Properties

Label 231.2.a.d.1.3
Level $231$
Weight $2$
Character 231.1
Self dual yes
Analytic conductor $1.845$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 231.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.52892 q^{2} -1.00000 q^{3} +4.39543 q^{4} +0.133492 q^{5} -2.52892 q^{6} -1.00000 q^{7} +6.05784 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.52892 q^{2} -1.00000 q^{3} +4.39543 q^{4} +0.133492 q^{5} -2.52892 q^{6} -1.00000 q^{7} +6.05784 q^{8} +1.00000 q^{9} +0.337590 q^{10} +1.00000 q^{11} -4.39543 q^{12} +0.133492 q^{13} -2.52892 q^{14} -0.133492 q^{15} +6.52892 q^{16} -5.05784 q^{17} +2.52892 q^{18} -0.924344 q^{19} +0.586754 q^{20} +1.00000 q^{21} +2.52892 q^{22} -7.05784 q^{23} -6.05784 q^{24} -4.98218 q^{25} +0.337590 q^{26} -1.00000 q^{27} -4.39543 q^{28} +3.86651 q^{29} -0.337590 q^{30} +2.79085 q^{31} +4.39543 q^{32} -1.00000 q^{33} -12.7909 q^{34} -0.133492 q^{35} +4.39543 q^{36} +9.98218 q^{37} -2.33759 q^{38} -0.133492 q^{39} +0.808672 q^{40} +11.8487 q^{41} +2.52892 q^{42} -3.05784 q^{43} +4.39543 q^{44} +0.133492 q^{45} -17.8487 q^{46} -3.07566 q^{47} -6.52892 q^{48} +1.00000 q^{49} -12.5995 q^{50} +5.05784 q^{51} +0.586754 q^{52} -4.79085 q^{53} -2.52892 q^{54} +0.133492 q^{55} -6.05784 q^{56} +0.924344 q^{57} +9.77808 q^{58} -12.6574 q^{59} -0.586754 q^{60} +6.00000 q^{61} +7.05784 q^{62} -1.00000 q^{63} -1.94216 q^{64} +0.0178201 q^{65} -2.52892 q^{66} +8.92434 q^{67} -22.2313 q^{68} +7.05784 q^{69} -0.337590 q^{70} -6.11567 q^{71} +6.05784 q^{72} +7.86651 q^{73} +25.2441 q^{74} +4.98218 q^{75} -4.06289 q^{76} -1.00000 q^{77} -0.337590 q^{78} +14.1157 q^{79} +0.871558 q^{80} +1.00000 q^{81} +29.9644 q^{82} -1.20915 q^{83} +4.39543 q^{84} -0.675180 q^{85} -7.73302 q^{86} -3.86651 q^{87} +6.05784 q^{88} +15.5817 q^{89} +0.337590 q^{90} -0.133492 q^{91} -31.0222 q^{92} -2.79085 q^{93} -7.77808 q^{94} -0.123392 q^{95} -4.39543 q^{96} +12.7909 q^{97} +2.52892 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{7} + 3 q^{8} + 3 q^{9} + 9 q^{10} + 3 q^{11} - 6 q^{12} + 12 q^{16} + 12 q^{19} - 21 q^{20} + 3 q^{21} - 6 q^{23} - 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} - 6 q^{28} + 12 q^{29} - 9 q^{30} - 6 q^{31} + 6 q^{32} - 3 q^{33} - 24 q^{34} + 6 q^{36} - 15 q^{38} + 18 q^{40} + 6 q^{41} + 6 q^{43} + 6 q^{44} - 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} - 39 q^{50} - 21 q^{52} - 3 q^{56} - 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} + 18 q^{61} + 6 q^{62} - 3 q^{63} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} + 3 q^{72} + 24 q^{73} + 39 q^{74} - 15 q^{75} - 3 q^{76} - 3 q^{77} - 9 q^{78} + 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} - 18 q^{83} + 6 q^{84} - 18 q^{85} - 24 q^{86} - 12 q^{87} + 3 q^{88} + 18 q^{89} + 9 q^{90} - 18 q^{92} + 6 q^{93} + 15 q^{94} + 12 q^{95} - 6 q^{96} + 24 q^{97} + 3 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.52892 1.78822 0.894108 0.447852i \(-0.147811\pi\)
0.894108 + 0.447852i \(0.147811\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.39543 2.19771
\(5\) 0.133492 0.0596994 0.0298497 0.999554i \(-0.490497\pi\)
0.0298497 + 0.999554i \(0.490497\pi\)
\(6\) −2.52892 −1.03243
\(7\) −1.00000 −0.377964
\(8\) 6.05784 2.14177
\(9\) 1.00000 0.333333
\(10\) 0.337590 0.106755
\(11\) 1.00000 0.301511
\(12\) −4.39543 −1.26885
\(13\) 0.133492 0.0370240 0.0185120 0.999829i \(-0.494107\pi\)
0.0185120 + 0.999829i \(0.494107\pi\)
\(14\) −2.52892 −0.675882
\(15\) −0.133492 −0.0344675
\(16\) 6.52892 1.63223
\(17\) −5.05784 −1.22671 −0.613353 0.789809i \(-0.710180\pi\)
−0.613353 + 0.789809i \(0.710180\pi\)
\(18\) 2.52892 0.596072
\(19\) −0.924344 −0.212059 −0.106030 0.994363i \(-0.533814\pi\)
−0.106030 + 0.994363i \(0.533814\pi\)
\(20\) 0.586754 0.131202
\(21\) 1.00000 0.218218
\(22\) 2.52892 0.539167
\(23\) −7.05784 −1.47166 −0.735830 0.677166i \(-0.763208\pi\)
−0.735830 + 0.677166i \(0.763208\pi\)
\(24\) −6.05784 −1.23655
\(25\) −4.98218 −0.996436
\(26\) 0.337590 0.0662069
\(27\) −1.00000 −0.192450
\(28\) −4.39543 −0.830657
\(29\) 3.86651 0.717993 0.358996 0.933339i \(-0.383119\pi\)
0.358996 + 0.933339i \(0.383119\pi\)
\(30\) −0.337590 −0.0616352
\(31\) 2.79085 0.501252 0.250626 0.968084i \(-0.419364\pi\)
0.250626 + 0.968084i \(0.419364\pi\)
\(32\) 4.39543 0.777009
\(33\) −1.00000 −0.174078
\(34\) −12.7909 −2.19361
\(35\) −0.133492 −0.0225643
\(36\) 4.39543 0.732571
\(37\) 9.98218 1.64106 0.820530 0.571603i \(-0.193678\pi\)
0.820530 + 0.571603i \(0.193678\pi\)
\(38\) −2.33759 −0.379207
\(39\) −0.133492 −0.0213758
\(40\) 0.808672 0.127862
\(41\) 11.8487 1.85045 0.925227 0.379414i \(-0.123874\pi\)
0.925227 + 0.379414i \(0.123874\pi\)
\(42\) 2.52892 0.390221
\(43\) −3.05784 −0.466316 −0.233158 0.972439i \(-0.574906\pi\)
−0.233158 + 0.972439i \(0.574906\pi\)
\(44\) 4.39543 0.662635
\(45\) 0.133492 0.0198998
\(46\) −17.8487 −2.63165
\(47\) −3.07566 −0.448631 −0.224315 0.974517i \(-0.572015\pi\)
−0.224315 + 0.974517i \(0.572015\pi\)
\(48\) −6.52892 −0.942368
\(49\) 1.00000 0.142857
\(50\) −12.5995 −1.78184
\(51\) 5.05784 0.708239
\(52\) 0.586754 0.0813681
\(53\) −4.79085 −0.658074 −0.329037 0.944317i \(-0.606724\pi\)
−0.329037 + 0.944317i \(0.606724\pi\)
\(54\) −2.52892 −0.344142
\(55\) 0.133492 0.0180000
\(56\) −6.05784 −0.809512
\(57\) 0.924344 0.122432
\(58\) 9.77808 1.28393
\(59\) −12.6574 −1.64785 −0.823924 0.566700i \(-0.808220\pi\)
−0.823924 + 0.566700i \(0.808220\pi\)
\(60\) −0.586754 −0.0757496
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 7.05784 0.896346
\(63\) −1.00000 −0.125988
\(64\) −1.94216 −0.242771
\(65\) 0.0178201 0.00221031
\(66\) −2.52892 −0.311288
\(67\) 8.92434 1.09028 0.545141 0.838344i \(-0.316476\pi\)
0.545141 + 0.838344i \(0.316476\pi\)
\(68\) −22.2313 −2.69595
\(69\) 7.05784 0.849664
\(70\) −0.337590 −0.0403497
\(71\) −6.11567 −0.725797 −0.362898 0.931829i \(-0.618213\pi\)
−0.362898 + 0.931829i \(0.618213\pi\)
\(72\) 6.05784 0.713923
\(73\) 7.86651 0.920705 0.460353 0.887736i \(-0.347723\pi\)
0.460353 + 0.887736i \(0.347723\pi\)
\(74\) 25.2441 2.93457
\(75\) 4.98218 0.575293
\(76\) −4.06289 −0.466045
\(77\) −1.00000 −0.113961
\(78\) −0.337590 −0.0382246
\(79\) 14.1157 1.58814 0.794069 0.607828i \(-0.207959\pi\)
0.794069 + 0.607828i \(0.207959\pi\)
\(80\) 0.871558 0.0974431
\(81\) 1.00000 0.111111
\(82\) 29.9644 3.30901
\(83\) −1.20915 −0.132721 −0.0663606 0.997796i \(-0.521139\pi\)
−0.0663606 + 0.997796i \(0.521139\pi\)
\(84\) 4.39543 0.479580
\(85\) −0.675180 −0.0732336
\(86\) −7.73302 −0.833873
\(87\) −3.86651 −0.414533
\(88\) 6.05784 0.645767
\(89\) 15.5817 1.65166 0.825829 0.563921i \(-0.190708\pi\)
0.825829 + 0.563921i \(0.190708\pi\)
\(90\) 0.337590 0.0355851
\(91\) −0.133492 −0.0139938
\(92\) −31.0222 −3.23429
\(93\) −2.79085 −0.289398
\(94\) −7.77808 −0.802248
\(95\) −0.123392 −0.0126598
\(96\) −4.39543 −0.448606
\(97\) 12.7909 1.29871 0.649357 0.760484i \(-0.275038\pi\)
0.649357 + 0.760484i \(0.275038\pi\)
\(98\) 2.52892 0.255459
\(99\) 1.00000 0.100504
\(100\) −21.8988 −2.18988
\(101\) −9.59180 −0.954420 −0.477210 0.878789i \(-0.658352\pi\)
−0.477210 + 0.878789i \(0.658352\pi\)
\(102\) 12.7909 1.26648
\(103\) 9.84869 0.970420 0.485210 0.874398i \(-0.338743\pi\)
0.485210 + 0.874398i \(0.338743\pi\)
\(104\) 0.808672 0.0792968
\(105\) 0.133492 0.0130275
\(106\) −12.1157 −1.17678
\(107\) 0.924344 0.0893597 0.0446799 0.999001i \(-0.485773\pi\)
0.0446799 + 0.999001i \(0.485773\pi\)
\(108\) −4.39543 −0.422950
\(109\) −8.52387 −0.816439 −0.408219 0.912884i \(-0.633850\pi\)
−0.408219 + 0.912884i \(0.633850\pi\)
\(110\) 0.337590 0.0321880
\(111\) −9.98218 −0.947467
\(112\) −6.52892 −0.616925
\(113\) −12.1157 −1.13975 −0.569873 0.821733i \(-0.693008\pi\)
−0.569873 + 0.821733i \(0.693008\pi\)
\(114\) 2.33759 0.218935
\(115\) −0.942164 −0.0878573
\(116\) 16.9950 1.57794
\(117\) 0.133492 0.0123413
\(118\) −32.0094 −2.94671
\(119\) 5.05784 0.463651
\(120\) −0.808672 −0.0738213
\(121\) 1.00000 0.0909091
\(122\) 15.1735 1.37374
\(123\) −11.8487 −1.06836
\(124\) 12.2670 1.10161
\(125\) −1.33254 −0.119186
\(126\) −2.52892 −0.225294
\(127\) −8.90652 −0.790326 −0.395163 0.918611i \(-0.629312\pi\)
−0.395163 + 0.918611i \(0.629312\pi\)
\(128\) −13.7024 −1.21113
\(129\) 3.05784 0.269227
\(130\) 0.0450656 0.00395251
\(131\) −15.6974 −1.37149 −0.685743 0.727844i \(-0.740523\pi\)
−0.685743 + 0.727844i \(0.740523\pi\)
\(132\) −4.39543 −0.382573
\(133\) 0.924344 0.0801508
\(134\) 22.5689 1.94966
\(135\) −0.133492 −0.0114892
\(136\) −30.6395 −2.62732
\(137\) 14.6395 1.25074 0.625370 0.780328i \(-0.284948\pi\)
0.625370 + 0.780328i \(0.284948\pi\)
\(138\) 17.8487 1.51938
\(139\) 18.6496 1.58184 0.790921 0.611918i \(-0.209602\pi\)
0.790921 + 0.611918i \(0.209602\pi\)
\(140\) −0.586754 −0.0495898
\(141\) 3.07566 0.259017
\(142\) −15.4660 −1.29788
\(143\) 0.133492 0.0111632
\(144\) 6.52892 0.544076
\(145\) 0.516148 0.0428637
\(146\) 19.8938 1.64642
\(147\) −1.00000 −0.0824786
\(148\) 43.8759 3.60658
\(149\) 11.8665 0.972142 0.486071 0.873919i \(-0.338430\pi\)
0.486071 + 0.873919i \(0.338430\pi\)
\(150\) 12.5995 1.02875
\(151\) −11.3248 −0.921601 −0.460800 0.887504i \(-0.652438\pi\)
−0.460800 + 0.887504i \(0.652438\pi\)
\(152\) −5.59952 −0.454181
\(153\) −5.05784 −0.408902
\(154\) −2.52892 −0.203786
\(155\) 0.372556 0.0299244
\(156\) −0.586754 −0.0469779
\(157\) −20.3827 −1.62671 −0.813357 0.581766i \(-0.802362\pi\)
−0.813357 + 0.581766i \(0.802362\pi\)
\(158\) 35.6974 2.83993
\(159\) 4.79085 0.379939
\(160\) 0.586754 0.0463870
\(161\) 7.05784 0.556235
\(162\) 2.52892 0.198691
\(163\) −19.3070 −1.51224 −0.756120 0.654432i \(-0.772908\pi\)
−0.756120 + 0.654432i \(0.772908\pi\)
\(164\) 52.0800 4.06677
\(165\) −0.133492 −0.0103923
\(166\) −3.05784 −0.237334
\(167\) 12.6395 0.978077 0.489038 0.872262i \(-0.337348\pi\)
0.489038 + 0.872262i \(0.337348\pi\)
\(168\) 6.05784 0.467372
\(169\) −12.9822 −0.998629
\(170\) −1.70748 −0.130957
\(171\) −0.924344 −0.0706864
\(172\) −13.4405 −1.02483
\(173\) −19.8487 −1.50907 −0.754534 0.656261i \(-0.772137\pi\)
−0.754534 + 0.656261i \(0.772137\pi\)
\(174\) −9.77808 −0.741274
\(175\) 4.98218 0.376617
\(176\) 6.52892 0.492136
\(177\) 12.6574 0.951385
\(178\) 39.4049 2.95352
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0.586754 0.0437341
\(181\) 2.67518 0.198845 0.0994223 0.995045i \(-0.468301\pi\)
0.0994223 + 0.995045i \(0.468301\pi\)
\(182\) −0.337590 −0.0250238
\(183\) −6.00000 −0.443533
\(184\) −42.7552 −3.15196
\(185\) 1.33254 0.0979703
\(186\) −7.05784 −0.517506
\(187\) −5.05784 −0.369866
\(188\) −13.5188 −0.985961
\(189\) 1.00000 0.0727393
\(190\) −0.312049 −0.0226384
\(191\) 3.32482 0.240576 0.120288 0.992739i \(-0.461618\pi\)
0.120288 + 0.992739i \(0.461618\pi\)
\(192\) 1.94216 0.140164
\(193\) 18.3726 1.32249 0.661243 0.750172i \(-0.270029\pi\)
0.661243 + 0.750172i \(0.270029\pi\)
\(194\) 32.3470 2.32238
\(195\) −0.0178201 −0.00127612
\(196\) 4.39543 0.313959
\(197\) −8.11567 −0.578218 −0.289109 0.957296i \(-0.593359\pi\)
−0.289109 + 0.957296i \(0.593359\pi\)
\(198\) 2.52892 0.179722
\(199\) −6.52387 −0.462465 −0.231232 0.972899i \(-0.574276\pi\)
−0.231232 + 0.972899i \(0.574276\pi\)
\(200\) −30.1812 −2.13414
\(201\) −8.92434 −0.629475
\(202\) −24.2569 −1.70671
\(203\) −3.86651 −0.271376
\(204\) 22.2313 1.55651
\(205\) 1.58170 0.110471
\(206\) 24.9065 1.73532
\(207\) −7.05784 −0.490554
\(208\) 0.871558 0.0604317
\(209\) −0.924344 −0.0639382
\(210\) 0.337590 0.0232959
\(211\) 13.8487 0.953383 0.476691 0.879071i \(-0.341836\pi\)
0.476691 + 0.879071i \(0.341836\pi\)
\(212\) −21.0578 −1.44626
\(213\) 6.11567 0.419039
\(214\) 2.33759 0.159794
\(215\) −0.408196 −0.0278388
\(216\) −6.05784 −0.412184
\(217\) −2.79085 −0.189455
\(218\) −21.5562 −1.45997
\(219\) −7.86651 −0.531569
\(220\) 0.586754 0.0395589
\(221\) −0.675180 −0.0454175
\(222\) −25.2441 −1.69427
\(223\) −24.4983 −1.64053 −0.820265 0.571984i \(-0.806174\pi\)
−0.820265 + 0.571984i \(0.806174\pi\)
\(224\) −4.39543 −0.293682
\(225\) −4.98218 −0.332145
\(226\) −30.6395 −2.03811
\(227\) −7.73302 −0.513258 −0.256629 0.966510i \(-0.582612\pi\)
−0.256629 + 0.966510i \(0.582612\pi\)
\(228\) 4.06289 0.269071
\(229\) −4.79085 −0.316588 −0.158294 0.987392i \(-0.550599\pi\)
−0.158294 + 0.987392i \(0.550599\pi\)
\(230\) −2.38266 −0.157108
\(231\) 1.00000 0.0657952
\(232\) 23.4227 1.53777
\(233\) −9.69738 −0.635296 −0.317648 0.948209i \(-0.602893\pi\)
−0.317648 + 0.948209i \(0.602893\pi\)
\(234\) 0.337590 0.0220690
\(235\) −0.410575 −0.0267830
\(236\) −55.6345 −3.62150
\(237\) −14.1157 −0.916911
\(238\) 12.7909 0.829108
\(239\) 23.3070 1.50760 0.753802 0.657101i \(-0.228218\pi\)
0.753802 + 0.657101i \(0.228218\pi\)
\(240\) −0.871558 −0.0562588
\(241\) 9.71520 0.625811 0.312905 0.949784i \(-0.398698\pi\)
0.312905 + 0.949784i \(0.398698\pi\)
\(242\) 2.52892 0.162565
\(243\) −1.00000 −0.0641500
\(244\) 26.3726 1.68833
\(245\) 0.133492 0.00852849
\(246\) −29.9644 −1.91046
\(247\) −0.123392 −0.00785127
\(248\) 16.9065 1.07357
\(249\) 1.20915 0.0766266
\(250\) −3.36989 −0.213130
\(251\) −15.0400 −0.949317 −0.474659 0.880170i \(-0.657428\pi\)
−0.474659 + 0.880170i \(0.657428\pi\)
\(252\) −4.39543 −0.276886
\(253\) −7.05784 −0.443722
\(254\) −22.5239 −1.41327
\(255\) 0.675180 0.0422814
\(256\) −30.7680 −1.92300
\(257\) 15.8309 0.987502 0.493751 0.869603i \(-0.335625\pi\)
0.493751 + 0.869603i \(0.335625\pi\)
\(258\) 7.73302 0.481437
\(259\) −9.98218 −0.620262
\(260\) 0.0783269 0.00485763
\(261\) 3.86651 0.239331
\(262\) −39.6974 −2.45251
\(263\) −3.07566 −0.189653 −0.0948265 0.995494i \(-0.530230\pi\)
−0.0948265 + 0.995494i \(0.530230\pi\)
\(264\) −6.05784 −0.372834
\(265\) −0.639540 −0.0392866
\(266\) 2.33759 0.143327
\(267\) −15.5817 −0.953585
\(268\) 39.2263 2.39613
\(269\) −10.5340 −0.642267 −0.321134 0.947034i \(-0.604064\pi\)
−0.321134 + 0.947034i \(0.604064\pi\)
\(270\) −0.337590 −0.0205451
\(271\) −4.92434 −0.299133 −0.149566 0.988752i \(-0.547788\pi\)
−0.149566 + 0.988752i \(0.547788\pi\)
\(272\) −33.0222 −2.00226
\(273\) 0.133492 0.00807930
\(274\) 37.0222 2.23659
\(275\) −4.98218 −0.300437
\(276\) 31.0222 1.86732
\(277\) −0.151312 −0.00909146 −0.00454573 0.999990i \(-0.501447\pi\)
−0.00454573 + 0.999990i \(0.501447\pi\)
\(278\) 47.1634 2.82867
\(279\) 2.79085 0.167084
\(280\) −0.808672 −0.0483274
\(281\) 6.51615 0.388721 0.194360 0.980930i \(-0.437737\pi\)
0.194360 + 0.980930i \(0.437737\pi\)
\(282\) 7.77808 0.463178
\(283\) −0.390376 −0.0232055 −0.0116027 0.999933i \(-0.503693\pi\)
−0.0116027 + 0.999933i \(0.503693\pi\)
\(284\) −26.8810 −1.59509
\(285\) 0.123392 0.00730914
\(286\) 0.337590 0.0199621
\(287\) −11.8487 −0.699406
\(288\) 4.39543 0.259003
\(289\) 8.58170 0.504806
\(290\) 1.30529 0.0766496
\(291\) −12.7909 −0.749813
\(292\) 34.5767 2.02345
\(293\) 29.4304 1.71934 0.859671 0.510848i \(-0.170669\pi\)
0.859671 + 0.510848i \(0.170669\pi\)
\(294\) −2.52892 −0.147489
\(295\) −1.68966 −0.0983755
\(296\) 60.4704 3.51477
\(297\) −1.00000 −0.0580259
\(298\) 30.0094 1.73840
\(299\) −0.942164 −0.0544868
\(300\) 21.8988 1.26433
\(301\) 3.05784 0.176251
\(302\) −28.6395 −1.64802
\(303\) 9.59180 0.551035
\(304\) −6.03497 −0.346129
\(305\) 0.800952 0.0458624
\(306\) −12.7909 −0.731204
\(307\) 23.6974 1.35248 0.676240 0.736681i \(-0.263608\pi\)
0.676240 + 0.736681i \(0.263608\pi\)
\(308\) −4.39543 −0.250453
\(309\) −9.84869 −0.560272
\(310\) 0.942164 0.0535113
\(311\) −12.2313 −0.693576 −0.346788 0.937944i \(-0.612728\pi\)
−0.346788 + 0.937944i \(0.612728\pi\)
\(312\) −0.808672 −0.0457820
\(313\) −19.1735 −1.08375 −0.541875 0.840459i \(-0.682286\pi\)
−0.541875 + 0.840459i \(0.682286\pi\)
\(314\) −51.5461 −2.90891
\(315\) −0.133492 −0.00752142
\(316\) 62.0444 3.49027
\(317\) 29.5562 1.66004 0.830020 0.557734i \(-0.188329\pi\)
0.830020 + 0.557734i \(0.188329\pi\)
\(318\) 12.1157 0.679413
\(319\) 3.86651 0.216483
\(320\) −0.259263 −0.0144933
\(321\) −0.924344 −0.0515919
\(322\) 17.8487 0.994668
\(323\) 4.67518 0.260134
\(324\) 4.39543 0.244190
\(325\) −0.665081 −0.0368920
\(326\) −48.8258 −2.70421
\(327\) 8.52387 0.471371
\(328\) 71.7774 3.96324
\(329\) 3.07566 0.169566
\(330\) −0.337590 −0.0185837
\(331\) 18.6496 1.02508 0.512538 0.858664i \(-0.328705\pi\)
0.512538 + 0.858664i \(0.328705\pi\)
\(332\) −5.31472 −0.291683
\(333\) 9.98218 0.547020
\(334\) 31.9644 1.74901
\(335\) 1.19133 0.0650892
\(336\) 6.52892 0.356182
\(337\) −21.0222 −1.14515 −0.572576 0.819852i \(-0.694056\pi\)
−0.572576 + 0.819852i \(0.694056\pi\)
\(338\) −32.8309 −1.78576
\(339\) 12.1157 0.658033
\(340\) −2.96770 −0.160946
\(341\) 2.79085 0.151133
\(342\) −2.33759 −0.126402
\(343\) −1.00000 −0.0539949
\(344\) −18.5239 −0.998740
\(345\) 0.942164 0.0507244
\(346\) −50.1957 −2.69854
\(347\) 23.1634 1.24348 0.621738 0.783225i \(-0.286427\pi\)
0.621738 + 0.783225i \(0.286427\pi\)
\(348\) −16.9950 −0.911025
\(349\) −28.0979 −1.50404 −0.752022 0.659138i \(-0.770921\pi\)
−0.752022 + 0.659138i \(0.770921\pi\)
\(350\) 12.5995 0.673473
\(351\) −0.133492 −0.00712527
\(352\) 4.39543 0.234277
\(353\) −33.4126 −1.77837 −0.889186 0.457546i \(-0.848728\pi\)
−0.889186 + 0.457546i \(0.848728\pi\)
\(354\) 32.0094 1.70128
\(355\) −0.816393 −0.0433296
\(356\) 68.4882 3.62987
\(357\) −5.05784 −0.267689
\(358\) −30.3470 −1.60389
\(359\) 13.5817 0.716815 0.358407 0.933565i \(-0.383320\pi\)
0.358407 + 0.933565i \(0.383320\pi\)
\(360\) 0.808672 0.0426208
\(361\) −18.1456 −0.955031
\(362\) 6.76531 0.355577
\(363\) −1.00000 −0.0524864
\(364\) −0.586754 −0.0307543
\(365\) 1.05012 0.0549655
\(366\) −15.1735 −0.793132
\(367\) −3.73302 −0.194862 −0.0974309 0.995242i \(-0.531063\pi\)
−0.0974309 + 0.995242i \(0.531063\pi\)
\(368\) −46.0800 −2.40209
\(369\) 11.8487 0.616818
\(370\) 3.36989 0.175192
\(371\) 4.79085 0.248729
\(372\) −12.2670 −0.636013
\(373\) 6.94216 0.359452 0.179726 0.983717i \(-0.442479\pi\)
0.179726 + 0.983717i \(0.442479\pi\)
\(374\) −12.7909 −0.661399
\(375\) 1.33254 0.0688121
\(376\) −18.6318 −0.960863
\(377\) 0.516148 0.0265830
\(378\) 2.52892 0.130074
\(379\) −0.390376 −0.0200523 −0.0100261 0.999950i \(-0.503191\pi\)
−0.0100261 + 0.999950i \(0.503191\pi\)
\(380\) −0.542362 −0.0278226
\(381\) 8.90652 0.456295
\(382\) 8.40820 0.430201
\(383\) −0.533968 −0.0272845 −0.0136422 0.999907i \(-0.504343\pi\)
−0.0136422 + 0.999907i \(0.504343\pi\)
\(384\) 13.7024 0.699249
\(385\) −0.133492 −0.00680338
\(386\) 46.4627 2.36489
\(387\) −3.05784 −0.155439
\(388\) 56.2212 2.85420
\(389\) −10.4983 −0.532286 −0.266143 0.963934i \(-0.585749\pi\)
−0.266143 + 0.963934i \(0.585749\pi\)
\(390\) −0.0450656 −0.00228198
\(391\) 35.6974 1.80529
\(392\) 6.05784 0.305967
\(393\) 15.6974 0.791828
\(394\) −20.5239 −1.03398
\(395\) 1.88433 0.0948108
\(396\) 4.39543 0.220878
\(397\) −20.7552 −1.04167 −0.520837 0.853656i \(-0.674380\pi\)
−0.520837 + 0.853656i \(0.674380\pi\)
\(398\) −16.4983 −0.826986
\(399\) −0.924344 −0.0462751
\(400\) −32.5282 −1.62641
\(401\) 21.3248 1.06491 0.532455 0.846458i \(-0.321269\pi\)
0.532455 + 0.846458i \(0.321269\pi\)
\(402\) −22.5689 −1.12564
\(403\) 0.372556 0.0185583
\(404\) −42.1601 −2.09754
\(405\) 0.133492 0.00663327
\(406\) −9.77808 −0.485278
\(407\) 9.98218 0.494798
\(408\) 30.6395 1.51688
\(409\) 33.9287 1.67767 0.838834 0.544388i \(-0.183238\pi\)
0.838834 + 0.544388i \(0.183238\pi\)
\(410\) 4.00000 0.197546
\(411\) −14.6395 −0.722115
\(412\) 43.2892 2.13270
\(413\) 12.6574 0.622828
\(414\) −17.8487 −0.877215
\(415\) −0.161411 −0.00792338
\(416\) 0.586754 0.0287680
\(417\) −18.6496 −0.913277
\(418\) −2.33759 −0.114335
\(419\) −9.99228 −0.488155 −0.244077 0.969756i \(-0.578485\pi\)
−0.244077 + 0.969756i \(0.578485\pi\)
\(420\) 0.586754 0.0286307
\(421\) −29.9822 −1.46124 −0.730621 0.682783i \(-0.760769\pi\)
−0.730621 + 0.682783i \(0.760769\pi\)
\(422\) 35.0222 1.70485
\(423\) −3.07566 −0.149544
\(424\) −29.0222 −1.40944
\(425\) 25.1990 1.22233
\(426\) 15.4660 0.749332
\(427\) −6.00000 −0.290360
\(428\) 4.06289 0.196387
\(429\) −0.133492 −0.00644505
\(430\) −1.03230 −0.0497817
\(431\) −2.54169 −0.122429 −0.0612144 0.998125i \(-0.519497\pi\)
−0.0612144 + 0.998125i \(0.519497\pi\)
\(432\) −6.52892 −0.314123
\(433\) 20.3827 0.979528 0.489764 0.871855i \(-0.337083\pi\)
0.489764 + 0.871855i \(0.337083\pi\)
\(434\) −7.05784 −0.338787
\(435\) −0.516148 −0.0247474
\(436\) −37.4660 −1.79430
\(437\) 6.52387 0.312079
\(438\) −19.8938 −0.950560
\(439\) −5.45831 −0.260511 −0.130256 0.991480i \(-0.541580\pi\)
−0.130256 + 0.991480i \(0.541580\pi\)
\(440\) 0.808672 0.0385519
\(441\) 1.00000 0.0476190
\(442\) −1.70748 −0.0812163
\(443\) −10.6496 −0.505980 −0.252990 0.967469i \(-0.581414\pi\)
−0.252990 + 0.967469i \(0.581414\pi\)
\(444\) −43.8759 −2.08226
\(445\) 2.08003 0.0986030
\(446\) −61.9543 −2.93362
\(447\) −11.8665 −0.561267
\(448\) 1.94216 0.0917586
\(449\) 27.0323 1.27573 0.637866 0.770147i \(-0.279817\pi\)
0.637866 + 0.770147i \(0.279817\pi\)
\(450\) −12.5995 −0.593947
\(451\) 11.8487 0.557933
\(452\) −53.2535 −2.50484
\(453\) 11.3248 0.532086
\(454\) −19.5562 −0.917816
\(455\) −0.0178201 −0.000835419 0
\(456\) 5.59952 0.262222
\(457\) 4.79085 0.224107 0.112053 0.993702i \(-0.464257\pi\)
0.112053 + 0.993702i \(0.464257\pi\)
\(458\) −12.1157 −0.566128
\(459\) 5.05784 0.236080
\(460\) −4.14121 −0.193085
\(461\) 22.4983 1.04785 0.523926 0.851764i \(-0.324467\pi\)
0.523926 + 0.851764i \(0.324467\pi\)
\(462\) 2.52892 0.117656
\(463\) 25.6897 1.19390 0.596950 0.802279i \(-0.296379\pi\)
0.596950 + 0.802279i \(0.296379\pi\)
\(464\) 25.2441 1.17193
\(465\) −0.372556 −0.0172769
\(466\) −24.5239 −1.13605
\(467\) 4.62172 0.213868 0.106934 0.994266i \(-0.465897\pi\)
0.106934 + 0.994266i \(0.465897\pi\)
\(468\) 0.586754 0.0271227
\(469\) −8.92434 −0.412088
\(470\) −1.03831 −0.0478937
\(471\) 20.3827 0.939183
\(472\) −76.6762 −3.52931
\(473\) −3.05784 −0.140599
\(474\) −35.6974 −1.63963
\(475\) 4.60525 0.211303
\(476\) 22.2313 1.01897
\(477\) −4.79085 −0.219358
\(478\) 58.9415 2.69592
\(479\) −0.372556 −0.0170225 −0.00851126 0.999964i \(-0.502709\pi\)
−0.00851126 + 0.999964i \(0.502709\pi\)
\(480\) −0.586754 −0.0267815
\(481\) 1.33254 0.0607586
\(482\) 24.5689 1.11908
\(483\) −7.05784 −0.321143
\(484\) 4.39543 0.199792
\(485\) 1.70748 0.0775325
\(486\) −2.52892 −0.114714
\(487\) 4.30262 0.194971 0.0974853 0.995237i \(-0.468920\pi\)
0.0974853 + 0.995237i \(0.468920\pi\)
\(488\) 36.3470 1.64535
\(489\) 19.3070 0.873093
\(490\) 0.337590 0.0152508
\(491\) 25.4583 1.14892 0.574459 0.818534i \(-0.305213\pi\)
0.574459 + 0.818534i \(0.305213\pi\)
\(492\) −52.0800 −2.34795
\(493\) −19.5562 −0.880765
\(494\) −0.312049 −0.0140398
\(495\) 0.133492 0.00600002
\(496\) 18.2212 0.818158
\(497\) 6.11567 0.274325
\(498\) 3.05784 0.137025
\(499\) −8.39038 −0.375605 −0.187802 0.982207i \(-0.560136\pi\)
−0.187802 + 0.982207i \(0.560136\pi\)
\(500\) −5.85708 −0.261937
\(501\) −12.6395 −0.564693
\(502\) −38.0350 −1.69758
\(503\) −26.7552 −1.19296 −0.596478 0.802629i \(-0.703434\pi\)
−0.596478 + 0.802629i \(0.703434\pi\)
\(504\) −6.05784 −0.269837
\(505\) −1.28043 −0.0569783
\(506\) −17.8487 −0.793471
\(507\) 12.9822 0.576559
\(508\) −39.1480 −1.73691
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 1.70748 0.0756083
\(511\) −7.86651 −0.347994
\(512\) −50.4049 −2.22760
\(513\) 0.924344 0.0408108
\(514\) 40.0350 1.76587
\(515\) 1.31472 0.0579335
\(516\) 13.4405 0.591685
\(517\) −3.07566 −0.135267
\(518\) −25.2441 −1.10916
\(519\) 19.8487 0.871261
\(520\) 0.107951 0.00473397
\(521\) −1.44821 −0.0634473 −0.0317237 0.999497i \(-0.510100\pi\)
−0.0317237 + 0.999497i \(0.510100\pi\)
\(522\) 9.77808 0.427975
\(523\) 23.5740 1.03082 0.515409 0.856944i \(-0.327640\pi\)
0.515409 + 0.856944i \(0.327640\pi\)
\(524\) −68.9967 −3.01413
\(525\) −4.98218 −0.217440
\(526\) −7.77808 −0.339140
\(527\) −14.1157 −0.614888
\(528\) −6.52892 −0.284135
\(529\) 26.8130 1.16578
\(530\) −1.61734 −0.0702529
\(531\) −12.6574 −0.549283
\(532\) 4.06289 0.176148
\(533\) 1.58170 0.0685112
\(534\) −39.4049 −1.70521
\(535\) 0.123392 0.00533472
\(536\) 54.0622 2.33513
\(537\) 12.0000 0.517838
\(538\) −26.6395 −1.14851
\(539\) 1.00000 0.0430730
\(540\) −0.586754 −0.0252499
\(541\) −18.4983 −0.795305 −0.397653 0.917536i \(-0.630175\pi\)
−0.397653 + 0.917536i \(0.630175\pi\)
\(542\) −12.4533 −0.534913
\(543\) −2.67518 −0.114803
\(544\) −22.2313 −0.953161
\(545\) −1.13787 −0.0487409
\(546\) 0.337590 0.0144475
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 64.3470 2.74877
\(549\) 6.00000 0.256074
\(550\) −12.5995 −0.537246
\(551\) −3.57398 −0.152257
\(552\) 42.7552 1.81978
\(553\) −14.1157 −0.600259
\(554\) −0.382656 −0.0162575
\(555\) −1.33254 −0.0565632
\(556\) 81.9731 3.47643
\(557\) 1.98218 0.0839877 0.0419938 0.999118i \(-0.486629\pi\)
0.0419938 + 0.999118i \(0.486629\pi\)
\(558\) 7.05784 0.298782
\(559\) −0.408196 −0.0172649
\(560\) −0.871558 −0.0368300
\(561\) 5.05784 0.213542
\(562\) 16.4788 0.695116
\(563\) 1.98990 0.0838643 0.0419322 0.999120i \(-0.486649\pi\)
0.0419322 + 0.999120i \(0.486649\pi\)
\(564\) 13.5188 0.569245
\(565\) −1.61734 −0.0680422
\(566\) −0.987230 −0.0414964
\(567\) −1.00000 −0.0419961
\(568\) −37.0477 −1.55449
\(569\) −14.5340 −0.609296 −0.304648 0.952465i \(-0.598539\pi\)
−0.304648 + 0.952465i \(0.598539\pi\)
\(570\) 0.312049 0.0130703
\(571\) −29.2791 −1.22529 −0.612646 0.790358i \(-0.709895\pi\)
−0.612646 + 0.790358i \(0.709895\pi\)
\(572\) 0.586754 0.0245334
\(573\) −3.32482 −0.138896
\(574\) −29.9644 −1.25069
\(575\) 35.1634 1.46642
\(576\) −1.94216 −0.0809235
\(577\) −5.59180 −0.232790 −0.116395 0.993203i \(-0.537134\pi\)
−0.116395 + 0.993203i \(0.537134\pi\)
\(578\) 21.7024 0.902702
\(579\) −18.3726 −0.763537
\(580\) 2.26869 0.0942022
\(581\) 1.20915 0.0501639
\(582\) −32.3470 −1.34083
\(583\) −4.79085 −0.198417
\(584\) 47.6540 1.97194
\(585\) 0.0178201 0.000736770 0
\(586\) 74.4270 3.07455
\(587\) −2.80867 −0.115926 −0.0579632 0.998319i \(-0.518461\pi\)
−0.0579632 + 0.998319i \(0.518461\pi\)
\(588\) −4.39543 −0.181264
\(589\) −2.57971 −0.106295
\(590\) −4.27300 −0.175917
\(591\) 8.11567 0.333834
\(592\) 65.1728 2.67859
\(593\) −42.1957 −1.73277 −0.866385 0.499377i \(-0.833562\pi\)
−0.866385 + 0.499377i \(0.833562\pi\)
\(594\) −2.52892 −0.103763
\(595\) 0.675180 0.0276797
\(596\) 52.1584 2.13649
\(597\) 6.52387 0.267004
\(598\) −2.38266 −0.0974340
\(599\) 34.3470 1.40338 0.701691 0.712482i \(-0.252429\pi\)
0.701691 + 0.712482i \(0.252429\pi\)
\(600\) 30.1812 1.23214
\(601\) 5.98218 0.244018 0.122009 0.992529i \(-0.461066\pi\)
0.122009 + 0.992529i \(0.461066\pi\)
\(602\) 7.73302 0.315174
\(603\) 8.92434 0.363427
\(604\) −49.7774 −2.02541
\(605\) 0.133492 0.00542722
\(606\) 24.2569 0.985369
\(607\) 8.12339 0.329718 0.164859 0.986317i \(-0.447283\pi\)
0.164859 + 0.986317i \(0.447283\pi\)
\(608\) −4.06289 −0.164772
\(609\) 3.86651 0.156679
\(610\) 2.02554 0.0820117
\(611\) −0.410575 −0.0166101
\(612\) −22.2313 −0.898649
\(613\) −16.5239 −0.667393 −0.333696 0.942681i \(-0.608296\pi\)
−0.333696 + 0.942681i \(0.608296\pi\)
\(614\) 59.9287 2.41853
\(615\) −1.58170 −0.0637805
\(616\) −6.05784 −0.244077
\(617\) 11.8843 0.478445 0.239223 0.970965i \(-0.423107\pi\)
0.239223 + 0.970965i \(0.423107\pi\)
\(618\) −24.9065 −1.00189
\(619\) 43.0222 1.72921 0.864604 0.502454i \(-0.167569\pi\)
0.864604 + 0.502454i \(0.167569\pi\)
\(620\) 1.63754 0.0657653
\(621\) 7.05784 0.283221
\(622\) −30.9321 −1.24026
\(623\) −15.5817 −0.624268
\(624\) −0.871558 −0.0348902
\(625\) 24.7330 0.989321
\(626\) −48.4882 −1.93798
\(627\) 0.924344 0.0369147
\(628\) −89.5905 −3.57505
\(629\) −50.4882 −2.01310
\(630\) −0.337590 −0.0134499
\(631\) 13.5817 0.540679 0.270340 0.962765i \(-0.412864\pi\)
0.270340 + 0.962765i \(0.412864\pi\)
\(632\) 85.5104 3.40142
\(633\) −13.8487 −0.550436
\(634\) 74.7451 2.96851
\(635\) −1.18895 −0.0471820
\(636\) 21.0578 0.834998
\(637\) 0.133492 0.00528914
\(638\) 9.77808 0.387118
\(639\) −6.11567 −0.241932
\(640\) −1.82916 −0.0723040
\(641\) 36.7552 1.45174 0.725872 0.687830i \(-0.241437\pi\)
0.725872 + 0.687830i \(0.241437\pi\)
\(642\) −2.33759 −0.0922573
\(643\) −44.3369 −1.74848 −0.874239 0.485496i \(-0.838639\pi\)
−0.874239 + 0.485496i \(0.838639\pi\)
\(644\) 31.0222 1.22245
\(645\) 0.408196 0.0160727
\(646\) 11.8231 0.465176
\(647\) −17.9721 −0.706555 −0.353278 0.935519i \(-0.614933\pi\)
−0.353278 + 0.935519i \(0.614933\pi\)
\(648\) 6.05784 0.237974
\(649\) −12.6574 −0.496845
\(650\) −1.68193 −0.0659709
\(651\) 2.79085 0.109382
\(652\) −84.8625 −3.32347
\(653\) −22.1957 −0.868585 −0.434292 0.900772i \(-0.643002\pi\)
−0.434292 + 0.900772i \(0.643002\pi\)
\(654\) 21.5562 0.842913
\(655\) −2.09547 −0.0818769
\(656\) 77.3591 3.02037
\(657\) 7.86651 0.306902
\(658\) 7.77808 0.303221
\(659\) 9.99228 0.389244 0.194622 0.980878i \(-0.437652\pi\)
0.194622 + 0.980878i \(0.437652\pi\)
\(660\) −0.586754 −0.0228394
\(661\) 27.9543 1.08729 0.543647 0.839314i \(-0.317043\pi\)
0.543647 + 0.839314i \(0.317043\pi\)
\(662\) 47.1634 1.83306
\(663\) 0.675180 0.0262218
\(664\) −7.32482 −0.284258
\(665\) 0.123392 0.00478495
\(666\) 25.2441 0.978190
\(667\) −27.2892 −1.05664
\(668\) 55.5562 2.14953
\(669\) 24.4983 0.947160
\(670\) 3.01277 0.116393
\(671\) 6.00000 0.231627
\(672\) 4.39543 0.169557
\(673\) −43.4203 −1.67373 −0.836865 0.547410i \(-0.815614\pi\)
−0.836865 + 0.547410i \(0.815614\pi\)
\(674\) −53.1634 −2.04778
\(675\) 4.98218 0.191764
\(676\) −57.0622 −2.19470
\(677\) −12.3470 −0.474534 −0.237267 0.971444i \(-0.576252\pi\)
−0.237267 + 0.971444i \(0.576252\pi\)
\(678\) 30.6395 1.17670
\(679\) −12.7909 −0.490868
\(680\) −4.09013 −0.156849
\(681\) 7.73302 0.296330
\(682\) 7.05784 0.270259
\(683\) −27.8386 −1.06521 −0.532607 0.846363i \(-0.678788\pi\)
−0.532607 + 0.846363i \(0.678788\pi\)
\(684\) −4.06289 −0.155348
\(685\) 1.95426 0.0746684
\(686\) −2.52892 −0.0965545
\(687\) 4.79085 0.182782
\(688\) −19.9644 −0.761134
\(689\) −0.639540 −0.0243645
\(690\) 2.38266 0.0907062
\(691\) −16.8010 −0.639138 −0.319569 0.947563i \(-0.603538\pi\)
−0.319569 + 0.947563i \(0.603538\pi\)
\(692\) −87.2434 −3.31650
\(693\) −1.00000 −0.0379869
\(694\) 58.5784 2.22360
\(695\) 2.48958 0.0944350
\(696\) −23.4227 −0.887834
\(697\) −59.9287 −2.26996
\(698\) −71.0572 −2.68955
\(699\) 9.69738 0.366788
\(700\) 21.8988 0.827697
\(701\) 9.16341 0.346097 0.173049 0.984913i \(-0.444638\pi\)
0.173049 + 0.984913i \(0.444638\pi\)
\(702\) −0.337590 −0.0127415
\(703\) −9.22697 −0.348002
\(704\) −1.94216 −0.0731981
\(705\) 0.410575 0.0154632
\(706\) −84.4977 −3.18011
\(707\) 9.59180 0.360737
\(708\) 55.6345 2.09087
\(709\) −34.7475 −1.30497 −0.652485 0.757802i \(-0.726273\pi\)
−0.652485 + 0.757802i \(0.726273\pi\)
\(710\) −2.06459 −0.0774827
\(711\) 14.1157 0.529379
\(712\) 94.3914 3.53747
\(713\) −19.6974 −0.737673
\(714\) −12.7909 −0.478686
\(715\) 0.0178201 0.000666434 0
\(716\) −52.7451 −1.97118
\(717\) −23.3070 −0.870416
\(718\) 34.3470 1.28182
\(719\) −25.1557 −0.938149 −0.469074 0.883159i \(-0.655412\pi\)
−0.469074 + 0.883159i \(0.655412\pi\)
\(720\) 0.871558 0.0324810
\(721\) −9.84869 −0.366784
\(722\) −45.8887 −1.70780
\(723\) −9.71520 −0.361312
\(724\) 11.7586 0.437003
\(725\) −19.2636 −0.715434
\(726\) −2.52892 −0.0938569
\(727\) −23.5918 −0.874972 −0.437486 0.899225i \(-0.644131\pi\)
−0.437486 + 0.899225i \(0.644131\pi\)
\(728\) −0.808672 −0.0299714
\(729\) 1.00000 0.0370370
\(730\) 2.65566 0.0982902
\(731\) 15.4660 0.572032
\(732\) −26.3726 −0.974758
\(733\) 21.9287 0.809956 0.404978 0.914326i \(-0.367279\pi\)
0.404978 + 0.914326i \(0.367279\pi\)
\(734\) −9.44049 −0.348455
\(735\) −0.133492 −0.00492392
\(736\) −31.0222 −1.14349
\(737\) 8.92434 0.328732
\(738\) 29.9644 1.10300
\(739\) 35.0578 1.28962 0.644812 0.764341i \(-0.276936\pi\)
0.644812 + 0.764341i \(0.276936\pi\)
\(740\) 5.85708 0.215311
\(741\) 0.123392 0.00453294
\(742\) 12.1157 0.444780
\(743\) −31.3070 −1.14854 −0.574271 0.818665i \(-0.694715\pi\)
−0.574271 + 0.818665i \(0.694715\pi\)
\(744\) −16.9065 −0.619823
\(745\) 1.58408 0.0580363
\(746\) 17.5562 0.642777
\(747\) −1.20915 −0.0442404
\(748\) −22.2313 −0.812858
\(749\) −0.924344 −0.0337748
\(750\) 3.36989 0.123051
\(751\) −4.42602 −0.161508 −0.0807538 0.996734i \(-0.525733\pi\)
−0.0807538 + 0.996734i \(0.525733\pi\)
\(752\) −20.0807 −0.732268
\(753\) 15.0400 0.548089
\(754\) 1.30529 0.0475360
\(755\) −1.51177 −0.0550190
\(756\) 4.39543 0.159860
\(757\) 2.51615 0.0914509 0.0457255 0.998954i \(-0.485440\pi\)
0.0457255 + 0.998954i \(0.485440\pi\)
\(758\) −0.987230 −0.0358578
\(759\) 7.05784 0.256183
\(760\) −0.747491 −0.0271144
\(761\) 13.7330 0.497821 0.248911 0.968526i \(-0.419927\pi\)
0.248911 + 0.968526i \(0.419927\pi\)
\(762\) 22.5239 0.815954
\(763\) 8.52387 0.308585
\(764\) 14.6140 0.528716
\(765\) −0.675180 −0.0244112
\(766\) −1.35036 −0.0487905
\(767\) −1.68966 −0.0610099
\(768\) 30.7680 1.11024
\(769\) 46.4805 1.67613 0.838065 0.545570i \(-0.183687\pi\)
0.838065 + 0.545570i \(0.183687\pi\)
\(770\) −0.337590 −0.0121659
\(771\) −15.8309 −0.570135
\(772\) 80.7552 2.90644
\(773\) −45.9822 −1.65386 −0.826932 0.562302i \(-0.809916\pi\)
−0.826932 + 0.562302i \(0.809916\pi\)
\(774\) −7.73302 −0.277958
\(775\) −13.9045 −0.499465
\(776\) 77.4849 2.78155
\(777\) 9.98218 0.358109
\(778\) −26.5494 −0.951842
\(779\) −10.9523 −0.392406
\(780\) −0.0783269 −0.00280455
\(781\) −6.11567 −0.218836
\(782\) 90.2757 3.22825
\(783\) −3.86651 −0.138178
\(784\) 6.52892 0.233176
\(785\) −2.72092 −0.0971138
\(786\) 39.6974 1.41596
\(787\) 40.9243 1.45880 0.729398 0.684090i \(-0.239800\pi\)
0.729398 + 0.684090i \(0.239800\pi\)
\(788\) −35.6718 −1.27076
\(789\) 3.07566 0.109496
\(790\) 4.76531 0.169542
\(791\) 12.1157 0.430784
\(792\) 6.05784 0.215256
\(793\) 0.800952 0.0284426
\(794\) −52.4882 −1.86274
\(795\) 0.639540 0.0226821
\(796\) −28.6752 −1.01636
\(797\) −28.8786 −1.02293 −0.511466 0.859303i \(-0.670898\pi\)
−0.511466 + 0.859303i \(0.670898\pi\)
\(798\) −2.33759 −0.0827498
\(799\) 15.5562 0.550338
\(800\) −21.8988 −0.774240
\(801\) 15.5817 0.550552
\(802\) 53.9287 1.90429
\(803\) 7.86651 0.277603
\(804\) −39.2263 −1.38340
\(805\) 0.942164 0.0332069
\(806\) 0.942164 0.0331863
\(807\) 10.5340 0.370813
\(808\) −58.1056 −2.04415
\(809\) 6.51615 0.229096 0.114548 0.993418i \(-0.463458\pi\)
0.114548 + 0.993418i \(0.463458\pi\)
\(810\) 0.337590 0.0118617
\(811\) 38.5060 1.35213 0.676065 0.736842i \(-0.263684\pi\)
0.676065 + 0.736842i \(0.263684\pi\)
\(812\) −16.9950 −0.596406
\(813\) 4.92434 0.172704
\(814\) 25.2441 0.884806
\(815\) −2.57733 −0.0902799
\(816\) 33.0222 1.15601
\(817\) 2.82649 0.0988864
\(818\) 85.8029 3.00003
\(819\) −0.133492 −0.00466459
\(820\) 6.95226 0.242784
\(821\) −37.3769 −1.30446 −0.652232 0.758019i \(-0.726167\pi\)
−0.652232 + 0.758019i \(0.726167\pi\)
\(822\) −37.0222 −1.29130
\(823\) 43.0400 1.50028 0.750140 0.661279i \(-0.229986\pi\)
0.750140 + 0.661279i \(0.229986\pi\)
\(824\) 59.6617 2.07842
\(825\) 4.98218 0.173457
\(826\) 32.0094 1.11375
\(827\) 11.3426 0.394422 0.197211 0.980361i \(-0.436812\pi\)
0.197211 + 0.980361i \(0.436812\pi\)
\(828\) −31.0222 −1.07810
\(829\) 14.4983 0.503548 0.251774 0.967786i \(-0.418986\pi\)
0.251774 + 0.967786i \(0.418986\pi\)
\(830\) −0.408196 −0.0141687
\(831\) 0.151312 0.00524896
\(832\) −0.259263 −0.00898833
\(833\) −5.05784 −0.175244
\(834\) −47.1634 −1.63314
\(835\) 1.68728 0.0583906
\(836\) −4.06289 −0.140518
\(837\) −2.79085 −0.0964660
\(838\) −25.2697 −0.872926
\(839\) −3.60962 −0.124618 −0.0623090 0.998057i \(-0.519846\pi\)
−0.0623090 + 0.998057i \(0.519846\pi\)
\(840\) 0.808672 0.0279018
\(841\) −14.0501 −0.484487
\(842\) −75.8225 −2.61301
\(843\) −6.51615 −0.224428
\(844\) 60.8709 2.09526
\(845\) −1.73302 −0.0596176
\(846\) −7.77808 −0.267416
\(847\) −1.00000 −0.0343604
\(848\) −31.2791 −1.07413
\(849\) 0.390376 0.0133977
\(850\) 63.7263 2.18579
\(851\) −70.4526 −2.41508
\(852\) 26.8810 0.920927
\(853\) −24.6496 −0.843988 −0.421994 0.906599i \(-0.638670\pi\)
−0.421994 + 0.906599i \(0.638670\pi\)
\(854\) −15.1735 −0.519227
\(855\) −0.123392 −0.00421993
\(856\) 5.59952 0.191388
\(857\) 17.9287 0.612433 0.306217 0.951962i \(-0.400937\pi\)
0.306217 + 0.951962i \(0.400937\pi\)
\(858\) −0.337590 −0.0115251
\(859\) −15.6974 −0.535588 −0.267794 0.963476i \(-0.586295\pi\)
−0.267794 + 0.963476i \(0.586295\pi\)
\(860\) −1.79420 −0.0611816
\(861\) 11.8487 0.403802
\(862\) −6.42772 −0.218929
\(863\) −15.0578 −0.512575 −0.256287 0.966601i \(-0.582499\pi\)
−0.256287 + 0.966601i \(0.582499\pi\)
\(864\) −4.39543 −0.149535
\(865\) −2.64964 −0.0900904
\(866\) 51.5461 1.75161
\(867\) −8.58170 −0.291450
\(868\) −12.2670 −0.416369
\(869\) 14.1157 0.478841
\(870\) −1.30529 −0.0442536
\(871\) 1.19133 0.0403666
\(872\) −51.6362 −1.74862
\(873\) 12.7909 0.432905
\(874\) 16.4983 0.558064
\(875\) 1.33254 0.0450481
\(876\) −34.5767 −1.16824
\(877\) −43.8487 −1.48066 −0.740332 0.672241i \(-0.765332\pi\)
−0.740332 + 0.672241i \(0.765332\pi\)
\(878\) −13.8036 −0.465850
\(879\) −29.4304 −0.992662
\(880\) 0.871558 0.0293802
\(881\) 11.0656 0.372808 0.186404 0.982473i \(-0.440317\pi\)
0.186404 + 0.982473i \(0.440317\pi\)
\(882\) 2.52892 0.0851531
\(883\) 5.72530 0.192672 0.0963358 0.995349i \(-0.469288\pi\)
0.0963358 + 0.995349i \(0.469288\pi\)
\(884\) −2.96770 −0.0998147
\(885\) 1.68966 0.0567971
\(886\) −26.9321 −0.904800
\(887\) 15.0935 0.506789 0.253395 0.967363i \(-0.418453\pi\)
0.253395 + 0.967363i \(0.418453\pi\)
\(888\) −60.4704 −2.02925
\(889\) 8.90652 0.298715
\(890\) 5.26023 0.176323
\(891\) 1.00000 0.0335013
\(892\) −107.681 −3.60541
\(893\) 2.84296 0.0951362
\(894\) −30.0094 −1.00367
\(895\) −1.60190 −0.0535457
\(896\) 13.7024 0.457766
\(897\) 0.942164 0.0314579
\(898\) 68.3625 2.28128
\(899\) 10.7909 0.359895
\(900\) −21.8988 −0.729960
\(901\) 24.2313 0.807263
\(902\) 29.9644 0.997704
\(903\) −3.05784 −0.101758
\(904\) −73.3948 −2.44107
\(905\) 0.357115 0.0118709
\(906\) 28.6395 0.951485
\(907\) 41.2993 1.37132 0.685660 0.727922i \(-0.259514\pi\)
0.685660 + 0.727922i \(0.259514\pi\)
\(908\) −33.9899 −1.12799
\(909\) −9.59180 −0.318140
\(910\) −0.0450656 −0.00149391
\(911\) 3.45059 0.114323 0.0571616 0.998365i \(-0.481795\pi\)
0.0571616 + 0.998365i \(0.481795\pi\)
\(912\) 6.03497 0.199838
\(913\) −1.20915 −0.0400170
\(914\) 12.1157 0.400751
\(915\) −0.800952 −0.0264786
\(916\) −21.0578 −0.695770
\(917\) 15.6974 0.518373
\(918\) 12.7909 0.422161
\(919\) −10.8265 −0.357133 −0.178567 0.983928i \(-0.557146\pi\)
−0.178567 + 0.983928i \(0.557146\pi\)
\(920\) −5.70748 −0.188170
\(921\) −23.6974 −0.780855
\(922\) 56.8964 1.87378
\(923\) −0.816393 −0.0268719
\(924\) 4.39543 0.144599
\(925\) −49.7330 −1.63521
\(926\) 64.9670 2.13495
\(927\) 9.84869 0.323473
\(928\) 16.9950 0.557887
\(929\) 17.4684 0.573120 0.286560 0.958062i \(-0.407488\pi\)
0.286560 + 0.958062i \(0.407488\pi\)
\(930\) −0.942164 −0.0308948
\(931\) −0.924344 −0.0302942
\(932\) −42.6241 −1.39620
\(933\) 12.2313 0.400436
\(934\) 11.6880 0.382441
\(935\) −0.675180 −0.0220808
\(936\) 0.808672 0.0264323
\(937\) 10.5340 0.344130 0.172065 0.985086i \(-0.444956\pi\)
0.172065 + 0.985086i \(0.444956\pi\)
\(938\) −22.5689 −0.736902
\(939\) 19.1735 0.625704
\(940\) −1.80465 −0.0588613
\(941\) −16.1513 −0.526518 −0.263259 0.964725i \(-0.584797\pi\)
−0.263259 + 0.964725i \(0.584797\pi\)
\(942\) 51.5461 1.67946
\(943\) −83.6261 −2.72324
\(944\) −82.6389 −2.68967
\(945\) 0.133492 0.00434249
\(946\) −7.73302 −0.251422
\(947\) −6.91662 −0.224760 −0.112380 0.993665i \(-0.535847\pi\)
−0.112380 + 0.993665i \(0.535847\pi\)
\(948\) −62.0444 −2.01511
\(949\) 1.05012 0.0340882
\(950\) 11.6463 0.377856
\(951\) −29.5562 −0.958424
\(952\) 30.6395 0.993033
\(953\) −16.1335 −0.522615 −0.261308 0.965256i \(-0.584154\pi\)
−0.261308 + 0.965256i \(0.584154\pi\)
\(954\) −12.1157 −0.392259
\(955\) 0.443837 0.0143622
\(956\) 102.444 3.31328
\(957\) −3.86651 −0.124986
\(958\) −0.942164 −0.0304399
\(959\) −14.6395 −0.472735
\(960\) 0.259263 0.00836768
\(961\) −23.2111 −0.748747
\(962\) 3.36989 0.108649
\(963\) 0.924344 0.0297866
\(964\) 42.7024 1.37535
\(965\) 2.45259 0.0789516
\(966\) −17.8487 −0.574272
\(967\) 17.8487 0.573975 0.286988 0.957934i \(-0.407346\pi\)
0.286988 + 0.957934i \(0.407346\pi\)
\(968\) 6.05784 0.194706
\(969\) −4.67518 −0.150188
\(970\) 4.31807 0.138645
\(971\) 33.9566 1.08972 0.544860 0.838527i \(-0.316583\pi\)
0.544860 + 0.838527i \(0.316583\pi\)
\(972\) −4.39543 −0.140983
\(973\) −18.6496 −0.597880
\(974\) 10.8810 0.348649
\(975\) 0.665081 0.0212996
\(976\) 39.1735 1.25391
\(977\) 41.0222 1.31242 0.656208 0.754580i \(-0.272159\pi\)
0.656208 + 0.754580i \(0.272159\pi\)
\(978\) 48.8258 1.56128
\(979\) 15.5817 0.497993
\(980\) 0.586754 0.0187432
\(981\) −8.52387 −0.272146
\(982\) 64.3820 2.05451
\(983\) 42.3470 1.35066 0.675330 0.737516i \(-0.264001\pi\)
0.675330 + 0.737516i \(0.264001\pi\)
\(984\) −71.7774 −2.28818
\(985\) −1.08338 −0.0345192
\(986\) −49.4559 −1.57500
\(987\) −3.07566 −0.0978992
\(988\) −0.542362 −0.0172548
\(989\) 21.5817 0.686258
\(990\) 0.337590 0.0107293
\(991\) 50.9687 1.61908 0.809538 0.587068i \(-0.199718\pi\)
0.809538 + 0.587068i \(0.199718\pi\)
\(992\) 12.2670 0.389477
\(993\) −18.6496 −0.591828
\(994\) 15.4660 0.490553
\(995\) −0.870884 −0.0276089
\(996\) 5.31472 0.168403
\(997\) −32.8608 −1.04071 −0.520356 0.853950i \(-0.674201\pi\)
−0.520356 + 0.853950i \(0.674201\pi\)
\(998\) −21.2186 −0.671662
\(999\) −9.98218 −0.315822
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 231.2.a.d.1.3 3
3.2 odd 2 693.2.a.m.1.1 3
4.3 odd 2 3696.2.a.bp.1.2 3
5.4 even 2 5775.2.a.bw.1.1 3
7.6 odd 2 1617.2.a.s.1.3 3
11.10 odd 2 2541.2.a.bi.1.1 3
21.20 even 2 4851.2.a.bp.1.1 3
33.32 even 2 7623.2.a.cb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.3 3 1.1 even 1 trivial
693.2.a.m.1.1 3 3.2 odd 2
1617.2.a.s.1.3 3 7.6 odd 2
2541.2.a.bi.1.1 3 11.10 odd 2
3696.2.a.bp.1.2 3 4.3 odd 2
4851.2.a.bp.1.1 3 21.20 even 2
5775.2.a.bw.1.1 3 5.4 even 2
7623.2.a.cb.1.3 3 33.32 even 2