Properties

Label 1521.4.a.bf.1.7
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
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: no (minimal twist has level 507)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(5.06791\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +O(q^{10})\) \(q+2.82093 q^{2} -0.0423641 q^{4} +3.41089 q^{5} +13.3442 q^{7} -22.6869 q^{8} +9.62187 q^{10} -35.4529 q^{11} +37.6430 q^{14} -63.6593 q^{16} -69.6526 q^{17} +12.4014 q^{19} -0.144499 q^{20} -100.010 q^{22} +126.251 q^{23} -113.366 q^{25} -0.565314 q^{28} +179.060 q^{29} +255.935 q^{31} +1.91716 q^{32} -196.485 q^{34} +45.5155 q^{35} +207.235 q^{37} +34.9833 q^{38} -77.3825 q^{40} +117.701 q^{41} +553.224 q^{43} +1.50193 q^{44} +356.146 q^{46} -62.9185 q^{47} -164.933 q^{49} -319.797 q^{50} +147.031 q^{53} -120.926 q^{55} -302.738 q^{56} +505.115 q^{58} -274.087 q^{59} +603.039 q^{61} +721.974 q^{62} +514.683 q^{64} -741.019 q^{67} +2.95077 q^{68} +128.396 q^{70} -572.574 q^{71} +26.7155 q^{73} +584.595 q^{74} -0.525372 q^{76} -473.090 q^{77} -207.798 q^{79} -217.135 q^{80} +332.026 q^{82} +1031.37 q^{83} -237.577 q^{85} +1560.60 q^{86} +804.318 q^{88} -1229.66 q^{89} -5.34852 q^{92} -177.488 q^{94} +42.2996 q^{95} +1795.12 q^{97} -465.264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8} - 54 q^{10} - 85 q^{11} - 158 q^{14} + 216 q^{16} - 178 q^{17} + 352 q^{19} - 402 q^{20} - 630 q^{22} - 150 q^{23} - 20 q^{25} + 940 q^{28} + 97 q^{29} + 717 q^{31} - 707 q^{32} + 632 q^{34} + 418 q^{35} + 1108 q^{37} + 660 q^{38} - 1506 q^{40} - 334 q^{41} + 242 q^{43} + 307 q^{44} + 979 q^{46} + 184 q^{47} - 38 q^{49} + 2031 q^{50} + 151 q^{53} + 2064 q^{55} - 2276 q^{56} + 1161 q^{58} - 537 q^{59} - 1340 q^{61} - 347 q^{62} + 893 q^{64} + 2308 q^{67} - 2785 q^{68} - 1420 q^{70} - 96 q^{71} + 2505 q^{73} + 1191 q^{74} + 2409 q^{76} + 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 1517 q^{82} - 1539 q^{83} + 4296 q^{85} + 3763 q^{86} - 3716 q^{88} + 592 q^{89} - 515 q^{92} - 692 q^{94} - 4158 q^{95} + 1445 q^{97} - 1457 q^{98}+O(q^{100}) \) 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 \(-0.333820\pi\)
0.498674 + 0.866789i \(0.333820\pi\)
\(3\) 0 0
\(4\) −0.0423641 −0.00529552
\(5\) 3.41089 0.305079 0.152539 0.988297i \(-0.451255\pi\)
0.152539 + 0.988297i \(0.451255\pi\)
\(6\) 0 0
\(7\) 13.3442 0.720518 0.360259 0.932852i \(-0.382688\pi\)
0.360259 + 0.932852i \(0.382688\pi\)
\(8\) −22.6869 −1.00263
\(9\) 0 0
\(10\) 9.62187 0.304270
\(11\) −35.4529 −0.971769 −0.485885 0.874023i \(-0.661502\pi\)
−0.485885 + 0.874023i \(0.661502\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 37.6430 0.718607
\(15\) 0 0
\(16\) −63.6593 −0.994676
\(17\) −69.6526 −0.993720 −0.496860 0.867831i \(-0.665514\pi\)
−0.496860 + 0.867831i \(0.665514\pi\)
\(18\) 0 0
\(19\) 12.4014 0.149740 0.0748701 0.997193i \(-0.476146\pi\)
0.0748701 + 0.997193i \(0.476146\pi\)
\(20\) −0.144499 −0.00161555
\(21\) 0 0
\(22\) −100.010 −0.969193
\(23\) 126.251 1.14457 0.572287 0.820053i \(-0.306056\pi\)
0.572287 + 0.820053i \(0.306056\pi\)
\(24\) 0 0
\(25\) −113.366 −0.906927
\(26\) 0 0
\(27\) 0 0
\(28\) −0.565314 −0.00381551
\(29\) 179.060 1.14657 0.573286 0.819356i \(-0.305669\pi\)
0.573286 + 0.819356i \(0.305669\pi\)
\(30\) 0 0
\(31\) 255.935 1.48281 0.741407 0.671055i \(-0.234159\pi\)
0.741407 + 0.671055i \(0.234159\pi\)
\(32\) 1.91716 0.0105909
\(33\) 0 0
\(34\) −196.485 −0.991085
\(35\) 45.5155 0.219815
\(36\) 0 0
\(37\) 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 \(-0.428033\pi\)
0.224169 + 0.974550i \(0.428033\pi\)
\(42\) 0 0
\(43\) 553.224 1.96200 0.980998 0.194016i \(-0.0621515\pi\)
0.980998 + 0.194016i \(0.0621515\pi\)
\(44\) 1.50193 0.00514602
\(45\) 0 0
\(46\) 356.146 1.14154
\(47\) −62.9185 −0.195268 −0.0976341 0.995222i \(-0.531127\pi\)
−0.0976341 + 0.995222i \(0.531127\pi\)
\(48\) 0 0
\(49\) −164.933 −0.480854
\(50\) −319.797 −0.904522
\(51\) 0 0
\(52\) 0 0
\(53\) 147.031 0.381061 0.190531 0.981681i \(-0.438979\pi\)
0.190531 + 0.981681i \(0.438979\pi\)
\(54\) 0 0
\(55\) −120.926 −0.296466
\(56\) −302.738 −0.722413
\(57\) 0 0
\(58\) 505.115 1.14353
\(59\) −274.087 −0.604798 −0.302399 0.953181i \(-0.597787\pi\)
−0.302399 + 0.953181i \(0.597787\pi\)
\(60\) 0 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
\(63\) 0 0
\(64\) 514.683 1.00524
\(65\) 0 0
\(66\) 0 0
\(67\) −741.019 −1.35119 −0.675596 0.737272i \(-0.736113\pi\)
−0.675596 + 0.737272i \(0.736113\pi\)
\(68\) 2.95077 0.00526226
\(69\) 0 0
\(70\) 128.396 0.219232
\(71\) −572.574 −0.957071 −0.478536 0.878068i \(-0.658832\pi\)
−0.478536 + 0.878068i \(0.658832\pi\)
\(72\) 0 0
\(73\) 26.7155 0.0428330 0.0214165 0.999771i \(-0.493182\pi\)
0.0214165 + 0.999771i \(0.493182\pi\)
\(74\) 584.595 0.918349
\(75\) 0 0
\(76\) −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
\(81\) 0 0
\(82\) 332.026 0.447149
\(83\) 1031.37 1.36395 0.681976 0.731374i \(-0.261121\pi\)
0.681976 + 0.731374i \(0.261121\pi\)
\(84\) 0 0
\(85\) −237.577 −0.303163
\(86\) 1560.60 1.95679
\(87\) 0 0
\(88\) 804.318 0.974325
\(89\) −1229.66 −1.46453 −0.732266 0.681019i \(-0.761537\pi\)
−0.732266 + 0.681019i \(0.761537\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.34852 −0.00606111
\(93\) 0 0
\(94\) −177.488 −0.194750
\(95\) 42.2996 0.0456826
\(96\) 0 0
\(97\) 1795.12 1.87903 0.939517 0.342502i \(-0.111274\pi\)
0.939517 + 0.342502i \(0.111274\pi\)
\(98\) −465.264 −0.479579
\(99\) 0 0
\(100\) 4.80265 0.00480265
\(101\) 769.697 0.758295 0.379147 0.925336i \(-0.376217\pi\)
0.379147 + 0.925336i \(0.376217\pi\)
\(102\) 0 0
\(103\) 1543.66 1.47671 0.738357 0.674410i \(-0.235602\pi\)
0.738357 + 0.674410i \(0.235602\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 414.764 0.380051
\(107\) −2027.64 −1.83196 −0.915978 0.401228i \(-0.868583\pi\)
−0.915978 + 0.401228i \(0.868583\pi\)
\(108\) 0 0
\(109\) 1644.97 1.44550 0.722751 0.691108i \(-0.242877\pi\)
0.722751 + 0.691108i \(0.242877\pi\)
\(110\) −341.123 −0.295680
\(111\) 0 0
\(112\) −849.481 −0.716682
\(113\) 654.514 0.544880 0.272440 0.962173i \(-0.412169\pi\)
0.272440 + 0.962173i \(0.412169\pi\)
\(114\) 0 0
\(115\) 430.629 0.349186
\(116\) −7.58571 −0.00607169
\(117\) 0 0
\(118\) −773.179 −0.603194
\(119\) −929.456 −0.715993
\(120\) 0 0
\(121\) −74.0896 −0.0556646
\(122\) 1701.13 1.26240
\(123\) 0 0
\(124\) −10.8425 −0.00785227
\(125\) −813.039 −0.581763
\(126\) 0 0
\(127\) 1325.54 0.926161 0.463080 0.886316i \(-0.346744\pi\)
0.463080 + 0.886316i \(0.346744\pi\)
\(128\) 1436.55 0.991983
\(129\) 0 0
\(130\) 0 0
\(131\) −930.114 −0.620339 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(132\) 0 0
\(133\) 165.486 0.107891
\(134\) −2090.36 −1.34761
\(135\) 0 0
\(136\) 1580.20 0.996333
\(137\) 2751.03 1.71559 0.857797 0.513989i \(-0.171833\pi\)
0.857797 + 0.513989i \(0.171833\pi\)
\(138\) 0 0
\(139\) 2436.65 1.48686 0.743432 0.668812i \(-0.233197\pi\)
0.743432 + 0.668812i \(0.233197\pi\)
\(140\) −1.92822 −0.00116403
\(141\) 0 0
\(142\) −1615.19 −0.954534
\(143\) 0 0
\(144\) 0 0
\(145\) 610.753 0.349795
\(146\) 75.3624 0.0427194
\(147\) 0 0
\(148\) −8.77933 −0.00487606
\(149\) 1517.58 0.834397 0.417198 0.908815i \(-0.363012\pi\)
0.417198 + 0.908815i \(0.363012\pi\)
\(150\) 0 0
\(151\) 583.642 0.314544 0.157272 0.987555i \(-0.449730\pi\)
0.157272 + 0.987555i \(0.449730\pi\)
\(152\) −281.349 −0.150134
\(153\) 0 0
\(154\) −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
\(159\) 0 0
\(160\) 6.53922 0.00323107
\(161\) 1684.72 0.824686
\(162\) 0 0
\(163\) 17.0905 0.00821246 0.00410623 0.999992i \(-0.498693\pi\)
0.00410623 + 0.999992i \(0.498693\pi\)
\(164\) −4.98631 −0.00237418
\(165\) 0 0
\(166\) 2909.43 1.36034
\(167\) −3919.48 −1.81616 −0.908079 0.418799i \(-0.862451\pi\)
−0.908079 + 0.418799i \(0.862451\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −670.188 −0.302359
\(171\) 0 0
\(172\) −23.4368 −0.0103898
\(173\) 1976.35 0.868551 0.434276 0.900780i \(-0.357004\pi\)
0.434276 + 0.900780i \(0.357004\pi\)
\(174\) 0 0
\(175\) −1512.77 −0.653457
\(176\) 2256.91 0.966596
\(177\) 0 0
\(178\) −3468.77 −1.46065
\(179\) 2784.34 1.16263 0.581316 0.813678i \(-0.302538\pi\)
0.581316 + 0.813678i \(0.302538\pi\)
\(180\) 0 0
\(181\) −1886.53 −0.774723 −0.387361 0.921928i \(-0.626613\pi\)
−0.387361 + 0.921928i \(0.626613\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2864.25 −1.14758
\(185\) 706.855 0.280914
\(186\) 0 0
\(187\) 2469.39 0.965666
\(188\) 2.66549 0.00103405
\(189\) 0 0
\(190\) 119.324 0.0455615
\(191\) −2447.94 −0.927364 −0.463682 0.886002i \(-0.653472\pi\)
−0.463682 + 0.886002i \(0.653472\pi\)
\(192\) 0 0
\(193\) −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 \(-0.393254\pi\)
0.329102 + 0.944294i \(0.393254\pi\)
\(198\) 0 0
\(199\) −1273.97 −0.453816 −0.226908 0.973916i \(-0.572862\pi\)
−0.226908 + 0.973916i \(0.572862\pi\)
\(200\) 2571.92 0.909312
\(201\) 0 0
\(202\) 2171.26 0.756284
\(203\) 2389.40 0.826125
\(204\) 0 0
\(205\) 401.465 0.136778
\(206\) 4354.56 1.47280
\(207\) 0 0
\(208\) 0 0
\(209\) −439.664 −0.145513
\(210\) 0 0
\(211\) −1319.44 −0.430492 −0.215246 0.976560i \(-0.569055\pi\)
−0.215246 + 0.976560i \(0.569055\pi\)
\(212\) −6.22884 −0.00201792
\(213\) 0 0
\(214\) −5719.83 −1.82710
\(215\) 1886.98 0.598564
\(216\) 0 0
\(217\) 3415.24 1.06839
\(218\) 4640.35 1.44167
\(219\) 0 0
\(220\) 5.12292 0.00156994
\(221\) 0 0
\(222\) 0 0
\(223\) 1203.45 0.361384 0.180692 0.983540i \(-0.442166\pi\)
0.180692 + 0.983540i \(0.442166\pi\)
\(224\) 25.5829 0.00763095
\(225\) 0 0
\(226\) 1846.34 0.543436
\(227\) 4361.23 1.27518 0.637589 0.770377i \(-0.279932\pi\)
0.637589 + 0.770377i \(0.279932\pi\)
\(228\) 0 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
\(231\) 0 0
\(232\) −4062.32 −1.14959
\(233\) 4913.60 1.38155 0.690774 0.723070i \(-0.257270\pi\)
0.690774 + 0.723070i \(0.257270\pi\)
\(234\) 0 0
\(235\) −214.608 −0.0595722
\(236\) 11.6114 0.00320272
\(237\) 0 0
\(238\) −2621.93 −0.714094
\(239\) −963.718 −0.260827 −0.130414 0.991460i \(-0.541631\pi\)
−0.130414 + 0.991460i \(0.541631\pi\)
\(240\) 0 0
\(241\) 1544.48 0.412817 0.206409 0.978466i \(-0.433822\pi\)
0.206409 + 0.978466i \(0.433822\pi\)
\(242\) −209.001 −0.0555170
\(243\) 0 0
\(244\) −25.5472 −0.00670284
\(245\) −562.568 −0.146698
\(246\) 0 0
\(247\) 0 0
\(248\) −5806.38 −1.48671
\(249\) 0 0
\(250\) −2293.52 −0.580221
\(251\) −3768.17 −0.947588 −0.473794 0.880636i \(-0.657116\pi\)
−0.473794 + 0.880636i \(0.657116\pi\)
\(252\) 0 0
\(253\) −4475.98 −1.11226
\(254\) 3739.25 0.923705
\(255\) 0 0
\(256\) −65.0695 −0.0158861
\(257\) −1282.68 −0.311327 −0.155664 0.987810i \(-0.549752\pi\)
−0.155664 + 0.987810i \(0.549752\pi\)
\(258\) 0 0
\(259\) 2765.38 0.663445
\(260\) 0 0
\(261\) 0 0
\(262\) −2623.78 −0.618694
\(263\) −5029.74 −1.17927 −0.589633 0.807671i \(-0.700728\pi\)
−0.589633 + 0.807671i \(0.700728\pi\)
\(264\) 0 0
\(265\) 501.506 0.116254
\(266\) 466.824 0.107604
\(267\) 0 0
\(268\) 31.3926 0.00715526
\(269\) −5628.46 −1.27574 −0.637868 0.770146i \(-0.720184\pi\)
−0.637868 + 0.770146i \(0.720184\pi\)
\(270\) 0 0
\(271\) 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
\(276\) 0 0
\(277\) −6507.75 −1.41160 −0.705799 0.708412i \(-0.749412\pi\)
−0.705799 + 0.708412i \(0.749412\pi\)
\(278\) 6873.62 1.48292
\(279\) 0 0
\(280\) −1032.61 −0.220393
\(281\) 3625.51 0.769680 0.384840 0.922983i \(-0.374257\pi\)
0.384840 + 0.922983i \(0.374257\pi\)
\(282\) 0 0
\(283\) −4635.41 −0.973662 −0.486831 0.873496i \(-0.661847\pi\)
−0.486831 + 0.873496i \(0.661847\pi\)
\(284\) 24.2566 0.00506819
\(285\) 0 0
\(286\) 0 0
\(287\) 1570.62 0.323035
\(288\) 0 0
\(289\) −61.5178 −0.0125214
\(290\) 1722.89 0.348867
\(291\) 0 0
\(292\) −1.13178 −0.000226823 0
\(293\) 2907.07 0.579634 0.289817 0.957082i \(-0.406406\pi\)
0.289817 + 0.957082i \(0.406406\pi\)
\(294\) 0 0
\(295\) −934.879 −0.184511
\(296\) −4701.53 −0.923212
\(297\) 0 0
\(298\) 4280.99 0.832185
\(299\) 0 0
\(300\) 0 0
\(301\) 7382.32 1.41365
\(302\) 1646.41 0.313710
\(303\) 0 0
\(304\) −789.461 −0.148943
\(305\) 2056.90 0.386156
\(306\) 0 0
\(307\) 933.950 0.173627 0.0868133 0.996225i \(-0.472332\pi\)
0.0868133 + 0.996225i \(0.472332\pi\)
\(308\) 20.0421 0.00370780
\(309\) 0 0
\(310\) 2462.57 0.451176
\(311\) 3633.55 0.662508 0.331254 0.943542i \(-0.392528\pi\)
0.331254 + 0.943542i \(0.392528\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 8.80319 0.00156715
\(317\) −1214.60 −0.215202 −0.107601 0.994194i \(-0.534317\pi\)
−0.107601 + 0.994194i \(0.534317\pi\)
\(318\) 0 0
\(319\) −6348.19 −1.11420
\(320\) 1755.52 0.306677
\(321\) 0 0
\(322\) 4752.47 0.822500
\(323\) −863.786 −0.148800
\(324\) 0 0
\(325\) 0 0
\(326\) 48.2111 0.00819069
\(327\) 0 0
\(328\) −2670.28 −0.449517
\(329\) −839.595 −0.140694
\(330\) 0 0
\(331\) 2443.20 0.405712 0.202856 0.979209i \(-0.434978\pi\)
0.202856 + 0.979209i \(0.434978\pi\)
\(332\) −43.6933 −0.00722283
\(333\) 0 0
\(334\) −11056.6 −1.81134
\(335\) −2527.53 −0.412220
\(336\) 0 0
\(337\) 1890.96 0.305660 0.152830 0.988253i \(-0.451161\pi\)
0.152830 + 0.988253i \(0.451161\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 10.0647 0.00160540
\(341\) −9073.64 −1.44095
\(342\) 0 0
\(343\) −6777.95 −1.06698
\(344\) −12551.0 −1.96716
\(345\) 0 0
\(346\) 5575.15 0.866248
\(347\) 2791.57 0.431872 0.215936 0.976408i \(-0.430720\pi\)
0.215936 + 0.976408i \(0.430720\pi\)
\(348\) 0 0
\(349\) −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 \(-0.695328\pi\)
−0.575849 + 0.817556i \(0.695328\pi\)
\(354\) 0 0
\(355\) −1952.99 −0.291982
\(356\) 52.0933 0.00775545
\(357\) 0 0
\(358\) 7854.42 1.15955
\(359\) 5490.97 0.807249 0.403625 0.914925i \(-0.367750\pi\)
0.403625 + 0.914925i \(0.367750\pi\)
\(360\) 0 0
\(361\) −6705.21 −0.977578
\(362\) −5321.77 −0.772669
\(363\) 0 0
\(364\) 0 0
\(365\) 91.1234 0.0130674
\(366\) 0 0
\(367\) 6737.43 0.958287 0.479143 0.877737i \(-0.340947\pi\)
0.479143 + 0.877737i \(0.340947\pi\)
\(368\) −8037.07 −1.13848
\(369\) 0 0
\(370\) 1993.99 0.280169
\(371\) 1962.01 0.274561
\(372\) 0 0
\(373\) −3930.42 −0.545602 −0.272801 0.962070i \(-0.587950\pi\)
−0.272801 + 0.962070i \(0.587950\pi\)
\(374\) 6965.97 0.963106
\(375\) 0 0
\(376\) 1427.43 0.195782
\(377\) 0 0
\(378\) 0 0
\(379\) 11897.7 1.61252 0.806258 0.591563i \(-0.201489\pi\)
0.806258 + 0.591563i \(0.201489\pi\)
\(380\) −1.79199 −0.000241913 0
\(381\) 0 0
\(382\) −6905.46 −0.924906
\(383\) −4748.31 −0.633492 −0.316746 0.948510i \(-0.602590\pi\)
−0.316746 + 0.948510i \(0.602590\pi\)
\(384\) 0 0
\(385\) −1613.66 −0.213609
\(386\) −5432.14 −0.716292
\(387\) 0 0
\(388\) −76.0485 −0.00995046
\(389\) −11299.5 −1.47277 −0.736386 0.676561i \(-0.763469\pi\)
−0.736386 + 0.676561i \(0.763469\pi\)
\(390\) 0 0
\(391\) −8793.73 −1.13739
\(392\) 3741.82 0.482119
\(393\) 0 0
\(394\) 5133.95 0.656458
\(395\) −708.776 −0.0902845
\(396\) 0 0
\(397\) 13853.0 1.75129 0.875646 0.482953i \(-0.160436\pi\)
0.875646 + 0.482953i \(0.160436\pi\)
\(398\) −3593.78 −0.452613
\(399\) 0 0
\(400\) 7216.79 0.902099
\(401\) −5434.64 −0.676790 −0.338395 0.941004i \(-0.609884\pi\)
−0.338395 + 0.941004i \(0.609884\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −32.6076 −0.00401556
\(405\) 0 0
\(406\) 6740.34 0.823935
\(407\) −7347.09 −0.894795
\(408\) 0 0
\(409\) −3472.43 −0.419806 −0.209903 0.977722i \(-0.567315\pi\)
−0.209903 + 0.977722i \(0.567315\pi\)
\(410\) 1132.50 0.136416
\(411\) 0 0
\(412\) −65.3959 −0.00781996
\(413\) −3657.46 −0.435767
\(414\) 0 0
\(415\) 3517.90 0.416113
\(416\) 0 0
\(417\) 0 0
\(418\) −1240.26 −0.145127
\(419\) 4261.65 0.496886 0.248443 0.968647i \(-0.420081\pi\)
0.248443 + 0.968647i \(0.420081\pi\)
\(420\) 0 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
\(423\) 0 0
\(424\) −3335.68 −0.382064
\(425\) 7896.22 0.901231
\(426\) 0 0
\(427\) 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 \(-0.544983\pi\)
−0.140849 + 0.990031i \(0.544983\pi\)
\(432\) 0 0
\(433\) −1283.89 −0.142494 −0.0712468 0.997459i \(-0.522698\pi\)
−0.0712468 + 0.997459i \(0.522698\pi\)
\(434\) 9634.15 1.06556
\(435\) 0 0
\(436\) −69.6878 −0.00765468
\(437\) 1565.69 0.171389
\(438\) 0 0
\(439\) −16942.0 −1.84190 −0.920951 0.389677i \(-0.872586\pi\)
−0.920951 + 0.389677i \(0.872586\pi\)
\(440\) 2743.44 0.297246
\(441\) 0 0
\(442\) 0 0
\(443\) 2163.00 0.231980 0.115990 0.993250i \(-0.462996\pi\)
0.115990 + 0.993250i \(0.462996\pi\)
\(444\) 0 0
\(445\) −4194.22 −0.446798
\(446\) 3394.83 0.360426
\(447\) 0 0
\(448\) 6868.01 0.724293
\(449\) 1673.35 0.175880 0.0879400 0.996126i \(-0.471972\pi\)
0.0879400 + 0.996126i \(0.471972\pi\)
\(450\) 0 0
\(451\) −4172.85 −0.435680
\(452\) −27.7279 −0.00288542
\(453\) 0 0
\(454\) 12302.7 1.27180
\(455\) 0 0
\(456\) 0 0
\(457\) −4652.38 −0.476212 −0.238106 0.971239i \(-0.576527\pi\)
−0.238106 + 0.971239i \(0.576527\pi\)
\(458\) 15188.7 1.54961
\(459\) 0 0
\(460\) −18.2432 −0.00184912
\(461\) −3460.07 −0.349570 −0.174785 0.984607i \(-0.555923\pi\)
−0.174785 + 0.984607i \(0.555923\pi\)
\(462\) 0 0
\(463\) −2105.64 −0.211355 −0.105677 0.994400i \(-0.533701\pi\)
−0.105677 + 0.994400i \(0.533701\pi\)
\(464\) −11398.8 −1.14047
\(465\) 0 0
\(466\) 13860.9 1.37789
\(467\) 1415.88 0.140298 0.0701489 0.997537i \(-0.477653\pi\)
0.0701489 + 0.997537i \(0.477653\pi\)
\(468\) 0 0
\(469\) −9888.28 −0.973557
\(470\) −605.393 −0.0594143
\(471\) 0 0
\(472\) 6218.19 0.606388
\(473\) −19613.4 −1.90661
\(474\) 0 0
\(475\) −1405.89 −0.135803
\(476\) 39.3756 0.00379155
\(477\) 0 0
\(478\) −2718.58 −0.260136
\(479\) 985.594 0.0940145 0.0470073 0.998895i \(-0.485032\pi\)
0.0470073 + 0.998895i \(0.485032\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4356.88 0.411723
\(483\) 0 0
\(484\) 3.13874 0.000294773 0
\(485\) 6122.93 0.573254
\(486\) 0 0
\(487\) 14724.7 1.37011 0.685053 0.728494i \(-0.259779\pi\)
0.685053 + 0.728494i \(0.259779\pi\)
\(488\) −13681.1 −1.26909
\(489\) 0 0
\(490\) −1586.96 −0.146310
\(491\) 16301.2 1.49830 0.749149 0.662401i \(-0.230463\pi\)
0.749149 + 0.662401i \(0.230463\pi\)
\(492\) 0 0
\(493\) −12472.0 −1.13937
\(494\) 0 0
\(495\) 0 0
\(496\) −16292.6 −1.47492
\(497\) −7640.53 −0.689587
\(498\) 0 0
\(499\) 3230.42 0.289806 0.144903 0.989446i \(-0.453713\pi\)
0.144903 + 0.989446i \(0.453713\pi\)
\(500\) 34.4437 0.00308074
\(501\) 0 0
\(502\) −10629.7 −0.945076
\(503\) 3577.57 0.317129 0.158565 0.987349i \(-0.449313\pi\)
0.158565 + 0.987349i \(0.449313\pi\)
\(504\) 0 0
\(505\) 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 \(-0.293703\pi\)
0.603674 + 0.797231i \(0.293703\pi\)
\(510\) 0 0
\(511\) 356.496 0.0308619
\(512\) −11675.9 −1.00783
\(513\) 0 0
\(514\) −3618.34 −0.310502
\(515\) 5265.25 0.450514
\(516\) 0 0
\(517\) 2230.64 0.189756
\(518\) 7800.94 0.661686
\(519\) 0 0
\(520\) 0 0
\(521\) 9554.66 0.803449 0.401725 0.915760i \(-0.368411\pi\)
0.401725 + 0.915760i \(0.368411\pi\)
\(522\) 0 0
\(523\) −19809.1 −1.65620 −0.828098 0.560583i \(-0.810577\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(524\) 39.4035 0.00328502
\(525\) 0 0
\(526\) −14188.5 −1.17614
\(527\) −17826.5 −1.47350
\(528\) 0 0
\(529\) 3772.38 0.310050
\(530\) 1414.71 0.115946
\(531\) 0 0
\(532\) −7.01066 −0.000571336 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6916.05 −0.558891
\(536\) 16811.4 1.35475
\(537\) 0 0
\(538\) −15877.5 −1.27235
\(539\) 5847.36 0.467279
\(540\) 0 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
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 11337.7 0.878987
\(551\) 2220.58 0.171688
\(552\) 0 0
\(553\) −2772.89 −0.213229
\(554\) −18357.9 −1.40785
\(555\) 0 0
\(556\) −103.227 −0.00787371
\(557\) −8510.94 −0.647433 −0.323716 0.946154i \(-0.604932\pi\)
−0.323716 + 0.946154i \(0.604932\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2897.48 −0.218645
\(561\) 0 0
\(562\) 10227.3 0.767639
\(563\) −16764.1 −1.25493 −0.627463 0.778646i \(-0.715907\pi\)
−0.627463 + 0.778646i \(0.715907\pi\)
\(564\) 0 0
\(565\) 2232.47 0.166232
\(566\) −13076.1 −0.971080
\(567\) 0 0
\(568\) 12990.0 0.959589
\(569\) −20755.4 −1.52919 −0.764597 0.644508i \(-0.777062\pi\)
−0.764597 + 0.644508i \(0.777062\pi\)
\(570\) 0 0
\(571\) −23408.6 −1.71562 −0.857810 0.513968i \(-0.828175\pi\)
−0.857810 + 0.513968i \(0.828175\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 4430.62 0.322179
\(575\) −14312.6 −1.03805
\(576\) 0 0
\(577\) 10387.2 0.749436 0.374718 0.927139i \(-0.377739\pi\)
0.374718 + 0.927139i \(0.377739\pi\)
\(578\) −173.537 −0.0124882
\(579\) 0 0
\(580\) −25.8740 −0.00185234
\(581\) 13762.8 0.982752
\(582\) 0 0
\(583\) −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 \(-0.467508\pi\)
0.101900 + 0.994795i \(0.467508\pi\)
\(588\) 0 0
\(589\) 3173.94 0.222037
\(590\) −2637.23 −0.184022
\(591\) 0 0
\(592\) −13192.4 −0.915888
\(593\) −8805.33 −0.609766 −0.304883 0.952390i \(-0.598617\pi\)
−0.304883 + 0.952390i \(0.598617\pi\)
\(594\) 0 0
\(595\) −3170.27 −0.218434
\(596\) −64.2910 −0.00441856
\(597\) 0 0
\(598\) 0 0
\(599\) 20236.8 1.38039 0.690194 0.723624i \(-0.257525\pi\)
0.690194 + 0.723624i \(0.257525\pi\)
\(600\) 0 0
\(601\) −9884.92 −0.670906 −0.335453 0.942057i \(-0.608889\pi\)
−0.335453 + 0.942057i \(0.608889\pi\)
\(602\) 20825.0 1.40991
\(603\) 0 0
\(604\) −24.7255 −0.00166567
\(605\) −252.711 −0.0169821
\(606\) 0 0
\(607\) 22919.3 1.53256 0.766281 0.642505i \(-0.222105\pi\)
0.766281 + 0.642505i \(0.222105\pi\)
\(608\) 23.7754 0.00158589
\(609\) 0 0
\(610\) 5802.36 0.385132
\(611\) 0 0
\(612\) 0 0
\(613\) 916.052 0.0603572 0.0301786 0.999545i \(-0.490392\pi\)
0.0301786 + 0.999545i \(0.490392\pi\)
\(614\) 2634.61 0.173166
\(615\) 0 0
\(616\) 10733.0 0.702019
\(617\) 30179.4 1.96917 0.984584 0.174915i \(-0.0559650\pi\)
0.984584 + 0.174915i \(0.0559650\pi\)
\(618\) 0 0
\(619\) 5571.12 0.361748 0.180874 0.983506i \(-0.442107\pi\)
0.180874 + 0.983506i \(0.442107\pi\)
\(620\) −36.9824 −0.00239556
\(621\) 0 0
\(622\) 10250.0 0.660751
\(623\) −16408.8 −1.05522
\(624\) 0 0
\(625\) 11397.5 0.729443
\(626\) −21177.5 −1.35211
\(627\) 0 0
\(628\) −1.10542 −7.02403e−5 0
\(629\) −14434.5 −0.915007
\(630\) 0 0
\(631\) −11360.1 −0.716703 −0.358352 0.933587i \(-0.616661\pi\)
−0.358352 + 0.933587i \(0.616661\pi\)
\(632\) 4714.30 0.296717
\(633\) 0 0
\(634\) −3426.31 −0.214631
\(635\) 4521.26 0.282552
\(636\) 0 0
\(637\) 0 0
\(638\) −17907.8 −1.11125
\(639\) 0 0
\(640\) 4899.89 0.302633
\(641\) 24709.8 1.52259 0.761293 0.648408i \(-0.224565\pi\)
0.761293 + 0.648408i \(0.224565\pi\)
\(642\) 0 0
\(643\) 15226.6 0.933869 0.466934 0.884292i \(-0.345358\pi\)
0.466934 + 0.884292i \(0.345358\pi\)
\(644\) −71.3717 −0.00436714
\(645\) 0 0
\(646\) −2436.68 −0.148405
\(647\) −16706.6 −1.01515 −0.507576 0.861607i \(-0.669458\pi\)
−0.507576 + 0.861607i \(0.669458\pi\)
\(648\) 0 0
\(649\) 9717.18 0.587724
\(650\) 0 0
\(651\) 0 0
\(652\) −0.724024 −4.34892e−5 0
\(653\) 28205.8 1.69032 0.845158 0.534516i \(-0.179506\pi\)
0.845158 + 0.534516i \(0.179506\pi\)
\(654\) 0 0
\(655\) −3172.51 −0.189252
\(656\) −7492.77 −0.445951
\(657\) 0 0
\(658\) −2368.44 −0.140321
\(659\) 11424.2 0.675299 0.337649 0.941272i \(-0.390368\pi\)
0.337649 + 0.941272i \(0.390368\pi\)
\(660\) 0 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
\(666\) 0 0
\(667\) 22606.5 1.31234
\(668\) 166.045 0.00961749
\(669\) 0 0
\(670\) −7129.98 −0.411127
\(671\) −21379.5 −1.23002
\(672\) 0 0
\(673\) 22816.8 1.30687 0.653434 0.756984i \(-0.273328\pi\)
0.653434 + 0.756984i \(0.273328\pi\)
\(674\) 5334.27 0.304849
\(675\) 0 0
\(676\) 0 0
\(677\) −17737.6 −1.00696 −0.503478 0.864008i \(-0.667947\pi\)
−0.503478 + 0.864008i \(0.667947\pi\)
\(678\) 0 0
\(679\) 23954.3 1.35388
\(680\) 5389.89 0.303960
\(681\) 0 0
\(682\) −25596.1 −1.43713
\(683\) 18878.4 1.05763 0.528816 0.848737i \(-0.322636\pi\)
0.528816 + 0.848737i \(0.322636\pi\)
\(684\) 0 0
\(685\) 9383.45 0.523391
\(686\) −19120.1 −1.06415
\(687\) 0 0
\(688\) −35217.8 −1.95155
\(689\) 0 0
\(690\) 0 0
\(691\) 11692.8 0.643726 0.321863 0.946786i \(-0.395691\pi\)
0.321863 + 0.946786i \(0.395691\pi\)
\(692\) −83.7265 −0.00459943
\(693\) 0 0
\(694\) 7874.82 0.430726
\(695\) 8311.14 0.453611
\(696\) 0 0
\(697\) −8198.19 −0.445522
\(698\) −19513.3 −1.05815
\(699\) 0 0
\(700\) 64.0873 0.00346039
\(701\) −20658.1 −1.11304 −0.556522 0.830833i \(-0.687865\pi\)
−0.556522 + 0.830833i \(0.687865\pi\)
\(702\) 0 0
\(703\) 2569.99 0.137879
\(704\) −18247.0 −0.976861
\(705\) 0 0
\(706\) −21547.3 −1.14864
\(707\) 10271.0 0.546365
\(708\) 0 0
\(709\) −20695.8 −1.09626 −0.548128 0.836394i \(-0.684659\pi\)
−0.548128 + 0.836394i \(0.684659\pi\)
\(710\) −5509.23 −0.291208
\(711\) 0 0
\(712\) 27897.1 1.46838
\(713\) 32312.1 1.69719
\(714\) 0 0
\(715\) 0 0
\(716\) −117.956 −0.00615674
\(717\) 0 0
\(718\) 15489.6 0.805109
\(719\) 26299.2 1.36411 0.682053 0.731302i \(-0.261087\pi\)
0.682053 + 0.731302i \(0.261087\pi\)
\(720\) 0 0
\(721\) 20598.9 1.06400
\(722\) −18914.9 −0.974986
\(723\) 0 0
\(724\) 79.9213 0.00410256
\(725\) −20299.3 −1.03986
\(726\) 0 0
\(727\) −16915.6 −0.862952 −0.431476 0.902125i \(-0.642007\pi\)
−0.431476 + 0.902125i \(0.642007\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 257.053 0.0130328
\(731\) −38533.5 −1.94967
\(732\) 0 0
\(733\) −13203.1 −0.665302 −0.332651 0.943050i \(-0.607943\pi\)
−0.332651 + 0.943050i \(0.607943\pi\)
\(734\) 19005.8 0.955746
\(735\) 0 0
\(736\) 242.044 0.0121221
\(737\) 26271.3 1.31305
\(738\) 0 0
\(739\) −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 \(-0.201167\pi\)
0.806857 + 0.590747i \(0.201167\pi\)
\(744\) 0 0
\(745\) 5176.30 0.254557
\(746\) −11087.4 −0.544156
\(747\) 0 0
\(748\) −104.613 −0.00511370
\(749\) −27057.2 −1.31996
\(750\) 0 0
\(751\) 2157.76 0.104844 0.0524219 0.998625i \(-0.483306\pi\)
0.0524219 + 0.998625i \(0.483306\pi\)
\(752\) 4005.35 0.194229
\(753\) 0 0
\(754\) 0 0
\(755\) 1990.74 0.0959606
\(756\) 0 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
\(759\) 0 0
\(760\) −959.648 −0.0458028
\(761\) −16905.2 −0.805272 −0.402636 0.915360i \(-0.631906\pi\)
−0.402636 + 0.915360i \(0.631906\pi\)
\(762\) 0 0
\(763\) 21950.8 1.04151
\(764\) 103.705 0.00491087
\(765\) 0 0
\(766\) −13394.7 −0.631813
\(767\) 0 0
\(768\) 0 0
\(769\) −1153.35 −0.0540845 −0.0270422 0.999634i \(-0.508609\pi\)
−0.0270422 + 0.999634i \(0.508609\pi\)
\(770\) −4552.01 −0.213043
\(771\) 0 0
\(772\) 81.5787 0.00380322
\(773\) −20713.9 −0.963811 −0.481905 0.876223i \(-0.660055\pi\)
−0.481905 + 0.876223i \(0.660055\pi\)
\(774\) 0 0
\(775\) −29014.3 −1.34480
\(776\) −40725.7 −1.88398
\(777\) 0 0
\(778\) −31875.1 −1.46887
\(779\) 1459.65 0.0671341
\(780\) 0 0
\(781\) 20299.4 0.930053
\(782\) −24806.5 −1.13437
\(783\) 0 0
\(784\) 10499.5 0.478294
\(785\) 89.0010 0.00404660
\(786\) 0 0
\(787\) −10870.2 −0.492353 −0.246177 0.969225i \(-0.579174\pi\)
−0.246177 + 0.969225i \(0.579174\pi\)
\(788\) −77.1006 −0.00348553
\(789\) 0 0
\(790\) −1999.41 −0.0900451
\(791\) 8733.95 0.392596
\(792\) 0 0
\(793\) 0 0
\(794\) 39078.4 1.74665
\(795\) 0 0
\(796\) 53.9706 0.00240319
\(797\) −12026.6 −0.534511 −0.267255 0.963626i \(-0.586117\pi\)
−0.267255 + 0.963626i \(0.586117\pi\)
\(798\) 0 0
\(799\) 4382.43 0.194042
\(800\) −217.341 −0.00960519
\(801\) 0 0
\(802\) −15330.7 −0.674996
\(803\) −947.142 −0.0416238
\(804\) 0 0
\(805\) 5746.39 0.251594
\(806\) 0 0
\(807\) 0 0
\(808\) −17462.1 −0.760289
\(809\) 32384.0 1.40737 0.703685 0.710512i \(-0.251537\pi\)
0.703685 + 0.710512i \(0.251537\pi\)
\(810\) 0 0
\(811\) −42506.8 −1.84046 −0.920231 0.391375i \(-0.872000\pi\)
−0.920231 + 0.391375i \(0.872000\pi\)
\(812\) −101.225 −0.00437476
\(813\) 0 0
\(814\) −20725.6 −0.892423
\(815\) 58.2937 0.00250545
\(816\) 0 0
\(817\) 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 \(-0.640153\pi\)
−0.426213 + 0.904623i \(0.640153\pi\)
\(822\) 0 0
\(823\) −31625.5 −1.33948 −0.669742 0.742594i \(-0.733595\pi\)
−0.669742 + 0.742594i \(0.733595\pi\)
\(824\) −35020.9 −1.48060
\(825\) 0 0
\(826\) −10317.4 −0.434612
\(827\) 5440.97 0.228780 0.114390 0.993436i \(-0.463509\pi\)
0.114390 + 0.993436i \(0.463509\pi\)
\(828\) 0 0
\(829\) −24845.8 −1.04093 −0.520465 0.853883i \(-0.674241\pi\)
−0.520465 + 0.853883i \(0.674241\pi\)
\(830\) 9923.75 0.415010
\(831\) 0 0
\(832\) 0 0
\(833\) 11488.0 0.477834
\(834\) 0 0
\(835\) −13368.9 −0.554071
\(836\) 18.6260 0.000770566 0
\(837\) 0 0
\(838\) 12021.8 0.495568
\(839\) 45887.8 1.88823 0.944113 0.329621i \(-0.106921\pi\)
0.944113 + 0.329621i \(0.106921\pi\)
\(840\) 0 0
\(841\) 7673.40 0.314625
\(842\) 17588.7 0.719887
\(843\) 0 0
\(844\) 55.8969 0.00227968
\(845\) 0 0
\(846\) 0 0
\(847\) −988.665 −0.0401073
\(848\) −9359.88 −0.379033
\(849\) 0 0
\(850\) 22274.7 0.898842
\(851\) 26163.7 1.05391
\(852\) 0 0
\(853\) 33655.3 1.35092 0.675460 0.737397i \(-0.263945\pi\)
0.675460 + 0.737397i \(0.263945\pi\)
\(854\) 22700.2 0.909583
\(855\) 0 0
\(856\) 46000.9 1.83677
\(857\) −41365.6 −1.64880 −0.824399 0.566009i \(-0.808487\pi\)
−0.824399 + 0.566009i \(0.808487\pi\)
\(858\) 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
\(861\) 0 0
\(862\) −7110.37 −0.280952
\(863\) 15469.1 0.610167 0.305083 0.952326i \(-0.401316\pi\)
0.305083 + 0.952326i \(0.401316\pi\)
\(864\) 0 0
\(865\) 6741.12 0.264977
\(866\) −3621.76 −0.142116
\(867\) 0 0
\(868\) −144.684 −0.00565770
\(869\) 7367.05 0.287584
\(870\) 0 0
\(871\) 0 0
\(872\) −37319.4 −1.44930
\(873\) 0 0
\(874\) 4416.69 0.170934
\(875\) −10849.3 −0.419171
\(876\) 0 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
\(879\) 0 0
\(880\) 7698.06 0.294888
\(881\) −3879.84 −0.148371 −0.0741857 0.997244i \(-0.523636\pi\)
−0.0741857 + 0.997244i \(0.523636\pi\)
\(882\) 0 0
\(883\) 13046.5 0.497225 0.248613 0.968603i \(-0.420025\pi\)
0.248613 + 0.968603i \(0.420025\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6101.66 0.231365
\(887\) 32833.2 1.24288 0.621438 0.783463i \(-0.286549\pi\)
0.621438 + 0.783463i \(0.286549\pi\)
\(888\) 0 0
\(889\) 17688.2 0.667315
\(890\) −11831.6 −0.445613
\(891\) 0 0
\(892\) −50.9829 −0.00191372
\(893\) −780.274 −0.0292395
\(894\) 0 0
\(895\) 9497.06 0.354695
\(896\) 19169.5 0.714741
\(897\) 0 0
\(898\) 4720.39 0.175414
\(899\) 45827.6 1.70015
\(900\) 0 0
\(901\) −10241.1 −0.378668
\(902\) −11771.3 −0.434525
\(903\) 0 0
\(904\) −14848.9 −0.546314
\(905\) −6434.74 −0.236352
\(906\) 0 0
\(907\) 21346.3 0.781468 0.390734 0.920504i \(-0.372221\pi\)
0.390734 + 0.920504i \(0.372221\pi\)
\(908\) −184.760 −0.00675272
\(909\) 0 0
\(910\) 0 0
\(911\) −7792.71 −0.283407 −0.141704 0.989909i \(-0.545258\pi\)
−0.141704 + 0.989909i \(0.545258\pi\)
\(912\) 0 0
\(913\) −36565.2 −1.32545
\(914\) −13124.0 −0.474950
\(915\) 0 0
\(916\) −228.101 −0.00822779
\(917\) −12411.6 −0.446965
\(918\) 0 0
\(919\) −31352.0 −1.12536 −0.562681 0.826674i \(-0.690230\pi\)
−0.562681 + 0.826674i \(0.690230\pi\)
\(920\) −9769.64 −0.350104
\(921\) 0 0
\(922\) −9760.61 −0.348643
\(923\) 0 0
\(924\) 0 0
\(925\) −23493.4 −0.835089
\(926\) −5939.85 −0.210794
\(927\) 0 0
\(928\) 343.286 0.0121432
\(929\) −12675.3 −0.447646 −0.223823 0.974630i \(-0.571854\pi\)
−0.223823 + 0.974630i \(0.571854\pi\)
\(930\) 0 0
\(931\) −2045.39 −0.0720032
\(932\) −208.161 −0.00731601
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 9.09167 0.000315466 0
\(941\) −20465.2 −0.708978 −0.354489 0.935060i \(-0.615345\pi\)
−0.354489 + 0.935060i \(0.615345\pi\)
\(942\) 0 0
\(943\) 14859.9 0.513155
\(944\) 17448.2 0.601578
\(945\) 0 0
\(946\) −55328.0 −1.90155
\(947\) −11170.8 −0.383318 −0.191659 0.981462i \(-0.561387\pi\)
−0.191659 + 0.981462i \(0.561387\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −3965.91 −0.135443
\(951\) 0 0
\(952\) 21086.5 0.717876
\(953\) −14109.4 −0.479587 −0.239794 0.970824i \(-0.577080\pi\)
−0.239794 + 0.970824i \(0.577080\pi\)
\(954\) 0 0
\(955\) −8349.64 −0.282919
\(956\) 40.8271 0.00138122
\(957\) 0 0
\(958\) 2780.29 0.0937653
\(959\) 36710.2 1.23612
\(960\) 0 0
\(961\) 35711.6 1.19874
\(962\) 0 0
\(963\) 0 0
\(964\) −65.4307 −0.00218608
\(965\) −6568.19 −0.219106
\(966\) 0 0
\(967\) −40785.8 −1.35634 −0.678171 0.734904i \(-0.737227\pi\)
−0.678171 + 0.734904i \(0.737227\pi\)
\(968\) 1680.87 0.0558110
\(969\) 0 0
\(970\) 17272.4 0.571734
\(971\) −4230.29 −0.139811 −0.0699055 0.997554i \(-0.522270\pi\)
−0.0699055 + 0.997554i \(0.522270\pi\)
\(972\) 0 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 \(-0.0619004\pi\)
0.981151 + 0.193242i \(0.0619004\pi\)
\(978\) 0 0
\(979\) 43594.9 1.42319
\(980\) 23.8327 0.000776844 0
\(981\) 0 0
\(982\) 45984.6 1.49433
\(983\) 23333.3 0.757087 0.378543 0.925584i \(-0.376425\pi\)
0.378543 + 0.925584i \(0.376425\pi\)
\(984\) 0 0
\(985\) 6207.64 0.200804
\(986\) −35182.5 −1.13635
\(987\) 0 0
\(988\) 0 0
\(989\) 69845.2 2.24565
\(990\) 0 0
\(991\) −32523.5 −1.04253 −0.521263 0.853396i \(-0.674539\pi\)
−0.521263 + 0.853396i \(0.674539\pi\)
\(992\) 490.668 0.0157044
\(993\) 0 0
\(994\) −21553.4 −0.687759
\(995\) −4345.37 −0.138450
\(996\) 0 0
\(997\) −7114.35 −0.225992 −0.112996 0.993595i \(-0.536045\pi\)
−0.112996 + 0.993595i \(0.536045\pi\)
\(998\) 9112.77 0.289038
\(999\) 0 0
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 1521.4.a.bf.1.7 9
3.2 odd 2 507.4.a.p.1.3 yes 9
13.12 even 2 1521.4.a.bi.1.3 9
39.5 even 4 507.4.b.k.337.13 18
39.8 even 4 507.4.b.k.337.6 18
39.38 odd 2 507.4.a.o.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.7 9 39.38 odd 2
507.4.a.p.1.3 yes 9 3.2 odd 2
507.4.b.k.337.6 18 39.8 even 4
507.4.b.k.337.13 18 39.5 even 4
1521.4.a.bf.1.7 9 1.1 even 1 trivial
1521.4.a.bi.1.3 9 13.12 even 2