Properties

Label 507.4.b.k.337.14
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.14
Root \(2.37150i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.5

$q$-expansion

\(f(q)\) \(=\) \(q+3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} -6.74147i q^{5} +9.52033i q^{6} -14.1726i q^{7} +18.8162i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.17344i q^{2} +3.00000 q^{3} -2.07074 q^{4} -6.74147i q^{5} +9.52033i q^{6} -14.1726i q^{7} +18.8162i q^{8} +9.00000 q^{9} +21.3937 q^{10} +62.4956i q^{11} -6.21221 q^{12} +44.9761 q^{14} -20.2244i q^{15} -76.2779 q^{16} +58.6172 q^{17} +28.5610i q^{18} -64.1652i q^{19} +13.9598i q^{20} -42.5179i q^{21} -198.326 q^{22} -10.9221 q^{23} +56.4485i q^{24} +79.5526 q^{25} +27.0000 q^{27} +29.3478i q^{28} +216.316 q^{29} +64.1810 q^{30} +38.6271i q^{31} -91.5342i q^{32} +187.487i q^{33} +186.018i q^{34} -95.5445 q^{35} -18.6366 q^{36} +423.770i q^{37} +203.625 q^{38} +126.849 q^{40} +366.126i q^{41} +134.928 q^{42} +128.297 q^{43} -129.412i q^{44} -60.6732i q^{45} -34.6605i q^{46} -93.1169i q^{47} -228.834 q^{48} +142.136 q^{49} +252.455i q^{50} +175.852 q^{51} +131.909 q^{53} +85.6829i q^{54} +421.313 q^{55} +266.675 q^{56} -192.496i q^{57} +686.467i q^{58} +386.729i q^{59} +41.8794i q^{60} -621.077 q^{61} -122.581 q^{62} -127.554i q^{63} -319.745 q^{64} -594.979 q^{66} -865.273i q^{67} -121.381 q^{68} -32.7662 q^{69} -303.205i q^{70} +607.506i q^{71} +169.346i q^{72} +980.958i q^{73} -1344.81 q^{74} +238.658 q^{75} +132.869i q^{76} +885.728 q^{77} +1331.91 q^{79} +514.226i q^{80} +81.0000 q^{81} -1161.88 q^{82} -907.633i q^{83} +88.0434i q^{84} -395.166i q^{85} +407.142i q^{86} +648.949 q^{87} -1175.93 q^{88} -1033.67i q^{89} +192.543 q^{90} +22.6167 q^{92} +115.881i q^{93} +295.501 q^{94} -432.568 q^{95} -274.603i q^{96} -1046.17i q^{97} +451.061i q^{98} +562.461i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.17344i 1.12198i 0.827822 + 0.560991i \(0.189580\pi\)
−0.827822 + 0.560991i \(0.810420\pi\)
\(3\) 3.00000 0.577350
\(4\) −2.07074 −0.258842
\(5\) − 6.74147i − 0.602975i −0.953470 0.301488i \(-0.902517\pi\)
0.953470 0.301488i \(-0.0974832\pi\)
\(6\) 9.52033i 0.647776i
\(7\) − 14.1726i − 0.765251i −0.923904 0.382625i \(-0.875020\pi\)
0.923904 0.382625i \(-0.124980\pi\)
\(8\) 18.8162i 0.831565i
\(9\) 9.00000 0.333333
\(10\) 21.3937 0.676527
\(11\) 62.4956i 1.71301i 0.516136 + 0.856507i \(0.327370\pi\)
−0.516136 + 0.856507i \(0.672630\pi\)
\(12\) −6.21221 −0.149442
\(13\) 0 0
\(14\) 44.9761 0.858597
\(15\) − 20.2244i − 0.348128i
\(16\) −76.2779 −1.19184
\(17\) 58.6172 0.836280 0.418140 0.908383i \(-0.362682\pi\)
0.418140 + 0.908383i \(0.362682\pi\)
\(18\) 28.5610i 0.373994i
\(19\) − 64.1652i − 0.774764i −0.921919 0.387382i \(-0.873380\pi\)
0.921919 0.387382i \(-0.126620\pi\)
\(20\) 13.9598i 0.156075i
\(21\) − 42.5179i − 0.441818i
\(22\) −198.326 −1.92197
\(23\) −10.9221 −0.0990177 −0.0495088 0.998774i \(-0.515766\pi\)
−0.0495088 + 0.998774i \(0.515766\pi\)
\(24\) 56.4485i 0.480105i
\(25\) 79.5526 0.636421
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 29.3478i 0.198079i
\(29\) 216.316 1.38514 0.692568 0.721353i \(-0.256479\pi\)
0.692568 + 0.721353i \(0.256479\pi\)
\(30\) 64.1810 0.390593
\(31\) 38.6271i 0.223795i 0.993720 + 0.111897i \(0.0356928\pi\)
−0.993720 + 0.111897i \(0.964307\pi\)
\(32\) − 91.5342i − 0.505660i
\(33\) 187.487i 0.989009i
\(34\) 186.018i 0.938290i
\(35\) −95.5445 −0.461427
\(36\) −18.6366 −0.0862807
\(37\) 423.770i 1.88290i 0.337147 + 0.941452i \(0.390538\pi\)
−0.337147 + 0.941452i \(0.609462\pi\)
\(38\) 203.625 0.869270
\(39\) 0 0
\(40\) 126.849 0.501414
\(41\) 366.126i 1.39461i 0.716772 + 0.697307i \(0.245619\pi\)
−0.716772 + 0.697307i \(0.754381\pi\)
\(42\) 134.928 0.495711
\(43\) 128.297 0.455001 0.227501 0.973778i \(-0.426945\pi\)
0.227501 + 0.973778i \(0.426945\pi\)
\(44\) − 129.412i − 0.443400i
\(45\) − 60.6732i − 0.200992i
\(46\) − 34.6605i − 0.111096i
\(47\) − 93.1169i − 0.288989i −0.989506 0.144495i \(-0.953844\pi\)
0.989506 0.144495i \(-0.0461556\pi\)
\(48\) −228.834 −0.688111
\(49\) 142.136 0.414391
\(50\) 252.455i 0.714052i
\(51\) 175.852 0.482826
\(52\) 0 0
\(53\) 131.909 0.341869 0.170934 0.985282i \(-0.445321\pi\)
0.170934 + 0.985282i \(0.445321\pi\)
\(54\) 85.6829i 0.215925i
\(55\) 421.313 1.03290
\(56\) 266.675 0.636356
\(57\) − 192.496i − 0.447310i
\(58\) 686.467i 1.55410i
\(59\) 386.729i 0.853353i 0.904404 + 0.426677i \(0.140316\pi\)
−0.904404 + 0.426677i \(0.859684\pi\)
\(60\) 41.8794i 0.0901102i
\(61\) −621.077 −1.30362 −0.651810 0.758382i \(-0.725990\pi\)
−0.651810 + 0.758382i \(0.725990\pi\)
\(62\) −122.581 −0.251094
\(63\) − 127.554i − 0.255084i
\(64\) −319.745 −0.624502
\(65\) 0 0
\(66\) −594.979 −1.10965
\(67\) − 865.273i − 1.57776i −0.614547 0.788880i \(-0.710661\pi\)
0.614547 0.788880i \(-0.289339\pi\)
\(68\) −121.381 −0.216464
\(69\) −32.7662 −0.0571679
\(70\) − 303.205i − 0.517713i
\(71\) 607.506i 1.01546i 0.861516 + 0.507730i \(0.169515\pi\)
−0.861516 + 0.507730i \(0.830485\pi\)
\(72\) 169.346i 0.277188i
\(73\) 980.958i 1.57277i 0.617735 + 0.786387i \(0.288051\pi\)
−0.617735 + 0.786387i \(0.711949\pi\)
\(74\) −1344.81 −2.11258
\(75\) 238.658 0.367438
\(76\) 132.869i 0.200541i
\(77\) 885.728 1.31088
\(78\) 0 0
\(79\) 1331.91 1.89685 0.948425 0.317003i \(-0.102676\pi\)
0.948425 + 0.317003i \(0.102676\pi\)
\(80\) 514.226i 0.718652i
\(81\) 81.0000 0.111111
\(82\) −1161.88 −1.56473
\(83\) − 907.633i − 1.20031i −0.799884 0.600155i \(-0.795106\pi\)
0.799884 0.600155i \(-0.204894\pi\)
\(84\) 88.0434i 0.114361i
\(85\) − 395.166i − 0.504256i
\(86\) 407.142i 0.510503i
\(87\) 648.949 0.799708
\(88\) −1175.93 −1.42448
\(89\) − 1033.67i − 1.23110i −0.788096 0.615552i \(-0.788933\pi\)
0.788096 0.615552i \(-0.211067\pi\)
\(90\) 192.543 0.225509
\(91\) 0 0
\(92\) 22.6167 0.0256299
\(93\) 115.881i 0.129208i
\(94\) 295.501 0.324240
\(95\) −432.568 −0.467163
\(96\) − 274.603i − 0.291943i
\(97\) − 1046.17i − 1.09508i −0.836781 0.547538i \(-0.815565\pi\)
0.836781 0.547538i \(-0.184435\pi\)
\(98\) 451.061i 0.464939i
\(99\) 562.461i 0.571004i
\(100\) −164.732 −0.164732
\(101\) −1416.64 −1.39566 −0.697828 0.716265i \(-0.745850\pi\)
−0.697828 + 0.716265i \(0.745850\pi\)
\(102\) 558.055i 0.541722i
\(103\) −387.629 −0.370818 −0.185409 0.982661i \(-0.559361\pi\)
−0.185409 + 0.982661i \(0.559361\pi\)
\(104\) 0 0
\(105\) −286.633 −0.266405
\(106\) 418.605i 0.383570i
\(107\) 86.4526 0.0781092 0.0390546 0.999237i \(-0.487565\pi\)
0.0390546 + 0.999237i \(0.487565\pi\)
\(108\) −55.9099 −0.0498142
\(109\) 940.072i 0.826079i 0.910713 + 0.413039i \(0.135533\pi\)
−0.910713 + 0.413039i \(0.864467\pi\)
\(110\) 1337.01i 1.15890i
\(111\) 1271.31i 1.08709i
\(112\) 1081.06i 0.912059i
\(113\) 960.499 0.799612 0.399806 0.916600i \(-0.369078\pi\)
0.399806 + 0.916600i \(0.369078\pi\)
\(114\) 610.874 0.501873
\(115\) 73.6307i 0.0597052i
\(116\) −447.934 −0.358531
\(117\) 0 0
\(118\) −1227.26 −0.957447
\(119\) − 830.760i − 0.639964i
\(120\) 380.546 0.289491
\(121\) −2574.71 −1.93441
\(122\) − 1970.95i − 1.46264i
\(123\) 1098.38i 0.805181i
\(124\) − 79.9866i − 0.0579275i
\(125\) − 1378.99i − 0.986721i
\(126\) 404.785 0.286199
\(127\) −2022.18 −1.41291 −0.706456 0.707757i \(-0.749707\pi\)
−0.706456 + 0.707757i \(0.749707\pi\)
\(128\) − 1746.97i − 1.20634i
\(129\) 384.890 0.262695
\(130\) 0 0
\(131\) 1857.90 1.23912 0.619561 0.784948i \(-0.287310\pi\)
0.619561 + 0.784948i \(0.287310\pi\)
\(132\) − 388.236i − 0.255997i
\(133\) −909.390 −0.592888
\(134\) 2745.90 1.77022
\(135\) − 182.020i − 0.116043i
\(136\) 1102.95i 0.695421i
\(137\) 1894.12i 1.18121i 0.806961 + 0.590604i \(0.201110\pi\)
−0.806961 + 0.590604i \(0.798890\pi\)
\(138\) − 103.982i − 0.0641413i
\(139\) −1226.08 −0.748165 −0.374082 0.927395i \(-0.622042\pi\)
−0.374082 + 0.927395i \(0.622042\pi\)
\(140\) 197.847 0.119437
\(141\) − 279.351i − 0.166848i
\(142\) −1927.89 −1.13933
\(143\) 0 0
\(144\) −686.501 −0.397281
\(145\) − 1458.29i − 0.835203i
\(146\) −3113.01 −1.76462
\(147\) 426.409 0.239249
\(148\) − 877.517i − 0.487375i
\(149\) − 3195.65i − 1.75703i −0.477711 0.878517i \(-0.658533\pi\)
0.477711 0.878517i \(-0.341467\pi\)
\(150\) 757.366i 0.412258i
\(151\) 508.232i 0.273903i 0.990578 + 0.136951i \(0.0437304\pi\)
−0.990578 + 0.136951i \(0.956270\pi\)
\(152\) 1207.34 0.644267
\(153\) 527.555 0.278760
\(154\) 2810.81i 1.47079i
\(155\) 260.404 0.134943
\(156\) 0 0
\(157\) −1243.08 −0.631900 −0.315950 0.948776i \(-0.602323\pi\)
−0.315950 + 0.948776i \(0.602323\pi\)
\(158\) 4226.73i 2.12823i
\(159\) 395.726 0.197378
\(160\) −617.075 −0.304901
\(161\) 154.794i 0.0757734i
\(162\) 257.049i 0.124665i
\(163\) − 33.9996i − 0.0163378i −0.999967 0.00816888i \(-0.997400\pi\)
0.999967 0.00816888i \(-0.00260026\pi\)
\(164\) − 758.149i − 0.360985i
\(165\) 1263.94 0.596348
\(166\) 2880.32 1.34673
\(167\) − 2210.67i − 1.02435i −0.858880 0.512176i \(-0.828839\pi\)
0.858880 0.512176i \(-0.171161\pi\)
\(168\) 800.025 0.367400
\(169\) 0 0
\(170\) 1254.04 0.565766
\(171\) − 577.487i − 0.258255i
\(172\) −265.668 −0.117773
\(173\) 661.307 0.290626 0.145313 0.989386i \(-0.453581\pi\)
0.145313 + 0.989386i \(0.453581\pi\)
\(174\) 2059.40i 0.897258i
\(175\) − 1127.47i − 0.487021i
\(176\) − 4767.04i − 2.04164i
\(177\) 1160.19i 0.492684i
\(178\) 3280.28 1.38128
\(179\) 2325.05 0.970850 0.485425 0.874278i \(-0.338665\pi\)
0.485425 + 0.874278i \(0.338665\pi\)
\(180\) 125.638i 0.0520251i
\(181\) −2122.20 −0.871503 −0.435752 0.900067i \(-0.643517\pi\)
−0.435752 + 0.900067i \(0.643517\pi\)
\(182\) 0 0
\(183\) −1863.23 −0.752645
\(184\) − 205.511i − 0.0823397i
\(185\) 2856.84 1.13534
\(186\) −367.743 −0.144969
\(187\) 3663.32i 1.43256i
\(188\) 192.820i 0.0748025i
\(189\) − 382.661i − 0.147273i
\(190\) − 1372.73i − 0.524149i
\(191\) −2484.37 −0.941166 −0.470583 0.882356i \(-0.655956\pi\)
−0.470583 + 0.882356i \(0.655956\pi\)
\(192\) −959.235 −0.360556
\(193\) − 266.771i − 0.0994955i −0.998762 0.0497478i \(-0.984158\pi\)
0.998762 0.0497478i \(-0.0158417\pi\)
\(194\) 3319.95 1.22865
\(195\) 0 0
\(196\) −294.327 −0.107262
\(197\) 1231.03i 0.445216i 0.974908 + 0.222608i \(0.0714569\pi\)
−0.974908 + 0.222608i \(0.928543\pi\)
\(198\) −1784.94 −0.640656
\(199\) 3246.14 1.15635 0.578173 0.815914i \(-0.303766\pi\)
0.578173 + 0.815914i \(0.303766\pi\)
\(200\) 1496.88i 0.529225i
\(201\) − 2595.82i − 0.910921i
\(202\) − 4495.63i − 1.56590i
\(203\) − 3065.77i − 1.05998i
\(204\) −364.142 −0.124976
\(205\) 2468.23 0.840919
\(206\) − 1230.12i − 0.416051i
\(207\) −98.2985 −0.0330059
\(208\) 0 0
\(209\) 4010.05 1.32718
\(210\) − 909.614i − 0.298902i
\(211\) 330.708 0.107900 0.0539500 0.998544i \(-0.482819\pi\)
0.0539500 + 0.998544i \(0.482819\pi\)
\(212\) −273.148 −0.0884900
\(213\) 1822.52i 0.586277i
\(214\) 274.352i 0.0876371i
\(215\) − 864.908i − 0.274355i
\(216\) 508.037i 0.160035i
\(217\) 547.449 0.171259
\(218\) −2983.26 −0.926845
\(219\) 2942.87i 0.908041i
\(220\) −872.427 −0.267359
\(221\) 0 0
\(222\) −4034.43 −1.21970
\(223\) − 5785.86i − 1.73744i −0.495300 0.868722i \(-0.664942\pi\)
0.495300 0.868722i \(-0.335058\pi\)
\(224\) −1297.28 −0.386957
\(225\) 715.973 0.212140
\(226\) 3048.09i 0.897149i
\(227\) − 2945.35i − 0.861189i −0.902545 0.430595i \(-0.858304\pi\)
0.902545 0.430595i \(-0.141696\pi\)
\(228\) 398.608i 0.115783i
\(229\) − 3541.26i − 1.02189i −0.859613 0.510945i \(-0.829296\pi\)
0.859613 0.510945i \(-0.170704\pi\)
\(230\) −233.663 −0.0669882
\(231\) 2657.18 0.756840
\(232\) 4070.25i 1.15183i
\(233\) −2340.76 −0.658148 −0.329074 0.944304i \(-0.606737\pi\)
−0.329074 + 0.944304i \(0.606737\pi\)
\(234\) 0 0
\(235\) −627.745 −0.174253
\(236\) − 800.814i − 0.220884i
\(237\) 3995.72 1.09515
\(238\) 2636.37 0.718027
\(239\) 1515.70i 0.410218i 0.978739 + 0.205109i \(0.0657549\pi\)
−0.978739 + 0.205109i \(0.934245\pi\)
\(240\) 1542.68i 0.414914i
\(241\) 2392.47i 0.639472i 0.947507 + 0.319736i \(0.103594\pi\)
−0.947507 + 0.319736i \(0.896406\pi\)
\(242\) − 8170.68i − 2.17038i
\(243\) 243.000 0.0641500
\(244\) 1286.09 0.337432
\(245\) − 958.207i − 0.249868i
\(246\) −3485.64 −0.903398
\(247\) 0 0
\(248\) −726.815 −0.186100
\(249\) − 2722.90i − 0.692999i
\(250\) 4376.13 1.10708
\(251\) 2198.78 0.552931 0.276465 0.961024i \(-0.410837\pi\)
0.276465 + 0.961024i \(0.410837\pi\)
\(252\) 264.130i 0.0660263i
\(253\) − 682.581i − 0.169619i
\(254\) − 6417.29i − 1.58526i
\(255\) − 1185.50i − 0.291132i
\(256\) 2985.94 0.728988
\(257\) 6194.26 1.50345 0.751727 0.659475i \(-0.229221\pi\)
0.751727 + 0.659475i \(0.229221\pi\)
\(258\) 1221.43i 0.294739i
\(259\) 6005.95 1.44089
\(260\) 0 0
\(261\) 1946.85 0.461712
\(262\) 5895.92i 1.39027i
\(263\) −4181.74 −0.980445 −0.490222 0.871597i \(-0.663084\pi\)
−0.490222 + 0.871597i \(0.663084\pi\)
\(264\) −3527.79 −0.822425
\(265\) − 889.258i − 0.206139i
\(266\) − 2885.90i − 0.665210i
\(267\) − 3101.00i − 0.710778i
\(268\) 1791.75i 0.408391i
\(269\) 2767.69 0.627320 0.313660 0.949535i \(-0.398445\pi\)
0.313660 + 0.949535i \(0.398445\pi\)
\(270\) 577.629 0.130198
\(271\) − 7191.36i − 1.61197i −0.591935 0.805986i \(-0.701636\pi\)
0.591935 0.805986i \(-0.298364\pi\)
\(272\) −4471.20 −0.996714
\(273\) 0 0
\(274\) −6010.88 −1.32529
\(275\) 4971.69i 1.09020i
\(276\) 67.8501 0.0147974
\(277\) −1317.27 −0.285729 −0.142864 0.989742i \(-0.545631\pi\)
−0.142864 + 0.989742i \(0.545631\pi\)
\(278\) − 3890.90i − 0.839427i
\(279\) 347.644i 0.0745983i
\(280\) − 1797.78i − 0.383707i
\(281\) − 3948.92i − 0.838338i −0.907908 0.419169i \(-0.862321\pi\)
0.907908 0.419169i \(-0.137679\pi\)
\(282\) 886.503 0.187200
\(283\) 4981.52 1.04636 0.523181 0.852221i \(-0.324745\pi\)
0.523181 + 0.852221i \(0.324745\pi\)
\(284\) − 1257.99i − 0.262844i
\(285\) −1297.70 −0.269717
\(286\) 0 0
\(287\) 5188.97 1.06723
\(288\) − 823.808i − 0.168553i
\(289\) −1477.03 −0.300636
\(290\) 4627.80 0.937082
\(291\) − 3138.50i − 0.632242i
\(292\) − 2031.31i − 0.407100i
\(293\) − 3203.02i − 0.638644i −0.947646 0.319322i \(-0.896545\pi\)
0.947646 0.319322i \(-0.103455\pi\)
\(294\) 1353.18i 0.268433i
\(295\) 2607.12 0.514551
\(296\) −7973.74 −1.56576
\(297\) 1687.38i 0.329670i
\(298\) 10141.2 1.97136
\(299\) 0 0
\(300\) −494.197 −0.0951083
\(301\) − 1818.30i − 0.348190i
\(302\) −1612.84 −0.307314
\(303\) −4249.93 −0.805782
\(304\) 4894.39i 0.923396i
\(305\) 4186.97i 0.786051i
\(306\) 1674.16i 0.312763i
\(307\) 4795.67i 0.891542i 0.895147 + 0.445771i \(0.147070\pi\)
−0.895147 + 0.445771i \(0.852930\pi\)
\(308\) −1834.11 −0.339312
\(309\) −1162.89 −0.214092
\(310\) 826.376i 0.151403i
\(311\) −630.213 −0.114907 −0.0574535 0.998348i \(-0.518298\pi\)
−0.0574535 + 0.998348i \(0.518298\pi\)
\(312\) 0 0
\(313\) −9314.73 −1.68211 −0.841054 0.540952i \(-0.818064\pi\)
−0.841054 + 0.540952i \(0.818064\pi\)
\(314\) − 3944.83i − 0.708980i
\(315\) −859.900 −0.153809
\(316\) −2758.02 −0.490984
\(317\) − 576.333i − 0.102114i −0.998696 0.0510569i \(-0.983741\pi\)
0.998696 0.0510569i \(-0.0162590\pi\)
\(318\) 1255.81i 0.221455i
\(319\) 13518.8i 2.37275i
\(320\) 2155.55i 0.376559i
\(321\) 259.358 0.0450964
\(322\) −491.231 −0.0850163
\(323\) − 3761.18i − 0.647919i
\(324\) −167.730 −0.0287602
\(325\) 0 0
\(326\) 107.896 0.0183307
\(327\) 2820.22i 0.476937i
\(328\) −6889.08 −1.15971
\(329\) −1319.71 −0.221149
\(330\) 4011.03i 0.669091i
\(331\) − 1575.95i − 0.261699i −0.991402 0.130849i \(-0.958230\pi\)
0.991402 0.130849i \(-0.0417704\pi\)
\(332\) 1879.47i 0.310691i
\(333\) 3813.93i 0.627635i
\(334\) 7015.44 1.14930
\(335\) −5833.22 −0.951351
\(336\) 3243.18i 0.526577i
\(337\) −9289.32 −1.50155 −0.750774 0.660559i \(-0.770319\pi\)
−0.750774 + 0.660559i \(0.770319\pi\)
\(338\) 0 0
\(339\) 2881.50 0.461656
\(340\) 818.284i 0.130523i
\(341\) −2414.03 −0.383363
\(342\) 1832.62 0.289757
\(343\) − 6875.66i − 1.08236i
\(344\) 2414.05i 0.378363i
\(345\) 220.892i 0.0344708i
\(346\) 2098.62i 0.326077i
\(347\) −7701.82 −1.19151 −0.595757 0.803164i \(-0.703148\pi\)
−0.595757 + 0.803164i \(0.703148\pi\)
\(348\) −1343.80 −0.206998
\(349\) − 4972.89i − 0.762730i −0.924425 0.381365i \(-0.875454\pi\)
0.924425 0.381365i \(-0.124546\pi\)
\(350\) 3577.96 0.546429
\(351\) 0 0
\(352\) 5720.49 0.866202
\(353\) − 1575.34i − 0.237526i −0.992923 0.118763i \(-0.962107\pi\)
0.992923 0.118763i \(-0.0378929\pi\)
\(354\) −3681.79 −0.552782
\(355\) 4095.49 0.612298
\(356\) 2140.45i 0.318661i
\(357\) − 2492.28i − 0.369483i
\(358\) 7378.41i 1.08928i
\(359\) 7567.42i 1.11252i 0.831010 + 0.556258i \(0.187763\pi\)
−0.831010 + 0.556258i \(0.812237\pi\)
\(360\) 1141.64 0.167138
\(361\) 2741.83 0.399741
\(362\) − 6734.69i − 0.977810i
\(363\) −7724.12 −1.11683
\(364\) 0 0
\(365\) 6613.10 0.948344
\(366\) − 5912.86i − 0.844454i
\(367\) 4368.25 0.621310 0.310655 0.950523i \(-0.399452\pi\)
0.310655 + 0.950523i \(0.399452\pi\)
\(368\) 833.112 0.118014
\(369\) 3295.13i 0.464872i
\(370\) 9066.01i 1.27384i
\(371\) − 1869.49i − 0.261615i
\(372\) − 239.960i − 0.0334445i
\(373\) −801.944 −0.111322 −0.0556610 0.998450i \(-0.517727\pi\)
−0.0556610 + 0.998450i \(0.517727\pi\)
\(374\) −11625.3 −1.60730
\(375\) − 4136.96i − 0.569684i
\(376\) 1752.10 0.240313
\(377\) 0 0
\(378\) 1214.35 0.165237
\(379\) − 68.0819i − 0.00922727i −0.999989 0.00461363i \(-0.998531\pi\)
0.999989 0.00461363i \(-0.00146857\pi\)
\(380\) 895.734 0.120922
\(381\) −6066.55 −0.815745
\(382\) − 7884.01i − 1.05597i
\(383\) − 1549.01i − 0.206659i −0.994647 0.103330i \(-0.967050\pi\)
0.994647 0.103330i \(-0.0329497\pi\)
\(384\) − 5240.90i − 0.696480i
\(385\) − 5971.11i − 0.790431i
\(386\) 846.584 0.111632
\(387\) 1154.67 0.151667
\(388\) 2166.34i 0.283451i
\(389\) −7300.51 −0.951544 −0.475772 0.879569i \(-0.657831\pi\)
−0.475772 + 0.879569i \(0.657831\pi\)
\(390\) 0 0
\(391\) −640.220 −0.0828065
\(392\) 2674.46i 0.344594i
\(393\) 5573.69 0.715408
\(394\) −3906.61 −0.499524
\(395\) − 8979.00i − 1.14375i
\(396\) − 1164.71i − 0.147800i
\(397\) 6096.27i 0.770688i 0.922773 + 0.385344i \(0.125917\pi\)
−0.922773 + 0.385344i \(0.874083\pi\)
\(398\) 10301.4i 1.29740i
\(399\) −2728.17 −0.342304
\(400\) −6068.11 −0.758513
\(401\) 7592.37i 0.945498i 0.881197 + 0.472749i \(0.156738\pi\)
−0.881197 + 0.472749i \(0.843262\pi\)
\(402\) 8237.69 1.02204
\(403\) 0 0
\(404\) 2933.49 0.361254
\(405\) − 546.059i − 0.0669973i
\(406\) 9729.05 1.18927
\(407\) −26483.8 −3.22544
\(408\) 3308.85i 0.401502i
\(409\) 7233.86i 0.874551i 0.899328 + 0.437275i \(0.144057\pi\)
−0.899328 + 0.437275i \(0.855943\pi\)
\(410\) 7832.77i 0.943495i
\(411\) 5682.36i 0.681971i
\(412\) 802.678 0.0959832
\(413\) 5480.97 0.653029
\(414\) − 311.945i − 0.0370320i
\(415\) −6118.78 −0.723757
\(416\) 0 0
\(417\) −3678.25 −0.431953
\(418\) 12725.6i 1.48907i
\(419\) −5312.55 −0.619416 −0.309708 0.950832i \(-0.600231\pi\)
−0.309708 + 0.950832i \(0.600231\pi\)
\(420\) 593.542 0.0689569
\(421\) − 15028.1i − 1.73973i −0.493290 0.869865i \(-0.664206\pi\)
0.493290 0.869865i \(-0.335794\pi\)
\(422\) 1049.48i 0.121062i
\(423\) − 838.052i − 0.0963297i
\(424\) 2482.02i 0.284286i
\(425\) 4663.15 0.532226
\(426\) −5783.66 −0.657791
\(427\) 8802.31i 0.997596i
\(428\) −179.020 −0.0202179
\(429\) 0 0
\(430\) 2744.73 0.307821
\(431\) 7154.66i 0.799600i 0.916602 + 0.399800i \(0.130920\pi\)
−0.916602 + 0.399800i \(0.869080\pi\)
\(432\) −2059.50 −0.229370
\(433\) 9542.58 1.05909 0.529546 0.848281i \(-0.322362\pi\)
0.529546 + 0.848281i \(0.322362\pi\)
\(434\) 1737.30i 0.192150i
\(435\) − 4374.87i − 0.482204i
\(436\) − 1946.64i − 0.213824i
\(437\) 700.816i 0.0767153i
\(438\) −9339.04 −1.01881
\(439\) −7070.70 −0.768715 −0.384358 0.923184i \(-0.625577\pi\)
−0.384358 + 0.923184i \(0.625577\pi\)
\(440\) 7927.49i 0.858928i
\(441\) 1279.23 0.138130
\(442\) 0 0
\(443\) 2092.58 0.224428 0.112214 0.993684i \(-0.464206\pi\)
0.112214 + 0.993684i \(0.464206\pi\)
\(444\) − 2632.55i − 0.281386i
\(445\) −6968.42 −0.742326
\(446\) 18361.1 1.94938
\(447\) − 9586.96i − 1.01442i
\(448\) 4531.63i 0.477901i
\(449\) 5842.05i 0.614038i 0.951703 + 0.307019i \(0.0993316\pi\)
−0.951703 + 0.307019i \(0.900668\pi\)
\(450\) 2272.10i 0.238017i
\(451\) −22881.3 −2.38899
\(452\) −1988.94 −0.206973
\(453\) 1524.69i 0.158138i
\(454\) 9346.91 0.966238
\(455\) 0 0
\(456\) 3622.03 0.371967
\(457\) − 5954.40i − 0.609486i −0.952435 0.304743i \(-0.901429\pi\)
0.952435 0.304743i \(-0.0985705\pi\)
\(458\) 11238.0 1.14654
\(459\) 1582.66 0.160942
\(460\) − 152.470i − 0.0154542i
\(461\) 1865.94i 0.188515i 0.995548 + 0.0942576i \(0.0300477\pi\)
−0.995548 + 0.0942576i \(0.969952\pi\)
\(462\) 8432.42i 0.849160i
\(463\) − 6700.05i − 0.672522i −0.941769 0.336261i \(-0.890838\pi\)
0.941769 0.336261i \(-0.109162\pi\)
\(464\) −16500.2 −1.65086
\(465\) 781.211 0.0779093
\(466\) − 7428.28i − 0.738430i
\(467\) 16585.8 1.64347 0.821734 0.569871i \(-0.193007\pi\)
0.821734 + 0.569871i \(0.193007\pi\)
\(468\) 0 0
\(469\) −12263.2 −1.20738
\(470\) − 1992.11i − 0.195509i
\(471\) −3729.23 −0.364828
\(472\) −7276.77 −0.709619
\(473\) 8017.98i 0.779423i
\(474\) 12680.2i 1.22873i
\(475\) − 5104.51i − 0.493075i
\(476\) 1720.29i 0.165649i
\(477\) 1187.18 0.113956
\(478\) −4809.97 −0.460257
\(479\) − 6166.88i − 0.588250i −0.955767 0.294125i \(-0.904972\pi\)
0.955767 0.294125i \(-0.0950282\pi\)
\(480\) −1851.23 −0.176034
\(481\) 0 0
\(482\) −7592.38 −0.717476
\(483\) 464.383i 0.0437478i
\(484\) 5331.53 0.500708
\(485\) −7052.71 −0.660303
\(486\) 771.146i 0.0719751i
\(487\) − 5718.51i − 0.532095i −0.963960 0.266047i \(-0.914282\pi\)
0.963960 0.266047i \(-0.0857178\pi\)
\(488\) − 11686.3i − 1.08405i
\(489\) − 101.999i − 0.00943261i
\(490\) 3040.82 0.280347
\(491\) 21060.9 1.93578 0.967888 0.251383i \(-0.0808854\pi\)
0.967888 + 0.251383i \(0.0808854\pi\)
\(492\) − 2274.45i − 0.208415i
\(493\) 12679.8 1.15836
\(494\) 0 0
\(495\) 3791.81 0.344302
\(496\) − 2946.40i − 0.266728i
\(497\) 8609.97 0.777082
\(498\) 8640.97 0.777532
\(499\) 7863.87i 0.705481i 0.935721 + 0.352741i \(0.114750\pi\)
−0.935721 + 0.352741i \(0.885250\pi\)
\(500\) 2855.51i 0.255405i
\(501\) − 6632.01i − 0.591410i
\(502\) 6977.70i 0.620378i
\(503\) −6504.06 −0.576544 −0.288272 0.957549i \(-0.593081\pi\)
−0.288272 + 0.957549i \(0.593081\pi\)
\(504\) 2400.07 0.212119
\(505\) 9550.26i 0.841546i
\(506\) 2166.13 0.190309
\(507\) 0 0
\(508\) 4187.41 0.365721
\(509\) − 14799.3i − 1.28873i −0.764717 0.644367i \(-0.777121\pi\)
0.764717 0.644367i \(-0.222879\pi\)
\(510\) 3762.11 0.326645
\(511\) 13902.8 1.20357
\(512\) − 4500.03i − 0.388428i
\(513\) − 1732.46i − 0.149103i
\(514\) 19657.1i 1.68685i
\(515\) 2613.19i 0.223594i
\(516\) −797.005 −0.0679965
\(517\) 5819.40 0.495042
\(518\) 19059.5i 1.61666i
\(519\) 1983.92 0.167793
\(520\) 0 0
\(521\) −6633.65 −0.557822 −0.278911 0.960317i \(-0.589974\pi\)
−0.278911 + 0.960317i \(0.589974\pi\)
\(522\) 6178.20i 0.518032i
\(523\) 4527.41 0.378527 0.189264 0.981926i \(-0.439390\pi\)
0.189264 + 0.981926i \(0.439390\pi\)
\(524\) −3847.21 −0.320737
\(525\) − 3382.41i − 0.281182i
\(526\) − 13270.5i − 1.10004i
\(527\) 2264.21i 0.187155i
\(528\) − 14301.1i − 1.17874i
\(529\) −12047.7 −0.990195
\(530\) 2822.01 0.231284
\(531\) 3480.56i 0.284451i
\(532\) 1883.11 0.153464
\(533\) 0 0
\(534\) 9840.83 0.797480
\(535\) − 582.818i − 0.0470980i
\(536\) 16281.1 1.31201
\(537\) 6975.14 0.560521
\(538\) 8783.11i 0.703841i
\(539\) 8882.89i 0.709858i
\(540\) 376.915i 0.0300367i
\(541\) 8685.42i 0.690232i 0.938560 + 0.345116i \(0.112160\pi\)
−0.938560 + 0.345116i \(0.887840\pi\)
\(542\) 22821.4 1.80860
\(543\) −6366.61 −0.503162
\(544\) − 5365.48i − 0.422873i
\(545\) 6337.47 0.498105
\(546\) 0 0
\(547\) 24656.6 1.92731 0.963657 0.267142i \(-0.0860793\pi\)
0.963657 + 0.267142i \(0.0860793\pi\)
\(548\) − 3922.22i − 0.305746i
\(549\) −5589.70 −0.434540
\(550\) −15777.4 −1.22318
\(551\) − 13880.0i − 1.07315i
\(552\) − 616.534i − 0.0475388i
\(553\) − 18876.6i − 1.45157i
\(554\) − 4180.27i − 0.320582i
\(555\) 8570.51 0.655492
\(556\) 2538.89 0.193656
\(557\) 9215.90i 0.701059i 0.936552 + 0.350530i \(0.113998\pi\)
−0.936552 + 0.350530i \(0.886002\pi\)
\(558\) −1103.23 −0.0836979
\(559\) 0 0
\(560\) 7287.93 0.549949
\(561\) 10990.0i 0.827088i
\(562\) 12531.7 0.940600
\(563\) 19686.7 1.47370 0.736850 0.676056i \(-0.236312\pi\)
0.736850 + 0.676056i \(0.236312\pi\)
\(564\) 578.461i 0.0431873i
\(565\) − 6475.17i − 0.482146i
\(566\) 15808.6i 1.17400i
\(567\) − 1147.98i − 0.0850279i
\(568\) −11430.9 −0.844422
\(569\) 3559.36 0.262243 0.131121 0.991366i \(-0.458142\pi\)
0.131121 + 0.991366i \(0.458142\pi\)
\(570\) − 4118.19i − 0.302617i
\(571\) 710.968 0.0521070 0.0260535 0.999661i \(-0.491706\pi\)
0.0260535 + 0.999661i \(0.491706\pi\)
\(572\) 0 0
\(573\) −7453.11 −0.543383
\(574\) 16466.9i 1.19741i
\(575\) −868.878 −0.0630169
\(576\) −2877.70 −0.208167
\(577\) 8041.67i 0.580206i 0.956995 + 0.290103i \(0.0936896\pi\)
−0.956995 + 0.290103i \(0.906310\pi\)
\(578\) − 4687.26i − 0.337308i
\(579\) − 800.314i − 0.0574438i
\(580\) 3019.73i 0.216186i
\(581\) −12863.6 −0.918538
\(582\) 9959.86 0.709364
\(583\) 8243.72i 0.585626i
\(584\) −18457.9 −1.30786
\(585\) 0 0
\(586\) 10164.6 0.716546
\(587\) 14641.3i 1.02949i 0.857344 + 0.514744i \(0.172113\pi\)
−0.857344 + 0.514744i \(0.827887\pi\)
\(588\) −882.980 −0.0619277
\(589\) 2478.52 0.173388
\(590\) 8273.56i 0.577317i
\(591\) 3693.10i 0.257045i
\(592\) − 32324.3i − 2.24413i
\(593\) 5735.76i 0.397200i 0.980081 + 0.198600i \(0.0636394\pi\)
−0.980081 + 0.198600i \(0.936361\pi\)
\(594\) −5354.81 −0.369883
\(595\) −5600.55 −0.385882
\(596\) 6617.35i 0.454794i
\(597\) 9738.43 0.667617
\(598\) 0 0
\(599\) −16109.3 −1.09884 −0.549422 0.835545i \(-0.685152\pi\)
−0.549422 + 0.835545i \(0.685152\pi\)
\(600\) 4490.63i 0.305548i
\(601\) −21005.4 −1.42567 −0.712837 0.701330i \(-0.752590\pi\)
−0.712837 + 0.701330i \(0.752590\pi\)
\(602\) 5770.28 0.390663
\(603\) − 7787.46i − 0.525920i
\(604\) − 1052.41i − 0.0708975i
\(605\) 17357.3i 1.16640i
\(606\) − 13486.9i − 0.904073i
\(607\) 7478.16 0.500048 0.250024 0.968240i \(-0.419561\pi\)
0.250024 + 0.968240i \(0.419561\pi\)
\(608\) −5873.31 −0.391767
\(609\) − 9197.32i − 0.611977i
\(610\) −13287.1 −0.881934
\(611\) 0 0
\(612\) −1092.43 −0.0721548
\(613\) − 16435.5i − 1.08291i −0.840730 0.541455i \(-0.817874\pi\)
0.840730 0.541455i \(-0.182126\pi\)
\(614\) −15218.8 −1.00029
\(615\) 7404.68 0.485505
\(616\) 16666.0i 1.09009i
\(617\) − 1290.89i − 0.0842289i −0.999113 0.0421145i \(-0.986591\pi\)
0.999113 0.0421145i \(-0.0134094\pi\)
\(618\) − 3690.36i − 0.240207i
\(619\) − 26719.0i − 1.73494i −0.497490 0.867470i \(-0.665745\pi\)
0.497490 0.867470i \(-0.334255\pi\)
\(620\) −539.227 −0.0349289
\(621\) −294.896 −0.0190560
\(622\) − 1999.94i − 0.128924i
\(623\) −14649.8 −0.942103
\(624\) 0 0
\(625\) 647.683 0.0414517
\(626\) − 29559.8i − 1.88729i
\(627\) 12030.1 0.766248
\(628\) 2574.08 0.163562
\(629\) 24840.2i 1.57463i
\(630\) − 2728.84i − 0.172571i
\(631\) − 10697.4i − 0.674893i −0.941345 0.337447i \(-0.890437\pi\)
0.941345 0.337447i \(-0.109563\pi\)
\(632\) 25061.4i 1.57735i
\(633\) 992.125 0.0622961
\(634\) 1828.96 0.114570
\(635\) 13632.5i 0.851951i
\(636\) −819.444 −0.0510897
\(637\) 0 0
\(638\) −42901.2 −2.66219
\(639\) 5467.56i 0.338487i
\(640\) −11777.1 −0.727393
\(641\) −23572.8 −1.45253 −0.726264 0.687416i \(-0.758745\pi\)
−0.726264 + 0.687416i \(0.758745\pi\)
\(642\) 823.057i 0.0505973i
\(643\) 14000.3i 0.858661i 0.903147 + 0.429331i \(0.141250\pi\)
−0.903147 + 0.429331i \(0.858750\pi\)
\(644\) − 320.538i − 0.0196133i
\(645\) − 2594.72i − 0.158399i
\(646\) 11935.9 0.726953
\(647\) −614.196 −0.0373207 −0.0186604 0.999826i \(-0.505940\pi\)
−0.0186604 + 0.999826i \(0.505940\pi\)
\(648\) 1524.11i 0.0923962i
\(649\) −24168.9 −1.46181
\(650\) 0 0
\(651\) 1642.35 0.0988765
\(652\) 70.4042i 0.00422890i
\(653\) −5333.42 −0.319622 −0.159811 0.987148i \(-0.551088\pi\)
−0.159811 + 0.987148i \(0.551088\pi\)
\(654\) −8949.79 −0.535114
\(655\) − 12524.9i − 0.747161i
\(656\) − 27927.3i − 1.66216i
\(657\) 8828.62i 0.524258i
\(658\) − 4188.03i − 0.248125i
\(659\) 21396.8 1.26480 0.632398 0.774644i \(-0.282071\pi\)
0.632398 + 0.774644i \(0.282071\pi\)
\(660\) −2617.28 −0.154360
\(661\) − 16107.9i − 0.947841i −0.880568 0.473921i \(-0.842838\pi\)
0.880568 0.473921i \(-0.157162\pi\)
\(662\) 5001.20 0.293621
\(663\) 0 0
\(664\) 17078.2 0.998136
\(665\) 6130.63i 0.357497i
\(666\) −12103.3 −0.704194
\(667\) −2362.62 −0.137153
\(668\) 4577.72i 0.265145i
\(669\) − 17357.6i − 1.00311i
\(670\) − 18511.4i − 1.06740i
\(671\) − 38814.6i − 2.23312i
\(672\) −3891.85 −0.223410
\(673\) −20329.9 −1.16443 −0.582213 0.813036i \(-0.697813\pi\)
−0.582213 + 0.813036i \(0.697813\pi\)
\(674\) − 29479.1i − 1.68471i
\(675\) 2147.92 0.122479
\(676\) 0 0
\(677\) −24883.5 −1.41263 −0.706314 0.707899i \(-0.749643\pi\)
−0.706314 + 0.707899i \(0.749643\pi\)
\(678\) 9144.26i 0.517969i
\(679\) −14827.0 −0.838007
\(680\) 7435.51 0.419322
\(681\) − 8836.06i − 0.497208i
\(682\) − 7660.78i − 0.430127i
\(683\) − 258.953i − 0.0145074i −0.999974 0.00725369i \(-0.997691\pi\)
0.999974 0.00725369i \(-0.00230894\pi\)
\(684\) 1195.82i 0.0668471i
\(685\) 12769.1 0.712240
\(686\) 21819.5 1.21439
\(687\) − 10623.8i − 0.589989i
\(688\) −9786.20 −0.542290
\(689\) 0 0
\(690\) −700.989 −0.0386756
\(691\) 658.193i 0.0362357i 0.999836 + 0.0181178i \(0.00576740\pi\)
−0.999836 + 0.0181178i \(0.994233\pi\)
\(692\) −1369.39 −0.0752262
\(693\) 7971.55 0.436962
\(694\) − 24441.3i − 1.33686i
\(695\) 8265.60i 0.451125i
\(696\) 12210.7i 0.665010i
\(697\) 21461.2i 1.16629i
\(698\) 15781.2 0.855768
\(699\) −7022.29 −0.379982
\(700\) 2334.69i 0.126062i
\(701\) 8222.16 0.443005 0.221503 0.975160i \(-0.428904\pi\)
0.221503 + 0.975160i \(0.428904\pi\)
\(702\) 0 0
\(703\) 27191.3 1.45881
\(704\) − 19982.7i − 1.06978i
\(705\) −1883.23 −0.100605
\(706\) 4999.25 0.266500
\(707\) 20077.6i 1.06803i
\(708\) − 2402.44i − 0.127527i
\(709\) 6817.51i 0.361124i 0.983564 + 0.180562i \(0.0577917\pi\)
−0.983564 + 0.180562i \(0.942208\pi\)
\(710\) 12996.8i 0.686987i
\(711\) 11987.2 0.632283
\(712\) 19449.6 1.02374
\(713\) − 421.888i − 0.0221596i
\(714\) 7909.11 0.414553
\(715\) 0 0
\(716\) −4814.56 −0.251297
\(717\) 4547.09i 0.236840i
\(718\) −24014.8 −1.24822
\(719\) −23385.7 −1.21299 −0.606496 0.795087i \(-0.707425\pi\)
−0.606496 + 0.795087i \(0.707425\pi\)
\(720\) 4628.03i 0.239551i
\(721\) 5493.73i 0.283769i
\(722\) 8701.03i 0.448502i
\(723\) 7177.42i 0.369199i
\(724\) 4394.52 0.225582
\(725\) 17208.5 0.881529
\(726\) − 24512.0i − 1.25307i
\(727\) −20488.6 −1.04523 −0.522613 0.852570i \(-0.675043\pi\)
−0.522613 + 0.852570i \(0.675043\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 20986.3i 1.06402i
\(731\) 7520.38 0.380508
\(732\) 3858.26 0.194816
\(733\) 16993.4i 0.856299i 0.903708 + 0.428149i \(0.140834\pi\)
−0.903708 + 0.428149i \(0.859166\pi\)
\(734\) 13862.4i 0.697099i
\(735\) − 2874.62i − 0.144261i
\(736\) 999.742i 0.0500693i
\(737\) 54075.8 2.70272
\(738\) −10456.9 −0.521577
\(739\) − 14014.3i − 0.697600i −0.937197 0.348800i \(-0.886589\pi\)
0.937197 0.348800i \(-0.113411\pi\)
\(740\) −5915.75 −0.293875
\(741\) 0 0
\(742\) 5932.73 0.293528
\(743\) 141.560i 0.00698971i 0.999994 + 0.00349485i \(0.00111245\pi\)
−0.999994 + 0.00349485i \(0.998888\pi\)
\(744\) −2180.45 −0.107445
\(745\) −21543.4 −1.05945
\(746\) − 2544.92i − 0.124901i
\(747\) − 8168.70i − 0.400103i
\(748\) − 7585.76i − 0.370806i
\(749\) − 1225.26i − 0.0597731i
\(750\) 13128.4 0.639175
\(751\) −20734.3 −1.00746 −0.503732 0.863860i \(-0.668040\pi\)
−0.503732 + 0.863860i \(0.668040\pi\)
\(752\) 7102.76i 0.344430i
\(753\) 6596.34 0.319235
\(754\) 0 0
\(755\) 3426.23 0.165157
\(756\) 792.391i 0.0381203i
\(757\) 17449.8 0.837812 0.418906 0.908030i \(-0.362414\pi\)
0.418906 + 0.908030i \(0.362414\pi\)
\(758\) 216.054 0.0103528
\(759\) − 2047.74i − 0.0979293i
\(760\) − 8139.27i − 0.388477i
\(761\) − 424.121i − 0.0202028i −0.999949 0.0101014i \(-0.996785\pi\)
0.999949 0.0101014i \(-0.00321544\pi\)
\(762\) − 19251.9i − 0.915251i
\(763\) 13323.3 0.632157
\(764\) 5144.48 0.243613
\(765\) − 3556.49i − 0.168085i
\(766\) 4915.68 0.231868
\(767\) 0 0
\(768\) 8957.81 0.420882
\(769\) − 38060.7i − 1.78479i −0.451252 0.892396i \(-0.649023\pi\)
0.451252 0.892396i \(-0.350977\pi\)
\(770\) 18949.0 0.886849
\(771\) 18582.8 0.868019
\(772\) 552.413i 0.0257536i
\(773\) 16683.3i 0.776268i 0.921603 + 0.388134i \(0.126880\pi\)
−0.921603 + 0.388134i \(0.873120\pi\)
\(774\) 3664.28i 0.170168i
\(775\) 3072.89i 0.142428i
\(776\) 19684.9 0.910627
\(777\) 18017.8 0.831900
\(778\) − 23167.7i − 1.06761i
\(779\) 23492.5 1.08050
\(780\) 0 0
\(781\) −37966.5 −1.73950
\(782\) − 2031.70i − 0.0929073i
\(783\) 5840.54 0.266569
\(784\) −10841.9 −0.493889
\(785\) 8380.17i 0.381020i
\(786\) 17687.8i 0.802674i
\(787\) − 39105.2i − 1.77122i −0.464432 0.885609i \(-0.653741\pi\)
0.464432 0.885609i \(-0.346259\pi\)
\(788\) − 2549.14i − 0.115240i
\(789\) −12545.2 −0.566060
\(790\) 28494.3 1.28327
\(791\) − 13612.8i − 0.611903i
\(792\) −10583.4 −0.474827
\(793\) 0 0
\(794\) −19346.2 −0.864697
\(795\) − 2667.78i − 0.119014i
\(796\) −6721.90 −0.299311
\(797\) 4346.93 0.193195 0.0965974 0.995324i \(-0.469204\pi\)
0.0965974 + 0.995324i \(0.469204\pi\)
\(798\) − 8657.69i − 0.384059i
\(799\) − 5458.25i − 0.241676i
\(800\) − 7281.78i − 0.321812i
\(801\) − 9302.99i − 0.410368i
\(802\) −24093.9 −1.06083
\(803\) −61305.6 −2.69418
\(804\) 5375.26i 0.235784i
\(805\) 1043.54 0.0456895
\(806\) 0 0
\(807\) 8303.07 0.362183
\(808\) − 26655.8i − 1.16058i
\(809\) −23030.2 −1.00086 −0.500432 0.865776i \(-0.666826\pi\)
−0.500432 + 0.865776i \(0.666826\pi\)
\(810\) 1732.89 0.0751697
\(811\) − 7898.11i − 0.341973i −0.985273 0.170987i \(-0.945305\pi\)
0.985273 0.170987i \(-0.0546955\pi\)
\(812\) 6348.41i 0.274366i
\(813\) − 21574.1i − 0.930672i
\(814\) − 84044.8i − 3.61888i
\(815\) −229.207 −0.00985127
\(816\) −13413.6 −0.575453
\(817\) − 8232.18i − 0.352518i
\(818\) −22956.2 −0.981230
\(819\) 0 0
\(820\) −5111.04 −0.217665
\(821\) 3939.61i 0.167471i 0.996488 + 0.0837353i \(0.0266850\pi\)
−0.996488 + 0.0837353i \(0.973315\pi\)
\(822\) −18032.6 −0.765159
\(823\) 17599.8 0.745430 0.372715 0.927946i \(-0.378427\pi\)
0.372715 + 0.927946i \(0.378427\pi\)
\(824\) − 7293.70i − 0.308359i
\(825\) 14915.1i 0.629425i
\(826\) 17393.6i 0.732687i
\(827\) 12510.6i 0.526042i 0.964790 + 0.263021i \(0.0847189\pi\)
−0.964790 + 0.263021i \(0.915281\pi\)
\(828\) 203.550 0.00854331
\(829\) −28630.8 −1.19950 −0.599752 0.800186i \(-0.704734\pi\)
−0.599752 + 0.800186i \(0.704734\pi\)
\(830\) − 19417.6i − 0.812042i
\(831\) −3951.80 −0.164966
\(832\) 0 0
\(833\) 8331.62 0.346547
\(834\) − 11672.7i − 0.484643i
\(835\) −14903.2 −0.617660
\(836\) −8303.75 −0.343530
\(837\) 1042.93i 0.0430693i
\(838\) − 16859.1i − 0.694973i
\(839\) 21250.3i 0.874426i 0.899358 + 0.437213i \(0.144034\pi\)
−0.899358 + 0.437213i \(0.855966\pi\)
\(840\) − 5393.34i − 0.221533i
\(841\) 22403.7 0.918600
\(842\) 47690.9 1.95194
\(843\) − 11846.8i − 0.484015i
\(844\) −684.810 −0.0279291
\(845\) 0 0
\(846\) 2659.51 0.108080
\(847\) 36490.4i 1.48031i
\(848\) −10061.7 −0.407454
\(849\) 14944.6 0.604118
\(850\) 14798.2i 0.597147i
\(851\) − 4628.45i − 0.186441i
\(852\) − 3773.96i − 0.151753i
\(853\) 34721.1i 1.39370i 0.717215 + 0.696852i \(0.245416\pi\)
−0.717215 + 0.696852i \(0.754584\pi\)
\(854\) −27933.6 −1.11928
\(855\) −3893.11 −0.155721
\(856\) 1626.71i 0.0649529i
\(857\) −4898.06 −0.195233 −0.0976163 0.995224i \(-0.531122\pi\)
−0.0976163 + 0.995224i \(0.531122\pi\)
\(858\) 0 0
\(859\) −12564.2 −0.499051 −0.249526 0.968368i \(-0.580275\pi\)
−0.249526 + 0.968368i \(0.580275\pi\)
\(860\) 1791.00i 0.0710145i
\(861\) 15566.9 0.616166
\(862\) −22704.9 −0.897137
\(863\) 22826.6i 0.900380i 0.892933 + 0.450190i \(0.148644\pi\)
−0.892933 + 0.450190i \(0.851356\pi\)
\(864\) − 2471.42i − 0.0973143i
\(865\) − 4458.18i − 0.175240i
\(866\) 30282.8i 1.18828i
\(867\) −4431.08 −0.173572
\(868\) −1133.62 −0.0443291
\(869\) 83238.3i 3.24933i
\(870\) 13883.4 0.541024
\(871\) 0 0
\(872\) −17688.6 −0.686939
\(873\) − 9415.51i − 0.365025i
\(874\) −2224.00 −0.0860731
\(875\) −19543.9 −0.755089
\(876\) − 6093.92i − 0.235039i
\(877\) − 25212.7i − 0.970776i −0.874299 0.485388i \(-0.838678\pi\)
0.874299 0.485388i \(-0.161322\pi\)
\(878\) − 22438.5i − 0.862484i
\(879\) − 9609.07i − 0.368721i
\(880\) −32136.9 −1.23106
\(881\) 18026.2 0.689352 0.344676 0.938722i \(-0.387989\pi\)
0.344676 + 0.938722i \(0.387989\pi\)
\(882\) 4059.55i 0.154980i
\(883\) 18833.1 0.717764 0.358882 0.933383i \(-0.383158\pi\)
0.358882 + 0.933383i \(0.383158\pi\)
\(884\) 0 0
\(885\) 7821.37 0.297076
\(886\) 6640.68i 0.251803i
\(887\) 38451.4 1.45555 0.727775 0.685816i \(-0.240555\pi\)
0.727775 + 0.685816i \(0.240555\pi\)
\(888\) −23921.2 −0.903991
\(889\) 28659.7i 1.08123i
\(890\) − 22113.9i − 0.832876i
\(891\) 5062.15i 0.190335i
\(892\) 11981.0i 0.449723i
\(893\) −5974.86 −0.223898
\(894\) 30423.7 1.13816
\(895\) − 15674.2i − 0.585399i
\(896\) −24759.1 −0.923152
\(897\) 0 0
\(898\) −18539.4 −0.688940
\(899\) 8355.68i 0.309986i
\(900\) −1482.59 −0.0549108
\(901\) 7732.11 0.285898
\(902\) − 72612.3i − 2.68041i
\(903\) − 5454.91i − 0.201028i
\(904\) 18072.9i 0.664929i
\(905\) 14306.8i 0.525495i
\(906\) −4838.53 −0.177428
\(907\) 5531.31 0.202496 0.101248 0.994861i \(-0.467716\pi\)
0.101248 + 0.994861i \(0.467716\pi\)
\(908\) 6099.05i 0.222912i
\(909\) −12749.8 −0.465219
\(910\) 0 0
\(911\) 15695.2 0.570806 0.285403 0.958408i \(-0.407872\pi\)
0.285403 + 0.958408i \(0.407872\pi\)
\(912\) 14683.2i 0.533123i
\(913\) 56723.1 2.05615
\(914\) 18895.9 0.683832
\(915\) 12560.9i 0.453827i
\(916\) 7333.01i 0.264508i
\(917\) − 26331.3i − 0.948240i
\(918\) 5022.49i 0.180574i
\(919\) −51503.5 −1.84869 −0.924344 0.381561i \(-0.875386\pi\)
−0.924344 + 0.381561i \(0.875386\pi\)
\(920\) −1385.45 −0.0496488
\(921\) 14387.0i 0.514732i
\(922\) −5921.45 −0.211510
\(923\) 0 0
\(924\) −5502.33 −0.195902
\(925\) 33712.0i 1.19832i
\(926\) 21262.2 0.754557
\(927\) −3488.66 −0.123606
\(928\) − 19800.3i − 0.700407i
\(929\) 42927.0i 1.51603i 0.652238 + 0.758014i \(0.273830\pi\)
−0.652238 + 0.758014i \(0.726170\pi\)
\(930\) 2479.13i 0.0874127i
\(931\) − 9120.20i − 0.321055i
\(932\) 4847.10 0.170356
\(933\) −1890.64 −0.0663416
\(934\) 52634.1i 1.84394i
\(935\) 24696.2 0.863797
\(936\) 0 0
\(937\) 44206.6 1.54127 0.770633 0.637280i \(-0.219940\pi\)
0.770633 + 0.637280i \(0.219940\pi\)
\(938\) − 38916.6i − 1.35466i
\(939\) −27944.2 −0.971165
\(940\) 1299.89 0.0451041
\(941\) 44067.7i 1.52664i 0.646022 + 0.763319i \(0.276432\pi\)
−0.646022 + 0.763319i \(0.723568\pi\)
\(942\) − 11834.5i − 0.409330i
\(943\) − 3998.85i − 0.138092i
\(944\) − 29498.9i − 1.01706i
\(945\) −2579.70 −0.0888017
\(946\) −25444.6 −0.874498
\(947\) − 44402.5i − 1.52364i −0.647789 0.761820i \(-0.724306\pi\)
0.647789 0.761820i \(-0.275694\pi\)
\(948\) −8274.07 −0.283470
\(949\) 0 0
\(950\) 16198.9 0.553221
\(951\) − 1729.00i − 0.0589554i
\(952\) 15631.7 0.532172
\(953\) 10361.7 0.352202 0.176101 0.984372i \(-0.443651\pi\)
0.176101 + 0.984372i \(0.443651\pi\)
\(954\) 3767.44i 0.127857i
\(955\) 16748.3i 0.567500i
\(956\) − 3138.60i − 0.106182i
\(957\) 40556.5i 1.36991i
\(958\) 19570.2 0.660006
\(959\) 26844.7 0.903920
\(960\) 6466.65i 0.217407i
\(961\) 28298.9 0.949916
\(962\) 0 0
\(963\) 778.073 0.0260364
\(964\) − 4954.18i − 0.165522i
\(965\) −1798.43 −0.0599933
\(966\) −1473.69 −0.0490842
\(967\) − 8432.54i − 0.280426i −0.990121 0.140213i \(-0.955221\pi\)
0.990121 0.140213i \(-0.0447788\pi\)
\(968\) − 48446.1i − 1.60859i
\(969\) − 11283.5i − 0.374076i
\(970\) − 22381.4i − 0.740848i
\(971\) −36917.1 −1.22011 −0.610055 0.792359i \(-0.708853\pi\)
−0.610055 + 0.792359i \(0.708853\pi\)
\(972\) −503.189 −0.0166047
\(973\) 17376.8i 0.572534i
\(974\) 18147.4 0.597001
\(975\) 0 0
\(976\) 47374.5 1.55371
\(977\) 38306.9i 1.25440i 0.778859 + 0.627199i \(0.215799\pi\)
−0.778859 + 0.627199i \(0.784201\pi\)
\(978\) 323.687 0.0105832
\(979\) 64599.6 2.10890
\(980\) 1984.19i 0.0646763i
\(981\) 8460.65i 0.275360i
\(982\) 66835.6i 2.17190i
\(983\) − 18810.9i − 0.610350i −0.952296 0.305175i \(-0.901285\pi\)
0.952296 0.305175i \(-0.0987149\pi\)
\(984\) −20667.3 −0.669561
\(985\) 8298.97 0.268454
\(986\) 40238.8i 1.29966i
\(987\) −3959.14 −0.127681
\(988\) 0 0
\(989\) −1401.26 −0.0450532
\(990\) 12033.1i 0.386300i
\(991\) −6492.09 −0.208101 −0.104051 0.994572i \(-0.533180\pi\)
−0.104051 + 0.994572i \(0.533180\pi\)
\(992\) 3535.71 0.113164
\(993\) − 4727.86i − 0.151092i
\(994\) 27323.2i 0.871872i
\(995\) − 21883.8i − 0.697249i
\(996\) 5638.41i 0.179377i
\(997\) 48552.1 1.54229 0.771143 0.636662i \(-0.219685\pi\)
0.771143 + 0.636662i \(0.219685\pi\)
\(998\) −24955.5 −0.791537
\(999\) 11441.8i 0.362365i
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 507.4.b.k.337.14 18
13.5 odd 4 507.4.a.o.1.8 9
13.8 odd 4 507.4.a.p.1.2 yes 9
13.12 even 2 inner 507.4.b.k.337.5 18
39.5 even 4 1521.4.a.bi.1.2 9
39.8 even 4 1521.4.a.bf.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.8 9 13.5 odd 4
507.4.a.p.1.2 yes 9 13.8 odd 4
507.4.b.k.337.5 18 13.12 even 2 inner
507.4.b.k.337.14 18 1.1 even 1 trivial
1521.4.a.bf.1.8 9 39.8 even 4
1521.4.a.bi.1.2 9 39.5 even 4