Properties

Label 7728.2.a.bl.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.19258\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.19258 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} +3.19258 q^{13} -2.19258 q^{15} +1.00000 q^{17} +3.38516 q^{19} +1.00000 q^{21} -1.00000 q^{23} -0.192582 q^{25} +1.00000 q^{27} -5.00000 q^{29} +3.38516 q^{31} -5.00000 q^{33} -2.19258 q^{35} +5.38516 q^{37} +3.19258 q^{39} -1.00000 q^{41} -5.19258 q^{43} -2.19258 q^{45} -10.3852 q^{47} +1.00000 q^{49} +1.00000 q^{51} -10.5777 q^{53} +10.9629 q^{55} +3.38516 q^{57} +12.5777 q^{59} +9.19258 q^{61} +1.00000 q^{63} -7.00000 q^{65} -2.19258 q^{67} -1.00000 q^{69} -13.5777 q^{71} -9.00000 q^{73} -0.192582 q^{75} -5.00000 q^{77} +9.77033 q^{79} +1.00000 q^{81} +11.3852 q^{83} -2.19258 q^{85} -5.00000 q^{87} +3.57775 q^{89} +3.19258 q^{91} +3.38516 q^{93} -7.42225 q^{95} -17.7703 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 10 q^{11} + q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 4 q^{31} - 10 q^{33} + q^{35} + q^{39} - 2 q^{41} - 5 q^{43} + q^{45} - 10 q^{47} + 2 q^{49} + 2 q^{51} - 5 q^{53} - 5 q^{55} - 4 q^{57} + 9 q^{59} + 13 q^{61} + 2 q^{63} - 14 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} - 18 q^{73} + 5 q^{75} - 10 q^{77} - 2 q^{79} + 2 q^{81} + 12 q^{83} + q^{85} - 10 q^{87} - 9 q^{89} + q^{91} - 4 q^{93} - 31 q^{95} - 14 q^{97} - 10 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −2.19258 −0.980553 −0.490276 0.871567i \(-0.663104\pi\)
−0.490276 + 0.871567i \(0.663104\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 3.19258 0.885463 0.442732 0.896654i \(-0.354009\pi\)
0.442732 + 0.896654i \(0.354009\pi\)
\(14\) 0 0
\(15\) −2.19258 −0.566122
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 3.38516 0.776610 0.388305 0.921531i \(-0.373061\pi\)
0.388305 + 0.921531i \(0.373061\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −0.192582 −0.0385165
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.38516 0.607994 0.303997 0.952673i \(-0.401679\pi\)
0.303997 + 0.952673i \(0.401679\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −2.19258 −0.370614
\(36\) 0 0
\(37\) 5.38516 0.885316 0.442658 0.896691i \(-0.354036\pi\)
0.442658 + 0.896691i \(0.354036\pi\)
\(38\) 0 0
\(39\) 3.19258 0.511222
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) −5.19258 −0.791861 −0.395931 0.918280i \(-0.629578\pi\)
−0.395931 + 0.918280i \(0.629578\pi\)
\(44\) 0 0
\(45\) −2.19258 −0.326851
\(46\) 0 0
\(47\) −10.3852 −1.51483 −0.757416 0.652933i \(-0.773538\pi\)
−0.757416 + 0.652933i \(0.773538\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −10.5777 −1.45297 −0.726483 0.687185i \(-0.758846\pi\)
−0.726483 + 0.687185i \(0.758846\pi\)
\(54\) 0 0
\(55\) 10.9629 1.47824
\(56\) 0 0
\(57\) 3.38516 0.448376
\(58\) 0 0
\(59\) 12.5777 1.63748 0.818742 0.574162i \(-0.194672\pi\)
0.818742 + 0.574162i \(0.194672\pi\)
\(60\) 0 0
\(61\) 9.19258 1.17699 0.588495 0.808501i \(-0.299721\pi\)
0.588495 + 0.808501i \(0.299721\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −7.00000 −0.868243
\(66\) 0 0
\(67\) −2.19258 −0.267867 −0.133933 0.990990i \(-0.542761\pi\)
−0.133933 + 0.990990i \(0.542761\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −13.5777 −1.61138 −0.805691 0.592336i \(-0.798206\pi\)
−0.805691 + 0.592336i \(0.798206\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) −0.192582 −0.0222375
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 9.77033 1.09925 0.549624 0.835412i \(-0.314771\pi\)
0.549624 + 0.835412i \(0.314771\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.3852 1.24968 0.624842 0.780751i \(-0.285163\pi\)
0.624842 + 0.780751i \(0.285163\pi\)
\(84\) 0 0
\(85\) −2.19258 −0.237819
\(86\) 0 0
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) 3.57775 0.379240 0.189620 0.981858i \(-0.439274\pi\)
0.189620 + 0.981858i \(0.439274\pi\)
\(90\) 0 0
\(91\) 3.19258 0.334674
\(92\) 0 0
\(93\) 3.38516 0.351025
\(94\) 0 0
\(95\) −7.42225 −0.761507
\(96\) 0 0
\(97\) −17.7703 −1.80430 −0.902152 0.431419i \(-0.858013\pi\)
−0.902152 + 0.431419i \(0.858013\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 14.5777 1.45054 0.725270 0.688465i \(-0.241715\pi\)
0.725270 + 0.688465i \(0.241715\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −2.19258 −0.213974
\(106\) 0 0
\(107\) −17.5777 −1.69930 −0.849652 0.527343i \(-0.823188\pi\)
−0.849652 + 0.527343i \(0.823188\pi\)
\(108\) 0 0
\(109\) 3.42225 0.327792 0.163896 0.986478i \(-0.447594\pi\)
0.163896 + 0.986478i \(0.447594\pi\)
\(110\) 0 0
\(111\) 5.38516 0.511137
\(112\) 0 0
\(113\) 4.57775 0.430638 0.215319 0.976544i \(-0.430921\pi\)
0.215319 + 0.976544i \(0.430921\pi\)
\(114\) 0 0
\(115\) 2.19258 0.204459
\(116\) 0 0
\(117\) 3.19258 0.295154
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −1.00000 −0.0901670
\(124\) 0 0
\(125\) 11.3852 1.01832
\(126\) 0 0
\(127\) −9.19258 −0.815710 −0.407855 0.913047i \(-0.633723\pi\)
−0.407855 + 0.913047i \(0.633723\pi\)
\(128\) 0 0
\(129\) −5.19258 −0.457181
\(130\) 0 0
\(131\) −9.77033 −0.853638 −0.426819 0.904337i \(-0.640366\pi\)
−0.426819 + 0.904337i \(0.640366\pi\)
\(132\) 0 0
\(133\) 3.38516 0.293531
\(134\) 0 0
\(135\) −2.19258 −0.188707
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) −0.577747 −0.0490039 −0.0245019 0.999700i \(-0.507800\pi\)
−0.0245019 + 0.999700i \(0.507800\pi\)
\(140\) 0 0
\(141\) −10.3852 −0.874589
\(142\) 0 0
\(143\) −15.9629 −1.33489
\(144\) 0 0
\(145\) 10.9629 0.910420
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 10.3852 0.850786 0.425393 0.905009i \(-0.360136\pi\)
0.425393 + 0.905009i \(0.360136\pi\)
\(150\) 0 0
\(151\) 2.38516 0.194102 0.0970510 0.995279i \(-0.469059\pi\)
0.0970510 + 0.995279i \(0.469059\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −7.42225 −0.596170
\(156\) 0 0
\(157\) −7.61484 −0.607730 −0.303865 0.952715i \(-0.598277\pi\)
−0.303865 + 0.952715i \(0.598277\pi\)
\(158\) 0 0
\(159\) −10.5777 −0.838870
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −19.3481 −1.51546 −0.757729 0.652569i \(-0.773691\pi\)
−0.757729 + 0.652569i \(0.773691\pi\)
\(164\) 0 0
\(165\) 10.9629 0.853462
\(166\) 0 0
\(167\) −23.0000 −1.77979 −0.889897 0.456162i \(-0.849224\pi\)
−0.889897 + 0.456162i \(0.849224\pi\)
\(168\) 0 0
\(169\) −2.80742 −0.215955
\(170\) 0 0
\(171\) 3.38516 0.258870
\(172\) 0 0
\(173\) −15.7703 −1.19900 −0.599498 0.800376i \(-0.704633\pi\)
−0.599498 + 0.800376i \(0.704633\pi\)
\(174\) 0 0
\(175\) −0.192582 −0.0145579
\(176\) 0 0
\(177\) 12.5777 0.945401
\(178\) 0 0
\(179\) −20.5777 −1.53805 −0.769027 0.639217i \(-0.779259\pi\)
−0.769027 + 0.639217i \(0.779259\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 9.19258 0.679535
\(184\) 0 0
\(185\) −11.8074 −0.868099
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −10.3852 −0.751444 −0.375722 0.926732i \(-0.622605\pi\)
−0.375722 + 0.926732i \(0.622605\pi\)
\(192\) 0 0
\(193\) −8.38516 −0.603577 −0.301789 0.953375i \(-0.597584\pi\)
−0.301789 + 0.953375i \(0.597584\pi\)
\(194\) 0 0
\(195\) −7.00000 −0.501280
\(196\) 0 0
\(197\) 9.19258 0.654944 0.327472 0.944861i \(-0.393803\pi\)
0.327472 + 0.944861i \(0.393803\pi\)
\(198\) 0 0
\(199\) 13.3481 0.946220 0.473110 0.881003i \(-0.343131\pi\)
0.473110 + 0.881003i \(0.343131\pi\)
\(200\) 0 0
\(201\) −2.19258 −0.154653
\(202\) 0 0
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 2.19258 0.153137
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −16.9258 −1.17078
\(210\) 0 0
\(211\) −13.3852 −0.921473 −0.460736 0.887537i \(-0.652415\pi\)
−0.460736 + 0.887537i \(0.652415\pi\)
\(212\) 0 0
\(213\) −13.5777 −0.930332
\(214\) 0 0
\(215\) 11.3852 0.776462
\(216\) 0 0
\(217\) 3.38516 0.229800
\(218\) 0 0
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 3.19258 0.214756
\(222\) 0 0
\(223\) −12.5777 −0.842268 −0.421134 0.906998i \(-0.638368\pi\)
−0.421134 + 0.906998i \(0.638368\pi\)
\(224\) 0 0
\(225\) −0.192582 −0.0128388
\(226\) 0 0
\(227\) −0.577747 −0.0383464 −0.0191732 0.999816i \(-0.506103\pi\)
−0.0191732 + 0.999816i \(0.506103\pi\)
\(228\) 0 0
\(229\) 4.19258 0.277054 0.138527 0.990359i \(-0.455763\pi\)
0.138527 + 0.990359i \(0.455763\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −2.96291 −0.194107 −0.0970534 0.995279i \(-0.530942\pi\)
−0.0970534 + 0.995279i \(0.530942\pi\)
\(234\) 0 0
\(235\) 22.7703 1.48537
\(236\) 0 0
\(237\) 9.77033 0.634651
\(238\) 0 0
\(239\) −22.1926 −1.43552 −0.717759 0.696291i \(-0.754832\pi\)
−0.717759 + 0.696291i \(0.754832\pi\)
\(240\) 0 0
\(241\) −19.7703 −1.27352 −0.636759 0.771063i \(-0.719726\pi\)
−0.636759 + 0.771063i \(0.719726\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.19258 −0.140079
\(246\) 0 0
\(247\) 10.8074 0.687660
\(248\) 0 0
\(249\) 11.3852 0.721506
\(250\) 0 0
\(251\) −11.6148 −0.733122 −0.366561 0.930394i \(-0.619465\pi\)
−0.366561 + 0.930394i \(0.619465\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −2.19258 −0.137305
\(256\) 0 0
\(257\) −3.15549 −0.196834 −0.0984172 0.995145i \(-0.531378\pi\)
−0.0984172 + 0.995145i \(0.531378\pi\)
\(258\) 0 0
\(259\) 5.38516 0.334618
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) −24.1555 −1.48949 −0.744746 0.667348i \(-0.767429\pi\)
−0.744746 + 0.667348i \(0.767429\pi\)
\(264\) 0 0
\(265\) 23.1926 1.42471
\(266\) 0 0
\(267\) 3.57775 0.218955
\(268\) 0 0
\(269\) 29.9629 1.82687 0.913435 0.406984i \(-0.133419\pi\)
0.913435 + 0.406984i \(0.133419\pi\)
\(270\) 0 0
\(271\) 5.00000 0.303728 0.151864 0.988401i \(-0.451472\pi\)
0.151864 + 0.988401i \(0.451472\pi\)
\(272\) 0 0
\(273\) 3.19258 0.193224
\(274\) 0 0
\(275\) 0.962912 0.0580658
\(276\) 0 0
\(277\) −13.1926 −0.792665 −0.396333 0.918107i \(-0.629717\pi\)
−0.396333 + 0.918107i \(0.629717\pi\)
\(278\) 0 0
\(279\) 3.38516 0.202665
\(280\) 0 0
\(281\) −16.3852 −0.977457 −0.488728 0.872436i \(-0.662539\pi\)
−0.488728 + 0.872436i \(0.662539\pi\)
\(282\) 0 0
\(283\) 5.03709 0.299424 0.149712 0.988730i \(-0.452165\pi\)
0.149712 + 0.988730i \(0.452165\pi\)
\(284\) 0 0
\(285\) −7.42225 −0.439656
\(286\) 0 0
\(287\) −1.00000 −0.0590281
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −17.7703 −1.04172
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) −27.5777 −1.60564
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −3.19258 −0.184632
\(300\) 0 0
\(301\) −5.19258 −0.299295
\(302\) 0 0
\(303\) 14.5777 0.837470
\(304\) 0 0
\(305\) −20.1555 −1.15410
\(306\) 0 0
\(307\) 0.614835 0.0350905 0.0175452 0.999846i \(-0.494415\pi\)
0.0175452 + 0.999846i \(0.494415\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 3.96291 0.224716 0.112358 0.993668i \(-0.464160\pi\)
0.112358 + 0.993668i \(0.464160\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 0 0
\(315\) −2.19258 −0.123538
\(316\) 0 0
\(317\) 22.1926 1.24646 0.623230 0.782039i \(-0.285820\pi\)
0.623230 + 0.782039i \(0.285820\pi\)
\(318\) 0 0
\(319\) 25.0000 1.39973
\(320\) 0 0
\(321\) −17.5777 −0.981094
\(322\) 0 0
\(323\) 3.38516 0.188356
\(324\) 0 0
\(325\) −0.614835 −0.0341049
\(326\) 0 0
\(327\) 3.42225 0.189251
\(328\) 0 0
\(329\) −10.3852 −0.572553
\(330\) 0 0
\(331\) −13.2297 −0.727168 −0.363584 0.931561i \(-0.618447\pi\)
−0.363584 + 0.931561i \(0.618447\pi\)
\(332\) 0 0
\(333\) 5.38516 0.295105
\(334\) 0 0
\(335\) 4.80742 0.262657
\(336\) 0 0
\(337\) −11.9629 −0.651661 −0.325831 0.945428i \(-0.605644\pi\)
−0.325831 + 0.945428i \(0.605644\pi\)
\(338\) 0 0
\(339\) 4.57775 0.248629
\(340\) 0 0
\(341\) −16.9258 −0.916585
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.19258 0.118045
\(346\) 0 0
\(347\) 22.1555 1.18937 0.594685 0.803959i \(-0.297277\pi\)
0.594685 + 0.803959i \(0.297277\pi\)
\(348\) 0 0
\(349\) −4.19258 −0.224424 −0.112212 0.993684i \(-0.535794\pi\)
−0.112212 + 0.993684i \(0.535794\pi\)
\(350\) 0 0
\(351\) 3.19258 0.170407
\(352\) 0 0
\(353\) −29.7703 −1.58451 −0.792257 0.610187i \(-0.791094\pi\)
−0.792257 + 0.610187i \(0.791094\pi\)
\(354\) 0 0
\(355\) 29.7703 1.58005
\(356\) 0 0
\(357\) 1.00000 0.0529256
\(358\) 0 0
\(359\) 1.42225 0.0750636 0.0375318 0.999295i \(-0.488050\pi\)
0.0375318 + 0.999295i \(0.488050\pi\)
\(360\) 0 0
\(361\) −7.54066 −0.396877
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 19.7332 1.03289
\(366\) 0 0
\(367\) −17.9629 −0.937656 −0.468828 0.883289i \(-0.655324\pi\)
−0.468828 + 0.883289i \(0.655324\pi\)
\(368\) 0 0
\(369\) −1.00000 −0.0520579
\(370\) 0 0
\(371\) −10.5777 −0.549169
\(372\) 0 0
\(373\) −8.15549 −0.422275 −0.211138 0.977456i \(-0.567717\pi\)
−0.211138 + 0.977456i \(0.567717\pi\)
\(374\) 0 0
\(375\) 11.3852 0.587927
\(376\) 0 0
\(377\) −15.9629 −0.822132
\(378\) 0 0
\(379\) −29.5407 −1.51740 −0.758701 0.651439i \(-0.774166\pi\)
−0.758701 + 0.651439i \(0.774166\pi\)
\(380\) 0 0
\(381\) −9.19258 −0.470950
\(382\) 0 0
\(383\) 37.3852 1.91029 0.955146 0.296134i \(-0.0956976\pi\)
0.955146 + 0.296134i \(0.0956976\pi\)
\(384\) 0 0
\(385\) 10.9629 0.558722
\(386\) 0 0
\(387\) −5.19258 −0.263954
\(388\) 0 0
\(389\) 28.9258 1.46660 0.733299 0.679907i \(-0.237980\pi\)
0.733299 + 0.679907i \(0.237980\pi\)
\(390\) 0 0
\(391\) −1.00000 −0.0505722
\(392\) 0 0
\(393\) −9.77033 −0.492848
\(394\) 0 0
\(395\) −21.4223 −1.07787
\(396\) 0 0
\(397\) −19.9258 −1.00005 −0.500024 0.866011i \(-0.666676\pi\)
−0.500024 + 0.866011i \(0.666676\pi\)
\(398\) 0 0
\(399\) 3.38516 0.169470
\(400\) 0 0
\(401\) −6.61484 −0.330329 −0.165165 0.986266i \(-0.552816\pi\)
−0.165165 + 0.986266i \(0.552816\pi\)
\(402\) 0 0
\(403\) 10.8074 0.538356
\(404\) 0 0
\(405\) −2.19258 −0.108950
\(406\) 0 0
\(407\) −26.9258 −1.33466
\(408\) 0 0
\(409\) 20.9258 1.03472 0.517358 0.855769i \(-0.326916\pi\)
0.517358 + 0.855769i \(0.326916\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) 12.5777 0.618910
\(414\) 0 0
\(415\) −24.9629 −1.22538
\(416\) 0 0
\(417\) −0.577747 −0.0282924
\(418\) 0 0
\(419\) −20.1926 −0.986472 −0.493236 0.869895i \(-0.664186\pi\)
−0.493236 + 0.869895i \(0.664186\pi\)
\(420\) 0 0
\(421\) 8.42225 0.410475 0.205238 0.978712i \(-0.434203\pi\)
0.205238 + 0.978712i \(0.434203\pi\)
\(422\) 0 0
\(423\) −10.3852 −0.504944
\(424\) 0 0
\(425\) −0.192582 −0.00934162
\(426\) 0 0
\(427\) 9.19258 0.444860
\(428\) 0 0
\(429\) −15.9629 −0.770697
\(430\) 0 0
\(431\) −27.9629 −1.34693 −0.673463 0.739221i \(-0.735194\pi\)
−0.673463 + 0.739221i \(0.735194\pi\)
\(432\) 0 0
\(433\) −11.6148 −0.558173 −0.279087 0.960266i \(-0.590032\pi\)
−0.279087 + 0.960266i \(0.590032\pi\)
\(434\) 0 0
\(435\) 10.9629 0.525631
\(436\) 0 0
\(437\) −3.38516 −0.161934
\(438\) 0 0
\(439\) −34.9258 −1.66692 −0.833459 0.552581i \(-0.813643\pi\)
−0.833459 + 0.552581i \(0.813643\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 30.3110 1.44012 0.720059 0.693913i \(-0.244115\pi\)
0.720059 + 0.693913i \(0.244115\pi\)
\(444\) 0 0
\(445\) −7.84451 −0.371865
\(446\) 0 0
\(447\) 10.3852 0.491201
\(448\) 0 0
\(449\) 6.19258 0.292246 0.146123 0.989266i \(-0.453320\pi\)
0.146123 + 0.989266i \(0.453320\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 0 0
\(453\) 2.38516 0.112065
\(454\) 0 0
\(455\) −7.00000 −0.328165
\(456\) 0 0
\(457\) 29.3481 1.37285 0.686423 0.727203i \(-0.259180\pi\)
0.686423 + 0.727203i \(0.259180\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) −22.9629 −1.06949 −0.534745 0.845014i \(-0.679592\pi\)
−0.534745 + 0.845014i \(0.679592\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 0 0
\(465\) −7.42225 −0.344199
\(466\) 0 0
\(467\) −24.9258 −1.15343 −0.576715 0.816946i \(-0.695666\pi\)
−0.576715 + 0.816946i \(0.695666\pi\)
\(468\) 0 0
\(469\) −2.19258 −0.101244
\(470\) 0 0
\(471\) −7.61484 −0.350873
\(472\) 0 0
\(473\) 25.9629 1.19378
\(474\) 0 0
\(475\) −0.651923 −0.0299123
\(476\) 0 0
\(477\) −10.5777 −0.484322
\(478\) 0 0
\(479\) 11.0000 0.502603 0.251301 0.967909i \(-0.419141\pi\)
0.251301 + 0.967909i \(0.419141\pi\)
\(480\) 0 0
\(481\) 17.1926 0.783914
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 38.9629 1.76921
\(486\) 0 0
\(487\) 27.7703 1.25839 0.629197 0.777246i \(-0.283384\pi\)
0.629197 + 0.777246i \(0.283384\pi\)
\(488\) 0 0
\(489\) −19.3481 −0.874950
\(490\) 0 0
\(491\) −6.03709 −0.272450 −0.136225 0.990678i \(-0.543497\pi\)
−0.136225 + 0.990678i \(0.543497\pi\)
\(492\) 0 0
\(493\) −5.00000 −0.225189
\(494\) 0 0
\(495\) 10.9629 0.492746
\(496\) 0 0
\(497\) −13.5777 −0.609045
\(498\) 0 0
\(499\) 0.192582 0.00862117 0.00431059 0.999991i \(-0.498628\pi\)
0.00431059 + 0.999991i \(0.498628\pi\)
\(500\) 0 0
\(501\) −23.0000 −1.02756
\(502\) 0 0
\(503\) 7.42225 0.330942 0.165471 0.986215i \(-0.447086\pi\)
0.165471 + 0.986215i \(0.447086\pi\)
\(504\) 0 0
\(505\) −31.9629 −1.42233
\(506\) 0 0
\(507\) −2.80742 −0.124682
\(508\) 0 0
\(509\) 10.3852 0.460314 0.230157 0.973153i \(-0.426076\pi\)
0.230157 + 0.973153i \(0.426076\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 0 0
\(513\) 3.38516 0.149459
\(514\) 0 0
\(515\) 17.5407 0.772934
\(516\) 0 0
\(517\) 51.9258 2.28370
\(518\) 0 0
\(519\) −15.7703 −0.692241
\(520\) 0 0
\(521\) −1.22967 −0.0538728 −0.0269364 0.999637i \(-0.508575\pi\)
−0.0269364 + 0.999637i \(0.508575\pi\)
\(522\) 0 0
\(523\) 2.77033 0.121138 0.0605690 0.998164i \(-0.480708\pi\)
0.0605690 + 0.998164i \(0.480708\pi\)
\(524\) 0 0
\(525\) −0.192582 −0.00840499
\(526\) 0 0
\(527\) 3.38516 0.147460
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.5777 0.545828
\(532\) 0 0
\(533\) −3.19258 −0.138286
\(534\) 0 0
\(535\) 38.5407 1.66626
\(536\) 0 0
\(537\) −20.5777 −0.887995
\(538\) 0 0
\(539\) −5.00000 −0.215365
\(540\) 0 0
\(541\) −18.3852 −0.790440 −0.395220 0.918587i \(-0.629332\pi\)
−0.395220 + 0.918587i \(0.629332\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) −7.50357 −0.321418
\(546\) 0 0
\(547\) −20.5777 −0.879841 −0.439920 0.898037i \(-0.644993\pi\)
−0.439920 + 0.898037i \(0.644993\pi\)
\(548\) 0 0
\(549\) 9.19258 0.392330
\(550\) 0 0
\(551\) −16.9258 −0.721064
\(552\) 0 0
\(553\) 9.77033 0.415477
\(554\) 0 0
\(555\) −11.8074 −0.501197
\(556\) 0 0
\(557\) 9.15549 0.387931 0.193965 0.981008i \(-0.437865\pi\)
0.193965 + 0.981008i \(0.437865\pi\)
\(558\) 0 0
\(559\) −16.5777 −0.701164
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) 0 0
\(563\) 9.96291 0.419887 0.209943 0.977714i \(-0.432672\pi\)
0.209943 + 0.977714i \(0.432672\pi\)
\(564\) 0 0
\(565\) −10.0371 −0.422263
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 37.5407 1.57379 0.786893 0.617089i \(-0.211688\pi\)
0.786893 + 0.617089i \(0.211688\pi\)
\(570\) 0 0
\(571\) 7.77033 0.325178 0.162589 0.986694i \(-0.448016\pi\)
0.162589 + 0.986694i \(0.448016\pi\)
\(572\) 0 0
\(573\) −10.3852 −0.433846
\(574\) 0 0
\(575\) 0.192582 0.00803124
\(576\) 0 0
\(577\) −17.2297 −0.717281 −0.358640 0.933476i \(-0.616760\pi\)
−0.358640 + 0.933476i \(0.616760\pi\)
\(578\) 0 0
\(579\) −8.38516 −0.348476
\(580\) 0 0
\(581\) 11.3852 0.472336
\(582\) 0 0
\(583\) 52.8887 2.19043
\(584\) 0 0
\(585\) −7.00000 −0.289414
\(586\) 0 0
\(587\) −20.3481 −0.839855 −0.419928 0.907558i \(-0.637945\pi\)
−0.419928 + 0.907558i \(0.637945\pi\)
\(588\) 0 0
\(589\) 11.4593 0.472174
\(590\) 0 0
\(591\) 9.19258 0.378132
\(592\) 0 0
\(593\) −35.6148 −1.46253 −0.731263 0.682096i \(-0.761069\pi\)
−0.731263 + 0.682096i \(0.761069\pi\)
\(594\) 0 0
\(595\) −2.19258 −0.0898871
\(596\) 0 0
\(597\) 13.3481 0.546300
\(598\) 0 0
\(599\) 30.7332 1.25573 0.627863 0.778324i \(-0.283930\pi\)
0.627863 + 0.778324i \(0.283930\pi\)
\(600\) 0 0
\(601\) 26.9629 1.09984 0.549920 0.835217i \(-0.314658\pi\)
0.549920 + 0.835217i \(0.314658\pi\)
\(602\) 0 0
\(603\) −2.19258 −0.0892889
\(604\) 0 0
\(605\) −30.6962 −1.24798
\(606\) 0 0
\(607\) −0.348077 −0.0141280 −0.00706400 0.999975i \(-0.502249\pi\)
−0.00706400 + 0.999975i \(0.502249\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) −33.1555 −1.34133
\(612\) 0 0
\(613\) −8.54066 −0.344954 −0.172477 0.985014i \(-0.555177\pi\)
−0.172477 + 0.985014i \(0.555177\pi\)
\(614\) 0 0
\(615\) 2.19258 0.0884135
\(616\) 0 0
\(617\) −29.5777 −1.19076 −0.595378 0.803446i \(-0.702998\pi\)
−0.595378 + 0.803446i \(0.702998\pi\)
\(618\) 0 0
\(619\) −22.7332 −0.913726 −0.456863 0.889537i \(-0.651027\pi\)
−0.456863 + 0.889537i \(0.651027\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 3.57775 0.143339
\(624\) 0 0
\(625\) −24.0000 −0.960000
\(626\) 0 0
\(627\) −16.9258 −0.675952
\(628\) 0 0
\(629\) 5.38516 0.214721
\(630\) 0 0
\(631\) 3.84451 0.153047 0.0765237 0.997068i \(-0.475618\pi\)
0.0765237 + 0.997068i \(0.475618\pi\)
\(632\) 0 0
\(633\) −13.3852 −0.532013
\(634\) 0 0
\(635\) 20.1555 0.799846
\(636\) 0 0
\(637\) 3.19258 0.126495
\(638\) 0 0
\(639\) −13.5777 −0.537127
\(640\) 0 0
\(641\) 22.7332 0.897909 0.448955 0.893555i \(-0.351797\pi\)
0.448955 + 0.893555i \(0.351797\pi\)
\(642\) 0 0
\(643\) −39.5036 −1.55787 −0.778934 0.627105i \(-0.784240\pi\)
−0.778934 + 0.627105i \(0.784240\pi\)
\(644\) 0 0
\(645\) 11.3852 0.448290
\(646\) 0 0
\(647\) 36.8887 1.45025 0.725123 0.688619i \(-0.241783\pi\)
0.725123 + 0.688619i \(0.241783\pi\)
\(648\) 0 0
\(649\) −62.8887 −2.46860
\(650\) 0 0
\(651\) 3.38516 0.132675
\(652\) 0 0
\(653\) −30.9629 −1.21167 −0.605836 0.795589i \(-0.707161\pi\)
−0.605836 + 0.795589i \(0.707161\pi\)
\(654\) 0 0
\(655\) 21.4223 0.837037
\(656\) 0 0
\(657\) −9.00000 −0.351123
\(658\) 0 0
\(659\) −29.0000 −1.12968 −0.564840 0.825201i \(-0.691062\pi\)
−0.564840 + 0.825201i \(0.691062\pi\)
\(660\) 0 0
\(661\) 28.5407 1.11010 0.555051 0.831816i \(-0.312699\pi\)
0.555051 + 0.831816i \(0.312699\pi\)
\(662\) 0 0
\(663\) 3.19258 0.123990
\(664\) 0 0
\(665\) −7.42225 −0.287823
\(666\) 0 0
\(667\) 5.00000 0.193601
\(668\) 0 0
\(669\) −12.5777 −0.486284
\(670\) 0 0
\(671\) −45.9629 −1.77438
\(672\) 0 0
\(673\) −15.7703 −0.607902 −0.303951 0.952688i \(-0.598306\pi\)
−0.303951 + 0.952688i \(0.598306\pi\)
\(674\) 0 0
\(675\) −0.192582 −0.00741250
\(676\) 0 0
\(677\) −27.1184 −1.04225 −0.521123 0.853482i \(-0.674487\pi\)
−0.521123 + 0.853482i \(0.674487\pi\)
\(678\) 0 0
\(679\) −17.7703 −0.681963
\(680\) 0 0
\(681\) −0.577747 −0.0221393
\(682\) 0 0
\(683\) 19.3852 0.741753 0.370876 0.928682i \(-0.379057\pi\)
0.370876 + 0.928682i \(0.379057\pi\)
\(684\) 0 0
\(685\) −19.7332 −0.753968
\(686\) 0 0
\(687\) 4.19258 0.159957
\(688\) 0 0
\(689\) −33.7703 −1.28655
\(690\) 0 0
\(691\) −8.42225 −0.320398 −0.160199 0.987085i \(-0.551214\pi\)
−0.160199 + 0.987085i \(0.551214\pi\)
\(692\) 0 0
\(693\) −5.00000 −0.189934
\(694\) 0 0
\(695\) 1.26676 0.0480509
\(696\) 0 0
\(697\) −1.00000 −0.0378777
\(698\) 0 0
\(699\) −2.96291 −0.112068
\(700\) 0 0
\(701\) 2.57775 0.0973602 0.0486801 0.998814i \(-0.484499\pi\)
0.0486801 + 0.998814i \(0.484499\pi\)
\(702\) 0 0
\(703\) 18.2297 0.687545
\(704\) 0 0
\(705\) 22.7703 0.857580
\(706\) 0 0
\(707\) 14.5777 0.548253
\(708\) 0 0
\(709\) 15.1184 0.567784 0.283892 0.958856i \(-0.408374\pi\)
0.283892 + 0.958856i \(0.408374\pi\)
\(710\) 0 0
\(711\) 9.77033 0.366416
\(712\) 0 0
\(713\) −3.38516 −0.126775
\(714\) 0 0
\(715\) 35.0000 1.30893
\(716\) 0 0
\(717\) −22.1926 −0.828797
\(718\) 0 0
\(719\) −22.9258 −0.854989 −0.427494 0.904018i \(-0.640604\pi\)
−0.427494 + 0.904018i \(0.640604\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −19.7703 −0.735266
\(724\) 0 0
\(725\) 0.962912 0.0357617
\(726\) 0 0
\(727\) 52.4665 1.94587 0.972937 0.231070i \(-0.0742227\pi\)
0.972937 + 0.231070i \(0.0742227\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.19258 −0.192055
\(732\) 0 0
\(733\) −27.6148 −1.01998 −0.509989 0.860181i \(-0.670350\pi\)
−0.509989 + 0.860181i \(0.670350\pi\)
\(734\) 0 0
\(735\) −2.19258 −0.0808746
\(736\) 0 0
\(737\) 10.9629 0.403824
\(738\) 0 0
\(739\) 28.9258 1.06405 0.532027 0.846728i \(-0.321431\pi\)
0.532027 + 0.846728i \(0.321431\pi\)
\(740\) 0 0
\(741\) 10.8074 0.397020
\(742\) 0 0
\(743\) 34.9629 1.28266 0.641332 0.767263i \(-0.278382\pi\)
0.641332 + 0.767263i \(0.278382\pi\)
\(744\) 0 0
\(745\) −22.7703 −0.834240
\(746\) 0 0
\(747\) 11.3852 0.416561
\(748\) 0 0
\(749\) −17.5777 −0.642277
\(750\) 0 0
\(751\) −18.1184 −0.661150 −0.330575 0.943780i \(-0.607243\pi\)
−0.330575 + 0.943780i \(0.607243\pi\)
\(752\) 0 0
\(753\) −11.6148 −0.423268
\(754\) 0 0
\(755\) −5.22967 −0.190327
\(756\) 0 0
\(757\) 46.4665 1.68885 0.844427 0.535671i \(-0.179941\pi\)
0.844427 + 0.535671i \(0.179941\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 16.7703 0.607924 0.303962 0.952684i \(-0.401690\pi\)
0.303962 + 0.952684i \(0.401690\pi\)
\(762\) 0 0
\(763\) 3.42225 0.123894
\(764\) 0 0
\(765\) −2.19258 −0.0792730
\(766\) 0 0
\(767\) 40.1555 1.44993
\(768\) 0 0
\(769\) −2.84451 −0.102575 −0.0512877 0.998684i \(-0.516333\pi\)
−0.0512877 + 0.998684i \(0.516333\pi\)
\(770\) 0 0
\(771\) −3.15549 −0.113642
\(772\) 0 0
\(773\) 40.1555 1.44429 0.722146 0.691740i \(-0.243156\pi\)
0.722146 + 0.691740i \(0.243156\pi\)
\(774\) 0 0
\(775\) −0.651923 −0.0234178
\(776\) 0 0
\(777\) 5.38516 0.193192
\(778\) 0 0
\(779\) −3.38516 −0.121286
\(780\) 0 0
\(781\) 67.8887 2.42925
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 16.6962 0.595911
\(786\) 0 0
\(787\) −39.2739 −1.39996 −0.699982 0.714161i \(-0.746809\pi\)
−0.699982 + 0.714161i \(0.746809\pi\)
\(788\) 0 0
\(789\) −24.1555 −0.859958
\(790\) 0 0
\(791\) 4.57775 0.162766
\(792\) 0 0
\(793\) 29.3481 1.04218
\(794\) 0 0
\(795\) 23.1926 0.822556
\(796\) 0 0
\(797\) 23.9258 0.847496 0.423748 0.905780i \(-0.360714\pi\)
0.423748 + 0.905780i \(0.360714\pi\)
\(798\) 0 0
\(799\) −10.3852 −0.367401
\(800\) 0 0
\(801\) 3.57775 0.126413
\(802\) 0 0
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) 2.19258 0.0772784
\(806\) 0 0
\(807\) 29.9629 1.05474
\(808\) 0 0
\(809\) 43.1184 1.51596 0.757981 0.652276i \(-0.226186\pi\)
0.757981 + 0.652276i \(0.226186\pi\)
\(810\) 0 0
\(811\) −35.3110 −1.23994 −0.619968 0.784627i \(-0.712855\pi\)
−0.619968 + 0.784627i \(0.712855\pi\)
\(812\) 0 0
\(813\) 5.00000 0.175358
\(814\) 0 0
\(815\) 42.4223 1.48599
\(816\) 0 0
\(817\) −17.5777 −0.614968
\(818\) 0 0
\(819\) 3.19258 0.111558
\(820\) 0 0
\(821\) −5.61484 −0.195959 −0.0979795 0.995188i \(-0.531238\pi\)
−0.0979795 + 0.995188i \(0.531238\pi\)
\(822\) 0 0
\(823\) 17.3481 0.604716 0.302358 0.953194i \(-0.402226\pi\)
0.302358 + 0.953194i \(0.402226\pi\)
\(824\) 0 0
\(825\) 0.962912 0.0335243
\(826\) 0 0
\(827\) 43.2739 1.50478 0.752390 0.658717i \(-0.228901\pi\)
0.752390 + 0.658717i \(0.228901\pi\)
\(828\) 0 0
\(829\) 10.9258 0.379470 0.189735 0.981835i \(-0.439237\pi\)
0.189735 + 0.981835i \(0.439237\pi\)
\(830\) 0 0
\(831\) −13.1926 −0.457646
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) 50.4294 1.74518
\(836\) 0 0
\(837\) 3.38516 0.117008
\(838\) 0 0
\(839\) 25.5036 0.880481 0.440241 0.897880i \(-0.354893\pi\)
0.440241 + 0.897880i \(0.354893\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −16.3852 −0.564335
\(844\) 0 0
\(845\) 6.15549 0.211755
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 5.03709 0.172872
\(850\) 0 0
\(851\) −5.38516 −0.184601
\(852\) 0 0
\(853\) 29.9258 1.02464 0.512320 0.858794i \(-0.328786\pi\)
0.512320 + 0.858794i \(0.328786\pi\)
\(854\) 0 0
\(855\) −7.42225 −0.253836
\(856\) 0 0
\(857\) 37.1555 1.26921 0.634604 0.772838i \(-0.281163\pi\)
0.634604 + 0.772838i \(0.281163\pi\)
\(858\) 0 0
\(859\) 50.7703 1.73226 0.866131 0.499818i \(-0.166600\pi\)
0.866131 + 0.499818i \(0.166600\pi\)
\(860\) 0 0
\(861\) −1.00000 −0.0340799
\(862\) 0 0
\(863\) −41.9258 −1.42717 −0.713586 0.700568i \(-0.752930\pi\)
−0.713586 + 0.700568i \(0.752930\pi\)
\(864\) 0 0
\(865\) 34.5777 1.17568
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −48.8516 −1.65718
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 0 0
\(873\) −17.7703 −0.601435
\(874\) 0 0
\(875\) 11.3852 0.384889
\(876\) 0 0
\(877\) 48.0000 1.62084 0.810422 0.585846i \(-0.199238\pi\)
0.810422 + 0.585846i \(0.199238\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 46.6962 1.57323 0.786617 0.617442i \(-0.211831\pi\)
0.786617 + 0.617442i \(0.211831\pi\)
\(882\) 0 0
\(883\) −31.8887 −1.07314 −0.536571 0.843855i \(-0.680281\pi\)
−0.536571 + 0.843855i \(0.680281\pi\)
\(884\) 0 0
\(885\) −27.5777 −0.927016
\(886\) 0 0
\(887\) −39.8887 −1.33933 −0.669666 0.742662i \(-0.733563\pi\)
−0.669666 + 0.742662i \(0.733563\pi\)
\(888\) 0 0
\(889\) −9.19258 −0.308309
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 0 0
\(893\) −35.1555 −1.17643
\(894\) 0 0
\(895\) 45.1184 1.50814
\(896\) 0 0
\(897\) −3.19258 −0.106597
\(898\) 0 0
\(899\) −16.9258 −0.564508
\(900\) 0 0
\(901\) −10.5777 −0.352396
\(902\) 0 0
\(903\) −5.19258 −0.172798
\(904\) 0 0
\(905\) −15.3481 −0.510187
\(906\) 0 0
\(907\) 52.1926 1.73303 0.866513 0.499154i \(-0.166356\pi\)
0.866513 + 0.499154i \(0.166356\pi\)
\(908\) 0 0
\(909\) 14.5777 0.483513
\(910\) 0 0
\(911\) 31.1555 1.03223 0.516114 0.856520i \(-0.327378\pi\)
0.516114 + 0.856520i \(0.327378\pi\)
\(912\) 0 0
\(913\) −56.9258 −1.88397
\(914\) 0 0
\(915\) −20.1555 −0.666320
\(916\) 0 0
\(917\) −9.77033 −0.322645
\(918\) 0 0
\(919\) −22.6148 −0.745995 −0.372997 0.927832i \(-0.621670\pi\)
−0.372997 + 0.927832i \(0.621670\pi\)
\(920\) 0 0
\(921\) 0.614835 0.0202595
\(922\) 0 0
\(923\) −43.3481 −1.42682
\(924\) 0 0
\(925\) −1.03709 −0.0340992
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −51.8074 −1.69975 −0.849873 0.526987i \(-0.823322\pi\)
−0.849873 + 0.526987i \(0.823322\pi\)
\(930\) 0 0
\(931\) 3.38516 0.110944
\(932\) 0 0
\(933\) 3.96291 0.129740
\(934\) 0 0
\(935\) 10.9629 0.358526
\(936\) 0 0
\(937\) −42.1555 −1.37716 −0.688580 0.725160i \(-0.741766\pi\)
−0.688580 + 0.725160i \(0.741766\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −8.77033 −0.285905 −0.142952 0.989730i \(-0.545660\pi\)
−0.142952 + 0.989730i \(0.545660\pi\)
\(942\) 0 0
\(943\) 1.00000 0.0325645
\(944\) 0 0
\(945\) −2.19258 −0.0713247
\(946\) 0 0
\(947\) −37.2297 −1.20980 −0.604901 0.796301i \(-0.706787\pi\)
−0.604901 + 0.796301i \(0.706787\pi\)
\(948\) 0 0
\(949\) −28.7332 −0.932720
\(950\) 0 0
\(951\) 22.1926 0.719644
\(952\) 0 0
\(953\) −32.8074 −1.06274 −0.531368 0.847141i \(-0.678322\pi\)
−0.531368 + 0.847141i \(0.678322\pi\)
\(954\) 0 0
\(955\) 22.7703 0.736831
\(956\) 0 0
\(957\) 25.0000 0.808135
\(958\) 0 0
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −19.5407 −0.630344
\(962\) 0 0
\(963\) −17.5777 −0.566435
\(964\) 0 0
\(965\) 18.3852 0.591839
\(966\) 0 0
\(967\) 37.0813 1.19245 0.596227 0.802816i \(-0.296666\pi\)
0.596227 + 0.802816i \(0.296666\pi\)
\(968\) 0 0
\(969\) 3.38516 0.108747
\(970\) 0 0
\(971\) −16.4223 −0.527015 −0.263508 0.964657i \(-0.584879\pi\)
−0.263508 + 0.964657i \(0.584879\pi\)
\(972\) 0 0
\(973\) −0.577747 −0.0185217
\(974\) 0 0
\(975\) −0.614835 −0.0196905
\(976\) 0 0
\(977\) 5.96291 0.190770 0.0953852 0.995440i \(-0.469592\pi\)
0.0953852 + 0.995440i \(0.469592\pi\)
\(978\) 0 0
\(979\) −17.8887 −0.571726
\(980\) 0 0
\(981\) 3.42225 0.109264
\(982\) 0 0
\(983\) −6.61484 −0.210980 −0.105490 0.994420i \(-0.533641\pi\)
−0.105490 + 0.994420i \(0.533641\pi\)
\(984\) 0 0
\(985\) −20.1555 −0.642207
\(986\) 0 0
\(987\) −10.3852 −0.330563
\(988\) 0 0
\(989\) 5.19258 0.165115
\(990\) 0 0
\(991\) 51.5036 1.63606 0.818032 0.575172i \(-0.195065\pi\)
0.818032 + 0.575172i \(0.195065\pi\)
\(992\) 0 0
\(993\) −13.2297 −0.419831
\(994\) 0 0
\(995\) −29.2668 −0.927819
\(996\) 0 0
\(997\) −25.6962 −0.813805 −0.406903 0.913472i \(-0.633391\pi\)
−0.406903 + 0.913472i \(0.633391\pi\)
\(998\) 0 0
\(999\) 5.38516 0.170379
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 7728.2.a.bl.1.1 2
4.3 odd 2 3864.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.h.1.1 2 4.3 odd 2
7728.2.a.bl.1.1 2 1.1 even 1 trivial