Properties

Label 4730.2.a.be.1.3
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.52230\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.52230 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.52230 q^{6} +4.63187 q^{7} +1.00000 q^{8} +3.36197 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.52230 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.52230 q^{6} +4.63187 q^{7} +1.00000 q^{8} +3.36197 q^{9} +1.00000 q^{10} -1.00000 q^{11} -2.52230 q^{12} -3.92208 q^{13} +4.63187 q^{14} -2.52230 q^{15} +1.00000 q^{16} -2.69717 q^{17} +3.36197 q^{18} -2.21477 q^{19} +1.00000 q^{20} -11.6829 q^{21} -1.00000 q^{22} +5.92208 q^{23} -2.52230 q^{24} +1.00000 q^{25} -3.92208 q^{26} -0.913006 q^{27} +4.63187 q^{28} +6.30944 q^{29} -2.52230 q^{30} +5.02749 q^{31} +1.00000 q^{32} +2.52230 q^{33} -2.69717 q^{34} +4.63187 q^{35} +3.36197 q^{36} -6.44231 q^{37} -2.21477 q^{38} +9.89264 q^{39} +1.00000 q^{40} -6.04125 q^{41} -11.6829 q^{42} +1.00000 q^{43} -1.00000 q^{44} +3.36197 q^{45} +5.92208 q^{46} +12.8214 q^{47} -2.52230 q^{48} +14.4542 q^{49} +1.00000 q^{50} +6.80306 q^{51} -3.92208 q^{52} +4.11880 q^{53} -0.913006 q^{54} -1.00000 q^{55} +4.63187 q^{56} +5.58631 q^{57} +6.30944 q^{58} -3.19129 q^{59} -2.52230 q^{60} +4.94068 q^{61} +5.02749 q^{62} +15.5722 q^{63} +1.00000 q^{64} -3.92208 q^{65} +2.52230 q^{66} -11.2025 q^{67} -2.69717 q^{68} -14.9372 q^{69} +4.63187 q^{70} +7.04905 q^{71} +3.36197 q^{72} +13.6863 q^{73} -6.44231 q^{74} -2.52230 q^{75} -2.21477 q^{76} -4.63187 q^{77} +9.89264 q^{78} -14.1493 q^{79} +1.00000 q^{80} -7.78305 q^{81} -6.04125 q^{82} +1.14363 q^{83} -11.6829 q^{84} -2.69717 q^{85} +1.00000 q^{86} -15.9143 q^{87} -1.00000 q^{88} +8.75478 q^{89} +3.36197 q^{90} -18.1666 q^{91} +5.92208 q^{92} -12.6808 q^{93} +12.8214 q^{94} -2.21477 q^{95} -2.52230 q^{96} -14.4102 q^{97} +14.4542 q^{98} -3.36197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + O(q^{10}) \) \( 12q + 12q^{2} + 3q^{3} + 12q^{4} + 12q^{5} + 3q^{6} + 8q^{7} + 12q^{8} + 25q^{9} + 12q^{10} - 12q^{11} + 3q^{12} + 16q^{13} + 8q^{14} + 3q^{15} + 12q^{16} + 18q^{17} + 25q^{18} - 4q^{19} + 12q^{20} + 4q^{21} - 12q^{22} + 8q^{23} + 3q^{24} + 12q^{25} + 16q^{26} + 6q^{27} + 8q^{28} + 20q^{29} + 3q^{30} + 5q^{31} + 12q^{32} - 3q^{33} + 18q^{34} + 8q^{35} + 25q^{36} + 19q^{37} - 4q^{38} + 6q^{39} + 12q^{40} + 16q^{41} + 4q^{42} + 12q^{43} - 12q^{44} + 25q^{45} + 8q^{46} - q^{47} + 3q^{48} + 52q^{49} + 12q^{50} + q^{51} + 16q^{52} + 11q^{53} + 6q^{54} - 12q^{55} + 8q^{56} + 9q^{57} + 20q^{58} - 11q^{59} + 3q^{60} + 18q^{61} + 5q^{62} + 15q^{63} + 12q^{64} + 16q^{65} - 3q^{66} - 10q^{67} + 18q^{68} + 8q^{70} - 2q^{71} + 25q^{72} + 29q^{73} + 19q^{74} + 3q^{75} - 4q^{76} - 8q^{77} + 6q^{78} + 2q^{79} + 12q^{80} - 8q^{81} + 16q^{82} + 26q^{83} + 4q^{84} + 18q^{85} + 12q^{86} - 4q^{87} - 12q^{88} + 41q^{89} + 25q^{90} - 4q^{91} + 8q^{92} + 5q^{93} - q^{94} - 4q^{95} + 3q^{96} - 7q^{97} + 52q^{98} - 25q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.52230 −1.45625 −0.728124 0.685445i \(-0.759607\pi\)
−0.728124 + 0.685445i \(0.759607\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.52230 −1.02972
\(7\) 4.63187 1.75068 0.875341 0.483505i \(-0.160637\pi\)
0.875341 + 0.483505i \(0.160637\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.36197 1.12066
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −2.52230 −0.728124
\(13\) −3.92208 −1.08779 −0.543895 0.839154i \(-0.683051\pi\)
−0.543895 + 0.839154i \(0.683051\pi\)
\(14\) 4.63187 1.23792
\(15\) −2.52230 −0.651254
\(16\) 1.00000 0.250000
\(17\) −2.69717 −0.654160 −0.327080 0.944997i \(-0.606065\pi\)
−0.327080 + 0.944997i \(0.606065\pi\)
\(18\) 3.36197 0.792425
\(19\) −2.21477 −0.508104 −0.254052 0.967191i \(-0.581763\pi\)
−0.254052 + 0.967191i \(0.581763\pi\)
\(20\) 1.00000 0.223607
\(21\) −11.6829 −2.54943
\(22\) −1.00000 −0.213201
\(23\) 5.92208 1.23484 0.617419 0.786634i \(-0.288178\pi\)
0.617419 + 0.786634i \(0.288178\pi\)
\(24\) −2.52230 −0.514861
\(25\) 1.00000 0.200000
\(26\) −3.92208 −0.769183
\(27\) −0.913006 −0.175708
\(28\) 4.63187 0.875341
\(29\) 6.30944 1.17163 0.585817 0.810443i \(-0.300774\pi\)
0.585817 + 0.810443i \(0.300774\pi\)
\(30\) −2.52230 −0.460506
\(31\) 5.02749 0.902963 0.451482 0.892280i \(-0.350896\pi\)
0.451482 + 0.892280i \(0.350896\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.52230 0.439075
\(34\) −2.69717 −0.462561
\(35\) 4.63187 0.782929
\(36\) 3.36197 0.560329
\(37\) −6.44231 −1.05911 −0.529555 0.848276i \(-0.677641\pi\)
−0.529555 + 0.848276i \(0.677641\pi\)
\(38\) −2.21477 −0.359284
\(39\) 9.89264 1.58409
\(40\) 1.00000 0.158114
\(41\) −6.04125 −0.943484 −0.471742 0.881737i \(-0.656375\pi\)
−0.471742 + 0.881737i \(0.656375\pi\)
\(42\) −11.6829 −1.80272
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) 3.36197 0.501174
\(46\) 5.92208 0.873163
\(47\) 12.8214 1.87019 0.935095 0.354396i \(-0.115314\pi\)
0.935095 + 0.354396i \(0.115314\pi\)
\(48\) −2.52230 −0.364062
\(49\) 14.4542 2.06489
\(50\) 1.00000 0.141421
\(51\) 6.80306 0.952619
\(52\) −3.92208 −0.543895
\(53\) 4.11880 0.565761 0.282880 0.959155i \(-0.408710\pi\)
0.282880 + 0.959155i \(0.408710\pi\)
\(54\) −0.913006 −0.124244
\(55\) −1.00000 −0.134840
\(56\) 4.63187 0.618960
\(57\) 5.58631 0.739925
\(58\) 6.30944 0.828470
\(59\) −3.19129 −0.415471 −0.207735 0.978185i \(-0.566609\pi\)
−0.207735 + 0.978185i \(0.566609\pi\)
\(60\) −2.52230 −0.325627
\(61\) 4.94068 0.632589 0.316295 0.948661i \(-0.397561\pi\)
0.316295 + 0.948661i \(0.397561\pi\)
\(62\) 5.02749 0.638492
\(63\) 15.5722 1.96192
\(64\) 1.00000 0.125000
\(65\) −3.92208 −0.486474
\(66\) 2.52230 0.310473
\(67\) −11.2025 −1.36860 −0.684301 0.729199i \(-0.739893\pi\)
−0.684301 + 0.729199i \(0.739893\pi\)
\(68\) −2.69717 −0.327080
\(69\) −14.9372 −1.79823
\(70\) 4.63187 0.553614
\(71\) 7.04905 0.836568 0.418284 0.908316i \(-0.362632\pi\)
0.418284 + 0.908316i \(0.362632\pi\)
\(72\) 3.36197 0.396212
\(73\) 13.6863 1.60186 0.800930 0.598758i \(-0.204339\pi\)
0.800930 + 0.598758i \(0.204339\pi\)
\(74\) −6.44231 −0.748904
\(75\) −2.52230 −0.291250
\(76\) −2.21477 −0.254052
\(77\) −4.63187 −0.527851
\(78\) 9.89264 1.12012
\(79\) −14.1493 −1.59193 −0.795963 0.605346i \(-0.793035\pi\)
−0.795963 + 0.605346i \(0.793035\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.78305 −0.864784
\(82\) −6.04125 −0.667144
\(83\) 1.14363 0.125529 0.0627646 0.998028i \(-0.480008\pi\)
0.0627646 + 0.998028i \(0.480008\pi\)
\(84\) −11.6829 −1.27471
\(85\) −2.69717 −0.292549
\(86\) 1.00000 0.107833
\(87\) −15.9143 −1.70619
\(88\) −1.00000 −0.106600
\(89\) 8.75478 0.928005 0.464002 0.885834i \(-0.346413\pi\)
0.464002 + 0.885834i \(0.346413\pi\)
\(90\) 3.36197 0.354383
\(91\) −18.1666 −1.90437
\(92\) 5.92208 0.617419
\(93\) −12.6808 −1.31494
\(94\) 12.8214 1.32242
\(95\) −2.21477 −0.227231
\(96\) −2.52230 −0.257431
\(97\) −14.4102 −1.46314 −0.731569 0.681767i \(-0.761212\pi\)
−0.731569 + 0.681767i \(0.761212\pi\)
\(98\) 14.4542 1.46010
\(99\) −3.36197 −0.337891
\(100\) 1.00000 0.100000
\(101\) −2.55687 −0.254418 −0.127209 0.991876i \(-0.540602\pi\)
−0.127209 + 0.991876i \(0.540602\pi\)
\(102\) 6.80306 0.673603
\(103\) 1.93523 0.190684 0.0953418 0.995445i \(-0.469606\pi\)
0.0953418 + 0.995445i \(0.469606\pi\)
\(104\) −3.92208 −0.384592
\(105\) −11.6829 −1.14014
\(106\) 4.11880 0.400053
\(107\) −14.1549 −1.36840 −0.684201 0.729294i \(-0.739849\pi\)
−0.684201 + 0.729294i \(0.739849\pi\)
\(108\) −0.913006 −0.0878540
\(109\) 15.2102 1.45687 0.728434 0.685116i \(-0.240248\pi\)
0.728434 + 0.685116i \(0.240248\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 16.2494 1.54233
\(112\) 4.63187 0.437671
\(113\) 14.3190 1.34702 0.673508 0.739180i \(-0.264787\pi\)
0.673508 + 0.739180i \(0.264787\pi\)
\(114\) 5.58631 0.523206
\(115\) 5.92208 0.552237
\(116\) 6.30944 0.585817
\(117\) −13.1859 −1.21904
\(118\) −3.19129 −0.293782
\(119\) −12.4929 −1.14523
\(120\) −2.52230 −0.230253
\(121\) 1.00000 0.0909091
\(122\) 4.94068 0.447308
\(123\) 15.2378 1.37395
\(124\) 5.02749 0.451482
\(125\) 1.00000 0.0894427
\(126\) 15.5722 1.38728
\(127\) 16.7232 1.48395 0.741973 0.670430i \(-0.233890\pi\)
0.741973 + 0.670430i \(0.233890\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.52230 −0.222076
\(130\) −3.92208 −0.343989
\(131\) −9.16306 −0.800580 −0.400290 0.916389i \(-0.631091\pi\)
−0.400290 + 0.916389i \(0.631091\pi\)
\(132\) 2.52230 0.219538
\(133\) −10.2585 −0.889528
\(134\) −11.2025 −0.967748
\(135\) −0.913006 −0.0785790
\(136\) −2.69717 −0.231280
\(137\) 10.1475 0.866961 0.433481 0.901163i \(-0.357285\pi\)
0.433481 + 0.901163i \(0.357285\pi\)
\(138\) −14.9372 −1.27154
\(139\) 17.2511 1.46322 0.731608 0.681725i \(-0.238770\pi\)
0.731608 + 0.681725i \(0.238770\pi\)
\(140\) 4.63187 0.391465
\(141\) −32.3393 −2.72346
\(142\) 7.04905 0.591543
\(143\) 3.92208 0.327981
\(144\) 3.36197 0.280165
\(145\) 6.30944 0.523971
\(146\) 13.6863 1.13269
\(147\) −36.4578 −3.00699
\(148\) −6.44231 −0.529555
\(149\) −8.91255 −0.730145 −0.365072 0.930979i \(-0.618956\pi\)
−0.365072 + 0.930979i \(0.618956\pi\)
\(150\) −2.52230 −0.205945
\(151\) 11.5887 0.943079 0.471539 0.881845i \(-0.343699\pi\)
0.471539 + 0.881845i \(0.343699\pi\)
\(152\) −2.21477 −0.179642
\(153\) −9.06781 −0.733089
\(154\) −4.63187 −0.373247
\(155\) 5.02749 0.403818
\(156\) 9.89264 0.792045
\(157\) −11.9998 −0.957690 −0.478845 0.877899i \(-0.658944\pi\)
−0.478845 + 0.877899i \(0.658944\pi\)
\(158\) −14.1493 −1.12566
\(159\) −10.3888 −0.823888
\(160\) 1.00000 0.0790569
\(161\) 27.4303 2.16181
\(162\) −7.78305 −0.611494
\(163\) 11.4512 0.896931 0.448466 0.893800i \(-0.351971\pi\)
0.448466 + 0.893800i \(0.351971\pi\)
\(164\) −6.04125 −0.471742
\(165\) 2.52230 0.196360
\(166\) 1.14363 0.0887625
\(167\) −3.19024 −0.246868 −0.123434 0.992353i \(-0.539391\pi\)
−0.123434 + 0.992353i \(0.539391\pi\)
\(168\) −11.6829 −0.901359
\(169\) 2.38270 0.183285
\(170\) −2.69717 −0.206863
\(171\) −7.44601 −0.569410
\(172\) 1.00000 0.0762493
\(173\) −1.22545 −0.0931695 −0.0465847 0.998914i \(-0.514834\pi\)
−0.0465847 + 0.998914i \(0.514834\pi\)
\(174\) −15.9143 −1.20646
\(175\) 4.63187 0.350137
\(176\) −1.00000 −0.0753778
\(177\) 8.04938 0.605028
\(178\) 8.75478 0.656199
\(179\) −2.01522 −0.150624 −0.0753122 0.997160i \(-0.523995\pi\)
−0.0753122 + 0.997160i \(0.523995\pi\)
\(180\) 3.36197 0.250587
\(181\) 14.7484 1.09624 0.548122 0.836399i \(-0.315343\pi\)
0.548122 + 0.836399i \(0.315343\pi\)
\(182\) −18.1666 −1.34660
\(183\) −12.4619 −0.921207
\(184\) 5.92208 0.436581
\(185\) −6.44231 −0.473648
\(186\) −12.6808 −0.929802
\(187\) 2.69717 0.197237
\(188\) 12.8214 0.935095
\(189\) −4.22892 −0.307609
\(190\) −2.21477 −0.160676
\(191\) −3.11661 −0.225510 −0.112755 0.993623i \(-0.535968\pi\)
−0.112755 + 0.993623i \(0.535968\pi\)
\(192\) −2.52230 −0.182031
\(193\) 16.5398 1.19056 0.595280 0.803518i \(-0.297041\pi\)
0.595280 + 0.803518i \(0.297041\pi\)
\(194\) −14.4102 −1.03460
\(195\) 9.89264 0.708427
\(196\) 14.4542 1.03245
\(197\) 26.3121 1.87466 0.937329 0.348445i \(-0.113290\pi\)
0.937329 + 0.348445i \(0.113290\pi\)
\(198\) −3.36197 −0.238925
\(199\) −9.23473 −0.654633 −0.327316 0.944915i \(-0.606144\pi\)
−0.327316 + 0.944915i \(0.606144\pi\)
\(200\) 1.00000 0.0707107
\(201\) 28.2560 1.99302
\(202\) −2.55687 −0.179901
\(203\) 29.2245 2.05116
\(204\) 6.80306 0.476309
\(205\) −6.04125 −0.421939
\(206\) 1.93523 0.134834
\(207\) 19.9099 1.38383
\(208\) −3.92208 −0.271947
\(209\) 2.21477 0.153199
\(210\) −11.6829 −0.806200
\(211\) 2.32602 0.160130 0.0800651 0.996790i \(-0.474487\pi\)
0.0800651 + 0.996790i \(0.474487\pi\)
\(212\) 4.11880 0.282880
\(213\) −17.7798 −1.21825
\(214\) −14.1549 −0.967606
\(215\) 1.00000 0.0681994
\(216\) −0.913006 −0.0621222
\(217\) 23.2867 1.58080
\(218\) 15.2102 1.03016
\(219\) −34.5209 −2.33270
\(220\) −1.00000 −0.0674200
\(221\) 10.5785 0.711588
\(222\) 16.2494 1.09059
\(223\) −25.7990 −1.72763 −0.863814 0.503811i \(-0.831931\pi\)
−0.863814 + 0.503811i \(0.831931\pi\)
\(224\) 4.63187 0.309480
\(225\) 3.36197 0.224132
\(226\) 14.3190 0.952484
\(227\) −19.1186 −1.26894 −0.634471 0.772946i \(-0.718782\pi\)
−0.634471 + 0.772946i \(0.718782\pi\)
\(228\) 5.58631 0.369962
\(229\) 7.28071 0.481123 0.240561 0.970634i \(-0.422668\pi\)
0.240561 + 0.970634i \(0.422668\pi\)
\(230\) 5.92208 0.390490
\(231\) 11.6829 0.768682
\(232\) 6.30944 0.414235
\(233\) 26.8808 1.76102 0.880509 0.474029i \(-0.157201\pi\)
0.880509 + 0.474029i \(0.157201\pi\)
\(234\) −13.1859 −0.861991
\(235\) 12.8214 0.836375
\(236\) −3.19129 −0.207735
\(237\) 35.6888 2.31824
\(238\) −12.4929 −0.809797
\(239\) −6.55613 −0.424081 −0.212041 0.977261i \(-0.568011\pi\)
−0.212041 + 0.977261i \(0.568011\pi\)
\(240\) −2.52230 −0.162813
\(241\) −20.1131 −1.29560 −0.647799 0.761811i \(-0.724310\pi\)
−0.647799 + 0.761811i \(0.724310\pi\)
\(242\) 1.00000 0.0642824
\(243\) 22.3702 1.43505
\(244\) 4.94068 0.316295
\(245\) 14.4542 0.923447
\(246\) 15.2378 0.971527
\(247\) 8.68651 0.552710
\(248\) 5.02749 0.319246
\(249\) −2.88456 −0.182802
\(250\) 1.00000 0.0632456
\(251\) −16.7913 −1.05986 −0.529928 0.848043i \(-0.677781\pi\)
−0.529928 + 0.848043i \(0.677781\pi\)
\(252\) 15.5722 0.980958
\(253\) −5.92208 −0.372318
\(254\) 16.7232 1.04931
\(255\) 6.80306 0.426024
\(256\) 1.00000 0.0625000
\(257\) −18.5623 −1.15789 −0.578943 0.815368i \(-0.696535\pi\)
−0.578943 + 0.815368i \(0.696535\pi\)
\(258\) −2.52230 −0.157031
\(259\) −29.8400 −1.85417
\(260\) −3.92208 −0.243237
\(261\) 21.2122 1.31300
\(262\) −9.16306 −0.566096
\(263\) 25.1701 1.55205 0.776026 0.630701i \(-0.217232\pi\)
0.776026 + 0.630701i \(0.217232\pi\)
\(264\) 2.52230 0.155237
\(265\) 4.11880 0.253016
\(266\) −10.2585 −0.628992
\(267\) −22.0821 −1.35141
\(268\) −11.2025 −0.684301
\(269\) −4.57383 −0.278872 −0.139436 0.990231i \(-0.544529\pi\)
−0.139436 + 0.990231i \(0.544529\pi\)
\(270\) −0.913006 −0.0555637
\(271\) 4.65356 0.282684 0.141342 0.989961i \(-0.454858\pi\)
0.141342 + 0.989961i \(0.454858\pi\)
\(272\) −2.69717 −0.163540
\(273\) 45.8214 2.77324
\(274\) 10.1475 0.613034
\(275\) −1.00000 −0.0603023
\(276\) −14.9372 −0.899116
\(277\) 33.2000 1.99479 0.997396 0.0721158i \(-0.0229751\pi\)
0.997396 + 0.0721158i \(0.0229751\pi\)
\(278\) 17.2511 1.03465
\(279\) 16.9023 1.01191
\(280\) 4.63187 0.276807
\(281\) 8.38605 0.500270 0.250135 0.968211i \(-0.419525\pi\)
0.250135 + 0.968211i \(0.419525\pi\)
\(282\) −32.3393 −1.92578
\(283\) 19.9325 1.18486 0.592431 0.805621i \(-0.298168\pi\)
0.592431 + 0.805621i \(0.298168\pi\)
\(284\) 7.04905 0.418284
\(285\) 5.58631 0.330904
\(286\) 3.92208 0.231917
\(287\) −27.9823 −1.65174
\(288\) 3.36197 0.198106
\(289\) −9.72527 −0.572075
\(290\) 6.30944 0.370503
\(291\) 36.3469 2.13069
\(292\) 13.6863 0.800930
\(293\) 10.0162 0.585152 0.292576 0.956242i \(-0.405488\pi\)
0.292576 + 0.956242i \(0.405488\pi\)
\(294\) −36.4578 −2.12626
\(295\) −3.19129 −0.185804
\(296\) −6.44231 −0.374452
\(297\) 0.913006 0.0529780
\(298\) −8.91255 −0.516290
\(299\) −23.2269 −1.34324
\(300\) −2.52230 −0.145625
\(301\) 4.63187 0.266977
\(302\) 11.5887 0.666857
\(303\) 6.44919 0.370496
\(304\) −2.21477 −0.127026
\(305\) 4.94068 0.282902
\(306\) −9.06781 −0.518372
\(307\) 10.0537 0.573794 0.286897 0.957962i \(-0.407376\pi\)
0.286897 + 0.957962i \(0.407376\pi\)
\(308\) −4.63187 −0.263925
\(309\) −4.88121 −0.277683
\(310\) 5.02749 0.285542
\(311\) 1.08738 0.0616599 0.0308299 0.999525i \(-0.490185\pi\)
0.0308299 + 0.999525i \(0.490185\pi\)
\(312\) 9.89264 0.560061
\(313\) 23.8551 1.34837 0.674185 0.738563i \(-0.264495\pi\)
0.674185 + 0.738563i \(0.264495\pi\)
\(314\) −11.9998 −0.677189
\(315\) 15.5722 0.877396
\(316\) −14.1493 −0.795963
\(317\) −14.7634 −0.829195 −0.414597 0.910005i \(-0.636078\pi\)
−0.414597 + 0.910005i \(0.636078\pi\)
\(318\) −10.3888 −0.582577
\(319\) −6.30944 −0.353261
\(320\) 1.00000 0.0559017
\(321\) 35.7027 1.99273
\(322\) 27.4303 1.52863
\(323\) 5.97362 0.332381
\(324\) −7.78305 −0.432392
\(325\) −3.92208 −0.217558
\(326\) 11.4512 0.634226
\(327\) −38.3645 −2.12156
\(328\) −6.04125 −0.333572
\(329\) 59.3870 3.27411
\(330\) 2.52230 0.138848
\(331\) 12.0944 0.664771 0.332385 0.943144i \(-0.392147\pi\)
0.332385 + 0.943144i \(0.392147\pi\)
\(332\) 1.14363 0.0627646
\(333\) −21.6589 −1.18690
\(334\) −3.19024 −0.174562
\(335\) −11.2025 −0.612058
\(336\) −11.6829 −0.637357
\(337\) −8.34846 −0.454769 −0.227385 0.973805i \(-0.573017\pi\)
−0.227385 + 0.973805i \(0.573017\pi\)
\(338\) 2.38270 0.129602
\(339\) −36.1167 −1.96159
\(340\) −2.69717 −0.146275
\(341\) −5.02749 −0.272254
\(342\) −7.44601 −0.402634
\(343\) 34.5270 1.86428
\(344\) 1.00000 0.0539164
\(345\) −14.9372 −0.804194
\(346\) −1.22545 −0.0658808
\(347\) 9.15805 0.491630 0.245815 0.969317i \(-0.420944\pi\)
0.245815 + 0.969317i \(0.420944\pi\)
\(348\) −15.9143 −0.853095
\(349\) 5.57565 0.298458 0.149229 0.988803i \(-0.452321\pi\)
0.149229 + 0.988803i \(0.452321\pi\)
\(350\) 4.63187 0.247584
\(351\) 3.58088 0.191133
\(352\) −1.00000 −0.0533002
\(353\) −23.8879 −1.27142 −0.635711 0.771927i \(-0.719293\pi\)
−0.635711 + 0.771927i \(0.719293\pi\)
\(354\) 8.04938 0.427820
\(355\) 7.04905 0.374125
\(356\) 8.75478 0.464002
\(357\) 31.5109 1.66773
\(358\) −2.01522 −0.106507
\(359\) −20.4519 −1.07941 −0.539704 0.841855i \(-0.681464\pi\)
−0.539704 + 0.841855i \(0.681464\pi\)
\(360\) 3.36197 0.177192
\(361\) −14.0948 −0.741831
\(362\) 14.7484 0.775161
\(363\) −2.52230 −0.132386
\(364\) −18.1666 −0.952187
\(365\) 13.6863 0.716373
\(366\) −12.4619 −0.651391
\(367\) 10.6239 0.554564 0.277282 0.960789i \(-0.410566\pi\)
0.277282 + 0.960789i \(0.410566\pi\)
\(368\) 5.92208 0.308710
\(369\) −20.3105 −1.05732
\(370\) −6.44231 −0.334920
\(371\) 19.0777 0.990467
\(372\) −12.6808 −0.657469
\(373\) 25.0492 1.29700 0.648500 0.761215i \(-0.275397\pi\)
0.648500 + 0.761215i \(0.275397\pi\)
\(374\) 2.69717 0.139467
\(375\) −2.52230 −0.130251
\(376\) 12.8214 0.661212
\(377\) −24.7461 −1.27449
\(378\) −4.22892 −0.217512
\(379\) −4.98512 −0.256069 −0.128034 0.991770i \(-0.540867\pi\)
−0.128034 + 0.991770i \(0.540867\pi\)
\(380\) −2.21477 −0.113615
\(381\) −42.1809 −2.16099
\(382\) −3.11661 −0.159460
\(383\) −15.5150 −0.792779 −0.396389 0.918083i \(-0.629737\pi\)
−0.396389 + 0.918083i \(0.629737\pi\)
\(384\) −2.52230 −0.128715
\(385\) −4.63187 −0.236062
\(386\) 16.5398 0.841854
\(387\) 3.36197 0.170899
\(388\) −14.4102 −0.731569
\(389\) 7.74070 0.392469 0.196234 0.980557i \(-0.437129\pi\)
0.196234 + 0.980557i \(0.437129\pi\)
\(390\) 9.89264 0.500933
\(391\) −15.9729 −0.807782
\(392\) 14.4542 0.730049
\(393\) 23.1119 1.16584
\(394\) 26.3121 1.32558
\(395\) −14.1493 −0.711931
\(396\) −3.36197 −0.168946
\(397\) 13.5642 0.680768 0.340384 0.940286i \(-0.389443\pi\)
0.340384 + 0.940286i \(0.389443\pi\)
\(398\) −9.23473 −0.462895
\(399\) 25.8751 1.29537
\(400\) 1.00000 0.0500000
\(401\) −3.21297 −0.160448 −0.0802240 0.996777i \(-0.525564\pi\)
−0.0802240 + 0.996777i \(0.525564\pi\)
\(402\) 28.2560 1.40928
\(403\) −19.7182 −0.982234
\(404\) −2.55687 −0.127209
\(405\) −7.78305 −0.386743
\(406\) 29.2245 1.45039
\(407\) 6.44231 0.319334
\(408\) 6.80306 0.336802
\(409\) 23.5186 1.16292 0.581459 0.813576i \(-0.302482\pi\)
0.581459 + 0.813576i \(0.302482\pi\)
\(410\) −6.04125 −0.298356
\(411\) −25.5950 −1.26251
\(412\) 1.93523 0.0953418
\(413\) −14.7817 −0.727357
\(414\) 19.9099 0.978517
\(415\) 1.14363 0.0561384
\(416\) −3.92208 −0.192296
\(417\) −43.5123 −2.13081
\(418\) 2.21477 0.108328
\(419\) 5.86828 0.286684 0.143342 0.989673i \(-0.454215\pi\)
0.143342 + 0.989673i \(0.454215\pi\)
\(420\) −11.6829 −0.570069
\(421\) 30.4948 1.48623 0.743113 0.669166i \(-0.233349\pi\)
0.743113 + 0.669166i \(0.233349\pi\)
\(422\) 2.32602 0.113229
\(423\) 43.1051 2.09584
\(424\) 4.11880 0.200027
\(425\) −2.69717 −0.130832
\(426\) −17.7798 −0.861433
\(427\) 22.8846 1.10746
\(428\) −14.1549 −0.684201
\(429\) −9.89264 −0.477621
\(430\) 1.00000 0.0482243
\(431\) −37.6639 −1.81421 −0.907103 0.420908i \(-0.861712\pi\)
−0.907103 + 0.420908i \(0.861712\pi\)
\(432\) −0.913006 −0.0439270
\(433\) 8.54423 0.410609 0.205305 0.978698i \(-0.434181\pi\)
0.205305 + 0.978698i \(0.434181\pi\)
\(434\) 23.2867 1.11780
\(435\) −15.9143 −0.763031
\(436\) 15.2102 0.728434
\(437\) −13.1161 −0.627426
\(438\) −34.5209 −1.64947
\(439\) 1.30117 0.0621017 0.0310508 0.999518i \(-0.490115\pi\)
0.0310508 + 0.999518i \(0.490115\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 48.5947 2.31404
\(442\) 10.5785 0.503169
\(443\) 17.3071 0.822286 0.411143 0.911571i \(-0.365130\pi\)
0.411143 + 0.911571i \(0.365130\pi\)
\(444\) 16.2494 0.771163
\(445\) 8.75478 0.415016
\(446\) −25.7990 −1.22162
\(447\) 22.4801 1.06327
\(448\) 4.63187 0.218835
\(449\) 21.8768 1.03243 0.516214 0.856460i \(-0.327341\pi\)
0.516214 + 0.856460i \(0.327341\pi\)
\(450\) 3.36197 0.158485
\(451\) 6.04125 0.284471
\(452\) 14.3190 0.673508
\(453\) −29.2302 −1.37336
\(454\) −19.1186 −0.897278
\(455\) −18.1666 −0.851662
\(456\) 5.58631 0.261603
\(457\) 24.1796 1.13107 0.565536 0.824723i \(-0.308669\pi\)
0.565536 + 0.824723i \(0.308669\pi\)
\(458\) 7.28071 0.340205
\(459\) 2.46253 0.114941
\(460\) 5.92208 0.276118
\(461\) −30.6950 −1.42961 −0.714803 0.699325i \(-0.753484\pi\)
−0.714803 + 0.699325i \(0.753484\pi\)
\(462\) 11.6829 0.543540
\(463\) 26.3386 1.22406 0.612029 0.790835i \(-0.290354\pi\)
0.612029 + 0.790835i \(0.290354\pi\)
\(464\) 6.30944 0.292908
\(465\) −12.6808 −0.588058
\(466\) 26.8808 1.24523
\(467\) −23.6445 −1.09414 −0.547069 0.837088i \(-0.684256\pi\)
−0.547069 + 0.837088i \(0.684256\pi\)
\(468\) −13.1859 −0.609520
\(469\) −51.8885 −2.39599
\(470\) 12.8214 0.591406
\(471\) 30.2671 1.39463
\(472\) −3.19129 −0.146891
\(473\) −1.00000 −0.0459800
\(474\) 35.6888 1.63924
\(475\) −2.21477 −0.101621
\(476\) −12.4929 −0.572613
\(477\) 13.8473 0.634024
\(478\) −6.55613 −0.299871
\(479\) 9.37940 0.428556 0.214278 0.976773i \(-0.431260\pi\)
0.214278 + 0.976773i \(0.431260\pi\)
\(480\) −2.52230 −0.115127
\(481\) 25.2673 1.15209
\(482\) −20.1131 −0.916126
\(483\) −69.1873 −3.14813
\(484\) 1.00000 0.0454545
\(485\) −14.4102 −0.654335
\(486\) 22.3702 1.01473
\(487\) −36.9369 −1.67377 −0.836886 0.547377i \(-0.815626\pi\)
−0.836886 + 0.547377i \(0.815626\pi\)
\(488\) 4.94068 0.223654
\(489\) −28.8834 −1.30615
\(490\) 14.4542 0.652976
\(491\) −16.5633 −0.747490 −0.373745 0.927531i \(-0.621927\pi\)
−0.373745 + 0.927531i \(0.621927\pi\)
\(492\) 15.2378 0.686973
\(493\) −17.0176 −0.766436
\(494\) 8.68651 0.390825
\(495\) −3.36197 −0.151109
\(496\) 5.02749 0.225741
\(497\) 32.6503 1.46457
\(498\) −2.88456 −0.129260
\(499\) −11.6915 −0.523382 −0.261691 0.965152i \(-0.584280\pi\)
−0.261691 + 0.965152i \(0.584280\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.04672 0.359501
\(502\) −16.7913 −0.749431
\(503\) −2.57093 −0.114632 −0.0573159 0.998356i \(-0.518254\pi\)
−0.0573159 + 0.998356i \(0.518254\pi\)
\(504\) 15.5722 0.693642
\(505\) −2.55687 −0.113779
\(506\) −5.92208 −0.263269
\(507\) −6.00989 −0.266908
\(508\) 16.7232 0.741973
\(509\) −8.84751 −0.392159 −0.196080 0.980588i \(-0.562821\pi\)
−0.196080 + 0.980588i \(0.562821\pi\)
\(510\) 6.80306 0.301245
\(511\) 63.3932 2.80435
\(512\) 1.00000 0.0441942
\(513\) 2.02210 0.0892779
\(514\) −18.5623 −0.818750
\(515\) 1.93523 0.0852763
\(516\) −2.52230 −0.111038
\(517\) −12.8214 −0.563884
\(518\) −29.8400 −1.31109
\(519\) 3.09095 0.135678
\(520\) −3.92208 −0.171995
\(521\) −0.0487768 −0.00213695 −0.00106847 0.999999i \(-0.500340\pi\)
−0.00106847 + 0.999999i \(0.500340\pi\)
\(522\) 21.2122 0.928432
\(523\) −26.6126 −1.16369 −0.581844 0.813301i \(-0.697668\pi\)
−0.581844 + 0.813301i \(0.697668\pi\)
\(524\) −9.16306 −0.400290
\(525\) −11.6829 −0.509886
\(526\) 25.1701 1.09747
\(527\) −13.5600 −0.590682
\(528\) 2.52230 0.109769
\(529\) 12.0710 0.524827
\(530\) 4.11880 0.178909
\(531\) −10.7290 −0.465601
\(532\) −10.2585 −0.444764
\(533\) 23.6942 1.02631
\(534\) −22.0821 −0.955588
\(535\) −14.1549 −0.611968
\(536\) −11.2025 −0.483874
\(537\) 5.08297 0.219346
\(538\) −4.57383 −0.197192
\(539\) −14.4542 −0.622588
\(540\) −0.913006 −0.0392895
\(541\) −27.3476 −1.17576 −0.587882 0.808947i \(-0.700038\pi\)
−0.587882 + 0.808947i \(0.700038\pi\)
\(542\) 4.65356 0.199888
\(543\) −37.1999 −1.59640
\(544\) −2.69717 −0.115640
\(545\) 15.2102 0.651531
\(546\) 45.8214 1.96098
\(547\) 23.2933 0.995951 0.497975 0.867191i \(-0.334077\pi\)
0.497975 + 0.867191i \(0.334077\pi\)
\(548\) 10.1475 0.433481
\(549\) 16.6104 0.708916
\(550\) −1.00000 −0.0426401
\(551\) −13.9740 −0.595312
\(552\) −14.9372 −0.635771
\(553\) −65.5379 −2.78696
\(554\) 33.2000 1.41053
\(555\) 16.2494 0.689749
\(556\) 17.2511 0.731608
\(557\) −7.95713 −0.337154 −0.168577 0.985688i \(-0.553917\pi\)
−0.168577 + 0.985688i \(0.553917\pi\)
\(558\) 16.9023 0.715531
\(559\) −3.92208 −0.165886
\(560\) 4.63187 0.195732
\(561\) −6.80306 −0.287225
\(562\) 8.38605 0.353744
\(563\) −32.7059 −1.37839 −0.689194 0.724577i \(-0.742035\pi\)
−0.689194 + 0.724577i \(0.742035\pi\)
\(564\) −32.3393 −1.36173
\(565\) 14.3190 0.602404
\(566\) 19.9325 0.837823
\(567\) −36.0501 −1.51396
\(568\) 7.04905 0.295771
\(569\) −7.68766 −0.322284 −0.161142 0.986931i \(-0.551518\pi\)
−0.161142 + 0.986931i \(0.551518\pi\)
\(570\) 5.58631 0.233985
\(571\) −36.5719 −1.53049 −0.765244 0.643741i \(-0.777382\pi\)
−0.765244 + 0.643741i \(0.777382\pi\)
\(572\) 3.92208 0.163990
\(573\) 7.86102 0.328399
\(574\) −27.9823 −1.16796
\(575\) 5.92208 0.246968
\(576\) 3.36197 0.140082
\(577\) −37.8528 −1.57583 −0.787917 0.615781i \(-0.788841\pi\)
−0.787917 + 0.615781i \(0.788841\pi\)
\(578\) −9.72527 −0.404518
\(579\) −41.7183 −1.73375
\(580\) 6.30944 0.261985
\(581\) 5.29713 0.219762
\(582\) 36.3469 1.50663
\(583\) −4.11880 −0.170583
\(584\) 13.6863 0.566343
\(585\) −13.1859 −0.545171
\(586\) 10.0162 0.413765
\(587\) −41.8305 −1.72653 −0.863265 0.504751i \(-0.831584\pi\)
−0.863265 + 0.504751i \(0.831584\pi\)
\(588\) −36.4578 −1.50350
\(589\) −11.1347 −0.458799
\(590\) −3.19129 −0.131383
\(591\) −66.3669 −2.72997
\(592\) −6.44231 −0.264777
\(593\) −37.9900 −1.56006 −0.780032 0.625740i \(-0.784797\pi\)
−0.780032 + 0.625740i \(0.784797\pi\)
\(594\) 0.913006 0.0374611
\(595\) −12.4929 −0.512161
\(596\) −8.91255 −0.365072
\(597\) 23.2927 0.953307
\(598\) −23.2269 −0.949817
\(599\) −10.6189 −0.433875 −0.216937 0.976186i \(-0.569607\pi\)
−0.216937 + 0.976186i \(0.569607\pi\)
\(600\) −2.52230 −0.102972
\(601\) 42.5331 1.73496 0.867482 0.497469i \(-0.165737\pi\)
0.867482 + 0.497469i \(0.165737\pi\)
\(602\) 4.63187 0.188781
\(603\) −37.6625 −1.53374
\(604\) 11.5887 0.471539
\(605\) 1.00000 0.0406558
\(606\) 6.44919 0.261981
\(607\) 18.4841 0.750245 0.375122 0.926975i \(-0.377601\pi\)
0.375122 + 0.926975i \(0.377601\pi\)
\(608\) −2.21477 −0.0898209
\(609\) −73.7129 −2.98700
\(610\) 4.94068 0.200042
\(611\) −50.2865 −2.03437
\(612\) −9.06781 −0.366545
\(613\) −12.2905 −0.496407 −0.248204 0.968708i \(-0.579840\pi\)
−0.248204 + 0.968708i \(0.579840\pi\)
\(614\) 10.0537 0.405733
\(615\) 15.2378 0.614448
\(616\) −4.63187 −0.186623
\(617\) −18.8883 −0.760415 −0.380207 0.924901i \(-0.624147\pi\)
−0.380207 + 0.924901i \(0.624147\pi\)
\(618\) −4.88121 −0.196351
\(619\) 17.1754 0.690338 0.345169 0.938541i \(-0.387822\pi\)
0.345169 + 0.938541i \(0.387822\pi\)
\(620\) 5.02749 0.201909
\(621\) −5.40689 −0.216971
\(622\) 1.08738 0.0436001
\(623\) 40.5510 1.62464
\(624\) 9.89264 0.396023
\(625\) 1.00000 0.0400000
\(626\) 23.8551 0.953441
\(627\) −5.58631 −0.223096
\(628\) −11.9998 −0.478845
\(629\) 17.3760 0.692827
\(630\) 15.5722 0.620413
\(631\) −11.6610 −0.464216 −0.232108 0.972690i \(-0.574562\pi\)
−0.232108 + 0.972690i \(0.574562\pi\)
\(632\) −14.1493 −0.562831
\(633\) −5.86692 −0.233189
\(634\) −14.7634 −0.586329
\(635\) 16.7232 0.663641
\(636\) −10.3888 −0.411944
\(637\) −56.6906 −2.24616
\(638\) −6.30944 −0.249793
\(639\) 23.6987 0.937507
\(640\) 1.00000 0.0395285
\(641\) −40.4607 −1.59810 −0.799052 0.601262i \(-0.794665\pi\)
−0.799052 + 0.601262i \(0.794665\pi\)
\(642\) 35.7027 1.40907
\(643\) −3.01098 −0.118742 −0.0593708 0.998236i \(-0.518909\pi\)
−0.0593708 + 0.998236i \(0.518909\pi\)
\(644\) 27.4303 1.08091
\(645\) −2.52230 −0.0993153
\(646\) 5.97362 0.235029
\(647\) 19.4547 0.764843 0.382422 0.923988i \(-0.375090\pi\)
0.382422 + 0.923988i \(0.375090\pi\)
\(648\) −7.78305 −0.305747
\(649\) 3.19129 0.125269
\(650\) −3.92208 −0.153837
\(651\) −58.7359 −2.30204
\(652\) 11.4512 0.448466
\(653\) 36.6440 1.43399 0.716994 0.697079i \(-0.245517\pi\)
0.716994 + 0.697079i \(0.245517\pi\)
\(654\) −38.3645 −1.50017
\(655\) −9.16306 −0.358030
\(656\) −6.04125 −0.235871
\(657\) 46.0130 1.79514
\(658\) 59.3870 2.31515
\(659\) −1.29024 −0.0502604 −0.0251302 0.999684i \(-0.508000\pi\)
−0.0251302 + 0.999684i \(0.508000\pi\)
\(660\) 2.52230 0.0981802
\(661\) −24.6149 −0.957410 −0.478705 0.877976i \(-0.658894\pi\)
−0.478705 + 0.877976i \(0.658894\pi\)
\(662\) 12.0944 0.470064
\(663\) −26.6821 −1.03625
\(664\) 1.14363 0.0443813
\(665\) −10.2585 −0.397809
\(666\) −21.6589 −0.839265
\(667\) 37.3650 1.44678
\(668\) −3.19024 −0.123434
\(669\) 65.0727 2.51585
\(670\) −11.2025 −0.432790
\(671\) −4.94068 −0.190733
\(672\) −11.6829 −0.450679
\(673\) 9.12109 0.351592 0.175796 0.984427i \(-0.443750\pi\)
0.175796 + 0.984427i \(0.443750\pi\)
\(674\) −8.34846 −0.321571
\(675\) −0.913006 −0.0351416
\(676\) 2.38270 0.0916425
\(677\) −2.91368 −0.111982 −0.0559910 0.998431i \(-0.517832\pi\)
−0.0559910 + 0.998431i \(0.517832\pi\)
\(678\) −36.1167 −1.38705
\(679\) −66.7464 −2.56149
\(680\) −2.69717 −0.103432
\(681\) 48.2226 1.84790
\(682\) −5.02749 −0.192512
\(683\) 21.2252 0.812159 0.406080 0.913838i \(-0.366896\pi\)
0.406080 + 0.913838i \(0.366896\pi\)
\(684\) −7.44601 −0.284705
\(685\) 10.1475 0.387717
\(686\) 34.5270 1.31825
\(687\) −18.3641 −0.700634
\(688\) 1.00000 0.0381246
\(689\) −16.1543 −0.615428
\(690\) −14.9372 −0.568651
\(691\) 47.7344 1.81590 0.907951 0.419076i \(-0.137646\pi\)
0.907951 + 0.419076i \(0.137646\pi\)
\(692\) −1.22545 −0.0465847
\(693\) −15.5722 −0.591540
\(694\) 9.15805 0.347635
\(695\) 17.2511 0.654371
\(696\) −15.9143 −0.603229
\(697\) 16.2943 0.617189
\(698\) 5.57565 0.211041
\(699\) −67.8013 −2.56448
\(700\) 4.63187 0.175068
\(701\) −36.8511 −1.39185 −0.695924 0.718115i \(-0.745005\pi\)
−0.695924 + 0.718115i \(0.745005\pi\)
\(702\) 3.58088 0.135152
\(703\) 14.2683 0.538138
\(704\) −1.00000 −0.0376889
\(705\) −32.3393 −1.21797
\(706\) −23.8879 −0.899031
\(707\) −11.8431 −0.445406
\(708\) 8.04938 0.302514
\(709\) 24.9895 0.938499 0.469249 0.883066i \(-0.344525\pi\)
0.469249 + 0.883066i \(0.344525\pi\)
\(710\) 7.04905 0.264546
\(711\) −47.5697 −1.78400
\(712\) 8.75478 0.328099
\(713\) 29.7732 1.11501
\(714\) 31.5109 1.17927
\(715\) 3.92208 0.146677
\(716\) −2.01522 −0.0753122
\(717\) 16.5365 0.617567
\(718\) −20.4519 −0.763257
\(719\) −41.8922 −1.56232 −0.781158 0.624334i \(-0.785370\pi\)
−0.781158 + 0.624334i \(0.785370\pi\)
\(720\) 3.36197 0.125293
\(721\) 8.96372 0.333826
\(722\) −14.0948 −0.524553
\(723\) 50.7311 1.88671
\(724\) 14.7484 0.548122
\(725\) 6.30944 0.234327
\(726\) −2.52230 −0.0936112
\(727\) 10.3572 0.384127 0.192063 0.981383i \(-0.438482\pi\)
0.192063 + 0.981383i \(0.438482\pi\)
\(728\) −18.1666 −0.673298
\(729\) −33.0750 −1.22500
\(730\) 13.6863 0.506553
\(731\) −2.69717 −0.0997584
\(732\) −12.4619 −0.460603
\(733\) 12.8827 0.475833 0.237916 0.971286i \(-0.423536\pi\)
0.237916 + 0.971286i \(0.423536\pi\)
\(734\) 10.6239 0.392136
\(735\) −36.4578 −1.34477
\(736\) 5.92208 0.218291
\(737\) 11.2025 0.412649
\(738\) −20.3105 −0.747640
\(739\) −2.89793 −0.106602 −0.0533010 0.998578i \(-0.516974\pi\)
−0.0533010 + 0.998578i \(0.516974\pi\)
\(740\) −6.44231 −0.236824
\(741\) −21.9100 −0.804882
\(742\) 19.0777 0.700366
\(743\) 37.7285 1.38412 0.692062 0.721838i \(-0.256703\pi\)
0.692062 + 0.721838i \(0.256703\pi\)
\(744\) −12.6808 −0.464901
\(745\) −8.91255 −0.326531
\(746\) 25.0492 0.917117
\(747\) 3.84484 0.140675
\(748\) 2.69717 0.0986183
\(749\) −65.5634 −2.39564
\(750\) −2.52230 −0.0921012
\(751\) −43.5827 −1.59036 −0.795178 0.606376i \(-0.792623\pi\)
−0.795178 + 0.606376i \(0.792623\pi\)
\(752\) 12.8214 0.467548
\(753\) 42.3526 1.54341
\(754\) −24.7461 −0.901201
\(755\) 11.5887 0.421758
\(756\) −4.22892 −0.153804
\(757\) 28.6284 1.04052 0.520259 0.854009i \(-0.325835\pi\)
0.520259 + 0.854009i \(0.325835\pi\)
\(758\) −4.98512 −0.181068
\(759\) 14.9372 0.542187
\(760\) −2.21477 −0.0803382
\(761\) −13.3213 −0.482895 −0.241448 0.970414i \(-0.577622\pi\)
−0.241448 + 0.970414i \(0.577622\pi\)
\(762\) −42.1809 −1.52805
\(763\) 70.4515 2.55051
\(764\) −3.11661 −0.112755
\(765\) −9.06781 −0.327848
\(766\) −15.5150 −0.560579
\(767\) 12.5165 0.451945
\(768\) −2.52230 −0.0910155
\(769\) 23.9167 0.862460 0.431230 0.902242i \(-0.358080\pi\)
0.431230 + 0.902242i \(0.358080\pi\)
\(770\) −4.63187 −0.166921
\(771\) 46.8197 1.68617
\(772\) 16.5398 0.595280
\(773\) −2.36210 −0.0849588 −0.0424794 0.999097i \(-0.513526\pi\)
−0.0424794 + 0.999097i \(0.513526\pi\)
\(774\) 3.36197 0.120844
\(775\) 5.02749 0.180593
\(776\) −14.4102 −0.517298
\(777\) 75.2652 2.70012
\(778\) 7.74070 0.277517
\(779\) 13.3800 0.479388
\(780\) 9.89264 0.354213
\(781\) −7.04905 −0.252235
\(782\) −15.9729 −0.571188
\(783\) −5.76055 −0.205865
\(784\) 14.4542 0.516223
\(785\) −11.9998 −0.428292
\(786\) 23.1119 0.824375
\(787\) −24.6344 −0.878121 −0.439060 0.898458i \(-0.644689\pi\)
−0.439060 + 0.898458i \(0.644689\pi\)
\(788\) 26.3121 0.937329
\(789\) −63.4863 −2.26017
\(790\) −14.1493 −0.503411
\(791\) 66.3236 2.35820
\(792\) −3.36197 −0.119463
\(793\) −19.3777 −0.688123
\(794\) 13.5642 0.481376
\(795\) −10.3888 −0.368454
\(796\) −9.23473 −0.327316
\(797\) −5.67534 −0.201031 −0.100515 0.994935i \(-0.532049\pi\)
−0.100515 + 0.994935i \(0.532049\pi\)
\(798\) 25.8751 0.915968
\(799\) −34.5814 −1.22340
\(800\) 1.00000 0.0353553
\(801\) 29.4333 1.03998
\(802\) −3.21297 −0.113454
\(803\) −13.6863 −0.482979
\(804\) 28.2560 0.996512
\(805\) 27.4303 0.966791
\(806\) −19.7182 −0.694544
\(807\) 11.5366 0.406106
\(808\) −2.55687 −0.0899505
\(809\) 51.0023 1.79315 0.896573 0.442897i \(-0.146049\pi\)
0.896573 + 0.442897i \(0.146049\pi\)
\(810\) −7.78305 −0.273469
\(811\) −32.0335 −1.12485 −0.562424 0.826849i \(-0.690131\pi\)
−0.562424 + 0.826849i \(0.690131\pi\)
\(812\) 29.2245 1.02558
\(813\) −11.7377 −0.411658
\(814\) 6.44231 0.225803
\(815\) 11.4512 0.401120
\(816\) 6.80306 0.238155
\(817\) −2.21477 −0.0774851
\(818\) 23.5186 0.822307
\(819\) −61.0755 −2.13415
\(820\) −6.04125 −0.210969
\(821\) −32.8989 −1.14818 −0.574089 0.818793i \(-0.694644\pi\)
−0.574089 + 0.818793i \(0.694644\pi\)
\(822\) −25.5950 −0.892730
\(823\) 4.63586 0.161596 0.0807980 0.996730i \(-0.474253\pi\)
0.0807980 + 0.996730i \(0.474253\pi\)
\(824\) 1.93523 0.0674168
\(825\) 2.52230 0.0878151
\(826\) −14.7817 −0.514319
\(827\) 9.67481 0.336426 0.168213 0.985751i \(-0.446200\pi\)
0.168213 + 0.985751i \(0.446200\pi\)
\(828\) 19.9099 0.691916
\(829\) −25.6440 −0.890654 −0.445327 0.895368i \(-0.646913\pi\)
−0.445327 + 0.895368i \(0.646913\pi\)
\(830\) 1.14363 0.0396958
\(831\) −83.7401 −2.90491
\(832\) −3.92208 −0.135974
\(833\) −38.9855 −1.35077
\(834\) −43.5123 −1.50671
\(835\) −3.19024 −0.110403
\(836\) 2.21477 0.0765995
\(837\) −4.59012 −0.158658
\(838\) 5.86828 0.202717
\(839\) −16.9831 −0.586321 −0.293160 0.956063i \(-0.594707\pi\)
−0.293160 + 0.956063i \(0.594707\pi\)
\(840\) −11.6829 −0.403100
\(841\) 10.8091 0.372726
\(842\) 30.4948 1.05092
\(843\) −21.1521 −0.728517
\(844\) 2.32602 0.0800651
\(845\) 2.38270 0.0819675
\(846\) 43.1051 1.48199
\(847\) 4.63187 0.159153
\(848\) 4.11880 0.141440
\(849\) −50.2755 −1.72545
\(850\) −2.69717 −0.0925122
\(851\) −38.1519 −1.30783
\(852\) −17.7798 −0.609125
\(853\) 20.7936 0.711959 0.355980 0.934494i \(-0.384147\pi\)
0.355980 + 0.934494i \(0.384147\pi\)
\(854\) 22.8846 0.783094
\(855\) −7.44601 −0.254648
\(856\) −14.1549 −0.483803
\(857\) 3.11620 0.106448 0.0532238 0.998583i \(-0.483050\pi\)
0.0532238 + 0.998583i \(0.483050\pi\)
\(858\) −9.89264 −0.337729
\(859\) −11.1163 −0.379282 −0.189641 0.981853i \(-0.560732\pi\)
−0.189641 + 0.981853i \(0.560732\pi\)
\(860\) 1.00000 0.0340997
\(861\) 70.5796 2.40535
\(862\) −37.6639 −1.28284
\(863\) 0.674845 0.0229720 0.0114860 0.999934i \(-0.496344\pi\)
0.0114860 + 0.999934i \(0.496344\pi\)
\(864\) −0.913006 −0.0310611
\(865\) −1.22545 −0.0416667
\(866\) 8.54423 0.290345
\(867\) 24.5300 0.833083
\(868\) 23.2867 0.790401
\(869\) 14.1493 0.479983
\(870\) −15.9143 −0.539544
\(871\) 43.9371 1.48875
\(872\) 15.2102 0.515081
\(873\) −48.4469 −1.63968
\(874\) −13.1161 −0.443657
\(875\) 4.63187 0.156586
\(876\) −34.5209 −1.16635
\(877\) −53.2460 −1.79799 −0.898995 0.437959i \(-0.855701\pi\)
−0.898995 + 0.437959i \(0.855701\pi\)
\(878\) 1.30117 0.0439125
\(879\) −25.2638 −0.852126
\(880\) −1.00000 −0.0337100
\(881\) −19.1905 −0.646545 −0.323273 0.946306i \(-0.604783\pi\)
−0.323273 + 0.946306i \(0.604783\pi\)
\(882\) 48.5947 1.63627
\(883\) 30.7947 1.03632 0.518162 0.855283i \(-0.326616\pi\)
0.518162 + 0.855283i \(0.326616\pi\)
\(884\) 10.5785 0.355794
\(885\) 8.04938 0.270577
\(886\) 17.3071 0.581444
\(887\) 41.7004 1.40016 0.700081 0.714064i \(-0.253147\pi\)
0.700081 + 0.714064i \(0.253147\pi\)
\(888\) 16.2494 0.545295
\(889\) 77.4598 2.59792
\(890\) 8.75478 0.293461
\(891\) 7.78305 0.260742
\(892\) −25.7990 −0.863814
\(893\) −28.3964 −0.950251
\(894\) 22.4801 0.751847
\(895\) −2.01522 −0.0673613
\(896\) 4.63187 0.154740
\(897\) 58.5850 1.95610
\(898\) 21.8768 0.730037
\(899\) 31.7206 1.05794
\(900\) 3.36197 0.112066
\(901\) −11.1091 −0.370098
\(902\) 6.04125 0.201152
\(903\) −11.6829 −0.388784
\(904\) 14.3190 0.476242
\(905\) 14.7484 0.490255
\(906\) −29.2302 −0.971109
\(907\) 39.4173 1.30883 0.654415 0.756136i \(-0.272915\pi\)
0.654415 + 0.756136i \(0.272915\pi\)
\(908\) −19.1186 −0.634471
\(909\) −8.59614 −0.285116
\(910\) −18.1666 −0.602216
\(911\) 20.2609 0.671274 0.335637 0.941991i \(-0.391048\pi\)
0.335637 + 0.941991i \(0.391048\pi\)
\(912\) 5.58631 0.184981
\(913\) −1.14363 −0.0378485
\(914\) 24.1796 0.799789
\(915\) −12.4619 −0.411976
\(916\) 7.28071 0.240561
\(917\) −42.4421 −1.40156
\(918\) 2.46253 0.0812756
\(919\) −52.3061 −1.72542 −0.862710 0.505699i \(-0.831235\pi\)
−0.862710 + 0.505699i \(0.831235\pi\)
\(920\) 5.92208 0.195245
\(921\) −25.3583 −0.835586
\(922\) −30.6950 −1.01088
\(923\) −27.6469 −0.910010
\(924\) 11.6829 0.384341
\(925\) −6.44231 −0.211822
\(926\) 26.3386 0.865540
\(927\) 6.50618 0.213691
\(928\) 6.30944 0.207118
\(929\) 15.0232 0.492894 0.246447 0.969156i \(-0.420737\pi\)
0.246447 + 0.969156i \(0.420737\pi\)
\(930\) −12.6808 −0.415820
\(931\) −32.0128 −1.04918
\(932\) 26.8808 0.880509
\(933\) −2.74270 −0.0897921
\(934\) −23.6445 −0.773672
\(935\) 2.69717 0.0882069
\(936\) −13.1859 −0.430996
\(937\) 29.1548 0.952447 0.476224 0.879324i \(-0.342005\pi\)
0.476224 + 0.879324i \(0.342005\pi\)
\(938\) −51.8885 −1.69422
\(939\) −60.1696 −1.96356
\(940\) 12.8214 0.418187
\(941\) −38.2638 −1.24736 −0.623682 0.781678i \(-0.714364\pi\)
−0.623682 + 0.781678i \(0.714364\pi\)
\(942\) 30.2671 0.986155
\(943\) −35.7767 −1.16505
\(944\) −3.19129 −0.103868
\(945\) −4.22892 −0.137567
\(946\) −1.00000 −0.0325128
\(947\) 22.9150 0.744638 0.372319 0.928105i \(-0.378563\pi\)
0.372319 + 0.928105i \(0.378563\pi\)
\(948\) 35.6888 1.15912
\(949\) −53.6787 −1.74249
\(950\) −2.21477 −0.0718567
\(951\) 37.2376 1.20751
\(952\) −12.4929 −0.404899
\(953\) −41.3442 −1.33927 −0.669634 0.742691i \(-0.733549\pi\)
−0.669634 + 0.742691i \(0.733549\pi\)
\(954\) 13.8473 0.448323
\(955\) −3.11661 −0.100851
\(956\) −6.55613 −0.212041
\(957\) 15.9143 0.514435
\(958\) 9.37940 0.303035
\(959\) 47.0020 1.51777
\(960\) −2.52230 −0.0814067
\(961\) −5.72437 −0.184657
\(962\) 25.2673 0.814649
\(963\) −47.5882 −1.53351
\(964\) −20.1131 −0.647799
\(965\) 16.5398 0.532435
\(966\) −69.1873 −2.22607
\(967\) 18.8045 0.604711 0.302356 0.953195i \(-0.402227\pi\)
0.302356 + 0.953195i \(0.402227\pi\)
\(968\) 1.00000 0.0321412
\(969\) −15.0672 −0.484029
\(970\) −14.4102 −0.462685
\(971\) 0.647774 0.0207880 0.0103940 0.999946i \(-0.496691\pi\)
0.0103940 + 0.999946i \(0.496691\pi\)
\(972\) 22.3702 0.717524
\(973\) 79.9048 2.56163
\(974\) −36.9369 −1.18354
\(975\) 9.89264 0.316818
\(976\) 4.94068 0.158147
\(977\) 7.00553 0.224127 0.112063 0.993701i \(-0.464254\pi\)
0.112063 + 0.993701i \(0.464254\pi\)
\(978\) −28.8834 −0.923590
\(979\) −8.75478 −0.279804
\(980\) 14.4542 0.461723
\(981\) 51.1361 1.63265
\(982\) −16.5633 −0.528555
\(983\) 21.2397 0.677442 0.338721 0.940887i \(-0.390006\pi\)
0.338721 + 0.940887i \(0.390006\pi\)
\(984\) 15.2378 0.485764
\(985\) 26.3121 0.838373
\(986\) −17.0176 −0.541952
\(987\) −149.792 −4.76792
\(988\) 8.68651 0.276355
\(989\) 5.92208 0.188311
\(990\) −3.36197 −0.106851
\(991\) −43.2153 −1.37278 −0.686389 0.727234i \(-0.740805\pi\)
−0.686389 + 0.727234i \(0.740805\pi\)
\(992\) 5.02749 0.159623
\(993\) −30.5058 −0.968071
\(994\) 32.6503 1.03560
\(995\) −9.23473 −0.292761
\(996\) −2.88456 −0.0914008
\(997\) 20.0117 0.633776 0.316888 0.948463i \(-0.397362\pi\)
0.316888 + 0.948463i \(0.397362\pi\)
\(998\) −11.6915 −0.370087
\(999\) 5.88187 0.186094
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 4730.2.a.be.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.be.1.3 12 1.1 even 1 trivial