Properties

Label 5775.2.a.bw.1.2
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.1136071673\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 5775.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.167449 q^{2} +1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.167449 q^{2} +1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.97196 q^{12} -3.80451 q^{13} +0.167449 q^{14} +3.83255 q^{16} -0.334898 q^{17} +0.167449 q^{18} +8.13941 q^{19} +1.00000 q^{21} +0.167449 q^{22} +1.66510 q^{23} -0.665102 q^{24} -0.637062 q^{26} +1.00000 q^{27} -1.97196 q^{28} +0.195488 q^{29} -9.94392 q^{31} +1.97196 q^{32} +1.00000 q^{33} -0.0560785 q^{34} -1.97196 q^{36} +4.47431 q^{37} +1.36294 q^{38} -3.80451 q^{39} -6.27882 q^{41} +0.167449 q^{42} -2.33490 q^{43} -1.97196 q^{44} +0.278820 q^{46} +12.1394 q^{47} +3.83255 q^{48} +1.00000 q^{49} -0.334898 q^{51} +7.50235 q^{52} -7.94392 q^{53} +0.167449 q^{54} -0.665102 q^{56} +8.13941 q^{57} +0.0327344 q^{58} +3.74843 q^{59} +6.00000 q^{61} -1.66510 q^{62} +1.00000 q^{63} -7.33490 q^{64} +0.167449 q^{66} +0.139410 q^{67} +0.660406 q^{68} +1.66510 q^{69} +4.66980 q^{71} -0.665102 q^{72} -4.19549 q^{73} +0.749219 q^{74} -16.0506 q^{76} +1.00000 q^{77} -0.637062 q^{78} +3.33020 q^{79} +1.00000 q^{81} -1.05138 q^{82} +13.9439 q^{83} -1.97196 q^{84} -0.390977 q^{86} +0.195488 q^{87} -0.665102 q^{88} -9.88784 q^{89} -3.80451 q^{91} -3.28352 q^{92} -9.94392 q^{93} +2.03273 q^{94} +1.97196 q^{96} -0.0560785 q^{97} +0.167449 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.167449 0.118404 0.0592022 0.998246i \(-0.481144\pi\)
0.0592022 + 0.998246i \(0.481144\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.97196 −0.985980
\(5\) 0 0
\(6\) 0.167449 0.0683608
\(7\) 1.00000 0.377964
\(8\) −0.665102 −0.235149
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.97196 −0.569256
\(13\) −3.80451 −1.05518 −0.527591 0.849499i \(-0.676905\pi\)
−0.527591 + 0.849499i \(0.676905\pi\)
\(14\) 0.167449 0.0447527
\(15\) 0 0
\(16\) 3.83255 0.958138
\(17\) −0.334898 −0.0812248 −0.0406124 0.999175i \(-0.512931\pi\)
−0.0406124 + 0.999175i \(0.512931\pi\)
\(18\) 0.167449 0.0394682
\(19\) 8.13941 1.86731 0.933654 0.358175i \(-0.116601\pi\)
0.933654 + 0.358175i \(0.116601\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0.167449 0.0357003
\(23\) 1.66510 0.347198 0.173599 0.984816i \(-0.444460\pi\)
0.173599 + 0.984816i \(0.444460\pi\)
\(24\) −0.665102 −0.135763
\(25\) 0 0
\(26\) −0.637062 −0.124938
\(27\) 1.00000 0.192450
\(28\) −1.97196 −0.372666
\(29\) 0.195488 0.0363013 0.0181506 0.999835i \(-0.494222\pi\)
0.0181506 + 0.999835i \(0.494222\pi\)
\(30\) 0 0
\(31\) −9.94392 −1.78598 −0.892991 0.450075i \(-0.851397\pi\)
−0.892991 + 0.450075i \(0.851397\pi\)
\(32\) 1.97196 0.348597
\(33\) 1.00000 0.174078
\(34\) −0.0560785 −0.00961738
\(35\) 0 0
\(36\) −1.97196 −0.328660
\(37\) 4.47431 0.735572 0.367786 0.929911i \(-0.380116\pi\)
0.367786 + 0.929911i \(0.380116\pi\)
\(38\) 1.36294 0.221098
\(39\) −3.80451 −0.609209
\(40\) 0 0
\(41\) −6.27882 −0.980587 −0.490293 0.871557i \(-0.663110\pi\)
−0.490293 + 0.871557i \(0.663110\pi\)
\(42\) 0.167449 0.0258380
\(43\) −2.33490 −0.356069 −0.178034 0.984024i \(-0.556974\pi\)
−0.178034 + 0.984024i \(0.556974\pi\)
\(44\) −1.97196 −0.297284
\(45\) 0 0
\(46\) 0.278820 0.0411098
\(47\) 12.1394 1.77071 0.885357 0.464911i \(-0.153914\pi\)
0.885357 + 0.464911i \(0.153914\pi\)
\(48\) 3.83255 0.553181
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.334898 −0.0468952
\(52\) 7.50235 1.04039
\(53\) −7.94392 −1.09118 −0.545591 0.838052i \(-0.683695\pi\)
−0.545591 + 0.838052i \(0.683695\pi\)
\(54\) 0.167449 0.0227869
\(55\) 0 0
\(56\) −0.665102 −0.0888779
\(57\) 8.13941 1.07809
\(58\) 0.0327344 0.00429823
\(59\) 3.74843 0.488004 0.244002 0.969775i \(-0.421540\pi\)
0.244002 + 0.969775i \(0.421540\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −1.66510 −0.211468
\(63\) 1.00000 0.125988
\(64\) −7.33490 −0.916862
\(65\) 0 0
\(66\) 0.167449 0.0206116
\(67\) 0.139410 0.0170316 0.00851582 0.999964i \(-0.497289\pi\)
0.00851582 + 0.999964i \(0.497289\pi\)
\(68\) 0.660406 0.0800860
\(69\) 1.66510 0.200455
\(70\) 0 0
\(71\) 4.66980 0.554203 0.277101 0.960841i \(-0.410626\pi\)
0.277101 + 0.960841i \(0.410626\pi\)
\(72\) −0.665102 −0.0783830
\(73\) −4.19549 −0.491045 −0.245522 0.969391i \(-0.578959\pi\)
−0.245522 + 0.969391i \(0.578959\pi\)
\(74\) 0.749219 0.0870950
\(75\) 0 0
\(76\) −16.0506 −1.84113
\(77\) 1.00000 0.113961
\(78\) −0.637062 −0.0721331
\(79\) 3.33020 0.374677 0.187339 0.982295i \(-0.440014\pi\)
0.187339 + 0.982295i \(0.440014\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.05138 −0.116106
\(83\) 13.9439 1.53054 0.765272 0.643707i \(-0.222604\pi\)
0.765272 + 0.643707i \(0.222604\pi\)
\(84\) −1.97196 −0.215159
\(85\) 0 0
\(86\) −0.390977 −0.0421601
\(87\) 0.195488 0.0209586
\(88\) −0.665102 −0.0709001
\(89\) −9.88784 −1.04811 −0.524055 0.851685i \(-0.675581\pi\)
−0.524055 + 0.851685i \(0.675581\pi\)
\(90\) 0 0
\(91\) −3.80451 −0.398821
\(92\) −3.28352 −0.342330
\(93\) −9.94392 −1.03114
\(94\) 2.03273 0.209661
\(95\) 0 0
\(96\) 1.97196 0.201262
\(97\) −0.0560785 −0.00569391 −0.00284695 0.999996i \(-0.500906\pi\)
−0.00284695 + 0.999996i \(0.500906\pi\)
\(98\) 0.167449 0.0169149
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −18.8831 −1.87894 −0.939472 0.342626i \(-0.888683\pi\)
−0.939472 + 0.342626i \(0.888683\pi\)
\(102\) −0.0560785 −0.00555260
\(103\) 8.27882 0.815736 0.407868 0.913041i \(-0.366272\pi\)
0.407868 + 0.913041i \(0.366272\pi\)
\(104\) 2.53039 0.248125
\(105\) 0 0
\(106\) −1.33020 −0.129201
\(107\) 8.13941 0.786866 0.393433 0.919353i \(-0.371287\pi\)
0.393433 + 0.919353i \(0.371287\pi\)
\(108\) −1.97196 −0.189752
\(109\) 11.5529 1.10657 0.553286 0.832992i \(-0.313374\pi\)
0.553286 + 0.832992i \(0.313374\pi\)
\(110\) 0 0
\(111\) 4.47431 0.424683
\(112\) 3.83255 0.362142
\(113\) 1.33020 0.125135 0.0625675 0.998041i \(-0.480071\pi\)
0.0625675 + 0.998041i \(0.480071\pi\)
\(114\) 1.36294 0.127651
\(115\) 0 0
\(116\) −0.385496 −0.0357924
\(117\) −3.80451 −0.351727
\(118\) 0.627672 0.0577819
\(119\) −0.334898 −0.0307001
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.00470 0.0909608
\(123\) −6.27882 −0.566142
\(124\) 19.6090 1.76094
\(125\) 0 0
\(126\) 0.167449 0.0149176
\(127\) −14.6137 −1.29676 −0.648379 0.761318i \(-0.724553\pi\)
−0.648379 + 0.761318i \(0.724553\pi\)
\(128\) −5.17214 −0.457157
\(129\) −2.33490 −0.205576
\(130\) 0 0
\(131\) 20.5576 1.79613 0.898065 0.439863i \(-0.144973\pi\)
0.898065 + 0.439863i \(0.144973\pi\)
\(132\) −1.97196 −0.171637
\(133\) 8.13941 0.705776
\(134\) 0.0233441 0.00201662
\(135\) 0 0
\(136\) 0.222741 0.0190999
\(137\) 16.2227 1.38600 0.693001 0.720936i \(-0.256288\pi\)
0.693001 + 0.720936i \(0.256288\pi\)
\(138\) 0.278820 0.0237347
\(139\) 22.5482 1.91252 0.956259 0.292522i \(-0.0944945\pi\)
0.956259 + 0.292522i \(0.0944945\pi\)
\(140\) 0 0
\(141\) 12.1394 1.02232
\(142\) 0.781954 0.0656201
\(143\) −3.80451 −0.318149
\(144\) 3.83255 0.319379
\(145\) 0 0
\(146\) −0.702531 −0.0581419
\(147\) 1.00000 0.0824786
\(148\) −8.82316 −0.725259
\(149\) 8.19549 0.671401 0.335700 0.941969i \(-0.391027\pi\)
0.335700 + 0.941969i \(0.391027\pi\)
\(150\) 0 0
\(151\) −13.2741 −1.08023 −0.540116 0.841590i \(-0.681620\pi\)
−0.540116 + 0.841590i \(0.681620\pi\)
\(152\) −5.41353 −0.439096
\(153\) −0.334898 −0.0270749
\(154\) 0.167449 0.0134934
\(155\) 0 0
\(156\) 7.50235 0.600669
\(157\) 16.9392 1.35190 0.675949 0.736949i \(-0.263734\pi\)
0.675949 + 0.736949i \(0.263734\pi\)
\(158\) 0.557640 0.0443634
\(159\) −7.94392 −0.629994
\(160\) 0 0
\(161\) 1.66510 0.131228
\(162\) 0.167449 0.0131561
\(163\) 6.79982 0.532603 0.266301 0.963890i \(-0.414198\pi\)
0.266301 + 0.963890i \(0.414198\pi\)
\(164\) 12.3816 0.966839
\(165\) 0 0
\(166\) 2.33490 0.181223
\(167\) 18.2227 1.41012 0.705059 0.709149i \(-0.250920\pi\)
0.705059 + 0.709149i \(0.250920\pi\)
\(168\) −0.665102 −0.0513137
\(169\) 1.47431 0.113408
\(170\) 0 0
\(171\) 8.13941 0.622436
\(172\) 4.60433 0.351077
\(173\) 1.72118 0.130859 0.0654294 0.997857i \(-0.479158\pi\)
0.0654294 + 0.997857i \(0.479158\pi\)
\(174\) 0.0327344 0.00248159
\(175\) 0 0
\(176\) 3.83255 0.288889
\(177\) 3.74843 0.281749
\(178\) −1.65571 −0.124101
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 0.725875 0.0539539 0.0269769 0.999636i \(-0.491412\pi\)
0.0269769 + 0.999636i \(0.491412\pi\)
\(182\) −0.637062 −0.0472222
\(183\) 6.00000 0.443533
\(184\) −1.10746 −0.0816432
\(185\) 0 0
\(186\) −1.66510 −0.122091
\(187\) −0.334898 −0.0244902
\(188\) −23.9384 −1.74589
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 5.27412 0.381622 0.190811 0.981627i \(-0.438888\pi\)
0.190811 + 0.981627i \(0.438888\pi\)
\(192\) −7.33490 −0.529351
\(193\) 19.8318 1.42752 0.713761 0.700390i \(-0.246990\pi\)
0.713761 + 0.700390i \(0.246990\pi\)
\(194\) −0.00939029 −0.000674184 0
\(195\) 0 0
\(196\) −1.97196 −0.140854
\(197\) −2.66980 −0.190215 −0.0951076 0.995467i \(-0.530320\pi\)
−0.0951076 + 0.995467i \(0.530320\pi\)
\(198\) 0.167449 0.0119001
\(199\) 13.5529 0.960743 0.480371 0.877065i \(-0.340502\pi\)
0.480371 + 0.877065i \(0.340502\pi\)
\(200\) 0 0
\(201\) 0.139410 0.00983322
\(202\) −3.16197 −0.222475
\(203\) 0.195488 0.0137206
\(204\) 0.660406 0.0462377
\(205\) 0 0
\(206\) 1.38628 0.0965868
\(207\) 1.66510 0.115733
\(208\) −14.5810 −1.01101
\(209\) 8.13941 0.563015
\(210\) 0 0
\(211\) −4.27882 −0.294566 −0.147283 0.989094i \(-0.547053\pi\)
−0.147283 + 0.989094i \(0.547053\pi\)
\(212\) 15.6651 1.07588
\(213\) 4.66980 0.319969
\(214\) 1.36294 0.0931685
\(215\) 0 0
\(216\) −0.665102 −0.0452544
\(217\) −9.94392 −0.675037
\(218\) 1.93453 0.131023
\(219\) −4.19549 −0.283505
\(220\) 0 0
\(221\) 1.27412 0.0857069
\(222\) 0.749219 0.0502843
\(223\) 10.2694 0.687692 0.343846 0.939026i \(-0.388270\pi\)
0.343846 + 0.939026i \(0.388270\pi\)
\(224\) 1.97196 0.131757
\(225\) 0 0
\(226\) 0.222741 0.0148165
\(227\) 0.390977 0.0259500 0.0129750 0.999916i \(-0.495870\pi\)
0.0129750 + 0.999916i \(0.495870\pi\)
\(228\) −16.0506 −1.06298
\(229\) 7.94392 0.524949 0.262475 0.964939i \(-0.415461\pi\)
0.262475 + 0.964939i \(0.415461\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −0.130020 −0.00853621
\(233\) −26.5576 −1.73985 −0.869924 0.493185i \(-0.835833\pi\)
−0.869924 + 0.493185i \(0.835833\pi\)
\(234\) −0.637062 −0.0416461
\(235\) 0 0
\(236\) −7.39176 −0.481163
\(237\) 3.33020 0.216320
\(238\) −0.0560785 −0.00363503
\(239\) 10.7998 0.698582 0.349291 0.937014i \(-0.386422\pi\)
0.349291 + 0.937014i \(0.386422\pi\)
\(240\) 0 0
\(241\) −12.0833 −0.778356 −0.389178 0.921163i \(-0.627241\pi\)
−0.389178 + 0.921163i \(0.627241\pi\)
\(242\) 0.167449 0.0107640
\(243\) 1.00000 0.0641500
\(244\) −11.8318 −0.757451
\(245\) 0 0
\(246\) −1.05138 −0.0670338
\(247\) −30.9665 −1.97035
\(248\) 6.61372 0.419972
\(249\) 13.9439 0.883660
\(250\) 0 0
\(251\) 4.80921 0.303554 0.151777 0.988415i \(-0.451500\pi\)
0.151777 + 0.988415i \(0.451500\pi\)
\(252\) −1.97196 −0.124222
\(253\) 1.66510 0.104684
\(254\) −2.44706 −0.153542
\(255\) 0 0
\(256\) 13.8037 0.862733
\(257\) 16.7531 1.04503 0.522516 0.852630i \(-0.324994\pi\)
0.522516 + 0.852630i \(0.324994\pi\)
\(258\) −0.390977 −0.0243412
\(259\) 4.47431 0.278020
\(260\) 0 0
\(261\) 0.195488 0.0121004
\(262\) 3.44236 0.212670
\(263\) 12.1394 0.748548 0.374274 0.927318i \(-0.377892\pi\)
0.374274 + 0.927318i \(0.377892\pi\)
\(264\) −0.665102 −0.0409342
\(265\) 0 0
\(266\) 1.36294 0.0835671
\(267\) −9.88784 −0.605126
\(268\) −0.274911 −0.0167929
\(269\) −25.2180 −1.53757 −0.768786 0.639506i \(-0.779139\pi\)
−0.768786 + 0.639506i \(0.779139\pi\)
\(270\) 0 0
\(271\) 4.13941 0.251451 0.125726 0.992065i \(-0.459874\pi\)
0.125726 + 0.992065i \(0.459874\pi\)
\(272\) −1.28352 −0.0778245
\(273\) −3.80451 −0.230260
\(274\) 2.71648 0.164109
\(275\) 0 0
\(276\) −3.28352 −0.197644
\(277\) 18.2788 1.09827 0.549134 0.835734i \(-0.314958\pi\)
0.549134 + 0.835734i \(0.314958\pi\)
\(278\) 3.77569 0.226451
\(279\) −9.94392 −0.595327
\(280\) 0 0
\(281\) 6.74374 0.402298 0.201149 0.979561i \(-0.435533\pi\)
0.201149 + 0.979561i \(0.435533\pi\)
\(282\) 2.03273 0.121048
\(283\) −23.3575 −1.38846 −0.694228 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(284\) −9.20866 −0.546433
\(285\) 0 0
\(286\) −0.637062 −0.0376703
\(287\) −6.27882 −0.370627
\(288\) 1.97196 0.116199
\(289\) −16.8878 −0.993403
\(290\) 0 0
\(291\) −0.0560785 −0.00328738
\(292\) 8.27334 0.484161
\(293\) 14.1667 0.827625 0.413813 0.910362i \(-0.364197\pi\)
0.413813 + 0.910362i \(0.364197\pi\)
\(294\) 0.167449 0.00976584
\(295\) 0 0
\(296\) −2.97587 −0.172969
\(297\) 1.00000 0.0580259
\(298\) 1.37233 0.0794968
\(299\) −6.33490 −0.366357
\(300\) 0 0
\(301\) −2.33490 −0.134581
\(302\) −2.22274 −0.127904
\(303\) −18.8831 −1.08481
\(304\) 31.1947 1.78914
\(305\) 0 0
\(306\) −0.0560785 −0.00320579
\(307\) 12.5576 0.716702 0.358351 0.933587i \(-0.383339\pi\)
0.358351 + 0.933587i \(0.383339\pi\)
\(308\) −1.97196 −0.112363
\(309\) 8.27882 0.470966
\(310\) 0 0
\(311\) 9.33959 0.529600 0.264800 0.964303i \(-0.414694\pi\)
0.264800 + 0.964303i \(0.414694\pi\)
\(312\) 2.53039 0.143255
\(313\) 2.99530 0.169305 0.0846523 0.996411i \(-0.473022\pi\)
0.0846523 + 0.996411i \(0.473022\pi\)
\(314\) 2.83646 0.160071
\(315\) 0 0
\(316\) −6.56703 −0.369424
\(317\) −9.93453 −0.557979 −0.278989 0.960294i \(-0.589999\pi\)
−0.278989 + 0.960294i \(0.589999\pi\)
\(318\) −1.33020 −0.0745941
\(319\) 0.195488 0.0109453
\(320\) 0 0
\(321\) 8.13941 0.454298
\(322\) 0.278820 0.0155380
\(323\) −2.72588 −0.151672
\(324\) −1.97196 −0.109553
\(325\) 0 0
\(326\) 1.13862 0.0630625
\(327\) 11.5529 0.638879
\(328\) 4.17605 0.230584
\(329\) 12.1394 0.669267
\(330\) 0 0
\(331\) 22.5482 1.23936 0.619682 0.784853i \(-0.287262\pi\)
0.619682 + 0.784853i \(0.287262\pi\)
\(332\) −27.4969 −1.50909
\(333\) 4.47431 0.245191
\(334\) 3.05138 0.166964
\(335\) 0 0
\(336\) 3.83255 0.209083
\(337\) −13.2835 −0.723599 −0.361800 0.932256i \(-0.617838\pi\)
−0.361800 + 0.932256i \(0.617838\pi\)
\(338\) 0.246872 0.0134281
\(339\) 1.33020 0.0722467
\(340\) 0 0
\(341\) −9.94392 −0.538494
\(342\) 1.36294 0.0736992
\(343\) 1.00000 0.0539949
\(344\) 1.55294 0.0837292
\(345\) 0 0
\(346\) 0.288210 0.0154943
\(347\) 27.7757 1.49108 0.745538 0.666463i \(-0.232192\pi\)
0.745538 + 0.666463i \(0.232192\pi\)
\(348\) −0.385496 −0.0206647
\(349\) −2.85589 −0.152873 −0.0764363 0.997074i \(-0.524354\pi\)
−0.0764363 + 0.997074i \(0.524354\pi\)
\(350\) 0 0
\(351\) −3.80451 −0.203070
\(352\) 1.97196 0.105106
\(353\) −24.6410 −1.31151 −0.655753 0.754975i \(-0.727649\pi\)
−0.655753 + 0.754975i \(0.727649\pi\)
\(354\) 0.627672 0.0333604
\(355\) 0 0
\(356\) 19.4984 1.03342
\(357\) −0.334898 −0.0177247
\(358\) −2.00939 −0.106200
\(359\) −11.8878 −0.627416 −0.313708 0.949519i \(-0.601571\pi\)
−0.313708 + 0.949519i \(0.601571\pi\)
\(360\) 0 0
\(361\) 47.2500 2.48684
\(362\) 0.121547 0.00638838
\(363\) 1.00000 0.0524864
\(364\) 7.50235 0.393230
\(365\) 0 0
\(366\) 1.00470 0.0525163
\(367\) −3.60902 −0.188389 −0.0941947 0.995554i \(-0.530028\pi\)
−0.0941947 + 0.995554i \(0.530028\pi\)
\(368\) 6.38159 0.332663
\(369\) −6.27882 −0.326862
\(370\) 0 0
\(371\) −7.94392 −0.412428
\(372\) 19.6090 1.01668
\(373\) −12.3349 −0.638677 −0.319338 0.947641i \(-0.603461\pi\)
−0.319338 + 0.947641i \(0.603461\pi\)
\(374\) −0.0560785 −0.00289975
\(375\) 0 0
\(376\) −8.07394 −0.416382
\(377\) −0.743738 −0.0383045
\(378\) 0.167449 0.00861266
\(379\) 23.3575 1.19979 0.599896 0.800078i \(-0.295209\pi\)
0.599896 + 0.800078i \(0.295209\pi\)
\(380\) 0 0
\(381\) −14.6137 −0.748683
\(382\) 0.883148 0.0451858
\(383\) 15.2180 0.777606 0.388803 0.921321i \(-0.372889\pi\)
0.388803 + 0.921321i \(0.372889\pi\)
\(384\) −5.17214 −0.263940
\(385\) 0 0
\(386\) 3.32081 0.169025
\(387\) −2.33490 −0.118690
\(388\) 0.110585 0.00561408
\(389\) 3.73057 0.189147 0.0945737 0.995518i \(-0.469851\pi\)
0.0945737 + 0.995518i \(0.469851\pi\)
\(390\) 0 0
\(391\) −0.557640 −0.0282011
\(392\) −0.665102 −0.0335927
\(393\) 20.5576 1.03700
\(394\) −0.447055 −0.0225223
\(395\) 0 0
\(396\) −1.97196 −0.0990948
\(397\) −20.8925 −1.04857 −0.524283 0.851544i \(-0.675667\pi\)
−0.524283 + 0.851544i \(0.675667\pi\)
\(398\) 2.26943 0.113756
\(399\) 8.13941 0.407480
\(400\) 0 0
\(401\) 23.2741 1.16225 0.581127 0.813813i \(-0.302612\pi\)
0.581127 + 0.813813i \(0.302612\pi\)
\(402\) 0.0233441 0.00116430
\(403\) 37.8318 1.88453
\(404\) 37.2368 1.85260
\(405\) 0 0
\(406\) 0.0327344 0.00162458
\(407\) 4.47431 0.221783
\(408\) 0.222741 0.0110273
\(409\) −23.8972 −1.18164 −0.590821 0.806803i \(-0.701196\pi\)
−0.590821 + 0.806803i \(0.701196\pi\)
\(410\) 0 0
\(411\) 16.2227 0.800209
\(412\) −16.3255 −0.804300
\(413\) 3.74843 0.184448
\(414\) 0.278820 0.0137033
\(415\) 0 0
\(416\) −7.50235 −0.367833
\(417\) 22.5482 1.10419
\(418\) 1.36294 0.0666635
\(419\) −30.2967 −1.48009 −0.740045 0.672557i \(-0.765196\pi\)
−0.740045 + 0.672557i \(0.765196\pi\)
\(420\) 0 0
\(421\) −15.5257 −0.756676 −0.378338 0.925668i \(-0.623504\pi\)
−0.378338 + 0.925668i \(0.623504\pi\)
\(422\) −0.716485 −0.0348779
\(423\) 12.1394 0.590238
\(424\) 5.28352 0.256590
\(425\) 0 0
\(426\) 0.781954 0.0378858
\(427\) 6.00000 0.290360
\(428\) −16.0506 −0.775835
\(429\) −3.80451 −0.183684
\(430\) 0 0
\(431\) 3.07864 0.148293 0.0741463 0.997247i \(-0.476377\pi\)
0.0741463 + 0.997247i \(0.476377\pi\)
\(432\) 3.83255 0.184394
\(433\) −16.9392 −0.814047 −0.407024 0.913418i \(-0.633433\pi\)
−0.407024 + 0.913418i \(0.633433\pi\)
\(434\) −1.66510 −0.0799274
\(435\) 0 0
\(436\) −22.7820 −1.09106
\(437\) 13.5529 0.648325
\(438\) −0.702531 −0.0335682
\(439\) −11.0786 −0.528754 −0.264377 0.964419i \(-0.585166\pi\)
−0.264377 + 0.964419i \(0.585166\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.213351 0.0101481
\(443\) 14.5482 0.691208 0.345604 0.938380i \(-0.387674\pi\)
0.345604 + 0.938380i \(0.387674\pi\)
\(444\) −8.82316 −0.418729
\(445\) 0 0
\(446\) 1.71961 0.0814258
\(447\) 8.19549 0.387633
\(448\) −7.33490 −0.346541
\(449\) 27.4875 1.29721 0.648607 0.761123i \(-0.275352\pi\)
0.648607 + 0.761123i \(0.275352\pi\)
\(450\) 0 0
\(451\) −6.27882 −0.295658
\(452\) −2.62311 −0.123381
\(453\) −13.2741 −0.623673
\(454\) 0.0654688 0.00307260
\(455\) 0 0
\(456\) −5.41353 −0.253512
\(457\) 7.94392 0.371601 0.185800 0.982587i \(-0.440512\pi\)
0.185800 + 0.982587i \(0.440512\pi\)
\(458\) 1.33020 0.0621563
\(459\) −0.334898 −0.0156317
\(460\) 0 0
\(461\) 8.26943 0.385146 0.192573 0.981283i \(-0.438317\pi\)
0.192573 + 0.981283i \(0.438317\pi\)
\(462\) 0.167449 0.00779044
\(463\) −9.73904 −0.452612 −0.226306 0.974056i \(-0.572665\pi\)
−0.226306 + 0.974056i \(0.572665\pi\)
\(464\) 0.749219 0.0347816
\(465\) 0 0
\(466\) −4.44706 −0.206006
\(467\) 40.6970 1.88323 0.941617 0.336685i \(-0.109306\pi\)
0.941617 + 0.336685i \(0.109306\pi\)
\(468\) 7.50235 0.346796
\(469\) 0.139410 0.00643735
\(470\) 0 0
\(471\) 16.9392 0.780518
\(472\) −2.49309 −0.114754
\(473\) −2.33490 −0.107359
\(474\) 0.557640 0.0256132
\(475\) 0 0
\(476\) 0.660406 0.0302697
\(477\) −7.94392 −0.363727
\(478\) 1.80842 0.0827152
\(479\) 37.8318 1.72858 0.864289 0.502996i \(-0.167769\pi\)
0.864289 + 0.502996i \(0.167769\pi\)
\(480\) 0 0
\(481\) −17.0226 −0.776162
\(482\) −2.02334 −0.0921608
\(483\) 1.66510 0.0757647
\(484\) −1.97196 −0.0896346
\(485\) 0 0
\(486\) 0.167449 0.00759565
\(487\) −40.5576 −1.83784 −0.918921 0.394442i \(-0.870938\pi\)
−0.918921 + 0.394442i \(0.870938\pi\)
\(488\) −3.99061 −0.180646
\(489\) 6.79982 0.307498
\(490\) 0 0
\(491\) 31.0786 1.40256 0.701280 0.712886i \(-0.252612\pi\)
0.701280 + 0.712886i \(0.252612\pi\)
\(492\) 12.3816 0.558205
\(493\) −0.0654688 −0.00294856
\(494\) −5.18531 −0.233298
\(495\) 0 0
\(496\) −38.1106 −1.71122
\(497\) 4.66980 0.209469
\(498\) 2.33490 0.104629
\(499\) 15.3575 0.687494 0.343747 0.939062i \(-0.388304\pi\)
0.343747 + 0.939062i \(0.388304\pi\)
\(500\) 0 0
\(501\) 18.2227 0.814132
\(502\) 0.805298 0.0359422
\(503\) −14.8925 −0.664025 −0.332013 0.943275i \(-0.607728\pi\)
−0.332013 + 0.943275i \(0.607728\pi\)
\(504\) −0.665102 −0.0296260
\(505\) 0 0
\(506\) 0.278820 0.0123951
\(507\) 1.47431 0.0654763
\(508\) 28.8177 1.27858
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −4.19549 −0.185597
\(512\) 12.6557 0.559309
\(513\) 8.13941 0.359364
\(514\) 2.80530 0.123736
\(515\) 0 0
\(516\) 4.60433 0.202694
\(517\) 12.1394 0.533891
\(518\) 0.749219 0.0329188
\(519\) 1.72118 0.0755514
\(520\) 0 0
\(521\) 27.6924 1.21322 0.606612 0.794998i \(-0.292528\pi\)
0.606612 + 0.794998i \(0.292528\pi\)
\(522\) 0.0327344 0.00143274
\(523\) −18.4088 −0.804962 −0.402481 0.915428i \(-0.631852\pi\)
−0.402481 + 0.915428i \(0.631852\pi\)
\(524\) −40.5389 −1.77095
\(525\) 0 0
\(526\) 2.03273 0.0886314
\(527\) 3.33020 0.145066
\(528\) 3.83255 0.166790
\(529\) −20.2274 −0.879454
\(530\) 0 0
\(531\) 3.74843 0.162668
\(532\) −16.0506 −0.695882
\(533\) 23.8878 1.03470
\(534\) −1.65571 −0.0716496
\(535\) 0 0
\(536\) −0.0927218 −0.00400497
\(537\) −12.0000 −0.517838
\(538\) −4.22274 −0.182055
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −4.26943 −0.183557 −0.0917786 0.995779i \(-0.529255\pi\)
−0.0917786 + 0.995779i \(0.529255\pi\)
\(542\) 0.693141 0.0297729
\(543\) 0.725875 0.0311503
\(544\) −0.660406 −0.0283147
\(545\) 0 0
\(546\) −0.637062 −0.0272638
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −31.9906 −1.36657
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 1.59116 0.0677857
\(552\) −1.10746 −0.0471367
\(553\) 3.33020 0.141615
\(554\) 3.06077 0.130040
\(555\) 0 0
\(556\) −44.4643 −1.88570
\(557\) 12.4743 0.528553 0.264277 0.964447i \(-0.414867\pi\)
0.264277 + 0.964447i \(0.414867\pi\)
\(558\) −1.66510 −0.0704894
\(559\) 8.88315 0.375717
\(560\) 0 0
\(561\) −0.334898 −0.0141394
\(562\) 1.12923 0.0476338
\(563\) 32.7710 1.38113 0.690566 0.723269i \(-0.257361\pi\)
0.690566 + 0.723269i \(0.257361\pi\)
\(564\) −23.9384 −1.00799
\(565\) 0 0
\(566\) −3.91119 −0.164399
\(567\) 1.00000 0.0419961
\(568\) −3.10589 −0.130320
\(569\) −29.2180 −1.22488 −0.612442 0.790515i \(-0.709813\pi\)
−0.612442 + 0.790515i \(0.709813\pi\)
\(570\) 0 0
\(571\) 32.4455 1.35780 0.678901 0.734230i \(-0.262457\pi\)
0.678901 + 0.734230i \(0.262457\pi\)
\(572\) 7.50235 0.313689
\(573\) 5.27412 0.220330
\(574\) −1.05138 −0.0438839
\(575\) 0 0
\(576\) −7.33490 −0.305621
\(577\) 14.8831 0.619594 0.309797 0.950803i \(-0.399739\pi\)
0.309797 + 0.950803i \(0.399739\pi\)
\(578\) −2.82786 −0.117623
\(579\) 19.8318 0.824180
\(580\) 0 0
\(581\) 13.9439 0.578491
\(582\) −0.00939029 −0.000389240 0
\(583\) −7.94392 −0.329004
\(584\) 2.79043 0.115469
\(585\) 0 0
\(586\) 2.37220 0.0979945
\(587\) 4.53039 0.186989 0.0934945 0.995620i \(-0.470196\pi\)
0.0934945 + 0.995620i \(0.470196\pi\)
\(588\) −1.97196 −0.0813223
\(589\) −80.9377 −3.33498
\(590\) 0 0
\(591\) −2.66980 −0.109821
\(592\) 17.1480 0.704779
\(593\) −8.28821 −0.340356 −0.170178 0.985413i \(-0.554434\pi\)
−0.170178 + 0.985413i \(0.554434\pi\)
\(594\) 0.167449 0.00687052
\(595\) 0 0
\(596\) −16.1612 −0.661988
\(597\) 13.5529 0.554685
\(598\) −1.06077 −0.0433783
\(599\) 1.99061 0.0813341 0.0406671 0.999173i \(-0.487052\pi\)
0.0406671 + 0.999173i \(0.487052\pi\)
\(600\) 0 0
\(601\) −8.47431 −0.345674 −0.172837 0.984950i \(-0.555293\pi\)
−0.172837 + 0.984950i \(0.555293\pi\)
\(602\) −0.390977 −0.0159350
\(603\) 0.139410 0.00567721
\(604\) 26.1761 1.06509
\(605\) 0 0
\(606\) −3.16197 −0.128446
\(607\) 22.9665 0.932181 0.466090 0.884737i \(-0.345662\pi\)
0.466090 + 0.884737i \(0.345662\pi\)
\(608\) 16.0506 0.650938
\(609\) 0.195488 0.00792159
\(610\) 0 0
\(611\) −46.1845 −1.86843
\(612\) 0.660406 0.0266953
\(613\) −3.55294 −0.143502 −0.0717510 0.997423i \(-0.522859\pi\)
−0.0717510 + 0.997423i \(0.522859\pi\)
\(614\) 2.10277 0.0848608
\(615\) 0 0
\(616\) −0.665102 −0.0267977
\(617\) −22.6698 −0.912652 −0.456326 0.889813i \(-0.650835\pi\)
−0.456326 + 0.889813i \(0.650835\pi\)
\(618\) 1.38628 0.0557644
\(619\) 8.71648 0.350345 0.175173 0.984538i \(-0.443952\pi\)
0.175173 + 0.984538i \(0.443952\pi\)
\(620\) 0 0
\(621\) 1.66510 0.0668182
\(622\) 1.56391 0.0627070
\(623\) −9.88784 −0.396148
\(624\) −14.5810 −0.583707
\(625\) 0 0
\(626\) 0.501561 0.0200464
\(627\) 8.13941 0.325057
\(628\) −33.4035 −1.33294
\(629\) −1.49844 −0.0597467
\(630\) 0 0
\(631\) −11.8878 −0.473248 −0.236624 0.971601i \(-0.576041\pi\)
−0.236624 + 0.971601i \(0.576041\pi\)
\(632\) −2.21492 −0.0881049
\(633\) −4.27882 −0.170068
\(634\) −1.66353 −0.0660672
\(635\) 0 0
\(636\) 15.6651 0.621162
\(637\) −3.80451 −0.150740
\(638\) 0.0327344 0.00129597
\(639\) 4.66980 0.184734
\(640\) 0 0
\(641\) −4.89254 −0.193244 −0.0966218 0.995321i \(-0.530804\pi\)
−0.0966218 + 0.995321i \(0.530804\pi\)
\(642\) 1.36294 0.0537909
\(643\) −22.7804 −0.898371 −0.449185 0.893439i \(-0.648286\pi\)
−0.449185 + 0.893439i \(0.648286\pi\)
\(644\) −3.28352 −0.129389
\(645\) 0 0
\(646\) −0.456446 −0.0179586
\(647\) −31.2453 −1.22838 −0.614190 0.789158i \(-0.710517\pi\)
−0.614190 + 0.789158i \(0.710517\pi\)
\(648\) −0.665102 −0.0261277
\(649\) 3.74843 0.147139
\(650\) 0 0
\(651\) −9.94392 −0.389733
\(652\) −13.4090 −0.525136
\(653\) −28.2882 −1.10700 −0.553502 0.832848i \(-0.686709\pi\)
−0.553502 + 0.832848i \(0.686709\pi\)
\(654\) 1.93453 0.0756462
\(655\) 0 0
\(656\) −24.0639 −0.939537
\(657\) −4.19549 −0.163682
\(658\) 2.03273 0.0792442
\(659\) 30.2967 1.18019 0.590096 0.807333i \(-0.299090\pi\)
0.590096 + 0.807333i \(0.299090\pi\)
\(660\) 0 0
\(661\) −35.7196 −1.38933 −0.694666 0.719333i \(-0.744448\pi\)
−0.694666 + 0.719333i \(0.744448\pi\)
\(662\) 3.77569 0.146746
\(663\) 1.27412 0.0494829
\(664\) −9.27412 −0.359906
\(665\) 0 0
\(666\) 0.749219 0.0290317
\(667\) 0.325508 0.0126037
\(668\) −35.9345 −1.39035
\(669\) 10.2694 0.397039
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 1.97196 0.0760700
\(673\) −34.9377 −1.34675 −0.673374 0.739302i \(-0.735156\pi\)
−0.673374 + 0.739302i \(0.735156\pi\)
\(674\) −2.22431 −0.0856774
\(675\) 0 0
\(676\) −2.90728 −0.111818
\(677\) −20.0094 −0.769023 −0.384512 0.923120i \(-0.625630\pi\)
−0.384512 + 0.923120i \(0.625630\pi\)
\(678\) 0.222741 0.00855433
\(679\) −0.0560785 −0.00215209
\(680\) 0 0
\(681\) 0.390977 0.0149823
\(682\) −1.66510 −0.0637600
\(683\) −25.0498 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(684\) −16.0506 −0.613710
\(685\) 0 0
\(686\) 0.167449 0.00639324
\(687\) 7.94392 0.303080
\(688\) −8.94862 −0.341163
\(689\) 30.2227 1.15139
\(690\) 0 0
\(691\) −38.8271 −1.47705 −0.738526 0.674225i \(-0.764478\pi\)
−0.738526 + 0.674225i \(0.764478\pi\)
\(692\) −3.39410 −0.129024
\(693\) 1.00000 0.0379869
\(694\) 4.65102 0.176550
\(695\) 0 0
\(696\) −0.130020 −0.00492838
\(697\) 2.10277 0.0796480
\(698\) −0.478217 −0.0181008
\(699\) −26.5576 −1.00450
\(700\) 0 0
\(701\) −41.7757 −1.57785 −0.788923 0.614492i \(-0.789361\pi\)
−0.788923 + 0.614492i \(0.789361\pi\)
\(702\) −0.637062 −0.0240444
\(703\) 36.4182 1.37354
\(704\) −7.33490 −0.276444
\(705\) 0 0
\(706\) −4.12611 −0.155288
\(707\) −18.8831 −0.710174
\(708\) −7.39176 −0.277799
\(709\) −13.4041 −0.503403 −0.251702 0.967805i \(-0.580990\pi\)
−0.251702 + 0.967805i \(0.580990\pi\)
\(710\) 0 0
\(711\) 3.33020 0.124892
\(712\) 6.57642 0.246462
\(713\) −16.5576 −0.620088
\(714\) −0.0560785 −0.00209868
\(715\) 0 0
\(716\) 23.6635 0.884348
\(717\) 10.7998 0.403327
\(718\) −1.99061 −0.0742889
\(719\) 5.47900 0.204332 0.102166 0.994767i \(-0.467423\pi\)
0.102166 + 0.994767i \(0.467423\pi\)
\(720\) 0 0
\(721\) 8.27882 0.308319
\(722\) 7.91197 0.294453
\(723\) −12.0833 −0.449384
\(724\) −1.43140 −0.0531975
\(725\) 0 0
\(726\) 0.167449 0.00621462
\(727\) 32.8831 1.21957 0.609784 0.792567i \(-0.291256\pi\)
0.609784 + 0.792567i \(0.291256\pi\)
\(728\) 2.53039 0.0937824
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.781954 0.0289216
\(732\) −11.8318 −0.437315
\(733\) 35.8972 1.32589 0.662947 0.748666i \(-0.269305\pi\)
0.662947 + 0.748666i \(0.269305\pi\)
\(734\) −0.604328 −0.0223062
\(735\) 0 0
\(736\) 3.28352 0.121032
\(737\) 0.139410 0.00513523
\(738\) −1.05138 −0.0387020
\(739\) 29.6651 1.09125 0.545624 0.838030i \(-0.316293\pi\)
0.545624 + 0.838030i \(0.316293\pi\)
\(740\) 0 0
\(741\) −30.9665 −1.13758
\(742\) −1.33020 −0.0488333
\(743\) 18.7998 0.689698 0.344849 0.938658i \(-0.387930\pi\)
0.344849 + 0.938658i \(0.387930\pi\)
\(744\) 6.61372 0.242471
\(745\) 0 0
\(746\) −2.06547 −0.0756222
\(747\) 13.9439 0.510181
\(748\) 0.660406 0.0241469
\(749\) 8.13941 0.297408
\(750\) 0 0
\(751\) −9.59116 −0.349986 −0.174993 0.984570i \(-0.555990\pi\)
−0.174993 + 0.984570i \(0.555990\pi\)
\(752\) 46.5249 1.69659
\(753\) 4.80921 0.175257
\(754\) −0.124538 −0.00453542
\(755\) 0 0
\(756\) −1.97196 −0.0717195
\(757\) −2.74374 −0.0997229 −0.0498614 0.998756i \(-0.515878\pi\)
−0.0498614 + 0.998756i \(0.515878\pi\)
\(758\) 3.91119 0.142061
\(759\) 1.66510 0.0604394
\(760\) 0 0
\(761\) 6.39098 0.231673 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(762\) −2.44706 −0.0886475
\(763\) 11.5529 0.418245
\(764\) −10.4004 −0.376272
\(765\) 0 0
\(766\) 2.54825 0.0920720
\(767\) −14.2610 −0.514933
\(768\) 13.8037 0.498099
\(769\) 17.7951 0.641708 0.320854 0.947129i \(-0.396030\pi\)
0.320854 + 0.947129i \(0.396030\pi\)
\(770\) 0 0
\(771\) 16.7531 0.603349
\(772\) −39.1075 −1.40751
\(773\) 31.5257 1.13390 0.566950 0.823752i \(-0.308123\pi\)
0.566950 + 0.823752i \(0.308123\pi\)
\(774\) −0.390977 −0.0140534
\(775\) 0 0
\(776\) 0.0372979 0.00133892
\(777\) 4.47431 0.160515
\(778\) 0.624681 0.0223959
\(779\) −51.1059 −1.83106
\(780\) 0 0
\(781\) 4.66980 0.167098
\(782\) −0.0933763 −0.00333913
\(783\) 0.195488 0.00698619
\(784\) 3.83255 0.136877
\(785\) 0 0
\(786\) 3.44236 0.122785
\(787\) −31.8606 −1.13571 −0.567854 0.823130i \(-0.692226\pi\)
−0.567854 + 0.823130i \(0.692226\pi\)
\(788\) 5.26473 0.187548
\(789\) 12.1394 0.432174
\(790\) 0 0
\(791\) 1.33020 0.0472966
\(792\) −0.665102 −0.0236334
\(793\) −22.8271 −0.810613
\(794\) −3.49844 −0.124155
\(795\) 0 0
\(796\) −26.7259 −0.947274
\(797\) −43.8590 −1.55357 −0.776783 0.629768i \(-0.783150\pi\)
−0.776783 + 0.629768i \(0.783150\pi\)
\(798\) 1.36294 0.0482475
\(799\) −4.06547 −0.143826
\(800\) 0 0
\(801\) −9.88784 −0.349370
\(802\) 3.89723 0.137616
\(803\) −4.19549 −0.148056
\(804\) −0.274911 −0.00969536
\(805\) 0 0
\(806\) 6.33490 0.223137
\(807\) −25.2180 −0.887717
\(808\) 12.5592 0.441832
\(809\) 6.74374 0.237097 0.118549 0.992948i \(-0.462176\pi\)
0.118549 + 0.992948i \(0.462176\pi\)
\(810\) 0 0
\(811\) 3.97275 0.139502 0.0697510 0.997564i \(-0.477780\pi\)
0.0697510 + 0.997564i \(0.477780\pi\)
\(812\) −0.385496 −0.0135282
\(813\) 4.13941 0.145175
\(814\) 0.749219 0.0262601
\(815\) 0 0
\(816\) −1.28352 −0.0449320
\(817\) −19.0047 −0.664890
\(818\) −4.00157 −0.139912
\(819\) −3.80451 −0.132940
\(820\) 0 0
\(821\) 49.5896 1.73069 0.865344 0.501178i \(-0.167100\pi\)
0.865344 + 0.501178i \(0.167100\pi\)
\(822\) 2.71648 0.0947483
\(823\) −23.1908 −0.808380 −0.404190 0.914675i \(-0.632447\pi\)
−0.404190 + 0.914675i \(0.632447\pi\)
\(824\) −5.50626 −0.191820
\(825\) 0 0
\(826\) 0.627672 0.0218395
\(827\) −27.7484 −0.964908 −0.482454 0.875921i \(-0.660254\pi\)
−0.482454 + 0.875921i \(0.660254\pi\)
\(828\) −3.28352 −0.114110
\(829\) 0.269430 0.00935768 0.00467884 0.999989i \(-0.498511\pi\)
0.00467884 + 0.999989i \(0.498511\pi\)
\(830\) 0 0
\(831\) 18.2788 0.634085
\(832\) 27.9057 0.967456
\(833\) −0.334898 −0.0116035
\(834\) 3.77569 0.130741
\(835\) 0 0
\(836\) −16.0506 −0.555122
\(837\) −9.94392 −0.343712
\(838\) −5.07316 −0.175249
\(839\) −27.3575 −0.944484 −0.472242 0.881469i \(-0.656555\pi\)
−0.472242 + 0.881469i \(0.656555\pi\)
\(840\) 0 0
\(841\) −28.9618 −0.998682
\(842\) −2.59976 −0.0895938
\(843\) 6.74374 0.232267
\(844\) 8.43767 0.290436
\(845\) 0 0
\(846\) 2.03273 0.0698868
\(847\) 1.00000 0.0343604
\(848\) −30.4455 −1.04550
\(849\) −23.3575 −0.801626
\(850\) 0 0
\(851\) 7.45018 0.255389
\(852\) −9.20866 −0.315483
\(853\) 28.5482 0.977473 0.488737 0.872431i \(-0.337458\pi\)
0.488737 + 0.872431i \(0.337458\pi\)
\(854\) 1.00470 0.0343800
\(855\) 0 0
\(856\) −5.41353 −0.185031
\(857\) 39.8972 1.36286 0.681432 0.731882i \(-0.261358\pi\)
0.681432 + 0.731882i \(0.261358\pi\)
\(858\) −0.637062 −0.0217490
\(859\) 20.5576 0.701418 0.350709 0.936485i \(-0.385941\pi\)
0.350709 + 0.936485i \(0.385941\pi\)
\(860\) 0 0
\(861\) −6.27882 −0.213982
\(862\) 0.515515 0.0175585
\(863\) 9.66510 0.329004 0.164502 0.986377i \(-0.447398\pi\)
0.164502 + 0.986377i \(0.447398\pi\)
\(864\) 1.97196 0.0670875
\(865\) 0 0
\(866\) −2.83646 −0.0963868
\(867\) −16.8878 −0.573541
\(868\) 19.6090 0.665574
\(869\) 3.33020 0.112969
\(870\) 0 0
\(871\) −0.530387 −0.0179715
\(872\) −7.68388 −0.260209
\(873\) −0.0560785 −0.00189797
\(874\) 2.26943 0.0767646
\(875\) 0 0
\(876\) 8.27334 0.279530
\(877\) 25.7212 0.868543 0.434271 0.900782i \(-0.357006\pi\)
0.434271 + 0.900782i \(0.357006\pi\)
\(878\) −1.85511 −0.0626069
\(879\) 14.1667 0.477830
\(880\) 0 0
\(881\) −14.6316 −0.492950 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(882\) 0.167449 0.00563831
\(883\) −18.6877 −0.628890 −0.314445 0.949276i \(-0.601818\pi\)
−0.314445 + 0.949276i \(0.601818\pi\)
\(884\) −2.51252 −0.0845053
\(885\) 0 0
\(886\) 2.43609 0.0818421
\(887\) −38.6137 −1.29652 −0.648261 0.761418i \(-0.724503\pi\)
−0.648261 + 0.761418i \(0.724503\pi\)
\(888\) −2.97587 −0.0998636
\(889\) −14.6137 −0.490128
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) −20.2509 −0.678051
\(893\) 98.8076 3.30647
\(894\) 1.37233 0.0458975
\(895\) 0 0
\(896\) −5.17214 −0.172789
\(897\) −6.33490 −0.211516
\(898\) 4.60276 0.153596
\(899\) −1.94392 −0.0648334
\(900\) 0 0
\(901\) 2.66041 0.0886310
\(902\) −1.05138 −0.0350072
\(903\) −2.33490 −0.0777006
\(904\) −0.884720 −0.0294254
\(905\) 0 0
\(906\) −2.22274 −0.0738456
\(907\) −49.0965 −1.63022 −0.815111 0.579304i \(-0.803324\pi\)
−0.815111 + 0.579304i \(0.803324\pi\)
\(908\) −0.770991 −0.0255862
\(909\) −18.8831 −0.626314
\(910\) 0 0
\(911\) 29.3753 0.973248 0.486624 0.873612i \(-0.338228\pi\)
0.486624 + 0.873612i \(0.338228\pi\)
\(912\) 31.1947 1.03296
\(913\) 13.9439 0.461476
\(914\) 1.33020 0.0439992
\(915\) 0 0
\(916\) −15.6651 −0.517590
\(917\) 20.5576 0.678873
\(918\) −0.0560785 −0.00185087
\(919\) −27.0047 −0.890803 −0.445401 0.895331i \(-0.646939\pi\)
−0.445401 + 0.895331i \(0.646939\pi\)
\(920\) 0 0
\(921\) 12.5576 0.413788
\(922\) 1.38471 0.0456030
\(923\) −17.7663 −0.584785
\(924\) −1.97196 −0.0648727
\(925\) 0 0
\(926\) −1.63079 −0.0535912
\(927\) 8.27882 0.271912
\(928\) 0.385496 0.0126545
\(929\) 57.8496 1.89798 0.948992 0.315299i \(-0.102105\pi\)
0.948992 + 0.315299i \(0.102105\pi\)
\(930\) 0 0
\(931\) 8.13941 0.266758
\(932\) 52.3706 1.71546
\(933\) 9.33959 0.305765
\(934\) 6.81469 0.222983
\(935\) 0 0
\(936\) 2.53039 0.0827083
\(937\) −25.2180 −0.823838 −0.411919 0.911221i \(-0.635141\pi\)
−0.411919 + 0.911221i \(0.635141\pi\)
\(938\) 0.0233441 0.000762211 0
\(939\) 2.99530 0.0977481
\(940\) 0 0
\(941\) −34.2788 −1.11746 −0.558729 0.829350i \(-0.688711\pi\)
−0.558729 + 0.829350i \(0.688711\pi\)
\(942\) 2.83646 0.0924169
\(943\) −10.4549 −0.340458
\(944\) 14.3661 0.467575
\(945\) 0 0
\(946\) −0.390977 −0.0127118
\(947\) 18.1573 0.590032 0.295016 0.955492i \(-0.404675\pi\)
0.295016 + 0.955492i \(0.404675\pi\)
\(948\) −6.56703 −0.213287
\(949\) 15.9618 0.518141
\(950\) 0 0
\(951\) −9.93453 −0.322149
\(952\) 0.222741 0.00721909
\(953\) 19.8045 0.641531 0.320766 0.947159i \(-0.396060\pi\)
0.320766 + 0.947159i \(0.396060\pi\)
\(954\) −1.33020 −0.0430669
\(955\) 0 0
\(956\) −21.2968 −0.688788
\(957\) 0.195488 0.00631924
\(958\) 6.33490 0.204671
\(959\) 16.2227 0.523860
\(960\) 0 0
\(961\) 67.8816 2.18973
\(962\) −2.85041 −0.0919010
\(963\) 8.13941 0.262289
\(964\) 23.8279 0.767444
\(965\) 0 0
\(966\) 0.278820 0.00897088
\(967\) 0.278820 0.00896624 0.00448312 0.999990i \(-0.498573\pi\)
0.00448312 + 0.999990i \(0.498573\pi\)
\(968\) −0.665102 −0.0213772
\(969\) −2.72588 −0.0875677
\(970\) 0 0
\(971\) 25.3481 0.813458 0.406729 0.913549i \(-0.366669\pi\)
0.406729 + 0.913549i \(0.366669\pi\)
\(972\) −1.97196 −0.0632507
\(973\) 22.5482 0.722864
\(974\) −6.79134 −0.217609
\(975\) 0 0
\(976\) 22.9953 0.736062
\(977\) −6.71648 −0.214879 −0.107440 0.994212i \(-0.534265\pi\)
−0.107440 + 0.994212i \(0.534265\pi\)
\(978\) 1.13862 0.0364092
\(979\) −9.88784 −0.316017
\(980\) 0 0
\(981\) 11.5529 0.368857
\(982\) 5.20409 0.166069
\(983\) −9.99061 −0.318651 −0.159325 0.987226i \(-0.550932\pi\)
−0.159325 + 0.987226i \(0.550932\pi\)
\(984\) 4.17605 0.133128
\(985\) 0 0
\(986\) −0.0109627 −0.000349123 0
\(987\) 12.1394 0.386402
\(988\) 61.0647 1.94273
\(989\) −3.88784 −0.123626
\(990\) 0 0
\(991\) −26.7064 −0.848358 −0.424179 0.905578i \(-0.639437\pi\)
−0.424179 + 0.905578i \(0.639437\pi\)
\(992\) −19.6090 −0.622587
\(993\) 22.5482 0.715547
\(994\) 0.781954 0.0248021
\(995\) 0 0
\(996\) −27.4969 −0.871272
\(997\) −54.3333 −1.72075 −0.860377 0.509658i \(-0.829772\pi\)
−0.860377 + 0.509658i \(0.829772\pi\)
\(998\) 2.57159 0.0814024
\(999\) 4.47431 0.141561
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 5775.2.a.bw.1.2 3
5.4 even 2 231.2.a.d.1.2 3
15.14 odd 2 693.2.a.m.1.2 3
20.19 odd 2 3696.2.a.bp.1.3 3
35.34 odd 2 1617.2.a.s.1.2 3
55.54 odd 2 2541.2.a.bi.1.2 3
105.104 even 2 4851.2.a.bp.1.2 3
165.164 even 2 7623.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 5.4 even 2
693.2.a.m.1.2 3 15.14 odd 2
1617.2.a.s.1.2 3 35.34 odd 2
2541.2.a.bi.1.2 3 55.54 odd 2
3696.2.a.bp.1.3 3 20.19 odd 2
4851.2.a.bp.1.2 3 105.104 even 2
5775.2.a.bw.1.2 3 1.1 even 1 trivial
7623.2.a.cb.1.2 3 165.164 even 2