Properties

Label 2541.2.a.v.1.2
Level 2541
Weight 2
Character 2541.1
Self dual yes
Analytic conductor 20.290
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\)
Character \(\chi\) = 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +1.61803 q^{5} -2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +1.61803 q^{5} -2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +3.61803 q^{10} -3.00000 q^{12} +1.23607 q^{13} -2.23607 q^{14} -1.61803 q^{15} -1.00000 q^{16} +1.85410 q^{17} +2.23607 q^{18} +3.85410 q^{19} +4.85410 q^{20} +1.00000 q^{21} +6.61803 q^{23} -2.23607 q^{24} -2.38197 q^{25} +2.76393 q^{26} -1.00000 q^{27} -3.00000 q^{28} +6.00000 q^{29} -3.61803 q^{30} +8.09017 q^{31} -6.70820 q^{32} +4.14590 q^{34} -1.61803 q^{35} +3.00000 q^{36} +2.38197 q^{37} +8.61803 q^{38} -1.23607 q^{39} +3.61803 q^{40} -9.61803 q^{41} +2.23607 q^{42} +3.70820 q^{43} +1.61803 q^{45} +14.7984 q^{46} -4.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} -5.32624 q^{50} -1.85410 q^{51} +3.70820 q^{52} +11.2361 q^{53} -2.23607 q^{54} -2.23607 q^{56} -3.85410 q^{57} +13.4164 q^{58} -1.23607 q^{59} -4.85410 q^{60} +18.0902 q^{62} -1.00000 q^{63} -13.0000 q^{64} +2.00000 q^{65} +5.56231 q^{68} -6.61803 q^{69} -3.61803 q^{70} +12.9443 q^{71} +2.23607 q^{72} -0.291796 q^{73} +5.32624 q^{74} +2.38197 q^{75} +11.5623 q^{76} -2.76393 q^{78} +10.0000 q^{79} -1.61803 q^{80} +1.00000 q^{81} -21.5066 q^{82} -7.52786 q^{83} +3.00000 q^{84} +3.00000 q^{85} +8.29180 q^{86} -6.00000 q^{87} +3.38197 q^{89} +3.61803 q^{90} -1.23607 q^{91} +19.8541 q^{92} -8.09017 q^{93} -10.0000 q^{94} +6.23607 q^{95} +6.70820 q^{96} -6.00000 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} + q^{5} - 2q^{7} + 2q^{9} + 5q^{10} - 6q^{12} - 2q^{13} - q^{15} - 2q^{16} - 3q^{17} + q^{19} + 3q^{20} + 2q^{21} + 11q^{23} - 7q^{25} + 10q^{26} - 2q^{27} - 6q^{28} + 12q^{29} - 5q^{30} + 5q^{31} + 15q^{34} - q^{35} + 6q^{36} + 7q^{37} + 15q^{38} + 2q^{39} + 5q^{40} - 17q^{41} - 6q^{43} + q^{45} + 5q^{46} + 2q^{48} + 2q^{49} + 5q^{50} + 3q^{51} - 6q^{52} + 18q^{53} - q^{57} + 2q^{59} - 3q^{60} + 25q^{62} - 2q^{63} - 26q^{64} + 4q^{65} - 9q^{68} - 11q^{69} - 5q^{70} + 8q^{71} - 14q^{73} - 5q^{74} + 7q^{75} + 3q^{76} - 10q^{78} + 20q^{79} - q^{80} + 2q^{81} - 5q^{82} - 24q^{83} + 6q^{84} + 6q^{85} + 30q^{86} - 12q^{87} + 9q^{89} + 5q^{90} + 2q^{91} + 33q^{92} - 5q^{93} - 20q^{94} + 8q^{95} - 12q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) 1.61803 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) −2.23607 −0.912871
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 3.61803 1.14412
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) −2.23607 −0.597614
\(15\) −1.61803 −0.417775
\(16\) −1.00000 −0.250000
\(17\) 1.85410 0.449686 0.224843 0.974395i \(-0.427813\pi\)
0.224843 + 0.974395i \(0.427813\pi\)
\(18\) 2.23607 0.527046
\(19\) 3.85410 0.884192 0.442096 0.896968i \(-0.354235\pi\)
0.442096 + 0.896968i \(0.354235\pi\)
\(20\) 4.85410 1.08541
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 6.61803 1.37996 0.689978 0.723831i \(-0.257620\pi\)
0.689978 + 0.723831i \(0.257620\pi\)
\(24\) −2.23607 −0.456435
\(25\) −2.38197 −0.476393
\(26\) 2.76393 0.542052
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −3.61803 −0.660560
\(31\) 8.09017 1.45304 0.726519 0.687147i \(-0.241137\pi\)
0.726519 + 0.687147i \(0.241137\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) 4.14590 0.711016
\(35\) −1.61803 −0.273498
\(36\) 3.00000 0.500000
\(37\) 2.38197 0.391593 0.195796 0.980645i \(-0.437271\pi\)
0.195796 + 0.980645i \(0.437271\pi\)
\(38\) 8.61803 1.39803
\(39\) −1.23607 −0.197929
\(40\) 3.61803 0.572061
\(41\) −9.61803 −1.50208 −0.751042 0.660254i \(-0.770449\pi\)
−0.751042 + 0.660254i \(0.770449\pi\)
\(42\) 2.23607 0.345033
\(43\) 3.70820 0.565496 0.282748 0.959194i \(-0.408754\pi\)
0.282748 + 0.959194i \(0.408754\pi\)
\(44\) 0 0
\(45\) 1.61803 0.241202
\(46\) 14.7984 2.18190
\(47\) −4.47214 −0.652328 −0.326164 0.945313i \(-0.605756\pi\)
−0.326164 + 0.945313i \(0.605756\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −5.32624 −0.753244
\(51\) −1.85410 −0.259626
\(52\) 3.70820 0.514235
\(53\) 11.2361 1.54339 0.771696 0.635991i \(-0.219409\pi\)
0.771696 + 0.635991i \(0.219409\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) −3.85410 −0.510488
\(58\) 13.4164 1.76166
\(59\) −1.23607 −0.160922 −0.0804612 0.996758i \(-0.525639\pi\)
−0.0804612 + 0.996758i \(0.525639\pi\)
\(60\) −4.85410 −0.626662
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 18.0902 2.29745
\(63\) −1.00000 −0.125988
\(64\) −13.0000 −1.62500
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 5.56231 0.674529
\(69\) −6.61803 −0.796718
\(70\) −3.61803 −0.432438
\(71\) 12.9443 1.53620 0.768101 0.640328i \(-0.221202\pi\)
0.768101 + 0.640328i \(0.221202\pi\)
\(72\) 2.23607 0.263523
\(73\) −0.291796 −0.0341521 −0.0170761 0.999854i \(-0.505436\pi\)
−0.0170761 + 0.999854i \(0.505436\pi\)
\(74\) 5.32624 0.619163
\(75\) 2.38197 0.275046
\(76\) 11.5623 1.32629
\(77\) 0 0
\(78\) −2.76393 −0.312954
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −1.61803 −0.180902
\(81\) 1.00000 0.111111
\(82\) −21.5066 −2.37500
\(83\) −7.52786 −0.826290 −0.413145 0.910665i \(-0.635570\pi\)
−0.413145 + 0.910665i \(0.635570\pi\)
\(84\) 3.00000 0.327327
\(85\) 3.00000 0.325396
\(86\) 8.29180 0.894127
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 3.38197 0.358488 0.179244 0.983805i \(-0.442635\pi\)
0.179244 + 0.983805i \(0.442635\pi\)
\(90\) 3.61803 0.381374
\(91\) −1.23607 −0.129575
\(92\) 19.8541 2.06993
\(93\) −8.09017 −0.838912
\(94\) −10.0000 −1.03142
\(95\) 6.23607 0.639807
\(96\) 6.70820 0.684653
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 2.23607 0.225877
\(99\) 0 0
\(100\) −7.14590 −0.714590
\(101\) −17.7984 −1.77100 −0.885502 0.464635i \(-0.846185\pi\)
−0.885502 + 0.464635i \(0.846185\pi\)
\(102\) −4.14590 −0.410505
\(103\) −14.3262 −1.41161 −0.705803 0.708408i \(-0.749414\pi\)
−0.705803 + 0.708408i \(0.749414\pi\)
\(104\) 2.76393 0.271026
\(105\) 1.61803 0.157904
\(106\) 25.1246 2.44032
\(107\) 8.79837 0.850571 0.425285 0.905059i \(-0.360174\pi\)
0.425285 + 0.905059i \(0.360174\pi\)
\(108\) −3.00000 −0.288675
\(109\) −2.56231 −0.245424 −0.122712 0.992442i \(-0.539159\pi\)
−0.122712 + 0.992442i \(0.539159\pi\)
\(110\) 0 0
\(111\) −2.38197 −0.226086
\(112\) 1.00000 0.0944911
\(113\) −0.763932 −0.0718647 −0.0359323 0.999354i \(-0.511440\pi\)
−0.0359323 + 0.999354i \(0.511440\pi\)
\(114\) −8.61803 −0.807153
\(115\) 10.7082 0.998545
\(116\) 18.0000 1.67126
\(117\) 1.23607 0.114275
\(118\) −2.76393 −0.254441
\(119\) −1.85410 −0.169965
\(120\) −3.61803 −0.330280
\(121\) 0 0
\(122\) 0 0
\(123\) 9.61803 0.867229
\(124\) 24.2705 2.17956
\(125\) −11.9443 −1.06833
\(126\) −2.23607 −0.199205
\(127\) −13.7082 −1.21641 −0.608203 0.793781i \(-0.708109\pi\)
−0.608203 + 0.793781i \(0.708109\pi\)
\(128\) −15.6525 −1.38350
\(129\) −3.70820 −0.326489
\(130\) 4.47214 0.392232
\(131\) −14.6525 −1.28019 −0.640096 0.768295i \(-0.721106\pi\)
−0.640096 + 0.768295i \(0.721106\pi\)
\(132\) 0 0
\(133\) −3.85410 −0.334193
\(134\) 0 0
\(135\) −1.61803 −0.139258
\(136\) 4.14590 0.355508
\(137\) −14.1803 −1.21151 −0.605754 0.795652i \(-0.707128\pi\)
−0.605754 + 0.795652i \(0.707128\pi\)
\(138\) −14.7984 −1.25972
\(139\) −19.8541 −1.68400 −0.842001 0.539475i \(-0.818623\pi\)
−0.842001 + 0.539475i \(0.818623\pi\)
\(140\) −4.85410 −0.410246
\(141\) 4.47214 0.376622
\(142\) 28.9443 2.42895
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 9.70820 0.806222
\(146\) −0.652476 −0.0539993
\(147\) −1.00000 −0.0824786
\(148\) 7.14590 0.587389
\(149\) −12.9443 −1.06044 −0.530218 0.847861i \(-0.677890\pi\)
−0.530218 + 0.847861i \(0.677890\pi\)
\(150\) 5.32624 0.434886
\(151\) 24.4721 1.99151 0.995757 0.0920207i \(-0.0293326\pi\)
0.995757 + 0.0920207i \(0.0293326\pi\)
\(152\) 8.61803 0.699015
\(153\) 1.85410 0.149895
\(154\) 0 0
\(155\) 13.0902 1.05143
\(156\) −3.70820 −0.296894
\(157\) 17.1246 1.36669 0.683346 0.730094i \(-0.260524\pi\)
0.683346 + 0.730094i \(0.260524\pi\)
\(158\) 22.3607 1.77892
\(159\) −11.2361 −0.891078
\(160\) −10.8541 −0.858092
\(161\) −6.61803 −0.521574
\(162\) 2.23607 0.175682
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −28.8541 −2.25313
\(165\) 0 0
\(166\) −16.8328 −1.30648
\(167\) 16.7639 1.29723 0.648616 0.761116i \(-0.275348\pi\)
0.648616 + 0.761116i \(0.275348\pi\)
\(168\) 2.23607 0.172516
\(169\) −11.4721 −0.882472
\(170\) 6.70820 0.514496
\(171\) 3.85410 0.294731
\(172\) 11.1246 0.848244
\(173\) 11.3820 0.865355 0.432677 0.901549i \(-0.357569\pi\)
0.432677 + 0.901549i \(0.357569\pi\)
\(174\) −13.4164 −1.01710
\(175\) 2.38197 0.180060
\(176\) 0 0
\(177\) 1.23607 0.0929086
\(178\) 7.56231 0.566819
\(179\) 9.79837 0.732365 0.366182 0.930543i \(-0.380665\pi\)
0.366182 + 0.930543i \(0.380665\pi\)
\(180\) 4.85410 0.361803
\(181\) 15.8885 1.18099 0.590493 0.807043i \(-0.298933\pi\)
0.590493 + 0.807043i \(0.298933\pi\)
\(182\) −2.76393 −0.204876
\(183\) 0 0
\(184\) 14.7984 1.09095
\(185\) 3.85410 0.283359
\(186\) −18.0902 −1.32644
\(187\) 0 0
\(188\) −13.4164 −0.978492
\(189\) 1.00000 0.0727393
\(190\) 13.9443 1.01162
\(191\) 16.0902 1.16424 0.582122 0.813102i \(-0.302223\pi\)
0.582122 + 0.813102i \(0.302223\pi\)
\(192\) 13.0000 0.938194
\(193\) 15.1459 1.09023 0.545113 0.838363i \(-0.316487\pi\)
0.545113 + 0.838363i \(0.316487\pi\)
\(194\) −13.4164 −0.963242
\(195\) −2.00000 −0.143223
\(196\) 3.00000 0.214286
\(197\) −13.5279 −0.963820 −0.481910 0.876221i \(-0.660057\pi\)
−0.481910 + 0.876221i \(0.660057\pi\)
\(198\) 0 0
\(199\) 19.8541 1.40742 0.703710 0.710487i \(-0.251525\pi\)
0.703710 + 0.710487i \(0.251525\pi\)
\(200\) −5.32624 −0.376622
\(201\) 0 0
\(202\) −39.7984 −2.80020
\(203\) −6.00000 −0.421117
\(204\) −5.56231 −0.389439
\(205\) −15.5623 −1.08692
\(206\) −32.0344 −2.23195
\(207\) 6.61803 0.459985
\(208\) −1.23607 −0.0857059
\(209\) 0 0
\(210\) 3.61803 0.249668
\(211\) −26.3607 −1.81474 −0.907372 0.420328i \(-0.861915\pi\)
−0.907372 + 0.420328i \(0.861915\pi\)
\(212\) 33.7082 2.31509
\(213\) −12.9443 −0.886927
\(214\) 19.6738 1.34487
\(215\) 6.00000 0.409197
\(216\) −2.23607 −0.152145
\(217\) −8.09017 −0.549197
\(218\) −5.72949 −0.388050
\(219\) 0.291796 0.0197178
\(220\) 0 0
\(221\) 2.29180 0.154163
\(222\) −5.32624 −0.357474
\(223\) 17.3820 1.16398 0.581991 0.813195i \(-0.302274\pi\)
0.581991 + 0.813195i \(0.302274\pi\)
\(224\) 6.70820 0.448211
\(225\) −2.38197 −0.158798
\(226\) −1.70820 −0.113628
\(227\) −3.52786 −0.234153 −0.117076 0.993123i \(-0.537352\pi\)
−0.117076 + 0.993123i \(0.537352\pi\)
\(228\) −11.5623 −0.765732
\(229\) −6.18034 −0.408408 −0.204204 0.978928i \(-0.565461\pi\)
−0.204204 + 0.978928i \(0.565461\pi\)
\(230\) 23.9443 1.57884
\(231\) 0 0
\(232\) 13.4164 0.880830
\(233\) 2.29180 0.150141 0.0750703 0.997178i \(-0.476082\pi\)
0.0750703 + 0.997178i \(0.476082\pi\)
\(234\) 2.76393 0.180684
\(235\) −7.23607 −0.472029
\(236\) −3.70820 −0.241384
\(237\) −10.0000 −0.649570
\(238\) −4.14590 −0.268739
\(239\) −10.0344 −0.649074 −0.324537 0.945873i \(-0.605208\pi\)
−0.324537 + 0.945873i \(0.605208\pi\)
\(240\) 1.61803 0.104444
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.61803 0.103372
\(246\) 21.5066 1.37121
\(247\) 4.76393 0.303122
\(248\) 18.0902 1.14873
\(249\) 7.52786 0.477059
\(250\) −26.7082 −1.68918
\(251\) 14.2918 0.902090 0.451045 0.892501i \(-0.351051\pi\)
0.451045 + 0.892501i \(0.351051\pi\)
\(252\) −3.00000 −0.188982
\(253\) 0 0
\(254\) −30.6525 −1.92331
\(255\) −3.00000 −0.187867
\(256\) −9.00000 −0.562500
\(257\) −31.4508 −1.96185 −0.980925 0.194386i \(-0.937728\pi\)
−0.980925 + 0.194386i \(0.937728\pi\)
\(258\) −8.29180 −0.516225
\(259\) −2.38197 −0.148008
\(260\) 6.00000 0.372104
\(261\) 6.00000 0.371391
\(262\) −32.7639 −2.02416
\(263\) −4.14590 −0.255647 −0.127824 0.991797i \(-0.540799\pi\)
−0.127824 + 0.991797i \(0.540799\pi\)
\(264\) 0 0
\(265\) 18.1803 1.11681
\(266\) −8.61803 −0.528406
\(267\) −3.38197 −0.206973
\(268\) 0 0
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) −3.61803 −0.220187
\(271\) −15.5066 −0.941958 −0.470979 0.882145i \(-0.656099\pi\)
−0.470979 + 0.882145i \(0.656099\pi\)
\(272\) −1.85410 −0.112421
\(273\) 1.23607 0.0748102
\(274\) −31.7082 −1.91556
\(275\) 0 0
\(276\) −19.8541 −1.19508
\(277\) −11.0902 −0.666344 −0.333172 0.942866i \(-0.608119\pi\)
−0.333172 + 0.942866i \(0.608119\pi\)
\(278\) −44.3951 −2.66264
\(279\) 8.09017 0.484346
\(280\) −3.61803 −0.216219
\(281\) 9.05573 0.540219 0.270110 0.962830i \(-0.412940\pi\)
0.270110 + 0.962830i \(0.412940\pi\)
\(282\) 10.0000 0.595491
\(283\) −16.2705 −0.967181 −0.483591 0.875294i \(-0.660668\pi\)
−0.483591 + 0.875294i \(0.660668\pi\)
\(284\) 38.8328 2.30430
\(285\) −6.23607 −0.369393
\(286\) 0 0
\(287\) 9.61803 0.567735
\(288\) −6.70820 −0.395285
\(289\) −13.5623 −0.797783
\(290\) 21.7082 1.27475
\(291\) 6.00000 0.351726
\(292\) −0.875388 −0.0512282
\(293\) −33.2148 −1.94043 −0.970214 0.242249i \(-0.922115\pi\)
−0.970214 + 0.242249i \(0.922115\pi\)
\(294\) −2.23607 −0.130410
\(295\) −2.00000 −0.116445
\(296\) 5.32624 0.309581
\(297\) 0 0
\(298\) −28.9443 −1.67670
\(299\) 8.18034 0.473081
\(300\) 7.14590 0.412569
\(301\) −3.70820 −0.213737
\(302\) 54.7214 3.14886
\(303\) 17.7984 1.02249
\(304\) −3.85410 −0.221048
\(305\) 0 0
\(306\) 4.14590 0.237005
\(307\) −2.56231 −0.146239 −0.0731193 0.997323i \(-0.523295\pi\)
−0.0731193 + 0.997323i \(0.523295\pi\)
\(308\) 0 0
\(309\) 14.3262 0.814991
\(310\) 29.2705 1.66245
\(311\) −10.6525 −0.604046 −0.302023 0.953301i \(-0.597662\pi\)
−0.302023 + 0.953301i \(0.597662\pi\)
\(312\) −2.76393 −0.156477
\(313\) −30.1803 −1.70589 −0.852947 0.521998i \(-0.825187\pi\)
−0.852947 + 0.521998i \(0.825187\pi\)
\(314\) 38.2918 2.16093
\(315\) −1.61803 −0.0911659
\(316\) 30.0000 1.68763
\(317\) 11.2361 0.631080 0.315540 0.948912i \(-0.397814\pi\)
0.315540 + 0.948912i \(0.397814\pi\)
\(318\) −25.1246 −1.40892
\(319\) 0 0
\(320\) −21.0344 −1.17586
\(321\) −8.79837 −0.491077
\(322\) −14.7984 −0.824681
\(323\) 7.14590 0.397608
\(324\) 3.00000 0.166667
\(325\) −2.94427 −0.163319
\(326\) −22.3607 −1.23844
\(327\) 2.56231 0.141696
\(328\) −21.5066 −1.18750
\(329\) 4.47214 0.246557
\(330\) 0 0
\(331\) −9.41641 −0.517573 −0.258786 0.965935i \(-0.583323\pi\)
−0.258786 + 0.965935i \(0.583323\pi\)
\(332\) −22.5836 −1.23944
\(333\) 2.38197 0.130531
\(334\) 37.4853 2.05110
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −15.9098 −0.866664 −0.433332 0.901234i \(-0.642662\pi\)
−0.433332 + 0.901234i \(0.642662\pi\)
\(338\) −25.6525 −1.39531
\(339\) 0.763932 0.0414911
\(340\) 9.00000 0.488094
\(341\) 0 0
\(342\) 8.61803 0.466010
\(343\) −1.00000 −0.0539949
\(344\) 8.29180 0.447064
\(345\) −10.7082 −0.576510
\(346\) 25.4508 1.36825
\(347\) −23.5066 −1.26190 −0.630950 0.775824i \(-0.717335\pi\)
−0.630950 + 0.775824i \(0.717335\pi\)
\(348\) −18.0000 −0.964901
\(349\) 8.29180 0.443850 0.221925 0.975064i \(-0.428766\pi\)
0.221925 + 0.975064i \(0.428766\pi\)
\(350\) 5.32624 0.284699
\(351\) −1.23607 −0.0659764
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 2.76393 0.146901
\(355\) 20.9443 1.11161
\(356\) 10.1459 0.537732
\(357\) 1.85410 0.0981295
\(358\) 21.9098 1.15797
\(359\) −26.5623 −1.40190 −0.700952 0.713208i \(-0.747241\pi\)
−0.700952 + 0.713208i \(0.747241\pi\)
\(360\) 3.61803 0.190687
\(361\) −4.14590 −0.218205
\(362\) 35.5279 1.86730
\(363\) 0 0
\(364\) −3.70820 −0.194363
\(365\) −0.472136 −0.0247127
\(366\) 0 0
\(367\) 36.2705 1.89331 0.946653 0.322256i \(-0.104441\pi\)
0.946653 + 0.322256i \(0.104441\pi\)
\(368\) −6.61803 −0.344989
\(369\) −9.61803 −0.500695
\(370\) 8.61803 0.448030
\(371\) −11.2361 −0.583348
\(372\) −24.2705 −1.25837
\(373\) −0.673762 −0.0348861 −0.0174430 0.999848i \(-0.505553\pi\)
−0.0174430 + 0.999848i \(0.505553\pi\)
\(374\) 0 0
\(375\) 11.9443 0.616800
\(376\) −10.0000 −0.515711
\(377\) 7.41641 0.381964
\(378\) 2.23607 0.115011
\(379\) 30.4721 1.56525 0.782624 0.622494i \(-0.213881\pi\)
0.782624 + 0.622494i \(0.213881\pi\)
\(380\) 18.7082 0.959711
\(381\) 13.7082 0.702293
\(382\) 35.9787 1.84083
\(383\) 32.8328 1.67768 0.838839 0.544379i \(-0.183235\pi\)
0.838839 + 0.544379i \(0.183235\pi\)
\(384\) 15.6525 0.798762
\(385\) 0 0
\(386\) 33.8673 1.72380
\(387\) 3.70820 0.188499
\(388\) −18.0000 −0.913812
\(389\) 10.7639 0.545753 0.272877 0.962049i \(-0.412025\pi\)
0.272877 + 0.962049i \(0.412025\pi\)
\(390\) −4.47214 −0.226455
\(391\) 12.2705 0.620546
\(392\) 2.23607 0.112938
\(393\) 14.6525 0.739120
\(394\) −30.2492 −1.52393
\(395\) 16.1803 0.814121
\(396\) 0 0
\(397\) −36.5410 −1.83394 −0.916971 0.398955i \(-0.869373\pi\)
−0.916971 + 0.398955i \(0.869373\pi\)
\(398\) 44.3951 2.22533
\(399\) 3.85410 0.192946
\(400\) 2.38197 0.119098
\(401\) −19.4164 −0.969609 −0.484805 0.874623i \(-0.661109\pi\)
−0.484805 + 0.874623i \(0.661109\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) −53.3951 −2.65651
\(405\) 1.61803 0.0804008
\(406\) −13.4164 −0.665845
\(407\) 0 0
\(408\) −4.14590 −0.205253
\(409\) −0.180340 −0.00891723 −0.00445862 0.999990i \(-0.501419\pi\)
−0.00445862 + 0.999990i \(0.501419\pi\)
\(410\) −34.7984 −1.71857
\(411\) 14.1803 0.699465
\(412\) −42.9787 −2.11741
\(413\) 1.23607 0.0608229
\(414\) 14.7984 0.727300
\(415\) −12.1803 −0.597909
\(416\) −8.29180 −0.406539
\(417\) 19.8541 0.972260
\(418\) 0 0
\(419\) 1.23607 0.0603859 0.0301929 0.999544i \(-0.490388\pi\)
0.0301929 + 0.999544i \(0.490388\pi\)
\(420\) 4.85410 0.236856
\(421\) −14.7984 −0.721229 −0.360614 0.932715i \(-0.617433\pi\)
−0.360614 + 0.932715i \(0.617433\pi\)
\(422\) −58.9443 −2.86936
\(423\) −4.47214 −0.217443
\(424\) 25.1246 1.22016
\(425\) −4.41641 −0.214227
\(426\) −28.9443 −1.40235
\(427\) 0 0
\(428\) 26.3951 1.27586
\(429\) 0 0
\(430\) 13.4164 0.646997
\(431\) −3.90983 −0.188330 −0.0941649 0.995557i \(-0.530018\pi\)
−0.0941649 + 0.995557i \(0.530018\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.7639 −1.28619 −0.643096 0.765785i \(-0.722350\pi\)
−0.643096 + 0.765785i \(0.722350\pi\)
\(434\) −18.0902 −0.868356
\(435\) −9.70820 −0.465473
\(436\) −7.68692 −0.368137
\(437\) 25.5066 1.22015
\(438\) 0.652476 0.0311765
\(439\) 4.85410 0.231674 0.115837 0.993268i \(-0.463045\pi\)
0.115837 + 0.993268i \(0.463045\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 5.12461 0.243753
\(443\) 22.6180 1.07462 0.537308 0.843386i \(-0.319441\pi\)
0.537308 + 0.843386i \(0.319441\pi\)
\(444\) −7.14590 −0.339129
\(445\) 5.47214 0.259404
\(446\) 38.8673 1.84042
\(447\) 12.9443 0.612243
\(448\) 13.0000 0.614192
\(449\) −6.11146 −0.288417 −0.144209 0.989547i \(-0.546064\pi\)
−0.144209 + 0.989547i \(0.546064\pi\)
\(450\) −5.32624 −0.251081
\(451\) 0 0
\(452\) −2.29180 −0.107797
\(453\) −24.4721 −1.14980
\(454\) −7.88854 −0.370228
\(455\) −2.00000 −0.0937614
\(456\) −8.61803 −0.403576
\(457\) 5.41641 0.253369 0.126684 0.991943i \(-0.459566\pi\)
0.126684 + 0.991943i \(0.459566\pi\)
\(458\) −13.8197 −0.645750
\(459\) −1.85410 −0.0865421
\(460\) 32.1246 1.49782
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 3.41641 0.158774 0.0793870 0.996844i \(-0.474704\pi\)
0.0793870 + 0.996844i \(0.474704\pi\)
\(464\) −6.00000 −0.278543
\(465\) −13.0902 −0.607042
\(466\) 5.12461 0.237393
\(467\) 3.81966 0.176753 0.0883764 0.996087i \(-0.471832\pi\)
0.0883764 + 0.996087i \(0.471832\pi\)
\(468\) 3.70820 0.171412
\(469\) 0 0
\(470\) −16.1803 −0.746343
\(471\) −17.1246 −0.789060
\(472\) −2.76393 −0.127220
\(473\) 0 0
\(474\) −22.3607 −1.02706
\(475\) −9.18034 −0.421223
\(476\) −5.56231 −0.254948
\(477\) 11.2361 0.514464
\(478\) −22.4377 −1.02628
\(479\) −16.9443 −0.774204 −0.387102 0.922037i \(-0.626524\pi\)
−0.387102 + 0.922037i \(0.626524\pi\)
\(480\) 10.8541 0.495420
\(481\) 2.94427 0.134247
\(482\) −26.8328 −1.22220
\(483\) 6.61803 0.301131
\(484\) 0 0
\(485\) −9.70820 −0.440827
\(486\) −2.23607 −0.101430
\(487\) 14.2918 0.647623 0.323812 0.946122i \(-0.395036\pi\)
0.323812 + 0.946122i \(0.395036\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 3.61803 0.163446
\(491\) 36.5066 1.64752 0.823759 0.566940i \(-0.191873\pi\)
0.823759 + 0.566940i \(0.191873\pi\)
\(492\) 28.8541 1.30084
\(493\) 11.1246 0.501027
\(494\) 10.6525 0.479278
\(495\) 0 0
\(496\) −8.09017 −0.363259
\(497\) −12.9443 −0.580630
\(498\) 16.8328 0.754297
\(499\) −19.1246 −0.856135 −0.428068 0.903747i \(-0.640805\pi\)
−0.428068 + 0.903747i \(0.640805\pi\)
\(500\) −35.8328 −1.60249
\(501\) −16.7639 −0.748957
\(502\) 31.9574 1.42633
\(503\) 5.52786 0.246475 0.123238 0.992377i \(-0.460672\pi\)
0.123238 + 0.992377i \(0.460672\pi\)
\(504\) −2.23607 −0.0996024
\(505\) −28.7984 −1.28151
\(506\) 0 0
\(507\) 11.4721 0.509495
\(508\) −41.1246 −1.82461
\(509\) −15.9098 −0.705191 −0.352595 0.935776i \(-0.614701\pi\)
−0.352595 + 0.935776i \(0.614701\pi\)
\(510\) −6.70820 −0.297044
\(511\) 0.291796 0.0129083
\(512\) 11.1803 0.494106
\(513\) −3.85410 −0.170163
\(514\) −70.3262 −3.10196
\(515\) −23.1803 −1.02145
\(516\) −11.1246 −0.489734
\(517\) 0 0
\(518\) −5.32624 −0.234021
\(519\) −11.3820 −0.499613
\(520\) 4.47214 0.196116
\(521\) 3.32624 0.145725 0.0728626 0.997342i \(-0.476787\pi\)
0.0728626 + 0.997342i \(0.476787\pi\)
\(522\) 13.4164 0.587220
\(523\) −19.8541 −0.868159 −0.434080 0.900875i \(-0.642926\pi\)
−0.434080 + 0.900875i \(0.642926\pi\)
\(524\) −43.9574 −1.92029
\(525\) −2.38197 −0.103958
\(526\) −9.27051 −0.404213
\(527\) 15.0000 0.653410
\(528\) 0 0
\(529\) 20.7984 0.904277
\(530\) 40.6525 1.76583
\(531\) −1.23607 −0.0536408
\(532\) −11.5623 −0.501290
\(533\) −11.8885 −0.514950
\(534\) −7.56231 −0.327253
\(535\) 14.2361 0.615479
\(536\) 0 0
\(537\) −9.79837 −0.422831
\(538\) 30.0000 1.29339
\(539\) 0 0
\(540\) −4.85410 −0.208887
\(541\) 0.673762 0.0289673 0.0144836 0.999895i \(-0.495390\pi\)
0.0144836 + 0.999895i \(0.495390\pi\)
\(542\) −34.6738 −1.48937
\(543\) −15.8885 −0.681843
\(544\) −12.4377 −0.533262
\(545\) −4.14590 −0.177591
\(546\) 2.76393 0.118285
\(547\) 27.5279 1.17701 0.588503 0.808495i \(-0.299717\pi\)
0.588503 + 0.808495i \(0.299717\pi\)
\(548\) −42.5410 −1.81726
\(549\) 0 0
\(550\) 0 0
\(551\) 23.1246 0.985142
\(552\) −14.7984 −0.629861
\(553\) −10.0000 −0.425243
\(554\) −24.7984 −1.05358
\(555\) −3.85410 −0.163598
\(556\) −59.5623 −2.52600
\(557\) 22.3607 0.947452 0.473726 0.880672i \(-0.342909\pi\)
0.473726 + 0.880672i \(0.342909\pi\)
\(558\) 18.0902 0.765818
\(559\) 4.58359 0.193865
\(560\) 1.61803 0.0683744
\(561\) 0 0
\(562\) 20.2492 0.854162
\(563\) 18.4721 0.778508 0.389254 0.921131i \(-0.372733\pi\)
0.389254 + 0.921131i \(0.372733\pi\)
\(564\) 13.4164 0.564933
\(565\) −1.23607 −0.0520018
\(566\) −36.3820 −1.52925
\(567\) −1.00000 −0.0419961
\(568\) 28.9443 1.21447
\(569\) 37.5967 1.57614 0.788069 0.615587i \(-0.211081\pi\)
0.788069 + 0.615587i \(0.211081\pi\)
\(570\) −13.9443 −0.584061
\(571\) −46.6525 −1.95235 −0.976173 0.216995i \(-0.930374\pi\)
−0.976173 + 0.216995i \(0.930374\pi\)
\(572\) 0 0
\(573\) −16.0902 −0.672176
\(574\) 21.5066 0.897667
\(575\) −15.7639 −0.657401
\(576\) −13.0000 −0.541667
\(577\) −22.2918 −0.928020 −0.464010 0.885830i \(-0.653590\pi\)
−0.464010 + 0.885830i \(0.653590\pi\)
\(578\) −30.3262 −1.26141
\(579\) −15.1459 −0.629442
\(580\) 29.1246 1.20933
\(581\) 7.52786 0.312308
\(582\) 13.4164 0.556128
\(583\) 0 0
\(584\) −0.652476 −0.0269996
\(585\) 2.00000 0.0826898
\(586\) −74.2705 −3.06809
\(587\) 25.4164 1.04905 0.524524 0.851396i \(-0.324243\pi\)
0.524524 + 0.851396i \(0.324243\pi\)
\(588\) −3.00000 −0.123718
\(589\) 31.1803 1.28476
\(590\) −4.47214 −0.184115
\(591\) 13.5279 0.556462
\(592\) −2.38197 −0.0978982
\(593\) 15.5066 0.636779 0.318389 0.947960i \(-0.396858\pi\)
0.318389 + 0.947960i \(0.396858\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −38.8328 −1.59065
\(597\) −19.8541 −0.812574
\(598\) 18.2918 0.748007
\(599\) −21.3820 −0.873643 −0.436822 0.899548i \(-0.643896\pi\)
−0.436822 + 0.899548i \(0.643896\pi\)
\(600\) 5.32624 0.217443
\(601\) −12.4721 −0.508749 −0.254375 0.967106i \(-0.581870\pi\)
−0.254375 + 0.967106i \(0.581870\pi\)
\(602\) −8.29180 −0.337948
\(603\) 0 0
\(604\) 73.4164 2.98727
\(605\) 0 0
\(606\) 39.7984 1.61670
\(607\) −34.9787 −1.41974 −0.709871 0.704332i \(-0.751247\pi\)
−0.709871 + 0.704332i \(0.751247\pi\)
\(608\) −25.8541 −1.04852
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) −5.52786 −0.223633
\(612\) 5.56231 0.224843
\(613\) −24.9787 −1.00888 −0.504440 0.863447i \(-0.668301\pi\)
−0.504440 + 0.863447i \(0.668301\pi\)
\(614\) −5.72949 −0.231223
\(615\) 15.5623 0.627533
\(616\) 0 0
\(617\) 44.0689 1.77415 0.887073 0.461629i \(-0.152735\pi\)
0.887073 + 0.461629i \(0.152735\pi\)
\(618\) 32.0344 1.28861
\(619\) 23.6869 0.952058 0.476029 0.879430i \(-0.342076\pi\)
0.476029 + 0.879430i \(0.342076\pi\)
\(620\) 39.2705 1.57714
\(621\) −6.61803 −0.265573
\(622\) −23.8197 −0.955081
\(623\) −3.38197 −0.135496
\(624\) 1.23607 0.0494823
\(625\) −7.41641 −0.296656
\(626\) −67.4853 −2.69725
\(627\) 0 0
\(628\) 51.3738 2.05004
\(629\) 4.41641 0.176094
\(630\) −3.61803 −0.144146
\(631\) 17.3050 0.688899 0.344450 0.938805i \(-0.388066\pi\)
0.344450 + 0.938805i \(0.388066\pi\)
\(632\) 22.3607 0.889460
\(633\) 26.3607 1.04774
\(634\) 25.1246 0.997826
\(635\) −22.1803 −0.880200
\(636\) −33.7082 −1.33662
\(637\) 1.23607 0.0489748
\(638\) 0 0
\(639\) 12.9443 0.512067
\(640\) −25.3262 −1.00111
\(641\) 17.2361 0.680784 0.340392 0.940284i \(-0.389440\pi\)
0.340392 + 0.940284i \(0.389440\pi\)
\(642\) −19.6738 −0.776461
\(643\) −23.2705 −0.917699 −0.458850 0.888514i \(-0.651738\pi\)
−0.458850 + 0.888514i \(0.651738\pi\)
\(644\) −19.8541 −0.782361
\(645\) −6.00000 −0.236250
\(646\) 15.9787 0.628674
\(647\) 27.5279 1.08223 0.541116 0.840948i \(-0.318002\pi\)
0.541116 + 0.840948i \(0.318002\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) −6.58359 −0.258230
\(651\) 8.09017 0.317079
\(652\) −30.0000 −1.17489
\(653\) −28.3607 −1.10984 −0.554920 0.831904i \(-0.687251\pi\)
−0.554920 + 0.831904i \(0.687251\pi\)
\(654\) 5.72949 0.224041
\(655\) −23.7082 −0.926356
\(656\) 9.61803 0.375521
\(657\) −0.291796 −0.0113840
\(658\) 10.0000 0.389841
\(659\) 19.1459 0.745818 0.372909 0.927868i \(-0.378360\pi\)
0.372909 + 0.927868i \(0.378360\pi\)
\(660\) 0 0
\(661\) 34.5410 1.34349 0.671745 0.740782i \(-0.265545\pi\)
0.671745 + 0.740782i \(0.265545\pi\)
\(662\) −21.0557 −0.818354
\(663\) −2.29180 −0.0890060
\(664\) −16.8328 −0.653240
\(665\) −6.23607 −0.241824
\(666\) 5.32624 0.206388
\(667\) 39.7082 1.53751
\(668\) 50.2918 1.94585
\(669\) −17.3820 −0.672026
\(670\) 0 0
\(671\) 0 0
\(672\) −6.70820 −0.258775
\(673\) −8.83282 −0.340480 −0.170240 0.985403i \(-0.554454\pi\)
−0.170240 + 0.985403i \(0.554454\pi\)
\(674\) −35.5755 −1.37032
\(675\) 2.38197 0.0916819
\(676\) −34.4164 −1.32371
\(677\) 10.3607 0.398193 0.199097 0.979980i \(-0.436199\pi\)
0.199097 + 0.979980i \(0.436199\pi\)
\(678\) 1.70820 0.0656032
\(679\) 6.00000 0.230259
\(680\) 6.70820 0.257248
\(681\) 3.52786 0.135188
\(682\) 0 0
\(683\) 2.20163 0.0842429 0.0421214 0.999112i \(-0.486588\pi\)
0.0421214 + 0.999112i \(0.486588\pi\)
\(684\) 11.5623 0.442096
\(685\) −22.9443 −0.876656
\(686\) −2.23607 −0.0853735
\(687\) 6.18034 0.235795
\(688\) −3.70820 −0.141374
\(689\) 13.8885 0.529111
\(690\) −23.9443 −0.911543
\(691\) 6.25735 0.238041 0.119020 0.992892i \(-0.462025\pi\)
0.119020 + 0.992892i \(0.462025\pi\)
\(692\) 34.1459 1.29803
\(693\) 0 0
\(694\) −52.5623 −1.99524
\(695\) −32.1246 −1.21856
\(696\) −13.4164 −0.508548
\(697\) −17.8328 −0.675466
\(698\) 18.5410 0.701788
\(699\) −2.29180 −0.0866837
\(700\) 7.14590 0.270090
\(701\) 9.59675 0.362464 0.181232 0.983440i \(-0.441991\pi\)
0.181232 + 0.983440i \(0.441991\pi\)
\(702\) −2.76393 −0.104318
\(703\) 9.18034 0.346243
\(704\) 0 0
\(705\) 7.23607 0.272526
\(706\) −40.2492 −1.51480
\(707\) 17.7984 0.669377
\(708\) 3.70820 0.139363
\(709\) 21.0344 0.789965 0.394983 0.918689i \(-0.370751\pi\)
0.394983 + 0.918689i \(0.370751\pi\)
\(710\) 46.8328 1.75760
\(711\) 10.0000 0.375029
\(712\) 7.56231 0.283409
\(713\) 53.5410 2.00513
\(714\) 4.14590 0.155156
\(715\) 0 0
\(716\) 29.3951 1.09855
\(717\) 10.0344 0.374743
\(718\) −59.3951 −2.21661
\(719\) −37.0132 −1.38036 −0.690179 0.723639i \(-0.742468\pi\)
−0.690179 + 0.723639i \(0.742468\pi\)
\(720\) −1.61803 −0.0603006
\(721\) 14.3262 0.533537
\(722\) −9.27051 −0.345013
\(723\) 12.0000 0.446285
\(724\) 47.6656 1.77148
\(725\) −14.2918 −0.530784
\(726\) 0 0
\(727\) −12.1459 −0.450466 −0.225233 0.974305i \(-0.572314\pi\)
−0.225233 + 0.974305i \(0.572314\pi\)
\(728\) −2.76393 −0.102438
\(729\) 1.00000 0.0370370
\(730\) −1.05573 −0.0390742
\(731\) 6.87539 0.254295
\(732\) 0 0
\(733\) −40.0689 −1.47998 −0.739989 0.672619i \(-0.765169\pi\)
−0.739989 + 0.672619i \(0.765169\pi\)
\(734\) 81.1033 2.99358
\(735\) −1.61803 −0.0596821
\(736\) −44.3951 −1.63643
\(737\) 0 0
\(738\) −21.5066 −0.791668
\(739\) −0.875388 −0.0322017 −0.0161008 0.999870i \(-0.505125\pi\)
−0.0161008 + 0.999870i \(0.505125\pi\)
\(740\) 11.5623 0.425039
\(741\) −4.76393 −0.175007
\(742\) −25.1246 −0.922354
\(743\) 0.673762 0.0247179 0.0123590 0.999924i \(-0.496066\pi\)
0.0123590 + 0.999924i \(0.496066\pi\)
\(744\) −18.0902 −0.663218
\(745\) −20.9443 −0.767339
\(746\) −1.50658 −0.0551597
\(747\) −7.52786 −0.275430
\(748\) 0 0
\(749\) −8.79837 −0.321486
\(750\) 26.7082 0.975246
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 4.47214 0.163082
\(753\) −14.2918 −0.520822
\(754\) 16.5836 0.603939
\(755\) 39.5967 1.44107
\(756\) 3.00000 0.109109
\(757\) −9.56231 −0.347548 −0.173774 0.984786i \(-0.555596\pi\)
−0.173774 + 0.984786i \(0.555596\pi\)
\(758\) 68.1378 2.47488
\(759\) 0 0
\(760\) 13.9443 0.505812
\(761\) 25.4164 0.921344 0.460672 0.887570i \(-0.347608\pi\)
0.460672 + 0.887570i \(0.347608\pi\)
\(762\) 30.6525 1.11042
\(763\) 2.56231 0.0927617
\(764\) 48.2705 1.74637
\(765\) 3.00000 0.108465
\(766\) 73.4164 2.65264
\(767\) −1.52786 −0.0551680
\(768\) 9.00000 0.324760
\(769\) −18.9443 −0.683148 −0.341574 0.939855i \(-0.610960\pi\)
−0.341574 + 0.939855i \(0.610960\pi\)
\(770\) 0 0
\(771\) 31.4508 1.13267
\(772\) 45.4377 1.63534
\(773\) −0.472136 −0.0169815 −0.00849077 0.999964i \(-0.502703\pi\)
−0.00849077 + 0.999964i \(0.502703\pi\)
\(774\) 8.29180 0.298042
\(775\) −19.2705 −0.692217
\(776\) −13.4164 −0.481621
\(777\) 2.38197 0.0854526
\(778\) 24.0689 0.862911
\(779\) −37.0689 −1.32813
\(780\) −6.00000 −0.214834
\(781\) 0 0
\(782\) 27.4377 0.981170
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) 27.7082 0.988948
\(786\) 32.7639 1.16865
\(787\) 19.1459 0.682478 0.341239 0.939977i \(-0.389154\pi\)
0.341239 + 0.939977i \(0.389154\pi\)
\(788\) −40.5836 −1.44573
\(789\) 4.14590 0.147598
\(790\) 36.1803 1.28724
\(791\) 0.763932 0.0271623
\(792\) 0 0
\(793\) 0 0
\(794\) −81.7082 −2.89972
\(795\) −18.1803 −0.644790
\(796\) 59.5623 2.11113
\(797\) 21.4377 0.759362 0.379681 0.925117i \(-0.376034\pi\)
0.379681 + 0.925117i \(0.376034\pi\)
\(798\) 8.61803 0.305075
\(799\) −8.29180 −0.293343
\(800\) 15.9787 0.564933
\(801\) 3.38197 0.119496
\(802\) −43.4164 −1.53309
\(803\) 0 0
\(804\) 0 0
\(805\) −10.7082 −0.377415
\(806\) 22.3607 0.787621
\(807\) −13.4164 −0.472280
\(808\) −39.7984 −1.40010
\(809\) −2.76393 −0.0971747 −0.0485873 0.998819i \(-0.515472\pi\)
−0.0485873 + 0.998819i \(0.515472\pi\)
\(810\) 3.61803 0.127125
\(811\) −35.4164 −1.24364 −0.621819 0.783161i \(-0.713606\pi\)
−0.621819 + 0.783161i \(0.713606\pi\)
\(812\) −18.0000 −0.631676
\(813\) 15.5066 0.543839
\(814\) 0 0
\(815\) −16.1803 −0.566773
\(816\) 1.85410 0.0649066
\(817\) 14.2918 0.500007
\(818\) −0.403252 −0.0140994
\(819\) −1.23607 −0.0431917
\(820\) −46.6869 −1.63038
\(821\) −6.94427 −0.242357 −0.121178 0.992631i \(-0.538667\pi\)
−0.121178 + 0.992631i \(0.538667\pi\)
\(822\) 31.7082 1.10595
\(823\) −48.8328 −1.70220 −0.851102 0.525000i \(-0.824065\pi\)
−0.851102 + 0.525000i \(0.824065\pi\)
\(824\) −32.0344 −1.11597
\(825\) 0 0
\(826\) 2.76393 0.0961695
\(827\) 52.8673 1.83837 0.919187 0.393821i \(-0.128847\pi\)
0.919187 + 0.393821i \(0.128847\pi\)
\(828\) 19.8541 0.689978
\(829\) 50.2492 1.74523 0.872614 0.488411i \(-0.162423\pi\)
0.872614 + 0.488411i \(0.162423\pi\)
\(830\) −27.2361 −0.945378
\(831\) 11.0902 0.384714
\(832\) −16.0689 −0.557088
\(833\) 1.85410 0.0642408
\(834\) 44.3951 1.53728
\(835\) 27.1246 0.938686
\(836\) 0 0
\(837\) −8.09017 −0.279637
\(838\) 2.76393 0.0954784
\(839\) 40.2492 1.38956 0.694779 0.719224i \(-0.255502\pi\)
0.694779 + 0.719224i \(0.255502\pi\)
\(840\) 3.61803 0.124834
\(841\) 7.00000 0.241379
\(842\) −33.0902 −1.14036
\(843\) −9.05573 −0.311896
\(844\) −79.0820 −2.72212
\(845\) −18.5623 −0.638563
\(846\) −10.0000 −0.343807
\(847\) 0 0
\(848\) −11.2361 −0.385848
\(849\) 16.2705 0.558402
\(850\) −9.87539 −0.338723
\(851\) 15.7639 0.540381
\(852\) −38.8328 −1.33039
\(853\) 28.2918 0.968693 0.484346 0.874876i \(-0.339057\pi\)
0.484346 + 0.874876i \(0.339057\pi\)
\(854\) 0 0
\(855\) 6.23607 0.213269
\(856\) 19.6738 0.672435
\(857\) −10.3607 −0.353914 −0.176957 0.984219i \(-0.556625\pi\)
−0.176957 + 0.984219i \(0.556625\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 18.0000 0.613795
\(861\) −9.61803 −0.327782
\(862\) −8.74265 −0.297776
\(863\) 3.32624 0.113226 0.0566132 0.998396i \(-0.481970\pi\)
0.0566132 + 0.998396i \(0.481970\pi\)
\(864\) 6.70820 0.228218
\(865\) 18.4164 0.626177
\(866\) −59.8460 −2.03365
\(867\) 13.5623 0.460600
\(868\) −24.2705 −0.823795
\(869\) 0 0
\(870\) −21.7082 −0.735977
\(871\) 0 0
\(872\) −5.72949 −0.194025
\(873\) −6.00000 −0.203069
\(874\) 57.0344 1.92922
\(875\) 11.9443 0.403790
\(876\) 0.875388 0.0295766
\(877\) −3.52786 −0.119128 −0.0595638 0.998225i \(-0.518971\pi\)
−0.0595638 + 0.998225i \(0.518971\pi\)
\(878\) 10.8541 0.366308
\(879\) 33.2148 1.12031
\(880\) 0 0
\(881\) −4.32624 −0.145755 −0.0728773 0.997341i \(-0.523218\pi\)
−0.0728773 + 0.997341i \(0.523218\pi\)
\(882\) 2.23607 0.0752923
\(883\) −29.5967 −0.996010 −0.498005 0.867174i \(-0.665934\pi\)
−0.498005 + 0.867174i \(0.665934\pi\)
\(884\) 6.87539 0.231244
\(885\) 2.00000 0.0672293
\(886\) 50.5755 1.69912
\(887\) 21.5967 0.725148 0.362574 0.931955i \(-0.381898\pi\)
0.362574 + 0.931955i \(0.381898\pi\)
\(888\) −5.32624 −0.178737
\(889\) 13.7082 0.459758
\(890\) 12.2361 0.410154
\(891\) 0 0
\(892\) 52.1459 1.74597
\(893\) −17.2361 −0.576783
\(894\) 28.9443 0.968041
\(895\) 15.8541 0.529944
\(896\) 15.6525 0.522913
\(897\) −8.18034 −0.273134
\(898\) −13.6656 −0.456028
\(899\) 48.5410 1.61893
\(900\) −7.14590 −0.238197
\(901\) 20.8328 0.694042
\(902\) 0 0
\(903\) 3.70820 0.123401
\(904\) −1.70820 −0.0568140
\(905\) 25.7082 0.854570
\(906\) −54.7214 −1.81800
\(907\) 29.7082 0.986445 0.493222 0.869903i \(-0.335819\pi\)
0.493222 + 0.869903i \(0.335819\pi\)
\(908\) −10.5836 −0.351229
\(909\) −17.7984 −0.590335
\(910\) −4.47214 −0.148250
\(911\) 16.9443 0.561389 0.280694 0.959797i \(-0.409435\pi\)
0.280694 + 0.959797i \(0.409435\pi\)
\(912\) 3.85410 0.127622
\(913\) 0 0
\(914\) 12.1115 0.400611
\(915\) 0 0
\(916\) −18.5410 −0.612613
\(917\) 14.6525 0.483867
\(918\) −4.14590 −0.136835
\(919\) 21.5279 0.710139 0.355069 0.934840i \(-0.384457\pi\)
0.355069 + 0.934840i \(0.384457\pi\)
\(920\) 23.9443 0.789419
\(921\) 2.56231 0.0844308
\(922\) 13.4164 0.441846
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −5.67376 −0.186552
\(926\) 7.63932 0.251044
\(927\) −14.3262 −0.470535
\(928\) −40.2492 −1.32125
\(929\) 57.2705 1.87898 0.939492 0.342570i \(-0.111297\pi\)
0.939492 + 0.342570i \(0.111297\pi\)
\(930\) −29.2705 −0.959818
\(931\) 3.85410 0.126313
\(932\) 6.87539 0.225211
\(933\) 10.6525 0.348746
\(934\) 8.54102 0.279471
\(935\) 0 0
\(936\) 2.76393 0.0903419
\(937\) 24.0689 0.786296 0.393148 0.919475i \(-0.371386\pi\)
0.393148 + 0.919475i \(0.371386\pi\)
\(938\) 0 0
\(939\) 30.1803 0.984898
\(940\) −21.7082 −0.708044
\(941\) 38.5066 1.25528 0.627639 0.778504i \(-0.284021\pi\)
0.627639 + 0.778504i \(0.284021\pi\)
\(942\) −38.2918 −1.24761
\(943\) −63.6525 −2.07281
\(944\) 1.23607 0.0402306
\(945\) 1.61803 0.0526346
\(946\) 0 0
\(947\) 7.14590 0.232210 0.116105 0.993237i \(-0.462959\pi\)
0.116105 + 0.993237i \(0.462959\pi\)
\(948\) −30.0000 −0.974355
\(949\) −0.360680 −0.0117082
\(950\) −20.5279 −0.666012
\(951\) −11.2361 −0.364354
\(952\) −4.14590 −0.134369
\(953\) 23.8885 0.773826 0.386913 0.922116i \(-0.373541\pi\)
0.386913 + 0.922116i \(0.373541\pi\)
\(954\) 25.1246 0.813439
\(955\) 26.0344 0.842455
\(956\) −30.1033 −0.973611
\(957\) 0 0
\(958\) −37.8885 −1.22412
\(959\) 14.1803 0.457907
\(960\) 21.0344 0.678884
\(961\) 34.4508 1.11132
\(962\) 6.58359 0.212264
\(963\) 8.79837 0.283524
\(964\) −36.0000 −1.15948
\(965\) 24.5066 0.788895
\(966\) 14.7984 0.476130
\(967\) 48.3607 1.55517 0.777587 0.628775i \(-0.216443\pi\)
0.777587 + 0.628775i \(0.216443\pi\)
\(968\) 0 0
\(969\) −7.14590 −0.229559
\(970\) −21.7082 −0.697008
\(971\) −12.5410 −0.402460 −0.201230 0.979544i \(-0.564494\pi\)
−0.201230 + 0.979544i \(0.564494\pi\)
\(972\) −3.00000 −0.0962250
\(973\) 19.8541 0.636493
\(974\) 31.9574 1.02398
\(975\) 2.94427 0.0942922
\(976\) 0 0
\(977\) −58.3607 −1.86712 −0.933562 0.358417i \(-0.883317\pi\)
−0.933562 + 0.358417i \(0.883317\pi\)
\(978\) 22.3607 0.715016
\(979\) 0 0
\(980\) 4.85410 0.155059
\(981\) −2.56231 −0.0818081
\(982\) 81.6312 2.60496
\(983\) 6.94427 0.221488 0.110744 0.993849i \(-0.464677\pi\)
0.110744 + 0.993849i \(0.464677\pi\)
\(984\) 21.5066 0.685605
\(985\) −21.8885 −0.697427
\(986\) 24.8754 0.792194
\(987\) −4.47214 −0.142350
\(988\) 14.2918 0.454683
\(989\) 24.5410 0.780359
\(990\) 0 0
\(991\) −28.7639 −0.913716 −0.456858 0.889540i \(-0.651025\pi\)
−0.456858 + 0.889540i \(0.651025\pi\)
\(992\) −54.2705 −1.72309
\(993\) 9.41641 0.298821
\(994\) −28.9443 −0.918057
\(995\) 32.1246 1.01842
\(996\) 22.5836 0.715588
\(997\) −58.4296 −1.85048 −0.925241 0.379379i \(-0.876138\pi\)
−0.925241 + 0.379379i \(0.876138\pi\)
\(998\) −42.7639 −1.35367
\(999\) −2.38197 −0.0753621
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 2541.2.a.v.1.2 2
3.2 odd 2 7623.2.a.bj.1.1 2
11.2 odd 10 231.2.j.e.169.1 4
11.6 odd 10 231.2.j.e.190.1 yes 4
11.10 odd 2 2541.2.a.w.1.1 2
33.2 even 10 693.2.m.a.631.1 4
33.17 even 10 693.2.m.a.190.1 4
33.32 even 2 7623.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.e.169.1 4 11.2 odd 10
231.2.j.e.190.1 yes 4 11.6 odd 10
693.2.m.a.190.1 4 33.17 even 10
693.2.m.a.631.1 4 33.2 even 10
2541.2.a.v.1.2 2 1.1 even 1 trivial
2541.2.a.w.1.1 2 11.10 odd 2
7623.2.a.bj.1.1 2 3.2 odd 2
7623.2.a.bk.1.2 2 33.32 even 2