Properties

Label 825.6.a.u.1.6
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 1.6
Root \(2.70074\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.70074 q^{2} -9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} +73.4900 q^{7} -153.148 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.70074 q^{2} -9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} +73.4900 q^{7} -153.148 q^{8} +81.0000 q^{9} +121.000 q^{11} +222.354 q^{12} -1058.30 q^{13} +198.478 q^{14} +376.977 q^{16} -1854.09 q^{17} +218.760 q^{18} +1753.76 q^{19} -661.410 q^{21} +326.790 q^{22} -572.513 q^{23} +1378.33 q^{24} -2858.21 q^{26} -729.000 q^{27} -1815.64 q^{28} +2446.73 q^{29} -6035.15 q^{31} +5918.86 q^{32} -1089.00 q^{33} -5007.41 q^{34} -2001.19 q^{36} +4859.75 q^{37} +4736.45 q^{38} +9524.74 q^{39} -12742.8 q^{41} -1786.30 q^{42} +1876.74 q^{43} -2989.42 q^{44} -1546.21 q^{46} -15242.3 q^{47} -3392.80 q^{48} -11406.2 q^{49} +16686.8 q^{51} +26146.5 q^{52} +397.221 q^{53} -1968.84 q^{54} -11254.9 q^{56} -15783.8 q^{57} +6607.98 q^{58} -14886.7 q^{59} +16880.6 q^{61} -16299.4 q^{62} +5952.69 q^{63} +3922.05 q^{64} -2941.11 q^{66} -19400.8 q^{67} +45807.0 q^{68} +5152.61 q^{69} -57338.6 q^{71} -12405.0 q^{72} -52587.5 q^{73} +13124.9 q^{74} -43328.3 q^{76} +8892.30 q^{77} +25723.9 q^{78} +57323.5 q^{79} +6561.00 q^{81} -34415.1 q^{82} -39354.7 q^{83} +16340.8 q^{84} +5068.58 q^{86} -22020.5 q^{87} -18530.9 q^{88} +7491.74 q^{89} -77774.8 q^{91} +14144.5 q^{92} +54316.4 q^{93} -41165.6 q^{94} -53269.8 q^{96} -46219.5 q^{97} -30805.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70074 0.477428 0.238714 0.971090i \(-0.423274\pi\)
0.238714 + 0.971090i \(0.423274\pi\)
\(3\) −9.00000 −0.577350
\(4\) −24.7060 −0.772062
\(5\) 0 0
\(6\) −24.3067 −0.275643
\(7\) 73.4900 0.566870 0.283435 0.958991i \(-0.408526\pi\)
0.283435 + 0.958991i \(0.408526\pi\)
\(8\) −153.148 −0.846033
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 222.354 0.445750
\(13\) −1058.30 −1.73681 −0.868405 0.495856i \(-0.834854\pi\)
−0.868405 + 0.495856i \(0.834854\pi\)
\(14\) 198.478 0.270640
\(15\) 0 0
\(16\) 376.977 0.368142
\(17\) −1854.09 −1.55599 −0.777996 0.628269i \(-0.783764\pi\)
−0.777996 + 0.628269i \(0.783764\pi\)
\(18\) 218.760 0.159143
\(19\) 1753.76 1.11451 0.557257 0.830340i \(-0.311854\pi\)
0.557257 + 0.830340i \(0.311854\pi\)
\(20\) 0 0
\(21\) −661.410 −0.327283
\(22\) 326.790 0.143950
\(23\) −572.513 −0.225666 −0.112833 0.993614i \(-0.535992\pi\)
−0.112833 + 0.993614i \(0.535992\pi\)
\(24\) 1378.33 0.488457
\(25\) 0 0
\(26\) −2858.21 −0.829202
\(27\) −729.000 −0.192450
\(28\) −1815.64 −0.437659
\(29\) 2446.73 0.540245 0.270122 0.962826i \(-0.412936\pi\)
0.270122 + 0.962826i \(0.412936\pi\)
\(30\) 0 0
\(31\) −6035.15 −1.12793 −0.563967 0.825797i \(-0.690726\pi\)
−0.563967 + 0.825797i \(0.690726\pi\)
\(32\) 5918.86 1.02179
\(33\) −1089.00 −0.174078
\(34\) −5007.41 −0.742875
\(35\) 0 0
\(36\) −2001.19 −0.257354
\(37\) 4859.75 0.583593 0.291796 0.956480i \(-0.405747\pi\)
0.291796 + 0.956480i \(0.405747\pi\)
\(38\) 4736.45 0.532101
\(39\) 9524.74 1.00275
\(40\) 0 0
\(41\) −12742.8 −1.18388 −0.591939 0.805983i \(-0.701637\pi\)
−0.591939 + 0.805983i \(0.701637\pi\)
\(42\) −1786.30 −0.156254
\(43\) 1876.74 0.154786 0.0773931 0.997001i \(-0.475340\pi\)
0.0773931 + 0.997001i \(0.475340\pi\)
\(44\) −2989.42 −0.232785
\(45\) 0 0
\(46\) −1546.21 −0.107739
\(47\) −15242.3 −1.00648 −0.503241 0.864146i \(-0.667859\pi\)
−0.503241 + 0.864146i \(0.667859\pi\)
\(48\) −3392.80 −0.212547
\(49\) −11406.2 −0.678658
\(50\) 0 0
\(51\) 16686.8 0.898353
\(52\) 26146.5 1.34092
\(53\) 397.221 0.0194242 0.00971209 0.999953i \(-0.496908\pi\)
0.00971209 + 0.999953i \(0.496908\pi\)
\(54\) −1968.84 −0.0918811
\(55\) 0 0
\(56\) −11254.9 −0.479591
\(57\) −15783.8 −0.643465
\(58\) 6607.98 0.257928
\(59\) −14886.7 −0.556759 −0.278380 0.960471i \(-0.589797\pi\)
−0.278380 + 0.960471i \(0.589797\pi\)
\(60\) 0 0
\(61\) 16880.6 0.580849 0.290425 0.956898i \(-0.406203\pi\)
0.290425 + 0.956898i \(0.406203\pi\)
\(62\) −16299.4 −0.538508
\(63\) 5952.69 0.188957
\(64\) 3922.05 0.119691
\(65\) 0 0
\(66\) −2941.11 −0.0831096
\(67\) −19400.8 −0.527999 −0.264000 0.964523i \(-0.585042\pi\)
−0.264000 + 0.964523i \(0.585042\pi\)
\(68\) 45807.0 1.20132
\(69\) 5152.61 0.130288
\(70\) 0 0
\(71\) −57338.6 −1.34990 −0.674949 0.737865i \(-0.735834\pi\)
−0.674949 + 0.737865i \(0.735834\pi\)
\(72\) −12405.0 −0.282011
\(73\) −52587.5 −1.15498 −0.577491 0.816397i \(-0.695968\pi\)
−0.577491 + 0.816397i \(0.695968\pi\)
\(74\) 13124.9 0.278624
\(75\) 0 0
\(76\) −43328.3 −0.860475
\(77\) 8892.30 0.170918
\(78\) 25723.9 0.478740
\(79\) 57323.5 1.03339 0.516695 0.856169i \(-0.327162\pi\)
0.516695 + 0.856169i \(0.327162\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −34415.1 −0.565217
\(83\) −39354.7 −0.627048 −0.313524 0.949580i \(-0.601510\pi\)
−0.313524 + 0.949580i \(0.601510\pi\)
\(84\) 16340.8 0.252682
\(85\) 0 0
\(86\) 5068.58 0.0738993
\(87\) −22020.5 −0.311910
\(88\) −18530.9 −0.255088
\(89\) 7491.74 0.100255 0.0501277 0.998743i \(-0.484037\pi\)
0.0501277 + 0.998743i \(0.484037\pi\)
\(90\) 0 0
\(91\) −77774.8 −0.984545
\(92\) 14144.5 0.174228
\(93\) 54316.4 0.651213
\(94\) −41165.6 −0.480523
\(95\) 0 0
\(96\) −53269.8 −0.589933
\(97\) −46219.5 −0.498765 −0.249382 0.968405i \(-0.580228\pi\)
−0.249382 + 0.968405i \(0.580228\pi\)
\(98\) −30805.2 −0.324011
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 137105. 1.33736 0.668682 0.743549i \(-0.266859\pi\)
0.668682 + 0.743549i \(0.266859\pi\)
\(102\) 45066.7 0.428899
\(103\) −1693.09 −0.0157248 −0.00786241 0.999969i \(-0.502503\pi\)
−0.00786241 + 0.999969i \(0.502503\pi\)
\(104\) 162077. 1.46940
\(105\) 0 0
\(106\) 1072.79 0.00927365
\(107\) 146935. 1.24070 0.620348 0.784327i \(-0.286991\pi\)
0.620348 + 0.784327i \(0.286991\pi\)
\(108\) 18010.7 0.148583
\(109\) 221030. 1.78191 0.890955 0.454092i \(-0.150036\pi\)
0.890955 + 0.454092i \(0.150036\pi\)
\(110\) 0 0
\(111\) −43737.8 −0.336937
\(112\) 27704.1 0.208689
\(113\) 134869. 0.993607 0.496803 0.867863i \(-0.334507\pi\)
0.496803 + 0.867863i \(0.334507\pi\)
\(114\) −42628.1 −0.307209
\(115\) 0 0
\(116\) −60448.8 −0.417102
\(117\) −85722.6 −0.578936
\(118\) −40205.0 −0.265813
\(119\) −136257. −0.882045
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 45590.2 0.277314
\(123\) 114686. 0.683512
\(124\) 149104. 0.870836
\(125\) 0 0
\(126\) 16076.7 0.0902133
\(127\) −171356. −0.942738 −0.471369 0.881936i \(-0.656240\pi\)
−0.471369 + 0.881936i \(0.656240\pi\)
\(128\) −178811. −0.964650
\(129\) −16890.6 −0.0893658
\(130\) 0 0
\(131\) 166556. 0.847973 0.423987 0.905668i \(-0.360630\pi\)
0.423987 + 0.905668i \(0.360630\pi\)
\(132\) 26904.8 0.134399
\(133\) 128884. 0.631785
\(134\) −52396.7 −0.252082
\(135\) 0 0
\(136\) 283950. 1.31642
\(137\) 94610.5 0.430663 0.215332 0.976541i \(-0.430917\pi\)
0.215332 + 0.976541i \(0.430917\pi\)
\(138\) 13915.9 0.0622032
\(139\) 301241. 1.32244 0.661222 0.750190i \(-0.270038\pi\)
0.661222 + 0.750190i \(0.270038\pi\)
\(140\) 0 0
\(141\) 137181. 0.581093
\(142\) −154857. −0.644479
\(143\) −128055. −0.523668
\(144\) 30535.2 0.122714
\(145\) 0 0
\(146\) −142025. −0.551421
\(147\) 102656. 0.391824
\(148\) −120065. −0.450570
\(149\) −39922.6 −0.147317 −0.0736586 0.997284i \(-0.523468\pi\)
−0.0736586 + 0.997284i \(0.523468\pi\)
\(150\) 0 0
\(151\) 114191. 0.407557 0.203778 0.979017i \(-0.434678\pi\)
0.203778 + 0.979017i \(0.434678\pi\)
\(152\) −268585. −0.942916
\(153\) −150181. −0.518664
\(154\) 24015.8 0.0816010
\(155\) 0 0
\(156\) −235318. −0.774183
\(157\) 597139. 1.93342 0.966711 0.255872i \(-0.0823624\pi\)
0.966711 + 0.255872i \(0.0823624\pi\)
\(158\) 154816. 0.493370
\(159\) −3574.99 −0.0112146
\(160\) 0 0
\(161\) −42074.0 −0.127923
\(162\) 17719.6 0.0530476
\(163\) 342738. 1.01040 0.505200 0.863002i \(-0.331419\pi\)
0.505200 + 0.863002i \(0.331419\pi\)
\(164\) 314825. 0.914027
\(165\) 0 0
\(166\) −106287. −0.299371
\(167\) −37584.6 −0.104284 −0.0521422 0.998640i \(-0.516605\pi\)
−0.0521422 + 0.998640i \(0.516605\pi\)
\(168\) 101294. 0.276892
\(169\) 748715. 2.01651
\(170\) 0 0
\(171\) 142054. 0.371505
\(172\) −46366.6 −0.119505
\(173\) 172329. 0.437767 0.218884 0.975751i \(-0.429758\pi\)
0.218884 + 0.975751i \(0.429758\pi\)
\(174\) −59471.8 −0.148915
\(175\) 0 0
\(176\) 45614.3 0.110999
\(177\) 133980. 0.321445
\(178\) 20233.3 0.0478648
\(179\) 422737. 0.986138 0.493069 0.869990i \(-0.335875\pi\)
0.493069 + 0.869990i \(0.335875\pi\)
\(180\) 0 0
\(181\) −584562. −1.32628 −0.663139 0.748497i \(-0.730776\pi\)
−0.663139 + 0.748497i \(0.730776\pi\)
\(182\) −210050. −0.470050
\(183\) −151925. −0.335354
\(184\) 87679.3 0.190921
\(185\) 0 0
\(186\) 146695. 0.310908
\(187\) −224344. −0.469149
\(188\) 376576. 0.777067
\(189\) −53574.2 −0.109094
\(190\) 0 0
\(191\) 52330.3 0.103793 0.0518967 0.998652i \(-0.483473\pi\)
0.0518967 + 0.998652i \(0.483473\pi\)
\(192\) −35298.5 −0.0691039
\(193\) 225739. 0.436227 0.218114 0.975923i \(-0.430010\pi\)
0.218114 + 0.975923i \(0.430010\pi\)
\(194\) −124827. −0.238124
\(195\) 0 0
\(196\) 281802. 0.523966
\(197\) 592389. 1.08753 0.543765 0.839238i \(-0.316998\pi\)
0.543765 + 0.839238i \(0.316998\pi\)
\(198\) 26470.0 0.0479834
\(199\) −108410. −0.194060 −0.0970299 0.995281i \(-0.530934\pi\)
−0.0970299 + 0.995281i \(0.530934\pi\)
\(200\) 0 0
\(201\) 174608. 0.304841
\(202\) 370285. 0.638495
\(203\) 179810. 0.306248
\(204\) −412263. −0.693584
\(205\) 0 0
\(206\) −4572.59 −0.00750748
\(207\) −46373.5 −0.0752219
\(208\) −398957. −0.639393
\(209\) 212205. 0.336039
\(210\) 0 0
\(211\) 514952. 0.796270 0.398135 0.917327i \(-0.369658\pi\)
0.398135 + 0.917327i \(0.369658\pi\)
\(212\) −9813.74 −0.0149967
\(213\) 516047. 0.779364
\(214\) 396833. 0.592344
\(215\) 0 0
\(216\) 111645. 0.162819
\(217\) −443524. −0.639392
\(218\) 596946. 0.850734
\(219\) 473287. 0.666829
\(220\) 0 0
\(221\) 1.96219e6 2.70246
\(222\) −118124. −0.160863
\(223\) 572648. 0.771127 0.385563 0.922681i \(-0.374007\pi\)
0.385563 + 0.922681i \(0.374007\pi\)
\(224\) 434978. 0.579224
\(225\) 0 0
\(226\) 364245. 0.474376
\(227\) −1.45788e6 −1.87784 −0.938918 0.344142i \(-0.888170\pi\)
−0.938918 + 0.344142i \(0.888170\pi\)
\(228\) 389955. 0.496795
\(229\) −979092. −1.23377 −0.616886 0.787053i \(-0.711606\pi\)
−0.616886 + 0.787053i \(0.711606\pi\)
\(230\) 0 0
\(231\) −80030.7 −0.0986794
\(232\) −374712. −0.457065
\(233\) 1.58285e6 1.91007 0.955036 0.296490i \(-0.0958161\pi\)
0.955036 + 0.296490i \(0.0958161\pi\)
\(234\) −231515. −0.276401
\(235\) 0 0
\(236\) 367790. 0.429853
\(237\) −515911. −0.596628
\(238\) −367995. −0.421114
\(239\) 342941. 0.388351 0.194176 0.980967i \(-0.437797\pi\)
0.194176 + 0.980967i \(0.437797\pi\)
\(240\) 0 0
\(241\) −1.12369e6 −1.24625 −0.623124 0.782123i \(-0.714137\pi\)
−0.623124 + 0.782123i \(0.714137\pi\)
\(242\) 39541.6 0.0434026
\(243\) −59049.0 −0.0641500
\(244\) −417052. −0.448452
\(245\) 0 0
\(246\) 309736. 0.326328
\(247\) −1.85601e6 −1.93570
\(248\) 924273. 0.954270
\(249\) 354192. 0.362026
\(250\) 0 0
\(251\) 767281. 0.768724 0.384362 0.923183i \(-0.374422\pi\)
0.384362 + 0.923183i \(0.374422\pi\)
\(252\) −147067. −0.145886
\(253\) −69274.0 −0.0680407
\(254\) −462790. −0.450090
\(255\) 0 0
\(256\) −608429. −0.580243
\(257\) −448786. −0.423844 −0.211922 0.977287i \(-0.567972\pi\)
−0.211922 + 0.977287i \(0.567972\pi\)
\(258\) −45617.2 −0.0426658
\(259\) 357143. 0.330821
\(260\) 0 0
\(261\) 198185. 0.180082
\(262\) 449825. 0.404847
\(263\) −921492. −0.821490 −0.410745 0.911750i \(-0.634731\pi\)
−0.410745 + 0.911750i \(0.634731\pi\)
\(264\) 166778. 0.147275
\(265\) 0 0
\(266\) 348082. 0.301632
\(267\) −67425.7 −0.0578825
\(268\) 479317. 0.407648
\(269\) 1.40136e6 1.18078 0.590388 0.807119i \(-0.298975\pi\)
0.590388 + 0.807119i \(0.298975\pi\)
\(270\) 0 0
\(271\) −2.02100e6 −1.67164 −0.835821 0.549003i \(-0.815008\pi\)
−0.835821 + 0.549003i \(0.815008\pi\)
\(272\) −698949. −0.572826
\(273\) 699974. 0.568427
\(274\) 255519. 0.205611
\(275\) 0 0
\(276\) −127300. −0.100591
\(277\) 74470.0 0.0583152 0.0291576 0.999575i \(-0.490718\pi\)
0.0291576 + 0.999575i \(0.490718\pi\)
\(278\) 813575. 0.631372
\(279\) −488847. −0.375978
\(280\) 0 0
\(281\) 951506. 0.718862 0.359431 0.933172i \(-0.382971\pi\)
0.359431 + 0.933172i \(0.382971\pi\)
\(282\) 370490. 0.277430
\(283\) −535770. −0.397660 −0.198830 0.980034i \(-0.563714\pi\)
−0.198830 + 0.980034i \(0.563714\pi\)
\(284\) 1.41661e6 1.04220
\(285\) 0 0
\(286\) −345843. −0.250014
\(287\) −936472. −0.671105
\(288\) 479428. 0.340598
\(289\) 2.01778e6 1.42111
\(290\) 0 0
\(291\) 415975. 0.287962
\(292\) 1.29923e6 0.891718
\(293\) −2.61667e6 −1.78065 −0.890326 0.455323i \(-0.849524\pi\)
−0.890326 + 0.455323i \(0.849524\pi\)
\(294\) 277247. 0.187068
\(295\) 0 0
\(296\) −744263. −0.493738
\(297\) −88209.0 −0.0580259
\(298\) −107821. −0.0703334
\(299\) 605892. 0.391938
\(300\) 0 0
\(301\) 137921. 0.0877436
\(302\) 308399. 0.194579
\(303\) −1.23394e6 −0.772127
\(304\) 661128. 0.410300
\(305\) 0 0
\(306\) −405600. −0.247625
\(307\) −2.42951e6 −1.47120 −0.735601 0.677415i \(-0.763100\pi\)
−0.735601 + 0.677415i \(0.763100\pi\)
\(308\) −219693. −0.131959
\(309\) 15237.8 0.00907873
\(310\) 0 0
\(311\) 571145. 0.334846 0.167423 0.985885i \(-0.446455\pi\)
0.167423 + 0.985885i \(0.446455\pi\)
\(312\) −1.45870e6 −0.848357
\(313\) 2.21026e6 1.27521 0.637605 0.770363i \(-0.279925\pi\)
0.637605 + 0.770363i \(0.279925\pi\)
\(314\) 1.61272e6 0.923070
\(315\) 0 0
\(316\) −1.41623e6 −0.797842
\(317\) −1.19719e6 −0.669137 −0.334568 0.942371i \(-0.608591\pi\)
−0.334568 + 0.942371i \(0.608591\pi\)
\(318\) −9655.13 −0.00535415
\(319\) 296054. 0.162890
\(320\) 0 0
\(321\) −1.32241e6 −0.716316
\(322\) −113631. −0.0610741
\(323\) −3.25162e6 −1.73418
\(324\) −162096. −0.0857847
\(325\) 0 0
\(326\) 925648. 0.482394
\(327\) −1.98927e6 −1.02879
\(328\) 1.95154e6 1.00160
\(329\) −1.12016e6 −0.570545
\(330\) 0 0
\(331\) 1.82817e6 0.917162 0.458581 0.888653i \(-0.348358\pi\)
0.458581 + 0.888653i \(0.348358\pi\)
\(332\) 972296. 0.484120
\(333\) 393640. 0.194531
\(334\) −101506. −0.0497883
\(335\) 0 0
\(336\) −249337. −0.120486
\(337\) −1.38711e6 −0.665329 −0.332665 0.943045i \(-0.607948\pi\)
−0.332665 + 0.943045i \(0.607948\pi\)
\(338\) 2.02209e6 0.962738
\(339\) −1.21382e6 −0.573659
\(340\) 0 0
\(341\) −730253. −0.340085
\(342\) 383653. 0.177367
\(343\) −2.07339e6 −0.951581
\(344\) −287419. −0.130954
\(345\) 0 0
\(346\) 465416. 0.209002
\(347\) 3.00936e6 1.34169 0.670843 0.741599i \(-0.265932\pi\)
0.670843 + 0.741599i \(0.265932\pi\)
\(348\) 544039. 0.240814
\(349\) 2.44051e6 1.07255 0.536275 0.844043i \(-0.319831\pi\)
0.536275 + 0.844043i \(0.319831\pi\)
\(350\) 0 0
\(351\) 771504. 0.334249
\(352\) 716183. 0.308083
\(353\) 1.65608e6 0.707368 0.353684 0.935365i \(-0.384929\pi\)
0.353684 + 0.935365i \(0.384929\pi\)
\(354\) 361845. 0.153467
\(355\) 0 0
\(356\) −185091. −0.0774034
\(357\) 1.22631e6 0.509249
\(358\) 1.14170e6 0.470810
\(359\) 1.24299e6 0.509018 0.254509 0.967070i \(-0.418086\pi\)
0.254509 + 0.967070i \(0.418086\pi\)
\(360\) 0 0
\(361\) 599571. 0.242143
\(362\) −1.57875e6 −0.633202
\(363\) −131769. −0.0524864
\(364\) 1.92150e6 0.760130
\(365\) 0 0
\(366\) −410312. −0.160107
\(367\) −2.26767e6 −0.878852 −0.439426 0.898279i \(-0.644818\pi\)
−0.439426 + 0.898279i \(0.644818\pi\)
\(368\) −215824. −0.0830770
\(369\) −1.03217e6 −0.394626
\(370\) 0 0
\(371\) 29191.8 0.0110110
\(372\) −1.34194e6 −0.502777
\(373\) 1.92708e6 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(374\) −605896. −0.223985
\(375\) 0 0
\(376\) 2.33433e6 0.851517
\(377\) −2.58938e6 −0.938302
\(378\) −144690. −0.0520847
\(379\) −4.49386e6 −1.60702 −0.803511 0.595291i \(-0.797037\pi\)
−0.803511 + 0.595291i \(0.797037\pi\)
\(380\) 0 0
\(381\) 1.54221e6 0.544290
\(382\) 141331. 0.0495539
\(383\) −2.69103e6 −0.937391 −0.468696 0.883360i \(-0.655276\pi\)
−0.468696 + 0.883360i \(0.655276\pi\)
\(384\) 1.60930e6 0.556941
\(385\) 0 0
\(386\) 609662. 0.208267
\(387\) 152016. 0.0515954
\(388\) 1.14190e6 0.385077
\(389\) −1.35749e6 −0.454843 −0.227422 0.973796i \(-0.573030\pi\)
−0.227422 + 0.973796i \(0.573030\pi\)
\(390\) 0 0
\(391\) 1.06149e6 0.351134
\(392\) 1.74684e6 0.574167
\(393\) −1.49900e6 −0.489578
\(394\) 1.59989e6 0.519218
\(395\) 0 0
\(396\) −242143. −0.0775952
\(397\) 3.44695e6 1.09764 0.548819 0.835941i \(-0.315078\pi\)
0.548819 + 0.835941i \(0.315078\pi\)
\(398\) −292787. −0.0926496
\(399\) −1.15995e6 −0.364761
\(400\) 0 0
\(401\) 5.61209e6 1.74287 0.871433 0.490515i \(-0.163191\pi\)
0.871433 + 0.490515i \(0.163191\pi\)
\(402\) 471570. 0.145540
\(403\) 6.38703e6 1.95901
\(404\) −3.38731e6 −1.03253
\(405\) 0 0
\(406\) 485621. 0.146212
\(407\) 588030. 0.175960
\(408\) −2.55555e6 −0.760036
\(409\) 4.41294e6 1.30443 0.652214 0.758035i \(-0.273840\pi\)
0.652214 + 0.758035i \(0.273840\pi\)
\(410\) 0 0
\(411\) −851494. −0.248644
\(412\) 41829.4 0.0121405
\(413\) −1.09402e6 −0.315610
\(414\) −125243. −0.0359131
\(415\) 0 0
\(416\) −6.26396e6 −1.77466
\(417\) −2.71117e6 −0.763513
\(418\) 573111. 0.160435
\(419\) 659874. 0.183623 0.0918113 0.995776i \(-0.470734\pi\)
0.0918113 + 0.995776i \(0.470734\pi\)
\(420\) 0 0
\(421\) 3.30334e6 0.908339 0.454170 0.890915i \(-0.349936\pi\)
0.454170 + 0.890915i \(0.349936\pi\)
\(422\) 1.39075e6 0.380162
\(423\) −1.23463e6 −0.335494
\(424\) −60833.7 −0.0164335
\(425\) 0 0
\(426\) 1.39371e6 0.372090
\(427\) 1.24056e6 0.329266
\(428\) −3.63017e6 −0.957895
\(429\) 1.15249e6 0.302340
\(430\) 0 0
\(431\) −1.04442e6 −0.270819 −0.135410 0.990790i \(-0.543235\pi\)
−0.135410 + 0.990790i \(0.543235\pi\)
\(432\) −274817. −0.0708490
\(433\) −3.34631e6 −0.857722 −0.428861 0.903371i \(-0.641085\pi\)
−0.428861 + 0.903371i \(0.641085\pi\)
\(434\) −1.19784e6 −0.305264
\(435\) 0 0
\(436\) −5.46077e6 −1.37575
\(437\) −1.00405e6 −0.251508
\(438\) 1.27823e6 0.318363
\(439\) 5.24252e6 1.29831 0.649155 0.760656i \(-0.275123\pi\)
0.649155 + 0.760656i \(0.275123\pi\)
\(440\) 0 0
\(441\) −923903. −0.226219
\(442\) 5.29936e6 1.29023
\(443\) −3.96633e6 −0.960239 −0.480119 0.877203i \(-0.659407\pi\)
−0.480119 + 0.877203i \(0.659407\pi\)
\(444\) 1.08058e6 0.260137
\(445\) 0 0
\(446\) 1.54658e6 0.368158
\(447\) 359304. 0.0850536
\(448\) 288232. 0.0678495
\(449\) −3.69739e6 −0.865524 −0.432762 0.901508i \(-0.642461\pi\)
−0.432762 + 0.901508i \(0.642461\pi\)
\(450\) 0 0
\(451\) −1.54188e6 −0.356952
\(452\) −3.33206e6 −0.767126
\(453\) −1.02772e6 −0.235303
\(454\) −3.93736e6 −0.896532
\(455\) 0 0
\(456\) 2.41727e6 0.544393
\(457\) 6.44119e6 1.44270 0.721349 0.692571i \(-0.243522\pi\)
0.721349 + 0.692571i \(0.243522\pi\)
\(458\) −2.64428e6 −0.589038
\(459\) 1.35163e6 0.299451
\(460\) 0 0
\(461\) 3.40133e6 0.745413 0.372706 0.927949i \(-0.378430\pi\)
0.372706 + 0.927949i \(0.378430\pi\)
\(462\) −216142. −0.0471123
\(463\) −6.02105e6 −1.30533 −0.652664 0.757647i \(-0.726349\pi\)
−0.652664 + 0.757647i \(0.726349\pi\)
\(464\) 922361. 0.198887
\(465\) 0 0
\(466\) 4.27487e6 0.911923
\(467\) −1.98722e6 −0.421652 −0.210826 0.977524i \(-0.567615\pi\)
−0.210826 + 0.977524i \(0.567615\pi\)
\(468\) 2.11786e6 0.446975
\(469\) −1.42577e6 −0.299307
\(470\) 0 0
\(471\) −5.37425e6 −1.11626
\(472\) 2.27987e6 0.471036
\(473\) 227085. 0.0466698
\(474\) −1.39334e6 −0.284847
\(475\) 0 0
\(476\) 3.36636e6 0.680994
\(477\) 32174.9 0.00647472
\(478\) 926196. 0.185410
\(479\) 1.78305e6 0.355078 0.177539 0.984114i \(-0.443186\pi\)
0.177539 + 0.984114i \(0.443186\pi\)
\(480\) 0 0
\(481\) −5.14310e6 −1.01359
\(482\) −3.03480e6 −0.594994
\(483\) 378666. 0.0738564
\(484\) −361720. −0.0701875
\(485\) 0 0
\(486\) −159476. −0.0306270
\(487\) 6.00734e6 1.14778 0.573891 0.818931i \(-0.305433\pi\)
0.573891 + 0.818931i \(0.305433\pi\)
\(488\) −2.58524e6 −0.491418
\(489\) −3.08465e6 −0.583355
\(490\) 0 0
\(491\) 3.33196e6 0.623729 0.311865 0.950127i \(-0.399046\pi\)
0.311865 + 0.950127i \(0.399046\pi\)
\(492\) −2.83342e6 −0.527714
\(493\) −4.53644e6 −0.840617
\(494\) −5.01261e6 −0.924158
\(495\) 0 0
\(496\) −2.27512e6 −0.415240
\(497\) −4.21381e6 −0.765216
\(498\) 956581. 0.172842
\(499\) 7.11288e6 1.27878 0.639388 0.768885i \(-0.279188\pi\)
0.639388 + 0.768885i \(0.279188\pi\)
\(500\) 0 0
\(501\) 338262. 0.0602086
\(502\) 2.07223e6 0.367010
\(503\) 2.89775e6 0.510671 0.255335 0.966853i \(-0.417814\pi\)
0.255335 + 0.966853i \(0.417814\pi\)
\(504\) −911645. −0.159864
\(505\) 0 0
\(506\) −187091. −0.0324846
\(507\) −6.73843e6 −1.16423
\(508\) 4.23353e6 0.727853
\(509\) 9.15158e6 1.56568 0.782838 0.622226i \(-0.213771\pi\)
0.782838 + 0.622226i \(0.213771\pi\)
\(510\) 0 0
\(511\) −3.86466e6 −0.654724
\(512\) 4.07875e6 0.687626
\(513\) −1.27849e6 −0.214488
\(514\) −1.21205e6 −0.202355
\(515\) 0 0
\(516\) 417300. 0.0689960
\(517\) −1.84432e6 −0.303466
\(518\) 964552. 0.157943
\(519\) −1.55096e6 −0.252745
\(520\) 0 0
\(521\) −1.17851e7 −1.90213 −0.951066 0.308989i \(-0.900009\pi\)
−0.951066 + 0.308989i \(0.900009\pi\)
\(522\) 535246. 0.0859760
\(523\) 2.89889e6 0.463423 0.231711 0.972785i \(-0.425567\pi\)
0.231711 + 0.972785i \(0.425567\pi\)
\(524\) −4.11493e6 −0.654688
\(525\) 0 0
\(526\) −2.48871e6 −0.392203
\(527\) 1.11897e7 1.75506
\(528\) −410528. −0.0640853
\(529\) −6.10857e6 −0.949075
\(530\) 0 0
\(531\) −1.20582e6 −0.185586
\(532\) −3.18420e6 −0.487777
\(533\) 1.34858e7 2.05617
\(534\) −182099. −0.0276347
\(535\) 0 0
\(536\) 2.97121e6 0.446705
\(537\) −3.80463e6 −0.569347
\(538\) 3.78470e6 0.563736
\(539\) −1.38015e6 −0.204623
\(540\) 0 0
\(541\) −9.80373e6 −1.44012 −0.720059 0.693913i \(-0.755885\pi\)
−0.720059 + 0.693913i \(0.755885\pi\)
\(542\) −5.45820e6 −0.798089
\(543\) 5.26106e6 0.765727
\(544\) −1.09741e7 −1.58990
\(545\) 0 0
\(546\) 1.89045e6 0.271383
\(547\) 7.81347e6 1.11654 0.558272 0.829658i \(-0.311465\pi\)
0.558272 + 0.829658i \(0.311465\pi\)
\(548\) −2.33745e6 −0.332499
\(549\) 1.36733e6 0.193616
\(550\) 0 0
\(551\) 4.29097e6 0.602111
\(552\) −789114. −0.110228
\(553\) 4.21270e6 0.585798
\(554\) 201124. 0.0278414
\(555\) 0 0
\(556\) −7.44246e6 −1.02101
\(557\) 1.08730e7 1.48494 0.742472 0.669877i \(-0.233653\pi\)
0.742472 + 0.669877i \(0.233653\pi\)
\(558\) −1.32025e6 −0.179503
\(559\) −1.98616e6 −0.268834
\(560\) 0 0
\(561\) 2.01910e6 0.270864
\(562\) 2.56977e6 0.343205
\(563\) −1.10093e7 −1.46382 −0.731910 0.681401i \(-0.761371\pi\)
−0.731910 + 0.681401i \(0.761371\pi\)
\(564\) −3.38919e6 −0.448640
\(565\) 0 0
\(566\) −1.44698e6 −0.189854
\(567\) 482168. 0.0629856
\(568\) 8.78130e6 1.14206
\(569\) −1.03276e7 −1.33727 −0.668635 0.743590i \(-0.733121\pi\)
−0.668635 + 0.743590i \(0.733121\pi\)
\(570\) 0 0
\(571\) 1.13717e7 1.45961 0.729803 0.683657i \(-0.239612\pi\)
0.729803 + 0.683657i \(0.239612\pi\)
\(572\) 3.16372e6 0.404304
\(573\) −470973. −0.0599252
\(574\) −2.52917e6 −0.320404
\(575\) 0 0
\(576\) 317686. 0.0398972
\(577\) 1.35247e7 1.69117 0.845585 0.533841i \(-0.179252\pi\)
0.845585 + 0.533841i \(0.179252\pi\)
\(578\) 5.44950e6 0.678480
\(579\) −2.03165e6 −0.251856
\(580\) 0 0
\(581\) −2.89218e6 −0.355455
\(582\) 1.12344e6 0.137481
\(583\) 48063.7 0.00585661
\(584\) 8.05368e6 0.977152
\(585\) 0 0
\(586\) −7.06694e6 −0.850134
\(587\) 5.40752e6 0.647743 0.323871 0.946101i \(-0.395015\pi\)
0.323871 + 0.946101i \(0.395015\pi\)
\(588\) −2.53622e6 −0.302512
\(589\) −1.05842e7 −1.25710
\(590\) 0 0
\(591\) −5.33150e6 −0.627886
\(592\) 1.83202e6 0.214845
\(593\) −1.18513e7 −1.38398 −0.691991 0.721906i \(-0.743266\pi\)
−0.691991 + 0.721906i \(0.743266\pi\)
\(594\) −238230. −0.0277032
\(595\) 0 0
\(596\) 986328. 0.113738
\(597\) 975687. 0.112040
\(598\) 1.63636e6 0.187122
\(599\) 8.77521e6 0.999287 0.499644 0.866231i \(-0.333464\pi\)
0.499644 + 0.866231i \(0.333464\pi\)
\(600\) 0 0
\(601\) −5.36530e6 −0.605910 −0.302955 0.953005i \(-0.597973\pi\)
−0.302955 + 0.953005i \(0.597973\pi\)
\(602\) 372490. 0.0418913
\(603\) −1.57147e6 −0.176000
\(604\) −2.82119e6 −0.314659
\(605\) 0 0
\(606\) −3.33257e6 −0.368635
\(607\) 1.28875e7 1.41971 0.709853 0.704350i \(-0.248761\pi\)
0.709853 + 0.704350i \(0.248761\pi\)
\(608\) 1.03803e7 1.13880
\(609\) −1.61829e6 −0.176813
\(610\) 0 0
\(611\) 1.61310e7 1.74807
\(612\) 3.71037e6 0.400441
\(613\) 2.15633e6 0.231773 0.115887 0.993262i \(-0.463029\pi\)
0.115887 + 0.993262i \(0.463029\pi\)
\(614\) −6.56148e6 −0.702394
\(615\) 0 0
\(616\) −1.36184e6 −0.144602
\(617\) −5.17630e6 −0.547402 −0.273701 0.961815i \(-0.588248\pi\)
−0.273701 + 0.961815i \(0.588248\pi\)
\(618\) 41153.3 0.00433445
\(619\) 5.56308e6 0.583564 0.291782 0.956485i \(-0.405752\pi\)
0.291782 + 0.956485i \(0.405752\pi\)
\(620\) 0 0
\(621\) 417362. 0.0434294
\(622\) 1.54252e6 0.159865
\(623\) 550568. 0.0568318
\(624\) 3.59061e6 0.369154
\(625\) 0 0
\(626\) 5.96934e6 0.608822
\(627\) −1.90984e6 −0.194012
\(628\) −1.47529e7 −1.49272
\(629\) −9.01040e6 −0.908066
\(630\) 0 0
\(631\) 4.18668e6 0.418597 0.209298 0.977852i \(-0.432882\pi\)
0.209298 + 0.977852i \(0.432882\pi\)
\(632\) −8.77899e6 −0.874283
\(633\) −4.63457e6 −0.459727
\(634\) −3.23330e6 −0.319465
\(635\) 0 0
\(636\) 88323.6 0.00865833
\(637\) 1.20712e7 1.17870
\(638\) 799565. 0.0777682
\(639\) −4.64442e6 −0.449966
\(640\) 0 0
\(641\) −1.85710e7 −1.78521 −0.892606 0.450837i \(-0.851125\pi\)
−0.892606 + 0.450837i \(0.851125\pi\)
\(642\) −3.57150e6 −0.341990
\(643\) 1.57847e7 1.50560 0.752799 0.658250i \(-0.228703\pi\)
0.752799 + 0.658250i \(0.228703\pi\)
\(644\) 1.03948e6 0.0987646
\(645\) 0 0
\(646\) −8.78179e6 −0.827945
\(647\) −1.26849e7 −1.19132 −0.595658 0.803238i \(-0.703109\pi\)
−0.595658 + 0.803238i \(0.703109\pi\)
\(648\) −1.00481e6 −0.0940036
\(649\) −1.80129e6 −0.167869
\(650\) 0 0
\(651\) 3.99171e6 0.369153
\(652\) −8.46769e6 −0.780092
\(653\) −1.85337e7 −1.70090 −0.850451 0.526055i \(-0.823671\pi\)
−0.850451 + 0.526055i \(0.823671\pi\)
\(654\) −5.37252e6 −0.491172
\(655\) 0 0
\(656\) −4.80377e6 −0.435835
\(657\) −4.25958e6 −0.384994
\(658\) −3.02526e6 −0.272394
\(659\) −1.80309e7 −1.61735 −0.808673 0.588258i \(-0.799814\pi\)
−0.808673 + 0.588258i \(0.799814\pi\)
\(660\) 0 0
\(661\) −1.68970e6 −0.150420 −0.0752099 0.997168i \(-0.523963\pi\)
−0.0752099 + 0.997168i \(0.523963\pi\)
\(662\) 4.93741e6 0.437879
\(663\) −1.76597e7 −1.56027
\(664\) 6.02710e6 0.530503
\(665\) 0 0
\(666\) 1.06312e6 0.0928746
\(667\) −1.40078e6 −0.121915
\(668\) 928566. 0.0805140
\(669\) −5.15384e6 −0.445210
\(670\) 0 0
\(671\) 2.04255e6 0.175133
\(672\) −3.91480e6 −0.334415
\(673\) −2.02346e7 −1.72209 −0.861046 0.508527i \(-0.830190\pi\)
−0.861046 + 0.508527i \(0.830190\pi\)
\(674\) −3.74623e6 −0.317647
\(675\) 0 0
\(676\) −1.84977e7 −1.55687
\(677\) −1.68122e7 −1.40978 −0.704892 0.709314i \(-0.749005\pi\)
−0.704892 + 0.709314i \(0.749005\pi\)
\(678\) −3.27821e6 −0.273881
\(679\) −3.39667e6 −0.282735
\(680\) 0 0
\(681\) 1.31209e7 1.08417
\(682\) −1.97223e6 −0.162366
\(683\) −1.07248e7 −0.879706 −0.439853 0.898070i \(-0.644969\pi\)
−0.439853 + 0.898070i \(0.644969\pi\)
\(684\) −3.50960e6 −0.286825
\(685\) 0 0
\(686\) −5.59969e6 −0.454312
\(687\) 8.81183e6 0.712319
\(688\) 707487. 0.0569833
\(689\) −420381. −0.0337361
\(690\) 0 0
\(691\) 6.49615e6 0.517561 0.258780 0.965936i \(-0.416679\pi\)
0.258780 + 0.965936i \(0.416679\pi\)
\(692\) −4.25756e6 −0.337983
\(693\) 720276. 0.0569726
\(694\) 8.12752e6 0.640559
\(695\) 0 0
\(696\) 3.37241e6 0.263886
\(697\) 2.36263e7 1.84210
\(698\) 6.59120e6 0.512066
\(699\) −1.42456e7 −1.10278
\(700\) 0 0
\(701\) −5.20523e6 −0.400078 −0.200039 0.979788i \(-0.564107\pi\)
−0.200039 + 0.979788i \(0.564107\pi\)
\(702\) 2.08363e6 0.159580
\(703\) 8.52283e6 0.650423
\(704\) 474568. 0.0360883
\(705\) 0 0
\(706\) 4.47266e6 0.337718
\(707\) 1.00758e7 0.758111
\(708\) −3.31011e6 −0.248176
\(709\) 1.89444e7 1.41536 0.707679 0.706534i \(-0.249742\pi\)
0.707679 + 0.706534i \(0.249742\pi\)
\(710\) 0 0
\(711\) 4.64320e6 0.344464
\(712\) −1.14735e6 −0.0848193
\(713\) 3.45520e6 0.254536
\(714\) 3.31195e6 0.243130
\(715\) 0 0
\(716\) −1.04441e7 −0.761360
\(717\) −3.08647e6 −0.224215
\(718\) 3.35701e6 0.243019
\(719\) −2.52342e7 −1.82040 −0.910199 0.414170i \(-0.864072\pi\)
−0.910199 + 0.414170i \(0.864072\pi\)
\(720\) 0 0
\(721\) −124425. −0.00891393
\(722\) 1.61929e6 0.115606
\(723\) 1.01132e7 0.719521
\(724\) 1.44422e7 1.02397
\(725\) 0 0
\(726\) −355874. −0.0250585
\(727\) 1.83133e7 1.28508 0.642541 0.766251i \(-0.277880\pi\)
0.642541 + 0.766251i \(0.277880\pi\)
\(728\) 1.19111e7 0.832957
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −3.47963e6 −0.240846
\(732\) 3.75347e6 0.258914
\(733\) −2.77295e7 −1.90626 −0.953130 0.302561i \(-0.902159\pi\)
−0.953130 + 0.302561i \(0.902159\pi\)
\(734\) −6.12441e6 −0.419589
\(735\) 0 0
\(736\) −3.38862e6 −0.230584
\(737\) −2.34750e6 −0.159198
\(738\) −2.78763e6 −0.188406
\(739\) 6.45701e6 0.434931 0.217466 0.976068i \(-0.430221\pi\)
0.217466 + 0.976068i \(0.430221\pi\)
\(740\) 0 0
\(741\) 1.67041e7 1.11758
\(742\) 78839.5 0.00525695
\(743\) −1.64452e7 −1.09287 −0.546435 0.837502i \(-0.684015\pi\)
−0.546435 + 0.837502i \(0.684015\pi\)
\(744\) −8.31846e6 −0.550948
\(745\) 0 0
\(746\) 5.20455e6 0.342402
\(747\) −3.18773e6 −0.209016
\(748\) 5.54265e6 0.362212
\(749\) 1.07983e7 0.703314
\(750\) 0 0
\(751\) −1.41589e7 −0.916072 −0.458036 0.888934i \(-0.651447\pi\)
−0.458036 + 0.888934i \(0.651447\pi\)
\(752\) −5.74601e6 −0.370529
\(753\) −6.90553e6 −0.443823
\(754\) −6.99325e6 −0.447972
\(755\) 0 0
\(756\) 1.32360e6 0.0842275
\(757\) −1.64472e7 −1.04316 −0.521581 0.853202i \(-0.674657\pi\)
−0.521581 + 0.853202i \(0.674657\pi\)
\(758\) −1.21368e7 −0.767237
\(759\) 623466. 0.0392833
\(760\) 0 0
\(761\) 2.78426e7 1.74280 0.871400 0.490573i \(-0.163213\pi\)
0.871400 + 0.490573i \(0.163213\pi\)
\(762\) 4.16511e6 0.259860
\(763\) 1.62435e7 1.01011
\(764\) −1.29287e6 −0.0801350
\(765\) 0 0
\(766\) −7.26777e6 −0.447537
\(767\) 1.57546e7 0.966984
\(768\) 5.47586e6 0.335003
\(769\) 1.96418e7 1.19775 0.598873 0.800844i \(-0.295615\pi\)
0.598873 + 0.800844i \(0.295615\pi\)
\(770\) 0 0
\(771\) 4.03907e6 0.244707
\(772\) −5.57710e6 −0.336794
\(773\) −7.95055e6 −0.478573 −0.239287 0.970949i \(-0.576914\pi\)
−0.239287 + 0.970949i \(0.576914\pi\)
\(774\) 410555. 0.0246331
\(775\) 0 0
\(776\) 7.07843e6 0.421971
\(777\) −3.21429e6 −0.191000
\(778\) −3.66622e6 −0.217155
\(779\) −2.23479e7 −1.31945
\(780\) 0 0
\(781\) −6.93796e6 −0.407009
\(782\) 2.86680e6 0.167641
\(783\) −1.78366e6 −0.103970
\(784\) −4.29989e6 −0.249843
\(785\) 0 0
\(786\) −4.04843e6 −0.233738
\(787\) −4.35212e6 −0.250475 −0.125237 0.992127i \(-0.539969\pi\)
−0.125237 + 0.992127i \(0.539969\pi\)
\(788\) −1.46355e7 −0.839641
\(789\) 8.29343e6 0.474287
\(790\) 0 0
\(791\) 9.91150e6 0.563246
\(792\) −1.50101e6 −0.0850295
\(793\) −1.78648e7 −1.00882
\(794\) 9.30933e6 0.524043
\(795\) 0 0
\(796\) 2.67837e6 0.149826
\(797\) −1.48951e7 −0.830609 −0.415305 0.909682i \(-0.636325\pi\)
−0.415305 + 0.909682i \(0.636325\pi\)
\(798\) −3.13274e6 −0.174147
\(799\) 2.82606e7 1.56608
\(800\) 0 0
\(801\) 606831. 0.0334185
\(802\) 1.51568e7 0.832094
\(803\) −6.36308e6 −0.348240
\(804\) −4.31385e6 −0.235356
\(805\) 0 0
\(806\) 1.72497e7 0.935286
\(807\) −1.26122e7 −0.681721
\(808\) −2.09974e7 −1.13145
\(809\) −1.06835e7 −0.573906 −0.286953 0.957945i \(-0.592642\pi\)
−0.286953 + 0.957945i \(0.592642\pi\)
\(810\) 0 0
\(811\) 1.64454e6 0.0877998 0.0438999 0.999036i \(-0.486022\pi\)
0.0438999 + 0.999036i \(0.486022\pi\)
\(812\) −4.44238e6 −0.236443
\(813\) 1.81890e7 0.965123
\(814\) 1.58812e6 0.0840082
\(815\) 0 0
\(816\) 6.29054e6 0.330721
\(817\) 3.29134e6 0.172511
\(818\) 1.19182e7 0.622771
\(819\) −6.29976e6 −0.328182
\(820\) 0 0
\(821\) −5.82113e6 −0.301404 −0.150702 0.988579i \(-0.548153\pi\)
−0.150702 + 0.988579i \(0.548153\pi\)
\(822\) −2.29967e6 −0.118709
\(823\) 2.30028e7 1.18381 0.591903 0.806009i \(-0.298377\pi\)
0.591903 + 0.806009i \(0.298377\pi\)
\(824\) 259293. 0.0133037
\(825\) 0 0
\(826\) −2.95467e6 −0.150681
\(827\) −1.19553e7 −0.607853 −0.303926 0.952696i \(-0.598298\pi\)
−0.303926 + 0.952696i \(0.598298\pi\)
\(828\) 1.14570e6 0.0580760
\(829\) 2.22758e7 1.12576 0.562880 0.826538i \(-0.309693\pi\)
0.562880 + 0.826538i \(0.309693\pi\)
\(830\) 0 0
\(831\) −670230. −0.0336683
\(832\) −4.15072e6 −0.207881
\(833\) 2.11481e7 1.05599
\(834\) −7.32217e6 −0.364523
\(835\) 0 0
\(836\) −5.24273e6 −0.259443
\(837\) 4.39962e6 0.217071
\(838\) 1.78215e6 0.0876666
\(839\) 1.07814e7 0.528773 0.264387 0.964417i \(-0.414830\pi\)
0.264387 + 0.964417i \(0.414830\pi\)
\(840\) 0 0
\(841\) −1.45247e7 −0.708136
\(842\) 8.92147e6 0.433667
\(843\) −8.56355e6 −0.415035
\(844\) −1.27224e7 −0.614770
\(845\) 0 0
\(846\) −3.33441e6 −0.160174
\(847\) 1.07597e6 0.0515336
\(848\) 149743. 0.00715086
\(849\) 4.82193e6 0.229589
\(850\) 0 0
\(851\) −2.78227e6 −0.131697
\(852\) −1.27495e7 −0.601717
\(853\) −1.90081e7 −0.894470 −0.447235 0.894416i \(-0.647591\pi\)
−0.447235 + 0.894416i \(0.647591\pi\)
\(854\) 3.35042e6 0.157201
\(855\) 0 0
\(856\) −2.25028e7 −1.04967
\(857\) 1.95218e7 0.907963 0.453981 0.891011i \(-0.350003\pi\)
0.453981 + 0.891011i \(0.350003\pi\)
\(858\) 3.11259e6 0.144346
\(859\) −1.63370e7 −0.755420 −0.377710 0.925924i \(-0.623288\pi\)
−0.377710 + 0.925924i \(0.623288\pi\)
\(860\) 0 0
\(861\) 8.42825e6 0.387462
\(862\) −2.82070e6 −0.129297
\(863\) −2.85401e7 −1.30445 −0.652226 0.758024i \(-0.726165\pi\)
−0.652226 + 0.758024i \(0.726165\pi\)
\(864\) −4.31485e6 −0.196644
\(865\) 0 0
\(866\) −9.03752e6 −0.409501
\(867\) −1.81600e7 −0.820480
\(868\) 1.09577e7 0.493650
\(869\) 6.93614e6 0.311579
\(870\) 0 0
\(871\) 2.05320e7 0.917034
\(872\) −3.38504e7 −1.50755
\(873\) −3.74378e6 −0.166255
\(874\) −2.71168e6 −0.120077
\(875\) 0 0
\(876\) −1.16930e7 −0.514833
\(877\) 1.20298e7 0.528153 0.264076 0.964502i \(-0.414933\pi\)
0.264076 + 0.964502i \(0.414933\pi\)
\(878\) 1.41587e7 0.619850
\(879\) 2.35500e7 1.02806
\(880\) 0 0
\(881\) 1.12631e7 0.488896 0.244448 0.969662i \(-0.421393\pi\)
0.244448 + 0.969662i \(0.421393\pi\)
\(882\) −2.49522e6 −0.108004
\(883\) −8.83933e6 −0.381520 −0.190760 0.981637i \(-0.561095\pi\)
−0.190760 + 0.981637i \(0.561095\pi\)
\(884\) −4.84778e7 −2.08647
\(885\) 0 0
\(886\) −1.07120e7 −0.458445
\(887\) −1.56404e7 −0.667481 −0.333741 0.942665i \(-0.608311\pi\)
−0.333741 + 0.942665i \(0.608311\pi\)
\(888\) 6.69836e6 0.285060
\(889\) −1.25930e7 −0.534410
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −1.41478e7 −0.595358
\(893\) −2.67313e7 −1.12174
\(894\) 970387. 0.0406070
\(895\) 0 0
\(896\) −1.31408e7 −0.546831
\(897\) −5.45303e6 −0.226286
\(898\) −9.98569e6 −0.413226
\(899\) −1.47664e7 −0.609360
\(900\) 0 0
\(901\) −736482. −0.0302239
\(902\) −4.16423e6 −0.170419
\(903\) −1.24129e6 −0.0506588
\(904\) −2.06549e7 −0.840624
\(905\) 0 0
\(906\) −2.77559e6 −0.112340
\(907\) 4.34658e7 1.75440 0.877202 0.480121i \(-0.159407\pi\)
0.877202 + 0.480121i \(0.159407\pi\)
\(908\) 3.60184e7 1.44981
\(909\) 1.11055e7 0.445788
\(910\) 0 0
\(911\) 2.58339e7 1.03132 0.515661 0.856792i \(-0.327546\pi\)
0.515661 + 0.856792i \(0.327546\pi\)
\(912\) −5.95015e6 −0.236887
\(913\) −4.76191e6 −0.189062
\(914\) 1.73960e7 0.688785
\(915\) 0 0
\(916\) 2.41894e7 0.952549
\(917\) 1.22402e7 0.480691
\(918\) 3.65040e6 0.142966
\(919\) 3.14288e7 1.22755 0.613775 0.789481i \(-0.289650\pi\)
0.613775 + 0.789481i \(0.289650\pi\)
\(920\) 0 0
\(921\) 2.18656e7 0.849399
\(922\) 9.18612e6 0.355881
\(923\) 6.06816e7 2.34451
\(924\) 1.97724e6 0.0761866
\(925\) 0 0
\(926\) −1.62613e7 −0.623201
\(927\) −137140. −0.00524161
\(928\) 1.44818e7 0.552019
\(929\) 3.42592e7 1.30238 0.651190 0.758915i \(-0.274270\pi\)
0.651190 + 0.758915i \(0.274270\pi\)
\(930\) 0 0
\(931\) −2.00037e7 −0.756375
\(932\) −3.91059e7 −1.47469
\(933\) −5.14030e6 −0.193323
\(934\) −5.36698e6 −0.201309
\(935\) 0 0
\(936\) 1.31283e7 0.489799
\(937\) 1.58561e7 0.589995 0.294997 0.955498i \(-0.404681\pi\)
0.294997 + 0.955498i \(0.404681\pi\)
\(938\) −3.85063e6 −0.142898
\(939\) −1.98923e7 −0.736243
\(940\) 0 0
\(941\) 1.34320e7 0.494500 0.247250 0.968952i \(-0.420473\pi\)
0.247250 + 0.968952i \(0.420473\pi\)
\(942\) −1.45145e7 −0.532935
\(943\) 7.29544e6 0.267160
\(944\) −5.61194e6 −0.204966
\(945\) 0 0
\(946\) 613298. 0.0222815
\(947\) −1.43177e7 −0.518796 −0.259398 0.965770i \(-0.583524\pi\)
−0.259398 + 0.965770i \(0.583524\pi\)
\(948\) 1.27461e7 0.460634
\(949\) 5.56535e7 2.00598
\(950\) 0 0
\(951\) 1.07747e7 0.386326
\(952\) 2.08675e7 0.746239
\(953\) −1.89929e6 −0.0677420 −0.0338710 0.999426i \(-0.510784\pi\)
−0.0338710 + 0.999426i \(0.510784\pi\)
\(954\) 86896.1 0.00309122
\(955\) 0 0
\(956\) −8.47270e6 −0.299831
\(957\) −2.66449e6 −0.0940445
\(958\) 4.81555e6 0.169524
\(959\) 6.95293e6 0.244130
\(960\) 0 0
\(961\) 7.79389e6 0.272236
\(962\) −1.38902e7 −0.483916
\(963\) 1.19017e7 0.413565
\(964\) 2.77619e7 0.962180
\(965\) 0 0
\(966\) 1.02268e6 0.0352611
\(967\) 2.40545e6 0.0827237 0.0413618 0.999144i \(-0.486830\pi\)
0.0413618 + 0.999144i \(0.486830\pi\)
\(968\) −2.24224e6 −0.0769121
\(969\) 2.92646e7 1.00123
\(970\) 0 0
\(971\) −1.46923e6 −0.0500083 −0.0250042 0.999687i \(-0.507960\pi\)
−0.0250042 + 0.999687i \(0.507960\pi\)
\(972\) 1.45886e6 0.0495278
\(973\) 2.21382e7 0.749654
\(974\) 1.62243e7 0.547984
\(975\) 0 0
\(976\) 6.36361e6 0.213835
\(977\) 2.02265e7 0.677929 0.338964 0.940799i \(-0.389923\pi\)
0.338964 + 0.940799i \(0.389923\pi\)
\(978\) −8.33083e6 −0.278510
\(979\) 906501. 0.0302281
\(980\) 0 0
\(981\) 1.79035e7 0.593970
\(982\) 8.99877e6 0.297786
\(983\) 1.06114e7 0.350257 0.175129 0.984546i \(-0.443966\pi\)
0.175129 + 0.984546i \(0.443966\pi\)
\(984\) −1.75639e7 −0.578274
\(985\) 0 0
\(986\) −1.22518e7 −0.401334
\(987\) 1.00814e7 0.329404
\(988\) 4.58546e7 1.49448
\(989\) −1.07446e6 −0.0349299
\(990\) 0 0
\(991\) −4.32715e6 −0.139965 −0.0699823 0.997548i \(-0.522294\pi\)
−0.0699823 + 0.997548i \(0.522294\pi\)
\(992\) −3.57212e7 −1.15252
\(993\) −1.64535e7 −0.529524
\(994\) −1.13804e7 −0.365336
\(995\) 0 0
\(996\) −8.75066e6 −0.279507
\(997\) 4.80656e7 1.53143 0.765714 0.643181i \(-0.222386\pi\)
0.765714 + 0.643181i \(0.222386\pi\)
\(998\) 1.92101e7 0.610524
\(999\) −3.54276e6 −0.112312
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.1.6 yes 10
5.4 even 2 825.6.a.t.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.5 10 5.4 even 2
825.6.a.u.1.6 yes 10 1.1 even 1 trivial