Properties

Label 5445.2.a.bs.1.4
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
Defining polynomial: \(x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.69696\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.69696 q^{2} +5.27358 q^{4} -1.00000 q^{5} +4.13176 q^{7} +8.82872 q^{8} +O(q^{10})\) \(q+2.69696 q^{2} +5.27358 q^{4} -1.00000 q^{5} +4.13176 q^{7} +8.82872 q^{8} -2.69696 q^{10} -2.73785 q^{13} +11.1432 q^{14} +13.2635 q^{16} +2.41541 q^{17} +1.15325 q^{19} -5.27358 q^{20} +8.54717 q^{23} +1.00000 q^{25} -7.38386 q^{26} +21.7892 q^{28} -1.67756 q^{29} -10.2635 q^{31} +18.1137 q^{32} +6.51425 q^{34} -4.13176 q^{35} +5.71636 q^{37} +3.11028 q^{38} -8.82872 q^{40} -9.11028 q^{41} -8.13176 q^{43} +23.0514 q^{46} -1.47569 q^{47} +10.0715 q^{49} +2.69696 q^{50} -14.4383 q^{52} +2.54717 q^{53} +36.4782 q^{56} -4.52431 q^{58} +8.24066 q^{59} -3.47569 q^{61} -27.6803 q^{62} +22.3249 q^{64} +2.73785 q^{65} +15.3350 q^{67} +12.7378 q^{68} -11.1432 q^{70} -2.54717 q^{71} -2.73785 q^{73} +15.4168 q^{74} +6.08178 q^{76} -5.15325 q^{79} -13.2635 q^{80} -24.5700 q^{82} +9.28502 q^{83} -2.41541 q^{85} -21.9310 q^{86} -4.78783 q^{89} -11.3121 q^{91} +45.0742 q^{92} -3.97989 q^{94} -1.15325 q^{95} +4.24066 q^{97} +27.1624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{4} - 4 q^{5} - 4 q^{7} + 6 q^{8} + O(q^{10}) \) \( 4 q + 2 q^{2} + 8 q^{4} - 4 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} - 8 q^{13} + 8 q^{14} + 12 q^{16} + 4 q^{17} - 4 q^{19} - 8 q^{20} + 8 q^{23} + 4 q^{25} + 16 q^{26} + 8 q^{28} - 4 q^{29} + 14 q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} - 20 q^{38} - 6 q^{40} - 4 q^{41} - 12 q^{43} + 16 q^{46} + 20 q^{49} + 2 q^{50} - 20 q^{52} - 16 q^{53} + 48 q^{56} - 24 q^{58} + 24 q^{59} - 8 q^{61} - 20 q^{62} + 8 q^{65} + 48 q^{68} - 8 q^{70} + 16 q^{71} - 8 q^{73} + 12 q^{74} + 36 q^{76} - 12 q^{79} - 12 q^{80} - 40 q^{82} + 8 q^{83} - 4 q^{85} - 16 q^{86} + 16 q^{89} - 16 q^{91} + 72 q^{92} + 40 q^{94} + 4 q^{95} + 8 q^{97} + 62 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.69696 1.90704 0.953519 0.301334i \(-0.0974317\pi\)
0.953519 + 0.301334i \(0.0974317\pi\)
\(3\) 0 0
\(4\) 5.27358 2.63679
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.13176 1.56166 0.780830 0.624744i \(-0.214796\pi\)
0.780830 + 0.624744i \(0.214796\pi\)
\(8\) 8.82872 3.12142
\(9\) 0 0
\(10\) −2.69696 −0.852853
\(11\) 0 0
\(12\) 0 0
\(13\) −2.73785 −0.759342 −0.379671 0.925122i \(-0.623963\pi\)
−0.379671 + 0.925122i \(0.623963\pi\)
\(14\) 11.1432 2.97814
\(15\) 0 0
\(16\) 13.2635 3.31588
\(17\) 2.41541 0.585822 0.292911 0.956140i \(-0.405376\pi\)
0.292911 + 0.956140i \(0.405376\pi\)
\(18\) 0 0
\(19\) 1.15325 0.264574 0.132287 0.991211i \(-0.457768\pi\)
0.132287 + 0.991211i \(0.457768\pi\)
\(20\) −5.27358 −1.17921
\(21\) 0 0
\(22\) 0 0
\(23\) 8.54717 1.78221 0.891104 0.453799i \(-0.149932\pi\)
0.891104 + 0.453799i \(0.149932\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −7.38386 −1.44809
\(27\) 0 0
\(28\) 21.7892 4.11777
\(29\) −1.67756 −0.311515 −0.155757 0.987795i \(-0.549782\pi\)
−0.155757 + 0.987795i \(0.549782\pi\)
\(30\) 0 0
\(31\) −10.2635 −1.84338 −0.921692 0.387922i \(-0.873193\pi\)
−0.921692 + 0.387922i \(0.873193\pi\)
\(32\) 18.1137 3.20209
\(33\) 0 0
\(34\) 6.51425 1.11718
\(35\) −4.13176 −0.698396
\(36\) 0 0
\(37\) 5.71636 0.939764 0.469882 0.882729i \(-0.344297\pi\)
0.469882 + 0.882729i \(0.344297\pi\)
\(38\) 3.11028 0.504553
\(39\) 0 0
\(40\) −8.82872 −1.39594
\(41\) −9.11028 −1.42279 −0.711393 0.702794i \(-0.751935\pi\)
−0.711393 + 0.702794i \(0.751935\pi\)
\(42\) 0 0
\(43\) −8.13176 −1.24008 −0.620041 0.784569i \(-0.712884\pi\)
−0.620041 + 0.784569i \(0.712884\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 23.0514 3.39874
\(47\) −1.47569 −0.215252 −0.107626 0.994191i \(-0.534325\pi\)
−0.107626 + 0.994191i \(0.534325\pi\)
\(48\) 0 0
\(49\) 10.0715 1.43878
\(50\) 2.69696 0.381408
\(51\) 0 0
\(52\) −14.4383 −2.00223
\(53\) 2.54717 0.349881 0.174940 0.984579i \(-0.444027\pi\)
0.174940 + 0.984579i \(0.444027\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 36.4782 4.87460
\(57\) 0 0
\(58\) −4.52431 −0.594070
\(59\) 8.24066 1.07284 0.536422 0.843950i \(-0.319776\pi\)
0.536422 + 0.843950i \(0.319776\pi\)
\(60\) 0 0
\(61\) −3.47569 −0.445017 −0.222509 0.974931i \(-0.571425\pi\)
−0.222509 + 0.974931i \(0.571425\pi\)
\(62\) −27.6803 −3.51540
\(63\) 0 0
\(64\) 22.3249 2.79062
\(65\) 2.73785 0.339588
\(66\) 0 0
\(67\) 15.3350 1.87347 0.936734 0.350041i \(-0.113832\pi\)
0.936734 + 0.350041i \(0.113832\pi\)
\(68\) 12.7378 1.54469
\(69\) 0 0
\(70\) −11.1432 −1.33187
\(71\) −2.54717 −0.302293 −0.151147 0.988511i \(-0.548297\pi\)
−0.151147 + 0.988511i \(0.548297\pi\)
\(72\) 0 0
\(73\) −2.73785 −0.320441 −0.160220 0.987081i \(-0.551220\pi\)
−0.160220 + 0.987081i \(0.551220\pi\)
\(74\) 15.4168 1.79216
\(75\) 0 0
\(76\) 6.08178 0.697628
\(77\) 0 0
\(78\) 0 0
\(79\) −5.15325 −0.579786 −0.289893 0.957059i \(-0.593620\pi\)
−0.289893 + 0.957059i \(0.593620\pi\)
\(80\) −13.2635 −1.48291
\(81\) 0 0
\(82\) −24.5700 −2.71331
\(83\) 9.28502 1.01916 0.509582 0.860422i \(-0.329800\pi\)
0.509582 + 0.860422i \(0.329800\pi\)
\(84\) 0 0
\(85\) −2.41541 −0.261988
\(86\) −21.9310 −2.36488
\(87\) 0 0
\(88\) 0 0
\(89\) −4.78783 −0.507509 −0.253755 0.967269i \(-0.581666\pi\)
−0.253755 + 0.967269i \(0.581666\pi\)
\(90\) 0 0
\(91\) −11.3121 −1.18583
\(92\) 45.0742 4.69931
\(93\) 0 0
\(94\) −3.97989 −0.410494
\(95\) −1.15325 −0.118321
\(96\) 0 0
\(97\) 4.24066 0.430574 0.215287 0.976551i \(-0.430931\pi\)
0.215287 + 0.976551i \(0.430931\pi\)
\(98\) 27.1624 2.74381
\(99\) 0 0
\(100\) 5.27358 0.527358
\(101\) −13.9411 −1.38719 −0.693595 0.720365i \(-0.743974\pi\)
−0.693595 + 0.720365i \(0.743974\pi\)
\(102\) 0 0
\(103\) −18.5042 −1.82327 −0.911636 0.410998i \(-0.865180\pi\)
−0.911636 + 0.410998i \(0.865180\pi\)
\(104\) −24.1717 −2.37023
\(105\) 0 0
\(106\) 6.86961 0.667236
\(107\) 13.7663 1.33084 0.665421 0.746468i \(-0.268252\pi\)
0.665421 + 0.746468i \(0.268252\pi\)
\(108\) 0 0
\(109\) −12.7878 −1.22485 −0.612426 0.790528i \(-0.709806\pi\)
−0.612426 + 0.790528i \(0.709806\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 54.8018 5.17828
\(113\) 14.8107 1.39327 0.696637 0.717424i \(-0.254679\pi\)
0.696637 + 0.717424i \(0.254679\pi\)
\(114\) 0 0
\(115\) −8.54717 −0.797028
\(116\) −8.84675 −0.821400
\(117\) 0 0
\(118\) 22.2247 2.04595
\(119\) 9.97989 0.914855
\(120\) 0 0
\(121\) 0 0
\(122\) −9.37380 −0.848664
\(123\) 0 0
\(124\) −54.1256 −4.86062
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.9626 −1.15024 −0.575121 0.818068i \(-0.695045\pi\)
−0.575121 + 0.818068i \(0.695045\pi\)
\(128\) 23.9820 2.11973
\(129\) 0 0
\(130\) 7.38386 0.647607
\(131\) −1.47569 −0.128932 −0.0644660 0.997920i \(-0.520534\pi\)
−0.0644660 + 0.997920i \(0.520534\pi\)
\(132\) 0 0
\(133\) 4.76497 0.413175
\(134\) 41.3579 3.57278
\(135\) 0 0
\(136\) 21.3249 1.82860
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 0.889725 0.0754655 0.0377327 0.999288i \(-0.487986\pi\)
0.0377327 + 0.999288i \(0.487986\pi\)
\(140\) −21.7892 −1.84152
\(141\) 0 0
\(142\) −6.86961 −0.576485
\(143\) 0 0
\(144\) 0 0
\(145\) 1.67756 0.139314
\(146\) −7.38386 −0.611093
\(147\) 0 0
\(148\) 30.1457 2.47796
\(149\) 12.7289 1.04279 0.521397 0.853314i \(-0.325411\pi\)
0.521397 + 0.853314i \(0.325411\pi\)
\(150\) 0 0
\(151\) 1.15325 0.0938504 0.0469252 0.998898i \(-0.485058\pi\)
0.0469252 + 0.998898i \(0.485058\pi\)
\(152\) 10.1818 0.825849
\(153\) 0 0
\(154\) 0 0
\(155\) 10.2635 0.824386
\(156\) 0 0
\(157\) −3.95702 −0.315805 −0.157902 0.987455i \(-0.550473\pi\)
−0.157902 + 0.987455i \(0.550473\pi\)
\(158\) −13.8981 −1.10567
\(159\) 0 0
\(160\) −18.1137 −1.43202
\(161\) 35.3149 2.78320
\(162\) 0 0
\(163\) 3.07148 0.240577 0.120288 0.992739i \(-0.461618\pi\)
0.120288 + 0.992739i \(0.461618\pi\)
\(164\) −48.0438 −3.75159
\(165\) 0 0
\(166\) 25.0413 1.94358
\(167\) 2.71498 0.210092 0.105046 0.994467i \(-0.466501\pi\)
0.105046 + 0.994467i \(0.466501\pi\)
\(168\) 0 0
\(169\) −5.50419 −0.423399
\(170\) −6.51425 −0.499620
\(171\) 0 0
\(172\) −42.8835 −3.26984
\(173\) 14.4154 1.09598 0.547992 0.836484i \(-0.315393\pi\)
0.547992 + 0.836484i \(0.315393\pi\)
\(174\) 0 0
\(175\) 4.13176 0.312332
\(176\) 0 0
\(177\) 0 0
\(178\) −12.9126 −0.967839
\(179\) −10.7878 −0.806321 −0.403160 0.915129i \(-0.632088\pi\)
−0.403160 + 0.915129i \(0.632088\pi\)
\(180\) 0 0
\(181\) −6.50419 −0.483453 −0.241726 0.970344i \(-0.577714\pi\)
−0.241726 + 0.970344i \(0.577714\pi\)
\(182\) −30.5084 −2.26143
\(183\) 0 0
\(184\) 75.4606 5.56303
\(185\) −5.71636 −0.420275
\(186\) 0 0
\(187\) 0 0
\(188\) −7.78220 −0.567575
\(189\) 0 0
\(190\) −3.11028 −0.225643
\(191\) −15.8822 −1.14919 −0.574597 0.818437i \(-0.694841\pi\)
−0.574597 + 0.818437i \(0.694841\pi\)
\(192\) 0 0
\(193\) 3.83219 0.275847 0.137923 0.990443i \(-0.455957\pi\)
0.137923 + 0.990443i \(0.455957\pi\)
\(194\) 11.4369 0.821121
\(195\) 0 0
\(196\) 53.1128 3.79377
\(197\) −2.67893 −0.190866 −0.0954331 0.995436i \(-0.530424\pi\)
−0.0954331 + 0.995436i \(0.530424\pi\)
\(198\) 0 0
\(199\) 6.78783 0.481177 0.240588 0.970627i \(-0.422660\pi\)
0.240588 + 0.970627i \(0.422660\pi\)
\(200\) 8.82872 0.624285
\(201\) 0 0
\(202\) −37.5985 −2.64542
\(203\) −6.93128 −0.486480
\(204\) 0 0
\(205\) 9.11028 0.636289
\(206\) −49.9050 −3.47705
\(207\) 0 0
\(208\) −36.3135 −2.51789
\(209\) 0 0
\(210\) 0 0
\(211\) −17.6803 −1.21716 −0.608581 0.793491i \(-0.708261\pi\)
−0.608581 + 0.793491i \(0.708261\pi\)
\(212\) 13.4327 0.922563
\(213\) 0 0
\(214\) 37.1273 2.53797
\(215\) 8.13176 0.554582
\(216\) 0 0
\(217\) −42.4065 −2.87874
\(218\) −34.4883 −2.33584
\(219\) 0 0
\(220\) 0 0
\(221\) −6.61301 −0.444839
\(222\) 0 0
\(223\) −8.83081 −0.591355 −0.295677 0.955288i \(-0.595545\pi\)
−0.295677 + 0.955288i \(0.595545\pi\)
\(224\) 74.8417 5.00057
\(225\) 0 0
\(226\) 39.9438 2.65702
\(227\) 10.4972 0.696723 0.348361 0.937360i \(-0.386738\pi\)
0.348361 + 0.937360i \(0.386738\pi\)
\(228\) 0 0
\(229\) 4.95139 0.327197 0.163598 0.986527i \(-0.447690\pi\)
0.163598 + 0.986527i \(0.447690\pi\)
\(230\) −23.0514 −1.51996
\(231\) 0 0
\(232\) −14.8107 −0.972370
\(233\) −3.89110 −0.254914 −0.127457 0.991844i \(-0.540682\pi\)
−0.127457 + 0.991844i \(0.540682\pi\)
\(234\) 0 0
\(235\) 1.47569 0.0962637
\(236\) 43.4578 2.82886
\(237\) 0 0
\(238\) 26.9153 1.74466
\(239\) −15.6186 −1.01029 −0.505143 0.863036i \(-0.668560\pi\)
−0.505143 + 0.863036i \(0.668560\pi\)
\(240\) 0 0
\(241\) 10.2635 0.661132 0.330566 0.943783i \(-0.392760\pi\)
0.330566 + 0.943783i \(0.392760\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −18.3294 −1.17342
\(245\) −10.0715 −0.643443
\(246\) 0 0
\(247\) −3.15743 −0.200902
\(248\) −90.6138 −5.75398
\(249\) 0 0
\(250\) −2.69696 −0.170571
\(251\) 19.6415 1.23976 0.619881 0.784696i \(-0.287181\pi\)
0.619881 + 0.784696i \(0.287181\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −34.9595 −2.19356
\(255\) 0 0
\(256\) 20.0285 1.25178
\(257\) 1.16919 0.0729320 0.0364660 0.999335i \(-0.488390\pi\)
0.0364660 + 0.999335i \(0.488390\pi\)
\(258\) 0 0
\(259\) 23.6186 1.46759
\(260\) 14.4383 0.895424
\(261\) 0 0
\(262\) −3.97989 −0.245878
\(263\) 8.93553 0.550989 0.275494 0.961303i \(-0.411158\pi\)
0.275494 + 0.961303i \(0.411158\pi\)
\(264\) 0 0
\(265\) −2.54717 −0.156471
\(266\) 12.8509 0.787941
\(267\) 0 0
\(268\) 80.8704 4.93995
\(269\) 14.9514 0.911602 0.455801 0.890082i \(-0.349353\pi\)
0.455801 + 0.890082i \(0.349353\pi\)
\(270\) 0 0
\(271\) 22.7289 1.38068 0.690342 0.723483i \(-0.257460\pi\)
0.690342 + 0.723483i \(0.257460\pi\)
\(272\) 32.0368 1.94252
\(273\) 0 0
\(274\) 16.1818 0.977575
\(275\) 0 0
\(276\) 0 0
\(277\) 22.1387 1.33019 0.665093 0.746761i \(-0.268392\pi\)
0.665093 + 0.746761i \(0.268392\pi\)
\(278\) 2.39955 0.143915
\(279\) 0 0
\(280\) −36.4782 −2.17999
\(281\) −19.9841 −1.19215 −0.596075 0.802929i \(-0.703274\pi\)
−0.596075 + 0.802929i \(0.703274\pi\)
\(282\) 0 0
\(283\) −24.2775 −1.44315 −0.721573 0.692339i \(-0.756580\pi\)
−0.721573 + 0.692339i \(0.756580\pi\)
\(284\) −13.4327 −0.797085
\(285\) 0 0
\(286\) 0 0
\(287\) −37.6415 −2.22191
\(288\) 0 0
\(289\) −11.1658 −0.656813
\(290\) 4.52431 0.265676
\(291\) 0 0
\(292\) −14.4383 −0.844936
\(293\) −16.4182 −0.959159 −0.479579 0.877498i \(-0.659211\pi\)
−0.479579 + 0.877498i \(0.659211\pi\)
\(294\) 0 0
\(295\) −8.24066 −0.479790
\(296\) 50.4681 2.93340
\(297\) 0 0
\(298\) 34.3294 1.98865
\(299\) −23.4008 −1.35331
\(300\) 0 0
\(301\) −33.5985 −1.93659
\(302\) 3.11028 0.178976
\(303\) 0 0
\(304\) 15.2962 0.877297
\(305\) 3.47569 0.199018
\(306\) 0 0
\(307\) −5.44390 −0.310700 −0.155350 0.987859i \(-0.549651\pi\)
−0.155350 + 0.987859i \(0.549651\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 27.6803 1.57214
\(311\) −28.2864 −1.60397 −0.801987 0.597341i \(-0.796224\pi\)
−0.801987 + 0.597341i \(0.796224\pi\)
\(312\) 0 0
\(313\) 22.8107 1.28934 0.644668 0.764462i \(-0.276995\pi\)
0.644668 + 0.764462i \(0.276995\pi\)
\(314\) −10.6719 −0.602252
\(315\) 0 0
\(316\) −27.1761 −1.52878
\(317\) −23.8593 −1.34007 −0.670036 0.742328i \(-0.733721\pi\)
−0.670036 + 0.742328i \(0.733721\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −22.3249 −1.24800
\(321\) 0 0
\(322\) 95.2428 5.30767
\(323\) 2.78557 0.154993
\(324\) 0 0
\(325\) −2.73785 −0.151868
\(326\) 8.28364 0.458788
\(327\) 0 0
\(328\) −80.4321 −4.44112
\(329\) −6.09722 −0.336151
\(330\) 0 0
\(331\) −18.9084 −1.03930 −0.519650 0.854379i \(-0.673938\pi\)
−0.519650 + 0.854379i \(0.673938\pi\)
\(332\) 48.9653 2.68732
\(333\) 0 0
\(334\) 7.32220 0.400653
\(335\) −15.3350 −0.837841
\(336\) 0 0
\(337\) −24.6630 −1.34348 −0.671740 0.740787i \(-0.734453\pi\)
−0.671740 + 0.740787i \(0.734453\pi\)
\(338\) −14.8446 −0.807439
\(339\) 0 0
\(340\) −12.7378 −0.690807
\(341\) 0 0
\(342\) 0 0
\(343\) 12.6906 0.685229
\(344\) −71.7931 −3.87082
\(345\) 0 0
\(346\) 38.8778 2.09008
\(347\) −15.5056 −0.832383 −0.416191 0.909277i \(-0.636635\pi\)
−0.416191 + 0.909277i \(0.636635\pi\)
\(348\) 0 0
\(349\) −20.4841 −1.09649 −0.548244 0.836319i \(-0.684703\pi\)
−0.548244 + 0.836319i \(0.684703\pi\)
\(350\) 11.1432 0.595629
\(351\) 0 0
\(352\) 0 0
\(353\) 21.9799 1.16987 0.584936 0.811080i \(-0.301120\pi\)
0.584936 + 0.811080i \(0.301120\pi\)
\(354\) 0 0
\(355\) 2.54717 0.135190
\(356\) −25.2490 −1.33820
\(357\) 0 0
\(358\) −29.0943 −1.53768
\(359\) −19.6962 −1.03953 −0.519764 0.854310i \(-0.673980\pi\)
−0.519764 + 0.854310i \(0.673980\pi\)
\(360\) 0 0
\(361\) −17.6700 −0.930000
\(362\) −17.5415 −0.921963
\(363\) 0 0
\(364\) −59.6555 −3.12680
\(365\) 2.73785 0.143305
\(366\) 0 0
\(367\) −28.2635 −1.47534 −0.737672 0.675159i \(-0.764075\pi\)
−0.737672 + 0.675159i \(0.764075\pi\)
\(368\) 113.366 5.90959
\(369\) 0 0
\(370\) −15.4168 −0.801480
\(371\) 10.5243 0.546395
\(372\) 0 0
\(373\) 9.78921 0.506866 0.253433 0.967353i \(-0.418440\pi\)
0.253433 + 0.967353i \(0.418440\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −13.0285 −0.671893
\(377\) 4.59290 0.236546
\(378\) 0 0
\(379\) 5.66162 0.290818 0.145409 0.989372i \(-0.453550\pi\)
0.145409 + 0.989372i \(0.453550\pi\)
\(380\) −6.08178 −0.311989
\(381\) 0 0
\(382\) −42.8336 −2.19156
\(383\) 26.5042 1.35430 0.677150 0.735845i \(-0.263215\pi\)
0.677150 + 0.735845i \(0.263215\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.3352 0.526050
\(387\) 0 0
\(388\) 22.3635 1.13533
\(389\) 15.3551 0.778535 0.389268 0.921125i \(-0.372728\pi\)
0.389268 + 0.921125i \(0.372728\pi\)
\(390\) 0 0
\(391\) 20.6449 1.04406
\(392\) 88.9183 4.49105
\(393\) 0 0
\(394\) −7.22497 −0.363989
\(395\) 5.15325 0.259288
\(396\) 0 0
\(397\) 16.7677 0.841548 0.420774 0.907166i \(-0.361759\pi\)
0.420774 + 0.907166i \(0.361759\pi\)
\(398\) 18.3065 0.917622
\(399\) 0 0
\(400\) 13.2635 0.663176
\(401\) 26.0457 1.30066 0.650331 0.759651i \(-0.274630\pi\)
0.650331 + 0.759651i \(0.274630\pi\)
\(402\) 0 0
\(403\) 28.1000 1.39976
\(404\) −73.5195 −3.65773
\(405\) 0 0
\(406\) −18.6934 −0.927736
\(407\) 0 0
\(408\) 0 0
\(409\) −22.4495 −1.11005 −0.555027 0.831832i \(-0.687292\pi\)
−0.555027 + 0.831832i \(0.687292\pi\)
\(410\) 24.5700 1.21343
\(411\) 0 0
\(412\) −97.5834 −4.80759
\(413\) 34.0485 1.67542
\(414\) 0 0
\(415\) −9.28502 −0.455784
\(416\) −49.5927 −2.43148
\(417\) 0 0
\(418\) 0 0
\(419\) −27.8822 −1.36213 −0.681067 0.732221i \(-0.738484\pi\)
−0.681067 + 0.732221i \(0.738484\pi\)
\(420\) 0 0
\(421\) 15.5985 0.760226 0.380113 0.924940i \(-0.375885\pi\)
0.380113 + 0.924940i \(0.375885\pi\)
\(422\) −47.6831 −2.32118
\(423\) 0 0
\(424\) 22.4883 1.09213
\(425\) 2.41541 0.117164
\(426\) 0 0
\(427\) −14.3608 −0.694965
\(428\) 72.5980 3.50916
\(429\) 0 0
\(430\) 21.9310 1.05761
\(431\) 2.95139 0.142163 0.0710817 0.997470i \(-0.477355\pi\)
0.0710817 + 0.997470i \(0.477355\pi\)
\(432\) 0 0
\(433\) 24.2864 1.16713 0.583565 0.812067i \(-0.301657\pi\)
0.583565 + 0.812067i \(0.301657\pi\)
\(434\) −114.369 −5.48986
\(435\) 0 0
\(436\) −67.4377 −3.22968
\(437\) 9.85705 0.471527
\(438\) 0 0
\(439\) 19.1103 0.912084 0.456042 0.889958i \(-0.349267\pi\)
0.456042 + 0.889958i \(0.349267\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −17.8350 −0.848325
\(443\) 1.97714 0.0939365 0.0469683 0.998896i \(-0.485044\pi\)
0.0469683 + 0.998896i \(0.485044\pi\)
\(444\) 0 0
\(445\) 4.78783 0.226965
\(446\) −23.8163 −1.12774
\(447\) 0 0
\(448\) 92.2414 4.35800
\(449\) −15.1719 −0.716008 −0.358004 0.933720i \(-0.616543\pi\)
−0.358004 + 0.933720i \(0.616543\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 78.1055 3.67377
\(453\) 0 0
\(454\) 28.3105 1.32868
\(455\) 11.3121 0.530321
\(456\) 0 0
\(457\) 6.17056 0.288647 0.144323 0.989531i \(-0.453899\pi\)
0.144323 + 0.989531i \(0.453899\pi\)
\(458\) 13.3537 0.623977
\(459\) 0 0
\(460\) −45.0742 −2.10160
\(461\) −23.1990 −1.08048 −0.540242 0.841510i \(-0.681667\pi\)
−0.540242 + 0.841510i \(0.681667\pi\)
\(462\) 0 0
\(463\) −1.39809 −0.0649750 −0.0324875 0.999472i \(-0.510343\pi\)
−0.0324875 + 0.999472i \(0.510343\pi\)
\(464\) −22.2503 −1.03295
\(465\) 0 0
\(466\) −10.4941 −0.486131
\(467\) 22.6901 1.04997 0.524987 0.851110i \(-0.324070\pi\)
0.524987 + 0.851110i \(0.324070\pi\)
\(468\) 0 0
\(469\) 63.3606 2.92572
\(470\) 3.97989 0.183578
\(471\) 0 0
\(472\) 72.7545 3.34880
\(473\) 0 0
\(474\) 0 0
\(475\) 1.15325 0.0529149
\(476\) 52.6298 2.41228
\(477\) 0 0
\(478\) −42.1228 −1.92665
\(479\) −24.7906 −1.13271 −0.566355 0.824161i \(-0.691647\pi\)
−0.566355 + 0.824161i \(0.691647\pi\)
\(480\) 0 0
\(481\) −15.6505 −0.713602
\(482\) 27.6803 1.26080
\(483\) 0 0
\(484\) 0 0
\(485\) −4.24066 −0.192559
\(486\) 0 0
\(487\) −33.7191 −1.52796 −0.763979 0.645241i \(-0.776757\pi\)
−0.763979 + 0.645241i \(0.776757\pi\)
\(488\) −30.6859 −1.38909
\(489\) 0 0
\(490\) −27.1624 −1.22707
\(491\) 6.90566 0.311648 0.155824 0.987785i \(-0.450197\pi\)
0.155824 + 0.987785i \(0.450197\pi\)
\(492\) 0 0
\(493\) −4.05198 −0.182492
\(494\) −8.51546 −0.383129
\(495\) 0 0
\(496\) −136.131 −6.11244
\(497\) −10.5243 −0.472080
\(498\) 0 0
\(499\) −2.26078 −0.101206 −0.0506031 0.998719i \(-0.516114\pi\)
−0.0506031 + 0.998719i \(0.516114\pi\)
\(500\) −5.27358 −0.235842
\(501\) 0 0
\(502\) 52.9723 2.36427
\(503\) 20.5999 0.918504 0.459252 0.888306i \(-0.348117\pi\)
0.459252 + 0.888306i \(0.348117\pi\)
\(504\) 0 0
\(505\) 13.9411 0.620370
\(506\) 0 0
\(507\) 0 0
\(508\) −68.3592 −3.03295
\(509\) 7.43272 0.329449 0.164725 0.986340i \(-0.447326\pi\)
0.164725 + 0.986340i \(0.447326\pi\)
\(510\) 0 0
\(511\) −11.3121 −0.500420
\(512\) 6.05208 0.267466
\(513\) 0 0
\(514\) 3.15325 0.139084
\(515\) 18.5042 0.815392
\(516\) 0 0
\(517\) 0 0
\(518\) 63.6985 2.79875
\(519\) 0 0
\(520\) 24.1717 1.06000
\(521\) 35.0084 1.53375 0.766873 0.641799i \(-0.221812\pi\)
0.766873 + 0.641799i \(0.221812\pi\)
\(522\) 0 0
\(523\) 1.52986 0.0668960 0.0334480 0.999440i \(-0.489351\pi\)
0.0334480 + 0.999440i \(0.489351\pi\)
\(524\) −7.78220 −0.339967
\(525\) 0 0
\(526\) 24.0988 1.05076
\(527\) −24.7906 −1.07989
\(528\) 0 0
\(529\) 50.0541 2.17627
\(530\) −6.86961 −0.298397
\(531\) 0 0
\(532\) 25.1285 1.08946
\(533\) 24.9425 1.08038
\(534\) 0 0
\(535\) −13.7663 −0.595171
\(536\) 135.388 5.84789
\(537\) 0 0
\(538\) 40.3233 1.73846
\(539\) 0 0
\(540\) 0 0
\(541\) −16.6700 −0.716700 −0.358350 0.933587i \(-0.616660\pi\)
−0.358350 + 0.933587i \(0.616660\pi\)
\(542\) 61.2990 2.63302
\(543\) 0 0
\(544\) 43.7520 1.87585
\(545\) 12.7878 0.547771
\(546\) 0 0
\(547\) 2.25234 0.0963032 0.0481516 0.998840i \(-0.484667\pi\)
0.0481516 + 0.998840i \(0.484667\pi\)
\(548\) 31.6415 1.35166
\(549\) 0 0
\(550\) 0 0
\(551\) −1.93465 −0.0824188
\(552\) 0 0
\(553\) −21.2920 −0.905429
\(554\) 59.7071 2.53671
\(555\) 0 0
\(556\) 4.69204 0.198987
\(557\) −11.9911 −0.508078 −0.254039 0.967194i \(-0.581759\pi\)
−0.254039 + 0.967194i \(0.581759\pi\)
\(558\) 0 0
\(559\) 22.2635 0.941647
\(560\) −54.8018 −2.31580
\(561\) 0 0
\(562\) −53.8962 −2.27347
\(563\) −19.4598 −0.820134 −0.410067 0.912055i \(-0.634495\pi\)
−0.410067 + 0.912055i \(0.634495\pi\)
\(564\) 0 0
\(565\) −14.8107 −0.623091
\(566\) −65.4753 −2.75213
\(567\) 0 0
\(568\) −22.4883 −0.943586
\(569\) 19.7205 0.826728 0.413364 0.910566i \(-0.364354\pi\)
0.413364 + 0.910566i \(0.364354\pi\)
\(570\) 0 0
\(571\) −5.41678 −0.226685 −0.113343 0.993556i \(-0.536156\pi\)
−0.113343 + 0.993556i \(0.536156\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −101.518 −4.23726
\(575\) 8.54717 0.356442
\(576\) 0 0
\(577\) 6.42708 0.267563 0.133781 0.991011i \(-0.457288\pi\)
0.133781 + 0.991011i \(0.457288\pi\)
\(578\) −30.1137 −1.25257
\(579\) 0 0
\(580\) 8.84675 0.367341
\(581\) 38.3635 1.59159
\(582\) 0 0
\(583\) 0 0
\(584\) −24.1717 −1.00023
\(585\) 0 0
\(586\) −44.2791 −1.82915
\(587\) −44.9883 −1.85686 −0.928432 0.371502i \(-0.878843\pi\)
−0.928432 + 0.371502i \(0.878843\pi\)
\(588\) 0 0
\(589\) −11.8364 −0.487712
\(590\) −22.2247 −0.914978
\(591\) 0 0
\(592\) 75.8191 3.11615
\(593\) 21.3892 0.878348 0.439174 0.898402i \(-0.355271\pi\)
0.439174 + 0.898402i \(0.355271\pi\)
\(594\) 0 0
\(595\) −9.97989 −0.409135
\(596\) 67.1270 2.74963
\(597\) 0 0
\(598\) −63.1111 −2.58081
\(599\) 26.9514 1.10120 0.550602 0.834768i \(-0.314398\pi\)
0.550602 + 0.834768i \(0.314398\pi\)
\(600\) 0 0
\(601\) −6.42708 −0.262166 −0.131083 0.991371i \(-0.541845\pi\)
−0.131083 + 0.991371i \(0.541845\pi\)
\(602\) −90.6138 −3.69314
\(603\) 0 0
\(604\) 6.08178 0.247464
\(605\) 0 0
\(606\) 0 0
\(607\) −12.9626 −0.526135 −0.263067 0.964777i \(-0.584734\pi\)
−0.263067 + 0.964777i \(0.584734\pi\)
\(608\) 20.8897 0.847190
\(609\) 0 0
\(610\) 9.37380 0.379534
\(611\) 4.04023 0.163450
\(612\) 0 0
\(613\) 24.4771 0.988620 0.494310 0.869286i \(-0.335421\pi\)
0.494310 + 0.869286i \(0.335421\pi\)
\(614\) −14.6820 −0.592517
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9453 0.480898 0.240449 0.970662i \(-0.422705\pi\)
0.240449 + 0.970662i \(0.422705\pi\)
\(618\) 0 0
\(619\) −18.9084 −0.759993 −0.379997 0.924988i \(-0.624075\pi\)
−0.379997 + 0.924988i \(0.624075\pi\)
\(620\) 54.1256 2.17374
\(621\) 0 0
\(622\) −76.2872 −3.05884
\(623\) −19.7822 −0.792557
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 61.5195 2.45881
\(627\) 0 0
\(628\) −20.8677 −0.832712
\(629\) 13.8073 0.550534
\(630\) 0 0
\(631\) −11.8364 −0.471201 −0.235601 0.971850i \(-0.575706\pi\)
−0.235601 + 0.971850i \(0.575706\pi\)
\(632\) −45.4966 −1.80976
\(633\) 0 0
\(634\) −64.3476 −2.55557
\(635\) 12.9626 0.514404
\(636\) 0 0
\(637\) −27.5742 −1.09253
\(638\) 0 0
\(639\) 0 0
\(640\) −23.9820 −0.947971
\(641\) 12.2206 0.482683 0.241341 0.970440i \(-0.422413\pi\)
0.241341 + 0.970440i \(0.422413\pi\)
\(642\) 0 0
\(643\) −8.84257 −0.348717 −0.174358 0.984682i \(-0.555785\pi\)
−0.174358 + 0.984682i \(0.555785\pi\)
\(644\) 186.236 7.33873
\(645\) 0 0
\(646\) 7.51258 0.295578
\(647\) −9.40985 −0.369939 −0.184970 0.982744i \(-0.559219\pi\)
−0.184970 + 0.982744i \(0.559219\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −7.38386 −0.289619
\(651\) 0 0
\(652\) 16.1977 0.634350
\(653\) −22.7449 −0.890075 −0.445038 0.895512i \(-0.646810\pi\)
−0.445038 + 0.895512i \(0.646810\pi\)
\(654\) 0 0
\(655\) 1.47569 0.0576602
\(656\) −120.834 −4.71779
\(657\) 0 0
\(658\) −16.4440 −0.641052
\(659\) −20.3635 −0.793249 −0.396625 0.917981i \(-0.629819\pi\)
−0.396625 + 0.917981i \(0.629819\pi\)
\(660\) 0 0
\(661\) 11.6616 0.453585 0.226792 0.973943i \(-0.427176\pi\)
0.226792 + 0.973943i \(0.427176\pi\)
\(662\) −50.9952 −1.98198
\(663\) 0 0
\(664\) 81.9748 3.18124
\(665\) −4.76497 −0.184778
\(666\) 0 0
\(667\) −14.3384 −0.555184
\(668\) 14.3177 0.553968
\(669\) 0 0
\(670\) −41.3579 −1.59779
\(671\) 0 0
\(672\) 0 0
\(673\) −30.0070 −1.15669 −0.578343 0.815794i \(-0.696300\pi\)
−0.578343 + 0.815794i \(0.696300\pi\)
\(674\) −66.5151 −2.56207
\(675\) 0 0
\(676\) −29.0268 −1.11642
\(677\) 51.4695 1.97813 0.989067 0.147466i \(-0.0471116\pi\)
0.989067 + 0.147466i \(0.0471116\pi\)
\(678\) 0 0
\(679\) 17.5214 0.672411
\(680\) −21.3249 −0.817774
\(681\) 0 0
\(682\) 0 0
\(683\) 27.6186 1.05680 0.528399 0.848996i \(-0.322793\pi\)
0.528399 + 0.848996i \(0.322793\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 34.2261 1.30676
\(687\) 0 0
\(688\) −107.856 −4.11197
\(689\) −6.97376 −0.265679
\(690\) 0 0
\(691\) −34.2344 −1.30234 −0.651169 0.758933i \(-0.725721\pi\)
−0.651169 + 0.758933i \(0.725721\pi\)
\(692\) 76.0209 2.88988
\(693\) 0 0
\(694\) −41.8179 −1.58738
\(695\) −0.889725 −0.0337492
\(696\) 0 0
\(697\) −22.0050 −0.833499
\(698\) −55.2447 −2.09104
\(699\) 0 0
\(700\) 21.7892 0.823555
\(701\) 11.8524 0.447658 0.223829 0.974628i \(-0.428144\pi\)
0.223829 + 0.974628i \(0.428144\pi\)
\(702\) 0 0
\(703\) 6.59241 0.248637
\(704\) 0 0
\(705\) 0 0
\(706\) 59.2788 2.23099
\(707\) −57.6013 −2.16632
\(708\) 0 0
\(709\) −28.1658 −1.05779 −0.528895 0.848687i \(-0.677393\pi\)
−0.528895 + 0.848687i \(0.677393\pi\)
\(710\) 6.86961 0.257812
\(711\) 0 0
\(712\) −42.2705 −1.58415
\(713\) −87.7241 −3.28529
\(714\) 0 0
\(715\) 0 0
\(716\) −56.8906 −2.12610
\(717\) 0 0
\(718\) −53.1200 −1.98242
\(719\) −26.2663 −0.979567 −0.489783 0.871844i \(-0.662924\pi\)
−0.489783 + 0.871844i \(0.662924\pi\)
\(720\) 0 0
\(721\) −76.4550 −2.84733
\(722\) −47.6553 −1.77355
\(723\) 0 0
\(724\) −34.3004 −1.27476
\(725\) −1.67756 −0.0623030
\(726\) 0 0
\(727\) 21.1401 0.784042 0.392021 0.919956i \(-0.371776\pi\)
0.392021 + 0.919956i \(0.371776\pi\)
\(728\) −99.8717 −3.70149
\(729\) 0 0
\(730\) 7.38386 0.273289
\(731\) −19.6415 −0.726467
\(732\) 0 0
\(733\) −3.86406 −0.142722 −0.0713611 0.997451i \(-0.522734\pi\)
−0.0713611 + 0.997451i \(0.522734\pi\)
\(734\) −76.2256 −2.81354
\(735\) 0 0
\(736\) 154.821 5.70679
\(737\) 0 0
\(738\) 0 0
\(739\) 14.0298 0.516094 0.258047 0.966132i \(-0.416921\pi\)
0.258047 + 0.966132i \(0.416921\pi\)
\(740\) −30.1457 −1.10818
\(741\) 0 0
\(742\) 28.3836 1.04200
\(743\) −26.2934 −0.964611 −0.482306 0.876003i \(-0.660201\pi\)
−0.482306 + 0.876003i \(0.660201\pi\)
\(744\) 0 0
\(745\) −12.7289 −0.466352
\(746\) 26.4011 0.966613
\(747\) 0 0
\(748\) 0 0
\(749\) 56.8793 2.07832
\(750\) 0 0
\(751\) 38.2206 1.39469 0.697344 0.716737i \(-0.254365\pi\)
0.697344 + 0.716737i \(0.254365\pi\)
\(752\) −19.5729 −0.713751
\(753\) 0 0
\(754\) 12.3869 0.451103
\(755\) −1.15325 −0.0419712
\(756\) 0 0
\(757\) −36.6158 −1.33082 −0.665411 0.746477i \(-0.731744\pi\)
−0.665411 + 0.746477i \(0.731744\pi\)
\(758\) 15.2692 0.554601
\(759\) 0 0
\(760\) −10.1818 −0.369331
\(761\) −51.6803 −1.87341 −0.936705 0.350120i \(-0.886141\pi\)
−0.936705 + 0.350120i \(0.886141\pi\)
\(762\) 0 0
\(763\) −52.8363 −1.91280
\(764\) −83.7560 −3.03019
\(765\) 0 0
\(766\) 71.4807 2.58270
\(767\) −22.5617 −0.814655
\(768\) 0 0
\(769\) −44.4841 −1.60414 −0.802068 0.597232i \(-0.796267\pi\)
−0.802068 + 0.597232i \(0.796267\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.2094 0.727351
\(773\) 0.973762 0.0350238 0.0175119 0.999847i \(-0.494426\pi\)
0.0175119 + 0.999847i \(0.494426\pi\)
\(774\) 0 0
\(775\) −10.2635 −0.368677
\(776\) 37.4396 1.34400
\(777\) 0 0
\(778\) 41.4121 1.48470
\(779\) −10.5065 −0.376433
\(780\) 0 0
\(781\) 0 0
\(782\) 55.6784 1.99106
\(783\) 0 0
\(784\) 133.583 4.77083
\(785\) 3.95702 0.141232
\(786\) 0 0
\(787\) 30.5382 1.08857 0.544285 0.838900i \(-0.316801\pi\)
0.544285 + 0.838900i \(0.316801\pi\)
\(788\) −14.1276 −0.503274
\(789\) 0 0
\(790\) 13.8981 0.494473
\(791\) 61.1943 2.17582
\(792\) 0 0
\(793\) 9.51592 0.337920
\(794\) 45.2218 1.60486
\(795\) 0 0
\(796\) 35.7962 1.26876
\(797\) 26.7247 0.946639 0.473319 0.880891i \(-0.343056\pi\)
0.473319 + 0.880891i \(0.343056\pi\)
\(798\) 0 0
\(799\) −3.56440 −0.126099
\(800\) 18.1137 0.640417
\(801\) 0 0
\(802\) 70.2442 2.48041
\(803\) 0 0
\(804\) 0 0
\(805\) −35.3149 −1.24469
\(806\) 75.7845 2.66939
\(807\) 0 0
\(808\) −123.082 −4.33001
\(809\) 17.0327 0.598837 0.299418 0.954122i \(-0.403207\pi\)
0.299418 + 0.954122i \(0.403207\pi\)
\(810\) 0 0
\(811\) −23.1103 −0.811512 −0.405756 0.913982i \(-0.632992\pi\)
−0.405756 + 0.913982i \(0.632992\pi\)
\(812\) −36.5527 −1.28275
\(813\) 0 0
\(814\) 0 0
\(815\) −3.07148 −0.107589
\(816\) 0 0
\(817\) −9.37798 −0.328094
\(818\) −60.5453 −2.11692
\(819\) 0 0
\(820\) 48.0438 1.67776
\(821\) 28.3476 0.989337 0.494668 0.869082i \(-0.335290\pi\)
0.494668 + 0.869082i \(0.335290\pi\)
\(822\) 0 0
\(823\) −10.4585 −0.364559 −0.182280 0.983247i \(-0.558348\pi\)
−0.182280 + 0.983247i \(0.558348\pi\)
\(824\) −163.368 −5.69121
\(825\) 0 0
\(826\) 91.8273 3.19508
\(827\) 45.2672 1.57409 0.787047 0.616893i \(-0.211609\pi\)
0.787047 + 0.616893i \(0.211609\pi\)
\(828\) 0 0
\(829\) −22.9855 −0.798320 −0.399160 0.916881i \(-0.630698\pi\)
−0.399160 + 0.916881i \(0.630698\pi\)
\(830\) −25.0413 −0.869196
\(831\) 0 0
\(832\) −61.1223 −2.11903
\(833\) 24.3267 0.842870
\(834\) 0 0
\(835\) −2.71498 −0.0939559
\(836\) 0 0
\(837\) 0 0
\(838\) −75.1971 −2.59764
\(839\) −20.1547 −0.695818 −0.347909 0.937528i \(-0.613108\pi\)
−0.347909 + 0.937528i \(0.613108\pi\)
\(840\) 0 0
\(841\) −26.1858 −0.902959
\(842\) 42.0686 1.44978
\(843\) 0 0
\(844\) −93.2386 −3.20941
\(845\) 5.50419 0.189350
\(846\) 0 0
\(847\) 0 0
\(848\) 33.7845 1.16016
\(849\) 0 0
\(850\) 6.51425 0.223437
\(851\) 48.8587 1.67485
\(852\) 0 0
\(853\) 38.2705 1.31036 0.655179 0.755474i \(-0.272593\pi\)
0.655179 + 0.755474i \(0.272593\pi\)
\(854\) −38.7303 −1.32533
\(855\) 0 0
\(856\) 121.539 4.15413
\(857\) 9.51250 0.324941 0.162470 0.986713i \(-0.448054\pi\)
0.162470 + 0.986713i \(0.448054\pi\)
\(858\) 0 0
\(859\) −1.61865 −0.0552275 −0.0276137 0.999619i \(-0.508791\pi\)
−0.0276137 + 0.999619i \(0.508791\pi\)
\(860\) 42.8835 1.46232
\(861\) 0 0
\(862\) 7.95977 0.271111
\(863\) 46.0229 1.56664 0.783318 0.621621i \(-0.213526\pi\)
0.783318 + 0.621621i \(0.213526\pi\)
\(864\) 0 0
\(865\) −14.4154 −0.490138
\(866\) 65.4994 2.22576
\(867\) 0 0
\(868\) −223.634 −7.59064
\(869\) 0 0
\(870\) 0 0
\(871\) −41.9849 −1.42260
\(872\) −112.900 −3.82328
\(873\) 0 0
\(874\) 26.5841 0.899219
\(875\) −4.13176 −0.139679
\(876\) 0 0
\(877\) 28.7630 0.971257 0.485628 0.874165i \(-0.338591\pi\)
0.485628 + 0.874165i \(0.338591\pi\)
\(878\) 51.5396 1.73938
\(879\) 0 0
\(880\) 0 0
\(881\) −23.3149 −0.785499 −0.392749 0.919646i \(-0.628476\pi\)
−0.392749 + 0.919646i \(0.628476\pi\)
\(882\) 0 0
\(883\) 29.6616 0.998193 0.499097 0.866546i \(-0.333665\pi\)
0.499097 + 0.866546i \(0.333665\pi\)
\(884\) −34.8743 −1.17295
\(885\) 0 0
\(886\) 5.33225 0.179141
\(887\) −20.3364 −0.682829 −0.341414 0.939913i \(-0.610906\pi\)
−0.341414 + 0.939913i \(0.610906\pi\)
\(888\) 0 0
\(889\) −53.5583 −1.79629
\(890\) 12.9126 0.432831
\(891\) 0 0
\(892\) −46.5700 −1.55928
\(893\) −1.70185 −0.0569502
\(894\) 0 0
\(895\) 10.7878 0.360598
\(896\) 99.0879 3.31029
\(897\) 0 0
\(898\) −40.9181 −1.36545
\(899\) 17.2177 0.574241
\(900\) 0 0
\(901\) 6.15245 0.204968
\(902\) 0 0
\(903\) 0 0
\(904\) 130.760 4.34900
\(905\) 6.50419 0.216207
\(906\) 0 0
\(907\) 9.87942 0.328041 0.164020 0.986457i \(-0.447554\pi\)
0.164020 + 0.986457i \(0.447554\pi\)
\(908\) 55.3578 1.83711
\(909\) 0 0
\(910\) 30.5084 1.01134
\(911\) 7.22343 0.239323 0.119662 0.992815i \(-0.461819\pi\)
0.119662 + 0.992815i \(0.461819\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 16.6418 0.550460
\(915\) 0 0
\(916\) 26.1116 0.862751
\(917\) −6.09722 −0.201348
\(918\) 0 0
\(919\) −35.7233 −1.17840 −0.589201 0.807986i \(-0.700557\pi\)
−0.589201 + 0.807986i \(0.700557\pi\)
\(920\) −75.4606 −2.48786
\(921\) 0 0
\(922\) −62.5667 −2.06052
\(923\) 6.97376 0.229544
\(924\) 0 0
\(925\) 5.71636 0.187953
\(926\) −3.77060 −0.123910
\(927\) 0 0
\(928\) −30.3869 −0.997497
\(929\) −1.12621 −0.0369498 −0.0184749 0.999829i \(-0.505881\pi\)
−0.0184749 + 0.999829i \(0.505881\pi\)
\(930\) 0 0
\(931\) 11.6150 0.380665
\(932\) −20.5200 −0.672157
\(933\) 0 0
\(934\) 61.1943 2.00234
\(935\) 0 0
\(936\) 0 0
\(937\) 0.127587 0.00416807 0.00208404 0.999998i \(-0.499337\pi\)
0.00208404 + 0.999998i \(0.499337\pi\)
\(938\) 170.881 5.57946
\(939\) 0 0
\(940\) 7.78220 0.253827
\(941\) −1.41403 −0.0460961 −0.0230480 0.999734i \(-0.507337\pi\)
−0.0230480 + 0.999734i \(0.507337\pi\)
\(942\) 0 0
\(943\) −77.8671 −2.53570
\(944\) 109.300 3.55742
\(945\) 0 0
\(946\) 0 0
\(947\) −37.1943 −1.20865 −0.604326 0.796737i \(-0.706558\pi\)
−0.604326 + 0.796737i \(0.706558\pi\)
\(948\) 0 0
\(949\) 7.49581 0.243324
\(950\) 3.11028 0.100911
\(951\) 0 0
\(952\) 88.1097 2.85565
\(953\) 16.5041 0.534621 0.267310 0.963610i \(-0.413865\pi\)
0.267310 + 0.963610i \(0.413865\pi\)
\(954\) 0 0
\(955\) 15.8822 0.513935
\(956\) −82.3663 −2.66391
\(957\) 0 0
\(958\) −66.8592 −2.16012
\(959\) 24.7906 0.800530
\(960\) 0 0
\(961\) 74.3400 2.39807
\(962\) −42.2088 −1.36087
\(963\) 0 0
\(964\) 54.1256 1.74327
\(965\) −3.83219 −0.123362
\(966\) 0 0
\(967\) 3.60471 0.115920 0.0579598 0.998319i \(-0.481540\pi\)
0.0579598 + 0.998319i \(0.481540\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −11.4369 −0.367217
\(971\) 29.6214 0.950596 0.475298 0.879825i \(-0.342340\pi\)
0.475298 + 0.879825i \(0.342340\pi\)
\(972\) 0 0
\(973\) 3.67613 0.117851
\(974\) −90.9390 −2.91387
\(975\) 0 0
\(976\) −46.1000 −1.47562
\(977\) −52.7157 −1.68653 −0.843263 0.537501i \(-0.819368\pi\)
−0.843263 + 0.537501i \(0.819368\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −53.1128 −1.69663
\(981\) 0 0
\(982\) 18.6243 0.594325
\(983\) 29.2602 0.933254 0.466627 0.884454i \(-0.345469\pi\)
0.466627 + 0.884454i \(0.345469\pi\)
\(984\) 0 0
\(985\) 2.67893 0.0853579
\(986\) −10.9280 −0.348019
\(987\) 0 0
\(988\) −16.6510 −0.529738
\(989\) −69.5036 −2.21008
\(990\) 0 0
\(991\) 38.1776 1.21275 0.606375 0.795179i \(-0.292623\pi\)
0.606375 + 0.795179i \(0.292623\pi\)
\(992\) −185.911 −5.90268
\(993\) 0 0
\(994\) −28.3836 −0.900274
\(995\) −6.78783 −0.215189
\(996\) 0 0
\(997\) 40.1500 1.27156 0.635781 0.771870i \(-0.280678\pi\)
0.635781 + 0.771870i \(0.280678\pi\)
\(998\) −6.09722 −0.193004
\(999\) 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.bs.1.4 4
3.2 odd 2 5445.2.a.bh.1.1 4
11.10 odd 2 495.2.a.f.1.1 4
33.32 even 2 495.2.a.g.1.4 yes 4
44.43 even 2 7920.2.a.cm.1.4 4
55.32 even 4 2475.2.c.t.199.1 8
55.43 even 4 2475.2.c.t.199.8 8
55.54 odd 2 2475.2.a.bj.1.4 4
132.131 odd 2 7920.2.a.cn.1.4 4
165.32 odd 4 2475.2.c.s.199.8 8
165.98 odd 4 2475.2.c.s.199.1 8
165.164 even 2 2475.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.a.f.1.1 4 11.10 odd 2
495.2.a.g.1.4 yes 4 33.32 even 2
2475.2.a.bf.1.1 4 165.164 even 2
2475.2.a.bj.1.4 4 55.54 odd 2
2475.2.c.s.199.1 8 165.98 odd 4
2475.2.c.s.199.8 8 165.32 odd 4
2475.2.c.t.199.1 8 55.32 even 4
2475.2.c.t.199.8 8 55.43 even 4
5445.2.a.bh.1.1 4 3.2 odd 2
5445.2.a.bs.1.4 4 1.1 even 1 trivial
7920.2.a.cm.1.4 4 44.43 even 2
7920.2.a.cn.1.4 4 132.131 odd 2