Properties

Label 6762.2.a.cj.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.23252.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.88474\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.26056 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.26056 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.26056 q^{10} -3.76948 q^{11} -1.00000 q^{12} +2.26056 q^{13} +4.26056 q^{15} +1.00000 q^{16} -0.352835 q^{17} +1.00000 q^{18} -2.00000 q^{19} -4.26056 q^{20} -3.76948 q^{22} -1.00000 q^{23} -1.00000 q^{24} +13.1523 q^{25} +2.26056 q^{26} -1.00000 q^{27} +9.13451 q^{29} +4.26056 q^{30} +7.89179 q^{31} +1.00000 q^{32} +3.76948 q^{33} -0.352835 q^{34} +1.00000 q^{36} +3.15608 q^{37} -2.00000 q^{38} -2.26056 q^{39} -4.26056 q^{40} -5.90772 q^{41} +9.67720 q^{43} -3.76948 q^{44} -4.26056 q^{45} -1.00000 q^{46} -9.27839 q^{47} -1.00000 q^{48} +13.1523 q^{50} +0.352835 q^{51} +2.26056 q^{52} -10.2906 q^{53} -1.00000 q^{54} +16.0601 q^{55} +2.00000 q^{57} +9.13451 q^{58} +9.74790 q^{59} +4.26056 q^{60} -1.12605 q^{61} +7.89179 q^{62} +1.00000 q^{64} -9.63123 q^{65} +3.76948 q^{66} -8.99626 q^{67} -0.352835 q^{68} +1.00000 q^{69} +3.29433 q^{71} +1.00000 q^{72} -3.01784 q^{73} +3.15608 q^{74} -13.1523 q^{75} -2.00000 q^{76} -2.26056 q^{78} +3.01784 q^{79} -4.26056 q^{80} +1.00000 q^{81} -5.90772 q^{82} -7.22678 q^{83} +1.50328 q^{85} +9.67720 q^{86} -9.13451 q^{87} -3.76948 q^{88} -4.35284 q^{89} -4.26056 q^{90} -1.00000 q^{92} -7.89179 q^{93} -9.27839 q^{94} +8.52111 q^{95} -1.00000 q^{96} -16.8173 q^{97} -3.76948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 3q^{5} - 4q^{6} + 4q^{8} + 4q^{9} - 3q^{10} + 2q^{11} - 4q^{12} - 5q^{13} + 3q^{15} + 4q^{16} - 10q^{17} + 4q^{18} - 8q^{19} - 3q^{20} + 2q^{22} - 4q^{23} - 4q^{24} + 13q^{25} - 5q^{26} - 4q^{27} + 3q^{29} + 3q^{30} + 6q^{31} + 4q^{32} - 2q^{33} - 10q^{34} + 4q^{36} + q^{37} - 8q^{38} + 5q^{39} - 3q^{40} - q^{41} - q^{43} + 2q^{44} - 3q^{45} - 4q^{46} - 17q^{47} - 4q^{48} + 13q^{50} + 10q^{51} - 5q^{52} + 4q^{53} - 4q^{54} + 2q^{55} + 8q^{57} + 3q^{58} + 3q^{60} - 24q^{61} + 6q^{62} + 4q^{64} - 27q^{65} - 2q^{66} - 8q^{67} - 10q^{68} + 4q^{69} - 4q^{71} + 4q^{72} - 6q^{73} + q^{74} - 13q^{75} - 8q^{76} + 5q^{78} + 6q^{79} - 3q^{80} + 4q^{81} - q^{82} - 18q^{83} - 16q^{85} - q^{86} - 3q^{87} + 2q^{88} - 26q^{89} - 3q^{90} - 4q^{92} - 6q^{93} - 17q^{94} + 6q^{95} - 4q^{96} - 13q^{97} + 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −4.26056 −1.90538 −0.952689 0.303945i \(-0.901696\pi\)
−0.952689 + 0.303945i \(0.901696\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −4.26056 −1.34731
\(11\) −3.76948 −1.13654 −0.568270 0.822842i \(-0.692387\pi\)
−0.568270 + 0.822842i \(0.692387\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.26056 0.626966 0.313483 0.949594i \(-0.398504\pi\)
0.313483 + 0.949594i \(0.398504\pi\)
\(14\) 0 0
\(15\) 4.26056 1.10007
\(16\) 1.00000 0.250000
\(17\) −0.352835 −0.0855752 −0.0427876 0.999084i \(-0.513624\pi\)
−0.0427876 + 0.999084i \(0.513624\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −4.26056 −0.952689
\(21\) 0 0
\(22\) −3.76948 −0.803655
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 13.1523 2.63047
\(26\) 2.26056 0.443332
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.13451 1.69624 0.848118 0.529808i \(-0.177736\pi\)
0.848118 + 0.529808i \(0.177736\pi\)
\(30\) 4.26056 0.777868
\(31\) 7.89179 1.41741 0.708703 0.705507i \(-0.249280\pi\)
0.708703 + 0.705507i \(0.249280\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.76948 0.656181
\(34\) −0.352835 −0.0605108
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.15608 0.518857 0.259428 0.965762i \(-0.416466\pi\)
0.259428 + 0.965762i \(0.416466\pi\)
\(38\) −2.00000 −0.324443
\(39\) −2.26056 −0.361979
\(40\) −4.26056 −0.673653
\(41\) −5.90772 −0.922631 −0.461316 0.887236i \(-0.652622\pi\)
−0.461316 + 0.887236i \(0.652622\pi\)
\(42\) 0 0
\(43\) 9.67720 1.47576 0.737879 0.674933i \(-0.235827\pi\)
0.737879 + 0.674933i \(0.235827\pi\)
\(44\) −3.76948 −0.568270
\(45\) −4.26056 −0.635126
\(46\) −1.00000 −0.147442
\(47\) −9.27839 −1.35339 −0.676696 0.736262i \(-0.736589\pi\)
−0.676696 + 0.736262i \(0.736589\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 13.1523 1.86002
\(51\) 0.352835 0.0494068
\(52\) 2.26056 0.313483
\(53\) −10.2906 −1.41352 −0.706761 0.707453i \(-0.749844\pi\)
−0.706761 + 0.707453i \(0.749844\pi\)
\(54\) −1.00000 −0.136083
\(55\) 16.0601 2.16554
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 9.13451 1.19942
\(59\) 9.74790 1.26907 0.634534 0.772895i \(-0.281192\pi\)
0.634534 + 0.772895i \(0.281192\pi\)
\(60\) 4.26056 0.550036
\(61\) −1.12605 −0.144176 −0.0720880 0.997398i \(-0.522966\pi\)
−0.0720880 + 0.997398i \(0.522966\pi\)
\(62\) 7.89179 1.00226
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.63123 −1.19461
\(66\) 3.76948 0.463990
\(67\) −8.99626 −1.09907 −0.549534 0.835471i \(-0.685195\pi\)
−0.549534 + 0.835471i \(0.685195\pi\)
\(68\) −0.352835 −0.0427876
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.29433 0.390965 0.195482 0.980707i \(-0.437373\pi\)
0.195482 + 0.980707i \(0.437373\pi\)
\(72\) 1.00000 0.117851
\(73\) −3.01784 −0.353211 −0.176606 0.984282i \(-0.556512\pi\)
−0.176606 + 0.984282i \(0.556512\pi\)
\(74\) 3.15608 0.366887
\(75\) −13.1523 −1.51870
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −2.26056 −0.255958
\(79\) 3.01784 0.339533 0.169767 0.985484i \(-0.445699\pi\)
0.169767 + 0.985484i \(0.445699\pi\)
\(80\) −4.26056 −0.476345
\(81\) 1.00000 0.111111
\(82\) −5.90772 −0.652399
\(83\) −7.22678 −0.793243 −0.396621 0.917982i \(-0.629817\pi\)
−0.396621 + 0.917982i \(0.629817\pi\)
\(84\) 0 0
\(85\) 1.50328 0.163053
\(86\) 9.67720 1.04352
\(87\) −9.13451 −0.979322
\(88\) −3.76948 −0.401827
\(89\) −4.35284 −0.461400 −0.230700 0.973025i \(-0.574102\pi\)
−0.230700 + 0.973025i \(0.574102\pi\)
\(90\) −4.26056 −0.449102
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −7.89179 −0.818340
\(94\) −9.27839 −0.956993
\(95\) 8.52111 0.874248
\(96\) −1.00000 −0.102062
\(97\) −16.8173 −1.70754 −0.853771 0.520648i \(-0.825690\pi\)
−0.853771 + 0.520648i \(0.825690\pi\)
\(98\) 0 0
\(99\) −3.76948 −0.378847
\(100\) 13.1523 1.31523
\(101\) −10.1007 −1.00506 −0.502530 0.864560i \(-0.667597\pi\)
−0.502530 + 0.864560i \(0.667597\pi\)
\(102\) 0.352835 0.0349359
\(103\) −9.90772 −0.976237 −0.488118 0.872777i \(-0.662317\pi\)
−0.488118 + 0.872777i \(0.662317\pi\)
\(104\) 2.26056 0.221666
\(105\) 0 0
\(106\) −10.2906 −0.999510
\(107\) 0.996260 0.0963121 0.0481560 0.998840i \(-0.484666\pi\)
0.0481560 + 0.998840i \(0.484666\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.4007 1.09199 0.545995 0.837789i \(-0.316152\pi\)
0.545995 + 0.837789i \(0.316152\pi\)
\(110\) 16.0601 1.53127
\(111\) −3.15608 −0.299562
\(112\) 0 0
\(113\) 3.90772 0.367607 0.183804 0.982963i \(-0.441159\pi\)
0.183804 + 0.982963i \(0.441159\pi\)
\(114\) 2.00000 0.187317
\(115\) 4.26056 0.397299
\(116\) 9.13451 0.848118
\(117\) 2.26056 0.208989
\(118\) 9.74790 0.897367
\(119\) 0 0
\(120\) 4.26056 0.388934
\(121\) 3.20895 0.291722
\(122\) −1.12605 −0.101948
\(123\) 5.90772 0.532681
\(124\) 7.89179 0.708703
\(125\) −34.7335 −3.10666
\(126\) 0 0
\(127\) −0.680937 −0.0604234 −0.0302117 0.999544i \(-0.509618\pi\)
−0.0302117 + 0.999544i \(0.509618\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.67720 −0.852030
\(130\) −9.63123 −0.844715
\(131\) 15.7836 1.37902 0.689509 0.724277i \(-0.257827\pi\)
0.689509 + 0.724277i \(0.257827\pi\)
\(132\) 3.76948 0.328091
\(133\) 0 0
\(134\) −8.99626 −0.777158
\(135\) 4.26056 0.366690
\(136\) −0.352835 −0.0302554
\(137\) −16.6735 −1.42451 −0.712255 0.701921i \(-0.752326\pi\)
−0.712255 + 0.701921i \(0.752326\pi\)
\(138\) 1.00000 0.0851257
\(139\) 2.09228 0.177465 0.0887324 0.996056i \(-0.471718\pi\)
0.0887324 + 0.996056i \(0.471718\pi\)
\(140\) 0 0
\(141\) 9.27839 0.781382
\(142\) 3.29433 0.276454
\(143\) −8.52111 −0.712571
\(144\) 1.00000 0.0833333
\(145\) −38.9181 −3.23197
\(146\) −3.01784 −0.249758
\(147\) 0 0
\(148\) 3.15608 0.259428
\(149\) −2.50702 −0.205383 −0.102691 0.994713i \(-0.532745\pi\)
−0.102691 + 0.994713i \(0.532745\pi\)
\(150\) −13.1523 −1.07388
\(151\) −14.4288 −1.17420 −0.587101 0.809514i \(-0.699731\pi\)
−0.587101 + 0.809514i \(0.699731\pi\)
\(152\) −2.00000 −0.162221
\(153\) −0.352835 −0.0285251
\(154\) 0 0
\(155\) −33.6234 −2.70070
\(156\) −2.26056 −0.180989
\(157\) 11.3951 0.909425 0.454712 0.890638i \(-0.349742\pi\)
0.454712 + 0.890638i \(0.349742\pi\)
\(158\) 3.01784 0.240086
\(159\) 10.2906 0.816097
\(160\) −4.26056 −0.336827
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −19.7836 −1.54957 −0.774785 0.632225i \(-0.782142\pi\)
−0.774785 + 0.632225i \(0.782142\pi\)
\(164\) −5.90772 −0.461316
\(165\) −16.0601 −1.25027
\(166\) −7.22678 −0.560907
\(167\) −1.85611 −0.143630 −0.0718151 0.997418i \(-0.522879\pi\)
−0.0718151 + 0.997418i \(0.522879\pi\)
\(168\) 0 0
\(169\) −7.88988 −0.606914
\(170\) 1.50328 0.115296
\(171\) −2.00000 −0.152944
\(172\) 9.67720 0.737879
\(173\) 15.9519 1.21280 0.606398 0.795161i \(-0.292614\pi\)
0.606398 + 0.795161i \(0.292614\pi\)
\(174\) −9.13451 −0.692485
\(175\) 0 0
\(176\) −3.76948 −0.284135
\(177\) −9.74790 −0.732697
\(178\) −4.35284 −0.326259
\(179\) 22.6735 1.69469 0.847347 0.531040i \(-0.178199\pi\)
0.847347 + 0.531040i \(0.178199\pi\)
\(180\) −4.26056 −0.317563
\(181\) 18.6894 1.38917 0.694586 0.719410i \(-0.255588\pi\)
0.694586 + 0.719410i \(0.255588\pi\)
\(182\) 0 0
\(183\) 1.12605 0.0832401
\(184\) −1.00000 −0.0737210
\(185\) −13.4467 −0.988619
\(186\) −7.89179 −0.578654
\(187\) 1.33000 0.0972596
\(188\) −9.27839 −0.676696
\(189\) 0 0
\(190\) 8.52111 0.618187
\(191\) −17.7479 −1.28419 −0.642096 0.766624i \(-0.721935\pi\)
−0.642096 + 0.766624i \(0.721935\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.9678 −1.72524 −0.862619 0.505853i \(-0.831178\pi\)
−0.862619 + 0.505853i \(0.831178\pi\)
\(194\) −16.8173 −1.20742
\(195\) 9.63123 0.689707
\(196\) 0 0
\(197\) 1.31906 0.0939794 0.0469897 0.998895i \(-0.485037\pi\)
0.0469897 + 0.998895i \(0.485037\pi\)
\(198\) −3.76948 −0.267885
\(199\) 5.20205 0.368764 0.184382 0.982855i \(-0.440972\pi\)
0.184382 + 0.982855i \(0.440972\pi\)
\(200\) 13.1523 0.930011
\(201\) 8.99626 0.634547
\(202\) −10.1007 −0.710685
\(203\) 0 0
\(204\) 0.352835 0.0247034
\(205\) 25.1702 1.75796
\(206\) −9.90772 −0.690304
\(207\) −1.00000 −0.0695048
\(208\) 2.26056 0.156741
\(209\) 7.53895 0.521480
\(210\) 0 0
\(211\) −0.982162 −0.0676148 −0.0338074 0.999428i \(-0.510763\pi\)
−0.0338074 + 0.999428i \(0.510763\pi\)
\(212\) −10.2906 −0.706761
\(213\) −3.29433 −0.225724
\(214\) 0.996260 0.0681029
\(215\) −41.2302 −2.81188
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 11.4007 0.772153
\(219\) 3.01784 0.203927
\(220\) 16.0601 1.08277
\(221\) −0.797605 −0.0536527
\(222\) −3.15608 −0.211822
\(223\) 29.4551 1.97246 0.986231 0.165376i \(-0.0528838\pi\)
0.986231 + 0.165376i \(0.0528838\pi\)
\(224\) 0 0
\(225\) 13.1523 0.876823
\(226\) 3.90772 0.259938
\(227\) 22.9856 1.52561 0.762805 0.646629i \(-0.223822\pi\)
0.762805 + 0.646629i \(0.223822\pi\)
\(228\) 2.00000 0.132453
\(229\) −9.12605 −0.603066 −0.301533 0.953456i \(-0.597498\pi\)
−0.301533 + 0.953456i \(0.597498\pi\)
\(230\) 4.26056 0.280933
\(231\) 0 0
\(232\) 9.13451 0.599710
\(233\) 2.06006 0.134959 0.0674797 0.997721i \(-0.478504\pi\)
0.0674797 + 0.997721i \(0.478504\pi\)
\(234\) 2.26056 0.147777
\(235\) 39.5311 2.57873
\(236\) 9.74790 0.634534
\(237\) −3.01784 −0.196030
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 4.26056 0.275018
\(241\) −13.3385 −0.859206 −0.429603 0.903018i \(-0.641346\pi\)
−0.429603 + 0.903018i \(0.641346\pi\)
\(242\) 3.20895 0.206279
\(243\) −1.00000 −0.0641500
\(244\) −1.12605 −0.0720880
\(245\) 0 0
\(246\) 5.90772 0.376663
\(247\) −4.52111 −0.287672
\(248\) 7.89179 0.501129
\(249\) 7.22678 0.457979
\(250\) −34.7335 −2.19674
\(251\) 14.3613 0.906477 0.453238 0.891389i \(-0.350269\pi\)
0.453238 + 0.891389i \(0.350269\pi\)
\(252\) 0 0
\(253\) 3.76948 0.236985
\(254\) −0.680937 −0.0427258
\(255\) −1.50328 −0.0941388
\(256\) 1.00000 0.0625000
\(257\) −28.5211 −1.77910 −0.889549 0.456840i \(-0.848981\pi\)
−0.889549 + 0.456840i \(0.848981\pi\)
\(258\) −9.67720 −0.602476
\(259\) 0 0
\(260\) −9.63123 −0.597304
\(261\) 9.13451 0.565412
\(262\) 15.7836 0.975112
\(263\) −3.13451 −0.193282 −0.0966409 0.995319i \(-0.530810\pi\)
−0.0966409 + 0.995319i \(0.530810\pi\)
\(264\) 3.76948 0.231995
\(265\) 43.8436 2.69329
\(266\) 0 0
\(267\) 4.35284 0.266389
\(268\) −8.99626 −0.549534
\(269\) −10.9415 −0.667115 −0.333557 0.942730i \(-0.608249\pi\)
−0.333557 + 0.942730i \(0.608249\pi\)
\(270\) 4.26056 0.259289
\(271\) −7.12605 −0.432877 −0.216438 0.976296i \(-0.569444\pi\)
−0.216438 + 0.976296i \(0.569444\pi\)
\(272\) −0.352835 −0.0213938
\(273\) 0 0
\(274\) −16.6735 −1.00728
\(275\) −49.5774 −2.98963
\(276\) 1.00000 0.0601929
\(277\) −27.9925 −1.68191 −0.840954 0.541107i \(-0.818005\pi\)
−0.840954 + 0.541107i \(0.818005\pi\)
\(278\) 2.09228 0.125487
\(279\) 7.89179 0.472469
\(280\) 0 0
\(281\) −17.9434 −1.07041 −0.535207 0.844721i \(-0.679766\pi\)
−0.535207 + 0.844721i \(0.679766\pi\)
\(282\) 9.27839 0.552520
\(283\) 18.3047 1.08810 0.544050 0.839053i \(-0.316890\pi\)
0.544050 + 0.839053i \(0.316890\pi\)
\(284\) 3.29433 0.195482
\(285\) −8.52111 −0.504747
\(286\) −8.52111 −0.503864
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.8755 −0.992677
\(290\) −38.9181 −2.28535
\(291\) 16.8173 0.985850
\(292\) −3.01784 −0.176606
\(293\) 4.38851 0.256380 0.128190 0.991750i \(-0.459083\pi\)
0.128190 + 0.991750i \(0.459083\pi\)
\(294\) 0 0
\(295\) −41.5315 −2.41806
\(296\) 3.15608 0.183444
\(297\) 3.76948 0.218727
\(298\) −2.50702 −0.145228
\(299\) −2.26056 −0.130731
\(300\) −13.1523 −0.759351
\(301\) 0 0
\(302\) −14.4288 −0.830286
\(303\) 10.1007 0.580272
\(304\) −2.00000 −0.114708
\(305\) 4.79760 0.274710
\(306\) −0.352835 −0.0201703
\(307\) −23.9603 −1.36749 −0.683743 0.729722i \(-0.739649\pi\)
−0.683743 + 0.729722i \(0.739649\pi\)
\(308\) 0 0
\(309\) 9.90772 0.563631
\(310\) −33.6234 −1.90968
\(311\) −11.4307 −0.648178 −0.324089 0.946027i \(-0.605058\pi\)
−0.324089 + 0.946027i \(0.605058\pi\)
\(312\) −2.26056 −0.127979
\(313\) 11.9519 0.675559 0.337779 0.941225i \(-0.390324\pi\)
0.337779 + 0.941225i \(0.390324\pi\)
\(314\) 11.3951 0.643061
\(315\) 0 0
\(316\) 3.01784 0.169767
\(317\) −17.1345 −0.962370 −0.481185 0.876619i \(-0.659793\pi\)
−0.481185 + 0.876619i \(0.659793\pi\)
\(318\) 10.2906 0.577068
\(319\) −34.4323 −1.92784
\(320\) −4.26056 −0.238172
\(321\) −0.996260 −0.0556058
\(322\) 0 0
\(323\) 0.705671 0.0392646
\(324\) 1.00000 0.0555556
\(325\) 29.7316 1.64921
\(326\) −19.7836 −1.09571
\(327\) −11.4007 −0.630460
\(328\) −5.90772 −0.326199
\(329\) 0 0
\(330\) −16.0601 −0.884077
\(331\) −5.04223 −0.277146 −0.138573 0.990352i \(-0.544252\pi\)
−0.138573 + 0.990352i \(0.544252\pi\)
\(332\) −7.22678 −0.396621
\(333\) 3.15608 0.172952
\(334\) −1.85611 −0.101562
\(335\) 38.3291 2.09414
\(336\) 0 0
\(337\) −12.4536 −0.678389 −0.339195 0.940716i \(-0.610154\pi\)
−0.339195 + 0.940716i \(0.610154\pi\)
\(338\) −7.88988 −0.429153
\(339\) −3.90772 −0.212238
\(340\) 1.50328 0.0815266
\(341\) −29.7479 −1.61094
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 9.67720 0.521759
\(345\) −4.26056 −0.229381
\(346\) 15.9519 0.857577
\(347\) 14.4288 0.774580 0.387290 0.921958i \(-0.373411\pi\)
0.387290 + 0.921958i \(0.373411\pi\)
\(348\) −9.13451 −0.489661
\(349\) 3.12605 0.167334 0.0836668 0.996494i \(-0.473337\pi\)
0.0836668 + 0.996494i \(0.473337\pi\)
\(350\) 0 0
\(351\) −2.26056 −0.120660
\(352\) −3.76948 −0.200914
\(353\) 5.44667 0.289897 0.144949 0.989439i \(-0.453698\pi\)
0.144949 + 0.989439i \(0.453698\pi\)
\(354\) −9.74790 −0.518095
\(355\) −14.0357 −0.744936
\(356\) −4.35284 −0.230700
\(357\) 0 0
\(358\) 22.6735 1.19833
\(359\) −21.2021 −1.11900 −0.559501 0.828830i \(-0.689007\pi\)
−0.559501 + 0.828830i \(0.689007\pi\)
\(360\) −4.26056 −0.224551
\(361\) −15.0000 −0.789474
\(362\) 18.6894 0.982293
\(363\) −3.20895 −0.168426
\(364\) 0 0
\(365\) 12.8577 0.673001
\(366\) 1.12605 0.0588596
\(367\) −20.8824 −1.09005 −0.545026 0.838419i \(-0.683480\pi\)
−0.545026 + 0.838419i \(0.683480\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −5.90772 −0.307544
\(370\) −13.4467 −0.699059
\(371\) 0 0
\(372\) −7.89179 −0.409170
\(373\) 1.70941 0.0885099 0.0442550 0.999020i \(-0.485909\pi\)
0.0442550 + 0.999020i \(0.485909\pi\)
\(374\) 1.33000 0.0687729
\(375\) 34.7335 1.79363
\(376\) −9.27839 −0.478497
\(377\) 20.6491 1.06348
\(378\) 0 0
\(379\) −2.13825 −0.109834 −0.0549171 0.998491i \(-0.517489\pi\)
−0.0549171 + 0.998491i \(0.517489\pi\)
\(380\) 8.52111 0.437124
\(381\) 0.680937 0.0348855
\(382\) −17.7479 −0.908061
\(383\) −30.2690 −1.54667 −0.773337 0.633995i \(-0.781414\pi\)
−0.773337 + 0.633995i \(0.781414\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −23.9678 −1.21993
\(387\) 9.67720 0.491920
\(388\) −16.8173 −0.853771
\(389\) 28.2831 1.43401 0.717005 0.697068i \(-0.245512\pi\)
0.717005 + 0.697068i \(0.245512\pi\)
\(390\) 9.63123 0.487696
\(391\) 0.352835 0.0178437
\(392\) 0 0
\(393\) −15.7836 −0.796176
\(394\) 1.31906 0.0664534
\(395\) −12.8577 −0.646940
\(396\) −3.76948 −0.189423
\(397\) 6.20395 0.311368 0.155684 0.987807i \(-0.450242\pi\)
0.155684 + 0.987807i \(0.450242\pi\)
\(398\) 5.20205 0.260755
\(399\) 0 0
\(400\) 13.1523 0.657617
\(401\) −31.9925 −1.59763 −0.798815 0.601577i \(-0.794539\pi\)
−0.798815 + 0.601577i \(0.794539\pi\)
\(402\) 8.99626 0.448693
\(403\) 17.8398 0.888665
\(404\) −10.1007 −0.502530
\(405\) −4.26056 −0.211709
\(406\) 0 0
\(407\) −11.8968 −0.589701
\(408\) 0.352835 0.0174680
\(409\) −20.3047 −1.00400 −0.502001 0.864867i \(-0.667403\pi\)
−0.502001 + 0.864867i \(0.667403\pi\)
\(410\) 25.1702 1.24307
\(411\) 16.6735 0.822441
\(412\) −9.90772 −0.488118
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 30.7901 1.51143
\(416\) 2.26056 0.110833
\(417\) −2.09228 −0.102459
\(418\) 7.53895 0.368742
\(419\) −14.3047 −0.698830 −0.349415 0.936968i \(-0.613620\pi\)
−0.349415 + 0.936968i \(0.613620\pi\)
\(420\) 0 0
\(421\) −26.1308 −1.27354 −0.636768 0.771056i \(-0.719729\pi\)
−0.636768 + 0.771056i \(0.719729\pi\)
\(422\) −0.982162 −0.0478109
\(423\) −9.27839 −0.451131
\(424\) −10.2906 −0.499755
\(425\) −4.64061 −0.225103
\(426\) −3.29433 −0.159611
\(427\) 0 0
\(428\) 0.996260 0.0481560
\(429\) 8.52111 0.411403
\(430\) −41.2302 −1.98830
\(431\) −16.7091 −0.804851 −0.402425 0.915453i \(-0.631833\pi\)
−0.402425 + 0.915453i \(0.631833\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 10.9981 0.528535 0.264267 0.964449i \(-0.414870\pi\)
0.264267 + 0.964449i \(0.414870\pi\)
\(434\) 0 0
\(435\) 38.9181 1.86598
\(436\) 11.4007 0.545995
\(437\) 2.00000 0.0956730
\(438\) 3.01784 0.144198
\(439\) 4.35284 0.207750 0.103875 0.994590i \(-0.466876\pi\)
0.103875 + 0.994590i \(0.466876\pi\)
\(440\) 16.0601 0.765634
\(441\) 0 0
\(442\) −0.797605 −0.0379382
\(443\) −15.8402 −0.752589 −0.376295 0.926500i \(-0.622802\pi\)
−0.376295 + 0.926500i \(0.622802\pi\)
\(444\) −3.15608 −0.149781
\(445\) 18.5455 0.879141
\(446\) 29.4551 1.39474
\(447\) 2.50702 0.118578
\(448\) 0 0
\(449\) 13.0779 0.617184 0.308592 0.951194i \(-0.400142\pi\)
0.308592 + 0.951194i \(0.400142\pi\)
\(450\) 13.1523 0.620007
\(451\) 22.2690 1.04861
\(452\) 3.90772 0.183804
\(453\) 14.4288 0.677926
\(454\) 22.9856 1.07877
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 39.3187 1.83925 0.919626 0.392795i \(-0.128492\pi\)
0.919626 + 0.392795i \(0.128492\pi\)
\(458\) −9.12605 −0.426432
\(459\) 0.352835 0.0164689
\(460\) 4.26056 0.198649
\(461\) 13.6115 0.633950 0.316975 0.948434i \(-0.397333\pi\)
0.316975 + 0.948434i \(0.397333\pi\)
\(462\) 0 0
\(463\) −3.45415 −0.160528 −0.0802640 0.996774i \(-0.525576\pi\)
−0.0802640 + 0.996774i \(0.525576\pi\)
\(464\) 9.13451 0.424059
\(465\) 33.6234 1.55925
\(466\) 2.06006 0.0954307
\(467\) −3.41848 −0.158188 −0.0790941 0.996867i \(-0.525203\pi\)
−0.0790941 + 0.996867i \(0.525203\pi\)
\(468\) 2.26056 0.104494
\(469\) 0 0
\(470\) 39.5311 1.82343
\(471\) −11.3951 −0.525057
\(472\) 9.74790 0.448683
\(473\) −36.4780 −1.67726
\(474\) −3.01784 −0.138614
\(475\) −26.3047 −1.20694
\(476\) 0 0
\(477\) −10.2906 −0.471174
\(478\) −8.00000 −0.365911
\(479\) −10.5568 −0.482352 −0.241176 0.970481i \(-0.577533\pi\)
−0.241176 + 0.970481i \(0.577533\pi\)
\(480\) 4.26056 0.194467
\(481\) 7.13451 0.325305
\(482\) −13.3385 −0.607550
\(483\) 0 0
\(484\) 3.20895 0.145861
\(485\) 71.6513 3.25352
\(486\) −1.00000 −0.0453609
\(487\) 34.7017 1.57248 0.786241 0.617920i \(-0.212024\pi\)
0.786241 + 0.617920i \(0.212024\pi\)
\(488\) −1.12605 −0.0509739
\(489\) 19.7836 0.894644
\(490\) 0 0
\(491\) −30.3009 −1.36746 −0.683730 0.729735i \(-0.739643\pi\)
−0.683730 + 0.729735i \(0.739643\pi\)
\(492\) 5.90772 0.266341
\(493\) −3.22298 −0.145156
\(494\) −4.52111 −0.203415
\(495\) 16.0601 0.721846
\(496\) 7.89179 0.354352
\(497\) 0 0
\(498\) 7.22678 0.323840
\(499\) −10.9747 −0.491294 −0.245647 0.969359i \(-0.579000\pi\)
−0.245647 + 0.969359i \(0.579000\pi\)
\(500\) −34.7335 −1.55333
\(501\) 1.85611 0.0829249
\(502\) 14.3613 0.640976
\(503\) −20.8258 −0.928577 −0.464288 0.885684i \(-0.653690\pi\)
−0.464288 + 0.885684i \(0.653690\pi\)
\(504\) 0 0
\(505\) 43.0347 1.91502
\(506\) 3.76948 0.167574
\(507\) 7.88988 0.350402
\(508\) −0.680937 −0.0302117
\(509\) −31.2462 −1.38496 −0.692481 0.721436i \(-0.743482\pi\)
−0.692481 + 0.721436i \(0.743482\pi\)
\(510\) −1.50328 −0.0665662
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −28.5211 −1.25801
\(515\) 42.2124 1.86010
\(516\) −9.67720 −0.426015
\(517\) 34.9747 1.53818
\(518\) 0 0
\(519\) −15.9519 −0.700209
\(520\) −9.63123 −0.422357
\(521\) 39.1111 1.71349 0.856744 0.515742i \(-0.172484\pi\)
0.856744 + 0.515742i \(0.172484\pi\)
\(522\) 9.13451 0.399806
\(523\) 37.0779 1.62130 0.810652 0.585529i \(-0.199113\pi\)
0.810652 + 0.585529i \(0.199113\pi\)
\(524\) 15.7836 0.689509
\(525\) 0 0
\(526\) −3.13451 −0.136671
\(527\) −2.78450 −0.121295
\(528\) 3.76948 0.164045
\(529\) 1.00000 0.0434783
\(530\) 43.8436 1.90445
\(531\) 9.74790 0.423023
\(532\) 0 0
\(533\) −13.3547 −0.578458
\(534\) 4.35284 0.188366
\(535\) −4.24462 −0.183511
\(536\) −8.99626 −0.388579
\(537\) −22.6735 −0.978432
\(538\) −10.9415 −0.471721
\(539\) 0 0
\(540\) 4.26056 0.183345
\(541\) 34.2765 1.47366 0.736831 0.676077i \(-0.236321\pi\)
0.736831 + 0.676077i \(0.236321\pi\)
\(542\) −7.12605 −0.306090
\(543\) −18.6894 −0.802039
\(544\) −0.352835 −0.0151277
\(545\) −48.5734 −2.08065
\(546\) 0 0
\(547\) −33.3469 −1.42581 −0.712906 0.701260i \(-0.752621\pi\)
−0.712906 + 0.701260i \(0.752621\pi\)
\(548\) −16.6735 −0.712255
\(549\) −1.12605 −0.0480587
\(550\) −49.5774 −2.11399
\(551\) −18.2690 −0.778286
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −27.9925 −1.18929
\(555\) 13.4467 0.570779
\(556\) 2.09228 0.0887324
\(557\) −13.3084 −0.563896 −0.281948 0.959430i \(-0.590981\pi\)
−0.281948 + 0.959430i \(0.590981\pi\)
\(558\) 7.89179 0.334086
\(559\) 21.8759 0.925250
\(560\) 0 0
\(561\) −1.33000 −0.0561528
\(562\) −17.9434 −0.756897
\(563\) 41.5743 1.75215 0.876074 0.482178i \(-0.160154\pi\)
0.876074 + 0.482178i \(0.160154\pi\)
\(564\) 9.27839 0.390691
\(565\) −16.6491 −0.700432
\(566\) 18.3047 0.769403
\(567\) 0 0
\(568\) 3.29433 0.138227
\(569\) 34.1336 1.43095 0.715477 0.698636i \(-0.246209\pi\)
0.715477 + 0.698636i \(0.246209\pi\)
\(570\) −8.52111 −0.356910
\(571\) −18.0385 −0.754887 −0.377444 0.926033i \(-0.623197\pi\)
−0.377444 + 0.926033i \(0.623197\pi\)
\(572\) −8.52111 −0.356286
\(573\) 17.7479 0.741429
\(574\) 0 0
\(575\) −13.1523 −0.548491
\(576\) 1.00000 0.0416667
\(577\) 3.01784 0.125634 0.0628171 0.998025i \(-0.479992\pi\)
0.0628171 + 0.998025i \(0.479992\pi\)
\(578\) −16.8755 −0.701929
\(579\) 23.9678 0.996067
\(580\) −38.9181 −1.61599
\(581\) 0 0
\(582\) 16.8173 0.697101
\(583\) 38.7901 1.60652
\(584\) −3.01784 −0.124879
\(585\) −9.63123 −0.398202
\(586\) 4.38851 0.181288
\(587\) 28.8577 1.19108 0.595542 0.803324i \(-0.296937\pi\)
0.595542 + 0.803324i \(0.296937\pi\)
\(588\) 0 0
\(589\) −15.7836 −0.650351
\(590\) −41.5315 −1.70982
\(591\) −1.31906 −0.0542590
\(592\) 3.15608 0.129714
\(593\) −8.02473 −0.329536 −0.164768 0.986332i \(-0.552688\pi\)
−0.164768 + 0.986332i \(0.552688\pi\)
\(594\) 3.76948 0.154663
\(595\) 0 0
\(596\) −2.50702 −0.102691
\(597\) −5.20205 −0.212906
\(598\) −2.26056 −0.0924410
\(599\) −3.81544 −0.155895 −0.0779474 0.996957i \(-0.524837\pi\)
−0.0779474 + 0.996957i \(0.524837\pi\)
\(600\) −13.1523 −0.536942
\(601\) 4.37971 0.178652 0.0893261 0.996002i \(-0.471529\pi\)
0.0893261 + 0.996002i \(0.471529\pi\)
\(602\) 0 0
\(603\) −8.99626 −0.366356
\(604\) −14.4288 −0.587101
\(605\) −13.6719 −0.555842
\(606\) 10.1007 0.410314
\(607\) −27.4908 −1.11582 −0.557909 0.829902i \(-0.688396\pi\)
−0.557909 + 0.829902i \(0.688396\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 4.79760 0.194249
\(611\) −20.9743 −0.848531
\(612\) −0.352835 −0.0142625
\(613\) −22.4504 −0.906764 −0.453382 0.891316i \(-0.649783\pi\)
−0.453382 + 0.891316i \(0.649783\pi\)
\(614\) −23.9603 −0.966959
\(615\) −25.1702 −1.01496
\(616\) 0 0
\(617\) 26.6094 1.07125 0.535627 0.844455i \(-0.320075\pi\)
0.535627 + 0.844455i \(0.320075\pi\)
\(618\) 9.90772 0.398547
\(619\) −41.9356 −1.68553 −0.842766 0.538279i \(-0.819075\pi\)
−0.842766 + 0.538279i \(0.819075\pi\)
\(620\) −33.6234 −1.35035
\(621\) 1.00000 0.0401286
\(622\) −11.4307 −0.458331
\(623\) 0 0
\(624\) −2.26056 −0.0904947
\(625\) 82.2224 3.28890
\(626\) 11.9519 0.477692
\(627\) −7.53895 −0.301077
\(628\) 11.3951 0.454712
\(629\) −1.11358 −0.0444013
\(630\) 0 0
\(631\) 21.5633 0.858423 0.429212 0.903204i \(-0.358791\pi\)
0.429212 + 0.903204i \(0.358791\pi\)
\(632\) 3.01784 0.120043
\(633\) 0.982162 0.0390374
\(634\) −17.1345 −0.680498
\(635\) 2.90117 0.115129
\(636\) 10.2906 0.408048
\(637\) 0 0
\(638\) −34.4323 −1.36319
\(639\) 3.29433 0.130322
\(640\) −4.26056 −0.168413
\(641\) 26.7091 1.05495 0.527474 0.849571i \(-0.323139\pi\)
0.527474 + 0.849571i \(0.323139\pi\)
\(642\) −0.996260 −0.0393192
\(643\) −22.1201 −0.872333 −0.436166 0.899866i \(-0.643664\pi\)
−0.436166 + 0.899866i \(0.643664\pi\)
\(644\) 0 0
\(645\) 41.2302 1.62344
\(646\) 0.705671 0.0277643
\(647\) 13.7642 0.541126 0.270563 0.962702i \(-0.412790\pi\)
0.270563 + 0.962702i \(0.412790\pi\)
\(648\) 1.00000 0.0392837
\(649\) −36.7445 −1.44235
\(650\) 29.7316 1.16617
\(651\) 0 0
\(652\) −19.7836 −0.774785
\(653\) 10.0848 0.394649 0.197324 0.980338i \(-0.436775\pi\)
0.197324 + 0.980338i \(0.436775\pi\)
\(654\) −11.4007 −0.445803
\(655\) −67.2468 −2.62755
\(656\) −5.90772 −0.230658
\(657\) −3.01784 −0.117737
\(658\) 0 0
\(659\) 7.72632 0.300975 0.150487 0.988612i \(-0.451916\pi\)
0.150487 + 0.988612i \(0.451916\pi\)
\(660\) −16.0601 −0.625137
\(661\) −12.9415 −0.503366 −0.251683 0.967810i \(-0.580984\pi\)
−0.251683 + 0.967810i \(0.580984\pi\)
\(662\) −5.04223 −0.195972
\(663\) 0.797605 0.0309764
\(664\) −7.22678 −0.280454
\(665\) 0 0
\(666\) 3.15608 0.122296
\(667\) −9.13451 −0.353689
\(668\) −1.85611 −0.0718151
\(669\) −29.4551 −1.13880
\(670\) 38.3291 1.48078
\(671\) 4.24462 0.163862
\(672\) 0 0
\(673\) 51.5236 1.98609 0.993045 0.117733i \(-0.0375627\pi\)
0.993045 + 0.117733i \(0.0375627\pi\)
\(674\) −12.4536 −0.479694
\(675\) −13.1523 −0.506234
\(676\) −7.88988 −0.303457
\(677\) −0.317160 −0.0121894 −0.00609472 0.999981i \(-0.501940\pi\)
−0.00609472 + 0.999981i \(0.501940\pi\)
\(678\) −3.90772 −0.150075
\(679\) 0 0
\(680\) 1.50328 0.0576480
\(681\) −22.9856 −0.880811
\(682\) −29.7479 −1.13911
\(683\) −24.8690 −0.951584 −0.475792 0.879558i \(-0.657839\pi\)
−0.475792 + 0.879558i \(0.657839\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 71.0382 2.71423
\(686\) 0 0
\(687\) 9.12605 0.348181
\(688\) 9.67720 0.368940
\(689\) −23.2625 −0.886229
\(690\) −4.26056 −0.162197
\(691\) −35.7439 −1.35976 −0.679881 0.733323i \(-0.737968\pi\)
−0.679881 + 0.733323i \(0.737968\pi\)
\(692\) 15.9519 0.606398
\(693\) 0 0
\(694\) 14.4288 0.547711
\(695\) −8.91427 −0.338138
\(696\) −9.13451 −0.346243
\(697\) 2.08445 0.0789543
\(698\) 3.12605 0.118323
\(699\) −2.06006 −0.0779188
\(700\) 0 0
\(701\) −33.8896 −1.27999 −0.639996 0.768378i \(-0.721064\pi\)
−0.639996 + 0.768378i \(0.721064\pi\)
\(702\) −2.26056 −0.0853192
\(703\) −6.31217 −0.238068
\(704\) −3.76948 −0.142067
\(705\) −39.5311 −1.48883
\(706\) 5.44667 0.204988
\(707\) 0 0
\(708\) −9.74790 −0.366348
\(709\) 48.3750 1.81676 0.908381 0.418143i \(-0.137319\pi\)
0.908381 + 0.418143i \(0.137319\pi\)
\(710\) −14.0357 −0.526750
\(711\) 3.01784 0.113178
\(712\) −4.35284 −0.163129
\(713\) −7.89179 −0.295550
\(714\) 0 0
\(715\) 36.3047 1.35772
\(716\) 22.6735 0.847347
\(717\) 8.00000 0.298765
\(718\) −21.2021 −0.791253
\(719\) 35.3028 1.31657 0.658286 0.752768i \(-0.271282\pi\)
0.658286 + 0.752768i \(0.271282\pi\)
\(720\) −4.26056 −0.158782
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) 13.3385 0.496063
\(724\) 18.6894 0.694586
\(725\) 120.140 4.46189
\(726\) −3.20895 −0.119095
\(727\) −9.25865 −0.343384 −0.171692 0.985151i \(-0.554923\pi\)
−0.171692 + 0.985151i \(0.554923\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.8577 0.475884
\(731\) −3.41446 −0.126288
\(732\) 1.12605 0.0416200
\(733\) 46.2565 1.70852 0.854262 0.519843i \(-0.174009\pi\)
0.854262 + 0.519843i \(0.174009\pi\)
\(734\) −20.8824 −0.770784
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 33.9112 1.24913
\(738\) −5.90772 −0.217466
\(739\) 3.29433 0.121184 0.0605919 0.998163i \(-0.480701\pi\)
0.0605919 + 0.998163i \(0.480701\pi\)
\(740\) −13.4467 −0.494310
\(741\) 4.52111 0.166087
\(742\) 0 0
\(743\) −16.7976 −0.616244 −0.308122 0.951347i \(-0.599701\pi\)
−0.308122 + 0.951347i \(0.599701\pi\)
\(744\) −7.89179 −0.289327
\(745\) 10.6813 0.391332
\(746\) 1.70941 0.0625860
\(747\) −7.22678 −0.264414
\(748\) 1.33000 0.0486298
\(749\) 0 0
\(750\) 34.7335 1.26829
\(751\) −3.35439 −0.122404 −0.0612018 0.998125i \(-0.519493\pi\)
−0.0612018 + 0.998125i \(0.519493\pi\)
\(752\) −9.27839 −0.338348
\(753\) −14.3613 −0.523355
\(754\) 20.6491 0.751995
\(755\) 61.4749 2.23730
\(756\) 0 0
\(757\) −9.03194 −0.328271 −0.164136 0.986438i \(-0.552483\pi\)
−0.164136 + 0.986438i \(0.552483\pi\)
\(758\) −2.13825 −0.0776646
\(759\) −3.76948 −0.136823
\(760\) 8.52111 0.309093
\(761\) 1.97561 0.0716158 0.0358079 0.999359i \(-0.488600\pi\)
0.0358079 + 0.999359i \(0.488600\pi\)
\(762\) 0.680937 0.0246677
\(763\) 0 0
\(764\) −17.7479 −0.642096
\(765\) 1.50328 0.0543510
\(766\) −30.2690 −1.09366
\(767\) 22.0357 0.795662
\(768\) −1.00000 −0.0360844
\(769\) 16.0085 0.577280 0.288640 0.957438i \(-0.406797\pi\)
0.288640 + 0.957438i \(0.406797\pi\)
\(770\) 0 0
\(771\) 28.5211 1.02716
\(772\) −23.9678 −0.862619
\(773\) −38.0085 −1.36707 −0.683535 0.729918i \(-0.739558\pi\)
−0.683535 + 0.729918i \(0.739558\pi\)
\(774\) 9.67720 0.347840
\(775\) 103.795 3.72844
\(776\) −16.8173 −0.603708
\(777\) 0 0
\(778\) 28.2831 1.01400
\(779\) 11.8154 0.423332
\(780\) 9.63123 0.344853
\(781\) −12.4179 −0.444347
\(782\) 0.352835 0.0126174
\(783\) −9.13451 −0.326441
\(784\) 0 0
\(785\) −48.5493 −1.73280
\(786\) −15.7836 −0.562981
\(787\) −55.1624 −1.96633 −0.983163 0.182732i \(-0.941506\pi\)
−0.983163 + 0.182732i \(0.941506\pi\)
\(788\) 1.31906 0.0469897
\(789\) 3.13451 0.111591
\(790\) −12.8577 −0.457455
\(791\) 0 0
\(792\) −3.76948 −0.133942
\(793\) −2.54550 −0.0903934
\(794\) 6.20395 0.220170
\(795\) −43.8436 −1.55497
\(796\) 5.20205 0.184382
\(797\) −18.1435 −0.642677 −0.321339 0.946964i \(-0.604133\pi\)
−0.321339 + 0.946964i \(0.604133\pi\)
\(798\) 0 0
\(799\) 3.27375 0.115817
\(800\) 13.1523 0.465006
\(801\) −4.35284 −0.153800
\(802\) −31.9925 −1.12970
\(803\) 11.3757 0.401439
\(804\) 8.99626 0.317274
\(805\) 0 0
\(806\) 17.8398 0.628381
\(807\) 10.9415 0.385159
\(808\) −10.1007 −0.355343
\(809\) −40.3291 −1.41789 −0.708947 0.705261i \(-0.750830\pi\)
−0.708947 + 0.705261i \(0.750830\pi\)
\(810\) −4.26056 −0.149701
\(811\) −7.10264 −0.249407 −0.124704 0.992194i \(-0.539798\pi\)
−0.124704 + 0.992194i \(0.539798\pi\)
\(812\) 0 0
\(813\) 7.12605 0.249922
\(814\) −11.8968 −0.416982
\(815\) 84.2890 2.95252
\(816\) 0.352835 0.0123517
\(817\) −19.3544 −0.677125
\(818\) −20.3047 −0.709937
\(819\) 0 0
\(820\) 25.1702 0.878981
\(821\) −36.6169 −1.27794 −0.638969 0.769233i \(-0.720639\pi\)
−0.638969 + 0.769233i \(0.720639\pi\)
\(822\) 16.6735 0.581554
\(823\) 49.3222 1.71926 0.859632 0.510914i \(-0.170693\pi\)
0.859632 + 0.510914i \(0.170693\pi\)
\(824\) −9.90772 −0.345152
\(825\) 49.5774 1.72606
\(826\) 0 0
\(827\) 47.8183 1.66280 0.831402 0.555672i \(-0.187539\pi\)
0.831402 + 0.555672i \(0.187539\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −23.6641 −0.821887 −0.410944 0.911661i \(-0.634801\pi\)
−0.410944 + 0.911661i \(0.634801\pi\)
\(830\) 30.7901 1.06874
\(831\) 27.9925 0.971050
\(832\) 2.26056 0.0783707
\(833\) 0 0
\(834\) −2.09228 −0.0724497
\(835\) 7.90807 0.273670
\(836\) 7.53895 0.260740
\(837\) −7.89179 −0.272780
\(838\) −14.3047 −0.494147
\(839\) 23.6309 0.815829 0.407914 0.913020i \(-0.366256\pi\)
0.407914 + 0.913020i \(0.366256\pi\)
\(840\) 0 0
\(841\) 54.4392 1.87721
\(842\) −26.1308 −0.900526
\(843\) 17.9434 0.618003
\(844\) −0.982162 −0.0338074
\(845\) 33.6153 1.15640
\(846\) −9.27839 −0.318998
\(847\) 0 0
\(848\) −10.2906 −0.353380
\(849\) −18.3047 −0.628215
\(850\) −4.64061 −0.159172
\(851\) −3.15608 −0.108189
\(852\) −3.29433 −0.112862
\(853\) −34.0441 −1.16565 −0.582824 0.812598i \(-0.698052\pi\)
−0.582824 + 0.812598i \(0.698052\pi\)
\(854\) 0 0
\(855\) 8.52111 0.291416
\(856\) 0.996260 0.0340515
\(857\) −35.4823 −1.21205 −0.606027 0.795444i \(-0.707238\pi\)
−0.606027 + 0.795444i \(0.707238\pi\)
\(858\) 8.52111 0.290906
\(859\) 32.8824 1.12193 0.560967 0.827838i \(-0.310430\pi\)
0.560967 + 0.827838i \(0.310430\pi\)
\(860\) −41.2302 −1.40594
\(861\) 0 0
\(862\) −16.7091 −0.569115
\(863\) 30.5023 1.03831 0.519156 0.854680i \(-0.326246\pi\)
0.519156 + 0.854680i \(0.326246\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −67.9638 −2.31084
\(866\) 10.9981 0.373731
\(867\) 16.8755 0.573122
\(868\) 0 0
\(869\) −11.3757 −0.385893
\(870\) 38.9181 1.31945
\(871\) −20.3366 −0.689078
\(872\) 11.4007 0.386077
\(873\) −16.8173 −0.569181
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 3.01784 0.101963
\(877\) −31.8793 −1.07649 −0.538244 0.842789i \(-0.680912\pi\)
−0.538244 + 0.842789i \(0.680912\pi\)
\(878\) 4.35284 0.146901
\(879\) −4.38851 −0.148021
\(880\) 16.0601 0.541385
\(881\) −37.4120 −1.26044 −0.630221 0.776416i \(-0.717036\pi\)
−0.630221 + 0.776416i \(0.717036\pi\)
\(882\) 0 0
\(883\) 2.34404 0.0788831 0.0394415 0.999222i \(-0.487442\pi\)
0.0394415 + 0.999222i \(0.487442\pi\)
\(884\) −0.797605 −0.0268263
\(885\) 41.5315 1.39607
\(886\) −15.8402 −0.532161
\(887\) 42.6819 1.43312 0.716559 0.697526i \(-0.245716\pi\)
0.716559 + 0.697526i \(0.245716\pi\)
\(888\) −3.15608 −0.105911
\(889\) 0 0
\(890\) 18.5455 0.621647
\(891\) −3.76948 −0.126282
\(892\) 29.4551 0.986231
\(893\) 18.5568 0.620979
\(894\) 2.50702 0.0838471
\(895\) −96.6015 −3.22903
\(896\) 0 0
\(897\) 2.26056 0.0754778
\(898\) 13.0779 0.436415
\(899\) 72.0876 2.40426
\(900\) 13.1523 0.438411
\(901\) 3.63088 0.120962
\(902\) 22.2690 0.741477
\(903\) 0 0
\(904\) 3.90772 0.129969
\(905\) −79.6272 −2.64690
\(906\) 14.4288 0.479366
\(907\) 46.9322 1.55836 0.779179 0.626802i \(-0.215636\pi\)
0.779179 + 0.626802i \(0.215636\pi\)
\(908\) 22.9856 0.762805
\(909\) −10.1007 −0.335020
\(910\) 0 0
\(911\) 17.5067 0.580024 0.290012 0.957023i \(-0.406341\pi\)
0.290012 + 0.957023i \(0.406341\pi\)
\(912\) 2.00000 0.0662266
\(913\) 27.2412 0.901552
\(914\) 39.3187 1.30055
\(915\) −4.79760 −0.158604
\(916\) −9.12605 −0.301533
\(917\) 0 0
\(918\) 0.352835 0.0116453
\(919\) −24.2446 −0.799756 −0.399878 0.916568i \(-0.630948\pi\)
−0.399878 + 0.916568i \(0.630948\pi\)
\(920\) 4.26056 0.140466
\(921\) 23.9603 0.789519
\(922\) 13.6115 0.448271
\(923\) 7.44702 0.245122
\(924\) 0 0
\(925\) 41.5099 1.36484
\(926\) −3.45415 −0.113510
\(927\) −9.90772 −0.325412
\(928\) 9.13451 0.299855
\(929\) 36.7297 1.20506 0.602531 0.798095i \(-0.294159\pi\)
0.602531 + 0.798095i \(0.294159\pi\)
\(930\) 33.6234 1.10255
\(931\) 0 0
\(932\) 2.06006 0.0674797
\(933\) 11.4307 0.374226
\(934\) −3.41848 −0.111856
\(935\) −5.66656 −0.185316
\(936\) 2.26056 0.0738886
\(937\) −18.3638 −0.599918 −0.299959 0.953952i \(-0.596973\pi\)
−0.299959 + 0.953952i \(0.596973\pi\)
\(938\) 0 0
\(939\) −11.9519 −0.390034
\(940\) 39.5311 1.28936
\(941\) 47.4417 1.54655 0.773277 0.634068i \(-0.218616\pi\)
0.773277 + 0.634068i \(0.218616\pi\)
\(942\) −11.3951 −0.371271
\(943\) 5.90772 0.192382
\(944\) 9.74790 0.317267
\(945\) 0 0
\(946\) −36.4780 −1.18600
\(947\) 3.16637 0.102893 0.0514467 0.998676i \(-0.483617\pi\)
0.0514467 + 0.998676i \(0.483617\pi\)
\(948\) −3.01784 −0.0980148
\(949\) −6.82199 −0.221451
\(950\) −26.3047 −0.853437
\(951\) 17.1345 0.555624
\(952\) 0 0
\(953\) −36.1126 −1.16980 −0.584902 0.811104i \(-0.698867\pi\)
−0.584902 + 0.811104i \(0.698867\pi\)
\(954\) −10.2906 −0.333170
\(955\) 75.6159 2.44687
\(956\) −8.00000 −0.258738
\(957\) 34.4323 1.11304
\(958\) −10.5568 −0.341074
\(959\) 0 0
\(960\) 4.26056 0.137509
\(961\) 31.2803 1.00904
\(962\) 7.13451 0.230026
\(963\) 0.996260 0.0321040
\(964\) −13.3385 −0.429603
\(965\) 102.116 3.28723
\(966\) 0 0
\(967\) −53.9882 −1.73614 −0.868071 0.496440i \(-0.834640\pi\)
−0.868071 + 0.496440i \(0.834640\pi\)
\(968\) 3.20895 0.103139
\(969\) −0.705671 −0.0226694
\(970\) 71.6513 2.30058
\(971\) 19.7798 0.634763 0.317381 0.948298i \(-0.397196\pi\)
0.317381 + 0.948298i \(0.397196\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 34.7017 1.11191
\(975\) −29.7316 −0.952174
\(976\) −1.12605 −0.0360440
\(977\) −10.1485 −0.324680 −0.162340 0.986735i \(-0.551904\pi\)
−0.162340 + 0.986735i \(0.551904\pi\)
\(978\) 19.7836 0.632609
\(979\) 16.4079 0.524399
\(980\) 0 0
\(981\) 11.4007 0.363997
\(982\) −30.3009 −0.966940
\(983\) 22.7901 0.726892 0.363446 0.931615i \(-0.381600\pi\)
0.363446 + 0.931615i \(0.381600\pi\)
\(984\) 5.90772 0.188331
\(985\) −5.61994 −0.179066
\(986\) −3.22298 −0.102641
\(987\) 0 0
\(988\) −4.52111 −0.143836
\(989\) −9.67720 −0.307717
\(990\) 16.0601 0.510422
\(991\) 28.3366 0.900140 0.450070 0.892993i \(-0.351399\pi\)
0.450070 + 0.892993i \(0.351399\pi\)
\(992\) 7.89179 0.250564
\(993\) 5.04223 0.160010
\(994\) 0 0
\(995\) −22.1636 −0.702634
\(996\) 7.22678 0.228989
\(997\) −37.2105 −1.17847 −0.589234 0.807962i \(-0.700570\pi\)
−0.589234 + 0.807962i \(0.700570\pi\)
\(998\) −10.9747 −0.347397
\(999\) −3.15608 −0.0998541
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 6762.2.a.cj.1.1 4
7.6 odd 2 6762.2.a.cs.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cj.1.1 4 1.1 even 1 trivial
6762.2.a.cs.1.4 yes 4 7.6 odd 2