Properties

Label 1568.2.a.v.1.1
Level $1568$
Weight $2$
Character 1568.1
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1568.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.23607 q^{3} -3.23607 q^{5} -1.47214 q^{9} +O(q^{10})\) \(q-1.23607 q^{3} -3.23607 q^{5} -1.47214 q^{9} -6.47214 q^{11} -0.763932 q^{13} +4.00000 q^{15} -4.47214 q^{17} +1.23607 q^{19} +4.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} -4.47214 q^{29} +2.47214 q^{31} +8.00000 q^{33} -4.47214 q^{37} +0.944272 q^{39} +8.47214 q^{41} +6.47214 q^{43} +4.76393 q^{45} -10.4721 q^{47} +5.52786 q^{51} -10.0000 q^{53} +20.9443 q^{55} -1.52786 q^{57} +9.23607 q^{59} -11.2361 q^{61} +2.47214 q^{65} +4.00000 q^{67} -4.94427 q^{69} -4.94427 q^{71} +2.94427 q^{73} -6.76393 q^{75} +12.9443 q^{79} -2.41641 q^{81} +9.23607 q^{83} +14.4721 q^{85} +5.52786 q^{87} +6.00000 q^{89} -3.05573 q^{93} -4.00000 q^{95} -12.4721 q^{97} +9.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 6q^{9} - 4q^{11} - 6q^{13} + 8q^{15} - 2q^{19} + 8q^{23} + 2q^{25} + 20q^{27} - 4q^{31} + 16q^{33} - 16q^{39} + 8q^{41} + 4q^{43} + 14q^{45} - 12q^{47} + 20q^{51} - 20q^{53} + 24q^{55} - 12q^{57} + 14q^{59} - 18q^{61} - 4q^{65} + 8q^{67} + 8q^{69} + 8q^{71} - 12q^{73} - 18q^{75} + 8q^{79} + 22q^{81} + 14q^{83} + 20q^{85} + 20q^{87} + 12q^{89} - 24q^{93} - 8q^{95} - 16q^{97} + 28q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) −6.47214 −1.95142 −0.975711 0.219061i \(-0.929701\pi\)
−0.975711 + 0.219061i \(0.929701\pi\)
\(12\) 0 0
\(13\) −0.763932 −0.211877 −0.105938 0.994373i \(-0.533785\pi\)
−0.105938 + 0.994373i \(0.533785\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 1.23607 0.283573 0.141787 0.989897i \(-0.454715\pi\)
0.141787 + 0.989897i \(0.454715\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 0.944272 0.151205
\(40\) 0 0
\(41\) 8.47214 1.32313 0.661563 0.749890i \(-0.269894\pi\)
0.661563 + 0.749890i \(0.269894\pi\)
\(42\) 0 0
\(43\) 6.47214 0.986991 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) 0 0
\(45\) 4.76393 0.710165
\(46\) 0 0
\(47\) −10.4721 −1.52752 −0.763759 0.645501i \(-0.776648\pi\)
−0.763759 + 0.645501i \(0.776648\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.52786 0.774056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 20.9443 2.82413
\(56\) 0 0
\(57\) −1.52786 −0.202371
\(58\) 0 0
\(59\) 9.23607 1.20243 0.601217 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) −11.2361 −1.43863 −0.719316 0.694683i \(-0.755544\pi\)
−0.719316 + 0.694683i \(0.755544\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.47214 0.306631
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −4.94427 −0.595220
\(70\) 0 0
\(71\) −4.94427 −0.586777 −0.293389 0.955993i \(-0.594783\pi\)
−0.293389 + 0.955993i \(0.594783\pi\)
\(72\) 0 0
\(73\) 2.94427 0.344601 0.172300 0.985044i \(-0.444880\pi\)
0.172300 + 0.985044i \(0.444880\pi\)
\(74\) 0 0
\(75\) −6.76393 −0.781032
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) 9.23607 1.01379 0.506895 0.862008i \(-0.330793\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(84\) 0 0
\(85\) 14.4721 1.56972
\(86\) 0 0
\(87\) 5.52786 0.592649
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.05573 −0.316864
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −12.4721 −1.26635 −0.633177 0.774007i \(-0.718249\pi\)
−0.633177 + 0.774007i \(0.718249\pi\)
\(98\) 0 0
\(99\) 9.52786 0.957586
\(100\) 0 0
\(101\) 1.70820 0.169973 0.0849863 0.996382i \(-0.472915\pi\)
0.0849863 + 0.996382i \(0.472915\pi\)
\(102\) 0 0
\(103\) 5.52786 0.544677 0.272338 0.962202i \(-0.412203\pi\)
0.272338 + 0.962202i \(0.412203\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.94427 0.864675 0.432338 0.901712i \(-0.357689\pi\)
0.432338 + 0.901712i \(0.357689\pi\)
\(108\) 0 0
\(109\) 8.47214 0.811483 0.405742 0.913988i \(-0.367013\pi\)
0.405742 + 0.913988i \(0.367013\pi\)
\(110\) 0 0
\(111\) 5.52786 0.524682
\(112\) 0 0
\(113\) −12.4721 −1.17328 −0.586640 0.809848i \(-0.699550\pi\)
−0.586640 + 0.809848i \(0.699550\pi\)
\(114\) 0 0
\(115\) −12.9443 −1.20706
\(116\) 0 0
\(117\) 1.12461 0.103970
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 30.8885 2.80805
\(122\) 0 0
\(123\) −10.4721 −0.944241
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 8.94427 0.793676 0.396838 0.917889i \(-0.370108\pi\)
0.396838 + 0.917889i \(0.370108\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −11.7082 −1.02295 −0.511475 0.859298i \(-0.670901\pi\)
−0.511475 + 0.859298i \(0.670901\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −17.8885 −1.53960
\(136\) 0 0
\(137\) 14.9443 1.27678 0.638388 0.769715i \(-0.279602\pi\)
0.638388 + 0.769715i \(0.279602\pi\)
\(138\) 0 0
\(139\) −1.23607 −0.104842 −0.0524210 0.998625i \(-0.516694\pi\)
−0.0524210 + 0.998625i \(0.516694\pi\)
\(140\) 0 0
\(141\) 12.9443 1.09010
\(142\) 0 0
\(143\) 4.94427 0.413461
\(144\) 0 0
\(145\) 14.4721 1.20185
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.94427 0.241204 0.120602 0.992701i \(-0.461517\pi\)
0.120602 + 0.992701i \(0.461517\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 0 0
\(153\) 6.58359 0.532252
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −0.763932 −0.0609684 −0.0304842 0.999535i \(-0.509705\pi\)
−0.0304842 + 0.999535i \(0.509705\pi\)
\(158\) 0 0
\(159\) 12.3607 0.980266
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.41641 −0.267594 −0.133797 0.991009i \(-0.542717\pi\)
−0.133797 + 0.991009i \(0.542717\pi\)
\(164\) 0 0
\(165\) −25.8885 −2.01542
\(166\) 0 0
\(167\) −23.4164 −1.81202 −0.906008 0.423261i \(-0.860886\pi\)
−0.906008 + 0.423261i \(0.860886\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) −1.81966 −0.139153
\(172\) 0 0
\(173\) −5.70820 −0.433987 −0.216993 0.976173i \(-0.569625\pi\)
−0.216993 + 0.976173i \(0.569625\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −11.4164 −0.858110
\(178\) 0 0
\(179\) −7.05573 −0.527370 −0.263685 0.964609i \(-0.584938\pi\)
−0.263685 + 0.964609i \(0.584938\pi\)
\(180\) 0 0
\(181\) 12.1803 0.905358 0.452679 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(182\) 0 0
\(183\) 13.8885 1.02667
\(184\) 0 0
\(185\) 14.4721 1.06401
\(186\) 0 0
\(187\) 28.9443 2.11661
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.94427 0.357755 0.178877 0.983871i \(-0.442753\pi\)
0.178877 + 0.983871i \(0.442753\pi\)
\(192\) 0 0
\(193\) 0.472136 0.0339851 0.0169925 0.999856i \(-0.494591\pi\)
0.0169925 + 0.999856i \(0.494591\pi\)
\(194\) 0 0
\(195\) −3.05573 −0.218825
\(196\) 0 0
\(197\) 10.9443 0.779747 0.389874 0.920868i \(-0.372519\pi\)
0.389874 + 0.920868i \(0.372519\pi\)
\(198\) 0 0
\(199\) −15.4164 −1.09284 −0.546420 0.837511i \(-0.684010\pi\)
−0.546420 + 0.837511i \(0.684010\pi\)
\(200\) 0 0
\(201\) −4.94427 −0.348742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −27.4164 −1.91484
\(206\) 0 0
\(207\) −5.88854 −0.409282
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 6.11146 0.418750
\(214\) 0 0
\(215\) −20.9443 −1.42839
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.63932 −0.245922
\(220\) 0 0
\(221\) 3.41641 0.229812
\(222\) 0 0
\(223\) −12.9443 −0.866813 −0.433406 0.901199i \(-0.642688\pi\)
−0.433406 + 0.901199i \(0.642688\pi\)
\(224\) 0 0
\(225\) −8.05573 −0.537049
\(226\) 0 0
\(227\) 17.2361 1.14400 0.571999 0.820254i \(-0.306168\pi\)
0.571999 + 0.820254i \(0.306168\pi\)
\(228\) 0 0
\(229\) −23.5967 −1.55932 −0.779658 0.626205i \(-0.784607\pi\)
−0.779658 + 0.626205i \(0.784607\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.8885 −1.04089 −0.520447 0.853894i \(-0.674235\pi\)
−0.520447 + 0.853894i \(0.674235\pi\)
\(234\) 0 0
\(235\) 33.8885 2.21064
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −13.8885 −0.898375 −0.449188 0.893437i \(-0.648287\pi\)
−0.449188 + 0.893437i \(0.648287\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 0 0
\(243\) −13.5967 −0.872232
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.944272 −0.0600826
\(248\) 0 0
\(249\) −11.4164 −0.723485
\(250\) 0 0
\(251\) −4.29180 −0.270896 −0.135448 0.990784i \(-0.543247\pi\)
−0.135448 + 0.990784i \(0.543247\pi\)
\(252\) 0 0
\(253\) −25.8885 −1.62760
\(254\) 0 0
\(255\) −17.8885 −1.12022
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.58359 0.407514
\(262\) 0 0
\(263\) 11.0557 0.681725 0.340863 0.940113i \(-0.389281\pi\)
0.340863 + 0.940113i \(0.389281\pi\)
\(264\) 0 0
\(265\) 32.3607 1.98790
\(266\) 0 0
\(267\) −7.41641 −0.453877
\(268\) 0 0
\(269\) 4.18034 0.254880 0.127440 0.991846i \(-0.459324\pi\)
0.127440 + 0.991846i \(0.459324\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −35.4164 −2.13569
\(276\) 0 0
\(277\) 7.88854 0.473977 0.236988 0.971512i \(-0.423840\pi\)
0.236988 + 0.971512i \(0.423840\pi\)
\(278\) 0 0
\(279\) −3.63932 −0.217880
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 6.18034 0.367383 0.183692 0.982984i \(-0.441195\pi\)
0.183692 + 0.982984i \(0.441195\pi\)
\(284\) 0 0
\(285\) 4.94427 0.292873
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 15.4164 0.903726
\(292\) 0 0
\(293\) 12.7639 0.745677 0.372838 0.927896i \(-0.378385\pi\)
0.372838 + 0.927896i \(0.378385\pi\)
\(294\) 0 0
\(295\) −29.8885 −1.74018
\(296\) 0 0
\(297\) −35.7771 −2.07600
\(298\) 0 0
\(299\) −3.05573 −0.176717
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.11146 −0.121300
\(304\) 0 0
\(305\) 36.3607 2.08201
\(306\) 0 0
\(307\) −1.81966 −0.103853 −0.0519267 0.998651i \(-0.516536\pi\)
−0.0519267 + 0.998651i \(0.516536\pi\)
\(308\) 0 0
\(309\) −6.83282 −0.388705
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 8.47214 0.478873 0.239437 0.970912i \(-0.423037\pi\)
0.239437 + 0.970912i \(0.423037\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.05573 0.508620 0.254310 0.967123i \(-0.418152\pi\)
0.254310 + 0.967123i \(0.418152\pi\)
\(318\) 0 0
\(319\) 28.9443 1.62057
\(320\) 0 0
\(321\) −11.0557 −0.617071
\(322\) 0 0
\(323\) −5.52786 −0.307579
\(324\) 0 0
\(325\) −4.18034 −0.231884
\(326\) 0 0
\(327\) −10.4721 −0.579110
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 22.4721 1.23518 0.617590 0.786500i \(-0.288109\pi\)
0.617590 + 0.786500i \(0.288109\pi\)
\(332\) 0 0
\(333\) 6.58359 0.360779
\(334\) 0 0
\(335\) −12.9443 −0.707221
\(336\) 0 0
\(337\) 10.3607 0.564382 0.282191 0.959358i \(-0.408939\pi\)
0.282191 + 0.959358i \(0.408939\pi\)
\(338\) 0 0
\(339\) 15.4164 0.837304
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) 6.47214 0.347442 0.173721 0.984795i \(-0.444421\pi\)
0.173721 + 0.984795i \(0.444421\pi\)
\(348\) 0 0
\(349\) −26.6525 −1.42667 −0.713337 0.700821i \(-0.752817\pi\)
−0.713337 + 0.700821i \(0.752817\pi\)
\(350\) 0 0
\(351\) −4.22291 −0.225402
\(352\) 0 0
\(353\) 15.8885 0.845662 0.422831 0.906209i \(-0.361036\pi\)
0.422831 + 0.906209i \(0.361036\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.9443 −0.894284 −0.447142 0.894463i \(-0.647558\pi\)
−0.447142 + 0.894463i \(0.647558\pi\)
\(360\) 0 0
\(361\) −17.4721 −0.919586
\(362\) 0 0
\(363\) −38.1803 −2.00395
\(364\) 0 0
\(365\) −9.52786 −0.498711
\(366\) 0 0
\(367\) 22.8328 1.19186 0.595932 0.803035i \(-0.296783\pi\)
0.595932 + 0.803035i \(0.296783\pi\)
\(368\) 0 0
\(369\) −12.4721 −0.649273
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.94427 0.152449 0.0762243 0.997091i \(-0.475713\pi\)
0.0762243 + 0.997091i \(0.475713\pi\)
\(374\) 0 0
\(375\) 1.88854 0.0975240
\(376\) 0 0
\(377\) 3.41641 0.175954
\(378\) 0 0
\(379\) −4.58359 −0.235443 −0.117722 0.993047i \(-0.537559\pi\)
−0.117722 + 0.993047i \(0.537559\pi\)
\(380\) 0 0
\(381\) −11.0557 −0.566402
\(382\) 0 0
\(383\) 15.4164 0.787742 0.393871 0.919166i \(-0.371136\pi\)
0.393871 + 0.919166i \(0.371136\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.52786 −0.484329
\(388\) 0 0
\(389\) −4.47214 −0.226746 −0.113373 0.993552i \(-0.536166\pi\)
−0.113373 + 0.993552i \(0.536166\pi\)
\(390\) 0 0
\(391\) −17.8885 −0.904663
\(392\) 0 0
\(393\) 14.4721 0.730023
\(394\) 0 0
\(395\) −41.8885 −2.10764
\(396\) 0 0
\(397\) 15.2361 0.764676 0.382338 0.924022i \(-0.375119\pi\)
0.382338 + 0.924022i \(0.375119\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −23.5279 −1.17493 −0.587463 0.809251i \(-0.699873\pi\)
−0.587463 + 0.809251i \(0.699873\pi\)
\(402\) 0 0
\(403\) −1.88854 −0.0940751
\(404\) 0 0
\(405\) 7.81966 0.388562
\(406\) 0 0
\(407\) 28.9443 1.43471
\(408\) 0 0
\(409\) 21.4164 1.05897 0.529487 0.848318i \(-0.322385\pi\)
0.529487 + 0.848318i \(0.322385\pi\)
\(410\) 0 0
\(411\) −18.4721 −0.911163
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −29.8885 −1.46717
\(416\) 0 0
\(417\) 1.52786 0.0748198
\(418\) 0 0
\(419\) −22.1803 −1.08358 −0.541790 0.840514i \(-0.682253\pi\)
−0.541790 + 0.840514i \(0.682253\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 15.4164 0.749571
\(424\) 0 0
\(425\) −24.4721 −1.18707
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.11146 −0.295064
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) −9.41641 −0.452524 −0.226262 0.974067i \(-0.572651\pi\)
−0.226262 + 0.974067i \(0.572651\pi\)
\(434\) 0 0
\(435\) −17.8885 −0.857690
\(436\) 0 0
\(437\) 4.94427 0.236517
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13.8885 −0.659865 −0.329932 0.944005i \(-0.607026\pi\)
−0.329932 + 0.944005i \(0.607026\pi\)
\(444\) 0 0
\(445\) −19.4164 −0.920426
\(446\) 0 0
\(447\) −3.63932 −0.172134
\(448\) 0 0
\(449\) −7.88854 −0.372283 −0.186142 0.982523i \(-0.559598\pi\)
−0.186142 + 0.982523i \(0.559598\pi\)
\(450\) 0 0
\(451\) −54.8328 −2.58198
\(452\) 0 0
\(453\) 11.0557 0.519443
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.52786 −0.352139 −0.176069 0.984378i \(-0.556338\pi\)
−0.176069 + 0.984378i \(0.556338\pi\)
\(458\) 0 0
\(459\) −24.7214 −1.15389
\(460\) 0 0
\(461\) −21.7082 −1.01105 −0.505526 0.862811i \(-0.668702\pi\)
−0.505526 + 0.862811i \(0.668702\pi\)
\(462\) 0 0
\(463\) −35.7771 −1.66270 −0.831351 0.555748i \(-0.812432\pi\)
−0.831351 + 0.555748i \(0.812432\pi\)
\(464\) 0 0
\(465\) 9.88854 0.458570
\(466\) 0 0
\(467\) 32.0689 1.48397 0.741985 0.670416i \(-0.233884\pi\)
0.741985 + 0.670416i \(0.233884\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.944272 0.0435098
\(472\) 0 0
\(473\) −41.8885 −1.92604
\(474\) 0 0
\(475\) 6.76393 0.310350
\(476\) 0 0
\(477\) 14.7214 0.674045
\(478\) 0 0
\(479\) 8.58359 0.392194 0.196097 0.980584i \(-0.437173\pi\)
0.196097 + 0.980584i \(0.437173\pi\)
\(480\) 0 0
\(481\) 3.41641 0.155775
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 40.3607 1.83268
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 4.22291 0.190967
\(490\) 0 0
\(491\) 37.8885 1.70989 0.854943 0.518722i \(-0.173592\pi\)
0.854943 + 0.518722i \(0.173592\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −30.8328 −1.38583
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 21.8885 0.979866 0.489933 0.871760i \(-0.337021\pi\)
0.489933 + 0.871760i \(0.337021\pi\)
\(500\) 0 0
\(501\) 28.9443 1.29313
\(502\) 0 0
\(503\) 4.94427 0.220454 0.110227 0.993906i \(-0.464842\pi\)
0.110227 + 0.993906i \(0.464842\pi\)
\(504\) 0 0
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) 15.3475 0.681607
\(508\) 0 0
\(509\) 41.1246 1.82282 0.911408 0.411503i \(-0.134996\pi\)
0.911408 + 0.411503i \(0.134996\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.83282 0.301676
\(514\) 0 0
\(515\) −17.8885 −0.788263
\(516\) 0 0
\(517\) 67.7771 2.98083
\(518\) 0 0
\(519\) 7.05573 0.309712
\(520\) 0 0
\(521\) 6.58359 0.288432 0.144216 0.989546i \(-0.453934\pi\)
0.144216 + 0.989546i \(0.453934\pi\)
\(522\) 0 0
\(523\) 4.29180 0.187667 0.0938336 0.995588i \(-0.470088\pi\)
0.0938336 + 0.995588i \(0.470088\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.0557 −0.481595
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −13.5967 −0.590049
\(532\) 0 0
\(533\) −6.47214 −0.280339
\(534\) 0 0
\(535\) −28.9443 −1.25137
\(536\) 0 0
\(537\) 8.72136 0.376354
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.05573 −0.217363 −0.108681 0.994077i \(-0.534663\pi\)
−0.108681 + 0.994077i \(0.534663\pi\)
\(542\) 0 0
\(543\) −15.0557 −0.646103
\(544\) 0 0
\(545\) −27.4164 −1.17439
\(546\) 0 0
\(547\) −4.58359 −0.195980 −0.0979901 0.995187i \(-0.531241\pi\)
−0.0979901 + 0.995187i \(0.531241\pi\)
\(548\) 0 0
\(549\) 16.5410 0.705954
\(550\) 0 0
\(551\) −5.52786 −0.235495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −17.8885 −0.759326
\(556\) 0 0
\(557\) 9.05573 0.383704 0.191852 0.981424i \(-0.438551\pi\)
0.191852 + 0.981424i \(0.438551\pi\)
\(558\) 0 0
\(559\) −4.94427 −0.209120
\(560\) 0 0
\(561\) −35.7771 −1.51051
\(562\) 0 0
\(563\) 17.8197 0.751009 0.375505 0.926821i \(-0.377469\pi\)
0.375505 + 0.926821i \(0.377469\pi\)
\(564\) 0 0
\(565\) 40.3607 1.69799
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26.3607 1.10510 0.552549 0.833481i \(-0.313655\pi\)
0.552549 + 0.833481i \(0.313655\pi\)
\(570\) 0 0
\(571\) −14.4721 −0.605640 −0.302820 0.953048i \(-0.597928\pi\)
−0.302820 + 0.953048i \(0.597928\pi\)
\(572\) 0 0
\(573\) −6.11146 −0.255310
\(574\) 0 0
\(575\) 21.8885 0.912815
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) −0.583592 −0.0242533
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 64.7214 2.68048
\(584\) 0 0
\(585\) −3.63932 −0.150467
\(586\) 0 0
\(587\) 3.70820 0.153054 0.0765270 0.997068i \(-0.475617\pi\)
0.0765270 + 0.997068i \(0.475617\pi\)
\(588\) 0 0
\(589\) 3.05573 0.125909
\(590\) 0 0
\(591\) −13.5279 −0.556462
\(592\) 0 0
\(593\) −32.8328 −1.34828 −0.674141 0.738603i \(-0.735486\pi\)
−0.674141 + 0.738603i \(0.735486\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.0557 0.779899
\(598\) 0 0
\(599\) 17.8885 0.730906 0.365453 0.930830i \(-0.380914\pi\)
0.365453 + 0.930830i \(0.380914\pi\)
\(600\) 0 0
\(601\) −29.7771 −1.21463 −0.607316 0.794460i \(-0.707754\pi\)
−0.607316 + 0.794460i \(0.707754\pi\)
\(602\) 0 0
\(603\) −5.88854 −0.239800
\(604\) 0 0
\(605\) −99.9574 −4.06385
\(606\) 0 0
\(607\) −9.88854 −0.401364 −0.200682 0.979656i \(-0.564316\pi\)
−0.200682 + 0.979656i \(0.564316\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 29.4164 1.18812 0.594059 0.804422i \(-0.297525\pi\)
0.594059 + 0.804422i \(0.297525\pi\)
\(614\) 0 0
\(615\) 33.8885 1.36652
\(616\) 0 0
\(617\) 34.3607 1.38331 0.691654 0.722229i \(-0.256882\pi\)
0.691654 + 0.722229i \(0.256882\pi\)
\(618\) 0 0
\(619\) 48.0689 1.93205 0.966026 0.258446i \(-0.0832103\pi\)
0.966026 + 0.258446i \(0.0832103\pi\)
\(620\) 0 0
\(621\) 22.1115 0.887302
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 9.88854 0.394910
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −44.9443 −1.78920 −0.894602 0.446865i \(-0.852541\pi\)
−0.894602 + 0.446865i \(0.852541\pi\)
\(632\) 0 0
\(633\) 14.8328 0.589551
\(634\) 0 0
\(635\) −28.9443 −1.14862
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 7.27864 0.287939
\(640\) 0 0
\(641\) 14.5836 0.576017 0.288009 0.957628i \(-0.407007\pi\)
0.288009 + 0.957628i \(0.407007\pi\)
\(642\) 0 0
\(643\) 43.7082 1.72368 0.861842 0.507177i \(-0.169311\pi\)
0.861842 + 0.507177i \(0.169311\pi\)
\(644\) 0 0
\(645\) 25.8885 1.01936
\(646\) 0 0
\(647\) 20.3607 0.800461 0.400230 0.916415i \(-0.368930\pi\)
0.400230 + 0.916415i \(0.368930\pi\)
\(648\) 0 0
\(649\) −59.7771 −2.34646
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.41641 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(654\) 0 0
\(655\) 37.8885 1.48043
\(656\) 0 0
\(657\) −4.33437 −0.169100
\(658\) 0 0
\(659\) 37.3050 1.45319 0.726597 0.687064i \(-0.241101\pi\)
0.726597 + 0.687064i \(0.241101\pi\)
\(660\) 0 0
\(661\) 22.6525 0.881079 0.440540 0.897733i \(-0.354787\pi\)
0.440540 + 0.897733i \(0.354787\pi\)
\(662\) 0 0
\(663\) −4.22291 −0.164004
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −17.8885 −0.692647
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 72.7214 2.80738
\(672\) 0 0
\(673\) −2.94427 −0.113493 −0.0567467 0.998389i \(-0.518073\pi\)
−0.0567467 + 0.998389i \(0.518073\pi\)
\(674\) 0 0
\(675\) 30.2492 1.16429
\(676\) 0 0
\(677\) −19.8197 −0.761731 −0.380866 0.924630i \(-0.624374\pi\)
−0.380866 + 0.924630i \(0.624374\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −21.3050 −0.816408
\(682\) 0 0
\(683\) 23.7771 0.909805 0.454902 0.890541i \(-0.349674\pi\)
0.454902 + 0.890541i \(0.349674\pi\)
\(684\) 0 0
\(685\) −48.3607 −1.84777
\(686\) 0 0
\(687\) 29.1672 1.11280
\(688\) 0 0
\(689\) 7.63932 0.291035
\(690\) 0 0
\(691\) 14.1803 0.539446 0.269723 0.962938i \(-0.413068\pi\)
0.269723 + 0.962938i \(0.413068\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −37.8885 −1.43513
\(698\) 0 0
\(699\) 19.6393 0.742827
\(700\) 0 0
\(701\) −41.4164 −1.56428 −0.782138 0.623105i \(-0.785871\pi\)
−0.782138 + 0.623105i \(0.785871\pi\)
\(702\) 0 0
\(703\) −5.52786 −0.208487
\(704\) 0 0
\(705\) −41.8885 −1.57761
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.63932 0.0615660 0.0307830 0.999526i \(-0.490200\pi\)
0.0307830 + 0.999526i \(0.490200\pi\)
\(710\) 0 0
\(711\) −19.0557 −0.714646
\(712\) 0 0
\(713\) 9.88854 0.370329
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 17.1672 0.641120
\(718\) 0 0
\(719\) −51.1935 −1.90920 −0.954598 0.297898i \(-0.903714\pi\)
−0.954598 + 0.297898i \(0.903714\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 15.4164 0.573342
\(724\) 0 0
\(725\) −24.4721 −0.908872
\(726\) 0 0
\(727\) −12.3607 −0.458432 −0.229216 0.973376i \(-0.573616\pi\)
−0.229216 + 0.973376i \(0.573616\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) −28.9443 −1.07054
\(732\) 0 0
\(733\) 4.76393 0.175960 0.0879799 0.996122i \(-0.471959\pi\)
0.0879799 + 0.996122i \(0.471959\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.8885 −0.953617
\(738\) 0 0
\(739\) −3.41641 −0.125675 −0.0628373 0.998024i \(-0.520015\pi\)
−0.0628373 + 0.998024i \(0.520015\pi\)
\(740\) 0 0
\(741\) 1.16718 0.0428776
\(742\) 0 0
\(743\) 24.9443 0.915117 0.457558 0.889180i \(-0.348724\pi\)
0.457558 + 0.889180i \(0.348724\pi\)
\(744\) 0 0
\(745\) −9.52786 −0.349074
\(746\) 0 0
\(747\) −13.5967 −0.497479
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 36.0000 1.31366 0.656829 0.754039i \(-0.271897\pi\)
0.656829 + 0.754039i \(0.271897\pi\)
\(752\) 0 0
\(753\) 5.30495 0.193323
\(754\) 0 0
\(755\) 28.9443 1.05339
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) 0 0
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 3.52786 0.127885 0.0639425 0.997954i \(-0.479633\pi\)
0.0639425 + 0.997954i \(0.479633\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −21.3050 −0.770282
\(766\) 0 0
\(767\) −7.05573 −0.254768
\(768\) 0 0
\(769\) 18.3607 0.662103 0.331052 0.943613i \(-0.392597\pi\)
0.331052 + 0.943613i \(0.392597\pi\)
\(770\) 0 0
\(771\) −17.3050 −0.623223
\(772\) 0 0
\(773\) −40.1803 −1.44519 −0.722593 0.691274i \(-0.757050\pi\)
−0.722593 + 0.691274i \(0.757050\pi\)
\(774\) 0 0
\(775\) 13.5279 0.485935
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10.4721 0.375203
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) −24.7214 −0.883469
\(784\) 0 0
\(785\) 2.47214 0.0882343
\(786\) 0 0
\(787\) 12.2918 0.438155 0.219078 0.975707i \(-0.429695\pi\)
0.219078 + 0.975707i \(0.429695\pi\)
\(788\) 0 0
\(789\) −13.6656 −0.486509
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.58359 0.304812
\(794\) 0 0
\(795\) −40.0000 −1.41865
\(796\) 0 0
\(797\) 52.1803 1.84832 0.924161 0.382003i \(-0.124765\pi\)
0.924161 + 0.382003i \(0.124765\pi\)
\(798\) 0 0
\(799\) 46.8328 1.65683
\(800\) 0 0
\(801\) −8.83282 −0.312092
\(802\) 0 0
\(803\) −19.0557 −0.672462
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.16718 −0.181894
\(808\) 0 0
\(809\) −17.4164 −0.612328 −0.306164 0.951979i \(-0.599046\pi\)
−0.306164 + 0.951979i \(0.599046\pi\)
\(810\) 0 0
\(811\) 53.0132 1.86154 0.930772 0.365601i \(-0.119136\pi\)
0.930772 + 0.365601i \(0.119136\pi\)
\(812\) 0 0
\(813\) −29.6656 −1.04042
\(814\) 0 0
\(815\) 11.0557 0.387265
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.11146 0.143491 0.0717454 0.997423i \(-0.477143\pi\)
0.0717454 + 0.997423i \(0.477143\pi\)
\(822\) 0 0
\(823\) −19.7771 −0.689386 −0.344693 0.938715i \(-0.612017\pi\)
−0.344693 + 0.938715i \(0.612017\pi\)
\(824\) 0 0
\(825\) 43.7771 1.52412
\(826\) 0 0
\(827\) 15.0557 0.523539 0.261769 0.965130i \(-0.415694\pi\)
0.261769 + 0.965130i \(0.415694\pi\)
\(828\) 0 0
\(829\) −11.2361 −0.390245 −0.195122 0.980779i \(-0.562510\pi\)
−0.195122 + 0.980779i \(0.562510\pi\)
\(830\) 0 0
\(831\) −9.75078 −0.338251
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 75.7771 2.62237
\(836\) 0 0
\(837\) 13.6656 0.472353
\(838\) 0 0
\(839\) 21.5279 0.743224 0.371612 0.928388i \(-0.378805\pi\)
0.371612 + 0.928388i \(0.378805\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) −32.1378 −1.10688
\(844\) 0 0
\(845\) 40.1803 1.38225
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −7.63932 −0.262181
\(850\) 0 0
\(851\) −17.8885 −0.613211
\(852\) 0 0
\(853\) −13.7082 −0.469360 −0.234680 0.972073i \(-0.575404\pi\)
−0.234680 + 0.972073i \(0.575404\pi\)
\(854\) 0 0
\(855\) 5.88854 0.201384
\(856\) 0 0
\(857\) 27.5279 0.940334 0.470167 0.882577i \(-0.344194\pi\)
0.470167 + 0.882577i \(0.344194\pi\)
\(858\) 0 0
\(859\) 33.8197 1.15391 0.576956 0.816775i \(-0.304240\pi\)
0.576956 + 0.816775i \(0.304240\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.9443 −0.712951 −0.356476 0.934305i \(-0.616022\pi\)
−0.356476 + 0.934305i \(0.616022\pi\)
\(864\) 0 0
\(865\) 18.4721 0.628071
\(866\) 0 0
\(867\) −3.70820 −0.125937
\(868\) 0 0
\(869\) −83.7771 −2.84194
\(870\) 0 0
\(871\) −3.05573 −0.103539
\(872\) 0 0
\(873\) 18.3607 0.621415
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.4164 0.453040 0.226520 0.974007i \(-0.427265\pi\)
0.226520 + 0.974007i \(0.427265\pi\)
\(878\) 0 0
\(879\) −15.7771 −0.532148
\(880\) 0 0
\(881\) −24.8328 −0.836639 −0.418319 0.908300i \(-0.637381\pi\)
−0.418319 + 0.908300i \(0.637381\pi\)
\(882\) 0 0
\(883\) 23.0557 0.775887 0.387944 0.921683i \(-0.373186\pi\)
0.387944 + 0.921683i \(0.373186\pi\)
\(884\) 0 0
\(885\) 36.9443 1.24187
\(886\) 0 0
\(887\) 33.3050 1.11827 0.559135 0.829076i \(-0.311133\pi\)
0.559135 + 0.829076i \(0.311133\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 15.6393 0.523937
\(892\) 0 0
\(893\) −12.9443 −0.433164
\(894\) 0 0
\(895\) 22.8328 0.763217
\(896\) 0 0
\(897\) 3.77709 0.126113
\(898\) 0 0
\(899\) −11.0557 −0.368729
\(900\) 0 0
\(901\) 44.7214 1.48988
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −39.4164 −1.31025
\(906\) 0 0
\(907\) 0.944272 0.0313540 0.0156770 0.999877i \(-0.495010\pi\)
0.0156770 + 0.999877i \(0.495010\pi\)
\(908\) 0 0
\(909\) −2.51471 −0.0834076
\(910\) 0 0
\(911\) −34.8328 −1.15406 −0.577031 0.816722i \(-0.695789\pi\)
−0.577031 + 0.816722i \(0.695789\pi\)
\(912\) 0 0
\(913\) −59.7771 −1.97833
\(914\) 0 0
\(915\) −44.9443 −1.48581
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −35.7771 −1.18018 −0.590089 0.807338i \(-0.700907\pi\)
−0.590089 + 0.807338i \(0.700907\pi\)
\(920\) 0 0
\(921\) 2.24922 0.0741144
\(922\) 0 0
\(923\) 3.77709 0.124324
\(924\) 0 0
\(925\) −24.4721 −0.804639
\(926\) 0 0
\(927\) −8.13777 −0.267279
\(928\) 0 0
\(929\) 47.3050 1.55203 0.776013 0.630717i \(-0.217239\pi\)
0.776013 + 0.630717i \(0.217239\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.88854 0.323736
\(934\) 0 0
\(935\) −93.6656 −3.06319
\(936\) 0 0
\(937\) 9.05573 0.295838 0.147919 0.988999i \(-0.452743\pi\)
0.147919 + 0.988999i \(0.452743\pi\)
\(938\) 0 0
\(939\) −10.4721 −0.341745
\(940\) 0 0
\(941\) 35.5967 1.16042 0.580210 0.814467i \(-0.302970\pi\)
0.580210 + 0.814467i \(0.302970\pi\)
\(942\) 0 0
\(943\) 33.8885 1.10356
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.58359 −0.148947 −0.0744734 0.997223i \(-0.523728\pi\)
−0.0744734 + 0.997223i \(0.523728\pi\)
\(948\) 0 0
\(949\) −2.24922 −0.0730129
\(950\) 0 0
\(951\) −11.1935 −0.362974
\(952\) 0 0
\(953\) 51.8885 1.68083 0.840417 0.541940i \(-0.182310\pi\)
0.840417 + 0.541940i \(0.182310\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) −35.7771 −1.15651
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) −13.1672 −0.424307
\(964\) 0 0
\(965\) −1.52786 −0.0491837
\(966\) 0 0
\(967\) 29.8885 0.961151 0.480575 0.876953i \(-0.340428\pi\)
0.480575 + 0.876953i \(0.340428\pi\)
\(968\) 0 0
\(969\) 6.83282 0.219502
\(970\) 0 0
\(971\) −22.7639 −0.730529 −0.365265 0.930904i \(-0.619022\pi\)
−0.365265 + 0.930904i \(0.619022\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 5.16718 0.165482
\(976\) 0 0
\(977\) −12.8328 −0.410558 −0.205279 0.978703i \(-0.565810\pi\)
−0.205279 + 0.978703i \(0.565810\pi\)
\(978\) 0 0
\(979\) −38.8328 −1.24110
\(980\) 0 0
\(981\) −12.4721 −0.398205
\(982\) 0 0
\(983\) 5.52786 0.176311 0.0881557 0.996107i \(-0.471903\pi\)
0.0881557 + 0.996107i \(0.471903\pi\)
\(984\) 0 0
\(985\) −35.4164 −1.12846
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25.8885 0.823208
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) −27.7771 −0.881479
\(994\) 0 0
\(995\) 49.8885 1.58157
\(996\) 0 0
\(997\) −18.0689 −0.572247 −0.286124 0.958193i \(-0.592367\pi\)
−0.286124 + 0.958193i \(0.592367\pi\)
\(998\) 0 0
\(999\) −24.7214 −0.782149
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 1568.2.a.v.1.1 2
4.3 odd 2 1568.2.a.k.1.2 2
7.2 even 3 1568.2.i.n.1537.2 4
7.3 odd 6 1568.2.i.v.961.1 4
7.4 even 3 1568.2.i.n.961.2 4
7.5 odd 6 1568.2.i.v.1537.1 4
7.6 odd 2 224.2.a.c.1.2 2
8.3 odd 2 3136.2.a.by.1.1 2
8.5 even 2 3136.2.a.bf.1.2 2
21.20 even 2 2016.2.a.r.1.1 2
28.3 even 6 1568.2.i.m.961.2 4
28.11 odd 6 1568.2.i.w.961.1 4
28.19 even 6 1568.2.i.m.1537.2 4
28.23 odd 6 1568.2.i.w.1537.1 4
28.27 even 2 224.2.a.d.1.1 yes 2
35.34 odd 2 5600.2.a.bk.1.1 2
56.13 odd 2 448.2.a.j.1.1 2
56.27 even 2 448.2.a.i.1.2 2
84.83 odd 2 2016.2.a.o.1.1 2
112.13 odd 4 1792.2.b.k.897.3 4
112.27 even 4 1792.2.b.m.897.3 4
112.69 odd 4 1792.2.b.k.897.2 4
112.83 even 4 1792.2.b.m.897.2 4
140.139 even 2 5600.2.a.z.1.2 2
168.83 odd 2 4032.2.a.bv.1.2 2
168.125 even 2 4032.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.2 2 7.6 odd 2
224.2.a.d.1.1 yes 2 28.27 even 2
448.2.a.i.1.2 2 56.27 even 2
448.2.a.j.1.1 2 56.13 odd 2
1568.2.a.k.1.2 2 4.3 odd 2
1568.2.a.v.1.1 2 1.1 even 1 trivial
1568.2.i.m.961.2 4 28.3 even 6
1568.2.i.m.1537.2 4 28.19 even 6
1568.2.i.n.961.2 4 7.4 even 3
1568.2.i.n.1537.2 4 7.2 even 3
1568.2.i.v.961.1 4 7.3 odd 6
1568.2.i.v.1537.1 4 7.5 odd 6
1568.2.i.w.961.1 4 28.11 odd 6
1568.2.i.w.1537.1 4 28.23 odd 6
1792.2.b.k.897.2 4 112.69 odd 4
1792.2.b.k.897.3 4 112.13 odd 4
1792.2.b.m.897.2 4 112.83 even 4
1792.2.b.m.897.3 4 112.27 even 4
2016.2.a.o.1.1 2 84.83 odd 2
2016.2.a.r.1.1 2 21.20 even 2
3136.2.a.bf.1.2 2 8.5 even 2
3136.2.a.by.1.1 2 8.3 odd 2
4032.2.a.bv.1.2 2 168.83 odd 2
4032.2.a.bw.1.2 2 168.125 even 2
5600.2.a.z.1.2 2 140.139 even 2
5600.2.a.bk.1.1 2 35.34 odd 2