Properties

Label 9075.2.a.bc.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{12} +3.23607 q^{13} +8.47214 q^{14} -4.85410 q^{16} +2.00000 q^{17} -1.61803 q^{18} -5.00000 q^{19} -5.23607 q^{21} -4.61803 q^{23} +2.23607 q^{24} -5.23607 q^{26} +1.00000 q^{27} -3.23607 q^{28} -0.854102 q^{29} +7.00000 q^{31} +3.38197 q^{32} -3.23607 q^{34} +0.618034 q^{36} -1.47214 q^{37} +8.09017 q^{38} +3.23607 q^{39} -11.4721 q^{41} +8.47214 q^{42} +9.09017 q^{43} +7.47214 q^{46} +6.09017 q^{47} -4.85410 q^{48} +20.4164 q^{49} +2.00000 q^{51} +2.00000 q^{52} +5.38197 q^{53} -1.61803 q^{54} -11.7082 q^{56} -5.00000 q^{57} +1.38197 q^{58} +6.70820 q^{59} -7.00000 q^{61} -11.3262 q^{62} -5.23607 q^{63} +4.23607 q^{64} +11.6180 q^{67} +1.23607 q^{68} -4.61803 q^{69} -8.00000 q^{71} +2.23607 q^{72} -4.52786 q^{73} +2.38197 q^{74} -3.09017 q^{76} -5.23607 q^{78} -3.09017 q^{79} +1.00000 q^{81} +18.5623 q^{82} +7.70820 q^{83} -3.23607 q^{84} -14.7082 q^{86} -0.854102 q^{87} +4.14590 q^{89} -16.9443 q^{91} -2.85410 q^{92} +7.00000 q^{93} -9.85410 q^{94} +3.38197 q^{96} -2.52786 q^{97} -33.0344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 10 q^{19} - 6 q^{21} - 7 q^{23} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} + 6 q^{37} + 5 q^{38} + 2 q^{39} - 14 q^{41} + 8 q^{42} + 7 q^{43} + 6 q^{46} + q^{47} - 3 q^{48} + 14 q^{49} + 4 q^{51} + 4 q^{52} + 13 q^{53} - q^{54} - 10 q^{56} - 10 q^{57} + 5 q^{58} - 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} + 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} + 7 q^{74} + 5 q^{76} - 6 q^{78} + 5 q^{79} + 2 q^{81} + 17 q^{82} + 2 q^{83} - 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} + q^{92} + 14 q^{93} - 13 q^{94} + 9 q^{96} - 14 q^{97} - 37 q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) −1.61803 −0.660560
\(7\) −5.23607 −1.97905 −0.989524 0.144370i \(-0.953885\pi\)
−0.989524 + 0.144370i \(0.953885\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 8.47214 2.26427
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.61803 −0.381374
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −5.23607 −1.14260
\(22\) 0 0
\(23\) −4.61803 −0.962927 −0.481463 0.876466i \(-0.659895\pi\)
−0.481463 + 0.876466i \(0.659895\pi\)
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) −5.23607 −1.02688
\(27\) 1.00000 0.192450
\(28\) −3.23607 −0.611559
\(29\) −0.854102 −0.158603 −0.0793014 0.996851i \(-0.525269\pi\)
−0.0793014 + 0.996851i \(0.525269\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 3.38197 0.597853
\(33\) 0 0
\(34\) −3.23607 −0.554981
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) −1.47214 −0.242018 −0.121009 0.992651i \(-0.538613\pi\)
−0.121009 + 0.992651i \(0.538613\pi\)
\(38\) 8.09017 1.31240
\(39\) 3.23607 0.518186
\(40\) 0 0
\(41\) −11.4721 −1.79165 −0.895823 0.444410i \(-0.853413\pi\)
−0.895823 + 0.444410i \(0.853413\pi\)
\(42\) 8.47214 1.30728
\(43\) 9.09017 1.38624 0.693119 0.720823i \(-0.256236\pi\)
0.693119 + 0.720823i \(0.256236\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 7.47214 1.10171
\(47\) 6.09017 0.888343 0.444171 0.895942i \(-0.353498\pi\)
0.444171 + 0.895942i \(0.353498\pi\)
\(48\) −4.85410 −0.700629
\(49\) 20.4164 2.91663
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 2.00000 0.277350
\(53\) 5.38197 0.739270 0.369635 0.929177i \(-0.379483\pi\)
0.369635 + 0.929177i \(0.379483\pi\)
\(54\) −1.61803 −0.220187
\(55\) 0 0
\(56\) −11.7082 −1.56457
\(57\) −5.00000 −0.662266
\(58\) 1.38197 0.181461
\(59\) 6.70820 0.873334 0.436667 0.899623i \(-0.356159\pi\)
0.436667 + 0.899623i \(0.356159\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −11.3262 −1.43843
\(63\) −5.23607 −0.659683
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) 11.6180 1.41937 0.709684 0.704520i \(-0.248838\pi\)
0.709684 + 0.704520i \(0.248838\pi\)
\(68\) 1.23607 0.149895
\(69\) −4.61803 −0.555946
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 2.23607 0.263523
\(73\) −4.52786 −0.529946 −0.264973 0.964256i \(-0.585363\pi\)
−0.264973 + 0.964256i \(0.585363\pi\)
\(74\) 2.38197 0.276898
\(75\) 0 0
\(76\) −3.09017 −0.354467
\(77\) 0 0
\(78\) −5.23607 −0.592868
\(79\) −3.09017 −0.347671 −0.173836 0.984775i \(-0.555616\pi\)
−0.173836 + 0.984775i \(0.555616\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 18.5623 2.04986
\(83\) 7.70820 0.846085 0.423043 0.906110i \(-0.360962\pi\)
0.423043 + 0.906110i \(0.360962\pi\)
\(84\) −3.23607 −0.353084
\(85\) 0 0
\(86\) −14.7082 −1.58603
\(87\) −0.854102 −0.0915693
\(88\) 0 0
\(89\) 4.14590 0.439464 0.219732 0.975560i \(-0.429482\pi\)
0.219732 + 0.975560i \(0.429482\pi\)
\(90\) 0 0
\(91\) −16.9443 −1.77624
\(92\) −2.85410 −0.297561
\(93\) 7.00000 0.725866
\(94\) −9.85410 −1.01637
\(95\) 0 0
\(96\) 3.38197 0.345170
\(97\) −2.52786 −0.256666 −0.128333 0.991731i \(-0.540963\pi\)
−0.128333 + 0.991731i \(0.540963\pi\)
\(98\) −33.0344 −3.33698
\(99\) 0 0
\(100\) 0 0
\(101\) −19.5623 −1.94652 −0.973261 0.229702i \(-0.926225\pi\)
−0.973261 + 0.229702i \(0.926225\pi\)
\(102\) −3.23607 −0.320418
\(103\) −4.61803 −0.455028 −0.227514 0.973775i \(-0.573060\pi\)
−0.227514 + 0.973775i \(0.573060\pi\)
\(104\) 7.23607 0.709555
\(105\) 0 0
\(106\) −8.70820 −0.845816
\(107\) −3.85410 −0.372590 −0.186295 0.982494i \(-0.559648\pi\)
−0.186295 + 0.982494i \(0.559648\pi\)
\(108\) 0.618034 0.0594703
\(109\) −10.8541 −1.03963 −0.519817 0.854278i \(-0.674000\pi\)
−0.519817 + 0.854278i \(0.674000\pi\)
\(110\) 0 0
\(111\) −1.47214 −0.139729
\(112\) 25.4164 2.40162
\(113\) 3.47214 0.326631 0.163316 0.986574i \(-0.447781\pi\)
0.163316 + 0.986574i \(0.447781\pi\)
\(114\) 8.09017 0.757714
\(115\) 0 0
\(116\) −0.527864 −0.0490109
\(117\) 3.23607 0.299175
\(118\) −10.8541 −0.999201
\(119\) −10.4721 −0.959979
\(120\) 0 0
\(121\) 0 0
\(122\) 11.3262 1.02543
\(123\) −11.4721 −1.03441
\(124\) 4.32624 0.388508
\(125\) 0 0
\(126\) 8.47214 0.754758
\(127\) −1.94427 −0.172526 −0.0862631 0.996272i \(-0.527493\pi\)
−0.0862631 + 0.996272i \(0.527493\pi\)
\(128\) −13.6180 −1.20368
\(129\) 9.09017 0.800345
\(130\) 0 0
\(131\) 9.18034 0.802090 0.401045 0.916058i \(-0.368647\pi\)
0.401045 + 0.916058i \(0.368647\pi\)
\(132\) 0 0
\(133\) 26.1803 2.27012
\(134\) −18.7984 −1.62393
\(135\) 0 0
\(136\) 4.47214 0.383482
\(137\) −5.29180 −0.452109 −0.226054 0.974115i \(-0.572583\pi\)
−0.226054 + 0.974115i \(0.572583\pi\)
\(138\) 7.47214 0.636070
\(139\) 1.18034 0.100115 0.0500576 0.998746i \(-0.484060\pi\)
0.0500576 + 0.998746i \(0.484060\pi\)
\(140\) 0 0
\(141\) 6.09017 0.512885
\(142\) 12.9443 1.08626
\(143\) 0 0
\(144\) −4.85410 −0.404508
\(145\) 0 0
\(146\) 7.32624 0.606324
\(147\) 20.4164 1.68392
\(148\) −0.909830 −0.0747876
\(149\) 20.1246 1.64867 0.824336 0.566101i \(-0.191549\pi\)
0.824336 + 0.566101i \(0.191549\pi\)
\(150\) 0 0
\(151\) 15.2361 1.23989 0.619947 0.784644i \(-0.287154\pi\)
0.619947 + 0.784644i \(0.287154\pi\)
\(152\) −11.1803 −0.906845
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −12.3262 −0.983741 −0.491870 0.870668i \(-0.663687\pi\)
−0.491870 + 0.870668i \(0.663687\pi\)
\(158\) 5.00000 0.397779
\(159\) 5.38197 0.426818
\(160\) 0 0
\(161\) 24.1803 1.90568
\(162\) −1.61803 −0.127125
\(163\) 6.56231 0.513999 0.257000 0.966411i \(-0.417266\pi\)
0.257000 + 0.966411i \(0.417266\pi\)
\(164\) −7.09017 −0.553649
\(165\) 0 0
\(166\) −12.4721 −0.968025
\(167\) 9.03444 0.699106 0.349553 0.936917i \(-0.386333\pi\)
0.349553 + 0.936917i \(0.386333\pi\)
\(168\) −11.7082 −0.903308
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 5.61803 0.428371
\(173\) −16.5623 −1.25921 −0.629604 0.776916i \(-0.716783\pi\)
−0.629604 + 0.776916i \(0.716783\pi\)
\(174\) 1.38197 0.104767
\(175\) 0 0
\(176\) 0 0
\(177\) 6.70820 0.504219
\(178\) −6.70820 −0.502801
\(179\) −10.5279 −0.786890 −0.393445 0.919348i \(-0.628717\pi\)
−0.393445 + 0.919348i \(0.628717\pi\)
\(180\) 0 0
\(181\) −0.763932 −0.0567826 −0.0283913 0.999597i \(-0.509038\pi\)
−0.0283913 + 0.999597i \(0.509038\pi\)
\(182\) 27.4164 2.03224
\(183\) −7.00000 −0.517455
\(184\) −10.3262 −0.761260
\(185\) 0 0
\(186\) −11.3262 −0.830480
\(187\) 0 0
\(188\) 3.76393 0.274513
\(189\) −5.23607 −0.380868
\(190\) 0 0
\(191\) −16.4164 −1.18785 −0.593925 0.804521i \(-0.702422\pi\)
−0.593925 + 0.804521i \(0.702422\pi\)
\(192\) 4.23607 0.305712
\(193\) 8.03444 0.578332 0.289166 0.957279i \(-0.406622\pi\)
0.289166 + 0.957279i \(0.406622\pi\)
\(194\) 4.09017 0.293657
\(195\) 0 0
\(196\) 12.6180 0.901288
\(197\) 9.76393 0.695651 0.347826 0.937559i \(-0.386920\pi\)
0.347826 + 0.937559i \(0.386920\pi\)
\(198\) 0 0
\(199\) −4.79837 −0.340148 −0.170074 0.985431i \(-0.554401\pi\)
−0.170074 + 0.985431i \(0.554401\pi\)
\(200\) 0 0
\(201\) 11.6180 0.819473
\(202\) 31.6525 2.22706
\(203\) 4.47214 0.313882
\(204\) 1.23607 0.0865421
\(205\) 0 0
\(206\) 7.47214 0.520608
\(207\) −4.61803 −0.320976
\(208\) −15.7082 −1.08917
\(209\) 0 0
\(210\) 0 0
\(211\) −2.85410 −0.196484 −0.0982422 0.995163i \(-0.531322\pi\)
−0.0982422 + 0.995163i \(0.531322\pi\)
\(212\) 3.32624 0.228447
\(213\) −8.00000 −0.548151
\(214\) 6.23607 0.426289
\(215\) 0 0
\(216\) 2.23607 0.152145
\(217\) −36.6525 −2.48813
\(218\) 17.5623 1.18947
\(219\) −4.52786 −0.305965
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 2.38197 0.159867
\(223\) 21.0344 1.40857 0.704285 0.709917i \(-0.251268\pi\)
0.704285 + 0.709917i \(0.251268\pi\)
\(224\) −17.7082 −1.18318
\(225\) 0 0
\(226\) −5.61803 −0.373706
\(227\) 8.05573 0.534677 0.267339 0.963603i \(-0.413856\pi\)
0.267339 + 0.963603i \(0.413856\pi\)
\(228\) −3.09017 −0.204652
\(229\) 6.90983 0.456614 0.228307 0.973589i \(-0.426681\pi\)
0.228307 + 0.973589i \(0.426681\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.90983 −0.125386
\(233\) 10.2705 0.672843 0.336422 0.941711i \(-0.390783\pi\)
0.336422 + 0.941711i \(0.390783\pi\)
\(234\) −5.23607 −0.342292
\(235\) 0 0
\(236\) 4.14590 0.269875
\(237\) −3.09017 −0.200728
\(238\) 16.9443 1.09833
\(239\) −22.0344 −1.42529 −0.712645 0.701525i \(-0.752503\pi\)
−0.712645 + 0.701525i \(0.752503\pi\)
\(240\) 0 0
\(241\) −0.618034 −0.0398111 −0.0199055 0.999802i \(-0.506337\pi\)
−0.0199055 + 0.999802i \(0.506337\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −4.32624 −0.276959
\(245\) 0 0
\(246\) 18.5623 1.18349
\(247\) −16.1803 −1.02953
\(248\) 15.6525 0.993933
\(249\) 7.70820 0.488488
\(250\) 0 0
\(251\) 16.4721 1.03971 0.519856 0.854254i \(-0.325986\pi\)
0.519856 + 0.854254i \(0.325986\pi\)
\(252\) −3.23607 −0.203853
\(253\) 0 0
\(254\) 3.14590 0.197391
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −9.56231 −0.596480 −0.298240 0.954491i \(-0.596400\pi\)
−0.298240 + 0.954491i \(0.596400\pi\)
\(258\) −14.7082 −0.915693
\(259\) 7.70820 0.478964
\(260\) 0 0
\(261\) −0.854102 −0.0528676
\(262\) −14.8541 −0.917689
\(263\) 24.2148 1.49315 0.746574 0.665303i \(-0.231698\pi\)
0.746574 + 0.665303i \(0.231698\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −42.3607 −2.59730
\(267\) 4.14590 0.253725
\(268\) 7.18034 0.438609
\(269\) 0.854102 0.0520755 0.0260378 0.999661i \(-0.491711\pi\)
0.0260378 + 0.999661i \(0.491711\pi\)
\(270\) 0 0
\(271\) −18.3820 −1.11662 −0.558312 0.829631i \(-0.688551\pi\)
−0.558312 + 0.829631i \(0.688551\pi\)
\(272\) −9.70820 −0.588646
\(273\) −16.9443 −1.02551
\(274\) 8.56231 0.517268
\(275\) 0 0
\(276\) −2.85410 −0.171797
\(277\) 22.6525 1.36106 0.680528 0.732722i \(-0.261751\pi\)
0.680528 + 0.732722i \(0.261751\pi\)
\(278\) −1.90983 −0.114544
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 1.09017 0.0650341 0.0325170 0.999471i \(-0.489648\pi\)
0.0325170 + 0.999471i \(0.489648\pi\)
\(282\) −9.85410 −0.586803
\(283\) −16.0344 −0.953149 −0.476574 0.879134i \(-0.658122\pi\)
−0.476574 + 0.879134i \(0.658122\pi\)
\(284\) −4.94427 −0.293389
\(285\) 0 0
\(286\) 0 0
\(287\) 60.0689 3.54575
\(288\) 3.38197 0.199284
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −2.52786 −0.148186
\(292\) −2.79837 −0.163762
\(293\) 25.4721 1.48810 0.744049 0.668125i \(-0.232903\pi\)
0.744049 + 0.668125i \(0.232903\pi\)
\(294\) −33.0344 −1.92661
\(295\) 0 0
\(296\) −3.29180 −0.191332
\(297\) 0 0
\(298\) −32.5623 −1.88628
\(299\) −14.9443 −0.864250
\(300\) 0 0
\(301\) −47.5967 −2.74343
\(302\) −24.6525 −1.41859
\(303\) −19.5623 −1.12383
\(304\) 24.2705 1.39201
\(305\) 0 0
\(306\) −3.23607 −0.184994
\(307\) −28.1246 −1.60516 −0.802578 0.596547i \(-0.796539\pi\)
−0.802578 + 0.596547i \(0.796539\pi\)
\(308\) 0 0
\(309\) −4.61803 −0.262711
\(310\) 0 0
\(311\) 12.5279 0.710390 0.355195 0.934792i \(-0.384414\pi\)
0.355195 + 0.934792i \(0.384414\pi\)
\(312\) 7.23607 0.409662
\(313\) 8.47214 0.478873 0.239437 0.970912i \(-0.423037\pi\)
0.239437 + 0.970912i \(0.423037\pi\)
\(314\) 19.9443 1.12552
\(315\) 0 0
\(316\) −1.90983 −0.107436
\(317\) −3.58359 −0.201275 −0.100637 0.994923i \(-0.532088\pi\)
−0.100637 + 0.994923i \(0.532088\pi\)
\(318\) −8.70820 −0.488332
\(319\) 0 0
\(320\) 0 0
\(321\) −3.85410 −0.215115
\(322\) −39.1246 −2.18033
\(323\) −10.0000 −0.556415
\(324\) 0.618034 0.0343352
\(325\) 0 0
\(326\) −10.6180 −0.588079
\(327\) −10.8541 −0.600233
\(328\) −25.6525 −1.41642
\(329\) −31.8885 −1.75807
\(330\) 0 0
\(331\) −27.5967 −1.51685 −0.758427 0.651758i \(-0.774032\pi\)
−0.758427 + 0.651758i \(0.774032\pi\)
\(332\) 4.76393 0.261455
\(333\) −1.47214 −0.0806726
\(334\) −14.6180 −0.799863
\(335\) 0 0
\(336\) 25.4164 1.38658
\(337\) −28.8541 −1.57178 −0.785892 0.618364i \(-0.787796\pi\)
−0.785892 + 0.618364i \(0.787796\pi\)
\(338\) 4.09017 0.222476
\(339\) 3.47214 0.188581
\(340\) 0 0
\(341\) 0 0
\(342\) 8.09017 0.437466
\(343\) −70.2492 −3.79310
\(344\) 20.3262 1.09592
\(345\) 0 0
\(346\) 26.7984 1.44069
\(347\) 28.1803 1.51280 0.756400 0.654109i \(-0.226956\pi\)
0.756400 + 0.654109i \(0.226956\pi\)
\(348\) −0.527864 −0.0282965
\(349\) −32.2361 −1.72556 −0.862779 0.505582i \(-0.831278\pi\)
−0.862779 + 0.505582i \(0.831278\pi\)
\(350\) 0 0
\(351\) 3.23607 0.172729
\(352\) 0 0
\(353\) −17.8328 −0.949145 −0.474573 0.880216i \(-0.657397\pi\)
−0.474573 + 0.880216i \(0.657397\pi\)
\(354\) −10.8541 −0.576889
\(355\) 0 0
\(356\) 2.56231 0.135802
\(357\) −10.4721 −0.554244
\(358\) 17.0344 0.900298
\(359\) −3.29180 −0.173734 −0.0868672 0.996220i \(-0.527686\pi\)
−0.0868672 + 0.996220i \(0.527686\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 1.23607 0.0649663
\(363\) 0 0
\(364\) −10.4721 −0.548889
\(365\) 0 0
\(366\) 11.3262 0.592032
\(367\) −22.1246 −1.15490 −0.577448 0.816428i \(-0.695951\pi\)
−0.577448 + 0.816428i \(0.695951\pi\)
\(368\) 22.4164 1.16854
\(369\) −11.4721 −0.597216
\(370\) 0 0
\(371\) −28.1803 −1.46305
\(372\) 4.32624 0.224305
\(373\) −32.7426 −1.69535 −0.847675 0.530516i \(-0.821998\pi\)
−0.847675 + 0.530516i \(0.821998\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 13.6180 0.702296
\(377\) −2.76393 −0.142350
\(378\) 8.47214 0.435760
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −1.94427 −0.0996081
\(382\) 26.5623 1.35905
\(383\) −13.3607 −0.682699 −0.341349 0.939936i \(-0.610884\pi\)
−0.341349 + 0.939936i \(0.610884\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) −13.0000 −0.661683
\(387\) 9.09017 0.462079
\(388\) −1.56231 −0.0793141
\(389\) −21.0557 −1.06757 −0.533784 0.845621i \(-0.679230\pi\)
−0.533784 + 0.845621i \(0.679230\pi\)
\(390\) 0 0
\(391\) −9.23607 −0.467088
\(392\) 45.6525 2.30580
\(393\) 9.18034 0.463087
\(394\) −15.7984 −0.795911
\(395\) 0 0
\(396\) 0 0
\(397\) −2.72949 −0.136989 −0.0684946 0.997651i \(-0.521820\pi\)
−0.0684946 + 0.997651i \(0.521820\pi\)
\(398\) 7.76393 0.389171
\(399\) 26.1803 1.31066
\(400\) 0 0
\(401\) −30.8885 −1.54250 −0.771250 0.636532i \(-0.780368\pi\)
−0.771250 + 0.636532i \(0.780368\pi\)
\(402\) −18.7984 −0.937578
\(403\) 22.6525 1.12840
\(404\) −12.0902 −0.601508
\(405\) 0 0
\(406\) −7.23607 −0.359120
\(407\) 0 0
\(408\) 4.47214 0.221404
\(409\) 26.3050 1.30070 0.650348 0.759636i \(-0.274623\pi\)
0.650348 + 0.759636i \(0.274623\pi\)
\(410\) 0 0
\(411\) −5.29180 −0.261025
\(412\) −2.85410 −0.140612
\(413\) −35.1246 −1.72837
\(414\) 7.47214 0.367235
\(415\) 0 0
\(416\) 10.9443 0.536587
\(417\) 1.18034 0.0578015
\(418\) 0 0
\(419\) −24.5967 −1.20163 −0.600815 0.799388i \(-0.705157\pi\)
−0.600815 + 0.799388i \(0.705157\pi\)
\(420\) 0 0
\(421\) −3.85410 −0.187837 −0.0939187 0.995580i \(-0.529939\pi\)
−0.0939187 + 0.995580i \(0.529939\pi\)
\(422\) 4.61803 0.224802
\(423\) 6.09017 0.296114
\(424\) 12.0344 0.584444
\(425\) 0 0
\(426\) 12.9443 0.627152
\(427\) 36.6525 1.77374
\(428\) −2.38197 −0.115137
\(429\) 0 0
\(430\) 0 0
\(431\) 15.8885 0.765324 0.382662 0.923888i \(-0.375007\pi\)
0.382662 + 0.923888i \(0.375007\pi\)
\(432\) −4.85410 −0.233543
\(433\) 1.03444 0.0497121 0.0248561 0.999691i \(-0.492087\pi\)
0.0248561 + 0.999691i \(0.492087\pi\)
\(434\) 59.3050 2.84673
\(435\) 0 0
\(436\) −6.70820 −0.321265
\(437\) 23.0902 1.10455
\(438\) 7.32624 0.350061
\(439\) −25.3262 −1.20876 −0.604378 0.796698i \(-0.706578\pi\)
−0.604378 + 0.796698i \(0.706578\pi\)
\(440\) 0 0
\(441\) 20.4164 0.972210
\(442\) −10.4721 −0.498109
\(443\) 8.27051 0.392944 0.196472 0.980509i \(-0.437052\pi\)
0.196472 + 0.980509i \(0.437052\pi\)
\(444\) −0.909830 −0.0431786
\(445\) 0 0
\(446\) −34.0344 −1.61158
\(447\) 20.1246 0.951861
\(448\) −22.1803 −1.04792
\(449\) −34.2705 −1.61733 −0.808663 0.588273i \(-0.799808\pi\)
−0.808663 + 0.588273i \(0.799808\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.14590 0.100935
\(453\) 15.2361 0.715853
\(454\) −13.0344 −0.611737
\(455\) 0 0
\(456\) −11.1803 −0.523567
\(457\) −7.47214 −0.349532 −0.174766 0.984610i \(-0.555917\pi\)
−0.174766 + 0.984610i \(0.555917\pi\)
\(458\) −11.1803 −0.522423
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −8.05573 −0.375193 −0.187596 0.982246i \(-0.560070\pi\)
−0.187596 + 0.982246i \(0.560070\pi\)
\(462\) 0 0
\(463\) −0.270510 −0.0125717 −0.00628583 0.999980i \(-0.502001\pi\)
−0.00628583 + 0.999980i \(0.502001\pi\)
\(464\) 4.14590 0.192468
\(465\) 0 0
\(466\) −16.6180 −0.769816
\(467\) 30.8885 1.42935 0.714676 0.699456i \(-0.246574\pi\)
0.714676 + 0.699456i \(0.246574\pi\)
\(468\) 2.00000 0.0924500
\(469\) −60.8328 −2.80900
\(470\) 0 0
\(471\) −12.3262 −0.567963
\(472\) 15.0000 0.690431
\(473\) 0 0
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) −6.47214 −0.296650
\(477\) 5.38197 0.246423
\(478\) 35.6525 1.63071
\(479\) −6.58359 −0.300812 −0.150406 0.988624i \(-0.548058\pi\)
−0.150406 + 0.988624i \(0.548058\pi\)
\(480\) 0 0
\(481\) −4.76393 −0.217217
\(482\) 1.00000 0.0455488
\(483\) 24.1803 1.10024
\(484\) 0 0
\(485\) 0 0
\(486\) −1.61803 −0.0733955
\(487\) −8.18034 −0.370687 −0.185343 0.982674i \(-0.559340\pi\)
−0.185343 + 0.982674i \(0.559340\pi\)
\(488\) −15.6525 −0.708554
\(489\) 6.56231 0.296758
\(490\) 0 0
\(491\) −7.20163 −0.325005 −0.162502 0.986708i \(-0.551957\pi\)
−0.162502 + 0.986708i \(0.551957\pi\)
\(492\) −7.09017 −0.319650
\(493\) −1.70820 −0.0769336
\(494\) 26.1803 1.17791
\(495\) 0 0
\(496\) −33.9787 −1.52569
\(497\) 41.8885 1.87896
\(498\) −12.4721 −0.558890
\(499\) 11.1803 0.500501 0.250250 0.968181i \(-0.419487\pi\)
0.250250 + 0.968181i \(0.419487\pi\)
\(500\) 0 0
\(501\) 9.03444 0.403629
\(502\) −26.6525 −1.18956
\(503\) 13.8885 0.619260 0.309630 0.950857i \(-0.399795\pi\)
0.309630 + 0.950857i \(0.399795\pi\)
\(504\) −11.7082 −0.521525
\(505\) 0 0
\(506\) 0 0
\(507\) −2.52786 −0.112266
\(508\) −1.20163 −0.0533135
\(509\) 9.47214 0.419845 0.209923 0.977718i \(-0.432679\pi\)
0.209923 + 0.977718i \(0.432679\pi\)
\(510\) 0 0
\(511\) 23.7082 1.04879
\(512\) 5.29180 0.233867
\(513\) −5.00000 −0.220755
\(514\) 15.4721 0.682447
\(515\) 0 0
\(516\) 5.61803 0.247320
\(517\) 0 0
\(518\) −12.4721 −0.547994
\(519\) −16.5623 −0.727005
\(520\) 0 0
\(521\) 29.3607 1.28631 0.643157 0.765734i \(-0.277624\pi\)
0.643157 + 0.765734i \(0.277624\pi\)
\(522\) 1.38197 0.0604870
\(523\) 3.76393 0.164585 0.0822926 0.996608i \(-0.473776\pi\)
0.0822926 + 0.996608i \(0.473776\pi\)
\(524\) 5.67376 0.247859
\(525\) 0 0
\(526\) −39.1803 −1.70834
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) −1.67376 −0.0727723
\(530\) 0 0
\(531\) 6.70820 0.291111
\(532\) 16.1803 0.701507
\(533\) −37.1246 −1.60805
\(534\) −6.70820 −0.290292
\(535\) 0 0
\(536\) 25.9787 1.12211
\(537\) −10.5279 −0.454311
\(538\) −1.38197 −0.0595808
\(539\) 0 0
\(540\) 0 0
\(541\) 1.61803 0.0695647 0.0347824 0.999395i \(-0.488926\pi\)
0.0347824 + 0.999395i \(0.488926\pi\)
\(542\) 29.7426 1.27756
\(543\) −0.763932 −0.0327835
\(544\) 6.76393 0.290001
\(545\) 0 0
\(546\) 27.4164 1.17331
\(547\) 19.3607 0.827803 0.413901 0.910322i \(-0.364166\pi\)
0.413901 + 0.910322i \(0.364166\pi\)
\(548\) −3.27051 −0.139709
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 4.27051 0.181930
\(552\) −10.3262 −0.439514
\(553\) 16.1803 0.688058
\(554\) −36.6525 −1.55721
\(555\) 0 0
\(556\) 0.729490 0.0309373
\(557\) −42.3951 −1.79634 −0.898169 0.439649i \(-0.855103\pi\)
−0.898169 + 0.439649i \(0.855103\pi\)
\(558\) −11.3262 −0.479478
\(559\) 29.4164 1.24418
\(560\) 0 0
\(561\) 0 0
\(562\) −1.76393 −0.0744070
\(563\) 20.6738 0.871295 0.435648 0.900117i \(-0.356519\pi\)
0.435648 + 0.900117i \(0.356519\pi\)
\(564\) 3.76393 0.158490
\(565\) 0 0
\(566\) 25.9443 1.09052
\(567\) −5.23607 −0.219894
\(568\) −17.8885 −0.750587
\(569\) 22.5623 0.945861 0.472931 0.881100i \(-0.343196\pi\)
0.472931 + 0.881100i \(0.343196\pi\)
\(570\) 0 0
\(571\) 11.0902 0.464109 0.232055 0.972703i \(-0.425455\pi\)
0.232055 + 0.972703i \(0.425455\pi\)
\(572\) 0 0
\(573\) −16.4164 −0.685805
\(574\) −97.1935 −4.05678
\(575\) 0 0
\(576\) 4.23607 0.176503
\(577\) −30.0132 −1.24946 −0.624732 0.780839i \(-0.714792\pi\)
−0.624732 + 0.780839i \(0.714792\pi\)
\(578\) 21.0344 0.874917
\(579\) 8.03444 0.333900
\(580\) 0 0
\(581\) −40.3607 −1.67444
\(582\) 4.09017 0.169543
\(583\) 0 0
\(584\) −10.1246 −0.418959
\(585\) 0 0
\(586\) −41.2148 −1.70257
\(587\) 21.6180 0.892272 0.446136 0.894965i \(-0.352800\pi\)
0.446136 + 0.894965i \(0.352800\pi\)
\(588\) 12.6180 0.520359
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) 9.76393 0.401634
\(592\) 7.14590 0.293695
\(593\) 3.11146 0.127772 0.0638861 0.997957i \(-0.479651\pi\)
0.0638861 + 0.997957i \(0.479651\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.4377 0.509468
\(597\) −4.79837 −0.196384
\(598\) 24.1803 0.988808
\(599\) −17.7639 −0.725815 −0.362907 0.931825i \(-0.618216\pi\)
−0.362907 + 0.931825i \(0.618216\pi\)
\(600\) 0 0
\(601\) 10.3607 0.422621 0.211310 0.977419i \(-0.432227\pi\)
0.211310 + 0.977419i \(0.432227\pi\)
\(602\) 77.0132 3.13882
\(603\) 11.6180 0.473123
\(604\) 9.41641 0.383148
\(605\) 0 0
\(606\) 31.6525 1.28579
\(607\) −33.0000 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(608\) −16.9098 −0.685784
\(609\) 4.47214 0.181220
\(610\) 0 0
\(611\) 19.7082 0.797309
\(612\) 1.23607 0.0499651
\(613\) 3.76393 0.152024 0.0760119 0.997107i \(-0.475781\pi\)
0.0760119 + 0.997107i \(0.475781\pi\)
\(614\) 45.5066 1.83650
\(615\) 0 0
\(616\) 0 0
\(617\) −42.2492 −1.70089 −0.850445 0.526064i \(-0.823667\pi\)
−0.850445 + 0.526064i \(0.823667\pi\)
\(618\) 7.47214 0.300573
\(619\) 42.8885 1.72384 0.861918 0.507048i \(-0.169263\pi\)
0.861918 + 0.507048i \(0.169263\pi\)
\(620\) 0 0
\(621\) −4.61803 −0.185315
\(622\) −20.2705 −0.812773
\(623\) −21.7082 −0.869721
\(624\) −15.7082 −0.628831
\(625\) 0 0
\(626\) −13.7082 −0.547890
\(627\) 0 0
\(628\) −7.61803 −0.303993
\(629\) −2.94427 −0.117396
\(630\) 0 0
\(631\) −39.6312 −1.57769 −0.788846 0.614590i \(-0.789321\pi\)
−0.788846 + 0.614590i \(0.789321\pi\)
\(632\) −6.90983 −0.274858
\(633\) −2.85410 −0.113440
\(634\) 5.79837 0.230283
\(635\) 0 0
\(636\) 3.32624 0.131894
\(637\) 66.0689 2.61774
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 4.23607 0.167315 0.0836573 0.996495i \(-0.473340\pi\)
0.0836573 + 0.996495i \(0.473340\pi\)
\(642\) 6.23607 0.246118
\(643\) 13.6738 0.539241 0.269620 0.962967i \(-0.413102\pi\)
0.269620 + 0.962967i \(0.413102\pi\)
\(644\) 14.9443 0.588887
\(645\) 0 0
\(646\) 16.1803 0.636607
\(647\) −31.5967 −1.24220 −0.621098 0.783733i \(-0.713313\pi\)
−0.621098 + 0.783733i \(0.713313\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) −36.6525 −1.43652
\(652\) 4.05573 0.158835
\(653\) −43.5623 −1.70472 −0.852362 0.522952i \(-0.824831\pi\)
−0.852362 + 0.522952i \(0.824831\pi\)
\(654\) 17.5623 0.686741
\(655\) 0 0
\(656\) 55.6869 2.17421
\(657\) −4.52786 −0.176649
\(658\) 51.5967 2.01145
\(659\) −42.0344 −1.63743 −0.818715 0.574201i \(-0.805313\pi\)
−0.818715 + 0.574201i \(0.805313\pi\)
\(660\) 0 0
\(661\) 15.0902 0.586940 0.293470 0.955968i \(-0.405190\pi\)
0.293470 + 0.955968i \(0.405190\pi\)
\(662\) 44.6525 1.73547
\(663\) 6.47214 0.251357
\(664\) 17.2361 0.668889
\(665\) 0 0
\(666\) 2.38197 0.0922993
\(667\) 3.94427 0.152723
\(668\) 5.58359 0.216036
\(669\) 21.0344 0.813239
\(670\) 0 0
\(671\) 0 0
\(672\) −17.7082 −0.683109
\(673\) −35.3050 −1.36091 −0.680453 0.732792i \(-0.738217\pi\)
−0.680453 + 0.732792i \(0.738217\pi\)
\(674\) 46.6869 1.79831
\(675\) 0 0
\(676\) −1.56231 −0.0600887
\(677\) 5.94427 0.228457 0.114228 0.993455i \(-0.463560\pi\)
0.114228 + 0.993455i \(0.463560\pi\)
\(678\) −5.61803 −0.215759
\(679\) 13.2361 0.507954
\(680\) 0 0
\(681\) 8.05573 0.308696
\(682\) 0 0
\(683\) 15.7082 0.601058 0.300529 0.953773i \(-0.402837\pi\)
0.300529 + 0.953773i \(0.402837\pi\)
\(684\) −3.09017 −0.118156
\(685\) 0 0
\(686\) 113.666 4.33977
\(687\) 6.90983 0.263626
\(688\) −44.1246 −1.68224
\(689\) 17.4164 0.663512
\(690\) 0 0
\(691\) −29.3050 −1.11481 −0.557406 0.830240i \(-0.688203\pi\)
−0.557406 + 0.830240i \(0.688203\pi\)
\(692\) −10.2361 −0.389117
\(693\) 0 0
\(694\) −45.5967 −1.73083
\(695\) 0 0
\(696\) −1.90983 −0.0723919
\(697\) −22.9443 −0.869076
\(698\) 52.1591 1.97425
\(699\) 10.2705 0.388466
\(700\) 0 0
\(701\) −40.4164 −1.52651 −0.763253 0.646099i \(-0.776399\pi\)
−0.763253 + 0.646099i \(0.776399\pi\)
\(702\) −5.23607 −0.197623
\(703\) 7.36068 0.277613
\(704\) 0 0
\(705\) 0 0
\(706\) 28.8541 1.08594
\(707\) 102.430 3.85226
\(708\) 4.14590 0.155812
\(709\) 18.8197 0.706787 0.353394 0.935475i \(-0.385028\pi\)
0.353394 + 0.935475i \(0.385028\pi\)
\(710\) 0 0
\(711\) −3.09017 −0.115890
\(712\) 9.27051 0.347427
\(713\) −32.3262 −1.21063
\(714\) 16.9443 0.634123
\(715\) 0 0
\(716\) −6.50658 −0.243162
\(717\) −22.0344 −0.822891
\(718\) 5.32624 0.198773
\(719\) 26.9098 1.00357 0.501784 0.864993i \(-0.332677\pi\)
0.501784 + 0.864993i \(0.332677\pi\)
\(720\) 0 0
\(721\) 24.1803 0.900523
\(722\) −9.70820 −0.361302
\(723\) −0.618034 −0.0229849
\(724\) −0.472136 −0.0175468
\(725\) 0 0
\(726\) 0 0
\(727\) −21.6738 −0.803835 −0.401918 0.915676i \(-0.631656\pi\)
−0.401918 + 0.915676i \(0.631656\pi\)
\(728\) −37.8885 −1.40424
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.1803 0.672424
\(732\) −4.32624 −0.159902
\(733\) −4.85410 −0.179290 −0.0896452 0.995974i \(-0.528573\pi\)
−0.0896452 + 0.995974i \(0.528573\pi\)
\(734\) 35.7984 1.32134
\(735\) 0 0
\(736\) −15.6180 −0.575688
\(737\) 0 0
\(738\) 18.5623 0.683288
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −16.1803 −0.594400
\(742\) 45.5967 1.67391
\(743\) 31.6525 1.16122 0.580608 0.814183i \(-0.302815\pi\)
0.580608 + 0.814183i \(0.302815\pi\)
\(744\) 15.6525 0.573848
\(745\) 0 0
\(746\) 52.9787 1.93969
\(747\) 7.70820 0.282028
\(748\) 0 0
\(749\) 20.1803 0.737374
\(750\) 0 0
\(751\) −10.1115 −0.368972 −0.184486 0.982835i \(-0.559062\pi\)
−0.184486 + 0.982835i \(0.559062\pi\)
\(752\) −29.5623 −1.07803
\(753\) 16.4721 0.600278
\(754\) 4.47214 0.162866
\(755\) 0 0
\(756\) −3.23607 −0.117695
\(757\) 1.81966 0.0661367 0.0330683 0.999453i \(-0.489472\pi\)
0.0330683 + 0.999453i \(0.489472\pi\)
\(758\) 8.09017 0.293848
\(759\) 0 0
\(760\) 0 0
\(761\) −9.96556 −0.361251 −0.180626 0.983552i \(-0.557812\pi\)
−0.180626 + 0.983552i \(0.557812\pi\)
\(762\) 3.14590 0.113964
\(763\) 56.8328 2.05749
\(764\) −10.1459 −0.367066
\(765\) 0 0
\(766\) 21.6180 0.781091
\(767\) 21.7082 0.783838
\(768\) 13.5623 0.489388
\(769\) −17.2361 −0.621549 −0.310774 0.950484i \(-0.600588\pi\)
−0.310774 + 0.950484i \(0.600588\pi\)
\(770\) 0 0
\(771\) −9.56231 −0.344378
\(772\) 4.96556 0.178714
\(773\) −27.5066 −0.989343 −0.494671 0.869080i \(-0.664712\pi\)
−0.494671 + 0.869080i \(0.664712\pi\)
\(774\) −14.7082 −0.528675
\(775\) 0 0
\(776\) −5.65248 −0.202912
\(777\) 7.70820 0.276530
\(778\) 34.0689 1.22143
\(779\) 57.3607 2.05516
\(780\) 0 0
\(781\) 0 0
\(782\) 14.9443 0.534406
\(783\) −0.854102 −0.0305231
\(784\) −99.1033 −3.53940
\(785\) 0 0
\(786\) −14.8541 −0.529828
\(787\) −32.7984 −1.16914 −0.584568 0.811345i \(-0.698736\pi\)
−0.584568 + 0.811345i \(0.698736\pi\)
\(788\) 6.03444 0.214968
\(789\) 24.2148 0.862069
\(790\) 0 0
\(791\) −18.1803 −0.646418
\(792\) 0 0
\(793\) −22.6525 −0.804413
\(794\) 4.41641 0.156732
\(795\) 0 0
\(796\) −2.96556 −0.105111
\(797\) −15.2918 −0.541663 −0.270832 0.962627i \(-0.587299\pi\)
−0.270832 + 0.962627i \(0.587299\pi\)
\(798\) −42.3607 −1.49955
\(799\) 12.1803 0.430909
\(800\) 0 0
\(801\) 4.14590 0.146488
\(802\) 49.9787 1.76481
\(803\) 0 0
\(804\) 7.18034 0.253231
\(805\) 0 0
\(806\) −36.6525 −1.29103
\(807\) 0.854102 0.0300658
\(808\) −43.7426 −1.53886
\(809\) −29.6738 −1.04327 −0.521637 0.853168i \(-0.674678\pi\)
−0.521637 + 0.853168i \(0.674678\pi\)
\(810\) 0 0
\(811\) −43.6312 −1.53210 −0.766049 0.642782i \(-0.777780\pi\)
−0.766049 + 0.642782i \(0.777780\pi\)
\(812\) 2.76393 0.0969950
\(813\) −18.3820 −0.644684
\(814\) 0 0
\(815\) 0 0
\(816\) −9.70820 −0.339855
\(817\) −45.4508 −1.59012
\(818\) −42.5623 −1.48816
\(819\) −16.9443 −0.592081
\(820\) 0 0
\(821\) −20.8197 −0.726611 −0.363306 0.931670i \(-0.618352\pi\)
−0.363306 + 0.931670i \(0.618352\pi\)
\(822\) 8.56231 0.298645
\(823\) −16.5279 −0.576125 −0.288063 0.957612i \(-0.593011\pi\)
−0.288063 + 0.957612i \(0.593011\pi\)
\(824\) −10.3262 −0.359732
\(825\) 0 0
\(826\) 56.8328 1.97747
\(827\) −51.4164 −1.78792 −0.893962 0.448143i \(-0.852086\pi\)
−0.893962 + 0.448143i \(0.852086\pi\)
\(828\) −2.85410 −0.0991869
\(829\) −32.8885 −1.14227 −0.571133 0.820857i \(-0.693496\pi\)
−0.571133 + 0.820857i \(0.693496\pi\)
\(830\) 0 0
\(831\) 22.6525 0.785806
\(832\) 13.7082 0.475246
\(833\) 40.8328 1.41477
\(834\) −1.90983 −0.0661320
\(835\) 0 0
\(836\) 0 0
\(837\) 7.00000 0.241955
\(838\) 39.7984 1.37481
\(839\) 36.3050 1.25339 0.626693 0.779266i \(-0.284408\pi\)
0.626693 + 0.779266i \(0.284408\pi\)
\(840\) 0 0
\(841\) −28.2705 −0.974845
\(842\) 6.23607 0.214909
\(843\) 1.09017 0.0375474
\(844\) −1.76393 −0.0607170
\(845\) 0 0
\(846\) −9.85410 −0.338791
\(847\) 0 0
\(848\) −26.1246 −0.897123
\(849\) −16.0344 −0.550301
\(850\) 0 0
\(851\) 6.79837 0.233045
\(852\) −4.94427 −0.169388
\(853\) 23.3607 0.799854 0.399927 0.916547i \(-0.369035\pi\)
0.399927 + 0.916547i \(0.369035\pi\)
\(854\) −59.3050 −2.02937
\(855\) 0 0
\(856\) −8.61803 −0.294558
\(857\) −41.0132 −1.40098 −0.700491 0.713661i \(-0.747036\pi\)
−0.700491 + 0.713661i \(0.747036\pi\)
\(858\) 0 0
\(859\) −39.2705 −1.33989 −0.669946 0.742410i \(-0.733683\pi\)
−0.669946 + 0.742410i \(0.733683\pi\)
\(860\) 0 0
\(861\) 60.0689 2.04714
\(862\) −25.7082 −0.875625
\(863\) −23.0344 −0.784102 −0.392051 0.919944i \(-0.628234\pi\)
−0.392051 + 0.919944i \(0.628234\pi\)
\(864\) 3.38197 0.115057
\(865\) 0 0
\(866\) −1.67376 −0.0568768
\(867\) −13.0000 −0.441503
\(868\) −22.6525 −0.768875
\(869\) 0 0
\(870\) 0 0
\(871\) 37.5967 1.27392
\(872\) −24.2705 −0.821903
\(873\) −2.52786 −0.0855552
\(874\) −37.3607 −1.26374
\(875\) 0 0
\(876\) −2.79837 −0.0945483
\(877\) 7.52786 0.254198 0.127099 0.991890i \(-0.459433\pi\)
0.127099 + 0.991890i \(0.459433\pi\)
\(878\) 40.9787 1.38296
\(879\) 25.4721 0.859154
\(880\) 0 0
\(881\) −17.1459 −0.577660 −0.288830 0.957380i \(-0.593266\pi\)
−0.288830 + 0.957380i \(0.593266\pi\)
\(882\) −33.0344 −1.11233
\(883\) −23.2361 −0.781956 −0.390978 0.920400i \(-0.627863\pi\)
−0.390978 + 0.920400i \(0.627863\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −13.3820 −0.449576
\(887\) 0.167184 0.00561350 0.00280675 0.999996i \(-0.499107\pi\)
0.00280675 + 0.999996i \(0.499107\pi\)
\(888\) −3.29180 −0.110465
\(889\) 10.1803 0.341438
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0000 0.435272
\(893\) −30.4508 −1.01900
\(894\) −32.5623 −1.08905
\(895\) 0 0
\(896\) 71.3050 2.38213
\(897\) −14.9443 −0.498975
\(898\) 55.4508 1.85042
\(899\) −5.97871 −0.199401
\(900\) 0 0
\(901\) 10.7639 0.358599
\(902\) 0 0
\(903\) −47.5967 −1.58392
\(904\) 7.76393 0.258225
\(905\) 0 0
\(906\) −24.6525 −0.819024
\(907\) −44.8885 −1.49050 −0.745250 0.666785i \(-0.767670\pi\)
−0.745250 + 0.666785i \(0.767670\pi\)
\(908\) 4.97871 0.165224
\(909\) −19.5623 −0.648841
\(910\) 0 0
\(911\) 52.9017 1.75271 0.876356 0.481664i \(-0.159968\pi\)
0.876356 + 0.481664i \(0.159968\pi\)
\(912\) 24.2705 0.803677
\(913\) 0 0
\(914\) 12.0902 0.399907
\(915\) 0 0
\(916\) 4.27051 0.141102
\(917\) −48.0689 −1.58737
\(918\) −3.23607 −0.106806
\(919\) 22.2361 0.733500 0.366750 0.930319i \(-0.380470\pi\)
0.366750 + 0.930319i \(0.380470\pi\)
\(920\) 0 0
\(921\) −28.1246 −0.926737
\(922\) 13.0344 0.429266
\(923\) −25.8885 −0.852132
\(924\) 0 0
\(925\) 0 0
\(926\) 0.437694 0.0143835
\(927\) −4.61803 −0.151676
\(928\) −2.88854 −0.0948211
\(929\) −10.8541 −0.356112 −0.178056 0.984020i \(-0.556981\pi\)
−0.178056 + 0.984020i \(0.556981\pi\)
\(930\) 0 0
\(931\) −102.082 −3.34560
\(932\) 6.34752 0.207920
\(933\) 12.5279 0.410144
\(934\) −49.9787 −1.63535
\(935\) 0 0
\(936\) 7.23607 0.236518
\(937\) 17.2016 0.561953 0.280976 0.959715i \(-0.409342\pi\)
0.280976 + 0.959715i \(0.409342\pi\)
\(938\) 98.4296 3.21384
\(939\) 8.47214 0.276478
\(940\) 0 0
\(941\) 9.70820 0.316478 0.158239 0.987401i \(-0.449418\pi\)
0.158239 + 0.987401i \(0.449418\pi\)
\(942\) 19.9443 0.649819
\(943\) 52.9787 1.72522
\(944\) −32.5623 −1.05981
\(945\) 0 0
\(946\) 0 0
\(947\) −4.56231 −0.148255 −0.0741275 0.997249i \(-0.523617\pi\)
−0.0741275 + 0.997249i \(0.523617\pi\)
\(948\) −1.90983 −0.0620284
\(949\) −14.6525 −0.475639
\(950\) 0 0
\(951\) −3.58359 −0.116206
\(952\) −23.4164 −0.758930
\(953\) 49.0902 1.59019 0.795093 0.606487i \(-0.207422\pi\)
0.795093 + 0.606487i \(0.207422\pi\)
\(954\) −8.70820 −0.281939
\(955\) 0 0
\(956\) −13.6180 −0.440439
\(957\) 0 0
\(958\) 10.6525 0.344166
\(959\) 27.7082 0.894745
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 7.70820 0.248522
\(963\) −3.85410 −0.124197
\(964\) −0.381966 −0.0123023
\(965\) 0 0
\(966\) −39.1246 −1.25881
\(967\) −32.6738 −1.05072 −0.525359 0.850881i \(-0.676069\pi\)
−0.525359 + 0.850881i \(0.676069\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) −31.5410 −1.01220 −0.506100 0.862475i \(-0.668913\pi\)
−0.506100 + 0.862475i \(0.668913\pi\)
\(972\) 0.618034 0.0198234
\(973\) −6.18034 −0.198133
\(974\) 13.2361 0.424111
\(975\) 0 0
\(976\) 33.9787 1.08763
\(977\) 46.0902 1.47456 0.737278 0.675590i \(-0.236111\pi\)
0.737278 + 0.675590i \(0.236111\pi\)
\(978\) −10.6180 −0.339527
\(979\) 0 0
\(980\) 0 0
\(981\) −10.8541 −0.346545
\(982\) 11.6525 0.371845
\(983\) 19.0000 0.606006 0.303003 0.952990i \(-0.402011\pi\)
0.303003 + 0.952990i \(0.402011\pi\)
\(984\) −25.6525 −0.817771
\(985\) 0 0
\(986\) 2.76393 0.0880215
\(987\) −31.8885 −1.01502
\(988\) −10.0000 −0.318142
\(989\) −41.9787 −1.33485
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 23.6738 0.751643
\(993\) −27.5967 −0.875756
\(994\) −67.7771 −2.14976
\(995\) 0 0
\(996\) 4.76393 0.150951
\(997\) −11.2918 −0.357615 −0.178807 0.983884i \(-0.557224\pi\)
−0.178807 + 0.983884i \(0.557224\pi\)
\(998\) −18.0902 −0.572634
\(999\) −1.47214 −0.0465763
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 9075.2.a.bc.1.1 2
5.4 even 2 9075.2.a.bt.1.2 2
11.7 odd 10 825.2.n.d.676.1 yes 4
11.8 odd 10 825.2.n.d.526.1 yes 4
11.10 odd 2 9075.2.a.by.1.2 2
55.7 even 20 825.2.bx.c.49.1 8
55.8 even 20 825.2.bx.c.724.1 8
55.18 even 20 825.2.bx.c.49.2 8
55.19 odd 10 825.2.n.b.526.1 4
55.29 odd 10 825.2.n.b.676.1 yes 4
55.52 even 20 825.2.bx.c.724.2 8
55.54 odd 2 9075.2.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.n.b.526.1 4 55.19 odd 10
825.2.n.b.676.1 yes 4 55.29 odd 10
825.2.n.d.526.1 yes 4 11.8 odd 10
825.2.n.d.676.1 yes 4 11.7 odd 10
825.2.bx.c.49.1 8 55.7 even 20
825.2.bx.c.49.2 8 55.18 even 20
825.2.bx.c.724.1 8 55.8 even 20
825.2.bx.c.724.2 8 55.52 even 20
9075.2.a.z.1.1 2 55.54 odd 2
9075.2.a.bc.1.1 2 1.1 even 1 trivial
9075.2.a.bt.1.2 2 5.4 even 2
9075.2.a.by.1.2 2 11.10 odd 2