Properties

Label 825.6.a.u.1.5
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.444648\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.444648 q^{2} -9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} -205.804 q^{7} +28.3695 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-0.444648 q^{2} -9.00000 q^{3} -31.8023 q^{4} +4.00183 q^{6} -205.804 q^{7} +28.3695 q^{8} +81.0000 q^{9} +121.000 q^{11} +286.221 q^{12} +893.005 q^{13} +91.5105 q^{14} +1005.06 q^{16} -836.291 q^{17} -36.0165 q^{18} +816.231 q^{19} +1852.24 q^{21} -53.8024 q^{22} -3125.01 q^{23} -255.326 q^{24} -397.073 q^{26} -729.000 q^{27} +6545.05 q^{28} -7307.37 q^{29} +152.387 q^{31} -1354.72 q^{32} -1089.00 q^{33} +371.855 q^{34} -2575.99 q^{36} -8734.38 q^{37} -362.935 q^{38} -8037.05 q^{39} -9608.76 q^{41} -823.594 q^{42} -9208.31 q^{43} -3848.08 q^{44} +1389.53 q^{46} -11806.9 q^{47} -9045.53 q^{48} +25548.5 q^{49} +7526.62 q^{51} -28399.6 q^{52} -12879.4 q^{53} +324.148 q^{54} -5838.58 q^{56} -7346.08 q^{57} +3249.21 q^{58} -517.608 q^{59} -50201.8 q^{61} -67.7587 q^{62} -16670.2 q^{63} -31559.5 q^{64} +484.221 q^{66} -4050.14 q^{67} +26596.0 q^{68} +28125.1 q^{69} +16467.7 q^{71} +2297.93 q^{72} +55131.2 q^{73} +3883.72 q^{74} -25958.0 q^{76} -24902.3 q^{77} +3573.66 q^{78} +25009.6 q^{79} +6561.00 q^{81} +4272.51 q^{82} +76147.7 q^{83} -58905.5 q^{84} +4094.45 q^{86} +65766.3 q^{87} +3432.72 q^{88} +104461. q^{89} -183784. q^{91} +99382.6 q^{92} -1371.49 q^{93} +5249.89 q^{94} +12192.5 q^{96} -39347.8 q^{97} -11360.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.444648 −0.0786034 −0.0393017 0.999227i \(-0.512513\pi\)
−0.0393017 + 0.999227i \(0.512513\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.8023 −0.993822
\(5\) 0 0
\(6\) 4.00183 0.0453817
\(7\) −205.804 −1.58748 −0.793742 0.608254i \(-0.791870\pi\)
−0.793742 + 0.608254i \(0.791870\pi\)
\(8\) 28.3695 0.156721
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 286.221 0.573783
\(13\) 893.005 1.46553 0.732767 0.680480i \(-0.238229\pi\)
0.732767 + 0.680480i \(0.238229\pi\)
\(14\) 91.5105 0.124782
\(15\) 0 0
\(16\) 1005.06 0.981503
\(17\) −836.291 −0.701835 −0.350918 0.936406i \(-0.614130\pi\)
−0.350918 + 0.936406i \(0.614130\pi\)
\(18\) −36.0165 −0.0262011
\(19\) 816.231 0.518715 0.259358 0.965781i \(-0.416489\pi\)
0.259358 + 0.965781i \(0.416489\pi\)
\(20\) 0 0
\(21\) 1852.24 0.916535
\(22\) −53.8024 −0.0236998
\(23\) −3125.01 −1.23178 −0.615889 0.787833i \(-0.711203\pi\)
−0.615889 + 0.787833i \(0.711203\pi\)
\(24\) −255.326 −0.0904830
\(25\) 0 0
\(26\) −397.073 −0.115196
\(27\) −729.000 −0.192450
\(28\) 6545.05 1.57768
\(29\) −7307.37 −1.61349 −0.806745 0.590900i \(-0.798773\pi\)
−0.806745 + 0.590900i \(0.798773\pi\)
\(30\) 0 0
\(31\) 152.387 0.0284803 0.0142401 0.999899i \(-0.495467\pi\)
0.0142401 + 0.999899i \(0.495467\pi\)
\(32\) −1354.72 −0.233871
\(33\) −1089.00 −0.174078
\(34\) 371.855 0.0551666
\(35\) 0 0
\(36\) −2575.99 −0.331274
\(37\) −8734.38 −1.04888 −0.524442 0.851446i \(-0.675726\pi\)
−0.524442 + 0.851446i \(0.675726\pi\)
\(38\) −362.935 −0.0407728
\(39\) −8037.05 −0.846126
\(40\) 0 0
\(41\) −9608.76 −0.892704 −0.446352 0.894857i \(-0.647277\pi\)
−0.446352 + 0.894857i \(0.647277\pi\)
\(42\) −823.594 −0.0720427
\(43\) −9208.31 −0.759467 −0.379733 0.925096i \(-0.623984\pi\)
−0.379733 + 0.925096i \(0.623984\pi\)
\(44\) −3848.08 −0.299648
\(45\) 0 0
\(46\) 1389.53 0.0968219
\(47\) −11806.9 −0.779632 −0.389816 0.920893i \(-0.627461\pi\)
−0.389816 + 0.920893i \(0.627461\pi\)
\(48\) −9045.53 −0.566671
\(49\) 25548.5 1.52011
\(50\) 0 0
\(51\) 7526.62 0.405205
\(52\) −28399.6 −1.45648
\(53\) −12879.4 −0.629807 −0.314904 0.949124i \(-0.601972\pi\)
−0.314904 + 0.949124i \(0.601972\pi\)
\(54\) 324.148 0.0151272
\(55\) 0 0
\(56\) −5838.58 −0.248792
\(57\) −7346.08 −0.299480
\(58\) 3249.21 0.126826
\(59\) −517.608 −0.0193585 −0.00967924 0.999953i \(-0.503081\pi\)
−0.00967924 + 0.999953i \(0.503081\pi\)
\(60\) 0 0
\(61\) −50201.8 −1.72741 −0.863704 0.503999i \(-0.831861\pi\)
−0.863704 + 0.503999i \(0.831861\pi\)
\(62\) −67.7587 −0.00223865
\(63\) −16670.2 −0.529162
\(64\) −31559.5 −0.963120
\(65\) 0 0
\(66\) 484.221 0.0136831
\(67\) −4050.14 −0.110226 −0.0551129 0.998480i \(-0.517552\pi\)
−0.0551129 + 0.998480i \(0.517552\pi\)
\(68\) 26596.0 0.697499
\(69\) 28125.1 0.711167
\(70\) 0 0
\(71\) 16467.7 0.387692 0.193846 0.981032i \(-0.437904\pi\)
0.193846 + 0.981032i \(0.437904\pi\)
\(72\) 2297.93 0.0522404
\(73\) 55131.2 1.21085 0.605425 0.795903i \(-0.293003\pi\)
0.605425 + 0.795903i \(0.293003\pi\)
\(74\) 3883.72 0.0824458
\(75\) 0 0
\(76\) −25958.0 −0.515510
\(77\) −24902.3 −0.478645
\(78\) 3573.66 0.0665084
\(79\) 25009.6 0.450858 0.225429 0.974260i \(-0.427622\pi\)
0.225429 + 0.974260i \(0.427622\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 4272.51 0.0701696
\(83\) 76147.7 1.21328 0.606641 0.794976i \(-0.292517\pi\)
0.606641 + 0.794976i \(0.292517\pi\)
\(84\) −58905.5 −0.910872
\(85\) 0 0
\(86\) 4094.45 0.0596966
\(87\) 65766.3 0.931548
\(88\) 3432.72 0.0472532
\(89\) 104461. 1.39790 0.698952 0.715169i \(-0.253650\pi\)
0.698952 + 0.715169i \(0.253650\pi\)
\(90\) 0 0
\(91\) −183784. −2.32651
\(92\) 99382.6 1.22417
\(93\) −1371.49 −0.0164431
\(94\) 5249.89 0.0612817
\(95\) 0 0
\(96\) 12192.5 0.135025
\(97\) −39347.8 −0.424611 −0.212306 0.977203i \(-0.568097\pi\)
−0.212306 + 0.977203i \(0.568097\pi\)
\(98\) −11360.1 −0.119486
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −96213.1 −0.938493 −0.469246 0.883067i \(-0.655474\pi\)
−0.469246 + 0.883067i \(0.655474\pi\)
\(102\) −3346.70 −0.0318505
\(103\) −54269.8 −0.504040 −0.252020 0.967722i \(-0.581095\pi\)
−0.252020 + 0.967722i \(0.581095\pi\)
\(104\) 25334.2 0.229680
\(105\) 0 0
\(106\) 5726.82 0.0495050
\(107\) 46180.6 0.389942 0.194971 0.980809i \(-0.437539\pi\)
0.194971 + 0.980809i \(0.437539\pi\)
\(108\) 23183.9 0.191261
\(109\) 162090. 1.30675 0.653373 0.757036i \(-0.273353\pi\)
0.653373 + 0.757036i \(0.273353\pi\)
\(110\) 0 0
\(111\) 78609.4 0.605574
\(112\) −206846. −1.55812
\(113\) −252815. −1.86255 −0.931274 0.364321i \(-0.881301\pi\)
−0.931274 + 0.364321i \(0.881301\pi\)
\(114\) 3266.42 0.0235402
\(115\) 0 0
\(116\) 232391. 1.60352
\(117\) 72333.4 0.488511
\(118\) 230.153 0.00152164
\(119\) 172112. 1.11415
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 22322.1 0.135780
\(123\) 86478.8 0.515403
\(124\) −4846.26 −0.0283043
\(125\) 0 0
\(126\) 7412.35 0.0415939
\(127\) −171792. −0.945135 −0.472568 0.881294i \(-0.656673\pi\)
−0.472568 + 0.881294i \(0.656673\pi\)
\(128\) 57384.0 0.309575
\(129\) 82874.8 0.438478
\(130\) 0 0
\(131\) 140420. 0.714911 0.357455 0.933930i \(-0.383644\pi\)
0.357455 + 0.933930i \(0.383644\pi\)
\(132\) 34632.7 0.173002
\(133\) −167984. −0.823453
\(134\) 1800.89 0.00866411
\(135\) 0 0
\(136\) −23725.2 −0.109992
\(137\) −194429. −0.885034 −0.442517 0.896760i \(-0.645914\pi\)
−0.442517 + 0.896760i \(0.645914\pi\)
\(138\) −12505.8 −0.0559001
\(139\) −246396. −1.08168 −0.540838 0.841127i \(-0.681893\pi\)
−0.540838 + 0.841127i \(0.681893\pi\)
\(140\) 0 0
\(141\) 106262. 0.450121
\(142\) −7322.32 −0.0304739
\(143\) 108054. 0.441875
\(144\) 81409.8 0.327168
\(145\) 0 0
\(146\) −24514.0 −0.0951768
\(147\) −229936. −0.877635
\(148\) 277773. 1.04240
\(149\) −329064. −1.21427 −0.607135 0.794599i \(-0.707681\pi\)
−0.607135 + 0.794599i \(0.707681\pi\)
\(150\) 0 0
\(151\) 125794. 0.448971 0.224486 0.974477i \(-0.427930\pi\)
0.224486 + 0.974477i \(0.427930\pi\)
\(152\) 23156.1 0.0812936
\(153\) −67739.6 −0.233945
\(154\) 11072.8 0.0376231
\(155\) 0 0
\(156\) 255597. 0.840898
\(157\) −21521.1 −0.0696810 −0.0348405 0.999393i \(-0.511092\pi\)
−0.0348405 + 0.999393i \(0.511092\pi\)
\(158\) −11120.5 −0.0354389
\(159\) 115915. 0.363619
\(160\) 0 0
\(161\) 643142. 1.95543
\(162\) −2917.33 −0.00873371
\(163\) −383784. −1.13140 −0.565702 0.824610i \(-0.691395\pi\)
−0.565702 + 0.824610i \(0.691395\pi\)
\(164\) 305580. 0.887189
\(165\) 0 0
\(166\) −33858.9 −0.0953681
\(167\) −512066. −1.42081 −0.710403 0.703795i \(-0.751487\pi\)
−0.710403 + 0.703795i \(0.751487\pi\)
\(168\) 52547.2 0.143640
\(169\) 426166. 1.14779
\(170\) 0 0
\(171\) 66114.7 0.172905
\(172\) 292845. 0.754774
\(173\) 260098. 0.660726 0.330363 0.943854i \(-0.392829\pi\)
0.330363 + 0.943854i \(0.392829\pi\)
\(174\) −29242.8 −0.0732228
\(175\) 0 0
\(176\) 121612. 0.295934
\(177\) 4658.47 0.0111766
\(178\) −46448.2 −0.109880
\(179\) 112495. 0.262421 0.131211 0.991354i \(-0.458114\pi\)
0.131211 + 0.991354i \(0.458114\pi\)
\(180\) 0 0
\(181\) 56459.6 0.128098 0.0640488 0.997947i \(-0.479599\pi\)
0.0640488 + 0.997947i \(0.479599\pi\)
\(182\) 81719.4 0.182872
\(183\) 451817. 0.997320
\(184\) −88655.2 −0.193046
\(185\) 0 0
\(186\) 609.828 0.00129248
\(187\) −101191. −0.211611
\(188\) 375485. 0.774815
\(189\) 150031. 0.305512
\(190\) 0 0
\(191\) 356386. 0.706865 0.353433 0.935460i \(-0.385014\pi\)
0.353433 + 0.935460i \(0.385014\pi\)
\(192\) 284036. 0.556057
\(193\) −607666. −1.17428 −0.587140 0.809486i \(-0.699746\pi\)
−0.587140 + 0.809486i \(0.699746\pi\)
\(194\) 17495.9 0.0333759
\(195\) 0 0
\(196\) −812500. −1.51072
\(197\) 91261.2 0.167541 0.0837704 0.996485i \(-0.473304\pi\)
0.0837704 + 0.996485i \(0.473304\pi\)
\(198\) −4357.99 −0.00789994
\(199\) 840148. 1.50392 0.751958 0.659212i \(-0.229110\pi\)
0.751958 + 0.659212i \(0.229110\pi\)
\(200\) 0 0
\(201\) 36451.3 0.0636389
\(202\) 42781.0 0.0737687
\(203\) 1.50389e6 2.56139
\(204\) −239364. −0.402701
\(205\) 0 0
\(206\) 24130.9 0.0396192
\(207\) −253126. −0.410593
\(208\) 897523. 1.43843
\(209\) 98763.9 0.156399
\(210\) 0 0
\(211\) 945258. 1.46165 0.730826 0.682563i \(-0.239135\pi\)
0.730826 + 0.682563i \(0.239135\pi\)
\(212\) 409596. 0.625916
\(213\) −148209. −0.223834
\(214\) −20534.1 −0.0306508
\(215\) 0 0
\(216\) −20681.4 −0.0301610
\(217\) −31362.0 −0.0452120
\(218\) −72073.2 −0.102715
\(219\) −496181. −0.699084
\(220\) 0 0
\(221\) −746813. −1.02856
\(222\) −34953.5 −0.0476001
\(223\) 951062. 1.28070 0.640349 0.768084i \(-0.278790\pi\)
0.640349 + 0.768084i \(0.278790\pi\)
\(224\) 278808. 0.371266
\(225\) 0 0
\(226\) 112414. 0.146402
\(227\) −1.13419e6 −1.46090 −0.730448 0.682968i \(-0.760689\pi\)
−0.730448 + 0.682968i \(0.760689\pi\)
\(228\) 233622. 0.297630
\(229\) 1.57670e6 1.98682 0.993412 0.114595i \(-0.0365569\pi\)
0.993412 + 0.114595i \(0.0365569\pi\)
\(230\) 0 0
\(231\) 224121. 0.276346
\(232\) −207307. −0.252868
\(233\) −1.06285e6 −1.28258 −0.641289 0.767299i \(-0.721600\pi\)
−0.641289 + 0.767299i \(0.721600\pi\)
\(234\) −32162.9 −0.0383986
\(235\) 0 0
\(236\) 16461.1 0.0192389
\(237\) −225087. −0.260303
\(238\) −76529.4 −0.0875762
\(239\) −62094.6 −0.0703168 −0.0351584 0.999382i \(-0.511194\pi\)
−0.0351584 + 0.999382i \(0.511194\pi\)
\(240\) 0 0
\(241\) 90536.8 0.100411 0.0502056 0.998739i \(-0.484012\pi\)
0.0502056 + 0.998739i \(0.484012\pi\)
\(242\) −6510.09 −0.00714576
\(243\) −59049.0 −0.0641500
\(244\) 1.59653e6 1.71674
\(245\) 0 0
\(246\) −38452.6 −0.0405124
\(247\) 728899. 0.760194
\(248\) 4323.16 0.00446346
\(249\) −685330. −0.700489
\(250\) 0 0
\(251\) −1.35882e6 −1.36138 −0.680689 0.732572i \(-0.738320\pi\)
−0.680689 + 0.732572i \(0.738320\pi\)
\(252\) 530149. 0.525892
\(253\) −378127. −0.371395
\(254\) 76387.0 0.0742908
\(255\) 0 0
\(256\) 984389. 0.938786
\(257\) −648294. −0.612265 −0.306132 0.951989i \(-0.599035\pi\)
−0.306132 + 0.951989i \(0.599035\pi\)
\(258\) −36850.1 −0.0344659
\(259\) 1.79757e6 1.66509
\(260\) 0 0
\(261\) −591897. −0.537830
\(262\) −62437.6 −0.0561944
\(263\) 1.78919e6 1.59502 0.797510 0.603306i \(-0.206150\pi\)
0.797510 + 0.603306i \(0.206150\pi\)
\(264\) −30894.4 −0.0272816
\(265\) 0 0
\(266\) 74693.7 0.0647261
\(267\) −940145. −0.807080
\(268\) 128804. 0.109545
\(269\) 1.39327e6 1.17397 0.586984 0.809599i \(-0.300315\pi\)
0.586984 + 0.809599i \(0.300315\pi\)
\(270\) 0 0
\(271\) −694086. −0.574104 −0.287052 0.957915i \(-0.592675\pi\)
−0.287052 + 0.957915i \(0.592675\pi\)
\(272\) −840522. −0.688853
\(273\) 1.65406e6 1.34321
\(274\) 86452.4 0.0695666
\(275\) 0 0
\(276\) −894443. −0.706773
\(277\) −1.92718e6 −1.50911 −0.754557 0.656234i \(-0.772148\pi\)
−0.754557 + 0.656234i \(0.772148\pi\)
\(278\) 109559. 0.0850233
\(279\) 12343.4 0.00949343
\(280\) 0 0
\(281\) 473885. 0.358020 0.179010 0.983847i \(-0.442711\pi\)
0.179010 + 0.983847i \(0.442711\pi\)
\(282\) −47249.0 −0.0353810
\(283\) −568257. −0.421773 −0.210887 0.977511i \(-0.567635\pi\)
−0.210887 + 0.977511i \(0.567635\pi\)
\(284\) −523710. −0.385297
\(285\) 0 0
\(286\) −48045.8 −0.0347329
\(287\) 1.97752e6 1.41715
\(288\) −109733. −0.0779568
\(289\) −720474. −0.507427
\(290\) 0 0
\(291\) 354130. 0.245149
\(292\) −1.75330e6 −1.20337
\(293\) 1.76731e6 1.20266 0.601331 0.799000i \(-0.294637\pi\)
0.601331 + 0.799000i \(0.294637\pi\)
\(294\) 102241. 0.0689851
\(295\) 0 0
\(296\) −247790. −0.164382
\(297\) −88209.0 −0.0580259
\(298\) 146318. 0.0954457
\(299\) −2.79065e6 −1.80521
\(300\) 0 0
\(301\) 1.89511e6 1.20564
\(302\) −55934.1 −0.0352906
\(303\) 865918. 0.541839
\(304\) 820360. 0.509120
\(305\) 0 0
\(306\) 30120.3 0.0183889
\(307\) 2.67297e6 1.61863 0.809316 0.587374i \(-0.199838\pi\)
0.809316 + 0.587374i \(0.199838\pi\)
\(308\) 791951. 0.475687
\(309\) 488428. 0.291008
\(310\) 0 0
\(311\) 2.23621e6 1.31102 0.655512 0.755185i \(-0.272453\pi\)
0.655512 + 0.755185i \(0.272453\pi\)
\(312\) −228007. −0.132606
\(313\) 1.55507e6 0.897199 0.448600 0.893733i \(-0.351923\pi\)
0.448600 + 0.893733i \(0.351923\pi\)
\(314\) 9569.29 0.00547716
\(315\) 0 0
\(316\) −795363. −0.448072
\(317\) −1.64671e6 −0.920383 −0.460192 0.887820i \(-0.652219\pi\)
−0.460192 + 0.887820i \(0.652219\pi\)
\(318\) −51541.4 −0.0285817
\(319\) −884192. −0.486485
\(320\) 0 0
\(321\) −415626. −0.225133
\(322\) −285972. −0.153703
\(323\) −682607. −0.364053
\(324\) −208655. −0.110425
\(325\) 0 0
\(326\) 170649. 0.0889322
\(327\) −1.45881e6 −0.754450
\(328\) −272596. −0.139906
\(329\) 2.42990e6 1.23765
\(330\) 0 0
\(331\) 249065. 0.124952 0.0624760 0.998046i \(-0.480100\pi\)
0.0624760 + 0.998046i \(0.480100\pi\)
\(332\) −2.42167e6 −1.20579
\(333\) −707484. −0.349628
\(334\) 227689. 0.111680
\(335\) 0 0
\(336\) 1.86161e6 0.899581
\(337\) −712382. −0.341695 −0.170847 0.985298i \(-0.554650\pi\)
−0.170847 + 0.985298i \(0.554650\pi\)
\(338\) −189494. −0.0902200
\(339\) 2.27534e6 1.07534
\(340\) 0 0
\(341\) 18438.9 0.00858713
\(342\) −29397.8 −0.0135909
\(343\) −1.79903e6 −0.825664
\(344\) −261236. −0.119024
\(345\) 0 0
\(346\) −115652. −0.0519353
\(347\) 1.05963e6 0.472421 0.236210 0.971702i \(-0.424095\pi\)
0.236210 + 0.971702i \(0.424095\pi\)
\(348\) −2.09152e6 −0.925793
\(349\) −2.96255e6 −1.30197 −0.650986 0.759090i \(-0.725644\pi\)
−0.650986 + 0.759090i \(0.725644\pi\)
\(350\) 0 0
\(351\) −651001. −0.282042
\(352\) −163921. −0.0705146
\(353\) −2.45696e6 −1.04945 −0.524725 0.851272i \(-0.675832\pi\)
−0.524725 + 0.851272i \(0.675832\pi\)
\(354\) −2071.38 −0.000878520 0
\(355\) 0 0
\(356\) −3.32209e6 −1.38927
\(357\) −1.54901e6 −0.643257
\(358\) −50020.5 −0.0206272
\(359\) 3.72854e6 1.52687 0.763436 0.645884i \(-0.223511\pi\)
0.763436 + 0.645884i \(0.223511\pi\)
\(360\) 0 0
\(361\) −1.80987e6 −0.730935
\(362\) −25104.6 −0.0100689
\(363\) −131769. −0.0524864
\(364\) 5.84477e6 2.31214
\(365\) 0 0
\(366\) −200899. −0.0783927
\(367\) 141287. 0.0547566 0.0273783 0.999625i \(-0.491284\pi\)
0.0273783 + 0.999625i \(0.491284\pi\)
\(368\) −3.14082e6 −1.20899
\(369\) −778309. −0.297568
\(370\) 0 0
\(371\) 2.65065e6 0.999809
\(372\) 43616.4 0.0163415
\(373\) 826181. 0.307470 0.153735 0.988112i \(-0.450870\pi\)
0.153735 + 0.988112i \(0.450870\pi\)
\(374\) 44994.5 0.0166334
\(375\) 0 0
\(376\) −334955. −0.122185
\(377\) −6.52552e6 −2.36462
\(378\) −66711.1 −0.0240142
\(379\) −1.51195e6 −0.540678 −0.270339 0.962765i \(-0.587136\pi\)
−0.270339 + 0.962765i \(0.587136\pi\)
\(380\) 0 0
\(381\) 1.54613e6 0.545674
\(382\) −158466. −0.0555620
\(383\) 3.81853e6 1.33015 0.665074 0.746778i \(-0.268400\pi\)
0.665074 + 0.746778i \(0.268400\pi\)
\(384\) −516456. −0.178733
\(385\) 0 0
\(386\) 270197. 0.0923023
\(387\) −745873. −0.253156
\(388\) 1.25135e6 0.421988
\(389\) 3.87738e6 1.29916 0.649582 0.760292i \(-0.274944\pi\)
0.649582 + 0.760292i \(0.274944\pi\)
\(390\) 0 0
\(391\) 2.61342e6 0.864505
\(392\) 724798. 0.238233
\(393\) −1.26378e6 −0.412754
\(394\) −40579.1 −0.0131693
\(395\) 0 0
\(396\) −311694. −0.0998828
\(397\) 4.70336e6 1.49772 0.748862 0.662725i \(-0.230600\pi\)
0.748862 + 0.662725i \(0.230600\pi\)
\(398\) −373570. −0.118213
\(399\) 1.51186e6 0.475421
\(400\) 0 0
\(401\) −2.21418e6 −0.687625 −0.343812 0.939038i \(-0.611718\pi\)
−0.343812 + 0.939038i \(0.611718\pi\)
\(402\) −16208.0 −0.00500223
\(403\) 136083. 0.0417388
\(404\) 3.05980e6 0.932694
\(405\) 0 0
\(406\) −668701. −0.201334
\(407\) −1.05686e6 −0.316251
\(408\) 213527. 0.0635041
\(409\) −2.24429e6 −0.663394 −0.331697 0.943386i \(-0.607621\pi\)
−0.331697 + 0.943386i \(0.607621\pi\)
\(410\) 0 0
\(411\) 1.74986e6 0.510974
\(412\) 1.72590e6 0.500926
\(413\) 106526. 0.0307313
\(414\) 112552. 0.0322740
\(415\) 0 0
\(416\) −1.20977e6 −0.342745
\(417\) 2.21757e6 0.624505
\(418\) −43915.2 −0.0122935
\(419\) −1.31098e6 −0.364806 −0.182403 0.983224i \(-0.558388\pi\)
−0.182403 + 0.983224i \(0.558388\pi\)
\(420\) 0 0
\(421\) 898192. 0.246981 0.123491 0.992346i \(-0.460591\pi\)
0.123491 + 0.992346i \(0.460591\pi\)
\(422\) −420307. −0.114891
\(423\) −956356. −0.259877
\(424\) −365384. −0.0987040
\(425\) 0 0
\(426\) 65900.9 0.0175941
\(427\) 1.03318e7 2.74223
\(428\) −1.46865e6 −0.387533
\(429\) −972483. −0.255117
\(430\) 0 0
\(431\) 6.21890e6 1.61258 0.806288 0.591523i \(-0.201473\pi\)
0.806288 + 0.591523i \(0.201473\pi\)
\(432\) −732688. −0.188890
\(433\) 1.00093e6 0.256557 0.128279 0.991738i \(-0.459055\pi\)
0.128279 + 0.991738i \(0.459055\pi\)
\(434\) 13945.0 0.00355382
\(435\) 0 0
\(436\) −5.15485e6 −1.29867
\(437\) −2.55073e6 −0.638942
\(438\) 220626. 0.0549504
\(439\) 4.86510e6 1.20484 0.602422 0.798178i \(-0.294203\pi\)
0.602422 + 0.798178i \(0.294203\pi\)
\(440\) 0 0
\(441\) 2.06943e6 0.506703
\(442\) 332069. 0.0808485
\(443\) −7.54601e6 −1.82687 −0.913435 0.406984i \(-0.866580\pi\)
−0.913435 + 0.406984i \(0.866580\pi\)
\(444\) −2.49996e6 −0.601832
\(445\) 0 0
\(446\) −422888. −0.100667
\(447\) 2.96158e6 0.701059
\(448\) 6.49509e6 1.52894
\(449\) −2.70457e6 −0.633115 −0.316557 0.948573i \(-0.602527\pi\)
−0.316557 + 0.948573i \(0.602527\pi\)
\(450\) 0 0
\(451\) −1.16266e6 −0.269160
\(452\) 8.04011e6 1.85104
\(453\) −1.13215e6 −0.259214
\(454\) 504313. 0.114831
\(455\) 0 0
\(456\) −208405. −0.0469349
\(457\) 3.69700e6 0.828055 0.414027 0.910264i \(-0.364122\pi\)
0.414027 + 0.910264i \(0.364122\pi\)
\(458\) −701075. −0.156171
\(459\) 609656. 0.135068
\(460\) 0 0
\(461\) 8.68496e6 1.90334 0.951669 0.307127i \(-0.0993676\pi\)
0.951669 + 0.307127i \(0.0993676\pi\)
\(462\) −99654.9 −0.0217217
\(463\) 2.75398e6 0.597047 0.298523 0.954402i \(-0.403506\pi\)
0.298523 + 0.954402i \(0.403506\pi\)
\(464\) −7.34433e6 −1.58364
\(465\) 0 0
\(466\) 472596. 0.100815
\(467\) 7.31776e6 1.55269 0.776347 0.630306i \(-0.217071\pi\)
0.776347 + 0.630306i \(0.217071\pi\)
\(468\) −2.30037e6 −0.485493
\(469\) 833537. 0.174982
\(470\) 0 0
\(471\) 193690. 0.0402304
\(472\) −14684.3 −0.00303388
\(473\) −1.11421e6 −0.228988
\(474\) 100084. 0.0204607
\(475\) 0 0
\(476\) −5.47357e6 −1.10727
\(477\) −1.04324e6 −0.209936
\(478\) 27610.2 0.00552713
\(479\) −230118. −0.0458260 −0.0229130 0.999737i \(-0.507294\pi\)
−0.0229130 + 0.999737i \(0.507294\pi\)
\(480\) 0 0
\(481\) −7.79985e6 −1.53718
\(482\) −40257.0 −0.00789267
\(483\) −5.78827e6 −1.12897
\(484\) −465617. −0.0903474
\(485\) 0 0
\(486\) 26256.0 0.00504241
\(487\) −8.18489e6 −1.56383 −0.781917 0.623383i \(-0.785758\pi\)
−0.781917 + 0.623383i \(0.785758\pi\)
\(488\) −1.42420e6 −0.270721
\(489\) 3.45405e6 0.653216
\(490\) 0 0
\(491\) −1.27094e6 −0.237915 −0.118957 0.992899i \(-0.537955\pi\)
−0.118957 + 0.992899i \(0.537955\pi\)
\(492\) −2.75022e6 −0.512219
\(493\) 6.11109e6 1.13240
\(494\) −324103. −0.0597538
\(495\) 0 0
\(496\) 153158. 0.0279535
\(497\) −3.38912e6 −0.615455
\(498\) 304730. 0.0550608
\(499\) −9.67837e6 −1.74001 −0.870003 0.493046i \(-0.835883\pi\)
−0.870003 + 0.493046i \(0.835883\pi\)
\(500\) 0 0
\(501\) 4.60859e6 0.820303
\(502\) 604198. 0.107009
\(503\) 6.99048e6 1.23193 0.615966 0.787773i \(-0.288766\pi\)
0.615966 + 0.787773i \(0.288766\pi\)
\(504\) −472925. −0.0829308
\(505\) 0 0
\(506\) 168133. 0.0291929
\(507\) −3.83549e6 −0.662676
\(508\) 5.46338e6 0.939296
\(509\) −6.03866e6 −1.03311 −0.516555 0.856254i \(-0.672786\pi\)
−0.516555 + 0.856254i \(0.672786\pi\)
\(510\) 0 0
\(511\) −1.13462e7 −1.92221
\(512\) −2.27399e6 −0.383367
\(513\) −595032. −0.0998268
\(514\) 288262. 0.0481261
\(515\) 0 0
\(516\) −2.63561e6 −0.435769
\(517\) −1.42863e6 −0.235068
\(518\) −799287. −0.130882
\(519\) −2.34088e6 −0.381470
\(520\) 0 0
\(521\) 6.75056e6 1.08955 0.544773 0.838584i \(-0.316616\pi\)
0.544773 + 0.838584i \(0.316616\pi\)
\(522\) 263186. 0.0422752
\(523\) 3.61437e6 0.577801 0.288900 0.957359i \(-0.406710\pi\)
0.288900 + 0.957359i \(0.406710\pi\)
\(524\) −4.46569e6 −0.710494
\(525\) 0 0
\(526\) −795557. −0.125374
\(527\) −127440. −0.0199885
\(528\) −1.09451e6 −0.170858
\(529\) 3.32937e6 0.517276
\(530\) 0 0
\(531\) −41926.3 −0.00645282
\(532\) 5.34227e6 0.818365
\(533\) −8.58067e6 −1.30829
\(534\) 418033. 0.0634392
\(535\) 0 0
\(536\) −114901. −0.0172747
\(537\) −1.01245e6 −0.151509
\(538\) −619516. −0.0922778
\(539\) 3.09136e6 0.458330
\(540\) 0 0
\(541\) 7.17998e6 1.05470 0.527351 0.849647i \(-0.323185\pi\)
0.527351 + 0.849647i \(0.323185\pi\)
\(542\) 308624. 0.0451265
\(543\) −508136. −0.0739572
\(544\) 1.13294e6 0.164139
\(545\) 0 0
\(546\) −735474. −0.105581
\(547\) 5.53996e6 0.791659 0.395830 0.918324i \(-0.370457\pi\)
0.395830 + 0.918324i \(0.370457\pi\)
\(548\) 6.18329e6 0.879565
\(549\) −4.06635e6 −0.575803
\(550\) 0 0
\(551\) −5.96450e6 −0.836941
\(552\) 797897. 0.111455
\(553\) −5.14709e6 −0.715730
\(554\) 856915. 0.118621
\(555\) 0 0
\(556\) 7.83596e6 1.07499
\(557\) −1.07664e7 −1.47039 −0.735197 0.677854i \(-0.762910\pi\)
−0.735197 + 0.677854i \(0.762910\pi\)
\(558\) −5488.45 −0.000746216 0
\(559\) −8.22307e6 −1.11302
\(560\) 0 0
\(561\) 910721. 0.122174
\(562\) −210712. −0.0281416
\(563\) −6.29087e6 −0.836449 −0.418224 0.908344i \(-0.637347\pi\)
−0.418224 + 0.908344i \(0.637347\pi\)
\(564\) −3.37937e6 −0.447340
\(565\) 0 0
\(566\) 252674. 0.0331528
\(567\) −1.35028e6 −0.176387
\(568\) 467181. 0.0607595
\(569\) −1.25297e7 −1.62241 −0.811205 0.584761i \(-0.801188\pi\)
−0.811205 + 0.584761i \(0.801188\pi\)
\(570\) 0 0
\(571\) −300370. −0.0385538 −0.0192769 0.999814i \(-0.506136\pi\)
−0.0192769 + 0.999814i \(0.506136\pi\)
\(572\) −3.43635e6 −0.439145
\(573\) −3.20747e6 −0.408109
\(574\) −879302. −0.111393
\(575\) 0 0
\(576\) −2.55632e6 −0.321040
\(577\) −4.91623e6 −0.614742 −0.307371 0.951590i \(-0.599449\pi\)
−0.307371 + 0.951590i \(0.599449\pi\)
\(578\) 320357. 0.0398855
\(579\) 5.46899e6 0.677971
\(580\) 0 0
\(581\) −1.56715e7 −1.92607
\(582\) −157463. −0.0192696
\(583\) −1.55841e6 −0.189894
\(584\) 1.56405e6 0.189766
\(585\) 0 0
\(586\) −785830. −0.0945333
\(587\) 1.27769e7 1.53048 0.765242 0.643742i \(-0.222619\pi\)
0.765242 + 0.643742i \(0.222619\pi\)
\(588\) 7.31250e6 0.872213
\(589\) 124383. 0.0147732
\(590\) 0 0
\(591\) −821351. −0.0967298
\(592\) −8.77856e6 −1.02948
\(593\) −2.84036e6 −0.331694 −0.165847 0.986152i \(-0.553036\pi\)
−0.165847 + 0.986152i \(0.553036\pi\)
\(594\) 39221.9 0.00456103
\(595\) 0 0
\(596\) 1.04650e7 1.20677
\(597\) −7.56134e6 −0.868286
\(598\) 1.24086e6 0.141896
\(599\) −1.61252e7 −1.83628 −0.918140 0.396257i \(-0.870309\pi\)
−0.918140 + 0.396257i \(0.870309\pi\)
\(600\) 0 0
\(601\) −6.84876e6 −0.773438 −0.386719 0.922198i \(-0.626392\pi\)
−0.386719 + 0.922198i \(0.626392\pi\)
\(602\) −842657. −0.0947675
\(603\) −328061. −0.0367419
\(604\) −4.00055e6 −0.446197
\(605\) 0 0
\(606\) −385029. −0.0425904
\(607\) 1.39670e7 1.53862 0.769312 0.638874i \(-0.220599\pi\)
0.769312 + 0.638874i \(0.220599\pi\)
\(608\) −1.10577e6 −0.121312
\(609\) −1.35350e7 −1.47882
\(610\) 0 0
\(611\) −1.05436e7 −1.14258
\(612\) 2.15427e6 0.232500
\(613\) −7.54369e6 −0.810835 −0.405418 0.914132i \(-0.632874\pi\)
−0.405418 + 0.914132i \(0.632874\pi\)
\(614\) −1.18853e6 −0.127230
\(615\) 0 0
\(616\) −706468. −0.0750137
\(617\) −3.93986e6 −0.416647 −0.208323 0.978060i \(-0.566801\pi\)
−0.208323 + 0.978060i \(0.566801\pi\)
\(618\) −217178. −0.0228742
\(619\) −5.93266e6 −0.622333 −0.311166 0.950355i \(-0.600720\pi\)
−0.311166 + 0.950355i \(0.600720\pi\)
\(620\) 0 0
\(621\) 2.27813e6 0.237056
\(622\) −994324. −0.103051
\(623\) −2.14984e7 −2.21915
\(624\) −8.07771e6 −0.830475
\(625\) 0 0
\(626\) −691458. −0.0705229
\(627\) −888875. −0.0902967
\(628\) 684419. 0.0692505
\(629\) 7.30448e6 0.736144
\(630\) 0 0
\(631\) −4.19698e6 −0.419627 −0.209813 0.977741i \(-0.567286\pi\)
−0.209813 + 0.977741i \(0.567286\pi\)
\(632\) 709512. 0.0706589
\(633\) −8.50732e6 −0.843886
\(634\) 732205. 0.0723452
\(635\) 0 0
\(636\) −3.68636e6 −0.361373
\(637\) 2.28149e7 2.22777
\(638\) 393154. 0.0382394
\(639\) 1.33388e6 0.129231
\(640\) 0 0
\(641\) 1.95566e7 1.87995 0.939977 0.341236i \(-0.110846\pi\)
0.939977 + 0.341236i \(0.110846\pi\)
\(642\) 184807. 0.0176962
\(643\) 1.89046e7 1.80318 0.901591 0.432590i \(-0.142400\pi\)
0.901591 + 0.432590i \(0.142400\pi\)
\(644\) −2.04534e7 −1.94335
\(645\) 0 0
\(646\) 303520. 0.0286158
\(647\) −1.35066e7 −1.26849 −0.634244 0.773133i \(-0.718689\pi\)
−0.634244 + 0.773133i \(0.718689\pi\)
\(648\) 186133. 0.0174135
\(649\) −62630.6 −0.00583680
\(650\) 0 0
\(651\) 282258. 0.0261032
\(652\) 1.22052e7 1.12441
\(653\) 1.11108e7 1.01968 0.509840 0.860269i \(-0.329705\pi\)
0.509840 + 0.860269i \(0.329705\pi\)
\(654\) 648658. 0.0593023
\(655\) 0 0
\(656\) −9.65736e6 −0.876192
\(657\) 4.46563e6 0.403616
\(658\) −1.08045e6 −0.0972838
\(659\) 1.35954e7 1.21949 0.609745 0.792597i \(-0.291272\pi\)
0.609745 + 0.792597i \(0.291272\pi\)
\(660\) 0 0
\(661\) −856154. −0.0762163 −0.0381082 0.999274i \(-0.512133\pi\)
−0.0381082 + 0.999274i \(0.512133\pi\)
\(662\) −110746. −0.00982164
\(663\) 6.72131e6 0.593841
\(664\) 2.16028e6 0.190147
\(665\) 0 0
\(666\) 314581. 0.0274819
\(667\) 2.28356e7 1.98746
\(668\) 1.62849e7 1.41203
\(669\) −8.55956e6 −0.739411
\(670\) 0 0
\(671\) −6.07442e6 −0.520833
\(672\) −2.50927e6 −0.214350
\(673\) −1.21181e7 −1.03133 −0.515665 0.856790i \(-0.672455\pi\)
−0.515665 + 0.856790i \(0.672455\pi\)
\(674\) 316759. 0.0268583
\(675\) 0 0
\(676\) −1.35530e7 −1.14070
\(677\) 8.70607e6 0.730046 0.365023 0.930998i \(-0.381061\pi\)
0.365023 + 0.930998i \(0.381061\pi\)
\(678\) −1.01172e6 −0.0845255
\(679\) 8.09796e6 0.674064
\(680\) 0 0
\(681\) 1.02077e7 0.843449
\(682\) −8198.80 −0.000674977 0
\(683\) −3.49312e6 −0.286525 −0.143262 0.989685i \(-0.545759\pi\)
−0.143262 + 0.989685i \(0.545759\pi\)
\(684\) −2.10260e6 −0.171837
\(685\) 0 0
\(686\) 799936. 0.0649000
\(687\) −1.41903e7 −1.14709
\(688\) −9.25489e6 −0.745419
\(689\) −1.15014e7 −0.923003
\(690\) 0 0
\(691\) −2.98454e6 −0.237784 −0.118892 0.992907i \(-0.537934\pi\)
−0.118892 + 0.992907i \(0.537934\pi\)
\(692\) −8.27170e6 −0.656643
\(693\) −2.01709e6 −0.159548
\(694\) −471160. −0.0371339
\(695\) 0 0
\(696\) 1.86576e6 0.145993
\(697\) 8.03572e6 0.626531
\(698\) 1.31729e6 0.102339
\(699\) 9.56569e6 0.740497
\(700\) 0 0
\(701\) −3.32551e6 −0.255601 −0.127801 0.991800i \(-0.540792\pi\)
−0.127801 + 0.991800i \(0.540792\pi\)
\(702\) 289466. 0.0221695
\(703\) −7.12927e6 −0.544072
\(704\) −3.81870e6 −0.290392
\(705\) 0 0
\(706\) 1.09248e6 0.0824903
\(707\) 1.98011e7 1.48984
\(708\) −148150. −0.0111076
\(709\) 1.53451e7 1.14645 0.573225 0.819398i \(-0.305692\pi\)
0.573225 + 0.819398i \(0.305692\pi\)
\(710\) 0 0
\(711\) 2.02578e6 0.150286
\(712\) 2.96350e6 0.219081
\(713\) −476212. −0.0350814
\(714\) 688765. 0.0505621
\(715\) 0 0
\(716\) −3.57759e6 −0.260800
\(717\) 558851. 0.0405974
\(718\) −1.65789e6 −0.120017
\(719\) 2.09235e7 1.50943 0.754714 0.656054i \(-0.227776\pi\)
0.754714 + 0.656054i \(0.227776\pi\)
\(720\) 0 0
\(721\) 1.11690e7 0.800156
\(722\) 804753. 0.0574539
\(723\) −814831. −0.0579725
\(724\) −1.79554e6 −0.127306
\(725\) 0 0
\(726\) 58590.8 0.00412561
\(727\) 1.75630e7 1.23243 0.616217 0.787576i \(-0.288664\pi\)
0.616217 + 0.787576i \(0.288664\pi\)
\(728\) −5.21388e6 −0.364614
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 7.70083e6 0.533021
\(732\) −1.43688e7 −0.991158
\(733\) −1.52765e7 −1.05018 −0.525090 0.851047i \(-0.675968\pi\)
−0.525090 + 0.851047i \(0.675968\pi\)
\(734\) −62822.8 −0.00430405
\(735\) 0 0
\(736\) 4.23353e6 0.288076
\(737\) −490067. −0.0332343
\(738\) 346074. 0.0233899
\(739\) 2.22683e7 1.49995 0.749974 0.661467i \(-0.230066\pi\)
0.749974 + 0.661467i \(0.230066\pi\)
\(740\) 0 0
\(741\) −6.56009e6 −0.438898
\(742\) −1.17860e6 −0.0785884
\(743\) 840214. 0.0558365 0.0279182 0.999610i \(-0.491112\pi\)
0.0279182 + 0.999610i \(0.491112\pi\)
\(744\) −38908.4 −0.00257698
\(745\) 0 0
\(746\) −367360. −0.0241682
\(747\) 6.16797e6 0.404427
\(748\) 3.21811e6 0.210304
\(749\) −9.50418e6 −0.619028
\(750\) 0 0
\(751\) −2.63571e7 −1.70529 −0.852644 0.522493i \(-0.825002\pi\)
−0.852644 + 0.522493i \(0.825002\pi\)
\(752\) −1.18666e7 −0.765211
\(753\) 1.22294e7 0.785992
\(754\) 2.90156e6 0.185867
\(755\) 0 0
\(756\) −4.77134e6 −0.303624
\(757\) 1.71589e7 1.08830 0.544150 0.838988i \(-0.316852\pi\)
0.544150 + 0.838988i \(0.316852\pi\)
\(758\) 672284. 0.0424991
\(759\) 3.40314e6 0.214425
\(760\) 0 0
\(761\) −2.23433e7 −1.39857 −0.699287 0.714841i \(-0.746499\pi\)
−0.699287 + 0.714841i \(0.746499\pi\)
\(762\) −687483. −0.0428918
\(763\) −3.33589e7 −2.07444
\(764\) −1.13339e7 −0.702498
\(765\) 0 0
\(766\) −1.69790e6 −0.104554
\(767\) −462227. −0.0283705
\(768\) −8.85950e6 −0.542008
\(769\) −8.69614e6 −0.530287 −0.265143 0.964209i \(-0.585419\pi\)
−0.265143 + 0.964209i \(0.585419\pi\)
\(770\) 0 0
\(771\) 5.83464e6 0.353491
\(772\) 1.93252e7 1.16702
\(773\) −5.79015e6 −0.348531 −0.174265 0.984699i \(-0.555755\pi\)
−0.174265 + 0.984699i \(0.555755\pi\)
\(774\) 331651. 0.0198989
\(775\) 0 0
\(776\) −1.11628e6 −0.0665455
\(777\) −1.61782e7 −0.961339
\(778\) −1.72407e6 −0.102119
\(779\) −7.84296e6 −0.463059
\(780\) 0 0
\(781\) 1.99259e6 0.116894
\(782\) −1.16205e6 −0.0679530
\(783\) 5.32707e6 0.310516
\(784\) 2.56777e7 1.49199
\(785\) 0 0
\(786\) 561938. 0.0324438
\(787\) 3.16605e7 1.82213 0.911067 0.412258i \(-0.135260\pi\)
0.911067 + 0.412258i \(0.135260\pi\)
\(788\) −2.90232e6 −0.166506
\(789\) −1.61027e7 −0.920885
\(790\) 0 0
\(791\) 5.20305e7 2.95677
\(792\) 278050. 0.0157511
\(793\) −4.48305e7 −2.53157
\(794\) −2.09134e6 −0.117726
\(795\) 0 0
\(796\) −2.67186e7 −1.49462
\(797\) 1.21137e7 0.675511 0.337755 0.941234i \(-0.390332\pi\)
0.337755 + 0.941234i \(0.390332\pi\)
\(798\) −672243. −0.0373697
\(799\) 9.87397e6 0.547173
\(800\) 0 0
\(801\) 8.46131e6 0.465968
\(802\) 984529. 0.0540496
\(803\) 6.67087e6 0.365085
\(804\) −1.15923e6 −0.0632457
\(805\) 0 0
\(806\) −60508.9 −0.00328081
\(807\) −1.25395e7 −0.677790
\(808\) −2.72952e6 −0.147082
\(809\) −1.66584e7 −0.894875 −0.447438 0.894315i \(-0.647663\pi\)
−0.447438 + 0.894315i \(0.647663\pi\)
\(810\) 0 0
\(811\) −2.63057e7 −1.40442 −0.702211 0.711969i \(-0.747804\pi\)
−0.702211 + 0.711969i \(0.747804\pi\)
\(812\) −4.78271e7 −2.54556
\(813\) 6.24678e6 0.331459
\(814\) 469930. 0.0248584
\(815\) 0 0
\(816\) 7.56470e6 0.397710
\(817\) −7.51610e6 −0.393947
\(818\) 997920. 0.0521450
\(819\) −1.48865e7 −0.775504
\(820\) 0 0
\(821\) 2.92543e6 0.151472 0.0757358 0.997128i \(-0.475869\pi\)
0.0757358 + 0.997128i \(0.475869\pi\)
\(822\) −778072. −0.0401643
\(823\) 2.99240e7 1.54000 0.770000 0.638044i \(-0.220256\pi\)
0.770000 + 0.638044i \(0.220256\pi\)
\(824\) −1.53961e6 −0.0789937
\(825\) 0 0
\(826\) −47366.6 −0.00241558
\(827\) −1.24304e6 −0.0632008 −0.0316004 0.999501i \(-0.510060\pi\)
−0.0316004 + 0.999501i \(0.510060\pi\)
\(828\) 8.04999e6 0.408056
\(829\) 7.42673e6 0.375328 0.187664 0.982233i \(-0.439908\pi\)
0.187664 + 0.982233i \(0.439908\pi\)
\(830\) 0 0
\(831\) 1.73446e7 0.871288
\(832\) −2.81828e7 −1.41148
\(833\) −2.13660e7 −1.06687
\(834\) −986035. −0.0490882
\(835\) 0 0
\(836\) −3.14092e6 −0.155432
\(837\) −111090. −0.00548103
\(838\) 582926. 0.0286750
\(839\) −9.30289e6 −0.456260 −0.228130 0.973631i \(-0.573261\pi\)
−0.228130 + 0.973631i \(0.573261\pi\)
\(840\) 0 0
\(841\) 3.28865e7 1.60335
\(842\) −399379. −0.0194136
\(843\) −4.26497e6 −0.206703
\(844\) −3.00614e7 −1.45262
\(845\) 0 0
\(846\) 425241. 0.0204272
\(847\) −3.01318e6 −0.144317
\(848\) −1.29446e7 −0.618157
\(849\) 5.11431e6 0.243511
\(850\) 0 0
\(851\) 2.72950e7 1.29199
\(852\) 4.71339e6 0.222451
\(853\) 3.06103e7 1.44044 0.720219 0.693746i \(-0.244041\pi\)
0.720219 + 0.693746i \(0.244041\pi\)
\(854\) −4.59399e6 −0.215549
\(855\) 0 0
\(856\) 1.31012e6 0.0611122
\(857\) −8.04257e6 −0.374061 −0.187031 0.982354i \(-0.559886\pi\)
−0.187031 + 0.982354i \(0.559886\pi\)
\(858\) 432412. 0.0200530
\(859\) 2.02293e7 0.935403 0.467702 0.883886i \(-0.345082\pi\)
0.467702 + 0.883886i \(0.345082\pi\)
\(860\) 0 0
\(861\) −1.77977e7 −0.818194
\(862\) −2.76522e6 −0.126754
\(863\) −2.06034e7 −0.941700 −0.470850 0.882213i \(-0.656053\pi\)
−0.470850 + 0.882213i \(0.656053\pi\)
\(864\) 987593. 0.0450084
\(865\) 0 0
\(866\) −445062. −0.0201663
\(867\) 6.48427e6 0.292963
\(868\) 997382. 0.0449327
\(869\) 3.02616e6 0.135939
\(870\) 0 0
\(871\) −3.61680e6 −0.161540
\(872\) 4.59843e6 0.204795
\(873\) −3.18717e6 −0.141537
\(874\) 1.13418e6 0.0502230
\(875\) 0 0
\(876\) 1.57797e7 0.694765
\(877\) −2.65867e7 −1.16726 −0.583628 0.812021i \(-0.698367\pi\)
−0.583628 + 0.812021i \(0.698367\pi\)
\(878\) −2.16326e6 −0.0947047
\(879\) −1.59058e7 −0.694357
\(880\) 0 0
\(881\) 4.01120e7 1.74114 0.870572 0.492041i \(-0.163749\pi\)
0.870572 + 0.492041i \(0.163749\pi\)
\(882\) −920166. −0.0398285
\(883\) 3.61504e7 1.56031 0.780155 0.625586i \(-0.215140\pi\)
0.780155 + 0.625586i \(0.215140\pi\)
\(884\) 2.37504e7 1.02221
\(885\) 0 0
\(886\) 3.35531e6 0.143598
\(887\) 2.25484e7 0.962290 0.481145 0.876641i \(-0.340221\pi\)
0.481145 + 0.876641i \(0.340221\pi\)
\(888\) 2.23011e6 0.0949062
\(889\) 3.53556e7 1.50039
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −3.02459e7 −1.27278
\(893\) −9.63712e6 −0.404407
\(894\) −1.31686e6 −0.0551056
\(895\) 0 0
\(896\) −1.18099e7 −0.491446
\(897\) 2.51159e7 1.04224
\(898\) 1.20258e6 0.0497650
\(899\) −1.11355e6 −0.0459526
\(900\) 0 0
\(901\) 1.07710e7 0.442021
\(902\) 516974. 0.0211569
\(903\) −1.70560e7 −0.696078
\(904\) −7.17226e6 −0.291900
\(905\) 0 0
\(906\) 503407. 0.0203751
\(907\) −1.61295e7 −0.651032 −0.325516 0.945537i \(-0.605538\pi\)
−0.325516 + 0.945537i \(0.605538\pi\)
\(908\) 3.60697e7 1.45187
\(909\) −7.79326e6 −0.312831
\(910\) 0 0
\(911\) −1.72749e7 −0.689637 −0.344819 0.938669i \(-0.612060\pi\)
−0.344819 + 0.938669i \(0.612060\pi\)
\(912\) −7.38324e6 −0.293941
\(913\) 9.21388e6 0.365818
\(914\) −1.64386e6 −0.0650879
\(915\) 0 0
\(916\) −5.01426e7 −1.97455
\(917\) −2.88991e7 −1.13491
\(918\) −271082. −0.0106168
\(919\) 966289. 0.0377414 0.0188707 0.999822i \(-0.493993\pi\)
0.0188707 + 0.999822i \(0.493993\pi\)
\(920\) 0 0
\(921\) −2.40567e7 −0.934517
\(922\) −3.86175e6 −0.149609
\(923\) 1.47057e7 0.568176
\(924\) −7.12756e6 −0.274638
\(925\) 0 0
\(926\) −1.22455e6 −0.0469299
\(927\) −4.39585e6 −0.168013
\(928\) 9.89946e6 0.377348
\(929\) −1.54446e7 −0.587133 −0.293567 0.955939i \(-0.594842\pi\)
−0.293567 + 0.955939i \(0.594842\pi\)
\(930\) 0 0
\(931\) 2.08534e7 0.788503
\(932\) 3.38012e7 1.27465
\(933\) −2.01259e7 −0.756920
\(934\) −3.25382e6 −0.122047
\(935\) 0 0
\(936\) 2.05207e6 0.0765600
\(937\) 3.01699e7 1.12260 0.561300 0.827612i \(-0.310301\pi\)
0.561300 + 0.827612i \(0.310301\pi\)
\(938\) −370630. −0.0137542
\(939\) −1.39956e7 −0.517998
\(940\) 0 0
\(941\) −1.74463e7 −0.642286 −0.321143 0.947031i \(-0.604067\pi\)
−0.321143 + 0.947031i \(0.604067\pi\)
\(942\) −86123.6 −0.00316224
\(943\) 3.00275e7 1.09961
\(944\) −520227. −0.0190004
\(945\) 0 0
\(946\) 495429. 0.0179992
\(947\) −3.47258e7 −1.25828 −0.629140 0.777292i \(-0.716593\pi\)
−0.629140 + 0.777292i \(0.716593\pi\)
\(948\) 7.15827e6 0.258694
\(949\) 4.92324e7 1.77454
\(950\) 0 0
\(951\) 1.48204e7 0.531383
\(952\) 4.88275e6 0.174611
\(953\) 5.30473e7 1.89204 0.946021 0.324104i \(-0.105063\pi\)
0.946021 + 0.324104i \(0.105063\pi\)
\(954\) 463872. 0.0165017
\(955\) 0 0
\(956\) 1.97475e6 0.0698823
\(957\) 7.95772e6 0.280872
\(958\) 102322. 0.00360208
\(959\) 4.00144e7 1.40498
\(960\) 0 0
\(961\) −2.86059e7 −0.999189
\(962\) 3.46818e6 0.120827
\(963\) 3.74063e6 0.129981
\(964\) −2.87928e6 −0.0997909
\(965\) 0 0
\(966\) 2.57374e6 0.0887406
\(967\) −8.74388e6 −0.300703 −0.150352 0.988633i \(-0.548041\pi\)
−0.150352 + 0.988633i \(0.548041\pi\)
\(968\) 415359. 0.0142474
\(969\) 6.14346e6 0.210186
\(970\) 0 0
\(971\) 2.90606e7 0.989136 0.494568 0.869139i \(-0.335326\pi\)
0.494568 + 0.869139i \(0.335326\pi\)
\(972\) 1.87789e6 0.0637537
\(973\) 5.07094e7 1.71714
\(974\) 3.63939e6 0.122923
\(975\) 0 0
\(976\) −5.04558e7 −1.69546
\(977\) −4.53644e6 −0.152047 −0.0760236 0.997106i \(-0.524222\pi\)
−0.0760236 + 0.997106i \(0.524222\pi\)
\(978\) −1.53584e6 −0.0513450
\(979\) 1.26397e7 0.421484
\(980\) 0 0
\(981\) 1.31293e7 0.435582
\(982\) 565121. 0.0187009
\(983\) −2.56053e7 −0.845174 −0.422587 0.906322i \(-0.638878\pi\)
−0.422587 + 0.906322i \(0.638878\pi\)
\(984\) 2.45336e6 0.0807745
\(985\) 0 0
\(986\) −2.71728e6 −0.0890107
\(987\) −2.18691e7 −0.714560
\(988\) −2.31806e7 −0.755498
\(989\) 2.87761e7 0.935494
\(990\) 0 0
\(991\) −7.43675e6 −0.240547 −0.120273 0.992741i \(-0.538377\pi\)
−0.120273 + 0.992741i \(0.538377\pi\)
\(992\) −206442. −0.00666070
\(993\) −2.24159e6 −0.0721410
\(994\) 1.50697e6 0.0483769
\(995\) 0 0
\(996\) 2.17951e7 0.696161
\(997\) 2.77690e7 0.884752 0.442376 0.896830i \(-0.354136\pi\)
0.442376 + 0.896830i \(0.354136\pi\)
\(998\) 4.30347e6 0.136770
\(999\) 6.36736e6 0.201858
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 825.6.a.u.1.5 yes 10
5.4 even 2 825.6.a.t.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.6 10 5.4 even 2
825.6.a.u.1.5 yes 10 1.1 even 1 trivial