Properties

Label 1225.4.a.p.1.2
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +46.2665 q^{11} +13.1662 q^{12} +61.3325 q^{13} -36.0000 q^{16} -101.332 q^{17} -4.63325 q^{18} +3.66750 q^{19} +107.182 q^{22} -84.8655 q^{23} +123.166 q^{24} +142.084 q^{26} +145.000 q^{27} +30.1980 q^{29} +188.997 q^{31} +113.668 q^{32} -231.332 q^{33} -234.749 q^{34} +5.26650 q^{36} -18.0685 q^{37} +8.49623 q^{38} -306.662 q^{39} +481.662 q^{41} +97.7995 q^{43} -121.831 q^{44} -196.602 q^{46} -117.665 q^{47} +180.000 q^{48} +506.662 q^{51} -161.504 q^{52} -667.995 q^{53} +335.911 q^{54} -18.3375 q^{57} +69.9574 q^{58} +57.3350 q^{59} +738.997 q^{61} +437.836 q^{62} +551.325 q^{64} -535.911 q^{66} -552.396 q^{67} +266.834 q^{68} +424.327 q^{69} -740.264 q^{71} +49.2665 q^{72} -233.325 q^{73} -41.8580 q^{74} -9.65745 q^{76} -710.422 q^{78} -1075.19 q^{79} -671.000 q^{81} +1115.83 q^{82} -683.325 q^{83} +226.565 q^{86} -150.990 q^{87} -1139.69 q^{88} -1380.32 q^{89} +223.472 q^{92} -944.987 q^{93} -272.586 q^{94} -568.338 q^{96} -218.008 q^{97} -92.5330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} - 40 q^{12} - 10 q^{13} - 72 q^{16} - 70 q^{17} + 4 q^{18} + 140 q^{19} + 22 q^{22} + 16 q^{23} + 180 q^{24} + 450 q^{26} + 290 q^{27} - 258 q^{29} - 20 q^{31} + 360 q^{32} - 330 q^{33} - 370 q^{34} - 16 q^{36} - 328 q^{37} - 580 q^{38} + 50 q^{39} + 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 350 q^{51} - 920 q^{52} - 540 q^{53} - 290 q^{54} - 700 q^{57} + 1314 q^{58} + 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 110 q^{66} - 468 q^{67} + 600 q^{68} - 80 q^{69} - 1056 q^{71} + 72 q^{72} + 860 q^{73} + 1296 q^{74} + 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 148 q^{86} + 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 100 q^{93} - 910 q^{94} - 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.31662 0.819051 0.409525 0.912299i \(-0.365694\pi\)
0.409525 + 0.912299i \(0.365694\pi\)
\(3\) −5.00000 −0.962250 −0.481125 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) −2.63325 −0.329156
\(5\) 0 0
\(6\) −11.5831 −0.788132
\(7\) 0 0
\(8\) −24.6332 −1.08865
\(9\) −2.00000 −0.0740741
\(10\) 0 0
\(11\) 46.2665 1.26817 0.634085 0.773263i \(-0.281377\pi\)
0.634085 + 0.773263i \(0.281377\pi\)
\(12\) 13.1662 0.316731
\(13\) 61.3325 1.30851 0.654253 0.756276i \(-0.272983\pi\)
0.654253 + 0.756276i \(0.272983\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −36.0000 −0.562500
\(17\) −101.332 −1.44569 −0.722845 0.691010i \(-0.757166\pi\)
−0.722845 + 0.691010i \(0.757166\pi\)
\(18\) −4.63325 −0.0606704
\(19\) 3.66750 0.0442833 0.0221417 0.999755i \(-0.492952\pi\)
0.0221417 + 0.999755i \(0.492952\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 107.182 1.03870
\(23\) −84.8655 −0.769377 −0.384689 0.923046i \(-0.625691\pi\)
−0.384689 + 0.923046i \(0.625691\pi\)
\(24\) 123.166 1.04755
\(25\) 0 0
\(26\) 142.084 1.07173
\(27\) 145.000 1.03353
\(28\) 0 0
\(29\) 30.1980 0.193366 0.0966832 0.995315i \(-0.469177\pi\)
0.0966832 + 0.995315i \(0.469177\pi\)
\(30\) 0 0
\(31\) 188.997 1.09500 0.547499 0.836806i \(-0.315580\pi\)
0.547499 + 0.836806i \(0.315580\pi\)
\(32\) 113.668 0.627930
\(33\) −231.332 −1.22030
\(34\) −234.749 −1.18409
\(35\) 0 0
\(36\) 5.26650 0.0243819
\(37\) −18.0685 −0.0802823 −0.0401411 0.999194i \(-0.512781\pi\)
−0.0401411 + 0.999194i \(0.512781\pi\)
\(38\) 8.49623 0.0362703
\(39\) −306.662 −1.25911
\(40\) 0 0
\(41\) 481.662 1.83471 0.917354 0.398072i \(-0.130321\pi\)
0.917354 + 0.398072i \(0.130321\pi\)
\(42\) 0 0
\(43\) 97.7995 0.346844 0.173422 0.984848i \(-0.444518\pi\)
0.173422 + 0.984848i \(0.444518\pi\)
\(44\) −121.831 −0.417426
\(45\) 0 0
\(46\) −196.602 −0.630159
\(47\) −117.665 −0.365175 −0.182587 0.983190i \(-0.558447\pi\)
−0.182587 + 0.983190i \(0.558447\pi\)
\(48\) 180.000 0.541266
\(49\) 0 0
\(50\) 0 0
\(51\) 506.662 1.39112
\(52\) −161.504 −0.430703
\(53\) −667.995 −1.73125 −0.865624 0.500694i \(-0.833078\pi\)
−0.865624 + 0.500694i \(0.833078\pi\)
\(54\) 335.911 0.846512
\(55\) 0 0
\(56\) 0 0
\(57\) −18.3375 −0.0426116
\(58\) 69.9574 0.158377
\(59\) 57.3350 0.126515 0.0632575 0.997997i \(-0.479851\pi\)
0.0632575 + 0.997997i \(0.479851\pi\)
\(60\) 0 0
\(61\) 738.997 1.55113 0.775565 0.631268i \(-0.217465\pi\)
0.775565 + 0.631268i \(0.217465\pi\)
\(62\) 437.836 0.896859
\(63\) 0 0
\(64\) 551.325 1.07681
\(65\) 0 0
\(66\) −535.911 −0.999485
\(67\) −552.396 −1.00725 −0.503626 0.863922i \(-0.668001\pi\)
−0.503626 + 0.863922i \(0.668001\pi\)
\(68\) 266.834 0.475858
\(69\) 424.327 0.740334
\(70\) 0 0
\(71\) −740.264 −1.23737 −0.618684 0.785640i \(-0.712334\pi\)
−0.618684 + 0.785640i \(0.712334\pi\)
\(72\) 49.2665 0.0806405
\(73\) −233.325 −0.374091 −0.187045 0.982351i \(-0.559891\pi\)
−0.187045 + 0.982351i \(0.559891\pi\)
\(74\) −41.8580 −0.0657553
\(75\) 0 0
\(76\) −9.65745 −0.0145761
\(77\) 0 0
\(78\) −710.422 −1.03127
\(79\) −1075.19 −1.53124 −0.765619 0.643294i \(-0.777567\pi\)
−0.765619 + 0.643294i \(0.777567\pi\)
\(80\) 0 0
\(81\) −671.000 −0.920439
\(82\) 1115.83 1.50272
\(83\) −683.325 −0.903671 −0.451835 0.892101i \(-0.649231\pi\)
−0.451835 + 0.892101i \(0.649231\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 226.565 0.284083
\(87\) −150.990 −0.186067
\(88\) −1139.69 −1.38059
\(89\) −1380.32 −1.64397 −0.821985 0.569509i \(-0.807133\pi\)
−0.821985 + 0.569509i \(0.807133\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 223.472 0.253245
\(93\) −944.987 −1.05366
\(94\) −272.586 −0.299096
\(95\) 0 0
\(96\) −568.338 −0.604226
\(97\) −218.008 −0.228199 −0.114100 0.993469i \(-0.536398\pi\)
−0.114100 + 0.993469i \(0.536398\pi\)
\(98\) 0 0
\(99\) −92.5330 −0.0939385
\(100\) 0 0
\(101\) −1474.33 −1.45249 −0.726243 0.687438i \(-0.758735\pi\)
−0.726243 + 0.687438i \(0.758735\pi\)
\(102\) 1173.75 1.13939
\(103\) 810.990 0.775818 0.387909 0.921698i \(-0.373198\pi\)
0.387909 + 0.921698i \(0.373198\pi\)
\(104\) −1510.82 −1.42450
\(105\) 0 0
\(106\) −1547.49 −1.41798
\(107\) −440.660 −0.398133 −0.199066 0.979986i \(-0.563791\pi\)
−0.199066 + 0.979986i \(0.563791\pi\)
\(108\) −381.821 −0.340192
\(109\) −1906.19 −1.67504 −0.837522 0.546404i \(-0.815996\pi\)
−0.837522 + 0.546404i \(0.815996\pi\)
\(110\) 0 0
\(111\) 90.3425 0.0772517
\(112\) 0 0
\(113\) −962.470 −0.801252 −0.400626 0.916242i \(-0.631207\pi\)
−0.400626 + 0.916242i \(0.631207\pi\)
\(114\) −42.4812 −0.0349011
\(115\) 0 0
\(116\) −79.5188 −0.0636478
\(117\) −122.665 −0.0969263
\(118\) 132.824 0.103622
\(119\) 0 0
\(120\) 0 0
\(121\) 809.589 0.608256
\(122\) 1711.98 1.27045
\(123\) −2408.31 −1.76545
\(124\) −497.678 −0.360426
\(125\) 0 0
\(126\) 0 0
\(127\) 1621.74 1.13312 0.566558 0.824022i \(-0.308275\pi\)
0.566558 + 0.824022i \(0.308275\pi\)
\(128\) 367.873 0.254029
\(129\) −488.997 −0.333751
\(130\) 0 0
\(131\) −1380.32 −0.920602 −0.460301 0.887763i \(-0.652259\pi\)
−0.460301 + 0.887763i \(0.652259\pi\)
\(132\) 609.156 0.401668
\(133\) 0 0
\(134\) −1279.69 −0.824991
\(135\) 0 0
\(136\) 2496.15 1.57385
\(137\) 1949.66 1.21584 0.607921 0.793997i \(-0.292004\pi\)
0.607921 + 0.793997i \(0.292004\pi\)
\(138\) 983.008 0.606371
\(139\) 2800.00 1.70858 0.854291 0.519795i \(-0.173992\pi\)
0.854291 + 0.519795i \(0.173992\pi\)
\(140\) 0 0
\(141\) 588.325 0.351389
\(142\) −1714.91 −1.01347
\(143\) 2837.64 1.65941
\(144\) 72.0000 0.0416667
\(145\) 0 0
\(146\) −540.526 −0.306399
\(147\) 0 0
\(148\) 47.5789 0.0264254
\(149\) −1434.12 −0.788506 −0.394253 0.919002i \(-0.628997\pi\)
−0.394253 + 0.919002i \(0.628997\pi\)
\(150\) 0 0
\(151\) −1985.58 −1.07009 −0.535047 0.844822i \(-0.679706\pi\)
−0.535047 + 0.844822i \(0.679706\pi\)
\(152\) −90.3425 −0.0482089
\(153\) 202.665 0.107088
\(154\) 0 0
\(155\) 0 0
\(156\) 807.519 0.414444
\(157\) −40.6600 −0.0206689 −0.0103345 0.999947i \(-0.503290\pi\)
−0.0103345 + 0.999947i \(0.503290\pi\)
\(158\) −2490.80 −1.25416
\(159\) 3339.97 1.66589
\(160\) 0 0
\(161\) 0 0
\(162\) −1554.46 −0.753886
\(163\) 3953.98 1.90000 0.950000 0.312250i \(-0.101083\pi\)
0.950000 + 0.312250i \(0.101083\pi\)
\(164\) −1268.34 −0.603906
\(165\) 0 0
\(166\) −1583.01 −0.740152
\(167\) −3380.30 −1.56632 −0.783161 0.621819i \(-0.786394\pi\)
−0.783161 + 0.621819i \(0.786394\pi\)
\(168\) 0 0
\(169\) 1564.68 0.712187
\(170\) 0 0
\(171\) −7.33501 −0.00328025
\(172\) −257.530 −0.114166
\(173\) −3206.66 −1.40924 −0.704619 0.709586i \(-0.748882\pi\)
−0.704619 + 0.709586i \(0.748882\pi\)
\(174\) −349.787 −0.152398
\(175\) 0 0
\(176\) −1665.59 −0.713346
\(177\) −286.675 −0.121739
\(178\) −3197.68 −1.34649
\(179\) 1442.65 0.602395 0.301198 0.953562i \(-0.402614\pi\)
0.301198 + 0.953562i \(0.402614\pi\)
\(180\) 0 0
\(181\) −908.680 −0.373158 −0.186579 0.982440i \(-0.559740\pi\)
−0.186579 + 0.982440i \(0.559740\pi\)
\(182\) 0 0
\(183\) −3694.99 −1.49258
\(184\) 2090.51 0.837580
\(185\) 0 0
\(186\) −2189.18 −0.863003
\(187\) −4688.30 −1.83338
\(188\) 309.841 0.120199
\(189\) 0 0
\(190\) 0 0
\(191\) 2474.64 0.937479 0.468739 0.883336i \(-0.344708\pi\)
0.468739 + 0.883336i \(0.344708\pi\)
\(192\) −2756.62 −1.03616
\(193\) −3533.52 −1.31787 −0.658934 0.752201i \(-0.728992\pi\)
−0.658934 + 0.752201i \(0.728992\pi\)
\(194\) −505.042 −0.186907
\(195\) 0 0
\(196\) 0 0
\(197\) 1952.57 0.706165 0.353083 0.935592i \(-0.385133\pi\)
0.353083 + 0.935592i \(0.385133\pi\)
\(198\) −214.364 −0.0769404
\(199\) 4064.67 1.44792 0.723962 0.689840i \(-0.242319\pi\)
0.723962 + 0.689840i \(0.242319\pi\)
\(200\) 0 0
\(201\) 2761.98 0.969229
\(202\) −3415.46 −1.18966
\(203\) 0 0
\(204\) −1334.17 −0.457895
\(205\) 0 0
\(206\) 1878.76 0.635434
\(207\) 169.731 0.0569909
\(208\) −2207.97 −0.736034
\(209\) 169.683 0.0561588
\(210\) 0 0
\(211\) −4325.34 −1.41123 −0.705613 0.708598i \(-0.749328\pi\)
−0.705613 + 0.708598i \(0.749328\pi\)
\(212\) 1759.00 0.569851
\(213\) 3701.32 1.19066
\(214\) −1020.84 −0.326091
\(215\) 0 0
\(216\) −3571.82 −1.12515
\(217\) 0 0
\(218\) −4415.92 −1.37194
\(219\) 1166.62 0.359969
\(220\) 0 0
\(221\) −6214.97 −1.89169
\(222\) 209.290 0.0632730
\(223\) 982.970 0.295177 0.147589 0.989049i \(-0.452849\pi\)
0.147589 + 0.989049i \(0.452849\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2229.68 −0.656266
\(227\) 1660.96 0.485648 0.242824 0.970070i \(-0.421926\pi\)
0.242824 + 0.970070i \(0.421926\pi\)
\(228\) 48.2873 0.0140259
\(229\) −574.327 −0.165732 −0.0828660 0.996561i \(-0.526407\pi\)
−0.0828660 + 0.996561i \(0.526407\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −743.875 −0.210508
\(233\) 2316.48 0.651320 0.325660 0.945487i \(-0.394414\pi\)
0.325660 + 0.945487i \(0.394414\pi\)
\(234\) −284.169 −0.0793876
\(235\) 0 0
\(236\) −150.977 −0.0416432
\(237\) 5375.93 1.47343
\(238\) 0 0
\(239\) −3659.31 −0.990382 −0.495191 0.868784i \(-0.664902\pi\)
−0.495191 + 0.868784i \(0.664902\pi\)
\(240\) 0 0
\(241\) 2446.33 0.653868 0.326934 0.945047i \(-0.393985\pi\)
0.326934 + 0.945047i \(0.393985\pi\)
\(242\) 1875.51 0.498193
\(243\) −560.000 −0.147835
\(244\) −1945.96 −0.510564
\(245\) 0 0
\(246\) −5579.16 −1.44599
\(247\) 224.937 0.0579450
\(248\) −4655.62 −1.19207
\(249\) 3416.62 0.869557
\(250\) 0 0
\(251\) −2909.29 −0.731605 −0.365802 0.930693i \(-0.619205\pi\)
−0.365802 + 0.930693i \(0.619205\pi\)
\(252\) 0 0
\(253\) −3926.43 −0.975702
\(254\) 3756.95 0.928080
\(255\) 0 0
\(256\) −3558.38 −0.868744
\(257\) −168.680 −0.0409415 −0.0204708 0.999790i \(-0.506517\pi\)
−0.0204708 + 0.999790i \(0.506517\pi\)
\(258\) −1132.82 −0.273359
\(259\) 0 0
\(260\) 0 0
\(261\) −60.3960 −0.0143234
\(262\) −3197.68 −0.754020
\(263\) 3244.47 0.760695 0.380347 0.924844i \(-0.375804\pi\)
0.380347 + 0.924844i \(0.375804\pi\)
\(264\) 5698.47 1.32847
\(265\) 0 0
\(266\) 0 0
\(267\) 6901.59 1.58191
\(268\) 1454.60 0.331543
\(269\) −2848.65 −0.645671 −0.322836 0.946455i \(-0.604636\pi\)
−0.322836 + 0.946455i \(0.604636\pi\)
\(270\) 0 0
\(271\) −2850.98 −0.639057 −0.319529 0.947577i \(-0.603525\pi\)
−0.319529 + 0.947577i \(0.603525\pi\)
\(272\) 3647.97 0.813201
\(273\) 0 0
\(274\) 4516.62 0.995837
\(275\) 0 0
\(276\) −1117.36 −0.243685
\(277\) −2298.63 −0.498597 −0.249298 0.968427i \(-0.580200\pi\)
−0.249298 + 0.968427i \(0.580200\pi\)
\(278\) 6486.55 1.39942
\(279\) −377.995 −0.0811110
\(280\) 0 0
\(281\) 6109.20 1.29695 0.648477 0.761234i \(-0.275406\pi\)
0.648477 + 0.761234i \(0.275406\pi\)
\(282\) 1362.93 0.287806
\(283\) −5854.95 −1.22983 −0.614913 0.788595i \(-0.710809\pi\)
−0.614913 + 0.788595i \(0.710809\pi\)
\(284\) 1949.30 0.407288
\(285\) 0 0
\(286\) 6573.75 1.35914
\(287\) 0 0
\(288\) −227.335 −0.0465133
\(289\) 5355.27 1.09002
\(290\) 0 0
\(291\) 1090.04 0.219585
\(292\) 614.403 0.123134
\(293\) 5135.34 1.02392 0.511962 0.859008i \(-0.328919\pi\)
0.511962 + 0.859008i \(0.328919\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 445.086 0.0873990
\(297\) 6708.64 1.31069
\(298\) −3322.31 −0.645827
\(299\) −5205.01 −1.00673
\(300\) 0 0
\(301\) 0 0
\(302\) −4599.85 −0.876462
\(303\) 7371.64 1.39766
\(304\) −132.030 −0.0249094
\(305\) 0 0
\(306\) 469.499 0.0877106
\(307\) −2102.97 −0.390954 −0.195477 0.980708i \(-0.562625\pi\)
−0.195477 + 0.980708i \(0.562625\pi\)
\(308\) 0 0
\(309\) −4054.95 −0.746531
\(310\) 0 0
\(311\) −5764.30 −1.05101 −0.525504 0.850791i \(-0.676123\pi\)
−0.525504 + 0.850791i \(0.676123\pi\)
\(312\) 7554.09 1.37073
\(313\) 1360.01 0.245599 0.122799 0.992432i \(-0.460813\pi\)
0.122799 + 0.992432i \(0.460813\pi\)
\(314\) −94.1939 −0.0169289
\(315\) 0 0
\(316\) 2831.23 0.504017
\(317\) −5138.95 −0.910511 −0.455256 0.890361i \(-0.650452\pi\)
−0.455256 + 0.890361i \(0.650452\pi\)
\(318\) 7737.47 1.36445
\(319\) 1397.16 0.245222
\(320\) 0 0
\(321\) 2203.30 0.383103
\(322\) 0 0
\(323\) −371.637 −0.0640200
\(324\) 1766.91 0.302968
\(325\) 0 0
\(326\) 9159.90 1.55620
\(327\) 9530.94 1.61181
\(328\) −11864.9 −1.99735
\(329\) 0 0
\(330\) 0 0
\(331\) 1825.70 0.303171 0.151585 0.988444i \(-0.451562\pi\)
0.151585 + 0.988444i \(0.451562\pi\)
\(332\) 1799.37 0.297449
\(333\) 36.1370 0.00594684
\(334\) −7830.90 −1.28290
\(335\) 0 0
\(336\) 0 0
\(337\) −153.985 −0.0248905 −0.0124452 0.999923i \(-0.503962\pi\)
−0.0124452 + 0.999923i \(0.503962\pi\)
\(338\) 3624.76 0.583317
\(339\) 4812.35 0.771005
\(340\) 0 0
\(341\) 8744.25 1.38864
\(342\) −16.9925 −0.00268669
\(343\) 0 0
\(344\) −2409.12 −0.377590
\(345\) 0 0
\(346\) −7428.63 −1.15424
\(347\) −4359.39 −0.674421 −0.337211 0.941429i \(-0.609483\pi\)
−0.337211 + 0.941429i \(0.609483\pi\)
\(348\) 397.594 0.0612451
\(349\) 1689.00 0.259054 0.129527 0.991576i \(-0.458654\pi\)
0.129527 + 0.991576i \(0.458654\pi\)
\(350\) 0 0
\(351\) 8893.21 1.35238
\(352\) 5259.00 0.796322
\(353\) 3921.36 0.591254 0.295627 0.955303i \(-0.404471\pi\)
0.295627 + 0.955303i \(0.404471\pi\)
\(354\) −664.119 −0.0997105
\(355\) 0 0
\(356\) 3634.72 0.541123
\(357\) 0 0
\(358\) 3342.08 0.493392
\(359\) −2867.86 −0.421616 −0.210808 0.977528i \(-0.567609\pi\)
−0.210808 + 0.977528i \(0.567609\pi\)
\(360\) 0 0
\(361\) −6845.55 −0.998039
\(362\) −2105.07 −0.305636
\(363\) −4047.94 −0.585295
\(364\) 0 0
\(365\) 0 0
\(366\) −8559.90 −1.22249
\(367\) −11503.0 −1.63611 −0.818054 0.575142i \(-0.804947\pi\)
−0.818054 + 0.575142i \(0.804947\pi\)
\(368\) 3055.16 0.432775
\(369\) −963.325 −0.135904
\(370\) 0 0
\(371\) 0 0
\(372\) 2488.39 0.346820
\(373\) −5086.43 −0.706073 −0.353037 0.935610i \(-0.614851\pi\)
−0.353037 + 0.935610i \(0.614851\pi\)
\(374\) −10861.0 −1.50163
\(375\) 0 0
\(376\) 2898.47 0.397546
\(377\) 1852.12 0.253021
\(378\) 0 0
\(379\) 954.827 0.129409 0.0647047 0.997904i \(-0.479389\pi\)
0.0647047 + 0.997904i \(0.479389\pi\)
\(380\) 0 0
\(381\) −8108.68 −1.09034
\(382\) 5732.81 0.767843
\(383\) −3083.91 −0.411437 −0.205719 0.978611i \(-0.565953\pi\)
−0.205719 + 0.978611i \(0.565953\pi\)
\(384\) −1839.37 −0.244439
\(385\) 0 0
\(386\) −8185.84 −1.07940
\(387\) −195.599 −0.0256921
\(388\) 574.068 0.0751131
\(389\) −6331.15 −0.825198 −0.412599 0.910913i \(-0.635379\pi\)
−0.412599 + 0.910913i \(0.635379\pi\)
\(390\) 0 0
\(391\) 8599.63 1.11228
\(392\) 0 0
\(393\) 6901.59 0.885850
\(394\) 4523.36 0.578385
\(395\) 0 0
\(396\) 243.662 0.0309205
\(397\) 12133.2 1.53388 0.766939 0.641720i \(-0.221779\pi\)
0.766939 + 0.641720i \(0.221779\pi\)
\(398\) 9416.32 1.18592
\(399\) 0 0
\(400\) 0 0
\(401\) −270.669 −0.0337072 −0.0168536 0.999858i \(-0.505365\pi\)
−0.0168536 + 0.999858i \(0.505365\pi\)
\(402\) 6398.47 0.793848
\(403\) 11591.7 1.43281
\(404\) 3882.27 0.478095
\(405\) 0 0
\(406\) 0 0
\(407\) −835.967 −0.101812
\(408\) −12480.7 −1.51443
\(409\) 4019.92 0.485996 0.242998 0.970027i \(-0.421869\pi\)
0.242998 + 0.970027i \(0.421869\pi\)
\(410\) 0 0
\(411\) −9748.29 −1.16995
\(412\) −2135.54 −0.255365
\(413\) 0 0
\(414\) 393.203 0.0466784
\(415\) 0 0
\(416\) 6971.51 0.821650
\(417\) −14000.0 −1.64408
\(418\) 393.091 0.0459969
\(419\) 2437.28 0.284175 0.142087 0.989854i \(-0.454619\pi\)
0.142087 + 0.989854i \(0.454619\pi\)
\(420\) 0 0
\(421\) −4751.36 −0.550041 −0.275020 0.961438i \(-0.588685\pi\)
−0.275020 + 0.961438i \(0.588685\pi\)
\(422\) −10020.2 −1.15586
\(423\) 235.330 0.0270500
\(424\) 16454.9 1.88472
\(425\) 0 0
\(426\) 8574.57 0.975210
\(427\) 0 0
\(428\) 1160.37 0.131048
\(429\) −14188.2 −1.59677
\(430\) 0 0
\(431\) 7925.19 0.885714 0.442857 0.896592i \(-0.353965\pi\)
0.442857 + 0.896592i \(0.353965\pi\)
\(432\) −5220.00 −0.581360
\(433\) 11487.3 1.27492 0.637462 0.770481i \(-0.279984\pi\)
0.637462 + 0.770481i \(0.279984\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 5019.47 0.551351
\(437\) −311.245 −0.0340706
\(438\) 2702.63 0.294833
\(439\) −9147.92 −0.994548 −0.497274 0.867594i \(-0.665666\pi\)
−0.497274 + 0.867594i \(0.665666\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14397.8 −1.54939
\(443\) −1864.35 −0.199950 −0.0999752 0.994990i \(-0.531876\pi\)
−0.0999752 + 0.994990i \(0.531876\pi\)
\(444\) −237.894 −0.0254279
\(445\) 0 0
\(446\) 2277.17 0.241765
\(447\) 7170.58 0.758741
\(448\) 0 0
\(449\) 4490.88 0.472022 0.236011 0.971750i \(-0.424160\pi\)
0.236011 + 0.971750i \(0.424160\pi\)
\(450\) 0 0
\(451\) 22284.8 2.32672
\(452\) 2534.42 0.263737
\(453\) 9927.91 1.02970
\(454\) 3847.83 0.397770
\(455\) 0 0
\(456\) 451.713 0.0463890
\(457\) 14343.8 1.46822 0.734109 0.679032i \(-0.237600\pi\)
0.734109 + 0.679032i \(0.237600\pi\)
\(458\) −1330.50 −0.135743
\(459\) −14693.2 −1.49416
\(460\) 0 0
\(461\) −14558.7 −1.47086 −0.735429 0.677602i \(-0.763019\pi\)
−0.735429 + 0.677602i \(0.763019\pi\)
\(462\) 0 0
\(463\) 1809.56 0.181636 0.0908178 0.995868i \(-0.471052\pi\)
0.0908178 + 0.995868i \(0.471052\pi\)
\(464\) −1087.13 −0.108769
\(465\) 0 0
\(466\) 5366.41 0.533464
\(467\) −5981.65 −0.592715 −0.296357 0.955077i \(-0.595772\pi\)
−0.296357 + 0.955077i \(0.595772\pi\)
\(468\) 323.008 0.0319039
\(469\) 0 0
\(470\) 0 0
\(471\) 203.300 0.0198887
\(472\) −1412.35 −0.137730
\(473\) 4524.84 0.439857
\(474\) 12454.0 1.20682
\(475\) 0 0
\(476\) 0 0
\(477\) 1335.99 0.128241
\(478\) −8477.25 −0.811173
\(479\) 11527.5 1.09959 0.549796 0.835299i \(-0.314705\pi\)
0.549796 + 0.835299i \(0.314705\pi\)
\(480\) 0 0
\(481\) −1108.19 −0.105050
\(482\) 5667.23 0.535551
\(483\) 0 0
\(484\) −2131.85 −0.200211
\(485\) 0 0
\(486\) −1297.31 −0.121085
\(487\) −15791.5 −1.46936 −0.734682 0.678411i \(-0.762669\pi\)
−0.734682 + 0.678411i \(0.762669\pi\)
\(488\) −18203.9 −1.68863
\(489\) −19769.9 −1.82828
\(490\) 0 0
\(491\) 13064.9 1.20083 0.600417 0.799687i \(-0.295001\pi\)
0.600417 + 0.799687i \(0.295001\pi\)
\(492\) 6341.69 0.581108
\(493\) −3060.04 −0.279548
\(494\) 521.095 0.0474599
\(495\) 0 0
\(496\) −6803.91 −0.615937
\(497\) 0 0
\(498\) 7915.04 0.712211
\(499\) −20135.8 −1.80642 −0.903209 0.429201i \(-0.858795\pi\)
−0.903209 + 0.429201i \(0.858795\pi\)
\(500\) 0 0
\(501\) 16901.5 1.50719
\(502\) −6739.73 −0.599221
\(503\) −751.675 −0.0666313 −0.0333156 0.999445i \(-0.510607\pi\)
−0.0333156 + 0.999445i \(0.510607\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −9096.06 −0.799149
\(507\) −7823.38 −0.685302
\(508\) −4270.44 −0.372972
\(509\) 12334.5 1.07410 0.537049 0.843551i \(-0.319539\pi\)
0.537049 + 0.843551i \(0.319539\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11186.4 −0.965574
\(513\) 531.788 0.0457681
\(514\) −390.768 −0.0335332
\(515\) 0 0
\(516\) 1287.65 0.109856
\(517\) −5443.95 −0.463104
\(518\) 0 0
\(519\) 16033.3 1.35604
\(520\) 0 0
\(521\) 1736.43 0.146016 0.0730082 0.997331i \(-0.476740\pi\)
0.0730082 + 0.997331i \(0.476740\pi\)
\(522\) −139.915 −0.0117316
\(523\) −1421.42 −0.118842 −0.0594210 0.998233i \(-0.518925\pi\)
−0.0594210 + 0.998233i \(0.518925\pi\)
\(524\) 3634.72 0.303022
\(525\) 0 0
\(526\) 7516.23 0.623048
\(527\) −19151.6 −1.58303
\(528\) 8327.97 0.686417
\(529\) −4964.85 −0.408059
\(530\) 0 0
\(531\) −114.670 −0.00937148
\(532\) 0 0
\(533\) 29541.6 2.40073
\(534\) 15988.4 1.29567
\(535\) 0 0
\(536\) 13607.3 1.09654
\(537\) −7213.25 −0.579655
\(538\) −6599.26 −0.528837
\(539\) 0 0
\(540\) 0 0
\(541\) 5773.27 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(542\) −6604.64 −0.523420
\(543\) 4543.40 0.359072
\(544\) −11518.2 −0.907793
\(545\) 0 0
\(546\) 0 0
\(547\) 3941.30 0.308076 0.154038 0.988065i \(-0.450772\pi\)
0.154038 + 0.988065i \(0.450772\pi\)
\(548\) −5133.93 −0.400202
\(549\) −1477.99 −0.114899
\(550\) 0 0
\(551\) 110.751 0.00856291
\(552\) −10452.6 −0.805961
\(553\) 0 0
\(554\) −5325.06 −0.408376
\(555\) 0 0
\(556\) −7373.10 −0.562390
\(557\) 6951.74 0.528823 0.264412 0.964410i \(-0.414822\pi\)
0.264412 + 0.964410i \(0.414822\pi\)
\(558\) −875.673 −0.0664340
\(559\) 5998.29 0.453847
\(560\) 0 0
\(561\) 23441.5 1.76417
\(562\) 14152.7 1.06227
\(563\) −24284.6 −1.81789 −0.908946 0.416913i \(-0.863112\pi\)
−0.908946 + 0.416913i \(0.863112\pi\)
\(564\) −1549.21 −0.115662
\(565\) 0 0
\(566\) −13563.7 −1.00729
\(567\) 0 0
\(568\) 18235.1 1.34706
\(569\) −21563.4 −1.58873 −0.794363 0.607443i \(-0.792195\pi\)
−0.794363 + 0.607443i \(0.792195\pi\)
\(570\) 0 0
\(571\) −3689.56 −0.270409 −0.135204 0.990818i \(-0.543169\pi\)
−0.135204 + 0.990818i \(0.543169\pi\)
\(572\) −7472.21 −0.546204
\(573\) −12373.2 −0.902090
\(574\) 0 0
\(575\) 0 0
\(576\) −1102.65 −0.0797634
\(577\) 22183.9 1.60057 0.800285 0.599620i \(-0.204682\pi\)
0.800285 + 0.599620i \(0.204682\pi\)
\(578\) 12406.2 0.892783
\(579\) 17667.6 1.26812
\(580\) 0 0
\(581\) 0 0
\(582\) 2525.21 0.179851
\(583\) −30905.8 −2.19552
\(584\) 5747.55 0.407252
\(585\) 0 0
\(586\) 11896.7 0.838646
\(587\) −10605.3 −0.745705 −0.372852 0.927891i \(-0.621620\pi\)
−0.372852 + 0.927891i \(0.621620\pi\)
\(588\) 0 0
\(589\) 693.149 0.0484902
\(590\) 0 0
\(591\) −9762.83 −0.679508
\(592\) 650.466 0.0451588
\(593\) −6277.25 −0.434698 −0.217349 0.976094i \(-0.569741\pi\)
−0.217349 + 0.976094i \(0.569741\pi\)
\(594\) 15541.4 1.07352
\(595\) 0 0
\(596\) 3776.39 0.259542
\(597\) −20323.4 −1.39327
\(598\) −12058.1 −0.824567
\(599\) −9970.73 −0.680122 −0.340061 0.940403i \(-0.610448\pi\)
−0.340061 + 0.940403i \(0.610448\pi\)
\(600\) 0 0
\(601\) −24619.2 −1.67094 −0.835472 0.549533i \(-0.814806\pi\)
−0.835472 + 0.549533i \(0.814806\pi\)
\(602\) 0 0
\(603\) 1104.79 0.0746113
\(604\) 5228.53 0.352228
\(605\) 0 0
\(606\) 17077.3 1.14475
\(607\) −11252.9 −0.752460 −0.376230 0.926526i \(-0.622780\pi\)
−0.376230 + 0.926526i \(0.622780\pi\)
\(608\) 416.876 0.0278068
\(609\) 0 0
\(610\) 0 0
\(611\) −7216.69 −0.477833
\(612\) −533.668 −0.0352487
\(613\) 15293.2 1.00765 0.503824 0.863807i \(-0.331926\pi\)
0.503824 + 0.863807i \(0.331926\pi\)
\(614\) −4871.79 −0.320211
\(615\) 0 0
\(616\) 0 0
\(617\) 17589.4 1.14769 0.573843 0.818966i \(-0.305452\pi\)
0.573843 + 0.818966i \(0.305452\pi\)
\(618\) −9393.80 −0.611446
\(619\) −23467.4 −1.52380 −0.761900 0.647694i \(-0.775733\pi\)
−0.761900 + 0.647694i \(0.775733\pi\)
\(620\) 0 0
\(621\) −12305.5 −0.795173
\(622\) −13353.7 −0.860829
\(623\) 0 0
\(624\) 11039.8 0.708249
\(625\) 0 0
\(626\) 3150.64 0.201158
\(627\) −848.413 −0.0540388
\(628\) 107.068 0.00680330
\(629\) 1830.93 0.116063
\(630\) 0 0
\(631\) −6040.86 −0.381114 −0.190557 0.981676i \(-0.561029\pi\)
−0.190557 + 0.981676i \(0.561029\pi\)
\(632\) 26485.3 1.66698
\(633\) 21626.7 1.35795
\(634\) −11905.0 −0.745755
\(635\) 0 0
\(636\) −8794.99 −0.548340
\(637\) 0 0
\(638\) 3236.68 0.200849
\(639\) 1480.53 0.0916569
\(640\) 0 0
\(641\) 25111.6 1.54735 0.773673 0.633586i \(-0.218418\pi\)
0.773673 + 0.633586i \(0.218418\pi\)
\(642\) 5104.22 0.313781
\(643\) −3095.03 −0.189823 −0.0949113 0.995486i \(-0.530257\pi\)
−0.0949113 + 0.995486i \(0.530257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −860.944 −0.0524356
\(647\) −9178.63 −0.557727 −0.278863 0.960331i \(-0.589958\pi\)
−0.278863 + 0.960331i \(0.589958\pi\)
\(648\) 16528.9 1.00203
\(649\) 2652.69 0.160443
\(650\) 0 0
\(651\) 0 0
\(652\) −10411.8 −0.625397
\(653\) 14438.4 0.865265 0.432632 0.901570i \(-0.357585\pi\)
0.432632 + 0.901570i \(0.357585\pi\)
\(654\) 22079.6 1.32015
\(655\) 0 0
\(656\) −17339.8 −1.03202
\(657\) 466.650 0.0277104
\(658\) 0 0
\(659\) −2900.64 −0.171461 −0.0857305 0.996318i \(-0.527322\pi\)
−0.0857305 + 0.996318i \(0.527322\pi\)
\(660\) 0 0
\(661\) 9976.52 0.587053 0.293526 0.955951i \(-0.405171\pi\)
0.293526 + 0.955951i \(0.405171\pi\)
\(662\) 4229.46 0.248312
\(663\) 31074.9 1.82028
\(664\) 16832.5 0.983777
\(665\) 0 0
\(666\) 83.7159 0.00487076
\(667\) −2562.77 −0.148772
\(668\) 8901.19 0.515565
\(669\) −4914.85 −0.284034
\(670\) 0 0
\(671\) 34190.8 1.96710
\(672\) 0 0
\(673\) −20760.8 −1.18911 −0.594554 0.804055i \(-0.702671\pi\)
−0.594554 + 0.804055i \(0.702671\pi\)
\(674\) −356.725 −0.0203866
\(675\) 0 0
\(676\) −4120.18 −0.234421
\(677\) −3209.13 −0.182181 −0.0910907 0.995843i \(-0.529035\pi\)
−0.0910907 + 0.995843i \(0.529035\pi\)
\(678\) 11148.4 0.631492
\(679\) 0 0
\(680\) 0 0
\(681\) −8304.82 −0.467315
\(682\) 20257.2 1.13737
\(683\) −4333.57 −0.242781 −0.121391 0.992605i \(-0.538735\pi\)
−0.121391 + 0.992605i \(0.538735\pi\)
\(684\) 19.3149 0.00107971
\(685\) 0 0
\(686\) 0 0
\(687\) 2871.64 0.159476
\(688\) −3520.78 −0.195100
\(689\) −40969.8 −2.26535
\(690\) 0 0
\(691\) −14446.0 −0.795297 −0.397649 0.917538i \(-0.630174\pi\)
−0.397649 + 0.917538i \(0.630174\pi\)
\(692\) 8443.94 0.463859
\(693\) 0 0
\(694\) −10099.1 −0.552385
\(695\) 0 0
\(696\) 3719.37 0.202561
\(697\) −48808.1 −2.65242
\(698\) 3912.77 0.212179
\(699\) −11582.4 −0.626733
\(700\) 0 0
\(701\) −859.801 −0.0463256 −0.0231628 0.999732i \(-0.507374\pi\)
−0.0231628 + 0.999732i \(0.507374\pi\)
\(702\) 20602.2 1.10767
\(703\) −66.2663 −0.00355517
\(704\) 25507.9 1.36557
\(705\) 0 0
\(706\) 9084.31 0.484267
\(707\) 0 0
\(708\) 754.887 0.0400712
\(709\) −7979.13 −0.422655 −0.211327 0.977415i \(-0.567779\pi\)
−0.211327 + 0.977415i \(0.567779\pi\)
\(710\) 0 0
\(711\) 2150.37 0.113425
\(712\) 34001.7 1.78970
\(713\) −16039.4 −0.842467
\(714\) 0 0
\(715\) 0 0
\(716\) −3798.86 −0.198282
\(717\) 18296.6 0.952995
\(718\) −6643.76 −0.345325
\(719\) −33703.3 −1.74815 −0.874076 0.485790i \(-0.838532\pi\)
−0.874076 + 0.485790i \(0.838532\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15858.6 −0.817444
\(723\) −12231.7 −0.629185
\(724\) 2392.78 0.122827
\(725\) 0 0
\(726\) −9377.57 −0.479386
\(727\) 30277.0 1.54458 0.772290 0.635270i \(-0.219111\pi\)
0.772290 + 0.635270i \(0.219111\pi\)
\(728\) 0 0
\(729\) 20917.0 1.06269
\(730\) 0 0
\(731\) −9910.27 −0.501429
\(732\) 9729.82 0.491291
\(733\) −19363.9 −0.975749 −0.487874 0.872914i \(-0.662228\pi\)
−0.487874 + 0.872914i \(0.662228\pi\)
\(734\) −26648.1 −1.34005
\(735\) 0 0
\(736\) −9646.45 −0.483115
\(737\) −25557.4 −1.27737
\(738\) −2231.66 −0.111313
\(739\) 24952.4 1.24207 0.621035 0.783783i \(-0.286713\pi\)
0.621035 + 0.783783i \(0.286713\pi\)
\(740\) 0 0
\(741\) −1124.69 −0.0557576
\(742\) 0 0
\(743\) 8154.54 0.402640 0.201320 0.979526i \(-0.435477\pi\)
0.201320 + 0.979526i \(0.435477\pi\)
\(744\) 23278.1 1.14707
\(745\) 0 0
\(746\) −11783.3 −0.578310
\(747\) 1366.65 0.0669386
\(748\) 12345.5 0.603469
\(749\) 0 0
\(750\) 0 0
\(751\) −4311.26 −0.209481 −0.104740 0.994500i \(-0.533401\pi\)
−0.104740 + 0.994500i \(0.533401\pi\)
\(752\) 4235.94 0.205411
\(753\) 14546.4 0.703987
\(754\) 4290.66 0.207237
\(755\) 0 0
\(756\) 0 0
\(757\) −3624.79 −0.174036 −0.0870179 0.996207i \(-0.527734\pi\)
−0.0870179 + 0.996207i \(0.527734\pi\)
\(758\) 2211.98 0.105993
\(759\) 19632.1 0.938869
\(760\) 0 0
\(761\) 20576.4 0.980150 0.490075 0.871680i \(-0.336969\pi\)
0.490075 + 0.871680i \(0.336969\pi\)
\(762\) −18784.8 −0.893045
\(763\) 0 0
\(764\) −6516.34 −0.308577
\(765\) 0 0
\(766\) −7144.26 −0.336988
\(767\) 3516.50 0.165546
\(768\) 17791.9 0.835949
\(769\) −3066.14 −0.143781 −0.0718907 0.997413i \(-0.522903\pi\)
−0.0718907 + 0.997413i \(0.522903\pi\)
\(770\) 0 0
\(771\) 843.400 0.0393960
\(772\) 9304.64 0.433784
\(773\) 19387.0 0.902074 0.451037 0.892505i \(-0.351054\pi\)
0.451037 + 0.892505i \(0.351054\pi\)
\(774\) −453.129 −0.0210432
\(775\) 0 0
\(776\) 5370.23 0.248428
\(777\) 0 0
\(778\) −14666.9 −0.675879
\(779\) 1766.50 0.0812470
\(780\) 0 0
\(781\) −34249.4 −1.56919
\(782\) 19922.1 0.911015
\(783\) 4378.71 0.199850
\(784\) 0 0
\(785\) 0 0
\(786\) 15988.4 0.725556
\(787\) −43363.4 −1.96409 −0.982044 0.188651i \(-0.939588\pi\)
−0.982044 + 0.188651i \(0.939588\pi\)
\(788\) −5141.59 −0.232439
\(789\) −16222.4 −0.731979
\(790\) 0 0
\(791\) 0 0
\(792\) 2279.39 0.102266
\(793\) 45324.6 2.02966
\(794\) 28108.2 1.25632
\(795\) 0 0
\(796\) −10703.3 −0.476593
\(797\) −17132.6 −0.761439 −0.380720 0.924691i \(-0.624324\pi\)
−0.380720 + 0.924691i \(0.624324\pi\)
\(798\) 0 0
\(799\) 11923.3 0.527929
\(800\) 0 0
\(801\) 2760.63 0.121776
\(802\) −627.039 −0.0276079
\(803\) −10795.1 −0.474411
\(804\) −7272.98 −0.319028
\(805\) 0 0
\(806\) 26853.6 1.17355
\(807\) 14243.3 0.621297
\(808\) 36317.5 1.58124
\(809\) −1080.49 −0.0469566 −0.0234783 0.999724i \(-0.507474\pi\)
−0.0234783 + 0.999724i \(0.507474\pi\)
\(810\) 0 0
\(811\) 19593.9 0.848378 0.424189 0.905574i \(-0.360559\pi\)
0.424189 + 0.905574i \(0.360559\pi\)
\(812\) 0 0
\(813\) 14254.9 0.614933
\(814\) −1936.62 −0.0833889
\(815\) 0 0
\(816\) −18239.8 −0.782503
\(817\) 358.680 0.0153594
\(818\) 9312.66 0.398056
\(819\) 0 0
\(820\) 0 0
\(821\) 5123.80 0.217810 0.108905 0.994052i \(-0.465266\pi\)
0.108905 + 0.994052i \(0.465266\pi\)
\(822\) −22583.1 −0.958244
\(823\) 13184.1 0.558405 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(824\) −19977.3 −0.844591
\(825\) 0 0
\(826\) 0 0
\(827\) −24658.7 −1.03684 −0.518421 0.855126i \(-0.673480\pi\)
−0.518421 + 0.855126i \(0.673480\pi\)
\(828\) −446.944 −0.0187589
\(829\) −28562.3 −1.19664 −0.598318 0.801259i \(-0.704164\pi\)
−0.598318 + 0.801259i \(0.704164\pi\)
\(830\) 0 0
\(831\) 11493.1 0.479775
\(832\) 33814.1 1.40901
\(833\) 0 0
\(834\) −32432.7 −1.34659
\(835\) 0 0
\(836\) −446.817 −0.0184850
\(837\) 27404.6 1.13171
\(838\) 5646.27 0.232753
\(839\) 31106.0 1.27997 0.639987 0.768386i \(-0.278940\pi\)
0.639987 + 0.768386i \(0.278940\pi\)
\(840\) 0 0
\(841\) −23477.1 −0.962609
\(842\) −11007.1 −0.450511
\(843\) −30546.0 −1.24800
\(844\) 11389.7 0.464514
\(845\) 0 0
\(846\) 545.171 0.0221553
\(847\) 0 0
\(848\) 24047.8 0.973827
\(849\) 29274.7 1.18340
\(850\) 0 0
\(851\) 1533.39 0.0617674
\(852\) −9746.50 −0.391913
\(853\) 20567.9 0.825596 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10854.9 0.433426
\(857\) 6459.44 0.257468 0.128734 0.991679i \(-0.458909\pi\)
0.128734 + 0.991679i \(0.458909\pi\)
\(858\) −32868.7 −1.30783
\(859\) 48214.4 1.91508 0.957541 0.288298i \(-0.0930895\pi\)
0.957541 + 0.288298i \(0.0930895\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18359.7 0.725445
\(863\) 31709.5 1.25076 0.625378 0.780322i \(-0.284945\pi\)
0.625378 + 0.780322i \(0.284945\pi\)
\(864\) 16481.8 0.648984
\(865\) 0 0
\(866\) 26611.7 1.04423
\(867\) −26776.4 −1.04887
\(868\) 0 0
\(869\) −49745.1 −1.94187
\(870\) 0 0
\(871\) −33879.8 −1.31800
\(872\) 46955.6 1.82353
\(873\) 436.015 0.0169036
\(874\) −721.037 −0.0279055
\(875\) 0 0
\(876\) −3072.01 −0.118486
\(877\) −25654.8 −0.987799 −0.493900 0.869519i \(-0.664429\pi\)
−0.493900 + 0.869519i \(0.664429\pi\)
\(878\) −21192.3 −0.814585
\(879\) −25676.7 −0.985272
\(880\) 0 0
\(881\) 11470.4 0.438647 0.219323 0.975652i \(-0.429615\pi\)
0.219323 + 0.975652i \(0.429615\pi\)
\(882\) 0 0
\(883\) −39124.0 −1.49108 −0.745542 0.666459i \(-0.767809\pi\)
−0.745542 + 0.666459i \(0.767809\pi\)
\(884\) 16365.6 0.622663
\(885\) 0 0
\(886\) −4319.01 −0.163770
\(887\) 15585.8 0.589987 0.294994 0.955499i \(-0.404682\pi\)
0.294994 + 0.955499i \(0.404682\pi\)
\(888\) −2225.43 −0.0840997
\(889\) 0 0
\(890\) 0 0
\(891\) −31044.8 −1.16727
\(892\) −2588.40 −0.0971594
\(893\) −431.537 −0.0161711
\(894\) 16611.6 0.621447
\(895\) 0 0
\(896\) 0 0
\(897\) 26025.1 0.968731
\(898\) 10403.7 0.386610
\(899\) 5707.34 0.211736
\(900\) 0 0
\(901\) 67689.6 2.50285
\(902\) 51625.6 1.90570
\(903\) 0 0
\(904\) 23708.8 0.872280
\(905\) 0 0
\(906\) 22999.2 0.843376
\(907\) −27596.1 −1.01027 −0.505134 0.863041i \(-0.668557\pi\)
−0.505134 + 0.863041i \(0.668557\pi\)
\(908\) −4373.73 −0.159854
\(909\) 2948.65 0.107592
\(910\) 0 0
\(911\) −14396.2 −0.523565 −0.261782 0.965127i \(-0.584310\pi\)
−0.261782 + 0.965127i \(0.584310\pi\)
\(912\) 660.151 0.0239691
\(913\) −31615.1 −1.14601
\(914\) 33229.2 1.20254
\(915\) 0 0
\(916\) 1512.35 0.0545517
\(917\) 0 0
\(918\) −34038.7 −1.22379
\(919\) 10279.6 0.368978 0.184489 0.982835i \(-0.440937\pi\)
0.184489 + 0.982835i \(0.440937\pi\)
\(920\) 0 0
\(921\) 10514.8 0.376196
\(922\) −33727.0 −1.20471
\(923\) −45402.2 −1.61910
\(924\) 0 0
\(925\) 0 0
\(926\) 4192.06 0.148769
\(927\) −1621.98 −0.0574680
\(928\) 3432.53 0.121421
\(929\) 6499.87 0.229552 0.114776 0.993391i \(-0.463385\pi\)
0.114776 + 0.993391i \(0.463385\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6099.86 −0.214386
\(933\) 28821.5 1.01133
\(934\) −13857.2 −0.485463
\(935\) 0 0
\(936\) 3021.64 0.105518
\(937\) 10269.8 0.358056 0.179028 0.983844i \(-0.442705\pi\)
0.179028 + 0.983844i \(0.442705\pi\)
\(938\) 0 0
\(939\) −6800.06 −0.236328
\(940\) 0 0
\(941\) 34396.2 1.19159 0.595794 0.803137i \(-0.296837\pi\)
0.595794 + 0.803137i \(0.296837\pi\)
\(942\) 470.969 0.0162898
\(943\) −40876.5 −1.41158
\(944\) −2064.06 −0.0711647
\(945\) 0 0
\(946\) 10482.4 0.360265
\(947\) −27192.1 −0.933078 −0.466539 0.884501i \(-0.654499\pi\)
−0.466539 + 0.884501i \(0.654499\pi\)
\(948\) −14156.2 −0.484990
\(949\) −14310.4 −0.489500
\(950\) 0 0
\(951\) 25694.7 0.876140
\(952\) 0 0
\(953\) −49965.2 −1.69836 −0.849178 0.528107i \(-0.822902\pi\)
−0.849178 + 0.528107i \(0.822902\pi\)
\(954\) 3094.99 0.105036
\(955\) 0 0
\(956\) 9635.88 0.325990
\(957\) −6985.78 −0.235965
\(958\) 26704.9 0.900622
\(959\) 0 0
\(960\) 0 0
\(961\) 5929.05 0.199022
\(962\) −2567.25 −0.0860411
\(963\) 881.320 0.0294913
\(964\) −6441.80 −0.215225
\(965\) 0 0
\(966\) 0 0
\(967\) 16755.5 0.557208 0.278604 0.960406i \(-0.410128\pi\)
0.278604 + 0.960406i \(0.410128\pi\)
\(968\) −19942.8 −0.662176
\(969\) 1858.19 0.0616033
\(970\) 0 0
\(971\) −37617.4 −1.24325 −0.621626 0.783314i \(-0.713528\pi\)
−0.621626 + 0.783314i \(0.713528\pi\)
\(972\) 1474.62 0.0486610
\(973\) 0 0
\(974\) −36583.0 −1.20348
\(975\) 0 0
\(976\) −26603.9 −0.872511
\(977\) 27690.9 0.906767 0.453384 0.891315i \(-0.350217\pi\)
0.453384 + 0.891315i \(0.350217\pi\)
\(978\) −45799.5 −1.49745
\(979\) −63862.5 −2.08483
\(980\) 0 0
\(981\) 3812.38 0.124077
\(982\) 30266.4 0.983544
\(983\) 22754.2 0.738299 0.369149 0.929370i \(-0.379649\pi\)
0.369149 + 0.929370i \(0.379649\pi\)
\(984\) 59324.6 1.92195
\(985\) 0 0
\(986\) −7088.96 −0.228964
\(987\) 0 0
\(988\) −592.316 −0.0190729
\(989\) −8299.80 −0.266854
\(990\) 0 0
\(991\) −55470.9 −1.77809 −0.889047 0.457816i \(-0.848632\pi\)
−0.889047 + 0.457816i \(0.848632\pi\)
\(992\) 21482.9 0.687583
\(993\) −9128.50 −0.291726
\(994\) 0 0
\(995\) 0 0
\(996\) −8996.83 −0.286220
\(997\) 15181.9 0.482264 0.241132 0.970492i \(-0.422481\pi\)
0.241132 + 0.970492i \(0.422481\pi\)
\(998\) −46647.1 −1.47955
\(999\) −2619.93 −0.0829740
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 1225.4.a.p.1.2 2
5.4 even 2 245.4.a.j.1.1 yes 2
7.6 odd 2 1225.4.a.q.1.2 2
15.14 odd 2 2205.4.a.w.1.2 2
35.4 even 6 245.4.e.j.226.2 4
35.9 even 6 245.4.e.j.116.2 4
35.19 odd 6 245.4.e.k.116.2 4
35.24 odd 6 245.4.e.k.226.2 4
35.34 odd 2 245.4.a.i.1.1 2
105.104 even 2 2205.4.a.x.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.1 2 35.34 odd 2
245.4.a.j.1.1 yes 2 5.4 even 2
245.4.e.j.116.2 4 35.9 even 6
245.4.e.j.226.2 4 35.4 even 6
245.4.e.k.116.2 4 35.19 odd 6
245.4.e.k.226.2 4 35.24 odd 6
1225.4.a.p.1.2 2 1.1 even 1 trivial
1225.4.a.q.1.2 2 7.6 odd 2
2205.4.a.w.1.2 2 15.14 odd 2
2205.4.a.x.1.2 2 105.104 even 2