Properties

Label 177.6.a.b.1.9
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
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 1.9
Root \(-5.27208\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.27208 q^{2} -9.00000 q^{3} -4.20512 q^{4} -17.5332 q^{5} -47.4488 q^{6} +172.711 q^{7} -190.876 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.27208 q^{2} -9.00000 q^{3} -4.20512 q^{4} -17.5332 q^{5} -47.4488 q^{6} +172.711 q^{7} -190.876 q^{8} +81.0000 q^{9} -92.4366 q^{10} +249.532 q^{11} +37.8461 q^{12} +282.746 q^{13} +910.545 q^{14} +157.799 q^{15} -871.753 q^{16} -1347.49 q^{17} +427.039 q^{18} -2886.53 q^{19} +73.7293 q^{20} -1554.40 q^{21} +1315.55 q^{22} +692.702 q^{23} +1717.89 q^{24} -2817.59 q^{25} +1490.66 q^{26} -729.000 q^{27} -726.269 q^{28} -2778.00 q^{29} +831.930 q^{30} -4109.10 q^{31} +1512.09 q^{32} -2245.79 q^{33} -7104.11 q^{34} -3028.17 q^{35} -340.615 q^{36} -10616.6 q^{37} -15218.0 q^{38} -2544.72 q^{39} +3346.68 q^{40} -828.466 q^{41} -8194.91 q^{42} +8995.66 q^{43} -1049.31 q^{44} -1420.19 q^{45} +3651.98 q^{46} -4346.77 q^{47} +7845.78 q^{48} +13022.0 q^{49} -14854.6 q^{50} +12127.5 q^{51} -1188.98 q^{52} -1799.96 q^{53} -3843.35 q^{54} -4375.10 q^{55} -32966.4 q^{56} +25978.8 q^{57} -14645.9 q^{58} -3481.00 q^{59} -663.564 q^{60} -28023.9 q^{61} -21663.5 q^{62} +13989.6 q^{63} +35868.0 q^{64} -4957.46 q^{65} -11840.0 q^{66} +30957.2 q^{67} +5666.38 q^{68} -6234.32 q^{69} -15964.8 q^{70} +3945.44 q^{71} -15461.0 q^{72} +17740.6 q^{73} -55971.8 q^{74} +25358.3 q^{75} +12138.2 q^{76} +43096.8 q^{77} -13416.0 q^{78} -57063.7 q^{79} +15284.6 q^{80} +6561.00 q^{81} -4367.74 q^{82} +17500.9 q^{83} +6536.42 q^{84} +23625.9 q^{85} +47425.9 q^{86} +25002.0 q^{87} -47629.8 q^{88} -53616.7 q^{89} -7487.37 q^{90} +48833.3 q^{91} -2912.89 q^{92} +36981.9 q^{93} -22916.6 q^{94} +50610.2 q^{95} -13608.8 q^{96} -115705. q^{97} +68653.0 q^{98} +20212.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.27208 0.931982 0.465991 0.884790i \(-0.345698\pi\)
0.465991 + 0.884790i \(0.345698\pi\)
\(3\) −9.00000 −0.577350
\(4\) −4.20512 −0.131410
\(5\) −17.5332 −0.313644 −0.156822 0.987627i \(-0.550125\pi\)
−0.156822 + 0.987627i \(0.550125\pi\)
\(6\) −47.4488 −0.538080
\(7\) 172.711 1.33221 0.666107 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(8\) −190.876 −1.05445
\(9\) 81.0000 0.333333
\(10\) −92.4366 −0.292310
\(11\) 249.532 0.621791 0.310895 0.950444i \(-0.399371\pi\)
0.310895 + 0.950444i \(0.399371\pi\)
\(12\) 37.8461 0.0758696
\(13\) 282.746 0.464022 0.232011 0.972713i \(-0.425469\pi\)
0.232011 + 0.972713i \(0.425469\pi\)
\(14\) 910.545 1.24160
\(15\) 157.799 0.181082
\(16\) −871.753 −0.851321
\(17\) −1347.49 −1.13085 −0.565425 0.824800i \(-0.691288\pi\)
−0.565425 + 0.824800i \(0.691288\pi\)
\(18\) 427.039 0.310661
\(19\) −2886.53 −1.83439 −0.917196 0.398436i \(-0.869553\pi\)
−0.917196 + 0.398436i \(0.869553\pi\)
\(20\) 73.7293 0.0412159
\(21\) −1554.40 −0.769154
\(22\) 1315.55 0.579498
\(23\) 692.702 0.273040 0.136520 0.990637i \(-0.456408\pi\)
0.136520 + 0.990637i \(0.456408\pi\)
\(24\) 1717.89 0.608789
\(25\) −2817.59 −0.901628
\(26\) 1490.66 0.432460
\(27\) −729.000 −0.192450
\(28\) −726.269 −0.175066
\(29\) −2778.00 −0.613391 −0.306695 0.951808i \(-0.599223\pi\)
−0.306695 + 0.951808i \(0.599223\pi\)
\(30\) 831.930 0.168765
\(31\) −4109.10 −0.767967 −0.383983 0.923340i \(-0.625448\pi\)
−0.383983 + 0.923340i \(0.625448\pi\)
\(32\) 1512.09 0.261038
\(33\) −2245.79 −0.358991
\(34\) −7104.11 −1.05393
\(35\) −3028.17 −0.417841
\(36\) −340.615 −0.0438033
\(37\) −10616.6 −1.27492 −0.637459 0.770484i \(-0.720015\pi\)
−0.637459 + 0.770484i \(0.720015\pi\)
\(38\) −15218.0 −1.70962
\(39\) −2544.72 −0.267903
\(40\) 3346.68 0.330723
\(41\) −828.466 −0.0769688 −0.0384844 0.999259i \(-0.512253\pi\)
−0.0384844 + 0.999259i \(0.512253\pi\)
\(42\) −8194.91 −0.716838
\(43\) 8995.66 0.741929 0.370964 0.928647i \(-0.379027\pi\)
0.370964 + 0.928647i \(0.379027\pi\)
\(44\) −1049.31 −0.0817096
\(45\) −1420.19 −0.104548
\(46\) 3651.98 0.254469
\(47\) −4346.77 −0.287027 −0.143513 0.989648i \(-0.545840\pi\)
−0.143513 + 0.989648i \(0.545840\pi\)
\(48\) 7845.78 0.491511
\(49\) 13022.0 0.774795
\(50\) −14854.6 −0.840300
\(51\) 12127.5 0.652896
\(52\) −1188.98 −0.0609772
\(53\) −1799.96 −0.0880185 −0.0440092 0.999031i \(-0.514013\pi\)
−0.0440092 + 0.999031i \(0.514013\pi\)
\(54\) −3843.35 −0.179360
\(55\) −4375.10 −0.195021
\(56\) −32966.4 −1.40476
\(57\) 25978.8 1.05909
\(58\) −14645.9 −0.571669
\(59\) −3481.00 −0.130189
\(60\) −663.564 −0.0237960
\(61\) −28023.9 −0.964283 −0.482142 0.876093i \(-0.660141\pi\)
−0.482142 + 0.876093i \(0.660141\pi\)
\(62\) −21663.5 −0.715731
\(63\) 13989.6 0.444071
\(64\) 35868.0 1.09460
\(65\) −4957.46 −0.145538
\(66\) −11840.0 −0.334573
\(67\) 30957.2 0.842509 0.421254 0.906943i \(-0.361590\pi\)
0.421254 + 0.906943i \(0.361590\pi\)
\(68\) 5666.38 0.148605
\(69\) −6234.32 −0.157640
\(70\) −15964.8 −0.389420
\(71\) 3945.44 0.0928858 0.0464429 0.998921i \(-0.485211\pi\)
0.0464429 + 0.998921i \(0.485211\pi\)
\(72\) −15461.0 −0.351484
\(73\) 17740.6 0.389637 0.194819 0.980839i \(-0.437588\pi\)
0.194819 + 0.980839i \(0.437588\pi\)
\(74\) −55971.8 −1.18820
\(75\) 25358.3 0.520555
\(76\) 12138.2 0.241058
\(77\) 43096.8 0.828359
\(78\) −13416.0 −0.249681
\(79\) −57063.7 −1.02871 −0.514354 0.857578i \(-0.671968\pi\)
−0.514354 + 0.857578i \(0.671968\pi\)
\(80\) 15284.6 0.267012
\(81\) 6561.00 0.111111
\(82\) −4367.74 −0.0717335
\(83\) 17500.9 0.278846 0.139423 0.990233i \(-0.455475\pi\)
0.139423 + 0.990233i \(0.455475\pi\)
\(84\) 6536.42 0.101075
\(85\) 23625.9 0.354684
\(86\) 47425.9 0.691464
\(87\) 25002.0 0.354141
\(88\) −47629.8 −0.655650
\(89\) −53616.7 −0.717505 −0.358753 0.933433i \(-0.616798\pi\)
−0.358753 + 0.933433i \(0.616798\pi\)
\(90\) −7487.37 −0.0974368
\(91\) 48833.3 0.618177
\(92\) −2912.89 −0.0358802
\(93\) 36981.9 0.443386
\(94\) −22916.6 −0.267504
\(95\) 50610.2 0.575346
\(96\) −13608.8 −0.150710
\(97\) −115705. −1.24859 −0.624297 0.781187i \(-0.714615\pi\)
−0.624297 + 0.781187i \(0.714615\pi\)
\(98\) 68653.0 0.722094
\(99\) 20212.1 0.207264
\(100\) 11848.3 0.118483
\(101\) 28439.9 0.277412 0.138706 0.990334i \(-0.455706\pi\)
0.138706 + 0.990334i \(0.455706\pi\)
\(102\) 63936.9 0.608487
\(103\) −25331.5 −0.235271 −0.117636 0.993057i \(-0.537531\pi\)
−0.117636 + 0.993057i \(0.537531\pi\)
\(104\) −53969.6 −0.489290
\(105\) 27253.6 0.241240
\(106\) −9489.56 −0.0820316
\(107\) 10368.8 0.0875529 0.0437764 0.999041i \(-0.486061\pi\)
0.0437764 + 0.999041i \(0.486061\pi\)
\(108\) 3065.53 0.0252899
\(109\) −37934.4 −0.305820 −0.152910 0.988240i \(-0.548865\pi\)
−0.152910 + 0.988240i \(0.548865\pi\)
\(110\) −23065.9 −0.181756
\(111\) 95549.7 0.736075
\(112\) −150561. −1.13414
\(113\) 2314.90 0.0170544 0.00852719 0.999964i \(-0.497286\pi\)
0.00852719 + 0.999964i \(0.497286\pi\)
\(114\) 136962. 0.987050
\(115\) −12145.3 −0.0856374
\(116\) 11681.8 0.0806057
\(117\) 22902.5 0.154674
\(118\) −18352.1 −0.121334
\(119\) −232727. −1.50653
\(120\) −30120.1 −0.190943
\(121\) −98784.8 −0.613376
\(122\) −147745. −0.898694
\(123\) 7456.19 0.0444380
\(124\) 17279.3 0.100919
\(125\) 104193. 0.596434
\(126\) 73754.2 0.413866
\(127\) 123281. 0.678245 0.339122 0.940742i \(-0.389870\pi\)
0.339122 + 0.940742i \(0.389870\pi\)
\(128\) 140712. 0.759113
\(129\) −80961.0 −0.428353
\(130\) −26136.1 −0.135638
\(131\) −114697. −0.583947 −0.291973 0.956426i \(-0.594312\pi\)
−0.291973 + 0.956426i \(0.594312\pi\)
\(132\) 9443.81 0.0471750
\(133\) −498535. −2.44380
\(134\) 163209. 0.785203
\(135\) 12781.7 0.0603608
\(136\) 257205. 1.19243
\(137\) −26527.5 −0.120752 −0.0603760 0.998176i \(-0.519230\pi\)
−0.0603760 + 0.998176i \(0.519230\pi\)
\(138\) −32867.8 −0.146917
\(139\) 178087. 0.781799 0.390900 0.920433i \(-0.372164\pi\)
0.390900 + 0.920433i \(0.372164\pi\)
\(140\) 12733.8 0.0549085
\(141\) 39121.0 0.165715
\(142\) 20800.7 0.0865679
\(143\) 70554.2 0.288525
\(144\) −70612.0 −0.283774
\(145\) 48707.3 0.192386
\(146\) 93529.8 0.363135
\(147\) −117198. −0.447328
\(148\) 44644.2 0.167537
\(149\) 387487. 1.42985 0.714926 0.699200i \(-0.246460\pi\)
0.714926 + 0.699200i \(0.246460\pi\)
\(150\) 133691. 0.485148
\(151\) 428825. 1.53052 0.765258 0.643724i \(-0.222611\pi\)
0.765258 + 0.643724i \(0.222611\pi\)
\(152\) 550971. 1.93428
\(153\) −109147. −0.376950
\(154\) 227210. 0.772015
\(155\) 72045.7 0.240868
\(156\) 10700.8 0.0352052
\(157\) 255345. 0.826759 0.413380 0.910559i \(-0.364348\pi\)
0.413380 + 0.910559i \(0.364348\pi\)
\(158\) −300845. −0.958737
\(159\) 16199.7 0.0508175
\(160\) −26511.8 −0.0818728
\(161\) 119637. 0.363748
\(162\) 34590.1 0.103554
\(163\) −344110. −1.01444 −0.507222 0.861816i \(-0.669328\pi\)
−0.507222 + 0.861816i \(0.669328\pi\)
\(164\) 3483.80 0.0101145
\(165\) 39375.9 0.112595
\(166\) 92266.1 0.259880
\(167\) 489467. 1.35810 0.679051 0.734091i \(-0.262391\pi\)
0.679051 + 0.734091i \(0.262391\pi\)
\(168\) 296698. 0.811037
\(169\) −291347. −0.784683
\(170\) 124558. 0.330559
\(171\) −233809. −0.611464
\(172\) −37827.9 −0.0974969
\(173\) 154041. 0.391310 0.195655 0.980673i \(-0.437317\pi\)
0.195655 + 0.980673i \(0.437317\pi\)
\(174\) 131813. 0.330053
\(175\) −486627. −1.20116
\(176\) −217530. −0.529344
\(177\) 31329.0 0.0751646
\(178\) −282672. −0.668702
\(179\) −309320. −0.721565 −0.360782 0.932650i \(-0.617490\pi\)
−0.360782 + 0.932650i \(0.617490\pi\)
\(180\) 5972.08 0.0137386
\(181\) −120297. −0.272935 −0.136467 0.990645i \(-0.543575\pi\)
−0.136467 + 0.990645i \(0.543575\pi\)
\(182\) 257453. 0.576130
\(183\) 252215. 0.556729
\(184\) −132220. −0.287908
\(185\) 186144. 0.399870
\(186\) 194972. 0.413227
\(187\) −336243. −0.703152
\(188\) 18278.7 0.0377182
\(189\) −125906. −0.256385
\(190\) 266821. 0.536212
\(191\) 868356. 1.72232 0.861161 0.508332i \(-0.169738\pi\)
0.861161 + 0.508332i \(0.169738\pi\)
\(192\) −322812. −0.631970
\(193\) −865143. −1.67184 −0.835920 0.548851i \(-0.815065\pi\)
−0.835920 + 0.548851i \(0.815065\pi\)
\(194\) −610004. −1.16367
\(195\) 44617.1 0.0840262
\(196\) −54759.0 −0.101816
\(197\) 1.03866e6 1.90682 0.953410 0.301677i \(-0.0975464\pi\)
0.953410 + 0.301677i \(0.0975464\pi\)
\(198\) 106560. 0.193166
\(199\) −518573. −0.928277 −0.464138 0.885763i \(-0.653636\pi\)
−0.464138 + 0.885763i \(0.653636\pi\)
\(200\) 537811. 0.950724
\(201\) −278615. −0.486423
\(202\) 149938. 0.258542
\(203\) −479790. −0.817168
\(204\) −50997.4 −0.0857971
\(205\) 14525.7 0.0241408
\(206\) −133550. −0.219268
\(207\) 56108.8 0.0910134
\(208\) −246485. −0.395032
\(209\) −720282. −1.14061
\(210\) 143683. 0.224832
\(211\) 268913. 0.415820 0.207910 0.978148i \(-0.433334\pi\)
0.207910 + 0.978148i \(0.433334\pi\)
\(212\) 7569.06 0.0115665
\(213\) −35508.9 −0.0536276
\(214\) 54665.4 0.0815977
\(215\) −157723. −0.232701
\(216\) 139149. 0.202930
\(217\) −709685. −1.02310
\(218\) −199993. −0.285019
\(219\) −159665. −0.224957
\(220\) 18397.8 0.0256277
\(221\) −380999. −0.524739
\(222\) 503746. 0.686008
\(223\) −1.29991e6 −1.75046 −0.875231 0.483705i \(-0.839291\pi\)
−0.875231 + 0.483705i \(0.839291\pi\)
\(224\) 261154. 0.347758
\(225\) −228224. −0.300543
\(226\) 12204.3 0.0158944
\(227\) 648463. 0.835257 0.417629 0.908618i \(-0.362861\pi\)
0.417629 + 0.908618i \(0.362861\pi\)
\(228\) −109244. −0.139175
\(229\) −1.20958e6 −1.52421 −0.762106 0.647452i \(-0.775835\pi\)
−0.762106 + 0.647452i \(0.775835\pi\)
\(230\) −64031.0 −0.0798125
\(231\) −387871. −0.478253
\(232\) 530255. 0.646792
\(233\) 766201. 0.924598 0.462299 0.886724i \(-0.347025\pi\)
0.462299 + 0.886724i \(0.347025\pi\)
\(234\) 120744. 0.144153
\(235\) 76213.0 0.0900242
\(236\) 14638.0 0.0171081
\(237\) 513573. 0.593925
\(238\) −1.22695e6 −1.40406
\(239\) −764937. −0.866226 −0.433113 0.901340i \(-0.642585\pi\)
−0.433113 + 0.901340i \(0.642585\pi\)
\(240\) −137562. −0.154159
\(241\) 982809. 1.09000 0.545000 0.838436i \(-0.316530\pi\)
0.545000 + 0.838436i \(0.316530\pi\)
\(242\) −520802. −0.571655
\(243\) −59049.0 −0.0641500
\(244\) 117844. 0.126717
\(245\) −228317. −0.243010
\(246\) 39309.7 0.0414154
\(247\) −816156. −0.851199
\(248\) 784330. 0.809785
\(249\) −157508. −0.160992
\(250\) 549313. 0.555865
\(251\) −1.96682e6 −1.97052 −0.985258 0.171077i \(-0.945275\pi\)
−0.985258 + 0.171077i \(0.945275\pi\)
\(252\) −58827.8 −0.0583554
\(253\) 172851. 0.169774
\(254\) 649947. 0.632112
\(255\) −212633. −0.204777
\(256\) −405929. −0.387124
\(257\) 80164.1 0.0757089 0.0378545 0.999283i \(-0.487948\pi\)
0.0378545 + 0.999283i \(0.487948\pi\)
\(258\) −426833. −0.399217
\(259\) −1.83361e6 −1.69847
\(260\) 20846.7 0.0191251
\(261\) −225018. −0.204464
\(262\) −604692. −0.544228
\(263\) −1.67260e6 −1.49108 −0.745542 0.666459i \(-0.767809\pi\)
−0.745542 + 0.666459i \(0.767809\pi\)
\(264\) 428668. 0.378539
\(265\) 31559.1 0.0276065
\(266\) −2.62832e6 −2.27758
\(267\) 482550. 0.414252
\(268\) −130179. −0.110714
\(269\) 218494. 0.184102 0.0920512 0.995754i \(-0.470658\pi\)
0.0920512 + 0.995754i \(0.470658\pi\)
\(270\) 67386.3 0.0562551
\(271\) 1.22825e6 1.01593 0.507966 0.861377i \(-0.330397\pi\)
0.507966 + 0.861377i \(0.330397\pi\)
\(272\) 1.17468e6 0.962716
\(273\) −439500. −0.356905
\(274\) −139855. −0.112539
\(275\) −703078. −0.560624
\(276\) 26216.1 0.0207155
\(277\) 1.66772e6 1.30594 0.652969 0.757384i \(-0.273523\pi\)
0.652969 + 0.757384i \(0.273523\pi\)
\(278\) 938890. 0.728623
\(279\) −332837. −0.255989
\(280\) 578007. 0.440594
\(281\) −2.37484e6 −1.79419 −0.897095 0.441837i \(-0.854327\pi\)
−0.897095 + 0.441837i \(0.854327\pi\)
\(282\) 206249. 0.154443
\(283\) 573803. 0.425889 0.212945 0.977064i \(-0.431695\pi\)
0.212945 + 0.977064i \(0.431695\pi\)
\(284\) −16591.0 −0.0122061
\(285\) −455492. −0.332176
\(286\) 371968. 0.268900
\(287\) −143085. −0.102539
\(288\) 122479. 0.0870125
\(289\) 395884. 0.278820
\(290\) 256789. 0.179300
\(291\) 1.04134e6 0.720876
\(292\) −74601.3 −0.0512023
\(293\) −196310. −0.133590 −0.0667949 0.997767i \(-0.521277\pi\)
−0.0667949 + 0.997767i \(0.521277\pi\)
\(294\) −617877. −0.416901
\(295\) 61033.1 0.0408329
\(296\) 2.02647e6 1.34434
\(297\) −181909. −0.119664
\(298\) 2.04286e6 1.33260
\(299\) 195859. 0.126697
\(300\) −106635. −0.0684061
\(301\) 1.55365e6 0.988408
\(302\) 2.26080e6 1.42641
\(303\) −255959. −0.160164
\(304\) 2.51634e6 1.56166
\(305\) 491350. 0.302441
\(306\) −575433. −0.351310
\(307\) −2.15114e6 −1.30263 −0.651317 0.758806i \(-0.725783\pi\)
−0.651317 + 0.758806i \(0.725783\pi\)
\(308\) −181227. −0.108855
\(309\) 227984. 0.135834
\(310\) 379831. 0.224485
\(311\) 2.39479e6 1.40400 0.701998 0.712179i \(-0.252291\pi\)
0.701998 + 0.712179i \(0.252291\pi\)
\(312\) 485727. 0.282492
\(313\) 1.14808e6 0.662388 0.331194 0.943563i \(-0.392549\pi\)
0.331194 + 0.943563i \(0.392549\pi\)
\(314\) 1.34620e6 0.770524
\(315\) −245282. −0.139280
\(316\) 239960. 0.135183
\(317\) 1.27600e6 0.713188 0.356594 0.934259i \(-0.383938\pi\)
0.356594 + 0.934259i \(0.383938\pi\)
\(318\) 85406.0 0.0473610
\(319\) −693200. −0.381401
\(320\) −628881. −0.343316
\(321\) −93319.5 −0.0505487
\(322\) 630736. 0.339007
\(323\) 3.88958e6 2.07442
\(324\) −27589.8 −0.0146011
\(325\) −796662. −0.418375
\(326\) −1.81418e6 −0.945443
\(327\) 341409. 0.176565
\(328\) 158135. 0.0811600
\(329\) −750734. −0.382381
\(330\) 207593. 0.104937
\(331\) −2.54695e6 −1.27777 −0.638883 0.769304i \(-0.720603\pi\)
−0.638883 + 0.769304i \(0.720603\pi\)
\(332\) −73593.3 −0.0366432
\(333\) −859947. −0.424973
\(334\) 2.58051e6 1.26573
\(335\) −542779. −0.264248
\(336\) 1.35505e6 0.654797
\(337\) −569867. −0.273337 −0.136669 0.990617i \(-0.543640\pi\)
−0.136669 + 0.990617i \(0.543640\pi\)
\(338\) −1.53601e6 −0.731311
\(339\) −20834.1 −0.00984635
\(340\) −99349.9 −0.0466090
\(341\) −1.02535e6 −0.477515
\(342\) −1.23266e6 −0.569873
\(343\) −653714. −0.300022
\(344\) −1.71706e6 −0.782329
\(345\) 109308. 0.0494428
\(346\) 812117. 0.364694
\(347\) 49490.3 0.0220646 0.0110323 0.999939i \(-0.496488\pi\)
0.0110323 + 0.999939i \(0.496488\pi\)
\(348\) −105136. −0.0465377
\(349\) 291186. 0.127970 0.0639849 0.997951i \(-0.479619\pi\)
0.0639849 + 0.997951i \(0.479619\pi\)
\(350\) −2.56554e6 −1.11946
\(351\) −206122. −0.0893011
\(352\) 377315. 0.162311
\(353\) 2.92278e6 1.24842 0.624208 0.781258i \(-0.285422\pi\)
0.624208 + 0.781258i \(0.285422\pi\)
\(354\) 165169. 0.0700520
\(355\) −69176.2 −0.0291331
\(356\) 225465. 0.0942874
\(357\) 2.09454e6 0.869797
\(358\) −1.63076e6 −0.672485
\(359\) −3.89236e6 −1.59396 −0.796978 0.604008i \(-0.793570\pi\)
−0.796978 + 0.604008i \(0.793570\pi\)
\(360\) 271081. 0.110241
\(361\) 5.85596e6 2.36500
\(362\) −634217. −0.254370
\(363\) 889064. 0.354133
\(364\) −205350. −0.0812346
\(365\) −311049. −0.122207
\(366\) 1.32970e6 0.518861
\(367\) 2.27009e6 0.879787 0.439893 0.898050i \(-0.355016\pi\)
0.439893 + 0.898050i \(0.355016\pi\)
\(368\) −603865. −0.232445
\(369\) −67105.7 −0.0256563
\(370\) 981366. 0.372672
\(371\) −310873. −0.117259
\(372\) −155513. −0.0582653
\(373\) 2.65018e6 0.986287 0.493143 0.869948i \(-0.335848\pi\)
0.493143 + 0.869948i \(0.335848\pi\)
\(374\) −1.77270e6 −0.655325
\(375\) −937734. −0.344351
\(376\) 829697. 0.302656
\(377\) −785470. −0.284627
\(378\) −663788. −0.238946
\(379\) −870790. −0.311398 −0.155699 0.987805i \(-0.549763\pi\)
−0.155699 + 0.987805i \(0.549763\pi\)
\(380\) −212822. −0.0756062
\(381\) −1.10953e6 −0.391585
\(382\) 4.57805e6 1.60517
\(383\) 2.50857e6 0.873834 0.436917 0.899502i \(-0.356070\pi\)
0.436917 + 0.899502i \(0.356070\pi\)
\(384\) −1.26641e6 −0.438274
\(385\) −755626. −0.259810
\(386\) −4.56111e6 −1.55812
\(387\) 728649. 0.247310
\(388\) 486552. 0.164078
\(389\) 2.35346e6 0.788555 0.394278 0.918991i \(-0.370995\pi\)
0.394278 + 0.918991i \(0.370995\pi\)
\(390\) 235225. 0.0783109
\(391\) −933412. −0.308767
\(392\) −2.48559e6 −0.816985
\(393\) 1.03227e6 0.337142
\(394\) 5.47593e6 1.77712
\(395\) 1.00051e6 0.322648
\(396\) −84994.3 −0.0272365
\(397\) 3.69134e6 1.17546 0.587730 0.809057i \(-0.300022\pi\)
0.587730 + 0.809057i \(0.300022\pi\)
\(398\) −2.73396e6 −0.865137
\(399\) 4.48681e6 1.41093
\(400\) 2.45624e6 0.767575
\(401\) 4.26272e6 1.32381 0.661905 0.749588i \(-0.269748\pi\)
0.661905 + 0.749588i \(0.269748\pi\)
\(402\) −1.46888e6 −0.453337
\(403\) −1.16183e6 −0.356353
\(404\) −119593. −0.0364547
\(405\) −115035. −0.0348493
\(406\) −2.52949e6 −0.761586
\(407\) −2.64919e6 −0.792733
\(408\) −2.31484e6 −0.688449
\(409\) 2.90876e6 0.859806 0.429903 0.902875i \(-0.358548\pi\)
0.429903 + 0.902875i \(0.358548\pi\)
\(410\) 76580.6 0.0224988
\(411\) 238747. 0.0697162
\(412\) 106522. 0.0309170
\(413\) −601206. −0.173440
\(414\) 295811. 0.0848228
\(415\) −306847. −0.0874584
\(416\) 427538. 0.121127
\(417\) −1.60278e6 −0.451372
\(418\) −3.79739e6 −1.06303
\(419\) −2.77871e6 −0.773230 −0.386615 0.922241i \(-0.626356\pi\)
−0.386615 + 0.922241i \(0.626356\pi\)
\(420\) −114605. −0.0317014
\(421\) −2.25028e6 −0.618774 −0.309387 0.950936i \(-0.600124\pi\)
−0.309387 + 0.950936i \(0.600124\pi\)
\(422\) 1.41773e6 0.387536
\(423\) −352089. −0.0956756
\(424\) 343571. 0.0928114
\(425\) 3.79668e6 1.01960
\(426\) −187206. −0.0499800
\(427\) −4.84003e6 −1.28463
\(428\) −43602.2 −0.0115053
\(429\) −634988. −0.166580
\(430\) −831529. −0.216873
\(431\) 1.51833e6 0.393708 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(432\) 635508. 0.163837
\(433\) −1.99352e6 −0.510975 −0.255488 0.966812i \(-0.582236\pi\)
−0.255488 + 0.966812i \(0.582236\pi\)
\(434\) −3.74152e6 −0.953507
\(435\) −438366. −0.111074
\(436\) 159519. 0.0401879
\(437\) −1.99951e6 −0.500863
\(438\) −841768. −0.209656
\(439\) −5.57781e6 −1.38134 −0.690672 0.723168i \(-0.742685\pi\)
−0.690672 + 0.723168i \(0.742685\pi\)
\(440\) 835103. 0.205640
\(441\) 1.05478e6 0.258265
\(442\) −2.00866e6 −0.489047
\(443\) 6.89006e6 1.66807 0.834034 0.551713i \(-0.186026\pi\)
0.834034 + 0.551713i \(0.186026\pi\)
\(444\) −401798. −0.0967276
\(445\) 940074. 0.225041
\(446\) −6.85326e6 −1.63140
\(447\) −3.48738e6 −0.825525
\(448\) 6.19478e6 1.45825
\(449\) −6.32514e6 −1.48066 −0.740328 0.672246i \(-0.765330\pi\)
−0.740328 + 0.672246i \(0.765330\pi\)
\(450\) −1.20322e6 −0.280100
\(451\) −206729. −0.0478585
\(452\) −9734.43 −0.00224112
\(453\) −3.85943e6 −0.883644
\(454\) 3.41875e6 0.778445
\(455\) −856205. −0.193887
\(456\) −4.95874e6 −1.11676
\(457\) −3.82920e6 −0.857664 −0.428832 0.903384i \(-0.641075\pi\)
−0.428832 + 0.903384i \(0.641075\pi\)
\(458\) −6.37700e6 −1.42054
\(459\) 982323. 0.217632
\(460\) 51072.4 0.0112536
\(461\) 1.78400e6 0.390969 0.195484 0.980707i \(-0.437372\pi\)
0.195484 + 0.980707i \(0.437372\pi\)
\(462\) −2.04489e6 −0.445723
\(463\) −5.66499e6 −1.22814 −0.614068 0.789253i \(-0.710468\pi\)
−0.614068 + 0.789253i \(0.710468\pi\)
\(464\) 2.42173e6 0.522193
\(465\) −648412. −0.139065
\(466\) 4.03948e6 0.861709
\(467\) −5.63789e6 −1.19626 −0.598128 0.801400i \(-0.704089\pi\)
−0.598128 + 0.801400i \(0.704089\pi\)
\(468\) −96307.6 −0.0203257
\(469\) 5.34663e6 1.12240
\(470\) 401801. 0.0839009
\(471\) −2.29811e6 −0.477330
\(472\) 664441. 0.137278
\(473\) 2.24470e6 0.461324
\(474\) 2.70760e6 0.553527
\(475\) 8.13305e6 1.65394
\(476\) 978644. 0.197974
\(477\) −145797. −0.0293395
\(478\) −4.03281e6 −0.807306
\(479\) −548448. −0.109219 −0.0546094 0.998508i \(-0.517391\pi\)
−0.0546094 + 0.998508i \(0.517391\pi\)
\(480\) 238606. 0.0472693
\(481\) −3.00182e6 −0.591591
\(482\) 5.18145e6 1.01586
\(483\) −1.07673e6 −0.210010
\(484\) 415402. 0.0806038
\(485\) 2.02867e6 0.391614
\(486\) −311311. −0.0597867
\(487\) 3.97227e6 0.758956 0.379478 0.925201i \(-0.376104\pi\)
0.379478 + 0.925201i \(0.376104\pi\)
\(488\) 5.34911e6 1.01679
\(489\) 3.09699e6 0.585689
\(490\) −1.20371e6 −0.226480
\(491\) 1.01781e7 1.90530 0.952648 0.304077i \(-0.0983480\pi\)
0.952648 + 0.304077i \(0.0983480\pi\)
\(492\) −31354.2 −0.00583960
\(493\) 3.74334e6 0.693652
\(494\) −4.30285e6 −0.793302
\(495\) −354383. −0.0650069
\(496\) 3.58212e6 0.653786
\(497\) 681419. 0.123744
\(498\) −830395. −0.150042
\(499\) 1.92114e6 0.345389 0.172694 0.984975i \(-0.444753\pi\)
0.172694 + 0.984975i \(0.444753\pi\)
\(500\) −438143. −0.0783774
\(501\) −4.40521e6 −0.784101
\(502\) −1.03692e7 −1.83648
\(503\) −7.27930e6 −1.28283 −0.641416 0.767193i \(-0.721653\pi\)
−0.641416 + 0.767193i \(0.721653\pi\)
\(504\) −2.67028e6 −0.468253
\(505\) −498643. −0.0870084
\(506\) 911286. 0.158226
\(507\) 2.62213e6 0.453037
\(508\) −518411. −0.0891281
\(509\) 4.24954e6 0.727021 0.363511 0.931590i \(-0.381578\pi\)
0.363511 + 0.931590i \(0.381578\pi\)
\(510\) −1.12102e6 −0.190848
\(511\) 3.06399e6 0.519080
\(512\) −6.64288e6 −1.11991
\(513\) 2.10428e6 0.353029
\(514\) 422632. 0.0705593
\(515\) 444144. 0.0737913
\(516\) 340451. 0.0562898
\(517\) −1.08466e6 −0.178471
\(518\) −9.66693e6 −1.58294
\(519\) −1.38637e6 −0.225923
\(520\) 946262. 0.153463
\(521\) 4.44795e6 0.717903 0.358951 0.933356i \(-0.383134\pi\)
0.358951 + 0.933356i \(0.383134\pi\)
\(522\) −1.18631e6 −0.190556
\(523\) −8.10718e6 −1.29603 −0.648016 0.761627i \(-0.724401\pi\)
−0.648016 + 0.761627i \(0.724401\pi\)
\(524\) 482314. 0.0767365
\(525\) 4.37964e6 0.693491
\(526\) −8.81808e6 −1.38966
\(527\) 5.53699e6 0.868454
\(528\) 1.95777e6 0.305617
\(529\) −5.95651e6 −0.925449
\(530\) 166383. 0.0257287
\(531\) −281961. −0.0433963
\(532\) 2.09640e6 0.321140
\(533\) −234246. −0.0357152
\(534\) 2.54405e6 0.386075
\(535\) −181799. −0.0274604
\(536\) −5.90900e6 −0.888386
\(537\) 2.78388e6 0.416596
\(538\) 1.15192e6 0.171580
\(539\) 3.24940e6 0.481760
\(540\) −53748.7 −0.00793201
\(541\) −2.00354e6 −0.294310 −0.147155 0.989113i \(-0.547012\pi\)
−0.147155 + 0.989113i \(0.547012\pi\)
\(542\) 6.47546e6 0.946830
\(543\) 1.08268e6 0.157579
\(544\) −2.03753e6 −0.295194
\(545\) 665111. 0.0959187
\(546\) −2.31708e6 −0.332629
\(547\) −1.34714e6 −0.192506 −0.0962529 0.995357i \(-0.530686\pi\)
−0.0962529 + 0.995357i \(0.530686\pi\)
\(548\) 111551. 0.0158680
\(549\) −2.26994e6 −0.321428
\(550\) −3.70668e6 −0.522491
\(551\) 8.01878e6 1.12520
\(552\) 1.18998e6 0.166224
\(553\) −9.85551e6 −1.37046
\(554\) 8.79234e6 1.21711
\(555\) −1.67529e6 −0.230865
\(556\) −748878. −0.102736
\(557\) 3.66674e6 0.500774 0.250387 0.968146i \(-0.419442\pi\)
0.250387 + 0.968146i \(0.419442\pi\)
\(558\) −1.75474e6 −0.238577
\(559\) 2.54349e6 0.344271
\(560\) 2.63982e6 0.355717
\(561\) 3.02619e6 0.405965
\(562\) −1.25204e7 −1.67215
\(563\) −5.44490e6 −0.723967 −0.361983 0.932185i \(-0.617900\pi\)
−0.361983 + 0.932185i \(0.617900\pi\)
\(564\) −164508. −0.0217766
\(565\) −40587.6 −0.00534900
\(566\) 3.02514e6 0.396921
\(567\) 1.13315e6 0.148024
\(568\) −753091. −0.0979438
\(569\) −1.26085e7 −1.63261 −0.816304 0.577623i \(-0.803981\pi\)
−0.816304 + 0.577623i \(0.803981\pi\)
\(570\) −2.40139e6 −0.309582
\(571\) 4.88397e6 0.626878 0.313439 0.949608i \(-0.398519\pi\)
0.313439 + 0.949608i \(0.398519\pi\)
\(572\) −296689. −0.0379150
\(573\) −7.81521e6 −0.994383
\(574\) −754355. −0.0955644
\(575\) −1.95175e6 −0.246181
\(576\) 2.90531e6 0.364868
\(577\) 1.67375e6 0.209291 0.104646 0.994510i \(-0.466629\pi\)
0.104646 + 0.994510i \(0.466629\pi\)
\(578\) 2.08714e6 0.259855
\(579\) 7.78629e6 0.965237
\(580\) −204820. −0.0252815
\(581\) 3.02259e6 0.371483
\(582\) 5.49004e6 0.671843
\(583\) −449148. −0.0547291
\(584\) −3.38626e6 −0.410854
\(585\) −401554. −0.0485126
\(586\) −1.03496e6 −0.124503
\(587\) 1.02763e7 1.23095 0.615477 0.788155i \(-0.288963\pi\)
0.615477 + 0.788155i \(0.288963\pi\)
\(588\) 492831. 0.0587834
\(589\) 1.18610e7 1.40875
\(590\) 321772. 0.0380556
\(591\) −9.34798e6 −1.10090
\(592\) 9.25508e6 1.08537
\(593\) −3.79932e6 −0.443679 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(594\) −959038. −0.111524
\(595\) 4.08045e6 0.472515
\(596\) −1.62943e6 −0.187897
\(597\) 4.66716e6 0.535941
\(598\) 1.03258e6 0.118079
\(599\) 8.22274e6 0.936375 0.468187 0.883629i \(-0.344907\pi\)
0.468187 + 0.883629i \(0.344907\pi\)
\(600\) −4.84030e6 −0.548901
\(601\) −1.05611e7 −1.19268 −0.596339 0.802733i \(-0.703378\pi\)
−0.596339 + 0.802733i \(0.703378\pi\)
\(602\) 8.19096e6 0.921178
\(603\) 2.50753e6 0.280836
\(604\) −1.80326e6 −0.201125
\(605\) 1.73202e6 0.192382
\(606\) −1.34944e6 −0.149270
\(607\) 4.34859e6 0.479045 0.239523 0.970891i \(-0.423009\pi\)
0.239523 + 0.970891i \(0.423009\pi\)
\(608\) −4.36470e6 −0.478845
\(609\) 4.31811e6 0.471792
\(610\) 2.59044e6 0.281870
\(611\) −1.22903e6 −0.133187
\(612\) 458977. 0.0495350
\(613\) 1.02071e7 1.09712 0.548558 0.836113i \(-0.315177\pi\)
0.548558 + 0.836113i \(0.315177\pi\)
\(614\) −1.13410e7 −1.21403
\(615\) −130731. −0.0139377
\(616\) −8.22617e6 −0.873466
\(617\) 1.49495e7 1.58093 0.790465 0.612507i \(-0.209839\pi\)
0.790465 + 0.612507i \(0.209839\pi\)
\(618\) 1.20195e6 0.126595
\(619\) −1.32070e7 −1.38541 −0.692703 0.721223i \(-0.743580\pi\)
−0.692703 + 0.721223i \(0.743580\pi\)
\(620\) −302961. −0.0316525
\(621\) −504980. −0.0525466
\(622\) 1.26255e7 1.30850
\(623\) −9.26018e6 −0.955871
\(624\) 2.21837e6 0.228072
\(625\) 6.97812e6 0.714560
\(626\) 6.05279e6 0.617333
\(627\) 6.48253e6 0.658531
\(628\) −1.07376e6 −0.108644
\(629\) 1.43059e7 1.44174
\(630\) −1.29315e6 −0.129807
\(631\) 1.29466e6 0.129444 0.0647220 0.997903i \(-0.479384\pi\)
0.0647220 + 0.997903i \(0.479384\pi\)
\(632\) 1.08921e7 1.08472
\(633\) −2.42021e6 −0.240074
\(634\) 6.72721e6 0.664678
\(635\) −2.16151e6 −0.212727
\(636\) −68121.6 −0.00667793
\(637\) 3.68192e6 0.359522
\(638\) −3.65461e6 −0.355459
\(639\) 319581. 0.0309619
\(640\) −2.46714e6 −0.238091
\(641\) −1.26928e7 −1.22015 −0.610076 0.792343i \(-0.708861\pi\)
−0.610076 + 0.792343i \(0.708861\pi\)
\(642\) −491988. −0.0471104
\(643\) −1.44334e7 −1.37671 −0.688353 0.725375i \(-0.741666\pi\)
−0.688353 + 0.725375i \(0.741666\pi\)
\(644\) −503088. −0.0478002
\(645\) 1.41951e6 0.134350
\(646\) 2.05062e7 1.93332
\(647\) −1.82857e7 −1.71731 −0.858657 0.512551i \(-0.828701\pi\)
−0.858657 + 0.512551i \(0.828701\pi\)
\(648\) −1.25234e6 −0.117161
\(649\) −868620. −0.0809503
\(650\) −4.20007e6 −0.389918
\(651\) 6.38717e6 0.590685
\(652\) 1.44702e6 0.133308
\(653\) 3.51962e6 0.323008 0.161504 0.986872i \(-0.448366\pi\)
0.161504 + 0.986872i \(0.448366\pi\)
\(654\) 1.79994e6 0.164556
\(655\) 2.01101e6 0.183151
\(656\) 722217. 0.0655252
\(657\) 1.43699e6 0.129879
\(658\) −3.95794e6 −0.356372
\(659\) 1.63394e7 1.46562 0.732811 0.680432i \(-0.238208\pi\)
0.732811 + 0.680432i \(0.238208\pi\)
\(660\) −165580. −0.0147962
\(661\) −1.96074e7 −1.74549 −0.872743 0.488179i \(-0.837661\pi\)
−0.872743 + 0.488179i \(0.837661\pi\)
\(662\) −1.34278e7 −1.19085
\(663\) 3.42899e6 0.302958
\(664\) −3.34051e6 −0.294030
\(665\) 8.74092e6 0.766484
\(666\) −4.53372e6 −0.396067
\(667\) −1.92433e6 −0.167480
\(668\) −2.05827e6 −0.178468
\(669\) 1.16992e7 1.01063
\(670\) −2.86158e6 −0.246274
\(671\) −6.99287e6 −0.599583
\(672\) −2.35039e6 −0.200778
\(673\) 1.03079e7 0.877268 0.438634 0.898666i \(-0.355462\pi\)
0.438634 + 0.898666i \(0.355462\pi\)
\(674\) −3.00439e6 −0.254745
\(675\) 2.05402e6 0.173518
\(676\) 1.22515e6 0.103115
\(677\) −5.71024e6 −0.478832 −0.239416 0.970917i \(-0.576956\pi\)
−0.239416 + 0.970917i \(0.576956\pi\)
\(678\) −109839. −0.00917662
\(679\) −1.99834e7 −1.66339
\(680\) −4.50963e6 −0.373998
\(681\) −5.83616e6 −0.482236
\(682\) −5.40574e6 −0.445035
\(683\) 1.66857e6 0.136865 0.0684325 0.997656i \(-0.478200\pi\)
0.0684325 + 0.997656i \(0.478200\pi\)
\(684\) 983195. 0.0803525
\(685\) 465112. 0.0378731
\(686\) −3.44644e6 −0.279615
\(687\) 1.08862e7 0.880004
\(688\) −7.84200e6 −0.631620
\(689\) −508933. −0.0408425
\(690\) 576279. 0.0460798
\(691\) −8.48113e6 −0.675707 −0.337854 0.941199i \(-0.609701\pi\)
−0.337854 + 0.941199i \(0.609701\pi\)
\(692\) −647761. −0.0514221
\(693\) 3.49084e6 0.276120
\(694\) 260917. 0.0205638
\(695\) −3.12244e6 −0.245207
\(696\) −4.77229e6 −0.373426
\(697\) 1.11635e6 0.0870401
\(698\) 1.53516e6 0.119265
\(699\) −6.89581e6 −0.533817
\(700\) 2.04633e6 0.157845
\(701\) 2.15565e7 1.65685 0.828426 0.560099i \(-0.189237\pi\)
0.828426 + 0.560099i \(0.189237\pi\)
\(702\) −1.08669e6 −0.0832270
\(703\) 3.06452e7 2.33870
\(704\) 8.95020e6 0.680614
\(705\) −685917. −0.0519755
\(706\) 1.54092e7 1.16350
\(707\) 4.91187e6 0.369572
\(708\) −131742. −0.00987738
\(709\) −1.62242e7 −1.21213 −0.606063 0.795417i \(-0.707252\pi\)
−0.606063 + 0.795417i \(0.707252\pi\)
\(710\) −364703. −0.0271515
\(711\) −4.62216e6 −0.342903
\(712\) 1.02342e7 0.756576
\(713\) −2.84638e6 −0.209686
\(714\) 1.10426e7 0.810635
\(715\) −1.23704e6 −0.0904940
\(716\) 1.30073e6 0.0948208
\(717\) 6.88443e6 0.500116
\(718\) −2.05208e7 −1.48554
\(719\) −6.26793e6 −0.452170 −0.226085 0.974108i \(-0.572593\pi\)
−0.226085 + 0.974108i \(0.572593\pi\)
\(720\) 1.23806e6 0.0890039
\(721\) −4.37503e6 −0.313432
\(722\) 3.08731e7 2.20413
\(723\) −8.84528e6 −0.629312
\(724\) 505864. 0.0358664
\(725\) 7.82725e6 0.553050
\(726\) 4.68722e6 0.330045
\(727\) −1.51303e7 −1.06172 −0.530862 0.847458i \(-0.678132\pi\)
−0.530862 + 0.847458i \(0.678132\pi\)
\(728\) −9.32113e6 −0.651839
\(729\) 531441. 0.0370370
\(730\) −1.63988e6 −0.113895
\(731\) −1.21216e7 −0.839009
\(732\) −1.06060e6 −0.0731598
\(733\) −1.09692e7 −0.754073 −0.377037 0.926198i \(-0.623057\pi\)
−0.377037 + 0.926198i \(0.623057\pi\)
\(734\) 1.19681e7 0.819945
\(735\) 2.05485e6 0.140302
\(736\) 1.04743e6 0.0712738
\(737\) 7.72480e6 0.523864
\(738\) −353787. −0.0239112
\(739\) 8.31352e6 0.559982 0.279991 0.960003i \(-0.409669\pi\)
0.279991 + 0.960003i \(0.409669\pi\)
\(740\) −782757. −0.0525470
\(741\) 7.34541e6 0.491440
\(742\) −1.63895e6 −0.109284
\(743\) −2.57946e7 −1.71418 −0.857091 0.515165i \(-0.827731\pi\)
−0.857091 + 0.515165i \(0.827731\pi\)
\(744\) −7.05897e6 −0.467530
\(745\) −6.79389e6 −0.448464
\(746\) 1.39720e7 0.919201
\(747\) 1.41757e6 0.0929487
\(748\) 1.41394e6 0.0924012
\(749\) 1.79081e6 0.116639
\(750\) −4.94381e6 −0.320929
\(751\) 2.42931e7 1.57175 0.785874 0.618387i \(-0.212214\pi\)
0.785874 + 0.618387i \(0.212214\pi\)
\(752\) 3.78931e6 0.244352
\(753\) 1.77014e7 1.13768
\(754\) −4.14106e6 −0.265267
\(755\) −7.51869e6 −0.480037
\(756\) 529450. 0.0336915
\(757\) 2.12215e7 1.34597 0.672987 0.739654i \(-0.265011\pi\)
0.672987 + 0.739654i \(0.265011\pi\)
\(758\) −4.59088e6 −0.290217
\(759\) −1.55566e6 −0.0980190
\(760\) −9.66029e6 −0.606675
\(761\) −2.57870e7 −1.61414 −0.807068 0.590459i \(-0.798947\pi\)
−0.807068 + 0.590459i \(0.798947\pi\)
\(762\) −5.84952e6 −0.364950
\(763\) −6.55167e6 −0.407418
\(764\) −3.65154e6 −0.226330
\(765\) 1.91370e6 0.118228
\(766\) 1.32254e7 0.814397
\(767\) −984240. −0.0604105
\(768\) 3.65336e6 0.223506
\(769\) −2.71632e7 −1.65640 −0.828200 0.560433i \(-0.810635\pi\)
−0.828200 + 0.560433i \(0.810635\pi\)
\(770\) −3.98372e6 −0.242138
\(771\) −721477. −0.0437106
\(772\) 3.63803e6 0.219697
\(773\) 6.81267e6 0.410080 0.205040 0.978754i \(-0.434268\pi\)
0.205040 + 0.978754i \(0.434268\pi\)
\(774\) 3.84150e6 0.230488
\(775\) 1.15777e7 0.692420
\(776\) 2.20853e7 1.31658
\(777\) 1.65025e7 0.980609
\(778\) 1.24076e7 0.734919
\(779\) 2.39139e6 0.141191
\(780\) −187620. −0.0110419
\(781\) 984513. 0.0577555
\(782\) −4.92103e6 −0.287766
\(783\) 2.02516e6 0.118047
\(784\) −1.13519e7 −0.659599
\(785\) −4.47703e6 −0.259308
\(786\) 5.44222e6 0.314210
\(787\) −1.38930e7 −0.799575 −0.399788 0.916608i \(-0.630916\pi\)
−0.399788 + 0.916608i \(0.630916\pi\)
\(788\) −4.36771e6 −0.250575
\(789\) 1.50534e7 0.860878
\(790\) 5.27477e6 0.300702
\(791\) 399808. 0.0227201
\(792\) −3.85801e6 −0.218550
\(793\) −7.92367e6 −0.447449
\(794\) 1.94611e7 1.09551
\(795\) −284032. −0.0159386
\(796\) 2.18066e6 0.121985
\(797\) 5.90905e6 0.329512 0.164756 0.986334i \(-0.447316\pi\)
0.164756 + 0.986334i \(0.447316\pi\)
\(798\) 2.36549e7 1.31496
\(799\) 5.85726e6 0.324584
\(800\) −4.26045e6 −0.235359
\(801\) −4.34295e6 −0.239168
\(802\) 2.24734e7 1.23377
\(803\) 4.42684e6 0.242273
\(804\) 1.17161e6 0.0639208
\(805\) −2.09762e6 −0.114087
\(806\) −6.12528e6 −0.332115
\(807\) −1.96645e6 −0.106292
\(808\) −5.42851e6 −0.292518
\(809\) −1.34744e7 −0.723832 −0.361916 0.932211i \(-0.617877\pi\)
−0.361916 + 0.932211i \(0.617877\pi\)
\(810\) −606477. −0.0324789
\(811\) 1.21144e6 0.0646770 0.0323385 0.999477i \(-0.489705\pi\)
0.0323385 + 0.999477i \(0.489705\pi\)
\(812\) 2.01758e6 0.107384
\(813\) −1.10543e7 −0.586549
\(814\) −1.39667e7 −0.738813
\(815\) 6.03335e6 0.318174
\(816\) −1.05721e7 −0.555824
\(817\) −2.59663e7 −1.36099
\(818\) 1.53353e7 0.801323
\(819\) 3.95550e6 0.206059
\(820\) −61082.2 −0.00317234
\(821\) 1.88231e7 0.974615 0.487308 0.873230i \(-0.337979\pi\)
0.487308 + 0.873230i \(0.337979\pi\)
\(822\) 1.25870e6 0.0649743
\(823\) −2.94279e7 −1.51446 −0.757232 0.653146i \(-0.773449\pi\)
−0.757232 + 0.653146i \(0.773449\pi\)
\(824\) 4.83520e6 0.248082
\(825\) 6.32770e6 0.323676
\(826\) −3.16961e6 −0.161642
\(827\) −2.79462e7 −1.42089 −0.710443 0.703755i \(-0.751505\pi\)
−0.710443 + 0.703755i \(0.751505\pi\)
\(828\) −235944. −0.0119601
\(829\) 2.14227e6 0.108265 0.0541324 0.998534i \(-0.482761\pi\)
0.0541324 + 0.998534i \(0.482761\pi\)
\(830\) −1.61772e6 −0.0815096
\(831\) −1.50095e7 −0.753984
\(832\) 1.01415e7 0.507920
\(833\) −1.75470e7 −0.876176
\(834\) −8.45001e6 −0.420671
\(835\) −8.58194e6 −0.425960
\(836\) 3.02887e6 0.149887
\(837\) 2.99553e6 0.147795
\(838\) −1.46496e7 −0.720636
\(839\) −3.02878e7 −1.48547 −0.742733 0.669588i \(-0.766471\pi\)
−0.742733 + 0.669588i \(0.766471\pi\)
\(840\) −5.20207e6 −0.254377
\(841\) −1.27939e7 −0.623752
\(842\) −1.18637e7 −0.576686
\(843\) 2.13736e7 1.03588
\(844\) −1.13081e6 −0.0546429
\(845\) 5.10826e6 0.246111
\(846\) −1.85624e6 −0.0891679
\(847\) −1.70612e7 −0.817148
\(848\) 1.56912e6 0.0749320
\(849\) −5.16422e6 −0.245887
\(850\) 2.00164e7 0.950253
\(851\) −7.35416e6 −0.348104
\(852\) 149319. 0.00704721
\(853\) −2.43537e7 −1.14602 −0.573011 0.819548i \(-0.694225\pi\)
−0.573011 + 0.819548i \(0.694225\pi\)
\(854\) −2.55171e7 −1.19725
\(855\) 4.09943e6 0.191782
\(856\) −1.97917e6 −0.0923204
\(857\) 8.50412e6 0.395528 0.197764 0.980250i \(-0.436632\pi\)
0.197764 + 0.980250i \(0.436632\pi\)
\(858\) −3.34771e6 −0.155249
\(859\) 2.11370e7 0.977372 0.488686 0.872460i \(-0.337476\pi\)
0.488686 + 0.872460i \(0.337476\pi\)
\(860\) 663244. 0.0305793
\(861\) 1.28776e6 0.0592009
\(862\) 8.00478e6 0.366928
\(863\) −2.26285e7 −1.03426 −0.517128 0.855908i \(-0.672999\pi\)
−0.517128 + 0.855908i \(0.672999\pi\)
\(864\) −1.10231e6 −0.0502367
\(865\) −2.70084e6 −0.122732
\(866\) −1.05100e7 −0.476220
\(867\) −3.56296e6 −0.160977
\(868\) 2.98431e6 0.134445
\(869\) −1.42392e7 −0.639641
\(870\) −2.31110e6 −0.103519
\(871\) 8.75303e6 0.390943
\(872\) 7.24077e6 0.322473
\(873\) −9.37207e6 −0.416198
\(874\) −1.05416e7 −0.466795
\(875\) 1.79952e7 0.794577
\(876\) 671411. 0.0295616
\(877\) 1.62481e7 0.713351 0.356675 0.934228i \(-0.383910\pi\)
0.356675 + 0.934228i \(0.383910\pi\)
\(878\) −2.94067e7 −1.28739
\(879\) 1.76679e6 0.0771281
\(880\) 3.81400e6 0.166025
\(881\) 1.94120e7 0.842617 0.421308 0.906917i \(-0.361571\pi\)
0.421308 + 0.906917i \(0.361571\pi\)
\(882\) 5.56089e6 0.240698
\(883\) −2.20769e7 −0.952875 −0.476437 0.879208i \(-0.658072\pi\)
−0.476437 + 0.879208i \(0.658072\pi\)
\(884\) 1.60215e6 0.0689560
\(885\) −549298. −0.0235749
\(886\) 3.63250e7 1.55461
\(887\) −1.75467e7 −0.748834 −0.374417 0.927260i \(-0.622157\pi\)
−0.374417 + 0.927260i \(0.622157\pi\)
\(888\) −1.82382e7 −0.776157
\(889\) 2.12919e7 0.903567
\(890\) 4.95615e6 0.209734
\(891\) 1.63718e6 0.0690879
\(892\) 5.46630e6 0.230028
\(893\) 1.25471e7 0.526520
\(894\) −1.83858e7 −0.769374
\(895\) 5.42337e6 0.226314
\(896\) 2.43025e7 1.01130
\(897\) −1.76273e6 −0.0731484
\(898\) −3.33467e7 −1.37994
\(899\) 1.14151e7 0.471064
\(900\) 959712. 0.0394943
\(901\) 2.42544e6 0.0995356
\(902\) −1.08989e6 −0.0446033
\(903\) −1.39828e7 −0.570657
\(904\) −441860. −0.0179831
\(905\) 2.10920e6 0.0856044
\(906\) −2.03472e7 −0.823540
\(907\) −2.33262e7 −0.941514 −0.470757 0.882263i \(-0.656019\pi\)
−0.470757 + 0.882263i \(0.656019\pi\)
\(908\) −2.72686e6 −0.109761
\(909\) 2.30363e6 0.0924705
\(910\) −4.51399e6 −0.180699
\(911\) −1.17853e7 −0.470483 −0.235242 0.971937i \(-0.575588\pi\)
−0.235242 + 0.971937i \(0.575588\pi\)
\(912\) −2.26471e7 −0.901623
\(913\) 4.36703e6 0.173384
\(914\) −2.01879e7 −0.799327
\(915\) −4.42215e6 −0.174615
\(916\) 5.08643e6 0.200297
\(917\) −1.98094e7 −0.777942
\(918\) 5.17889e6 0.202829
\(919\) 1.39182e7 0.543619 0.271810 0.962351i \(-0.412378\pi\)
0.271810 + 0.962351i \(0.412378\pi\)
\(920\) 2.31825e6 0.0903006
\(921\) 1.93602e7 0.752076
\(922\) 9.40539e6 0.364376
\(923\) 1.11556e6 0.0431011
\(924\) 1.63105e6 0.0628473
\(925\) 2.99133e7 1.14950
\(926\) −2.98663e7 −1.14460
\(927\) −2.05186e6 −0.0784237
\(928\) −4.20059e6 −0.160118
\(929\) −3.32567e7 −1.26427 −0.632136 0.774857i \(-0.717822\pi\)
−0.632136 + 0.774857i \(0.717822\pi\)
\(930\) −3.41848e6 −0.129606
\(931\) −3.75883e7 −1.42128
\(932\) −3.22197e6 −0.121501
\(933\) −2.15531e7 −0.810598
\(934\) −2.97234e7 −1.11489
\(935\) 5.89542e6 0.220539
\(936\) −4.37154e6 −0.163097
\(937\) −1.26083e7 −0.469144 −0.234572 0.972099i \(-0.575369\pi\)
−0.234572 + 0.972099i \(0.575369\pi\)
\(938\) 2.81879e7 1.04606
\(939\) −1.03327e7 −0.382430
\(940\) −320485. −0.0118301
\(941\) 2.76092e6 0.101644 0.0508218 0.998708i \(-0.483816\pi\)
0.0508218 + 0.998708i \(0.483816\pi\)
\(942\) −1.21158e7 −0.444862
\(943\) −573880. −0.0210156
\(944\) 3.03457e6 0.110833
\(945\) 2.20754e6 0.0804135
\(946\) 1.18343e7 0.429946
\(947\) 2.29370e6 0.0831117 0.0415559 0.999136i \(-0.486769\pi\)
0.0415559 + 0.999136i \(0.486769\pi\)
\(948\) −2.15964e6 −0.0780477
\(949\) 5.01608e6 0.180800
\(950\) 4.28781e7 1.54144
\(951\) −1.14840e7 −0.411759
\(952\) 4.44220e7 1.58857
\(953\) −3.80168e6 −0.135595 −0.0677975 0.997699i \(-0.521597\pi\)
−0.0677975 + 0.997699i \(0.521597\pi\)
\(954\) −768654. −0.0273439
\(955\) −1.52251e7 −0.540196
\(956\) 3.21665e6 0.113831
\(957\) 6.23880e6 0.220202
\(958\) −2.89147e6 −0.101790
\(959\) −4.58158e6 −0.160868
\(960\) 5.65993e6 0.198213
\(961\) −1.17445e7 −0.410227
\(962\) −1.58258e7 −0.551352
\(963\) 839875. 0.0291843
\(964\) −4.13283e6 −0.143237
\(965\) 1.51687e7 0.524362
\(966\) −5.67663e6 −0.195726
\(967\) −1.18301e7 −0.406838 −0.203419 0.979092i \(-0.565205\pi\)
−0.203419 + 0.979092i \(0.565205\pi\)
\(968\) 1.88557e7 0.646777
\(969\) −3.50063e7 −1.19767
\(970\) 1.06953e7 0.364977
\(971\) −4.88523e7 −1.66279 −0.831394 0.555683i \(-0.812457\pi\)
−0.831394 + 0.555683i \(0.812457\pi\)
\(972\) 248308. 0.00842996
\(973\) 3.07575e7 1.04152
\(974\) 2.09421e7 0.707333
\(975\) 7.16996e6 0.241549
\(976\) 2.44300e7 0.820915
\(977\) 4.77067e7 1.59898 0.799489 0.600681i \(-0.205104\pi\)
0.799489 + 0.600681i \(0.205104\pi\)
\(978\) 1.63276e7 0.545852
\(979\) −1.33791e7 −0.446138
\(980\) 960101. 0.0319339
\(981\) −3.07268e6 −0.101940
\(982\) 5.36597e7 1.77570
\(983\) 4.35421e7 1.43723 0.718613 0.695410i \(-0.244777\pi\)
0.718613 + 0.695410i \(0.244777\pi\)
\(984\) −1.42321e6 −0.0468578
\(985\) −1.82111e7 −0.598062
\(986\) 1.97352e7 0.646471
\(987\) 6.75661e6 0.220768
\(988\) 3.43204e6 0.111856
\(989\) 6.23131e6 0.202576
\(990\) −1.86834e6 −0.0605853
\(991\) 1.39948e7 0.452672 0.226336 0.974049i \(-0.427325\pi\)
0.226336 + 0.974049i \(0.427325\pi\)
\(992\) −6.21333e6 −0.200468
\(993\) 2.29226e7 0.737718
\(994\) 3.59250e6 0.115327
\(995\) 9.09226e6 0.291148
\(996\) 662340. 0.0211560
\(997\) 1.81460e7 0.578152 0.289076 0.957306i \(-0.406652\pi\)
0.289076 + 0.957306i \(0.406652\pi\)
\(998\) 1.01284e7 0.321896
\(999\) 7.73953e6 0.245358
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.b.1.9 12
3.2 odd 2 531.6.a.d.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.9 12 1.1 even 1 trivial
531.6.a.d.1.4 12 3.2 odd 2