Properties

Label 507.4.a.p.1.3
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.06791\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} -3.41089 q^{5} -8.46278 q^{6} +13.3442 q^{7} +22.6869 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.82093 q^{2} +3.00000 q^{3} -0.0423641 q^{4} -3.41089 q^{5} -8.46278 q^{6} +13.3442 q^{7} +22.6869 q^{8} +9.00000 q^{9} +9.62187 q^{10} +35.4529 q^{11} -0.127092 q^{12} -37.6430 q^{14} -10.2327 q^{15} -63.6593 q^{16} +69.6526 q^{17} -25.3884 q^{18} +12.4014 q^{19} +0.144499 q^{20} +40.0325 q^{21} -100.010 q^{22} -126.251 q^{23} +68.0608 q^{24} -113.366 q^{25} +27.0000 q^{27} -0.565314 q^{28} -179.060 q^{29} +28.8656 q^{30} +255.935 q^{31} -1.91716 q^{32} +106.359 q^{33} -196.485 q^{34} -45.5155 q^{35} -0.381277 q^{36} +207.235 q^{37} -34.9833 q^{38} -77.3825 q^{40} -117.701 q^{41} -112.929 q^{42} +553.224 q^{43} -1.50193 q^{44} -30.6980 q^{45} +356.146 q^{46} +62.9185 q^{47} -190.978 q^{48} -164.933 q^{49} +319.797 q^{50} +208.958 q^{51} -147.031 q^{53} -76.1651 q^{54} -120.926 q^{55} +302.738 q^{56} +37.2041 q^{57} +505.115 q^{58} +274.087 q^{59} +0.433498 q^{60} +603.039 q^{61} -721.974 q^{62} +120.098 q^{63} +514.683 q^{64} -300.031 q^{66} -741.019 q^{67} -2.95077 q^{68} -378.754 q^{69} +128.396 q^{70} +572.574 q^{71} +204.182 q^{72} +26.7155 q^{73} -584.595 q^{74} -340.098 q^{75} -0.525372 q^{76} +473.090 q^{77} -207.798 q^{79} +217.135 q^{80} +81.0000 q^{81} +332.026 q^{82} -1031.37 q^{83} -1.69594 q^{84} -237.577 q^{85} -1560.60 q^{86} -537.179 q^{87} +804.318 q^{88} +1229.66 q^{89} +86.5968 q^{90} +5.34852 q^{92} +767.805 q^{93} -177.488 q^{94} -42.2996 q^{95} -5.75148 q^{96} +1795.12 q^{97} +465.264 q^{98} +319.076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{2} + 27 q^{3} + 44 q^{4} + 33 q^{5} + 18 q^{6} + 83 q^{7} + 87 q^{8} + 81 q^{9} - 54 q^{10} + 85 q^{11} + 132 q^{12} + 158 q^{14} + 99 q^{15} + 216 q^{16} + 178 q^{17} + 54 q^{18} + 352 q^{19} + 402 q^{20} + 249 q^{21} - 630 q^{22} + 150 q^{23} + 261 q^{24} - 20 q^{25} + 243 q^{27} + 940 q^{28} - 97 q^{29} - 162 q^{30} + 717 q^{31} + 707 q^{32} + 255 q^{33} + 632 q^{34} - 418 q^{35} + 396 q^{36} + 1108 q^{37} - 660 q^{38} - 1506 q^{40} + 334 q^{41} + 474 q^{42} + 242 q^{43} - 307 q^{44} + 297 q^{45} + 979 q^{46} - 184 q^{47} + 648 q^{48} - 38 q^{49} - 2031 q^{50} + 534 q^{51} - 151 q^{53} + 162 q^{54} + 2064 q^{55} + 2276 q^{56} + 1056 q^{57} + 1161 q^{58} + 537 q^{59} + 1206 q^{60} - 1340 q^{61} + 347 q^{62} + 747 q^{63} + 893 q^{64} - 1890 q^{66} + 2308 q^{67} + 2785 q^{68} + 450 q^{69} - 1420 q^{70} + 96 q^{71} + 783 q^{72} + 2505 q^{73} - 1191 q^{74} - 60 q^{75} + 2409 q^{76} - 2142 q^{77} - 1591 q^{79} - 2671 q^{80} + 729 q^{81} + 1517 q^{82} + 1539 q^{83} + 2820 q^{84} + 4296 q^{85} - 3763 q^{86} - 291 q^{87} - 3716 q^{88} - 592 q^{89} - 486 q^{90} + 515 q^{92} + 2151 q^{93} - 692 q^{94} + 4158 q^{95} + 2121 q^{96} + 1445 q^{97} + 1457 q^{98} + 765 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82093 −0.997349 −0.498674 0.866789i \(-0.666180\pi\)
−0.498674 + 0.866789i \(0.666180\pi\)
\(3\) 3.00000 0.577350
\(4\) −0.0423641 −0.00529552
\(5\) −3.41089 −0.305079 −0.152539 0.988297i \(-0.548745\pi\)
−0.152539 + 0.988297i \(0.548745\pi\)
\(6\) −8.46278 −0.575820
\(7\) 13.3442 0.720518 0.360259 0.932852i \(-0.382688\pi\)
0.360259 + 0.932852i \(0.382688\pi\)
\(8\) 22.6869 1.00263
\(9\) 9.00000 0.333333
\(10\) 9.62187 0.304270
\(11\) 35.4529 0.971769 0.485885 0.874023i \(-0.338498\pi\)
0.485885 + 0.874023i \(0.338498\pi\)
\(12\) −0.127092 −0.00305737
\(13\) 0 0
\(14\) −37.6430 −0.718607
\(15\) −10.2327 −0.176137
\(16\) −63.6593 −0.994676
\(17\) 69.6526 0.993720 0.496860 0.867831i \(-0.334486\pi\)
0.496860 + 0.867831i \(0.334486\pi\)
\(18\) −25.3884 −0.332450
\(19\) 12.4014 0.149740 0.0748701 0.997193i \(-0.476146\pi\)
0.0748701 + 0.997193i \(0.476146\pi\)
\(20\) 0.144499 0.00161555
\(21\) 40.0325 0.415991
\(22\) −100.010 −0.969193
\(23\) −126.251 −1.14457 −0.572287 0.820053i \(-0.693944\pi\)
−0.572287 + 0.820053i \(0.693944\pi\)
\(24\) 68.0608 0.578869
\(25\) −113.366 −0.906927
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) −0.565314 −0.00381551
\(29\) −179.060 −1.14657 −0.573286 0.819356i \(-0.694331\pi\)
−0.573286 + 0.819356i \(0.694331\pi\)
\(30\) 28.8656 0.175670
\(31\) 255.935 1.48281 0.741407 0.671055i \(-0.234159\pi\)
0.741407 + 0.671055i \(0.234159\pi\)
\(32\) −1.91716 −0.0105909
\(33\) 106.359 0.561051
\(34\) −196.485 −0.991085
\(35\) −45.5155 −0.219815
\(36\) −0.381277 −0.00176517
\(37\) 207.235 0.920790 0.460395 0.887714i \(-0.347708\pi\)
0.460395 + 0.887714i \(0.347708\pi\)
\(38\) −34.9833 −0.149343
\(39\) 0 0
\(40\) −77.3825 −0.305881
\(41\) −117.701 −0.448337 −0.224169 0.974550i \(-0.571967\pi\)
−0.224169 + 0.974550i \(0.571967\pi\)
\(42\) −112.929 −0.414888
\(43\) 553.224 1.96200 0.980998 0.194016i \(-0.0621515\pi\)
0.980998 + 0.194016i \(0.0621515\pi\)
\(44\) −1.50193 −0.00514602
\(45\) −30.6980 −0.101693
\(46\) 356.146 1.14154
\(47\) 62.9185 0.195268 0.0976341 0.995222i \(-0.468873\pi\)
0.0976341 + 0.995222i \(0.468873\pi\)
\(48\) −190.978 −0.574277
\(49\) −164.933 −0.480854
\(50\) 319.797 0.904522
\(51\) 208.958 0.573724
\(52\) 0 0
\(53\) −147.031 −0.381061 −0.190531 0.981681i \(-0.561021\pi\)
−0.190531 + 0.981681i \(0.561021\pi\)
\(54\) −76.1651 −0.191940
\(55\) −120.926 −0.296466
\(56\) 302.738 0.722413
\(57\) 37.2041 0.0864526
\(58\) 505.115 1.14353
\(59\) 274.087 0.604798 0.302399 0.953181i \(-0.402213\pi\)
0.302399 + 0.953181i \(0.402213\pi\)
\(60\) 0.433498 0.000932738 0
\(61\) 603.039 1.26576 0.632879 0.774251i \(-0.281873\pi\)
0.632879 + 0.774251i \(0.281873\pi\)
\(62\) −721.974 −1.47888
\(63\) 120.098 0.240173
\(64\) 514.683 1.00524
\(65\) 0 0
\(66\) −300.031 −0.559564
\(67\) −741.019 −1.35119 −0.675596 0.737272i \(-0.736113\pi\)
−0.675596 + 0.737272i \(0.736113\pi\)
\(68\) −2.95077 −0.00526226
\(69\) −378.754 −0.660820
\(70\) 128.396 0.219232
\(71\) 572.574 0.957071 0.478536 0.878068i \(-0.341168\pi\)
0.478536 + 0.878068i \(0.341168\pi\)
\(72\) 204.182 0.334210
\(73\) 26.7155 0.0428330 0.0214165 0.999771i \(-0.493182\pi\)
0.0214165 + 0.999771i \(0.493182\pi\)
\(74\) −584.595 −0.918349
\(75\) −340.098 −0.523614
\(76\) −0.525372 −0.000792952 0
\(77\) 473.090 0.700177
\(78\) 0 0
\(79\) −207.798 −0.295938 −0.147969 0.988992i \(-0.547274\pi\)
−0.147969 + 0.988992i \(0.547274\pi\)
\(80\) 217.135 0.303455
\(81\) 81.0000 0.111111
\(82\) 332.026 0.447149
\(83\) −1031.37 −1.36395 −0.681976 0.731374i \(-0.738879\pi\)
−0.681976 + 0.731374i \(0.738879\pi\)
\(84\) −1.69594 −0.00220289
\(85\) −237.577 −0.303163
\(86\) −1560.60 −1.95679
\(87\) −537.179 −0.661973
\(88\) 804.318 0.974325
\(89\) 1229.66 1.46453 0.732266 0.681019i \(-0.238463\pi\)
0.732266 + 0.681019i \(0.238463\pi\)
\(90\) 86.5968 0.101423
\(91\) 0 0
\(92\) 5.34852 0.00606111
\(93\) 767.805 0.856103
\(94\) −177.488 −0.194750
\(95\) −42.2996 −0.0456826
\(96\) −5.75148 −0.00611467
\(97\) 1795.12 1.87903 0.939517 0.342502i \(-0.111274\pi\)
0.939517 + 0.342502i \(0.111274\pi\)
\(98\) 465.264 0.479579
\(99\) 319.076 0.323923
\(100\) 4.80265 0.00480265
\(101\) −769.697 −0.758295 −0.379147 0.925336i \(-0.623783\pi\)
−0.379147 + 0.925336i \(0.623783\pi\)
\(102\) −589.455 −0.572203
\(103\) 1543.66 1.47671 0.738357 0.674410i \(-0.235602\pi\)
0.738357 + 0.674410i \(0.235602\pi\)
\(104\) 0 0
\(105\) −136.546 −0.126910
\(106\) 414.764 0.380051
\(107\) 2027.64 1.83196 0.915978 0.401228i \(-0.131417\pi\)
0.915978 + 0.401228i \(0.131417\pi\)
\(108\) −1.14383 −0.00101912
\(109\) 1644.97 1.44550 0.722751 0.691108i \(-0.242877\pi\)
0.722751 + 0.691108i \(0.242877\pi\)
\(110\) 341.123 0.295680
\(111\) 621.705 0.531618
\(112\) −849.481 −0.716682
\(113\) −654.514 −0.544880 −0.272440 0.962173i \(-0.587831\pi\)
−0.272440 + 0.962173i \(0.587831\pi\)
\(114\) −104.950 −0.0862234
\(115\) 430.629 0.349186
\(116\) 7.58571 0.00607169
\(117\) 0 0
\(118\) −773.179 −0.603194
\(119\) 929.456 0.715993
\(120\) −232.148 −0.176601
\(121\) −74.0896 −0.0556646
\(122\) −1701.13 −1.26240
\(123\) −353.103 −0.258848
\(124\) −10.8425 −0.00785227
\(125\) 813.039 0.581763
\(126\) −338.787 −0.239536
\(127\) 1325.54 0.926161 0.463080 0.886316i \(-0.346744\pi\)
0.463080 + 0.886316i \(0.346744\pi\)
\(128\) −1436.55 −0.991983
\(129\) 1659.67 1.13276
\(130\) 0 0
\(131\) 930.114 0.620339 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(132\) −4.50580 −0.00297106
\(133\) 165.486 0.107891
\(134\) 2090.36 1.34761
\(135\) −92.0939 −0.0587125
\(136\) 1580.20 0.996333
\(137\) −2751.03 −1.71559 −0.857797 0.513989i \(-0.828167\pi\)
−0.857797 + 0.513989i \(0.828167\pi\)
\(138\) 1068.44 0.659068
\(139\) 2436.65 1.48686 0.743432 0.668812i \(-0.233197\pi\)
0.743432 + 0.668812i \(0.233197\pi\)
\(140\) 1.92822 0.00116403
\(141\) 188.755 0.112738
\(142\) −1615.19 −0.954534
\(143\) 0 0
\(144\) −572.934 −0.331559
\(145\) 610.753 0.349795
\(146\) −75.3624 −0.0427194
\(147\) −494.799 −0.277621
\(148\) −8.77933 −0.00487606
\(149\) −1517.58 −0.834397 −0.417198 0.908815i \(-0.636988\pi\)
−0.417198 + 0.908815i \(0.636988\pi\)
\(150\) 959.391 0.522226
\(151\) 583.642 0.314544 0.157272 0.987555i \(-0.449730\pi\)
0.157272 + 0.987555i \(0.449730\pi\)
\(152\) 281.349 0.150134
\(153\) 626.873 0.331240
\(154\) −1334.55 −0.698321
\(155\) −872.965 −0.452376
\(156\) 0 0
\(157\) 26.0932 0.0132641 0.00663206 0.999978i \(-0.497889\pi\)
0.00663206 + 0.999978i \(0.497889\pi\)
\(158\) 586.184 0.295154
\(159\) −441.093 −0.220006
\(160\) 6.53922 0.00323107
\(161\) −1684.72 −0.824686
\(162\) −228.495 −0.110817
\(163\) 17.0905 0.00821246 0.00410623 0.999992i \(-0.498693\pi\)
0.00410623 + 0.999992i \(0.498693\pi\)
\(164\) 4.98631 0.00237418
\(165\) −362.778 −0.171165
\(166\) 2909.43 1.36034
\(167\) 3919.48 1.81616 0.908079 0.418799i \(-0.137549\pi\)
0.908079 + 0.418799i \(0.137549\pi\)
\(168\) 908.215 0.417085
\(169\) 0 0
\(170\) 670.188 0.302359
\(171\) 111.612 0.0499134
\(172\) −23.4368 −0.0103898
\(173\) −1976.35 −0.868551 −0.434276 0.900780i \(-0.642996\pi\)
−0.434276 + 0.900780i \(0.642996\pi\)
\(174\) 1515.34 0.660218
\(175\) −1512.77 −0.653457
\(176\) −2256.91 −0.966596
\(177\) 822.260 0.349180
\(178\) −3468.77 −1.46065
\(179\) −2784.34 −1.16263 −0.581316 0.813678i \(-0.697462\pi\)
−0.581316 + 0.813678i \(0.697462\pi\)
\(180\) 1.30049 0.000538517 0
\(181\) −1886.53 −0.774723 −0.387361 0.921928i \(-0.626613\pi\)
−0.387361 + 0.921928i \(0.626613\pi\)
\(182\) 0 0
\(183\) 1809.12 0.730786
\(184\) −2864.25 −1.14758
\(185\) −706.855 −0.280914
\(186\) −2165.92 −0.853834
\(187\) 2469.39 0.965666
\(188\) −2.66549 −0.00103405
\(189\) 360.293 0.138664
\(190\) 119.324 0.0455615
\(191\) 2447.94 0.927364 0.463682 0.886002i \(-0.346528\pi\)
0.463682 + 0.886002i \(0.346528\pi\)
\(192\) 1544.05 0.580375
\(193\) −1925.66 −0.718196 −0.359098 0.933300i \(-0.616916\pi\)
−0.359098 + 0.933300i \(0.616916\pi\)
\(194\) −5063.89 −1.87405
\(195\) 0 0
\(196\) 6.98724 0.00254637
\(197\) −1819.95 −0.658203 −0.329102 0.944294i \(-0.606746\pi\)
−0.329102 + 0.944294i \(0.606746\pi\)
\(198\) −900.092 −0.323064
\(199\) −1273.97 −0.453816 −0.226908 0.973916i \(-0.572862\pi\)
−0.226908 + 0.973916i \(0.572862\pi\)
\(200\) −2571.92 −0.909312
\(201\) −2223.06 −0.780111
\(202\) 2171.26 0.756284
\(203\) −2389.40 −0.826125
\(204\) −8.85231 −0.00303817
\(205\) 401.465 0.136778
\(206\) −4354.56 −1.47280
\(207\) −1136.26 −0.381525
\(208\) 0 0
\(209\) 439.664 0.145513
\(210\) 385.188 0.126574
\(211\) −1319.44 −0.430492 −0.215246 0.976560i \(-0.569055\pi\)
−0.215246 + 0.976560i \(0.569055\pi\)
\(212\) 6.22884 0.00201792
\(213\) 1717.72 0.552565
\(214\) −5719.83 −1.82710
\(215\) −1886.98 −0.598564
\(216\) 612.547 0.192956
\(217\) 3415.24 1.06839
\(218\) −4640.35 −1.44167
\(219\) 80.1464 0.0247296
\(220\) 5.12292 0.00156994
\(221\) 0 0
\(222\) −1753.79 −0.530209
\(223\) 1203.45 0.361384 0.180692 0.983540i \(-0.442166\pi\)
0.180692 + 0.983540i \(0.442166\pi\)
\(224\) −25.5829 −0.00763095
\(225\) −1020.29 −0.302309
\(226\) 1846.34 0.543436
\(227\) −4361.23 −1.27518 −0.637589 0.770377i \(-0.720068\pi\)
−0.637589 + 0.770377i \(0.720068\pi\)
\(228\) −1.57612 −0.000457811 0
\(229\) 5384.29 1.55373 0.776864 0.629669i \(-0.216809\pi\)
0.776864 + 0.629669i \(0.216809\pi\)
\(230\) −1214.77 −0.348260
\(231\) 1419.27 0.404247
\(232\) −4062.32 −1.14959
\(233\) −4913.60 −1.38155 −0.690774 0.723070i \(-0.742730\pi\)
−0.690774 + 0.723070i \(0.742730\pi\)
\(234\) 0 0
\(235\) −214.608 −0.0595722
\(236\) −11.6114 −0.00320272
\(237\) −623.394 −0.170860
\(238\) −2621.93 −0.714094
\(239\) 963.718 0.260827 0.130414 0.991460i \(-0.458369\pi\)
0.130414 + 0.991460i \(0.458369\pi\)
\(240\) 651.404 0.175200
\(241\) 1544.48 0.412817 0.206409 0.978466i \(-0.433822\pi\)
0.206409 + 0.978466i \(0.433822\pi\)
\(242\) 209.001 0.0555170
\(243\) 243.000 0.0641500
\(244\) −25.5472 −0.00670284
\(245\) 562.568 0.146698
\(246\) 996.079 0.258161
\(247\) 0 0
\(248\) 5806.38 1.48671
\(249\) −3094.12 −0.787478
\(250\) −2293.52 −0.580221
\(251\) 3768.17 0.947588 0.473794 0.880636i \(-0.342884\pi\)
0.473794 + 0.880636i \(0.342884\pi\)
\(252\) −5.08783 −0.00127184
\(253\) −4475.98 −1.11226
\(254\) −3739.25 −0.923705
\(255\) −712.731 −0.175031
\(256\) −65.0695 −0.0158861
\(257\) 1282.68 0.311327 0.155664 0.987810i \(-0.450248\pi\)
0.155664 + 0.987810i \(0.450248\pi\)
\(258\) −4681.81 −1.12976
\(259\) 2765.38 0.663445
\(260\) 0 0
\(261\) −1611.54 −0.382190
\(262\) −2623.78 −0.618694
\(263\) 5029.74 1.17927 0.589633 0.807671i \(-0.299272\pi\)
0.589633 + 0.807671i \(0.299272\pi\)
\(264\) 2412.95 0.562527
\(265\) 501.506 0.116254
\(266\) −466.824 −0.107604
\(267\) 3688.97 0.845548
\(268\) 31.3926 0.00715526
\(269\) 5628.46 1.27574 0.637868 0.770146i \(-0.279816\pi\)
0.637868 + 0.770146i \(0.279816\pi\)
\(270\) 259.790 0.0585568
\(271\) 3368.39 0.755037 0.377518 0.926002i \(-0.376778\pi\)
0.377518 + 0.926002i \(0.376778\pi\)
\(272\) −4434.03 −0.988429
\(273\) 0 0
\(274\) 7760.45 1.71104
\(275\) −4019.15 −0.881324
\(276\) 16.0456 0.00349938
\(277\) −6507.75 −1.41160 −0.705799 0.708412i \(-0.749412\pi\)
−0.705799 + 0.708412i \(0.749412\pi\)
\(278\) −6873.62 −1.48292
\(279\) 2303.41 0.494272
\(280\) −1032.61 −0.220393
\(281\) −3625.51 −0.769680 −0.384840 0.922983i \(-0.625743\pi\)
−0.384840 + 0.922983i \(0.625743\pi\)
\(282\) −532.465 −0.112439
\(283\) −4635.41 −0.973662 −0.486831 0.873496i \(-0.661847\pi\)
−0.486831 + 0.873496i \(0.661847\pi\)
\(284\) −24.2566 −0.00506819
\(285\) −126.899 −0.0263749
\(286\) 0 0
\(287\) −1570.62 −0.323035
\(288\) −17.2545 −0.00353031
\(289\) −61.5178 −0.0125214
\(290\) −1722.89 −0.348867
\(291\) 5385.35 1.08486
\(292\) −1.13178 −0.000226823 0
\(293\) −2907.07 −0.579634 −0.289817 0.957082i \(-0.593594\pi\)
−0.289817 + 0.957082i \(0.593594\pi\)
\(294\) 1395.79 0.276885
\(295\) −934.879 −0.184511
\(296\) 4701.53 0.923212
\(297\) 957.229 0.187017
\(298\) 4280.99 0.832185
\(299\) 0 0
\(300\) 14.4079 0.00277281
\(301\) 7382.32 1.41365
\(302\) −1646.41 −0.313710
\(303\) −2309.09 −0.437802
\(304\) −789.461 −0.148943
\(305\) −2056.90 −0.386156
\(306\) −1768.36 −0.330362
\(307\) 933.950 0.173627 0.0868133 0.996225i \(-0.472332\pi\)
0.0868133 + 0.996225i \(0.472332\pi\)
\(308\) −20.0421 −0.00370780
\(309\) 4630.98 0.852581
\(310\) 2462.57 0.451176
\(311\) −3633.55 −0.662508 −0.331254 0.943542i \(-0.607472\pi\)
−0.331254 + 0.943542i \(0.607472\pi\)
\(312\) 0 0
\(313\) −7507.26 −1.35570 −0.677852 0.735198i \(-0.737089\pi\)
−0.677852 + 0.735198i \(0.737089\pi\)
\(314\) −73.6071 −0.0132289
\(315\) −409.639 −0.0732716
\(316\) 8.80319 0.00156715
\(317\) 1214.60 0.215202 0.107601 0.994194i \(-0.465683\pi\)
0.107601 + 0.994194i \(0.465683\pi\)
\(318\) 1244.29 0.219423
\(319\) −6348.19 −1.11420
\(320\) −1755.52 −0.306677
\(321\) 6082.92 1.05768
\(322\) 4752.47 0.822500
\(323\) 863.786 0.148800
\(324\) −3.43149 −0.000588391 0
\(325\) 0 0
\(326\) −48.2111 −0.00819069
\(327\) 4934.92 0.834561
\(328\) −2670.28 −0.449517
\(329\) 839.595 0.140694
\(330\) 1023.37 0.170711
\(331\) 2443.20 0.405712 0.202856 0.979209i \(-0.434978\pi\)
0.202856 + 0.979209i \(0.434978\pi\)
\(332\) 43.6933 0.00722283
\(333\) 1865.12 0.306930
\(334\) −11056.6 −1.81134
\(335\) 2527.53 0.412220
\(336\) −2548.44 −0.413777
\(337\) 1890.96 0.305660 0.152830 0.988253i \(-0.451161\pi\)
0.152830 + 0.988253i \(0.451161\pi\)
\(338\) 0 0
\(339\) −1963.54 −0.314587
\(340\) 10.0647 0.00160540
\(341\) 9073.64 1.44095
\(342\) −314.850 −0.0497811
\(343\) −6777.95 −1.06698
\(344\) 12551.0 1.96716
\(345\) 1291.89 0.201602
\(346\) 5575.15 0.866248
\(347\) −2791.57 −0.431872 −0.215936 0.976408i \(-0.569280\pi\)
−0.215936 + 0.976408i \(0.569280\pi\)
\(348\) 22.7571 0.00350549
\(349\) −6917.33 −1.06096 −0.530482 0.847696i \(-0.677989\pi\)
−0.530482 + 0.847696i \(0.677989\pi\)
\(350\) 4267.43 0.651724
\(351\) 0 0
\(352\) −67.9690 −0.0102919
\(353\) 7638.37 1.15170 0.575849 0.817556i \(-0.304672\pi\)
0.575849 + 0.817556i \(0.304672\pi\)
\(354\) −2319.54 −0.348254
\(355\) −1952.99 −0.291982
\(356\) −52.0933 −0.00775545
\(357\) 2788.37 0.413379
\(358\) 7854.42 1.15955
\(359\) −5490.97 −0.807249 −0.403625 0.914925i \(-0.632250\pi\)
−0.403625 + 0.914925i \(0.632250\pi\)
\(360\) −696.443 −0.101960
\(361\) −6705.21 −0.977578
\(362\) 5321.77 0.772669
\(363\) −222.269 −0.0321380
\(364\) 0 0
\(365\) −91.1234 −0.0130674
\(366\) −5103.39 −0.728848
\(367\) 6737.43 0.958287 0.479143 0.877737i \(-0.340947\pi\)
0.479143 + 0.877737i \(0.340947\pi\)
\(368\) 8037.07 1.13848
\(369\) −1059.31 −0.149446
\(370\) 1993.99 0.280169
\(371\) −1962.01 −0.274561
\(372\) −32.5274 −0.00453351
\(373\) −3930.42 −0.545602 −0.272801 0.962070i \(-0.587950\pi\)
−0.272801 + 0.962070i \(0.587950\pi\)
\(374\) −6965.97 −0.963106
\(375\) 2439.12 0.335881
\(376\) 1427.43 0.195782
\(377\) 0 0
\(378\) −1016.36 −0.138296
\(379\) 11897.7 1.61252 0.806258 0.591563i \(-0.201489\pi\)
0.806258 + 0.591563i \(0.201489\pi\)
\(380\) 1.79199 0.000241913 0
\(381\) 3976.61 0.534719
\(382\) −6905.46 −0.924906
\(383\) 4748.31 0.633492 0.316746 0.948510i \(-0.397410\pi\)
0.316746 + 0.948510i \(0.397410\pi\)
\(384\) −4309.64 −0.572722
\(385\) −1613.66 −0.213609
\(386\) 5432.14 0.716292
\(387\) 4979.02 0.653999
\(388\) −76.0485 −0.00995046
\(389\) 11299.5 1.47277 0.736386 0.676561i \(-0.236531\pi\)
0.736386 + 0.676561i \(0.236531\pi\)
\(390\) 0 0
\(391\) −8793.73 −1.13739
\(392\) −3741.82 −0.482119
\(393\) 2790.34 0.358153
\(394\) 5133.95 0.656458
\(395\) 708.776 0.0902845
\(396\) −13.5174 −0.00171534
\(397\) 13853.0 1.75129 0.875646 0.482953i \(-0.160436\pi\)
0.875646 + 0.482953i \(0.160436\pi\)
\(398\) 3593.78 0.452613
\(399\) 496.457 0.0622906
\(400\) 7216.79 0.902099
\(401\) 5434.64 0.676790 0.338395 0.941004i \(-0.390116\pi\)
0.338395 + 0.941004i \(0.390116\pi\)
\(402\) 6271.08 0.778042
\(403\) 0 0
\(404\) 32.6076 0.00401556
\(405\) −276.282 −0.0338977
\(406\) 6740.34 0.823935
\(407\) 7347.09 0.894795
\(408\) 4740.61 0.575233
\(409\) −3472.43 −0.419806 −0.209903 0.977722i \(-0.567315\pi\)
−0.209903 + 0.977722i \(0.567315\pi\)
\(410\) −1132.50 −0.136416
\(411\) −8253.09 −0.990498
\(412\) −65.3959 −0.00781996
\(413\) 3657.46 0.435767
\(414\) 3205.31 0.380513
\(415\) 3517.90 0.416113
\(416\) 0 0
\(417\) 7309.95 0.858441
\(418\) −1240.26 −0.145127
\(419\) −4261.65 −0.496886 −0.248443 0.968647i \(-0.579919\pi\)
−0.248443 + 0.968647i \(0.579919\pi\)
\(420\) 5.78467 0.000672055 0
\(421\) 6235.06 0.721801 0.360900 0.932604i \(-0.382469\pi\)
0.360900 + 0.932604i \(0.382469\pi\)
\(422\) 3722.04 0.429351
\(423\) 566.266 0.0650894
\(424\) −3335.68 −0.382064
\(425\) −7896.22 −0.901231
\(426\) −4845.57 −0.551100
\(427\) 8047.06 0.912001
\(428\) −85.8992 −0.00970115
\(429\) 0 0
\(430\) 5323.05 0.596977
\(431\) 2520.58 0.281698 0.140849 0.990031i \(-0.455017\pi\)
0.140849 + 0.990031i \(0.455017\pi\)
\(432\) −1718.80 −0.191426
\(433\) −1283.89 −0.142494 −0.0712468 0.997459i \(-0.522698\pi\)
−0.0712468 + 0.997459i \(0.522698\pi\)
\(434\) −9634.15 −1.06556
\(435\) 1832.26 0.201954
\(436\) −69.6878 −0.00765468
\(437\) −1565.69 −0.171389
\(438\) −226.087 −0.0246641
\(439\) −16942.0 −1.84190 −0.920951 0.389677i \(-0.872586\pi\)
−0.920951 + 0.389677i \(0.872586\pi\)
\(440\) −2743.44 −0.297246
\(441\) −1484.40 −0.160285
\(442\) 0 0
\(443\) −2163.00 −0.231980 −0.115990 0.993250i \(-0.537004\pi\)
−0.115990 + 0.993250i \(0.537004\pi\)
\(444\) −26.3380 −0.00281519
\(445\) −4194.22 −0.446798
\(446\) −3394.83 −0.360426
\(447\) −4552.75 −0.481739
\(448\) 6868.01 0.724293
\(449\) −1673.35 −0.175880 −0.0879400 0.996126i \(-0.528028\pi\)
−0.0879400 + 0.996126i \(0.528028\pi\)
\(450\) 2878.17 0.301507
\(451\) −4172.85 −0.435680
\(452\) 27.7279 0.00288542
\(453\) 1750.92 0.181602
\(454\) 12302.7 1.27180
\(455\) 0 0
\(456\) 844.046 0.0866800
\(457\) −4652.38 −0.476212 −0.238106 0.971239i \(-0.576527\pi\)
−0.238106 + 0.971239i \(0.576527\pi\)
\(458\) −15188.7 −1.54961
\(459\) 1880.62 0.191241
\(460\) −18.2432 −0.00184912
\(461\) 3460.07 0.349570 0.174785 0.984607i \(-0.444077\pi\)
0.174785 + 0.984607i \(0.444077\pi\)
\(462\) −4003.66 −0.403176
\(463\) −2105.64 −0.211355 −0.105677 0.994400i \(-0.533701\pi\)
−0.105677 + 0.994400i \(0.533701\pi\)
\(464\) 11398.8 1.14047
\(465\) −2618.89 −0.261179
\(466\) 13860.9 1.37789
\(467\) −1415.88 −0.140298 −0.0701489 0.997537i \(-0.522347\pi\)
−0.0701489 + 0.997537i \(0.522347\pi\)
\(468\) 0 0
\(469\) −9888.28 −0.973557
\(470\) 605.393 0.0594143
\(471\) 78.2797 0.00765804
\(472\) 6218.19 0.606388
\(473\) 19613.4 1.90661
\(474\) 1758.55 0.170407
\(475\) −1405.89 −0.135803
\(476\) −39.3756 −0.00379155
\(477\) −1323.28 −0.127020
\(478\) −2718.58 −0.260136
\(479\) −985.594 −0.0940145 −0.0470073 0.998895i \(-0.514968\pi\)
−0.0470073 + 0.998895i \(0.514968\pi\)
\(480\) 19.6177 0.00186546
\(481\) 0 0
\(482\) −4356.88 −0.411723
\(483\) −5054.16 −0.476133
\(484\) 3.13874 0.000294773 0
\(485\) −6122.93 −0.573254
\(486\) −685.486 −0.0639800
\(487\) 14724.7 1.37011 0.685053 0.728494i \(-0.259779\pi\)
0.685053 + 0.728494i \(0.259779\pi\)
\(488\) 13681.1 1.26909
\(489\) 51.2715 0.00474147
\(490\) −1586.96 −0.146310
\(491\) −16301.2 −1.49830 −0.749149 0.662401i \(-0.769537\pi\)
−0.749149 + 0.662401i \(0.769537\pi\)
\(492\) 14.9589 0.00137073
\(493\) −12472.0 −1.13937
\(494\) 0 0
\(495\) −1088.33 −0.0988221
\(496\) −16292.6 −1.47492
\(497\) 7640.53 0.689587
\(498\) 8728.30 0.785391
\(499\) 3230.42 0.289806 0.144903 0.989446i \(-0.453713\pi\)
0.144903 + 0.989446i \(0.453713\pi\)
\(500\) −34.4437 −0.00308074
\(501\) 11758.4 1.04856
\(502\) −10629.7 −0.945076
\(503\) −3577.57 −0.317129 −0.158565 0.987349i \(-0.550687\pi\)
−0.158565 + 0.987349i \(0.550687\pi\)
\(504\) 2724.65 0.240804
\(505\) 2625.35 0.231340
\(506\) 12626.4 1.10931
\(507\) 0 0
\(508\) −56.1552 −0.00490450
\(509\) −13864.7 −1.20735 −0.603674 0.797231i \(-0.706297\pi\)
−0.603674 + 0.797231i \(0.706297\pi\)
\(510\) 2010.56 0.174567
\(511\) 356.496 0.0308619
\(512\) 11675.9 1.00783
\(513\) 334.836 0.0288175
\(514\) −3618.34 −0.310502
\(515\) −5265.25 −0.450514
\(516\) −70.3105 −0.00599854
\(517\) 2230.64 0.189756
\(518\) −7800.94 −0.661686
\(519\) −5929.06 −0.501458
\(520\) 0 0
\(521\) −9554.66 −0.803449 −0.401725 0.915760i \(-0.631589\pi\)
−0.401725 + 0.915760i \(0.631589\pi\)
\(522\) 4546.03 0.381177
\(523\) −19809.1 −1.65620 −0.828098 0.560583i \(-0.810577\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(524\) −39.4035 −0.00328502
\(525\) −4538.32 −0.377274
\(526\) −14188.5 −1.17614
\(527\) 17826.5 1.47350
\(528\) −6770.73 −0.558064
\(529\) 3772.38 0.310050
\(530\) −1414.71 −0.115946
\(531\) 2466.78 0.201599
\(532\) −7.01066 −0.000571336 0
\(533\) 0 0
\(534\) −10406.3 −0.843306
\(535\) −6916.05 −0.558891
\(536\) −16811.4 −1.35475
\(537\) −8353.02 −0.671246
\(538\) −15877.5 −1.27235
\(539\) −5847.36 −0.467279
\(540\) 3.90148 0.000310913 0
\(541\) −1011.14 −0.0803551 −0.0401775 0.999193i \(-0.512792\pi\)
−0.0401775 + 0.999193i \(0.512792\pi\)
\(542\) −9501.98 −0.753035
\(543\) −5659.59 −0.447286
\(544\) −133.535 −0.0105244
\(545\) −5610.81 −0.440992
\(546\) 0 0
\(547\) 15745.2 1.23074 0.615372 0.788237i \(-0.289006\pi\)
0.615372 + 0.788237i \(0.289006\pi\)
\(548\) 116.545 0.00908495
\(549\) 5427.35 0.421919
\(550\) 11337.7 0.878987
\(551\) −2220.58 −0.171688
\(552\) −8592.76 −0.662558
\(553\) −2772.89 −0.213229
\(554\) 18357.9 1.40785
\(555\) −2120.57 −0.162186
\(556\) −103.227 −0.00787371
\(557\) 8510.94 0.647433 0.323716 0.946154i \(-0.395068\pi\)
0.323716 + 0.946154i \(0.395068\pi\)
\(558\) −6497.76 −0.492961
\(559\) 0 0
\(560\) 2897.48 0.218645
\(561\) 7408.16 0.557528
\(562\) 10227.3 0.767639
\(563\) 16764.1 1.25493 0.627463 0.778646i \(-0.284093\pi\)
0.627463 + 0.778646i \(0.284093\pi\)
\(564\) −7.99646 −0.000597006 0
\(565\) 2232.47 0.166232
\(566\) 13076.1 0.971080
\(567\) 1080.88 0.0800575
\(568\) 12990.0 0.959589
\(569\) 20755.4 1.52919 0.764597 0.644508i \(-0.222938\pi\)
0.764597 + 0.644508i \(0.222938\pi\)
\(570\) 357.972 0.0263049
\(571\) −23408.6 −1.71562 −0.857810 0.513968i \(-0.828175\pi\)
−0.857810 + 0.513968i \(0.828175\pi\)
\(572\) 0 0
\(573\) 7343.81 0.535414
\(574\) 4430.62 0.322179
\(575\) 14312.6 1.03805
\(576\) 4632.14 0.335080
\(577\) 10387.2 0.749436 0.374718 0.927139i \(-0.377739\pi\)
0.374718 + 0.927139i \(0.377739\pi\)
\(578\) 173.537 0.0124882
\(579\) −5776.97 −0.414650
\(580\) −25.8740 −0.00185234
\(581\) −13762.8 −0.982752
\(582\) −15191.7 −1.08198
\(583\) −5212.68 −0.370304
\(584\) 606.092 0.0429457
\(585\) 0 0
\(586\) 8200.63 0.578097
\(587\) −2898.42 −0.203800 −0.101900 0.994795i \(-0.532492\pi\)
−0.101900 + 0.994795i \(0.532492\pi\)
\(588\) 20.9617 0.00147015
\(589\) 3173.94 0.222037
\(590\) 2637.23 0.184022
\(591\) −5459.85 −0.380014
\(592\) −13192.4 −0.915888
\(593\) 8805.33 0.609766 0.304883 0.952390i \(-0.401383\pi\)
0.304883 + 0.952390i \(0.401383\pi\)
\(594\) −2700.27 −0.186521
\(595\) −3170.27 −0.218434
\(596\) 64.2910 0.00441856
\(597\) −3821.91 −0.262011
\(598\) 0 0
\(599\) −20236.8 −1.38039 −0.690194 0.723624i \(-0.742475\pi\)
−0.690194 + 0.723624i \(0.742475\pi\)
\(600\) −7715.77 −0.524992
\(601\) −9884.92 −0.670906 −0.335453 0.942057i \(-0.608889\pi\)
−0.335453 + 0.942057i \(0.608889\pi\)
\(602\) −20825.0 −1.40991
\(603\) −6669.17 −0.450397
\(604\) −24.7255 −0.00166567
\(605\) 252.711 0.0169821
\(606\) 6513.78 0.436641
\(607\) 22919.3 1.53256 0.766281 0.642505i \(-0.222105\pi\)
0.766281 + 0.642505i \(0.222105\pi\)
\(608\) −23.7754 −0.00158589
\(609\) −7168.21 −0.476963
\(610\) 5802.36 0.385132
\(611\) 0 0
\(612\) −26.5569 −0.00175409
\(613\) 916.052 0.0603572 0.0301786 0.999545i \(-0.490392\pi\)
0.0301786 + 0.999545i \(0.490392\pi\)
\(614\) −2634.61 −0.173166
\(615\) 1204.40 0.0789690
\(616\) 10733.0 0.702019
\(617\) −30179.4 −1.96917 −0.984584 0.174915i \(-0.944035\pi\)
−0.984584 + 0.174915i \(0.944035\pi\)
\(618\) −13063.7 −0.850320
\(619\) 5571.12 0.361748 0.180874 0.983506i \(-0.442107\pi\)
0.180874 + 0.983506i \(0.442107\pi\)
\(620\) 36.9824 0.00239556
\(621\) −3408.78 −0.220273
\(622\) 10250.0 0.660751
\(623\) 16408.8 1.05522
\(624\) 0 0
\(625\) 11397.5 0.729443
\(626\) 21177.5 1.35211
\(627\) 1318.99 0.0840120
\(628\) −1.10542 −7.02403e−5 0
\(629\) 14434.5 0.915007
\(630\) 1155.56 0.0730773
\(631\) −11360.1 −0.716703 −0.358352 0.933587i \(-0.616661\pi\)
−0.358352 + 0.933587i \(0.616661\pi\)
\(632\) −4714.30 −0.296717
\(633\) −3958.32 −0.248545
\(634\) −3426.31 −0.214631
\(635\) −4521.26 −0.282552
\(636\) 18.6865 0.00116504
\(637\) 0 0
\(638\) 17907.8 1.11125
\(639\) 5153.17 0.319024
\(640\) 4899.89 0.302633
\(641\) −24709.8 −1.52259 −0.761293 0.648408i \(-0.775435\pi\)
−0.761293 + 0.648408i \(0.775435\pi\)
\(642\) −17159.5 −1.05488
\(643\) 15226.6 0.933869 0.466934 0.884292i \(-0.345358\pi\)
0.466934 + 0.884292i \(0.345358\pi\)
\(644\) 71.3717 0.00436714
\(645\) −5660.95 −0.345581
\(646\) −2436.68 −0.148405
\(647\) 16706.6 1.01515 0.507576 0.861607i \(-0.330542\pi\)
0.507576 + 0.861607i \(0.330542\pi\)
\(648\) 1837.64 0.111403
\(649\) 9717.18 0.587724
\(650\) 0 0
\(651\) 10245.7 0.616838
\(652\) −0.724024 −4.34892e−5 0
\(653\) −28205.8 −1.69032 −0.845158 0.534516i \(-0.820494\pi\)
−0.845158 + 0.534516i \(0.820494\pi\)
\(654\) −13921.0 −0.832349
\(655\) −3172.51 −0.189252
\(656\) 7492.77 0.445951
\(657\) 240.439 0.0142777
\(658\) −2368.44 −0.140321
\(659\) −11424.2 −0.675299 −0.337649 0.941272i \(-0.609632\pi\)
−0.337649 + 0.941272i \(0.609632\pi\)
\(660\) 15.3688 0.000906407 0
\(661\) −26518.2 −1.56042 −0.780211 0.625516i \(-0.784888\pi\)
−0.780211 + 0.625516i \(0.784888\pi\)
\(662\) −6892.10 −0.404636
\(663\) 0 0
\(664\) −23398.7 −1.36754
\(665\) −564.453 −0.0329151
\(666\) −5261.36 −0.306116
\(667\) 22606.5 1.31234
\(668\) −166.045 −0.00961749
\(669\) 3610.34 0.208645
\(670\) −7129.98 −0.411127
\(671\) 21379.5 1.23002
\(672\) −76.7488 −0.00440573
\(673\) 22816.8 1.30687 0.653434 0.756984i \(-0.273328\pi\)
0.653434 + 0.756984i \(0.273328\pi\)
\(674\) −5334.27 −0.304849
\(675\) −3060.88 −0.174538
\(676\) 0 0
\(677\) 17737.6 1.00696 0.503478 0.864008i \(-0.332053\pi\)
0.503478 + 0.864008i \(0.332053\pi\)
\(678\) 5539.01 0.313753
\(679\) 23954.3 1.35388
\(680\) −5389.89 −0.303960
\(681\) −13083.7 −0.736224
\(682\) −25596.1 −1.43713
\(683\) −18878.4 −1.05763 −0.528816 0.848737i \(-0.677364\pi\)
−0.528816 + 0.848737i \(0.677364\pi\)
\(684\) −4.72835 −0.000264317 0
\(685\) 9383.45 0.523391
\(686\) 19120.1 1.06415
\(687\) 16152.9 0.897045
\(688\) −35217.8 −1.95155
\(689\) 0 0
\(690\) −3644.32 −0.201068
\(691\) 11692.8 0.643726 0.321863 0.946786i \(-0.395691\pi\)
0.321863 + 0.946786i \(0.395691\pi\)
\(692\) 83.7265 0.00459943
\(693\) 4257.81 0.233392
\(694\) 7874.82 0.430726
\(695\) −8311.14 −0.453611
\(696\) −12186.9 −0.663714
\(697\) −8198.19 −0.445522
\(698\) 19513.3 1.05815
\(699\) −14740.8 −0.797638
\(700\) 64.0873 0.00346039
\(701\) 20658.1 1.11304 0.556522 0.830833i \(-0.312135\pi\)
0.556522 + 0.830833i \(0.312135\pi\)
\(702\) 0 0
\(703\) 2569.99 0.137879
\(704\) 18247.0 0.976861
\(705\) −643.823 −0.0343940
\(706\) −21547.3 −1.14864
\(707\) −10271.0 −0.546365
\(708\) −34.8343 −0.00184909
\(709\) −20695.8 −1.09626 −0.548128 0.836394i \(-0.684659\pi\)
−0.548128 + 0.836394i \(0.684659\pi\)
\(710\) 5509.23 0.291208
\(711\) −1870.18 −0.0986461
\(712\) 27897.1 1.46838
\(713\) −32312.1 −1.69719
\(714\) −7865.79 −0.412283
\(715\) 0 0
\(716\) 117.956 0.00615674
\(717\) 2891.16 0.150589
\(718\) 15489.6 0.805109
\(719\) −26299.2 −1.36411 −0.682053 0.731302i \(-0.738913\pi\)
−0.682053 + 0.731302i \(0.738913\pi\)
\(720\) 1954.21 0.101152
\(721\) 20598.9 1.06400
\(722\) 18914.9 0.974986
\(723\) 4633.45 0.238340
\(724\) 79.9213 0.00410256
\(725\) 20299.3 1.03986
\(726\) 627.004 0.0320528
\(727\) −16915.6 −0.862952 −0.431476 0.902125i \(-0.642007\pi\)
−0.431476 + 0.902125i \(0.642007\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 257.053 0.0130328
\(731\) 38533.5 1.94967
\(732\) −76.6417 −0.00386989
\(733\) −13203.1 −0.665302 −0.332651 0.943050i \(-0.607943\pi\)
−0.332651 + 0.943050i \(0.607943\pi\)
\(734\) −19005.8 −0.955746
\(735\) 1687.70 0.0846964
\(736\) 242.044 0.0121221
\(737\) −26271.3 −1.31305
\(738\) 2988.24 0.149050
\(739\) −16813.9 −0.836954 −0.418477 0.908227i \(-0.637436\pi\)
−0.418477 + 0.908227i \(0.637436\pi\)
\(740\) 29.9453 0.00148758
\(741\) 0 0
\(742\) 5534.68 0.273834
\(743\) −32682.1 −1.61371 −0.806857 0.590747i \(-0.798833\pi\)
−0.806857 + 0.590747i \(0.798833\pi\)
\(744\) 17419.1 0.858355
\(745\) 5176.30 0.254557
\(746\) 11087.4 0.544156
\(747\) −9282.37 −0.454651
\(748\) −104.613 −0.00511370
\(749\) 27057.2 1.31996
\(750\) −6880.57 −0.334991
\(751\) 2157.76 0.104844 0.0524219 0.998625i \(-0.483306\pi\)
0.0524219 + 0.998625i \(0.483306\pi\)
\(752\) −4005.35 −0.194229
\(753\) 11304.5 0.547090
\(754\) 0 0
\(755\) −1990.74 −0.0959606
\(756\) −15.2635 −0.000734296 0
\(757\) 13769.6 0.661117 0.330558 0.943786i \(-0.392763\pi\)
0.330558 + 0.943786i \(0.392763\pi\)
\(758\) −33562.6 −1.60824
\(759\) −13427.9 −0.642165
\(760\) −959.648 −0.0458028
\(761\) 16905.2 0.805272 0.402636 0.915360i \(-0.368094\pi\)
0.402636 + 0.915360i \(0.368094\pi\)
\(762\) −11217.7 −0.533302
\(763\) 21950.8 1.04151
\(764\) −103.705 −0.00491087
\(765\) −2138.19 −0.101054
\(766\) −13394.7 −0.631813
\(767\) 0 0
\(768\) −195.208 −0.00917185
\(769\) −1153.35 −0.0540845 −0.0270422 0.999634i \(-0.508609\pi\)
−0.0270422 + 0.999634i \(0.508609\pi\)
\(770\) 4552.01 0.213043
\(771\) 3848.03 0.179745
\(772\) 81.5787 0.00380322
\(773\) 20713.9 0.963811 0.481905 0.876223i \(-0.339945\pi\)
0.481905 + 0.876223i \(0.339945\pi\)
\(774\) −14045.4 −0.652265
\(775\) −29014.3 −1.34480
\(776\) 40725.7 1.88398
\(777\) 8296.14 0.383040
\(778\) −31875.1 −1.46887
\(779\) −1459.65 −0.0671341
\(780\) 0 0
\(781\) 20299.4 0.930053
\(782\) 24806.5 1.13437
\(783\) −4834.61 −0.220658
\(784\) 10499.5 0.478294
\(785\) −89.0010 −0.00404660
\(786\) −7871.35 −0.357203
\(787\) −10870.2 −0.492353 −0.246177 0.969225i \(-0.579174\pi\)
−0.246177 + 0.969225i \(0.579174\pi\)
\(788\) 77.1006 0.00348553
\(789\) 15089.2 0.680850
\(790\) −1999.41 −0.0900451
\(791\) −8733.95 −0.392596
\(792\) 7238.86 0.324775
\(793\) 0 0
\(794\) −39078.4 −1.74665
\(795\) 1504.52 0.0671192
\(796\) 53.9706 0.00240319
\(797\) 12026.6 0.534511 0.267255 0.963626i \(-0.413883\pi\)
0.267255 + 0.963626i \(0.413883\pi\)
\(798\) −1400.47 −0.0621255
\(799\) 4382.43 0.194042
\(800\) 217.341 0.00960519
\(801\) 11066.9 0.488177
\(802\) −15330.7 −0.674996
\(803\) 947.142 0.0416238
\(804\) 94.1778 0.00413109
\(805\) 5746.39 0.251594
\(806\) 0 0
\(807\) 16885.4 0.736547
\(808\) −17462.1 −0.760289
\(809\) −32384.0 −1.40737 −0.703685 0.710512i \(-0.748463\pi\)
−0.703685 + 0.710512i \(0.748463\pi\)
\(810\) 779.371 0.0338078
\(811\) −42506.8 −1.84046 −0.920231 0.391375i \(-0.872000\pi\)
−0.920231 + 0.391375i \(0.872000\pi\)
\(812\) 101.225 0.00437476
\(813\) 10105.2 0.435921
\(814\) −20725.6 −0.892423
\(815\) −58.2937 −0.00250545
\(816\) −13302.1 −0.570670
\(817\) 6860.72 0.293790
\(818\) 9795.47 0.418693
\(819\) 0 0
\(820\) −17.0077 −0.000724312 0
\(821\) 20052.6 0.852427 0.426213 0.904623i \(-0.359847\pi\)
0.426213 + 0.904623i \(0.359847\pi\)
\(822\) 23281.4 0.987872
\(823\) −31625.5 −1.33948 −0.669742 0.742594i \(-0.733595\pi\)
−0.669742 + 0.742594i \(0.733595\pi\)
\(824\) 35020.9 1.48060
\(825\) −12057.5 −0.508832
\(826\) −10317.4 −0.434612
\(827\) −5440.97 −0.228780 −0.114390 0.993436i \(-0.536491\pi\)
−0.114390 + 0.993436i \(0.536491\pi\)
\(828\) 48.1367 0.00202037
\(829\) −24845.8 −1.04093 −0.520465 0.853883i \(-0.674241\pi\)
−0.520465 + 0.853883i \(0.674241\pi\)
\(830\) −9923.75 −0.415010
\(831\) −19523.2 −0.814986
\(832\) 0 0
\(833\) −11488.0 −0.477834
\(834\) −20620.9 −0.856165
\(835\) −13368.9 −0.554071
\(836\) −18.6260 −0.000770566 0
\(837\) 6910.24 0.285368
\(838\) 12021.8 0.495568
\(839\) −45887.8 −1.88823 −0.944113 0.329621i \(-0.893079\pi\)
−0.944113 + 0.329621i \(0.893079\pi\)
\(840\) −3097.82 −0.127244
\(841\) 7673.40 0.314625
\(842\) −17588.7 −0.719887
\(843\) −10876.5 −0.444375
\(844\) 55.8969 0.00227968
\(845\) 0 0
\(846\) −1597.40 −0.0649168
\(847\) −988.665 −0.0401073
\(848\) 9359.88 0.379033
\(849\) −13906.2 −0.562144
\(850\) 22274.7 0.898842
\(851\) −26163.7 −1.05391
\(852\) −72.7698 −0.00292612
\(853\) 33655.3 1.35092 0.675460 0.737397i \(-0.263945\pi\)
0.675460 + 0.737397i \(0.263945\pi\)
\(854\) −22700.2 −0.909583
\(855\) −380.696 −0.0152275
\(856\) 46000.9 1.83677
\(857\) 41365.6 1.64880 0.824399 0.566009i \(-0.191513\pi\)
0.824399 + 0.566009i \(0.191513\pi\)
\(858\) 0 0
\(859\) 14866.0 0.590480 0.295240 0.955423i \(-0.404600\pi\)
0.295240 + 0.955423i \(0.404600\pi\)
\(860\) 79.9404 0.00316970
\(861\) −4711.87 −0.186504
\(862\) −7110.37 −0.280952
\(863\) −15469.1 −0.610167 −0.305083 0.952326i \(-0.598684\pi\)
−0.305083 + 0.952326i \(0.598684\pi\)
\(864\) −51.7634 −0.00203822
\(865\) 6741.12 0.264977
\(866\) 3621.76 0.142116
\(867\) −184.553 −0.00722925
\(868\) −144.684 −0.00565770
\(869\) −7367.05 −0.287584
\(870\) −5168.67 −0.201419
\(871\) 0 0
\(872\) 37319.4 1.44930
\(873\) 16156.0 0.626345
\(874\) 4416.69 0.170934
\(875\) 10849.3 0.419171
\(876\) −3.39533 −0.000130956 0
\(877\) −38100.7 −1.46701 −0.733505 0.679684i \(-0.762117\pi\)
−0.733505 + 0.679684i \(0.762117\pi\)
\(878\) 47792.0 1.83702
\(879\) −8721.21 −0.334652
\(880\) 7698.06 0.294888
\(881\) 3879.84 0.148371 0.0741857 0.997244i \(-0.476364\pi\)
0.0741857 + 0.997244i \(0.476364\pi\)
\(882\) 4187.38 0.159860
\(883\) 13046.5 0.497225 0.248613 0.968603i \(-0.420025\pi\)
0.248613 + 0.968603i \(0.420025\pi\)
\(884\) 0 0
\(885\) −2804.64 −0.106528
\(886\) 6101.66 0.231365
\(887\) −32833.2 −1.24288 −0.621438 0.783463i \(-0.713451\pi\)
−0.621438 + 0.783463i \(0.713451\pi\)
\(888\) 14104.6 0.533017
\(889\) 17688.2 0.667315
\(890\) 11831.6 0.445613
\(891\) 2871.69 0.107974
\(892\) −50.9829 −0.00191372
\(893\) 780.274 0.0292395
\(894\) 12843.0 0.480462
\(895\) 9497.06 0.354695
\(896\) −19169.5 −0.714741
\(897\) 0 0
\(898\) 4720.39 0.175414
\(899\) −45827.6 −1.70015
\(900\) 43.2238 0.00160088
\(901\) −10241.1 −0.378668
\(902\) 11771.3 0.434525
\(903\) 22147.0 0.816173
\(904\) −14848.9 −0.546314
\(905\) 6434.74 0.236352
\(906\) −4939.23 −0.181120
\(907\) 21346.3 0.781468 0.390734 0.920504i \(-0.372221\pi\)
0.390734 + 0.920504i \(0.372221\pi\)
\(908\) 184.760 0.00675272
\(909\) −6927.28 −0.252765
\(910\) 0 0
\(911\) 7792.71 0.283407 0.141704 0.989909i \(-0.454742\pi\)
0.141704 + 0.989909i \(0.454742\pi\)
\(912\) −2368.38 −0.0859923
\(913\) −36565.2 −1.32545
\(914\) 13124.0 0.474950
\(915\) −6170.69 −0.222947
\(916\) −228.101 −0.00822779
\(917\) 12411.6 0.446965
\(918\) −5305.09 −0.190734
\(919\) −31352.0 −1.12536 −0.562681 0.826674i \(-0.690230\pi\)
−0.562681 + 0.826674i \(0.690230\pi\)
\(920\) 9769.64 0.350104
\(921\) 2801.85 0.100243
\(922\) −9760.61 −0.348643
\(923\) 0 0
\(924\) −60.1262 −0.00214070
\(925\) −23493.4 −0.835089
\(926\) 5939.85 0.210794
\(927\) 13893.0 0.492238
\(928\) 343.286 0.0121432
\(929\) 12675.3 0.447646 0.223823 0.974630i \(-0.428146\pi\)
0.223823 + 0.974630i \(0.428146\pi\)
\(930\) 7387.71 0.260487
\(931\) −2045.39 −0.0720032
\(932\) 208.161 0.00731601
\(933\) −10900.7 −0.382499
\(934\) 3994.09 0.139926
\(935\) −8422.80 −0.294604
\(936\) 0 0
\(937\) −12308.5 −0.429138 −0.214569 0.976709i \(-0.568835\pi\)
−0.214569 + 0.976709i \(0.568835\pi\)
\(938\) 27894.1 0.970976
\(939\) −22521.8 −0.782717
\(940\) 9.09167 0.000315466 0
\(941\) 20465.2 0.708978 0.354489 0.935060i \(-0.384655\pi\)
0.354489 + 0.935060i \(0.384655\pi\)
\(942\) −220.821 −0.00763774
\(943\) 14859.9 0.513155
\(944\) −17448.2 −0.601578
\(945\) −1228.92 −0.0423034
\(946\) −55328.0 −1.90155
\(947\) 11170.8 0.383318 0.191659 0.981462i \(-0.438613\pi\)
0.191659 + 0.981462i \(0.438613\pi\)
\(948\) 26.4096 0.000904792 0
\(949\) 0 0
\(950\) 3965.91 0.135443
\(951\) 3643.81 0.124247
\(952\) 21086.5 0.717876
\(953\) 14109.4 0.479587 0.239794 0.970824i \(-0.422920\pi\)
0.239794 + 0.970824i \(0.422920\pi\)
\(954\) 3732.87 0.126684
\(955\) −8349.64 −0.282919
\(956\) −40.8271 −0.00138122
\(957\) −19044.6 −0.643285
\(958\) 2780.29 0.0937653
\(959\) −36710.2 −1.23612
\(960\) −5266.57 −0.177060
\(961\) 35711.6 1.19874
\(962\) 0 0
\(963\) 18248.8 0.610652
\(964\) −65.4307 −0.00218608
\(965\) 6568.19 0.219106
\(966\) 14257.4 0.474870
\(967\) −40785.8 −1.35634 −0.678171 0.734904i \(-0.737227\pi\)
−0.678171 + 0.734904i \(0.737227\pi\)
\(968\) −1680.87 −0.0558110
\(969\) 2591.36 0.0859096
\(970\) 17272.4 0.571734
\(971\) 4230.29 0.139811 0.0699055 0.997554i \(-0.477730\pi\)
0.0699055 + 0.997554i \(0.477730\pi\)
\(972\) −10.2945 −0.000339708 0
\(973\) 32515.1 1.07131
\(974\) −41537.4 −1.36647
\(975\) 0 0
\(976\) −38389.0 −1.25902
\(977\) −59925.0 −1.96230 −0.981151 0.193242i \(-0.938100\pi\)
−0.981151 + 0.193242i \(0.938100\pi\)
\(978\) −144.633 −0.00472890
\(979\) 43594.9 1.42319
\(980\) −23.8327 −0.000776844 0
\(981\) 14804.8 0.481834
\(982\) 45984.6 1.49433
\(983\) −23333.3 −0.757087 −0.378543 0.925584i \(-0.623575\pi\)
−0.378543 + 0.925584i \(0.623575\pi\)
\(984\) −8010.83 −0.259528
\(985\) 6207.64 0.200804
\(986\) 35182.5 1.13635
\(987\) 2518.79 0.0812298
\(988\) 0 0
\(989\) −69845.2 −2.24565
\(990\) 3070.11 0.0985601
\(991\) −32523.5 −1.04253 −0.521263 0.853396i \(-0.674539\pi\)
−0.521263 + 0.853396i \(0.674539\pi\)
\(992\) −490.668 −0.0157044
\(993\) 7329.61 0.234238
\(994\) −21553.4 −0.687759
\(995\) 4345.37 0.138450
\(996\) 131.080 0.00417010
\(997\) −7114.35 −0.225992 −0.112996 0.993595i \(-0.536045\pi\)
−0.112996 + 0.993595i \(0.536045\pi\)
\(998\) −9112.77 −0.289038
\(999\) 5595.35 0.177206
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.a.p.1.3 yes 9
3.2 odd 2 1521.4.a.bf.1.7 9
13.5 odd 4 507.4.b.k.337.13 18
13.8 odd 4 507.4.b.k.337.6 18
13.12 even 2 507.4.a.o.1.7 9
39.38 odd 2 1521.4.a.bi.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.7 9 13.12 even 2
507.4.a.p.1.3 yes 9 1.1 even 1 trivial
507.4.b.k.337.6 18 13.8 odd 4
507.4.b.k.337.13 18 13.5 odd 4
1521.4.a.bf.1.7 9 3.2 odd 2
1521.4.a.bi.1.3 9 39.38 odd 2