Properties

Label 177.6.a.c.1.7
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 1.7
Root \(-0.689340\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.68934 q^{2} +9.00000 q^{3} -24.7675 q^{4} -83.5049 q^{5} +24.2041 q^{6} -48.3401 q^{7} -152.667 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.68934 q^{2} +9.00000 q^{3} -24.7675 q^{4} -83.5049 q^{5} +24.2041 q^{6} -48.3401 q^{7} -152.667 q^{8} +81.0000 q^{9} -224.573 q^{10} +19.9993 q^{11} -222.907 q^{12} +611.920 q^{13} -130.003 q^{14} -751.544 q^{15} +381.985 q^{16} +1525.25 q^{17} +217.837 q^{18} -1272.41 q^{19} +2068.20 q^{20} -435.061 q^{21} +53.7850 q^{22} +3308.46 q^{23} -1374.00 q^{24} +3848.06 q^{25} +1645.66 q^{26} +729.000 q^{27} +1197.26 q^{28} -2171.14 q^{29} -2021.16 q^{30} +7163.58 q^{31} +5912.63 q^{32} +179.994 q^{33} +4101.91 q^{34} +4036.64 q^{35} -2006.16 q^{36} -5908.05 q^{37} -3421.93 q^{38} +5507.28 q^{39} +12748.4 q^{40} +2583.15 q^{41} -1170.03 q^{42} -5333.76 q^{43} -495.332 q^{44} -6763.89 q^{45} +8897.58 q^{46} -10520.3 q^{47} +3437.87 q^{48} -14470.2 q^{49} +10348.7 q^{50} +13727.2 q^{51} -15155.7 q^{52} +6041.06 q^{53} +1960.53 q^{54} -1670.04 q^{55} +7379.94 q^{56} -11451.7 q^{57} -5838.93 q^{58} -3481.00 q^{59} +18613.8 q^{60} +134.207 q^{61} +19265.3 q^{62} -3915.55 q^{63} +3677.55 q^{64} -51098.3 q^{65} +484.065 q^{66} +49806.2 q^{67} -37776.5 q^{68} +29776.2 q^{69} +10855.9 q^{70} +64703.4 q^{71} -12366.0 q^{72} +43618.4 q^{73} -15888.8 q^{74} +34632.5 q^{75} +31514.3 q^{76} -966.770 q^{77} +14810.9 q^{78} +74841.5 q^{79} -31897.6 q^{80} +6561.00 q^{81} +6946.96 q^{82} +5270.76 q^{83} +10775.4 q^{84} -127366. q^{85} -14344.3 q^{86} -19540.2 q^{87} -3053.23 q^{88} +85402.3 q^{89} -18190.4 q^{90} -29580.3 q^{91} -81942.2 q^{92} +64472.3 q^{93} -28292.8 q^{94} +106252. q^{95} +53213.7 q^{96} +47381.9 q^{97} -38915.4 q^{98} +1619.94 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + O(q^{10}) \) \( 12q + 22q^{2} + 108q^{3} + 198q^{4} + 158q^{5} + 198q^{6} + 413q^{7} + 723q^{8} + 972q^{9} + 601q^{10} + 1480q^{11} + 1782q^{12} + 472q^{13} + 1065q^{14} + 1422q^{15} + 6370q^{16} + 1565q^{17} + 1782q^{18} + 3939q^{19} + 8033q^{20} + 3717q^{21} - 1738q^{22} + 7245q^{23} + 6507q^{24} + 9690q^{25} + 3764q^{26} + 8748q^{27} + 12154q^{28} + 10003q^{29} + 5409q^{30} + 7295q^{31} + 11628q^{32} + 13320q^{33} - 16344q^{34} + 11015q^{35} + 16038q^{36} + 6741q^{37} + 3035q^{38} + 4248q^{39} + 5572q^{40} + 34025q^{41} + 9585q^{42} - 6336q^{43} + 41168q^{44} + 12798q^{45} + 2345q^{46} + 66167q^{47} + 57330q^{48} + 28319q^{49} + 31173q^{50} + 14085q^{51} + 16440q^{52} + 62290q^{53} + 16038q^{54} + 55764q^{55} + 107306q^{56} + 35451q^{57} + 37952q^{58} - 41772q^{59} + 72297q^{60} + 68469q^{61} + 99190q^{62} + 33453q^{63} + 68525q^{64} + 80156q^{65} - 15642q^{66} + 113310q^{67} + 33887q^{68} + 65205q^{69} + 32034q^{70} + 84520q^{71} + 58563q^{72} + 135895q^{73} - 31962q^{74} + 87210q^{75} - 61848q^{76} - 3799q^{77} + 33876q^{78} + 14122q^{79} + 77609q^{80} + 78732q^{81} - 1501q^{82} + 114463q^{83} + 109386q^{84} - 101097q^{85} - 203536q^{86} + 90027q^{87} - 244967q^{88} + 189109q^{89} + 48681q^{90} - 168249q^{91} - 71946q^{92} + 65655q^{93} - 472284q^{94} + 21923q^{95} + 104652q^{96} - 76192q^{97} - 17544q^{98} + 119880q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.68934 0.475413 0.237706 0.971337i \(-0.423604\pi\)
0.237706 + 0.971337i \(0.423604\pi\)
\(3\) 9.00000 0.577350
\(4\) −24.7675 −0.773983
\(5\) −83.5049 −1.49378 −0.746890 0.664948i \(-0.768454\pi\)
−0.746890 + 0.664948i \(0.768454\pi\)
\(6\) 24.2041 0.274480
\(7\) −48.3401 −0.372875 −0.186437 0.982467i \(-0.559694\pi\)
−0.186437 + 0.982467i \(0.559694\pi\)
\(8\) −152.667 −0.843374
\(9\) 81.0000 0.333333
\(10\) −224.573 −0.710162
\(11\) 19.9993 0.0498349 0.0249174 0.999690i \(-0.492068\pi\)
0.0249174 + 0.999690i \(0.492068\pi\)
\(12\) −222.907 −0.446859
\(13\) 611.920 1.00424 0.502119 0.864799i \(-0.332554\pi\)
0.502119 + 0.864799i \(0.332554\pi\)
\(14\) −130.003 −0.177269
\(15\) −751.544 −0.862434
\(16\) 381.985 0.373032
\(17\) 1525.25 1.28002 0.640012 0.768365i \(-0.278929\pi\)
0.640012 + 0.768365i \(0.278929\pi\)
\(18\) 217.837 0.158471
\(19\) −1272.41 −0.808615 −0.404308 0.914623i \(-0.632487\pi\)
−0.404308 + 0.914623i \(0.632487\pi\)
\(20\) 2068.20 1.15616
\(21\) −435.061 −0.215279
\(22\) 53.7850 0.0236921
\(23\) 3308.46 1.30409 0.652044 0.758181i \(-0.273912\pi\)
0.652044 + 0.758181i \(0.273912\pi\)
\(24\) −1374.00 −0.486922
\(25\) 3848.06 1.23138
\(26\) 1645.66 0.477427
\(27\) 729.000 0.192450
\(28\) 1197.26 0.288599
\(29\) −2171.14 −0.479394 −0.239697 0.970848i \(-0.577048\pi\)
−0.239697 + 0.970848i \(0.577048\pi\)
\(30\) −2021.16 −0.410012
\(31\) 7163.58 1.33883 0.669416 0.742888i \(-0.266544\pi\)
0.669416 + 0.742888i \(0.266544\pi\)
\(32\) 5912.63 1.02072
\(33\) 179.994 0.0287722
\(34\) 4101.91 0.608540
\(35\) 4036.64 0.556993
\(36\) −2006.16 −0.257994
\(37\) −5908.05 −0.709480 −0.354740 0.934965i \(-0.615431\pi\)
−0.354740 + 0.934965i \(0.615431\pi\)
\(38\) −3421.93 −0.384426
\(39\) 5507.28 0.579797
\(40\) 12748.4 1.25982
\(41\) 2583.15 0.239988 0.119994 0.992775i \(-0.461712\pi\)
0.119994 + 0.992775i \(0.461712\pi\)
\(42\) −1170.03 −0.102346
\(43\) −5333.76 −0.439908 −0.219954 0.975510i \(-0.570591\pi\)
−0.219954 + 0.975510i \(0.570591\pi\)
\(44\) −495.332 −0.0385713
\(45\) −6763.89 −0.497927
\(46\) 8897.58 0.619980
\(47\) −10520.3 −0.694680 −0.347340 0.937739i \(-0.612915\pi\)
−0.347340 + 0.937739i \(0.612915\pi\)
\(48\) 3437.87 0.215370
\(49\) −14470.2 −0.860965
\(50\) 10348.7 0.585413
\(51\) 13727.2 0.739022
\(52\) −15155.7 −0.777262
\(53\) 6041.06 0.295409 0.147705 0.989032i \(-0.452812\pi\)
0.147705 + 0.989032i \(0.452812\pi\)
\(54\) 1960.53 0.0914932
\(55\) −1670.04 −0.0744424
\(56\) 7379.94 0.314473
\(57\) −11451.7 −0.466854
\(58\) −5838.93 −0.227910
\(59\) −3481.00 −0.130189
\(60\) 18613.8 0.667509
\(61\) 134.207 0.00461798 0.00230899 0.999997i \(-0.499265\pi\)
0.00230899 + 0.999997i \(0.499265\pi\)
\(62\) 19265.3 0.636498
\(63\) −3915.55 −0.124292
\(64\) 3677.55 0.112230
\(65\) −51098.3 −1.50011
\(66\) 484.065 0.0136787
\(67\) 49806.2 1.35549 0.677745 0.735297i \(-0.262957\pi\)
0.677745 + 0.735297i \(0.262957\pi\)
\(68\) −37776.5 −0.990717
\(69\) 29776.2 0.752915
\(70\) 10855.9 0.264801
\(71\) 64703.4 1.52329 0.761643 0.647997i \(-0.224393\pi\)
0.761643 + 0.647997i \(0.224393\pi\)
\(72\) −12366.0 −0.281125
\(73\) 43618.4 0.957993 0.478996 0.877817i \(-0.341001\pi\)
0.478996 + 0.877817i \(0.341001\pi\)
\(74\) −15888.8 −0.337296
\(75\) 34632.5 0.710937
\(76\) 31514.3 0.625854
\(77\) −966.770 −0.0185822
\(78\) 14810.9 0.275643
\(79\) 74841.5 1.34920 0.674598 0.738185i \(-0.264317\pi\)
0.674598 + 0.738185i \(0.264317\pi\)
\(80\) −31897.6 −0.557228
\(81\) 6561.00 0.111111
\(82\) 6946.96 0.114093
\(83\) 5270.76 0.0839804 0.0419902 0.999118i \(-0.486630\pi\)
0.0419902 + 0.999118i \(0.486630\pi\)
\(84\) 10775.4 0.166622
\(85\) −127366. −1.91208
\(86\) −14344.3 −0.209138
\(87\) −19540.2 −0.276778
\(88\) −3053.23 −0.0420294
\(89\) 85402.3 1.14286 0.571432 0.820649i \(-0.306388\pi\)
0.571432 + 0.820649i \(0.306388\pi\)
\(90\) −18190.4 −0.236721
\(91\) −29580.3 −0.374455
\(92\) −81942.2 −1.00934
\(93\) 64472.3 0.772975
\(94\) −28292.8 −0.330260
\(95\) 106252. 1.20789
\(96\) 53213.7 0.589312
\(97\) 47381.9 0.511309 0.255654 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(98\) −38915.4 −0.409313
\(99\) 1619.94 0.0166116
\(100\) −95306.6 −0.953066
\(101\) 111845. 1.09097 0.545484 0.838121i \(-0.316346\pi\)
0.545484 + 0.838121i \(0.316346\pi\)
\(102\) 36917.2 0.351341
\(103\) −161601. −1.50089 −0.750447 0.660930i \(-0.770162\pi\)
−0.750447 + 0.660930i \(0.770162\pi\)
\(104\) −93420.0 −0.846947
\(105\) 36329.7 0.321580
\(106\) 16246.5 0.140441
\(107\) −181209. −1.53010 −0.765052 0.643969i \(-0.777287\pi\)
−0.765052 + 0.643969i \(0.777287\pi\)
\(108\) −18055.5 −0.148953
\(109\) −69858.3 −0.563186 −0.281593 0.959534i \(-0.590863\pi\)
−0.281593 + 0.959534i \(0.590863\pi\)
\(110\) −4491.30 −0.0353908
\(111\) −53172.5 −0.409618
\(112\) −18465.2 −0.139094
\(113\) 152141. 1.12086 0.560428 0.828203i \(-0.310637\pi\)
0.560428 + 0.828203i \(0.310637\pi\)
\(114\) −30797.4 −0.221948
\(115\) −276273. −1.94802
\(116\) 53773.6 0.371043
\(117\) 49565.5 0.334746
\(118\) −9361.59 −0.0618934
\(119\) −73730.7 −0.477289
\(120\) 114736. 0.727355
\(121\) −160651. −0.997516
\(122\) 360.929 0.00219545
\(123\) 23248.3 0.138557
\(124\) −177424. −1.03623
\(125\) −60379.0 −0.345630
\(126\) −10530.2 −0.0590898
\(127\) 275370. 1.51498 0.757491 0.652846i \(-0.226425\pi\)
0.757491 + 0.652846i \(0.226425\pi\)
\(128\) −179314. −0.967363
\(129\) −48003.8 −0.253981
\(130\) −137421. −0.713171
\(131\) 10248.5 0.0521775 0.0260888 0.999660i \(-0.491695\pi\)
0.0260888 + 0.999660i \(0.491695\pi\)
\(132\) −4457.99 −0.0222692
\(133\) 61508.3 0.301512
\(134\) 133946. 0.644417
\(135\) −60875.0 −0.287478
\(136\) −232855. −1.07954
\(137\) 391127. 1.78039 0.890197 0.455576i \(-0.150567\pi\)
0.890197 + 0.455576i \(0.150567\pi\)
\(138\) 80078.2 0.357945
\(139\) −45851.0 −0.201285 −0.100643 0.994923i \(-0.532090\pi\)
−0.100643 + 0.994923i \(0.532090\pi\)
\(140\) −99977.2 −0.431103
\(141\) −94683.0 −0.401074
\(142\) 174009. 0.724189
\(143\) 12238.0 0.0500460
\(144\) 30940.8 0.124344
\(145\) 181301. 0.716109
\(146\) 117305. 0.455442
\(147\) −130232. −0.497078
\(148\) 146327. 0.549125
\(149\) 188439. 0.695354 0.347677 0.937614i \(-0.386971\pi\)
0.347677 + 0.937614i \(0.386971\pi\)
\(150\) 93138.7 0.337988
\(151\) −70211.4 −0.250591 −0.125295 0.992119i \(-0.539988\pi\)
−0.125295 + 0.992119i \(0.539988\pi\)
\(152\) 194254. 0.681965
\(153\) 123545. 0.426675
\(154\) −2599.97 −0.00883419
\(155\) −598194. −1.99992
\(156\) −136401. −0.448753
\(157\) −409601. −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(158\) 201274. 0.641425
\(159\) 54369.6 0.170554
\(160\) −493733. −1.52473
\(161\) −159932. −0.486261
\(162\) 17644.8 0.0528236
\(163\) −447613. −1.31957 −0.659787 0.751452i \(-0.729354\pi\)
−0.659787 + 0.751452i \(0.729354\pi\)
\(164\) −63978.0 −0.185747
\(165\) −15030.4 −0.0429793
\(166\) 14174.9 0.0399254
\(167\) −64777.2 −0.179734 −0.0898672 0.995954i \(-0.528644\pi\)
−0.0898672 + 0.995954i \(0.528644\pi\)
\(168\) 66419.5 0.181561
\(169\) 3153.10 0.00849222
\(170\) −342529. −0.909025
\(171\) −103065. −0.269538
\(172\) 132104. 0.340481
\(173\) 436181. 1.10803 0.554014 0.832507i \(-0.313095\pi\)
0.554014 + 0.832507i \(0.313095\pi\)
\(174\) −52550.4 −0.131584
\(175\) −186016. −0.459150
\(176\) 7639.44 0.0185900
\(177\) −31329.0 −0.0751646
\(178\) 229676. 0.543332
\(179\) 248687. 0.580123 0.290061 0.957008i \(-0.406324\pi\)
0.290061 + 0.957008i \(0.406324\pi\)
\(180\) 167524. 0.385387
\(181\) −509737. −1.15651 −0.578256 0.815856i \(-0.696266\pi\)
−0.578256 + 0.815856i \(0.696266\pi\)
\(182\) −79551.5 −0.178020
\(183\) 1207.87 0.00266619
\(184\) −505093. −1.09983
\(185\) 493351. 1.05981
\(186\) 173388. 0.367482
\(187\) 30503.9 0.0637899
\(188\) 260562. 0.537671
\(189\) −35240.0 −0.0717598
\(190\) 285748. 0.574248
\(191\) 582176. 1.15471 0.577353 0.816495i \(-0.304086\pi\)
0.577353 + 0.816495i \(0.304086\pi\)
\(192\) 33097.9 0.0647960
\(193\) 33922.3 0.0655529 0.0327765 0.999463i \(-0.489565\pi\)
0.0327765 + 0.999463i \(0.489565\pi\)
\(194\) 127426. 0.243083
\(195\) −459885. −0.866089
\(196\) 358391. 0.666372
\(197\) 51381.6 0.0943283 0.0471642 0.998887i \(-0.484982\pi\)
0.0471642 + 0.998887i \(0.484982\pi\)
\(198\) 4356.58 0.00789738
\(199\) 293133. 0.524725 0.262363 0.964969i \(-0.415498\pi\)
0.262363 + 0.964969i \(0.415498\pi\)
\(200\) −587472. −1.03851
\(201\) 448256. 0.782593
\(202\) 300789. 0.518660
\(203\) 104953. 0.178754
\(204\) −339989. −0.571991
\(205\) −215705. −0.358489
\(206\) −434599. −0.713544
\(207\) 267985. 0.434696
\(208\) 233744. 0.374613
\(209\) −25447.3 −0.0402972
\(210\) 97703.0 0.152883
\(211\) −257285. −0.397839 −0.198920 0.980016i \(-0.563743\pi\)
−0.198920 + 0.980016i \(0.563743\pi\)
\(212\) −149622. −0.228642
\(213\) 582331. 0.879469
\(214\) −487333. −0.727430
\(215\) 445395. 0.657126
\(216\) −111294. −0.162307
\(217\) −346289. −0.499217
\(218\) −187873. −0.267746
\(219\) 392565. 0.553098
\(220\) 41362.6 0.0576171
\(221\) 933330. 1.28545
\(222\) −142999. −0.194738
\(223\) −269708. −0.363188 −0.181594 0.983374i \(-0.558126\pi\)
−0.181594 + 0.983374i \(0.558126\pi\)
\(224\) −285817. −0.380600
\(225\) 311693. 0.410460
\(226\) 409159. 0.532869
\(227\) 474834. 0.611614 0.305807 0.952094i \(-0.401074\pi\)
0.305807 + 0.952094i \(0.401074\pi\)
\(228\) 283628. 0.361337
\(229\) 735574. 0.926910 0.463455 0.886120i \(-0.346610\pi\)
0.463455 + 0.886120i \(0.346610\pi\)
\(230\) −742991. −0.926113
\(231\) −8700.93 −0.0107284
\(232\) 331461. 0.404308
\(233\) 147473. 0.177960 0.0889800 0.996033i \(-0.471639\pi\)
0.0889800 + 0.996033i \(0.471639\pi\)
\(234\) 133299. 0.159142
\(235\) 878499. 1.03770
\(236\) 86215.5 0.100764
\(237\) 673574. 0.778959
\(238\) −198287. −0.226909
\(239\) −951198. −1.07715 −0.538575 0.842577i \(-0.681037\pi\)
−0.538575 + 0.842577i \(0.681037\pi\)
\(240\) −287078. −0.321716
\(241\) 341256. 0.378475 0.189238 0.981931i \(-0.439398\pi\)
0.189238 + 0.981931i \(0.439398\pi\)
\(242\) −432045. −0.474232
\(243\) 59049.0 0.0641500
\(244\) −3323.98 −0.00357424
\(245\) 1.20833e6 1.28609
\(246\) 62522.7 0.0658718
\(247\) −778611. −0.812042
\(248\) −1.09364e6 −1.12914
\(249\) 47436.8 0.0484861
\(250\) −162380. −0.164317
\(251\) 1.54946e6 1.55238 0.776188 0.630502i \(-0.217151\pi\)
0.776188 + 0.630502i \(0.217151\pi\)
\(252\) 96978.2 0.0961995
\(253\) 66167.0 0.0649890
\(254\) 740564. 0.720242
\(255\) −1.14629e6 −1.10394
\(256\) −599918. −0.572126
\(257\) −819098. −0.773576 −0.386788 0.922169i \(-0.626416\pi\)
−0.386788 + 0.922169i \(0.626416\pi\)
\(258\) −129099. −0.120746
\(259\) 285596. 0.264547
\(260\) 1.26557e6 1.16106
\(261\) −175862. −0.159798
\(262\) 27561.8 0.0248058
\(263\) 1.22245e6 1.08979 0.544894 0.838505i \(-0.316570\pi\)
0.544894 + 0.838505i \(0.316570\pi\)
\(264\) −27479.1 −0.0242657
\(265\) −504458. −0.441276
\(266\) 165417. 0.143343
\(267\) 768621. 0.659833
\(268\) −1.23357e6 −1.04913
\(269\) −1.60210e6 −1.34992 −0.674961 0.737853i \(-0.735840\pi\)
−0.674961 + 0.737853i \(0.735840\pi\)
\(270\) −163714. −0.136671
\(271\) 1.62601e6 1.34493 0.672465 0.740129i \(-0.265235\pi\)
0.672465 + 0.740129i \(0.265235\pi\)
\(272\) 582622. 0.477490
\(273\) −266223. −0.216191
\(274\) 1.05187e6 0.846422
\(275\) 76958.6 0.0613656
\(276\) −737480. −0.582743
\(277\) 1.36348e6 1.06770 0.533851 0.845579i \(-0.320744\pi\)
0.533851 + 0.845579i \(0.320744\pi\)
\(278\) −123309. −0.0956935
\(279\) 580250. 0.446277
\(280\) −616261. −0.469753
\(281\) 1.74237e6 1.31636 0.658179 0.752862i \(-0.271327\pi\)
0.658179 + 0.752862i \(0.271327\pi\)
\(282\) −254635. −0.190676
\(283\) −1.23647e6 −0.917735 −0.458868 0.888505i \(-0.651745\pi\)
−0.458868 + 0.888505i \(0.651745\pi\)
\(284\) −1.60254e6 −1.17900
\(285\) 956269. 0.697378
\(286\) 32912.1 0.0237925
\(287\) −124870. −0.0894854
\(288\) 478923. 0.340239
\(289\) 906525. 0.638462
\(290\) 487579. 0.340447
\(291\) 426437. 0.295204
\(292\) −1.08032e6 −0.741470
\(293\) 2.68981e6 1.83042 0.915212 0.402972i \(-0.132023\pi\)
0.915212 + 0.402972i \(0.132023\pi\)
\(294\) −350238. −0.236317
\(295\) 290680. 0.194474
\(296\) 901964. 0.598357
\(297\) 14579.5 0.00959073
\(298\) 506777. 0.330580
\(299\) 2.02451e6 1.30961
\(300\) −857760. −0.550253
\(301\) 257835. 0.164031
\(302\) −188822. −0.119134
\(303\) 1.00660e6 0.629871
\(304\) −486040. −0.301640
\(305\) −11207.0 −0.00689825
\(306\) 332255. 0.202847
\(307\) −1.87388e6 −1.13474 −0.567369 0.823463i \(-0.692039\pi\)
−0.567369 + 0.823463i \(0.692039\pi\)
\(308\) 23944.4 0.0143823
\(309\) −1.45441e6 −0.866542
\(310\) −1.60875e6 −0.950788
\(311\) −1.66629e6 −0.976898 −0.488449 0.872592i \(-0.662437\pi\)
−0.488449 + 0.872592i \(0.662437\pi\)
\(312\) −840780. −0.488985
\(313\) −1.52243e6 −0.878368 −0.439184 0.898397i \(-0.644732\pi\)
−0.439184 + 0.898397i \(0.644732\pi\)
\(314\) −1.10156e6 −0.630497
\(315\) 326968. 0.185664
\(316\) −1.85363e6 −1.04425
\(317\) −1.08948e6 −0.608933 −0.304466 0.952523i \(-0.598478\pi\)
−0.304466 + 0.952523i \(0.598478\pi\)
\(318\) 146218. 0.0810837
\(319\) −43421.3 −0.0238905
\(320\) −307093. −0.167647
\(321\) −1.63088e6 −0.883405
\(322\) −430110. −0.231175
\(323\) −1.94074e6 −1.03505
\(324\) −162499. −0.0859981
\(325\) 2.35471e6 1.23660
\(326\) −1.20378e6 −0.627342
\(327\) −628725. −0.325156
\(328\) −394361. −0.202400
\(329\) 508555. 0.259029
\(330\) −40421.7 −0.0204329
\(331\) 3.01617e6 1.51317 0.756583 0.653898i \(-0.226867\pi\)
0.756583 + 0.653898i \(0.226867\pi\)
\(332\) −130543. −0.0649994
\(333\) −478552. −0.236493
\(334\) −174208. −0.0854480
\(335\) −4.15906e6 −2.02480
\(336\) −166187. −0.0803061
\(337\) −2.89211e6 −1.38720 −0.693602 0.720359i \(-0.743977\pi\)
−0.693602 + 0.720359i \(0.743977\pi\)
\(338\) 8479.77 0.00403731
\(339\) 1.36927e6 0.647127
\(340\) 3.15452e6 1.47991
\(341\) 143267. 0.0667205
\(342\) −277177. −0.128142
\(343\) 1.51195e6 0.693906
\(344\) 814289. 0.371007
\(345\) −2.48645e6 −1.12469
\(346\) 1.17304e6 0.526771
\(347\) −2.45959e6 −1.09658 −0.548289 0.836289i \(-0.684721\pi\)
−0.548289 + 0.836289i \(0.684721\pi\)
\(348\) 483962. 0.214222
\(349\) 1.57913e6 0.693992 0.346996 0.937867i \(-0.387202\pi\)
0.346996 + 0.937867i \(0.387202\pi\)
\(350\) −500260. −0.218286
\(351\) 446090. 0.193266
\(352\) 118249. 0.0508674
\(353\) −932705. −0.398389 −0.199195 0.979960i \(-0.563833\pi\)
−0.199195 + 0.979960i \(0.563833\pi\)
\(354\) −84254.3 −0.0357342
\(355\) −5.40305e6 −2.27545
\(356\) −2.11520e6 −0.884557
\(357\) −663576. −0.275563
\(358\) 668803. 0.275798
\(359\) −94646.8 −0.0387587 −0.0193794 0.999812i \(-0.506169\pi\)
−0.0193794 + 0.999812i \(0.506169\pi\)
\(360\) 1.03262e6 0.419938
\(361\) −857080. −0.346141
\(362\) −1.37086e6 −0.549820
\(363\) −1.44586e6 −0.575916
\(364\) 732629. 0.289821
\(365\) −3.64235e6 −1.43103
\(366\) 3248.36 0.00126754
\(367\) 3.17308e6 1.22975 0.614873 0.788626i \(-0.289207\pi\)
0.614873 + 0.788626i \(0.289207\pi\)
\(368\) 1.26378e6 0.486467
\(369\) 209235. 0.0799960
\(370\) 1.32679e6 0.503845
\(371\) −292026. −0.110151
\(372\) −1.59681e6 −0.598269
\(373\) −2.35041e6 −0.874723 −0.437362 0.899286i \(-0.644087\pi\)
−0.437362 + 0.899286i \(0.644087\pi\)
\(374\) 82035.4 0.0303265
\(375\) −543411. −0.199549
\(376\) 1.60611e6 0.585875
\(377\) −1.32856e6 −0.481425
\(378\) −94772.2 −0.0341155
\(379\) 998577. 0.357095 0.178547 0.983931i \(-0.442860\pi\)
0.178547 + 0.983931i \(0.442860\pi\)
\(380\) −2.63159e6 −0.934889
\(381\) 2.47833e6 0.874675
\(382\) 1.56567e6 0.548961
\(383\) −3.24360e6 −1.12987 −0.564937 0.825134i \(-0.691100\pi\)
−0.564937 + 0.825134i \(0.691100\pi\)
\(384\) −1.61383e6 −0.558507
\(385\) 80730.0 0.0277577
\(386\) 91228.6 0.0311647
\(387\) −432034. −0.146636
\(388\) −1.17353e6 −0.395744
\(389\) −434588. −0.145614 −0.0728070 0.997346i \(-0.523196\pi\)
−0.0728070 + 0.997346i \(0.523196\pi\)
\(390\) −1.23679e6 −0.411749
\(391\) 5.04623e6 1.66926
\(392\) 2.20913e6 0.726115
\(393\) 92236.8 0.0301247
\(394\) 138183. 0.0448449
\(395\) −6.24963e6 −2.01540
\(396\) −40121.9 −0.0128571
\(397\) 2.29030e6 0.729315 0.364658 0.931142i \(-0.381186\pi\)
0.364658 + 0.931142i \(0.381186\pi\)
\(398\) 788334. 0.249461
\(399\) 553575. 0.174078
\(400\) 1.46990e6 0.459344
\(401\) 1.95111e6 0.605928 0.302964 0.953002i \(-0.402024\pi\)
0.302964 + 0.953002i \(0.402024\pi\)
\(402\) 1.20551e6 0.372054
\(403\) 4.38354e6 1.34450
\(404\) −2.77011e6 −0.844391
\(405\) −547875. −0.165976
\(406\) 282255. 0.0849818
\(407\) −118157. −0.0353568
\(408\) −2.09570e6 −0.623272
\(409\) 1.46908e6 0.434247 0.217123 0.976144i \(-0.430333\pi\)
0.217123 + 0.976144i \(0.430333\pi\)
\(410\) −580105. −0.170430
\(411\) 3.52014e6 1.02791
\(412\) 4.00244e6 1.16167
\(413\) 168272. 0.0485441
\(414\) 720704. 0.206660
\(415\) −440134. −0.125448
\(416\) 3.61806e6 1.02504
\(417\) −412659. −0.116212
\(418\) −68436.3 −0.0191578
\(419\) −3.97963e6 −1.10741 −0.553703 0.832714i \(-0.686786\pi\)
−0.553703 + 0.832714i \(0.686786\pi\)
\(420\) −899795. −0.248897
\(421\) 828909. 0.227930 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(422\) −691926. −0.189138
\(423\) −852147. −0.231560
\(424\) −922271. −0.249140
\(425\) 5.86925e6 1.57620
\(426\) 1.56608e6 0.418111
\(427\) −6487.61 −0.00172193
\(428\) 4.48809e6 1.18427
\(429\) 110142. 0.0288941
\(430\) 1.19782e6 0.312406
\(431\) 6.65314e6 1.72518 0.862588 0.505907i \(-0.168842\pi\)
0.862588 + 0.505907i \(0.168842\pi\)
\(432\) 278467. 0.0717901
\(433\) −532462. −0.136480 −0.0682400 0.997669i \(-0.521738\pi\)
−0.0682400 + 0.997669i \(0.521738\pi\)
\(434\) −931288. −0.237334
\(435\) 1.63171e6 0.413446
\(436\) 1.73021e6 0.435896
\(437\) −4.20971e6 −1.05450
\(438\) 1.05574e6 0.262950
\(439\) 4.04363e6 1.00141 0.500703 0.865619i \(-0.333075\pi\)
0.500703 + 0.865619i \(0.333075\pi\)
\(440\) 254960. 0.0627827
\(441\) −1.17209e6 −0.286988
\(442\) 2.51004e6 0.611118
\(443\) 1.66963e6 0.404214 0.202107 0.979363i \(-0.435221\pi\)
0.202107 + 0.979363i \(0.435221\pi\)
\(444\) 1.31695e6 0.317038
\(445\) −7.13151e6 −1.70719
\(446\) −725336. −0.172664
\(447\) 1.69595e6 0.401463
\(448\) −177773. −0.0418477
\(449\) −4.07437e6 −0.953772 −0.476886 0.878965i \(-0.658235\pi\)
−0.476886 + 0.878965i \(0.658235\pi\)
\(450\) 838248. 0.195138
\(451\) 51661.2 0.0119598
\(452\) −3.76814e6 −0.867524
\(453\) −631902. −0.144679
\(454\) 1.27699e6 0.290769
\(455\) 2.47010e6 0.559353
\(456\) 1.74829e6 0.393733
\(457\) −6.80572e6 −1.52435 −0.762173 0.647373i \(-0.775868\pi\)
−0.762173 + 0.647373i \(0.775868\pi\)
\(458\) 1.97821e6 0.440665
\(459\) 1.11191e6 0.246341
\(460\) 6.84257e6 1.50773
\(461\) 6.87814e6 1.50737 0.753683 0.657239i \(-0.228276\pi\)
0.753683 + 0.657239i \(0.228276\pi\)
\(462\) −23399.8 −0.00510042
\(463\) −2.39491e6 −0.519202 −0.259601 0.965716i \(-0.583591\pi\)
−0.259601 + 0.965716i \(0.583591\pi\)
\(464\) −829343. −0.178829
\(465\) −5.38375e6 −1.15465
\(466\) 396605. 0.0846044
\(467\) 5.32072e6 1.12896 0.564480 0.825447i \(-0.309077\pi\)
0.564480 + 0.825447i \(0.309077\pi\)
\(468\) −1.22761e6 −0.259087
\(469\) −2.40764e6 −0.505428
\(470\) 2.36258e6 0.493336
\(471\) −3.68641e6 −0.765687
\(472\) 531434. 0.109798
\(473\) −106671. −0.0219228
\(474\) 1.81147e6 0.370327
\(475\) −4.89630e6 −0.995712
\(476\) 1.82612e6 0.369413
\(477\) 489326. 0.0984697
\(478\) −2.55810e6 −0.512091
\(479\) −476392. −0.0948693 −0.0474347 0.998874i \(-0.515105\pi\)
−0.0474347 + 0.998874i \(0.515105\pi\)
\(480\) −4.44360e6 −0.880302
\(481\) −3.61526e6 −0.712486
\(482\) 917753. 0.179932
\(483\) −1.43938e6 −0.280743
\(484\) 3.97892e6 0.772061
\(485\) −3.95662e6 −0.763783
\(486\) 158803. 0.0304977
\(487\) −3.24904e6 −0.620774 −0.310387 0.950610i \(-0.600459\pi\)
−0.310387 + 0.950610i \(0.600459\pi\)
\(488\) −20489.0 −0.00389468
\(489\) −4.02852e6 −0.761857
\(490\) 3.24962e6 0.611424
\(491\) −1.12407e6 −0.210421 −0.105211 0.994450i \(-0.533552\pi\)
−0.105211 + 0.994450i \(0.533552\pi\)
\(492\) −575802. −0.107241
\(493\) −3.31153e6 −0.613636
\(494\) −2.09395e6 −0.386055
\(495\) −135273. −0.0248141
\(496\) 2.73638e6 0.499428
\(497\) −3.12777e6 −0.567994
\(498\) 127574. 0.0230509
\(499\) 876516. 0.157583 0.0787913 0.996891i \(-0.474894\pi\)
0.0787913 + 0.996891i \(0.474894\pi\)
\(500\) 1.49543e6 0.267512
\(501\) −582995. −0.103770
\(502\) 4.16703e6 0.738019
\(503\) −1.53970e6 −0.271342 −0.135671 0.990754i \(-0.543319\pi\)
−0.135671 + 0.990754i \(0.543319\pi\)
\(504\) 597775. 0.104824
\(505\) −9.33958e6 −1.62967
\(506\) 177946. 0.0308966
\(507\) 28377.9 0.00490299
\(508\) −6.82022e6 −1.17257
\(509\) 3.23654e6 0.553715 0.276858 0.960911i \(-0.410707\pi\)
0.276858 + 0.960911i \(0.410707\pi\)
\(510\) −3.08277e6 −0.524826
\(511\) −2.10852e6 −0.357211
\(512\) 4.12467e6 0.695367
\(513\) −927585. −0.155618
\(514\) −2.20283e6 −0.367768
\(515\) 1.34944e7 2.24201
\(516\) 1.18893e6 0.196577
\(517\) −210400. −0.0346193
\(518\) 768065. 0.125769
\(519\) 3.92562e6 0.639721
\(520\) 7.80102e6 1.26515
\(521\) 8.43608e6 1.36159 0.680795 0.732474i \(-0.261634\pi\)
0.680795 + 0.732474i \(0.261634\pi\)
\(522\) −472953. −0.0759700
\(523\) 1.04546e7 1.67129 0.835645 0.549270i \(-0.185094\pi\)
0.835645 + 0.549270i \(0.185094\pi\)
\(524\) −253830. −0.0403845
\(525\) −1.67414e6 −0.265090
\(526\) 3.28759e6 0.518099
\(527\) 1.09262e7 1.71374
\(528\) 68755.0 0.0107330
\(529\) 4.50958e6 0.700644
\(530\) −1.35666e6 −0.209788
\(531\) −281961. −0.0433963
\(532\) −1.52340e6 −0.233365
\(533\) 1.58068e6 0.241005
\(534\) 2.06708e6 0.313693
\(535\) 1.51318e7 2.28564
\(536\) −7.60376e6 −1.14318
\(537\) 2.23818e6 0.334934
\(538\) −4.30859e6 −0.641770
\(539\) −289395. −0.0429061
\(540\) 1.50772e6 0.222503
\(541\) −7.11362e6 −1.04496 −0.522478 0.852653i \(-0.674992\pi\)
−0.522478 + 0.852653i \(0.674992\pi\)
\(542\) 4.37289e6 0.639397
\(543\) −4.58764e6 −0.667712
\(544\) 9.01823e6 1.30654
\(545\) 5.83351e6 0.841276
\(546\) −715963. −0.102780
\(547\) −3.99245e6 −0.570520 −0.285260 0.958450i \(-0.592080\pi\)
−0.285260 + 0.958450i \(0.592080\pi\)
\(548\) −9.68721e6 −1.37799
\(549\) 10870.8 0.00153933
\(550\) 206968. 0.0291740
\(551\) 2.76257e6 0.387645
\(552\) −4.54584e6 −0.634989
\(553\) −3.61785e6 −0.503081
\(554\) 3.66686e6 0.507599
\(555\) 4.44016e6 0.611880
\(556\) 1.13561e6 0.155791
\(557\) −1.13329e6 −0.154776 −0.0773881 0.997001i \(-0.524658\pi\)
−0.0773881 + 0.997001i \(0.524658\pi\)
\(558\) 1.56049e6 0.212166
\(559\) −3.26383e6 −0.441772
\(560\) 1.54193e6 0.207776
\(561\) 274535. 0.0368291
\(562\) 4.68582e6 0.625813
\(563\) −1.33775e7 −1.77871 −0.889353 0.457222i \(-0.848844\pi\)
−0.889353 + 0.457222i \(0.848844\pi\)
\(564\) 2.34506e6 0.310424
\(565\) −1.27045e7 −1.67431
\(566\) −3.32529e6 −0.436303
\(567\) −317160. −0.0414305
\(568\) −9.87807e6 −1.28470
\(569\) −9.96665e6 −1.29053 −0.645266 0.763958i \(-0.723253\pi\)
−0.645266 + 0.763958i \(0.723253\pi\)
\(570\) 2.57173e6 0.331542
\(571\) −5.30435e6 −0.680835 −0.340417 0.940274i \(-0.610568\pi\)
−0.340417 + 0.940274i \(0.610568\pi\)
\(572\) −303104. −0.0387348
\(573\) 5.23959e6 0.666669
\(574\) −335817. −0.0425425
\(575\) 1.27312e7 1.60583
\(576\) 297882. 0.0374100
\(577\) 1.35522e7 1.69461 0.847307 0.531103i \(-0.178222\pi\)
0.847307 + 0.531103i \(0.178222\pi\)
\(578\) 2.43795e6 0.303533
\(579\) 305301. 0.0378470
\(580\) −4.49035e6 −0.554256
\(581\) −254789. −0.0313142
\(582\) 1.14683e6 0.140344
\(583\) 120817. 0.0147217
\(584\) −6.65909e6 −0.807946
\(585\) −4.13896e6 −0.500037
\(586\) 7.23380e6 0.870207
\(587\) −1.53895e7 −1.84344 −0.921720 0.387857i \(-0.873215\pi\)
−0.921720 + 0.387857i \(0.873215\pi\)
\(588\) 3.22552e6 0.384730
\(589\) −9.11499e6 −1.08260
\(590\) 781738. 0.0924552
\(591\) 462435. 0.0544605
\(592\) −2.25679e6 −0.264659
\(593\) 9.50339e6 1.10979 0.554896 0.831920i \(-0.312758\pi\)
0.554896 + 0.831920i \(0.312758\pi\)
\(594\) 39209.2 0.00455955
\(595\) 6.15687e6 0.712964
\(596\) −4.66716e6 −0.538192
\(597\) 2.63820e6 0.302950
\(598\) 5.44461e6 0.622606
\(599\) 5.58939e6 0.636498 0.318249 0.948007i \(-0.396905\pi\)
0.318249 + 0.948007i \(0.396905\pi\)
\(600\) −5.28725e6 −0.599586
\(601\) −1.67571e7 −1.89240 −0.946200 0.323581i \(-0.895113\pi\)
−0.946200 + 0.323581i \(0.895113\pi\)
\(602\) 693405. 0.0779822
\(603\) 4.03430e6 0.451830
\(604\) 1.73896e6 0.193953
\(605\) 1.34151e7 1.49007
\(606\) 2.70710e6 0.299449
\(607\) −1.00054e7 −1.10220 −0.551101 0.834439i \(-0.685792\pi\)
−0.551101 + 0.834439i \(0.685792\pi\)
\(608\) −7.52327e6 −0.825368
\(609\) 944578. 0.103204
\(610\) −30139.4 −0.00327951
\(611\) −6.43761e6 −0.697624
\(612\) −3.05990e6 −0.330239
\(613\) 1.22753e7 1.31941 0.659704 0.751525i \(-0.270682\pi\)
0.659704 + 0.751525i \(0.270682\pi\)
\(614\) −5.03950e6 −0.539469
\(615\) −1.94135e6 −0.206974
\(616\) 147594. 0.0156717
\(617\) −3.39106e6 −0.358611 −0.179305 0.983793i \(-0.557385\pi\)
−0.179305 + 0.983793i \(0.557385\pi\)
\(618\) −3.91139e6 −0.411965
\(619\) −1.36685e7 −1.43382 −0.716908 0.697168i \(-0.754443\pi\)
−0.716908 + 0.697168i \(0.754443\pi\)
\(620\) 1.48157e7 1.54790
\(621\) 2.41187e6 0.250972
\(622\) −4.48122e6 −0.464430
\(623\) −4.12836e6 −0.426145
\(624\) 2.10370e6 0.216283
\(625\) −6.98325e6 −0.715084
\(626\) −4.09433e6 −0.417587
\(627\) −229025. −0.0232656
\(628\) 1.01448e7 1.02646
\(629\) −9.01125e6 −0.908151
\(630\) 879327. 0.0882671
\(631\) 1.06988e7 1.06969 0.534847 0.844949i \(-0.320369\pi\)
0.534847 + 0.844949i \(0.320369\pi\)
\(632\) −1.14258e7 −1.13788
\(633\) −2.31556e6 −0.229693
\(634\) −2.92997e6 −0.289494
\(635\) −2.29947e7 −2.26305
\(636\) −1.34660e6 −0.132006
\(637\) −8.85462e6 −0.864613
\(638\) −116775. −0.0113579
\(639\) 5.24097e6 0.507762
\(640\) 1.49736e7 1.44503
\(641\) −5.09674e6 −0.489945 −0.244972 0.969530i \(-0.578779\pi\)
−0.244972 + 0.969530i \(0.578779\pi\)
\(642\) −4.38600e6 −0.419982
\(643\) 540230. 0.0515289 0.0257645 0.999668i \(-0.491798\pi\)
0.0257645 + 0.999668i \(0.491798\pi\)
\(644\) 3.96110e6 0.376358
\(645\) 4.00855e6 0.379392
\(646\) −5.21930e6 −0.492075
\(647\) 1.70813e6 0.160421 0.0802105 0.996778i \(-0.474441\pi\)
0.0802105 + 0.996778i \(0.474441\pi\)
\(648\) −1.00165e6 −0.0937082
\(649\) −69617.6 −0.00648795
\(650\) 6.33260e6 0.587894
\(651\) −3.11660e6 −0.288223
\(652\) 1.10862e7 1.02133
\(653\) −1.69268e6 −0.155343 −0.0776716 0.996979i \(-0.524749\pi\)
−0.0776716 + 0.996979i \(0.524749\pi\)
\(654\) −1.69086e6 −0.154583
\(655\) −855802. −0.0779417
\(656\) 986724. 0.0895233
\(657\) 3.53309e6 0.319331
\(658\) 1.36768e6 0.123145
\(659\) 1.08698e7 0.975005 0.487502 0.873122i \(-0.337908\pi\)
0.487502 + 0.873122i \(0.337908\pi\)
\(660\) 372264. 0.0332653
\(661\) 1.45109e7 1.29178 0.645892 0.763429i \(-0.276486\pi\)
0.645892 + 0.763429i \(0.276486\pi\)
\(662\) 8.11152e6 0.719378
\(663\) 8.39997e6 0.742154
\(664\) −804671. −0.0708269
\(665\) −5.13624e6 −0.450393
\(666\) −1.28699e6 −0.112432
\(667\) −7.18313e6 −0.625172
\(668\) 1.60437e6 0.139111
\(669\) −2.42737e6 −0.209687
\(670\) −1.11851e7 −0.962618
\(671\) 2684.06 0.000230136 0
\(672\) −2.57236e6 −0.219739
\(673\) −1.88349e7 −1.60297 −0.801484 0.598016i \(-0.795956\pi\)
−0.801484 + 0.598016i \(0.795956\pi\)
\(674\) −7.77787e6 −0.659494
\(675\) 2.80524e6 0.236979
\(676\) −78094.3 −0.00657284
\(677\) −8.52740e6 −0.715064 −0.357532 0.933901i \(-0.616382\pi\)
−0.357532 + 0.933901i \(0.616382\pi\)
\(678\) 3.68243e6 0.307652
\(679\) −2.29045e6 −0.190654
\(680\) 1.94445e7 1.61259
\(681\) 4.27351e6 0.353115
\(682\) 385293. 0.0317198
\(683\) 1.61614e7 1.32564 0.662821 0.748778i \(-0.269359\pi\)
0.662821 + 0.748778i \(0.269359\pi\)
\(684\) 2.55266e6 0.208618
\(685\) −3.26610e7 −2.65952
\(686\) 4.06614e6 0.329892
\(687\) 6.62016e6 0.535152
\(688\) −2.03742e6 −0.164100
\(689\) 3.69665e6 0.296661
\(690\) −6.68692e6 −0.534692
\(691\) −2.32191e7 −1.84991 −0.924953 0.380081i \(-0.875896\pi\)
−0.924953 + 0.380081i \(0.875896\pi\)
\(692\) −1.08031e7 −0.857595
\(693\) −78308.3 −0.00619405
\(694\) −6.61468e6 −0.521327
\(695\) 3.82878e6 0.300676
\(696\) 2.98315e6 0.233427
\(697\) 3.93994e6 0.307191
\(698\) 4.24682e6 0.329932
\(699\) 1.32726e6 0.102745
\(700\) 4.60714e6 0.355374
\(701\) −1.45476e7 −1.11814 −0.559070 0.829121i \(-0.688842\pi\)
−0.559070 + 0.829121i \(0.688842\pi\)
\(702\) 1.19969e6 0.0918809
\(703\) 7.51744e6 0.573696
\(704\) 73548.5 0.00559296
\(705\) 7.90649e6 0.599116
\(706\) −2.50836e6 −0.189399
\(707\) −5.40659e6 −0.406795
\(708\) 775939. 0.0581761
\(709\) 2.04913e7 1.53092 0.765462 0.643481i \(-0.222510\pi\)
0.765462 + 0.643481i \(0.222510\pi\)
\(710\) −1.45306e7 −1.08178
\(711\) 6.06216e6 0.449732
\(712\) −1.30381e7 −0.963861
\(713\) 2.37005e7 1.74595
\(714\) −1.78458e6 −0.131006
\(715\) −1.02193e6 −0.0747578
\(716\) −6.15933e6 −0.449005
\(717\) −8.56079e6 −0.621893
\(718\) −254537. −0.0184264
\(719\) 5.74561e6 0.414490 0.207245 0.978289i \(-0.433550\pi\)
0.207245 + 0.978289i \(0.433550\pi\)
\(720\) −2.58371e6 −0.185743
\(721\) 7.81180e6 0.559645
\(722\) −2.30498e6 −0.164560
\(723\) 3.07130e6 0.218513
\(724\) 1.26249e7 0.895120
\(725\) −8.35467e6 −0.590316
\(726\) −3.88841e6 −0.273798
\(727\) −2.22394e7 −1.56058 −0.780292 0.625416i \(-0.784929\pi\)
−0.780292 + 0.625416i \(0.784929\pi\)
\(728\) 4.51593e6 0.315805
\(729\) 531441. 0.0370370
\(730\) −9.79551e6 −0.680330
\(731\) −8.13530e6 −0.563093
\(732\) −29915.8 −0.00206359
\(733\) 1.66373e7 1.14373 0.571866 0.820347i \(-0.306220\pi\)
0.571866 + 0.820347i \(0.306220\pi\)
\(734\) 8.53349e6 0.584637
\(735\) 1.08750e7 0.742525
\(736\) 1.95617e7 1.33111
\(737\) 996090. 0.0675507
\(738\) 562704. 0.0380311
\(739\) −6.69365e6 −0.450871 −0.225435 0.974258i \(-0.572380\pi\)
−0.225435 + 0.974258i \(0.572380\pi\)
\(740\) −1.22190e7 −0.820272
\(741\) −7.00750e6 −0.468832
\(742\) −785357. −0.0523669
\(743\) 2.73131e7 1.81509 0.907547 0.419951i \(-0.137953\pi\)
0.907547 + 0.419951i \(0.137953\pi\)
\(744\) −9.84278e6 −0.651907
\(745\) −1.57356e7 −1.03871
\(746\) −6.32104e6 −0.415854
\(747\) 426932. 0.0279935
\(748\) −755504. −0.0493723
\(749\) 8.75968e6 0.570537
\(750\) −1.46142e6 −0.0948683
\(751\) −3.14794e6 −0.203670 −0.101835 0.994801i \(-0.532471\pi\)
−0.101835 + 0.994801i \(0.532471\pi\)
\(752\) −4.01861e6 −0.259138
\(753\) 1.39452e7 0.896264
\(754\) −3.57296e6 −0.228876
\(755\) 5.86299e6 0.374328
\(756\) 872804. 0.0555408
\(757\) 8.68258e6 0.550692 0.275346 0.961345i \(-0.411208\pi\)
0.275346 + 0.961345i \(0.411208\pi\)
\(758\) 2.68551e6 0.169767
\(759\) 595503. 0.0375214
\(760\) −1.62212e7 −1.01871
\(761\) −9.78953e6 −0.612774 −0.306387 0.951907i \(-0.599120\pi\)
−0.306387 + 0.951907i \(0.599120\pi\)
\(762\) 6.66507e6 0.415832
\(763\) 3.37696e6 0.209998
\(764\) −1.44190e7 −0.893722
\(765\) −1.03166e7 −0.637358
\(766\) −8.72314e6 −0.537157
\(767\) −2.13009e6 −0.130741
\(768\) −5.39926e6 −0.330317
\(769\) −2.96608e7 −1.80870 −0.904351 0.426789i \(-0.859645\pi\)
−0.904351 + 0.426789i \(0.859645\pi\)
\(770\) 217110. 0.0131963
\(771\) −7.37188e6 −0.446624
\(772\) −840169. −0.0507368
\(773\) −7.78504e6 −0.468611 −0.234305 0.972163i \(-0.575282\pi\)
−0.234305 + 0.972163i \(0.575282\pi\)
\(774\) −1.16189e6 −0.0697126
\(775\) 2.75659e7 1.64861
\(776\) −7.23365e6 −0.431224
\(777\) 2.57036e6 0.152736
\(778\) −1.16875e6 −0.0692268
\(779\) −3.28681e6 −0.194058
\(780\) 1.13902e7 0.670338
\(781\) 1.29402e6 0.0759127
\(782\) 1.35710e7 0.793589
\(783\) −1.58276e6 −0.0922594
\(784\) −5.52741e6 −0.321168
\(785\) 3.42037e7 1.98106
\(786\) 248056. 0.0143217
\(787\) −2.25929e6 −0.130028 −0.0650138 0.997884i \(-0.520709\pi\)
−0.0650138 + 0.997884i \(0.520709\pi\)
\(788\) −1.27259e6 −0.0730085
\(789\) 1.10021e7 0.629190
\(790\) −1.68074e7 −0.958148
\(791\) −7.35451e6 −0.417939
\(792\) −247312. −0.0140098
\(793\) 82124.2 0.00463755
\(794\) 6.15938e6 0.346726
\(795\) −4.54012e6 −0.254771
\(796\) −7.26016e6 −0.406128
\(797\) −1.95399e7 −1.08962 −0.544812 0.838558i \(-0.683399\pi\)
−0.544812 + 0.838558i \(0.683399\pi\)
\(798\) 1.48875e6 0.0827589
\(799\) −1.60461e7 −0.889208
\(800\) 2.27522e7 1.25689
\(801\) 6.91759e6 0.380955
\(802\) 5.24720e6 0.288066
\(803\) 872338. 0.0477415
\(804\) −1.11022e7 −0.605713
\(805\) 1.33551e7 0.726367
\(806\) 1.17888e7 0.639195
\(807\) −1.44189e7 −0.779378
\(808\) −1.70750e7 −0.920094
\(809\) −2.92703e7 −1.57237 −0.786187 0.617989i \(-0.787948\pi\)
−0.786187 + 0.617989i \(0.787948\pi\)
\(810\) −1.47342e6 −0.0789069
\(811\) 3.63592e6 0.194116 0.0970581 0.995279i \(-0.469057\pi\)
0.0970581 + 0.995279i \(0.469057\pi\)
\(812\) −2.59942e6 −0.138352
\(813\) 1.46341e7 0.776496
\(814\) −317764. −0.0168091
\(815\) 3.73779e7 1.97115
\(816\) 5.24360e6 0.275679
\(817\) 6.78671e6 0.355717
\(818\) 3.95085e6 0.206446
\(819\) −2.39600e6 −0.124818
\(820\) 5.34247e6 0.277465
\(821\) 1.26564e6 0.0655318 0.0327659 0.999463i \(-0.489568\pi\)
0.0327659 + 0.999463i \(0.489568\pi\)
\(822\) 9.46685e6 0.488682
\(823\) 2.48803e7 1.28043 0.640216 0.768195i \(-0.278845\pi\)
0.640216 + 0.768195i \(0.278845\pi\)
\(824\) 2.46711e7 1.26581
\(825\) 692627. 0.0354295
\(826\) 452541. 0.0230785
\(827\) 1.44783e7 0.736131 0.368065 0.929800i \(-0.380020\pi\)
0.368065 + 0.929800i \(0.380020\pi\)
\(828\) −6.63732e6 −0.336447
\(829\) 476856. 0.0240991 0.0120495 0.999927i \(-0.496164\pi\)
0.0120495 + 0.999927i \(0.496164\pi\)
\(830\) −1.18367e6 −0.0596397
\(831\) 1.22713e7 0.616438
\(832\) 2.25037e6 0.112705
\(833\) −2.20707e7 −1.10206
\(834\) −1.10978e6 −0.0552486
\(835\) 5.40921e6 0.268484
\(836\) 630264. 0.0311894
\(837\) 5.22225e6 0.257658
\(838\) −1.07026e7 −0.526475
\(839\) 2.73963e7 1.34365 0.671827 0.740708i \(-0.265510\pi\)
0.671827 + 0.740708i \(0.265510\pi\)
\(840\) −5.54635e6 −0.271212
\(841\) −1.57973e7 −0.770181
\(842\) 2.22922e6 0.108361
\(843\) 1.56813e7 0.759999
\(844\) 6.37228e6 0.307921
\(845\) −263299. −0.0126855
\(846\) −2.29171e6 −0.110087
\(847\) 7.76589e6 0.371949
\(848\) 2.30760e6 0.110197
\(849\) −1.11282e7 −0.529855
\(850\) 1.57844e7 0.749343
\(851\) −1.95466e7 −0.925223
\(852\) −1.44228e7 −0.680694
\(853\) 2.87120e6 0.135111 0.0675556 0.997716i \(-0.478480\pi\)
0.0675556 + 0.997716i \(0.478480\pi\)
\(854\) −17447.4 −0.000818626 0
\(855\) 8.60642e6 0.402631
\(856\) 2.76647e7 1.29045
\(857\) 2.66310e7 1.23861 0.619305 0.785150i \(-0.287414\pi\)
0.619305 + 0.785150i \(0.287414\pi\)
\(858\) 296209. 0.0137366
\(859\) 7.47800e6 0.345782 0.172891 0.984941i \(-0.444689\pi\)
0.172891 + 0.984941i \(0.444689\pi\)
\(860\) −1.10313e7 −0.508604
\(861\) −1.12383e6 −0.0516644
\(862\) 1.78926e7 0.820171
\(863\) −3.33536e7 −1.52446 −0.762230 0.647306i \(-0.775895\pi\)
−0.762230 + 0.647306i \(0.775895\pi\)
\(864\) 4.31031e6 0.196437
\(865\) −3.64232e7 −1.65515
\(866\) −1.43197e6 −0.0648843
\(867\) 8.15873e6 0.368616
\(868\) 8.57669e6 0.386385
\(869\) 1.49678e6 0.0672370
\(870\) 4.38821e6 0.196557
\(871\) 3.04774e7 1.36123
\(872\) 1.06651e7 0.474976
\(873\) 3.83793e6 0.170436
\(874\) −1.13213e7 −0.501325
\(875\) 2.91873e6 0.128877
\(876\) −9.72284e6 −0.428088
\(877\) 2.97763e7 1.30729 0.653645 0.756802i \(-0.273239\pi\)
0.653645 + 0.756802i \(0.273239\pi\)
\(878\) 1.08747e7 0.476081
\(879\) 2.42083e7 1.05680
\(880\) −637930. −0.0277694
\(881\) 1.74483e7 0.757380 0.378690 0.925523i \(-0.376375\pi\)
0.378690 + 0.925523i \(0.376375\pi\)
\(882\) −3.15214e6 −0.136438
\(883\) 3.08101e7 1.32981 0.664907 0.746926i \(-0.268471\pi\)
0.664907 + 0.746926i \(0.268471\pi\)
\(884\) −2.31162e7 −0.994915
\(885\) 2.61612e6 0.112279
\(886\) 4.49021e6 0.192169
\(887\) −110509. −0.00471615 −0.00235807 0.999997i \(-0.500751\pi\)
−0.00235807 + 0.999997i \(0.500751\pi\)
\(888\) 8.11768e6 0.345461
\(889\) −1.33114e7 −0.564898
\(890\) −1.91790e7 −0.811618
\(891\) 131216. 0.00553721
\(892\) 6.67997e6 0.281101
\(893\) 1.33861e7 0.561729
\(894\) 4.56100e6 0.190860
\(895\) −2.07665e7 −0.866576
\(896\) 8.66806e6 0.360705
\(897\) 1.82206e7 0.756105
\(898\) −1.09574e7 −0.453435
\(899\) −1.55531e7 −0.641828
\(900\) −7.71984e6 −0.317689
\(901\) 9.21412e6 0.378131
\(902\) 138934. 0.00568583
\(903\) 2.32051e6 0.0947031
\(904\) −2.32269e7 −0.945301
\(905\) 4.25655e7 1.72757
\(906\) −1.69940e6 −0.0687821
\(907\) 3.31400e7 1.33763 0.668813 0.743431i \(-0.266803\pi\)
0.668813 + 0.743431i \(0.266803\pi\)
\(908\) −1.17604e7 −0.473379
\(909\) 9.05943e6 0.363656
\(910\) 6.64293e6 0.265923
\(911\) −2.55670e7 −1.02067 −0.510334 0.859976i \(-0.670478\pi\)
−0.510334 + 0.859976i \(0.670478\pi\)
\(912\) −4.37436e6 −0.174152
\(913\) 105412. 0.00418515
\(914\) −1.83029e7 −0.724694
\(915\) −100863. −0.00398270
\(916\) −1.82183e7 −0.717412
\(917\) −495415. −0.0194557
\(918\) 2.99029e6 0.117114
\(919\) 4.08635e7 1.59605 0.798026 0.602623i \(-0.205878\pi\)
0.798026 + 0.602623i \(0.205878\pi\)
\(920\) 4.21777e7 1.64291
\(921\) −1.68649e7 −0.655142
\(922\) 1.84976e7 0.716620
\(923\) 3.95933e7 1.52974
\(924\) 215500. 0.00830361
\(925\) −2.27345e7 −0.873639
\(926\) −6.44072e6 −0.246835
\(927\) −1.30897e7 −0.500298
\(928\) −1.28371e7 −0.489326
\(929\) −1.44490e7 −0.549287 −0.274644 0.961546i \(-0.588560\pi\)
−0.274644 + 0.961546i \(0.588560\pi\)
\(930\) −1.44787e7 −0.548938
\(931\) 1.84120e7 0.696189
\(932\) −3.65253e6 −0.137738
\(933\) −1.49966e7 −0.564013
\(934\) 1.43092e7 0.536722
\(935\) −2.54723e6 −0.0952880
\(936\) −7.56702e6 −0.282316
\(937\) −9.49392e6 −0.353262 −0.176631 0.984277i \(-0.556520\pi\)
−0.176631 + 0.984277i \(0.556520\pi\)
\(938\) −6.47496e6 −0.240287
\(939\) −1.37019e7 −0.507126
\(940\) −2.17582e7 −0.803162
\(941\) 4.83451e6 0.177983 0.0889915 0.996032i \(-0.471636\pi\)
0.0889915 + 0.996032i \(0.471636\pi\)
\(942\) −9.91401e6 −0.364017
\(943\) 8.54625e6 0.312965
\(944\) −1.32969e6 −0.0485647
\(945\) 2.94271e6 0.107193
\(946\) −286876. −0.0104224
\(947\) −2.50321e7 −0.907030 −0.453515 0.891249i \(-0.649830\pi\)
−0.453515 + 0.891249i \(0.649830\pi\)
\(948\) −1.66827e7 −0.602901
\(949\) 2.66910e7 0.962052
\(950\) −1.31678e7 −0.473374
\(951\) −9.80528e6 −0.351567
\(952\) 1.12562e7 0.402533
\(953\) −1.73815e7 −0.619949 −0.309975 0.950745i \(-0.600321\pi\)
−0.309975 + 0.950745i \(0.600321\pi\)
\(954\) 1.31596e6 0.0468137
\(955\) −4.86145e7 −1.72488
\(956\) 2.35588e7 0.833696
\(957\) −390792. −0.0137932
\(958\) −1.28118e6 −0.0451021
\(959\) −1.89071e7 −0.663864
\(960\) −2.76384e6 −0.0967909
\(961\) 2.26878e7 0.792472
\(962\) −9.72265e6 −0.338725
\(963\) −1.46779e7 −0.510034
\(964\) −8.45204e6 −0.292933
\(965\) −2.83268e6 −0.0979216
\(966\) −3.87099e6 −0.133469
\(967\) −5.09719e6 −0.175293 −0.0876465 0.996152i \(-0.527935\pi\)
−0.0876465 + 0.996152i \(0.527935\pi\)
\(968\) 2.45261e7 0.841279
\(969\) −1.74666e7 −0.597585
\(970\) −1.06407e7 −0.363112
\(971\) 393529. 0.0133946 0.00669729 0.999978i \(-0.497868\pi\)
0.00669729 + 0.999978i \(0.497868\pi\)
\(972\) −1.46249e6 −0.0496510
\(973\) 2.21644e6 0.0750541
\(974\) −8.73778e6 −0.295124
\(975\) 2.11923e7 0.713949
\(976\) 51265.2 0.00172266
\(977\) −5.44154e7 −1.82384 −0.911918 0.410373i \(-0.865398\pi\)
−0.911918 + 0.410373i \(0.865398\pi\)
\(978\) −1.08341e7 −0.362196
\(979\) 1.70799e6 0.0569545
\(980\) −2.99274e7 −0.995413
\(981\) −5.65852e6 −0.187729
\(982\) −3.02301e6 −0.100037
\(983\) −6.93627e6 −0.228951 −0.114475 0.993426i \(-0.536519\pi\)
−0.114475 + 0.993426i \(0.536519\pi\)
\(984\) −3.54925e6 −0.116855
\(985\) −4.29061e6 −0.140906
\(986\) −8.90582e6 −0.291730
\(987\) 4.57699e6 0.149550
\(988\) 1.92842e7 0.628506
\(989\) −1.76465e7 −0.573679
\(990\) −363796. −0.0117969
\(991\) 2.32589e7 0.752326 0.376163 0.926554i \(-0.377243\pi\)
0.376163 + 0.926554i \(0.377243\pi\)
\(992\) 4.23556e7 1.36657
\(993\) 2.71456e7 0.873626
\(994\) −8.41164e6 −0.270032
\(995\) −2.44780e7 −0.783824
\(996\) −1.17489e6 −0.0375274
\(997\) −4.77014e7 −1.51982 −0.759912 0.650026i \(-0.774758\pi\)
−0.759912 + 0.650026i \(0.774758\pi\)
\(998\) 2.35725e6 0.0749168
\(999\) −4.30697e6 −0.136539
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.1.7 12
3.2 odd 2 531.6.a.c.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.7 12 1.1 even 1 trivial
531.6.a.c.1.6 12 3.2 odd 2