Properties

Label 825.2.a.g.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} +4.82843 q^{7} +1.58579 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.82843 q^{12} -5.65685 q^{13} -2.00000 q^{14} +3.00000 q^{16} +6.82843 q^{17} -0.414214 q^{18} -1.17157 q^{19} +4.82843 q^{21} +0.414214 q^{22} +4.00000 q^{23} +1.58579 q^{24} +2.34315 q^{26} +1.00000 q^{27} -8.82843 q^{28} +0.828427 q^{29} -4.41421 q^{32} -1.00000 q^{33} -2.82843 q^{34} -1.82843 q^{36} -0.343146 q^{37} +0.485281 q^{38} -5.65685 q^{39} -0.828427 q^{41} -2.00000 q^{42} +3.17157 q^{43} +1.82843 q^{44} -1.65685 q^{46} +4.00000 q^{47} +3.00000 q^{48} +16.3137 q^{49} +6.82843 q^{51} +10.3431 q^{52} +13.3137 q^{53} -0.414214 q^{54} +7.65685 q^{56} -1.17157 q^{57} -0.343146 q^{58} -4.00000 q^{59} -0.343146 q^{61} +4.82843 q^{63} -4.17157 q^{64} +0.414214 q^{66} -5.65685 q^{67} -12.4853 q^{68} +4.00000 q^{69} +13.6569 q^{71} +1.58579 q^{72} +11.3137 q^{73} +0.142136 q^{74} +2.14214 q^{76} -4.82843 q^{77} +2.34315 q^{78} -8.48528 q^{79} +1.00000 q^{81} +0.343146 q^{82} +10.0000 q^{83} -8.82843 q^{84} -1.31371 q^{86} +0.828427 q^{87} -1.58579 q^{88} -7.65685 q^{89} -27.3137 q^{91} -7.31371 q^{92} -1.65685 q^{94} -4.41421 q^{96} -0.343146 q^{97} -6.75736 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 6q^{8} + 2q^{9} - 2q^{11} + 2q^{12} - 4q^{14} + 6q^{16} + 8q^{17} + 2q^{18} - 8q^{19} + 4q^{21} - 2q^{22} + 8q^{23} + 6q^{24} + 16q^{26} + 2q^{27} - 12q^{28} - 4q^{29} - 6q^{32} - 2q^{33} + 2q^{36} - 12q^{37} - 16q^{38} + 4q^{41} - 4q^{42} + 12q^{43} - 2q^{44} + 8q^{46} + 8q^{47} + 6q^{48} + 10q^{49} + 8q^{51} + 32q^{52} + 4q^{53} + 2q^{54} + 4q^{56} - 8q^{57} - 12q^{58} - 8q^{59} - 12q^{61} + 4q^{63} - 14q^{64} - 2q^{66} - 8q^{68} + 8q^{69} + 16q^{71} + 6q^{72} - 28q^{74} - 24q^{76} - 4q^{77} + 16q^{78} + 2q^{81} + 12q^{82} + 20q^{83} - 12q^{84} + 20q^{86} - 4q^{87} - 6q^{88} - 4q^{89} - 32q^{91} + 8q^{92} + 8q^{94} - 6q^{96} - 12q^{97} - 22q^{98} - 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 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 \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 1.58579 0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.82843 −0.527821
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) −0.414214 −0.0976311
\(19\) −1.17157 −0.268777 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(20\) 0 0
\(21\) 4.82843 1.05365
\(22\) 0.414214 0.0883106
\(23\) 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 \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −4.41421 −0.780330
\(33\) −1.00000 −0.174078
\(34\) −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 \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) −2.00000 −0.308607
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) 1.82843 0.275646
\(45\) 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 \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) 4.82843 0.608325
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0.414214 0.0509862
\(67\) −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 \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0.142136 0.0165229
\(75\) 0 0
\(76\) 2.14214 0.245720
\(77\) −4.82843 −0.550250
\(78\) 2.34315 0.265309
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.343146 0.0378941
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) −8.82843 −0.963260
\(85\) 0 0
\(86\) −1.31371 −0.141661
\(87\) 0.828427 0.0888167
\(88\) −1.58579 −0.169045
\(89\) −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
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) −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 \(-0.417316\pi\)
0.256848 + 0.966452i \(0.417316\pi\)
\(108\) −1.82843 −0.175940
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 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
\(121\) 1.00000 0.0909091
\(122\) 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 \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) 10.5563 0.933058
\(129\) 3.17157 0.279241
\(130\) 0 0
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 1.82843 0.159144
\(133\) −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 \(-0.746430\pi\)
−0.699132 + 0.714993i \(0.746430\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −5.65685 −0.474713
\(143\) 5.65685 0.473050
\(144\) 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 \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) 0 0
\(151\) −0.485281 −0.0394916 −0.0197458 0.999805i \(-0.506286\pi\)
−0.0197458 + 0.999805i \(0.506286\pi\)
\(152\) −1.85786 −0.150693
\(153\) 6.82843 0.552046
\(154\) 2.00000 0.161165
\(155\) 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 \(-0.617346\pi\)
−0.360358 + 0.932814i \(0.617346\pi\)
\(168\) 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 \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) −0.343146 −0.0260138
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −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
\(187\) −6.82843 −0.499344
\(188\) −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 \(-0.526876\pi\)
−0.0843317 + 0.996438i \(0.526876\pi\)
\(194\) 0.142136 0.0102047
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) −8.48528 −0.604551 −0.302276 0.953221i \(-0.597746\pi\)
−0.302276 + 0.953221i \(0.597746\pi\)
\(198\) 0.414214 0.0294369
\(199\) −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
\(209\) 1.17157 0.0810394
\(210\) 0 0
\(211\) 6.82843 0.470088 0.235044 0.971985i \(-0.424477\pi\)
0.235044 + 0.971985i \(0.424477\pi\)
\(212\) −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 \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 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
\(231\) −4.82843 −0.317687
\(232\) 1.31371 0.0862492
\(233\) 13.1716 0.862898 0.431449 0.902137i \(-0.358002\pi\)
0.431449 + 0.902137i \(0.358002\pi\)
\(234\) 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 \(-0.434231\pi\)
0.205152 + 0.978730i \(0.434231\pi\)
\(240\) 0 0
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 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
\(253\) −4.00000 −0.251478
\(254\) 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 \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) −1.58579 −0.0975984
\(265\) 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 \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) 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 \(-0.365431\pi\)
0.410280 + 0.911959i \(0.365431\pi\)
\(278\) 6.82843 0.409542
\(279\) 0 0
\(280\) 0 0
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) −1.65685 −0.0986642
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) −24.9706 −1.48173
\(285\) 0 0
\(286\) −2.34315 −0.138553
\(287\) −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 \(-0.489105\pi\)
0.0342220 + 0.999414i \(0.489105\pi\)
\(294\) −6.75736 −0.394097
\(295\) 0 0
\(296\) −0.544156 −0.0316284
\(297\) −1.00000 −0.0580259
\(298\) −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 \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) 8.82843 0.503046
\(309\) −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
\(319\) −0.828427 −0.0463830
\(320\) 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 \(-0.323675\pi\)
0.526042 + 0.850458i \(0.323675\pi\)
\(338\) −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 \(-0.442562\pi\)
0.179469 + 0.983764i \(0.442562\pi\)
\(348\) −1.51472 −0.0811974
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) −5.65685 −0.301941
\(352\) 4.41421 0.235278
\(353\) −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 \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 5.79899 0.304788
\(363\) 1.00000 0.0524864
\(364\) 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 \(-0.146123\pi\)
0.896470 + 0.443105i \(0.146123\pi\)
\(374\) 2.82843 0.146254
\(375\) 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
\(396\) 1.82843 0.0918819
\(397\) −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
\(407\) 0.343146 0.0170091
\(408\) 10.8284 0.536087
\(409\) −8.34315 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(410\) 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
\(418\) −0.485281 −0.0237359
\(419\) −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
\(429\) 5.65685 0.273115
\(430\) 0 0
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 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 \(-0.526729\pi\)
−0.0838742 + 0.996476i \(0.526729\pi\)
\(440\) 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
\(451\) 0.828427 0.0390091
\(452\) 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 \(-0.494890\pi\)
0.0160517 + 0.999871i \(0.494890\pi\)
\(458\) 0.828427 0.0387099
\(459\) 6.82843 0.318724
\(460\) 0 0
\(461\) −28.1421 −1.31071 −0.655355 0.755321i \(-0.727481\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(462\) 2.00000 0.0930484
\(463\) 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
\(473\) −3.17157 −0.145829
\(474\) 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 \(-0.477952\pi\)
0.0692093 + 0.997602i \(0.477952\pi\)
\(480\) 0 0
\(481\) 1.94113 0.0885077
\(482\) 9.79899 0.446332
\(483\) 19.3137 0.878804
\(484\) −1.82843 −0.0831103
\(485\) 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 \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) 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 \(-0.462203\pi\)
0.118463 + 0.992958i \(0.462203\pi\)
\(504\) 7.65685 0.341063
\(505\) 0 0
\(506\) 1.65685 0.0736562
\(507\) 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
\(517\) −4.00000 −0.175920
\(518\) 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 \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) 35.3137 1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −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
\(539\) −16.3137 −0.702681
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) −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 \(-0.641701\pi\)
−0.430608 + 0.902539i \(0.641701\pi\)
\(548\) 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 \(-0.573679\pi\)
−0.229408 + 0.973330i \(0.573679\pi\)
\(558\) 0 0
\(559\) −17.9411 −0.758829
\(560\) 0 0
\(561\) −6.82843 −0.288296
\(562\) 6.97056 0.294035
\(563\) −20.3431 −0.857361 −0.428681 0.903456i \(-0.641021\pi\)
−0.428681 + 0.903456i \(0.641021\pi\)
\(564\) −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 \(-0.394982\pi\)
0.323971 + 0.946067i \(0.394982\pi\)
\(570\) 0 0
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) −10.3431 −0.432469
\(573\) −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
\(583\) −13.3137 −0.551397
\(584\) 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 \(-0.385306\pi\)
0.352576 + 0.935783i \(0.385306\pi\)
\(594\) 0.414214 0.0169954
\(595\) 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 \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) −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 \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) 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 \(-0.646120\pi\)
−0.443097 + 0.896474i \(0.646120\pi\)
\(614\) −3.65685 −0.147579
\(615\) 0 0
\(616\) −7.65685 −0.308503
\(617\) −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
\(627\) 1.17157 0.0467881
\(628\) 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
\(638\) 0.343146 0.0135853
\(639\) 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
\(649\) 4.00000 0.157014
\(650\) 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 \(-0.454502\pi\)
0.142451 + 0.989802i \(0.454502\pi\)
\(660\) 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
\(671\) 0.343146 0.0132470
\(672\) −21.3137 −0.822194
\(673\) −29.6569 −1.14319 −0.571594 0.820537i \(-0.693675\pi\)
−0.571594 + 0.820537i \(0.693675\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −34.7401 −1.33616
\(677\) 21.4558 0.824615 0.412308 0.911045i \(-0.364723\pi\)
0.412308 + 0.911045i \(0.364723\pi\)
\(678\) 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
\(693\) −4.82843 −0.183417
\(694\) −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 \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) 2.34315 0.0884363
\(703\) 0.402020 0.0151625
\(704\) 4.17157 0.157222
\(705\) 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
\(726\) −0.414214 −0.0153729
\(727\) 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 \(-0.394270\pi\)
0.326085 + 0.945340i \(0.394270\pi\)
\(734\) −0.686292 −0.0253315
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) 5.65685 0.208373
\(738\) 0.343146 0.0126314
\(739\) 47.1127 1.73307 0.866534 0.499118i \(-0.166342\pi\)
0.866534 + 0.499118i \(0.166342\pi\)
\(740\) 0 0
\(741\) 6.62742 0.243464
\(742\) −26.6274 −0.977523
\(743\) −47.6569 −1.74836 −0.874180 0.485602i \(-0.838601\pi\)
−0.874180 + 0.485602i \(0.838601\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.3431 −0.525140
\(747\) 10.0000 0.365881
\(748\) 12.4853 0.456507
\(749\) 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
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −23.1716 −0.839969 −0.419984 0.907531i \(-0.637964\pi\)
−0.419984 + 0.907531i \(0.637964\pi\)
\(762\) 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 \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) 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
\(781\) −13.6569 −0.488681
\(782\) −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 \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) 15.5147 0.552689
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) −72.2843 −2.57013
\(792\) −1.58579 −0.0563485
\(793\) 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
\(803\) −11.3137 −0.399252
\(804\) 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 \(-0.959517\pi\)
−0.991923 + 0.126838i \(0.959517\pi\)
\(810\) 0 0
\(811\) −16.4853 −0.578877 −0.289438 0.957197i \(-0.593468\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(812\) −7.31371 −0.256661
\(813\) 27.7990 0.974953
\(814\) −0.142136 −0.00498185
\(815\) 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 \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) 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 \(-0.605328\pi\)
−0.324893 + 0.945751i \(0.605328\pi\)
\(828\) −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
\(836\) −2.14214 −0.0740873
\(837\) 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
\(847\) 4.82843 0.165907
\(848\) 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 \(-0.319904\pi\)
0.536080 + 0.844167i \(0.319904\pi\)
\(854\) 0.686292 0.0234844
\(855\) 0 0
\(856\) 8.42641 0.288009
\(857\) 11.5147 0.393335 0.196668 0.980470i \(-0.436988\pi\)
0.196668 + 0.980470i \(0.436988\pi\)
\(858\) −2.34315 −0.0799937
\(859\) −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
\(869\) 8.48528 0.287843
\(870\) 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 \(-0.244273\pi\)
0.719713 + 0.694272i \(0.244273\pi\)
\(878\) 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 \(-0.476770\pi\)
0.0729143 + 0.997338i \(0.476770\pi\)
\(888\) −0.544156 −0.0182607
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −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
\(902\) −0.343146 −0.0114255
\(903\) 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
\(913\) −10.0000 −0.330952
\(914\) −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 \(-0.344318\pi\)
0.469821 + 0.882762i \(0.344318\pi\)
\(920\) 0 0
\(921\) 8.82843 0.290907
\(922\) 11.6569 0.383898
\(923\) −77.2548 −2.54287
\(924\) 8.82843 0.290434
\(925\) 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 \(-0.762612\pi\)
−0.734562 + 0.678541i \(0.762612\pi\)
\(938\) 11.3137 0.369406
\(939\) −4.34315 −0.141733
\(940\) 0 0
\(941\) 38.7696 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(942\) 7.45584 0.242925
\(943\) −3.31371 −0.107909
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 1.31371 0.0427123
\(947\) 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 \(-0.648665\pi\)
−0.450249 + 0.892903i \(0.648665\pi\)
\(954\) −5.51472 −0.178546
\(955\) 0 0
\(956\) −11.5980 −0.375105
\(957\) −0.828427 −0.0267792
\(958\) −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 \(-0.281244\pi\)
0.634407 + 0.772999i \(0.281244\pi\)
\(968\) 1.58579 0.0509691
\(969\) −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
\(979\) 7.65685 0.244714
\(980\) 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 \(-0.414882\pi\)
0.264230 + 0.964460i \(0.414882\pi\)
\(998\) 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 825.2.a.g.1.1 2
3.2 odd 2 2475.2.a.m.1.2 2
5.2 odd 4 825.2.c.e.199.2 4
5.3 odd 4 825.2.c.e.199.3 4
5.4 even 2 165.2.a.a.1.2 2
11.10 odd 2 9075.2.a.v.1.2 2
15.2 even 4 2475.2.c.m.199.3 4
15.8 even 4 2475.2.c.m.199.2 4
15.14 odd 2 495.2.a.d.1.1 2
20.19 odd 2 2640.2.a.bb.1.2 2
35.34 odd 2 8085.2.a.ba.1.2 2
55.54 odd 2 1815.2.a.k.1.1 2
60.59 even 2 7920.2.a.cg.1.2 2
165.164 even 2 5445.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 5.4 even 2
495.2.a.d.1.1 2 15.14 odd 2
825.2.a.g.1.1 2 1.1 even 1 trivial
825.2.c.e.199.2 4 5.2 odd 4
825.2.c.e.199.3 4 5.3 odd 4
1815.2.a.k.1.1 2 55.54 odd 2
2475.2.a.m.1.2 2 3.2 odd 2
2475.2.c.m.199.2 4 15.8 even 4
2475.2.c.m.199.3 4 15.2 even 4
2640.2.a.bb.1.2 2 20.19 odd 2
5445.2.a.m.1.2 2 165.164 even 2
7920.2.a.cg.1.2 2 60.59 even 2
8085.2.a.ba.1.2 2 35.34 odd 2
9075.2.a.v.1.2 2 11.10 odd 2