Properties

Label 2541.2.a.bi.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
Dimension $3$
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: \(1\)
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 \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} -0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} +3.80451 q^{5} -0.167449 q^{6} +1.00000 q^{7} -0.665102 q^{8} +1.00000 q^{9} +0.637062 q^{10} +1.97196 q^{12} -3.80451 q^{13} +0.167449 q^{14} -3.80451 q^{15} +3.83255 q^{16} -0.334898 q^{17} +0.167449 q^{18} -8.13941 q^{19} -7.50235 q^{20} -1.00000 q^{21} -1.66510 q^{23} +0.665102 q^{24} +9.47431 q^{25} -0.637062 q^{26} -1.00000 q^{27} -1.97196 q^{28} -0.195488 q^{29} -0.637062 q^{30} -9.94392 q^{31} +1.97196 q^{32} -0.0560785 q^{34} +3.80451 q^{35} -1.97196 q^{36} -4.47431 q^{37} -1.36294 q^{38} +3.80451 q^{39} -2.53039 q^{40} +6.27882 q^{41} -0.167449 q^{42} -2.33490 q^{43} +3.80451 q^{45} -0.278820 q^{46} -12.1394 q^{47} -3.83255 q^{48} +1.00000 q^{49} +1.58647 q^{50} +0.334898 q^{51} +7.50235 q^{52} +7.94392 q^{53} -0.167449 q^{54} -0.665102 q^{56} +8.13941 q^{57} -0.0327344 q^{58} +3.74843 q^{59} +7.50235 q^{60} -6.00000 q^{61} -1.66510 q^{62} +1.00000 q^{63} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} +0.660406 q^{68} +1.66510 q^{69} +0.637062 q^{70} +4.66980 q^{71} -0.665102 q^{72} -4.19549 q^{73} -0.749219 q^{74} -9.47431 q^{75} +16.0506 q^{76} +0.637062 q^{78} -3.33020 q^{79} +14.5810 q^{80} +1.00000 q^{81} +1.05138 q^{82} +13.9439 q^{83} +1.97196 q^{84} -1.27412 q^{85} -0.390977 q^{86} +0.195488 q^{87} -9.88784 q^{89} +0.637062 q^{90} -3.80451 q^{91} +3.28352 q^{92} +9.94392 q^{93} -2.03273 q^{94} -30.9665 q^{95} -1.97196 q^{96} +0.0560785 q^{97} +0.167449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 9 q^{10} - 6 q^{12} + 12 q^{16} - 12 q^{19} - 21 q^{20} - 3 q^{21} - 6 q^{23} + 3 q^{24} + 15 q^{25} + 9 q^{26} - 3 q^{27} + 6 q^{28} - 12 q^{29} + 9 q^{30} - 6 q^{31} - 6 q^{32} - 24 q^{34} + 6 q^{36} - 15 q^{38} - 18 q^{40} - 6 q^{41} - 6 q^{43} + 24 q^{46} - 24 q^{47} - 12 q^{48} + 3 q^{49} + 39 q^{50} + 21 q^{52} - 3 q^{56} + 12 q^{57} - 9 q^{58} - 24 q^{59} + 21 q^{60} - 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 30 q^{65} + 12 q^{67} + 6 q^{68} + 6 q^{69} - 9 q^{70} + 12 q^{71} - 3 q^{72} - 24 q^{73} - 39 q^{74} - 15 q^{75} + 3 q^{76} - 9 q^{78} - 12 q^{79} + 9 q^{80} + 3 q^{81} + 30 q^{82} + 18 q^{83} - 6 q^{84} + 18 q^{85} - 24 q^{86} + 12 q^{87} + 18 q^{89} - 9 q^{90} - 18 q^{92} + 6 q^{93} - 15 q^{94} - 12 q^{95} + 6 q^{96} + 24 q^{97}+O(q^{100}) \) 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
\(5\) 3.80451 1.70143 0.850715 0.525628i \(-0.176170\pi\)
0.850715 + 0.525628i \(0.176170\pi\)
\(6\) −0.167449 −0.0683608
\(7\) 1.00000 0.377964
\(8\) −0.665102 −0.235149
\(9\) 1.00000 0.333333
\(10\) 0.637062 0.201457
\(11\) 0 0
\(12\) 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
\(15\) −3.80451 −0.982321
\(16\) 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 \(-0.883399\pi\)
−0.933654 + 0.358175i \(0.883399\pi\)
\(20\) −7.50235 −1.67758
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.66510 −0.347198 −0.173599 0.984816i \(-0.555540\pi\)
−0.173599 + 0.984816i \(0.555540\pi\)
\(24\) 0.665102 0.135763
\(25\) 9.47431 1.89486
\(26\) −0.637062 −0.124938
\(27\) −1.00000 −0.192450
\(28\) −1.97196 −0.372666
\(29\) −0.195488 −0.0363013 −0.0181506 0.999835i \(-0.505778\pi\)
−0.0181506 + 0.999835i \(0.505778\pi\)
\(30\) −0.637062 −0.116311
\(31\) −9.94392 −1.78598 −0.892991 0.450075i \(-0.851397\pi\)
−0.892991 + 0.450075i \(0.851397\pi\)
\(32\) 1.97196 0.348597
\(33\) 0 0
\(34\) −0.0560785 −0.00961738
\(35\) 3.80451 0.643080
\(36\) −1.97196 −0.328660
\(37\) −4.47431 −0.735572 −0.367786 0.929911i \(-0.619884\pi\)
−0.367786 + 0.929911i \(0.619884\pi\)
\(38\) −1.36294 −0.221098
\(39\) 3.80451 0.609209
\(40\) −2.53039 −0.400089
\(41\) 6.27882 0.980587 0.490293 0.871557i \(-0.336890\pi\)
0.490293 + 0.871557i \(0.336890\pi\)
\(42\) −0.167449 −0.0258380
\(43\) −2.33490 −0.356069 −0.178034 0.984024i \(-0.556974\pi\)
−0.178034 + 0.984024i \(0.556974\pi\)
\(44\) 0 0
\(45\) 3.80451 0.567143
\(46\) −0.278820 −0.0411098
\(47\) −12.1394 −1.77071 −0.885357 0.464911i \(-0.846086\pi\)
−0.885357 + 0.464911i \(0.846086\pi\)
\(48\) −3.83255 −0.553181
\(49\) 1.00000 0.142857
\(50\) 1.58647 0.224360
\(51\) 0.334898 0.0468952
\(52\) 7.50235 1.04039
\(53\) 7.94392 1.09118 0.545591 0.838052i \(-0.316305\pi\)
0.545591 + 0.838052i \(0.316305\pi\)
\(54\) −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 \(0.421540\pi\)
\(60\) 7.50235 0.968549
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −1.66510 −0.211468
\(63\) 1.00000 0.125988
\(64\) −7.33490 −0.916862
\(65\) −14.4743 −1.79532
\(66\) 0 0
\(67\) −0.139410 −0.0170316 −0.00851582 0.999964i \(-0.502711\pi\)
−0.00851582 + 0.999964i \(0.502711\pi\)
\(68\) 0.660406 0.0800860
\(69\) 1.66510 0.200455
\(70\) 0.637062 0.0761435
\(71\) 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
\(75\) −9.47431 −1.09400
\(76\) 16.0506 1.84113
\(77\) 0 0
\(78\) 0.637062 0.0721331
\(79\) −3.33020 −0.374677 −0.187339 0.982295i \(-0.559986\pi\)
−0.187339 + 0.982295i \(0.559986\pi\)
\(80\) 14.5810 1.63020
\(81\) 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
\(85\) −1.27412 −0.138198
\(86\) −0.390977 −0.0421601
\(87\) 0.195488 0.0209586
\(88\) 0 0
\(89\) −9.88784 −1.04811 −0.524055 0.851685i \(-0.675581\pi\)
−0.524055 + 0.851685i \(0.675581\pi\)
\(90\) 0.637062 0.0671523
\(91\) −3.80451 −0.398821
\(92\) 3.28352 0.342330
\(93\) 9.94392 1.03114
\(94\) −2.03273 −0.209661
\(95\) −30.9665 −3.17709
\(96\) −1.97196 −0.201262
\(97\) 0.0560785 0.00569391 0.00284695 0.999996i \(-0.499094\pi\)
0.00284695 + 0.999996i \(0.499094\pi\)
\(98\) 0.167449 0.0169149
\(99\) 0 0
\(100\) −18.6830 −1.86830
\(101\) 18.8831 1.87894 0.939472 0.342626i \(-0.111317\pi\)
0.939472 + 0.342626i \(0.111317\pi\)
\(102\) 0.0560785 0.00555260
\(103\) −8.27882 −0.815736 −0.407868 0.913041i \(-0.633728\pi\)
−0.407868 + 0.913041i \(0.633728\pi\)
\(104\) 2.53039 0.248125
\(105\) −3.80451 −0.371282
\(106\) 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 \(-0.686626\pi\)
−0.553286 + 0.832992i \(0.686626\pi\)
\(110\) 0 0
\(111\) 4.47431 0.424683
\(112\) 3.83255 0.362142
\(113\) −1.33020 −0.125135 −0.0625675 0.998041i \(-0.519929\pi\)
−0.0625675 + 0.998041i \(0.519929\pi\)
\(114\) 1.36294 0.127651
\(115\) −6.33490 −0.590732
\(116\) 0.385496 0.0357924
\(117\) −3.80451 −0.351727
\(118\) 0.627672 0.0577819
\(119\) −0.334898 −0.0307001
\(120\) 2.53039 0.230992
\(121\) 0 0
\(122\) −1.00470 −0.0909608
\(123\) −6.27882 −0.566142
\(124\) 19.6090 1.76094
\(125\) 17.0226 1.52254
\(126\) 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
\(130\) −2.42371 −0.212574
\(131\) −20.5576 −1.79613 −0.898065 0.439863i \(-0.855027\pi\)
−0.898065 + 0.439863i \(0.855027\pi\)
\(132\) 0 0
\(133\) −8.13941 −0.705776
\(134\) −0.0233441 −0.00201662
\(135\) −3.80451 −0.327440
\(136\) 0.222741 0.0190999
\(137\) −16.2227 −1.38600 −0.693001 0.720936i \(-0.743712\pi\)
−0.693001 + 0.720936i \(0.743712\pi\)
\(138\) 0.278820 0.0237347
\(139\) −22.5482 −1.91252 −0.956259 0.292522i \(-0.905506\pi\)
−0.956259 + 0.292522i \(0.905506\pi\)
\(140\) −7.50235 −0.634064
\(141\) 12.1394 1.02232
\(142\) 0.781954 0.0656201
\(143\) 0 0
\(144\) 3.83255 0.319379
\(145\) −0.743738 −0.0617641
\(146\) −0.702531 −0.0581419
\(147\) −1.00000 −0.0824786
\(148\) 8.82316 0.725259
\(149\) −8.19549 −0.671401 −0.335700 0.941969i \(-0.608973\pi\)
−0.335700 + 0.941969i \(0.608973\pi\)
\(150\) −1.58647 −0.129534
\(151\) 13.2741 1.08023 0.540116 0.841590i \(-0.318380\pi\)
0.540116 + 0.841590i \(0.318380\pi\)
\(152\) 5.41353 0.439096
\(153\) −0.334898 −0.0270749
\(154\) 0 0
\(155\) −37.8318 −3.03872
\(156\) −7.50235 −0.600669
\(157\) −16.9392 −1.35190 −0.675949 0.736949i \(-0.736266\pi\)
−0.675949 + 0.736949i \(0.736266\pi\)
\(158\) −0.557640 −0.0443634
\(159\) −7.94392 −0.629994
\(160\) 7.50235 0.593113
\(161\) −1.66510 −0.131228
\(162\) 0.167449 0.0131561
\(163\) −6.79982 −0.532603 −0.266301 0.963890i \(-0.585802\pi\)
−0.266301 + 0.963890i \(0.585802\pi\)
\(164\) −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
\(170\) −0.213351 −0.0163633
\(171\) −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
\(175\) 9.47431 0.716190
\(176\) 0 0
\(177\) −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 \(0.648028\pi\)
\(180\) −7.50235 −0.559192
\(181\) 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
\(185\) −17.0226 −1.25152
\(186\) 1.66510 0.122091
\(187\) 0 0
\(188\) 23.9384 1.74589
\(189\) −1.00000 −0.0727393
\(190\) −5.18531 −0.376182
\(191\) 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
\(195\) 14.4743 1.03653
\(196\) −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 0
\(199\) 13.5529 0.960743 0.480371 0.877065i \(-0.340502\pi\)
0.480371 + 0.877065i \(0.340502\pi\)
\(200\) −6.30138 −0.445575
\(201\) 0.139410 0.00983322
\(202\) 3.16197 0.222475
\(203\) −0.195488 −0.0137206
\(204\) −0.660406 −0.0462377
\(205\) 23.8878 1.66840
\(206\) −1.38628 −0.0965868
\(207\) −1.66510 −0.115733
\(208\) −14.5810 −1.01101
\(209\) 0 0
\(210\) −0.637062 −0.0439615
\(211\) 4.27882 0.294566 0.147283 0.989094i \(-0.452947\pi\)
0.147283 + 0.989094i \(0.452947\pi\)
\(212\) −15.6651 −1.07588
\(213\) −4.66980 −0.319969
\(214\) 1.36294 0.0931685
\(215\) −8.88315 −0.605826
\(216\) 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 \(-0.611730\pi\)
−0.343846 + 0.939026i \(0.611730\pi\)
\(224\) 1.97196 0.131757
\(225\) 9.47431 0.631621
\(226\) −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 \(0.415461\pi\)
\(230\) −1.06077 −0.0699453
\(231\) 0 0
\(232\) 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
\(235\) −46.1845 −3.01275
\(236\) −7.39176 −0.481163
\(237\) 3.33020 0.216320
\(238\) −0.0560785 −0.00363503
\(239\) −10.7998 −0.698582 −0.349291 0.937014i \(-0.613578\pi\)
−0.349291 + 0.937014i \(0.613578\pi\)
\(240\) −14.5810 −0.941198
\(241\) 12.0833 0.778356 0.389178 0.921163i \(-0.372759\pi\)
0.389178 + 0.921163i \(0.372759\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 11.8318 0.757451
\(245\) 3.80451 0.243061
\(246\) −1.05138 −0.0670338
\(247\) 30.9665 1.97035
\(248\) 6.61372 0.419972
\(249\) −13.9439 −0.883660
\(250\) 2.85041 0.180276
\(251\) 4.80921 0.303554 0.151777 0.988415i \(-0.451500\pi\)
0.151777 + 0.988415i \(0.451500\pi\)
\(252\) −1.97196 −0.124222
\(253\) 0 0
\(254\) −2.44706 −0.153542
\(255\) 1.27412 0.0797888
\(256\) 13.8037 0.862733
\(257\) −16.7531 −1.04503 −0.522516 0.852630i \(-0.675006\pi\)
−0.522516 + 0.852630i \(0.675006\pi\)
\(258\) 0.390977 0.0243412
\(259\) −4.47431 −0.278020
\(260\) 28.5428 1.77015
\(261\) −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 0
\(265\) 30.2227 1.85657
\(266\) −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 \(0.779139\pi\)
\(270\) −0.637062 −0.0387704
\(271\) −4.13941 −0.251451 −0.125726 0.992065i \(-0.540126\pi\)
−0.125726 + 0.992065i \(0.540126\pi\)
\(272\) −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
\(280\) −2.53039 −0.151220
\(281\) −6.74374 −0.402298 −0.201149 0.979561i \(-0.564467\pi\)
−0.201149 + 0.979561i \(0.564467\pi\)
\(282\) 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
\(285\) 30.9665 1.83430
\(286\) 0 0
\(287\) 6.27882 0.370627
\(288\) 1.97196 0.116199
\(289\) −16.8878 −0.993403
\(290\) −0.124538 −0.00731314
\(291\) −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
\(295\) 14.2610 0.830305
\(296\) 2.97587 0.172969
\(297\) 0 0
\(298\) −1.37233 −0.0794968
\(299\) 6.33490 0.366357
\(300\) 18.6830 1.07866
\(301\) −2.33490 −0.134581
\(302\) 2.22274 0.127904
\(303\) −18.8831 −1.08481
\(304\) −31.1947 −1.78914
\(305\) −22.8271 −1.30707
\(306\) −0.0560785 −0.00320579
\(307\) 12.5576 0.716702 0.358351 0.933587i \(-0.383339\pi\)
0.358351 + 0.933587i \(0.383339\pi\)
\(308\) 0 0
\(309\) 8.27882 0.470966
\(310\) −6.33490 −0.359798
\(311\) 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 \(-0.526978\pi\)
−0.0846523 + 0.996411i \(0.526978\pi\)
\(314\) −2.83646 −0.160071
\(315\) 3.80451 0.214360
\(316\) 6.56703 0.369424
\(317\) 9.93453 0.557979 0.278989 0.960294i \(-0.410001\pi\)
0.278989 + 0.960294i \(0.410001\pi\)
\(318\) −1.33020 −0.0745941
\(319\) 0 0
\(320\) −27.9057 −1.55998
\(321\) −8.13941 −0.454298
\(322\) −0.278820 −0.0155380
\(323\) 2.72588 0.151672
\(324\) −1.97196 −0.109553
\(325\) −36.0451 −1.99942
\(326\) −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
\(335\) −0.530387 −0.0289781
\(336\) −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
\(340\) 2.51252 0.136261
\(341\) 0 0
\(342\) −1.36294 −0.0736992
\(343\) 1.00000 0.0539949
\(344\) 1.55294 0.0837292
\(345\) 6.33490 0.341059
\(346\) 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 \(-0.475646\pi\)
0.0764363 + 0.997074i \(0.475646\pi\)
\(350\) 1.58647 0.0848001
\(351\) 3.80451 0.203070
\(352\) 0 0
\(353\) 24.6410 1.31151 0.655753 0.754975i \(-0.272351\pi\)
0.655753 + 0.754975i \(0.272351\pi\)
\(354\) −0.627672 −0.0333604
\(355\) 17.7663 0.942937
\(356\) 19.4984 1.03342
\(357\) 0.334898 0.0177247
\(358\) −2.00939 −0.106200
\(359\) 11.8878 0.627416 0.313708 0.949519i \(-0.398429\pi\)
0.313708 + 0.949519i \(0.398429\pi\)
\(360\) −2.53039 −0.133363
\(361\) 47.2500 2.48684
\(362\) 0.121547 0.00638838
\(363\) 0 0
\(364\) 7.50235 0.393230
\(365\) −15.9618 −0.835478
\(366\) 1.00470 0.0525163
\(367\) 3.60902 0.188389 0.0941947 0.995554i \(-0.469972\pi\)
0.0941947 + 0.995554i \(0.469972\pi\)
\(368\) −6.38159 −0.332663
\(369\) 6.27882 0.326862
\(370\) −2.85041 −0.148186
\(371\) 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 0
\(375\) −17.0226 −0.879041
\(376\) 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 \(0.295209\pi\)
\(380\) 61.0647 3.13255
\(381\) 14.6137 0.748683
\(382\) 0.883148 0.0451858
\(383\) −15.2180 −0.777606 −0.388803 0.921321i \(-0.627111\pi\)
−0.388803 + 0.921321i \(0.627111\pi\)
\(384\) 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 \(0.469851\pi\)
\(390\) 2.42371 0.122729
\(391\) 0.557640 0.0282011
\(392\) −0.665102 −0.0335927
\(393\) 20.5576 1.03700
\(394\) −0.447055 −0.0225223
\(395\) −12.6698 −0.637487
\(396\) 0 0
\(397\) 20.8925 1.04857 0.524283 0.851544i \(-0.324333\pi\)
0.524283 + 0.851544i \(0.324333\pi\)
\(398\) 2.26943 0.113756
\(399\) 8.13941 0.407480
\(400\) 36.3108 1.81554
\(401\) 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
\(405\) 3.80451 0.189048
\(406\) −0.0327344 −0.00162458
\(407\) 0 0
\(408\) −0.222741 −0.0110273
\(409\) 23.8972 1.18164 0.590821 0.806803i \(-0.298804\pi\)
0.590821 + 0.806803i \(0.298804\pi\)
\(410\) 4.00000 0.197546
\(411\) 16.2227 0.800209
\(412\) 16.3255 0.804300
\(413\) 3.74843 0.184448
\(414\) −0.278820 −0.0137033
\(415\) 53.0498 2.60411
\(416\) −7.50235 −0.367833
\(417\) 22.5482 1.10419
\(418\) 0 0
\(419\) −30.2967 −1.48009 −0.740045 0.672557i \(-0.765196\pi\)
−0.740045 + 0.672557i \(0.765196\pi\)
\(420\) 7.50235 0.366077
\(421\) −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
\(425\) −3.17293 −0.153910
\(426\) −0.781954 −0.0378858
\(427\) −6.00000 −0.290360
\(428\) −16.0506 −0.775835
\(429\) 0 0
\(430\) −1.48748 −0.0717325
\(431\) −3.07864 −0.148293 −0.0741463 0.997247i \(-0.523623\pi\)
−0.0741463 + 0.997247i \(0.523623\pi\)
\(432\) −3.83255 −0.184394
\(433\) 16.9392 0.814047 0.407024 0.913418i \(-0.366567\pi\)
0.407024 + 0.913418i \(0.366567\pi\)
\(434\) −1.66510 −0.0799274
\(435\) 0.743738 0.0356595
\(436\) 22.7820 1.09106
\(437\) 13.5529 0.648325
\(438\) 0.702531 0.0335682
\(439\) 11.0786 0.528754 0.264377 0.964419i \(-0.414834\pi\)
0.264377 + 0.964419i \(0.414834\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0.213351 0.0101481
\(443\) −14.5482 −0.691208 −0.345604 0.938380i \(-0.612326\pi\)
−0.345604 + 0.938380i \(0.612326\pi\)
\(444\) −8.82316 −0.418729
\(445\) −37.6184 −1.78328
\(446\) −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 \(0.275352\pi\)
\(450\) 1.58647 0.0747867
\(451\) 0 0
\(452\) 2.62311 0.123381
\(453\) −13.2741 −0.623673
\(454\) 0.0654688 0.00307260
\(455\) −14.4743 −0.678566
\(456\) −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
\(460\) 12.4922 0.582450
\(461\) −8.26943 −0.385146 −0.192573 0.981283i \(-0.561683\pi\)
−0.192573 + 0.981283i \(0.561683\pi\)
\(462\) 0 0
\(463\) 9.73904 0.452612 0.226306 0.974056i \(-0.427335\pi\)
0.226306 + 0.974056i \(0.427335\pi\)
\(464\) −0.749219 −0.0347816
\(465\) 37.8318 1.75441
\(466\) −4.44706 −0.206006
\(467\) −40.6970 −1.88323 −0.941617 0.336685i \(-0.890694\pi\)
−0.941617 + 0.336685i \(0.890694\pi\)
\(468\) 7.50235 0.346796
\(469\) −0.139410 −0.00643735
\(470\) −7.73356 −0.356723
\(471\) 16.9392 0.780518
\(472\) −2.49309 −0.114754
\(473\) 0 0
\(474\) 0.557640 0.0256132
\(475\) −77.1153 −3.53829
\(476\) 0.660406 0.0302697
\(477\) 7.94392 0.363727
\(478\) −1.80842 −0.0827152
\(479\) −37.8318 −1.72858 −0.864289 0.502996i \(-0.832231\pi\)
−0.864289 + 0.502996i \(0.832231\pi\)
\(480\) −7.50235 −0.342434
\(481\) 17.0226 0.776162
\(482\) 2.02334 0.0921608
\(483\) 1.66510 0.0757647
\(484\) 0 0
\(485\) 0.213351 0.00968778
\(486\) −0.167449 −0.00759565
\(487\) 40.5576 1.83784 0.918921 0.394442i \(-0.129062\pi\)
0.918921 + 0.394442i \(0.129062\pi\)
\(488\) 3.99061 0.180646
\(489\) 6.79982 0.307498
\(490\) 0.637062 0.0287795
\(491\) −31.0786 −1.40256 −0.701280 0.712886i \(-0.747388\pi\)
−0.701280 + 0.712886i \(0.747388\pi\)
\(492\) 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 \(0.388304\pi\)
\(500\) −33.5678 −1.50120
\(501\) −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
\(505\) 71.8412 3.19689
\(506\) 0 0
\(507\) −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 \(0.630614\pi\)
\(510\) 0.213351 0.00944735
\(511\) −4.19549 −0.185597
\(512\) 12.6557 0.559309
\(513\) 8.13941 0.359364
\(514\) −2.80530 −0.123736
\(515\) −31.4969 −1.38792
\(516\) −4.60433 −0.202694
\(517\) 0 0
\(518\) −0.749219 −0.0329188
\(519\) −1.72118 −0.0755514
\(520\) 9.62689 0.422167
\(521\) 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
\(525\) −9.47431 −0.413493
\(526\) 2.03273 0.0886314
\(527\) 3.33020 0.145066
\(528\) 0 0
\(529\) −20.2274 −0.879454
\(530\) 5.06077 0.219826
\(531\) 3.74843 0.162668
\(532\) 16.0506 0.695882
\(533\) −23.8878 −1.03470
\(534\) 1.65571 0.0716496
\(535\) 30.9665 1.33880
\(536\) 0.0927218 0.00400497
\(537\) 12.0000 0.517838
\(538\) −4.22274 −0.182055
\(539\) 0 0
\(540\) 7.50235 0.322850
\(541\) 4.26943 0.183557 0.0917786 0.995779i \(-0.470745\pi\)
0.0917786 + 0.995779i \(0.470745\pi\)
\(542\) −0.693141 −0.0297729
\(543\) −0.725875 −0.0311503
\(544\) −0.660406 −0.0283147
\(545\) −43.9533 −1.88275
\(546\) 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
\(555\) 17.0226 0.722567
\(556\) 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
\(560\) 14.5810 0.616159
\(561\) 0 0
\(562\) −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
\(565\) −5.06077 −0.212908
\(566\) −3.91119 −0.164399
\(567\) 1.00000 0.0419961
\(568\) −3.10589 −0.130320
\(569\) 29.2180 1.22488 0.612442 0.790515i \(-0.290187\pi\)
0.612442 + 0.790515i \(0.290187\pi\)
\(570\) 5.18531 0.217189
\(571\) −32.4455 −1.35780 −0.678901 0.734230i \(-0.737543\pi\)
−0.678901 + 0.734230i \(0.737543\pi\)
\(572\) 0 0
\(573\) −5.27412 −0.220330
\(574\) 1.05138 0.0438839
\(575\) −15.7757 −0.657892
\(576\) −7.33490 −0.305621
\(577\) −14.8831 −0.619594 −0.309797 0.950803i \(-0.600261\pi\)
−0.309797 + 0.950803i \(0.600261\pi\)
\(578\) −2.82786 −0.117623
\(579\) −19.8318 −0.824180
\(580\) 1.46662 0.0608982
\(581\) 13.9439 0.578491
\(582\) −0.00939029 −0.000389240 0
\(583\) 0 0
\(584\) 2.79043 0.115469
\(585\) −14.4743 −0.598439
\(586\) 2.37220 0.0979945
\(587\) −4.53039 −0.186989 −0.0934945 0.995620i \(-0.529804\pi\)
−0.0934945 + 0.995620i \(0.529804\pi\)
\(588\) 1.97196 0.0813223
\(589\) 80.9377 3.33498
\(590\) 2.38799 0.0983118
\(591\) 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 0
\(595\) −1.27412 −0.0522340
\(596\) 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 \(0.487052\pi\)
\(600\) 6.30138 0.257253
\(601\) 8.47431 0.345674 0.172837 0.984950i \(-0.444707\pi\)
0.172837 + 0.984950i \(0.444707\pi\)
\(602\) −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
\(610\) −3.82237 −0.154763
\(611\) 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
\(615\) −23.8878 −0.963251
\(616\) 0 0
\(617\) 22.6698 0.912652 0.456326 0.889813i \(-0.349165\pi\)
0.456326 + 0.889813i \(0.349165\pi\)
\(618\) 1.38628 0.0557644
\(619\) 8.71648 0.350345 0.175173 0.984538i \(-0.443952\pi\)
0.175173 + 0.984538i \(0.443952\pi\)
\(620\) 74.6028 2.99612
\(621\) 1.66510 0.0668182
\(622\) 1.56391 0.0627070
\(623\) −9.88784 −0.396148
\(624\) 14.5810 0.583707
\(625\) 17.3910 0.695639
\(626\) −0.501561 −0.0200464
\(627\) 0 0
\(628\) 33.4035 1.33294
\(629\) 1.49844 0.0597467
\(630\) 0.637062 0.0253812
\(631\) −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
\(635\) −55.5981 −2.20634
\(636\) 15.6651 0.621162
\(637\) −3.80451 −0.150740
\(638\) 0 0
\(639\) 4.66980 0.184734
\(640\) −19.6775 −0.777821
\(641\) −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 \(-0.351714\pi\)
0.449185 + 0.893439i \(0.351714\pi\)
\(644\) 3.28352 0.129389
\(645\) 8.88315 0.349774
\(646\) 0.456446 0.0179586
\(647\) 31.2453 1.22838 0.614190 0.789158i \(-0.289483\pi\)
0.614190 + 0.789158i \(0.289483\pi\)
\(648\) −0.665102 −0.0261277
\(649\) 0 0
\(650\) −6.03573 −0.236741
\(651\) 9.94392 0.389733
\(652\) 13.4090 0.525136
\(653\) 28.2882 1.10700 0.553502 0.832848i \(-0.313291\pi\)
0.553502 + 0.832848i \(0.313291\pi\)
\(654\) 1.93453 0.0756462
\(655\) −78.2118 −3.05599
\(656\) 24.0639 0.939537
\(657\) −4.19549 −0.163682
\(658\) −2.03273 −0.0792442
\(659\) −30.2967 −1.18019 −0.590096 0.807333i \(-0.700910\pi\)
−0.590096 + 0.807333i \(0.700910\pi\)
\(660\) 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
\(665\) −30.9665 −1.20083
\(666\) −0.749219 −0.0290317
\(667\) 0.325508 0.0126037
\(668\) −35.9345 −1.39035
\(669\) 10.2694 0.397039
\(670\) −0.0888128 −0.00343114
\(671\) 0 0
\(672\) −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
\(675\) −9.47431 −0.364666
\(676\) −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
\(680\) 0.847422 0.0324972
\(681\) −0.390977 −0.0149823
\(682\) 0 0
\(683\) 25.0498 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(684\) 16.0506 0.613710
\(685\) −61.7196 −2.35818
\(686\) 0.167449 0.00639324
\(687\) −7.94392 −0.303080
\(688\) −8.94862 −0.341163
\(689\) −30.2227 −1.15139
\(690\) 1.06077 0.0403830
\(691\) −38.8271 −1.47705 −0.738526 0.674225i \(-0.764478\pi\)
−0.738526 + 0.674225i \(0.764478\pi\)
\(692\) −3.39410 −0.129024
\(693\) 0 0
\(694\) 4.65102 0.176550
\(695\) −85.7851 −3.25401
\(696\) −0.130020 −0.00492838
\(697\) −2.10277 −0.0796480
\(698\) 0.478217 0.0181008
\(699\) 26.5576 1.00450
\(700\) −18.6830 −0.706150
\(701\) 41.7757 1.57785 0.788923 0.614492i \(-0.210639\pi\)
0.788923 + 0.614492i \(0.210639\pi\)
\(702\) 0.637062 0.0240444
\(703\) 36.4182 1.37354
\(704\) 0 0
\(705\) 46.1845 1.73941
\(706\) 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 \(0.580990\pi\)
\(710\) 2.97495 0.111648
\(711\) −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 \(0.467423\pi\)
\(720\) 14.5810 0.543401
\(721\) −8.27882 −0.308319
\(722\) 7.91197 0.294453
\(723\) −12.0833 −0.449384
\(724\) −1.43140 −0.0531975
\(725\) −1.85212 −0.0687859
\(726\) 0 0
\(727\) −32.8831 −1.21957 −0.609784 0.792567i \(-0.708744\pi\)
−0.609784 + 0.792567i \(0.708744\pi\)
\(728\) 2.53039 0.0937824
\(729\) 1.00000 0.0370370
\(730\) −2.67279 −0.0989243
\(731\) 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
\(735\) −3.80451 −0.140332
\(736\) −3.28352 −0.121032
\(737\) 0 0
\(738\) 1.05138 0.0387020
\(739\) −29.6651 −1.09125 −0.545624 0.838030i \(-0.683707\pi\)
−0.545624 + 0.838030i \(0.683707\pi\)
\(740\) 33.5678 1.23398
\(741\) −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
\(745\) −31.1798 −1.14234
\(746\) −2.06547 −0.0756222
\(747\) 13.9439 0.510181
\(748\) 0 0
\(749\) 8.13941 0.297408
\(750\) −2.85041 −0.104082
\(751\) −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
\(755\) 50.5016 1.83794
\(756\) 1.97196 0.0717195
\(757\) 2.74374 0.0997229 0.0498614 0.998756i \(-0.484122\pi\)
0.0498614 + 0.998756i \(0.484122\pi\)
\(758\) 3.91119 0.142061
\(759\) 0 0
\(760\) 20.5959 0.747090
\(761\) −6.39098 −0.231673 −0.115836 0.993268i \(-0.536955\pi\)
−0.115836 + 0.993268i \(0.536955\pi\)
\(762\) 2.44706 0.0886475
\(763\) −11.5529 −0.418245
\(764\) −10.4004 −0.376272
\(765\) −1.27412 −0.0460661
\(766\) −2.54825 −0.0920720
\(767\) −14.2610 −0.514933
\(768\) −13.8037 −0.498099
\(769\) −17.7951 −0.641708 −0.320854 0.947129i \(-0.603970\pi\)
−0.320854 + 0.947129i \(0.603970\pi\)
\(770\) 0 0
\(771\) 16.7531 0.603349
\(772\) −39.1075 −1.40751
\(773\) −31.5257 −1.13390 −0.566950 0.823752i \(-0.691877\pi\)
−0.566950 + 0.823752i \(0.691877\pi\)
\(774\) −0.390977 −0.0140534
\(775\) −94.2118 −3.38419
\(776\) −0.0372979 −0.00133892
\(777\) 4.47431 0.160515
\(778\) 0.624681 0.0223959
\(779\) −51.1059 −1.83106
\(780\) −28.5428 −1.02200
\(781\) 0 0
\(782\) 0.0933763 0.00333913
\(783\) 0.195488 0.00698619
\(784\) 3.83255 0.136877
\(785\) −64.4455 −2.30016
\(786\) 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
\(790\) −2.12155 −0.0754813
\(791\) −1.33020 −0.0472966
\(792\) 0 0
\(793\) 22.8271 0.810613
\(794\) 3.49844 0.124155
\(795\) −30.2227 −1.07189
\(796\) −26.7259 −0.947274
\(797\) 43.8590 1.55357 0.776783 0.629768i \(-0.216850\pi\)
0.776783 + 0.629768i \(0.216850\pi\)
\(798\) 1.36294 0.0482475
\(799\) 4.06547 0.143826
\(800\) 18.6830 0.660543
\(801\) −9.88784 −0.349370
\(802\) 3.89723 0.137616
\(803\) 0 0
\(804\) −0.274911 −0.00969536
\(805\) −6.33490 −0.223276
\(806\) 6.33490 0.223137
\(807\) 25.2180 0.887717
\(808\) −12.5592 −0.441832
\(809\) −6.74374 −0.237097 −0.118549 0.992948i \(-0.537824\pi\)
−0.118549 + 0.992948i \(0.537824\pi\)
\(810\) 0.637062 0.0223841
\(811\) −3.97275 −0.139502 −0.0697510 0.997564i \(-0.522220\pi\)
−0.0697510 + 0.997564i \(0.522220\pi\)
\(812\) 0.385496 0.0135282
\(813\) 4.13941 0.145175
\(814\) 0 0
\(815\) −25.8700 −0.906186
\(816\) 1.28352 0.0449320
\(817\) 19.0047 0.664890
\(818\) 4.00157 0.139912
\(819\) −3.80451 −0.132940
\(820\) −47.1059 −1.64501
\(821\) −49.5896 −1.73069 −0.865344 0.501178i \(-0.832900\pi\)
−0.865344 + 0.501178i \(0.832900\pi\)
\(822\) 2.71648 0.0947483
\(823\) 23.1908 0.808380 0.404190 0.914675i \(-0.367553\pi\)
0.404190 + 0.914675i \(0.367553\pi\)
\(824\) 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 \(0.498511\pi\)
\(830\) 8.88315 0.308339
\(831\) −18.2788 −0.634085
\(832\) 27.9057 0.967456
\(833\) −0.334898 −0.0116035
\(834\) 3.77569 0.130741
\(835\) 69.3286 2.39922
\(836\) 0 0
\(837\) 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 \(0.656555\pi\)
\(840\) 2.53039 0.0873066
\(841\) −28.9618 −0.998682
\(842\) −2.59976 −0.0895938
\(843\) 6.74374 0.232267
\(844\) −8.43767 −0.290436
\(845\) 5.60902 0.192956
\(846\) −2.03273 −0.0698868
\(847\) 0 0
\(848\) 30.4455 1.04550
\(849\) 23.3575 0.801626
\(850\) −0.531305 −0.0182236
\(851\) 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
\(855\) −30.9665 −1.05903
\(856\) −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 0
\(859\) 20.5576 0.701418 0.350709 0.936485i \(-0.385941\pi\)
0.350709 + 0.936485i \(0.385941\pi\)
\(860\) 17.5172 0.597332
\(861\) −6.27882 −0.213982
\(862\) −0.515515 −0.0175585
\(863\) −9.66510 −0.329004 −0.164502 0.986377i \(-0.552602\pi\)
−0.164502 + 0.986377i \(0.552602\pi\)
\(864\) −1.97196 −0.0670875
\(865\) 6.54825 0.222647
\(866\) 2.83646 0.0963868
\(867\) 16.8878 0.573541
\(868\) 19.6090 0.665574
\(869\) 0 0
\(870\) 0.124538 0.00422224
\(871\) 0.530387 0.0179715
\(872\) 7.68388 0.260209
\(873\) 0.0560785 0.00189797
\(874\) 2.26943 0.0767646
\(875\) 17.0226 0.575467
\(876\) −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 \(-0.398182\pi\)
0.314445 + 0.949276i \(0.398182\pi\)
\(884\) −2.51252 −0.0845053
\(885\) −14.2610 −0.479377
\(886\) −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
\(890\) −6.29917 −0.211149
\(891\) 0 0
\(892\) 20.2509 0.678051
\(893\) 98.8076 3.30647
\(894\) 1.37233 0.0458975
\(895\) −45.6541 −1.52605
\(896\) −5.17214 −0.172789
\(897\) −6.33490 −0.211516
\(898\) 4.60276 0.153596
\(899\) 1.94392 0.0648334
\(900\) −18.6830 −0.622765
\(901\) −2.66041 −0.0886310
\(902\) 0 0
\(903\) 2.33490 0.0777006
\(904\) 0.884720 0.0294254
\(905\) 2.76160 0.0917987
\(906\) −2.22274 −0.0738456
\(907\) 49.0965 1.63022 0.815111 0.579304i \(-0.196676\pi\)
0.815111 + 0.579304i \(0.196676\pi\)
\(908\) −0.770991 −0.0255862
\(909\) 18.8831 0.626314
\(910\) −2.42371 −0.0803452
\(911\) 29.3753 0.973248 0.486624 0.873612i \(-0.338228\pi\)
0.486624 + 0.873612i \(0.338228\pi\)
\(912\) 31.1947 1.03296
\(913\) 0 0
\(914\) 1.33020 0.0439992
\(915\) 22.8271 0.754640
\(916\) −15.6651 −0.517590
\(917\) −20.5576 −0.678873
\(918\) 0.0560785 0.00185087
\(919\) 27.0047 0.890803 0.445401 0.895331i \(-0.353061\pi\)
0.445401 + 0.895331i \(0.353061\pi\)
\(920\) 4.21335 0.138910
\(921\) −12.5576 −0.413788
\(922\) −1.38471 −0.0456030
\(923\) −17.7663 −0.584785
\(924\) 0 0
\(925\) −42.3910 −1.39381
\(926\) 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 \(0.102105\pi\)
\(930\) 6.33490 0.207730
\(931\) −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
\(940\) 91.0741 2.97051
\(941\) 34.2788 1.11746 0.558729 0.829350i \(-0.311289\pi\)
0.558729 + 0.829350i \(0.311289\pi\)
\(942\) 2.83646 0.0924169
\(943\) −10.4549 −0.340458
\(944\) 14.3661 0.467575
\(945\) −3.80451 −0.123761
\(946\) 0 0
\(947\) −18.1573 −0.590032 −0.295016 0.955492i \(-0.595325\pi\)
−0.295016 + 0.955492i \(0.595325\pi\)
\(948\) −6.56703 −0.213287
\(949\) 15.9618 0.518141
\(950\) −12.9129 −0.418950
\(951\) −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
\(955\) 20.0655 0.649303
\(956\) 21.2968 0.688788
\(957\) 0 0
\(958\) −6.33490 −0.204671
\(959\) −16.2227 −0.523860
\(960\) 27.9057 0.900653
\(961\) 67.8816 2.18973
\(962\) 2.85041 0.0919010
\(963\) 8.13941 0.262289
\(964\) −23.8279 −0.767444
\(965\) 75.4502 2.42883
\(966\) 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 0
\(969\) −2.72588 −0.0875677
\(970\) 0.0357255 0.00114708
\(971\) 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
\(975\) 36.0451 1.15437
\(976\) −22.9953 −0.736062
\(977\) 6.71648 0.214879 0.107440 0.994212i \(-0.465735\pi\)
0.107440 + 0.994212i \(0.465735\pi\)
\(978\) 1.13862 0.0364092
\(979\) 0 0
\(980\) −7.50235 −0.239654
\(981\) −11.5529 −0.368857
\(982\) −5.20409 −0.166069
\(983\) 9.99061 0.318651 0.159325 0.987226i \(-0.449068\pi\)
0.159325 + 0.987226i \(0.449068\pi\)
\(984\) 4.17605 0.133128
\(985\) −10.1573 −0.323638
\(986\) 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
\(995\) 51.5623 1.63464
\(996\) 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 2541.2.a.bi.1.2 3
3.2 odd 2 7623.2.a.cb.1.2 3
11.10 odd 2 231.2.a.d.1.2 3
33.32 even 2 693.2.a.m.1.2 3
44.43 even 2 3696.2.a.bp.1.3 3
55.54 odd 2 5775.2.a.bw.1.2 3
77.76 even 2 1617.2.a.s.1.2 3
231.230 odd 2 4851.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 11.10 odd 2
693.2.a.m.1.2 3 33.32 even 2
1617.2.a.s.1.2 3 77.76 even 2
2541.2.a.bi.1.2 3 1.1 even 1 trivial
3696.2.a.bp.1.3 3 44.43 even 2
4851.2.a.bp.1.2 3 231.230 odd 2
5775.2.a.bw.1.2 3 55.54 odd 2
7623.2.a.cb.1.2 3 3.2 odd 2