Properties

Label 1521.4.a.t.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.56155 q^{2} +12.8078 q^{4} -2.80776 q^{5} +9.56155 q^{7} +21.9309 q^{8} +O(q^{10})\) \(q+4.56155 q^{2} +12.8078 q^{4} -2.80776 q^{5} +9.56155 q^{7} +21.9309 q^{8} -12.8078 q^{10} -39.4233 q^{11} +43.6155 q^{14} -2.42329 q^{16} -2.01515 q^{17} -60.1922 q^{19} -35.9612 q^{20} -179.831 q^{22} -4.46876 q^{23} -117.116 q^{25} +122.462 q^{28} -140.693 q^{29} +136.155 q^{31} -186.501 q^{32} -9.19224 q^{34} -26.8466 q^{35} -185.708 q^{37} -274.570 q^{38} -61.5767 q^{40} -310.231 q^{41} +427.471 q^{43} -504.924 q^{44} -20.3845 q^{46} +258.617 q^{47} -251.577 q^{49} -534.233 q^{50} -612.656 q^{53} +110.691 q^{55} +209.693 q^{56} -641.779 q^{58} +517.885 q^{59} -161.311 q^{61} +621.080 q^{62} -831.348 q^{64} -49.8987 q^{67} -25.8096 q^{68} -122.462 q^{70} -279.963 q^{71} +467.732 q^{73} -847.118 q^{74} -770.928 q^{76} -376.948 q^{77} +37.5379 q^{79} +6.80403 q^{80} -1415.14 q^{82} +76.1553 q^{83} +5.65808 q^{85} +1949.93 q^{86} -864.587 q^{88} -202.806 q^{89} -57.2348 q^{92} +1179.70 q^{94} +169.006 q^{95} -1174.37 q^{97} -1147.58 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.56155 1.61275 0.806376 0.591403i \(-0.201426\pi\)
0.806376 + 0.591403i \(0.201426\pi\)
\(3\) 0 0
\(4\) 12.8078 1.60097
\(5\) −2.80776 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 9.56155 0.516275 0.258138 0.966108i \(-0.416891\pi\)
0.258138 + 0.966108i \(0.416891\pi\)
\(8\) 21.9309 0.969217
\(9\) 0 0
\(10\) −12.8078 −0.405017
\(11\) −39.4233 −1.08060 −0.540299 0.841473i \(-0.681689\pi\)
−0.540299 + 0.841473i \(0.681689\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 43.6155 0.832624
\(15\) 0 0
\(16\) −2.42329 −0.0378639
\(17\) −2.01515 −0.0287498 −0.0143749 0.999897i \(-0.504576\pi\)
−0.0143749 + 0.999897i \(0.504576\pi\)
\(18\) 0 0
\(19\) −60.1922 −0.726792 −0.363396 0.931635i \(-0.618383\pi\)
−0.363396 + 0.931635i \(0.618383\pi\)
\(20\) −35.9612 −0.402058
\(21\) 0 0
\(22\) −179.831 −1.74274
\(23\) −4.46876 −0.0405131 −0.0202565 0.999795i \(-0.506448\pi\)
−0.0202565 + 0.999795i \(0.506448\pi\)
\(24\) 0 0
\(25\) −117.116 −0.936932
\(26\) 0 0
\(27\) 0 0
\(28\) 122.462 0.826542
\(29\) −140.693 −0.900899 −0.450449 0.892802i \(-0.648736\pi\)
−0.450449 + 0.892802i \(0.648736\pi\)
\(30\) 0 0
\(31\) 136.155 0.788845 0.394423 0.918929i \(-0.370945\pi\)
0.394423 + 0.918929i \(0.370945\pi\)
\(32\) −186.501 −1.03028
\(33\) 0 0
\(34\) −9.19224 −0.0463663
\(35\) −26.8466 −0.129654
\(36\) 0 0
\(37\) −185.708 −0.825142 −0.412571 0.910925i \(-0.635369\pi\)
−0.412571 + 0.910925i \(0.635369\pi\)
\(38\) −274.570 −1.17214
\(39\) 0 0
\(40\) −61.5767 −0.243403
\(41\) −310.231 −1.18171 −0.590853 0.806779i \(-0.701209\pi\)
−0.590853 + 0.806779i \(0.701209\pi\)
\(42\) 0 0
\(43\) 427.471 1.51602 0.758008 0.652246i \(-0.226173\pi\)
0.758008 + 0.652246i \(0.226173\pi\)
\(44\) −504.924 −1.73000
\(45\) 0 0
\(46\) −20.3845 −0.0653375
\(47\) 258.617 0.802622 0.401311 0.915942i \(-0.368555\pi\)
0.401311 + 0.915942i \(0.368555\pi\)
\(48\) 0 0
\(49\) −251.577 −0.733460
\(50\) −534.233 −1.51104
\(51\) 0 0
\(52\) 0 0
\(53\) −612.656 −1.58783 −0.793913 0.608031i \(-0.791960\pi\)
−0.793913 + 0.608031i \(0.791960\pi\)
\(54\) 0 0
\(55\) 110.691 0.271375
\(56\) 209.693 0.500383
\(57\) 0 0
\(58\) −641.779 −1.45293
\(59\) 517.885 1.14276 0.571381 0.820685i \(-0.306408\pi\)
0.571381 + 0.820685i \(0.306408\pi\)
\(60\) 0 0
\(61\) −161.311 −0.338585 −0.169293 0.985566i \(-0.554148\pi\)
−0.169293 + 0.985566i \(0.554148\pi\)
\(62\) 621.080 1.27221
\(63\) 0 0
\(64\) −831.348 −1.62373
\(65\) 0 0
\(66\) 0 0
\(67\) −49.8987 −0.0909865 −0.0454933 0.998965i \(-0.514486\pi\)
−0.0454933 + 0.998965i \(0.514486\pi\)
\(68\) −25.8096 −0.0460276
\(69\) 0 0
\(70\) −122.462 −0.209100
\(71\) −279.963 −0.467965 −0.233982 0.972241i \(-0.575176\pi\)
−0.233982 + 0.972241i \(0.575176\pi\)
\(72\) 0 0
\(73\) 467.732 0.749916 0.374958 0.927042i \(-0.377657\pi\)
0.374958 + 0.927042i \(0.377657\pi\)
\(74\) −847.118 −1.33075
\(75\) 0 0
\(76\) −770.928 −1.16357
\(77\) −376.948 −0.557886
\(78\) 0 0
\(79\) 37.5379 0.0534600 0.0267300 0.999643i \(-0.491491\pi\)
0.0267300 + 0.999643i \(0.491491\pi\)
\(80\) 6.80403 0.00950892
\(81\) 0 0
\(82\) −1415.14 −1.90580
\(83\) 76.1553 0.100712 0.0503562 0.998731i \(-0.483964\pi\)
0.0503562 + 0.998731i \(0.483964\pi\)
\(84\) 0 0
\(85\) 5.65808 0.00722006
\(86\) 1949.93 2.44496
\(87\) 0 0
\(88\) −864.587 −1.04733
\(89\) −202.806 −0.241544 −0.120772 0.992680i \(-0.538537\pi\)
−0.120772 + 0.992680i \(0.538537\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −57.2348 −0.0648602
\(93\) 0 0
\(94\) 1179.70 1.29443
\(95\) 169.006 0.182522
\(96\) 0 0
\(97\) −1174.37 −1.22927 −0.614634 0.788812i \(-0.710696\pi\)
−0.614634 + 0.788812i \(0.710696\pi\)
\(98\) −1147.58 −1.18289
\(99\) 0 0
\(100\) −1500.00 −1.50000
\(101\) −970.697 −0.956316 −0.478158 0.878274i \(-0.658695\pi\)
−0.478158 + 0.878274i \(0.658695\pi\)
\(102\) 0 0
\(103\) −1899.70 −1.81731 −0.908654 0.417550i \(-0.862889\pi\)
−0.908654 + 0.417550i \(0.862889\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2794.66 −2.56077
\(107\) 1906.49 1.72250 0.861251 0.508180i \(-0.169682\pi\)
0.861251 + 0.508180i \(0.169682\pi\)
\(108\) 0 0
\(109\) −896.004 −0.787354 −0.393677 0.919249i \(-0.628797\pi\)
−0.393677 + 0.919249i \(0.628797\pi\)
\(110\) 504.924 0.437660
\(111\) 0 0
\(112\) −23.1704 −0.0195482
\(113\) 334.882 0.278788 0.139394 0.990237i \(-0.455484\pi\)
0.139394 + 0.990237i \(0.455484\pi\)
\(114\) 0 0
\(115\) 12.5472 0.0101742
\(116\) −1801.96 −1.44231
\(117\) 0 0
\(118\) 2362.36 1.84299
\(119\) −19.2680 −0.0148428
\(120\) 0 0
\(121\) 223.196 0.167690
\(122\) −735.827 −0.546054
\(123\) 0 0
\(124\) 1743.84 1.26292
\(125\) 679.806 0.486430
\(126\) 0 0
\(127\) 620.893 0.433822 0.216911 0.976191i \(-0.430402\pi\)
0.216911 + 0.976191i \(0.430402\pi\)
\(128\) −2300.23 −1.58839
\(129\) 0 0
\(130\) 0 0
\(131\) 1331.70 0.888180 0.444090 0.895982i \(-0.353527\pi\)
0.444090 + 0.895982i \(0.353527\pi\)
\(132\) 0 0
\(133\) −575.531 −0.375225
\(134\) −227.616 −0.146739
\(135\) 0 0
\(136\) −44.1941 −0.0278648
\(137\) −622.015 −0.387900 −0.193950 0.981011i \(-0.562130\pi\)
−0.193950 + 0.981011i \(0.562130\pi\)
\(138\) 0 0
\(139\) 330.580 0.201723 0.100861 0.994900i \(-0.467840\pi\)
0.100861 + 0.994900i \(0.467840\pi\)
\(140\) −343.845 −0.207573
\(141\) 0 0
\(142\) −1277.07 −0.754711
\(143\) 0 0
\(144\) 0 0
\(145\) 395.033 0.226246
\(146\) 2133.58 1.20943
\(147\) 0 0
\(148\) −2378.51 −1.32103
\(149\) 1810.54 0.995470 0.497735 0.867329i \(-0.334165\pi\)
0.497735 + 0.867329i \(0.334165\pi\)
\(150\) 0 0
\(151\) −423.239 −0.228097 −0.114049 0.993475i \(-0.536382\pi\)
−0.114049 + 0.993475i \(0.536382\pi\)
\(152\) −1320.07 −0.704419
\(153\) 0 0
\(154\) −1719.47 −0.899732
\(155\) −382.292 −0.198106
\(156\) 0 0
\(157\) 1322.17 0.672105 0.336052 0.941843i \(-0.390908\pi\)
0.336052 + 0.941843i \(0.390908\pi\)
\(158\) 171.231 0.0862178
\(159\) 0 0
\(160\) 523.651 0.258739
\(161\) −42.7283 −0.0209159
\(162\) 0 0
\(163\) −3606.39 −1.73297 −0.866486 0.499201i \(-0.833627\pi\)
−0.866486 + 0.499201i \(0.833627\pi\)
\(164\) −3973.37 −1.89188
\(165\) 0 0
\(166\) 347.386 0.162424
\(167\) 3415.43 1.58260 0.791300 0.611429i \(-0.209405\pi\)
0.791300 + 0.611429i \(0.209405\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 25.8096 0.0116442
\(171\) 0 0
\(172\) 5474.94 2.42710
\(173\) −2342.23 −1.02934 −0.514671 0.857388i \(-0.672086\pi\)
−0.514671 + 0.857388i \(0.672086\pi\)
\(174\) 0 0
\(175\) −1119.82 −0.483715
\(176\) 95.5342 0.0409157
\(177\) 0 0
\(178\) −925.110 −0.389550
\(179\) 666.891 0.278468 0.139234 0.990260i \(-0.455536\pi\)
0.139234 + 0.990260i \(0.455536\pi\)
\(180\) 0 0
\(181\) −701.037 −0.287888 −0.143944 0.989586i \(-0.545978\pi\)
−0.143944 + 0.989586i \(0.545978\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −98.0037 −0.0392659
\(185\) 521.425 0.207221
\(186\) 0 0
\(187\) 79.4440 0.0310670
\(188\) 3312.31 1.28497
\(189\) 0 0
\(190\) 770.928 0.294363
\(191\) −1300.88 −0.492819 −0.246409 0.969166i \(-0.579251\pi\)
−0.246409 + 0.969166i \(0.579251\pi\)
\(192\) 0 0
\(193\) −519.333 −0.193691 −0.0968457 0.995299i \(-0.530875\pi\)
−0.0968457 + 0.995299i \(0.530875\pi\)
\(194\) −5356.94 −1.98251
\(195\) 0 0
\(196\) −3222.14 −1.17425
\(197\) −3121.05 −1.12876 −0.564379 0.825516i \(-0.690884\pi\)
−0.564379 + 0.825516i \(0.690884\pi\)
\(198\) 0 0
\(199\) 1237.06 0.440667 0.220333 0.975425i \(-0.429285\pi\)
0.220333 + 0.975425i \(0.429285\pi\)
\(200\) −2568.47 −0.908090
\(201\) 0 0
\(202\) −4427.89 −1.54230
\(203\) −1345.25 −0.465112
\(204\) 0 0
\(205\) 871.056 0.296767
\(206\) −8665.57 −2.93087
\(207\) 0 0
\(208\) 0 0
\(209\) 2372.98 0.785369
\(210\) 0 0
\(211\) −2531.67 −0.826006 −0.413003 0.910730i \(-0.635520\pi\)
−0.413003 + 0.910730i \(0.635520\pi\)
\(212\) −7846.76 −2.54206
\(213\) 0 0
\(214\) 8696.57 2.77797
\(215\) −1200.24 −0.380723
\(216\) 0 0
\(217\) 1301.86 0.407261
\(218\) −4087.17 −1.26981
\(219\) 0 0
\(220\) 1417.71 0.434463
\(221\) 0 0
\(222\) 0 0
\(223\) −1195.53 −0.359008 −0.179504 0.983757i \(-0.557449\pi\)
−0.179504 + 0.983757i \(0.557449\pi\)
\(224\) −1783.24 −0.531909
\(225\) 0 0
\(226\) 1527.58 0.449617
\(227\) 869.192 0.254142 0.127071 0.991894i \(-0.459442\pi\)
0.127071 + 0.991894i \(0.459442\pi\)
\(228\) 0 0
\(229\) 4684.64 1.35183 0.675916 0.736978i \(-0.263748\pi\)
0.675916 + 0.736978i \(0.263748\pi\)
\(230\) 57.2348 0.0164085
\(231\) 0 0
\(232\) −3085.52 −0.873166
\(233\) 4868.99 1.36900 0.684502 0.729011i \(-0.260020\pi\)
0.684502 + 0.729011i \(0.260020\pi\)
\(234\) 0 0
\(235\) −726.137 −0.201566
\(236\) 6632.95 1.82953
\(237\) 0 0
\(238\) −87.8920 −0.0239378
\(239\) −4807.53 −1.30114 −0.650572 0.759444i \(-0.725471\pi\)
−0.650572 + 0.759444i \(0.725471\pi\)
\(240\) 0 0
\(241\) 5875.96 1.57056 0.785278 0.619143i \(-0.212520\pi\)
0.785278 + 0.619143i \(0.212520\pi\)
\(242\) 1018.12 0.270443
\(243\) 0 0
\(244\) −2066.03 −0.542065
\(245\) 706.368 0.184197
\(246\) 0 0
\(247\) 0 0
\(248\) 2986.00 0.764562
\(249\) 0 0
\(250\) 3100.97 0.784490
\(251\) −5806.27 −1.46011 −0.730057 0.683386i \(-0.760506\pi\)
−0.730057 + 0.683386i \(0.760506\pi\)
\(252\) 0 0
\(253\) 176.173 0.0437783
\(254\) 2832.24 0.699647
\(255\) 0 0
\(256\) −3841.83 −0.937947
\(257\) 1195.86 0.290256 0.145128 0.989413i \(-0.453641\pi\)
0.145128 + 0.989413i \(0.453641\pi\)
\(258\) 0 0
\(259\) −1775.66 −0.426001
\(260\) 0 0
\(261\) 0 0
\(262\) 6074.64 1.43241
\(263\) −234.184 −0.0549066 −0.0274533 0.999623i \(-0.508740\pi\)
−0.0274533 + 0.999623i \(0.508740\pi\)
\(264\) 0 0
\(265\) 1720.19 0.398757
\(266\) −2625.32 −0.605145
\(267\) 0 0
\(268\) −639.091 −0.145667
\(269\) 2668.27 0.604785 0.302393 0.953183i \(-0.402215\pi\)
0.302393 + 0.953183i \(0.402215\pi\)
\(270\) 0 0
\(271\) 5701.28 1.27796 0.638982 0.769222i \(-0.279356\pi\)
0.638982 + 0.769222i \(0.279356\pi\)
\(272\) 4.88331 0.00108858
\(273\) 0 0
\(274\) −2837.35 −0.625587
\(275\) 4617.12 1.01245
\(276\) 0 0
\(277\) −7152.49 −1.55145 −0.775725 0.631072i \(-0.782615\pi\)
−0.775725 + 0.631072i \(0.782615\pi\)
\(278\) 1507.96 0.325329
\(279\) 0 0
\(280\) −588.769 −0.125663
\(281\) 6132.87 1.30198 0.650990 0.759086i \(-0.274354\pi\)
0.650990 + 0.759086i \(0.274354\pi\)
\(282\) 0 0
\(283\) 3377.15 0.709367 0.354683 0.934986i \(-0.384589\pi\)
0.354683 + 0.934986i \(0.384589\pi\)
\(284\) −3585.70 −0.749198
\(285\) 0 0
\(286\) 0 0
\(287\) −2966.29 −0.610086
\(288\) 0 0
\(289\) −4908.94 −0.999173
\(290\) 1801.96 0.364879
\(291\) 0 0
\(292\) 5990.60 1.20059
\(293\) 4704.77 0.938073 0.469037 0.883179i \(-0.344601\pi\)
0.469037 + 0.883179i \(0.344601\pi\)
\(294\) 0 0
\(295\) −1454.10 −0.286986
\(296\) −4072.75 −0.799742
\(297\) 0 0
\(298\) 8258.86 1.60545
\(299\) 0 0
\(300\) 0 0
\(301\) 4087.28 0.782681
\(302\) −1930.62 −0.367864
\(303\) 0 0
\(304\) 145.863 0.0275192
\(305\) 452.922 0.0850303
\(306\) 0 0
\(307\) 5130.49 0.953787 0.476894 0.878961i \(-0.341763\pi\)
0.476894 + 0.878961i \(0.341763\pi\)
\(308\) −4827.86 −0.893159
\(309\) 0 0
\(310\) −1743.84 −0.319496
\(311\) −7948.94 −1.44933 −0.724667 0.689099i \(-0.758006\pi\)
−0.724667 + 0.689099i \(0.758006\pi\)
\(312\) 0 0
\(313\) −8521.87 −1.53893 −0.769465 0.638689i \(-0.779477\pi\)
−0.769465 + 0.638689i \(0.779477\pi\)
\(314\) 6031.14 1.08394
\(315\) 0 0
\(316\) 480.776 0.0855879
\(317\) 6662.46 1.18044 0.590222 0.807241i \(-0.299040\pi\)
0.590222 + 0.807241i \(0.299040\pi\)
\(318\) 0 0
\(319\) 5546.59 0.973509
\(320\) 2334.23 0.407773
\(321\) 0 0
\(322\) −194.907 −0.0337322
\(323\) 121.297 0.0208951
\(324\) 0 0
\(325\) 0 0
\(326\) −16450.8 −2.79486
\(327\) 0 0
\(328\) −6803.64 −1.14533
\(329\) 2472.78 0.414374
\(330\) 0 0
\(331\) −3911.77 −0.649578 −0.324789 0.945786i \(-0.605293\pi\)
−0.324789 + 0.945786i \(0.605293\pi\)
\(332\) 975.379 0.161238
\(333\) 0 0
\(334\) 15579.7 2.55234
\(335\) 140.104 0.0228498
\(336\) 0 0
\(337\) −627.211 −0.101384 −0.0506919 0.998714i \(-0.516143\pi\)
−0.0506919 + 0.998714i \(0.516143\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 72.4674 0.0115591
\(341\) −5367.69 −0.852424
\(342\) 0 0
\(343\) −5685.08 −0.894943
\(344\) 9374.80 1.46935
\(345\) 0 0
\(346\) −10684.2 −1.66007
\(347\) 3823.02 0.591442 0.295721 0.955274i \(-0.404440\pi\)
0.295721 + 0.955274i \(0.404440\pi\)
\(348\) 0 0
\(349\) 3410.67 0.523120 0.261560 0.965187i \(-0.415763\pi\)
0.261560 + 0.965187i \(0.415763\pi\)
\(350\) −5108.10 −0.780112
\(351\) 0 0
\(352\) 7352.48 1.11332
\(353\) 5587.64 0.842492 0.421246 0.906946i \(-0.361593\pi\)
0.421246 + 0.906946i \(0.361593\pi\)
\(354\) 0 0
\(355\) 786.070 0.117522
\(356\) −2597.49 −0.386704
\(357\) 0 0
\(358\) 3042.06 0.449100
\(359\) −2230.14 −0.327861 −0.163931 0.986472i \(-0.552417\pi\)
−0.163931 + 0.986472i \(0.552417\pi\)
\(360\) 0 0
\(361\) −3235.89 −0.471774
\(362\) −3197.82 −0.464292
\(363\) 0 0
\(364\) 0 0
\(365\) −1313.28 −0.188330
\(366\) 0 0
\(367\) −8699.14 −1.23731 −0.618653 0.785664i \(-0.712321\pi\)
−0.618653 + 0.785664i \(0.712321\pi\)
\(368\) 10.8291 0.00153398
\(369\) 0 0
\(370\) 2378.51 0.334197
\(371\) −5857.94 −0.819756
\(372\) 0 0
\(373\) 10964.2 1.52199 0.760997 0.648755i \(-0.224710\pi\)
0.760997 + 0.648755i \(0.224710\pi\)
\(374\) 362.388 0.0501033
\(375\) 0 0
\(376\) 5671.70 0.777914
\(377\) 0 0
\(378\) 0 0
\(379\) 13910.1 1.88526 0.942631 0.333835i \(-0.108343\pi\)
0.942631 + 0.333835i \(0.108343\pi\)
\(380\) 2164.58 0.292213
\(381\) 0 0
\(382\) −5934.03 −0.794794
\(383\) 494.753 0.0660071 0.0330035 0.999455i \(-0.489493\pi\)
0.0330035 + 0.999455i \(0.489493\pi\)
\(384\) 0 0
\(385\) 1058.38 0.140104
\(386\) −2368.97 −0.312376
\(387\) 0 0
\(388\) −15041.0 −1.96802
\(389\) 4140.47 0.539666 0.269833 0.962907i \(-0.413032\pi\)
0.269833 + 0.962907i \(0.413032\pi\)
\(390\) 0 0
\(391\) 9.00524 0.00116474
\(392\) −5517.30 −0.710881
\(393\) 0 0
\(394\) −14236.8 −1.82041
\(395\) −105.398 −0.0134256
\(396\) 0 0
\(397\) −1881.79 −0.237895 −0.118948 0.992901i \(-0.537952\pi\)
−0.118948 + 0.992901i \(0.537952\pi\)
\(398\) 5642.90 0.710686
\(399\) 0 0
\(400\) 283.807 0.0354759
\(401\) −421.765 −0.0525236 −0.0262618 0.999655i \(-0.508360\pi\)
−0.0262618 + 0.999655i \(0.508360\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −12432.5 −1.53103
\(405\) 0 0
\(406\) −6136.41 −0.750110
\(407\) 7321.23 0.891646
\(408\) 0 0
\(409\) 2550.22 0.308313 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(410\) 3973.37 0.478611
\(411\) 0 0
\(412\) −24330.9 −2.90946
\(413\) 4951.79 0.589980
\(414\) 0 0
\(415\) −213.826 −0.0252923
\(416\) 0 0
\(417\) 0 0
\(418\) 10824.5 1.26661
\(419\) −12384.8 −1.44400 −0.722002 0.691891i \(-0.756778\pi\)
−0.722002 + 0.691891i \(0.756778\pi\)
\(420\) 0 0
\(421\) 10463.0 1.21124 0.605622 0.795752i \(-0.292924\pi\)
0.605622 + 0.795752i \(0.292924\pi\)
\(422\) −11548.3 −1.33214
\(423\) 0 0
\(424\) −13436.1 −1.53895
\(425\) 236.008 0.0269366
\(426\) 0 0
\(427\) −1542.38 −0.174803
\(428\) 24417.9 2.75767
\(429\) 0 0
\(430\) −5474.94 −0.614012
\(431\) 3962.39 0.442834 0.221417 0.975179i \(-0.428932\pi\)
0.221417 + 0.975179i \(0.428932\pi\)
\(432\) 0 0
\(433\) −8394.14 −0.931632 −0.465816 0.884882i \(-0.654239\pi\)
−0.465816 + 0.884882i \(0.654239\pi\)
\(434\) 5938.48 0.656812
\(435\) 0 0
\(436\) −11475.8 −1.26053
\(437\) 268.984 0.0294446
\(438\) 0 0
\(439\) 10174.5 1.10616 0.553079 0.833129i \(-0.313453\pi\)
0.553079 + 0.833129i \(0.313453\pi\)
\(440\) 2427.56 0.263021
\(441\) 0 0
\(442\) 0 0
\(443\) 5880.74 0.630705 0.315353 0.948975i \(-0.397877\pi\)
0.315353 + 0.948975i \(0.397877\pi\)
\(444\) 0 0
\(445\) 569.431 0.0606598
\(446\) −5453.48 −0.578990
\(447\) 0 0
\(448\) −7948.97 −0.838289
\(449\) −10664.9 −1.12095 −0.560475 0.828172i \(-0.689381\pi\)
−0.560475 + 0.828172i \(0.689381\pi\)
\(450\) 0 0
\(451\) 12230.3 1.27695
\(452\) 4289.10 0.446332
\(453\) 0 0
\(454\) 3964.87 0.409869
\(455\) 0 0
\(456\) 0 0
\(457\) −14828.9 −1.51787 −0.758933 0.651169i \(-0.774279\pi\)
−0.758933 + 0.651169i \(0.774279\pi\)
\(458\) 21369.2 2.18017
\(459\) 0 0
\(460\) 160.702 0.0162886
\(461\) 9711.70 0.981169 0.490585 0.871394i \(-0.336783\pi\)
0.490585 + 0.871394i \(0.336783\pi\)
\(462\) 0 0
\(463\) 11353.5 1.13962 0.569809 0.821777i \(-0.307017\pi\)
0.569809 + 0.821777i \(0.307017\pi\)
\(464\) 340.941 0.0341116
\(465\) 0 0
\(466\) 22210.1 2.20787
\(467\) −6451.31 −0.639252 −0.319626 0.947544i \(-0.603557\pi\)
−0.319626 + 0.947544i \(0.603557\pi\)
\(468\) 0 0
\(469\) −477.109 −0.0469741
\(470\) −3312.31 −0.325076
\(471\) 0 0
\(472\) 11357.7 1.10758
\(473\) −16852.3 −1.63820
\(474\) 0 0
\(475\) 7049.50 0.680954
\(476\) −246.780 −0.0237629
\(477\) 0 0
\(478\) −21929.8 −2.09842
\(479\) 9566.46 0.912531 0.456266 0.889844i \(-0.349187\pi\)
0.456266 + 0.889844i \(0.349187\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 26803.5 2.53292
\(483\) 0 0
\(484\) 2858.64 0.268467
\(485\) 3297.35 0.308711
\(486\) 0 0
\(487\) 4917.11 0.457527 0.228764 0.973482i \(-0.426532\pi\)
0.228764 + 0.973482i \(0.426532\pi\)
\(488\) −3537.68 −0.328162
\(489\) 0 0
\(490\) 3222.14 0.297064
\(491\) −2950.82 −0.271220 −0.135610 0.990762i \(-0.543299\pi\)
−0.135610 + 0.990762i \(0.543299\pi\)
\(492\) 0 0
\(493\) 283.519 0.0259007
\(494\) 0 0
\(495\) 0 0
\(496\) −329.944 −0.0298688
\(497\) −2676.88 −0.241599
\(498\) 0 0
\(499\) 13430.1 1.20484 0.602418 0.798180i \(-0.294204\pi\)
0.602418 + 0.798180i \(0.294204\pi\)
\(500\) 8706.79 0.778759
\(501\) 0 0
\(502\) −26485.6 −2.35480
\(503\) −1320.29 −0.117035 −0.0585175 0.998286i \(-0.518637\pi\)
−0.0585175 + 0.998286i \(0.518637\pi\)
\(504\) 0 0
\(505\) 2725.49 0.240164
\(506\) 803.623 0.0706036
\(507\) 0 0
\(508\) 7952.25 0.694536
\(509\) −20916.4 −1.82143 −0.910713 0.413041i \(-0.864467\pi\)
−0.910713 + 0.413041i \(0.864467\pi\)
\(510\) 0 0
\(511\) 4472.24 0.387163
\(512\) 877.105 0.0757089
\(513\) 0 0
\(514\) 5454.98 0.468110
\(515\) 5333.90 0.456388
\(516\) 0 0
\(517\) −10195.5 −0.867311
\(518\) −8099.77 −0.687033
\(519\) 0 0
\(520\) 0 0
\(521\) 10104.2 0.849661 0.424831 0.905273i \(-0.360334\pi\)
0.424831 + 0.905273i \(0.360334\pi\)
\(522\) 0 0
\(523\) 7131.22 0.596227 0.298113 0.954530i \(-0.403643\pi\)
0.298113 + 0.954530i \(0.403643\pi\)
\(524\) 17056.2 1.42195
\(525\) 0 0
\(526\) −1068.24 −0.0885507
\(527\) −274.374 −0.0226792
\(528\) 0 0
\(529\) −12147.0 −0.998359
\(530\) 7846.76 0.643097
\(531\) 0 0
\(532\) −7371.27 −0.600724
\(533\) 0 0
\(534\) 0 0
\(535\) −5352.98 −0.432579
\(536\) −1094.32 −0.0881856
\(537\) 0 0
\(538\) 12171.5 0.975369
\(539\) 9917.98 0.792575
\(540\) 0 0
\(541\) 16831.7 1.33762 0.668809 0.743435i \(-0.266805\pi\)
0.668809 + 0.743435i \(0.266805\pi\)
\(542\) 26006.7 2.06104
\(543\) 0 0
\(544\) 375.828 0.0296204
\(545\) 2515.77 0.197731
\(546\) 0 0
\(547\) −9560.55 −0.747312 −0.373656 0.927567i \(-0.621896\pi\)
−0.373656 + 0.927567i \(0.621896\pi\)
\(548\) −7966.62 −0.621017
\(549\) 0 0
\(550\) 21061.2 1.63282
\(551\) 8468.64 0.654766
\(552\) 0 0
\(553\) 358.920 0.0276001
\(554\) −32626.5 −2.50210
\(555\) 0 0
\(556\) 4234.00 0.322952
\(557\) 22827.9 1.73653 0.868267 0.496097i \(-0.165234\pi\)
0.868267 + 0.496097i \(0.165234\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 65.0571 0.00490922
\(561\) 0 0
\(562\) 27975.4 2.09977
\(563\) −21629.7 −1.61916 −0.809578 0.587013i \(-0.800304\pi\)
−0.809578 + 0.587013i \(0.800304\pi\)
\(564\) 0 0
\(565\) −940.271 −0.0700133
\(566\) 15405.1 1.14403
\(567\) 0 0
\(568\) −6139.83 −0.453559
\(569\) 10589.9 0.780229 0.390114 0.920766i \(-0.372436\pi\)
0.390114 + 0.920766i \(0.372436\pi\)
\(570\) 0 0
\(571\) −1757.27 −0.128791 −0.0643954 0.997924i \(-0.520512\pi\)
−0.0643954 + 0.997924i \(0.520512\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −13530.9 −0.983917
\(575\) 523.365 0.0379580
\(576\) 0 0
\(577\) −13580.6 −0.979840 −0.489920 0.871767i \(-0.662974\pi\)
−0.489920 + 0.871767i \(0.662974\pi\)
\(578\) −22392.4 −1.61142
\(579\) 0 0
\(580\) 5059.49 0.362214
\(581\) 728.163 0.0519953
\(582\) 0 0
\(583\) 24152.9 1.71580
\(584\) 10257.8 0.726831
\(585\) 0 0
\(586\) 21461.0 1.51288
\(587\) 957.326 0.0673136 0.0336568 0.999433i \(-0.489285\pi\)
0.0336568 + 0.999433i \(0.489285\pi\)
\(588\) 0 0
\(589\) −8195.49 −0.573327
\(590\) −6632.95 −0.462838
\(591\) 0 0
\(592\) 450.026 0.0312431
\(593\) −6729.49 −0.466015 −0.233007 0.972475i \(-0.574857\pi\)
−0.233007 + 0.972475i \(0.574857\pi\)
\(594\) 0 0
\(595\) 54.1000 0.00372754
\(596\) 23188.9 1.59372
\(597\) 0 0
\(598\) 0 0
\(599\) −2281.52 −0.155626 −0.0778132 0.996968i \(-0.524794\pi\)
−0.0778132 + 0.996968i \(0.524794\pi\)
\(600\) 0 0
\(601\) 6401.42 0.434475 0.217237 0.976119i \(-0.430295\pi\)
0.217237 + 0.976119i \(0.430295\pi\)
\(602\) 18644.4 1.26227
\(603\) 0 0
\(604\) −5420.74 −0.365177
\(605\) −626.682 −0.0421128
\(606\) 0 0
\(607\) 2779.24 0.185841 0.0929207 0.995674i \(-0.470380\pi\)
0.0929207 + 0.995674i \(0.470380\pi\)
\(608\) 11225.9 0.748800
\(609\) 0 0
\(610\) 2066.03 0.137133
\(611\) 0 0
\(612\) 0 0
\(613\) 22620.8 1.49045 0.745226 0.666812i \(-0.232342\pi\)
0.745226 + 0.666812i \(0.232342\pi\)
\(614\) 23403.0 1.53822
\(615\) 0 0
\(616\) −8266.80 −0.540712
\(617\) −21974.0 −1.43378 −0.716889 0.697187i \(-0.754435\pi\)
−0.716889 + 0.697187i \(0.754435\pi\)
\(618\) 0 0
\(619\) 7145.19 0.463957 0.231979 0.972721i \(-0.425480\pi\)
0.231979 + 0.972721i \(0.425480\pi\)
\(620\) −4896.30 −0.317162
\(621\) 0 0
\(622\) −36259.5 −2.33742
\(623\) −1939.14 −0.124703
\(624\) 0 0
\(625\) 12730.8 0.814773
\(626\) −38873.0 −2.48191
\(627\) 0 0
\(628\) 16934.0 1.07602
\(629\) 374.231 0.0237227
\(630\) 0 0
\(631\) −18883.2 −1.19133 −0.595666 0.803232i \(-0.703112\pi\)
−0.595666 + 0.803232i \(0.703112\pi\)
\(632\) 823.239 0.0518144
\(633\) 0 0
\(634\) 30391.1 1.90376
\(635\) −1743.32 −0.108947
\(636\) 0 0
\(637\) 0 0
\(638\) 25301.1 1.57003
\(639\) 0 0
\(640\) 6458.50 0.398898
\(641\) 3631.08 0.223743 0.111871 0.993723i \(-0.464316\pi\)
0.111871 + 0.993723i \(0.464316\pi\)
\(642\) 0 0
\(643\) −10772.0 −0.660660 −0.330330 0.943866i \(-0.607160\pi\)
−0.330330 + 0.943866i \(0.607160\pi\)
\(644\) −547.253 −0.0334857
\(645\) 0 0
\(646\) 553.301 0.0336987
\(647\) −15148.3 −0.920464 −0.460232 0.887799i \(-0.652234\pi\)
−0.460232 + 0.887799i \(0.652234\pi\)
\(648\) 0 0
\(649\) −20416.7 −1.23487
\(650\) 0 0
\(651\) 0 0
\(652\) −46189.8 −2.77444
\(653\) −7358.89 −0.441004 −0.220502 0.975387i \(-0.570770\pi\)
−0.220502 + 0.975387i \(0.570770\pi\)
\(654\) 0 0
\(655\) −3739.11 −0.223052
\(656\) 751.780 0.0447441
\(657\) 0 0
\(658\) 11279.7 0.668282
\(659\) −28333.3 −1.67482 −0.837411 0.546574i \(-0.815932\pi\)
−0.837411 + 0.546574i \(0.815932\pi\)
\(660\) 0 0
\(661\) 1109.68 0.0652975 0.0326488 0.999467i \(-0.489606\pi\)
0.0326488 + 0.999467i \(0.489606\pi\)
\(662\) −17843.8 −1.04761
\(663\) 0 0
\(664\) 1670.15 0.0976121
\(665\) 1615.96 0.0942317
\(666\) 0 0
\(667\) 628.724 0.0364982
\(668\) 43744.1 2.53369
\(669\) 0 0
\(670\) 639.091 0.0368511
\(671\) 6359.39 0.365874
\(672\) 0 0
\(673\) 20979.1 1.20161 0.600806 0.799395i \(-0.294846\pi\)
0.600806 + 0.799395i \(0.294846\pi\)
\(674\) −2861.06 −0.163507
\(675\) 0 0
\(676\) 0 0
\(677\) −30941.9 −1.75656 −0.878282 0.478142i \(-0.841310\pi\)
−0.878282 + 0.478142i \(0.841310\pi\)
\(678\) 0 0
\(679\) −11228.8 −0.634641
\(680\) 124.087 0.00699780
\(681\) 0 0
\(682\) −24485.0 −1.37475
\(683\) −5426.21 −0.303995 −0.151997 0.988381i \(-0.548571\pi\)
−0.151997 + 0.988381i \(0.548571\pi\)
\(684\) 0 0
\(685\) 1746.47 0.0974150
\(686\) −25932.8 −1.44332
\(687\) 0 0
\(688\) −1035.89 −0.0574023
\(689\) 0 0
\(690\) 0 0
\(691\) −33792.7 −1.86040 −0.930199 0.367056i \(-0.880366\pi\)
−0.930199 + 0.367056i \(0.880366\pi\)
\(692\) −29998.7 −1.64795
\(693\) 0 0
\(694\) 17438.9 0.953849
\(695\) −928.192 −0.0506595
\(696\) 0 0
\(697\) 625.164 0.0339738
\(698\) 15557.9 0.843662
\(699\) 0 0
\(700\) −14342.3 −0.774413
\(701\) −6905.96 −0.372089 −0.186045 0.982541i \(-0.559567\pi\)
−0.186045 + 0.982541i \(0.559567\pi\)
\(702\) 0 0
\(703\) 11178.2 0.599707
\(704\) 32774.5 1.75459
\(705\) 0 0
\(706\) 25488.3 1.35873
\(707\) −9281.37 −0.493723
\(708\) 0 0
\(709\) −2007.13 −0.106318 −0.0531589 0.998586i \(-0.516929\pi\)
−0.0531589 + 0.998586i \(0.516929\pi\)
\(710\) 3585.70 0.189534
\(711\) 0 0
\(712\) −4447.71 −0.234108
\(713\) −608.445 −0.0319585
\(714\) 0 0
\(715\) 0 0
\(716\) 8541.38 0.445819
\(717\) 0 0
\(718\) −10172.9 −0.528759
\(719\) 12787.4 0.663270 0.331635 0.943408i \(-0.392400\pi\)
0.331635 + 0.943408i \(0.392400\pi\)
\(720\) 0 0
\(721\) −18164.1 −0.938231
\(722\) −14760.7 −0.760854
\(723\) 0 0
\(724\) −8978.72 −0.460900
\(725\) 16477.5 0.844081
\(726\) 0 0
\(727\) −6090.70 −0.310717 −0.155359 0.987858i \(-0.549653\pi\)
−0.155359 + 0.987858i \(0.549653\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −5990.60 −0.303729
\(731\) −861.420 −0.0435852
\(732\) 0 0
\(733\) −38846.5 −1.95747 −0.978737 0.205117i \(-0.934243\pi\)
−0.978737 + 0.205117i \(0.934243\pi\)
\(734\) −39681.6 −1.99547
\(735\) 0 0
\(736\) 833.427 0.0417399
\(737\) 1967.17 0.0983198
\(738\) 0 0
\(739\) 14457.5 0.719661 0.359830 0.933018i \(-0.382835\pi\)
0.359830 + 0.933018i \(0.382835\pi\)
\(740\) 6678.29 0.331755
\(741\) 0 0
\(742\) −26721.3 −1.32206
\(743\) 1277.80 0.0630929 0.0315464 0.999502i \(-0.489957\pi\)
0.0315464 + 0.999502i \(0.489957\pi\)
\(744\) 0 0
\(745\) −5083.56 −0.249996
\(746\) 50013.7 2.45460
\(747\) 0 0
\(748\) 1017.50 0.0497373
\(749\) 18229.0 0.889285
\(750\) 0 0
\(751\) −13007.9 −0.632042 −0.316021 0.948752i \(-0.602347\pi\)
−0.316021 + 0.948752i \(0.602347\pi\)
\(752\) −626.706 −0.0303904
\(753\) 0 0
\(754\) 0 0
\(755\) 1188.35 0.0572829
\(756\) 0 0
\(757\) −10723.2 −0.514850 −0.257425 0.966298i \(-0.582874\pi\)
−0.257425 + 0.966298i \(0.582874\pi\)
\(758\) 63451.7 3.04046
\(759\) 0 0
\(760\) 3706.44 0.176904
\(761\) −13621.8 −0.648870 −0.324435 0.945908i \(-0.605174\pi\)
−0.324435 + 0.945908i \(0.605174\pi\)
\(762\) 0 0
\(763\) −8567.19 −0.406491
\(764\) −16661.4 −0.788988
\(765\) 0 0
\(766\) 2256.84 0.106453
\(767\) 0 0
\(768\) 0 0
\(769\) −8495.15 −0.398365 −0.199183 0.979962i \(-0.563829\pi\)
−0.199183 + 0.979962i \(0.563829\pi\)
\(770\) 4827.86 0.225953
\(771\) 0 0
\(772\) −6651.50 −0.310094
\(773\) −34262.5 −1.59423 −0.797113 0.603830i \(-0.793641\pi\)
−0.797113 + 0.603830i \(0.793641\pi\)
\(774\) 0 0
\(775\) −15946.0 −0.739094
\(776\) −25754.9 −1.19143
\(777\) 0 0
\(778\) 18887.0 0.870348
\(779\) 18673.5 0.858854
\(780\) 0 0
\(781\) 11037.1 0.505681
\(782\) 41.0779 0.00187844
\(783\) 0 0
\(784\) 609.644 0.0277717
\(785\) −3712.33 −0.168788
\(786\) 0 0
\(787\) −12642.6 −0.572629 −0.286315 0.958136i \(-0.592430\pi\)
−0.286315 + 0.958136i \(0.592430\pi\)
\(788\) −39973.6 −1.80711
\(789\) 0 0
\(790\) −480.776 −0.0216522
\(791\) 3202.00 0.143932
\(792\) 0 0
\(793\) 0 0
\(794\) −8583.89 −0.383666
\(795\) 0 0
\(796\) 15843.9 0.705494
\(797\) −19084.4 −0.848184 −0.424092 0.905619i \(-0.639407\pi\)
−0.424092 + 0.905619i \(0.639407\pi\)
\(798\) 0 0
\(799\) −521.154 −0.0230752
\(800\) 21842.3 0.965304
\(801\) 0 0
\(802\) −1923.90 −0.0847075
\(803\) −18439.5 −0.810357
\(804\) 0 0
\(805\) 119.971 0.00525269
\(806\) 0 0
\(807\) 0 0
\(808\) −21288.2 −0.926878
\(809\) −11610.0 −0.504558 −0.252279 0.967655i \(-0.581180\pi\)
−0.252279 + 0.967655i \(0.581180\pi\)
\(810\) 0 0
\(811\) 9613.36 0.416240 0.208120 0.978103i \(-0.433266\pi\)
0.208120 + 0.978103i \(0.433266\pi\)
\(812\) −17229.6 −0.744630
\(813\) 0 0
\(814\) 33396.2 1.43800
\(815\) 10125.9 0.435208
\(816\) 0 0
\(817\) −25730.4 −1.10183
\(818\) 11632.9 0.497233
\(819\) 0 0
\(820\) 11156.3 0.475115
\(821\) 26481.5 1.12571 0.562856 0.826555i \(-0.309703\pi\)
0.562856 + 0.826555i \(0.309703\pi\)
\(822\) 0 0
\(823\) 13814.5 0.585107 0.292553 0.956249i \(-0.405495\pi\)
0.292553 + 0.956249i \(0.405495\pi\)
\(824\) −41662.0 −1.76136
\(825\) 0 0
\(826\) 22587.8 0.951491
\(827\) 44401.0 1.86696 0.933479 0.358633i \(-0.116757\pi\)
0.933479 + 0.358633i \(0.116757\pi\)
\(828\) 0 0
\(829\) 24337.4 1.01963 0.509815 0.860284i \(-0.329714\pi\)
0.509815 + 0.860284i \(0.329714\pi\)
\(830\) −975.379 −0.0407902
\(831\) 0 0
\(832\) 0 0
\(833\) 506.966 0.0210868
\(834\) 0 0
\(835\) −9589.73 −0.397445
\(836\) 30392.5 1.25735
\(837\) 0 0
\(838\) −56494.0 −2.32882
\(839\) 24680.1 1.01556 0.507778 0.861488i \(-0.330467\pi\)
0.507778 + 0.861488i \(0.330467\pi\)
\(840\) 0 0
\(841\) −4594.43 −0.188381
\(842\) 47727.4 1.95344
\(843\) 0 0
\(844\) −32425.0 −1.32241
\(845\) 0 0
\(846\) 0 0
\(847\) 2134.10 0.0865744
\(848\) 1484.65 0.0601214
\(849\) 0 0
\(850\) 1076.56 0.0434421
\(851\) 829.885 0.0334290
\(852\) 0 0
\(853\) −10151.7 −0.407490 −0.203745 0.979024i \(-0.565311\pi\)
−0.203745 + 0.979024i \(0.565311\pi\)
\(854\) −7035.65 −0.281914
\(855\) 0 0
\(856\) 41811.1 1.66948
\(857\) −2028.92 −0.0808713 −0.0404357 0.999182i \(-0.512875\pi\)
−0.0404357 + 0.999182i \(0.512875\pi\)
\(858\) 0 0
\(859\) 6655.76 0.264367 0.132184 0.991225i \(-0.457801\pi\)
0.132184 + 0.991225i \(0.457801\pi\)
\(860\) −15372.3 −0.609526
\(861\) 0 0
\(862\) 18074.6 0.714182
\(863\) −45690.8 −1.80224 −0.901121 0.433568i \(-0.857254\pi\)
−0.901121 + 0.433568i \(0.857254\pi\)
\(864\) 0 0
\(865\) 6576.42 0.258503
\(866\) −38290.3 −1.50249
\(867\) 0 0
\(868\) 16673.9 0.652014
\(869\) −1479.87 −0.0577688
\(870\) 0 0
\(871\) 0 0
\(872\) −19650.1 −0.763117
\(873\) 0 0
\(874\) 1226.99 0.0474868
\(875\) 6500.00 0.251132
\(876\) 0 0
\(877\) 30447.5 1.17234 0.586168 0.810189i \(-0.300636\pi\)
0.586168 + 0.810189i \(0.300636\pi\)
\(878\) 46411.6 1.78396
\(879\) 0 0
\(880\) −268.237 −0.0102753
\(881\) −32542.0 −1.24446 −0.622230 0.782835i \(-0.713773\pi\)
−0.622230 + 0.782835i \(0.713773\pi\)
\(882\) 0 0
\(883\) 27641.9 1.05348 0.526741 0.850026i \(-0.323414\pi\)
0.526741 + 0.850026i \(0.323414\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 26825.3 1.01717
\(887\) −40099.9 −1.51795 −0.758976 0.651119i \(-0.774300\pi\)
−0.758976 + 0.651119i \(0.774300\pi\)
\(888\) 0 0
\(889\) 5936.70 0.223971
\(890\) 2597.49 0.0978293
\(891\) 0 0
\(892\) −15312.1 −0.574761
\(893\) −15566.8 −0.583339
\(894\) 0 0
\(895\) −1872.47 −0.0699328
\(896\) −21993.8 −0.820044
\(897\) 0 0
\(898\) −48648.4 −1.80781
\(899\) −19156.1 −0.710670
\(900\) 0 0
\(901\) 1234.60 0.0456497
\(902\) 55789.3 2.05940
\(903\) 0 0
\(904\) 7344.26 0.270206
\(905\) 1968.35 0.0722984
\(906\) 0 0
\(907\) −36824.9 −1.34812 −0.674062 0.738674i \(-0.735452\pi\)
−0.674062 + 0.738674i \(0.735452\pi\)
\(908\) 11132.4 0.406874
\(909\) 0 0
\(910\) 0 0
\(911\) −34520.5 −1.25545 −0.627725 0.778435i \(-0.716014\pi\)
−0.627725 + 0.778435i \(0.716014\pi\)
\(912\) 0 0
\(913\) −3002.29 −0.108830
\(914\) −67642.6 −2.44794
\(915\) 0 0
\(916\) 59999.8 2.16424
\(917\) 12733.2 0.458545
\(918\) 0 0
\(919\) 23522.8 0.844336 0.422168 0.906518i \(-0.361269\pi\)
0.422168 + 0.906518i \(0.361269\pi\)
\(920\) 275.171 0.00986101
\(921\) 0 0
\(922\) 44300.4 1.58238
\(923\) 0 0
\(924\) 0 0
\(925\) 21749.5 0.773102
\(926\) 51789.7 1.83792
\(927\) 0 0
\(928\) 26239.4 0.928180
\(929\) 24563.2 0.867482 0.433741 0.901038i \(-0.357193\pi\)
0.433741 + 0.901038i \(0.357193\pi\)
\(930\) 0 0
\(931\) 15143.0 0.533073
\(932\) 62360.9 2.19174
\(933\) 0 0
\(934\) −29428.0 −1.03096
\(935\) −223.060 −0.00780197
\(936\) 0 0
\(937\) −12115.6 −0.422411 −0.211206 0.977442i \(-0.567739\pi\)
−0.211206 + 0.977442i \(0.567739\pi\)
\(938\) −2176.36 −0.0757576
\(939\) 0 0
\(940\) −9300.19 −0.322701
\(941\) −14898.3 −0.516123 −0.258062 0.966128i \(-0.583084\pi\)
−0.258062 + 0.966128i \(0.583084\pi\)
\(942\) 0 0
\(943\) 1386.35 0.0478745
\(944\) −1254.99 −0.0432695
\(945\) 0 0
\(946\) −76872.7 −2.64201
\(947\) −7434.32 −0.255103 −0.127552 0.991832i \(-0.540712\pi\)
−0.127552 + 0.991832i \(0.540712\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 32156.7 1.09821
\(951\) 0 0
\(952\) −422.564 −0.0143859
\(953\) 23528.6 0.799754 0.399877 0.916569i \(-0.369053\pi\)
0.399877 + 0.916569i \(0.369053\pi\)
\(954\) 0 0
\(955\) 3652.56 0.123764
\(956\) −61573.8 −2.08309
\(957\) 0 0
\(958\) 43637.9 1.47169
\(959\) −5947.43 −0.200263
\(960\) 0 0
\(961\) −11252.7 −0.377723
\(962\) 0 0
\(963\) 0 0
\(964\) 75257.9 2.51441
\(965\) 1458.17 0.0486425
\(966\) 0 0
\(967\) 23558.0 0.783427 0.391713 0.920087i \(-0.371882\pi\)
0.391713 + 0.920087i \(0.371882\pi\)
\(968\) 4894.88 0.162528
\(969\) 0 0
\(970\) 15041.0 0.497875
\(971\) 262.338 0.00867026 0.00433513 0.999991i \(-0.498620\pi\)
0.00433513 + 0.999991i \(0.498620\pi\)
\(972\) 0 0
\(973\) 3160.86 0.104144
\(974\) 22429.7 0.737878
\(975\) 0 0
\(976\) 390.903 0.0128202
\(977\) −33144.4 −1.08535 −0.542673 0.839944i \(-0.682588\pi\)
−0.542673 + 0.839944i \(0.682588\pi\)
\(978\) 0 0
\(979\) 7995.28 0.261011
\(980\) 9047.00 0.294894
\(981\) 0 0
\(982\) −13460.3 −0.437410
\(983\) 4866.80 0.157911 0.0789557 0.996878i \(-0.474841\pi\)
0.0789557 + 0.996878i \(0.474841\pi\)
\(984\) 0 0
\(985\) 8763.17 0.283470
\(986\) 1293.28 0.0417714
\(987\) 0 0
\(988\) 0 0
\(989\) −1910.26 −0.0614184
\(990\) 0 0
\(991\) 12533.9 0.401770 0.200885 0.979615i \(-0.435618\pi\)
0.200885 + 0.979615i \(0.435618\pi\)
\(992\) −25393.1 −0.812733
\(993\) 0 0
\(994\) −12210.7 −0.389639
\(995\) −3473.37 −0.110666
\(996\) 0 0
\(997\) −3560.92 −0.113115 −0.0565574 0.998399i \(-0.518012\pi\)
−0.0565574 + 0.998399i \(0.518012\pi\)
\(998\) 61262.1 1.94310
\(999\) 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.t.1.2 2
3.2 odd 2 169.4.a.f.1.1 2
13.3 even 3 117.4.g.d.100.1 4
13.9 even 3 117.4.g.d.55.1 4
13.12 even 2 1521.4.a.l.1.1 2
39.2 even 12 169.4.e.g.147.1 8
39.5 even 4 169.4.b.e.168.4 4
39.8 even 4 169.4.b.e.168.1 4
39.11 even 12 169.4.e.g.147.4 8
39.17 odd 6 169.4.c.f.146.1 4
39.20 even 12 169.4.e.g.23.1 8
39.23 odd 6 169.4.c.f.22.1 4
39.29 odd 6 13.4.c.b.9.2 yes 4
39.32 even 12 169.4.e.g.23.4 8
39.35 odd 6 13.4.c.b.3.2 4
39.38 odd 2 169.4.a.j.1.2 2
156.35 even 6 208.4.i.e.81.2 4
156.107 even 6 208.4.i.e.113.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.2 4 39.35 odd 6
13.4.c.b.9.2 yes 4 39.29 odd 6
117.4.g.d.55.1 4 13.9 even 3
117.4.g.d.100.1 4 13.3 even 3
169.4.a.f.1.1 2 3.2 odd 2
169.4.a.j.1.2 2 39.38 odd 2
169.4.b.e.168.1 4 39.8 even 4
169.4.b.e.168.4 4 39.5 even 4
169.4.c.f.22.1 4 39.23 odd 6
169.4.c.f.146.1 4 39.17 odd 6
169.4.e.g.23.1 8 39.20 even 12
169.4.e.g.23.4 8 39.32 even 12
169.4.e.g.147.1 8 39.2 even 12
169.4.e.g.147.4 8 39.11 even 12
208.4.i.e.81.2 4 156.35 even 6
208.4.i.e.113.2 4 156.107 even 6
1521.4.a.l.1.1 2 13.12 even 2
1521.4.a.t.1.2 2 1.1 even 1 trivial