −0.753077
\(135\) 24245.9 0.114500
\(136\) −749471. −3.47462
\(137\) −128339. −0.584196 −0.292098 0.956388i \(-0.594353\pi\)
−0.292098 + 0.956388i \(0.594353\pi\)
\(138\) −13377.8 −0.0597979
\(139\) 155483. 0.682566 0.341283 0.939961i \(-0.389139\pi\)
0.341283 + 0.939961i \(0.389139\pi\)
\(140\) −143135. −0.617201
\(141\) 133287. 0.564601
\(142\) −137648. −0.572860
\(143\) 77006.3 0.314910
\(144\) 183726. 0.738355
\(145\) 9155.09 0.0361612
\(146\) −440756. −1.71126
\(147\) −122072. −0.465932
\(148\) −84649.5 −0.317665
\(149\) 444140. 1.63891 0.819454 0.573146i \(-0.194277\pi\)
0.819454 + 0.573146i \(0.194277\pi\)
\(150\) 188444. 0.683838
\(151\) −68940.1 −0.246054 −0.123027 0.992403i \(-0.539260\pi\)
−0.123027 + 0.992403i \(0.539260\pi\)
\(152\) 76288.6 0.267825
\(153\) 134351. 0.463996
\(154\) 255443. 0.867946
\(155\) −75623.0 −0.252828
\(156\) 121101. 0.398416
\(157\) −69283.9 −0.224328 −0.112164 0.993690i \(-0.535778\pi\)
−0.112164 + 0.993690i \(0.535778\pi\)
\(158\) 576504. 1.83721
\(159\) 86868.0 0.272500
\(160\) −301512. −0.931117
\(161\) −8162.15 −0.0248165
\(162\) −68047.1 −0.203715
\(163\) −267280. −0.787946 −0.393973 0.919122i \(-0.628900\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(164\) −159780. −0.463888
\(165\) 129451. 0.370166
\(166\) −840538. −2.36749
\(167\) 387809. 1.07604 0.538018 0.842934i \(-0.319173\pi\)
0.538018 + 0.842934i \(0.319173\pi\)
\(168\) 231602. 0.633096
\(169\) −339587. −0.914605
\(170\) −572148. −1.51840
\(171\) −13675.6 −0.0357649
\(172\) 290182. 0.747910
\(173\) −190512. −0.483958 −0.241979 0.970281i \(-0.577797\pi\)
−0.241979 + 0.970281i \(0.577797\pi\)
\(174\) −25694.1 −0.0643369
\(175\) 114975. 0.283796
\(176\) 980932. 2.38703
\(177\) −31329.0 −0.0751646
\(178\) −1.11397e6 −2.63527
\(179\) −173534. −0.404811 −0.202406 0.979302i \(-0.564876\pi\)
−0.202406 + 0.979302i \(0.564876\pi\)
\(180\) 203577. 0.468325
\(181\) 717762. 1.62849 0.814243 0.580524i \(-0.197152\pi\)
0.814243 + 0.580524i \(0.197152\pi\)
\(182\) 105176. 0.235363
\(183\) 263287. 0.581168
\(184\) −64758.8 −0.141011
\(185\) −37256.6 −0.0800339
\(186\) 212239. 0.449824
\(187\) 717315. 1.50005
\(188\) 1.11913e6 2.30932
\(189\) −41517.4 −0.0845427
\(190\) 58238.9 0.117039
\(191\) 336630. 0.667681 0.333841 0.942630i \(-0.391655\pi\)
0.333841 + 0.942630i \(0.391655\pi\)
\(192\) 192954. 0.377746
\(193\) 293107. 0.566413 0.283206 0.959059i \(-0.408602\pi\)
0.283206 + 0.959059i \(0.408602\pi\)
\(194\) 958699. 1.82885
\(195\) 53300.1 0.100379
\(196\) −1.02496e6 −1.90575
\(197\) −254135. −0.466551 −0.233276 0.972411i \(-0.574944\pi\)
−0.233276 + 0.972411i \(0.574944\pi\)
\(198\) −363310. −0.658588
\(199\) 65290.6 0.116874 0.0584370 0.998291i \(-0.481388\pi\)
0.0584370 + 0.998291i \(0.481388\pi\)
\(200\) 912213. 1.61258
\(201\) 135833. 0.237145
\(202\) 1.39775e6 2.41020
\(203\) −15676.7 −0.0267002
\(204\) 1.12806e6 1.89783
\(205\) −70323.8 −0.116874
\(206\) −1.72317e6 −2.82918
\(207\) 11608.8 0.0188305
\(208\) 403888. 0.647295
\(209\) −73015.4 −0.115624
\(210\) 176806. 0.276661
\(211\) 93142.8 0.144027 0.0720134 0.997404i \(-0.477058\pi\)
0.0720134 + 0.997404i \(0.477058\pi\)
\(212\) 729372. 1.11458
\(213\) 119446. 0.180395
\(214\) −1.24468e6 −1.85790
\(215\) 127717. 0.188432
\(216\) −329401. −0.480386
\(217\) 129493. 0.186679
\(218\) −1.95015e6 −2.77925
\(219\) 382474. 0.538879
\(220\) 1.08692e6 1.51405
\(221\) 295346. 0.406772
\(222\) 104562. 0.142394
\(223\) 30528.4 0.0411095 0.0205548 0.999789i \(-0.493457\pi\)
0.0205548 + 0.999789i \(0.493457\pi\)
\(224\) 516292. 0.687504
\(225\) −163525. −0.215342
\(226\) −1.18310e6 −1.54082
\(227\) −777882. −1.00196 −0.500979 0.865460i \(-0.667027\pi\)
−0.500979 + 0.865460i \(0.667027\pi\)
\(228\) −114825. −0.146285
\(229\) −763186. −0.961705 −0.480852 0.876802i \(-0.659673\pi\)
−0.480852 + 0.876802i \(0.659673\pi\)
\(230\) −49437.1 −0.0616216
\(231\) −221665. −0.273318
\(232\) −124379. −0.151715
\(233\) −1.27683e6 −1.54079 −0.770393 0.637570i \(-0.779940\pi\)
−0.770393 + 0.637570i \(0.779940\pi\)
\(234\) −149589. −0.178591
\(235\) 492559. 0.581820
\(236\) −263049. −0.307437
\(237\) −500271. −0.578542
\(238\) 979715. 1.12113
\(239\) 71356.0 0.0808045 0.0404023 0.999183i \(-0.487136\pi\)
0.0404023 + 0.999183i \(0.487136\pi\)
\(240\) 678954. 0.760874
\(241\) −1.25147e6 −1.38797 −0.693984 0.719991i \(-0.744146\pi\)
−0.693984 + 0.719991i \(0.744146\pi\)
\(242\) −269410. −0.295716
\(243\) 59049.0 0.0641500
\(244\) 2.21065e6 2.37709
\(245\) −451113. −0.480142
\(246\) 197366. 0.207939
\(247\) −30063.3 −0.0313540
\(248\) 1.02740e6 1.06074
\(249\) 729391. 0.745525
\(250\) 1.77434e6 1.79551
\(251\) −1.22669e6 −1.22900 −0.614501 0.788916i \(-0.710642\pi\)
−0.614501 + 0.788916i \(0.710642\pi\)
\(252\) −348594. −0.345795
\(253\) 61980.3 0.0608769
\(254\) −3.16640e6 −3.07951
\(255\) 496491. 0.478146
\(256\) −1.38862e6 −1.32429
\(257\) −1.22989e6 −1.16154 −0.580770 0.814068i \(-0.697248\pi\)
−0.580770 + 0.814068i \(0.697248\pi\)
\(258\) −358443. −0.335252
\(259\) 63796.2 0.0590943
\(260\) 447525. 0.410567
\(261\) 22296.5 0.0202598
\(262\) 1.58426e6 1.42585
\(263\) −1.03240e6 −0.920364 −0.460182 0.887825i \(-0.652216\pi\)
−0.460182 + 0.887825i \(0.652216\pi\)
\(264\) −1.75870e6 −1.55304
\(265\) 321017. 0.280811
\(266\) −99725.1 −0.0864173
\(267\) 966670. 0.829851
\(268\) 1.14050e6 0.969967
\(269\) −565251. −0.476278 −0.238139 0.971231i \(-0.576537\pi\)
−0.238139 + 0.971231i \(0.576537\pi\)
\(270\) −251466. −0.209927
\(271\) −1.35862e6 −1.12376 −0.561880 0.827219i \(-0.689922\pi\)
−0.561880 + 0.827219i \(0.689922\pi\)
\(272\) 3.76222e6 3.08334
\(273\) −91268.1 −0.0741161
\(274\) 1.33107e6 1.07108
\(275\) −873075. −0.696177
\(276\) 97471.2 0.0770200
\(277\) 219170. 0.171625 0.0858127 0.996311i \(-0.472651\pi\)
0.0858127 + 0.996311i \(0.472651\pi\)
\(278\) −1.61258e6 −1.25144
\(279\) −184174. −0.141650
\(280\) 855877. 0.652404
\(281\) 1.54978e6 1.17086 0.585431 0.810722i \(-0.300925\pi\)
0.585431 + 0.810722i \(0.300925\pi\)
\(282\) −1.38238e6 −1.03516
\(283\) −1.39782e6 −1.03749 −0.518746 0.854928i \(-0.673601\pi\)
−0.518746 + 0.854928i \(0.673601\pi\)
\(284\) 1.00291e6 0.737847
\(285\) −50537.8 −0.0368556
\(286\) −798667. −0.577365
\(287\) 120419. 0.0862957
\(288\) −734307. −0.521671
\(289\) 1.33130e6 0.937628
\(290\) −94951.6 −0.0662990
\(291\) −831927. −0.575908
\(292\) 3.21138e6 2.20411
\(293\) 2.33918e6 1.59182 0.795910 0.605415i \(-0.206993\pi\)
0.795910 + 0.605415i \(0.206993\pi\)
\(294\) 1.26606e6 0.854255
\(295\) −115775. −0.0774569
\(296\) 506162. 0.335784
\(297\) 315268. 0.207390
\(298\) −4.60638e6 −3.00482
\(299\) 25519.7 0.0165081
\(300\) −1.37301e6 −0.880787
\(301\) −218696. −0.139131
\(302\) 715009. 0.451122
\(303\) −1.21292e6 −0.758975
\(304\) −382956. −0.237665
\(305\) 972968. 0.598893
\(306\) −1.39342e6 −0.850704
\(307\) 1.73450e6 1.05034 0.525170 0.850998i \(-0.324002\pi\)
0.525170 + 0.850998i \(0.324002\pi\)
\(308\) −1.86118e6 −1.11792
\(309\) 1.49531e6 0.890913
\(310\) 784320. 0.463542
\(311\) −1.36982e6 −0.803086 −0.401543 0.915840i \(-0.631526\pi\)
−0.401543 + 0.915840i \(0.631526\pi\)
\(312\) −724125. −0.421140
\(313\) −2.30372e6 −1.32914 −0.664569 0.747227i \(-0.731385\pi\)
−0.664569 + 0.747227i \(0.731385\pi\)
\(314\) 718575. 0.411290
\(315\) −153426. −0.0871210
\(316\) −4.20045e6 −2.36634
\(317\) 2.31279e6 1.29267 0.646337 0.763052i \(-0.276300\pi\)
0.646337 + 0.763052i \(0.276300\pi\)
\(318\) −900947. −0.499610
\(319\) 119043. 0.0654978
\(320\) 713053. 0.389266
\(321\) 1.08009e6 0.585056
\(322\) 84653.3 0.0454993
\(323\) −280040. −0.149353
\(324\) 495795. 0.262385
\(325\) −359479. −0.188784
\(326\) 2.77208e6 1.44465
\(327\) 1.69228e6 0.875190
\(328\) 955406. 0.490347
\(329\) −843431. −0.429596
\(330\) −1.34260e6 −0.678673
\(331\) 1.01902e6 0.511224 0.255612 0.966779i \(-0.417723\pi\)
0.255612 + 0.966779i \(0.417723\pi\)
\(332\) 6.12421e6 3.04933
\(333\) −90735.5 −0.0448401
\(334\) −4.02214e6 −1.97284
\(335\) 501965. 0.244377
\(336\) −1.16260e6 −0.561803
\(337\) 1.96163e6 0.940899 0.470450 0.882427i \(-0.344092\pi\)
0.470450 + 0.882427i \(0.344092\pi\)
\(338\) 3.52201e6 1.67687
\(339\) 1.02666e6 0.485206
\(340\) 4.16870e6 1.95571
\(341\) −983320. −0.457940
\(342\) 141836. 0.0655725
\(343\) 1.72964e6 0.793817
\(344\) −1.73514e6 −0.790569
\(345\) 42899.8 0.0194047
\(346\) 1.97589e6 0.887304
\(347\) 567054. 0.252814 0.126407 0.991978i \(-0.459656\pi\)
0.126407 + 0.991978i \(0.459656\pi\)
\(348\) 187209. 0.0828663
\(349\) 2.79317e6 1.22753 0.613767 0.789487i \(-0.289653\pi\)
0.613767 + 0.789487i \(0.289653\pi\)
\(350\) −1.19245e6 −0.520321
\(351\) 129808. 0.0562385
\(352\) −3.92053e6 −1.68651
\(353\) 2.82854e6 1.20816 0.604081 0.796923i \(-0.293540\pi\)
0.604081 + 0.796923i \(0.293540\pi\)
\(354\) 324927. 0.137809
\(355\) 441409. 0.185896
\(356\) 8.11648e6 3.39424
\(357\) −850164. −0.353047
\(358\) 1.79980e6 0.742194
\(359\) −200396. −0.0820641 −0.0410321 0.999158i \(-0.513065\pi\)
−0.0410321 + 0.999158i \(0.513065\pi\)
\(360\) −1.21729e6 −0.495037
\(361\) −2.44759e6 −0.988488
\(362\) −7.44424e6 −2.98572
\(363\) 233785. 0.0931214
\(364\) −766318. −0.303148
\(365\) 1.41342e6 0.555313
\(366\) −2.73067e6 −1.06553
\(367\) −1.05311e6 −0.408140 −0.204070 0.978956i \(-0.565417\pi\)
−0.204070 + 0.978956i \(0.565417\pi\)
\(368\) 325079. 0.125132
\(369\) −171268. −0.0654802
\(370\) 386405. 0.146737
\(371\) −549693. −0.207341
\(372\) −1.54638e6 −0.579375
\(373\) −1.20089e6 −0.446922 −0.223461 0.974713i \(-0.571735\pi\)
−0.223461 + 0.974713i \(0.571735\pi\)
\(374\) −7.43959e6 −2.75024
\(375\) −1.53971e6 −0.565408
\(376\) −6.69182e6 −2.44104
\(377\) 49014.5 0.0177612
\(378\) 430596. 0.155003
\(379\) −3.95845e6 −1.41556 −0.707778 0.706435i \(-0.750302\pi\)
−0.707778 + 0.706435i \(0.750302\pi\)
\(380\) −424332. −0.150746
\(381\) 2.74770e6 0.969743
\(382\) −3.49134e6 −1.22415
\(383\) −3.58271e6 −1.24800 −0.624000 0.781424i \(-0.714494\pi\)
−0.624000 + 0.781424i \(0.714494\pi\)
\(384\) 609661. 0.210989
\(385\) −819156. −0.281653
\(386\) −3.03994e6 −1.03848
\(387\) 311045. 0.105571
\(388\) −6.98514e6 −2.35557
\(389\) −2.75954e6 −0.924619 −0.462309 0.886719i \(-0.652979\pi\)
−0.462309 + 0.886719i \(0.652979\pi\)
\(390\) −552799. −0.184037
\(391\) 237716. 0.0786352
\(392\) 6.12874e6 2.01445
\(393\) −1.37477e6 −0.449003
\(394\) 2.63575e6 0.855390
\(395\) −1.84873e6 −0.596186
\(396\) 2.64709e6 0.848265
\(397\) 4.77985e6 1.52208 0.761041 0.648704i \(-0.224689\pi\)
0.761041 + 0.648704i \(0.224689\pi\)
\(398\) −677158. −0.214280
\(399\) 86538.2 0.0272129
\(400\) −4.57916e6 −1.43099
\(401\) −4.79242e6 −1.48831 −0.744156 0.668005i \(-0.767148\pi\)
−0.744156 + 0.668005i \(0.767148\pi\)
\(402\) −1.40878e6 −0.434789
\(403\) −404871. −0.124181
\(404\) −1.01841e7 −3.10435
\(405\) 218213. 0.0661065
\(406\) 162590. 0.0489529
\(407\) −484445. −0.144963
\(408\) −6.74524e6 −2.00607
\(409\) 804704. 0.237864 0.118932 0.992902i \(-0.462053\pi\)
0.118932 + 0.992902i \(0.462053\pi\)
\(410\) 729360. 0.214280
\(411\) −1.15505e6 −0.337286
\(412\) 1.25551e7 3.64400
\(413\) 198247. 0.0571915
\(414\) −120400. −0.0345243
\(415\) 2.69544e6 0.768262
\(416\) −1.61423e6 −0.457334
\(417\) 1.39934e6 0.394080
\(418\) 757276. 0.211989
\(419\) 5.46503e6 1.52075 0.760374 0.649486i \(-0.225016\pi\)
0.760374 + 0.649486i \(0.225016\pi\)
\(420\) −1.28822e6 −0.356341
\(421\) −1.96598e6 −0.540597 −0.270298 0.962777i \(-0.587122\pi\)
−0.270298 + 0.962777i \(0.587122\pi\)
\(422\) −966026. −0.264063
\(423\) 1.19959e6 0.325972
\(424\) −4.36128e6 −1.17815
\(425\) −3.34855e6 −0.899258
\(426\) −1.23883e6 −0.330741
\(427\) −1.66606e6 −0.442202
\(428\) 9.06880e6 2.39299
\(429\) 693056. 0.181813
\(430\) −1.32461e6 −0.345476
\(431\) 6.18098e6 1.60274 0.801372 0.598167i \(-0.204104\pi\)
0.801372 + 0.598167i \(0.204104\pi\)
\(432\) 1.65354e6 0.426290
\(433\) −6.81186e6 −1.74601 −0.873003 0.487715i \(-0.837831\pi\)
−0.873003 + 0.487715i \(0.837831\pi\)
\(434\) −1.34303e6 −0.342264
\(435\) 82395.8 0.0208777
\(436\) 1.42089e7 3.57969
\(437\) −24197.1 −0.00606122
\(438\) −3.96681e6 −0.987997
\(439\) 5.97720e6 1.48026 0.740128 0.672466i \(-0.234765\pi\)
0.740128 + 0.672466i \(0.234765\pi\)
\(440\) −6.49921e6 −1.60040
\(441\) −1.09865e6 −0.269006
\(442\) −3.06317e6 −0.745787
\(443\) −490983. −0.118866 −0.0594330 0.998232i \(-0.518929\pi\)
−0.0594330 + 0.998232i \(0.518929\pi\)
\(444\) −761845. −0.183404
\(445\) 3.57229e6 0.855159
\(446\) −316624. −0.0753715
\(447\) 3.99726e6 0.946223
\(448\) −1.22099e6 −0.287421
\(449\) 7.25924e6 1.69932 0.849660 0.527331i \(-0.176807\pi\)
0.849660 + 0.527331i \(0.176807\pi\)
\(450\) 1.69599e6 0.394814
\(451\) −914415. −0.211691
\(452\) 8.62015e6 1.98458
\(453\) −620461. −0.142059
\(454\) 8.06777e6 1.83702
\(455\) −337278. −0.0763765
\(456\) 686597. 0.154629
\(457\) −2.78034e6 −0.622740 −0.311370 0.950289i \(-0.600788\pi\)
−0.311370 + 0.950289i \(0.600788\pi\)
\(458\) 7.91535e6 1.76322
\(459\) 1.20916e6 0.267888
\(460\) 360201. 0.0793690
\(461\) −6.01371e6 −1.31792 −0.658962 0.752176i \(-0.729004\pi\)
−0.658962 + 0.752176i \(0.729004\pi\)
\(462\) 2.29899e6 0.501109
\(463\) −7.10787e6 −1.54094 −0.770472 0.637474i \(-0.779979\pi\)
−0.770472 + 0.637474i \(0.779979\pi\)
\(464\) 624364. 0.134630
\(465\) −680607. −0.145970
\(466\) 1.32425e7 2.82492
\(467\) 5.51961e6 1.17116 0.585579 0.810615i \(-0.300867\pi\)
0.585579 + 0.810615i \(0.300867\pi\)
\(468\) 1.08991e6 0.230026
\(469\) −859537. −0.180440
\(470\) −5.10855e6 −1.06673
\(471\) −623555. −0.129516
\(472\) 1.57290e6 0.324972
\(473\) 1.66070e6 0.341301
\(474\) 5.18854e6 1.06072
\(475\) 340849. 0.0693150
\(476\) −7.13826e6 −1.44403
\(477\) 781812. 0.157328
\(478\) −740065. −0.148150
\(479\) −3.18017e6 −0.633304 −0.316652 0.948542i \(-0.602559\pi\)
−0.316652 + 0.948542i \(0.602559\pi\)
\(480\) −2.71360e6 −0.537580
\(481\) −199465. −0.0393100
\(482\) 1.29796e7 2.54474
\(483\) −73459.3 −0.0143278
\(484\) 1.96293e6 0.380884
\(485\) −3.07436e6 −0.593472
\(486\) −612424. −0.117615
\(487\) −4.01718e6 −0.767537 −0.383768 0.923429i \(-0.625374\pi\)
−0.383768 + 0.923429i \(0.625374\pi\)
\(488\) −1.32186e7 −2.51267
\(489\) −2.40552e6 −0.454921
\(490\) 4.67869e6 0.880308
\(491\) 2.25211e6 0.421585 0.210793 0.977531i \(-0.432396\pi\)
0.210793 + 0.977531i \(0.432396\pi\)
\(492\) −1.43802e6 −0.267826
\(493\) 456571. 0.0846041
\(494\) 311800. 0.0574855
\(495\) 1.16506e6 0.213715
\(496\) −5.15738e6 −0.941293
\(497\) −755845. −0.137259
\(498\) −7.56484e6 −1.36687
\(499\) 4.87566e6 0.876561 0.438281 0.898838i \(-0.355588\pi\)
0.438281 + 0.898838i \(0.355588\pi\)
\(500\) −1.29280e7 −2.31262
\(501\) 3.49028e6 0.621249
\(502\) 1.27226e7 2.25329
\(503\) −3.03011e6 −0.533996 −0.266998 0.963697i \(-0.586032\pi\)
−0.266998 + 0.963697i \(0.586032\pi\)
\(504\) 2.08442e6 0.365518
\(505\) −4.48232e6 −0.782122
\(506\) −642826. −0.111614
\(507\) −3.05628e6 −0.528048
\(508\) 2.30706e7 3.96643
\(509\) −6.31223e6 −1.07991 −0.539956 0.841693i \(-0.681559\pi\)
−0.539956 + 0.841693i \(0.681559\pi\)
\(510\) −5.14933e6 −0.876648
\(511\) −2.42026e6 −0.410024
\(512\) 1.22343e7 2.06255
\(513\) −123081. −0.0206489
\(514\) 1.27558e7 2.12960
\(515\) 5.52586e6 0.918083
\(516\) 2.61164e6 0.431806
\(517\) 6.40470e6 1.05384
\(518\) −661659. −0.108345
\(519\) −1.71461e6 −0.279414
\(520\) −2.67598e6 −0.433984
\(521\) −1.00384e7 −1.62021 −0.810106 0.586283i \(-0.800591\pi\)
−0.810106 + 0.586283i \(0.800591\pi\)
\(522\) −231247. −0.0371449
\(523\) 4.29564e6 0.686710 0.343355 0.939206i \(-0.388437\pi\)
0.343355 + 0.939206i \(0.388437\pi\)
\(524\) −1.15430e7 −1.83650
\(525\) 1.03477e6 0.163850
\(526\) 1.07075e7 1.68742
\(527\) −3.77138e6 −0.591525
\(528\) 8.82839e6 1.37815
\(529\) −6.41580e6 −0.996809
\(530\) −3.32942e6 −0.514847
\(531\) −281961. −0.0433963
\(532\) 726603. 0.111306
\(533\) −376500. −0.0574046
\(534\) −1.00258e7 −1.52147
\(535\) 3.99144e6 0.602899
\(536\) −6.81960e6 −1.02529
\(537\) −1.56181e6 −0.233718
\(538\) 5.86247e6 0.873223
\(539\) −5.86578e6 −0.869669
\(540\) 1.83219e6 0.270388
\(541\) −6.40724e6 −0.941190 −0.470595 0.882349i \(-0.655961\pi\)
−0.470595 + 0.882349i \(0.655961\pi\)
\(542\) 1.40908e7 2.06033
\(543\) 6.45986e6 0.940207
\(544\) −1.50366e7 −2.17848
\(545\) 6.25375e6 0.901882
\(546\) 946583. 0.135887
\(547\) 2.87214e6 0.410429 0.205214 0.978717i \(-0.434211\pi\)
0.205214 + 0.978717i \(0.434211\pi\)
\(548\) −9.69822e6 −1.37956
\(549\) 2.36958e6 0.335538
\(550\) 9.05505e6 1.27639
\(551\) −46474.4 −0.00652130
\(552\) −582829. −0.0814130
\(553\) 3.16567e6 0.440203
\(554\) −2.27311e6 −0.314663
\(555\) −335310. −0.0462076
\(556\) 1.17494e7 1.61186
\(557\) −1.93992e6 −0.264939 −0.132470 0.991187i \(-0.542291\pi\)
−0.132470 + 0.991187i \(0.542291\pi\)
\(558\) 1.91015e6 0.259706
\(559\) 683774. 0.0925514
\(560\) −4.29636e6 −0.578937
\(561\) 6.45583e6 0.866054
\(562\) −1.60735e7 −2.14669
\(563\) 2.17438e6 0.289110 0.144555 0.989497i \(-0.453825\pi\)
0.144555 + 0.989497i \(0.453825\pi\)
\(564\) 1.00721e7 1.33329
\(565\) 3.79397e6 0.500004
\(566\) 1.44974e7 1.90217
\(567\) −373657. −0.0488107
\(568\) −5.99690e6 −0.779931
\(569\) −4.07789e6 −0.528025 −0.264012 0.964519i \(-0.585046\pi\)
−0.264012 + 0.964519i \(0.585046\pi\)
\(570\) 524150. 0.0675723
\(571\) 7.41602e6 0.951876 0.475938 0.879479i \(-0.342109\pi\)
0.475938 + 0.879479i \(0.342109\pi\)
\(572\) 5.81913e6 0.743649
\(573\) 3.02967e6 0.385486
\(574\) −1.24892e6 −0.158217
\(575\) −289335. −0.0364948
\(576\) 1.73658e6 0.218092
\(577\) 1.29497e6 0.161927 0.0809636 0.996717i \(-0.474200\pi\)
0.0809636 + 0.996717i \(0.474200\pi\)
\(578\) −1.38075e7 −1.71908
\(579\) 2.63796e6 0.327018
\(580\) 691823. 0.0853935
\(581\) −4.61552e6 −0.567258
\(582\) 8.62829e6 1.05589
\(583\) 4.17416e6 0.508625
\(584\) −1.92024e7 −2.32983
\(585\) 479701. 0.0579536
\(586\) −2.42607e7 −2.91849
\(587\) 9.79730e6 1.17358 0.586788 0.809741i \(-0.300392\pi\)
0.586788 + 0.809741i \(0.300392\pi\)
\(588\) −9.22462e6 −1.10028
\(589\) 383888. 0.0455949
\(590\) 1.20076e6 0.142012
\(591\) −2.28722e6 −0.269364
\(592\) −2.54085e6 −0.297971
\(593\) 2.89314e6 0.337857 0.168929 0.985628i \(-0.445969\pi\)
0.168929 + 0.985628i \(0.445969\pi\)
\(594\) −3.26979e6 −0.380236
\(595\) −3.14175e6 −0.363814
\(596\) 3.35623e7 3.87023
\(597\) 587615. 0.0674772
\(598\) −264676. −0.0302665
\(599\) 1.07146e6 0.122013 0.0610066 0.998137i \(-0.480569\pi\)
0.0610066 + 0.998137i \(0.480569\pi\)
\(600\) 8.20992e6 0.931024
\(601\) −8.77640e6 −0.991129 −0.495564 0.868571i \(-0.665039\pi\)
−0.495564 + 0.868571i \(0.665039\pi\)
\(602\) 2.26820e6 0.255088
\(603\) 1.22249e6 0.136916
\(604\) −5.20960e6 −0.581048
\(605\) 863943. 0.0959614
\(606\) 1.25798e7 1.39153
\(607\) −1.38832e7 −1.52939 −0.764696 0.644391i \(-0.777111\pi\)
−0.764696 + 0.644391i \(0.777111\pi\)
\(608\) 1.53058e6 0.167917
\(609\) −141090. −0.0154153
\(610\) −1.00911e7 −1.09803
\(611\) 2.63706e6 0.285771
\(612\) 1.01525e7 1.09571
\(613\) −2.96162e6 −0.318330 −0.159165 0.987252i \(-0.550880\pi\)
−0.159165 + 0.987252i \(0.550880\pi\)
\(614\) −1.79893e7 −1.92572
\(615\) −632914. −0.0674772
\(616\) 1.11289e7 1.18168
\(617\) −1.63962e7 −1.73393 −0.866964 0.498370i \(-0.833932\pi\)
−0.866964 + 0.498370i \(0.833932\pi\)
\(618\) −1.55085e7 −1.63343
\(619\) −1.35257e7 −1.41884 −0.709418 0.704788i \(-0.751042\pi\)
−0.709418 + 0.704788i \(0.751042\pi\)
\(620\) −5.71460e6 −0.597045
\(621\) 104479. 0.0108718
\(622\) 1.42070e7 1.47240
\(623\) −6.11700e6 −0.631420
\(624\) 3.63499e6 0.373716
\(625\) 618876. 0.0633729
\(626\) 2.38930e7 2.43688
\(627\) −657139. −0.0667557
\(628\) −5.23558e6 −0.529743
\(629\) −1.85802e6 −0.187250
\(630\) 1.59125e6 0.159730
\(631\) −1.19825e7 −1.19805 −0.599025 0.800731i \(-0.704445\pi\)
−0.599025 + 0.800731i \(0.704445\pi\)
\(632\) 2.51166e7 2.50131
\(633\) 838285. 0.0831539
\(634\) −2.39870e7 −2.37003
\(635\) 1.01540e7 0.999318
\(636\) 6.56435e6 0.643501
\(637\) −2.41517e6 −0.235830
\(638\) −1.23465e6 −0.120086
\(639\) 1.07502e6 0.104151
\(640\) 2.25298e6 0.217424
\(641\) −7.00957e6 −0.673823 −0.336912 0.941536i \(-0.609382\pi\)
−0.336912 + 0.941536i \(0.609382\pi\)
\(642\) −1.12021e7 −1.07266
\(643\) 605033. 0.0577101 0.0288550 0.999584i \(-0.490814\pi\)
0.0288550 + 0.999584i \(0.490814\pi\)
\(644\) −616789. −0.0586033
\(645\) 1.14946e6 0.108791
\(646\) 2.90442e6 0.273828
\(647\) 1.46414e7 1.37506 0.687531 0.726155i \(-0.258695\pi\)
0.687531 + 0.726155i \(0.258695\pi\)
\(648\) −2.96461e6 −0.277351
\(649\) −1.50542e6 −0.140296
\(650\) 3.72831e6 0.346122
\(651\) 1.16543e6 0.107779
\(652\) −2.01975e7 −1.86071
\(653\) 3.99497e6 0.366632 0.183316 0.983054i \(-0.441317\pi\)
0.183316 + 0.983054i \(0.441317\pi\)
\(654\) −1.75514e7 −1.60460
\(655\) −5.08041e6 −0.462696
\(656\) −4.79598e6 −0.435129
\(657\) 3.44226e6 0.311122
\(658\) 8.74761e6 0.787634
\(659\) 274018. 0.0245791 0.0122895 0.999924i \(-0.496088\pi\)
0.0122895 + 0.999924i \(0.496088\pi\)
\(660\) 9.78224e6 0.874135
\(661\) −1.92782e7 −1.71618 −0.858091 0.513497i \(-0.828350\pi\)
−0.858091 + 0.513497i \(0.828350\pi\)
\(662\) −1.05687e7 −0.937294
\(663\) 2.65811e6 0.234850
\(664\) −3.66197e7 −3.22326
\(665\) 319799. 0.0280429
\(666\) 941058. 0.0822112
\(667\) 39450.5 0.00343351
\(668\) 2.93056e7 2.54102
\(669\) 274756. 0.0237346
\(670\) −5.20610e6 −0.448049
\(671\) 1.26514e7 1.08476
\(672\) 4.64663e6 0.396931
\(673\) −1.19222e7 −1.01466 −0.507329 0.861753i \(-0.669367\pi\)
−0.507329 + 0.861753i \(0.669367\pi\)
\(674\) −2.03450e7 −1.72507
\(675\) −1.47173e6 −0.124328
\(676\) −2.56615e7 −2.15981
\(677\) −3.81809e6 −0.320166 −0.160083 0.987104i \(-0.551176\pi\)
−0.160083 + 0.987104i \(0.551176\pi\)
\(678\) −1.06479e7 −0.889591
\(679\) 5.26436e6 0.438199
\(680\) −2.49268e7 −2.06725
\(681\) −7.00094e6 −0.578480
\(682\) 1.01985e7 0.839602
\(683\) 1.71086e7 1.40334 0.701672 0.712500i \(-0.252437\pi\)
0.701672 + 0.712500i \(0.252437\pi\)
\(684\) −1.03343e6 −0.0844577
\(685\) −4.26846e6 −0.347572
\(686\) −1.79389e7 −1.45541
\(687\) −6.86867e6 −0.555240
\(688\) 8.71014e6 0.701543
\(689\) 1.71866e6 0.137925
\(690\) −444933. −0.0355772
\(691\) 1.16432e6 0.0927639 0.0463820 0.998924i \(-0.485231\pi\)
0.0463820 + 0.998924i \(0.485231\pi\)
\(692\) −1.43965e7 −1.14285
\(693\) −1.99499e6 −0.157800
\(694\) −5.88117e6 −0.463516
\(695\) 5.17122e6 0.406098
\(696\) −1.11941e6 −0.0875927
\(697\) −3.50710e6 −0.273443
\(698\) −2.89692e7 −2.25060
\(699\) −1.14914e7 −0.889573
\(700\) 8.68829e6 0.670177
\(701\) −2.35243e6 −0.180810 −0.0904050 0.995905i \(-0.528816\pi\)
−0.0904050 + 0.995905i \(0.528816\pi\)
\(702\) −1.34630e6 −0.103109
\(703\) 189127. 0.0144333
\(704\) 9.27177e6 0.705068
\(705\) 4.43303e6 0.335914
\(706\) −2.93360e7 −2.21508
\(707\) 7.67528e6 0.577492
\(708\) −2.36744e6 −0.177499
\(709\) −2.37540e7 −1.77468 −0.887342 0.461112i \(-0.847451\pi\)
−0.887342 + 0.461112i \(0.847451\pi\)
\(710\) −4.57805e6 −0.340828
\(711\) −4.50244e6 −0.334021
\(712\) −4.85325e7 −3.58784
\(713\) −325870. −0.0240060
\(714\) 8.81744e6 0.647287
\(715\) 2.56116e6 0.187358
\(716\) −1.31135e7 −0.955949
\(717\) 642204. 0.0466525
\(718\) 2.07840e6 0.150459
\(719\) 7.28156e6 0.525294 0.262647 0.964892i \(-0.415405\pi\)
0.262647 + 0.964892i \(0.415405\pi\)
\(720\) 6.11059e6 0.439291
\(721\) −9.46219e6 −0.677881
\(722\) 2.53851e7 1.81232
\(723\) −1.12633e7 −0.801344
\(724\) 5.42392e7 3.84562
\(725\) −555713. −0.0392650
\(726\) −2.42469e6 −0.170732
\(727\) 400663. 0.0281153 0.0140577 0.999901i \(-0.495525\pi\)
0.0140577 + 0.999901i \(0.495525\pi\)
\(728\) 4.58220e6 0.320439
\(729\) 531441. 0.0370370
\(730\) −1.46592e7 −1.01813
\(731\) 6.36936e6 0.440862
\(732\) 1.98958e7 1.37241
\(733\) 2.06252e7 1.41787 0.708937 0.705272i \(-0.249175\pi\)
0.708937 + 0.705272i \(0.249175\pi\)
\(734\) 1.09223e7 0.748297
\(735\) −4.06001e6 −0.277210
\(736\) −1.29925e6 −0.0884097
\(737\) 6.52701e6 0.442634
\(738\) 1.77630e6 0.120053
\(739\) −1.36694e7 −0.920744 −0.460372 0.887726i \(-0.652284\pi\)
−0.460372 + 0.887726i \(0.652284\pi\)
\(740\) −2.81537e6 −0.188998
\(741\) −270569. −0.0181023
\(742\) 5.70111e6 0.380145
\(743\) 1.13736e7 0.755833 0.377917 0.925840i \(-0.376641\pi\)
0.377917 + 0.925840i \(0.376641\pi\)
\(744\) 9.24660e6 0.612421
\(745\) 1.47717e7 0.975081
\(746\) 1.24550e7 0.819400
\(747\) 6.56452e6 0.430429
\(748\) 5.42053e7 3.54232
\(749\) −6.83472e6 −0.445160
\(750\) 1.59691e7 1.03664
\(751\) −1.25684e6 −0.0813171 −0.0406585 0.999173i \(-0.512946\pi\)
−0.0406585 + 0.999173i \(0.512946\pi\)
\(752\) 3.35918e7 2.16615
\(753\) −1.10403e7 −0.709564
\(754\) −508352. −0.0325639
\(755\) −2.29289e6 −0.146392
\(756\) −3.13735e6 −0.199645
\(757\) −1.34819e7 −0.855087 −0.427544 0.903995i \(-0.640621\pi\)
−0.427544 + 0.903995i \(0.640621\pi\)
\(758\) 4.10549e7 2.59533
\(759\) 557823. 0.0351473
\(760\) 2.53729e6 0.159344
\(761\) −865364. −0.0541673 −0.0270837 0.999633i \(-0.508622\pi\)
−0.0270837 + 0.999633i \(0.508622\pi\)
\(762\) −2.84976e7 −1.77796
\(763\) −1.07086e7 −0.665918
\(764\) 2.54381e7 1.57671
\(765\) 4.46842e6 0.276058
\(766\) 3.71579e7 2.28812
\(767\) −619837. −0.0380443
\(768\) −1.24976e7 −0.764580
\(769\) −2.08760e7 −1.27301 −0.636503 0.771274i \(-0.719620\pi\)
−0.636503 + 0.771274i \(0.719620\pi\)
\(770\) 8.49584e6 0.516392
\(771\) −1.10690e7 −0.670615
\(772\) 2.21492e7 1.33757
\(773\) 2.75458e7 1.65809 0.829043 0.559185i \(-0.188886\pi\)
0.829043 + 0.559185i \(0.188886\pi\)
\(774\) −3.22599e6 −0.193558
\(775\) 4.59031e6 0.274529
\(776\) 4.17677e7 2.48992
\(777\) 574166. 0.0341181
\(778\) 2.86204e7 1.69522
\(779\) 356988. 0.0210770
\(780\) 4.02773e6 0.237041
\(781\) 5.73961e6 0.336709
\(782\) −2.46546e6 −0.144172
\(783\) 200668. 0.0116970
\(784\) −3.07652e7 −1.78760
\(785\) −2.30433e6 −0.133466
\(786\) 1.42584e7 0.823215
\(787\) 1.86224e7 1.07176 0.535881 0.844293i \(-0.319979\pi\)
0.535881 + 0.844293i \(0.319979\pi\)
\(788\) −1.92042e7 −1.10175
\(789\) −9.29162e6 −0.531373
\(790\) 1.91741e7 1.09307
\(791\) −6.49659e6 −0.369185
\(792\) −1.58283e7 −0.896647
\(793\) 5.20908e6 0.294156
\(794\) −4.95740e7 −2.79063
\(795\) 2.88916e6 0.162126
\(796\) 4.93381e6 0.275994
\(797\) −1.01313e7 −0.564965 −0.282482 0.959272i \(-0.591158\pi\)
−0.282482 + 0.959272i \(0.591158\pi\)
\(798\) −897526. −0.0498930
\(799\) 2.45643e7 1.36125
\(800\) 1.83017e7 1.01104
\(801\) 8.70003e6 0.479115
\(802\) 4.97044e7 2.72872
\(803\) 1.83785e7 1.00582
\(804\) 1.02645e7 0.560011
\(805\) −271466. −0.0147648
\(806\) 4.19910e6 0.227677
\(807\) −5.08726e6 −0.274979
\(808\) 6.08960e7 3.28141
\(809\) 1.76196e7 0.946507 0.473254 0.880926i \(-0.343079\pi\)
0.473254 + 0.880926i \(0.343079\pi\)
\(810\) −2.26319e6 −0.121202
\(811\) −1.37418e6 −0.0733654 −0.0366827 0.999327i \(-0.511679\pi\)
−0.0366827 + 0.999327i \(0.511679\pi\)
\(812\) −1.18464e6 −0.0630516
\(813\) −1.22275e7 −0.648803
\(814\) 5.02440e6 0.265780
\(815\) −8.88950e6 −0.468795
\(816\) 3.38600e7 1.78017
\(817\) −648337. −0.0339817
\(818\) −8.34595e6 −0.436107
\(819\) −821413. −0.0427909
\(820\) −5.31416e6 −0.275994
\(821\) −1.13057e7 −0.585382 −0.292691 0.956207i \(-0.594551\pi\)
−0.292691 + 0.956207i \(0.594551\pi\)
\(822\) 1.19796e7 0.618390
\(823\) 2.63250e7 1.35478 0.677389 0.735625i \(-0.263111\pi\)
0.677389 + 0.735625i \(0.263111\pi\)
\(824\) −7.50734e7 −3.85184
\(825\) −7.85767e6 −0.401938
\(826\) −2.05611e6 −0.104857
\(827\) 2.02274e6 0.102843 0.0514217 0.998677i \(-0.483625\pi\)
0.0514217 + 0.998677i \(0.483625\pi\)
\(828\) 877241. 0.0444675
\(829\) −1.96942e7 −0.995297 −0.497649 0.867379i \(-0.665803\pi\)
−0.497649 + 0.867379i \(0.665803\pi\)
\(830\) −2.79556e7 −1.40855
\(831\) 1.97253e6 0.0990880
\(832\) 3.81754e6 0.191195
\(833\) −2.24973e7 −1.12336
\(834\) −1.45132e7 −0.722518
\(835\) 1.28982e7 0.640196
\(836\) −5.51756e6 −0.273043
\(837\) −1.65756e6 −0.0817818
\(838\) −5.66803e7 −2.78819
\(839\) −3.39690e6 −0.166601 −0.0833006 0.996524i \(-0.526546\pi\)
−0.0833006 + 0.996524i \(0.526546\pi\)
\(840\) 7.70290e6 0.376665
\(841\) −2.04354e7 −0.996306
\(842\) 2.03900e7 0.991147
\(843\) 1.39481e7 0.675997
\(844\) 7.03852e6 0.340115
\(845\) −1.12944e7 −0.544152
\(846\) −1.24415e7 −0.597648
\(847\) −1.47937e6 −0.0708546
\(848\) 2.18929e7 1.04548
\(849\) −1.25804e7 −0.598997
\(850\) 3.47293e7 1.64873
\(851\) −160544. −0.00759923
\(852\) 9.02619e6 0.425996
\(853\) −8.12673e6 −0.382422 −0.191211 0.981549i \(-0.561242\pi\)
−0.191211 + 0.981549i \(0.561242\pi\)
\(854\) 1.72794e7 0.810746
\(855\) −454840. −0.0212786
\(856\) −5.42269e7 −2.52948
\(857\) 2.17376e7 1.01102 0.505509 0.862821i \(-0.331305\pi\)
0.505509 + 0.862821i \(0.331305\pi\)
\(858\) −7.18800e6 −0.333342
\(859\) 180961. 0.00836764 0.00418382 0.999991i \(-0.498668\pi\)
0.00418382 + 0.999991i \(0.498668\pi\)
\(860\) 9.65122e6 0.444975
\(861\) 1.08377e6 0.0498228
\(862\) −6.41057e7 −2.93852
\(863\) 3.07976e7 1.40764 0.703818 0.710381i \(-0.251477\pi\)
0.703818 + 0.710381i \(0.251477\pi\)
\(864\) −6.60876e6 −0.301187
\(865\) −6.33629e6 −0.287935
\(866\) 7.06488e7 3.20118
\(867\) 1.19817e7 0.541340
\(868\) 9.78538e6 0.440837
\(869\) −2.40389e7 −1.07986
\(870\) −854564. −0.0382778
\(871\) 2.68742e6 0.120030
\(872\) −8.49624e7 −3.78386
\(873\) −7.48735e6 −0.332501
\(874\) 250959. 0.0111128
\(875\) 9.74318e6 0.430210
\(876\) 2.89024e7 1.27255
\(877\) 1.10924e7 0.486998 0.243499 0.969901i \(-0.421705\pi\)
0.243499 + 0.969901i \(0.421705\pi\)
\(878\) −6.19923e7 −2.71395
\(879\) 2.10526e7 0.919038
\(880\) 3.26250e7 1.42018
\(881\) 3.34548e7 1.45217 0.726086 0.687603i \(-0.241337\pi\)
0.726086 + 0.687603i \(0.241337\pi\)
\(882\) 1.13946e7 0.493204
\(883\) −2.73782e7 −1.18169 −0.590844 0.806786i \(-0.701205\pi\)
−0.590844 + 0.806786i \(0.701205\pi\)
\(884\) 2.23184e7 0.960578
\(885\) −1.04198e6 −0.0447198
\(886\) 5.09221e6 0.217932
\(887\) −2.42168e7 −1.03350 −0.516748 0.856138i \(-0.672857\pi\)
−0.516748 + 0.856138i \(0.672857\pi\)
\(888\) 4.55546e6 0.193865
\(889\) −1.73872e7 −0.737862
\(890\) −3.70499e7 −1.56788
\(891\) 2.83741e6 0.119737
\(892\) 2.30694e6 0.0970788
\(893\) −2.50040e6 −0.104925
\(894\) −4.14574e7 −1.73484
\(895\) −5.77161e6 −0.240846
\(896\) −3.85788e6 −0.160538
\(897\) 229677. 0.00953096
\(898\) −7.52888e7 −3.11559
\(899\) −625884. −0.0258282
\(900\) −1.23571e7 −0.508523
\(901\) 1.60094e7 0.656996
\(902\) 9.48381e6 0.388120
\(903\) −1.96827e6 −0.0803276
\(904\) −5.15442e7 −2.09778
\(905\) 2.38722e7 0.968881
\(906\) 6.43508e6 0.260455
\(907\) −2.31796e7 −0.935595 −0.467797 0.883836i \(-0.654952\pi\)
−0.467797 + 0.883836i \(0.654952\pi\)
\(908\) −5.87822e7 −2.36609
\(909\) −1.09163e7 −0.438194
\(910\) 3.49806e6 0.140031
\(911\) 1.41080e7 0.563209 0.281605 0.959531i \(-0.409133\pi\)
0.281605 + 0.959531i \(0.409133\pi\)
\(912\) −3.44660e6 −0.137216
\(913\) 3.50486e7 1.39153
\(914\) 2.88361e7 1.14175
\(915\) 8.75671e6 0.345771
\(916\) −5.76717e7 −2.27104
\(917\) 8.69943e6 0.341639
\(918\) −1.25408e7 −0.491154
\(919\) 3.48252e7 1.36020 0.680102 0.733117i \(-0.261935\pi\)
0.680102 + 0.733117i \(0.261935\pi\)
\(920\) −2.15382e6 −0.0838959
\(921\) 1.56105e7 0.606414
\(922\) 6.23709e7 2.41632
\(923\) 2.36322e6 0.0913060
\(924\) −1.67506e7 −0.645431
\(925\) 2.26147e6 0.0869034
\(926\) 7.37189e7 2.82521
\(927\) 1.34578e7 0.514369
\(928\) −2.49542e6 −0.0951204
\(929\) 8.18223e6 0.311052 0.155526 0.987832i \(-0.450293\pi\)
0.155526 + 0.987832i \(0.450293\pi\)
\(930\) 7.05888e6 0.267626
\(931\) 2.29000e6 0.0865888
\(932\) −9.64860e7 −3.63852
\(933\) −1.23284e7 −0.463662
\(934\) −5.72463e7 −2.14724
\(935\) 2.38573e7 0.892467
\(936\) −6.51712e6 −0.243146
\(937\) −3.29144e7 −1.22472 −0.612361 0.790578i \(-0.709780\pi\)
−0.612361 + 0.790578i \(0.709780\pi\)
\(938\) 8.91465e6 0.330824
\(939\) −2.07335e7 −0.767378
\(940\) 3.72212e7 1.37395
\(941\) 2.12414e7 0.782004 0.391002 0.920390i \(-0.372128\pi\)
0.391002 + 0.920390i \(0.372128\pi\)
\(942\) 6.46717e6 0.237458
\(943\) −303035. −0.0110972
\(944\) −7.89570e6 −0.288377
\(945\) −1.38084e6 −0.0502993
\(946\) −1.72239e7 −0.625752
\(947\) 4.07231e7 1.47559 0.737795 0.675025i \(-0.235867\pi\)
0.737795 + 0.675025i \(0.235867\pi\)
\(948\) −3.78040e7 −1.36621
\(949\) 7.56716e6 0.272752
\(950\) −3.53509e6 −0.127084
\(951\) 2.08151e7 0.746325
\(952\) 4.26833e7 1.52639
\(953\) −3.41302e7 −1.21733 −0.608663 0.793429i \(-0.708294\pi\)
−0.608663 + 0.793429i \(0.708294\pi\)
\(954\) −8.10852e6 −0.288450
\(955\) 1.11960e7 0.397242
\(956\) 5.39216e6 0.190817
\(957\) 1.07139e6 0.0378152
\(958\) 3.29830e7 1.16112
\(959\) 7.30908e6 0.256635
\(960\) 6.41748e6 0.224743
\(961\) −2.34592e7 −0.819417
\(962\) 2.06874e6 0.0720722
\(963\) 9.72082e6 0.337782
\(964\) −9.45702e7 −3.27764
\(965\) 9.74849e6 0.336992
\(966\) 761880. 0.0262690
\(967\) 5.84049e6 0.200855 0.100428 0.994944i \(-0.467979\pi\)
0.100428 + 0.994944i \(0.467979\pi\)
\(968\) −1.17374e7 −0.402608
\(969\) −2.52036e6 −0.0862289
\(970\) 3.18855e7 1.08809
\(971\) −3.95959e7 −1.34773 −0.673864 0.738856i \(-0.735367\pi\)
−0.673864 + 0.738856i \(0.735367\pi\)
\(972\) 4.46216e6 0.151488
\(973\) −8.85492e6 −0.299849
\(974\) 4.16640e7 1.40723
\(975\) −3.23531e6 −0.108994
\(976\) 6.63550e7 2.22971
\(977\) 2.78950e7 0.934955 0.467477 0.884005i \(-0.345163\pi\)
0.467477 + 0.884005i \(0.345163\pi\)
\(978\) 2.49487e7 0.834066
\(979\) 4.64502e7 1.54893
\(980\) −3.40892e7 −1.13384
\(981\) 1.52305e7 0.505291
\(982\) −2.33576e7 −0.772948
\(983\) −2.02573e7 −0.668650 −0.334325 0.942458i \(-0.608508\pi\)
−0.334325 + 0.942458i \(0.608508\pi\)
\(984\) 8.59866e6 0.283102
\(985\) −8.45233e6 −0.277579
\(986\) −4.73531e6 −0.155116
\(987\) −7.59088e6 −0.248027
\(988\) −2.27179e6 −0.0740416
\(989\) 550352. 0.0178916
\(990\) −1.20834e7 −0.391832
\(991\) 4.46979e6 0.144578 0.0722891 0.997384i \(-0.476970\pi\)
0.0722891 + 0.997384i \(0.476970\pi\)
\(992\) 2.06127e7 0.665053
\(993\) 9.17115e6 0.295155
\(994\) 7.83921e6 0.251655
\(995\) 2.17151e6 0.0695351
\(996\) 5.51179e7 1.76053
\(997\) −482842. −0.0153839 −0.00769196 0.999970i \(-0.502448\pi\)
−0.00769196 + 0.999970i \(0.502448\pi\)
\(998\) −5.05677e7 −1.60711
\(999\) −816619. −0.0258884
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 177.6.a.c.1.1 12
3.2 odd 2 531.6.a.c.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.1 12 1.1 even 1 trivial
531.6.a.c.1.12 12 3.2 odd 2