Properties

Label 177.6.a.a.1.6
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + 13849341 x^{3} - 23890558 x^{2} - 74443300 x - 14846072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.216241\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.21624 q^{2} +9.00000 q^{3} -30.5208 q^{4} +0.914506 q^{5} -10.9462 q^{6} +61.7584 q^{7} +76.0403 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.21624 q^{2} +9.00000 q^{3} -30.5208 q^{4} +0.914506 q^{5} -10.9462 q^{6} +61.7584 q^{7} +76.0403 q^{8} +81.0000 q^{9} -1.11226 q^{10} -311.532 q^{11} -274.687 q^{12} -150.260 q^{13} -75.1131 q^{14} +8.23055 q^{15} +884.181 q^{16} +1232.93 q^{17} -98.5155 q^{18} -266.956 q^{19} -27.9114 q^{20} +555.825 q^{21} +378.898 q^{22} -1920.26 q^{23} +684.363 q^{24} -3124.16 q^{25} +182.752 q^{26} +729.000 q^{27} -1884.91 q^{28} -2981.52 q^{29} -10.0103 q^{30} -8356.31 q^{31} -3508.67 q^{32} -2803.79 q^{33} -1499.54 q^{34} +56.4784 q^{35} -2472.18 q^{36} +8500.15 q^{37} +324.682 q^{38} -1352.34 q^{39} +69.5393 q^{40} +13953.0 q^{41} -676.018 q^{42} -11289.0 q^{43} +9508.19 q^{44} +74.0750 q^{45} +2335.50 q^{46} -24855.4 q^{47} +7957.63 q^{48} -12992.9 q^{49} +3799.74 q^{50} +11096.3 q^{51} +4586.04 q^{52} -35346.4 q^{53} -886.640 q^{54} -284.898 q^{55} +4696.13 q^{56} -2402.60 q^{57} +3626.25 q^{58} +3481.00 q^{59} -251.203 q^{60} -46879.0 q^{61} +10163.3 q^{62} +5002.43 q^{63} -24026.4 q^{64} -137.413 q^{65} +3410.08 q^{66} +37659.9 q^{67} -37629.8 q^{68} -17282.3 q^{69} -68.6913 q^{70} -56413.7 q^{71} +6159.26 q^{72} +66253.6 q^{73} -10338.2 q^{74} -28117.5 q^{75} +8147.69 q^{76} -19239.7 q^{77} +1644.77 q^{78} -51579.2 q^{79} +808.588 q^{80} +6561.00 q^{81} -16970.2 q^{82} -28845.7 q^{83} -16964.2 q^{84} +1127.52 q^{85} +13730.1 q^{86} -26833.7 q^{87} -23689.0 q^{88} -9518.85 q^{89} -90.0930 q^{90} -9279.79 q^{91} +58607.8 q^{92} -75206.8 q^{93} +30230.2 q^{94} -244.132 q^{95} -31578.0 q^{96} +96360.6 q^{97} +15802.5 q^{98} -25234.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} + O(q^{10}) \) \( 11q - 6q^{2} + 99q^{3} + 150q^{4} - 192q^{5} - 54q^{6} - 371q^{7} - 621q^{8} + 891q^{9} - 399q^{10} - 698q^{11} + 1350q^{12} - 1556q^{13} - 1679q^{14} - 1728q^{15} - 2662q^{16} - 4793q^{17} - 486q^{18} - 3753q^{19} - 11023q^{20} - 3339q^{21} - 9534q^{22} - 7323q^{23} - 5589q^{24} + 7867q^{25} - 4844q^{26} + 8019q^{27} + 3650q^{28} - 15467q^{29} - 3591q^{30} - 5151q^{31} - 15368q^{32} - 6282q^{33} + 8452q^{34} - 23285q^{35} + 12150q^{36} + 8623q^{37} + 15205q^{38} - 14004q^{39} + 41530q^{40} - 6369q^{41} - 15111q^{42} - 20506q^{43} - 55632q^{44} - 15552q^{45} - 45191q^{46} - 47899q^{47} - 23958q^{48} - 10322q^{49} - 102147q^{50} - 43137q^{51} - 292q^{52} - 80048q^{53} - 4374q^{54} - 2114q^{55} - 108126q^{56} - 33777q^{57} - 58294q^{58} + 38291q^{59} - 99207q^{60} - 82527q^{61} - 67438q^{62} - 30051q^{63} - 51411q^{64} - 167646q^{65} - 85806q^{66} - 166976q^{67} - 136533q^{68} - 65907q^{69} + 76140q^{70} - 183560q^{71} - 50301q^{72} - 36809q^{73} - 116686q^{74} + 70803q^{75} + 55580q^{76} - 164885q^{77} - 43596q^{78} - 281518q^{79} - 32683q^{80} + 72171q^{81} + 178815q^{82} - 254691q^{83} + 32850q^{84} + 4763q^{85} + 349324q^{86} - 139203q^{87} + 251285q^{88} - 89687q^{89} - 32319q^{90} + 34897q^{91} - 20240q^{92} - 46359q^{93} + 96548q^{94} - 155113q^{95} - 138312q^{96} - 45828q^{97} + 465864q^{98} - 56538q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21624 −0.215003 −0.107502 0.994205i \(-0.534285\pi\)
−0.107502 + 0.994205i \(0.534285\pi\)
\(3\) 9.00000 0.577350
\(4\) −30.5208 −0.953774
\(5\) 0.914506 0.0163592 0.00817959 0.999967i \(-0.497396\pi\)
0.00817959 + 0.999967i \(0.497396\pi\)
\(6\) −10.9462 −0.124132
\(7\) 61.7584 0.476377 0.238188 0.971219i \(-0.423446\pi\)
0.238188 + 0.971219i \(0.423446\pi\)
\(8\) 76.0403 0.420067
\(9\) 81.0000 0.333333
\(10\) −1.11226 −0.00351727
\(11\) −311.532 −0.776284 −0.388142 0.921599i \(-0.626883\pi\)
−0.388142 + 0.921599i \(0.626883\pi\)
\(12\) −274.687 −0.550661
\(13\) −150.260 −0.246595 −0.123297 0.992370i \(-0.539347\pi\)
−0.123297 + 0.992370i \(0.539347\pi\)
\(14\) −75.1131 −0.102422
\(15\) 8.23055 0.00944497
\(16\) 884.181 0.863458
\(17\) 1232.93 1.03470 0.517350 0.855774i \(-0.326918\pi\)
0.517350 + 0.855774i \(0.326918\pi\)
\(18\) −98.5155 −0.0716677
\(19\) −266.956 −0.169651 −0.0848253 0.996396i \(-0.527033\pi\)
−0.0848253 + 0.996396i \(0.527033\pi\)
\(20\) −27.9114 −0.0156029
\(21\) 555.825 0.275036
\(22\) 378.898 0.166904
\(23\) −1920.26 −0.756903 −0.378451 0.925621i \(-0.623543\pi\)
−0.378451 + 0.925621i \(0.623543\pi\)
\(24\) 684.363 0.242526
\(25\) −3124.16 −0.999732
\(26\) 182.752 0.0530187
\(27\) 729.000 0.192450
\(28\) −1884.91 −0.454356
\(29\) −2981.52 −0.658329 −0.329165 0.944273i \(-0.606767\pi\)
−0.329165 + 0.944273i \(0.606767\pi\)
\(30\) −10.0103 −0.00203070
\(31\) −8356.31 −1.56175 −0.780873 0.624689i \(-0.785226\pi\)
−0.780873 + 0.624689i \(0.785226\pi\)
\(32\) −3508.67 −0.605713
\(33\) −2803.79 −0.448188
\(34\) −1499.54 −0.222464
\(35\) 56.4784 0.00779313
\(36\) −2472.18 −0.317925
\(37\) 8500.15 1.02076 0.510378 0.859950i \(-0.329505\pi\)
0.510378 + 0.859950i \(0.329505\pi\)
\(38\) 324.682 0.0364754
\(39\) −1352.34 −0.142372
\(40\) 69.5393 0.00687195
\(41\) 13953.0 1.29631 0.648155 0.761509i \(-0.275541\pi\)
0.648155 + 0.761509i \(0.275541\pi\)
\(42\) −676.018 −0.0591337
\(43\) −11289.0 −0.931071 −0.465535 0.885029i \(-0.654138\pi\)
−0.465535 + 0.885029i \(0.654138\pi\)
\(44\) 9508.19 0.740400
\(45\) 74.0750 0.00545306
\(46\) 2335.50 0.162736
\(47\) −24855.4 −1.64126 −0.820628 0.571463i \(-0.806376\pi\)
−0.820628 + 0.571463i \(0.806376\pi\)
\(48\) 7957.63 0.498518
\(49\) −12992.9 −0.773065
\(50\) 3799.74 0.214946
\(51\) 11096.3 0.597385
\(52\) 4586.04 0.235196
\(53\) −35346.4 −1.72844 −0.864222 0.503111i \(-0.832189\pi\)
−0.864222 + 0.503111i \(0.832189\pi\)
\(54\) −886.640 −0.0413774
\(55\) −284.898 −0.0126994
\(56\) 4696.13 0.200110
\(57\) −2402.60 −0.0979478
\(58\) 3626.25 0.141543
\(59\) 3481.00 0.130189
\(60\) −251.203 −0.00900837
\(61\) −46879.0 −1.61307 −0.806536 0.591185i \(-0.798660\pi\)
−0.806536 + 0.591185i \(0.798660\pi\)
\(62\) 10163.3 0.335780
\(63\) 5002.43 0.158792
\(64\) −24026.4 −0.733228
\(65\) −137.413 −0.00403409
\(66\) 3410.08 0.0963618
\(67\) 37659.9 1.02492 0.512462 0.858710i \(-0.328734\pi\)
0.512462 + 0.858710i \(0.328734\pi\)
\(68\) −37629.8 −0.986870
\(69\) −17282.3 −0.436998
\(70\) −68.6913 −0.00167555
\(71\) −56413.7 −1.32812 −0.664062 0.747677i \(-0.731169\pi\)
−0.664062 + 0.747677i \(0.731169\pi\)
\(72\) 6159.26 0.140022
\(73\) 66253.6 1.45513 0.727566 0.686037i \(-0.240651\pi\)
0.727566 + 0.686037i \(0.240651\pi\)
\(74\) −10338.2 −0.219466
\(75\) −28117.5 −0.577196
\(76\) 8147.69 0.161808
\(77\) −19239.7 −0.369804
\(78\) 1644.77 0.0306103
\(79\) −51579.2 −0.929837 −0.464918 0.885354i \(-0.653916\pi\)
−0.464918 + 0.885354i \(0.653916\pi\)
\(80\) 808.588 0.0141255
\(81\) 6561.00 0.111111
\(82\) −16970.2 −0.278710
\(83\) −28845.7 −0.459606 −0.229803 0.973237i \(-0.573808\pi\)
−0.229803 + 0.973237i \(0.573808\pi\)
\(84\) −16964.2 −0.262322
\(85\) 1127.52 0.0169269
\(86\) 13730.1 0.200183
\(87\) −26833.7 −0.380087
\(88\) −23689.0 −0.326092
\(89\) −9518.85 −0.127382 −0.0636912 0.997970i \(-0.520287\pi\)
−0.0636912 + 0.997970i \(0.520287\pi\)
\(90\) −90.0930 −0.00117242
\(91\) −9279.79 −0.117472
\(92\) 58607.8 0.721914
\(93\) −75206.8 −0.901675
\(94\) 30230.2 0.352875
\(95\) −244.132 −0.00277534
\(96\) −31578.0 −0.349709
\(97\) 96360.6 1.03985 0.519924 0.854212i \(-0.325960\pi\)
0.519924 + 0.854212i \(0.325960\pi\)
\(98\) 15802.5 0.166211
\(99\) −25234.1 −0.258761
\(100\) 95351.8 0.953518
\(101\) −2640.59 −0.0257571 −0.0128785 0.999917i \(-0.504099\pi\)
−0.0128785 + 0.999917i \(0.504099\pi\)
\(102\) −13495.8 −0.128440
\(103\) 117451. 1.09085 0.545425 0.838160i \(-0.316368\pi\)
0.545425 + 0.838160i \(0.316368\pi\)
\(104\) −11425.8 −0.103586
\(105\) 508.305 0.00449937
\(106\) 42989.7 0.371621
\(107\) −228868. −1.93253 −0.966263 0.257558i \(-0.917082\pi\)
−0.966263 + 0.257558i \(0.917082\pi\)
\(108\) −22249.6 −0.183554
\(109\) 163809. 1.32060 0.660301 0.751001i \(-0.270429\pi\)
0.660301 + 0.751001i \(0.270429\pi\)
\(110\) 346.504 0.00273040
\(111\) 76501.3 0.589334
\(112\) 54605.6 0.411331
\(113\) 16245.7 0.119686 0.0598430 0.998208i \(-0.480940\pi\)
0.0598430 + 0.998208i \(0.480940\pi\)
\(114\) 2922.14 0.0210591
\(115\) −1756.09 −0.0123823
\(116\) 90998.4 0.627897
\(117\) −12171.0 −0.0821983
\(118\) −4233.73 −0.0279910
\(119\) 76143.5 0.492908
\(120\) 625.854 0.00396752
\(121\) −63998.8 −0.397382
\(122\) 57016.2 0.346816
\(123\) 125577. 0.748424
\(124\) 255041. 1.48955
\(125\) −5714.90 −0.0327140
\(126\) −6084.16 −0.0341408
\(127\) 285457. 1.57047 0.785237 0.619195i \(-0.212541\pi\)
0.785237 + 0.619195i \(0.212541\pi\)
\(128\) 141499. 0.763360
\(129\) −101601. −0.537554
\(130\) 167.128 0.000867341 0
\(131\) 153837. 0.783217 0.391609 0.920132i \(-0.371919\pi\)
0.391609 + 0.920132i \(0.371919\pi\)
\(132\) 85573.7 0.427470
\(133\) −16486.8 −0.0808176
\(134\) −45803.5 −0.220362
\(135\) 666.675 0.00314832
\(136\) 93752.1 0.434644
\(137\) −281641. −1.28202 −0.641010 0.767532i \(-0.721484\pi\)
−0.641010 + 0.767532i \(0.721484\pi\)
\(138\) 21019.5 0.0939559
\(139\) 171456. 0.752689 0.376344 0.926480i \(-0.377181\pi\)
0.376344 + 0.926480i \(0.377181\pi\)
\(140\) −1723.76 −0.00743289
\(141\) −223699. −0.947580
\(142\) 68612.7 0.285551
\(143\) 46810.7 0.191428
\(144\) 71618.7 0.287819
\(145\) −2726.62 −0.0107697
\(146\) −80580.4 −0.312858
\(147\) −116936. −0.446329
\(148\) −259431. −0.973570
\(149\) 246577. 0.909885 0.454942 0.890521i \(-0.349660\pi\)
0.454942 + 0.890521i \(0.349660\pi\)
\(150\) 34197.6 0.124099
\(151\) 161787. 0.577433 0.288717 0.957415i \(-0.406771\pi\)
0.288717 + 0.957415i \(0.406771\pi\)
\(152\) −20299.4 −0.0712646
\(153\) 99867.0 0.344900
\(154\) 23400.1 0.0795090
\(155\) −7641.90 −0.0255489
\(156\) 41274.4 0.135790
\(157\) −212289. −0.687351 −0.343675 0.939089i \(-0.611672\pi\)
−0.343675 + 0.939089i \(0.611672\pi\)
\(158\) 62732.7 0.199918
\(159\) −318117. −0.997917
\(160\) −3208.70 −0.00990897
\(161\) −118592. −0.360571
\(162\) −7979.76 −0.0238892
\(163\) 428153. 1.26221 0.631103 0.775699i \(-0.282603\pi\)
0.631103 + 0.775699i \(0.282603\pi\)
\(164\) −425857. −1.23639
\(165\) −2564.08 −0.00733199
\(166\) 35083.3 0.0988166
\(167\) 260524. 0.722863 0.361432 0.932399i \(-0.382288\pi\)
0.361432 + 0.932399i \(0.382288\pi\)
\(168\) 42265.1 0.115534
\(169\) −348715. −0.939191
\(170\) −1371.33 −0.00363932
\(171\) −21623.4 −0.0565502
\(172\) 344547. 0.888031
\(173\) 375123. 0.952924 0.476462 0.879195i \(-0.341919\pi\)
0.476462 + 0.879195i \(0.341919\pi\)
\(174\) 32636.3 0.0817198
\(175\) −192943. −0.476249
\(176\) −275451. −0.670289
\(177\) 31329.0 0.0751646
\(178\) 11577.2 0.0273876
\(179\) −569801. −1.32920 −0.664601 0.747199i \(-0.731398\pi\)
−0.664601 + 0.747199i \(0.731398\pi\)
\(180\) −2260.82 −0.00520098
\(181\) −532297. −1.20770 −0.603848 0.797099i \(-0.706367\pi\)
−0.603848 + 0.797099i \(0.706367\pi\)
\(182\) 11286.5 0.0252569
\(183\) −421911. −0.931308
\(184\) −146017. −0.317950
\(185\) 7773.43 0.0166987
\(186\) 91469.6 0.193863
\(187\) −384096. −0.803222
\(188\) 758606. 1.56539
\(189\) 45021.9 0.0916788
\(190\) 296.924 0.000596707 0
\(191\) −756470. −1.50040 −0.750202 0.661209i \(-0.770044\pi\)
−0.750202 + 0.661209i \(0.770044\pi\)
\(192\) −216238. −0.423329
\(193\) 715629. 1.38291 0.691456 0.722418i \(-0.256969\pi\)
0.691456 + 0.722418i \(0.256969\pi\)
\(194\) −117198. −0.223571
\(195\) −1236.72 −0.00232908
\(196\) 396553. 0.737329
\(197\) 330788. 0.607273 0.303637 0.952788i \(-0.401799\pi\)
0.303637 + 0.952788i \(0.401799\pi\)
\(198\) 30690.7 0.0556345
\(199\) −145072. −0.259687 −0.129844 0.991534i \(-0.541448\pi\)
−0.129844 + 0.991534i \(0.541448\pi\)
\(200\) −237562. −0.419955
\(201\) 338939. 0.591740
\(202\) 3211.59 0.00553785
\(203\) −184134. −0.313613
\(204\) −338669. −0.569770
\(205\) 12760.1 0.0212065
\(206\) −142849. −0.234536
\(207\) −155541. −0.252301
\(208\) −132857. −0.212924
\(209\) 83165.2 0.131697
\(210\) −618.222 −0.000967378 0
\(211\) 836441. 1.29339 0.646695 0.762749i \(-0.276151\pi\)
0.646695 + 0.762749i \(0.276151\pi\)
\(212\) 1.07880e6 1.64854
\(213\) −507724. −0.766793
\(214\) 278358. 0.415499
\(215\) −10323.8 −0.0152315
\(216\) 55433.4 0.0808420
\(217\) −516072. −0.743980
\(218\) −199231. −0.283933
\(219\) 596283. 0.840121
\(220\) 8695.29 0.0121123
\(221\) −185259. −0.255152
\(222\) −93044.0 −0.126709
\(223\) 734281. 0.988781 0.494391 0.869240i \(-0.335391\pi\)
0.494391 + 0.869240i \(0.335391\pi\)
\(224\) −216690. −0.288548
\(225\) −253057. −0.333244
\(226\) −19758.7 −0.0257329
\(227\) 370264. 0.476922 0.238461 0.971152i \(-0.423357\pi\)
0.238461 + 0.971152i \(0.423357\pi\)
\(228\) 73329.2 0.0934200
\(229\) −690656. −0.870309 −0.435154 0.900356i \(-0.643306\pi\)
−0.435154 + 0.900356i \(0.643306\pi\)
\(230\) 2135.83 0.00266223
\(231\) −173157. −0.213506
\(232\) −226716. −0.276543
\(233\) −924763. −1.11594 −0.557970 0.829861i \(-0.688419\pi\)
−0.557970 + 0.829861i \(0.688419\pi\)
\(234\) 14802.9 0.0176729
\(235\) −22730.4 −0.0268496
\(236\) −106243. −0.124171
\(237\) −464213. −0.536841
\(238\) −92608.9 −0.105977
\(239\) −568443. −0.643713 −0.321856 0.946788i \(-0.604307\pi\)
−0.321856 + 0.946788i \(0.604307\pi\)
\(240\) 7277.30 0.00815534
\(241\) −292121. −0.323981 −0.161990 0.986792i \(-0.551791\pi\)
−0.161990 + 0.986792i \(0.551791\pi\)
\(242\) 77838.0 0.0854384
\(243\) 59049.0 0.0641500
\(244\) 1.43078e6 1.53851
\(245\) −11882.1 −0.0126467
\(246\) −152732. −0.160914
\(247\) 40112.7 0.0418350
\(248\) −635417. −0.656039
\(249\) −259611. −0.265353
\(250\) 6950.69 0.00703360
\(251\) −1.76450e6 −1.76781 −0.883907 0.467663i \(-0.845096\pi\)
−0.883907 + 0.467663i \(0.845096\pi\)
\(252\) −152678. −0.151452
\(253\) 598222. 0.587572
\(254\) −347184. −0.337657
\(255\) 10147.7 0.00977272
\(256\) 596748. 0.569103
\(257\) −824695. −0.778862 −0.389431 0.921056i \(-0.627328\pi\)
−0.389431 + 0.921056i \(0.627328\pi\)
\(258\) 123571. 0.115576
\(259\) 524955. 0.486265
\(260\) 4193.96 0.00384761
\(261\) −241503. −0.219443
\(262\) −187103. −0.168394
\(263\) −915456. −0.816109 −0.408055 0.912958i \(-0.633793\pi\)
−0.408055 + 0.912958i \(0.633793\pi\)
\(264\) −213201. −0.188269
\(265\) −32324.4 −0.0282759
\(266\) 20051.9 0.0173760
\(267\) −85669.7 −0.0735443
\(268\) −1.14941e6 −0.977546
\(269\) 1.00139e6 0.843769 0.421884 0.906650i \(-0.361369\pi\)
0.421884 + 0.906650i \(0.361369\pi\)
\(270\) −810.837 −0.000676899 0
\(271\) −2.25170e6 −1.86246 −0.931232 0.364428i \(-0.881265\pi\)
−0.931232 + 0.364428i \(0.881265\pi\)
\(272\) 1.09013e6 0.893421
\(273\) −83518.2 −0.0678226
\(274\) 342544. 0.275638
\(275\) 973277. 0.776077
\(276\) 527470. 0.416797
\(277\) 5282.15 0.00413629 0.00206815 0.999998i \(-0.499342\pi\)
0.00206815 + 0.999998i \(0.499342\pi\)
\(278\) −208532. −0.161830
\(279\) −676861. −0.520582
\(280\) 4294.63 0.00327364
\(281\) −2.48246e6 −1.87550 −0.937748 0.347317i \(-0.887093\pi\)
−0.937748 + 0.347317i \(0.887093\pi\)
\(282\) 272072. 0.203733
\(283\) −87005.1 −0.0645771 −0.0322886 0.999479i \(-0.510280\pi\)
−0.0322886 + 0.999479i \(0.510280\pi\)
\(284\) 1.72179e6 1.26673
\(285\) −2197.19 −0.00160234
\(286\) −56933.1 −0.0411576
\(287\) 861716. 0.617532
\(288\) −284202. −0.201904
\(289\) 100250. 0.0706059
\(290\) 3316.23 0.00231552
\(291\) 867245. 0.600357
\(292\) −2.02211e6 −1.38787
\(293\) 165504. 0.112626 0.0563131 0.998413i \(-0.482065\pi\)
0.0563131 + 0.998413i \(0.482065\pi\)
\(294\) 142223. 0.0959622
\(295\) 3183.39 0.00212978
\(296\) 646354. 0.428786
\(297\) −227107. −0.149396
\(298\) −299897. −0.195628
\(299\) 288538. 0.186648
\(300\) 858167. 0.550514
\(301\) −697188. −0.443541
\(302\) −196772. −0.124150
\(303\) −23765.3 −0.0148709
\(304\) −236037. −0.146486
\(305\) −42871.1 −0.0263885
\(306\) −121462. −0.0741546
\(307\) 746565. 0.452087 0.226043 0.974117i \(-0.427421\pi\)
0.226043 + 0.974117i \(0.427421\pi\)
\(308\) 587210. 0.352709
\(309\) 1.05706e6 0.629802
\(310\) 9294.39 0.00549309
\(311\) 638206. 0.374162 0.187081 0.982344i \(-0.440097\pi\)
0.187081 + 0.982344i \(0.440097\pi\)
\(312\) −102832. −0.0598057
\(313\) 1.55232e6 0.895615 0.447808 0.894130i \(-0.352205\pi\)
0.447808 + 0.894130i \(0.352205\pi\)
\(314\) 258195. 0.147782
\(315\) 4574.75 0.00259771
\(316\) 1.57424e6 0.886854
\(317\) 1.97404e6 1.10334 0.551669 0.834063i \(-0.313991\pi\)
0.551669 + 0.834063i \(0.313991\pi\)
\(318\) 386907. 0.214555
\(319\) 928840. 0.511051
\(320\) −21972.3 −0.0119950
\(321\) −2.05981e6 −1.11574
\(322\) 144237. 0.0775239
\(323\) −329137. −0.175538
\(324\) −200247. −0.105975
\(325\) 469436. 0.246529
\(326\) −520737. −0.271378
\(327\) 1.47428e6 0.762450
\(328\) 1.06099e6 0.544537
\(329\) −1.53503e6 −0.781857
\(330\) 3118.54 0.00157640
\(331\) −2.40736e6 −1.20774 −0.603868 0.797084i \(-0.706375\pi\)
−0.603868 + 0.797084i \(0.706375\pi\)
\(332\) 880391. 0.438360
\(333\) 688512. 0.340252
\(334\) −316860. −0.155418
\(335\) 34440.2 0.0167669
\(336\) 491450. 0.237482
\(337\) 1.90636e6 0.914389 0.457194 0.889367i \(-0.348854\pi\)
0.457194 + 0.889367i \(0.348854\pi\)
\(338\) 424121. 0.201929
\(339\) 146212. 0.0691007
\(340\) −34412.7 −0.0161444
\(341\) 2.60326e6 1.21236
\(342\) 26299.3 0.0121585
\(343\) −1.84039e6 −0.844647
\(344\) −858416. −0.391112
\(345\) −15804.8 −0.00714893
\(346\) −456240. −0.204882
\(347\) 3.72382e6 1.66022 0.830110 0.557600i \(-0.188278\pi\)
0.830110 + 0.557600i \(0.188278\pi\)
\(348\) 818985. 0.362517
\(349\) −99542.3 −0.0437466 −0.0218733 0.999761i \(-0.506963\pi\)
−0.0218733 + 0.999761i \(0.506963\pi\)
\(350\) 234665. 0.102395
\(351\) −109539. −0.0474572
\(352\) 1.09306e6 0.470206
\(353\) 528656. 0.225806 0.112903 0.993606i \(-0.463985\pi\)
0.112903 + 0.993606i \(0.463985\pi\)
\(354\) −38103.6 −0.0161606
\(355\) −51590.7 −0.0217270
\(356\) 290523. 0.121494
\(357\) 685292. 0.284580
\(358\) 693016. 0.285782
\(359\) −1.49351e6 −0.611605 −0.305803 0.952095i \(-0.598925\pi\)
−0.305803 + 0.952095i \(0.598925\pi\)
\(360\) 5632.68 0.00229065
\(361\) −2.40483e6 −0.971219
\(362\) 647402. 0.259658
\(363\) −575990. −0.229429
\(364\) 283226. 0.112042
\(365\) 60589.3 0.0238048
\(366\) 513146. 0.200234
\(367\) 154208. 0.0597642 0.0298821 0.999553i \(-0.490487\pi\)
0.0298821 + 0.999553i \(0.490487\pi\)
\(368\) −1.69786e6 −0.653554
\(369\) 1.13019e6 0.432103
\(370\) −9454.36 −0.00359028
\(371\) −2.18293e6 −0.823390
\(372\) 2.29537e6 0.859994
\(373\) −443333. −0.164990 −0.0824951 0.996591i \(-0.526289\pi\)
−0.0824951 + 0.996591i \(0.526289\pi\)
\(374\) 467153. 0.172695
\(375\) −51434.1 −0.0188874
\(376\) −1.89001e6 −0.689438
\(377\) 448003. 0.162341
\(378\) −54757.4 −0.0197112
\(379\) 2.52046e6 0.901324 0.450662 0.892695i \(-0.351188\pi\)
0.450662 + 0.892695i \(0.351188\pi\)
\(380\) 7451.11 0.00264705
\(381\) 2.56911e6 0.906714
\(382\) 920049. 0.322591
\(383\) 3.56075e6 1.24035 0.620175 0.784463i \(-0.287062\pi\)
0.620175 + 0.784463i \(0.287062\pi\)
\(384\) 1.27349e6 0.440726
\(385\) −17594.8 −0.00604969
\(386\) −870377. −0.297330
\(387\) −914405. −0.310357
\(388\) −2.94100e6 −0.991780
\(389\) 1.66815e6 0.558933 0.279467 0.960155i \(-0.409842\pi\)
0.279467 + 0.960155i \(0.409842\pi\)
\(390\) 1504.15 0.000500760 0
\(391\) −2.36754e6 −0.783168
\(392\) −987984. −0.324739
\(393\) 1.38453e6 0.452191
\(394\) −402318. −0.130566
\(395\) −47169.4 −0.0152114
\(396\) 770163. 0.246800
\(397\) −4.98663e6 −1.58793 −0.793964 0.607964i \(-0.791986\pi\)
−0.793964 + 0.607964i \(0.791986\pi\)
\(398\) 176443. 0.0558336
\(399\) −148381. −0.0466601
\(400\) −2.76233e6 −0.863227
\(401\) 1.72114e6 0.534509 0.267255 0.963626i \(-0.413884\pi\)
0.267255 + 0.963626i \(0.413884\pi\)
\(402\) −412231. −0.127226
\(403\) 1.25562e6 0.385119
\(404\) 80592.7 0.0245664
\(405\) 6000.07 0.00181769
\(406\) 223951. 0.0674277
\(407\) −2.64807e6 −0.792397
\(408\) 843769. 0.250942
\(409\) 4.38543e6 1.29630 0.648148 0.761515i \(-0.275544\pi\)
0.648148 + 0.761515i \(0.275544\pi\)
\(410\) −15519.4 −0.00455947
\(411\) −2.53477e6 −0.740175
\(412\) −3.58470e6 −1.04042
\(413\) 214981. 0.0620190
\(414\) 189175. 0.0542455
\(415\) −26379.5 −0.00751877
\(416\) 527211. 0.149366
\(417\) 1.54310e6 0.434565
\(418\) −101149. −0.0283153
\(419\) 5.77647e6 1.60741 0.803706 0.595026i \(-0.202858\pi\)
0.803706 + 0.595026i \(0.202858\pi\)
\(420\) −15513.9 −0.00429138
\(421\) −624265. −0.171658 −0.0858290 0.996310i \(-0.527354\pi\)
−0.0858290 + 0.996310i \(0.527354\pi\)
\(422\) −1.01731e6 −0.278083
\(423\) −2.01329e6 −0.547085
\(424\) −2.68775e6 −0.726062
\(425\) −3.85186e6 −1.03442
\(426\) 617514. 0.164863
\(427\) −2.89517e6 −0.768431
\(428\) 6.98522e6 1.84319
\(429\) 421296. 0.110521
\(430\) 12556.2 0.00327483
\(431\) 1.51781e6 0.393573 0.196787 0.980446i \(-0.436949\pi\)
0.196787 + 0.980446i \(0.436949\pi\)
\(432\) 644568. 0.166173
\(433\) 6.06106e6 1.55356 0.776781 0.629770i \(-0.216851\pi\)
0.776781 + 0.629770i \(0.216851\pi\)
\(434\) 627668. 0.159958
\(435\) −24539.6 −0.00621790
\(436\) −4.99958e6 −1.25956
\(437\) 512624. 0.128409
\(438\) −725223. −0.180629
\(439\) 4.36235e6 1.08034 0.540168 0.841557i \(-0.318361\pi\)
0.540168 + 0.841557i \(0.318361\pi\)
\(440\) −21663.7 −0.00533459
\(441\) −1.05243e6 −0.257688
\(442\) 225320. 0.0548585
\(443\) −6.36182e6 −1.54018 −0.770091 0.637934i \(-0.779789\pi\)
−0.770091 + 0.637934i \(0.779789\pi\)
\(444\) −2.33488e6 −0.562091
\(445\) −8705.04 −0.00208387
\(446\) −893063. −0.212591
\(447\) 2.21919e6 0.525322
\(448\) −1.48383e6 −0.349293
\(449\) −4.13291e6 −0.967475 −0.483737 0.875213i \(-0.660721\pi\)
−0.483737 + 0.875213i \(0.660721\pi\)
\(450\) 307779. 0.0716485
\(451\) −4.34681e6 −1.00630
\(452\) −495832. −0.114153
\(453\) 1.45609e6 0.333381
\(454\) −450330. −0.102540
\(455\) −8486.42 −0.00192175
\(456\) −182695. −0.0411447
\(457\) −4.22694e6 −0.946751 −0.473375 0.880861i \(-0.656965\pi\)
−0.473375 + 0.880861i \(0.656965\pi\)
\(458\) 840005. 0.187119
\(459\) 898803. 0.199128
\(460\) 53597.1 0.0118099
\(461\) 5.51414e6 1.20844 0.604220 0.796817i \(-0.293485\pi\)
0.604220 + 0.796817i \(0.293485\pi\)
\(462\) 210601. 0.0459045
\(463\) −1.86964e6 −0.405327 −0.202664 0.979248i \(-0.564960\pi\)
−0.202664 + 0.979248i \(0.564960\pi\)
\(464\) −2.63621e6 −0.568440
\(465\) −68777.1 −0.0147507
\(466\) 1.12473e6 0.239930
\(467\) 4.32669e6 0.918045 0.459022 0.888425i \(-0.348200\pi\)
0.459022 + 0.888425i \(0.348200\pi\)
\(468\) 371469. 0.0783986
\(469\) 2.32581e6 0.488250
\(470\) 27645.7 0.00577274
\(471\) −1.91060e6 −0.396842
\(472\) 264696. 0.0546881
\(473\) 3.51687e6 0.722776
\(474\) 564594. 0.115423
\(475\) 834013. 0.169605
\(476\) −2.32396e6 −0.470122
\(477\) −2.86306e6 −0.576148
\(478\) 691364. 0.138400
\(479\) −3.34586e6 −0.666298 −0.333149 0.942874i \(-0.608111\pi\)
−0.333149 + 0.942874i \(0.608111\pi\)
\(480\) −28878.3 −0.00572095
\(481\) −1.27723e6 −0.251713
\(482\) 355289. 0.0696569
\(483\) −1.06733e6 −0.208176
\(484\) 1.95329e6 0.379013
\(485\) 88122.3 0.0170111
\(486\) −71817.8 −0.0137925
\(487\) 1.41911e6 0.271139 0.135570 0.990768i \(-0.456714\pi\)
0.135570 + 0.990768i \(0.456714\pi\)
\(488\) −3.56469e6 −0.677599
\(489\) 3.85338e6 0.728735
\(490\) 14451.5 0.00271908
\(491\) −3.66412e6 −0.685908 −0.342954 0.939352i \(-0.611427\pi\)
−0.342954 + 0.939352i \(0.611427\pi\)
\(492\) −3.83271e6 −0.713827
\(493\) −3.67600e6 −0.681174
\(494\) −48786.7 −0.00899464
\(495\) −23076.7 −0.00423312
\(496\) −7.38849e6 −1.34850
\(497\) −3.48402e6 −0.632688
\(498\) 315749. 0.0570518
\(499\) −1.94309e6 −0.349335 −0.174667 0.984627i \(-0.555885\pi\)
−0.174667 + 0.984627i \(0.555885\pi\)
\(500\) 174423. 0.0312017
\(501\) 2.34471e6 0.417345
\(502\) 2.14605e6 0.380085
\(503\) 2.10114e6 0.370285 0.185142 0.982712i \(-0.440725\pi\)
0.185142 + 0.982712i \(0.440725\pi\)
\(504\) 380386. 0.0667035
\(505\) −2414.83 −0.000421365 0
\(506\) −727582. −0.126330
\(507\) −3.13844e6 −0.542242
\(508\) −8.71235e6 −1.49788
\(509\) −2.83832e6 −0.485587 −0.242793 0.970078i \(-0.578064\pi\)
−0.242793 + 0.970078i \(0.578064\pi\)
\(510\) −12342.0 −0.00210117
\(511\) 4.09172e6 0.693192
\(512\) −5.25376e6 −0.885718
\(513\) −194611. −0.0326493
\(514\) 1.00303e6 0.167458
\(515\) 107410. 0.0178454
\(516\) 3.10093e6 0.512705
\(517\) 7.74326e6 1.27408
\(518\) −638472. −0.104548
\(519\) 3.37611e6 0.550171
\(520\) −10449.0 −0.00169459
\(521\) 6.04781e6 0.976121 0.488060 0.872810i \(-0.337705\pi\)
0.488060 + 0.872810i \(0.337705\pi\)
\(522\) 293726. 0.0471809
\(523\) 1.00745e7 1.61053 0.805265 0.592915i \(-0.202023\pi\)
0.805265 + 0.592915i \(0.202023\pi\)
\(524\) −4.69522e6 −0.747012
\(525\) −1.73649e6 −0.274963
\(526\) 1.11342e6 0.175466
\(527\) −1.03027e7 −1.61594
\(528\) −2.47906e6 −0.386992
\(529\) −2.74895e6 −0.427098
\(530\) 39314.3 0.00607940
\(531\) 281961. 0.0433963
\(532\) 503188. 0.0770817
\(533\) −2.09658e6 −0.319663
\(534\) 104195. 0.0158122
\(535\) −209301. −0.0316145
\(536\) 2.86367e6 0.430537
\(537\) −5.12821e6 −0.767415
\(538\) −1.21793e6 −0.181413
\(539\) 4.04770e6 0.600118
\(540\) −20347.4 −0.00300279
\(541\) 2.90078e6 0.426110 0.213055 0.977040i \(-0.431659\pi\)
0.213055 + 0.977040i \(0.431659\pi\)
\(542\) 2.73861e6 0.400435
\(543\) −4.79068e6 −0.697264
\(544\) −4.32593e6 −0.626732
\(545\) 149804. 0.0216040
\(546\) 101578. 0.0145821
\(547\) −6.58103e6 −0.940428 −0.470214 0.882552i \(-0.655823\pi\)
−0.470214 + 0.882552i \(0.655823\pi\)
\(548\) 8.59591e6 1.22276
\(549\) −3.79720e6 −0.537691
\(550\) −1.18374e6 −0.166859
\(551\) 795935. 0.111686
\(552\) −1.31415e6 −0.183569
\(553\) −3.18545e6 −0.442953
\(554\) −6424.37 −0.000889316 0
\(555\) 69960.9 0.00964102
\(556\) −5.23296e6 −0.717895
\(557\) −8.57629e6 −1.17128 −0.585641 0.810571i \(-0.699157\pi\)
−0.585641 + 0.810571i \(0.699157\pi\)
\(558\) 823227. 0.111927
\(559\) 1.69628e6 0.229597
\(560\) 49937.1 0.00672904
\(561\) −3.45686e6 −0.463741
\(562\) 3.01927e6 0.403237
\(563\) −6.42359e6 −0.854096 −0.427048 0.904229i \(-0.640446\pi\)
−0.427048 + 0.904229i \(0.640446\pi\)
\(564\) 6.82745e6 0.903776
\(565\) 14856.8 0.00195796
\(566\) 105819. 0.0138843
\(567\) 405197. 0.0529308
\(568\) −4.28972e6 −0.557902
\(569\) 1.17161e7 1.51706 0.758531 0.651638i \(-0.225918\pi\)
0.758531 + 0.651638i \(0.225918\pi\)
\(570\) 2672.32 0.000344509 0
\(571\) −4.99504e6 −0.641134 −0.320567 0.947226i \(-0.603873\pi\)
−0.320567 + 0.947226i \(0.603873\pi\)
\(572\) −1.42870e6 −0.182579
\(573\) −6.80823e6 −0.866258
\(574\) −1.04805e6 −0.132771
\(575\) 5.99920e6 0.756700
\(576\) −1.94614e6 −0.244409
\(577\) 495976. 0.0620185 0.0310093 0.999519i \(-0.490128\pi\)
0.0310093 + 0.999519i \(0.490128\pi\)
\(578\) −121929. −0.0151805
\(579\) 6.44066e6 0.798425
\(580\) 83218.5 0.0102719
\(581\) −1.78146e6 −0.218945
\(582\) −1.05478e6 −0.129079
\(583\) 1.10115e7 1.34176
\(584\) 5.03795e6 0.611254
\(585\) −11130.5 −0.00134470
\(586\) −201293. −0.0242150
\(587\) −202428. −0.0242480 −0.0121240 0.999927i \(-0.503859\pi\)
−0.0121240 + 0.999927i \(0.503859\pi\)
\(588\) 3.56898e6 0.425697
\(589\) 2.23077e6 0.264951
\(590\) −3871.77 −0.000457910 0
\(591\) 2.97709e6 0.350609
\(592\) 7.51567e6 0.881380
\(593\) 7.95364e6 0.928815 0.464407 0.885622i \(-0.346267\pi\)
0.464407 + 0.885622i \(0.346267\pi\)
\(594\) 276217. 0.0321206
\(595\) 69633.7 0.00806356
\(596\) −7.52571e6 −0.867824
\(597\) −1.30565e6 −0.149931
\(598\) −350931. −0.0401300
\(599\) −1.68592e7 −1.91986 −0.959930 0.280241i \(-0.909586\pi\)
−0.959930 + 0.280241i \(0.909586\pi\)
\(600\) −2.13806e6 −0.242461
\(601\) 5.03561e6 0.568677 0.284339 0.958724i \(-0.408226\pi\)
0.284339 + 0.958724i \(0.408226\pi\)
\(602\) 847948. 0.0953626
\(603\) 3.05045e6 0.341641
\(604\) −4.93787e6 −0.550741
\(605\) −58527.3 −0.00650085
\(606\) 28904.3 0.00319728
\(607\) 9.93216e6 1.09414 0.547069 0.837088i \(-0.315744\pi\)
0.547069 + 0.837088i \(0.315744\pi\)
\(608\) 936659. 0.102760
\(609\) −1.65721e6 −0.181065
\(610\) 52141.6 0.00567362
\(611\) 3.73477e6 0.404725
\(612\) −3.04802e6 −0.328957
\(613\) 1.09871e7 1.18095 0.590477 0.807054i \(-0.298940\pi\)
0.590477 + 0.807054i \(0.298940\pi\)
\(614\) −908003. −0.0972000
\(615\) 114841. 0.0122436
\(616\) −1.46299e6 −0.155343
\(617\) 6.30095e6 0.666336 0.333168 0.942867i \(-0.391882\pi\)
0.333168 + 0.942867i \(0.391882\pi\)
\(618\) −1.28564e6 −0.135409
\(619\) 5.08049e6 0.532941 0.266470 0.963843i \(-0.414143\pi\)
0.266470 + 0.963843i \(0.414143\pi\)
\(620\) 233236. 0.0243679
\(621\) −1.39987e6 −0.145666
\(622\) −776212. −0.0804460
\(623\) −587869. −0.0606820
\(624\) −1.19571e6 −0.122932
\(625\) 9.75779e6 0.999197
\(626\) −1.88800e6 −0.192560
\(627\) 748487. 0.0760353
\(628\) 6.47922e6 0.655577
\(629\) 1.04801e7 1.05618
\(630\) −5564.00 −0.000558516 0
\(631\) −7.62504e6 −0.762376 −0.381188 0.924498i \(-0.624485\pi\)
−0.381188 + 0.924498i \(0.624485\pi\)
\(632\) −3.92210e6 −0.390594
\(633\) 7.52797e6 0.746739
\(634\) −2.40091e6 −0.237221
\(635\) 261052. 0.0256917
\(636\) 9.70918e6 0.951787
\(637\) 1.95231e6 0.190634
\(638\) −1.12969e6 −0.109878
\(639\) −4.56951e6 −0.442708
\(640\) 129402. 0.0124879
\(641\) −7.34805e6 −0.706362 −0.353181 0.935555i \(-0.614900\pi\)
−0.353181 + 0.935555i \(0.614900\pi\)
\(642\) 2.50523e6 0.239888
\(643\) −4.91274e6 −0.468594 −0.234297 0.972165i \(-0.575279\pi\)
−0.234297 + 0.972165i \(0.575279\pi\)
\(644\) 3.61952e6 0.343903
\(645\) −92914.3 −0.00879394
\(646\) 400310. 0.0377411
\(647\) 5.02847e6 0.472254 0.236127 0.971722i \(-0.424122\pi\)
0.236127 + 0.971722i \(0.424122\pi\)
\(648\) 498900. 0.0466741
\(649\) −1.08444e6 −0.101064
\(650\) −570947. −0.0530045
\(651\) −4.64465e6 −0.429537
\(652\) −1.30676e7 −1.20386
\(653\) 5.06290e6 0.464639 0.232320 0.972639i \(-0.425368\pi\)
0.232320 + 0.972639i \(0.425368\pi\)
\(654\) −1.79308e6 −0.163929
\(655\) 140685. 0.0128128
\(656\) 1.23370e7 1.11931
\(657\) 5.36654e6 0.485044
\(658\) 1.86697e6 0.168102
\(659\) −176951. −0.0158723 −0.00793615 0.999969i \(-0.502526\pi\)
−0.00793615 + 0.999969i \(0.502526\pi\)
\(660\) 78257.6 0.00699305
\(661\) 1.83120e7 1.63016 0.815082 0.579346i \(-0.196692\pi\)
0.815082 + 0.579346i \(0.196692\pi\)
\(662\) 2.92794e6 0.259667
\(663\) −1.66733e6 −0.147312
\(664\) −2.19343e6 −0.193065
\(665\) −15077.2 −0.00132211
\(666\) −837396. −0.0731552
\(667\) 5.72530e6 0.498291
\(668\) −7.95138e6 −0.689448
\(669\) 6.60853e6 0.570873
\(670\) −41887.5 −0.00360494
\(671\) 1.46043e7 1.25220
\(672\) −1.95021e6 −0.166593
\(673\) 1.31732e7 1.12112 0.560561 0.828113i \(-0.310586\pi\)
0.560561 + 0.828113i \(0.310586\pi\)
\(674\) −2.31860e6 −0.196596
\(675\) −2.27752e6 −0.192399
\(676\) 1.06430e7 0.895776
\(677\) −5.29079e6 −0.443658 −0.221829 0.975086i \(-0.571203\pi\)
−0.221829 + 0.975086i \(0.571203\pi\)
\(678\) −177829. −0.0148569
\(679\) 5.95107e6 0.495360
\(680\) 85736.8 0.00711042
\(681\) 3.33238e6 0.275351
\(682\) −3.16619e6 −0.260661
\(683\) −1.28101e7 −1.05076 −0.525378 0.850869i \(-0.676076\pi\)
−0.525378 + 0.850869i \(0.676076\pi\)
\(684\) 659963. 0.0539361
\(685\) −257563. −0.0209728
\(686\) 2.23836e6 0.181602
\(687\) −6.21591e6 −0.502473
\(688\) −9.98148e6 −0.803940
\(689\) 5.31113e6 0.426225
\(690\) 19222.4 0.00153704
\(691\) −2.41425e7 −1.92348 −0.961739 0.273969i \(-0.911664\pi\)
−0.961739 + 0.273969i \(0.911664\pi\)
\(692\) −1.14490e7 −0.908874
\(693\) −1.55842e6 −0.123268
\(694\) −4.52907e6 −0.356952
\(695\) 156797. 0.0123134
\(696\) −2.04044e6 −0.159662
\(697\) 1.72030e7 1.34129
\(698\) 121067. 0.00940565
\(699\) −8.32286e6 −0.644288
\(700\) 5.88877e6 0.454234
\(701\) −1.34383e7 −1.03288 −0.516441 0.856323i \(-0.672743\pi\)
−0.516441 + 0.856323i \(0.672743\pi\)
\(702\) 133226. 0.0102034
\(703\) −2.26916e6 −0.173172
\(704\) 7.48499e6 0.569193
\(705\) −204574. −0.0155016
\(706\) −642973. −0.0485491
\(707\) −163078. −0.0122701
\(708\) −956185. −0.0716900
\(709\) 4.97439e6 0.371642 0.185821 0.982584i \(-0.440506\pi\)
0.185821 + 0.982584i \(0.440506\pi\)
\(710\) 62746.7 0.00467138
\(711\) −4.17791e6 −0.309946
\(712\) −723816. −0.0535092
\(713\) 1.60463e7 1.18209
\(714\) −833480. −0.0611856
\(715\) 42808.6 0.00313160
\(716\) 1.73908e7 1.26776
\(717\) −5.11599e6 −0.371648
\(718\) 1.81647e6 0.131497
\(719\) −2.75626e6 −0.198837 −0.0994185 0.995046i \(-0.531698\pi\)
−0.0994185 + 0.995046i \(0.531698\pi\)
\(720\) 65495.7 0.00470849
\(721\) 7.25360e6 0.519655
\(722\) 2.92486e6 0.208815
\(723\) −2.62908e6 −0.187050
\(724\) 1.62461e7 1.15187
\(725\) 9.31477e6 0.658153
\(726\) 700542. 0.0493279
\(727\) −1.11365e7 −0.781470 −0.390735 0.920503i \(-0.627779\pi\)
−0.390735 + 0.920503i \(0.627779\pi\)
\(728\) −705638. −0.0493462
\(729\) 531441. 0.0370370
\(730\) −73691.2 −0.00511810
\(731\) −1.39185e7 −0.963380
\(732\) 1.28770e7 0.888257
\(733\) 2.89889e6 0.199284 0.0996419 0.995023i \(-0.468230\pi\)
0.0996419 + 0.995023i \(0.468230\pi\)
\(734\) −187554. −0.0128495
\(735\) −106939. −0.00730158
\(736\) 6.73755e6 0.458466
\(737\) −1.17322e7 −0.795633
\(738\) −1.37459e6 −0.0929035
\(739\) 2.76654e7 1.86348 0.931741 0.363123i \(-0.118290\pi\)
0.931741 + 0.363123i \(0.118290\pi\)
\(740\) −237251. −0.0159268
\(741\) 361014. 0.0241534
\(742\) 2.65497e6 0.177031
\(743\) −1.08048e7 −0.718033 −0.359017 0.933331i \(-0.616888\pi\)
−0.359017 + 0.933331i \(0.616888\pi\)
\(744\) −5.71875e6 −0.378764
\(745\) 225496. 0.0148850
\(746\) 539200. 0.0354734
\(747\) −2.33650e6 −0.153202
\(748\) 1.17229e7 0.766092
\(749\) −1.41345e7 −0.920611
\(750\) 62556.2 0.00406085
\(751\) −2.36060e7 −1.52729 −0.763645 0.645636i \(-0.776592\pi\)
−0.763645 + 0.645636i \(0.776592\pi\)
\(752\) −2.19767e7 −1.41716
\(753\) −1.58805e7 −1.02065
\(754\) −544879. −0.0349037
\(755\) 147955. 0.00944633
\(756\) −1.37410e6 −0.0874408
\(757\) 2.12366e7 1.34693 0.673464 0.739220i \(-0.264806\pi\)
0.673464 + 0.739220i \(0.264806\pi\)
\(758\) −3.06548e6 −0.193787
\(759\) 5.38400e6 0.339235
\(760\) −18563.9 −0.00116583
\(761\) −2.37074e7 −1.48396 −0.741982 0.670420i \(-0.766114\pi\)
−0.741982 + 0.670420i \(0.766114\pi\)
\(762\) −3.12466e6 −0.194946
\(763\) 1.01166e7 0.629104
\(764\) 2.30880e7 1.43105
\(765\) 91329.0 0.00564228
\(766\) −4.33073e6 −0.266679
\(767\) −523054. −0.0321039
\(768\) 5.37073e6 0.328572
\(769\) 1.33593e7 0.814644 0.407322 0.913285i \(-0.366463\pi\)
0.407322 + 0.913285i \(0.366463\pi\)
\(770\) 21399.5 0.00130070
\(771\) −7.42226e6 −0.449676
\(772\) −2.18415e7 −1.31899
\(773\) −1.28438e7 −0.773117 −0.386558 0.922265i \(-0.626336\pi\)
−0.386558 + 0.922265i \(0.626336\pi\)
\(774\) 1.11214e6 0.0667277
\(775\) 2.61065e7 1.56133
\(776\) 7.32729e6 0.436806
\(777\) 4.72460e6 0.280745
\(778\) −2.02887e6 −0.120172
\(779\) −3.72484e6 −0.219920
\(780\) 37745.6 0.00222142
\(781\) 1.75747e7 1.03100
\(782\) 2.87950e6 0.168384
\(783\) −2.17353e6 −0.126696
\(784\) −1.14881e7 −0.667509
\(785\) −194139. −0.0112445
\(786\) −1.68392e6 −0.0972224
\(787\) −5.78070e6 −0.332693 −0.166347 0.986067i \(-0.553197\pi\)
−0.166347 + 0.986067i \(0.553197\pi\)
\(788\) −1.00959e7 −0.579201
\(789\) −8.23911e6 −0.471181
\(790\) 57369.4 0.00327049
\(791\) 1.00331e6 0.0570157
\(792\) −1.91881e6 −0.108697
\(793\) 7.04403e6 0.397776
\(794\) 6.06495e6 0.341410
\(795\) −290920. −0.0163251
\(796\) 4.42771e6 0.247683
\(797\) −4.78906e6 −0.267057 −0.133529 0.991045i \(-0.542631\pi\)
−0.133529 + 0.991045i \(0.542631\pi\)
\(798\) 180467. 0.0100321
\(799\) −3.06449e7 −1.69821
\(800\) 1.09616e7 0.605551
\(801\) −771027. −0.0424608
\(802\) −2.09332e6 −0.114921
\(803\) −2.06401e7 −1.12960
\(804\) −1.03447e7 −0.564386
\(805\) −108453. −0.00589864
\(806\) −1.52713e6 −0.0828017
\(807\) 9.01253e6 0.487150
\(808\) −200791. −0.0108197
\(809\) −1.17461e7 −0.630987 −0.315494 0.948928i \(-0.602170\pi\)
−0.315494 + 0.948928i \(0.602170\pi\)
\(810\) −7297.53 −0.000390808 0
\(811\) 2.70056e6 0.144179 0.0720893 0.997398i \(-0.477033\pi\)
0.0720893 + 0.997398i \(0.477033\pi\)
\(812\) 5.61991e6 0.299116
\(813\) −2.02653e7 −1.07529
\(814\) 3.22069e6 0.170368
\(815\) 391548. 0.0206486
\(816\) 9.81117e6 0.515817
\(817\) 3.01365e6 0.157957
\(818\) −5.33374e6 −0.278708
\(819\) −751663. −0.0391574
\(820\) −389448. −0.0202262
\(821\) 2.05695e7 1.06504 0.532520 0.846418i \(-0.321245\pi\)
0.532520 + 0.846418i \(0.321245\pi\)
\(822\) 3.08289e6 0.159140
\(823\) −3.85189e7 −1.98232 −0.991162 0.132654i \(-0.957650\pi\)
−0.991162 + 0.132654i \(0.957650\pi\)
\(824\) 8.93103e6 0.458230
\(825\) 8.75949e6 0.448068
\(826\) −261469. −0.0133343
\(827\) 726695. 0.0369478 0.0184739 0.999829i \(-0.494119\pi\)
0.0184739 + 0.999829i \(0.494119\pi\)
\(828\) 4.74723e6 0.240638
\(829\) 1.92752e7 0.974120 0.487060 0.873369i \(-0.338069\pi\)
0.487060 + 0.873369i \(0.338069\pi\)
\(830\) 32083.8 0.00161656
\(831\) 47539.4 0.00238809
\(832\) 3.61020e6 0.180810
\(833\) −1.60193e7 −0.799891
\(834\) −1.87679e6 −0.0934328
\(835\) 238250. 0.0118254
\(836\) −2.53827e6 −0.125609
\(837\) −6.09175e6 −0.300558
\(838\) −7.02558e6 −0.345599
\(839\) −2.47117e7 −1.21199 −0.605994 0.795470i \(-0.707224\pi\)
−0.605994 + 0.795470i \(0.707224\pi\)
\(840\) 38651.7 0.00189004
\(841\) −1.16217e7 −0.566602
\(842\) 759257. 0.0369070
\(843\) −2.23421e7 −1.08282
\(844\) −2.55288e7 −1.23360
\(845\) −318902. −0.0153644
\(846\) 2.44864e6 0.117625
\(847\) −3.95246e6 −0.189304
\(848\) −3.12526e7 −1.49244
\(849\) −783046. −0.0372836
\(850\) 4.68479e6 0.222404
\(851\) −1.63225e7 −0.772613
\(852\) 1.54961e7 0.731347
\(853\) 2.66298e7 1.25313 0.626563 0.779370i \(-0.284461\pi\)
0.626563 + 0.779370i \(0.284461\pi\)
\(854\) 3.52123e6 0.165215
\(855\) −19774.7 −0.000925114 0
\(856\) −1.74032e7 −0.811791
\(857\) 5.12844e6 0.238525 0.119262 0.992863i \(-0.461947\pi\)
0.119262 + 0.992863i \(0.461947\pi\)
\(858\) −512398. −0.0237623
\(859\) 2.30168e7 1.06430 0.532148 0.846651i \(-0.321385\pi\)
0.532148 + 0.846651i \(0.321385\pi\)
\(860\) 315091. 0.0145274
\(861\) 7.75544e6 0.356532
\(862\) −1.84603e6 −0.0846194
\(863\) −3.81545e7 −1.74389 −0.871944 0.489606i \(-0.837141\pi\)
−0.871944 + 0.489606i \(0.837141\pi\)
\(864\) −2.55782e6 −0.116570
\(865\) 343052. 0.0155891
\(866\) −7.37171e6 −0.334021
\(867\) 902253. 0.0407644
\(868\) 1.57509e7 0.709589
\(869\) 1.60686e7 0.721818
\(870\) 29846.0 0.00133687
\(871\) −5.65876e6 −0.252741
\(872\) 1.24561e7 0.554742
\(873\) 7.80521e6 0.346616
\(874\) −623474. −0.0276083
\(875\) −352943. −0.0155842
\(876\) −1.81990e7 −0.801285
\(877\) −3.38162e7 −1.48466 −0.742328 0.670036i \(-0.766278\pi\)
−0.742328 + 0.670036i \(0.766278\pi\)
\(878\) −5.30566e6 −0.232275
\(879\) 1.48954e6 0.0650248
\(880\) −251901. −0.0109654
\(881\) 2.86617e7 1.24412 0.622059 0.782970i \(-0.286296\pi\)
0.622059 + 0.782970i \(0.286296\pi\)
\(882\) 1.28000e6 0.0554038
\(883\) −4.01713e7 −1.73386 −0.866930 0.498430i \(-0.833910\pi\)
−0.866930 + 0.498430i \(0.833910\pi\)
\(884\) 5.65425e6 0.243357
\(885\) 28650.5 0.00122963
\(886\) 7.73751e6 0.331144
\(887\) −1.88137e7 −0.802906 −0.401453 0.915880i \(-0.631495\pi\)
−0.401453 + 0.915880i \(0.631495\pi\)
\(888\) 5.81718e6 0.247560
\(889\) 1.76293e7 0.748138
\(890\) 10587.4 0.000448039 0
\(891\) −2.04396e6 −0.0862538
\(892\) −2.24108e7 −0.943074
\(893\) 6.63529e6 0.278440
\(894\) −2.69907e6 −0.112946
\(895\) −521086. −0.0217446
\(896\) 8.73876e6 0.363647
\(897\) 2.59684e6 0.107762
\(898\) 5.02661e6 0.208010
\(899\) 2.49146e7 1.02814
\(900\) 7.72350e6 0.317839
\(901\) −4.35795e7 −1.78842
\(902\) 5.28677e6 0.216359
\(903\) −6.27469e6 −0.256078
\(904\) 1.23533e6 0.0502762
\(905\) −486789. −0.0197569
\(906\) −1.77095e6 −0.0716780
\(907\) −8.85338e6 −0.357348 −0.178674 0.983908i \(-0.557181\pi\)
−0.178674 + 0.983908i \(0.557181\pi\)
\(908\) −1.13007e7 −0.454875
\(909\) −213887. −0.00858570
\(910\) 10321.5 0.000413181 0
\(911\) 1.12187e7 0.447864 0.223932 0.974605i \(-0.428111\pi\)
0.223932 + 0.974605i \(0.428111\pi\)
\(912\) −2.12433e6 −0.0845738
\(913\) 8.98634e6 0.356785
\(914\) 5.14098e6 0.203554
\(915\) −385840. −0.0152354
\(916\) 2.10794e7 0.830078
\(917\) 9.50072e6 0.373107
\(918\) −1.09316e6 −0.0428132
\(919\) 2.15141e7 0.840299 0.420149 0.907455i \(-0.361978\pi\)
0.420149 + 0.907455i \(0.361978\pi\)
\(920\) −133533. −0.00520140
\(921\) 6.71909e6 0.261012
\(922\) −6.70652e6 −0.259818
\(923\) 8.47671e6 0.327509
\(924\) 5.28489e6 0.203637
\(925\) −2.65558e7 −1.02048
\(926\) 2.27393e6 0.0871466
\(927\) 9.51355e6 0.363616
\(928\) 1.04612e7 0.398759
\(929\) 1.34111e7 0.509831 0.254916 0.966963i \(-0.417952\pi\)
0.254916 + 0.966963i \(0.417952\pi\)
\(930\) 83649.5 0.00317144
\(931\) 3.46853e6 0.131151
\(932\) 2.82245e7 1.06435
\(933\) 5.74385e6 0.216023
\(934\) −5.26230e6 −0.197382
\(935\) −351258. −0.0131401
\(936\) −925489. −0.0345288
\(937\) −3.24101e7 −1.20596 −0.602978 0.797758i \(-0.706019\pi\)
−0.602978 + 0.797758i \(0.706019\pi\)
\(938\) −2.82875e6 −0.104975
\(939\) 1.39709e7 0.517084
\(940\) 693749. 0.0256084
\(941\) −3.91775e6 −0.144232 −0.0721162 0.997396i \(-0.522975\pi\)
−0.0721162 + 0.997396i \(0.522975\pi\)
\(942\) 2.32375e6 0.0853223
\(943\) −2.67934e7 −0.981180
\(944\) 3.07783e6 0.112413
\(945\) 41172.7 0.00149979
\(946\) −4.27736e6 −0.155399
\(947\) −3.42000e7 −1.23923 −0.619614 0.784907i \(-0.712711\pi\)
−0.619614 + 0.784907i \(0.712711\pi\)
\(948\) 1.41681e7 0.512025
\(949\) −9.95525e6 −0.358828
\(950\) −1.01436e6 −0.0364656
\(951\) 1.77664e7 0.637013
\(952\) 5.78998e6 0.207054
\(953\) 4.25014e7 1.51590 0.757951 0.652312i \(-0.226201\pi\)
0.757951 + 0.652312i \(0.226201\pi\)
\(954\) 3.48217e6 0.123874
\(955\) −691796. −0.0245454
\(956\) 1.73493e7 0.613956
\(957\) 8.35956e6 0.295055
\(958\) 4.06937e6 0.143256
\(959\) −1.73937e7 −0.610725
\(960\) −197751. −0.00692532
\(961\) 4.11988e7 1.43905
\(962\) 1.55342e6 0.0541191
\(963\) −1.85383e7 −0.644175
\(964\) 8.91574e6 0.309005
\(965\) 654447. 0.0226233
\(966\) 1.29813e6 0.0447584
\(967\) 2.74628e7 0.944450 0.472225 0.881478i \(-0.343451\pi\)
0.472225 + 0.881478i \(0.343451\pi\)
\(968\) −4.86649e6 −0.166927
\(969\) −2.96223e6 −0.101347
\(970\) −107178. −0.00365743
\(971\) 4.50046e7 1.53183 0.765913 0.642945i \(-0.222287\pi\)
0.765913 + 0.642945i \(0.222287\pi\)
\(972\) −1.80222e6 −0.0611846
\(973\) 1.05888e7 0.358564
\(974\) −1.72598e6 −0.0582958
\(975\) 4.22492e6 0.142334
\(976\) −4.14495e7 −1.39282
\(977\) −1.02124e7 −0.342287 −0.171144 0.985246i \(-0.554746\pi\)
−0.171144 + 0.985246i \(0.554746\pi\)
\(978\) −4.68664e6 −0.156680
\(979\) 2.96543e6 0.0988850
\(980\) 362650. 0.0120621
\(981\) 1.32685e7 0.440201
\(982\) 4.45645e6 0.147472
\(983\) 1.34161e7 0.442835 0.221418 0.975179i \(-0.428932\pi\)
0.221418 + 0.975179i \(0.428932\pi\)
\(984\) 9.54893e6 0.314389
\(985\) 302507. 0.00993449
\(986\) 4.47090e6 0.146455
\(987\) −1.38153e7 −0.451405
\(988\) −1.22427e6 −0.0399011
\(989\) 2.16777e7 0.704730
\(990\) 28066.8 0.000910135 0
\(991\) −4.37148e7 −1.41398 −0.706992 0.707222i \(-0.749948\pi\)
−0.706992 + 0.707222i \(0.749948\pi\)
\(992\) 2.93195e7 0.945971
\(993\) −2.16663e7 −0.697287
\(994\) 4.23741e6 0.136030
\(995\) −132669. −0.00424827
\(996\) 7.92352e6 0.253087
\(997\) −3.11427e7 −0.992243 −0.496121 0.868253i \(-0.665243\pi\)
−0.496121 + 0.868253i \(0.665243\pi\)
\(998\) 2.36327e6 0.0751081
\(999\) 6.19661e6 0.196445
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.a.1.6 11
3.2 odd 2 531.6.a.b.1.6 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.a.1.6 11 1.1 even 1 trivial
531.6.a.b.1.6 11 3.2 odd 2