Properties

Label 3381.2.a.bi.1.10
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(2.79962\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.79962 q^{2} -1.00000 q^{3} +5.83786 q^{4} -1.94708 q^{5} -2.79962 q^{6} +10.7446 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.79962 q^{2} -1.00000 q^{3} +5.83786 q^{4} -1.94708 q^{5} -2.79962 q^{6} +10.7446 q^{8} +1.00000 q^{9} -5.45107 q^{10} -2.68769 q^{11} -5.83786 q^{12} +4.19039 q^{13} +1.94708 q^{15} +18.4049 q^{16} -0.632396 q^{17} +2.79962 q^{18} +5.55426 q^{19} -11.3668 q^{20} -7.52450 q^{22} +1.00000 q^{23} -10.7446 q^{24} -1.20890 q^{25} +11.7315 q^{26} -1.00000 q^{27} +5.57812 q^{29} +5.45107 q^{30} -5.59818 q^{31} +30.0377 q^{32} +2.68769 q^{33} -1.77047 q^{34} +5.83786 q^{36} +4.11612 q^{37} +15.5498 q^{38} -4.19039 q^{39} -20.9205 q^{40} +5.19481 q^{41} -1.98626 q^{43} -15.6903 q^{44} -1.94708 q^{45} +2.79962 q^{46} -0.292587 q^{47} -18.4049 q^{48} -3.38445 q^{50} +0.632396 q^{51} +24.4629 q^{52} -10.1722 q^{53} -2.79962 q^{54} +5.23313 q^{55} -5.55426 q^{57} +15.6166 q^{58} +7.31960 q^{59} +11.3668 q^{60} -6.28692 q^{61} -15.6728 q^{62} +47.2842 q^{64} -8.15900 q^{65} +7.52450 q^{66} +3.14875 q^{67} -3.69184 q^{68} -1.00000 q^{69} +6.00227 q^{71} +10.7446 q^{72} +13.8625 q^{73} +11.5236 q^{74} +1.20890 q^{75} +32.4250 q^{76} -11.7315 q^{78} +15.7902 q^{79} -35.8358 q^{80} +1.00000 q^{81} +14.5435 q^{82} +12.4983 q^{83} +1.23132 q^{85} -5.56077 q^{86} -5.57812 q^{87} -28.8780 q^{88} +1.81155 q^{89} -5.45107 q^{90} +5.83786 q^{92} +5.59818 q^{93} -0.819132 q^{94} -10.8146 q^{95} -30.0377 q^{96} -2.95190 q^{97} -2.68769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.79962 1.97963 0.989815 0.142362i \(-0.0454697\pi\)
0.989815 + 0.142362i \(0.0454697\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.83786 2.91893
\(5\) −1.94708 −0.870759 −0.435379 0.900247i \(-0.643386\pi\)
−0.435379 + 0.900247i \(0.643386\pi\)
\(6\) −2.79962 −1.14294
\(7\) 0 0
\(8\) 10.7446 3.79878
\(9\) 1.00000 0.333333
\(10\) −5.45107 −1.72378
\(11\) −2.68769 −0.810368 −0.405184 0.914235i \(-0.632793\pi\)
−0.405184 + 0.914235i \(0.632793\pi\)
\(12\) −5.83786 −1.68525
\(13\) 4.19039 1.16220 0.581102 0.813831i \(-0.302622\pi\)
0.581102 + 0.813831i \(0.302622\pi\)
\(14\) 0 0
\(15\) 1.94708 0.502733
\(16\) 18.4049 4.60123
\(17\) −0.632396 −0.153379 −0.0766893 0.997055i \(-0.524435\pi\)
−0.0766893 + 0.997055i \(0.524435\pi\)
\(18\) 2.79962 0.659876
\(19\) 5.55426 1.27423 0.637117 0.770767i \(-0.280127\pi\)
0.637117 + 0.770767i \(0.280127\pi\)
\(20\) −11.3668 −2.54169
\(21\) 0 0
\(22\) −7.52450 −1.60423
\(23\) 1.00000 0.208514
\(24\) −10.7446 −2.19322
\(25\) −1.20890 −0.241779
\(26\) 11.7315 2.30073
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.57812 1.03583 0.517915 0.855432i \(-0.326708\pi\)
0.517915 + 0.855432i \(0.326708\pi\)
\(30\) 5.45107 0.995225
\(31\) −5.59818 −1.00546 −0.502732 0.864443i \(-0.667672\pi\)
−0.502732 + 0.864443i \(0.667672\pi\)
\(32\) 30.0377 5.30996
\(33\) 2.68769 0.467866
\(34\) −1.77047 −0.303633
\(35\) 0 0
\(36\) 5.83786 0.972977
\(37\) 4.11612 0.676686 0.338343 0.941023i \(-0.390134\pi\)
0.338343 + 0.941023i \(0.390134\pi\)
\(38\) 15.5498 2.52251
\(39\) −4.19039 −0.670999
\(40\) −20.9205 −3.30782
\(41\) 5.19481 0.811293 0.405647 0.914030i \(-0.367046\pi\)
0.405647 + 0.914030i \(0.367046\pi\)
\(42\) 0 0
\(43\) −1.98626 −0.302902 −0.151451 0.988465i \(-0.548395\pi\)
−0.151451 + 0.988465i \(0.548395\pi\)
\(44\) −15.6903 −2.36541
\(45\) −1.94708 −0.290253
\(46\) 2.79962 0.412781
\(47\) −0.292587 −0.0426782 −0.0213391 0.999772i \(-0.506793\pi\)
−0.0213391 + 0.999772i \(0.506793\pi\)
\(48\) −18.4049 −2.65652
\(49\) 0 0
\(50\) −3.38445 −0.478633
\(51\) 0.632396 0.0885532
\(52\) 24.4629 3.39240
\(53\) −10.1722 −1.39725 −0.698627 0.715486i \(-0.746205\pi\)
−0.698627 + 0.715486i \(0.746205\pi\)
\(54\) −2.79962 −0.380980
\(55\) 5.23313 0.705635
\(56\) 0 0
\(57\) −5.55426 −0.735679
\(58\) 15.6166 2.05056
\(59\) 7.31960 0.952931 0.476466 0.879193i \(-0.341918\pi\)
0.476466 + 0.879193i \(0.341918\pi\)
\(60\) 11.3668 1.46744
\(61\) −6.28692 −0.804958 −0.402479 0.915429i \(-0.631851\pi\)
−0.402479 + 0.915429i \(0.631851\pi\)
\(62\) −15.6728 −1.99044
\(63\) 0 0
\(64\) 47.2842 5.91053
\(65\) −8.15900 −1.01200
\(66\) 7.52450 0.926201
\(67\) 3.14875 0.384681 0.192340 0.981328i \(-0.438392\pi\)
0.192340 + 0.981328i \(0.438392\pi\)
\(68\) −3.69184 −0.447702
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.00227 0.712338 0.356169 0.934422i \(-0.384083\pi\)
0.356169 + 0.934422i \(0.384083\pi\)
\(72\) 10.7446 1.26626
\(73\) 13.8625 1.62249 0.811243 0.584709i \(-0.198791\pi\)
0.811243 + 0.584709i \(0.198791\pi\)
\(74\) 11.5236 1.33959
\(75\) 1.20890 0.139591
\(76\) 32.4250 3.71940
\(77\) 0 0
\(78\) −11.7315 −1.32833
\(79\) 15.7902 1.77654 0.888268 0.459326i \(-0.151909\pi\)
0.888268 + 0.459326i \(0.151909\pi\)
\(80\) −35.8358 −4.00656
\(81\) 1.00000 0.111111
\(82\) 14.5435 1.60606
\(83\) 12.4983 1.37186 0.685931 0.727666i \(-0.259395\pi\)
0.685931 + 0.727666i \(0.259395\pi\)
\(84\) 0 0
\(85\) 1.23132 0.133556
\(86\) −5.56077 −0.599633
\(87\) −5.57812 −0.598037
\(88\) −28.8780 −3.07841
\(89\) 1.81155 0.192024 0.0960119 0.995380i \(-0.469391\pi\)
0.0960119 + 0.995380i \(0.469391\pi\)
\(90\) −5.45107 −0.574593
\(91\) 0 0
\(92\) 5.83786 0.608639
\(93\) 5.59818 0.580504
\(94\) −0.819132 −0.0844870
\(95\) −10.8146 −1.10955
\(96\) −30.0377 −3.06571
\(97\) −2.95190 −0.299720 −0.149860 0.988707i \(-0.547882\pi\)
−0.149860 + 0.988707i \(0.547882\pi\)
\(98\) 0 0
\(99\) −2.68769 −0.270123
\(100\) −7.05738 −0.705738
\(101\) −12.8013 −1.27378 −0.636889 0.770955i \(-0.719779\pi\)
−0.636889 + 0.770955i \(0.719779\pi\)
\(102\) 1.77047 0.175302
\(103\) 13.0581 1.28665 0.643325 0.765593i \(-0.277554\pi\)
0.643325 + 0.765593i \(0.277554\pi\)
\(104\) 45.0239 4.41495
\(105\) 0 0
\(106\) −28.4782 −2.76605
\(107\) −13.7217 −1.32653 −0.663263 0.748387i \(-0.730829\pi\)
−0.663263 + 0.748387i \(0.730829\pi\)
\(108\) −5.83786 −0.561749
\(109\) −4.50591 −0.431588 −0.215794 0.976439i \(-0.569234\pi\)
−0.215794 + 0.976439i \(0.569234\pi\)
\(110\) 14.6508 1.39690
\(111\) −4.11612 −0.390685
\(112\) 0 0
\(113\) −16.4013 −1.54290 −0.771451 0.636288i \(-0.780469\pi\)
−0.771451 + 0.636288i \(0.780469\pi\)
\(114\) −15.5498 −1.45637
\(115\) −1.94708 −0.181566
\(116\) 32.5643 3.02352
\(117\) 4.19039 0.387401
\(118\) 20.4921 1.88645
\(119\) 0 0
\(120\) 20.9205 1.90977
\(121\) −3.77634 −0.343304
\(122\) −17.6010 −1.59352
\(123\) −5.19481 −0.468400
\(124\) −32.6814 −2.93488
\(125\) 12.0892 1.08129
\(126\) 0 0
\(127\) −18.2031 −1.61527 −0.807634 0.589685i \(-0.799252\pi\)
−0.807634 + 0.589685i \(0.799252\pi\)
\(128\) 72.3024 6.39069
\(129\) 1.98626 0.174880
\(130\) −22.8421 −2.00338
\(131\) −8.32841 −0.727656 −0.363828 0.931466i \(-0.618530\pi\)
−0.363828 + 0.931466i \(0.618530\pi\)
\(132\) 15.6903 1.36567
\(133\) 0 0
\(134\) 8.81530 0.761526
\(135\) 1.94708 0.167578
\(136\) −6.79482 −0.582651
\(137\) −13.3043 −1.13667 −0.568333 0.822798i \(-0.692412\pi\)
−0.568333 + 0.822798i \(0.692412\pi\)
\(138\) −2.79962 −0.238319
\(139\) −5.93319 −0.503247 −0.251623 0.967825i \(-0.580964\pi\)
−0.251623 + 0.967825i \(0.580964\pi\)
\(140\) 0 0
\(141\) 0.292587 0.0246403
\(142\) 16.8041 1.41017
\(143\) −11.2624 −0.941813
\(144\) 18.4049 1.53374
\(145\) −10.8610 −0.901959
\(146\) 38.8098 3.21192
\(147\) 0 0
\(148\) 24.0293 1.97520
\(149\) −4.83066 −0.395743 −0.197872 0.980228i \(-0.563403\pi\)
−0.197872 + 0.980228i \(0.563403\pi\)
\(150\) 3.38445 0.276339
\(151\) 20.9847 1.70771 0.853854 0.520513i \(-0.174259\pi\)
0.853854 + 0.520513i \(0.174259\pi\)
\(152\) 59.6780 4.84053
\(153\) −0.632396 −0.0511262
\(154\) 0 0
\(155\) 10.9001 0.875516
\(156\) −24.4629 −1.95860
\(157\) 1.59444 0.127250 0.0636250 0.997974i \(-0.479734\pi\)
0.0636250 + 0.997974i \(0.479734\pi\)
\(158\) 44.2065 3.51688
\(159\) 10.1722 0.806705
\(160\) −58.4857 −4.62370
\(161\) 0 0
\(162\) 2.79962 0.219959
\(163\) −10.8977 −0.853571 −0.426786 0.904353i \(-0.640354\pi\)
−0.426786 + 0.904353i \(0.640354\pi\)
\(164\) 30.3266 2.36811
\(165\) −5.23313 −0.407398
\(166\) 34.9904 2.71578
\(167\) −15.5931 −1.20663 −0.603313 0.797504i \(-0.706153\pi\)
−0.603313 + 0.797504i \(0.706153\pi\)
\(168\) 0 0
\(169\) 4.55935 0.350719
\(170\) 3.44723 0.264391
\(171\) 5.55426 0.424745
\(172\) −11.5955 −0.884149
\(173\) −11.2721 −0.857002 −0.428501 0.903541i \(-0.640958\pi\)
−0.428501 + 0.903541i \(0.640958\pi\)
\(174\) −15.6166 −1.18389
\(175\) 0 0
\(176\) −49.4667 −3.72869
\(177\) −7.31960 −0.550175
\(178\) 5.07165 0.380136
\(179\) 8.17508 0.611034 0.305517 0.952187i \(-0.401171\pi\)
0.305517 + 0.952187i \(0.401171\pi\)
\(180\) −11.3668 −0.847229
\(181\) −3.00094 −0.223058 −0.111529 0.993761i \(-0.535575\pi\)
−0.111529 + 0.993761i \(0.535575\pi\)
\(182\) 0 0
\(183\) 6.28692 0.464743
\(184\) 10.7446 0.792099
\(185\) −8.01439 −0.589230
\(186\) 15.6728 1.14918
\(187\) 1.69968 0.124293
\(188\) −1.70808 −0.124575
\(189\) 0 0
\(190\) −30.2766 −2.19650
\(191\) −11.5788 −0.837814 −0.418907 0.908029i \(-0.637587\pi\)
−0.418907 + 0.908029i \(0.637587\pi\)
\(192\) −47.2842 −3.41244
\(193\) 5.69929 0.410244 0.205122 0.978736i \(-0.434241\pi\)
0.205122 + 0.978736i \(0.434241\pi\)
\(194\) −8.26419 −0.593334
\(195\) 8.15900 0.584278
\(196\) 0 0
\(197\) −12.9883 −0.925379 −0.462689 0.886521i \(-0.653115\pi\)
−0.462689 + 0.886521i \(0.653115\pi\)
\(198\) −7.52450 −0.534743
\(199\) −21.3255 −1.51173 −0.755864 0.654729i \(-0.772783\pi\)
−0.755864 + 0.654729i \(0.772783\pi\)
\(200\) −12.9891 −0.918465
\(201\) −3.14875 −0.222096
\(202\) −35.8388 −2.52161
\(203\) 0 0
\(204\) 3.69184 0.258481
\(205\) −10.1147 −0.706441
\(206\) 36.5576 2.54709
\(207\) 1.00000 0.0695048
\(208\) 77.1238 5.34758
\(209\) −14.9281 −1.03260
\(210\) 0 0
\(211\) 7.69191 0.529533 0.264766 0.964313i \(-0.414705\pi\)
0.264766 + 0.964313i \(0.414705\pi\)
\(212\) −59.3837 −4.07849
\(213\) −6.00227 −0.411269
\(214\) −38.4155 −2.62603
\(215\) 3.86740 0.263754
\(216\) −10.7446 −0.731075
\(217\) 0 0
\(218\) −12.6148 −0.854384
\(219\) −13.8625 −0.936743
\(220\) 30.5503 2.05970
\(221\) −2.64998 −0.178257
\(222\) −11.5236 −0.773411
\(223\) 20.9689 1.40418 0.702091 0.712087i \(-0.252250\pi\)
0.702091 + 0.712087i \(0.252250\pi\)
\(224\) 0 0
\(225\) −1.20890 −0.0805931
\(226\) −45.9173 −3.05438
\(227\) −24.1758 −1.60460 −0.802302 0.596918i \(-0.796392\pi\)
−0.802302 + 0.596918i \(0.796392\pi\)
\(228\) −32.4250 −2.14740
\(229\) −8.67855 −0.573495 −0.286747 0.958006i \(-0.592574\pi\)
−0.286747 + 0.958006i \(0.592574\pi\)
\(230\) −5.45107 −0.359433
\(231\) 0 0
\(232\) 59.9344 3.93489
\(233\) 2.69180 0.176346 0.0881729 0.996105i \(-0.471897\pi\)
0.0881729 + 0.996105i \(0.471897\pi\)
\(234\) 11.7315 0.766911
\(235\) 0.569689 0.0371624
\(236\) 42.7308 2.78154
\(237\) −15.7902 −1.02568
\(238\) 0 0
\(239\) −0.0523580 −0.00338675 −0.00169338 0.999999i \(-0.500539\pi\)
−0.00169338 + 0.999999i \(0.500539\pi\)
\(240\) 35.8358 2.31319
\(241\) −0.299551 −0.0192958 −0.00964789 0.999953i \(-0.503071\pi\)
−0.00964789 + 0.999953i \(0.503071\pi\)
\(242\) −10.5723 −0.679615
\(243\) −1.00000 −0.0641500
\(244\) −36.7022 −2.34962
\(245\) 0 0
\(246\) −14.5435 −0.927259
\(247\) 23.2745 1.48092
\(248\) −60.1500 −3.81953
\(249\) −12.4983 −0.792045
\(250\) 33.8451 2.14055
\(251\) −5.72848 −0.361579 −0.180789 0.983522i \(-0.557865\pi\)
−0.180789 + 0.983522i \(0.557865\pi\)
\(252\) 0 0
\(253\) −2.68769 −0.168973
\(254\) −50.9618 −3.19763
\(255\) −1.23132 −0.0771084
\(256\) 107.851 6.74066
\(257\) −17.2297 −1.07476 −0.537379 0.843341i \(-0.680585\pi\)
−0.537379 + 0.843341i \(0.680585\pi\)
\(258\) 5.56077 0.346198
\(259\) 0 0
\(260\) −47.6312 −2.95396
\(261\) 5.57812 0.345277
\(262\) −23.3164 −1.44049
\(263\) 23.5216 1.45040 0.725202 0.688537i \(-0.241747\pi\)
0.725202 + 0.688537i \(0.241747\pi\)
\(264\) 28.8780 1.77732
\(265\) 19.8060 1.21667
\(266\) 0 0
\(267\) −1.81155 −0.110865
\(268\) 18.3820 1.12286
\(269\) −13.8092 −0.841965 −0.420982 0.907069i \(-0.638315\pi\)
−0.420982 + 0.907069i \(0.638315\pi\)
\(270\) 5.45107 0.331742
\(271\) −22.4422 −1.36327 −0.681634 0.731693i \(-0.738730\pi\)
−0.681634 + 0.731693i \(0.738730\pi\)
\(272\) −11.6392 −0.705731
\(273\) 0 0
\(274\) −37.2471 −2.25018
\(275\) 3.24913 0.195930
\(276\) −5.83786 −0.351398
\(277\) 22.2276 1.33553 0.667765 0.744373i \(-0.267251\pi\)
0.667765 + 0.744373i \(0.267251\pi\)
\(278\) −16.6107 −0.996242
\(279\) −5.59818 −0.335154
\(280\) 0 0
\(281\) 6.99680 0.417394 0.208697 0.977980i \(-0.433078\pi\)
0.208697 + 0.977980i \(0.433078\pi\)
\(282\) 0.819132 0.0487786
\(283\) 8.06377 0.479341 0.239671 0.970854i \(-0.422961\pi\)
0.239671 + 0.970854i \(0.422961\pi\)
\(284\) 35.0404 2.07927
\(285\) 10.8146 0.640599
\(286\) −31.5306 −1.86444
\(287\) 0 0
\(288\) 30.0377 1.76999
\(289\) −16.6001 −0.976475
\(290\) −30.4067 −1.78554
\(291\) 2.95190 0.173043
\(292\) 80.9276 4.73593
\(293\) −13.1458 −0.767986 −0.383993 0.923336i \(-0.625451\pi\)
−0.383993 + 0.923336i \(0.625451\pi\)
\(294\) 0 0
\(295\) −14.2518 −0.829773
\(296\) 44.2259 2.57058
\(297\) 2.68769 0.155955
\(298\) −13.5240 −0.783425
\(299\) 4.19039 0.242336
\(300\) 7.05738 0.407458
\(301\) 0 0
\(302\) 58.7490 3.38063
\(303\) 12.8013 0.735416
\(304\) 102.226 5.86305
\(305\) 12.2411 0.700924
\(306\) −1.77047 −0.101211
\(307\) 15.1086 0.862293 0.431147 0.902282i \(-0.358109\pi\)
0.431147 + 0.902282i \(0.358109\pi\)
\(308\) 0 0
\(309\) −13.0581 −0.742848
\(310\) 30.5161 1.73320
\(311\) −21.4828 −1.21818 −0.609090 0.793101i \(-0.708465\pi\)
−0.609090 + 0.793101i \(0.708465\pi\)
\(312\) −45.0239 −2.54897
\(313\) 14.8382 0.838707 0.419353 0.907823i \(-0.362257\pi\)
0.419353 + 0.907823i \(0.362257\pi\)
\(314\) 4.46382 0.251908
\(315\) 0 0
\(316\) 92.1810 5.18559
\(317\) −4.70173 −0.264075 −0.132038 0.991245i \(-0.542152\pi\)
−0.132038 + 0.991245i \(0.542152\pi\)
\(318\) 28.4782 1.59698
\(319\) −14.9922 −0.839404
\(320\) −92.0659 −5.14664
\(321\) 13.7217 0.765870
\(322\) 0 0
\(323\) −3.51249 −0.195440
\(324\) 5.83786 0.324326
\(325\) −5.06575 −0.280997
\(326\) −30.5093 −1.68975
\(327\) 4.50591 0.249177
\(328\) 55.8160 3.08192
\(329\) 0 0
\(330\) −14.6508 −0.806498
\(331\) −7.87864 −0.433049 −0.216524 0.976277i \(-0.569472\pi\)
−0.216524 + 0.976277i \(0.569472\pi\)
\(332\) 72.9632 4.00437
\(333\) 4.11612 0.225562
\(334\) −43.6546 −2.38867
\(335\) −6.13085 −0.334964
\(336\) 0 0
\(337\) −18.3857 −1.00153 −0.500767 0.865582i \(-0.666949\pi\)
−0.500767 + 0.865582i \(0.666949\pi\)
\(338\) 12.7644 0.694294
\(339\) 16.4013 0.890795
\(340\) 7.18830 0.389840
\(341\) 15.0462 0.814795
\(342\) 15.5498 0.840837
\(343\) 0 0
\(344\) −21.3415 −1.15066
\(345\) 1.94708 0.104827
\(346\) −31.5576 −1.69655
\(347\) 6.02078 0.323212 0.161606 0.986855i \(-0.448333\pi\)
0.161606 + 0.986855i \(0.448333\pi\)
\(348\) −32.5643 −1.74563
\(349\) 13.3378 0.713956 0.356978 0.934113i \(-0.383807\pi\)
0.356978 + 0.934113i \(0.383807\pi\)
\(350\) 0 0
\(351\) −4.19039 −0.223666
\(352\) −80.7319 −4.30302
\(353\) 17.4870 0.930739 0.465369 0.885117i \(-0.345922\pi\)
0.465369 + 0.885117i \(0.345922\pi\)
\(354\) −20.4921 −1.08914
\(355\) −11.6869 −0.620275
\(356\) 10.5756 0.560505
\(357\) 0 0
\(358\) 22.8871 1.20962
\(359\) −8.56723 −0.452161 −0.226081 0.974109i \(-0.572591\pi\)
−0.226081 + 0.974109i \(0.572591\pi\)
\(360\) −20.9205 −1.10261
\(361\) 11.8498 0.623671
\(362\) −8.40150 −0.441573
\(363\) 3.77634 0.198207
\(364\) 0 0
\(365\) −26.9914 −1.41279
\(366\) 17.6010 0.920018
\(367\) −19.7115 −1.02893 −0.514467 0.857510i \(-0.672010\pi\)
−0.514467 + 0.857510i \(0.672010\pi\)
\(368\) 18.4049 0.959424
\(369\) 5.19481 0.270431
\(370\) −22.4372 −1.16646
\(371\) 0 0
\(372\) 32.6814 1.69445
\(373\) 21.9259 1.13528 0.567639 0.823278i \(-0.307857\pi\)
0.567639 + 0.823278i \(0.307857\pi\)
\(374\) 4.75846 0.246054
\(375\) −12.0892 −0.624283
\(376\) −3.14372 −0.162125
\(377\) 23.3745 1.20385
\(378\) 0 0
\(379\) 9.89943 0.508500 0.254250 0.967139i \(-0.418171\pi\)
0.254250 + 0.967139i \(0.418171\pi\)
\(380\) −63.1339 −3.23870
\(381\) 18.2031 0.932575
\(382\) −32.4163 −1.65856
\(383\) −17.5379 −0.896143 −0.448071 0.893998i \(-0.647889\pi\)
−0.448071 + 0.893998i \(0.647889\pi\)
\(384\) −72.3024 −3.68966
\(385\) 0 0
\(386\) 15.9558 0.812131
\(387\) −1.98626 −0.100967
\(388\) −17.2328 −0.874862
\(389\) 19.2476 0.975894 0.487947 0.872873i \(-0.337746\pi\)
0.487947 + 0.872873i \(0.337746\pi\)
\(390\) 22.8421 1.15665
\(391\) −0.632396 −0.0319816
\(392\) 0 0
\(393\) 8.32841 0.420113
\(394\) −36.3623 −1.83191
\(395\) −30.7447 −1.54693
\(396\) −15.6903 −0.788470
\(397\) −24.5994 −1.23461 −0.617304 0.786724i \(-0.711775\pi\)
−0.617304 + 0.786724i \(0.711775\pi\)
\(398\) −59.7034 −2.99266
\(399\) 0 0
\(400\) −22.2497 −1.11248
\(401\) −7.88512 −0.393764 −0.196882 0.980427i \(-0.563082\pi\)
−0.196882 + 0.980427i \(0.563082\pi\)
\(402\) −8.81530 −0.439667
\(403\) −23.4586 −1.16855
\(404\) −74.7323 −3.71807
\(405\) −1.94708 −0.0967510
\(406\) 0 0
\(407\) −11.0628 −0.548364
\(408\) 6.79482 0.336394
\(409\) 4.80677 0.237680 0.118840 0.992913i \(-0.462083\pi\)
0.118840 + 0.992913i \(0.462083\pi\)
\(410\) −28.3173 −1.39849
\(411\) 13.3043 0.656255
\(412\) 76.2313 3.75565
\(413\) 0 0
\(414\) 2.79962 0.137594
\(415\) −24.3351 −1.19456
\(416\) 125.870 6.17126
\(417\) 5.93319 0.290550
\(418\) −41.7930 −2.04416
\(419\) 4.37312 0.213641 0.106820 0.994278i \(-0.465933\pi\)
0.106820 + 0.994278i \(0.465933\pi\)
\(420\) 0 0
\(421\) −15.6894 −0.764655 −0.382327 0.924027i \(-0.624877\pi\)
−0.382327 + 0.924027i \(0.624877\pi\)
\(422\) 21.5344 1.04828
\(423\) −0.292587 −0.0142261
\(424\) −109.295 −5.30785
\(425\) 0.764501 0.0370838
\(426\) −16.8041 −0.814159
\(427\) 0 0
\(428\) −80.1053 −3.87204
\(429\) 11.2624 0.543756
\(430\) 10.8272 0.522136
\(431\) −3.34633 −0.161187 −0.0805935 0.996747i \(-0.525682\pi\)
−0.0805935 + 0.996747i \(0.525682\pi\)
\(432\) −18.4049 −0.885508
\(433\) −15.2273 −0.731779 −0.365889 0.930658i \(-0.619235\pi\)
−0.365889 + 0.930658i \(0.619235\pi\)
\(434\) 0 0
\(435\) 10.8610 0.520746
\(436\) −26.3049 −1.25978
\(437\) 5.55426 0.265696
\(438\) −38.8098 −1.85440
\(439\) 21.4914 1.02573 0.512865 0.858469i \(-0.328584\pi\)
0.512865 + 0.858469i \(0.328584\pi\)
\(440\) 56.2276 2.68055
\(441\) 0 0
\(442\) −7.41895 −0.352883
\(443\) 23.7846 1.13004 0.565019 0.825078i \(-0.308869\pi\)
0.565019 + 0.825078i \(0.308869\pi\)
\(444\) −24.0293 −1.14038
\(445\) −3.52722 −0.167206
\(446\) 58.7050 2.77976
\(447\) 4.83066 0.228482
\(448\) 0 0
\(449\) −8.36232 −0.394642 −0.197321 0.980339i \(-0.563224\pi\)
−0.197321 + 0.980339i \(0.563224\pi\)
\(450\) −3.38445 −0.159544
\(451\) −13.9620 −0.657446
\(452\) −95.7485 −4.50363
\(453\) −20.9847 −0.985945
\(454\) −67.6831 −3.17652
\(455\) 0 0
\(456\) −59.6780 −2.79468
\(457\) −15.9314 −0.745238 −0.372619 0.927984i \(-0.621540\pi\)
−0.372619 + 0.927984i \(0.621540\pi\)
\(458\) −24.2966 −1.13531
\(459\) 0.632396 0.0295177
\(460\) −11.3668 −0.529978
\(461\) −11.6774 −0.543873 −0.271936 0.962315i \(-0.587664\pi\)
−0.271936 + 0.962315i \(0.587664\pi\)
\(462\) 0 0
\(463\) −6.74188 −0.313322 −0.156661 0.987652i \(-0.550073\pi\)
−0.156661 + 0.987652i \(0.550073\pi\)
\(464\) 102.665 4.76610
\(465\) −10.9001 −0.505479
\(466\) 7.53602 0.349099
\(467\) 15.4800 0.716328 0.358164 0.933659i \(-0.383403\pi\)
0.358164 + 0.933659i \(0.383403\pi\)
\(468\) 24.4629 1.13080
\(469\) 0 0
\(470\) 1.59491 0.0735678
\(471\) −1.59444 −0.0734679
\(472\) 78.6459 3.61997
\(473\) 5.33844 0.245462
\(474\) −44.2065 −2.03047
\(475\) −6.71452 −0.308083
\(476\) 0 0
\(477\) −10.1722 −0.465751
\(478\) −0.146582 −0.00670452
\(479\) 18.5720 0.848574 0.424287 0.905528i \(-0.360525\pi\)
0.424287 + 0.905528i \(0.360525\pi\)
\(480\) 58.4857 2.66949
\(481\) 17.2481 0.786447
\(482\) −0.838629 −0.0381985
\(483\) 0 0
\(484\) −22.0458 −1.00208
\(485\) 5.74757 0.260984
\(486\) −2.79962 −0.126993
\(487\) 18.1617 0.822984 0.411492 0.911413i \(-0.365008\pi\)
0.411492 + 0.911413i \(0.365008\pi\)
\(488\) −67.5502 −3.05785
\(489\) 10.8977 0.492810
\(490\) 0 0
\(491\) 14.1905 0.640410 0.320205 0.947348i \(-0.396248\pi\)
0.320205 + 0.947348i \(0.396248\pi\)
\(492\) −30.3266 −1.36723
\(493\) −3.52758 −0.158874
\(494\) 65.1597 2.93167
\(495\) 5.23313 0.235212
\(496\) −103.034 −4.62637
\(497\) 0 0
\(498\) −34.9904 −1.56796
\(499\) 14.3387 0.641890 0.320945 0.947098i \(-0.396000\pi\)
0.320945 + 0.947098i \(0.396000\pi\)
\(500\) 70.5751 3.15621
\(501\) 15.5931 0.696646
\(502\) −16.0376 −0.715791
\(503\) 10.8254 0.482681 0.241341 0.970440i \(-0.422413\pi\)
0.241341 + 0.970440i \(0.422413\pi\)
\(504\) 0 0
\(505\) 24.9251 1.10915
\(506\) −7.52450 −0.334505
\(507\) −4.55935 −0.202488
\(508\) −106.267 −4.71486
\(509\) 27.5502 1.22114 0.610570 0.791962i \(-0.290940\pi\)
0.610570 + 0.791962i \(0.290940\pi\)
\(510\) −3.44723 −0.152646
\(511\) 0 0
\(512\) 157.336 6.95333
\(513\) −5.55426 −0.245226
\(514\) −48.2365 −2.12762
\(515\) −25.4251 −1.12036
\(516\) 11.5955 0.510464
\(517\) 0.786382 0.0345850
\(518\) 0 0
\(519\) 11.2721 0.494791
\(520\) −87.6649 −3.84436
\(521\) −13.6364 −0.597423 −0.298712 0.954343i \(-0.596557\pi\)
−0.298712 + 0.954343i \(0.596557\pi\)
\(522\) 15.6166 0.683520
\(523\) −10.5636 −0.461912 −0.230956 0.972964i \(-0.574185\pi\)
−0.230956 + 0.972964i \(0.574185\pi\)
\(524\) −48.6201 −2.12398
\(525\) 0 0
\(526\) 65.8515 2.87126
\(527\) 3.54027 0.154217
\(528\) 49.4667 2.15276
\(529\) 1.00000 0.0434783
\(530\) 55.4492 2.40856
\(531\) 7.31960 0.317644
\(532\) 0 0
\(533\) 21.7683 0.942889
\(534\) −5.07165 −0.219472
\(535\) 26.7172 1.15508
\(536\) 33.8319 1.46132
\(537\) −8.17508 −0.352781
\(538\) −38.6606 −1.66678
\(539\) 0 0
\(540\) 11.3668 0.489148
\(541\) 31.6652 1.36139 0.680696 0.732566i \(-0.261677\pi\)
0.680696 + 0.732566i \(0.261677\pi\)
\(542\) −62.8297 −2.69877
\(543\) 3.00094 0.128783
\(544\) −18.9957 −0.814435
\(545\) 8.77335 0.375809
\(546\) 0 0
\(547\) −30.4882 −1.30358 −0.651792 0.758398i \(-0.725982\pi\)
−0.651792 + 0.758398i \(0.725982\pi\)
\(548\) −77.6689 −3.31785
\(549\) −6.28692 −0.268319
\(550\) 9.09634 0.387869
\(551\) 30.9823 1.31989
\(552\) −10.7446 −0.457319
\(553\) 0 0
\(554\) 62.2289 2.64385
\(555\) 8.01439 0.340192
\(556\) −34.6372 −1.46894
\(557\) 14.1228 0.598401 0.299200 0.954190i \(-0.403280\pi\)
0.299200 + 0.954190i \(0.403280\pi\)
\(558\) −15.6728 −0.663482
\(559\) −8.32319 −0.352034
\(560\) 0 0
\(561\) −1.69968 −0.0717606
\(562\) 19.5884 0.826285
\(563\) −17.2841 −0.728437 −0.364219 0.931313i \(-0.618664\pi\)
−0.364219 + 0.931313i \(0.618664\pi\)
\(564\) 1.70808 0.0719233
\(565\) 31.9345 1.34350
\(566\) 22.5755 0.948918
\(567\) 0 0
\(568\) 64.4917 2.70601
\(569\) 2.74157 0.114932 0.0574662 0.998347i \(-0.481698\pi\)
0.0574662 + 0.998347i \(0.481698\pi\)
\(570\) 30.2766 1.26815
\(571\) 31.1908 1.30529 0.652647 0.757662i \(-0.273659\pi\)
0.652647 + 0.757662i \(0.273659\pi\)
\(572\) −65.7486 −2.74909
\(573\) 11.5788 0.483712
\(574\) 0 0
\(575\) −1.20890 −0.0504145
\(576\) 47.2842 1.97018
\(577\) −4.24819 −0.176855 −0.0884273 0.996083i \(-0.528184\pi\)
−0.0884273 + 0.996083i \(0.528184\pi\)
\(578\) −46.4739 −1.93306
\(579\) −5.69929 −0.236854
\(580\) −63.4052 −2.63276
\(581\) 0 0
\(582\) 8.26419 0.342562
\(583\) 27.3396 1.13229
\(584\) 148.947 6.16346
\(585\) −8.15900 −0.337333
\(586\) −36.8032 −1.52033
\(587\) −27.9648 −1.15423 −0.577116 0.816662i \(-0.695822\pi\)
−0.577116 + 0.816662i \(0.695822\pi\)
\(588\) 0 0
\(589\) −31.0937 −1.28119
\(590\) −39.8997 −1.64264
\(591\) 12.9883 0.534268
\(592\) 75.7569 3.11359
\(593\) −24.4826 −1.00538 −0.502690 0.864467i \(-0.667656\pi\)
−0.502690 + 0.864467i \(0.667656\pi\)
\(594\) 7.52450 0.308734
\(595\) 0 0
\(596\) −28.2007 −1.15515
\(597\) 21.3255 0.872796
\(598\) 11.7315 0.479736
\(599\) −43.9902 −1.79739 −0.898696 0.438573i \(-0.855484\pi\)
−0.898696 + 0.438573i \(0.855484\pi\)
\(600\) 12.9891 0.530276
\(601\) 9.57098 0.390408 0.195204 0.980763i \(-0.437463\pi\)
0.195204 + 0.980763i \(0.437463\pi\)
\(602\) 0 0
\(603\) 3.14875 0.128227
\(604\) 122.506 4.98468
\(605\) 7.35283 0.298935
\(606\) 35.8388 1.45585
\(607\) −1.59084 −0.0645701 −0.0322851 0.999479i \(-0.510278\pi\)
−0.0322851 + 0.999479i \(0.510278\pi\)
\(608\) 166.837 6.76613
\(609\) 0 0
\(610\) 34.2705 1.38757
\(611\) −1.22605 −0.0496008
\(612\) −3.69184 −0.149234
\(613\) 17.2292 0.695881 0.347941 0.937517i \(-0.386881\pi\)
0.347941 + 0.937517i \(0.386881\pi\)
\(614\) 42.2983 1.70702
\(615\) 10.1147 0.407864
\(616\) 0 0
\(617\) −28.9242 −1.16444 −0.582222 0.813030i \(-0.697817\pi\)
−0.582222 + 0.813030i \(0.697817\pi\)
\(618\) −36.5576 −1.47056
\(619\) −21.8645 −0.878809 −0.439404 0.898289i \(-0.644810\pi\)
−0.439404 + 0.898289i \(0.644810\pi\)
\(620\) 63.6332 2.55557
\(621\) −1.00000 −0.0401286
\(622\) −60.1438 −2.41154
\(623\) 0 0
\(624\) −77.1238 −3.08742
\(625\) −17.4941 −0.699763
\(626\) 41.5414 1.66033
\(627\) 14.9281 0.596171
\(628\) 9.30812 0.371434
\(629\) −2.60302 −0.103789
\(630\) 0 0
\(631\) −11.5152 −0.458412 −0.229206 0.973378i \(-0.573613\pi\)
−0.229206 + 0.973378i \(0.573613\pi\)
\(632\) 169.659 6.74866
\(633\) −7.69191 −0.305726
\(634\) −13.1630 −0.522771
\(635\) 35.4429 1.40651
\(636\) 59.3837 2.35472
\(637\) 0 0
\(638\) −41.9725 −1.66171
\(639\) 6.00227 0.237446
\(640\) −140.778 −5.56475
\(641\) 9.26801 0.366064 0.183032 0.983107i \(-0.441409\pi\)
0.183032 + 0.983107i \(0.441409\pi\)
\(642\) 38.4155 1.51614
\(643\) 13.8712 0.547025 0.273513 0.961868i \(-0.411814\pi\)
0.273513 + 0.961868i \(0.411814\pi\)
\(644\) 0 0
\(645\) −3.86740 −0.152279
\(646\) −9.83363 −0.386899
\(647\) −7.30063 −0.287017 −0.143509 0.989649i \(-0.545838\pi\)
−0.143509 + 0.989649i \(0.545838\pi\)
\(648\) 10.7446 0.422086
\(649\) −19.6728 −0.772225
\(650\) −14.1822 −0.556270
\(651\) 0 0
\(652\) −63.6191 −2.49152
\(653\) −28.2777 −1.10659 −0.553296 0.832984i \(-0.686630\pi\)
−0.553296 + 0.832984i \(0.686630\pi\)
\(654\) 12.6148 0.493279
\(655\) 16.2160 0.633613
\(656\) 95.6102 3.73295
\(657\) 13.8625 0.540829
\(658\) 0 0
\(659\) 26.3606 1.02686 0.513431 0.858131i \(-0.328374\pi\)
0.513431 + 0.858131i \(0.328374\pi\)
\(660\) −30.5503 −1.18917
\(661\) −16.9293 −0.658474 −0.329237 0.944247i \(-0.606792\pi\)
−0.329237 + 0.944247i \(0.606792\pi\)
\(662\) −22.0572 −0.857276
\(663\) 2.64998 0.102917
\(664\) 134.288 5.21140
\(665\) 0 0
\(666\) 11.5236 0.446529
\(667\) 5.57812 0.215986
\(668\) −91.0302 −3.52206
\(669\) −20.9689 −0.810705
\(670\) −17.1640 −0.663105
\(671\) 16.8973 0.652312
\(672\) 0 0
\(673\) −23.7746 −0.916444 −0.458222 0.888838i \(-0.651513\pi\)
−0.458222 + 0.888838i \(0.651513\pi\)
\(674\) −51.4730 −1.98267
\(675\) 1.20890 0.0465305
\(676\) 26.6169 1.02373
\(677\) 22.7084 0.872755 0.436377 0.899764i \(-0.356261\pi\)
0.436377 + 0.899764i \(0.356261\pi\)
\(678\) 45.9173 1.76344
\(679\) 0 0
\(680\) 13.2300 0.507348
\(681\) 24.1758 0.926419
\(682\) 42.1235 1.61299
\(683\) 10.8416 0.414841 0.207421 0.978252i \(-0.433493\pi\)
0.207421 + 0.978252i \(0.433493\pi\)
\(684\) 32.4250 1.23980
\(685\) 25.9046 0.989762
\(686\) 0 0
\(687\) 8.67855 0.331107
\(688\) −36.5570 −1.39372
\(689\) −42.6253 −1.62390
\(690\) 5.45107 0.207519
\(691\) 15.0020 0.570704 0.285352 0.958423i \(-0.407889\pi\)
0.285352 + 0.958423i \(0.407889\pi\)
\(692\) −65.8050 −2.50153
\(693\) 0 0
\(694\) 16.8559 0.639841
\(695\) 11.5524 0.438207
\(696\) −59.9344 −2.27181
\(697\) −3.28518 −0.124435
\(698\) 37.3408 1.41337
\(699\) −2.69180 −0.101813
\(700\) 0 0
\(701\) −8.60412 −0.324973 −0.162486 0.986711i \(-0.551951\pi\)
−0.162486 + 0.986711i \(0.551951\pi\)
\(702\) −11.7315 −0.442776
\(703\) 22.8620 0.862256
\(704\) −127.085 −4.78970
\(705\) −0.569689 −0.0214557
\(706\) 48.9569 1.84252
\(707\) 0 0
\(708\) −42.7308 −1.60592
\(709\) −29.1209 −1.09366 −0.546829 0.837244i \(-0.684165\pi\)
−0.546829 + 0.837244i \(0.684165\pi\)
\(710\) −32.7188 −1.22791
\(711\) 15.7902 0.592178
\(712\) 19.4643 0.729455
\(713\) −5.59818 −0.209654
\(714\) 0 0
\(715\) 21.9288 0.820092
\(716\) 47.7250 1.78357
\(717\) 0.0523580 0.00195534
\(718\) −23.9850 −0.895112
\(719\) −15.3158 −0.571182 −0.285591 0.958352i \(-0.592190\pi\)
−0.285591 + 0.958352i \(0.592190\pi\)
\(720\) −35.8358 −1.33552
\(721\) 0 0
\(722\) 33.1748 1.23464
\(723\) 0.299551 0.0111404
\(724\) −17.5191 −0.651093
\(725\) −6.74337 −0.250443
\(726\) 10.5723 0.392376
\(727\) 32.6484 1.21086 0.605432 0.795897i \(-0.293000\pi\)
0.605432 + 0.795897i \(0.293000\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −75.5656 −2.79681
\(731\) 1.25610 0.0464586
\(732\) 36.7022 1.35655
\(733\) −43.4532 −1.60498 −0.802490 0.596665i \(-0.796492\pi\)
−0.802490 + 0.596665i \(0.796492\pi\)
\(734\) −55.1848 −2.03691
\(735\) 0 0
\(736\) 30.0377 1.10720
\(737\) −8.46285 −0.311733
\(738\) 14.5435 0.535353
\(739\) 3.48869 0.128333 0.0641667 0.997939i \(-0.479561\pi\)
0.0641667 + 0.997939i \(0.479561\pi\)
\(740\) −46.7869 −1.71992
\(741\) −23.2745 −0.855010
\(742\) 0 0
\(743\) 22.0130 0.807577 0.403789 0.914852i \(-0.367693\pi\)
0.403789 + 0.914852i \(0.367693\pi\)
\(744\) 60.1500 2.20521
\(745\) 9.40566 0.344597
\(746\) 61.3840 2.24743
\(747\) 12.4983 0.457288
\(748\) 9.92251 0.362803
\(749\) 0 0
\(750\) −33.8451 −1.23585
\(751\) 15.4339 0.563192 0.281596 0.959533i \(-0.409136\pi\)
0.281596 + 0.959533i \(0.409136\pi\)
\(752\) −5.38504 −0.196372
\(753\) 5.72848 0.208757
\(754\) 65.4397 2.38317
\(755\) −40.8587 −1.48700
\(756\) 0 0
\(757\) −13.1778 −0.478955 −0.239478 0.970902i \(-0.576976\pi\)
−0.239478 + 0.970902i \(0.576976\pi\)
\(758\) 27.7146 1.00664
\(759\) 2.68769 0.0975568
\(760\) −116.198 −4.21493
\(761\) 10.7069 0.388123 0.194062 0.980989i \(-0.437834\pi\)
0.194062 + 0.980989i \(0.437834\pi\)
\(762\) 50.9618 1.84615
\(763\) 0 0
\(764\) −67.5956 −2.44552
\(765\) 1.23132 0.0445186
\(766\) −49.0993 −1.77403
\(767\) 30.6720 1.10750
\(768\) −107.851 −3.89172
\(769\) 30.1544 1.08739 0.543697 0.839282i \(-0.317024\pi\)
0.543697 + 0.839282i \(0.317024\pi\)
\(770\) 0 0
\(771\) 17.2297 0.620512
\(772\) 33.2717 1.19747
\(773\) 46.2383 1.66307 0.831537 0.555469i \(-0.187461\pi\)
0.831537 + 0.555469i \(0.187461\pi\)
\(774\) −5.56077 −0.199878
\(775\) 6.76762 0.243100
\(776\) −31.7169 −1.13857
\(777\) 0 0
\(778\) 53.8860 1.93191
\(779\) 28.8533 1.03378
\(780\) 47.6312 1.70547
\(781\) −16.1322 −0.577256
\(782\) −1.77047 −0.0633118
\(783\) −5.57812 −0.199346
\(784\) 0 0
\(785\) −3.10449 −0.110804
\(786\) 23.3164 0.831667
\(787\) 36.7305 1.30930 0.654651 0.755931i \(-0.272816\pi\)
0.654651 + 0.755931i \(0.272816\pi\)
\(788\) −75.8240 −2.70112
\(789\) −23.5216 −0.837391
\(790\) −86.0734 −3.06236
\(791\) 0 0
\(792\) −28.8780 −1.02614
\(793\) −26.3446 −0.935526
\(794\) −68.8690 −2.44407
\(795\) −19.8060 −0.702445
\(796\) −124.496 −4.41263
\(797\) −32.2468 −1.14224 −0.571120 0.820867i \(-0.693491\pi\)
−0.571120 + 0.820867i \(0.693491\pi\)
\(798\) 0 0
\(799\) 0.185031 0.00654592
\(800\) −36.3125 −1.28384
\(801\) 1.81155 0.0640080
\(802\) −22.0753 −0.779507
\(803\) −37.2581 −1.31481
\(804\) −18.3820 −0.648282
\(805\) 0 0
\(806\) −65.6750 −2.31330
\(807\) 13.8092 0.486108
\(808\) −137.544 −4.83880
\(809\) 6.80062 0.239097 0.119548 0.992828i \(-0.461855\pi\)
0.119548 + 0.992828i \(0.461855\pi\)
\(810\) −5.45107 −0.191531
\(811\) −32.1358 −1.12844 −0.564220 0.825625i \(-0.690823\pi\)
−0.564220 + 0.825625i \(0.690823\pi\)
\(812\) 0 0
\(813\) 22.4422 0.787083
\(814\) −30.9717 −1.08556
\(815\) 21.2186 0.743255
\(816\) 11.6392 0.407454
\(817\) −11.0322 −0.385967
\(818\) 13.4571 0.470517
\(819\) 0 0
\(820\) −59.0482 −2.06205
\(821\) −30.7924 −1.07466 −0.537331 0.843372i \(-0.680567\pi\)
−0.537331 + 0.843372i \(0.680567\pi\)
\(822\) 37.2471 1.29914
\(823\) −55.2351 −1.92537 −0.962687 0.270618i \(-0.912772\pi\)
−0.962687 + 0.270618i \(0.912772\pi\)
\(824\) 140.303 4.88770
\(825\) −3.24913 −0.113120
\(826\) 0 0
\(827\) 25.1429 0.874303 0.437152 0.899388i \(-0.355987\pi\)
0.437152 + 0.899388i \(0.355987\pi\)
\(828\) 5.83786 0.202880
\(829\) 19.7977 0.687604 0.343802 0.939042i \(-0.388285\pi\)
0.343802 + 0.939042i \(0.388285\pi\)
\(830\) −68.1289 −2.36479
\(831\) −22.2276 −0.771068
\(832\) 198.139 6.86924
\(833\) 0 0
\(834\) 16.6107 0.575181
\(835\) 30.3609 1.05068
\(836\) −87.1482 −3.01408
\(837\) 5.59818 0.193501
\(838\) 12.2431 0.422930
\(839\) 32.6430 1.12696 0.563480 0.826130i \(-0.309462\pi\)
0.563480 + 0.826130i \(0.309462\pi\)
\(840\) 0 0
\(841\) 2.11543 0.0729458
\(842\) −43.9243 −1.51373
\(843\) −6.99680 −0.240982
\(844\) 44.9043 1.54567
\(845\) −8.87740 −0.305392
\(846\) −0.819132 −0.0281623
\(847\) 0 0
\(848\) −187.218 −6.42909
\(849\) −8.06377 −0.276748
\(850\) 2.14031 0.0734121
\(851\) 4.11612 0.141099
\(852\) −35.0404 −1.20047
\(853\) 50.0984 1.71534 0.857668 0.514204i \(-0.171913\pi\)
0.857668 + 0.514204i \(0.171913\pi\)
\(854\) 0 0
\(855\) −10.8146 −0.369850
\(856\) −147.433 −5.03917
\(857\) 3.60431 0.123121 0.0615605 0.998103i \(-0.480392\pi\)
0.0615605 + 0.998103i \(0.480392\pi\)
\(858\) 31.5306 1.07644
\(859\) 4.40656 0.150350 0.0751749 0.997170i \(-0.476048\pi\)
0.0751749 + 0.997170i \(0.476048\pi\)
\(860\) 22.5773 0.769881
\(861\) 0 0
\(862\) −9.36845 −0.319091
\(863\) 47.1708 1.60571 0.802857 0.596172i \(-0.203312\pi\)
0.802857 + 0.596172i \(0.203312\pi\)
\(864\) −30.0377 −1.02190
\(865\) 21.9476 0.746242
\(866\) −42.6307 −1.44865
\(867\) 16.6001 0.563768
\(868\) 0 0
\(869\) −42.4391 −1.43965
\(870\) 30.4067 1.03088
\(871\) 13.1945 0.447078
\(872\) −48.4140 −1.63951
\(873\) −2.95190 −0.0999066
\(874\) 15.5498 0.525980
\(875\) 0 0
\(876\) −80.9276 −2.73429
\(877\) 41.0405 1.38584 0.692920 0.721014i \(-0.256324\pi\)
0.692920 + 0.721014i \(0.256324\pi\)
\(878\) 60.1678 2.03057
\(879\) 13.1458 0.443397
\(880\) 96.3154 3.24679
\(881\) 50.3825 1.69743 0.848715 0.528851i \(-0.177377\pi\)
0.848715 + 0.528851i \(0.177377\pi\)
\(882\) 0 0
\(883\) 7.65081 0.257470 0.128735 0.991679i \(-0.458908\pi\)
0.128735 + 0.991679i \(0.458908\pi\)
\(884\) −15.4703 −0.520321
\(885\) 14.2518 0.479070
\(886\) 66.5877 2.23706
\(887\) −37.3128 −1.25284 −0.626421 0.779485i \(-0.715481\pi\)
−0.626421 + 0.779485i \(0.715481\pi\)
\(888\) −44.2259 −1.48412
\(889\) 0 0
\(890\) −9.87488 −0.331007
\(891\) −2.68769 −0.0900409
\(892\) 122.414 4.09871
\(893\) −1.62510 −0.0543820
\(894\) 13.5240 0.452311
\(895\) −15.9175 −0.532064
\(896\) 0 0
\(897\) −4.19039 −0.139913
\(898\) −23.4113 −0.781245
\(899\) −31.2273 −1.04149
\(900\) −7.05738 −0.235246
\(901\) 6.43284 0.214309
\(902\) −39.0883 −1.30150
\(903\) 0 0
\(904\) −176.225 −5.86114
\(905\) 5.84307 0.194230
\(906\) −58.7490 −1.95181
\(907\) 50.1224 1.66429 0.832144 0.554560i \(-0.187113\pi\)
0.832144 + 0.554560i \(0.187113\pi\)
\(908\) −141.135 −4.68373
\(909\) −12.8013 −0.424593
\(910\) 0 0
\(911\) 33.1001 1.09666 0.548328 0.836264i \(-0.315265\pi\)
0.548328 + 0.836264i \(0.315265\pi\)
\(912\) −102.226 −3.38503
\(913\) −33.5914 −1.11171
\(914\) −44.6018 −1.47530
\(915\) −12.2411 −0.404679
\(916\) −50.6642 −1.67399
\(917\) 0 0
\(918\) 1.77047 0.0584341
\(919\) −45.9534 −1.51586 −0.757931 0.652334i \(-0.773790\pi\)
−0.757931 + 0.652334i \(0.773790\pi\)
\(920\) −20.9205 −0.689727
\(921\) −15.1086 −0.497845
\(922\) −32.6924 −1.07667
\(923\) 25.1518 0.827883
\(924\) 0 0
\(925\) −4.97596 −0.163609
\(926\) −18.8747 −0.620261
\(927\) 13.0581 0.428884
\(928\) 167.554 5.50022
\(929\) −22.5360 −0.739383 −0.369692 0.929155i \(-0.620537\pi\)
−0.369692 + 0.929155i \(0.620537\pi\)
\(930\) −30.5161 −1.00066
\(931\) 0 0
\(932\) 15.7144 0.514741
\(933\) 21.4828 0.703317
\(934\) 43.3381 1.41806
\(935\) −3.30941 −0.108229
\(936\) 45.0239 1.47165
\(937\) −20.5555 −0.671520 −0.335760 0.941948i \(-0.608993\pi\)
−0.335760 + 0.941948i \(0.608993\pi\)
\(938\) 0 0
\(939\) −14.8382 −0.484227
\(940\) 3.32577 0.108475
\(941\) −39.8841 −1.30019 −0.650093 0.759855i \(-0.725270\pi\)
−0.650093 + 0.759855i \(0.725270\pi\)
\(942\) −4.46382 −0.145439
\(943\) 5.19481 0.169166
\(944\) 134.717 4.38466
\(945\) 0 0
\(946\) 14.9456 0.485923
\(947\) 2.73802 0.0889738 0.0444869 0.999010i \(-0.485835\pi\)
0.0444869 + 0.999010i \(0.485835\pi\)
\(948\) −92.1810 −2.99390
\(949\) 58.0894 1.88566
\(950\) −18.7981 −0.609891
\(951\) 4.70173 0.152464
\(952\) 0 0
\(953\) 20.6225 0.668027 0.334014 0.942568i \(-0.391597\pi\)
0.334014 + 0.942568i \(0.391597\pi\)
\(954\) −28.4782 −0.922015
\(955\) 22.5448 0.729534
\(956\) −0.305659 −0.00988571
\(957\) 14.9922 0.484630
\(958\) 51.9944 1.67986
\(959\) 0 0
\(960\) 92.0659 2.97141
\(961\) 0.339647 0.0109563
\(962\) 48.2882 1.55687
\(963\) −13.7217 −0.442175
\(964\) −1.74874 −0.0563231
\(965\) −11.0969 −0.357223
\(966\) 0 0
\(967\) 46.6896 1.50144 0.750719 0.660622i \(-0.229707\pi\)
0.750719 + 0.660622i \(0.229707\pi\)
\(968\) −40.5752 −1.30413
\(969\) 3.51249 0.112837
\(970\) 16.0910 0.516651
\(971\) −39.8063 −1.27745 −0.638723 0.769437i \(-0.720537\pi\)
−0.638723 + 0.769437i \(0.720537\pi\)
\(972\) −5.83786 −0.187250
\(973\) 0 0
\(974\) 50.8458 1.62920
\(975\) 5.06575 0.162234
\(976\) −115.710 −3.70380
\(977\) −32.2999 −1.03337 −0.516683 0.856177i \(-0.672833\pi\)
−0.516683 + 0.856177i \(0.672833\pi\)
\(978\) 30.5093 0.975580
\(979\) −4.86888 −0.155610
\(980\) 0 0
\(981\) −4.50591 −0.143863
\(982\) 39.7281 1.26777
\(983\) 57.0785 1.82052 0.910261 0.414035i \(-0.135881\pi\)
0.910261 + 0.414035i \(0.135881\pi\)
\(984\) −55.8160 −1.77935
\(985\) 25.2892 0.805781
\(986\) −9.87588 −0.314512
\(987\) 0 0
\(988\) 135.873 4.32271
\(989\) −1.98626 −0.0631594
\(990\) 14.6508 0.465632
\(991\) −37.0328 −1.17639 −0.588193 0.808721i \(-0.700160\pi\)
−0.588193 + 0.808721i \(0.700160\pi\)
\(992\) −168.156 −5.33897
\(993\) 7.87864 0.250021
\(994\) 0 0
\(995\) 41.5224 1.31635
\(996\) −72.9632 −2.31193
\(997\) −47.0188 −1.48910 −0.744550 0.667567i \(-0.767336\pi\)
−0.744550 + 0.667567i \(0.767336\pi\)
\(998\) 40.1430 1.27070
\(999\) −4.11612 −0.130228
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 3381.2.a.bi.1.10 10
7.2 even 3 483.2.i.h.277.1 20
7.4 even 3 483.2.i.h.415.1 yes 20
7.6 odd 2 3381.2.a.bj.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.1 20 7.2 even 3
483.2.i.h.415.1 yes 20 7.4 even 3
3381.2.a.bi.1.10 10 1.1 even 1 trivial
3381.2.a.bj.1.10 10 7.6 odd 2