Properties

Label 7728.2.a.ce.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.32973\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.73589 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.73589 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.50407 q^{11} +1.48511 q^{13} -2.73589 q^{15} +0.902098 q^{17} -2.50407 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.48511 q^{25} +1.00000 q^{27} +6.68466 q^{29} -1.09790 q^{31} +2.50407 q^{33} -2.73589 q^{35} +9.91023 q^{37} +1.48511 q^{39} -2.30826 q^{41} -5.83380 q^{43} -2.73589 q^{45} +6.74671 q^{47} +1.00000 q^{49} +0.902098 q^{51} -4.91648 q^{53} -6.85086 q^{55} -2.50407 q^{57} +7.32973 q^{59} -1.00625 q^{61} +1.00000 q^{63} -4.06311 q^{65} -11.2929 q^{67} -1.00000 q^{69} +0.362008 q^{71} -5.34411 q^{73} +2.48511 q^{75} +2.50407 q^{77} -10.4672 q^{79} +1.00000 q^{81} -7.59946 q^{83} -2.46805 q^{85} +6.68466 q^{87} +10.6245 q^{89} +1.48511 q^{91} -1.09790 q^{93} +6.85086 q^{95} +13.9102 q^{97} +2.50407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} + 4q^{7} + 4q^{9} - q^{11} + 7q^{13} + 5q^{15} + 2q^{17} + q^{19} + 4q^{21} - 4q^{23} + 11q^{25} + 4q^{27} + 2q^{29} - 6q^{31} - q^{33} + 5q^{35} + 16q^{37} + 7q^{39} + 5q^{41} - 9q^{43} + 5q^{45} + 21q^{47} + 4q^{49} + 2q^{51} + 10q^{53} - 17q^{55} + q^{57} + 26q^{59} + 2q^{61} + 4q^{63} + 26q^{65} - 5q^{67} - 4q^{69} + 19q^{71} + 10q^{73} + 11q^{75} - q^{77} + 6q^{79} + 4q^{81} + 2q^{83} - 7q^{85} + 2q^{87} - 17q^{89} + 7q^{91} - 6q^{93} + 17q^{95} + 32q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) −2.73589 −1.22353 −0.611764 0.791040i \(-0.709540\pi\)
−0.611764 + 0.791040i \(0.709540\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.50407 0.755005 0.377502 0.926009i \(-0.376783\pi\)
0.377502 + 0.926009i \(0.376783\pi\)
\(12\) 0 0
\(13\) 1.48511 0.411896 0.205948 0.978563i \(-0.433972\pi\)
0.205948 + 0.978563i \(0.433972\pi\)
\(14\) 0 0
\(15\) −2.73589 −0.706405
\(16\) 0 0
\(17\) 0.902098 0.218791 0.109396 0.993998i \(-0.465108\pi\)
0.109396 + 0.993998i \(0.465108\pi\)
\(18\) 0 0
\(19\) −2.50407 −0.574473 −0.287236 0.957860i \(-0.592736\pi\)
−0.287236 + 0.957860i \(0.592736\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.48511 0.497023
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.68466 1.24131 0.620655 0.784084i \(-0.286867\pi\)
0.620655 + 0.784084i \(0.286867\pi\)
\(30\) 0 0
\(31\) −1.09790 −0.197189 −0.0985945 0.995128i \(-0.531435\pi\)
−0.0985945 + 0.995128i \(0.531435\pi\)
\(32\) 0 0
\(33\) 2.50407 0.435902
\(34\) 0 0
\(35\) −2.73589 −0.462450
\(36\) 0 0
\(37\) 9.91023 1.62923 0.814616 0.580000i \(-0.196948\pi\)
0.814616 + 0.580000i \(0.196948\pi\)
\(38\) 0 0
\(39\) 1.48511 0.237808
\(40\) 0 0
\(41\) −2.30826 −0.360490 −0.180245 0.983622i \(-0.557689\pi\)
−0.180245 + 0.983622i \(0.557689\pi\)
\(42\) 0 0
\(43\) −5.83380 −0.889645 −0.444823 0.895619i \(-0.646733\pi\)
−0.444823 + 0.895619i \(0.646733\pi\)
\(44\) 0 0
\(45\) −2.73589 −0.407843
\(46\) 0 0
\(47\) 6.74671 0.984109 0.492055 0.870564i \(-0.336246\pi\)
0.492055 + 0.870564i \(0.336246\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.902098 0.126319
\(52\) 0 0
\(53\) −4.91648 −0.675331 −0.337666 0.941266i \(-0.609637\pi\)
−0.337666 + 0.941266i \(0.609637\pi\)
\(54\) 0 0
\(55\) −6.85086 −0.923770
\(56\) 0 0
\(57\) −2.50407 −0.331672
\(58\) 0 0
\(59\) 7.32973 0.954249 0.477125 0.878836i \(-0.341679\pi\)
0.477125 + 0.878836i \(0.341679\pi\)
\(60\) 0 0
\(61\) −1.00625 −0.128837 −0.0644185 0.997923i \(-0.520519\pi\)
−0.0644185 + 0.997923i \(0.520519\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.06311 −0.503967
\(66\) 0 0
\(67\) −11.2929 −1.37964 −0.689822 0.723979i \(-0.742311\pi\)
−0.689822 + 0.723979i \(0.742311\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0.362008 0.0429624 0.0214812 0.999769i \(-0.493162\pi\)
0.0214812 + 0.999769i \(0.493162\pi\)
\(72\) 0 0
\(73\) −5.34411 −0.625481 −0.312741 0.949839i \(-0.601247\pi\)
−0.312741 + 0.949839i \(0.601247\pi\)
\(74\) 0 0
\(75\) 2.48511 0.286956
\(76\) 0 0
\(77\) 2.50407 0.285365
\(78\) 0 0
\(79\) −10.4672 −1.17765 −0.588827 0.808259i \(-0.700410\pi\)
−0.588827 + 0.808259i \(0.700410\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.59946 −0.834149 −0.417075 0.908872i \(-0.636945\pi\)
−0.417075 + 0.908872i \(0.636945\pi\)
\(84\) 0 0
\(85\) −2.46805 −0.267697
\(86\) 0 0
\(87\) 6.68466 0.716671
\(88\) 0 0
\(89\) 10.6245 1.12619 0.563097 0.826391i \(-0.309610\pi\)
0.563097 + 0.826391i \(0.309610\pi\)
\(90\) 0 0
\(91\) 1.48511 0.155682
\(92\) 0 0
\(93\) −1.09790 −0.113847
\(94\) 0 0
\(95\) 6.85086 0.702884
\(96\) 0 0
\(97\) 13.9102 1.41237 0.706185 0.708027i \(-0.250415\pi\)
0.706185 + 0.708027i \(0.250415\pi\)
\(98\) 0 0
\(99\) 2.50407 0.251668
\(100\) 0 0
\(101\) 1.81502 0.180601 0.0903004 0.995915i \(-0.471217\pi\)
0.0903004 + 0.995915i \(0.471217\pi\)
\(102\) 0 0
\(103\) 16.6605 1.64161 0.820805 0.571209i \(-0.193525\pi\)
0.820805 + 0.571209i \(0.193525\pi\)
\(104\) 0 0
\(105\) −2.73589 −0.266996
\(106\) 0 0
\(107\) −1.68092 −0.162500 −0.0812502 0.996694i \(-0.525891\pi\)
−0.0812502 + 0.996694i \(0.525891\pi\)
\(108\) 0 0
\(109\) 11.4333 1.09511 0.547554 0.836771i \(-0.315559\pi\)
0.547554 + 0.836771i \(0.315559\pi\)
\(110\) 0 0
\(111\) 9.91023 0.940638
\(112\) 0 0
\(113\) 3.14914 0.296246 0.148123 0.988969i \(-0.452677\pi\)
0.148123 + 0.988969i \(0.452677\pi\)
\(114\) 0 0
\(115\) 2.73589 0.255123
\(116\) 0 0
\(117\) 1.48511 0.137299
\(118\) 0 0
\(119\) 0.902098 0.0826952
\(120\) 0 0
\(121\) −4.72964 −0.429968
\(122\) 0 0
\(123\) −2.30826 −0.208129
\(124\) 0 0
\(125\) 6.88046 0.615407
\(126\) 0 0
\(127\) −0.449266 −0.0398659 −0.0199329 0.999801i \(-0.506345\pi\)
−0.0199329 + 0.999801i \(0.506345\pi\)
\(128\) 0 0
\(129\) −5.83380 −0.513637
\(130\) 0 0
\(131\) 19.9759 1.74530 0.872649 0.488347i \(-0.162400\pi\)
0.872649 + 0.488347i \(0.162400\pi\)
\(132\) 0 0
\(133\) −2.50407 −0.217130
\(134\) 0 0
\(135\) −2.73589 −0.235468
\(136\) 0 0
\(137\) 0.270356 0.0230981 0.0115490 0.999933i \(-0.496324\pi\)
0.0115490 + 0.999933i \(0.496324\pi\)
\(138\) 0 0
\(139\) −4.42512 −0.375334 −0.187667 0.982233i \(-0.560093\pi\)
−0.187667 + 0.982233i \(0.560093\pi\)
\(140\) 0 0
\(141\) 6.74671 0.568176
\(142\) 0 0
\(143\) 3.71883 0.310984
\(144\) 0 0
\(145\) −18.2885 −1.51878
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −3.69530 −0.302731 −0.151365 0.988478i \(-0.548367\pi\)
−0.151365 + 0.988478i \(0.548367\pi\)
\(150\) 0 0
\(151\) −8.63174 −0.702441 −0.351221 0.936293i \(-0.614233\pi\)
−0.351221 + 0.936293i \(0.614233\pi\)
\(152\) 0 0
\(153\) 0.902098 0.0729303
\(154\) 0 0
\(155\) 3.00374 0.241266
\(156\) 0 0
\(157\) 9.93438 0.792850 0.396425 0.918067i \(-0.370251\pi\)
0.396425 + 0.918067i \(0.370251\pi\)
\(158\) 0 0
\(159\) −4.91648 −0.389903
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −14.7082 −1.15203 −0.576017 0.817438i \(-0.695394\pi\)
−0.576017 + 0.817438i \(0.695394\pi\)
\(164\) 0 0
\(165\) −6.85086 −0.533339
\(166\) 0 0
\(167\) 5.47430 0.423614 0.211807 0.977312i \(-0.432065\pi\)
0.211807 + 0.977312i \(0.432065\pi\)
\(168\) 0 0
\(169\) −10.7944 −0.830341
\(170\) 0 0
\(171\) −2.50407 −0.191491
\(172\) 0 0
\(173\) 18.1727 1.38165 0.690823 0.723024i \(-0.257248\pi\)
0.690823 + 0.723024i \(0.257248\pi\)
\(174\) 0 0
\(175\) 2.48511 0.187857
\(176\) 0 0
\(177\) 7.32973 0.550936
\(178\) 0 0
\(179\) −10.1860 −0.761341 −0.380670 0.924711i \(-0.624307\pi\)
−0.380670 + 0.924711i \(0.624307\pi\)
\(180\) 0 0
\(181\) 13.5186 1.00483 0.502416 0.864626i \(-0.332445\pi\)
0.502416 + 0.864626i \(0.332445\pi\)
\(182\) 0 0
\(183\) −1.00625 −0.0743841
\(184\) 0 0
\(185\) −27.1133 −1.99341
\(186\) 0 0
\(187\) 2.25892 0.165188
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0.631742 0.0457113 0.0228556 0.999739i \(-0.492724\pi\)
0.0228556 + 0.999739i \(0.492724\pi\)
\(192\) 0 0
\(193\) 1.36826 0.0984893 0.0492447 0.998787i \(-0.484319\pi\)
0.0492447 + 0.998787i \(0.484319\pi\)
\(194\) 0 0
\(195\) −4.06311 −0.290966
\(196\) 0 0
\(197\) 24.7611 1.76416 0.882078 0.471104i \(-0.156144\pi\)
0.882078 + 0.471104i \(0.156144\pi\)
\(198\) 0 0
\(199\) −1.44470 −0.102412 −0.0512059 0.998688i \(-0.516306\pi\)
−0.0512059 + 0.998688i \(0.516306\pi\)
\(200\) 0 0
\(201\) −11.2929 −0.796538
\(202\) 0 0
\(203\) 6.68466 0.469171
\(204\) 0 0
\(205\) 6.31517 0.441070
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −6.27036 −0.433730
\(210\) 0 0
\(211\) 23.0990 1.59020 0.795099 0.606480i \(-0.207419\pi\)
0.795099 + 0.606480i \(0.207419\pi\)
\(212\) 0 0
\(213\) 0.362008 0.0248044
\(214\) 0 0
\(215\) 15.9606 1.08851
\(216\) 0 0
\(217\) −1.09790 −0.0745304
\(218\) 0 0
\(219\) −5.34411 −0.361122
\(220\) 0 0
\(221\) 1.33972 0.0901192
\(222\) 0 0
\(223\) 18.6497 1.24888 0.624438 0.781074i \(-0.285328\pi\)
0.624438 + 0.781074i \(0.285328\pi\)
\(224\) 0 0
\(225\) 2.48511 0.165674
\(226\) 0 0
\(227\) −20.6334 −1.36949 −0.684744 0.728783i \(-0.740086\pi\)
−0.684744 + 0.728783i \(0.740086\pi\)
\(228\) 0 0
\(229\) 14.9544 0.988214 0.494107 0.869401i \(-0.335495\pi\)
0.494107 + 0.869401i \(0.335495\pi\)
\(230\) 0 0
\(231\) 2.50407 0.164756
\(232\) 0 0
\(233\) 28.1052 1.84123 0.920617 0.390467i \(-0.127687\pi\)
0.920617 + 0.390467i \(0.127687\pi\)
\(234\) 0 0
\(235\) −18.4583 −1.20409
\(236\) 0 0
\(237\) −10.4672 −0.679919
\(238\) 0 0
\(239\) −25.8536 −1.67233 −0.836166 0.548476i \(-0.815208\pi\)
−0.836166 + 0.548476i \(0.815208\pi\)
\(240\) 0 0
\(241\) −14.8311 −0.955356 −0.477678 0.878535i \(-0.658521\pi\)
−0.477678 + 0.878535i \(0.658521\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.73589 −0.174790
\(246\) 0 0
\(247\) −3.71883 −0.236623
\(248\) 0 0
\(249\) −7.59946 −0.481596
\(250\) 0 0
\(251\) 16.2841 1.02784 0.513922 0.857837i \(-0.328192\pi\)
0.513922 + 0.857837i \(0.328192\pi\)
\(252\) 0 0
\(253\) −2.50407 −0.157429
\(254\) 0 0
\(255\) −2.46805 −0.154555
\(256\) 0 0
\(257\) 22.9446 1.43125 0.715623 0.698486i \(-0.246143\pi\)
0.715623 + 0.698486i \(0.246143\pi\)
\(258\) 0 0
\(259\) 9.91023 0.615792
\(260\) 0 0
\(261\) 6.68466 0.413770
\(262\) 0 0
\(263\) 0.776932 0.0479077 0.0239538 0.999713i \(-0.492375\pi\)
0.0239538 + 0.999713i \(0.492375\pi\)
\(264\) 0 0
\(265\) 13.4510 0.826287
\(266\) 0 0
\(267\) 10.6245 0.650208
\(268\) 0 0
\(269\) 3.05937 0.186533 0.0932666 0.995641i \(-0.470269\pi\)
0.0932666 + 0.995641i \(0.470269\pi\)
\(270\) 0 0
\(271\) 16.0631 0.975765 0.487882 0.872909i \(-0.337770\pi\)
0.487882 + 0.872909i \(0.337770\pi\)
\(272\) 0 0
\(273\) 1.48511 0.0898832
\(274\) 0 0
\(275\) 6.22289 0.375255
\(276\) 0 0
\(277\) −11.6103 −0.697594 −0.348797 0.937198i \(-0.613410\pi\)
−0.348797 + 0.937198i \(0.613410\pi\)
\(278\) 0 0
\(279\) −1.09790 −0.0657296
\(280\) 0 0
\(281\) 9.89753 0.590437 0.295219 0.955430i \(-0.404608\pi\)
0.295219 + 0.955430i \(0.404608\pi\)
\(282\) 0 0
\(283\) −13.9994 −0.832177 −0.416088 0.909324i \(-0.636599\pi\)
−0.416088 + 0.909324i \(0.636599\pi\)
\(284\) 0 0
\(285\) 6.85086 0.405810
\(286\) 0 0
\(287\) −2.30826 −0.136253
\(288\) 0 0
\(289\) −16.1862 −0.952130
\(290\) 0 0
\(291\) 13.9102 0.815432
\(292\) 0 0
\(293\) −8.14038 −0.475566 −0.237783 0.971318i \(-0.576421\pi\)
−0.237783 + 0.971318i \(0.576421\pi\)
\(294\) 0 0
\(295\) −20.0534 −1.16755
\(296\) 0 0
\(297\) 2.50407 0.145301
\(298\) 0 0
\(299\) −1.48511 −0.0858863
\(300\) 0 0
\(301\) −5.83380 −0.336254
\(302\) 0 0
\(303\) 1.81502 0.104270
\(304\) 0 0
\(305\) 2.75299 0.157636
\(306\) 0 0
\(307\) −13.5095 −0.771027 −0.385514 0.922702i \(-0.625976\pi\)
−0.385514 + 0.922702i \(0.625976\pi\)
\(308\) 0 0
\(309\) 16.6605 0.947783
\(310\) 0 0
\(311\) 24.3035 1.37813 0.689063 0.724701i \(-0.258022\pi\)
0.689063 + 0.724701i \(0.258022\pi\)
\(312\) 0 0
\(313\) −33.7836 −1.90956 −0.954782 0.297308i \(-0.903911\pi\)
−0.954782 + 0.297308i \(0.903911\pi\)
\(314\) 0 0
\(315\) −2.73589 −0.154150
\(316\) 0 0
\(317\) 32.1594 1.80625 0.903126 0.429376i \(-0.141267\pi\)
0.903126 + 0.429376i \(0.141267\pi\)
\(318\) 0 0
\(319\) 16.7388 0.937195
\(320\) 0 0
\(321\) −1.68092 −0.0938196
\(322\) 0 0
\(323\) −2.25892 −0.125689
\(324\) 0 0
\(325\) 3.69068 0.204722
\(326\) 0 0
\(327\) 11.4333 0.632261
\(328\) 0 0
\(329\) 6.74671 0.371958
\(330\) 0 0
\(331\) 4.84004 0.266033 0.133016 0.991114i \(-0.457534\pi\)
0.133016 + 0.991114i \(0.457534\pi\)
\(332\) 0 0
\(333\) 9.91023 0.543077
\(334\) 0 0
\(335\) 30.8961 1.68803
\(336\) 0 0
\(337\) −2.59195 −0.141192 −0.0705962 0.997505i \(-0.522490\pi\)
−0.0705962 + 0.997505i \(0.522490\pi\)
\(338\) 0 0
\(339\) 3.14914 0.171038
\(340\) 0 0
\(341\) −2.74922 −0.148879
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.73589 0.147296
\(346\) 0 0
\(347\) 15.9390 0.855651 0.427825 0.903861i \(-0.359280\pi\)
0.427825 + 0.903861i \(0.359280\pi\)
\(348\) 0 0
\(349\) −8.02603 −0.429624 −0.214812 0.976655i \(-0.568914\pi\)
−0.214812 + 0.976655i \(0.568914\pi\)
\(350\) 0 0
\(351\) 1.48511 0.0792695
\(352\) 0 0
\(353\) 10.7871 0.574141 0.287070 0.957909i \(-0.407319\pi\)
0.287070 + 0.957909i \(0.407319\pi\)
\(354\) 0 0
\(355\) −0.990415 −0.0525658
\(356\) 0 0
\(357\) 0.902098 0.0477441
\(358\) 0 0
\(359\) 2.17891 0.114998 0.0574992 0.998346i \(-0.481687\pi\)
0.0574992 + 0.998346i \(0.481687\pi\)
\(360\) 0 0
\(361\) −12.7296 −0.669981
\(362\) 0 0
\(363\) −4.72964 −0.248242
\(364\) 0 0
\(365\) 14.6209 0.765294
\(366\) 0 0
\(367\) −27.0388 −1.41141 −0.705707 0.708504i \(-0.749370\pi\)
−0.705707 + 0.708504i \(0.749370\pi\)
\(368\) 0 0
\(369\) −2.30826 −0.120163
\(370\) 0 0
\(371\) −4.91648 −0.255251
\(372\) 0 0
\(373\) 16.4189 0.850140 0.425070 0.905161i \(-0.360249\pi\)
0.425070 + 0.905161i \(0.360249\pi\)
\(374\) 0 0
\(375\) 6.88046 0.355306
\(376\) 0 0
\(377\) 9.92748 0.511291
\(378\) 0 0
\(379\) 34.6399 1.77933 0.889667 0.456610i \(-0.150936\pi\)
0.889667 + 0.456610i \(0.150936\pi\)
\(380\) 0 0
\(381\) −0.449266 −0.0230166
\(382\) 0 0
\(383\) 10.6722 0.545322 0.272661 0.962110i \(-0.412096\pi\)
0.272661 + 0.962110i \(0.412096\pi\)
\(384\) 0 0
\(385\) −6.85086 −0.349152
\(386\) 0 0
\(387\) −5.83380 −0.296548
\(388\) 0 0
\(389\) 0.0302201 0.00153222 0.000766110 1.00000i \(-0.499756\pi\)
0.000766110 1.00000i \(0.499756\pi\)
\(390\) 0 0
\(391\) −0.902098 −0.0456211
\(392\) 0 0
\(393\) 19.9759 1.00765
\(394\) 0 0
\(395\) 28.6372 1.44089
\(396\) 0 0
\(397\) 20.4749 1.02761 0.513803 0.857908i \(-0.328236\pi\)
0.513803 + 0.857908i \(0.328236\pi\)
\(398\) 0 0
\(399\) −2.50407 −0.125360
\(400\) 0 0
\(401\) −36.4304 −1.81925 −0.909623 0.415435i \(-0.863629\pi\)
−0.909623 + 0.415435i \(0.863629\pi\)
\(402\) 0 0
\(403\) −1.63051 −0.0812214
\(404\) 0 0
\(405\) −2.73589 −0.135948
\(406\) 0 0
\(407\) 24.8159 1.23008
\(408\) 0 0
\(409\) 37.4054 1.84958 0.924790 0.380479i \(-0.124241\pi\)
0.924790 + 0.380479i \(0.124241\pi\)
\(410\) 0 0
\(411\) 0.270356 0.0133357
\(412\) 0 0
\(413\) 7.32973 0.360672
\(414\) 0 0
\(415\) 20.7913 1.02061
\(416\) 0 0
\(417\) −4.42512 −0.216699
\(418\) 0 0
\(419\) 37.2017 1.81742 0.908710 0.417428i \(-0.137068\pi\)
0.908710 + 0.417428i \(0.137068\pi\)
\(420\) 0 0
\(421\) 27.6326 1.34673 0.673366 0.739309i \(-0.264848\pi\)
0.673366 + 0.739309i \(0.264848\pi\)
\(422\) 0 0
\(423\) 6.74671 0.328036
\(424\) 0 0
\(425\) 2.24182 0.108744
\(426\) 0 0
\(427\) −1.00625 −0.0486958
\(428\) 0 0
\(429\) 3.71883 0.179547
\(430\) 0 0
\(431\) −35.0676 −1.68915 −0.844573 0.535441i \(-0.820145\pi\)
−0.844573 + 0.535441i \(0.820145\pi\)
\(432\) 0 0
\(433\) −1.00708 −0.0483970 −0.0241985 0.999707i \(-0.507703\pi\)
−0.0241985 + 0.999707i \(0.507703\pi\)
\(434\) 0 0
\(435\) −18.2885 −0.876867
\(436\) 0 0
\(437\) 2.50407 0.119786
\(438\) 0 0
\(439\) 4.04649 0.193129 0.0965643 0.995327i \(-0.469215\pi\)
0.0965643 + 0.995327i \(0.469215\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 29.3947 1.39659 0.698293 0.715812i \(-0.253943\pi\)
0.698293 + 0.715812i \(0.253943\pi\)
\(444\) 0 0
\(445\) −29.0675 −1.37793
\(446\) 0 0
\(447\) −3.69530 −0.174782
\(448\) 0 0
\(449\) −4.87627 −0.230126 −0.115063 0.993358i \(-0.536707\pi\)
−0.115063 + 0.993358i \(0.536707\pi\)
\(450\) 0 0
\(451\) −5.78005 −0.272172
\(452\) 0 0
\(453\) −8.63174 −0.405555
\(454\) 0 0
\(455\) −4.06311 −0.190482
\(456\) 0 0
\(457\) −35.5212 −1.66161 −0.830806 0.556562i \(-0.812120\pi\)
−0.830806 + 0.556562i \(0.812120\pi\)
\(458\) 0 0
\(459\) 0.902098 0.0421064
\(460\) 0 0
\(461\) −18.9442 −0.882319 −0.441160 0.897429i \(-0.645433\pi\)
−0.441160 + 0.897429i \(0.645433\pi\)
\(462\) 0 0
\(463\) −14.1933 −0.659618 −0.329809 0.944048i \(-0.606984\pi\)
−0.329809 + 0.944048i \(0.606984\pi\)
\(464\) 0 0
\(465\) 3.00374 0.139295
\(466\) 0 0
\(467\) 31.0355 1.43615 0.718075 0.695966i \(-0.245024\pi\)
0.718075 + 0.695966i \(0.245024\pi\)
\(468\) 0 0
\(469\) −11.2929 −0.521457
\(470\) 0 0
\(471\) 9.93438 0.457752
\(472\) 0 0
\(473\) −14.6082 −0.671687
\(474\) 0 0
\(475\) −6.22289 −0.285526
\(476\) 0 0
\(477\) −4.91648 −0.225110
\(478\) 0 0
\(479\) −10.5214 −0.480735 −0.240367 0.970682i \(-0.577268\pi\)
−0.240367 + 0.970682i \(0.577268\pi\)
\(480\) 0 0
\(481\) 14.7178 0.671075
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −38.0569 −1.72808
\(486\) 0 0
\(487\) 31.7378 1.43818 0.719089 0.694918i \(-0.244559\pi\)
0.719089 + 0.694918i \(0.244559\pi\)
\(488\) 0 0
\(489\) −14.7082 −0.665127
\(490\) 0 0
\(491\) 31.8788 1.43867 0.719336 0.694662i \(-0.244446\pi\)
0.719336 + 0.694662i \(0.244446\pi\)
\(492\) 0 0
\(493\) 6.03022 0.271587
\(494\) 0 0
\(495\) −6.85086 −0.307923
\(496\) 0 0
\(497\) 0.362008 0.0162383
\(498\) 0 0
\(499\) 39.4279 1.76503 0.882517 0.470280i \(-0.155847\pi\)
0.882517 + 0.470280i \(0.155847\pi\)
\(500\) 0 0
\(501\) 5.47430 0.244573
\(502\) 0 0
\(503\) −24.4589 −1.09057 −0.545284 0.838251i \(-0.683578\pi\)
−0.545284 + 0.838251i \(0.683578\pi\)
\(504\) 0 0
\(505\) −4.96569 −0.220970
\(506\) 0 0
\(507\) −10.7944 −0.479398
\(508\) 0 0
\(509\) 20.3927 0.903889 0.451944 0.892046i \(-0.350731\pi\)
0.451944 + 0.892046i \(0.350731\pi\)
\(510\) 0 0
\(511\) −5.34411 −0.236410
\(512\) 0 0
\(513\) −2.50407 −0.110557
\(514\) 0 0
\(515\) −45.5814 −2.00856
\(516\) 0 0
\(517\) 16.8942 0.743007
\(518\) 0 0
\(519\) 18.1727 0.797694
\(520\) 0 0
\(521\) 26.3058 1.15248 0.576238 0.817282i \(-0.304520\pi\)
0.576238 + 0.817282i \(0.304520\pi\)
\(522\) 0 0
\(523\) 40.4256 1.76769 0.883843 0.467783i \(-0.154947\pi\)
0.883843 + 0.467783i \(0.154947\pi\)
\(524\) 0 0
\(525\) 2.48511 0.108459
\(526\) 0 0
\(527\) −0.990415 −0.0431432
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.32973 0.318083
\(532\) 0 0
\(533\) −3.42804 −0.148485
\(534\) 0 0
\(535\) 4.59881 0.198824
\(536\) 0 0
\(537\) −10.1860 −0.439560
\(538\) 0 0
\(539\) 2.50407 0.107858
\(540\) 0 0
\(541\) 0.360122 0.0154828 0.00774142 0.999970i \(-0.497536\pi\)
0.00774142 + 0.999970i \(0.497536\pi\)
\(542\) 0 0
\(543\) 13.5186 0.580140
\(544\) 0 0
\(545\) −31.2802 −1.33990
\(546\) 0 0
\(547\) −20.2023 −0.863789 −0.431894 0.901924i \(-0.642155\pi\)
−0.431894 + 0.901924i \(0.642155\pi\)
\(548\) 0 0
\(549\) −1.00625 −0.0429457
\(550\) 0 0
\(551\) −16.7388 −0.713099
\(552\) 0 0
\(553\) −10.4672 −0.445111
\(554\) 0 0
\(555\) −27.1133 −1.15090
\(556\) 0 0
\(557\) 9.71086 0.411463 0.205731 0.978609i \(-0.434043\pi\)
0.205731 + 0.978609i \(0.434043\pi\)
\(558\) 0 0
\(559\) −8.66385 −0.366442
\(560\) 0 0
\(561\) 2.25892 0.0953715
\(562\) 0 0
\(563\) −27.2789 −1.14967 −0.574835 0.818269i \(-0.694934\pi\)
−0.574835 + 0.818269i \(0.694934\pi\)
\(564\) 0 0
\(565\) −8.61570 −0.362465
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 19.6132 0.822229 0.411115 0.911584i \(-0.365140\pi\)
0.411115 + 0.911584i \(0.365140\pi\)
\(570\) 0 0
\(571\) −21.1859 −0.886601 −0.443301 0.896373i \(-0.646193\pi\)
−0.443301 + 0.896373i \(0.646193\pi\)
\(572\) 0 0
\(573\) 0.631742 0.0263914
\(574\) 0 0
\(575\) −2.48511 −0.103636
\(576\) 0 0
\(577\) −12.4079 −0.516547 −0.258273 0.966072i \(-0.583154\pi\)
−0.258273 + 0.966072i \(0.583154\pi\)
\(578\) 0 0
\(579\) 1.36826 0.0568629
\(580\) 0 0
\(581\) −7.59946 −0.315279
\(582\) 0 0
\(583\) −12.3112 −0.509878
\(584\) 0 0
\(585\) −4.06311 −0.167989
\(586\) 0 0
\(587\) 9.35493 0.386119 0.193060 0.981187i \(-0.438159\pi\)
0.193060 + 0.981187i \(0.438159\pi\)
\(588\) 0 0
\(589\) 2.74922 0.113280
\(590\) 0 0
\(591\) 24.7611 1.01854
\(592\) 0 0
\(593\) −16.7836 −0.689218 −0.344609 0.938746i \(-0.611989\pi\)
−0.344609 + 0.938746i \(0.611989\pi\)
\(594\) 0 0
\(595\) −2.46805 −0.101180
\(596\) 0 0
\(597\) −1.44470 −0.0591275
\(598\) 0 0
\(599\) 16.0512 0.655836 0.327918 0.944706i \(-0.393653\pi\)
0.327918 + 0.944706i \(0.393653\pi\)
\(600\) 0 0
\(601\) −24.4364 −0.996781 −0.498390 0.866953i \(-0.666075\pi\)
−0.498390 + 0.866953i \(0.666075\pi\)
\(602\) 0 0
\(603\) −11.2929 −0.459882
\(604\) 0 0
\(605\) 12.9398 0.526078
\(606\) 0 0
\(607\) −28.2329 −1.14594 −0.572969 0.819577i \(-0.694208\pi\)
−0.572969 + 0.819577i \(0.694208\pi\)
\(608\) 0 0
\(609\) 6.68466 0.270876
\(610\) 0 0
\(611\) 10.0196 0.405351
\(612\) 0 0
\(613\) 17.9873 0.726500 0.363250 0.931692i \(-0.381667\pi\)
0.363250 + 0.931692i \(0.381667\pi\)
\(614\) 0 0
\(615\) 6.31517 0.254652
\(616\) 0 0
\(617\) 12.9998 0.523353 0.261677 0.965156i \(-0.415725\pi\)
0.261677 + 0.965156i \(0.415725\pi\)
\(618\) 0 0
\(619\) 1.26266 0.0507505 0.0253752 0.999678i \(-0.491922\pi\)
0.0253752 + 0.999678i \(0.491922\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 10.6245 0.425661
\(624\) 0 0
\(625\) −31.2498 −1.24999
\(626\) 0 0
\(627\) −6.27036 −0.250414
\(628\) 0 0
\(629\) 8.94001 0.356461
\(630\) 0 0
\(631\) 24.4568 0.973611 0.486806 0.873510i \(-0.338162\pi\)
0.486806 + 0.873510i \(0.338162\pi\)
\(632\) 0 0
\(633\) 23.0990 0.918101
\(634\) 0 0
\(635\) 1.22914 0.0487771
\(636\) 0 0
\(637\) 1.48511 0.0588423
\(638\) 0 0
\(639\) 0.362008 0.0143208
\(640\) 0 0
\(641\) −1.29037 −0.0509665 −0.0254833 0.999675i \(-0.508112\pi\)
−0.0254833 + 0.999675i \(0.508112\pi\)
\(642\) 0 0
\(643\) −34.3110 −1.35309 −0.676547 0.736399i \(-0.736524\pi\)
−0.676547 + 0.736399i \(0.736524\pi\)
\(644\) 0 0
\(645\) 15.9606 0.628450
\(646\) 0 0
\(647\) 2.62092 0.103039 0.0515196 0.998672i \(-0.483594\pi\)
0.0515196 + 0.998672i \(0.483594\pi\)
\(648\) 0 0
\(649\) 18.3541 0.720463
\(650\) 0 0
\(651\) −1.09790 −0.0430302
\(652\) 0 0
\(653\) 7.53697 0.294944 0.147472 0.989066i \(-0.452886\pi\)
0.147472 + 0.989066i \(0.452886\pi\)
\(654\) 0 0
\(655\) −54.6518 −2.13542
\(656\) 0 0
\(657\) −5.34411 −0.208494
\(658\) 0 0
\(659\) −22.8213 −0.888990 −0.444495 0.895781i \(-0.646617\pi\)
−0.444495 + 0.895781i \(0.646617\pi\)
\(660\) 0 0
\(661\) 27.3668 1.06445 0.532223 0.846604i \(-0.321357\pi\)
0.532223 + 0.846604i \(0.321357\pi\)
\(662\) 0 0
\(663\) 1.33972 0.0520304
\(664\) 0 0
\(665\) 6.85086 0.265665
\(666\) 0 0
\(667\) −6.68466 −0.258831
\(668\) 0 0
\(669\) 18.6497 0.721039
\(670\) 0 0
\(671\) −2.51972 −0.0972726
\(672\) 0 0
\(673\) 22.9790 0.885774 0.442887 0.896577i \(-0.353954\pi\)
0.442887 + 0.896577i \(0.353954\pi\)
\(674\) 0 0
\(675\) 2.48511 0.0956521
\(676\) 0 0
\(677\) −8.80814 −0.338524 −0.169262 0.985571i \(-0.554138\pi\)
−0.169262 + 0.985571i \(0.554138\pi\)
\(678\) 0 0
\(679\) 13.9102 0.533826
\(680\) 0 0
\(681\) −20.6334 −0.790674
\(682\) 0 0
\(683\) −34.4889 −1.31968 −0.659840 0.751406i \(-0.729376\pi\)
−0.659840 + 0.751406i \(0.729376\pi\)
\(684\) 0 0
\(685\) −0.739666 −0.0282612
\(686\) 0 0
\(687\) 14.9544 0.570546
\(688\) 0 0
\(689\) −7.30154 −0.278166
\(690\) 0 0
\(691\) −37.6110 −1.43079 −0.715395 0.698721i \(-0.753753\pi\)
−0.715395 + 0.698721i \(0.753753\pi\)
\(692\) 0 0
\(693\) 2.50407 0.0951217
\(694\) 0 0
\(695\) 12.1067 0.459232
\(696\) 0 0
\(697\) −2.08228 −0.0788721
\(698\) 0 0
\(699\) 28.1052 1.06304
\(700\) 0 0
\(701\) −47.5348 −1.79536 −0.897682 0.440644i \(-0.854750\pi\)
−0.897682 + 0.440644i \(0.854750\pi\)
\(702\) 0 0
\(703\) −24.8159 −0.935949
\(704\) 0 0
\(705\) −18.4583 −0.695179
\(706\) 0 0
\(707\) 1.81502 0.0682607
\(708\) 0 0
\(709\) −41.1332 −1.54479 −0.772395 0.635143i \(-0.780941\pi\)
−0.772395 + 0.635143i \(0.780941\pi\)
\(710\) 0 0
\(711\) −10.4672 −0.392551
\(712\) 0 0
\(713\) 1.09790 0.0411167
\(714\) 0 0
\(715\) −10.1743 −0.380498
\(716\) 0 0
\(717\) −25.8536 −0.965522
\(718\) 0 0
\(719\) −8.25857 −0.307993 −0.153996 0.988071i \(-0.549214\pi\)
−0.153996 + 0.988071i \(0.549214\pi\)
\(720\) 0 0
\(721\) 16.6605 0.620470
\(722\) 0 0
\(723\) −14.8311 −0.551575
\(724\) 0 0
\(725\) 16.6121 0.616959
\(726\) 0 0
\(727\) −0.985235 −0.0365403 −0.0182702 0.999833i \(-0.505816\pi\)
−0.0182702 + 0.999833i \(0.505816\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −5.26266 −0.194646
\(732\) 0 0
\(733\) 27.7847 1.02625 0.513125 0.858314i \(-0.328488\pi\)
0.513125 + 0.858314i \(0.328488\pi\)
\(734\) 0 0
\(735\) −2.73589 −0.100915
\(736\) 0 0
\(737\) −28.2781 −1.04164
\(738\) 0 0
\(739\) −30.0540 −1.10555 −0.552777 0.833329i \(-0.686432\pi\)
−0.552777 + 0.833329i \(0.686432\pi\)
\(740\) 0 0
\(741\) −3.71883 −0.136614
\(742\) 0 0
\(743\) −40.1617 −1.47339 −0.736695 0.676225i \(-0.763615\pi\)
−0.736695 + 0.676225i \(0.763615\pi\)
\(744\) 0 0
\(745\) 10.1100 0.370400
\(746\) 0 0
\(747\) −7.59946 −0.278050
\(748\) 0 0
\(749\) −1.68092 −0.0614194
\(750\) 0 0
\(751\) 2.15350 0.0785823 0.0392912 0.999228i \(-0.487490\pi\)
0.0392912 + 0.999228i \(0.487490\pi\)
\(752\) 0 0
\(753\) 16.2841 0.593426
\(754\) 0 0
\(755\) 23.6155 0.859457
\(756\) 0 0
\(757\) −32.0577 −1.16516 −0.582579 0.812774i \(-0.697956\pi\)
−0.582579 + 0.812774i \(0.697956\pi\)
\(758\) 0 0
\(759\) −2.50407 −0.0908919
\(760\) 0 0
\(761\) 0.669651 0.0242748 0.0121374 0.999926i \(-0.496136\pi\)
0.0121374 + 0.999926i \(0.496136\pi\)
\(762\) 0 0
\(763\) 11.4333 0.413912
\(764\) 0 0
\(765\) −2.46805 −0.0892324
\(766\) 0 0
\(767\) 10.8855 0.393052
\(768\) 0 0
\(769\) 43.5115 1.56906 0.784532 0.620089i \(-0.212903\pi\)
0.784532 + 0.620089i \(0.212903\pi\)
\(770\) 0 0
\(771\) 22.9446 0.826331
\(772\) 0 0
\(773\) 21.0046 0.755482 0.377741 0.925911i \(-0.376701\pi\)
0.377741 + 0.925911i \(0.376701\pi\)
\(774\) 0 0
\(775\) −2.72841 −0.0980074
\(776\) 0 0
\(777\) 9.91023 0.355528
\(778\) 0 0
\(779\) 5.78005 0.207092
\(780\) 0 0
\(781\) 0.906493 0.0324369
\(782\) 0 0
\(783\) 6.68466 0.238890
\(784\) 0 0
\(785\) −27.1794 −0.970075
\(786\) 0 0
\(787\) −22.1975 −0.791255 −0.395627 0.918411i \(-0.629473\pi\)
−0.395627 + 0.918411i \(0.629473\pi\)
\(788\) 0 0
\(789\) 0.776932 0.0276595
\(790\) 0 0
\(791\) 3.14914 0.111970
\(792\) 0 0
\(793\) −1.49440 −0.0530675
\(794\) 0 0
\(795\) 13.4510 0.477057
\(796\) 0 0
\(797\) 22.8502 0.809397 0.404699 0.914450i \(-0.367376\pi\)
0.404699 + 0.914450i \(0.367376\pi\)
\(798\) 0 0
\(799\) 6.08620 0.215314
\(800\) 0 0
\(801\) 10.6245 0.375398
\(802\) 0 0
\(803\) −13.3820 −0.472241
\(804\) 0 0
\(805\) 2.73589 0.0964276
\(806\) 0 0
\(807\) 3.05937 0.107695
\(808\) 0 0
\(809\) −38.0621 −1.33819 −0.669097 0.743175i \(-0.733319\pi\)
−0.669097 + 0.743175i \(0.733319\pi\)
\(810\) 0 0
\(811\) −21.0208 −0.738142 −0.369071 0.929401i \(-0.620324\pi\)
−0.369071 + 0.929401i \(0.620324\pi\)
\(812\) 0 0
\(813\) 16.0631 0.563358
\(814\) 0 0
\(815\) 40.2400 1.40955
\(816\) 0 0
\(817\) 14.6082 0.511077
\(818\) 0 0
\(819\) 1.48511 0.0518941
\(820\) 0 0
\(821\) −10.4132 −0.363425 −0.181712 0.983352i \(-0.558164\pi\)
−0.181712 + 0.983352i \(0.558164\pi\)
\(822\) 0 0
\(823\) −30.0396 −1.04711 −0.523557 0.851991i \(-0.675395\pi\)
−0.523557 + 0.851991i \(0.675395\pi\)
\(824\) 0 0
\(825\) 6.22289 0.216653
\(826\) 0 0
\(827\) −28.6978 −0.997920 −0.498960 0.866625i \(-0.666285\pi\)
−0.498960 + 0.866625i \(0.666285\pi\)
\(828\) 0 0
\(829\) 16.4384 0.570931 0.285465 0.958389i \(-0.407852\pi\)
0.285465 + 0.958389i \(0.407852\pi\)
\(830\) 0 0
\(831\) −11.6103 −0.402756
\(832\) 0 0
\(833\) 0.902098 0.0312559
\(834\) 0 0
\(835\) −14.9771 −0.518304
\(836\) 0 0
\(837\) −1.09790 −0.0379490
\(838\) 0 0
\(839\) 12.5437 0.433055 0.216528 0.976277i \(-0.430527\pi\)
0.216528 + 0.976277i \(0.430527\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 0 0
\(843\) 9.89753 0.340889
\(844\) 0 0
\(845\) 29.5324 1.01595
\(846\) 0 0
\(847\) −4.72964 −0.162512
\(848\) 0 0
\(849\) −13.9994 −0.480457
\(850\) 0 0
\(851\) −9.91023 −0.339718
\(852\) 0 0
\(853\) −22.0613 −0.755365 −0.377683 0.925935i \(-0.623279\pi\)
−0.377683 + 0.925935i \(0.623279\pi\)
\(854\) 0 0
\(855\) 6.85086 0.234295
\(856\) 0 0
\(857\) 6.15359 0.210203 0.105101 0.994462i \(-0.466483\pi\)
0.105101 + 0.994462i \(0.466483\pi\)
\(858\) 0 0
\(859\) −23.0698 −0.787131 −0.393566 0.919296i \(-0.628759\pi\)
−0.393566 + 0.919296i \(0.628759\pi\)
\(860\) 0 0
\(861\) −2.30826 −0.0786655
\(862\) 0 0
\(863\) 5.22145 0.177740 0.0888702 0.996043i \(-0.471674\pi\)
0.0888702 + 0.996043i \(0.471674\pi\)
\(864\) 0 0
\(865\) −49.7186 −1.69048
\(866\) 0 0
\(867\) −16.1862 −0.549713
\(868\) 0 0
\(869\) −26.2106 −0.889135
\(870\) 0 0
\(871\) −16.7712 −0.568271
\(872\) 0 0
\(873\) 13.9102 0.470790
\(874\) 0 0
\(875\) 6.88046 0.232602
\(876\) 0 0
\(877\) 49.8716 1.68404 0.842022 0.539443i \(-0.181365\pi\)
0.842022 + 0.539443i \(0.181365\pi\)
\(878\) 0 0
\(879\) −8.14038 −0.274568
\(880\) 0 0
\(881\) −20.3054 −0.684107 −0.342053 0.939681i \(-0.611122\pi\)
−0.342053 + 0.939681i \(0.611122\pi\)
\(882\) 0 0
\(883\) 50.3500 1.69441 0.847206 0.531265i \(-0.178283\pi\)
0.847206 + 0.531265i \(0.178283\pi\)
\(884\) 0 0
\(885\) −20.0534 −0.674086
\(886\) 0 0
\(887\) −47.2694 −1.58715 −0.793576 0.608471i \(-0.791783\pi\)
−0.793576 + 0.608471i \(0.791783\pi\)
\(888\) 0 0
\(889\) −0.449266 −0.0150679
\(890\) 0 0
\(891\) 2.50407 0.0838894
\(892\) 0 0
\(893\) −16.8942 −0.565344
\(894\) 0 0
\(895\) 27.8679 0.931522
\(896\) 0 0
\(897\) −1.48511 −0.0495865
\(898\) 0 0
\(899\) −7.33910 −0.244773
\(900\) 0 0
\(901\) −4.43515 −0.147756
\(902\) 0 0
\(903\) −5.83380 −0.194137
\(904\) 0 0
\(905\) −36.9855 −1.22944
\(906\) 0 0
\(907\) 5.65035 0.187617 0.0938084 0.995590i \(-0.470096\pi\)
0.0938084 + 0.995590i \(0.470096\pi\)
\(908\) 0 0
\(909\) 1.81502 0.0602003
\(910\) 0 0
\(911\) −18.2196 −0.603641 −0.301820 0.953365i \(-0.597594\pi\)
−0.301820 + 0.953365i \(0.597594\pi\)
\(912\) 0 0
\(913\) −19.0296 −0.629787
\(914\) 0 0
\(915\) 2.75299 0.0910111
\(916\) 0 0
\(917\) 19.9759 0.659661
\(918\) 0 0
\(919\) −7.79561 −0.257154 −0.128577 0.991700i \(-0.541041\pi\)
−0.128577 + 0.991700i \(0.541041\pi\)
\(920\) 0 0
\(921\) −13.5095 −0.445153
\(922\) 0 0
\(923\) 0.537623 0.0176961
\(924\) 0 0
\(925\) 24.6281 0.809766
\(926\) 0 0
\(927\) 16.6605 0.547203
\(928\) 0 0
\(929\) 11.5912 0.380293 0.190147 0.981756i \(-0.439104\pi\)
0.190147 + 0.981756i \(0.439104\pi\)
\(930\) 0 0
\(931\) −2.50407 −0.0820675
\(932\) 0 0
\(933\) 24.3035 0.795662
\(934\) 0 0
\(935\) −6.18015 −0.202113
\(936\) 0 0
\(937\) −32.0050 −1.04556 −0.522778 0.852469i \(-0.675105\pi\)
−0.522778 + 0.852469i \(0.675105\pi\)
\(938\) 0 0
\(939\) −33.7836 −1.10249
\(940\) 0 0
\(941\) 11.1090 0.362141 0.181071 0.983470i \(-0.442044\pi\)
0.181071 + 0.983470i \(0.442044\pi\)
\(942\) 0 0
\(943\) 2.30826 0.0751674
\(944\) 0 0
\(945\) −2.73589 −0.0889986
\(946\) 0 0
\(947\) 12.1262 0.394049 0.197025 0.980399i \(-0.436872\pi\)
0.197025 + 0.980399i \(0.436872\pi\)
\(948\) 0 0
\(949\) −7.93661 −0.257633
\(950\) 0 0
\(951\) 32.1594 1.04284
\(952\) 0 0
\(953\) 27.6645 0.896140 0.448070 0.893999i \(-0.352112\pi\)
0.448070 + 0.893999i \(0.352112\pi\)
\(954\) 0 0
\(955\) −1.72838 −0.0559291
\(956\) 0 0
\(957\) 16.7388 0.541090
\(958\) 0 0
\(959\) 0.270356 0.00873026
\(960\) 0 0
\(961\) −29.7946 −0.961117
\(962\) 0 0
\(963\) −1.68092 −0.0541668
\(964\) 0 0
\(965\) −3.74341 −0.120505
\(966\) 0 0
\(967\) 56.1838 1.80675 0.903374 0.428853i \(-0.141082\pi\)
0.903374 + 0.428853i \(0.141082\pi\)
\(968\) 0 0
\(969\) −2.25892 −0.0725668
\(970\) 0 0
\(971\) 43.9884 1.41166 0.705828 0.708383i \(-0.250575\pi\)
0.705828 + 0.708383i \(0.250575\pi\)
\(972\) 0 0
\(973\) −4.42512 −0.141863
\(974\) 0 0
\(975\) 3.69068 0.118196
\(976\) 0 0
\(977\) −6.47804 −0.207251 −0.103625 0.994616i \(-0.533044\pi\)
−0.103625 + 0.994616i \(0.533044\pi\)
\(978\) 0 0
\(979\) 26.6044 0.850282
\(980\) 0 0
\(981\) 11.4333 0.365036
\(982\) 0 0
\(983\) −37.8961 −1.20870 −0.604349 0.796720i \(-0.706567\pi\)
−0.604349 + 0.796720i \(0.706567\pi\)
\(984\) 0 0
\(985\) −67.7437 −2.15849
\(986\) 0 0
\(987\) 6.74671 0.214750
\(988\) 0 0
\(989\) 5.83380 0.185504
\(990\) 0 0
\(991\) −53.3499 −1.69472 −0.847358 0.531022i \(-0.821808\pi\)
−0.847358 + 0.531022i \(0.821808\pi\)
\(992\) 0 0
\(993\) 4.84004 0.153594
\(994\) 0 0
\(995\) 3.95254 0.125304
\(996\) 0 0
\(997\) 24.8754 0.787813 0.393907 0.919150i \(-0.371123\pi\)
0.393907 + 0.919150i \(0.371123\pi\)
\(998\) 0 0
\(999\) 9.91023 0.313546
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.ce.1.1 4
4.3 odd 2 483.2.a.j.1.2 4
12.11 even 2 1449.2.a.o.1.3 4
28.27 even 2 3381.2.a.x.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.j.1.2 4 4.3 odd 2
1449.2.a.o.1.3 4 12.11 even 2
3381.2.a.x.1.2 4 28.27 even 2
7728.2.a.ce.1.1 4 1.1 even 1 trivial