Properties

Label 177.8.a.b.1.1
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-20.3182\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-22.3182 q^{2} +27.0000 q^{3} +370.104 q^{4} -84.4166 q^{5} -602.593 q^{6} +1532.13 q^{7} -5403.34 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-22.3182 q^{2} +27.0000 q^{3} +370.104 q^{4} -84.4166 q^{5} -602.593 q^{6} +1532.13 q^{7} -5403.34 q^{8} +729.000 q^{9} +1884.03 q^{10} +4639.41 q^{11} +9992.81 q^{12} -1710.47 q^{13} -34194.5 q^{14} -2279.25 q^{15} +73219.7 q^{16} -14226.6 q^{17} -16270.0 q^{18} -21902.8 q^{19} -31242.9 q^{20} +41367.5 q^{21} -103543. q^{22} -107697. q^{23} -145890. q^{24} -70998.8 q^{25} +38174.6 q^{26} +19683.0 q^{27} +567048. q^{28} -195131. q^{29} +50868.8 q^{30} -68117.7 q^{31} -942507. q^{32} +125264. q^{33} +317512. q^{34} -129337. q^{35} +269806. q^{36} -309640. q^{37} +488831. q^{38} -46182.6 q^{39} +456131. q^{40} +873409. q^{41} -923251. q^{42} +159630. q^{43} +1.71706e6 q^{44} -61539.7 q^{45} +2.40360e6 q^{46} -615386. q^{47} +1.97693e6 q^{48} +1.52388e6 q^{49} +1.58457e6 q^{50} -384117. q^{51} -633051. q^{52} -745946. q^{53} -439290. q^{54} -391643. q^{55} -8.27862e6 q^{56} -591374. q^{57} +4.35498e6 q^{58} -205379. q^{59} -843558. q^{60} -2.69167e6 q^{61} +1.52027e6 q^{62} +1.11692e6 q^{63} +1.16630e7 q^{64} +144392. q^{65} -2.79567e6 q^{66} +2.11862e6 q^{67} -5.26531e6 q^{68} -2.90781e6 q^{69} +2.88658e6 q^{70} -1.64750e6 q^{71} -3.93903e6 q^{72} -2.91422e6 q^{73} +6.91062e6 q^{74} -1.91697e6 q^{75} -8.10630e6 q^{76} +7.10818e6 q^{77} +1.03071e6 q^{78} +171784. q^{79} -6.18095e6 q^{80} +531441. q^{81} -1.94930e7 q^{82} +4.94728e6 q^{83} +1.53103e7 q^{84} +1.20096e6 q^{85} -3.56266e6 q^{86} -5.26854e6 q^{87} -2.50683e7 q^{88} -1.83391e6 q^{89} +1.37346e6 q^{90} -2.62066e6 q^{91} -3.98589e7 q^{92} -1.83918e6 q^{93} +1.37343e7 q^{94} +1.84896e6 q^{95} -2.54477e7 q^{96} -1.09662e7 q^{97} -3.40104e7 q^{98} +3.38213e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.3182 −1.97267 −0.986336 0.164744i \(-0.947320\pi\)
−0.986336 + 0.164744i \(0.947320\pi\)
\(3\) 27.0000 0.577350
\(4\) 370.104 2.89144
\(5\) −84.4166 −0.302018 −0.151009 0.988532i \(-0.548252\pi\)
−0.151009 + 0.988532i \(0.548252\pi\)
\(6\) −602.593 −1.13892
\(7\) 1532.13 1.68831 0.844156 0.536097i \(-0.180102\pi\)
0.844156 + 0.536097i \(0.180102\pi\)
\(8\) −5403.34 −3.73119
\(9\) 729.000 0.333333
\(10\) 1884.03 0.595782
\(11\) 4639.41 1.05097 0.525483 0.850804i \(-0.323885\pi\)
0.525483 + 0.850804i \(0.323885\pi\)
\(12\) 9992.81 1.66937
\(13\) −1710.47 −0.215930 −0.107965 0.994155i \(-0.534433\pi\)
−0.107965 + 0.994155i \(0.534433\pi\)
\(14\) −34194.5 −3.33049
\(15\) −2279.25 −0.174370
\(16\) 73219.7 4.46897
\(17\) −14226.6 −0.702310 −0.351155 0.936317i \(-0.614211\pi\)
−0.351155 + 0.936317i \(0.614211\pi\)
\(18\) −16270.0 −0.657558
\(19\) −21902.8 −0.732590 −0.366295 0.930499i \(-0.619374\pi\)
−0.366295 + 0.930499i \(0.619374\pi\)
\(20\) −31242.9 −0.873266
\(21\) 41367.5 0.974748
\(22\) −103543. −2.07321
\(23\) −107697. −1.84567 −0.922836 0.385194i \(-0.874135\pi\)
−0.922836 + 0.385194i \(0.874135\pi\)
\(24\) −145890. −2.15420
\(25\) −70998.8 −0.908785
\(26\) 38174.6 0.425959
\(27\) 19683.0 0.192450
\(28\) 567048. 4.88165
\(29\) −195131. −1.48571 −0.742853 0.669454i \(-0.766528\pi\)
−0.742853 + 0.669454i \(0.766528\pi\)
\(30\) 50868.8 0.343975
\(31\) −68117.7 −0.410671 −0.205336 0.978692i \(-0.565829\pi\)
−0.205336 + 0.978692i \(0.565829\pi\)
\(32\) −942507. −5.08463
\(33\) 125264. 0.606775
\(34\) 317512. 1.38543
\(35\) −129337. −0.509901
\(36\) 269806. 0.963812
\(37\) −309640. −1.00496 −0.502482 0.864588i \(-0.667580\pi\)
−0.502482 + 0.864588i \(0.667580\pi\)
\(38\) 488831. 1.44516
\(39\) −46182.6 −0.124667
\(40\) 456131. 1.12689
\(41\) 873409. 1.97913 0.989565 0.144086i \(-0.0460242\pi\)
0.989565 + 0.144086i \(0.0460242\pi\)
\(42\) −923251. −1.92286
\(43\) 159630. 0.306179 0.153089 0.988212i \(-0.451078\pi\)
0.153089 + 0.988212i \(0.451078\pi\)
\(44\) 1.71706e6 3.03880
\(45\) −61539.7 −0.100673
\(46\) 2.40360e6 3.64091
\(47\) −615386. −0.864580 −0.432290 0.901735i \(-0.642294\pi\)
−0.432290 + 0.901735i \(0.642294\pi\)
\(48\) 1.97693e6 2.58016
\(49\) 1.52388e6 1.85040
\(50\) 1.58457e6 1.79274
\(51\) −384117. −0.405479
\(52\) −633051. −0.624348
\(53\) −745946. −0.688243 −0.344121 0.938925i \(-0.611823\pi\)
−0.344121 + 0.938925i \(0.611823\pi\)
\(54\) −439290. −0.379641
\(55\) −391643. −0.317410
\(56\) −8.27862e6 −6.29941
\(57\) −591374. −0.422961
\(58\) 4.35498e6 2.93081
\(59\) −205379. −0.130189
\(60\) −843558. −0.504180
\(61\) −2.69167e6 −1.51833 −0.759167 0.650896i \(-0.774393\pi\)
−0.759167 + 0.650896i \(0.774393\pi\)
\(62\) 1.52027e6 0.810119
\(63\) 1.11692e6 0.562771
\(64\) 1.16630e7 5.56135
\(65\) 144392. 0.0652147
\(66\) −2.79567e6 −1.19697
\(67\) 2.11862e6 0.860581 0.430290 0.902691i \(-0.358411\pi\)
0.430290 + 0.902691i \(0.358411\pi\)
\(68\) −5.26531e6 −2.03068
\(69\) −2.90781e6 −1.06560
\(70\) 2.88658e6 1.00587
\(71\) −1.64750e6 −0.546289 −0.273144 0.961973i \(-0.588064\pi\)
−0.273144 + 0.961973i \(0.588064\pi\)
\(72\) −3.93903e6 −1.24373
\(73\) −2.91422e6 −0.876783 −0.438391 0.898784i \(-0.644452\pi\)
−0.438391 + 0.898784i \(0.644452\pi\)
\(74\) 6.91062e6 1.98247
\(75\) −1.91697e6 −0.524687
\(76\) −8.10630e6 −2.11824
\(77\) 7.10818e6 1.77436
\(78\) 1.03071e6 0.245928
\(79\) 171784. 0.0392002 0.0196001 0.999808i \(-0.493761\pi\)
0.0196001 + 0.999808i \(0.493761\pi\)
\(80\) −6.18095e6 −1.34971
\(81\) 531441. 0.111111
\(82\) −1.94930e7 −3.90418
\(83\) 4.94728e6 0.949714 0.474857 0.880063i \(-0.342500\pi\)
0.474857 + 0.880063i \(0.342500\pi\)
\(84\) 1.53103e7 2.81842
\(85\) 1.20096e6 0.212110
\(86\) −3.56266e6 −0.603990
\(87\) −5.26854e6 −0.857773
\(88\) −2.50683e7 −3.92135
\(89\) −1.83391e6 −0.275748 −0.137874 0.990450i \(-0.544027\pi\)
−0.137874 + 0.990450i \(0.544027\pi\)
\(90\) 1.37346e6 0.198594
\(91\) −2.62066e6 −0.364557
\(92\) −3.98589e7 −5.33664
\(93\) −1.83918e6 −0.237101
\(94\) 1.37343e7 1.70553
\(95\) 1.84896e6 0.221255
\(96\) −2.54477e7 −2.93561
\(97\) −1.09662e7 −1.21998 −0.609992 0.792408i \(-0.708827\pi\)
−0.609992 + 0.792408i \(0.708827\pi\)
\(98\) −3.40104e7 −3.65023
\(99\) 3.38213e6 0.350322
\(100\) −2.62770e7 −2.62770
\(101\) −7.80317e6 −0.753610 −0.376805 0.926293i \(-0.622977\pi\)
−0.376805 + 0.926293i \(0.622977\pi\)
\(102\) 8.57282e6 0.799877
\(103\) −150244. −0.0135477 −0.00677386 0.999977i \(-0.502156\pi\)
−0.00677386 + 0.999977i \(0.502156\pi\)
\(104\) 9.24223e6 0.805675
\(105\) −3.49211e6 −0.294391
\(106\) 1.66482e7 1.35768
\(107\) 8.15970e6 0.643918 0.321959 0.946754i \(-0.395659\pi\)
0.321959 + 0.946754i \(0.395659\pi\)
\(108\) 7.28476e6 0.556457
\(109\) 6.65386e6 0.492131 0.246066 0.969253i \(-0.420862\pi\)
0.246066 + 0.969253i \(0.420862\pi\)
\(110\) 8.74078e6 0.626147
\(111\) −8.36027e6 −0.580217
\(112\) 1.12182e8 7.54502
\(113\) 1.76235e7 1.14899 0.574496 0.818507i \(-0.305198\pi\)
0.574496 + 0.818507i \(0.305198\pi\)
\(114\) 1.31984e7 0.834364
\(115\) 9.09137e6 0.557426
\(116\) −7.22187e7 −4.29583
\(117\) −1.24693e6 −0.0719767
\(118\) 4.58370e6 0.256820
\(119\) −2.17970e7 −1.18572
\(120\) 1.23155e7 0.650607
\(121\) 2.03695e6 0.104528
\(122\) 6.00733e7 2.99517
\(123\) 2.35821e7 1.14265
\(124\) −2.52106e7 −1.18743
\(125\) 1.25885e7 0.576487
\(126\) −2.49278e7 −1.11016
\(127\) −2.94003e7 −1.27362 −0.636808 0.771023i \(-0.719745\pi\)
−0.636808 + 0.771023i \(0.719745\pi\)
\(128\) −1.39656e8 −5.88608
\(129\) 4.31001e6 0.176772
\(130\) −3.22257e6 −0.128647
\(131\) 4.56665e6 0.177480 0.0887398 0.996055i \(-0.471716\pi\)
0.0887398 + 0.996055i \(0.471716\pi\)
\(132\) 4.63607e7 1.75445
\(133\) −3.35579e7 −1.23684
\(134\) −4.72839e7 −1.69764
\(135\) −1.66157e6 −0.0581234
\(136\) 7.68709e7 2.62045
\(137\) 3.33611e6 0.110846 0.0554228 0.998463i \(-0.482349\pi\)
0.0554228 + 0.998463i \(0.482349\pi\)
\(138\) 6.48971e7 2.10208
\(139\) 2.57204e7 0.812317 0.406158 0.913803i \(-0.366868\pi\)
0.406158 + 0.913803i \(0.366868\pi\)
\(140\) −4.78682e7 −1.47435
\(141\) −1.66154e7 −0.499166
\(142\) 3.67694e7 1.07765
\(143\) −7.93556e6 −0.226935
\(144\) 5.33771e7 1.48966
\(145\) 1.64723e7 0.448710
\(146\) 6.50402e7 1.72961
\(147\) 4.11448e7 1.06833
\(148\) −1.14599e8 −2.90579
\(149\) −7.15213e7 −1.77127 −0.885633 0.464387i \(-0.846275\pi\)
−0.885633 + 0.464387i \(0.846275\pi\)
\(150\) 4.27834e7 1.03504
\(151\) 3.70730e7 0.876271 0.438135 0.898909i \(-0.355639\pi\)
0.438135 + 0.898909i \(0.355639\pi\)
\(152\) 1.18348e8 2.73343
\(153\) −1.03712e7 −0.234103
\(154\) −1.58642e8 −3.50023
\(155\) 5.75026e6 0.124030
\(156\) −1.70924e7 −0.360467
\(157\) −5.84600e7 −1.20562 −0.602810 0.797885i \(-0.705952\pi\)
−0.602810 + 0.797885i \(0.705952\pi\)
\(158\) −3.83392e6 −0.0773291
\(159\) −2.01405e7 −0.397357
\(160\) 7.95632e7 1.53565
\(161\) −1.65005e8 −3.11607
\(162\) −1.18608e7 −0.219186
\(163\) 1.08299e8 1.95869 0.979346 0.202190i \(-0.0648057\pi\)
0.979346 + 0.202190i \(0.0648057\pi\)
\(164\) 3.23252e8 5.72253
\(165\) −1.05744e7 −0.183257
\(166\) −1.10415e8 −1.87347
\(167\) −6.99062e6 −0.116147 −0.0580735 0.998312i \(-0.518496\pi\)
−0.0580735 + 0.998312i \(0.518496\pi\)
\(168\) −2.23523e8 −3.63697
\(169\) −5.98228e7 −0.953374
\(170\) −2.68033e7 −0.418424
\(171\) −1.59671e7 −0.244197
\(172\) 5.90797e7 0.885296
\(173\) −9.78233e7 −1.43642 −0.718209 0.695827i \(-0.755038\pi\)
−0.718209 + 0.695827i \(0.755038\pi\)
\(174\) 1.17584e8 1.69211
\(175\) −1.08780e8 −1.53431
\(176\) 3.39696e8 4.69674
\(177\) −5.54523e6 −0.0751646
\(178\) 4.09296e7 0.543961
\(179\) 7.71690e7 1.00567 0.502837 0.864381i \(-0.332290\pi\)
0.502837 + 0.864381i \(0.332290\pi\)
\(180\) −2.27761e7 −0.291089
\(181\) −6.26316e7 −0.785088 −0.392544 0.919733i \(-0.628405\pi\)
−0.392544 + 0.919733i \(0.628405\pi\)
\(182\) 5.84885e7 0.719152
\(183\) −7.26750e7 −0.876610
\(184\) 5.81921e8 6.88655
\(185\) 2.61387e7 0.303517
\(186\) 4.10472e7 0.467723
\(187\) −6.60028e7 −0.738103
\(188\) −2.27757e8 −2.49988
\(189\) 3.01569e7 0.324916
\(190\) −4.12654e7 −0.436464
\(191\) −2.82767e7 −0.293638 −0.146819 0.989163i \(-0.546903\pi\)
−0.146819 + 0.989163i \(0.546903\pi\)
\(192\) 3.14901e8 3.21084
\(193\) −3.31268e7 −0.331688 −0.165844 0.986152i \(-0.553035\pi\)
−0.165844 + 0.986152i \(0.553035\pi\)
\(194\) 2.44746e8 2.40663
\(195\) 3.89858e6 0.0376517
\(196\) 5.63995e8 5.35031
\(197\) −1.09797e8 −1.02320 −0.511598 0.859225i \(-0.670946\pi\)
−0.511598 + 0.859225i \(0.670946\pi\)
\(198\) −7.54832e7 −0.691070
\(199\) 1.14654e8 1.03134 0.515671 0.856787i \(-0.327543\pi\)
0.515671 + 0.856787i \(0.327543\pi\)
\(200\) 3.83631e8 3.39085
\(201\) 5.72028e7 0.496856
\(202\) 1.74153e8 1.48663
\(203\) −2.98966e8 −2.50834
\(204\) −1.42163e8 −1.17242
\(205\) −7.37302e7 −0.597733
\(206\) 3.35318e6 0.0267252
\(207\) −7.85108e7 −0.615224
\(208\) −1.25240e8 −0.964985
\(209\) −1.01616e8 −0.769927
\(210\) 7.79377e7 0.580737
\(211\) −1.35295e8 −0.991502 −0.495751 0.868465i \(-0.665107\pi\)
−0.495751 + 0.868465i \(0.665107\pi\)
\(212\) −2.76077e8 −1.99001
\(213\) −4.44826e7 −0.315400
\(214\) −1.82110e8 −1.27024
\(215\) −1.34754e7 −0.0924714
\(216\) −1.06354e8 −0.718067
\(217\) −1.04365e8 −0.693341
\(218\) −1.48502e8 −0.970814
\(219\) −7.86839e7 −0.506211
\(220\) −1.44949e8 −0.917772
\(221\) 2.43341e7 0.151650
\(222\) 1.86587e8 1.14458
\(223\) 9.14456e7 0.552199 0.276100 0.961129i \(-0.410958\pi\)
0.276100 + 0.961129i \(0.410958\pi\)
\(224\) −1.44404e9 −8.58445
\(225\) −5.17582e7 −0.302928
\(226\) −3.93325e8 −2.26659
\(227\) −8.92462e7 −0.506406 −0.253203 0.967413i \(-0.581484\pi\)
−0.253203 + 0.967413i \(0.581484\pi\)
\(228\) −2.18870e8 −1.22297
\(229\) 2.51758e8 1.38535 0.692673 0.721251i \(-0.256433\pi\)
0.692673 + 0.721251i \(0.256433\pi\)
\(230\) −2.02903e8 −1.09962
\(231\) 1.91921e8 1.02443
\(232\) 1.05436e9 5.54345
\(233\) 3.50359e8 1.81455 0.907273 0.420542i \(-0.138160\pi\)
0.907273 + 0.420542i \(0.138160\pi\)
\(234\) 2.78293e7 0.141986
\(235\) 5.19488e7 0.261119
\(236\) −7.60116e7 −0.376433
\(237\) 4.63817e6 0.0226322
\(238\) 4.86470e8 2.33903
\(239\) 1.94400e8 0.921092 0.460546 0.887636i \(-0.347654\pi\)
0.460546 + 0.887636i \(0.347654\pi\)
\(240\) −1.66886e8 −0.779255
\(241\) 6.19386e7 0.285038 0.142519 0.989792i \(-0.454480\pi\)
0.142519 + 0.989792i \(0.454480\pi\)
\(242\) −4.54612e7 −0.206199
\(243\) 1.43489e7 0.0641500
\(244\) −9.96197e8 −4.39017
\(245\) −1.28641e8 −0.558853
\(246\) −5.26310e8 −2.25408
\(247\) 3.74639e7 0.158188
\(248\) 3.68063e8 1.53229
\(249\) 1.33576e8 0.548318
\(250\) −2.80954e8 −1.13722
\(251\) 2.42156e8 0.966577 0.483289 0.875461i \(-0.339442\pi\)
0.483289 + 0.875461i \(0.339442\pi\)
\(252\) 4.13378e8 1.62722
\(253\) −4.99648e8 −1.93974
\(254\) 6.56162e8 2.51243
\(255\) 3.24259e7 0.122462
\(256\) 1.62403e9 6.04996
\(257\) −3.80530e8 −1.39837 −0.699187 0.714939i \(-0.746454\pi\)
−0.699187 + 0.714939i \(0.746454\pi\)
\(258\) −9.61918e7 −0.348714
\(259\) −4.74409e8 −1.69669
\(260\) 5.34400e7 0.188564
\(261\) −1.42250e8 −0.495236
\(262\) −1.01920e8 −0.350109
\(263\) −1.88914e8 −0.640353 −0.320176 0.947358i \(-0.603742\pi\)
−0.320176 + 0.947358i \(0.603742\pi\)
\(264\) −6.76844e8 −2.26399
\(265\) 6.29702e7 0.207862
\(266\) 7.48953e8 2.43988
\(267\) −4.95155e7 −0.159203
\(268\) 7.84110e8 2.48832
\(269\) 2.07575e8 0.650194 0.325097 0.945681i \(-0.394603\pi\)
0.325097 + 0.945681i \(0.394603\pi\)
\(270\) 3.70834e7 0.114658
\(271\) −7.81832e7 −0.238628 −0.119314 0.992857i \(-0.538069\pi\)
−0.119314 + 0.992857i \(0.538069\pi\)
\(272\) −1.04166e9 −3.13860
\(273\) −7.07578e7 −0.210477
\(274\) −7.44562e7 −0.218662
\(275\) −3.29393e8 −0.955102
\(276\) −1.07619e9 −3.08111
\(277\) −1.30983e8 −0.370284 −0.185142 0.982712i \(-0.559274\pi\)
−0.185142 + 0.982712i \(0.559274\pi\)
\(278\) −5.74033e8 −1.60243
\(279\) −4.96578e7 −0.136890
\(280\) 6.98853e8 1.90253
\(281\) 1.69682e8 0.456209 0.228105 0.973637i \(-0.426747\pi\)
0.228105 + 0.973637i \(0.426747\pi\)
\(282\) 3.70827e8 0.984691
\(283\) 2.01304e8 0.527958 0.263979 0.964528i \(-0.414965\pi\)
0.263979 + 0.964528i \(0.414965\pi\)
\(284\) −6.09748e8 −1.57956
\(285\) 4.99218e7 0.127742
\(286\) 1.77108e8 0.447668
\(287\) 1.33818e9 3.34139
\(288\) −6.87088e8 −1.69488
\(289\) −2.07944e8 −0.506761
\(290\) −3.67632e8 −0.885158
\(291\) −2.96087e8 −0.704358
\(292\) −1.07856e9 −2.53516
\(293\) −4.63019e7 −0.107538 −0.0537691 0.998553i \(-0.517123\pi\)
−0.0537691 + 0.998553i \(0.517123\pi\)
\(294\) −9.18280e8 −2.10746
\(295\) 1.73374e7 0.0393194
\(296\) 1.67309e9 3.74971
\(297\) 9.13175e7 0.202258
\(298\) 1.59623e9 3.49413
\(299\) 1.84211e8 0.398536
\(300\) −7.09478e8 −1.51710
\(301\) 2.44574e8 0.516925
\(302\) −8.27404e8 −1.72859
\(303\) −2.10686e8 −0.435097
\(304\) −1.60371e9 −3.27393
\(305\) 2.27221e8 0.458564
\(306\) 2.31466e8 0.461809
\(307\) −8.24546e8 −1.62641 −0.813206 0.581975i \(-0.802280\pi\)
−0.813206 + 0.581975i \(0.802280\pi\)
\(308\) 2.63077e9 5.13044
\(309\) −4.05658e6 −0.00782178
\(310\) −1.28336e8 −0.244671
\(311\) 8.21297e8 1.54824 0.774121 0.633038i \(-0.218192\pi\)
0.774121 + 0.633038i \(0.218192\pi\)
\(312\) 2.49540e8 0.465157
\(313\) 4.00974e8 0.739113 0.369557 0.929208i \(-0.379510\pi\)
0.369557 + 0.929208i \(0.379510\pi\)
\(314\) 1.30472e9 2.37829
\(315\) −9.42868e7 −0.169967
\(316\) 6.35779e7 0.113345
\(317\) 2.03257e8 0.358375 0.179187 0.983815i \(-0.442653\pi\)
0.179187 + 0.983815i \(0.442653\pi\)
\(318\) 4.49501e8 0.783855
\(319\) −9.05292e8 −1.56143
\(320\) −9.84549e8 −1.67963
\(321\) 2.20312e8 0.371766
\(322\) 3.68263e9 6.14699
\(323\) 3.11601e8 0.514505
\(324\) 1.96688e8 0.321271
\(325\) 1.21441e8 0.196234
\(326\) −2.41704e9 −3.86386
\(327\) 1.79654e8 0.284132
\(328\) −4.71932e9 −7.38451
\(329\) −9.42853e8 −1.45968
\(330\) 2.36001e8 0.361506
\(331\) 4.59053e8 0.695769 0.347884 0.937537i \(-0.386900\pi\)
0.347884 + 0.937537i \(0.386900\pi\)
\(332\) 1.83101e9 2.74604
\(333\) −2.25727e8 −0.334988
\(334\) 1.56018e8 0.229120
\(335\) −1.78847e8 −0.259911
\(336\) 3.02892e9 4.35612
\(337\) −1.03100e9 −1.46741 −0.733706 0.679467i \(-0.762211\pi\)
−0.733706 + 0.679467i \(0.762211\pi\)
\(338\) 1.33514e9 1.88070
\(339\) 4.75834e8 0.663371
\(340\) 4.44479e8 0.613303
\(341\) −3.16026e8 −0.431601
\(342\) 3.56358e8 0.481720
\(343\) 1.07301e9 1.43574
\(344\) −8.62534e8 −1.14241
\(345\) 2.45467e8 0.321830
\(346\) 2.18324e9 2.83358
\(347\) 8.50881e8 1.09324 0.546620 0.837381i \(-0.315914\pi\)
0.546620 + 0.837381i \(0.315914\pi\)
\(348\) −1.94991e9 −2.48020
\(349\) −7.28476e8 −0.917331 −0.458666 0.888609i \(-0.651672\pi\)
−0.458666 + 0.888609i \(0.651672\pi\)
\(350\) 2.42777e9 3.02670
\(351\) −3.36671e7 −0.0415557
\(352\) −4.37268e9 −5.34377
\(353\) 1.14502e9 1.38548 0.692741 0.721186i \(-0.256403\pi\)
0.692741 + 0.721186i \(0.256403\pi\)
\(354\) 1.23760e8 0.148275
\(355\) 1.39077e8 0.164989
\(356\) −6.78737e8 −0.797309
\(357\) −5.88518e8 −0.684575
\(358\) −1.72228e9 −1.98387
\(359\) 4.66157e8 0.531743 0.265871 0.964008i \(-0.414340\pi\)
0.265871 + 0.964008i \(0.414340\pi\)
\(360\) 3.32520e8 0.375628
\(361\) −4.14141e8 −0.463312
\(362\) 1.39783e9 1.54872
\(363\) 5.49977e7 0.0603491
\(364\) −9.69917e8 −1.05409
\(365\) 2.46008e8 0.264804
\(366\) 1.62198e9 1.72926
\(367\) 1.98730e8 0.209861 0.104930 0.994480i \(-0.466538\pi\)
0.104930 + 0.994480i \(0.466538\pi\)
\(368\) −7.88551e9 −8.24826
\(369\) 6.36716e8 0.659710
\(370\) −5.83370e8 −0.598740
\(371\) −1.14289e9 −1.16197
\(372\) −6.80687e8 −0.685563
\(373\) 9.45866e8 0.943732 0.471866 0.881670i \(-0.343581\pi\)
0.471866 + 0.881670i \(0.343581\pi\)
\(374\) 1.47307e9 1.45604
\(375\) 3.39890e8 0.332835
\(376\) 3.32514e9 3.22591
\(377\) 3.33765e8 0.320809
\(378\) −6.73050e8 −0.640953
\(379\) −1.07925e9 −1.01832 −0.509162 0.860671i \(-0.670044\pi\)
−0.509162 + 0.860671i \(0.670044\pi\)
\(380\) 6.84306e8 0.639746
\(381\) −7.93807e8 −0.735322
\(382\) 6.31086e8 0.579251
\(383\) −5.46503e8 −0.497046 −0.248523 0.968626i \(-0.579945\pi\)
−0.248523 + 0.968626i \(0.579945\pi\)
\(384\) −3.77073e9 −3.39833
\(385\) −6.00048e8 −0.535888
\(386\) 7.39333e8 0.654311
\(387\) 1.16370e8 0.102060
\(388\) −4.05863e9 −3.52751
\(389\) −6.83446e8 −0.588682 −0.294341 0.955700i \(-0.595100\pi\)
−0.294341 + 0.955700i \(0.595100\pi\)
\(390\) −8.70094e7 −0.0742745
\(391\) 1.53215e9 1.29623
\(392\) −8.23405e9 −6.90418
\(393\) 1.23300e8 0.102468
\(394\) 2.45048e9 2.01843
\(395\) −1.45014e7 −0.0118391
\(396\) 1.25174e9 1.01293
\(397\) −1.21932e9 −0.978030 −0.489015 0.872275i \(-0.662644\pi\)
−0.489015 + 0.872275i \(0.662644\pi\)
\(398\) −2.55887e9 −2.03450
\(399\) −9.06063e8 −0.714091
\(400\) −5.19851e9 −4.06134
\(401\) −1.27738e9 −0.989268 −0.494634 0.869101i \(-0.664698\pi\)
−0.494634 + 0.869101i \(0.664698\pi\)
\(402\) −1.27667e9 −0.980135
\(403\) 1.16513e8 0.0886762
\(404\) −2.88799e9 −2.17902
\(405\) −4.48624e7 −0.0335575
\(406\) 6.67240e9 4.94813
\(407\) −1.43655e9 −1.05618
\(408\) 2.07551e9 1.51292
\(409\) −1.52758e9 −1.10401 −0.552004 0.833842i \(-0.686137\pi\)
−0.552004 + 0.833842i \(0.686137\pi\)
\(410\) 1.64553e9 1.17913
\(411\) 9.00750e7 0.0639968
\(412\) −5.56058e7 −0.0391724
\(413\) −3.14668e8 −0.219800
\(414\) 1.75222e9 1.21364
\(415\) −4.17632e8 −0.286831
\(416\) 1.61213e9 1.09792
\(417\) 6.94450e8 0.468991
\(418\) 2.26789e9 1.51881
\(419\) 2.15816e9 1.43329 0.716645 0.697438i \(-0.245677\pi\)
0.716645 + 0.697438i \(0.245677\pi\)
\(420\) −1.29244e9 −0.851214
\(421\) −4.86121e8 −0.317509 −0.158755 0.987318i \(-0.550748\pi\)
−0.158755 + 0.987318i \(0.550748\pi\)
\(422\) 3.01955e9 1.95591
\(423\) −4.48617e8 −0.288193
\(424\) 4.03059e9 2.56796
\(425\) 1.01007e9 0.638249
\(426\) 9.92773e8 0.622181
\(427\) −4.12399e9 −2.56342
\(428\) 3.01994e9 1.86185
\(429\) −2.14260e8 −0.131021
\(430\) 3.00747e8 0.182416
\(431\) 8.97584e8 0.540013 0.270007 0.962858i \(-0.412974\pi\)
0.270007 + 0.962858i \(0.412974\pi\)
\(432\) 1.44118e9 0.860054
\(433\) −1.07413e8 −0.0635843 −0.0317921 0.999495i \(-0.510121\pi\)
−0.0317921 + 0.999495i \(0.510121\pi\)
\(434\) 2.32925e9 1.36773
\(435\) 4.44752e8 0.259063
\(436\) 2.46262e9 1.42297
\(437\) 2.35885e9 1.35212
\(438\) 1.75609e9 0.998588
\(439\) −4.23839e8 −0.239097 −0.119549 0.992828i \(-0.538145\pi\)
−0.119549 + 0.992828i \(0.538145\pi\)
\(440\) 2.11618e9 1.18432
\(441\) 1.11091e9 0.616799
\(442\) −5.43094e8 −0.299155
\(443\) −1.56918e9 −0.857552 −0.428776 0.903411i \(-0.641055\pi\)
−0.428776 + 0.903411i \(0.641055\pi\)
\(444\) −3.09417e9 −1.67766
\(445\) 1.54812e8 0.0832809
\(446\) −2.04091e9 −1.08931
\(447\) −1.93108e9 −1.02264
\(448\) 1.78692e10 9.38929
\(449\) −6.52660e8 −0.340271 −0.170135 0.985421i \(-0.554420\pi\)
−0.170135 + 0.985421i \(0.554420\pi\)
\(450\) 1.15515e9 0.597579
\(451\) 4.05210e9 2.08000
\(452\) 6.52252e9 3.32224
\(453\) 1.00097e9 0.505915
\(454\) 1.99182e9 0.998974
\(455\) 2.21227e8 0.110103
\(456\) 3.19539e9 1.57815
\(457\) −3.14784e9 −1.54278 −0.771392 0.636360i \(-0.780439\pi\)
−0.771392 + 0.636360i \(0.780439\pi\)
\(458\) −5.61879e9 −2.73284
\(459\) −2.80021e8 −0.135160
\(460\) 3.36475e9 1.61176
\(461\) 3.81603e9 1.81409 0.907044 0.421037i \(-0.138334\pi\)
0.907044 + 0.421037i \(0.138334\pi\)
\(462\) −4.28334e9 −2.02086
\(463\) 7.32034e8 0.342766 0.171383 0.985204i \(-0.445176\pi\)
0.171383 + 0.985204i \(0.445176\pi\)
\(464\) −1.42874e10 −6.63958
\(465\) 1.55257e8 0.0716087
\(466\) −7.81941e9 −3.57951
\(467\) −9.56937e8 −0.434785 −0.217392 0.976084i \(-0.569755\pi\)
−0.217392 + 0.976084i \(0.569755\pi\)
\(468\) −4.61494e8 −0.208116
\(469\) 3.24601e9 1.45293
\(470\) −1.15941e9 −0.515102
\(471\) −1.57842e9 −0.696065
\(472\) 1.10973e9 0.485759
\(473\) 7.40589e8 0.321783
\(474\) −1.03516e8 −0.0446460
\(475\) 1.55507e9 0.665767
\(476\) −8.06714e9 −3.42843
\(477\) −5.43794e8 −0.229414
\(478\) −4.33866e9 −1.81701
\(479\) 3.19983e9 1.33031 0.665154 0.746706i \(-0.268366\pi\)
0.665154 + 0.746706i \(0.268366\pi\)
\(480\) 2.14821e9 0.886608
\(481\) 5.29629e8 0.217002
\(482\) −1.38236e9 −0.562286
\(483\) −4.45514e9 −1.79906
\(484\) 7.53884e8 0.302236
\(485\) 9.25727e8 0.368457
\(486\) −3.20242e8 −0.126547
\(487\) −3.55741e8 −0.139567 −0.0697834 0.997562i \(-0.522231\pi\)
−0.0697834 + 0.997562i \(0.522231\pi\)
\(488\) 1.45440e10 5.66518
\(489\) 2.92406e9 1.13085
\(490\) 2.87104e9 1.10243
\(491\) −1.77516e7 −0.00676788 −0.00338394 0.999994i \(-0.501077\pi\)
−0.00338394 + 0.999994i \(0.501077\pi\)
\(492\) 8.72781e9 3.30391
\(493\) 2.77604e9 1.04343
\(494\) −8.36129e8 −0.312054
\(495\) −2.85508e8 −0.105803
\(496\) −4.98755e9 −1.83528
\(497\) −2.52419e9 −0.922306
\(498\) −2.98119e9 −1.08165
\(499\) 1.97045e9 0.709927 0.354963 0.934880i \(-0.384493\pi\)
0.354963 + 0.934880i \(0.384493\pi\)
\(500\) 4.65906e9 1.66688
\(501\) −1.88747e8 −0.0670575
\(502\) −5.40449e9 −1.90674
\(503\) −4.77882e9 −1.67430 −0.837148 0.546976i \(-0.815779\pi\)
−0.837148 + 0.546976i \(0.815779\pi\)
\(504\) −6.03511e9 −2.09980
\(505\) 6.58717e8 0.227604
\(506\) 1.11513e10 3.82647
\(507\) −1.61522e9 −0.550431
\(508\) −1.08812e10 −3.68258
\(509\) −2.21746e9 −0.745321 −0.372660 0.927968i \(-0.621554\pi\)
−0.372660 + 0.927968i \(0.621554\pi\)
\(510\) −7.23688e8 −0.241577
\(511\) −4.46496e9 −1.48028
\(512\) −1.83694e10 −6.04852
\(513\) −4.31112e8 −0.140987
\(514\) 8.49277e9 2.75853
\(515\) 1.26831e7 0.00409165
\(516\) 1.59515e9 0.511126
\(517\) −2.85503e9 −0.908644
\(518\) 1.05880e10 3.34702
\(519\) −2.64123e9 −0.829317
\(520\) −7.80197e8 −0.243328
\(521\) −2.05916e9 −0.637908 −0.318954 0.947770i \(-0.603332\pi\)
−0.318954 + 0.947770i \(0.603332\pi\)
\(522\) 3.17478e9 0.976938
\(523\) 4.60384e9 1.40723 0.703614 0.710582i \(-0.251568\pi\)
0.703614 + 0.710582i \(0.251568\pi\)
\(524\) 1.69014e9 0.513171
\(525\) −2.93705e9 −0.885836
\(526\) 4.21623e9 1.26321
\(527\) 9.69081e8 0.288418
\(528\) 9.17179e9 2.71166
\(529\) 8.19372e9 2.40650
\(530\) −1.40538e9 −0.410043
\(531\) −1.49721e8 −0.0433963
\(532\) −1.24199e10 −3.57625
\(533\) −1.49394e9 −0.427354
\(534\) 1.10510e9 0.314056
\(535\) −6.88814e8 −0.194475
\(536\) −1.14476e10 −3.21099
\(537\) 2.08356e9 0.580626
\(538\) −4.63272e9 −1.28262
\(539\) 7.06992e9 1.94470
\(540\) −6.14954e8 −0.168060
\(541\) −5.44524e9 −1.47852 −0.739258 0.673422i \(-0.764824\pi\)
−0.739258 + 0.673422i \(0.764824\pi\)
\(542\) 1.74491e9 0.470735
\(543\) −1.69105e9 −0.453271
\(544\) 1.34086e10 3.57099
\(545\) −5.61696e8 −0.148632
\(546\) 1.57919e9 0.415203
\(547\) −2.09397e9 −0.547035 −0.273518 0.961867i \(-0.588187\pi\)
−0.273518 + 0.961867i \(0.588187\pi\)
\(548\) 1.23471e9 0.320503
\(549\) −1.96223e9 −0.506111
\(550\) 7.35147e9 1.88410
\(551\) 4.27390e9 1.08841
\(552\) 1.57119e10 3.97595
\(553\) 2.63196e8 0.0661821
\(554\) 2.92330e9 0.730449
\(555\) 7.05746e8 0.175236
\(556\) 9.51921e9 2.34876
\(557\) 4.55263e9 1.11627 0.558135 0.829750i \(-0.311517\pi\)
0.558135 + 0.829750i \(0.311517\pi\)
\(558\) 1.10827e9 0.270040
\(559\) −2.73042e8 −0.0661131
\(560\) −9.47003e9 −2.27873
\(561\) −1.78208e9 −0.426144
\(562\) −3.78701e9 −0.899952
\(563\) −2.34649e9 −0.554166 −0.277083 0.960846i \(-0.589368\pi\)
−0.277083 + 0.960846i \(0.589368\pi\)
\(564\) −6.14944e9 −1.44331
\(565\) −1.48771e9 −0.347016
\(566\) −4.49274e9 −1.04149
\(567\) 8.14237e8 0.187590
\(568\) 8.90202e9 2.03831
\(569\) −2.75800e9 −0.627626 −0.313813 0.949485i \(-0.601606\pi\)
−0.313813 + 0.949485i \(0.601606\pi\)
\(570\) −1.11417e9 −0.251993
\(571\) −1.49430e9 −0.335901 −0.167950 0.985795i \(-0.553715\pi\)
−0.167950 + 0.985795i \(0.553715\pi\)
\(572\) −2.93698e9 −0.656168
\(573\) −7.63470e8 −0.169532
\(574\) −2.98658e10 −6.59147
\(575\) 7.64633e9 1.67732
\(576\) 8.50232e9 1.85378
\(577\) −5.27606e8 −0.114339 −0.0571695 0.998364i \(-0.518208\pi\)
−0.0571695 + 0.998364i \(0.518208\pi\)
\(578\) 4.64094e9 0.999673
\(579\) −8.94425e8 −0.191500
\(580\) 6.09646e9 1.29742
\(581\) 7.57987e9 1.60341
\(582\) 6.60814e9 1.38947
\(583\) −3.46075e9 −0.723319
\(584\) 1.57465e10 3.27144
\(585\) 1.05262e8 0.0217382
\(586\) 1.03338e9 0.212138
\(587\) −3.02681e9 −0.617664 −0.308832 0.951117i \(-0.599938\pi\)
−0.308832 + 0.951117i \(0.599938\pi\)
\(588\) 1.52279e10 3.08900
\(589\) 1.49197e9 0.300854
\(590\) −3.86940e8 −0.0775643
\(591\) −2.96452e9 −0.590743
\(592\) −2.26717e10 −4.49116
\(593\) −7.55687e9 −1.48816 −0.744081 0.668089i \(-0.767113\pi\)
−0.744081 + 0.668089i \(0.767113\pi\)
\(594\) −2.03805e9 −0.398990
\(595\) 1.84002e9 0.358108
\(596\) −2.64703e10 −5.12150
\(597\) 3.09565e9 0.595445
\(598\) −4.11127e9 −0.786181
\(599\) −7.21826e9 −1.37227 −0.686133 0.727476i \(-0.740693\pi\)
−0.686133 + 0.727476i \(0.740693\pi\)
\(600\) 1.03580e10 1.95771
\(601\) −4.11700e9 −0.773607 −0.386803 0.922162i \(-0.626421\pi\)
−0.386803 + 0.922162i \(0.626421\pi\)
\(602\) −5.45846e9 −1.01972
\(603\) 1.54448e9 0.286860
\(604\) 1.37209e10 2.53368
\(605\) −1.71952e8 −0.0315693
\(606\) 4.70213e9 0.858303
\(607\) 3.42535e9 0.621649 0.310824 0.950467i \(-0.399395\pi\)
0.310824 + 0.950467i \(0.399395\pi\)
\(608\) 2.06435e10 3.72495
\(609\) −8.07209e9 −1.44819
\(610\) −5.07118e9 −0.904596
\(611\) 1.05260e9 0.186689
\(612\) −3.83841e9 −0.676895
\(613\) −8.07359e9 −1.41565 −0.707823 0.706390i \(-0.750323\pi\)
−0.707823 + 0.706390i \(0.750323\pi\)
\(614\) 1.84024e10 3.20838
\(615\) −1.99072e9 −0.345101
\(616\) −3.84079e10 −6.62046
\(617\) −7.36421e9 −1.26220 −0.631100 0.775702i \(-0.717396\pi\)
−0.631100 + 0.775702i \(0.717396\pi\)
\(618\) 9.05358e7 0.0154298
\(619\) 9.33576e9 1.58210 0.791048 0.611755i \(-0.209536\pi\)
0.791048 + 0.611755i \(0.209536\pi\)
\(620\) 2.12820e9 0.358625
\(621\) −2.11979e9 −0.355200
\(622\) −1.83299e10 −3.05417
\(623\) −2.80979e9 −0.465549
\(624\) −3.38147e9 −0.557134
\(625\) 4.48410e9 0.734676
\(626\) −8.94903e9 −1.45803
\(627\) −2.74363e9 −0.444517
\(628\) −2.16363e10 −3.48597
\(629\) 4.40511e9 0.705796
\(630\) 2.10432e9 0.335289
\(631\) 1.58549e9 0.251225 0.125612 0.992079i \(-0.459910\pi\)
0.125612 + 0.992079i \(0.459910\pi\)
\(632\) −9.28206e8 −0.146263
\(633\) −3.65297e9 −0.572444
\(634\) −4.53633e9 −0.706956
\(635\) 2.48187e9 0.384655
\(636\) −7.45409e9 −1.14893
\(637\) −2.60655e9 −0.399556
\(638\) 2.02045e10 3.08018
\(639\) −1.20103e9 −0.182096
\(640\) 1.17893e10 1.77770
\(641\) 7.99377e9 1.19880 0.599402 0.800448i \(-0.295405\pi\)
0.599402 + 0.800448i \(0.295405\pi\)
\(642\) −4.91697e9 −0.733373
\(643\) −2.26242e9 −0.335610 −0.167805 0.985820i \(-0.553668\pi\)
−0.167805 + 0.985820i \(0.553668\pi\)
\(644\) −6.10691e10 −9.00992
\(645\) −3.63836e8 −0.0533884
\(646\) −6.95438e9 −1.01495
\(647\) 4.58044e8 0.0664879 0.0332439 0.999447i \(-0.489416\pi\)
0.0332439 + 0.999447i \(0.489416\pi\)
\(648\) −2.87155e9 −0.414576
\(649\) −9.52837e8 −0.136824
\(650\) −2.71035e9 −0.387105
\(651\) −2.81786e9 −0.400301
\(652\) 4.00818e10 5.66344
\(653\) 8.67155e9 1.21871 0.609355 0.792897i \(-0.291428\pi\)
0.609355 + 0.792897i \(0.291428\pi\)
\(654\) −4.00957e9 −0.560500
\(655\) −3.85501e8 −0.0536020
\(656\) 6.39507e10 8.84468
\(657\) −2.12446e9 −0.292261
\(658\) 2.10428e10 2.87947
\(659\) 8.76494e9 1.19303 0.596513 0.802603i \(-0.296552\pi\)
0.596513 + 0.802603i \(0.296552\pi\)
\(660\) −3.91361e9 −0.529876
\(661\) 1.58045e9 0.212850 0.106425 0.994321i \(-0.466060\pi\)
0.106425 + 0.994321i \(0.466060\pi\)
\(662\) −1.02453e10 −1.37252
\(663\) 6.57020e8 0.0875550
\(664\) −2.67318e10 −3.54356
\(665\) 2.83284e9 0.373548
\(666\) 5.03784e9 0.660822
\(667\) 2.10149e10 2.74213
\(668\) −2.58726e9 −0.335832
\(669\) 2.46903e9 0.318812
\(670\) 3.99155e9 0.512719
\(671\) −1.24877e10 −1.59572
\(672\) −3.89892e10 −4.95623
\(673\) −2.26086e9 −0.285905 −0.142952 0.989730i \(-0.545660\pi\)
−0.142952 + 0.989730i \(0.545660\pi\)
\(674\) 2.30100e10 2.89472
\(675\) −1.39747e9 −0.174896
\(676\) −2.21407e10 −2.75662
\(677\) 6.78561e9 0.840483 0.420241 0.907412i \(-0.361945\pi\)
0.420241 + 0.907412i \(0.361945\pi\)
\(678\) −1.06198e10 −1.30861
\(679\) −1.68016e10 −2.05971
\(680\) −6.48918e9 −0.791423
\(681\) −2.40965e9 −0.292374
\(682\) 7.05314e9 0.851407
\(683\) −7.74627e9 −0.930294 −0.465147 0.885234i \(-0.653998\pi\)
−0.465147 + 0.885234i \(0.653998\pi\)
\(684\) −5.90949e9 −0.706080
\(685\) −2.81623e8 −0.0334774
\(686\) −2.39477e10 −2.83224
\(687\) 6.79746e9 0.799830
\(688\) 1.16880e10 1.36830
\(689\) 1.27592e9 0.148612
\(690\) −5.47839e9 −0.634865
\(691\) −5.90227e9 −0.680528 −0.340264 0.940330i \(-0.610516\pi\)
−0.340264 + 0.940330i \(0.610516\pi\)
\(692\) −3.62048e10 −4.15331
\(693\) 5.18187e9 0.591452
\(694\) −1.89902e10 −2.15660
\(695\) −2.17122e9 −0.245334
\(696\) 2.84677e10 3.20051
\(697\) −1.24256e10 −1.38996
\(698\) 1.62583e10 1.80959
\(699\) 9.45970e9 1.04763
\(700\) −4.02597e10 −4.43637
\(701\) −1.01648e10 −1.11452 −0.557259 0.830339i \(-0.688147\pi\)
−0.557259 + 0.830339i \(0.688147\pi\)
\(702\) 7.51391e8 0.0819759
\(703\) 6.78196e9 0.736227
\(704\) 5.41094e10 5.84478
\(705\) 1.40262e9 0.150757
\(706\) −2.55548e10 −2.73310
\(707\) −1.19555e10 −1.27233
\(708\) −2.05231e9 −0.217334
\(709\) −7.24332e9 −0.763266 −0.381633 0.924314i \(-0.624638\pi\)
−0.381633 + 0.924314i \(0.624638\pi\)
\(710\) −3.10395e9 −0.325469
\(711\) 1.25231e8 0.0130667
\(712\) 9.90922e9 1.02887
\(713\) 7.33604e9 0.757964
\(714\) 1.31347e10 1.35044
\(715\) 6.69892e8 0.0685384
\(716\) 2.85606e10 2.90784
\(717\) 5.24879e9 0.531792
\(718\) −1.04038e10 −1.04895
\(719\) 1.37810e10 1.38270 0.691350 0.722520i \(-0.257016\pi\)
0.691350 + 0.722520i \(0.257016\pi\)
\(720\) −4.50591e9 −0.449903
\(721\) −2.30193e8 −0.0228728
\(722\) 9.24290e9 0.913962
\(723\) 1.67234e9 0.164567
\(724\) −2.31802e10 −2.27003
\(725\) 1.38541e10 1.35019
\(726\) −1.22745e9 −0.119049
\(727\) −1.04638e10 −1.00999 −0.504996 0.863122i \(-0.668506\pi\)
−0.504996 + 0.863122i \(0.668506\pi\)
\(728\) 1.41603e10 1.36023
\(729\) 3.87420e8 0.0370370
\(730\) −5.49047e9 −0.522372
\(731\) −2.27099e9 −0.215032
\(732\) −2.68973e10 −2.53466
\(733\) −6.09625e9 −0.571740 −0.285870 0.958268i \(-0.592283\pi\)
−0.285870 + 0.958268i \(0.592283\pi\)
\(734\) −4.43530e9 −0.413987
\(735\) −3.47331e9 −0.322654
\(736\) 1.01505e11 9.38456
\(737\) 9.82915e9 0.904440
\(738\) −1.42104e10 −1.30139
\(739\) 1.62223e10 1.47862 0.739308 0.673367i \(-0.235153\pi\)
0.739308 + 0.673367i \(0.235153\pi\)
\(740\) 9.67405e9 0.877601
\(741\) 1.01153e9 0.0913300
\(742\) 2.55072e10 2.29218
\(743\) −2.03461e10 −1.81978 −0.909891 0.414847i \(-0.863835\pi\)
−0.909891 + 0.414847i \(0.863835\pi\)
\(744\) 9.93770e9 0.884668
\(745\) 6.03758e9 0.534954
\(746\) −2.11101e10 −1.86167
\(747\) 3.60656e9 0.316571
\(748\) −2.44279e10 −2.13418
\(749\) 1.25017e10 1.08714
\(750\) −7.58575e9 −0.656575
\(751\) 6.71015e9 0.578086 0.289043 0.957316i \(-0.406663\pi\)
0.289043 + 0.957316i \(0.406663\pi\)
\(752\) −4.50584e10 −3.86379
\(753\) 6.53820e9 0.558054
\(754\) −7.44905e9 −0.632850
\(755\) −3.12957e9 −0.264649
\(756\) 1.11612e10 0.939474
\(757\) 7.51213e9 0.629401 0.314700 0.949191i \(-0.398096\pi\)
0.314700 + 0.949191i \(0.398096\pi\)
\(758\) 2.40870e10 2.00882
\(759\) −1.34905e10 −1.11991
\(760\) −9.99053e9 −0.825545
\(761\) 1.17043e10 0.962721 0.481361 0.876523i \(-0.340143\pi\)
0.481361 + 0.876523i \(0.340143\pi\)
\(762\) 1.77164e10 1.45055
\(763\) 1.01946e10 0.830871
\(764\) −1.04653e10 −0.849035
\(765\) 8.75498e8 0.0707034
\(766\) 1.21970e10 0.980509
\(767\) 3.51294e8 0.0281117
\(768\) 4.38487e10 3.49295
\(769\) −2.01241e9 −0.159579 −0.0797894 0.996812i \(-0.525425\pi\)
−0.0797894 + 0.996812i \(0.525425\pi\)
\(770\) 1.33920e10 1.05713
\(771\) −1.02743e10 −0.807351
\(772\) −1.22604e10 −0.959055
\(773\) −5.06105e9 −0.394105 −0.197053 0.980393i \(-0.563137\pi\)
−0.197053 + 0.980393i \(0.563137\pi\)
\(774\) −2.59718e9 −0.201330
\(775\) 4.83628e9 0.373212
\(776\) 5.92539e10 4.55199
\(777\) −1.28090e10 −0.979587
\(778\) 1.52533e10 1.16128
\(779\) −1.91301e10 −1.44989
\(780\) 1.44288e9 0.108868
\(781\) −7.64344e9 −0.574130
\(782\) −3.41949e10 −2.55704
\(783\) −3.84076e9 −0.285924
\(784\) 1.11578e11 8.26938
\(785\) 4.93499e9 0.364119
\(786\) −2.75183e9 −0.202136
\(787\) −1.29130e10 −0.944311 −0.472155 0.881515i \(-0.656524\pi\)
−0.472155 + 0.881515i \(0.656524\pi\)
\(788\) −4.06364e10 −2.95851
\(789\) −5.10068e9 −0.369708
\(790\) 3.23646e8 0.0233548
\(791\) 2.70015e10 1.93986
\(792\) −1.82748e10 −1.30712
\(793\) 4.60401e9 0.327854
\(794\) 2.72132e10 1.92933
\(795\) 1.70019e9 0.120009
\(796\) 4.24338e10 2.98206
\(797\) −1.75345e10 −1.22684 −0.613422 0.789755i \(-0.710208\pi\)
−0.613422 + 0.789755i \(0.710208\pi\)
\(798\) 2.02217e10 1.40867
\(799\) 8.75483e9 0.607203
\(800\) 6.69169e10 4.62084
\(801\) −1.33692e9 −0.0919161
\(802\) 2.85088e10 1.95150
\(803\) −1.35203e10 −0.921468
\(804\) 2.11710e10 1.43663
\(805\) 1.39292e10 0.941109
\(806\) −2.60037e9 −0.174929
\(807\) 5.60454e9 0.375390
\(808\) 4.21632e10 2.81186
\(809\) 3.03994e9 0.201858 0.100929 0.994894i \(-0.467819\pi\)
0.100929 + 0.994894i \(0.467819\pi\)
\(810\) 1.00125e9 0.0661980
\(811\) 1.35418e10 0.891465 0.445733 0.895166i \(-0.352943\pi\)
0.445733 + 0.895166i \(0.352943\pi\)
\(812\) −1.10649e11 −7.25270
\(813\) −2.11095e9 −0.137772
\(814\) 3.20612e10 2.08350
\(815\) −9.14220e9 −0.591560
\(816\) −2.81249e10 −1.81207
\(817\) −3.49634e9 −0.224303
\(818\) 3.40929e10 2.17785
\(819\) −1.91046e9 −0.121519
\(820\) −2.72879e10 −1.72831
\(821\) −1.44424e10 −0.910834 −0.455417 0.890278i \(-0.650510\pi\)
−0.455417 + 0.890278i \(0.650510\pi\)
\(822\) −2.01032e9 −0.126245
\(823\) 1.58161e10 0.989009 0.494504 0.869175i \(-0.335350\pi\)
0.494504 + 0.869175i \(0.335350\pi\)
\(824\) 8.11818e8 0.0505491
\(825\) −8.89360e9 −0.551428
\(826\) 7.02283e9 0.433593
\(827\) −2.91022e10 −1.78919 −0.894594 0.446879i \(-0.852535\pi\)
−0.894594 + 0.446879i \(0.852535\pi\)
\(828\) −2.90572e10 −1.77888
\(829\) −1.38090e10 −0.841825 −0.420913 0.907101i \(-0.638290\pi\)
−0.420913 + 0.907101i \(0.638290\pi\)
\(830\) 9.32081e9 0.565823
\(831\) −3.53653e9 −0.213784
\(832\) −1.99492e10 −1.20086
\(833\) −2.16796e10 −1.29955
\(834\) −1.54989e10 −0.925166
\(835\) 5.90124e8 0.0350785
\(836\) −3.76084e10 −2.22620
\(837\) −1.34076e9 −0.0790337
\(838\) −4.81663e10 −2.82741
\(839\) 2.26960e10 1.32673 0.663366 0.748295i \(-0.269127\pi\)
0.663366 + 0.748295i \(0.269127\pi\)
\(840\) 1.88690e10 1.09843
\(841\) 2.08262e10 1.20732
\(842\) 1.08494e10 0.626342
\(843\) 4.58142e9 0.263393
\(844\) −5.00733e10 −2.86687
\(845\) 5.05004e9 0.287936
\(846\) 1.00123e10 0.568511
\(847\) 3.12088e9 0.176476
\(848\) −5.46179e10 −3.07574
\(849\) 5.43520e9 0.304817
\(850\) −2.25430e10 −1.25906
\(851\) 3.33471e10 1.85483
\(852\) −1.64632e10 −0.911959
\(853\) −2.76986e9 −0.152804 −0.0764022 0.997077i \(-0.524343\pi\)
−0.0764022 + 0.997077i \(0.524343\pi\)
\(854\) 9.20402e10 5.05679
\(855\) 1.34789e9 0.0737518
\(856\) −4.40896e10 −2.40258
\(857\) 8.53606e9 0.463260 0.231630 0.972804i \(-0.425594\pi\)
0.231630 + 0.972804i \(0.425594\pi\)
\(858\) 4.78191e9 0.258461
\(859\) −7.27081e9 −0.391387 −0.195694 0.980665i \(-0.562696\pi\)
−0.195694 + 0.980665i \(0.562696\pi\)
\(860\) −4.98730e9 −0.267375
\(861\) 3.61308e10 1.92915
\(862\) −2.00325e10 −1.06527
\(863\) 2.06289e10 1.09254 0.546270 0.837609i \(-0.316047\pi\)
0.546270 + 0.837609i \(0.316047\pi\)
\(864\) −1.85514e10 −0.978538
\(865\) 8.25791e9 0.433824
\(866\) 2.39727e9 0.125431
\(867\) −5.61448e9 −0.292579
\(868\) −3.86260e10 −2.00475
\(869\) 7.96976e8 0.0411980
\(870\) −9.92607e9 −0.511046
\(871\) −3.62383e9 −0.185825
\(872\) −3.59530e10 −1.83623
\(873\) −7.99434e9 −0.406661
\(874\) −5.26454e10 −2.66729
\(875\) 1.92873e10 0.973291
\(876\) −2.91212e10 −1.46368
\(877\) 1.04224e10 0.521760 0.260880 0.965371i \(-0.415987\pi\)
0.260880 + 0.965371i \(0.415987\pi\)
\(878\) 9.45934e9 0.471661
\(879\) −1.25015e9 −0.0620872
\(880\) −2.86760e10 −1.41850
\(881\) 3.28267e10 1.61738 0.808690 0.588235i \(-0.200177\pi\)
0.808690 + 0.588235i \(0.200177\pi\)
\(882\) −2.47936e10 −1.21674
\(883\) 2.09738e10 1.02522 0.512608 0.858623i \(-0.328680\pi\)
0.512608 + 0.858623i \(0.328680\pi\)
\(884\) 9.00613e9 0.438486
\(885\) 4.68110e8 0.0227011
\(886\) 3.50214e10 1.69167
\(887\) 4.36272e9 0.209906 0.104953 0.994477i \(-0.466531\pi\)
0.104953 + 0.994477i \(0.466531\pi\)
\(888\) 4.51734e10 2.16490
\(889\) −4.50451e10 −2.15026
\(890\) −3.45514e9 −0.164286
\(891\) 2.46557e9 0.116774
\(892\) 3.38444e10 1.59665
\(893\) 1.34787e10 0.633383
\(894\) 4.30982e10 2.01733
\(895\) −6.51434e9 −0.303732
\(896\) −2.13972e11 −9.93754
\(897\) 4.97371e9 0.230095
\(898\) 1.45662e10 0.671243
\(899\) 1.32919e10 0.610137
\(900\) −1.91559e10 −0.875899
\(901\) 1.06122e10 0.483360
\(902\) −9.04359e10 −4.10315
\(903\) 6.60350e9 0.298447
\(904\) −9.52256e10 −4.28711
\(905\) 5.28714e9 0.237111
\(906\) −2.23399e10 −0.998005
\(907\) 9.67343e9 0.430482 0.215241 0.976561i \(-0.430946\pi\)
0.215241 + 0.976561i \(0.430946\pi\)
\(908\) −3.30304e10 −1.46424
\(909\) −5.68851e9 −0.251203
\(910\) −4.93740e9 −0.217197
\(911\) 1.13862e10 0.498959 0.249480 0.968380i \(-0.419740\pi\)
0.249480 + 0.968380i \(0.419740\pi\)
\(912\) −4.33002e10 −1.89020
\(913\) 2.29524e10 0.998116
\(914\) 7.02542e10 3.04341
\(915\) 6.13498e9 0.264752
\(916\) 9.31765e10 4.00564
\(917\) 6.99671e9 0.299641
\(918\) 6.24959e9 0.266626
\(919\) −1.74980e9 −0.0743676 −0.0371838 0.999308i \(-0.511839\pi\)
−0.0371838 + 0.999308i \(0.511839\pi\)
\(920\) −4.91237e10 −2.07986
\(921\) −2.22628e10 −0.939010
\(922\) −8.51670e10 −3.57860
\(923\) 2.81800e9 0.117960
\(924\) 7.10307e10 2.96206
\(925\) 2.19841e10 0.913297
\(926\) −1.63377e10 −0.676165
\(927\) −1.09528e8 −0.00451591
\(928\) 1.83912e11 7.55428
\(929\) −5.01633e9 −0.205273 −0.102636 0.994719i \(-0.532728\pi\)
−0.102636 + 0.994719i \(0.532728\pi\)
\(930\) −3.46507e9 −0.141261
\(931\) −3.33772e10 −1.35558
\(932\) 1.29669e11 5.24665
\(933\) 2.21750e10 0.893878
\(934\) 2.13572e10 0.857688
\(935\) 5.57173e9 0.222920
\(936\) 6.73758e9 0.268558
\(937\) 2.41581e10 0.959343 0.479672 0.877448i \(-0.340756\pi\)
0.479672 + 0.877448i \(0.340756\pi\)
\(938\) −7.24452e10 −2.86615
\(939\) 1.08263e10 0.426727
\(940\) 1.92265e10 0.755009
\(941\) −2.00603e10 −0.784826 −0.392413 0.919789i \(-0.628360\pi\)
−0.392413 + 0.919789i \(0.628360\pi\)
\(942\) 3.52276e10 1.37311
\(943\) −9.40632e10 −3.65282
\(944\) −1.50378e10 −0.581811
\(945\) −2.54574e9 −0.0981304
\(946\) −1.65286e10 −0.634772
\(947\) −1.33454e10 −0.510632 −0.255316 0.966858i \(-0.582180\pi\)
−0.255316 + 0.966858i \(0.582180\pi\)
\(948\) 1.71660e9 0.0654396
\(949\) 4.98467e9 0.189324
\(950\) −3.47064e10 −1.31334
\(951\) 5.48793e9 0.206908
\(952\) 1.17776e11 4.42414
\(953\) −4.58477e10 −1.71590 −0.857950 0.513733i \(-0.828262\pi\)
−0.857950 + 0.513733i \(0.828262\pi\)
\(954\) 1.21365e10 0.452559
\(955\) 2.38702e9 0.0886838
\(956\) 7.19481e10 2.66328
\(957\) −2.44429e10 −0.901490
\(958\) −7.14146e10 −2.62426
\(959\) 5.11136e9 0.187142
\(960\) −2.65828e10 −0.969732
\(961\) −2.28726e10 −0.831349
\(962\) −1.18204e10 −0.428074
\(963\) 5.94842e9 0.214639
\(964\) 2.29237e10 0.824168
\(965\) 2.79645e9 0.100176
\(966\) 9.94309e10 3.54896
\(967\) 2.81641e10 1.00162 0.500810 0.865557i \(-0.333036\pi\)
0.500810 + 0.865557i \(0.333036\pi\)
\(968\) −1.10063e10 −0.390013
\(969\) 8.41322e9 0.297050
\(970\) −2.06606e10 −0.726845
\(971\) 2.91527e10 1.02191 0.510954 0.859608i \(-0.329292\pi\)
0.510954 + 0.859608i \(0.329292\pi\)
\(972\) 5.31059e9 0.185486
\(973\) 3.94070e10 1.37144
\(974\) 7.93951e9 0.275320
\(975\) 3.27891e9 0.113296
\(976\) −1.97083e11 −6.78539
\(977\) 2.18798e10 0.750607 0.375303 0.926902i \(-0.377539\pi\)
0.375303 + 0.926902i \(0.377539\pi\)
\(978\) −6.52599e10 −2.23080
\(979\) −8.50825e9 −0.289802
\(980\) −4.76105e10 −1.61589
\(981\) 4.85066e9 0.164044
\(982\) 3.96185e8 0.0133508
\(983\) 7.16781e9 0.240685 0.120342 0.992732i \(-0.461601\pi\)
0.120342 + 0.992732i \(0.461601\pi\)
\(984\) −1.27422e11 −4.26345
\(985\) 9.26870e9 0.309024
\(986\) −6.19564e10 −2.05834
\(987\) −2.54570e10 −0.842748
\(988\) 1.38656e10 0.457391
\(989\) −1.71916e10 −0.565105
\(990\) 6.37203e9 0.208716
\(991\) −3.83800e9 −0.125270 −0.0626350 0.998037i \(-0.519950\pi\)
−0.0626350 + 0.998037i \(0.519950\pi\)
\(992\) 6.42014e10 2.08811
\(993\) 1.23944e10 0.401702
\(994\) 5.63355e10 1.81941
\(995\) −9.67868e9 −0.311484
\(996\) 4.94372e10 1.58543
\(997\) 5.63360e10 1.80033 0.900166 0.435547i \(-0.143445\pi\)
0.900166 + 0.435547i \(0.143445\pi\)
\(998\) −4.39770e10 −1.40045
\(999\) −6.09464e9 −0.193406
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 177.8.a.b.1.1 17
3.2 odd 2 531.8.a.d.1.17 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.1 17 1.1 even 1 trivial
531.8.a.d.1.17 17 3.2 odd 2