Properties

Label 768.4.a.e.1.2
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} +10.4222 q^{5} +6.42221 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +10.4222 q^{5} +6.42221 q^{7} +9.00000 q^{9} -61.6888 q^{11} +64.8444 q^{13} -31.2666 q^{15} -75.6888 q^{17} +10.3112 q^{19} -19.2666 q^{21} -156.844 q^{23} -16.3776 q^{25} -27.0000 q^{27} -53.7998 q^{29} +227.489 q^{31} +185.066 q^{33} +66.9335 q^{35} +10.3112 q^{37} -194.533 q^{39} +70.4441 q^{41} -298.311 q^{43} +93.7998 q^{45} -89.9109 q^{47} -301.755 q^{49} +227.066 q^{51} +388.333 q^{53} -642.934 q^{55} -30.9335 q^{57} -324.000 q^{59} +324.000 q^{61} +57.7998 q^{63} +675.822 q^{65} +920.266 q^{67} +470.533 q^{69} -995.156 q^{71} -362.266 q^{73} +49.1329 q^{75} -396.178 q^{77} -1098.91 q^{79} +81.0000 q^{81} +791.822 q^{83} -788.844 q^{85} +161.400 q^{87} +150.622 q^{89} +416.444 q^{91} -682.466 q^{93} +107.465 q^{95} -1879.15 q^{97} -555.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{5} - 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{5} - 16 q^{7} + 18 q^{9} - 8 q^{11} + 72 q^{13} + 24 q^{15} - 36 q^{17} + 136 q^{19} + 48 q^{21} - 256 q^{23} + 198 q^{25} - 54 q^{27} + 152 q^{29} + 80 q^{31} + 24 q^{33} + 480 q^{35} + 136 q^{37} - 216 q^{39} - 436 q^{41} - 712 q^{43} - 72 q^{45} + 224 q^{47} - 142 q^{49} + 108 q^{51} + 344 q^{53} - 1632 q^{55} - 408 q^{57} - 648 q^{59} + 648 q^{61} - 144 q^{63} + 544 q^{65} + 456 q^{67} + 768 q^{69} - 2048 q^{71} + 660 q^{73} - 594 q^{75} - 1600 q^{77} - 496 q^{79} + 162 q^{81} + 776 q^{83} - 1520 q^{85} - 456 q^{87} + 532 q^{89} + 256 q^{91} - 240 q^{93} - 2208 q^{95} - 1220 q^{97} - 72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 10.4222 0.932190 0.466095 0.884735i \(-0.345660\pi\)
0.466095 + 0.884735i \(0.345660\pi\)
\(6\) 0 0
\(7\) 6.42221 0.346766 0.173383 0.984854i \(-0.444530\pi\)
0.173383 + 0.984854i \(0.444530\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −61.6888 −1.69090 −0.845449 0.534056i \(-0.820667\pi\)
−0.845449 + 0.534056i \(0.820667\pi\)
\(12\) 0 0
\(13\) 64.8444 1.38343 0.691716 0.722170i \(-0.256855\pi\)
0.691716 + 0.722170i \(0.256855\pi\)
\(14\) 0 0
\(15\) −31.2666 −0.538200
\(16\) 0 0
\(17\) −75.6888 −1.07984 −0.539919 0.841717i \(-0.681545\pi\)
−0.539919 + 0.841717i \(0.681545\pi\)
\(18\) 0 0
\(19\) 10.3112 0.124502 0.0622512 0.998061i \(-0.480172\pi\)
0.0622512 + 0.998061i \(0.480172\pi\)
\(20\) 0 0
\(21\) −19.2666 −0.200206
\(22\) 0 0
\(23\) −156.844 −1.42193 −0.710963 0.703229i \(-0.751741\pi\)
−0.710963 + 0.703229i \(0.751741\pi\)
\(24\) 0 0
\(25\) −16.3776 −0.131021
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −53.7998 −0.344496 −0.172248 0.985054i \(-0.555103\pi\)
−0.172248 + 0.985054i \(0.555103\pi\)
\(30\) 0 0
\(31\) 227.489 1.31801 0.659003 0.752141i \(-0.270979\pi\)
0.659003 + 0.752141i \(0.270979\pi\)
\(32\) 0 0
\(33\) 185.066 0.976240
\(34\) 0 0
\(35\) 66.9335 0.323252
\(36\) 0 0
\(37\) 10.3112 0.0458148 0.0229074 0.999738i \(-0.492708\pi\)
0.0229074 + 0.999738i \(0.492708\pi\)
\(38\) 0 0
\(39\) −194.533 −0.798724
\(40\) 0 0
\(41\) 70.4441 0.268330 0.134165 0.990959i \(-0.457165\pi\)
0.134165 + 0.990959i \(0.457165\pi\)
\(42\) 0 0
\(43\) −298.311 −1.05795 −0.528977 0.848636i \(-0.677424\pi\)
−0.528977 + 0.848636i \(0.677424\pi\)
\(44\) 0 0
\(45\) 93.7998 0.310730
\(46\) 0 0
\(47\) −89.9109 −0.279039 −0.139520 0.990219i \(-0.544556\pi\)
−0.139520 + 0.990219i \(0.544556\pi\)
\(48\) 0 0
\(49\) −301.755 −0.879753
\(50\) 0 0
\(51\) 227.066 0.623444
\(52\) 0 0
\(53\) 388.333 1.00645 0.503223 0.864157i \(-0.332147\pi\)
0.503223 + 0.864157i \(0.332147\pi\)
\(54\) 0 0
\(55\) −642.934 −1.57624
\(56\) 0 0
\(57\) −30.9335 −0.0718815
\(58\) 0 0
\(59\) −324.000 −0.714936 −0.357468 0.933925i \(-0.616360\pi\)
−0.357468 + 0.933925i \(0.616360\pi\)
\(60\) 0 0
\(61\) 324.000 0.680065 0.340032 0.940414i \(-0.389562\pi\)
0.340032 + 0.940414i \(0.389562\pi\)
\(62\) 0 0
\(63\) 57.7998 0.115589
\(64\) 0 0
\(65\) 675.822 1.28962
\(66\) 0 0
\(67\) 920.266 1.67804 0.839018 0.544104i \(-0.183130\pi\)
0.839018 + 0.544104i \(0.183130\pi\)
\(68\) 0 0
\(69\) 470.533 0.820950
\(70\) 0 0
\(71\) −995.156 −1.66343 −0.831713 0.555206i \(-0.812639\pi\)
−0.831713 + 0.555206i \(0.812639\pi\)
\(72\) 0 0
\(73\) −362.266 −0.580822 −0.290411 0.956902i \(-0.593792\pi\)
−0.290411 + 0.956902i \(0.593792\pi\)
\(74\) 0 0
\(75\) 49.1329 0.0756451
\(76\) 0 0
\(77\) −396.178 −0.586347
\(78\) 0 0
\(79\) −1098.91 −1.56503 −0.782513 0.622634i \(-0.786062\pi\)
−0.782513 + 0.622634i \(0.786062\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 791.822 1.04715 0.523577 0.851979i \(-0.324597\pi\)
0.523577 + 0.851979i \(0.324597\pi\)
\(84\) 0 0
\(85\) −788.844 −1.00661
\(86\) 0 0
\(87\) 161.400 0.198895
\(88\) 0 0
\(89\) 150.622 0.179393 0.0896963 0.995969i \(-0.471410\pi\)
0.0896963 + 0.995969i \(0.471410\pi\)
\(90\) 0 0
\(91\) 416.444 0.479728
\(92\) 0 0
\(93\) −682.466 −0.760951
\(94\) 0 0
\(95\) 107.465 0.116060
\(96\) 0 0
\(97\) −1879.15 −1.96700 −0.983501 0.180903i \(-0.942098\pi\)
−0.983501 + 0.180903i \(0.942098\pi\)
\(98\) 0 0
\(99\) −555.199 −0.563633
\(100\) 0 0
\(101\) −1722.51 −1.69699 −0.848496 0.529202i \(-0.822491\pi\)
−0.848496 + 0.529202i \(0.822491\pi\)
\(102\) 0 0
\(103\) −1908.82 −1.82604 −0.913018 0.407919i \(-0.866255\pi\)
−0.913018 + 0.407919i \(0.866255\pi\)
\(104\) 0 0
\(105\) −200.801 −0.186630
\(106\) 0 0
\(107\) −128.622 −0.116209 −0.0581046 0.998310i \(-0.518506\pi\)
−0.0581046 + 0.998310i \(0.518506\pi\)
\(108\) 0 0
\(109\) 758.267 0.666320 0.333160 0.942870i \(-0.391885\pi\)
0.333160 + 0.942870i \(0.391885\pi\)
\(110\) 0 0
\(111\) −30.9335 −0.0264512
\(112\) 0 0
\(113\) 921.643 0.767265 0.383632 0.923486i \(-0.374673\pi\)
0.383632 + 0.923486i \(0.374673\pi\)
\(114\) 0 0
\(115\) −1634.66 −1.32551
\(116\) 0 0
\(117\) 583.600 0.461144
\(118\) 0 0
\(119\) −486.089 −0.374451
\(120\) 0 0
\(121\) 2474.51 1.85914
\(122\) 0 0
\(123\) −211.332 −0.154920
\(124\) 0 0
\(125\) −1473.47 −1.05433
\(126\) 0 0
\(127\) −1647.22 −1.15092 −0.575462 0.817828i \(-0.695178\pi\)
−0.575462 + 0.817828i \(0.695178\pi\)
\(128\) 0 0
\(129\) 894.934 0.610810
\(130\) 0 0
\(131\) −2000.27 −1.33408 −0.667038 0.745023i \(-0.732438\pi\)
−0.667038 + 0.745023i \(0.732438\pi\)
\(132\) 0 0
\(133\) 66.2205 0.0431733
\(134\) 0 0
\(135\) −281.400 −0.179400
\(136\) 0 0
\(137\) −2574.71 −1.60564 −0.802819 0.596223i \(-0.796667\pi\)
−0.802819 + 0.596223i \(0.796667\pi\)
\(138\) 0 0
\(139\) −1331.29 −0.812362 −0.406181 0.913793i \(-0.633140\pi\)
−0.406181 + 0.913793i \(0.633140\pi\)
\(140\) 0 0
\(141\) 269.733 0.161103
\(142\) 0 0
\(143\) −4000.18 −2.33924
\(144\) 0 0
\(145\) −560.713 −0.321136
\(146\) 0 0
\(147\) 905.266 0.507926
\(148\) 0 0
\(149\) 552.377 0.303708 0.151854 0.988403i \(-0.451476\pi\)
0.151854 + 0.988403i \(0.451476\pi\)
\(150\) 0 0
\(151\) 1307.13 0.704456 0.352228 0.935914i \(-0.385424\pi\)
0.352228 + 0.935914i \(0.385424\pi\)
\(152\) 0 0
\(153\) −681.199 −0.359946
\(154\) 0 0
\(155\) 2370.93 1.22863
\(156\) 0 0
\(157\) 3527.11 1.79295 0.896477 0.443089i \(-0.146118\pi\)
0.896477 + 0.443089i \(0.146118\pi\)
\(158\) 0 0
\(159\) −1165.00 −0.581072
\(160\) 0 0
\(161\) −1007.29 −0.493077
\(162\) 0 0
\(163\) −926.045 −0.444991 −0.222495 0.974934i \(-0.571420\pi\)
−0.222495 + 0.974934i \(0.571420\pi\)
\(164\) 0 0
\(165\) 1928.80 0.910042
\(166\) 0 0
\(167\) 1393.33 0.645624 0.322812 0.946463i \(-0.395372\pi\)
0.322812 + 0.946463i \(0.395372\pi\)
\(168\) 0 0
\(169\) 2007.80 0.913881
\(170\) 0 0
\(171\) 92.8006 0.0415008
\(172\) 0 0
\(173\) −3311.71 −1.45540 −0.727701 0.685894i \(-0.759411\pi\)
−0.727701 + 0.685894i \(0.759411\pi\)
\(174\) 0 0
\(175\) −105.181 −0.0454337
\(176\) 0 0
\(177\) 972.000 0.412768
\(178\) 0 0
\(179\) 323.287 0.134992 0.0674961 0.997720i \(-0.478499\pi\)
0.0674961 + 0.997720i \(0.478499\pi\)
\(180\) 0 0
\(181\) −3066.80 −1.25941 −0.629705 0.776834i \(-0.716824\pi\)
−0.629705 + 0.776834i \(0.716824\pi\)
\(182\) 0 0
\(183\) −972.000 −0.392636
\(184\) 0 0
\(185\) 107.465 0.0427081
\(186\) 0 0
\(187\) 4669.15 1.82589
\(188\) 0 0
\(189\) −173.400 −0.0667352
\(190\) 0 0
\(191\) −1856.44 −0.703286 −0.351643 0.936134i \(-0.614377\pi\)
−0.351643 + 0.936134i \(0.614377\pi\)
\(192\) 0 0
\(193\) 3008.84 1.12218 0.561091 0.827754i \(-0.310382\pi\)
0.561091 + 0.827754i \(0.310382\pi\)
\(194\) 0 0
\(195\) −2027.47 −0.744563
\(196\) 0 0
\(197\) 909.309 0.328861 0.164430 0.986389i \(-0.447421\pi\)
0.164430 + 0.986389i \(0.447421\pi\)
\(198\) 0 0
\(199\) 2780.24 0.990383 0.495191 0.868784i \(-0.335098\pi\)
0.495191 + 0.868784i \(0.335098\pi\)
\(200\) 0 0
\(201\) −2760.80 −0.968814
\(202\) 0 0
\(203\) −345.514 −0.119460
\(204\) 0 0
\(205\) 734.183 0.250134
\(206\) 0 0
\(207\) −1411.60 −0.473976
\(208\) 0 0
\(209\) −636.085 −0.210521
\(210\) 0 0
\(211\) 385.511 0.125780 0.0628901 0.998020i \(-0.479968\pi\)
0.0628901 + 0.998020i \(0.479968\pi\)
\(212\) 0 0
\(213\) 2985.47 0.960379
\(214\) 0 0
\(215\) −3109.06 −0.986215
\(216\) 0 0
\(217\) 1460.98 0.457040
\(218\) 0 0
\(219\) 1086.80 0.335338
\(220\) 0 0
\(221\) −4908.00 −1.49388
\(222\) 0 0
\(223\) −601.975 −0.180768 −0.0903839 0.995907i \(-0.528809\pi\)
−0.0903839 + 0.995907i \(0.528809\pi\)
\(224\) 0 0
\(225\) −147.399 −0.0436737
\(226\) 0 0
\(227\) −185.779 −0.0543199 −0.0271600 0.999631i \(-0.508646\pi\)
−0.0271600 + 0.999631i \(0.508646\pi\)
\(228\) 0 0
\(229\) 1237.11 0.356989 0.178495 0.983941i \(-0.442877\pi\)
0.178495 + 0.983941i \(0.442877\pi\)
\(230\) 0 0
\(231\) 1188.53 0.338527
\(232\) 0 0
\(233\) −217.378 −0.0611197 −0.0305598 0.999533i \(-0.509729\pi\)
−0.0305598 + 0.999533i \(0.509729\pi\)
\(234\) 0 0
\(235\) −937.070 −0.260118
\(236\) 0 0
\(237\) 3296.73 0.903568
\(238\) 0 0
\(239\) 6055.99 1.63904 0.819518 0.573053i \(-0.194241\pi\)
0.819518 + 0.573053i \(0.194241\pi\)
\(240\) 0 0
\(241\) −6815.69 −1.82173 −0.910865 0.412705i \(-0.864584\pi\)
−0.910865 + 0.412705i \(0.864584\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −3144.96 −0.820097
\(246\) 0 0
\(247\) 668.622 0.172241
\(248\) 0 0
\(249\) −2375.47 −0.604574
\(250\) 0 0
\(251\) 2448.53 0.615738 0.307869 0.951429i \(-0.400384\pi\)
0.307869 + 0.951429i \(0.400384\pi\)
\(252\) 0 0
\(253\) 9675.55 2.40433
\(254\) 0 0
\(255\) 2366.53 0.581169
\(256\) 0 0
\(257\) 2949.55 0.715907 0.357953 0.933739i \(-0.383475\pi\)
0.357953 + 0.933739i \(0.383475\pi\)
\(258\) 0 0
\(259\) 66.2205 0.0158870
\(260\) 0 0
\(261\) −484.199 −0.114832
\(262\) 0 0
\(263\) 4831.64 1.13282 0.566410 0.824124i \(-0.308332\pi\)
0.566410 + 0.824124i \(0.308332\pi\)
\(264\) 0 0
\(265\) 4047.29 0.938199
\(266\) 0 0
\(267\) −451.867 −0.103572
\(268\) 0 0
\(269\) 1093.84 0.247928 0.123964 0.992287i \(-0.460439\pi\)
0.123964 + 0.992287i \(0.460439\pi\)
\(270\) 0 0
\(271\) 3079.80 0.690349 0.345175 0.938539i \(-0.387820\pi\)
0.345175 + 0.938539i \(0.387820\pi\)
\(272\) 0 0
\(273\) −1249.33 −0.276971
\(274\) 0 0
\(275\) 1010.32 0.221543
\(276\) 0 0
\(277\) 5600.75 1.21486 0.607430 0.794373i \(-0.292200\pi\)
0.607430 + 0.794373i \(0.292200\pi\)
\(278\) 0 0
\(279\) 2047.40 0.439335
\(280\) 0 0
\(281\) 203.426 0.0431864 0.0215932 0.999767i \(-0.493126\pi\)
0.0215932 + 0.999767i \(0.493126\pi\)
\(282\) 0 0
\(283\) −2214.84 −0.465225 −0.232612 0.972570i \(-0.574727\pi\)
−0.232612 + 0.972570i \(0.574727\pi\)
\(284\) 0 0
\(285\) −322.396 −0.0670073
\(286\) 0 0
\(287\) 452.406 0.0930478
\(288\) 0 0
\(289\) 815.798 0.166049
\(290\) 0 0
\(291\) 5637.46 1.13565
\(292\) 0 0
\(293\) −5812.37 −1.15892 −0.579458 0.815002i \(-0.696736\pi\)
−0.579458 + 0.815002i \(0.696736\pi\)
\(294\) 0 0
\(295\) −3376.79 −0.666456
\(296\) 0 0
\(297\) 1665.60 0.325413
\(298\) 0 0
\(299\) −10170.5 −1.96714
\(300\) 0 0
\(301\) −1915.82 −0.366863
\(302\) 0 0
\(303\) 5167.53 0.979758
\(304\) 0 0
\(305\) 3376.79 0.633950
\(306\) 0 0
\(307\) 7337.33 1.36405 0.682025 0.731329i \(-0.261099\pi\)
0.682025 + 0.731329i \(0.261099\pi\)
\(308\) 0 0
\(309\) 5726.46 1.05426
\(310\) 0 0
\(311\) 6575.91 1.19899 0.599494 0.800379i \(-0.295368\pi\)
0.599494 + 0.800379i \(0.295368\pi\)
\(312\) 0 0
\(313\) −1556.67 −0.281113 −0.140556 0.990073i \(-0.544889\pi\)
−0.140556 + 0.990073i \(0.544889\pi\)
\(314\) 0 0
\(315\) 602.402 0.107751
\(316\) 0 0
\(317\) −9457.27 −1.67562 −0.837812 0.545959i \(-0.816166\pi\)
−0.837812 + 0.545959i \(0.816166\pi\)
\(318\) 0 0
\(319\) 3318.85 0.582507
\(320\) 0 0
\(321\) 385.867 0.0670935
\(322\) 0 0
\(323\) −780.441 −0.134442
\(324\) 0 0
\(325\) −1062.00 −0.181259
\(326\) 0 0
\(327\) −2274.80 −0.384700
\(328\) 0 0
\(329\) −577.426 −0.0967615
\(330\) 0 0
\(331\) −10907.6 −1.81129 −0.905647 0.424032i \(-0.860614\pi\)
−0.905647 + 0.424032i \(0.860614\pi\)
\(332\) 0 0
\(333\) 92.8006 0.0152716
\(334\) 0 0
\(335\) 9591.20 1.56425
\(336\) 0 0
\(337\) 4256.22 0.687986 0.343993 0.938972i \(-0.388220\pi\)
0.343993 + 0.938972i \(0.388220\pi\)
\(338\) 0 0
\(339\) −2764.93 −0.442981
\(340\) 0 0
\(341\) −14033.5 −2.22861
\(342\) 0 0
\(343\) −4140.75 −0.651835
\(344\) 0 0
\(345\) 4903.99 0.765282
\(346\) 0 0
\(347\) 4217.07 0.652403 0.326202 0.945300i \(-0.394231\pi\)
0.326202 + 0.945300i \(0.394231\pi\)
\(348\) 0 0
\(349\) −2986.57 −0.458073 −0.229037 0.973418i \(-0.573558\pi\)
−0.229037 + 0.973418i \(0.573558\pi\)
\(350\) 0 0
\(351\) −1750.80 −0.266241
\(352\) 0 0
\(353\) −8231.42 −1.24112 −0.620558 0.784160i \(-0.713094\pi\)
−0.620558 + 0.784160i \(0.713094\pi\)
\(354\) 0 0
\(355\) −10371.7 −1.55063
\(356\) 0 0
\(357\) 1458.27 0.216190
\(358\) 0 0
\(359\) −5857.19 −0.861088 −0.430544 0.902569i \(-0.641678\pi\)
−0.430544 + 0.902569i \(0.641678\pi\)
\(360\) 0 0
\(361\) −6752.68 −0.984499
\(362\) 0 0
\(363\) −7423.53 −1.07337
\(364\) 0 0
\(365\) −3775.61 −0.541437
\(366\) 0 0
\(367\) 10715.7 1.52412 0.762061 0.647505i \(-0.224188\pi\)
0.762061 + 0.647505i \(0.224188\pi\)
\(368\) 0 0
\(369\) 633.997 0.0894433
\(370\) 0 0
\(371\) 2493.95 0.349002
\(372\) 0 0
\(373\) 1817.52 0.252299 0.126149 0.992011i \(-0.459738\pi\)
0.126149 + 0.992011i \(0.459738\pi\)
\(374\) 0 0
\(375\) 4420.40 0.608716
\(376\) 0 0
\(377\) −3488.62 −0.476586
\(378\) 0 0
\(379\) 9789.68 1.32681 0.663407 0.748259i \(-0.269110\pi\)
0.663407 + 0.748259i \(0.269110\pi\)
\(380\) 0 0
\(381\) 4941.67 0.664486
\(382\) 0 0
\(383\) 11502.3 1.53457 0.767285 0.641306i \(-0.221607\pi\)
0.767285 + 0.641306i \(0.221607\pi\)
\(384\) 0 0
\(385\) −4129.05 −0.546587
\(386\) 0 0
\(387\) −2684.80 −0.352651
\(388\) 0 0
\(389\) 6.06261 0.000790197 0 0.000395098 1.00000i \(-0.499874\pi\)
0.000395098 1.00000i \(0.499874\pi\)
\(390\) 0 0
\(391\) 11871.4 1.53545
\(392\) 0 0
\(393\) 6000.80 0.770230
\(394\) 0 0
\(395\) −11453.1 −1.45890
\(396\) 0 0
\(397\) −5982.31 −0.756281 −0.378141 0.925748i \(-0.623436\pi\)
−0.378141 + 0.925748i \(0.623436\pi\)
\(398\) 0 0
\(399\) −198.662 −0.0249261
\(400\) 0 0
\(401\) 10443.6 1.30057 0.650284 0.759691i \(-0.274650\pi\)
0.650284 + 0.759691i \(0.274650\pi\)
\(402\) 0 0
\(403\) 14751.4 1.82337
\(404\) 0 0
\(405\) 844.199 0.103577
\(406\) 0 0
\(407\) −636.085 −0.0774682
\(408\) 0 0
\(409\) −8141.01 −0.984223 −0.492112 0.870532i \(-0.663775\pi\)
−0.492112 + 0.870532i \(0.663775\pi\)
\(410\) 0 0
\(411\) 7724.13 0.927015
\(412\) 0 0
\(413\) −2080.79 −0.247916
\(414\) 0 0
\(415\) 8252.53 0.976146
\(416\) 0 0
\(417\) 3993.86 0.469017
\(418\) 0 0
\(419\) 8716.96 1.01635 0.508176 0.861253i \(-0.330320\pi\)
0.508176 + 0.861253i \(0.330320\pi\)
\(420\) 0 0
\(421\) 13437.5 1.55560 0.777798 0.628514i \(-0.216337\pi\)
0.777798 + 0.628514i \(0.216337\pi\)
\(422\) 0 0
\(423\) −809.198 −0.0930131
\(424\) 0 0
\(425\) 1239.60 0.141482
\(426\) 0 0
\(427\) 2080.79 0.235824
\(428\) 0 0
\(429\) 12000.5 1.35056
\(430\) 0 0
\(431\) −7343.86 −0.820745 −0.410373 0.911918i \(-0.634601\pi\)
−0.410373 + 0.911918i \(0.634601\pi\)
\(432\) 0 0
\(433\) 4490.80 0.498416 0.249208 0.968450i \(-0.419830\pi\)
0.249208 + 0.968450i \(0.419830\pi\)
\(434\) 0 0
\(435\) 1682.14 0.185408
\(436\) 0 0
\(437\) −1617.25 −0.177033
\(438\) 0 0
\(439\) 9437.93 1.02608 0.513038 0.858366i \(-0.328520\pi\)
0.513038 + 0.858366i \(0.328520\pi\)
\(440\) 0 0
\(441\) −2715.80 −0.293251
\(442\) 0 0
\(443\) 12668.3 1.35866 0.679331 0.733832i \(-0.262270\pi\)
0.679331 + 0.733832i \(0.262270\pi\)
\(444\) 0 0
\(445\) 1569.82 0.167228
\(446\) 0 0
\(447\) −1657.13 −0.175346
\(448\) 0 0
\(449\) −13052.0 −1.37186 −0.685929 0.727669i \(-0.740604\pi\)
−0.685929 + 0.727669i \(0.740604\pi\)
\(450\) 0 0
\(451\) −4345.61 −0.453718
\(452\) 0 0
\(453\) −3921.40 −0.406718
\(454\) 0 0
\(455\) 4340.27 0.447197
\(456\) 0 0
\(457\) −1313.64 −0.134463 −0.0672316 0.997737i \(-0.521417\pi\)
−0.0672316 + 0.997737i \(0.521417\pi\)
\(458\) 0 0
\(459\) 2043.60 0.207815
\(460\) 0 0
\(461\) −627.883 −0.0634347 −0.0317174 0.999497i \(-0.510098\pi\)
−0.0317174 + 0.999497i \(0.510098\pi\)
\(462\) 0 0
\(463\) −7315.03 −0.734251 −0.367126 0.930171i \(-0.619658\pi\)
−0.367126 + 0.930171i \(0.619658\pi\)
\(464\) 0 0
\(465\) −7112.80 −0.709351
\(466\) 0 0
\(467\) 759.997 0.0753072 0.0376536 0.999291i \(-0.488012\pi\)
0.0376536 + 0.999291i \(0.488012\pi\)
\(468\) 0 0
\(469\) 5910.14 0.581886
\(470\) 0 0
\(471\) −10581.3 −1.03516
\(472\) 0 0
\(473\) 18402.5 1.78889
\(474\) 0 0
\(475\) −168.873 −0.0163125
\(476\) 0 0
\(477\) 3495.00 0.335482
\(478\) 0 0
\(479\) −7403.51 −0.706211 −0.353106 0.935583i \(-0.614874\pi\)
−0.353106 + 0.935583i \(0.614874\pi\)
\(480\) 0 0
\(481\) 668.622 0.0633816
\(482\) 0 0
\(483\) 3021.86 0.284678
\(484\) 0 0
\(485\) −19584.9 −1.83362
\(486\) 0 0
\(487\) 3488.11 0.324561 0.162281 0.986745i \(-0.448115\pi\)
0.162281 + 0.986745i \(0.448115\pi\)
\(488\) 0 0
\(489\) 2778.14 0.256915
\(490\) 0 0
\(491\) 6575.73 0.604396 0.302198 0.953245i \(-0.402280\pi\)
0.302198 + 0.953245i \(0.402280\pi\)
\(492\) 0 0
\(493\) 4072.05 0.372000
\(494\) 0 0
\(495\) −5786.40 −0.525413
\(496\) 0 0
\(497\) −6391.09 −0.576820
\(498\) 0 0
\(499\) −5187.82 −0.465408 −0.232704 0.972548i \(-0.574757\pi\)
−0.232704 + 0.972548i \(0.574757\pi\)
\(500\) 0 0
\(501\) −4180.00 −0.372751
\(502\) 0 0
\(503\) 7248.38 0.642524 0.321262 0.946990i \(-0.395893\pi\)
0.321262 + 0.946990i \(0.395893\pi\)
\(504\) 0 0
\(505\) −17952.4 −1.58192
\(506\) 0 0
\(507\) −6023.39 −0.527630
\(508\) 0 0
\(509\) −6613.44 −0.575905 −0.287952 0.957645i \(-0.592974\pi\)
−0.287952 + 0.957645i \(0.592974\pi\)
\(510\) 0 0
\(511\) −2326.55 −0.201410
\(512\) 0 0
\(513\) −278.402 −0.0239605
\(514\) 0 0
\(515\) −19894.1 −1.70221
\(516\) 0 0
\(517\) 5546.50 0.471827
\(518\) 0 0
\(519\) 9935.13 0.840277
\(520\) 0 0
\(521\) 20761.3 1.74581 0.872906 0.487888i \(-0.162233\pi\)
0.872906 + 0.487888i \(0.162233\pi\)
\(522\) 0 0
\(523\) −13494.2 −1.12822 −0.564112 0.825698i \(-0.690781\pi\)
−0.564112 + 0.825698i \(0.690781\pi\)
\(524\) 0 0
\(525\) 315.542 0.0262312
\(526\) 0 0
\(527\) −17218.3 −1.42323
\(528\) 0 0
\(529\) 12433.2 1.02188
\(530\) 0 0
\(531\) −2916.00 −0.238312
\(532\) 0 0
\(533\) 4567.91 0.371216
\(534\) 0 0
\(535\) −1340.53 −0.108329
\(536\) 0 0
\(537\) −969.861 −0.0779378
\(538\) 0 0
\(539\) 18614.9 1.48757
\(540\) 0 0
\(541\) −2708.58 −0.215251 −0.107626 0.994191i \(-0.534325\pi\)
−0.107626 + 0.994191i \(0.534325\pi\)
\(542\) 0 0
\(543\) 9200.39 0.727121
\(544\) 0 0
\(545\) 7902.82 0.621137
\(546\) 0 0
\(547\) −15783.5 −1.23373 −0.616866 0.787068i \(-0.711598\pi\)
−0.616866 + 0.787068i \(0.711598\pi\)
\(548\) 0 0
\(549\) 2916.00 0.226688
\(550\) 0 0
\(551\) −554.740 −0.0428906
\(552\) 0 0
\(553\) −7057.43 −0.542699
\(554\) 0 0
\(555\) −322.396 −0.0246575
\(556\) 0 0
\(557\) −1892.77 −0.143984 −0.0719922 0.997405i \(-0.522936\pi\)
−0.0719922 + 0.997405i \(0.522936\pi\)
\(558\) 0 0
\(559\) −19343.8 −1.46361
\(560\) 0 0
\(561\) −14007.5 −1.05418
\(562\) 0 0
\(563\) −3876.26 −0.290169 −0.145084 0.989419i \(-0.546345\pi\)
−0.145084 + 0.989419i \(0.546345\pi\)
\(564\) 0 0
\(565\) 9605.56 0.715237
\(566\) 0 0
\(567\) 520.199 0.0385296
\(568\) 0 0
\(569\) −14900.9 −1.09785 −0.548927 0.835870i \(-0.684963\pi\)
−0.548927 + 0.835870i \(0.684963\pi\)
\(570\) 0 0
\(571\) 24926.0 1.82683 0.913414 0.407032i \(-0.133436\pi\)
0.913414 + 0.407032i \(0.133436\pi\)
\(572\) 0 0
\(573\) 5569.33 0.406042
\(574\) 0 0
\(575\) 2568.74 0.186302
\(576\) 0 0
\(577\) 21022.4 1.51677 0.758384 0.651808i \(-0.225989\pi\)
0.758384 + 0.651808i \(0.225989\pi\)
\(578\) 0 0
\(579\) −9026.52 −0.647892
\(580\) 0 0
\(581\) 5085.24 0.363118
\(582\) 0 0
\(583\) −23955.8 −1.70180
\(584\) 0 0
\(585\) 6082.40 0.429874
\(586\) 0 0
\(587\) −13847.0 −0.973641 −0.486821 0.873502i \(-0.661843\pi\)
−0.486821 + 0.873502i \(0.661843\pi\)
\(588\) 0 0
\(589\) 2345.68 0.164095
\(590\) 0 0
\(591\) −2727.93 −0.189868
\(592\) 0 0
\(593\) −4113.01 −0.284825 −0.142413 0.989807i \(-0.545486\pi\)
−0.142413 + 0.989807i \(0.545486\pi\)
\(594\) 0 0
\(595\) −5066.12 −0.349060
\(596\) 0 0
\(597\) −8340.73 −0.571798
\(598\) 0 0
\(599\) −4863.60 −0.331755 −0.165878 0.986146i \(-0.553046\pi\)
−0.165878 + 0.986146i \(0.553046\pi\)
\(600\) 0 0
\(601\) 6827.76 0.463411 0.231706 0.972786i \(-0.425569\pi\)
0.231706 + 0.972786i \(0.425569\pi\)
\(602\) 0 0
\(603\) 8282.39 0.559345
\(604\) 0 0
\(605\) 25789.9 1.73307
\(606\) 0 0
\(607\) −18178.3 −1.21554 −0.607770 0.794113i \(-0.707936\pi\)
−0.607770 + 0.794113i \(0.707936\pi\)
\(608\) 0 0
\(609\) 1036.54 0.0689700
\(610\) 0 0
\(611\) −5830.22 −0.386032
\(612\) 0 0
\(613\) 15687.2 1.03360 0.516802 0.856105i \(-0.327122\pi\)
0.516802 + 0.856105i \(0.327122\pi\)
\(614\) 0 0
\(615\) −2202.55 −0.144415
\(616\) 0 0
\(617\) −16420.2 −1.07140 −0.535700 0.844409i \(-0.679952\pi\)
−0.535700 + 0.844409i \(0.679952\pi\)
\(618\) 0 0
\(619\) 5517.78 0.358285 0.179142 0.983823i \(-0.442668\pi\)
0.179142 + 0.983823i \(0.442668\pi\)
\(620\) 0 0
\(621\) 4234.80 0.273650
\(622\) 0 0
\(623\) 967.328 0.0622073
\(624\) 0 0
\(625\) −13309.6 −0.851812
\(626\) 0 0
\(627\) 1908.25 0.121544
\(628\) 0 0
\(629\) −780.441 −0.0494725
\(630\) 0 0
\(631\) −1559.17 −0.0983670 −0.0491835 0.998790i \(-0.515662\pi\)
−0.0491835 + 0.998790i \(0.515662\pi\)
\(632\) 0 0
\(633\) −1156.53 −0.0726193
\(634\) 0 0
\(635\) −17167.7 −1.07288
\(636\) 0 0
\(637\) −19567.1 −1.21708
\(638\) 0 0
\(639\) −8956.40 −0.554475
\(640\) 0 0
\(641\) −15188.6 −0.935900 −0.467950 0.883755i \(-0.655007\pi\)
−0.467950 + 0.883755i \(0.655007\pi\)
\(642\) 0 0
\(643\) 16666.6 1.02219 0.511095 0.859524i \(-0.329240\pi\)
0.511095 + 0.859524i \(0.329240\pi\)
\(644\) 0 0
\(645\) 9327.18 0.569391
\(646\) 0 0
\(647\) −3038.27 −0.184616 −0.0923081 0.995730i \(-0.529424\pi\)
−0.0923081 + 0.995730i \(0.529424\pi\)
\(648\) 0 0
\(649\) 19987.2 1.20888
\(650\) 0 0
\(651\) −4382.94 −0.263872
\(652\) 0 0
\(653\) −9078.99 −0.544086 −0.272043 0.962285i \(-0.587699\pi\)
−0.272043 + 0.962285i \(0.587699\pi\)
\(654\) 0 0
\(655\) −20847.2 −1.24361
\(656\) 0 0
\(657\) −3260.39 −0.193607
\(658\) 0 0
\(659\) 16892.0 0.998511 0.499255 0.866455i \(-0.333607\pi\)
0.499255 + 0.866455i \(0.333607\pi\)
\(660\) 0 0
\(661\) 5064.16 0.297992 0.148996 0.988838i \(-0.452396\pi\)
0.148996 + 0.988838i \(0.452396\pi\)
\(662\) 0 0
\(663\) 14724.0 0.862492
\(664\) 0 0
\(665\) 690.164 0.0402457
\(666\) 0 0
\(667\) 8438.21 0.489848
\(668\) 0 0
\(669\) 1805.93 0.104366
\(670\) 0 0
\(671\) −19987.2 −1.14992
\(672\) 0 0
\(673\) 28625.9 1.63959 0.819797 0.572654i \(-0.194086\pi\)
0.819797 + 0.572654i \(0.194086\pi\)
\(674\) 0 0
\(675\) 442.196 0.0252150
\(676\) 0 0
\(677\) −16061.2 −0.911791 −0.455895 0.890033i \(-0.650681\pi\)
−0.455895 + 0.890033i \(0.650681\pi\)
\(678\) 0 0
\(679\) −12068.3 −0.682090
\(680\) 0 0
\(681\) 557.338 0.0313616
\(682\) 0 0
\(683\) −7868.09 −0.440796 −0.220398 0.975410i \(-0.570736\pi\)
−0.220398 + 0.975410i \(0.570736\pi\)
\(684\) 0 0
\(685\) −26834.2 −1.49676
\(686\) 0 0
\(687\) −3711.33 −0.206108
\(688\) 0 0
\(689\) 25181.2 1.39235
\(690\) 0 0
\(691\) −16886.2 −0.929641 −0.464820 0.885405i \(-0.653881\pi\)
−0.464820 + 0.885405i \(0.653881\pi\)
\(692\) 0 0
\(693\) −3565.60 −0.195449
\(694\) 0 0
\(695\) −13874.9 −0.757276
\(696\) 0 0
\(697\) −5331.83 −0.289753
\(698\) 0 0
\(699\) 652.133 0.0352875
\(700\) 0 0
\(701\) 18293.6 0.985647 0.492824 0.870129i \(-0.335965\pi\)
0.492824 + 0.870129i \(0.335965\pi\)
\(702\) 0 0
\(703\) 106.320 0.00570406
\(704\) 0 0
\(705\) 2811.21 0.150179
\(706\) 0 0
\(707\) −11062.3 −0.588460
\(708\) 0 0
\(709\) −21555.6 −1.14180 −0.570900 0.821019i \(-0.693406\pi\)
−0.570900 + 0.821019i \(0.693406\pi\)
\(710\) 0 0
\(711\) −9890.19 −0.521675
\(712\) 0 0
\(713\) −35680.3 −1.87411
\(714\) 0 0
\(715\) −41690.6 −2.18062
\(716\) 0 0
\(717\) −18168.0 −0.946298
\(718\) 0 0
\(719\) 9059.31 0.469896 0.234948 0.972008i \(-0.424508\pi\)
0.234948 + 0.972008i \(0.424508\pi\)
\(720\) 0 0
\(721\) −12258.8 −0.633208
\(722\) 0 0
\(723\) 20447.1 1.05178
\(724\) 0 0
\(725\) 881.115 0.0451362
\(726\) 0 0
\(727\) −7074.83 −0.360923 −0.180462 0.983582i \(-0.557759\pi\)
−0.180462 + 0.983582i \(0.557759\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 22578.8 1.14242
\(732\) 0 0
\(733\) −17095.9 −0.861459 −0.430730 0.902481i \(-0.641744\pi\)
−0.430730 + 0.902481i \(0.641744\pi\)
\(734\) 0 0
\(735\) 9434.87 0.473483
\(736\) 0 0
\(737\) −56770.1 −2.83739
\(738\) 0 0
\(739\) 28157.4 1.40161 0.700803 0.713355i \(-0.252825\pi\)
0.700803 + 0.713355i \(0.252825\pi\)
\(740\) 0 0
\(741\) −2005.87 −0.0994431
\(742\) 0 0
\(743\) −34033.2 −1.68043 −0.840214 0.542255i \(-0.817571\pi\)
−0.840214 + 0.542255i \(0.817571\pi\)
\(744\) 0 0
\(745\) 5756.99 0.283114
\(746\) 0 0
\(747\) 7126.40 0.349051
\(748\) 0 0
\(749\) −826.039 −0.0402975
\(750\) 0 0
\(751\) 34356.8 1.66937 0.834685 0.550728i \(-0.185650\pi\)
0.834685 + 0.550728i \(0.185650\pi\)
\(752\) 0 0
\(753\) −7345.60 −0.355496
\(754\) 0 0
\(755\) 13623.2 0.656687
\(756\) 0 0
\(757\) −29464.6 −1.41467 −0.707337 0.706876i \(-0.750104\pi\)
−0.707337 + 0.706876i \(0.750104\pi\)
\(758\) 0 0
\(759\) −29026.6 −1.38814
\(760\) 0 0
\(761\) 27284.5 1.29969 0.649843 0.760069i \(-0.274835\pi\)
0.649843 + 0.760069i \(0.274835\pi\)
\(762\) 0 0
\(763\) 4869.75 0.231057
\(764\) 0 0
\(765\) −7099.60 −0.335538
\(766\) 0 0
\(767\) −21009.6 −0.989064
\(768\) 0 0
\(769\) 2320.13 0.108798 0.0543992 0.998519i \(-0.482676\pi\)
0.0543992 + 0.998519i \(0.482676\pi\)
\(770\) 0 0
\(771\) −8848.66 −0.413329
\(772\) 0 0
\(773\) 4217.83 0.196254 0.0981272 0.995174i \(-0.468715\pi\)
0.0981272 + 0.995174i \(0.468715\pi\)
\(774\) 0 0
\(775\) −3725.73 −0.172687
\(776\) 0 0
\(777\) −198.662 −0.00917238
\(778\) 0 0
\(779\) 726.362 0.0334077
\(780\) 0 0
\(781\) 61390.0 2.81268
\(782\) 0 0
\(783\) 1452.60 0.0662983
\(784\) 0 0
\(785\) 36760.3 1.67138
\(786\) 0 0
\(787\) 11436.3 0.517991 0.258995 0.965879i \(-0.416609\pi\)
0.258995 + 0.965879i \(0.416609\pi\)
\(788\) 0 0
\(789\) −14494.9 −0.654034
\(790\) 0 0
\(791\) 5918.98 0.266062
\(792\) 0 0
\(793\) 21009.6 0.940823
\(794\) 0 0
\(795\) −12141.9 −0.541670
\(796\) 0 0
\(797\) −14102.3 −0.626761 −0.313381 0.949628i \(-0.601462\pi\)
−0.313381 + 0.949628i \(0.601462\pi\)
\(798\) 0 0
\(799\) 6805.25 0.301317
\(800\) 0 0
\(801\) 1355.60 0.0597975
\(802\) 0 0
\(803\) 22347.8 0.982111
\(804\) 0 0
\(805\) −10498.2 −0.459641
\(806\) 0 0
\(807\) −3281.52 −0.143141
\(808\) 0 0
\(809\) 18791.3 0.816647 0.408323 0.912837i \(-0.366114\pi\)
0.408323 + 0.912837i \(0.366114\pi\)
\(810\) 0 0
\(811\) 14452.6 0.625771 0.312886 0.949791i \(-0.398704\pi\)
0.312886 + 0.949791i \(0.398704\pi\)
\(812\) 0 0
\(813\) −9239.40 −0.398573
\(814\) 0 0
\(815\) −9651.43 −0.414816
\(816\) 0 0
\(817\) −3075.94 −0.131718
\(818\) 0 0
\(819\) 3748.00 0.159909
\(820\) 0 0
\(821\) 27786.1 1.18117 0.590585 0.806975i \(-0.298897\pi\)
0.590585 + 0.806975i \(0.298897\pi\)
\(822\) 0 0
\(823\) −39205.8 −1.66055 −0.830273 0.557357i \(-0.811815\pi\)
−0.830273 + 0.557357i \(0.811815\pi\)
\(824\) 0 0
\(825\) −3030.95 −0.127908
\(826\) 0 0
\(827\) −5929.15 −0.249307 −0.124653 0.992200i \(-0.539782\pi\)
−0.124653 + 0.992200i \(0.539782\pi\)
\(828\) 0 0
\(829\) −1269.44 −0.0531840 −0.0265920 0.999646i \(-0.508465\pi\)
−0.0265920 + 0.999646i \(0.508465\pi\)
\(830\) 0 0
\(831\) −16802.3 −0.701400
\(832\) 0 0
\(833\) 22839.5 0.949990
\(834\) 0 0
\(835\) 14521.6 0.601845
\(836\) 0 0
\(837\) −6142.19 −0.253650
\(838\) 0 0
\(839\) −17884.1 −0.735910 −0.367955 0.929844i \(-0.619942\pi\)
−0.367955 + 0.929844i \(0.619942\pi\)
\(840\) 0 0
\(841\) −21494.6 −0.881323
\(842\) 0 0
\(843\) −610.278 −0.0249337
\(844\) 0 0
\(845\) 20925.7 0.851911
\(846\) 0 0
\(847\) 15891.8 0.644686
\(848\) 0 0
\(849\) 6644.52 0.268598
\(850\) 0 0
\(851\) −1617.25 −0.0651453
\(852\) 0 0
\(853\) 16083.4 0.645585 0.322793 0.946470i \(-0.395378\pi\)
0.322793 + 0.946470i \(0.395378\pi\)
\(854\) 0 0
\(855\) 967.187 0.0386867
\(856\) 0 0
\(857\) 17203.6 0.685722 0.342861 0.939386i \(-0.388604\pi\)
0.342861 + 0.939386i \(0.388604\pi\)
\(858\) 0 0
\(859\) −24427.5 −0.970264 −0.485132 0.874441i \(-0.661228\pi\)
−0.485132 + 0.874441i \(0.661228\pi\)
\(860\) 0 0
\(861\) −1357.22 −0.0537212
\(862\) 0 0
\(863\) 46584.1 1.83747 0.918737 0.394870i \(-0.129210\pi\)
0.918737 + 0.394870i \(0.129210\pi\)
\(864\) 0 0
\(865\) −34515.3 −1.35671
\(866\) 0 0
\(867\) −2447.39 −0.0958683
\(868\) 0 0
\(869\) 67790.5 2.64630
\(870\) 0 0
\(871\) 59674.1 2.32145
\(872\) 0 0
\(873\) −16912.4 −0.655667
\(874\) 0 0
\(875\) −9462.91 −0.365605
\(876\) 0 0
\(877\) −7402.05 −0.285005 −0.142503 0.989794i \(-0.545515\pi\)
−0.142503 + 0.989794i \(0.545515\pi\)
\(878\) 0 0
\(879\) 17437.1 0.669101
\(880\) 0 0
\(881\) 12044.7 0.460608 0.230304 0.973119i \(-0.426028\pi\)
0.230304 + 0.973119i \(0.426028\pi\)
\(882\) 0 0
\(883\) −7150.02 −0.272500 −0.136250 0.990674i \(-0.543505\pi\)
−0.136250 + 0.990674i \(0.543505\pi\)
\(884\) 0 0
\(885\) 10130.4 0.384779
\(886\) 0 0
\(887\) −21897.6 −0.828916 −0.414458 0.910068i \(-0.636029\pi\)
−0.414458 + 0.910068i \(0.636029\pi\)
\(888\) 0 0
\(889\) −10578.8 −0.399102
\(890\) 0 0
\(891\) −4996.79 −0.187878
\(892\) 0 0
\(893\) −927.087 −0.0347411
\(894\) 0 0
\(895\) 3369.36 0.125838
\(896\) 0 0
\(897\) 30511.4 1.13573
\(898\) 0 0
\(899\) −12238.9 −0.454047
\(900\) 0 0
\(901\) −29392.5 −1.08680
\(902\) 0 0
\(903\) 5747.45 0.211808
\(904\) 0 0
\(905\) −31962.8 −1.17401
\(906\) 0 0
\(907\) 800.885 0.0293197 0.0146598 0.999893i \(-0.495333\pi\)
0.0146598 + 0.999893i \(0.495333\pi\)
\(908\) 0 0
\(909\) −15502.6 −0.565664
\(910\) 0 0
\(911\) 26742.9 0.972593 0.486296 0.873794i \(-0.338348\pi\)
0.486296 + 0.873794i \(0.338348\pi\)
\(912\) 0 0
\(913\) −48846.5 −1.77063
\(914\) 0 0
\(915\) −10130.4 −0.366011
\(916\) 0 0
\(917\) −12846.1 −0.462613
\(918\) 0 0
\(919\) 3744.72 0.134414 0.0672072 0.997739i \(-0.478591\pi\)
0.0672072 + 0.997739i \(0.478591\pi\)
\(920\) 0 0
\(921\) −22012.0 −0.787535
\(922\) 0 0
\(923\) −64530.3 −2.30124
\(924\) 0 0
\(925\) −168.873 −0.00600271
\(926\) 0 0
\(927\) −17179.4 −0.608679
\(928\) 0 0
\(929\) −25667.3 −0.906477 −0.453238 0.891389i \(-0.649731\pi\)
−0.453238 + 0.891389i \(0.649731\pi\)
\(930\) 0 0
\(931\) −3111.45 −0.109531
\(932\) 0 0
\(933\) −19727.7 −0.692236
\(934\) 0 0
\(935\) 48662.9 1.70208
\(936\) 0 0
\(937\) 17978.7 0.626829 0.313414 0.949616i \(-0.398527\pi\)
0.313414 + 0.949616i \(0.398527\pi\)
\(938\) 0 0
\(939\) 4670.01 0.162300
\(940\) 0 0
\(941\) 8120.01 0.281302 0.140651 0.990059i \(-0.455081\pi\)
0.140651 + 0.990059i \(0.455081\pi\)
\(942\) 0 0
\(943\) −11048.8 −0.381545
\(944\) 0 0
\(945\) −1807.21 −0.0622099
\(946\) 0 0
\(947\) −7930.95 −0.272145 −0.136072 0.990699i \(-0.543448\pi\)
−0.136072 + 0.990699i \(0.543448\pi\)
\(948\) 0 0
\(949\) −23490.9 −0.803527
\(950\) 0 0
\(951\) 28371.8 0.967422
\(952\) 0 0
\(953\) 29833.3 1.01406 0.507028 0.861930i \(-0.330744\pi\)
0.507028 + 0.861930i \(0.330744\pi\)
\(954\) 0 0
\(955\) −19348.2 −0.655596
\(956\) 0 0
\(957\) −9956.55 −0.336311
\(958\) 0 0
\(959\) −16535.3 −0.556781
\(960\) 0 0
\(961\) 21960.1 0.737139
\(962\) 0 0
\(963\) −1157.60 −0.0387364
\(964\) 0 0
\(965\) 31358.7 1.04609
\(966\) 0 0
\(967\) −7650.56 −0.254421 −0.127211 0.991876i \(-0.540602\pi\)
−0.127211 + 0.991876i \(0.540602\pi\)
\(968\) 0 0
\(969\) 2341.32 0.0776204
\(970\) 0 0
\(971\) 5634.51 0.186220 0.0931102 0.995656i \(-0.470319\pi\)
0.0931102 + 0.995656i \(0.470319\pi\)
\(972\) 0 0
\(973\) −8549.80 −0.281700
\(974\) 0 0
\(975\) 3186.00 0.104650
\(976\) 0 0
\(977\) 43983.5 1.44028 0.720142 0.693827i \(-0.244077\pi\)
0.720142 + 0.693827i \(0.244077\pi\)
\(978\) 0 0
\(979\) −9291.72 −0.303335
\(980\) 0 0
\(981\) 6824.41 0.222107
\(982\) 0 0
\(983\) 7759.30 0.251763 0.125882 0.992045i \(-0.459824\pi\)
0.125882 + 0.992045i \(0.459824\pi\)
\(984\) 0 0
\(985\) 9477.00 0.306561
\(986\) 0 0
\(987\) 1732.28 0.0558653
\(988\) 0 0
\(989\) 46788.4 1.50433
\(990\) 0 0
\(991\) −10114.8 −0.324226 −0.162113 0.986772i \(-0.551831\pi\)
−0.162113 + 0.986772i \(0.551831\pi\)
\(992\) 0 0
\(993\) 32722.9 1.04575
\(994\) 0 0
\(995\) 28976.3 0.923225
\(996\) 0 0
\(997\) −43842.2 −1.39267 −0.696337 0.717715i \(-0.745188\pi\)
−0.696337 + 0.717715i \(0.745188\pi\)
\(998\) 0 0
\(999\) −278.402 −0.00881706
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 768.4.a.e.1.2 2
3.2 odd 2 2304.4.a.bp.1.1 2
4.3 odd 2 768.4.a.k.1.2 2
8.3 odd 2 768.4.a.j.1.1 2
8.5 even 2 768.4.a.p.1.1 2
12.11 even 2 2304.4.a.bq.1.1 2
16.3 odd 4 384.4.d.c.193.3 yes 4
16.5 even 4 384.4.d.e.193.4 yes 4
16.11 odd 4 384.4.d.c.193.2 4
16.13 even 4 384.4.d.e.193.1 yes 4
24.5 odd 2 2304.4.a.s.1.2 2
24.11 even 2 2304.4.a.t.1.2 2
48.5 odd 4 1152.4.d.o.577.2 4
48.11 even 4 1152.4.d.i.577.2 4
48.29 odd 4 1152.4.d.o.577.3 4
48.35 even 4 1152.4.d.i.577.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.2 4 16.11 odd 4
384.4.d.c.193.3 yes 4 16.3 odd 4
384.4.d.e.193.1 yes 4 16.13 even 4
384.4.d.e.193.4 yes 4 16.5 even 4
768.4.a.e.1.2 2 1.1 even 1 trivial
768.4.a.j.1.1 2 8.3 odd 2
768.4.a.k.1.2 2 4.3 odd 2
768.4.a.p.1.1 2 8.5 even 2
1152.4.d.i.577.2 4 48.11 even 4
1152.4.d.i.577.3 4 48.35 even 4
1152.4.d.o.577.2 4 48.5 odd 4
1152.4.d.o.577.3 4 48.29 odd 4
2304.4.a.s.1.2 2 24.5 odd 2
2304.4.a.t.1.2 2 24.11 even 2
2304.4.a.bp.1.1 2 3.2 odd 2
2304.4.a.bq.1.1 2 12.11 even 2