Properties

Label 4235.2.a.bc.1.3
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.8166452560\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.288385.1
Defining polynomial: \(x^{5} - 7 x^{3} - 2 x^{2} + 10 x + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.300703\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.300703 q^{2} +2.68115 q^{3} -1.90958 q^{4} +1.00000 q^{5} +0.806228 q^{6} -1.00000 q^{7} -1.17562 q^{8} +4.18855 q^{9} +O(q^{10})\) \(q+0.300703 q^{2} +2.68115 q^{3} -1.90958 q^{4} +1.00000 q^{5} +0.806228 q^{6} -1.00000 q^{7} -1.17562 q^{8} +4.18855 q^{9} +0.300703 q^{10} -5.11986 q^{12} -3.71222 q^{13} -0.300703 q^{14} +2.68115 q^{15} +3.46564 q^{16} -2.57780 q^{17} +1.25951 q^{18} -4.28255 q^{19} -1.90958 q^{20} -2.68115 q^{21} -6.57257 q^{23} -3.15201 q^{24} +1.00000 q^{25} -1.11628 q^{26} +3.18667 q^{27} +1.90958 q^{28} +5.39695 q^{29} +0.806228 q^{30} +0.259508 q^{31} +3.39337 q^{32} -0.775151 q^{34} -1.00000 q^{35} -7.99836 q^{36} +8.65646 q^{37} -1.28778 q^{38} -9.95302 q^{39} -1.17562 q^{40} -5.56152 q^{41} -0.806228 q^{42} -7.87197 q^{43} +4.18855 q^{45} -1.97639 q^{46} -7.41642 q^{47} +9.29190 q^{48} +1.00000 q^{49} +0.300703 q^{50} -6.91145 q^{51} +7.08878 q^{52} -3.03108 q^{53} +0.958241 q^{54} +1.17562 q^{56} -11.4822 q^{57} +1.62288 q^{58} +9.55630 q^{59} -5.11986 q^{60} -4.10335 q^{61} +0.0780349 q^{62} -4.18855 q^{63} -5.91089 q^{64} -3.71222 q^{65} -12.5164 q^{67} +4.92250 q^{68} -17.6220 q^{69} -0.300703 q^{70} -5.34471 q^{71} -4.92415 q^{72} -3.79876 q^{73} +2.60302 q^{74} +2.68115 q^{75} +8.17787 q^{76} -2.99290 q^{78} -16.1672 q^{79} +3.46564 q^{80} -4.02171 q^{81} -1.67237 q^{82} +6.53457 q^{83} +5.11986 q^{84} -2.57780 q^{85} -2.36712 q^{86} +14.4700 q^{87} -14.7054 q^{89} +1.25951 q^{90} +3.71222 q^{91} +12.5508 q^{92} +0.695780 q^{93} -2.23014 q^{94} -4.28255 q^{95} +9.09813 q^{96} -3.69766 q^{97} +0.300703 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{3} + 4 q^{4} + 5 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9} + O(q^{10}) \) \( 5 q + 2 q^{3} + 4 q^{4} + 5 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9} + 3 q^{12} - 6 q^{13} + 2 q^{15} - 6 q^{16} - 2 q^{17} + 3 q^{18} - 7 q^{19} + 4 q^{20} - 2 q^{21} + 5 q^{23} - 8 q^{24} + 5 q^{25} + 9 q^{26} - 7 q^{27} - 4 q^{28} - 11 q^{29} - 9 q^{30} - 2 q^{31} - 7 q^{32} + 8 q^{34} - 5 q^{35} + q^{36} + 2 q^{37} - 19 q^{38} - 2 q^{39} - 6 q^{40} - 13 q^{41} + 9 q^{42} - 10 q^{43} + 7 q^{45} - 2 q^{46} - 2 q^{47} + 32 q^{48} + 5 q^{49} - 30 q^{51} + 8 q^{52} - 14 q^{53} - 16 q^{54} + 6 q^{56} - 6 q^{57} + 4 q^{58} + 6 q^{59} + 3 q^{60} - 20 q^{61} + 15 q^{62} - 7 q^{63} - 6 q^{64} - 6 q^{65} - 9 q^{67} - 3 q^{68} - 30 q^{69} - 10 q^{71} - 38 q^{72} - 15 q^{73} + 19 q^{74} + 2 q^{75} + 5 q^{76} + 21 q^{78} - 23 q^{79} - 6 q^{80} + 13 q^{81} - 16 q^{82} - 8 q^{83} - 3 q^{84} - 2 q^{85} - 29 q^{86} + 22 q^{87} + 7 q^{89} + 3 q^{90} + 6 q^{91} + 26 q^{92} - 45 q^{93} + 14 q^{94} - 7 q^{95} + 18 q^{96} + 21 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.300703 0.212629 0.106315 0.994333i \(-0.466095\pi\)
0.106315 + 0.994333i \(0.466095\pi\)
\(3\) 2.68115 1.54796 0.773980 0.633210i \(-0.218263\pi\)
0.773980 + 0.633210i \(0.218263\pi\)
\(4\) −1.90958 −0.954789
\(5\) 1.00000 0.447214
\(6\) 0.806228 0.329141
\(7\) −1.00000 −0.377964
\(8\) −1.17562 −0.415645
\(9\) 4.18855 1.39618
\(10\) 0.300703 0.0950906
\(11\) 0 0
\(12\) −5.11986 −1.47798
\(13\) −3.71222 −1.02959 −0.514793 0.857315i \(-0.672131\pi\)
−0.514793 + 0.857315i \(0.672131\pi\)
\(14\) −0.300703 −0.0803662
\(15\) 2.68115 0.692269
\(16\) 3.46564 0.866411
\(17\) −2.57780 −0.625208 −0.312604 0.949884i \(-0.601201\pi\)
−0.312604 + 0.949884i \(0.601201\pi\)
\(18\) 1.25951 0.296869
\(19\) −4.28255 −0.982485 −0.491242 0.871023i \(-0.663457\pi\)
−0.491242 + 0.871023i \(0.663457\pi\)
\(20\) −1.90958 −0.426995
\(21\) −2.68115 −0.585074
\(22\) 0 0
\(23\) −6.57257 −1.37048 −0.685238 0.728319i \(-0.740302\pi\)
−0.685238 + 0.728319i \(0.740302\pi\)
\(24\) −3.15201 −0.643402
\(25\) 1.00000 0.200000
\(26\) −1.11628 −0.218920
\(27\) 3.18667 0.613275
\(28\) 1.90958 0.360876
\(29\) 5.39695 1.00219 0.501095 0.865393i \(-0.332931\pi\)
0.501095 + 0.865393i \(0.332931\pi\)
\(30\) 0.806228 0.147197
\(31\) 0.259508 0.0466091 0.0233045 0.999728i \(-0.492581\pi\)
0.0233045 + 0.999728i \(0.492581\pi\)
\(32\) 3.39337 0.599869
\(33\) 0 0
\(34\) −0.775151 −0.132937
\(35\) −1.00000 −0.169031
\(36\) −7.99836 −1.33306
\(37\) 8.65646 1.42311 0.711557 0.702629i \(-0.247990\pi\)
0.711557 + 0.702629i \(0.247990\pi\)
\(38\) −1.28778 −0.208905
\(39\) −9.95302 −1.59376
\(40\) −1.17562 −0.185882
\(41\) −5.56152 −0.868564 −0.434282 0.900777i \(-0.642998\pi\)
−0.434282 + 0.900777i \(0.642998\pi\)
\(42\) −0.806228 −0.124404
\(43\) −7.87197 −1.20046 −0.600232 0.799826i \(-0.704925\pi\)
−0.600232 + 0.799826i \(0.704925\pi\)
\(44\) 0 0
\(45\) 4.18855 0.624392
\(46\) −1.97639 −0.291403
\(47\) −7.41642 −1.08180 −0.540898 0.841088i \(-0.681915\pi\)
−0.540898 + 0.841088i \(0.681915\pi\)
\(48\) 9.29190 1.34117
\(49\) 1.00000 0.142857
\(50\) 0.300703 0.0425258
\(51\) −6.91145 −0.967797
\(52\) 7.08878 0.983037
\(53\) −3.03108 −0.416351 −0.208175 0.978092i \(-0.566752\pi\)
−0.208175 + 0.978092i \(0.566752\pi\)
\(54\) 0.958241 0.130400
\(55\) 0 0
\(56\) 1.17562 0.157099
\(57\) −11.4822 −1.52085
\(58\) 1.62288 0.213094
\(59\) 9.55630 1.24412 0.622062 0.782968i \(-0.286295\pi\)
0.622062 + 0.782968i \(0.286295\pi\)
\(60\) −5.11986 −0.660971
\(61\) −4.10335 −0.525380 −0.262690 0.964880i \(-0.584610\pi\)
−0.262690 + 0.964880i \(0.584610\pi\)
\(62\) 0.0780349 0.00991044
\(63\) −4.18855 −0.527707
\(64\) −5.91089 −0.738861
\(65\) −3.71222 −0.460445
\(66\) 0 0
\(67\) −12.5164 −1.52912 −0.764562 0.644550i \(-0.777045\pi\)
−0.764562 + 0.644550i \(0.777045\pi\)
\(68\) 4.92250 0.596941
\(69\) −17.6220 −2.12144
\(70\) −0.300703 −0.0359409
\(71\) −5.34471 −0.634300 −0.317150 0.948375i \(-0.602726\pi\)
−0.317150 + 0.948375i \(0.602726\pi\)
\(72\) −4.92415 −0.580316
\(73\) −3.79876 −0.444611 −0.222306 0.974977i \(-0.571358\pi\)
−0.222306 + 0.974977i \(0.571358\pi\)
\(74\) 2.60302 0.302595
\(75\) 2.68115 0.309592
\(76\) 8.17787 0.938066
\(77\) 0 0
\(78\) −2.99290 −0.338879
\(79\) −16.1672 −1.81895 −0.909475 0.415758i \(-0.863516\pi\)
−0.909475 + 0.415758i \(0.863516\pi\)
\(80\) 3.46564 0.387471
\(81\) −4.02171 −0.446856
\(82\) −1.67237 −0.184682
\(83\) 6.53457 0.717262 0.358631 0.933479i \(-0.383244\pi\)
0.358631 + 0.933479i \(0.383244\pi\)
\(84\) 5.11986 0.558622
\(85\) −2.57780 −0.279601
\(86\) −2.36712 −0.255253
\(87\) 14.4700 1.55135
\(88\) 0 0
\(89\) −14.7054 −1.55877 −0.779383 0.626548i \(-0.784467\pi\)
−0.779383 + 0.626548i \(0.784467\pi\)
\(90\) 1.25951 0.132764
\(91\) 3.71222 0.389147
\(92\) 12.5508 1.30852
\(93\) 0.695780 0.0721490
\(94\) −2.23014 −0.230021
\(95\) −4.28255 −0.439381
\(96\) 9.09813 0.928574
\(97\) −3.69766 −0.375440 −0.187720 0.982223i \(-0.560110\pi\)
−0.187720 + 0.982223i \(0.560110\pi\)
\(98\) 0.300703 0.0303756
\(99\) 0 0
\(100\) −1.90958 −0.190958
\(101\) 0.578032 0.0575163 0.0287582 0.999586i \(-0.490845\pi\)
0.0287582 + 0.999586i \(0.490845\pi\)
\(102\) −2.07829 −0.205782
\(103\) 19.5745 1.92873 0.964367 0.264568i \(-0.0852292\pi\)
0.964367 + 0.264568i \(0.0852292\pi\)
\(104\) 4.36417 0.427942
\(105\) −2.68115 −0.261653
\(106\) −0.911454 −0.0885282
\(107\) 13.7510 1.32936 0.664680 0.747128i \(-0.268568\pi\)
0.664680 + 0.747128i \(0.268568\pi\)
\(108\) −6.08520 −0.585549
\(109\) −16.7462 −1.60399 −0.801997 0.597328i \(-0.796229\pi\)
−0.801997 + 0.597328i \(0.796229\pi\)
\(110\) 0 0
\(111\) 23.2092 2.20292
\(112\) −3.46564 −0.327472
\(113\) 0.440329 0.0414226 0.0207113 0.999785i \(-0.493407\pi\)
0.0207113 + 0.999785i \(0.493407\pi\)
\(114\) −3.45272 −0.323376
\(115\) −6.57257 −0.612896
\(116\) −10.3059 −0.956879
\(117\) −15.5488 −1.43749
\(118\) 2.87361 0.264537
\(119\) 2.57780 0.236306
\(120\) −3.15201 −0.287738
\(121\) 0 0
\(122\) −1.23389 −0.111711
\(123\) −14.9113 −1.34450
\(124\) −0.495551 −0.0445018
\(125\) 1.00000 0.0894427
\(126\) −1.25951 −0.112206
\(127\) 11.2721 1.00024 0.500119 0.865957i \(-0.333290\pi\)
0.500119 + 0.865957i \(0.333290\pi\)
\(128\) −8.56416 −0.756972
\(129\) −21.1059 −1.85827
\(130\) −1.11628 −0.0979039
\(131\) 13.1216 1.14644 0.573218 0.819403i \(-0.305695\pi\)
0.573218 + 0.819403i \(0.305695\pi\)
\(132\) 0 0
\(133\) 4.28255 0.371344
\(134\) −3.76372 −0.325136
\(135\) 3.18667 0.274265
\(136\) 3.03051 0.259864
\(137\) 8.60021 0.734765 0.367383 0.930070i \(-0.380254\pi\)
0.367383 + 0.930070i \(0.380254\pi\)
\(138\) −5.29900 −0.451081
\(139\) −8.21253 −0.696577 −0.348289 0.937387i \(-0.613237\pi\)
−0.348289 + 0.937387i \(0.613237\pi\)
\(140\) 1.90958 0.161389
\(141\) −19.8845 −1.67458
\(142\) −1.60717 −0.134871
\(143\) 0 0
\(144\) 14.5160 1.20967
\(145\) 5.39695 0.448193
\(146\) −1.14230 −0.0945372
\(147\) 2.68115 0.221137
\(148\) −16.5302 −1.35877
\(149\) −12.8156 −1.04989 −0.524946 0.851135i \(-0.675915\pi\)
−0.524946 + 0.851135i \(0.675915\pi\)
\(150\) 0.806228 0.0658283
\(151\) 13.3123 1.08334 0.541670 0.840592i \(-0.317792\pi\)
0.541670 + 0.840592i \(0.317792\pi\)
\(152\) 5.03466 0.408365
\(153\) −10.7972 −0.872904
\(154\) 0 0
\(155\) 0.259508 0.0208442
\(156\) 19.0061 1.52170
\(157\) 4.90529 0.391485 0.195742 0.980655i \(-0.437288\pi\)
0.195742 + 0.980655i \(0.437288\pi\)
\(158\) −4.86152 −0.386762
\(159\) −8.12676 −0.644494
\(160\) 3.39337 0.268270
\(161\) 6.57257 0.517991
\(162\) −1.20934 −0.0950147
\(163\) 22.8933 1.79315 0.896573 0.442897i \(-0.146049\pi\)
0.896573 + 0.442897i \(0.146049\pi\)
\(164\) 10.6202 0.829295
\(165\) 0 0
\(166\) 1.96496 0.152511
\(167\) −3.06761 −0.237379 −0.118689 0.992931i \(-0.537869\pi\)
−0.118689 + 0.992931i \(0.537869\pi\)
\(168\) 3.15201 0.243183
\(169\) 0.780609 0.0600468
\(170\) −0.775151 −0.0594514
\(171\) −17.9377 −1.37173
\(172\) 15.0321 1.14619
\(173\) −16.7396 −1.27269 −0.636345 0.771405i \(-0.719555\pi\)
−0.636345 + 0.771405i \(0.719555\pi\)
\(174\) 4.35118 0.329862
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 25.6218 1.92586
\(178\) −4.42194 −0.331439
\(179\) −9.63091 −0.719848 −0.359924 0.932982i \(-0.617197\pi\)
−0.359924 + 0.932982i \(0.617197\pi\)
\(180\) −7.99836 −0.596162
\(181\) 7.92382 0.588973 0.294486 0.955656i \(-0.404851\pi\)
0.294486 + 0.955656i \(0.404851\pi\)
\(182\) 1.11628 0.0827439
\(183\) −11.0017 −0.813268
\(184\) 7.72686 0.569631
\(185\) 8.65646 0.636436
\(186\) 0.209223 0.0153410
\(187\) 0 0
\(188\) 14.1622 1.03289
\(189\) −3.18667 −0.231796
\(190\) −1.28778 −0.0934251
\(191\) −12.3576 −0.894165 −0.447083 0.894493i \(-0.647537\pi\)
−0.447083 + 0.894493i \(0.647537\pi\)
\(192\) −15.8480 −1.14373
\(193\) 0.316278 0.0227662 0.0113831 0.999935i \(-0.496377\pi\)
0.0113831 + 0.999935i \(0.496377\pi\)
\(194\) −1.11190 −0.0798295
\(195\) −9.95302 −0.712750
\(196\) −1.90958 −0.136398
\(197\) 0.671102 0.0478140 0.0239070 0.999714i \(-0.492389\pi\)
0.0239070 + 0.999714i \(0.492389\pi\)
\(198\) 0 0
\(199\) −2.70355 −0.191650 −0.0958250 0.995398i \(-0.530549\pi\)
−0.0958250 + 0.995398i \(0.530549\pi\)
\(200\) −1.17562 −0.0831290
\(201\) −33.5584 −2.36702
\(202\) 0.173816 0.0122296
\(203\) −5.39695 −0.378792
\(204\) 13.1980 0.924042
\(205\) −5.56152 −0.388434
\(206\) 5.88611 0.410105
\(207\) −27.5295 −1.91344
\(208\) −12.8652 −0.892044
\(209\) 0 0
\(210\) −0.806228 −0.0556351
\(211\) 15.4069 1.06065 0.530327 0.847793i \(-0.322069\pi\)
0.530327 + 0.847793i \(0.322069\pi\)
\(212\) 5.78808 0.397527
\(213\) −14.3299 −0.981872
\(214\) 4.13497 0.282660
\(215\) −7.87197 −0.536864
\(216\) −3.74632 −0.254905
\(217\) −0.259508 −0.0176166
\(218\) −5.03563 −0.341056
\(219\) −10.1850 −0.688241
\(220\) 0 0
\(221\) 9.56936 0.643705
\(222\) 6.97909 0.468406
\(223\) 15.7486 1.05460 0.527302 0.849678i \(-0.323204\pi\)
0.527302 + 0.849678i \(0.323204\pi\)
\(224\) −3.39337 −0.226729
\(225\) 4.18855 0.279237
\(226\) 0.132408 0.00880765
\(227\) 20.1493 1.33735 0.668677 0.743553i \(-0.266861\pi\)
0.668677 + 0.743553i \(0.266861\pi\)
\(228\) 21.9261 1.45209
\(229\) −1.54976 −0.102411 −0.0512057 0.998688i \(-0.516306\pi\)
−0.0512057 + 0.998688i \(0.516306\pi\)
\(230\) −1.97639 −0.130319
\(231\) 0 0
\(232\) −6.34477 −0.416555
\(233\) −18.2298 −1.19427 −0.597137 0.802139i \(-0.703695\pi\)
−0.597137 + 0.802139i \(0.703695\pi\)
\(234\) −4.67558 −0.305652
\(235\) −7.41642 −0.483794
\(236\) −18.2485 −1.18788
\(237\) −43.3466 −2.81566
\(238\) 0.775151 0.0502456
\(239\) −27.6129 −1.78613 −0.893066 0.449926i \(-0.851450\pi\)
−0.893066 + 0.449926i \(0.851450\pi\)
\(240\) 9.29190 0.599789
\(241\) −1.26342 −0.0813840 −0.0406920 0.999172i \(-0.512956\pi\)
−0.0406920 + 0.999172i \(0.512956\pi\)
\(242\) 0 0
\(243\) −20.3428 −1.30499
\(244\) 7.83566 0.501627
\(245\) 1.00000 0.0638877
\(246\) −4.48386 −0.285880
\(247\) 15.8978 1.01155
\(248\) −0.305084 −0.0193728
\(249\) 17.5201 1.11029
\(250\) 0.300703 0.0190181
\(251\) 17.4533 1.10164 0.550821 0.834623i \(-0.314315\pi\)
0.550821 + 0.834623i \(0.314315\pi\)
\(252\) 7.99836 0.503849
\(253\) 0 0
\(254\) 3.38955 0.212680
\(255\) −6.91145 −0.432812
\(256\) 9.24651 0.577907
\(257\) −5.27117 −0.328807 −0.164403 0.986393i \(-0.552570\pi\)
−0.164403 + 0.986393i \(0.552570\pi\)
\(258\) −6.34660 −0.395122
\(259\) −8.65646 −0.537886
\(260\) 7.08878 0.439628
\(261\) 22.6054 1.39924
\(262\) 3.94569 0.243766
\(263\) −28.0145 −1.72745 −0.863724 0.503964i \(-0.831874\pi\)
−0.863724 + 0.503964i \(0.831874\pi\)
\(264\) 0 0
\(265\) −3.03108 −0.186198
\(266\) 1.28778 0.0789586
\(267\) −39.4272 −2.41291
\(268\) 23.9011 1.45999
\(269\) 7.95674 0.485131 0.242566 0.970135i \(-0.422011\pi\)
0.242566 + 0.970135i \(0.422011\pi\)
\(270\) 0.958241 0.0583167
\(271\) −3.85304 −0.234056 −0.117028 0.993129i \(-0.537337\pi\)
−0.117028 + 0.993129i \(0.537337\pi\)
\(272\) −8.93373 −0.541687
\(273\) 9.95302 0.602384
\(274\) 2.58611 0.156232
\(275\) 0 0
\(276\) 33.6506 2.02553
\(277\) 11.3954 0.684681 0.342340 0.939576i \(-0.388780\pi\)
0.342340 + 0.939576i \(0.388780\pi\)
\(278\) −2.46953 −0.148113
\(279\) 1.08696 0.0650748
\(280\) 1.17562 0.0702568
\(281\) 13.0913 0.780963 0.390482 0.920611i \(-0.372308\pi\)
0.390482 + 0.920611i \(0.372308\pi\)
\(282\) −5.97933 −0.356064
\(283\) 15.1856 0.902690 0.451345 0.892350i \(-0.350944\pi\)
0.451345 + 0.892350i \(0.350944\pi\)
\(284\) 10.2061 0.605623
\(285\) −11.4822 −0.680144
\(286\) 0 0
\(287\) 5.56152 0.328286
\(288\) 14.2133 0.837527
\(289\) −10.3550 −0.609115
\(290\) 1.62288 0.0952987
\(291\) −9.91396 −0.581167
\(292\) 7.25403 0.424510
\(293\) −16.0307 −0.936526 −0.468263 0.883589i \(-0.655120\pi\)
−0.468263 + 0.883589i \(0.655120\pi\)
\(294\) 0.806228 0.0470202
\(295\) 9.55630 0.556389
\(296\) −10.1767 −0.591510
\(297\) 0 0
\(298\) −3.85368 −0.223238
\(299\) 24.3989 1.41102
\(300\) −5.11986 −0.295595
\(301\) 7.87197 0.453733
\(302\) 4.00304 0.230349
\(303\) 1.54979 0.0890331
\(304\) −14.8418 −0.851235
\(305\) −4.10335 −0.234957
\(306\) −3.24676 −0.185605
\(307\) 16.8799 0.963387 0.481694 0.876340i \(-0.340022\pi\)
0.481694 + 0.876340i \(0.340022\pi\)
\(308\) 0 0
\(309\) 52.4821 2.98561
\(310\) 0.0780349 0.00443209
\(311\) −8.28047 −0.469542 −0.234771 0.972051i \(-0.575434\pi\)
−0.234771 + 0.972051i \(0.575434\pi\)
\(312\) 11.7010 0.662437
\(313\) −33.6412 −1.90151 −0.950755 0.309942i \(-0.899690\pi\)
−0.950755 + 0.309942i \(0.899690\pi\)
\(314\) 1.47503 0.0832410
\(315\) −4.18855 −0.235998
\(316\) 30.8725 1.73671
\(317\) 18.4021 1.03356 0.516782 0.856117i \(-0.327130\pi\)
0.516782 + 0.856117i \(0.327130\pi\)
\(318\) −2.44374 −0.137038
\(319\) 0 0
\(320\) −5.91089 −0.330429
\(321\) 36.8685 2.05780
\(322\) 1.97639 0.110140
\(323\) 11.0396 0.614257
\(324\) 7.67976 0.426654
\(325\) −3.71222 −0.205917
\(326\) 6.88409 0.381275
\(327\) −44.8990 −2.48292
\(328\) 6.53824 0.361014
\(329\) 7.41642 0.408880
\(330\) 0 0
\(331\) 2.16461 0.118978 0.0594889 0.998229i \(-0.481053\pi\)
0.0594889 + 0.998229i \(0.481053\pi\)
\(332\) −12.4783 −0.684834
\(333\) 36.2580 1.98693
\(334\) −0.922440 −0.0504737
\(335\) −12.5164 −0.683845
\(336\) −9.29190 −0.506915
\(337\) −19.8686 −1.08231 −0.541155 0.840923i \(-0.682013\pi\)
−0.541155 + 0.840923i \(0.682013\pi\)
\(338\) 0.234731 0.0127677
\(339\) 1.18059 0.0641206
\(340\) 4.92250 0.266960
\(341\) 0 0
\(342\) −5.39391 −0.291669
\(343\) −1.00000 −0.0539949
\(344\) 9.25445 0.498966
\(345\) −17.6220 −0.948739
\(346\) −5.03365 −0.270611
\(347\) −17.1645 −0.921437 −0.460718 0.887546i \(-0.652408\pi\)
−0.460718 + 0.887546i \(0.652408\pi\)
\(348\) −27.6316 −1.48121
\(349\) 23.0282 1.23267 0.616337 0.787483i \(-0.288616\pi\)
0.616337 + 0.787483i \(0.288616\pi\)
\(350\) −0.300703 −0.0160732
\(351\) −11.8296 −0.631420
\(352\) 0 0
\(353\) −0.804658 −0.0428276 −0.0214138 0.999771i \(-0.506817\pi\)
−0.0214138 + 0.999771i \(0.506817\pi\)
\(354\) 7.70456 0.409493
\(355\) −5.34471 −0.283668
\(356\) 28.0810 1.48829
\(357\) 6.91145 0.365793
\(358\) −2.89604 −0.153061
\(359\) 26.4792 1.39752 0.698761 0.715356i \(-0.253735\pi\)
0.698761 + 0.715356i \(0.253735\pi\)
\(360\) −4.92415 −0.259525
\(361\) −0.659744 −0.0347234
\(362\) 2.38271 0.125233
\(363\) 0 0
\(364\) −7.08878 −0.371553
\(365\) −3.79876 −0.198836
\(366\) −3.30824 −0.172924
\(367\) −22.0904 −1.15311 −0.576555 0.817059i \(-0.695603\pi\)
−0.576555 + 0.817059i \(0.695603\pi\)
\(368\) −22.7782 −1.18740
\(369\) −23.2947 −1.21267
\(370\) 2.60302 0.135325
\(371\) 3.03108 0.157366
\(372\) −1.32865 −0.0688871
\(373\) 21.9021 1.13405 0.567023 0.823702i \(-0.308095\pi\)
0.567023 + 0.823702i \(0.308095\pi\)
\(374\) 0 0
\(375\) 2.68115 0.138454
\(376\) 8.71889 0.449643
\(377\) −20.0347 −1.03184
\(378\) −0.958241 −0.0492866
\(379\) −20.6422 −1.06032 −0.530159 0.847898i \(-0.677868\pi\)
−0.530159 + 0.847898i \(0.677868\pi\)
\(380\) 8.17787 0.419516
\(381\) 30.2222 1.54833
\(382\) −3.71597 −0.190125
\(383\) −13.0515 −0.666899 −0.333450 0.942768i \(-0.608213\pi\)
−0.333450 + 0.942768i \(0.608213\pi\)
\(384\) −22.9618 −1.17176
\(385\) 0 0
\(386\) 0.0951056 0.00484075
\(387\) −32.9721 −1.67607
\(388\) 7.06096 0.358466
\(389\) 5.56270 0.282040 0.141020 0.990007i \(-0.454962\pi\)
0.141020 + 0.990007i \(0.454962\pi\)
\(390\) −2.99290 −0.151551
\(391\) 16.9428 0.856832
\(392\) −1.17562 −0.0593778
\(393\) 35.1808 1.77464
\(394\) 0.201802 0.0101667
\(395\) −16.1672 −0.813459
\(396\) 0 0
\(397\) 23.2318 1.16597 0.582986 0.812482i \(-0.301884\pi\)
0.582986 + 0.812482i \(0.301884\pi\)
\(398\) −0.812967 −0.0407503
\(399\) 11.4822 0.574827
\(400\) 3.46564 0.173282
\(401\) 17.2555 0.861700 0.430850 0.902424i \(-0.358214\pi\)
0.430850 + 0.902424i \(0.358214\pi\)
\(402\) −10.0911 −0.503298
\(403\) −0.963353 −0.0479881
\(404\) −1.10380 −0.0549160
\(405\) −4.02171 −0.199840
\(406\) −1.62288 −0.0805421
\(407\) 0 0
\(408\) 8.12525 0.402260
\(409\) −19.8948 −0.983733 −0.491867 0.870671i \(-0.663685\pi\)
−0.491867 + 0.870671i \(0.663685\pi\)
\(410\) −1.67237 −0.0825923
\(411\) 23.0584 1.13739
\(412\) −37.3791 −1.84153
\(413\) −9.55630 −0.470235
\(414\) −8.27821 −0.406852
\(415\) 6.53457 0.320769
\(416\) −12.5970 −0.617617
\(417\) −22.0190 −1.07827
\(418\) 0 0
\(419\) −35.1138 −1.71542 −0.857711 0.514132i \(-0.828114\pi\)
−0.857711 + 0.514132i \(0.828114\pi\)
\(420\) 5.11986 0.249824
\(421\) −19.2133 −0.936397 −0.468198 0.883623i \(-0.655097\pi\)
−0.468198 + 0.883623i \(0.655097\pi\)
\(422\) 4.63290 0.225526
\(423\) −31.0640 −1.51038
\(424\) 3.56340 0.173054
\(425\) −2.57780 −0.125042
\(426\) −4.30906 −0.208774
\(427\) 4.10335 0.198575
\(428\) −26.2586 −1.26926
\(429\) 0 0
\(430\) −2.36712 −0.114153
\(431\) −25.0503 −1.20663 −0.603315 0.797503i \(-0.706154\pi\)
−0.603315 + 0.797503i \(0.706154\pi\)
\(432\) 11.0439 0.531348
\(433\) −0.137355 −0.00660088 −0.00330044 0.999995i \(-0.501051\pi\)
−0.00330044 + 0.999995i \(0.501051\pi\)
\(434\) −0.0780349 −0.00374580
\(435\) 14.4700 0.693785
\(436\) 31.9781 1.53148
\(437\) 28.1474 1.34647
\(438\) −3.06267 −0.146340
\(439\) 1.33995 0.0639523 0.0319761 0.999489i \(-0.489820\pi\)
0.0319761 + 0.999489i \(0.489820\pi\)
\(440\) 0 0
\(441\) 4.18855 0.199455
\(442\) 2.87753 0.136870
\(443\) 24.6896 1.17304 0.586520 0.809935i \(-0.300498\pi\)
0.586520 + 0.809935i \(0.300498\pi\)
\(444\) −44.3199 −2.10333
\(445\) −14.7054 −0.697101
\(446\) 4.73565 0.224239
\(447\) −34.3604 −1.62519
\(448\) 5.91089 0.279263
\(449\) −4.81239 −0.227111 −0.113555 0.993532i \(-0.536224\pi\)
−0.113555 + 0.993532i \(0.536224\pi\)
\(450\) 1.25951 0.0593738
\(451\) 0 0
\(452\) −0.840842 −0.0395499
\(453\) 35.6922 1.67697
\(454\) 6.05894 0.284360
\(455\) 3.71222 0.174032
\(456\) 13.4987 0.632133
\(457\) 9.68909 0.453236 0.226618 0.973984i \(-0.427233\pi\)
0.226618 + 0.973984i \(0.427233\pi\)
\(458\) −0.466019 −0.0217756
\(459\) −8.21460 −0.383425
\(460\) 12.5508 0.585186
\(461\) 27.0491 1.25980 0.629900 0.776676i \(-0.283096\pi\)
0.629900 + 0.776676i \(0.283096\pi\)
\(462\) 0 0
\(463\) −40.6992 −1.89145 −0.945725 0.324967i \(-0.894647\pi\)
−0.945725 + 0.324967i \(0.894647\pi\)
\(464\) 18.7039 0.868307
\(465\) 0.695780 0.0322660
\(466\) −5.48176 −0.253937
\(467\) 24.3282 1.12577 0.562887 0.826534i \(-0.309691\pi\)
0.562887 + 0.826534i \(0.309691\pi\)
\(468\) 29.6917 1.37250
\(469\) 12.5164 0.577954
\(470\) −2.23014 −0.102869
\(471\) 13.1518 0.606003
\(472\) −11.2346 −0.517114
\(473\) 0 0
\(474\) −13.0344 −0.598692
\(475\) −4.28255 −0.196497
\(476\) −4.92250 −0.225623
\(477\) −12.6958 −0.581301
\(478\) −8.30329 −0.379783
\(479\) 19.0405 0.869982 0.434991 0.900435i \(-0.356752\pi\)
0.434991 + 0.900435i \(0.356752\pi\)
\(480\) 9.09813 0.415271
\(481\) −32.1347 −1.46522
\(482\) −0.379914 −0.0173046
\(483\) 17.6220 0.801830
\(484\) 0 0
\(485\) −3.69766 −0.167902
\(486\) −6.11714 −0.277479
\(487\) 36.9451 1.67414 0.837070 0.547095i \(-0.184267\pi\)
0.837070 + 0.547095i \(0.184267\pi\)
\(488\) 4.82398 0.218372
\(489\) 61.3804 2.77572
\(490\) 0.300703 0.0135844
\(491\) 33.2950 1.50258 0.751291 0.659971i \(-0.229432\pi\)
0.751291 + 0.659971i \(0.229432\pi\)
\(492\) 28.4742 1.28372
\(493\) −13.9123 −0.626576
\(494\) 4.78051 0.215085
\(495\) 0 0
\(496\) 0.899363 0.0403826
\(497\) 5.34471 0.239743
\(498\) 5.26835 0.236081
\(499\) 7.54204 0.337628 0.168814 0.985648i \(-0.446006\pi\)
0.168814 + 0.985648i \(0.446006\pi\)
\(500\) −1.90958 −0.0853989
\(501\) −8.22472 −0.367453
\(502\) 5.24826 0.234241
\(503\) 23.1318 1.03140 0.515699 0.856770i \(-0.327532\pi\)
0.515699 + 0.856770i \(0.327532\pi\)
\(504\) 4.92415 0.219339
\(505\) 0.578032 0.0257221
\(506\) 0 0
\(507\) 2.09293 0.0929502
\(508\) −21.5250 −0.955016
\(509\) −38.8695 −1.72286 −0.861429 0.507877i \(-0.830430\pi\)
−0.861429 + 0.507877i \(0.830430\pi\)
\(510\) −2.07829 −0.0920284
\(511\) 3.79876 0.168047
\(512\) 19.9088 0.879852
\(513\) −13.6471 −0.602534
\(514\) −1.58506 −0.0699138
\(515\) 19.5745 0.862556
\(516\) 40.3033 1.77426
\(517\) 0 0
\(518\) −2.60302 −0.114370
\(519\) −44.8814 −1.97007
\(520\) 4.36417 0.191381
\(521\) 11.3257 0.496189 0.248095 0.968736i \(-0.420196\pi\)
0.248095 + 0.968736i \(0.420196\pi\)
\(522\) 6.79751 0.297519
\(523\) 33.7878 1.47744 0.738719 0.674014i \(-0.235431\pi\)
0.738719 + 0.674014i \(0.235431\pi\)
\(524\) −25.0566 −1.09460
\(525\) −2.68115 −0.117015
\(526\) −8.42404 −0.367306
\(527\) −0.668960 −0.0291404
\(528\) 0 0
\(529\) 20.1987 0.878206
\(530\) −0.911454 −0.0395910
\(531\) 40.0270 1.73702
\(532\) −8.17787 −0.354556
\(533\) 20.6456 0.894261
\(534\) −11.8559 −0.513054
\(535\) 13.7510 0.594508
\(536\) 14.7146 0.635572
\(537\) −25.8219 −1.11430
\(538\) 2.39262 0.103153
\(539\) 0 0
\(540\) −6.08520 −0.261865
\(541\) −23.7564 −1.02137 −0.510685 0.859768i \(-0.670608\pi\)
−0.510685 + 0.859768i \(0.670608\pi\)
\(542\) −1.15862 −0.0497670
\(543\) 21.2449 0.911707
\(544\) −8.74742 −0.375043
\(545\) −16.7462 −0.717328
\(546\) 2.99290 0.128084
\(547\) −0.643211 −0.0275017 −0.0137508 0.999905i \(-0.504377\pi\)
−0.0137508 + 0.999905i \(0.504377\pi\)
\(548\) −16.4228 −0.701546
\(549\) −17.1871 −0.733527
\(550\) 0 0
\(551\) −23.1127 −0.984636
\(552\) 20.7168 0.881767
\(553\) 16.1672 0.687499
\(554\) 3.42662 0.145583
\(555\) 23.2092 0.985178
\(556\) 15.6825 0.665084
\(557\) 41.7961 1.77096 0.885479 0.464680i \(-0.153831\pi\)
0.885479 + 0.464680i \(0.153831\pi\)
\(558\) 0.326853 0.0138368
\(559\) 29.2225 1.23598
\(560\) −3.46564 −0.146450
\(561\) 0 0
\(562\) 3.93660 0.166055
\(563\) −17.6396 −0.743419 −0.371709 0.928349i \(-0.621228\pi\)
−0.371709 + 0.928349i \(0.621228\pi\)
\(564\) 37.9710 1.59887
\(565\) 0.440329 0.0185248
\(566\) 4.56635 0.191938
\(567\) 4.02171 0.168896
\(568\) 6.28335 0.263644
\(569\) −30.8220 −1.29213 −0.646064 0.763284i \(-0.723586\pi\)
−0.646064 + 0.763284i \(0.723586\pi\)
\(570\) −3.45272 −0.144618
\(571\) 38.2527 1.60082 0.800412 0.599450i \(-0.204614\pi\)
0.800412 + 0.599450i \(0.204614\pi\)
\(572\) 0 0
\(573\) −33.1326 −1.38413
\(574\) 1.67237 0.0698032
\(575\) −6.57257 −0.274095
\(576\) −24.7580 −1.03159
\(577\) 31.4711 1.31016 0.655080 0.755559i \(-0.272635\pi\)
0.655080 + 0.755559i \(0.272635\pi\)
\(578\) −3.11377 −0.129516
\(579\) 0.847987 0.0352411
\(580\) −10.3059 −0.427929
\(581\) −6.53457 −0.271100
\(582\) −2.98116 −0.123573
\(583\) 0 0
\(584\) 4.46590 0.184800
\(585\) −15.5488 −0.642865
\(586\) −4.82049 −0.199133
\(587\) −44.3220 −1.82936 −0.914682 0.404174i \(-0.867559\pi\)
−0.914682 + 0.404174i \(0.867559\pi\)
\(588\) −5.11986 −0.211139
\(589\) −1.11136 −0.0457927
\(590\) 2.87361 0.118305
\(591\) 1.79932 0.0740143
\(592\) 30.0002 1.23300
\(593\) 47.1384 1.93574 0.967871 0.251447i \(-0.0809066\pi\)
0.967871 + 0.251447i \(0.0809066\pi\)
\(594\) 0 0
\(595\) 2.57780 0.105679
\(596\) 24.4723 1.00243
\(597\) −7.24863 −0.296667
\(598\) 7.33681 0.300024
\(599\) 25.2437 1.03143 0.515716 0.856760i \(-0.327526\pi\)
0.515716 + 0.856760i \(0.327526\pi\)
\(600\) −3.15201 −0.128680
\(601\) 19.4806 0.794629 0.397314 0.917683i \(-0.369942\pi\)
0.397314 + 0.917683i \(0.369942\pi\)
\(602\) 2.36712 0.0964767
\(603\) −52.4256 −2.13494
\(604\) −25.4209 −1.03436
\(605\) 0 0
\(606\) 0.466026 0.0189310
\(607\) 34.6714 1.40727 0.703634 0.710563i \(-0.251560\pi\)
0.703634 + 0.710563i \(0.251560\pi\)
\(608\) −14.5323 −0.589362
\(609\) −14.4700 −0.586355
\(610\) −1.23389 −0.0499587
\(611\) 27.5314 1.11380
\(612\) 20.6181 0.833439
\(613\) −28.2413 −1.14065 −0.570327 0.821418i \(-0.693183\pi\)
−0.570327 + 0.821418i \(0.693183\pi\)
\(614\) 5.07584 0.204844
\(615\) −14.9113 −0.601280
\(616\) 0 0
\(617\) 17.7098 0.712968 0.356484 0.934301i \(-0.383975\pi\)
0.356484 + 0.934301i \(0.383975\pi\)
\(618\) 15.7815 0.634826
\(619\) −14.8725 −0.597778 −0.298889 0.954288i \(-0.596616\pi\)
−0.298889 + 0.954288i \(0.596616\pi\)
\(620\) −0.495551 −0.0199018
\(621\) −20.9446 −0.840480
\(622\) −2.48996 −0.0998383
\(623\) 14.7054 0.589158
\(624\) −34.4936 −1.38085
\(625\) 1.00000 0.0400000
\(626\) −10.1160 −0.404316
\(627\) 0 0
\(628\) −9.36703 −0.373785
\(629\) −22.3146 −0.889742
\(630\) −1.25951 −0.0501800
\(631\) 18.2768 0.727589 0.363795 0.931479i \(-0.381481\pi\)
0.363795 + 0.931479i \(0.381481\pi\)
\(632\) 19.0065 0.756037
\(633\) 41.3082 1.64185
\(634\) 5.53356 0.219766
\(635\) 11.2721 0.447320
\(636\) 15.5187 0.615356
\(637\) −3.71222 −0.147084
\(638\) 0 0
\(639\) −22.3866 −0.885599
\(640\) −8.56416 −0.338528
\(641\) −19.6674 −0.776816 −0.388408 0.921487i \(-0.626975\pi\)
−0.388408 + 0.921487i \(0.626975\pi\)
\(642\) 11.0865 0.437547
\(643\) 15.5620 0.613705 0.306853 0.951757i \(-0.400724\pi\)
0.306853 + 0.951757i \(0.400724\pi\)
\(644\) −12.5508 −0.494572
\(645\) −21.1059 −0.831044
\(646\) 3.31963 0.130609
\(647\) −42.4134 −1.66744 −0.833721 0.552186i \(-0.813794\pi\)
−0.833721 + 0.552186i \(0.813794\pi\)
\(648\) 4.72801 0.185734
\(649\) 0 0
\(650\) −1.11628 −0.0437840
\(651\) −0.695780 −0.0272698
\(652\) −43.7166 −1.71208
\(653\) 28.7703 1.12587 0.562933 0.826502i \(-0.309673\pi\)
0.562933 + 0.826502i \(0.309673\pi\)
\(654\) −13.5013 −0.527941
\(655\) 13.1216 0.512702
\(656\) −19.2743 −0.752533
\(657\) −15.9113 −0.620758
\(658\) 2.23014 0.0869398
\(659\) 25.8627 1.00747 0.503734 0.863859i \(-0.331959\pi\)
0.503734 + 0.863859i \(0.331959\pi\)
\(660\) 0 0
\(661\) −42.9119 −1.66908 −0.834539 0.550949i \(-0.814266\pi\)
−0.834539 + 0.550949i \(0.814266\pi\)
\(662\) 0.650905 0.0252981
\(663\) 25.6569 0.996430
\(664\) −7.68218 −0.298126
\(665\) 4.28255 0.166070
\(666\) 10.9029 0.422478
\(667\) −35.4719 −1.37348
\(668\) 5.85785 0.226647
\(669\) 42.2243 1.63249
\(670\) −3.76372 −0.145405
\(671\) 0 0
\(672\) −9.09813 −0.350968
\(673\) −2.43117 −0.0937146 −0.0468573 0.998902i \(-0.514921\pi\)
−0.0468573 + 0.998902i \(0.514921\pi\)
\(674\) −5.97454 −0.230131
\(675\) 3.18667 0.122655
\(676\) −1.49063 −0.0573321
\(677\) 3.00264 0.115401 0.0577004 0.998334i \(-0.481623\pi\)
0.0577004 + 0.998334i \(0.481623\pi\)
\(678\) 0.355005 0.0136339
\(679\) 3.69766 0.141903
\(680\) 3.03051 0.116215
\(681\) 54.0232 2.07017
\(682\) 0 0
\(683\) 0.170585 0.00652724 0.00326362 0.999995i \(-0.498961\pi\)
0.00326362 + 0.999995i \(0.498961\pi\)
\(684\) 34.2534 1.30971
\(685\) 8.60021 0.328597
\(686\) −0.300703 −0.0114809
\(687\) −4.15515 −0.158529
\(688\) −27.2814 −1.04009
\(689\) 11.2520 0.428669
\(690\) −5.29900 −0.201729
\(691\) 31.3760 1.19360 0.596800 0.802390i \(-0.296439\pi\)
0.596800 + 0.802390i \(0.296439\pi\)
\(692\) 31.9656 1.21515
\(693\) 0 0
\(694\) −5.16140 −0.195924
\(695\) −8.21253 −0.311519
\(696\) −17.0113 −0.644810
\(697\) 14.3365 0.543033
\(698\) 6.92466 0.262102
\(699\) −48.8768 −1.84869
\(700\) 1.90958 0.0721753
\(701\) −39.6624 −1.49803 −0.749014 0.662554i \(-0.769473\pi\)
−0.749014 + 0.662554i \(0.769473\pi\)
\(702\) −3.55721 −0.134258
\(703\) −37.0718 −1.39819
\(704\) 0 0
\(705\) −19.8845 −0.748894
\(706\) −0.241963 −0.00910639
\(707\) −0.578032 −0.0217391
\(708\) −48.9269 −1.83879
\(709\) −9.87077 −0.370705 −0.185352 0.982672i \(-0.559343\pi\)
−0.185352 + 0.982672i \(0.559343\pi\)
\(710\) −1.60717 −0.0603160
\(711\) −67.7170 −2.53959
\(712\) 17.2879 0.647893
\(713\) −1.70564 −0.0638767
\(714\) 2.07829 0.0777782
\(715\) 0 0
\(716\) 18.3910 0.687303
\(717\) −74.0343 −2.76486
\(718\) 7.96239 0.297154
\(719\) 19.3147 0.720318 0.360159 0.932891i \(-0.382722\pi\)
0.360159 + 0.932891i \(0.382722\pi\)
\(720\) 14.5160 0.540980
\(721\) −19.5745 −0.728993
\(722\) −0.198387 −0.00738319
\(723\) −3.38741 −0.125979
\(724\) −15.1311 −0.562345
\(725\) 5.39695 0.200438
\(726\) 0 0
\(727\) −40.6707 −1.50839 −0.754197 0.656648i \(-0.771974\pi\)
−0.754197 + 0.656648i \(0.771974\pi\)
\(728\) −4.36417 −0.161747
\(729\) −42.4769 −1.57322
\(730\) −1.14230 −0.0422783
\(731\) 20.2923 0.750539
\(732\) 21.0086 0.776499
\(733\) −0.738739 −0.0272860 −0.0136430 0.999907i \(-0.504343\pi\)
−0.0136430 + 0.999907i \(0.504343\pi\)
\(734\) −6.64264 −0.245184
\(735\) 2.68115 0.0988956
\(736\) −22.3032 −0.822106
\(737\) 0 0
\(738\) −7.00478 −0.257850
\(739\) −48.4080 −1.78072 −0.890358 0.455261i \(-0.849546\pi\)
−0.890358 + 0.455261i \(0.849546\pi\)
\(740\) −16.5302 −0.607662
\(741\) 42.6243 1.56584
\(742\) 0.911454 0.0334605
\(743\) −29.3346 −1.07618 −0.538092 0.842886i \(-0.680855\pi\)
−0.538092 + 0.842886i \(0.680855\pi\)
\(744\) −0.817974 −0.0299884
\(745\) −12.8156 −0.469526
\(746\) 6.58602 0.241131
\(747\) 27.3704 1.00143
\(748\) 0 0
\(749\) −13.7510 −0.502451
\(750\) 0.806228 0.0294393
\(751\) 52.7991 1.92667 0.963334 0.268305i \(-0.0864634\pi\)
0.963334 + 0.268305i \(0.0864634\pi\)
\(752\) −25.7026 −0.937279
\(753\) 46.7949 1.70530
\(754\) −6.02449 −0.219399
\(755\) 13.3123 0.484484
\(756\) 6.08520 0.221317
\(757\) −25.9231 −0.942191 −0.471096 0.882082i \(-0.656141\pi\)
−0.471096 + 0.882082i \(0.656141\pi\)
\(758\) −6.20716 −0.225454
\(759\) 0 0
\(760\) 5.03466 0.182626
\(761\) −30.3537 −1.10032 −0.550160 0.835059i \(-0.685433\pi\)
−0.550160 + 0.835059i \(0.685433\pi\)
\(762\) 9.08789 0.329220
\(763\) 16.7462 0.606253
\(764\) 23.5978 0.853739
\(765\) −10.7972 −0.390375
\(766\) −3.92462 −0.141802
\(767\) −35.4751 −1.28093
\(768\) 24.7913 0.894577
\(769\) −6.95155 −0.250679 −0.125340 0.992114i \(-0.540002\pi\)
−0.125340 + 0.992114i \(0.540002\pi\)
\(770\) 0 0
\(771\) −14.1328 −0.508980
\(772\) −0.603957 −0.0217369
\(773\) −10.3245 −0.371347 −0.185674 0.982611i \(-0.559447\pi\)
−0.185674 + 0.982611i \(0.559447\pi\)
\(774\) −9.91481 −0.356380
\(775\) 0.259508 0.00932182
\(776\) 4.34704 0.156050
\(777\) −23.2092 −0.832627
\(778\) 1.67272 0.0599699
\(779\) 23.8175 0.853351
\(780\) 19.0061 0.680526
\(781\) 0 0
\(782\) 5.09474 0.182187
\(783\) 17.1983 0.614618
\(784\) 3.46564 0.123773
\(785\) 4.90529 0.175077
\(786\) 10.5790 0.377340
\(787\) 6.94028 0.247394 0.123697 0.992320i \(-0.460525\pi\)
0.123697 + 0.992320i \(0.460525\pi\)
\(788\) −1.28152 −0.0456523
\(789\) −75.1110 −2.67402
\(790\) −4.86152 −0.172965
\(791\) −0.440329 −0.0156563
\(792\) 0 0
\(793\) 15.2326 0.540924
\(794\) 6.98588 0.247920
\(795\) −8.12676 −0.288227
\(796\) 5.16265 0.182985
\(797\) −47.4444 −1.68057 −0.840283 0.542149i \(-0.817611\pi\)
−0.840283 + 0.542149i \(0.817611\pi\)
\(798\) 3.45272 0.122225
\(799\) 19.1180 0.676347
\(800\) 3.39337 0.119974
\(801\) −61.5941 −2.17632
\(802\) 5.18879 0.183222
\(803\) 0 0
\(804\) 64.0823 2.26001
\(805\) 6.57257 0.231653
\(806\) −0.289683 −0.0102037
\(807\) 21.3332 0.750964
\(808\) −0.679547 −0.0239064
\(809\) 18.7678 0.659841 0.329921 0.944009i \(-0.392978\pi\)
0.329921 + 0.944009i \(0.392978\pi\)
\(810\) −1.20934 −0.0424918
\(811\) 12.7534 0.447832 0.223916 0.974608i \(-0.428116\pi\)
0.223916 + 0.974608i \(0.428116\pi\)
\(812\) 10.3059 0.361666
\(813\) −10.3306 −0.362309
\(814\) 0 0
\(815\) 22.8933 0.801919
\(816\) −23.9526 −0.838510
\(817\) 33.7121 1.17944
\(818\) −5.98242 −0.209170
\(819\) 15.5488 0.543320
\(820\) 10.6202 0.370872
\(821\) −25.9816 −0.906764 −0.453382 0.891316i \(-0.649783\pi\)
−0.453382 + 0.891316i \(0.649783\pi\)
\(822\) 6.93373 0.241842
\(823\) −17.1651 −0.598338 −0.299169 0.954200i \(-0.596709\pi\)
−0.299169 + 0.954200i \(0.596709\pi\)
\(824\) −23.0122 −0.801668
\(825\) 0 0
\(826\) −2.87361 −0.0999856
\(827\) 10.4088 0.361951 0.180975 0.983488i \(-0.442075\pi\)
0.180975 + 0.983488i \(0.442075\pi\)
\(828\) 52.5698 1.82693
\(829\) −2.59315 −0.0900640 −0.0450320 0.998986i \(-0.514339\pi\)
−0.0450320 + 0.998986i \(0.514339\pi\)
\(830\) 1.96496 0.0682049
\(831\) 30.5526 1.05986
\(832\) 21.9425 0.760721
\(833\) −2.57780 −0.0893154
\(834\) −6.62117 −0.229272
\(835\) −3.06761 −0.106159
\(836\) 0 0
\(837\) 0.826968 0.0285842
\(838\) −10.5588 −0.364748
\(839\) −30.9632 −1.06897 −0.534484 0.845179i \(-0.679494\pi\)
−0.534484 + 0.845179i \(0.679494\pi\)
\(840\) 3.15201 0.108755
\(841\) 0.127103 0.00438286
\(842\) −5.77748 −0.199105
\(843\) 35.0998 1.20890
\(844\) −29.4207 −1.01270
\(845\) 0.780609 0.0268538
\(846\) −9.34104 −0.321151
\(847\) 0 0
\(848\) −10.5046 −0.360731
\(849\) 40.7148 1.39733
\(850\) −0.775151 −0.0265875
\(851\) −56.8952 −1.95034
\(852\) 27.3641 0.937480
\(853\) −0.713929 −0.0244445 −0.0122222 0.999925i \(-0.503891\pi\)
−0.0122222 + 0.999925i \(0.503891\pi\)
\(854\) 1.23389 0.0422228
\(855\) −17.9377 −0.613456
\(856\) −16.1660 −0.552542
\(857\) 43.4895 1.48557 0.742786 0.669529i \(-0.233504\pi\)
0.742786 + 0.669529i \(0.233504\pi\)
\(858\) 0 0
\(859\) −23.4136 −0.798862 −0.399431 0.916763i \(-0.630792\pi\)
−0.399431 + 0.916763i \(0.630792\pi\)
\(860\) 15.0321 0.512591
\(861\) 14.9113 0.508174
\(862\) −7.53269 −0.256564
\(863\) −0.482491 −0.0164242 −0.00821210 0.999966i \(-0.502614\pi\)
−0.00821210 + 0.999966i \(0.502614\pi\)
\(864\) 10.8136 0.367885
\(865\) −16.7396 −0.569164
\(866\) −0.0413032 −0.00140354
\(867\) −27.7632 −0.942887
\(868\) 0.495551 0.0168201
\(869\) 0 0
\(870\) 4.35118 0.147519
\(871\) 46.4637 1.57436
\(872\) 19.6872 0.666692
\(873\) −15.4878 −0.524183
\(874\) 8.46400 0.286299
\(875\) −1.00000 −0.0338062
\(876\) 19.4491 0.657125
\(877\) 11.2489 0.379848 0.189924 0.981799i \(-0.439176\pi\)
0.189924 + 0.981799i \(0.439176\pi\)
\(878\) 0.402927 0.0135981
\(879\) −42.9808 −1.44971
\(880\) 0 0
\(881\) −34.5391 −1.16365 −0.581826 0.813313i \(-0.697662\pi\)
−0.581826 + 0.813313i \(0.697662\pi\)
\(882\) 1.25951 0.0424099
\(883\) 1.23309 0.0414967 0.0207484 0.999785i \(-0.493395\pi\)
0.0207484 + 0.999785i \(0.493395\pi\)
\(884\) −18.2734 −0.614602
\(885\) 25.6218 0.861269
\(886\) 7.42424 0.249422
\(887\) −43.8838 −1.47347 −0.736737 0.676180i \(-0.763634\pi\)
−0.736737 + 0.676180i \(0.763634\pi\)
\(888\) −27.2853 −0.915634
\(889\) −11.2721 −0.378054
\(890\) −4.42194 −0.148224
\(891\) 0 0
\(892\) −30.0732 −1.00692
\(893\) 31.7612 1.06285
\(894\) −10.3323 −0.345563
\(895\) −9.63091 −0.321926
\(896\) 8.56416 0.286109
\(897\) 65.4170 2.18421
\(898\) −1.44710 −0.0482904
\(899\) 1.40055 0.0467111
\(900\) −7.99836 −0.266612
\(901\) 7.81350 0.260306
\(902\) 0 0
\(903\) 21.1059 0.702360
\(904\) −0.517660 −0.0172171
\(905\) 7.92382 0.263397
\(906\) 10.7328 0.356572
\(907\) −34.7542 −1.15399 −0.576997 0.816747i \(-0.695775\pi\)
−0.576997 + 0.816747i \(0.695775\pi\)
\(908\) −38.4766 −1.27689
\(909\) 2.42112 0.0803033
\(910\) 1.11628 0.0370042
\(911\) −41.4019 −1.37171 −0.685854 0.727739i \(-0.740571\pi\)
−0.685854 + 0.727739i \(0.740571\pi\)
\(912\) −39.7930 −1.31768
\(913\) 0 0
\(914\) 2.91354 0.0963712
\(915\) −11.0017 −0.363704
\(916\) 2.95940 0.0977812
\(917\) −13.1216 −0.433312
\(918\) −2.47015 −0.0815272
\(919\) 4.92654 0.162511 0.0812557 0.996693i \(-0.474107\pi\)
0.0812557 + 0.996693i \(0.474107\pi\)
\(920\) 7.72686 0.254747
\(921\) 45.2575 1.49129
\(922\) 8.13373 0.267870
\(923\) 19.8408 0.653066
\(924\) 0 0
\(925\) 8.65646 0.284623
\(926\) −12.2384 −0.402177
\(927\) 81.9888 2.69287
\(928\) 18.3139 0.601182
\(929\) 46.1669 1.51469 0.757343 0.653017i \(-0.226497\pi\)
0.757343 + 0.653017i \(0.226497\pi\)
\(930\) 0.209223 0.00686069
\(931\) −4.28255 −0.140355
\(932\) 34.8112 1.14028
\(933\) −22.2012 −0.726833
\(934\) 7.31555 0.239372
\(935\) 0 0
\(936\) 18.2795 0.597485
\(937\) 32.4292 1.05942 0.529708 0.848180i \(-0.322301\pi\)
0.529708 + 0.848180i \(0.322301\pi\)
\(938\) 3.76372 0.122890
\(939\) −90.1969 −2.94346
\(940\) 14.1622 0.461921
\(941\) −36.0325 −1.17463 −0.587313 0.809360i \(-0.699814\pi\)
−0.587313 + 0.809360i \(0.699814\pi\)
\(942\) 3.95478 0.128854
\(943\) 36.5535 1.19035
\(944\) 33.1187 1.07792
\(945\) −3.18667 −0.103662
\(946\) 0 0
\(947\) 18.8299 0.611890 0.305945 0.952049i \(-0.401028\pi\)
0.305945 + 0.952049i \(0.401028\pi\)
\(948\) 82.7737 2.68836
\(949\) 14.1018 0.457765
\(950\) −1.28778 −0.0417810
\(951\) 49.3387 1.59992
\(952\) −3.03051 −0.0982195
\(953\) 34.5345 1.11868 0.559342 0.828937i \(-0.311054\pi\)
0.559342 + 0.828937i \(0.311054\pi\)
\(954\) −3.81767 −0.123602
\(955\) −12.3576 −0.399883
\(956\) 52.7290 1.70538
\(957\) 0 0
\(958\) 5.72553 0.184983
\(959\) −8.60021 −0.277715
\(960\) −15.8480 −0.511491
\(961\) −30.9327 −0.997828
\(962\) −9.66300 −0.311548
\(963\) 57.5968 1.85603
\(964\) 2.41260 0.0777046
\(965\) 0.316278 0.0101813
\(966\) 5.29900 0.170492
\(967\) −28.2598 −0.908773 −0.454387 0.890805i \(-0.650142\pi\)
−0.454387 + 0.890805i \(0.650142\pi\)
\(968\) 0 0
\(969\) 29.5987 0.950846
\(970\) −1.11190 −0.0357008
\(971\) 53.2128 1.70768 0.853840 0.520535i \(-0.174267\pi\)
0.853840 + 0.520535i \(0.174267\pi\)
\(972\) 38.8462 1.24599
\(973\) 8.21253 0.263282
\(974\) 11.1095 0.355971
\(975\) −9.95302 −0.318752
\(976\) −14.2207 −0.455195
\(977\) −47.1433 −1.50825 −0.754125 0.656731i \(-0.771939\pi\)
−0.754125 + 0.656731i \(0.771939\pi\)
\(978\) 18.4573 0.590198
\(979\) 0 0
\(980\) −1.90958 −0.0609992
\(981\) −70.1422 −2.23947
\(982\) 10.0119 0.319493
\(983\) 15.9235 0.507881 0.253941 0.967220i \(-0.418273\pi\)
0.253941 + 0.967220i \(0.418273\pi\)
\(984\) 17.5300 0.558836
\(985\) 0.671102 0.0213831
\(986\) −4.18345 −0.133228
\(987\) 19.8845 0.632931
\(988\) −30.3581 −0.965819
\(989\) 51.7391 1.64521
\(990\) 0 0
\(991\) −47.8509 −1.52003 −0.760017 0.649904i \(-0.774809\pi\)
−0.760017 + 0.649904i \(0.774809\pi\)
\(992\) 0.880608 0.0279593
\(993\) 5.80364 0.184173
\(994\) 1.60717 0.0509763
\(995\) −2.70355 −0.0857084
\(996\) −33.4561 −1.06010
\(997\) 1.60157 0.0507222 0.0253611 0.999678i \(-0.491926\pi\)
0.0253611 + 0.999678i \(0.491926\pi\)
\(998\) 2.26791 0.0717895
\(999\) 27.5853 0.872761
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 4235.2.a.bc.1.3 5
11.10 odd 2 4235.2.a.bd.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.bc.1.3 5 1.1 even 1 trivial
4235.2.a.bd.1.3 yes 5 11.10 odd 2