Properties

Label 8281.2.a.co.1.12
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.58860\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} -1.61205 q^{5} -1.34210 q^{6} +6.99143 q^{8} -2.73119 q^{9} +O(q^{10})\) \(q+2.58860 q^{2} -0.518466 q^{3} +4.70085 q^{4} -1.61205 q^{5} -1.34210 q^{6} +6.99143 q^{8} -2.73119 q^{9} -4.17296 q^{10} +2.70496 q^{11} -2.43723 q^{12} +0.835795 q^{15} +8.69632 q^{16} -3.12661 q^{17} -7.06997 q^{18} -3.68150 q^{19} -7.57803 q^{20} +7.00205 q^{22} +1.98604 q^{23} -3.62482 q^{24} -2.40128 q^{25} +2.97143 q^{27} -5.37271 q^{29} +2.16354 q^{30} -10.4780 q^{31} +8.52843 q^{32} -1.40243 q^{33} -8.09354 q^{34} -12.8389 q^{36} +5.95346 q^{37} -9.52994 q^{38} -11.2706 q^{40} +7.70150 q^{41} -3.35600 q^{43} +12.7156 q^{44} +4.40283 q^{45} +5.14106 q^{46} -1.05508 q^{47} -4.50874 q^{48} -6.21596 q^{50} +1.62104 q^{51} +7.26568 q^{53} +7.69184 q^{54} -4.36054 q^{55} +1.90873 q^{57} -13.9078 q^{58} -11.4241 q^{59} +3.92895 q^{60} +2.92507 q^{61} -27.1235 q^{62} +4.68406 q^{64} -3.63033 q^{66} -13.5818 q^{67} -14.6977 q^{68} -1.02969 q^{69} -1.35111 q^{71} -19.0949 q^{72} -9.10335 q^{73} +15.4111 q^{74} +1.24498 q^{75} -17.3062 q^{76} -6.20578 q^{79} -14.0189 q^{80} +6.65300 q^{81} +19.9361 q^{82} +2.69672 q^{83} +5.04026 q^{85} -8.68734 q^{86} +2.78557 q^{87} +18.9115 q^{88} -1.75988 q^{89} +11.3972 q^{90} +9.33607 q^{92} +5.43251 q^{93} -2.73119 q^{94} +5.93478 q^{95} -4.42170 q^{96} -15.4820 q^{97} -7.38776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} + O(q^{10}) \) \( 12 q - 6 q^{3} + 8 q^{4} + 2 q^{9} - 24 q^{10} + 2 q^{12} + 16 q^{16} - 34 q^{17} + 30 q^{22} + 6 q^{23} - 10 q^{25} - 12 q^{27} + 2 q^{29} + 22 q^{30} - 26 q^{36} - 38 q^{38} - 2 q^{40} + 22 q^{43} + 38 q^{48} + 8 q^{51} + 16 q^{53} - 30 q^{55} + 10 q^{61} - 82 q^{62} - 2 q^{64} - 68 q^{66} - 22 q^{68} - 14 q^{69} + 66 q^{74} - 2 q^{75} + 70 q^{79} - 28 q^{81} - 10 q^{82} + 20 q^{87} - 28 q^{88} - 66 q^{92} + 2 q^{94} + 4 q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58860 1.83042 0.915209 0.402981i \(-0.132026\pi\)
0.915209 + 0.402981i \(0.132026\pi\)
\(3\) −0.518466 −0.299336 −0.149668 0.988736i \(-0.547821\pi\)
−0.149668 + 0.988736i \(0.547821\pi\)
\(4\) 4.70085 2.35043
\(5\) −1.61205 −0.720932 −0.360466 0.932772i \(-0.617382\pi\)
−0.360466 + 0.932772i \(0.617382\pi\)
\(6\) −1.34210 −0.547910
\(7\) 0 0
\(8\) 6.99143 2.47184
\(9\) −2.73119 −0.910398
\(10\) −4.17296 −1.31961
\(11\) 2.70496 0.815575 0.407788 0.913077i \(-0.366300\pi\)
0.407788 + 0.913077i \(0.366300\pi\)
\(12\) −2.43723 −0.703568
\(13\) 0 0
\(14\) 0 0
\(15\) 0.835795 0.215801
\(16\) 8.69632 2.17408
\(17\) −3.12661 −0.758314 −0.379157 0.925332i \(-0.623786\pi\)
−0.379157 + 0.925332i \(0.623786\pi\)
\(18\) −7.06997 −1.66641
\(19\) −3.68150 −0.844595 −0.422297 0.906457i \(-0.638776\pi\)
−0.422297 + 0.906457i \(0.638776\pi\)
\(20\) −7.57803 −1.69450
\(21\) 0 0
\(22\) 7.00205 1.49284
\(23\) 1.98604 0.414117 0.207059 0.978329i \(-0.433611\pi\)
0.207059 + 0.978329i \(0.433611\pi\)
\(24\) −3.62482 −0.739913
\(25\) −2.40128 −0.480257
\(26\) 0 0
\(27\) 2.97143 0.571852
\(28\) 0 0
\(29\) −5.37271 −0.997687 −0.498844 0.866692i \(-0.666242\pi\)
−0.498844 + 0.866692i \(0.666242\pi\)
\(30\) 2.16354 0.395006
\(31\) −10.4780 −1.88191 −0.940956 0.338529i \(-0.890071\pi\)
−0.940956 + 0.338529i \(0.890071\pi\)
\(32\) 8.52843 1.50763
\(33\) −1.40243 −0.244131
\(34\) −8.09354 −1.38803
\(35\) 0 0
\(36\) −12.8389 −2.13982
\(37\) 5.95346 0.978743 0.489371 0.872075i \(-0.337226\pi\)
0.489371 + 0.872075i \(0.337226\pi\)
\(38\) −9.52994 −1.54596
\(39\) 0 0
\(40\) −11.2706 −1.78203
\(41\) 7.70150 1.20277 0.601386 0.798958i \(-0.294615\pi\)
0.601386 + 0.798958i \(0.294615\pi\)
\(42\) 0 0
\(43\) −3.35600 −0.511785 −0.255892 0.966705i \(-0.582369\pi\)
−0.255892 + 0.966705i \(0.582369\pi\)
\(44\) 12.7156 1.91695
\(45\) 4.40283 0.656335
\(46\) 5.14106 0.758008
\(47\) −1.05508 −0.153900 −0.0769500 0.997035i \(-0.524518\pi\)
−0.0769500 + 0.997035i \(0.524518\pi\)
\(48\) −4.50874 −0.650781
\(49\) 0 0
\(50\) −6.21596 −0.879070
\(51\) 1.62104 0.226991
\(52\) 0 0
\(53\) 7.26568 0.998017 0.499009 0.866597i \(-0.333698\pi\)
0.499009 + 0.866597i \(0.333698\pi\)
\(54\) 7.69184 1.04673
\(55\) −4.36054 −0.587975
\(56\) 0 0
\(57\) 1.90873 0.252818
\(58\) −13.9078 −1.82618
\(59\) −11.4241 −1.48729 −0.743643 0.668577i \(-0.766904\pi\)
−0.743643 + 0.668577i \(0.766904\pi\)
\(60\) 3.92895 0.507225
\(61\) 2.92507 0.374517 0.187259 0.982311i \(-0.440040\pi\)
0.187259 + 0.982311i \(0.440040\pi\)
\(62\) −27.1235 −3.44468
\(63\) 0 0
\(64\) 4.68406 0.585507
\(65\) 0 0
\(66\) −3.63033 −0.446862
\(67\) −13.5818 −1.65928 −0.829642 0.558296i \(-0.811455\pi\)
−0.829642 + 0.558296i \(0.811455\pi\)
\(68\) −14.6977 −1.78236
\(69\) −1.02969 −0.123960
\(70\) 0 0
\(71\) −1.35111 −0.160347 −0.0801736 0.996781i \(-0.525547\pi\)
−0.0801736 + 0.996781i \(0.525547\pi\)
\(72\) −19.0949 −2.25036
\(73\) −9.10335 −1.06547 −0.532733 0.846283i \(-0.678835\pi\)
−0.532733 + 0.846283i \(0.678835\pi\)
\(74\) 15.4111 1.79151
\(75\) 1.24498 0.143758
\(76\) −17.3062 −1.98516
\(77\) 0 0
\(78\) 0 0
\(79\) −6.20578 −0.698205 −0.349102 0.937085i \(-0.613513\pi\)
−0.349102 + 0.937085i \(0.613513\pi\)
\(80\) −14.0189 −1.56736
\(81\) 6.65300 0.739222
\(82\) 19.9361 2.20158
\(83\) 2.69672 0.296003 0.148002 0.988987i \(-0.452716\pi\)
0.148002 + 0.988987i \(0.452716\pi\)
\(84\) 0 0
\(85\) 5.04026 0.546693
\(86\) −8.68734 −0.936780
\(87\) 2.78557 0.298644
\(88\) 18.9115 2.01597
\(89\) −1.75988 −0.186546 −0.0932732 0.995641i \(-0.529733\pi\)
−0.0932732 + 0.995641i \(0.529733\pi\)
\(90\) 11.3972 1.20137
\(91\) 0 0
\(92\) 9.33607 0.973353
\(93\) 5.43251 0.563325
\(94\) −2.73119 −0.281701
\(95\) 5.93478 0.608896
\(96\) −4.42170 −0.451288
\(97\) −15.4820 −1.57196 −0.785981 0.618250i \(-0.787842\pi\)
−0.785981 + 0.618250i \(0.787842\pi\)
\(98\) 0 0
\(99\) −7.38776 −0.742498
\(100\) −11.2881 −1.12881
\(101\) 1.27930 0.127295 0.0636477 0.997972i \(-0.479727\pi\)
0.0636477 + 0.997972i \(0.479727\pi\)
\(102\) 4.19622 0.415488
\(103\) 11.4673 1.12991 0.564956 0.825121i \(-0.308893\pi\)
0.564956 + 0.825121i \(0.308893\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 18.8079 1.82679
\(107\) −5.13525 −0.496444 −0.248222 0.968703i \(-0.579846\pi\)
−0.248222 + 0.968703i \(0.579846\pi\)
\(108\) 13.9682 1.34410
\(109\) 1.72783 0.165496 0.0827481 0.996570i \(-0.473630\pi\)
0.0827481 + 0.996570i \(0.473630\pi\)
\(110\) −11.2877 −1.07624
\(111\) −3.08667 −0.292973
\(112\) 0 0
\(113\) −8.59113 −0.808185 −0.404093 0.914718i \(-0.632413\pi\)
−0.404093 + 0.914718i \(0.632413\pi\)
\(114\) 4.94095 0.462762
\(115\) −3.20160 −0.298551
\(116\) −25.2563 −2.34499
\(117\) 0 0
\(118\) −29.5723 −2.72235
\(119\) 0 0
\(120\) 5.84340 0.533427
\(121\) −3.68321 −0.334837
\(122\) 7.57184 0.685522
\(123\) −3.99297 −0.360034
\(124\) −49.2557 −4.42330
\(125\) 11.9313 1.06716
\(126\) 0 0
\(127\) −3.12412 −0.277221 −0.138610 0.990347i \(-0.544264\pi\)
−0.138610 + 0.990347i \(0.544264\pi\)
\(128\) −4.93170 −0.435904
\(129\) 1.73997 0.153196
\(130\) 0 0
\(131\) −10.2092 −0.891982 −0.445991 0.895038i \(-0.647149\pi\)
−0.445991 + 0.895038i \(0.647149\pi\)
\(132\) −6.59261 −0.573813
\(133\) 0 0
\(134\) −35.1579 −3.03718
\(135\) −4.79010 −0.412266
\(136\) −21.8595 −1.87443
\(137\) −9.99261 −0.853726 −0.426863 0.904316i \(-0.640381\pi\)
−0.426863 + 0.904316i \(0.640381\pi\)
\(138\) −2.66546 −0.226899
\(139\) 1.66420 0.141156 0.0705778 0.997506i \(-0.477516\pi\)
0.0705778 + 0.997506i \(0.477516\pi\)
\(140\) 0 0
\(141\) 0.547025 0.0460679
\(142\) −3.49748 −0.293502
\(143\) 0 0
\(144\) −23.7513 −1.97928
\(145\) 8.66110 0.719265
\(146\) −23.5649 −1.95025
\(147\) 0 0
\(148\) 27.9863 2.30046
\(149\) 19.7980 1.62192 0.810959 0.585103i \(-0.198946\pi\)
0.810959 + 0.585103i \(0.198946\pi\)
\(150\) 3.22276 0.263138
\(151\) 7.53493 0.613184 0.306592 0.951841i \(-0.400811\pi\)
0.306592 + 0.951841i \(0.400811\pi\)
\(152\) −25.7390 −2.08771
\(153\) 8.53937 0.690367
\(154\) 0 0
\(155\) 16.8912 1.35673
\(156\) 0 0
\(157\) −14.0045 −1.11768 −0.558839 0.829276i \(-0.688753\pi\)
−0.558839 + 0.829276i \(0.688753\pi\)
\(158\) −16.0643 −1.27801
\(159\) −3.76700 −0.298743
\(160\) −13.7483 −1.08690
\(161\) 0 0
\(162\) 17.2219 1.35308
\(163\) −7.16995 −0.561594 −0.280797 0.959767i \(-0.590599\pi\)
−0.280797 + 0.959767i \(0.590599\pi\)
\(164\) 36.2036 2.82703
\(165\) 2.26079 0.176002
\(166\) 6.98072 0.541809
\(167\) 17.9805 1.39138 0.695688 0.718344i \(-0.255099\pi\)
0.695688 + 0.718344i \(0.255099\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 13.0472 1.00068
\(171\) 10.0549 0.768917
\(172\) −15.7761 −1.20291
\(173\) 12.8116 0.974047 0.487023 0.873389i \(-0.338083\pi\)
0.487023 + 0.873389i \(0.338083\pi\)
\(174\) 7.21072 0.546643
\(175\) 0 0
\(176\) 23.5232 1.77312
\(177\) 5.92298 0.445199
\(178\) −4.55561 −0.341458
\(179\) 1.84022 0.137545 0.0687723 0.997632i \(-0.478092\pi\)
0.0687723 + 0.997632i \(0.478092\pi\)
\(180\) 20.6971 1.54267
\(181\) −3.29928 −0.245234 −0.122617 0.992454i \(-0.539129\pi\)
−0.122617 + 0.992454i \(0.539129\pi\)
\(182\) 0 0
\(183\) −1.51655 −0.112107
\(184\) 13.8852 1.02363
\(185\) −9.59730 −0.705607
\(186\) 14.0626 1.03112
\(187\) −8.45734 −0.618462
\(188\) −4.95980 −0.361731
\(189\) 0 0
\(190\) 15.3628 1.11453
\(191\) 4.89614 0.354272 0.177136 0.984186i \(-0.443317\pi\)
0.177136 + 0.984186i \(0.443317\pi\)
\(192\) −2.42852 −0.175264
\(193\) 3.01910 0.217320 0.108660 0.994079i \(-0.465344\pi\)
0.108660 + 0.994079i \(0.465344\pi\)
\(194\) −40.0768 −2.87735
\(195\) 0 0
\(196\) 0 0
\(197\) 4.64991 0.331292 0.165646 0.986185i \(-0.447029\pi\)
0.165646 + 0.986185i \(0.447029\pi\)
\(198\) −19.1240 −1.35908
\(199\) −0.410721 −0.0291152 −0.0145576 0.999894i \(-0.504634\pi\)
−0.0145576 + 0.999894i \(0.504634\pi\)
\(200\) −16.7884 −1.18712
\(201\) 7.04171 0.496684
\(202\) 3.31160 0.233004
\(203\) 0 0
\(204\) 7.62027 0.533525
\(205\) −12.4152 −0.867118
\(206\) 29.6844 2.06821
\(207\) −5.42425 −0.377012
\(208\) 0 0
\(209\) −9.95831 −0.688831
\(210\) 0 0
\(211\) −7.51600 −0.517423 −0.258711 0.965955i \(-0.583298\pi\)
−0.258711 + 0.965955i \(0.583298\pi\)
\(212\) 34.1549 2.34577
\(213\) 0.700504 0.0479977
\(214\) −13.2931 −0.908699
\(215\) 5.41005 0.368962
\(216\) 20.7745 1.41353
\(217\) 0 0
\(218\) 4.47267 0.302927
\(219\) 4.71978 0.318933
\(220\) −20.4982 −1.38199
\(221\) 0 0
\(222\) −7.99014 −0.536263
\(223\) 22.5794 1.51203 0.756016 0.654553i \(-0.227143\pi\)
0.756016 + 0.654553i \(0.227143\pi\)
\(224\) 0 0
\(225\) 6.55837 0.437225
\(226\) −22.2390 −1.47932
\(227\) −13.6717 −0.907424 −0.453712 0.891148i \(-0.649901\pi\)
−0.453712 + 0.891148i \(0.649901\pi\)
\(228\) 8.97268 0.594230
\(229\) −7.93086 −0.524086 −0.262043 0.965056i \(-0.584396\pi\)
−0.262043 + 0.965056i \(0.584396\pi\)
\(230\) −8.28766 −0.546472
\(231\) 0 0
\(232\) −37.5629 −2.46613
\(233\) −6.57171 −0.430527 −0.215263 0.976556i \(-0.569061\pi\)
−0.215263 + 0.976556i \(0.569061\pi\)
\(234\) 0 0
\(235\) 1.70085 0.110951
\(236\) −53.7028 −3.49576
\(237\) 3.21749 0.208998
\(238\) 0 0
\(239\) −9.39284 −0.607572 −0.303786 0.952740i \(-0.598251\pi\)
−0.303786 + 0.952740i \(0.598251\pi\)
\(240\) 7.26833 0.469169
\(241\) 10.0858 0.649686 0.324843 0.945768i \(-0.394689\pi\)
0.324843 + 0.945768i \(0.394689\pi\)
\(242\) −9.53435 −0.612891
\(243\) −12.3636 −0.793128
\(244\) 13.7503 0.880275
\(245\) 0 0
\(246\) −10.3362 −0.659012
\(247\) 0 0
\(248\) −73.2565 −4.65179
\(249\) −1.39816 −0.0886045
\(250\) 30.8853 1.95336
\(251\) 10.3485 0.653194 0.326597 0.945164i \(-0.394098\pi\)
0.326597 + 0.945164i \(0.394098\pi\)
\(252\) 0 0
\(253\) 5.37215 0.337744
\(254\) −8.08709 −0.507429
\(255\) −2.61320 −0.163645
\(256\) −22.1343 −1.38339
\(257\) −7.98658 −0.498189 −0.249095 0.968479i \(-0.580133\pi\)
−0.249095 + 0.968479i \(0.580133\pi\)
\(258\) 4.50409 0.280412
\(259\) 0 0
\(260\) 0 0
\(261\) 14.6739 0.908292
\(262\) −26.4275 −1.63270
\(263\) 5.05934 0.311972 0.155986 0.987759i \(-0.450144\pi\)
0.155986 + 0.987759i \(0.450144\pi\)
\(264\) −9.80498 −0.603455
\(265\) −11.7127 −0.719503
\(266\) 0 0
\(267\) 0.912435 0.0558401
\(268\) −63.8461 −3.90002
\(269\) −13.8902 −0.846902 −0.423451 0.905919i \(-0.639181\pi\)
−0.423451 + 0.905919i \(0.639181\pi\)
\(270\) −12.3997 −0.754619
\(271\) 8.32721 0.505842 0.252921 0.967487i \(-0.418609\pi\)
0.252921 + 0.967487i \(0.418609\pi\)
\(272\) −27.1900 −1.64863
\(273\) 0 0
\(274\) −25.8669 −1.56267
\(275\) −6.49537 −0.391685
\(276\) −4.84043 −0.291360
\(277\) −23.2116 −1.39465 −0.697325 0.716755i \(-0.745626\pi\)
−0.697325 + 0.716755i \(0.745626\pi\)
\(278\) 4.30795 0.258374
\(279\) 28.6176 1.71329
\(280\) 0 0
\(281\) 27.1595 1.62020 0.810100 0.586292i \(-0.199413\pi\)
0.810100 + 0.586292i \(0.199413\pi\)
\(282\) 1.41603 0.0843234
\(283\) 16.1513 0.960092 0.480046 0.877243i \(-0.340620\pi\)
0.480046 + 0.877243i \(0.340620\pi\)
\(284\) −6.35136 −0.376884
\(285\) −3.07698 −0.182265
\(286\) 0 0
\(287\) 0 0
\(288\) −23.2928 −1.37254
\(289\) −7.22433 −0.424960
\(290\) 22.4201 1.31656
\(291\) 8.02691 0.470546
\(292\) −42.7935 −2.50430
\(293\) 14.6452 0.855582 0.427791 0.903878i \(-0.359292\pi\)
0.427791 + 0.903878i \(0.359292\pi\)
\(294\) 0 0
\(295\) 18.4162 1.07223
\(296\) 41.6232 2.41930
\(297\) 8.03758 0.466388
\(298\) 51.2492 2.96879
\(299\) 0 0
\(300\) 5.85248 0.337893
\(301\) 0 0
\(302\) 19.5049 1.12238
\(303\) −0.663274 −0.0381041
\(304\) −32.0155 −1.83622
\(305\) −4.71537 −0.270001
\(306\) 22.1050 1.26366
\(307\) 8.97844 0.512427 0.256213 0.966620i \(-0.417525\pi\)
0.256213 + 0.966620i \(0.417525\pi\)
\(308\) 0 0
\(309\) −5.94543 −0.338224
\(310\) 43.7245 2.48338
\(311\) −12.1816 −0.690755 −0.345378 0.938464i \(-0.612249\pi\)
−0.345378 + 0.938464i \(0.612249\pi\)
\(312\) 0 0
\(313\) −13.1240 −0.741810 −0.370905 0.928671i \(-0.620953\pi\)
−0.370905 + 0.928671i \(0.620953\pi\)
\(314\) −36.2520 −2.04582
\(315\) 0 0
\(316\) −29.1725 −1.64108
\(317\) −16.7155 −0.938836 −0.469418 0.882976i \(-0.655536\pi\)
−0.469418 + 0.882976i \(0.655536\pi\)
\(318\) −9.75127 −0.546824
\(319\) −14.5330 −0.813689
\(320\) −7.55095 −0.422111
\(321\) 2.66245 0.148604
\(322\) 0 0
\(323\) 11.5106 0.640468
\(324\) 31.2748 1.73749
\(325\) 0 0
\(326\) −18.5601 −1.02795
\(327\) −0.895821 −0.0495390
\(328\) 53.8445 2.97307
\(329\) 0 0
\(330\) 5.85228 0.322157
\(331\) −3.96665 −0.218027 −0.109013 0.994040i \(-0.534769\pi\)
−0.109013 + 0.994040i \(0.534769\pi\)
\(332\) 12.6769 0.695733
\(333\) −16.2600 −0.891045
\(334\) 46.5445 2.54680
\(335\) 21.8946 1.19623
\(336\) 0 0
\(337\) 13.7032 0.746461 0.373230 0.927739i \(-0.378250\pi\)
0.373230 + 0.927739i \(0.378250\pi\)
\(338\) 0 0
\(339\) 4.45421 0.241919
\(340\) 23.6935 1.28496
\(341\) −28.3426 −1.53484
\(342\) 26.0281 1.40744
\(343\) 0 0
\(344\) −23.4632 −1.26505
\(345\) 1.65992 0.0893671
\(346\) 33.1641 1.78291
\(347\) −26.3979 −1.41711 −0.708556 0.705655i \(-0.750653\pi\)
−0.708556 + 0.705655i \(0.750653\pi\)
\(348\) 13.0945 0.701941
\(349\) −4.89024 −0.261769 −0.130884 0.991398i \(-0.541782\pi\)
−0.130884 + 0.991398i \(0.541782\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 23.0690 1.22958
\(353\) −13.5577 −0.721605 −0.360802 0.932642i \(-0.617497\pi\)
−0.360802 + 0.932642i \(0.617497\pi\)
\(354\) 15.3322 0.814899
\(355\) 2.17806 0.115599
\(356\) −8.27291 −0.438464
\(357\) 0 0
\(358\) 4.76360 0.251764
\(359\) 8.58568 0.453135 0.226567 0.973996i \(-0.427250\pi\)
0.226567 + 0.973996i \(0.427250\pi\)
\(360\) 30.7821 1.62236
\(361\) −5.44653 −0.286659
\(362\) −8.54053 −0.448880
\(363\) 1.90962 0.100229
\(364\) 0 0
\(365\) 14.6751 0.768130
\(366\) −3.92574 −0.205202
\(367\) 1.66322 0.0868196 0.0434098 0.999057i \(-0.486178\pi\)
0.0434098 + 0.999057i \(0.486178\pi\)
\(368\) 17.2712 0.900324
\(369\) −21.0343 −1.09500
\(370\) −24.8436 −1.29156
\(371\) 0 0
\(372\) 25.5374 1.32405
\(373\) 13.9635 0.723002 0.361501 0.932372i \(-0.382264\pi\)
0.361501 + 0.932372i \(0.382264\pi\)
\(374\) −21.8927 −1.13204
\(375\) −6.18595 −0.319441
\(376\) −7.37655 −0.380417
\(377\) 0 0
\(378\) 0 0
\(379\) −31.5758 −1.62194 −0.810969 0.585089i \(-0.801059\pi\)
−0.810969 + 0.585089i \(0.801059\pi\)
\(380\) 27.8985 1.43116
\(381\) 1.61975 0.0829822
\(382\) 12.6741 0.648466
\(383\) 31.9082 1.63043 0.815217 0.579156i \(-0.196618\pi\)
0.815217 + 0.579156i \(0.196618\pi\)
\(384\) 2.55692 0.130482
\(385\) 0 0
\(386\) 7.81525 0.397786
\(387\) 9.16588 0.465928
\(388\) −72.7788 −3.69478
\(389\) −25.4150 −1.28859 −0.644296 0.764776i \(-0.722850\pi\)
−0.644296 + 0.764776i \(0.722850\pi\)
\(390\) 0 0
\(391\) −6.20956 −0.314031
\(392\) 0 0
\(393\) 5.29312 0.267003
\(394\) 12.0368 0.606403
\(395\) 10.0041 0.503358
\(396\) −34.7288 −1.74519
\(397\) −4.15897 −0.208733 −0.104366 0.994539i \(-0.533281\pi\)
−0.104366 + 0.994539i \(0.533281\pi\)
\(398\) −1.06319 −0.0532930
\(399\) 0 0
\(400\) −20.8823 −1.04412
\(401\) −19.6013 −0.978844 −0.489422 0.872047i \(-0.662792\pi\)
−0.489422 + 0.872047i \(0.662792\pi\)
\(402\) 18.2282 0.909139
\(403\) 0 0
\(404\) 6.01381 0.299198
\(405\) −10.7250 −0.532929
\(406\) 0 0
\(407\) 16.1039 0.798238
\(408\) 11.3334 0.561086
\(409\) 17.6337 0.871930 0.435965 0.899964i \(-0.356407\pi\)
0.435965 + 0.899964i \(0.356407\pi\)
\(410\) −32.1381 −1.58719
\(411\) 5.18083 0.255551
\(412\) 53.9063 2.65577
\(413\) 0 0
\(414\) −14.0412 −0.690088
\(415\) −4.34725 −0.213398
\(416\) 0 0
\(417\) −0.862831 −0.0422530
\(418\) −25.7781 −1.26085
\(419\) −29.8911 −1.46027 −0.730137 0.683301i \(-0.760544\pi\)
−0.730137 + 0.683301i \(0.760544\pi\)
\(420\) 0 0
\(421\) −12.8528 −0.626407 −0.313203 0.949686i \(-0.601402\pi\)
−0.313203 + 0.949686i \(0.601402\pi\)
\(422\) −19.4559 −0.947100
\(423\) 2.88164 0.140110
\(424\) 50.7975 2.46694
\(425\) 7.50787 0.364185
\(426\) 1.81332 0.0878559
\(427\) 0 0
\(428\) −24.1401 −1.16685
\(429\) 0 0
\(430\) 14.0045 0.675355
\(431\) −8.97060 −0.432098 −0.216049 0.976382i \(-0.569317\pi\)
−0.216049 + 0.976382i \(0.569317\pi\)
\(432\) 25.8405 1.24325
\(433\) 3.45062 0.165826 0.0829132 0.996557i \(-0.473578\pi\)
0.0829132 + 0.996557i \(0.473578\pi\)
\(434\) 0 0
\(435\) −4.49048 −0.215302
\(436\) 8.12228 0.388987
\(437\) −7.31161 −0.349762
\(438\) 12.2176 0.583780
\(439\) 38.5144 1.83819 0.919096 0.394034i \(-0.128921\pi\)
0.919096 + 0.394034i \(0.128921\pi\)
\(440\) −30.4864 −1.45338
\(441\) 0 0
\(442\) 0 0
\(443\) −15.0399 −0.714569 −0.357284 0.933996i \(-0.616297\pi\)
−0.357284 + 0.933996i \(0.616297\pi\)
\(444\) −14.5100 −0.688612
\(445\) 2.83701 0.134487
\(446\) 58.4492 2.76765
\(447\) −10.2646 −0.485499
\(448\) 0 0
\(449\) 38.9235 1.83691 0.918456 0.395522i \(-0.129436\pi\)
0.918456 + 0.395522i \(0.129436\pi\)
\(450\) 16.9770 0.800303
\(451\) 20.8322 0.980952
\(452\) −40.3856 −1.89958
\(453\) −3.90660 −0.183548
\(454\) −35.3906 −1.66097
\(455\) 0 0
\(456\) 13.3448 0.624927
\(457\) 13.9396 0.652069 0.326034 0.945358i \(-0.394287\pi\)
0.326034 + 0.945358i \(0.394287\pi\)
\(458\) −20.5298 −0.959295
\(459\) −9.29049 −0.433643
\(460\) −15.0502 −0.701721
\(461\) 37.4635 1.74485 0.872424 0.488749i \(-0.162547\pi\)
0.872424 + 0.488749i \(0.162547\pi\)
\(462\) 0 0
\(463\) −6.75275 −0.313827 −0.156913 0.987612i \(-0.550154\pi\)
−0.156913 + 0.987612i \(0.550154\pi\)
\(464\) −46.7228 −2.16905
\(465\) −8.75749 −0.406119
\(466\) −17.0115 −0.788044
\(467\) −5.05032 −0.233701 −0.116851 0.993150i \(-0.537280\pi\)
−0.116851 + 0.993150i \(0.537280\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.40283 0.203087
\(471\) 7.26084 0.334562
\(472\) −79.8705 −3.67634
\(473\) −9.07783 −0.417399
\(474\) 8.32878 0.382554
\(475\) 8.84033 0.405622
\(476\) 0 0
\(477\) −19.8440 −0.908593
\(478\) −24.3143 −1.11211
\(479\) −9.45319 −0.431927 −0.215964 0.976401i \(-0.569289\pi\)
−0.215964 + 0.976401i \(0.569289\pi\)
\(480\) 7.12801 0.325348
\(481\) 0 0
\(482\) 26.1082 1.18920
\(483\) 0 0
\(484\) −17.3142 −0.787010
\(485\) 24.9579 1.13328
\(486\) −32.0045 −1.45175
\(487\) 39.9996 1.81255 0.906277 0.422684i \(-0.138912\pi\)
0.906277 + 0.422684i \(0.138912\pi\)
\(488\) 20.4504 0.925748
\(489\) 3.71737 0.168105
\(490\) 0 0
\(491\) −6.76097 −0.305118 −0.152559 0.988294i \(-0.548751\pi\)
−0.152559 + 0.988294i \(0.548751\pi\)
\(492\) −18.7704 −0.846233
\(493\) 16.7984 0.756560
\(494\) 0 0
\(495\) 11.9095 0.535291
\(496\) −91.1203 −4.09142
\(497\) 0 0
\(498\) −3.61927 −0.162183
\(499\) 11.3575 0.508433 0.254217 0.967147i \(-0.418182\pi\)
0.254217 + 0.967147i \(0.418182\pi\)
\(500\) 56.0871 2.50829
\(501\) −9.32230 −0.416490
\(502\) 26.7882 1.19562
\(503\) 13.9285 0.621040 0.310520 0.950567i \(-0.399497\pi\)
0.310520 + 0.950567i \(0.399497\pi\)
\(504\) 0 0
\(505\) −2.06230 −0.0917713
\(506\) 13.9063 0.618212
\(507\) 0 0
\(508\) −14.6860 −0.651587
\(509\) 19.8149 0.878281 0.439141 0.898418i \(-0.355283\pi\)
0.439141 + 0.898418i \(0.355283\pi\)
\(510\) −6.76454 −0.299539
\(511\) 0 0
\(512\) −47.4335 −2.09628
\(513\) −10.9393 −0.482983
\(514\) −20.6741 −0.911894
\(515\) −18.4860 −0.814590
\(516\) 8.17934 0.360076
\(517\) −2.85396 −0.125517
\(518\) 0 0
\(519\) −6.64237 −0.291568
\(520\) 0 0
\(521\) 31.0951 1.36230 0.681151 0.732143i \(-0.261480\pi\)
0.681151 + 0.732143i \(0.261480\pi\)
\(522\) 37.9849 1.66255
\(523\) −22.7202 −0.993485 −0.496742 0.867898i \(-0.665471\pi\)
−0.496742 + 0.867898i \(0.665471\pi\)
\(524\) −47.9919 −2.09654
\(525\) 0 0
\(526\) 13.0966 0.571040
\(527\) 32.7607 1.42708
\(528\) −12.1960 −0.530761
\(529\) −19.0557 −0.828507
\(530\) −30.3194 −1.31699
\(531\) 31.2013 1.35402
\(532\) 0 0
\(533\) 0 0
\(534\) 2.36193 0.102211
\(535\) 8.27830 0.357902
\(536\) −94.9564 −4.10149
\(537\) −0.954091 −0.0411721
\(538\) −35.9563 −1.55018
\(539\) 0 0
\(540\) −22.5176 −0.969002
\(541\) 2.09872 0.0902310 0.0451155 0.998982i \(-0.485634\pi\)
0.0451155 + 0.998982i \(0.485634\pi\)
\(542\) 21.5558 0.925902
\(543\) 1.71057 0.0734074
\(544\) −26.6650 −1.14325
\(545\) −2.78536 −0.119312
\(546\) 0 0
\(547\) 25.3770 1.08504 0.542521 0.840042i \(-0.317470\pi\)
0.542521 + 0.840042i \(0.317470\pi\)
\(548\) −46.9738 −2.00662
\(549\) −7.98894 −0.340959
\(550\) −16.8139 −0.716948
\(551\) 19.7797 0.842642
\(552\) −7.19902 −0.306411
\(553\) 0 0
\(554\) −60.0855 −2.55279
\(555\) 4.97587 0.211214
\(556\) 7.82316 0.331776
\(557\) −44.2503 −1.87495 −0.937473 0.348058i \(-0.886841\pi\)
−0.937473 + 0.348058i \(0.886841\pi\)
\(558\) 74.0794 3.13603
\(559\) 0 0
\(560\) 0 0
\(561\) 4.38484 0.185128
\(562\) 70.3051 2.96564
\(563\) 38.8907 1.63905 0.819523 0.573046i \(-0.194238\pi\)
0.819523 + 0.573046i \(0.194238\pi\)
\(564\) 2.57149 0.108279
\(565\) 13.8494 0.582647
\(566\) 41.8092 1.75737
\(567\) 0 0
\(568\) −9.44618 −0.396353
\(569\) 46.1579 1.93504 0.967520 0.252796i \(-0.0813500\pi\)
0.967520 + 0.252796i \(0.0813500\pi\)
\(570\) −7.96508 −0.333620
\(571\) 21.1368 0.884548 0.442274 0.896880i \(-0.354172\pi\)
0.442274 + 0.896880i \(0.354172\pi\)
\(572\) 0 0
\(573\) −2.53848 −0.106047
\(574\) 0 0
\(575\) −4.76904 −0.198883
\(576\) −12.7931 −0.533045
\(577\) −25.3304 −1.05452 −0.527259 0.849705i \(-0.676780\pi\)
−0.527259 + 0.849705i \(0.676780\pi\)
\(578\) −18.7009 −0.777855
\(579\) −1.56530 −0.0650517
\(580\) 40.7146 1.69058
\(581\) 0 0
\(582\) 20.7785 0.861295
\(583\) 19.6533 0.813958
\(584\) −63.6455 −2.63367
\(585\) 0 0
\(586\) 37.9106 1.56607
\(587\) −3.56287 −0.147056 −0.0735278 0.997293i \(-0.523426\pi\)
−0.0735278 + 0.997293i \(0.523426\pi\)
\(588\) 0 0
\(589\) 38.5749 1.58945
\(590\) 47.6722 1.96263
\(591\) −2.41082 −0.0991679
\(592\) 51.7732 2.12786
\(593\) −25.3536 −1.04115 −0.520573 0.853817i \(-0.674282\pi\)
−0.520573 + 0.853817i \(0.674282\pi\)
\(594\) 20.8061 0.853685
\(595\) 0 0
\(596\) 93.0677 3.81220
\(597\) 0.212945 0.00871524
\(598\) 0 0
\(599\) 10.9216 0.446243 0.223122 0.974791i \(-0.428375\pi\)
0.223122 + 0.974791i \(0.428375\pi\)
\(600\) 8.70421 0.355348
\(601\) −24.2564 −0.989439 −0.494720 0.869053i \(-0.664729\pi\)
−0.494720 + 0.869053i \(0.664729\pi\)
\(602\) 0 0
\(603\) 37.0946 1.51061
\(604\) 35.4206 1.44124
\(605\) 5.93753 0.241395
\(606\) −1.71695 −0.0697464
\(607\) 9.85447 0.399981 0.199990 0.979798i \(-0.435909\pi\)
0.199990 + 0.979798i \(0.435909\pi\)
\(608\) −31.3974 −1.27333
\(609\) 0 0
\(610\) −12.2062 −0.494215
\(611\) 0 0
\(612\) 40.1423 1.62266
\(613\) 3.67688 0.148508 0.0742540 0.997239i \(-0.476342\pi\)
0.0742540 + 0.997239i \(0.476342\pi\)
\(614\) 23.2416 0.937955
\(615\) 6.43688 0.259560
\(616\) 0 0
\(617\) −18.7468 −0.754718 −0.377359 0.926067i \(-0.623168\pi\)
−0.377359 + 0.926067i \(0.623168\pi\)
\(618\) −15.3903 −0.619090
\(619\) 15.8945 0.638854 0.319427 0.947611i \(-0.396510\pi\)
0.319427 + 0.947611i \(0.396510\pi\)
\(620\) 79.4029 3.18890
\(621\) 5.90137 0.236814
\(622\) −31.5333 −1.26437
\(623\) 0 0
\(624\) 0 0
\(625\) −7.22743 −0.289097
\(626\) −33.9727 −1.35782
\(627\) 5.16304 0.206192
\(628\) −65.8330 −2.62702
\(629\) −18.6141 −0.742194
\(630\) 0 0
\(631\) −19.7451 −0.786040 −0.393020 0.919530i \(-0.628570\pi\)
−0.393020 + 0.919530i \(0.628570\pi\)
\(632\) −43.3873 −1.72585
\(633\) 3.89679 0.154883
\(634\) −43.2698 −1.71846
\(635\) 5.03624 0.199857
\(636\) −17.7081 −0.702173
\(637\) 0 0
\(638\) −37.6200 −1.48939
\(639\) 3.69014 0.145980
\(640\) 7.95016 0.314258
\(641\) −29.7786 −1.17618 −0.588092 0.808794i \(-0.700121\pi\)
−0.588092 + 0.808794i \(0.700121\pi\)
\(642\) 6.89203 0.272007
\(643\) 11.5725 0.456373 0.228187 0.973617i \(-0.426720\pi\)
0.228187 + 0.973617i \(0.426720\pi\)
\(644\) 0 0
\(645\) −2.80493 −0.110444
\(646\) 29.7964 1.17232
\(647\) −25.5065 −1.00276 −0.501382 0.865226i \(-0.667175\pi\)
−0.501382 + 0.865226i \(0.667175\pi\)
\(648\) 46.5140 1.82724
\(649\) −30.9016 −1.21299
\(650\) 0 0
\(651\) 0 0
\(652\) −33.7049 −1.31999
\(653\) −44.8293 −1.75430 −0.877152 0.480212i \(-0.840560\pi\)
−0.877152 + 0.480212i \(0.840560\pi\)
\(654\) −2.31892 −0.0906771
\(655\) 16.4578 0.643058
\(656\) 66.9747 2.61492
\(657\) 24.8630 0.969999
\(658\) 0 0
\(659\) 41.1734 1.60389 0.801944 0.597399i \(-0.203799\pi\)
0.801944 + 0.597399i \(0.203799\pi\)
\(660\) 10.6276 0.413680
\(661\) 21.8938 0.851569 0.425785 0.904825i \(-0.359998\pi\)
0.425785 + 0.904825i \(0.359998\pi\)
\(662\) −10.2681 −0.399080
\(663\) 0 0
\(664\) 18.8539 0.731673
\(665\) 0 0
\(666\) −42.0908 −1.63098
\(667\) −10.6704 −0.413160
\(668\) 84.5239 3.27033
\(669\) −11.7067 −0.452606
\(670\) 56.6764 2.18960
\(671\) 7.91219 0.305447
\(672\) 0 0
\(673\) 35.6688 1.37493 0.687466 0.726217i \(-0.258723\pi\)
0.687466 + 0.726217i \(0.258723\pi\)
\(674\) 35.4721 1.36633
\(675\) −7.13524 −0.274635
\(676\) 0 0
\(677\) −2.55532 −0.0982089 −0.0491044 0.998794i \(-0.515637\pi\)
−0.0491044 + 0.998794i \(0.515637\pi\)
\(678\) 11.5302 0.442813
\(679\) 0 0
\(680\) 35.2386 1.35134
\(681\) 7.08832 0.271625
\(682\) −73.3678 −2.80940
\(683\) 35.7399 1.36755 0.683775 0.729693i \(-0.260337\pi\)
0.683775 + 0.729693i \(0.260337\pi\)
\(684\) 47.2666 1.80728
\(685\) 16.1086 0.615479
\(686\) 0 0
\(687\) 4.11188 0.156878
\(688\) −29.1848 −1.11266
\(689\) 0 0
\(690\) 4.29687 0.163579
\(691\) 26.0292 0.990197 0.495099 0.868837i \(-0.335132\pi\)
0.495099 + 0.868837i \(0.335132\pi\)
\(692\) 60.2254 2.28943
\(693\) 0 0
\(694\) −68.3335 −2.59391
\(695\) −2.68278 −0.101764
\(696\) 19.4751 0.738202
\(697\) −24.0796 −0.912079
\(698\) −12.6589 −0.479146
\(699\) 3.40721 0.128872
\(700\) 0 0
\(701\) 1.12731 0.0425779 0.0212890 0.999773i \(-0.493223\pi\)
0.0212890 + 0.999773i \(0.493223\pi\)
\(702\) 0 0
\(703\) −21.9177 −0.826641
\(704\) 12.6702 0.477525
\(705\) −0.881834 −0.0332118
\(706\) −35.0955 −1.32084
\(707\) 0 0
\(708\) 27.8431 1.04641
\(709\) 6.05031 0.227224 0.113612 0.993525i \(-0.463758\pi\)
0.113612 + 0.993525i \(0.463758\pi\)
\(710\) 5.63813 0.211595
\(711\) 16.9492 0.635644
\(712\) −12.3040 −0.461114
\(713\) −20.8098 −0.779332
\(714\) 0 0
\(715\) 0 0
\(716\) 8.65061 0.323288
\(717\) 4.86986 0.181868
\(718\) 22.2249 0.829425
\(719\) −47.1177 −1.75719 −0.878597 0.477563i \(-0.841520\pi\)
−0.878597 + 0.477563i \(0.841520\pi\)
\(720\) 38.2884 1.42692
\(721\) 0 0
\(722\) −14.0989 −0.524706
\(723\) −5.22916 −0.194475
\(724\) −15.5095 −0.576404
\(725\) 12.9014 0.479146
\(726\) 4.94324 0.183461
\(727\) 17.9215 0.664671 0.332335 0.943161i \(-0.392163\pi\)
0.332335 + 0.943161i \(0.392163\pi\)
\(728\) 0 0
\(729\) −13.5489 −0.501810
\(730\) 37.9880 1.40600
\(731\) 10.4929 0.388093
\(732\) −7.12908 −0.263498
\(733\) −45.2685 −1.67203 −0.836016 0.548705i \(-0.815121\pi\)
−0.836016 + 0.548705i \(0.815121\pi\)
\(734\) 4.30542 0.158916
\(735\) 0 0
\(736\) 16.9378 0.624335
\(737\) −36.7382 −1.35327
\(738\) −54.4494 −2.00431
\(739\) 19.2613 0.708539 0.354270 0.935143i \(-0.384730\pi\)
0.354270 + 0.935143i \(0.384730\pi\)
\(740\) −45.1155 −1.65848
\(741\) 0 0
\(742\) 0 0
\(743\) 34.8853 1.27982 0.639908 0.768452i \(-0.278972\pi\)
0.639908 + 0.768452i \(0.278972\pi\)
\(744\) 37.9810 1.39245
\(745\) −31.9155 −1.16929
\(746\) 36.1459 1.32339
\(747\) −7.36525 −0.269480
\(748\) −39.7567 −1.45365
\(749\) 0 0
\(750\) −16.0130 −0.584711
\(751\) −24.9668 −0.911051 −0.455526 0.890223i \(-0.650549\pi\)
−0.455526 + 0.890223i \(0.650549\pi\)
\(752\) −9.17535 −0.334591
\(753\) −5.36536 −0.195525
\(754\) 0 0
\(755\) −12.1467 −0.442064
\(756\) 0 0
\(757\) −10.6049 −0.385440 −0.192720 0.981254i \(-0.561731\pi\)
−0.192720 + 0.981254i \(0.561731\pi\)
\(758\) −81.7370 −2.96882
\(759\) −2.78527 −0.101099
\(760\) 41.4926 1.50510
\(761\) 32.6388 1.18316 0.591578 0.806248i \(-0.298505\pi\)
0.591578 + 0.806248i \(0.298505\pi\)
\(762\) 4.19288 0.151892
\(763\) 0 0
\(764\) 23.0160 0.832691
\(765\) −13.7659 −0.497708
\(766\) 82.5976 2.98437
\(767\) 0 0
\(768\) 11.4759 0.414100
\(769\) 52.1752 1.88149 0.940744 0.339119i \(-0.110129\pi\)
0.940744 + 0.339119i \(0.110129\pi\)
\(770\) 0 0
\(771\) 4.14077 0.149126
\(772\) 14.1924 0.510794
\(773\) 35.7057 1.28425 0.642123 0.766602i \(-0.278054\pi\)
0.642123 + 0.766602i \(0.278054\pi\)
\(774\) 23.7268 0.852842
\(775\) 25.1607 0.903800
\(776\) −108.242 −3.88565
\(777\) 0 0
\(778\) −65.7893 −2.35866
\(779\) −28.3531 −1.01586
\(780\) 0 0
\(781\) −3.65469 −0.130775
\(782\) −16.0741 −0.574808
\(783\) −15.9646 −0.570529
\(784\) 0 0
\(785\) 22.5760 0.805770
\(786\) 13.7018 0.488726
\(787\) 6.10621 0.217663 0.108831 0.994060i \(-0.465289\pi\)
0.108831 + 0.994060i \(0.465289\pi\)
\(788\) 21.8586 0.778679
\(789\) −2.62310 −0.0933847
\(790\) 25.8965 0.921356
\(791\) 0 0
\(792\) −51.6510 −1.83534
\(793\) 0 0
\(794\) −10.7659 −0.382068
\(795\) 6.07261 0.215373
\(796\) −1.93074 −0.0684332
\(797\) 46.2299 1.63755 0.818773 0.574117i \(-0.194654\pi\)
0.818773 + 0.574117i \(0.194654\pi\)
\(798\) 0 0
\(799\) 3.29884 0.116704
\(800\) −20.4792 −0.724048
\(801\) 4.80656 0.169831
\(802\) −50.7400 −1.79169
\(803\) −24.6242 −0.868968
\(804\) 33.1020 1.16742
\(805\) 0 0
\(806\) 0 0
\(807\) 7.20161 0.253509
\(808\) 8.94415 0.314654
\(809\) 39.2879 1.38129 0.690644 0.723195i \(-0.257327\pi\)
0.690644 + 0.723195i \(0.257327\pi\)
\(810\) −27.7627 −0.975482
\(811\) 6.90664 0.242525 0.121262 0.992620i \(-0.461306\pi\)
0.121262 + 0.992620i \(0.461306\pi\)
\(812\) 0 0
\(813\) −4.31738 −0.151417
\(814\) 41.6864 1.46111
\(815\) 11.5583 0.404871
\(816\) 14.0971 0.493496
\(817\) 12.3551 0.432251
\(818\) 45.6466 1.59600
\(819\) 0 0
\(820\) −58.3622 −2.03810
\(821\) 1.91049 0.0666765 0.0333382 0.999444i \(-0.489386\pi\)
0.0333382 + 0.999444i \(0.489386\pi\)
\(822\) 13.4111 0.467765
\(823\) 1.57969 0.0550645 0.0275322 0.999621i \(-0.491235\pi\)
0.0275322 + 0.999621i \(0.491235\pi\)
\(824\) 80.1732 2.79296
\(825\) 3.36763 0.117246
\(826\) 0 0
\(827\) 32.5050 1.13031 0.565155 0.824985i \(-0.308816\pi\)
0.565155 + 0.824985i \(0.308816\pi\)
\(828\) −25.4986 −0.886138
\(829\) 35.0538 1.21747 0.608735 0.793373i \(-0.291677\pi\)
0.608735 + 0.793373i \(0.291677\pi\)
\(830\) −11.2533 −0.390608
\(831\) 12.0344 0.417469
\(832\) 0 0
\(833\) 0 0
\(834\) −2.23353 −0.0773407
\(835\) −28.9856 −1.00309
\(836\) −46.8126 −1.61905
\(837\) −31.1347 −1.07617
\(838\) −77.3760 −2.67291
\(839\) −5.35487 −0.184871 −0.0924354 0.995719i \(-0.529465\pi\)
−0.0924354 + 0.995719i \(0.529465\pi\)
\(840\) 0 0
\(841\) −0.133978 −0.00461993
\(842\) −33.2708 −1.14659
\(843\) −14.0813 −0.484985
\(844\) −35.3316 −1.21616
\(845\) 0 0
\(846\) 7.45942 0.256460
\(847\) 0 0
\(848\) 63.1846 2.16977
\(849\) −8.37387 −0.287391
\(850\) 19.4349 0.666611
\(851\) 11.8238 0.405314
\(852\) 3.29297 0.112815
\(853\) −49.6270 −1.69920 −0.849598 0.527431i \(-0.823155\pi\)
−0.849598 + 0.527431i \(0.823155\pi\)
\(854\) 0 0
\(855\) −16.2090 −0.554337
\(856\) −35.9028 −1.22713
\(857\) −5.88392 −0.200991 −0.100496 0.994938i \(-0.532043\pi\)
−0.100496 + 0.994938i \(0.532043\pi\)
\(858\) 0 0
\(859\) −43.3862 −1.48032 −0.740159 0.672432i \(-0.765250\pi\)
−0.740159 + 0.672432i \(0.765250\pi\)
\(860\) 25.4318 0.867219
\(861\) 0 0
\(862\) −23.2213 −0.790920
\(863\) −31.1272 −1.05958 −0.529792 0.848128i \(-0.677730\pi\)
−0.529792 + 0.848128i \(0.677730\pi\)
\(864\) 25.3416 0.862139
\(865\) −20.6530 −0.702222
\(866\) 8.93228 0.303531
\(867\) 3.74557 0.127206
\(868\) 0 0
\(869\) −16.7864 −0.569439
\(870\) −11.6241 −0.394093
\(871\) 0 0
\(872\) 12.0800 0.409081
\(873\) 42.2844 1.43111
\(874\) −18.9268 −0.640209
\(875\) 0 0
\(876\) 22.1870 0.749629
\(877\) 29.9106 1.01001 0.505004 0.863117i \(-0.331491\pi\)
0.505004 + 0.863117i \(0.331491\pi\)
\(878\) 99.6984 3.36466
\(879\) −7.59304 −0.256107
\(880\) −37.9206 −1.27830
\(881\) −14.5695 −0.490860 −0.245430 0.969414i \(-0.578929\pi\)
−0.245430 + 0.969414i \(0.578929\pi\)
\(882\) 0 0
\(883\) −48.9296 −1.64661 −0.823307 0.567597i \(-0.807873\pi\)
−0.823307 + 0.567597i \(0.807873\pi\)
\(884\) 0 0
\(885\) −9.54817 −0.320958
\(886\) −38.9324 −1.30796
\(887\) −54.5902 −1.83296 −0.916480 0.400080i \(-0.868982\pi\)
−0.916480 + 0.400080i \(0.868982\pi\)
\(888\) −21.5802 −0.724184
\(889\) 0 0
\(890\) 7.34389 0.246168
\(891\) 17.9961 0.602891
\(892\) 106.143 3.55392
\(893\) 3.88430 0.129983
\(894\) −26.5710 −0.888666
\(895\) −2.96653 −0.0991603
\(896\) 0 0
\(897\) 0 0
\(898\) 100.757 3.36232
\(899\) 56.2955 1.87756
\(900\) 30.8299 1.02766
\(901\) −22.7169 −0.756810
\(902\) 53.9263 1.79555
\(903\) 0 0
\(904\) −60.0643 −1.99771
\(905\) 5.31862 0.176797
\(906\) −10.1126 −0.335970
\(907\) 22.7255 0.754589 0.377295 0.926093i \(-0.376854\pi\)
0.377295 + 0.926093i \(0.376854\pi\)
\(908\) −64.2688 −2.13283
\(909\) −3.49402 −0.115889
\(910\) 0 0
\(911\) −42.2359 −1.39934 −0.699669 0.714467i \(-0.746669\pi\)
−0.699669 + 0.714467i \(0.746669\pi\)
\(912\) 16.5990 0.549646
\(913\) 7.29450 0.241413
\(914\) 36.0842 1.19356
\(915\) 2.44476 0.0808213
\(916\) −37.2818 −1.23182
\(917\) 0 0
\(918\) −24.0494 −0.793747
\(919\) 30.6940 1.01250 0.506251 0.862386i \(-0.331031\pi\)
0.506251 + 0.862386i \(0.331031\pi\)
\(920\) −22.3838 −0.737971
\(921\) −4.65502 −0.153388
\(922\) 96.9780 3.19380
\(923\) 0 0
\(924\) 0 0
\(925\) −14.2959 −0.470048
\(926\) −17.4802 −0.574434
\(927\) −31.3195 −1.02867
\(928\) −45.8208 −1.50414
\(929\) 37.4250 1.22787 0.613936 0.789355i \(-0.289585\pi\)
0.613936 + 0.789355i \(0.289585\pi\)
\(930\) −22.6696 −0.743367
\(931\) 0 0
\(932\) −30.8926 −1.01192
\(933\) 6.31574 0.206768
\(934\) −13.0733 −0.427770
\(935\) 13.6337 0.445869
\(936\) 0 0
\(937\) −44.3386 −1.44848 −0.724239 0.689549i \(-0.757809\pi\)
−0.724239 + 0.689549i \(0.757809\pi\)
\(938\) 0 0
\(939\) 6.80433 0.222051
\(940\) 7.99546 0.260783
\(941\) 27.5052 0.896646 0.448323 0.893872i \(-0.352022\pi\)
0.448323 + 0.893872i \(0.352022\pi\)
\(942\) 18.7954 0.612388
\(943\) 15.2955 0.498089
\(944\) −99.3472 −3.23348
\(945\) 0 0
\(946\) −23.4989 −0.764014
\(947\) 5.08330 0.165185 0.0825925 0.996583i \(-0.473680\pi\)
0.0825925 + 0.996583i \(0.473680\pi\)
\(948\) 15.1249 0.491235
\(949\) 0 0
\(950\) 22.8841 0.742458
\(951\) 8.66642 0.281028
\(952\) 0 0
\(953\) 9.81437 0.317919 0.158959 0.987285i \(-0.449186\pi\)
0.158959 + 0.987285i \(0.449186\pi\)
\(954\) −51.3681 −1.66310
\(955\) −7.89284 −0.255406
\(956\) −44.1543 −1.42805
\(957\) 7.53484 0.243567
\(958\) −24.4705 −0.790607
\(959\) 0 0
\(960\) 3.91491 0.126353
\(961\) 78.7893 2.54159
\(962\) 0 0
\(963\) 14.0254 0.451961
\(964\) 47.4121 1.52704
\(965\) −4.86695 −0.156673
\(966\) 0 0
\(967\) −2.69619 −0.0867036 −0.0433518 0.999060i \(-0.513804\pi\)
−0.0433518 + 0.999060i \(0.513804\pi\)
\(968\) −25.7509 −0.827665
\(969\) −5.96786 −0.191715
\(970\) 64.6060 2.07437
\(971\) −24.9240 −0.799850 −0.399925 0.916548i \(-0.630964\pi\)
−0.399925 + 0.916548i \(0.630964\pi\)
\(972\) −58.1196 −1.86419
\(973\) 0 0
\(974\) 103.543 3.31773
\(975\) 0 0
\(976\) 25.4373 0.814230
\(977\) −28.3129 −0.905811 −0.452906 0.891558i \(-0.649613\pi\)
−0.452906 + 0.891558i \(0.649613\pi\)
\(978\) 9.62280 0.307703
\(979\) −4.76039 −0.152143
\(980\) 0 0
\(981\) −4.71904 −0.150667
\(982\) −17.5015 −0.558494
\(983\) −37.8517 −1.20728 −0.603641 0.797256i \(-0.706284\pi\)
−0.603641 + 0.797256i \(0.706284\pi\)
\(984\) −27.9166 −0.889947
\(985\) −7.49591 −0.238839
\(986\) 43.4842 1.38482
\(987\) 0 0
\(988\) 0 0
\(989\) −6.66514 −0.211939
\(990\) 30.8289 0.979805
\(991\) 58.4158 1.85564 0.927820 0.373028i \(-0.121681\pi\)
0.927820 + 0.373028i \(0.121681\pi\)
\(992\) −89.3612 −2.83722
\(993\) 2.05657 0.0652633
\(994\) 0 0
\(995\) 0.662104 0.0209901
\(996\) −6.57252 −0.208258
\(997\) −28.0588 −0.888632 −0.444316 0.895870i \(-0.646553\pi\)
−0.444316 + 0.895870i \(0.646553\pi\)
\(998\) 29.4001 0.930645
\(999\) 17.6903 0.559696
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.co.1.12 12
7.3 odd 6 1183.2.e.j.170.1 24
7.5 odd 6 1183.2.e.j.508.1 24
7.6 odd 2 8281.2.a.cp.1.12 12
13.2 odd 12 637.2.q.i.589.6 12
13.7 odd 12 637.2.q.i.491.6 12
13.12 even 2 inner 8281.2.a.co.1.1 12
91.2 odd 12 637.2.k.i.459.1 12
91.12 odd 6 1183.2.e.j.508.12 24
91.20 even 12 637.2.q.g.491.6 12
91.33 even 12 91.2.u.b.88.1 yes 12
91.38 odd 6 1183.2.e.j.170.12 24
91.41 even 12 637.2.q.g.589.6 12
91.46 odd 12 637.2.k.i.569.6 12
91.54 even 12 91.2.k.b.4.1 12
91.59 even 12 91.2.k.b.23.6 yes 12
91.67 odd 12 637.2.u.g.30.1 12
91.72 odd 12 637.2.u.g.361.1 12
91.80 even 12 91.2.u.b.30.1 yes 12
91.90 odd 2 8281.2.a.cp.1.1 12
273.59 odd 12 819.2.bm.f.478.1 12
273.80 odd 12 819.2.do.e.667.6 12
273.215 odd 12 819.2.do.e.361.6 12
273.236 odd 12 819.2.bm.f.550.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.1 12 91.54 even 12
91.2.k.b.23.6 yes 12 91.59 even 12
91.2.u.b.30.1 yes 12 91.80 even 12
91.2.u.b.88.1 yes 12 91.33 even 12
637.2.k.i.459.1 12 91.2 odd 12
637.2.k.i.569.6 12 91.46 odd 12
637.2.q.g.491.6 12 91.20 even 12
637.2.q.g.589.6 12 91.41 even 12
637.2.q.i.491.6 12 13.7 odd 12
637.2.q.i.589.6 12 13.2 odd 12
637.2.u.g.30.1 12 91.67 odd 12
637.2.u.g.361.1 12 91.72 odd 12
819.2.bm.f.478.1 12 273.59 odd 12
819.2.bm.f.550.6 12 273.236 odd 12
819.2.do.e.361.6 12 273.215 odd 12
819.2.do.e.667.6 12 273.80 odd 12
1183.2.e.j.170.1 24 7.3 odd 6
1183.2.e.j.170.12 24 91.38 odd 6
1183.2.e.j.508.1 24 7.5 odd 6
1183.2.e.j.508.12 24 91.12 odd 6
8281.2.a.co.1.1 12 13.12 even 2 inner
8281.2.a.co.1.12 12 1.1 even 1 trivial
8281.2.a.cp.1.1 12 91.90 odd 2
8281.2.a.cp.1.12 12 7.6 odd 2