Properties

Label 69.6.a.e.1.5
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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 1.5
Root \(3.40352\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.34908 q^{2} +9.00000 q^{3} +55.4053 q^{4} +70.5203 q^{5} +84.1417 q^{6} -89.7132 q^{7} +218.818 q^{8} +81.0000 q^{9} +659.300 q^{10} -436.763 q^{11} +498.647 q^{12} -258.865 q^{13} -838.736 q^{14} +634.683 q^{15} +272.775 q^{16} +591.819 q^{17} +757.275 q^{18} +1663.69 q^{19} +3907.20 q^{20} -807.419 q^{21} -4083.33 q^{22} -529.000 q^{23} +1969.36 q^{24} +1848.12 q^{25} -2420.15 q^{26} +729.000 q^{27} -4970.58 q^{28} -5121.48 q^{29} +5933.70 q^{30} +1871.29 q^{31} -4451.97 q^{32} -3930.87 q^{33} +5532.96 q^{34} -6326.60 q^{35} +4487.83 q^{36} +884.534 q^{37} +15554.0 q^{38} -2329.78 q^{39} +15431.1 q^{40} -18410.5 q^{41} -7548.62 q^{42} +12334.9 q^{43} -24199.0 q^{44} +5712.15 q^{45} -4945.66 q^{46} +23871.6 q^{47} +2454.98 q^{48} -8758.55 q^{49} +17278.2 q^{50} +5326.37 q^{51} -14342.5 q^{52} -14594.8 q^{53} +6815.48 q^{54} -30800.7 q^{55} -19630.8 q^{56} +14973.2 q^{57} -47881.1 q^{58} +47739.1 q^{59} +35164.8 q^{60} -15351.5 q^{61} +17494.8 q^{62} -7266.77 q^{63} -50350.6 q^{64} -18255.2 q^{65} -36750.0 q^{66} -41174.4 q^{67} +32789.9 q^{68} -4761.00 q^{69} -59147.9 q^{70} -45608.3 q^{71} +17724.2 q^{72} -7898.24 q^{73} +8269.57 q^{74} +16633.1 q^{75} +92177.1 q^{76} +39183.4 q^{77} -21781.3 q^{78} +71435.2 q^{79} +19236.2 q^{80} +6561.00 q^{81} -172121. q^{82} +112580. q^{83} -44735.2 q^{84} +41735.3 q^{85} +115320. q^{86} -46093.3 q^{87} -95571.5 q^{88} +137845. q^{89} +53403.3 q^{90} +23223.6 q^{91} -29309.4 q^{92} +16841.6 q^{93} +223177. q^{94} +117324. q^{95} -40067.7 q^{96} +41679.8 q^{97} -81884.3 q^{98} -35377.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} + O(q^{10}) \) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.34908 1.65270 0.826350 0.563157i \(-0.190414\pi\)
0.826350 + 0.563157i \(0.190414\pi\)
\(3\) 9.00000 0.577350
\(4\) 55.4053 1.73141
\(5\) 70.5203 1.26151 0.630753 0.775984i \(-0.282746\pi\)
0.630753 + 0.775984i \(0.282746\pi\)
\(6\) 84.1417 0.954186
\(7\) −89.7132 −0.692008 −0.346004 0.938233i \(-0.612462\pi\)
−0.346004 + 0.938233i \(0.612462\pi\)
\(8\) 218.818 1.20881
\(9\) 81.0000 0.333333
\(10\) 659.300 2.08489
\(11\) −436.763 −1.08834 −0.544170 0.838975i \(-0.683155\pi\)
−0.544170 + 0.838975i \(0.683155\pi\)
\(12\) 498.647 0.999633
\(13\) −258.865 −0.424830 −0.212415 0.977180i \(-0.568133\pi\)
−0.212415 + 0.977180i \(0.568133\pi\)
\(14\) −838.736 −1.14368
\(15\) 634.683 0.728331
\(16\) 272.775 0.266382
\(17\) 591.819 0.496669 0.248334 0.968674i \(-0.420117\pi\)
0.248334 + 0.968674i \(0.420117\pi\)
\(18\) 757.275 0.550900
\(19\) 1663.69 1.05727 0.528637 0.848848i \(-0.322703\pi\)
0.528637 + 0.848848i \(0.322703\pi\)
\(20\) 3907.20 2.18419
\(21\) −807.419 −0.399531
\(22\) −4083.33 −1.79870
\(23\) −529.000 −0.208514
\(24\) 1969.36 0.697906
\(25\) 1848.12 0.591398
\(26\) −2420.15 −0.702116
\(27\) 729.000 0.192450
\(28\) −4970.58 −1.19815
\(29\) −5121.48 −1.13084 −0.565419 0.824804i \(-0.691286\pi\)
−0.565419 + 0.824804i \(0.691286\pi\)
\(30\) 5933.70 1.20371
\(31\) 1871.29 0.349733 0.174867 0.984592i \(-0.444051\pi\)
0.174867 + 0.984592i \(0.444051\pi\)
\(32\) −4451.97 −0.768559
\(33\) −3930.87 −0.628353
\(34\) 5532.96 0.820844
\(35\) −6326.60 −0.872972
\(36\) 4487.83 0.577138
\(37\) 884.534 0.106221 0.0531105 0.998589i \(-0.483086\pi\)
0.0531105 + 0.998589i \(0.483086\pi\)
\(38\) 15554.0 1.74736
\(39\) −2329.78 −0.245275
\(40\) 15431.1 1.52492
\(41\) −18410.5 −1.71043 −0.855216 0.518271i \(-0.826576\pi\)
−0.855216 + 0.518271i \(0.826576\pi\)
\(42\) −7548.62 −0.660305
\(43\) 12334.9 1.01734 0.508669 0.860962i \(-0.330138\pi\)
0.508669 + 0.860962i \(0.330138\pi\)
\(44\) −24199.0 −1.88437
\(45\) 5712.15 0.420502
\(46\) −4945.66 −0.344612
\(47\) 23871.6 1.57629 0.788146 0.615489i \(-0.211041\pi\)
0.788146 + 0.615489i \(0.211041\pi\)
\(48\) 2454.98 0.153796
\(49\) −8758.55 −0.521125
\(50\) 17278.2 0.977403
\(51\) 5326.37 0.286752
\(52\) −14342.5 −0.735556
\(53\) −14594.8 −0.713690 −0.356845 0.934164i \(-0.616147\pi\)
−0.356845 + 0.934164i \(0.616147\pi\)
\(54\) 6815.48 0.318062
\(55\) −30800.7 −1.37295
\(56\) −19630.8 −0.836505
\(57\) 14973.2 0.610418
\(58\) −47881.1 −1.86894
\(59\) 47739.1 1.78544 0.892718 0.450616i \(-0.148795\pi\)
0.892718 + 0.450616i \(0.148795\pi\)
\(60\) 35164.8 1.26104
\(61\) −15351.5 −0.528233 −0.264116 0.964491i \(-0.585080\pi\)
−0.264116 + 0.964491i \(0.585080\pi\)
\(62\) 17494.8 0.578004
\(63\) −7266.77 −0.230669
\(64\) −50350.6 −1.53658
\(65\) −18255.2 −0.535925
\(66\) −36750.0 −1.03848
\(67\) −41174.4 −1.12057 −0.560287 0.828299i \(-0.689309\pi\)
−0.560287 + 0.828299i \(0.689309\pi\)
\(68\) 32789.9 0.859939
\(69\) −4761.00 −0.120386
\(70\) −59147.9 −1.44276
\(71\) −45608.3 −1.07374 −0.536869 0.843666i \(-0.680393\pi\)
−0.536869 + 0.843666i \(0.680393\pi\)
\(72\) 17724.2 0.402936
\(73\) −7898.24 −0.173470 −0.0867348 0.996231i \(-0.527643\pi\)
−0.0867348 + 0.996231i \(0.527643\pi\)
\(74\) 8269.57 0.175551
\(75\) 16633.1 0.341444
\(76\) 92177.1 1.83058
\(77\) 39183.4 0.753140
\(78\) −21781.3 −0.405367
\(79\) 71435.2 1.28779 0.643894 0.765115i \(-0.277318\pi\)
0.643894 + 0.765115i \(0.277318\pi\)
\(80\) 19236.2 0.336043
\(81\) 6561.00 0.111111
\(82\) −172121. −2.82683
\(83\) 112580. 1.79377 0.896885 0.442264i \(-0.145825\pi\)
0.896885 + 0.442264i \(0.145825\pi\)
\(84\) −44735.2 −0.691754
\(85\) 41735.3 0.626551
\(86\) 115320. 1.68136
\(87\) −46093.3 −0.652890
\(88\) −95571.5 −1.31559
\(89\) 137845. 1.84466 0.922330 0.386404i \(-0.126283\pi\)
0.922330 + 0.386404i \(0.126283\pi\)
\(90\) 53403.3 0.694963
\(91\) 23223.6 0.293985
\(92\) −29309.4 −0.361025
\(93\) 16841.6 0.201918
\(94\) 223177. 2.60514
\(95\) 117324. 1.33376
\(96\) −40067.7 −0.443728
\(97\) 41679.8 0.449776 0.224888 0.974385i \(-0.427798\pi\)
0.224888 + 0.974385i \(0.427798\pi\)
\(98\) −81884.3 −0.861263
\(99\) −35377.8 −0.362780
\(100\) 102395. 1.02395
\(101\) 37100.3 0.361888 0.180944 0.983493i \(-0.442085\pi\)
0.180944 + 0.983493i \(0.442085\pi\)
\(102\) 49796.7 0.473914
\(103\) −123323. −1.14538 −0.572692 0.819771i \(-0.694101\pi\)
−0.572692 + 0.819771i \(0.694101\pi\)
\(104\) −56644.2 −0.513538
\(105\) −56939.4 −0.504011
\(106\) −136448. −1.17951
\(107\) 107455. 0.907338 0.453669 0.891170i \(-0.350115\pi\)
0.453669 + 0.891170i \(0.350115\pi\)
\(108\) 40390.4 0.333211
\(109\) 96652.7 0.779198 0.389599 0.920985i \(-0.372614\pi\)
0.389599 + 0.920985i \(0.372614\pi\)
\(110\) −287958. −2.26907
\(111\) 7960.80 0.0613267
\(112\) −24471.5 −0.184338
\(113\) 55788.7 0.411008 0.205504 0.978656i \(-0.434117\pi\)
0.205504 + 0.978656i \(0.434117\pi\)
\(114\) 139986. 1.00884
\(115\) −37305.3 −0.263042
\(116\) −283757. −1.95795
\(117\) −20968.1 −0.141610
\(118\) 446317. 2.95079
\(119\) −53094.0 −0.343699
\(120\) 138880. 0.880413
\(121\) 29711.1 0.184483
\(122\) −143522. −0.873010
\(123\) −165695. −0.987519
\(124\) 103679. 0.605533
\(125\) −90046.2 −0.515454
\(126\) −67937.6 −0.381227
\(127\) 167839. 0.923388 0.461694 0.887039i \(-0.347242\pi\)
0.461694 + 0.887039i \(0.347242\pi\)
\(128\) −328269. −1.77094
\(129\) 111014. 0.587361
\(130\) −170670. −0.885723
\(131\) 129729. 0.660476 0.330238 0.943898i \(-0.392871\pi\)
0.330238 + 0.943898i \(0.392871\pi\)
\(132\) −217791. −1.08794
\(133\) −149255. −0.731643
\(134\) −384943. −1.85197
\(135\) 51409.3 0.242777
\(136\) 129501. 0.600377
\(137\) −226702. −1.03194 −0.515970 0.856607i \(-0.672568\pi\)
−0.515970 + 0.856607i \(0.672568\pi\)
\(138\) −44511.0 −0.198962
\(139\) −419175. −1.84017 −0.920086 0.391716i \(-0.871882\pi\)
−0.920086 + 0.391716i \(0.871882\pi\)
\(140\) −350527. −1.51148
\(141\) 214844. 0.910072
\(142\) −426396. −1.77457
\(143\) 113063. 0.462359
\(144\) 22094.8 0.0887940
\(145\) −361169. −1.42656
\(146\) −73841.3 −0.286693
\(147\) −78826.9 −0.300872
\(148\) 49007.8 0.183912
\(149\) 503270. 1.85710 0.928551 0.371206i \(-0.121055\pi\)
0.928551 + 0.371206i \(0.121055\pi\)
\(150\) 155504. 0.564304
\(151\) 517943. 1.84858 0.924292 0.381685i \(-0.124656\pi\)
0.924292 + 0.381685i \(0.124656\pi\)
\(152\) 364044. 1.27804
\(153\) 47937.4 0.165556
\(154\) 366329. 1.24471
\(155\) 131964. 0.441190
\(156\) −129082. −0.424674
\(157\) −39747.9 −0.128696 −0.0643480 0.997928i \(-0.520497\pi\)
−0.0643480 + 0.997928i \(0.520497\pi\)
\(158\) 667853. 2.12833
\(159\) −131353. −0.412049
\(160\) −313954. −0.969542
\(161\) 47458.3 0.144294
\(162\) 61339.3 0.183633
\(163\) −157828. −0.465279 −0.232640 0.972563i \(-0.574736\pi\)
−0.232640 + 0.972563i \(0.574736\pi\)
\(164\) −1.02004e6 −2.96147
\(165\) −277206. −0.792671
\(166\) 1.05252e6 2.96456
\(167\) −101000. −0.280241 −0.140120 0.990134i \(-0.544749\pi\)
−0.140120 + 0.990134i \(0.544749\pi\)
\(168\) −176677. −0.482956
\(169\) −304282. −0.819520
\(170\) 390187. 1.03550
\(171\) 134759. 0.352425
\(172\) 683420. 1.76144
\(173\) 61568.5 0.156402 0.0782012 0.996938i \(-0.475082\pi\)
0.0782012 + 0.996938i \(0.475082\pi\)
\(174\) −430930. −1.07903
\(175\) −165801. −0.409252
\(176\) −119138. −0.289914
\(177\) 429652. 1.03082
\(178\) 1.28872e6 3.04867
\(179\) −610536. −1.42423 −0.712113 0.702065i \(-0.752262\pi\)
−0.712113 + 0.702065i \(0.752262\pi\)
\(180\) 316483. 0.728063
\(181\) 121378. 0.275387 0.137694 0.990475i \(-0.456031\pi\)
0.137694 + 0.990475i \(0.456031\pi\)
\(182\) 217119. 0.485870
\(183\) −138163. −0.304975
\(184\) −115755. −0.252054
\(185\) 62377.6 0.133998
\(186\) 157453. 0.333710
\(187\) −258485. −0.540544
\(188\) 1.32261e6 2.72921
\(189\) −65400.9 −0.133177
\(190\) 1.09687e6 2.20430
\(191\) −294581. −0.584280 −0.292140 0.956376i \(-0.594367\pi\)
−0.292140 + 0.956376i \(0.594367\pi\)
\(192\) −453156. −0.887144
\(193\) −8895.53 −0.0171901 −0.00859505 0.999963i \(-0.502736\pi\)
−0.00859505 + 0.999963i \(0.502736\pi\)
\(194\) 389668. 0.743345
\(195\) −164297. −0.309417
\(196\) −485270. −0.902283
\(197\) −698113. −1.28162 −0.640811 0.767699i \(-0.721402\pi\)
−0.640811 + 0.767699i \(0.721402\pi\)
\(198\) −330750. −0.599566
\(199\) −533271. −0.954586 −0.477293 0.878744i \(-0.658382\pi\)
−0.477293 + 0.878744i \(0.658382\pi\)
\(200\) 404401. 0.714887
\(201\) −370570. −0.646963
\(202\) 346854. 0.598092
\(203\) 459464. 0.782550
\(204\) 295109. 0.496486
\(205\) −1.29831e6 −2.15772
\(206\) −1.15296e6 −1.89298
\(207\) −42849.0 −0.0695048
\(208\) −70611.9 −0.113167
\(209\) −726638. −1.15067
\(210\) −532331. −0.832978
\(211\) −658407. −1.01809 −0.509047 0.860739i \(-0.670002\pi\)
−0.509047 + 0.860739i \(0.670002\pi\)
\(212\) −808630. −1.23569
\(213\) −410475. −0.619923
\(214\) 1.00461e6 1.49956
\(215\) 869864. 1.28338
\(216\) 159518. 0.232635
\(217\) −167879. −0.242018
\(218\) 903614. 1.28778
\(219\) −71084.2 −0.100153
\(220\) −1.70652e6 −2.37714
\(221\) −153201. −0.211000
\(222\) 74426.2 0.101355
\(223\) 79895.6 0.107587 0.0537936 0.998552i \(-0.482869\pi\)
0.0537936 + 0.998552i \(0.482869\pi\)
\(224\) 399400. 0.531849
\(225\) 149698. 0.197133
\(226\) 521573. 0.679273
\(227\) 716111. 0.922393 0.461196 0.887298i \(-0.347420\pi\)
0.461196 + 0.887298i \(0.347420\pi\)
\(228\) 829594. 1.05689
\(229\) −581299. −0.732506 −0.366253 0.930515i \(-0.619360\pi\)
−0.366253 + 0.930515i \(0.619360\pi\)
\(230\) −348770. −0.434730
\(231\) 352651. 0.434825
\(232\) −1.12067e6 −1.36697
\(233\) 122734. 0.148107 0.0740534 0.997254i \(-0.476406\pi\)
0.0740534 + 0.997254i \(0.476406\pi\)
\(234\) −196032. −0.234039
\(235\) 1.68343e6 1.98850
\(236\) 2.64500e6 3.09133
\(237\) 642917. 0.743505
\(238\) −496380. −0.568031
\(239\) 210076. 0.237893 0.118946 0.992901i \(-0.462048\pi\)
0.118946 + 0.992901i \(0.462048\pi\)
\(240\) 173126. 0.194014
\(241\) 1.23224e6 1.36664 0.683319 0.730120i \(-0.260536\pi\)
0.683319 + 0.730120i \(0.260536\pi\)
\(242\) 277772. 0.304894
\(243\) 59049.0 0.0641500
\(244\) −850553. −0.914590
\(245\) −617656. −0.657402
\(246\) −1.54909e6 −1.63207
\(247\) −430670. −0.449162
\(248\) 409471. 0.422760
\(249\) 1.01322e6 1.03563
\(250\) −841849. −0.851891
\(251\) −545986. −0.547012 −0.273506 0.961870i \(-0.588183\pi\)
−0.273506 + 0.961870i \(0.588183\pi\)
\(252\) −402617. −0.399384
\(253\) 231048. 0.226934
\(254\) 1.56914e6 1.52608
\(255\) 375618. 0.361739
\(256\) −1.45779e6 −1.39026
\(257\) 556705. 0.525765 0.262883 0.964828i \(-0.415327\pi\)
0.262883 + 0.964828i \(0.415327\pi\)
\(258\) 1.03788e6 0.970731
\(259\) −79354.3 −0.0735057
\(260\) −1.01144e6 −0.927909
\(261\) −414840. −0.376946
\(262\) 1.21284e6 1.09157
\(263\) 1.65392e6 1.47443 0.737217 0.675656i \(-0.236140\pi\)
0.737217 + 0.675656i \(0.236140\pi\)
\(264\) −860144. −0.759558
\(265\) −1.02923e6 −0.900324
\(266\) −1.39539e6 −1.20919
\(267\) 1.24061e6 1.06501
\(268\) −2.28128e6 −1.94018
\(269\) −2.06545e6 −1.74034 −0.870168 0.492755i \(-0.835990\pi\)
−0.870168 + 0.492755i \(0.835990\pi\)
\(270\) 480630. 0.401237
\(271\) 708326. 0.585881 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(272\) 161434. 0.132304
\(273\) 209012. 0.169733
\(274\) −2.11946e6 −1.70549
\(275\) −807190. −0.643641
\(276\) −263784. −0.208438
\(277\) −17729.4 −0.0138833 −0.00694167 0.999976i \(-0.502210\pi\)
−0.00694167 + 0.999976i \(0.502210\pi\)
\(278\) −3.91890e6 −3.04125
\(279\) 151574. 0.116578
\(280\) −1.38437e6 −1.05526
\(281\) 754660. 0.570145 0.285073 0.958506i \(-0.407982\pi\)
0.285073 + 0.958506i \(0.407982\pi\)
\(282\) 2.00860e6 1.50408
\(283\) −417633. −0.309976 −0.154988 0.987916i \(-0.549534\pi\)
−0.154988 + 0.987916i \(0.549534\pi\)
\(284\) −2.52694e6 −1.85909
\(285\) 1.05591e6 0.770046
\(286\) 1.05703e6 0.764140
\(287\) 1.65166e6 1.18363
\(288\) −360610. −0.256186
\(289\) −1.06961e6 −0.753320
\(290\) −3.37659e6 −2.35767
\(291\) 375118. 0.259678
\(292\) −437604. −0.300348
\(293\) −2.68010e6 −1.82382 −0.911908 0.410394i \(-0.865391\pi\)
−0.911908 + 0.410394i \(0.865391\pi\)
\(294\) −736959. −0.497250
\(295\) 3.36658e6 2.25234
\(296\) 193552. 0.128401
\(297\) −318400. −0.209451
\(298\) 4.70511e6 3.06923
\(299\) 136940. 0.0885831
\(300\) 921559. 0.591181
\(301\) −1.10661e6 −0.704007
\(302\) 4.84229e6 3.05515
\(303\) 333903. 0.208936
\(304\) 453813. 0.281639
\(305\) −1.08259e6 −0.666369
\(306\) 448170. 0.273615
\(307\) −2.42927e6 −1.47106 −0.735528 0.677494i \(-0.763066\pi\)
−0.735528 + 0.677494i \(0.763066\pi\)
\(308\) 2.17097e6 1.30400
\(309\) −1.10991e6 −0.661288
\(310\) 1.23374e6 0.729155
\(311\) 1.71783e6 1.00712 0.503559 0.863961i \(-0.332024\pi\)
0.503559 + 0.863961i \(0.332024\pi\)
\(312\) −509798. −0.296491
\(313\) 1.56583e6 0.903406 0.451703 0.892168i \(-0.350817\pi\)
0.451703 + 0.892168i \(0.350817\pi\)
\(314\) −371606. −0.212696
\(315\) −512455. −0.290991
\(316\) 3.95789e6 2.22970
\(317\) 673750. 0.376574 0.188287 0.982114i \(-0.439706\pi\)
0.188287 + 0.982114i \(0.439706\pi\)
\(318\) −1.22803e6 −0.680993
\(319\) 2.23688e6 1.23074
\(320\) −3.55074e6 −1.93840
\(321\) 967099. 0.523852
\(322\) 443691. 0.238474
\(323\) 984603. 0.525115
\(324\) 363514. 0.192379
\(325\) −478413. −0.251243
\(326\) −1.47554e6 −0.768967
\(327\) 869874. 0.449870
\(328\) −4.02854e6 −2.06759
\(329\) −2.14159e6 −1.09081
\(330\) −2.59162e6 −1.31005
\(331\) 1.69768e6 0.851699 0.425850 0.904794i \(-0.359975\pi\)
0.425850 + 0.904794i \(0.359975\pi\)
\(332\) 6.23754e6 3.10576
\(333\) 71647.2 0.0354070
\(334\) −944259. −0.463153
\(335\) −2.90363e6 −1.41361
\(336\) −220244. −0.106428
\(337\) −2.37281e6 −1.13812 −0.569061 0.822295i \(-0.692693\pi\)
−0.569061 + 0.822295i \(0.692693\pi\)
\(338\) −2.84476e6 −1.35442
\(339\) 502099. 0.237296
\(340\) 2.31236e6 1.08482
\(341\) −817310. −0.380628
\(342\) 1.25987e6 0.582452
\(343\) 2.29357e6 1.05263
\(344\) 2.69910e6 1.22977
\(345\) −335747. −0.151867
\(346\) 575609. 0.258486
\(347\) 3.98316e6 1.77584 0.887921 0.459996i \(-0.152149\pi\)
0.887921 + 0.459996i \(0.152149\pi\)
\(348\) −2.55381e6 −1.13042
\(349\) −2.10463e6 −0.924935 −0.462468 0.886636i \(-0.653036\pi\)
−0.462468 + 0.886636i \(0.653036\pi\)
\(350\) −1.55008e6 −0.676370
\(351\) −188712. −0.0817585
\(352\) 1.94446e6 0.836453
\(353\) 3.41107e6 1.45698 0.728490 0.685056i \(-0.240222\pi\)
0.728490 + 0.685056i \(0.240222\pi\)
\(354\) 4.01685e6 1.70364
\(355\) −3.21631e6 −1.35453
\(356\) 7.63734e6 3.19387
\(357\) −477846. −0.198435
\(358\) −5.70795e6 −2.35382
\(359\) −1.86760e6 −0.764800 −0.382400 0.923997i \(-0.624902\pi\)
−0.382400 + 0.923997i \(0.624902\pi\)
\(360\) 1.24992e6 0.508306
\(361\) 291759. 0.117830
\(362\) 1.13477e6 0.455132
\(363\) 267400. 0.106511
\(364\) 1.28671e6 0.509011
\(365\) −556987. −0.218833
\(366\) −1.29170e6 −0.504033
\(367\) −437210. −0.169444 −0.0847218 0.996405i \(-0.527000\pi\)
−0.0847218 + 0.996405i \(0.527000\pi\)
\(368\) −144298. −0.0555445
\(369\) −1.49125e6 −0.570144
\(370\) 583173. 0.221459
\(371\) 1.30935e6 0.493879
\(372\) 933113. 0.349605
\(373\) −921734. −0.343031 −0.171516 0.985181i \(-0.554866\pi\)
−0.171516 + 0.985181i \(0.554866\pi\)
\(374\) −2.41660e6 −0.893357
\(375\) −810415. −0.297598
\(376\) 5.22352e6 1.90543
\(377\) 1.32577e6 0.480414
\(378\) −611438. −0.220102
\(379\) −3.80141e6 −1.35940 −0.679699 0.733491i \(-0.737889\pi\)
−0.679699 + 0.733491i \(0.737889\pi\)
\(380\) 6.50036e6 2.30929
\(381\) 1.51055e6 0.533118
\(382\) −2.75406e6 −0.965639
\(383\) −4.52311e6 −1.57558 −0.787790 0.615944i \(-0.788775\pi\)
−0.787790 + 0.615944i \(0.788775\pi\)
\(384\) −2.95442e6 −1.02245
\(385\) 2.76323e6 0.950090
\(386\) −83165.0 −0.0284101
\(387\) 999130. 0.339113
\(388\) 2.30928e6 0.778749
\(389\) 4.48168e6 1.50164 0.750822 0.660505i \(-0.229658\pi\)
0.750822 + 0.660505i \(0.229658\pi\)
\(390\) −1.53603e6 −0.511372
\(391\) −313072. −0.103563
\(392\) −1.91652e6 −0.629940
\(393\) 1.16756e6 0.381326
\(394\) −6.52671e6 −2.11814
\(395\) 5.03763e6 1.62455
\(396\) −1.96012e6 −0.628122
\(397\) 1.76656e6 0.562539 0.281269 0.959629i \(-0.409245\pi\)
0.281269 + 0.959629i \(0.409245\pi\)
\(398\) −4.98559e6 −1.57764
\(399\) −1.34329e6 −0.422414
\(400\) 504121. 0.157538
\(401\) −522373. −0.162226 −0.0811128 0.996705i \(-0.525847\pi\)
−0.0811128 + 0.996705i \(0.525847\pi\)
\(402\) −3.46448e6 −1.06924
\(403\) −484411. −0.148577
\(404\) 2.05555e6 0.626579
\(405\) 462684. 0.140167
\(406\) 4.29557e6 1.29332
\(407\) −386332. −0.115604
\(408\) 1.16550e6 0.346628
\(409\) −1.60642e6 −0.474845 −0.237422 0.971407i \(-0.576303\pi\)
−0.237422 + 0.971407i \(0.576303\pi\)
\(410\) −1.21380e7 −3.56606
\(411\) −2.04032e6 −0.595790
\(412\) −6.83275e6 −1.98314
\(413\) −4.28283e6 −1.23554
\(414\) −400599. −0.114871
\(415\) 7.93919e6 2.26285
\(416\) 1.15246e6 0.326507
\(417\) −3.77258e6 −1.06242
\(418\) −6.79339e6 −1.90172
\(419\) −5.30658e6 −1.47666 −0.738329 0.674441i \(-0.764385\pi\)
−0.738329 + 0.674441i \(0.764385\pi\)
\(420\) −3.15474e6 −0.872652
\(421\) −2.30575e6 −0.634025 −0.317012 0.948421i \(-0.602680\pi\)
−0.317012 + 0.948421i \(0.602680\pi\)
\(422\) −6.15549e6 −1.68260
\(423\) 1.93360e6 0.525430
\(424\) −3.19361e6 −0.862714
\(425\) 1.09375e6 0.293729
\(426\) −3.83756e6 −1.02455
\(427\) 1.37723e6 0.365541
\(428\) 5.95360e6 1.57098
\(429\) 1.01756e6 0.266943
\(430\) 8.13242e6 2.12104
\(431\) −1.27091e6 −0.329549 −0.164775 0.986331i \(-0.552690\pi\)
−0.164775 + 0.986331i \(0.552690\pi\)
\(432\) 198853. 0.0512652
\(433\) −3.71880e6 −0.953199 −0.476599 0.879121i \(-0.658131\pi\)
−0.476599 + 0.879121i \(0.658131\pi\)
\(434\) −1.56952e6 −0.399983
\(435\) −3.25052e6 −0.823625
\(436\) 5.35507e6 1.34911
\(437\) −880091. −0.220457
\(438\) −664571. −0.165522
\(439\) −3.51069e6 −0.869423 −0.434711 0.900570i \(-0.643150\pi\)
−0.434711 + 0.900570i \(0.643150\pi\)
\(440\) −6.73974e6 −1.65963
\(441\) −709442. −0.173708
\(442\) −1.43229e6 −0.348719
\(443\) 2.67040e6 0.646497 0.323248 0.946314i \(-0.395225\pi\)
0.323248 + 0.946314i \(0.395225\pi\)
\(444\) 441070. 0.106182
\(445\) 9.72088e6 2.32705
\(446\) 746950. 0.177809
\(447\) 4.52943e6 1.07220
\(448\) 4.51711e6 1.06332
\(449\) 808771. 0.189326 0.0946628 0.995509i \(-0.469823\pi\)
0.0946628 + 0.995509i \(0.469823\pi\)
\(450\) 1.39953e6 0.325801
\(451\) 8.04103e6 1.86153
\(452\) 3.09099e6 0.711625
\(453\) 4.66148e6 1.06728
\(454\) 6.69498e6 1.52444
\(455\) 1.63774e6 0.370864
\(456\) 3.27640e6 0.737878
\(457\) −6.03865e6 −1.35254 −0.676269 0.736655i \(-0.736404\pi\)
−0.676269 + 0.736655i \(0.736404\pi\)
\(458\) −5.43461e6 −1.21061
\(459\) 431436. 0.0955839
\(460\) −2.06691e6 −0.455435
\(461\) −6.06334e6 −1.32880 −0.664399 0.747378i \(-0.731313\pi\)
−0.664399 + 0.747378i \(0.731313\pi\)
\(462\) 3.29696e6 0.718636
\(463\) −987185. −0.214016 −0.107008 0.994258i \(-0.534127\pi\)
−0.107008 + 0.994258i \(0.534127\pi\)
\(464\) −1.39701e6 −0.301235
\(465\) 1.18768e6 0.254721
\(466\) 1.14745e6 0.244776
\(467\) −2.75869e6 −0.585344 −0.292672 0.956213i \(-0.594544\pi\)
−0.292672 + 0.956213i \(0.594544\pi\)
\(468\) −1.16174e6 −0.245185
\(469\) 3.69389e6 0.775446
\(470\) 1.57385e7 3.28639
\(471\) −357731. −0.0743027
\(472\) 1.04462e7 2.15825
\(473\) −5.38745e6 −1.10721
\(474\) 6.01068e6 1.22879
\(475\) 3.07469e6 0.625270
\(476\) −2.94169e6 −0.595085
\(477\) −1.18218e6 −0.237897
\(478\) 1.96402e6 0.393165
\(479\) −1.43871e6 −0.286507 −0.143254 0.989686i \(-0.545756\pi\)
−0.143254 + 0.989686i \(0.545756\pi\)
\(480\) −2.82559e6 −0.559765
\(481\) −228975. −0.0451258
\(482\) 1.15203e7 2.25864
\(483\) 427124. 0.0833080
\(484\) 1.64615e6 0.319416
\(485\) 2.93927e6 0.567395
\(486\) 552054. 0.106021
\(487\) −9.54793e6 −1.82426 −0.912130 0.409901i \(-0.865563\pi\)
−0.912130 + 0.409901i \(0.865563\pi\)
\(488\) −3.35917e6 −0.638532
\(489\) −1.42045e6 −0.268629
\(490\) −5.77451e6 −1.08649
\(491\) 8.94131e6 1.67378 0.836889 0.547373i \(-0.184372\pi\)
0.836889 + 0.547373i \(0.184372\pi\)
\(492\) −9.18035e6 −1.70980
\(493\) −3.03099e6 −0.561652
\(494\) −4.02637e6 −0.742329
\(495\) −2.49486e6 −0.457649
\(496\) 510441. 0.0931626
\(497\) 4.09167e6 0.743035
\(498\) 9.47269e6 1.71159
\(499\) 2.34135e6 0.420934 0.210467 0.977601i \(-0.432501\pi\)
0.210467 + 0.977601i \(0.432501\pi\)
\(500\) −4.98903e6 −0.892465
\(501\) −909002. −0.161797
\(502\) −5.10446e6 −0.904046
\(503\) 41307.7 0.00727966 0.00363983 0.999993i \(-0.498841\pi\)
0.00363983 + 0.999993i \(0.498841\pi\)
\(504\) −1.59010e6 −0.278835
\(505\) 2.61633e6 0.456524
\(506\) 2.16008e6 0.375054
\(507\) −2.73854e6 −0.473150
\(508\) 9.29918e6 1.59877
\(509\) 3.09166e6 0.528929 0.264464 0.964395i \(-0.414805\pi\)
0.264464 + 0.964395i \(0.414805\pi\)
\(510\) 3.51168e6 0.597846
\(511\) 708576. 0.120042
\(512\) −3.12440e6 −0.526735
\(513\) 1.21283e6 0.203473
\(514\) 5.20467e6 0.868932
\(515\) −8.69678e6 −1.44491
\(516\) 6.15078e6 1.01697
\(517\) −1.04262e7 −1.71554
\(518\) −741890. −0.121483
\(519\) 554117. 0.0902990
\(520\) −3.99457e6 −0.647831
\(521\) 6.02002e6 0.971635 0.485818 0.874060i \(-0.338522\pi\)
0.485818 + 0.874060i \(0.338522\pi\)
\(522\) −3.87837e6 −0.622979
\(523\) −6.39458e6 −1.02225 −0.511125 0.859506i \(-0.670771\pi\)
−0.511125 + 0.859506i \(0.670771\pi\)
\(524\) 7.18764e6 1.14356
\(525\) −1.49220e6 −0.236282
\(526\) 1.54626e7 2.43679
\(527\) 1.10746e6 0.173701
\(528\) −1.07224e6 −0.167382
\(529\) 279841. 0.0434783
\(530\) −9.62237e6 −1.48796
\(531\) 3.86687e6 0.595145
\(532\) −8.26950e6 −1.26678
\(533\) 4.76583e6 0.726642
\(534\) 1.15985e7 1.76015
\(535\) 7.57779e6 1.14461
\(536\) −9.00969e6 −1.35456
\(537\) −5.49483e6 −0.822278
\(538\) −1.93100e7 −2.87625
\(539\) 3.82541e6 0.567161
\(540\) 2.84835e6 0.420348
\(541\) −3.24248e6 −0.476304 −0.238152 0.971228i \(-0.576542\pi\)
−0.238152 + 0.971228i \(0.576542\pi\)
\(542\) 6.62219e6 0.968286
\(543\) 1.09240e6 0.158995
\(544\) −2.63476e6 −0.381719
\(545\) 6.81598e6 0.982963
\(546\) 1.95407e6 0.280517
\(547\) 9.39508e6 1.34255 0.671277 0.741206i \(-0.265746\pi\)
0.671277 + 0.741206i \(0.265746\pi\)
\(548\) −1.25605e7 −1.78671
\(549\) −1.24347e6 −0.176078
\(550\) −7.54648e6 −1.06375
\(551\) −8.52055e6 −1.19561
\(552\) −1.04179e6 −0.145523
\(553\) −6.40868e6 −0.891160
\(554\) −165753. −0.0229450
\(555\) 561398. 0.0773640
\(556\) −2.32245e7 −3.18610
\(557\) 2.95696e6 0.403839 0.201919 0.979402i \(-0.435282\pi\)
0.201919 + 0.979402i \(0.435282\pi\)
\(558\) 1.41708e6 0.192668
\(559\) −3.19308e6 −0.432196
\(560\) −1.72574e6 −0.232544
\(561\) −2.32636e6 −0.312083
\(562\) 7.05538e6 0.942279
\(563\) −1.35314e7 −1.79917 −0.899584 0.436747i \(-0.856130\pi\)
−0.899584 + 0.436747i \(0.856130\pi\)
\(564\) 1.19035e7 1.57571
\(565\) 3.93424e6 0.518489
\(566\) −3.90448e6 −0.512297
\(567\) −588608. −0.0768898
\(568\) −9.97991e6 −1.29794
\(569\) 1.32776e7 1.71924 0.859622 0.510931i \(-0.170699\pi\)
0.859622 + 0.510931i \(0.170699\pi\)
\(570\) 9.87183e6 1.27265
\(571\) 7.74960e6 0.994693 0.497346 0.867552i \(-0.334308\pi\)
0.497346 + 0.867552i \(0.334308\pi\)
\(572\) 6.26427e6 0.800535
\(573\) −2.65123e6 −0.337334
\(574\) 1.54415e7 1.95619
\(575\) −977654. −0.123315
\(576\) −4.07840e6 −0.512193
\(577\) −4.83479e6 −0.604558 −0.302279 0.953220i \(-0.597747\pi\)
−0.302279 + 0.953220i \(0.597747\pi\)
\(578\) −9.99984e6 −1.24501
\(579\) −80059.7 −0.00992471
\(580\) −2.00106e7 −2.46997
\(581\) −1.00999e7 −1.24130
\(582\) 3.50701e6 0.429170
\(583\) 6.37448e6 0.776736
\(584\) −1.72827e6 −0.209691
\(585\) −1.47867e6 −0.178642
\(586\) −2.50564e7 −3.01422
\(587\) 5.13380e6 0.614955 0.307478 0.951555i \(-0.400515\pi\)
0.307478 + 0.951555i \(0.400515\pi\)
\(588\) −4.36743e6 −0.520934
\(589\) 3.11324e6 0.369764
\(590\) 3.14744e7 3.72244
\(591\) −6.28301e6 −0.739945
\(592\) 241279. 0.0282953
\(593\) 1.29134e7 1.50801 0.754003 0.656871i \(-0.228121\pi\)
0.754003 + 0.656871i \(0.228121\pi\)
\(594\) −2.97675e6 −0.346160
\(595\) −3.74421e6 −0.433578
\(596\) 2.78838e7 3.21541
\(597\) −4.79944e6 −0.551130
\(598\) 1.28026e6 0.146401
\(599\) −5.79980e6 −0.660460 −0.330230 0.943901i \(-0.607126\pi\)
−0.330230 + 0.943901i \(0.607126\pi\)
\(600\) 3.63961e6 0.412740
\(601\) 1.06993e7 1.20829 0.604144 0.796875i \(-0.293515\pi\)
0.604144 + 0.796875i \(0.293515\pi\)
\(602\) −1.03457e7 −1.16351
\(603\) −3.33513e6 −0.373525
\(604\) 2.86968e7 3.20067
\(605\) 2.09524e6 0.232726
\(606\) 3.12169e6 0.345309
\(607\) −1.48509e7 −1.63600 −0.817998 0.575221i \(-0.804916\pi\)
−0.817998 + 0.575221i \(0.804916\pi\)
\(608\) −7.40669e6 −0.812578
\(609\) 4.13518e6 0.451805
\(610\) −1.01212e7 −1.10131
\(611\) −6.17951e6 −0.669655
\(612\) 2.65598e6 0.286646
\(613\) −7.64415e6 −0.821633 −0.410816 0.911718i \(-0.634756\pi\)
−0.410816 + 0.911718i \(0.634756\pi\)
\(614\) −2.27114e7 −2.43121
\(615\) −1.16848e7 −1.24576
\(616\) 8.57402e6 0.910401
\(617\) −8.02025e6 −0.848155 −0.424077 0.905626i \(-0.639402\pi\)
−0.424077 + 0.905626i \(0.639402\pi\)
\(618\) −1.03766e7 −1.09291
\(619\) 9.46459e6 0.992831 0.496416 0.868085i \(-0.334649\pi\)
0.496416 + 0.868085i \(0.334649\pi\)
\(620\) 7.31150e6 0.763883
\(621\) −385641. −0.0401286
\(622\) 1.60602e7 1.66446
\(623\) −1.23665e7 −1.27652
\(624\) −635507. −0.0653370
\(625\) −1.21255e7 −1.24165
\(626\) 1.46390e7 1.49306
\(627\) −6.53974e6 −0.664342
\(628\) −2.20224e6 −0.222826
\(629\) 523484. 0.0527566
\(630\) −4.79098e6 −0.480920
\(631\) 8.76525e6 0.876377 0.438189 0.898883i \(-0.355620\pi\)
0.438189 + 0.898883i \(0.355620\pi\)
\(632\) 1.56313e7 1.55669
\(633\) −5.92566e6 −0.587797
\(634\) 6.29894e6 0.622364
\(635\) 1.18361e7 1.16486
\(636\) −7.27767e6 −0.713427
\(637\) 2.26728e6 0.221389
\(638\) 2.09127e7 2.03404
\(639\) −3.69427e6 −0.357913
\(640\) −2.31496e7 −2.23406
\(641\) −8.04677e6 −0.773529 −0.386764 0.922179i \(-0.626407\pi\)
−0.386764 + 0.922179i \(0.626407\pi\)
\(642\) 9.04148e6 0.865769
\(643\) 1.81157e7 1.72794 0.863970 0.503543i \(-0.167970\pi\)
0.863970 + 0.503543i \(0.167970\pi\)
\(644\) 2.62944e6 0.249832
\(645\) 7.82877e6 0.740959
\(646\) 9.20513e6 0.867858
\(647\) 6.09047e6 0.571992 0.285996 0.958231i \(-0.407676\pi\)
0.285996 + 0.958231i \(0.407676\pi\)
\(648\) 1.43566e6 0.134312
\(649\) −2.08507e7 −1.94316
\(650\) −4.47272e6 −0.415230
\(651\) −1.51091e6 −0.139729
\(652\) −8.74448e6 −0.805591
\(653\) 1.73408e7 1.59142 0.795712 0.605675i \(-0.207097\pi\)
0.795712 + 0.605675i \(0.207097\pi\)
\(654\) 8.13252e6 0.743500
\(655\) 9.14850e6 0.833195
\(656\) −5.02193e6 −0.455629
\(657\) −639757. −0.0578232
\(658\) −2.00219e7 −1.80277
\(659\) 1.27618e7 1.14471 0.572357 0.820004i \(-0.306029\pi\)
0.572357 + 0.820004i \(0.306029\pi\)
\(660\) −1.53587e7 −1.37244
\(661\) −6.44195e6 −0.573474 −0.286737 0.958009i \(-0.592571\pi\)
−0.286737 + 0.958009i \(0.592571\pi\)
\(662\) 1.58718e7 1.40760
\(663\) −1.37881e6 −0.121821
\(664\) 2.46345e7 2.16832
\(665\) −1.05255e7 −0.922972
\(666\) 669836. 0.0585171
\(667\) 2.70926e6 0.235796
\(668\) −5.59594e6 −0.485213
\(669\) 719060. 0.0621155
\(670\) −2.71463e7 −2.33627
\(671\) 6.70496e6 0.574897
\(672\) 3.59460e6 0.307063
\(673\) 8.64864e6 0.736055 0.368028 0.929815i \(-0.380033\pi\)
0.368028 + 0.929815i \(0.380033\pi\)
\(674\) −2.21836e7 −1.88097
\(675\) 1.34728e6 0.113815
\(676\) −1.68588e7 −1.41893
\(677\) −1.22020e7 −1.02320 −0.511600 0.859223i \(-0.670947\pi\)
−0.511600 + 0.859223i \(0.670947\pi\)
\(678\) 4.69416e6 0.392178
\(679\) −3.73923e6 −0.311249
\(680\) 9.13242e6 0.757380
\(681\) 6.44500e6 0.532544
\(682\) −7.64110e6 −0.629064
\(683\) −1.10784e7 −0.908708 −0.454354 0.890821i \(-0.650130\pi\)
−0.454354 + 0.890821i \(0.650130\pi\)
\(684\) 7.46634e6 0.610194
\(685\) −1.59871e7 −1.30180
\(686\) 2.14427e7 1.73968
\(687\) −5.23170e6 −0.422913
\(688\) 3.36466e6 0.271001
\(689\) 3.77809e6 0.303196
\(690\) −3.13893e6 −0.250991
\(691\) −1.24814e7 −0.994415 −0.497207 0.867632i \(-0.665641\pi\)
−0.497207 + 0.867632i \(0.665641\pi\)
\(692\) 3.41122e6 0.270797
\(693\) 3.17386e6 0.251047
\(694\) 3.72389e7 2.93493
\(695\) −2.95604e7 −2.32139
\(696\) −1.00860e7 −0.789219
\(697\) −1.08957e7 −0.849518
\(698\) −1.96763e7 −1.52864
\(699\) 1.10461e6 0.0855094
\(700\) −9.18622e6 −0.708585
\(701\) −4.71570e6 −0.362452 −0.181226 0.983441i \(-0.558007\pi\)
−0.181226 + 0.983441i \(0.558007\pi\)
\(702\) −1.76429e6 −0.135122
\(703\) 1.47159e6 0.112305
\(704\) 2.19913e7 1.67232
\(705\) 1.51509e7 1.14806
\(706\) 3.18903e7 2.40795
\(707\) −3.32839e6 −0.250430
\(708\) 2.38050e7 1.78478
\(709\) 1.05958e7 0.791620 0.395810 0.918332i \(-0.370464\pi\)
0.395810 + 0.918332i \(0.370464\pi\)
\(710\) −3.00696e7 −2.23863
\(711\) 5.78625e6 0.429263
\(712\) 3.01629e7 2.22984
\(713\) −989912. −0.0729244
\(714\) −4.46742e6 −0.327953
\(715\) 7.97322e6 0.583268
\(716\) −3.38269e7 −2.46593
\(717\) 1.89068e6 0.137347
\(718\) −1.74603e7 −1.26398
\(719\) 2.95783e6 0.213379 0.106689 0.994292i \(-0.465975\pi\)
0.106689 + 0.994292i \(0.465975\pi\)
\(720\) 1.55813e6 0.112014
\(721\) 1.10637e7 0.792615
\(722\) 2.72768e6 0.194738
\(723\) 1.10902e7 0.789029
\(724\) 6.72499e6 0.476810
\(725\) −9.46510e6 −0.668776
\(726\) 2.49995e6 0.176031
\(727\) −1.30768e7 −0.917625 −0.458813 0.888533i \(-0.651725\pi\)
−0.458813 + 0.888533i \(0.651725\pi\)
\(728\) 5.08173e6 0.355372
\(729\) 531441. 0.0370370
\(730\) −5.20731e6 −0.361665
\(731\) 7.30005e6 0.505280
\(732\) −7.65497e6 −0.528039
\(733\) 5.06819e6 0.348412 0.174206 0.984709i \(-0.444264\pi\)
0.174206 + 0.984709i \(0.444264\pi\)
\(734\) −4.08751e6 −0.280039
\(735\) −5.55890e6 −0.379551
\(736\) 2.35509e6 0.160256
\(737\) 1.79835e7 1.21956
\(738\) −1.39418e7 −0.942277
\(739\) 5.39832e6 0.363620 0.181810 0.983334i \(-0.441804\pi\)
0.181810 + 0.983334i \(0.441804\pi\)
\(740\) 3.45605e6 0.232007
\(741\) −3.87603e6 −0.259324
\(742\) 1.22412e7 0.816233
\(743\) 3.71690e6 0.247007 0.123503 0.992344i \(-0.460587\pi\)
0.123503 + 0.992344i \(0.460587\pi\)
\(744\) 3.68524e6 0.244081
\(745\) 3.54908e7 2.34274
\(746\) −8.61736e6 −0.566927
\(747\) 9.11900e6 0.597923
\(748\) −1.43214e7 −0.935906
\(749\) −9.64017e6 −0.627885
\(750\) −7.57664e6 −0.491839
\(751\) 1.00711e7 0.651594 0.325797 0.945440i \(-0.394367\pi\)
0.325797 + 0.945440i \(0.394367\pi\)
\(752\) 6.51157e6 0.419896
\(753\) −4.91387e6 −0.315818
\(754\) 1.23947e7 0.793979
\(755\) 3.65255e7 2.33200
\(756\) −3.62355e6 −0.230585
\(757\) −8.49996e6 −0.539110 −0.269555 0.962985i \(-0.586877\pi\)
−0.269555 + 0.962985i \(0.586877\pi\)
\(758\) −3.55397e7 −2.24667
\(759\) 2.07943e6 0.131021
\(760\) 2.56725e7 1.61226
\(761\) 2.62780e7 1.64487 0.822434 0.568861i \(-0.192616\pi\)
0.822434 + 0.568861i \(0.192616\pi\)
\(762\) 1.41223e7 0.881084
\(763\) −8.67102e6 −0.539211
\(764\) −1.63213e7 −1.01163
\(765\) 3.38056e6 0.208850
\(766\) −4.22869e7 −2.60396
\(767\) −1.23580e7 −0.758506
\(768\) −1.31201e7 −0.802666
\(769\) 899889. 0.0548748 0.0274374 0.999624i \(-0.491265\pi\)
0.0274374 + 0.999624i \(0.491265\pi\)
\(770\) 2.58336e7 1.57021
\(771\) 5.01034e6 0.303551
\(772\) −492859. −0.0297632
\(773\) −1.33758e7 −0.805140 −0.402570 0.915389i \(-0.631883\pi\)
−0.402570 + 0.915389i \(0.631883\pi\)
\(774\) 9.34094e6 0.560452
\(775\) 3.45836e6 0.206831
\(776\) 9.12028e6 0.543693
\(777\) −714189. −0.0424385
\(778\) 4.18996e7 2.48177
\(779\) −3.06293e7 −1.80840
\(780\) −9.10293e6 −0.535728
\(781\) 1.99200e7 1.16859
\(782\) −2.92694e6 −0.171158
\(783\) −3.73356e6 −0.217630
\(784\) −2.38911e6 −0.138818
\(785\) −2.80304e6 −0.162351
\(786\) 1.09156e7 0.630217
\(787\) 1.67690e7 0.965098 0.482549 0.875869i \(-0.339711\pi\)
0.482549 + 0.875869i \(0.339711\pi\)
\(788\) −3.86791e7 −2.21902
\(789\) 1.48853e7 0.851264
\(790\) 4.70972e7 2.68490
\(791\) −5.00498e6 −0.284421
\(792\) −7.74129e6 −0.438531
\(793\) 3.97396e6 0.224409
\(794\) 1.65157e7 0.929707
\(795\) −9.26309e6 −0.519802
\(796\) −2.95460e7 −1.65278
\(797\) 2.78280e7 1.55180 0.775902 0.630854i \(-0.217295\pi\)
0.775902 + 0.630854i \(0.217295\pi\)
\(798\) −1.25585e7 −0.698123
\(799\) 1.41277e7 0.782894
\(800\) −8.22776e6 −0.454524
\(801\) 1.11654e7 0.614886
\(802\) −4.88370e6 −0.268110
\(803\) 3.44966e6 0.188794
\(804\) −2.05315e7 −1.12016
\(805\) 3.34677e6 0.182027
\(806\) −4.52880e6 −0.245553
\(807\) −1.85890e7 −1.00478
\(808\) 8.11821e6 0.437454
\(809\) −1.98606e7 −1.06689 −0.533447 0.845834i \(-0.679103\pi\)
−0.533447 + 0.845834i \(0.679103\pi\)
\(810\) 4.32567e6 0.231654
\(811\) 1.95245e7 1.04238 0.521192 0.853439i \(-0.325488\pi\)
0.521192 + 0.853439i \(0.325488\pi\)
\(812\) 2.54568e7 1.35492
\(813\) 6.37493e6 0.338259
\(814\) −3.61185e6 −0.191059
\(815\) −1.11301e7 −0.586953
\(816\) 1.45290e6 0.0763855
\(817\) 2.05215e7 1.07561
\(818\) −1.50186e7 −0.784775
\(819\) 1.88111e6 0.0979952
\(820\) −7.19335e7 −3.73591
\(821\) −2.39444e7 −1.23978 −0.619891 0.784688i \(-0.712823\pi\)
−0.619891 + 0.784688i \(0.712823\pi\)
\(822\) −1.90751e7 −0.984662
\(823\) −2.78310e7 −1.43229 −0.716143 0.697953i \(-0.754094\pi\)
−0.716143 + 0.697953i \(0.754094\pi\)
\(824\) −2.69853e7 −1.38455
\(825\) −7.26471e6 −0.371607
\(826\) −4.00405e7 −2.04197
\(827\) −1.94204e7 −0.987405 −0.493703 0.869631i \(-0.664357\pi\)
−0.493703 + 0.869631i \(0.664357\pi\)
\(828\) −2.37406e6 −0.120342
\(829\) 2.66039e7 1.34449 0.672247 0.740327i \(-0.265329\pi\)
0.672247 + 0.740327i \(0.265329\pi\)
\(830\) 7.42241e7 3.73981
\(831\) −159564. −0.00801555
\(832\) 1.30340e7 0.652784
\(833\) −5.18348e6 −0.258826
\(834\) −3.52701e7 −1.75587
\(835\) −7.12257e6 −0.353525
\(836\) −4.02596e7 −1.99229
\(837\) 1.36417e6 0.0673062
\(838\) −4.96117e7 −2.44047
\(839\) −1.41085e7 −0.691954 −0.345977 0.938243i \(-0.612453\pi\)
−0.345977 + 0.938243i \(0.612453\pi\)
\(840\) −1.24594e7 −0.609253
\(841\) 5.71843e6 0.278796
\(842\) −2.15566e7 −1.04785
\(843\) 6.79194e6 0.329174
\(844\) −3.64792e7 −1.76274
\(845\) −2.14581e7 −1.03383
\(846\) 1.80774e7 0.868378
\(847\) −2.66548e6 −0.127664
\(848\) −3.98111e6 −0.190114
\(849\) −3.75869e6 −0.178965
\(850\) 1.02256e7 0.485445
\(851\) −467918. −0.0221486
\(852\) −2.27425e7 −1.07334
\(853\) −1.87882e7 −0.884122 −0.442061 0.896985i \(-0.645752\pi\)
−0.442061 + 0.896985i \(0.645752\pi\)
\(854\) 1.28758e7 0.604130
\(855\) 9.50323e6 0.444586
\(856\) 2.35131e7 1.09680
\(857\) 3.67369e7 1.70864 0.854320 0.519748i \(-0.173974\pi\)
0.854320 + 0.519748i \(0.173974\pi\)
\(858\) 9.51329e6 0.441176
\(859\) 2.03228e6 0.0939723 0.0469861 0.998896i \(-0.485038\pi\)
0.0469861 + 0.998896i \(0.485038\pi\)
\(860\) 4.81950e7 2.22206
\(861\) 1.48650e7 0.683371
\(862\) −1.18818e7 −0.544646
\(863\) −1.21735e7 −0.556400 −0.278200 0.960523i \(-0.589738\pi\)
−0.278200 + 0.960523i \(0.589738\pi\)
\(864\) −3.24549e6 −0.147909
\(865\) 4.34183e6 0.197303
\(866\) −3.47674e7 −1.57535
\(867\) −9.62646e6 −0.434930
\(868\) −9.30140e6 −0.419034
\(869\) −3.12003e7 −1.40155
\(870\) −3.03893e7 −1.36120
\(871\) 1.06586e7 0.476053
\(872\) 2.11493e7 0.941901
\(873\) 3.37606e6 0.149925
\(874\) −8.22804e6 −0.364349
\(875\) 8.07833e6 0.356698
\(876\) −3.93844e6 −0.173406
\(877\) 4.13453e7 1.81521 0.907605 0.419825i \(-0.137909\pi\)
0.907605 + 0.419825i \(0.137909\pi\)
\(878\) −3.28217e7 −1.43689
\(879\) −2.41209e7 −1.05298
\(880\) −8.40166e6 −0.365728
\(881\) 1.92417e7 0.835227 0.417613 0.908625i \(-0.362867\pi\)
0.417613 + 0.908625i \(0.362867\pi\)
\(882\) −6.63263e6 −0.287088
\(883\) 4.59010e6 0.198117 0.0990583 0.995082i \(-0.468417\pi\)
0.0990583 + 0.995082i \(0.468417\pi\)
\(884\) −8.48815e6 −0.365328
\(885\) 3.02992e7 1.30039
\(886\) 2.49657e7 1.06846
\(887\) −1.34822e7 −0.575374 −0.287687 0.957724i \(-0.592886\pi\)
−0.287687 + 0.957724i \(0.592886\pi\)
\(888\) 1.74196e6 0.0741322
\(889\) −1.50574e7 −0.638992
\(890\) 9.08813e7 3.84591
\(891\) −2.86560e6 −0.120927
\(892\) 4.42663e6 0.186278
\(893\) 3.97149e7 1.66657
\(894\) 4.23460e7 1.77202
\(895\) −4.30552e7 −1.79667
\(896\) 2.94500e7 1.22551
\(897\) 1.23246e6 0.0511435
\(898\) 7.56126e6 0.312898
\(899\) −9.58377e6 −0.395492
\(900\) 8.29403e6 0.341318
\(901\) −8.63750e6 −0.354467
\(902\) 7.51762e7 3.07655
\(903\) −9.95945e6 −0.406458
\(904\) 1.22076e7 0.496830
\(905\) 8.55962e6 0.347403
\(906\) 4.35806e7 1.76389
\(907\) 5.13039e6 0.207077 0.103539 0.994625i \(-0.466983\pi\)
0.103539 + 0.994625i \(0.466983\pi\)
\(908\) 3.96763e7 1.59704
\(909\) 3.00513e6 0.120629
\(910\) 1.53113e7 0.612927
\(911\) 3.88471e7 1.55082 0.775411 0.631456i \(-0.217543\pi\)
0.775411 + 0.631456i \(0.217543\pi\)
\(912\) 4.08432e6 0.162604
\(913\) −4.91709e7 −1.95223
\(914\) −5.64558e7 −2.23534
\(915\) −9.74332e6 −0.384728
\(916\) −3.22071e7 −1.26827
\(917\) −1.16384e7 −0.457055
\(918\) 4.03353e6 0.157971
\(919\) 2.43482e7 0.950993 0.475496 0.879718i \(-0.342268\pi\)
0.475496 + 0.879718i \(0.342268\pi\)
\(920\) −8.16305e6 −0.317968
\(921\) −2.18634e7 −0.849315
\(922\) −5.66866e7 −2.19610
\(923\) 1.18064e7 0.456156
\(924\) 1.95387e7 0.752863
\(925\) 1.63472e6 0.0628188
\(926\) −9.22927e6 −0.353704
\(927\) −9.98917e6 −0.381795
\(928\) 2.28007e7 0.869116
\(929\) 2.17027e7 0.825040 0.412520 0.910948i \(-0.364649\pi\)
0.412520 + 0.910948i \(0.364649\pi\)
\(930\) 1.11037e7 0.420978
\(931\) −1.45715e7 −0.550972
\(932\) 6.80010e6 0.256434
\(933\) 1.54605e7 0.581460
\(934\) −2.57912e7 −0.967397
\(935\) −1.82284e7 −0.681900
\(936\) −4.58818e6 −0.171179
\(937\) −2.01218e6 −0.0748718 −0.0374359 0.999299i \(-0.511919\pi\)
−0.0374359 + 0.999299i \(0.511919\pi\)
\(938\) 3.45344e7 1.28158
\(939\) 1.40925e7 0.521582
\(940\) 9.32710e7 3.44292
\(941\) −1.17136e7 −0.431237 −0.215618 0.976478i \(-0.569177\pi\)
−0.215618 + 0.976478i \(0.569177\pi\)
\(942\) −3.34446e6 −0.122800
\(943\) 9.73916e6 0.356650
\(944\) 1.30220e7 0.475608
\(945\) −4.61209e6 −0.168004
\(946\) −5.03676e7 −1.82989
\(947\) 571455. 0.0207065 0.0103533 0.999946i \(-0.496704\pi\)
0.0103533 + 0.999946i \(0.496704\pi\)
\(948\) 3.56210e7 1.28732
\(949\) 2.04458e6 0.0736950
\(950\) 2.87455e7 1.03338
\(951\) 6.06375e6 0.217415
\(952\) −1.16179e7 −0.415466
\(953\) 2.03678e7 0.726461 0.363231 0.931699i \(-0.381674\pi\)
0.363231 + 0.931699i \(0.381674\pi\)
\(954\) −1.10523e7 −0.393171
\(955\) −2.07739e7 −0.737073
\(956\) 1.16393e7 0.411891
\(957\) 2.01319e7 0.710566
\(958\) −1.34506e7 −0.473510
\(959\) 2.03382e7 0.714110
\(960\) −3.19567e7 −1.11914
\(961\) −2.51274e7 −0.877687
\(962\) −2.14070e6 −0.0745794
\(963\) 8.70389e6 0.302446
\(964\) 6.82727e7 2.36622
\(965\) −627316. −0.0216854
\(966\) 3.99322e6 0.137683
\(967\) −3.93183e7 −1.35216 −0.676081 0.736827i \(-0.736323\pi\)
−0.676081 + 0.736827i \(0.736323\pi\)
\(968\) 6.50132e6 0.223004
\(969\) 8.86142e6 0.303175
\(970\) 2.74795e7 0.937734
\(971\) 4.86930e7 1.65736 0.828682 0.559719i \(-0.189091\pi\)
0.828682 + 0.559719i \(0.189091\pi\)
\(972\) 3.27163e6 0.111070
\(973\) 3.76055e7 1.27341
\(974\) −8.92643e7 −3.01495
\(975\) −4.30572e6 −0.145055
\(976\) −4.18750e6 −0.140712
\(977\) −1.11758e7 −0.374577 −0.187288 0.982305i \(-0.559970\pi\)
−0.187288 + 0.982305i \(0.559970\pi\)
\(978\) −1.32799e7 −0.443963
\(979\) −6.02056e7 −2.00762
\(980\) −3.42214e7 −1.13824
\(981\) 7.82887e6 0.259733
\(982\) 8.35930e7 2.76625
\(983\) 2.09439e7 0.691310 0.345655 0.938362i \(-0.387657\pi\)
0.345655 + 0.938362i \(0.387657\pi\)
\(984\) −3.62569e7 −1.19372
\(985\) −4.92311e7 −1.61677
\(986\) −2.83370e7 −0.928242
\(987\) −1.92744e7 −0.629777
\(988\) −2.38614e7 −0.777685
\(989\) −6.52518e6 −0.212130
\(990\) −2.33246e7 −0.756356
\(991\) −1.95125e7 −0.631144 −0.315572 0.948902i \(-0.602196\pi\)
−0.315572 + 0.948902i \(0.602196\pi\)
\(992\) −8.33092e6 −0.268790
\(993\) 1.52791e7 0.491729
\(994\) 3.82533e7 1.22801
\(995\) −3.76064e7 −1.20422
\(996\) 5.61378e7 1.79311
\(997\) 1.32696e7 0.422786 0.211393 0.977401i \(-0.432200\pi\)
0.211393 + 0.977401i \(0.432200\pi\)
\(998\) 2.18894e7 0.695678
\(999\) 644825. 0.0204422
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.1.5 5
3.2 odd 2 207.6.a.f.1.1 5
4.3 odd 2 1104.6.a.r.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.5 5 1.1 even 1 trivial
207.6.a.f.1.1 5 3.2 odd 2
1104.6.a.r.1.4 5 4.3 odd 2