Properties

Label 9702.2.a.dz.1.4
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.27133\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.79793 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.79793 q^{5} -1.00000 q^{8} -1.79793 q^{10} -1.00000 q^{11} +3.63899 q^{13} +1.00000 q^{16} -6.11582 q^{17} +1.84106 q^{19} +1.79793 q^{20} +1.00000 q^{22} +7.37109 q^{23} -1.76744 q^{25} -3.63899 q^{26} +10.4243 q^{29} +7.97958 q^{31} -1.00000 q^{32} +6.11582 q^{34} +10.1995 q^{37} -1.84106 q^{38} -1.79793 q^{40} -8.17680 q^{41} -2.28577 q^{43} -1.00000 q^{44} -7.37109 q^{46} +0.669485 q^{47} +1.76744 q^{50} +3.63899 q^{52} -11.2848 q^{53} -1.79793 q^{55} -10.4243 q^{58} +13.5706 q^{59} -0.274758 q^{61} -7.97958 q^{62} +1.00000 q^{64} +6.54266 q^{65} +3.88163 q^{67} -6.11582 q^{68} -10.8738 q^{71} +3.85892 q^{73} -10.1995 q^{74} +1.84106 q^{76} -12.5174 q^{79} +1.79793 q^{80} +8.17680 q^{82} -5.27314 q^{83} -10.9958 q^{85} +2.28577 q^{86} +1.00000 q^{88} -1.98214 q^{89} +7.37109 q^{92} -0.669485 q^{94} +3.31010 q^{95} +4.12066 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + O(q^{10}) \) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.79793 0.804060 0.402030 0.915627i \(-0.368305\pi\)
0.402030 + 0.915627i \(0.368305\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.79793 −0.568556
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.63899 1.00927 0.504637 0.863331i \(-0.331626\pi\)
0.504637 + 0.863331i \(0.331626\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.11582 −1.48330 −0.741652 0.670785i \(-0.765957\pi\)
−0.741652 + 0.670785i \(0.765957\pi\)
\(18\) 0 0
\(19\) 1.84106 0.422368 0.211184 0.977446i \(-0.432268\pi\)
0.211184 + 0.977446i \(0.432268\pi\)
\(20\) 1.79793 0.402030
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.37109 1.53698 0.768489 0.639863i \(-0.221009\pi\)
0.768489 + 0.639863i \(0.221009\pi\)
\(24\) 0 0
\(25\) −1.76744 −0.353488
\(26\) −3.63899 −0.713665
\(27\) 0 0
\(28\) 0 0
\(29\) 10.4243 1.93574 0.967871 0.251446i \(-0.0809062\pi\)
0.967871 + 0.251446i \(0.0809062\pi\)
\(30\) 0 0
\(31\) 7.97958 1.43318 0.716588 0.697497i \(-0.245703\pi\)
0.716588 + 0.697497i \(0.245703\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.11582 1.04885
\(35\) 0 0
\(36\) 0 0
\(37\) 10.1995 1.67679 0.838395 0.545063i \(-0.183494\pi\)
0.838395 + 0.545063i \(0.183494\pi\)
\(38\) −1.84106 −0.298659
\(39\) 0 0
\(40\) −1.79793 −0.284278
\(41\) −8.17680 −1.27700 −0.638501 0.769621i \(-0.720445\pi\)
−0.638501 + 0.769621i \(0.720445\pi\)
\(42\) 0 0
\(43\) −2.28577 −0.348576 −0.174288 0.984695i \(-0.555762\pi\)
−0.174288 + 0.984695i \(0.555762\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −7.37109 −1.08681
\(47\) 0.669485 0.0976545 0.0488272 0.998807i \(-0.484452\pi\)
0.0488272 + 0.998807i \(0.484452\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.76744 0.249954
\(51\) 0 0
\(52\) 3.63899 0.504637
\(53\) −11.2848 −1.55009 −0.775046 0.631905i \(-0.782273\pi\)
−0.775046 + 0.631905i \(0.782273\pi\)
\(54\) 0 0
\(55\) −1.79793 −0.242433
\(56\) 0 0
\(57\) 0 0
\(58\) −10.4243 −1.36878
\(59\) 13.5706 1.76674 0.883371 0.468674i \(-0.155268\pi\)
0.883371 + 0.468674i \(0.155268\pi\)
\(60\) 0 0
\(61\) −0.274758 −0.0351791 −0.0175896 0.999845i \(-0.505599\pi\)
−0.0175896 + 0.999845i \(0.505599\pi\)
\(62\) −7.97958 −1.01341
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.54266 0.811517
\(66\) 0 0
\(67\) 3.88163 0.474217 0.237108 0.971483i \(-0.423800\pi\)
0.237108 + 0.971483i \(0.423800\pi\)
\(68\) −6.11582 −0.741652
\(69\) 0 0
\(70\) 0 0
\(71\) −10.8738 −1.29049 −0.645244 0.763976i \(-0.723244\pi\)
−0.645244 + 0.763976i \(0.723244\pi\)
\(72\) 0 0
\(73\) 3.85892 0.451653 0.225826 0.974168i \(-0.427492\pi\)
0.225826 + 0.974168i \(0.427492\pi\)
\(74\) −10.1995 −1.18567
\(75\) 0 0
\(76\) 1.84106 0.211184
\(77\) 0 0
\(78\) 0 0
\(79\) −12.5174 −1.40832 −0.704159 0.710043i \(-0.748676\pi\)
−0.704159 + 0.710043i \(0.748676\pi\)
\(80\) 1.79793 0.201015
\(81\) 0 0
\(82\) 8.17680 0.902977
\(83\) −5.27314 −0.578802 −0.289401 0.957208i \(-0.593456\pi\)
−0.289401 + 0.957208i \(0.593456\pi\)
\(84\) 0 0
\(85\) −10.9958 −1.19266
\(86\) 2.28577 0.246481
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −1.98214 −0.210106 −0.105053 0.994467i \(-0.533501\pi\)
−0.105053 + 0.994467i \(0.533501\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.37109 0.768489
\(93\) 0 0
\(94\) −0.669485 −0.0690522
\(95\) 3.31010 0.339609
\(96\) 0 0
\(97\) 4.12066 0.418390 0.209195 0.977874i \(-0.432916\pi\)
0.209195 + 0.977874i \(0.432916\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.76744 −0.176744
\(101\) −6.90367 −0.686941 −0.343470 0.939163i \(-0.611603\pi\)
−0.343470 + 0.939163i \(0.611603\pi\)
\(102\) 0 0
\(103\) 12.6154 1.24303 0.621514 0.783403i \(-0.286518\pi\)
0.621514 + 0.783403i \(0.286518\pi\)
\(104\) −3.63899 −0.356832
\(105\) 0 0
\(106\) 11.2848 1.09608
\(107\) −16.9417 −1.63782 −0.818908 0.573925i \(-0.805420\pi\)
−0.818908 + 0.573925i \(0.805420\pi\)
\(108\) 0 0
\(109\) 9.76326 0.935151 0.467576 0.883953i \(-0.345128\pi\)
0.467576 + 0.883953i \(0.345128\pi\)
\(110\) 1.79793 0.171426
\(111\) 0 0
\(112\) 0 0
\(113\) 12.9348 1.21681 0.608404 0.793628i \(-0.291810\pi\)
0.608404 + 0.793628i \(0.291810\pi\)
\(114\) 0 0
\(115\) 13.2527 1.23582
\(116\) 10.4243 0.967871
\(117\) 0 0
\(118\) −13.5706 −1.24928
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.274758 0.0248754
\(123\) 0 0
\(124\) 7.97958 0.716588
\(125\) −12.1674 −1.08829
\(126\) 0 0
\(127\) −7.43208 −0.659490 −0.329745 0.944070i \(-0.606963\pi\)
−0.329745 + 0.944070i \(0.606963\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.54266 −0.573829
\(131\) −1.64154 −0.143422 −0.0717112 0.997425i \(-0.522846\pi\)
−0.0717112 + 0.997425i \(0.522846\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3.88163 −0.335322
\(135\) 0 0
\(136\) 6.11582 0.524427
\(137\) −14.5307 −1.24144 −0.620721 0.784032i \(-0.713160\pi\)
−0.620721 + 0.784032i \(0.713160\pi\)
\(138\) 0 0
\(139\) −8.26535 −0.701058 −0.350529 0.936552i \(-0.613998\pi\)
−0.350529 + 0.936552i \(0.613998\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.8738 0.912513
\(143\) −3.63899 −0.304308
\(144\) 0 0
\(145\) 18.7422 1.55645
\(146\) −3.85892 −0.319367
\(147\) 0 0
\(148\) 10.1995 0.838395
\(149\) 7.55045 0.618557 0.309278 0.950972i \(-0.399913\pi\)
0.309278 + 0.950972i \(0.399913\pi\)
\(150\) 0 0
\(151\) 12.3990 1.00902 0.504509 0.863406i \(-0.331673\pi\)
0.504509 + 0.863406i \(0.331673\pi\)
\(152\) −1.84106 −0.149330
\(153\) 0 0
\(154\) 0 0
\(155\) 14.3468 1.15236
\(156\) 0 0
\(157\) 4.54011 0.362340 0.181170 0.983452i \(-0.442012\pi\)
0.181170 + 0.983452i \(0.442012\pi\)
\(158\) 12.5174 0.995831
\(159\) 0 0
\(160\) −1.79793 −0.142139
\(161\) 0 0
\(162\) 0 0
\(163\) 15.8275 1.23971 0.619853 0.784718i \(-0.287192\pi\)
0.619853 + 0.784718i \(0.287192\pi\)
\(164\) −8.17680 −0.638501
\(165\) 0 0
\(166\) 5.27314 0.409275
\(167\) 22.0844 1.70894 0.854471 0.519499i \(-0.173882\pi\)
0.854471 + 0.519499i \(0.173882\pi\)
\(168\) 0 0
\(169\) 0.242256 0.0186350
\(170\) 10.9958 0.843341
\(171\) 0 0
\(172\) −2.28577 −0.174288
\(173\) 10.2628 0.780266 0.390133 0.920758i \(-0.372429\pi\)
0.390133 + 0.920758i \(0.372429\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 1.98214 0.148567
\(179\) 12.3954 0.926477 0.463239 0.886234i \(-0.346687\pi\)
0.463239 + 0.886234i \(0.346687\pi\)
\(180\) 0 0
\(181\) −22.4506 −1.66874 −0.834370 0.551204i \(-0.814169\pi\)
−0.834370 + 0.551204i \(0.814169\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.37109 −0.543404
\(185\) 18.3380 1.34824
\(186\) 0 0
\(187\) 6.11582 0.447233
\(188\) 0.669485 0.0488272
\(189\) 0 0
\(190\) −3.31010 −0.240140
\(191\) 13.7142 0.992327 0.496164 0.868229i \(-0.334742\pi\)
0.496164 + 0.868229i \(0.334742\pi\)
\(192\) 0 0
\(193\) 3.77883 0.272006 0.136003 0.990708i \(-0.456574\pi\)
0.136003 + 0.990708i \(0.456574\pi\)
\(194\) −4.12066 −0.295846
\(195\) 0 0
\(196\) 0 0
\(197\) 21.6316 1.54119 0.770594 0.637327i \(-0.219960\pi\)
0.770594 + 0.637327i \(0.219960\pi\)
\(198\) 0 0
\(199\) −17.8323 −1.26410 −0.632051 0.774927i \(-0.717787\pi\)
−0.632051 + 0.774927i \(0.717787\pi\)
\(200\) 1.76744 0.124977
\(201\) 0 0
\(202\) 6.90367 0.485741
\(203\) 0 0
\(204\) 0 0
\(205\) −14.7013 −1.02679
\(206\) −12.6154 −0.878953
\(207\) 0 0
\(208\) 3.63899 0.252319
\(209\) −1.84106 −0.127349
\(210\) 0 0
\(211\) −8.27020 −0.569344 −0.284672 0.958625i \(-0.591885\pi\)
−0.284672 + 0.958625i \(0.591885\pi\)
\(212\) −11.2848 −0.775046
\(213\) 0 0
\(214\) 16.9417 1.15811
\(215\) −4.10965 −0.280276
\(216\) 0 0
\(217\) 0 0
\(218\) −9.76326 −0.661252
\(219\) 0 0
\(220\) −1.79793 −0.121217
\(221\) −22.2554 −1.49706
\(222\) 0 0
\(223\) −9.10574 −0.609765 −0.304883 0.952390i \(-0.598617\pi\)
−0.304883 + 0.952390i \(0.598617\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −12.9348 −0.860413
\(227\) 0.620456 0.0411811 0.0205905 0.999788i \(-0.493445\pi\)
0.0205905 + 0.999788i \(0.493445\pi\)
\(228\) 0 0
\(229\) 18.0011 1.18954 0.594772 0.803895i \(-0.297242\pi\)
0.594772 + 0.803895i \(0.297242\pi\)
\(230\) −13.2527 −0.873858
\(231\) 0 0
\(232\) −10.4243 −0.684388
\(233\) 19.8697 1.30171 0.650853 0.759204i \(-0.274412\pi\)
0.650853 + 0.759204i \(0.274412\pi\)
\(234\) 0 0
\(235\) 1.20369 0.0785201
\(236\) 13.5706 0.883371
\(237\) 0 0
\(238\) 0 0
\(239\) 17.4201 1.12681 0.563407 0.826180i \(-0.309490\pi\)
0.563407 + 0.826180i \(0.309490\pi\)
\(240\) 0 0
\(241\) 18.4149 1.18621 0.593104 0.805126i \(-0.297902\pi\)
0.593104 + 0.805126i \(0.297902\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −0.274758 −0.0175896
\(245\) 0 0
\(246\) 0 0
\(247\) 6.69959 0.426285
\(248\) −7.97958 −0.506704
\(249\) 0 0
\(250\) 12.1674 0.769534
\(251\) −21.8022 −1.37614 −0.688072 0.725642i \(-0.741543\pi\)
−0.688072 + 0.725642i \(0.741543\pi\)
\(252\) 0 0
\(253\) −7.37109 −0.463416
\(254\) 7.43208 0.466330
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.00229 0.249656 0.124828 0.992178i \(-0.460162\pi\)
0.124828 + 0.992178i \(0.460162\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.54266 0.405759
\(261\) 0 0
\(262\) 1.64154 0.101415
\(263\) 1.13946 0.0702619 0.0351309 0.999383i \(-0.488815\pi\)
0.0351309 + 0.999383i \(0.488815\pi\)
\(264\) 0 0
\(265\) −20.2894 −1.24637
\(266\) 0 0
\(267\) 0 0
\(268\) 3.88163 0.237108
\(269\) −3.85892 −0.235283 −0.117641 0.993056i \(-0.537533\pi\)
−0.117641 + 0.993056i \(0.537533\pi\)
\(270\) 0 0
\(271\) −22.0202 −1.33763 −0.668815 0.743429i \(-0.733198\pi\)
−0.668815 + 0.743429i \(0.733198\pi\)
\(272\) −6.11582 −0.370826
\(273\) 0 0
\(274\) 14.5307 0.877832
\(275\) 1.76744 0.106581
\(276\) 0 0
\(277\) −13.1307 −0.788950 −0.394475 0.918907i \(-0.629073\pi\)
−0.394475 + 0.918907i \(0.629073\pi\)
\(278\) 8.26535 0.495723
\(279\) 0 0
\(280\) 0 0
\(281\) 16.1064 0.960828 0.480414 0.877042i \(-0.340486\pi\)
0.480414 + 0.877042i \(0.340486\pi\)
\(282\) 0 0
\(283\) −10.6085 −0.630610 −0.315305 0.948990i \(-0.602107\pi\)
−0.315305 + 0.948990i \(0.602107\pi\)
\(284\) −10.8738 −0.645244
\(285\) 0 0
\(286\) 3.63899 0.215178
\(287\) 0 0
\(288\) 0 0
\(289\) 20.4032 1.20019
\(290\) −18.7422 −1.10058
\(291\) 0 0
\(292\) 3.85892 0.225826
\(293\) 3.69313 0.215755 0.107877 0.994164i \(-0.465595\pi\)
0.107877 + 0.994164i \(0.465595\pi\)
\(294\) 0 0
\(295\) 24.3990 1.42057
\(296\) −10.1995 −0.592835
\(297\) 0 0
\(298\) −7.55045 −0.437386
\(299\) 26.8233 1.55123
\(300\) 0 0
\(301\) 0 0
\(302\) −12.3990 −0.713484
\(303\) 0 0
\(304\) 1.84106 0.105592
\(305\) −0.493996 −0.0282861
\(306\) 0 0
\(307\) 19.1223 1.09137 0.545683 0.837992i \(-0.316270\pi\)
0.545683 + 0.837992i \(0.316270\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −14.3468 −0.814841
\(311\) 13.1190 0.743913 0.371956 0.928250i \(-0.378687\pi\)
0.371956 + 0.928250i \(0.378687\pi\)
\(312\) 0 0
\(313\) −12.6248 −0.713594 −0.356797 0.934182i \(-0.616131\pi\)
−0.356797 + 0.934182i \(0.616131\pi\)
\(314\) −4.54011 −0.256213
\(315\) 0 0
\(316\) −12.5174 −0.704159
\(317\) 29.4555 1.65438 0.827192 0.561919i \(-0.189937\pi\)
0.827192 + 0.561919i \(0.189937\pi\)
\(318\) 0 0
\(319\) −10.4243 −0.583648
\(320\) 1.79793 0.100507
\(321\) 0 0
\(322\) 0 0
\(323\) −11.2596 −0.626499
\(324\) 0 0
\(325\) −6.43169 −0.356766
\(326\) −15.8275 −0.876604
\(327\) 0 0
\(328\) 8.17680 0.451489
\(329\) 0 0
\(330\) 0 0
\(331\) 2.28901 0.125815 0.0629077 0.998019i \(-0.479963\pi\)
0.0629077 + 0.998019i \(0.479963\pi\)
\(332\) −5.27314 −0.289401
\(333\) 0 0
\(334\) −22.0844 −1.20840
\(335\) 6.97891 0.381299
\(336\) 0 0
\(337\) 5.35093 0.291484 0.145742 0.989323i \(-0.453443\pi\)
0.145742 + 0.989323i \(0.453443\pi\)
\(338\) −0.242256 −0.0131770
\(339\) 0 0
\(340\) −10.9958 −0.596332
\(341\) −7.97958 −0.432119
\(342\) 0 0
\(343\) 0 0
\(344\) 2.28577 0.123240
\(345\) 0 0
\(346\) −10.2628 −0.551731
\(347\) −20.8197 −1.11766 −0.558830 0.829282i \(-0.688750\pi\)
−0.558830 + 0.829282i \(0.688750\pi\)
\(348\) 0 0
\(349\) −23.1843 −1.24103 −0.620514 0.784195i \(-0.713076\pi\)
−0.620514 + 0.784195i \(0.713076\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −12.1243 −0.645310 −0.322655 0.946517i \(-0.604575\pi\)
−0.322655 + 0.946517i \(0.604575\pi\)
\(354\) 0 0
\(355\) −19.5504 −1.03763
\(356\) −1.98214 −0.105053
\(357\) 0 0
\(358\) −12.3954 −0.655118
\(359\) −12.3312 −0.650815 −0.325408 0.945574i \(-0.605501\pi\)
−0.325408 + 0.945574i \(0.605501\pi\)
\(360\) 0 0
\(361\) −15.6105 −0.821605
\(362\) 22.4506 1.17998
\(363\) 0 0
\(364\) 0 0
\(365\) 6.93808 0.363156
\(366\) 0 0
\(367\) −18.0860 −0.944081 −0.472041 0.881577i \(-0.656482\pi\)
−0.472041 + 0.881577i \(0.656482\pi\)
\(368\) 7.37109 0.384245
\(369\) 0 0
\(370\) −18.3380 −0.953349
\(371\) 0 0
\(372\) 0 0
\(373\) 6.95365 0.360046 0.180023 0.983662i \(-0.442383\pi\)
0.180023 + 0.983662i \(0.442383\pi\)
\(374\) −6.11582 −0.316241
\(375\) 0 0
\(376\) −0.669485 −0.0345261
\(377\) 37.9339 1.95370
\(378\) 0 0
\(379\) 5.14307 0.264182 0.132091 0.991238i \(-0.457831\pi\)
0.132091 + 0.991238i \(0.457831\pi\)
\(380\) 3.31010 0.169804
\(381\) 0 0
\(382\) −13.7142 −0.701681
\(383\) 18.8810 0.964772 0.482386 0.875959i \(-0.339770\pi\)
0.482386 + 0.875959i \(0.339770\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.77883 −0.192337
\(387\) 0 0
\(388\) 4.12066 0.209195
\(389\) 30.3990 1.54129 0.770646 0.637263i \(-0.219934\pi\)
0.770646 + 0.637263i \(0.219934\pi\)
\(390\) 0 0
\(391\) −45.0802 −2.27980
\(392\) 0 0
\(393\) 0 0
\(394\) −21.6316 −1.08978
\(395\) −22.5054 −1.13237
\(396\) 0 0
\(397\) −2.08009 −0.104397 −0.0521983 0.998637i \(-0.516623\pi\)
−0.0521983 + 0.998637i \(0.516623\pi\)
\(398\) 17.8323 0.893855
\(399\) 0 0
\(400\) −1.76744 −0.0883719
\(401\) 21.8444 1.09086 0.545429 0.838157i \(-0.316367\pi\)
0.545429 + 0.838157i \(0.316367\pi\)
\(402\) 0 0
\(403\) 29.0376 1.44647
\(404\) −6.90367 −0.343470
\(405\) 0 0
\(406\) 0 0
\(407\) −10.1995 −0.505571
\(408\) 0 0
\(409\) 33.7571 1.66918 0.834591 0.550871i \(-0.185704\pi\)
0.834591 + 0.550871i \(0.185704\pi\)
\(410\) 14.7013 0.726048
\(411\) 0 0
\(412\) 12.6154 0.621514
\(413\) 0 0
\(414\) 0 0
\(415\) −9.48074 −0.465391
\(416\) −3.63899 −0.178416
\(417\) 0 0
\(418\) 1.84106 0.0900491
\(419\) −22.7371 −1.11078 −0.555389 0.831590i \(-0.687431\pi\)
−0.555389 + 0.831590i \(0.687431\pi\)
\(420\) 0 0
\(421\) −21.0192 −1.02441 −0.512207 0.858862i \(-0.671172\pi\)
−0.512207 + 0.858862i \(0.671172\pi\)
\(422\) 8.27020 0.402587
\(423\) 0 0
\(424\) 11.2848 0.548040
\(425\) 10.8093 0.524329
\(426\) 0 0
\(427\) 0 0
\(428\) −16.9417 −0.818908
\(429\) 0 0
\(430\) 4.10965 0.198185
\(431\) −7.03305 −0.338770 −0.169385 0.985550i \(-0.554178\pi\)
−0.169385 + 0.985550i \(0.554178\pi\)
\(432\) 0 0
\(433\) 5.77293 0.277429 0.138715 0.990332i \(-0.455703\pi\)
0.138715 + 0.990332i \(0.455703\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.76326 0.467576
\(437\) 13.5706 0.649170
\(438\) 0 0
\(439\) 31.2275 1.49041 0.745203 0.666838i \(-0.232353\pi\)
0.745203 + 0.666838i \(0.232353\pi\)
\(440\) 1.79793 0.0857131
\(441\) 0 0
\(442\) 22.2554 1.05858
\(443\) 35.8953 1.70544 0.852720 0.522369i \(-0.174952\pi\)
0.852720 + 0.522369i \(0.174952\pi\)
\(444\) 0 0
\(445\) −3.56375 −0.168938
\(446\) 9.10574 0.431169
\(447\) 0 0
\(448\) 0 0
\(449\) 24.5495 1.15856 0.579282 0.815127i \(-0.303333\pi\)
0.579282 + 0.815127i \(0.303333\pi\)
\(450\) 0 0
\(451\) 8.17680 0.385031
\(452\) 12.9348 0.608404
\(453\) 0 0
\(454\) −0.620456 −0.0291194
\(455\) 0 0
\(456\) 0 0
\(457\) 26.2771 1.22919 0.614594 0.788843i \(-0.289320\pi\)
0.614594 + 0.788843i \(0.289320\pi\)
\(458\) −18.0011 −0.841134
\(459\) 0 0
\(460\) 13.2527 0.617911
\(461\) −20.8330 −0.970289 −0.485144 0.874434i \(-0.661233\pi\)
−0.485144 + 0.874434i \(0.661233\pi\)
\(462\) 0 0
\(463\) 17.8697 0.830474 0.415237 0.909713i \(-0.363699\pi\)
0.415237 + 0.909713i \(0.363699\pi\)
\(464\) 10.4243 0.483936
\(465\) 0 0
\(466\) −19.8697 −0.920445
\(467\) 26.8642 1.24312 0.621562 0.783365i \(-0.286498\pi\)
0.621562 + 0.783365i \(0.286498\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.20369 −0.0555221
\(471\) 0 0
\(472\) −13.5706 −0.624638
\(473\) 2.28577 0.105100
\(474\) 0 0
\(475\) −3.25396 −0.149302
\(476\) 0 0
\(477\) 0 0
\(478\) −17.4201 −0.796778
\(479\) 15.5050 0.708443 0.354221 0.935162i \(-0.384746\pi\)
0.354221 + 0.935162i \(0.384746\pi\)
\(480\) 0 0
\(481\) 37.1159 1.69234
\(482\) −18.4149 −0.838775
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 7.40867 0.336411
\(486\) 0 0
\(487\) 10.8858 0.493283 0.246641 0.969107i \(-0.420673\pi\)
0.246641 + 0.969107i \(0.420673\pi\)
\(488\) 0.274758 0.0124377
\(489\) 0 0
\(490\) 0 0
\(491\) −7.06975 −0.319053 −0.159527 0.987194i \(-0.550997\pi\)
−0.159527 + 0.987194i \(0.550997\pi\)
\(492\) 0 0
\(493\) −63.7531 −2.87129
\(494\) −6.69959 −0.301429
\(495\) 0 0
\(496\) 7.97958 0.358294
\(497\) 0 0
\(498\) 0 0
\(499\) 18.8678 0.844637 0.422319 0.906447i \(-0.361216\pi\)
0.422319 + 0.906447i \(0.361216\pi\)
\(500\) −12.1674 −0.544143
\(501\) 0 0
\(502\) 21.8022 0.973081
\(503\) −28.9752 −1.29194 −0.645969 0.763364i \(-0.723547\pi\)
−0.645969 + 0.763364i \(0.723547\pi\)
\(504\) 0 0
\(505\) −12.4123 −0.552342
\(506\) 7.37109 0.327685
\(507\) 0 0
\(508\) −7.43208 −0.329745
\(509\) 16.3539 0.724874 0.362437 0.932008i \(-0.381945\pi\)
0.362437 + 0.932008i \(0.381945\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −4.00229 −0.176534
\(515\) 22.6816 0.999469
\(516\) 0 0
\(517\) −0.669485 −0.0294439
\(518\) 0 0
\(519\) 0 0
\(520\) −6.54266 −0.286915
\(521\) 21.6949 0.950470 0.475235 0.879859i \(-0.342363\pi\)
0.475235 + 0.879859i \(0.342363\pi\)
\(522\) 0 0
\(523\) −37.8171 −1.65363 −0.826814 0.562475i \(-0.809849\pi\)
−0.826814 + 0.562475i \(0.809849\pi\)
\(524\) −1.64154 −0.0717112
\(525\) 0 0
\(526\) −1.13946 −0.0496826
\(527\) −48.8017 −2.12583
\(528\) 0 0
\(529\) 31.3329 1.36230
\(530\) 20.2894 0.881314
\(531\) 0 0
\(532\) 0 0
\(533\) −29.7553 −1.28885
\(534\) 0 0
\(535\) −30.4600 −1.31690
\(536\) −3.88163 −0.167661
\(537\) 0 0
\(538\) 3.85892 0.166370
\(539\) 0 0
\(540\) 0 0
\(541\) 1.68212 0.0723198 0.0361599 0.999346i \(-0.488487\pi\)
0.0361599 + 0.999346i \(0.488487\pi\)
\(542\) 22.0202 0.945847
\(543\) 0 0
\(544\) 6.11582 0.262213
\(545\) 17.5537 0.751917
\(546\) 0 0
\(547\) 13.6665 0.584339 0.292170 0.956366i \(-0.405623\pi\)
0.292170 + 0.956366i \(0.405623\pi\)
\(548\) −14.5307 −0.620721
\(549\) 0 0
\(550\) −1.76744 −0.0753638
\(551\) 19.1917 0.817595
\(552\) 0 0
\(553\) 0 0
\(554\) 13.1307 0.557872
\(555\) 0 0
\(556\) −8.26535 −0.350529
\(557\) 8.99583 0.381165 0.190583 0.981671i \(-0.438962\pi\)
0.190583 + 0.981671i \(0.438962\pi\)
\(558\) 0 0
\(559\) −8.31788 −0.351809
\(560\) 0 0
\(561\) 0 0
\(562\) −16.1064 −0.679408
\(563\) −21.7461 −0.916489 −0.458244 0.888826i \(-0.651522\pi\)
−0.458244 + 0.888826i \(0.651522\pi\)
\(564\) 0 0
\(565\) 23.2560 0.978386
\(566\) 10.6085 0.445908
\(567\) 0 0
\(568\) 10.8738 0.456256
\(569\) −32.4549 −1.36058 −0.680290 0.732943i \(-0.738146\pi\)
−0.680290 + 0.732943i \(0.738146\pi\)
\(570\) 0 0
\(571\) −20.2031 −0.845474 −0.422737 0.906252i \(-0.638931\pi\)
−0.422737 + 0.906252i \(0.638931\pi\)
\(572\) −3.63899 −0.152154
\(573\) 0 0
\(574\) 0 0
\(575\) −13.0279 −0.543303
\(576\) 0 0
\(577\) 24.4458 1.01769 0.508845 0.860858i \(-0.330073\pi\)
0.508845 + 0.860858i \(0.330073\pi\)
\(578\) −20.4032 −0.848661
\(579\) 0 0
\(580\) 18.7422 0.778226
\(581\) 0 0
\(582\) 0 0
\(583\) 11.2848 0.467370
\(584\) −3.85892 −0.159683
\(585\) 0 0
\(586\) −3.69313 −0.152562
\(587\) −16.8894 −0.697101 −0.348550 0.937290i \(-0.613326\pi\)
−0.348550 + 0.937290i \(0.613326\pi\)
\(588\) 0 0
\(589\) 14.6909 0.605327
\(590\) −24.3990 −1.00449
\(591\) 0 0
\(592\) 10.1995 0.419197
\(593\) 22.2781 0.914852 0.457426 0.889248i \(-0.348771\pi\)
0.457426 + 0.889248i \(0.348771\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.55045 0.309278
\(597\) 0 0
\(598\) −26.8233 −1.09689
\(599\) 6.25179 0.255441 0.127721 0.991810i \(-0.459234\pi\)
0.127721 + 0.991810i \(0.459234\pi\)
\(600\) 0 0
\(601\) 31.3043 1.27693 0.638465 0.769651i \(-0.279570\pi\)
0.638465 + 0.769651i \(0.279570\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 12.3990 0.504509
\(605\) 1.79793 0.0730964
\(606\) 0 0
\(607\) −25.0348 −1.01613 −0.508066 0.861318i \(-0.669639\pi\)
−0.508066 + 0.861318i \(0.669639\pi\)
\(608\) −1.84106 −0.0746648
\(609\) 0 0
\(610\) 0.493996 0.0200013
\(611\) 2.43625 0.0985602
\(612\) 0 0
\(613\) 1.17157 0.0473194 0.0236597 0.999720i \(-0.492468\pi\)
0.0236597 + 0.999720i \(0.492468\pi\)
\(614\) −19.1223 −0.771713
\(615\) 0 0
\(616\) 0 0
\(617\) −5.06651 −0.203970 −0.101985 0.994786i \(-0.532519\pi\)
−0.101985 + 0.994786i \(0.532519\pi\)
\(618\) 0 0
\(619\) −24.6812 −0.992021 −0.496010 0.868317i \(-0.665202\pi\)
−0.496010 + 0.868317i \(0.665202\pi\)
\(620\) 14.3468 0.576180
\(621\) 0 0
\(622\) −13.1190 −0.526026
\(623\) 0 0
\(624\) 0 0
\(625\) −13.0390 −0.521559
\(626\) 12.6248 0.504587
\(627\) 0 0
\(628\) 4.54011 0.181170
\(629\) −62.3784 −2.48719
\(630\) 0 0
\(631\) −25.8408 −1.02871 −0.514353 0.857579i \(-0.671968\pi\)
−0.514353 + 0.857579i \(0.671968\pi\)
\(632\) 12.5174 0.497915
\(633\) 0 0
\(634\) −29.4555 −1.16983
\(635\) −13.3624 −0.530270
\(636\) 0 0
\(637\) 0 0
\(638\) 10.4243 0.412702
\(639\) 0 0
\(640\) −1.79793 −0.0710695
\(641\) 3.19684 0.126267 0.0631337 0.998005i \(-0.479891\pi\)
0.0631337 + 0.998005i \(0.479891\pi\)
\(642\) 0 0
\(643\) 20.7623 0.818787 0.409393 0.912358i \(-0.365740\pi\)
0.409393 + 0.912358i \(0.365740\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 11.2596 0.443002
\(647\) −28.4056 −1.11674 −0.558370 0.829592i \(-0.688573\pi\)
−0.558370 + 0.829592i \(0.688573\pi\)
\(648\) 0 0
\(649\) −13.5706 −0.532693
\(650\) 6.43169 0.252272
\(651\) 0 0
\(652\) 15.8275 0.619853
\(653\) −16.7825 −0.656750 −0.328375 0.944548i \(-0.606501\pi\)
−0.328375 + 0.944548i \(0.606501\pi\)
\(654\) 0 0
\(655\) −2.95138 −0.115320
\(656\) −8.17680 −0.319251
\(657\) 0 0
\(658\) 0 0
\(659\) 15.3426 0.597662 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(660\) 0 0
\(661\) −6.68735 −0.260108 −0.130054 0.991507i \(-0.541515\pi\)
−0.130054 + 0.991507i \(0.541515\pi\)
\(662\) −2.28901 −0.0889649
\(663\) 0 0
\(664\) 5.27314 0.204637
\(665\) 0 0
\(666\) 0 0
\(667\) 76.8384 2.97519
\(668\) 22.0844 0.854471
\(669\) 0 0
\(670\) −6.97891 −0.269619
\(671\) 0.274758 0.0106069
\(672\) 0 0
\(673\) 12.4921 0.481536 0.240768 0.970583i \(-0.422601\pi\)
0.240768 + 0.970583i \(0.422601\pi\)
\(674\) −5.35093 −0.206110
\(675\) 0 0
\(676\) 0.242256 0.00931752
\(677\) −16.8830 −0.648866 −0.324433 0.945909i \(-0.605174\pi\)
−0.324433 + 0.945909i \(0.605174\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 10.9958 0.421671
\(681\) 0 0
\(682\) 7.97958 0.305554
\(683\) 6.76970 0.259036 0.129518 0.991577i \(-0.458657\pi\)
0.129518 + 0.991577i \(0.458657\pi\)
\(684\) 0 0
\(685\) −26.1252 −0.998193
\(686\) 0 0
\(687\) 0 0
\(688\) −2.28577 −0.0871440
\(689\) −41.0654 −1.56447
\(690\) 0 0
\(691\) −42.8876 −1.63152 −0.815760 0.578391i \(-0.803681\pi\)
−0.815760 + 0.578391i \(0.803681\pi\)
\(692\) 10.2628 0.390133
\(693\) 0 0
\(694\) 20.8197 0.790305
\(695\) −14.8605 −0.563693
\(696\) 0 0
\(697\) 50.0078 1.89418
\(698\) 23.1843 0.877540
\(699\) 0 0
\(700\) 0 0
\(701\) 33.8990 1.28035 0.640173 0.768231i \(-0.278863\pi\)
0.640173 + 0.768231i \(0.278863\pi\)
\(702\) 0 0
\(703\) 18.7779 0.708222
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 12.1243 0.456303
\(707\) 0 0
\(708\) 0 0
\(709\) −28.6407 −1.07562 −0.537812 0.843065i \(-0.680749\pi\)
−0.537812 + 0.843065i \(0.680749\pi\)
\(710\) 19.5504 0.733715
\(711\) 0 0
\(712\) 1.98214 0.0742837
\(713\) 58.8182 2.20276
\(714\) 0 0
\(715\) −6.54266 −0.244682
\(716\) 12.3954 0.463239
\(717\) 0 0
\(718\) 12.3312 0.460196
\(719\) −2.52225 −0.0940639 −0.0470319 0.998893i \(-0.514976\pi\)
−0.0470319 + 0.998893i \(0.514976\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 15.6105 0.580963
\(723\) 0 0
\(724\) −22.4506 −0.834370
\(725\) −18.4243 −0.684261
\(726\) 0 0
\(727\) −30.0452 −1.11431 −0.557157 0.830407i \(-0.688108\pi\)
−0.557157 + 0.830407i \(0.688108\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6.93808 −0.256790
\(731\) 13.9793 0.517044
\(732\) 0 0
\(733\) −33.5866 −1.24055 −0.620275 0.784385i \(-0.712979\pi\)
−0.620275 + 0.784385i \(0.712979\pi\)
\(734\) 18.0860 0.667566
\(735\) 0 0
\(736\) −7.37109 −0.271702
\(737\) −3.88163 −0.142982
\(738\) 0 0
\(739\) −29.7380 −1.09393 −0.546965 0.837155i \(-0.684217\pi\)
−0.546965 + 0.837155i \(0.684217\pi\)
\(740\) 18.3380 0.674120
\(741\) 0 0
\(742\) 0 0
\(743\) −1.59262 −0.0584276 −0.0292138 0.999573i \(-0.509300\pi\)
−0.0292138 + 0.999573i \(0.509300\pi\)
\(744\) 0 0
\(745\) 13.5752 0.497357
\(746\) −6.95365 −0.254591
\(747\) 0 0
\(748\) 6.11582 0.223616
\(749\) 0 0
\(750\) 0 0
\(751\) 0.305924 0.0111633 0.00558166 0.999984i \(-0.498223\pi\)
0.00558166 + 0.999984i \(0.498223\pi\)
\(752\) 0.669485 0.0244136
\(753\) 0 0
\(754\) −37.9339 −1.38147
\(755\) 22.2926 0.811312
\(756\) 0 0
\(757\) 40.4921 1.47171 0.735856 0.677138i \(-0.236780\pi\)
0.735856 + 0.677138i \(0.236780\pi\)
\(758\) −5.14307 −0.186805
\(759\) 0 0
\(760\) −3.31010 −0.120070
\(761\) −48.6887 −1.76496 −0.882482 0.470346i \(-0.844129\pi\)
−0.882482 + 0.470346i \(0.844129\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 13.7142 0.496164
\(765\) 0 0
\(766\) −18.8810 −0.682197
\(767\) 49.3833 1.78313
\(768\) 0 0
\(769\) 14.7255 0.531017 0.265508 0.964109i \(-0.414460\pi\)
0.265508 + 0.964109i \(0.414460\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.77883 0.136003
\(773\) −28.9612 −1.04166 −0.520830 0.853660i \(-0.674378\pi\)
−0.520830 + 0.853660i \(0.674378\pi\)
\(774\) 0 0
\(775\) −14.1034 −0.506610
\(776\) −4.12066 −0.147923
\(777\) 0 0
\(778\) −30.3990 −1.08986
\(779\) −15.0540 −0.539365
\(780\) 0 0
\(781\) 10.8738 0.389097
\(782\) 45.0802 1.61207
\(783\) 0 0
\(784\) 0 0
\(785\) 8.16281 0.291343
\(786\) 0 0
\(787\) −2.14203 −0.0763551 −0.0381775 0.999271i \(-0.512155\pi\)
−0.0381775 + 0.999271i \(0.512155\pi\)
\(788\) 21.6316 0.770594
\(789\) 0 0
\(790\) 22.5054 0.800708
\(791\) 0 0
\(792\) 0 0
\(793\) −0.999840 −0.0355054
\(794\) 2.08009 0.0738196
\(795\) 0 0
\(796\) −17.8323 −0.632051
\(797\) −7.38735 −0.261673 −0.130837 0.991404i \(-0.541766\pi\)
−0.130837 + 0.991404i \(0.541766\pi\)
\(798\) 0 0
\(799\) −4.09445 −0.144851
\(800\) 1.76744 0.0624884
\(801\) 0 0
\(802\) −21.8444 −0.771353
\(803\) −3.85892 −0.136178
\(804\) 0 0
\(805\) 0 0
\(806\) −29.0376 −1.02281
\(807\) 0 0
\(808\) 6.90367 0.242870
\(809\) −8.49214 −0.298568 −0.149284 0.988794i \(-0.547697\pi\)
−0.149284 + 0.988794i \(0.547697\pi\)
\(810\) 0 0
\(811\) −5.97995 −0.209984 −0.104992 0.994473i \(-0.533482\pi\)
−0.104992 + 0.994473i \(0.533482\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 10.1995 0.357493
\(815\) 28.4568 0.996797
\(816\) 0 0
\(817\) −4.20823 −0.147227
\(818\) −33.7571 −1.18029
\(819\) 0 0
\(820\) −14.7013 −0.513393
\(821\) −20.6430 −0.720445 −0.360223 0.932866i \(-0.617299\pi\)
−0.360223 + 0.932866i \(0.617299\pi\)
\(822\) 0 0
\(823\) −32.8269 −1.14427 −0.572137 0.820158i \(-0.693886\pi\)
−0.572137 + 0.820158i \(0.693886\pi\)
\(824\) −12.6154 −0.439477
\(825\) 0 0
\(826\) 0 0
\(827\) −7.22287 −0.251164 −0.125582 0.992083i \(-0.540080\pi\)
−0.125582 + 0.992083i \(0.540080\pi\)
\(828\) 0 0
\(829\) 52.2625 1.81515 0.907577 0.419887i \(-0.137930\pi\)
0.907577 + 0.419887i \(0.137930\pi\)
\(830\) 9.48074 0.329081
\(831\) 0 0
\(832\) 3.63899 0.126159
\(833\) 0 0
\(834\) 0 0
\(835\) 39.7062 1.37409
\(836\) −1.84106 −0.0636743
\(837\) 0 0
\(838\) 22.7371 0.785439
\(839\) 20.5644 0.709963 0.354981 0.934873i \(-0.384487\pi\)
0.354981 + 0.934873i \(0.384487\pi\)
\(840\) 0 0
\(841\) 79.6659 2.74710
\(842\) 21.0192 0.724370
\(843\) 0 0
\(844\) −8.27020 −0.284672
\(845\) 0.435559 0.0149837
\(846\) 0 0
\(847\) 0 0
\(848\) −11.2848 −0.387523
\(849\) 0 0
\(850\) −10.8093 −0.370757
\(851\) 75.1815 2.57719
\(852\) 0 0
\(853\) 24.8004 0.849148 0.424574 0.905393i \(-0.360424\pi\)
0.424574 + 0.905393i \(0.360424\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 16.9417 0.579055
\(857\) 30.1658 1.03044 0.515222 0.857056i \(-0.327709\pi\)
0.515222 + 0.857056i \(0.327709\pi\)
\(858\) 0 0
\(859\) 26.3550 0.899219 0.449610 0.893225i \(-0.351563\pi\)
0.449610 + 0.893225i \(0.351563\pi\)
\(860\) −4.10965 −0.140138
\(861\) 0 0
\(862\) 7.03305 0.239547
\(863\) 16.8489 0.573545 0.286772 0.957999i \(-0.407418\pi\)
0.286772 + 0.957999i \(0.407418\pi\)
\(864\) 0 0
\(865\) 18.4518 0.627381
\(866\) −5.77293 −0.196172
\(867\) 0 0
\(868\) 0 0
\(869\) 12.5174 0.424624
\(870\) 0 0
\(871\) 14.1252 0.478615
\(872\) −9.76326 −0.330626
\(873\) 0 0
\(874\) −13.5706 −0.459032
\(875\) 0 0
\(876\) 0 0
\(877\) 39.3054 1.32725 0.663624 0.748066i \(-0.269018\pi\)
0.663624 + 0.748066i \(0.269018\pi\)
\(878\) −31.2275 −1.05388
\(879\) 0 0
\(880\) −1.79793 −0.0606083
\(881\) −12.1055 −0.407843 −0.203922 0.978987i \(-0.565369\pi\)
−0.203922 + 0.978987i \(0.565369\pi\)
\(882\) 0 0
\(883\) 21.0614 0.708773 0.354386 0.935099i \(-0.384690\pi\)
0.354386 + 0.935099i \(0.384690\pi\)
\(884\) −22.2554 −0.748530
\(885\) 0 0
\(886\) −35.8953 −1.20593
\(887\) 45.1762 1.51687 0.758434 0.651750i \(-0.225965\pi\)
0.758434 + 0.651750i \(0.225965\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 3.56375 0.119457
\(891\) 0 0
\(892\) −9.10574 −0.304883
\(893\) 1.23256 0.0412461
\(894\) 0 0
\(895\) 22.2861 0.744943
\(896\) 0 0
\(897\) 0 0
\(898\) −24.5495 −0.819228
\(899\) 83.1815 2.77426
\(900\) 0 0
\(901\) 69.0160 2.29926
\(902\) −8.17680 −0.272258
\(903\) 0 0
\(904\) −12.9348 −0.430206
\(905\) −40.3647 −1.34177
\(906\) 0 0
\(907\) −33.5283 −1.11329 −0.556644 0.830751i \(-0.687911\pi\)
−0.556644 + 0.830751i \(0.687911\pi\)
\(908\) 0.620456 0.0205905
\(909\) 0 0
\(910\) 0 0
\(911\) 27.3400 0.905813 0.452906 0.891558i \(-0.350387\pi\)
0.452906 + 0.891558i \(0.350387\pi\)
\(912\) 0 0
\(913\) 5.27314 0.174515
\(914\) −26.2771 −0.869168
\(915\) 0 0
\(916\) 18.0011 0.594772
\(917\) 0 0
\(918\) 0 0
\(919\) −12.6118 −0.416026 −0.208013 0.978126i \(-0.566700\pi\)
−0.208013 + 0.978126i \(0.566700\pi\)
\(920\) −13.2527 −0.436929
\(921\) 0 0
\(922\) 20.8330 0.686098
\(923\) −39.5698 −1.30246
\(924\) 0 0
\(925\) −18.0270 −0.592725
\(926\) −17.8697 −0.587234
\(927\) 0 0
\(928\) −10.4243 −0.342194
\(929\) 53.3407 1.75005 0.875027 0.484075i \(-0.160844\pi\)
0.875027 + 0.484075i \(0.160844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 19.8697 0.650853
\(933\) 0 0
\(934\) −26.8642 −0.879022
\(935\) 10.9958 0.359602
\(936\) 0 0
\(937\) −20.5927 −0.672736 −0.336368 0.941731i \(-0.609199\pi\)
−0.336368 + 0.941731i \(0.609199\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.20369 0.0392600
\(941\) 27.9554 0.911319 0.455660 0.890154i \(-0.349403\pi\)
0.455660 + 0.890154i \(0.349403\pi\)
\(942\) 0 0
\(943\) −60.2719 −1.96272
\(944\) 13.5706 0.441686
\(945\) 0 0
\(946\) −2.28577 −0.0743167
\(947\) 44.7266 1.45342 0.726710 0.686945i \(-0.241049\pi\)
0.726710 + 0.686945i \(0.241049\pi\)
\(948\) 0 0
\(949\) 14.0426 0.455841
\(950\) 3.25396 0.105572
\(951\) 0 0
\(952\) 0 0
\(953\) 51.6527 1.67320 0.836598 0.547817i \(-0.184541\pi\)
0.836598 + 0.547817i \(0.184541\pi\)
\(954\) 0 0
\(955\) 24.6573 0.797890
\(956\) 17.4201 0.563407
\(957\) 0 0
\(958\) −15.5050 −0.500945
\(959\) 0 0
\(960\) 0 0
\(961\) 32.6738 1.05399
\(962\) −37.1159 −1.19667
\(963\) 0 0
\(964\) 18.4149 0.593104
\(965\) 6.79409 0.218709
\(966\) 0 0
\(967\) −50.7320 −1.63143 −0.815715 0.578454i \(-0.803656\pi\)
−0.815715 + 0.578454i \(0.803656\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −7.40867 −0.237878
\(971\) 22.7299 0.729436 0.364718 0.931118i \(-0.381165\pi\)
0.364718 + 0.931118i \(0.381165\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −10.8858 −0.348804
\(975\) 0 0
\(976\) −0.274758 −0.00879478
\(977\) 27.4811 0.879198 0.439599 0.898194i \(-0.355120\pi\)
0.439599 + 0.898194i \(0.355120\pi\)
\(978\) 0 0
\(979\) 1.98214 0.0633494
\(980\) 0 0
\(981\) 0 0
\(982\) 7.06975 0.225605
\(983\) −34.2044 −1.09095 −0.545475 0.838127i \(-0.683651\pi\)
−0.545475 + 0.838127i \(0.683651\pi\)
\(984\) 0 0
\(985\) 38.8921 1.23921
\(986\) 63.7531 2.03031
\(987\) 0 0
\(988\) 6.69959 0.213143
\(989\) −16.8486 −0.535754
\(990\) 0 0
\(991\) −33.3890 −1.06064 −0.530318 0.847799i \(-0.677927\pi\)
−0.530318 + 0.847799i \(0.677927\pi\)
\(992\) −7.97958 −0.253352
\(993\) 0 0
\(994\) 0 0
\(995\) −32.0614 −1.01641
\(996\) 0 0
\(997\) 7.74406 0.245257 0.122628 0.992453i \(-0.460868\pi\)
0.122628 + 0.992453i \(0.460868\pi\)
\(998\) −18.8678 −0.597249
\(999\) 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 9702.2.a.dz.1.4 4
3.2 odd 2 3234.2.a.bm.1.1 yes 4
7.6 odd 2 9702.2.a.ea.1.1 4
21.20 even 2 3234.2.a.bl.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bl.1.4 4 21.20 even 2
3234.2.a.bm.1.1 yes 4 3.2 odd 2
9702.2.a.dz.1.4 4 1.1 even 1 trivial
9702.2.a.ea.1.1 4 7.6 odd 2