Properties

Label 33.6.a.d.1.2
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{313}) \)
Defining polynomial: \(x^{2} - x - 78\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.34590\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.34590 q^{2} -9.00000 q^{3} +55.3459 q^{4} +69.4590 q^{5} -84.1131 q^{6} +8.69181 q^{7} +218.189 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.34590 q^{2} -9.00000 q^{3} +55.3459 q^{4} +69.4590 q^{5} -84.1131 q^{6} +8.69181 q^{7} +218.189 q^{8} +81.0000 q^{9} +649.157 q^{10} +121.000 q^{11} -498.113 q^{12} -970.666 q^{13} +81.2328 q^{14} -625.131 q^{15} +268.100 q^{16} -424.616 q^{17} +757.018 q^{18} -1430.62 q^{19} +3844.27 q^{20} -78.2263 q^{21} +1130.85 q^{22} +2852.99 q^{23} -1963.70 q^{24} +1699.56 q^{25} -9071.75 q^{26} -729.000 q^{27} +481.056 q^{28} -7467.66 q^{29} -5842.42 q^{30} +10346.3 q^{31} -4476.40 q^{32} -1089.00 q^{33} -3968.42 q^{34} +603.724 q^{35} +4483.02 q^{36} +167.311 q^{37} -13370.4 q^{38} +8735.99 q^{39} +15155.2 q^{40} +5682.18 q^{41} -731.095 q^{42} +21148.9 q^{43} +6696.85 q^{44} +5626.18 q^{45} +26663.8 q^{46} -9785.11 q^{47} -2412.90 q^{48} -16731.5 q^{49} +15883.9 q^{50} +3821.55 q^{51} -53722.4 q^{52} +25639.5 q^{53} -6813.16 q^{54} +8404.54 q^{55} +1896.45 q^{56} +12875.5 q^{57} -69792.0 q^{58} -23411.5 q^{59} -34598.5 q^{60} +18591.8 q^{61} +96695.6 q^{62} +704.036 q^{63} -50415.2 q^{64} -67421.5 q^{65} -10177.7 q^{66} +39477.7 q^{67} -23500.8 q^{68} -25676.9 q^{69} +5642.35 q^{70} +3283.01 q^{71} +17673.3 q^{72} +29536.7 q^{73} +1563.67 q^{74} -15296.0 q^{75} -79178.8 q^{76} +1051.71 q^{77} +81645.7 q^{78} -10280.8 q^{79} +18622.0 q^{80} +6561.00 q^{81} +53105.1 q^{82} -38360.3 q^{83} -4329.50 q^{84} -29493.4 q^{85} +197656. q^{86} +67208.9 q^{87} +26400.8 q^{88} -2319.99 q^{89} +52581.7 q^{90} -8436.84 q^{91} +157902. q^{92} -93116.8 q^{93} -91450.7 q^{94} -99369.2 q^{95} +40287.6 q^{96} -81868.3 q^{97} -156371. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + O(q^{10}) \) \( 2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + 1546q^{10} + 242q^{11} - 837q^{12} - 66q^{13} + 304q^{14} + 342q^{15} - 543q^{16} - 920q^{17} + 81q^{18} - 2932q^{19} - 202q^{20} + 162q^{21} + 121q^{22} + 5246q^{23} - 1539q^{24} + 10122q^{25} - 16622q^{26} - 1458q^{27} - 524q^{28} - 12600q^{29} - 13914q^{30} + 9936q^{31} + 3803q^{32} - 2178q^{33} + 166q^{34} + 3472q^{35} + 7533q^{36} + 5996q^{37} - 840q^{38} + 594q^{39} + 20226q^{40} + 24244q^{41} - 2736q^{42} + 20360q^{43} + 11253q^{44} - 3078q^{45} + 6692q^{46} - 5806q^{47} + 4887q^{48} - 32826q^{49} - 54409q^{50} + 8280q^{51} - 19658q^{52} + 40770q^{53} - 729q^{54} - 4598q^{55} + 3156q^{56} + 26388q^{57} - 26958q^{58} + 18212q^{59} + 1818q^{60} - 11398q^{61} + 100120q^{62} - 1458q^{63} - 93559q^{64} - 164636q^{65} - 1089q^{66} + 65368q^{67} - 42154q^{68} - 47214q^{69} - 18296q^{70} + 61446q^{71} + 13851q^{72} + 53412q^{73} - 47082q^{74} - 91098q^{75} - 135712q^{76} - 2178q^{77} + 149598q^{78} + 17122q^{79} + 105782q^{80} + 13122q^{81} - 101810q^{82} - 14304q^{83} + 4716q^{84} + 23740q^{85} + 204240q^{86} + 113400q^{87} + 20691q^{88} - 58140q^{89} + 125226q^{90} - 32584q^{91} + 248008q^{92} - 89424q^{93} - 124660q^{94} + 61968q^{95} - 34227q^{96} - 183056q^{97} - 22047q^{98} + 19602q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.34590 1.65214 0.826069 0.563569i \(-0.190572\pi\)
0.826069 + 0.563569i \(0.190572\pi\)
\(3\) −9.00000 −0.577350
\(4\) 55.3459 1.72956
\(5\) 69.4590 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(6\) −84.1131 −0.953862
\(7\) 8.69181 0.0670448 0.0335224 0.999438i \(-0.489327\pi\)
0.0335224 + 0.999438i \(0.489327\pi\)
\(8\) 218.189 1.20533
\(9\) 81.0000 0.333333
\(10\) 649.157 2.05282
\(11\) 121.000 0.301511
\(12\) −498.113 −0.998562
\(13\) −970.666 −1.59298 −0.796492 0.604649i \(-0.793313\pi\)
−0.796492 + 0.604649i \(0.793313\pi\)
\(14\) 81.2328 0.110767
\(15\) −625.131 −0.717370
\(16\) 268.100 0.261816
\(17\) −424.616 −0.356348 −0.178174 0.983999i \(-0.557019\pi\)
−0.178174 + 0.983999i \(0.557019\pi\)
\(18\) 757.018 0.550713
\(19\) −1430.62 −0.909158 −0.454579 0.890707i \(-0.650210\pi\)
−0.454579 + 0.890707i \(0.650210\pi\)
\(20\) 3844.27 2.14901
\(21\) −78.2263 −0.0387083
\(22\) 1130.85 0.498138
\(23\) 2852.99 1.12456 0.562278 0.826948i \(-0.309925\pi\)
0.562278 + 0.826948i \(0.309925\pi\)
\(24\) −1963.70 −0.695899
\(25\) 1699.56 0.543858
\(26\) −9071.75 −2.63183
\(27\) −729.000 −0.192450
\(28\) 481.056 0.115958
\(29\) −7467.66 −1.64888 −0.824441 0.565948i \(-0.808510\pi\)
−0.824441 + 0.565948i \(0.808510\pi\)
\(30\) −5842.42 −1.18519
\(31\) 10346.3 1.93366 0.966832 0.255411i \(-0.0822109\pi\)
0.966832 + 0.255411i \(0.0822109\pi\)
\(32\) −4476.40 −0.772776
\(33\) −1089.00 −0.174078
\(34\) −3968.42 −0.588736
\(35\) 603.724 0.0833045
\(36\) 4483.02 0.576520
\(37\) 167.311 0.0200919 0.0100459 0.999950i \(-0.496802\pi\)
0.0100459 + 0.999950i \(0.496802\pi\)
\(38\) −13370.4 −1.50205
\(39\) 8735.99 0.919710
\(40\) 15155.2 1.49765
\(41\) 5682.18 0.527905 0.263952 0.964536i \(-0.414974\pi\)
0.263952 + 0.964536i \(0.414974\pi\)
\(42\) −731.095 −0.0639515
\(43\) 21148.9 1.74428 0.872142 0.489253i \(-0.162731\pi\)
0.872142 + 0.489253i \(0.162731\pi\)
\(44\) 6696.85 0.521482
\(45\) 5626.18 0.414174
\(46\) 26663.8 1.85792
\(47\) −9785.11 −0.646132 −0.323066 0.946376i \(-0.604714\pi\)
−0.323066 + 0.946376i \(0.604714\pi\)
\(48\) −2412.90 −0.151160
\(49\) −16731.5 −0.995505
\(50\) 15883.9 0.898529
\(51\) 3821.55 0.205738
\(52\) −53722.4 −2.75516
\(53\) 25639.5 1.25377 0.626887 0.779110i \(-0.284329\pi\)
0.626887 + 0.779110i \(0.284329\pi\)
\(54\) −6813.16 −0.317954
\(55\) 8404.54 0.374634
\(56\) 1896.45 0.0808113
\(57\) 12875.5 0.524902
\(58\) −69792.0 −2.72418
\(59\) −23411.5 −0.875588 −0.437794 0.899075i \(-0.644240\pi\)
−0.437794 + 0.899075i \(0.644240\pi\)
\(60\) −34598.5 −1.24073
\(61\) 18591.8 0.639732 0.319866 0.947463i \(-0.396362\pi\)
0.319866 + 0.947463i \(0.396362\pi\)
\(62\) 96695.6 3.19468
\(63\) 704.036 0.0223483
\(64\) −50415.2 −1.53855
\(65\) −67421.5 −1.97932
\(66\) −10177.7 −0.287600
\(67\) 39477.7 1.07440 0.537198 0.843456i \(-0.319483\pi\)
0.537198 + 0.843456i \(0.319483\pi\)
\(68\) −23500.8 −0.616325
\(69\) −25676.9 −0.649263
\(70\) 5642.35 0.137631
\(71\) 3283.01 0.0772905 0.0386453 0.999253i \(-0.487696\pi\)
0.0386453 + 0.999253i \(0.487696\pi\)
\(72\) 17673.3 0.401778
\(73\) 29536.7 0.648716 0.324358 0.945934i \(-0.394852\pi\)
0.324358 + 0.945934i \(0.394852\pi\)
\(74\) 1563.67 0.0331945
\(75\) −15296.0 −0.313997
\(76\) −79178.8 −1.57244
\(77\) 1051.71 0.0202148
\(78\) 81645.7 1.51949
\(79\) −10280.8 −0.185335 −0.0926676 0.995697i \(-0.529539\pi\)
−0.0926676 + 0.995697i \(0.529539\pi\)
\(80\) 18622.0 0.325312
\(81\) 6561.00 0.111111
\(82\) 53105.1 0.872171
\(83\) −38360.3 −0.611206 −0.305603 0.952159i \(-0.598858\pi\)
−0.305603 + 0.952159i \(0.598858\pi\)
\(84\) −4329.50 −0.0669483
\(85\) −29493.4 −0.442770
\(86\) 197656. 2.88180
\(87\) 67208.9 0.951982
\(88\) 26400.8 0.363422
\(89\) −2319.99 −0.0310464 −0.0155232 0.999880i \(-0.504941\pi\)
−0.0155232 + 0.999880i \(0.504941\pi\)
\(90\) 52581.7 0.684272
\(91\) −8436.84 −0.106801
\(92\) 157902. 1.94499
\(93\) −93116.8 −1.11640
\(94\) −91450.7 −1.06750
\(95\) −99369.2 −1.12965
\(96\) 40287.6 0.446162
\(97\) −81868.3 −0.883459 −0.441729 0.897148i \(-0.645635\pi\)
−0.441729 + 0.897148i \(0.645635\pi\)
\(98\) −156371. −1.64471
\(99\) 9801.00 0.100504
\(100\) 94063.5 0.940635
\(101\) −180138. −1.75712 −0.878560 0.477631i \(-0.841496\pi\)
−0.878560 + 0.477631i \(0.841496\pi\)
\(102\) 35715.8 0.339907
\(103\) −109911. −1.02082 −0.510410 0.859931i \(-0.670506\pi\)
−0.510410 + 0.859931i \(0.670506\pi\)
\(104\) −211788. −1.92008
\(105\) −5433.52 −0.0480959
\(106\) 239624. 2.07141
\(107\) 65683.5 0.554621 0.277311 0.960780i \(-0.410557\pi\)
0.277311 + 0.960780i \(0.410557\pi\)
\(108\) −40347.2 −0.332854
\(109\) 146333. 1.17971 0.589854 0.807510i \(-0.299185\pi\)
0.589854 + 0.807510i \(0.299185\pi\)
\(110\) 78548.0 0.618947
\(111\) −1505.80 −0.0116000
\(112\) 2330.27 0.0175534
\(113\) 261872. 1.92927 0.964636 0.263585i \(-0.0849048\pi\)
0.964636 + 0.263585i \(0.0849048\pi\)
\(114\) 120334. 0.867211
\(115\) 198166. 1.39728
\(116\) −413304. −2.85184
\(117\) −78623.9 −0.530995
\(118\) −218802. −1.44659
\(119\) −3690.68 −0.0238913
\(120\) −136396. −0.864669
\(121\) 14641.0 0.0909091
\(122\) 173758. 1.05693
\(123\) −51139.6 −0.304786
\(124\) 572626. 3.34439
\(125\) −99009.9 −0.566766
\(126\) 6579.85 0.0369224
\(127\) −22466.6 −0.123603 −0.0618015 0.998088i \(-0.519685\pi\)
−0.0618015 + 0.998088i \(0.519685\pi\)
\(128\) −327931. −1.76912
\(129\) −190340. −1.00706
\(130\) −630115. −3.27010
\(131\) 119595. 0.608886 0.304443 0.952531i \(-0.401530\pi\)
0.304443 + 0.952531i \(0.401530\pi\)
\(132\) −60271.7 −0.301078
\(133\) −12434.6 −0.0609543
\(134\) 368954. 1.77505
\(135\) −50635.6 −0.239123
\(136\) −92646.4 −0.429518
\(137\) 176443. 0.803160 0.401580 0.915824i \(-0.368461\pi\)
0.401580 + 0.915824i \(0.368461\pi\)
\(138\) −239974. −1.07267
\(139\) 288218. 1.26527 0.632636 0.774450i \(-0.281973\pi\)
0.632636 + 0.774450i \(0.281973\pi\)
\(140\) 33413.7 0.144080
\(141\) 88066.0 0.373044
\(142\) 30682.7 0.127695
\(143\) −117451. −0.480303
\(144\) 21716.1 0.0872722
\(145\) −518696. −2.04877
\(146\) 276047. 1.07177
\(147\) 150583. 0.574755
\(148\) 9259.98 0.0347501
\(149\) −133635. −0.493123 −0.246561 0.969127i \(-0.579301\pi\)
−0.246561 + 0.969127i \(0.579301\pi\)
\(150\) −142955. −0.518766
\(151\) 406122. 1.44949 0.724744 0.689019i \(-0.241958\pi\)
0.724744 + 0.689019i \(0.241958\pi\)
\(152\) −312144. −1.09584
\(153\) −34393.9 −0.118783
\(154\) 9829.17 0.0333976
\(155\) 718645. 2.40262
\(156\) 483501. 1.59069
\(157\) −158746. −0.513988 −0.256994 0.966413i \(-0.582732\pi\)
−0.256994 + 0.966413i \(0.582732\pi\)
\(158\) −96083.1 −0.306199
\(159\) −230755. −0.723867
\(160\) −310926. −0.960190
\(161\) 24797.7 0.0753956
\(162\) 61318.5 0.183571
\(163\) 174460. 0.514311 0.257155 0.966370i \(-0.417215\pi\)
0.257155 + 0.966370i \(0.417215\pi\)
\(164\) 314486. 0.913043
\(165\) −75640.9 −0.216295
\(166\) −358512. −1.00980
\(167\) 105879. 0.293777 0.146889 0.989153i \(-0.453074\pi\)
0.146889 + 0.989153i \(0.453074\pi\)
\(168\) −17068.1 −0.0466564
\(169\) 570899. 1.53760
\(170\) −275643. −0.731517
\(171\) −115880. −0.303053
\(172\) 1.17051e6 3.01684
\(173\) −432459. −1.09857 −0.549287 0.835634i \(-0.685101\pi\)
−0.549287 + 0.835634i \(0.685101\pi\)
\(174\) 628128. 1.57281
\(175\) 14772.2 0.0364628
\(176\) 32440.1 0.0789406
\(177\) 210704. 0.505521
\(178\) −21682.4 −0.0512929
\(179\) −624448. −1.45668 −0.728340 0.685216i \(-0.759708\pi\)
−0.728340 + 0.685216i \(0.759708\pi\)
\(180\) 311386. 0.716338
\(181\) 10602.4 0.0240551 0.0120275 0.999928i \(-0.496171\pi\)
0.0120275 + 0.999928i \(0.496171\pi\)
\(182\) −78849.9 −0.176450
\(183\) −167327. −0.369349
\(184\) 622490. 1.35546
\(185\) 11621.3 0.0249646
\(186\) −870260. −1.84445
\(187\) −51378.6 −0.107443
\(188\) −541566. −1.11752
\(189\) −6336.33 −0.0129028
\(190\) −928695. −1.86633
\(191\) 441955. 0.876586 0.438293 0.898832i \(-0.355583\pi\)
0.438293 + 0.898832i \(0.355583\pi\)
\(192\) 453736. 0.888282
\(193\) −444354. −0.858690 −0.429345 0.903141i \(-0.641255\pi\)
−0.429345 + 0.903141i \(0.641255\pi\)
\(194\) −765133. −1.45960
\(195\) 606793. 1.14276
\(196\) −926017. −1.72179
\(197\) −416576. −0.764766 −0.382383 0.924004i \(-0.624897\pi\)
−0.382383 + 0.924004i \(0.624897\pi\)
\(198\) 91599.2 0.166046
\(199\) −502973. −0.900351 −0.450176 0.892940i \(-0.648639\pi\)
−0.450176 + 0.892940i \(0.648639\pi\)
\(200\) 370824. 0.655530
\(201\) −355299. −0.620303
\(202\) −1.68355e6 −2.90301
\(203\) −64907.4 −0.110549
\(204\) 211507. 0.355836
\(205\) 394679. 0.655933
\(206\) −1.02722e6 −1.68653
\(207\) 231092. 0.374852
\(208\) −260236. −0.417069
\(209\) −173105. −0.274121
\(210\) −50781.1 −0.0794610
\(211\) −897392. −1.38764 −0.693819 0.720149i \(-0.744073\pi\)
−0.693819 + 0.720149i \(0.744073\pi\)
\(212\) 1.41904e6 2.16848
\(213\) −29547.1 −0.0446237
\(214\) 613871. 0.916311
\(215\) 1.46898e6 2.16731
\(216\) −159059. −0.231966
\(217\) 89928.1 0.129642
\(218\) 1.36761e6 1.94904
\(219\) −265830. −0.374536
\(220\) 465157. 0.647952
\(221\) 412161. 0.567657
\(222\) −14073.1 −0.0191649
\(223\) 117328. 0.157994 0.0789971 0.996875i \(-0.474828\pi\)
0.0789971 + 0.996875i \(0.474828\pi\)
\(224\) −38908.0 −0.0518106
\(225\) 137664. 0.181286
\(226\) 2.44743e6 3.18742
\(227\) 75051.3 0.0966704 0.0483352 0.998831i \(-0.484608\pi\)
0.0483352 + 0.998831i \(0.484608\pi\)
\(228\) 712609. 0.907850
\(229\) 634950. 0.800112 0.400056 0.916491i \(-0.368991\pi\)
0.400056 + 0.916491i \(0.368991\pi\)
\(230\) 1.85204e6 2.30851
\(231\) −9465.38 −0.0116710
\(232\) −1.62936e6 −1.98745
\(233\) −375281. −0.452863 −0.226431 0.974027i \(-0.572706\pi\)
−0.226431 + 0.974027i \(0.572706\pi\)
\(234\) −734812. −0.877276
\(235\) −679664. −0.802832
\(236\) −1.29573e6 −1.51438
\(237\) 92527.0 0.107003
\(238\) −34492.8 −0.0394717
\(239\) −859529. −0.973343 −0.486671 0.873585i \(-0.661789\pi\)
−0.486671 + 0.873585i \(0.661789\pi\)
\(240\) −167598. −0.187819
\(241\) −552284. −0.612520 −0.306260 0.951948i \(-0.599078\pi\)
−0.306260 + 0.951948i \(0.599078\pi\)
\(242\) 136833. 0.150194
\(243\) −59049.0 −0.0641500
\(244\) 1.02898e6 1.10645
\(245\) −1.16215e6 −1.23694
\(246\) −477946. −0.503548
\(247\) 1.38865e6 1.44827
\(248\) 2.25745e6 2.33071
\(249\) 345243. 0.352880
\(250\) −925337. −0.936375
\(251\) 198740. 0.199114 0.0995569 0.995032i \(-0.468257\pi\)
0.0995569 + 0.995032i \(0.468257\pi\)
\(252\) 38965.5 0.0386526
\(253\) 345212. 0.339066
\(254\) −209971. −0.204209
\(255\) 265441. 0.255633
\(256\) −1.45152e6 −1.38428
\(257\) −76585.7 −0.0723294 −0.0361647 0.999346i \(-0.511514\pi\)
−0.0361647 + 0.999346i \(0.511514\pi\)
\(258\) −1.77890e6 −1.66381
\(259\) 1454.24 0.00134705
\(260\) −3.73150e6 −3.42334
\(261\) −604880. −0.549627
\(262\) 1.11773e6 1.00596
\(263\) 2.10812e6 1.87935 0.939673 0.342073i \(-0.111129\pi\)
0.939673 + 0.342073i \(0.111129\pi\)
\(264\) −237607. −0.209822
\(265\) 1.78089e6 1.55784
\(266\) −116213. −0.100705
\(267\) 20879.9 0.0179246
\(268\) 2.18493e6 1.85823
\(269\) −1.97320e6 −1.66261 −0.831306 0.555816i \(-0.812406\pi\)
−0.831306 + 0.555816i \(0.812406\pi\)
\(270\) −473236. −0.395065
\(271\) −926426. −0.766280 −0.383140 0.923690i \(-0.625157\pi\)
−0.383140 + 0.923690i \(0.625157\pi\)
\(272\) −113840. −0.0932978
\(273\) 75931.5 0.0616617
\(274\) 1.64902e6 1.32693
\(275\) 205646. 0.163979
\(276\) −1.42111e6 −1.12294
\(277\) −1.60640e6 −1.25792 −0.628962 0.777436i \(-0.716520\pi\)
−0.628962 + 0.777436i \(0.716520\pi\)
\(278\) 2.69365e6 2.09040
\(279\) 838051. 0.644555
\(280\) 131726. 0.100410
\(281\) −814868. −0.615632 −0.307816 0.951446i \(-0.599598\pi\)
−0.307816 + 0.951446i \(0.599598\pi\)
\(282\) 823056. 0.616321
\(283\) −61465.6 −0.0456211 −0.0228106 0.999740i \(-0.507261\pi\)
−0.0228106 + 0.999740i \(0.507261\pi\)
\(284\) 181701. 0.133679
\(285\) 894323. 0.652202
\(286\) −1.09768e6 −0.793526
\(287\) 49388.4 0.0353933
\(288\) −362588. −0.257592
\(289\) −1.23956e6 −0.873016
\(290\) −4.84769e6 −3.38485
\(291\) 736814. 0.510065
\(292\) 1.63473e6 1.12199
\(293\) −298524. −0.203147 −0.101573 0.994828i \(-0.532388\pi\)
−0.101573 + 0.994828i \(0.532388\pi\)
\(294\) 1.40733e6 0.949575
\(295\) −1.62614e6 −1.08794
\(296\) 36505.4 0.0242174
\(297\) −88209.0 −0.0580259
\(298\) −1.24894e6 −0.814707
\(299\) −2.76930e6 −1.79140
\(300\) −846572. −0.543076
\(301\) 183822. 0.116945
\(302\) 3.79558e6 2.39475
\(303\) 1.62124e6 1.01447
\(304\) −383548. −0.238032
\(305\) 1.29137e6 0.794880
\(306\) −321442. −0.196245
\(307\) −1.35385e6 −0.819834 −0.409917 0.912123i \(-0.634442\pi\)
−0.409917 + 0.912123i \(0.634442\pi\)
\(308\) 58207.8 0.0349626
\(309\) 989201. 0.589370
\(310\) 6.71638e6 3.96946
\(311\) −2.68320e6 −1.57309 −0.786544 0.617535i \(-0.788132\pi\)
−0.786544 + 0.617535i \(0.788132\pi\)
\(312\) 1.90609e6 1.10856
\(313\) 2.85968e6 1.64989 0.824947 0.565210i \(-0.191205\pi\)
0.824947 + 0.565210i \(0.191205\pi\)
\(314\) −1.48362e6 −0.849180
\(315\) 48901.7 0.0277682
\(316\) −568999. −0.320548
\(317\) −111110. −0.0621020 −0.0310510 0.999518i \(-0.509885\pi\)
−0.0310510 + 0.999518i \(0.509885\pi\)
\(318\) −2.15662e6 −1.19593
\(319\) −903587. −0.497157
\(320\) −3.50179e6 −1.91168
\(321\) −591151. −0.320211
\(322\) 231757. 0.124564
\(323\) 607463. 0.323977
\(324\) 363124. 0.192173
\(325\) −1.64970e6 −0.866357
\(326\) 1.63048e6 0.849712
\(327\) −1.31699e6 −0.681105
\(328\) 1.23979e6 0.636301
\(329\) −85050.3 −0.0433198
\(330\) −706932. −0.357349
\(331\) −1.36346e6 −0.684028 −0.342014 0.939695i \(-0.611109\pi\)
−0.342014 + 0.939695i \(0.611109\pi\)
\(332\) −2.12309e6 −1.05712
\(333\) 13552.2 0.00669729
\(334\) 989534. 0.485361
\(335\) 2.74208e6 1.33496
\(336\) −20972.5 −0.0101345
\(337\) −621625. −0.298163 −0.149082 0.988825i \(-0.547632\pi\)
−0.149082 + 0.988825i \(0.547632\pi\)
\(338\) 5.33557e6 2.54032
\(339\) −2.35685e6 −1.11387
\(340\) −1.63234e6 −0.765797
\(341\) 1.25190e6 0.583022
\(342\) −1.08300e6 −0.500685
\(343\) −291510. −0.133788
\(344\) 4.61445e6 2.10244
\(345\) −1.78350e6 −0.806723
\(346\) −4.04172e6 −1.81500
\(347\) 125580. 0.0559883 0.0279942 0.999608i \(-0.491088\pi\)
0.0279942 + 0.999608i \(0.491088\pi\)
\(348\) 3.71974e6 1.64651
\(349\) 3.24565e6 1.42639 0.713195 0.700966i \(-0.247247\pi\)
0.713195 + 0.700966i \(0.247247\pi\)
\(350\) 138060. 0.0602417
\(351\) 707615. 0.306570
\(352\) −541644. −0.233001
\(353\) 1.67773e6 0.716614 0.358307 0.933604i \(-0.383354\pi\)
0.358307 + 0.933604i \(0.383354\pi\)
\(354\) 1.96922e6 0.835191
\(355\) 228035. 0.0960351
\(356\) −128402. −0.0536966
\(357\) 33216.1 0.0137936
\(358\) −5.83603e6 −2.40663
\(359\) −453973. −0.185906 −0.0929531 0.995670i \(-0.529631\pi\)
−0.0929531 + 0.995670i \(0.529631\pi\)
\(360\) 1.22757e6 0.499217
\(361\) −429436. −0.173432
\(362\) 99088.7 0.0397423
\(363\) −131769. −0.0524864
\(364\) −466944. −0.184719
\(365\) 2.05159e6 0.806043
\(366\) −1.56382e6 −0.610216
\(367\) 4.87840e6 1.89066 0.945328 0.326121i \(-0.105742\pi\)
0.945328 + 0.326121i \(0.105742\pi\)
\(368\) 764888. 0.294427
\(369\) 460257. 0.175968
\(370\) 108611. 0.0412449
\(371\) 222853. 0.0840590
\(372\) −5.15363e6 −1.93088
\(373\) −605344. −0.225284 −0.112642 0.993636i \(-0.535931\pi\)
−0.112642 + 0.993636i \(0.535931\pi\)
\(374\) −480179. −0.177511
\(375\) 891089. 0.327222
\(376\) −2.13500e6 −0.778804
\(377\) 7.24860e6 2.62664
\(378\) −59218.7 −0.0213172
\(379\) −3.90340e6 −1.39587 −0.697936 0.716160i \(-0.745898\pi\)
−0.697936 + 0.716160i \(0.745898\pi\)
\(380\) −5.49968e6 −1.95379
\(381\) 202200. 0.0713622
\(382\) 4.13047e6 1.44824
\(383\) 2.80018e6 0.975413 0.487707 0.873008i \(-0.337834\pi\)
0.487707 + 0.873008i \(0.337834\pi\)
\(384\) 2.95138e6 1.02140
\(385\) 73050.7 0.0251173
\(386\) −4.15289e6 −1.41867
\(387\) 1.71306e6 0.581428
\(388\) −4.53107e6 −1.52799
\(389\) 3.71874e6 1.24601 0.623006 0.782217i \(-0.285911\pi\)
0.623006 + 0.782217i \(0.285911\pi\)
\(390\) 5.67103e6 1.88799
\(391\) −1.21143e6 −0.400733
\(392\) −3.65061e6 −1.19991
\(393\) −1.07636e6 −0.351540
\(394\) −3.89328e6 −1.26350
\(395\) −714093. −0.230283
\(396\) 542445. 0.173827
\(397\) 3.85343e6 1.22708 0.613538 0.789665i \(-0.289746\pi\)
0.613538 + 0.789665i \(0.289746\pi\)
\(398\) −4.70074e6 −1.48750
\(399\) 111912. 0.0351920
\(400\) 455651. 0.142391
\(401\) 1.08196e6 0.336009 0.168004 0.985786i \(-0.446268\pi\)
0.168004 + 0.985786i \(0.446268\pi\)
\(402\) −3.32059e6 −1.02483
\(403\) −1.00428e7 −3.08030
\(404\) −9.96989e6 −3.03905
\(405\) 455721. 0.138058
\(406\) −606619. −0.182642
\(407\) 20244.6 0.00605793
\(408\) 833818. 0.247982
\(409\) 1.98167e6 0.585765 0.292883 0.956148i \(-0.405385\pi\)
0.292883 + 0.956148i \(0.405385\pi\)
\(410\) 3.68863e6 1.08369
\(411\) −1.58798e6 −0.463705
\(412\) −6.08313e6 −1.76557
\(413\) −203489. −0.0587036
\(414\) 2.15977e6 0.619307
\(415\) −2.66447e6 −0.759436
\(416\) 4.34508e6 1.23102
\(417\) −2.59396e6 −0.730505
\(418\) −1.61782e6 −0.452886
\(419\) −997473. −0.277566 −0.138783 0.990323i \(-0.544319\pi\)
−0.138783 + 0.990323i \(0.544319\pi\)
\(420\) −300723. −0.0831847
\(421\) 5.36616e6 1.47556 0.737782 0.675039i \(-0.235873\pi\)
0.737782 + 0.675039i \(0.235873\pi\)
\(422\) −8.38694e6 −2.29257
\(423\) −792594. −0.215377
\(424\) 5.59424e6 1.51122
\(425\) −721660. −0.193803
\(426\) −276144. −0.0737245
\(427\) 161597. 0.0428907
\(428\) 3.63531e6 0.959250
\(429\) 1.05705e6 0.277303
\(430\) 1.37290e7 3.58069
\(431\) −3.37897e6 −0.876175 −0.438088 0.898932i \(-0.644344\pi\)
−0.438088 + 0.898932i \(0.644344\pi\)
\(432\) −195445. −0.0503866
\(433\) 3.06239e6 0.784948 0.392474 0.919763i \(-0.371619\pi\)
0.392474 + 0.919763i \(0.371619\pi\)
\(434\) 840459. 0.214187
\(435\) 4.66827e6 1.18286
\(436\) 8.09891e6 2.04038
\(437\) −4.08154e6 −1.02240
\(438\) −2.48442e6 −0.618786
\(439\) 2.56020e6 0.634035 0.317018 0.948420i \(-0.397319\pi\)
0.317018 + 0.948420i \(0.397319\pi\)
\(440\) 1.83377e6 0.451559
\(441\) −1.35525e6 −0.331835
\(442\) 3.85201e6 0.937847
\(443\) −1.10849e6 −0.268363 −0.134181 0.990957i \(-0.542840\pi\)
−0.134181 + 0.990957i \(0.542840\pi\)
\(444\) −83339.8 −0.0200630
\(445\) −161144. −0.0385758
\(446\) 1.09654e6 0.261028
\(447\) 1.20272e6 0.284705
\(448\) −438199. −0.103152
\(449\) −5.98494e6 −1.40102 −0.700509 0.713643i \(-0.747044\pi\)
−0.700509 + 0.713643i \(0.747044\pi\)
\(450\) 1.28660e6 0.299510
\(451\) 687544. 0.159169
\(452\) 1.44936e7 3.33679
\(453\) −3.65510e6 −0.836862
\(454\) 701422. 0.159713
\(455\) −586015. −0.132703
\(456\) 2.80930e6 0.632682
\(457\) 3.91747e6 0.877436 0.438718 0.898625i \(-0.355433\pi\)
0.438718 + 0.898625i \(0.355433\pi\)
\(458\) 5.93418e6 1.32190
\(459\) 309545. 0.0685792
\(460\) 1.09677e7 2.41669
\(461\) −2.16827e6 −0.475184 −0.237592 0.971365i \(-0.576358\pi\)
−0.237592 + 0.971365i \(0.576358\pi\)
\(462\) −88462.5 −0.0192821
\(463\) −5.01423e6 −1.08706 −0.543528 0.839391i \(-0.682912\pi\)
−0.543528 + 0.839391i \(0.682912\pi\)
\(464\) −2.00208e6 −0.431704
\(465\) −6.46780e6 −1.38715
\(466\) −3.50734e6 −0.748191
\(467\) −2.57098e6 −0.545514 −0.272757 0.962083i \(-0.587936\pi\)
−0.272757 + 0.962083i \(0.587936\pi\)
\(468\) −4.35151e6 −0.918387
\(469\) 343132. 0.0720326
\(470\) −6.35208e6 −1.32639
\(471\) 1.42871e6 0.296751
\(472\) −5.10813e6 −1.05538
\(473\) 2.55902e6 0.525921
\(474\) 864748. 0.176784
\(475\) −2.43141e6 −0.494453
\(476\) −204264. −0.0413214
\(477\) 2.07680e6 0.417925
\(478\) −8.03307e6 −1.60810
\(479\) 5.16179e6 1.02793 0.513963 0.857813i \(-0.328177\pi\)
0.513963 + 0.857813i \(0.328177\pi\)
\(480\) 2.79834e6 0.554366
\(481\) −162403. −0.0320060
\(482\) −5.16160e6 −1.01197
\(483\) −223179. −0.0435297
\(484\) 810319. 0.157233
\(485\) −5.68649e6 −1.09772
\(486\) −551866. −0.105985
\(487\) 5.85746e6 1.11915 0.559573 0.828781i \(-0.310965\pi\)
0.559573 + 0.828781i \(0.310965\pi\)
\(488\) 4.05653e6 0.771090
\(489\) −1.57014e6 −0.296937
\(490\) −1.08613e7 −2.04359
\(491\) 956557. 0.179064 0.0895318 0.995984i \(-0.471463\pi\)
0.0895318 + 0.995984i \(0.471463\pi\)
\(492\) −2.83037e6 −0.527145
\(493\) 3.17089e6 0.587576
\(494\) 1.29782e7 2.39275
\(495\) 680768. 0.124878
\(496\) 2.77385e6 0.506265
\(497\) 28535.3 0.00518192
\(498\) 3.22661e6 0.583006
\(499\) −1.47579e6 −0.265322 −0.132661 0.991161i \(-0.542352\pi\)
−0.132661 + 0.991161i \(0.542352\pi\)
\(500\) −5.47979e6 −0.980255
\(501\) −952911. −0.169612
\(502\) 1.85741e6 0.328963
\(503\) 991048. 0.174652 0.0873262 0.996180i \(-0.472168\pi\)
0.0873262 + 0.996180i \(0.472168\pi\)
\(504\) 153613. 0.0269371
\(505\) −1.25122e7 −2.18326
\(506\) 3.22632e6 0.560185
\(507\) −5.13809e6 −0.887732
\(508\) −1.24344e6 −0.213779
\(509\) −8.24580e6 −1.41071 −0.705356 0.708854i \(-0.749213\pi\)
−0.705356 + 0.708854i \(0.749213\pi\)
\(510\) 2.48079e6 0.422342
\(511\) 256727. 0.0434930
\(512\) −3.07201e6 −0.517901
\(513\) 1.04292e6 0.174967
\(514\) −715762. −0.119498
\(515\) −7.63432e6 −1.26839
\(516\) −1.05346e7 −1.74177
\(517\) −1.18400e6 −0.194816
\(518\) 13591.1 0.00222552
\(519\) 3.89213e6 0.634262
\(520\) −1.47106e7 −2.38573
\(521\) −1.18607e6 −0.191432 −0.0957161 0.995409i \(-0.530514\pi\)
−0.0957161 + 0.995409i \(0.530514\pi\)
\(522\) −5.65315e6 −0.908060
\(523\) 2.80341e6 0.448159 0.224079 0.974571i \(-0.428062\pi\)
0.224079 + 0.974571i \(0.428062\pi\)
\(524\) 6.61911e6 1.05310
\(525\) −132950. −0.0210518
\(526\) 1.97023e7 3.10494
\(527\) −4.39321e6 −0.689058
\(528\) −291961. −0.0455764
\(529\) 1.70323e6 0.264627
\(530\) 1.66440e7 2.57377
\(531\) −1.89633e6 −0.291863
\(532\) −688206. −0.105424
\(533\) −5.51550e6 −0.840944
\(534\) 195142. 0.0296140
\(535\) 4.56231e6 0.689129
\(536\) 8.61357e6 1.29500
\(537\) 5.62003e6 0.841014
\(538\) −1.84413e7 −2.74686
\(539\) −2.02451e6 −0.300156
\(540\) −2.80247e6 −0.413578
\(541\) 5.30698e6 0.779569 0.389785 0.920906i \(-0.372549\pi\)
0.389785 + 0.920906i \(0.372549\pi\)
\(542\) −8.65829e6 −1.26600
\(543\) −95421.3 −0.0138882
\(544\) 1.90075e6 0.275377
\(545\) 1.01641e7 1.46581
\(546\) 709649. 0.101874
\(547\) 7.62305e6 1.08933 0.544666 0.838653i \(-0.316656\pi\)
0.544666 + 0.838653i \(0.316656\pi\)
\(548\) 9.76538e6 1.38911
\(549\) 1.50594e6 0.213244
\(550\) 1.92195e6 0.270917
\(551\) 1.06834e7 1.49909
\(552\) −5.60241e6 −0.782578
\(553\) −89358.5 −0.0124258
\(554\) −1.50133e7 −2.07826
\(555\) −104591. −0.0144133
\(556\) 1.59517e7 2.18836
\(557\) 7.81748e6 1.06765 0.533825 0.845595i \(-0.320754\pi\)
0.533825 + 0.845595i \(0.320754\pi\)
\(558\) 7.83234e6 1.06489
\(559\) −2.05285e7 −2.77861
\(560\) 161859. 0.0218105
\(561\) 462407. 0.0620322
\(562\) −7.61567e6 −1.01711
\(563\) 4.46693e6 0.593933 0.296967 0.954888i \(-0.404025\pi\)
0.296967 + 0.954888i \(0.404025\pi\)
\(564\) 4.87409e6 0.645203
\(565\) 1.81894e7 2.39716
\(566\) −574452. −0.0753724
\(567\) 57026.9 0.00744942
\(568\) 716315. 0.0931608
\(569\) −1.03786e7 −1.34387 −0.671933 0.740612i \(-0.734536\pi\)
−0.671933 + 0.740612i \(0.734536\pi\)
\(570\) 8.35826e6 1.07753
\(571\) 5.85319e6 0.751280 0.375640 0.926766i \(-0.377423\pi\)
0.375640 + 0.926766i \(0.377423\pi\)
\(572\) −6.50041e6 −0.830712
\(573\) −3.97760e6 −0.506097
\(574\) 461579. 0.0584745
\(575\) 4.84882e6 0.611599
\(576\) −4.08363e6 −0.512850
\(577\) 1.01944e7 1.27475 0.637373 0.770555i \(-0.280021\pi\)
0.637373 + 0.770555i \(0.280021\pi\)
\(578\) −1.15848e7 −1.44234
\(579\) 3.99919e6 0.495765
\(580\) −2.87077e7 −3.54347
\(581\) −333421. −0.0409781
\(582\) 6.88620e6 0.842698
\(583\) 3.10238e6 0.378027
\(584\) 6.44457e6 0.781919
\(585\) −5.46114e6 −0.659772
\(586\) −2.78997e6 −0.335626
\(587\) −3.10970e6 −0.372497 −0.186249 0.982503i \(-0.559633\pi\)
−0.186249 + 0.982503i \(0.559633\pi\)
\(588\) 8.33416e6 0.994073
\(589\) −1.48016e7 −1.75801
\(590\) −1.51978e7 −1.79742
\(591\) 3.74918e6 0.441538
\(592\) 44856.1 0.00526038
\(593\) −1.59890e7 −1.86717 −0.933587 0.358352i \(-0.883339\pi\)
−0.933587 + 0.358352i \(0.883339\pi\)
\(594\) −824393. −0.0958668
\(595\) −256351. −0.0296854
\(596\) −7.39616e6 −0.852885
\(597\) 4.52676e6 0.519818
\(598\) −2.58816e7 −2.95964
\(599\) −4.21384e6 −0.479856 −0.239928 0.970791i \(-0.577124\pi\)
−0.239928 + 0.970791i \(0.577124\pi\)
\(600\) −3.33741e6 −0.378470
\(601\) 4.68871e6 0.529502 0.264751 0.964317i \(-0.414710\pi\)
0.264751 + 0.964317i \(0.414710\pi\)
\(602\) 1.71799e6 0.193209
\(603\) 3.19769e6 0.358132
\(604\) 2.24772e7 2.50697
\(605\) 1.01695e6 0.112956
\(606\) 1.51520e7 1.67605
\(607\) 4.29265e6 0.472883 0.236441 0.971646i \(-0.424019\pi\)
0.236441 + 0.971646i \(0.424019\pi\)
\(608\) 6.40401e6 0.702575
\(609\) 584167. 0.0638254
\(610\) 1.20690e7 1.31325
\(611\) 9.49807e6 1.02928
\(612\) −1.90356e6 −0.205442
\(613\) 1.67420e7 1.79951 0.899757 0.436392i \(-0.143744\pi\)
0.899757 + 0.436392i \(0.143744\pi\)
\(614\) −1.26530e7 −1.35448
\(615\) −3.55211e6 −0.378703
\(616\) 229471. 0.0243655
\(617\) −2.10611e6 −0.222724 −0.111362 0.993780i \(-0.535521\pi\)
−0.111362 + 0.993780i \(0.535521\pi\)
\(618\) 9.24497e6 0.973721
\(619\) 2.52452e6 0.264821 0.132410 0.991195i \(-0.457728\pi\)
0.132410 + 0.991195i \(0.457728\pi\)
\(620\) 3.97740e7 4.15547
\(621\) −2.07983e6 −0.216421
\(622\) −2.50770e7 −2.59896
\(623\) −20164.9 −0.00208150
\(624\) 2.34212e6 0.240795
\(625\) −1.21882e7 −1.24808
\(626\) 2.67263e7 2.72585
\(627\) 1.55794e6 0.158264
\(628\) −8.78594e6 −0.888974
\(629\) −71043.0 −0.00715970
\(630\) 457030. 0.0458769
\(631\) 1.11540e6 0.111521 0.0557607 0.998444i \(-0.482242\pi\)
0.0557607 + 0.998444i \(0.482242\pi\)
\(632\) −2.24315e6 −0.223391
\(633\) 8.07653e6 0.801153
\(634\) −1.03843e6 −0.102601
\(635\) −1.56051e6 −0.153579
\(636\) −1.27714e7 −1.25197
\(637\) 1.62406e7 1.58582
\(638\) −8.44483e6 −0.821371
\(639\) 265924. 0.0257635
\(640\) −2.27777e7 −2.19817
\(641\) 9.80630e6 0.942671 0.471336 0.881954i \(-0.343772\pi\)
0.471336 + 0.881954i \(0.343772\pi\)
\(642\) −5.52484e6 −0.529032
\(643\) −5.97068e6 −0.569504 −0.284752 0.958601i \(-0.591911\pi\)
−0.284752 + 0.958601i \(0.591911\pi\)
\(644\) 1.37245e6 0.130401
\(645\) −1.32209e7 −1.25130
\(646\) 5.67729e6 0.535254
\(647\) 8.55002e6 0.802984 0.401492 0.915863i \(-0.368492\pi\)
0.401492 + 0.915863i \(0.368492\pi\)
\(648\) 1.43154e6 0.133926
\(649\) −2.83280e6 −0.264000
\(650\) −1.54180e7 −1.43134
\(651\) −809353. −0.0748489
\(652\) 9.65562e6 0.889531
\(653\) −4.56279e6 −0.418743 −0.209371 0.977836i \(-0.567142\pi\)
−0.209371 + 0.977836i \(0.567142\pi\)
\(654\) −1.23085e7 −1.12528
\(655\) 8.30697e6 0.756553
\(656\) 1.52339e6 0.138214
\(657\) 2.39247e6 0.216239
\(658\) −794872. −0.0715702
\(659\) −1.97288e6 −0.176965 −0.0884825 0.996078i \(-0.528202\pi\)
−0.0884825 + 0.996078i \(0.528202\pi\)
\(660\) −4.18641e6 −0.374095
\(661\) −1.78964e7 −1.59317 −0.796585 0.604527i \(-0.793362\pi\)
−0.796585 + 0.604527i \(0.793362\pi\)
\(662\) −1.27428e7 −1.13011
\(663\) −3.70945e6 −0.327737
\(664\) −8.36979e6 −0.736706
\(665\) −863698. −0.0757370
\(666\) 126657. 0.0110648
\(667\) −2.13052e7 −1.85426
\(668\) 5.85997e6 0.508106
\(669\) −1.05596e6 −0.0912180
\(670\) 2.56272e7 2.20554
\(671\) 2.24961e6 0.192886
\(672\) 350172. 0.0299129
\(673\) −7.40111e6 −0.629882 −0.314941 0.949111i \(-0.601985\pi\)
−0.314941 + 0.949111i \(0.601985\pi\)
\(674\) −5.80965e6 −0.492607
\(675\) −1.23898e6 −0.104666
\(676\) 3.15969e7 2.65937
\(677\) −6.94001e6 −0.581954 −0.290977 0.956730i \(-0.593980\pi\)
−0.290977 + 0.956730i \(0.593980\pi\)
\(678\) −2.20269e7 −1.84026
\(679\) −711583. −0.0592313
\(680\) −6.43513e6 −0.533685
\(681\) −675461. −0.0558127
\(682\) 1.17002e7 0.963233
\(683\) 5.97233e6 0.489882 0.244941 0.969538i \(-0.421231\pi\)
0.244941 + 0.969538i \(0.421231\pi\)
\(684\) −6.41348e6 −0.524147
\(685\) 1.22555e7 0.997943
\(686\) −2.72442e6 −0.221037
\(687\) −5.71455e6 −0.461945
\(688\) 5.67003e6 0.456682
\(689\) −2.48874e7 −1.99724
\(690\) −1.66684e7 −1.33282
\(691\) −2.05711e7 −1.63893 −0.819467 0.573126i \(-0.805731\pi\)
−0.819467 + 0.573126i \(0.805731\pi\)
\(692\) −2.39348e7 −1.90005
\(693\) 85188.4 0.00673825
\(694\) 1.17366e6 0.0925004
\(695\) 2.00193e7 1.57213
\(696\) 1.46642e7 1.14746
\(697\) −2.41275e6 −0.188118
\(698\) 3.03335e7 2.35659
\(699\) 3.37753e6 0.261460
\(700\) 817582. 0.0630647
\(701\) 9.61010e6 0.738640 0.369320 0.929302i \(-0.379591\pi\)
0.369320 + 0.929302i \(0.379591\pi\)
\(702\) 6.61330e6 0.506496
\(703\) −239358. −0.0182667
\(704\) −6.10024e6 −0.463890
\(705\) 6.11698e6 0.463515
\(706\) 1.56799e7 1.18394
\(707\) −1.56572e6 −0.117806
\(708\) 1.16616e7 0.874329
\(709\) 2.13666e6 0.159632 0.0798161 0.996810i \(-0.474567\pi\)
0.0798161 + 0.996810i \(0.474567\pi\)
\(710\) 2.13119e6 0.158663
\(711\) −832743. −0.0617784
\(712\) −506195. −0.0374212
\(713\) 2.95180e7 2.17451
\(714\) 310435. 0.0227890
\(715\) −8.15800e6 −0.596786
\(716\) −3.45607e7 −2.51941
\(717\) 7.73576e6 0.561960
\(718\) −4.24279e6 −0.307143
\(719\) 1.57303e7 1.13479 0.567395 0.823446i \(-0.307951\pi\)
0.567395 + 0.823446i \(0.307951\pi\)
\(720\) 1.50838e6 0.108437
\(721\) −955327. −0.0684406
\(722\) −4.01346e6 −0.286534
\(723\) 4.97056e6 0.353639
\(724\) 586797. 0.0416046
\(725\) −1.26917e7 −0.896758
\(726\) −1.23150e6 −0.0867147
\(727\) 1.39695e7 0.980271 0.490136 0.871646i \(-0.336947\pi\)
0.490136 + 0.871646i \(0.336947\pi\)
\(728\) −1.84082e6 −0.128731
\(729\) 531441. 0.0370370
\(730\) 1.91740e7 1.33169
\(731\) −8.98018e6 −0.621572
\(732\) −9.26084e6 −0.638812
\(733\) 2.03921e6 0.140185 0.0700927 0.997540i \(-0.477671\pi\)
0.0700927 + 0.997540i \(0.477671\pi\)
\(734\) 4.55931e7 3.12362
\(735\) 1.04594e7 0.714145
\(736\) −1.27711e7 −0.869030
\(737\) 4.77680e6 0.323943
\(738\) 4.30152e6 0.290724
\(739\) −1.13456e7 −0.764216 −0.382108 0.924118i \(-0.624802\pi\)
−0.382108 + 0.924118i \(0.624802\pi\)
\(740\) 643189. 0.0431777
\(741\) −1.24979e7 −0.836161
\(742\) 2.08277e6 0.138877
\(743\) −1.17404e7 −0.780206 −0.390103 0.920771i \(-0.627561\pi\)
−0.390103 + 0.920771i \(0.627561\pi\)
\(744\) −2.03170e7 −1.34564
\(745\) −9.28217e6 −0.612715
\(746\) −5.65749e6 −0.372200
\(747\) −3.10719e6 −0.203735
\(748\) −2.84359e6 −0.185829
\(749\) 570908. 0.0371845
\(750\) 8.32803e6 0.540616
\(751\) 2.23069e7 1.44324 0.721620 0.692289i \(-0.243398\pi\)
0.721620 + 0.692289i \(0.243398\pi\)
\(752\) −2.62339e6 −0.169168
\(753\) −1.78866e6 −0.114958
\(754\) 6.77447e7 4.33957
\(755\) 2.82089e7 1.80102
\(756\) −350690. −0.0223161
\(757\) −2.86550e7 −1.81744 −0.908721 0.417403i \(-0.862940\pi\)
−0.908721 + 0.417403i \(0.862940\pi\)
\(758\) −3.64808e7 −2.30617
\(759\) −3.10691e6 −0.195760
\(760\) −2.16812e7 −1.36160
\(761\) −1.61728e7 −1.01233 −0.506167 0.862435i \(-0.668938\pi\)
−0.506167 + 0.862435i \(0.668938\pi\)
\(762\) 1.88974e6 0.117900
\(763\) 1.27189e6 0.0790933
\(764\) 2.44604e7 1.51611
\(765\) −2.38897e6 −0.147590
\(766\) 2.61702e7 1.61152
\(767\) 2.27248e7 1.39480
\(768\) 1.30637e7 0.799214
\(769\) 1.31452e7 0.801588 0.400794 0.916168i \(-0.368734\pi\)
0.400794 + 0.916168i \(0.368734\pi\)
\(770\) 682724. 0.0414972
\(771\) 689271. 0.0417594
\(772\) −2.45932e7 −1.48515
\(773\) −2.45719e7 −1.47907 −0.739537 0.673116i \(-0.764955\pi\)
−0.739537 + 0.673116i \(0.764955\pi\)
\(774\) 1.60101e7 0.960599
\(775\) 1.75841e7 1.05164
\(776\) −1.78627e7 −1.06486
\(777\) −13088.1 −0.000777722 0
\(778\) 3.47550e7 2.05858
\(779\) −8.12902e6 −0.479949
\(780\) 3.35835e7 1.97647
\(781\) 397244. 0.0233040
\(782\) −1.13219e7 −0.662067
\(783\) 5.44392e6 0.317327
\(784\) −4.48570e6 −0.260640
\(785\) −1.10263e7 −0.638641
\(786\) −1.00595e7 −0.580793
\(787\) −2.62366e7 −1.50998 −0.754989 0.655737i \(-0.772358\pi\)
−0.754989 + 0.655737i \(0.772358\pi\)
\(788\) −2.30558e7 −1.32271
\(789\) −1.89731e7 −1.08504
\(790\) −6.67384e6 −0.380459
\(791\) 2.27614e6 0.129348
\(792\) 2.13847e6 0.121141
\(793\) −1.80465e7 −1.01908
\(794\) 3.60138e7 2.02730
\(795\) −1.60280e7 −0.899420
\(796\) −2.78375e7 −1.55721
\(797\) −333892. −0.0186192 −0.00930959 0.999957i \(-0.502963\pi\)
−0.00930959 + 0.999957i \(0.502963\pi\)
\(798\) 1.04592e6 0.0581420
\(799\) 4.15492e6 0.230248
\(800\) −7.60789e6 −0.420280
\(801\) −187919. −0.0103488
\(802\) 1.01119e7 0.555133
\(803\) 3.57394e6 0.195595
\(804\) −1.96643e7 −1.07285
\(805\) 1.72242e6 0.0936806
\(806\) −9.38591e7 −5.08907
\(807\) 1.77588e7 0.959909
\(808\) −3.93040e7 −2.11792
\(809\) 2.96083e6 0.159053 0.0795265 0.996833i \(-0.474659\pi\)
0.0795265 + 0.996833i \(0.474659\pi\)
\(810\) 4.25912e6 0.228091
\(811\) −2.24739e7 −1.19985 −0.599925 0.800056i \(-0.704803\pi\)
−0.599925 + 0.800056i \(0.704803\pi\)
\(812\) −3.59236e6 −0.191201
\(813\) 8.33783e6 0.442412
\(814\) 189204. 0.0100085
\(815\) 1.21178e7 0.639042
\(816\) 1.02456e6 0.0538655
\(817\) −3.02560e7 −1.58583
\(818\) 1.85205e7 0.967765
\(819\) −683384. −0.0356004
\(820\) 2.18439e7 1.13447
\(821\) 2.07336e7 1.07353 0.536767 0.843730i \(-0.319645\pi\)
0.536767 + 0.843730i \(0.319645\pi\)
\(822\) −1.48411e7 −0.766104
\(823\) 2.99008e7 1.53880 0.769401 0.638766i \(-0.220555\pi\)
0.769401 + 0.638766i \(0.220555\pi\)
\(824\) −2.39814e7 −1.23043
\(825\) −1.85082e6 −0.0946736
\(826\) −1.90178e6 −0.0969865
\(827\) −3.09628e7 −1.57426 −0.787130 0.616787i \(-0.788434\pi\)
−0.787130 + 0.616787i \(0.788434\pi\)
\(828\) 1.27900e7 0.648329
\(829\) −2.79498e7 −1.41251 −0.706255 0.707957i \(-0.749617\pi\)
−0.706255 + 0.707957i \(0.749617\pi\)
\(830\) −2.49019e7 −1.25469
\(831\) 1.44576e7 0.726263
\(832\) 4.89363e7 2.45088
\(833\) 7.10445e6 0.354746
\(834\) −2.42429e7 −1.20689
\(835\) 7.35425e6 0.365025
\(836\) −9.58063e6 −0.474109
\(837\) −7.54246e6 −0.372134
\(838\) −9.32228e6 −0.458577
\(839\) −5.17408e6 −0.253763 −0.126882 0.991918i \(-0.540497\pi\)
−0.126882 + 0.991918i \(0.540497\pi\)
\(840\) −1.18553e6 −0.0579716
\(841\) 3.52548e7 1.71881
\(842\) 5.01516e7 2.43784
\(843\) 7.33381e6 0.355435
\(844\) −4.96670e7 −2.40000
\(845\) 3.96541e7 1.91050
\(846\) −7.40751e6 −0.355833
\(847\) 127257. 0.00609498
\(848\) 6.87394e6 0.328259
\(849\) 553190. 0.0263394
\(850\) −6.74456e6 −0.320189
\(851\) 477337. 0.0225944
\(852\) −1.63531e6 −0.0771793
\(853\) 4.50230e6 0.211866 0.105933 0.994373i \(-0.466217\pi\)
0.105933 + 0.994373i \(0.466217\pi\)
\(854\) 1.51027e6 0.0708613
\(855\) −8.04891e6 −0.376549
\(856\) 1.43314e7 0.668503
\(857\) 2.97267e7 1.38259 0.691296 0.722572i \(-0.257040\pi\)
0.691296 + 0.722572i \(0.257040\pi\)
\(858\) 9.87913e6 0.458143
\(859\) 2.65435e7 1.22737 0.613685 0.789551i \(-0.289686\pi\)
0.613685 + 0.789551i \(0.289686\pi\)
\(860\) 8.13022e7 3.74849
\(861\) −444496. −0.0204343
\(862\) −3.15795e7 −1.44756
\(863\) −1.61893e7 −0.739949 −0.369974 0.929042i \(-0.620634\pi\)
−0.369974 + 0.929042i \(0.620634\pi\)
\(864\) 3.26329e6 0.148721
\(865\) −3.00382e7 −1.36500
\(866\) 2.86208e7 1.29684
\(867\) 1.11560e7 0.504036
\(868\) 4.97715e6 0.224224
\(869\) −1.24397e6 −0.0558807
\(870\) 4.36292e7 1.95424
\(871\) −3.83196e7 −1.71150
\(872\) 3.19281e7 1.42194
\(873\) −6.63133e6 −0.294486
\(874\) −3.81457e7 −1.68914
\(875\) −860575. −0.0379987
\(876\) −1.47126e7 −0.647783
\(877\) −1.71684e6 −0.0753757 −0.0376879 0.999290i \(-0.511999\pi\)
−0.0376879 + 0.999290i \(0.511999\pi\)
\(878\) 2.39274e7 1.04751
\(879\) 2.68671e6 0.117287
\(880\) 2.25326e6 0.0980854
\(881\) 1.59472e6 0.0692221 0.0346110 0.999401i \(-0.488981\pi\)
0.0346110 + 0.999401i \(0.488981\pi\)
\(882\) −1.26660e7 −0.548237
\(883\) 3.27757e7 1.41465 0.707327 0.706887i \(-0.249901\pi\)
0.707327 + 0.706887i \(0.249901\pi\)
\(884\) 2.28114e7 0.981796
\(885\) 1.46353e7 0.628121
\(886\) −1.03598e7 −0.443372
\(887\) 2.30536e7 0.983853 0.491926 0.870637i \(-0.336293\pi\)
0.491926 + 0.870637i \(0.336293\pi\)
\(888\) −328548. −0.0139819
\(889\) −195276. −0.00828693
\(890\) −1.50604e6 −0.0637325
\(891\) 793881. 0.0335013
\(892\) 6.49365e6 0.273260
\(893\) 1.39987e7 0.587436
\(894\) 1.12405e7 0.470371
\(895\) −4.33736e7 −1.80995
\(896\) −2.85031e6 −0.118610
\(897\) 2.49237e7 1.03427
\(898\) −5.59347e7 −2.31468
\(899\) −7.72627e7 −3.18838
\(900\) 7.61914e6 0.313545
\(901\) −1.08869e7 −0.446780
\(902\) 6.42572e6 0.262970
\(903\) −1.65440e6 −0.0675183
\(904\) 5.71376e7 2.32542
\(905\) 736430. 0.0298889
\(906\) −3.41602e7 −1.38261
\(907\) −2.13981e7 −0.863689 −0.431845 0.901948i \(-0.642137\pi\)
−0.431845 + 0.901948i \(0.642137\pi\)
\(908\) 4.15378e6 0.167197
\(909\) −1.45912e7 −0.585707
\(910\) −5.47684e6 −0.219243
\(911\) −2.14014e7 −0.854369 −0.427184 0.904165i \(-0.640494\pi\)
−0.427184 + 0.904165i \(0.640494\pi\)
\(912\) 3.45194e6 0.137428
\(913\) −4.64160e6 −0.184285
\(914\) 3.66123e7 1.44964
\(915\) −1.16223e7 −0.458924
\(916\) 3.51419e7 1.38384
\(917\) 1.03950e6 0.0408226
\(918\) 2.89298e6 0.113302
\(919\) 5.88304e6 0.229781 0.114890 0.993378i \(-0.463348\pi\)
0.114890 + 0.993378i \(0.463348\pi\)
\(920\) 4.32376e7 1.68419
\(921\) 1.21847e7 0.473331
\(922\) −2.02645e7 −0.785069
\(923\) −3.18670e6 −0.123123
\(924\) −523870. −0.0201857
\(925\) 284355. 0.0109271
\(926\) −4.68625e7 −1.79597
\(927\) −8.90281e6 −0.340273
\(928\) 3.34282e7 1.27422
\(929\) 1.32356e7 0.503158 0.251579 0.967837i \(-0.419050\pi\)
0.251579 + 0.967837i \(0.419050\pi\)
\(930\) −6.04474e7 −2.29177
\(931\) 2.39363e7 0.905071
\(932\) −2.07703e7 −0.783253
\(933\) 2.41488e7 0.908222
\(934\) −2.40281e7 −0.901265
\(935\) −3.56871e6 −0.133500
\(936\) −1.71548e7 −0.640025
\(937\) 4.19890e7 1.56238 0.781189 0.624294i \(-0.214613\pi\)
0.781189 + 0.624294i \(0.214613\pi\)
\(938\) 3.20688e6 0.119008
\(939\) −2.57371e7 −0.952567
\(940\) −3.76166e7 −1.38855
\(941\) −4.99313e7 −1.83823 −0.919113 0.393993i \(-0.871093\pi\)
−0.919113 + 0.393993i \(0.871093\pi\)
\(942\) 1.33526e7 0.490274
\(943\) 1.62112e7 0.593659
\(944\) −6.27664e6 −0.229243
\(945\) −440115. −0.0160320
\(946\) 2.39163e7 0.868894
\(947\) −1.72916e7 −0.626557 −0.313278 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(948\) 5.12099e6 0.185069
\(949\) −2.86703e7 −1.03339
\(950\) −2.27238e7 −0.816904
\(951\) 999992. 0.0358546
\(952\) −805265. −0.0287969
\(953\) −1.17729e7 −0.419905 −0.209953 0.977712i \(-0.567331\pi\)
−0.209953 + 0.977712i \(0.567331\pi\)
\(954\) 1.94095e7 0.690469
\(955\) 3.06978e7 1.08918
\(956\) −4.75714e7 −1.68345
\(957\) 8.13228e6 0.287033
\(958\) 4.82416e7 1.69827
\(959\) 1.53360e6 0.0538477
\(960\) 3.15161e7 1.10371
\(961\) 7.84170e7 2.73906
\(962\) −1.51780e6 −0.0528783
\(963\) 5.32036e6 0.184874
\(964\) −3.05667e7 −1.05939
\(965\) −3.08644e7 −1.06694
\(966\) −2.08581e6 −0.0719170
\(967\) 4.31614e7 1.48433 0.742164 0.670218i \(-0.233800\pi\)
0.742164 + 0.670218i \(0.233800\pi\)
\(968\) 3.19450e6 0.109576
\(969\) −5.46717e6 −0.187048
\(970\) −5.31454e7 −1.81358
\(971\) 1.06735e7 0.363293 0.181647 0.983364i \(-0.441857\pi\)
0.181647 + 0.983364i \(0.441857\pi\)
\(972\) −3.26812e6 −0.110951
\(973\) 2.50513e6 0.0848298
\(974\) 5.47432e7 1.84898
\(975\) 1.48473e7 0.500192
\(976\) 4.98448e6 0.167492
\(977\) −3.55410e7 −1.19122 −0.595612 0.803272i \(-0.703091\pi\)
−0.595612 + 0.803272i \(0.703091\pi\)
\(978\) −1.46743e7 −0.490582
\(979\) −280719. −0.00936084
\(980\) −6.43203e7 −2.13935
\(981\) 1.18529e7 0.393236
\(982\) 8.93989e6 0.295838
\(983\) −1.30764e6 −0.0431623 −0.0215811 0.999767i \(-0.506870\pi\)
−0.0215811 + 0.999767i \(0.506870\pi\)
\(984\) −1.11581e7 −0.367368
\(985\) −2.89350e7 −0.950238
\(986\) 2.96348e7 0.970756
\(987\) 765453. 0.0250107
\(988\) 7.68561e7 2.50487
\(989\) 6.03377e7 1.96154
\(990\) 6.36239e6 0.206316
\(991\) −2.32452e7 −0.751882 −0.375941 0.926644i \(-0.622680\pi\)
−0.375941 + 0.926644i \(0.622680\pi\)
\(992\) −4.63142e7 −1.49429
\(993\) 1.22712e7 0.394924
\(994\) 266688. 0.00856125
\(995\) −3.49360e7 −1.11871
\(996\) 1.91078e7 0.610326
\(997\) 6.20462e6 0.197686 0.0988432 0.995103i \(-0.468486\pi\)
0.0988432 + 0.995103i \(0.468486\pi\)
\(998\) −1.37926e7 −0.438348
\(999\) −121970. −0.00386668
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 33.6.a.d.1.2 2
3.2 odd 2 99.6.a.e.1.1 2
4.3 odd 2 528.6.a.q.1.2 2
5.4 even 2 825.6.a.d.1.1 2
11.10 odd 2 363.6.a.g.1.1 2
33.32 even 2 1089.6.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.d.1.2 2 1.1 even 1 trivial
99.6.a.e.1.1 2 3.2 odd 2
363.6.a.g.1.1 2 11.10 odd 2
528.6.a.q.1.2 2 4.3 odd 2
825.6.a.d.1.1 2 5.4 even 2
1089.6.a.o.1.2 2 33.32 even 2