Properties

Label 1815.2.a.w.1.3
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 1.3
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.47726 q^{2} +1.00000 q^{3} +0.182297 q^{4} +1.00000 q^{5} +1.47726 q^{6} +2.29496 q^{7} -2.68522 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.47726 q^{2} +1.00000 q^{3} +0.182297 q^{4} +1.00000 q^{5} +1.47726 q^{6} +2.29496 q^{7} -2.68522 q^{8} +1.00000 q^{9} +1.47726 q^{10} +0.182297 q^{12} +2.14077 q^{13} +3.39026 q^{14} +1.00000 q^{15} -4.33136 q^{16} -0.544446 q^{17} +1.47726 q^{18} +2.18230 q^{19} +0.182297 q^{20} +2.29496 q^{21} +2.03908 q^{23} -2.68522 q^{24} +1.00000 q^{25} +3.16248 q^{26} +1.00000 q^{27} +0.418365 q^{28} +9.94544 q^{29} +1.47726 q^{30} +6.77143 q^{31} -1.02811 q^{32} -0.804288 q^{34} +2.29496 q^{35} +0.182297 q^{36} -8.81959 q^{37} +3.22382 q^{38} +2.14077 q^{39} -2.68522 q^{40} +1.82283 q^{41} +3.39026 q^{42} +0.620713 q^{43} +1.00000 q^{45} +3.01225 q^{46} -0.378009 q^{47} -4.33136 q^{48} -1.73315 q^{49} +1.47726 q^{50} -0.544446 q^{51} +0.390257 q^{52} +11.5931 q^{53} +1.47726 q^{54} -6.16248 q^{56} +2.18230 q^{57} +14.6920 q^{58} -8.07792 q^{59} +0.182297 q^{60} -8.72595 q^{61} +10.0032 q^{62} +2.29496 q^{63} +7.14394 q^{64} +2.14077 q^{65} -9.75802 q^{67} -0.0992509 q^{68} +2.03908 q^{69} +3.39026 q^{70} -14.9968 q^{71} -2.68522 q^{72} -7.85844 q^{73} -13.0288 q^{74} +1.00000 q^{75} +0.397826 q^{76} +3.16248 q^{78} +9.22454 q^{79} -4.33136 q^{80} +1.00000 q^{81} +2.69279 q^{82} +9.45232 q^{83} +0.418365 q^{84} -0.544446 q^{85} +0.916954 q^{86} +9.94544 q^{87} +0.583290 q^{89} +1.47726 q^{90} +4.91300 q^{91} +0.371718 q^{92} +6.77143 q^{93} -0.558418 q^{94} +2.18230 q^{95} -1.02811 q^{96} -5.37801 q^{97} -2.56031 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + 3q^{10} + q^{12} + 7q^{13} + 3q^{14} + 4q^{15} - q^{16} + 10q^{17} + 3q^{18} + 9q^{19} + q^{20} + 6q^{21} - 3q^{23} + 3q^{24} + 4q^{25} - 4q^{26} + 4q^{27} - 7q^{28} + 15q^{29} + 3q^{30} - 13q^{31} - 6q^{32} - 3q^{34} + 6q^{35} + q^{36} - 3q^{37} + 15q^{38} + 7q^{39} + 3q^{40} + 22q^{41} + 3q^{42} + 4q^{45} + q^{46} - 2q^{47} - q^{48} - 12q^{49} + 3q^{50} + 10q^{51} - 9q^{52} + 10q^{53} + 3q^{54} - 8q^{56} + 9q^{57} + 39q^{58} - 21q^{59} + q^{60} + 11q^{61} + 10q^{62} + 6q^{63} - 3q^{64} + 7q^{65} + q^{67} + 3q^{68} - 3q^{69} + 3q^{70} - 13q^{71} + 3q^{72} + q^{73} - 11q^{74} + 4q^{75} + 19q^{76} - 4q^{78} - 4q^{79} - q^{80} + 4q^{81} + 25q^{82} + 3q^{83} - 7q^{84} + 10q^{85} + 15q^{87} - 10q^{89} + 3q^{90} + 12q^{91} - 24q^{92} - 13q^{93} - 35q^{94} + 9q^{95} - 6q^{96} - 22q^{97} - 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.47726 1.04458 0.522290 0.852768i \(-0.325078\pi\)
0.522290 + 0.852768i \(0.325078\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.182297 0.0911485
\(5\) 1.00000 0.447214
\(6\) 1.47726 0.603089
\(7\) 2.29496 0.867414 0.433707 0.901054i \(-0.357205\pi\)
0.433707 + 0.901054i \(0.357205\pi\)
\(8\) −2.68522 −0.949369
\(9\) 1.00000 0.333333
\(10\) 1.47726 0.467151
\(11\) 0 0
\(12\) 0.182297 0.0526246
\(13\) 2.14077 0.593744 0.296872 0.954917i \(-0.404057\pi\)
0.296872 + 0.954917i \(0.404057\pi\)
\(14\) 3.39026 0.906084
\(15\) 1.00000 0.258199
\(16\) −4.33136 −1.08284
\(17\) −0.544446 −0.132048 −0.0660238 0.997818i \(-0.521031\pi\)
−0.0660238 + 0.997818i \(0.521031\pi\)
\(18\) 1.47726 0.348194
\(19\) 2.18230 0.500653 0.250327 0.968161i \(-0.419462\pi\)
0.250327 + 0.968161i \(0.419462\pi\)
\(20\) 0.182297 0.0407629
\(21\) 2.29496 0.500802
\(22\) 0 0
\(23\) 2.03908 0.425177 0.212589 0.977142i \(-0.431811\pi\)
0.212589 + 0.977142i \(0.431811\pi\)
\(24\) −2.68522 −0.548118
\(25\) 1.00000 0.200000
\(26\) 3.16248 0.620213
\(27\) 1.00000 0.192450
\(28\) 0.418365 0.0790636
\(29\) 9.94544 1.84682 0.923411 0.383813i \(-0.125389\pi\)
0.923411 + 0.383813i \(0.125389\pi\)
\(30\) 1.47726 0.269710
\(31\) 6.77143 1.21619 0.608093 0.793866i \(-0.291935\pi\)
0.608093 + 0.793866i \(0.291935\pi\)
\(32\) −1.02811 −0.181746
\(33\) 0 0
\(34\) −0.804288 −0.137934
\(35\) 2.29496 0.387920
\(36\) 0.182297 0.0303828
\(37\) −8.81959 −1.44993 −0.724966 0.688785i \(-0.758145\pi\)
−0.724966 + 0.688785i \(0.758145\pi\)
\(38\) 3.22382 0.522973
\(39\) 2.14077 0.342798
\(40\) −2.68522 −0.424571
\(41\) 1.82283 0.284678 0.142339 0.989818i \(-0.454538\pi\)
0.142339 + 0.989818i \(0.454538\pi\)
\(42\) 3.39026 0.523128
\(43\) 0.620713 0.0946578 0.0473289 0.998879i \(-0.484929\pi\)
0.0473289 + 0.998879i \(0.484929\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 3.01225 0.444132
\(47\) −0.378009 −0.0551383 −0.0275691 0.999620i \(-0.508777\pi\)
−0.0275691 + 0.999620i \(0.508777\pi\)
\(48\) −4.33136 −0.625178
\(49\) −1.73315 −0.247592
\(50\) 1.47726 0.208916
\(51\) −0.544446 −0.0762377
\(52\) 0.390257 0.0541189
\(53\) 11.5931 1.59243 0.796217 0.605011i \(-0.206831\pi\)
0.796217 + 0.605011i \(0.206831\pi\)
\(54\) 1.47726 0.201030
\(55\) 0 0
\(56\) −6.16248 −0.823496
\(57\) 2.18230 0.289052
\(58\) 14.6920 1.92915
\(59\) −8.07792 −1.05166 −0.525828 0.850591i \(-0.676244\pi\)
−0.525828 + 0.850591i \(0.676244\pi\)
\(60\) 0.182297 0.0235345
\(61\) −8.72595 −1.11724 −0.558622 0.829422i \(-0.688670\pi\)
−0.558622 + 0.829422i \(0.688670\pi\)
\(62\) 10.0032 1.27040
\(63\) 2.29496 0.289138
\(64\) 7.14394 0.892993
\(65\) 2.14077 0.265530
\(66\) 0 0
\(67\) −9.75802 −1.19213 −0.596066 0.802936i \(-0.703270\pi\)
−0.596066 + 0.802936i \(0.703270\pi\)
\(68\) −0.0992509 −0.0120359
\(69\) 2.03908 0.245476
\(70\) 3.39026 0.405213
\(71\) −14.9968 −1.77979 −0.889894 0.456167i \(-0.849222\pi\)
−0.889894 + 0.456167i \(0.849222\pi\)
\(72\) −2.68522 −0.316456
\(73\) −7.85844 −0.919760 −0.459880 0.887981i \(-0.652108\pi\)
−0.459880 + 0.887981i \(0.652108\pi\)
\(74\) −13.0288 −1.51457
\(75\) 1.00000 0.115470
\(76\) 0.397826 0.0456338
\(77\) 0 0
\(78\) 3.16248 0.358080
\(79\) 9.22454 1.03784 0.518921 0.854822i \(-0.326334\pi\)
0.518921 + 0.854822i \(0.326334\pi\)
\(80\) −4.33136 −0.484261
\(81\) 1.00000 0.111111
\(82\) 2.69279 0.297369
\(83\) 9.45232 1.03753 0.518763 0.854918i \(-0.326393\pi\)
0.518763 + 0.854918i \(0.326393\pi\)
\(84\) 0.418365 0.0456474
\(85\) −0.544446 −0.0590534
\(86\) 0.916954 0.0988777
\(87\) 9.94544 1.06626
\(88\) 0 0
\(89\) 0.583290 0.0618287 0.0309143 0.999522i \(-0.490158\pi\)
0.0309143 + 0.999522i \(0.490158\pi\)
\(90\) 1.47726 0.155717
\(91\) 4.91300 0.515022
\(92\) 0.371718 0.0387543
\(93\) 6.77143 0.702165
\(94\) −0.558418 −0.0575964
\(95\) 2.18230 0.223899
\(96\) −1.02811 −0.104931
\(97\) −5.37801 −0.546054 −0.273027 0.962006i \(-0.588025\pi\)
−0.273027 + 0.962006i \(0.588025\pi\)
\(98\) −2.56031 −0.258630
\(99\) 0 0
\(100\) 0.182297 0.0182297
\(101\) 19.5291 1.94322 0.971608 0.236598i \(-0.0760323\pi\)
0.971608 + 0.236598i \(0.0760323\pi\)
\(102\) −0.804288 −0.0796364
\(103\) 13.0062 1.28154 0.640771 0.767732i \(-0.278615\pi\)
0.640771 + 0.767732i \(0.278615\pi\)
\(104\) −5.74845 −0.563682
\(105\) 2.29496 0.223965
\(106\) 17.1260 1.66343
\(107\) −1.97355 −0.190790 −0.0953950 0.995439i \(-0.530411\pi\)
−0.0953950 + 0.995439i \(0.530411\pi\)
\(108\) 0.182297 0.0175415
\(109\) −10.6212 −1.01733 −0.508663 0.860966i \(-0.669860\pi\)
−0.508663 + 0.860966i \(0.669860\pi\)
\(110\) 0 0
\(111\) −8.81959 −0.837119
\(112\) −9.94032 −0.939271
\(113\) −16.8956 −1.58940 −0.794700 0.607002i \(-0.792372\pi\)
−0.794700 + 0.607002i \(0.792372\pi\)
\(114\) 3.22382 0.301938
\(115\) 2.03908 0.190145
\(116\) 1.81302 0.168335
\(117\) 2.14077 0.197915
\(118\) −11.9332 −1.09854
\(119\) −1.24948 −0.114540
\(120\) −2.68522 −0.245126
\(121\) 0 0
\(122\) −12.8905 −1.16705
\(123\) 1.82283 0.164359
\(124\) 1.23441 0.110854
\(125\) 1.00000 0.0894427
\(126\) 3.39026 0.302028
\(127\) −4.13248 −0.366699 −0.183349 0.983048i \(-0.558694\pi\)
−0.183349 + 0.983048i \(0.558694\pi\)
\(128\) 12.6097 1.11455
\(129\) 0.620713 0.0546507
\(130\) 3.16248 0.277368
\(131\) 0.436527 0.0381395 0.0190698 0.999818i \(-0.493930\pi\)
0.0190698 + 0.999818i \(0.493930\pi\)
\(132\) 0 0
\(133\) 5.00829 0.434274
\(134\) −14.4151 −1.24528
\(135\) 1.00000 0.0860663
\(136\) 1.46196 0.125362
\(137\) −7.86789 −0.672200 −0.336100 0.941826i \(-0.609108\pi\)
−0.336100 + 0.941826i \(0.609108\pi\)
\(138\) 3.01225 0.256420
\(139\) −17.6431 −1.49647 −0.748236 0.663433i \(-0.769099\pi\)
−0.748236 + 0.663433i \(0.769099\pi\)
\(140\) 0.418365 0.0353583
\(141\) −0.378009 −0.0318341
\(142\) −22.1541 −1.85913
\(143\) 0 0
\(144\) −4.33136 −0.360947
\(145\) 9.94544 0.825924
\(146\) −11.6090 −0.960764
\(147\) −1.73315 −0.142947
\(148\) −1.60779 −0.132159
\(149\) 10.9601 0.897889 0.448945 0.893560i \(-0.351800\pi\)
0.448945 + 0.893560i \(0.351800\pi\)
\(150\) 1.47726 0.120618
\(151\) 20.0324 1.63021 0.815106 0.579312i \(-0.196679\pi\)
0.815106 + 0.579312i \(0.196679\pi\)
\(152\) −5.85995 −0.475304
\(153\) −0.544446 −0.0440158
\(154\) 0 0
\(155\) 6.77143 0.543895
\(156\) 0.390257 0.0312456
\(157\) 9.15468 0.730623 0.365311 0.930885i \(-0.380963\pi\)
0.365311 + 0.930885i \(0.380963\pi\)
\(158\) 13.6270 1.08411
\(159\) 11.5931 0.919392
\(160\) −1.02811 −0.0812791
\(161\) 4.67961 0.368805
\(162\) 1.47726 0.116065
\(163\) −11.3321 −0.887597 −0.443799 0.896127i \(-0.646369\pi\)
−0.443799 + 0.896127i \(0.646369\pi\)
\(164\) 0.332296 0.0259480
\(165\) 0 0
\(166\) 13.9635 1.08378
\(167\) −15.1001 −1.16848 −0.584241 0.811580i \(-0.698608\pi\)
−0.584241 + 0.811580i \(0.698608\pi\)
\(168\) −6.16248 −0.475446
\(169\) −8.41709 −0.647468
\(170\) −0.804288 −0.0616861
\(171\) 2.18230 0.166884
\(172\) 0.113154 0.00862792
\(173\) −18.2641 −1.38859 −0.694297 0.719688i \(-0.744285\pi\)
−0.694297 + 0.719688i \(0.744285\pi\)
\(174\) 14.6920 1.11380
\(175\) 2.29496 0.173483
\(176\) 0 0
\(177\) −8.07792 −0.607174
\(178\) 0.861672 0.0645850
\(179\) −0.525419 −0.0392716 −0.0196358 0.999807i \(-0.506251\pi\)
−0.0196358 + 0.999807i \(0.506251\pi\)
\(180\) 0.182297 0.0135876
\(181\) −17.7106 −1.31642 −0.658211 0.752833i \(-0.728687\pi\)
−0.658211 + 0.752833i \(0.728687\pi\)
\(182\) 7.25777 0.537982
\(183\) −8.72595 −0.645041
\(184\) −5.47537 −0.403650
\(185\) −8.81959 −0.648429
\(186\) 10.0032 0.733468
\(187\) 0 0
\(188\) −0.0689100 −0.00502578
\(189\) 2.29496 0.166934
\(190\) 3.22382 0.233880
\(191\) −16.5519 −1.19766 −0.598828 0.800877i \(-0.704367\pi\)
−0.598828 + 0.800877i \(0.704367\pi\)
\(192\) 7.14394 0.515570
\(193\) 24.5032 1.76378 0.881891 0.471454i \(-0.156271\pi\)
0.881891 + 0.471454i \(0.156271\pi\)
\(194\) −7.94472 −0.570397
\(195\) 2.14077 0.153304
\(196\) −0.315947 −0.0225677
\(197\) 7.50877 0.534978 0.267489 0.963561i \(-0.413806\pi\)
0.267489 + 0.963561i \(0.413806\pi\)
\(198\) 0 0
\(199\) −20.0956 −1.42454 −0.712270 0.701906i \(-0.752333\pi\)
−0.712270 + 0.701906i \(0.752333\pi\)
\(200\) −2.68522 −0.189874
\(201\) −9.75802 −0.688278
\(202\) 28.8495 2.02985
\(203\) 22.8244 1.60196
\(204\) −0.0992509 −0.00694895
\(205\) 1.82283 0.127312
\(206\) 19.2136 1.33867
\(207\) 2.03908 0.141726
\(208\) −9.27247 −0.642930
\(209\) 0 0
\(210\) 3.39026 0.233950
\(211\) −5.41860 −0.373032 −0.186516 0.982452i \(-0.559720\pi\)
−0.186516 + 0.982452i \(0.559720\pi\)
\(212\) 2.11339 0.145148
\(213\) −14.9968 −1.02756
\(214\) −2.91544 −0.199296
\(215\) 0.620713 0.0423322
\(216\) −2.68522 −0.182706
\(217\) 15.5402 1.05494
\(218\) −15.6903 −1.06268
\(219\) −7.85844 −0.531024
\(220\) 0 0
\(221\) −1.16554 −0.0784024
\(222\) −13.0288 −0.874438
\(223\) 15.7074 1.05185 0.525923 0.850532i \(-0.323720\pi\)
0.525923 + 0.850532i \(0.323720\pi\)
\(224\) −2.35947 −0.157649
\(225\) 1.00000 0.0666667
\(226\) −24.9591 −1.66026
\(227\) 1.47141 0.0976612 0.0488306 0.998807i \(-0.484451\pi\)
0.0488306 + 0.998807i \(0.484451\pi\)
\(228\) 0.397826 0.0263467
\(229\) −15.9751 −1.05566 −0.527831 0.849350i \(-0.676994\pi\)
−0.527831 + 0.849350i \(0.676994\pi\)
\(230\) 3.01225 0.198622
\(231\) 0 0
\(232\) −26.7057 −1.75331
\(233\) 25.6382 1.67961 0.839806 0.542887i \(-0.182669\pi\)
0.839806 + 0.542887i \(0.182669\pi\)
\(234\) 3.16248 0.206738
\(235\) −0.378009 −0.0246586
\(236\) −1.47258 −0.0958569
\(237\) 9.22454 0.599198
\(238\) −1.84581 −0.119646
\(239\) 4.18206 0.270515 0.135258 0.990810i \(-0.456814\pi\)
0.135258 + 0.990810i \(0.456814\pi\)
\(240\) −4.33136 −0.279588
\(241\) 3.29180 0.212043 0.106022 0.994364i \(-0.466189\pi\)
0.106022 + 0.994364i \(0.466189\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −1.59072 −0.101835
\(245\) −1.73315 −0.110727
\(246\) 2.69279 0.171686
\(247\) 4.67180 0.297260
\(248\) −18.1828 −1.15461
\(249\) 9.45232 0.599016
\(250\) 1.47726 0.0934301
\(251\) −9.36459 −0.591088 −0.295544 0.955329i \(-0.595501\pi\)
−0.295544 + 0.955329i \(0.595501\pi\)
\(252\) 0.418365 0.0263545
\(253\) 0 0
\(254\) −6.10475 −0.383046
\(255\) −0.544446 −0.0340945
\(256\) 4.33989 0.271243
\(257\) −12.5714 −0.784182 −0.392091 0.919927i \(-0.628248\pi\)
−0.392091 + 0.919927i \(0.628248\pi\)
\(258\) 0.916954 0.0570870
\(259\) −20.2406 −1.25769
\(260\) 0.390257 0.0242027
\(261\) 9.94544 0.615607
\(262\) 0.644864 0.0398398
\(263\) −4.82946 −0.297797 −0.148899 0.988852i \(-0.547573\pi\)
−0.148899 + 0.988852i \(0.547573\pi\)
\(264\) 0 0
\(265\) 11.5931 0.712158
\(266\) 7.39855 0.453634
\(267\) 0.583290 0.0356968
\(268\) −1.77886 −0.108661
\(269\) −5.22929 −0.318835 −0.159418 0.987211i \(-0.550962\pi\)
−0.159418 + 0.987211i \(0.550962\pi\)
\(270\) 1.47726 0.0899032
\(271\) −30.3311 −1.84248 −0.921240 0.388994i \(-0.872823\pi\)
−0.921240 + 0.388994i \(0.872823\pi\)
\(272\) 2.35819 0.142986
\(273\) 4.91300 0.297348
\(274\) −11.6229 −0.702167
\(275\) 0 0
\(276\) 0.371718 0.0223748
\(277\) −15.9694 −0.959511 −0.479756 0.877402i \(-0.659275\pi\)
−0.479756 + 0.877402i \(0.659275\pi\)
\(278\) −26.0635 −1.56319
\(279\) 6.77143 0.405395
\(280\) −6.16248 −0.368279
\(281\) 1.39067 0.0829604 0.0414802 0.999139i \(-0.486793\pi\)
0.0414802 + 0.999139i \(0.486793\pi\)
\(282\) −0.558418 −0.0332533
\(283\) −5.87453 −0.349205 −0.174602 0.984639i \(-0.555864\pi\)
−0.174602 + 0.984639i \(0.555864\pi\)
\(284\) −2.73387 −0.162225
\(285\) 2.18230 0.129268
\(286\) 0 0
\(287\) 4.18332 0.246934
\(288\) −1.02811 −0.0605819
\(289\) −16.7036 −0.982563
\(290\) 14.6920 0.862744
\(291\) −5.37801 −0.315264
\(292\) −1.43257 −0.0838348
\(293\) −19.9745 −1.16692 −0.583461 0.812141i \(-0.698302\pi\)
−0.583461 + 0.812141i \(0.698302\pi\)
\(294\) −2.56031 −0.149320
\(295\) −8.07792 −0.470315
\(296\) 23.6825 1.37652
\(297\) 0 0
\(298\) 16.1910 0.937917
\(299\) 4.36520 0.252446
\(300\) 0.182297 0.0105249
\(301\) 1.42451 0.0821075
\(302\) 29.5930 1.70289
\(303\) 19.5291 1.12192
\(304\) −9.45232 −0.542128
\(305\) −8.72595 −0.499647
\(306\) −0.804288 −0.0459781
\(307\) −19.4372 −1.10934 −0.554671 0.832070i \(-0.687156\pi\)
−0.554671 + 0.832070i \(0.687156\pi\)
\(308\) 0 0
\(309\) 13.0062 0.739898
\(310\) 10.0032 0.568142
\(311\) 6.02182 0.341466 0.170733 0.985317i \(-0.445386\pi\)
0.170733 + 0.985317i \(0.445386\pi\)
\(312\) −5.74845 −0.325442
\(313\) −5.10833 −0.288740 −0.144370 0.989524i \(-0.546116\pi\)
−0.144370 + 0.989524i \(0.546116\pi\)
\(314\) 13.5238 0.763194
\(315\) 2.29496 0.129307
\(316\) 1.68161 0.0945978
\(317\) 13.6788 0.768279 0.384139 0.923275i \(-0.374498\pi\)
0.384139 + 0.923275i \(0.374498\pi\)
\(318\) 17.1260 0.960379
\(319\) 0 0
\(320\) 7.14394 0.399358
\(321\) −1.97355 −0.110153
\(322\) 6.91300 0.385246
\(323\) −1.18814 −0.0661100
\(324\) 0.182297 0.0101276
\(325\) 2.14077 0.118749
\(326\) −16.7404 −0.927167
\(327\) −10.6212 −0.587354
\(328\) −4.89469 −0.270264
\(329\) −0.867517 −0.0478278
\(330\) 0 0
\(331\) 11.4695 0.630418 0.315209 0.949022i \(-0.397925\pi\)
0.315209 + 0.949022i \(0.397925\pi\)
\(332\) 1.72313 0.0945691
\(333\) −8.81959 −0.483311
\(334\) −22.3068 −1.22057
\(335\) −9.75802 −0.533137
\(336\) −9.94032 −0.542289
\(337\) −3.76404 −0.205041 −0.102520 0.994731i \(-0.532691\pi\)
−0.102520 + 0.994731i \(0.532691\pi\)
\(338\) −12.4342 −0.676333
\(339\) −16.8956 −0.917641
\(340\) −0.0992509 −0.00538264
\(341\) 0 0
\(342\) 3.22382 0.174324
\(343\) −20.0422 −1.08218
\(344\) −1.66675 −0.0898651
\(345\) 2.03908 0.109780
\(346\) −26.9808 −1.45050
\(347\) 0.395980 0.0212573 0.0106287 0.999944i \(-0.496617\pi\)
0.0106287 + 0.999944i \(0.496617\pi\)
\(348\) 1.81302 0.0971883
\(349\) 13.1708 0.705015 0.352508 0.935809i \(-0.385329\pi\)
0.352508 + 0.935809i \(0.385329\pi\)
\(350\) 3.39026 0.181217
\(351\) 2.14077 0.114266
\(352\) 0 0
\(353\) 11.3853 0.605977 0.302989 0.952994i \(-0.402016\pi\)
0.302989 + 0.952994i \(0.402016\pi\)
\(354\) −11.9332 −0.634242
\(355\) −14.9968 −0.795946
\(356\) 0.106332 0.00563559
\(357\) −1.24948 −0.0661296
\(358\) −0.776180 −0.0410224
\(359\) 14.3682 0.758326 0.379163 0.925330i \(-0.376212\pi\)
0.379163 + 0.925330i \(0.376212\pi\)
\(360\) −2.68522 −0.141524
\(361\) −14.2376 −0.749346
\(362\) −26.1632 −1.37511
\(363\) 0 0
\(364\) 0.895625 0.0469435
\(365\) −7.85844 −0.411329
\(366\) −12.8905 −0.673797
\(367\) −26.2640 −1.37097 −0.685485 0.728087i \(-0.740410\pi\)
−0.685485 + 0.728087i \(0.740410\pi\)
\(368\) −8.83198 −0.460399
\(369\) 1.82283 0.0948926
\(370\) −13.0288 −0.677337
\(371\) 26.6057 1.38130
\(372\) 1.23441 0.0640013
\(373\) −21.8951 −1.13368 −0.566842 0.823827i \(-0.691835\pi\)
−0.566842 + 0.823827i \(0.691835\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 1.01504 0.0523466
\(377\) 21.2909 1.09654
\(378\) 3.39026 0.174376
\(379\) 25.5596 1.31291 0.656455 0.754365i \(-0.272055\pi\)
0.656455 + 0.754365i \(0.272055\pi\)
\(380\) 0.397826 0.0204081
\(381\) −4.13248 −0.211714
\(382\) −24.4515 −1.25105
\(383\) 19.0519 0.973508 0.486754 0.873539i \(-0.338181\pi\)
0.486754 + 0.873539i \(0.338181\pi\)
\(384\) 12.6097 0.643485
\(385\) 0 0
\(386\) 36.1976 1.84241
\(387\) 0.620713 0.0315526
\(388\) −0.980395 −0.0497720
\(389\) 16.3925 0.831134 0.415567 0.909563i \(-0.363583\pi\)
0.415567 + 0.909563i \(0.363583\pi\)
\(390\) 3.16248 0.160138
\(391\) −1.11017 −0.0561436
\(392\) 4.65388 0.235056
\(393\) 0.436527 0.0220199
\(394\) 11.0924 0.558827
\(395\) 9.22454 0.464137
\(396\) 0 0
\(397\) 30.9826 1.55497 0.777485 0.628901i \(-0.216495\pi\)
0.777485 + 0.628901i \(0.216495\pi\)
\(398\) −29.6864 −1.48805
\(399\) 5.00829 0.250728
\(400\) −4.33136 −0.216568
\(401\) −6.36611 −0.317908 −0.158954 0.987286i \(-0.550812\pi\)
−0.158954 + 0.987286i \(0.550812\pi\)
\(402\) −14.4151 −0.718961
\(403\) 14.4961 0.722103
\(404\) 3.56009 0.177121
\(405\) 1.00000 0.0496904
\(406\) 33.7176 1.67338
\(407\) 0 0
\(408\) 1.46196 0.0723776
\(409\) 6.26874 0.309969 0.154985 0.987917i \(-0.450467\pi\)
0.154985 + 0.987917i \(0.450467\pi\)
\(410\) 2.69279 0.132987
\(411\) −7.86789 −0.388095
\(412\) 2.37100 0.116811
\(413\) −18.5385 −0.912222
\(414\) 3.01225 0.148044
\(415\) 9.45232 0.463996
\(416\) −2.20095 −0.107910
\(417\) −17.6431 −0.863988
\(418\) 0 0
\(419\) 3.90332 0.190689 0.0953447 0.995444i \(-0.469605\pi\)
0.0953447 + 0.995444i \(0.469605\pi\)
\(420\) 0.418365 0.0204141
\(421\) 17.3715 0.846637 0.423318 0.905981i \(-0.360865\pi\)
0.423318 + 0.905981i \(0.360865\pi\)
\(422\) −8.00468 −0.389662
\(423\) −0.378009 −0.0183794
\(424\) −31.1300 −1.51181
\(425\) −0.544446 −0.0264095
\(426\) −22.1541 −1.07337
\(427\) −20.0257 −0.969113
\(428\) −0.359772 −0.0173902
\(429\) 0 0
\(430\) 0.916954 0.0442194
\(431\) 7.03651 0.338937 0.169468 0.985536i \(-0.445795\pi\)
0.169468 + 0.985536i \(0.445795\pi\)
\(432\) −4.33136 −0.208393
\(433\) 2.75229 0.132267 0.0661334 0.997811i \(-0.478934\pi\)
0.0661334 + 0.997811i \(0.478934\pi\)
\(434\) 22.9569 1.10197
\(435\) 9.94544 0.476847
\(436\) −1.93621 −0.0927278
\(437\) 4.44987 0.212866
\(438\) −11.6090 −0.554697
\(439\) 2.73703 0.130631 0.0653157 0.997865i \(-0.479195\pi\)
0.0653157 + 0.997865i \(0.479195\pi\)
\(440\) 0 0
\(441\) −1.73315 −0.0825307
\(442\) −1.72180 −0.0818976
\(443\) 11.1042 0.527576 0.263788 0.964581i \(-0.415028\pi\)
0.263788 + 0.964581i \(0.415028\pi\)
\(444\) −1.60779 −0.0763021
\(445\) 0.583290 0.0276506
\(446\) 23.2039 1.09874
\(447\) 10.9601 0.518396
\(448\) 16.3951 0.774595
\(449\) 14.8268 0.699722 0.349861 0.936802i \(-0.386229\pi\)
0.349861 + 0.936802i \(0.386229\pi\)
\(450\) 1.47726 0.0696387
\(451\) 0 0
\(452\) −3.08001 −0.144872
\(453\) 20.0324 0.941203
\(454\) 2.17366 0.102015
\(455\) 4.91300 0.230325
\(456\) −5.85995 −0.274417
\(457\) −29.1825 −1.36510 −0.682549 0.730840i \(-0.739129\pi\)
−0.682549 + 0.730840i \(0.739129\pi\)
\(458\) −23.5993 −1.10272
\(459\) −0.544446 −0.0254126
\(460\) 0.371718 0.0173314
\(461\) 31.1798 1.45219 0.726094 0.687595i \(-0.241334\pi\)
0.726094 + 0.687595i \(0.241334\pi\)
\(462\) 0 0
\(463\) 41.1642 1.91306 0.956531 0.291631i \(-0.0941978\pi\)
0.956531 + 0.291631i \(0.0941978\pi\)
\(464\) −43.0773 −1.99981
\(465\) 6.77143 0.314018
\(466\) 37.8742 1.75449
\(467\) 38.7766 1.79437 0.897184 0.441656i \(-0.145609\pi\)
0.897184 + 0.441656i \(0.145609\pi\)
\(468\) 0.390257 0.0180396
\(469\) −22.3943 −1.03407
\(470\) −0.558418 −0.0257579
\(471\) 9.15468 0.421825
\(472\) 21.6910 0.998409
\(473\) 0 0
\(474\) 13.6270 0.625911
\(475\) 2.18230 0.100131
\(476\) −0.227777 −0.0104401
\(477\) 11.5931 0.530811
\(478\) 6.17800 0.282575
\(479\) −16.6876 −0.762476 −0.381238 0.924477i \(-0.624502\pi\)
−0.381238 + 0.924477i \(0.624502\pi\)
\(480\) −1.02811 −0.0469265
\(481\) −18.8808 −0.860888
\(482\) 4.86284 0.221496
\(483\) 4.67961 0.212930
\(484\) 0 0
\(485\) −5.37801 −0.244203
\(486\) 1.47726 0.0670099
\(487\) 2.51193 0.113827 0.0569133 0.998379i \(-0.481874\pi\)
0.0569133 + 0.998379i \(0.481874\pi\)
\(488\) 23.4311 1.06068
\(489\) −11.3321 −0.512455
\(490\) −2.56031 −0.115663
\(491\) 7.55865 0.341117 0.170559 0.985348i \(-0.445443\pi\)
0.170559 + 0.985348i \(0.445443\pi\)
\(492\) 0.332296 0.0149811
\(493\) −5.41475 −0.243868
\(494\) 6.90147 0.310512
\(495\) 0 0
\(496\) −29.3295 −1.31693
\(497\) −34.4170 −1.54381
\(498\) 13.9635 0.625721
\(499\) 16.7964 0.751911 0.375955 0.926638i \(-0.377315\pi\)
0.375955 + 0.926638i \(0.377315\pi\)
\(500\) 0.182297 0.00815257
\(501\) −15.1001 −0.674623
\(502\) −13.8339 −0.617439
\(503\) −7.63258 −0.340320 −0.170160 0.985416i \(-0.554428\pi\)
−0.170160 + 0.985416i \(0.554428\pi\)
\(504\) −6.16248 −0.274499
\(505\) 19.5291 0.869032
\(506\) 0 0
\(507\) −8.41709 −0.373816
\(508\) −0.753340 −0.0334240
\(509\) 32.9039 1.45844 0.729219 0.684280i \(-0.239883\pi\)
0.729219 + 0.684280i \(0.239883\pi\)
\(510\) −0.804288 −0.0356145
\(511\) −18.0348 −0.797813
\(512\) −18.8082 −0.831213
\(513\) 2.18230 0.0963508
\(514\) −18.5712 −0.819141
\(515\) 13.0062 0.573123
\(516\) 0.113154 0.00498133
\(517\) 0 0
\(518\) −29.9007 −1.31376
\(519\) −18.2641 −0.801705
\(520\) −5.74845 −0.252086
\(521\) 0.797954 0.0349590 0.0174795 0.999847i \(-0.494436\pi\)
0.0174795 + 0.999847i \(0.494436\pi\)
\(522\) 14.6920 0.643051
\(523\) −4.30514 −0.188251 −0.0941254 0.995560i \(-0.530005\pi\)
−0.0941254 + 0.995560i \(0.530005\pi\)
\(524\) 0.0795776 0.00347636
\(525\) 2.29496 0.100160
\(526\) −7.13437 −0.311073
\(527\) −3.68668 −0.160594
\(528\) 0 0
\(529\) −18.8422 −0.819224
\(530\) 17.1260 0.743906
\(531\) −8.07792 −0.350552
\(532\) 0.912997 0.0395834
\(533\) 3.90226 0.169026
\(534\) 0.861672 0.0372882
\(535\) −1.97355 −0.0853239
\(536\) 26.2024 1.13177
\(537\) −0.525419 −0.0226735
\(538\) −7.72502 −0.333049
\(539\) 0 0
\(540\) 0.182297 0.00784482
\(541\) 22.7081 0.976299 0.488149 0.872760i \(-0.337672\pi\)
0.488149 + 0.872760i \(0.337672\pi\)
\(542\) −44.8069 −1.92462
\(543\) −17.7106 −0.760037
\(544\) 0.559749 0.0239990
\(545\) −10.6212 −0.454962
\(546\) 7.25777 0.310604
\(547\) −32.1551 −1.37485 −0.687425 0.726255i \(-0.741259\pi\)
−0.687425 + 0.726255i \(0.741259\pi\)
\(548\) −1.43429 −0.0612700
\(549\) −8.72595 −0.372415
\(550\) 0 0
\(551\) 21.7039 0.924617
\(552\) −5.47537 −0.233047
\(553\) 21.1700 0.900239
\(554\) −23.5910 −1.00229
\(555\) −8.81959 −0.374371
\(556\) −3.21629 −0.136401
\(557\) −37.7151 −1.59804 −0.799021 0.601303i \(-0.794648\pi\)
−0.799021 + 0.601303i \(0.794648\pi\)
\(558\) 10.0032 0.423468
\(559\) 1.32881 0.0562025
\(560\) −9.94032 −0.420055
\(561\) 0 0
\(562\) 2.05438 0.0866588
\(563\) −21.9871 −0.926645 −0.463322 0.886190i \(-0.653343\pi\)
−0.463322 + 0.886190i \(0.653343\pi\)
\(564\) −0.0689100 −0.00290163
\(565\) −16.8956 −0.710801
\(566\) −8.67821 −0.364772
\(567\) 2.29496 0.0963794
\(568\) 40.2696 1.68968
\(569\) 20.4804 0.858584 0.429292 0.903166i \(-0.358763\pi\)
0.429292 + 0.903166i \(0.358763\pi\)
\(570\) 3.22382 0.135031
\(571\) −17.4373 −0.729728 −0.364864 0.931061i \(-0.618884\pi\)
−0.364864 + 0.931061i \(0.618884\pi\)
\(572\) 0 0
\(573\) −16.5519 −0.691467
\(574\) 6.17985 0.257942
\(575\) 2.03908 0.0850354
\(576\) 7.14394 0.297664
\(577\) −33.2349 −1.38359 −0.691793 0.722096i \(-0.743179\pi\)
−0.691793 + 0.722096i \(0.743179\pi\)
\(578\) −24.6755 −1.02637
\(579\) 24.5032 1.01832
\(580\) 1.81302 0.0752818
\(581\) 21.6927 0.899966
\(582\) −7.94472 −0.329319
\(583\) 0 0
\(584\) 21.1016 0.873192
\(585\) 2.14077 0.0885101
\(586\) −29.5075 −1.21894
\(587\) −42.7954 −1.76635 −0.883177 0.469040i \(-0.844600\pi\)
−0.883177 + 0.469040i \(0.844600\pi\)
\(588\) −0.315947 −0.0130294
\(589\) 14.7773 0.608887
\(590\) −11.9332 −0.491282
\(591\) 7.50877 0.308869
\(592\) 38.2008 1.57004
\(593\) 42.6570 1.75171 0.875857 0.482570i \(-0.160297\pi\)
0.875857 + 0.482570i \(0.160297\pi\)
\(594\) 0 0
\(595\) −1.24948 −0.0512238
\(596\) 1.99800 0.0818413
\(597\) −20.0956 −0.822458
\(598\) 6.44854 0.263700
\(599\) 28.3204 1.15714 0.578570 0.815633i \(-0.303611\pi\)
0.578570 + 0.815633i \(0.303611\pi\)
\(600\) −2.68522 −0.109624
\(601\) 7.65458 0.312237 0.156118 0.987738i \(-0.450102\pi\)
0.156118 + 0.987738i \(0.450102\pi\)
\(602\) 2.10437 0.0857679
\(603\) −9.75802 −0.397377
\(604\) 3.65184 0.148591
\(605\) 0 0
\(606\) 28.8495 1.17193
\(607\) −11.5986 −0.470771 −0.235386 0.971902i \(-0.575635\pi\)
−0.235386 + 0.971902i \(0.575635\pi\)
\(608\) −2.24364 −0.0909915
\(609\) 22.8244 0.924892
\(610\) −12.8905 −0.521921
\(611\) −0.809232 −0.0327380
\(612\) −0.0992509 −0.00401198
\(613\) 32.1077 1.29682 0.648410 0.761292i \(-0.275434\pi\)
0.648410 + 0.761292i \(0.275434\pi\)
\(614\) −28.7139 −1.15880
\(615\) 1.82283 0.0735035
\(616\) 0 0
\(617\) −18.7392 −0.754414 −0.377207 0.926129i \(-0.623115\pi\)
−0.377207 + 0.926129i \(0.623115\pi\)
\(618\) 19.2136 0.772883
\(619\) −39.0600 −1.56995 −0.784977 0.619525i \(-0.787325\pi\)
−0.784977 + 0.619525i \(0.787325\pi\)
\(620\) 1.23441 0.0495752
\(621\) 2.03908 0.0818254
\(622\) 8.89579 0.356689
\(623\) 1.33863 0.0536311
\(624\) −9.27247 −0.371196
\(625\) 1.00000 0.0400000
\(626\) −7.54633 −0.301612
\(627\) 0 0
\(628\) 1.66887 0.0665952
\(629\) 4.80179 0.191460
\(630\) 3.39026 0.135071
\(631\) 45.0561 1.79366 0.896828 0.442380i \(-0.145866\pi\)
0.896828 + 0.442380i \(0.145866\pi\)
\(632\) −24.7699 −0.985295
\(633\) −5.41860 −0.215370
\(634\) 20.2072 0.802529
\(635\) −4.13248 −0.163993
\(636\) 2.11339 0.0838013
\(637\) −3.71027 −0.147006
\(638\) 0 0
\(639\) −14.9968 −0.593263
\(640\) 12.6097 0.498441
\(641\) 25.8928 1.02270 0.511351 0.859372i \(-0.329145\pi\)
0.511351 + 0.859372i \(0.329145\pi\)
\(642\) −2.91544 −0.115063
\(643\) 30.6381 1.20825 0.604125 0.796890i \(-0.293523\pi\)
0.604125 + 0.796890i \(0.293523\pi\)
\(644\) 0.853079 0.0336160
\(645\) 0.620713 0.0244405
\(646\) −1.75520 −0.0690572
\(647\) −15.1084 −0.593972 −0.296986 0.954882i \(-0.595981\pi\)
−0.296986 + 0.954882i \(0.595981\pi\)
\(648\) −2.68522 −0.105485
\(649\) 0 0
\(650\) 3.16248 0.124043
\(651\) 15.5402 0.609068
\(652\) −2.06581 −0.0809032
\(653\) −5.23528 −0.204872 −0.102436 0.994740i \(-0.532664\pi\)
−0.102436 + 0.994740i \(0.532664\pi\)
\(654\) −15.6903 −0.613538
\(655\) 0.436527 0.0170565
\(656\) −7.89532 −0.308261
\(657\) −7.85844 −0.306587
\(658\) −1.28155 −0.0499599
\(659\) −14.9207 −0.581229 −0.290615 0.956840i \(-0.593860\pi\)
−0.290615 + 0.956840i \(0.593860\pi\)
\(660\) 0 0
\(661\) 45.7403 1.77909 0.889545 0.456847i \(-0.151021\pi\)
0.889545 + 0.456847i \(0.151021\pi\)
\(662\) 16.9434 0.658523
\(663\) −1.16554 −0.0452656
\(664\) −25.3816 −0.984995
\(665\) 5.00829 0.194213
\(666\) −13.0288 −0.504857
\(667\) 20.2795 0.785226
\(668\) −2.75271 −0.106505
\(669\) 15.7074 0.607284
\(670\) −14.4151 −0.556905
\(671\) 0 0
\(672\) −2.35947 −0.0910185
\(673\) −21.1087 −0.813680 −0.406840 0.913500i \(-0.633369\pi\)
−0.406840 + 0.913500i \(0.633369\pi\)
\(674\) −5.56047 −0.214181
\(675\) 1.00000 0.0384900
\(676\) −1.53441 −0.0590158
\(677\) 44.0695 1.69373 0.846863 0.531810i \(-0.178488\pi\)
0.846863 + 0.531810i \(0.178488\pi\)
\(678\) −24.9591 −0.958550
\(679\) −12.3423 −0.473655
\(680\) 1.46196 0.0560635
\(681\) 1.47141 0.0563847
\(682\) 0 0
\(683\) −42.5318 −1.62743 −0.813717 0.581261i \(-0.802560\pi\)
−0.813717 + 0.581261i \(0.802560\pi\)
\(684\) 0.397826 0.0152113
\(685\) −7.86789 −0.300617
\(686\) −29.6076 −1.13042
\(687\) −15.9751 −0.609487
\(688\) −2.68853 −0.102499
\(689\) 24.8182 0.945498
\(690\) 3.01225 0.114674
\(691\) −5.74570 −0.218577 −0.109288 0.994010i \(-0.534857\pi\)
−0.109288 + 0.994010i \(0.534857\pi\)
\(692\) −3.32949 −0.126568
\(693\) 0 0
\(694\) 0.584966 0.0222050
\(695\) −17.6431 −0.669242
\(696\) −26.7057 −1.01228
\(697\) −0.992430 −0.0375910
\(698\) 19.4567 0.736445
\(699\) 25.6382 0.969724
\(700\) 0.418365 0.0158127
\(701\) 1.21594 0.0459255 0.0229628 0.999736i \(-0.492690\pi\)
0.0229628 + 0.999736i \(0.492690\pi\)
\(702\) 3.16248 0.119360
\(703\) −19.2470 −0.725913
\(704\) 0 0
\(705\) −0.378009 −0.0142366
\(706\) 16.8190 0.632992
\(707\) 44.8185 1.68557
\(708\) −1.47258 −0.0553430
\(709\) −26.4100 −0.991850 −0.495925 0.868365i \(-0.665171\pi\)
−0.495925 + 0.868365i \(0.665171\pi\)
\(710\) −22.1541 −0.831429
\(711\) 9.22454 0.345947
\(712\) −1.56626 −0.0586982
\(713\) 13.8075 0.517094
\(714\) −1.84581 −0.0690777
\(715\) 0 0
\(716\) −0.0957823 −0.00357955
\(717\) 4.18206 0.156182
\(718\) 21.2256 0.792133
\(719\) 49.4435 1.84393 0.921967 0.387269i \(-0.126582\pi\)
0.921967 + 0.387269i \(0.126582\pi\)
\(720\) −4.33136 −0.161420
\(721\) 29.8488 1.11163
\(722\) −21.0326 −0.782753
\(723\) 3.29180 0.122423
\(724\) −3.22860 −0.119990
\(725\) 9.94544 0.369364
\(726\) 0 0
\(727\) −39.2447 −1.45551 −0.727753 0.685839i \(-0.759435\pi\)
−0.727753 + 0.685839i \(0.759435\pi\)
\(728\) −13.1925 −0.488946
\(729\) 1.00000 0.0370370
\(730\) −11.6090 −0.429667
\(731\) −0.337944 −0.0124993
\(732\) −1.59072 −0.0587946
\(733\) −5.19920 −0.192037 −0.0960185 0.995380i \(-0.530611\pi\)
−0.0960185 + 0.995380i \(0.530611\pi\)
\(734\) −38.7988 −1.43209
\(735\) −1.73315 −0.0639280
\(736\) −2.09639 −0.0772740
\(737\) 0 0
\(738\) 2.69279 0.0991229
\(739\) −29.0090 −1.06711 −0.533557 0.845764i \(-0.679145\pi\)
−0.533557 + 0.845764i \(0.679145\pi\)
\(740\) −1.60779 −0.0591034
\(741\) 4.67180 0.171623
\(742\) 39.3036 1.44288
\(743\) −8.50677 −0.312083 −0.156042 0.987750i \(-0.549873\pi\)
−0.156042 + 0.987750i \(0.549873\pi\)
\(744\) −18.1828 −0.666613
\(745\) 10.9601 0.401548
\(746\) −32.3447 −1.18422
\(747\) 9.45232 0.345842
\(748\) 0 0
\(749\) −4.52922 −0.165494
\(750\) 1.47726 0.0539419
\(751\) −35.1051 −1.28100 −0.640502 0.767957i \(-0.721274\pi\)
−0.640502 + 0.767957i \(0.721274\pi\)
\(752\) 1.63729 0.0597060
\(753\) −9.36459 −0.341265
\(754\) 31.4522 1.14542
\(755\) 20.0324 0.729053
\(756\) 0.418365 0.0152158
\(757\) −7.18316 −0.261076 −0.130538 0.991443i \(-0.541671\pi\)
−0.130538 + 0.991443i \(0.541671\pi\)
\(758\) 37.7582 1.37144
\(759\) 0 0
\(760\) −5.85995 −0.212563
\(761\) 43.3866 1.57276 0.786382 0.617741i \(-0.211952\pi\)
0.786382 + 0.617741i \(0.211952\pi\)
\(762\) −6.10475 −0.221152
\(763\) −24.3753 −0.882444
\(764\) −3.01737 −0.109165
\(765\) −0.544446 −0.0196845
\(766\) 28.1447 1.01691
\(767\) −17.2930 −0.624414
\(768\) 4.33989 0.156602
\(769\) 19.6548 0.708771 0.354385 0.935099i \(-0.384690\pi\)
0.354385 + 0.935099i \(0.384690\pi\)
\(770\) 0 0
\(771\) −12.5714 −0.452747
\(772\) 4.46687 0.160766
\(773\) −24.4359 −0.878899 −0.439450 0.898267i \(-0.644826\pi\)
−0.439450 + 0.898267i \(0.644826\pi\)
\(774\) 0.916954 0.0329592
\(775\) 6.77143 0.243237
\(776\) 14.4411 0.518407
\(777\) −20.2406 −0.726129
\(778\) 24.2160 0.868186
\(779\) 3.97795 0.142525
\(780\) 0.390257 0.0139734
\(781\) 0 0
\(782\) −1.64001 −0.0586465
\(783\) 9.94544 0.355421
\(784\) 7.50688 0.268103
\(785\) 9.15468 0.326744
\(786\) 0.644864 0.0230015
\(787\) 34.1432 1.21707 0.608536 0.793526i \(-0.291757\pi\)
0.608536 + 0.793526i \(0.291757\pi\)
\(788\) 1.36883 0.0487624
\(789\) −4.82946 −0.171933
\(790\) 13.6270 0.484829
\(791\) −38.7747 −1.37867
\(792\) 0 0
\(793\) −18.6803 −0.663357
\(794\) 45.7693 1.62429
\(795\) 11.5931 0.411165
\(796\) −3.66337 −0.129845
\(797\) −3.04303 −0.107790 −0.0538949 0.998547i \(-0.517164\pi\)
−0.0538949 + 0.998547i \(0.517164\pi\)
\(798\) 7.39855 0.261906
\(799\) 0.205805 0.00728087
\(800\) −1.02811 −0.0363491
\(801\) 0.583290 0.0206096
\(802\) −9.40439 −0.332081
\(803\) 0 0
\(804\) −1.77886 −0.0627355
\(805\) 4.67961 0.164934
\(806\) 21.4145 0.754294
\(807\) −5.22929 −0.184080
\(808\) −52.4399 −1.84483
\(809\) 11.1011 0.390294 0.195147 0.980774i \(-0.437482\pi\)
0.195147 + 0.980774i \(0.437482\pi\)
\(810\) 1.47726 0.0519056
\(811\) −41.3262 −1.45116 −0.725579 0.688139i \(-0.758428\pi\)
−0.725579 + 0.688139i \(0.758428\pi\)
\(812\) 4.16082 0.146016
\(813\) −30.3311 −1.06376
\(814\) 0 0
\(815\) −11.3321 −0.396946
\(816\) 2.35819 0.0825532
\(817\) 1.35458 0.0473907
\(818\) 9.26056 0.323788
\(819\) 4.91300 0.171674
\(820\) 0.332296 0.0116043
\(821\) −8.17883 −0.285443 −0.142722 0.989763i \(-0.545585\pi\)
−0.142722 + 0.989763i \(0.545585\pi\)
\(822\) −11.6229 −0.405396
\(823\) 13.1973 0.460029 0.230014 0.973187i \(-0.426123\pi\)
0.230014 + 0.973187i \(0.426123\pi\)
\(824\) −34.9246 −1.21665
\(825\) 0 0
\(826\) −27.3862 −0.952889
\(827\) 30.7370 1.06883 0.534415 0.845222i \(-0.320532\pi\)
0.534415 + 0.845222i \(0.320532\pi\)
\(828\) 0.371718 0.0129181
\(829\) 3.68790 0.128086 0.0640430 0.997947i \(-0.479601\pi\)
0.0640430 + 0.997947i \(0.479601\pi\)
\(830\) 13.9635 0.484681
\(831\) −15.9694 −0.553974
\(832\) 15.2936 0.530209
\(833\) 0.943604 0.0326939
\(834\) −26.0635 −0.902505
\(835\) −15.1001 −0.522561
\(836\) 0 0
\(837\) 6.77143 0.234055
\(838\) 5.76621 0.199191
\(839\) 43.5517 1.50357 0.751786 0.659407i \(-0.229192\pi\)
0.751786 + 0.659407i \(0.229192\pi\)
\(840\) −6.16248 −0.212626
\(841\) 69.9118 2.41075
\(842\) 25.6623 0.884381
\(843\) 1.39067 0.0478972
\(844\) −0.987795 −0.0340013
\(845\) −8.41709 −0.289557
\(846\) −0.558418 −0.0191988
\(847\) 0 0
\(848\) −50.2139 −1.72435
\(849\) −5.87453 −0.201613
\(850\) −0.804288 −0.0275869
\(851\) −17.9838 −0.616478
\(852\) −2.73387 −0.0936607
\(853\) 30.3434 1.03894 0.519469 0.854489i \(-0.326130\pi\)
0.519469 + 0.854489i \(0.326130\pi\)
\(854\) −29.5832 −1.01232
\(855\) 2.18230 0.0746330
\(856\) 5.29941 0.181130
\(857\) −12.5402 −0.428365 −0.214182 0.976794i \(-0.568709\pi\)
−0.214182 + 0.976794i \(0.568709\pi\)
\(858\) 0 0
\(859\) −8.67783 −0.296084 −0.148042 0.988981i \(-0.547297\pi\)
−0.148042 + 0.988981i \(0.547297\pi\)
\(860\) 0.113154 0.00385852
\(861\) 4.18332 0.142567
\(862\) 10.3948 0.354047
\(863\) −38.3918 −1.30687 −0.653437 0.756981i \(-0.726673\pi\)
−0.653437 + 0.756981i \(0.726673\pi\)
\(864\) −1.02811 −0.0349770
\(865\) −18.2641 −0.620998
\(866\) 4.06585 0.138163
\(867\) −16.7036 −0.567283
\(868\) 2.83293 0.0961559
\(869\) 0 0
\(870\) 14.6920 0.498105
\(871\) −20.8897 −0.707821
\(872\) 28.5203 0.965818
\(873\) −5.37801 −0.182018
\(874\) 6.57362 0.222356
\(875\) 2.29496 0.0775839
\(876\) −1.43257 −0.0484021
\(877\) −31.2898 −1.05658 −0.528291 0.849064i \(-0.677167\pi\)
−0.528291 + 0.849064i \(0.677167\pi\)
\(878\) 4.04331 0.136455
\(879\) −19.9745 −0.673723
\(880\) 0 0
\(881\) 49.2703 1.65996 0.829979 0.557795i \(-0.188353\pi\)
0.829979 + 0.557795i \(0.188353\pi\)
\(882\) −2.56031 −0.0862100
\(883\) 28.1861 0.948539 0.474270 0.880380i \(-0.342712\pi\)
0.474270 + 0.880380i \(0.342712\pi\)
\(884\) −0.212474 −0.00714626
\(885\) −8.07792 −0.271536
\(886\) 16.4038 0.551096
\(887\) 25.4943 0.856016 0.428008 0.903775i \(-0.359216\pi\)
0.428008 + 0.903775i \(0.359216\pi\)
\(888\) 23.6825 0.794734
\(889\) −9.48390 −0.318080
\(890\) 0.861672 0.0288833
\(891\) 0 0
\(892\) 2.86342 0.0958743
\(893\) −0.824928 −0.0276052
\(894\) 16.1910 0.541507
\(895\) −0.525419 −0.0175628
\(896\) 28.9387 0.966775
\(897\) 4.36520 0.145750
\(898\) 21.9031 0.730916
\(899\) 67.3449 2.24608
\(900\) 0.182297 0.00607657
\(901\) −6.31181 −0.210277
\(902\) 0 0
\(903\) 1.42451 0.0474048
\(904\) 45.3683 1.50893
\(905\) −17.7106 −0.588722
\(906\) 29.5930 0.983162
\(907\) 2.48054 0.0823649 0.0411825 0.999152i \(-0.486888\pi\)
0.0411825 + 0.999152i \(0.486888\pi\)
\(908\) 0.268235 0.00890168
\(909\) 19.5291 0.647738
\(910\) 7.25777 0.240593
\(911\) 2.71559 0.0899716 0.0449858 0.998988i \(-0.485676\pi\)
0.0449858 + 0.998988i \(0.485676\pi\)
\(912\) −9.45232 −0.312998
\(913\) 0 0
\(914\) −43.1101 −1.42595
\(915\) −8.72595 −0.288471
\(916\) −2.91221 −0.0962220
\(917\) 1.00181 0.0330828
\(918\) −0.804288 −0.0265455
\(919\) −3.23900 −0.106845 −0.0534224 0.998572i \(-0.517013\pi\)
−0.0534224 + 0.998572i \(0.517013\pi\)
\(920\) −5.47537 −0.180518
\(921\) −19.4372 −0.640479
\(922\) 46.0607 1.51693
\(923\) −32.1047 −1.05674
\(924\) 0 0
\(925\) −8.81959 −0.289986
\(926\) 60.8102 1.99835
\(927\) 13.0062 0.427180
\(928\) −10.2250 −0.335652
\(929\) 7.18207 0.235636 0.117818 0.993035i \(-0.462410\pi\)
0.117818 + 0.993035i \(0.462410\pi\)
\(930\) 10.0032 0.328017
\(931\) −3.78224 −0.123958
\(932\) 4.67376 0.153094
\(933\) 6.02182 0.197145
\(934\) 57.2832 1.87436
\(935\) 0 0
\(936\) −5.74845 −0.187894
\(937\) −20.3127 −0.663586 −0.331793 0.943352i \(-0.607654\pi\)
−0.331793 + 0.943352i \(0.607654\pi\)
\(938\) −33.0822 −1.08017
\(939\) −5.10833 −0.166704
\(940\) −0.0689100 −0.00224760
\(941\) 25.4246 0.828818 0.414409 0.910091i \(-0.363988\pi\)
0.414409 + 0.910091i \(0.363988\pi\)
\(942\) 13.5238 0.440630
\(943\) 3.71689 0.121038
\(944\) 34.9884 1.13878
\(945\) 2.29496 0.0746551
\(946\) 0 0
\(947\) −4.55536 −0.148029 −0.0740147 0.997257i \(-0.523581\pi\)
−0.0740147 + 0.997257i \(0.523581\pi\)
\(948\) 1.68161 0.0546161
\(949\) −16.8231 −0.546102
\(950\) 3.22382 0.104595
\(951\) 13.6788 0.443566
\(952\) 3.35514 0.108741
\(953\) 41.5855 1.34709 0.673543 0.739148i \(-0.264772\pi\)
0.673543 + 0.739148i \(0.264772\pi\)
\(954\) 17.1260 0.554475
\(955\) −16.5519 −0.535608
\(956\) 0.762378 0.0246571
\(957\) 0 0
\(958\) −24.6519 −0.796467
\(959\) −18.0565 −0.583076
\(960\) 7.14394 0.230570
\(961\) 14.8523 0.479107
\(962\) −27.8918 −0.899267
\(963\) −1.97355 −0.0635967
\(964\) 0.600085 0.0193274
\(965\) 24.5032 0.788787
\(966\) 6.91300 0.222422
\(967\) 16.2161 0.521476 0.260738 0.965410i \(-0.416034\pi\)
0.260738 + 0.965410i \(0.416034\pi\)
\(968\) 0 0
\(969\) −1.18814 −0.0381686
\(970\) −7.94472 −0.255090
\(971\) −13.4705 −0.432290 −0.216145 0.976361i \(-0.569348\pi\)
−0.216145 + 0.976361i \(0.569348\pi\)
\(972\) 0.182297 0.00584718
\(973\) −40.4904 −1.29806
\(974\) 3.71078 0.118901
\(975\) 2.14077 0.0685596
\(976\) 37.7953 1.20980
\(977\) −14.5400 −0.465177 −0.232588 0.972575i \(-0.574719\pi\)
−0.232588 + 0.972575i \(0.574719\pi\)
\(978\) −16.7404 −0.535300
\(979\) 0 0
\(980\) −0.315947 −0.0100926
\(981\) −10.6212 −0.339109
\(982\) 11.1661 0.356324
\(983\) −24.5938 −0.784422 −0.392211 0.919875i \(-0.628290\pi\)
−0.392211 + 0.919875i \(0.628290\pi\)
\(984\) −4.89469 −0.156037
\(985\) 7.50877 0.239249
\(986\) −7.99900 −0.254740
\(987\) −0.867517 −0.0276134
\(988\) 0.851656 0.0270948
\(989\) 1.26568 0.0402463
\(990\) 0 0
\(991\) 4.43775 0.140970 0.0704848 0.997513i \(-0.477545\pi\)
0.0704848 + 0.997513i \(0.477545\pi\)
\(992\) −6.96177 −0.221036
\(993\) 11.4695 0.363972
\(994\) −50.8429 −1.61264
\(995\) −20.0956 −0.637073
\(996\) 1.72313 0.0545995
\(997\) −0.0104705 −0.000331605 0 −0.000165802 1.00000i \(-0.500053\pi\)
−0.000165802 1.00000i \(0.500053\pi\)
\(998\) 24.8127 0.785431
\(999\) −8.81959 −0.279040
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.1.3 4
3.2 odd 2 5445.2.a.bf.1.2 4
5.4 even 2 9075.2.a.cm.1.2 4
11.7 odd 10 165.2.m.d.16.1 8
11.8 odd 10 165.2.m.d.31.1 yes 8
11.10 odd 2 1815.2.a.p.1.2 4
33.8 even 10 495.2.n.a.361.2 8
33.29 even 10 495.2.n.a.181.2 8
33.32 even 2 5445.2.a.bt.1.3 4
55.7 even 20 825.2.bx.f.49.4 16
55.8 even 20 825.2.bx.f.724.4 16
55.18 even 20 825.2.bx.f.49.1 16
55.19 odd 10 825.2.n.g.526.2 8
55.29 odd 10 825.2.n.g.676.2 8
55.52 even 20 825.2.bx.f.724.1 16
55.54 odd 2 9075.2.a.di.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.16.1 8 11.7 odd 10
165.2.m.d.31.1 yes 8 11.8 odd 10
495.2.n.a.181.2 8 33.29 even 10
495.2.n.a.361.2 8 33.8 even 10
825.2.n.g.526.2 8 55.19 odd 10
825.2.n.g.676.2 8 55.29 odd 10
825.2.bx.f.49.1 16 55.18 even 20
825.2.bx.f.49.4 16 55.7 even 20
825.2.bx.f.724.1 16 55.52 even 20
825.2.bx.f.724.4 16 55.8 even 20
1815.2.a.p.1.2 4 11.10 odd 2
1815.2.a.w.1.3 4 1.1 even 1 trivial
5445.2.a.bf.1.2 4 3.2 odd 2
5445.2.a.bt.1.3 4 33.32 even 2
9075.2.a.cm.1.2 4 5.4 even 2
9075.2.a.di.1.3 4 55.54 odd 2