Properties

Label 9075.2.a.v.1.2
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} +0.414214 q^{6} -4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.82843 q^{12} +5.65685 q^{13} -2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} +1.17157 q^{19} -4.82843 q^{21} +4.00000 q^{23} -1.58579 q^{24} +2.34315 q^{26} +1.00000 q^{27} +8.82843 q^{28} -0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} -1.82843 q^{36} -0.343146 q^{37} +0.485281 q^{38} +5.65685 q^{39} +0.828427 q^{41} -2.00000 q^{42} -3.17157 q^{43} +1.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} +16.3137 q^{49} -6.82843 q^{51} -10.3431 q^{52} +13.3137 q^{53} +0.414214 q^{54} +7.65685 q^{56} +1.17157 q^{57} -0.343146 q^{58} -4.00000 q^{59} +0.343146 q^{61} -4.82843 q^{63} -4.17157 q^{64} -5.65685 q^{67} +12.4853 q^{68} +4.00000 q^{69} +13.6569 q^{71} -1.58579 q^{72} -11.3137 q^{73} -0.142136 q^{74} -2.14214 q^{76} +2.34315 q^{78} +8.48528 q^{79} +1.00000 q^{81} +0.343146 q^{82} -10.0000 q^{83} +8.82843 q^{84} -1.31371 q^{86} -0.828427 q^{87} -7.65685 q^{89} -27.3137 q^{91} -7.31371 q^{92} +1.65685 q^{94} +4.41421 q^{96} -0.343146 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 4q^{7} - 6q^{8} + 2q^{9} + 2q^{12} - 4q^{14} + 6q^{16} - 8q^{17} - 2q^{18} + 8q^{19} - 4q^{21} + 8q^{23} - 6q^{24} + 16q^{26} + 2q^{27} + 12q^{28} + 4q^{29} + 6q^{32} + 2q^{36} - 12q^{37} - 16q^{38} - 4q^{41} - 4q^{42} - 12q^{43} - 8q^{46} + 8q^{47} + 6q^{48} + 10q^{49} - 8q^{51} - 32q^{52} + 4q^{53} - 2q^{54} + 4q^{56} + 8q^{57} - 12q^{58} - 8q^{59} + 12q^{61} - 4q^{63} - 14q^{64} + 8q^{68} + 8q^{69} + 16q^{71} - 6q^{72} + 28q^{74} + 24q^{76} + 16q^{78} + 2q^{81} + 12q^{82} - 20q^{83} + 12q^{84} + 20q^{86} + 4q^{87} - 4q^{89} - 32q^{91} + 8q^{92} - 8q^{94} + 6q^{96} - 12q^{97} + 22q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0.414214 0.169102
\(7\) −4.82843 −1.82497 −0.912487 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.82843 −0.527821
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0.414214 0.0976311
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) −4.82843 −1.05365
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.58579 −0.323697
\(25\) 0 0
\(26\) 2.34315 0.459529
\(27\) 1.00000 0.192450
\(28\) 8.82843 1.66842
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) −0.343146 −0.0564128 −0.0282064 0.999602i \(-0.508980\pi\)
−0.0282064 + 0.999602i \(0.508980\pi\)
\(38\) 0.485281 0.0787230
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) 0.828427 0.129379 0.0646893 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) −2.00000 −0.308607
\(43\) −3.17157 −0.483660 −0.241830 0.970319i \(-0.577748\pi\)
−0.241830 + 0.970319i \(0.577748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.65685 0.244290
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 3.00000 0.433013
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) −6.82843 −0.956171
\(52\) −10.3431 −1.43434
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) 7.65685 1.02319
\(57\) 1.17157 0.155179
\(58\) −0.343146 −0.0450572
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0.343146 0.0439353 0.0219677 0.999759i \(-0.493007\pi\)
0.0219677 + 0.999759i \(0.493007\pi\)
\(62\) 0 0
\(63\) −4.82843 −0.608325
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 12.4853 1.51406
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) −1.58579 −0.186887
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) −0.142136 −0.0165229
\(75\) 0 0
\(76\) −2.14214 −0.245720
\(77\) 0 0
\(78\) 2.34315 0.265309
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.343146 0.0378941
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 8.82843 0.963260
\(85\) 0 0
\(86\) −1.31371 −0.141661
\(87\) −0.828427 −0.0888167
\(88\) 0 0
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) −27.3137 −2.86325
\(92\) −7.31371 −0.762507
\(93\) 0 0
\(94\) 1.65685 0.170891
\(95\) 0 0
\(96\) 4.41421 0.450524
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 6.75736 0.682596
\(99\) 0 0
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) −2.82843 −0.280056
\(103\) −19.3137 −1.90304 −0.951518 0.307593i \(-0.900477\pi\)
−0.951518 + 0.307593i \(0.900477\pi\)
\(104\) −8.97056 −0.879636
\(105\) 0 0
\(106\) 5.51472 0.535637
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) −1.82843 −0.175940
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) −0.343146 −0.0325700
\(112\) −14.4853 −1.36873
\(113\) −14.9706 −1.40831 −0.704156 0.710045i \(-0.748674\pi\)
−0.704156 + 0.710045i \(0.748674\pi\)
\(114\) 0.485281 0.0454508
\(115\) 0 0
\(116\) 1.51472 0.140638
\(117\) 5.65685 0.522976
\(118\) −1.65685 −0.152526
\(119\) 32.9706 3.02241
\(120\) 0 0
\(121\) 0 0
\(122\) 0.142136 0.0128684
\(123\) 0.828427 0.0746968
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 2.48528 0.220533 0.110267 0.993902i \(-0.464830\pi\)
0.110267 + 0.993902i \(0.464830\pi\)
\(128\) −10.5563 −0.933058
\(129\) −3.17157 −0.279241
\(130\) 0 0
\(131\) 19.3137 1.68745 0.843723 0.536778i \(-0.180359\pi\)
0.843723 + 0.536778i \(0.180359\pi\)
\(132\) 0 0
\(133\) −5.65685 −0.490511
\(134\) −2.34315 −0.202417
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) −9.31371 −0.795724 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(138\) 1.65685 0.141041
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 5.65685 0.474713
\(143\) 0 0
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −4.68629 −0.387840
\(147\) 16.3137 1.34553
\(148\) 0.627417 0.0515734
\(149\) −18.4853 −1.51437 −0.757187 0.653199i \(-0.773427\pi\)
−0.757187 + 0.653199i \(0.773427\pi\)
\(150\) 0 0
\(151\) 0.485281 0.0394916 0.0197458 0.999805i \(-0.493714\pi\)
0.0197458 + 0.999805i \(0.493714\pi\)
\(152\) −1.85786 −0.150693
\(153\) −6.82843 −0.552046
\(154\) 0 0
\(155\) 0 0
\(156\) −10.3431 −0.828114
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 3.51472 0.279616
\(159\) 13.3137 1.05585
\(160\) 0 0
\(161\) −19.3137 −1.52213
\(162\) 0.414214 0.0325437
\(163\) −15.3137 −1.19946 −0.599731 0.800202i \(-0.704726\pi\)
−0.599731 + 0.800202i \(0.704726\pi\)
\(164\) −1.51472 −0.118280
\(165\) 0 0
\(166\) −4.14214 −0.321492
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) 7.65685 0.590739
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 1.17157 0.0895924
\(172\) 5.79899 0.442169
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) −0.343146 −0.0260138
\(175\) 0 0
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) −3.17157 −0.237719
\(179\) −6.34315 −0.474109 −0.237054 0.971496i \(-0.576182\pi\)
−0.237054 + 0.971496i \(0.576182\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) 0.343146 0.0253661
\(184\) −6.34315 −0.467623
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −7.31371 −0.533407
\(189\) −4.82843 −0.351216
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) −4.17157 −0.301057
\(193\) 2.34315 0.168663 0.0843317 0.996438i \(-0.473124\pi\)
0.0843317 + 0.996438i \(0.473124\pi\)
\(194\) −0.142136 −0.0102047
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) −2.00000 −0.140720
\(203\) 4.00000 0.280745
\(204\) 12.4853 0.874145
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) 16.9706 1.17670
\(209\) 0 0
\(210\) 0 0
\(211\) −6.82843 −0.470088 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(212\) −24.3431 −1.67189
\(213\) 13.6569 0.935752
\(214\) −2.20101 −0.150458
\(215\) 0 0
\(216\) −1.58579 −0.107899
\(217\) 0 0
\(218\) 2.20101 0.149071
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) −38.6274 −2.59836
\(222\) −0.142136 −0.00953952
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) −21.3137 −1.42408
\(225\) 0 0
\(226\) −6.20101 −0.412485
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) −2.14214 −0.141866
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.31371 0.0862492
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) 2.34315 0.153176
\(235\) 0 0
\(236\) 7.31371 0.476082
\(237\) 8.48528 0.551178
\(238\) 13.6569 0.885242
\(239\) −6.34315 −0.410304 −0.205152 0.978730i \(-0.565769\pi\)
−0.205152 + 0.978730i \(0.565769\pi\)
\(240\) 0 0
\(241\) 23.6569 1.52387 0.761936 0.647652i \(-0.224249\pi\)
0.761936 + 0.647652i \(0.224249\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −0.627417 −0.0401663
\(245\) 0 0
\(246\) 0.343146 0.0218782
\(247\) 6.62742 0.421692
\(248\) 0 0
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) 8.82843 0.556139
\(253\) 0 0
\(254\) 1.02944 0.0645926
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 27.6569 1.72519 0.862594 0.505898i \(-0.168839\pi\)
0.862594 + 0.505898i \(0.168839\pi\)
\(258\) −1.31371 −0.0817879
\(259\) 1.65685 0.102952
\(260\) 0 0
\(261\) −0.828427 −0.0512784
\(262\) 8.00000 0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.34315 −0.143667
\(267\) −7.65685 −0.468592
\(268\) 10.3431 0.631808
\(269\) −24.6274 −1.50156 −0.750780 0.660552i \(-0.770322\pi\)
−0.750780 + 0.660552i \(0.770322\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) −20.4853 −1.24210
\(273\) −27.3137 −1.65310
\(274\) −3.85786 −0.233062
\(275\) 0 0
\(276\) −7.31371 −0.440234
\(277\) −13.6569 −0.820561 −0.410280 0.911959i \(-0.634569\pi\)
−0.410280 + 0.911959i \(0.634569\pi\)
\(278\) 6.82843 0.409542
\(279\) 0 0
\(280\) 0 0
\(281\) 16.8284 1.00390 0.501950 0.864897i \(-0.332616\pi\)
0.501950 + 0.864897i \(0.332616\pi\)
\(282\) 1.65685 0.0986642
\(283\) −3.17157 −0.188530 −0.0942652 0.995547i \(-0.530050\pi\)
−0.0942652 + 0.995547i \(0.530050\pi\)
\(284\) −24.9706 −1.48173
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) −0.343146 −0.0201156
\(292\) 20.6863 1.21057
\(293\) −1.17157 −0.0684440 −0.0342220 0.999414i \(-0.510895\pi\)
−0.0342220 + 0.999414i \(0.510895\pi\)
\(294\) 6.75736 0.394097
\(295\) 0 0
\(296\) 0.544156 0.0316284
\(297\) 0 0
\(298\) −7.65685 −0.443550
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) 0.201010 0.0115668
\(303\) −4.82843 −0.277386
\(304\) 3.51472 0.201583
\(305\) 0 0
\(306\) −2.82843 −0.161690
\(307\) −8.82843 −0.503865 −0.251932 0.967745i \(-0.581066\pi\)
−0.251932 + 0.967745i \(0.581066\pi\)
\(308\) 0 0
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) 19.3137 1.09518 0.547590 0.836747i \(-0.315545\pi\)
0.547590 + 0.836747i \(0.315545\pi\)
\(312\) −8.97056 −0.507858
\(313\) −4.34315 −0.245489 −0.122745 0.992438i \(-0.539170\pi\)
−0.122745 + 0.992438i \(0.539170\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0 0
\(316\) −15.5147 −0.872771
\(317\) −30.2843 −1.70093 −0.850467 0.526028i \(-0.823681\pi\)
−0.850467 + 0.526028i \(0.823681\pi\)
\(318\) 5.51472 0.309250
\(319\) 0 0
\(320\) 0 0
\(321\) −5.31371 −0.296582
\(322\) −8.00000 −0.445823
\(323\) −8.00000 −0.445132
\(324\) −1.82843 −0.101579
\(325\) 0 0
\(326\) −6.34315 −0.351314
\(327\) 5.31371 0.293849
\(328\) −1.31371 −0.0725374
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) 18.2843 1.00348
\(333\) −0.343146 −0.0188043
\(334\) 3.85786 0.211093
\(335\) 0 0
\(336\) −14.4853 −0.790237
\(337\) −19.3137 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(338\) 7.87006 0.428075
\(339\) −14.9706 −0.813089
\(340\) 0 0
\(341\) 0 0
\(342\) 0.485281 0.0262410
\(343\) −44.9706 −2.42818
\(344\) 5.02944 0.271169
\(345\) 0 0
\(346\) −1.17157 −0.0629841
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) 1.51472 0.0811974
\(349\) 22.9706 1.22959 0.614793 0.788688i \(-0.289240\pi\)
0.614793 + 0.788688i \(0.289240\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) −1.65685 −0.0880608
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 32.9706 1.74499
\(358\) −2.62742 −0.138863
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) −5.79899 −0.304788
\(363\) 0 0
\(364\) 49.9411 2.61763
\(365\) 0 0
\(366\) 0.142136 0.00742955
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) 12.0000 0.625543
\(369\) 0.828427 0.0431262
\(370\) 0 0
\(371\) −64.2843 −3.33747
\(372\) 0 0
\(373\) −34.6274 −1.79294 −0.896470 0.443105i \(-0.853877\pi\)
−0.896470 + 0.443105i \(0.853877\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.34315 −0.327123
\(377\) −4.68629 −0.241356
\(378\) −2.00000 −0.102869
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) 2.48528 0.127325
\(382\) −2.34315 −0.119886
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −10.5563 −0.538701
\(385\) 0 0
\(386\) 0.970563 0.0494003
\(387\) −3.17157 −0.161220
\(388\) 0.627417 0.0318523
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) 0 0
\(391\) −27.3137 −1.38131
\(392\) −25.8701 −1.30664
\(393\) 19.3137 0.974248
\(394\) 3.51472 0.177069
\(395\) 0 0
\(396\) 0 0
\(397\) −18.9706 −0.952105 −0.476053 0.879417i \(-0.657933\pi\)
−0.476053 + 0.879417i \(0.657933\pi\)
\(398\) −4.28427 −0.214751
\(399\) −5.65685 −0.283197
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) −2.34315 −0.116865
\(403\) 0 0
\(404\) 8.82843 0.439231
\(405\) 0 0
\(406\) 1.65685 0.0822283
\(407\) 0 0
\(408\) 10.8284 0.536087
\(409\) 8.34315 0.412542 0.206271 0.978495i \(-0.433867\pi\)
0.206271 + 0.978495i \(0.433867\pi\)
\(410\) 0 0
\(411\) −9.31371 −0.459411
\(412\) 35.3137 1.73978
\(413\) 19.3137 0.950365
\(414\) 1.65685 0.0814299
\(415\) 0 0
\(416\) 24.9706 1.22428
\(417\) 16.4853 0.807288
\(418\) 0 0
\(419\) −3.02944 −0.147998 −0.0739988 0.997258i \(-0.523576\pi\)
−0.0739988 + 0.997258i \(0.523576\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −2.82843 −0.137686
\(423\) 4.00000 0.194487
\(424\) −21.1127 −1.02532
\(425\) 0 0
\(426\) 5.65685 0.274075
\(427\) −1.65685 −0.0801808
\(428\) 9.71573 0.469627
\(429\) 0 0
\(430\) 0 0
\(431\) −10.3431 −0.498212 −0.249106 0.968476i \(-0.580137\pi\)
−0.249106 + 0.968476i \(0.580137\pi\)
\(432\) 3.00000 0.144338
\(433\) 4.34315 0.208718 0.104359 0.994540i \(-0.466721\pi\)
0.104359 + 0.994540i \(0.466721\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.71573 −0.465299
\(437\) 4.68629 0.224176
\(438\) −4.68629 −0.223920
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) −16.0000 −0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0.627417 0.0297759
\(445\) 0 0
\(446\) 7.31371 0.346314
\(447\) −18.4853 −0.874324
\(448\) 20.1421 0.951626
\(449\) −2.97056 −0.140190 −0.0700948 0.997540i \(-0.522330\pi\)
−0.0700948 + 0.997540i \(0.522330\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 27.3726 1.28750
\(453\) 0.485281 0.0228005
\(454\) −5.79899 −0.272160
\(455\) 0 0
\(456\) −1.85786 −0.0870025
\(457\) −0.686292 −0.0321034 −0.0160517 0.999871i \(-0.505110\pi\)
−0.0160517 + 0.999871i \(0.505110\pi\)
\(458\) −0.828427 −0.0387099
\(459\) −6.82843 −0.318724
\(460\) 0 0
\(461\) 28.1421 1.31071 0.655355 0.755321i \(-0.272519\pi\)
0.655355 + 0.755321i \(0.272519\pi\)
\(462\) 0 0
\(463\) 28.9706 1.34638 0.673188 0.739471i \(-0.264924\pi\)
0.673188 + 0.739471i \(0.264924\pi\)
\(464\) −2.48528 −0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) −10.3431 −0.478112
\(469\) 27.3137 1.26123
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 6.34315 0.291967
\(473\) 0 0
\(474\) 3.51472 0.161436
\(475\) 0 0
\(476\) −60.2843 −2.76313
\(477\) 13.3137 0.609593
\(478\) −2.62742 −0.120175
\(479\) −3.02944 −0.138419 −0.0692093 0.997602i \(-0.522048\pi\)
−0.0692093 + 0.997602i \(0.522048\pi\)
\(480\) 0 0
\(481\) −1.94113 −0.0885077
\(482\) 9.79899 0.446332
\(483\) −19.3137 −0.878804
\(484\) 0 0
\(485\) 0 0
\(486\) 0.414214 0.0187891
\(487\) −20.9706 −0.950267 −0.475133 0.879914i \(-0.657600\pi\)
−0.475133 + 0.879914i \(0.657600\pi\)
\(488\) −0.544156 −0.0246328
\(489\) −15.3137 −0.692510
\(490\) 0 0
\(491\) −25.6569 −1.15788 −0.578939 0.815371i \(-0.696533\pi\)
−0.578939 + 0.815371i \(0.696533\pi\)
\(492\) −1.51472 −0.0682888
\(493\) 5.65685 0.254772
\(494\) 2.74517 0.123511
\(495\) 0 0
\(496\) 0 0
\(497\) −65.9411 −2.95786
\(498\) −4.14214 −0.185614
\(499\) −33.6569 −1.50669 −0.753344 0.657627i \(-0.771560\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(500\) 0 0
\(501\) 9.31371 0.416106
\(502\) −5.37258 −0.239790
\(503\) −5.31371 −0.236927 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(504\) 7.65685 0.341063
\(505\) 0 0
\(506\) 0 0
\(507\) 19.0000 0.843820
\(508\) −4.54416 −0.201614
\(509\) 41.3137 1.83120 0.915599 0.402093i \(-0.131717\pi\)
0.915599 + 0.402093i \(0.131717\pi\)
\(510\) 0 0
\(511\) 54.6274 2.41657
\(512\) 22.7574 1.00574
\(513\) 1.17157 0.0517262
\(514\) 11.4558 0.505296
\(515\) 0 0
\(516\) 5.79899 0.255286
\(517\) 0 0
\(518\) 0.686292 0.0301539
\(519\) −2.82843 −0.124154
\(520\) 0 0
\(521\) 12.6274 0.553217 0.276609 0.960983i \(-0.410789\pi\)
0.276609 + 0.960983i \(0.410789\pi\)
\(522\) −0.343146 −0.0150191
\(523\) −26.4853 −1.15812 −0.579060 0.815285i \(-0.696580\pi\)
−0.579060 + 0.815285i \(0.696580\pi\)
\(524\) −35.3137 −1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 10.3431 0.448432
\(533\) 4.68629 0.202986
\(534\) −3.17157 −0.137247
\(535\) 0 0
\(536\) 8.97056 0.387469
\(537\) −6.34315 −0.273727
\(538\) −10.2010 −0.439797
\(539\) 0 0
\(540\) 0 0
\(541\) 5.31371 0.228454 0.114227 0.993455i \(-0.463561\pi\)
0.114227 + 0.993455i \(0.463561\pi\)
\(542\) −11.5147 −0.494600
\(543\) −14.0000 −0.600798
\(544\) −30.1421 −1.29233
\(545\) 0 0
\(546\) −11.3137 −0.484182
\(547\) 20.1421 0.861216 0.430608 0.902539i \(-0.358299\pi\)
0.430608 + 0.902539i \(0.358299\pi\)
\(548\) 17.0294 0.727462
\(549\) 0.343146 0.0146451
\(550\) 0 0
\(551\) −0.970563 −0.0413474
\(552\) −6.34315 −0.269982
\(553\) −40.9706 −1.74225
\(554\) −5.65685 −0.240337
\(555\) 0 0
\(556\) −30.1421 −1.27831
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) −17.9411 −0.758829
\(560\) 0 0
\(561\) 0 0
\(562\) 6.97056 0.294035
\(563\) 20.3431 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(564\) −7.31371 −0.307963
\(565\) 0 0
\(566\) −1.31371 −0.0552193
\(567\) −4.82843 −0.202775
\(568\) −21.6569 −0.908701
\(569\) −15.4558 −0.647943 −0.323971 0.946067i \(-0.605018\pi\)
−0.323971 + 0.946067i \(0.605018\pi\)
\(570\) 0 0
\(571\) −0.485281 −0.0203084 −0.0101542 0.999948i \(-0.503232\pi\)
−0.0101542 + 0.999948i \(0.503232\pi\)
\(572\) 0 0
\(573\) −5.65685 −0.236318
\(574\) −1.65685 −0.0691558
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 12.2721 0.510451
\(579\) 2.34315 0.0973778
\(580\) 0 0
\(581\) 48.2843 2.00317
\(582\) −0.142136 −0.00589171
\(583\) 0 0
\(584\) 17.9411 0.742409
\(585\) 0 0
\(586\) −0.485281 −0.0200468
\(587\) 30.6274 1.26413 0.632064 0.774916i \(-0.282208\pi\)
0.632064 + 0.774916i \(0.282208\pi\)
\(588\) −29.8284 −1.23010
\(589\) 0 0
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) −1.02944 −0.0423096
\(593\) −17.1716 −0.705152 −0.352576 0.935783i \(-0.614694\pi\)
−0.352576 + 0.935783i \(0.614694\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 33.7990 1.38446
\(597\) −10.3431 −0.423317
\(598\) 9.37258 0.383273
\(599\) 4.68629 0.191477 0.0957383 0.995407i \(-0.469479\pi\)
0.0957383 + 0.995407i \(0.469479\pi\)
\(600\) 0 0
\(601\) −17.3137 −0.706241 −0.353120 0.935578i \(-0.614879\pi\)
−0.353120 + 0.935578i \(0.614879\pi\)
\(602\) 6.34315 0.258527
\(603\) −5.65685 −0.230365
\(604\) −0.887302 −0.0361038
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 18.4853 0.750294 0.375147 0.926965i \(-0.377592\pi\)
0.375147 + 0.926965i \(0.377592\pi\)
\(608\) 5.17157 0.209735
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 12.4853 0.504688
\(613\) 21.9411 0.886194 0.443097 0.896474i \(-0.353880\pi\)
0.443097 + 0.896474i \(0.353880\pi\)
\(614\) −3.65685 −0.147579
\(615\) 0 0
\(616\) 0 0
\(617\) −11.6569 −0.469287 −0.234644 0.972081i \(-0.575392\pi\)
−0.234644 + 0.972081i \(0.575392\pi\)
\(618\) −8.00000 −0.321807
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000 0.320771
\(623\) 36.9706 1.48119
\(624\) 16.9706 0.679366
\(625\) 0 0
\(626\) −1.79899 −0.0719021
\(627\) 0 0
\(628\) 32.9117 1.31332
\(629\) 2.34315 0.0934273
\(630\) 0 0
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) −13.4558 −0.535245
\(633\) −6.82843 −0.271406
\(634\) −12.5442 −0.498192
\(635\) 0 0
\(636\) −24.3431 −0.965269
\(637\) 92.2843 3.65644
\(638\) 0 0
\(639\) 13.6569 0.540257
\(640\) 0 0
\(641\) −26.9706 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(642\) −2.20101 −0.0868669
\(643\) 29.9411 1.18076 0.590381 0.807124i \(-0.298977\pi\)
0.590381 + 0.807124i \(0.298977\pi\)
\(644\) 35.3137 1.39156
\(645\) 0 0
\(646\) −3.31371 −0.130376
\(647\) −27.3137 −1.07381 −0.536906 0.843642i \(-0.680407\pi\)
−0.536906 + 0.843642i \(0.680407\pi\)
\(648\) −1.58579 −0.0622956
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000 1.09656
\(653\) 26.9706 1.05544 0.527720 0.849418i \(-0.323047\pi\)
0.527720 + 0.849418i \(0.323047\pi\)
\(654\) 2.20101 0.0860663
\(655\) 0 0
\(656\) 2.48528 0.0970339
\(657\) −11.3137 −0.441390
\(658\) −8.00000 −0.311872
\(659\) −7.31371 −0.284902 −0.142451 0.989802i \(-0.545498\pi\)
−0.142451 + 0.989802i \(0.545498\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 7.31371 0.284255
\(663\) −38.6274 −1.50016
\(664\) 15.8579 0.615404
\(665\) 0 0
\(666\) −0.142136 −0.00550764
\(667\) −3.31371 −0.128307
\(668\) −17.0294 −0.658889
\(669\) 17.6569 0.682653
\(670\) 0 0
\(671\) 0 0
\(672\) −21.3137 −0.822194
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −34.7401 −1.33616
\(677\) −21.4558 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(678\) −6.20101 −0.238148
\(679\) 1.65685 0.0635842
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) −2.14214 −0.0819066
\(685\) 0 0
\(686\) −18.6274 −0.711198
\(687\) −2.00000 −0.0763048
\(688\) −9.51472 −0.362745
\(689\) 75.3137 2.86922
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 5.17157 0.196594
\(693\) 0 0
\(694\) −2.76955 −0.105131
\(695\) 0 0
\(696\) 1.31371 0.0497960
\(697\) −5.65685 −0.214269
\(698\) 9.51472 0.360137
\(699\) −13.1716 −0.498195
\(700\) 0 0
\(701\) −7.85786 −0.296787 −0.148394 0.988928i \(-0.547410\pi\)
−0.148394 + 0.988928i \(0.547410\pi\)
\(702\) 2.34315 0.0884363
\(703\) −0.402020 −0.0151625
\(704\) 0 0
\(705\) 0 0
\(706\) −10.7696 −0.405317
\(707\) 23.3137 0.876802
\(708\) 7.31371 0.274866
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 12.1421 0.455046
\(713\) 0 0
\(714\) 13.6569 0.511095
\(715\) 0 0
\(716\) 11.5980 0.433437
\(717\) −6.34315 −0.236889
\(718\) −4.97056 −0.185500
\(719\) 31.5980 1.17841 0.589203 0.807985i \(-0.299442\pi\)
0.589203 + 0.807985i \(0.299442\pi\)
\(720\) 0 0
\(721\) 93.2548 3.47299
\(722\) −7.30152 −0.271734
\(723\) 23.6569 0.879808
\(724\) 25.5980 0.951341
\(725\) 0 0
\(726\) 0 0
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 43.3137 1.60531
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.6569 0.801008
\(732\) −0.627417 −0.0231900
\(733\) −17.6569 −0.652171 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(734\) 0.686292 0.0253315
\(735\) 0 0
\(736\) 17.6569 0.650840
\(737\) 0 0
\(738\) 0.343146 0.0126314
\(739\) −47.1127 −1.73307 −0.866534 0.499118i \(-0.833658\pi\)
−0.866534 + 0.499118i \(0.833658\pi\)
\(740\) 0 0
\(741\) 6.62742 0.243464
\(742\) −26.6274 −0.977523
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.3431 −0.525140
\(747\) −10.0000 −0.365881
\(748\) 0 0
\(749\) 25.6569 0.937481
\(750\) 0 0
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) 12.0000 0.437595
\(753\) −12.9706 −0.472674
\(754\) −1.94113 −0.0706916
\(755\) 0 0
\(756\) 8.82843 0.321087
\(757\) −8.62742 −0.313569 −0.156784 0.987633i \(-0.550113\pi\)
−0.156784 + 0.987633i \(0.550113\pi\)
\(758\) −0.284271 −0.0103252
\(759\) 0 0
\(760\) 0 0
\(761\) 23.1716 0.839969 0.419984 0.907531i \(-0.362036\pi\)
0.419984 + 0.907531i \(0.362036\pi\)
\(762\) 1.02944 0.0372926
\(763\) −25.6569 −0.928840
\(764\) 10.3431 0.374202
\(765\) 0 0
\(766\) 3.31371 0.119729
\(767\) −22.6274 −0.817029
\(768\) 3.97056 0.143275
\(769\) −33.3137 −1.20132 −0.600662 0.799503i \(-0.705096\pi\)
−0.600662 + 0.799503i \(0.705096\pi\)
\(770\) 0 0
\(771\) 27.6569 0.996037
\(772\) −4.28427 −0.154194
\(773\) 7.65685 0.275398 0.137699 0.990474i \(-0.456029\pi\)
0.137699 + 0.990474i \(0.456029\pi\)
\(774\) −1.31371 −0.0472203
\(775\) 0 0
\(776\) 0.544156 0.0195341
\(777\) 1.65685 0.0594393
\(778\) −5.11270 −0.183299
\(779\) 0.970563 0.0347740
\(780\) 0 0
\(781\) 0 0
\(782\) −11.3137 −0.404577
\(783\) −0.828427 −0.0296056
\(784\) 48.9411 1.74790
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) 8.14214 0.290236 0.145118 0.989414i \(-0.453644\pi\)
0.145118 + 0.989414i \(0.453644\pi\)
\(788\) −15.5147 −0.552689
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 72.2843 2.57013
\(792\) 0 0
\(793\) 1.94113 0.0689314
\(794\) −7.85786 −0.278865
\(795\) 0 0
\(796\) 18.9117 0.670307
\(797\) −1.02944 −0.0364645 −0.0182323 0.999834i \(-0.505804\pi\)
−0.0182323 + 0.999834i \(0.505804\pi\)
\(798\) −2.34315 −0.0829465
\(799\) −27.3137 −0.966290
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) −12.1421 −0.428754
\(803\) 0 0
\(804\) 10.3431 0.364775
\(805\) 0 0
\(806\) 0 0
\(807\) −24.6274 −0.866926
\(808\) 7.65685 0.269367
\(809\) 56.4264 1.98385 0.991923 0.126838i \(-0.0404829\pi\)
0.991923 + 0.126838i \(0.0404829\pi\)
\(810\) 0 0
\(811\) 16.4853 0.578877 0.289438 0.957197i \(-0.406532\pi\)
0.289438 + 0.957197i \(0.406532\pi\)
\(812\) −7.31371 −0.256661
\(813\) −27.7990 −0.974953
\(814\) 0 0
\(815\) 0 0
\(816\) −20.4853 −0.717128
\(817\) −3.71573 −0.129997
\(818\) 3.45584 0.120831
\(819\) −27.3137 −0.954418
\(820\) 0 0
\(821\) 7.17157 0.250290 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(822\) −3.85786 −0.134558
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 30.6274 1.06696
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) −7.31371 −0.254169
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −13.6569 −0.473751
\(832\) −23.5980 −0.818113
\(833\) −111.397 −3.85968
\(834\) 6.82843 0.236449
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −1.25483 −0.0433475
\(839\) 22.6274 0.781185 0.390593 0.920564i \(-0.372270\pi\)
0.390593 + 0.920564i \(0.372270\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) −2.48528 −0.0856485
\(843\) 16.8284 0.579602
\(844\) 12.4853 0.429761
\(845\) 0 0
\(846\) 1.65685 0.0569638
\(847\) 0 0
\(848\) 39.9411 1.37158
\(849\) −3.17157 −0.108848
\(850\) 0 0
\(851\) −1.37258 −0.0470515
\(852\) −24.9706 −0.855477
\(853\) −31.3137 −1.07216 −0.536080 0.844167i \(-0.680096\pi\)
−0.536080 + 0.844167i \(0.680096\pi\)
\(854\) −0.686292 −0.0234844
\(855\) 0 0
\(856\) 8.42641 0.288009
\(857\) −11.5147 −0.393335 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(858\) 0 0
\(859\) −19.0294 −0.649276 −0.324638 0.945838i \(-0.605242\pi\)
−0.324638 + 0.945838i \(0.605242\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) −4.28427 −0.145923
\(863\) 43.3137 1.47442 0.737208 0.675666i \(-0.236144\pi\)
0.737208 + 0.675666i \(0.236144\pi\)
\(864\) 4.41421 0.150175
\(865\) 0 0
\(866\) 1.79899 0.0611322
\(867\) 29.6274 1.00620
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) −8.42641 −0.285354
\(873\) −0.343146 −0.0116137
\(874\) 1.94113 0.0656595
\(875\) 0 0
\(876\) 20.6863 0.698925
\(877\) −42.6274 −1.43943 −0.719713 0.694272i \(-0.755727\pi\)
−0.719713 + 0.694272i \(0.755727\pi\)
\(878\) 1.45584 0.0491324
\(879\) −1.17157 −0.0395162
\(880\) 0 0
\(881\) −13.0294 −0.438973 −0.219486 0.975616i \(-0.570438\pi\)
−0.219486 + 0.975616i \(0.570438\pi\)
\(882\) 6.75736 0.227532
\(883\) 50.6274 1.70375 0.851874 0.523747i \(-0.175466\pi\)
0.851874 + 0.523747i \(0.175466\pi\)
\(884\) 70.6274 2.37546
\(885\) 0 0
\(886\) −4.97056 −0.166989
\(887\) −4.34315 −0.145829 −0.0729143 0.997338i \(-0.523230\pi\)
−0.0729143 + 0.997338i \(0.523230\pi\)
\(888\) 0.544156 0.0182607
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) −32.2843 −1.08096
\(893\) 4.68629 0.156821
\(894\) −7.65685 −0.256084
\(895\) 0 0
\(896\) 50.9706 1.70281
\(897\) 22.6274 0.755507
\(898\) −1.23045 −0.0410606
\(899\) 0 0
\(900\) 0 0
\(901\) −90.9117 −3.02871
\(902\) 0 0
\(903\) 15.3137 0.509608
\(904\) 23.7401 0.789584
\(905\) 0 0
\(906\) 0.201010 0.00667811
\(907\) −7.02944 −0.233409 −0.116704 0.993167i \(-0.537233\pi\)
−0.116704 + 0.993167i \(0.537233\pi\)
\(908\) 25.5980 0.849499
\(909\) −4.82843 −0.160149
\(910\) 0 0
\(911\) −15.0294 −0.497947 −0.248974 0.968510i \(-0.580093\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(912\) 3.51472 0.116384
\(913\) 0 0
\(914\) −0.284271 −0.00940286
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) −93.2548 −3.07955
\(918\) −2.82843 −0.0933520
\(919\) −28.4853 −0.939643 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(920\) 0 0
\(921\) −8.82843 −0.290907
\(922\) 11.6569 0.383898
\(923\) 77.2548 2.54287
\(924\) 0 0
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) −19.3137 −0.634345
\(928\) −3.65685 −0.120042
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) 19.1127 0.626393
\(932\) 24.0833 0.788873
\(933\) 19.3137 0.632302
\(934\) −9.37258 −0.306680
\(935\) 0 0
\(936\) −8.97056 −0.293212
\(937\) 44.9706 1.46912 0.734562 0.678541i \(-0.237388\pi\)
0.734562 + 0.678541i \(0.237388\pi\)
\(938\) 11.3137 0.369406
\(939\) −4.34315 −0.141733
\(940\) 0 0
\(941\) −38.7696 −1.26385 −0.631926 0.775029i \(-0.717735\pi\)
−0.631926 + 0.775029i \(0.717735\pi\)
\(942\) −7.45584 −0.242925
\(943\) 3.31371 0.107909
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 38.6274 1.25522 0.627611 0.778527i \(-0.284033\pi\)
0.627611 + 0.778527i \(0.284033\pi\)
\(948\) −15.5147 −0.503895
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) −30.2843 −0.982035
\(952\) −52.2843 −1.69454
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) 5.51472 0.178546
\(955\) 0 0
\(956\) 11.5980 0.375105
\(957\) 0 0
\(958\) −1.25483 −0.0405418
\(959\) 44.9706 1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −0.804041 −0.0259233
\(963\) −5.31371 −0.171232
\(964\) −43.2548 −1.39314
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) −39.4558 −1.26881 −0.634407 0.772999i \(-0.718756\pi\)
−0.634407 + 0.772999i \(0.718756\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 10.6274 0.341050 0.170525 0.985353i \(-0.445454\pi\)
0.170525 + 0.985353i \(0.445454\pi\)
\(972\) −1.82843 −0.0586468
\(973\) −79.5980 −2.55179
\(974\) −8.68629 −0.278327
\(975\) 0 0
\(976\) 1.02944 0.0329515
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) −6.34315 −0.202831
\(979\) 0 0
\(980\) 0 0
\(981\) 5.31371 0.169654
\(982\) −10.6274 −0.339135
\(983\) −14.6274 −0.466542 −0.233271 0.972412i \(-0.574943\pi\)
−0.233271 + 0.972412i \(0.574943\pi\)
\(984\) −1.31371 −0.0418795
\(985\) 0 0
\(986\) 2.34315 0.0746210
\(987\) −19.3137 −0.614762
\(988\) −12.1177 −0.385517
\(989\) −12.6863 −0.403401
\(990\) 0 0
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) 17.6569 0.560323
\(994\) −27.3137 −0.866338
\(995\) 0 0
\(996\) 18.2843 0.579359
\(997\) −16.6863 −0.528460 −0.264230 0.964460i \(-0.585118\pi\)
−0.264230 + 0.964460i \(0.585118\pi\)
\(998\) −13.9411 −0.441299
\(999\) −0.343146 −0.0108567
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 9075.2.a.v.1.2 2
5.4 even 2 1815.2.a.k.1.1 2
11.10 odd 2 825.2.a.g.1.1 2
15.14 odd 2 5445.2.a.m.1.2 2
33.32 even 2 2475.2.a.m.1.2 2
55.32 even 4 825.2.c.e.199.2 4
55.43 even 4 825.2.c.e.199.3 4
55.54 odd 2 165.2.a.a.1.2 2
165.32 odd 4 2475.2.c.m.199.3 4
165.98 odd 4 2475.2.c.m.199.2 4
165.164 even 2 495.2.a.d.1.1 2
220.219 even 2 2640.2.a.bb.1.2 2
385.384 even 2 8085.2.a.ba.1.2 2
660.659 odd 2 7920.2.a.cg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 55.54 odd 2
495.2.a.d.1.1 2 165.164 even 2
825.2.a.g.1.1 2 11.10 odd 2
825.2.c.e.199.2 4 55.32 even 4
825.2.c.e.199.3 4 55.43 even 4
1815.2.a.k.1.1 2 5.4 even 2
2475.2.a.m.1.2 2 33.32 even 2
2475.2.c.m.199.2 4 165.98 odd 4
2475.2.c.m.199.3 4 165.32 odd 4
2640.2.a.bb.1.2 2 220.219 even 2
5445.2.a.m.1.2 2 15.14 odd 2
7920.2.a.cg.1.2 2 660.659 odd 2
8085.2.a.ba.1.2 2 385.384 even 2
9075.2.a.v.1.2 2 1.1 even 1 trivial