Properties

Label 7098.2.a.bu.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.27492 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.27492 q^{10} +2.27492 q^{11} -1.00000 q^{12} -1.00000 q^{14} -4.27492 q^{15} +1.00000 q^{16} +0.274917 q^{17} +1.00000 q^{18} +2.27492 q^{19} +4.27492 q^{20} +1.00000 q^{21} +2.27492 q^{22} +2.27492 q^{23} -1.00000 q^{24} +13.2749 q^{25} -1.00000 q^{27} -1.00000 q^{28} +8.27492 q^{29} -4.27492 q^{30} -8.00000 q^{31} +1.00000 q^{32} -2.27492 q^{33} +0.274917 q^{34} -4.27492 q^{35} +1.00000 q^{36} +4.27492 q^{37} +2.27492 q^{38} +4.27492 q^{40} -6.54983 q^{41} +1.00000 q^{42} -2.27492 q^{43} +2.27492 q^{44} +4.27492 q^{45} +2.27492 q^{46} -1.00000 q^{48} +1.00000 q^{49} +13.2749 q^{50} -0.274917 q^{51} +10.0000 q^{53} -1.00000 q^{54} +9.72508 q^{55} -1.00000 q^{56} -2.27492 q^{57} +8.27492 q^{58} -8.00000 q^{59} -4.27492 q^{60} -12.2749 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.27492 q^{66} -12.5498 q^{67} +0.274917 q^{68} -2.27492 q^{69} -4.27492 q^{70} +1.00000 q^{72} +12.8248 q^{73} +4.27492 q^{74} -13.2749 q^{75} +2.27492 q^{76} -2.27492 q^{77} +12.5498 q^{79} +4.27492 q^{80} +1.00000 q^{81} -6.54983 q^{82} +4.54983 q^{83} +1.00000 q^{84} +1.17525 q^{85} -2.27492 q^{86} -8.27492 q^{87} +2.27492 q^{88} +14.0000 q^{89} +4.27492 q^{90} +2.27492 q^{92} +8.00000 q^{93} +9.72508 q^{95} -1.00000 q^{96} -15.0997 q^{97} +1.00000 q^{98} +2.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + q^{10} - 3q^{11} - 2q^{12} - 2q^{14} - q^{15} + 2q^{16} - 7q^{17} + 2q^{18} - 3q^{19} + q^{20} + 2q^{21} - 3q^{22} - 3q^{23} - 2q^{24} + 19q^{25} - 2q^{27} - 2q^{28} + 9q^{29} - q^{30} - 16q^{31} + 2q^{32} + 3q^{33} - 7q^{34} - q^{35} + 2q^{36} + q^{37} - 3q^{38} + q^{40} + 2q^{41} + 2q^{42} + 3q^{43} - 3q^{44} + q^{45} - 3q^{46} - 2q^{48} + 2q^{49} + 19q^{50} + 7q^{51} + 20q^{53} - 2q^{54} + 27q^{55} - 2q^{56} + 3q^{57} + 9q^{58} - 16q^{59} - q^{60} - 17q^{61} - 16q^{62} - 2q^{63} + 2q^{64} + 3q^{66} - 10q^{67} - 7q^{68} + 3q^{69} - q^{70} + 2q^{72} + 3q^{73} + q^{74} - 19q^{75} - 3q^{76} + 3q^{77} + 10q^{79} + q^{80} + 2q^{81} + 2q^{82} - 6q^{83} + 2q^{84} + 25q^{85} + 3q^{86} - 9q^{87} - 3q^{88} + 28q^{89} + q^{90} - 3q^{92} + 16q^{93} + 27q^{95} - 2q^{96} + 2q^{98} - 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 4.27492 1.91180 0.955901 0.293691i \(-0.0948835\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.27492 1.35185
\(11\) 2.27492 0.685913 0.342957 0.939351i \(-0.388572\pi\)
0.342957 + 0.939351i \(0.388572\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −4.27492 −1.10378
\(16\) 1.00000 0.250000
\(17\) 0.274917 0.0666772 0.0333386 0.999444i \(-0.489386\pi\)
0.0333386 + 0.999444i \(0.489386\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.27492 0.521902 0.260951 0.965352i \(-0.415964\pi\)
0.260951 + 0.965352i \(0.415964\pi\)
\(20\) 4.27492 0.955901
\(21\) 1.00000 0.218218
\(22\) 2.27492 0.485014
\(23\) 2.27492 0.474353 0.237177 0.971467i \(-0.423778\pi\)
0.237177 + 0.971467i \(0.423778\pi\)
\(24\) −1.00000 −0.204124
\(25\) 13.2749 2.65498
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 8.27492 1.53661 0.768307 0.640082i \(-0.221100\pi\)
0.768307 + 0.640082i \(0.221100\pi\)
\(30\) −4.27492 −0.780490
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.27492 −0.396012
\(34\) 0.274917 0.0471479
\(35\) −4.27492 −0.722593
\(36\) 1.00000 0.166667
\(37\) 4.27492 0.702792 0.351396 0.936227i \(-0.385707\pi\)
0.351396 + 0.936227i \(0.385707\pi\)
\(38\) 2.27492 0.369040
\(39\) 0 0
\(40\) 4.27492 0.675924
\(41\) −6.54983 −1.02291 −0.511456 0.859309i \(-0.670894\pi\)
−0.511456 + 0.859309i \(0.670894\pi\)
\(42\) 1.00000 0.154303
\(43\) −2.27492 −0.346922 −0.173461 0.984841i \(-0.555495\pi\)
−0.173461 + 0.984841i \(0.555495\pi\)
\(44\) 2.27492 0.342957
\(45\) 4.27492 0.637267
\(46\) 2.27492 0.335418
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 13.2749 1.87736
\(51\) −0.274917 −0.0384961
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −1.00000 −0.136083
\(55\) 9.72508 1.31133
\(56\) −1.00000 −0.133631
\(57\) −2.27492 −0.301320
\(58\) 8.27492 1.08655
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −4.27492 −0.551889
\(61\) −12.2749 −1.57164 −0.785821 0.618454i \(-0.787759\pi\)
−0.785821 + 0.618454i \(0.787759\pi\)
\(62\) −8.00000 −1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.27492 −0.280023
\(67\) −12.5498 −1.53321 −0.766603 0.642121i \(-0.778055\pi\)
−0.766603 + 0.642121i \(0.778055\pi\)
\(68\) 0.274917 0.0333386
\(69\) −2.27492 −0.273868
\(70\) −4.27492 −0.510950
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.8248 1.50102 0.750512 0.660857i \(-0.229807\pi\)
0.750512 + 0.660857i \(0.229807\pi\)
\(74\) 4.27492 0.496949
\(75\) −13.2749 −1.53286
\(76\) 2.27492 0.260951
\(77\) −2.27492 −0.259251
\(78\) 0 0
\(79\) 12.5498 1.41197 0.705983 0.708228i \(-0.250505\pi\)
0.705983 + 0.708228i \(0.250505\pi\)
\(80\) 4.27492 0.477950
\(81\) 1.00000 0.111111
\(82\) −6.54983 −0.723308
\(83\) 4.54983 0.499409 0.249705 0.968322i \(-0.419666\pi\)
0.249705 + 0.968322i \(0.419666\pi\)
\(84\) 1.00000 0.109109
\(85\) 1.17525 0.127474
\(86\) −2.27492 −0.245311
\(87\) −8.27492 −0.887164
\(88\) 2.27492 0.242507
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 4.27492 0.450616
\(91\) 0 0
\(92\) 2.27492 0.237177
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 9.72508 0.997772
\(96\) −1.00000 −0.102062
\(97\) −15.0997 −1.53314 −0.766570 0.642161i \(-0.778038\pi\)
−0.766570 + 0.642161i \(0.778038\pi\)
\(98\) 1.00000 0.101015
\(99\) 2.27492 0.228638
\(100\) 13.2749 1.32749
\(101\) 2.54983 0.253718 0.126859 0.991921i \(-0.459510\pi\)
0.126859 + 0.991921i \(0.459510\pi\)
\(102\) −0.274917 −0.0272209
\(103\) 10.2749 1.01242 0.506209 0.862411i \(-0.331046\pi\)
0.506209 + 0.862411i \(0.331046\pi\)
\(104\) 0 0
\(105\) 4.27492 0.417189
\(106\) 10.0000 0.971286
\(107\) 0.549834 0.0531545 0.0265773 0.999647i \(-0.491539\pi\)
0.0265773 + 0.999647i \(0.491539\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −0.274917 −0.0263323 −0.0131661 0.999913i \(-0.504191\pi\)
−0.0131661 + 0.999913i \(0.504191\pi\)
\(110\) 9.72508 0.927250
\(111\) −4.27492 −0.405757
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −2.27492 −0.213066
\(115\) 9.72508 0.906869
\(116\) 8.27492 0.768307
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −0.274917 −0.0252016
\(120\) −4.27492 −0.390245
\(121\) −5.82475 −0.529523
\(122\) −12.2749 −1.11132
\(123\) 6.54983 0.590579
\(124\) −8.00000 −0.718421
\(125\) 35.3746 3.16400
\(126\) −1.00000 −0.0890871
\(127\) −20.5498 −1.82350 −0.911751 0.410742i \(-0.865270\pi\)
−0.911751 + 0.410742i \(0.865270\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.27492 0.200295
\(130\) 0 0
\(131\) 6.82475 0.596281 0.298141 0.954522i \(-0.403634\pi\)
0.298141 + 0.954522i \(0.403634\pi\)
\(132\) −2.27492 −0.198006
\(133\) −2.27492 −0.197260
\(134\) −12.5498 −1.08414
\(135\) −4.27492 −0.367926
\(136\) 0.274917 0.0235740
\(137\) 17.3746 1.48441 0.742206 0.670172i \(-0.233780\pi\)
0.742206 + 0.670172i \(0.233780\pi\)
\(138\) −2.27492 −0.193654
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −4.27492 −0.361296
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 35.3746 2.93770
\(146\) 12.8248 1.06138
\(147\) −1.00000 −0.0824786
\(148\) 4.27492 0.351396
\(149\) −22.5498 −1.84735 −0.923677 0.383172i \(-0.874832\pi\)
−0.923677 + 0.383172i \(0.874832\pi\)
\(150\) −13.2749 −1.08389
\(151\) −6.27492 −0.510646 −0.255323 0.966856i \(-0.582182\pi\)
−0.255323 + 0.966856i \(0.582182\pi\)
\(152\) 2.27492 0.184520
\(153\) 0.274917 0.0222257
\(154\) −2.27492 −0.183318
\(155\) −34.1993 −2.74696
\(156\) 0 0
\(157\) −13.3746 −1.06741 −0.533704 0.845671i \(-0.679200\pi\)
−0.533704 + 0.845671i \(0.679200\pi\)
\(158\) 12.5498 0.998411
\(159\) −10.0000 −0.793052
\(160\) 4.27492 0.337962
\(161\) −2.27492 −0.179289
\(162\) 1.00000 0.0785674
\(163\) −12.5498 −0.982979 −0.491489 0.870884i \(-0.663547\pi\)
−0.491489 + 0.870884i \(0.663547\pi\)
\(164\) −6.54983 −0.511456
\(165\) −9.72508 −0.757097
\(166\) 4.54983 0.353136
\(167\) 1.72508 0.133491 0.0667455 0.997770i \(-0.478738\pi\)
0.0667455 + 0.997770i \(0.478738\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 1.17525 0.0901374
\(171\) 2.27492 0.173967
\(172\) −2.27492 −0.173461
\(173\) 7.09967 0.539778 0.269889 0.962891i \(-0.413013\pi\)
0.269889 + 0.962891i \(0.413013\pi\)
\(174\) −8.27492 −0.627320
\(175\) −13.2749 −1.00349
\(176\) 2.27492 0.171478
\(177\) 8.00000 0.601317
\(178\) 14.0000 1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 4.27492 0.318634
\(181\) 19.0997 1.41967 0.709834 0.704369i \(-0.248770\pi\)
0.709834 + 0.704369i \(0.248770\pi\)
\(182\) 0 0
\(183\) 12.2749 0.907388
\(184\) 2.27492 0.167709
\(185\) 18.2749 1.34360
\(186\) 8.00000 0.586588
\(187\) 0.625414 0.0457348
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 9.72508 0.705532
\(191\) 2.27492 0.164607 0.0823036 0.996607i \(-0.473772\pi\)
0.0823036 + 0.996607i \(0.473772\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.5498 1.33525 0.667623 0.744499i \(-0.267312\pi\)
0.667623 + 0.744499i \(0.267312\pi\)
\(194\) −15.0997 −1.08409
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.54983 0.181668 0.0908341 0.995866i \(-0.471047\pi\)
0.0908341 + 0.995866i \(0.471047\pi\)
\(198\) 2.27492 0.161671
\(199\) 6.82475 0.483794 0.241897 0.970302i \(-0.422230\pi\)
0.241897 + 0.970302i \(0.422230\pi\)
\(200\) 13.2749 0.938678
\(201\) 12.5498 0.885197
\(202\) 2.54983 0.179406
\(203\) −8.27492 −0.580785
\(204\) −0.274917 −0.0192481
\(205\) −28.0000 −1.95560
\(206\) 10.2749 0.715887
\(207\) 2.27492 0.158118
\(208\) 0 0
\(209\) 5.17525 0.357979
\(210\) 4.27492 0.294997
\(211\) 1.17525 0.0809074 0.0404537 0.999181i \(-0.487120\pi\)
0.0404537 + 0.999181i \(0.487120\pi\)
\(212\) 10.0000 0.686803
\(213\) 0 0
\(214\) 0.549834 0.0375859
\(215\) −9.72508 −0.663245
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) −0.274917 −0.0186197
\(219\) −12.8248 −0.866616
\(220\) 9.72508 0.655665
\(221\) 0 0
\(222\) −4.27492 −0.286914
\(223\) −21.6495 −1.44976 −0.724879 0.688876i \(-0.758104\pi\)
−0.724879 + 0.688876i \(0.758104\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 13.2749 0.884994
\(226\) −6.00000 −0.399114
\(227\) 20.5498 1.36394 0.681970 0.731380i \(-0.261123\pi\)
0.681970 + 0.731380i \(0.261123\pi\)
\(228\) −2.27492 −0.150660
\(229\) −19.0997 −1.26214 −0.631071 0.775725i \(-0.717384\pi\)
−0.631071 + 0.775725i \(0.717384\pi\)
\(230\) 9.72508 0.641253
\(231\) 2.27492 0.149679
\(232\) 8.27492 0.543275
\(233\) 5.45017 0.357052 0.178526 0.983935i \(-0.442867\pi\)
0.178526 + 0.983935i \(0.442867\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) −12.5498 −0.815199
\(238\) −0.274917 −0.0178202
\(239\) −25.0997 −1.62356 −0.811781 0.583962i \(-0.801502\pi\)
−0.811781 + 0.583962i \(0.801502\pi\)
\(240\) −4.27492 −0.275945
\(241\) −15.0997 −0.972655 −0.486328 0.873777i \(-0.661664\pi\)
−0.486328 + 0.873777i \(0.661664\pi\)
\(242\) −5.82475 −0.374429
\(243\) −1.00000 −0.0641500
\(244\) −12.2749 −0.785821
\(245\) 4.27492 0.273114
\(246\) 6.54983 0.417602
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −4.54983 −0.288334
\(250\) 35.3746 2.23729
\(251\) 23.9244 1.51010 0.755048 0.655669i \(-0.227614\pi\)
0.755048 + 0.655669i \(0.227614\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 5.17525 0.325365
\(254\) −20.5498 −1.28941
\(255\) −1.17525 −0.0735969
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 2.27492 0.141630
\(259\) −4.27492 −0.265630
\(260\) 0 0
\(261\) 8.27492 0.512205
\(262\) 6.82475 0.421635
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −2.27492 −0.140011
\(265\) 42.7492 2.62606
\(266\) −2.27492 −0.139484
\(267\) −14.0000 −0.856786
\(268\) −12.5498 −0.766603
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −4.27492 −0.260163
\(271\) −28.5498 −1.73428 −0.867139 0.498065i \(-0.834044\pi\)
−0.867139 + 0.498065i \(0.834044\pi\)
\(272\) 0.274917 0.0166693
\(273\) 0 0
\(274\) 17.3746 1.04964
\(275\) 30.1993 1.82109
\(276\) −2.27492 −0.136934
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −4.27492 −0.255475
\(281\) −15.0997 −0.900771 −0.450385 0.892834i \(-0.648713\pi\)
−0.450385 + 0.892834i \(0.648713\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −9.72508 −0.576064
\(286\) 0 0
\(287\) 6.54983 0.386625
\(288\) 1.00000 0.0589256
\(289\) −16.9244 −0.995554
\(290\) 35.3746 2.07727
\(291\) 15.0997 0.885158
\(292\) 12.8248 0.750512
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −34.1993 −1.99116
\(296\) 4.27492 0.248475
\(297\) −2.27492 −0.132004
\(298\) −22.5498 −1.30628
\(299\) 0 0
\(300\) −13.2749 −0.766428
\(301\) 2.27492 0.131124
\(302\) −6.27492 −0.361081
\(303\) −2.54983 −0.146484
\(304\) 2.27492 0.130475
\(305\) −52.4743 −3.00467
\(306\) 0.274917 0.0157160
\(307\) −21.0997 −1.20422 −0.602111 0.798412i \(-0.705673\pi\)
−0.602111 + 0.798412i \(0.705673\pi\)
\(308\) −2.27492 −0.129625
\(309\) −10.2749 −0.584520
\(310\) −34.1993 −1.94239
\(311\) 3.45017 0.195641 0.0978205 0.995204i \(-0.468813\pi\)
0.0978205 + 0.995204i \(0.468813\pi\)
\(312\) 0 0
\(313\) −27.6495 −1.56284 −0.781421 0.624004i \(-0.785505\pi\)
−0.781421 + 0.624004i \(0.785505\pi\)
\(314\) −13.3746 −0.754772
\(315\) −4.27492 −0.240864
\(316\) 12.5498 0.705983
\(317\) 1.45017 0.0814494 0.0407247 0.999170i \(-0.487033\pi\)
0.0407247 + 0.999170i \(0.487033\pi\)
\(318\) −10.0000 −0.560772
\(319\) 18.8248 1.05398
\(320\) 4.27492 0.238975
\(321\) −0.549834 −0.0306888
\(322\) −2.27492 −0.126776
\(323\) 0.625414 0.0347990
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.5498 −0.695071
\(327\) 0.274917 0.0152030
\(328\) −6.54983 −0.361654
\(329\) 0 0
\(330\) −9.72508 −0.535348
\(331\) 29.6495 1.62968 0.814842 0.579683i \(-0.196824\pi\)
0.814842 + 0.579683i \(0.196824\pi\)
\(332\) 4.54983 0.249705
\(333\) 4.27492 0.234264
\(334\) 1.72508 0.0943923
\(335\) −53.6495 −2.93119
\(336\) 1.00000 0.0545545
\(337\) 9.37459 0.510666 0.255333 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 1.17525 0.0637368
\(341\) −18.1993 −0.985549
\(342\) 2.27492 0.123013
\(343\) −1.00000 −0.0539949
\(344\) −2.27492 −0.122655
\(345\) −9.72508 −0.523581
\(346\) 7.09967 0.381681
\(347\) 5.09967 0.273765 0.136882 0.990587i \(-0.456292\pi\)
0.136882 + 0.990587i \(0.456292\pi\)
\(348\) −8.27492 −0.443582
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −13.2749 −0.709574
\(351\) 0 0
\(352\) 2.27492 0.121253
\(353\) −27.0997 −1.44237 −0.721185 0.692743i \(-0.756402\pi\)
−0.721185 + 0.692743i \(0.756402\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0.274917 0.0145502
\(358\) 12.0000 0.634220
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 4.27492 0.225308
\(361\) −13.8248 −0.727619
\(362\) 19.0997 1.00386
\(363\) 5.82475 0.305720
\(364\) 0 0
\(365\) 54.8248 2.86966
\(366\) 12.2749 0.641620
\(367\) −2.90033 −0.151396 −0.0756980 0.997131i \(-0.524119\pi\)
−0.0756980 + 0.997131i \(0.524119\pi\)
\(368\) 2.27492 0.118588
\(369\) −6.54983 −0.340971
\(370\) 18.2749 0.950068
\(371\) −10.0000 −0.519174
\(372\) 8.00000 0.414781
\(373\) 26.5498 1.37470 0.687349 0.726327i \(-0.258774\pi\)
0.687349 + 0.726327i \(0.258774\pi\)
\(374\) 0.625414 0.0323394
\(375\) −35.3746 −1.82674
\(376\) 0 0
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 33.0997 1.70022 0.850108 0.526609i \(-0.176537\pi\)
0.850108 + 0.526609i \(0.176537\pi\)
\(380\) 9.72508 0.498886
\(381\) 20.5498 1.05280
\(382\) 2.27492 0.116395
\(383\) −30.2749 −1.54698 −0.773488 0.633811i \(-0.781490\pi\)
−0.773488 + 0.633811i \(0.781490\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −9.72508 −0.495636
\(386\) 18.5498 0.944162
\(387\) −2.27492 −0.115641
\(388\) −15.0997 −0.766570
\(389\) 28.1993 1.42976 0.714882 0.699246i \(-0.246481\pi\)
0.714882 + 0.699246i \(0.246481\pi\)
\(390\) 0 0
\(391\) 0.625414 0.0316285
\(392\) 1.00000 0.0505076
\(393\) −6.82475 −0.344263
\(394\) 2.54983 0.128459
\(395\) 53.6495 2.69940
\(396\) 2.27492 0.114319
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 6.82475 0.342094
\(399\) 2.27492 0.113888
\(400\) 13.2749 0.663746
\(401\) 3.09967 0.154790 0.0773950 0.997001i \(-0.475340\pi\)
0.0773950 + 0.997001i \(0.475340\pi\)
\(402\) 12.5498 0.625929
\(403\) 0 0
\(404\) 2.54983 0.126859
\(405\) 4.27492 0.212422
\(406\) −8.27492 −0.410677
\(407\) 9.72508 0.482054
\(408\) −0.274917 −0.0136104
\(409\) 7.17525 0.354793 0.177397 0.984139i \(-0.443232\pi\)
0.177397 + 0.984139i \(0.443232\pi\)
\(410\) −28.0000 −1.38282
\(411\) −17.3746 −0.857025
\(412\) 10.2749 0.506209
\(413\) 8.00000 0.393654
\(414\) 2.27492 0.111806
\(415\) 19.4502 0.954771
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 5.17525 0.253130
\(419\) 2.27492 0.111137 0.0555685 0.998455i \(-0.482303\pi\)
0.0555685 + 0.998455i \(0.482303\pi\)
\(420\) 4.27492 0.208595
\(421\) −3.09967 −0.151069 −0.0755343 0.997143i \(-0.524066\pi\)
−0.0755343 + 0.997143i \(0.524066\pi\)
\(422\) 1.17525 0.0572102
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 3.64950 0.177027
\(426\) 0 0
\(427\) 12.2749 0.594025
\(428\) 0.549834 0.0265773
\(429\) 0 0
\(430\) −9.72508 −0.468985
\(431\) 11.4502 0.551535 0.275768 0.961224i \(-0.411068\pi\)
0.275768 + 0.961224i \(0.411068\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 23.6495 1.13652 0.568261 0.822848i \(-0.307616\pi\)
0.568261 + 0.822848i \(0.307616\pi\)
\(434\) 8.00000 0.384012
\(435\) −35.3746 −1.69608
\(436\) −0.274917 −0.0131661
\(437\) 5.17525 0.247566
\(438\) −12.8248 −0.612790
\(439\) −14.8248 −0.707547 −0.353773 0.935331i \(-0.615102\pi\)
−0.353773 + 0.935331i \(0.615102\pi\)
\(440\) 9.72508 0.463625
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.45017 0.353968 0.176984 0.984214i \(-0.443366\pi\)
0.176984 + 0.984214i \(0.443366\pi\)
\(444\) −4.27492 −0.202879
\(445\) 59.8488 2.83711
\(446\) −21.6495 −1.02513
\(447\) 22.5498 1.06657
\(448\) −1.00000 −0.0472456
\(449\) 0.274917 0.0129741 0.00648707 0.999979i \(-0.497935\pi\)
0.00648707 + 0.999979i \(0.497935\pi\)
\(450\) 13.2749 0.625786
\(451\) −14.9003 −0.701629
\(452\) −6.00000 −0.282216
\(453\) 6.27492 0.294821
\(454\) 20.5498 0.964452
\(455\) 0 0
\(456\) −2.27492 −0.106533
\(457\) 18.5498 0.867725 0.433862 0.900979i \(-0.357150\pi\)
0.433862 + 0.900979i \(0.357150\pi\)
\(458\) −19.0997 −0.892469
\(459\) −0.274917 −0.0128320
\(460\) 9.72508 0.453434
\(461\) 17.9244 0.834823 0.417412 0.908717i \(-0.362937\pi\)
0.417412 + 0.908717i \(0.362937\pi\)
\(462\) 2.27492 0.105839
\(463\) −30.2749 −1.40699 −0.703497 0.710698i \(-0.748379\pi\)
−0.703497 + 0.710698i \(0.748379\pi\)
\(464\) 8.27492 0.384153
\(465\) 34.1993 1.58596
\(466\) 5.45017 0.252474
\(467\) −26.2749 −1.21586 −0.607929 0.793991i \(-0.708000\pi\)
−0.607929 + 0.793991i \(0.708000\pi\)
\(468\) 0 0
\(469\) 12.5498 0.579498
\(470\) 0 0
\(471\) 13.3746 0.616268
\(472\) −8.00000 −0.368230
\(473\) −5.17525 −0.237958
\(474\) −12.5498 −0.576433
\(475\) 30.1993 1.38564
\(476\) −0.274917 −0.0126008
\(477\) 10.0000 0.457869
\(478\) −25.0997 −1.14803
\(479\) 23.3746 1.06801 0.534006 0.845481i \(-0.320686\pi\)
0.534006 + 0.845481i \(0.320686\pi\)
\(480\) −4.27492 −0.195122
\(481\) 0 0
\(482\) −15.0997 −0.687771
\(483\) 2.27492 0.103512
\(484\) −5.82475 −0.264761
\(485\) −64.5498 −2.93106
\(486\) −1.00000 −0.0453609
\(487\) 18.1993 0.824691 0.412345 0.911028i \(-0.364710\pi\)
0.412345 + 0.911028i \(0.364710\pi\)
\(488\) −12.2749 −0.555659
\(489\) 12.5498 0.567523
\(490\) 4.27492 0.193121
\(491\) 8.54983 0.385849 0.192924 0.981214i \(-0.438203\pi\)
0.192924 + 0.981214i \(0.438203\pi\)
\(492\) 6.54983 0.295289
\(493\) 2.27492 0.102457
\(494\) 0 0
\(495\) 9.72508 0.437110
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.54983 −0.203883
\(499\) −25.0997 −1.12362 −0.561808 0.827268i \(-0.689894\pi\)
−0.561808 + 0.827268i \(0.689894\pi\)
\(500\) 35.3746 1.58200
\(501\) −1.72508 −0.0770710
\(502\) 23.9244 1.06780
\(503\) −3.45017 −0.153835 −0.0769176 0.997037i \(-0.524508\pi\)
−0.0769176 + 0.997037i \(0.524508\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 10.9003 0.485058
\(506\) 5.17525 0.230068
\(507\) 0 0
\(508\) −20.5498 −0.911751
\(509\) 9.92442 0.439892 0.219946 0.975512i \(-0.429412\pi\)
0.219946 + 0.975512i \(0.429412\pi\)
\(510\) −1.17525 −0.0520409
\(511\) −12.8248 −0.567334
\(512\) 1.00000 0.0441942
\(513\) −2.27492 −0.100440
\(514\) −14.0000 −0.617514
\(515\) 43.9244 1.93554
\(516\) 2.27492 0.100148
\(517\) 0 0
\(518\) −4.27492 −0.187829
\(519\) −7.09967 −0.311641
\(520\) 0 0
\(521\) −4.27492 −0.187288 −0.0936438 0.995606i \(-0.529851\pi\)
−0.0936438 + 0.995606i \(0.529851\pi\)
\(522\) 8.27492 0.362183
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 6.82475 0.298141
\(525\) 13.2749 0.579365
\(526\) 28.0000 1.22086
\(527\) −2.19934 −0.0958047
\(528\) −2.27492 −0.0990031
\(529\) −17.8248 −0.774989
\(530\) 42.7492 1.85691
\(531\) −8.00000 −0.347170
\(532\) −2.27492 −0.0986302
\(533\) 0 0
\(534\) −14.0000 −0.605839
\(535\) 2.35050 0.101621
\(536\) −12.5498 −0.542070
\(537\) −12.0000 −0.517838
\(538\) 6.00000 0.258678
\(539\) 2.27492 0.0979876
\(540\) −4.27492 −0.183963
\(541\) −42.4743 −1.82611 −0.913055 0.407835i \(-0.866284\pi\)
−0.913055 + 0.407835i \(0.866284\pi\)
\(542\) −28.5498 −1.22632
\(543\) −19.0997 −0.819645
\(544\) 0.274917 0.0117870
\(545\) −1.17525 −0.0503421
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 17.3746 0.742206
\(549\) −12.2749 −0.523881
\(550\) 30.1993 1.28770
\(551\) 18.8248 0.801961
\(552\) −2.27492 −0.0968269
\(553\) −12.5498 −0.533673
\(554\) −10.0000 −0.424859
\(555\) −18.2749 −0.775727
\(556\) 4.00000 0.169638
\(557\) 44.7492 1.89608 0.948042 0.318146i \(-0.103060\pi\)
0.948042 + 0.318146i \(0.103060\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) −4.27492 −0.180648
\(561\) −0.625414 −0.0264050
\(562\) −15.0997 −0.636941
\(563\) −35.3746 −1.49086 −0.745431 0.666583i \(-0.767756\pi\)
−0.745431 + 0.666583i \(0.767756\pi\)
\(564\) 0 0
\(565\) −25.6495 −1.07908
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 3.09967 0.129945 0.0649724 0.997887i \(-0.479304\pi\)
0.0649724 + 0.997887i \(0.479304\pi\)
\(570\) −9.72508 −0.407339
\(571\) −37.0997 −1.55257 −0.776286 0.630380i \(-0.782899\pi\)
−0.776286 + 0.630380i \(0.782899\pi\)
\(572\) 0 0
\(573\) −2.27492 −0.0950360
\(574\) 6.54983 0.273385
\(575\) 30.1993 1.25940
\(576\) 1.00000 0.0416667
\(577\) 8.90033 0.370526 0.185263 0.982689i \(-0.440686\pi\)
0.185263 + 0.982689i \(0.440686\pi\)
\(578\) −16.9244 −0.703963
\(579\) −18.5498 −0.770905
\(580\) 35.3746 1.46885
\(581\) −4.54983 −0.188759
\(582\) 15.0997 0.625901
\(583\) 22.7492 0.942174
\(584\) 12.8248 0.530692
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −46.7492 −1.92954 −0.964772 0.263086i \(-0.915260\pi\)
−0.964772 + 0.263086i \(0.915260\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −18.1993 −0.749891
\(590\) −34.1993 −1.40796
\(591\) −2.54983 −0.104886
\(592\) 4.27492 0.175698
\(593\) 7.09967 0.291548 0.145774 0.989318i \(-0.453433\pi\)
0.145774 + 0.989318i \(0.453433\pi\)
\(594\) −2.27492 −0.0933410
\(595\) −1.17525 −0.0481805
\(596\) −22.5498 −0.923677
\(597\) −6.82475 −0.279318
\(598\) 0 0
\(599\) 21.7251 0.887663 0.443831 0.896110i \(-0.353619\pi\)
0.443831 + 0.896110i \(0.353619\pi\)
\(600\) −13.2749 −0.541946
\(601\) 23.6495 0.964683 0.482342 0.875983i \(-0.339786\pi\)
0.482342 + 0.875983i \(0.339786\pi\)
\(602\) 2.27492 0.0927187
\(603\) −12.5498 −0.511069
\(604\) −6.27492 −0.255323
\(605\) −24.9003 −1.01234
\(606\) −2.54983 −0.103580
\(607\) 9.17525 0.372412 0.186206 0.982511i \(-0.440381\pi\)
0.186206 + 0.982511i \(0.440381\pi\)
\(608\) 2.27492 0.0922601
\(609\) 8.27492 0.335317
\(610\) −52.4743 −2.12462
\(611\) 0 0
\(612\) 0.274917 0.0111129
\(613\) −16.2749 −0.657338 −0.328669 0.944445i \(-0.606600\pi\)
−0.328669 + 0.944445i \(0.606600\pi\)
\(614\) −21.0997 −0.851513
\(615\) 28.0000 1.12907
\(616\) −2.27492 −0.0916590
\(617\) 20.8248 0.838373 0.419186 0.907900i \(-0.362315\pi\)
0.419186 + 0.907900i \(0.362315\pi\)
\(618\) −10.2749 −0.413318
\(619\) −29.7251 −1.19475 −0.597376 0.801961i \(-0.703790\pi\)
−0.597376 + 0.801961i \(0.703790\pi\)
\(620\) −34.1993 −1.37348
\(621\) −2.27492 −0.0912893
\(622\) 3.45017 0.138339
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 84.8488 3.39395
\(626\) −27.6495 −1.10510
\(627\) −5.17525 −0.206680
\(628\) −13.3746 −0.533704
\(629\) 1.17525 0.0468602
\(630\) −4.27492 −0.170317
\(631\) 10.8248 0.430927 0.215463 0.976512i \(-0.430874\pi\)
0.215463 + 0.976512i \(0.430874\pi\)
\(632\) 12.5498 0.499206
\(633\) −1.17525 −0.0467119
\(634\) 1.45017 0.0575934
\(635\) −87.8488 −3.48617
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 18.8248 0.745279
\(639\) 0 0
\(640\) 4.27492 0.168981
\(641\) −15.0997 −0.596401 −0.298201 0.954503i \(-0.596386\pi\)
−0.298201 + 0.954503i \(0.596386\pi\)
\(642\) −0.549834 −0.0217002
\(643\) 23.9244 0.943487 0.471744 0.881736i \(-0.343625\pi\)
0.471744 + 0.881736i \(0.343625\pi\)
\(644\) −2.27492 −0.0896443
\(645\) 9.72508 0.382925
\(646\) 0.625414 0.0246066
\(647\) −18.1993 −0.715490 −0.357745 0.933819i \(-0.616454\pi\)
−0.357745 + 0.933819i \(0.616454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −18.1993 −0.714386
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) −12.5498 −0.491489
\(653\) −5.37459 −0.210324 −0.105162 0.994455i \(-0.533536\pi\)
−0.105162 + 0.994455i \(0.533536\pi\)
\(654\) 0.274917 0.0107501
\(655\) 29.1752 1.13997
\(656\) −6.54983 −0.255728
\(657\) 12.8248 0.500341
\(658\) 0 0
\(659\) −7.45017 −0.290217 −0.145109 0.989416i \(-0.546353\pi\)
−0.145109 + 0.989416i \(0.546353\pi\)
\(660\) −9.72508 −0.378548
\(661\) 40.1993 1.56357 0.781787 0.623546i \(-0.214309\pi\)
0.781787 + 0.623546i \(0.214309\pi\)
\(662\) 29.6495 1.15236
\(663\) 0 0
\(664\) 4.54983 0.176568
\(665\) −9.72508 −0.377123
\(666\) 4.27492 0.165650
\(667\) 18.8248 0.728897
\(668\) 1.72508 0.0667455
\(669\) 21.6495 0.837018
\(670\) −53.6495 −2.07266
\(671\) −27.9244 −1.07801
\(672\) 1.00000 0.0385758
\(673\) 16.2749 0.627352 0.313676 0.949530i \(-0.398439\pi\)
0.313676 + 0.949530i \(0.398439\pi\)
\(674\) 9.37459 0.361096
\(675\) −13.2749 −0.510952
\(676\) 0 0
\(677\) 16.1993 0.622591 0.311296 0.950313i \(-0.399237\pi\)
0.311296 + 0.950313i \(0.399237\pi\)
\(678\) 6.00000 0.230429
\(679\) 15.0997 0.579472
\(680\) 1.17525 0.0450687
\(681\) −20.5498 −0.787471
\(682\) −18.1993 −0.696889
\(683\) −3.37459 −0.129125 −0.0645625 0.997914i \(-0.520565\pi\)
−0.0645625 + 0.997914i \(0.520565\pi\)
\(684\) 2.27492 0.0869836
\(685\) 74.2749 2.83790
\(686\) −1.00000 −0.0381802
\(687\) 19.0997 0.728698
\(688\) −2.27492 −0.0867304
\(689\) 0 0
\(690\) −9.72508 −0.370228
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 7.09967 0.269889
\(693\) −2.27492 −0.0864170
\(694\) 5.09967 0.193581
\(695\) 17.0997 0.648627
\(696\) −8.27492 −0.313660
\(697\) −1.80066 −0.0682049
\(698\) −2.00000 −0.0757011
\(699\) −5.45017 −0.206144
\(700\) −13.2749 −0.501745
\(701\) −15.0997 −0.570307 −0.285153 0.958482i \(-0.592045\pi\)
−0.285153 + 0.958482i \(0.592045\pi\)
\(702\) 0 0
\(703\) 9.72508 0.366788
\(704\) 2.27492 0.0857392
\(705\) 0 0
\(706\) −27.0997 −1.01991
\(707\) −2.54983 −0.0958964
\(708\) 8.00000 0.300658
\(709\) 23.0997 0.867526 0.433763 0.901027i \(-0.357185\pi\)
0.433763 + 0.901027i \(0.357185\pi\)
\(710\) 0 0
\(711\) 12.5498 0.470656
\(712\) 14.0000 0.524672
\(713\) −18.1993 −0.681571
\(714\) 0.274917 0.0102885
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 25.0997 0.937364
\(718\) 32.0000 1.19423
\(719\) −29.6495 −1.10574 −0.552870 0.833268i \(-0.686467\pi\)
−0.552870 + 0.833268i \(0.686467\pi\)
\(720\) 4.27492 0.159317
\(721\) −10.2749 −0.382658
\(722\) −13.8248 −0.514504
\(723\) 15.0997 0.561563
\(724\) 19.0997 0.709834
\(725\) 109.849 4.07968
\(726\) 5.82475 0.216177
\(727\) 35.3746 1.31197 0.655985 0.754774i \(-0.272253\pi\)
0.655985 + 0.754774i \(0.272253\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 54.8248 2.02916
\(731\) −0.625414 −0.0231318
\(732\) 12.2749 0.453694
\(733\) −14.5498 −0.537410 −0.268705 0.963222i \(-0.586596\pi\)
−0.268705 + 0.963222i \(0.586596\pi\)
\(734\) −2.90033 −0.107053
\(735\) −4.27492 −0.157683
\(736\) 2.27492 0.0838546
\(737\) −28.5498 −1.05165
\(738\) −6.54983 −0.241103
\(739\) −37.6495 −1.38496 −0.692480 0.721437i \(-0.743482\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(740\) 18.2749 0.671799
\(741\) 0 0
\(742\) −10.0000 −0.367112
\(743\) −11.4502 −0.420066 −0.210033 0.977694i \(-0.567357\pi\)
−0.210033 + 0.977694i \(0.567357\pi\)
\(744\) 8.00000 0.293294
\(745\) −96.3987 −3.53177
\(746\) 26.5498 0.972059
\(747\) 4.54983 0.166470
\(748\) 0.625414 0.0228674
\(749\) −0.549834 −0.0200905
\(750\) −35.3746 −1.29170
\(751\) 10.1993 0.372179 0.186090 0.982533i \(-0.440419\pi\)
0.186090 + 0.982533i \(0.440419\pi\)
\(752\) 0 0
\(753\) −23.9244 −0.871854
\(754\) 0 0
\(755\) −26.8248 −0.976253
\(756\) 1.00000 0.0363696
\(757\) −35.0997 −1.27572 −0.637860 0.770153i \(-0.720180\pi\)
−0.637860 + 0.770153i \(0.720180\pi\)
\(758\) 33.0997 1.20223
\(759\) −5.17525 −0.187850
\(760\) 9.72508 0.352766
\(761\) −38.5498 −1.39743 −0.698715 0.715400i \(-0.746245\pi\)
−0.698715 + 0.715400i \(0.746245\pi\)
\(762\) 20.5498 0.744442
\(763\) 0.274917 0.00995267
\(764\) 2.27492 0.0823036
\(765\) 1.17525 0.0424912
\(766\) −30.2749 −1.09388
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −32.8248 −1.18369 −0.591845 0.806051i \(-0.701600\pi\)
−0.591845 + 0.806051i \(0.701600\pi\)
\(770\) −9.72508 −0.350468
\(771\) 14.0000 0.504198
\(772\) 18.5498 0.667623
\(773\) 9.92442 0.356957 0.178478 0.983944i \(-0.442883\pi\)
0.178478 + 0.983944i \(0.442883\pi\)
\(774\) −2.27492 −0.0817702
\(775\) −106.199 −3.81479
\(776\) −15.0997 −0.542047
\(777\) 4.27492 0.153362
\(778\) 28.1993 1.01100
\(779\) −14.9003 −0.533860
\(780\) 0 0
\(781\) 0 0
\(782\) 0.625414 0.0223648
\(783\) −8.27492 −0.295721
\(784\) 1.00000 0.0357143
\(785\) −57.1752 −2.04067
\(786\) −6.82475 −0.243431
\(787\) 10.2749 0.366261 0.183131 0.983089i \(-0.441377\pi\)
0.183131 + 0.983089i \(0.441377\pi\)
\(788\) 2.54983 0.0908341
\(789\) −28.0000 −0.996826
\(790\) 53.6495 1.90876
\(791\) 6.00000 0.213335
\(792\) 2.27492 0.0808357
\(793\) 0 0
\(794\) 14.0000 0.496841
\(795\) −42.7492 −1.51616
\(796\) 6.82475 0.241897
\(797\) −15.6495 −0.554334 −0.277167 0.960822i \(-0.589396\pi\)
−0.277167 + 0.960822i \(0.589396\pi\)
\(798\) 2.27492 0.0805312
\(799\) 0 0
\(800\) 13.2749 0.469339
\(801\) 14.0000 0.494666
\(802\) 3.09967 0.109453
\(803\) 29.1752 1.02957
\(804\) 12.5498 0.442599
\(805\) −9.72508 −0.342764
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 2.54983 0.0897029
\(809\) −56.1993 −1.97586 −0.987932 0.154890i \(-0.950498\pi\)
−0.987932 + 0.154890i \(0.950498\pi\)
\(810\) 4.27492 0.150205
\(811\) 11.3746 0.399416 0.199708 0.979855i \(-0.436001\pi\)
0.199708 + 0.979855i \(0.436001\pi\)
\(812\) −8.27492 −0.290393
\(813\) 28.5498 1.00129
\(814\) 9.72508 0.340864
\(815\) −53.6495 −1.87926
\(816\) −0.274917 −0.00962403
\(817\) −5.17525 −0.181059
\(818\) 7.17525 0.250877
\(819\) 0 0
\(820\) −28.0000 −0.977802
\(821\) −31.6495 −1.10458 −0.552288 0.833654i \(-0.686245\pi\)
−0.552288 + 0.833654i \(0.686245\pi\)
\(822\) −17.3746 −0.606008
\(823\) −25.0997 −0.874919 −0.437460 0.899238i \(-0.644122\pi\)
−0.437460 + 0.899238i \(0.644122\pi\)
\(824\) 10.2749 0.357944
\(825\) −30.1993 −1.05141
\(826\) 8.00000 0.278356
\(827\) 26.2749 0.913668 0.456834 0.889552i \(-0.348983\pi\)
0.456834 + 0.889552i \(0.348983\pi\)
\(828\) 2.27492 0.0790588
\(829\) 11.7251 0.407229 0.203614 0.979051i \(-0.434731\pi\)
0.203614 + 0.979051i \(0.434731\pi\)
\(830\) 19.4502 0.675125
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 0.274917 0.00952532
\(834\) −4.00000 −0.138509
\(835\) 7.37459 0.255208
\(836\) 5.17525 0.178990
\(837\) 8.00000 0.276520
\(838\) 2.27492 0.0785857
\(839\) −22.9003 −0.790607 −0.395304 0.918551i \(-0.629361\pi\)
−0.395304 + 0.918551i \(0.629361\pi\)
\(840\) 4.27492 0.147499
\(841\) 39.4743 1.36118
\(842\) −3.09967 −0.106822
\(843\) 15.0997 0.520060
\(844\) 1.17525 0.0404537
\(845\) 0 0
\(846\) 0 0
\(847\) 5.82475 0.200141
\(848\) 10.0000 0.343401
\(849\) 4.00000 0.137280
\(850\) 3.64950 0.125177
\(851\) 9.72508 0.333372
\(852\) 0 0
\(853\) 27.6495 0.946701 0.473350 0.880874i \(-0.343044\pi\)
0.473350 + 0.880874i \(0.343044\pi\)
\(854\) 12.2749 0.420039
\(855\) 9.72508 0.332591
\(856\) 0.549834 0.0187930
\(857\) 19.0997 0.652432 0.326216 0.945295i \(-0.394226\pi\)
0.326216 + 0.945295i \(0.394226\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −9.72508 −0.331623
\(861\) −6.54983 −0.223218
\(862\) 11.4502 0.389994
\(863\) −29.6495 −1.00928 −0.504640 0.863330i \(-0.668375\pi\)
−0.504640 + 0.863330i \(0.668375\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 30.3505 1.03195
\(866\) 23.6495 0.803643
\(867\) 16.9244 0.574783
\(868\) 8.00000 0.271538
\(869\) 28.5498 0.968487
\(870\) −35.3746 −1.19931
\(871\) 0 0
\(872\) −0.274917 −0.00930987
\(873\) −15.0997 −0.511046
\(874\) 5.17525 0.175055
\(875\) −35.3746 −1.19588
\(876\) −12.8248 −0.433308
\(877\) −51.0997 −1.72551 −0.862757 0.505619i \(-0.831264\pi\)
−0.862757 + 0.505619i \(0.831264\pi\)
\(878\) −14.8248 −0.500311
\(879\) −6.00000 −0.202375
\(880\) 9.72508 0.327832
\(881\) −31.7251 −1.06885 −0.534423 0.845217i \(-0.679471\pi\)
−0.534423 + 0.845217i \(0.679471\pi\)
\(882\) 1.00000 0.0336718
\(883\) 31.9244 1.07434 0.537171 0.843473i \(-0.319493\pi\)
0.537171 + 0.843473i \(0.319493\pi\)
\(884\) 0 0
\(885\) 34.1993 1.14960
\(886\) 7.45017 0.250293
\(887\) −33.0997 −1.11138 −0.555689 0.831390i \(-0.687545\pi\)
−0.555689 + 0.831390i \(0.687545\pi\)
\(888\) −4.27492 −0.143457
\(889\) 20.5498 0.689219
\(890\) 59.8488 2.00614
\(891\) 2.27492 0.0762126
\(892\) −21.6495 −0.724879
\(893\) 0 0
\(894\) 22.5498 0.754179
\(895\) 51.2990 1.71474
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 0.274917 0.00917411
\(899\) −66.1993 −2.20787
\(900\) 13.2749 0.442497
\(901\) 2.74917 0.0915882
\(902\) −14.9003 −0.496127
\(903\) −2.27492 −0.0757045
\(904\) −6.00000 −0.199557
\(905\) 81.6495 2.71412
\(906\) 6.27492 0.208470
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 20.5498 0.681970
\(909\) 2.54983 0.0845727
\(910\) 0 0
\(911\) −52.4743 −1.73855 −0.869275 0.494329i \(-0.835414\pi\)
−0.869275 + 0.494329i \(0.835414\pi\)
\(912\) −2.27492 −0.0753300
\(913\) 10.3505 0.342551
\(914\) 18.5498 0.613574
\(915\) 52.4743 1.73475
\(916\) −19.0997 −0.631071
\(917\) −6.82475 −0.225373
\(918\) −0.274917 −0.00907362
\(919\) −12.5498 −0.413981 −0.206990 0.978343i \(-0.566367\pi\)
−0.206990 + 0.978343i \(0.566367\pi\)
\(920\) 9.72508 0.320626
\(921\) 21.0997 0.695258
\(922\) 17.9244 0.590309
\(923\) 0 0
\(924\) 2.27492 0.0748393
\(925\) 56.7492 1.86590
\(926\) −30.2749 −0.994896
\(927\) 10.2749 0.337473
\(928\) 8.27492 0.271637
\(929\) −14.5498 −0.477365 −0.238682 0.971098i \(-0.576715\pi\)
−0.238682 + 0.971098i \(0.576715\pi\)
\(930\) 34.1993 1.12144
\(931\) 2.27492 0.0745574
\(932\) 5.45017 0.178526
\(933\) −3.45017 −0.112953
\(934\) −26.2749 −0.859742
\(935\) 2.67359 0.0874358
\(936\) 0 0
\(937\) 16.9003 0.552110 0.276055 0.961142i \(-0.410973\pi\)
0.276055 + 0.961142i \(0.410973\pi\)
\(938\) 12.5498 0.409767
\(939\) 27.6495 0.902307
\(940\) 0 0
\(941\) −51.0997 −1.66580 −0.832901 0.553422i \(-0.813322\pi\)
−0.832901 + 0.553422i \(0.813322\pi\)
\(942\) 13.3746 0.435768
\(943\) −14.9003 −0.485222
\(944\) −8.00000 −0.260378
\(945\) 4.27492 0.139063
\(946\) −5.17525 −0.168262
\(947\) 17.1752 0.558121 0.279060 0.960274i \(-0.409977\pi\)
0.279060 + 0.960274i \(0.409977\pi\)
\(948\) −12.5498 −0.407600
\(949\) 0 0
\(950\) 30.1993 0.979796
\(951\) −1.45017 −0.0470248
\(952\) −0.274917 −0.00891012
\(953\) 31.6495 1.02523 0.512614 0.858619i \(-0.328677\pi\)
0.512614 + 0.858619i \(0.328677\pi\)
\(954\) 10.0000 0.323762
\(955\) 9.72508 0.314696
\(956\) −25.0997 −0.811781
\(957\) −18.8248 −0.608518
\(958\) 23.3746 0.755199
\(959\) −17.3746 −0.561055
\(960\) −4.27492 −0.137972
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0.549834 0.0177182
\(964\) −15.0997 −0.486328
\(965\) 79.2990 2.55273
\(966\) 2.27492 0.0731943
\(967\) −42.8248 −1.37715 −0.688576 0.725165i \(-0.741764\pi\)
−0.688576 + 0.725165i \(0.741764\pi\)
\(968\) −5.82475 −0.187215
\(969\) −0.625414 −0.0200912
\(970\) −64.5498 −2.07257
\(971\) −29.0997 −0.933853 −0.466926 0.884296i \(-0.654639\pi\)
−0.466926 + 0.884296i \(0.654639\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 18.1993 0.583144
\(975\) 0 0
\(976\) −12.2749 −0.392911
\(977\) 13.9244 0.445482 0.222741 0.974878i \(-0.428500\pi\)
0.222741 + 0.974878i \(0.428500\pi\)
\(978\) 12.5498 0.401299
\(979\) 31.8488 1.01789
\(980\) 4.27492 0.136557
\(981\) −0.274917 −0.00877743
\(982\) 8.54983 0.272836
\(983\) −61.0241 −1.94637 −0.973183 0.230032i \(-0.926117\pi\)
−0.973183 + 0.230032i \(0.926117\pi\)
\(984\) 6.54983 0.208801
\(985\) 10.9003 0.347313
\(986\) 2.27492 0.0724481
\(987\) 0 0
\(988\) 0 0
\(989\) −5.17525 −0.164563
\(990\) 9.72508 0.309083
\(991\) −46.7492 −1.48504 −0.742518 0.669826i \(-0.766369\pi\)
−0.742518 + 0.669826i \(0.766369\pi\)
\(992\) −8.00000 −0.254000
\(993\) −29.6495 −0.940899
\(994\) 0 0
\(995\) 29.1752 0.924918
\(996\) −4.54983 −0.144167
\(997\) −12.9003 −0.408558 −0.204279 0.978913i \(-0.565485\pi\)
−0.204279 + 0.978913i \(0.565485\pi\)
\(998\) −25.0997 −0.794516
\(999\) −4.27492 −0.135252
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 7098.2.a.bu.1.2 2
13.12 even 2 546.2.a.h.1.1 2
39.38 odd 2 1638.2.a.y.1.2 2
52.51 odd 2 4368.2.a.bh.1.1 2
91.90 odd 2 3822.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.h.1.1 2 13.12 even 2
1638.2.a.y.1.2 2 39.38 odd 2
3822.2.a.bm.1.2 2 91.90 odd 2
4368.2.a.bh.1.1 2 52.51 odd 2
7098.2.a.bu.1.2 2 1.1 even 1 trivial