Properties

Label 7623.2.a.cz.1.8
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(0.870105\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.870105 q^{2} -1.24292 q^{4} -0.709862 q^{5} -1.00000 q^{7} -2.82168 q^{8} +O(q^{10})\) \(q+0.870105 q^{2} -1.24292 q^{4} -0.709862 q^{5} -1.00000 q^{7} -2.82168 q^{8} -0.617654 q^{10} +3.47182 q^{13} -0.870105 q^{14} +0.0306816 q^{16} +1.34682 q^{17} -0.503149 q^{19} +0.882300 q^{20} +3.69178 q^{23} -4.49610 q^{25} +3.02085 q^{26} +1.24292 q^{28} -9.21603 q^{29} +1.39006 q^{31} +5.67005 q^{32} +1.17188 q^{34} +0.709862 q^{35} +2.24922 q^{37} -0.437792 q^{38} +2.00300 q^{40} +12.4700 q^{41} -5.60495 q^{43} +3.21224 q^{46} -5.79140 q^{47} +1.00000 q^{49} -3.91207 q^{50} -4.31519 q^{52} +3.60444 q^{53} +2.82168 q^{56} -8.01891 q^{58} -7.64892 q^{59} -5.52488 q^{61} +1.20950 q^{62} +4.87217 q^{64} -2.46451 q^{65} +3.25948 q^{67} -1.67399 q^{68} +0.617654 q^{70} +8.53290 q^{71} +2.35102 q^{73} +1.95705 q^{74} +0.625372 q^{76} +4.00104 q^{79} -0.0217797 q^{80} +10.8502 q^{82} +14.6924 q^{83} -0.956057 q^{85} -4.87689 q^{86} -1.42869 q^{89} -3.47182 q^{91} -4.58858 q^{92} -5.03913 q^{94} +0.357166 q^{95} -8.56194 q^{97} +0.870105 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} - 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} - 12q^{7} - 20q^{10} - 20q^{13} + 28q^{16} - 12q^{19} + 32q^{25} - 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} - 48q^{40} - 16q^{43} - 24q^{46} + 12q^{49} - 96q^{52} + 20q^{58} - 44q^{61} + 76q^{64} + 20q^{70} - 52q^{73} + 8q^{79} - 68q^{82} - 72q^{85} + 20q^{91} - 20q^{94} - 32q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.870105 0.615257 0.307628 0.951507i \(-0.400465\pi\)
0.307628 + 0.951507i \(0.400465\pi\)
\(3\) 0 0
\(4\) −1.24292 −0.621459
\(5\) −0.709862 −0.317460 −0.158730 0.987322i \(-0.550740\pi\)
−0.158730 + 0.987322i \(0.550740\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.82168 −0.997614
\(9\) 0 0
\(10\) −0.617654 −0.195319
\(11\) 0 0
\(12\) 0 0
\(13\) 3.47182 0.962910 0.481455 0.876471i \(-0.340109\pi\)
0.481455 + 0.876471i \(0.340109\pi\)
\(14\) −0.870105 −0.232545
\(15\) 0 0
\(16\) 0.0306816 0.00767039
\(17\) 1.34682 0.326652 0.163326 0.986572i \(-0.447778\pi\)
0.163326 + 0.986572i \(0.447778\pi\)
\(18\) 0 0
\(19\) −0.503149 −0.115430 −0.0577151 0.998333i \(-0.518382\pi\)
−0.0577151 + 0.998333i \(0.518382\pi\)
\(20\) 0.882300 0.197288
\(21\) 0 0
\(22\) 0 0
\(23\) 3.69178 0.769790 0.384895 0.922960i \(-0.374238\pi\)
0.384895 + 0.922960i \(0.374238\pi\)
\(24\) 0 0
\(25\) −4.49610 −0.899219
\(26\) 3.02085 0.592437
\(27\) 0 0
\(28\) 1.24292 0.234889
\(29\) −9.21603 −1.71137 −0.855687 0.517494i \(-0.826865\pi\)
−0.855687 + 0.517494i \(0.826865\pi\)
\(30\) 0 0
\(31\) 1.39006 0.249663 0.124832 0.992178i \(-0.460161\pi\)
0.124832 + 0.992178i \(0.460161\pi\)
\(32\) 5.67005 1.00233
\(33\) 0 0
\(34\) 1.17188 0.200975
\(35\) 0.709862 0.119989
\(36\) 0 0
\(37\) 2.24922 0.369769 0.184884 0.982760i \(-0.440809\pi\)
0.184884 + 0.982760i \(0.440809\pi\)
\(38\) −0.437792 −0.0710192
\(39\) 0 0
\(40\) 2.00300 0.316702
\(41\) 12.4700 1.94749 0.973745 0.227641i \(-0.0731012\pi\)
0.973745 + 0.227641i \(0.0731012\pi\)
\(42\) 0 0
\(43\) −5.60495 −0.854747 −0.427374 0.904075i \(-0.640561\pi\)
−0.427374 + 0.904075i \(0.640561\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 3.21224 0.473618
\(47\) −5.79140 −0.844763 −0.422381 0.906418i \(-0.638806\pi\)
−0.422381 + 0.906418i \(0.638806\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.91207 −0.553251
\(51\) 0 0
\(52\) −4.31519 −0.598409
\(53\) 3.60444 0.495108 0.247554 0.968874i \(-0.420373\pi\)
0.247554 + 0.968874i \(0.420373\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.82168 0.377063
\(57\) 0 0
\(58\) −8.01891 −1.05293
\(59\) −7.64892 −0.995804 −0.497902 0.867233i \(-0.665896\pi\)
−0.497902 + 0.867233i \(0.665896\pi\)
\(60\) 0 0
\(61\) −5.52488 −0.707389 −0.353694 0.935361i \(-0.615075\pi\)
−0.353694 + 0.935361i \(0.615075\pi\)
\(62\) 1.20950 0.153607
\(63\) 0 0
\(64\) 4.87217 0.609022
\(65\) −2.46451 −0.305685
\(66\) 0 0
\(67\) 3.25948 0.398208 0.199104 0.979978i \(-0.436197\pi\)
0.199104 + 0.979978i \(0.436197\pi\)
\(68\) −1.67399 −0.203001
\(69\) 0 0
\(70\) 0.617654 0.0738237
\(71\) 8.53290 1.01267 0.506335 0.862337i \(-0.331000\pi\)
0.506335 + 0.862337i \(0.331000\pi\)
\(72\) 0 0
\(73\) 2.35102 0.275166 0.137583 0.990490i \(-0.456067\pi\)
0.137583 + 0.990490i \(0.456067\pi\)
\(74\) 1.95705 0.227503
\(75\) 0 0
\(76\) 0.625372 0.0717351
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00104 0.450152 0.225076 0.974341i \(-0.427737\pi\)
0.225076 + 0.974341i \(0.427737\pi\)
\(80\) −0.0217797 −0.00243504
\(81\) 0 0
\(82\) 10.8502 1.19821
\(83\) 14.6924 1.61270 0.806352 0.591436i \(-0.201439\pi\)
0.806352 + 0.591436i \(0.201439\pi\)
\(84\) 0 0
\(85\) −0.956057 −0.103699
\(86\) −4.87689 −0.525889
\(87\) 0 0
\(88\) 0 0
\(89\) −1.42869 −0.151440 −0.0757202 0.997129i \(-0.524126\pi\)
−0.0757202 + 0.997129i \(0.524126\pi\)
\(90\) 0 0
\(91\) −3.47182 −0.363946
\(92\) −4.58858 −0.478393
\(93\) 0 0
\(94\) −5.03913 −0.519746
\(95\) 0.357166 0.0366444
\(96\) 0 0
\(97\) −8.56194 −0.869333 −0.434667 0.900591i \(-0.643134\pi\)
−0.434667 + 0.900591i \(0.643134\pi\)
\(98\) 0.870105 0.0878938
\(99\) 0 0
\(100\) 5.58828 0.558828
\(101\) 3.97425 0.395452 0.197726 0.980257i \(-0.436644\pi\)
0.197726 + 0.980257i \(0.436644\pi\)
\(102\) 0 0
\(103\) 12.2729 1.20929 0.604644 0.796496i \(-0.293316\pi\)
0.604644 + 0.796496i \(0.293316\pi\)
\(104\) −9.79636 −0.960612
\(105\) 0 0
\(106\) 3.13624 0.304619
\(107\) −12.0377 −1.16373 −0.581865 0.813286i \(-0.697677\pi\)
−0.581865 + 0.813286i \(0.697677\pi\)
\(108\) 0 0
\(109\) −7.18589 −0.688284 −0.344142 0.938918i \(-0.611830\pi\)
−0.344142 + 0.938918i \(0.611830\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.0306816 −0.00289914
\(113\) −8.67927 −0.816477 −0.408238 0.912875i \(-0.633857\pi\)
−0.408238 + 0.912875i \(0.633857\pi\)
\(114\) 0 0
\(115\) −2.62065 −0.244377
\(116\) 11.4548 1.06355
\(117\) 0 0
\(118\) −6.65536 −0.612675
\(119\) −1.34682 −0.123463
\(120\) 0 0
\(121\) 0 0
\(122\) −4.80723 −0.435226
\(123\) 0 0
\(124\) −1.72774 −0.155155
\(125\) 6.74091 0.602926
\(126\) 0 0
\(127\) −21.5942 −1.91618 −0.958089 0.286472i \(-0.907518\pi\)
−0.958089 + 0.286472i \(0.907518\pi\)
\(128\) −7.10080 −0.627628
\(129\) 0 0
\(130\) −2.14438 −0.188075
\(131\) −8.93177 −0.780372 −0.390186 0.920736i \(-0.627589\pi\)
−0.390186 + 0.920736i \(0.627589\pi\)
\(132\) 0 0
\(133\) 0.503149 0.0436285
\(134\) 2.83608 0.245000
\(135\) 0 0
\(136\) −3.80030 −0.325873
\(137\) −8.70309 −0.743555 −0.371778 0.928322i \(-0.621252\pi\)
−0.371778 + 0.928322i \(0.621252\pi\)
\(138\) 0 0
\(139\) 8.54256 0.724571 0.362285 0.932067i \(-0.381997\pi\)
0.362285 + 0.932067i \(0.381997\pi\)
\(140\) −0.882300 −0.0745679
\(141\) 0 0
\(142\) 7.42452 0.623052
\(143\) 0 0
\(144\) 0 0
\(145\) 6.54211 0.543292
\(146\) 2.04563 0.169298
\(147\) 0 0
\(148\) −2.79559 −0.229796
\(149\) −2.22541 −0.182313 −0.0911563 0.995837i \(-0.529056\pi\)
−0.0911563 + 0.995837i \(0.529056\pi\)
\(150\) 0 0
\(151\) −13.0845 −1.06480 −0.532400 0.846493i \(-0.678710\pi\)
−0.532400 + 0.846493i \(0.678710\pi\)
\(152\) 1.41972 0.115155
\(153\) 0 0
\(154\) 0 0
\(155\) −0.986754 −0.0792580
\(156\) 0 0
\(157\) −16.7874 −1.33978 −0.669889 0.742461i \(-0.733658\pi\)
−0.669889 + 0.742461i \(0.733658\pi\)
\(158\) 3.48132 0.276959
\(159\) 0 0
\(160\) −4.02495 −0.318200
\(161\) −3.69178 −0.290953
\(162\) 0 0
\(163\) −4.70967 −0.368890 −0.184445 0.982843i \(-0.559049\pi\)
−0.184445 + 0.982843i \(0.559049\pi\)
\(164\) −15.4992 −1.21029
\(165\) 0 0
\(166\) 12.7839 0.992227
\(167\) −3.59383 −0.278099 −0.139050 0.990285i \(-0.544405\pi\)
−0.139050 + 0.990285i \(0.544405\pi\)
\(168\) 0 0
\(169\) −0.946464 −0.0728049
\(170\) −0.831870 −0.0638015
\(171\) 0 0
\(172\) 6.96650 0.531190
\(173\) −9.26569 −0.704457 −0.352229 0.935914i \(-0.614576\pi\)
−0.352229 + 0.935914i \(0.614576\pi\)
\(174\) 0 0
\(175\) 4.49610 0.339873
\(176\) 0 0
\(177\) 0 0
\(178\) −1.24311 −0.0931747
\(179\) −6.25994 −0.467890 −0.233945 0.972250i \(-0.575164\pi\)
−0.233945 + 0.972250i \(0.575164\pi\)
\(180\) 0 0
\(181\) 16.1467 1.20017 0.600087 0.799935i \(-0.295133\pi\)
0.600087 + 0.799935i \(0.295133\pi\)
\(182\) −3.02085 −0.223920
\(183\) 0 0
\(184\) −10.4170 −0.767953
\(185\) −1.59663 −0.117387
\(186\) 0 0
\(187\) 0 0
\(188\) 7.19824 0.524986
\(189\) 0 0
\(190\) 0.310772 0.0225457
\(191\) −18.5412 −1.34159 −0.670797 0.741641i \(-0.734048\pi\)
−0.670797 + 0.741641i \(0.734048\pi\)
\(192\) 0 0
\(193\) −8.23992 −0.593122 −0.296561 0.955014i \(-0.595840\pi\)
−0.296561 + 0.955014i \(0.595840\pi\)
\(194\) −7.44978 −0.534863
\(195\) 0 0
\(196\) −1.24292 −0.0887799
\(197\) 3.60845 0.257092 0.128546 0.991704i \(-0.458969\pi\)
0.128546 + 0.991704i \(0.458969\pi\)
\(198\) 0 0
\(199\) 7.74954 0.549350 0.274675 0.961537i \(-0.411430\pi\)
0.274675 + 0.961537i \(0.411430\pi\)
\(200\) 12.6865 0.897074
\(201\) 0 0
\(202\) 3.45801 0.243305
\(203\) 9.21603 0.646839
\(204\) 0 0
\(205\) −8.85199 −0.618250
\(206\) 10.6787 0.744022
\(207\) 0 0
\(208\) 0.106521 0.00738590
\(209\) 0 0
\(210\) 0 0
\(211\) 0.130589 0.00899014 0.00449507 0.999990i \(-0.498569\pi\)
0.00449507 + 0.999990i \(0.498569\pi\)
\(212\) −4.48002 −0.307689
\(213\) 0 0
\(214\) −10.4741 −0.715992
\(215\) 3.97874 0.271348
\(216\) 0 0
\(217\) −1.39006 −0.0943638
\(218\) −6.25248 −0.423471
\(219\) 0 0
\(220\) 0 0
\(221\) 4.67593 0.314537
\(222\) 0 0
\(223\) −12.3141 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(224\) −5.67005 −0.378846
\(225\) 0 0
\(226\) −7.55187 −0.502343
\(227\) 13.9626 0.926733 0.463367 0.886167i \(-0.346641\pi\)
0.463367 + 0.886167i \(0.346641\pi\)
\(228\) 0 0
\(229\) −19.9419 −1.31780 −0.658898 0.752233i \(-0.728977\pi\)
−0.658898 + 0.752233i \(0.728977\pi\)
\(230\) −2.28024 −0.150355
\(231\) 0 0
\(232\) 26.0047 1.70729
\(233\) −4.58513 −0.300382 −0.150191 0.988657i \(-0.547989\pi\)
−0.150191 + 0.988657i \(0.547989\pi\)
\(234\) 0 0
\(235\) 4.11109 0.268178
\(236\) 9.50698 0.618852
\(237\) 0 0
\(238\) −1.17188 −0.0759615
\(239\) 2.46796 0.159639 0.0798195 0.996809i \(-0.474566\pi\)
0.0798195 + 0.996809i \(0.474566\pi\)
\(240\) 0 0
\(241\) −21.1525 −1.36255 −0.681276 0.732027i \(-0.738575\pi\)
−0.681276 + 0.732027i \(0.738575\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.86698 0.439613
\(245\) −0.709862 −0.0453514
\(246\) 0 0
\(247\) −1.74684 −0.111149
\(248\) −3.92232 −0.249067
\(249\) 0 0
\(250\) 5.86530 0.370954
\(251\) −18.8125 −1.18743 −0.593717 0.804674i \(-0.702340\pi\)
−0.593717 + 0.804674i \(0.702340\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −18.7892 −1.17894
\(255\) 0 0
\(256\) −15.9228 −0.995174
\(257\) −29.1877 −1.82068 −0.910339 0.413862i \(-0.864179\pi\)
−0.910339 + 0.413862i \(0.864179\pi\)
\(258\) 0 0
\(259\) −2.24922 −0.139759
\(260\) 3.06319 0.189971
\(261\) 0 0
\(262\) −7.77157 −0.480129
\(263\) −16.5583 −1.02103 −0.510516 0.859868i \(-0.670545\pi\)
−0.510516 + 0.859868i \(0.670545\pi\)
\(264\) 0 0
\(265\) −2.55865 −0.157177
\(266\) 0.437792 0.0268427
\(267\) 0 0
\(268\) −4.05126 −0.247470
\(269\) −22.2756 −1.35817 −0.679084 0.734061i \(-0.737623\pi\)
−0.679084 + 0.734061i \(0.737623\pi\)
\(270\) 0 0
\(271\) −18.3230 −1.11304 −0.556521 0.830834i \(-0.687864\pi\)
−0.556521 + 0.830834i \(0.687864\pi\)
\(272\) 0.0413226 0.00250555
\(273\) 0 0
\(274\) −7.57260 −0.457478
\(275\) 0 0
\(276\) 0 0
\(277\) −19.2766 −1.15822 −0.579110 0.815249i \(-0.696600\pi\)
−0.579110 + 0.815249i \(0.696600\pi\)
\(278\) 7.43292 0.445797
\(279\) 0 0
\(280\) −2.00300 −0.119702
\(281\) 16.9012 1.00824 0.504120 0.863634i \(-0.331817\pi\)
0.504120 + 0.863634i \(0.331817\pi\)
\(282\) 0 0
\(283\) −13.0611 −0.776400 −0.388200 0.921575i \(-0.626903\pi\)
−0.388200 + 0.921575i \(0.626903\pi\)
\(284\) −10.6057 −0.629332
\(285\) 0 0
\(286\) 0 0
\(287\) −12.4700 −0.736082
\(288\) 0 0
\(289\) −15.1861 −0.893298
\(290\) 5.69232 0.334264
\(291\) 0 0
\(292\) −2.92212 −0.171004
\(293\) 12.7793 0.746577 0.373288 0.927715i \(-0.378230\pi\)
0.373288 + 0.927715i \(0.378230\pi\)
\(294\) 0 0
\(295\) 5.42967 0.316128
\(296\) −6.34656 −0.368886
\(297\) 0 0
\(298\) −1.93634 −0.112169
\(299\) 12.8172 0.741238
\(300\) 0 0
\(301\) 5.60495 0.323064
\(302\) −11.3849 −0.655126
\(303\) 0 0
\(304\) −0.0154374 −0.000885395 0
\(305\) 3.92190 0.224567
\(306\) 0 0
\(307\) −27.9531 −1.59537 −0.797685 0.603074i \(-0.793942\pi\)
−0.797685 + 0.603074i \(0.793942\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.858579 −0.0487640
\(311\) −2.60259 −0.147579 −0.0737896 0.997274i \(-0.523509\pi\)
−0.0737896 + 0.997274i \(0.523509\pi\)
\(312\) 0 0
\(313\) 13.3114 0.752405 0.376202 0.926537i \(-0.377230\pi\)
0.376202 + 0.926537i \(0.377230\pi\)
\(314\) −14.6068 −0.824308
\(315\) 0 0
\(316\) −4.97297 −0.279751
\(317\) 28.1411 1.58056 0.790282 0.612743i \(-0.209934\pi\)
0.790282 + 0.612743i \(0.209934\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.45857 −0.193340
\(321\) 0 0
\(322\) −3.21224 −0.179011
\(323\) −0.677652 −0.0377056
\(324\) 0 0
\(325\) −15.6096 −0.865867
\(326\) −4.09790 −0.226962
\(327\) 0 0
\(328\) −35.1864 −1.94284
\(329\) 5.79140 0.319290
\(330\) 0 0
\(331\) 26.1522 1.43746 0.718728 0.695291i \(-0.244725\pi\)
0.718728 + 0.695291i \(0.244725\pi\)
\(332\) −18.2615 −1.00223
\(333\) 0 0
\(334\) −3.12701 −0.171102
\(335\) −2.31378 −0.126415
\(336\) 0 0
\(337\) −18.8676 −1.02779 −0.513893 0.857854i \(-0.671797\pi\)
−0.513893 + 0.857854i \(0.671797\pi\)
\(338\) −0.823522 −0.0447937
\(339\) 0 0
\(340\) 1.18830 0.0644447
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 15.8154 0.852707
\(345\) 0 0
\(346\) −8.06211 −0.433422
\(347\) 11.8666 0.637033 0.318517 0.947917i \(-0.396815\pi\)
0.318517 + 0.947917i \(0.396815\pi\)
\(348\) 0 0
\(349\) −25.7072 −1.37607 −0.688037 0.725675i \(-0.741527\pi\)
−0.688037 + 0.725675i \(0.741527\pi\)
\(350\) 3.91207 0.209109
\(351\) 0 0
\(352\) 0 0
\(353\) 25.4391 1.35399 0.676993 0.735990i \(-0.263283\pi\)
0.676993 + 0.735990i \(0.263283\pi\)
\(354\) 0 0
\(355\) −6.05718 −0.321482
\(356\) 1.77574 0.0941140
\(357\) 0 0
\(358\) −5.44680 −0.287872
\(359\) −16.9602 −0.895126 −0.447563 0.894252i \(-0.647708\pi\)
−0.447563 + 0.894252i \(0.647708\pi\)
\(360\) 0 0
\(361\) −18.7468 −0.986676
\(362\) 14.0493 0.738415
\(363\) 0 0
\(364\) 4.31519 0.226177
\(365\) −1.66890 −0.0873541
\(366\) 0 0
\(367\) 11.6589 0.608587 0.304294 0.952578i \(-0.401580\pi\)
0.304294 + 0.952578i \(0.401580\pi\)
\(368\) 0.113270 0.00590459
\(369\) 0 0
\(370\) −1.38924 −0.0722230
\(371\) −3.60444 −0.187133
\(372\) 0 0
\(373\) 16.1414 0.835771 0.417886 0.908500i \(-0.362771\pi\)
0.417886 + 0.908500i \(0.362771\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 16.3415 0.842747
\(377\) −31.9964 −1.64790
\(378\) 0 0
\(379\) 3.64587 0.187276 0.0936380 0.995606i \(-0.470150\pi\)
0.0936380 + 0.995606i \(0.470150\pi\)
\(380\) −0.443928 −0.0227730
\(381\) 0 0
\(382\) −16.1328 −0.825424
\(383\) 27.9141 1.42634 0.713172 0.700989i \(-0.247258\pi\)
0.713172 + 0.700989i \(0.247258\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.16959 −0.364922
\(387\) 0 0
\(388\) 10.6418 0.540255
\(389\) 32.3483 1.64013 0.820063 0.572274i \(-0.193939\pi\)
0.820063 + 0.572274i \(0.193939\pi\)
\(390\) 0 0
\(391\) 4.97217 0.251454
\(392\) −2.82168 −0.142516
\(393\) 0 0
\(394\) 3.13973 0.158177
\(395\) −2.84019 −0.142905
\(396\) 0 0
\(397\) 22.4932 1.12890 0.564451 0.825467i \(-0.309088\pi\)
0.564451 + 0.825467i \(0.309088\pi\)
\(398\) 6.74291 0.337992
\(399\) 0 0
\(400\) −0.137947 −0.00689737
\(401\) 13.2813 0.663236 0.331618 0.943414i \(-0.392406\pi\)
0.331618 + 0.943414i \(0.392406\pi\)
\(402\) 0 0
\(403\) 4.82606 0.240403
\(404\) −4.93966 −0.245757
\(405\) 0 0
\(406\) 8.01891 0.397972
\(407\) 0 0
\(408\) 0 0
\(409\) −6.75052 −0.333792 −0.166896 0.985975i \(-0.553374\pi\)
−0.166896 + 0.985975i \(0.553374\pi\)
\(410\) −7.70216 −0.380382
\(411\) 0 0
\(412\) −15.2542 −0.751522
\(413\) 7.64892 0.376379
\(414\) 0 0
\(415\) −10.4296 −0.511968
\(416\) 19.6854 0.965156
\(417\) 0 0
\(418\) 0 0
\(419\) 25.5259 1.24702 0.623511 0.781815i \(-0.285706\pi\)
0.623511 + 0.781815i \(0.285706\pi\)
\(420\) 0 0
\(421\) −14.5994 −0.711529 −0.355765 0.934576i \(-0.615780\pi\)
−0.355765 + 0.934576i \(0.615780\pi\)
\(422\) 0.113626 0.00553124
\(423\) 0 0
\(424\) −10.1706 −0.493926
\(425\) −6.05544 −0.293732
\(426\) 0 0
\(427\) 5.52488 0.267368
\(428\) 14.9619 0.723210
\(429\) 0 0
\(430\) 3.46192 0.166949
\(431\) −3.80384 −0.183225 −0.0916123 0.995795i \(-0.529202\pi\)
−0.0916123 + 0.995795i \(0.529202\pi\)
\(432\) 0 0
\(433\) −2.18094 −0.104809 −0.0524046 0.998626i \(-0.516689\pi\)
−0.0524046 + 0.998626i \(0.516689\pi\)
\(434\) −1.20950 −0.0580579
\(435\) 0 0
\(436\) 8.93148 0.427740
\(437\) −1.85752 −0.0888570
\(438\) 0 0
\(439\) −34.6200 −1.65232 −0.826162 0.563433i \(-0.809480\pi\)
−0.826162 + 0.563433i \(0.809480\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 4.06854 0.193521
\(443\) 30.4185 1.44523 0.722614 0.691252i \(-0.242940\pi\)
0.722614 + 0.691252i \(0.242940\pi\)
\(444\) 0 0
\(445\) 1.01417 0.0480762
\(446\) −10.7146 −0.507350
\(447\) 0 0
\(448\) −4.87217 −0.230189
\(449\) −5.65261 −0.266763 −0.133382 0.991065i \(-0.542584\pi\)
−0.133382 + 0.991065i \(0.542584\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 10.7876 0.507407
\(453\) 0 0
\(454\) 12.1490 0.570179
\(455\) 2.46451 0.115538
\(456\) 0 0
\(457\) −19.7755 −0.925060 −0.462530 0.886604i \(-0.653058\pi\)
−0.462530 + 0.886604i \(0.653058\pi\)
\(458\) −17.3515 −0.810782
\(459\) 0 0
\(460\) 3.25726 0.151870
\(461\) −0.336395 −0.0156675 −0.00783374 0.999969i \(-0.502494\pi\)
−0.00783374 + 0.999969i \(0.502494\pi\)
\(462\) 0 0
\(463\) 21.0213 0.976941 0.488471 0.872580i \(-0.337555\pi\)
0.488471 + 0.872580i \(0.337555\pi\)
\(464\) −0.282762 −0.0131269
\(465\) 0 0
\(466\) −3.98955 −0.184812
\(467\) −11.5498 −0.534462 −0.267231 0.963633i \(-0.586109\pi\)
−0.267231 + 0.963633i \(0.586109\pi\)
\(468\) 0 0
\(469\) −3.25948 −0.150509
\(470\) 3.57708 0.164998
\(471\) 0 0
\(472\) 21.5828 0.993428
\(473\) 0 0
\(474\) 0 0
\(475\) 2.26220 0.103797
\(476\) 1.67399 0.0767272
\(477\) 0 0
\(478\) 2.14738 0.0982190
\(479\) −33.3819 −1.52526 −0.762628 0.646837i \(-0.776091\pi\)
−0.762628 + 0.646837i \(0.776091\pi\)
\(480\) 0 0
\(481\) 7.80887 0.356054
\(482\) −18.4049 −0.838319
\(483\) 0 0
\(484\) 0 0
\(485\) 6.07779 0.275978
\(486\) 0 0
\(487\) 2.53639 0.114935 0.0574673 0.998347i \(-0.481697\pi\)
0.0574673 + 0.998347i \(0.481697\pi\)
\(488\) 15.5894 0.705701
\(489\) 0 0
\(490\) −0.617654 −0.0279028
\(491\) −4.17150 −0.188257 −0.0941286 0.995560i \(-0.530006\pi\)
−0.0941286 + 0.995560i \(0.530006\pi\)
\(492\) 0 0
\(493\) −12.4124 −0.559024
\(494\) −1.51993 −0.0683851
\(495\) 0 0
\(496\) 0.0426494 0.00191501
\(497\) −8.53290 −0.382753
\(498\) 0 0
\(499\) 37.2831 1.66902 0.834509 0.550994i \(-0.185751\pi\)
0.834509 + 0.550994i \(0.185751\pi\)
\(500\) −8.37840 −0.374694
\(501\) 0 0
\(502\) −16.3688 −0.730576
\(503\) 9.14838 0.407906 0.203953 0.978981i \(-0.434621\pi\)
0.203953 + 0.978981i \(0.434621\pi\)
\(504\) 0 0
\(505\) −2.82117 −0.125540
\(506\) 0 0
\(507\) 0 0
\(508\) 26.8398 1.19083
\(509\) 21.1205 0.936151 0.468075 0.883689i \(-0.344948\pi\)
0.468075 + 0.883689i \(0.344948\pi\)
\(510\) 0 0
\(511\) −2.35102 −0.104003
\(512\) 0.347113 0.0153404
\(513\) 0 0
\(514\) −25.3964 −1.12019
\(515\) −8.71208 −0.383900
\(516\) 0 0
\(517\) 0 0
\(518\) −1.95705 −0.0859879
\(519\) 0 0
\(520\) 6.95406 0.304956
\(521\) −20.2060 −0.885239 −0.442619 0.896710i \(-0.645951\pi\)
−0.442619 + 0.896710i \(0.645951\pi\)
\(522\) 0 0
\(523\) −11.0877 −0.484830 −0.242415 0.970173i \(-0.577940\pi\)
−0.242415 + 0.970173i \(0.577940\pi\)
\(524\) 11.1015 0.484969
\(525\) 0 0
\(526\) −14.4075 −0.628196
\(527\) 1.87217 0.0815530
\(528\) 0 0
\(529\) −9.37074 −0.407424
\(530\) −2.22630 −0.0967041
\(531\) 0 0
\(532\) −0.625372 −0.0271133
\(533\) 43.2937 1.87526
\(534\) 0 0
\(535\) 8.54511 0.369437
\(536\) −9.19719 −0.397258
\(537\) 0 0
\(538\) −19.3821 −0.835622
\(539\) 0 0
\(540\) 0 0
\(541\) −25.8592 −1.11177 −0.555887 0.831258i \(-0.687621\pi\)
−0.555887 + 0.831258i \(0.687621\pi\)
\(542\) −15.9429 −0.684806
\(543\) 0 0
\(544\) 7.63655 0.327414
\(545\) 5.10099 0.218502
\(546\) 0 0
\(547\) 10.9691 0.469006 0.234503 0.972115i \(-0.424654\pi\)
0.234503 + 0.972115i \(0.424654\pi\)
\(548\) 10.8172 0.462089
\(549\) 0 0
\(550\) 0 0
\(551\) 4.63703 0.197544
\(552\) 0 0
\(553\) −4.00104 −0.170142
\(554\) −16.7727 −0.712603
\(555\) 0 0
\(556\) −10.6177 −0.450291
\(557\) 18.0302 0.763964 0.381982 0.924170i \(-0.375242\pi\)
0.381982 + 0.924170i \(0.375242\pi\)
\(558\) 0 0
\(559\) −19.4594 −0.823044
\(560\) 0.0217797 0.000920359 0
\(561\) 0 0
\(562\) 14.7058 0.620326
\(563\) −21.5405 −0.907822 −0.453911 0.891047i \(-0.649972\pi\)
−0.453911 + 0.891047i \(0.649972\pi\)
\(564\) 0 0
\(565\) 6.16108 0.259198
\(566\) −11.3645 −0.477685
\(567\) 0 0
\(568\) −24.0771 −1.01025
\(569\) 21.8568 0.916286 0.458143 0.888879i \(-0.348515\pi\)
0.458143 + 0.888879i \(0.348515\pi\)
\(570\) 0 0
\(571\) −5.01476 −0.209861 −0.104930 0.994480i \(-0.533462\pi\)
−0.104930 + 0.994480i \(0.533462\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −10.8502 −0.452880
\(575\) −16.5986 −0.692210
\(576\) 0 0
\(577\) 11.8613 0.493793 0.246897 0.969042i \(-0.420589\pi\)
0.246897 + 0.969042i \(0.420589\pi\)
\(578\) −13.2135 −0.549608
\(579\) 0 0
\(580\) −8.13130 −0.337634
\(581\) −14.6924 −0.609545
\(582\) 0 0
\(583\) 0 0
\(584\) −6.63381 −0.274509
\(585\) 0 0
\(586\) 11.1194 0.459336
\(587\) 30.1344 1.24378 0.621890 0.783104i \(-0.286365\pi\)
0.621890 + 0.783104i \(0.286365\pi\)
\(588\) 0 0
\(589\) −0.699409 −0.0288187
\(590\) 4.72438 0.194500
\(591\) 0 0
\(592\) 0.0690095 0.00283627
\(593\) −0.154283 −0.00633564 −0.00316782 0.999995i \(-0.501008\pi\)
−0.00316782 + 0.999995i \(0.501008\pi\)
\(594\) 0 0
\(595\) 0.956057 0.0391945
\(596\) 2.76600 0.113300
\(597\) 0 0
\(598\) 11.1523 0.456052
\(599\) 33.8158 1.38168 0.690838 0.723010i \(-0.257242\pi\)
0.690838 + 0.723010i \(0.257242\pi\)
\(600\) 0 0
\(601\) 47.2993 1.92938 0.964688 0.263394i \(-0.0848418\pi\)
0.964688 + 0.263394i \(0.0848418\pi\)
\(602\) 4.87689 0.198767
\(603\) 0 0
\(604\) 16.2629 0.661730
\(605\) 0 0
\(606\) 0 0
\(607\) −9.76035 −0.396160 −0.198080 0.980186i \(-0.563471\pi\)
−0.198080 + 0.980186i \(0.563471\pi\)
\(608\) −2.85288 −0.115700
\(609\) 0 0
\(610\) 3.41247 0.138167
\(611\) −20.1067 −0.813430
\(612\) 0 0
\(613\) −41.4957 −1.67600 −0.837998 0.545673i \(-0.816274\pi\)
−0.837998 + 0.545673i \(0.816274\pi\)
\(614\) −24.3222 −0.981562
\(615\) 0 0
\(616\) 0 0
\(617\) −23.2181 −0.934725 −0.467362 0.884066i \(-0.654796\pi\)
−0.467362 + 0.884066i \(0.654796\pi\)
\(618\) 0 0
\(619\) −20.3371 −0.817416 −0.408708 0.912665i \(-0.634021\pi\)
−0.408708 + 0.912665i \(0.634021\pi\)
\(620\) 1.22645 0.0492556
\(621\) 0 0
\(622\) −2.26452 −0.0907991
\(623\) 1.42869 0.0572391
\(624\) 0 0
\(625\) 17.6954 0.707815
\(626\) 11.5823 0.462922
\(627\) 0 0
\(628\) 20.8653 0.832617
\(629\) 3.02929 0.120786
\(630\) 0 0
\(631\) 33.1617 1.32015 0.660073 0.751202i \(-0.270525\pi\)
0.660073 + 0.751202i \(0.270525\pi\)
\(632\) −11.2896 −0.449078
\(633\) 0 0
\(634\) 24.4857 0.972453
\(635\) 15.3289 0.608309
\(636\) 0 0
\(637\) 3.47182 0.137559
\(638\) 0 0
\(639\) 0 0
\(640\) 5.04059 0.199247
\(641\) 24.7317 0.976843 0.488422 0.872608i \(-0.337573\pi\)
0.488422 + 0.872608i \(0.337573\pi\)
\(642\) 0 0
\(643\) 5.06615 0.199789 0.0998947 0.994998i \(-0.468149\pi\)
0.0998947 + 0.994998i \(0.468149\pi\)
\(644\) 4.58858 0.180816
\(645\) 0 0
\(646\) −0.589628 −0.0231986
\(647\) 35.4931 1.39538 0.697690 0.716400i \(-0.254211\pi\)
0.697690 + 0.716400i \(0.254211\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −13.5820 −0.532731
\(651\) 0 0
\(652\) 5.85373 0.229250
\(653\) 9.63770 0.377152 0.188576 0.982059i \(-0.439613\pi\)
0.188576 + 0.982059i \(0.439613\pi\)
\(654\) 0 0
\(655\) 6.34032 0.247737
\(656\) 0.382600 0.0149380
\(657\) 0 0
\(658\) 5.03913 0.196446
\(659\) 15.4794 0.602992 0.301496 0.953467i \(-0.402514\pi\)
0.301496 + 0.953467i \(0.402514\pi\)
\(660\) 0 0
\(661\) −21.2397 −0.826128 −0.413064 0.910702i \(-0.635541\pi\)
−0.413064 + 0.910702i \(0.635541\pi\)
\(662\) 22.7552 0.884405
\(663\) 0 0
\(664\) −41.4573 −1.60886
\(665\) −0.357166 −0.0138503
\(666\) 0 0
\(667\) −34.0236 −1.31740
\(668\) 4.46684 0.172827
\(669\) 0 0
\(670\) −2.01323 −0.0777778
\(671\) 0 0
\(672\) 0 0
\(673\) 2.16841 0.0835861 0.0417931 0.999126i \(-0.486693\pi\)
0.0417931 + 0.999126i \(0.486693\pi\)
\(674\) −16.4168 −0.632352
\(675\) 0 0
\(676\) 1.17638 0.0452453
\(677\) −1.79264 −0.0688966 −0.0344483 0.999406i \(-0.510967\pi\)
−0.0344483 + 0.999406i \(0.510967\pi\)
\(678\) 0 0
\(679\) 8.56194 0.328577
\(680\) 2.69769 0.103452
\(681\) 0 0
\(682\) 0 0
\(683\) −17.8000 −0.681099 −0.340549 0.940227i \(-0.610613\pi\)
−0.340549 + 0.940227i \(0.610613\pi\)
\(684\) 0 0
\(685\) 6.17799 0.236049
\(686\) −0.870105 −0.0332207
\(687\) 0 0
\(688\) −0.171969 −0.00655625
\(689\) 12.5140 0.476744
\(690\) 0 0
\(691\) 31.8957 1.21337 0.606686 0.794942i \(-0.292499\pi\)
0.606686 + 0.794942i \(0.292499\pi\)
\(692\) 11.5165 0.437791
\(693\) 0 0
\(694\) 10.3252 0.391939
\(695\) −6.06404 −0.230022
\(696\) 0 0
\(697\) 16.7949 0.636152
\(698\) −22.3679 −0.846639
\(699\) 0 0
\(700\) −5.58828 −0.211217
\(701\) −34.6298 −1.30795 −0.653975 0.756516i \(-0.726900\pi\)
−0.653975 + 0.756516i \(0.726900\pi\)
\(702\) 0 0
\(703\) −1.13169 −0.0426825
\(704\) 0 0
\(705\) 0 0
\(706\) 22.1347 0.833049
\(707\) −3.97425 −0.149467
\(708\) 0 0
\(709\) −24.5584 −0.922310 −0.461155 0.887320i \(-0.652565\pi\)
−0.461155 + 0.887320i \(0.652565\pi\)
\(710\) −5.27038 −0.197794
\(711\) 0 0
\(712\) 4.03129 0.151079
\(713\) 5.13182 0.192188
\(714\) 0 0
\(715\) 0 0
\(716\) 7.78059 0.290774
\(717\) 0 0
\(718\) −14.7572 −0.550732
\(719\) 30.8151 1.14921 0.574604 0.818431i \(-0.305156\pi\)
0.574604 + 0.818431i \(0.305156\pi\)
\(720\) 0 0
\(721\) −12.2729 −0.457068
\(722\) −16.3117 −0.607059
\(723\) 0 0
\(724\) −20.0690 −0.745858
\(725\) 41.4362 1.53890
\(726\) 0 0
\(727\) −25.5902 −0.949090 −0.474545 0.880231i \(-0.657387\pi\)
−0.474545 + 0.880231i \(0.657387\pi\)
\(728\) 9.79636 0.363077
\(729\) 0 0
\(730\) −1.45212 −0.0537452
\(731\) −7.54887 −0.279205
\(732\) 0 0
\(733\) −25.5577 −0.943995 −0.471998 0.881600i \(-0.656467\pi\)
−0.471998 + 0.881600i \(0.656467\pi\)
\(734\) 10.1444 0.374437
\(735\) 0 0
\(736\) 20.9326 0.771586
\(737\) 0 0
\(738\) 0 0
\(739\) −22.5651 −0.830072 −0.415036 0.909805i \(-0.636231\pi\)
−0.415036 + 0.909805i \(0.636231\pi\)
\(740\) 1.98448 0.0729510
\(741\) 0 0
\(742\) −3.13624 −0.115135
\(743\) −5.20368 −0.190905 −0.0954523 0.995434i \(-0.530430\pi\)
−0.0954523 + 0.995434i \(0.530430\pi\)
\(744\) 0 0
\(745\) 1.57973 0.0578769
\(746\) 14.0447 0.514214
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0377 0.439848
\(750\) 0 0
\(751\) −43.3856 −1.58316 −0.791581 0.611065i \(-0.790742\pi\)
−0.791581 + 0.611065i \(0.790742\pi\)
\(752\) −0.177689 −0.00647966
\(753\) 0 0
\(754\) −27.8402 −1.01388
\(755\) 9.28818 0.338031
\(756\) 0 0
\(757\) 16.2472 0.590513 0.295256 0.955418i \(-0.404595\pi\)
0.295256 + 0.955418i \(0.404595\pi\)
\(758\) 3.17229 0.115223
\(759\) 0 0
\(760\) −1.00781 −0.0365570
\(761\) −27.7710 −1.00670 −0.503349 0.864083i \(-0.667899\pi\)
−0.503349 + 0.864083i \(0.667899\pi\)
\(762\) 0 0
\(763\) 7.18589 0.260147
\(764\) 23.0452 0.833745
\(765\) 0 0
\(766\) 24.2882 0.877568
\(767\) −26.5557 −0.958870
\(768\) 0 0
\(769\) −47.5109 −1.71329 −0.856643 0.515910i \(-0.827454\pi\)
−0.856643 + 0.515910i \(0.827454\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.2415 0.368601
\(773\) 53.7721 1.93405 0.967023 0.254689i \(-0.0819730\pi\)
0.967023 + 0.254689i \(0.0819730\pi\)
\(774\) 0 0
\(775\) −6.24987 −0.224502
\(776\) 24.1590 0.867259
\(777\) 0 0
\(778\) 28.1464 1.00910
\(779\) −6.27427 −0.224799
\(780\) 0 0
\(781\) 0 0
\(782\) 4.32631 0.154709
\(783\) 0 0
\(784\) 0.0306816 0.00109577
\(785\) 11.9167 0.425326
\(786\) 0 0
\(787\) 47.9658 1.70980 0.854898 0.518796i \(-0.173619\pi\)
0.854898 + 0.518796i \(0.173619\pi\)
\(788\) −4.48501 −0.159772
\(789\) 0 0
\(790\) −2.47126 −0.0879234
\(791\) 8.67927 0.308599
\(792\) 0 0
\(793\) −19.1814 −0.681152
\(794\) 19.5714 0.694564
\(795\) 0 0
\(796\) −9.63204 −0.341399
\(797\) 21.2494 0.752691 0.376346 0.926479i \(-0.377181\pi\)
0.376346 + 0.926479i \(0.377181\pi\)
\(798\) 0 0
\(799\) −7.79999 −0.275944
\(800\) −25.4931 −0.901317
\(801\) 0 0
\(802\) 11.5561 0.408060
\(803\) 0 0
\(804\) 0 0
\(805\) 2.62065 0.0923659
\(806\) 4.19917 0.147910
\(807\) 0 0
\(808\) −11.2140 −0.394509
\(809\) 35.7295 1.25618 0.628092 0.778139i \(-0.283836\pi\)
0.628092 + 0.778139i \(0.283836\pi\)
\(810\) 0 0
\(811\) −12.0383 −0.422721 −0.211360 0.977408i \(-0.567789\pi\)
−0.211360 + 0.977408i \(0.567789\pi\)
\(812\) −11.4548 −0.401984
\(813\) 0 0
\(814\) 0 0
\(815\) 3.34321 0.117108
\(816\) 0 0
\(817\) 2.82012 0.0986636
\(818\) −5.87366 −0.205368
\(819\) 0 0
\(820\) 11.0023 0.384217
\(821\) 47.0391 1.64168 0.820838 0.571162i \(-0.193507\pi\)
0.820838 + 0.571162i \(0.193507\pi\)
\(822\) 0 0
\(823\) −37.9888 −1.32421 −0.662103 0.749413i \(-0.730336\pi\)
−0.662103 + 0.749413i \(0.730336\pi\)
\(824\) −34.6302 −1.20640
\(825\) 0 0
\(826\) 6.65536 0.231570
\(827\) 1.05716 0.0367611 0.0183805 0.999831i \(-0.494149\pi\)
0.0183805 + 0.999831i \(0.494149\pi\)
\(828\) 0 0
\(829\) −35.4048 −1.22966 −0.614831 0.788659i \(-0.710776\pi\)
−0.614831 + 0.788659i \(0.710776\pi\)
\(830\) −9.07483 −0.314992
\(831\) 0 0
\(832\) 16.9153 0.586433
\(833\) 1.34682 0.0466646
\(834\) 0 0
\(835\) 2.55112 0.0882853
\(836\) 0 0
\(837\) 0 0
\(838\) 22.2102 0.767238
\(839\) 41.3140 1.42632 0.713159 0.701002i \(-0.247264\pi\)
0.713159 + 0.701002i \(0.247264\pi\)
\(840\) 0 0
\(841\) 55.9353 1.92880
\(842\) −12.7030 −0.437773
\(843\) 0 0
\(844\) −0.162312 −0.00558700
\(845\) 0.671858 0.0231126
\(846\) 0 0
\(847\) 0 0
\(848\) 0.110590 0.00379767
\(849\) 0 0
\(850\) −5.26887 −0.180721
\(851\) 8.30361 0.284644
\(852\) 0 0
\(853\) 0.244685 0.00837784 0.00418892 0.999991i \(-0.498667\pi\)
0.00418892 + 0.999991i \(0.498667\pi\)
\(854\) 4.80723 0.164500
\(855\) 0 0
\(856\) 33.9665 1.16095
\(857\) −50.2741 −1.71733 −0.858666 0.512536i \(-0.828706\pi\)
−0.858666 + 0.512536i \(0.828706\pi\)
\(858\) 0 0
\(859\) 13.3223 0.454552 0.227276 0.973830i \(-0.427018\pi\)
0.227276 + 0.973830i \(0.427018\pi\)
\(860\) −4.94525 −0.168632
\(861\) 0 0
\(862\) −3.30974 −0.112730
\(863\) −29.5005 −1.00421 −0.502105 0.864807i \(-0.667441\pi\)
−0.502105 + 0.864807i \(0.667441\pi\)
\(864\) 0 0
\(865\) 6.57735 0.223637
\(866\) −1.89764 −0.0644845
\(867\) 0 0
\(868\) 1.72774 0.0586432
\(869\) 0 0
\(870\) 0 0
\(871\) 11.3163 0.383439
\(872\) 20.2763 0.686641
\(873\) 0 0
\(874\) −1.61623 −0.0546699
\(875\) −6.74091 −0.227884
\(876\) 0 0
\(877\) 22.6648 0.765335 0.382667 0.923886i \(-0.375006\pi\)
0.382667 + 0.923886i \(0.375006\pi\)
\(878\) −30.1230 −1.01660
\(879\) 0 0
\(880\) 0 0
\(881\) −45.1422 −1.52088 −0.760440 0.649409i \(-0.775016\pi\)
−0.760440 + 0.649409i \(0.775016\pi\)
\(882\) 0 0
\(883\) −15.7815 −0.531089 −0.265545 0.964099i \(-0.585552\pi\)
−0.265545 + 0.964099i \(0.585552\pi\)
\(884\) −5.81179 −0.195472
\(885\) 0 0
\(886\) 26.4673 0.889186
\(887\) 37.8474 1.27079 0.635396 0.772187i \(-0.280837\pi\)
0.635396 + 0.772187i \(0.280837\pi\)
\(888\) 0 0
\(889\) 21.5942 0.724247
\(890\) 0.882433 0.0295792
\(891\) 0 0
\(892\) 15.3055 0.512465
\(893\) 2.91394 0.0975112
\(894\) 0 0
\(895\) 4.44369 0.148536
\(896\) 7.10080 0.237221
\(897\) 0 0
\(898\) −4.91836 −0.164128
\(899\) −12.8109 −0.427267
\(900\) 0 0
\(901\) 4.85454 0.161728
\(902\) 0 0
\(903\) 0 0
\(904\) 24.4901 0.814528
\(905\) −11.4619 −0.381007
\(906\) 0 0
\(907\) −15.1259 −0.502246 −0.251123 0.967955i \(-0.580800\pi\)
−0.251123 + 0.967955i \(0.580800\pi\)
\(908\) −17.3544 −0.575927
\(909\) 0 0
\(910\) 2.14438 0.0710856
\(911\) −51.5380 −1.70753 −0.853765 0.520658i \(-0.825687\pi\)
−0.853765 + 0.520658i \(0.825687\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.2068 −0.569149
\(915\) 0 0
\(916\) 24.7861 0.818956
\(917\) 8.93177 0.294953
\(918\) 0 0
\(919\) −12.1376 −0.400384 −0.200192 0.979757i \(-0.564157\pi\)
−0.200192 + 0.979757i \(0.564157\pi\)
\(920\) 7.39464 0.243794
\(921\) 0 0
\(922\) −0.292699 −0.00963953
\(923\) 29.6247 0.975109
\(924\) 0 0
\(925\) −10.1127 −0.332503
\(926\) 18.2907 0.601070
\(927\) 0 0
\(928\) −52.2554 −1.71537
\(929\) 39.7034 1.30263 0.651314 0.758808i \(-0.274218\pi\)
0.651314 + 0.758808i \(0.274218\pi\)
\(930\) 0 0
\(931\) −0.503149 −0.0164900
\(932\) 5.69895 0.186675
\(933\) 0 0
\(934\) −10.0495 −0.328831
\(935\) 0 0
\(936\) 0 0
\(937\) −10.5516 −0.344705 −0.172353 0.985035i \(-0.555137\pi\)
−0.172353 + 0.985035i \(0.555137\pi\)
\(938\) −2.83608 −0.0926014
\(939\) 0 0
\(940\) −5.10975 −0.166662
\(941\) −27.3882 −0.892829 −0.446415 0.894826i \(-0.647299\pi\)
−0.446415 + 0.894826i \(0.647299\pi\)
\(942\) 0 0
\(943\) 46.0366 1.49916
\(944\) −0.234681 −0.00763821
\(945\) 0 0
\(946\) 0 0
\(947\) 13.1492 0.427292 0.213646 0.976911i \(-0.431466\pi\)
0.213646 + 0.976911i \(0.431466\pi\)
\(948\) 0 0
\(949\) 8.16231 0.264960
\(950\) 1.96835 0.0638619
\(951\) 0 0
\(952\) 3.80030 0.123168
\(953\) 30.3710 0.983815 0.491907 0.870648i \(-0.336300\pi\)
0.491907 + 0.870648i \(0.336300\pi\)
\(954\) 0 0
\(955\) 13.1617 0.425902
\(956\) −3.06747 −0.0992091
\(957\) 0 0
\(958\) −29.0457 −0.938425
\(959\) 8.70309 0.281038
\(960\) 0 0
\(961\) −29.0677 −0.937668
\(962\) 6.79453 0.219065
\(963\) 0 0
\(964\) 26.2908 0.846770
\(965\) 5.84920 0.188292
\(966\) 0 0
\(967\) 49.6355 1.59617 0.798085 0.602545i \(-0.205847\pi\)
0.798085 + 0.602545i \(0.205847\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 5.28832 0.169798
\(971\) −48.5870 −1.55923 −0.779615 0.626259i \(-0.784585\pi\)
−0.779615 + 0.626259i \(0.784585\pi\)
\(972\) 0 0
\(973\) −8.54256 −0.273862
\(974\) 2.20692 0.0707143
\(975\) 0 0
\(976\) −0.169512 −0.00542595
\(977\) −43.9332 −1.40555 −0.702773 0.711414i \(-0.748055\pi\)
−0.702773 + 0.711414i \(0.748055\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.882300 0.0281840
\(981\) 0 0
\(982\) −3.62964 −0.115827
\(983\) −33.4481 −1.06683 −0.533415 0.845854i \(-0.679092\pi\)
−0.533415 + 0.845854i \(0.679092\pi\)
\(984\) 0 0
\(985\) −2.56150 −0.0816162
\(986\) −10.8000 −0.343944
\(987\) 0 0
\(988\) 2.17118 0.0690745
\(989\) −20.6923 −0.657976
\(990\) 0 0
\(991\) −42.7034 −1.35652 −0.678260 0.734822i \(-0.737266\pi\)
−0.678260 + 0.734822i \(0.737266\pi\)
\(992\) 7.88174 0.250245
\(993\) 0 0
\(994\) −7.42452 −0.235491
\(995\) −5.50110 −0.174397
\(996\) 0 0
\(997\) −17.0468 −0.539879 −0.269939 0.962877i \(-0.587004\pi\)
−0.269939 + 0.962877i \(0.587004\pi\)
\(998\) 32.4402 1.02688
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.cz.1.8 yes 12
3.2 odd 2 inner 7623.2.a.cz.1.5 12
11.10 odd 2 7623.2.a.da.1.5 yes 12
33.32 even 2 7623.2.a.da.1.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cz.1.5 12 3.2 odd 2 inner
7623.2.a.cz.1.8 yes 12 1.1 even 1 trivial
7623.2.a.da.1.5 yes 12 11.10 odd 2
7623.2.a.da.1.8 yes 12 33.32 even 2