Properties

Label 177.4.a.b.1.2
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44\)
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 1.2
Root \(3.06012\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.06012 q^{2} -3.00000 q^{3} +1.36432 q^{4} +0.675250 q^{5} +9.18035 q^{6} -3.38755 q^{7} +20.3060 q^{8} +9.00000 q^{9} -2.06635 q^{10} -8.79097 q^{11} -4.09296 q^{12} +59.4118 q^{13} +10.3663 q^{14} -2.02575 q^{15} -73.0532 q^{16} +31.2380 q^{17} -27.5411 q^{18} -77.5189 q^{19} +0.921258 q^{20} +10.1626 q^{21} +26.9014 q^{22} +73.1225 q^{23} -60.9179 q^{24} -124.544 q^{25} -181.807 q^{26} -27.0000 q^{27} -4.62171 q^{28} -61.5678 q^{29} +6.19904 q^{30} -278.253 q^{31} +61.1037 q^{32} +26.3729 q^{33} -95.5920 q^{34} -2.28744 q^{35} +12.2789 q^{36} -326.365 q^{37} +237.217 q^{38} -178.235 q^{39} +13.7116 q^{40} +453.551 q^{41} -31.0989 q^{42} -174.901 q^{43} -11.9937 q^{44} +6.07725 q^{45} -223.763 q^{46} -480.883 q^{47} +219.160 q^{48} -331.525 q^{49} +381.119 q^{50} -93.7140 q^{51} +81.0568 q^{52} -185.568 q^{53} +82.6232 q^{54} -5.93610 q^{55} -68.7874 q^{56} +232.557 q^{57} +188.405 q^{58} +59.0000 q^{59} -2.76378 q^{60} +807.823 q^{61} +851.486 q^{62} -30.4879 q^{63} +397.441 q^{64} +40.1178 q^{65} -80.7042 q^{66} -129.081 q^{67} +42.6187 q^{68} -219.367 q^{69} +6.99985 q^{70} -272.032 q^{71} +182.754 q^{72} -610.054 q^{73} +998.716 q^{74} +373.632 q^{75} -105.761 q^{76} +29.7798 q^{77} +545.422 q^{78} +886.175 q^{79} -49.3292 q^{80} +81.0000 q^{81} -1387.92 q^{82} +51.4871 q^{83} +13.8651 q^{84} +21.0935 q^{85} +535.217 q^{86} +184.703 q^{87} -178.509 q^{88} +182.350 q^{89} -18.5971 q^{90} -201.261 q^{91} +99.7625 q^{92} +834.758 q^{93} +1471.56 q^{94} -52.3447 q^{95} -183.311 q^{96} -1353.44 q^{97} +1014.50 q^{98} -79.1187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} - 71q^{10} - 5q^{11} - 78q^{12} - 67q^{13} - 65q^{14} + 6q^{15} - 94q^{16} - 23q^{17} - 176q^{19} - 207q^{20} + 177q^{21} - 704q^{22} - 218q^{23} + 63q^{24} - 183q^{25} + 58q^{26} - 189q^{27} - 938q^{28} + 168q^{29} + 213q^{30} - 604q^{31} - 448q^{32} + 15q^{33} - 610q^{34} - 336q^{35} + 234q^{36} - 505q^{37} - 453q^{38} + 201q^{39} - 1080q^{40} - 265q^{41} + 195q^{42} - 493q^{43} + 504q^{44} - 18q^{45} + 381q^{46} - 244q^{47} + 282q^{48} + 770q^{49} + 1639q^{50} + 69q^{51} + 160q^{52} + 686q^{53} - 116q^{55} + 2190q^{56} + 528q^{57} + 1584q^{58} + 413q^{59} + 621q^{60} - 838q^{61} + 286q^{62} - 531q^{63} + 205q^{64} + 490q^{65} + 2112q^{66} - 1504q^{67} + 3047q^{68} + 654q^{69} + 1530q^{70} - 1267q^{71} - 189q^{72} - 666q^{73} + 528q^{74} + 549q^{75} - 64q^{76} + 1109q^{77} - 174q^{78} - 2741q^{79} + 1213q^{80} + 567q^{81} + 953q^{82} - 2025q^{83} + 2814q^{84} - 1274q^{85} + 4394q^{86} - 504q^{87} - 1639q^{88} + 616q^{89} - 639q^{90} - 2415q^{91} + 218q^{92} + 1812q^{93} + 900q^{94} + 2554q^{95} + 1344q^{96} - 1298q^{97} - 172q^{98} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.06012 −1.08192 −0.540958 0.841050i \(-0.681938\pi\)
−0.540958 + 0.841050i \(0.681938\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.36432 0.170540
\(5\) 0.675250 0.0603962 0.0301981 0.999544i \(-0.490386\pi\)
0.0301981 + 0.999544i \(0.490386\pi\)
\(6\) 9.18035 0.624644
\(7\) −3.38755 −0.182910 −0.0914552 0.995809i \(-0.529152\pi\)
−0.0914552 + 0.995809i \(0.529152\pi\)
\(8\) 20.3060 0.897405
\(9\) 9.00000 0.333333
\(10\) −2.06635 −0.0653436
\(11\) −8.79097 −0.240962 −0.120481 0.992716i \(-0.538444\pi\)
−0.120481 + 0.992716i \(0.538444\pi\)
\(12\) −4.09296 −0.0984614
\(13\) 59.4118 1.26753 0.633764 0.773526i \(-0.281509\pi\)
0.633764 + 0.773526i \(0.281509\pi\)
\(14\) 10.3663 0.197894
\(15\) −2.02575 −0.0348698
\(16\) −73.0532 −1.14146
\(17\) 31.2380 0.445666 0.222833 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(18\) −27.5411 −0.360638
\(19\) −77.5189 −0.936003 −0.468002 0.883728i \(-0.655026\pi\)
−0.468002 + 0.883728i \(0.655026\pi\)
\(20\) 0.921258 0.0103000
\(21\) 10.1626 0.105603
\(22\) 26.9014 0.260700
\(23\) 73.1225 0.662917 0.331458 0.943470i \(-0.392459\pi\)
0.331458 + 0.943470i \(0.392459\pi\)
\(24\) −60.9179 −0.518117
\(25\) −124.544 −0.996352
\(26\) −181.807 −1.37136
\(27\) −27.0000 −0.192450
\(28\) −4.62171 −0.0311936
\(29\) −61.5678 −0.394236 −0.197118 0.980380i \(-0.563158\pi\)
−0.197118 + 0.980380i \(0.563158\pi\)
\(30\) 6.19904 0.0377261
\(31\) −278.253 −1.61212 −0.806059 0.591835i \(-0.798404\pi\)
−0.806059 + 0.591835i \(0.798404\pi\)
\(32\) 61.1037 0.337554
\(33\) 26.3729 0.139119
\(34\) −95.5920 −0.482173
\(35\) −2.28744 −0.0110471
\(36\) 12.2789 0.0568467
\(37\) −326.365 −1.45011 −0.725055 0.688691i \(-0.758186\pi\)
−0.725055 + 0.688691i \(0.758186\pi\)
\(38\) 237.217 1.01268
\(39\) −178.235 −0.731808
\(40\) 13.7116 0.0541999
\(41\) 453.551 1.72763 0.863815 0.503809i \(-0.168069\pi\)
0.863815 + 0.503809i \(0.168069\pi\)
\(42\) −31.0989 −0.114254
\(43\) −174.901 −0.620282 −0.310141 0.950691i \(-0.600376\pi\)
−0.310141 + 0.950691i \(0.600376\pi\)
\(44\) −11.9937 −0.0410936
\(45\) 6.07725 0.0201321
\(46\) −223.763 −0.717220
\(47\) −480.883 −1.49243 −0.746213 0.665707i \(-0.768130\pi\)
−0.746213 + 0.665707i \(0.768130\pi\)
\(48\) 219.160 0.659020
\(49\) −331.525 −0.966544
\(50\) 381.119 1.07797
\(51\) −93.7140 −0.257306
\(52\) 81.0568 0.216165
\(53\) −185.568 −0.480938 −0.240469 0.970657i \(-0.577301\pi\)
−0.240469 + 0.970657i \(0.577301\pi\)
\(54\) 82.6232 0.208215
\(55\) −5.93610 −0.0145532
\(56\) −68.7874 −0.164145
\(57\) 232.557 0.540402
\(58\) 188.405 0.426530
\(59\) 59.0000 0.130189
\(60\) −2.76378 −0.00594670
\(61\) 807.823 1.69559 0.847796 0.530322i \(-0.177929\pi\)
0.847796 + 0.530322i \(0.177929\pi\)
\(62\) 851.486 1.74417
\(63\) −30.4879 −0.0609702
\(64\) 397.441 0.776252
\(65\) 40.1178 0.0765539
\(66\) −80.7042 −0.150515
\(67\) −129.081 −0.235369 −0.117685 0.993051i \(-0.537547\pi\)
−0.117685 + 0.993051i \(0.537547\pi\)
\(68\) 42.6187 0.0760040
\(69\) −219.367 −0.382735
\(70\) 6.99985 0.0119520
\(71\) −272.032 −0.454708 −0.227354 0.973812i \(-0.573007\pi\)
−0.227354 + 0.973812i \(0.573007\pi\)
\(72\) 182.754 0.299135
\(73\) −610.054 −0.978102 −0.489051 0.872255i \(-0.662657\pi\)
−0.489051 + 0.872255i \(0.662657\pi\)
\(74\) 998.716 1.56890
\(75\) 373.632 0.575244
\(76\) −105.761 −0.159626
\(77\) 29.7798 0.0440744
\(78\) 545.422 0.791754
\(79\) 886.175 1.26206 0.631028 0.775760i \(-0.282633\pi\)
0.631028 + 0.775760i \(0.282633\pi\)
\(80\) −49.3292 −0.0689396
\(81\) 81.0000 0.111111
\(82\) −1387.92 −1.86915
\(83\) 51.4871 0.0680896 0.0340448 0.999420i \(-0.489161\pi\)
0.0340448 + 0.999420i \(0.489161\pi\)
\(84\) 13.8651 0.0180096
\(85\) 21.0935 0.0269166
\(86\) 535.217 0.671092
\(87\) 184.703 0.227612
\(88\) −178.509 −0.216240
\(89\) 182.350 0.217180 0.108590 0.994087i \(-0.465366\pi\)
0.108590 + 0.994087i \(0.465366\pi\)
\(90\) −18.5971 −0.0217812
\(91\) −201.261 −0.231844
\(92\) 99.7625 0.113054
\(93\) 834.758 0.930757
\(94\) 1471.56 1.61468
\(95\) −52.3447 −0.0565311
\(96\) −183.311 −0.194887
\(97\) −1353.44 −1.41671 −0.708354 0.705857i \(-0.750562\pi\)
−0.708354 + 0.705857i \(0.750562\pi\)
\(98\) 1014.50 1.04572
\(99\) −79.1187 −0.0803205
\(100\) −169.918 −0.169918
\(101\) −406.406 −0.400385 −0.200192 0.979757i \(-0.564157\pi\)
−0.200192 + 0.979757i \(0.564157\pi\)
\(102\) 286.776 0.278383
\(103\) −304.606 −0.291395 −0.145697 0.989329i \(-0.546543\pi\)
−0.145697 + 0.989329i \(0.546543\pi\)
\(104\) 1206.41 1.13749
\(105\) 6.86233 0.00637805
\(106\) 567.859 0.520334
\(107\) −1515.70 −1.36942 −0.684711 0.728815i \(-0.740071\pi\)
−0.684711 + 0.728815i \(0.740071\pi\)
\(108\) −36.8367 −0.0328205
\(109\) −1.57624 −0.00138511 −0.000692553 1.00000i \(-0.500220\pi\)
−0.000692553 1.00000i \(0.500220\pi\)
\(110\) 18.1652 0.0157453
\(111\) 979.095 0.837222
\(112\) 247.471 0.208784
\(113\) −577.813 −0.481027 −0.240514 0.970646i \(-0.577316\pi\)
−0.240514 + 0.970646i \(0.577316\pi\)
\(114\) −711.651 −0.584669
\(115\) 49.3760 0.0400377
\(116\) −83.9983 −0.0672331
\(117\) 534.706 0.422510
\(118\) −180.547 −0.140853
\(119\) −105.820 −0.0815171
\(120\) −41.1348 −0.0312923
\(121\) −1253.72 −0.941938
\(122\) −2472.03 −1.83449
\(123\) −1360.65 −0.997448
\(124\) −379.626 −0.274931
\(125\) −168.505 −0.120572
\(126\) 93.2967 0.0659645
\(127\) 291.667 0.203789 0.101895 0.994795i \(-0.467510\pi\)
0.101895 + 0.994795i \(0.467510\pi\)
\(128\) −1705.05 −1.17739
\(129\) 524.702 0.358120
\(130\) −122.765 −0.0828249
\(131\) −1160.30 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(132\) 35.9811 0.0237254
\(133\) 262.599 0.171205
\(134\) 395.003 0.254649
\(135\) −18.2318 −0.0116233
\(136\) 634.318 0.399943
\(137\) 571.419 0.356348 0.178174 0.983999i \(-0.442981\pi\)
0.178174 + 0.983999i \(0.442981\pi\)
\(138\) 671.290 0.414087
\(139\) −576.322 −0.351677 −0.175838 0.984419i \(-0.556264\pi\)
−0.175838 + 0.984419i \(0.556264\pi\)
\(140\) −3.12081 −0.00188397
\(141\) 1442.65 0.861653
\(142\) 832.450 0.491956
\(143\) −522.287 −0.305426
\(144\) −657.479 −0.380485
\(145\) −41.5737 −0.0238104
\(146\) 1866.84 1.05822
\(147\) 994.574 0.558034
\(148\) −445.267 −0.247302
\(149\) −1845.62 −1.01476 −0.507380 0.861723i \(-0.669386\pi\)
−0.507380 + 0.861723i \(0.669386\pi\)
\(150\) −1143.36 −0.622365
\(151\) −1957.64 −1.05503 −0.527517 0.849545i \(-0.676877\pi\)
−0.527517 + 0.849545i \(0.676877\pi\)
\(152\) −1574.10 −0.839974
\(153\) 281.142 0.148555
\(154\) −91.1298 −0.0476847
\(155\) −187.890 −0.0973658
\(156\) −243.170 −0.124803
\(157\) 791.623 0.402410 0.201205 0.979549i \(-0.435514\pi\)
0.201205 + 0.979549i \(0.435514\pi\)
\(158\) −2711.80 −1.36544
\(159\) 556.703 0.277669
\(160\) 41.2603 0.0203870
\(161\) −247.706 −0.121254
\(162\) −247.870 −0.120213
\(163\) −3364.00 −1.61650 −0.808248 0.588843i \(-0.799584\pi\)
−0.808248 + 0.588843i \(0.799584\pi\)
\(164\) 618.790 0.294630
\(165\) 17.8083 0.00840227
\(166\) −157.556 −0.0736672
\(167\) 514.992 0.238630 0.119315 0.992856i \(-0.461930\pi\)
0.119315 + 0.992856i \(0.461930\pi\)
\(168\) 206.362 0.0947690
\(169\) 1332.76 0.606629
\(170\) −64.5485 −0.0291214
\(171\) −697.670 −0.312001
\(172\) −238.621 −0.105783
\(173\) 2625.47 1.15382 0.576910 0.816808i \(-0.304258\pi\)
0.576910 + 0.816808i \(0.304258\pi\)
\(174\) −565.214 −0.246257
\(175\) 421.899 0.182243
\(176\) 642.208 0.275047
\(177\) −177.000 −0.0751646
\(178\) −558.012 −0.234971
\(179\) −165.711 −0.0691947 −0.0345973 0.999401i \(-0.511015\pi\)
−0.0345973 + 0.999401i \(0.511015\pi\)
\(180\) 8.29133 0.00343333
\(181\) 1680.47 0.690100 0.345050 0.938584i \(-0.387862\pi\)
0.345050 + 0.938584i \(0.387862\pi\)
\(182\) 615.881 0.250836
\(183\) −2423.47 −0.978951
\(184\) 1484.82 0.594905
\(185\) −220.378 −0.0875812
\(186\) −2554.46 −1.00700
\(187\) −274.612 −0.107388
\(188\) −656.079 −0.254519
\(189\) 91.4638 0.0352011
\(190\) 160.181 0.0611618
\(191\) −825.467 −0.312716 −0.156358 0.987700i \(-0.549975\pi\)
−0.156358 + 0.987700i \(0.549975\pi\)
\(192\) −1192.32 −0.448169
\(193\) 3110.12 1.15995 0.579977 0.814633i \(-0.303061\pi\)
0.579977 + 0.814633i \(0.303061\pi\)
\(194\) 4141.68 1.53276
\(195\) −120.354 −0.0441984
\(196\) −452.306 −0.164835
\(197\) −1252.31 −0.452909 −0.226455 0.974022i \(-0.572713\pi\)
−0.226455 + 0.974022i \(0.572713\pi\)
\(198\) 242.113 0.0869000
\(199\) 854.395 0.304354 0.152177 0.988353i \(-0.451372\pi\)
0.152177 + 0.988353i \(0.451372\pi\)
\(200\) −2528.99 −0.894132
\(201\) 387.242 0.135890
\(202\) 1243.65 0.433182
\(203\) 208.564 0.0721099
\(204\) −127.856 −0.0438809
\(205\) 306.261 0.104342
\(206\) 932.129 0.315265
\(207\) 658.102 0.220972
\(208\) −4340.22 −1.44683
\(209\) 681.466 0.225541
\(210\) −20.9995 −0.00690051
\(211\) 5555.73 1.81266 0.906332 0.422566i \(-0.138870\pi\)
0.906332 + 0.422566i \(0.138870\pi\)
\(212\) −253.174 −0.0820192
\(213\) 816.096 0.262526
\(214\) 4638.22 1.48160
\(215\) −118.102 −0.0374627
\(216\) −548.261 −0.172706
\(217\) 942.595 0.294873
\(218\) 4.82349 0.00149857
\(219\) 1830.16 0.564707
\(220\) −8.09875 −0.00248190
\(221\) 1855.91 0.564895
\(222\) −2996.15 −0.905803
\(223\) −2120.53 −0.636776 −0.318388 0.947961i \(-0.603141\pi\)
−0.318388 + 0.947961i \(0.603141\pi\)
\(224\) −206.992 −0.0617421
\(225\) −1120.90 −0.332117
\(226\) 1768.18 0.520431
\(227\) 3373.06 0.986247 0.493123 0.869959i \(-0.335855\pi\)
0.493123 + 0.869959i \(0.335855\pi\)
\(228\) 317.282 0.0921602
\(229\) 1545.12 0.445872 0.222936 0.974833i \(-0.428436\pi\)
0.222936 + 0.974833i \(0.428436\pi\)
\(230\) −151.096 −0.0433173
\(231\) −89.3395 −0.0254464
\(232\) −1250.19 −0.353790
\(233\) 4860.03 1.36649 0.683243 0.730191i \(-0.260569\pi\)
0.683243 + 0.730191i \(0.260569\pi\)
\(234\) −1636.26 −0.457120
\(235\) −324.717 −0.0901369
\(236\) 80.4950 0.0222024
\(237\) −2658.52 −0.728648
\(238\) 323.823 0.0881945
\(239\) 6957.12 1.88292 0.941462 0.337119i \(-0.109452\pi\)
0.941462 + 0.337119i \(0.109452\pi\)
\(240\) 147.988 0.0398023
\(241\) −463.881 −0.123988 −0.0619942 0.998077i \(-0.519746\pi\)
−0.0619942 + 0.998077i \(0.519746\pi\)
\(242\) 3836.53 1.01910
\(243\) −243.000 −0.0641500
\(244\) 1102.13 0.289167
\(245\) −223.862 −0.0583756
\(246\) 4163.76 1.07915
\(247\) −4605.54 −1.18641
\(248\) −5650.19 −1.44672
\(249\) −154.461 −0.0393116
\(250\) 515.644 0.130449
\(251\) −7550.40 −1.89871 −0.949356 0.314202i \(-0.898263\pi\)
−0.949356 + 0.314202i \(0.898263\pi\)
\(252\) −41.5954 −0.0103979
\(253\) −642.817 −0.159737
\(254\) −892.534 −0.220483
\(255\) −63.2804 −0.0155403
\(256\) 2038.11 0.497586
\(257\) 69.0755 0.0167658 0.00838290 0.999965i \(-0.497332\pi\)
0.00838290 + 0.999965i \(0.497332\pi\)
\(258\) −1605.65 −0.387455
\(259\) 1105.58 0.265240
\(260\) 54.7336 0.0130555
\(261\) −554.110 −0.131412
\(262\) 3550.66 0.837254
\(263\) −6387.24 −1.49754 −0.748772 0.662828i \(-0.769356\pi\)
−0.748772 + 0.662828i \(0.769356\pi\)
\(264\) 535.527 0.124846
\(265\) −125.305 −0.0290468
\(266\) −803.585 −0.185229
\(267\) −547.049 −0.125389
\(268\) −176.108 −0.0401399
\(269\) −1359.76 −0.308201 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(270\) 55.7913 0.0125754
\(271\) −3184.33 −0.713780 −0.356890 0.934146i \(-0.616163\pi\)
−0.356890 + 0.934146i \(0.616163\pi\)
\(272\) −2282.04 −0.508709
\(273\) 603.782 0.133855
\(274\) −1748.61 −0.385538
\(275\) 1094.86 0.240083
\(276\) −299.288 −0.0652717
\(277\) 2086.69 0.452626 0.226313 0.974055i \(-0.427333\pi\)
0.226313 + 0.974055i \(0.427333\pi\)
\(278\) 1763.61 0.380484
\(279\) −2504.27 −0.537373
\(280\) −46.4487 −0.00991372
\(281\) 6730.70 1.42890 0.714448 0.699688i \(-0.246678\pi\)
0.714448 + 0.699688i \(0.246678\pi\)
\(282\) −4414.68 −0.932235
\(283\) 6977.08 1.46553 0.732764 0.680483i \(-0.238230\pi\)
0.732764 + 0.680483i \(0.238230\pi\)
\(284\) −371.139 −0.0775460
\(285\) 157.034 0.0326382
\(286\) 1598.26 0.330445
\(287\) −1536.43 −0.316002
\(288\) 549.934 0.112518
\(289\) −3937.19 −0.801382
\(290\) 127.220 0.0257608
\(291\) 4060.31 0.817937
\(292\) −832.310 −0.166806
\(293\) 1173.58 0.233999 0.116999 0.993132i \(-0.462672\pi\)
0.116999 + 0.993132i \(0.462672\pi\)
\(294\) −3043.51 −0.603746
\(295\) 39.8398 0.00786292
\(296\) −6627.16 −1.30134
\(297\) 237.356 0.0463731
\(298\) 5647.82 1.09788
\(299\) 4344.34 0.840266
\(300\) 509.754 0.0981023
\(301\) 592.485 0.113456
\(302\) 5990.60 1.14146
\(303\) 1219.22 0.231162
\(304\) 5663.01 1.06841
\(305\) 545.483 0.102407
\(306\) −860.328 −0.160724
\(307\) −6437.77 −1.19682 −0.598409 0.801191i \(-0.704200\pi\)
−0.598409 + 0.801191i \(0.704200\pi\)
\(308\) 40.6293 0.00751646
\(309\) 913.817 0.168237
\(310\) 574.966 0.105342
\(311\) −3665.91 −0.668407 −0.334204 0.942501i \(-0.608467\pi\)
−0.334204 + 0.942501i \(0.608467\pi\)
\(312\) −3619.24 −0.656728
\(313\) −1795.66 −0.324270 −0.162135 0.986769i \(-0.551838\pi\)
−0.162135 + 0.986769i \(0.551838\pi\)
\(314\) −2422.46 −0.435373
\(315\) −20.5870 −0.00368237
\(316\) 1209.03 0.215231
\(317\) 793.810 0.140646 0.0703230 0.997524i \(-0.477597\pi\)
0.0703230 + 0.997524i \(0.477597\pi\)
\(318\) −1703.58 −0.300415
\(319\) 541.240 0.0949958
\(320\) 268.372 0.0468827
\(321\) 4547.09 0.790636
\(322\) 758.009 0.131187
\(323\) −2421.54 −0.417145
\(324\) 110.510 0.0189489
\(325\) −7399.39 −1.26291
\(326\) 10294.2 1.74891
\(327\) 4.72873 0.000799692 0
\(328\) 9209.80 1.55038
\(329\) 1629.02 0.272980
\(330\) −54.4955 −0.00909055
\(331\) 3322.57 0.551737 0.275868 0.961195i \(-0.411035\pi\)
0.275868 + 0.961195i \(0.411035\pi\)
\(332\) 70.2449 0.0116120
\(333\) −2937.29 −0.483370
\(334\) −1575.94 −0.258178
\(335\) −87.1619 −0.0142154
\(336\) −742.414 −0.120542
\(337\) 2092.35 0.338213 0.169106 0.985598i \(-0.445912\pi\)
0.169106 + 0.985598i \(0.445912\pi\)
\(338\) −4078.42 −0.656322
\(339\) 1733.44 0.277721
\(340\) 28.7783 0.00459036
\(341\) 2446.11 0.388458
\(342\) 2134.95 0.337559
\(343\) 2284.99 0.359701
\(344\) −3551.53 −0.556644
\(345\) −148.128 −0.0231158
\(346\) −8034.25 −1.24834
\(347\) −9919.25 −1.53456 −0.767281 0.641311i \(-0.778391\pi\)
−0.767281 + 0.641311i \(0.778391\pi\)
\(348\) 251.995 0.0388171
\(349\) 1359.74 0.208554 0.104277 0.994548i \(-0.466747\pi\)
0.104277 + 0.994548i \(0.466747\pi\)
\(350\) −1291.06 −0.197172
\(351\) −1604.12 −0.243936
\(352\) −537.161 −0.0813374
\(353\) −1123.35 −0.169376 −0.0846880 0.996408i \(-0.526989\pi\)
−0.0846880 + 0.996408i \(0.526989\pi\)
\(354\) 541.641 0.0813217
\(355\) −183.690 −0.0274626
\(356\) 248.784 0.0370380
\(357\) 317.461 0.0470639
\(358\) 507.096 0.0748628
\(359\) 4438.64 0.652541 0.326271 0.945276i \(-0.394208\pi\)
0.326271 + 0.945276i \(0.394208\pi\)
\(360\) 123.404 0.0180666
\(361\) −849.815 −0.123898
\(362\) −5142.43 −0.746630
\(363\) 3761.16 0.543828
\(364\) −274.584 −0.0395388
\(365\) −411.939 −0.0590736
\(366\) 7416.10 1.05914
\(367\) −12343.3 −1.75563 −0.877817 0.478996i \(-0.841001\pi\)
−0.877817 + 0.478996i \(0.841001\pi\)
\(368\) −5341.83 −0.756690
\(369\) 4081.96 0.575877
\(370\) 674.383 0.0947554
\(371\) 628.620 0.0879685
\(372\) 1138.88 0.158731
\(373\) 8458.96 1.17423 0.587116 0.809503i \(-0.300263\pi\)
0.587116 + 0.809503i \(0.300263\pi\)
\(374\) 840.346 0.116185
\(375\) 505.514 0.0696123
\(376\) −9764.79 −1.33931
\(377\) −3657.85 −0.499706
\(378\) −279.890 −0.0380846
\(379\) −7901.32 −1.07088 −0.535440 0.844573i \(-0.679854\pi\)
−0.535440 + 0.844573i \(0.679854\pi\)
\(380\) −71.4150 −0.00964082
\(381\) −875.000 −0.117658
\(382\) 2526.03 0.338332
\(383\) 1775.56 0.236885 0.118443 0.992961i \(-0.462210\pi\)
0.118443 + 0.992961i \(0.462210\pi\)
\(384\) 5115.14 0.679768
\(385\) 20.1088 0.00266193
\(386\) −9517.33 −1.25497
\(387\) −1574.11 −0.206761
\(388\) −1846.52 −0.241606
\(389\) 12845.2 1.67423 0.837115 0.547027i \(-0.184241\pi\)
0.837115 + 0.547027i \(0.184241\pi\)
\(390\) 368.296 0.0478190
\(391\) 2284.20 0.295440
\(392\) −6731.92 −0.867381
\(393\) 3480.91 0.446790
\(394\) 3832.20 0.490009
\(395\) 598.390 0.0762234
\(396\) −107.943 −0.0136979
\(397\) 3824.54 0.483497 0.241749 0.970339i \(-0.422279\pi\)
0.241749 + 0.970339i \(0.422279\pi\)
\(398\) −2614.55 −0.329285
\(399\) −787.798 −0.0988452
\(400\) 9098.34 1.13729
\(401\) −10107.1 −1.25867 −0.629334 0.777135i \(-0.716672\pi\)
−0.629334 + 0.777135i \(0.716672\pi\)
\(402\) −1185.01 −0.147022
\(403\) −16531.5 −2.04341
\(404\) −554.468 −0.0682817
\(405\) 54.6953 0.00671069
\(406\) −638.230 −0.0780168
\(407\) 2869.07 0.349421
\(408\) −1902.95 −0.230907
\(409\) −863.888 −0.104441 −0.0522207 0.998636i \(-0.516630\pi\)
−0.0522207 + 0.998636i \(0.516630\pi\)
\(410\) −937.194 −0.112890
\(411\) −1714.26 −0.205737
\(412\) −415.580 −0.0496945
\(413\) −199.865 −0.0238129
\(414\) −2013.87 −0.239073
\(415\) 34.7666 0.00411235
\(416\) 3630.28 0.427859
\(417\) 1728.97 0.203041
\(418\) −2085.37 −0.244016
\(419\) 10373.1 1.20945 0.604725 0.796434i \(-0.293283\pi\)
0.604725 + 0.796434i \(0.293283\pi\)
\(420\) 9.36243 0.00108771
\(421\) 4058.69 0.469853 0.234927 0.972013i \(-0.424515\pi\)
0.234927 + 0.972013i \(0.424515\pi\)
\(422\) −17001.2 −1.96115
\(423\) −4327.95 −0.497475
\(424\) −3768.13 −0.431596
\(425\) −3890.51 −0.444041
\(426\) −2497.35 −0.284031
\(427\) −2736.54 −0.310142
\(428\) −2067.90 −0.233541
\(429\) 1566.86 0.176338
\(430\) 361.405 0.0405314
\(431\) −13813.1 −1.54375 −0.771873 0.635777i \(-0.780680\pi\)
−0.771873 + 0.635777i \(0.780680\pi\)
\(432\) 1972.44 0.219673
\(433\) −14065.7 −1.56110 −0.780551 0.625093i \(-0.785061\pi\)
−0.780551 + 0.625093i \(0.785061\pi\)
\(434\) −2884.45 −0.319028
\(435\) 124.721 0.0137469
\(436\) −2.15050 −0.000236216 0
\(437\) −5668.37 −0.620492
\(438\) −5600.51 −0.610965
\(439\) 15470.0 1.68187 0.840937 0.541133i \(-0.182004\pi\)
0.840937 + 0.541133i \(0.182004\pi\)
\(440\) −120.538 −0.0130601
\(441\) −2983.72 −0.322181
\(442\) −5679.29 −0.611168
\(443\) 805.886 0.0864307 0.0432153 0.999066i \(-0.486240\pi\)
0.0432153 + 0.999066i \(0.486240\pi\)
\(444\) 1335.80 0.142780
\(445\) 123.132 0.0131169
\(446\) 6489.06 0.688937
\(447\) 5536.86 0.585871
\(448\) −1346.35 −0.141985
\(449\) 15047.6 1.58160 0.790800 0.612074i \(-0.209665\pi\)
0.790800 + 0.612074i \(0.209665\pi\)
\(450\) 3430.07 0.359323
\(451\) −3987.16 −0.416292
\(452\) −788.323 −0.0820345
\(453\) 5872.91 0.609124
\(454\) −10322.0 −1.06704
\(455\) −135.901 −0.0140025
\(456\) 4722.29 0.484959
\(457\) 2132.43 0.218273 0.109137 0.994027i \(-0.465191\pi\)
0.109137 + 0.994027i \(0.465191\pi\)
\(458\) −4728.26 −0.482395
\(459\) −843.426 −0.0857685
\(460\) 67.3647 0.00682803
\(461\) −7140.37 −0.721389 −0.360695 0.932684i \(-0.617460\pi\)
−0.360695 + 0.932684i \(0.617460\pi\)
\(462\) 273.390 0.0275308
\(463\) 10919.2 1.09602 0.548011 0.836471i \(-0.315385\pi\)
0.548011 + 0.836471i \(0.315385\pi\)
\(464\) 4497.72 0.450003
\(465\) 563.671 0.0562142
\(466\) −14872.3 −1.47842
\(467\) 13946.8 1.38197 0.690987 0.722867i \(-0.257176\pi\)
0.690987 + 0.722867i \(0.257176\pi\)
\(468\) 729.511 0.0720549
\(469\) 437.268 0.0430515
\(470\) 993.671 0.0975205
\(471\) −2374.87 −0.232332
\(472\) 1198.05 0.116832
\(473\) 1537.55 0.149464
\(474\) 8135.40 0.788336
\(475\) 9654.52 0.932589
\(476\) −144.373 −0.0139019
\(477\) −1670.11 −0.160313
\(478\) −21289.6 −2.03716
\(479\) 18107.2 1.72723 0.863613 0.504155i \(-0.168196\pi\)
0.863613 + 0.504155i \(0.168196\pi\)
\(480\) −123.781 −0.0117704
\(481\) −19389.9 −1.83806
\(482\) 1419.53 0.134145
\(483\) 743.118 0.0700063
\(484\) −1710.48 −0.160638
\(485\) −913.908 −0.0855638
\(486\) 743.609 0.0694049
\(487\) −15930.5 −1.48230 −0.741150 0.671340i \(-0.765719\pi\)
−0.741150 + 0.671340i \(0.765719\pi\)
\(488\) 16403.6 1.52163
\(489\) 10092.0 0.933284
\(490\) 685.044 0.0631574
\(491\) 2619.27 0.240745 0.120373 0.992729i \(-0.461591\pi\)
0.120373 + 0.992729i \(0.461591\pi\)
\(492\) −1856.37 −0.170105
\(493\) −1923.25 −0.175698
\(494\) 14093.5 1.28360
\(495\) −53.4249 −0.00485105
\(496\) 20327.2 1.84016
\(497\) 921.522 0.0831709
\(498\) 472.669 0.0425318
\(499\) −9350.73 −0.838870 −0.419435 0.907785i \(-0.637772\pi\)
−0.419435 + 0.907785i \(0.637772\pi\)
\(500\) −229.895 −0.0205624
\(501\) −1544.98 −0.137773
\(502\) 23105.1 2.05425
\(503\) −7300.43 −0.647137 −0.323568 0.946205i \(-0.604883\pi\)
−0.323568 + 0.946205i \(0.604883\pi\)
\(504\) −619.087 −0.0547149
\(505\) −274.426 −0.0241817
\(506\) 1967.10 0.172822
\(507\) −3998.29 −0.350238
\(508\) 397.927 0.0347542
\(509\) 16241.5 1.41432 0.707162 0.707052i \(-0.249975\pi\)
0.707162 + 0.707052i \(0.249975\pi\)
\(510\) 193.645 0.0168133
\(511\) 2066.59 0.178905
\(512\) 7403.50 0.639046
\(513\) 2093.01 0.180134
\(514\) −211.379 −0.0181392
\(515\) −205.685 −0.0175991
\(516\) 715.862 0.0610738
\(517\) 4227.43 0.359617
\(518\) −3383.20 −0.286968
\(519\) −7876.41 −0.666158
\(520\) 814.631 0.0686999
\(521\) 21182.1 1.78120 0.890598 0.454792i \(-0.150286\pi\)
0.890598 + 0.454792i \(0.150286\pi\)
\(522\) 1695.64 0.142177
\(523\) 19761.6 1.65222 0.826112 0.563505i \(-0.190548\pi\)
0.826112 + 0.563505i \(0.190548\pi\)
\(524\) −1583.03 −0.131975
\(525\) −1265.70 −0.105218
\(526\) 19545.7 1.62022
\(527\) −8692.06 −0.718467
\(528\) −1926.63 −0.158798
\(529\) −6820.11 −0.560541
\(530\) 383.447 0.0314262
\(531\) 531.000 0.0433963
\(532\) 358.270 0.0291973
\(533\) 26946.3 2.18982
\(534\) 1674.04 0.135660
\(535\) −1023.48 −0.0827079
\(536\) −2621.11 −0.211221
\(537\) 497.134 0.0399496
\(538\) 4161.02 0.333447
\(539\) 2914.42 0.232900
\(540\) −24.8740 −0.00198223
\(541\) 1805.49 0.143482 0.0717412 0.997423i \(-0.477144\pi\)
0.0717412 + 0.997423i \(0.477144\pi\)
\(542\) 9744.44 0.772250
\(543\) −5041.40 −0.398429
\(544\) 1908.76 0.150436
\(545\) −1.06436 −8.36552e−5 0
\(546\) −1847.64 −0.144820
\(547\) −2017.11 −0.157670 −0.0788349 0.996888i \(-0.525120\pi\)
−0.0788349 + 0.996888i \(0.525120\pi\)
\(548\) 779.599 0.0607716
\(549\) 7270.41 0.565198
\(550\) −3350.41 −0.259749
\(551\) 4772.67 0.369006
\(552\) −4454.46 −0.343468
\(553\) −3001.96 −0.230843
\(554\) −6385.53 −0.489703
\(555\) 661.134 0.0505650
\(556\) −786.289 −0.0599750
\(557\) 22836.9 1.73722 0.868611 0.495495i \(-0.165013\pi\)
0.868611 + 0.495495i \(0.165013\pi\)
\(558\) 7663.37 0.581392
\(559\) −10391.2 −0.786225
\(560\) 167.105 0.0126098
\(561\) 823.837 0.0620007
\(562\) −20596.7 −1.54594
\(563\) −11521.8 −0.862494 −0.431247 0.902234i \(-0.641926\pi\)
−0.431247 + 0.902234i \(0.641926\pi\)
\(564\) 1968.24 0.146946
\(565\) −390.168 −0.0290522
\(566\) −21350.7 −1.58558
\(567\) −274.392 −0.0203234
\(568\) −5523.87 −0.408057
\(569\) −431.235 −0.0317721 −0.0158860 0.999874i \(-0.505057\pi\)
−0.0158860 + 0.999874i \(0.505057\pi\)
\(570\) −480.543 −0.0353118
\(571\) 7544.48 0.552937 0.276468 0.961023i \(-0.410836\pi\)
0.276468 + 0.961023i \(0.410836\pi\)
\(572\) −712.568 −0.0520874
\(573\) 2476.40 0.180547
\(574\) 4701.65 0.341887
\(575\) −9106.97 −0.660499
\(576\) 3576.97 0.258751
\(577\) −12507.5 −0.902416 −0.451208 0.892419i \(-0.649007\pi\)
−0.451208 + 0.892419i \(0.649007\pi\)
\(578\) 12048.3 0.867027
\(579\) −9330.36 −0.669700
\(580\) −56.7198 −0.00406063
\(581\) −174.415 −0.0124543
\(582\) −12425.0 −0.884938
\(583\) 1631.32 0.115887
\(584\) −12387.7 −0.877753
\(585\) 361.061 0.0255180
\(586\) −3591.31 −0.253167
\(587\) 12553.1 0.882659 0.441330 0.897345i \(-0.354507\pi\)
0.441330 + 0.897345i \(0.354507\pi\)
\(588\) 1356.92 0.0951673
\(589\) 21569.9 1.50895
\(590\) −121.914 −0.00850701
\(591\) 3756.92 0.261487
\(592\) 23842.0 1.65524
\(593\) −4198.56 −0.290749 −0.145374 0.989377i \(-0.546439\pi\)
−0.145374 + 0.989377i \(0.546439\pi\)
\(594\) −726.338 −0.0501717
\(595\) −71.4552 −0.00492332
\(596\) −2518.02 −0.173057
\(597\) −2563.18 −0.175719
\(598\) −13294.2 −0.909097
\(599\) −4991.05 −0.340449 −0.170224 0.985405i \(-0.554449\pi\)
−0.170224 + 0.985405i \(0.554449\pi\)
\(600\) 7586.96 0.516227
\(601\) −8600.34 −0.583719 −0.291859 0.956461i \(-0.594274\pi\)
−0.291859 + 0.956461i \(0.594274\pi\)
\(602\) −1813.07 −0.122750
\(603\) −1161.73 −0.0784564
\(604\) −2670.84 −0.179926
\(605\) −846.574 −0.0568895
\(606\) −3730.95 −0.250098
\(607\) −17833.1 −1.19246 −0.596230 0.802814i \(-0.703335\pi\)
−0.596230 + 0.802814i \(0.703335\pi\)
\(608\) −4736.70 −0.315951
\(609\) −625.692 −0.0416327
\(610\) −1669.24 −0.110796
\(611\) −28570.2 −1.89169
\(612\) 383.568 0.0253347
\(613\) −22215.0 −1.46371 −0.731856 0.681459i \(-0.761346\pi\)
−0.731856 + 0.681459i \(0.761346\pi\)
\(614\) 19700.3 1.29485
\(615\) −918.782 −0.0602421
\(616\) 604.708 0.0395526
\(617\) 18251.1 1.19086 0.595432 0.803406i \(-0.296981\pi\)
0.595432 + 0.803406i \(0.296981\pi\)
\(618\) −2796.39 −0.182018
\(619\) 9872.40 0.641042 0.320521 0.947241i \(-0.396142\pi\)
0.320521 + 0.947241i \(0.396142\pi\)
\(620\) −256.343 −0.0166048
\(621\) −1974.31 −0.127578
\(622\) 11218.1 0.723160
\(623\) −617.719 −0.0397245
\(624\) 13020.7 0.835327
\(625\) 15454.2 0.989070
\(626\) 5494.92 0.350833
\(627\) −2044.40 −0.130216
\(628\) 1080.03 0.0686271
\(629\) −10195.0 −0.646265
\(630\) 62.9986 0.00398401
\(631\) −9843.54 −0.621023 −0.310511 0.950570i \(-0.600500\pi\)
−0.310511 + 0.950570i \(0.600500\pi\)
\(632\) 17994.6 1.13258
\(633\) −16667.2 −1.04654
\(634\) −2429.15 −0.152167
\(635\) 196.948 0.0123081
\(636\) 759.522 0.0473538
\(637\) −19696.5 −1.22512
\(638\) −1656.26 −0.102777
\(639\) −2448.29 −0.151569
\(640\) −1151.33 −0.0711100
\(641\) −16239.8 −1.00068 −0.500338 0.865830i \(-0.666791\pi\)
−0.500338 + 0.865830i \(0.666791\pi\)
\(642\) −13914.6 −0.855401
\(643\) −4102.04 −0.251584 −0.125792 0.992057i \(-0.540147\pi\)
−0.125792 + 0.992057i \(0.540147\pi\)
\(644\) −337.951 −0.0206788
\(645\) 354.305 0.0216291
\(646\) 7410.19 0.451316
\(647\) −29339.0 −1.78274 −0.891371 0.453274i \(-0.850256\pi\)
−0.891371 + 0.453274i \(0.850256\pi\)
\(648\) 1644.78 0.0997117
\(649\) −518.667 −0.0313705
\(650\) 22643.0 1.36636
\(651\) −2827.78 −0.170245
\(652\) −4589.58 −0.275677
\(653\) −22467.1 −1.34641 −0.673206 0.739455i \(-0.735083\pi\)
−0.673206 + 0.739455i \(0.735083\pi\)
\(654\) −14.4705 −0.000865198 0
\(655\) −783.494 −0.0467384
\(656\) −33133.4 −1.97201
\(657\) −5490.49 −0.326034
\(658\) −4984.98 −0.295342
\(659\) 14150.3 0.836447 0.418223 0.908344i \(-0.362653\pi\)
0.418223 + 0.908344i \(0.362653\pi\)
\(660\) 24.2963 0.00143293
\(661\) −10577.7 −0.622428 −0.311214 0.950340i \(-0.600736\pi\)
−0.311214 + 0.950340i \(0.600736\pi\)
\(662\) −10167.4 −0.596932
\(663\) −5567.72 −0.326142
\(664\) 1045.49 0.0611040
\(665\) 177.320 0.0103401
\(666\) 8988.44 0.522965
\(667\) −4501.99 −0.261346
\(668\) 702.615 0.0406961
\(669\) 6361.58 0.367643
\(670\) 266.726 0.0153799
\(671\) −7101.55 −0.408573
\(672\) 620.976 0.0356468
\(673\) −9785.95 −0.560506 −0.280253 0.959926i \(-0.590418\pi\)
−0.280253 + 0.959926i \(0.590418\pi\)
\(674\) −6402.85 −0.365918
\(675\) 3362.69 0.191748
\(676\) 1818.32 0.103455
\(677\) −25327.9 −1.43786 −0.718929 0.695084i \(-0.755367\pi\)
−0.718929 + 0.695084i \(0.755367\pi\)
\(678\) −5304.53 −0.300471
\(679\) 4584.83 0.259131
\(680\) 428.323 0.0241551
\(681\) −10119.2 −0.569410
\(682\) −7485.39 −0.420279
\(683\) 11813.3 0.661819 0.330909 0.943662i \(-0.392645\pi\)
0.330909 + 0.943662i \(0.392645\pi\)
\(684\) −951.847 −0.0532087
\(685\) 385.851 0.0215220
\(686\) −6992.32 −0.389166
\(687\) −4635.37 −0.257424
\(688\) 12777.1 0.708024
\(689\) −11024.9 −0.609602
\(690\) 453.289 0.0250093
\(691\) 10486.4 0.577309 0.288655 0.957433i \(-0.406792\pi\)
0.288655 + 0.957433i \(0.406792\pi\)
\(692\) 3581.99 0.196773
\(693\) 268.019 0.0146915
\(694\) 30354.1 1.66027
\(695\) −389.162 −0.0212399
\(696\) 3750.58 0.204261
\(697\) 14168.0 0.769946
\(698\) −4160.97 −0.225637
\(699\) −14580.1 −0.788942
\(700\) 575.606 0.0310798
\(701\) 10841.7 0.584144 0.292072 0.956396i \(-0.405655\pi\)
0.292072 + 0.956396i \(0.405655\pi\)
\(702\) 4908.79 0.263918
\(703\) 25299.5 1.35731
\(704\) −3493.89 −0.187047
\(705\) 974.150 0.0520406
\(706\) 3437.57 0.183250
\(707\) 1376.72 0.0732346
\(708\) −241.485 −0.0128186
\(709\) −15139.3 −0.801930 −0.400965 0.916093i \(-0.631325\pi\)
−0.400965 + 0.916093i \(0.631325\pi\)
\(710\) 562.112 0.0297123
\(711\) 7975.57 0.420685
\(712\) 3702.79 0.194899
\(713\) −20346.5 −1.06870
\(714\) −971.468 −0.0509191
\(715\) −352.675 −0.0184466
\(716\) −226.084 −0.0118005
\(717\) −20871.4 −1.08711
\(718\) −13582.8 −0.705994
\(719\) −26507.9 −1.37494 −0.687468 0.726215i \(-0.741278\pi\)
−0.687468 + 0.726215i \(0.741278\pi\)
\(720\) −443.963 −0.0229799
\(721\) 1031.87 0.0532992
\(722\) 2600.53 0.134047
\(723\) 1391.64 0.0715847
\(724\) 2292.70 0.117690
\(725\) 7667.90 0.392798
\(726\) −11509.6 −0.588376
\(727\) −4196.55 −0.214087 −0.107044 0.994254i \(-0.534138\pi\)
−0.107044 + 0.994254i \(0.534138\pi\)
\(728\) −4086.79 −0.208058
\(729\) 729.000 0.0370370
\(730\) 1260.58 0.0639127
\(731\) −5463.55 −0.276439
\(732\) −3306.39 −0.166950
\(733\) 19609.3 0.988110 0.494055 0.869431i \(-0.335514\pi\)
0.494055 + 0.869431i \(0.335514\pi\)
\(734\) 37772.1 1.89945
\(735\) 671.586 0.0337032
\(736\) 4468.05 0.223770
\(737\) 1134.75 0.0567149
\(738\) −12491.3 −0.623050
\(739\) 2865.95 0.142660 0.0713299 0.997453i \(-0.477276\pi\)
0.0713299 + 0.997453i \(0.477276\pi\)
\(740\) −300.667 −0.0149361
\(741\) 13816.6 0.684975
\(742\) −1923.65 −0.0951745
\(743\) 6771.12 0.334332 0.167166 0.985929i \(-0.446538\pi\)
0.167166 + 0.985929i \(0.446538\pi\)
\(744\) 16950.6 0.835266
\(745\) −1246.26 −0.0612876
\(746\) −25885.4 −1.27042
\(747\) 463.383 0.0226965
\(748\) −374.659 −0.0183140
\(749\) 5134.50 0.250482
\(750\) −1546.93 −0.0753146
\(751\) −15205.4 −0.738819 −0.369409 0.929267i \(-0.620440\pi\)
−0.369409 + 0.929267i \(0.620440\pi\)
\(752\) 35130.1 1.70354
\(753\) 22651.2 1.09622
\(754\) 11193.5 0.540639
\(755\) −1321.89 −0.0637201
\(756\) 124.786 0.00600321
\(757\) 11748.5 0.564077 0.282038 0.959403i \(-0.408989\pi\)
0.282038 + 0.959403i \(0.408989\pi\)
\(758\) 24179.0 1.15860
\(759\) 1928.45 0.0922244
\(760\) −1062.91 −0.0507313
\(761\) −33071.4 −1.57535 −0.787673 0.616094i \(-0.788714\pi\)
−0.787673 + 0.616094i \(0.788714\pi\)
\(762\) 2677.60 0.127296
\(763\) 5.33960 0.000253351 0
\(764\) −1126.20 −0.0533306
\(765\) 189.841 0.00897219
\(766\) −5433.43 −0.256290
\(767\) 3505.30 0.165018
\(768\) −6114.34 −0.287282
\(769\) 32399.7 1.51933 0.759665 0.650315i \(-0.225363\pi\)
0.759665 + 0.650315i \(0.225363\pi\)
\(770\) −61.5354 −0.00287998
\(771\) −207.227 −0.00967974
\(772\) 4243.20 0.197819
\(773\) −35967.3 −1.67355 −0.836774 0.547548i \(-0.815561\pi\)
−0.836774 + 0.547548i \(0.815561\pi\)
\(774\) 4816.95 0.223697
\(775\) 34654.7 1.60624
\(776\) −27482.8 −1.27136
\(777\) −3316.73 −0.153137
\(778\) −39307.7 −1.81137
\(779\) −35158.8 −1.61707
\(780\) −164.201 −0.00753761
\(781\) 2391.43 0.109567
\(782\) −6989.92 −0.319641
\(783\) 1662.33 0.0758708
\(784\) 24218.9 1.10327
\(785\) 534.543 0.0243040
\(786\) −10652.0 −0.483389
\(787\) −2130.77 −0.0965106 −0.0482553 0.998835i \(-0.515366\pi\)
−0.0482553 + 0.998835i \(0.515366\pi\)
\(788\) −1708.55 −0.0772392
\(789\) 19161.7 0.864607
\(790\) −1831.14 −0.0824673
\(791\) 1957.37 0.0879849
\(792\) −1606.58 −0.0720800
\(793\) 47994.3 2.14921
\(794\) −11703.6 −0.523103
\(795\) 375.914 0.0167702
\(796\) 1165.67 0.0519046
\(797\) 25863.7 1.14949 0.574743 0.818334i \(-0.305102\pi\)
0.574743 + 0.818334i \(0.305102\pi\)
\(798\) 2410.75 0.106942
\(799\) −15021.8 −0.665124
\(800\) −7610.11 −0.336322
\(801\) 1641.15 0.0723934
\(802\) 30929.0 1.36177
\(803\) 5362.97 0.235685
\(804\) 528.323 0.0231748
\(805\) −167.263 −0.00732331
\(806\) 50588.3 2.21079
\(807\) 4079.28 0.177940
\(808\) −8252.46 −0.359307
\(809\) 26720.2 1.16123 0.580614 0.814179i \(-0.302813\pi\)
0.580614 + 0.814179i \(0.302813\pi\)
\(810\) −167.374 −0.00726040
\(811\) 23248.5 1.00662 0.503308 0.864107i \(-0.332116\pi\)
0.503308 + 0.864107i \(0.332116\pi\)
\(812\) 284.548 0.0122976
\(813\) 9553.00 0.412101
\(814\) −8779.68 −0.378044
\(815\) −2271.54 −0.0976302
\(816\) 6846.11 0.293703
\(817\) 13558.1 0.580586
\(818\) 2643.60 0.112997
\(819\) −1811.34 −0.0772814
\(820\) 417.838 0.0177946
\(821\) 7799.17 0.331538 0.165769 0.986165i \(-0.446989\pi\)
0.165769 + 0.986165i \(0.446989\pi\)
\(822\) 5245.83 0.222590
\(823\) 2499.10 0.105848 0.0529242 0.998599i \(-0.483146\pi\)
0.0529242 + 0.998599i \(0.483146\pi\)
\(824\) −6185.31 −0.261499
\(825\) −3284.59 −0.138612
\(826\) 611.612 0.0257636
\(827\) −39871.3 −1.67649 −0.838246 0.545292i \(-0.816419\pi\)
−0.838246 + 0.545292i \(0.816419\pi\)
\(828\) 897.863 0.0376846
\(829\) 8721.76 0.365403 0.182702 0.983168i \(-0.441516\pi\)
0.182702 + 0.983168i \(0.441516\pi\)
\(830\) −106.390 −0.00444922
\(831\) −6260.08 −0.261324
\(832\) 23612.7 0.983922
\(833\) −10356.2 −0.430756
\(834\) −5290.84 −0.219673
\(835\) 347.749 0.0144124
\(836\) 929.739 0.0384638
\(837\) 7512.82 0.310252
\(838\) −31742.9 −1.30852
\(839\) −11850.6 −0.487637 −0.243818 0.969821i \(-0.578400\pi\)
−0.243818 + 0.969821i \(0.578400\pi\)
\(840\) 139.346 0.00572369
\(841\) −20598.4 −0.844578
\(842\) −12420.1 −0.508341
\(843\) −20192.1 −0.824974
\(844\) 7579.80 0.309132
\(845\) 899.950 0.0366381
\(846\) 13244.0 0.538226
\(847\) 4247.04 0.172290
\(848\) 13556.3 0.548969
\(849\) −20931.2 −0.846123
\(850\) 11905.4 0.480414
\(851\) −23864.6 −0.961303
\(852\) 1113.42 0.0447712
\(853\) −25857.7 −1.03793 −0.518963 0.854797i \(-0.673682\pi\)
−0.518963 + 0.854797i \(0.673682\pi\)
\(854\) 8374.14 0.335547
\(855\) −471.102 −0.0188437
\(856\) −30777.7 −1.22893
\(857\) −13217.7 −0.526847 −0.263424 0.964680i \(-0.584852\pi\)
−0.263424 + 0.964680i \(0.584852\pi\)
\(858\) −4794.78 −0.190782
\(859\) 45527.8 1.80837 0.904184 0.427142i \(-0.140480\pi\)
0.904184 + 0.427142i \(0.140480\pi\)
\(860\) −161.129 −0.00638889
\(861\) 4609.28 0.182444
\(862\) 42269.8 1.67020
\(863\) 5044.12 0.198961 0.0994807 0.995039i \(-0.468282\pi\)
0.0994807 + 0.995039i \(0.468282\pi\)
\(864\) −1649.80 −0.0649622
\(865\) 1772.85 0.0696864
\(866\) 43042.9 1.68898
\(867\) 11811.6 0.462678
\(868\) 1286.00 0.0502877
\(869\) −7790.33 −0.304107
\(870\) −381.661 −0.0148730
\(871\) −7668.93 −0.298337
\(872\) −32.0071 −0.00124300
\(873\) −12180.9 −0.472236
\(874\) 17345.9 0.671320
\(875\) 570.818 0.0220539
\(876\) 2496.93 0.0963053
\(877\) −16427.0 −0.632498 −0.316249 0.948676i \(-0.602424\pi\)
−0.316249 + 0.948676i \(0.602424\pi\)
\(878\) −47340.0 −1.81965
\(879\) −3520.75 −0.135099
\(880\) 433.651 0.0166118
\(881\) 49555.6 1.89508 0.947542 0.319631i \(-0.103559\pi\)
0.947542 + 0.319631i \(0.103559\pi\)
\(882\) 9130.54 0.348573
\(883\) −10096.6 −0.384798 −0.192399 0.981317i \(-0.561627\pi\)
−0.192399 + 0.981317i \(0.561627\pi\)
\(884\) 2532.05 0.0963373
\(885\) −119.519 −0.00453966
\(886\) −2466.11 −0.0935106
\(887\) 17531.6 0.663647 0.331824 0.943341i \(-0.392336\pi\)
0.331824 + 0.943341i \(0.392336\pi\)
\(888\) 19881.5 0.751327
\(889\) −988.035 −0.0372752
\(890\) −376.798 −0.0141913
\(891\) −712.068 −0.0267735
\(892\) −2893.08 −0.108596
\(893\) 37277.6 1.39692
\(894\) −16943.4 −0.633863
\(895\) −111.897 −0.00417910
\(896\) 5775.93 0.215357
\(897\) −13033.0 −0.485128
\(898\) −46047.3 −1.71116
\(899\) 17131.4 0.635555
\(900\) −1529.26 −0.0566394
\(901\) −5796.77 −0.214338
\(902\) 12201.2 0.450393
\(903\) −1777.45 −0.0655039
\(904\) −11733.0 −0.431676
\(905\) 1134.74 0.0416794
\(906\) −17971.8 −0.659021
\(907\) 14205.5 0.520052 0.260026 0.965602i \(-0.416269\pi\)
0.260026 + 0.965602i \(0.416269\pi\)
\(908\) 4601.94 0.168195
\(909\) −3657.65 −0.133462
\(910\) 415.874 0.0151495
\(911\) −6382.81 −0.232132 −0.116066 0.993242i \(-0.537028\pi\)
−0.116066 + 0.993242i \(0.537028\pi\)
\(912\) −16989.0 −0.616845
\(913\) −452.621 −0.0164070
\(914\) −6525.48 −0.236153
\(915\) −1636.45 −0.0591249
\(916\) 2108.04 0.0760390
\(917\) 3930.58 0.141548
\(918\) 2580.98 0.0927943
\(919\) −4515.26 −0.162073 −0.0810363 0.996711i \(-0.525823\pi\)
−0.0810363 + 0.996711i \(0.525823\pi\)
\(920\) 1002.63 0.0359300
\(921\) 19313.3 0.690983
\(922\) 21850.4 0.780482
\(923\) −16161.9 −0.576356
\(924\) −121.888 −0.00433963
\(925\) 40646.8 1.44482
\(926\) −33414.0 −1.18580
\(927\) −2741.45 −0.0971316
\(928\) −3762.02 −0.133076
\(929\) −33860.3 −1.19583 −0.597913 0.801561i \(-0.704003\pi\)
−0.597913 + 0.801561i \(0.704003\pi\)
\(930\) −1724.90 −0.0608190
\(931\) 25699.4 0.904688
\(932\) 6630.65 0.233041
\(933\) 10997.7 0.385905
\(934\) −42678.9 −1.49518
\(935\) −185.432 −0.00648586
\(936\) 10857.7 0.379162
\(937\) −22085.0 −0.769996 −0.384998 0.922917i \(-0.625798\pi\)
−0.384998 + 0.922917i \(0.625798\pi\)
\(938\) −1338.09 −0.0465780
\(939\) 5386.97 0.187217
\(940\) −443.018 −0.0153720
\(941\) −14767.9 −0.511603 −0.255801 0.966729i \(-0.582339\pi\)
−0.255801 + 0.966729i \(0.582339\pi\)
\(942\) 7267.38 0.251363
\(943\) 33164.8 1.14527
\(944\) −4310.14 −0.148605
\(945\) 61.7610 0.00212602
\(946\) −4705.07 −0.161707
\(947\) 39640.1 1.36022 0.680111 0.733109i \(-0.261932\pi\)
0.680111 + 0.733109i \(0.261932\pi\)
\(948\) −3627.08 −0.124264
\(949\) −36244.4 −1.23977
\(950\) −29544.0 −1.00898
\(951\) −2381.43 −0.0812020
\(952\) −2148.78 −0.0731538
\(953\) −10159.5 −0.345328 −0.172664 0.984981i \(-0.555237\pi\)
−0.172664 + 0.984981i \(0.555237\pi\)
\(954\) 5110.73 0.173445
\(955\) −557.397 −0.0188869
\(956\) 9491.75 0.321114
\(957\) −1623.72 −0.0548458
\(958\) −55410.3 −1.86871
\(959\) −1935.71 −0.0651797
\(960\) −805.116 −0.0270677
\(961\) 47633.6 1.59892
\(962\) 59335.5 1.98862
\(963\) −13641.3 −0.456474
\(964\) −632.882 −0.0211450
\(965\) 2100.11 0.0700569
\(966\) −2274.03 −0.0757408
\(967\) 2430.67 0.0808325 0.0404163 0.999183i \(-0.487132\pi\)
0.0404163 + 0.999183i \(0.487132\pi\)
\(968\) −25458.0 −0.845300
\(969\) 7264.61 0.240839
\(970\) 2796.67 0.0925728
\(971\) 7053.09 0.233105 0.116552 0.993185i \(-0.462816\pi\)
0.116552 + 0.993185i \(0.462816\pi\)
\(972\) −331.530 −0.0109402
\(973\) 1952.32 0.0643253
\(974\) 48749.2 1.60372
\(975\) 22198.2 0.729139
\(976\) −59014.1 −1.93545
\(977\) 36193.6 1.18519 0.592597 0.805499i \(-0.298103\pi\)
0.592597 + 0.805499i \(0.298103\pi\)
\(978\) −30882.7 −1.00973
\(979\) −1603.03 −0.0523321
\(980\) −305.420 −0.00995538
\(981\) −14.1862 −0.000461702 0
\(982\) −8015.27 −0.260466
\(983\) −12172.1 −0.394944 −0.197472 0.980309i \(-0.563273\pi\)
−0.197472 + 0.980309i \(0.563273\pi\)
\(984\) −27629.4 −0.895115
\(985\) −845.620 −0.0273540
\(986\) 5885.39 0.190090
\(987\) −4887.05 −0.157605
\(988\) −6283.44 −0.202331
\(989\) −12789.2 −0.411195
\(990\) 163.487 0.00524843
\(991\) 37357.4 1.19747 0.598737 0.800946i \(-0.295670\pi\)
0.598737 + 0.800946i \(0.295670\pi\)
\(992\) −17002.3 −0.544176
\(993\) −9967.70 −0.318545
\(994\) −2819.97 −0.0899838
\(995\) 576.930 0.0183818
\(996\) −210.735 −0.00670420
\(997\) −43827.1 −1.39220 −0.696098 0.717947i \(-0.745082\pi\)
−0.696098 + 0.717947i \(0.745082\pi\)
\(998\) 28614.3 0.907586
\(999\) 8811.86 0.279074
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.b.1.2 7
3.2 odd 2 531.4.a.c.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.b.1.2 7 1.1 even 1 trivial
531.4.a.c.1.6 7 3.2 odd 2