Properties

Label 1815.2.a.w.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
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 1.2
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.262360 q^{2} +1.00000 q^{3} -1.93117 q^{4} +1.00000 q^{5} +0.262360 q^{6} +3.19353 q^{7} -1.03138 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.262360 q^{2} +1.00000 q^{3} -1.93117 q^{4} +1.00000 q^{5} +0.262360 q^{6} +3.19353 q^{7} -1.03138 q^{8} +1.00000 q^{9} +0.262360 q^{10} -1.93117 q^{12} +1.11961 q^{13} +0.837853 q^{14} +1.00000 q^{15} +3.59174 q^{16} +0.0882264 q^{17} +0.262360 q^{18} +0.0688326 q^{19} -1.93117 q^{20} +3.19353 q^{21} +6.65450 q^{23} -1.03138 q^{24} +1.00000 q^{25} +0.293740 q^{26} +1.00000 q^{27} -6.16724 q^{28} -3.73583 q^{29} +0.262360 q^{30} -9.58484 q^{31} +3.00509 q^{32} +0.0231471 q^{34} +3.19353 q^{35} -1.93117 q^{36} -8.33021 q^{37} +0.0180589 q^{38} +1.11961 q^{39} -1.03138 q^{40} +11.6657 q^{41} +0.837853 q^{42} +11.8217 q^{43} +1.00000 q^{45} +1.74587 q^{46} +0.908020 q^{47} +3.59174 q^{48} +3.19862 q^{49} +0.262360 q^{50} +0.0882264 q^{51} -2.16215 q^{52} +0.872377 q^{53} +0.262360 q^{54} -3.29374 q^{56} +0.0688326 q^{57} -0.980131 q^{58} +1.83604 q^{59} -1.93117 q^{60} +10.0601 q^{61} -2.51468 q^{62} +3.19353 q^{63} -6.39507 q^{64} +1.11961 q^{65} +9.53916 q^{67} -0.170380 q^{68} +6.65450 q^{69} +0.837853 q^{70} -4.66454 q^{71} -1.03138 q^{72} +7.16034 q^{73} -2.18551 q^{74} +1.00000 q^{75} -0.132927 q^{76} +0.293740 q^{78} -0.791342 q^{79} +3.59174 q^{80} +1.00000 q^{81} +3.06060 q^{82} -0.247229 q^{83} -6.16724 q^{84} +0.0882264 q^{85} +3.10155 q^{86} -3.73583 q^{87} -14.5788 q^{89} +0.262360 q^{90} +3.57549 q^{91} -12.8510 q^{92} -9.58484 q^{93} +0.238228 q^{94} +0.0688326 q^{95} +3.00509 q^{96} -4.09198 q^{97} +0.839188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + 3q^{10} + q^{12} + 7q^{13} + 3q^{14} + 4q^{15} - q^{16} + 10q^{17} + 3q^{18} + 9q^{19} + q^{20} + 6q^{21} - 3q^{23} + 3q^{24} + 4q^{25} - 4q^{26} + 4q^{27} - 7q^{28} + 15q^{29} + 3q^{30} - 13q^{31} - 6q^{32} - 3q^{34} + 6q^{35} + q^{36} - 3q^{37} + 15q^{38} + 7q^{39} + 3q^{40} + 22q^{41} + 3q^{42} + 4q^{45} + q^{46} - 2q^{47} - q^{48} - 12q^{49} + 3q^{50} + 10q^{51} - 9q^{52} + 10q^{53} + 3q^{54} - 8q^{56} + 9q^{57} + 39q^{58} - 21q^{59} + q^{60} + 11q^{61} + 10q^{62} + 6q^{63} - 3q^{64} + 7q^{65} + q^{67} + 3q^{68} - 3q^{69} + 3q^{70} - 13q^{71} + 3q^{72} + q^{73} - 11q^{74} + 4q^{75} + 19q^{76} - 4q^{78} - 4q^{79} - q^{80} + 4q^{81} + 25q^{82} + 3q^{83} - 7q^{84} + 10q^{85} + 15q^{87} - 10q^{89} + 3q^{90} + 12q^{91} - 24q^{92} - 13q^{93} - 35q^{94} + 9q^{95} - 6q^{96} - 22q^{97} - 11q^{98} + O(q^{100}) \)

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 \(-0.470432\pi\)
0.0927582 + 0.995689i \(0.470432\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.93117 −0.965584
\(5\) 1.00000 0.447214
\(6\) 0.262360 0.107108
\(7\) 3.19353 1.20704 0.603520 0.797348i \(-0.293764\pi\)
0.603520 + 0.797348i \(0.293764\pi\)
\(8\) −1.03138 −0.364648
\(9\) 1.00000 0.333333
\(10\) 0.262360 0.0829654
\(11\) 0 0
\(12\) −1.93117 −0.557480
\(13\) 1.11961 0.310523 0.155261 0.987873i \(-0.450378\pi\)
0.155261 + 0.987873i \(0.450378\pi\)
\(14\) 0.837853 0.223926
\(15\) 1.00000 0.258199
\(16\) 3.59174 0.897936
\(17\) 0.0882264 0.0213981 0.0106990 0.999943i \(-0.496594\pi\)
0.0106990 + 0.999943i \(0.496594\pi\)
\(18\) 0.262360 0.0618388
\(19\) 0.0688326 0.0157913 0.00789564 0.999969i \(-0.497487\pi\)
0.00789564 + 0.999969i \(0.497487\pi\)
\(20\) −1.93117 −0.431822
\(21\) 3.19353 0.696885
\(22\) 0 0
\(23\) 6.65450 1.38756 0.693780 0.720187i \(-0.255944\pi\)
0.693780 + 0.720187i \(0.255944\pi\)
\(24\) −1.03138 −0.210530
\(25\) 1.00000 0.200000
\(26\) 0.293740 0.0576071
\(27\) 1.00000 0.192450
\(28\) −6.16724 −1.16550
\(29\) −3.73583 −0.693726 −0.346863 0.937916i \(-0.612753\pi\)
−0.346863 + 0.937916i \(0.612753\pi\)
\(30\) 0.262360 0.0479001
\(31\) −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
\(36\) −1.93117 −0.321861
\(37\) −8.33021 −1.36948 −0.684739 0.728789i \(-0.740084\pi\)
−0.684739 + 0.728789i \(0.740084\pi\)
\(38\) 0.0180589 0.00292954
\(39\) 1.11961 0.179280
\(40\) −1.03138 −0.163075
\(41\) 11.6657 1.82187 0.910935 0.412549i \(-0.135362\pi\)
0.910935 + 0.412549i \(0.135362\pi\)
\(42\) 0.837853 0.129283
\(43\) 11.8217 1.80280 0.901399 0.432989i \(-0.142541\pi\)
0.901399 + 0.432989i \(0.142541\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 1.74587 0.257415
\(47\) 0.908020 0.132448 0.0662242 0.997805i \(-0.478905\pi\)
0.0662242 + 0.997805i \(0.478905\pi\)
\(48\) 3.59174 0.518423
\(49\) 3.19862 0.456945
\(50\) 0.262360 0.0371033
\(51\) 0.0882264 0.0123542
\(52\) −2.16215 −0.299836
\(53\) 0.872377 0.119830 0.0599151 0.998203i \(-0.480917\pi\)
0.0599151 + 0.998203i \(0.480917\pi\)
\(54\) 0.262360 0.0357026
\(55\) 0 0
\(56\) −3.29374 −0.440144
\(57\) 0.0688326 0.00911710
\(58\) −0.980131 −0.128698
\(59\) 1.83604 0.239032 0.119516 0.992832i \(-0.461866\pi\)
0.119516 + 0.992832i \(0.461866\pi\)
\(60\) −1.93117 −0.249313
\(61\) 10.0601 1.28807 0.644034 0.764997i \(-0.277260\pi\)
0.644034 + 0.764997i \(0.277260\pi\)
\(62\) −2.51468 −0.319364
\(63\) 3.19353 0.402347
\(64\) −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
\(69\) 6.65450 0.801108
\(70\) 0.837853 0.100143
\(71\) −4.66454 −0.553580 −0.276790 0.960930i \(-0.589271\pi\)
−0.276790 + 0.960930i \(0.589271\pi\)
\(72\) −1.03138 −0.121549
\(73\) 7.16034 0.838054 0.419027 0.907974i \(-0.362371\pi\)
0.419027 + 0.907974i \(0.362371\pi\)
\(74\) −2.18551 −0.254060
\(75\) 1.00000 0.115470
\(76\) −0.132927 −0.0152478
\(77\) 0 0
\(78\) 0.293740 0.0332595
\(79\) −0.791342 −0.0890330 −0.0445165 0.999009i \(-0.514175\pi\)
−0.0445165 + 0.999009i \(0.514175\pi\)
\(80\) 3.59174 0.401569
\(81\) 1.00000 0.111111
\(82\) 3.06060 0.337987
\(83\) −0.247229 −0.0271369 −0.0135685 0.999908i \(-0.504319\pi\)
−0.0135685 + 0.999908i \(0.504319\pi\)
\(84\) −6.16724 −0.672901
\(85\) 0.0882264 0.00956950
\(86\) 3.10155 0.334448
\(87\) −3.73583 −0.400523
\(88\) 0 0
\(89\) −14.5788 −1.54535 −0.772673 0.634804i \(-0.781081\pi\)
−0.772673 + 0.634804i \(0.781081\pi\)
\(90\) 0.262360 0.0276551
\(91\) 3.57549 0.374814
\(92\) −12.8510 −1.33980
\(93\) −9.58484 −0.993902
\(94\) 0.238228 0.0245713
\(95\) 0.0688326 0.00706208
\(96\) 3.00509 0.306706
\(97\) −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 \(-0.568493\pi\)
−0.213521 + 0.976939i \(0.568493\pi\)
\(102\) 0.0231471 0.00229190
\(103\) 2.09280 0.206210 0.103105 0.994670i \(-0.467122\pi\)
0.103105 + 0.994670i \(0.467122\pi\)
\(104\) −1.15474 −0.113232
\(105\) 3.19353 0.311656
\(106\) 0.228877 0.0222305
\(107\) 15.7409 1.52173 0.760866 0.648909i \(-0.224774\pi\)
0.760866 + 0.648909i \(0.224774\pi\)
\(108\) −1.93117 −0.185827
\(109\) 4.13271 0.395842 0.197921 0.980218i \(-0.436581\pi\)
0.197921 + 0.980218i \(0.436581\pi\)
\(110\) 0 0
\(111\) −8.33021 −0.790668
\(112\) 11.4703 1.08384
\(113\) 13.7550 1.29396 0.646981 0.762506i \(-0.276031\pi\)
0.646981 + 0.762506i \(0.276031\pi\)
\(114\) 0.0180589 0.00169137
\(115\) 6.65450 0.620536
\(116\) 7.21451 0.669851
\(117\) 1.11961 0.103508
\(118\) 0.481704 0.0443444
\(119\) 0.281754 0.0258283
\(120\) −1.03138 −0.0941517
\(121\) 0 0
\(122\) 2.63937 0.238957
\(123\) 11.6657 1.05186
\(124\) 18.5099 1.66224
\(125\) 1.00000 0.0894427
\(126\) 0.837853 0.0746419
\(127\) −7.89979 −0.700993 −0.350496 0.936564i \(-0.613987\pi\)
−0.350496 + 0.936564i \(0.613987\pi\)
\(128\) −7.68799 −0.679528
\(129\) 11.8217 1.04085
\(130\) 0.293740 0.0257627
\(131\) 16.3539 1.42884 0.714422 0.699715i \(-0.246690\pi\)
0.714422 + 0.699715i \(0.246690\pi\)
\(132\) 0 0
\(133\) 0.219819 0.0190607
\(134\) 2.50269 0.216200
\(135\) 1.00000 0.0860663
\(136\) −0.0909950 −0.00780275
\(137\) 7.73208 0.660596 0.330298 0.943877i \(-0.392851\pi\)
0.330298 + 0.943877i \(0.392851\pi\)
\(138\) 1.74587 0.148619
\(139\) −20.1865 −1.71219 −0.856097 0.516815i \(-0.827117\pi\)
−0.856097 + 0.516815i \(0.827117\pi\)
\(140\) −6.16724 −0.521227
\(141\) 0.908020 0.0764691
\(142\) −1.22379 −0.102698
\(143\) 0 0
\(144\) 3.59174 0.299312
\(145\) −3.73583 −0.310244
\(146\) 1.87858 0.155473
\(147\) 3.19862 0.263817
\(148\) 16.0870 1.32235
\(149\) −9.69523 −0.794264 −0.397132 0.917761i \(-0.629995\pi\)
−0.397132 + 0.917761i \(0.629995\pi\)
\(150\) 0.262360 0.0214216
\(151\) −15.1615 −1.23382 −0.616911 0.787033i \(-0.711616\pi\)
−0.616911 + 0.787033i \(0.711616\pi\)
\(152\) −0.0709926 −0.00575826
\(153\) 0.0882264 0.00713268
\(154\) 0 0
\(155\) −9.58484 −0.769873
\(156\) −2.16215 −0.173110
\(157\) −14.8805 −1.18760 −0.593798 0.804614i \(-0.702372\pi\)
−0.593798 + 0.804614i \(0.702372\pi\)
\(158\) −0.207616 −0.0165171
\(159\) 0.872377 0.0691840
\(160\) 3.00509 0.237573
\(161\) 21.2513 1.67484
\(162\) 0.262360 0.0206129
\(163\) 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 \(-0.160826\pi\)
0.875053 + 0.484027i \(0.160826\pi\)
\(168\) −3.29374 −0.254118
\(169\) −11.7465 −0.903576
\(170\) 0.0231471 0.00177530
\(171\) 0.0688326 0.00526376
\(172\) −22.8298 −1.74075
\(173\) 13.0913 0.995312 0.497656 0.867374i \(-0.334194\pi\)
0.497656 + 0.867374i \(0.334194\pi\)
\(174\) −0.980131 −0.0743036
\(175\) 3.19353 0.241408
\(176\) 0 0
\(177\) 1.83604 0.138005
\(178\) −3.82488 −0.286687
\(179\) −15.1774 −1.13441 −0.567207 0.823576i \(-0.691976\pi\)
−0.567207 + 0.823576i \(0.691976\pi\)
\(180\) −1.93117 −0.143941
\(181\) 1.41348 0.105063 0.0525316 0.998619i \(-0.483271\pi\)
0.0525316 + 0.998619i \(0.483271\pi\)
\(182\) 0.938065 0.0695340
\(183\) 10.0601 0.743666
\(184\) −6.86332 −0.505971
\(185\) −8.33021 −0.612449
\(186\) −2.51468 −0.184385
\(187\) 0 0
\(188\) −1.75354 −0.127890
\(189\) 3.19353 0.232295
\(190\) 0.0180589 0.00131013
\(191\) 4.90914 0.355213 0.177606 0.984102i \(-0.443165\pi\)
0.177606 + 0.984102i \(0.443165\pi\)
\(192\) −6.39507 −0.461524
\(193\) −9.71987 −0.699652 −0.349826 0.936815i \(-0.613759\pi\)
−0.349826 + 0.936815i \(0.613759\pi\)
\(194\) −1.07357 −0.0770779
\(195\) 1.11961 0.0801767
\(196\) −6.17706 −0.441219
\(197\) 8.88764 0.633218 0.316609 0.948556i \(-0.397456\pi\)
0.316609 + 0.948556i \(0.397456\pi\)
\(198\) 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
\(201\) 9.53916 0.672840
\(202\) −1.12597 −0.0792232
\(203\) −11.9305 −0.837355
\(204\) −0.170380 −0.0119290
\(205\) 11.6657 0.814765
\(206\) 0.549068 0.0382554
\(207\) 6.65450 0.462520
\(208\) 4.02134 0.278830
\(209\) 0 0
\(210\) 0.837853 0.0578173
\(211\) −17.9778 −1.23764 −0.618822 0.785531i \(-0.712390\pi\)
−0.618822 + 0.785531i \(0.712390\pi\)
\(212\) −1.68471 −0.115706
\(213\) −4.66454 −0.319609
\(214\) 4.12978 0.282306
\(215\) 11.8217 0.806236
\(216\) −1.03138 −0.0701765
\(217\) −30.6095 −2.07791
\(218\) 1.08426 0.0734351
\(219\) 7.16034 0.483851
\(220\) 0 0
\(221\) 0.0987789 0.00664459
\(222\) −2.18551 −0.146682
\(223\) −13.7489 −0.920697 −0.460348 0.887738i \(-0.652276\pi\)
−0.460348 + 0.887738i \(0.652276\pi\)
\(224\) 9.59683 0.641215
\(225\) 1.00000 0.0666667
\(226\) 3.60876 0.240051
\(227\) −0.662735 −0.0439873 −0.0219936 0.999758i \(-0.507001\pi\)
−0.0219936 + 0.999758i \(0.507001\pi\)
\(228\) −0.132927 −0.00880332
\(229\) −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 \(-0.612043\pi\)
−0.344770 + 0.938687i \(0.612043\pi\)
\(234\) 0.293740 0.0192024
\(235\) 0.908020 0.0592327
\(236\) −3.54571 −0.230806
\(237\) −0.791342 −0.0514032
\(238\) 0.0739208 0.00479157
\(239\) −17.0762 −1.10457 −0.552284 0.833656i \(-0.686244\pi\)
−0.552284 + 0.833656i \(0.686244\pi\)
\(240\) 3.59174 0.231846
\(241\) 16.7082 1.07627 0.538135 0.842859i \(-0.319129\pi\)
0.538135 + 0.842859i \(0.319129\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −19.4278 −1.24374
\(245\) 3.19862 0.204352
\(246\) 3.06060 0.195137
\(247\) 0.0770654 0.00490356
\(248\) 9.88562 0.627737
\(249\) −0.247229 −0.0156675
\(250\) 0.262360 0.0165931
\(251\) −5.13767 −0.324287 −0.162143 0.986767i \(-0.551841\pi\)
−0.162143 + 0.986767i \(0.551841\pi\)
\(252\) −6.16724 −0.388499
\(253\) 0 0
\(254\) −2.07259 −0.130046
\(255\) 0.0882264 0.00552495
\(256\) 10.7731 0.673320
\(257\) −3.69824 −0.230690 −0.115345 0.993325i \(-0.536797\pi\)
−0.115345 + 0.993325i \(0.536797\pi\)
\(258\) 3.10155 0.193094
\(259\) −26.6027 −1.65301
\(260\) −2.16215 −0.134091
\(261\) −3.73583 −0.231242
\(262\) 4.29059 0.265074
\(263\) −31.6315 −1.95048 −0.975241 0.221147i \(-0.929020\pi\)
−0.975241 + 0.221147i \(0.929020\pi\)
\(264\) 0 0
\(265\) 0.872377 0.0535897
\(266\) 0.0576716 0.00353607
\(267\) −14.5788 −0.892206
\(268\) −18.4217 −1.12529
\(269\) −14.7754 −0.900875 −0.450437 0.892808i \(-0.648732\pi\)
−0.450437 + 0.892808i \(0.648732\pi\)
\(270\) 0.262360 0.0159667
\(271\) 19.5869 1.18982 0.594910 0.803792i \(-0.297188\pi\)
0.594910 + 0.803792i \(0.297188\pi\)
\(272\) 0.316887 0.0192141
\(273\) 3.57549 0.216399
\(274\) 2.02859 0.122551
\(275\) 0 0
\(276\) −12.8510 −0.773537
\(277\) −25.7104 −1.54479 −0.772393 0.635145i \(-0.780940\pi\)
−0.772393 + 0.635145i \(0.780940\pi\)
\(278\) −5.29612 −0.317640
\(279\) −9.58484 −0.573830
\(280\) −3.29374 −0.196839
\(281\) 26.7109 1.59344 0.796719 0.604350i \(-0.206567\pi\)
0.796719 + 0.604350i \(0.206567\pi\)
\(282\) 0.238228 0.0141863
\(283\) −7.23375 −0.430002 −0.215001 0.976614i \(-0.568975\pi\)
−0.215001 + 0.976614i \(0.568975\pi\)
\(284\) 9.00802 0.534527
\(285\) 0.0688326 0.00407729
\(286\) 0 0
\(287\) 37.2546 2.19907
\(288\) 3.00509 0.177077
\(289\) −16.9922 −0.999542
\(290\) −0.980131 −0.0575553
\(291\) −4.09198 −0.239876
\(292\) −13.8278 −0.809211
\(293\) −14.5947 −0.852633 −0.426317 0.904574i \(-0.640189\pi\)
−0.426317 + 0.904574i \(0.640189\pi\)
\(294\) 0.839188 0.0489424
\(295\) 1.83604 0.106899
\(296\) 8.59161 0.499377
\(297\) 0 0
\(298\) −2.54364 −0.147349
\(299\) 7.45042 0.430869
\(300\) −1.93117 −0.111496
\(301\) 37.7530 2.17605
\(302\) −3.97775 −0.228894
\(303\) −4.29171 −0.246552
\(304\) 0.247229 0.0141796
\(305\) 10.0601 0.576041
\(306\) 0.0231471 0.00132323
\(307\) −28.5445 −1.62912 −0.814559 0.580080i \(-0.803021\pi\)
−0.814559 + 0.580080i \(0.803021\pi\)
\(308\) 0 0
\(309\) 2.09280 0.119055
\(310\) −2.51468 −0.142824
\(311\) −9.94803 −0.564101 −0.282050 0.959400i \(-0.591015\pi\)
−0.282050 + 0.959400i \(0.591015\pi\)
\(312\) −1.15474 −0.0653742
\(313\) −16.4309 −0.928731 −0.464366 0.885644i \(-0.653718\pi\)
−0.464366 + 0.885644i \(0.653718\pi\)
\(314\) −3.90405 −0.220318
\(315\) 3.19353 0.179935
\(316\) 1.52821 0.0859688
\(317\) 14.2106 0.798147 0.399073 0.916919i \(-0.369332\pi\)
0.399073 + 0.916919i \(0.369332\pi\)
\(318\) 0.228877 0.0128348
\(319\) 0 0
\(320\) −6.39507 −0.357495
\(321\) 15.7409 0.878572
\(322\) 5.57549 0.310710
\(323\) 0.00607286 0.000337903 0
\(324\) −1.93117 −0.107287
\(325\) 1.11961 0.0621046
\(326\) 0.892323 0.0494212
\(327\) 4.13271 0.228539
\(328\) −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
\(333\) −8.33021 −0.456493
\(334\) 5.93362 0.324673
\(335\) 9.53916 0.521180
\(336\) 11.4703 0.625758
\(337\) 5.88609 0.320636 0.160318 0.987065i \(-0.448748\pi\)
0.160318 + 0.987065i \(0.448748\pi\)
\(338\) −3.08180 −0.167628
\(339\) 13.7550 0.747069
\(340\) −0.170380 −0.00924015
\(341\) 0 0
\(342\) 0.0180589 0.000976514 0
\(343\) −12.1398 −0.655489
\(344\) −12.1927 −0.657386
\(345\) 6.65450 0.358266
\(346\) 3.43463 0.184647
\(347\) −34.5043 −1.85229 −0.926143 0.377173i \(-0.876896\pi\)
−0.926143 + 0.377173i \(0.876896\pi\)
\(348\) 7.21451 0.386739
\(349\) 5.51356 0.295134 0.147567 0.989052i \(-0.452856\pi\)
0.147567 + 0.989052i \(0.452856\pi\)
\(350\) 0.837853 0.0447851
\(351\) 1.11961 0.0597602
\(352\) 0 0
\(353\) −29.8740 −1.59003 −0.795016 0.606589i \(-0.792537\pi\)
−0.795016 + 0.606589i \(0.792537\pi\)
\(354\) 0.481704 0.0256023
\(355\) −4.66454 −0.247568
\(356\) 28.1540 1.49216
\(357\) 0.281754 0.0149120
\(358\) −3.98194 −0.210452
\(359\) 35.9131 1.89542 0.947710 0.319133i \(-0.103392\pi\)
0.947710 + 0.319133i \(0.103392\pi\)
\(360\) −1.03138 −0.0543585
\(361\) −18.9953 −0.999751
\(362\) 0.370840 0.0194909
\(363\) 0 0
\(364\) −6.90488 −0.361914
\(365\) 7.16034 0.374789
\(366\) 2.63937 0.137962
\(367\) −9.03087 −0.471408 −0.235704 0.971825i \(-0.575740\pi\)
−0.235704 + 0.971825i \(0.575740\pi\)
\(368\) 23.9013 1.24594
\(369\) 11.6657 0.607290
\(370\) −2.18551 −0.113619
\(371\) 2.78596 0.144640
\(372\) 18.5099 0.959696
\(373\) −17.1994 −0.890553 −0.445277 0.895393i \(-0.646895\pi\)
−0.445277 + 0.895393i \(0.646895\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −0.936514 −0.0482970
\(377\) −4.18266 −0.215418
\(378\) 0.837853 0.0430945
\(379\) −8.00208 −0.411039 −0.205520 0.978653i \(-0.565888\pi\)
−0.205520 + 0.978653i \(0.565888\pi\)
\(380\) −0.132927 −0.00681903
\(381\) −7.89979 −0.404718
\(382\) 1.28796 0.0658978
\(383\) 29.7131 1.51827 0.759135 0.650933i \(-0.225622\pi\)
0.759135 + 0.650933i \(0.225622\pi\)
\(384\) −7.68799 −0.392326
\(385\) 0 0
\(386\) −2.55010 −0.129797
\(387\) 11.8217 0.600933
\(388\) 7.90230 0.401178
\(389\) −0.595397 −0.0301878 −0.0150939 0.999886i \(-0.504805\pi\)
−0.0150939 + 0.999886i \(0.504805\pi\)
\(390\) 0.293740 0.0148741
\(391\) 0.587103 0.0296911
\(392\) −3.29899 −0.166624
\(393\) 16.3539 0.824943
\(394\) 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
\(399\) 0.219819 0.0110047
\(400\) 3.59174 0.179587
\(401\) −11.3690 −0.567741 −0.283870 0.958863i \(-0.591619\pi\)
−0.283870 + 0.958863i \(0.591619\pi\)
\(402\) 2.50269 0.124823
\(403\) −10.7313 −0.534562
\(404\) 8.28802 0.412344
\(405\) 1.00000 0.0496904
\(406\) −3.13008 −0.155343
\(407\) 0 0
\(408\) −0.0909950 −0.00450492
\(409\) 8.59766 0.425127 0.212563 0.977147i \(-0.431819\pi\)
0.212563 + 0.977147i \(0.431819\pi\)
\(410\) 3.06060 0.151152
\(411\) 7.73208 0.381395
\(412\) −4.04156 −0.199113
\(413\) 5.86345 0.288522
\(414\) 1.74587 0.0858050
\(415\) −0.247229 −0.0121360
\(416\) 3.36452 0.164959
\(417\) −20.1865 −0.988536
\(418\) 0 0
\(419\) 31.3915 1.53358 0.766789 0.641899i \(-0.221853\pi\)
0.766789 + 0.641899i \(0.221853\pi\)
\(420\) −6.16724 −0.300930
\(421\) −4.57893 −0.223163 −0.111582 0.993755i \(-0.535592\pi\)
−0.111582 + 0.993755i \(0.535592\pi\)
\(422\) −4.71665 −0.229603
\(423\) 0.908020 0.0441495
\(424\) −0.899752 −0.0436958
\(425\) 0.0882264 0.00427961
\(426\) −1.22379 −0.0592927
\(427\) 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 \(-0.625149\pi\)
−0.383117 + 0.923700i \(0.625149\pi\)
\(432\) 3.59174 0.172808
\(433\) 15.8030 0.759441 0.379721 0.925101i \(-0.376020\pi\)
0.379721 + 0.925101i \(0.376020\pi\)
\(434\) −8.03069 −0.385485
\(435\) −3.73583 −0.179119
\(436\) −7.98096 −0.382218
\(437\) 0.458047 0.0219113
\(438\) 1.87858 0.0897622
\(439\) −21.5227 −1.02722 −0.513611 0.858023i \(-0.671693\pi\)
−0.513611 + 0.858023i \(0.671693\pi\)
\(440\) 0 0
\(441\) 3.19862 0.152315
\(442\) 0.0259156 0.00123268
\(443\) 10.1769 0.483519 0.241760 0.970336i \(-0.422275\pi\)
0.241760 + 0.970336i \(0.422275\pi\)
\(444\) 16.0870 0.763456
\(445\) −14.5788 −0.691100
\(446\) −3.60717 −0.170804
\(447\) −9.69523 −0.458569
\(448\) −20.4228 −0.964888
\(449\) 38.6088 1.82206 0.911031 0.412338i \(-0.135288\pi\)
0.911031 + 0.412338i \(0.135288\pi\)
\(450\) 0.262360 0.0123678
\(451\) 0 0
\(452\) −26.5632 −1.24943
\(453\) −15.1615 −0.712347
\(454\) −0.173875 −0.00816035
\(455\) 3.57549 0.167622
\(456\) −0.0709926 −0.00332453
\(457\) −23.3637 −1.09291 −0.546455 0.837488i \(-0.684023\pi\)
−0.546455 + 0.837488i \(0.684023\pi\)
\(458\) −1.96518 −0.0918269
\(459\) 0.0882264 0.00411806
\(460\) −12.8510 −0.599179
\(461\) −4.93120 −0.229669 −0.114834 0.993385i \(-0.536634\pi\)
−0.114834 + 0.993385i \(0.536634\pi\)
\(462\) 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
\(465\) −9.58484 −0.444487
\(466\) −2.76143 −0.127921
\(467\) −23.6778 −1.09568 −0.547839 0.836584i \(-0.684550\pi\)
−0.547839 + 0.836584i \(0.684550\pi\)
\(468\) −2.16215 −0.0999453
\(469\) 30.4636 1.40668
\(470\) 0.238228 0.0109886
\(471\) −14.8805 −0.685659
\(472\) −1.89366 −0.0871627
\(473\) 0 0
\(474\) −0.207616 −0.00953613
\(475\) 0.0688326 0.00315826
\(476\) −0.544113 −0.0249394
\(477\) 0.872377 0.0399434
\(478\) −4.48011 −0.204915
\(479\) 13.5240 0.617929 0.308964 0.951074i \(-0.400018\pi\)
0.308964 + 0.951074i \(0.400018\pi\)
\(480\) 3.00509 0.137163
\(481\) −9.32655 −0.425254
\(482\) 4.38356 0.199666
\(483\) 21.2513 0.966969
\(484\) 0 0
\(485\) −4.09198 −0.185807
\(486\) 0.262360 0.0119009
\(487\) −8.62704 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(488\) −10.3758 −0.469691
\(489\) 3.40114 0.153805
\(490\) 0.839188 0.0379106
\(491\) 25.9068 1.16916 0.584579 0.811337i \(-0.301260\pi\)
0.584579 + 0.811337i \(0.301260\pi\)
\(492\) −22.5283 −1.01566
\(493\) −0.329599 −0.0148444
\(494\) 0.0202189 0.000909690 0
\(495\) 0 0
\(496\) −34.4263 −1.54579
\(497\) −14.8963 −0.668193
\(498\) −0.0648629 −0.00290658
\(499\) −28.0475 −1.25558 −0.627790 0.778383i \(-0.716040\pi\)
−0.627790 + 0.778383i \(0.716040\pi\)
\(500\) −1.93117 −0.0863644
\(501\) 22.6164 1.01042
\(502\) −1.34792 −0.0601604
\(503\) −29.3999 −1.31088 −0.655438 0.755249i \(-0.727516\pi\)
−0.655438 + 0.755249i \(0.727516\pi\)
\(504\) −3.29374 −0.146715
\(505\) −4.29171 −0.190979
\(506\) 0 0
\(507\) −11.7465 −0.521680
\(508\) 15.2558 0.676867
\(509\) −19.3904 −0.859463 −0.429731 0.902957i \(-0.641392\pi\)
−0.429731 + 0.902957i \(0.641392\pi\)
\(510\) 0.0231471 0.00102497
\(511\) 22.8667 1.01156
\(512\) 18.2024 0.804440
\(513\) 0.0688326 0.00303903
\(514\) −0.970270 −0.0427968
\(515\) 2.09280 0.0922200
\(516\) −22.8298 −1.00502
\(517\) 0 0
\(518\) −6.97949 −0.306661
\(519\) 13.0913 0.574644
\(520\) −1.15474 −0.0506387
\(521\) 25.0062 1.09554 0.547771 0.836628i \(-0.315477\pi\)
0.547771 + 0.836628i \(0.315477\pi\)
\(522\) −0.980131 −0.0428992
\(523\) 2.18761 0.0956577 0.0478288 0.998856i \(-0.484770\pi\)
0.0478288 + 0.998856i \(0.484770\pi\)
\(524\) −31.5820 −1.37967
\(525\) 3.19353 0.139377
\(526\) −8.29883 −0.361846
\(527\) −0.845637 −0.0368365
\(528\) 0 0
\(529\) 21.2824 0.925322
\(530\) 0.228877 0.00994177
\(531\) 1.83604 0.0796775
\(532\) −0.424507 −0.0184047
\(533\) 13.0610 0.565733
\(534\) −3.82488 −0.165519
\(535\) 15.7409 0.680539
\(536\) −9.83850 −0.424958
\(537\) −15.1774 −0.654954
\(538\) −3.87648 −0.167127
\(539\) 0 0
\(540\) −1.93117 −0.0831042
\(541\) −13.5583 −0.582918 −0.291459 0.956583i \(-0.594141\pi\)
−0.291459 + 0.956583i \(0.594141\pi\)
\(542\) 5.13882 0.220731
\(543\) 1.41348 0.0606582
\(544\) 0.265128 0.0113673
\(545\) 4.13271 0.177026
\(546\) 0.938065 0.0401455
\(547\) 3.71282 0.158749 0.0793743 0.996845i \(-0.474708\pi\)
0.0793743 + 0.996845i \(0.474708\pi\)
\(548\) −14.9319 −0.637861
\(549\) 10.0601 0.429356
\(550\) 0 0
\(551\) −0.257147 −0.0109548
\(552\) −6.86332 −0.292122
\(553\) −2.52717 −0.107466
\(554\) −6.74536 −0.286583
\(555\) −8.33021 −0.353598
\(556\) 38.9835 1.65327
\(557\) −6.57520 −0.278600 −0.139300 0.990250i \(-0.544485\pi\)
−0.139300 + 0.990250i \(0.544485\pi\)
\(558\) −2.51468 −0.106455
\(559\) 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 \(-0.453494\pi\)
0.145585 + 0.989346i \(0.453494\pi\)
\(564\) −1.75354 −0.0738373
\(565\) 13.7550 0.578678
\(566\) −1.89784 −0.0797723
\(567\) 3.19353 0.134116
\(568\) 4.81092 0.201862
\(569\) 6.74768 0.282878 0.141439 0.989947i \(-0.454827\pi\)
0.141439 + 0.989947i \(0.454827\pi\)
\(570\) 0.0180589 0.000756404 0
\(571\) −9.68928 −0.405484 −0.202742 0.979232i \(-0.564985\pi\)
−0.202742 + 0.979232i \(0.564985\pi\)
\(572\) 0 0
\(573\) 4.90914 0.205082
\(574\) 9.77411 0.407963
\(575\) 6.65450 0.277512
\(576\) −6.39507 −0.266461
\(577\) 15.1498 0.630695 0.315348 0.948976i \(-0.397879\pi\)
0.315348 + 0.948976i \(0.397879\pi\)
\(578\) −4.45807 −0.185431
\(579\) −9.71987 −0.403944
\(580\) 7.21451 0.299566
\(581\) −0.789532 −0.0327553
\(582\) −1.07357 −0.0445009
\(583\) 0 0
\(584\) −7.38503 −0.305595
\(585\) 1.11961 0.0462900
\(586\) −3.82907 −0.158177
\(587\) −27.0112 −1.11487 −0.557436 0.830220i \(-0.688215\pi\)
−0.557436 + 0.830220i \(0.688215\pi\)
\(588\) −6.17706 −0.254738
\(589\) −0.659750 −0.0271845
\(590\) 0.481704 0.0198314
\(591\) 8.88764 0.365589
\(592\) −29.9200 −1.22970
\(593\) 22.1863 0.911084 0.455542 0.890214i \(-0.349446\pi\)
0.455542 + 0.890214i \(0.349446\pi\)
\(594\) 0 0
\(595\) 0.281754 0.0115508
\(596\) 18.7231 0.766929
\(597\) 18.0381 0.738250
\(598\) 1.95469 0.0799332
\(599\) 11.7487 0.480038 0.240019 0.970768i \(-0.422846\pi\)
0.240019 + 0.970768i \(0.422846\pi\)
\(600\) −1.03138 −0.0421059
\(601\) −34.3806 −1.40242 −0.701208 0.712957i \(-0.747355\pi\)
−0.701208 + 0.712957i \(0.747355\pi\)
\(602\) 9.90488 0.403693
\(603\) 9.53916 0.388465
\(604\) 29.2793 1.19136
\(605\) 0 0
\(606\) −1.12597 −0.0457395
\(607\) −13.6298 −0.553215 −0.276607 0.960983i \(-0.589210\pi\)
−0.276607 + 0.960983i \(0.589210\pi\)
\(608\) 0.206848 0.00838880
\(609\) −11.9305 −0.483447
\(610\) 2.63937 0.106865
\(611\) 1.01663 0.0411283
\(612\) −0.170380 −0.00688720
\(613\) 6.82994 0.275858 0.137929 0.990442i \(-0.455955\pi\)
0.137929 + 0.990442i \(0.455955\pi\)
\(614\) −7.48892 −0.302228
\(615\) 11.6657 0.470405
\(616\) 0 0
\(617\) 30.3730 1.22277 0.611386 0.791333i \(-0.290612\pi\)
0.611386 + 0.791333i \(0.290612\pi\)
\(618\) 0.549068 0.0220867
\(619\) −25.7879 −1.03650 −0.518252 0.855228i \(-0.673417\pi\)
−0.518252 + 0.855228i \(0.673417\pi\)
\(620\) 18.5099 0.743377
\(621\) 6.65450 0.267036
\(622\) −2.60996 −0.104650
\(623\) −46.5577 −1.86529
\(624\) 4.02134 0.160982
\(625\) 1.00000 0.0400000
\(626\) −4.31081 −0.172295
\(627\) 0 0
\(628\) 28.7368 1.14672
\(629\) −0.734945 −0.0293042
\(630\) 0.837853 0.0333809
\(631\) −25.5656 −1.01775 −0.508876 0.860840i \(-0.669939\pi\)
−0.508876 + 0.860840i \(0.669939\pi\)
\(632\) 0.816174 0.0324657
\(633\) −17.9778 −0.714554
\(634\) 3.72829 0.148069
\(635\) −7.89979 −0.313494
\(636\) −1.68471 −0.0668030
\(637\) 3.58119 0.141892
\(638\) 0 0
\(639\) −4.66454 −0.184527
\(640\) −7.68799 −0.303894
\(641\) −4.07262 −0.160859 −0.0804294 0.996760i \(-0.525629\pi\)
−0.0804294 + 0.996760i \(0.525629\pi\)
\(642\) 4.12978 0.162990
\(643\) 11.3298 0.446804 0.223402 0.974726i \(-0.428284\pi\)
0.223402 + 0.974726i \(0.428284\pi\)
\(644\) −41.0399 −1.61720
\(645\) 11.8217 0.465480
\(646\) 0.00159327 6.26865e−5 0
\(647\) −36.8480 −1.44864 −0.724322 0.689462i \(-0.757847\pi\)
−0.724322 + 0.689462i \(0.757847\pi\)
\(648\) −1.03138 −0.0405164
\(649\) 0 0
\(650\) 0.293740 0.0115214
\(651\) −30.6095 −1.19968
\(652\) −6.56818 −0.257230
\(653\) 15.2768 0.597827 0.298914 0.954280i \(-0.403376\pi\)
0.298914 + 0.954280i \(0.403376\pi\)
\(654\) 1.08426 0.0423978
\(655\) 16.3539 0.638998
\(656\) 41.9001 1.63592
\(657\) 7.16034 0.279351
\(658\) 0.760787 0.0296586
\(659\) −11.8996 −0.463544 −0.231772 0.972770i \(-0.574452\pi\)
−0.231772 + 0.972770i \(0.574452\pi\)
\(660\) 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
\(663\) 0.0987789 0.00383625
\(664\) 0.254987 0.00989542
\(665\) 0.219819 0.00852421
\(666\) −2.18551 −0.0846868
\(667\) −24.8601 −0.962587
\(668\) −43.6760 −1.68987
\(669\) −13.7489 −0.531565
\(670\) 2.50269 0.0966874
\(671\) 0 0
\(672\) 9.59683 0.370206
\(673\) 19.1067 0.736508 0.368254 0.929725i \(-0.379956\pi\)
0.368254 + 0.929725i \(0.379956\pi\)
\(674\) 1.54427 0.0594832
\(675\) 1.00000 0.0384900
\(676\) 22.6844 0.872478
\(677\) −46.4614 −1.78566 −0.892829 0.450396i \(-0.851283\pi\)
−0.892829 + 0.450396i \(0.851283\pi\)
\(678\) 3.60876 0.138594
\(679\) −13.0678 −0.501498
\(680\) −0.0909950 −0.00348950
\(681\) −0.662735 −0.0253961
\(682\) 0 0
\(683\) −42.5651 −1.62871 −0.814355 0.580367i \(-0.802909\pi\)
−0.814355 + 0.580367i \(0.802909\pi\)
\(684\) −0.132927 −0.00508260
\(685\) 7.73208 0.295427
\(686\) −3.18500 −0.121604
\(687\) −7.49041 −0.285777
\(688\) 42.4606 1.61880
\(689\) 0.976719 0.0372100
\(690\) 1.74587 0.0664643
\(691\) 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
\(696\) 3.85306 0.146050
\(697\) 1.02922 0.0389845
\(698\) 1.44654 0.0547522
\(699\) −10.5254 −0.398106
\(700\) −6.16724 −0.233100
\(701\) 30.6713 1.15844 0.579219 0.815172i \(-0.303358\pi\)
0.579219 + 0.815172i \(0.303358\pi\)
\(702\) 0.293740 0.0110865
\(703\) −0.573390 −0.0216258
\(704\) 0 0
\(705\) 0.908020 0.0341980
\(706\) −7.83773 −0.294977
\(707\) −13.7057 −0.515456
\(708\) −3.54571 −0.133256
\(709\) 15.0924 0.566806 0.283403 0.959001i \(-0.408537\pi\)
0.283403 + 0.959001i \(0.408537\pi\)
\(710\) −1.22379 −0.0459280
\(711\) −0.791342 −0.0296777
\(712\) 15.0363 0.563507
\(713\) −63.7824 −2.38867
\(714\) 0.0739208 0.00276642
\(715\) 0 0
\(716\) 29.3101 1.09537
\(717\) −17.0762 −0.637723
\(718\) 9.42215 0.351631
\(719\) −6.19020 −0.230856 −0.115428 0.993316i \(-0.536824\pi\)
−0.115428 + 0.993316i \(0.536824\pi\)
\(720\) 3.59174 0.133856
\(721\) 6.68343 0.248904
\(722\) −4.98359 −0.185470
\(723\) 16.7082 0.621385
\(724\) −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
\(729\) 1.00000 0.0370370
\(730\) 1.87858 0.0695295
\(731\) 1.04299 0.0385764
\(732\) −19.4278 −0.718072
\(733\) 28.7409 1.06157 0.530785 0.847507i \(-0.321897\pi\)
0.530785 + 0.847507i \(0.321897\pi\)
\(734\) −2.36934 −0.0874538
\(735\) 3.19862 0.117983
\(736\) 19.9974 0.737113
\(737\) 0 0
\(738\) 3.06060 0.112662
\(739\) 46.8341 1.72282 0.861411 0.507909i \(-0.169581\pi\)
0.861411 + 0.507909i \(0.169581\pi\)
\(740\) 16.0870 0.591371
\(741\) 0.0770654 0.00283107
\(742\) 0.730924 0.0268331
\(743\) −26.6108 −0.976254 −0.488127 0.872773i \(-0.662320\pi\)
−0.488127 + 0.872773i \(0.662320\pi\)
\(744\) 9.88562 0.362424
\(745\) −9.69523 −0.355206
\(746\) −4.51244 −0.165212
\(747\) −0.247229 −0.00904564
\(748\) 0 0
\(749\) 50.2691 1.83679
\(750\) 0.262360 0.00958002
\(751\) 40.5821 1.48086 0.740431 0.672132i \(-0.234621\pi\)
0.740431 + 0.672132i \(0.234621\pi\)
\(752\) 3.26137 0.118930
\(753\) −5.13767 −0.187227
\(754\) −1.09736 −0.0399635
\(755\) −15.1615 −0.551782
\(756\) −6.16724 −0.224300
\(757\) 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 \(-0.484824\pi\)
0.0476574 + 0.998864i \(0.484824\pi\)
\(762\) −2.07259 −0.0750819
\(763\) 13.1979 0.477797
\(764\) −9.48037 −0.342988
\(765\) 0.0882264 0.00318983
\(766\) 7.79553 0.281664
\(767\) 2.05565 0.0742251
\(768\) 10.7731 0.388742
\(769\) −3.23559 −0.116678 −0.0583392 0.998297i \(-0.518581\pi\)
−0.0583392 + 0.998297i \(0.518581\pi\)
\(770\) 0 0
\(771\) −3.69824 −0.133189
\(772\) 18.7707 0.675572
\(773\) 43.6365 1.56950 0.784748 0.619815i \(-0.212792\pi\)
0.784748 + 0.619815i \(0.212792\pi\)
\(774\) 3.10155 0.111483
\(775\) −9.58484 −0.344298
\(776\) 4.22039 0.151503
\(777\) −26.6027 −0.954368
\(778\) −0.156208 −0.00560033
\(779\) 0.802978 0.0287697
\(780\) −2.16215 −0.0774173
\(781\) 0 0
\(782\) 0.154032 0.00550818
\(783\) −3.73583 −0.133508
\(784\) 11.4886 0.410307
\(785\) −14.8805 −0.531109
\(786\) 4.29059 0.153040
\(787\) 4.56420 0.162696 0.0813480 0.996686i \(-0.474077\pi\)
0.0813480 + 0.996686i \(0.474077\pi\)
\(788\) −17.1635 −0.611425
\(789\) −31.6315 −1.12611
\(790\) −0.207616 −0.00738666
\(791\) 43.9270 1.56186
\(792\) 0 0
\(793\) 11.2634 0.399974
\(794\) 0.438009 0.0155444
\(795\) 0.872377 0.0309400
\(796\) −34.8346 −1.23468
\(797\) −11.1806 −0.396036 −0.198018 0.980198i \(-0.563450\pi\)
−0.198018 + 0.980198i \(0.563450\pi\)
\(798\) 0.0576716 0.00204155
\(799\) 0.0801114 0.00283414
\(800\) 3.00509 0.106246
\(801\) −14.5788 −0.515116
\(802\) −2.98277 −0.105325
\(803\) 0 0
\(804\) −18.4217 −0.649684
\(805\) 21.2513 0.749011
\(806\) −2.81545 −0.0991699
\(807\) −14.7754 −0.520120
\(808\) 4.42639 0.155720
\(809\) 45.5417 1.60116 0.800581 0.599225i \(-0.204525\pi\)
0.800581 + 0.599225i \(0.204525\pi\)
\(810\) 0.262360 0.00921838
\(811\) −2.82363 −0.0991511 −0.0495755 0.998770i \(-0.515787\pi\)
−0.0495755 + 0.998770i \(0.515787\pi\)
\(812\) 23.0397 0.808537
\(813\) 19.5869 0.686943
\(814\) 0 0
\(815\) 3.40114 0.119137
\(816\) 0.316887 0.0110933
\(817\) 0.813721 0.0284685
\(818\) 2.25568 0.0788679
\(819\) 3.57549 0.124938
\(820\) −22.5283 −0.786724
\(821\) 23.4117 0.817073 0.408536 0.912742i \(-0.366039\pi\)
0.408536 + 0.912742i \(0.366039\pi\)
\(822\) 2.02859 0.0707550
\(823\) 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 \(-0.227421\pi\)
0.755445 + 0.655212i \(0.227421\pi\)
\(828\) −12.8510 −0.446602
\(829\) 15.4712 0.537335 0.268668 0.963233i \(-0.413417\pi\)
0.268668 + 0.963233i \(0.413417\pi\)
\(830\) −0.0648629 −0.00225142
\(831\) −25.7104 −0.891883
\(832\) −7.15996 −0.248227
\(833\) 0.282202 0.00977773
\(834\) −5.29612 −0.183389
\(835\) 22.6164 0.782671
\(836\) 0 0
\(837\) −9.58484 −0.331301
\(838\) 8.23588 0.284504
\(839\) 30.2181 1.04324 0.521622 0.853177i \(-0.325327\pi\)
0.521622 + 0.853177i \(0.325327\pi\)
\(840\) −3.29374 −0.113645
\(841\) −15.0436 −0.518744
\(842\) −1.20133 −0.0414005
\(843\) 26.7109 0.919971
\(844\) 34.7182 1.19505
\(845\) −11.7465 −0.404091
\(846\) 0.238228 0.00819045
\(847\) 0 0
\(848\) 3.13335 0.107600
\(849\) −7.23375 −0.248262
\(850\) 0.0231471 0.000793938 0
\(851\) −55.4334 −1.90023
\(852\) 9.00802 0.308610
\(853\) 2.00908 0.0687897 0.0343949 0.999408i \(-0.489050\pi\)
0.0343949 + 0.999408i \(0.489050\pi\)
\(854\) 8.42890 0.288431
\(855\) 0.0688326 0.00235403
\(856\) −16.2349 −0.554896
\(857\) 33.6095 1.14808 0.574039 0.818828i \(-0.305376\pi\)
0.574039 + 0.818828i \(0.305376\pi\)
\(858\) 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
\(861\) 37.2546 1.26963
\(862\) −4.17347 −0.142149
\(863\) 8.75829 0.298136 0.149068 0.988827i \(-0.452373\pi\)
0.149068 + 0.988827i \(0.452373\pi\)
\(864\) 3.00509 0.102235
\(865\) 13.0913 0.445117
\(866\) 4.14606 0.140889
\(867\) −16.9922 −0.577086
\(868\) 59.1120 2.00639
\(869\) 0 0
\(870\) −0.980131 −0.0332296
\(871\) 10.6801 0.361881
\(872\) −4.26239 −0.144343
\(873\) −4.09198 −0.138493
\(874\) 0.120173 0.00406491
\(875\) 3.19353 0.107961
\(876\) −13.8278 −0.467198
\(877\) 15.2463 0.514830 0.257415 0.966301i \(-0.417129\pi\)
0.257415 + 0.966301i \(0.417129\pi\)
\(878\) −5.64669 −0.190566
\(879\) −14.5947 −0.492268
\(880\) 0 0
\(881\) −15.8486 −0.533953 −0.266977 0.963703i \(-0.586025\pi\)
−0.266977 + 0.963703i \(0.586025\pi\)
\(882\) 0.839188 0.0282569
\(883\) −32.9606 −1.10921 −0.554606 0.832113i \(-0.687131\pi\)
−0.554606 + 0.832113i \(0.687131\pi\)
\(884\) −0.190759 −0.00641590
\(885\) 1.83604 0.0617179
\(886\) 2.67001 0.0897007
\(887\) −11.2525 −0.377820 −0.188910 0.981994i \(-0.560495\pi\)
−0.188910 + 0.981994i \(0.560495\pi\)
\(888\) 8.59161 0.288316
\(889\) −25.2282 −0.846126
\(890\) −3.82488 −0.128210
\(891\) 0 0
\(892\) 26.5515 0.889010
\(893\) 0.0625014 0.00209153
\(894\) −2.54364 −0.0850720
\(895\) −15.1774 −0.507325
\(896\) −24.5518 −0.820218
\(897\) 7.45042 0.248762
\(898\) 10.1294 0.338022
\(899\) 35.8074 1.19424
\(900\) −1.93117 −0.0643722
\(901\) 0.0769667 0.00256413
\(902\) 0 0
\(903\) 37.7530 1.25634
\(904\) −14.1866 −0.471841
\(905\) 1.41348 0.0469857
\(906\) −3.97775 −0.132152
\(907\) −25.3745 −0.842546 −0.421273 0.906934i \(-0.638417\pi\)
−0.421273 + 0.906934i \(0.638417\pi\)
\(908\) 1.27985 0.0424734
\(909\) −4.29171 −0.142347
\(910\) 0.938065 0.0310966
\(911\) −17.4070 −0.576718 −0.288359 0.957522i \(-0.593110\pi\)
−0.288359 + 0.957522i \(0.593110\pi\)
\(912\) 0.247229 0.00818657
\(913\) 0 0
\(914\) −6.12971 −0.202753
\(915\) 10.0601 0.332577
\(916\) 14.4652 0.477945
\(917\) 52.2265 1.72467
\(918\) 0.0231471 0.000763967 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
\(921\) −28.5445 −0.940572
\(922\) −1.29375 −0.0426073
\(923\) −5.22245 −0.171899
\(924\) 0 0
\(925\) −8.33021 −0.273896
\(926\) 7.07448 0.232482
\(927\) 2.09280 0.0687367
\(928\) −11.2265 −0.368528
\(929\) 33.3131 1.09297 0.546484 0.837469i \(-0.315966\pi\)
0.546484 + 0.837469i \(0.315966\pi\)
\(930\) −2.51468 −0.0824595
\(931\) 0.220169 0.00721575
\(932\) 20.3262 0.665808
\(933\) −9.94803 −0.325684
\(934\) −6.21211 −0.203266
\(935\) 0 0
\(936\) −1.15474 −0.0377438
\(937\) 23.8637 0.779593 0.389797 0.920901i \(-0.372545\pi\)
0.389797 + 0.920901i \(0.372545\pi\)
\(938\) 7.99241 0.260961
\(939\) −16.4309 −0.536203
\(940\) −1.75354 −0.0571941
\(941\) 7.92557 0.258366 0.129183 0.991621i \(-0.458765\pi\)
0.129183 + 0.991621i \(0.458765\pi\)
\(942\) −3.90405 −0.127201
\(943\) 77.6292 2.52795
\(944\) 6.59459 0.214636
\(945\) 3.19353 0.103885
\(946\) 0 0
\(947\) −2.15429 −0.0700050 −0.0350025 0.999387i \(-0.511144\pi\)
−0.0350025 + 0.999387i \(0.511144\pi\)
\(948\) 1.52821 0.0496341
\(949\) 8.01676 0.260235
\(950\) 0.0180589 0.000585908 0
\(951\) 14.2106 0.460810
\(952\) −0.290595 −0.00941824
\(953\) 5.99303 0.194133 0.0970666 0.995278i \(-0.469054\pi\)
0.0970666 + 0.995278i \(0.469054\pi\)
\(954\) 0.228877 0.00741016
\(955\) 4.90914 0.158856
\(956\) 32.9770 1.06655
\(957\) 0 0
\(958\) 3.54816 0.114636
\(959\) 24.6926 0.797366
\(960\) −6.39507 −0.206400
\(961\) 60.8692 1.96352
\(962\) −2.44691 −0.0788916
\(963\) 15.7409 0.507244
\(964\) −32.2663 −1.03923
\(965\) −9.71987 −0.312894
\(966\) 5.57549 0.179389
\(967\) −22.2784 −0.716424 −0.358212 0.933640i \(-0.616613\pi\)
−0.358212 + 0.933640i \(0.616613\pi\)
\(968\) 0 0
\(969\) 0.00607286 0.000195088 0
\(970\) −1.07357 −0.0344703
\(971\) −36.6909 −1.17747 −0.588734 0.808327i \(-0.700374\pi\)
−0.588734 + 0.808327i \(0.700374\pi\)
\(972\) −1.93117 −0.0619422
\(973\) −64.4660 −2.06669
\(974\) −2.26339 −0.0725236
\(975\) 1.11961 0.0358561
\(976\) 36.1334 1.15660
\(977\) 27.9044 0.892740 0.446370 0.894848i \(-0.352717\pi\)
0.446370 + 0.894848i \(0.352717\pi\)
\(978\) 0.892323 0.0285333
\(979\) 0 0
\(980\) −6.17706 −0.197319
\(981\) 4.13271 0.131947
\(982\) 6.79691 0.216898
\(983\) 9.79220 0.312323 0.156161 0.987732i \(-0.450088\pi\)
0.156161 + 0.987732i \(0.450088\pi\)
\(984\) −12.0317 −0.383558
\(985\) 8.88764 0.283184
\(986\) −0.0864735 −0.00275388
\(987\) 2.89979 0.0923013
\(988\) −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
\(993\) −10.9119 −0.346279
\(994\) −3.90820 −0.123961
\(995\) 18.0381 0.571846
\(996\) 0.477441 0.0151283
\(997\) 12.7753 0.404599 0.202299 0.979324i \(-0.435159\pi\)
0.202299 + 0.979324i \(0.435159\pi\)
\(998\) −7.35855 −0.232931
\(999\) −8.33021 −0.263556
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 1815.2.a.w.1.2 4
3.2 odd 2 5445.2.a.bf.1.3 4
5.4 even 2 9075.2.a.cm.1.3 4
11.2 odd 10 165.2.m.d.136.1 yes 8
11.6 odd 10 165.2.m.d.91.1 8
11.10 odd 2 1815.2.a.p.1.3 4
33.2 even 10 495.2.n.a.136.2 8
33.17 even 10 495.2.n.a.91.2 8
33.32 even 2 5445.2.a.bt.1.2 4
55.2 even 20 825.2.bx.f.499.2 16
55.13 even 20 825.2.bx.f.499.3 16
55.17 even 20 825.2.bx.f.124.3 16
55.24 odd 10 825.2.n.g.301.2 8
55.28 even 20 825.2.bx.f.124.2 16
55.39 odd 10 825.2.n.g.751.2 8
55.54 odd 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 11.6 odd 10
165.2.m.d.136.1 yes 8 11.2 odd 10
495.2.n.a.91.2 8 33.17 even 10
495.2.n.a.136.2 8 33.2 even 10
825.2.n.g.301.2 8 55.24 odd 10
825.2.n.g.751.2 8 55.39 odd 10
825.2.bx.f.124.2 16 55.28 even 20
825.2.bx.f.124.3 16 55.17 even 20
825.2.bx.f.499.2 16 55.2 even 20
825.2.bx.f.499.3 16 55.13 even 20
1815.2.a.p.1.3 4 11.10 odd 2
1815.2.a.w.1.2 4 1.1 even 1 trivial
5445.2.a.bf.1.3 4 3.2 odd 2
5445.2.a.bt.1.2 4 33.32 even 2
9075.2.a.cm.1.3 4 5.4 even 2
9075.2.a.di.1.2 4 55.54 odd 2