Properties

Label 177.4.a.d.1.2
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.77847\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.77847 q^{2} +3.00000 q^{3} +6.27681 q^{4} -5.74283 q^{5} -11.3354 q^{6} -24.4226 q^{7} +6.51103 q^{8} +9.00000 q^{9} +21.6991 q^{10} +9.31943 q^{11} +18.8304 q^{12} +17.0721 q^{13} +92.2799 q^{14} -17.2285 q^{15} -74.8162 q^{16} +19.6787 q^{17} -34.0062 q^{18} -15.1270 q^{19} -36.0466 q^{20} -73.2678 q^{21} -35.2132 q^{22} +108.469 q^{23} +19.5331 q^{24} -92.0199 q^{25} -64.5065 q^{26} +27.0000 q^{27} -153.296 q^{28} +237.137 q^{29} +65.0973 q^{30} +184.628 q^{31} +230.602 q^{32} +27.9583 q^{33} -74.3553 q^{34} +140.255 q^{35} +56.4913 q^{36} +155.472 q^{37} +57.1569 q^{38} +51.2164 q^{39} -37.3917 q^{40} +261.745 q^{41} +276.840 q^{42} +127.647 q^{43} +58.4963 q^{44} -51.6855 q^{45} -409.848 q^{46} +555.816 q^{47} -224.448 q^{48} +253.463 q^{49} +347.694 q^{50} +59.0361 q^{51} +107.158 q^{52} -220.668 q^{53} -102.019 q^{54} -53.5199 q^{55} -159.016 q^{56} -45.3810 q^{57} -896.014 q^{58} +59.0000 q^{59} -108.140 q^{60} -436.694 q^{61} -697.610 q^{62} -219.803 q^{63} -272.793 q^{64} -98.0424 q^{65} -105.639 q^{66} +924.006 q^{67} +123.519 q^{68} +325.408 q^{69} -529.948 q^{70} -937.019 q^{71} +58.5992 q^{72} +13.0348 q^{73} -587.446 q^{74} -276.060 q^{75} -94.9493 q^{76} -227.605 q^{77} -193.519 q^{78} +140.714 q^{79} +429.657 q^{80} +81.0000 q^{81} -988.996 q^{82} -26.4265 q^{83} -459.888 q^{84} -113.011 q^{85} -482.310 q^{86} +711.411 q^{87} +60.6791 q^{88} -136.748 q^{89} +195.292 q^{90} -416.946 q^{91} +680.842 q^{92} +553.883 q^{93} -2100.13 q^{94} +86.8719 q^{95} +691.806 q^{96} -360.785 q^{97} -957.701 q^{98} +83.8749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + 21q^{10} + 67q^{11} + 102q^{12} + 33q^{13} + 79q^{14} + 126q^{15} - 30q^{16} + 139q^{17} + 54q^{18} + 64q^{19} + 117q^{20} + 159q^{21} - 84q^{22} + 226q^{23} + 153q^{24} + 96q^{25} + 24q^{26} + 216q^{27} + 34q^{28} + 456q^{29} + 63q^{30} + 124q^{31} + 174q^{32} + 201q^{33} - 114q^{34} + 556q^{35} + 306q^{36} + 127q^{37} + 237q^{38} + 99q^{39} - 188q^{40} + 425q^{41} + 237q^{42} - 115q^{43} + 510q^{44} + 378q^{45} - 711q^{46} + 420q^{47} - 90q^{48} + 171q^{49} - 137q^{50} + 417q^{51} - 922q^{52} + 98q^{53} + 162q^{54} - 616q^{55} - 412q^{56} + 192q^{57} - 1548q^{58} + 472q^{59} + 351q^{60} - 1254q^{61} - 766q^{62} + 477q^{63} - 2019q^{64} - 734q^{65} - 252q^{66} - 1010q^{67} - 503q^{68} + 678q^{69} - 2956q^{70} - 17q^{71} + 459q^{72} - 1180q^{73} - 1228q^{74} + 288q^{75} - 2008q^{76} + 441q^{77} + 72q^{78} - 873q^{79} - 865q^{80} + 648q^{81} - 3645q^{82} + 759q^{83} + 102q^{84} - 850q^{85} - 1226q^{86} + 1368q^{87} - 3047q^{88} + 988q^{89} + 189q^{90} - 2111q^{91} - 1062q^{92} + 372q^{93} - 2240q^{94} + 1822q^{95} + 522q^{96} - 668q^{97} - 1368q^{98} + 603q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.77847 −1.33589 −0.667945 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(3\) 3.00000 0.577350
\(4\) 6.27681 0.784601
\(5\) −5.74283 −0.513655 −0.256827 0.966457i \(-0.582677\pi\)
−0.256827 + 0.966457i \(0.582677\pi\)
\(6\) −11.3354 −0.771276
\(7\) −24.4226 −1.31870 −0.659348 0.751838i \(-0.729168\pi\)
−0.659348 + 0.751838i \(0.729168\pi\)
\(8\) 6.51103 0.287749
\(9\) 9.00000 0.333333
\(10\) 21.6991 0.686186
\(11\) 9.31943 0.255447 0.127723 0.991810i \(-0.459233\pi\)
0.127723 + 0.991810i \(0.459233\pi\)
\(12\) 18.8304 0.452990
\(13\) 17.0721 0.364227 0.182114 0.983277i \(-0.441706\pi\)
0.182114 + 0.983277i \(0.441706\pi\)
\(14\) 92.2799 1.76163
\(15\) −17.2285 −0.296559
\(16\) −74.8162 −1.16900
\(17\) 19.6787 0.280752 0.140376 0.990098i \(-0.455169\pi\)
0.140376 + 0.990098i \(0.455169\pi\)
\(18\) −34.0062 −0.445297
\(19\) −15.1270 −0.182651 −0.0913256 0.995821i \(-0.529110\pi\)
−0.0913256 + 0.995821i \(0.529110\pi\)
\(20\) −36.0466 −0.403014
\(21\) −73.2678 −0.761349
\(22\) −35.2132 −0.341249
\(23\) 108.469 0.983367 0.491683 0.870774i \(-0.336382\pi\)
0.491683 + 0.870774i \(0.336382\pi\)
\(24\) 19.5331 0.166132
\(25\) −92.0199 −0.736159
\(26\) −64.5065 −0.486568
\(27\) 27.0000 0.192450
\(28\) −153.296 −1.03465
\(29\) 237.137 1.51846 0.759228 0.650825i \(-0.225577\pi\)
0.759228 + 0.650825i \(0.225577\pi\)
\(30\) 65.0973 0.396169
\(31\) 184.628 1.06968 0.534841 0.844953i \(-0.320372\pi\)
0.534841 + 0.844953i \(0.320372\pi\)
\(32\) 230.602 1.27391
\(33\) 27.9583 0.147482
\(34\) −74.3553 −0.375054
\(35\) 140.255 0.677354
\(36\) 56.4913 0.261534
\(37\) 155.472 0.690795 0.345398 0.938456i \(-0.387744\pi\)
0.345398 + 0.938456i \(0.387744\pi\)
\(38\) 57.1569 0.244002
\(39\) 51.2164 0.210287
\(40\) −37.3917 −0.147804
\(41\) 261.745 0.997019 0.498509 0.866884i \(-0.333881\pi\)
0.498509 + 0.866884i \(0.333881\pi\)
\(42\) 276.840 1.01708
\(43\) 127.647 0.452697 0.226349 0.974046i \(-0.427321\pi\)
0.226349 + 0.974046i \(0.427321\pi\)
\(44\) 58.4963 0.200424
\(45\) −51.6855 −0.171218
\(46\) −409.848 −1.31367
\(47\) 555.816 1.72498 0.862490 0.506075i \(-0.168904\pi\)
0.862490 + 0.506075i \(0.168904\pi\)
\(48\) −224.448 −0.674924
\(49\) 253.463 0.738959
\(50\) 347.694 0.983427
\(51\) 59.0361 0.162092
\(52\) 107.158 0.285773
\(53\) −220.668 −0.571908 −0.285954 0.958243i \(-0.592311\pi\)
−0.285954 + 0.958243i \(0.592311\pi\)
\(54\) −102.019 −0.257092
\(55\) −53.5199 −0.131211
\(56\) −159.016 −0.379454
\(57\) −45.3810 −0.105454
\(58\) −896.014 −2.02849
\(59\) 59.0000 0.130189
\(60\) −108.140 −0.232680
\(61\) −436.694 −0.916606 −0.458303 0.888796i \(-0.651543\pi\)
−0.458303 + 0.888796i \(0.651543\pi\)
\(62\) −697.610 −1.42898
\(63\) −219.803 −0.439565
\(64\) −272.793 −0.532799
\(65\) −98.0424 −0.187087
\(66\) −105.639 −0.197020
\(67\) 924.006 1.68486 0.842428 0.538809i \(-0.181126\pi\)
0.842428 + 0.538809i \(0.181126\pi\)
\(68\) 123.519 0.220278
\(69\) 325.408 0.567747
\(70\) −529.948 −0.904870
\(71\) −937.019 −1.56625 −0.783125 0.621865i \(-0.786375\pi\)
−0.783125 + 0.621865i \(0.786375\pi\)
\(72\) 58.5992 0.0959165
\(73\) 13.0348 0.0208988 0.0104494 0.999945i \(-0.496674\pi\)
0.0104494 + 0.999945i \(0.496674\pi\)
\(74\) −587.446 −0.922826
\(75\) −276.060 −0.425022
\(76\) −94.9493 −0.143308
\(77\) −227.605 −0.336857
\(78\) −193.519 −0.280920
\(79\) 140.714 0.200400 0.100200 0.994967i \(-0.468052\pi\)
0.100200 + 0.994967i \(0.468052\pi\)
\(80\) 429.657 0.600463
\(81\) 81.0000 0.111111
\(82\) −988.996 −1.33191
\(83\) −26.4265 −0.0349480 −0.0174740 0.999847i \(-0.505562\pi\)
−0.0174740 + 0.999847i \(0.505562\pi\)
\(84\) −459.888 −0.597355
\(85\) −113.011 −0.144210
\(86\) −482.310 −0.604753
\(87\) 711.411 0.876681
\(88\) 60.6791 0.0735047
\(89\) −136.748 −0.162869 −0.0814343 0.996679i \(-0.525950\pi\)
−0.0814343 + 0.996679i \(0.525950\pi\)
\(90\) 195.292 0.228729
\(91\) −416.946 −0.480305
\(92\) 680.842 0.771551
\(93\) 553.883 0.617581
\(94\) −2100.13 −2.30438
\(95\) 86.8719 0.0938196
\(96\) 691.806 0.735491
\(97\) −360.785 −0.377651 −0.188826 0.982011i \(-0.560468\pi\)
−0.188826 + 0.982011i \(0.560468\pi\)
\(98\) −957.701 −0.987168
\(99\) 83.8749 0.0851489
\(100\) −577.591 −0.577591
\(101\) 421.340 0.415098 0.207549 0.978225i \(-0.433451\pi\)
0.207549 + 0.978225i \(0.433451\pi\)
\(102\) −223.066 −0.216537
\(103\) −1389.28 −1.32903 −0.664515 0.747275i \(-0.731362\pi\)
−0.664515 + 0.747275i \(0.731362\pi\)
\(104\) 111.157 0.104806
\(105\) 420.765 0.391071
\(106\) 833.788 0.764006
\(107\) −339.530 −0.306763 −0.153382 0.988167i \(-0.549016\pi\)
−0.153382 + 0.988167i \(0.549016\pi\)
\(108\) 169.474 0.150997
\(109\) 297.022 0.261005 0.130503 0.991448i \(-0.458341\pi\)
0.130503 + 0.991448i \(0.458341\pi\)
\(110\) 202.223 0.175284
\(111\) 466.416 0.398831
\(112\) 1827.20 1.54156
\(113\) 1553.81 1.29354 0.646769 0.762686i \(-0.276120\pi\)
0.646769 + 0.762686i \(0.276120\pi\)
\(114\) 171.471 0.140875
\(115\) −622.922 −0.505111
\(116\) 1488.46 1.19138
\(117\) 153.649 0.121409
\(118\) −222.930 −0.173918
\(119\) −480.605 −0.370227
\(120\) −112.175 −0.0853346
\(121\) −1244.15 −0.934747
\(122\) 1650.03 1.22448
\(123\) 785.236 0.575629
\(124\) 1158.87 0.839273
\(125\) 1246.31 0.891786
\(126\) 830.519 0.587211
\(127\) 1320.50 0.922642 0.461321 0.887233i \(-0.347376\pi\)
0.461321 + 0.887233i \(0.347376\pi\)
\(128\) −814.078 −0.562148
\(129\) 382.941 0.261365
\(130\) 370.450 0.249928
\(131\) −1738.41 −1.15943 −0.579716 0.814819i \(-0.696836\pi\)
−0.579716 + 0.814819i \(0.696836\pi\)
\(132\) 175.489 0.115715
\(133\) 369.441 0.240861
\(134\) −3491.33 −2.25078
\(135\) −155.056 −0.0988529
\(136\) 128.129 0.0807863
\(137\) −776.721 −0.484378 −0.242189 0.970229i \(-0.577865\pi\)
−0.242189 + 0.970229i \(0.577865\pi\)
\(138\) −1229.54 −0.758447
\(139\) 855.546 0.522061 0.261030 0.965331i \(-0.415938\pi\)
0.261030 + 0.965331i \(0.415938\pi\)
\(140\) 880.353 0.531453
\(141\) 1667.45 0.995917
\(142\) 3540.49 2.09234
\(143\) 159.103 0.0930407
\(144\) −673.345 −0.389667
\(145\) −1361.84 −0.779962
\(146\) −49.2517 −0.0279185
\(147\) 760.389 0.426638
\(148\) 975.867 0.541999
\(149\) 2001.36 1.10039 0.550194 0.835037i \(-0.314554\pi\)
0.550194 + 0.835037i \(0.314554\pi\)
\(150\) 1043.08 0.567782
\(151\) 1656.52 0.892750 0.446375 0.894846i \(-0.352715\pi\)
0.446375 + 0.894846i \(0.352715\pi\)
\(152\) −98.4924 −0.0525578
\(153\) 177.108 0.0935841
\(154\) 859.996 0.450003
\(155\) −1060.29 −0.549447
\(156\) 321.475 0.164991
\(157\) 649.443 0.330135 0.165068 0.986282i \(-0.447216\pi\)
0.165068 + 0.986282i \(0.447216\pi\)
\(158\) −531.685 −0.267713
\(159\) −662.005 −0.330191
\(160\) −1324.31 −0.654349
\(161\) −2649.10 −1.29676
\(162\) −306.056 −0.148432
\(163\) 508.548 0.244372 0.122186 0.992507i \(-0.461010\pi\)
0.122186 + 0.992507i \(0.461010\pi\)
\(164\) 1642.93 0.782262
\(165\) −160.560 −0.0757549
\(166\) 99.8517 0.0466867
\(167\) 1419.46 0.657733 0.328867 0.944376i \(-0.393333\pi\)
0.328867 + 0.944376i \(0.393333\pi\)
\(168\) −477.048 −0.219078
\(169\) −1905.54 −0.867338
\(170\) 427.010 0.192648
\(171\) −136.143 −0.0608837
\(172\) 801.215 0.355187
\(173\) −929.169 −0.408343 −0.204172 0.978935i \(-0.565450\pi\)
−0.204172 + 0.978935i \(0.565450\pi\)
\(174\) −2688.04 −1.17115
\(175\) 2247.36 0.970770
\(176\) −697.244 −0.298618
\(177\) 177.000 0.0751646
\(178\) 516.699 0.217574
\(179\) −2212.72 −0.923948 −0.461974 0.886893i \(-0.652859\pi\)
−0.461974 + 0.886893i \(0.652859\pi\)
\(180\) −324.420 −0.134338
\(181\) −46.5937 −0.0191342 −0.00956709 0.999954i \(-0.503045\pi\)
−0.00956709 + 0.999954i \(0.503045\pi\)
\(182\) 1575.41 0.641635
\(183\) −1310.08 −0.529203
\(184\) 706.247 0.282963
\(185\) −892.849 −0.354830
\(186\) −2092.83 −0.825020
\(187\) 183.394 0.0717172
\(188\) 3488.75 1.35342
\(189\) −659.410 −0.253783
\(190\) −328.242 −0.125333
\(191\) −2035.98 −0.771302 −0.385651 0.922645i \(-0.626023\pi\)
−0.385651 + 0.922645i \(0.626023\pi\)
\(192\) −818.379 −0.307611
\(193\) −1530.18 −0.570697 −0.285349 0.958424i \(-0.592109\pi\)
−0.285349 + 0.958424i \(0.592109\pi\)
\(194\) 1363.21 0.504500
\(195\) −294.127 −0.108015
\(196\) 1590.94 0.579788
\(197\) 2461.81 0.890337 0.445169 0.895447i \(-0.353144\pi\)
0.445169 + 0.895447i \(0.353144\pi\)
\(198\) −316.918 −0.113750
\(199\) 1442.27 0.513768 0.256884 0.966442i \(-0.417304\pi\)
0.256884 + 0.966442i \(0.417304\pi\)
\(200\) −599.144 −0.211829
\(201\) 2772.02 0.972752
\(202\) −1592.02 −0.554525
\(203\) −5791.50 −2.00238
\(204\) 370.558 0.127178
\(205\) −1503.16 −0.512123
\(206\) 5249.36 1.77544
\(207\) 976.225 0.327789
\(208\) −1277.27 −0.425783
\(209\) −140.975 −0.0466577
\(210\) −1589.84 −0.522427
\(211\) −732.346 −0.238942 −0.119471 0.992838i \(-0.538120\pi\)
−0.119471 + 0.992838i \(0.538120\pi\)
\(212\) −1385.09 −0.448720
\(213\) −2811.06 −0.904274
\(214\) 1282.90 0.409802
\(215\) −733.055 −0.232530
\(216\) 175.798 0.0553774
\(217\) −4509.09 −1.41058
\(218\) −1122.29 −0.348674
\(219\) 39.1045 0.0120659
\(220\) −335.934 −0.102949
\(221\) 335.957 0.102258
\(222\) −1762.34 −0.532794
\(223\) 2928.26 0.879332 0.439666 0.898161i \(-0.355097\pi\)
0.439666 + 0.898161i \(0.355097\pi\)
\(224\) −5631.90 −1.67990
\(225\) −828.179 −0.245386
\(226\) −5871.00 −1.72802
\(227\) 3909.72 1.14316 0.571580 0.820546i \(-0.306331\pi\)
0.571580 + 0.820546i \(0.306331\pi\)
\(228\) −284.848 −0.0827391
\(229\) 5580.68 1.61040 0.805200 0.593003i \(-0.202058\pi\)
0.805200 + 0.593003i \(0.202058\pi\)
\(230\) 2353.69 0.674772
\(231\) −682.814 −0.194484
\(232\) 1544.01 0.436935
\(233\) −3793.50 −1.06661 −0.533306 0.845923i \(-0.679050\pi\)
−0.533306 + 0.845923i \(0.679050\pi\)
\(234\) −580.558 −0.162189
\(235\) −3191.96 −0.886043
\(236\) 370.332 0.102146
\(237\) 422.143 0.115701
\(238\) 1815.95 0.494582
\(239\) 4995.54 1.35203 0.676013 0.736889i \(-0.263706\pi\)
0.676013 + 0.736889i \(0.263706\pi\)
\(240\) 1288.97 0.346678
\(241\) 1674.51 0.447572 0.223786 0.974638i \(-0.428158\pi\)
0.223786 + 0.974638i \(0.428158\pi\)
\(242\) 4700.97 1.24872
\(243\) 243.000 0.0641500
\(244\) −2741.05 −0.719170
\(245\) −1455.60 −0.379570
\(246\) −2966.99 −0.768977
\(247\) −258.250 −0.0665266
\(248\) 1202.12 0.307800
\(249\) −79.2795 −0.0201773
\(250\) −4709.14 −1.19133
\(251\) 4489.35 1.12895 0.564473 0.825452i \(-0.309080\pi\)
0.564473 + 0.825452i \(0.309080\pi\)
\(252\) −1379.66 −0.344883
\(253\) 1010.87 0.251198
\(254\) −4989.47 −1.23255
\(255\) −339.034 −0.0832595
\(256\) 5258.31 1.28377
\(257\) 5099.27 1.23768 0.618840 0.785517i \(-0.287603\pi\)
0.618840 + 0.785517i \(0.287603\pi\)
\(258\) −1446.93 −0.349154
\(259\) −3797.03 −0.910949
\(260\) −615.393 −0.146789
\(261\) 2134.23 0.506152
\(262\) 6568.52 1.54887
\(263\) 7949.30 1.86378 0.931892 0.362737i \(-0.118158\pi\)
0.931892 + 0.362737i \(0.118158\pi\)
\(264\) 182.037 0.0424379
\(265\) 1267.26 0.293763
\(266\) −1395.92 −0.321764
\(267\) −410.245 −0.0940322
\(268\) 5799.81 1.32194
\(269\) −3831.43 −0.868426 −0.434213 0.900810i \(-0.642973\pi\)
−0.434213 + 0.900810i \(0.642973\pi\)
\(270\) 585.876 0.132056
\(271\) −6824.86 −1.52982 −0.764909 0.644139i \(-0.777216\pi\)
−0.764909 + 0.644139i \(0.777216\pi\)
\(272\) −1472.28 −0.328200
\(273\) −1250.84 −0.277304
\(274\) 2934.82 0.647075
\(275\) −857.573 −0.188049
\(276\) 2042.52 0.445455
\(277\) −40.1056 −0.00869931 −0.00434966 0.999991i \(-0.501385\pi\)
−0.00434966 + 0.999991i \(0.501385\pi\)
\(278\) −3232.65 −0.697416
\(279\) 1661.65 0.356560
\(280\) 913.203 0.194908
\(281\) −2405.83 −0.510746 −0.255373 0.966843i \(-0.582198\pi\)
−0.255373 + 0.966843i \(0.582198\pi\)
\(282\) −6300.39 −1.33044
\(283\) 8232.15 1.72915 0.864577 0.502501i \(-0.167587\pi\)
0.864577 + 0.502501i \(0.167587\pi\)
\(284\) −5881.49 −1.22888
\(285\) 260.616 0.0541668
\(286\) −601.164 −0.124292
\(287\) −6392.50 −1.31476
\(288\) 2075.42 0.424636
\(289\) −4525.75 −0.921178
\(290\) 5145.66 1.04194
\(291\) −1082.36 −0.218037
\(292\) 81.8171 0.0163972
\(293\) 1902.22 0.379280 0.189640 0.981854i \(-0.439268\pi\)
0.189640 + 0.981854i \(0.439268\pi\)
\(294\) −2873.10 −0.569941
\(295\) −338.827 −0.0668721
\(296\) 1012.28 0.198776
\(297\) 251.625 0.0491608
\(298\) −7562.07 −1.47000
\(299\) 1851.80 0.358169
\(300\) −1732.77 −0.333472
\(301\) −3117.47 −0.596970
\(302\) −6259.09 −1.19262
\(303\) 1264.02 0.239657
\(304\) 1131.74 0.213520
\(305\) 2507.86 0.470819
\(306\) −669.198 −0.125018
\(307\) 9226.20 1.71520 0.857601 0.514315i \(-0.171954\pi\)
0.857601 + 0.514315i \(0.171954\pi\)
\(308\) −1428.63 −0.264298
\(309\) −4167.85 −0.767316
\(310\) 4006.26 0.734000
\(311\) −2395.89 −0.436844 −0.218422 0.975854i \(-0.570091\pi\)
−0.218422 + 0.975854i \(0.570091\pi\)
\(312\) 333.471 0.0605099
\(313\) −10071.9 −1.81884 −0.909421 0.415878i \(-0.863474\pi\)
−0.909421 + 0.415878i \(0.863474\pi\)
\(314\) −2453.90 −0.441024
\(315\) 1262.29 0.225785
\(316\) 883.238 0.157234
\(317\) 1068.39 0.189296 0.0946478 0.995511i \(-0.469828\pi\)
0.0946478 + 0.995511i \(0.469828\pi\)
\(318\) 2501.36 0.441099
\(319\) 2209.98 0.387885
\(320\) 1566.60 0.273674
\(321\) −1018.59 −0.177110
\(322\) 10009.6 1.73233
\(323\) −297.680 −0.0512797
\(324\) 508.421 0.0871779
\(325\) −1570.98 −0.268129
\(326\) −1921.53 −0.326454
\(327\) 891.066 0.150691
\(328\) 1704.23 0.286892
\(329\) −13574.5 −2.27472
\(330\) 606.670 0.101200
\(331\) −9649.80 −1.60242 −0.801210 0.598384i \(-0.795810\pi\)
−0.801210 + 0.598384i \(0.795810\pi\)
\(332\) −165.874 −0.0274203
\(333\) 1399.25 0.230265
\(334\) −5363.40 −0.878659
\(335\) −5306.41 −0.865434
\(336\) 5481.61 0.890019
\(337\) −6617.95 −1.06974 −0.534871 0.844934i \(-0.679640\pi\)
−0.534871 + 0.844934i \(0.679640\pi\)
\(338\) 7200.03 1.15867
\(339\) 4661.42 0.746824
\(340\) −709.351 −0.113147
\(341\) 1720.63 0.273247
\(342\) 514.412 0.0813340
\(343\) 2186.73 0.344234
\(344\) 831.113 0.130263
\(345\) −1868.77 −0.291626
\(346\) 3510.83 0.545502
\(347\) 3888.20 0.601525 0.300763 0.953699i \(-0.402759\pi\)
0.300763 + 0.953699i \(0.402759\pi\)
\(348\) 4465.39 0.687845
\(349\) −8651.21 −1.32690 −0.663451 0.748220i \(-0.730909\pi\)
−0.663451 + 0.748220i \(0.730909\pi\)
\(350\) −8491.59 −1.29684
\(351\) 460.948 0.0700956
\(352\) 2149.08 0.325416
\(353\) 1996.55 0.301036 0.150518 0.988607i \(-0.451906\pi\)
0.150518 + 0.988607i \(0.451906\pi\)
\(354\) −668.789 −0.100412
\(355\) 5381.14 0.804511
\(356\) −858.343 −0.127787
\(357\) −1441.81 −0.213751
\(358\) 8360.70 1.23429
\(359\) 11674.1 1.71626 0.858128 0.513436i \(-0.171627\pi\)
0.858128 + 0.513436i \(0.171627\pi\)
\(360\) −336.526 −0.0492679
\(361\) −6630.17 −0.966639
\(362\) 176.053 0.0255611
\(363\) −3732.44 −0.539676
\(364\) −2617.09 −0.376848
\(365\) −74.8569 −0.0107348
\(366\) 4950.10 0.706956
\(367\) 11122.7 1.58201 0.791005 0.611810i \(-0.209558\pi\)
0.791005 + 0.611810i \(0.209558\pi\)
\(368\) −8115.27 −1.14956
\(369\) 2355.71 0.332340
\(370\) 3373.60 0.474014
\(371\) 5389.30 0.754173
\(372\) 3476.62 0.484554
\(373\) 7447.39 1.03381 0.516905 0.856043i \(-0.327084\pi\)
0.516905 + 0.856043i \(0.327084\pi\)
\(374\) −692.949 −0.0958063
\(375\) 3738.93 0.514873
\(376\) 3618.93 0.496362
\(377\) 4048.43 0.553063
\(378\) 2491.56 0.339026
\(379\) −2736.87 −0.370933 −0.185467 0.982651i \(-0.559380\pi\)
−0.185467 + 0.982651i \(0.559380\pi\)
\(380\) 545.278 0.0736110
\(381\) 3961.51 0.532688
\(382\) 7692.90 1.03037
\(383\) −3711.27 −0.495136 −0.247568 0.968871i \(-0.579631\pi\)
−0.247568 + 0.968871i \(0.579631\pi\)
\(384\) −2442.23 −0.324556
\(385\) 1307.10 0.173028
\(386\) 5781.72 0.762388
\(387\) 1148.82 0.150899
\(388\) −2264.58 −0.296306
\(389\) 68.7597 0.00896209 0.00448105 0.999990i \(-0.498574\pi\)
0.00448105 + 0.999990i \(0.498574\pi\)
\(390\) 1111.35 0.144296
\(391\) 2134.54 0.276082
\(392\) 1650.30 0.212635
\(393\) −5215.23 −0.669398
\(394\) −9301.85 −1.18939
\(395\) −808.100 −0.102936
\(396\) 526.466 0.0668079
\(397\) −9361.13 −1.18343 −0.591715 0.806147i \(-0.701549\pi\)
−0.591715 + 0.806147i \(0.701549\pi\)
\(398\) −5449.57 −0.686337
\(399\) 1108.32 0.139061
\(400\) 6884.57 0.860572
\(401\) −95.5960 −0.0119048 −0.00595241 0.999982i \(-0.501895\pi\)
−0.00595241 + 0.999982i \(0.501895\pi\)
\(402\) −10474.0 −1.29949
\(403\) 3151.99 0.389607
\(404\) 2644.67 0.325686
\(405\) −465.169 −0.0570727
\(406\) 21883.0 2.67496
\(407\) 1448.91 0.176461
\(408\) 384.386 0.0466420
\(409\) −11054.7 −1.33648 −0.668240 0.743946i \(-0.732952\pi\)
−0.668240 + 0.743946i \(0.732952\pi\)
\(410\) 5679.64 0.684140
\(411\) −2330.16 −0.279656
\(412\) −8720.26 −1.04276
\(413\) −1440.93 −0.171680
\(414\) −3688.63 −0.437890
\(415\) 151.763 0.0179512
\(416\) 3936.87 0.463992
\(417\) 2566.64 0.301412
\(418\) 532.670 0.0623295
\(419\) −4854.35 −0.565992 −0.282996 0.959121i \(-0.591328\pi\)
−0.282996 + 0.959121i \(0.591328\pi\)
\(420\) 2641.06 0.306834
\(421\) −5480.81 −0.634485 −0.317242 0.948344i \(-0.602757\pi\)
−0.317242 + 0.948344i \(0.602757\pi\)
\(422\) 2767.14 0.319200
\(423\) 5002.34 0.574993
\(424\) −1436.78 −0.164566
\(425\) −1810.83 −0.206678
\(426\) 10621.5 1.20801
\(427\) 10665.2 1.20872
\(428\) −2131.17 −0.240687
\(429\) 477.308 0.0537171
\(430\) 2769.82 0.310634
\(431\) 7366.56 0.823282 0.411641 0.911346i \(-0.364956\pi\)
0.411641 + 0.911346i \(0.364956\pi\)
\(432\) −2020.04 −0.224975
\(433\) 228.362 0.0253450 0.0126725 0.999920i \(-0.495966\pi\)
0.0126725 + 0.999920i \(0.495966\pi\)
\(434\) 17037.4 1.88439
\(435\) −4085.51 −0.450311
\(436\) 1864.35 0.204785
\(437\) −1640.82 −0.179613
\(438\) −147.755 −0.0161187
\(439\) −16897.8 −1.83710 −0.918550 0.395305i \(-0.870639\pi\)
−0.918550 + 0.395305i \(0.870639\pi\)
\(440\) −348.470 −0.0377560
\(441\) 2281.17 0.246320
\(442\) −1269.40 −0.136605
\(443\) 8663.70 0.929176 0.464588 0.885527i \(-0.346202\pi\)
0.464588 + 0.885527i \(0.346202\pi\)
\(444\) 2927.60 0.312923
\(445\) 785.323 0.0836582
\(446\) −11064.3 −1.17469
\(447\) 6004.08 0.635309
\(448\) 6662.31 0.702599
\(449\) −12014.8 −1.26284 −0.631418 0.775442i \(-0.717527\pi\)
−0.631418 + 0.775442i \(0.717527\pi\)
\(450\) 3129.25 0.327809
\(451\) 2439.32 0.254685
\(452\) 9752.94 1.01491
\(453\) 4969.55 0.515430
\(454\) −14772.8 −1.52714
\(455\) 2394.45 0.246711
\(456\) −295.477 −0.0303443
\(457\) 15590.7 1.59585 0.797925 0.602756i \(-0.205931\pi\)
0.797925 + 0.602756i \(0.205931\pi\)
\(458\) −21086.4 −2.15132
\(459\) 531.325 0.0540308
\(460\) −3909.96 −0.396310
\(461\) 1475.12 0.149031 0.0745155 0.997220i \(-0.476259\pi\)
0.0745155 + 0.997220i \(0.476259\pi\)
\(462\) 2579.99 0.259809
\(463\) 1335.14 0.134015 0.0670077 0.997752i \(-0.478655\pi\)
0.0670077 + 0.997752i \(0.478655\pi\)
\(464\) −17741.7 −1.77508
\(465\) −3180.86 −0.317223
\(466\) 14333.6 1.42488
\(467\) −13910.9 −1.37841 −0.689205 0.724566i \(-0.742040\pi\)
−0.689205 + 0.724566i \(0.742040\pi\)
\(468\) 964.426 0.0952577
\(469\) −22566.6 −2.22181
\(470\) 12060.7 1.18366
\(471\) 1948.33 0.190604
\(472\) 384.151 0.0374618
\(473\) 1189.60 0.115640
\(474\) −1595.05 −0.154564
\(475\) 1391.99 0.134460
\(476\) −3016.66 −0.290480
\(477\) −1986.02 −0.190636
\(478\) −18875.5 −1.80616
\(479\) 15483.9 1.47699 0.738495 0.674259i \(-0.235537\pi\)
0.738495 + 0.674259i \(0.235537\pi\)
\(480\) −3972.93 −0.377789
\(481\) 2654.24 0.251607
\(482\) −6327.09 −0.597906
\(483\) −7947.31 −0.748686
\(484\) −7809.28 −0.733403
\(485\) 2071.93 0.193982
\(486\) −918.167 −0.0856974
\(487\) −12901.8 −1.20049 −0.600243 0.799817i \(-0.704930\pi\)
−0.600243 + 0.799817i \(0.704930\pi\)
\(488\) −2843.33 −0.263753
\(489\) 1525.65 0.141088
\(490\) 5499.92 0.507063
\(491\) −4348.11 −0.399649 −0.199824 0.979832i \(-0.564037\pi\)
−0.199824 + 0.979832i \(0.564037\pi\)
\(492\) 4928.78 0.451639
\(493\) 4666.55 0.426310
\(494\) 975.790 0.0888722
\(495\) −481.679 −0.0437371
\(496\) −13813.1 −1.25046
\(497\) 22884.4 2.06541
\(498\) 299.555 0.0269546
\(499\) −12051.7 −1.08118 −0.540589 0.841287i \(-0.681798\pi\)
−0.540589 + 0.841287i \(0.681798\pi\)
\(500\) 7822.84 0.699696
\(501\) 4258.39 0.379742
\(502\) −16962.9 −1.50815
\(503\) 16522.2 1.46459 0.732293 0.680990i \(-0.238450\pi\)
0.732293 + 0.680990i \(0.238450\pi\)
\(504\) −1431.15 −0.126485
\(505\) −2419.69 −0.213217
\(506\) −3819.55 −0.335573
\(507\) −5716.63 −0.500758
\(508\) 8288.53 0.723906
\(509\) −15147.5 −1.31906 −0.659530 0.751678i \(-0.729245\pi\)
−0.659530 + 0.751678i \(0.729245\pi\)
\(510\) 1281.03 0.111225
\(511\) −318.344 −0.0275592
\(512\) −13355.7 −1.15282
\(513\) −408.429 −0.0351512
\(514\) −19267.4 −1.65340
\(515\) 7978.42 0.682662
\(516\) 2403.65 0.205067
\(517\) 5179.88 0.440640
\(518\) 14346.9 1.21693
\(519\) −2787.51 −0.235757
\(520\) −638.357 −0.0538342
\(521\) 14442.1 1.21443 0.607215 0.794538i \(-0.292287\pi\)
0.607215 + 0.794538i \(0.292287\pi\)
\(522\) −8064.13 −0.676163
\(523\) −4796.97 −0.401065 −0.200532 0.979687i \(-0.564267\pi\)
−0.200532 + 0.979687i \(0.564267\pi\)
\(524\) −10911.7 −0.909691
\(525\) 6742.09 0.560474
\(526\) −30036.2 −2.48981
\(527\) 3633.24 0.300315
\(528\) −2091.73 −0.172407
\(529\) −401.383 −0.0329895
\(530\) −4788.31 −0.392435
\(531\) 531.000 0.0433963
\(532\) 2318.91 0.188980
\(533\) 4468.55 0.363142
\(534\) 1550.10 0.125617
\(535\) 1949.87 0.157570
\(536\) 6016.23 0.484816
\(537\) −6638.17 −0.533442
\(538\) 14476.9 1.16012
\(539\) 2362.13 0.188765
\(540\) −973.259 −0.0775600
\(541\) −833.411 −0.0662313 −0.0331157 0.999452i \(-0.510543\pi\)
−0.0331157 + 0.999452i \(0.510543\pi\)
\(542\) 25787.5 2.04367
\(543\) −139.781 −0.0110471
\(544\) 4537.95 0.357653
\(545\) −1705.75 −0.134066
\(546\) 4726.24 0.370448
\(547\) −19682.1 −1.53848 −0.769239 0.638961i \(-0.779365\pi\)
−0.769239 + 0.638961i \(0.779365\pi\)
\(548\) −4875.33 −0.380043
\(549\) −3930.25 −0.305535
\(550\) 3240.31 0.251213
\(551\) −3587.17 −0.277348
\(552\) 2118.74 0.163369
\(553\) −3436.61 −0.264267
\(554\) 151.538 0.0116213
\(555\) −2678.55 −0.204861
\(556\) 5370.10 0.409609
\(557\) −10287.7 −0.782594 −0.391297 0.920264i \(-0.627974\pi\)
−0.391297 + 0.920264i \(0.627974\pi\)
\(558\) −6278.49 −0.476325
\(559\) 2179.21 0.164885
\(560\) −10493.3 −0.791829
\(561\) 550.183 0.0414060
\(562\) 9090.34 0.682301
\(563\) −8172.20 −0.611754 −0.305877 0.952071i \(-0.598950\pi\)
−0.305877 + 0.952071i \(0.598950\pi\)
\(564\) 10466.2 0.781398
\(565\) −8923.25 −0.664431
\(566\) −31104.9 −2.30996
\(567\) −1978.23 −0.146522
\(568\) −6100.96 −0.450687
\(569\) 9993.92 0.736322 0.368161 0.929762i \(-0.379988\pi\)
0.368161 + 0.929762i \(0.379988\pi\)
\(570\) −984.727 −0.0723608
\(571\) −8399.41 −0.615594 −0.307797 0.951452i \(-0.599592\pi\)
−0.307797 + 0.951452i \(0.599592\pi\)
\(572\) 998.656 0.0729998
\(573\) −6107.95 −0.445311
\(574\) 24153.9 1.75638
\(575\) −9981.34 −0.723914
\(576\) −2455.14 −0.177600
\(577\) −9807.97 −0.707645 −0.353822 0.935313i \(-0.615118\pi\)
−0.353822 + 0.935313i \(0.615118\pi\)
\(578\) 17100.4 1.23059
\(579\) −4590.53 −0.329492
\(580\) −8547.99 −0.611959
\(581\) 645.404 0.0460858
\(582\) 4089.64 0.291273
\(583\) −2056.50 −0.146092
\(584\) 84.8702 0.00601362
\(585\) −882.381 −0.0623624
\(586\) −7187.48 −0.506676
\(587\) −6955.54 −0.489073 −0.244537 0.969640i \(-0.578636\pi\)
−0.244537 + 0.969640i \(0.578636\pi\)
\(588\) 4772.81 0.334741
\(589\) −2792.87 −0.195379
\(590\) 1280.25 0.0893338
\(591\) 7385.42 0.514036
\(592\) −11631.8 −0.807542
\(593\) −13988.6 −0.968710 −0.484355 0.874872i \(-0.660946\pi\)
−0.484355 + 0.874872i \(0.660946\pi\)
\(594\) −950.755 −0.0656733
\(595\) 2760.03 0.190169
\(596\) 12562.1 0.863365
\(597\) 4326.81 0.296624
\(598\) −6996.98 −0.478475
\(599\) 26538.4 1.81023 0.905117 0.425163i \(-0.139783\pi\)
0.905117 + 0.425163i \(0.139783\pi\)
\(600\) −1797.43 −0.122300
\(601\) −16540.3 −1.12261 −0.561307 0.827608i \(-0.689701\pi\)
−0.561307 + 0.827608i \(0.689701\pi\)
\(602\) 11779.3 0.797486
\(603\) 8316.06 0.561619
\(604\) 10397.6 0.700453
\(605\) 7144.93 0.480137
\(606\) −4776.06 −0.320155
\(607\) 2733.68 0.182795 0.0913974 0.995814i \(-0.470867\pi\)
0.0913974 + 0.995814i \(0.470867\pi\)
\(608\) −3488.32 −0.232681
\(609\) −17374.5 −1.15608
\(610\) −9475.87 −0.628962
\(611\) 9488.95 0.628285
\(612\) 1111.67 0.0734261
\(613\) 9935.81 0.654655 0.327328 0.944911i \(-0.393852\pi\)
0.327328 + 0.944911i \(0.393852\pi\)
\(614\) −34860.9 −2.29132
\(615\) −4509.48 −0.295674
\(616\) −1481.94 −0.0969303
\(617\) 4126.63 0.269257 0.134629 0.990896i \(-0.457016\pi\)
0.134629 + 0.990896i \(0.457016\pi\)
\(618\) 15748.1 1.02505
\(619\) 13867.2 0.900436 0.450218 0.892919i \(-0.351346\pi\)
0.450218 + 0.892919i \(0.351346\pi\)
\(620\) −6655.21 −0.431096
\(621\) 2928.67 0.189249
\(622\) 9052.79 0.583576
\(623\) 3339.75 0.214774
\(624\) −3831.81 −0.245826
\(625\) 4345.14 0.278089
\(626\) 38056.3 2.42977
\(627\) −422.925 −0.0269378
\(628\) 4076.43 0.259024
\(629\) 3059.49 0.193942
\(630\) −4769.53 −0.301623
\(631\) 2950.74 0.186160 0.0930801 0.995659i \(-0.470329\pi\)
0.0930801 + 0.995659i \(0.470329\pi\)
\(632\) 916.196 0.0576651
\(633\) −2197.04 −0.137953
\(634\) −4036.87 −0.252878
\(635\) −7583.42 −0.473919
\(636\) −4155.28 −0.259068
\(637\) 4327.15 0.269149
\(638\) −8350.34 −0.518171
\(639\) −8433.17 −0.522083
\(640\) 4675.11 0.288750
\(641\) −11795.4 −0.726820 −0.363410 0.931629i \(-0.618388\pi\)
−0.363410 + 0.931629i \(0.618388\pi\)
\(642\) 3848.71 0.236599
\(643\) 19304.7 1.18398 0.591992 0.805944i \(-0.298342\pi\)
0.591992 + 0.805944i \(0.298342\pi\)
\(644\) −16627.9 −1.01744
\(645\) −2199.16 −0.134251
\(646\) 1124.77 0.0685041
\(647\) 21361.3 1.29799 0.648995 0.760793i \(-0.275190\pi\)
0.648995 + 0.760793i \(0.275190\pi\)
\(648\) 527.393 0.0319722
\(649\) 549.846 0.0332563
\(650\) 5935.88 0.358191
\(651\) −13527.3 −0.814401
\(652\) 3192.06 0.191734
\(653\) 5795.08 0.347288 0.173644 0.984809i \(-0.444446\pi\)
0.173644 + 0.984809i \(0.444446\pi\)
\(654\) −3366.86 −0.201307
\(655\) 9983.39 0.595547
\(656\) −19582.8 −1.16552
\(657\) 117.313 0.00696626
\(658\) 51290.6 3.03878
\(659\) 11950.9 0.706434 0.353217 0.935542i \(-0.385088\pi\)
0.353217 + 0.935542i \(0.385088\pi\)
\(660\) −1007.80 −0.0594374
\(661\) 17906.5 1.05368 0.526838 0.849965i \(-0.323377\pi\)
0.526838 + 0.849965i \(0.323377\pi\)
\(662\) 36461.4 2.14065
\(663\) 1007.87 0.0590385
\(664\) −172.064 −0.0100563
\(665\) −2121.64 −0.123720
\(666\) −5287.01 −0.307609
\(667\) 25722.1 1.49320
\(668\) 8909.70 0.516058
\(669\) 8784.79 0.507683
\(670\) 20050.1 1.15612
\(671\) −4069.74 −0.234144
\(672\) −16895.7 −0.969890
\(673\) 12825.6 0.734608 0.367304 0.930101i \(-0.380281\pi\)
0.367304 + 0.930101i \(0.380281\pi\)
\(674\) 25005.7 1.42906
\(675\) −2484.54 −0.141674
\(676\) −11960.7 −0.680514
\(677\) 11752.0 0.667156 0.333578 0.942722i \(-0.391744\pi\)
0.333578 + 0.942722i \(0.391744\pi\)
\(678\) −17613.0 −0.997675
\(679\) 8811.31 0.498007
\(680\) −735.821 −0.0414962
\(681\) 11729.2 0.660004
\(682\) −6501.33 −0.365027
\(683\) −10731.7 −0.601223 −0.300612 0.953747i \(-0.597191\pi\)
−0.300612 + 0.953747i \(0.597191\pi\)
\(684\) −854.544 −0.0477694
\(685\) 4460.58 0.248803
\(686\) −8262.47 −0.459858
\(687\) 16742.0 0.929765
\(688\) −9550.05 −0.529204
\(689\) −3767.28 −0.208305
\(690\) 7061.07 0.389580
\(691\) 28804.7 1.58579 0.792895 0.609358i \(-0.208573\pi\)
0.792895 + 0.609358i \(0.208573\pi\)
\(692\) −5832.21 −0.320387
\(693\) −2048.44 −0.112286
\(694\) −14691.4 −0.803572
\(695\) −4913.26 −0.268159
\(696\) 4632.02 0.252265
\(697\) 5150.81 0.279915
\(698\) 32688.3 1.77259
\(699\) −11380.5 −0.615808
\(700\) 14106.3 0.761667
\(701\) −1663.10 −0.0896070 −0.0448035 0.998996i \(-0.514266\pi\)
−0.0448035 + 0.998996i \(0.514266\pi\)
\(702\) −1741.67 −0.0936400
\(703\) −2351.83 −0.126175
\(704\) −2542.28 −0.136102
\(705\) −9575.87 −0.511557
\(706\) −7543.90 −0.402151
\(707\) −10290.2 −0.547388
\(708\) 1110.99 0.0589742
\(709\) 32357.6 1.71399 0.856993 0.515328i \(-0.172330\pi\)
0.856993 + 0.515328i \(0.172330\pi\)
\(710\) −20332.5 −1.07474
\(711\) 1266.43 0.0668001
\(712\) −890.372 −0.0468653
\(713\) 20026.5 1.05189
\(714\) 5447.85 0.285547
\(715\) −913.699 −0.0477908
\(716\) −13888.8 −0.724931
\(717\) 14986.6 0.780593
\(718\) −44110.2 −2.29273
\(719\) −9360.97 −0.485543 −0.242771 0.970084i \(-0.578057\pi\)
−0.242771 + 0.970084i \(0.578057\pi\)
\(720\) 3866.91 0.200154
\(721\) 33929.9 1.75259
\(722\) 25051.9 1.29132
\(723\) 5023.54 0.258406
\(724\) −292.460 −0.0150127
\(725\) −21821.3 −1.11783
\(726\) 14102.9 0.720948
\(727\) −18574.0 −0.947556 −0.473778 0.880644i \(-0.657110\pi\)
−0.473778 + 0.880644i \(0.657110\pi\)
\(728\) −2714.74 −0.138208
\(729\) 729.000 0.0370370
\(730\) 282.844 0.0143405
\(731\) 2511.93 0.127096
\(732\) −8223.14 −0.415213
\(733\) 26673.5 1.34408 0.672038 0.740516i \(-0.265419\pi\)
0.672038 + 0.740516i \(0.265419\pi\)
\(734\) −42026.6 −2.11339
\(735\) −4366.79 −0.219145
\(736\) 25013.3 1.25272
\(737\) 8611.21 0.430391
\(738\) −8900.97 −0.443969
\(739\) 25484.9 1.26857 0.634287 0.773098i \(-0.281294\pi\)
0.634287 + 0.773098i \(0.281294\pi\)
\(740\) −5604.24 −0.278400
\(741\) −774.751 −0.0384091
\(742\) −20363.3 −1.00749
\(743\) −14039.9 −0.693237 −0.346619 0.938006i \(-0.612670\pi\)
−0.346619 + 0.938006i \(0.612670\pi\)
\(744\) 3606.35 0.177709
\(745\) −11493.5 −0.565219
\(746\) −28139.7 −1.38106
\(747\) −237.839 −0.0116493
\(748\) 1151.13 0.0562694
\(749\) 8292.21 0.404527
\(750\) −14127.4 −0.687813
\(751\) 34000.2 1.65205 0.826023 0.563637i \(-0.190598\pi\)
0.826023 + 0.563637i \(0.190598\pi\)
\(752\) −41584.0 −2.01650
\(753\) 13468.1 0.651797
\(754\) −15296.9 −0.738832
\(755\) −9513.09 −0.458565
\(756\) −4138.99 −0.199118
\(757\) −37552.9 −1.80302 −0.901508 0.432763i \(-0.857539\pi\)
−0.901508 + 0.432763i \(0.857539\pi\)
\(758\) 10341.2 0.495526
\(759\) 3032.62 0.145029
\(760\) 565.625 0.0269966
\(761\) −18976.0 −0.903917 −0.451959 0.892039i \(-0.649275\pi\)
−0.451959 + 0.892039i \(0.649275\pi\)
\(762\) −14968.4 −0.711612
\(763\) −7254.05 −0.344186
\(764\) −12779.5 −0.605164
\(765\) −1017.10 −0.0480699
\(766\) 14022.9 0.661447
\(767\) 1007.26 0.0474184
\(768\) 15774.9 0.741183
\(769\) −14782.6 −0.693206 −0.346603 0.938012i \(-0.612665\pi\)
−0.346603 + 0.938012i \(0.612665\pi\)
\(770\) −4938.82 −0.231146
\(771\) 15297.8 0.714575
\(772\) −9604.62 −0.447769
\(773\) −32518.7 −1.51309 −0.756543 0.653944i \(-0.773113\pi\)
−0.756543 + 0.653944i \(0.773113\pi\)
\(774\) −4340.79 −0.201584
\(775\) −16989.4 −0.787456
\(776\) −2349.08 −0.108669
\(777\) −11391.1 −0.525937
\(778\) −259.806 −0.0119724
\(779\) −3959.43 −0.182107
\(780\) −1846.18 −0.0847485
\(781\) −8732.48 −0.400093
\(782\) −8065.28 −0.368816
\(783\) 6402.70 0.292227
\(784\) −18963.1 −0.863845
\(785\) −3729.64 −0.169575
\(786\) 19705.6 0.894242
\(787\) −7602.30 −0.344336 −0.172168 0.985068i \(-0.555077\pi\)
−0.172168 + 0.985068i \(0.555077\pi\)
\(788\) 15452.3 0.698559
\(789\) 23847.9 1.07606
\(790\) 3053.38 0.137512
\(791\) −37948.0 −1.70578
\(792\) 546.112 0.0245016
\(793\) −7455.30 −0.333853
\(794\) 35370.7 1.58093
\(795\) 3801.79 0.169604
\(796\) 9052.85 0.403103
\(797\) −7713.91 −0.342836 −0.171418 0.985198i \(-0.554835\pi\)
−0.171418 + 0.985198i \(0.554835\pi\)
\(798\) −4187.76 −0.185771
\(799\) 10937.7 0.484292
\(800\) −21220.0 −0.937799
\(801\) −1230.74 −0.0542895
\(802\) 361.206 0.0159035
\(803\) 121.477 0.00533853
\(804\) 17399.4 0.763222
\(805\) 15213.4 0.666088
\(806\) −11909.7 −0.520472
\(807\) −11494.3 −0.501386
\(808\) 2743.36 0.119444
\(809\) 25690.3 1.11647 0.558234 0.829683i \(-0.311479\pi\)
0.558234 + 0.829683i \(0.311479\pi\)
\(810\) 1757.63 0.0762429
\(811\) 28467.8 1.23260 0.616302 0.787510i \(-0.288630\pi\)
0.616302 + 0.787510i \(0.288630\pi\)
\(812\) −36352.1 −1.57107
\(813\) −20474.6 −0.883241
\(814\) −5474.66 −0.235733
\(815\) −2920.51 −0.125523
\(816\) −4416.85 −0.189486
\(817\) −1930.92 −0.0826857
\(818\) 41769.8 1.78539
\(819\) −3752.51 −0.160102
\(820\) −9435.05 −0.401812
\(821\) −38119.4 −1.62043 −0.810217 0.586130i \(-0.800651\pi\)
−0.810217 + 0.586130i \(0.800651\pi\)
\(822\) 8804.45 0.373589
\(823\) −26656.3 −1.12902 −0.564508 0.825427i \(-0.690934\pi\)
−0.564508 + 0.825427i \(0.690934\pi\)
\(824\) −9045.66 −0.382428
\(825\) −2572.72 −0.108570
\(826\) 5444.52 0.229345
\(827\) −19016.7 −0.799606 −0.399803 0.916601i \(-0.630922\pi\)
−0.399803 + 0.916601i \(0.630922\pi\)
\(828\) 6127.57 0.257184
\(829\) −36260.4 −1.51915 −0.759575 0.650420i \(-0.774593\pi\)
−0.759575 + 0.650420i \(0.774593\pi\)
\(830\) −573.432 −0.0239808
\(831\) −120.317 −0.00502255
\(832\) −4657.16 −0.194060
\(833\) 4987.82 0.207464
\(834\) −9697.95 −0.402653
\(835\) −8151.75 −0.337848
\(836\) −884.873 −0.0366076
\(837\) 4984.95 0.205860
\(838\) 18342.0 0.756102
\(839\) 18034.3 0.742088 0.371044 0.928615i \(-0.379000\pi\)
0.371044 + 0.928615i \(0.379000\pi\)
\(840\) 2739.61 0.112530
\(841\) 31845.0 1.30571
\(842\) 20709.0 0.847602
\(843\) −7217.49 −0.294880
\(844\) −4596.79 −0.187474
\(845\) 10943.2 0.445512
\(846\) −18901.2 −0.768127
\(847\) 30385.3 1.23265
\(848\) 16509.6 0.668562
\(849\) 24696.4 0.998327
\(850\) 6842.17 0.276099
\(851\) 16864.0 0.679305
\(852\) −17644.5 −0.709494
\(853\) −17485.1 −0.701851 −0.350926 0.936403i \(-0.614133\pi\)
−0.350926 + 0.936403i \(0.614133\pi\)
\(854\) −40298.1 −1.61472
\(855\) 781.847 0.0312732
\(856\) −2210.69 −0.0882709
\(857\) 3337.89 0.133046 0.0665228 0.997785i \(-0.478809\pi\)
0.0665228 + 0.997785i \(0.478809\pi\)
\(858\) −1803.49 −0.0717601
\(859\) −41566.7 −1.65103 −0.825517 0.564377i \(-0.809117\pi\)
−0.825517 + 0.564377i \(0.809117\pi\)
\(860\) −4601.24 −0.182443
\(861\) −19177.5 −0.759080
\(862\) −27834.3 −1.09981
\(863\) 21883.2 0.863165 0.431583 0.902073i \(-0.357955\pi\)
0.431583 + 0.902073i \(0.357955\pi\)
\(864\) 6226.26 0.245164
\(865\) 5336.06 0.209747
\(866\) −862.859 −0.0338581
\(867\) −13577.2 −0.531842
\(868\) −28302.7 −1.10675
\(869\) 1311.38 0.0511916
\(870\) 15437.0 0.601566
\(871\) 15774.8 0.613671
\(872\) 1933.92 0.0751041
\(873\) −3247.07 −0.125884
\(874\) 6199.77 0.239943
\(875\) −30438.1 −1.17599
\(876\) 245.451 0.00946693
\(877\) −18985.8 −0.731019 −0.365509 0.930808i \(-0.619105\pi\)
−0.365509 + 0.930808i \(0.619105\pi\)
\(878\) 63847.7 2.45416
\(879\) 5706.66 0.218977
\(880\) 4004.16 0.153386
\(881\) −47938.8 −1.83326 −0.916628 0.399742i \(-0.869100\pi\)
−0.916628 + 0.399742i \(0.869100\pi\)
\(882\) −8619.31 −0.329056
\(883\) 45963.2 1.75174 0.875868 0.482550i \(-0.160290\pi\)
0.875868 + 0.482550i \(0.160290\pi\)
\(884\) 2108.74 0.0802314
\(885\) −1016.48 −0.0386086
\(886\) −32735.5 −1.24128
\(887\) −25211.9 −0.954378 −0.477189 0.878801i \(-0.658344\pi\)
−0.477189 + 0.878801i \(0.658344\pi\)
\(888\) 3036.85 0.114763
\(889\) −32250.1 −1.21668
\(890\) −2967.32 −0.111758
\(891\) 754.874 0.0283830
\(892\) 18380.2 0.689925
\(893\) −8407.83 −0.315070
\(894\) −22686.2 −0.848703
\(895\) 12707.3 0.474590
\(896\) 19881.9 0.741303
\(897\) 5555.41 0.206789
\(898\) 45397.5 1.68701
\(899\) 43782.1 1.62426
\(900\) −5198.32 −0.192530
\(901\) −4342.47 −0.160565
\(902\) −9216.88 −0.340231
\(903\) −9352.41 −0.344661
\(904\) 10116.9 0.372215
\(905\) 267.580 0.00982836
\(906\) −18777.3 −0.688557
\(907\) −9607.36 −0.351717 −0.175858 0.984415i \(-0.556270\pi\)
−0.175858 + 0.984415i \(0.556270\pi\)
\(908\) 24540.6 0.896924
\(909\) 3792.06 0.138366
\(910\) −9047.34 −0.329579
\(911\) −44171.3 −1.60643 −0.803217 0.595686i \(-0.796880\pi\)
−0.803217 + 0.595686i \(0.796880\pi\)
\(912\) 3395.23 0.123276
\(913\) −246.280 −0.00892736
\(914\) −58909.1 −2.13188
\(915\) 7523.59 0.271827
\(916\) 35028.8 1.26352
\(917\) 42456.4 1.52894
\(918\) −2007.59 −0.0721792
\(919\) 33636.3 1.20736 0.603678 0.797228i \(-0.293701\pi\)
0.603678 + 0.797228i \(0.293701\pi\)
\(920\) −4055.86 −0.145345
\(921\) 27678.6 0.990273
\(922\) −5573.70 −0.199089
\(923\) −15996.9 −0.570471
\(924\) −4285.89 −0.152593
\(925\) −14306.5 −0.508535
\(926\) −5044.78 −0.179030
\(927\) −12503.5 −0.443010
\(928\) 54684.3 1.93437
\(929\) 11890.9 0.419945 0.209972 0.977707i \(-0.432663\pi\)
0.209972 + 0.977707i \(0.432663\pi\)
\(930\) 12018.8 0.423775
\(931\) −3834.14 −0.134972
\(932\) −23811.1 −0.836864
\(933\) −7187.67 −0.252212
\(934\) 52561.7 1.84140
\(935\) −1053.20 −0.0368379
\(936\) 1000.41 0.0349354
\(937\) −18281.5 −0.637385 −0.318692 0.947858i \(-0.603244\pi\)
−0.318692 + 0.947858i \(0.603244\pi\)
\(938\) 85267.2 2.96810
\(939\) −30215.7 −1.05011
\(940\) −20035.3 −0.695190
\(941\) 40341.0 1.39753 0.698767 0.715349i \(-0.253732\pi\)
0.698767 + 0.715349i \(0.253732\pi\)
\(942\) −7361.70 −0.254625
\(943\) 28391.4 0.980435
\(944\) −4414.15 −0.152191
\(945\) 3786.88 0.130357
\(946\) −4494.85 −0.154482
\(947\) −44404.6 −1.52371 −0.761856 0.647746i \(-0.775712\pi\)
−0.761856 + 0.647746i \(0.775712\pi\)
\(948\) 2649.71 0.0907792
\(949\) 222.532 0.00761191
\(950\) −5259.57 −0.179624
\(951\) 3205.17 0.109290
\(952\) −3129.23 −0.106533
\(953\) 34851.1 1.18461 0.592307 0.805713i \(-0.298217\pi\)
0.592307 + 0.805713i \(0.298217\pi\)
\(954\) 7504.09 0.254669
\(955\) 11692.3 0.396183
\(956\) 31356.0 1.06080
\(957\) 6629.95 0.223945
\(958\) −58505.5 −1.97310
\(959\) 18969.5 0.638747
\(960\) 4699.81 0.158006
\(961\) 4296.41 0.144218
\(962\) −10028.9 −0.336119
\(963\) −3055.77 −0.102254
\(964\) 10510.6 0.351165
\(965\) 8787.55 0.293141
\(966\) 30028.7 1.00016
\(967\) 2971.34 0.0988128 0.0494064 0.998779i \(-0.484267\pi\)
0.0494064 + 0.998779i \(0.484267\pi\)
\(968\) −8100.68 −0.268973
\(969\) −893.040 −0.0296064
\(970\) −7828.71 −0.259139
\(971\) 34741.0 1.14819 0.574094 0.818789i \(-0.305354\pi\)
0.574094 + 0.818789i \(0.305354\pi\)
\(972\) 1525.26 0.0503322
\(973\) −20894.6 −0.688440
\(974\) 48749.1 1.60372
\(975\) −4712.93 −0.154805
\(976\) 32671.8 1.07151
\(977\) −31701.1 −1.03808 −0.519042 0.854749i \(-0.673711\pi\)
−0.519042 + 0.854749i \(0.673711\pi\)
\(978\) −5764.60 −0.188478
\(979\) −1274.42 −0.0416042
\(980\) −9136.49 −0.297811
\(981\) 2673.20 0.0870017
\(982\) 16429.2 0.533887
\(983\) −47493.5 −1.54100 −0.770502 0.637437i \(-0.779995\pi\)
−0.770502 + 0.637437i \(0.779995\pi\)
\(984\) 5112.70 0.165637
\(985\) −14137.7 −0.457326
\(986\) −17632.4 −0.569503
\(987\) −40723.4 −1.31331
\(988\) −1620.99 −0.0521968
\(989\) 13845.8 0.445167
\(990\) 1820.01 0.0584280
\(991\) −48621.1 −1.55853 −0.779263 0.626697i \(-0.784406\pi\)
−0.779263 + 0.626697i \(0.784406\pi\)
\(992\) 42575.5 1.36268
\(993\) −28949.4 −0.925157
\(994\) −86468.0 −2.75915
\(995\) −8282.71 −0.263899
\(996\) −497.622 −0.0158311
\(997\) 37128.0 1.17939 0.589697 0.807625i \(-0.299247\pi\)
0.589697 + 0.807625i \(0.299247\pi\)
\(998\) 45536.9 1.44433
\(999\) 4197.74 0.132944
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 177.4.a.d.1.2 8
3.2 odd 2 531.4.a.e.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.d.1.2 8 1.1 even 1 trivial
531.4.a.e.1.7 8 3.2 odd 2