Properties

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

$q$-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 \(-0.810420\pi\)
0.827822 0.560991i \(-0.189580\pi\)
\(3\) 3.00000 0.577350
\(4\) −2.07074 −0.258842
\(5\) 6.74147i 0.602975i 0.953470 + 0.301488i \(0.0974832\pi\)
−0.953470 + 0.301488i \(0.902517\pi\)
\(6\) − 9.52033i − 0.647776i
\(7\) 14.1726i 0.765251i 0.923904 + 0.382625i \(0.124980\pi\)
−0.923904 + 0.382625i \(0.875020\pi\)
\(8\) − 18.8162i − 0.831565i
\(9\) 9.00000 0.333333
\(10\) 21.3937 0.676527
\(11\) − 62.4956i − 1.71301i −0.516136 0.856507i \(-0.672630\pi\)
0.516136 0.856507i \(-0.327370\pi\)
\(12\) −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 \(0.126620\pi\)
−0.921919 + 0.387382i \(0.873380\pi\)
\(20\) − 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 \(-0.964307\pi\)
0.993720 0.111897i \(-0.0356928\pi\)
\(32\) 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 \(-0.609462\pi\)
0.337147 0.941452i \(-0.390538\pi\)
\(38\) 203.625 0.869270
\(39\) 0 0
\(40\) 126.849 0.501414
\(41\) − 366.126i − 1.39461i −0.716772 0.697307i \(-0.754381\pi\)
0.716772 0.697307i \(-0.245619\pi\)
\(42\) 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 \(0.0461556\pi\)
−0.989506 + 0.144495i \(0.953844\pi\)
\(48\) −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 \(-0.859684\pi\)
0.904404 0.426677i \(-0.140316\pi\)
\(60\) − 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 \(0.289339\pi\)
−0.614547 + 0.788880i \(0.710661\pi\)
\(68\) −121.381 −0.216464
\(69\) −32.7662 −0.0571679
\(70\) 303.205i 0.517713i
\(71\) − 607.506i − 1.01546i −0.861516 0.507730i \(-0.830485\pi\)
0.861516 0.507730i \(-0.169515\pi\)
\(72\) − 169.346i − 0.277188i
\(73\) − 980.958i − 1.57277i −0.617735 0.786387i \(-0.711949\pi\)
0.617735 0.786387i \(-0.288051\pi\)
\(74\) −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 \(0.204894\pi\)
−0.799884 + 0.600155i \(0.795106\pi\)
\(84\) − 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 \(0.211067\pi\)
−0.788096 + 0.615552i \(0.788933\pi\)
\(90\) 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 \(0.184435\pi\)
−0.836781 + 0.547538i \(0.815565\pi\)
\(98\) − 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 \(-0.864467\pi\)
0.910713 0.413039i \(-0.135533\pi\)
\(110\) − 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 \(-0.798890\pi\)
0.806961 0.590604i \(-0.201110\pi\)
\(138\) 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 \(0.341467\pi\)
−0.477711 + 0.878517i \(0.658533\pi\)
\(150\) − 757.366i − 0.412258i
\(151\) − 508.232i − 0.273903i −0.990578 0.136951i \(-0.956270\pi\)
0.990578 0.136951i \(-0.0437304\pi\)
\(152\) 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 \(0.00260026\pi\)
−0.999967 + 0.00816888i \(0.997400\pi\)
\(164\) 758.149i 0.360985i
\(165\) 1263.94 0.596348
\(166\) 2880.32 1.34673
\(167\) 2210.67i 1.02435i 0.858880 + 0.512176i \(0.171161\pi\)
−0.858880 + 0.512176i \(0.828839\pi\)
\(168\) 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 \(0.0158417\pi\)
−0.998762 + 0.0497478i \(0.984158\pi\)
\(194\) 3319.95 1.22865
\(195\) 0 0
\(196\) −294.327 −0.107262
\(197\) − 1231.03i − 0.445216i −0.974908 0.222608i \(-0.928543\pi\)
0.974908 0.222608i \(-0.0714569\pi\)
\(198\) −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 \(0.335058\pi\)
−0.495300 + 0.868722i \(0.664942\pi\)
\(224\) −1297.28 −0.386957
\(225\) 715.973 0.212140
\(226\) − 3048.09i − 0.897149i
\(227\) 2945.35i 0.861189i 0.902545 + 0.430595i \(0.141696\pi\)
−0.902545 + 0.430595i \(0.858304\pi\)
\(228\) − 398.608i − 0.115783i
\(229\) 3541.26i 1.02189i 0.859613 + 0.510945i \(0.170704\pi\)
−0.859613 + 0.510945i \(0.829296\pi\)
\(230\) −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 \(-0.934245\pi\)
0.978739 0.205109i \(-0.0657549\pi\)
\(240\) − 1542.68i − 0.414914i
\(241\) − 2392.47i − 0.639472i −0.947507 0.319736i \(-0.896406\pi\)
0.947507 0.319736i \(-0.103594\pi\)
\(242\) 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 \(0.298364\pi\)
−0.591935 + 0.805986i \(0.701636\pi\)
\(272\) −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 \(0.137679\pi\)
−0.907908 + 0.419169i \(0.862321\pi\)
\(282\) 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 \(0.103455\pi\)
−0.947646 + 0.319322i \(0.896545\pi\)
\(294\) − 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 \(-0.852930\pi\)
0.895147 0.445771i \(-0.147070\pi\)
\(308\) −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 \(0.0162590\pi\)
−0.998696 + 0.0510569i \(0.983741\pi\)
\(318\) − 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 \(0.0417704\pi\)
−0.991402 + 0.130849i \(0.958230\pi\)
\(332\) − 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 \(0.124546\pi\)
−0.924425 + 0.381365i \(0.875454\pi\)
\(350\) 3577.96 0.546429
\(351\) 0 0
\(352\) 5720.49 0.866202
\(353\) 1575.34i 0.237526i 0.992923 + 0.118763i \(0.0378929\pi\)
−0.992923 + 0.118763i \(0.962107\pi\)
\(354\) −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 \(-0.812237\pi\)
0.831010 0.556258i \(-0.187763\pi\)
\(360\) 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 \(0.00146857\pi\)
−0.999989 + 0.00461363i \(0.998531\pi\)
\(380\) 895.734 0.120922
\(381\) −6066.55 −0.815745
\(382\) 7884.01i 1.05597i
\(383\) 1549.01i 0.206659i 0.994647 + 0.103330i \(0.0329497\pi\)
−0.994647 + 0.103330i \(0.967050\pi\)
\(384\) 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 \(-0.874083\pi\)
0.922773 0.385344i \(-0.125917\pi\)
\(398\) − 10301.4i − 1.29740i
\(399\) −2728.17 −0.342304
\(400\) −6068.11 −0.758513
\(401\) − 7592.37i − 0.945498i −0.881197 0.472749i \(-0.843262\pi\)
0.881197 0.472749i \(-0.156738\pi\)
\(402\) 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 \(-0.855943\pi\)
0.899328 0.437275i \(-0.144057\pi\)
\(410\) − 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 \(0.335794\pi\)
−0.493290 + 0.869865i \(0.664206\pi\)
\(422\) − 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 \(-0.869080\pi\)
0.916602 0.399800i \(-0.130920\pi\)
\(432\) −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 \(-0.900668\pi\)
0.951703 0.307019i \(-0.0993316\pi\)
\(450\) − 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 \(0.0985705\pi\)
−0.952435 + 0.304743i \(0.901429\pi\)
\(458\) 11238.0 1.14654
\(459\) 1582.66 0.160942
\(460\) 152.470i 0.0154542i
\(461\) − 1865.94i − 0.188515i −0.995548 0.0942576i \(-0.969952\pi\)
0.995548 0.0942576i \(-0.0300477\pi\)
\(462\) − 8432.42i − 0.849160i
\(463\) 6700.05i 0.672522i 0.941769 + 0.336261i \(0.109162\pi\)
−0.941769 + 0.336261i \(0.890838\pi\)
\(464\) −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 \(0.0950282\pi\)
−0.955767 + 0.294125i \(0.904972\pi\)
\(480\) −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 \(0.0857178\pi\)
−0.963960 + 0.266047i \(0.914282\pi\)
\(488\) 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 \(-0.885250\pi\)
0.935721 0.352741i \(-0.114750\pi\)
\(500\) − 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 \(0.222879\pi\)
−0.764717 + 0.644367i \(0.777121\pi\)
\(510\) 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 \(-0.887840\pi\)
0.938560 0.345116i \(-0.112160\pi\)
\(542\) 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 \(-0.886002\pi\)
0.936552 0.350530i \(-0.113998\pi\)
\(558\) −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 \(-0.906310\pi\)
0.956995 0.290103i \(-0.0936896\pi\)
\(578\) 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 \(-0.827887\pi\)
0.857344 0.514744i \(-0.172113\pi\)
\(588\) −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 \(-0.936361\pi\)
0.980081 0.198600i \(-0.0636394\pi\)
\(594\) −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 \(0.182126\pi\)
−0.840730 + 0.541455i \(0.817874\pi\)
\(614\) −15218.8 −1.00029
\(615\) 7404.68 0.485505
\(616\) − 16666.0i − 1.09009i
\(617\) 1290.89i 0.0842289i 0.999113 + 0.0421145i \(0.0134094\pi\)
−0.999113 + 0.0421145i \(0.986591\pi\)
\(618\) 3690.36i 0.240207i
\(619\) 26719.0i 1.73494i 0.497490 + 0.867470i \(0.334255\pi\)
−0.497490 + 0.867470i \(0.665745\pi\)
\(620\) −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 \(0.109563\pi\)
−0.941345 + 0.337447i \(0.890437\pi\)
\(632\) − 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 \(-0.858750\pi\)
0.903147 0.429331i \(-0.141250\pi\)
\(644\) 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 \(0.157162\pi\)
−0.880568 + 0.473921i \(0.842838\pi\)
\(662\) 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 \(0.00230894\pi\)
−0.999974 + 0.00725369i \(0.997691\pi\)
\(684\) − 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 \(-0.994233\pi\)
0.999836 0.0181178i \(-0.00576740\pi\)
\(692\) −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 \(-0.942208\pi\)
0.983564 0.180562i \(-0.0577917\pi\)
\(710\) − 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 \(-0.859166\pi\)
0.903708 0.428149i \(-0.140834\pi\)
\(734\) − 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 \(0.113411\pi\)
−0.937197 + 0.348800i \(0.886589\pi\)
\(740\) −5915.75 −0.293875
\(741\) 0 0
\(742\) 5932.73 0.293528
\(743\) − 141.560i − 0.00698971i −0.999994 0.00349485i \(-0.998888\pi\)
0.999994 0.00349485i \(-0.00111245\pi\)
\(744\) −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 \(0.00321544\pi\)
−0.999949 + 0.0101014i \(0.996785\pi\)
\(762\) 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 \(0.350977\pi\)
−0.451252 + 0.892396i \(0.649023\pi\)
\(770\) 18949.0 0.886849
\(771\) 18582.8 0.868019
\(772\) − 552.413i − 0.0257536i
\(773\) − 16683.3i − 0.776268i −0.921603 0.388134i \(-0.873120\pi\)
0.921603 0.388134i \(-0.126880\pi\)
\(774\) − 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 \(0.346259\pi\)
−0.464432 + 0.885609i \(0.653741\pi\)
\(788\) 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 \(0.0546955\pi\)
−0.985273 + 0.170987i \(0.945305\pi\)
\(812\) − 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 \(-0.973315\pi\)
0.996488 0.0837353i \(-0.0266850\pi\)
\(822\) −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 \(-0.915281\pi\)
0.964790 0.263021i \(-0.0847189\pi\)
\(828\) 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 \(-0.855966\pi\)
0.899358 0.437213i \(-0.144034\pi\)
\(840\) 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 \(-0.754584\pi\)
0.717215 0.696852i \(-0.245416\pi\)
\(854\) −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 \(-0.851356\pi\)
0.892933 0.450190i \(-0.148644\pi\)
\(864\) 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 \(0.161322\pi\)
−0.874299 + 0.485388i \(0.838678\pi\)
\(878\) 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 \(-0.726170\pi\)
0.652238 0.758014i \(-0.273830\pi\)
\(930\) − 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 \(-0.723568\pi\)
0.646022 0.763319i \(-0.276432\pi\)
\(942\) 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 \(0.275694\pi\)
−0.647789 + 0.761820i \(0.724306\pi\)
\(948\) −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 \(0.0447788\pi\)
−0.990121 + 0.140213i \(0.955221\pi\)
\(968\) 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 \(-0.784201\pi\)
0.778859 0.627199i \(-0.215799\pi\)
\(978\) 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 \(0.0987149\pi\)
−0.952296 + 0.305175i \(0.901285\pi\)
\(984\) −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.337.5 18
13.5 odd 4 507.4.a.p.1.2 yes 9
13.8 odd 4 507.4.a.o.1.8 9
13.12 even 2 inner 507.4.b.k.337.14 18
39.5 even 4 1521.4.a.bf.1.8 9
39.8 even 4 1521.4.a.bi.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.8 9 13.8 odd 4
507.4.a.p.1.2 yes 9 13.5 odd 4
507.4.b.k.337.5 18 1.1 even 1 trivial
507.4.b.k.337.14 18 13.12 even 2 inner
1521.4.a.bf.1.8 9 39.5 even 4
1521.4.a.bi.1.2 9 39.8 even 4