Properties

Label 8034.2.a.z.1.8
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.343018\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.343018 q^{5} +1.00000 q^{6} +2.39676 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.343018 q^{5} +1.00000 q^{6} +2.39676 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.343018 q^{10} +1.78117 q^{11} -1.00000 q^{12} +1.00000 q^{13} -2.39676 q^{14} +0.343018 q^{15} +1.00000 q^{16} +5.22357 q^{17} -1.00000 q^{18} -3.94679 q^{19} -0.343018 q^{20} -2.39676 q^{21} -1.78117 q^{22} +2.02924 q^{23} +1.00000 q^{24} -4.88234 q^{25} -1.00000 q^{26} -1.00000 q^{27} +2.39676 q^{28} -7.94406 q^{29} -0.343018 q^{30} +2.26086 q^{31} -1.00000 q^{32} -1.78117 q^{33} -5.22357 q^{34} -0.822130 q^{35} +1.00000 q^{36} +8.14049 q^{37} +3.94679 q^{38} -1.00000 q^{39} +0.343018 q^{40} -0.158645 q^{41} +2.39676 q^{42} +2.52136 q^{43} +1.78117 q^{44} -0.343018 q^{45} -2.02924 q^{46} +7.07689 q^{47} -1.00000 q^{48} -1.25555 q^{49} +4.88234 q^{50} -5.22357 q^{51} +1.00000 q^{52} +6.10766 q^{53} +1.00000 q^{54} -0.610974 q^{55} -2.39676 q^{56} +3.94679 q^{57} +7.94406 q^{58} -8.74158 q^{59} +0.343018 q^{60} +3.19752 q^{61} -2.26086 q^{62} +2.39676 q^{63} +1.00000 q^{64} -0.343018 q^{65} +1.78117 q^{66} +7.84847 q^{67} +5.22357 q^{68} -2.02924 q^{69} +0.822130 q^{70} +7.09487 q^{71} -1.00000 q^{72} +17.0401 q^{73} -8.14049 q^{74} +4.88234 q^{75} -3.94679 q^{76} +4.26904 q^{77} +1.00000 q^{78} -3.08307 q^{79} -0.343018 q^{80} +1.00000 q^{81} +0.158645 q^{82} -9.65424 q^{83} -2.39676 q^{84} -1.79178 q^{85} -2.52136 q^{86} +7.94406 q^{87} -1.78117 q^{88} +6.89933 q^{89} +0.343018 q^{90} +2.39676 q^{91} +2.02924 q^{92} -2.26086 q^{93} -7.07689 q^{94} +1.35382 q^{95} +1.00000 q^{96} +7.83476 q^{97} +1.25555 q^{98} +1.78117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) −0.343018 −0.153402 −0.0767010 0.997054i \(-0.524439\pi\)
−0.0767010 + 0.997054i \(0.524439\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.39676 0.905889 0.452945 0.891539i \(-0.350373\pi\)
0.452945 + 0.891539i \(0.350373\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.343018 0.108472
\(11\) 1.78117 0.537044 0.268522 0.963274i \(-0.413465\pi\)
0.268522 + 0.963274i \(0.413465\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −2.39676 −0.640561
\(15\) 0.343018 0.0885667
\(16\) 1.00000 0.250000
\(17\) 5.22357 1.26690 0.633451 0.773783i \(-0.281638\pi\)
0.633451 + 0.773783i \(0.281638\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.94679 −0.905456 −0.452728 0.891649i \(-0.649549\pi\)
−0.452728 + 0.891649i \(0.649549\pi\)
\(20\) −0.343018 −0.0767010
\(21\) −2.39676 −0.523016
\(22\) −1.78117 −0.379748
\(23\) 2.02924 0.423125 0.211562 0.977364i \(-0.432145\pi\)
0.211562 + 0.977364i \(0.432145\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.88234 −0.976468
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 2.39676 0.452945
\(29\) −7.94406 −1.47518 −0.737588 0.675251i \(-0.764035\pi\)
−0.737588 + 0.675251i \(0.764035\pi\)
\(30\) −0.343018 −0.0626261
\(31\) 2.26086 0.406063 0.203031 0.979172i \(-0.434921\pi\)
0.203031 + 0.979172i \(0.434921\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.78117 −0.310063
\(34\) −5.22357 −0.895835
\(35\) −0.822130 −0.138965
\(36\) 1.00000 0.166667
\(37\) 8.14049 1.33829 0.669144 0.743133i \(-0.266661\pi\)
0.669144 + 0.743133i \(0.266661\pi\)
\(38\) 3.94679 0.640254
\(39\) −1.00000 −0.160128
\(40\) 0.343018 0.0542358
\(41\) −0.158645 −0.0247762 −0.0123881 0.999923i \(-0.503943\pi\)
−0.0123881 + 0.999923i \(0.503943\pi\)
\(42\) 2.39676 0.369828
\(43\) 2.52136 0.384504 0.192252 0.981346i \(-0.438421\pi\)
0.192252 + 0.981346i \(0.438421\pi\)
\(44\) 1.78117 0.268522
\(45\) −0.343018 −0.0511340
\(46\) −2.02924 −0.299194
\(47\) 7.07689 1.03227 0.516135 0.856507i \(-0.327370\pi\)
0.516135 + 0.856507i \(0.327370\pi\)
\(48\) −1.00000 −0.144338
\(49\) −1.25555 −0.179364
\(50\) 4.88234 0.690467
\(51\) −5.22357 −0.731446
\(52\) 1.00000 0.138675
\(53\) 6.10766 0.838951 0.419476 0.907767i \(-0.362214\pi\)
0.419476 + 0.907767i \(0.362214\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.610974 −0.0823837
\(56\) −2.39676 −0.320280
\(57\) 3.94679 0.522765
\(58\) 7.94406 1.04311
\(59\) −8.74158 −1.13806 −0.569028 0.822318i \(-0.692681\pi\)
−0.569028 + 0.822318i \(0.692681\pi\)
\(60\) 0.343018 0.0442834
\(61\) 3.19752 0.409400 0.204700 0.978825i \(-0.434378\pi\)
0.204700 + 0.978825i \(0.434378\pi\)
\(62\) −2.26086 −0.287130
\(63\) 2.39676 0.301963
\(64\) 1.00000 0.125000
\(65\) −0.343018 −0.0425461
\(66\) 1.78117 0.219247
\(67\) 7.84847 0.958843 0.479422 0.877585i \(-0.340846\pi\)
0.479422 + 0.877585i \(0.340846\pi\)
\(68\) 5.22357 0.633451
\(69\) −2.02924 −0.244291
\(70\) 0.822130 0.0982633
\(71\) 7.09487 0.842006 0.421003 0.907059i \(-0.361678\pi\)
0.421003 + 0.907059i \(0.361678\pi\)
\(72\) −1.00000 −0.117851
\(73\) 17.0401 1.99439 0.997197 0.0748173i \(-0.0238374\pi\)
0.997197 + 0.0748173i \(0.0238374\pi\)
\(74\) −8.14049 −0.946312
\(75\) 4.88234 0.563764
\(76\) −3.94679 −0.452728
\(77\) 4.26904 0.486503
\(78\) 1.00000 0.113228
\(79\) −3.08307 −0.346873 −0.173436 0.984845i \(-0.555487\pi\)
−0.173436 + 0.984845i \(0.555487\pi\)
\(80\) −0.343018 −0.0383505
\(81\) 1.00000 0.111111
\(82\) 0.158645 0.0175194
\(83\) −9.65424 −1.05969 −0.529845 0.848094i \(-0.677750\pi\)
−0.529845 + 0.848094i \(0.677750\pi\)
\(84\) −2.39676 −0.261508
\(85\) −1.79178 −0.194345
\(86\) −2.52136 −0.271885
\(87\) 7.94406 0.851693
\(88\) −1.78117 −0.189874
\(89\) 6.89933 0.731327 0.365664 0.930747i \(-0.380842\pi\)
0.365664 + 0.930747i \(0.380842\pi\)
\(90\) 0.343018 0.0361572
\(91\) 2.39676 0.251249
\(92\) 2.02924 0.211562
\(93\) −2.26086 −0.234440
\(94\) −7.07689 −0.729925
\(95\) 1.35382 0.138899
\(96\) 1.00000 0.102062
\(97\) 7.83476 0.795499 0.397750 0.917494i \(-0.369791\pi\)
0.397750 + 0.917494i \(0.369791\pi\)
\(98\) 1.25555 0.126830
\(99\) 1.78117 0.179015
\(100\) −4.88234 −0.488234
\(101\) −5.96014 −0.593056 −0.296528 0.955024i \(-0.595829\pi\)
−0.296528 + 0.955024i \(0.595829\pi\)
\(102\) 5.22357 0.517211
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 0.822130 0.0802317
\(106\) −6.10766 −0.593228
\(107\) 0.562160 0.0543461 0.0271730 0.999631i \(-0.491349\pi\)
0.0271730 + 0.999631i \(0.491349\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.55415 −0.723557 −0.361778 0.932264i \(-0.617830\pi\)
−0.361778 + 0.932264i \(0.617830\pi\)
\(110\) 0.610974 0.0582541
\(111\) −8.14049 −0.772661
\(112\) 2.39676 0.226472
\(113\) −6.21659 −0.584807 −0.292404 0.956295i \(-0.594455\pi\)
−0.292404 + 0.956295i \(0.594455\pi\)
\(114\) −3.94679 −0.369651
\(115\) −0.696063 −0.0649082
\(116\) −7.94406 −0.737588
\(117\) 1.00000 0.0924500
\(118\) 8.74158 0.804728
\(119\) 12.5196 1.14767
\(120\) −0.343018 −0.0313131
\(121\) −7.82742 −0.711583
\(122\) −3.19752 −0.289490
\(123\) 0.158645 0.0143045
\(124\) 2.26086 0.203031
\(125\) 3.38982 0.303194
\(126\) −2.39676 −0.213520
\(127\) 13.9651 1.23920 0.619600 0.784918i \(-0.287295\pi\)
0.619600 + 0.784918i \(0.287295\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.52136 −0.221993
\(130\) 0.343018 0.0300846
\(131\) −3.65169 −0.319050 −0.159525 0.987194i \(-0.550996\pi\)
−0.159525 + 0.987194i \(0.550996\pi\)
\(132\) −1.78117 −0.155031
\(133\) −9.45951 −0.820243
\(134\) −7.84847 −0.678004
\(135\) 0.343018 0.0295222
\(136\) −5.22357 −0.447918
\(137\) −0.558208 −0.0476909 −0.0238454 0.999716i \(-0.507591\pi\)
−0.0238454 + 0.999716i \(0.507591\pi\)
\(138\) 2.02924 0.172740
\(139\) 19.2351 1.63150 0.815748 0.578407i \(-0.196325\pi\)
0.815748 + 0.578407i \(0.196325\pi\)
\(140\) −0.822130 −0.0694827
\(141\) −7.07689 −0.595982
\(142\) −7.09487 −0.595388
\(143\) 1.78117 0.148949
\(144\) 1.00000 0.0833333
\(145\) 2.72495 0.226295
\(146\) −17.0401 −1.41025
\(147\) 1.25555 0.103556
\(148\) 8.14049 0.669144
\(149\) −23.8755 −1.95596 −0.977980 0.208698i \(-0.933077\pi\)
−0.977980 + 0.208698i \(0.933077\pi\)
\(150\) −4.88234 −0.398641
\(151\) 2.70857 0.220420 0.110210 0.993908i \(-0.464848\pi\)
0.110210 + 0.993908i \(0.464848\pi\)
\(152\) 3.94679 0.320127
\(153\) 5.22357 0.422301
\(154\) −4.26904 −0.344009
\(155\) −0.775515 −0.0622909
\(156\) −1.00000 −0.0800641
\(157\) −7.82902 −0.624824 −0.312412 0.949947i \(-0.601137\pi\)
−0.312412 + 0.949947i \(0.601137\pi\)
\(158\) 3.08307 0.245276
\(159\) −6.10766 −0.484369
\(160\) 0.343018 0.0271179
\(161\) 4.86359 0.383304
\(162\) −1.00000 −0.0785674
\(163\) −6.11690 −0.479112 −0.239556 0.970882i \(-0.577002\pi\)
−0.239556 + 0.970882i \(0.577002\pi\)
\(164\) −0.158645 −0.0123881
\(165\) 0.610974 0.0475643
\(166\) 9.65424 0.749314
\(167\) −0.442298 −0.0342261 −0.0171130 0.999854i \(-0.505448\pi\)
−0.0171130 + 0.999854i \(0.505448\pi\)
\(168\) 2.39676 0.184914
\(169\) 1.00000 0.0769231
\(170\) 1.79178 0.137423
\(171\) −3.94679 −0.301819
\(172\) 2.52136 0.192252
\(173\) 8.46037 0.643230 0.321615 0.946871i \(-0.395774\pi\)
0.321615 + 0.946871i \(0.395774\pi\)
\(174\) −7.94406 −0.602238
\(175\) −11.7018 −0.884572
\(176\) 1.78117 0.134261
\(177\) 8.74158 0.657057
\(178\) −6.89933 −0.517127
\(179\) −15.7783 −1.17933 −0.589663 0.807649i \(-0.700739\pi\)
−0.589663 + 0.807649i \(0.700739\pi\)
\(180\) −0.343018 −0.0255670
\(181\) 24.0105 1.78469 0.892343 0.451358i \(-0.149060\pi\)
0.892343 + 0.451358i \(0.149060\pi\)
\(182\) −2.39676 −0.177660
\(183\) −3.19752 −0.236367
\(184\) −2.02924 −0.149597
\(185\) −2.79233 −0.205296
\(186\) 2.26086 0.165774
\(187\) 9.30409 0.680383
\(188\) 7.07689 0.516135
\(189\) −2.39676 −0.174339
\(190\) −1.35382 −0.0982163
\(191\) −9.36830 −0.677866 −0.338933 0.940811i \(-0.610066\pi\)
−0.338933 + 0.940811i \(0.610066\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.10133 0.655128 0.327564 0.944829i \(-0.393772\pi\)
0.327564 + 0.944829i \(0.393772\pi\)
\(194\) −7.83476 −0.562503
\(195\) 0.343018 0.0245640
\(196\) −1.25555 −0.0896821
\(197\) −17.7399 −1.26391 −0.631956 0.775004i \(-0.717748\pi\)
−0.631956 + 0.775004i \(0.717748\pi\)
\(198\) −1.78117 −0.126583
\(199\) 3.44401 0.244139 0.122070 0.992522i \(-0.461047\pi\)
0.122070 + 0.992522i \(0.461047\pi\)
\(200\) 4.88234 0.345234
\(201\) −7.84847 −0.553588
\(202\) 5.96014 0.419354
\(203\) −19.0400 −1.33635
\(204\) −5.22357 −0.365723
\(205\) 0.0544180 0.00380072
\(206\) 1.00000 0.0696733
\(207\) 2.02924 0.141042
\(208\) 1.00000 0.0693375
\(209\) −7.02993 −0.486270
\(210\) −0.822130 −0.0567324
\(211\) 16.1940 1.11484 0.557419 0.830231i \(-0.311792\pi\)
0.557419 + 0.830231i \(0.311792\pi\)
\(212\) 6.10766 0.419476
\(213\) −7.09487 −0.486133
\(214\) −0.562160 −0.0384285
\(215\) −0.864871 −0.0589837
\(216\) 1.00000 0.0680414
\(217\) 5.41874 0.367848
\(218\) 7.55415 0.511632
\(219\) −17.0401 −1.15146
\(220\) −0.610974 −0.0411919
\(221\) 5.22357 0.351375
\(222\) 8.14049 0.546354
\(223\) 2.60140 0.174202 0.0871012 0.996199i \(-0.472240\pi\)
0.0871012 + 0.996199i \(0.472240\pi\)
\(224\) −2.39676 −0.160140
\(225\) −4.88234 −0.325489
\(226\) 6.21659 0.413521
\(227\) 0.153884 0.0102136 0.00510682 0.999987i \(-0.498374\pi\)
0.00510682 + 0.999987i \(0.498374\pi\)
\(228\) 3.94679 0.261383
\(229\) 10.5875 0.699640 0.349820 0.936817i \(-0.386243\pi\)
0.349820 + 0.936817i \(0.386243\pi\)
\(230\) 0.696063 0.0458971
\(231\) −4.26904 −0.280883
\(232\) 7.94406 0.521553
\(233\) 12.4251 0.813996 0.406998 0.913429i \(-0.366576\pi\)
0.406998 + 0.913429i \(0.366576\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −2.42750 −0.158352
\(236\) −8.74158 −0.569028
\(237\) 3.08307 0.200267
\(238\) −12.5196 −0.811528
\(239\) −10.7307 −0.694114 −0.347057 0.937844i \(-0.612819\pi\)
−0.347057 + 0.937844i \(0.612819\pi\)
\(240\) 0.343018 0.0221417
\(241\) −8.36037 −0.538539 −0.269269 0.963065i \(-0.586782\pi\)
−0.269269 + 0.963065i \(0.586782\pi\)
\(242\) 7.82742 0.503165
\(243\) −1.00000 −0.0641500
\(244\) 3.19752 0.204700
\(245\) 0.430676 0.0275149
\(246\) −0.158645 −0.0101148
\(247\) −3.94679 −0.251128
\(248\) −2.26086 −0.143565
\(249\) 9.65424 0.611812
\(250\) −3.38982 −0.214391
\(251\) −5.05823 −0.319273 −0.159636 0.987176i \(-0.551032\pi\)
−0.159636 + 0.987176i \(0.551032\pi\)
\(252\) 2.39676 0.150982
\(253\) 3.61442 0.227237
\(254\) −13.9651 −0.876247
\(255\) 1.79178 0.112205
\(256\) 1.00000 0.0625000
\(257\) 10.1908 0.635684 0.317842 0.948144i \(-0.397042\pi\)
0.317842 + 0.948144i \(0.397042\pi\)
\(258\) 2.52136 0.156973
\(259\) 19.5108 1.21234
\(260\) −0.343018 −0.0212730
\(261\) −7.94406 −0.491725
\(262\) 3.65169 0.225602
\(263\) −17.1554 −1.05784 −0.528922 0.848670i \(-0.677404\pi\)
−0.528922 + 0.848670i \(0.677404\pi\)
\(264\) 1.78117 0.109624
\(265\) −2.09503 −0.128697
\(266\) 9.45951 0.580000
\(267\) −6.89933 −0.422232
\(268\) 7.84847 0.479422
\(269\) −0.814999 −0.0496913 −0.0248457 0.999691i \(-0.507909\pi\)
−0.0248457 + 0.999691i \(0.507909\pi\)
\(270\) −0.343018 −0.0208754
\(271\) 9.45379 0.574277 0.287138 0.957889i \(-0.407296\pi\)
0.287138 + 0.957889i \(0.407296\pi\)
\(272\) 5.22357 0.316726
\(273\) −2.39676 −0.145058
\(274\) 0.558208 0.0337226
\(275\) −8.69630 −0.524406
\(276\) −2.02924 −0.122146
\(277\) −1.05069 −0.0631299 −0.0315650 0.999502i \(-0.510049\pi\)
−0.0315650 + 0.999502i \(0.510049\pi\)
\(278\) −19.2351 −1.15364
\(279\) 2.26086 0.135354
\(280\) 0.822130 0.0491317
\(281\) 31.5105 1.87976 0.939879 0.341508i \(-0.110938\pi\)
0.939879 + 0.341508i \(0.110938\pi\)
\(282\) 7.07689 0.421423
\(283\) −10.4757 −0.622714 −0.311357 0.950293i \(-0.600784\pi\)
−0.311357 + 0.950293i \(0.600784\pi\)
\(284\) 7.09487 0.421003
\(285\) −1.35382 −0.0801933
\(286\) −1.78117 −0.105323
\(287\) −0.380234 −0.0224445
\(288\) −1.00000 −0.0589256
\(289\) 10.2857 0.605041
\(290\) −2.72495 −0.160015
\(291\) −7.83476 −0.459282
\(292\) 17.0401 0.997197
\(293\) −4.79493 −0.280123 −0.140061 0.990143i \(-0.544730\pi\)
−0.140061 + 0.990143i \(0.544730\pi\)
\(294\) −1.25555 −0.0732252
\(295\) 2.99851 0.174580
\(296\) −8.14049 −0.473156
\(297\) −1.78117 −0.103354
\(298\) 23.8755 1.38307
\(299\) 2.02924 0.117354
\(300\) 4.88234 0.281882
\(301\) 6.04309 0.348318
\(302\) −2.70857 −0.155861
\(303\) 5.96014 0.342401
\(304\) −3.94679 −0.226364
\(305\) −1.09681 −0.0628029
\(306\) −5.22357 −0.298612
\(307\) 19.2898 1.10092 0.550462 0.834860i \(-0.314452\pi\)
0.550462 + 0.834860i \(0.314452\pi\)
\(308\) 4.26904 0.243251
\(309\) 1.00000 0.0568880
\(310\) 0.775515 0.0440463
\(311\) 21.2835 1.20687 0.603437 0.797411i \(-0.293797\pi\)
0.603437 + 0.797411i \(0.293797\pi\)
\(312\) 1.00000 0.0566139
\(313\) −11.9002 −0.672637 −0.336318 0.941748i \(-0.609182\pi\)
−0.336318 + 0.941748i \(0.609182\pi\)
\(314\) 7.82902 0.441817
\(315\) −0.822130 −0.0463218
\(316\) −3.08307 −0.173436
\(317\) −21.6584 −1.21646 −0.608229 0.793761i \(-0.708120\pi\)
−0.608229 + 0.793761i \(0.708120\pi\)
\(318\) 6.10766 0.342500
\(319\) −14.1498 −0.792234
\(320\) −0.343018 −0.0191753
\(321\) −0.562160 −0.0313767
\(322\) −4.86359 −0.271037
\(323\) −20.6163 −1.14712
\(324\) 1.00000 0.0555556
\(325\) −4.88234 −0.270823
\(326\) 6.11690 0.338784
\(327\) 7.55415 0.417746
\(328\) 0.158645 0.00875971
\(329\) 16.9616 0.935123
\(330\) −0.610974 −0.0336330
\(331\) 26.4528 1.45398 0.726989 0.686650i \(-0.240919\pi\)
0.726989 + 0.686650i \(0.240919\pi\)
\(332\) −9.65424 −0.529845
\(333\) 8.14049 0.446096
\(334\) 0.442298 0.0242015
\(335\) −2.69216 −0.147089
\(336\) −2.39676 −0.130754
\(337\) 12.3203 0.671127 0.335564 0.942018i \(-0.391073\pi\)
0.335564 + 0.942018i \(0.391073\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 6.21659 0.337639
\(340\) −1.79178 −0.0971727
\(341\) 4.02699 0.218074
\(342\) 3.94679 0.213418
\(343\) −19.7866 −1.06837
\(344\) −2.52136 −0.135943
\(345\) 0.696063 0.0374748
\(346\) −8.46037 −0.454832
\(347\) 7.66523 0.411491 0.205746 0.978606i \(-0.434038\pi\)
0.205746 + 0.978606i \(0.434038\pi\)
\(348\) 7.94406 0.425846
\(349\) 18.8662 1.00988 0.504942 0.863153i \(-0.331514\pi\)
0.504942 + 0.863153i \(0.331514\pi\)
\(350\) 11.7018 0.625487
\(351\) −1.00000 −0.0533761
\(352\) −1.78117 −0.0949369
\(353\) −1.98763 −0.105791 −0.0528954 0.998600i \(-0.516845\pi\)
−0.0528954 + 0.998600i \(0.516845\pi\)
\(354\) −8.74158 −0.464610
\(355\) −2.43367 −0.129166
\(356\) 6.89933 0.365664
\(357\) −12.5196 −0.662609
\(358\) 15.7783 0.833910
\(359\) 32.8713 1.73488 0.867439 0.497543i \(-0.165764\pi\)
0.867439 + 0.497543i \(0.165764\pi\)
\(360\) 0.343018 0.0180786
\(361\) −3.42283 −0.180149
\(362\) −24.0105 −1.26196
\(363\) 7.82742 0.410833
\(364\) 2.39676 0.125624
\(365\) −5.84506 −0.305944
\(366\) 3.19752 0.167137
\(367\) 3.74369 0.195419 0.0977095 0.995215i \(-0.468848\pi\)
0.0977095 + 0.995215i \(0.468848\pi\)
\(368\) 2.02924 0.105781
\(369\) −0.158645 −0.00825873
\(370\) 2.79233 0.145166
\(371\) 14.6386 0.759997
\(372\) −2.26086 −0.117220
\(373\) −0.977914 −0.0506345 −0.0253172 0.999679i \(-0.508060\pi\)
−0.0253172 + 0.999679i \(0.508060\pi\)
\(374\) −9.30409 −0.481103
\(375\) −3.38982 −0.175049
\(376\) −7.07689 −0.364963
\(377\) −7.94406 −0.409140
\(378\) 2.39676 0.123276
\(379\) −15.2708 −0.784410 −0.392205 0.919878i \(-0.628288\pi\)
−0.392205 + 0.919878i \(0.628288\pi\)
\(380\) 1.35382 0.0694494
\(381\) −13.9651 −0.715452
\(382\) 9.36830 0.479324
\(383\) −25.2275 −1.28907 −0.644533 0.764576i \(-0.722948\pi\)
−0.644533 + 0.764576i \(0.722948\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.46436 −0.0746305
\(386\) −9.10133 −0.463245
\(387\) 2.52136 0.128168
\(388\) 7.83476 0.397750
\(389\) 23.3407 1.18342 0.591711 0.806151i \(-0.298453\pi\)
0.591711 + 0.806151i \(0.298453\pi\)
\(390\) −0.343018 −0.0173694
\(391\) 10.5999 0.536058
\(392\) 1.25555 0.0634148
\(393\) 3.65169 0.184204
\(394\) 17.7399 0.893721
\(395\) 1.05755 0.0532110
\(396\) 1.78117 0.0895074
\(397\) −33.7019 −1.69145 −0.845726 0.533617i \(-0.820832\pi\)
−0.845726 + 0.533617i \(0.820832\pi\)
\(398\) −3.44401 −0.172633
\(399\) 9.45951 0.473568
\(400\) −4.88234 −0.244117
\(401\) −17.7330 −0.885543 −0.442772 0.896634i \(-0.646005\pi\)
−0.442772 + 0.896634i \(0.646005\pi\)
\(402\) 7.84847 0.391446
\(403\) 2.26086 0.112622
\(404\) −5.96014 −0.296528
\(405\) −0.343018 −0.0170447
\(406\) 19.0400 0.944939
\(407\) 14.4996 0.718720
\(408\) 5.22357 0.258605
\(409\) 0.866427 0.0428420 0.0214210 0.999771i \(-0.493181\pi\)
0.0214210 + 0.999771i \(0.493181\pi\)
\(410\) −0.0544180 −0.00268752
\(411\) 0.558208 0.0275343
\(412\) −1.00000 −0.0492665
\(413\) −20.9515 −1.03095
\(414\) −2.02924 −0.0997315
\(415\) 3.31157 0.162559
\(416\) −1.00000 −0.0490290
\(417\) −19.2351 −0.941945
\(418\) 7.02993 0.343845
\(419\) 4.23789 0.207035 0.103517 0.994628i \(-0.466990\pi\)
0.103517 + 0.994628i \(0.466990\pi\)
\(420\) 0.822130 0.0401158
\(421\) 12.9281 0.630075 0.315038 0.949079i \(-0.397983\pi\)
0.315038 + 0.949079i \(0.397983\pi\)
\(422\) −16.1940 −0.788310
\(423\) 7.07689 0.344090
\(424\) −6.10766 −0.296614
\(425\) −25.5032 −1.23709
\(426\) 7.09487 0.343748
\(427\) 7.66368 0.370872
\(428\) 0.562160 0.0271730
\(429\) −1.78117 −0.0859959
\(430\) 0.864871 0.0417078
\(431\) 2.33069 0.112266 0.0561328 0.998423i \(-0.482123\pi\)
0.0561328 + 0.998423i \(0.482123\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 11.0775 0.532350 0.266175 0.963925i \(-0.414240\pi\)
0.266175 + 0.963925i \(0.414240\pi\)
\(434\) −5.41874 −0.260108
\(435\) −2.72495 −0.130651
\(436\) −7.55415 −0.361778
\(437\) −8.00897 −0.383121
\(438\) 17.0401 0.814208
\(439\) −22.3671 −1.06752 −0.533761 0.845635i \(-0.679222\pi\)
−0.533761 + 0.845635i \(0.679222\pi\)
\(440\) 0.610974 0.0291270
\(441\) −1.25555 −0.0597881
\(442\) −5.22357 −0.248460
\(443\) −0.372988 −0.0177212 −0.00886060 0.999961i \(-0.502820\pi\)
−0.00886060 + 0.999961i \(0.502820\pi\)
\(444\) −8.14049 −0.386330
\(445\) −2.36659 −0.112187
\(446\) −2.60140 −0.123180
\(447\) 23.8755 1.12927
\(448\) 2.39676 0.113236
\(449\) −9.55036 −0.450709 −0.225355 0.974277i \(-0.572354\pi\)
−0.225355 + 0.974277i \(0.572354\pi\)
\(450\) 4.88234 0.230156
\(451\) −0.282575 −0.0133059
\(452\) −6.21659 −0.292404
\(453\) −2.70857 −0.127260
\(454\) −0.153884 −0.00722213
\(455\) −0.822130 −0.0385421
\(456\) −3.94679 −0.184825
\(457\) 8.43762 0.394695 0.197348 0.980334i \(-0.436767\pi\)
0.197348 + 0.980334i \(0.436767\pi\)
\(458\) −10.5875 −0.494720
\(459\) −5.22357 −0.243815
\(460\) −0.696063 −0.0324541
\(461\) 0.632681 0.0294669 0.0147335 0.999891i \(-0.495310\pi\)
0.0147335 + 0.999891i \(0.495310\pi\)
\(462\) 4.26904 0.198614
\(463\) −18.6646 −0.867417 −0.433709 0.901053i \(-0.642795\pi\)
−0.433709 + 0.901053i \(0.642795\pi\)
\(464\) −7.94406 −0.368794
\(465\) 0.775515 0.0359636
\(466\) −12.4251 −0.575582
\(467\) 30.1881 1.39694 0.698470 0.715639i \(-0.253864\pi\)
0.698470 + 0.715639i \(0.253864\pi\)
\(468\) 1.00000 0.0462250
\(469\) 18.8109 0.868606
\(470\) 2.42750 0.111972
\(471\) 7.82902 0.360742
\(472\) 8.74158 0.402364
\(473\) 4.49098 0.206496
\(474\) −3.08307 −0.141610
\(475\) 19.2696 0.884149
\(476\) 12.5196 0.573837
\(477\) 6.10766 0.279650
\(478\) 10.7307 0.490812
\(479\) 3.54195 0.161836 0.0809179 0.996721i \(-0.474215\pi\)
0.0809179 + 0.996721i \(0.474215\pi\)
\(480\) −0.343018 −0.0156565
\(481\) 8.14049 0.371174
\(482\) 8.36037 0.380804
\(483\) −4.86359 −0.221301
\(484\) −7.82742 −0.355792
\(485\) −2.68746 −0.122031
\(486\) 1.00000 0.0453609
\(487\) 23.7808 1.07761 0.538805 0.842431i \(-0.318876\pi\)
0.538805 + 0.842431i \(0.318876\pi\)
\(488\) −3.19752 −0.144745
\(489\) 6.11690 0.276616
\(490\) −0.430676 −0.0194559
\(491\) 21.9035 0.988491 0.494246 0.869322i \(-0.335444\pi\)
0.494246 + 0.869322i \(0.335444\pi\)
\(492\) 0.158645 0.00715227
\(493\) −41.4964 −1.86890
\(494\) 3.94679 0.177575
\(495\) −0.610974 −0.0274612
\(496\) 2.26086 0.101516
\(497\) 17.0047 0.762765
\(498\) −9.65424 −0.432617
\(499\) 12.0837 0.540939 0.270470 0.962728i \(-0.412821\pi\)
0.270470 + 0.962728i \(0.412821\pi\)
\(500\) 3.38982 0.151597
\(501\) 0.442298 0.0197604
\(502\) 5.05823 0.225760
\(503\) −9.05121 −0.403574 −0.201787 0.979429i \(-0.564675\pi\)
−0.201787 + 0.979429i \(0.564675\pi\)
\(504\) −2.39676 −0.106760
\(505\) 2.04443 0.0909761
\(506\) −3.61442 −0.160681
\(507\) −1.00000 −0.0444116
\(508\) 13.9651 0.619600
\(509\) −40.8400 −1.81020 −0.905100 0.425200i \(-0.860204\pi\)
−0.905100 + 0.425200i \(0.860204\pi\)
\(510\) −1.79178 −0.0793412
\(511\) 40.8410 1.80670
\(512\) −1.00000 −0.0441942
\(513\) 3.94679 0.174255
\(514\) −10.1908 −0.449496
\(515\) 0.343018 0.0151152
\(516\) −2.52136 −0.110997
\(517\) 12.6052 0.554375
\(518\) −19.5108 −0.857254
\(519\) −8.46037 −0.371369
\(520\) 0.343018 0.0150423
\(521\) −2.46115 −0.107825 −0.0539126 0.998546i \(-0.517169\pi\)
−0.0539126 + 0.998546i \(0.517169\pi\)
\(522\) 7.94406 0.347702
\(523\) 0.745614 0.0326034 0.0163017 0.999867i \(-0.494811\pi\)
0.0163017 + 0.999867i \(0.494811\pi\)
\(524\) −3.65169 −0.159525
\(525\) 11.7018 0.510708
\(526\) 17.1554 0.748009
\(527\) 11.8098 0.514442
\(528\) −1.78117 −0.0775157
\(529\) −18.8822 −0.820965
\(530\) 2.09503 0.0910024
\(531\) −8.74158 −0.379352
\(532\) −9.45951 −0.410122
\(533\) −0.158645 −0.00687168
\(534\) 6.89933 0.298563
\(535\) −0.192831 −0.00833680
\(536\) −7.84847 −0.339002
\(537\) 15.7783 0.680884
\(538\) 0.814999 0.0351371
\(539\) −2.23635 −0.0963266
\(540\) 0.343018 0.0147611
\(541\) 21.4494 0.922180 0.461090 0.887353i \(-0.347459\pi\)
0.461090 + 0.887353i \(0.347459\pi\)
\(542\) −9.45379 −0.406075
\(543\) −24.0105 −1.03039
\(544\) −5.22357 −0.223959
\(545\) 2.59121 0.110995
\(546\) 2.39676 0.102572
\(547\) 6.04582 0.258500 0.129250 0.991612i \(-0.458743\pi\)
0.129250 + 0.991612i \(0.458743\pi\)
\(548\) −0.558208 −0.0238454
\(549\) 3.19752 0.136467
\(550\) 8.69630 0.370811
\(551\) 31.3536 1.33571
\(552\) 2.02924 0.0863700
\(553\) −7.38938 −0.314228
\(554\) 1.05069 0.0446396
\(555\) 2.79233 0.118528
\(556\) 19.2351 0.815748
\(557\) 26.3842 1.11793 0.558967 0.829190i \(-0.311198\pi\)
0.558967 + 0.829190i \(0.311198\pi\)
\(558\) −2.26086 −0.0957099
\(559\) 2.52136 0.106642
\(560\) −0.822130 −0.0347413
\(561\) −9.30409 −0.392819
\(562\) −31.5105 −1.32919
\(563\) 21.6247 0.911374 0.455687 0.890140i \(-0.349394\pi\)
0.455687 + 0.890140i \(0.349394\pi\)
\(564\) −7.07689 −0.297991
\(565\) 2.13240 0.0897107
\(566\) 10.4757 0.440325
\(567\) 2.39676 0.100654
\(568\) −7.09487 −0.297694
\(569\) 23.8137 0.998322 0.499161 0.866509i \(-0.333642\pi\)
0.499161 + 0.866509i \(0.333642\pi\)
\(570\) 1.35382 0.0567052
\(571\) 18.1098 0.757870 0.378935 0.925423i \(-0.376290\pi\)
0.378935 + 0.925423i \(0.376290\pi\)
\(572\) 1.78117 0.0744746
\(573\) 9.36830 0.391366
\(574\) 0.380234 0.0158707
\(575\) −9.90742 −0.413168
\(576\) 1.00000 0.0416667
\(577\) 31.1375 1.29627 0.648136 0.761525i \(-0.275549\pi\)
0.648136 + 0.761525i \(0.275549\pi\)
\(578\) −10.2857 −0.427828
\(579\) −9.10133 −0.378238
\(580\) 2.72495 0.113147
\(581\) −23.1389 −0.959962
\(582\) 7.83476 0.324761
\(583\) 10.8788 0.450554
\(584\) −17.0401 −0.705125
\(585\) −0.343018 −0.0141820
\(586\) 4.79493 0.198077
\(587\) −14.5846 −0.601969 −0.300985 0.953629i \(-0.597315\pi\)
−0.300985 + 0.953629i \(0.597315\pi\)
\(588\) 1.25555 0.0517780
\(589\) −8.92315 −0.367672
\(590\) −2.99851 −0.123447
\(591\) 17.7399 0.729720
\(592\) 8.14049 0.334572
\(593\) 0.370789 0.0152265 0.00761324 0.999971i \(-0.497577\pi\)
0.00761324 + 0.999971i \(0.497577\pi\)
\(594\) 1.78117 0.0730825
\(595\) −4.29445 −0.176055
\(596\) −23.8755 −0.977980
\(597\) −3.44401 −0.140954
\(598\) −2.02924 −0.0829816
\(599\) 14.1950 0.579993 0.289997 0.957028i \(-0.406346\pi\)
0.289997 + 0.957028i \(0.406346\pi\)
\(600\) −4.88234 −0.199321
\(601\) −6.50391 −0.265300 −0.132650 0.991163i \(-0.542349\pi\)
−0.132650 + 0.991163i \(0.542349\pi\)
\(602\) −6.04309 −0.246298
\(603\) 7.84847 0.319614
\(604\) 2.70857 0.110210
\(605\) 2.68494 0.109158
\(606\) −5.96014 −0.242114
\(607\) 24.1093 0.978566 0.489283 0.872125i \(-0.337259\pi\)
0.489283 + 0.872125i \(0.337259\pi\)
\(608\) 3.94679 0.160064
\(609\) 19.0400 0.771540
\(610\) 1.09681 0.0444083
\(611\) 7.07689 0.286300
\(612\) 5.22357 0.211150
\(613\) −1.14964 −0.0464333 −0.0232167 0.999730i \(-0.507391\pi\)
−0.0232167 + 0.999730i \(0.507391\pi\)
\(614\) −19.2898 −0.778471
\(615\) −0.0544180 −0.00219435
\(616\) −4.26904 −0.172005
\(617\) −4.66642 −0.187863 −0.0939315 0.995579i \(-0.529943\pi\)
−0.0939315 + 0.995579i \(0.529943\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 11.0487 0.444085 0.222043 0.975037i \(-0.428728\pi\)
0.222043 + 0.975037i \(0.428728\pi\)
\(620\) −0.775515 −0.0311454
\(621\) −2.02924 −0.0814304
\(622\) −21.2835 −0.853389
\(623\) 16.5360 0.662502
\(624\) −1.00000 −0.0400320
\(625\) 23.2489 0.929957
\(626\) 11.9002 0.475626
\(627\) 7.02993 0.280748
\(628\) −7.82902 −0.312412
\(629\) 42.5224 1.69548
\(630\) 0.822130 0.0327544
\(631\) 36.5947 1.45681 0.728405 0.685146i \(-0.240262\pi\)
0.728405 + 0.685146i \(0.240262\pi\)
\(632\) 3.08307 0.122638
\(633\) −16.1940 −0.643653
\(634\) 21.6584 0.860166
\(635\) −4.79026 −0.190096
\(636\) −6.10766 −0.242184
\(637\) −1.25555 −0.0497467
\(638\) 14.1498 0.560194
\(639\) 7.09487 0.280669
\(640\) 0.343018 0.0135590
\(641\) 14.3213 0.565657 0.282828 0.959171i \(-0.408727\pi\)
0.282828 + 0.959171i \(0.408727\pi\)
\(642\) 0.562160 0.0221867
\(643\) 42.7131 1.68444 0.842220 0.539135i \(-0.181249\pi\)
0.842220 + 0.539135i \(0.181249\pi\)
\(644\) 4.86359 0.191652
\(645\) 0.864871 0.0340543
\(646\) 20.6163 0.811139
\(647\) 14.3611 0.564592 0.282296 0.959327i \(-0.408904\pi\)
0.282296 + 0.959327i \(0.408904\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −15.5703 −0.611187
\(650\) 4.88234 0.191501
\(651\) −5.41874 −0.212377
\(652\) −6.11690 −0.239556
\(653\) 45.2621 1.77124 0.885621 0.464409i \(-0.153733\pi\)
0.885621 + 0.464409i \(0.153733\pi\)
\(654\) −7.55415 −0.295391
\(655\) 1.25260 0.0489429
\(656\) −0.158645 −0.00619405
\(657\) 17.0401 0.664798
\(658\) −16.9616 −0.661232
\(659\) 35.4677 1.38163 0.690813 0.723033i \(-0.257253\pi\)
0.690813 + 0.723033i \(0.257253\pi\)
\(660\) 0.610974 0.0237821
\(661\) −17.9913 −0.699780 −0.349890 0.936791i \(-0.613781\pi\)
−0.349890 + 0.936791i \(0.613781\pi\)
\(662\) −26.4528 −1.02812
\(663\) −5.22357 −0.202867
\(664\) 9.65424 0.374657
\(665\) 3.24478 0.125827
\(666\) −8.14049 −0.315437
\(667\) −16.1204 −0.624183
\(668\) −0.442298 −0.0171130
\(669\) −2.60140 −0.100576
\(670\) 2.69216 0.104007
\(671\) 5.69534 0.219866
\(672\) 2.39676 0.0924570
\(673\) −7.41406 −0.285791 −0.142895 0.989738i \(-0.545641\pi\)
−0.142895 + 0.989738i \(0.545641\pi\)
\(674\) −12.3203 −0.474558
\(675\) 4.88234 0.187921
\(676\) 1.00000 0.0384615
\(677\) 0.502213 0.0193016 0.00965080 0.999953i \(-0.496928\pi\)
0.00965080 + 0.999953i \(0.496928\pi\)
\(678\) −6.21659 −0.238747
\(679\) 18.7780 0.720634
\(680\) 1.79178 0.0687115
\(681\) −0.153884 −0.00589684
\(682\) −4.02699 −0.154201
\(683\) 17.8495 0.682991 0.341495 0.939883i \(-0.389067\pi\)
0.341495 + 0.939883i \(0.389067\pi\)
\(684\) −3.94679 −0.150909
\(685\) 0.191475 0.00731588
\(686\) 19.7866 0.755454
\(687\) −10.5875 −0.403937
\(688\) 2.52136 0.0961260
\(689\) 6.10766 0.232683
\(690\) −0.696063 −0.0264987
\(691\) −28.8752 −1.09846 −0.549232 0.835670i \(-0.685080\pi\)
−0.549232 + 0.835670i \(0.685080\pi\)
\(692\) 8.46037 0.321615
\(693\) 4.26904 0.162168
\(694\) −7.66523 −0.290968
\(695\) −6.59796 −0.250275
\(696\) −7.94406 −0.301119
\(697\) −0.828694 −0.0313890
\(698\) −18.8662 −0.714096
\(699\) −12.4251 −0.469961
\(700\) −11.7018 −0.442286
\(701\) 24.3254 0.918758 0.459379 0.888240i \(-0.348072\pi\)
0.459379 + 0.888240i \(0.348072\pi\)
\(702\) 1.00000 0.0377426
\(703\) −32.1288 −1.21176
\(704\) 1.78117 0.0671305
\(705\) 2.42750 0.0914248
\(706\) 1.98763 0.0748054
\(707\) −14.2850 −0.537244
\(708\) 8.74158 0.328529
\(709\) −33.8429 −1.27100 −0.635499 0.772102i \(-0.719206\pi\)
−0.635499 + 0.772102i \(0.719206\pi\)
\(710\) 2.43367 0.0913338
\(711\) −3.08307 −0.115624
\(712\) −6.89933 −0.258563
\(713\) 4.58782 0.171815
\(714\) 12.5196 0.468536
\(715\) −0.610974 −0.0228491
\(716\) −15.7783 −0.589663
\(717\) 10.7307 0.400747
\(718\) −32.8713 −1.22674
\(719\) −47.6357 −1.77651 −0.888256 0.459348i \(-0.848083\pi\)
−0.888256 + 0.459348i \(0.848083\pi\)
\(720\) −0.343018 −0.0127835
\(721\) −2.39676 −0.0892599
\(722\) 3.42283 0.127385
\(723\) 8.36037 0.310925
\(724\) 24.0105 0.892343
\(725\) 38.7856 1.44046
\(726\) −7.82742 −0.290503
\(727\) −31.8772 −1.18226 −0.591130 0.806576i \(-0.701318\pi\)
−0.591130 + 0.806576i \(0.701318\pi\)
\(728\) −2.39676 −0.0888298
\(729\) 1.00000 0.0370370
\(730\) 5.84506 0.216335
\(731\) 13.1705 0.487129
\(732\) −3.19752 −0.118184
\(733\) 3.97359 0.146768 0.0733840 0.997304i \(-0.476620\pi\)
0.0733840 + 0.997304i \(0.476620\pi\)
\(734\) −3.74369 −0.138182
\(735\) −0.430676 −0.0158857
\(736\) −2.02924 −0.0747986
\(737\) 13.9795 0.514941
\(738\) 0.158645 0.00583981
\(739\) 15.9452 0.586555 0.293278 0.956027i \(-0.405254\pi\)
0.293278 + 0.956027i \(0.405254\pi\)
\(740\) −2.79233 −0.102648
\(741\) 3.94679 0.144989
\(742\) −14.6386 −0.537399
\(743\) 35.0364 1.28536 0.642680 0.766134i \(-0.277822\pi\)
0.642680 + 0.766134i \(0.277822\pi\)
\(744\) 2.26086 0.0828872
\(745\) 8.18973 0.300048
\(746\) 0.977914 0.0358040
\(747\) −9.65424 −0.353230
\(748\) 9.30409 0.340191
\(749\) 1.34736 0.0492315
\(750\) 3.38982 0.123779
\(751\) 13.8824 0.506576 0.253288 0.967391i \(-0.418488\pi\)
0.253288 + 0.967391i \(0.418488\pi\)
\(752\) 7.07689 0.258068
\(753\) 5.05823 0.184332
\(754\) 7.94406 0.289306
\(755\) −0.929086 −0.0338129
\(756\) −2.39676 −0.0871693
\(757\) −37.8619 −1.37611 −0.688057 0.725657i \(-0.741536\pi\)
−0.688057 + 0.725657i \(0.741536\pi\)
\(758\) 15.2708 0.554662
\(759\) −3.61442 −0.131195
\(760\) −1.35382 −0.0491082
\(761\) 22.2610 0.806961 0.403481 0.914988i \(-0.367800\pi\)
0.403481 + 0.914988i \(0.367800\pi\)
\(762\) 13.9651 0.505901
\(763\) −18.1055 −0.655462
\(764\) −9.36830 −0.338933
\(765\) −1.79178 −0.0647818
\(766\) 25.2275 0.911508
\(767\) −8.74158 −0.315640
\(768\) −1.00000 −0.0360844
\(769\) 0.172135 0.00620734 0.00310367 0.999995i \(-0.499012\pi\)
0.00310367 + 0.999995i \(0.499012\pi\)
\(770\) 1.46436 0.0527718
\(771\) −10.1908 −0.367012
\(772\) 9.10133 0.327564
\(773\) −6.30578 −0.226803 −0.113402 0.993549i \(-0.536175\pi\)
−0.113402 + 0.993549i \(0.536175\pi\)
\(774\) −2.52136 −0.0906285
\(775\) −11.0383 −0.396507
\(776\) −7.83476 −0.281251
\(777\) −19.5108 −0.699945
\(778\) −23.3407 −0.836805
\(779\) 0.626139 0.0224338
\(780\) 0.343018 0.0122820
\(781\) 12.6372 0.452195
\(782\) −10.5999 −0.379050
\(783\) 7.94406 0.283898
\(784\) −1.25555 −0.0448411
\(785\) 2.68549 0.0958493
\(786\) −3.65169 −0.130252
\(787\) −37.3651 −1.33192 −0.665961 0.745986i \(-0.731978\pi\)
−0.665961 + 0.745986i \(0.731978\pi\)
\(788\) −17.7399 −0.631956
\(789\) 17.1554 0.610747
\(790\) −1.05755 −0.0376258
\(791\) −14.8997 −0.529771
\(792\) −1.78117 −0.0632913
\(793\) 3.19752 0.113547
\(794\) 33.7019 1.19604
\(795\) 2.09503 0.0743032
\(796\) 3.44401 0.122070
\(797\) 16.4811 0.583791 0.291896 0.956450i \(-0.405714\pi\)
0.291896 + 0.956450i \(0.405714\pi\)
\(798\) −9.45951 −0.334863
\(799\) 36.9666 1.30779
\(800\) 4.88234 0.172617
\(801\) 6.89933 0.243776
\(802\) 17.7330 0.626174
\(803\) 30.3514 1.07108
\(804\) −7.84847 −0.276794
\(805\) −1.66830 −0.0587997
\(806\) −2.26086 −0.0796354
\(807\) 0.814999 0.0286893
\(808\) 5.96014 0.209677
\(809\) −0.895485 −0.0314836 −0.0157418 0.999876i \(-0.505011\pi\)
−0.0157418 + 0.999876i \(0.505011\pi\)
\(810\) 0.343018 0.0120524
\(811\) −36.3541 −1.27657 −0.638283 0.769802i \(-0.720355\pi\)
−0.638283 + 0.769802i \(0.720355\pi\)
\(812\) −19.0400 −0.668173
\(813\) −9.45379 −0.331559
\(814\) −14.4996 −0.508212
\(815\) 2.09820 0.0734969
\(816\) −5.22357 −0.182862
\(817\) −9.95129 −0.348152
\(818\) −0.866427 −0.0302939
\(819\) 2.39676 0.0837495
\(820\) 0.0544180 0.00190036
\(821\) −23.6636 −0.825867 −0.412934 0.910761i \(-0.635496\pi\)
−0.412934 + 0.910761i \(0.635496\pi\)
\(822\) −0.558208 −0.0194697
\(823\) −12.6850 −0.442172 −0.221086 0.975254i \(-0.570960\pi\)
−0.221086 + 0.975254i \(0.570960\pi\)
\(824\) 1.00000 0.0348367
\(825\) 8.69630 0.302766
\(826\) 20.9515 0.728994
\(827\) 2.08401 0.0724680 0.0362340 0.999343i \(-0.488464\pi\)
0.0362340 + 0.999343i \(0.488464\pi\)
\(828\) 2.02924 0.0705208
\(829\) 15.9265 0.553149 0.276574 0.960993i \(-0.410801\pi\)
0.276574 + 0.960993i \(0.410801\pi\)
\(830\) −3.31157 −0.114946
\(831\) 1.05069 0.0364481
\(832\) 1.00000 0.0346688
\(833\) −6.55845 −0.227237
\(834\) 19.2351 0.666056
\(835\) 0.151716 0.00525035
\(836\) −7.02993 −0.243135
\(837\) −2.26086 −0.0781468
\(838\) −4.23789 −0.146396
\(839\) 52.0067 1.79547 0.897736 0.440534i \(-0.145211\pi\)
0.897736 + 0.440534i \(0.145211\pi\)
\(840\) −0.822130 −0.0283662
\(841\) 34.1081 1.17614
\(842\) −12.9281 −0.445530
\(843\) −31.5105 −1.08528
\(844\) 16.1940 0.557419
\(845\) −0.343018 −0.0118002
\(846\) −7.07689 −0.243308
\(847\) −18.7604 −0.644616
\(848\) 6.10766 0.209738
\(849\) 10.4757 0.359524
\(850\) 25.5032 0.874754
\(851\) 16.5190 0.566263
\(852\) −7.09487 −0.243066
\(853\) 1.52195 0.0521107 0.0260553 0.999661i \(-0.491705\pi\)
0.0260553 + 0.999661i \(0.491705\pi\)
\(854\) −7.66368 −0.262246
\(855\) 1.35382 0.0462996
\(856\) −0.562160 −0.0192142
\(857\) −0.192010 −0.00655892 −0.00327946 0.999995i \(-0.501044\pi\)
−0.00327946 + 0.999995i \(0.501044\pi\)
\(858\) 1.78117 0.0608083
\(859\) 20.9645 0.715299 0.357649 0.933856i \(-0.383578\pi\)
0.357649 + 0.933856i \(0.383578\pi\)
\(860\) −0.864871 −0.0294919
\(861\) 0.380234 0.0129583
\(862\) −2.33069 −0.0793838
\(863\) 20.7125 0.705060 0.352530 0.935800i \(-0.385321\pi\)
0.352530 + 0.935800i \(0.385321\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.90205 −0.0986728
\(866\) −11.0775 −0.376428
\(867\) −10.2857 −0.349320
\(868\) 5.41874 0.183924
\(869\) −5.49149 −0.186286
\(870\) 2.72495 0.0923845
\(871\) 7.84847 0.265935
\(872\) 7.55415 0.255816
\(873\) 7.83476 0.265166
\(874\) 8.00897 0.270907
\(875\) 8.12457 0.274661
\(876\) −17.0401 −0.575732
\(877\) 44.6845 1.50889 0.754445 0.656363i \(-0.227906\pi\)
0.754445 + 0.656363i \(0.227906\pi\)
\(878\) 22.3671 0.754852
\(879\) 4.79493 0.161729
\(880\) −0.610974 −0.0205959
\(881\) −7.06710 −0.238097 −0.119048 0.992888i \(-0.537984\pi\)
−0.119048 + 0.992888i \(0.537984\pi\)
\(882\) 1.25555 0.0422766
\(883\) 0.499721 0.0168170 0.00840848 0.999965i \(-0.497323\pi\)
0.00840848 + 0.999965i \(0.497323\pi\)
\(884\) 5.22357 0.175688
\(885\) −2.99851 −0.100794
\(886\) 0.372988 0.0125308
\(887\) −38.7309 −1.30046 −0.650228 0.759739i \(-0.725327\pi\)
−0.650228 + 0.759739i \(0.725327\pi\)
\(888\) 8.14049 0.273177
\(889\) 33.4709 1.12258
\(890\) 2.36659 0.0793283
\(891\) 1.78117 0.0596716
\(892\) 2.60140 0.0871012
\(893\) −27.9310 −0.934676
\(894\) −23.8755 −0.798517
\(895\) 5.41224 0.180911
\(896\) −2.39676 −0.0800701
\(897\) −2.02924 −0.0677542
\(898\) 9.55036 0.318700
\(899\) −17.9604 −0.599014
\(900\) −4.88234 −0.162745
\(901\) 31.9038 1.06287
\(902\) 0.282575 0.00940871
\(903\) −6.04309 −0.201102
\(904\) 6.21659 0.206761
\(905\) −8.23602 −0.273775
\(906\) 2.70857 0.0899861
\(907\) 21.6452 0.718718 0.359359 0.933199i \(-0.382995\pi\)
0.359359 + 0.933199i \(0.382995\pi\)
\(908\) 0.153884 0.00510682
\(909\) −5.96014 −0.197685
\(910\) 0.822130 0.0272533
\(911\) −6.10346 −0.202217 −0.101108 0.994875i \(-0.532239\pi\)
−0.101108 + 0.994875i \(0.532239\pi\)
\(912\) 3.94679 0.130691
\(913\) −17.1959 −0.569101
\(914\) −8.43762 −0.279092
\(915\) 1.09681 0.0362593
\(916\) 10.5875 0.349820
\(917\) −8.75223 −0.289024
\(918\) 5.22357 0.172404
\(919\) 48.6005 1.60318 0.801591 0.597873i \(-0.203987\pi\)
0.801591 + 0.597873i \(0.203987\pi\)
\(920\) 0.696063 0.0229485
\(921\) −19.2898 −0.635619
\(922\) −0.632681 −0.0208363
\(923\) 7.09487 0.233531
\(924\) −4.26904 −0.140441
\(925\) −39.7446 −1.30679
\(926\) 18.6646 0.613357
\(927\) −1.00000 −0.0328443
\(928\) 7.94406 0.260777
\(929\) 40.9361 1.34307 0.671535 0.740972i \(-0.265635\pi\)
0.671535 + 0.740972i \(0.265635\pi\)
\(930\) −0.775515 −0.0254301
\(931\) 4.95539 0.162406
\(932\) 12.4251 0.406998
\(933\) −21.2835 −0.696789
\(934\) −30.1881 −0.987786
\(935\) −3.19147 −0.104372
\(936\) −1.00000 −0.0326860
\(937\) −20.7616 −0.678253 −0.339126 0.940741i \(-0.610131\pi\)
−0.339126 + 0.940741i \(0.610131\pi\)
\(938\) −18.8109 −0.614197
\(939\) 11.9002 0.388347
\(940\) −2.42750 −0.0791762
\(941\) −32.0125 −1.04358 −0.521789 0.853075i \(-0.674735\pi\)
−0.521789 + 0.853075i \(0.674735\pi\)
\(942\) −7.82902 −0.255083
\(943\) −0.321928 −0.0104834
\(944\) −8.74158 −0.284514
\(945\) 0.822130 0.0267439
\(946\) −4.49098 −0.146014
\(947\) −4.85737 −0.157843 −0.0789216 0.996881i \(-0.525148\pi\)
−0.0789216 + 0.996881i \(0.525148\pi\)
\(948\) 3.08307 0.100133
\(949\) 17.0401 0.553146
\(950\) −19.2696 −0.625188
\(951\) 21.6584 0.702323
\(952\) −12.5196 −0.405764
\(953\) 4.91973 0.159366 0.0796828 0.996820i \(-0.474609\pi\)
0.0796828 + 0.996820i \(0.474609\pi\)
\(954\) −6.10766 −0.197743
\(955\) 3.21349 0.103986
\(956\) −10.7307 −0.347057
\(957\) 14.1498 0.457397
\(958\) −3.54195 −0.114435
\(959\) −1.33789 −0.0432027
\(960\) 0.343018 0.0110708
\(961\) −25.8885 −0.835113
\(962\) −8.14049 −0.262460
\(963\) 0.562160 0.0181154
\(964\) −8.36037 −0.269269
\(965\) −3.12192 −0.100498
\(966\) 4.86359 0.156483
\(967\) −37.5026 −1.20600 −0.603001 0.797741i \(-0.706028\pi\)
−0.603001 + 0.797741i \(0.706028\pi\)
\(968\) 7.82742 0.251583
\(969\) 20.6163 0.662293
\(970\) 2.68746 0.0862891
\(971\) −18.4820 −0.593114 −0.296557 0.955015i \(-0.595839\pi\)
−0.296557 + 0.955015i \(0.595839\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 46.1018 1.47796
\(974\) −23.7808 −0.761985
\(975\) 4.88234 0.156360
\(976\) 3.19752 0.102350
\(977\) 10.7816 0.344934 0.172467 0.985015i \(-0.444826\pi\)
0.172467 + 0.985015i \(0.444826\pi\)
\(978\) −6.11690 −0.195597
\(979\) 12.2889 0.392755
\(980\) 0.430676 0.0137574
\(981\) −7.55415 −0.241186
\(982\) −21.9035 −0.698969
\(983\) 47.5111 1.51537 0.757684 0.652622i \(-0.226331\pi\)
0.757684 + 0.652622i \(0.226331\pi\)
\(984\) −0.158645 −0.00505742
\(985\) 6.08508 0.193887
\(986\) 41.4964 1.32151
\(987\) −16.9616 −0.539893
\(988\) −3.94679 −0.125564
\(989\) 5.11644 0.162693
\(990\) 0.610974 0.0194180
\(991\) 36.7868 1.16857 0.584285 0.811549i \(-0.301375\pi\)
0.584285 + 0.811549i \(0.301375\pi\)
\(992\) −2.26086 −0.0717824
\(993\) −26.4528 −0.839454
\(994\) −17.0047 −0.539356
\(995\) −1.18136 −0.0374515
\(996\) 9.65424 0.305906
\(997\) 5.44037 0.172298 0.0861491 0.996282i \(-0.472544\pi\)
0.0861491 + 0.996282i \(0.472544\pi\)
\(998\) −12.0837 −0.382502
\(999\) −8.14049 −0.257554
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 8034.2.a.z.1.8 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.8 14 1.1 even 1 trivial