Properties

Label 177.6.a.b.1.4
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 1.4
Root \(4.14510\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.14510 q^{2} -9.00000 q^{3} -14.8181 q^{4} -34.9684 q^{5} +37.3059 q^{6} -110.249 q^{7} +194.066 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.14510 q^{2} -9.00000 q^{3} -14.8181 q^{4} -34.9684 q^{5} +37.3059 q^{6} -110.249 q^{7} +194.066 q^{8} +81.0000 q^{9} +144.948 q^{10} +176.487 q^{11} +133.363 q^{12} +818.757 q^{13} +456.993 q^{14} +314.716 q^{15} -330.244 q^{16} -1403.91 q^{17} -335.753 q^{18} +2005.24 q^{19} +518.166 q^{20} +992.239 q^{21} -731.559 q^{22} +846.668 q^{23} -1746.59 q^{24} -1902.21 q^{25} -3393.83 q^{26} -729.000 q^{27} +1633.68 q^{28} +3649.85 q^{29} -1304.53 q^{30} +5743.05 q^{31} -4841.22 q^{32} -1588.39 q^{33} +5819.34 q^{34} +3855.23 q^{35} -1200.27 q^{36} -12323.7 q^{37} -8311.92 q^{38} -7368.81 q^{39} -6786.18 q^{40} +11909.1 q^{41} -4112.93 q^{42} -13792.7 q^{43} -2615.21 q^{44} -2832.44 q^{45} -3509.53 q^{46} +7639.87 q^{47} +2972.19 q^{48} -4652.21 q^{49} +7884.86 q^{50} +12635.2 q^{51} -12132.4 q^{52} +6035.29 q^{53} +3021.78 q^{54} -6171.49 q^{55} -21395.5 q^{56} -18047.1 q^{57} -15129.0 q^{58} -3481.00 q^{59} -4663.50 q^{60} -40205.0 q^{61} -23805.6 q^{62} -8930.15 q^{63} +30635.1 q^{64} -28630.6 q^{65} +6584.03 q^{66} -54222.1 q^{67} +20803.3 q^{68} -7620.02 q^{69} -15980.3 q^{70} -15242.5 q^{71} +15719.3 q^{72} +68611.8 q^{73} +51082.8 q^{74} +17119.9 q^{75} -29713.8 q^{76} -19457.5 q^{77} +30544.5 q^{78} -85026.3 q^{79} +11548.1 q^{80} +6561.00 q^{81} -49364.6 q^{82} +62362.4 q^{83} -14703.1 q^{84} +49092.4 q^{85} +57172.3 q^{86} -32848.6 q^{87} +34250.2 q^{88} -136085. q^{89} +11740.8 q^{90} -90267.0 q^{91} -12546.0 q^{92} -51687.5 q^{93} -31668.1 q^{94} -70120.0 q^{95} +43570.9 q^{96} +77595.8 q^{97} +19283.9 q^{98} +14295.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.14510 −0.732758 −0.366379 0.930466i \(-0.619403\pi\)
−0.366379 + 0.930466i \(0.619403\pi\)
\(3\) −9.00000 −0.577350
\(4\) −14.8181 −0.463066
\(5\) −34.9684 −0.625534 −0.312767 0.949830i \(-0.601256\pi\)
−0.312767 + 0.949830i \(0.601256\pi\)
\(6\) 37.3059 0.423058
\(7\) −110.249 −0.850411 −0.425205 0.905097i \(-0.639798\pi\)
−0.425205 + 0.905097i \(0.639798\pi\)
\(8\) 194.066 1.07207
\(9\) 81.0000 0.333333
\(10\) 144.948 0.458365
\(11\) 176.487 0.439777 0.219888 0.975525i \(-0.429431\pi\)
0.219888 + 0.975525i \(0.429431\pi\)
\(12\) 133.363 0.267351
\(13\) 818.757 1.34368 0.671841 0.740695i \(-0.265504\pi\)
0.671841 + 0.740695i \(0.265504\pi\)
\(14\) 456.993 0.623145
\(15\) 314.716 0.361152
\(16\) −330.244 −0.322504
\(17\) −1403.91 −1.17819 −0.589096 0.808063i \(-0.700516\pi\)
−0.589096 + 0.808063i \(0.700516\pi\)
\(18\) −335.753 −0.244253
\(19\) 2005.24 1.27433 0.637165 0.770728i \(-0.280107\pi\)
0.637165 + 0.770728i \(0.280107\pi\)
\(20\) 518.166 0.289664
\(21\) 992.239 0.490985
\(22\) −731.559 −0.322250
\(23\) 846.668 0.333729 0.166864 0.985980i \(-0.446636\pi\)
0.166864 + 0.985980i \(0.446636\pi\)
\(24\) −1746.59 −0.618962
\(25\) −1902.21 −0.608707
\(26\) −3393.83 −0.984594
\(27\) −729.000 −0.192450
\(28\) 1633.68 0.393796
\(29\) 3649.85 0.805897 0.402949 0.915223i \(-0.367985\pi\)
0.402949 + 0.915223i \(0.367985\pi\)
\(30\) −1304.53 −0.264637
\(31\) 5743.05 1.07334 0.536672 0.843791i \(-0.319681\pi\)
0.536672 + 0.843791i \(0.319681\pi\)
\(32\) −4841.22 −0.835756
\(33\) −1588.39 −0.253905
\(34\) 5819.34 0.863330
\(35\) 3855.23 0.531961
\(36\) −1200.27 −0.154355
\(37\) −12323.7 −1.47991 −0.739955 0.672656i \(-0.765153\pi\)
−0.739955 + 0.672656i \(0.765153\pi\)
\(38\) −8311.92 −0.933775
\(39\) −7368.81 −0.775775
\(40\) −6786.18 −0.670618
\(41\) 11909.1 1.10642 0.553211 0.833041i \(-0.313402\pi\)
0.553211 + 0.833041i \(0.313402\pi\)
\(42\) −4112.93 −0.359773
\(43\) −13792.7 −1.13757 −0.568787 0.822485i \(-0.692587\pi\)
−0.568787 + 0.822485i \(0.692587\pi\)
\(44\) −2615.21 −0.203646
\(45\) −2832.44 −0.208511
\(46\) −3509.53 −0.244542
\(47\) 7639.87 0.504477 0.252239 0.967665i \(-0.418833\pi\)
0.252239 + 0.967665i \(0.418833\pi\)
\(48\) 2972.19 0.186198
\(49\) −4652.21 −0.276802
\(50\) 7884.86 0.446035
\(51\) 12635.2 0.680230
\(52\) −12132.4 −0.622214
\(53\) 6035.29 0.295127 0.147563 0.989053i \(-0.452857\pi\)
0.147563 + 0.989053i \(0.452857\pi\)
\(54\) 3021.78 0.141019
\(55\) −6171.49 −0.275095
\(56\) −21395.5 −0.911702
\(57\) −18047.1 −0.735734
\(58\) −15129.0 −0.590527
\(59\) −3481.00 −0.130189
\(60\) −4663.50 −0.167237
\(61\) −40205.0 −1.38342 −0.691712 0.722174i \(-0.743143\pi\)
−0.691712 + 0.722174i \(0.743143\pi\)
\(62\) −23805.6 −0.786501
\(63\) −8930.15 −0.283470
\(64\) 30635.1 0.934910
\(65\) −28630.6 −0.840519
\(66\) 6584.03 0.186051
\(67\) −54222.1 −1.47567 −0.737836 0.674980i \(-0.764152\pi\)
−0.737836 + 0.674980i \(0.764152\pi\)
\(68\) 20803.3 0.545581
\(69\) −7620.02 −0.192678
\(70\) −15980.3 −0.389798
\(71\) −15242.5 −0.358848 −0.179424 0.983772i \(-0.557423\pi\)
−0.179424 + 0.983772i \(0.557423\pi\)
\(72\) 15719.3 0.357358
\(73\) 68611.8 1.50693 0.753463 0.657490i \(-0.228382\pi\)
0.753463 + 0.657490i \(0.228382\pi\)
\(74\) 51082.8 1.08442
\(75\) 17119.9 0.351437
\(76\) −29713.8 −0.590099
\(77\) −19457.5 −0.373991
\(78\) 30544.5 0.568455
\(79\) −85026.3 −1.53280 −0.766401 0.642363i \(-0.777954\pi\)
−0.766401 + 0.642363i \(0.777954\pi\)
\(80\) 11548.1 0.201737
\(81\) 6561.00 0.111111
\(82\) −49364.6 −0.810739
\(83\) 62362.4 0.993636 0.496818 0.867855i \(-0.334502\pi\)
0.496818 + 0.867855i \(0.334502\pi\)
\(84\) −14703.1 −0.227358
\(85\) 49092.4 0.737000
\(86\) 57172.3 0.833566
\(87\) −32848.6 −0.465285
\(88\) 34250.2 0.471473
\(89\) −136085. −1.82111 −0.910555 0.413389i \(-0.864345\pi\)
−0.910555 + 0.413389i \(0.864345\pi\)
\(90\) 11740.8 0.152788
\(91\) −90267.0 −1.14268
\(92\) −12546.0 −0.154539
\(93\) −51687.5 −0.619695
\(94\) −31668.1 −0.369660
\(95\) −70120.0 −0.797137
\(96\) 43570.9 0.482524
\(97\) 77595.8 0.837353 0.418677 0.908135i \(-0.362494\pi\)
0.418677 + 0.908135i \(0.362494\pi\)
\(98\) 19283.9 0.202829
\(99\) 14295.5 0.146592
\(100\) 28187.2 0.281872
\(101\) 172130. 1.67901 0.839503 0.543355i \(-0.182846\pi\)
0.839503 + 0.543355i \(0.182846\pi\)
\(102\) −52374.1 −0.498444
\(103\) 58120.5 0.539804 0.269902 0.962888i \(-0.413009\pi\)
0.269902 + 0.962888i \(0.413009\pi\)
\(104\) 158893. 1.44053
\(105\) −34697.0 −0.307128
\(106\) −25016.9 −0.216256
\(107\) −142810. −1.20587 −0.602935 0.797791i \(-0.706002\pi\)
−0.602935 + 0.797791i \(0.706002\pi\)
\(108\) 10802.4 0.0891171
\(109\) 15204.1 0.122573 0.0612866 0.998120i \(-0.480480\pi\)
0.0612866 + 0.998120i \(0.480480\pi\)
\(110\) 25581.5 0.201578
\(111\) 110913. 0.854426
\(112\) 36409.0 0.274261
\(113\) 145209. 1.06978 0.534892 0.844920i \(-0.320352\pi\)
0.534892 + 0.844920i \(0.320352\pi\)
\(114\) 74807.2 0.539115
\(115\) −29606.7 −0.208759
\(116\) −54083.9 −0.373184
\(117\) 66319.3 0.447894
\(118\) 14429.1 0.0953969
\(119\) 154779. 1.00195
\(120\) 61075.6 0.387182
\(121\) −129903. −0.806597
\(122\) 166654. 1.01371
\(123\) −107182. −0.638793
\(124\) −85101.2 −0.497029
\(125\) 175794. 1.00630
\(126\) 37016.4 0.207715
\(127\) −21609.6 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(128\) 27933.1 0.150693
\(129\) 124135. 0.656778
\(130\) 118677. 0.615897
\(131\) 3428.28 0.0174541 0.00872707 0.999962i \(-0.497222\pi\)
0.00872707 + 0.999962i \(0.497222\pi\)
\(132\) 23536.9 0.117575
\(133\) −221075. −1.08370
\(134\) 224756. 1.08131
\(135\) 25492.0 0.120384
\(136\) −272451. −1.26311
\(137\) 53929.4 0.245484 0.122742 0.992439i \(-0.460831\pi\)
0.122742 + 0.992439i \(0.460831\pi\)
\(138\) 31585.8 0.141187
\(139\) −46431.4 −0.203833 −0.101917 0.994793i \(-0.532497\pi\)
−0.101917 + 0.994793i \(0.532497\pi\)
\(140\) −57127.2 −0.246333
\(141\) −68758.9 −0.291260
\(142\) 63181.7 0.262948
\(143\) 144500. 0.590920
\(144\) −26749.7 −0.107501
\(145\) −127629. −0.504116
\(146\) −284403. −1.10421
\(147\) 41869.9 0.159812
\(148\) 182613. 0.685296
\(149\) 395001. 1.45758 0.728790 0.684737i \(-0.240083\pi\)
0.728790 + 0.684737i \(0.240083\pi\)
\(150\) −70963.7 −0.257518
\(151\) 359613. 1.28349 0.641746 0.766918i \(-0.278211\pi\)
0.641746 + 0.766918i \(0.278211\pi\)
\(152\) 389148. 1.36617
\(153\) −113717. −0.392731
\(154\) 80653.5 0.274045
\(155\) −200826. −0.671413
\(156\) 109192. 0.359235
\(157\) −602958. −1.95226 −0.976130 0.217185i \(-0.930312\pi\)
−0.976130 + 0.217185i \(0.930312\pi\)
\(158\) 352443. 1.12317
\(159\) −54317.6 −0.170392
\(160\) 169290. 0.522794
\(161\) −93344.2 −0.283807
\(162\) −27196.0 −0.0814175
\(163\) −571386. −1.68446 −0.842230 0.539118i \(-0.818758\pi\)
−0.842230 + 0.539118i \(0.818758\pi\)
\(164\) −176471. −0.512346
\(165\) 55543.4 0.158826
\(166\) −258498. −0.728094
\(167\) −484312. −1.34380 −0.671899 0.740643i \(-0.734521\pi\)
−0.671899 + 0.740643i \(0.734521\pi\)
\(168\) 192560. 0.526372
\(169\) 299070. 0.805483
\(170\) −203493. −0.540042
\(171\) 162424. 0.424777
\(172\) 204382. 0.526772
\(173\) −79449.3 −0.201825 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(174\) 136161. 0.340941
\(175\) 209716. 0.517651
\(176\) −58283.9 −0.141830
\(177\) 31329.0 0.0751646
\(178\) 564087. 1.33443
\(179\) −417149. −0.973103 −0.486552 0.873652i \(-0.661745\pi\)
−0.486552 + 0.873652i \(0.661745\pi\)
\(180\) 41971.5 0.0965546
\(181\) −349894. −0.793854 −0.396927 0.917850i \(-0.629923\pi\)
−0.396927 + 0.917850i \(0.629923\pi\)
\(182\) 374166. 0.837309
\(183\) 361845. 0.798720
\(184\) 164310. 0.357782
\(185\) 430939. 0.925734
\(186\) 214250. 0.454086
\(187\) −247772. −0.518141
\(188\) −113209. −0.233606
\(189\) 80371.4 0.163662
\(190\) 290655. 0.584108
\(191\) −162965. −0.323229 −0.161614 0.986854i \(-0.551670\pi\)
−0.161614 + 0.986854i \(0.551670\pi\)
\(192\) −275716. −0.539771
\(193\) −408375. −0.789162 −0.394581 0.918861i \(-0.629110\pi\)
−0.394581 + 0.918861i \(0.629110\pi\)
\(194\) −321642. −0.613577
\(195\) 257676. 0.485274
\(196\) 68936.9 0.128178
\(197\) −639162. −1.17340 −0.586699 0.809805i \(-0.699573\pi\)
−0.586699 + 0.809805i \(0.699573\pi\)
\(198\) −59256.3 −0.107417
\(199\) −137487. −0.246111 −0.123055 0.992400i \(-0.539269\pi\)
−0.123055 + 0.992400i \(0.539269\pi\)
\(200\) −369154. −0.652578
\(201\) 487999. 0.851979
\(202\) −713495. −1.23030
\(203\) −402391. −0.685344
\(204\) −187229. −0.314991
\(205\) −416444. −0.692105
\(206\) −240916. −0.395546
\(207\) 68580.1 0.111243
\(208\) −270389. −0.433343
\(209\) 353899. 0.560420
\(210\) 143823. 0.225050
\(211\) −146412. −0.226397 −0.113199 0.993572i \(-0.536110\pi\)
−0.113199 + 0.993572i \(0.536110\pi\)
\(212\) −89431.6 −0.136663
\(213\) 137182. 0.207181
\(214\) 591964. 0.883610
\(215\) 482310. 0.711591
\(216\) −141474. −0.206321
\(217\) −633165. −0.912782
\(218\) −63022.7 −0.0898164
\(219\) −617507. −0.870024
\(220\) 91449.8 0.127387
\(221\) −1.14946e6 −1.58312
\(222\) −459746. −0.626088
\(223\) 212713. 0.286439 0.143220 0.989691i \(-0.454254\pi\)
0.143220 + 0.989691i \(0.454254\pi\)
\(224\) 533738. 0.710736
\(225\) −154079. −0.202902
\(226\) −601905. −0.783893
\(227\) 56603.0 0.0729079 0.0364540 0.999335i \(-0.488394\pi\)
0.0364540 + 0.999335i \(0.488394\pi\)
\(228\) 267425. 0.340694
\(229\) −893019. −1.12531 −0.562655 0.826692i \(-0.690220\pi\)
−0.562655 + 0.826692i \(0.690220\pi\)
\(230\) 122723. 0.152970
\(231\) 175118. 0.215924
\(232\) 708311. 0.863981
\(233\) −1.08789e6 −1.31280 −0.656398 0.754415i \(-0.727921\pi\)
−0.656398 + 0.754415i \(0.727921\pi\)
\(234\) −274900. −0.328198
\(235\) −267154. −0.315568
\(236\) 51581.9 0.0602861
\(237\) 765237. 0.884963
\(238\) −641575. −0.734185
\(239\) −330858. −0.374668 −0.187334 0.982296i \(-0.559985\pi\)
−0.187334 + 0.982296i \(0.559985\pi\)
\(240\) −103933. −0.116473
\(241\) 53121.9 0.0589157 0.0294578 0.999566i \(-0.490622\pi\)
0.0294578 + 0.999566i \(0.490622\pi\)
\(242\) 538462. 0.591040
\(243\) −59049.0 −0.0641500
\(244\) 595762. 0.640616
\(245\) 162680. 0.173149
\(246\) 444282. 0.468080
\(247\) 1.64180e6 1.71229
\(248\) 1.11453e6 1.15070
\(249\) −561261. −0.573676
\(250\) −728683. −0.737375
\(251\) −1.41412e6 −1.41678 −0.708390 0.705821i \(-0.750578\pi\)
−0.708390 + 0.705821i \(0.750578\pi\)
\(252\) 132328. 0.131265
\(253\) 149426. 0.146766
\(254\) 89573.9 0.0871158
\(255\) −441832. −0.425507
\(256\) −1.09611e6 −1.04533
\(257\) 4655.81 0.00439706 0.00219853 0.999998i \(-0.499300\pi\)
0.00219853 + 0.999998i \(0.499300\pi\)
\(258\) −514551. −0.481259
\(259\) 1.35867e6 1.25853
\(260\) 424252. 0.389216
\(261\) 295638. 0.268632
\(262\) −14210.6 −0.0127897
\(263\) −147942. −0.131887 −0.0659433 0.997823i \(-0.521006\pi\)
−0.0659433 + 0.997823i \(0.521006\pi\)
\(264\) −308252. −0.272205
\(265\) −211045. −0.184612
\(266\) 916379. 0.794092
\(267\) 1.22477e6 1.05142
\(268\) 803470. 0.683333
\(269\) 1.11352e6 0.938245 0.469122 0.883133i \(-0.344570\pi\)
0.469122 + 0.883133i \(0.344570\pi\)
\(270\) −105667. −0.0882124
\(271\) −427188. −0.353343 −0.176671 0.984270i \(-0.556533\pi\)
−0.176671 + 0.984270i \(0.556533\pi\)
\(272\) 463632. 0.379971
\(273\) 812403. 0.659728
\(274\) −223543. −0.179881
\(275\) −335716. −0.267695
\(276\) 112914. 0.0892229
\(277\) −1.69355e6 −1.32616 −0.663082 0.748547i \(-0.730752\pi\)
−0.663082 + 0.748547i \(0.730752\pi\)
\(278\) 192463. 0.149360
\(279\) 465187. 0.357781
\(280\) 748168. 0.570301
\(281\) 2.23022e6 1.68493 0.842466 0.538750i \(-0.181103\pi\)
0.842466 + 0.538750i \(0.181103\pi\)
\(282\) 285013. 0.213423
\(283\) −1.84337e6 −1.36819 −0.684097 0.729391i \(-0.739803\pi\)
−0.684097 + 0.729391i \(0.739803\pi\)
\(284\) 225865. 0.166170
\(285\) 631080. 0.460227
\(286\) −598969. −0.433001
\(287\) −1.31297e6 −0.940913
\(288\) −392138. −0.278585
\(289\) 551100. 0.388138
\(290\) 529037. 0.369395
\(291\) −698362. −0.483446
\(292\) −1.01670e6 −0.697806
\(293\) 326532. 0.222206 0.111103 0.993809i \(-0.464562\pi\)
0.111103 + 0.993809i \(0.464562\pi\)
\(294\) −173555. −0.117103
\(295\) 121725. 0.0814376
\(296\) −2.39160e6 −1.58657
\(297\) −128659. −0.0846350
\(298\) −1.63732e6 −1.06805
\(299\) 693216. 0.448426
\(300\) −253684. −0.162739
\(301\) 1.52063e6 0.967404
\(302\) −1.49063e6 −0.940488
\(303\) −1.54917e6 −0.969375
\(304\) −662217. −0.410976
\(305\) 1.40590e6 0.865379
\(306\) 471367. 0.287777
\(307\) −1.14971e6 −0.696216 −0.348108 0.937454i \(-0.613176\pi\)
−0.348108 + 0.937454i \(0.613176\pi\)
\(308\) 288324. 0.173182
\(309\) −523085. −0.311656
\(310\) 832443. 0.491983
\(311\) −1.82214e6 −1.06827 −0.534133 0.845400i \(-0.679362\pi\)
−0.534133 + 0.845400i \(0.679362\pi\)
\(312\) −1.43004e6 −0.831688
\(313\) −1.40324e6 −0.809603 −0.404801 0.914405i \(-0.632659\pi\)
−0.404801 + 0.914405i \(0.632659\pi\)
\(314\) 2.49932e6 1.43053
\(315\) 312273. 0.177320
\(316\) 1.25993e6 0.709788
\(317\) 2.02187e6 1.13007 0.565034 0.825068i \(-0.308863\pi\)
0.565034 + 0.825068i \(0.308863\pi\)
\(318\) 225152. 0.124856
\(319\) 644152. 0.354415
\(320\) −1.07126e6 −0.584818
\(321\) 1.28529e6 0.696209
\(322\) 386921. 0.207961
\(323\) −2.81517e6 −1.50141
\(324\) −97221.7 −0.0514518
\(325\) −1.55745e6 −0.817909
\(326\) 2.36846e6 1.23430
\(327\) −136837. −0.0707676
\(328\) 2.31116e6 1.18616
\(329\) −842287. −0.429013
\(330\) −230233. −0.116381
\(331\) 2.67961e6 1.34432 0.672159 0.740407i \(-0.265367\pi\)
0.672159 + 0.740407i \(0.265367\pi\)
\(332\) −924093. −0.460119
\(333\) −998216. −0.493303
\(334\) 2.00752e6 0.984678
\(335\) 1.89606e6 0.923083
\(336\) −327681. −0.158344
\(337\) 3.08781e6 1.48107 0.740536 0.672017i \(-0.234572\pi\)
0.740536 + 0.672017i \(0.234572\pi\)
\(338\) −1.23968e6 −0.590224
\(339\) −1.30688e6 −0.617641
\(340\) −727457. −0.341280
\(341\) 1.01358e6 0.472031
\(342\) −673265. −0.311258
\(343\) 2.36585e6 1.08581
\(344\) −2.67670e6 −1.21956
\(345\) 266460. 0.120527
\(346\) 329326. 0.147889
\(347\) 1.59388e6 0.710612 0.355306 0.934750i \(-0.384377\pi\)
0.355306 + 0.934750i \(0.384377\pi\)
\(348\) 486755. 0.215458
\(349\) 4.02815e6 1.77028 0.885141 0.465323i \(-0.154062\pi\)
0.885141 + 0.465323i \(0.154062\pi\)
\(350\) −869296. −0.379313
\(351\) −596874. −0.258592
\(352\) −854414. −0.367546
\(353\) −1.83928e6 −0.785618 −0.392809 0.919620i \(-0.628497\pi\)
−0.392809 + 0.919620i \(0.628497\pi\)
\(354\) −129862. −0.0550774
\(355\) 533006. 0.224471
\(356\) 2.01653e6 0.843294
\(357\) −1.39301e6 −0.578475
\(358\) 1.72913e6 0.713049
\(359\) 2.26891e6 0.929140 0.464570 0.885536i \(-0.346209\pi\)
0.464570 + 0.885536i \(0.346209\pi\)
\(360\) −549681. −0.223539
\(361\) 1.54488e6 0.623916
\(362\) 1.45035e6 0.581703
\(363\) 1.16913e6 0.465689
\(364\) 1.33759e6 0.529137
\(365\) −2.39925e6 −0.942634
\(366\) −1.49988e6 −0.585268
\(367\) −4.64590e6 −1.80055 −0.900273 0.435325i \(-0.856633\pi\)
−0.900273 + 0.435325i \(0.856633\pi\)
\(368\) −279607. −0.107629
\(369\) 964640. 0.368807
\(370\) −1.78629e6 −0.678339
\(371\) −665383. −0.250979
\(372\) 765911. 0.286960
\(373\) −417633. −0.155426 −0.0777128 0.996976i \(-0.524762\pi\)
−0.0777128 + 0.996976i \(0.524762\pi\)
\(374\) 1.02704e6 0.379672
\(375\) −1.58214e6 −0.580988
\(376\) 1.48264e6 0.540836
\(377\) 2.98834e6 1.08287
\(378\) −333148. −0.119924
\(379\) −137274. −0.0490897 −0.0245449 0.999699i \(-0.507814\pi\)
−0.0245449 + 0.999699i \(0.507814\pi\)
\(380\) 1.03905e6 0.369127
\(381\) 194486. 0.0686398
\(382\) 675505. 0.236848
\(383\) −3.45107e6 −1.20214 −0.601072 0.799195i \(-0.705260\pi\)
−0.601072 + 0.799195i \(0.705260\pi\)
\(384\) −251398. −0.0870028
\(385\) 680399. 0.233944
\(386\) 1.69276e6 0.578264
\(387\) −1.11721e6 −0.379191
\(388\) −1.14982e6 −0.387750
\(389\) −4.61323e6 −1.54572 −0.772860 0.634576i \(-0.781175\pi\)
−0.772860 + 0.634576i \(0.781175\pi\)
\(390\) −1.06809e6 −0.355588
\(391\) −1.18864e6 −0.393197
\(392\) −902835. −0.296752
\(393\) −30854.6 −0.0100772
\(394\) 2.64939e6 0.859816
\(395\) 2.97324e6 0.958819
\(396\) −211832. −0.0678819
\(397\) 2.12195e6 0.675707 0.337854 0.941199i \(-0.390299\pi\)
0.337854 + 0.941199i \(0.390299\pi\)
\(398\) 569900. 0.180339
\(399\) 1.98967e6 0.625676
\(400\) 628193. 0.196310
\(401\) −138495. −0.0430105 −0.0215052 0.999769i \(-0.506846\pi\)
−0.0215052 + 0.999769i \(0.506846\pi\)
\(402\) −2.02281e6 −0.624294
\(403\) 4.70217e6 1.44223
\(404\) −2.55064e6 −0.777491
\(405\) −229428. −0.0695038
\(406\) 1.66795e6 0.502191
\(407\) −2.17497e6 −0.650830
\(408\) 2.45206e6 0.729256
\(409\) −379673. −0.112228 −0.0561141 0.998424i \(-0.517871\pi\)
−0.0561141 + 0.998424i \(0.517871\pi\)
\(410\) 1.72620e6 0.507145
\(411\) −485364. −0.141730
\(412\) −861236. −0.249965
\(413\) 383776. 0.110714
\(414\) −284272. −0.0815141
\(415\) −2.18071e6 −0.621553
\(416\) −3.96378e6 −1.12299
\(417\) 417883. 0.117683
\(418\) −1.46695e6 −0.410652
\(419\) 2.94021e6 0.818170 0.409085 0.912496i \(-0.365848\pi\)
0.409085 + 0.912496i \(0.365848\pi\)
\(420\) 514145. 0.142220
\(421\) −2.46278e6 −0.677204 −0.338602 0.940930i \(-0.609954\pi\)
−0.338602 + 0.940930i \(0.609954\pi\)
\(422\) 606894. 0.165894
\(423\) 618830. 0.168159
\(424\) 1.17124e6 0.316397
\(425\) 2.67053e6 0.717174
\(426\) −568635. −0.151813
\(427\) 4.43255e6 1.17648
\(428\) 2.11618e6 0.558397
\(429\) −1.30050e6 −0.341168
\(430\) −1.99923e6 −0.521424
\(431\) 894076. 0.231836 0.115918 0.993259i \(-0.463019\pi\)
0.115918 + 0.993259i \(0.463019\pi\)
\(432\) 240748. 0.0620659
\(433\) −2.26975e6 −0.581779 −0.290890 0.956757i \(-0.593951\pi\)
−0.290890 + 0.956757i \(0.593951\pi\)
\(434\) 2.62453e6 0.668848
\(435\) 1.14866e6 0.291052
\(436\) −225297. −0.0567595
\(437\) 1.69777e6 0.425281
\(438\) 2.55963e6 0.637517
\(439\) −5.23628e6 −1.29677 −0.648383 0.761314i \(-0.724555\pi\)
−0.648383 + 0.761314i \(0.724555\pi\)
\(440\) −1.19768e6 −0.294922
\(441\) −376829. −0.0922673
\(442\) 4.76463e6 1.16004
\(443\) 87708.4 0.0212340 0.0106170 0.999944i \(-0.496620\pi\)
0.0106170 + 0.999944i \(0.496620\pi\)
\(444\) −1.64352e6 −0.395656
\(445\) 4.75869e6 1.13917
\(446\) −881719. −0.209891
\(447\) −3.55501e6 −0.841535
\(448\) −3.37749e6 −0.795058
\(449\) −2.20132e6 −0.515308 −0.257654 0.966237i \(-0.582949\pi\)
−0.257654 + 0.966237i \(0.582949\pi\)
\(450\) 638673. 0.148678
\(451\) 2.10181e6 0.486578
\(452\) −2.15172e6 −0.495381
\(453\) −3.23652e6 −0.741024
\(454\) −234625. −0.0534239
\(455\) 3.15649e6 0.714787
\(456\) −3.50233e6 −0.788761
\(457\) 6.51177e6 1.45851 0.729254 0.684243i \(-0.239867\pi\)
0.729254 + 0.684243i \(0.239867\pi\)
\(458\) 3.70166e6 0.824580
\(459\) 1.02345e6 0.226743
\(460\) 438715. 0.0966691
\(461\) 3.62362e6 0.794128 0.397064 0.917791i \(-0.370029\pi\)
0.397064 + 0.917791i \(0.370029\pi\)
\(462\) −725881. −0.158220
\(463\) −1.17608e6 −0.254967 −0.127484 0.991841i \(-0.540690\pi\)
−0.127484 + 0.991841i \(0.540690\pi\)
\(464\) −1.20534e6 −0.259905
\(465\) 1.80743e6 0.387640
\(466\) 4.50944e6 0.961961
\(467\) −2.36743e6 −0.502324 −0.251162 0.967945i \(-0.580813\pi\)
−0.251162 + 0.967945i \(0.580813\pi\)
\(468\) −982727. −0.207405
\(469\) 5.97792e6 1.25493
\(470\) 1.10738e6 0.231235
\(471\) 5.42662e6 1.12714
\(472\) −675544. −0.139572
\(473\) −2.43425e6 −0.500278
\(474\) −3.17199e6 −0.648464
\(475\) −3.81438e6 −0.775693
\(476\) −2.29353e6 −0.463968
\(477\) 488859. 0.0983756
\(478\) 1.37144e6 0.274541
\(479\) −9.93768e6 −1.97900 −0.989501 0.144525i \(-0.953835\pi\)
−0.989501 + 0.144525i \(0.953835\pi\)
\(480\) −1.52361e6 −0.301835
\(481\) −1.00901e7 −1.98853
\(482\) −220196. −0.0431709
\(483\) 840097. 0.163856
\(484\) 1.92492e6 0.373508
\(485\) −2.71340e6 −0.523793
\(486\) 244764. 0.0470064
\(487\) 1.20632e6 0.230483 0.115242 0.993337i \(-0.463236\pi\)
0.115242 + 0.993337i \(0.463236\pi\)
\(488\) −7.80241e6 −1.48313
\(489\) 5.14248e6 0.972524
\(490\) −674327. −0.126876
\(491\) −7.58775e6 −1.42040 −0.710198 0.704002i \(-0.751395\pi\)
−0.710198 + 0.704002i \(0.751395\pi\)
\(492\) 1.58824e6 0.295803
\(493\) −5.12405e6 −0.949502
\(494\) −6.80544e6 −1.25470
\(495\) −499890. −0.0916984
\(496\) −1.89661e6 −0.346157
\(497\) 1.68047e6 0.305168
\(498\) 2.32649e6 0.420365
\(499\) 1.92717e6 0.346472 0.173236 0.984880i \(-0.444578\pi\)
0.173236 + 0.984880i \(0.444578\pi\)
\(500\) −2.60493e6 −0.465984
\(501\) 4.35881e6 0.775842
\(502\) 5.86168e6 1.03816
\(503\) −8.43785e6 −1.48700 −0.743502 0.668734i \(-0.766836\pi\)
−0.743502 + 0.668734i \(0.766836\pi\)
\(504\) −1.73304e6 −0.303901
\(505\) −6.01910e6 −1.05028
\(506\) −619388. −0.107544
\(507\) −2.69163e6 −0.465046
\(508\) 320213. 0.0550528
\(509\) 6.51732e6 1.11500 0.557500 0.830177i \(-0.311761\pi\)
0.557500 + 0.830177i \(0.311761\pi\)
\(510\) 1.83144e6 0.311794
\(511\) −7.56437e6 −1.28151
\(512\) 3.64963e6 0.615282
\(513\) −1.46182e6 −0.245245
\(514\) −19298.8 −0.00322198
\(515\) −2.03238e6 −0.337666
\(516\) −1.83944e6 −0.304132
\(517\) 1.34834e6 0.221857
\(518\) −5.63182e6 −0.922198
\(519\) 715044. 0.116524
\(520\) −5.55623e6 −0.901098
\(521\) −252152. −0.0406975 −0.0203487 0.999793i \(-0.506478\pi\)
−0.0203487 + 0.999793i \(0.506478\pi\)
\(522\) −1.22545e6 −0.196842
\(523\) −7.96238e6 −1.27288 −0.636442 0.771325i \(-0.719594\pi\)
−0.636442 + 0.771325i \(0.719594\pi\)
\(524\) −50800.7 −0.00808242
\(525\) −1.88745e6 −0.298866
\(526\) 613233. 0.0966410
\(527\) −8.06272e6 −1.26461
\(528\) 524555. 0.0818853
\(529\) −5.71950e6 −0.888625
\(530\) 874802. 0.135276
\(531\) −281961. −0.0433963
\(532\) 3.27591e6 0.501826
\(533\) 9.75069e6 1.48668
\(534\) −5.07679e6 −0.770435
\(535\) 4.99385e6 0.754312
\(536\) −1.05227e7 −1.58203
\(537\) 3.75434e6 0.561822
\(538\) −4.61564e6 −0.687506
\(539\) −821056. −0.121731
\(540\) −377743. −0.0557458
\(541\) −1.58336e6 −0.232587 −0.116294 0.993215i \(-0.537101\pi\)
−0.116294 + 0.993215i \(0.537101\pi\)
\(542\) 1.77074e6 0.258915
\(543\) 3.14905e6 0.458332
\(544\) 6.79662e6 0.984681
\(545\) −531664. −0.0766737
\(546\) −3.36749e6 −0.483421
\(547\) −5.75298e6 −0.822099 −0.411050 0.911613i \(-0.634838\pi\)
−0.411050 + 0.911613i \(0.634838\pi\)
\(548\) −799132. −0.113676
\(549\) −3.25660e6 −0.461141
\(550\) 1.39158e6 0.196156
\(551\) 7.31881e6 1.02698
\(552\) −1.47879e6 −0.206565
\(553\) 9.37405e6 1.30351
\(554\) 7.01992e6 0.971757
\(555\) −3.87845e6 −0.534473
\(556\) 688026. 0.0943882
\(557\) 1.88965e6 0.258074 0.129037 0.991640i \(-0.458811\pi\)
0.129037 + 0.991640i \(0.458811\pi\)
\(558\) −1.92825e6 −0.262167
\(559\) −1.12929e7 −1.52854
\(560\) −1.27316e6 −0.171559
\(561\) 2.22995e6 0.299149
\(562\) −9.24450e6 −1.23465
\(563\) 1.46556e7 1.94864 0.974322 0.225159i \(-0.0722902\pi\)
0.974322 + 0.225159i \(0.0722902\pi\)
\(564\) 1.01888e6 0.134873
\(565\) −5.07772e6 −0.669187
\(566\) 7.64098e6 1.00255
\(567\) −723342. −0.0944901
\(568\) −2.95805e6 −0.384711
\(569\) 3.30520e6 0.427973 0.213987 0.976837i \(-0.431355\pi\)
0.213987 + 0.976837i \(0.431355\pi\)
\(570\) −2.61589e6 −0.337235
\(571\) 9.00066e6 1.15527 0.577636 0.816295i \(-0.303975\pi\)
0.577636 + 0.816295i \(0.303975\pi\)
\(572\) −2.14122e6 −0.273635
\(573\) 1.46668e6 0.186616
\(574\) 5.44239e6 0.689461
\(575\) −1.61054e6 −0.203143
\(576\) 2.48145e6 0.311637
\(577\) 4.23516e6 0.529579 0.264789 0.964306i \(-0.414698\pi\)
0.264789 + 0.964306i \(0.414698\pi\)
\(578\) −2.28437e6 −0.284411
\(579\) 3.67538e6 0.455623
\(580\) 1.89123e6 0.233439
\(581\) −6.87537e6 −0.844999
\(582\) 2.89478e6 0.354249
\(583\) 1.06515e6 0.129790
\(584\) 1.33152e7 1.61553
\(585\) −2.31908e6 −0.280173
\(586\) −1.35351e6 −0.162823
\(587\) −1.09727e7 −1.31438 −0.657188 0.753727i \(-0.728254\pi\)
−0.657188 + 0.753727i \(0.728254\pi\)
\(588\) −620432. −0.0740033
\(589\) 1.15162e7 1.36779
\(590\) −504563. −0.0596740
\(591\) 5.75245e6 0.677461
\(592\) 4.06981e6 0.477276
\(593\) −1.11353e7 −1.30036 −0.650182 0.759779i \(-0.725307\pi\)
−0.650182 + 0.759779i \(0.725307\pi\)
\(594\) 533306. 0.0620170
\(595\) −5.41238e6 −0.626752
\(596\) −5.85317e6 −0.674956
\(597\) 1.23739e6 0.142092
\(598\) −2.87345e6 −0.328587
\(599\) 6.17540e6 0.703231 0.351615 0.936145i \(-0.385632\pi\)
0.351615 + 0.936145i \(0.385632\pi\)
\(600\) 3.32239e6 0.376766
\(601\) 1.17003e7 1.32133 0.660666 0.750680i \(-0.270274\pi\)
0.660666 + 0.750680i \(0.270274\pi\)
\(602\) −6.30318e6 −0.708873
\(603\) −4.39199e6 −0.491890
\(604\) −5.32879e6 −0.594341
\(605\) 4.54251e6 0.504554
\(606\) 6.42146e6 0.710317
\(607\) 7.60296e6 0.837551 0.418775 0.908090i \(-0.362459\pi\)
0.418775 + 0.908090i \(0.362459\pi\)
\(608\) −9.70779e6 −1.06503
\(609\) 3.62152e6 0.395683
\(610\) −5.82762e6 −0.634113
\(611\) 6.25520e6 0.677857
\(612\) 1.68506e6 0.181860
\(613\) −5.69046e6 −0.611640 −0.305820 0.952089i \(-0.598931\pi\)
−0.305820 + 0.952089i \(0.598931\pi\)
\(614\) 4.76569e6 0.510158
\(615\) 3.74799e6 0.399587
\(616\) −3.77604e6 −0.400945
\(617\) 1.04223e7 1.10218 0.551089 0.834446i \(-0.314212\pi\)
0.551089 + 0.834446i \(0.314212\pi\)
\(618\) 2.16824e6 0.228368
\(619\) 5.95972e6 0.625172 0.312586 0.949889i \(-0.398805\pi\)
0.312586 + 0.949889i \(0.398805\pi\)
\(620\) 2.97586e6 0.310909
\(621\) −617221. −0.0642262
\(622\) 7.55294e6 0.782781
\(623\) 1.50032e7 1.54869
\(624\) 2.43350e6 0.250190
\(625\) −202820. −0.0207688
\(626\) 5.81659e6 0.593243
\(627\) −3.18509e6 −0.323559
\(628\) 8.93470e6 0.904026
\(629\) 1.73013e7 1.74362
\(630\) −1.29441e6 −0.129933
\(631\) −8.46233e6 −0.846090 −0.423045 0.906109i \(-0.639039\pi\)
−0.423045 + 0.906109i \(0.639039\pi\)
\(632\) −1.65007e7 −1.64327
\(633\) 1.31771e6 0.130711
\(634\) −8.38085e6 −0.828065
\(635\) 755652. 0.0743683
\(636\) 804885. 0.0789025
\(637\) −3.80903e6 −0.371934
\(638\) −2.67008e6 −0.259700
\(639\) −1.23464e6 −0.119616
\(640\) −976775. −0.0942638
\(641\) 1.51067e7 1.45219 0.726096 0.687593i \(-0.241333\pi\)
0.726096 + 0.687593i \(0.241333\pi\)
\(642\) −5.32767e6 −0.510152
\(643\) −2.65460e6 −0.253205 −0.126602 0.991954i \(-0.540407\pi\)
−0.126602 + 0.991954i \(0.540407\pi\)
\(644\) 1.38318e6 0.131421
\(645\) −4.34079e6 −0.410837
\(646\) 1.16692e7 1.10017
\(647\) −9.81722e6 −0.921993 −0.460997 0.887402i \(-0.652508\pi\)
−0.460997 + 0.887402i \(0.652508\pi\)
\(648\) 1.27327e6 0.119119
\(649\) −614353. −0.0572540
\(650\) 6.45578e6 0.599329
\(651\) 5.69848e6 0.526995
\(652\) 8.46687e6 0.780017
\(653\) −1.23492e7 −1.13333 −0.566664 0.823949i \(-0.691766\pi\)
−0.566664 + 0.823949i \(0.691766\pi\)
\(654\) 567204. 0.0518555
\(655\) −119882. −0.0109182
\(656\) −3.93292e6 −0.356825
\(657\) 5.55756e6 0.502309
\(658\) 3.49137e6 0.314362
\(659\) 1.03862e7 0.931627 0.465814 0.884883i \(-0.345762\pi\)
0.465814 + 0.884883i \(0.345762\pi\)
\(660\) −823048. −0.0735471
\(661\) 5.33228e6 0.474689 0.237345 0.971426i \(-0.423723\pi\)
0.237345 + 0.971426i \(0.423723\pi\)
\(662\) −1.11073e7 −0.985060
\(663\) 1.03451e7 0.914013
\(664\) 1.21024e7 1.06525
\(665\) 7.73064e6 0.677893
\(666\) 4.13771e6 0.361472
\(667\) 3.09021e6 0.268951
\(668\) 7.17659e6 0.622267
\(669\) −1.91442e6 −0.165376
\(670\) −7.85938e6 −0.676396
\(671\) −7.09567e6 −0.608397
\(672\) −4.80364e6 −0.410343
\(673\) 1.40382e7 1.19474 0.597371 0.801965i \(-0.296212\pi\)
0.597371 + 0.801965i \(0.296212\pi\)
\(674\) −1.27993e7 −1.08527
\(675\) 1.38671e6 0.117146
\(676\) −4.43166e6 −0.372992
\(677\) 2.13392e7 1.78940 0.894698 0.446671i \(-0.147390\pi\)
0.894698 + 0.446671i \(0.147390\pi\)
\(678\) 5.41714e6 0.452581
\(679\) −8.55484e6 −0.712094
\(680\) 9.52717e6 0.790117
\(681\) −509427. −0.0420934
\(682\) −4.20138e6 −0.345885
\(683\) 6.51986e6 0.534794 0.267397 0.963586i \(-0.413836\pi\)
0.267397 + 0.963586i \(0.413836\pi\)
\(684\) −2.40682e6 −0.196700
\(685\) −1.88583e6 −0.153559
\(686\) −9.80670e6 −0.795633
\(687\) 8.03717e6 0.649698
\(688\) 4.55497e6 0.366872
\(689\) 4.94144e6 0.396557
\(690\) −1.10450e6 −0.0883171
\(691\) 8.62667e6 0.687302 0.343651 0.939097i \(-0.388336\pi\)
0.343651 + 0.939097i \(0.388336\pi\)
\(692\) 1.17729e6 0.0934583
\(693\) −1.57606e6 −0.124664
\(694\) −6.60681e6 −0.520707
\(695\) 1.62363e6 0.127505
\(696\) −6.37480e6 −0.498819
\(697\) −1.67193e7 −1.30358
\(698\) −1.66971e7 −1.29719
\(699\) 9.79105e6 0.757943
\(700\) −3.10760e6 −0.239707
\(701\) 2.00997e7 1.54488 0.772440 0.635088i \(-0.219036\pi\)
0.772440 + 0.635088i \(0.219036\pi\)
\(702\) 2.47410e6 0.189485
\(703\) −2.47119e7 −1.88589
\(704\) 5.40672e6 0.411152
\(705\) 2.40439e6 0.182193
\(706\) 7.62401e6 0.575667
\(707\) −1.89771e7 −1.42784
\(708\) −464237. −0.0348062
\(709\) −5.65773e6 −0.422695 −0.211347 0.977411i \(-0.567785\pi\)
−0.211347 + 0.977411i \(0.567785\pi\)
\(710\) −2.20937e6 −0.164483
\(711\) −6.88713e6 −0.510934
\(712\) −2.64095e7 −1.95236
\(713\) 4.86246e6 0.358206
\(714\) 5.77418e6 0.423882
\(715\) −5.05295e6 −0.369641
\(716\) 6.18137e6 0.450611
\(717\) 2.97772e6 0.216314
\(718\) −9.40487e6 −0.680834
\(719\) 1.35569e7 0.977996 0.488998 0.872285i \(-0.337363\pi\)
0.488998 + 0.872285i \(0.337363\pi\)
\(720\) 935396. 0.0672457
\(721\) −6.40771e6 −0.459055
\(722\) −6.40368e6 −0.457179
\(723\) −478097. −0.0340150
\(724\) 5.18478e6 0.367607
\(725\) −6.94277e6 −0.490555
\(726\) −4.84616e6 −0.341237
\(727\) −1.97077e7 −1.38293 −0.691465 0.722410i \(-0.743034\pi\)
−0.691465 + 0.722410i \(0.743034\pi\)
\(728\) −1.75177e7 −1.22504
\(729\) 531441. 0.0370370
\(730\) 9.94513e6 0.690722
\(731\) 1.93637e7 1.34028
\(732\) −5.36186e6 −0.369860
\(733\) −475415. −0.0326823 −0.0163412 0.999866i \(-0.505202\pi\)
−0.0163412 + 0.999866i \(0.505202\pi\)
\(734\) 1.92577e7 1.31936
\(735\) −1.46412e6 −0.0999676
\(736\) −4.09890e6 −0.278916
\(737\) −9.56953e6 −0.648966
\(738\) −3.99853e6 −0.270246
\(739\) 1.72886e7 1.16453 0.582264 0.813000i \(-0.302167\pi\)
0.582264 + 0.813000i \(0.302167\pi\)
\(740\) −6.38570e6 −0.428676
\(741\) −1.47762e7 −0.988594
\(742\) 2.75808e6 0.183907
\(743\) 2.09012e7 1.38899 0.694494 0.719499i \(-0.255628\pi\)
0.694494 + 0.719499i \(0.255628\pi\)
\(744\) −1.00308e7 −0.664358
\(745\) −1.38126e7 −0.911767
\(746\) 1.73113e6 0.113889
\(747\) 5.05135e6 0.331212
\(748\) 3.67152e6 0.239934
\(749\) 1.57447e7 1.02548
\(750\) 6.55814e6 0.425724
\(751\) −1.10366e7 −0.714063 −0.357031 0.934092i \(-0.616211\pi\)
−0.357031 + 0.934092i \(0.616211\pi\)
\(752\) −2.52302e6 −0.162696
\(753\) 1.27271e7 0.817978
\(754\) −1.23870e7 −0.793481
\(755\) −1.25751e7 −0.802868
\(756\) −1.19095e6 −0.0757861
\(757\) −4.46889e6 −0.283439 −0.141720 0.989907i \(-0.545263\pi\)
−0.141720 + 0.989907i \(0.545263\pi\)
\(758\) 569016. 0.0359709
\(759\) −1.34484e6 −0.0847355
\(760\) −1.36079e7 −0.854589
\(761\) 1.50552e7 0.942379 0.471190 0.882032i \(-0.343825\pi\)
0.471190 + 0.882032i \(0.343825\pi\)
\(762\) −806165. −0.0502963
\(763\) −1.67624e6 −0.104237
\(764\) 2.41483e6 0.149676
\(765\) 3.97649e6 0.245667
\(766\) 1.43050e7 0.880881
\(767\) −2.85009e6 −0.174933
\(768\) 9.86499e6 0.603523
\(769\) −2.11014e7 −1.28675 −0.643377 0.765549i \(-0.722467\pi\)
−0.643377 + 0.765549i \(0.722467\pi\)
\(770\) −2.82032e6 −0.171424
\(771\) −41902.3 −0.00253864
\(772\) 6.05135e6 0.365434
\(773\) −2.11957e7 −1.27585 −0.637924 0.770099i \(-0.720207\pi\)
−0.637924 + 0.770099i \(0.720207\pi\)
\(774\) 4.63096e6 0.277855
\(775\) −1.09245e7 −0.653352
\(776\) 1.50587e7 0.897704
\(777\) −1.22280e7 −0.726613
\(778\) 1.91223e7 1.13264
\(779\) 2.38806e7 1.40995
\(780\) −3.81827e6 −0.224714
\(781\) −2.69011e6 −0.157813
\(782\) 4.92705e6 0.288118
\(783\) −2.66074e6 −0.155095
\(784\) 1.53636e6 0.0892696
\(785\) 2.10845e7 1.22121
\(786\) 127895. 0.00738411
\(787\) 9.79595e6 0.563780 0.281890 0.959447i \(-0.409039\pi\)
0.281890 + 0.959447i \(0.409039\pi\)
\(788\) 9.47117e6 0.543360
\(789\) 1.33147e6 0.0761448
\(790\) −1.23244e7 −0.702582
\(791\) −1.60091e7 −0.909756
\(792\) 2.77427e6 0.157158
\(793\) −3.29181e7 −1.85888
\(794\) −8.79569e6 −0.495130
\(795\) 1.89940e6 0.106586
\(796\) 2.03730e6 0.113965
\(797\) −8.45495e6 −0.471482 −0.235741 0.971816i \(-0.575752\pi\)
−0.235741 + 0.971816i \(0.575752\pi\)
\(798\) −8.24741e6 −0.458469
\(799\) −1.07257e7 −0.594371
\(800\) 9.20901e6 0.508730
\(801\) −1.10229e7 −0.607036
\(802\) 574078. 0.0315163
\(803\) 1.21091e7 0.662711
\(804\) −7.23123e6 −0.394523
\(805\) 3.26410e6 0.177531
\(806\) −1.94910e7 −1.05681
\(807\) −1.00217e7 −0.541696
\(808\) 3.34045e7 1.80002
\(809\) −2.42480e7 −1.30258 −0.651291 0.758828i \(-0.725772\pi\)
−0.651291 + 0.758828i \(0.725772\pi\)
\(810\) 951002. 0.0509294
\(811\) 4.71726e6 0.251848 0.125924 0.992040i \(-0.459811\pi\)
0.125924 + 0.992040i \(0.459811\pi\)
\(812\) 5.96268e6 0.317359
\(813\) 3.84469e6 0.204002
\(814\) 9.01548e6 0.476901
\(815\) 1.99805e7 1.05369
\(816\) −4.17269e6 −0.219377
\(817\) −2.76577e7 −1.44964
\(818\) 1.57379e6 0.0822361
\(819\) −7.31162e6 −0.380894
\(820\) 6.17091e6 0.320490
\(821\) −2.51555e7 −1.30249 −0.651246 0.758867i \(-0.725753\pi\)
−0.651246 + 0.758867i \(0.725753\pi\)
\(822\) 2.01189e6 0.103854
\(823\) 3.88001e6 0.199679 0.0998397 0.995004i \(-0.468167\pi\)
0.0998397 + 0.995004i \(0.468167\pi\)
\(824\) 1.12792e7 0.578710
\(825\) 3.02144e6 0.154554
\(826\) −1.59079e6 −0.0811266
\(827\) −1.75611e7 −0.892871 −0.446435 0.894816i \(-0.647307\pi\)
−0.446435 + 0.894816i \(0.647307\pi\)
\(828\) −1.01623e6 −0.0515128
\(829\) −1.13811e7 −0.575173 −0.287587 0.957755i \(-0.592853\pi\)
−0.287587 + 0.957755i \(0.592853\pi\)
\(830\) 9.03928e6 0.455448
\(831\) 1.52419e7 0.765661
\(832\) 2.50827e7 1.25622
\(833\) 6.53127e6 0.326126
\(834\) −1.73217e6 −0.0862332
\(835\) 1.69356e7 0.840591
\(836\) −5.24412e6 −0.259512
\(837\) −4.18669e6 −0.206565
\(838\) −1.21875e7 −0.599520
\(839\) 1.12946e7 0.553943 0.276972 0.960878i \(-0.410669\pi\)
0.276972 + 0.960878i \(0.410669\pi\)
\(840\) −6.73351e6 −0.329263
\(841\) −7.18977e6 −0.350530
\(842\) 1.02085e7 0.496227
\(843\) −2.00720e7 −0.972796
\(844\) 2.16955e6 0.104837
\(845\) −1.04580e7 −0.503857
\(846\) −2.56511e6 −0.123220
\(847\) 1.43217e7 0.685938
\(848\) −1.99312e6 −0.0951795
\(849\) 1.65904e7 0.789927
\(850\) −1.10696e7 −0.525515
\(851\) −1.04341e7 −0.493889
\(852\) −2.03279e6 −0.0959384
\(853\) 1.36639e7 0.642985 0.321493 0.946912i \(-0.395815\pi\)
0.321493 + 0.946912i \(0.395815\pi\)
\(854\) −1.83734e7 −0.862073
\(855\) −5.67972e6 −0.265712
\(856\) −2.77146e7 −1.29278
\(857\) −2.02331e7 −0.941044 −0.470522 0.882388i \(-0.655934\pi\)
−0.470522 + 0.882388i \(0.655934\pi\)
\(858\) 5.39072e6 0.249993
\(859\) 2.02393e7 0.935861 0.467931 0.883765i \(-0.345000\pi\)
0.467931 + 0.883765i \(0.345000\pi\)
\(860\) −7.14693e6 −0.329514
\(861\) 1.18167e7 0.543236
\(862\) −3.70604e6 −0.169880
\(863\) −2.16287e7 −0.988560 −0.494280 0.869303i \(-0.664568\pi\)
−0.494280 + 0.869303i \(0.664568\pi\)
\(864\) 3.52925e6 0.160841
\(865\) 2.77822e6 0.126248
\(866\) 9.40835e6 0.426303
\(867\) −4.95990e6 −0.224091
\(868\) 9.38231e6 0.422679
\(869\) −1.50061e7 −0.674090
\(870\) −4.76133e6 −0.213270
\(871\) −4.43948e7 −1.98283
\(872\) 2.95060e6 0.131407
\(873\) 6.28526e6 0.279118
\(874\) −7.03744e6 −0.311628
\(875\) −1.93810e7 −0.855769
\(876\) 9.15028e6 0.402879
\(877\) −1.99308e7 −0.875034 −0.437517 0.899210i \(-0.644142\pi\)
−0.437517 + 0.899210i \(0.644142\pi\)
\(878\) 2.17049e7 0.950216
\(879\) −2.93879e6 −0.128291
\(880\) 2.03810e6 0.0887192
\(881\) −9.32845e6 −0.404920 −0.202460 0.979290i \(-0.564894\pi\)
−0.202460 + 0.979290i \(0.564894\pi\)
\(882\) 1.56199e6 0.0676095
\(883\) −1.36534e7 −0.589302 −0.294651 0.955605i \(-0.595203\pi\)
−0.294651 + 0.955605i \(0.595203\pi\)
\(884\) 1.70328e7 0.733088
\(885\) −1.09553e6 −0.0470180
\(886\) −363560. −0.0155594
\(887\) −1.94974e7 −0.832085 −0.416042 0.909345i \(-0.636583\pi\)
−0.416042 + 0.909345i \(0.636583\pi\)
\(888\) 2.15244e7 0.916008
\(889\) 2.38243e6 0.101103
\(890\) −1.97252e7 −0.834733
\(891\) 1.15793e6 0.0488641
\(892\) −3.15201e6 −0.132640
\(893\) 1.53198e7 0.642870
\(894\) 1.47359e7 0.616641
\(895\) 1.45871e7 0.608709
\(896\) −3.07959e6 −0.128151
\(897\) −6.23894e6 −0.258899
\(898\) 9.12468e6 0.377596
\(899\) 2.09613e7 0.865004
\(900\) 2.28316e6 0.0939572
\(901\) −8.47299e6 −0.347716
\(902\) −8.71223e6 −0.356544
\(903\) −1.36857e7 −0.558531
\(904\) 2.81801e7 1.14689
\(905\) 1.22353e7 0.496583
\(906\) 1.34157e7 0.542991
\(907\) −2.93504e7 −1.18467 −0.592333 0.805693i \(-0.701793\pi\)
−0.592333 + 0.805693i \(0.701793\pi\)
\(908\) −838750. −0.0337612
\(909\) 1.39425e7 0.559669
\(910\) −1.30840e7 −0.523765
\(911\) −4.70892e7 −1.87986 −0.939929 0.341370i \(-0.889109\pi\)
−0.939929 + 0.341370i \(0.889109\pi\)
\(912\) 5.95995e6 0.237277
\(913\) 1.10062e7 0.436978
\(914\) −2.69920e7 −1.06873
\(915\) −1.26531e7 −0.499627
\(916\) 1.32329e7 0.521093
\(917\) −377964. −0.0148432
\(918\) −4.24230e6 −0.166148
\(919\) 658785. 0.0257309 0.0128654 0.999917i \(-0.495905\pi\)
0.0128654 + 0.999917i \(0.495905\pi\)
\(920\) −5.74564e6 −0.223805
\(921\) 1.03474e7 0.401961
\(922\) −1.50203e7 −0.581903
\(923\) −1.24799e7 −0.482177
\(924\) −2.59491e6 −0.0999869
\(925\) 2.34422e7 0.900832
\(926\) 4.87498e6 0.186829
\(927\) 4.70776e6 0.179935
\(928\) −1.76697e7 −0.673533
\(929\) 4.12467e7 1.56801 0.784006 0.620753i \(-0.213173\pi\)
0.784006 + 0.620753i \(0.213173\pi\)
\(930\) −7.49198e6 −0.284046
\(931\) −9.32878e6 −0.352737
\(932\) 1.61205e7 0.607911
\(933\) 1.63992e7 0.616764
\(934\) 9.81323e6 0.368082
\(935\) 8.66420e6 0.324115
\(936\) 1.28703e7 0.480175
\(937\) −2.66656e7 −0.992208 −0.496104 0.868263i \(-0.665236\pi\)
−0.496104 + 0.868263i \(0.665236\pi\)
\(938\) −2.47791e7 −0.919557
\(939\) 1.26292e7 0.467424
\(940\) 3.95872e6 0.146129
\(941\) −6.50842e6 −0.239608 −0.119804 0.992798i \(-0.538227\pi\)
−0.119804 + 0.992798i \(0.538227\pi\)
\(942\) −2.24939e7 −0.825919
\(943\) 1.00831e7 0.369245
\(944\) 1.14958e6 0.0419864
\(945\) −2.81046e6 −0.102376
\(946\) 1.00902e7 0.366583
\(947\) −3.13849e7 −1.13722 −0.568612 0.822606i \(-0.692519\pi\)
−0.568612 + 0.822606i \(0.692519\pi\)
\(948\) −1.13394e7 −0.409796
\(949\) 5.61764e7 2.02483
\(950\) 1.58110e7 0.568395
\(951\) −1.81968e7 −0.652445
\(952\) 3.00374e7 1.07416
\(953\) 1.79573e7 0.640485 0.320242 0.947336i \(-0.396236\pi\)
0.320242 + 0.947336i \(0.396236\pi\)
\(954\) −2.02637e6 −0.0720855
\(955\) 5.69862e6 0.202191
\(956\) 4.90268e6 0.173496
\(957\) −5.79737e6 −0.204621
\(958\) 4.11927e7 1.45013
\(959\) −5.94565e6 −0.208763
\(960\) 9.64136e6 0.337645
\(961\) 4.35351e6 0.152066
\(962\) 4.18244e7 1.45711
\(963\) −1.15676e7 −0.401956
\(964\) −787166. −0.0272818
\(965\) 1.42802e7 0.493648
\(966\) −3.48229e6 −0.120067
\(967\) 2.71548e7 0.933858 0.466929 0.884295i \(-0.345360\pi\)
0.466929 + 0.884295i \(0.345360\pi\)
\(968\) −2.52098e7 −0.864730
\(969\) 2.53365e7 0.866837
\(970\) 1.12473e7 0.383813
\(971\) −5.34589e7 −1.81958 −0.909792 0.415065i \(-0.863759\pi\)
−0.909792 + 0.415065i \(0.863759\pi\)
\(972\) 874995. 0.0297057
\(973\) 5.11900e6 0.173342
\(974\) −5.00031e6 −0.168888
\(975\) 1.40170e7 0.472220
\(976\) 1.32774e7 0.446159
\(977\) 5.95524e6 0.199601 0.0998006 0.995007i \(-0.468180\pi\)
0.0998006 + 0.995007i \(0.468180\pi\)
\(978\) −2.13161e7 −0.712624
\(979\) −2.40173e7 −0.800881
\(980\) −2.41062e6 −0.0801794
\(981\) 1.23153e6 0.0408577
\(982\) 3.14520e7 1.04081
\(983\) −3.53361e7 −1.16636 −0.583182 0.812341i \(-0.698193\pi\)
−0.583182 + 0.812341i \(0.698193\pi\)
\(984\) −2.08004e7 −0.684833
\(985\) 2.23505e7 0.734000
\(986\) 2.12397e7 0.695755
\(987\) 7.58058e6 0.247691
\(988\) −2.43284e7 −0.792906
\(989\) −1.16779e7 −0.379641
\(990\) 2.07210e6 0.0671927
\(991\) 7.02723e6 0.227300 0.113650 0.993521i \(-0.463746\pi\)
0.113650 + 0.993521i \(0.463746\pi\)
\(992\) −2.78034e7 −0.897053
\(993\) −2.41165e7 −0.776143
\(994\) −6.96571e6 −0.223614
\(995\) 4.80772e6 0.153951
\(996\) 8.31683e6 0.265650
\(997\) 2.73040e6 0.0869939 0.0434970 0.999054i \(-0.486150\pi\)
0.0434970 + 0.999054i \(0.486150\pi\)
\(998\) −7.98831e6 −0.253880
\(999\) 8.98395e6 0.284809
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.1.4 12
3.2 odd 2 531.6.a.d.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.4 12 1.1 even 1 trivial
531.6.a.d.1.9 12 3.2 odd 2