Properties

Label 69.6.a.e.1.2
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 113 x^{3} - 257 x^{2} + 1404 x + 2197\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(11.0973\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-3.54286 q^{2} +9.00000 q^{3} -19.4482 q^{4} -92.1306 q^{5} -31.8857 q^{6} -8.98894 q^{7} +182.273 q^{8} +81.0000 q^{9} +326.406 q^{10} +612.746 q^{11} -175.034 q^{12} -496.413 q^{13} +31.8465 q^{14} -829.175 q^{15} -23.4274 q^{16} +745.481 q^{17} -286.971 q^{18} +1299.52 q^{19} +1791.77 q^{20} -80.9005 q^{21} -2170.87 q^{22} -529.000 q^{23} +1640.46 q^{24} +5363.05 q^{25} +1758.72 q^{26} +729.000 q^{27} +174.818 q^{28} +7754.82 q^{29} +2937.65 q^{30} -1066.62 q^{31} -5749.75 q^{32} +5514.72 q^{33} -2641.13 q^{34} +828.157 q^{35} -1575.30 q^{36} -840.279 q^{37} -4604.02 q^{38} -4467.72 q^{39} -16793.0 q^{40} +9821.00 q^{41} +286.619 q^{42} -13529.8 q^{43} -11916.8 q^{44} -7462.58 q^{45} +1874.17 q^{46} +20485.1 q^{47} -210.846 q^{48} -16726.2 q^{49} -19000.5 q^{50} +6709.33 q^{51} +9654.33 q^{52} +4816.85 q^{53} -2582.74 q^{54} -56452.7 q^{55} -1638.45 q^{56} +11695.7 q^{57} -27474.2 q^{58} +30528.3 q^{59} +16125.9 q^{60} +9.53050 q^{61} +3778.88 q^{62} -728.104 q^{63} +21120.2 q^{64} +45734.8 q^{65} -19537.8 q^{66} +9996.72 q^{67} -14498.2 q^{68} -4761.00 q^{69} -2934.04 q^{70} -73725.9 q^{71} +14764.2 q^{72} +55169.6 q^{73} +2976.99 q^{74} +48267.4 q^{75} -25273.3 q^{76} -5507.94 q^{77} +15828.5 q^{78} +62397.4 q^{79} +2158.38 q^{80} +6561.00 q^{81} -34794.4 q^{82} -82435.2 q^{83} +1573.37 q^{84} -68681.6 q^{85} +47934.2 q^{86} +69793.4 q^{87} +111687. q^{88} +95173.0 q^{89} +26438.8 q^{90} +4462.23 q^{91} +10288.1 q^{92} -9599.57 q^{93} -72575.8 q^{94} -119726. q^{95} -51747.8 q^{96} +47134.4 q^{97} +59258.5 q^{98} +49632.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} + O(q^{10}) \) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.54286 −0.626294 −0.313147 0.949705i \(-0.601383\pi\)
−0.313147 + 0.949705i \(0.601383\pi\)
\(3\) 9.00000 0.577350
\(4\) −19.4482 −0.607755
\(5\) −92.1306 −1.64808 −0.824041 0.566530i \(-0.808286\pi\)
−0.824041 + 0.566530i \(0.808286\pi\)
\(6\) −31.8857 −0.361591
\(7\) −8.98894 −0.0693368 −0.0346684 0.999399i \(-0.511038\pi\)
−0.0346684 + 0.999399i \(0.511038\pi\)
\(8\) 182.273 1.00693
\(9\) 81.0000 0.333333
\(10\) 326.406 1.03218
\(11\) 612.746 1.52686 0.763430 0.645891i \(-0.223514\pi\)
0.763430 + 0.645891i \(0.223514\pi\)
\(12\) −175.034 −0.350888
\(13\) −496.413 −0.814676 −0.407338 0.913278i \(-0.633543\pi\)
−0.407338 + 0.913278i \(0.633543\pi\)
\(14\) 31.8465 0.0434252
\(15\) −829.175 −0.951521
\(16\) −23.4274 −0.0228783
\(17\) 745.481 0.625625 0.312813 0.949815i \(-0.398729\pi\)
0.312813 + 0.949815i \(0.398729\pi\)
\(18\) −286.971 −0.208765
\(19\) 1299.52 0.825847 0.412924 0.910766i \(-0.364508\pi\)
0.412924 + 0.910766i \(0.364508\pi\)
\(20\) 1791.77 1.00163
\(21\) −80.9005 −0.0400316
\(22\) −2170.87 −0.956263
\(23\) −529.000 −0.208514
\(24\) 1640.46 0.581350
\(25\) 5363.05 1.71618
\(26\) 1758.72 0.510227
\(27\) 729.000 0.192450
\(28\) 174.818 0.0421398
\(29\) 7754.82 1.71229 0.856144 0.516738i \(-0.172854\pi\)
0.856144 + 0.516738i \(0.172854\pi\)
\(30\) 2937.65 0.595932
\(31\) −1066.62 −0.199345 −0.0996724 0.995020i \(-0.531779\pi\)
−0.0996724 + 0.995020i \(0.531779\pi\)
\(32\) −5749.75 −0.992600
\(33\) 5514.72 0.881533
\(34\) −2641.13 −0.391826
\(35\) 828.157 0.114273
\(36\) −1575.30 −0.202585
\(37\) −840.279 −0.100907 −0.0504533 0.998726i \(-0.516067\pi\)
−0.0504533 + 0.998726i \(0.516067\pi\)
\(38\) −4604.02 −0.517224
\(39\) −4467.72 −0.470353
\(40\) −16793.0 −1.65950
\(41\) 9821.00 0.912422 0.456211 0.889872i \(-0.349206\pi\)
0.456211 + 0.889872i \(0.349206\pi\)
\(42\) 286.619 0.0250716
\(43\) −13529.8 −1.11589 −0.557944 0.829879i \(-0.688410\pi\)
−0.557944 + 0.829879i \(0.688410\pi\)
\(44\) −11916.8 −0.927957
\(45\) −7462.58 −0.549361
\(46\) 1874.17 0.130591
\(47\) 20485.1 1.35268 0.676338 0.736591i \(-0.263566\pi\)
0.676338 + 0.736591i \(0.263566\pi\)
\(48\) −210.846 −0.0132088
\(49\) −16726.2 −0.995192
\(50\) −19000.5 −1.07483
\(51\) 6709.33 0.361205
\(52\) 9654.33 0.495124
\(53\) 4816.85 0.235545 0.117772 0.993041i \(-0.462425\pi\)
0.117772 + 0.993041i \(0.462425\pi\)
\(54\) −2582.74 −0.120530
\(55\) −56452.7 −2.51639
\(56\) −1638.45 −0.0698171
\(57\) 11695.7 0.476803
\(58\) −27474.2 −1.07240
\(59\) 30528.3 1.14175 0.570877 0.821036i \(-0.306603\pi\)
0.570877 + 0.821036i \(0.306603\pi\)
\(60\) 16125.9 0.578292
\(61\) 9.53050 0.000327937 0 0.000163969 1.00000i \(-0.499948\pi\)
0.000163969 1.00000i \(0.499948\pi\)
\(62\) 3778.88 0.124849
\(63\) −728.104 −0.0231123
\(64\) 21120.2 0.644538
\(65\) 45734.8 1.34265
\(66\) −19537.8 −0.552099
\(67\) 9996.72 0.272064 0.136032 0.990704i \(-0.456565\pi\)
0.136032 + 0.990704i \(0.456565\pi\)
\(68\) −14498.2 −0.380227
\(69\) −4761.00 −0.120386
\(70\) −2934.04 −0.0715684
\(71\) −73725.9 −1.73570 −0.867849 0.496828i \(-0.834498\pi\)
−0.867849 + 0.496828i \(0.834498\pi\)
\(72\) 14764.2 0.335643
\(73\) 55169.6 1.21169 0.605847 0.795581i \(-0.292834\pi\)
0.605847 + 0.795581i \(0.292834\pi\)
\(74\) 2976.99 0.0631972
\(75\) 48267.4 0.990834
\(76\) −25273.3 −0.501913
\(77\) −5507.94 −0.105867
\(78\) 15828.5 0.294580
\(79\) 62397.4 1.12486 0.562431 0.826844i \(-0.309866\pi\)
0.562431 + 0.826844i \(0.309866\pi\)
\(80\) 2158.38 0.0377053
\(81\) 6561.00 0.111111
\(82\) −34794.4 −0.571445
\(83\) −82435.2 −1.31346 −0.656731 0.754125i \(-0.728061\pi\)
−0.656731 + 0.754125i \(0.728061\pi\)
\(84\) 1573.37 0.0243294
\(85\) −68681.6 −1.03108
\(86\) 47934.2 0.698875
\(87\) 69793.4 0.988589
\(88\) 111687. 1.53744
\(89\) 95173.0 1.27362 0.636808 0.771022i \(-0.280254\pi\)
0.636808 + 0.771022i \(0.280254\pi\)
\(90\) 26438.8 0.344062
\(91\) 4462.23 0.0564870
\(92\) 10288.1 0.126726
\(93\) −9599.57 −0.115092
\(94\) −72575.8 −0.847174
\(95\) −119726. −1.36106
\(96\) −51747.8 −0.573078
\(97\) 47134.4 0.508638 0.254319 0.967120i \(-0.418149\pi\)
0.254319 + 0.967120i \(0.418149\pi\)
\(98\) 59258.5 0.623283
\(99\) 49632.4 0.508953
\(100\) −104301. −1.04301
\(101\) −15830.2 −0.154413 −0.0772065 0.997015i \(-0.524600\pi\)
−0.0772065 + 0.997015i \(0.524600\pi\)
\(102\) −23770.2 −0.226221
\(103\) −90782.5 −0.843158 −0.421579 0.906792i \(-0.638524\pi\)
−0.421579 + 0.906792i \(0.638524\pi\)
\(104\) −90482.9 −0.820320
\(105\) 7453.41 0.0659754
\(106\) −17065.4 −0.147520
\(107\) 209377. 1.76795 0.883973 0.467538i \(-0.154859\pi\)
0.883973 + 0.467538i \(0.154859\pi\)
\(108\) −14177.7 −0.116963
\(109\) 40116.0 0.323408 0.161704 0.986839i \(-0.448301\pi\)
0.161704 + 0.986839i \(0.448301\pi\)
\(110\) 200004. 1.57600
\(111\) −7562.51 −0.0582584
\(112\) 210.587 0.00158631
\(113\) 248398. 1.83000 0.915001 0.403453i \(-0.132190\pi\)
0.915001 + 0.403453i \(0.132190\pi\)
\(114\) −41436.2 −0.298619
\(115\) 48737.1 0.343649
\(116\) −150817. −1.04065
\(117\) −40209.5 −0.271559
\(118\) −108157. −0.715074
\(119\) −6701.09 −0.0433788
\(120\) −151137. −0.958113
\(121\) 214407. 1.33130
\(122\) −33.7652 −0.000205385 0
\(123\) 88389.0 0.526787
\(124\) 20743.8 0.121153
\(125\) −206193. −1.18032
\(126\) 2579.57 0.0144751
\(127\) 128385. 0.706328 0.353164 0.935561i \(-0.385106\pi\)
0.353164 + 0.935561i \(0.385106\pi\)
\(128\) 109166. 0.588929
\(129\) −121768. −0.644258
\(130\) −162032. −0.840896
\(131\) −361743. −1.84171 −0.920857 0.389901i \(-0.872509\pi\)
−0.920857 + 0.389901i \(0.872509\pi\)
\(132\) −107251. −0.535756
\(133\) −11681.3 −0.0572616
\(134\) −35416.9 −0.170392
\(135\) −67163.2 −0.317174
\(136\) 135881. 0.629960
\(137\) −116630. −0.530897 −0.265448 0.964125i \(-0.585520\pi\)
−0.265448 + 0.964125i \(0.585520\pi\)
\(138\) 16867.5 0.0753970
\(139\) 94540.9 0.415033 0.207517 0.978231i \(-0.433462\pi\)
0.207517 + 0.978231i \(0.433462\pi\)
\(140\) −16106.1 −0.0694498
\(141\) 184366. 0.780968
\(142\) 261200. 1.08706
\(143\) −304175. −1.24390
\(144\) −1897.62 −0.00762610
\(145\) −714456. −2.82199
\(146\) −195458. −0.758877
\(147\) −150536. −0.574575
\(148\) 16341.9 0.0613265
\(149\) 275055. 1.01497 0.507486 0.861660i \(-0.330575\pi\)
0.507486 + 0.861660i \(0.330575\pi\)
\(150\) −171005. −0.620554
\(151\) −47405.4 −0.169194 −0.0845971 0.996415i \(-0.526960\pi\)
−0.0845971 + 0.996415i \(0.526960\pi\)
\(152\) 236868. 0.831569
\(153\) 60383.9 0.208542
\(154\) 19513.8 0.0663042
\(155\) 98268.2 0.328537
\(156\) 86888.9 0.285860
\(157\) 88519.4 0.286609 0.143304 0.989679i \(-0.454227\pi\)
0.143304 + 0.989679i \(0.454227\pi\)
\(158\) −221065. −0.704494
\(159\) 43351.7 0.135992
\(160\) 529728. 1.63589
\(161\) 4755.15 0.0144577
\(162\) −23244.7 −0.0695883
\(163\) 522220. 1.53952 0.769759 0.638335i \(-0.220377\pi\)
0.769759 + 0.638335i \(0.220377\pi\)
\(164\) −191000. −0.554530
\(165\) −508074. −1.45284
\(166\) 292056. 0.822614
\(167\) −234704. −0.651223 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(168\) −14746.0 −0.0403089
\(169\) −124867. −0.336303
\(170\) 243329. 0.645761
\(171\) 105261. 0.275282
\(172\) 263130. 0.678187
\(173\) −249399. −0.633547 −0.316773 0.948501i \(-0.602599\pi\)
−0.316773 + 0.948501i \(0.602599\pi\)
\(174\) −247268. −0.619148
\(175\) −48208.1 −0.118994
\(176\) −14355.0 −0.0349319
\(177\) 274755. 0.659192
\(178\) −337184. −0.797659
\(179\) 353019. 0.823503 0.411751 0.911296i \(-0.364917\pi\)
0.411751 + 0.911296i \(0.364917\pi\)
\(180\) 145133. 0.333877
\(181\) −188804. −0.428367 −0.214183 0.976793i \(-0.568709\pi\)
−0.214183 + 0.976793i \(0.568709\pi\)
\(182\) −15809.0 −0.0353775
\(183\) 85.7745 0.000189335 0
\(184\) −96422.7 −0.209959
\(185\) 77415.4 0.166302
\(186\) 34009.9 0.0720813
\(187\) 456791. 0.955241
\(188\) −398398. −0.822096
\(189\) −6552.94 −0.0133439
\(190\) 424171. 0.852427
\(191\) 374966. 0.743719 0.371859 0.928289i \(-0.378720\pi\)
0.371859 + 0.928289i \(0.378720\pi\)
\(192\) 190082. 0.372124
\(193\) −761894. −1.47232 −0.736158 0.676810i \(-0.763362\pi\)
−0.736158 + 0.676810i \(0.763362\pi\)
\(194\) −166991. −0.318557
\(195\) 411614. 0.775181
\(196\) 325294. 0.604833
\(197\) −772963. −1.41904 −0.709518 0.704688i \(-0.751087\pi\)
−0.709518 + 0.704688i \(0.751087\pi\)
\(198\) −175841. −0.318754
\(199\) 863431. 1.54559 0.772796 0.634654i \(-0.218857\pi\)
0.772796 + 0.634654i \(0.218857\pi\)
\(200\) 977541. 1.72807
\(201\) 89970.5 0.157076
\(202\) 56084.2 0.0967080
\(203\) −69707.6 −0.118724
\(204\) −130484. −0.219524
\(205\) −904814. −1.50375
\(206\) 321629. 0.528065
\(207\) −42849.0 −0.0695048
\(208\) 11629.7 0.0186384
\(209\) 796278. 1.26095
\(210\) −26406.4 −0.0413200
\(211\) 973725. 1.50567 0.752835 0.658209i \(-0.228686\pi\)
0.752835 + 0.658209i \(0.228686\pi\)
\(212\) −93679.0 −0.143154
\(213\) −663533. −1.00211
\(214\) −741792. −1.10725
\(215\) 1.24651e6 1.83908
\(216\) 132877. 0.193783
\(217\) 9587.78 0.0138219
\(218\) −142125. −0.202549
\(219\) 496527. 0.699572
\(220\) 1.09790e6 1.52935
\(221\) −370066. −0.509682
\(222\) 26792.9 0.0364869
\(223\) −1.07660e6 −1.44974 −0.724870 0.688885i \(-0.758100\pi\)
−0.724870 + 0.688885i \(0.758100\pi\)
\(224\) 51684.2 0.0688236
\(225\) 434407. 0.572059
\(226\) −880037. −1.14612
\(227\) −313794. −0.404185 −0.202092 0.979366i \(-0.564774\pi\)
−0.202092 + 0.979366i \(0.564774\pi\)
\(228\) −227460. −0.289780
\(229\) 1.11820e6 1.40907 0.704534 0.709671i \(-0.251156\pi\)
0.704534 + 0.709671i \(0.251156\pi\)
\(230\) −172669. −0.215225
\(231\) −49571.5 −0.0611226
\(232\) 1.41350e6 1.72415
\(233\) −562380. −0.678641 −0.339321 0.940671i \(-0.610197\pi\)
−0.339321 + 0.940671i \(0.610197\pi\)
\(234\) 142456. 0.170076
\(235\) −1.88731e6 −2.22932
\(236\) −593719. −0.693907
\(237\) 561577. 0.649439
\(238\) 23741.0 0.0271679
\(239\) −560026. −0.634181 −0.317090 0.948395i \(-0.602706\pi\)
−0.317090 + 0.948395i \(0.602706\pi\)
\(240\) 19425.4 0.0217692
\(241\) −1.62461e6 −1.80180 −0.900902 0.434023i \(-0.857093\pi\)
−0.900902 + 0.434023i \(0.857093\pi\)
\(242\) −759613. −0.833785
\(243\) 59049.0 0.0641500
\(244\) −185.351 −0.000199306 0
\(245\) 1.54099e6 1.64016
\(246\) −313149. −0.329924
\(247\) −645100. −0.672798
\(248\) −194416. −0.200726
\(249\) −741917. −0.758327
\(250\) 730511. 0.739226
\(251\) −1.09921e6 −1.10127 −0.550637 0.834745i \(-0.685615\pi\)
−0.550637 + 0.834745i \(0.685615\pi\)
\(252\) 14160.3 0.0140466
\(253\) −324143. −0.318372
\(254\) −454851. −0.442369
\(255\) −618134. −0.595295
\(256\) −1.06261e6 −1.01338
\(257\) −1.36203e6 −1.28633 −0.643167 0.765726i \(-0.722380\pi\)
−0.643167 + 0.765726i \(0.722380\pi\)
\(258\) 431408. 0.403495
\(259\) 7553.22 0.00699653
\(260\) −889459. −0.816004
\(261\) 628140. 0.570762
\(262\) 1.28160e6 1.15346
\(263\) −802156. −0.715104 −0.357552 0.933893i \(-0.616389\pi\)
−0.357552 + 0.933893i \(0.616389\pi\)
\(264\) 1.00519e6 0.887640
\(265\) −443780. −0.388197
\(266\) 41385.3 0.0358626
\(267\) 856557. 0.735323
\(268\) −194418. −0.165348
\(269\) −352123. −0.296698 −0.148349 0.988935i \(-0.547396\pi\)
−0.148349 + 0.988935i \(0.547396\pi\)
\(270\) 237950. 0.198644
\(271\) −1.51322e6 −1.25164 −0.625819 0.779968i \(-0.715235\pi\)
−0.625819 + 0.779968i \(0.715235\pi\)
\(272\) −17464.7 −0.0143132
\(273\) 40160.1 0.0326128
\(274\) 413205. 0.332498
\(275\) 3.28619e6 2.62036
\(276\) 92592.7 0.0731651
\(277\) −1.26262e6 −0.988720 −0.494360 0.869257i \(-0.664598\pi\)
−0.494360 + 0.869257i \(0.664598\pi\)
\(278\) −334945. −0.259933
\(279\) −86396.1 −0.0664483
\(280\) 150951. 0.115064
\(281\) 1.24571e6 0.941133 0.470567 0.882365i \(-0.344050\pi\)
0.470567 + 0.882365i \(0.344050\pi\)
\(282\) −653183. −0.489116
\(283\) 372353. 0.276368 0.138184 0.990407i \(-0.455873\pi\)
0.138184 + 0.990407i \(0.455873\pi\)
\(284\) 1.43383e6 1.05488
\(285\) −1.07753e6 −0.785811
\(286\) 1.07765e6 0.779045
\(287\) −88280.4 −0.0632644
\(288\) −465730. −0.330867
\(289\) −864115. −0.608593
\(290\) 2.53122e6 1.76740
\(291\) 424210. 0.293662
\(292\) −1.07295e6 −0.736413
\(293\) 1.74374e6 1.18662 0.593311 0.804974i \(-0.297821\pi\)
0.593311 + 0.804974i \(0.297821\pi\)
\(294\) 533327. 0.359853
\(295\) −2.81259e6 −1.88170
\(296\) −153161. −0.101606
\(297\) 446692. 0.293844
\(298\) −974480. −0.635671
\(299\) 262603. 0.169872
\(300\) −938713. −0.602185
\(301\) 121619. 0.0773721
\(302\) 167951. 0.105965
\(303\) −142472. −0.0891504
\(304\) −30444.4 −0.0188940
\(305\) −878.050 −0.000540468 0
\(306\) −213932. −0.130609
\(307\) −1.60710e6 −0.973188 −0.486594 0.873628i \(-0.661761\pi\)
−0.486594 + 0.873628i \(0.661761\pi\)
\(308\) 107119. 0.0643415
\(309\) −817042. −0.486797
\(310\) −348150. −0.205761
\(311\) 2.21364e6 1.29780 0.648898 0.760876i \(-0.275230\pi\)
0.648898 + 0.760876i \(0.275230\pi\)
\(312\) −814346. −0.473612
\(313\) 480737. 0.277362 0.138681 0.990337i \(-0.455714\pi\)
0.138681 + 0.990337i \(0.455714\pi\)
\(314\) −313612. −0.179501
\(315\) 67080.7 0.0380909
\(316\) −1.21352e6 −0.683640
\(317\) 3.23334e6 1.80719 0.903593 0.428393i \(-0.140920\pi\)
0.903593 + 0.428393i \(0.140920\pi\)
\(318\) −153589. −0.0851710
\(319\) 4.75174e6 2.61442
\(320\) −1.94582e6 −1.06225
\(321\) 1.88439e6 1.02072
\(322\) −16846.8 −0.00905479
\(323\) 968769. 0.516671
\(324\) −127599. −0.0675284
\(325\) −2.66229e6 −1.39813
\(326\) −1.85015e6 −0.964191
\(327\) 361044. 0.186720
\(328\) 1.79011e6 0.918744
\(329\) −184140. −0.0937902
\(330\) 1.80003e6 0.909905
\(331\) 2.61134e6 1.31007 0.655034 0.755600i \(-0.272654\pi\)
0.655034 + 0.755600i \(0.272654\pi\)
\(332\) 1.60321e6 0.798263
\(333\) −68062.6 −0.0336355
\(334\) 831523. 0.407857
\(335\) −921004. −0.448383
\(336\) 1895.29 0.000915855 0
\(337\) 2.76804e6 1.32769 0.663846 0.747870i \(-0.268923\pi\)
0.663846 + 0.747870i \(0.268923\pi\)
\(338\) 442386. 0.210625
\(339\) 2.23558e6 1.05655
\(340\) 1.33573e6 0.626645
\(341\) −653567. −0.304371
\(342\) −372926. −0.172408
\(343\) 301428. 0.138340
\(344\) −2.46613e6 −1.12362
\(345\) 438634. 0.198406
\(346\) 883583. 0.396787
\(347\) 1.39621e6 0.622484 0.311242 0.950331i \(-0.399255\pi\)
0.311242 + 0.950331i \(0.399255\pi\)
\(348\) −1.35735e6 −0.600820
\(349\) −1.00509e6 −0.441714 −0.220857 0.975306i \(-0.570885\pi\)
−0.220857 + 0.975306i \(0.570885\pi\)
\(350\) 170795. 0.0745253
\(351\) −361885. −0.156784
\(352\) −3.52314e6 −1.51556
\(353\) −4.24269e6 −1.81219 −0.906096 0.423073i \(-0.860952\pi\)
−0.906096 + 0.423073i \(0.860952\pi\)
\(354\) −973416. −0.412848
\(355\) 6.79241e6 2.86057
\(356\) −1.85094e6 −0.774047
\(357\) −60309.8 −0.0250448
\(358\) −1.25069e6 −0.515755
\(359\) 313836. 0.128519 0.0642595 0.997933i \(-0.479531\pi\)
0.0642595 + 0.997933i \(0.479531\pi\)
\(360\) −1.36023e6 −0.553167
\(361\) −787340. −0.317976
\(362\) 668907. 0.268284
\(363\) 1.92966e6 0.768626
\(364\) −86782.2 −0.0343303
\(365\) −5.08281e6 −1.99697
\(366\) −303.887 −0.000118579 0
\(367\) 290172. 0.112458 0.0562290 0.998418i \(-0.482092\pi\)
0.0562290 + 0.998418i \(0.482092\pi\)
\(368\) 12393.1 0.00477045
\(369\) 795501. 0.304141
\(370\) −274272. −0.104154
\(371\) −43298.4 −0.0163319
\(372\) 186694. 0.0699476
\(373\) −5.14109e6 −1.91330 −0.956649 0.291243i \(-0.905931\pi\)
−0.956649 + 0.291243i \(0.905931\pi\)
\(374\) −1.61834e6 −0.598262
\(375\) −1.85573e6 −0.681456
\(376\) 3.73389e6 1.36205
\(377\) −3.84959e6 −1.39496
\(378\) 23216.1 0.00835719
\(379\) −631911. −0.225974 −0.112987 0.993596i \(-0.536042\pi\)
−0.112987 + 0.993596i \(0.536042\pi\)
\(380\) 2.32845e6 0.827194
\(381\) 1.15547e6 0.407799
\(382\) −1.32845e6 −0.465787
\(383\) −118799. −0.0413823 −0.0206911 0.999786i \(-0.506587\pi\)
−0.0206911 + 0.999786i \(0.506587\pi\)
\(384\) 982495. 0.340018
\(385\) 507450. 0.174478
\(386\) 2.69928e6 0.922103
\(387\) −1.09591e6 −0.371963
\(388\) −916678. −0.309128
\(389\) 2.58745e6 0.866957 0.433478 0.901164i \(-0.357286\pi\)
0.433478 + 0.901164i \(0.357286\pi\)
\(390\) −1.45829e6 −0.485492
\(391\) −394359. −0.130452
\(392\) −3.04874e6 −1.00209
\(393\) −3.25569e6 −1.06331
\(394\) 2.73850e6 0.888734
\(395\) −5.74871e6 −1.85386
\(396\) −965260. −0.309319
\(397\) 3.56801e6 1.13619 0.568093 0.822964i \(-0.307681\pi\)
0.568093 + 0.822964i \(0.307681\pi\)
\(398\) −3.05901e6 −0.967996
\(399\) −105132. −0.0330600
\(400\) −125642. −0.0392632
\(401\) 4.28994e6 1.33226 0.666132 0.745833i \(-0.267949\pi\)
0.666132 + 0.745833i \(0.267949\pi\)
\(402\) −318753. −0.0983759
\(403\) 529483. 0.162401
\(404\) 307869. 0.0938453
\(405\) −604469. −0.183120
\(406\) 246964. 0.0743565
\(407\) −514878. −0.154070
\(408\) 1.22293e6 0.363707
\(409\) 3.14276e6 0.928973 0.464486 0.885580i \(-0.346239\pi\)
0.464486 + 0.885580i \(0.346239\pi\)
\(410\) 3.20563e6 0.941789
\(411\) −1.04967e6 −0.306513
\(412\) 1.76555e6 0.512434
\(413\) −274417. −0.0791655
\(414\) 151808. 0.0435305
\(415\) 7.59480e6 2.16469
\(416\) 2.85425e6 0.808647
\(417\) 850868. 0.239620
\(418\) −2.82110e6 −0.789728
\(419\) −257827. −0.0717454 −0.0358727 0.999356i \(-0.511421\pi\)
−0.0358727 + 0.999356i \(0.511421\pi\)
\(420\) −144955. −0.0400969
\(421\) −1.48632e6 −0.408701 −0.204351 0.978898i \(-0.565508\pi\)
−0.204351 + 0.978898i \(0.565508\pi\)
\(422\) −3.44977e6 −0.942993
\(423\) 1.65929e6 0.450892
\(424\) 877985. 0.237177
\(425\) 3.99805e6 1.07368
\(426\) 2.35080e6 0.627613
\(427\) −85.6691 −2.27381e−5 0
\(428\) −4.07199e6 −1.07448
\(429\) −2.73758e6 −0.718163
\(430\) −4.41621e6 −1.15180
\(431\) 4.98067e6 1.29150 0.645750 0.763549i \(-0.276545\pi\)
0.645750 + 0.763549i \(0.276545\pi\)
\(432\) −17078.6 −0.00440293
\(433\) 1.44742e6 0.371001 0.185500 0.982644i \(-0.440609\pi\)
0.185500 + 0.982644i \(0.440609\pi\)
\(434\) −33968.1 −0.00865659
\(435\) −6.43010e6 −1.62928
\(436\) −780182. −0.196553
\(437\) −687447. −0.172201
\(438\) −1.75912e6 −0.438138
\(439\) −4.31837e6 −1.06944 −0.534722 0.845028i \(-0.679584\pi\)
−0.534722 + 0.845028i \(0.679584\pi\)
\(440\) −1.02898e7 −2.53382
\(441\) −1.35482e6 −0.331731
\(442\) 1.31109e6 0.319211
\(443\) 5.65863e6 1.36994 0.684971 0.728570i \(-0.259815\pi\)
0.684971 + 0.728570i \(0.259815\pi\)
\(444\) 147077. 0.0354068
\(445\) −8.76834e6 −2.09902
\(446\) 3.81422e6 0.907965
\(447\) 2.47550e6 0.585994
\(448\) −189848. −0.0446902
\(449\) −1.72793e6 −0.404492 −0.202246 0.979335i \(-0.564824\pi\)
−0.202246 + 0.979335i \(0.564824\pi\)
\(450\) −1.53904e6 −0.358277
\(451\) 6.01778e6 1.39314
\(452\) −4.83088e6 −1.11219
\(453\) −426649. −0.0976843
\(454\) 1.11173e6 0.253139
\(455\) −411108. −0.0930952
\(456\) 2.13182e6 0.480107
\(457\) −3.19765e6 −0.716210 −0.358105 0.933681i \(-0.616577\pi\)
−0.358105 + 0.933681i \(0.616577\pi\)
\(458\) −3.96163e6 −0.882491
\(459\) 543456. 0.120402
\(460\) −947847. −0.208854
\(461\) 7.02590e6 1.53975 0.769874 0.638196i \(-0.220319\pi\)
0.769874 + 0.638196i \(0.220319\pi\)
\(462\) 175625. 0.0382808
\(463\) −9.07013e6 −1.96635 −0.983176 0.182663i \(-0.941528\pi\)
−0.983176 + 0.182663i \(0.941528\pi\)
\(464\) −181675. −0.0391742
\(465\) 884414. 0.189681
\(466\) 1.99243e6 0.425029
\(467\) −446092. −0.0946525 −0.0473263 0.998879i \(-0.515070\pi\)
−0.0473263 + 0.998879i \(0.515070\pi\)
\(468\) 782000. 0.165041
\(469\) −89860.0 −0.0188640
\(470\) 6.68646e6 1.39621
\(471\) 796675. 0.165474
\(472\) 5.56450e6 1.14966
\(473\) −8.29034e6 −1.70380
\(474\) −1.98959e6 −0.406740
\(475\) 6.96940e6 1.41730
\(476\) 130324. 0.0263637
\(477\) 390165. 0.0785150
\(478\) 1.98409e6 0.397184
\(479\) −2.86711e6 −0.570960 −0.285480 0.958385i \(-0.592153\pi\)
−0.285480 + 0.958385i \(0.592153\pi\)
\(480\) 4.76755e6 0.944479
\(481\) 417125. 0.0822061
\(482\) 5.75577e6 1.12846
\(483\) 42796.4 0.00834717
\(484\) −4.16982e6 −0.809104
\(485\) −4.34252e6 −0.838278
\(486\) −209202. −0.0401768
\(487\) 3.38408e6 0.646574 0.323287 0.946301i \(-0.395212\pi\)
0.323287 + 0.946301i \(0.395212\pi\)
\(488\) 1737.16 0.000330209 0
\(489\) 4.69998e6 0.888841
\(490\) −5.45952e6 −1.02722
\(491\) 2.36070e6 0.441913 0.220956 0.975284i \(-0.429082\pi\)
0.220956 + 0.975284i \(0.429082\pi\)
\(492\) −1.71900e6 −0.320158
\(493\) 5.78107e6 1.07125
\(494\) 2.28550e6 0.421370
\(495\) −4.57267e6 −0.838797
\(496\) 24988.1 0.00456067
\(497\) 662718. 0.120348
\(498\) 2.62850e6 0.474936
\(499\) −3.42412e6 −0.615598 −0.307799 0.951451i \(-0.599592\pi\)
−0.307799 + 0.951451i \(0.599592\pi\)
\(500\) 4.01007e6 0.717343
\(501\) −2.11234e6 −0.375984
\(502\) 3.89434e6 0.689722
\(503\) 7.49119e6 1.32017 0.660087 0.751190i \(-0.270520\pi\)
0.660087 + 0.751190i \(0.270520\pi\)
\(504\) −132714. −0.0232724
\(505\) 1.45845e6 0.254485
\(506\) 1.14839e6 0.199395
\(507\) −1.12380e6 −0.194165
\(508\) −2.49686e6 −0.429275
\(509\) −5.34876e6 −0.915079 −0.457539 0.889189i \(-0.651269\pi\)
−0.457539 + 0.889189i \(0.651269\pi\)
\(510\) 2.18996e6 0.372830
\(511\) −495917. −0.0840149
\(512\) 271348. 0.0457458
\(513\) 947352. 0.158934
\(514\) 4.82548e6 0.805624
\(515\) 8.36384e6 1.38959
\(516\) 2.36817e6 0.391551
\(517\) 1.25522e7 2.06535
\(518\) −26760.0 −0.00438189
\(519\) −2.24459e6 −0.365778
\(520\) 8.33625e6 1.35196
\(521\) −4.73330e6 −0.763958 −0.381979 0.924171i \(-0.624757\pi\)
−0.381979 + 0.924171i \(0.624757\pi\)
\(522\) −2.22541e6 −0.357465
\(523\) −2.28134e6 −0.364701 −0.182350 0.983234i \(-0.558370\pi\)
−0.182350 + 0.983234i \(0.558370\pi\)
\(524\) 7.03524e6 1.11931
\(525\) −433873. −0.0687013
\(526\) 2.84192e6 0.447866
\(527\) −795144. −0.124715
\(528\) −129195. −0.0201680
\(529\) 279841. 0.0434783
\(530\) 1.57225e6 0.243126
\(531\) 2.47279e6 0.380584
\(532\) 227181. 0.0348010
\(533\) −4.87527e6 −0.743329
\(534\) −3.03466e6 −0.460528
\(535\) −1.92900e7 −2.91372
\(536\) 1.82214e6 0.273949
\(537\) 3.17717e6 0.475449
\(538\) 1.24752e6 0.185820
\(539\) −1.02489e7 −1.51952
\(540\) 1.30620e6 0.192764
\(541\) −2.57849e6 −0.378767 −0.189383 0.981903i \(-0.560649\pi\)
−0.189383 + 0.981903i \(0.560649\pi\)
\(542\) 5.36112e6 0.783894
\(543\) −1.69924e6 −0.247318
\(544\) −4.28633e6 −0.620995
\(545\) −3.69591e6 −0.533004
\(546\) −142281. −0.0204252
\(547\) 6.20267e6 0.886360 0.443180 0.896433i \(-0.353850\pi\)
0.443180 + 0.896433i \(0.353850\pi\)
\(548\) 2.26825e6 0.322655
\(549\) 771.970 0.000109312 0
\(550\) −1.16425e7 −1.64112
\(551\) 1.00776e7 1.41409
\(552\) −867804. −0.121220
\(553\) −560887. −0.0779942
\(554\) 4.47328e6 0.619230
\(555\) 696739. 0.0960146
\(556\) −1.83865e6 −0.252239
\(557\) −2.67729e6 −0.365643 −0.182821 0.983146i \(-0.558523\pi\)
−0.182821 + 0.983146i \(0.558523\pi\)
\(558\) 306089. 0.0416162
\(559\) 6.71638e6 0.909087
\(560\) −19401.5 −0.00261436
\(561\) 4.11112e6 0.551509
\(562\) −4.41337e6 −0.589426
\(563\) −7.74164e6 −1.02935 −0.514673 0.857386i \(-0.672087\pi\)
−0.514673 + 0.857386i \(0.672087\pi\)
\(564\) −3.58558e6 −0.474637
\(565\) −2.28850e7 −3.01599
\(566\) −1.31919e6 −0.173088
\(567\) −58976.5 −0.00770408
\(568\) −1.34383e7 −1.74772
\(569\) 1.38677e7 1.79565 0.897827 0.440348i \(-0.145145\pi\)
0.897827 + 0.440348i \(0.145145\pi\)
\(570\) 3.81754e6 0.492149
\(571\) −3.83164e6 −0.491807 −0.245904 0.969294i \(-0.579085\pi\)
−0.245904 + 0.969294i \(0.579085\pi\)
\(572\) 5.91565e6 0.755984
\(573\) 3.37470e6 0.429386
\(574\) 312765. 0.0396222
\(575\) −2.83705e6 −0.357847
\(576\) 1.71074e6 0.214846
\(577\) −2.35210e6 −0.294114 −0.147057 0.989128i \(-0.546980\pi\)
−0.147057 + 0.989128i \(0.546980\pi\)
\(578\) 3.06144e6 0.381159
\(579\) −6.85704e6 −0.850042
\(580\) 1.38949e7 1.71508
\(581\) 741005. 0.0910712
\(582\) −1.50291e6 −0.183919
\(583\) 2.95151e6 0.359644
\(584\) 1.00560e7 1.22009
\(585\) 3.70452e6 0.447551
\(586\) −6.17781e6 −0.743174
\(587\) −5.80653e6 −0.695539 −0.347770 0.937580i \(-0.613061\pi\)
−0.347770 + 0.937580i \(0.613061\pi\)
\(588\) 2.92765e6 0.349201
\(589\) −1.38610e6 −0.164628
\(590\) 9.96460e6 1.17850
\(591\) −6.95667e6 −0.819281
\(592\) 19685.5 0.00230857
\(593\) −5.28179e6 −0.616800 −0.308400 0.951257i \(-0.599794\pi\)
−0.308400 + 0.951257i \(0.599794\pi\)
\(594\) −1.58257e6 −0.184033
\(595\) 617375. 0.0714919
\(596\) −5.34932e6 −0.616854
\(597\) 7.77088e6 0.892348
\(598\) −930363. −0.106390
\(599\) −9.88981e6 −1.12621 −0.563107 0.826384i \(-0.690394\pi\)
−0.563107 + 0.826384i \(0.690394\pi\)
\(600\) 8.79787e6 0.997699
\(601\) 4.37165e6 0.493695 0.246848 0.969054i \(-0.420605\pi\)
0.246848 + 0.969054i \(0.420605\pi\)
\(602\) −430878. −0.0484577
\(603\) 809734. 0.0906879
\(604\) 921948. 0.102829
\(605\) −1.97534e7 −2.19409
\(606\) 504758. 0.0558344
\(607\) −778883. −0.0858026 −0.0429013 0.999079i \(-0.513660\pi\)
−0.0429013 + 0.999079i \(0.513660\pi\)
\(608\) −7.47193e6 −0.819736
\(609\) −627369. −0.0685456
\(610\) 3110.81 0.000338492 0
\(611\) −1.01691e7 −1.10199
\(612\) −1.17436e6 −0.126742
\(613\) −5.29772e6 −0.569427 −0.284713 0.958613i \(-0.591898\pi\)
−0.284713 + 0.958613i \(0.591898\pi\)
\(614\) 5.69372e6 0.609502
\(615\) −8.14333e6 −0.868189
\(616\) −1.00395e6 −0.106601
\(617\) −1.13961e7 −1.20515 −0.602577 0.798060i \(-0.705860\pi\)
−0.602577 + 0.798060i \(0.705860\pi\)
\(618\) 2.89466e6 0.304878
\(619\) −1.65105e7 −1.73194 −0.865971 0.500094i \(-0.833299\pi\)
−0.865971 + 0.500094i \(0.833299\pi\)
\(620\) −1.91114e6 −0.199670
\(621\) −385641. −0.0401286
\(622\) −7.84261e6 −0.812802
\(623\) −855504. −0.0883084
\(624\) 104667. 0.0107609
\(625\) 2.23714e6 0.229083
\(626\) −1.70318e6 −0.173710
\(627\) 7.16650e6 0.728011
\(628\) −1.72154e6 −0.174188
\(629\) −626412. −0.0631296
\(630\) −237657. −0.0238561
\(631\) 1.21667e7 1.21646 0.608232 0.793760i \(-0.291879\pi\)
0.608232 + 0.793760i \(0.291879\pi\)
\(632\) 1.13734e7 1.13265
\(633\) 8.76352e6 0.869299
\(634\) −1.14552e7 −1.13183
\(635\) −1.18282e7 −1.16409
\(636\) −843111. −0.0826498
\(637\) 8.30310e6 0.810759
\(638\) −1.68347e7 −1.63740
\(639\) −5.97180e6 −0.578566
\(640\) −1.00575e7 −0.970604
\(641\) 1.41113e7 1.35651 0.678253 0.734829i \(-0.262737\pi\)
0.678253 + 0.734829i \(0.262737\pi\)
\(642\) −6.67613e6 −0.639274
\(643\) 5.63004e6 0.537012 0.268506 0.963278i \(-0.413470\pi\)
0.268506 + 0.963278i \(0.413470\pi\)
\(644\) −92479.0 −0.00878675
\(645\) 1.12186e7 1.06179
\(646\) −3.43221e6 −0.323588
\(647\) 7.21927e6 0.678004 0.339002 0.940786i \(-0.389911\pi\)
0.339002 + 0.940786i \(0.389911\pi\)
\(648\) 1.19590e6 0.111881
\(649\) 1.87061e7 1.74330
\(650\) 9.43210e6 0.875639
\(651\) 86290.0 0.00798009
\(652\) −1.01562e7 −0.935650
\(653\) 1.37204e7 1.25917 0.629586 0.776931i \(-0.283225\pi\)
0.629586 + 0.776931i \(0.283225\pi\)
\(654\) −1.27913e6 −0.116942
\(655\) 3.33276e7 3.03530
\(656\) −230080. −0.0208747
\(657\) 4.46874e6 0.403898
\(658\) 652380. 0.0587403
\(659\) 1.35053e7 1.21141 0.605705 0.795689i \(-0.292891\pi\)
0.605705 + 0.795689i \(0.292891\pi\)
\(660\) 9.88111e6 0.882970
\(661\) 6.74829e6 0.600745 0.300373 0.953822i \(-0.402889\pi\)
0.300373 + 0.953822i \(0.402889\pi\)
\(662\) −9.25161e6 −0.820488
\(663\) −3.33060e6 −0.294265
\(664\) −1.50257e7 −1.32256
\(665\) 1.07621e6 0.0943718
\(666\) 241136. 0.0210657
\(667\) −4.10230e6 −0.357037
\(668\) 4.56456e6 0.395784
\(669\) −9.68936e6 −0.837008
\(670\) 3.26298e6 0.280820
\(671\) 5839.78 0.000500714 0
\(672\) 465158. 0.0397354
\(673\) −1.24941e7 −1.06333 −0.531665 0.846955i \(-0.678434\pi\)
−0.531665 + 0.846955i \(0.678434\pi\)
\(674\) −9.80676e6 −0.831526
\(675\) 3.90966e6 0.330278
\(676\) 2.42844e6 0.204390
\(677\) 9.64155e6 0.808491 0.404246 0.914650i \(-0.367534\pi\)
0.404246 + 0.914650i \(0.367534\pi\)
\(678\) −7.92033e6 −0.661712
\(679\) −423689. −0.0352673
\(680\) −1.25188e7 −1.03823
\(681\) −2.82415e6 −0.233356
\(682\) 2.31549e6 0.190626
\(683\) 4.95512e6 0.406445 0.203223 0.979133i \(-0.434858\pi\)
0.203223 + 0.979133i \(0.434858\pi\)
\(684\) −2.04714e6 −0.167304
\(685\) 1.07452e7 0.874962
\(686\) −1.06792e6 −0.0866417
\(687\) 1.00638e7 0.813525
\(688\) 316968. 0.0255296
\(689\) −2.39115e6 −0.191893
\(690\) −1.55402e6 −0.124260
\(691\) −1.18245e7 −0.942079 −0.471039 0.882112i \(-0.656121\pi\)
−0.471039 + 0.882112i \(0.656121\pi\)
\(692\) 4.85035e6 0.385041
\(693\) −446143. −0.0352892
\(694\) −4.94658e6 −0.389858
\(695\) −8.71011e6 −0.684009
\(696\) 1.27215e7 0.995439
\(697\) 7.32137e6 0.570834
\(698\) 3.56089e6 0.276643
\(699\) −5.06142e6 −0.391814
\(700\) 937560. 0.0723193
\(701\) 8.50355e6 0.653590 0.326795 0.945095i \(-0.394031\pi\)
0.326795 + 0.945095i \(0.394031\pi\)
\(702\) 1.28211e6 0.0981932
\(703\) −1.09196e6 −0.0833334
\(704\) 1.29413e7 0.984119
\(705\) −1.69858e7 −1.28710
\(706\) 1.50312e7 1.13497
\(707\) 142297. 0.0107065
\(708\) −5.34347e6 −0.400627
\(709\) 1.18862e7 0.888029 0.444015 0.896020i \(-0.353554\pi\)
0.444015 + 0.896020i \(0.353554\pi\)
\(710\) −2.40645e7 −1.79156
\(711\) 5.05419e6 0.374954
\(712\) 1.73475e7 1.28244
\(713\) 564241. 0.0415663
\(714\) 213669. 0.0156854
\(715\) 2.80239e7 2.05004
\(716\) −6.86556e6 −0.500488
\(717\) −5.04023e6 −0.366145
\(718\) −1.11188e6 −0.0804907
\(719\) −3.08137e6 −0.222291 −0.111145 0.993804i \(-0.535452\pi\)
−0.111145 + 0.993804i \(0.535452\pi\)
\(720\) 174829. 0.0125684
\(721\) 816038. 0.0584618
\(722\) 2.78943e6 0.199147
\(723\) −1.46215e7 −1.04027
\(724\) 3.67190e6 0.260342
\(725\) 4.15895e7 2.93859
\(726\) −6.83652e6 −0.481386
\(727\) −4.11888e6 −0.289030 −0.144515 0.989503i \(-0.546162\pi\)
−0.144515 + 0.989503i \(0.546162\pi\)
\(728\) 813346. 0.0568783
\(729\) 531441. 0.0370370
\(730\) 1.80077e7 1.25069
\(731\) −1.00862e7 −0.698128
\(732\) −1668.16 −0.000115069 0
\(733\) −2.07922e6 −0.142935 −0.0714677 0.997443i \(-0.522768\pi\)
−0.0714677 + 0.997443i \(0.522768\pi\)
\(734\) −1.02804e6 −0.0704318
\(735\) 1.38690e7 0.946946
\(736\) 3.04162e6 0.206971
\(737\) 6.12545e6 0.415403
\(738\) −2.81835e6 −0.190482
\(739\) −9.96819e6 −0.671437 −0.335719 0.941962i \(-0.608979\pi\)
−0.335719 + 0.941962i \(0.608979\pi\)
\(740\) −1.50559e6 −0.101071
\(741\) −5.80590e6 −0.388440
\(742\) 153400. 0.0102286
\(743\) −1.21883e7 −0.809977 −0.404988 0.914322i \(-0.632724\pi\)
−0.404988 + 0.914322i \(0.632724\pi\)
\(744\) −1.74975e6 −0.115889
\(745\) −2.53410e7 −1.67276
\(746\) 1.82141e7 1.19829
\(747\) −6.67725e6 −0.437821
\(748\) −8.88374e6 −0.580553
\(749\) −1.88208e6 −0.122584
\(750\) 6.57460e6 0.426792
\(751\) 1.30656e7 0.845334 0.422667 0.906285i \(-0.361094\pi\)
0.422667 + 0.906285i \(0.361094\pi\)
\(752\) −479913. −0.0309469
\(753\) −9.89287e6 −0.635821
\(754\) 1.36386e7 0.873655
\(755\) 4.36749e6 0.278846
\(756\) 127443. 0.00810980
\(757\) 1.30510e7 0.827761 0.413881 0.910331i \(-0.364173\pi\)
0.413881 + 0.910331i \(0.364173\pi\)
\(758\) 2.23877e6 0.141526
\(759\) −2.91729e6 −0.183812
\(760\) −2.18228e7 −1.37049
\(761\) 8.03065e6 0.502677 0.251338 0.967899i \(-0.419129\pi\)
0.251338 + 0.967899i \(0.419129\pi\)
\(762\) −4.09366e6 −0.255402
\(763\) −360600. −0.0224241
\(764\) −7.29241e6 −0.451999
\(765\) −5.56321e6 −0.343694
\(766\) 420886. 0.0259175
\(767\) −1.51546e7 −0.930159
\(768\) −9.56346e6 −0.585076
\(769\) 6.12813e6 0.373691 0.186845 0.982389i \(-0.440174\pi\)
0.186845 + 0.982389i \(0.440174\pi\)
\(770\) −1.79782e6 −0.109275
\(771\) −1.22583e7 −0.742665
\(772\) 1.48174e7 0.894808
\(773\) 8.04760e6 0.484415 0.242208 0.970224i \(-0.422128\pi\)
0.242208 + 0.970224i \(0.422128\pi\)
\(774\) 3.88267e6 0.232958
\(775\) −5.72033e6 −0.342111
\(776\) 8.59136e6 0.512162
\(777\) 67979.0 0.00403945
\(778\) −9.16695e6 −0.542970
\(779\) 1.27626e7 0.753522
\(780\) −8.00513e6 −0.471120
\(781\) −4.51753e7 −2.65017
\(782\) 1.39716e6 0.0817013
\(783\) 5.65326e6 0.329530
\(784\) 391851. 0.0227683
\(785\) −8.15535e6 −0.472355
\(786\) 1.15344e7 0.665948
\(787\) −3.49427e6 −0.201103 −0.100552 0.994932i \(-0.532061\pi\)
−0.100552 + 0.994932i \(0.532061\pi\)
\(788\) 1.50327e7 0.862426
\(789\) −7.21940e6 −0.412866
\(790\) 2.03669e7 1.16106
\(791\) −2.23283e6 −0.126886
\(792\) 9.04668e6 0.512479
\(793\) −4731.06 −0.000267163 0
\(794\) −1.26409e7 −0.711587
\(795\) −3.99402e6 −0.224126
\(796\) −1.67922e7 −0.939342
\(797\) −1.33411e7 −0.743951 −0.371976 0.928242i \(-0.621320\pi\)
−0.371976 + 0.928242i \(0.621320\pi\)
\(798\) 372468. 0.0207053
\(799\) 1.52713e7 0.846268
\(800\) −3.08362e7 −1.70348
\(801\) 7.70901e6 0.424539
\(802\) −1.51987e7 −0.834390
\(803\) 3.38050e7 1.85009
\(804\) −1.74976e6 −0.0954638
\(805\) −438095. −0.0238275
\(806\) −1.87588e6 −0.101711
\(807\) −3.16911e6 −0.171298
\(808\) −2.88543e6 −0.155483
\(809\) −5.57231e6 −0.299339 −0.149670 0.988736i \(-0.547821\pi\)
−0.149670 + 0.988736i \(0.547821\pi\)
\(810\) 2.14155e6 0.114687
\(811\) 8.30913e6 0.443612 0.221806 0.975091i \(-0.428805\pi\)
0.221806 + 0.975091i \(0.428805\pi\)
\(812\) 1.35569e6 0.0721554
\(813\) −1.36190e7 −0.722633
\(814\) 1.82414e6 0.0964932
\(815\) −4.81125e7 −2.53725
\(816\) −157182. −0.00826375
\(817\) −1.75823e7 −0.921553
\(818\) −1.11344e7 −0.581811
\(819\) 361441. 0.0188290
\(820\) 1.75970e7 0.913910
\(821\) 1.35323e7 0.700669 0.350334 0.936625i \(-0.386068\pi\)
0.350334 + 0.936625i \(0.386068\pi\)
\(822\) 3.71884e6 0.191968
\(823\) 1.20061e7 0.617879 0.308939 0.951082i \(-0.400026\pi\)
0.308939 + 0.951082i \(0.400026\pi\)
\(824\) −1.65472e7 −0.848999
\(825\) 2.95757e7 1.51286
\(826\) 972220. 0.0495809
\(827\) −2.71374e7 −1.37976 −0.689881 0.723923i \(-0.742337\pi\)
−0.689881 + 0.723923i \(0.742337\pi\)
\(828\) 833335. 0.0422419
\(829\) −6.90219e6 −0.348819 −0.174410 0.984673i \(-0.555802\pi\)
−0.174410 + 0.984673i \(0.555802\pi\)
\(830\) −2.69073e7 −1.35574
\(831\) −1.13636e7 −0.570838
\(832\) −1.04844e7 −0.525090
\(833\) −1.24691e7 −0.622617
\(834\) −3.01450e6 −0.150072
\(835\) 2.16234e7 1.07327
\(836\) −1.54861e7 −0.766351
\(837\) −777565. −0.0383639
\(838\) 913445. 0.0449337
\(839\) −1.57032e6 −0.0770164 −0.0385082 0.999258i \(-0.512261\pi\)
−0.0385082 + 0.999258i \(0.512261\pi\)
\(840\) 1.35856e6 0.0664325
\(841\) 3.96260e7 1.93193
\(842\) 5.26580e6 0.255967
\(843\) 1.12114e7 0.543363
\(844\) −1.89372e7 −0.915079
\(845\) 1.15041e7 0.554255
\(846\) −5.87864e6 −0.282391
\(847\) −1.92729e6 −0.0923079
\(848\) −112846. −0.00538887
\(849\) 3.35118e6 0.159561
\(850\) −1.41645e7 −0.672441
\(851\) 444508. 0.0210405
\(852\) 1.29045e7 0.609035
\(853\) −2.20053e7 −1.03551 −0.517755 0.855529i \(-0.673232\pi\)
−0.517755 + 0.855529i \(0.673232\pi\)
\(854\) 303.513 1.42408e−5 0
\(855\) −9.69779e6 −0.453688
\(856\) 3.81638e7 1.78019
\(857\) 9.95578e6 0.463045 0.231523 0.972830i \(-0.425629\pi\)
0.231523 + 0.972830i \(0.425629\pi\)
\(858\) 9.69884e6 0.449782
\(859\) 1.38922e7 0.642374 0.321187 0.947016i \(-0.395918\pi\)
0.321187 + 0.947016i \(0.395918\pi\)
\(860\) −2.42423e7 −1.11771
\(861\) −794524. −0.0365257
\(862\) −1.76458e7 −0.808860
\(863\) 7.55200e6 0.345171 0.172586 0.984995i \(-0.444788\pi\)
0.172586 + 0.984995i \(0.444788\pi\)
\(864\) −4.19157e6 −0.191026
\(865\) 2.29772e7 1.04414
\(866\) −5.12800e6 −0.232356
\(867\) −7.77704e6 −0.351371
\(868\) −186465. −0.00840035
\(869\) 3.82338e7 1.71750
\(870\) 2.27809e7 1.02041
\(871\) −4.96250e6 −0.221644
\(872\) 7.31208e6 0.325649
\(873\) 3.81789e6 0.169546
\(874\) 2.43553e6 0.107849
\(875\) 1.85346e6 0.0818393
\(876\) −9.65653e6 −0.425168
\(877\) −2.70277e6 −0.118662 −0.0593309 0.998238i \(-0.518897\pi\)
−0.0593309 + 0.998238i \(0.518897\pi\)
\(878\) 1.52994e7 0.669787
\(879\) 1.56936e7 0.685096
\(880\) 1.32254e6 0.0575707
\(881\) −4.41216e7 −1.91519 −0.957595 0.288117i \(-0.906971\pi\)
−0.957595 + 0.288117i \(0.906971\pi\)
\(882\) 4.79994e6 0.207761
\(883\) 2.72497e7 1.17614 0.588071 0.808809i \(-0.299888\pi\)
0.588071 + 0.808809i \(0.299888\pi\)
\(884\) 7.19711e6 0.309762
\(885\) −2.53133e7 −1.08640
\(886\) −2.00477e7 −0.857987
\(887\) 1.11988e7 0.477930 0.238965 0.971028i \(-0.423192\pi\)
0.238965 + 0.971028i \(0.423192\pi\)
\(888\) −1.37845e6 −0.0586620
\(889\) −1.15405e6 −0.0489745
\(890\) 3.10650e7 1.31461
\(891\) 4.02023e6 0.169651
\(892\) 2.09378e7 0.881088
\(893\) 2.66209e7 1.11710
\(894\) −8.77032e6 −0.367005
\(895\) −3.25238e7 −1.35720
\(896\) −981288. −0.0408344
\(897\) 2.36342e6 0.0980754
\(898\) 6.12181e6 0.253331
\(899\) −8.27143e6 −0.341336
\(900\) −8.44842e6 −0.347672
\(901\) 3.59087e6 0.147363
\(902\) −2.13201e7 −0.872516
\(903\) 1.09457e6 0.0446708
\(904\) 4.52763e7 1.84268
\(905\) 1.73947e7 0.705984
\(906\) 1.51156e6 0.0611792
\(907\) 2.65141e7 1.07018 0.535092 0.844794i \(-0.320277\pi\)
0.535092 + 0.844794i \(0.320277\pi\)
\(908\) 6.10272e6 0.245645
\(909\) −1.28225e6 −0.0514710
\(910\) 1.45650e6 0.0583050
\(911\) −1.62499e7 −0.648715 −0.324357 0.945935i \(-0.605148\pi\)
−0.324357 + 0.945935i \(0.605148\pi\)
\(912\) −274000. −0.0109084
\(913\) −5.05119e7 −2.00547
\(914\) 1.13288e7 0.448558
\(915\) −7902.45 −0.000312039 0
\(916\) −2.17470e7 −0.856368
\(917\) 3.25169e6 0.127698
\(918\) −1.92538e6 −0.0754069
\(919\) −2.66141e7 −1.03950 −0.519748 0.854319i \(-0.673974\pi\)
−0.519748 + 0.854319i \(0.673974\pi\)
\(920\) 8.88348e6 0.346030
\(921\) −1.44639e7 −0.561870
\(922\) −2.48917e7 −0.964335
\(923\) 3.65985e7 1.41403
\(924\) 964074. 0.0371476
\(925\) −4.50646e6 −0.173173
\(926\) 3.21342e7 1.23151
\(927\) −7.35338e6 −0.281053
\(928\) −4.45883e7 −1.69962
\(929\) −1.98073e7 −0.752984 −0.376492 0.926420i \(-0.622870\pi\)
−0.376492 + 0.926420i \(0.622870\pi\)
\(930\) −3.13335e6 −0.118796
\(931\) −2.17361e7 −0.821877
\(932\) 1.09373e7 0.412448
\(933\) 1.99228e7 0.749282
\(934\) 1.58044e6 0.0592804
\(935\) −4.20844e7 −1.57432
\(936\) −7.32912e6 −0.273440
\(937\) −9.15888e6 −0.340795 −0.170398 0.985375i \(-0.554505\pi\)
−0.170398 + 0.985375i \(0.554505\pi\)
\(938\) 318361. 0.0118144
\(939\) 4.32664e6 0.160135
\(940\) 3.67047e7 1.35488
\(941\) 5.38492e7 1.98247 0.991233 0.132128i \(-0.0421811\pi\)
0.991233 + 0.132128i \(0.0421811\pi\)
\(942\) −2.82250e6 −0.103635
\(943\) −5.19531e6 −0.190253
\(944\) −715198. −0.0261214
\(945\) 603726. 0.0219918
\(946\) 2.93715e7 1.06708
\(947\) −1.89117e7 −0.685261 −0.342631 0.939470i \(-0.611318\pi\)
−0.342631 + 0.939470i \(0.611318\pi\)
\(948\) −1.09216e7 −0.394700
\(949\) −2.73869e7 −0.987138
\(950\) −2.46916e7 −0.887647
\(951\) 2.91000e7 1.04338
\(952\) −1.22143e6 −0.0436794
\(953\) 3.40143e7 1.21319 0.606595 0.795011i \(-0.292535\pi\)
0.606595 + 0.795011i \(0.292535\pi\)
\(954\) −1.38230e6 −0.0491735
\(955\) −3.45459e7 −1.22571
\(956\) 1.08915e7 0.385427
\(957\) 4.27656e7 1.50944
\(958\) 1.01578e7 0.357589
\(959\) 1.04838e6 0.0368107
\(960\) −1.75124e7 −0.613291
\(961\) −2.74915e7 −0.960262
\(962\) −1.47782e6 −0.0514852
\(963\) 1.69595e7 0.589315
\(964\) 3.15957e7 1.09506
\(965\) 7.01937e7 2.42650
\(966\) −151621. −0.00522778
\(967\) −6.27887e6 −0.215931 −0.107966 0.994155i \(-0.534434\pi\)
−0.107966 + 0.994155i \(0.534434\pi\)
\(968\) 3.90807e7 1.34052
\(969\) 8.71892e6 0.298300
\(970\) 1.53849e7 0.525009
\(971\) −2.00595e7 −0.682765 −0.341383 0.939924i \(-0.610895\pi\)
−0.341383 + 0.939924i \(0.610895\pi\)
\(972\) −1.14839e6 −0.0389875
\(973\) −849823. −0.0287771
\(974\) −1.19893e7 −0.404945
\(975\) −2.39606e7 −0.807209
\(976\) −223.274 −7.50265e−6 0
\(977\) −4.81426e7 −1.61359 −0.806795 0.590831i \(-0.798800\pi\)
−0.806795 + 0.590831i \(0.798800\pi\)
\(978\) −1.66514e7 −0.556676
\(979\) 5.83169e7 1.94463
\(980\) −2.99695e7 −0.996815
\(981\) 3.24939e6 0.107803
\(982\) −8.36361e6 −0.276768
\(983\) −5.45980e7 −1.80216 −0.901079 0.433656i \(-0.857223\pi\)
−0.901079 + 0.433656i \(0.857223\pi\)
\(984\) 1.61110e7 0.530437
\(985\) 7.12136e7 2.33869
\(986\) −2.04815e7 −0.670918
\(987\) −1.65726e6 −0.0541498
\(988\) 1.25460e7 0.408896
\(989\) 7.15727e6 0.232679
\(990\) 1.62003e7 0.525334
\(991\) 1.21280e7 0.392288 0.196144 0.980575i \(-0.437158\pi\)
0.196144 + 0.980575i \(0.437158\pi\)
\(992\) 6.13279e6 0.197870
\(993\) 2.35021e7 0.756368
\(994\) −2.34791e6 −0.0753731
\(995\) −7.95484e7 −2.54726
\(996\) 1.44289e7 0.460877
\(997\) −4.49074e6 −0.143080 −0.0715401 0.997438i \(-0.522791\pi\)
−0.0715401 + 0.997438i \(0.522791\pi\)
\(998\) 1.21312e7 0.385546
\(999\) −612563. −0.0194195
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 69.6.a.e.1.2 5
3.2 odd 2 207.6.a.f.1.4 5
4.3 odd 2 1104.6.a.r.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.2 5 1.1 even 1 trivial
207.6.a.f.1.4 5 3.2 odd 2
1104.6.a.r.1.1 5 4.3 odd 2