Properties

Label 2169.2.a.e.1.7
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.60363\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +O(q^{10})\) \(q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +4.40364 q^{10} +3.27094 q^{11} +4.30649 q^{13} -3.39998 q^{14} +9.27984 q^{16} +1.02456 q^{17} -7.01250 q^{19} +8.08274 q^{20} +8.51631 q^{22} -0.835873 q^{23} -2.13934 q^{25} +11.2125 q^{26} -6.24054 q^{28} +1.11761 q^{29} -3.97344 q^{31} +9.69098 q^{32} +2.66758 q^{34} -2.20867 q^{35} -11.3098 q^{37} -18.2579 q^{38} +12.2371 q^{40} -1.22869 q^{41} +10.8406 q^{43} +15.6314 q^{44} -2.17630 q^{46} +0.151820 q^{47} -5.29473 q^{49} -5.57003 q^{50} +20.5801 q^{52} -3.02053 q^{53} +5.53231 q^{55} -9.44808 q^{56} +2.90984 q^{58} +4.15373 q^{59} +5.62714 q^{61} -10.3454 q^{62} +6.67199 q^{64} +7.28378 q^{65} +12.9934 q^{67} +4.89626 q^{68} -5.75055 q^{70} +11.2862 q^{71} +11.7148 q^{73} -29.4464 q^{74} -33.5118 q^{76} -4.27140 q^{77} -1.66517 q^{79} +15.6955 q^{80} -3.19906 q^{82} +2.34322 q^{83} +1.73290 q^{85} +28.2250 q^{86} +23.6657 q^{88} -18.1099 q^{89} -5.62368 q^{91} -3.99453 q^{92} +0.395283 q^{94} -11.8606 q^{95} -7.17873 q^{97} -13.7855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.60363 1.84104 0.920521 0.390694i \(-0.127765\pi\)
0.920521 + 0.390694i \(0.127765\pi\)
\(3\) 0 0
\(4\) 4.77887 2.38943
\(5\) 1.69135 0.756395 0.378197 0.925725i \(-0.376544\pi\)
0.378197 + 0.925725i \(0.376544\pi\)
\(6\) 0 0
\(7\) −1.30586 −0.493569 −0.246785 0.969070i \(-0.579374\pi\)
−0.246785 + 0.969070i \(0.579374\pi\)
\(8\) 7.23513 2.55801
\(9\) 0 0
\(10\) 4.40364 1.39255
\(11\) 3.27094 0.986226 0.493113 0.869965i \(-0.335859\pi\)
0.493113 + 0.869965i \(0.335859\pi\)
\(12\) 0 0
\(13\) 4.30649 1.19441 0.597203 0.802090i \(-0.296279\pi\)
0.597203 + 0.802090i \(0.296279\pi\)
\(14\) −3.39998 −0.908682
\(15\) 0 0
\(16\) 9.27984 2.31996
\(17\) 1.02456 0.248493 0.124247 0.992251i \(-0.460349\pi\)
0.124247 + 0.992251i \(0.460349\pi\)
\(18\) 0 0
\(19\) −7.01250 −1.60878 −0.804388 0.594104i \(-0.797507\pi\)
−0.804388 + 0.594104i \(0.797507\pi\)
\(20\) 8.08274 1.80736
\(21\) 0 0
\(22\) 8.51631 1.81568
\(23\) −0.835873 −0.174292 −0.0871458 0.996196i \(-0.527775\pi\)
−0.0871458 + 0.996196i \(0.527775\pi\)
\(24\) 0 0
\(25\) −2.13934 −0.427867
\(26\) 11.2125 2.19895
\(27\) 0 0
\(28\) −6.24054 −1.17935
\(29\) 1.11761 0.207535 0.103767 0.994602i \(-0.466910\pi\)
0.103767 + 0.994602i \(0.466910\pi\)
\(30\) 0 0
\(31\) −3.97344 −0.713652 −0.356826 0.934171i \(-0.616141\pi\)
−0.356826 + 0.934171i \(0.616141\pi\)
\(32\) 9.69098 1.71314
\(33\) 0 0
\(34\) 2.66758 0.457486
\(35\) −2.20867 −0.373333
\(36\) 0 0
\(37\) −11.3098 −1.85932 −0.929658 0.368423i \(-0.879898\pi\)
−0.929658 + 0.368423i \(0.879898\pi\)
\(38\) −18.2579 −2.96182
\(39\) 0 0
\(40\) 12.2371 1.93486
\(41\) −1.22869 −0.191890 −0.0959448 0.995387i \(-0.530587\pi\)
−0.0959448 + 0.995387i \(0.530587\pi\)
\(42\) 0 0
\(43\) 10.8406 1.65318 0.826591 0.562804i \(-0.190277\pi\)
0.826591 + 0.562804i \(0.190277\pi\)
\(44\) 15.6314 2.35652
\(45\) 0 0
\(46\) −2.17630 −0.320878
\(47\) 0.151820 0.0221453 0.0110726 0.999939i \(-0.496475\pi\)
0.0110726 + 0.999939i \(0.496475\pi\)
\(48\) 0 0
\(49\) −5.29473 −0.756389
\(50\) −5.57003 −0.787721
\(51\) 0 0
\(52\) 20.5801 2.85395
\(53\) −3.02053 −0.414902 −0.207451 0.978245i \(-0.566517\pi\)
−0.207451 + 0.978245i \(0.566517\pi\)
\(54\) 0 0
\(55\) 5.53231 0.745976
\(56\) −9.44808 −1.26255
\(57\) 0 0
\(58\) 2.90984 0.382080
\(59\) 4.15373 0.540769 0.270385 0.962752i \(-0.412849\pi\)
0.270385 + 0.962752i \(0.412849\pi\)
\(60\) 0 0
\(61\) 5.62714 0.720482 0.360241 0.932859i \(-0.382694\pi\)
0.360241 + 0.932859i \(0.382694\pi\)
\(62\) −10.3454 −1.31386
\(63\) 0 0
\(64\) 6.67199 0.833999
\(65\) 7.28378 0.903442
\(66\) 0 0
\(67\) 12.9934 1.58740 0.793700 0.608309i \(-0.208152\pi\)
0.793700 + 0.608309i \(0.208152\pi\)
\(68\) 4.89626 0.593758
\(69\) 0 0
\(70\) −5.75055 −0.687322
\(71\) 11.2862 1.33942 0.669711 0.742622i \(-0.266418\pi\)
0.669711 + 0.742622i \(0.266418\pi\)
\(72\) 0 0
\(73\) 11.7148 1.37112 0.685560 0.728017i \(-0.259558\pi\)
0.685560 + 0.728017i \(0.259558\pi\)
\(74\) −29.4464 −3.42308
\(75\) 0 0
\(76\) −33.5118 −3.84407
\(77\) −4.27140 −0.486771
\(78\) 0 0
\(79\) −1.66517 −0.187346 −0.0936732 0.995603i \(-0.529861\pi\)
−0.0936732 + 0.995603i \(0.529861\pi\)
\(80\) 15.6955 1.75481
\(81\) 0 0
\(82\) −3.19906 −0.353277
\(83\) 2.34322 0.257201 0.128601 0.991696i \(-0.458951\pi\)
0.128601 + 0.991696i \(0.458951\pi\)
\(84\) 0 0
\(85\) 1.73290 0.187959
\(86\) 28.2250 3.04358
\(87\) 0 0
\(88\) 23.6657 2.52277
\(89\) −18.1099 −1.91965 −0.959825 0.280598i \(-0.909467\pi\)
−0.959825 + 0.280598i \(0.909467\pi\)
\(90\) 0 0
\(91\) −5.62368 −0.589522
\(92\) −3.99453 −0.416458
\(93\) 0 0
\(94\) 0.395283 0.0407703
\(95\) −11.8606 −1.21687
\(96\) 0 0
\(97\) −7.17873 −0.728890 −0.364445 0.931225i \(-0.618741\pi\)
−0.364445 + 0.931225i \(0.618741\pi\)
\(98\) −13.7855 −1.39254
\(99\) 0 0
\(100\) −10.2236 −1.02236
\(101\) 5.48113 0.545393 0.272696 0.962100i \(-0.412085\pi\)
0.272696 + 0.962100i \(0.412085\pi\)
\(102\) 0 0
\(103\) −15.7371 −1.55062 −0.775311 0.631580i \(-0.782407\pi\)
−0.775311 + 0.631580i \(0.782407\pi\)
\(104\) 31.1580 3.05530
\(105\) 0 0
\(106\) −7.86433 −0.763851
\(107\) 11.1816 1.08097 0.540483 0.841355i \(-0.318242\pi\)
0.540483 + 0.841355i \(0.318242\pi\)
\(108\) 0 0
\(109\) −0.296424 −0.0283922 −0.0141961 0.999899i \(-0.504519\pi\)
−0.0141961 + 0.999899i \(0.504519\pi\)
\(110\) 14.4041 1.37337
\(111\) 0 0
\(112\) −12.1182 −1.14506
\(113\) −10.5881 −0.996045 −0.498023 0.867164i \(-0.665940\pi\)
−0.498023 + 0.867164i \(0.665940\pi\)
\(114\) 0 0
\(115\) −1.41375 −0.131833
\(116\) 5.34091 0.495891
\(117\) 0 0
\(118\) 10.8147 0.995578
\(119\) −1.33794 −0.122649
\(120\) 0 0
\(121\) −0.300940 −0.0273582
\(122\) 14.6510 1.32644
\(123\) 0 0
\(124\) −18.9886 −1.70522
\(125\) −12.0751 −1.08003
\(126\) 0 0
\(127\) 14.6989 1.30432 0.652160 0.758081i \(-0.273863\pi\)
0.652160 + 0.758081i \(0.273863\pi\)
\(128\) −2.01059 −0.177713
\(129\) 0 0
\(130\) 18.9642 1.66327
\(131\) −9.63853 −0.842122 −0.421061 0.907032i \(-0.638342\pi\)
−0.421061 + 0.907032i \(0.638342\pi\)
\(132\) 0 0
\(133\) 9.15735 0.794043
\(134\) 33.8300 2.92247
\(135\) 0 0
\(136\) 7.41286 0.635647
\(137\) −23.2376 −1.98532 −0.992661 0.120933i \(-0.961411\pi\)
−0.992661 + 0.120933i \(0.961411\pi\)
\(138\) 0 0
\(139\) −1.29376 −0.109736 −0.0548678 0.998494i \(-0.517474\pi\)
−0.0548678 + 0.998494i \(0.517474\pi\)
\(140\) −10.5549 −0.892055
\(141\) 0 0
\(142\) 29.3850 2.46593
\(143\) 14.0863 1.17795
\(144\) 0 0
\(145\) 1.89027 0.156978
\(146\) 30.5011 2.52429
\(147\) 0 0
\(148\) −54.0480 −4.44271
\(149\) −17.5316 −1.43625 −0.718123 0.695916i \(-0.754998\pi\)
−0.718123 + 0.695916i \(0.754998\pi\)
\(150\) 0 0
\(151\) 5.26990 0.428858 0.214429 0.976740i \(-0.431211\pi\)
0.214429 + 0.976740i \(0.431211\pi\)
\(152\) −50.7363 −4.11526
\(153\) 0 0
\(154\) −11.1211 −0.896166
\(155\) −6.72048 −0.539802
\(156\) 0 0
\(157\) −14.2165 −1.13460 −0.567299 0.823512i \(-0.692012\pi\)
−0.567299 + 0.823512i \(0.692012\pi\)
\(158\) −4.33548 −0.344912
\(159\) 0 0
\(160\) 16.3908 1.29581
\(161\) 1.09153 0.0860250
\(162\) 0 0
\(163\) −15.9688 −1.25077 −0.625386 0.780315i \(-0.715059\pi\)
−0.625386 + 0.780315i \(0.715059\pi\)
\(164\) −5.87176 −0.458508
\(165\) 0 0
\(166\) 6.10086 0.473518
\(167\) 21.5275 1.66585 0.832925 0.553386i \(-0.186665\pi\)
0.832925 + 0.553386i \(0.186665\pi\)
\(168\) 0 0
\(169\) 5.54585 0.426604
\(170\) 4.51181 0.346040
\(171\) 0 0
\(172\) 51.8060 3.95017
\(173\) 5.82531 0.442890 0.221445 0.975173i \(-0.428923\pi\)
0.221445 + 0.975173i \(0.428923\pi\)
\(174\) 0 0
\(175\) 2.79368 0.211182
\(176\) 30.3538 2.28801
\(177\) 0 0
\(178\) −47.1515 −3.53416
\(179\) 6.03932 0.451400 0.225700 0.974197i \(-0.427533\pi\)
0.225700 + 0.974197i \(0.427533\pi\)
\(180\) 0 0
\(181\) −4.03706 −0.300073 −0.150036 0.988680i \(-0.547939\pi\)
−0.150036 + 0.988680i \(0.547939\pi\)
\(182\) −14.6420 −1.08533
\(183\) 0 0
\(184\) −6.04765 −0.445839
\(185\) −19.1288 −1.40638
\(186\) 0 0
\(187\) 3.35129 0.245071
\(188\) 0.725529 0.0529146
\(189\) 0 0
\(190\) −30.8805 −2.24031
\(191\) 16.4803 1.19247 0.596235 0.802810i \(-0.296663\pi\)
0.596235 + 0.802810i \(0.296663\pi\)
\(192\) 0 0
\(193\) −23.0631 −1.66012 −0.830060 0.557674i \(-0.811694\pi\)
−0.830060 + 0.557674i \(0.811694\pi\)
\(194\) −18.6907 −1.34192
\(195\) 0 0
\(196\) −25.3028 −1.80734
\(197\) 13.5540 0.965680 0.482840 0.875709i \(-0.339605\pi\)
0.482840 + 0.875709i \(0.339605\pi\)
\(198\) 0 0
\(199\) −25.6725 −1.81988 −0.909938 0.414745i \(-0.863871\pi\)
−0.909938 + 0.414745i \(0.863871\pi\)
\(200\) −15.4784 −1.09449
\(201\) 0 0
\(202\) 14.2708 1.00409
\(203\) −1.45944 −0.102433
\(204\) 0 0
\(205\) −2.07815 −0.145144
\(206\) −40.9735 −2.85476
\(207\) 0 0
\(208\) 39.9635 2.77097
\(209\) −22.9375 −1.58662
\(210\) 0 0
\(211\) 5.18301 0.356813 0.178407 0.983957i \(-0.442906\pi\)
0.178407 + 0.983957i \(0.442906\pi\)
\(212\) −14.4347 −0.991380
\(213\) 0 0
\(214\) 29.1127 1.99010
\(215\) 18.3353 1.25046
\(216\) 0 0
\(217\) 5.18877 0.352237
\(218\) −0.771777 −0.0522713
\(219\) 0 0
\(220\) 26.4382 1.78246
\(221\) 4.41227 0.296802
\(222\) 0 0
\(223\) −6.50914 −0.435884 −0.217942 0.975962i \(-0.569934\pi\)
−0.217942 + 0.975962i \(0.569934\pi\)
\(224\) −12.6551 −0.845553
\(225\) 0 0
\(226\) −27.5675 −1.83376
\(227\) −11.0460 −0.733150 −0.366575 0.930389i \(-0.619470\pi\)
−0.366575 + 0.930389i \(0.619470\pi\)
\(228\) 0 0
\(229\) 8.95679 0.591881 0.295941 0.955206i \(-0.404367\pi\)
0.295941 + 0.955206i \(0.404367\pi\)
\(230\) −3.68089 −0.242710
\(231\) 0 0
\(232\) 8.08605 0.530875
\(233\) 6.46647 0.423632 0.211816 0.977310i \(-0.432062\pi\)
0.211816 + 0.977310i \(0.432062\pi\)
\(234\) 0 0
\(235\) 0.256781 0.0167506
\(236\) 19.8501 1.29213
\(237\) 0 0
\(238\) −3.48349 −0.225801
\(239\) 25.0588 1.62092 0.810460 0.585794i \(-0.199217\pi\)
0.810460 + 0.585794i \(0.199217\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) −0.783536 −0.0503676
\(243\) 0 0
\(244\) 26.8914 1.72154
\(245\) −8.95523 −0.572129
\(246\) 0 0
\(247\) −30.1992 −1.92153
\(248\) −28.7484 −1.82552
\(249\) 0 0
\(250\) −31.4391 −1.98838
\(251\) 12.5611 0.792847 0.396424 0.918068i \(-0.370251\pi\)
0.396424 + 0.918068i \(0.370251\pi\)
\(252\) 0 0
\(253\) −2.73409 −0.171891
\(254\) 38.2706 2.40131
\(255\) 0 0
\(256\) −18.5788 −1.16117
\(257\) −16.0367 −1.00034 −0.500171 0.865927i \(-0.666730\pi\)
−0.500171 + 0.865927i \(0.666730\pi\)
\(258\) 0 0
\(259\) 14.7690 0.917702
\(260\) 34.8082 2.15871
\(261\) 0 0
\(262\) −25.0951 −1.55038
\(263\) −3.33210 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(264\) 0 0
\(265\) −5.10877 −0.313829
\(266\) 23.8423 1.46187
\(267\) 0 0
\(268\) 62.0939 3.79299
\(269\) 7.44101 0.453686 0.226843 0.973931i \(-0.427160\pi\)
0.226843 + 0.973931i \(0.427160\pi\)
\(270\) 0 0
\(271\) −20.6031 −1.25155 −0.625774 0.780005i \(-0.715217\pi\)
−0.625774 + 0.780005i \(0.715217\pi\)
\(272\) 9.50779 0.576495
\(273\) 0 0
\(274\) −60.5020 −3.65506
\(275\) −6.99764 −0.421974
\(276\) 0 0
\(277\) 5.17667 0.311036 0.155518 0.987833i \(-0.450295\pi\)
0.155518 + 0.987833i \(0.450295\pi\)
\(278\) −3.36848 −0.202028
\(279\) 0 0
\(280\) −15.9800 −0.954988
\(281\) 14.1821 0.846031 0.423015 0.906123i \(-0.360972\pi\)
0.423015 + 0.906123i \(0.360972\pi\)
\(282\) 0 0
\(283\) −7.05196 −0.419196 −0.209598 0.977788i \(-0.567215\pi\)
−0.209598 + 0.977788i \(0.567215\pi\)
\(284\) 53.9351 3.20046
\(285\) 0 0
\(286\) 36.6754 2.16866
\(287\) 1.60450 0.0947109
\(288\) 0 0
\(289\) −15.9503 −0.938251
\(290\) 4.92155 0.289004
\(291\) 0 0
\(292\) 55.9837 3.27620
\(293\) 2.78966 0.162974 0.0814869 0.996674i \(-0.474033\pi\)
0.0814869 + 0.996674i \(0.474033\pi\)
\(294\) 0 0
\(295\) 7.02540 0.409035
\(296\) −81.8278 −4.75614
\(297\) 0 0
\(298\) −45.6458 −2.64419
\(299\) −3.59968 −0.208175
\(300\) 0 0
\(301\) −14.1564 −0.815960
\(302\) 13.7208 0.789546
\(303\) 0 0
\(304\) −65.0749 −3.73230
\(305\) 9.51747 0.544969
\(306\) 0 0
\(307\) 19.8285 1.13167 0.565837 0.824517i \(-0.308553\pi\)
0.565837 + 0.824517i \(0.308553\pi\)
\(308\) −20.4124 −1.16311
\(309\) 0 0
\(310\) −17.4976 −0.993798
\(311\) −33.5109 −1.90023 −0.950114 0.311902i \(-0.899034\pi\)
−0.950114 + 0.311902i \(0.899034\pi\)
\(312\) 0 0
\(313\) 19.4152 1.09741 0.548705 0.836016i \(-0.315121\pi\)
0.548705 + 0.836016i \(0.315121\pi\)
\(314\) −37.0144 −2.08884
\(315\) 0 0
\(316\) −7.95763 −0.447652
\(317\) 11.7255 0.658569 0.329284 0.944231i \(-0.393193\pi\)
0.329284 + 0.944231i \(0.393193\pi\)
\(318\) 0 0
\(319\) 3.65564 0.204676
\(320\) 11.2847 0.630832
\(321\) 0 0
\(322\) 2.84195 0.158376
\(323\) −7.18475 −0.399770
\(324\) 0 0
\(325\) −9.21303 −0.511047
\(326\) −41.5768 −2.30272
\(327\) 0 0
\(328\) −8.88976 −0.490855
\(329\) −0.198256 −0.0109302
\(330\) 0 0
\(331\) 4.93072 0.271017 0.135509 0.990776i \(-0.456733\pi\)
0.135509 + 0.990776i \(0.456733\pi\)
\(332\) 11.1979 0.614566
\(333\) 0 0
\(334\) 56.0496 3.06690
\(335\) 21.9764 1.20070
\(336\) 0 0
\(337\) 34.4425 1.87620 0.938100 0.346364i \(-0.112584\pi\)
0.938100 + 0.346364i \(0.112584\pi\)
\(338\) 14.4393 0.785395
\(339\) 0 0
\(340\) 8.28128 0.449116
\(341\) −12.9969 −0.703822
\(342\) 0 0
\(343\) 16.0552 0.866900
\(344\) 78.4334 4.22885
\(345\) 0 0
\(346\) 15.1669 0.815378
\(347\) 23.4553 1.25915 0.629574 0.776940i \(-0.283229\pi\)
0.629574 + 0.776940i \(0.283229\pi\)
\(348\) 0 0
\(349\) 15.8727 0.849648 0.424824 0.905276i \(-0.360336\pi\)
0.424824 + 0.905276i \(0.360336\pi\)
\(350\) 7.27369 0.388795
\(351\) 0 0
\(352\) 31.6986 1.68954
\(353\) 28.4598 1.51476 0.757382 0.652972i \(-0.226478\pi\)
0.757382 + 0.652972i \(0.226478\pi\)
\(354\) 0 0
\(355\) 19.0889 1.01313
\(356\) −86.5450 −4.58688
\(357\) 0 0
\(358\) 15.7241 0.831047
\(359\) 0.772338 0.0407625 0.0203812 0.999792i \(-0.493512\pi\)
0.0203812 + 0.999792i \(0.493512\pi\)
\(360\) 0 0
\(361\) 30.1751 1.58816
\(362\) −10.5110 −0.552446
\(363\) 0 0
\(364\) −26.8748 −1.40862
\(365\) 19.8139 1.03711
\(366\) 0 0
\(367\) −23.3631 −1.21954 −0.609772 0.792577i \(-0.708739\pi\)
−0.609772 + 0.792577i \(0.708739\pi\)
\(368\) −7.75677 −0.404350
\(369\) 0 0
\(370\) −49.8042 −2.58920
\(371\) 3.94439 0.204783
\(372\) 0 0
\(373\) 27.7831 1.43855 0.719277 0.694724i \(-0.244473\pi\)
0.719277 + 0.694724i \(0.244473\pi\)
\(374\) 8.72550 0.451185
\(375\) 0 0
\(376\) 1.09844 0.0566477
\(377\) 4.81297 0.247881
\(378\) 0 0
\(379\) 7.33328 0.376685 0.188343 0.982103i \(-0.439688\pi\)
0.188343 + 0.982103i \(0.439688\pi\)
\(380\) −56.6802 −2.90763
\(381\) 0 0
\(382\) 42.9084 2.19539
\(383\) 33.4931 1.71142 0.855708 0.517459i \(-0.173122\pi\)
0.855708 + 0.517459i \(0.173122\pi\)
\(384\) 0 0
\(385\) −7.22443 −0.368191
\(386\) −60.0477 −3.05635
\(387\) 0 0
\(388\) −34.3062 −1.74163
\(389\) −15.6882 −0.795425 −0.397713 0.917510i \(-0.630196\pi\)
−0.397713 + 0.917510i \(0.630196\pi\)
\(390\) 0 0
\(391\) −0.856405 −0.0433103
\(392\) −38.3080 −1.93485
\(393\) 0 0
\(394\) 35.2894 1.77786
\(395\) −2.81639 −0.141708
\(396\) 0 0
\(397\) 8.34483 0.418815 0.209408 0.977828i \(-0.432846\pi\)
0.209408 + 0.977828i \(0.432846\pi\)
\(398\) −66.8416 −3.35047
\(399\) 0 0
\(400\) −19.8527 −0.992635
\(401\) 38.5766 1.92642 0.963211 0.268747i \(-0.0866094\pi\)
0.963211 + 0.268747i \(0.0866094\pi\)
\(402\) 0 0
\(403\) −17.1116 −0.852389
\(404\) 26.1936 1.30318
\(405\) 0 0
\(406\) −3.79985 −0.188583
\(407\) −36.9936 −1.83371
\(408\) 0 0
\(409\) −19.5415 −0.966266 −0.483133 0.875547i \(-0.660501\pi\)
−0.483133 + 0.875547i \(0.660501\pi\)
\(410\) −5.41073 −0.267217
\(411\) 0 0
\(412\) −75.2055 −3.70511
\(413\) −5.42419 −0.266907
\(414\) 0 0
\(415\) 3.96320 0.194546
\(416\) 41.7341 2.04618
\(417\) 0 0
\(418\) −59.7206 −2.92103
\(419\) 29.7748 1.45459 0.727297 0.686323i \(-0.240776\pi\)
0.727297 + 0.686323i \(0.240776\pi\)
\(420\) 0 0
\(421\) −4.78920 −0.233411 −0.116706 0.993167i \(-0.537233\pi\)
−0.116706 + 0.993167i \(0.537233\pi\)
\(422\) 13.4946 0.656908
\(423\) 0 0
\(424\) −21.8539 −1.06132
\(425\) −2.19189 −0.106322
\(426\) 0 0
\(427\) −7.34827 −0.355608
\(428\) 53.4354 2.58290
\(429\) 0 0
\(430\) 47.7383 2.30214
\(431\) −5.35939 −0.258153 −0.129076 0.991635i \(-0.541201\pi\)
−0.129076 + 0.991635i \(0.541201\pi\)
\(432\) 0 0
\(433\) 21.5028 1.03336 0.516680 0.856179i \(-0.327168\pi\)
0.516680 + 0.856179i \(0.327168\pi\)
\(434\) 13.5096 0.648482
\(435\) 0 0
\(436\) −1.41657 −0.0678414
\(437\) 5.86156 0.280396
\(438\) 0 0
\(439\) 4.22306 0.201556 0.100778 0.994909i \(-0.467867\pi\)
0.100778 + 0.994909i \(0.467867\pi\)
\(440\) 40.0270 1.90821
\(441\) 0 0
\(442\) 11.4879 0.546424
\(443\) −8.34030 −0.396260 −0.198130 0.980176i \(-0.563487\pi\)
−0.198130 + 0.980176i \(0.563487\pi\)
\(444\) 0 0
\(445\) −30.6303 −1.45201
\(446\) −16.9474 −0.802481
\(447\) 0 0
\(448\) −8.71269 −0.411636
\(449\) −15.7796 −0.744684 −0.372342 0.928096i \(-0.621445\pi\)
−0.372342 + 0.928096i \(0.621445\pi\)
\(450\) 0 0
\(451\) −4.01898 −0.189247
\(452\) −50.5992 −2.37998
\(453\) 0 0
\(454\) −28.7597 −1.34976
\(455\) −9.51161 −0.445911
\(456\) 0 0
\(457\) −8.71419 −0.407633 −0.203816 0.979009i \(-0.565335\pi\)
−0.203816 + 0.979009i \(0.565335\pi\)
\(458\) 23.3201 1.08968
\(459\) 0 0
\(460\) −6.75614 −0.315007
\(461\) −24.9140 −1.16036 −0.580181 0.814487i \(-0.697018\pi\)
−0.580181 + 0.814487i \(0.697018\pi\)
\(462\) 0 0
\(463\) −2.87272 −0.133507 −0.0667534 0.997770i \(-0.521264\pi\)
−0.0667534 + 0.997770i \(0.521264\pi\)
\(464\) 10.3712 0.481473
\(465\) 0 0
\(466\) 16.8363 0.779925
\(467\) −4.87474 −0.225576 −0.112788 0.993619i \(-0.535978\pi\)
−0.112788 + 0.993619i \(0.535978\pi\)
\(468\) 0 0
\(469\) −16.9676 −0.783492
\(470\) 0.668562 0.0308385
\(471\) 0 0
\(472\) 30.0528 1.38329
\(473\) 35.4591 1.63041
\(474\) 0 0
\(475\) 15.0021 0.688343
\(476\) −6.39383 −0.293061
\(477\) 0 0
\(478\) 65.2438 2.98418
\(479\) −0.194055 −0.00886661 −0.00443331 0.999990i \(-0.501411\pi\)
−0.00443331 + 0.999990i \(0.501411\pi\)
\(480\) 0 0
\(481\) −48.7055 −2.22078
\(482\) −2.60363 −0.118592
\(483\) 0 0
\(484\) −1.43815 −0.0653707
\(485\) −12.1417 −0.551328
\(486\) 0 0
\(487\) −30.7592 −1.39383 −0.696915 0.717154i \(-0.745445\pi\)
−0.696915 + 0.717154i \(0.745445\pi\)
\(488\) 40.7131 1.84300
\(489\) 0 0
\(490\) −23.3161 −1.05331
\(491\) 21.5528 0.972663 0.486332 0.873774i \(-0.338335\pi\)
0.486332 + 0.873774i \(0.338335\pi\)
\(492\) 0 0
\(493\) 1.14506 0.0515710
\(494\) −78.6275 −3.53762
\(495\) 0 0
\(496\) −36.8729 −1.65564
\(497\) −14.7382 −0.661097
\(498\) 0 0
\(499\) 21.8992 0.980341 0.490170 0.871627i \(-0.336935\pi\)
0.490170 + 0.871627i \(0.336935\pi\)
\(500\) −57.7054 −2.58066
\(501\) 0 0
\(502\) 32.7043 1.45966
\(503\) −4.44114 −0.198021 −0.0990104 0.995086i \(-0.531568\pi\)
−0.0990104 + 0.995086i \(0.531568\pi\)
\(504\) 0 0
\(505\) 9.27050 0.412532
\(506\) −7.11855 −0.316458
\(507\) 0 0
\(508\) 70.2443 3.11659
\(509\) −29.6966 −1.31628 −0.658140 0.752895i \(-0.728657\pi\)
−0.658140 + 0.752895i \(0.728657\pi\)
\(510\) 0 0
\(511\) −15.2980 −0.676742
\(512\) −44.3511 −1.96006
\(513\) 0 0
\(514\) −41.7536 −1.84167
\(515\) −26.6169 −1.17288
\(516\) 0 0
\(517\) 0.496595 0.0218402
\(518\) 38.4530 1.68953
\(519\) 0 0
\(520\) 52.6991 2.31101
\(521\) 32.5336 1.42532 0.712661 0.701509i \(-0.247490\pi\)
0.712661 + 0.701509i \(0.247490\pi\)
\(522\) 0 0
\(523\) −21.6812 −0.948053 −0.474027 0.880510i \(-0.657200\pi\)
−0.474027 + 0.880510i \(0.657200\pi\)
\(524\) −46.0612 −2.01219
\(525\) 0 0
\(526\) −8.67555 −0.378272
\(527\) −4.07105 −0.177338
\(528\) 0 0
\(529\) −22.3013 −0.969622
\(530\) −13.3013 −0.577773
\(531\) 0 0
\(532\) 43.7618 1.89731
\(533\) −5.29135 −0.229194
\(534\) 0 0
\(535\) 18.9120 0.817637
\(536\) 94.0092 4.06058
\(537\) 0 0
\(538\) 19.3736 0.835255
\(539\) −17.3187 −0.745971
\(540\) 0 0
\(541\) 6.08311 0.261533 0.130767 0.991413i \(-0.458256\pi\)
0.130767 + 0.991413i \(0.458256\pi\)
\(542\) −53.6427 −2.30415
\(543\) 0 0
\(544\) 9.92903 0.425703
\(545\) −0.501356 −0.0214757
\(546\) 0 0
\(547\) −12.1956 −0.521447 −0.260723 0.965414i \(-0.583961\pi\)
−0.260723 + 0.965414i \(0.583961\pi\)
\(548\) −111.049 −4.74379
\(549\) 0 0
\(550\) −18.2192 −0.776871
\(551\) −7.83723 −0.333877
\(552\) 0 0
\(553\) 2.17448 0.0924684
\(554\) 13.4781 0.572630
\(555\) 0 0
\(556\) −6.18273 −0.262206
\(557\) 37.5982 1.59309 0.796544 0.604581i \(-0.206659\pi\)
0.796544 + 0.604581i \(0.206659\pi\)
\(558\) 0 0
\(559\) 46.6851 1.97457
\(560\) −20.4961 −0.866118
\(561\) 0 0
\(562\) 36.9248 1.55758
\(563\) −20.6037 −0.868342 −0.434171 0.900830i \(-0.642959\pi\)
−0.434171 + 0.900830i \(0.642959\pi\)
\(564\) 0 0
\(565\) −17.9082 −0.753403
\(566\) −18.3607 −0.771756
\(567\) 0 0
\(568\) 81.6569 3.42625
\(569\) −20.5762 −0.862601 −0.431301 0.902208i \(-0.641945\pi\)
−0.431301 + 0.902208i \(0.641945\pi\)
\(570\) 0 0
\(571\) 37.1320 1.55392 0.776962 0.629547i \(-0.216760\pi\)
0.776962 + 0.629547i \(0.216760\pi\)
\(572\) 67.3165 2.81464
\(573\) 0 0
\(574\) 4.17753 0.174367
\(575\) 1.78821 0.0745736
\(576\) 0 0
\(577\) −34.7064 −1.44484 −0.722422 0.691452i \(-0.756971\pi\)
−0.722422 + 0.691452i \(0.756971\pi\)
\(578\) −41.5285 −1.72736
\(579\) 0 0
\(580\) 9.03335 0.375089
\(581\) −3.05992 −0.126947
\(582\) 0 0
\(583\) −9.87998 −0.409187
\(584\) 84.7585 3.50733
\(585\) 0 0
\(586\) 7.26324 0.300042
\(587\) 1.14455 0.0472405 0.0236203 0.999721i \(-0.492481\pi\)
0.0236203 + 0.999721i \(0.492481\pi\)
\(588\) 0 0
\(589\) 27.8638 1.14811
\(590\) 18.2915 0.753050
\(591\) 0 0
\(592\) −104.953 −4.31354
\(593\) −3.60735 −0.148136 −0.0740680 0.997253i \(-0.523598\pi\)
−0.0740680 + 0.997253i \(0.523598\pi\)
\(594\) 0 0
\(595\) −2.26292 −0.0927708
\(596\) −83.7813 −3.43181
\(597\) 0 0
\(598\) −9.37222 −0.383258
\(599\) 31.6723 1.29410 0.647048 0.762449i \(-0.276003\pi\)
0.647048 + 0.762449i \(0.276003\pi\)
\(600\) 0 0
\(601\) 6.55665 0.267451 0.133726 0.991018i \(-0.457306\pi\)
0.133726 + 0.991018i \(0.457306\pi\)
\(602\) −36.8579 −1.50222
\(603\) 0 0
\(604\) 25.1841 1.02473
\(605\) −0.508996 −0.0206936
\(606\) 0 0
\(607\) −9.16648 −0.372056 −0.186028 0.982544i \(-0.559562\pi\)
−0.186028 + 0.982544i \(0.559562\pi\)
\(608\) −67.9579 −2.75606
\(609\) 0 0
\(610\) 24.7799 1.00331
\(611\) 0.653812 0.0264504
\(612\) 0 0
\(613\) 7.92345 0.320025 0.160012 0.987115i \(-0.448847\pi\)
0.160012 + 0.987115i \(0.448847\pi\)
\(614\) 51.6260 2.08346
\(615\) 0 0
\(616\) −30.9041 −1.24516
\(617\) −16.5959 −0.668126 −0.334063 0.942551i \(-0.608420\pi\)
−0.334063 + 0.942551i \(0.608420\pi\)
\(618\) 0 0
\(619\) 4.49086 0.180503 0.0902514 0.995919i \(-0.471233\pi\)
0.0902514 + 0.995919i \(0.471233\pi\)
\(620\) −32.1163 −1.28982
\(621\) 0 0
\(622\) −87.2498 −3.49840
\(623\) 23.6491 0.947481
\(624\) 0 0
\(625\) −9.72656 −0.389063
\(626\) 50.5499 2.02038
\(627\) 0 0
\(628\) −67.9387 −2.71105
\(629\) −11.5876 −0.462028
\(630\) 0 0
\(631\) −5.96553 −0.237484 −0.118742 0.992925i \(-0.537886\pi\)
−0.118742 + 0.992925i \(0.537886\pi\)
\(632\) −12.0477 −0.479233
\(633\) 0 0
\(634\) 30.5288 1.21245
\(635\) 24.8611 0.986581
\(636\) 0 0
\(637\) −22.8017 −0.903435
\(638\) 9.51791 0.376818
\(639\) 0 0
\(640\) −3.40061 −0.134421
\(641\) 1.73375 0.0684789 0.0342395 0.999414i \(-0.489099\pi\)
0.0342395 + 0.999414i \(0.489099\pi\)
\(642\) 0 0
\(643\) 38.3312 1.51163 0.755817 0.654783i \(-0.227240\pi\)
0.755817 + 0.654783i \(0.227240\pi\)
\(644\) 5.21630 0.205551
\(645\) 0 0
\(646\) −18.7064 −0.735994
\(647\) −29.2098 −1.14836 −0.574178 0.818730i \(-0.694678\pi\)
−0.574178 + 0.818730i \(0.694678\pi\)
\(648\) 0 0
\(649\) 13.5866 0.533321
\(650\) −23.9873 −0.940858
\(651\) 0 0
\(652\) −76.3127 −2.98864
\(653\) 42.7952 1.67471 0.837353 0.546662i \(-0.184102\pi\)
0.837353 + 0.546662i \(0.184102\pi\)
\(654\) 0 0
\(655\) −16.3021 −0.636976
\(656\) −11.4021 −0.445177
\(657\) 0 0
\(658\) −0.516185 −0.0201230
\(659\) 23.7004 0.923238 0.461619 0.887078i \(-0.347269\pi\)
0.461619 + 0.887078i \(0.347269\pi\)
\(660\) 0 0
\(661\) 24.1228 0.938268 0.469134 0.883127i \(-0.344566\pi\)
0.469134 + 0.883127i \(0.344566\pi\)
\(662\) 12.8378 0.498954
\(663\) 0 0
\(664\) 16.9535 0.657923
\(665\) 15.4883 0.600610
\(666\) 0 0
\(667\) −0.934180 −0.0361716
\(668\) 102.877 3.98044
\(669\) 0 0
\(670\) 57.2184 2.21054
\(671\) 18.4061 0.710558
\(672\) 0 0
\(673\) −3.67522 −0.141669 −0.0708346 0.997488i \(-0.522566\pi\)
−0.0708346 + 0.997488i \(0.522566\pi\)
\(674\) 89.6753 3.45416
\(675\) 0 0
\(676\) 26.5029 1.01934
\(677\) −45.7769 −1.75935 −0.879675 0.475575i \(-0.842240\pi\)
−0.879675 + 0.475575i \(0.842240\pi\)
\(678\) 0 0
\(679\) 9.37443 0.359758
\(680\) 12.5377 0.480800
\(681\) 0 0
\(682\) −33.8391 −1.29577
\(683\) 1.52419 0.0583215 0.0291608 0.999575i \(-0.490717\pi\)
0.0291608 + 0.999575i \(0.490717\pi\)
\(684\) 0 0
\(685\) −39.3029 −1.50169
\(686\) 41.8018 1.59600
\(687\) 0 0
\(688\) 100.599 3.83532
\(689\) −13.0079 −0.495561
\(690\) 0 0
\(691\) 20.3036 0.772386 0.386193 0.922418i \(-0.373790\pi\)
0.386193 + 0.922418i \(0.373790\pi\)
\(692\) 27.8384 1.05826
\(693\) 0 0
\(694\) 61.0689 2.31814
\(695\) −2.18821 −0.0830034
\(696\) 0 0
\(697\) −1.25888 −0.0476833
\(698\) 41.3267 1.56424
\(699\) 0 0
\(700\) 13.3506 0.504606
\(701\) 10.3863 0.392285 0.196143 0.980575i \(-0.437158\pi\)
0.196143 + 0.980575i \(0.437158\pi\)
\(702\) 0 0
\(703\) 79.3098 2.99123
\(704\) 21.8237 0.822511
\(705\) 0 0
\(706\) 74.0988 2.78874
\(707\) −7.15759 −0.269189
\(708\) 0 0
\(709\) 34.9053 1.31090 0.655448 0.755240i \(-0.272480\pi\)
0.655448 + 0.755240i \(0.272480\pi\)
\(710\) 49.7002 1.86522
\(711\) 0 0
\(712\) −131.028 −4.91048
\(713\) 3.32129 0.124383
\(714\) 0 0
\(715\) 23.8248 0.890998
\(716\) 28.8611 1.07859
\(717\) 0 0
\(718\) 2.01088 0.0750454
\(719\) −5.17696 −0.193068 −0.0965341 0.995330i \(-0.530776\pi\)
−0.0965341 + 0.995330i \(0.530776\pi\)
\(720\) 0 0
\(721\) 20.5505 0.765339
\(722\) 78.5646 2.92387
\(723\) 0 0
\(724\) −19.2926 −0.717003
\(725\) −2.39094 −0.0887974
\(726\) 0 0
\(727\) −7.00384 −0.259758 −0.129879 0.991530i \(-0.541459\pi\)
−0.129879 + 0.991530i \(0.541459\pi\)
\(728\) −40.6881 −1.50800
\(729\) 0 0
\(730\) 51.5880 1.90936
\(731\) 11.1069 0.410804
\(732\) 0 0
\(733\) −1.22518 −0.0452530 −0.0226265 0.999744i \(-0.507203\pi\)
−0.0226265 + 0.999744i \(0.507203\pi\)
\(734\) −60.8288 −2.24523
\(735\) 0 0
\(736\) −8.10042 −0.298586
\(737\) 42.5008 1.56554
\(738\) 0 0
\(739\) −31.4569 −1.15716 −0.578580 0.815625i \(-0.696393\pi\)
−0.578580 + 0.815625i \(0.696393\pi\)
\(740\) −91.4140 −3.36045
\(741\) 0 0
\(742\) 10.2697 0.377014
\(743\) −17.5176 −0.642659 −0.321330 0.946967i \(-0.604130\pi\)
−0.321330 + 0.946967i \(0.604130\pi\)
\(744\) 0 0
\(745\) −29.6521 −1.08637
\(746\) 72.3368 2.64844
\(747\) 0 0
\(748\) 16.0154 0.585580
\(749\) −14.6016 −0.533532
\(750\) 0 0
\(751\) −35.2893 −1.28773 −0.643863 0.765141i \(-0.722669\pi\)
−0.643863 + 0.765141i \(0.722669\pi\)
\(752\) 1.40887 0.0513761
\(753\) 0 0
\(754\) 12.5312 0.456359
\(755\) 8.91324 0.324386
\(756\) 0 0
\(757\) −17.2974 −0.628686 −0.314343 0.949309i \(-0.601784\pi\)
−0.314343 + 0.949309i \(0.601784\pi\)
\(758\) 19.0931 0.693493
\(759\) 0 0
\(760\) −85.8129 −3.11276
\(761\) −9.53250 −0.345553 −0.172776 0.984961i \(-0.555274\pi\)
−0.172776 + 0.984961i \(0.555274\pi\)
\(762\) 0 0
\(763\) 0.387088 0.0140135
\(764\) 78.7570 2.84933
\(765\) 0 0
\(766\) 87.2034 3.15079
\(767\) 17.8880 0.645897
\(768\) 0 0
\(769\) −37.0399 −1.33569 −0.667847 0.744299i \(-0.732784\pi\)
−0.667847 + 0.744299i \(0.732784\pi\)
\(770\) −18.8097 −0.677855
\(771\) 0 0
\(772\) −110.216 −3.96675
\(773\) 6.97450 0.250855 0.125428 0.992103i \(-0.459970\pi\)
0.125428 + 0.992103i \(0.459970\pi\)
\(774\) 0 0
\(775\) 8.50053 0.305348
\(776\) −51.9391 −1.86450
\(777\) 0 0
\(778\) −40.8463 −1.46441
\(779\) 8.61621 0.308708
\(780\) 0 0
\(781\) 36.9164 1.32097
\(782\) −2.22976 −0.0797360
\(783\) 0 0
\(784\) −49.1342 −1.75479
\(785\) −24.0450 −0.858204
\(786\) 0 0
\(787\) 29.9359 1.06710 0.533550 0.845768i \(-0.320857\pi\)
0.533550 + 0.845768i \(0.320857\pi\)
\(788\) 64.7726 2.30743
\(789\) 0 0
\(790\) −7.33281 −0.260890
\(791\) 13.8266 0.491617
\(792\) 0 0
\(793\) 24.2332 0.860547
\(794\) 21.7268 0.771056
\(795\) 0 0
\(796\) −122.685 −4.34847
\(797\) 10.7103 0.379380 0.189690 0.981844i \(-0.439252\pi\)
0.189690 + 0.981844i \(0.439252\pi\)
\(798\) 0 0
\(799\) 0.155550 0.00550295
\(800\) −20.7323 −0.732996
\(801\) 0 0
\(802\) 100.439 3.54662
\(803\) 38.3186 1.35223
\(804\) 0 0
\(805\) 1.84617 0.0650688
\(806\) −44.5522 −1.56928
\(807\) 0 0
\(808\) 39.6567 1.39512
\(809\) 30.4625 1.07100 0.535502 0.844534i \(-0.320122\pi\)
0.535502 + 0.844534i \(0.320122\pi\)
\(810\) 0 0
\(811\) −6.38159 −0.224088 −0.112044 0.993703i \(-0.535740\pi\)
−0.112044 + 0.993703i \(0.535740\pi\)
\(812\) −6.97449 −0.244757
\(813\) 0 0
\(814\) −96.3176 −3.37593
\(815\) −27.0088 −0.946077
\(816\) 0 0
\(817\) −76.0199 −2.65960
\(818\) −50.8788 −1.77894
\(819\) 0 0
\(820\) −9.93121 −0.346813
\(821\) 31.3814 1.09522 0.547609 0.836734i \(-0.315538\pi\)
0.547609 + 0.836734i \(0.315538\pi\)
\(822\) 0 0
\(823\) −8.61720 −0.300377 −0.150188 0.988657i \(-0.547988\pi\)
−0.150188 + 0.988657i \(0.547988\pi\)
\(824\) −113.860 −3.96650
\(825\) 0 0
\(826\) −14.1226 −0.491387
\(827\) 18.2047 0.633039 0.316520 0.948586i \(-0.397486\pi\)
0.316520 + 0.948586i \(0.397486\pi\)
\(828\) 0 0
\(829\) 38.0162 1.32036 0.660179 0.751108i \(-0.270480\pi\)
0.660179 + 0.751108i \(0.270480\pi\)
\(830\) 10.3187 0.358167
\(831\) 0 0
\(832\) 28.7328 0.996132
\(833\) −5.42479 −0.187958
\(834\) 0 0
\(835\) 36.4106 1.26004
\(836\) −109.615 −3.79112
\(837\) 0 0
\(838\) 77.5224 2.67797
\(839\) −45.6228 −1.57507 −0.787537 0.616267i \(-0.788644\pi\)
−0.787537 + 0.616267i \(0.788644\pi\)
\(840\) 0 0
\(841\) −27.7509 −0.956929
\(842\) −12.4693 −0.429720
\(843\) 0 0
\(844\) 24.7689 0.852582
\(845\) 9.37997 0.322681
\(846\) 0 0
\(847\) 0.392987 0.0135032
\(848\) −28.0300 −0.962556
\(849\) 0 0
\(850\) −5.70685 −0.195743
\(851\) 9.45354 0.324063
\(852\) 0 0
\(853\) 17.2406 0.590307 0.295153 0.955450i \(-0.404629\pi\)
0.295153 + 0.955450i \(0.404629\pi\)
\(854\) −19.1321 −0.654689
\(855\) 0 0
\(856\) 80.9003 2.76512
\(857\) −9.87638 −0.337371 −0.168685 0.985670i \(-0.553952\pi\)
−0.168685 + 0.985670i \(0.553952\pi\)
\(858\) 0 0
\(859\) 25.8904 0.883369 0.441685 0.897170i \(-0.354381\pi\)
0.441685 + 0.897170i \(0.354381\pi\)
\(860\) 87.6220 2.98789
\(861\) 0 0
\(862\) −13.9538 −0.475270
\(863\) 14.8445 0.505311 0.252656 0.967556i \(-0.418696\pi\)
0.252656 + 0.967556i \(0.418696\pi\)
\(864\) 0 0
\(865\) 9.85263 0.334999
\(866\) 55.9853 1.90246
\(867\) 0 0
\(868\) 24.7964 0.841646
\(869\) −5.44668 −0.184766
\(870\) 0 0
\(871\) 55.9561 1.89600
\(872\) −2.14467 −0.0726275
\(873\) 0 0
\(874\) 15.2613 0.516221
\(875\) 15.7684 0.533070
\(876\) 0 0
\(877\) 42.0781 1.42088 0.710439 0.703759i \(-0.248497\pi\)
0.710439 + 0.703759i \(0.248497\pi\)
\(878\) 10.9953 0.371072
\(879\) 0 0
\(880\) 51.3389 1.73064
\(881\) 19.0418 0.641534 0.320767 0.947158i \(-0.396059\pi\)
0.320767 + 0.947158i \(0.396059\pi\)
\(882\) 0 0
\(883\) 6.51087 0.219108 0.109554 0.993981i \(-0.465058\pi\)
0.109554 + 0.993981i \(0.465058\pi\)
\(884\) 21.0857 0.709188
\(885\) 0 0
\(886\) −21.7150 −0.729531
\(887\) −5.59140 −0.187741 −0.0938704 0.995584i \(-0.529924\pi\)
−0.0938704 + 0.995584i \(0.529924\pi\)
\(888\) 0 0
\(889\) −19.1948 −0.643773
\(890\) −79.7497 −2.67322
\(891\) 0 0
\(892\) −31.1063 −1.04152
\(893\) −1.06464 −0.0356268
\(894\) 0 0
\(895\) 10.2146 0.341437
\(896\) 2.62555 0.0877135
\(897\) 0 0
\(898\) −41.0841 −1.37099
\(899\) −4.44076 −0.148108
\(900\) 0 0
\(901\) −3.09473 −0.103100
\(902\) −10.4639 −0.348411
\(903\) 0 0
\(904\) −76.6063 −2.54789
\(905\) −6.82809 −0.226973
\(906\) 0 0
\(907\) 18.3915 0.610681 0.305340 0.952243i \(-0.401230\pi\)
0.305340 + 0.952243i \(0.401230\pi\)
\(908\) −52.7874 −1.75181
\(909\) 0 0
\(910\) −24.7647 −0.820941
\(911\) 38.2665 1.26783 0.633913 0.773404i \(-0.281448\pi\)
0.633913 + 0.773404i \(0.281448\pi\)
\(912\) 0 0
\(913\) 7.66452 0.253659
\(914\) −22.6885 −0.750469
\(915\) 0 0
\(916\) 42.8033 1.41426
\(917\) 12.5866 0.415646
\(918\) 0 0
\(919\) −4.89962 −0.161624 −0.0808118 0.996729i \(-0.525751\pi\)
−0.0808118 + 0.996729i \(0.525751\pi\)
\(920\) −10.2287 −0.337230
\(921\) 0 0
\(922\) −64.8669 −2.13628
\(923\) 48.6038 1.59981
\(924\) 0 0
\(925\) 24.1954 0.795541
\(926\) −7.47949 −0.245791
\(927\) 0 0
\(928\) 10.8307 0.355536
\(929\) −17.9381 −0.588528 −0.294264 0.955724i \(-0.595075\pi\)
−0.294264 + 0.955724i \(0.595075\pi\)
\(930\) 0 0
\(931\) 37.1292 1.21686
\(932\) 30.9024 1.01224
\(933\) 0 0
\(934\) −12.6920 −0.415295
\(935\) 5.66820 0.185370
\(936\) 0 0
\(937\) 29.8369 0.974730 0.487365 0.873198i \(-0.337958\pi\)
0.487365 + 0.873198i \(0.337958\pi\)
\(938\) −44.1773 −1.44244
\(939\) 0 0
\(940\) 1.22712 0.0400243
\(941\) −26.8047 −0.873809 −0.436904 0.899508i \(-0.643925\pi\)
−0.436904 + 0.899508i \(0.643925\pi\)
\(942\) 0 0
\(943\) 1.02703 0.0334448
\(944\) 38.5459 1.25456
\(945\) 0 0
\(946\) 92.3222 3.00165
\(947\) −2.94622 −0.0957393 −0.0478697 0.998854i \(-0.515243\pi\)
−0.0478697 + 0.998854i \(0.515243\pi\)
\(948\) 0 0
\(949\) 50.4499 1.63767
\(950\) 39.0598 1.26727
\(951\) 0 0
\(952\) −9.68017 −0.313736
\(953\) 4.62861 0.149935 0.0749676 0.997186i \(-0.476115\pi\)
0.0749676 + 0.997186i \(0.476115\pi\)
\(954\) 0 0
\(955\) 27.8739 0.901978
\(956\) 119.753 3.87308
\(957\) 0 0
\(958\) −0.505247 −0.0163238
\(959\) 30.3451 0.979894
\(960\) 0 0
\(961\) −15.2117 −0.490702
\(962\) −126.811 −4.08854
\(963\) 0 0
\(964\) −4.77887 −0.153917
\(965\) −39.0078 −1.25571
\(966\) 0 0
\(967\) 20.1721 0.648689 0.324345 0.945939i \(-0.394856\pi\)
0.324345 + 0.945939i \(0.394856\pi\)
\(968\) −2.17734 −0.0699825
\(969\) 0 0
\(970\) −31.6126 −1.01502
\(971\) −30.0251 −0.963550 −0.481775 0.876295i \(-0.660008\pi\)
−0.481775 + 0.876295i \(0.660008\pi\)
\(972\) 0 0
\(973\) 1.68948 0.0541621
\(974\) −80.0853 −2.56610
\(975\) 0 0
\(976\) 52.2190 1.67149
\(977\) −57.7096 −1.84629 −0.923147 0.384448i \(-0.874392\pi\)
−0.923147 + 0.384448i \(0.874392\pi\)
\(978\) 0 0
\(979\) −59.2366 −1.89321
\(980\) −42.7959 −1.36706
\(981\) 0 0
\(982\) 56.1154 1.79071
\(983\) 3.33817 0.106471 0.0532356 0.998582i \(-0.483047\pi\)
0.0532356 + 0.998582i \(0.483047\pi\)
\(984\) 0 0
\(985\) 22.9245 0.730435
\(986\) 2.98132 0.0949444
\(987\) 0 0
\(988\) −144.318 −4.59137
\(989\) −9.06139 −0.288136
\(990\) 0 0
\(991\) 4.48757 0.142552 0.0712762 0.997457i \(-0.477293\pi\)
0.0712762 + 0.997457i \(0.477293\pi\)
\(992\) −38.5065 −1.22258
\(993\) 0 0
\(994\) −38.3727 −1.21711
\(995\) −43.4212 −1.37654
\(996\) 0 0
\(997\) −47.4116 −1.50154 −0.750770 0.660564i \(-0.770317\pi\)
−0.750770 + 0.660564i \(0.770317\pi\)
\(998\) 57.0172 1.80485
\(999\) 0 0
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 2169.2.a.e.1.7 7
3.2 odd 2 241.2.a.a.1.1 7
12.11 even 2 3856.2.a.j.1.2 7
15.14 odd 2 6025.2.a.f.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.1 7 3.2 odd 2
2169.2.a.e.1.7 7 1.1 even 1 trivial
3856.2.a.j.1.2 7 12.11 even 2
6025.2.a.f.1.7 7 15.14 odd 2