Properties

Label 825.6.a.h.1.3
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.476452\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.64018 q^{2} +9.00000 q^{3} -0.188384 q^{4} +50.7616 q^{6} -145.021 q^{7} -181.548 q^{8} +81.0000 q^{9} -121.000 q^{11} -1.69545 q^{12} +69.9067 q^{13} -817.946 q^{14} -1017.94 q^{16} +500.495 q^{17} +456.854 q^{18} -670.685 q^{19} -1305.19 q^{21} -682.462 q^{22} -791.025 q^{23} -1633.93 q^{24} +394.287 q^{26} +729.000 q^{27} +27.3196 q^{28} +1545.91 q^{29} +2703.09 q^{31} +68.2013 q^{32} -1089.00 q^{33} +2822.88 q^{34} -15.2591 q^{36} +2667.13 q^{37} -3782.78 q^{38} +629.161 q^{39} +9622.09 q^{41} -7361.51 q^{42} +6681.57 q^{43} +22.7944 q^{44} -4461.52 q^{46} +1167.06 q^{47} -9161.43 q^{48} +4224.17 q^{49} +4504.46 q^{51} -13.1693 q^{52} -28872.9 q^{53} +4111.69 q^{54} +26328.4 q^{56} -6036.16 q^{57} +8719.22 q^{58} +23599.3 q^{59} +17601.3 q^{61} +15245.9 q^{62} -11746.7 q^{63} +32958.6 q^{64} -6142.15 q^{66} -16501.5 q^{67} -94.2851 q^{68} -7119.22 q^{69} +72059.4 q^{71} -14705.4 q^{72} +45480.8 q^{73} +15043.1 q^{74} +126.346 q^{76} +17547.6 q^{77} +3548.58 q^{78} -21688.7 q^{79} +6561.00 q^{81} +54270.3 q^{82} -6934.34 q^{83} +245.877 q^{84} +37685.2 q^{86} +13913.2 q^{87} +21967.3 q^{88} +42779.8 q^{89} -10138.0 q^{91} +149.016 q^{92} +24327.8 q^{93} +6582.41 q^{94} +613.812 q^{96} +20992.6 q^{97} +23825.1 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.64018 0.997052 0.498526 0.866875i \(-0.333875\pi\)
0.498526 + 0.866875i \(0.333875\pi\)
\(3\) 9.00000 0.577350
\(4\) −0.188384 −0.00588699
\(5\) 0 0
\(6\) 50.7616 0.575648
\(7\) −145.021 −1.11863 −0.559315 0.828955i \(-0.688936\pi\)
−0.559315 + 0.828955i \(0.688936\pi\)
\(8\) −181.548 −1.00292
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −1.69545 −0.00339886
\(13\) 69.9067 0.114726 0.0573628 0.998353i \(-0.481731\pi\)
0.0573628 + 0.998353i \(0.481731\pi\)
\(14\) −817.946 −1.11533
\(15\) 0 0
\(16\) −1017.94 −0.994078
\(17\) 500.495 0.420027 0.210014 0.977698i \(-0.432649\pi\)
0.210014 + 0.977698i \(0.432649\pi\)
\(18\) 456.854 0.332351
\(19\) −670.685 −0.426221 −0.213110 0.977028i \(-0.568359\pi\)
−0.213110 + 0.977028i \(0.568359\pi\)
\(20\) 0 0
\(21\) −1305.19 −0.645842
\(22\) −682.462 −0.300623
\(23\) −791.025 −0.311796 −0.155898 0.987773i \(-0.549827\pi\)
−0.155898 + 0.987773i \(0.549827\pi\)
\(24\) −1633.93 −0.579037
\(25\) 0 0
\(26\) 394.287 0.114388
\(27\) 729.000 0.192450
\(28\) 27.3196 0.00658537
\(29\) 1545.91 0.341342 0.170671 0.985328i \(-0.445406\pi\)
0.170671 + 0.985328i \(0.445406\pi\)
\(30\) 0 0
\(31\) 2703.09 0.505191 0.252596 0.967572i \(-0.418716\pi\)
0.252596 + 0.967572i \(0.418716\pi\)
\(32\) 68.2013 0.0117738
\(33\) −1089.00 −0.174078
\(34\) 2822.88 0.418789
\(35\) 0 0
\(36\) −15.2591 −0.00196233
\(37\) 2667.13 0.320287 0.160144 0.987094i \(-0.448804\pi\)
0.160144 + 0.987094i \(0.448804\pi\)
\(38\) −3782.78 −0.424964
\(39\) 629.161 0.0662369
\(40\) 0 0
\(41\) 9622.09 0.893942 0.446971 0.894548i \(-0.352503\pi\)
0.446971 + 0.894548i \(0.352503\pi\)
\(42\) −7361.51 −0.643938
\(43\) 6681.57 0.551070 0.275535 0.961291i \(-0.411145\pi\)
0.275535 + 0.961291i \(0.411145\pi\)
\(44\) 22.7944 0.00177499
\(45\) 0 0
\(46\) −4461.52 −0.310877
\(47\) 1167.06 0.0770632 0.0385316 0.999257i \(-0.487732\pi\)
0.0385316 + 0.999257i \(0.487732\pi\)
\(48\) −9161.43 −0.573931
\(49\) 4224.17 0.251334
\(50\) 0 0
\(51\) 4504.46 0.242503
\(52\) −13.1693 −0.000675389 0
\(53\) −28872.9 −1.41189 −0.705944 0.708268i \(-0.749477\pi\)
−0.705944 + 0.708268i \(0.749477\pi\)
\(54\) 4111.69 0.191883
\(55\) 0 0
\(56\) 26328.4 1.12190
\(57\) −6036.16 −0.246079
\(58\) 8719.22 0.340336
\(59\) 23599.3 0.882610 0.441305 0.897357i \(-0.354516\pi\)
0.441305 + 0.897357i \(0.354516\pi\)
\(60\) 0 0
\(61\) 17601.3 0.605648 0.302824 0.953046i \(-0.402070\pi\)
0.302824 + 0.953046i \(0.402070\pi\)
\(62\) 15245.9 0.503702
\(63\) −11746.7 −0.372877
\(64\) 32958.6 1.00582
\(65\) 0 0
\(66\) −6142.15 −0.173565
\(67\) −16501.5 −0.449092 −0.224546 0.974463i \(-0.572090\pi\)
−0.224546 + 0.974463i \(0.572090\pi\)
\(68\) −94.2851 −0.00247270
\(69\) −7119.22 −0.180016
\(70\) 0 0
\(71\) 72059.4 1.69646 0.848232 0.529625i \(-0.177667\pi\)
0.848232 + 0.529625i \(0.177667\pi\)
\(72\) −14705.4 −0.334307
\(73\) 45480.8 0.998898 0.499449 0.866343i \(-0.333536\pi\)
0.499449 + 0.866343i \(0.333536\pi\)
\(74\) 15043.1 0.319343
\(75\) 0 0
\(76\) 126.346 0.00250916
\(77\) 17547.6 0.337280
\(78\) 3548.58 0.0660417
\(79\) −21688.7 −0.390990 −0.195495 0.980705i \(-0.562631\pi\)
−0.195495 + 0.980705i \(0.562631\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 54270.3 0.891307
\(83\) −6934.34 −0.110487 −0.0552434 0.998473i \(-0.517593\pi\)
−0.0552434 + 0.998473i \(0.517593\pi\)
\(84\) 245.877 0.00380206
\(85\) 0 0
\(86\) 37685.2 0.549446
\(87\) 13913.2 0.197074
\(88\) 21967.3 0.302392
\(89\) 42779.8 0.572484 0.286242 0.958157i \(-0.407594\pi\)
0.286242 + 0.958157i \(0.407594\pi\)
\(90\) 0 0
\(91\) −10138.0 −0.128336
\(92\) 149.016 0.00183554
\(93\) 24327.8 0.291672
\(94\) 6582.41 0.0768361
\(95\) 0 0
\(96\) 613.812 0.00679762
\(97\) 20992.6 0.226535 0.113268 0.993565i \(-0.463868\pi\)
0.113268 + 0.993565i \(0.463868\pi\)
\(98\) 23825.1 0.250593
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 189654. 1.84995 0.924973 0.380033i \(-0.124087\pi\)
0.924973 + 0.380033i \(0.124087\pi\)
\(102\) 25405.9 0.241788
\(103\) 160899. 1.49438 0.747190 0.664611i \(-0.231403\pi\)
0.747190 + 0.664611i \(0.231403\pi\)
\(104\) −12691.4 −0.115061
\(105\) 0 0
\(106\) −162848. −1.40773
\(107\) −113244. −0.956216 −0.478108 0.878301i \(-0.658677\pi\)
−0.478108 + 0.878301i \(0.658677\pi\)
\(108\) −137.332 −0.00113295
\(109\) −1768.32 −0.0142559 −0.00712797 0.999975i \(-0.502269\pi\)
−0.00712797 + 0.999975i \(0.502269\pi\)
\(110\) 0 0
\(111\) 24004.1 0.184918
\(112\) 147622. 1.11201
\(113\) 77273.2 0.569289 0.284645 0.958633i \(-0.408124\pi\)
0.284645 + 0.958633i \(0.408124\pi\)
\(114\) −34045.0 −0.245353
\(115\) 0 0
\(116\) −291.224 −0.00200948
\(117\) 5662.45 0.0382419
\(118\) 133104. 0.880008
\(119\) −72582.5 −0.469855
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 99274.6 0.603863
\(123\) 86598.8 0.516118
\(124\) −509.218 −0.00297406
\(125\) 0 0
\(126\) −66253.6 −0.371778
\(127\) −104707. −0.576058 −0.288029 0.957622i \(-0.593000\pi\)
−0.288029 + 0.957622i \(0.593000\pi\)
\(128\) 183710. 0.991079
\(129\) 60134.1 0.318161
\(130\) 0 0
\(131\) 101248. 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(132\) 205.150 0.00102479
\(133\) 97263.6 0.476783
\(134\) −93071.2 −0.447768
\(135\) 0 0
\(136\) −90864.0 −0.421255
\(137\) −128311. −0.584068 −0.292034 0.956408i \(-0.594332\pi\)
−0.292034 + 0.956408i \(0.594332\pi\)
\(138\) −40153.7 −0.179485
\(139\) 161754. 0.710100 0.355050 0.934847i \(-0.384464\pi\)
0.355050 + 0.934847i \(0.384464\pi\)
\(140\) 0 0
\(141\) 10503.5 0.0444925
\(142\) 406428. 1.69146
\(143\) −8458.72 −0.0345911
\(144\) −82452.8 −0.331359
\(145\) 0 0
\(146\) 256520. 0.995954
\(147\) 38017.5 0.145108
\(148\) −502.443 −0.00188553
\(149\) −404421. −1.49234 −0.746170 0.665755i \(-0.768110\pi\)
−0.746170 + 0.665755i \(0.768110\pi\)
\(150\) 0 0
\(151\) 416180. 1.48538 0.742692 0.669633i \(-0.233549\pi\)
0.742692 + 0.669633i \(0.233549\pi\)
\(152\) 121762. 0.427466
\(153\) 40540.1 0.140009
\(154\) 98971.5 0.336286
\(155\) 0 0
\(156\) −118.524 −0.000389936 0
\(157\) −332834. −1.07765 −0.538827 0.842417i \(-0.681132\pi\)
−0.538827 + 0.842417i \(0.681132\pi\)
\(158\) −122328. −0.389837
\(159\) −259856. −0.815154
\(160\) 0 0
\(161\) 114715. 0.348785
\(162\) 37005.2 0.110784
\(163\) 14764.9 0.0435272 0.0217636 0.999763i \(-0.493072\pi\)
0.0217636 + 0.999763i \(0.493072\pi\)
\(164\) −1812.64 −0.00526263
\(165\) 0 0
\(166\) −39110.9 −0.110161
\(167\) −169960. −0.471579 −0.235790 0.971804i \(-0.575768\pi\)
−0.235790 + 0.971804i \(0.575768\pi\)
\(168\) 236955. 0.647729
\(169\) −366406. −0.986838
\(170\) 0 0
\(171\) −54325.5 −0.142074
\(172\) −1258.70 −0.00324415
\(173\) −725869. −1.84392 −0.921962 0.387281i \(-0.873414\pi\)
−0.921962 + 0.387281i \(0.873414\pi\)
\(174\) 78472.9 0.196493
\(175\) 0 0
\(176\) 123170. 0.299726
\(177\) 212394. 0.509575
\(178\) 241286. 0.570797
\(179\) 69311.4 0.161686 0.0808429 0.996727i \(-0.474239\pi\)
0.0808429 + 0.996727i \(0.474239\pi\)
\(180\) 0 0
\(181\) 790396. 1.79328 0.896640 0.442761i \(-0.146001\pi\)
0.896640 + 0.442761i \(0.146001\pi\)
\(182\) −57179.9 −0.127957
\(183\) 158412. 0.349671
\(184\) 143609. 0.312707
\(185\) 0 0
\(186\) 137213. 0.290813
\(187\) −60559.9 −0.126643
\(188\) −219.854 −0.000453670 0
\(189\) −105721. −0.215281
\(190\) 0 0
\(191\) 418979. 0.831015 0.415507 0.909590i \(-0.363604\pi\)
0.415507 + 0.909590i \(0.363604\pi\)
\(192\) 296628. 0.580709
\(193\) −265524. −0.513110 −0.256555 0.966530i \(-0.582587\pi\)
−0.256555 + 0.966530i \(0.582587\pi\)
\(194\) 118402. 0.225868
\(195\) 0 0
\(196\) −795.765 −0.00147960
\(197\) 87754.1 0.161102 0.0805512 0.996750i \(-0.474332\pi\)
0.0805512 + 0.996750i \(0.474332\pi\)
\(198\) −55279.4 −0.100208
\(199\) 380988. 0.681992 0.340996 0.940065i \(-0.389236\pi\)
0.340996 + 0.940065i \(0.389236\pi\)
\(200\) 0 0
\(201\) −148513. −0.259284
\(202\) 1.06968e6 1.84449
\(203\) −224190. −0.381835
\(204\) −848.566 −0.00142761
\(205\) 0 0
\(206\) 907501. 1.48997
\(207\) −64073.0 −0.103932
\(208\) −71160.6 −0.114046
\(209\) 81152.9 0.128510
\(210\) 0 0
\(211\) 624404. 0.965516 0.482758 0.875754i \(-0.339635\pi\)
0.482758 + 0.875754i \(0.339635\pi\)
\(212\) 5439.18 0.00831177
\(213\) 648534. 0.979454
\(214\) −638717. −0.953398
\(215\) 0 0
\(216\) −132349. −0.193012
\(217\) −392005. −0.565123
\(218\) −9973.67 −0.0142139
\(219\) 409328. 0.576714
\(220\) 0 0
\(221\) 34988.0 0.0481879
\(222\) 135388. 0.184373
\(223\) 658298. 0.886462 0.443231 0.896407i \(-0.353832\pi\)
0.443231 + 0.896407i \(0.353832\pi\)
\(224\) −9890.64 −0.0131706
\(225\) 0 0
\(226\) 435835. 0.567611
\(227\) −219954. −0.283313 −0.141657 0.989916i \(-0.545243\pi\)
−0.141657 + 0.989916i \(0.545243\pi\)
\(228\) 1137.11 0.00144866
\(229\) −313324. −0.394825 −0.197412 0.980321i \(-0.563254\pi\)
−0.197412 + 0.980321i \(0.563254\pi\)
\(230\) 0 0
\(231\) 157928. 0.194729
\(232\) −280657. −0.342339
\(233\) 1.01608e6 1.22614 0.613069 0.790030i \(-0.289935\pi\)
0.613069 + 0.790030i \(0.289935\pi\)
\(234\) 31937.2 0.0381292
\(235\) 0 0
\(236\) −4445.72 −0.00519592
\(237\) −195198. −0.225738
\(238\) −409378. −0.468470
\(239\) 762420. 0.863375 0.431688 0.902023i \(-0.357918\pi\)
0.431688 + 0.902023i \(0.357918\pi\)
\(240\) 0 0
\(241\) −22909.2 −0.0254078 −0.0127039 0.999919i \(-0.504044\pi\)
−0.0127039 + 0.999919i \(0.504044\pi\)
\(242\) 82577.9 0.0906411
\(243\) 59049.0 0.0641500
\(244\) −3315.80 −0.00356545
\(245\) 0 0
\(246\) 488433. 0.514596
\(247\) −46885.4 −0.0488985
\(248\) −490741. −0.506668
\(249\) −62409.1 −0.0637895
\(250\) 0 0
\(251\) −691159. −0.692458 −0.346229 0.938150i \(-0.612538\pi\)
−0.346229 + 0.938150i \(0.612538\pi\)
\(252\) 2212.89 0.00219512
\(253\) 95714.0 0.0940100
\(254\) −590566. −0.574360
\(255\) 0 0
\(256\) −18518.2 −0.0176604
\(257\) 1.34845e6 1.27351 0.636753 0.771068i \(-0.280277\pi\)
0.636753 + 0.771068i \(0.280277\pi\)
\(258\) 339167. 0.317223
\(259\) −386790. −0.358283
\(260\) 0 0
\(261\) 125219. 0.113781
\(262\) 571057. 0.513957
\(263\) −1.34541e6 −1.19940 −0.599700 0.800225i \(-0.704713\pi\)
−0.599700 + 0.800225i \(0.704713\pi\)
\(264\) 197706. 0.174586
\(265\) 0 0
\(266\) 548584. 0.475378
\(267\) 385018. 0.330524
\(268\) 3108.61 0.00264380
\(269\) −2.03803e6 −1.71724 −0.858620 0.512613i \(-0.828678\pi\)
−0.858620 + 0.512613i \(0.828678\pi\)
\(270\) 0 0
\(271\) −218053. −0.180360 −0.0901799 0.995925i \(-0.528744\pi\)
−0.0901799 + 0.995925i \(0.528744\pi\)
\(272\) −509472. −0.417540
\(273\) −91241.7 −0.0740946
\(274\) −723698. −0.582346
\(275\) 0 0
\(276\) 1341.15 0.00105975
\(277\) 410179. 0.321199 0.160599 0.987020i \(-0.448657\pi\)
0.160599 + 0.987020i \(0.448657\pi\)
\(278\) 912324. 0.708006
\(279\) 218950. 0.168397
\(280\) 0 0
\(281\) −406589. −0.307178 −0.153589 0.988135i \(-0.549083\pi\)
−0.153589 + 0.988135i \(0.549083\pi\)
\(282\) 59241.7 0.0443613
\(283\) −1.26416e6 −0.938291 −0.469146 0.883121i \(-0.655438\pi\)
−0.469146 + 0.883121i \(0.655438\pi\)
\(284\) −13574.8 −0.00998707
\(285\) 0 0
\(286\) −47708.7 −0.0344891
\(287\) −1.39541e6 −0.999991
\(288\) 5524.30 0.00392461
\(289\) −1.16936e6 −0.823577
\(290\) 0 0
\(291\) 188933. 0.130790
\(292\) −8567.85 −0.00588050
\(293\) −351099. −0.238924 −0.119462 0.992839i \(-0.538117\pi\)
−0.119462 + 0.992839i \(0.538117\pi\)
\(294\) 214426. 0.144680
\(295\) 0 0
\(296\) −484212. −0.321223
\(297\) −88209.0 −0.0580259
\(298\) −2.28101e6 −1.48794
\(299\) −55298.0 −0.0357710
\(300\) 0 0
\(301\) −968969. −0.616444
\(302\) 2.34733e6 1.48101
\(303\) 1.70689e6 1.06807
\(304\) 682714. 0.423697
\(305\) 0 0
\(306\) 228654. 0.139596
\(307\) 1.46228e6 0.885493 0.442747 0.896647i \(-0.354004\pi\)
0.442747 + 0.896647i \(0.354004\pi\)
\(308\) −3305.68 −0.00198556
\(309\) 1.44809e6 0.862781
\(310\) 0 0
\(311\) 2.20292e6 1.29151 0.645755 0.763545i \(-0.276543\pi\)
0.645755 + 0.763545i \(0.276543\pi\)
\(312\) −114223. −0.0664304
\(313\) −647848. −0.373777 −0.186888 0.982381i \(-0.559840\pi\)
−0.186888 + 0.982381i \(0.559840\pi\)
\(314\) −1.87725e6 −1.07448
\(315\) 0 0
\(316\) 4085.80 0.00230175
\(317\) 1.62933e6 0.910670 0.455335 0.890320i \(-0.349519\pi\)
0.455335 + 0.890320i \(0.349519\pi\)
\(318\) −1.46563e6 −0.812751
\(319\) −187055. −0.102918
\(320\) 0 0
\(321\) −1.01920e6 −0.552072
\(322\) 647016. 0.347756
\(323\) −335675. −0.179024
\(324\) −1235.99 −0.000654110 0
\(325\) 0 0
\(326\) 83276.6 0.0433989
\(327\) −15914.9 −0.00823067
\(328\) −1.74687e6 −0.896554
\(329\) −169248. −0.0862053
\(330\) 0 0
\(331\) 799339. 0.401016 0.200508 0.979692i \(-0.435741\pi\)
0.200508 + 0.979692i \(0.435741\pi\)
\(332\) 1306.32 0.000650434 0
\(333\) 216037. 0.106762
\(334\) −958603. −0.470189
\(335\) 0 0
\(336\) 1.32860e6 0.642017
\(337\) 3.76736e6 1.80702 0.903508 0.428570i \(-0.140983\pi\)
0.903508 + 0.428570i \(0.140983\pi\)
\(338\) −2.06660e6 −0.983929
\(339\) 695459. 0.328679
\(340\) 0 0
\(341\) −327074. −0.152321
\(342\) −306405. −0.141655
\(343\) 1.82478e6 0.837481
\(344\) −1.21303e6 −0.552681
\(345\) 0 0
\(346\) −4.09403e6 −1.83849
\(347\) −923519. −0.411739 −0.205869 0.978579i \(-0.566002\pi\)
−0.205869 + 0.978579i \(0.566002\pi\)
\(348\) −2621.02 −0.00116017
\(349\) 2.61022e6 1.14713 0.573567 0.819159i \(-0.305559\pi\)
0.573567 + 0.819159i \(0.305559\pi\)
\(350\) 0 0
\(351\) 50962.0 0.0220790
\(352\) −8252.36 −0.00354994
\(353\) −1.25644e6 −0.536669 −0.268335 0.963326i \(-0.586473\pi\)
−0.268335 + 0.963326i \(0.586473\pi\)
\(354\) 1.19794e6 0.508073
\(355\) 0 0
\(356\) −8059.01 −0.00337021
\(357\) −653242. −0.271271
\(358\) 390928. 0.161209
\(359\) −2.38200e6 −0.975451 −0.487726 0.872997i \(-0.662173\pi\)
−0.487726 + 0.872997i \(0.662173\pi\)
\(360\) 0 0
\(361\) −2.02628e6 −0.818336
\(362\) 4.45797e6 1.78799
\(363\) 131769. 0.0524864
\(364\) 1909.83 0.000755511 0
\(365\) 0 0
\(366\) 893471. 0.348641
\(367\) 3.46320e6 1.34219 0.671093 0.741373i \(-0.265825\pi\)
0.671093 + 0.741373i \(0.265825\pi\)
\(368\) 805213. 0.309950
\(369\) 779389. 0.297981
\(370\) 0 0
\(371\) 4.18718e6 1.57938
\(372\) −4582.96 −0.00171707
\(373\) −811517. −0.302013 −0.151006 0.988533i \(-0.548251\pi\)
−0.151006 + 0.988533i \(0.548251\pi\)
\(374\) −341569. −0.126270
\(375\) 0 0
\(376\) −211877. −0.0772884
\(377\) 108070. 0.0391607
\(378\) −596283. −0.214646
\(379\) −522481. −0.186841 −0.0934206 0.995627i \(-0.529780\pi\)
−0.0934206 + 0.995627i \(0.529780\pi\)
\(380\) 0 0
\(381\) −942362. −0.332587
\(382\) 2.36312e6 0.828565
\(383\) 2.24960e6 0.783625 0.391813 0.920045i \(-0.371848\pi\)
0.391813 + 0.920045i \(0.371848\pi\)
\(384\) 1.65339e6 0.572200
\(385\) 0 0
\(386\) −1.49760e6 −0.511598
\(387\) 541207. 0.183690
\(388\) −3954.65 −0.00133361
\(389\) −2.42055e6 −0.811035 −0.405518 0.914087i \(-0.632909\pi\)
−0.405518 + 0.914087i \(0.632909\pi\)
\(390\) 0 0
\(391\) −395904. −0.130963
\(392\) −766891. −0.252068
\(393\) 911233. 0.297610
\(394\) 494949. 0.160627
\(395\) 0 0
\(396\) 1846.35 0.000591665 0
\(397\) −1.96075e6 −0.624375 −0.312187 0.950021i \(-0.601062\pi\)
−0.312187 + 0.950021i \(0.601062\pi\)
\(398\) 2.14884e6 0.679981
\(399\) 875372. 0.275271
\(400\) 0 0
\(401\) −3.86313e6 −1.19972 −0.599858 0.800107i \(-0.704776\pi\)
−0.599858 + 0.800107i \(0.704776\pi\)
\(402\) −837641. −0.258519
\(403\) 188964. 0.0579584
\(404\) −35727.8 −0.0108906
\(405\) 0 0
\(406\) −1.26447e6 −0.380710
\(407\) −322722. −0.0965702
\(408\) −817776. −0.243212
\(409\) 1.75292e6 0.518149 0.259075 0.965857i \(-0.416582\pi\)
0.259075 + 0.965857i \(0.416582\pi\)
\(410\) 0 0
\(411\) −1.15480e6 −0.337212
\(412\) −30310.8 −0.00879740
\(413\) −3.42240e6 −0.987314
\(414\) −361383. −0.103626
\(415\) 0 0
\(416\) 4767.73 0.00135076
\(417\) 1.45579e6 0.409976
\(418\) 457717. 0.128132
\(419\) −4.28414e6 −1.19214 −0.596071 0.802932i \(-0.703272\pi\)
−0.596071 + 0.802932i \(0.703272\pi\)
\(420\) 0 0
\(421\) 5.74050e6 1.57850 0.789250 0.614072i \(-0.210470\pi\)
0.789250 + 0.614072i \(0.210470\pi\)
\(422\) 3.52175e6 0.962670
\(423\) 94531.6 0.0256877
\(424\) 5.24182e6 1.41601
\(425\) 0 0
\(426\) 3.65785e6 0.976567
\(427\) −2.55257e6 −0.677497
\(428\) 21333.3 0.00562924
\(429\) −76128.4 −0.0199712
\(430\) 0 0
\(431\) 5.29228e6 1.37230 0.686150 0.727460i \(-0.259299\pi\)
0.686150 + 0.727460i \(0.259299\pi\)
\(432\) −742076. −0.191310
\(433\) 2.37840e6 0.609628 0.304814 0.952412i \(-0.401406\pi\)
0.304814 + 0.952412i \(0.401406\pi\)
\(434\) −2.21098e6 −0.563457
\(435\) 0 0
\(436\) 333.123 8.39245e−5 0
\(437\) 530528. 0.132894
\(438\) 2.30868e6 0.575014
\(439\) 6.44393e6 1.59584 0.797921 0.602762i \(-0.205933\pi\)
0.797921 + 0.602762i \(0.205933\pi\)
\(440\) 0 0
\(441\) 342158. 0.0837780
\(442\) 197339. 0.0480459
\(443\) −2501.14 −0.000605520 0 −0.000302760 1.00000i \(-0.500096\pi\)
−0.000302760 1.00000i \(0.500096\pi\)
\(444\) −4521.99 −0.00108861
\(445\) 0 0
\(446\) 3.71292e6 0.883849
\(447\) −3.63979e6 −0.861603
\(448\) −4.77970e6 −1.12514
\(449\) −5.04731e6 −1.18153 −0.590764 0.806844i \(-0.701174\pi\)
−0.590764 + 0.806844i \(0.701174\pi\)
\(450\) 0 0
\(451\) −1.16427e6 −0.269534
\(452\) −14557.0 −0.00335140
\(453\) 3.74562e6 0.857587
\(454\) −1.24058e6 −0.282478
\(455\) 0 0
\(456\) 1.09585e6 0.246798
\(457\) −4.62692e6 −1.03634 −0.518169 0.855278i \(-0.673386\pi\)
−0.518169 + 0.855278i \(0.673386\pi\)
\(458\) −1.76720e6 −0.393661
\(459\) 364861. 0.0808343
\(460\) 0 0
\(461\) 1.43209e6 0.313846 0.156923 0.987611i \(-0.449842\pi\)
0.156923 + 0.987611i \(0.449842\pi\)
\(462\) 890743. 0.194155
\(463\) −1.58115e6 −0.342785 −0.171393 0.985203i \(-0.554827\pi\)
−0.171393 + 0.985203i \(0.554827\pi\)
\(464\) −1.57364e6 −0.339321
\(465\) 0 0
\(466\) 5.73089e6 1.22252
\(467\) −8724.92 −0.00185127 −0.000925634 1.00000i \(-0.500295\pi\)
−0.000925634 1.00000i \(0.500295\pi\)
\(468\) −1066.71 −0.000225130 0
\(469\) 2.39306e6 0.502368
\(470\) 0 0
\(471\) −2.99551e6 −0.622183
\(472\) −4.28441e6 −0.885189
\(473\) −808470. −0.166154
\(474\) −1.10095e6 −0.225073
\(475\) 0 0
\(476\) 13673.4 0.00276603
\(477\) −2.33870e6 −0.470629
\(478\) 4.30018e6 0.860830
\(479\) 8.15482e6 1.62396 0.811981 0.583684i \(-0.198389\pi\)
0.811981 + 0.583684i \(0.198389\pi\)
\(480\) 0 0
\(481\) 186450. 0.0367452
\(482\) −129212. −0.0253329
\(483\) 1.03244e6 0.201371
\(484\) −2758.13 −0.000535181 0
\(485\) 0 0
\(486\) 333047. 0.0639609
\(487\) −9.00123e6 −1.71981 −0.859903 0.510457i \(-0.829476\pi\)
−0.859903 + 0.510457i \(0.829476\pi\)
\(488\) −3.19549e6 −0.607418
\(489\) 132884. 0.0251305
\(490\) 0 0
\(491\) 6.03644e6 1.13000 0.564999 0.825092i \(-0.308877\pi\)
0.564999 + 0.825092i \(0.308877\pi\)
\(492\) −16313.8 −0.00303838
\(493\) 773721. 0.143373
\(494\) −264442. −0.0487543
\(495\) 0 0
\(496\) −2.75157e6 −0.502200
\(497\) −1.04501e7 −1.89772
\(498\) −351998. −0.0636015
\(499\) 233452. 0.0419707 0.0209853 0.999780i \(-0.493320\pi\)
0.0209853 + 0.999780i \(0.493320\pi\)
\(500\) 0 0
\(501\) −1.52964e6 −0.272267
\(502\) −3.89826e6 −0.690417
\(503\) 7.79728e6 1.37412 0.687058 0.726603i \(-0.258902\pi\)
0.687058 + 0.726603i \(0.258902\pi\)
\(504\) 2.13260e6 0.373966
\(505\) 0 0
\(506\) 539844. 0.0937329
\(507\) −3.29765e6 −0.569751
\(508\) 19725.1 0.00339125
\(509\) 5.32415e6 0.910869 0.455434 0.890269i \(-0.349484\pi\)
0.455434 + 0.890269i \(0.349484\pi\)
\(510\) 0 0
\(511\) −6.59569e6 −1.11740
\(512\) −5.98317e6 −1.00869
\(513\) −488929. −0.0820262
\(514\) 7.60548e6 1.26975
\(515\) 0 0
\(516\) −11328.3 −0.00187301
\(517\) −141214. −0.0232354
\(518\) −2.18157e6 −0.357227
\(519\) −6.53282e6 −1.06459
\(520\) 0 0
\(521\) −8.42076e6 −1.35912 −0.679559 0.733621i \(-0.737829\pi\)
−0.679559 + 0.733621i \(0.737829\pi\)
\(522\) 706257. 0.113445
\(523\) 6.81759e6 1.08988 0.544938 0.838476i \(-0.316553\pi\)
0.544938 + 0.838476i \(0.316553\pi\)
\(524\) −19073.5 −0.00303460
\(525\) 0 0
\(526\) −7.58833e6 −1.19586
\(527\) 1.35288e6 0.212194
\(528\) 1.10853e6 0.173047
\(529\) −5.81062e6 −0.902783
\(530\) 0 0
\(531\) 1.91154e6 0.294203
\(532\) −18322.9 −0.00280682
\(533\) 672649. 0.102558
\(534\) 2.17157e6 0.329550
\(535\) 0 0
\(536\) 2.99581e6 0.450404
\(537\) 623802. 0.0933493
\(538\) −1.14949e7 −1.71218
\(539\) −511125. −0.0757801
\(540\) 0 0
\(541\) −1.00313e7 −1.47354 −0.736770 0.676143i \(-0.763650\pi\)
−0.736770 + 0.676143i \(0.763650\pi\)
\(542\) −1.22986e6 −0.179828
\(543\) 7.11356e6 1.03535
\(544\) 34134.4 0.00494533
\(545\) 0 0
\(546\) −514619. −0.0738762
\(547\) −341056. −0.0487368 −0.0243684 0.999703i \(-0.507757\pi\)
−0.0243684 + 0.999703i \(0.507757\pi\)
\(548\) 24171.7 0.00343840
\(549\) 1.42571e6 0.201883
\(550\) 0 0
\(551\) −1.03682e6 −0.145487
\(552\) 1.29248e6 0.180541
\(553\) 3.14532e6 0.437373
\(554\) 2.31348e6 0.320252
\(555\) 0 0
\(556\) −30471.9 −0.00418035
\(557\) −8.70989e6 −1.18953 −0.594764 0.803901i \(-0.702754\pi\)
−0.594764 + 0.803901i \(0.702754\pi\)
\(558\) 1.23492e6 0.167901
\(559\) 467087. 0.0632219
\(560\) 0 0
\(561\) −545039. −0.0731174
\(562\) −2.29324e6 −0.306272
\(563\) 1.05570e7 1.40369 0.701845 0.712330i \(-0.252360\pi\)
0.701845 + 0.712330i \(0.252360\pi\)
\(564\) −1978.69 −0.000261927 0
\(565\) 0 0
\(566\) −7.13011e6 −0.935525
\(567\) −951485. −0.124292
\(568\) −1.30823e7 −1.70142
\(569\) −2.88092e6 −0.373035 −0.186518 0.982452i \(-0.559720\pi\)
−0.186518 + 0.982452i \(0.559720\pi\)
\(570\) 0 0
\(571\) −8.00484e6 −1.02745 −0.513727 0.857954i \(-0.671735\pi\)
−0.513727 + 0.857954i \(0.671735\pi\)
\(572\) 1593.48 0.000203637 0
\(573\) 3.77081e6 0.479786
\(574\) −7.87035e6 −0.997043
\(575\) 0 0
\(576\) 2.66965e6 0.335272
\(577\) 1.42967e7 1.78770 0.893852 0.448361i \(-0.147992\pi\)
0.893852 + 0.448361i \(0.147992\pi\)
\(578\) −6.59541e6 −0.821149
\(579\) −2.38972e6 −0.296244
\(580\) 0 0
\(581\) 1.00563e6 0.123594
\(582\) 1.06562e6 0.130405
\(583\) 3.49362e6 0.425700
\(584\) −8.25697e6 −1.00182
\(585\) 0 0
\(586\) −1.98026e6 −0.238220
\(587\) −9.83619e6 −1.17823 −0.589117 0.808047i \(-0.700524\pi\)
−0.589117 + 0.808047i \(0.700524\pi\)
\(588\) −7161.88 −0.000854248 0
\(589\) −1.81292e6 −0.215323
\(590\) 0 0
\(591\) 789787. 0.0930125
\(592\) −2.71497e6 −0.318390
\(593\) 9.84429e6 1.14960 0.574801 0.818293i \(-0.305079\pi\)
0.574801 + 0.818293i \(0.305079\pi\)
\(594\) −497515. −0.0578548
\(595\) 0 0
\(596\) 76186.3 0.00878540
\(597\) 3.42890e6 0.393748
\(598\) −311890. −0.0356656
\(599\) −3.94699e6 −0.449468 −0.224734 0.974420i \(-0.572151\pi\)
−0.224734 + 0.974420i \(0.572151\pi\)
\(600\) 0 0
\(601\) −7.63150e6 −0.861834 −0.430917 0.902392i \(-0.641810\pi\)
−0.430917 + 0.902392i \(0.641810\pi\)
\(602\) −5.46516e6 −0.614627
\(603\) −1.33662e6 −0.149697
\(604\) −78401.5 −0.00874444
\(605\) 0 0
\(606\) 9.62715e6 1.06492
\(607\) 1.12292e7 1.23703 0.618513 0.785775i \(-0.287736\pi\)
0.618513 + 0.785775i \(0.287736\pi\)
\(608\) −45741.6 −0.00501825
\(609\) −2.01771e6 −0.220453
\(610\) 0 0
\(611\) 81585.1 0.00884113
\(612\) −7637.10 −0.000824232 0
\(613\) −2.62390e6 −0.282031 −0.141016 0.990007i \(-0.545037\pi\)
−0.141016 + 0.990007i \(0.545037\pi\)
\(614\) 8.24753e6 0.882883
\(615\) 0 0
\(616\) −3.18573e6 −0.338265
\(617\) 3.59228e6 0.379889 0.189945 0.981795i \(-0.439169\pi\)
0.189945 + 0.981795i \(0.439169\pi\)
\(618\) 8.16750e6 0.860237
\(619\) −1.19777e7 −1.25646 −0.628228 0.778029i \(-0.716219\pi\)
−0.628228 + 0.778029i \(0.716219\pi\)
\(620\) 0 0
\(621\) −576657. −0.0600052
\(622\) 1.24249e7 1.28770
\(623\) −6.20398e6 −0.640398
\(624\) −640445. −0.0658447
\(625\) 0 0
\(626\) −3.65398e6 −0.372675
\(627\) 730376. 0.0741955
\(628\) 62700.6 0.00634413
\(629\) 1.33488e6 0.134529
\(630\) 0 0
\(631\) 2.50844e6 0.250801 0.125401 0.992106i \(-0.459978\pi\)
0.125401 + 0.992106i \(0.459978\pi\)
\(632\) 3.93754e6 0.392132
\(633\) 5.61963e6 0.557441
\(634\) 9.18972e6 0.907986
\(635\) 0 0
\(636\) 48952.6 0.00479880
\(637\) 295298. 0.0288345
\(638\) −1.05503e6 −0.102615
\(639\) 5.83681e6 0.565488
\(640\) 0 0
\(641\) 1.12431e7 1.08079 0.540395 0.841411i \(-0.318275\pi\)
0.540395 + 0.841411i \(0.318275\pi\)
\(642\) −5.74845e6 −0.550444
\(643\) −5.61271e6 −0.535360 −0.267680 0.963508i \(-0.586257\pi\)
−0.267680 + 0.963508i \(0.586257\pi\)
\(644\) −21610.5 −0.00205329
\(645\) 0 0
\(646\) −1.89326e6 −0.178497
\(647\) −1.10739e7 −1.04002 −0.520008 0.854162i \(-0.674071\pi\)
−0.520008 + 0.854162i \(0.674071\pi\)
\(648\) −1.19114e6 −0.111436
\(649\) −2.85551e6 −0.266117
\(650\) 0 0
\(651\) −3.52805e6 −0.326274
\(652\) −2781.46 −0.000256244 0
\(653\) 7.83486e6 0.719032 0.359516 0.933139i \(-0.382942\pi\)
0.359516 + 0.933139i \(0.382942\pi\)
\(654\) −89763.0 −0.00820640
\(655\) 0 0
\(656\) −9.79467e6 −0.888649
\(657\) 3.68395e6 0.332966
\(658\) −954589. −0.0859511
\(659\) 1.02774e7 0.921868 0.460934 0.887434i \(-0.347514\pi\)
0.460934 + 0.887434i \(0.347514\pi\)
\(660\) 0 0
\(661\) −1.42467e7 −1.26827 −0.634134 0.773223i \(-0.718643\pi\)
−0.634134 + 0.773223i \(0.718643\pi\)
\(662\) 4.50842e6 0.399833
\(663\) 314892. 0.0278213
\(664\) 1.25892e6 0.110810
\(665\) 0 0
\(666\) 1.21849e6 0.106448
\(667\) −1.22285e6 −0.106429
\(668\) 32017.6 0.00277618
\(669\) 5.92468e6 0.511799
\(670\) 0 0
\(671\) −2.12976e6 −0.182610
\(672\) −89015.7 −0.00760403
\(673\) 3.85183e6 0.327815 0.163908 0.986476i \(-0.447590\pi\)
0.163908 + 0.986476i \(0.447590\pi\)
\(674\) 2.12486e7 1.80169
\(675\) 0 0
\(676\) 69024.9 0.00580951
\(677\) −1.39430e7 −1.16919 −0.584594 0.811326i \(-0.698746\pi\)
−0.584594 + 0.811326i \(0.698746\pi\)
\(678\) 3.92251e6 0.327710
\(679\) −3.04437e6 −0.253409
\(680\) 0 0
\(681\) −1.97959e6 −0.163571
\(682\) −1.84475e6 −0.151872
\(683\) 1.24329e7 1.01981 0.509905 0.860231i \(-0.329681\pi\)
0.509905 + 0.860231i \(0.329681\pi\)
\(684\) 10234.0 0.000836386 0
\(685\) 0 0
\(686\) 1.02921e7 0.835012
\(687\) −2.81991e6 −0.227952
\(688\) −6.80141e6 −0.547807
\(689\) −2.01841e6 −0.161980
\(690\) 0 0
\(691\) 1.46905e7 1.17042 0.585208 0.810883i \(-0.301013\pi\)
0.585208 + 0.810883i \(0.301013\pi\)
\(692\) 136742. 0.0108552
\(693\) 1.42135e6 0.112427
\(694\) −5.20881e6 −0.410525
\(695\) 0 0
\(696\) −2.52592e6 −0.197650
\(697\) 4.81581e6 0.375480
\(698\) 1.47221e7 1.14375
\(699\) 9.14474e6 0.707911
\(700\) 0 0
\(701\) −3.55096e6 −0.272930 −0.136465 0.990645i \(-0.543574\pi\)
−0.136465 + 0.990645i \(0.543574\pi\)
\(702\) 287435. 0.0220139
\(703\) −1.78880e6 −0.136513
\(704\) −3.98799e6 −0.303265
\(705\) 0 0
\(706\) −7.08657e6 −0.535087
\(707\) −2.75039e7 −2.06941
\(708\) −40011.5 −0.00299986
\(709\) −8.56408e6 −0.639831 −0.319915 0.947446i \(-0.603654\pi\)
−0.319915 + 0.947446i \(0.603654\pi\)
\(710\) 0 0
\(711\) −1.75678e6 −0.130330
\(712\) −7.76659e6 −0.574157
\(713\) −2.13821e6 −0.157517
\(714\) −3.68440e6 −0.270472
\(715\) 0 0
\(716\) −13057.1 −0.000951843 0
\(717\) 6.86178e6 0.498470
\(718\) −1.34349e7 −0.972576
\(719\) 1.07318e7 0.774195 0.387097 0.922039i \(-0.373478\pi\)
0.387097 + 0.922039i \(0.373478\pi\)
\(720\) 0 0
\(721\) −2.33338e7 −1.67166
\(722\) −1.14286e7 −0.815924
\(723\) −206182. −0.0146692
\(724\) −148898. −0.0105570
\(725\) 0 0
\(726\) 743201. 0.0523317
\(727\) −1.49237e7 −1.04722 −0.523611 0.851957i \(-0.675416\pi\)
−0.523611 + 0.851957i \(0.675416\pi\)
\(728\) 1.84053e6 0.128711
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.34409e6 0.231465
\(732\) −29842.2 −0.00205851
\(733\) −1.21777e7 −0.837155 −0.418577 0.908181i \(-0.637471\pi\)
−0.418577 + 0.908181i \(0.637471\pi\)
\(734\) 1.95331e7 1.33823
\(735\) 0 0
\(736\) −53948.9 −0.00367103
\(737\) 1.99668e6 0.135406
\(738\) 4.39589e6 0.297102
\(739\) 4.10708e6 0.276644 0.138322 0.990387i \(-0.455829\pi\)
0.138322 + 0.990387i \(0.455829\pi\)
\(740\) 0 0
\(741\) −421969. −0.0282315
\(742\) 2.36164e7 1.57472
\(743\) −2.73077e7 −1.81474 −0.907368 0.420337i \(-0.861912\pi\)
−0.907368 + 0.420337i \(0.861912\pi\)
\(744\) −4.41667e6 −0.292525
\(745\) 0 0
\(746\) −4.57710e6 −0.301123
\(747\) −561682. −0.0368289
\(748\) 11408.5 0.000745546 0
\(749\) 1.64228e7 1.06965
\(750\) 0 0
\(751\) −1.54004e7 −0.996398 −0.498199 0.867063i \(-0.666005\pi\)
−0.498199 + 0.867063i \(0.666005\pi\)
\(752\) −1.18799e6 −0.0766069
\(753\) −6.22043e6 −0.399791
\(754\) 609532. 0.0390452
\(755\) 0 0
\(756\) 19916.0 0.00126735
\(757\) −8.23327e6 −0.522195 −0.261097 0.965312i \(-0.584084\pi\)
−0.261097 + 0.965312i \(0.584084\pi\)
\(758\) −2.94689e6 −0.186290
\(759\) 861426. 0.0542767
\(760\) 0 0
\(761\) 1.69020e7 1.05798 0.528989 0.848629i \(-0.322571\pi\)
0.528989 + 0.848629i \(0.322571\pi\)
\(762\) −5.31509e6 −0.331607
\(763\) 256445. 0.0159471
\(764\) −78928.8 −0.00489217
\(765\) 0 0
\(766\) 1.26881e7 0.781315
\(767\) 1.64975e6 0.101258
\(768\) −166664. −0.0101962
\(769\) −1.72204e7 −1.05009 −0.525045 0.851075i \(-0.675951\pi\)
−0.525045 + 0.851075i \(0.675951\pi\)
\(770\) 0 0
\(771\) 1.21360e7 0.735259
\(772\) 50020.4 0.00302067
\(773\) −1.86902e7 −1.12503 −0.562517 0.826786i \(-0.690167\pi\)
−0.562517 + 0.826786i \(0.690167\pi\)
\(774\) 3.05250e6 0.183149
\(775\) 0 0
\(776\) −3.81116e6 −0.227197
\(777\) −3.48111e6 −0.206855
\(778\) −1.36523e7 −0.808644
\(779\) −6.45339e6 −0.381017
\(780\) 0 0
\(781\) −8.71919e6 −0.511503
\(782\) −2.23297e6 −0.130577
\(783\) 1.12697e6 0.0656913
\(784\) −4.29994e6 −0.249846
\(785\) 0 0
\(786\) 5.13952e6 0.296733
\(787\) 8.17563e6 0.470527 0.235264 0.971932i \(-0.424405\pi\)
0.235264 + 0.971932i \(0.424405\pi\)
\(788\) −16531.4 −0.000948408 0
\(789\) −1.21087e7 −0.692474
\(790\) 0 0
\(791\) −1.12063e7 −0.636824
\(792\) 1.77935e6 0.100797
\(793\) 1.23045e6 0.0694834
\(794\) −1.10590e7 −0.622534
\(795\) 0 0
\(796\) −71772.0 −0.00401488
\(797\) 3.39844e7 1.89511 0.947554 0.319595i \(-0.103547\pi\)
0.947554 + 0.319595i \(0.103547\pi\)
\(798\) 4.93726e6 0.274460
\(799\) 584106. 0.0323687
\(800\) 0 0
\(801\) 3.46516e6 0.190828
\(802\) −2.17887e7 −1.19618
\(803\) −5.50318e6 −0.301179
\(804\) 27977.5 0.00152640
\(805\) 0 0
\(806\) 1.06579e6 0.0577876
\(807\) −1.83423e7 −0.991449
\(808\) −3.44314e7 −1.85535
\(809\) 2.50619e7 1.34630 0.673151 0.739505i \(-0.264940\pi\)
0.673151 + 0.739505i \(0.264940\pi\)
\(810\) 0 0
\(811\) −1.73791e7 −0.927844 −0.463922 0.885876i \(-0.653558\pi\)
−0.463922 + 0.885876i \(0.653558\pi\)
\(812\) 42233.7 0.00224786
\(813\) −1.96248e6 −0.104131
\(814\) −1.82021e6 −0.0962855
\(815\) 0 0
\(816\) −4.58525e6 −0.241067
\(817\) −4.48123e6 −0.234878
\(818\) 9.88681e6 0.516622
\(819\) −821175. −0.0427786
\(820\) 0 0
\(821\) −1.21448e7 −0.628827 −0.314413 0.949286i \(-0.601808\pi\)
−0.314413 + 0.949286i \(0.601808\pi\)
\(822\) −6.51329e6 −0.336218
\(823\) −9.33503e6 −0.480415 −0.240207 0.970722i \(-0.577215\pi\)
−0.240207 + 0.970722i \(0.577215\pi\)
\(824\) −2.92110e7 −1.49875
\(825\) 0 0
\(826\) −1.93029e7 −0.984404
\(827\) 1.15378e7 0.586622 0.293311 0.956017i \(-0.405243\pi\)
0.293311 + 0.956017i \(0.405243\pi\)
\(828\) 12070.3 0.000611847 0
\(829\) −1.25927e7 −0.636404 −0.318202 0.948023i \(-0.603079\pi\)
−0.318202 + 0.948023i \(0.603079\pi\)
\(830\) 0 0
\(831\) 3.69161e6 0.185444
\(832\) 2.30403e6 0.115393
\(833\) 2.11418e6 0.105567
\(834\) 8.21092e6 0.408768
\(835\) 0 0
\(836\) −15287.9 −0.000756539 0
\(837\) 1.97055e6 0.0972241
\(838\) −2.41633e7 −1.18863
\(839\) 3.69853e7 1.81395 0.906973 0.421189i \(-0.138387\pi\)
0.906973 + 0.421189i \(0.138387\pi\)
\(840\) 0 0
\(841\) −1.81213e7 −0.883486
\(842\) 3.23775e7 1.57385
\(843\) −3.65930e6 −0.177349
\(844\) −117627. −0.00568398
\(845\) 0 0
\(846\) 533175. 0.0256120
\(847\) −2.12326e6 −0.101694
\(848\) 2.93907e7 1.40353
\(849\) −1.13775e7 −0.541723
\(850\) 0 0
\(851\) −2.10976e6 −0.0998642
\(852\) −122173. −0.00576604
\(853\) −3.49513e7 −1.64471 −0.822357 0.568972i \(-0.807341\pi\)
−0.822357 + 0.568972i \(0.807341\pi\)
\(854\) −1.43969e7 −0.675500
\(855\) 0 0
\(856\) 2.05593e7 0.959010
\(857\) 2.24410e7 1.04373 0.521867 0.853027i \(-0.325236\pi\)
0.521867 + 0.853027i \(0.325236\pi\)
\(858\) −429378. −0.0199123
\(859\) 3.30887e7 1.53002 0.765010 0.644019i \(-0.222734\pi\)
0.765010 + 0.644019i \(0.222734\pi\)
\(860\) 0 0
\(861\) −1.25587e7 −0.577345
\(862\) 2.98494e7 1.36826
\(863\) −1.58950e6 −0.0726495 −0.0363248 0.999340i \(-0.511565\pi\)
−0.0363248 + 0.999340i \(0.511565\pi\)
\(864\) 49718.7 0.00226587
\(865\) 0 0
\(866\) 1.34146e7 0.607830
\(867\) −1.05243e7 −0.475492
\(868\) 73847.4 0.00332687
\(869\) 2.62433e6 0.117888
\(870\) 0 0
\(871\) −1.15356e6 −0.0515224
\(872\) 321036. 0.0142976
\(873\) 1.70040e6 0.0755118
\(874\) 2.99228e6 0.132502
\(875\) 0 0
\(876\) −77110.6 −0.00339511
\(877\) 1.78401e7 0.783246 0.391623 0.920126i \(-0.371914\pi\)
0.391623 + 0.920126i \(0.371914\pi\)
\(878\) 3.63449e7 1.59114
\(879\) −3.15989e6 −0.137943
\(880\) 0 0
\(881\) −1.53213e7 −0.665054 −0.332527 0.943094i \(-0.607901\pi\)
−0.332527 + 0.943094i \(0.607901\pi\)
\(882\) 1.92983e6 0.0835310
\(883\) −1.92226e7 −0.829680 −0.414840 0.909894i \(-0.636162\pi\)
−0.414840 + 0.909894i \(0.636162\pi\)
\(884\) −6591.17 −0.000283682 0
\(885\) 0 0
\(886\) −14106.9 −0.000603735 0
\(887\) 1.78729e7 0.762756 0.381378 0.924419i \(-0.375450\pi\)
0.381378 + 0.924419i \(0.375450\pi\)
\(888\) −4.35791e6 −0.185458
\(889\) 1.51847e7 0.644396
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −124013. −0.00521860
\(893\) −782727. −0.0328459
\(894\) −2.05291e7 −0.859063
\(895\) 0 0
\(896\) −2.66419e7 −1.10865
\(897\) −497682. −0.0206524
\(898\) −2.84677e7 −1.17804
\(899\) 4.17874e6 0.172443
\(900\) 0 0
\(901\) −1.44507e7 −0.593032
\(902\) −6.56670e6 −0.268739
\(903\) −8.72072e6 −0.355904
\(904\) −1.40288e7 −0.570953
\(905\) 0 0
\(906\) 2.11260e7 0.855059
\(907\) −2.36602e7 −0.954993 −0.477497 0.878634i \(-0.658456\pi\)
−0.477497 + 0.878634i \(0.658456\pi\)
\(908\) 41435.7 0.00166786
\(909\) 1.53620e7 0.616649
\(910\) 0 0
\(911\) 2.62399e7 1.04753 0.523766 0.851862i \(-0.324527\pi\)
0.523766 + 0.851862i \(0.324527\pi\)
\(912\) 6.14443e6 0.244621
\(913\) 839056. 0.0333130
\(914\) −2.60966e7 −1.03328
\(915\) 0 0
\(916\) 59025.0 0.00232433
\(917\) −1.46831e7 −0.576627
\(918\) 2.05788e6 0.0805960
\(919\) 1.05336e7 0.411422 0.205711 0.978613i \(-0.434049\pi\)
0.205711 + 0.978613i \(0.434049\pi\)
\(920\) 0 0
\(921\) 1.31605e7 0.511240
\(922\) 8.07723e6 0.312921
\(923\) 5.03744e6 0.194628
\(924\) −29751.1 −0.00114637
\(925\) 0 0
\(926\) −8.91800e6 −0.341775
\(927\) 1.30328e7 0.498127
\(928\) 105433. 0.00401890
\(929\) 4.93218e7 1.87499 0.937497 0.347993i \(-0.113137\pi\)
0.937497 + 0.347993i \(0.113137\pi\)
\(930\) 0 0
\(931\) −2.83309e6 −0.107124
\(932\) −191413. −0.00721826
\(933\) 1.98263e7 0.745653
\(934\) −49210.1 −0.00184581
\(935\) 0 0
\(936\) −1.02801e6 −0.0383536
\(937\) −3.96100e6 −0.147386 −0.0736929 0.997281i \(-0.523478\pi\)
−0.0736929 + 0.997281i \(0.523478\pi\)
\(938\) 1.34973e7 0.500887
\(939\) −5.83063e6 −0.215800
\(940\) 0 0
\(941\) 3.82868e7 1.40953 0.704766 0.709440i \(-0.251052\pi\)
0.704766 + 0.709440i \(0.251052\pi\)
\(942\) −1.68952e7 −0.620349
\(943\) −7.61131e6 −0.278728
\(944\) −2.40226e7 −0.877383
\(945\) 0 0
\(946\) −4.55991e6 −0.165664
\(947\) 1.34832e7 0.488562 0.244281 0.969705i \(-0.421448\pi\)
0.244281 + 0.969705i \(0.421448\pi\)
\(948\) 36772.2 0.00132892
\(949\) 3.17942e6 0.114599
\(950\) 0 0
\(951\) 1.46640e7 0.525776
\(952\) 1.31772e7 0.471228
\(953\) 9.20696e6 0.328385 0.164193 0.986428i \(-0.447498\pi\)
0.164193 + 0.986428i \(0.447498\pi\)
\(954\) −1.31907e7 −0.469242
\(955\) 0 0
\(956\) −143627. −0.00508268
\(957\) −1.68350e6 −0.0594200
\(958\) 4.59947e7 1.61917
\(959\) 1.86079e7 0.653356
\(960\) 0 0
\(961\) −2.13225e7 −0.744782
\(962\) 1.05161e6 0.0366368
\(963\) −9.17277e6 −0.318739
\(964\) 4315.71 0.000149575 0
\(965\) 0 0
\(966\) 5.82314e6 0.200777
\(967\) −2.95285e6 −0.101549 −0.0507745 0.998710i \(-0.516169\pi\)
−0.0507745 + 0.998710i \(0.516169\pi\)
\(968\) −2.65805e6 −0.0911747
\(969\) −3.02107e6 −0.103360
\(970\) 0 0
\(971\) −3.09624e7 −1.05387 −0.526935 0.849906i \(-0.676659\pi\)
−0.526935 + 0.849906i \(0.676659\pi\)
\(972\) −11123.9 −0.000377651 0
\(973\) −2.34578e7 −0.794339
\(974\) −5.07686e7 −1.71474
\(975\) 0 0
\(976\) −1.79170e7 −0.602062
\(977\) 2.85400e7 0.956573 0.478287 0.878204i \(-0.341258\pi\)
0.478287 + 0.878204i \(0.341258\pi\)
\(978\) 749489. 0.0250564
\(979\) −5.17635e6 −0.172610
\(980\) 0 0
\(981\) −143234. −0.00475198
\(982\) 3.40466e7 1.12667
\(983\) 4.46869e7 1.47501 0.737507 0.675340i \(-0.236003\pi\)
0.737507 + 0.675340i \(0.236003\pi\)
\(984\) −1.57219e7 −0.517626
\(985\) 0 0
\(986\) 4.36393e6 0.142950
\(987\) −1.52323e6 −0.0497706
\(988\) 8832.44 0.000287865 0
\(989\) −5.28529e6 −0.171822
\(990\) 0 0
\(991\) 5.85115e7 1.89259 0.946296 0.323302i \(-0.104793\pi\)
0.946296 + 0.323302i \(0.104793\pi\)
\(992\) 184354. 0.00594804
\(993\) 7.19405e6 0.231526
\(994\) −5.89407e7 −1.89212
\(995\) 0 0
\(996\) 11756.9 0.000375528 0
\(997\) 4.32684e7 1.37858 0.689292 0.724483i \(-0.257922\pi\)
0.689292 + 0.724483i \(0.257922\pi\)
\(998\) 1.31671e6 0.0418469
\(999\) 1.94434e6 0.0616393
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.h.1.3 3
5.4 even 2 165.6.a.d.1.1 3
15.14 odd 2 495.6.a.c.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.d.1.1 3 5.4 even 2
495.6.a.c.1.3 3 15.14 odd 2
825.6.a.h.1.3 3 1.1 even 1 trivial