Properties

Label 4598.2.a.cb.1.8
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.75320\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.37860 q^{3} +1.00000 q^{4} -4.17017 q^{5} +3.37860 q^{6} -3.60700 q^{7} +1.00000 q^{8} +8.41492 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.37860 q^{3} +1.00000 q^{4} -4.17017 q^{5} +3.37860 q^{6} -3.60700 q^{7} +1.00000 q^{8} +8.41492 q^{9} -4.17017 q^{10} +3.37860 q^{12} -0.669426 q^{13} -3.60700 q^{14} -14.0893 q^{15} +1.00000 q^{16} +4.10493 q^{17} +8.41492 q^{18} -1.00000 q^{19} -4.17017 q^{20} -12.1866 q^{21} +6.35872 q^{23} +3.37860 q^{24} +12.3903 q^{25} -0.669426 q^{26} +18.2948 q^{27} -3.60700 q^{28} +3.27213 q^{29} -14.0893 q^{30} -1.55576 q^{31} +1.00000 q^{32} +4.10493 q^{34} +15.0418 q^{35} +8.41492 q^{36} +3.51430 q^{37} -1.00000 q^{38} -2.26172 q^{39} -4.17017 q^{40} +7.30933 q^{41} -12.1866 q^{42} -6.04669 q^{43} -35.0916 q^{45} +6.35872 q^{46} +0.862845 q^{47} +3.37860 q^{48} +6.01048 q^{49} +12.3903 q^{50} +13.8689 q^{51} -0.669426 q^{52} +6.79423 q^{53} +18.2948 q^{54} -3.60700 q^{56} -3.37860 q^{57} +3.27213 q^{58} -9.41317 q^{59} -14.0893 q^{60} -4.40660 q^{61} -1.55576 q^{62} -30.3526 q^{63} +1.00000 q^{64} +2.79162 q^{65} +1.80340 q^{67} +4.10493 q^{68} +21.4836 q^{69} +15.0418 q^{70} +11.9933 q^{71} +8.41492 q^{72} +9.91252 q^{73} +3.51430 q^{74} +41.8618 q^{75} -1.00000 q^{76} -2.26172 q^{78} +1.04552 q^{79} -4.17017 q^{80} +36.5661 q^{81} +7.30933 q^{82} +0.503831 q^{83} -12.1866 q^{84} -17.1183 q^{85} -6.04669 q^{86} +11.0552 q^{87} +8.23674 q^{89} -35.0916 q^{90} +2.41462 q^{91} +6.35872 q^{92} -5.25629 q^{93} +0.862845 q^{94} +4.17017 q^{95} +3.37860 q^{96} +0.809369 q^{97} +6.01048 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + 8 q^{12} - 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} - 4 q^{17} + 22 q^{18} - 8 q^{19} - 20 q^{21} + 14 q^{23} + 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} + 4 q^{28} - 2 q^{29} + 4 q^{30} + 8 q^{32} - 4 q^{34} + 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} + 16 q^{39} + 8 q^{41} - 20 q^{42} + 8 q^{43} + 16 q^{45} + 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} + 36 q^{50} + 18 q^{51} - 12 q^{52} + 36 q^{53} + 32 q^{54} + 4 q^{56} - 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} + 12 q^{61} + 24 q^{63} + 8 q^{64} + 16 q^{65} + 16 q^{67} - 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} + 22 q^{72} - 20 q^{73} + 24 q^{74} + 40 q^{75} - 8 q^{76} + 16 q^{78} - 12 q^{79} + 40 q^{81} + 8 q^{82} + 20 q^{83} - 20 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} - 16 q^{94} + 8 q^{96} + 4 q^{97} + 34 q^{98} + O(q^{100}) \)

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
\(3\) 3.37860 1.95063 0.975317 0.220810i \(-0.0708701\pi\)
0.975317 + 0.220810i \(0.0708701\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.17017 −1.86496 −0.932478 0.361228i \(-0.882358\pi\)
−0.932478 + 0.361228i \(0.882358\pi\)
\(6\) 3.37860 1.37931
\(7\) −3.60700 −1.36332 −0.681660 0.731669i \(-0.738742\pi\)
−0.681660 + 0.731669i \(0.738742\pi\)
\(8\) 1.00000 0.353553
\(9\) 8.41492 2.80497
\(10\) −4.17017 −1.31872
\(11\) 0 0
\(12\) 3.37860 0.975317
\(13\) −0.669426 −0.185665 −0.0928326 0.995682i \(-0.529592\pi\)
−0.0928326 + 0.995682i \(0.529592\pi\)
\(14\) −3.60700 −0.964012
\(15\) −14.0893 −3.63784
\(16\) 1.00000 0.250000
\(17\) 4.10493 0.995592 0.497796 0.867294i \(-0.334143\pi\)
0.497796 + 0.867294i \(0.334143\pi\)
\(18\) 8.41492 1.98341
\(19\) −1.00000 −0.229416
\(20\) −4.17017 −0.932478
\(21\) −12.1866 −2.65934
\(22\) 0 0
\(23\) 6.35872 1.32589 0.662943 0.748670i \(-0.269307\pi\)
0.662943 + 0.748670i \(0.269307\pi\)
\(24\) 3.37860 0.689653
\(25\) 12.3903 2.47806
\(26\) −0.669426 −0.131285
\(27\) 18.2948 3.52084
\(28\) −3.60700 −0.681660
\(29\) 3.27213 0.607619 0.303809 0.952733i \(-0.401741\pi\)
0.303809 + 0.952733i \(0.401741\pi\)
\(30\) −14.0893 −2.57234
\(31\) −1.55576 −0.279423 −0.139712 0.990192i \(-0.544618\pi\)
−0.139712 + 0.990192i \(0.544618\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.10493 0.703990
\(35\) 15.0418 2.54253
\(36\) 8.41492 1.40249
\(37\) 3.51430 0.577747 0.288873 0.957367i \(-0.406719\pi\)
0.288873 + 0.957367i \(0.406719\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.26172 −0.362165
\(40\) −4.17017 −0.659361
\(41\) 7.30933 1.14153 0.570763 0.821115i \(-0.306648\pi\)
0.570763 + 0.821115i \(0.306648\pi\)
\(42\) −12.1866 −1.88044
\(43\) −6.04669 −0.922112 −0.461056 0.887371i \(-0.652529\pi\)
−0.461056 + 0.887371i \(0.652529\pi\)
\(44\) 0 0
\(45\) −35.0916 −5.23115
\(46\) 6.35872 0.937543
\(47\) 0.862845 0.125859 0.0629294 0.998018i \(-0.479956\pi\)
0.0629294 + 0.998018i \(0.479956\pi\)
\(48\) 3.37860 0.487658
\(49\) 6.01048 0.858640
\(50\) 12.3903 1.75225
\(51\) 13.8689 1.94204
\(52\) −0.669426 −0.0928326
\(53\) 6.79423 0.933260 0.466630 0.884453i \(-0.345468\pi\)
0.466630 + 0.884453i \(0.345468\pi\)
\(54\) 18.2948 2.48961
\(55\) 0 0
\(56\) −3.60700 −0.482006
\(57\) −3.37860 −0.447506
\(58\) 3.27213 0.429651
\(59\) −9.41317 −1.22549 −0.612745 0.790280i \(-0.709935\pi\)
−0.612745 + 0.790280i \(0.709935\pi\)
\(60\) −14.0893 −1.81892
\(61\) −4.40660 −0.564207 −0.282103 0.959384i \(-0.591032\pi\)
−0.282103 + 0.959384i \(0.591032\pi\)
\(62\) −1.55576 −0.197582
\(63\) −30.3526 −3.82407
\(64\) 1.00000 0.125000
\(65\) 2.79162 0.346257
\(66\) 0 0
\(67\) 1.80340 0.220320 0.110160 0.993914i \(-0.464864\pi\)
0.110160 + 0.993914i \(0.464864\pi\)
\(68\) 4.10493 0.497796
\(69\) 21.4836 2.58632
\(70\) 15.0418 1.79784
\(71\) 11.9933 1.42334 0.711669 0.702514i \(-0.247939\pi\)
0.711669 + 0.702514i \(0.247939\pi\)
\(72\) 8.41492 0.991707
\(73\) 9.91252 1.16017 0.580087 0.814555i \(-0.303019\pi\)
0.580087 + 0.814555i \(0.303019\pi\)
\(74\) 3.51430 0.408529
\(75\) 41.8618 4.83378
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −2.26172 −0.256089
\(79\) 1.04552 0.117630 0.0588152 0.998269i \(-0.481268\pi\)
0.0588152 + 0.998269i \(0.481268\pi\)
\(80\) −4.17017 −0.466239
\(81\) 36.5661 4.06290
\(82\) 7.30933 0.807180
\(83\) 0.503831 0.0553026 0.0276513 0.999618i \(-0.491197\pi\)
0.0276513 + 0.999618i \(0.491197\pi\)
\(84\) −12.1866 −1.32967
\(85\) −17.1183 −1.85674
\(86\) −6.04669 −0.652032
\(87\) 11.0552 1.18524
\(88\) 0 0
\(89\) 8.23674 0.873093 0.436546 0.899682i \(-0.356201\pi\)
0.436546 + 0.899682i \(0.356201\pi\)
\(90\) −35.0916 −3.69898
\(91\) 2.41462 0.253121
\(92\) 6.35872 0.662943
\(93\) −5.25629 −0.545052
\(94\) 0.862845 0.0889957
\(95\) 4.17017 0.427850
\(96\) 3.37860 0.344827
\(97\) 0.809369 0.0821790 0.0410895 0.999155i \(-0.486917\pi\)
0.0410895 + 0.999155i \(0.486917\pi\)
\(98\) 6.01048 0.607150
\(99\) 0 0
\(100\) 12.3903 1.23903
\(101\) −12.0493 −1.19895 −0.599475 0.800393i \(-0.704624\pi\)
−0.599475 + 0.800393i \(0.704624\pi\)
\(102\) 13.8689 1.37323
\(103\) 5.64096 0.555820 0.277910 0.960607i \(-0.410358\pi\)
0.277910 + 0.960607i \(0.410358\pi\)
\(104\) −0.669426 −0.0656426
\(105\) 50.8202 4.95954
\(106\) 6.79423 0.659914
\(107\) 15.1059 1.46034 0.730172 0.683264i \(-0.239440\pi\)
0.730172 + 0.683264i \(0.239440\pi\)
\(108\) 18.2948 1.76042
\(109\) −11.6030 −1.11137 −0.555685 0.831393i \(-0.687544\pi\)
−0.555685 + 0.831393i \(0.687544\pi\)
\(110\) 0 0
\(111\) 11.8734 1.12697
\(112\) −3.60700 −0.340830
\(113\) −6.03660 −0.567876 −0.283938 0.958843i \(-0.591641\pi\)
−0.283938 + 0.958843i \(0.591641\pi\)
\(114\) −3.37860 −0.316435
\(115\) −26.5169 −2.47272
\(116\) 3.27213 0.303809
\(117\) −5.63316 −0.520786
\(118\) −9.41317 −0.866553
\(119\) −14.8065 −1.35731
\(120\) −14.0893 −1.28617
\(121\) 0 0
\(122\) −4.40660 −0.398954
\(123\) 24.6953 2.22670
\(124\) −1.55576 −0.139712
\(125\) −30.8187 −2.75651
\(126\) −30.3526 −2.70403
\(127\) −7.06535 −0.626949 −0.313474 0.949597i \(-0.601493\pi\)
−0.313474 + 0.949597i \(0.601493\pi\)
\(128\) 1.00000 0.0883883
\(129\) −20.4293 −1.79870
\(130\) 2.79162 0.244841
\(131\) −3.26224 −0.285023 −0.142512 0.989793i \(-0.545518\pi\)
−0.142512 + 0.989793i \(0.545518\pi\)
\(132\) 0 0
\(133\) 3.60700 0.312767
\(134\) 1.80340 0.155790
\(135\) −76.2924 −6.56621
\(136\) 4.10493 0.351995
\(137\) 1.25379 0.107118 0.0535592 0.998565i \(-0.482943\pi\)
0.0535592 + 0.998565i \(0.482943\pi\)
\(138\) 21.4836 1.82880
\(139\) 1.52427 0.129287 0.0646433 0.997908i \(-0.479409\pi\)
0.0646433 + 0.997908i \(0.479409\pi\)
\(140\) 15.0418 1.27127
\(141\) 2.91521 0.245505
\(142\) 11.9933 1.00645
\(143\) 0 0
\(144\) 8.41492 0.701243
\(145\) −13.6453 −1.13318
\(146\) 9.91252 0.820366
\(147\) 20.3070 1.67489
\(148\) 3.51430 0.288873
\(149\) −15.6872 −1.28515 −0.642575 0.766223i \(-0.722134\pi\)
−0.642575 + 0.766223i \(0.722134\pi\)
\(150\) 41.8618 3.41800
\(151\) 18.9865 1.54510 0.772548 0.634957i \(-0.218982\pi\)
0.772548 + 0.634957i \(0.218982\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 34.5427 2.79261
\(154\) 0 0
\(155\) 6.48779 0.521112
\(156\) −2.26172 −0.181082
\(157\) −8.89570 −0.709954 −0.354977 0.934875i \(-0.615511\pi\)
−0.354977 + 0.934875i \(0.615511\pi\)
\(158\) 1.04552 0.0831772
\(159\) 22.9550 1.82045
\(160\) −4.17017 −0.329681
\(161\) −22.9359 −1.80761
\(162\) 36.5661 2.87290
\(163\) −4.84499 −0.379489 −0.189744 0.981834i \(-0.560766\pi\)
−0.189744 + 0.981834i \(0.560766\pi\)
\(164\) 7.30933 0.570763
\(165\) 0 0
\(166\) 0.503831 0.0391049
\(167\) −4.61536 −0.357148 −0.178574 0.983927i \(-0.557148\pi\)
−0.178574 + 0.983927i \(0.557148\pi\)
\(168\) −12.1866 −0.940218
\(169\) −12.5519 −0.965528
\(170\) −17.1183 −1.31291
\(171\) −8.41492 −0.643505
\(172\) −6.04669 −0.461056
\(173\) −9.33836 −0.709982 −0.354991 0.934870i \(-0.615516\pi\)
−0.354991 + 0.934870i \(0.615516\pi\)
\(174\) 11.0552 0.838092
\(175\) −44.6918 −3.37839
\(176\) 0 0
\(177\) −31.8033 −2.39048
\(178\) 8.23674 0.617370
\(179\) 16.7881 1.25480 0.627401 0.778696i \(-0.284119\pi\)
0.627401 + 0.778696i \(0.284119\pi\)
\(180\) −35.0916 −2.61557
\(181\) 17.5877 1.30729 0.653643 0.756803i \(-0.273240\pi\)
0.653643 + 0.756803i \(0.273240\pi\)
\(182\) 2.41462 0.178984
\(183\) −14.8881 −1.10056
\(184\) 6.35872 0.468771
\(185\) −14.6552 −1.07747
\(186\) −5.25629 −0.385410
\(187\) 0 0
\(188\) 0.862845 0.0629294
\(189\) −65.9895 −4.80003
\(190\) 4.17017 0.302536
\(191\) −2.76249 −0.199887 −0.0999435 0.994993i \(-0.531866\pi\)
−0.0999435 + 0.994993i \(0.531866\pi\)
\(192\) 3.37860 0.243829
\(193\) 21.4775 1.54598 0.772991 0.634417i \(-0.218760\pi\)
0.772991 + 0.634417i \(0.218760\pi\)
\(194\) 0.809369 0.0581093
\(195\) 9.43175 0.675421
\(196\) 6.01048 0.429320
\(197\) −18.9032 −1.34680 −0.673399 0.739280i \(-0.735166\pi\)
−0.673399 + 0.739280i \(0.735166\pi\)
\(198\) 0 0
\(199\) 7.76283 0.550292 0.275146 0.961402i \(-0.411274\pi\)
0.275146 + 0.961402i \(0.411274\pi\)
\(200\) 12.3903 0.876126
\(201\) 6.09296 0.429764
\(202\) −12.0493 −0.847786
\(203\) −11.8026 −0.828378
\(204\) 13.8689 0.971018
\(205\) −30.4811 −2.12889
\(206\) 5.64096 0.393024
\(207\) 53.5081 3.71907
\(208\) −0.669426 −0.0464163
\(209\) 0 0
\(210\) 50.8202 3.50693
\(211\) −6.88377 −0.473898 −0.236949 0.971522i \(-0.576147\pi\)
−0.236949 + 0.971522i \(0.576147\pi\)
\(212\) 6.79423 0.466630
\(213\) 40.5204 2.77641
\(214\) 15.1059 1.03262
\(215\) 25.2157 1.71970
\(216\) 18.2948 1.24480
\(217\) 5.61164 0.380943
\(218\) −11.6030 −0.785857
\(219\) 33.4904 2.26307
\(220\) 0 0
\(221\) −2.74795 −0.184847
\(222\) 11.8734 0.796890
\(223\) 14.5212 0.972409 0.486204 0.873845i \(-0.338381\pi\)
0.486204 + 0.873845i \(0.338381\pi\)
\(224\) −3.60700 −0.241003
\(225\) 104.263 6.95088
\(226\) −6.03660 −0.401549
\(227\) −18.1073 −1.20182 −0.600912 0.799315i \(-0.705196\pi\)
−0.600912 + 0.799315i \(0.705196\pi\)
\(228\) −3.37860 −0.223753
\(229\) −25.3580 −1.67570 −0.837850 0.545900i \(-0.816188\pi\)
−0.837850 + 0.545900i \(0.816188\pi\)
\(230\) −26.5169 −1.74848
\(231\) 0 0
\(232\) 3.27213 0.214826
\(233\) −21.9886 −1.44052 −0.720261 0.693704i \(-0.755978\pi\)
−0.720261 + 0.693704i \(0.755978\pi\)
\(234\) −5.63316 −0.368251
\(235\) −3.59821 −0.234721
\(236\) −9.41317 −0.612745
\(237\) 3.53240 0.229454
\(238\) −14.8065 −0.959764
\(239\) 15.2971 0.989488 0.494744 0.869039i \(-0.335262\pi\)
0.494744 + 0.869039i \(0.335262\pi\)
\(240\) −14.0893 −0.909461
\(241\) 16.0409 1.03328 0.516642 0.856201i \(-0.327182\pi\)
0.516642 + 0.856201i \(0.327182\pi\)
\(242\) 0 0
\(243\) 68.6575 4.40438
\(244\) −4.40660 −0.282103
\(245\) −25.0647 −1.60133
\(246\) 24.6953 1.57451
\(247\) 0.669426 0.0425945
\(248\) −1.55576 −0.0987910
\(249\) 1.70224 0.107875
\(250\) −30.8187 −1.94915
\(251\) −6.91197 −0.436279 −0.218140 0.975918i \(-0.569999\pi\)
−0.218140 + 0.975918i \(0.569999\pi\)
\(252\) −30.3526 −1.91204
\(253\) 0 0
\(254\) −7.06535 −0.443320
\(255\) −57.8357 −3.62181
\(256\) 1.00000 0.0625000
\(257\) −24.7512 −1.54394 −0.771968 0.635661i \(-0.780727\pi\)
−0.771968 + 0.635661i \(0.780727\pi\)
\(258\) −20.4293 −1.27187
\(259\) −12.6761 −0.787654
\(260\) 2.79162 0.173129
\(261\) 27.5347 1.70435
\(262\) −3.26224 −0.201542
\(263\) 16.8155 1.03689 0.518444 0.855112i \(-0.326512\pi\)
0.518444 + 0.855112i \(0.326512\pi\)
\(264\) 0 0
\(265\) −28.3331 −1.74049
\(266\) 3.60700 0.221160
\(267\) 27.8286 1.70308
\(268\) 1.80340 0.110160
\(269\) 17.8288 1.08704 0.543521 0.839395i \(-0.317091\pi\)
0.543521 + 0.839395i \(0.317091\pi\)
\(270\) −76.2924 −4.64301
\(271\) −30.2063 −1.83490 −0.917450 0.397850i \(-0.869756\pi\)
−0.917450 + 0.397850i \(0.869756\pi\)
\(272\) 4.10493 0.248898
\(273\) 8.15803 0.493747
\(274\) 1.25379 0.0757441
\(275\) 0 0
\(276\) 21.4836 1.29316
\(277\) 32.7642 1.96861 0.984306 0.176473i \(-0.0564688\pi\)
0.984306 + 0.176473i \(0.0564688\pi\)
\(278\) 1.52427 0.0914194
\(279\) −13.0916 −0.783774
\(280\) 15.0418 0.898920
\(281\) −9.70846 −0.579158 −0.289579 0.957154i \(-0.593515\pi\)
−0.289579 + 0.957154i \(0.593515\pi\)
\(282\) 2.91521 0.173598
\(283\) 14.5739 0.866326 0.433163 0.901316i \(-0.357397\pi\)
0.433163 + 0.901316i \(0.357397\pi\)
\(284\) 11.9933 0.711669
\(285\) 14.0893 0.834579
\(286\) 0 0
\(287\) −26.3648 −1.55626
\(288\) 8.41492 0.495854
\(289\) −0.149526 −0.00879562
\(290\) −13.6453 −0.801281
\(291\) 2.73453 0.160301
\(292\) 9.91252 0.580087
\(293\) −10.5241 −0.614824 −0.307412 0.951576i \(-0.599463\pi\)
−0.307412 + 0.951576i \(0.599463\pi\)
\(294\) 20.3070 1.18433
\(295\) 39.2545 2.28549
\(296\) 3.51430 0.204264
\(297\) 0 0
\(298\) −15.6872 −0.908738
\(299\) −4.25669 −0.246171
\(300\) 41.8618 2.41689
\(301\) 21.8104 1.25713
\(302\) 18.9865 1.09255
\(303\) −40.7097 −2.33871
\(304\) −1.00000 −0.0573539
\(305\) 18.3762 1.05222
\(306\) 34.5427 1.97467
\(307\) −21.4131 −1.22211 −0.611054 0.791589i \(-0.709254\pi\)
−0.611054 + 0.791589i \(0.709254\pi\)
\(308\) 0 0
\(309\) 19.0585 1.08420
\(310\) 6.48779 0.368482
\(311\) 8.91440 0.505489 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(312\) −2.26172 −0.128045
\(313\) −29.2280 −1.65206 −0.826031 0.563624i \(-0.809407\pi\)
−0.826031 + 0.563624i \(0.809407\pi\)
\(314\) −8.89570 −0.502014
\(315\) 126.576 7.13173
\(316\) 1.04552 0.0588152
\(317\) −3.86251 −0.216940 −0.108470 0.994100i \(-0.534595\pi\)
−0.108470 + 0.994100i \(0.534595\pi\)
\(318\) 22.9550 1.28725
\(319\) 0 0
\(320\) −4.17017 −0.233119
\(321\) 51.0368 2.84860
\(322\) −22.9359 −1.27817
\(323\) −4.10493 −0.228405
\(324\) 36.5661 2.03145
\(325\) −8.29438 −0.460089
\(326\) −4.84499 −0.268339
\(327\) −39.2020 −2.16788
\(328\) 7.30933 0.403590
\(329\) −3.11229 −0.171586
\(330\) 0 0
\(331\) 30.7968 1.69275 0.846373 0.532590i \(-0.178781\pi\)
0.846373 + 0.532590i \(0.178781\pi\)
\(332\) 0.503831 0.0276513
\(333\) 29.5725 1.62056
\(334\) −4.61536 −0.252542
\(335\) −7.52048 −0.410888
\(336\) −12.1866 −0.664834
\(337\) −16.0941 −0.876704 −0.438352 0.898803i \(-0.644438\pi\)
−0.438352 + 0.898803i \(0.644438\pi\)
\(338\) −12.5519 −0.682732
\(339\) −20.3953 −1.10772
\(340\) −17.1183 −0.928368
\(341\) 0 0
\(342\) −8.41492 −0.455027
\(343\) 3.56920 0.192719
\(344\) −6.04669 −0.326016
\(345\) −89.5900 −4.82337
\(346\) −9.33836 −0.502033
\(347\) −31.7933 −1.70676 −0.853378 0.521292i \(-0.825450\pi\)
−0.853378 + 0.521292i \(0.825450\pi\)
\(348\) 11.0552 0.592621
\(349\) 11.5331 0.617350 0.308675 0.951168i \(-0.400114\pi\)
0.308675 + 0.951168i \(0.400114\pi\)
\(350\) −44.6918 −2.38888
\(351\) −12.2470 −0.653697
\(352\) 0 0
\(353\) −3.56825 −0.189919 −0.0949593 0.995481i \(-0.530272\pi\)
−0.0949593 + 0.995481i \(0.530272\pi\)
\(354\) −31.8033 −1.69033
\(355\) −50.0139 −2.65446
\(356\) 8.23674 0.436546
\(357\) −50.0252 −2.64762
\(358\) 16.7881 0.887279
\(359\) 21.8282 1.15205 0.576025 0.817432i \(-0.304603\pi\)
0.576025 + 0.817432i \(0.304603\pi\)
\(360\) −35.0916 −1.84949
\(361\) 1.00000 0.0526316
\(362\) 17.5877 0.924391
\(363\) 0 0
\(364\) 2.41462 0.126561
\(365\) −41.3369 −2.16367
\(366\) −14.8881 −0.778214
\(367\) 7.86931 0.410774 0.205387 0.978681i \(-0.434155\pi\)
0.205387 + 0.978681i \(0.434155\pi\)
\(368\) 6.35872 0.331471
\(369\) 61.5074 3.20195
\(370\) −14.6552 −0.761888
\(371\) −24.5068 −1.27233
\(372\) −5.25629 −0.272526
\(373\) −2.90718 −0.150528 −0.0752639 0.997164i \(-0.523980\pi\)
−0.0752639 + 0.997164i \(0.523980\pi\)
\(374\) 0 0
\(375\) −104.124 −5.37695
\(376\) 0.862845 0.0444978
\(377\) −2.19045 −0.112814
\(378\) −65.9895 −3.39413
\(379\) 4.89402 0.251389 0.125694 0.992069i \(-0.459884\pi\)
0.125694 + 0.992069i \(0.459884\pi\)
\(380\) 4.17017 0.213925
\(381\) −23.8710 −1.22295
\(382\) −2.76249 −0.141341
\(383\) −12.3879 −0.632994 −0.316497 0.948593i \(-0.602507\pi\)
−0.316497 + 0.948593i \(0.602507\pi\)
\(384\) 3.37860 0.172413
\(385\) 0 0
\(386\) 21.4775 1.09317
\(387\) −50.8824 −2.58650
\(388\) 0.809369 0.0410895
\(389\) −13.7527 −0.697291 −0.348645 0.937255i \(-0.613358\pi\)
−0.348645 + 0.937255i \(0.613358\pi\)
\(390\) 9.43175 0.477595
\(391\) 26.1021 1.32004
\(392\) 6.01048 0.303575
\(393\) −11.0218 −0.555975
\(394\) −18.9032 −0.952330
\(395\) −4.36000 −0.219375
\(396\) 0 0
\(397\) 10.0289 0.503338 0.251669 0.967813i \(-0.419021\pi\)
0.251669 + 0.967813i \(0.419021\pi\)
\(398\) 7.76283 0.389115
\(399\) 12.1866 0.610094
\(400\) 12.3903 0.619515
\(401\) 22.2869 1.11296 0.556479 0.830862i \(-0.312152\pi\)
0.556479 + 0.830862i \(0.312152\pi\)
\(402\) 6.09296 0.303889
\(403\) 1.04147 0.0518792
\(404\) −12.0493 −0.599475
\(405\) −152.487 −7.57712
\(406\) −11.8026 −0.585752
\(407\) 0 0
\(408\) 13.8689 0.686613
\(409\) 5.88120 0.290807 0.145403 0.989372i \(-0.453552\pi\)
0.145403 + 0.989372i \(0.453552\pi\)
\(410\) −30.4811 −1.50536
\(411\) 4.23604 0.208949
\(412\) 5.64096 0.277910
\(413\) 33.9534 1.67074
\(414\) 53.5081 2.62978
\(415\) −2.10106 −0.103137
\(416\) −0.669426 −0.0328213
\(417\) 5.14988 0.252191
\(418\) 0 0
\(419\) 3.24502 0.158530 0.0792648 0.996854i \(-0.474743\pi\)
0.0792648 + 0.996854i \(0.474743\pi\)
\(420\) 50.8202 2.47977
\(421\) −19.2545 −0.938405 −0.469203 0.883091i \(-0.655459\pi\)
−0.469203 + 0.883091i \(0.655459\pi\)
\(422\) −6.88377 −0.335097
\(423\) 7.26077 0.353031
\(424\) 6.79423 0.329957
\(425\) 50.8613 2.46714
\(426\) 40.5204 1.96322
\(427\) 15.8946 0.769194
\(428\) 15.1059 0.730172
\(429\) 0 0
\(430\) 25.2157 1.21601
\(431\) −23.7828 −1.14558 −0.572789 0.819703i \(-0.694139\pi\)
−0.572789 + 0.819703i \(0.694139\pi\)
\(432\) 18.2948 0.880210
\(433\) −17.1402 −0.823707 −0.411853 0.911250i \(-0.635118\pi\)
−0.411853 + 0.911250i \(0.635118\pi\)
\(434\) 5.61164 0.269367
\(435\) −46.1020 −2.21042
\(436\) −11.6030 −0.555685
\(437\) −6.35872 −0.304179
\(438\) 33.4904 1.60023
\(439\) −0.144467 −0.00689503 −0.00344752 0.999994i \(-0.501097\pi\)
−0.00344752 + 0.999994i \(0.501097\pi\)
\(440\) 0 0
\(441\) 50.5777 2.40846
\(442\) −2.74795 −0.130707
\(443\) −19.6940 −0.935692 −0.467846 0.883810i \(-0.654970\pi\)
−0.467846 + 0.883810i \(0.654970\pi\)
\(444\) 11.8734 0.563486
\(445\) −34.3486 −1.62828
\(446\) 14.5212 0.687597
\(447\) −53.0009 −2.50686
\(448\) −3.60700 −0.170415
\(449\) −30.0122 −1.41636 −0.708182 0.706030i \(-0.750484\pi\)
−0.708182 + 0.706030i \(0.750484\pi\)
\(450\) 104.263 4.91502
\(451\) 0 0
\(452\) −6.03660 −0.283938
\(453\) 64.1476 3.01392
\(454\) −18.1073 −0.849817
\(455\) −10.0694 −0.472060
\(456\) −3.37860 −0.158217
\(457\) 0.810437 0.0379106 0.0189553 0.999820i \(-0.493966\pi\)
0.0189553 + 0.999820i \(0.493966\pi\)
\(458\) −25.3580 −1.18490
\(459\) 75.0990 3.50532
\(460\) −26.5169 −1.23636
\(461\) 29.4451 1.37139 0.685697 0.727888i \(-0.259498\pi\)
0.685697 + 0.727888i \(0.259498\pi\)
\(462\) 0 0
\(463\) 24.9830 1.16106 0.580529 0.814239i \(-0.302846\pi\)
0.580529 + 0.814239i \(0.302846\pi\)
\(464\) 3.27213 0.151905
\(465\) 21.9196 1.01650
\(466\) −21.9886 −1.01860
\(467\) −21.6826 −1.00335 −0.501676 0.865056i \(-0.667283\pi\)
−0.501676 + 0.865056i \(0.667283\pi\)
\(468\) −5.63316 −0.260393
\(469\) −6.50487 −0.300367
\(470\) −3.59821 −0.165973
\(471\) −30.0550 −1.38486
\(472\) −9.41317 −0.433276
\(473\) 0 0
\(474\) 3.53240 0.162248
\(475\) −12.3903 −0.568506
\(476\) −14.8065 −0.678655
\(477\) 57.1729 2.61777
\(478\) 15.2971 0.699674
\(479\) 30.8590 1.40998 0.704992 0.709215i \(-0.250951\pi\)
0.704992 + 0.709215i \(0.250951\pi\)
\(480\) −14.0893 −0.643086
\(481\) −2.35256 −0.107268
\(482\) 16.0409 0.730643
\(483\) −77.4913 −3.52598
\(484\) 0 0
\(485\) −3.37520 −0.153260
\(486\) 68.6575 3.11437
\(487\) −5.68001 −0.257386 −0.128693 0.991685i \(-0.541078\pi\)
−0.128693 + 0.991685i \(0.541078\pi\)
\(488\) −4.40660 −0.199477
\(489\) −16.3693 −0.740244
\(490\) −25.0647 −1.13231
\(491\) 29.9204 1.35029 0.675145 0.737685i \(-0.264081\pi\)
0.675145 + 0.737685i \(0.264081\pi\)
\(492\) 24.6953 1.11335
\(493\) 13.4319 0.604941
\(494\) 0.669426 0.0301189
\(495\) 0 0
\(496\) −1.55576 −0.0698558
\(497\) −43.2598 −1.94047
\(498\) 1.70224 0.0762793
\(499\) −2.89019 −0.129383 −0.0646913 0.997905i \(-0.520606\pi\)
−0.0646913 + 0.997905i \(0.520606\pi\)
\(500\) −30.8187 −1.37826
\(501\) −15.5935 −0.696664
\(502\) −6.91197 −0.308496
\(503\) −27.9653 −1.24691 −0.623456 0.781858i \(-0.714272\pi\)
−0.623456 + 0.781858i \(0.714272\pi\)
\(504\) −30.3526 −1.35201
\(505\) 50.2476 2.23599
\(506\) 0 0
\(507\) −42.4077 −1.88339
\(508\) −7.06535 −0.313474
\(509\) −34.7436 −1.53998 −0.769992 0.638054i \(-0.779740\pi\)
−0.769992 + 0.638054i \(0.779740\pi\)
\(510\) −57.8357 −2.56101
\(511\) −35.7545 −1.58169
\(512\) 1.00000 0.0441942
\(513\) −18.2948 −0.807736
\(514\) −24.7512 −1.09173
\(515\) −23.5237 −1.03658
\(516\) −20.4293 −0.899351
\(517\) 0 0
\(518\) −12.6761 −0.556955
\(519\) −31.5506 −1.38492
\(520\) 2.79162 0.122420
\(521\) 26.9219 1.17947 0.589734 0.807597i \(-0.299232\pi\)
0.589734 + 0.807597i \(0.299232\pi\)
\(522\) 27.5347 1.20516
\(523\) −2.23979 −0.0979394 −0.0489697 0.998800i \(-0.515594\pi\)
−0.0489697 + 0.998800i \(0.515594\pi\)
\(524\) −3.26224 −0.142512
\(525\) −150.996 −6.58999
\(526\) 16.8155 0.733190
\(527\) −6.38630 −0.278192
\(528\) 0 0
\(529\) 17.4334 0.757973
\(530\) −28.3331 −1.23071
\(531\) −79.2110 −3.43747
\(532\) 3.60700 0.156383
\(533\) −4.89305 −0.211942
\(534\) 27.8286 1.20426
\(535\) −62.9942 −2.72348
\(536\) 1.80340 0.0778950
\(537\) 56.7203 2.44766
\(538\) 17.8288 0.768655
\(539\) 0 0
\(540\) −76.2924 −3.28310
\(541\) −9.52299 −0.409425 −0.204713 0.978822i \(-0.565626\pi\)
−0.204713 + 0.978822i \(0.565626\pi\)
\(542\) −30.2063 −1.29747
\(543\) 59.4219 2.55004
\(544\) 4.10493 0.175998
\(545\) 48.3866 2.07265
\(546\) 8.15803 0.349132
\(547\) −17.0265 −0.727999 −0.364000 0.931399i \(-0.618589\pi\)
−0.364000 + 0.931399i \(0.618589\pi\)
\(548\) 1.25379 0.0535592
\(549\) −37.0811 −1.58258
\(550\) 0 0
\(551\) −3.27213 −0.139397
\(552\) 21.4836 0.914401
\(553\) −3.77120 −0.160368
\(554\) 32.7642 1.39202
\(555\) −49.5140 −2.10175
\(556\) 1.52427 0.0646433
\(557\) −14.4105 −0.610593 −0.305296 0.952257i \(-0.598756\pi\)
−0.305296 + 0.952257i \(0.598756\pi\)
\(558\) −13.0916 −0.554212
\(559\) 4.04781 0.171204
\(560\) 15.0418 0.635633
\(561\) 0 0
\(562\) −9.70846 −0.409527
\(563\) −14.3589 −0.605156 −0.302578 0.953125i \(-0.597847\pi\)
−0.302578 + 0.953125i \(0.597847\pi\)
\(564\) 2.91521 0.122752
\(565\) 25.1736 1.05906
\(566\) 14.5739 0.612585
\(567\) −131.894 −5.53902
\(568\) 11.9933 0.503226
\(569\) 27.8880 1.16912 0.584562 0.811349i \(-0.301266\pi\)
0.584562 + 0.811349i \(0.301266\pi\)
\(570\) 14.0893 0.590136
\(571\) −40.1622 −1.68073 −0.840367 0.542018i \(-0.817660\pi\)
−0.840367 + 0.542018i \(0.817660\pi\)
\(572\) 0 0
\(573\) −9.33335 −0.389906
\(574\) −26.3648 −1.10044
\(575\) 78.7864 3.28562
\(576\) 8.41492 0.350621
\(577\) 3.68171 0.153271 0.0766357 0.997059i \(-0.475582\pi\)
0.0766357 + 0.997059i \(0.475582\pi\)
\(578\) −0.149526 −0.00621944
\(579\) 72.5637 3.01564
\(580\) −13.6453 −0.566591
\(581\) −1.81732 −0.0753952
\(582\) 2.73453 0.113350
\(583\) 0 0
\(584\) 9.91252 0.410183
\(585\) 23.4912 0.971242
\(586\) −10.5241 −0.434746
\(587\) −44.1246 −1.82122 −0.910608 0.413272i \(-0.864386\pi\)
−0.910608 + 0.413272i \(0.864386\pi\)
\(588\) 20.3070 0.837446
\(589\) 1.55576 0.0641041
\(590\) 39.2545 1.61608
\(591\) −63.8663 −2.62711
\(592\) 3.51430 0.144437
\(593\) −2.42622 −0.0996330 −0.0498165 0.998758i \(-0.515864\pi\)
−0.0498165 + 0.998758i \(0.515864\pi\)
\(594\) 0 0
\(595\) 61.7456 2.53132
\(596\) −15.6872 −0.642575
\(597\) 26.2275 1.07342
\(598\) −4.25669 −0.174069
\(599\) 30.0748 1.22882 0.614412 0.788986i \(-0.289393\pi\)
0.614412 + 0.788986i \(0.289393\pi\)
\(600\) 41.8618 1.70900
\(601\) 43.6023 1.77858 0.889288 0.457348i \(-0.151201\pi\)
0.889288 + 0.457348i \(0.151201\pi\)
\(602\) 21.8104 0.888927
\(603\) 15.1755 0.617992
\(604\) 18.9865 0.772548
\(605\) 0 0
\(606\) −40.7097 −1.65372
\(607\) −3.95907 −0.160694 −0.0803468 0.996767i \(-0.525603\pi\)
−0.0803468 + 0.996767i \(0.525603\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −39.8761 −1.61586
\(610\) 18.3762 0.744032
\(611\) −0.577611 −0.0233676
\(612\) 34.5427 1.39630
\(613\) 33.3417 1.34666 0.673329 0.739343i \(-0.264864\pi\)
0.673329 + 0.739343i \(0.264864\pi\)
\(614\) −21.4131 −0.864161
\(615\) −102.983 −4.15269
\(616\) 0 0
\(617\) −46.8084 −1.88443 −0.942217 0.335004i \(-0.891262\pi\)
−0.942217 + 0.335004i \(0.891262\pi\)
\(618\) 19.0585 0.766646
\(619\) −3.44952 −0.138648 −0.0693239 0.997594i \(-0.522084\pi\)
−0.0693239 + 0.997594i \(0.522084\pi\)
\(620\) 6.48779 0.260556
\(621\) 116.332 4.66823
\(622\) 8.91440 0.357435
\(623\) −29.7100 −1.19030
\(624\) −2.26172 −0.0905412
\(625\) 66.5679 2.66271
\(626\) −29.2280 −1.16818
\(627\) 0 0
\(628\) −8.89570 −0.354977
\(629\) 14.4260 0.575200
\(630\) 126.576 5.04289
\(631\) −3.45479 −0.137533 −0.0687665 0.997633i \(-0.521906\pi\)
−0.0687665 + 0.997633i \(0.521906\pi\)
\(632\) 1.04552 0.0415886
\(633\) −23.2575 −0.924402
\(634\) −3.86251 −0.153400
\(635\) 29.4637 1.16923
\(636\) 22.9550 0.910224
\(637\) −4.02357 −0.159420
\(638\) 0 0
\(639\) 100.922 3.99243
\(640\) −4.17017 −0.164840
\(641\) 4.16142 0.164366 0.0821831 0.996617i \(-0.473811\pi\)
0.0821831 + 0.996617i \(0.473811\pi\)
\(642\) 51.0368 2.01426
\(643\) −27.0695 −1.06752 −0.533759 0.845637i \(-0.679221\pi\)
−0.533759 + 0.845637i \(0.679221\pi\)
\(644\) −22.9359 −0.903803
\(645\) 85.1937 3.35450
\(646\) −4.10493 −0.161506
\(647\) −4.31680 −0.169711 −0.0848554 0.996393i \(-0.527043\pi\)
−0.0848554 + 0.996393i \(0.527043\pi\)
\(648\) 36.5661 1.43645
\(649\) 0 0
\(650\) −8.29438 −0.325332
\(651\) 18.9595 0.743080
\(652\) −4.84499 −0.189744
\(653\) −5.85277 −0.229037 −0.114518 0.993421i \(-0.536532\pi\)
−0.114518 + 0.993421i \(0.536532\pi\)
\(654\) −39.2020 −1.53292
\(655\) 13.6041 0.531555
\(656\) 7.30933 0.285381
\(657\) 83.4131 3.25425
\(658\) −3.11229 −0.121330
\(659\) 16.4116 0.639307 0.319653 0.947535i \(-0.396434\pi\)
0.319653 + 0.947535i \(0.396434\pi\)
\(660\) 0 0
\(661\) −25.8425 −1.00516 −0.502578 0.864532i \(-0.667615\pi\)
−0.502578 + 0.864532i \(0.667615\pi\)
\(662\) 30.7968 1.19695
\(663\) −9.28421 −0.360569
\(664\) 0.503831 0.0195524
\(665\) −15.0418 −0.583296
\(666\) 29.5725 1.14591
\(667\) 20.8066 0.805633
\(668\) −4.61536 −0.178574
\(669\) 49.0611 1.89681
\(670\) −7.52048 −0.290541
\(671\) 0 0
\(672\) −12.1866 −0.470109
\(673\) 4.23989 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(674\) −16.0941 −0.619923
\(675\) 226.678 8.72484
\(676\) −12.5519 −0.482764
\(677\) 49.7388 1.91162 0.955809 0.293988i \(-0.0949826\pi\)
0.955809 + 0.293988i \(0.0949826\pi\)
\(678\) −20.3953 −0.783275
\(679\) −2.91940 −0.112036
\(680\) −17.1183 −0.656455
\(681\) −61.1772 −2.34432
\(682\) 0 0
\(683\) 38.4329 1.47059 0.735297 0.677745i \(-0.237043\pi\)
0.735297 + 0.677745i \(0.237043\pi\)
\(684\) −8.41492 −0.321752
\(685\) −5.22851 −0.199771
\(686\) 3.56920 0.136273
\(687\) −85.6743 −3.26868
\(688\) −6.04669 −0.230528
\(689\) −4.54823 −0.173274
\(690\) −89.5900 −3.41063
\(691\) 26.5880 1.01145 0.505727 0.862693i \(-0.331224\pi\)
0.505727 + 0.862693i \(0.331224\pi\)
\(692\) −9.33836 −0.354991
\(693\) 0 0
\(694\) −31.7933 −1.20686
\(695\) −6.35644 −0.241114
\(696\) 11.0552 0.419046
\(697\) 30.0043 1.13649
\(698\) 11.5331 0.436532
\(699\) −74.2906 −2.80993
\(700\) −44.6918 −1.68919
\(701\) −14.6368 −0.552826 −0.276413 0.961039i \(-0.589146\pi\)
−0.276413 + 0.961039i \(0.589146\pi\)
\(702\) −12.2470 −0.462234
\(703\) −3.51430 −0.132544
\(704\) 0 0
\(705\) −12.1569 −0.457855
\(706\) −3.56825 −0.134293
\(707\) 43.4619 1.63455
\(708\) −31.8033 −1.19524
\(709\) 14.7490 0.553911 0.276956 0.960883i \(-0.410674\pi\)
0.276956 + 0.960883i \(0.410674\pi\)
\(710\) −50.0139 −1.87699
\(711\) 8.79798 0.329950
\(712\) 8.23674 0.308685
\(713\) −9.89266 −0.370483
\(714\) −50.0252 −1.87215
\(715\) 0 0
\(716\) 16.7881 0.627401
\(717\) 51.6828 1.93013
\(718\) 21.8282 0.814622
\(719\) 8.82348 0.329060 0.164530 0.986372i \(-0.447389\pi\)
0.164530 + 0.986372i \(0.447389\pi\)
\(720\) −35.0916 −1.30779
\(721\) −20.3470 −0.757760
\(722\) 1.00000 0.0372161
\(723\) 54.1957 2.01556
\(724\) 17.5877 0.653643
\(725\) 40.5426 1.50571
\(726\) 0 0
\(727\) 0.938013 0.0347890 0.0173945 0.999849i \(-0.494463\pi\)
0.0173945 + 0.999849i \(0.494463\pi\)
\(728\) 2.41462 0.0894918
\(729\) 122.268 4.52844
\(730\) −41.3369 −1.52995
\(731\) −24.8213 −0.918048
\(732\) −14.8881 −0.550280
\(733\) 2.58558 0.0955007 0.0477504 0.998859i \(-0.484795\pi\)
0.0477504 + 0.998859i \(0.484795\pi\)
\(734\) 7.86931 0.290461
\(735\) −84.6835 −3.12360
\(736\) 6.35872 0.234386
\(737\) 0 0
\(738\) 61.5074 2.26412
\(739\) −1.56553 −0.0575888 −0.0287944 0.999585i \(-0.509167\pi\)
−0.0287944 + 0.999585i \(0.509167\pi\)
\(740\) −14.6552 −0.538736
\(741\) 2.26172 0.0830863
\(742\) −24.5068 −0.899674
\(743\) 35.3325 1.29623 0.648113 0.761545i \(-0.275559\pi\)
0.648113 + 0.761545i \(0.275559\pi\)
\(744\) −5.25629 −0.192705
\(745\) 65.4184 2.39675
\(746\) −2.90718 −0.106439
\(747\) 4.23970 0.155122
\(748\) 0 0
\(749\) −54.4871 −1.99092
\(750\) −104.124 −3.80208
\(751\) −11.3947 −0.415799 −0.207900 0.978150i \(-0.566663\pi\)
−0.207900 + 0.978150i \(0.566663\pi\)
\(752\) 0.862845 0.0314647
\(753\) −23.3527 −0.851021
\(754\) −2.19045 −0.0797713
\(755\) −79.1767 −2.88153
\(756\) −65.9895 −2.40001
\(757\) −6.68211 −0.242865 −0.121433 0.992600i \(-0.538749\pi\)
−0.121433 + 0.992600i \(0.538749\pi\)
\(758\) 4.89402 0.177759
\(759\) 0 0
\(760\) 4.17017 0.151268
\(761\) 2.33169 0.0845239 0.0422619 0.999107i \(-0.486544\pi\)
0.0422619 + 0.999107i \(0.486544\pi\)
\(762\) −23.8710 −0.864754
\(763\) 41.8522 1.51515
\(764\) −2.76249 −0.0999435
\(765\) −144.049 −5.20809
\(766\) −12.3879 −0.447594
\(767\) 6.30142 0.227531
\(768\) 3.37860 0.121915
\(769\) 28.8994 1.04214 0.521070 0.853514i \(-0.325533\pi\)
0.521070 + 0.853514i \(0.325533\pi\)
\(770\) 0 0
\(771\) −83.6242 −3.01165
\(772\) 21.4775 0.772991
\(773\) −25.5506 −0.918991 −0.459495 0.888180i \(-0.651970\pi\)
−0.459495 + 0.888180i \(0.651970\pi\)
\(774\) −50.8824 −1.82893
\(775\) −19.2763 −0.692427
\(776\) 0.809369 0.0290546
\(777\) −42.8274 −1.53642
\(778\) −13.7527 −0.493059
\(779\) −7.30933 −0.261884
\(780\) 9.43175 0.337711
\(781\) 0 0
\(782\) 26.1021 0.933411
\(783\) 59.8630 2.13933
\(784\) 6.01048 0.214660
\(785\) 37.0966 1.32403
\(786\) −11.0218 −0.393134
\(787\) 10.5042 0.374436 0.187218 0.982318i \(-0.440053\pi\)
0.187218 + 0.982318i \(0.440053\pi\)
\(788\) −18.9032 −0.673399
\(789\) 56.8128 2.02259
\(790\) −4.36000 −0.155122
\(791\) 21.7741 0.774196
\(792\) 0 0
\(793\) 2.94989 0.104754
\(794\) 10.0289 0.355914
\(795\) −95.7261 −3.39505
\(796\) 7.76283 0.275146
\(797\) −6.50041 −0.230256 −0.115128 0.993351i \(-0.536728\pi\)
−0.115128 + 0.993351i \(0.536728\pi\)
\(798\) 12.1866 0.431401
\(799\) 3.54192 0.125304
\(800\) 12.3903 0.438063
\(801\) 69.3115 2.44900
\(802\) 22.2869 0.786979
\(803\) 0 0
\(804\) 6.09296 0.214882
\(805\) 95.6467 3.37110
\(806\) 1.04147 0.0366841
\(807\) 60.2364 2.12042
\(808\) −12.0493 −0.423893
\(809\) −30.5543 −1.07423 −0.537116 0.843509i \(-0.680486\pi\)
−0.537116 + 0.843509i \(0.680486\pi\)
\(810\) −152.487 −5.35783
\(811\) 5.81000 0.204017 0.102008 0.994784i \(-0.467473\pi\)
0.102008 + 0.994784i \(0.467473\pi\)
\(812\) −11.8026 −0.414189
\(813\) −102.055 −3.57922
\(814\) 0 0
\(815\) 20.2044 0.707730
\(816\) 13.8689 0.485509
\(817\) 6.04669 0.211547
\(818\) 5.88120 0.205631
\(819\) 20.3188 0.709997
\(820\) −30.4811 −1.06445
\(821\) −43.0690 −1.50312 −0.751559 0.659666i \(-0.770698\pi\)
−0.751559 + 0.659666i \(0.770698\pi\)
\(822\) 4.23604 0.147749
\(823\) −20.6760 −0.720722 −0.360361 0.932813i \(-0.617346\pi\)
−0.360361 + 0.932813i \(0.617346\pi\)
\(824\) 5.64096 0.196512
\(825\) 0 0
\(826\) 33.9534 1.18139
\(827\) 35.8535 1.24675 0.623374 0.781924i \(-0.285761\pi\)
0.623374 + 0.781924i \(0.285761\pi\)
\(828\) 53.5081 1.85954
\(829\) 10.6519 0.369956 0.184978 0.982743i \(-0.440779\pi\)
0.184978 + 0.982743i \(0.440779\pi\)
\(830\) −2.10106 −0.0729288
\(831\) 110.697 3.84004
\(832\) −0.669426 −0.0232082
\(833\) 24.6726 0.854856
\(834\) 5.14988 0.178326
\(835\) 19.2468 0.666064
\(836\) 0 0
\(837\) −28.4624 −0.983804
\(838\) 3.24502 0.112097
\(839\) −46.8329 −1.61685 −0.808425 0.588599i \(-0.799680\pi\)
−0.808425 + 0.588599i \(0.799680\pi\)
\(840\) 50.8202 1.75346
\(841\) −18.2932 −0.630799
\(842\) −19.2545 −0.663553
\(843\) −32.8010 −1.12973
\(844\) −6.88377 −0.236949
\(845\) 52.3434 1.80067
\(846\) 7.26077 0.249630
\(847\) 0 0
\(848\) 6.79423 0.233315
\(849\) 49.2392 1.68988
\(850\) 50.8613 1.74453
\(851\) 22.3464 0.766026
\(852\) 40.5204 1.38821
\(853\) 35.3351 1.20985 0.604926 0.796282i \(-0.293203\pi\)
0.604926 + 0.796282i \(0.293203\pi\)
\(854\) 15.8946 0.543902
\(855\) 35.0916 1.20011
\(856\) 15.1059 0.516309
\(857\) −4.94229 −0.168826 −0.0844128 0.996431i \(-0.526901\pi\)
−0.0844128 + 0.996431i \(0.526901\pi\)
\(858\) 0 0
\(859\) 10.1131 0.345054 0.172527 0.985005i \(-0.444807\pi\)
0.172527 + 0.985005i \(0.444807\pi\)
\(860\) 25.2157 0.859849
\(861\) −89.0760 −3.03570
\(862\) −23.7828 −0.810046
\(863\) 12.9844 0.441995 0.220998 0.975274i \(-0.429069\pi\)
0.220998 + 0.975274i \(0.429069\pi\)
\(864\) 18.2948 0.622402
\(865\) 38.9425 1.32409
\(866\) −17.1402 −0.582449
\(867\) −0.505186 −0.0171570
\(868\) 5.61164 0.190471
\(869\) 0 0
\(870\) −46.1020 −1.56300
\(871\) −1.20724 −0.0409058
\(872\) −11.6030 −0.392929
\(873\) 6.81077 0.230510
\(874\) −6.35872 −0.215087
\(875\) 111.163 3.75801
\(876\) 33.4904 1.13154
\(877\) 44.8363 1.51401 0.757007 0.653407i \(-0.226661\pi\)
0.757007 + 0.653407i \(0.226661\pi\)
\(878\) −0.144467 −0.00487552
\(879\) −35.5567 −1.19930
\(880\) 0 0
\(881\) −21.0954 −0.710722 −0.355361 0.934729i \(-0.615642\pi\)
−0.355361 + 0.934729i \(0.615642\pi\)
\(882\) 50.5777 1.70304
\(883\) 53.8142 1.81099 0.905496 0.424355i \(-0.139499\pi\)
0.905496 + 0.424355i \(0.139499\pi\)
\(884\) −2.74795 −0.0924235
\(885\) 132.625 4.45814
\(886\) −19.6940 −0.661634
\(887\) −28.4930 −0.956701 −0.478351 0.878169i \(-0.658765\pi\)
−0.478351 + 0.878169i \(0.658765\pi\)
\(888\) 11.8734 0.398445
\(889\) 25.4848 0.854731
\(890\) −34.3486 −1.15137
\(891\) 0 0
\(892\) 14.5212 0.486204
\(893\) −0.862845 −0.0288740
\(894\) −53.0009 −1.77261
\(895\) −70.0092 −2.34015
\(896\) −3.60700 −0.120502
\(897\) −14.3816 −0.480189
\(898\) −30.0122 −1.00152
\(899\) −5.09065 −0.169783
\(900\) 104.263 3.47544
\(901\) 27.8899 0.929146
\(902\) 0 0
\(903\) 73.6887 2.45221
\(904\) −6.03660 −0.200774
\(905\) −73.3438 −2.43803
\(906\) 64.1476 2.13116
\(907\) −19.0581 −0.632815 −0.316408 0.948623i \(-0.602477\pi\)
−0.316408 + 0.948623i \(0.602477\pi\)
\(908\) −18.1073 −0.600912
\(909\) −101.394 −3.36302
\(910\) −10.0694 −0.333796
\(911\) −8.41671 −0.278858 −0.139429 0.990232i \(-0.544527\pi\)
−0.139429 + 0.990232i \(0.544527\pi\)
\(912\) −3.37860 −0.111877
\(913\) 0 0
\(914\) 0.810437 0.0268069
\(915\) 62.0859 2.05250
\(916\) −25.3580 −0.837850
\(917\) 11.7669 0.388577
\(918\) 75.0990 2.47864
\(919\) −10.5289 −0.347315 −0.173658 0.984806i \(-0.555559\pi\)
−0.173658 + 0.984806i \(0.555559\pi\)
\(920\) −26.5169 −0.874238
\(921\) −72.3461 −2.38388
\(922\) 29.4451 0.969721
\(923\) −8.02860 −0.264265
\(924\) 0 0
\(925\) 43.5432 1.43169
\(926\) 24.9830 0.820992
\(927\) 47.4682 1.55906
\(928\) 3.27213 0.107413
\(929\) −19.5004 −0.639787 −0.319893 0.947454i \(-0.603647\pi\)
−0.319893 + 0.947454i \(0.603647\pi\)
\(930\) 21.9196 0.718773
\(931\) −6.01048 −0.196986
\(932\) −21.9886 −0.720261
\(933\) 30.1182 0.986024
\(934\) −21.6826 −0.709477
\(935\) 0 0
\(936\) −5.63316 −0.184126
\(937\) 46.4018 1.51588 0.757941 0.652323i \(-0.226206\pi\)
0.757941 + 0.652323i \(0.226206\pi\)
\(938\) −6.50487 −0.212392
\(939\) −98.7495 −3.22257
\(940\) −3.59821 −0.117361
\(941\) −17.4235 −0.567991 −0.283996 0.958826i \(-0.591660\pi\)
−0.283996 + 0.958826i \(0.591660\pi\)
\(942\) −30.0550 −0.979245
\(943\) 46.4780 1.51353
\(944\) −9.41317 −0.306373
\(945\) 275.187 8.95184
\(946\) 0 0
\(947\) −18.8103 −0.611254 −0.305627 0.952151i \(-0.598866\pi\)
−0.305627 + 0.952151i \(0.598866\pi\)
\(948\) 3.53240 0.114727
\(949\) −6.63570 −0.215404
\(950\) −12.3903 −0.401994
\(951\) −13.0499 −0.423171
\(952\) −14.8065 −0.479882
\(953\) 54.4674 1.76437 0.882187 0.470900i \(-0.156071\pi\)
0.882187 + 0.470900i \(0.156071\pi\)
\(954\) 57.1729 1.85104
\(955\) 11.5201 0.372780
\(956\) 15.2971 0.494744
\(957\) 0 0
\(958\) 30.8590 0.997009
\(959\) −4.52242 −0.146037
\(960\) −14.0893 −0.454731
\(961\) −28.5796 −0.921923
\(962\) −2.35256 −0.0758496
\(963\) 127.115 4.09622
\(964\) 16.0409 0.516642
\(965\) −89.5646 −2.88319
\(966\) −77.4913 −2.49324
\(967\) 51.7050 1.66272 0.831361 0.555733i \(-0.187562\pi\)
0.831361 + 0.555733i \(0.187562\pi\)
\(968\) 0 0
\(969\) −13.8689 −0.445534
\(970\) −3.37520 −0.108371
\(971\) 41.0651 1.31784 0.658920 0.752213i \(-0.271013\pi\)
0.658920 + 0.752213i \(0.271013\pi\)
\(972\) 68.6575 2.20219
\(973\) −5.49803 −0.176259
\(974\) −5.68001 −0.181999
\(975\) −28.0234 −0.897466
\(976\) −4.40660 −0.141052
\(977\) −15.0675 −0.482052 −0.241026 0.970519i \(-0.577484\pi\)
−0.241026 + 0.970519i \(0.577484\pi\)
\(978\) −16.3693 −0.523431
\(979\) 0 0
\(980\) −25.0647 −0.800663
\(981\) −97.6386 −3.11736
\(982\) 29.9204 0.954799
\(983\) −24.2754 −0.774263 −0.387132 0.922024i \(-0.626534\pi\)
−0.387132 + 0.922024i \(0.626534\pi\)
\(984\) 24.6953 0.787257
\(985\) 78.8295 2.51172
\(986\) 13.4319 0.427758
\(987\) −10.5152 −0.334701
\(988\) 0.669426 0.0212973
\(989\) −38.4492 −1.22261
\(990\) 0 0
\(991\) −18.0803 −0.574339 −0.287169 0.957880i \(-0.592714\pi\)
−0.287169 + 0.957880i \(0.592714\pi\)
\(992\) −1.55576 −0.0493955
\(993\) 104.050 3.30193
\(994\) −43.2598 −1.37212
\(995\) −32.3723 −1.02627
\(996\) 1.70224 0.0539376
\(997\) 17.0357 0.539527 0.269764 0.962927i \(-0.413054\pi\)
0.269764 + 0.962927i \(0.413054\pi\)
\(998\) −2.89019 −0.0914873
\(999\) 64.2934 2.03415
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 4598.2.a.cb.1.8 yes 8
11.10 odd 2 4598.2.a.by.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.8 8 11.10 odd 2
4598.2.a.cb.1.8 yes 8 1.1 even 1 trivial