Properties

Label 8281.2.a.bv.1.2
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{23})\)
Defining polynomial: \(x^{4} - 24 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.09827\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.41421 q^{3} -1.00000 q^{4} +2.68406 q^{5} -1.41421 q^{6} -3.00000 q^{8} -1.00000 q^{9} +2.68406 q^{10} +5.79583 q^{11} +1.41421 q^{12} -3.79583 q^{15} -1.00000 q^{16} -5.51249 q^{17} -1.00000 q^{18} +2.82843 q^{19} -2.68406 q^{20} +5.79583 q^{22} +1.79583 q^{23} +4.24264 q^{24} +2.20417 q^{25} +5.65685 q^{27} -8.79583 q^{29} -3.79583 q^{30} +1.41421 q^{31} +5.00000 q^{32} -8.19654 q^{33} -5.51249 q^{34} +1.00000 q^{36} -6.79583 q^{37} +2.82843 q^{38} -8.05217 q^{40} -9.75513 q^{41} -1.79583 q^{43} -5.79583 q^{44} -2.68406 q^{45} +1.79583 q^{46} -2.82843 q^{47} +1.41421 q^{48} +2.20417 q^{50} +7.79583 q^{51} +6.59166 q^{53} +5.65685 q^{54} +15.5563 q^{55} -4.00000 q^{57} -8.79583 q^{58} -1.12548 q^{59} +3.79583 q^{60} -1.55858 q^{61} +1.41421 q^{62} +7.00000 q^{64} -8.19654 q^{66} +5.79583 q^{67} +5.51249 q^{68} -2.53969 q^{69} +6.00000 q^{71} +3.00000 q^{72} +5.80122 q^{73} -6.79583 q^{74} -3.11716 q^{75} -2.82843 q^{76} -11.7958 q^{79} -2.68406 q^{80} -5.00000 q^{81} -9.75513 q^{82} +9.89949 q^{83} -14.7958 q^{85} -1.79583 q^{86} +12.4392 q^{87} -17.3875 q^{88} -12.1504 q^{89} -2.68406 q^{90} -1.79583 q^{92} -2.00000 q^{93} -2.82843 q^{94} +7.59166 q^{95} -7.07107 q^{96} +4.24264 q^{97} -5.79583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{16} - 4 q^{18} + 4 q^{22} - 12 q^{23} + 28 q^{25} - 16 q^{29} + 4 q^{30} + 20 q^{32} + 4 q^{36} - 8 q^{37} + 12 q^{43} - 4 q^{44} - 12 q^{46} + 28 q^{50} + 12 q^{51} - 12 q^{53} - 16 q^{57} - 16 q^{58} - 4 q^{60} + 28 q^{64} + 4 q^{67} + 24 q^{71} + 12 q^{72} - 8 q^{74} - 28 q^{79} - 20 q^{81} - 40 q^{85} + 12 q^{86} - 12 q^{88} + 12 q^{92} - 8 q^{93} - 8 q^{95} - 4 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.68406 1.20035 0.600174 0.799870i \(-0.295098\pi\)
0.600174 + 0.799870i \(0.295098\pi\)
\(6\) −1.41421 −0.577350
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) −1.00000 −0.333333
\(10\) 2.68406 0.848774
\(11\) 5.79583 1.74751 0.873754 0.486367i \(-0.161678\pi\)
0.873754 + 0.486367i \(0.161678\pi\)
\(12\) 1.41421 0.408248
\(13\) 0 0
\(14\) 0 0
\(15\) −3.79583 −0.980079
\(16\) −1.00000 −0.250000
\(17\) −5.51249 −1.33697 −0.668487 0.743724i \(-0.733058\pi\)
−0.668487 + 0.743724i \(0.733058\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −2.68406 −0.600174
\(21\) 0 0
\(22\) 5.79583 1.23568
\(23\) 1.79583 0.374457 0.187228 0.982316i \(-0.440050\pi\)
0.187228 + 0.982316i \(0.440050\pi\)
\(24\) 4.24264 0.866025
\(25\) 2.20417 0.440834
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −8.79583 −1.63334 −0.816672 0.577101i \(-0.804184\pi\)
−0.816672 + 0.577101i \(0.804184\pi\)
\(30\) −3.79583 −0.693021
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 5.00000 0.883883
\(33\) −8.19654 −1.42684
\(34\) −5.51249 −0.945383
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.79583 −1.11723 −0.558614 0.829428i \(-0.688667\pi\)
−0.558614 + 0.829428i \(0.688667\pi\)
\(38\) 2.82843 0.458831
\(39\) 0 0
\(40\) −8.05217 −1.27316
\(41\) −9.75513 −1.52349 −0.761747 0.647874i \(-0.775658\pi\)
−0.761747 + 0.647874i \(0.775658\pi\)
\(42\) 0 0
\(43\) −1.79583 −0.273862 −0.136931 0.990581i \(-0.543724\pi\)
−0.136931 + 0.990581i \(0.543724\pi\)
\(44\) −5.79583 −0.873754
\(45\) −2.68406 −0.400116
\(46\) 1.79583 0.264781
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 1.41421 0.204124
\(49\) 0 0
\(50\) 2.20417 0.311716
\(51\) 7.79583 1.09163
\(52\) 0 0
\(53\) 6.59166 0.905435 0.452717 0.891654i \(-0.350455\pi\)
0.452717 + 0.891654i \(0.350455\pi\)
\(54\) 5.65685 0.769800
\(55\) 15.5563 2.09762
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) −8.79583 −1.15495
\(59\) −1.12548 −0.146524 −0.0732622 0.997313i \(-0.523341\pi\)
−0.0732622 + 0.997313i \(0.523341\pi\)
\(60\) 3.79583 0.490040
\(61\) −1.55858 −0.199556 −0.0997780 0.995010i \(-0.531813\pi\)
−0.0997780 + 0.995010i \(0.531813\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −8.19654 −1.00892
\(67\) 5.79583 0.708074 0.354037 0.935232i \(-0.384809\pi\)
0.354037 + 0.935232i \(0.384809\pi\)
\(68\) 5.51249 0.668487
\(69\) −2.53969 −0.305743
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 3.00000 0.353553
\(73\) 5.80122 0.678982 0.339491 0.940609i \(-0.389745\pi\)
0.339491 + 0.940609i \(0.389745\pi\)
\(74\) −6.79583 −0.789999
\(75\) −3.11716 −0.359939
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) 0 0
\(79\) −11.7958 −1.32713 −0.663567 0.748117i \(-0.730958\pi\)
−0.663567 + 0.748117i \(0.730958\pi\)
\(80\) −2.68406 −0.300087
\(81\) −5.00000 −0.555556
\(82\) −9.75513 −1.07727
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) −14.7958 −1.60483
\(86\) −1.79583 −0.193649
\(87\) 12.4392 1.33362
\(88\) −17.3875 −1.85351
\(89\) −12.1504 −1.28794 −0.643972 0.765049i \(-0.722715\pi\)
−0.643972 + 0.765049i \(0.722715\pi\)
\(90\) −2.68406 −0.282925
\(91\) 0 0
\(92\) −1.79583 −0.187228
\(93\) −2.00000 −0.207390
\(94\) −2.82843 −0.291730
\(95\) 7.59166 0.778888
\(96\) −7.07107 −0.721688
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 0 0
\(99\) −5.79583 −0.582503
\(100\) −2.20417 −0.220417
\(101\) 2.97280 0.295804 0.147902 0.989002i \(-0.452748\pi\)
0.147902 + 0.989002i \(0.452748\pi\)
\(102\) 7.79583 0.771902
\(103\) 8.19654 0.807629 0.403815 0.914841i \(-0.367684\pi\)
0.403815 + 0.914841i \(0.367684\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.59166 0.640239
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −5.65685 −0.544331
\(109\) −17.5917 −1.68498 −0.842488 0.538715i \(-0.818910\pi\)
−0.842488 + 0.538715i \(0.818910\pi\)
\(110\) 15.5563 1.48324
\(111\) 9.61076 0.912213
\(112\) 0 0
\(113\) −16.5917 −1.56081 −0.780406 0.625273i \(-0.784988\pi\)
−0.780406 + 0.625273i \(0.784988\pi\)
\(114\) −4.00000 −0.374634
\(115\) 4.82012 0.449478
\(116\) 8.79583 0.816672
\(117\) 0 0
\(118\) −1.12548 −0.103608
\(119\) 0 0
\(120\) 11.3875 1.03953
\(121\) 22.5917 2.05379
\(122\) −1.55858 −0.141107
\(123\) 13.7958 1.24393
\(124\) −1.41421 −0.127000
\(125\) −7.50417 −0.671194
\(126\) 0 0
\(127\) 7.59166 0.673651 0.336826 0.941567i \(-0.390647\pi\)
0.336826 + 0.941567i \(0.390647\pi\)
\(128\) −3.00000 −0.265165
\(129\) 2.53969 0.223607
\(130\) 0 0
\(131\) 8.19654 0.716135 0.358068 0.933696i \(-0.383436\pi\)
0.358068 + 0.933696i \(0.383436\pi\)
\(132\) 8.19654 0.713418
\(133\) 0 0
\(134\) 5.79583 0.500684
\(135\) 15.1833 1.30677
\(136\) 16.5375 1.41808
\(137\) −9.20417 −0.786365 −0.393183 0.919460i \(-0.628626\pi\)
−0.393183 + 0.919460i \(0.628626\pi\)
\(138\) −2.53969 −0.216193
\(139\) 15.2676 1.29498 0.647491 0.762073i \(-0.275818\pi\)
0.647491 + 0.762073i \(0.275818\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −23.6085 −1.96058
\(146\) 5.80122 0.480113
\(147\) 0 0
\(148\) 6.79583 0.558614
\(149\) 4.59166 0.376164 0.188082 0.982153i \(-0.439773\pi\)
0.188082 + 0.982153i \(0.439773\pi\)
\(150\) −3.11716 −0.254515
\(151\) −17.5917 −1.43159 −0.715795 0.698311i \(-0.753935\pi\)
−0.715795 + 0.698311i \(0.753935\pi\)
\(152\) −8.48528 −0.688247
\(153\) 5.51249 0.445658
\(154\) 0 0
\(155\) 3.79583 0.304889
\(156\) 0 0
\(157\) 6.92670 0.552811 0.276405 0.961041i \(-0.410857\pi\)
0.276405 + 0.961041i \(0.410857\pi\)
\(158\) −11.7958 −0.938426
\(159\) −9.32202 −0.739284
\(160\) 13.4203 1.06097
\(161\) 0 0
\(162\) −5.00000 −0.392837
\(163\) −13.3875 −1.04859 −0.524295 0.851537i \(-0.675671\pi\)
−0.524295 + 0.851537i \(0.675671\pi\)
\(164\) 9.75513 0.761747
\(165\) −22.0000 −1.71270
\(166\) 9.89949 0.768350
\(167\) −15.2676 −1.18144 −0.590722 0.806875i \(-0.701157\pi\)
−0.590722 + 0.806875i \(0.701157\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −14.7958 −1.13479
\(171\) −2.82843 −0.216295
\(172\) 1.79583 0.136931
\(173\) −9.32202 −0.708740 −0.354370 0.935105i \(-0.615305\pi\)
−0.354370 + 0.935105i \(0.615305\pi\)
\(174\) 12.4392 0.943012
\(175\) 0 0
\(176\) −5.79583 −0.436877
\(177\) 1.59166 0.119637
\(178\) −12.1504 −0.910714
\(179\) −0.408337 −0.0305205 −0.0152603 0.999884i \(-0.504858\pi\)
−0.0152603 + 0.999884i \(0.504858\pi\)
\(180\) 2.68406 0.200058
\(181\) 16.5375 1.22922 0.614610 0.788831i \(-0.289314\pi\)
0.614610 + 0.788831i \(0.289314\pi\)
\(182\) 0 0
\(183\) 2.20417 0.162937
\(184\) −5.38749 −0.397171
\(185\) −18.2404 −1.34106
\(186\) −2.00000 −0.146647
\(187\) −31.9494 −2.33637
\(188\) 2.82843 0.206284
\(189\) 0 0
\(190\) 7.59166 0.550757
\(191\) −25.7958 −1.86652 −0.933260 0.359200i \(-0.883049\pi\)
−0.933260 + 0.359200i \(0.883049\pi\)
\(192\) −9.89949 −0.714435
\(193\) 3.40834 0.245337 0.122669 0.992448i \(-0.460855\pi\)
0.122669 + 0.992448i \(0.460855\pi\)
\(194\) 4.24264 0.304604
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −5.79583 −0.411892
\(199\) 22.0499 1.56308 0.781539 0.623856i \(-0.214435\pi\)
0.781539 + 0.623856i \(0.214435\pi\)
\(200\) −6.61251 −0.467575
\(201\) −8.19654 −0.578140
\(202\) 2.97280 0.209165
\(203\) 0 0
\(204\) −7.79583 −0.545817
\(205\) −26.1833 −1.82872
\(206\) 8.19654 0.571080
\(207\) −1.79583 −0.124819
\(208\) 0 0
\(209\) 16.3931 1.13393
\(210\) 0 0
\(211\) −1.79583 −0.123630 −0.0618151 0.998088i \(-0.519689\pi\)
−0.0618151 + 0.998088i \(0.519689\pi\)
\(212\) −6.59166 −0.452717
\(213\) −8.48528 −0.581402
\(214\) −6.00000 −0.410152
\(215\) −4.82012 −0.328729
\(216\) −16.9706 −1.15470
\(217\) 0 0
\(218\) −17.5917 −1.19146
\(219\) −8.20417 −0.554386
\(220\) −15.5563 −1.04881
\(221\) 0 0
\(222\) 9.61076 0.645032
\(223\) 5.65685 0.378811 0.189405 0.981899i \(-0.439344\pi\)
0.189405 + 0.981899i \(0.439344\pi\)
\(224\) 0 0
\(225\) −2.20417 −0.146945
\(226\) −16.5917 −1.10366
\(227\) −20.9245 −1.38881 −0.694403 0.719587i \(-0.744331\pi\)
−0.694403 + 0.719587i \(0.744331\pi\)
\(228\) 4.00000 0.264906
\(229\) −12.7279 −0.841085 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(230\) 4.82012 0.317829
\(231\) 0 0
\(232\) 26.3875 1.73242
\(233\) −21.1833 −1.38777 −0.693883 0.720088i \(-0.744101\pi\)
−0.693883 + 0.720088i \(0.744101\pi\)
\(234\) 0 0
\(235\) −7.59166 −0.495225
\(236\) 1.12548 0.0732622
\(237\) 16.6818 1.08360
\(238\) 0 0
\(239\) −19.7958 −1.28049 −0.640243 0.768172i \(-0.721166\pi\)
−0.640243 + 0.768172i \(0.721166\pi\)
\(240\) 3.79583 0.245020
\(241\) −4.38701 −0.282592 −0.141296 0.989967i \(-0.545127\pi\)
−0.141296 + 0.989967i \(0.545127\pi\)
\(242\) 22.5917 1.45225
\(243\) −9.89949 −0.635053
\(244\) 1.55858 0.0997780
\(245\) 0 0
\(246\) 13.7958 0.879590
\(247\) 0 0
\(248\) −4.24264 −0.269408
\(249\) −14.0000 −0.887214
\(250\) −7.50417 −0.474606
\(251\) 3.11716 0.196754 0.0983769 0.995149i \(-0.468635\pi\)
0.0983769 + 0.995149i \(0.468635\pi\)
\(252\) 0 0
\(253\) 10.4083 0.654367
\(254\) 7.59166 0.476343
\(255\) 20.9245 1.31034
\(256\) −17.0000 −1.06250
\(257\) −15.4120 −0.961373 −0.480686 0.876893i \(-0.659612\pi\)
−0.480686 + 0.876893i \(0.659612\pi\)
\(258\) 2.53969 0.158114
\(259\) 0 0
\(260\) 0 0
\(261\) 8.79583 0.544448
\(262\) 8.19654 0.506384
\(263\) −25.3875 −1.56546 −0.782730 0.622361i \(-0.786173\pi\)
−0.782730 + 0.622361i \(0.786173\pi\)
\(264\) 24.5896 1.51339
\(265\) 17.6924 1.08684
\(266\) 0 0
\(267\) 17.1833 1.05160
\(268\) −5.79583 −0.354037
\(269\) 1.12548 0.0686215 0.0343107 0.999411i \(-0.489076\pi\)
0.0343107 + 0.999411i \(0.489076\pi\)
\(270\) 15.1833 0.924028
\(271\) 23.1754 1.40781 0.703903 0.710296i \(-0.251439\pi\)
0.703903 + 0.710296i \(0.251439\pi\)
\(272\) 5.51249 0.334244
\(273\) 0 0
\(274\) −9.20417 −0.556044
\(275\) 12.7750 0.770361
\(276\) 2.53969 0.152871
\(277\) −6.18333 −0.371520 −0.185760 0.982595i \(-0.559475\pi\)
−0.185760 + 0.982595i \(0.559475\pi\)
\(278\) 15.2676 0.915690
\(279\) −1.41421 −0.0846668
\(280\) 0 0
\(281\) 15.2042 0.907005 0.453502 0.891255i \(-0.350174\pi\)
0.453502 + 0.891255i \(0.350174\pi\)
\(282\) 4.00000 0.238197
\(283\) 29.4097 1.74823 0.874114 0.485721i \(-0.161443\pi\)
0.874114 + 0.485721i \(0.161443\pi\)
\(284\) −6.00000 −0.356034
\(285\) −10.7362 −0.635960
\(286\) 0 0
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) 13.3875 0.787500
\(290\) −23.6085 −1.38634
\(291\) −6.00000 −0.351726
\(292\) −5.80122 −0.339491
\(293\) 9.75513 0.569901 0.284950 0.958542i \(-0.408023\pi\)
0.284950 + 0.958542i \(0.408023\pi\)
\(294\) 0 0
\(295\) −3.02084 −0.175880
\(296\) 20.3875 1.18500
\(297\) 32.7862 1.90245
\(298\) 4.59166 0.265988
\(299\) 0 0
\(300\) 3.11716 0.179970
\(301\) 0 0
\(302\) −17.5917 −1.01229
\(303\) −4.20417 −0.241523
\(304\) −2.82843 −0.162221
\(305\) −4.18333 −0.239537
\(306\) 5.51249 0.315128
\(307\) −34.2004 −1.95192 −0.975960 0.217951i \(-0.930063\pi\)
−0.975960 + 0.217951i \(0.930063\pi\)
\(308\) 0 0
\(309\) −11.5917 −0.659427
\(310\) 3.79583 0.215589
\(311\) −11.8617 −0.672616 −0.336308 0.941752i \(-0.609178\pi\)
−0.336308 + 0.941752i \(0.609178\pi\)
\(312\) 0 0
\(313\) −1.70295 −0.0962565 −0.0481283 0.998841i \(-0.515326\pi\)
−0.0481283 + 0.998841i \(0.515326\pi\)
\(314\) 6.92670 0.390896
\(315\) 0 0
\(316\) 11.7958 0.663567
\(317\) −12.5917 −0.707218 −0.353609 0.935393i \(-0.615046\pi\)
−0.353609 + 0.935393i \(0.615046\pi\)
\(318\) −9.32202 −0.522753
\(319\) −50.9792 −2.85428
\(320\) 18.7884 1.05030
\(321\) 8.48528 0.473602
\(322\) 0 0
\(323\) −15.5917 −0.867543
\(324\) 5.00000 0.277778
\(325\) 0 0
\(326\) −13.3875 −0.741465
\(327\) 24.8784 1.37578
\(328\) 29.2654 1.61591
\(329\) 0 0
\(330\) −22.0000 −1.21106
\(331\) 0.612505 0.0336663 0.0168332 0.999858i \(-0.494642\pi\)
0.0168332 + 0.999858i \(0.494642\pi\)
\(332\) −9.89949 −0.543305
\(333\) 6.79583 0.372409
\(334\) −15.2676 −0.835407
\(335\) 15.5563 0.849934
\(336\) 0 0
\(337\) −29.9792 −1.63307 −0.816534 0.577297i \(-0.804108\pi\)
−0.816534 + 0.577297i \(0.804108\pi\)
\(338\) 0 0
\(339\) 23.4642 1.27440
\(340\) 14.7958 0.802417
\(341\) 8.19654 0.443868
\(342\) −2.82843 −0.152944
\(343\) 0 0
\(344\) 5.38749 0.290474
\(345\) −6.81667 −0.366997
\(346\) −9.32202 −0.501155
\(347\) 14.9792 0.804123 0.402062 0.915613i \(-0.368294\pi\)
0.402062 + 0.915613i \(0.368294\pi\)
\(348\) −12.4392 −0.666810
\(349\) −13.0167 −0.696766 −0.348383 0.937352i \(-0.613269\pi\)
−0.348383 + 0.937352i \(0.613269\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 28.9792 1.54459
\(353\) 15.1232 0.804929 0.402464 0.915436i \(-0.368154\pi\)
0.402464 + 0.915436i \(0.368154\pi\)
\(354\) 1.59166 0.0845959
\(355\) 16.1043 0.854730
\(356\) 12.1504 0.643972
\(357\) 0 0
\(358\) −0.408337 −0.0215813
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 8.05217 0.424387
\(361\) −11.0000 −0.578947
\(362\) 16.5375 0.869189
\(363\) −31.9494 −1.67691
\(364\) 0 0
\(365\) 15.5708 0.815014
\(366\) 2.20417 0.115214
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) −1.79583 −0.0936142
\(369\) 9.75513 0.507832
\(370\) −18.2404 −0.948274
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −12.5917 −0.651972 −0.325986 0.945375i \(-0.605696\pi\)
−0.325986 + 0.945375i \(0.605696\pi\)
\(374\) −31.9494 −1.65207
\(375\) 10.6125 0.548027
\(376\) 8.48528 0.437595
\(377\) 0 0
\(378\) 0 0
\(379\) −3.38749 −0.174004 −0.0870020 0.996208i \(-0.527729\pi\)
−0.0870020 + 0.996208i \(0.527729\pi\)
\(380\) −7.59166 −0.389444
\(381\) −10.7362 −0.550034
\(382\) −25.7958 −1.31983
\(383\) 14.9789 0.765385 0.382692 0.923876i \(-0.374997\pi\)
0.382692 + 0.923876i \(0.374997\pi\)
\(384\) 4.24264 0.216506
\(385\) 0 0
\(386\) 3.40834 0.173480
\(387\) 1.79583 0.0912872
\(388\) −4.24264 −0.215387
\(389\) 28.3875 1.43930 0.719652 0.694335i \(-0.244302\pi\)
0.719652 + 0.694335i \(0.244302\pi\)
\(390\) 0 0
\(391\) −9.89949 −0.500639
\(392\) 0 0
\(393\) −11.5917 −0.584722
\(394\) −8.00000 −0.403034
\(395\) −31.6607 −1.59302
\(396\) 5.79583 0.291251
\(397\) −7.07107 −0.354887 −0.177443 0.984131i \(-0.556783\pi\)
−0.177443 + 0.984131i \(0.556783\pi\)
\(398\) 22.0499 1.10526
\(399\) 0 0
\(400\) −2.20417 −0.110208
\(401\) 24.3875 1.21785 0.608927 0.793227i \(-0.291600\pi\)
0.608927 + 0.793227i \(0.291600\pi\)
\(402\) −8.19654 −0.408806
\(403\) 0 0
\(404\) −2.97280 −0.147902
\(405\) −13.4203 −0.666860
\(406\) 0 0
\(407\) −39.3875 −1.95237
\(408\) −23.3875 −1.15785
\(409\) 28.6879 1.41853 0.709263 0.704944i \(-0.249028\pi\)
0.709263 + 0.704944i \(0.249028\pi\)
\(410\) −26.1833 −1.29310
\(411\) 13.0167 0.642064
\(412\) −8.19654 −0.403815
\(413\) 0 0
\(414\) −1.79583 −0.0882603
\(415\) 26.5708 1.30431
\(416\) 0 0
\(417\) −21.5917 −1.05735
\(418\) 16.3931 0.801812
\(419\) −28.5435 −1.39444 −0.697221 0.716856i \(-0.745581\pi\)
−0.697221 + 0.716856i \(0.745581\pi\)
\(420\) 0 0
\(421\) −6.59166 −0.321258 −0.160629 0.987015i \(-0.551352\pi\)
−0.160629 + 0.987015i \(0.551352\pi\)
\(422\) −1.79583 −0.0874197
\(423\) 2.82843 0.137523
\(424\) −19.7750 −0.960358
\(425\) −12.1504 −0.589383
\(426\) −8.48528 −0.411113
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −4.82012 −0.232447
\(431\) −5.18333 −0.249672 −0.124836 0.992177i \(-0.539840\pi\)
−0.124836 + 0.992177i \(0.539840\pi\)
\(432\) −5.65685 −0.272166
\(433\) −13.4203 −0.644938 −0.322469 0.946580i \(-0.604513\pi\)
−0.322469 + 0.946580i \(0.604513\pi\)
\(434\) 0 0
\(435\) 33.3875 1.60081
\(436\) 17.5917 0.842488
\(437\) 5.07938 0.242980
\(438\) −8.20417 −0.392010
\(439\) −34.2004 −1.63230 −0.816148 0.577843i \(-0.803894\pi\)
−0.816148 + 0.577843i \(0.803894\pi\)
\(440\) −46.6690 −2.22486
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000 0.475114 0.237557 0.971374i \(-0.423653\pi\)
0.237557 + 0.971374i \(0.423653\pi\)
\(444\) −9.61076 −0.456106
\(445\) −32.6125 −1.54598
\(446\) 5.65685 0.267860
\(447\) −6.49359 −0.307136
\(448\) 0 0
\(449\) 4.40834 0.208042 0.104021 0.994575i \(-0.466829\pi\)
0.104021 + 0.994575i \(0.466829\pi\)
\(450\) −2.20417 −0.103905
\(451\) −56.5391 −2.66232
\(452\) 16.5917 0.780406
\(453\) 24.8784 1.16889
\(454\) −20.9245 −0.982034
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −14.5917 −0.682569 −0.341285 0.939960i \(-0.610862\pi\)
−0.341285 + 0.939960i \(0.610862\pi\)
\(458\) −12.7279 −0.594737
\(459\) −31.1833 −1.45551
\(460\) −4.82012 −0.224739
\(461\) −30.9683 −1.44234 −0.721169 0.692759i \(-0.756395\pi\)
−0.721169 + 0.692759i \(0.756395\pi\)
\(462\) 0 0
\(463\) 11.3875 0.529222 0.264611 0.964355i \(-0.414756\pi\)
0.264611 + 0.964355i \(0.414756\pi\)
\(464\) 8.79583 0.408336
\(465\) −5.36812 −0.248940
\(466\) −21.1833 −0.981299
\(467\) 4.82012 0.223048 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −7.59166 −0.350177
\(471\) −9.79583 −0.451368
\(472\) 3.37643 0.155413
\(473\) −10.4083 −0.478576
\(474\) 16.6818 0.766222
\(475\) 6.23433 0.286051
\(476\) 0 0
\(477\) −6.59166 −0.301812
\(478\) −19.7958 −0.905440
\(479\) 7.35981 0.336278 0.168139 0.985763i \(-0.446224\pi\)
0.168139 + 0.985763i \(0.446224\pi\)
\(480\) −18.9792 −0.866276
\(481\) 0 0
\(482\) −4.38701 −0.199823
\(483\) 0 0
\(484\) −22.5917 −1.02689
\(485\) 11.3875 0.517079
\(486\) −9.89949 −0.449050
\(487\) 19.5917 0.887783 0.443891 0.896081i \(-0.353598\pi\)
0.443891 + 0.896081i \(0.353598\pi\)
\(488\) 4.67575 0.211661
\(489\) 18.9328 0.856170
\(490\) 0 0
\(491\) −9.59166 −0.432866 −0.216433 0.976298i \(-0.569442\pi\)
−0.216433 + 0.976298i \(0.569442\pi\)
\(492\) −13.7958 −0.621964
\(493\) 48.4869 2.18374
\(494\) 0 0
\(495\) −15.5563 −0.699206
\(496\) −1.41421 −0.0635001
\(497\) 0 0
\(498\) −14.0000 −0.627355
\(499\) 16.2042 0.725398 0.362699 0.931906i \(-0.381855\pi\)
0.362699 + 0.931906i \(0.381855\pi\)
\(500\) 7.50417 0.335597
\(501\) 21.5917 0.964644
\(502\) 3.11716 0.139126
\(503\) 28.5435 1.27269 0.636347 0.771403i \(-0.280445\pi\)
0.636347 + 0.771403i \(0.280445\pi\)
\(504\) 0 0
\(505\) 7.97916 0.355068
\(506\) 10.4083 0.462707
\(507\) 0 0
\(508\) −7.59166 −0.336826
\(509\) −26.4370 −1.17180 −0.585899 0.810384i \(-0.699258\pi\)
−0.585899 + 0.810384i \(0.699258\pi\)
\(510\) 20.9245 0.926551
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 16.0000 0.706417
\(514\) −15.4120 −0.679793
\(515\) 22.0000 0.969436
\(516\) −2.53969 −0.111804
\(517\) −16.3931 −0.720967
\(518\) 0 0
\(519\) 13.1833 0.578684
\(520\) 0 0
\(521\) −25.0227 −1.09627 −0.548133 0.836391i \(-0.684661\pi\)
−0.548133 + 0.836391i \(0.684661\pi\)
\(522\) 8.79583 0.384983
\(523\) −28.5730 −1.24941 −0.624705 0.780861i \(-0.714781\pi\)
−0.624705 + 0.780861i \(0.714781\pi\)
\(524\) −8.19654 −0.358068
\(525\) 0 0
\(526\) −25.3875 −1.10695
\(527\) −7.79583 −0.339592
\(528\) 8.19654 0.356709
\(529\) −19.7750 −0.859782
\(530\) 17.6924 0.768509
\(531\) 1.12548 0.0488415
\(532\) 0 0
\(533\) 0 0
\(534\) 17.1833 0.743595
\(535\) −16.1043 −0.696252
\(536\) −17.3875 −0.751025
\(537\) 0.577476 0.0249199
\(538\) 1.12548 0.0485227
\(539\) 0 0
\(540\) −15.1833 −0.653386
\(541\) −12.5917 −0.541358 −0.270679 0.962670i \(-0.587248\pi\)
−0.270679 + 0.962670i \(0.587248\pi\)
\(542\) 23.1754 0.995469
\(543\) −23.3875 −1.00365
\(544\) −27.5624 −1.18173
\(545\) −47.2170 −2.02256
\(546\) 0 0
\(547\) −36.9792 −1.58111 −0.790557 0.612388i \(-0.790209\pi\)
−0.790557 + 0.612388i \(0.790209\pi\)
\(548\) 9.20417 0.393183
\(549\) 1.55858 0.0665187
\(550\) 12.7750 0.544727
\(551\) −24.8784 −1.05985
\(552\) 7.61907 0.324289
\(553\) 0 0
\(554\) −6.18333 −0.262704
\(555\) 25.7958 1.09497
\(556\) −15.2676 −0.647491
\(557\) 20.5917 0.872497 0.436248 0.899826i \(-0.356307\pi\)
0.436248 + 0.899826i \(0.356307\pi\)
\(558\) −1.41421 −0.0598684
\(559\) 0 0
\(560\) 0 0
\(561\) 45.1833 1.90764
\(562\) 15.2042 0.641349
\(563\) 1.12548 0.0474331 0.0237166 0.999719i \(-0.492450\pi\)
0.0237166 + 0.999719i \(0.492450\pi\)
\(564\) −4.00000 −0.168430
\(565\) −44.5330 −1.87352
\(566\) 29.4097 1.23618
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) 6.40834 0.268651 0.134326 0.990937i \(-0.457113\pi\)
0.134326 + 0.990937i \(0.457113\pi\)
\(570\) −10.7362 −0.449691
\(571\) 15.1833 0.635402 0.317701 0.948191i \(-0.397089\pi\)
0.317701 + 0.948191i \(0.397089\pi\)
\(572\) 0 0
\(573\) 36.4808 1.52401
\(574\) 0 0
\(575\) 3.95832 0.165073
\(576\) −7.00000 −0.291667
\(577\) 9.17765 0.382071 0.191035 0.981583i \(-0.438816\pi\)
0.191035 + 0.981583i \(0.438816\pi\)
\(578\) 13.3875 0.556846
\(579\) −4.82012 −0.200317
\(580\) 23.6085 0.980291
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 38.2042 1.58225
\(584\) −17.4037 −0.720169
\(585\) 0 0
\(586\) 9.75513 0.402981
\(587\) −3.11716 −0.128659 −0.0643296 0.997929i \(-0.520491\pi\)
−0.0643296 + 0.997929i \(0.520491\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) −3.02084 −0.124366
\(591\) 11.3137 0.465384
\(592\) 6.79583 0.279307
\(593\) 1.55858 0.0640033 0.0320017 0.999488i \(-0.489812\pi\)
0.0320017 + 0.999488i \(0.489812\pi\)
\(594\) 32.7862 1.34523
\(595\) 0 0
\(596\) −4.59166 −0.188082
\(597\) −31.1833 −1.27625
\(598\) 0 0
\(599\) −14.4083 −0.588709 −0.294354 0.955696i \(-0.595105\pi\)
−0.294354 + 0.955696i \(0.595105\pi\)
\(600\) 9.35149 0.381773
\(601\) 30.9683 1.26322 0.631612 0.775284i \(-0.282393\pi\)
0.631612 + 0.775284i \(0.282393\pi\)
\(602\) 0 0
\(603\) −5.79583 −0.236025
\(604\) 17.5917 0.715795
\(605\) 60.6373 2.46526
\(606\) −4.20417 −0.170783
\(607\) 35.9033 1.45727 0.728636 0.684901i \(-0.240155\pi\)
0.728636 + 0.684901i \(0.240155\pi\)
\(608\) 14.1421 0.573539
\(609\) 0 0
\(610\) −4.18333 −0.169378
\(611\) 0 0
\(612\) −5.51249 −0.222829
\(613\) −5.97916 −0.241496 −0.120748 0.992683i \(-0.538529\pi\)
−0.120748 + 0.992683i \(0.538529\pi\)
\(614\) −34.2004 −1.38022
\(615\) 37.0288 1.49315
\(616\) 0 0
\(617\) −24.3875 −0.981804 −0.490902 0.871215i \(-0.663333\pi\)
−0.490902 + 0.871215i \(0.663333\pi\)
\(618\) −11.5917 −0.466285
\(619\) −33.9116 −1.36302 −0.681512 0.731807i \(-0.738677\pi\)
−0.681512 + 0.731807i \(0.738677\pi\)
\(620\) −3.79583 −0.152444
\(621\) 10.1588 0.407657
\(622\) −11.8617 −0.475611
\(623\) 0 0
\(624\) 0 0
\(625\) −31.1625 −1.24650
\(626\) −1.70295 −0.0680636
\(627\) −23.1833 −0.925853
\(628\) −6.92670 −0.276405
\(629\) 37.4619 1.49370
\(630\) 0 0
\(631\) −20.4083 −0.812443 −0.406222 0.913775i \(-0.633154\pi\)
−0.406222 + 0.913775i \(0.633154\pi\)
\(632\) 35.3875 1.40764
\(633\) 2.53969 0.100944
\(634\) −12.5917 −0.500079
\(635\) 20.3765 0.808615
\(636\) 9.32202 0.369642
\(637\) 0 0
\(638\) −50.9792 −2.01828
\(639\) −6.00000 −0.237356
\(640\) −8.05217 −0.318290
\(641\) 4.79583 0.189424 0.0947120 0.995505i \(-0.469807\pi\)
0.0947120 + 0.995505i \(0.469807\pi\)
\(642\) 8.48528 0.334887
\(643\) −5.65685 −0.223085 −0.111542 0.993760i \(-0.535579\pi\)
−0.111542 + 0.993760i \(0.535579\pi\)
\(644\) 0 0
\(645\) 6.81667 0.268406
\(646\) −15.5917 −0.613446
\(647\) 1.96221 0.0771426 0.0385713 0.999256i \(-0.487719\pi\)
0.0385713 + 0.999256i \(0.487719\pi\)
\(648\) 15.0000 0.589256
\(649\) −6.52307 −0.256053
\(650\) 0 0
\(651\) 0 0
\(652\) 13.3875 0.524295
\(653\) 25.1833 0.985500 0.492750 0.870171i \(-0.335992\pi\)
0.492750 + 0.870171i \(0.335992\pi\)
\(654\) 24.8784 0.972821
\(655\) 22.0000 0.859611
\(656\) 9.75513 0.380874
\(657\) −5.80122 −0.226327
\(658\) 0 0
\(659\) −37.5917 −1.46436 −0.732182 0.681109i \(-0.761498\pi\)
−0.732182 + 0.681109i \(0.761498\pi\)
\(660\) 22.0000 0.856349
\(661\) −27.8512 −1.08328 −0.541642 0.840609i \(-0.682197\pi\)
−0.541642 + 0.840609i \(0.682197\pi\)
\(662\) 0.612505 0.0238057
\(663\) 0 0
\(664\) −29.6985 −1.15252
\(665\) 0 0
\(666\) 6.79583 0.263333
\(667\) −15.7958 −0.611617
\(668\) 15.2676 0.590722
\(669\) −8.00000 −0.309298
\(670\) 15.5563 0.600994
\(671\) −9.03328 −0.348726
\(672\) 0 0
\(673\) 23.9792 0.924329 0.462164 0.886794i \(-0.347073\pi\)
0.462164 + 0.886794i \(0.347073\pi\)
\(674\) −29.9792 −1.15475
\(675\) 12.4687 0.479919
\(676\) 0 0
\(677\) −34.7779 −1.33662 −0.668311 0.743882i \(-0.732982\pi\)
−0.668311 + 0.743882i \(0.732982\pi\)
\(678\) 23.4642 0.901135
\(679\) 0 0
\(680\) 44.3875 1.70218
\(681\) 29.5917 1.13395
\(682\) 8.19654 0.313862
\(683\) 50.7750 1.94285 0.971425 0.237345i \(-0.0762771\pi\)
0.971425 + 0.237345i \(0.0762771\pi\)
\(684\) 2.82843 0.108148
\(685\) −24.7045 −0.943911
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 1.79583 0.0684654
\(689\) 0 0
\(690\) −6.81667 −0.259506
\(691\) −11.0250 −0.419410 −0.209705 0.977765i \(-0.567250\pi\)
−0.209705 + 0.977765i \(0.567250\pi\)
\(692\) 9.32202 0.354370
\(693\) 0 0
\(694\) 14.9792 0.568601
\(695\) 40.9792 1.55443
\(696\) −37.3176 −1.41452
\(697\) 53.7750 2.03687
\(698\) −13.0167 −0.492688
\(699\) 29.9577 1.13311
\(700\) 0 0
\(701\) −10.4083 −0.393117 −0.196559 0.980492i \(-0.562977\pi\)
−0.196559 + 0.980492i \(0.562977\pi\)
\(702\) 0 0
\(703\) −19.2215 −0.724953
\(704\) 40.5708 1.52907
\(705\) 10.7362 0.404350
\(706\) 15.1232 0.569171
\(707\) 0 0
\(708\) −1.59166 −0.0598184
\(709\) −18.7958 −0.705892 −0.352946 0.935644i \(-0.614820\pi\)
−0.352946 + 0.935644i \(0.614820\pi\)
\(710\) 16.1043 0.604385
\(711\) 11.7958 0.442378
\(712\) 36.4513 1.36607
\(713\) 2.53969 0.0951121
\(714\) 0 0
\(715\) 0 0
\(716\) 0.408337 0.0152603
\(717\) 27.9955 1.04551
\(718\) −4.00000 −0.149279
\(719\) −29.1210 −1.08603 −0.543015 0.839723i \(-0.682717\pi\)
−0.543015 + 0.839723i \(0.682717\pi\)
\(720\) 2.68406 0.100029
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 6.20417 0.230736
\(724\) −16.5375 −0.614610
\(725\) −19.3875 −0.720033
\(726\) −31.9494 −1.18575
\(727\) 35.3259 1.31016 0.655082 0.755558i \(-0.272634\pi\)
0.655082 + 0.755558i \(0.272634\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 15.5708 0.576302
\(731\) 9.89949 0.366146
\(732\) −2.20417 −0.0814684
\(733\) −0.692369 −0.0255732 −0.0127866 0.999918i \(-0.504070\pi\)
−0.0127866 + 0.999918i \(0.504070\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) 8.97916 0.330976
\(737\) 33.5917 1.23736
\(738\) 9.75513 0.359091
\(739\) 15.1833 0.558528 0.279264 0.960214i \(-0.409910\pi\)
0.279264 + 0.960214i \(0.409910\pi\)
\(740\) 18.2404 0.670531
\(741\) 0 0
\(742\) 0 0
\(743\) 9.18333 0.336904 0.168452 0.985710i \(-0.446123\pi\)
0.168452 + 0.985710i \(0.446123\pi\)
\(744\) 6.00000 0.219971
\(745\) 12.3243 0.451527
\(746\) −12.5917 −0.461014
\(747\) −9.89949 −0.362204
\(748\) 31.9494 1.16819
\(749\) 0 0
\(750\) 10.6125 0.387514
\(751\) 1.79583 0.0655308 0.0327654 0.999463i \(-0.489569\pi\)
0.0327654 + 0.999463i \(0.489569\pi\)
\(752\) 2.82843 0.103142
\(753\) −4.40834 −0.160649
\(754\) 0 0
\(755\) −47.2170 −1.71840
\(756\) 0 0
\(757\) −25.1833 −0.915304 −0.457652 0.889132i \(-0.651309\pi\)
−0.457652 + 0.889132i \(0.651309\pi\)
\(758\) −3.38749 −0.123039
\(759\) −14.7196 −0.534288
\(760\) −22.7750 −0.826136
\(761\) −9.32202 −0.337923 −0.168961 0.985623i \(-0.554041\pi\)
−0.168961 + 0.985623i \(0.554041\pi\)
\(762\) −10.7362 −0.388933
\(763\) 0 0
\(764\) 25.7958 0.933260
\(765\) 14.7958 0.534944
\(766\) 14.9789 0.541209
\(767\) 0 0
\(768\) 24.0416 0.867528
\(769\) −4.24264 −0.152994 −0.0764968 0.997070i \(-0.524373\pi\)
−0.0764968 + 0.997070i \(0.524373\pi\)
\(770\) 0 0
\(771\) 21.7958 0.784958
\(772\) −3.40834 −0.122669
\(773\) −7.64854 −0.275099 −0.137549 0.990495i \(-0.543923\pi\)
−0.137549 + 0.990495i \(0.543923\pi\)
\(774\) 1.79583 0.0645498
\(775\) 3.11716 0.111972
\(776\) −12.7279 −0.456906
\(777\) 0 0
\(778\) 28.3875 1.01774
\(779\) −27.5917 −0.988574
\(780\) 0 0
\(781\) 34.7750 1.24435
\(782\) −9.89949 −0.354005
\(783\) −49.7567 −1.77816
\(784\) 0 0
\(785\) 18.5917 0.663565
\(786\) −11.5917 −0.413461
\(787\) −25.4264 −0.906352 −0.453176 0.891421i \(-0.649709\pi\)
−0.453176 + 0.891421i \(0.649709\pi\)
\(788\) 8.00000 0.284988
\(789\) 35.9033 1.27819
\(790\) −31.6607 −1.12644
\(791\) 0 0
\(792\) 17.3875 0.617838
\(793\) 0 0
\(794\) −7.07107 −0.250943
\(795\) −25.0208 −0.887398
\(796\) −22.0499 −0.781539
\(797\) 26.2926 0.931331 0.465666 0.884961i \(-0.345815\pi\)
0.465666 + 0.884961i \(0.345815\pi\)
\(798\) 0 0
\(799\) 15.5917 0.551593
\(800\) 11.0208 0.389646
\(801\) 12.1504 0.429315
\(802\) 24.3875 0.861152
\(803\) 33.6229 1.18653
\(804\) 8.19654 0.289070
\(805\) 0 0
\(806\) 0 0
\(807\) −1.59166 −0.0560292
\(808\) −8.91839 −0.313748
\(809\) 39.3667 1.38406 0.692029 0.721870i \(-0.256717\pi\)
0.692029 + 0.721870i \(0.256717\pi\)
\(810\) −13.4203 −0.471541
\(811\) 38.4725 1.35095 0.675476 0.737382i \(-0.263938\pi\)
0.675476 + 0.737382i \(0.263938\pi\)
\(812\) 0 0
\(813\) −32.7750 −1.14947
\(814\) −39.3875 −1.38053
\(815\) −35.9328 −1.25867
\(816\) −7.79583 −0.272909
\(817\) −5.07938 −0.177705
\(818\) 28.6879 1.00305
\(819\) 0 0
\(820\) 26.1833 0.914361
\(821\) −40.3667 −1.40881 −0.704403 0.709800i \(-0.748785\pi\)
−0.704403 + 0.709800i \(0.748785\pi\)
\(822\) 13.0167 0.454008
\(823\) 36.7750 1.28190 0.640948 0.767584i \(-0.278542\pi\)
0.640948 + 0.767584i \(0.278542\pi\)
\(824\) −24.5896 −0.856620
\(825\) −18.0666 −0.628997
\(826\) 0 0
\(827\) 26.9792 0.938157 0.469079 0.883156i \(-0.344586\pi\)
0.469079 + 0.883156i \(0.344586\pi\)
\(828\) 1.79583 0.0624095
\(829\) −20.2026 −0.701666 −0.350833 0.936438i \(-0.614102\pi\)
−0.350833 + 0.936438i \(0.614102\pi\)
\(830\) 26.5708 0.922287
\(831\) 8.74454 0.303345
\(832\) 0 0
\(833\) 0 0
\(834\) −21.5917 −0.747658
\(835\) −40.9792 −1.41814
\(836\) −16.3931 −0.566967
\(837\) 8.00000 0.276520
\(838\) −28.5435 −0.986020
\(839\) −5.65685 −0.195296 −0.0976481 0.995221i \(-0.531132\pi\)
−0.0976481 + 0.995221i \(0.531132\pi\)
\(840\) 0 0
\(841\) 48.3667 1.66782
\(842\) −6.59166 −0.227164
\(843\) −21.5019 −0.740566
\(844\) 1.79583 0.0618151
\(845\) 0 0
\(846\) 2.82843 0.0972433
\(847\) 0 0
\(848\) −6.59166 −0.226359
\(849\) −41.5917 −1.42742
\(850\) −12.1504 −0.416757
\(851\) −12.2042 −0.418354
\(852\) 8.48528 0.290701
\(853\) 13.1610 0.450625 0.225313 0.974287i \(-0.427660\pi\)
0.225313 + 0.974287i \(0.427660\pi\)
\(854\) 0 0
\(855\) −7.59166 −0.259629
\(856\) 18.0000 0.615227
\(857\) 37.7507 1.28954 0.644769 0.764377i \(-0.276954\pi\)
0.644769 + 0.764377i \(0.276954\pi\)
\(858\) 0 0
\(859\) −23.7529 −0.810438 −0.405219 0.914220i \(-0.632805\pi\)
−0.405219 + 0.914220i \(0.632805\pi\)
\(860\) 4.82012 0.164365
\(861\) 0 0
\(862\) −5.18333 −0.176545
\(863\) 17.7958 0.605777 0.302889 0.953026i \(-0.402049\pi\)
0.302889 + 0.953026i \(0.402049\pi\)
\(864\) 28.2843 0.962250
\(865\) −25.0208 −0.850734
\(866\) −13.4203 −0.456040
\(867\) −18.9328 −0.642991
\(868\) 0 0
\(869\) −68.3667 −2.31918
\(870\) 33.3875 1.13194
\(871\) 0 0
\(872\) 52.7750 1.78719
\(873\) −4.24264 −0.143592
\(874\) 5.07938 0.171813
\(875\) 0 0
\(876\) 8.20417 0.277193
\(877\) −4.79583 −0.161944 −0.0809719 0.996716i \(-0.525802\pi\)
−0.0809719 + 0.996716i \(0.525802\pi\)
\(878\) −34.2004 −1.15421
\(879\) −13.7958 −0.465322
\(880\) −15.5563 −0.524404
\(881\) 23.3198 0.785664 0.392832 0.919610i \(-0.371495\pi\)
0.392832 + 0.919610i \(0.371495\pi\)
\(882\) 0 0
\(883\) −16.6125 −0.559055 −0.279528 0.960138i \(-0.590178\pi\)
−0.279528 + 0.960138i \(0.590178\pi\)
\(884\) 0 0
\(885\) 4.27212 0.143606
\(886\) 10.0000 0.335957
\(887\) 28.5730 0.959388 0.479694 0.877436i \(-0.340748\pi\)
0.479694 + 0.877436i \(0.340748\pi\)
\(888\) −28.8323 −0.967548
\(889\) 0 0
\(890\) −32.6125 −1.09317
\(891\) −28.9792 −0.970838
\(892\) −5.65685 −0.189405
\(893\) −8.00000 −0.267710
\(894\) −6.49359 −0.217178
\(895\) −1.09600 −0.0366352
\(896\) 0 0
\(897\) 0 0
\(898\) 4.40834 0.147108
\(899\) −12.4392 −0.414870
\(900\) 2.20417 0.0734723
\(901\) −36.3364 −1.21054
\(902\) −56.5391 −1.88255
\(903\) 0 0
\(904\) 49.7750 1.65549
\(905\) 44.3875 1.47549
\(906\) 24.8784 0.826528
\(907\) 49.3875 1.63988 0.819942 0.572446i \(-0.194005\pi\)
0.819942 + 0.572446i \(0.194005\pi\)
\(908\) 20.9245 0.694403
\(909\) −2.97280 −0.0986014
\(910\) 0 0
\(911\) −43.1833 −1.43073 −0.715364 0.698752i \(-0.753739\pi\)
−0.715364 + 0.698752i \(0.753739\pi\)
\(912\) 4.00000 0.132453
\(913\) 57.3758 1.89886
\(914\) −14.5917 −0.482649
\(915\) 5.91612 0.195581
\(916\) 12.7279 0.420542
\(917\) 0 0
\(918\) −31.1833 −1.02920
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) −14.4603 −0.476744
\(921\) 48.3667 1.59374
\(922\) −30.9683 −1.01989
\(923\) 0 0
\(924\) 0 0
\(925\) −14.9792 −0.492512
\(926\) 11.3875 0.374216
\(927\) −8.19654 −0.269210
\(928\) −43.9792 −1.44369
\(929\) 28.1399 0.923240 0.461620 0.887078i \(-0.347268\pi\)
0.461620 + 0.887078i \(0.347268\pi\)
\(930\) −5.36812 −0.176027
\(931\) 0 0
\(932\) 21.1833 0.693883
\(933\) 16.7750 0.549188
\(934\) 4.82012 0.157719
\(935\) −85.7541 −2.80446
\(936\) 0 0
\(937\) −6.63796 −0.216853 −0.108426 0.994104i \(-0.534581\pi\)
−0.108426 + 0.994104i \(0.534581\pi\)
\(938\) 0 0
\(939\) 2.40834 0.0785931
\(940\) 7.59166 0.247613
\(941\) 23.4642 0.764910 0.382455 0.923974i \(-0.375079\pi\)
0.382455 + 0.923974i \(0.375079\pi\)
\(942\) −9.79583 −0.319165
\(943\) −17.5186 −0.570483
\(944\) 1.12548 0.0366311
\(945\) 0 0
\(946\) −10.4083 −0.338404
\(947\) 28.2042 0.916512 0.458256 0.888820i \(-0.348474\pi\)
0.458256 + 0.888820i \(0.348474\pi\)
\(948\) −16.6818 −0.541800
\(949\) 0 0
\(950\) 6.23433 0.202268
\(951\) 17.8073 0.577441
\(952\) 0 0
\(953\) −4.81667 −0.156027 −0.0780137 0.996952i \(-0.524858\pi\)
−0.0780137 + 0.996952i \(0.524858\pi\)
\(954\) −6.59166 −0.213413
\(955\) −69.2375 −2.24047
\(956\) 19.7958 0.640243
\(957\) 72.0954 2.33051
\(958\) 7.35981 0.237785
\(959\) 0 0
\(960\) −26.5708 −0.857570
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 4.38701 0.141296
\(965\) 9.14817 0.294490
\(966\) 0 0
\(967\) 11.3875 0.366197 0.183099 0.983095i \(-0.441387\pi\)
0.183099 + 0.983095i \(0.441387\pi\)
\(968\) −67.7750 −2.17837
\(969\) 22.0499 0.708346
\(970\) 11.3875 0.365630
\(971\) 22.6274 0.726148 0.363074 0.931760i \(-0.381727\pi\)
0.363074 + 0.931760i \(0.381727\pi\)
\(972\) 9.89949 0.317526
\(973\) 0 0
\(974\) 19.5917 0.627757
\(975\) 0 0
\(976\) 1.55858 0.0498890
\(977\) 0.387495 0.0123970 0.00619852 0.999981i \(-0.498027\pi\)
0.00619852 + 0.999981i \(0.498027\pi\)
\(978\) 18.9328 0.605403
\(979\) −70.4219 −2.25069
\(980\) 0 0
\(981\) 17.5917 0.561659
\(982\) −9.59166 −0.306082
\(983\) −41.5602 −1.32556 −0.662782 0.748812i \(-0.730624\pi\)
−0.662782 + 0.748812i \(0.730624\pi\)
\(984\) −41.3875 −1.31939
\(985\) −21.4725 −0.684170
\(986\) 48.4869 1.54414
\(987\) 0 0
\(988\) 0 0
\(989\) −3.22501 −0.102549
\(990\) −15.5563 −0.494413
\(991\) 18.2042 0.578274 0.289137 0.957288i \(-0.406632\pi\)
0.289137 + 0.957288i \(0.406632\pi\)
\(992\) 7.07107 0.224507
\(993\) −0.866213 −0.0274885
\(994\) 0 0
\(995\) 59.1833 1.87624
\(996\) 14.0000 0.443607
\(997\) 5.51249 0.174582 0.0872911 0.996183i \(-0.472179\pi\)
0.0872911 + 0.996183i \(0.472179\pi\)
\(998\) 16.2042 0.512934
\(999\) −38.4430 −1.21628
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 8281.2.a.bv.1.2 4
7.6 odd 2 inner 8281.2.a.bv.1.3 4
13.3 even 3 637.2.f.g.295.4 yes 8
13.9 even 3 637.2.f.g.393.4 yes 8
13.12 even 2 8281.2.a.bn.1.1 4
91.3 odd 6 637.2.g.h.373.4 8
91.9 even 3 637.2.g.h.263.1 8
91.16 even 3 637.2.h.k.165.3 8
91.48 odd 6 637.2.f.g.393.1 yes 8
91.55 odd 6 637.2.f.g.295.1 8
91.61 odd 6 637.2.g.h.263.4 8
91.68 odd 6 637.2.h.k.165.2 8
91.74 even 3 637.2.h.k.471.3 8
91.81 even 3 637.2.g.h.373.1 8
91.87 odd 6 637.2.h.k.471.2 8
91.90 odd 2 8281.2.a.bn.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.f.g.295.1 8 91.55 odd 6
637.2.f.g.295.4 yes 8 13.3 even 3
637.2.f.g.393.1 yes 8 91.48 odd 6
637.2.f.g.393.4 yes 8 13.9 even 3
637.2.g.h.263.1 8 91.9 even 3
637.2.g.h.263.4 8 91.61 odd 6
637.2.g.h.373.1 8 91.81 even 3
637.2.g.h.373.4 8 91.3 odd 6
637.2.h.k.165.2 8 91.68 odd 6
637.2.h.k.165.3 8 91.16 even 3
637.2.h.k.471.2 8 91.87 odd 6
637.2.h.k.471.3 8 91.74 even 3
8281.2.a.bn.1.1 4 13.12 even 2
8281.2.a.bn.1.4 4 91.90 odd 2
8281.2.a.bv.1.2 4 1.1 even 1 trivial
8281.2.a.bv.1.3 4 7.6 odd 2 inner