Properties

Label 5445.2.a.bf.1.3
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +O(q^{10})\) \(q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +0.262360 q^{10} +1.11961 q^{13} -0.837853 q^{14} +3.59174 q^{16} -0.0882264 q^{17} +0.0688326 q^{19} +1.93117 q^{20} -6.65450 q^{23} +1.00000 q^{25} -0.293740 q^{26} -6.16724 q^{28} +3.73583 q^{29} -9.58484 q^{31} -3.00509 q^{32} +0.0231471 q^{34} -3.19353 q^{35} -8.33021 q^{37} -0.0180589 q^{38} -1.03138 q^{40} -11.6657 q^{41} +11.8217 q^{43} +1.74587 q^{46} -0.908020 q^{47} +3.19862 q^{49} -0.262360 q^{50} -2.16215 q^{52} -0.872377 q^{53} +3.29374 q^{56} -0.980131 q^{58} -1.83604 q^{59} +10.0601 q^{61} +2.51468 q^{62} -6.39507 q^{64} -1.11961 q^{65} +9.53916 q^{67} +0.170380 q^{68} +0.837853 q^{70} +4.66454 q^{71} +7.16034 q^{73} +2.18551 q^{74} -0.132927 q^{76} -0.791342 q^{79} -3.59174 q^{80} +3.06060 q^{82} +0.247229 q^{83} +0.0882264 q^{85} -3.10155 q^{86} +14.5788 q^{89} +3.57549 q^{91} +12.8510 q^{92} +0.238228 q^{94} -0.0688326 q^{95} -4.09198 q^{97} -0.839188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + O(q^{10}) \) \( 4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + 3q^{10} + 7q^{13} - 3q^{14} - q^{16} - 10q^{17} + 9q^{19} - q^{20} + 3q^{23} + 4q^{25} + 4q^{26} - 7q^{28} - 15q^{29} - 13q^{31} + 6q^{32} - 3q^{34} - 6q^{35} - 3q^{37} - 15q^{38} + 3q^{40} - 22q^{41} + q^{46} + 2q^{47} - 12q^{49} - 3q^{50} - 9q^{52} - 10q^{53} + 8q^{56} + 39q^{58} + 21q^{59} + 11q^{61} - 10q^{62} - 3q^{64} - 7q^{65} + q^{67} - 3q^{68} + 3q^{70} + 13q^{71} + q^{73} + 11q^{74} + 19q^{76} - 4q^{79} + q^{80} + 25q^{82} - 3q^{83} + 10q^{85} + 10q^{89} + 12q^{91} + 24q^{92} - 35q^{94} - 9q^{95} - 22q^{97} + 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.262360 −0.185516 −0.0927582 0.995689i \(-0.529568\pi\)
−0.0927582 + 0.995689i \(0.529568\pi\)
\(3\) 0 0
\(4\) −1.93117 −0.965584
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.19353 1.20704 0.603520 0.797348i \(-0.293764\pi\)
0.603520 + 0.797348i \(0.293764\pi\)
\(8\) 1.03138 0.364648
\(9\) 0 0
\(10\) 0.262360 0.0829654
\(11\) 0 0
\(12\) 0 0
\(13\) 1.11961 0.310523 0.155261 0.987873i \(-0.450378\pi\)
0.155261 + 0.987873i \(0.450378\pi\)
\(14\) −0.837853 −0.223926
\(15\) 0 0
\(16\) 3.59174 0.897936
\(17\) −0.0882264 −0.0213981 −0.0106990 0.999943i \(-0.503406\pi\)
−0.0106990 + 0.999943i \(0.503406\pi\)
\(18\) 0 0
\(19\) 0.0688326 0.0157913 0.00789564 0.999969i \(-0.497487\pi\)
0.00789564 + 0.999969i \(0.497487\pi\)
\(20\) 1.93117 0.431822
\(21\) 0 0
\(22\) 0 0
\(23\) −6.65450 −1.38756 −0.693780 0.720187i \(-0.744056\pi\)
−0.693780 + 0.720187i \(0.744056\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.293740 −0.0576071
\(27\) 0 0
\(28\) −6.16724 −1.16550
\(29\) 3.73583 0.693726 0.346863 0.937916i \(-0.387247\pi\)
0.346863 + 0.937916i \(0.387247\pi\)
\(30\) 0 0
\(31\) −9.58484 −1.72149 −0.860744 0.509037i \(-0.830002\pi\)
−0.860744 + 0.509037i \(0.830002\pi\)
\(32\) −3.00509 −0.531230
\(33\) 0 0
\(34\) 0.0231471 0.00396969
\(35\) −3.19353 −0.539805
\(36\) 0 0
\(37\) −8.33021 −1.36948 −0.684739 0.728789i \(-0.740084\pi\)
−0.684739 + 0.728789i \(0.740084\pi\)
\(38\) −0.0180589 −0.00292954
\(39\) 0 0
\(40\) −1.03138 −0.163075
\(41\) −11.6657 −1.82187 −0.910935 0.412549i \(-0.864638\pi\)
−0.910935 + 0.412549i \(0.864638\pi\)
\(42\) 0 0
\(43\) 11.8217 1.80280 0.901399 0.432989i \(-0.142541\pi\)
0.901399 + 0.432989i \(0.142541\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.74587 0.257415
\(47\) −0.908020 −0.132448 −0.0662242 0.997805i \(-0.521095\pi\)
−0.0662242 + 0.997805i \(0.521095\pi\)
\(48\) 0 0
\(49\) 3.19862 0.456945
\(50\) −0.262360 −0.0371033
\(51\) 0 0
\(52\) −2.16215 −0.299836
\(53\) −0.872377 −0.119830 −0.0599151 0.998203i \(-0.519083\pi\)
−0.0599151 + 0.998203i \(0.519083\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.29374 0.440144
\(57\) 0 0
\(58\) −0.980131 −0.128698
\(59\) −1.83604 −0.239032 −0.119516 0.992832i \(-0.538134\pi\)
−0.119516 + 0.992832i \(0.538134\pi\)
\(60\) 0 0
\(61\) 10.0601 1.28807 0.644034 0.764997i \(-0.277260\pi\)
0.644034 + 0.764997i \(0.277260\pi\)
\(62\) 2.51468 0.319364
\(63\) 0 0
\(64\) −6.39507 −0.799384
\(65\) −1.11961 −0.138870
\(66\) 0 0
\(67\) 9.53916 1.16539 0.582697 0.812690i \(-0.301997\pi\)
0.582697 + 0.812690i \(0.301997\pi\)
\(68\) 0.170380 0.0206616
\(69\) 0 0
\(70\) 0.837853 0.100143
\(71\) 4.66454 0.553580 0.276790 0.960930i \(-0.410729\pi\)
0.276790 + 0.960930i \(0.410729\pi\)
\(72\) 0 0
\(73\) 7.16034 0.838054 0.419027 0.907974i \(-0.362371\pi\)
0.419027 + 0.907974i \(0.362371\pi\)
\(74\) 2.18551 0.254060
\(75\) 0 0
\(76\) −0.132927 −0.0152478
\(77\) 0 0
\(78\) 0 0
\(79\) −0.791342 −0.0890330 −0.0445165 0.999009i \(-0.514175\pi\)
−0.0445165 + 0.999009i \(0.514175\pi\)
\(80\) −3.59174 −0.401569
\(81\) 0 0
\(82\) 3.06060 0.337987
\(83\) 0.247229 0.0271369 0.0135685 0.999908i \(-0.495681\pi\)
0.0135685 + 0.999908i \(0.495681\pi\)
\(84\) 0 0
\(85\) 0.0882264 0.00956950
\(86\) −3.10155 −0.334448
\(87\) 0 0
\(88\) 0 0
\(89\) 14.5788 1.54535 0.772673 0.634804i \(-0.218919\pi\)
0.772673 + 0.634804i \(0.218919\pi\)
\(90\) 0 0
\(91\) 3.57549 0.374814
\(92\) 12.8510 1.33980
\(93\) 0 0
\(94\) 0.238228 0.0245713
\(95\) −0.0688326 −0.00706208
\(96\) 0 0
\(97\) −4.09198 −0.415478 −0.207739 0.978184i \(-0.566610\pi\)
−0.207739 + 0.978184i \(0.566610\pi\)
\(98\) −0.839188 −0.0847708
\(99\) 0 0
\(100\) −1.93117 −0.193117
\(101\) 4.29171 0.427041 0.213521 0.976939i \(-0.431507\pi\)
0.213521 + 0.976939i \(0.431507\pi\)
\(102\) 0 0
\(103\) 2.09280 0.206210 0.103105 0.994670i \(-0.467122\pi\)
0.103105 + 0.994670i \(0.467122\pi\)
\(104\) 1.15474 0.113232
\(105\) 0 0
\(106\) 0.228877 0.0222305
\(107\) −15.7409 −1.52173 −0.760866 0.648909i \(-0.775226\pi\)
−0.760866 + 0.648909i \(0.775226\pi\)
\(108\) 0 0
\(109\) 4.13271 0.395842 0.197921 0.980218i \(-0.436581\pi\)
0.197921 + 0.980218i \(0.436581\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 11.4703 1.08384
\(113\) −13.7550 −1.29396 −0.646981 0.762506i \(-0.723969\pi\)
−0.646981 + 0.762506i \(0.723969\pi\)
\(114\) 0 0
\(115\) 6.65450 0.620536
\(116\) −7.21451 −0.669851
\(117\) 0 0
\(118\) 0.481704 0.0443444
\(119\) −0.281754 −0.0258283
\(120\) 0 0
\(121\) 0 0
\(122\) −2.63937 −0.238957
\(123\) 0 0
\(124\) 18.5099 1.66224
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.89979 −0.700993 −0.350496 0.936564i \(-0.613987\pi\)
−0.350496 + 0.936564i \(0.613987\pi\)
\(128\) 7.68799 0.679528
\(129\) 0 0
\(130\) 0.293740 0.0257627
\(131\) −16.3539 −1.42884 −0.714422 0.699715i \(-0.753310\pi\)
−0.714422 + 0.699715i \(0.753310\pi\)
\(132\) 0 0
\(133\) 0.219819 0.0190607
\(134\) −2.50269 −0.216200
\(135\) 0 0
\(136\) −0.0909950 −0.00780275
\(137\) −7.73208 −0.660596 −0.330298 0.943877i \(-0.607149\pi\)
−0.330298 + 0.943877i \(0.607149\pi\)
\(138\) 0 0
\(139\) −20.1865 −1.71219 −0.856097 0.516815i \(-0.827117\pi\)
−0.856097 + 0.516815i \(0.827117\pi\)
\(140\) 6.16724 0.521227
\(141\) 0 0
\(142\) −1.22379 −0.102698
\(143\) 0 0
\(144\) 0 0
\(145\) −3.73583 −0.310244
\(146\) −1.87858 −0.155473
\(147\) 0 0
\(148\) 16.0870 1.32235
\(149\) 9.69523 0.794264 0.397132 0.917761i \(-0.370005\pi\)
0.397132 + 0.917761i \(0.370005\pi\)
\(150\) 0 0
\(151\) −15.1615 −1.23382 −0.616911 0.787033i \(-0.711616\pi\)
−0.616911 + 0.787033i \(0.711616\pi\)
\(152\) 0.0709926 0.00575826
\(153\) 0 0
\(154\) 0 0
\(155\) 9.58484 0.769873
\(156\) 0 0
\(157\) −14.8805 −1.18760 −0.593798 0.804614i \(-0.702372\pi\)
−0.593798 + 0.804614i \(0.702372\pi\)
\(158\) 0.207616 0.0165171
\(159\) 0 0
\(160\) 3.00509 0.237573
\(161\) −21.2513 −1.67484
\(162\) 0 0
\(163\) 3.40114 0.266398 0.133199 0.991089i \(-0.457475\pi\)
0.133199 + 0.991089i \(0.457475\pi\)
\(164\) 22.5283 1.75917
\(165\) 0 0
\(166\) −0.0648629 −0.00503434
\(167\) −22.6164 −1.75011 −0.875053 0.484027i \(-0.839174\pi\)
−0.875053 + 0.484027i \(0.839174\pi\)
\(168\) 0 0
\(169\) −11.7465 −0.903576
\(170\) −0.0231471 −0.00177530
\(171\) 0 0
\(172\) −22.8298 −1.74075
\(173\) −13.0913 −0.995312 −0.497656 0.867374i \(-0.665806\pi\)
−0.497656 + 0.867374i \(0.665806\pi\)
\(174\) 0 0
\(175\) 3.19353 0.241408
\(176\) 0 0
\(177\) 0 0
\(178\) −3.82488 −0.286687
\(179\) 15.1774 1.13441 0.567207 0.823576i \(-0.308024\pi\)
0.567207 + 0.823576i \(0.308024\pi\)
\(180\) 0 0
\(181\) 1.41348 0.105063 0.0525316 0.998619i \(-0.483271\pi\)
0.0525316 + 0.998619i \(0.483271\pi\)
\(182\) −0.938065 −0.0695340
\(183\) 0 0
\(184\) −6.86332 −0.505971
\(185\) 8.33021 0.612449
\(186\) 0 0
\(187\) 0 0
\(188\) 1.75354 0.127890
\(189\) 0 0
\(190\) 0.0180589 0.00131013
\(191\) −4.90914 −0.355213 −0.177606 0.984102i \(-0.556835\pi\)
−0.177606 + 0.984102i \(0.556835\pi\)
\(192\) 0 0
\(193\) −9.71987 −0.699652 −0.349826 0.936815i \(-0.613759\pi\)
−0.349826 + 0.936815i \(0.613759\pi\)
\(194\) 1.07357 0.0770779
\(195\) 0 0
\(196\) −6.17706 −0.441219
\(197\) −8.88764 −0.633218 −0.316609 0.948556i \(-0.602544\pi\)
−0.316609 + 0.948556i \(0.602544\pi\)
\(198\) 0 0
\(199\) 18.0381 1.27869 0.639343 0.768921i \(-0.279206\pi\)
0.639343 + 0.768921i \(0.279206\pi\)
\(200\) 1.03138 0.0729296
\(201\) 0 0
\(202\) −1.12597 −0.0792232
\(203\) 11.9305 0.837355
\(204\) 0 0
\(205\) 11.6657 0.814765
\(206\) −0.549068 −0.0382554
\(207\) 0 0
\(208\) 4.02134 0.278830
\(209\) 0 0
\(210\) 0 0
\(211\) −17.9778 −1.23764 −0.618822 0.785531i \(-0.712390\pi\)
−0.618822 + 0.785531i \(0.712390\pi\)
\(212\) 1.68471 0.115706
\(213\) 0 0
\(214\) 4.12978 0.282306
\(215\) −11.8217 −0.806236
\(216\) 0 0
\(217\) −30.6095 −2.07791
\(218\) −1.08426 −0.0734351
\(219\) 0 0
\(220\) 0 0
\(221\) −0.0987789 −0.00664459
\(222\) 0 0
\(223\) −13.7489 −0.920697 −0.460348 0.887738i \(-0.652276\pi\)
−0.460348 + 0.887738i \(0.652276\pi\)
\(224\) −9.59683 −0.641215
\(225\) 0 0
\(226\) 3.60876 0.240051
\(227\) 0.662735 0.0439873 0.0219936 0.999758i \(-0.492999\pi\)
0.0219936 + 0.999758i \(0.492999\pi\)
\(228\) 0 0
\(229\) −7.49041 −0.494980 −0.247490 0.968890i \(-0.579606\pi\)
−0.247490 + 0.968890i \(0.579606\pi\)
\(230\) −1.74587 −0.115119
\(231\) 0 0
\(232\) 3.85306 0.252966
\(233\) 10.5254 0.689539 0.344770 0.938687i \(-0.387957\pi\)
0.344770 + 0.938687i \(0.387957\pi\)
\(234\) 0 0
\(235\) 0.908020 0.0592327
\(236\) 3.54571 0.230806
\(237\) 0 0
\(238\) 0.0739208 0.00479157
\(239\) 17.0762 1.10457 0.552284 0.833656i \(-0.313756\pi\)
0.552284 + 0.833656i \(0.313756\pi\)
\(240\) 0 0
\(241\) 16.7082 1.07627 0.538135 0.842859i \(-0.319129\pi\)
0.538135 + 0.842859i \(0.319129\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −19.4278 −1.24374
\(245\) −3.19862 −0.204352
\(246\) 0 0
\(247\) 0.0770654 0.00490356
\(248\) −9.88562 −0.627737
\(249\) 0 0
\(250\) 0.262360 0.0165931
\(251\) 5.13767 0.324287 0.162143 0.986767i \(-0.448159\pi\)
0.162143 + 0.986767i \(0.448159\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.07259 0.130046
\(255\) 0 0
\(256\) 10.7731 0.673320
\(257\) 3.69824 0.230690 0.115345 0.993325i \(-0.463203\pi\)
0.115345 + 0.993325i \(0.463203\pi\)
\(258\) 0 0
\(259\) −26.6027 −1.65301
\(260\) 2.16215 0.134091
\(261\) 0 0
\(262\) 4.29059 0.265074
\(263\) 31.6315 1.95048 0.975241 0.221147i \(-0.0709799\pi\)
0.975241 + 0.221147i \(0.0709799\pi\)
\(264\) 0 0
\(265\) 0.872377 0.0535897
\(266\) −0.0576716 −0.00353607
\(267\) 0 0
\(268\) −18.4217 −1.12529
\(269\) 14.7754 0.900875 0.450437 0.892808i \(-0.351268\pi\)
0.450437 + 0.892808i \(0.351268\pi\)
\(270\) 0 0
\(271\) 19.5869 1.18982 0.594910 0.803792i \(-0.297188\pi\)
0.594910 + 0.803792i \(0.297188\pi\)
\(272\) −0.316887 −0.0192141
\(273\) 0 0
\(274\) 2.02859 0.122551
\(275\) 0 0
\(276\) 0 0
\(277\) −25.7104 −1.54479 −0.772393 0.635145i \(-0.780940\pi\)
−0.772393 + 0.635145i \(0.780940\pi\)
\(278\) 5.29612 0.317640
\(279\) 0 0
\(280\) −3.29374 −0.196839
\(281\) −26.7109 −1.59344 −0.796719 0.604350i \(-0.793433\pi\)
−0.796719 + 0.604350i \(0.793433\pi\)
\(282\) 0 0
\(283\) −7.23375 −0.430002 −0.215001 0.976614i \(-0.568975\pi\)
−0.215001 + 0.976614i \(0.568975\pi\)
\(284\) −9.00802 −0.534527
\(285\) 0 0
\(286\) 0 0
\(287\) −37.2546 −2.19907
\(288\) 0 0
\(289\) −16.9922 −0.999542
\(290\) 0.980131 0.0575553
\(291\) 0 0
\(292\) −13.8278 −0.809211
\(293\) 14.5947 0.852633 0.426317 0.904574i \(-0.359811\pi\)
0.426317 + 0.904574i \(0.359811\pi\)
\(294\) 0 0
\(295\) 1.83604 0.106899
\(296\) −8.59161 −0.499377
\(297\) 0 0
\(298\) −2.54364 −0.147349
\(299\) −7.45042 −0.430869
\(300\) 0 0
\(301\) 37.7530 2.17605
\(302\) 3.97775 0.228894
\(303\) 0 0
\(304\) 0.247229 0.0141796
\(305\) −10.0601 −0.576041
\(306\) 0 0
\(307\) −28.5445 −1.62912 −0.814559 0.580080i \(-0.803021\pi\)
−0.814559 + 0.580080i \(0.803021\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.51468 −0.142824
\(311\) 9.94803 0.564101 0.282050 0.959400i \(-0.408985\pi\)
0.282050 + 0.959400i \(0.408985\pi\)
\(312\) 0 0
\(313\) −16.4309 −0.928731 −0.464366 0.885644i \(-0.653718\pi\)
−0.464366 + 0.885644i \(0.653718\pi\)
\(314\) 3.90405 0.220318
\(315\) 0 0
\(316\) 1.52821 0.0859688
\(317\) −14.2106 −0.798147 −0.399073 0.916919i \(-0.630668\pi\)
−0.399073 + 0.916919i \(0.630668\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 6.39507 0.357495
\(321\) 0 0
\(322\) 5.57549 0.310710
\(323\) −0.00607286 −0.000337903 0
\(324\) 0 0
\(325\) 1.11961 0.0621046
\(326\) −0.892323 −0.0494212
\(327\) 0 0
\(328\) −12.0317 −0.664341
\(329\) −2.89979 −0.159870
\(330\) 0 0
\(331\) −10.9119 −0.599773 −0.299886 0.953975i \(-0.596949\pi\)
−0.299886 + 0.953975i \(0.596949\pi\)
\(332\) −0.477441 −0.0262030
\(333\) 0 0
\(334\) 5.93362 0.324673
\(335\) −9.53916 −0.521180
\(336\) 0 0
\(337\) 5.88609 0.320636 0.160318 0.987065i \(-0.448748\pi\)
0.160318 + 0.987065i \(0.448748\pi\)
\(338\) 3.08180 0.167628
\(339\) 0 0
\(340\) −0.170380 −0.00924015
\(341\) 0 0
\(342\) 0 0
\(343\) −12.1398 −0.655489
\(344\) 12.1927 0.657386
\(345\) 0 0
\(346\) 3.43463 0.184647
\(347\) 34.5043 1.85229 0.926143 0.377173i \(-0.123104\pi\)
0.926143 + 0.377173i \(0.123104\pi\)
\(348\) 0 0
\(349\) 5.51356 0.295134 0.147567 0.989052i \(-0.452856\pi\)
0.147567 + 0.989052i \(0.452856\pi\)
\(350\) −0.837853 −0.0447851
\(351\) 0 0
\(352\) 0 0
\(353\) 29.8740 1.59003 0.795016 0.606589i \(-0.207463\pi\)
0.795016 + 0.606589i \(0.207463\pi\)
\(354\) 0 0
\(355\) −4.66454 −0.247568
\(356\) −28.1540 −1.49216
\(357\) 0 0
\(358\) −3.98194 −0.210452
\(359\) −35.9131 −1.89542 −0.947710 0.319133i \(-0.896608\pi\)
−0.947710 + 0.319133i \(0.896608\pi\)
\(360\) 0 0
\(361\) −18.9953 −0.999751
\(362\) −0.370840 −0.0194909
\(363\) 0 0
\(364\) −6.90488 −0.361914
\(365\) −7.16034 −0.374789
\(366\) 0 0
\(367\) −9.03087 −0.471408 −0.235704 0.971825i \(-0.575740\pi\)
−0.235704 + 0.971825i \(0.575740\pi\)
\(368\) −23.9013 −1.24594
\(369\) 0 0
\(370\) −2.18551 −0.113619
\(371\) −2.78596 −0.144640
\(372\) 0 0
\(373\) −17.1994 −0.890553 −0.445277 0.895393i \(-0.646895\pi\)
−0.445277 + 0.895393i \(0.646895\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −0.936514 −0.0482970
\(377\) 4.18266 0.215418
\(378\) 0 0
\(379\) −8.00208 −0.411039 −0.205520 0.978653i \(-0.565888\pi\)
−0.205520 + 0.978653i \(0.565888\pi\)
\(380\) 0.132927 0.00681903
\(381\) 0 0
\(382\) 1.28796 0.0658978
\(383\) −29.7131 −1.51827 −0.759135 0.650933i \(-0.774378\pi\)
−0.759135 + 0.650933i \(0.774378\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.55010 0.129797
\(387\) 0 0
\(388\) 7.90230 0.401178
\(389\) 0.595397 0.0301878 0.0150939 0.999886i \(-0.495195\pi\)
0.0150939 + 0.999886i \(0.495195\pi\)
\(390\) 0 0
\(391\) 0.587103 0.0296911
\(392\) 3.29899 0.166624
\(393\) 0 0
\(394\) 2.33176 0.117472
\(395\) 0.791342 0.0398167
\(396\) 0 0
\(397\) 1.66950 0.0837898 0.0418949 0.999122i \(-0.486661\pi\)
0.0418949 + 0.999122i \(0.486661\pi\)
\(398\) −4.73247 −0.237217
\(399\) 0 0
\(400\) 3.59174 0.179587
\(401\) 11.3690 0.567741 0.283870 0.958863i \(-0.408381\pi\)
0.283870 + 0.958863i \(0.408381\pi\)
\(402\) 0 0
\(403\) −10.7313 −0.534562
\(404\) −8.28802 −0.412344
\(405\) 0 0
\(406\) −3.13008 −0.155343
\(407\) 0 0
\(408\) 0 0
\(409\) 8.59766 0.425127 0.212563 0.977147i \(-0.431819\pi\)
0.212563 + 0.977147i \(0.431819\pi\)
\(410\) −3.06060 −0.151152
\(411\) 0 0
\(412\) −4.04156 −0.199113
\(413\) −5.86345 −0.288522
\(414\) 0 0
\(415\) −0.247229 −0.0121360
\(416\) −3.36452 −0.164959
\(417\) 0 0
\(418\) 0 0
\(419\) −31.3915 −1.53358 −0.766789 0.641899i \(-0.778147\pi\)
−0.766789 + 0.641899i \(0.778147\pi\)
\(420\) 0 0
\(421\) −4.57893 −0.223163 −0.111582 0.993755i \(-0.535592\pi\)
−0.111582 + 0.993755i \(0.535592\pi\)
\(422\) 4.71665 0.229603
\(423\) 0 0
\(424\) −0.899752 −0.0436958
\(425\) −0.0882264 −0.00427961
\(426\) 0 0
\(427\) 32.1273 1.55475
\(428\) 30.3983 1.46936
\(429\) 0 0
\(430\) 3.10155 0.149570
\(431\) 15.9074 0.766234 0.383117 0.923700i \(-0.374851\pi\)
0.383117 + 0.923700i \(0.374851\pi\)
\(432\) 0 0
\(433\) 15.8030 0.759441 0.379721 0.925101i \(-0.376020\pi\)
0.379721 + 0.925101i \(0.376020\pi\)
\(434\) 8.03069 0.385485
\(435\) 0 0
\(436\) −7.98096 −0.382218
\(437\) −0.458047 −0.0219113
\(438\) 0 0
\(439\) −21.5227 −1.02722 −0.513611 0.858023i \(-0.671693\pi\)
−0.513611 + 0.858023i \(0.671693\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.0259156 0.00123268
\(443\) −10.1769 −0.483519 −0.241760 0.970336i \(-0.577725\pi\)
−0.241760 + 0.970336i \(0.577725\pi\)
\(444\) 0 0
\(445\) −14.5788 −0.691100
\(446\) 3.60717 0.170804
\(447\) 0 0
\(448\) −20.4228 −0.964888
\(449\) −38.6088 −1.82206 −0.911031 0.412338i \(-0.864712\pi\)
−0.911031 + 0.412338i \(0.864712\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 26.5632 1.24943
\(453\) 0 0
\(454\) −0.173875 −0.00816035
\(455\) −3.57549 −0.167622
\(456\) 0 0
\(457\) −23.3637 −1.09291 −0.546455 0.837488i \(-0.684023\pi\)
−0.546455 + 0.837488i \(0.684023\pi\)
\(458\) 1.96518 0.0918269
\(459\) 0 0
\(460\) −12.8510 −0.599179
\(461\) 4.93120 0.229669 0.114834 0.993385i \(-0.463366\pi\)
0.114834 + 0.993385i \(0.463366\pi\)
\(462\) 0 0
\(463\) 26.9648 1.25316 0.626580 0.779357i \(-0.284454\pi\)
0.626580 + 0.779357i \(0.284454\pi\)
\(464\) 13.4181 0.622922
\(465\) 0 0
\(466\) −2.76143 −0.127921
\(467\) 23.6778 1.09568 0.547839 0.836584i \(-0.315450\pi\)
0.547839 + 0.836584i \(0.315450\pi\)
\(468\) 0 0
\(469\) 30.4636 1.40668
\(470\) −0.238228 −0.0109886
\(471\) 0 0
\(472\) −1.89366 −0.0871627
\(473\) 0 0
\(474\) 0 0
\(475\) 0.0688326 0.00315826
\(476\) 0.544113 0.0249394
\(477\) 0 0
\(478\) −4.48011 −0.204915
\(479\) −13.5240 −0.617929 −0.308964 0.951074i \(-0.599982\pi\)
−0.308964 + 0.951074i \(0.599982\pi\)
\(480\) 0 0
\(481\) −9.32655 −0.425254
\(482\) −4.38356 −0.199666
\(483\) 0 0
\(484\) 0 0
\(485\) 4.09198 0.185807
\(486\) 0 0
\(487\) −8.62704 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(488\) 10.3758 0.469691
\(489\) 0 0
\(490\) 0.839188 0.0379106
\(491\) −25.9068 −1.16916 −0.584579 0.811337i \(-0.698740\pi\)
−0.584579 + 0.811337i \(0.698740\pi\)
\(492\) 0 0
\(493\) −0.329599 −0.0148444
\(494\) −0.0202189 −0.000909690 0
\(495\) 0 0
\(496\) −34.4263 −1.54579
\(497\) 14.8963 0.668193
\(498\) 0 0
\(499\) −28.0475 −1.25558 −0.627790 0.778383i \(-0.716040\pi\)
−0.627790 + 0.778383i \(0.716040\pi\)
\(500\) 1.93117 0.0863644
\(501\) 0 0
\(502\) −1.34792 −0.0601604
\(503\) 29.3999 1.31088 0.655438 0.755249i \(-0.272484\pi\)
0.655438 + 0.755249i \(0.272484\pi\)
\(504\) 0 0
\(505\) −4.29171 −0.190979
\(506\) 0 0
\(507\) 0 0
\(508\) 15.2558 0.676867
\(509\) 19.3904 0.859463 0.429731 0.902957i \(-0.358608\pi\)
0.429731 + 0.902957i \(0.358608\pi\)
\(510\) 0 0
\(511\) 22.8667 1.01156
\(512\) −18.2024 −0.804440
\(513\) 0 0
\(514\) −0.970270 −0.0427968
\(515\) −2.09280 −0.0922200
\(516\) 0 0
\(517\) 0 0
\(518\) 6.97949 0.306661
\(519\) 0 0
\(520\) −1.15474 −0.0506387
\(521\) −25.0062 −1.09554 −0.547771 0.836628i \(-0.684523\pi\)
−0.547771 + 0.836628i \(0.684523\pi\)
\(522\) 0 0
\(523\) 2.18761 0.0956577 0.0478288 0.998856i \(-0.484770\pi\)
0.0478288 + 0.998856i \(0.484770\pi\)
\(524\) 31.5820 1.37967
\(525\) 0 0
\(526\) −8.29883 −0.361846
\(527\) 0.845637 0.0368365
\(528\) 0 0
\(529\) 21.2824 0.925322
\(530\) −0.228877 −0.00994177
\(531\) 0 0
\(532\) −0.424507 −0.0184047
\(533\) −13.0610 −0.565733
\(534\) 0 0
\(535\) 15.7409 0.680539
\(536\) 9.83850 0.424958
\(537\) 0 0
\(538\) −3.87648 −0.167127
\(539\) 0 0
\(540\) 0 0
\(541\) −13.5583 −0.582918 −0.291459 0.956583i \(-0.594141\pi\)
−0.291459 + 0.956583i \(0.594141\pi\)
\(542\) −5.13882 −0.220731
\(543\) 0 0
\(544\) 0.265128 0.0113673
\(545\) −4.13271 −0.177026
\(546\) 0 0
\(547\) 3.71282 0.158749 0.0793743 0.996845i \(-0.474708\pi\)
0.0793743 + 0.996845i \(0.474708\pi\)
\(548\) 14.9319 0.637861
\(549\) 0 0
\(550\) 0 0
\(551\) 0.257147 0.0109548
\(552\) 0 0
\(553\) −2.52717 −0.107466
\(554\) 6.74536 0.286583
\(555\) 0 0
\(556\) 38.9835 1.65327
\(557\) 6.57520 0.278600 0.139300 0.990250i \(-0.455515\pi\)
0.139300 + 0.990250i \(0.455515\pi\)
\(558\) 0 0
\(559\) 13.2357 0.559810
\(560\) −11.4703 −0.484710
\(561\) 0 0
\(562\) 7.00786 0.295609
\(563\) −6.90876 −0.291170 −0.145585 0.989346i \(-0.546506\pi\)
−0.145585 + 0.989346i \(0.546506\pi\)
\(564\) 0 0
\(565\) 13.7550 0.578678
\(566\) 1.89784 0.0797723
\(567\) 0 0
\(568\) 4.81092 0.201862
\(569\) −6.74768 −0.282878 −0.141439 0.989947i \(-0.545173\pi\)
−0.141439 + 0.989947i \(0.545173\pi\)
\(570\) 0 0
\(571\) −9.68928 −0.405484 −0.202742 0.979232i \(-0.564985\pi\)
−0.202742 + 0.979232i \(0.564985\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.77411 0.407963
\(575\) −6.65450 −0.277512
\(576\) 0 0
\(577\) 15.1498 0.630695 0.315348 0.948976i \(-0.397879\pi\)
0.315348 + 0.948976i \(0.397879\pi\)
\(578\) 4.45807 0.185431
\(579\) 0 0
\(580\) 7.21451 0.299566
\(581\) 0.789532 0.0327553
\(582\) 0 0
\(583\) 0 0
\(584\) 7.38503 0.305595
\(585\) 0 0
\(586\) −3.82907 −0.158177
\(587\) 27.0112 1.11487 0.557436 0.830220i \(-0.311785\pi\)
0.557436 + 0.830220i \(0.311785\pi\)
\(588\) 0 0
\(589\) −0.659750 −0.0271845
\(590\) −0.481704 −0.0198314
\(591\) 0 0
\(592\) −29.9200 −1.22970
\(593\) −22.1863 −0.911084 −0.455542 0.890214i \(-0.650554\pi\)
−0.455542 + 0.890214i \(0.650554\pi\)
\(594\) 0 0
\(595\) 0.281754 0.0115508
\(596\) −18.7231 −0.766929
\(597\) 0 0
\(598\) 1.95469 0.0799332
\(599\) −11.7487 −0.480038 −0.240019 0.970768i \(-0.577154\pi\)
−0.240019 + 0.970768i \(0.577154\pi\)
\(600\) 0 0
\(601\) −34.3806 −1.40242 −0.701208 0.712957i \(-0.747355\pi\)
−0.701208 + 0.712957i \(0.747355\pi\)
\(602\) −9.90488 −0.403693
\(603\) 0 0
\(604\) 29.2793 1.19136
\(605\) 0 0
\(606\) 0 0
\(607\) −13.6298 −0.553215 −0.276607 0.960983i \(-0.589210\pi\)
−0.276607 + 0.960983i \(0.589210\pi\)
\(608\) −0.206848 −0.00838880
\(609\) 0 0
\(610\) 2.63937 0.106865
\(611\) −1.01663 −0.0411283
\(612\) 0 0
\(613\) 6.82994 0.275858 0.137929 0.990442i \(-0.455955\pi\)
0.137929 + 0.990442i \(0.455955\pi\)
\(614\) 7.48892 0.302228
\(615\) 0 0
\(616\) 0 0
\(617\) −30.3730 −1.22277 −0.611386 0.791333i \(-0.709388\pi\)
−0.611386 + 0.791333i \(0.709388\pi\)
\(618\) 0 0
\(619\) −25.7879 −1.03650 −0.518252 0.855228i \(-0.673417\pi\)
−0.518252 + 0.855228i \(0.673417\pi\)
\(620\) −18.5099 −0.743377
\(621\) 0 0
\(622\) −2.60996 −0.104650
\(623\) 46.5577 1.86529
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 4.31081 0.172295
\(627\) 0 0
\(628\) 28.7368 1.14672
\(629\) 0.734945 0.0293042
\(630\) 0 0
\(631\) −25.5656 −1.01775 −0.508876 0.860840i \(-0.669939\pi\)
−0.508876 + 0.860840i \(0.669939\pi\)
\(632\) −0.816174 −0.0324657
\(633\) 0 0
\(634\) 3.72829 0.148069
\(635\) 7.89979 0.313494
\(636\) 0 0
\(637\) 3.58119 0.141892
\(638\) 0 0
\(639\) 0 0
\(640\) −7.68799 −0.303894
\(641\) 4.07262 0.160859 0.0804294 0.996760i \(-0.474371\pi\)
0.0804294 + 0.996760i \(0.474371\pi\)
\(642\) 0 0
\(643\) 11.3298 0.446804 0.223402 0.974726i \(-0.428284\pi\)
0.223402 + 0.974726i \(0.428284\pi\)
\(644\) 41.0399 1.61720
\(645\) 0 0
\(646\) 0.00159327 6.26865e−5 0
\(647\) 36.8480 1.44864 0.724322 0.689462i \(-0.242153\pi\)
0.724322 + 0.689462i \(0.242153\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.293740 −0.0115214
\(651\) 0 0
\(652\) −6.56818 −0.257230
\(653\) −15.2768 −0.597827 −0.298914 0.954280i \(-0.596624\pi\)
−0.298914 + 0.954280i \(0.596624\pi\)
\(654\) 0 0
\(655\) 16.3539 0.638998
\(656\) −41.9001 −1.63592
\(657\) 0 0
\(658\) 0.760787 0.0296586
\(659\) 11.8996 0.463544 0.231772 0.972770i \(-0.425548\pi\)
0.231772 + 0.972770i \(0.425548\pi\)
\(660\) 0 0
\(661\) −44.8648 −1.74504 −0.872520 0.488578i \(-0.837516\pi\)
−0.872520 + 0.488578i \(0.837516\pi\)
\(662\) 2.86284 0.111268
\(663\) 0 0
\(664\) 0.254987 0.00989542
\(665\) −0.219819 −0.00852421
\(666\) 0 0
\(667\) −24.8601 −0.962587
\(668\) 43.6760 1.68987
\(669\) 0 0
\(670\) 2.50269 0.0966874
\(671\) 0 0
\(672\) 0 0
\(673\) 19.1067 0.736508 0.368254 0.929725i \(-0.379956\pi\)
0.368254 + 0.929725i \(0.379956\pi\)
\(674\) −1.54427 −0.0594832
\(675\) 0 0
\(676\) 22.6844 0.872478
\(677\) 46.4614 1.78566 0.892829 0.450396i \(-0.148717\pi\)
0.892829 + 0.450396i \(0.148717\pi\)
\(678\) 0 0
\(679\) −13.0678 −0.501498
\(680\) 0.0909950 0.00348950
\(681\) 0 0
\(682\) 0 0
\(683\) 42.5651 1.62871 0.814355 0.580367i \(-0.197091\pi\)
0.814355 + 0.580367i \(0.197091\pi\)
\(684\) 0 0
\(685\) 7.73208 0.295427
\(686\) 3.18500 0.121604
\(687\) 0 0
\(688\) 42.4606 1.61880
\(689\) −0.976719 −0.0372100
\(690\) 0 0
\(691\) 35.1352 1.33660 0.668302 0.743890i \(-0.267021\pi\)
0.668302 + 0.743890i \(0.267021\pi\)
\(692\) 25.2815 0.961057
\(693\) 0 0
\(694\) −9.05253 −0.343629
\(695\) 20.1865 0.765716
\(696\) 0 0
\(697\) 1.02922 0.0389845
\(698\) −1.44654 −0.0547522
\(699\) 0 0
\(700\) −6.16724 −0.233100
\(701\) −30.6713 −1.15844 −0.579219 0.815172i \(-0.696642\pi\)
−0.579219 + 0.815172i \(0.696642\pi\)
\(702\) 0 0
\(703\) −0.573390 −0.0216258
\(704\) 0 0
\(705\) 0 0
\(706\) −7.83773 −0.294977
\(707\) 13.7057 0.515456
\(708\) 0 0
\(709\) 15.0924 0.566806 0.283403 0.959001i \(-0.408537\pi\)
0.283403 + 0.959001i \(0.408537\pi\)
\(710\) 1.22379 0.0459280
\(711\) 0 0
\(712\) 15.0363 0.563507
\(713\) 63.7824 2.38867
\(714\) 0 0
\(715\) 0 0
\(716\) −29.3101 −1.09537
\(717\) 0 0
\(718\) 9.42215 0.351631
\(719\) 6.19020 0.230856 0.115428 0.993316i \(-0.463176\pi\)
0.115428 + 0.993316i \(0.463176\pi\)
\(720\) 0 0
\(721\) 6.68343 0.248904
\(722\) 4.98359 0.185470
\(723\) 0 0
\(724\) −2.72967 −0.101447
\(725\) 3.73583 0.138745
\(726\) 0 0
\(727\) −18.1515 −0.673200 −0.336600 0.941648i \(-0.609277\pi\)
−0.336600 + 0.941648i \(0.609277\pi\)
\(728\) 3.68769 0.136675
\(729\) 0 0
\(730\) 1.87858 0.0695295
\(731\) −1.04299 −0.0385764
\(732\) 0 0
\(733\) 28.7409 1.06157 0.530785 0.847507i \(-0.321897\pi\)
0.530785 + 0.847507i \(0.321897\pi\)
\(734\) 2.36934 0.0874538
\(735\) 0 0
\(736\) 19.9974 0.737113
\(737\) 0 0
\(738\) 0 0
\(739\) 46.8341 1.72282 0.861411 0.507909i \(-0.169581\pi\)
0.861411 + 0.507909i \(0.169581\pi\)
\(740\) −16.0870 −0.591371
\(741\) 0 0
\(742\) 0.730924 0.0268331
\(743\) 26.6108 0.976254 0.488127 0.872773i \(-0.337680\pi\)
0.488127 + 0.872773i \(0.337680\pi\)
\(744\) 0 0
\(745\) −9.69523 −0.355206
\(746\) 4.51244 0.165212
\(747\) 0 0
\(748\) 0 0
\(749\) −50.2691 −1.83679
\(750\) 0 0
\(751\) 40.5821 1.48086 0.740431 0.672132i \(-0.234621\pi\)
0.740431 + 0.672132i \(0.234621\pi\)
\(752\) −3.26137 −0.118930
\(753\) 0 0
\(754\) −1.09736 −0.0399635
\(755\) 15.1615 0.551782
\(756\) 0 0
\(757\) 32.7179 1.18915 0.594576 0.804039i \(-0.297320\pi\)
0.594576 + 0.804039i \(0.297320\pi\)
\(758\) 2.09942 0.0762545
\(759\) 0 0
\(760\) −0.0709926 −0.00257517
\(761\) −2.62938 −0.0953148 −0.0476574 0.998864i \(-0.515176\pi\)
−0.0476574 + 0.998864i \(0.515176\pi\)
\(762\) 0 0
\(763\) 13.1979 0.477797
\(764\) 9.48037 0.342988
\(765\) 0 0
\(766\) 7.79553 0.281664
\(767\) −2.05565 −0.0742251
\(768\) 0 0
\(769\) −3.23559 −0.116678 −0.0583392 0.998297i \(-0.518581\pi\)
−0.0583392 + 0.998297i \(0.518581\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.7707 0.675572
\(773\) −43.6365 −1.56950 −0.784748 0.619815i \(-0.787208\pi\)
−0.784748 + 0.619815i \(0.787208\pi\)
\(774\) 0 0
\(775\) −9.58484 −0.344298
\(776\) −4.22039 −0.151503
\(777\) 0 0
\(778\) −0.156208 −0.00560033
\(779\) −0.802978 −0.0287697
\(780\) 0 0
\(781\) 0 0
\(782\) −0.154032 −0.00550818
\(783\) 0 0
\(784\) 11.4886 0.410307
\(785\) 14.8805 0.531109
\(786\) 0 0
\(787\) 4.56420 0.162696 0.0813480 0.996686i \(-0.474077\pi\)
0.0813480 + 0.996686i \(0.474077\pi\)
\(788\) 17.1635 0.611425
\(789\) 0 0
\(790\) −0.207616 −0.00738666
\(791\) −43.9270 −1.56186
\(792\) 0 0
\(793\) 11.2634 0.399974
\(794\) −0.438009 −0.0155444
\(795\) 0 0
\(796\) −34.8346 −1.23468
\(797\) 11.1806 0.396036 0.198018 0.980198i \(-0.436550\pi\)
0.198018 + 0.980198i \(0.436550\pi\)
\(798\) 0 0
\(799\) 0.0801114 0.00283414
\(800\) −3.00509 −0.106246
\(801\) 0 0
\(802\) −2.98277 −0.105325
\(803\) 0 0
\(804\) 0 0
\(805\) 21.2513 0.749011
\(806\) 2.81545 0.0991699
\(807\) 0 0
\(808\) 4.42639 0.155720
\(809\) −45.5417 −1.60116 −0.800581 0.599225i \(-0.795475\pi\)
−0.800581 + 0.599225i \(0.795475\pi\)
\(810\) 0 0
\(811\) −2.82363 −0.0991511 −0.0495755 0.998770i \(-0.515787\pi\)
−0.0495755 + 0.998770i \(0.515787\pi\)
\(812\) −23.0397 −0.808537
\(813\) 0 0
\(814\) 0 0
\(815\) −3.40114 −0.119137
\(816\) 0 0
\(817\) 0.813721 0.0284685
\(818\) −2.25568 −0.0788679
\(819\) 0 0
\(820\) −22.5283 −0.786724
\(821\) −23.4117 −0.817073 −0.408536 0.912742i \(-0.633961\pi\)
−0.408536 + 0.912742i \(0.633961\pi\)
\(822\) 0 0
\(823\) 46.1046 1.60711 0.803553 0.595233i \(-0.202940\pi\)
0.803553 + 0.595233i \(0.202940\pi\)
\(824\) 2.15848 0.0751941
\(825\) 0 0
\(826\) 1.53833 0.0535255
\(827\) −43.4496 −1.51089 −0.755445 0.655212i \(-0.772579\pi\)
−0.755445 + 0.655212i \(0.772579\pi\)
\(828\) 0 0
\(829\) 15.4712 0.537335 0.268668 0.963233i \(-0.413417\pi\)
0.268668 + 0.963233i \(0.413417\pi\)
\(830\) 0.0648629 0.00225142
\(831\) 0 0
\(832\) −7.15996 −0.248227
\(833\) −0.282202 −0.00977773
\(834\) 0 0
\(835\) 22.6164 0.782671
\(836\) 0 0
\(837\) 0 0
\(838\) 8.23588 0.284504
\(839\) −30.2181 −1.04324 −0.521622 0.853177i \(-0.674673\pi\)
−0.521622 + 0.853177i \(0.674673\pi\)
\(840\) 0 0
\(841\) −15.0436 −0.518744
\(842\) 1.20133 0.0414005
\(843\) 0 0
\(844\) 34.7182 1.19505
\(845\) 11.7465 0.404091
\(846\) 0 0
\(847\) 0 0
\(848\) −3.13335 −0.107600
\(849\) 0 0
\(850\) 0.0231471 0.000793938 0
\(851\) 55.4334 1.90023
\(852\) 0 0
\(853\) 2.00908 0.0687897 0.0343949 0.999408i \(-0.489050\pi\)
0.0343949 + 0.999408i \(0.489050\pi\)
\(854\) −8.42890 −0.288431
\(855\) 0 0
\(856\) −16.2349 −0.554896
\(857\) −33.6095 −1.14808 −0.574039 0.818828i \(-0.694624\pi\)
−0.574039 + 0.818828i \(0.694624\pi\)
\(858\) 0 0
\(859\) −13.7301 −0.468466 −0.234233 0.972180i \(-0.575258\pi\)
−0.234233 + 0.972180i \(0.575258\pi\)
\(860\) 22.8298 0.778488
\(861\) 0 0
\(862\) −4.17347 −0.142149
\(863\) −8.75829 −0.298136 −0.149068 0.988827i \(-0.547627\pi\)
−0.149068 + 0.988827i \(0.547627\pi\)
\(864\) 0 0
\(865\) 13.0913 0.445117
\(866\) −4.14606 −0.140889
\(867\) 0 0
\(868\) 59.1120 2.00639
\(869\) 0 0
\(870\) 0 0
\(871\) 10.6801 0.361881
\(872\) 4.26239 0.144343
\(873\) 0 0
\(874\) 0.120173 0.00406491
\(875\) −3.19353 −0.107961
\(876\) 0 0
\(877\) 15.2463 0.514830 0.257415 0.966301i \(-0.417129\pi\)
0.257415 + 0.966301i \(0.417129\pi\)
\(878\) 5.64669 0.190566
\(879\) 0 0
\(880\) 0 0
\(881\) 15.8486 0.533953 0.266977 0.963703i \(-0.413975\pi\)
0.266977 + 0.963703i \(0.413975\pi\)
\(882\) 0 0
\(883\) −32.9606 −1.10921 −0.554606 0.832113i \(-0.687131\pi\)
−0.554606 + 0.832113i \(0.687131\pi\)
\(884\) 0.190759 0.00641590
\(885\) 0 0
\(886\) 2.67001 0.0897007
\(887\) 11.2525 0.377820 0.188910 0.981994i \(-0.439505\pi\)
0.188910 + 0.981994i \(0.439505\pi\)
\(888\) 0 0
\(889\) −25.2282 −0.846126
\(890\) 3.82488 0.128210
\(891\) 0 0
\(892\) 26.5515 0.889010
\(893\) −0.0625014 −0.00209153
\(894\) 0 0
\(895\) −15.1774 −0.507325
\(896\) 24.5518 0.820218
\(897\) 0 0
\(898\) 10.1294 0.338022
\(899\) −35.8074 −1.19424
\(900\) 0 0
\(901\) 0.0769667 0.00256413
\(902\) 0 0
\(903\) 0 0
\(904\) −14.1866 −0.471841
\(905\) −1.41348 −0.0469857
\(906\) 0 0
\(907\) −25.3745 −0.842546 −0.421273 0.906934i \(-0.638417\pi\)
−0.421273 + 0.906934i \(0.638417\pi\)
\(908\) −1.27985 −0.0424734
\(909\) 0 0
\(910\) 0.938065 0.0310966
\(911\) 17.4070 0.576718 0.288359 0.957522i \(-0.406890\pi\)
0.288359 + 0.957522i \(0.406890\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.12971 0.202753
\(915\) 0 0
\(916\) 14.4652 0.477945
\(917\) −52.2265 −1.72467
\(918\) 0 0
\(919\) 32.8958 1.08513 0.542566 0.840013i \(-0.317453\pi\)
0.542566 + 0.840013i \(0.317453\pi\)
\(920\) 6.86332 0.226277
\(921\) 0 0
\(922\) −1.29375 −0.0426073
\(923\) 5.22245 0.171899
\(924\) 0 0
\(925\) −8.33021 −0.273896
\(926\) −7.07448 −0.232482
\(927\) 0 0
\(928\) −11.2265 −0.368528
\(929\) −33.3131 −1.09297 −0.546484 0.837469i \(-0.684034\pi\)
−0.546484 + 0.837469i \(0.684034\pi\)
\(930\) 0 0
\(931\) 0.220169 0.00721575
\(932\) −20.3262 −0.665808
\(933\) 0 0
\(934\) −6.21211 −0.203266
\(935\) 0 0
\(936\) 0 0
\(937\) 23.8637 0.779593 0.389797 0.920901i \(-0.372545\pi\)
0.389797 + 0.920901i \(0.372545\pi\)
\(938\) −7.99241 −0.260961
\(939\) 0 0
\(940\) −1.75354 −0.0571941
\(941\) −7.92557 −0.258366 −0.129183 0.991621i \(-0.541235\pi\)
−0.129183 + 0.991621i \(0.541235\pi\)
\(942\) 0 0
\(943\) 77.6292 2.52795
\(944\) −6.59459 −0.214636
\(945\) 0 0
\(946\) 0 0
\(947\) 2.15429 0.0700050 0.0350025 0.999387i \(-0.488856\pi\)
0.0350025 + 0.999387i \(0.488856\pi\)
\(948\) 0 0
\(949\) 8.01676 0.260235
\(950\) −0.0180589 −0.000585908 0
\(951\) 0 0
\(952\) −0.290595 −0.00941824
\(953\) −5.99303 −0.194133 −0.0970666 0.995278i \(-0.530946\pi\)
−0.0970666 + 0.995278i \(0.530946\pi\)
\(954\) 0 0
\(955\) 4.90914 0.158856
\(956\) −32.9770 −1.06655
\(957\) 0 0
\(958\) 3.54816 0.114636
\(959\) −24.6926 −0.797366
\(960\) 0 0
\(961\) 60.8692 1.96352
\(962\) 2.44691 0.0788916
\(963\) 0 0
\(964\) −32.2663 −1.03923
\(965\) 9.71987 0.312894
\(966\) 0 0
\(967\) −22.2784 −0.716424 −0.358212 0.933640i \(-0.616613\pi\)
−0.358212 + 0.933640i \(0.616613\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1.07357 −0.0344703
\(971\) 36.6909 1.17747 0.588734 0.808327i \(-0.299626\pi\)
0.588734 + 0.808327i \(0.299626\pi\)
\(972\) 0 0
\(973\) −64.4660 −2.06669
\(974\) 2.26339 0.0725236
\(975\) 0 0
\(976\) 36.1334 1.15660
\(977\) −27.9044 −0.892740 −0.446370 0.894848i \(-0.647283\pi\)
−0.446370 + 0.894848i \(0.647283\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.17706 0.197319
\(981\) 0 0
\(982\) 6.79691 0.216898
\(983\) −9.79220 −0.312323 −0.156161 0.987732i \(-0.549912\pi\)
−0.156161 + 0.987732i \(0.549912\pi\)
\(984\) 0 0
\(985\) 8.88764 0.283184
\(986\) 0.0864735 0.00275388
\(987\) 0 0
\(988\) −0.148826 −0.00473479
\(989\) −78.6678 −2.50149
\(990\) 0 0
\(991\) 34.9794 1.11116 0.555578 0.831464i \(-0.312497\pi\)
0.555578 + 0.831464i \(0.312497\pi\)
\(992\) 28.8033 0.914506
\(993\) 0 0
\(994\) −3.90820 −0.123961
\(995\) −18.0381 −0.571846
\(996\) 0 0
\(997\) 12.7753 0.404599 0.202299 0.979324i \(-0.435159\pi\)
0.202299 + 0.979324i \(0.435159\pi\)
\(998\) 7.35855 0.232931
\(999\) 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 5445.2.a.bf.1.3 4
3.2 odd 2 1815.2.a.w.1.2 4
11.2 odd 10 495.2.n.a.136.2 8
11.6 odd 10 495.2.n.a.91.2 8
11.10 odd 2 5445.2.a.bt.1.2 4
15.14 odd 2 9075.2.a.cm.1.3 4
33.2 even 10 165.2.m.d.136.1 yes 8
33.17 even 10 165.2.m.d.91.1 8
33.32 even 2 1815.2.a.p.1.3 4
165.2 odd 20 825.2.bx.f.499.2 16
165.17 odd 20 825.2.bx.f.124.3 16
165.68 odd 20 825.2.bx.f.499.3 16
165.83 odd 20 825.2.bx.f.124.2 16
165.134 even 10 825.2.n.g.301.2 8
165.149 even 10 825.2.n.g.751.2 8
165.164 even 2 9075.2.a.di.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.91.1 8 33.17 even 10
165.2.m.d.136.1 yes 8 33.2 even 10
495.2.n.a.91.2 8 11.6 odd 10
495.2.n.a.136.2 8 11.2 odd 10
825.2.n.g.301.2 8 165.134 even 10
825.2.n.g.751.2 8 165.149 even 10
825.2.bx.f.124.2 16 165.83 odd 20
825.2.bx.f.124.3 16 165.17 odd 20
825.2.bx.f.499.2 16 165.2 odd 20
825.2.bx.f.499.3 16 165.68 odd 20
1815.2.a.p.1.3 4 33.32 even 2
1815.2.a.w.1.2 4 3.2 odd 2
5445.2.a.bf.1.3 4 1.1 even 1 trivial
5445.2.a.bt.1.2 4 11.10 odd 2
9075.2.a.cm.1.3 4 15.14 odd 2
9075.2.a.di.1.2 4 165.164 even 2