Properties

Label 1521.4.a.bf.1.5
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $9$
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: \(0\)
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 1.5
Root \(0.107680\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +O(q^{10})\) \(q-0.447278 q^{2} -7.79994 q^{4} -1.93073 q^{5} -8.14537 q^{7} +7.06697 q^{8} +0.863573 q^{10} -8.40842 q^{11} +3.64325 q^{14} +59.2386 q^{16} +52.1271 q^{17} +48.8304 q^{19} +15.0596 q^{20} +3.76090 q^{22} -88.9229 q^{23} -121.272 q^{25} +63.5334 q^{28} -191.979 q^{29} +115.257 q^{31} -83.0319 q^{32} -23.3153 q^{34} +15.7265 q^{35} -136.716 q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077 q^{41} +202.048 q^{43} +65.5852 q^{44} +39.7733 q^{46} -618.160 q^{47} -276.653 q^{49} +54.2425 q^{50} +453.170 q^{53} +16.2344 q^{55} -57.5631 q^{56} +85.8679 q^{58} -500.044 q^{59} +480.502 q^{61} -51.5522 q^{62} -436.771 q^{64} +886.769 q^{67} -406.589 q^{68} -7.03412 q^{70} -123.732 q^{71} +673.168 q^{73} +61.1501 q^{74} -380.875 q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374 q^{80} +195.048 q^{82} +939.418 q^{83} -100.643 q^{85} -90.3716 q^{86} -59.4220 q^{88} +754.979 q^{89} +693.594 q^{92} +276.489 q^{94} -94.2783 q^{95} +1051.10 q^{97} +123.741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 44 q^{4} - 33 q^{5} + 83 q^{7} - 87 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.447278 −0.158137 −0.0790684 0.996869i \(-0.525195\pi\)
−0.0790684 + 0.996869i \(0.525195\pi\)
\(3\) 0 0
\(4\) −7.79994 −0.974993
\(5\) −1.93073 −0.172690 −0.0863448 0.996265i \(-0.527519\pi\)
−0.0863448 + 0.996265i \(0.527519\pi\)
\(6\) 0 0
\(7\) −8.14537 −0.439809 −0.219904 0.975521i \(-0.570575\pi\)
−0.219904 + 0.975521i \(0.570575\pi\)
\(8\) 7.06697 0.312319
\(9\) 0 0
\(10\) 0.863573 0.0273086
\(11\) −8.40842 −0.230476 −0.115238 0.993338i \(-0.536763\pi\)
−0.115238 + 0.993338i \(0.536763\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 3.64325 0.0695499
\(15\) 0 0
\(16\) 59.2386 0.925604
\(17\) 52.1271 0.743688 0.371844 0.928295i \(-0.378726\pi\)
0.371844 + 0.928295i \(0.378726\pi\)
\(18\) 0 0
\(19\) 48.8304 0.589604 0.294802 0.955558i \(-0.404746\pi\)
0.294802 + 0.955558i \(0.404746\pi\)
\(20\) 15.0596 0.168371
\(21\) 0 0
\(22\) 3.76090 0.0364467
\(23\) −88.9229 −0.806162 −0.403081 0.915164i \(-0.632061\pi\)
−0.403081 + 0.915164i \(0.632061\pi\)
\(24\) 0 0
\(25\) −121.272 −0.970178
\(26\) 0 0
\(27\) 0 0
\(28\) 63.5334 0.428810
\(29\) −191.979 −1.22930 −0.614648 0.788802i \(-0.710702\pi\)
−0.614648 + 0.788802i \(0.710702\pi\)
\(30\) 0 0
\(31\) 115.257 0.667769 0.333885 0.942614i \(-0.391640\pi\)
0.333885 + 0.942614i \(0.391640\pi\)
\(32\) −83.0319 −0.458691
\(33\) 0 0
\(34\) −23.3153 −0.117604
\(35\) 15.7265 0.0759504
\(36\) 0 0
\(37\) −136.716 −0.607459 −0.303729 0.952758i \(-0.598232\pi\)
−0.303729 + 0.952758i \(0.598232\pi\)
\(38\) −21.8408 −0.0932380
\(39\) 0 0
\(40\) −13.6444 −0.0539342
\(41\) −436.077 −1.66107 −0.830535 0.556967i \(-0.811965\pi\)
−0.830535 + 0.556967i \(0.811965\pi\)
\(42\) 0 0
\(43\) 202.048 0.716559 0.358279 0.933614i \(-0.383364\pi\)
0.358279 + 0.933614i \(0.383364\pi\)
\(44\) 65.5852 0.224712
\(45\) 0 0
\(46\) 39.7733 0.127484
\(47\) −618.160 −1.91847 −0.959233 0.282617i \(-0.908798\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(48\) 0 0
\(49\) −276.653 −0.806568
\(50\) 54.2425 0.153421
\(51\) 0 0
\(52\) 0 0
\(53\) 453.170 1.17448 0.587242 0.809411i \(-0.300214\pi\)
0.587242 + 0.809411i \(0.300214\pi\)
\(54\) 0 0
\(55\) 16.2344 0.0398008
\(56\) −57.5631 −0.137361
\(57\) 0 0
\(58\) 85.8679 0.194397
\(59\) −500.044 −1.10339 −0.551697 0.834045i \(-0.686019\pi\)
−0.551697 + 0.834045i \(0.686019\pi\)
\(60\) 0 0
\(61\) 480.502 1.00856 0.504279 0.863541i \(-0.331758\pi\)
0.504279 + 0.863541i \(0.331758\pi\)
\(62\) −51.5522 −0.105599
\(63\) 0 0
\(64\) −436.771 −0.853068
\(65\) 0 0
\(66\) 0 0
\(67\) 886.769 1.61696 0.808478 0.588526i \(-0.200292\pi\)
0.808478 + 0.588526i \(0.200292\pi\)
\(68\) −406.589 −0.725090
\(69\) 0 0
\(70\) −7.03412 −0.0120105
\(71\) −123.732 −0.206821 −0.103410 0.994639i \(-0.532975\pi\)
−0.103410 + 0.994639i \(0.532975\pi\)
\(72\) 0 0
\(73\) 673.168 1.07929 0.539646 0.841892i \(-0.318558\pi\)
0.539646 + 0.841892i \(0.318558\pi\)
\(74\) 61.1501 0.0960616
\(75\) 0 0
\(76\) −380.875 −0.574859
\(77\) 68.4897 0.101365
\(78\) 0 0
\(79\) −681.298 −0.970279 −0.485139 0.874437i \(-0.661231\pi\)
−0.485139 + 0.874437i \(0.661231\pi\)
\(80\) −114.374 −0.159842
\(81\) 0 0
\(82\) 195.048 0.262676
\(83\) 939.418 1.24234 0.621172 0.783675i \(-0.286657\pi\)
0.621172 + 0.783675i \(0.286657\pi\)
\(84\) 0 0
\(85\) −100.643 −0.128427
\(86\) −90.3716 −0.113314
\(87\) 0 0
\(88\) −59.4220 −0.0719819
\(89\) 754.979 0.899187 0.449593 0.893233i \(-0.351569\pi\)
0.449593 + 0.893233i \(0.351569\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 693.594 0.786002
\(93\) 0 0
\(94\) 276.489 0.303380
\(95\) −94.2783 −0.101818
\(96\) 0 0
\(97\) 1051.10 1.10024 0.550118 0.835087i \(-0.314583\pi\)
0.550118 + 0.835087i \(0.314583\pi\)
\(98\) 123.741 0.127548
\(99\) 0 0
\(100\) 945.917 0.945917
\(101\) −599.873 −0.590986 −0.295493 0.955345i \(-0.595484\pi\)
−0.295493 + 0.955345i \(0.595484\pi\)
\(102\) 0 0
\(103\) −293.312 −0.280591 −0.140295 0.990110i \(-0.544805\pi\)
−0.140295 + 0.990110i \(0.544805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −202.693 −0.185729
\(107\) −1533.69 −1.38568 −0.692839 0.721092i \(-0.743640\pi\)
−0.692839 + 0.721092i \(0.743640\pi\)
\(108\) 0 0
\(109\) 590.718 0.519088 0.259544 0.965731i \(-0.416428\pi\)
0.259544 + 0.965731i \(0.416428\pi\)
\(110\) −7.26128 −0.00629396
\(111\) 0 0
\(112\) −482.521 −0.407089
\(113\) 653.785 0.544274 0.272137 0.962259i \(-0.412270\pi\)
0.272137 + 0.962259i \(0.412270\pi\)
\(114\) 0 0
\(115\) 171.686 0.139216
\(116\) 1497.42 1.19855
\(117\) 0 0
\(118\) 223.659 0.174487
\(119\) −424.595 −0.327080
\(120\) 0 0
\(121\) −1260.30 −0.946881
\(122\) −214.918 −0.159490
\(123\) 0 0
\(124\) −899.002 −0.651070
\(125\) 475.485 0.340229
\(126\) 0 0
\(127\) −141.203 −0.0986595 −0.0493297 0.998783i \(-0.515709\pi\)
−0.0493297 + 0.998783i \(0.515709\pi\)
\(128\) 859.613 0.593592
\(129\) 0 0
\(130\) 0 0
\(131\) 731.910 0.488147 0.244074 0.969757i \(-0.421516\pi\)
0.244074 + 0.969757i \(0.421516\pi\)
\(132\) 0 0
\(133\) −397.742 −0.259313
\(134\) −396.633 −0.255700
\(135\) 0 0
\(136\) 368.381 0.232268
\(137\) −1762.27 −1.09898 −0.549492 0.835499i \(-0.685179\pi\)
−0.549492 + 0.835499i \(0.685179\pi\)
\(138\) 0 0
\(139\) −664.776 −0.405652 −0.202826 0.979215i \(-0.565013\pi\)
−0.202826 + 0.979215i \(0.565013\pi\)
\(140\) −122.666 −0.0740511
\(141\) 0 0
\(142\) 55.3425 0.0327059
\(143\) 0 0
\(144\) 0 0
\(145\) 370.659 0.212286
\(146\) −301.094 −0.170676
\(147\) 0 0
\(148\) 1066.38 0.592268
\(149\) 3300.71 1.81480 0.907399 0.420270i \(-0.138064\pi\)
0.907399 + 0.420270i \(0.138064\pi\)
\(150\) 0 0
\(151\) 1464.15 0.789079 0.394540 0.918879i \(-0.370904\pi\)
0.394540 + 0.918879i \(0.370904\pi\)
\(152\) 345.083 0.184144
\(153\) 0 0
\(154\) −30.6339 −0.0160296
\(155\) −222.531 −0.115317
\(156\) 0 0
\(157\) 1535.32 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(158\) 304.730 0.153437
\(159\) 0 0
\(160\) 160.312 0.0792111
\(161\) 724.310 0.354557
\(162\) 0 0
\(163\) 793.554 0.381325 0.190663 0.981656i \(-0.438936\pi\)
0.190663 + 0.981656i \(0.438936\pi\)
\(164\) 3401.38 1.61953
\(165\) 0 0
\(166\) −420.181 −0.196460
\(167\) 1336.87 0.619460 0.309730 0.950825i \(-0.399761\pi\)
0.309730 + 0.950825i \(0.399761\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 45.0156 0.0203090
\(171\) 0 0
\(172\) −1575.96 −0.698640
\(173\) −568.460 −0.249822 −0.124911 0.992168i \(-0.539865\pi\)
−0.124911 + 0.992168i \(0.539865\pi\)
\(174\) 0 0
\(175\) 987.808 0.426693
\(176\) −498.103 −0.213329
\(177\) 0 0
\(178\) −337.686 −0.142194
\(179\) −1546.00 −0.645552 −0.322776 0.946475i \(-0.604616\pi\)
−0.322776 + 0.946475i \(0.604616\pi\)
\(180\) 0 0
\(181\) 3408.96 1.39992 0.699960 0.714182i \(-0.253201\pi\)
0.699960 + 0.714182i \(0.253201\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −628.416 −0.251780
\(185\) 263.962 0.104902
\(186\) 0 0
\(187\) −438.307 −0.171402
\(188\) 4821.61 1.87049
\(189\) 0 0
\(190\) 42.1686 0.0161012
\(191\) 3464.71 1.31255 0.656277 0.754520i \(-0.272130\pi\)
0.656277 + 0.754520i \(0.272130\pi\)
\(192\) 0 0
\(193\) 4652.24 1.73510 0.867552 0.497346i \(-0.165692\pi\)
0.867552 + 0.497346i \(0.165692\pi\)
\(194\) −470.134 −0.173988
\(195\) 0 0
\(196\) 2157.88 0.786398
\(197\) 2870.98 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(198\) 0 0
\(199\) 25.0330 0.00891730 0.00445865 0.999990i \(-0.498581\pi\)
0.00445865 + 0.999990i \(0.498581\pi\)
\(200\) −857.028 −0.303005
\(201\) 0 0
\(202\) 268.310 0.0934565
\(203\) 1563.74 0.540655
\(204\) 0 0
\(205\) 841.947 0.286849
\(206\) 131.192 0.0443717
\(207\) 0 0
\(208\) 0 0
\(209\) −410.587 −0.135889
\(210\) 0 0
\(211\) −3605.91 −1.17650 −0.588249 0.808679i \(-0.700183\pi\)
−0.588249 + 0.808679i \(0.700183\pi\)
\(212\) −3534.70 −1.14511
\(213\) 0 0
\(214\) 685.987 0.219127
\(215\) −390.100 −0.123742
\(216\) 0 0
\(217\) −938.815 −0.293691
\(218\) −264.215 −0.0820868
\(219\) 0 0
\(220\) −126.627 −0.0388054
\(221\) 0 0
\(222\) 0 0
\(223\) −4304.37 −1.29256 −0.646282 0.763098i \(-0.723677\pi\)
−0.646282 + 0.763098i \(0.723677\pi\)
\(224\) 676.326 0.201736
\(225\) 0 0
\(226\) −292.424 −0.0860697
\(227\) 5475.19 1.60089 0.800443 0.599409i \(-0.204598\pi\)
0.800443 + 0.599409i \(0.204598\pi\)
\(228\) 0 0
\(229\) 378.108 0.109110 0.0545548 0.998511i \(-0.482626\pi\)
0.0545548 + 0.998511i \(0.482626\pi\)
\(230\) −76.7914 −0.0220151
\(231\) 0 0
\(232\) −1356.71 −0.383932
\(233\) −2547.96 −0.716405 −0.358202 0.933644i \(-0.616610\pi\)
−0.358202 + 0.933644i \(0.616610\pi\)
\(234\) 0 0
\(235\) 1193.50 0.331299
\(236\) 3900.32 1.07580
\(237\) 0 0
\(238\) 189.912 0.0517234
\(239\) 6313.91 1.70884 0.854420 0.519583i \(-0.173913\pi\)
0.854420 + 0.519583i \(0.173913\pi\)
\(240\) 0 0
\(241\) −6763.73 −1.80784 −0.903921 0.427699i \(-0.859324\pi\)
−0.903921 + 0.427699i \(0.859324\pi\)
\(242\) 563.704 0.149737
\(243\) 0 0
\(244\) −3747.89 −0.983336
\(245\) 534.142 0.139286
\(246\) 0 0
\(247\) 0 0
\(248\) 814.521 0.208557
\(249\) 0 0
\(250\) −212.674 −0.0538027
\(251\) 3416.00 0.859028 0.429514 0.903060i \(-0.358685\pi\)
0.429514 + 0.903060i \(0.358685\pi\)
\(252\) 0 0
\(253\) 747.701 0.185801
\(254\) 63.1571 0.0156017
\(255\) 0 0
\(256\) 3109.68 0.759199
\(257\) −7002.36 −1.69959 −0.849796 0.527112i \(-0.823275\pi\)
−0.849796 + 0.527112i \(0.823275\pi\)
\(258\) 0 0
\(259\) 1113.60 0.267166
\(260\) 0 0
\(261\) 0 0
\(262\) −327.367 −0.0771940
\(263\) −3369.76 −0.790071 −0.395035 0.918666i \(-0.629268\pi\)
−0.395035 + 0.918666i \(0.629268\pi\)
\(264\) 0 0
\(265\) −874.948 −0.202821
\(266\) 177.901 0.0410069
\(267\) 0 0
\(268\) −6916.75 −1.57652
\(269\) 7801.70 1.76832 0.884160 0.467185i \(-0.154732\pi\)
0.884160 + 0.467185i \(0.154732\pi\)
\(270\) 0 0
\(271\) 4410.05 0.988528 0.494264 0.869312i \(-0.335438\pi\)
0.494264 + 0.869312i \(0.335438\pi\)
\(272\) 3087.94 0.688360
\(273\) 0 0
\(274\) 788.225 0.173790
\(275\) 1019.71 0.223603
\(276\) 0 0
\(277\) −4636.44 −1.00569 −0.502846 0.864376i \(-0.667714\pi\)
−0.502846 + 0.864376i \(0.667714\pi\)
\(278\) 297.340 0.0641485
\(279\) 0 0
\(280\) 111.139 0.0237207
\(281\) −1090.50 −0.231508 −0.115754 0.993278i \(-0.536928\pi\)
−0.115754 + 0.993278i \(0.536928\pi\)
\(282\) 0 0
\(283\) 1140.52 0.239565 0.119782 0.992800i \(-0.461780\pi\)
0.119782 + 0.992800i \(0.461780\pi\)
\(284\) 965.101 0.201649
\(285\) 0 0
\(286\) 0 0
\(287\) 3552.01 0.730553
\(288\) 0 0
\(289\) −2195.76 −0.446929
\(290\) −165.788 −0.0335703
\(291\) 0 0
\(292\) −5250.67 −1.05230
\(293\) 335.079 0.0668106 0.0334053 0.999442i \(-0.489365\pi\)
0.0334053 + 0.999442i \(0.489365\pi\)
\(294\) 0 0
\(295\) 965.449 0.190545
\(296\) −966.168 −0.189721
\(297\) 0 0
\(298\) −1476.34 −0.286986
\(299\) 0 0
\(300\) 0 0
\(301\) −1645.76 −0.315149
\(302\) −654.883 −0.124782
\(303\) 0 0
\(304\) 2892.65 0.545739
\(305\) −927.719 −0.174167
\(306\) 0 0
\(307\) 2540.04 0.472207 0.236104 0.971728i \(-0.424130\pi\)
0.236104 + 0.971728i \(0.424130\pi\)
\(308\) −534.215 −0.0988303
\(309\) 0 0
\(310\) 99.5332 0.0182358
\(311\) −2376.36 −0.433283 −0.216642 0.976251i \(-0.569510\pi\)
−0.216642 + 0.976251i \(0.569510\pi\)
\(312\) 0 0
\(313\) 1315.78 0.237612 0.118806 0.992917i \(-0.462093\pi\)
0.118806 + 0.992917i \(0.462093\pi\)
\(314\) −686.713 −0.123419
\(315\) 0 0
\(316\) 5314.09 0.946015
\(317\) 6043.13 1.07071 0.535357 0.844626i \(-0.320177\pi\)
0.535357 + 0.844626i \(0.320177\pi\)
\(318\) 0 0
\(319\) 1614.24 0.283323
\(320\) 843.286 0.147316
\(321\) 0 0
\(322\) −323.968 −0.0560685
\(323\) 2545.39 0.438481
\(324\) 0 0
\(325\) 0 0
\(326\) −354.940 −0.0603015
\(327\) 0 0
\(328\) −3081.75 −0.518784
\(329\) 5035.14 0.843758
\(330\) 0 0
\(331\) 8168.18 1.35639 0.678193 0.734884i \(-0.262763\pi\)
0.678193 + 0.734884i \(0.262763\pi\)
\(332\) −7327.40 −1.21128
\(333\) 0 0
\(334\) −597.951 −0.0979594
\(335\) −1712.11 −0.279232
\(336\) 0 0
\(337\) 3076.32 0.497263 0.248631 0.968598i \(-0.420019\pi\)
0.248631 + 0.968598i \(0.420019\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 785.012 0.125215
\(341\) −969.133 −0.153905
\(342\) 0 0
\(343\) 5047.30 0.794544
\(344\) 1427.87 0.223795
\(345\) 0 0
\(346\) 254.260 0.0395061
\(347\) 8754.07 1.35430 0.677152 0.735844i \(-0.263214\pi\)
0.677152 + 0.735844i \(0.263214\pi\)
\(348\) 0 0
\(349\) 1064.97 0.163343 0.0816715 0.996659i \(-0.473974\pi\)
0.0816715 + 0.996659i \(0.473974\pi\)
\(350\) −441.825 −0.0674758
\(351\) 0 0
\(352\) 698.167 0.105717
\(353\) −5047.62 −0.761070 −0.380535 0.924767i \(-0.624260\pi\)
−0.380535 + 0.924767i \(0.624260\pi\)
\(354\) 0 0
\(355\) 238.893 0.0357158
\(356\) −5888.79 −0.876700
\(357\) 0 0
\(358\) 691.494 0.102085
\(359\) −8152.39 −1.19851 −0.599257 0.800557i \(-0.704537\pi\)
−0.599257 + 0.800557i \(0.704537\pi\)
\(360\) 0 0
\(361\) −4474.59 −0.652367
\(362\) −1524.75 −0.221379
\(363\) 0 0
\(364\) 0 0
\(365\) −1299.71 −0.186383
\(366\) 0 0
\(367\) 11636.3 1.65507 0.827537 0.561412i \(-0.189742\pi\)
0.827537 + 0.561412i \(0.189742\pi\)
\(368\) −5267.67 −0.746186
\(369\) 0 0
\(370\) −118.064 −0.0165888
\(371\) −3691.24 −0.516548
\(372\) 0 0
\(373\) 4720.59 0.655289 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(374\) 196.045 0.0271049
\(375\) 0 0
\(376\) −4368.52 −0.599173
\(377\) 0 0
\(378\) 0 0
\(379\) 12933.7 1.75293 0.876464 0.481467i \(-0.159896\pi\)
0.876464 + 0.481467i \(0.159896\pi\)
\(380\) 735.365 0.0992722
\(381\) 0 0
\(382\) −1549.69 −0.207563
\(383\) −554.945 −0.0740376 −0.0370188 0.999315i \(-0.511786\pi\)
−0.0370188 + 0.999315i \(0.511786\pi\)
\(384\) 0 0
\(385\) −132.235 −0.0175047
\(386\) −2080.84 −0.274384
\(387\) 0 0
\(388\) −8198.51 −1.07272
\(389\) −5091.35 −0.663604 −0.331802 0.943349i \(-0.607657\pi\)
−0.331802 + 0.943349i \(0.607657\pi\)
\(390\) 0 0
\(391\) −4635.30 −0.599532
\(392\) −1955.10 −0.251907
\(393\) 0 0
\(394\) −1284.13 −0.164196
\(395\) 1315.40 0.167557
\(396\) 0 0
\(397\) 10254.6 1.29638 0.648188 0.761480i \(-0.275527\pi\)
0.648188 + 0.761480i \(0.275527\pi\)
\(398\) −11.1967 −0.00141015
\(399\) 0 0
\(400\) −7184.00 −0.898001
\(401\) −505.816 −0.0629906 −0.0314953 0.999504i \(-0.510027\pi\)
−0.0314953 + 0.999504i \(0.510027\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4678.97 0.576207
\(405\) 0 0
\(406\) −699.426 −0.0854974
\(407\) 1149.57 0.140005
\(408\) 0 0
\(409\) 2219.21 0.268295 0.134148 0.990961i \(-0.457170\pi\)
0.134148 + 0.990961i \(0.457170\pi\)
\(410\) −376.585 −0.0453614
\(411\) 0 0
\(412\) 2287.82 0.273574
\(413\) 4073.04 0.485282
\(414\) 0 0
\(415\) −1813.76 −0.214540
\(416\) 0 0
\(417\) 0 0
\(418\) 183.646 0.0214891
\(419\) −10249.3 −1.19501 −0.597506 0.801864i \(-0.703842\pi\)
−0.597506 + 0.801864i \(0.703842\pi\)
\(420\) 0 0
\(421\) 97.9194 0.0113356 0.00566781 0.999984i \(-0.498196\pi\)
0.00566781 + 0.999984i \(0.498196\pi\)
\(422\) 1612.85 0.186048
\(423\) 0 0
\(424\) 3202.54 0.366814
\(425\) −6321.58 −0.721510
\(426\) 0 0
\(427\) −3913.87 −0.443572
\(428\) 11962.7 1.35103
\(429\) 0 0
\(430\) 174.483 0.0195682
\(431\) −11727.2 −1.31062 −0.655310 0.755360i \(-0.727462\pi\)
−0.655310 + 0.755360i \(0.727462\pi\)
\(432\) 0 0
\(433\) 6945.13 0.770812 0.385406 0.922747i \(-0.374061\pi\)
0.385406 + 0.922747i \(0.374061\pi\)
\(434\) 419.911 0.0464433
\(435\) 0 0
\(436\) −4607.57 −0.506107
\(437\) −4342.15 −0.475316
\(438\) 0 0
\(439\) 11446.3 1.24442 0.622210 0.782851i \(-0.286235\pi\)
0.622210 + 0.782851i \(0.286235\pi\)
\(440\) 114.728 0.0124305
\(441\) 0 0
\(442\) 0 0
\(443\) 10513.6 1.12758 0.563789 0.825919i \(-0.309343\pi\)
0.563789 + 0.825919i \(0.309343\pi\)
\(444\) 0 0
\(445\) −1457.66 −0.155280
\(446\) 1925.25 0.204402
\(447\) 0 0
\(448\) 3557.66 0.375187
\(449\) 9862.37 1.03660 0.518300 0.855199i \(-0.326565\pi\)
0.518300 + 0.855199i \(0.326565\pi\)
\(450\) 0 0
\(451\) 3666.72 0.382836
\(452\) −5099.49 −0.530663
\(453\) 0 0
\(454\) −2448.93 −0.253159
\(455\) 0 0
\(456\) 0 0
\(457\) −5990.48 −0.613179 −0.306590 0.951842i \(-0.599188\pi\)
−0.306590 + 0.951842i \(0.599188\pi\)
\(458\) −169.120 −0.0172542
\(459\) 0 0
\(460\) −1339.14 −0.135734
\(461\) −16579.5 −1.67502 −0.837510 0.546423i \(-0.815989\pi\)
−0.837510 + 0.546423i \(0.815989\pi\)
\(462\) 0 0
\(463\) 3606.65 0.362020 0.181010 0.983481i \(-0.442063\pi\)
0.181010 + 0.983481i \(0.442063\pi\)
\(464\) −11372.6 −1.13784
\(465\) 0 0
\(466\) 1139.65 0.113290
\(467\) 6789.61 0.672775 0.336387 0.941724i \(-0.390795\pi\)
0.336387 + 0.941724i \(0.390795\pi\)
\(468\) 0 0
\(469\) −7223.06 −0.711152
\(470\) −533.826 −0.0523906
\(471\) 0 0
\(472\) −3533.80 −0.344611
\(473\) −1698.90 −0.165149
\(474\) 0 0
\(475\) −5921.78 −0.572021
\(476\) 3311.82 0.318901
\(477\) 0 0
\(478\) −2824.07 −0.270230
\(479\) 12688.3 1.21032 0.605161 0.796103i \(-0.293109\pi\)
0.605161 + 0.796103i \(0.293109\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3025.27 0.285886
\(483\) 0 0
\(484\) 9830.26 0.923202
\(485\) −2029.39 −0.189999
\(486\) 0 0
\(487\) −1217.28 −0.113265 −0.0566325 0.998395i \(-0.518036\pi\)
−0.0566325 + 0.998395i \(0.518036\pi\)
\(488\) 3395.69 0.314991
\(489\) 0 0
\(490\) −238.910 −0.0220262
\(491\) 10496.3 0.964747 0.482374 0.875966i \(-0.339775\pi\)
0.482374 + 0.875966i \(0.339775\pi\)
\(492\) 0 0
\(493\) −10007.3 −0.914211
\(494\) 0 0
\(495\) 0 0
\(496\) 6827.69 0.618090
\(497\) 1007.84 0.0909615
\(498\) 0 0
\(499\) −10516.5 −0.943450 −0.471725 0.881746i \(-0.656368\pi\)
−0.471725 + 0.881746i \(0.656368\pi\)
\(500\) −3708.75 −0.331721
\(501\) 0 0
\(502\) −1527.90 −0.135844
\(503\) −13400.9 −1.18790 −0.593952 0.804500i \(-0.702433\pi\)
−0.593952 + 0.804500i \(0.702433\pi\)
\(504\) 0 0
\(505\) 1158.19 0.102057
\(506\) −334.430 −0.0293819
\(507\) 0 0
\(508\) 1101.38 0.0961923
\(509\) −4683.90 −0.407878 −0.203939 0.978984i \(-0.565374\pi\)
−0.203939 + 0.978984i \(0.565374\pi\)
\(510\) 0 0
\(511\) −5483.20 −0.474682
\(512\) −8267.80 −0.713649
\(513\) 0 0
\(514\) 3132.00 0.268768
\(515\) 566.305 0.0484551
\(516\) 0 0
\(517\) 5197.75 0.442160
\(518\) −498.090 −0.0422487
\(519\) 0 0
\(520\) 0 0
\(521\) −15034.6 −1.26426 −0.632128 0.774864i \(-0.717818\pi\)
−0.632128 + 0.774864i \(0.717818\pi\)
\(522\) 0 0
\(523\) 8009.33 0.669643 0.334822 0.942282i \(-0.391324\pi\)
0.334822 + 0.942282i \(0.391324\pi\)
\(524\) −5708.86 −0.475940
\(525\) 0 0
\(526\) 1507.22 0.124939
\(527\) 6008.04 0.496612
\(528\) 0 0
\(529\) −4259.71 −0.350103
\(530\) 391.345 0.0320735
\(531\) 0 0
\(532\) 3102.36 0.252828
\(533\) 0 0
\(534\) 0 0
\(535\) 2961.14 0.239292
\(536\) 6266.77 0.505006
\(537\) 0 0
\(538\) −3489.53 −0.279636
\(539\) 2326.21 0.185894
\(540\) 0 0
\(541\) 20833.2 1.65561 0.827807 0.561013i \(-0.189588\pi\)
0.827807 + 0.561013i \(0.189588\pi\)
\(542\) −1972.52 −0.156323
\(543\) 0 0
\(544\) −4328.22 −0.341123
\(545\) −1140.52 −0.0896411
\(546\) 0 0
\(547\) −14247.2 −1.11365 −0.556825 0.830630i \(-0.687981\pi\)
−0.556825 + 0.830630i \(0.687981\pi\)
\(548\) 13745.6 1.07150
\(549\) 0 0
\(550\) −456.093 −0.0353598
\(551\) −9374.41 −0.724797
\(552\) 0 0
\(553\) 5549.43 0.426737
\(554\) 2073.78 0.159037
\(555\) 0 0
\(556\) 5185.22 0.395508
\(557\) −2007.94 −0.152745 −0.0763725 0.997079i \(-0.524334\pi\)
−0.0763725 + 0.997079i \(0.524334\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 931.616 0.0703000
\(561\) 0 0
\(562\) 487.758 0.0366100
\(563\) −10562.1 −0.790659 −0.395330 0.918539i \(-0.629370\pi\)
−0.395330 + 0.918539i \(0.629370\pi\)
\(564\) 0 0
\(565\) −1262.28 −0.0939905
\(566\) −510.129 −0.0378840
\(567\) 0 0
\(568\) −874.409 −0.0645940
\(569\) 23207.2 1.70983 0.854916 0.518766i \(-0.173608\pi\)
0.854916 + 0.518766i \(0.173608\pi\)
\(570\) 0 0
\(571\) 7987.78 0.585426 0.292713 0.956200i \(-0.405442\pi\)
0.292713 + 0.956200i \(0.405442\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1588.74 −0.115527
\(575\) 10783.9 0.782120
\(576\) 0 0
\(577\) 3224.41 0.232641 0.116321 0.993212i \(-0.462890\pi\)
0.116321 + 0.993212i \(0.462890\pi\)
\(578\) 982.116 0.0706759
\(579\) 0 0
\(580\) −2891.12 −0.206978
\(581\) −7651.90 −0.546393
\(582\) 0 0
\(583\) −3810.44 −0.270690
\(584\) 4757.26 0.337084
\(585\) 0 0
\(586\) −149.873 −0.0105652
\(587\) 661.034 0.0464800 0.0232400 0.999730i \(-0.492602\pi\)
0.0232400 + 0.999730i \(0.492602\pi\)
\(588\) 0 0
\(589\) 5628.07 0.393719
\(590\) −431.824 −0.0301321
\(591\) 0 0
\(592\) −8098.87 −0.562266
\(593\) 10172.5 0.704443 0.352221 0.935917i \(-0.385426\pi\)
0.352221 + 0.935917i \(0.385426\pi\)
\(594\) 0 0
\(595\) 819.777 0.0564834
\(596\) −25745.4 −1.76942
\(597\) 0 0
\(598\) 0 0
\(599\) −23462.2 −1.60040 −0.800200 0.599733i \(-0.795274\pi\)
−0.800200 + 0.599733i \(0.795274\pi\)
\(600\) 0 0
\(601\) −10482.6 −0.711474 −0.355737 0.934586i \(-0.615770\pi\)
−0.355737 + 0.934586i \(0.615770\pi\)
\(602\) 736.111 0.0498366
\(603\) 0 0
\(604\) −11420.3 −0.769347
\(605\) 2433.29 0.163516
\(606\) 0 0
\(607\) −24949.8 −1.66834 −0.834169 0.551508i \(-0.814053\pi\)
−0.834169 + 0.551508i \(0.814053\pi\)
\(608\) −4054.48 −0.270446
\(609\) 0 0
\(610\) 414.949 0.0275423
\(611\) 0 0
\(612\) 0 0
\(613\) 8238.86 0.542846 0.271423 0.962460i \(-0.412506\pi\)
0.271423 + 0.962460i \(0.412506\pi\)
\(614\) −1136.10 −0.0746733
\(615\) 0 0
\(616\) 484.014 0.0316583
\(617\) −5248.85 −0.342481 −0.171241 0.985229i \(-0.554778\pi\)
−0.171241 + 0.985229i \(0.554778\pi\)
\(618\) 0 0
\(619\) −17820.9 −1.15716 −0.578580 0.815626i \(-0.696393\pi\)
−0.578580 + 0.815626i \(0.696393\pi\)
\(620\) 1735.73 0.112433
\(621\) 0 0
\(622\) 1062.89 0.0685180
\(623\) −6149.58 −0.395470
\(624\) 0 0
\(625\) 14241.0 0.911424
\(626\) −588.522 −0.0375752
\(627\) 0 0
\(628\) −11975.4 −0.760938
\(629\) −7126.62 −0.451760
\(630\) 0 0
\(631\) 25445.0 1.60531 0.802653 0.596446i \(-0.203421\pi\)
0.802653 + 0.596446i \(0.203421\pi\)
\(632\) −4814.71 −0.303036
\(633\) 0 0
\(634\) −2702.96 −0.169319
\(635\) 272.625 0.0170375
\(636\) 0 0
\(637\) 0 0
\(638\) −722.013 −0.0448037
\(639\) 0 0
\(640\) −1659.68 −0.102507
\(641\) 18701.0 1.15233 0.576167 0.817332i \(-0.304548\pi\)
0.576167 + 0.817332i \(0.304548\pi\)
\(642\) 0 0
\(643\) −28465.9 −1.74586 −0.872928 0.487849i \(-0.837782\pi\)
−0.872928 + 0.487849i \(0.837782\pi\)
\(644\) −5649.58 −0.345690
\(645\) 0 0
\(646\) −1138.50 −0.0693399
\(647\) 10924.6 0.663818 0.331909 0.943311i \(-0.392307\pi\)
0.331909 + 0.943311i \(0.392307\pi\)
\(648\) 0 0
\(649\) 4204.58 0.254305
\(650\) 0 0
\(651\) 0 0
\(652\) −6189.68 −0.371789
\(653\) 27110.9 1.62470 0.812351 0.583169i \(-0.198187\pi\)
0.812351 + 0.583169i \(0.198187\pi\)
\(654\) 0 0
\(655\) −1413.12 −0.0842980
\(656\) −25832.6 −1.53749
\(657\) 0 0
\(658\) −2252.11 −0.133429
\(659\) −1727.08 −0.102090 −0.0510452 0.998696i \(-0.516255\pi\)
−0.0510452 + 0.998696i \(0.516255\pi\)
\(660\) 0 0
\(661\) 20651.1 1.21518 0.607592 0.794249i \(-0.292136\pi\)
0.607592 + 0.794249i \(0.292136\pi\)
\(662\) −3653.45 −0.214494
\(663\) 0 0
\(664\) 6638.84 0.388007
\(665\) 767.932 0.0447806
\(666\) 0 0
\(667\) 17071.3 0.991010
\(668\) −10427.5 −0.603969
\(669\) 0 0
\(670\) 765.790 0.0441568
\(671\) −4040.26 −0.232448
\(672\) 0 0
\(673\) 10858.0 0.621907 0.310954 0.950425i \(-0.399352\pi\)
0.310954 + 0.950425i \(0.399352\pi\)
\(674\) −1375.97 −0.0786355
\(675\) 0 0
\(676\) 0 0
\(677\) −29132.9 −1.65387 −0.826933 0.562301i \(-0.809916\pi\)
−0.826933 + 0.562301i \(0.809916\pi\)
\(678\) 0 0
\(679\) −8561.59 −0.483894
\(680\) −711.244 −0.0401102
\(681\) 0 0
\(682\) 433.472 0.0243380
\(683\) 9443.65 0.529065 0.264532 0.964377i \(-0.414782\pi\)
0.264532 + 0.964377i \(0.414782\pi\)
\(684\) 0 0
\(685\) 3402.46 0.189783
\(686\) −2257.55 −0.125647
\(687\) 0 0
\(688\) 11969.0 0.663249
\(689\) 0 0
\(690\) 0 0
\(691\) −23700.1 −1.30477 −0.652384 0.757889i \(-0.726231\pi\)
−0.652384 + 0.757889i \(0.726231\pi\)
\(692\) 4433.96 0.243575
\(693\) 0 0
\(694\) −3915.51 −0.214165
\(695\) 1283.50 0.0700519
\(696\) 0 0
\(697\) −22731.5 −1.23532
\(698\) −476.340 −0.0258305
\(699\) 0 0
\(700\) −7704.84 −0.416022
\(701\) −21643.1 −1.16612 −0.583059 0.812430i \(-0.698144\pi\)
−0.583059 + 0.812430i \(0.698144\pi\)
\(702\) 0 0
\(703\) −6675.91 −0.358160
\(704\) 3672.55 0.196611
\(705\) 0 0
\(706\) 2257.69 0.120353
\(707\) 4886.18 0.259921
\(708\) 0 0
\(709\) −28965.3 −1.53429 −0.767146 0.641472i \(-0.778324\pi\)
−0.767146 + 0.641472i \(0.778324\pi\)
\(710\) −106.851 −0.00564798
\(711\) 0 0
\(712\) 5335.41 0.280833
\(713\) −10249.0 −0.538330
\(714\) 0 0
\(715\) 0 0
\(716\) 12058.7 0.629408
\(717\) 0 0
\(718\) 3646.39 0.189529
\(719\) 21256.5 1.10255 0.551276 0.834323i \(-0.314141\pi\)
0.551276 + 0.834323i \(0.314141\pi\)
\(720\) 0 0
\(721\) 2389.13 0.123406
\(722\) 2001.39 0.103163
\(723\) 0 0
\(724\) −26589.7 −1.36491
\(725\) 23281.7 1.19264
\(726\) 0 0
\(727\) −11868.9 −0.605494 −0.302747 0.953071i \(-0.597904\pi\)
−0.302747 + 0.953071i \(0.597904\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 581.330 0.0294739
\(731\) 10532.2 0.532896
\(732\) 0 0
\(733\) 27296.4 1.37546 0.687732 0.725965i \(-0.258607\pi\)
0.687732 + 0.725965i \(0.258607\pi\)
\(734\) −5204.68 −0.261728
\(735\) 0 0
\(736\) 7383.44 0.369779
\(737\) −7456.32 −0.372669
\(738\) 0 0
\(739\) −26424.6 −1.31535 −0.657674 0.753302i \(-0.728460\pi\)
−0.657674 + 0.753302i \(0.728460\pi\)
\(740\) −2058.89 −0.102279
\(741\) 0 0
\(742\) 1651.01 0.0816853
\(743\) 25974.8 1.28253 0.641267 0.767318i \(-0.278409\pi\)
0.641267 + 0.767318i \(0.278409\pi\)
\(744\) 0 0
\(745\) −6372.78 −0.313397
\(746\) −2111.42 −0.103625
\(747\) 0 0
\(748\) 3418.77 0.167116
\(749\) 12492.5 0.609433
\(750\) 0 0
\(751\) 20234.8 0.983195 0.491598 0.870823i \(-0.336413\pi\)
0.491598 + 0.870823i \(0.336413\pi\)
\(752\) −36619.0 −1.77574
\(753\) 0 0
\(754\) 0 0
\(755\) −2826.88 −0.136266
\(756\) 0 0
\(757\) −3363.48 −0.161490 −0.0807449 0.996735i \(-0.525730\pi\)
−0.0807449 + 0.996735i \(0.525730\pi\)
\(758\) −5784.97 −0.277202
\(759\) 0 0
\(760\) −666.262 −0.0317998
\(761\) 21311.3 1.01516 0.507579 0.861605i \(-0.330541\pi\)
0.507579 + 0.861605i \(0.330541\pi\)
\(762\) 0 0
\(763\) −4811.62 −0.228299
\(764\) −27024.5 −1.27973
\(765\) 0 0
\(766\) 248.215 0.0117081
\(767\) 0 0
\(768\) 0 0
\(769\) −34989.6 −1.64078 −0.820388 0.571807i \(-0.806242\pi\)
−0.820388 + 0.571807i \(0.806242\pi\)
\(770\) 59.1458 0.00276814
\(771\) 0 0
\(772\) −36287.2 −1.69171
\(773\) −20034.1 −0.932181 −0.466091 0.884737i \(-0.654338\pi\)
−0.466091 + 0.884737i \(0.654338\pi\)
\(774\) 0 0
\(775\) −13977.5 −0.647855
\(776\) 7428.09 0.343625
\(777\) 0 0
\(778\) 2277.25 0.104940
\(779\) −21293.9 −0.979373
\(780\) 0 0
\(781\) 1040.39 0.0476671
\(782\) 2073.27 0.0948081
\(783\) 0 0
\(784\) −16388.5 −0.746563
\(785\) −2964.28 −0.134777
\(786\) 0 0
\(787\) 5591.98 0.253282 0.126641 0.991949i \(-0.459580\pi\)
0.126641 + 0.991949i \(0.459580\pi\)
\(788\) −22393.5 −1.01235
\(789\) 0 0
\(790\) −588.351 −0.0264969
\(791\) −5325.32 −0.239376
\(792\) 0 0
\(793\) 0 0
\(794\) −4586.64 −0.205005
\(795\) 0 0
\(796\) −195.256 −0.00869430
\(797\) 12738.6 0.566152 0.283076 0.959098i \(-0.408645\pi\)
0.283076 + 0.959098i \(0.408645\pi\)
\(798\) 0 0
\(799\) −32222.9 −1.42674
\(800\) 10069.5 0.445012
\(801\) 0 0
\(802\) 226.240 0.00996113
\(803\) −5660.28 −0.248751
\(804\) 0 0
\(805\) −1398.45 −0.0612283
\(806\) 0 0
\(807\) 0 0
\(808\) −4239.28 −0.184576
\(809\) 8806.48 0.382719 0.191359 0.981520i \(-0.438710\pi\)
0.191359 + 0.981520i \(0.438710\pi\)
\(810\) 0 0
\(811\) −10565.8 −0.457478 −0.228739 0.973488i \(-0.573460\pi\)
−0.228739 + 0.973488i \(0.573460\pi\)
\(812\) −12197.1 −0.527134
\(813\) 0 0
\(814\) −514.176 −0.0221399
\(815\) −1532.14 −0.0658509
\(816\) 0 0
\(817\) 9866.09 0.422486
\(818\) −992.604 −0.0424273
\(819\) 0 0
\(820\) −6567.14 −0.279676
\(821\) −422.966 −0.0179800 −0.00899002 0.999960i \(-0.502862\pi\)
−0.00899002 + 0.999960i \(0.502862\pi\)
\(822\) 0 0
\(823\) 11148.8 0.472202 0.236101 0.971728i \(-0.424130\pi\)
0.236101 + 0.971728i \(0.424130\pi\)
\(824\) −2072.83 −0.0876339
\(825\) 0 0
\(826\) −1821.78 −0.0767409
\(827\) 813.158 0.0341914 0.0170957 0.999854i \(-0.494558\pi\)
0.0170957 + 0.999854i \(0.494558\pi\)
\(828\) 0 0
\(829\) −28320.8 −1.18652 −0.593258 0.805012i \(-0.702159\pi\)
−0.593258 + 0.805012i \(0.702159\pi\)
\(830\) 811.255 0.0339266
\(831\) 0 0
\(832\) 0 0
\(833\) −14421.1 −0.599835
\(834\) 0 0
\(835\) −2581.13 −0.106974
\(836\) 3202.55 0.132491
\(837\) 0 0
\(838\) 4584.28 0.188975
\(839\) 22139.3 0.911006 0.455503 0.890234i \(-0.349459\pi\)
0.455503 + 0.890234i \(0.349459\pi\)
\(840\) 0 0
\(841\) 12466.8 0.511166
\(842\) −43.7972 −0.00179258
\(843\) 0 0
\(844\) 28125.9 1.14708
\(845\) 0 0
\(846\) 0 0
\(847\) 10265.6 0.416446
\(848\) 26845.2 1.08711
\(849\) 0 0
\(850\) 2827.50 0.114097
\(851\) 12157.2 0.489710
\(852\) 0 0
\(853\) −14080.5 −0.565191 −0.282595 0.959239i \(-0.591195\pi\)
−0.282595 + 0.959239i \(0.591195\pi\)
\(854\) 1750.59 0.0701451
\(855\) 0 0
\(856\) −10838.6 −0.432774
\(857\) −10907.7 −0.434772 −0.217386 0.976086i \(-0.569753\pi\)
−0.217386 + 0.976086i \(0.569753\pi\)
\(858\) 0 0
\(859\) 7739.24 0.307403 0.153702 0.988117i \(-0.450881\pi\)
0.153702 + 0.988117i \(0.450881\pi\)
\(860\) 3042.76 0.120648
\(861\) 0 0
\(862\) 5245.30 0.207257
\(863\) 29072.5 1.14674 0.573372 0.819295i \(-0.305635\pi\)
0.573372 + 0.819295i \(0.305635\pi\)
\(864\) 0 0
\(865\) 1097.54 0.0431417
\(866\) −3106.40 −0.121894
\(867\) 0 0
\(868\) 7322.70 0.286346
\(869\) 5728.64 0.223626
\(870\) 0 0
\(871\) 0 0
\(872\) 4174.59 0.162121
\(873\) 0 0
\(874\) 1942.15 0.0751649
\(875\) −3873.00 −0.149636
\(876\) 0 0
\(877\) 2391.96 0.0920988 0.0460494 0.998939i \(-0.485337\pi\)
0.0460494 + 0.998939i \(0.485337\pi\)
\(878\) −5119.66 −0.196788
\(879\) 0 0
\(880\) 961.702 0.0368397
\(881\) 11975.8 0.457974 0.228987 0.973430i \(-0.426459\pi\)
0.228987 + 0.973430i \(0.426459\pi\)
\(882\) 0 0
\(883\) −44712.4 −1.70407 −0.852035 0.523485i \(-0.824632\pi\)
−0.852035 + 0.523485i \(0.824632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4702.51 −0.178311
\(887\) −17023.3 −0.644403 −0.322202 0.946671i \(-0.604423\pi\)
−0.322202 + 0.946671i \(0.604423\pi\)
\(888\) 0 0
\(889\) 1150.15 0.0433913
\(890\) 651.979 0.0245555
\(891\) 0 0
\(892\) 33573.8 1.26024
\(893\) −30185.0 −1.13113
\(894\) 0 0
\(895\) 2984.91 0.111480
\(896\) −7001.87 −0.261067
\(897\) 0 0
\(898\) −4411.22 −0.163925
\(899\) −22127.0 −0.820886
\(900\) 0 0
\(901\) 23622.5 0.873449
\(902\) −1640.04 −0.0605405
\(903\) 0 0
\(904\) 4620.28 0.169987
\(905\) −6581.77 −0.241752
\(906\) 0 0
\(907\) 2303.59 0.0843324 0.0421662 0.999111i \(-0.486574\pi\)
0.0421662 + 0.999111i \(0.486574\pi\)
\(908\) −42706.2 −1.56085
\(909\) 0 0
\(910\) 0 0
\(911\) 12897.0 0.469042 0.234521 0.972111i \(-0.424648\pi\)
0.234521 + 0.972111i \(0.424648\pi\)
\(912\) 0 0
\(913\) −7899.01 −0.286330
\(914\) 2679.41 0.0969662
\(915\) 0 0
\(916\) −2949.22 −0.106381
\(917\) −5961.68 −0.214691
\(918\) 0 0
\(919\) 16785.2 0.602496 0.301248 0.953546i \(-0.402597\pi\)
0.301248 + 0.953546i \(0.402597\pi\)
\(920\) 1213.30 0.0434797
\(921\) 0 0
\(922\) 7415.64 0.264882
\(923\) 0 0
\(924\) 0 0
\(925\) 16579.9 0.589344
\(926\) −1613.18 −0.0572487
\(927\) 0 0
\(928\) 15940.4 0.563866
\(929\) −16348.0 −0.577352 −0.288676 0.957427i \(-0.593215\pi\)
−0.288676 + 0.957427i \(0.593215\pi\)
\(930\) 0 0
\(931\) −13509.1 −0.475556
\(932\) 19873.9 0.698489
\(933\) 0 0
\(934\) −3036.85 −0.106390
\(935\) 846.251 0.0295993
\(936\) 0 0
\(937\) −26247.5 −0.915121 −0.457560 0.889179i \(-0.651277\pi\)
−0.457560 + 0.889179i \(0.651277\pi\)
\(938\) 3230.72 0.112459
\(939\) 0 0
\(940\) −9309.22 −0.323014
\(941\) −43838.4 −1.51869 −0.759347 0.650686i \(-0.774481\pi\)
−0.759347 + 0.650686i \(0.774481\pi\)
\(942\) 0 0
\(943\) 38777.3 1.33909
\(944\) −29621.9 −1.02130
\(945\) 0 0
\(946\) 759.882 0.0261162
\(947\) 45077.2 1.54679 0.773395 0.633924i \(-0.218557\pi\)
0.773395 + 0.633924i \(0.218557\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 2648.68 0.0904575
\(951\) 0 0
\(952\) −3000.60 −0.102153
\(953\) −26479.1 −0.900046 −0.450023 0.893017i \(-0.648584\pi\)
−0.450023 + 0.893017i \(0.648584\pi\)
\(954\) 0 0
\(955\) −6689.41 −0.226664
\(956\) −49248.1 −1.66611
\(957\) 0 0
\(958\) −5675.21 −0.191396
\(959\) 14354.3 0.483343
\(960\) 0 0
\(961\) −16506.7 −0.554084
\(962\) 0 0
\(963\) 0 0
\(964\) 52756.7 1.76263
\(965\) −8982.20 −0.299635
\(966\) 0 0
\(967\) −602.475 −0.0200355 −0.0100177 0.999950i \(-0.503189\pi\)
−0.0100177 + 0.999950i \(0.503189\pi\)
\(968\) −8906.49 −0.295729
\(969\) 0 0
\(970\) 907.701 0.0300459
\(971\) 11330.2 0.374464 0.187232 0.982316i \(-0.440048\pi\)
0.187232 + 0.982316i \(0.440048\pi\)
\(972\) 0 0
\(973\) 5414.85 0.178409
\(974\) 544.461 0.0179114
\(975\) 0 0
\(976\) 28464.3 0.933524
\(977\) −55633.6 −1.82178 −0.910888 0.412653i \(-0.864602\pi\)
−0.910888 + 0.412653i \(0.864602\pi\)
\(978\) 0 0
\(979\) −6348.18 −0.207241
\(980\) −4166.27 −0.135803
\(981\) 0 0
\(982\) −4694.76 −0.152562
\(983\) 6884.90 0.223392 0.111696 0.993742i \(-0.464372\pi\)
0.111696 + 0.993742i \(0.464372\pi\)
\(984\) 0 0
\(985\) −5543.08 −0.179307
\(986\) 4476.05 0.144570
\(987\) 0 0
\(988\) 0 0
\(989\) −17966.7 −0.577662
\(990\) 0 0
\(991\) 52646.1 1.68755 0.843773 0.536700i \(-0.180329\pi\)
0.843773 + 0.536700i \(0.180329\pi\)
\(992\) −9570.05 −0.306300
\(993\) 0 0
\(994\) −450.786 −0.0143844
\(995\) −48.3319 −0.00153992
\(996\) 0 0
\(997\) 26100.7 0.829106 0.414553 0.910025i \(-0.363938\pi\)
0.414553 + 0.910025i \(0.363938\pi\)
\(998\) 4703.78 0.149194
\(999\) 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.1.5 9
3.2 odd 2 507.4.a.p.1.5 yes 9
13.12 even 2 1521.4.a.bi.1.5 9
39.5 even 4 507.4.b.k.337.8 18
39.8 even 4 507.4.b.k.337.11 18
39.38 odd 2 507.4.a.o.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.5 9 39.38 odd 2
507.4.a.p.1.5 yes 9 3.2 odd 2
507.4.b.k.337.8 18 39.5 even 4
507.4.b.k.337.11 18 39.8 even 4
1521.4.a.bf.1.5 9 1.1 even 1 trivial
1521.4.a.bi.1.5 9 13.12 even 2