Properties

Label 3549.2.a.h.1.2
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Defining polynomial: \(x^{3} - x^{2} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.273891\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.27389 q^{2} -1.00000 q^{3} -0.377203 q^{4} +1.10331 q^{5} +1.27389 q^{6} -1.00000 q^{7} +3.02830 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.27389 q^{2} -1.00000 q^{3} -0.377203 q^{4} +1.10331 q^{5} +1.27389 q^{6} -1.00000 q^{7} +3.02830 q^{8} +1.00000 q^{9} -1.40550 q^{10} -0.348907 q^{11} +0.377203 q^{12} +1.27389 q^{14} -1.10331 q^{15} -3.10331 q^{16} +0.726109 q^{17} -1.27389 q^{18} +2.30219 q^{19} -0.416173 q^{20} +1.00000 q^{21} +0.444469 q^{22} +0.0750160 q^{23} -3.02830 q^{24} -3.78270 q^{25} -1.00000 q^{27} +0.377203 q^{28} +0.480515 q^{29} +1.40550 q^{30} -6.85772 q^{31} -2.10331 q^{32} +0.348907 q^{33} -0.924984 q^{34} -1.10331 q^{35} -0.377203 q^{36} -1.10331 q^{37} -2.93273 q^{38} +3.34116 q^{40} +3.09556 q^{41} -1.27389 q^{42} -4.62280 q^{43} +0.131609 q^{44} +1.10331 q^{45} -0.0955622 q^{46} +4.70769 q^{47} +3.10331 q^{48} +1.00000 q^{49} +4.81875 q^{50} -0.726109 q^{51} +5.54778 q^{53} +1.27389 q^{54} -0.384953 q^{55} -3.02830 q^{56} -2.30219 q^{57} -0.612124 q^{58} -3.92498 q^{59} +0.416173 q^{60} +5.54778 q^{61} +8.73598 q^{62} -1.00000 q^{63} +8.88601 q^{64} -0.444469 q^{66} +8.12386 q^{67} -0.273891 q^{68} -0.0750160 q^{69} +1.40550 q^{70} -9.52936 q^{71} +3.02830 q^{72} -10.4338 q^{73} +1.40550 q^{74} +3.78270 q^{75} -0.868391 q^{76} +0.348907 q^{77} +17.1054 q^{79} -3.42392 q^{80} +1.00000 q^{81} -3.94341 q^{82} -11.9143 q^{83} -0.377203 q^{84} +0.801125 q^{85} +5.88894 q^{86} -0.480515 q^{87} -1.05659 q^{88} -11.9610 q^{89} -1.40550 q^{90} -0.0282963 q^{92} +6.85772 q^{93} -5.99708 q^{94} +2.54003 q^{95} +2.10331 q^{96} +10.1033 q^{97} -1.27389 q^{98} -0.348907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + 13q^{10} - 8q^{11} - 4q^{12} + 2q^{14} - 6q^{16} + 4q^{17} - 2q^{18} - 7q^{19} - 13q^{20} + 3q^{21} + q^{22} + 9q^{23} + 3q^{24} + 11q^{25} - 3q^{27} - 4q^{28} - 7q^{29} - 13q^{30} - 7q^{31} - 3q^{32} + 8q^{33} + 6q^{34} + 4q^{36} - 4q^{38} + 13q^{40} + 2q^{41} - 2q^{42} - 19q^{43} - 15q^{44} + 7q^{46} - 17q^{47} + 6q^{48} + 3q^{49} - 16q^{50} - 4q^{51} + 13q^{53} + 2q^{54} + 3q^{56} + 7q^{57} + 22q^{58} - 3q^{59} + 13q^{60} + 13q^{61} - 17q^{62} - 3q^{63} + q^{64} - q^{66} + 5q^{67} + q^{68} - 9q^{69} - 13q^{70} + 8q^{71} - 3q^{72} - 2q^{73} - 13q^{74} - 11q^{75} - 18q^{76} + 8q^{77} - q^{79} - 26q^{80} + 3q^{81} - 36q^{82} + 2q^{83} + 4q^{84} + 13q^{85} + 17q^{86} + 7q^{87} + 21q^{88} - 19q^{89} + 13q^{90} + 12q^{92} + 7q^{93} + 7q^{94} + 3q^{96} + 27q^{97} - 2q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27389 −0.900777 −0.450388 0.892833i \(-0.648714\pi\)
−0.450388 + 0.892833i \(0.648714\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.377203 −0.188601
\(5\) 1.10331 0.493416 0.246708 0.969090i \(-0.420651\pi\)
0.246708 + 0.969090i \(0.420651\pi\)
\(6\) 1.27389 0.520064
\(7\) −1.00000 −0.377964
\(8\) 3.02830 1.07066
\(9\) 1.00000 0.333333
\(10\) −1.40550 −0.444458
\(11\) −0.348907 −0.105199 −0.0525996 0.998616i \(-0.516751\pi\)
−0.0525996 + 0.998616i \(0.516751\pi\)
\(12\) 0.377203 0.108889
\(13\) 0 0
\(14\) 1.27389 0.340462
\(15\) −1.10331 −0.284874
\(16\) −3.10331 −0.775828
\(17\) 0.726109 0.176107 0.0880537 0.996116i \(-0.471935\pi\)
0.0880537 + 0.996116i \(0.471935\pi\)
\(18\) −1.27389 −0.300259
\(19\) 2.30219 0.528158 0.264079 0.964501i \(-0.414932\pi\)
0.264079 + 0.964501i \(0.414932\pi\)
\(20\) −0.416173 −0.0930590
\(21\) 1.00000 0.218218
\(22\) 0.444469 0.0947611
\(23\) 0.0750160 0.0156419 0.00782096 0.999969i \(-0.497510\pi\)
0.00782096 + 0.999969i \(0.497510\pi\)
\(24\) −3.02830 −0.618148
\(25\) −3.78270 −0.756540
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0.377203 0.0712846
\(29\) 0.480515 0.0892294 0.0446147 0.999004i \(-0.485794\pi\)
0.0446147 + 0.999004i \(0.485794\pi\)
\(30\) 1.40550 0.256608
\(31\) −6.85772 −1.23168 −0.615841 0.787870i \(-0.711184\pi\)
−0.615841 + 0.787870i \(0.711184\pi\)
\(32\) −2.10331 −0.371817
\(33\) 0.348907 0.0607368
\(34\) −0.924984 −0.158633
\(35\) −1.10331 −0.186494
\(36\) −0.377203 −0.0628671
\(37\) −1.10331 −0.181383 −0.0906917 0.995879i \(-0.528908\pi\)
−0.0906917 + 0.995879i \(0.528908\pi\)
\(38\) −2.93273 −0.475752
\(39\) 0 0
\(40\) 3.34116 0.528283
\(41\) 3.09556 0.483446 0.241723 0.970345i \(-0.422288\pi\)
0.241723 + 0.970345i \(0.422288\pi\)
\(42\) −1.27389 −0.196566
\(43\) −4.62280 −0.704970 −0.352485 0.935817i \(-0.614663\pi\)
−0.352485 + 0.935817i \(0.614663\pi\)
\(44\) 0.131609 0.0198407
\(45\) 1.10331 0.164472
\(46\) −0.0955622 −0.0140899
\(47\) 4.70769 0.686687 0.343343 0.939210i \(-0.388441\pi\)
0.343343 + 0.939210i \(0.388441\pi\)
\(48\) 3.10331 0.447925
\(49\) 1.00000 0.142857
\(50\) 4.81875 0.681474
\(51\) −0.726109 −0.101676
\(52\) 0 0
\(53\) 5.54778 0.762046 0.381023 0.924565i \(-0.375572\pi\)
0.381023 + 0.924565i \(0.375572\pi\)
\(54\) 1.27389 0.173355
\(55\) −0.384953 −0.0519070
\(56\) −3.02830 −0.404673
\(57\) −2.30219 −0.304932
\(58\) −0.612124 −0.0803758
\(59\) −3.92498 −0.510989 −0.255495 0.966810i \(-0.582238\pi\)
−0.255495 + 0.966810i \(0.582238\pi\)
\(60\) 0.416173 0.0537276
\(61\) 5.54778 0.710321 0.355160 0.934805i \(-0.384426\pi\)
0.355160 + 0.934805i \(0.384426\pi\)
\(62\) 8.73598 1.10947
\(63\) −1.00000 −0.125988
\(64\) 8.88601 1.11075
\(65\) 0 0
\(66\) −0.444469 −0.0547103
\(67\) 8.12386 0.992487 0.496244 0.868183i \(-0.334712\pi\)
0.496244 + 0.868183i \(0.334712\pi\)
\(68\) −0.273891 −0.0332141
\(69\) −0.0750160 −0.00903087
\(70\) 1.40550 0.167989
\(71\) −9.52936 −1.13093 −0.565463 0.824773i \(-0.691303\pi\)
−0.565463 + 0.824773i \(0.691303\pi\)
\(72\) 3.02830 0.356888
\(73\) −10.4338 −1.22118 −0.610592 0.791946i \(-0.709068\pi\)
−0.610592 + 0.791946i \(0.709068\pi\)
\(74\) 1.40550 0.163386
\(75\) 3.78270 0.436789
\(76\) −0.868391 −0.0996113
\(77\) 0.348907 0.0397616
\(78\) 0 0
\(79\) 17.1054 1.92451 0.962256 0.272146i \(-0.0877335\pi\)
0.962256 + 0.272146i \(0.0877335\pi\)
\(80\) −3.42392 −0.382806
\(81\) 1.00000 0.111111
\(82\) −3.94341 −0.435477
\(83\) −11.9143 −1.30777 −0.653883 0.756596i \(-0.726861\pi\)
−0.653883 + 0.756596i \(0.726861\pi\)
\(84\) −0.377203 −0.0411562
\(85\) 0.801125 0.0868943
\(86\) 5.88894 0.635020
\(87\) −0.480515 −0.0515166
\(88\) −1.05659 −0.112633
\(89\) −11.9610 −1.26787 −0.633933 0.773388i \(-0.718561\pi\)
−0.633933 + 0.773388i \(0.718561\pi\)
\(90\) −1.40550 −0.148153
\(91\) 0 0
\(92\) −0.0282963 −0.00295009
\(93\) 6.85772 0.711112
\(94\) −5.99708 −0.618551
\(95\) 2.54003 0.260602
\(96\) 2.10331 0.214668
\(97\) 10.1033 1.02584 0.512918 0.858438i \(-0.328564\pi\)
0.512918 + 0.858438i \(0.328564\pi\)
\(98\) −1.27389 −0.128682
\(99\) −0.348907 −0.0350664
\(100\) 1.42685 0.142685
\(101\) 4.07502 0.405479 0.202740 0.979233i \(-0.435015\pi\)
0.202740 + 0.979233i \(0.435015\pi\)
\(102\) 0.924984 0.0915871
\(103\) −4.11106 −0.405075 −0.202538 0.979275i \(-0.564919\pi\)
−0.202538 + 0.979275i \(0.564919\pi\)
\(104\) 0 0
\(105\) 1.10331 0.107672
\(106\) −7.06727 −0.686434
\(107\) −6.17058 −0.596532 −0.298266 0.954483i \(-0.596408\pi\)
−0.298266 + 0.954483i \(0.596408\pi\)
\(108\) 0.377203 0.0362964
\(109\) −8.82167 −0.844963 −0.422481 0.906372i \(-0.638841\pi\)
−0.422481 + 0.906372i \(0.638841\pi\)
\(110\) 0.490388 0.0467567
\(111\) 1.10331 0.104722
\(112\) 3.10331 0.293235
\(113\) 12.0099 1.12979 0.564897 0.825161i \(-0.308916\pi\)
0.564897 + 0.825161i \(0.308916\pi\)
\(114\) 2.93273 0.274676
\(115\) 0.0827661 0.00771798
\(116\) −0.181252 −0.0168288
\(117\) 0 0
\(118\) 5.00000 0.460287
\(119\) −0.726109 −0.0665623
\(120\) −3.34116 −0.305004
\(121\) −10.8783 −0.988933
\(122\) −7.06727 −0.639840
\(123\) −3.09556 −0.279117
\(124\) 2.58675 0.232297
\(125\) −9.69006 −0.866706
\(126\) 1.27389 0.113487
\(127\) −5.85772 −0.519788 −0.259894 0.965637i \(-0.583688\pi\)
−0.259894 + 0.965637i \(0.583688\pi\)
\(128\) −7.11319 −0.628723
\(129\) 4.62280 0.407015
\(130\) 0 0
\(131\) 22.4055 1.95758 0.978789 0.204872i \(-0.0656778\pi\)
0.978789 + 0.204872i \(0.0656778\pi\)
\(132\) −0.131609 −0.0114551
\(133\) −2.30219 −0.199625
\(134\) −10.3489 −0.894009
\(135\) −1.10331 −0.0949580
\(136\) 2.19887 0.188552
\(137\) 20.2087 1.72655 0.863275 0.504734i \(-0.168409\pi\)
0.863275 + 0.504734i \(0.168409\pi\)
\(138\) 0.0955622 0.00813480
\(139\) 4.27389 0.362507 0.181253 0.983436i \(-0.441985\pi\)
0.181253 + 0.983436i \(0.441985\pi\)
\(140\) 0.416173 0.0351730
\(141\) −4.70769 −0.396459
\(142\) 12.1394 1.01871
\(143\) 0 0
\(144\) −3.10331 −0.258609
\(145\) 0.530158 0.0440272
\(146\) 13.2915 1.10001
\(147\) −1.00000 −0.0824786
\(148\) 0.416173 0.0342092
\(149\) −4.89669 −0.401152 −0.200576 0.979678i \(-0.564281\pi\)
−0.200576 + 0.979678i \(0.564281\pi\)
\(150\) −4.81875 −0.393449
\(151\) −8.94553 −0.727977 −0.363988 0.931403i \(-0.618585\pi\)
−0.363988 + 0.931403i \(0.618585\pi\)
\(152\) 6.97170 0.565480
\(153\) 0.726109 0.0587025
\(154\) −0.444469 −0.0358163
\(155\) −7.56620 −0.607732
\(156\) 0 0
\(157\) 6.59450 0.526298 0.263149 0.964755i \(-0.415239\pi\)
0.263149 + 0.964755i \(0.415239\pi\)
\(158\) −21.7905 −1.73356
\(159\) −5.54778 −0.439968
\(160\) −2.32061 −0.183460
\(161\) −0.0750160 −0.00591209
\(162\) −1.27389 −0.100086
\(163\) −9.90444 −0.775775 −0.387888 0.921707i \(-0.626795\pi\)
−0.387888 + 0.921707i \(0.626795\pi\)
\(164\) −1.16765 −0.0911785
\(165\) 0.384953 0.0299685
\(166\) 15.1775 1.17800
\(167\) −16.7565 −1.29666 −0.648330 0.761360i \(-0.724532\pi\)
−0.648330 + 0.761360i \(0.724532\pi\)
\(168\) 3.02830 0.233638
\(169\) 0 0
\(170\) −1.02055 −0.0782723
\(171\) 2.30219 0.176053
\(172\) 1.74373 0.132958
\(173\) 13.7184 1.04299 0.521494 0.853255i \(-0.325375\pi\)
0.521494 + 0.853255i \(0.325375\pi\)
\(174\) 0.612124 0.0464050
\(175\) 3.78270 0.285945
\(176\) 1.08277 0.0816166
\(177\) 3.92498 0.295020
\(178\) 15.2370 1.14206
\(179\) −22.0304 −1.64663 −0.823315 0.567584i \(-0.807878\pi\)
−0.823315 + 0.567584i \(0.807878\pi\)
\(180\) −0.416173 −0.0310197
\(181\) 9.43380 0.701208 0.350604 0.936524i \(-0.385976\pi\)
0.350604 + 0.936524i \(0.385976\pi\)
\(182\) 0 0
\(183\) −5.54778 −0.410104
\(184\) 0.227171 0.0167473
\(185\) −1.21730 −0.0894975
\(186\) −8.73598 −0.640553
\(187\) −0.253344 −0.0185264
\(188\) −1.77575 −0.129510
\(189\) 1.00000 0.0727393
\(190\) −3.23572 −0.234744
\(191\) −3.37720 −0.244366 −0.122183 0.992508i \(-0.538989\pi\)
−0.122183 + 0.992508i \(0.538989\pi\)
\(192\) −8.88601 −0.641293
\(193\) 22.3227 1.60683 0.803413 0.595423i \(-0.203015\pi\)
0.803413 + 0.595423i \(0.203015\pi\)
\(194\) −12.8705 −0.924049
\(195\) 0 0
\(196\) −0.377203 −0.0269431
\(197\) −25.4514 −1.81334 −0.906669 0.421842i \(-0.861384\pi\)
−0.906669 + 0.421842i \(0.861384\pi\)
\(198\) 0.444469 0.0315870
\(199\) −22.3404 −1.58367 −0.791833 0.610738i \(-0.790873\pi\)
−0.791833 + 0.610738i \(0.790873\pi\)
\(200\) −11.4551 −0.810001
\(201\) −8.12386 −0.573013
\(202\) −5.19112 −0.365246
\(203\) −0.480515 −0.0337256
\(204\) 0.273891 0.0191762
\(205\) 3.41537 0.238540
\(206\) 5.23704 0.364882
\(207\) 0.0750160 0.00521397
\(208\) 0 0
\(209\) −0.803248 −0.0555618
\(210\) −1.40550 −0.0969887
\(211\) 10.0382 0.691056 0.345528 0.938408i \(-0.387700\pi\)
0.345528 + 0.938408i \(0.387700\pi\)
\(212\) −2.09264 −0.143723
\(213\) 9.52936 0.652941
\(214\) 7.86064 0.537342
\(215\) −5.10039 −0.347844
\(216\) −3.02830 −0.206049
\(217\) 6.85772 0.465532
\(218\) 11.2378 0.761123
\(219\) 10.4338 0.705051
\(220\) 0.145205 0.00978974
\(221\) 0 0
\(222\) −1.40550 −0.0943309
\(223\) −5.13453 −0.343834 −0.171917 0.985111i \(-0.554996\pi\)
−0.171917 + 0.985111i \(0.554996\pi\)
\(224\) 2.10331 0.140533
\(225\) −3.78270 −0.252180
\(226\) −15.2993 −1.01769
\(227\) −28.9709 −1.92287 −0.961433 0.275039i \(-0.911309\pi\)
−0.961433 + 0.275039i \(0.911309\pi\)
\(228\) 0.868391 0.0575106
\(229\) 0.679390 0.0448953 0.0224477 0.999748i \(-0.492854\pi\)
0.0224477 + 0.999748i \(0.492854\pi\)
\(230\) −0.105435 −0.00695218
\(231\) −0.348907 −0.0229564
\(232\) 1.45514 0.0955348
\(233\) −18.3022 −1.19902 −0.599508 0.800369i \(-0.704637\pi\)
−0.599508 + 0.800369i \(0.704637\pi\)
\(234\) 0 0
\(235\) 5.19405 0.338822
\(236\) 1.48052 0.0963733
\(237\) −17.1054 −1.11112
\(238\) 0.924984 0.0599578
\(239\) −5.45222 −0.352675 −0.176337 0.984330i \(-0.556425\pi\)
−0.176337 + 0.984330i \(0.556425\pi\)
\(240\) 3.42392 0.221013
\(241\) −15.4621 −0.996001 −0.498000 0.867177i \(-0.665932\pi\)
−0.498000 + 0.867177i \(0.665932\pi\)
\(242\) 13.8577 0.890808
\(243\) −1.00000 −0.0641500
\(244\) −2.09264 −0.133967
\(245\) 1.10331 0.0704880
\(246\) 3.94341 0.251422
\(247\) 0 0
\(248\) −20.7672 −1.31872
\(249\) 11.9143 0.755039
\(250\) 12.3441 0.780708
\(251\) −8.29444 −0.523540 −0.261770 0.965130i \(-0.584306\pi\)
−0.261770 + 0.965130i \(0.584306\pi\)
\(252\) 0.377203 0.0237615
\(253\) −0.0261736 −0.00164552
\(254\) 7.46209 0.468213
\(255\) −0.801125 −0.0501684
\(256\) −8.71061 −0.544413
\(257\) −15.0382 −0.938055 −0.469028 0.883184i \(-0.655396\pi\)
−0.469028 + 0.883184i \(0.655396\pi\)
\(258\) −5.88894 −0.366629
\(259\) 1.10331 0.0685565
\(260\) 0 0
\(261\) 0.480515 0.0297431
\(262\) −28.5422 −1.76334
\(263\) −17.2448 −1.06336 −0.531680 0.846945i \(-0.678439\pi\)
−0.531680 + 0.846945i \(0.678439\pi\)
\(264\) 1.05659 0.0650288
\(265\) 6.12094 0.376006
\(266\) 2.93273 0.179817
\(267\) 11.9610 0.732003
\(268\) −3.06434 −0.187185
\(269\) −24.2944 −1.48126 −0.740629 0.671914i \(-0.765472\pi\)
−0.740629 + 0.671914i \(0.765472\pi\)
\(270\) 1.40550 0.0855360
\(271\) 18.3326 1.11363 0.556813 0.830638i \(-0.312024\pi\)
0.556813 + 0.830638i \(0.312024\pi\)
\(272\) −2.25334 −0.136629
\(273\) 0 0
\(274\) −25.7437 −1.55524
\(275\) 1.31981 0.0795875
\(276\) 0.0282963 0.00170323
\(277\) 8.45434 0.507972 0.253986 0.967208i \(-0.418258\pi\)
0.253986 + 0.967208i \(0.418258\pi\)
\(278\) −5.44447 −0.326538
\(279\) −6.85772 −0.410561
\(280\) −3.34116 −0.199672
\(281\) −25.0120 −1.49209 −0.746045 0.665895i \(-0.768050\pi\)
−0.746045 + 0.665895i \(0.768050\pi\)
\(282\) 5.99708 0.357121
\(283\) 27.1415 1.61339 0.806697 0.590966i \(-0.201253\pi\)
0.806697 + 0.590966i \(0.201253\pi\)
\(284\) 3.59450 0.213294
\(285\) −2.54003 −0.150458
\(286\) 0 0
\(287\) −3.09556 −0.182725
\(288\) −2.10331 −0.123939
\(289\) −16.4728 −0.968986
\(290\) −0.675364 −0.0396587
\(291\) −10.1033 −0.592267
\(292\) 3.93566 0.230317
\(293\) −10.5400 −0.615755 −0.307878 0.951426i \(-0.599619\pi\)
−0.307878 + 0.951426i \(0.599619\pi\)
\(294\) 1.27389 0.0742948
\(295\) −4.33048 −0.252130
\(296\) −3.34116 −0.194201
\(297\) 0.348907 0.0202456
\(298\) 6.23784 0.361349
\(299\) 0 0
\(300\) −1.42685 −0.0823790
\(301\) 4.62280 0.266454
\(302\) 11.3956 0.655745
\(303\) −4.07502 −0.234104
\(304\) −7.14440 −0.409760
\(305\) 6.12094 0.350484
\(306\) −0.924984 −0.0528778
\(307\) −4.40842 −0.251602 −0.125801 0.992055i \(-0.540150\pi\)
−0.125801 + 0.992055i \(0.540150\pi\)
\(308\) −0.131609 −0.00749909
\(309\) 4.11106 0.233870
\(310\) 9.63852 0.547431
\(311\) −2.32836 −0.132029 −0.0660146 0.997819i \(-0.521028\pi\)
−0.0660146 + 0.997819i \(0.521028\pi\)
\(312\) 0 0
\(313\) 12.6687 0.716078 0.358039 0.933707i \(-0.383445\pi\)
0.358039 + 0.933707i \(0.383445\pi\)
\(314\) −8.40067 −0.474077
\(315\) −1.10331 −0.0621646
\(316\) −6.45222 −0.362966
\(317\) −12.2915 −0.690360 −0.345180 0.938536i \(-0.612182\pi\)
−0.345180 + 0.938536i \(0.612182\pi\)
\(318\) 7.06727 0.396313
\(319\) −0.167655 −0.00938687
\(320\) 9.80405 0.548063
\(321\) 6.17058 0.344408
\(322\) 0.0955622 0.00532547
\(323\) 1.67164 0.0930125
\(324\) −0.377203 −0.0209557
\(325\) 0 0
\(326\) 12.6172 0.698800
\(327\) 8.82167 0.487840
\(328\) 9.37428 0.517608
\(329\) −4.70769 −0.259543
\(330\) −0.490388 −0.0269950
\(331\) 13.2915 0.730568 0.365284 0.930896i \(-0.380972\pi\)
0.365284 + 0.930896i \(0.380972\pi\)
\(332\) 4.49411 0.246646
\(333\) −1.10331 −0.0604611
\(334\) 21.3460 1.16800
\(335\) 8.96315 0.489709
\(336\) −3.10331 −0.169300
\(337\) −3.40550 −0.185509 −0.0927547 0.995689i \(-0.529567\pi\)
−0.0927547 + 0.995689i \(0.529567\pi\)
\(338\) 0 0
\(339\) −12.0099 −0.652287
\(340\) −0.302187 −0.0163884
\(341\) 2.39270 0.129572
\(342\) −2.93273 −0.158584
\(343\) −1.00000 −0.0539949
\(344\) −13.9992 −0.754786
\(345\) −0.0827661 −0.00445598
\(346\) −17.4757 −0.939499
\(347\) −33.4981 −1.79827 −0.899137 0.437667i \(-0.855805\pi\)
−0.899137 + 0.437667i \(0.855805\pi\)
\(348\) 0.181252 0.00971611
\(349\) −28.1805 −1.50846 −0.754232 0.656607i \(-0.771991\pi\)
−0.754232 + 0.656607i \(0.771991\pi\)
\(350\) −4.81875 −0.257573
\(351\) 0 0
\(352\) 0.733860 0.0391148
\(353\) −2.99225 −0.159261 −0.0796307 0.996824i \(-0.525374\pi\)
−0.0796307 + 0.996824i \(0.525374\pi\)
\(354\) −5.00000 −0.265747
\(355\) −10.5139 −0.558018
\(356\) 4.51173 0.239121
\(357\) 0.726109 0.0384298
\(358\) 28.0643 1.48325
\(359\) 3.80888 0.201025 0.100512 0.994936i \(-0.467952\pi\)
0.100512 + 0.994936i \(0.467952\pi\)
\(360\) 3.34116 0.176094
\(361\) −13.6999 −0.721049
\(362\) −12.0176 −0.631632
\(363\) 10.8783 0.570961
\(364\) 0 0
\(365\) −11.5117 −0.602552
\(366\) 7.06727 0.369412
\(367\) −26.4514 −1.38075 −0.690376 0.723450i \(-0.742555\pi\)
−0.690376 + 0.723450i \(0.742555\pi\)
\(368\) −0.232798 −0.0121354
\(369\) 3.09556 0.161149
\(370\) 1.55070 0.0806173
\(371\) −5.54778 −0.288026
\(372\) −2.58675 −0.134117
\(373\) 2.13161 0.110371 0.0551853 0.998476i \(-0.482425\pi\)
0.0551853 + 0.998476i \(0.482425\pi\)
\(374\) 0.322733 0.0166881
\(375\) 9.69006 0.500393
\(376\) 14.2563 0.735211
\(377\) 0 0
\(378\) −1.27389 −0.0655219
\(379\) −13.9434 −0.716225 −0.358112 0.933678i \(-0.616580\pi\)
−0.358112 + 0.933678i \(0.616580\pi\)
\(380\) −0.958107 −0.0491498
\(381\) 5.85772 0.300100
\(382\) 4.30219 0.220119
\(383\) −18.6044 −0.950639 −0.475320 0.879813i \(-0.657668\pi\)
−0.475320 + 0.879813i \(0.657668\pi\)
\(384\) 7.11319 0.362993
\(385\) 0.384953 0.0196190
\(386\) −28.4367 −1.44739
\(387\) −4.62280 −0.234990
\(388\) −3.81100 −0.193474
\(389\) −1.23009 −0.0623682 −0.0311841 0.999514i \(-0.509928\pi\)
−0.0311841 + 0.999514i \(0.509928\pi\)
\(390\) 0 0
\(391\) 0.0544699 0.00275466
\(392\) 3.02830 0.152952
\(393\) −22.4055 −1.13021
\(394\) 32.4223 1.63341
\(395\) 18.8726 0.949585
\(396\) 0.131609 0.00661358
\(397\) 29.6142 1.48630 0.743148 0.669127i \(-0.233332\pi\)
0.743148 + 0.669127i \(0.233332\pi\)
\(398\) 28.4592 1.42653
\(399\) 2.30219 0.115253
\(400\) 11.7389 0.586945
\(401\) 8.18045 0.408512 0.204256 0.978917i \(-0.434522\pi\)
0.204256 + 0.978917i \(0.434522\pi\)
\(402\) 10.3489 0.516157
\(403\) 0 0
\(404\) −1.53711 −0.0764740
\(405\) 1.10331 0.0548240
\(406\) 0.612124 0.0303792
\(407\) 0.384953 0.0190814
\(408\) −2.19887 −0.108861
\(409\) 10.0673 0.497794 0.248897 0.968530i \(-0.419932\pi\)
0.248897 + 0.968530i \(0.419932\pi\)
\(410\) −4.35081 −0.214871
\(411\) −20.2087 −0.996824
\(412\) 1.55070 0.0763977
\(413\) 3.92498 0.193136
\(414\) −0.0955622 −0.00469663
\(415\) −13.1452 −0.645273
\(416\) 0 0
\(417\) −4.27389 −0.209293
\(418\) 1.02325 0.0500488
\(419\) 9.45997 0.462150 0.231075 0.972936i \(-0.425776\pi\)
0.231075 + 0.972936i \(0.425776\pi\)
\(420\) −0.416173 −0.0203071
\(421\) 31.0643 1.51398 0.756992 0.653424i \(-0.226668\pi\)
0.756992 + 0.653424i \(0.226668\pi\)
\(422\) −12.7875 −0.622487
\(423\) 4.70769 0.228896
\(424\) 16.8003 0.815896
\(425\) −2.74666 −0.133232
\(426\) −12.1394 −0.588154
\(427\) −5.54778 −0.268476
\(428\) 2.32756 0.112507
\(429\) 0 0
\(430\) 6.49734 0.313329
\(431\) −25.2186 −1.21474 −0.607369 0.794420i \(-0.707775\pi\)
−0.607369 + 0.794420i \(0.707775\pi\)
\(432\) 3.10331 0.149308
\(433\) 3.76991 0.181170 0.0905851 0.995889i \(-0.471126\pi\)
0.0905851 + 0.995889i \(0.471126\pi\)
\(434\) −8.73598 −0.419341
\(435\) −0.530158 −0.0254191
\(436\) 3.32756 0.159361
\(437\) 0.172701 0.00826141
\(438\) −13.2915 −0.635093
\(439\) −22.6348 −1.08030 −0.540150 0.841569i \(-0.681632\pi\)
−0.540150 + 0.841569i \(0.681632\pi\)
\(440\) −1.16575 −0.0555750
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.64334 0.220612 0.110306 0.993898i \(-0.464817\pi\)
0.110306 + 0.993898i \(0.464817\pi\)
\(444\) −0.416173 −0.0197507
\(445\) −13.1968 −0.625586
\(446\) 6.54083 0.309717
\(447\) 4.89669 0.231605
\(448\) −8.88601 −0.419825
\(449\) 9.42312 0.444705 0.222352 0.974966i \(-0.428626\pi\)
0.222352 + 0.974966i \(0.428626\pi\)
\(450\) 4.81875 0.227158
\(451\) −1.08006 −0.0508581
\(452\) −4.53016 −0.213081
\(453\) 8.94553 0.420298
\(454\) 36.9058 1.73207
\(455\) 0 0
\(456\) −6.97170 −0.326480
\(457\) 12.7162 0.594840 0.297420 0.954747i \(-0.403874\pi\)
0.297420 + 0.954747i \(0.403874\pi\)
\(458\) −0.865468 −0.0404407
\(459\) −0.726109 −0.0338919
\(460\) −0.0312196 −0.00145562
\(461\) −29.7253 −1.38445 −0.692223 0.721684i \(-0.743368\pi\)
−0.692223 + 0.721684i \(0.743368\pi\)
\(462\) 0.444469 0.0206786
\(463\) −15.2838 −0.710297 −0.355148 0.934810i \(-0.615570\pi\)
−0.355148 + 0.934810i \(0.615570\pi\)
\(464\) −1.49119 −0.0692267
\(465\) 7.56620 0.350874
\(466\) 23.3150 1.08005
\(467\) −26.6503 −1.23323 −0.616614 0.787265i \(-0.711496\pi\)
−0.616614 + 0.787265i \(0.711496\pi\)
\(468\) 0 0
\(469\) −8.12386 −0.375125
\(470\) −6.61665 −0.305203
\(471\) −6.59450 −0.303859
\(472\) −11.8860 −0.547098
\(473\) 1.61292 0.0741623
\(474\) 21.7905 1.00087
\(475\) −8.70849 −0.399573
\(476\) 0.273891 0.0125538
\(477\) 5.54778 0.254015
\(478\) 6.94553 0.317681
\(479\) −22.2117 −1.01488 −0.507439 0.861688i \(-0.669408\pi\)
−0.507439 + 0.861688i \(0.669408\pi\)
\(480\) 2.32061 0.105921
\(481\) 0 0
\(482\) 19.6970 0.897174
\(483\) 0.0750160 0.00341335
\(484\) 4.10331 0.186514
\(485\) 11.1471 0.506164
\(486\) 1.27389 0.0577848
\(487\) −0.971704 −0.0440321 −0.0220160 0.999758i \(-0.507008\pi\)
−0.0220160 + 0.999758i \(0.507008\pi\)
\(488\) 16.8003 0.760515
\(489\) 9.90444 0.447894
\(490\) −1.40550 −0.0634940
\(491\) 26.9349 1.21555 0.607777 0.794108i \(-0.292062\pi\)
0.607777 + 0.794108i \(0.292062\pi\)
\(492\) 1.16765 0.0526419
\(493\) 0.348907 0.0157140
\(494\) 0 0
\(495\) −0.384953 −0.0173023
\(496\) 21.2816 0.955574
\(497\) 9.52936 0.427450
\(498\) −15.1775 −0.680121
\(499\) −21.1239 −0.945634 −0.472817 0.881161i \(-0.656763\pi\)
−0.472817 + 0.881161i \(0.656763\pi\)
\(500\) 3.65512 0.163462
\(501\) 16.7565 0.748626
\(502\) 10.5662 0.471593
\(503\) 12.5654 0.560264 0.280132 0.959962i \(-0.409622\pi\)
0.280132 + 0.959962i \(0.409622\pi\)
\(504\) −3.02830 −0.134891
\(505\) 4.49602 0.200070
\(506\) 0.0333423 0.00148225
\(507\) 0 0
\(508\) 2.20955 0.0980328
\(509\) 27.6065 1.22364 0.611818 0.790998i \(-0.290438\pi\)
0.611818 + 0.790998i \(0.290438\pi\)
\(510\) 1.02055 0.0451905
\(511\) 10.4338 0.461564
\(512\) 25.3227 1.11912
\(513\) −2.30219 −0.101644
\(514\) 19.1570 0.844978
\(515\) −4.53579 −0.199871
\(516\) −1.74373 −0.0767635
\(517\) −1.64254 −0.0722389
\(518\) −1.40550 −0.0617541
\(519\) −13.7184 −0.602169
\(520\) 0 0
\(521\) −36.8783 −1.61567 −0.807833 0.589411i \(-0.799360\pi\)
−0.807833 + 0.589411i \(0.799360\pi\)
\(522\) −0.612124 −0.0267919
\(523\) 30.8054 1.34702 0.673512 0.739176i \(-0.264785\pi\)
0.673512 + 0.739176i \(0.264785\pi\)
\(524\) −8.45142 −0.369202
\(525\) −3.78270 −0.165091
\(526\) 21.9680 0.957849
\(527\) −4.97945 −0.216908
\(528\) −1.08277 −0.0471213
\(529\) −22.9944 −0.999755
\(530\) −7.79740 −0.338697
\(531\) −3.92498 −0.170330
\(532\) 0.868391 0.0376495
\(533\) 0 0
\(534\) −15.2370 −0.659371
\(535\) −6.80807 −0.294339
\(536\) 24.6015 1.06262
\(537\) 22.0304 0.950683
\(538\) 30.9485 1.33428
\(539\) −0.348907 −0.0150285
\(540\) 0.416173 0.0179092
\(541\) −41.4981 −1.78414 −0.892072 0.451893i \(-0.850749\pi\)
−0.892072 + 0.451893i \(0.850749\pi\)
\(542\) −23.3537 −1.00313
\(543\) −9.43380 −0.404843
\(544\) −1.52723 −0.0654797
\(545\) −9.73306 −0.416918
\(546\) 0 0
\(547\) −41.3716 −1.76892 −0.884460 0.466615i \(-0.845473\pi\)
−0.884460 + 0.466615i \(0.845473\pi\)
\(548\) −7.62280 −0.325630
\(549\) 5.54778 0.236774
\(550\) −1.68129 −0.0716906
\(551\) 1.10624 0.0471272
\(552\) −0.227171 −0.00966903
\(553\) −17.1054 −0.727397
\(554\) −10.7699 −0.457569
\(555\) 1.21730 0.0516714
\(556\) −1.61212 −0.0683693
\(557\) 30.0149 1.27177 0.635886 0.771783i \(-0.280635\pi\)
0.635886 + 0.771783i \(0.280635\pi\)
\(558\) 8.73598 0.369824
\(559\) 0 0
\(560\) 3.42392 0.144687
\(561\) 0.253344 0.0106962
\(562\) 31.8625 1.34404
\(563\) 25.9349 1.09302 0.546512 0.837451i \(-0.315955\pi\)
0.546512 + 0.837451i \(0.315955\pi\)
\(564\) 1.77575 0.0747727
\(565\) 13.2506 0.557459
\(566\) −34.5753 −1.45331
\(567\) −1.00000 −0.0419961
\(568\) −28.8577 −1.21084
\(569\) −6.56620 −0.275270 −0.137635 0.990483i \(-0.543950\pi\)
−0.137635 + 0.990483i \(0.543950\pi\)
\(570\) 3.23572 0.135529
\(571\) −30.0539 −1.25772 −0.628858 0.777520i \(-0.716477\pi\)
−0.628858 + 0.777520i \(0.716477\pi\)
\(572\) 0 0
\(573\) 3.37720 0.141085
\(574\) 3.94341 0.164595
\(575\) −0.283763 −0.0118337
\(576\) 8.88601 0.370251
\(577\) 21.1706 0.881343 0.440671 0.897669i \(-0.354740\pi\)
0.440671 + 0.897669i \(0.354740\pi\)
\(578\) 20.9845 0.872840
\(579\) −22.3227 −0.927701
\(580\) −0.199977 −0.00830360
\(581\) 11.9143 0.494289
\(582\) 12.8705 0.533500
\(583\) −1.93566 −0.0801667
\(584\) −31.5966 −1.30748
\(585\) 0 0
\(586\) 13.4268 0.554658
\(587\) 0.0312196 0.00128857 0.000644286 1.00000i \(-0.499795\pi\)
0.000644286 1.00000i \(0.499795\pi\)
\(588\) 0.377203 0.0155556
\(589\) −15.7877 −0.650523
\(590\) 5.51656 0.227113
\(591\) 25.4514 1.04693
\(592\) 3.42392 0.140722
\(593\) −10.3150 −0.423586 −0.211793 0.977315i \(-0.567930\pi\)
−0.211793 + 0.977315i \(0.567930\pi\)
\(594\) −0.444469 −0.0182368
\(595\) −0.801125 −0.0328429
\(596\) 1.84704 0.0756579
\(597\) 22.3404 0.914330
\(598\) 0 0
\(599\) 44.0176 1.79851 0.899256 0.437423i \(-0.144109\pi\)
0.899256 + 0.437423i \(0.144109\pi\)
\(600\) 11.4551 0.467654
\(601\) 5.04672 0.205860 0.102930 0.994689i \(-0.467178\pi\)
0.102930 + 0.994689i \(0.467178\pi\)
\(602\) −5.88894 −0.240015
\(603\) 8.12386 0.330829
\(604\) 3.37428 0.137297
\(605\) −12.0021 −0.487956
\(606\) 5.19112 0.210875
\(607\) 16.6482 0.675728 0.337864 0.941195i \(-0.390296\pi\)
0.337864 + 0.941195i \(0.390296\pi\)
\(608\) −4.84222 −0.196378
\(609\) 0.480515 0.0194715
\(610\) −7.79740 −0.315708
\(611\) 0 0
\(612\) −0.273891 −0.0110714
\(613\) 38.0120 1.53529 0.767645 0.640875i \(-0.221428\pi\)
0.767645 + 0.640875i \(0.221428\pi\)
\(614\) 5.61585 0.226637
\(615\) −3.41537 −0.137721
\(616\) 1.05659 0.0425713
\(617\) 15.6666 0.630713 0.315357 0.948973i \(-0.397876\pi\)
0.315357 + 0.948973i \(0.397876\pi\)
\(618\) −5.23704 −0.210665
\(619\) 3.26109 0.131074 0.0655372 0.997850i \(-0.479124\pi\)
0.0655372 + 0.997850i \(0.479124\pi\)
\(620\) 2.85399 0.114619
\(621\) −0.0750160 −0.00301029
\(622\) 2.96608 0.118929
\(623\) 11.9610 0.479209
\(624\) 0 0
\(625\) 8.22234 0.328894
\(626\) −16.1386 −0.645027
\(627\) 0.803248 0.0320786
\(628\) −2.48746 −0.0992606
\(629\) −0.801125 −0.0319430
\(630\) 1.40550 0.0559964
\(631\) 1.59955 0.0636770 0.0318385 0.999493i \(-0.489864\pi\)
0.0318385 + 0.999493i \(0.489864\pi\)
\(632\) 51.8003 2.06051
\(633\) −10.0382 −0.398981
\(634\) 15.6580 0.621860
\(635\) −6.46289 −0.256472
\(636\) 2.09264 0.0829785
\(637\) 0 0
\(638\) 0.213574 0.00845548
\(639\) −9.52936 −0.376976
\(640\) −7.84806 −0.310222
\(641\) 31.3559 1.23848 0.619241 0.785201i \(-0.287440\pi\)
0.619241 + 0.785201i \(0.287440\pi\)
\(642\) −7.86064 −0.310235
\(643\) −18.6631 −0.736000 −0.368000 0.929826i \(-0.619957\pi\)
−0.368000 + 0.929826i \(0.619957\pi\)
\(644\) 0.0282963 0.00111503
\(645\) 5.10039 0.200828
\(646\) −2.12949 −0.0837835
\(647\) 6.53711 0.257000 0.128500 0.991709i \(-0.458984\pi\)
0.128500 + 0.991709i \(0.458984\pi\)
\(648\) 3.02830 0.118963
\(649\) 1.36945 0.0537557
\(650\) 0 0
\(651\) −6.85772 −0.268775
\(652\) 3.73598 0.146312
\(653\) 15.8521 0.620340 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(654\) −11.2378 −0.439434
\(655\) 24.7203 0.965901
\(656\) −9.60650 −0.375071
\(657\) −10.4338 −0.407061
\(658\) 5.99708 0.233790
\(659\) −49.7974 −1.93983 −0.969916 0.243441i \(-0.921724\pi\)
−0.969916 + 0.243441i \(0.921724\pi\)
\(660\) −0.145205 −0.00565211
\(661\) 3.57103 0.138897 0.0694485 0.997586i \(-0.477876\pi\)
0.0694485 + 0.997586i \(0.477876\pi\)
\(662\) −16.9319 −0.658078
\(663\) 0 0
\(664\) −36.0801 −1.40018
\(665\) −2.54003 −0.0984982
\(666\) 1.40550 0.0544620
\(667\) 0.0360463 0.00139572
\(668\) 6.32061 0.244552
\(669\) 5.13453 0.198512
\(670\) −11.4181 −0.441119
\(671\) −1.93566 −0.0747252
\(672\) −2.10331 −0.0811370
\(673\) 11.7339 0.452307 0.226154 0.974092i \(-0.427385\pi\)
0.226154 + 0.974092i \(0.427385\pi\)
\(674\) 4.33823 0.167102
\(675\) 3.78270 0.145596
\(676\) 0 0
\(677\) 17.3326 0.666146 0.333073 0.942901i \(-0.391914\pi\)
0.333073 + 0.942901i \(0.391914\pi\)
\(678\) 15.2993 0.587565
\(679\) −10.1033 −0.387730
\(680\) 2.42605 0.0930346
\(681\) 28.9709 1.11017
\(682\) −3.04804 −0.116716
\(683\) 16.9640 0.649108 0.324554 0.945867i \(-0.394786\pi\)
0.324554 + 0.945867i \(0.394786\pi\)
\(684\) −0.868391 −0.0332038
\(685\) 22.2966 0.851908
\(686\) 1.27389 0.0486374
\(687\) −0.679390 −0.0259203
\(688\) 14.3460 0.546935
\(689\) 0 0
\(690\) 0.105435 0.00401384
\(691\) 36.6425 1.39395 0.696974 0.717096i \(-0.254529\pi\)
0.696974 + 0.717096i \(0.254529\pi\)
\(692\) −5.17460 −0.196709
\(693\) 0.348907 0.0132539
\(694\) 42.6730 1.61984
\(695\) 4.71544 0.178867
\(696\) −1.45514 −0.0551570
\(697\) 2.24772 0.0851384
\(698\) 35.8988 1.35879
\(699\) 18.3022 0.692252
\(700\) −1.42685 −0.0539297
\(701\) −14.6092 −0.551782 −0.275891 0.961189i \(-0.588973\pi\)
−0.275891 + 0.961189i \(0.588973\pi\)
\(702\) 0 0
\(703\) −2.54003 −0.0957991
\(704\) −3.10039 −0.116850
\(705\) −5.19405 −0.195619
\(706\) 3.81180 0.143459
\(707\) −4.07502 −0.153257
\(708\) −1.48052 −0.0556412
\(709\) −23.5860 −0.885789 −0.442894 0.896574i \(-0.646048\pi\)
−0.442894 + 0.896574i \(0.646048\pi\)
\(710\) 13.3935 0.502649
\(711\) 17.1054 0.641504
\(712\) −36.2215 −1.35746
\(713\) −0.514439 −0.0192659
\(714\) −0.924984 −0.0346167
\(715\) 0 0
\(716\) 8.30994 0.310557
\(717\) 5.45222 0.203617
\(718\) −4.85209 −0.181078
\(719\) 34.1337 1.27297 0.636487 0.771288i \(-0.280387\pi\)
0.636487 + 0.771288i \(0.280387\pi\)
\(720\) −3.42392 −0.127602
\(721\) 4.11106 0.153104
\(722\) 17.4522 0.649504
\(723\) 15.4621 0.575041
\(724\) −3.55845 −0.132249
\(725\) −1.81765 −0.0675057
\(726\) −13.8577 −0.514308
\(727\) −15.3481 −0.569230 −0.284615 0.958642i \(-0.591866\pi\)
−0.284615 + 0.958642i \(0.591866\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.6647 0.542765
\(731\) −3.35666 −0.124150
\(732\) 2.09264 0.0773462
\(733\) 23.2894 0.860213 0.430107 0.902778i \(-0.358476\pi\)
0.430107 + 0.902778i \(0.358476\pi\)
\(734\) 33.6962 1.24375
\(735\) −1.10331 −0.0406963
\(736\) −0.157782 −0.00581593
\(737\) −2.83447 −0.104409
\(738\) −3.94341 −0.145159
\(739\) −10.2661 −0.377646 −0.188823 0.982011i \(-0.560467\pi\)
−0.188823 + 0.982011i \(0.560467\pi\)
\(740\) 0.459168 0.0168794
\(741\) 0 0
\(742\) 7.06727 0.259447
\(743\) −45.3481 −1.66366 −0.831830 0.555030i \(-0.812707\pi\)
−0.831830 + 0.555030i \(0.812707\pi\)
\(744\) 20.7672 0.761363
\(745\) −5.40258 −0.197935
\(746\) −2.71544 −0.0994192
\(747\) −11.9143 −0.435922
\(748\) 0.0955622 0.00349410
\(749\) 6.17058 0.225468
\(750\) −12.3441 −0.450742
\(751\) −29.6113 −1.08053 −0.540266 0.841494i \(-0.681676\pi\)
−0.540266 + 0.841494i \(0.681676\pi\)
\(752\) −14.6094 −0.532751
\(753\) 8.29444 0.302266
\(754\) 0 0
\(755\) −9.86971 −0.359196
\(756\) −0.377203 −0.0137187
\(757\) −25.9816 −0.944316 −0.472158 0.881514i \(-0.656525\pi\)
−0.472158 + 0.881514i \(0.656525\pi\)
\(758\) 17.7624 0.645159
\(759\) 0.0261736 0.000950041 0
\(760\) 7.69197 0.279017
\(761\) −54.0665 −1.95991 −0.979954 0.199224i \(-0.936158\pi\)
−0.979954 + 0.199224i \(0.936158\pi\)
\(762\) −7.46209 −0.270323
\(763\) 8.82167 0.319366
\(764\) 1.27389 0.0460877
\(765\) 0.801125 0.0289648
\(766\) 23.6999 0.856313
\(767\) 0 0
\(768\) 8.71061 0.314317
\(769\) 25.7934 0.930133 0.465066 0.885276i \(-0.346030\pi\)
0.465066 + 0.885276i \(0.346030\pi\)
\(770\) −0.490388 −0.0176724
\(771\) 15.0382 0.541586
\(772\) −8.42020 −0.303050
\(773\) −0.121736 −0.00437853 −0.00218927 0.999998i \(-0.500697\pi\)
−0.00218927 + 0.999998i \(0.500697\pi\)
\(774\) 5.88894 0.211673
\(775\) 25.9407 0.931818
\(776\) 30.5958 1.09833
\(777\) −1.10331 −0.0395811
\(778\) 1.56701 0.0561799
\(779\) 7.12656 0.255336
\(780\) 0 0
\(781\) 3.32486 0.118973
\(782\) −0.0693886 −0.00248133
\(783\) −0.480515 −0.0171722
\(784\) −3.10331 −0.110833
\(785\) 7.27579 0.259684
\(786\) 28.5422 1.01806
\(787\) −20.2215 −0.720820 −0.360410 0.932794i \(-0.617363\pi\)
−0.360410 + 0.932794i \(0.617363\pi\)
\(788\) 9.60035 0.341998
\(789\) 17.2448 0.613931
\(790\) −24.0417 −0.855364
\(791\) −12.0099 −0.427022
\(792\) −1.05659 −0.0375444
\(793\) 0 0
\(794\) −37.7253 −1.33882
\(795\) −6.12094 −0.217087
\(796\) 8.42685 0.298682
\(797\) 32.4124 1.14811 0.574054 0.818818i \(-0.305370\pi\)
0.574054 + 0.818818i \(0.305370\pi\)
\(798\) −2.93273 −0.103818
\(799\) 3.41830 0.120931
\(800\) 7.95620 0.281294
\(801\) −11.9610 −0.422622
\(802\) −10.4210 −0.367978
\(803\) 3.64042 0.128468
\(804\) 3.06434 0.108071
\(805\) −0.0827661 −0.00291712
\(806\) 0 0
\(807\) 24.2944 0.855205
\(808\) 12.3404 0.434132
\(809\) 24.4629 0.860069 0.430035 0.902812i \(-0.358501\pi\)
0.430035 + 0.902812i \(0.358501\pi\)
\(810\) −1.40550 −0.0493842
\(811\) −45.2448 −1.58876 −0.794380 0.607421i \(-0.792204\pi\)
−0.794380 + 0.607421i \(0.792204\pi\)
\(812\) 0.181252 0.00636069
\(813\) −18.3326 −0.642953
\(814\) −0.490388 −0.0171881
\(815\) −10.9277 −0.382780
\(816\) 2.25334 0.0788828
\(817\) −10.6425 −0.372335
\(818\) −12.8246 −0.448401
\(819\) 0 0
\(820\) −1.28829 −0.0449890
\(821\) 36.7331 1.28199 0.640996 0.767544i \(-0.278521\pi\)
0.640996 + 0.767544i \(0.278521\pi\)
\(822\) 25.7437 0.897916
\(823\) 42.6476 1.48660 0.743301 0.668957i \(-0.233259\pi\)
0.743301 + 0.668957i \(0.233259\pi\)
\(824\) −12.4495 −0.433699
\(825\) −1.31981 −0.0459499
\(826\) −5.00000 −0.173972
\(827\) −50.0635 −1.74088 −0.870440 0.492275i \(-0.836166\pi\)
−0.870440 + 0.492275i \(0.836166\pi\)
\(828\) −0.0282963 −0.000983363 0
\(829\) 25.8911 0.899234 0.449617 0.893222i \(-0.351561\pi\)
0.449617 + 0.893222i \(0.351561\pi\)
\(830\) 16.7456 0.581247
\(831\) −8.45434 −0.293278
\(832\) 0 0
\(833\) 0.726109 0.0251582
\(834\) 5.44447 0.188527
\(835\) −18.4877 −0.639793
\(836\) 0.302987 0.0104790
\(837\) 6.85772 0.237037
\(838\) −12.0510 −0.416294
\(839\) −28.9349 −0.998942 −0.499471 0.866331i \(-0.666472\pi\)
−0.499471 + 0.866331i \(0.666472\pi\)
\(840\) 3.34116 0.115281
\(841\) −28.7691 −0.992038
\(842\) −39.5726 −1.36376
\(843\) 25.0120 0.861459
\(844\) −3.78643 −0.130334
\(845\) 0 0
\(846\) −5.99708 −0.206184
\(847\) 10.8783 0.373782
\(848\) −17.2165 −0.591217
\(849\) −27.1415 −0.931493
\(850\) 3.49894 0.120013
\(851\) −0.0827661 −0.00283719
\(852\) −3.59450 −0.123146
\(853\) 40.3094 1.38017 0.690083 0.723730i \(-0.257574\pi\)
0.690083 + 0.723730i \(0.257574\pi\)
\(854\) 7.06727 0.241837
\(855\) 2.54003 0.0868672
\(856\) −18.6863 −0.638686
\(857\) 40.0459 1.36794 0.683971 0.729509i \(-0.260251\pi\)
0.683971 + 0.729509i \(0.260251\pi\)
\(858\) 0 0
\(859\) 6.66659 0.227461 0.113731 0.993512i \(-0.463720\pi\)
0.113731 + 0.993512i \(0.463720\pi\)
\(860\) 1.92388 0.0656038
\(861\) 3.09556 0.105496
\(862\) 32.1258 1.09421
\(863\) −5.85289 −0.199235 −0.0996174 0.995026i \(-0.531762\pi\)
−0.0996174 + 0.995026i \(0.531762\pi\)
\(864\) 2.10331 0.0715561
\(865\) 15.1356 0.514627
\(866\) −4.80245 −0.163194
\(867\) 16.4728 0.559444
\(868\) −2.58675 −0.0878000
\(869\) −5.96820 −0.202457
\(870\) 0.675364 0.0228970
\(871\) 0 0
\(872\) −26.7146 −0.904672
\(873\) 10.1033 0.341945
\(874\) −0.220002 −0.00744168
\(875\) 9.69006 0.327584
\(876\) −3.93566 −0.132974
\(877\) 4.36733 0.147474 0.0737371 0.997278i \(-0.476507\pi\)
0.0737371 + 0.997278i \(0.476507\pi\)
\(878\) 28.8342 0.973109
\(879\) 10.5400 0.355506
\(880\) 1.19463 0.0402709
\(881\) 3.14733 0.106036 0.0530181 0.998594i \(-0.483116\pi\)
0.0530181 + 0.998594i \(0.483116\pi\)
\(882\) −1.27389 −0.0428941
\(883\) 14.9554 0.503289 0.251645 0.967820i \(-0.419029\pi\)
0.251645 + 0.967820i \(0.419029\pi\)
\(884\) 0 0
\(885\) 4.33048 0.145568
\(886\) −5.91511 −0.198722
\(887\) 22.5264 0.756364 0.378182 0.925731i \(-0.376549\pi\)
0.378182 + 0.925731i \(0.376549\pi\)
\(888\) 3.34116 0.112122
\(889\) 5.85772 0.196462
\(890\) 16.8112 0.563513
\(891\) −0.348907 −0.0116888
\(892\) 1.93676 0.0648475
\(893\) 10.8380 0.362679
\(894\) −6.23784 −0.208625
\(895\) −24.3064 −0.812474
\(896\) 7.11319 0.237635
\(897\) 0 0
\(898\) −12.0040 −0.400580
\(899\) −3.29524 −0.109902
\(900\) 1.42685 0.0475615
\(901\) 4.02830 0.134202
\(902\) 1.37588 0.0458118
\(903\) −4.62280 −0.153837
\(904\) 36.3695 1.20963
\(905\) 10.4084 0.345988
\(906\) −11.3956 −0.378594
\(907\) −18.6532 −0.619370 −0.309685 0.950839i \(-0.600224\pi\)
−0.309685 + 0.950839i \(0.600224\pi\)
\(908\) 10.9279 0.362655
\(909\) 4.07502 0.135160
\(910\) 0 0
\(911\) 18.3278 0.607226 0.303613 0.952795i \(-0.401807\pi\)
0.303613 + 0.952795i \(0.401807\pi\)
\(912\) 7.14440 0.236575
\(913\) 4.15698 0.137576
\(914\) −16.1991 −0.535818
\(915\) −6.12094 −0.202352
\(916\) −0.256268 −0.00846732
\(917\) −22.4055 −0.739895
\(918\) 0.924984 0.0305290
\(919\) 15.1054 0.498282 0.249141 0.968467i \(-0.419852\pi\)
0.249141 + 0.968467i \(0.419852\pi\)
\(920\) 0.250640 0.00826337
\(921\) 4.40842 0.145262
\(922\) 37.8668 1.24708
\(923\) 0 0
\(924\) 0.131609 0.00432960
\(925\) 4.17350 0.137224
\(926\) 19.4698 0.639819
\(927\) −4.11106 −0.135025
\(928\) −1.01067 −0.0331770
\(929\) −6.25817 −0.205324 −0.102662 0.994716i \(-0.532736\pi\)
−0.102662 + 0.994716i \(0.532736\pi\)
\(930\) −9.63852 −0.316059
\(931\) 2.30219 0.0754511
\(932\) 6.90364 0.226136
\(933\) 2.32836 0.0762271
\(934\) 33.9496 1.11086
\(935\) −0.279518 −0.00914121
\(936\) 0 0
\(937\) −22.6610 −0.740301 −0.370151 0.928972i \(-0.620694\pi\)
−0.370151 + 0.928972i \(0.620694\pi\)
\(938\) 10.3489 0.337904
\(939\) −12.6687 −0.413428
\(940\) −1.95921 −0.0639024
\(941\) 17.2944 0.563783 0.281891 0.959446i \(-0.409038\pi\)
0.281891 + 0.959446i \(0.409038\pi\)
\(942\) 8.40067 0.273709
\(943\) 0.232217 0.00756202
\(944\) 12.1805 0.396440
\(945\) 1.10331 0.0358908
\(946\) −2.05469 −0.0668037
\(947\) −27.6121 −0.897273 −0.448637 0.893714i \(-0.648090\pi\)
−0.448637 + 0.893714i \(0.648090\pi\)
\(948\) 6.45222 0.209558
\(949\) 0 0
\(950\) 11.0937 0.359926
\(951\) 12.2915 0.398580
\(952\) −2.19887 −0.0712659
\(953\) 4.39080 0.142232 0.0711160 0.997468i \(-0.477344\pi\)
0.0711160 + 0.997468i \(0.477344\pi\)
\(954\) −7.06727 −0.228811
\(955\) −3.72611 −0.120574
\(956\) 2.05659 0.0665150
\(957\) 0.167655 0.00541951
\(958\) 28.2952 0.914178
\(959\) −20.2087 −0.652574
\(960\) −9.80405 −0.316424
\(961\) 16.0283 0.517042
\(962\) 0 0
\(963\) −6.17058 −0.198844
\(964\) 5.83235 0.187847
\(965\) 24.6289 0.792834
\(966\) −0.0955622 −0.00307466
\(967\) −18.4875 −0.594517 −0.297258 0.954797i \(-0.596072\pi\)
−0.297258 + 0.954797i \(0.596072\pi\)
\(968\) −32.9426 −1.05882
\(969\) −1.67164 −0.0537008
\(970\) −14.2002 −0.455941
\(971\) −52.3374 −1.67959 −0.839794 0.542905i \(-0.817324\pi\)
−0.839794 + 0.542905i \(0.817324\pi\)
\(972\) 0.377203 0.0120988
\(973\) −4.27389 −0.137015
\(974\) 1.23784 0.0396631
\(975\) 0 0
\(976\) −17.2165 −0.551087
\(977\) 51.6447 1.65226 0.826130 0.563480i \(-0.190538\pi\)
0.826130 + 0.563480i \(0.190538\pi\)
\(978\) −12.6172 −0.403453
\(979\) 4.17328 0.133379
\(980\) −0.416173 −0.0132941
\(981\) −8.82167 −0.281654
\(982\) −34.3121 −1.09494
\(983\) 37.5419 1.19740 0.598701 0.800973i \(-0.295684\pi\)
0.598701 + 0.800973i \(0.295684\pi\)
\(984\) −9.37428 −0.298841
\(985\) −28.0809 −0.894731
\(986\) −0.444469 −0.0141548
\(987\) 4.70769 0.149847
\(988\) 0 0
\(989\) −0.346784 −0.0110271
\(990\) 0.490388 0.0155856
\(991\) −50.5724 −1.60648 −0.803242 0.595653i \(-0.796893\pi\)
−0.803242 + 0.595653i \(0.796893\pi\)
\(992\) 14.4239 0.457960
\(993\) −13.2915 −0.421793
\(994\) −12.1394 −0.385037
\(995\) −24.6484 −0.781406
\(996\) −4.49411 −0.142401
\(997\) −25.9221 −0.820960 −0.410480 0.911870i \(-0.634639\pi\)
−0.410480 + 0.911870i \(0.634639\pi\)
\(998\) 26.9095 0.851805
\(999\) 1.10331 0.0349073
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 3549.2.a.h.1.2 3
13.3 even 3 273.2.k.d.22.2 6
13.9 even 3 273.2.k.d.211.2 yes 6
13.12 even 2 3549.2.a.s.1.2 3
39.29 odd 6 819.2.o.d.568.2 6
39.35 odd 6 819.2.o.d.757.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.k.d.22.2 6 13.3 even 3
273.2.k.d.211.2 yes 6 13.9 even 3
819.2.o.d.568.2 6 39.29 odd 6
819.2.o.d.757.2 6 39.35 odd 6
3549.2.a.h.1.2 3 1.1 even 1 trivial
3549.2.a.s.1.2 3 13.12 even 2