Properties

Label 33.6.a.d.1.1
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 1.1
Root \(-8.34590\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.34590 q^{2} -9.00000 q^{3} +37.6541 q^{4} -107.459 q^{5} +75.1131 q^{6} -26.6918 q^{7} -47.1885 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.34590 q^{2} -9.00000 q^{3} +37.6541 q^{4} -107.459 q^{5} +75.1131 q^{6} -26.6918 q^{7} -47.1885 q^{8} +81.0000 q^{9} +896.843 q^{10} +121.000 q^{11} -338.887 q^{12} +904.666 q^{13} +222.767 q^{14} +967.131 q^{15} -811.100 q^{16} -495.384 q^{17} -676.018 q^{18} -1501.38 q^{19} -4046.27 q^{20} +240.226 q^{21} -1009.85 q^{22} +2393.01 q^{23} +424.697 q^{24} +8422.44 q^{25} -7550.25 q^{26} -729.000 q^{27} -1005.06 q^{28} -5132.34 q^{29} -8071.58 q^{30} -410.309 q^{31} +8279.40 q^{32} -1089.00 q^{33} +4134.42 q^{34} +2868.28 q^{35} +3049.98 q^{36} +5828.69 q^{37} +12530.4 q^{38} -8141.99 q^{39} +5070.84 q^{40} +18561.8 q^{41} -2004.91 q^{42} -788.920 q^{43} +4556.15 q^{44} -8704.18 q^{45} -19971.8 q^{46} +3979.11 q^{47} +7299.90 q^{48} -16094.5 q^{49} -70292.9 q^{50} +4458.45 q^{51} +34064.4 q^{52} +15130.5 q^{53} +6084.16 q^{54} -13002.5 q^{55} +1259.55 q^{56} +13512.5 q^{57} +42834.0 q^{58} +41623.5 q^{59} +36416.5 q^{60} -29989.8 q^{61} +3424.40 q^{62} -2162.04 q^{63} -43143.8 q^{64} -97214.5 q^{65} +9088.69 q^{66} +25890.3 q^{67} -18653.2 q^{68} -21537.1 q^{69} -23938.3 q^{70} +58163.0 q^{71} -3822.27 q^{72} +23875.3 q^{73} -48645.7 q^{74} -75802.0 q^{75} -56533.2 q^{76} -3229.71 q^{77} +67952.3 q^{78} +27402.8 q^{79} +87160.0 q^{80} +6561.00 q^{81} -154915. q^{82} +24056.3 q^{83} +9045.50 q^{84} +53233.4 q^{85} +6584.25 q^{86} +46191.1 q^{87} -5709.81 q^{88} -55820.0 q^{89} +72644.3 q^{90} -24147.2 q^{91} +90106.5 q^{92} +3692.78 q^{93} -33209.3 q^{94} +161337. q^{95} -74514.6 q^{96} -101188. q^{97} +134324. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + O(q^{10}) \) \( 2q + q^{2} - 18q^{3} + 93q^{4} - 38q^{5} - 9q^{6} - 18q^{7} + 171q^{8} + 162q^{9} + 1546q^{10} + 242q^{11} - 837q^{12} - 66q^{13} + 304q^{14} + 342q^{15} - 543q^{16} - 920q^{17} + 81q^{18} - 2932q^{19} - 202q^{20} + 162q^{21} + 121q^{22} + 5246q^{23} - 1539q^{24} + 10122q^{25} - 16622q^{26} - 1458q^{27} - 524q^{28} - 12600q^{29} - 13914q^{30} + 9936q^{31} + 3803q^{32} - 2178q^{33} + 166q^{34} + 3472q^{35} + 7533q^{36} + 5996q^{37} - 840q^{38} + 594q^{39} + 20226q^{40} + 24244q^{41} - 2736q^{42} + 20360q^{43} + 11253q^{44} - 3078q^{45} + 6692q^{46} - 5806q^{47} + 4887q^{48} - 32826q^{49} - 54409q^{50} + 8280q^{51} - 19658q^{52} + 40770q^{53} - 729q^{54} - 4598q^{55} + 3156q^{56} + 26388q^{57} - 26958q^{58} + 18212q^{59} + 1818q^{60} - 11398q^{61} + 100120q^{62} - 1458q^{63} - 93559q^{64} - 164636q^{65} - 1089q^{66} + 65368q^{67} - 42154q^{68} - 47214q^{69} - 18296q^{70} + 61446q^{71} + 13851q^{72} + 53412q^{73} - 47082q^{74} - 91098q^{75} - 135712q^{76} - 2178q^{77} + 149598q^{78} + 17122q^{79} + 105782q^{80} + 13122q^{81} - 101810q^{82} - 14304q^{83} + 4716q^{84} + 23740q^{85} + 204240q^{86} + 113400q^{87} + 20691q^{88} - 58140q^{89} + 125226q^{90} - 32584q^{91} + 248008q^{92} - 89424q^{93} - 124660q^{94} + 61968q^{95} - 34227q^{96} - 183056q^{97} - 22047q^{98} + 19602q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.34590 −1.47536 −0.737681 0.675150i \(-0.764079\pi\)
−0.737681 + 0.675150i \(0.764079\pi\)
\(3\) −9.00000 −0.577350
\(4\) 37.6541 1.17669
\(5\) −107.459 −1.92229 −0.961143 0.276052i \(-0.910974\pi\)
−0.961143 + 0.276052i \(0.910974\pi\)
\(6\) 75.1131 0.851800
\(7\) −26.6918 −0.205889 −0.102944 0.994687i \(-0.532826\pi\)
−0.102944 + 0.994687i \(0.532826\pi\)
\(8\) −47.1885 −0.260682
\(9\) 81.0000 0.333333
\(10\) 896.843 2.83607
\(11\) 121.000 0.301511
\(12\) −338.887 −0.679363
\(13\) 904.666 1.48467 0.742335 0.670029i \(-0.233718\pi\)
0.742335 + 0.670029i \(0.233718\pi\)
\(14\) 222.767 0.303760
\(15\) 967.131 1.10983
\(16\) −811.100 −0.792090
\(17\) −495.384 −0.415738 −0.207869 0.978157i \(-0.566653\pi\)
−0.207869 + 0.978157i \(0.566653\pi\)
\(18\) −676.018 −0.491787
\(19\) −1501.38 −0.954130 −0.477065 0.878868i \(-0.658299\pi\)
−0.477065 + 0.878868i \(0.658299\pi\)
\(20\) −4046.27 −2.26194
\(21\) 240.226 0.118870
\(22\) −1009.85 −0.444838
\(23\) 2393.01 0.943245 0.471622 0.881801i \(-0.343669\pi\)
0.471622 + 0.881801i \(0.343669\pi\)
\(24\) 424.697 0.150505
\(25\) 8422.44 2.69518
\(26\) −7550.25 −2.19042
\(27\) −729.000 −0.192450
\(28\) −1005.06 −0.242267
\(29\) −5132.34 −1.13324 −0.566618 0.823980i \(-0.691749\pi\)
−0.566618 + 0.823980i \(0.691749\pi\)
\(30\) −8071.58 −1.63740
\(31\) −410.309 −0.0766844 −0.0383422 0.999265i \(-0.512208\pi\)
−0.0383422 + 0.999265i \(0.512208\pi\)
\(32\) 8279.40 1.42930
\(33\) −1089.00 −0.174078
\(34\) 4134.42 0.613363
\(35\) 2868.28 0.395777
\(36\) 3049.98 0.392230
\(37\) 5828.69 0.699949 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(38\) 12530.4 1.40769
\(39\) −8141.99 −0.857174
\(40\) 5070.84 0.501106
\(41\) 18561.8 1.72449 0.862245 0.506491i \(-0.169058\pi\)
0.862245 + 0.506491i \(0.169058\pi\)
\(42\) −2004.91 −0.175376
\(43\) −788.920 −0.0650671 −0.0325336 0.999471i \(-0.510358\pi\)
−0.0325336 + 0.999471i \(0.510358\pi\)
\(44\) 4556.15 0.354786
\(45\) −8704.18 −0.640762
\(46\) −19971.8 −1.39163
\(47\) 3979.11 0.262749 0.131375 0.991333i \(-0.458061\pi\)
0.131375 + 0.991333i \(0.458061\pi\)
\(48\) 7299.90 0.457313
\(49\) −16094.5 −0.957610
\(50\) −70292.9 −3.97637
\(51\) 4458.45 0.240026
\(52\) 34064.4 1.74700
\(53\) 15130.5 0.739886 0.369943 0.929055i \(-0.379377\pi\)
0.369943 + 0.929055i \(0.379377\pi\)
\(54\) 6084.16 0.283933
\(55\) −13002.5 −0.579591
\(56\) 1259.55 0.0536716
\(57\) 13512.5 0.550867
\(58\) 42834.0 1.67193
\(59\) 41623.5 1.55671 0.778357 0.627822i \(-0.216053\pi\)
0.778357 + 0.627822i \(0.216053\pi\)
\(60\) 36416.5 1.30593
\(61\) −29989.8 −1.03193 −0.515964 0.856610i \(-0.672566\pi\)
−0.515964 + 0.856610i \(0.672566\pi\)
\(62\) 3424.40 0.113137
\(63\) −2162.04 −0.0686296
\(64\) −43143.8 −1.31665
\(65\) −97214.5 −2.85396
\(66\) 9088.69 0.256827
\(67\) 25890.3 0.704613 0.352307 0.935885i \(-0.385397\pi\)
0.352307 + 0.935885i \(0.385397\pi\)
\(68\) −18653.2 −0.489195
\(69\) −21537.1 −0.544582
\(70\) −23938.3 −0.583914
\(71\) 58163.0 1.36931 0.684654 0.728869i \(-0.259954\pi\)
0.684654 + 0.728869i \(0.259954\pi\)
\(72\) −3822.27 −0.0868941
\(73\) 23875.3 0.524375 0.262187 0.965017i \(-0.415556\pi\)
0.262187 + 0.965017i \(0.415556\pi\)
\(74\) −48645.7 −1.03268
\(75\) −75802.0 −1.55606
\(76\) −56533.2 −1.12272
\(77\) −3229.71 −0.0620778
\(78\) 67952.3 1.26464
\(79\) 27402.8 0.494000 0.247000 0.969016i \(-0.420555\pi\)
0.247000 + 0.969016i \(0.420555\pi\)
\(80\) 87160.0 1.52262
\(81\) 6561.00 0.111111
\(82\) −154915. −2.54425
\(83\) 24056.3 0.383296 0.191648 0.981464i \(-0.438617\pi\)
0.191648 + 0.981464i \(0.438617\pi\)
\(84\) 9045.50 0.139873
\(85\) 53233.4 0.799166
\(86\) 6584.25 0.0959975
\(87\) 46191.1 0.654274
\(88\) −5709.81 −0.0785987
\(89\) −55820.0 −0.746990 −0.373495 0.927632i \(-0.621841\pi\)
−0.373495 + 0.927632i \(0.621841\pi\)
\(90\) 72644.3 0.945355
\(91\) −24147.2 −0.305677
\(92\) 90106.5 1.10991
\(93\) 3692.78 0.0442737
\(94\) −33209.3 −0.387650
\(95\) 161337. 1.83411
\(96\) −74514.6 −0.825207
\(97\) −101188. −1.09194 −0.545970 0.837805i \(-0.683839\pi\)
−0.545970 + 0.837805i \(0.683839\pi\)
\(98\) 134324. 1.41282
\(99\) 9801.00 0.100504
\(100\) 317139. 3.17139
\(101\) −75190.1 −0.733428 −0.366714 0.930334i \(-0.619517\pi\)
−0.366714 + 0.930334i \(0.619517\pi\)
\(102\) −37209.8 −0.354125
\(103\) 176059. 1.63518 0.817590 0.575801i \(-0.195310\pi\)
0.817590 + 0.575801i \(0.195310\pi\)
\(104\) −42689.9 −0.387027
\(105\) −25814.5 −0.228502
\(106\) −126278. −1.09160
\(107\) −124963. −1.05517 −0.527586 0.849501i \(-0.676903\pi\)
−0.527586 + 0.849501i \(0.676903\pi\)
\(108\) −27449.8 −0.226454
\(109\) 48001.5 0.386980 0.193490 0.981102i \(-0.438019\pi\)
0.193490 + 0.981102i \(0.438019\pi\)
\(110\) 108518. 0.855106
\(111\) −52458.2 −0.404116
\(112\) 21649.7 0.163082
\(113\) 38955.6 0.286995 0.143497 0.989651i \(-0.454165\pi\)
0.143497 + 0.989651i \(0.454165\pi\)
\(114\) −112774. −0.812728
\(115\) −257150. −1.81319
\(116\) −193254. −1.33347
\(117\) 73277.9 0.494890
\(118\) −347386. −2.29672
\(119\) 13222.7 0.0855957
\(120\) −45637.5 −0.289314
\(121\) 14641.0 0.0909091
\(122\) 250292. 1.52247
\(123\) −167056. −0.995635
\(124\) −15449.8 −0.0902338
\(125\) −569258. −3.25862
\(126\) 18044.1 0.101253
\(127\) 186757. 1.02746 0.513732 0.857951i \(-0.328262\pi\)
0.513732 + 0.857951i \(0.328262\pi\)
\(128\) 95133.6 0.513226
\(129\) 7100.28 0.0375665
\(130\) 811343. 4.21062
\(131\) −363603. −1.85118 −0.925592 0.378523i \(-0.876432\pi\)
−0.925592 + 0.378523i \(0.876432\pi\)
\(132\) −41005.3 −0.204836
\(133\) 40074.6 0.196445
\(134\) −216078. −1.03956
\(135\) 78337.6 0.369944
\(136\) 23376.4 0.108375
\(137\) −74002.6 −0.336857 −0.168429 0.985714i \(-0.553869\pi\)
−0.168429 + 0.985714i \(0.553869\pi\)
\(138\) 179746. 0.803456
\(139\) 70042.3 0.307485 0.153742 0.988111i \(-0.450867\pi\)
0.153742 + 0.988111i \(0.450867\pi\)
\(140\) 108002. 0.465707
\(141\) −35812.0 −0.151698
\(142\) −485423. −2.02022
\(143\) 109465. 0.447645
\(144\) −65699.1 −0.264030
\(145\) 551516. 2.17840
\(146\) −199261. −0.773642
\(147\) 144851. 0.552876
\(148\) 219474. 0.823624
\(149\) 327767. 1.20948 0.604741 0.796422i \(-0.293276\pi\)
0.604741 + 0.796422i \(0.293276\pi\)
\(150\) 632636. 2.29576
\(151\) 519.828 0.00185531 0.000927656 1.00000i \(-0.499705\pi\)
0.000927656 1.00000i \(0.499705\pi\)
\(152\) 70848.1 0.248725
\(153\) −40126.1 −0.138579
\(154\) 26954.8 0.0915872
\(155\) 44091.4 0.147409
\(156\) −306579. −1.00863
\(157\) −119470. −0.386821 −0.193411 0.981118i \(-0.561955\pi\)
−0.193411 + 0.981118i \(0.561955\pi\)
\(158\) −228701. −0.728828
\(159\) −136175. −0.427173
\(160\) −889696. −2.74752
\(161\) −63873.7 −0.194204
\(162\) −54757.5 −0.163929
\(163\) 314012. 0.925716 0.462858 0.886432i \(-0.346824\pi\)
0.462858 + 0.886432i \(0.346824\pi\)
\(164\) 698928. 2.02919
\(165\) 117023. 0.334627
\(166\) −200772. −0.565500
\(167\) 281877. 0.782111 0.391056 0.920367i \(-0.372110\pi\)
0.391056 + 0.920367i \(0.372110\pi\)
\(168\) −11335.9 −0.0309873
\(169\) 447127. 1.20424
\(170\) −444281. −1.17906
\(171\) −121612. −0.318043
\(172\) −29706.1 −0.0765639
\(173\) 154131. 0.391538 0.195769 0.980650i \(-0.437280\pi\)
0.195769 + 0.980650i \(0.437280\pi\)
\(174\) −385506. −0.965291
\(175\) −224810. −0.554908
\(176\) −98143.1 −0.238824
\(177\) −374612. −0.898770
\(178\) 465868. 1.10208
\(179\) 603292. 1.40733 0.703664 0.710533i \(-0.251546\pi\)
0.703664 + 0.710533i \(0.251546\pi\)
\(180\) −327748. −0.753978
\(181\) −560418. −1.27150 −0.635749 0.771896i \(-0.719309\pi\)
−0.635749 + 0.771896i \(0.719309\pi\)
\(182\) 201530. 0.450984
\(183\) 269909. 0.595784
\(184\) −112922. −0.245887
\(185\) −626345. −1.34550
\(186\) −30819.6 −0.0653198
\(187\) −59941.4 −0.125350
\(188\) 149830. 0.309175
\(189\) 19458.3 0.0396233
\(190\) −1.34650e6 −2.70598
\(191\) 337255. 0.668921 0.334461 0.942410i \(-0.391446\pi\)
0.334461 + 0.942410i \(0.391446\pi\)
\(192\) 388295. 0.760166
\(193\) 427002. 0.825158 0.412579 0.910922i \(-0.364628\pi\)
0.412579 + 0.910922i \(0.364628\pi\)
\(194\) 844503. 1.61100
\(195\) 874931. 1.64773
\(196\) −606026. −1.12681
\(197\) 387552. 0.711483 0.355741 0.934584i \(-0.384228\pi\)
0.355741 + 0.934584i \(0.384228\pi\)
\(198\) −81798.2 −0.148279
\(199\) 564409. 1.01033 0.505163 0.863024i \(-0.331432\pi\)
0.505163 + 0.863024i \(0.331432\pi\)
\(200\) −397443. −0.702586
\(201\) −233013. −0.406809
\(202\) 627529. 1.08207
\(203\) 136991. 0.233321
\(204\) 167879. 0.282437
\(205\) −1.99463e6 −3.31496
\(206\) −1.46937e6 −2.41248
\(207\) 193834. 0.314415
\(208\) −733774. −1.17599
\(209\) −181667. −0.287681
\(210\) 215445. 0.337123
\(211\) 495448. 0.766112 0.383056 0.923725i \(-0.374872\pi\)
0.383056 + 0.923725i \(0.374872\pi\)
\(212\) 569727. 0.870616
\(213\) −523467. −0.790570
\(214\) 1.04293e6 1.55676
\(215\) 84776.5 0.125078
\(216\) 34400.4 0.0501683
\(217\) 10951.9 0.0157885
\(218\) −400616. −0.570935
\(219\) −214878. −0.302748
\(220\) −489599. −0.681999
\(221\) −448157. −0.617233
\(222\) 437811. 0.596217
\(223\) 1.23517e6 1.66327 0.831637 0.555320i \(-0.187404\pi\)
0.831637 + 0.555320i \(0.187404\pi\)
\(224\) −220992. −0.294277
\(225\) 682218. 0.898394
\(226\) −325120. −0.423421
\(227\) −802179. −1.03325 −0.516627 0.856211i \(-0.672812\pi\)
−0.516627 + 0.856211i \(0.672812\pi\)
\(228\) 508799. 0.648200
\(229\) −162526. −0.204802 −0.102401 0.994743i \(-0.532652\pi\)
−0.102401 + 0.994743i \(0.532652\pi\)
\(230\) 2.14615e6 2.67510
\(231\) 29067.4 0.0358407
\(232\) 242188. 0.295415
\(233\) −1.45802e6 −1.75944 −0.879718 0.475496i \(-0.842269\pi\)
−0.879718 + 0.475496i \(0.842269\pi\)
\(234\) −611570. −0.730141
\(235\) −427592. −0.505079
\(236\) 1.56730e6 1.83177
\(237\) −246625. −0.285211
\(238\) −110355. −0.126285
\(239\) −582475. −0.659603 −0.329802 0.944050i \(-0.606982\pi\)
−0.329802 + 0.944050i \(0.606982\pi\)
\(240\) −784440. −0.879087
\(241\) 434564. 0.481961 0.240980 0.970530i \(-0.422531\pi\)
0.240980 + 0.970530i \(0.422531\pi\)
\(242\) −122192. −0.134124
\(243\) −59049.0 −0.0641500
\(244\) −1.12924e6 −1.21426
\(245\) 1.72950e6 1.84080
\(246\) 1.39424e6 1.46892
\(247\) −1.35825e6 −1.41657
\(248\) 19361.9 0.0199903
\(249\) −216507. −0.221296
\(250\) 4.75097e6 4.80765
\(251\) −948184. −0.949967 −0.474983 0.879995i \(-0.657546\pi\)
−0.474983 + 0.879995i \(0.657546\pi\)
\(252\) −81409.5 −0.0807558
\(253\) 289554. 0.284399
\(254\) −1.55865e6 −1.51588
\(255\) −479101. −0.461399
\(256\) 586627. 0.559451
\(257\) 1.14089e6 1.07749 0.538744 0.842470i \(-0.318899\pi\)
0.538744 + 0.842470i \(0.318899\pi\)
\(258\) −59258.2 −0.0554242
\(259\) −155578. −0.144112
\(260\) −3.66052e6 −3.35823
\(261\) −415720. −0.377745
\(262\) 3.03460e6 2.73116
\(263\) 280915. 0.250430 0.125215 0.992130i \(-0.460038\pi\)
0.125215 + 0.992130i \(0.460038\pi\)
\(264\) 51388.3 0.0453790
\(265\) −1.62591e6 −1.42227
\(266\) −334459. −0.289827
\(267\) 502380. 0.431275
\(268\) 974878. 0.829112
\(269\) 1.08759e6 0.916397 0.458198 0.888850i \(-0.348495\pi\)
0.458198 + 0.888850i \(0.348495\pi\)
\(270\) −653798. −0.545801
\(271\) 1.17929e6 0.975431 0.487716 0.873003i \(-0.337830\pi\)
0.487716 + 0.873003i \(0.337830\pi\)
\(272\) 401806. 0.329302
\(273\) 217324. 0.176483
\(274\) 617619. 0.496986
\(275\) 1.01912e6 0.812628
\(276\) −810958. −0.640805
\(277\) −621357. −0.486566 −0.243283 0.969955i \(-0.578224\pi\)
−0.243283 + 0.969955i \(0.578224\pi\)
\(278\) −584566. −0.453651
\(279\) −33235.0 −0.0255615
\(280\) −135350. −0.103172
\(281\) −243988. −0.184333 −0.0921666 0.995744i \(-0.529379\pi\)
−0.0921666 + 0.995744i \(0.529379\pi\)
\(282\) 298884. 0.223810
\(283\) −1.84579e6 −1.36999 −0.684993 0.728549i \(-0.740195\pi\)
−0.684993 + 0.728549i \(0.740195\pi\)
\(284\) 2.19007e6 1.61125
\(285\) −1.45204e6 −1.05892
\(286\) −913581. −0.660438
\(287\) −495448. −0.355053
\(288\) 670631. 0.476434
\(289\) −1.17445e6 −0.827162
\(290\) −4.60290e6 −3.21393
\(291\) 910690. 0.630431
\(292\) 899003. 0.617027
\(293\) −1.34899e6 −0.917995 −0.458998 0.888437i \(-0.651791\pi\)
−0.458998 + 0.888437i \(0.651791\pi\)
\(294\) −1.20891e6 −0.815692
\(295\) −4.47283e6 −2.99245
\(296\) −275047. −0.182464
\(297\) −88209.0 −0.0580259
\(298\) −2.73551e6 −1.78442
\(299\) 2.16487e6 1.40041
\(300\) −2.85426e6 −1.83101
\(301\) 21057.7 0.0133966
\(302\) −4338.43 −0.00273726
\(303\) 676711. 0.423445
\(304\) 1.21777e6 0.755757
\(305\) 3.22268e6 1.98366
\(306\) 334888. 0.204454
\(307\) −1.20545e6 −0.729970 −0.364985 0.931013i \(-0.618926\pi\)
−0.364985 + 0.931013i \(0.618926\pi\)
\(308\) −121612. −0.0730464
\(309\) −1.58453e6 −0.944072
\(310\) −367983. −0.217482
\(311\) 636875. 0.373382 0.186691 0.982419i \(-0.440224\pi\)
0.186691 + 0.982419i \(0.440224\pi\)
\(312\) 384209. 0.223450
\(313\) 1.86526e6 1.07616 0.538081 0.842893i \(-0.319150\pi\)
0.538081 + 0.842893i \(0.319150\pi\)
\(314\) 997086. 0.570701
\(315\) 232330. 0.131926
\(316\) 1.03183e6 0.581285
\(317\) 2.35417e6 1.31580 0.657900 0.753105i \(-0.271445\pi\)
0.657900 + 0.753105i \(0.271445\pi\)
\(318\) 1.13650e6 0.630235
\(319\) −621013. −0.341684
\(320\) 4.63619e6 2.53097
\(321\) 1.12467e6 0.609204
\(322\) 533083. 0.286520
\(323\) 743761. 0.396668
\(324\) 247049. 0.130743
\(325\) 7.61950e6 4.00145
\(326\) −2.62072e6 −1.36577
\(327\) −432013. −0.223423
\(328\) −875905. −0.449544
\(329\) −106210. −0.0540972
\(330\) −976662. −0.493696
\(331\) 1.97887e6 0.992768 0.496384 0.868103i \(-0.334661\pi\)
0.496384 + 0.868103i \(0.334661\pi\)
\(332\) 905820. 0.451021
\(333\) 472124. 0.233316
\(334\) −2.35252e6 −1.15390
\(335\) −2.78215e6 −1.35447
\(336\) −194848. −0.0941557
\(337\) 2.86225e6 1.37288 0.686439 0.727187i \(-0.259173\pi\)
0.686439 + 0.727187i \(0.259173\pi\)
\(338\) −3.73168e6 −1.77669
\(339\) −350601. −0.165697
\(340\) 2.00446e6 0.940372
\(341\) −49647.4 −0.0231212
\(342\) 1.01496e6 0.469229
\(343\) 878202. 0.403050
\(344\) 37228.0 0.0169619
\(345\) 2.31435e6 1.04684
\(346\) −1.28636e6 −0.577660
\(347\) −1.81514e6 −0.809257 −0.404629 0.914481i \(-0.632599\pi\)
−0.404629 + 0.914481i \(0.632599\pi\)
\(348\) 1.73928e6 0.769878
\(349\) −1.77907e6 −0.781861 −0.390930 0.920420i \(-0.627847\pi\)
−0.390930 + 0.920420i \(0.627847\pi\)
\(350\) 1.87624e6 0.818690
\(351\) −659501. −0.285725
\(352\) 1.00181e6 0.430950
\(353\) −4.18817e6 −1.78890 −0.894452 0.447164i \(-0.852434\pi\)
−0.894452 + 0.447164i \(0.852434\pi\)
\(354\) 3.12647e6 1.32601
\(355\) −6.25014e6 −2.63220
\(356\) −2.10185e6 −0.878976
\(357\) −119004. −0.0494187
\(358\) −5.03502e6 −2.07632
\(359\) 1.13143e6 0.463329 0.231665 0.972796i \(-0.425583\pi\)
0.231665 + 0.972796i \(0.425583\pi\)
\(360\) 410738. 0.167035
\(361\) −221946. −0.0896355
\(362\) 4.67720e6 1.87592
\(363\) −131769. −0.0524864
\(364\) −909240. −0.359687
\(365\) −2.56562e6 −1.00800
\(366\) −2.25263e6 −0.878997
\(367\) 1.31924e6 0.511279 0.255639 0.966772i \(-0.417714\pi\)
0.255639 + 0.966772i \(0.417714\pi\)
\(368\) −1.94097e6 −0.747134
\(369\) 1.50351e6 0.574830
\(370\) 5.22742e6 1.98510
\(371\) −403861. −0.152334
\(372\) 139048. 0.0520965
\(373\) 741.945 0.000276121 0 0.000138061 1.00000i \(-0.499956\pi\)
0.000138061 1.00000i \(0.499956\pi\)
\(374\) 500265. 0.184936
\(375\) 5.12332e6 1.88137
\(376\) −187769. −0.0684941
\(377\) −4.64305e6 −1.68248
\(378\) −162397. −0.0584587
\(379\) −567437. −0.202917 −0.101459 0.994840i \(-0.532351\pi\)
−0.101459 + 0.994840i \(0.532351\pi\)
\(380\) 6.07501e6 2.15818
\(381\) −1.68081e6 −0.593207
\(382\) −2.81470e6 −0.986900
\(383\) 3.54829e6 1.23601 0.618006 0.786174i \(-0.287941\pi\)
0.618006 + 0.786174i \(0.287941\pi\)
\(384\) −856202. −0.296311
\(385\) 347061. 0.119331
\(386\) −3.56372e6 −1.21741
\(387\) −63902.5 −0.0216890
\(388\) −3.81013e6 −1.28487
\(389\) 4.09335e6 1.37153 0.685764 0.727824i \(-0.259468\pi\)
0.685764 + 0.727824i \(0.259468\pi\)
\(390\) −7.30209e6 −2.43100
\(391\) −1.18546e6 −0.392142
\(392\) 759478. 0.249632
\(393\) 3.27243e6 1.06878
\(394\) −3.23447e6 −1.04969
\(395\) −2.94468e6 −0.949609
\(396\) 369048. 0.118262
\(397\) 153511. 0.0488835 0.0244417 0.999701i \(-0.492219\pi\)
0.0244417 + 0.999701i \(0.492219\pi\)
\(398\) −4.71050e6 −1.49059
\(399\) −360672. −0.113417
\(400\) −6.83144e6 −2.13483
\(401\) 474424. 0.147335 0.0736674 0.997283i \(-0.476530\pi\)
0.0736674 + 0.997283i \(0.476530\pi\)
\(402\) 1.94470e6 0.600190
\(403\) −371193. −0.113851
\(404\) −2.83122e6 −0.863017
\(405\) −705039. −0.213587
\(406\) −1.14332e6 −0.344232
\(407\) 705271. 0.211043
\(408\) −210388. −0.0625706
\(409\) 2.85246e6 0.843164 0.421582 0.906790i \(-0.361475\pi\)
0.421582 + 0.906790i \(0.361475\pi\)
\(410\) 1.66470e7 4.89077
\(411\) 666023. 0.194484
\(412\) 6.62935e6 1.92410
\(413\) −1.11101e6 −0.320510
\(414\) −1.61772e6 −0.463875
\(415\) −2.58507e6 −0.736805
\(416\) 7.49009e6 2.12204
\(417\) −630381. −0.177526
\(418\) 1.51618e6 0.424434
\(419\) 6.61913e6 1.84190 0.920950 0.389681i \(-0.127415\pi\)
0.920950 + 0.389681i \(0.127415\pi\)
\(420\) −972021. −0.268876
\(421\) 1.67579e6 0.460802 0.230401 0.973096i \(-0.425996\pi\)
0.230401 + 0.973096i \(0.425996\pi\)
\(422\) −4.13496e6 −1.13029
\(423\) 322308. 0.0875831
\(424\) −713988. −0.192875
\(425\) −4.17234e6 −1.12049
\(426\) 4.36880e6 1.16638
\(427\) 800483. 0.212463
\(428\) −4.70539e6 −1.24161
\(429\) −985181. −0.258448
\(430\) −707537. −0.184535
\(431\) −4.53856e6 −1.17686 −0.588430 0.808548i \(-0.700254\pi\)
−0.588430 + 0.808548i \(0.700254\pi\)
\(432\) 591292. 0.152438
\(433\) 379605. 0.0972999 0.0486500 0.998816i \(-0.484508\pi\)
0.0486500 + 0.998816i \(0.484508\pi\)
\(434\) −91403.4 −0.0232937
\(435\) −4.96365e6 −1.25770
\(436\) 1.80745e6 0.455355
\(437\) −3.59282e6 −0.899978
\(438\) 1.79335e6 0.446663
\(439\) −1.97778e6 −0.489797 −0.244899 0.969549i \(-0.578755\pi\)
−0.244899 + 0.969549i \(0.578755\pi\)
\(440\) 613571. 0.151089
\(441\) −1.30366e6 −0.319203
\(442\) 3.74027e6 0.910641
\(443\) 3.58373e6 0.867613 0.433807 0.901006i \(-0.357170\pi\)
0.433807 + 0.901006i \(0.357170\pi\)
\(444\) −1.97527e6 −0.475519
\(445\) 5.99836e6 1.43593
\(446\) −1.03086e7 −2.45393
\(447\) −2.94990e6 −0.698295
\(448\) 1.15159e6 0.271083
\(449\) −1.05735e6 −0.247515 −0.123758 0.992312i \(-0.539495\pi\)
−0.123758 + 0.992312i \(0.539495\pi\)
\(450\) −5.69372e6 −1.32546
\(451\) 2.24598e6 0.519954
\(452\) 1.46684e6 0.337704
\(453\) −4678.45 −0.00107117
\(454\) 6.69491e6 1.52442
\(455\) 2.59483e6 0.587598
\(456\) −637633. −0.143601
\(457\) 1.39016e6 0.311368 0.155684 0.987807i \(-0.450242\pi\)
0.155684 + 0.987807i \(0.450242\pi\)
\(458\) 1.35643e6 0.302157
\(459\) 361135. 0.0800087
\(460\) −9.68276e6 −2.13356
\(461\) −1.09360e6 −0.239666 −0.119833 0.992794i \(-0.538236\pi\)
−0.119833 + 0.992794i \(0.538236\pi\)
\(462\) −242594. −0.0528779
\(463\) 1.94891e6 0.422513 0.211256 0.977431i \(-0.432244\pi\)
0.211256 + 0.977431i \(0.432244\pi\)
\(464\) 4.16284e6 0.897625
\(465\) −396823. −0.0851068
\(466\) 1.21685e7 2.59580
\(467\) −4.69017e6 −0.995169 −0.497584 0.867416i \(-0.665780\pi\)
−0.497584 + 0.867416i \(0.665780\pi\)
\(468\) 2.75921e6 0.582332
\(469\) −691060. −0.145072
\(470\) 3.56864e6 0.745174
\(471\) 1.07523e6 0.223331
\(472\) −1.96415e6 −0.405808
\(473\) −95459.3 −0.0196185
\(474\) 2.05831e6 0.420789
\(475\) −1.26453e7 −2.57155
\(476\) 497888. 0.100720
\(477\) 1.22557e6 0.246629
\(478\) 4.86128e6 0.973153
\(479\) 4.15576e6 0.827584 0.413792 0.910372i \(-0.364204\pi\)
0.413792 + 0.910372i \(0.364204\pi\)
\(480\) 8.00726e6 1.58628
\(481\) 5.27302e6 1.03919
\(482\) −3.62683e6 −0.711066
\(483\) 574863. 0.112123
\(484\) 551294. 0.106972
\(485\) 1.08735e7 2.09902
\(486\) 492817. 0.0946445
\(487\) −2.37242e6 −0.453282 −0.226641 0.973978i \(-0.572774\pi\)
−0.226641 + 0.973978i \(0.572774\pi\)
\(488\) 1.41518e6 0.269006
\(489\) −2.82611e6 −0.534463
\(490\) −1.44343e7 −2.71584
\(491\) −1.18797e6 −0.222384 −0.111192 0.993799i \(-0.535467\pi\)
−0.111192 + 0.993799i \(0.535467\pi\)
\(492\) −6.29036e6 −1.17155
\(493\) 2.54248e6 0.471129
\(494\) 1.13358e7 2.08995
\(495\) −1.05321e6 −0.193197
\(496\) 332802. 0.0607409
\(497\) −1.55248e6 −0.281925
\(498\) 1.80695e6 0.326492
\(499\) 710493. 0.127735 0.0638673 0.997958i \(-0.479657\pi\)
0.0638673 + 0.997958i \(0.479657\pi\)
\(500\) −2.14349e7 −3.83439
\(501\) −2.53689e6 −0.451552
\(502\) 7.91345e6 1.40154
\(503\) 4.75381e6 0.837765 0.418882 0.908041i \(-0.362422\pi\)
0.418882 + 0.908041i \(0.362422\pi\)
\(504\) 102023. 0.0178905
\(505\) 8.07986e6 1.40986
\(506\) −2.41659e6 −0.419591
\(507\) −4.02414e6 −0.695270
\(508\) 7.03215e6 1.20901
\(509\) −5.05479e6 −0.864787 −0.432393 0.901685i \(-0.642331\pi\)
−0.432393 + 0.901685i \(0.642331\pi\)
\(510\) 3.99853e6 0.680730
\(511\) −637275. −0.107963
\(512\) −7.94021e6 −1.33862
\(513\) 1.09451e6 0.183622
\(514\) −9.52179e6 −1.58968
\(515\) −1.89191e7 −3.14328
\(516\) 267355. 0.0442042
\(517\) 481473. 0.0792219
\(518\) 1.29844e6 0.212617
\(519\) −1.38718e6 −0.226055
\(520\) 4.58741e6 0.743977
\(521\) −4.69775e6 −0.758220 −0.379110 0.925352i \(-0.623770\pi\)
−0.379110 + 0.925352i \(0.623770\pi\)
\(522\) 3.46956e6 0.557311
\(523\) −1.97083e6 −0.315062 −0.157531 0.987514i \(-0.550353\pi\)
−0.157531 + 0.987514i \(0.550353\pi\)
\(524\) −1.36912e7 −2.17827
\(525\) 2.02329e6 0.320376
\(526\) −2.34449e6 −0.369474
\(527\) 203260. 0.0318806
\(528\) 883288. 0.137885
\(529\) −709863. −0.110290
\(530\) 1.35697e7 2.09836
\(531\) 3.37151e6 0.518905
\(532\) 1.50897e6 0.231155
\(533\) 1.67922e7 2.56030
\(534\) −4.19282e6 −0.636286
\(535\) 1.34285e7 2.02834
\(536\) −1.22173e6 −0.183680
\(537\) −5.42963e6 −0.812521
\(538\) −9.07690e6 −1.35202
\(539\) −1.94744e6 −0.288730
\(540\) 2.94973e6 0.435310
\(541\) −5.76437e6 −0.846757 −0.423378 0.905953i \(-0.639156\pi\)
−0.423378 + 0.905953i \(0.639156\pi\)
\(542\) −9.84222e6 −1.43911
\(543\) 5.04377e6 0.734100
\(544\) −4.10148e6 −0.594214
\(545\) −5.15819e6 −0.743886
\(546\) −1.81377e6 −0.260376
\(547\) −4.65202e6 −0.664772 −0.332386 0.943143i \(-0.607854\pi\)
−0.332386 + 0.943143i \(0.607854\pi\)
\(548\) −2.78650e6 −0.396376
\(549\) −2.42918e6 −0.343976
\(550\) −8.50544e6 −1.19892
\(551\) 7.70561e6 1.08126
\(552\) 1.01630e6 0.141963
\(553\) −731430. −0.101709
\(554\) 5.18578e6 0.717860
\(555\) 5.63711e6 0.776826
\(556\) 2.63738e6 0.361814
\(557\) −4.24834e6 −0.580204 −0.290102 0.956996i \(-0.593689\pi\)
−0.290102 + 0.956996i \(0.593689\pi\)
\(558\) 277376. 0.0377124
\(559\) −713709. −0.0966032
\(560\) −2.32646e6 −0.313491
\(561\) 539473. 0.0723706
\(562\) 2.03630e6 0.271958
\(563\) −3.09512e6 −0.411534 −0.205767 0.978601i \(-0.565969\pi\)
−0.205767 + 0.978601i \(0.565969\pi\)
\(564\) −1.34847e6 −0.178502
\(565\) −4.18613e6 −0.551686
\(566\) 1.54048e7 2.02122
\(567\) −175125. −0.0228765
\(568\) −2.74463e6 −0.356954
\(569\) −7.60568e6 −0.984821 −0.492411 0.870363i \(-0.663884\pi\)
−0.492411 + 0.870363i \(0.663884\pi\)
\(570\) 1.21185e7 1.56230
\(571\) −2.28582e6 −0.293395 −0.146697 0.989181i \(-0.546864\pi\)
−0.146697 + 0.989181i \(0.546864\pi\)
\(572\) 4.12179e6 0.526739
\(573\) −3.03529e6 −0.386202
\(574\) 4.13496e6 0.523832
\(575\) 2.01550e7 2.54222
\(576\) −3.49465e6 −0.438882
\(577\) −6.27770e6 −0.784984 −0.392492 0.919755i \(-0.628387\pi\)
−0.392492 + 0.919755i \(0.628387\pi\)
\(578\) 9.80186e6 1.22036
\(579\) −3.84302e6 −0.476405
\(580\) 2.07669e7 2.56331
\(581\) −642107. −0.0789164
\(582\) −7.60053e6 −0.930114
\(583\) 1.83079e6 0.223084
\(584\) −1.12664e6 −0.136695
\(585\) −7.87437e6 −0.951320
\(586\) 1.12586e7 1.35437
\(587\) −1.48651e6 −0.178063 −0.0890314 0.996029i \(-0.528377\pi\)
−0.0890314 + 0.996029i \(0.528377\pi\)
\(588\) 5.45423e6 0.650564
\(589\) 616031. 0.0731669
\(590\) 3.73298e7 4.41494
\(591\) −3.48797e6 −0.410775
\(592\) −4.72765e6 −0.554423
\(593\) 4.72874e6 0.552216 0.276108 0.961127i \(-0.410955\pi\)
0.276108 + 0.961127i \(0.410955\pi\)
\(594\) 736184. 0.0856091
\(595\) −1.42090e6 −0.164539
\(596\) 1.23418e7 1.42319
\(597\) −5.07968e6 −0.583312
\(598\) −1.80678e7 −2.06611
\(599\) −325927. −0.0371153 −0.0185577 0.999828i \(-0.505907\pi\)
−0.0185577 + 0.999828i \(0.505907\pi\)
\(600\) 3.57699e6 0.405638
\(601\) −1.30291e7 −1.47140 −0.735698 0.677310i \(-0.763146\pi\)
−0.735698 + 0.677310i \(0.763146\pi\)
\(602\) −175745. −0.0197648
\(603\) 2.09712e6 0.234871
\(604\) 19573.6 0.00218313
\(605\) −1.57331e6 −0.174753
\(606\) −5.64776e6 −0.624734
\(607\) −1.45956e6 −0.160787 −0.0803933 0.996763i \(-0.525618\pi\)
−0.0803933 + 0.996763i \(0.525618\pi\)
\(608\) −1.24305e7 −1.36374
\(609\) −1.23292e6 −0.134708
\(610\) −2.68962e7 −2.92662
\(611\) 3.59977e6 0.390096
\(612\) −1.51091e6 −0.163065
\(613\) 1.26574e7 1.36048 0.680241 0.732989i \(-0.261875\pi\)
0.680241 + 0.732989i \(0.261875\pi\)
\(614\) 1.00606e7 1.07697
\(615\) 1.79517e7 1.91390
\(616\) 152405. 0.0161826
\(617\) 2.57472e6 0.272281 0.136140 0.990690i \(-0.456530\pi\)
0.136140 + 0.990690i \(0.456530\pi\)
\(618\) 1.32244e7 1.39285
\(619\) −1.82556e7 −1.91500 −0.957500 0.288434i \(-0.906866\pi\)
−0.957500 + 0.288434i \(0.906866\pi\)
\(620\) 1.66022e6 0.173455
\(621\) −1.74450e6 −0.181527
\(622\) −5.31530e6 −0.550873
\(623\) 1.48994e6 0.153797
\(624\) 6.60397e6 0.678959
\(625\) 3.48518e7 3.56882
\(626\) −1.55672e7 −1.58773
\(627\) 1.63501e6 0.166093
\(628\) −4.49854e6 −0.455169
\(629\) −2.88744e6 −0.290995
\(630\) −1.93901e6 −0.194638
\(631\) 7.91613e6 0.791480 0.395740 0.918363i \(-0.370488\pi\)
0.395740 + 0.918363i \(0.370488\pi\)
\(632\) −1.29310e6 −0.128777
\(633\) −4.45903e6 −0.442315
\(634\) −1.96477e7 −1.94128
\(635\) −2.00687e7 −1.97508
\(636\) −5.12754e6 −0.502651
\(637\) −1.45602e7 −1.42173
\(638\) 5.18292e6 0.504107
\(639\) 4.71120e6 0.456436
\(640\) −1.02230e7 −0.986568
\(641\) −3.40205e6 −0.327036 −0.163518 0.986540i \(-0.552284\pi\)
−0.163518 + 0.986540i \(0.552284\pi\)
\(642\) −9.38640e6 −0.898796
\(643\) −6.92901e6 −0.660912 −0.330456 0.943821i \(-0.607203\pi\)
−0.330456 + 0.943821i \(0.607203\pi\)
\(644\) −2.40511e6 −0.228517
\(645\) −762989. −0.0722136
\(646\) −6.20736e6 −0.585228
\(647\) 1.11125e7 1.04364 0.521819 0.853056i \(-0.325254\pi\)
0.521819 + 0.853056i \(0.325254\pi\)
\(648\) −309604. −0.0289647
\(649\) 5.03645e6 0.469367
\(650\) −6.35916e7 −5.90359
\(651\) −98567.0 −0.00911547
\(652\) 1.18239e7 1.08928
\(653\) 6.61748e6 0.607309 0.303654 0.952782i \(-0.401793\pi\)
0.303654 + 0.952782i \(0.401793\pi\)
\(654\) 3.60554e6 0.329629
\(655\) 3.90725e7 3.55850
\(656\) −1.50555e7 −1.36595
\(657\) 1.93390e6 0.174792
\(658\) 886416. 0.0798128
\(659\) 1.88838e7 1.69385 0.846926 0.531711i \(-0.178451\pi\)
0.846926 + 0.531711i \(0.178451\pi\)
\(660\) 4.40639e6 0.393752
\(661\) 7.10304e6 0.632326 0.316163 0.948705i \(-0.397605\pi\)
0.316163 + 0.948705i \(0.397605\pi\)
\(662\) −1.65155e7 −1.46469
\(663\) 4.03341e6 0.356360
\(664\) −1.13518e6 −0.0999185
\(665\) −4.30638e6 −0.377623
\(666\) −3.94030e6 −0.344226
\(667\) −1.22817e7 −1.06892
\(668\) 1.06138e7 0.920303
\(669\) −1.11165e7 −0.960291
\(670\) 2.32196e7 1.99833
\(671\) −3.62877e6 −0.311138
\(672\) 1.98893e6 0.169901
\(673\) −1.79739e7 −1.52970 −0.764849 0.644209i \(-0.777187\pi\)
−0.764849 + 0.644209i \(0.777187\pi\)
\(674\) −2.38880e7 −2.02549
\(675\) −6.13996e6 −0.518688
\(676\) 1.68362e7 1.41702
\(677\) −617034. −0.0517413 −0.0258706 0.999665i \(-0.508236\pi\)
−0.0258706 + 0.999665i \(0.508236\pi\)
\(678\) 2.92608e6 0.244462
\(679\) 2.70088e6 0.224818
\(680\) −2.51201e6 −0.208329
\(681\) 7.21961e6 0.596549
\(682\) 414352. 0.0341121
\(683\) 924287. 0.0758150 0.0379075 0.999281i \(-0.487931\pi\)
0.0379075 + 0.999281i \(0.487931\pi\)
\(684\) −4.57919e6 −0.374239
\(685\) 7.95225e6 0.647535
\(686\) −7.32939e6 −0.594644
\(687\) 1.46273e6 0.118242
\(688\) 639893. 0.0515390
\(689\) 1.36881e7 1.09849
\(690\) −1.93154e7 −1.54447
\(691\) 2.33550e7 1.86074 0.930368 0.366627i \(-0.119487\pi\)
0.930368 + 0.366627i \(0.119487\pi\)
\(692\) 5.80365e6 0.460719
\(693\) −261606. −0.0206926
\(694\) 1.51490e7 1.19395
\(695\) −7.52668e6 −0.591073
\(696\) −2.17969e6 −0.170558
\(697\) −9.19522e6 −0.716936
\(698\) 1.48479e7 1.15353
\(699\) 1.31222e7 1.01581
\(700\) −8.46503e6 −0.652955
\(701\) 1.15300e7 0.886206 0.443103 0.896471i \(-0.353878\pi\)
0.443103 + 0.896471i \(0.353878\pi\)
\(702\) 5.50413e6 0.421547
\(703\) −8.75110e6 −0.667843
\(704\) −5.22040e6 −0.396984
\(705\) 3.84832e6 0.291608
\(706\) 3.49540e7 2.63928
\(707\) 2.00696e6 0.151005
\(708\) −1.41057e7 −1.05757
\(709\) −1.13035e7 −0.844497 −0.422248 0.906480i \(-0.638759\pi\)
−0.422248 + 0.906480i \(0.638759\pi\)
\(710\) 5.21631e7 3.88344
\(711\) 2.21962e6 0.164667
\(712\) 2.63407e6 0.194727
\(713\) −981872. −0.0723321
\(714\) 993197. 0.0729105
\(715\) −1.17630e7 −0.860501
\(716\) 2.27164e7 1.65599
\(717\) 5.24228e6 0.380822
\(718\) −9.44276e6 −0.683578
\(719\) 406909. 0.0293545 0.0146773 0.999892i \(-0.495328\pi\)
0.0146773 + 0.999892i \(0.495328\pi\)
\(720\) 7.05996e6 0.507541
\(721\) −4.69934e6 −0.336665
\(722\) 1.85234e6 0.132245
\(723\) −3.91108e6 −0.278260
\(724\) −2.11020e7 −1.49616
\(725\) −4.32268e7 −3.05428
\(726\) 1.09973e6 0.0774364
\(727\) 1.93239e6 0.135600 0.0678000 0.997699i \(-0.478402\pi\)
0.0678000 + 0.997699i \(0.478402\pi\)
\(728\) 1.13947e6 0.0796846
\(729\) 531441. 0.0370370
\(730\) 2.14124e7 1.48716
\(731\) 390818. 0.0270509
\(732\) 1.01632e7 0.701054
\(733\) 357394. 0.0245690 0.0122845 0.999925i \(-0.496090\pi\)
0.0122845 + 0.999925i \(0.496090\pi\)
\(734\) −1.10102e7 −0.754320
\(735\) −1.55655e7 −1.06279
\(736\) 1.98126e7 1.34818
\(737\) 3.13273e6 0.212449
\(738\) −1.25481e7 −0.848082
\(739\) 2.22117e7 1.49613 0.748066 0.663624i \(-0.230983\pi\)
0.748066 + 0.663624i \(0.230983\pi\)
\(740\) −2.35845e7 −1.58324
\(741\) 1.22243e7 0.817856
\(742\) 3.37059e6 0.224748
\(743\) −2.10894e7 −1.40149 −0.700747 0.713410i \(-0.747150\pi\)
−0.700747 + 0.713410i \(0.747150\pi\)
\(744\) −174257. −0.0115414
\(745\) −3.52215e7 −2.32497
\(746\) −6192.20 −0.000407378 0
\(747\) 1.94856e6 0.127765
\(748\) −2.25704e6 −0.147498
\(749\) 3.33550e6 0.217248
\(750\) −4.27588e7 −2.77570
\(751\) −3.08277e7 −1.99453 −0.997267 0.0738774i \(-0.976463\pi\)
−0.997267 + 0.0738774i \(0.976463\pi\)
\(752\) −3.22746e6 −0.208121
\(753\) 8.53366e6 0.548464
\(754\) 3.87505e7 2.48227
\(755\) −55860.2 −0.00356644
\(756\) 732686. 0.0466244
\(757\) 3.26795e6 0.207270 0.103635 0.994615i \(-0.466953\pi\)
0.103635 + 0.994615i \(0.466953\pi\)
\(758\) 4.73577e6 0.299376
\(759\) −2.60598e6 −0.164198
\(760\) −7.61327e6 −0.478120
\(761\) 1.01120e7 0.632961 0.316480 0.948599i \(-0.397499\pi\)
0.316480 + 0.948599i \(0.397499\pi\)
\(762\) 1.40279e7 0.875194
\(763\) −1.28125e6 −0.0796748
\(764\) 1.26990e7 0.787113
\(765\) 4.31191e6 0.266389
\(766\) −2.96137e7 −1.82356
\(767\) 3.76554e7 2.31121
\(768\) −5.27964e6 −0.322999
\(769\) 2.14392e7 1.30735 0.653676 0.756775i \(-0.273226\pi\)
0.653676 + 0.756775i \(0.273226\pi\)
\(770\) −2.89654e6 −0.176057
\(771\) −1.02680e7 −0.622088
\(772\) 1.60784e7 0.970955
\(773\) −1.89394e7 −1.14003 −0.570017 0.821633i \(-0.693063\pi\)
−0.570017 + 0.821633i \(0.693063\pi\)
\(774\) 533324. 0.0319992
\(775\) −3.45580e6 −0.206678
\(776\) 4.77490e6 0.284649
\(777\) 1.40020e6 0.0832030
\(778\) −3.41627e7 −2.02350
\(779\) −2.78684e7 −1.64539
\(780\) 3.29447e7 1.93887
\(781\) 7.03772e6 0.412862
\(782\) 9.89370e6 0.578551
\(783\) 3.74148e6 0.218091
\(784\) 1.30543e7 0.758513
\(785\) 1.28381e7 0.743580
\(786\) −2.73114e7 −1.57684
\(787\) −1.10351e7 −0.635096 −0.317548 0.948242i \(-0.602859\pi\)
−0.317548 + 0.948242i \(0.602859\pi\)
\(788\) 1.45929e7 0.837195
\(789\) −2.52824e6 −0.144586
\(790\) 2.45760e7 1.40102
\(791\) −1.03980e6 −0.0590890
\(792\) −462495. −0.0261996
\(793\) −2.71308e7 −1.53207
\(794\) −1.28118e6 −0.0721208
\(795\) 1.46332e7 0.821149
\(796\) 2.12523e7 1.18884
\(797\) −1.22483e7 −0.683015 −0.341507 0.939879i \(-0.610937\pi\)
−0.341507 + 0.939879i \(0.610937\pi\)
\(798\) 3.01013e6 0.167332
\(799\) −1.97119e6 −0.109235
\(800\) 6.97327e7 3.85223
\(801\) −4.52142e6 −0.248997
\(802\) −3.95949e6 −0.217372
\(803\) 2.88891e6 0.158105
\(804\) −8.77390e6 −0.478688
\(805\) 6.86380e6 0.373315
\(806\) 3.09794e6 0.167971
\(807\) −9.78829e6 −0.529082
\(808\) 3.54811e6 0.191192
\(809\) −2.42528e7 −1.30284 −0.651421 0.758717i \(-0.725827\pi\)
−0.651421 + 0.758717i \(0.725827\pi\)
\(810\) 5.88418e6 0.315118
\(811\) 3.64893e7 1.94811 0.974055 0.226311i \(-0.0726664\pi\)
0.974055 + 0.226311i \(0.0726664\pi\)
\(812\) 5.15829e6 0.274546
\(813\) −1.06136e7 −0.563166
\(814\) −5.88613e6 −0.311364
\(815\) −3.37435e7 −1.77949
\(816\) −3.61625e6 −0.190122
\(817\) 1.18447e6 0.0620825
\(818\) −2.38064e7 −1.24397
\(819\) −1.95592e6 −0.101892
\(820\) −7.51062e7 −3.90069
\(821\) 2.40297e7 1.24420 0.622100 0.782938i \(-0.286280\pi\)
0.622100 + 0.782938i \(0.286280\pi\)
\(822\) −5.55857e6 −0.286935
\(823\) −1.16283e7 −0.598432 −0.299216 0.954185i \(-0.596725\pi\)
−0.299216 + 0.954185i \(0.596725\pi\)
\(824\) −8.30798e6 −0.426263
\(825\) −9.17204e6 −0.469171
\(826\) 9.27236e6 0.472868
\(827\) 1.01425e7 0.515680 0.257840 0.966188i \(-0.416989\pi\)
0.257840 + 0.966188i \(0.416989\pi\)
\(828\) 7.29863e6 0.369969
\(829\) 1.51928e7 0.767804 0.383902 0.923374i \(-0.374580\pi\)
0.383902 + 0.923374i \(0.374580\pi\)
\(830\) 2.15748e7 1.08705
\(831\) 5.59221e6 0.280919
\(832\) −3.90307e7 −1.95478
\(833\) 7.97298e6 0.398114
\(834\) 5.26110e6 0.261916
\(835\) −3.02902e7 −1.50344
\(836\) −6.84052e6 −0.338512
\(837\) 299115. 0.0147579
\(838\) −5.52426e7 −2.71747
\(839\) −9.62729e6 −0.472171 −0.236085 0.971732i \(-0.575864\pi\)
−0.236085 + 0.971732i \(0.575864\pi\)
\(840\) 1.21815e6 0.0595665
\(841\) 5.82977e6 0.284225
\(842\) −1.39860e7 −0.679849
\(843\) 2.19590e6 0.106425
\(844\) 1.86557e7 0.901477
\(845\) −4.80478e7 −2.31490
\(846\) −2.68995e6 −0.129217
\(847\) −390795. −0.0187172
\(848\) −1.22724e7 −0.586056
\(849\) 1.66121e7 0.790962
\(850\) 3.48219e7 1.65313
\(851\) 1.39481e7 0.660223
\(852\) −1.97107e7 −0.930256
\(853\) 7.49543e6 0.352715 0.176358 0.984326i \(-0.443568\pi\)
0.176358 + 0.984326i \(0.443568\pi\)
\(854\) −6.68076e6 −0.313459
\(855\) 1.30683e7 0.611370
\(856\) 5.89684e6 0.275065
\(857\) −2.69493e6 −0.125342 −0.0626708 0.998034i \(-0.519962\pi\)
−0.0626708 + 0.998034i \(0.519962\pi\)
\(858\) 8.22222e6 0.381304
\(859\) −3.28791e7 −1.52033 −0.760164 0.649731i \(-0.774882\pi\)
−0.760164 + 0.649731i \(0.774882\pi\)
\(860\) 3.19218e6 0.147178
\(861\) 4.45904e6 0.204990
\(862\) 3.78784e7 1.73629
\(863\) 3.03331e7 1.38640 0.693202 0.720744i \(-0.256199\pi\)
0.693202 + 0.720744i \(0.256199\pi\)
\(864\) −6.03568e6 −0.275069
\(865\) −1.65627e7 −0.752648
\(866\) −3.16815e6 −0.143553
\(867\) 1.05701e7 0.477562
\(868\) 412383. 0.0185781
\(869\) 3.31574e6 0.148947
\(870\) 4.14261e7 1.85556
\(871\) 2.34221e7 1.04612
\(872\) −2.26512e6 −0.100879
\(873\) −8.19621e6 −0.363980
\(874\) 2.99853e7 1.32779
\(875\) 1.51945e7 0.670914
\(876\) −8.09103e6 −0.356241
\(877\) −3.81826e7 −1.67636 −0.838178 0.545396i \(-0.816379\pi\)
−0.838178 + 0.545396i \(0.816379\pi\)
\(878\) 1.65064e7 0.722628
\(879\) 1.21409e7 0.530005
\(880\) 1.05464e7 0.459088
\(881\) −8.09402e6 −0.351338 −0.175669 0.984449i \(-0.556209\pi\)
−0.175669 + 0.984449i \(0.556209\pi\)
\(882\) 1.08802e7 0.470940
\(883\) 1.90914e7 0.824018 0.412009 0.911180i \(-0.364827\pi\)
0.412009 + 0.911180i \(0.364827\pi\)
\(884\) −1.68749e7 −0.726292
\(885\) 4.02554e7 1.72769
\(886\) −2.99095e7 −1.28004
\(887\) 1.40640e7 0.600205 0.300102 0.953907i \(-0.402979\pi\)
0.300102 + 0.953907i \(0.402979\pi\)
\(888\) 2.47543e6 0.105346
\(889\) −4.98487e6 −0.211543
\(890\) −5.00618e7 −2.11851
\(891\) 793881. 0.0335013
\(892\) 4.65091e7 1.95716
\(893\) −5.97417e6 −0.250697
\(894\) 2.46196e7 1.03024
\(895\) −6.48292e7 −2.70529
\(896\) −2.53929e6 −0.105668
\(897\) −1.94838e7 −0.808525
\(898\) 8.82452e6 0.365174
\(899\) 2.10585e6 0.0869015
\(900\) 2.56883e7 1.05713
\(901\) −7.49542e6 −0.307598
\(902\) −1.87447e7 −0.767119
\(903\) −189519. −0.00773453
\(904\) −1.83826e6 −0.0748145
\(905\) 6.02220e7 2.44418
\(906\) 39045.9 0.00158036
\(907\) 1.37464e7 0.554845 0.277423 0.960748i \(-0.410520\pi\)
0.277423 + 0.960748i \(0.410520\pi\)
\(908\) −3.02053e7 −1.21582
\(909\) −6.09040e6 −0.244476
\(910\) −2.16562e7 −0.866920
\(911\) 3.83931e7 1.53270 0.766350 0.642423i \(-0.222071\pi\)
0.766350 + 0.642423i \(0.222071\pi\)
\(912\) −1.09600e7 −0.436336
\(913\) 2.91082e6 0.115568
\(914\) −1.16021e7 −0.459381
\(915\) −2.90041e7 −1.14527
\(916\) −6.11977e6 −0.240988
\(917\) 9.70523e6 0.381138
\(918\) −3.01399e6 −0.118042
\(919\) 1.85935e7 0.726227 0.363114 0.931745i \(-0.381714\pi\)
0.363114 + 0.931745i \(0.381714\pi\)
\(920\) 1.21345e7 0.472665
\(921\) 1.08491e7 0.421448
\(922\) 9.12710e6 0.353595
\(923\) 5.26181e7 2.03297
\(924\) 1.09451e6 0.0421734
\(925\) 4.90918e7 1.88649
\(926\) −1.62654e7 −0.623359
\(927\) 1.42608e7 0.545060
\(928\) −4.24927e7 −1.61974
\(929\) −2.77987e7 −1.05678 −0.528391 0.849001i \(-0.677204\pi\)
−0.528391 + 0.849001i \(0.677204\pi\)
\(930\) 3.31184e6 0.125563
\(931\) 2.41641e7 0.913684
\(932\) −5.49004e7 −2.07031
\(933\) −5.73187e6 −0.215572
\(934\) 3.91437e7 1.46823
\(935\) 6.44125e6 0.240958
\(936\) −3.45788e6 −0.129009
\(937\) 4.55794e7 1.69598 0.847988 0.530015i \(-0.177814\pi\)
0.847988 + 0.530015i \(0.177814\pi\)
\(938\) 5.76752e6 0.214034
\(939\) −1.67873e7 −0.621322
\(940\) −1.61006e7 −0.594322
\(941\) −2.38944e7 −0.879676 −0.439838 0.898077i \(-0.644964\pi\)
−0.439838 + 0.898077i \(0.644964\pi\)
\(942\) −8.97377e6 −0.329494
\(943\) 4.44186e7 1.62662
\(944\) −3.37609e7 −1.23306
\(945\) −2.09097e6 −0.0761674
\(946\) 796694. 0.0289443
\(947\) 3.52637e7 1.27777 0.638885 0.769302i \(-0.279396\pi\)
0.638885 + 0.769302i \(0.279396\pi\)
\(948\) −9.28644e6 −0.335605
\(949\) 2.15992e7 0.778523
\(950\) 1.05537e8 3.79397
\(951\) −2.11875e7 −0.759678
\(952\) −623959. −0.0223133
\(953\) 5.08550e7 1.81385 0.906924 0.421293i \(-0.138424\pi\)
0.906924 + 0.421293i \(0.138424\pi\)
\(954\) −1.02285e7 −0.363866
\(955\) −3.62411e7 −1.28586
\(956\) −2.19326e7 −0.776149
\(957\) 5.58912e6 0.197271
\(958\) −3.46836e7 −1.22098
\(959\) 1.97526e6 0.0693551
\(960\) −4.17258e7 −1.46126
\(961\) −2.84608e7 −0.994120
\(962\) −4.40081e7 −1.53319
\(963\) −1.01220e7 −0.351724
\(964\) 1.63631e7 0.567119
\(965\) −4.58853e7 −1.58619
\(966\) −4.79775e6 −0.165423
\(967\) 2.33318e7 0.802384 0.401192 0.915994i \(-0.368596\pi\)
0.401192 + 0.915994i \(0.368596\pi\)
\(968\) −690887. −0.0236984
\(969\) −6.69385e6 −0.229016
\(970\) −9.07495e7 −3.09681
\(971\) −2.99725e7 −1.02017 −0.510087 0.860123i \(-0.670387\pi\)
−0.510087 + 0.860123i \(0.670387\pi\)
\(972\) −2.22344e6 −0.0754847
\(973\) −1.86956e6 −0.0633077
\(974\) 1.97999e7 0.668754
\(975\) −6.85755e7 −2.31024
\(976\) 2.43248e7 0.817380
\(977\) 3.06238e7 1.02642 0.513208 0.858264i \(-0.328457\pi\)
0.513208 + 0.858264i \(0.328457\pi\)
\(978\) 2.35865e7 0.788525
\(979\) −6.75422e6 −0.225226
\(980\) 6.51229e7 2.16605
\(981\) 3.88812e6 0.128993
\(982\) 9.91471e6 0.328096
\(983\) 2.10228e7 0.693915 0.346958 0.937881i \(-0.387215\pi\)
0.346958 + 0.937881i \(0.387215\pi\)
\(984\) 7.88315e6 0.259545
\(985\) −4.16460e7 −1.36767
\(986\) −2.12193e7 −0.695085
\(987\) 955887. 0.0312330
\(988\) −5.11437e7 −1.66686
\(989\) −1.88789e6 −0.0613742
\(990\) 8.78995e6 0.285035
\(991\) −2.43651e7 −0.788103 −0.394052 0.919088i \(-0.628927\pi\)
−0.394052 + 0.919088i \(0.628927\pi\)
\(992\) −3.39711e6 −0.109605
\(993\) −1.78098e7 −0.573175
\(994\) 1.29568e7 0.415941
\(995\) −6.06508e7 −1.94213
\(996\) −8.15238e6 −0.260397
\(997\) −1.04094e7 −0.331655 −0.165827 0.986155i \(-0.553029\pi\)
−0.165827 + 0.986155i \(0.553029\pi\)
\(998\) −5.92971e6 −0.188455
\(999\) −4.24911e6 −0.134705
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.1.1 2
3.2 odd 2 99.6.a.e.1.2 2
4.3 odd 2 528.6.a.q.1.1 2
5.4 even 2 825.6.a.d.1.2 2
11.10 odd 2 363.6.a.g.1.2 2
33.32 even 2 1089.6.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.d.1.1 2 1.1 even 1 trivial
99.6.a.e.1.2 2 3.2 odd 2
363.6.a.g.1.2 2 11.10 odd 2
528.6.a.q.1.1 2 4.3 odd 2
825.6.a.d.1.2 2 5.4 even 2
1089.6.a.o.1.1 2 33.32 even 2