Properties

Label 177.2.a.d.1.1
Level $177$
Weight $2$
Character 177.1
Self dual yes
Analytic conductor $1.413$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
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 1.1
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.86081 q^{2} -1.00000 q^{3} +1.46260 q^{4} -3.32340 q^{5} +1.86081 q^{6} +1.13919 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.86081 q^{2} -1.00000 q^{3} +1.46260 q^{4} -3.32340 q^{5} +1.86081 q^{6} +1.13919 q^{7} +1.00000 q^{8} +1.00000 q^{9} +6.18421 q^{10} +0.398207 q^{11} -1.46260 q^{12} +2.39821 q^{13} -2.11982 q^{14} +3.32340 q^{15} -4.78600 q^{16} +7.10941 q^{17} -1.86081 q^{18} +1.53740 q^{19} -4.86081 q^{20} -1.13919 q^{21} -0.740987 q^{22} +3.25901 q^{23} -1.00000 q^{24} +6.04502 q^{25} -4.46260 q^{26} -1.00000 q^{27} +1.66618 q^{28} -3.93561 q^{29} -6.18421 q^{30} +7.78600 q^{31} +6.90582 q^{32} -0.398207 q^{33} -13.2292 q^{34} -3.78600 q^{35} +1.46260 q^{36} +1.25901 q^{37} -2.86081 q^{38} -2.39821 q^{39} -3.32340 q^{40} -7.50761 q^{41} +2.11982 q^{42} -9.69182 q^{43} +0.582418 q^{44} -3.32340 q^{45} -6.06439 q^{46} +8.71120 q^{47} +4.78600 q^{48} -5.70224 q^{49} -11.2486 q^{50} -7.10941 q^{51} +3.50761 q^{52} +10.7666 q^{53} +1.86081 q^{54} -1.32340 q^{55} +1.13919 q^{56} -1.53740 q^{57} +7.32340 q^{58} +1.00000 q^{59} +4.86081 q^{60} +0.989588 q^{61} -14.4882 q^{62} +1.13919 q^{63} -3.27839 q^{64} -7.97021 q^{65} +0.740987 q^{66} +5.45219 q^{67} +10.3982 q^{68} -3.25901 q^{69} +7.04502 q^{70} +15.0450 q^{71} +1.00000 q^{72} +4.73202 q^{73} -2.34278 q^{74} -6.04502 q^{75} +2.24860 q^{76} +0.453636 q^{77} +4.46260 q^{78} +7.04502 q^{79} +15.9058 q^{80} +1.00000 q^{81} +13.9702 q^{82} +13.0796 q^{83} -1.66618 q^{84} -23.6274 q^{85} +18.0346 q^{86} +3.93561 q^{87} +0.398207 q^{88} -16.4328 q^{89} +6.18421 q^{90} +2.73202 q^{91} +4.76663 q^{92} -7.78600 q^{93} -16.2099 q^{94} -5.10941 q^{95} -6.90582 q^{96} +14.7666 q^{97} +10.6108 q^{98} +0.398207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + 5q^{10} - 2q^{11} - 2q^{12} + 4q^{13} + 8q^{14} + 2q^{15} - 4q^{16} + 3q^{17} + 7q^{19} - 9q^{20} - 9q^{21} - 11q^{22} + q^{23} - 3q^{24} - q^{25} - 11q^{26} - 3q^{27} + 9q^{28} - 11q^{29} - 5q^{30} + 13q^{31} - 4q^{32} + 2q^{33} - 7q^{34} - q^{35} + 2q^{36} - 5q^{37} - 3q^{38} - 4q^{39} - 2q^{40} - q^{41} - 8q^{42} + 6q^{43} - 15q^{44} - 2q^{45} - 19q^{46} + 11q^{47} + 4q^{48} + 14q^{49} - 21q^{50} - 3q^{51} - 11q^{52} + 2q^{53} + 4q^{55} + 9q^{56} - 7q^{57} + 14q^{58} + 3q^{59} + 9q^{60} - q^{61} - 2q^{62} + 9q^{63} - 21q^{64} + 11q^{66} + 10q^{67} + 28q^{68} - q^{69} + 2q^{70} + 26q^{71} + 3q^{72} + 7q^{73} - 19q^{74} + q^{75} - 6q^{76} - 17q^{77} + 11q^{78} + 2q^{79} + 23q^{80} + 3q^{81} + 18q^{82} - 3q^{83} - 9q^{84} - 35q^{85} + 31q^{86} + 11q^{87} - 2q^{88} - 23q^{89} + 5q^{90} + q^{91} - 16q^{92} - 13q^{93} + 4q^{94} + 3q^{95} + 4q^{96} + 14q^{97} + 51q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86081 −1.31579 −0.657894 0.753110i \(-0.728553\pi\)
−0.657894 + 0.753110i \(0.728553\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.46260 0.731299
\(5\) −3.32340 −1.48627 −0.743136 0.669141i \(-0.766662\pi\)
−0.743136 + 0.669141i \(0.766662\pi\)
\(6\) 1.86081 0.759671
\(7\) 1.13919 0.430575 0.215287 0.976551i \(-0.430931\pi\)
0.215287 + 0.976551i \(0.430931\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 6.18421 1.95562
\(11\) 0.398207 0.120064 0.0600320 0.998196i \(-0.480880\pi\)
0.0600320 + 0.998196i \(0.480880\pi\)
\(12\) −1.46260 −0.422216
\(13\) 2.39821 0.665143 0.332572 0.943078i \(-0.392084\pi\)
0.332572 + 0.943078i \(0.392084\pi\)
\(14\) −2.11982 −0.566545
\(15\) 3.32340 0.858099
\(16\) −4.78600 −1.19650
\(17\) 7.10941 1.72428 0.862142 0.506666i \(-0.169122\pi\)
0.862142 + 0.506666i \(0.169122\pi\)
\(18\) −1.86081 −0.438596
\(19\) 1.53740 0.352704 0.176352 0.984327i \(-0.443570\pi\)
0.176352 + 0.984327i \(0.443570\pi\)
\(20\) −4.86081 −1.08691
\(21\) −1.13919 −0.248593
\(22\) −0.740987 −0.157979
\(23\) 3.25901 0.679551 0.339776 0.940507i \(-0.389649\pi\)
0.339776 + 0.940507i \(0.389649\pi\)
\(24\) −1.00000 −0.204124
\(25\) 6.04502 1.20900
\(26\) −4.46260 −0.875188
\(27\) −1.00000 −0.192450
\(28\) 1.66618 0.314879
\(29\) −3.93561 −0.730824 −0.365412 0.930846i \(-0.619072\pi\)
−0.365412 + 0.930846i \(0.619072\pi\)
\(30\) −6.18421 −1.12908
\(31\) 7.78600 1.39841 0.699204 0.714923i \(-0.253538\pi\)
0.699204 + 0.714923i \(0.253538\pi\)
\(32\) 6.90582 1.22079
\(33\) −0.398207 −0.0693190
\(34\) −13.2292 −2.26879
\(35\) −3.78600 −0.639951
\(36\) 1.46260 0.243766
\(37\) 1.25901 0.206981 0.103490 0.994630i \(-0.466999\pi\)
0.103490 + 0.994630i \(0.466999\pi\)
\(38\) −2.86081 −0.464084
\(39\) −2.39821 −0.384021
\(40\) −3.32340 −0.525476
\(41\) −7.50761 −1.17249 −0.586246 0.810133i \(-0.699395\pi\)
−0.586246 + 0.810133i \(0.699395\pi\)
\(42\) 2.11982 0.327095
\(43\) −9.69182 −1.47799 −0.738995 0.673711i \(-0.764699\pi\)
−0.738995 + 0.673711i \(0.764699\pi\)
\(44\) 0.582418 0.0878028
\(45\) −3.32340 −0.495424
\(46\) −6.06439 −0.894146
\(47\) 8.71120 1.27066 0.635330 0.772241i \(-0.280864\pi\)
0.635330 + 0.772241i \(0.280864\pi\)
\(48\) 4.78600 0.690800
\(49\) −5.70224 −0.814605
\(50\) −11.2486 −1.59079
\(51\) −7.10941 −0.995516
\(52\) 3.50761 0.486419
\(53\) 10.7666 1.47891 0.739455 0.673206i \(-0.235083\pi\)
0.739455 + 0.673206i \(0.235083\pi\)
\(54\) 1.86081 0.253224
\(55\) −1.32340 −0.178448
\(56\) 1.13919 0.152231
\(57\) −1.53740 −0.203634
\(58\) 7.32340 0.961610
\(59\) 1.00000 0.130189
\(60\) 4.86081 0.627527
\(61\) 0.989588 0.126704 0.0633519 0.997991i \(-0.479821\pi\)
0.0633519 + 0.997991i \(0.479821\pi\)
\(62\) −14.4882 −1.84001
\(63\) 1.13919 0.143525
\(64\) −3.27839 −0.409799
\(65\) −7.97021 −0.988583
\(66\) 0.740987 0.0912092
\(67\) 5.45219 0.666091 0.333045 0.942911i \(-0.391924\pi\)
0.333045 + 0.942911i \(0.391924\pi\)
\(68\) 10.3982 1.26097
\(69\) −3.25901 −0.392339
\(70\) 7.04502 0.842040
\(71\) 15.0450 1.78551 0.892757 0.450538i \(-0.148768\pi\)
0.892757 + 0.450538i \(0.148768\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.73202 0.553842 0.276921 0.960893i \(-0.410686\pi\)
0.276921 + 0.960893i \(0.410686\pi\)
\(74\) −2.34278 −0.272343
\(75\) −6.04502 −0.698018
\(76\) 2.24860 0.257932
\(77\) 0.453636 0.0516966
\(78\) 4.46260 0.505290
\(79\) 7.04502 0.792626 0.396313 0.918115i \(-0.370289\pi\)
0.396313 + 0.918115i \(0.370289\pi\)
\(80\) 15.9058 1.77832
\(81\) 1.00000 0.111111
\(82\) 13.9702 1.54275
\(83\) 13.0796 1.43567 0.717837 0.696211i \(-0.245132\pi\)
0.717837 + 0.696211i \(0.245132\pi\)
\(84\) −1.66618 −0.181796
\(85\) −23.6274 −2.56275
\(86\) 18.0346 1.94472
\(87\) 3.93561 0.421942
\(88\) 0.398207 0.0424491
\(89\) −16.4328 −1.74187 −0.870937 0.491394i \(-0.836487\pi\)
−0.870937 + 0.491394i \(0.836487\pi\)
\(90\) 6.18421 0.651873
\(91\) 2.73202 0.286394
\(92\) 4.76663 0.496955
\(93\) −7.78600 −0.807371
\(94\) −16.2099 −1.67192
\(95\) −5.10941 −0.524214
\(96\) −6.90582 −0.704822
\(97\) 14.7666 1.49932 0.749662 0.661821i \(-0.230216\pi\)
0.749662 + 0.661821i \(0.230216\pi\)
\(98\) 10.6108 1.07185
\(99\) 0.398207 0.0400214
\(100\) 8.84143 0.884143
\(101\) −0.278388 −0.0277007 −0.0138503 0.999904i \(-0.504409\pi\)
−0.0138503 + 0.999904i \(0.504409\pi\)
\(102\) 13.2292 1.30989
\(103\) −15.0152 −1.47949 −0.739747 0.672885i \(-0.765055\pi\)
−0.739747 + 0.672885i \(0.765055\pi\)
\(104\) 2.39821 0.235164
\(105\) 3.78600 0.369476
\(106\) −20.0346 −1.94593
\(107\) 1.16898 0.113010 0.0565048 0.998402i \(-0.482004\pi\)
0.0565048 + 0.998402i \(0.482004\pi\)
\(108\) −1.46260 −0.140739
\(109\) −4.89059 −0.468434 −0.234217 0.972184i \(-0.575253\pi\)
−0.234217 + 0.972184i \(0.575253\pi\)
\(110\) 2.46260 0.234800
\(111\) −1.25901 −0.119500
\(112\) −5.45219 −0.515183
\(113\) −15.5630 −1.46405 −0.732024 0.681279i \(-0.761424\pi\)
−0.732024 + 0.681279i \(0.761424\pi\)
\(114\) 2.86081 0.267939
\(115\) −10.8310 −1.01000
\(116\) −5.75622 −0.534451
\(117\) 2.39821 0.221714
\(118\) −1.86081 −0.171301
\(119\) 8.09899 0.742434
\(120\) 3.32340 0.303384
\(121\) −10.8414 −0.985585
\(122\) −1.84143 −0.166715
\(123\) 7.50761 0.676939
\(124\) 11.3878 1.02265
\(125\) −3.47301 −0.310636
\(126\) −2.11982 −0.188848
\(127\) 4.39821 0.390278 0.195139 0.980776i \(-0.437484\pi\)
0.195139 + 0.980776i \(0.437484\pi\)
\(128\) −7.71120 −0.681580
\(129\) 9.69182 0.853318
\(130\) 14.8310 1.30077
\(131\) 17.6620 1.54314 0.771570 0.636145i \(-0.219472\pi\)
0.771570 + 0.636145i \(0.219472\pi\)
\(132\) −0.582418 −0.0506929
\(133\) 1.75140 0.151866
\(134\) −10.1455 −0.876434
\(135\) 3.32340 0.286033
\(136\) 7.10941 0.609627
\(137\) −10.7666 −0.919855 −0.459928 0.887956i \(-0.652125\pi\)
−0.459928 + 0.887956i \(0.652125\pi\)
\(138\) 6.06439 0.516235
\(139\) 9.82061 0.832973 0.416486 0.909142i \(-0.363261\pi\)
0.416486 + 0.909142i \(0.363261\pi\)
\(140\) −5.53740 −0.467996
\(141\) −8.71120 −0.733615
\(142\) −27.9959 −2.34936
\(143\) 0.954984 0.0798598
\(144\) −4.78600 −0.398834
\(145\) 13.0796 1.08620
\(146\) −8.80538 −0.728738
\(147\) 5.70224 0.470313
\(148\) 1.84143 0.151365
\(149\) −5.01938 −0.411203 −0.205602 0.978636i \(-0.565915\pi\)
−0.205602 + 0.978636i \(0.565915\pi\)
\(150\) 11.2486 0.918444
\(151\) 7.35801 0.598786 0.299393 0.954130i \(-0.403216\pi\)
0.299393 + 0.954130i \(0.403216\pi\)
\(152\) 1.53740 0.124700
\(153\) 7.10941 0.574761
\(154\) −0.844128 −0.0680218
\(155\) −25.8760 −2.07841
\(156\) −3.50761 −0.280834
\(157\) −1.73057 −0.138115 −0.0690574 0.997613i \(-0.521999\pi\)
−0.0690574 + 0.997613i \(0.521999\pi\)
\(158\) −13.1094 −1.04293
\(159\) −10.7666 −0.853849
\(160\) −22.9508 −1.81442
\(161\) 3.71265 0.292598
\(162\) −1.86081 −0.146199
\(163\) −8.58242 −0.672227 −0.336113 0.941822i \(-0.609113\pi\)
−0.336113 + 0.941822i \(0.609113\pi\)
\(164\) −10.9806 −0.857443
\(165\) 1.32340 0.103027
\(166\) −24.3386 −1.88904
\(167\) 6.49239 0.502396 0.251198 0.967936i \(-0.419175\pi\)
0.251198 + 0.967936i \(0.419175\pi\)
\(168\) −1.13919 −0.0878907
\(169\) −7.24860 −0.557585
\(170\) 43.9661 3.37204
\(171\) 1.53740 0.117568
\(172\) −14.1752 −1.08085
\(173\) −20.5437 −1.56191 −0.780953 0.624590i \(-0.785266\pi\)
−0.780953 + 0.624590i \(0.785266\pi\)
\(174\) −7.32340 −0.555186
\(175\) 6.88645 0.520566
\(176\) −1.90582 −0.143657
\(177\) −1.00000 −0.0751646
\(178\) 30.5783 2.29194
\(179\) −23.9702 −1.79162 −0.895809 0.444439i \(-0.853403\pi\)
−0.895809 + 0.444439i \(0.853403\pi\)
\(180\) −4.86081 −0.362303
\(181\) 14.7112 1.09347 0.546737 0.837304i \(-0.315870\pi\)
0.546737 + 0.837304i \(0.315870\pi\)
\(182\) −5.08377 −0.376834
\(183\) −0.989588 −0.0731524
\(184\) 3.25901 0.240258
\(185\) −4.18421 −0.307629
\(186\) 14.4882 1.06233
\(187\) 2.83102 0.207025
\(188\) 12.7410 0.929232
\(189\) −1.13919 −0.0828642
\(190\) 9.50761 0.689755
\(191\) 2.91623 0.211011 0.105506 0.994419i \(-0.466354\pi\)
0.105506 + 0.994419i \(0.466354\pi\)
\(192\) 3.27839 0.236597
\(193\) −0.646809 −0.0465583 −0.0232791 0.999729i \(-0.507411\pi\)
−0.0232791 + 0.999729i \(0.507411\pi\)
\(194\) −27.4778 −1.97279
\(195\) 7.97021 0.570759
\(196\) −8.34008 −0.595720
\(197\) −5.87122 −0.418307 −0.209153 0.977883i \(-0.567071\pi\)
−0.209153 + 0.977883i \(0.567071\pi\)
\(198\) −0.740987 −0.0526596
\(199\) 11.1004 0.786890 0.393445 0.919348i \(-0.371283\pi\)
0.393445 + 0.919348i \(0.371283\pi\)
\(200\) 6.04502 0.427447
\(201\) −5.45219 −0.384568
\(202\) 0.518027 0.0364482
\(203\) −4.48342 −0.314675
\(204\) −10.3982 −0.728020
\(205\) 24.9508 1.74264
\(206\) 27.9404 1.94670
\(207\) 3.25901 0.226517
\(208\) −11.4778 −0.795844
\(209\) 0.612205 0.0423471
\(210\) −7.04502 −0.486152
\(211\) −13.4778 −0.927852 −0.463926 0.885874i \(-0.653560\pi\)
−0.463926 + 0.885874i \(0.653560\pi\)
\(212\) 15.7473 1.08153
\(213\) −15.0450 −1.03087
\(214\) −2.17525 −0.148697
\(215\) 32.2099 2.19669
\(216\) −1.00000 −0.0680414
\(217\) 8.86977 0.602119
\(218\) 9.10044 0.616360
\(219\) −4.73202 −0.319761
\(220\) −1.93561 −0.130499
\(221\) 17.0498 1.14690
\(222\) 2.34278 0.157237
\(223\) 16.7368 1.12078 0.560391 0.828228i \(-0.310651\pi\)
0.560391 + 0.828228i \(0.310651\pi\)
\(224\) 7.86707 0.525641
\(225\) 6.04502 0.403001
\(226\) 28.9598 1.92638
\(227\) −14.0346 −0.931509 −0.465755 0.884914i \(-0.654217\pi\)
−0.465755 + 0.884914i \(0.654217\pi\)
\(228\) −2.24860 −0.148917
\(229\) −12.2445 −0.809136 −0.404568 0.914508i \(-0.632578\pi\)
−0.404568 + 0.914508i \(0.632578\pi\)
\(230\) 20.1544 1.32894
\(231\) −0.453636 −0.0298470
\(232\) −3.93561 −0.258385
\(233\) 1.87122 0.122588 0.0612938 0.998120i \(-0.480477\pi\)
0.0612938 + 0.998120i \(0.480477\pi\)
\(234\) −4.46260 −0.291729
\(235\) −28.9508 −1.88854
\(236\) 1.46260 0.0952070
\(237\) −7.04502 −0.457623
\(238\) −15.0707 −0.976886
\(239\) −13.9910 −0.905005 −0.452502 0.891763i \(-0.649469\pi\)
−0.452502 + 0.891763i \(0.649469\pi\)
\(240\) −15.9058 −1.02672
\(241\) 27.5333 1.77357 0.886786 0.462179i \(-0.152932\pi\)
0.886786 + 0.462179i \(0.152932\pi\)
\(242\) 20.1738 1.29682
\(243\) −1.00000 −0.0641500
\(244\) 1.44737 0.0926583
\(245\) 18.9508 1.21072
\(246\) −13.9702 −0.890708
\(247\) 3.68701 0.234599
\(248\) 7.78600 0.494412
\(249\) −13.0796 −0.828887
\(250\) 6.46260 0.408731
\(251\) −2.61702 −0.165185 −0.0825925 0.996583i \(-0.526320\pi\)
−0.0825925 + 0.996583i \(0.526320\pi\)
\(252\) 1.66618 0.104960
\(253\) 1.29776 0.0815897
\(254\) −8.18421 −0.513523
\(255\) 23.6274 1.47961
\(256\) 20.9058 1.30661
\(257\) 20.3892 1.27185 0.635923 0.771752i \(-0.280620\pi\)
0.635923 + 0.771752i \(0.280620\pi\)
\(258\) −18.0346 −1.12279
\(259\) 1.43426 0.0891206
\(260\) −11.6572 −0.722950
\(261\) −3.93561 −0.243608
\(262\) −32.8656 −2.03044
\(263\) −23.8552 −1.47098 −0.735488 0.677538i \(-0.763047\pi\)
−0.735488 + 0.677538i \(0.763047\pi\)
\(264\) −0.398207 −0.0245080
\(265\) −35.7819 −2.19806
\(266\) −3.25901 −0.199823
\(267\) 16.4328 1.00567
\(268\) 7.97436 0.487112
\(269\) −28.2999 −1.72547 −0.862737 0.505653i \(-0.831252\pi\)
−0.862737 + 0.505653i \(0.831252\pi\)
\(270\) −6.18421 −0.376359
\(271\) −23.4134 −1.42226 −0.711132 0.703058i \(-0.751817\pi\)
−0.711132 + 0.703058i \(0.751817\pi\)
\(272\) −34.0256 −2.06311
\(273\) −2.73202 −0.165350
\(274\) 20.0346 1.21033
\(275\) 2.40717 0.145158
\(276\) −4.76663 −0.286917
\(277\) 15.6918 0.942830 0.471415 0.881911i \(-0.343743\pi\)
0.471415 + 0.881911i \(0.343743\pi\)
\(278\) −18.2742 −1.09602
\(279\) 7.78600 0.466136
\(280\) −3.78600 −0.226257
\(281\) −5.63158 −0.335952 −0.167976 0.985791i \(-0.553723\pi\)
−0.167976 + 0.985791i \(0.553723\pi\)
\(282\) 16.2099 0.965283
\(283\) 10.8310 0.643837 0.321919 0.946767i \(-0.395672\pi\)
0.321919 + 0.946767i \(0.395672\pi\)
\(284\) 22.0048 1.30575
\(285\) 5.10941 0.302655
\(286\) −1.77704 −0.105079
\(287\) −8.55263 −0.504846
\(288\) 6.90582 0.406929
\(289\) 33.5437 1.97316
\(290\) −24.3386 −1.42921
\(291\) −14.7666 −0.865635
\(292\) 6.92105 0.405024
\(293\) 33.9959 1.98606 0.993029 0.117866i \(-0.0376054\pi\)
0.993029 + 0.117866i \(0.0376054\pi\)
\(294\) −10.6108 −0.618832
\(295\) −3.32340 −0.193496
\(296\) 1.25901 0.0731787
\(297\) −0.398207 −0.0231063
\(298\) 9.34008 0.541056
\(299\) 7.81579 0.451999
\(300\) −8.84143 −0.510460
\(301\) −11.0409 −0.636385
\(302\) −13.6918 −0.787876
\(303\) 0.278388 0.0159930
\(304\) −7.35801 −0.422011
\(305\) −3.28880 −0.188316
\(306\) −13.2292 −0.756265
\(307\) −4.25756 −0.242992 −0.121496 0.992592i \(-0.538769\pi\)
−0.121496 + 0.992592i \(0.538769\pi\)
\(308\) 0.663487 0.0378057
\(309\) 15.0152 0.854187
\(310\) 48.1503 2.73475
\(311\) 21.5374 1.22127 0.610637 0.791911i \(-0.290913\pi\)
0.610637 + 0.791911i \(0.290913\pi\)
\(312\) −2.39821 −0.135772
\(313\) −3.96540 −0.224137 −0.112069 0.993700i \(-0.535748\pi\)
−0.112069 + 0.993700i \(0.535748\pi\)
\(314\) 3.22026 0.181730
\(315\) −3.78600 −0.213317
\(316\) 10.3040 0.579647
\(317\) 7.58097 0.425790 0.212895 0.977075i \(-0.431711\pi\)
0.212895 + 0.977075i \(0.431711\pi\)
\(318\) 20.0346 1.12348
\(319\) −1.56719 −0.0877457
\(320\) 10.8954 0.609072
\(321\) −1.16898 −0.0652462
\(322\) −6.90852 −0.384997
\(323\) 10.9300 0.608162
\(324\) 1.46260 0.0812555
\(325\) 14.4972 0.804160
\(326\) 15.9702 0.884508
\(327\) 4.89059 0.270450
\(328\) −7.50761 −0.414539
\(329\) 9.92375 0.547114
\(330\) −2.46260 −0.135562
\(331\) −0.667633 −0.0366964 −0.0183482 0.999832i \(-0.505841\pi\)
−0.0183482 + 0.999832i \(0.505841\pi\)
\(332\) 19.1302 1.04991
\(333\) 1.25901 0.0689935
\(334\) −12.0811 −0.661047
\(335\) −18.1198 −0.989991
\(336\) 5.45219 0.297441
\(337\) 7.47783 0.407343 0.203672 0.979039i \(-0.434713\pi\)
0.203672 + 0.979039i \(0.434713\pi\)
\(338\) 13.4882 0.733664
\(339\) 15.5630 0.845268
\(340\) −34.5574 −1.87414
\(341\) 3.10044 0.167898
\(342\) −2.86081 −0.154695
\(343\) −14.4703 −0.781323
\(344\) −9.69182 −0.522548
\(345\) 10.8310 0.583122
\(346\) 38.2278 2.05514
\(347\) −29.0200 −1.55788 −0.778939 0.627100i \(-0.784242\pi\)
−0.778939 + 0.627100i \(0.784242\pi\)
\(348\) 5.75622 0.308566
\(349\) −16.8746 −0.903276 −0.451638 0.892201i \(-0.649160\pi\)
−0.451638 + 0.892201i \(0.649160\pi\)
\(350\) −12.8143 −0.684955
\(351\) −2.39821 −0.128007
\(352\) 2.74995 0.146573
\(353\) −2.98062 −0.158643 −0.0793213 0.996849i \(-0.525275\pi\)
−0.0793213 + 0.996849i \(0.525275\pi\)
\(354\) 1.86081 0.0989007
\(355\) −50.0007 −2.65376
\(356\) −24.0346 −1.27383
\(357\) −8.09899 −0.428644
\(358\) 44.6039 2.35739
\(359\) −1.85039 −0.0976600 −0.0488300 0.998807i \(-0.515549\pi\)
−0.0488300 + 0.998807i \(0.515549\pi\)
\(360\) −3.32340 −0.175159
\(361\) −16.6364 −0.875600
\(362\) −27.3747 −1.43878
\(363\) 10.8414 0.569028
\(364\) 3.99585 0.209440
\(365\) −15.7264 −0.823159
\(366\) 1.84143 0.0962531
\(367\) 7.97021 0.416042 0.208021 0.978124i \(-0.433298\pi\)
0.208021 + 0.978124i \(0.433298\pi\)
\(368\) −15.5976 −0.813084
\(369\) −7.50761 −0.390831
\(370\) 7.78600 0.404775
\(371\) 12.2653 0.636782
\(372\) −11.3878 −0.590430
\(373\) 1.31859 0.0682739 0.0341369 0.999417i \(-0.489132\pi\)
0.0341369 + 0.999417i \(0.489132\pi\)
\(374\) −5.26798 −0.272401
\(375\) 3.47301 0.179345
\(376\) 8.71120 0.449246
\(377\) −9.43841 −0.486103
\(378\) 2.11982 0.109032
\(379\) 20.5783 1.05703 0.528517 0.848922i \(-0.322748\pi\)
0.528517 + 0.848922i \(0.322748\pi\)
\(380\) −7.47301 −0.383357
\(381\) −4.39821 −0.225327
\(382\) −5.42655 −0.277646
\(383\) 21.8802 1.11803 0.559013 0.829159i \(-0.311180\pi\)
0.559013 + 0.829159i \(0.311180\pi\)
\(384\) 7.71120 0.393511
\(385\) −1.50761 −0.0768351
\(386\) 1.20359 0.0612609
\(387\) −9.69182 −0.492663
\(388\) 21.5976 1.09645
\(389\) −6.29921 −0.319383 −0.159691 0.987167i \(-0.551050\pi\)
−0.159691 + 0.987167i \(0.551050\pi\)
\(390\) −14.8310 −0.750998
\(391\) 23.1697 1.17174
\(392\) −5.70224 −0.288006
\(393\) −17.6620 −0.890932
\(394\) 10.9252 0.550403
\(395\) −23.4134 −1.17806
\(396\) 0.582418 0.0292676
\(397\) −12.1198 −0.608276 −0.304138 0.952628i \(-0.598368\pi\)
−0.304138 + 0.952628i \(0.598368\pi\)
\(398\) −20.6558 −1.03538
\(399\) −1.75140 −0.0876796
\(400\) −28.9315 −1.44657
\(401\) −16.3338 −0.815672 −0.407836 0.913055i \(-0.633716\pi\)
−0.407836 + 0.913055i \(0.633716\pi\)
\(402\) 10.1455 0.506010
\(403\) 18.6724 0.930141
\(404\) −0.407170 −0.0202575
\(405\) −3.32340 −0.165141
\(406\) 8.34278 0.414045
\(407\) 0.501348 0.0248509
\(408\) −7.10941 −0.351968
\(409\) 37.0665 1.83282 0.916410 0.400240i \(-0.131073\pi\)
0.916410 + 0.400240i \(0.131073\pi\)
\(410\) −46.4287 −2.29295
\(411\) 10.7666 0.531079
\(412\) −21.9612 −1.08195
\(413\) 1.13919 0.0560561
\(414\) −6.06439 −0.298049
\(415\) −43.4689 −2.13380
\(416\) 16.5616 0.811999
\(417\) −9.82061 −0.480917
\(418\) −1.13919 −0.0557198
\(419\) 14.6468 0.715543 0.357772 0.933809i \(-0.383537\pi\)
0.357772 + 0.933809i \(0.383537\pi\)
\(420\) 5.53740 0.270198
\(421\) −3.32340 −0.161973 −0.0809864 0.996715i \(-0.525807\pi\)
−0.0809864 + 0.996715i \(0.525807\pi\)
\(422\) 25.0796 1.22086
\(423\) 8.71120 0.423553
\(424\) 10.7666 0.522874
\(425\) 42.9765 2.08467
\(426\) 27.9959 1.35640
\(427\) 1.12733 0.0545555
\(428\) 1.70975 0.0826439
\(429\) −0.954984 −0.0461071
\(430\) −59.9363 −2.89038
\(431\) 2.73202 0.131597 0.0657985 0.997833i \(-0.479041\pi\)
0.0657985 + 0.997833i \(0.479041\pi\)
\(432\) 4.78600 0.230267
\(433\) −36.1038 −1.73504 −0.867519 0.497404i \(-0.834287\pi\)
−0.867519 + 0.497404i \(0.834287\pi\)
\(434\) −16.5049 −0.792261
\(435\) −13.0796 −0.627120
\(436\) −7.15297 −0.342565
\(437\) 5.01041 0.239681
\(438\) 8.80538 0.420737
\(439\) −37.9571 −1.81159 −0.905797 0.423712i \(-0.860727\pi\)
−0.905797 + 0.423712i \(0.860727\pi\)
\(440\) −1.32340 −0.0630908
\(441\) −5.70224 −0.271535
\(442\) −31.7264 −1.50907
\(443\) 1.84625 0.0877179 0.0438589 0.999038i \(-0.486035\pi\)
0.0438589 + 0.999038i \(0.486035\pi\)
\(444\) −1.84143 −0.0873904
\(445\) 54.6129 2.58890
\(446\) −31.1440 −1.47471
\(447\) 5.01938 0.237408
\(448\) −3.73472 −0.176449
\(449\) 27.9494 1.31901 0.659507 0.751699i \(-0.270765\pi\)
0.659507 + 0.751699i \(0.270765\pi\)
\(450\) −11.2486 −0.530264
\(451\) −2.98959 −0.140774
\(452\) −22.7625 −1.07066
\(453\) −7.35801 −0.345709
\(454\) 26.1157 1.22567
\(455\) −9.07962 −0.425659
\(456\) −1.53740 −0.0719954
\(457\) −23.2501 −1.08759 −0.543796 0.839218i \(-0.683013\pi\)
−0.543796 + 0.839218i \(0.683013\pi\)
\(458\) 22.7846 1.06465
\(459\) −7.10941 −0.331839
\(460\) −15.8414 −0.738611
\(461\) 20.4447 0.952203 0.476102 0.879390i \(-0.342049\pi\)
0.476102 + 0.879390i \(0.342049\pi\)
\(462\) 0.844128 0.0392724
\(463\) −12.6122 −0.586139 −0.293069 0.956091i \(-0.594677\pi\)
−0.293069 + 0.956091i \(0.594677\pi\)
\(464\) 18.8358 0.874432
\(465\) 25.8760 1.19997
\(466\) −3.48197 −0.161299
\(467\) −1.89204 −0.0875533 −0.0437766 0.999041i \(-0.513939\pi\)
−0.0437766 + 0.999041i \(0.513939\pi\)
\(468\) 3.50761 0.162140
\(469\) 6.21110 0.286802
\(470\) 53.8719 2.48492
\(471\) 1.73057 0.0797407
\(472\) 1.00000 0.0460287
\(473\) −3.85936 −0.177453
\(474\) 13.1094 0.602135
\(475\) 9.29362 0.426420
\(476\) 11.8456 0.542941
\(477\) 10.7666 0.492970
\(478\) 26.0346 1.19080
\(479\) −3.51658 −0.160677 −0.0803383 0.996768i \(-0.525600\pi\)
−0.0803383 + 0.996768i \(0.525600\pi\)
\(480\) 22.9508 1.04756
\(481\) 3.01938 0.137672
\(482\) −51.2340 −2.33365
\(483\) −3.71265 −0.168931
\(484\) −15.8567 −0.720757
\(485\) −49.0755 −2.22840
\(486\) 1.86081 0.0844079
\(487\) 21.2203 0.961582 0.480791 0.876835i \(-0.340350\pi\)
0.480791 + 0.876835i \(0.340350\pi\)
\(488\) 0.989588 0.0447965
\(489\) 8.58242 0.388110
\(490\) −35.2638 −1.59306
\(491\) 12.4536 0.562025 0.281012 0.959704i \(-0.409330\pi\)
0.281012 + 0.959704i \(0.409330\pi\)
\(492\) 10.9806 0.495045
\(493\) −27.9798 −1.26015
\(494\) −6.86081 −0.308682
\(495\) −1.32340 −0.0594826
\(496\) −37.2638 −1.67320
\(497\) 17.1392 0.768798
\(498\) 24.3386 1.09064
\(499\) 33.0755 1.48066 0.740331 0.672243i \(-0.234669\pi\)
0.740331 + 0.672243i \(0.234669\pi\)
\(500\) −5.07962 −0.227168
\(501\) −6.49239 −0.290058
\(502\) 4.86977 0.217348
\(503\) 12.5824 0.561022 0.280511 0.959851i \(-0.409496\pi\)
0.280511 + 0.959851i \(0.409496\pi\)
\(504\) 1.13919 0.0507437
\(505\) 0.925197 0.0411707
\(506\) −2.41489 −0.107355
\(507\) 7.24860 0.321922
\(508\) 6.43281 0.285410
\(509\) 28.0465 1.24314 0.621569 0.783360i \(-0.286496\pi\)
0.621569 + 0.783360i \(0.286496\pi\)
\(510\) −43.9661 −1.94685
\(511\) 5.39069 0.238470
\(512\) −23.4793 −1.03765
\(513\) −1.53740 −0.0678779
\(514\) −37.9404 −1.67348
\(515\) 49.9017 2.19893
\(516\) 14.1752 0.624030
\(517\) 3.46886 0.152560
\(518\) −2.66888 −0.117264
\(519\) 20.5437 0.901767
\(520\) −7.97021 −0.349517
\(521\) −37.7956 −1.65586 −0.827928 0.560834i \(-0.810481\pi\)
−0.827928 + 0.560834i \(0.810481\pi\)
\(522\) 7.32340 0.320537
\(523\) −17.1946 −0.751868 −0.375934 0.926646i \(-0.622678\pi\)
−0.375934 + 0.926646i \(0.622678\pi\)
\(524\) 25.8325 1.12850
\(525\) −6.88645 −0.300549
\(526\) 44.3899 1.93549
\(527\) 55.3539 2.41125
\(528\) 1.90582 0.0829402
\(529\) −12.3788 −0.538210
\(530\) 66.5831 2.89218
\(531\) 1.00000 0.0433963
\(532\) 2.56159 0.111059
\(533\) −18.0048 −0.779875
\(534\) −30.5783 −1.32325
\(535\) −3.88500 −0.167963
\(536\) 5.45219 0.235499
\(537\) 23.9702 1.03439
\(538\) 52.6606 2.27036
\(539\) −2.27067 −0.0978048
\(540\) 4.86081 0.209176
\(541\) −3.94939 −0.169797 −0.0848987 0.996390i \(-0.527057\pi\)
−0.0848987 + 0.996390i \(0.527057\pi\)
\(542\) 43.5679 1.87140
\(543\) −14.7112 −0.631318
\(544\) 49.0963 2.10499
\(545\) 16.2534 0.696220
\(546\) 5.08377 0.217565
\(547\) 17.1094 0.731545 0.365773 0.930704i \(-0.380805\pi\)
0.365773 + 0.930704i \(0.380805\pi\)
\(548\) −15.7473 −0.672689
\(549\) 0.989588 0.0422346
\(550\) −4.47928 −0.190997
\(551\) −6.05061 −0.257765
\(552\) −3.25901 −0.138713
\(553\) 8.02564 0.341285
\(554\) −29.1994 −1.24057
\(555\) 4.18421 0.177610
\(556\) 14.3636 0.609152
\(557\) 3.29362 0.139555 0.0697775 0.997563i \(-0.477771\pi\)
0.0697775 + 0.997563i \(0.477771\pi\)
\(558\) −14.4882 −0.613336
\(559\) −23.2430 −0.983074
\(560\) 18.1198 0.765702
\(561\) −2.83102 −0.119526
\(562\) 10.4793 0.442042
\(563\) 3.10459 0.130843 0.0654214 0.997858i \(-0.479161\pi\)
0.0654214 + 0.997858i \(0.479161\pi\)
\(564\) −12.7410 −0.536492
\(565\) 51.7223 2.17597
\(566\) −20.1544 −0.847154
\(567\) 1.13919 0.0478417
\(568\) 15.0450 0.631275
\(569\) 14.4024 0.603778 0.301889 0.953343i \(-0.402383\pi\)
0.301889 + 0.953343i \(0.402383\pi\)
\(570\) −9.50761 −0.398230
\(571\) −34.1198 −1.42787 −0.713935 0.700212i \(-0.753089\pi\)
−0.713935 + 0.700212i \(0.753089\pi\)
\(572\) 1.39676 0.0584014
\(573\) −2.91623 −0.121827
\(574\) 15.9148 0.664270
\(575\) 19.7008 0.821580
\(576\) −3.27839 −0.136600
\(577\) −43.6143 −1.81569 −0.907844 0.419308i \(-0.862273\pi\)
−0.907844 + 0.419308i \(0.862273\pi\)
\(578\) −62.4183 −2.59626
\(579\) 0.646809 0.0268804
\(580\) 19.1302 0.794340
\(581\) 14.9002 0.618166
\(582\) 27.4778 1.13899
\(583\) 4.28735 0.177564
\(584\) 4.73202 0.195813
\(585\) −7.97021 −0.329528
\(586\) −63.2597 −2.61323
\(587\) 10.6170 0.438211 0.219106 0.975701i \(-0.429686\pi\)
0.219106 + 0.975701i \(0.429686\pi\)
\(588\) 8.34008 0.343939
\(589\) 11.9702 0.493224
\(590\) 6.18421 0.254600
\(591\) 5.87122 0.241510
\(592\) −6.02564 −0.247652
\(593\) −0.621168 −0.0255083 −0.0127541 0.999919i \(-0.504060\pi\)
−0.0127541 + 0.999919i \(0.504060\pi\)
\(594\) 0.740987 0.0304031
\(595\) −26.9162 −1.10346
\(596\) −7.34133 −0.300713
\(597\) −11.1004 −0.454311
\(598\) −14.5437 −0.594735
\(599\) 3.01523 0.123199 0.0615995 0.998101i \(-0.480380\pi\)
0.0615995 + 0.998101i \(0.480380\pi\)
\(600\) −6.04502 −0.246787
\(601\) −6.24378 −0.254689 −0.127345 0.991859i \(-0.540645\pi\)
−0.127345 + 0.991859i \(0.540645\pi\)
\(602\) 20.5449 0.837348
\(603\) 5.45219 0.222030
\(604\) 10.7618 0.437892
\(605\) 36.0305 1.46485
\(606\) −0.518027 −0.0210434
\(607\) −0.526989 −0.0213898 −0.0106949 0.999943i \(-0.503404\pi\)
−0.0106949 + 0.999943i \(0.503404\pi\)
\(608\) 10.6170 0.430577
\(609\) 4.48342 0.181677
\(610\) 6.11982 0.247784
\(611\) 20.8913 0.845170
\(612\) 10.3982 0.420323
\(613\) 44.6564 1.80366 0.901828 0.432094i \(-0.142225\pi\)
0.901828 + 0.432094i \(0.142225\pi\)
\(614\) 7.92250 0.319726
\(615\) −24.9508 −1.00611
\(616\) 0.453636 0.0182775
\(617\) −44.5831 −1.79485 −0.897424 0.441170i \(-0.854564\pi\)
−0.897424 + 0.441170i \(0.854564\pi\)
\(618\) −27.9404 −1.12393
\(619\) −39.6531 −1.59379 −0.796896 0.604117i \(-0.793526\pi\)
−0.796896 + 0.604117i \(0.793526\pi\)
\(620\) −37.8462 −1.51994
\(621\) −3.25901 −0.130780
\(622\) −40.0769 −1.60694
\(623\) −18.7202 −0.750007
\(624\) 11.4778 0.459481
\(625\) −18.6829 −0.747314
\(626\) 7.37883 0.294917
\(627\) −0.612205 −0.0244491
\(628\) −2.53114 −0.101003
\(629\) 8.95084 0.356893
\(630\) 7.04502 0.280680
\(631\) −6.30818 −0.251125 −0.125562 0.992086i \(-0.540073\pi\)
−0.125562 + 0.992086i \(0.540073\pi\)
\(632\) 7.04502 0.280236
\(633\) 13.4778 0.535696
\(634\) −14.1067 −0.560249
\(635\) −14.6170 −0.580059
\(636\) −15.7473 −0.624419
\(637\) −13.6751 −0.541829
\(638\) 2.91623 0.115455
\(639\) 15.0450 0.595172
\(640\) 25.6274 1.01301
\(641\) 16.5062 0.651954 0.325977 0.945378i \(-0.394307\pi\)
0.325977 + 0.945378i \(0.394307\pi\)
\(642\) 2.17525 0.0858502
\(643\) −22.1455 −0.873332 −0.436666 0.899624i \(-0.643841\pi\)
−0.436666 + 0.899624i \(0.643841\pi\)
\(644\) 5.43011 0.213976
\(645\) −32.2099 −1.26826
\(646\) −20.3386 −0.800213
\(647\) −12.4376 −0.488974 −0.244487 0.969653i \(-0.578619\pi\)
−0.244487 + 0.969653i \(0.578619\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0.398207 0.0156310
\(650\) −26.9765 −1.05810
\(651\) −8.86977 −0.347634
\(652\) −12.5526 −0.491599
\(653\) −10.3941 −0.406751 −0.203376 0.979101i \(-0.565191\pi\)
−0.203376 + 0.979101i \(0.565191\pi\)
\(654\) −9.10044 −0.355856
\(655\) −58.6981 −2.29352
\(656\) 35.9315 1.40289
\(657\) 4.73202 0.184614
\(658\) −18.4662 −0.719886
\(659\) −46.5139 −1.81192 −0.905962 0.423360i \(-0.860851\pi\)
−0.905962 + 0.423360i \(0.860851\pi\)
\(660\) 1.93561 0.0753435
\(661\) −31.3788 −1.22050 −0.610248 0.792211i \(-0.708930\pi\)
−0.610248 + 0.792211i \(0.708930\pi\)
\(662\) 1.24234 0.0482847
\(663\) −17.0498 −0.662161
\(664\) 13.0796 0.507588
\(665\) −5.82061 −0.225713
\(666\) −2.34278 −0.0907809
\(667\) −12.8262 −0.496633
\(668\) 9.49575 0.367402
\(669\) −16.7368 −0.647084
\(670\) 33.7175 1.30262
\(671\) 0.394061 0.0152126
\(672\) −7.86707 −0.303479
\(673\) 23.2936 0.897903 0.448951 0.893556i \(-0.351798\pi\)
0.448951 + 0.893556i \(0.351798\pi\)
\(674\) −13.9148 −0.535977
\(675\) −6.04502 −0.232673
\(676\) −10.6018 −0.407761
\(677\) −0.775591 −0.0298084 −0.0149042 0.999889i \(-0.504744\pi\)
−0.0149042 + 0.999889i \(0.504744\pi\)
\(678\) −28.9598 −1.11219
\(679\) 16.8221 0.645571
\(680\) −23.6274 −0.906071
\(681\) 14.0346 0.537807
\(682\) −5.76932 −0.220919
\(683\) −13.5810 −0.519661 −0.259831 0.965654i \(-0.583667\pi\)
−0.259831 + 0.965654i \(0.583667\pi\)
\(684\) 2.24860 0.0859774
\(685\) 35.7819 1.36715
\(686\) 26.9264 1.02806
\(687\) 12.2445 0.467155
\(688\) 46.3851 1.76842
\(689\) 25.8206 0.983687
\(690\) −20.1544 −0.767266
\(691\) 18.6981 0.711309 0.355654 0.934618i \(-0.384258\pi\)
0.355654 + 0.934618i \(0.384258\pi\)
\(692\) −30.0471 −1.14222
\(693\) 0.453636 0.0172322
\(694\) 54.0007 2.04984
\(695\) −32.6378 −1.23802
\(696\) 3.93561 0.149179
\(697\) −53.3747 −2.02171
\(698\) 31.4003 1.18852
\(699\) −1.87122 −0.0707760
\(700\) 10.0721 0.380690
\(701\) −46.7625 −1.76619 −0.883097 0.469190i \(-0.844546\pi\)
−0.883097 + 0.469190i \(0.844546\pi\)
\(702\) 4.46260 0.168430
\(703\) 1.93561 0.0730029
\(704\) −1.30548 −0.0492021
\(705\) 28.9508 1.09035
\(706\) 5.54636 0.208740
\(707\) −0.317138 −0.0119272
\(708\) −1.46260 −0.0549678
\(709\) 2.98477 0.112095 0.0560477 0.998428i \(-0.482150\pi\)
0.0560477 + 0.998428i \(0.482150\pi\)
\(710\) 93.0415 3.49179
\(711\) 7.04502 0.264209
\(712\) −16.4328 −0.615846
\(713\) 25.3747 0.950289
\(714\) 15.0707 0.564005
\(715\) −3.17380 −0.118693
\(716\) −35.0588 −1.31021
\(717\) 13.9910 0.522505
\(718\) 3.44322 0.128500
\(719\) −33.0915 −1.23410 −0.617052 0.786922i \(-0.711673\pi\)
−0.617052 + 0.786922i \(0.711673\pi\)
\(720\) 15.9058 0.592775
\(721\) −17.1053 −0.637033
\(722\) 30.9571 1.15210
\(723\) −27.5333 −1.02397
\(724\) 21.5166 0.799657
\(725\) −23.7908 −0.883569
\(726\) −20.1738 −0.748720
\(727\) 25.9100 0.960948 0.480474 0.877009i \(-0.340465\pi\)
0.480474 + 0.877009i \(0.340465\pi\)
\(728\) 2.73202 0.101256
\(729\) 1.00000 0.0370370
\(730\) 29.2638 1.08310
\(731\) −68.9031 −2.54847
\(732\) −1.44737 −0.0534963
\(733\) −3.43426 −0.126847 −0.0634237 0.997987i \(-0.520202\pi\)
−0.0634237 + 0.997987i \(0.520202\pi\)
\(734\) −14.8310 −0.547423
\(735\) −18.9508 −0.699012
\(736\) 22.5062 0.829588
\(737\) 2.17110 0.0799735
\(738\) 13.9702 0.514251
\(739\) 19.0014 0.698980 0.349490 0.936940i \(-0.386355\pi\)
0.349490 + 0.936940i \(0.386355\pi\)
\(740\) −6.11982 −0.224969
\(741\) −3.68701 −0.135446
\(742\) −22.8233 −0.837870
\(743\) −13.1607 −0.482819 −0.241409 0.970423i \(-0.577610\pi\)
−0.241409 + 0.970423i \(0.577610\pi\)
\(744\) −7.78600 −0.285449
\(745\) 16.6814 0.611160
\(746\) −2.45364 −0.0898340
\(747\) 13.0796 0.478558
\(748\) 4.14064 0.151397
\(749\) 1.33170 0.0486591
\(750\) −6.46260 −0.235981
\(751\) −14.6981 −0.536341 −0.268170 0.963371i \(-0.586419\pi\)
−0.268170 + 0.963371i \(0.586419\pi\)
\(752\) −41.6918 −1.52034
\(753\) 2.61702 0.0953696
\(754\) 17.5630 0.639608
\(755\) −24.4536 −0.889959
\(756\) −1.66618 −0.0605985
\(757\) 3.27984 0.119208 0.0596039 0.998222i \(-0.481016\pi\)
0.0596039 + 0.998222i \(0.481016\pi\)
\(758\) −38.2922 −1.39083
\(759\) −1.29776 −0.0471058
\(760\) −5.10941 −0.185338
\(761\) −15.5810 −0.564810 −0.282405 0.959295i \(-0.591132\pi\)
−0.282405 + 0.959295i \(0.591132\pi\)
\(762\) 8.18421 0.296483
\(763\) −5.57133 −0.201696
\(764\) 4.26528 0.154312
\(765\) −23.6274 −0.854252
\(766\) −40.7148 −1.47108
\(767\) 2.39821 0.0865943
\(768\) −20.9058 −0.754374
\(769\) 11.0402 0.398120 0.199060 0.979987i \(-0.436211\pi\)
0.199060 + 0.979987i \(0.436211\pi\)
\(770\) 2.80538 0.101099
\(771\) −20.3892 −0.734301
\(772\) −0.946021 −0.0340480
\(773\) 39.8600 1.43367 0.716833 0.697245i \(-0.245591\pi\)
0.716833 + 0.697245i \(0.245591\pi\)
\(774\) 18.0346 0.648240
\(775\) 47.0665 1.69068
\(776\) 14.7666 0.530091
\(777\) −1.43426 −0.0514538
\(778\) 11.7216 0.420240
\(779\) −11.5422 −0.413543
\(780\) 11.6572 0.417395
\(781\) 5.99104 0.214376
\(782\) −43.1142 −1.54176
\(783\) 3.93561 0.140647
\(784\) 27.2909 0.974676
\(785\) 5.75140 0.205276
\(786\) 32.8656 1.17228
\(787\) 24.0692 0.857975 0.428987 0.903311i \(-0.358871\pi\)
0.428987 + 0.903311i \(0.358871\pi\)
\(788\) −8.58723 −0.305908
\(789\) 23.8552 0.849268
\(790\) 43.5679 1.55007
\(791\) −17.7293 −0.630382
\(792\) 0.398207 0.0141497
\(793\) 2.37324 0.0842761
\(794\) 22.5526 0.800363
\(795\) 35.7819 1.26905
\(796\) 16.2355 0.575452
\(797\) 10.3088 0.365158 0.182579 0.983191i \(-0.441555\pi\)
0.182579 + 0.983191i \(0.441555\pi\)
\(798\) 3.25901 0.115368
\(799\) 61.9315 2.19098
\(800\) 41.7458 1.47594
\(801\) −16.4328 −0.580625
\(802\) 30.3941 1.07325
\(803\) 1.88433 0.0664965
\(804\) −7.97436 −0.281234
\(805\) −12.3386 −0.434880
\(806\) −34.7458 −1.22387
\(807\) 28.2999 0.996203
\(808\) −0.278388 −0.00979367
\(809\) 24.7119 0.868823 0.434412 0.900715i \(-0.356956\pi\)
0.434412 + 0.900715i \(0.356956\pi\)
\(810\) 6.18421 0.217291
\(811\) −16.3088 −0.572681 −0.286341 0.958128i \(-0.592439\pi\)
−0.286341 + 0.958128i \(0.592439\pi\)
\(812\) −6.55745 −0.230121
\(813\) 23.4134 0.821145
\(814\) −0.932912 −0.0326986
\(815\) 28.5228 0.999112
\(816\) 34.0256 1.19114
\(817\) −14.9002 −0.521293
\(818\) −68.9736 −2.41160
\(819\) 2.73202 0.0954646
\(820\) 36.4931 1.27439
\(821\) −39.5076 −1.37883 −0.689413 0.724369i \(-0.742131\pi\)
−0.689413 + 0.724369i \(0.742131\pi\)
\(822\) −20.0346 −0.698787
\(823\) −0.655771 −0.0228588 −0.0114294 0.999935i \(-0.503638\pi\)
−0.0114294 + 0.999935i \(0.503638\pi\)
\(824\) −15.0152 −0.523080
\(825\) −2.40717 −0.0838069
\(826\) −2.11982 −0.0737579
\(827\) 33.5035 1.16503 0.582515 0.812820i \(-0.302069\pi\)
0.582515 + 0.812820i \(0.302069\pi\)
\(828\) 4.76663 0.165652
\(829\) 28.1752 0.978567 0.489283 0.872125i \(-0.337258\pi\)
0.489283 + 0.872125i \(0.337258\pi\)
\(830\) 80.8871 2.80763
\(831\) −15.6918 −0.544343
\(832\) −7.86226 −0.272575
\(833\) −40.5395 −1.40461
\(834\) 18.2742 0.632785
\(835\) −21.5768 −0.746697
\(836\) 0.895410 0.0309684
\(837\) −7.78600 −0.269124
\(838\) −27.2549 −0.941504
\(839\) −21.9315 −0.757158 −0.378579 0.925569i \(-0.623587\pi\)
−0.378579 + 0.925569i \(0.623587\pi\)
\(840\) 3.78600 0.130630
\(841\) −13.5110 −0.465896
\(842\) 6.18421 0.213122
\(843\) 5.63158 0.193962
\(844\) −19.7126 −0.678537
\(845\) 24.0900 0.828722
\(846\) −16.2099 −0.557306
\(847\) −12.3505 −0.424368
\(848\) −51.5291 −1.76952
\(849\) −10.8310 −0.371720
\(850\) −79.9709 −2.74298
\(851\) 4.10314 0.140654
\(852\) −22.0048 −0.753873
\(853\) −22.9903 −0.787171 −0.393586 0.919288i \(-0.628766\pi\)
−0.393586 + 0.919288i \(0.628766\pi\)
\(854\) −2.09775 −0.0717834
\(855\) −5.10941 −0.174738
\(856\) 1.16898 0.0399550
\(857\) −16.5574 −0.565592 −0.282796 0.959180i \(-0.591262\pi\)
−0.282796 + 0.959180i \(0.591262\pi\)
\(858\) 1.77704 0.0606671
\(859\) −3.50280 −0.119514 −0.0597570 0.998213i \(-0.519033\pi\)
−0.0597570 + 0.998213i \(0.519033\pi\)
\(860\) 47.1101 1.60644
\(861\) 8.55263 0.291473
\(862\) −5.08377 −0.173154
\(863\) −19.1994 −0.653557 −0.326778 0.945101i \(-0.605963\pi\)
−0.326778 + 0.945101i \(0.605963\pi\)
\(864\) −6.90582 −0.234941
\(865\) 68.2749 2.32142
\(866\) 67.1822 2.28294
\(867\) −33.5437 −1.13920
\(868\) 12.9729 0.440329
\(869\) 2.80538 0.0951659
\(870\) 24.3386 0.825157
\(871\) 13.0755 0.443046
\(872\) −4.89059 −0.165616
\(873\) 14.7666 0.499775
\(874\) −9.32340 −0.315369
\(875\) −3.95643 −0.133752
\(876\) −6.92105 −0.233841
\(877\) 52.8864 1.78585 0.892924 0.450207i \(-0.148650\pi\)
0.892924 + 0.450207i \(0.148650\pi\)
\(878\) 70.6308 2.38367
\(879\) −33.9959 −1.14665
\(880\) 6.33382 0.213513
\(881\) −6.86562 −0.231309 −0.115654 0.993290i \(-0.536896\pi\)
−0.115654 + 0.993290i \(0.536896\pi\)
\(882\) 10.6108 0.357283
\(883\) −51.6233 −1.73726 −0.868631 0.495460i \(-0.835000\pi\)
−0.868631 + 0.495460i \(0.835000\pi\)
\(884\) 24.9371 0.838724
\(885\) 3.32340 0.111715
\(886\) −3.43551 −0.115418
\(887\) 19.7126 0.661886 0.330943 0.943651i \(-0.392633\pi\)
0.330943 + 0.943651i \(0.392633\pi\)
\(888\) −1.25901 −0.0422497
\(889\) 5.01041 0.168044
\(890\) −101.624 −3.40644
\(891\) 0.398207 0.0133405
\(892\) 24.4793 0.819627
\(893\) 13.3926 0.448167
\(894\) −9.34008 −0.312379
\(895\) 79.6627 2.66283
\(896\) −8.78455 −0.293471
\(897\) −7.81579 −0.260962
\(898\) −52.0084 −1.73554
\(899\) −30.6427 −1.02199
\(900\) 8.84143 0.294714
\(901\) 76.5443 2.55006
\(902\) 5.56304 0.185229
\(903\) 11.0409 0.367417
\(904\) −15.5630 −0.517619
\(905\) −48.8913 −1.62520
\(906\) 13.6918 0.454880
\(907\) −40.5366 −1.34600 −0.672998 0.739644i \(-0.734994\pi\)
−0.672998 + 0.739644i \(0.734994\pi\)
\(908\) −20.5270 −0.681212
\(909\) −0.278388 −0.00923356
\(910\) 16.8954 0.560077
\(911\) 54.5949 1.80881 0.904406 0.426674i \(-0.140315\pi\)
0.904406 + 0.426674i \(0.140315\pi\)
\(912\) 7.35801 0.243648
\(913\) 5.20840 0.172373
\(914\) 43.2638 1.43104
\(915\) 3.28880 0.108724
\(916\) −17.9087 −0.591721
\(917\) 20.1205 0.664437
\(918\) 13.2292 0.436630
\(919\) −6.95565 −0.229446 −0.114723 0.993398i \(-0.536598\pi\)
−0.114723 + 0.993398i \(0.536598\pi\)
\(920\) −10.8310 −0.357088
\(921\) 4.25756 0.140292
\(922\) −38.0436 −1.25290
\(923\) 36.0811 1.18762
\(924\) −0.663487 −0.0218271
\(925\) 7.61076 0.250240
\(926\) 23.4689 0.771235
\(927\) −15.0152 −0.493165
\(928\) −27.1786 −0.892182
\(929\) 16.5478 0.542916 0.271458 0.962450i \(-0.412494\pi\)
0.271458 + 0.962450i \(0.412494\pi\)
\(930\) −48.1503 −1.57891
\(931\) −8.76663 −0.287315
\(932\) 2.73684 0.0896482
\(933\) −21.5374 −0.705103
\(934\) 3.52072 0.115202
\(935\) −9.40862 −0.307695
\(936\) 2.39821 0.0783879
\(937\) −28.6766 −0.936824 −0.468412 0.883510i \(-0.655174\pi\)
−0.468412 + 0.883510i \(0.655174\pi\)
\(938\) −11.5576 −0.377371
\(939\) 3.96540 0.129406
\(940\) −42.3434 −1.38109
\(941\) −24.1198 −0.786284 −0.393142 0.919478i \(-0.628612\pi\)
−0.393142 + 0.919478i \(0.628612\pi\)
\(942\) −3.22026 −0.104922
\(943\) −24.4674 −0.796769
\(944\) −4.78600 −0.155771
\(945\) 3.78600 0.123159
\(946\) 7.18151 0.233491
\(947\) 33.0617 1.07436 0.537180 0.843467i \(-0.319489\pi\)
0.537180 + 0.843467i \(0.319489\pi\)
\(948\) −10.3040 −0.334659
\(949\) 11.3484 0.368384
\(950\) −17.2936 −0.561079
\(951\) −7.58097 −0.245830
\(952\) 8.09899 0.262490
\(953\) −6.79641 −0.220157 −0.110079 0.993923i \(-0.535110\pi\)
−0.110079 + 0.993923i \(0.535110\pi\)
\(954\) −20.0346 −0.648644
\(955\) −9.69182 −0.313620
\(956\) −20.4633 −0.661829
\(957\) 1.56719 0.0506600
\(958\) 6.54367 0.211416
\(959\) −12.2653 −0.396067
\(960\) −10.8954 −0.351648
\(961\) 29.6218 0.955543
\(962\) −5.61847 −0.181147
\(963\) 1.16898 0.0376699
\(964\) 40.2701 1.29701
\(965\) 2.14961 0.0691983
\(966\) 6.90852 0.222278
\(967\) −43.7479 −1.40684 −0.703419 0.710775i \(-0.748344\pi\)
−0.703419 + 0.710775i \(0.748344\pi\)
\(968\) −10.8414 −0.348457
\(969\) −10.9300 −0.351123
\(970\) 91.3199 2.93211
\(971\) 30.3434 0.973768 0.486884 0.873467i \(-0.338134\pi\)
0.486884 + 0.873467i \(0.338134\pi\)
\(972\) −1.46260 −0.0469129
\(973\) 11.1876 0.358657
\(974\) −39.4868 −1.26524
\(975\) −14.4972 −0.464282
\(976\) −4.73617 −0.151601
\(977\) 47.2638 1.51210 0.756052 0.654512i \(-0.227126\pi\)
0.756052 + 0.654512i \(0.227126\pi\)
\(978\) −15.9702 −0.510671
\(979\) −6.54367 −0.209137
\(980\) 27.7175 0.885402
\(981\) −4.89059 −0.156145
\(982\) −23.1738 −0.739506
\(983\) 30.9854 0.988282 0.494141 0.869382i \(-0.335483\pi\)
0.494141 + 0.869382i \(0.335483\pi\)
\(984\) 7.50761 0.239334
\(985\) 19.5124 0.621718
\(986\) 52.0651 1.65809
\(987\) −9.92375 −0.315876
\(988\) 5.39261 0.171562
\(989\) −31.5858 −1.00437
\(990\) 2.46260 0.0782665
\(991\) −18.8131 −0.597618 −0.298809 0.954313i \(-0.596589\pi\)
−0.298809 + 0.954313i \(0.596589\pi\)
\(992\) 53.7687 1.70716
\(993\) 0.667633 0.0211867
\(994\) −31.8927 −1.01158
\(995\) −36.8913 −1.16953
\(996\) −19.1302 −0.606165
\(997\) 16.5020 0.522624 0.261312 0.965254i \(-0.415845\pi\)
0.261312 + 0.965254i \(0.415845\pi\)
\(998\) −61.5470 −1.94824
\(999\) −1.25901 −0.0398334
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 177.2.a.d.1.1 3
3.2 odd 2 531.2.a.d.1.3 3
4.3 odd 2 2832.2.a.t.1.1 3
5.4 even 2 4425.2.a.w.1.3 3
7.6 odd 2 8673.2.a.s.1.1 3
12.11 even 2 8496.2.a.bl.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.d.1.1 3 1.1 even 1 trivial
531.2.a.d.1.3 3 3.2 odd 2
2832.2.a.t.1.1 3 4.3 odd 2
4425.2.a.w.1.3 3 5.4 even 2
8496.2.a.bl.1.3 3 12.11 even 2
8673.2.a.s.1.1 3 7.6 odd 2