Properties

Label 825.6.a.t.1.1
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 1.1
Root \(11.0151\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-11.0151 q^{2} +9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} -62.7690 q^{7} -631.533 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-11.0151 q^{2} +9.00000 q^{3} +89.3332 q^{4} -99.1362 q^{6} -62.7690 q^{7} -631.533 q^{8} +81.0000 q^{9} +121.000 q^{11} +803.999 q^{12} -986.561 q^{13} +691.409 q^{14} +4097.76 q^{16} -577.936 q^{17} -892.226 q^{18} -234.681 q^{19} -564.921 q^{21} -1332.83 q^{22} +2024.23 q^{23} -5683.79 q^{24} +10867.1 q^{26} +729.000 q^{27} -5607.35 q^{28} -6877.68 q^{29} -689.224 q^{31} -24928.3 q^{32} +1089.00 q^{33} +6366.04 q^{34} +7235.99 q^{36} -12437.0 q^{37} +2585.04 q^{38} -8879.05 q^{39} -5372.90 q^{41} +6222.68 q^{42} -13722.5 q^{43} +10809.3 q^{44} -22297.2 q^{46} +22652.2 q^{47} +36879.8 q^{48} -12867.1 q^{49} -5201.42 q^{51} -88132.7 q^{52} +5321.80 q^{53} -8030.03 q^{54} +39640.7 q^{56} -2112.12 q^{57} +75758.6 q^{58} +23792.4 q^{59} +45926.5 q^{61} +7591.90 q^{62} -5084.29 q^{63} +143460. q^{64} -11995.5 q^{66} -50509.5 q^{67} -51628.9 q^{68} +18218.1 q^{69} -29316.4 q^{71} -51154.1 q^{72} +42433.0 q^{73} +136995. q^{74} -20964.8 q^{76} -7595.05 q^{77} +97804.0 q^{78} +63806.8 q^{79} +6561.00 q^{81} +59183.2 q^{82} +57952.6 q^{83} -50466.2 q^{84} +151155. q^{86} -61899.1 q^{87} -76415.5 q^{88} +75013.4 q^{89} +61925.5 q^{91} +180831. q^{92} -6203.02 q^{93} -249517. q^{94} -224355. q^{96} +10869.1 q^{97} +141732. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.0151 −1.94722 −0.973610 0.228220i \(-0.926709\pi\)
−0.973610 + 0.228220i \(0.926709\pi\)
\(3\) 9.00000 0.577350
\(4\) 89.3332 2.79166
\(5\) 0 0
\(6\) −99.1362 −1.12423
\(7\) −62.7690 −0.484173 −0.242086 0.970255i \(-0.577832\pi\)
−0.242086 + 0.970255i \(0.577832\pi\)
\(8\) −631.533 −3.48876
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 803.999 1.61177
\(13\) −986.561 −1.61907 −0.809535 0.587071i \(-0.800281\pi\)
−0.809535 + 0.587071i \(0.800281\pi\)
\(14\) 691.409 0.942790
\(15\) 0 0
\(16\) 4097.76 4.00171
\(17\) −577.936 −0.485018 −0.242509 0.970149i \(-0.577970\pi\)
−0.242509 + 0.970149i \(0.577970\pi\)
\(18\) −892.226 −0.649073
\(19\) −234.681 −0.149140 −0.0745698 0.997216i \(-0.523758\pi\)
−0.0745698 + 0.997216i \(0.523758\pi\)
\(20\) 0 0
\(21\) −564.921 −0.279537
\(22\) −1332.83 −0.587109
\(23\) 2024.23 0.797885 0.398943 0.916976i \(-0.369377\pi\)
0.398943 + 0.916976i \(0.369377\pi\)
\(24\) −5683.79 −2.01424
\(25\) 0 0
\(26\) 10867.1 3.15268
\(27\) 729.000 0.192450
\(28\) −5607.35 −1.35165
\(29\) −6877.68 −1.51861 −0.759306 0.650734i \(-0.774461\pi\)
−0.759306 + 0.650734i \(0.774461\pi\)
\(30\) 0 0
\(31\) −689.224 −0.128812 −0.0644060 0.997924i \(-0.520515\pi\)
−0.0644060 + 0.997924i \(0.520515\pi\)
\(32\) −24928.3 −4.30346
\(33\) 1089.00 0.174078
\(34\) 6366.04 0.944436
\(35\) 0 0
\(36\) 7235.99 0.930554
\(37\) −12437.0 −1.49352 −0.746761 0.665092i \(-0.768392\pi\)
−0.746761 + 0.665092i \(0.768392\pi\)
\(38\) 2585.04 0.290408
\(39\) −8879.05 −0.934771
\(40\) 0 0
\(41\) −5372.90 −0.499170 −0.249585 0.968353i \(-0.580294\pi\)
−0.249585 + 0.968353i \(0.580294\pi\)
\(42\) 6222.68 0.544320
\(43\) −13722.5 −1.13178 −0.565889 0.824482i \(-0.691467\pi\)
−0.565889 + 0.824482i \(0.691467\pi\)
\(44\) 10809.3 0.841718
\(45\) 0 0
\(46\) −22297.2 −1.55366
\(47\) 22652.2 1.49577 0.747887 0.663826i \(-0.231068\pi\)
0.747887 + 0.663826i \(0.231068\pi\)
\(48\) 36879.8 2.31039
\(49\) −12867.1 −0.765577
\(50\) 0 0
\(51\) −5201.42 −0.280025
\(52\) −88132.7 −4.51990
\(53\) 5321.80 0.260237 0.130118 0.991498i \(-0.458464\pi\)
0.130118 + 0.991498i \(0.458464\pi\)
\(54\) −8030.03 −0.374742
\(55\) 0 0
\(56\) 39640.7 1.68916
\(57\) −2112.12 −0.0861058
\(58\) 75758.6 2.95707
\(59\) 23792.4 0.889833 0.444916 0.895572i \(-0.353233\pi\)
0.444916 + 0.895572i \(0.353233\pi\)
\(60\) 0 0
\(61\) 45926.5 1.58030 0.790149 0.612915i \(-0.210003\pi\)
0.790149 + 0.612915i \(0.210003\pi\)
\(62\) 7591.90 0.250825
\(63\) −5084.29 −0.161391
\(64\) 143460. 4.37806
\(65\) 0 0
\(66\) −11995.5 −0.338967
\(67\) −50509.5 −1.37463 −0.687315 0.726359i \(-0.741211\pi\)
−0.687315 + 0.726359i \(0.741211\pi\)
\(68\) −51628.9 −1.35401
\(69\) 18218.1 0.460659
\(70\) 0 0
\(71\) −29316.4 −0.690184 −0.345092 0.938569i \(-0.612152\pi\)
−0.345092 + 0.938569i \(0.612152\pi\)
\(72\) −51154.1 −1.16292
\(73\) 42433.0 0.931959 0.465979 0.884796i \(-0.345702\pi\)
0.465979 + 0.884796i \(0.345702\pi\)
\(74\) 136995. 2.90822
\(75\) 0 0
\(76\) −20964.8 −0.416347
\(77\) −7595.05 −0.145984
\(78\) 97804.0 1.82020
\(79\) 63806.8 1.15027 0.575134 0.818059i \(-0.304950\pi\)
0.575134 + 0.818059i \(0.304950\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 59183.2 0.971994
\(83\) 57952.6 0.923374 0.461687 0.887043i \(-0.347244\pi\)
0.461687 + 0.887043i \(0.347244\pi\)
\(84\) −50466.2 −0.780373
\(85\) 0 0
\(86\) 151155. 2.20382
\(87\) −61899.1 −0.876771
\(88\) −76415.5 −1.05190
\(89\) 75013.4 1.00384 0.501919 0.864915i \(-0.332628\pi\)
0.501919 + 0.864915i \(0.332628\pi\)
\(90\) 0 0
\(91\) 61925.5 0.783909
\(92\) 180831. 2.22743
\(93\) −6203.02 −0.0743696
\(94\) −249517. −2.91260
\(95\) 0 0
\(96\) −224355. −2.48460
\(97\) 10869.1 0.117291 0.0586454 0.998279i \(-0.481322\pi\)
0.0586454 + 0.998279i \(0.481322\pi\)
\(98\) 141732. 1.49075
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 112358. 1.09597 0.547987 0.836487i \(-0.315394\pi\)
0.547987 + 0.836487i \(0.315394\pi\)
\(102\) 57294.4 0.545270
\(103\) −63393.0 −0.588774 −0.294387 0.955686i \(-0.595115\pi\)
−0.294387 + 0.955686i \(0.595115\pi\)
\(104\) 623046. 5.64855
\(105\) 0 0
\(106\) −58620.3 −0.506738
\(107\) −163095. −1.37715 −0.688574 0.725167i \(-0.741763\pi\)
−0.688574 + 0.725167i \(0.741763\pi\)
\(108\) 65123.9 0.537256
\(109\) −177552. −1.43139 −0.715697 0.698411i \(-0.753891\pi\)
−0.715697 + 0.698411i \(0.753891\pi\)
\(110\) 0 0
\(111\) −111933. −0.862286
\(112\) −257212. −1.93752
\(113\) 3009.41 0.0221710 0.0110855 0.999939i \(-0.496471\pi\)
0.0110855 + 0.999939i \(0.496471\pi\)
\(114\) 23265.3 0.167667
\(115\) 0 0
\(116\) −614405. −4.23945
\(117\) −79911.5 −0.539690
\(118\) −262077. −1.73270
\(119\) 36276.5 0.234832
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −505887. −3.07719
\(123\) −48356.1 −0.288196
\(124\) −61570.6 −0.359599
\(125\) 0 0
\(126\) 56004.1 0.314263
\(127\) −151116. −0.831382 −0.415691 0.909506i \(-0.636460\pi\)
−0.415691 + 0.909506i \(0.636460\pi\)
\(128\) −782528. −4.22158
\(129\) −123502. −0.653432
\(130\) 0 0
\(131\) −45401.3 −0.231148 −0.115574 0.993299i \(-0.536871\pi\)
−0.115574 + 0.993299i \(0.536871\pi\)
\(132\) 97283.8 0.485966
\(133\) 14730.7 0.0722093
\(134\) 556369. 2.67671
\(135\) 0 0
\(136\) 364986. 1.69211
\(137\) 297728. 1.35525 0.677624 0.735408i \(-0.263010\pi\)
0.677624 + 0.735408i \(0.263010\pi\)
\(138\) −200675. −0.897005
\(139\) −214343. −0.940962 −0.470481 0.882410i \(-0.655920\pi\)
−0.470481 + 0.882410i \(0.655920\pi\)
\(140\) 0 0
\(141\) 203870. 0.863586
\(142\) 322924. 1.34394
\(143\) −119374. −0.488168
\(144\) 331918. 1.33390
\(145\) 0 0
\(146\) −467405. −1.81473
\(147\) −115803. −0.442006
\(148\) −1.11104e6 −4.16941
\(149\) 234432. 0.865071 0.432536 0.901617i \(-0.357619\pi\)
0.432536 + 0.901617i \(0.357619\pi\)
\(150\) 0 0
\(151\) 60872.6 0.217260 0.108630 0.994082i \(-0.465354\pi\)
0.108630 + 0.994082i \(0.465354\pi\)
\(152\) 148208. 0.520312
\(153\) −46812.8 −0.161673
\(154\) 83660.5 0.284262
\(155\) 0 0
\(156\) −793194. −2.60956
\(157\) −112122. −0.363029 −0.181514 0.983388i \(-0.558100\pi\)
−0.181514 + 0.983388i \(0.558100\pi\)
\(158\) −702840. −2.23982
\(159\) 47896.2 0.150248
\(160\) 0 0
\(161\) −127059. −0.386314
\(162\) −72270.3 −0.216358
\(163\) −364063. −1.07327 −0.536634 0.843815i \(-0.680304\pi\)
−0.536634 + 0.843815i \(0.680304\pi\)
\(164\) −479978. −1.39351
\(165\) 0 0
\(166\) −638356. −1.79801
\(167\) 599451. 1.66327 0.831635 0.555323i \(-0.187405\pi\)
0.831635 + 0.555323i \(0.187405\pi\)
\(168\) 356766. 0.975237
\(169\) 602011. 1.62139
\(170\) 0 0
\(171\) −19009.1 −0.0497132
\(172\) −1.22587e6 −3.15954
\(173\) 546315. 1.38780 0.693902 0.720070i \(-0.255890\pi\)
0.693902 + 0.720070i \(0.255890\pi\)
\(174\) 681827. 1.70727
\(175\) 0 0
\(176\) 495828. 1.20656
\(177\) 214132. 0.513745
\(178\) −826282. −1.95469
\(179\) −482038. −1.12447 −0.562236 0.826977i \(-0.690059\pi\)
−0.562236 + 0.826977i \(0.690059\pi\)
\(180\) 0 0
\(181\) −481935. −1.09343 −0.546717 0.837318i \(-0.684123\pi\)
−0.546717 + 0.837318i \(0.684123\pi\)
\(182\) −682117. −1.52644
\(183\) 413339. 0.912385
\(184\) −1.27837e6 −2.78363
\(185\) 0 0
\(186\) 68327.1 0.144814
\(187\) −69930.3 −0.146238
\(188\) 2.02359e6 4.17570
\(189\) −45758.6 −0.0931790
\(190\) 0 0
\(191\) −300485. −0.595990 −0.297995 0.954567i \(-0.596318\pi\)
−0.297995 + 0.954567i \(0.596318\pi\)
\(192\) 1.29114e6 2.52767
\(193\) 860084. 1.66206 0.831032 0.556225i \(-0.187751\pi\)
0.831032 + 0.556225i \(0.187751\pi\)
\(194\) −119724. −0.228391
\(195\) 0 0
\(196\) −1.14945e6 −2.13723
\(197\) 623099. 1.14391 0.571954 0.820286i \(-0.306186\pi\)
0.571954 + 0.820286i \(0.306186\pi\)
\(198\) −107959. −0.195703
\(199\) −417370. −0.747116 −0.373558 0.927607i \(-0.621862\pi\)
−0.373558 + 0.927607i \(0.621862\pi\)
\(200\) 0 0
\(201\) −454585. −0.793643
\(202\) −1.23764e6 −2.13410
\(203\) 431705. 0.735270
\(204\) −464660. −0.781735
\(205\) 0 0
\(206\) 698283. 1.14647
\(207\) 163963. 0.265962
\(208\) −4.04269e6 −6.47906
\(209\) −28396.3 −0.0449673
\(210\) 0 0
\(211\) 1.24795e6 1.92970 0.964852 0.262795i \(-0.0846441\pi\)
0.964852 + 0.262795i \(0.0846441\pi\)
\(212\) 475413. 0.726493
\(213\) −263848. −0.398478
\(214\) 1.79651e6 2.68161
\(215\) 0 0
\(216\) −460387. −0.671412
\(217\) 43261.9 0.0623672
\(218\) 1.95576e6 2.78724
\(219\) 381897. 0.538067
\(220\) 0 0
\(221\) 570169. 0.785278
\(222\) 1.23296e6 1.67906
\(223\) 399761. 0.538317 0.269159 0.963096i \(-0.413254\pi\)
0.269159 + 0.963096i \(0.413254\pi\)
\(224\) 1.56472e6 2.08362
\(225\) 0 0
\(226\) −33149.1 −0.0431718
\(227\) −1.14478e6 −1.47454 −0.737269 0.675599i \(-0.763885\pi\)
−0.737269 + 0.675599i \(0.763885\pi\)
\(228\) −188683. −0.240378
\(229\) 24928.6 0.0314130 0.0157065 0.999877i \(-0.495000\pi\)
0.0157065 + 0.999877i \(0.495000\pi\)
\(230\) 0 0
\(231\) −68355.4 −0.0842836
\(232\) 4.34348e6 5.29807
\(233\) 626233. 0.755694 0.377847 0.925868i \(-0.376664\pi\)
0.377847 + 0.925868i \(0.376664\pi\)
\(234\) 880236. 1.05089
\(235\) 0 0
\(236\) 2.12545e6 2.48411
\(237\) 574261. 0.664108
\(238\) −399590. −0.457270
\(239\) −1.05033e6 −1.18941 −0.594704 0.803945i \(-0.702731\pi\)
−0.594704 + 0.803945i \(0.702731\pi\)
\(240\) 0 0
\(241\) 251724. 0.279178 0.139589 0.990210i \(-0.455422\pi\)
0.139589 + 0.990210i \(0.455422\pi\)
\(242\) −161273. −0.177020
\(243\) 59049.0 0.0641500
\(244\) 4.10276e6 4.41166
\(245\) 0 0
\(246\) 532649. 0.561181
\(247\) 231527. 0.241468
\(248\) 435268. 0.449394
\(249\) 521574. 0.533111
\(250\) 0 0
\(251\) 926617. 0.928359 0.464180 0.885741i \(-0.346349\pi\)
0.464180 + 0.885741i \(0.346349\pi\)
\(252\) −454196. −0.450549
\(253\) 244932. 0.240571
\(254\) 1.66456e6 1.61888
\(255\) 0 0
\(256\) 4.02893e6 3.84228
\(257\) −1.37226e6 −1.29600 −0.647998 0.761642i \(-0.724393\pi\)
−0.647998 + 0.761642i \(0.724393\pi\)
\(258\) 1.36039e6 1.27238
\(259\) 780659. 0.723123
\(260\) 0 0
\(261\) −557092. −0.506204
\(262\) 500101. 0.450095
\(263\) −1.81605e6 −1.61896 −0.809482 0.587144i \(-0.800252\pi\)
−0.809482 + 0.587144i \(0.800252\pi\)
\(264\) −687739. −0.607315
\(265\) 0 0
\(266\) −162260. −0.140607
\(267\) 675120. 0.579566
\(268\) −4.51217e6 −3.83750
\(269\) 1.27139e6 1.07127 0.535634 0.844450i \(-0.320073\pi\)
0.535634 + 0.844450i \(0.320073\pi\)
\(270\) 0 0
\(271\) 1.44540e6 1.19554 0.597769 0.801668i \(-0.296054\pi\)
0.597769 + 0.801668i \(0.296054\pi\)
\(272\) −2.36824e6 −1.94090
\(273\) 557329. 0.452590
\(274\) −3.27952e6 −2.63897
\(275\) 0 0
\(276\) 1.62748e6 1.28601
\(277\) −1.74706e6 −1.36807 −0.684034 0.729450i \(-0.739776\pi\)
−0.684034 + 0.729450i \(0.739776\pi\)
\(278\) 2.36102e6 1.83226
\(279\) −55827.1 −0.0429373
\(280\) 0 0
\(281\) −1.84101e6 −1.39088 −0.695441 0.718583i \(-0.744791\pi\)
−0.695441 + 0.718583i \(0.744791\pi\)
\(282\) −2.24565e6 −1.68159
\(283\) 986043. 0.731863 0.365931 0.930642i \(-0.380751\pi\)
0.365931 + 0.930642i \(0.380751\pi\)
\(284\) −2.61893e6 −1.92676
\(285\) 0 0
\(286\) 1.31492e6 0.950570
\(287\) 337251. 0.241685
\(288\) −2.01919e6 −1.43449
\(289\) −1.08585e6 −0.764758
\(290\) 0 0
\(291\) 97821.8 0.0677179
\(292\) 3.79068e6 2.60171
\(293\) 1.98564e6 1.35124 0.675618 0.737252i \(-0.263877\pi\)
0.675618 + 0.737252i \(0.263877\pi\)
\(294\) 1.27559e6 0.860683
\(295\) 0 0
\(296\) 7.85438e6 5.21054
\(297\) 88209.0 0.0580259
\(298\) −2.58230e6 −1.68448
\(299\) −1.99703e6 −1.29183
\(300\) 0 0
\(301\) 861346. 0.547976
\(302\) −670520. −0.423052
\(303\) 1.01122e6 0.632761
\(304\) −961663. −0.596814
\(305\) 0 0
\(306\) 515650. 0.314812
\(307\) 1.13983e6 0.690232 0.345116 0.938560i \(-0.387840\pi\)
0.345116 + 0.938560i \(0.387840\pi\)
\(308\) −678490. −0.407537
\(309\) −570537. −0.339929
\(310\) 0 0
\(311\) −131904. −0.0773319 −0.0386659 0.999252i \(-0.512311\pi\)
−0.0386659 + 0.999252i \(0.512311\pi\)
\(312\) 5.60741e6 3.26119
\(313\) 2.07325e6 1.19616 0.598082 0.801435i \(-0.295930\pi\)
0.598082 + 0.801435i \(0.295930\pi\)
\(314\) 1.23504e6 0.706896
\(315\) 0 0
\(316\) 5.70006e6 3.21116
\(317\) −95123.6 −0.0531667 −0.0265834 0.999647i \(-0.508463\pi\)
−0.0265834 + 0.999647i \(0.508463\pi\)
\(318\) −527583. −0.292565
\(319\) −832199. −0.457879
\(320\) 0 0
\(321\) −1.46785e6 −0.795096
\(322\) 1.39957e6 0.752238
\(323\) 135630. 0.0723353
\(324\) 586115. 0.310185
\(325\) 0 0
\(326\) 4.01021e6 2.08989
\(327\) −1.59797e6 −0.826415
\(328\) 3.39316e6 1.74148
\(329\) −1.42186e6 −0.724213
\(330\) 0 0
\(331\) −3.20153e6 −1.60616 −0.803078 0.595874i \(-0.796806\pi\)
−0.803078 + 0.595874i \(0.796806\pi\)
\(332\) 5.17709e6 2.57775
\(333\) −1.00740e6 −0.497841
\(334\) −6.60304e6 −3.23875
\(335\) 0 0
\(336\) −2.31491e6 −1.11863
\(337\) 2.16577e6 1.03881 0.519406 0.854527i \(-0.326153\pi\)
0.519406 + 0.854527i \(0.326153\pi\)
\(338\) −6.63123e6 −3.15720
\(339\) 27084.7 0.0128004
\(340\) 0 0
\(341\) −83396.1 −0.0388383
\(342\) 209388. 0.0968025
\(343\) 1.86261e6 0.854844
\(344\) 8.66619e6 3.94850
\(345\) 0 0
\(346\) −6.01773e6 −2.70236
\(347\) −2.76730e6 −1.23377 −0.616883 0.787055i \(-0.711605\pi\)
−0.616883 + 0.787055i \(0.711605\pi\)
\(348\) −5.52964e6 −2.44765
\(349\) 3.44895e6 1.51574 0.757868 0.652408i \(-0.226241\pi\)
0.757868 + 0.652408i \(0.226241\pi\)
\(350\) 0 0
\(351\) −719203. −0.311590
\(352\) −3.01632e6 −1.29754
\(353\) 4.34110e6 1.85423 0.927113 0.374781i \(-0.122282\pi\)
0.927113 + 0.374781i \(0.122282\pi\)
\(354\) −2.35869e6 −1.00037
\(355\) 0 0
\(356\) 6.70118e6 2.80238
\(357\) 326488. 0.135580
\(358\) 5.30971e6 2.18959
\(359\) 2.70091e6 1.10605 0.553024 0.833166i \(-0.313474\pi\)
0.553024 + 0.833166i \(0.313474\pi\)
\(360\) 0 0
\(361\) −2.42102e6 −0.977757
\(362\) 5.30858e6 2.12915
\(363\) 131769. 0.0524864
\(364\) 5.53200e6 2.18841
\(365\) 0 0
\(366\) −4.55298e6 −1.77661
\(367\) 3.59856e6 1.39465 0.697323 0.716757i \(-0.254374\pi\)
0.697323 + 0.716757i \(0.254374\pi\)
\(368\) 8.29480e6 3.19291
\(369\) −435205. −0.166390
\(370\) 0 0
\(371\) −334044. −0.126000
\(372\) −554135. −0.207615
\(373\) −4.71222e6 −1.75369 −0.876846 0.480772i \(-0.840357\pi\)
−0.876846 + 0.480772i \(0.840357\pi\)
\(374\) 770291. 0.284758
\(375\) 0 0
\(376\) −1.43056e7 −5.21839
\(377\) 6.78525e6 2.45874
\(378\) 504037. 0.181440
\(379\) 2.89500e6 1.03526 0.517631 0.855604i \(-0.326814\pi\)
0.517631 + 0.855604i \(0.326814\pi\)
\(380\) 0 0
\(381\) −1.36004e6 −0.479999
\(382\) 3.30988e6 1.16052
\(383\) 691654. 0.240931 0.120465 0.992718i \(-0.461561\pi\)
0.120465 + 0.992718i \(0.461561\pi\)
\(384\) −7.04275e6 −2.43733
\(385\) 0 0
\(386\) −9.47394e6 −3.23640
\(387\) −1.11152e6 −0.377259
\(388\) 970970. 0.327436
\(389\) −210206. −0.0704323 −0.0352161 0.999380i \(-0.511212\pi\)
−0.0352161 + 0.999380i \(0.511212\pi\)
\(390\) 0 0
\(391\) −1.16988e6 −0.386988
\(392\) 8.12596e6 2.67091
\(393\) −408611. −0.133453
\(394\) −6.86352e6 −2.22744
\(395\) 0 0
\(396\) 875555. 0.280573
\(397\) 2.04826e6 0.652242 0.326121 0.945328i \(-0.394258\pi\)
0.326121 + 0.945328i \(0.394258\pi\)
\(398\) 4.59738e6 1.45480
\(399\) 132576. 0.0416901
\(400\) 0 0
\(401\) 1.26605e6 0.393179 0.196589 0.980486i \(-0.437013\pi\)
0.196589 + 0.980486i \(0.437013\pi\)
\(402\) 5.00732e6 1.54540
\(403\) 679962. 0.208556
\(404\) 1.00373e7 3.05959
\(405\) 0 0
\(406\) −4.75529e6 −1.43173
\(407\) −1.50488e6 −0.450314
\(408\) 3.28487e6 0.976940
\(409\) 2.04510e6 0.604514 0.302257 0.953226i \(-0.402260\pi\)
0.302257 + 0.953226i \(0.402260\pi\)
\(410\) 0 0
\(411\) 2.67956e6 0.782453
\(412\) −5.66310e6 −1.64366
\(413\) −1.49343e6 −0.430832
\(414\) −1.80607e6 −0.517886
\(415\) 0 0
\(416\) 2.45933e7 6.96760
\(417\) −1.92909e6 −0.543265
\(418\) 312790. 0.0875612
\(419\) −1.00407e6 −0.279401 −0.139701 0.990194i \(-0.544614\pi\)
−0.139701 + 0.990194i \(0.544614\pi\)
\(420\) 0 0
\(421\) −4.99725e6 −1.37412 −0.687061 0.726599i \(-0.741100\pi\)
−0.687061 + 0.726599i \(0.741100\pi\)
\(422\) −1.37463e7 −3.75756
\(423\) 1.83483e6 0.498591
\(424\) −3.36089e6 −0.907903
\(425\) 0 0
\(426\) 2.90632e6 0.775924
\(427\) −2.88276e6 −0.765137
\(428\) −1.45698e7 −3.84453
\(429\) −1.07437e6 −0.281844
\(430\) 0 0
\(431\) −397569. −0.103091 −0.0515453 0.998671i \(-0.516415\pi\)
−0.0515453 + 0.998671i \(0.516415\pi\)
\(432\) 2.98726e6 0.770130
\(433\) 5.89453e6 1.51088 0.755439 0.655219i \(-0.227424\pi\)
0.755439 + 0.655219i \(0.227424\pi\)
\(434\) −476536. −0.121443
\(435\) 0 0
\(436\) −1.58613e7 −3.99597
\(437\) −475048. −0.118996
\(438\) −4.20665e6 −1.04773
\(439\) −2.26759e6 −0.561570 −0.280785 0.959771i \(-0.590595\pi\)
−0.280785 + 0.959771i \(0.590595\pi\)
\(440\) 0 0
\(441\) −1.04223e6 −0.255192
\(442\) −6.28049e6 −1.52911
\(443\) 1.42871e6 0.345888 0.172944 0.984932i \(-0.444672\pi\)
0.172944 + 0.984932i \(0.444672\pi\)
\(444\) −9.99934e6 −2.40721
\(445\) 0 0
\(446\) −4.40342e6 −1.04822
\(447\) 2.10989e6 0.499449
\(448\) −9.00485e6 −2.11974
\(449\) 4.49834e6 1.05302 0.526510 0.850169i \(-0.323500\pi\)
0.526510 + 0.850169i \(0.323500\pi\)
\(450\) 0 0
\(451\) −650120. −0.150506
\(452\) 268840. 0.0618940
\(453\) 547853. 0.125435
\(454\) 1.26099e7 2.87125
\(455\) 0 0
\(456\) 1.33388e6 0.300402
\(457\) 5.64865e6 1.26518 0.632592 0.774485i \(-0.281991\pi\)
0.632592 + 0.774485i \(0.281991\pi\)
\(458\) −274592. −0.0611680
\(459\) −421315. −0.0933417
\(460\) 0 0
\(461\) −8.44590e6 −1.85094 −0.925472 0.378815i \(-0.876332\pi\)
−0.925472 + 0.378815i \(0.876332\pi\)
\(462\) 752944. 0.164119
\(463\) 4.07836e6 0.884165 0.442082 0.896974i \(-0.354240\pi\)
0.442082 + 0.896974i \(0.354240\pi\)
\(464\) −2.81830e7 −6.07705
\(465\) 0 0
\(466\) −6.89804e6 −1.47150
\(467\) 3.43895e6 0.729682 0.364841 0.931070i \(-0.381123\pi\)
0.364841 + 0.931070i \(0.381123\pi\)
\(468\) −7.13875e6 −1.50663
\(469\) 3.17043e6 0.665558
\(470\) 0 0
\(471\) −1.00910e6 −0.209595
\(472\) −1.50257e7 −3.10441
\(473\) −1.66042e6 −0.341244
\(474\) −6.32556e6 −1.29316
\(475\) 0 0
\(476\) 3.24069e6 0.655572
\(477\) 431066. 0.0867456
\(478\) 1.15695e7 2.31604
\(479\) 3.52502e6 0.701978 0.350989 0.936380i \(-0.385845\pi\)
0.350989 + 0.936380i \(0.385845\pi\)
\(480\) 0 0
\(481\) 1.22699e7 2.41812
\(482\) −2.77277e6 −0.543621
\(483\) −1.14353e6 −0.223039
\(484\) 1.30793e6 0.253787
\(485\) 0 0
\(486\) −650433. −0.124914
\(487\) 1.06326e6 0.203151 0.101575 0.994828i \(-0.467612\pi\)
0.101575 + 0.994828i \(0.467612\pi\)
\(488\) −2.90041e7 −5.51328
\(489\) −3.27657e6 −0.619651
\(490\) 0 0
\(491\) 9.16347e6 1.71536 0.857682 0.514180i \(-0.171904\pi\)
0.857682 + 0.514180i \(0.171904\pi\)
\(492\) −4.31980e6 −0.804546
\(493\) 3.97486e6 0.736554
\(494\) −2.55030e6 −0.470190
\(495\) 0 0
\(496\) −2.82427e6 −0.515469
\(497\) 1.84016e6 0.334168
\(498\) −5.74520e6 −1.03808
\(499\) −4.49589e6 −0.808285 −0.404142 0.914696i \(-0.632430\pi\)
−0.404142 + 0.914696i \(0.632430\pi\)
\(500\) 0 0
\(501\) 5.39506e6 0.960289
\(502\) −1.02068e7 −1.80772
\(503\) 8.70281e6 1.53370 0.766848 0.641828i \(-0.221824\pi\)
0.766848 + 0.641828i \(0.221824\pi\)
\(504\) 3.21089e6 0.563054
\(505\) 0 0
\(506\) −2.69796e6 −0.468445
\(507\) 5.41809e6 0.936110
\(508\) −1.34997e7 −2.32094
\(509\) 5.96436e6 1.02040 0.510199 0.860057i \(-0.329572\pi\)
0.510199 + 0.860057i \(0.329572\pi\)
\(510\) 0 0
\(511\) −2.66348e6 −0.451229
\(512\) −1.93383e7 −3.26019
\(513\) −171082. −0.0287019
\(514\) 1.51156e7 2.52359
\(515\) 0 0
\(516\) −1.10328e7 −1.82416
\(517\) 2.74092e6 0.450993
\(518\) −8.59906e6 −1.40808
\(519\) 4.91683e6 0.801248
\(520\) 0 0
\(521\) −2.15902e6 −0.348468 −0.174234 0.984704i \(-0.555745\pi\)
−0.174234 + 0.984704i \(0.555745\pi\)
\(522\) 6.13644e6 0.985690
\(523\) −768246. −0.122814 −0.0614068 0.998113i \(-0.519559\pi\)
−0.0614068 + 0.998113i \(0.519559\pi\)
\(524\) −4.05584e6 −0.645286
\(525\) 0 0
\(526\) 2.00040e7 3.15248
\(527\) 398327. 0.0624761
\(528\) 4.46246e6 0.696609
\(529\) −2.33883e6 −0.363379
\(530\) 0 0
\(531\) 1.92718e6 0.296611
\(532\) 1.31594e6 0.201584
\(533\) 5.30069e6 0.808192
\(534\) −7.43654e6 −1.12854
\(535\) 0 0
\(536\) 3.18984e7 4.79575
\(537\) −4.33834e6 −0.649214
\(538\) −1.40045e7 −2.08599
\(539\) −1.55691e6 −0.230830
\(540\) 0 0
\(541\) −518319. −0.0761385 −0.0380693 0.999275i \(-0.512121\pi\)
−0.0380693 + 0.999275i \(0.512121\pi\)
\(542\) −1.59212e7 −2.32797
\(543\) −4.33742e6 −0.631294
\(544\) 1.44070e7 2.08725
\(545\) 0 0
\(546\) −6.13906e6 −0.881292
\(547\) 4.96112e6 0.708944 0.354472 0.935067i \(-0.384661\pi\)
0.354472 + 0.935067i \(0.384661\pi\)
\(548\) 2.65970e7 3.78340
\(549\) 3.72005e6 0.526766
\(550\) 0 0
\(551\) 1.61406e6 0.226485
\(552\) −1.15053e7 −1.60713
\(553\) −4.00509e6 −0.556928
\(554\) 1.92441e7 2.66393
\(555\) 0 0
\(556\) −1.91479e7 −2.62685
\(557\) −9.82979e6 −1.34248 −0.671238 0.741242i \(-0.734237\pi\)
−0.671238 + 0.741242i \(0.734237\pi\)
\(558\) 614944. 0.0836084
\(559\) 1.35381e7 1.83243
\(560\) 0 0
\(561\) −629372. −0.0844307
\(562\) 2.02790e7 2.70835
\(563\) 7.66483e6 1.01913 0.509567 0.860431i \(-0.329805\pi\)
0.509567 + 0.860431i \(0.329805\pi\)
\(564\) 1.82124e7 2.41084
\(565\) 0 0
\(566\) −1.08614e7 −1.42510
\(567\) −411827. −0.0537969
\(568\) 1.85143e7 2.40788
\(569\) −62102.5 −0.00804134 −0.00402067 0.999992i \(-0.501280\pi\)
−0.00402067 + 0.999992i \(0.501280\pi\)
\(570\) 0 0
\(571\) −3.48638e6 −0.447491 −0.223745 0.974648i \(-0.571828\pi\)
−0.223745 + 0.974648i \(0.571828\pi\)
\(572\) −1.06641e7 −1.36280
\(573\) −2.70436e6 −0.344095
\(574\) −3.71487e6 −0.470613
\(575\) 0 0
\(576\) 1.16203e7 1.45935
\(577\) −9.55413e6 −1.19468 −0.597340 0.801988i \(-0.703776\pi\)
−0.597340 + 0.801988i \(0.703776\pi\)
\(578\) 1.19607e7 1.48915
\(579\) 7.74076e6 0.959593
\(580\) 0 0
\(581\) −3.63763e6 −0.447073
\(582\) −1.07752e6 −0.131861
\(583\) 643938. 0.0784644
\(584\) −2.67978e7 −3.25138
\(585\) 0 0
\(586\) −2.18721e7 −2.63115
\(587\) −5.56273e6 −0.666335 −0.333168 0.942868i \(-0.608118\pi\)
−0.333168 + 0.942868i \(0.608118\pi\)
\(588\) −1.03451e7 −1.23393
\(589\) 161747. 0.0192110
\(590\) 0 0
\(591\) 5.60789e6 0.660436
\(592\) −5.09639e7 −5.97665
\(593\) 1.10409e7 1.28934 0.644669 0.764462i \(-0.276995\pi\)
0.644669 + 0.764462i \(0.276995\pi\)
\(594\) −971634. −0.112989
\(595\) 0 0
\(596\) 2.09426e7 2.41499
\(597\) −3.75633e6 −0.431348
\(598\) 2.19975e7 2.51548
\(599\) −1.47678e6 −0.168170 −0.0840852 0.996459i \(-0.526797\pi\)
−0.0840852 + 0.996459i \(0.526797\pi\)
\(600\) 0 0
\(601\) −7.70993e6 −0.870691 −0.435346 0.900263i \(-0.643374\pi\)
−0.435346 + 0.900263i \(0.643374\pi\)
\(602\) −9.48784e6 −1.06703
\(603\) −4.09127e6 −0.458210
\(604\) 5.43794e6 0.606516
\(605\) 0 0
\(606\) −1.11387e7 −1.23212
\(607\) 1.55229e7 1.71002 0.855009 0.518613i \(-0.173551\pi\)
0.855009 + 0.518613i \(0.173551\pi\)
\(608\) 5.85018e6 0.641816
\(609\) 3.88535e6 0.424509
\(610\) 0 0
\(611\) −2.23478e7 −2.42176
\(612\) −4.18194e6 −0.451335
\(613\) 7.98615e6 0.858394 0.429197 0.903211i \(-0.358797\pi\)
0.429197 + 0.903211i \(0.358797\pi\)
\(614\) −1.25554e7 −1.34403
\(615\) 0 0
\(616\) 4.79652e6 0.509301
\(617\) −8.91087e6 −0.942339 −0.471170 0.882043i \(-0.656168\pi\)
−0.471170 + 0.882043i \(0.656168\pi\)
\(618\) 6.28454e6 0.661915
\(619\) −4.54740e6 −0.477019 −0.238510 0.971140i \(-0.576659\pi\)
−0.238510 + 0.971140i \(0.576659\pi\)
\(620\) 0 0
\(621\) 1.47566e6 0.153553
\(622\) 1.45295e6 0.150582
\(623\) −4.70851e6 −0.486031
\(624\) −3.63842e7 −3.74069
\(625\) 0 0
\(626\) −2.28371e7 −2.32919
\(627\) −255567. −0.0259619
\(628\) −1.00162e7 −1.01345
\(629\) 7.18780e6 0.724385
\(630\) 0 0
\(631\) 1.26569e7 1.26548 0.632740 0.774365i \(-0.281930\pi\)
0.632740 + 0.774365i \(0.281930\pi\)
\(632\) −4.02961e7 −4.01301
\(633\) 1.12315e7 1.11411
\(634\) 1.04780e6 0.103527
\(635\) 0 0
\(636\) 4.27872e6 0.419441
\(637\) 1.26941e7 1.23952
\(638\) 9.16679e6 0.891590
\(639\) −2.37463e6 −0.230061
\(640\) 0 0
\(641\) 3.39088e6 0.325963 0.162981 0.986629i \(-0.447889\pi\)
0.162981 + 0.986629i \(0.447889\pi\)
\(642\) 1.61686e7 1.54823
\(643\) 5.58626e6 0.532837 0.266418 0.963858i \(-0.414160\pi\)
0.266418 + 0.963858i \(0.414160\pi\)
\(644\) −1.13506e7 −1.07846
\(645\) 0 0
\(646\) −1.49399e6 −0.140853
\(647\) 1.49548e7 1.40450 0.702248 0.711932i \(-0.252180\pi\)
0.702248 + 0.711932i \(0.252180\pi\)
\(648\) −4.14349e6 −0.387640
\(649\) 2.87888e6 0.268295
\(650\) 0 0
\(651\) 389357. 0.0360077
\(652\) −3.25229e7 −2.99620
\(653\) 3.85494e6 0.353782 0.176891 0.984230i \(-0.443396\pi\)
0.176891 + 0.984230i \(0.443396\pi\)
\(654\) 1.76018e7 1.60921
\(655\) 0 0
\(656\) −2.20168e7 −1.99754
\(657\) 3.43707e6 0.310653
\(658\) 1.56619e7 1.41020
\(659\) −8.60888e6 −0.772206 −0.386103 0.922456i \(-0.626179\pi\)
−0.386103 + 0.922456i \(0.626179\pi\)
\(660\) 0 0
\(661\) 3.93015e6 0.349869 0.174934 0.984580i \(-0.444029\pi\)
0.174934 + 0.984580i \(0.444029\pi\)
\(662\) 3.52653e7 3.12754
\(663\) 5.13153e6 0.453380
\(664\) −3.65990e7 −3.22143
\(665\) 0 0
\(666\) 1.10966e7 0.969405
\(667\) −1.39220e7 −1.21168
\(668\) 5.35509e7 4.64329
\(669\) 3.59785e6 0.310798
\(670\) 0 0
\(671\) 5.55711e6 0.476478
\(672\) 1.40825e7 1.20298
\(673\) −1.74231e7 −1.48282 −0.741408 0.671054i \(-0.765842\pi\)
−0.741408 + 0.671054i \(0.765842\pi\)
\(674\) −2.38562e7 −2.02280
\(675\) 0 0
\(676\) 5.37795e7 4.52637
\(677\) −1.74041e7 −1.45942 −0.729711 0.683756i \(-0.760345\pi\)
−0.729711 + 0.683756i \(0.760345\pi\)
\(678\) −298342. −0.0249253
\(679\) −682242. −0.0567890
\(680\) 0 0
\(681\) −1.03030e7 −0.851325
\(682\) 918619. 0.0756266
\(683\) 6.21136e6 0.509489 0.254745 0.967008i \(-0.418009\pi\)
0.254745 + 0.967008i \(0.418009\pi\)
\(684\) −1.69815e6 −0.138782
\(685\) 0 0
\(686\) −2.05169e7 −1.66457
\(687\) 224358. 0.0181363
\(688\) −5.62313e7 −4.52905
\(689\) −5.25028e6 −0.421342
\(690\) 0 0
\(691\) −3.49150e6 −0.278174 −0.139087 0.990280i \(-0.544417\pi\)
−0.139087 + 0.990280i \(0.544417\pi\)
\(692\) 4.88041e7 3.87428
\(693\) −615199. −0.0486612
\(694\) 3.04822e7 2.40241
\(695\) 0 0
\(696\) 3.90913e7 3.05884
\(697\) 3.10519e6 0.242106
\(698\) −3.79907e7 −2.95147
\(699\) 5.63610e6 0.436300
\(700\) 0 0
\(701\) 9.84192e6 0.756458 0.378229 0.925712i \(-0.376533\pi\)
0.378229 + 0.925712i \(0.376533\pi\)
\(702\) 7.92212e6 0.606734
\(703\) 2.91873e6 0.222743
\(704\) 1.73587e7 1.32003
\(705\) 0 0
\(706\) −4.78178e7 −3.61059
\(707\) −7.05260e6 −0.530641
\(708\) 1.91291e7 1.43420
\(709\) 2.04292e7 1.52629 0.763144 0.646228i \(-0.223655\pi\)
0.763144 + 0.646228i \(0.223655\pi\)
\(710\) 0 0
\(711\) 5.16835e6 0.383423
\(712\) −4.73734e7 −3.50215
\(713\) −1.39515e6 −0.102777
\(714\) −3.59631e6 −0.264005
\(715\) 0 0
\(716\) −4.30620e7 −3.13915
\(717\) −9.45296e6 −0.686705
\(718\) −2.97509e7 −2.15372
\(719\) 1.70520e7 1.23014 0.615069 0.788474i \(-0.289128\pi\)
0.615069 + 0.788474i \(0.289128\pi\)
\(720\) 0 0
\(721\) 3.97912e6 0.285068
\(722\) 2.66679e7 1.90391
\(723\) 2.26551e6 0.161184
\(724\) −4.30528e7 −3.05250
\(725\) 0 0
\(726\) −1.45145e6 −0.102202
\(727\) 6.52920e6 0.458167 0.229084 0.973407i \(-0.426427\pi\)
0.229084 + 0.973407i \(0.426427\pi\)
\(728\) −3.91080e7 −2.73487
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 7.93071e6 0.548932
\(732\) 3.69248e7 2.54707
\(733\) 1.96819e7 1.35303 0.676513 0.736430i \(-0.263490\pi\)
0.676513 + 0.736430i \(0.263490\pi\)
\(734\) −3.96386e7 −2.71568
\(735\) 0 0
\(736\) −5.04606e7 −3.43366
\(737\) −6.11165e6 −0.414467
\(738\) 4.79384e6 0.323998
\(739\) 5.50059e6 0.370509 0.185254 0.982691i \(-0.440689\pi\)
0.185254 + 0.982691i \(0.440689\pi\)
\(740\) 0 0
\(741\) 2.08374e6 0.139411
\(742\) 3.67954e6 0.245349
\(743\) −6.56551e6 −0.436311 −0.218156 0.975914i \(-0.570004\pi\)
−0.218156 + 0.975914i \(0.570004\pi\)
\(744\) 3.91741e6 0.259458
\(745\) 0 0
\(746\) 5.19057e7 3.41482
\(747\) 4.69416e6 0.307791
\(748\) −6.24709e6 −0.408248
\(749\) 1.02373e7 0.666777
\(750\) 0 0
\(751\) 2.04233e7 1.32137 0.660687 0.750662i \(-0.270265\pi\)
0.660687 + 0.750662i \(0.270265\pi\)
\(752\) 9.28232e7 5.98566
\(753\) 8.33956e6 0.535989
\(754\) −7.47405e7 −4.78771
\(755\) 0 0
\(756\) −4.08776e6 −0.260124
\(757\) −1.28683e7 −0.816174 −0.408087 0.912943i \(-0.633804\pi\)
−0.408087 + 0.912943i \(0.633804\pi\)
\(758\) −3.18888e7 −2.01588
\(759\) 2.20439e6 0.138894
\(760\) 0 0
\(761\) 1.16269e7 0.727782 0.363891 0.931442i \(-0.381448\pi\)
0.363891 + 0.931442i \(0.381448\pi\)
\(762\) 1.49811e7 0.934663
\(763\) 1.11448e7 0.693041
\(764\) −2.68432e7 −1.66380
\(765\) 0 0
\(766\) −7.61866e6 −0.469144
\(767\) −2.34727e7 −1.44070
\(768\) 3.62603e7 2.21834
\(769\) 2.67451e7 1.63091 0.815453 0.578823i \(-0.196488\pi\)
0.815453 + 0.578823i \(0.196488\pi\)
\(770\) 0 0
\(771\) −1.23503e7 −0.748243
\(772\) 7.68340e7 4.63992
\(773\) 1.62196e7 0.976320 0.488160 0.872754i \(-0.337668\pi\)
0.488160 + 0.872754i \(0.337668\pi\)
\(774\) 1.22435e7 0.734606
\(775\) 0 0
\(776\) −6.86419e6 −0.409199
\(777\) 7.02593e6 0.417495
\(778\) 2.31545e6 0.137147
\(779\) 1.26091e6 0.0744461
\(780\) 0 0
\(781\) −3.54728e6 −0.208098
\(782\) 1.28863e7 0.753551
\(783\) −5.01383e6 −0.292257
\(784\) −5.27260e7 −3.06362
\(785\) 0 0
\(786\) 4.50091e6 0.259863
\(787\) −3.30833e7 −1.90402 −0.952011 0.306063i \(-0.900988\pi\)
−0.952011 + 0.306063i \(0.900988\pi\)
\(788\) 5.56634e7 3.19341
\(789\) −1.63444e7 −0.934710
\(790\) 0 0
\(791\) −188898. −0.0107346
\(792\) −6.18965e6 −0.350633
\(793\) −4.53093e7 −2.55861
\(794\) −2.25619e7 −1.27006
\(795\) 0 0
\(796\) −3.72850e7 −2.08570
\(797\) −6.59997e6 −0.368041 −0.184020 0.982922i \(-0.558911\pi\)
−0.184020 + 0.982922i \(0.558911\pi\)
\(798\) −1.46034e6 −0.0811797
\(799\) −1.30915e7 −0.725477
\(800\) 0 0
\(801\) 6.07608e6 0.334613
\(802\) −1.39457e7 −0.765605
\(803\) 5.13439e6 0.280996
\(804\) −4.06096e7 −2.21558
\(805\) 0 0
\(806\) −7.48987e6 −0.406104
\(807\) 1.14425e7 0.618497
\(808\) −7.09577e7 −3.82359
\(809\) −2.14723e7 −1.15347 −0.576736 0.816931i \(-0.695674\pi\)
−0.576736 + 0.816931i \(0.695674\pi\)
\(810\) 0 0
\(811\) −1.13634e7 −0.606678 −0.303339 0.952883i \(-0.598101\pi\)
−0.303339 + 0.952883i \(0.598101\pi\)
\(812\) 3.85656e7 2.05263
\(813\) 1.30086e7 0.690244
\(814\) 1.65764e7 0.876860
\(815\) 0 0
\(816\) −2.13142e7 −1.12058
\(817\) 3.22040e6 0.168793
\(818\) −2.25271e7 −1.17712
\(819\) 5.01596e6 0.261303
\(820\) 0 0
\(821\) −6.33658e6 −0.328093 −0.164047 0.986453i \(-0.552455\pi\)
−0.164047 + 0.986453i \(0.552455\pi\)
\(822\) −2.95157e7 −1.52361
\(823\) 3.27073e7 1.68324 0.841619 0.540071i \(-0.181603\pi\)
0.841619 + 0.540071i \(0.181603\pi\)
\(824\) 4.00348e7 2.05409
\(825\) 0 0
\(826\) 1.64503e7 0.838925
\(827\) −3.83082e6 −0.194773 −0.0973863 0.995247i \(-0.531048\pi\)
−0.0973863 + 0.995247i \(0.531048\pi\)
\(828\) 1.46473e7 0.742475
\(829\) −2.45814e7 −1.24228 −0.621142 0.783698i \(-0.713331\pi\)
−0.621142 + 0.783698i \(0.713331\pi\)
\(830\) 0 0
\(831\) −1.57235e7 −0.789854
\(832\) −1.41532e8 −7.08838
\(833\) 7.43633e6 0.371318
\(834\) 2.12491e7 1.05786
\(835\) 0 0
\(836\) −2.53674e6 −0.125533
\(837\) −502444. −0.0247899
\(838\) 1.10599e7 0.544055
\(839\) 8.44908e6 0.414386 0.207193 0.978300i \(-0.433567\pi\)
0.207193 + 0.978300i \(0.433567\pi\)
\(840\) 0 0
\(841\) 2.67913e7 1.30618
\(842\) 5.50453e7 2.67572
\(843\) −1.65691e7 −0.803026
\(844\) 1.11483e8 5.38708
\(845\) 0 0
\(846\) −2.02109e7 −0.970867
\(847\) −919001. −0.0440157
\(848\) 2.18074e7 1.04139
\(849\) 8.87438e6 0.422541
\(850\) 0 0
\(851\) −2.51754e7 −1.19166
\(852\) −2.35703e7 −1.11242
\(853\) −1.03931e7 −0.489072 −0.244536 0.969640i \(-0.578636\pi\)
−0.244536 + 0.969640i \(0.578636\pi\)
\(854\) 3.17540e7 1.48989
\(855\) 0 0
\(856\) 1.03000e8 4.80453
\(857\) 1.78951e7 0.832302 0.416151 0.909295i \(-0.363379\pi\)
0.416151 + 0.909295i \(0.363379\pi\)
\(858\) 1.18343e7 0.548812
\(859\) −1.57750e7 −0.729435 −0.364717 0.931118i \(-0.618834\pi\)
−0.364717 + 0.931118i \(0.618834\pi\)
\(860\) 0 0
\(861\) 3.03526e6 0.139537
\(862\) 4.37928e6 0.200740
\(863\) −3.34930e7 −1.53083 −0.765416 0.643536i \(-0.777467\pi\)
−0.765416 + 0.643536i \(0.777467\pi\)
\(864\) −1.81727e7 −0.828201
\(865\) 0 0
\(866\) −6.49290e7 −2.94201
\(867\) −9.77262e6 −0.441533
\(868\) 3.86472e6 0.174108
\(869\) 7.72062e6 0.346819
\(870\) 0 0
\(871\) 4.98307e7 2.22562
\(872\) 1.12130e8 4.99379
\(873\) 880396. 0.0390969
\(874\) 5.23271e6 0.231712
\(875\) 0 0
\(876\) 3.41161e7 1.50210
\(877\) 3.05516e7 1.34133 0.670665 0.741760i \(-0.266009\pi\)
0.670665 + 0.741760i \(0.266009\pi\)
\(878\) 2.49778e7 1.09350
\(879\) 1.78708e7 0.780137
\(880\) 0 0
\(881\) −6.31867e6 −0.274275 −0.137137 0.990552i \(-0.543790\pi\)
−0.137137 + 0.990552i \(0.543790\pi\)
\(882\) 1.14803e7 0.496915
\(883\) 2.91551e7 1.25838 0.629191 0.777251i \(-0.283386\pi\)
0.629191 + 0.777251i \(0.283386\pi\)
\(884\) 5.09351e7 2.19223
\(885\) 0 0
\(886\) −1.57375e7 −0.673519
\(887\) 1.26078e7 0.538061 0.269030 0.963132i \(-0.413297\pi\)
0.269030 + 0.963132i \(0.413297\pi\)
\(888\) 7.06894e7 3.00831
\(889\) 9.48539e6 0.402532
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 3.57119e7 1.50280
\(893\) −5.31603e6 −0.223079
\(894\) −2.32407e7 −0.972537
\(895\) 0 0
\(896\) 4.91185e7 2.04397
\(897\) −1.79733e7 −0.745840
\(898\) −4.95499e7 −2.05046
\(899\) 4.74026e6 0.195615
\(900\) 0 0
\(901\) −3.07566e6 −0.126219
\(902\) 7.16116e6 0.293067
\(903\) 7.75211e6 0.316374
\(904\) −1.90054e6 −0.0773493
\(905\) 0 0
\(906\) −6.03468e6 −0.244249
\(907\) 2.75312e7 1.11124 0.555618 0.831437i \(-0.312482\pi\)
0.555618 + 0.831437i \(0.312482\pi\)
\(908\) −1.02267e8 −4.11641
\(909\) 9.10099e6 0.365325
\(910\) 0 0
\(911\) 2.45616e7 0.980531 0.490265 0.871573i \(-0.336900\pi\)
0.490265 + 0.871573i \(0.336900\pi\)
\(912\) −8.65497e6 −0.344571
\(913\) 7.01227e6 0.278408
\(914\) −6.22206e7 −2.46359
\(915\) 0 0
\(916\) 2.22695e6 0.0876945
\(917\) 2.84979e6 0.111915
\(918\) 4.64085e6 0.181757
\(919\) 1.79061e7 0.699379 0.349690 0.936866i \(-0.386287\pi\)
0.349690 + 0.936866i \(0.386287\pi\)
\(920\) 0 0
\(921\) 1.02585e7 0.398505
\(922\) 9.30327e7 3.60419
\(923\) 2.89224e7 1.11746
\(924\) −6.10641e6 −0.235291
\(925\) 0 0
\(926\) −4.49237e7 −1.72166
\(927\) −5.13483e6 −0.196258
\(928\) 1.71449e8 6.53528
\(929\) −1.63508e7 −0.621583 −0.310791 0.950478i \(-0.600594\pi\)
−0.310791 + 0.950478i \(0.600594\pi\)
\(930\) 0 0
\(931\) 3.01965e6 0.114178
\(932\) 5.59434e7 2.10964
\(933\) −1.18714e6 −0.0446476
\(934\) −3.78805e7 −1.42085
\(935\) 0 0
\(936\) 5.04667e7 1.88285
\(937\) −5.61949e6 −0.209097 −0.104548 0.994520i \(-0.533340\pi\)
−0.104548 + 0.994520i \(0.533340\pi\)
\(938\) −3.49227e7 −1.29599
\(939\) 1.86592e7 0.690605
\(940\) 0 0
\(941\) 5.03132e6 0.185228 0.0926142 0.995702i \(-0.470478\pi\)
0.0926142 + 0.995702i \(0.470478\pi\)
\(942\) 1.11153e7 0.408127
\(943\) −1.08760e7 −0.398281
\(944\) 9.74955e7 3.56086
\(945\) 0 0
\(946\) 1.82897e7 0.664476
\(947\) 1.16336e7 0.421540 0.210770 0.977536i \(-0.432403\pi\)
0.210770 + 0.977536i \(0.432403\pi\)
\(948\) 5.13006e7 1.85396
\(949\) −4.18628e7 −1.50891
\(950\) 0 0
\(951\) −856112. −0.0306958
\(952\) −2.29098e7 −0.819273
\(953\) −1.43027e7 −0.510135 −0.255068 0.966923i \(-0.582098\pi\)
−0.255068 + 0.966923i \(0.582098\pi\)
\(954\) −4.74825e6 −0.168913
\(955\) 0 0
\(956\) −9.38293e7 −3.32042
\(957\) −7.48979e6 −0.264356
\(958\) −3.88286e7 −1.36690
\(959\) −1.86881e7 −0.656174
\(960\) 0 0
\(961\) −2.81541e7 −0.983407
\(962\) −1.35154e8 −4.70861
\(963\) −1.32107e7 −0.459049
\(964\) 2.24873e7 0.779372
\(965\) 0 0
\(966\) 1.25961e7 0.434305
\(967\) −1.87953e7 −0.646372 −0.323186 0.946335i \(-0.604754\pi\)
−0.323186 + 0.946335i \(0.604754\pi\)
\(968\) −9.24627e6 −0.317160
\(969\) 1.22067e6 0.0417628
\(970\) 0 0
\(971\) 3.65105e7 1.24271 0.621355 0.783529i \(-0.286582\pi\)
0.621355 + 0.783529i \(0.286582\pi\)
\(972\) 5.27504e6 0.179085
\(973\) 1.34541e7 0.455588
\(974\) −1.17120e7 −0.395579
\(975\) 0 0
\(976\) 1.88196e8 6.32390
\(977\) −2.62418e7 −0.879542 −0.439771 0.898110i \(-0.644940\pi\)
−0.439771 + 0.898110i \(0.644940\pi\)
\(978\) 3.60919e7 1.20660
\(979\) 9.07662e6 0.302669
\(980\) 0 0
\(981\) −1.43817e7 −0.477131
\(982\) −1.00937e8 −3.34019
\(983\) 4.19211e7 1.38372 0.691861 0.722030i \(-0.256791\pi\)
0.691861 + 0.722030i \(0.256791\pi\)
\(984\) 3.05384e7 1.00545
\(985\) 0 0
\(986\) −4.37836e7 −1.43423
\(987\) −1.27967e7 −0.418124
\(988\) 2.06830e7 0.674096
\(989\) −2.77774e7 −0.903029
\(990\) 0 0
\(991\) 3.68229e7 1.19106 0.595530 0.803333i \(-0.296942\pi\)
0.595530 + 0.803333i \(0.296942\pi\)
\(992\) 1.71812e7 0.554337
\(993\) −2.88138e7 −0.927315
\(994\) −2.02696e7 −0.650698
\(995\) 0 0
\(996\) 4.65938e7 1.48826
\(997\) −5.96749e6 −0.190132 −0.0950658 0.995471i \(-0.530306\pi\)
−0.0950658 + 0.995471i \(0.530306\pi\)
\(998\) 4.95228e7 1.57391
\(999\) −9.06658e6 −0.287429
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.t.1.1 10
5.4 even 2 825.6.a.u.1.10 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.1 10 1.1 even 1 trivial
825.6.a.u.1.10 yes 10 5.4 even 2