Properties

Label 9075.2.a.bo.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 363)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.00000 q^{12} +1.73205 q^{13} -6.00000 q^{14} -5.00000 q^{16} -1.73205 q^{17} -1.73205 q^{18} +6.92820 q^{19} +3.46410 q^{21} +6.00000 q^{23} +1.73205 q^{24} -3.00000 q^{26} +1.00000 q^{27} +3.46410 q^{28} -1.73205 q^{29} +4.00000 q^{31} +5.19615 q^{32} +3.00000 q^{34} +1.00000 q^{36} +11.0000 q^{37} -12.0000 q^{38} +1.73205 q^{39} -1.73205 q^{41} -6.00000 q^{42} -3.46410 q^{43} -10.3923 q^{46} -5.00000 q^{48} +5.00000 q^{49} -1.73205 q^{51} +1.73205 q^{52} +9.00000 q^{53} -1.73205 q^{54} +6.00000 q^{56} +6.92820 q^{57} +3.00000 q^{58} -6.00000 q^{59} -6.92820 q^{62} +3.46410 q^{63} +1.00000 q^{64} +2.00000 q^{67} -1.73205 q^{68} +6.00000 q^{69} -6.00000 q^{71} +1.73205 q^{72} -6.92820 q^{73} -19.0526 q^{74} +6.92820 q^{76} -3.00000 q^{78} +1.00000 q^{81} +3.00000 q^{82} +3.46410 q^{84} +6.00000 q^{86} -1.73205 q^{87} +9.00000 q^{89} +6.00000 q^{91} +6.00000 q^{92} +4.00000 q^{93} +5.19615 q^{96} +7.00000 q^{97} -8.66025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} + 2 q^{12} - 12 q^{14} - 10 q^{16} + 12 q^{23} - 6 q^{26} + 2 q^{27} + 8 q^{31} + 6 q^{34} + 2 q^{36} + 22 q^{37} - 24 q^{38} - 12 q^{42} - 10 q^{48} + 10 q^{49} + 18 q^{53} + 12 q^{56} + 6 q^{58} - 12 q^{59} + 2 q^{64} + 4 q^{67} + 12 q^{69} - 12 q^{71} - 6 q^{78} + 2 q^{81} + 6 q^{82} + 12 q^{86} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 8 q^{93} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.73205 −0.707107
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 1.73205 0.480384 0.240192 0.970725i \(-0.422790\pi\)
0.240192 + 0.970725i \(0.422790\pi\)
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) −1.73205 −0.408248
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.73205 0.353553
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 1.00000 0.192450
\(28\) 3.46410 0.654654
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) −12.0000 −1.94666
\(39\) 1.73205 0.277350
\(40\) 0 0
\(41\) −1.73205 −0.270501 −0.135250 0.990811i \(-0.543184\pi\)
−0.135250 + 0.990811i \(0.543184\pi\)
\(42\) −6.00000 −0.925820
\(43\) −3.46410 −0.528271 −0.264135 0.964486i \(-0.585087\pi\)
−0.264135 + 0.964486i \(0.585087\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.3923 −1.53226
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −5.00000 −0.721688
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −1.73205 −0.242536
\(52\) 1.73205 0.240192
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −1.73205 −0.235702
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 6.92820 0.917663
\(58\) 3.00000 0.393919
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −6.92820 −0.879883
\(63\) 3.46410 0.436436
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −1.73205 −0.210042
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.73205 0.204124
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) −19.0526 −2.21481
\(75\) 0 0
\(76\) 6.92820 0.794719
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 3.46410 0.377964
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −1.73205 −0.185695
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 6.00000 0.625543
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 5.19615 0.530330
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −8.66025 −0.874818
\(99\) 0 0
\(100\) 0 0
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 3.00000 0.297044
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −15.5885 −1.51408
\(107\) −3.46410 −0.334887 −0.167444 0.985882i \(-0.553551\pi\)
−0.167444 + 0.985882i \(0.553551\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.5885 1.49310 0.746552 0.665327i \(-0.231708\pi\)
0.746552 + 0.665327i \(0.231708\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) −17.3205 −1.63663
\(113\) 21.0000 1.97551 0.987757 0.156001i \(-0.0498603\pi\)
0.987757 + 0.156001i \(0.0498603\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) −1.73205 −0.160817
\(117\) 1.73205 0.160128
\(118\) 10.3923 0.956689
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −1.73205 −0.156174
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −12.1244 −1.07165
\(129\) −3.46410 −0.304997
\(130\) 0 0
\(131\) −17.3205 −1.51330 −0.756650 0.653820i \(-0.773165\pi\)
−0.756650 + 0.653820i \(0.773165\pi\)
\(132\) 0 0
\(133\) 24.0000 2.08106
\(134\) −3.46410 −0.299253
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −10.3923 −0.884652
\(139\) 10.3923 0.881464 0.440732 0.897639i \(-0.354719\pi\)
0.440732 + 0.897639i \(0.354719\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.3923 0.872103
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 5.00000 0.412393
\(148\) 11.0000 0.904194
\(149\) −12.1244 −0.993266 −0.496633 0.867961i \(-0.665430\pi\)
−0.496633 + 0.867961i \(0.665430\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 12.0000 0.973329
\(153\) −1.73205 −0.140028
\(154\) 0 0
\(155\) 0 0
\(156\) 1.73205 0.138675
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 20.7846 1.63806
\(162\) −1.73205 −0.136083
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −1.73205 −0.135250
\(165\) 0 0
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 6.00000 0.462910
\(169\) −10.0000 −0.769231
\(170\) 0 0
\(171\) 6.92820 0.529813
\(172\) −3.46410 −0.264135
\(173\) −20.7846 −1.58022 −0.790112 0.612962i \(-0.789978\pi\)
−0.790112 + 0.612962i \(0.789978\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) −15.5885 −1.16840
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) −10.3923 −0.770329
\(183\) 0 0
\(184\) 10.3923 0.766131
\(185\) 0 0
\(186\) −6.92820 −0.508001
\(187\) 0 0
\(188\) 0 0
\(189\) 3.46410 0.251976
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) 5.19615 0.374027 0.187014 0.982357i \(-0.440119\pi\)
0.187014 + 0.982357i \(0.440119\pi\)
\(194\) −12.1244 −0.870478
\(195\) 0 0
\(196\) 5.00000 0.357143
\(197\) 19.0526 1.35744 0.678719 0.734398i \(-0.262535\pi\)
0.678719 + 0.734398i \(0.262535\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 24.0000 1.68863
\(203\) −6.00000 −0.421117
\(204\) −1.73205 −0.121268
\(205\) 0 0
\(206\) 24.2487 1.68949
\(207\) 6.00000 0.417029
\(208\) −8.66025 −0.600481
\(209\) 0 0
\(210\) 0 0
\(211\) −17.3205 −1.19239 −0.596196 0.802839i \(-0.703322\pi\)
−0.596196 + 0.802839i \(0.703322\pi\)
\(212\) 9.00000 0.618123
\(213\) −6.00000 −0.411113
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 1.73205 0.117851
\(217\) 13.8564 0.940634
\(218\) −27.0000 −1.82867
\(219\) −6.92820 −0.468165
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) −19.0526 −1.27872
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 18.0000 1.20268
\(225\) 0 0
\(226\) −36.3731 −2.41950
\(227\) 24.2487 1.60944 0.804722 0.593652i \(-0.202314\pi\)
0.804722 + 0.593652i \(0.202314\pi\)
\(228\) 6.92820 0.458831
\(229\) −23.0000 −1.51988 −0.759941 0.649992i \(-0.774772\pi\)
−0.759941 + 0.649992i \(0.774772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) −29.4449 −1.92900 −0.964499 0.264088i \(-0.914929\pi\)
−0.964499 + 0.264088i \(0.914929\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 10.3923 0.673633
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) −20.7846 −1.33885 −0.669427 0.742878i \(-0.733460\pi\)
−0.669427 + 0.742878i \(0.733460\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 3.00000 0.191273
\(247\) 12.0000 0.763542
\(248\) 6.92820 0.439941
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 3.46410 0.218218
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 6.00000 0.373544
\(259\) 38.1051 2.36774
\(260\) 0 0
\(261\) −1.73205 −0.107211
\(262\) 30.0000 1.85341
\(263\) −13.8564 −0.854423 −0.427211 0.904152i \(-0.640504\pi\)
−0.427211 + 0.904152i \(0.640504\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −41.5692 −2.54877
\(267\) 9.00000 0.550791
\(268\) 2.00000 0.122169
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −3.46410 −0.210429 −0.105215 0.994450i \(-0.533553\pi\)
−0.105215 + 0.994450i \(0.533553\pi\)
\(272\) 8.66025 0.525105
\(273\) 6.00000 0.363137
\(274\) −10.3923 −0.627822
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 5.19615 0.312207 0.156103 0.987741i \(-0.450107\pi\)
0.156103 + 0.987741i \(0.450107\pi\)
\(278\) −18.0000 −1.07957
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.92820 −0.413302 −0.206651 0.978415i \(-0.566256\pi\)
−0.206651 + 0.978415i \(0.566256\pi\)
\(282\) 0 0
\(283\) 31.1769 1.85328 0.926638 0.375956i \(-0.122686\pi\)
0.926638 + 0.375956i \(0.122686\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 5.19615 0.306186
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) 7.00000 0.410347
\(292\) −6.92820 −0.405442
\(293\) −19.0526 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(294\) −8.66025 −0.505076
\(295\) 0 0
\(296\) 19.0526 1.10741
\(297\) 0 0
\(298\) 21.0000 1.21650
\(299\) 10.3923 0.601003
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) −24.0000 −1.38104
\(303\) −13.8564 −0.796030
\(304\) −34.6410 −1.98680
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) −3.46410 −0.197707 −0.0988534 0.995102i \(-0.531517\pi\)
−0.0988534 + 0.995102i \(0.531517\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 3.00000 0.169842
\(313\) −7.00000 −0.395663 −0.197832 0.980236i \(-0.563390\pi\)
−0.197832 + 0.980236i \(0.563390\pi\)
\(314\) −24.2487 −1.36843
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −15.5885 −0.874157
\(319\) 0 0
\(320\) 0 0
\(321\) −3.46410 −0.193347
\(322\) −36.0000 −2.00620
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 3.46410 0.191859
\(327\) 15.5885 0.862044
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 11.0000 0.602796
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) −17.3205 −0.944911
\(337\) 12.1244 0.660456 0.330228 0.943901i \(-0.392874\pi\)
0.330228 + 0.943901i \(0.392874\pi\)
\(338\) 17.3205 0.942111
\(339\) 21.0000 1.14056
\(340\) 0 0
\(341\) 0 0
\(342\) −12.0000 −0.648886
\(343\) −6.92820 −0.374088
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) 36.0000 1.93537
\(347\) −27.7128 −1.48770 −0.743851 0.668346i \(-0.767003\pi\)
−0.743851 + 0.668346i \(0.767003\pi\)
\(348\) −1.73205 −0.0928477
\(349\) −1.73205 −0.0927146 −0.0463573 0.998925i \(-0.514761\pi\)
−0.0463573 + 0.998925i \(0.514761\pi\)
\(350\) 0 0
\(351\) 1.73205 0.0924500
\(352\) 0 0
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 10.3923 0.552345
\(355\) 0 0
\(356\) 9.00000 0.476999
\(357\) −6.00000 −0.317554
\(358\) −20.7846 −1.09850
\(359\) 31.1769 1.64545 0.822727 0.568436i \(-0.192451\pi\)
0.822727 + 0.568436i \(0.192451\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 12.1244 0.637242
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) −30.0000 −1.56386
\(369\) −1.73205 −0.0901670
\(370\) 0 0
\(371\) 31.1769 1.61862
\(372\) 4.00000 0.207390
\(373\) −20.7846 −1.07619 −0.538093 0.842885i \(-0.680855\pi\)
−0.538093 + 0.842885i \(0.680855\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) −6.00000 −0.308607
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.7846 −1.06343
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) −12.1244 −0.618718
\(385\) 0 0
\(386\) −9.00000 −0.458088
\(387\) −3.46410 −0.176090
\(388\) 7.00000 0.355371
\(389\) −27.0000 −1.36895 −0.684477 0.729034i \(-0.739969\pi\)
−0.684477 + 0.729034i \(0.739969\pi\)
\(390\) 0 0
\(391\) −10.3923 −0.525561
\(392\) 8.66025 0.437409
\(393\) −17.3205 −0.873704
\(394\) −33.0000 −1.66252
\(395\) 0 0
\(396\) 0 0
\(397\) −11.0000 −0.552074 −0.276037 0.961147i \(-0.589021\pi\)
−0.276037 + 0.961147i \(0.589021\pi\)
\(398\) −17.3205 −0.868199
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) −3.46410 −0.172774
\(403\) 6.92820 0.345118
\(404\) −13.8564 −0.689382
\(405\) 0 0
\(406\) 10.3923 0.515761
\(407\) 0 0
\(408\) −3.00000 −0.148522
\(409\) 1.73205 0.0856444 0.0428222 0.999083i \(-0.486365\pi\)
0.0428222 + 0.999083i \(0.486365\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) −14.0000 −0.689730
\(413\) −20.7846 −1.02274
\(414\) −10.3923 −0.510754
\(415\) 0 0
\(416\) 9.00000 0.441261
\(417\) 10.3923 0.508913
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 7.00000 0.341159 0.170580 0.985344i \(-0.445436\pi\)
0.170580 + 0.985344i \(0.445436\pi\)
\(422\) 30.0000 1.46038
\(423\) 0 0
\(424\) 15.5885 0.757042
\(425\) 0 0
\(426\) 10.3923 0.503509
\(427\) 0 0
\(428\) −3.46410 −0.167444
\(429\) 0 0
\(430\) 0 0
\(431\) −17.3205 −0.834300 −0.417150 0.908838i \(-0.636971\pi\)
−0.417150 + 0.908838i \(0.636971\pi\)
\(432\) −5.00000 −0.240563
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) −24.0000 −1.15204
\(435\) 0 0
\(436\) 15.5885 0.746552
\(437\) 41.5692 1.98853
\(438\) 12.0000 0.573382
\(439\) −6.92820 −0.330665 −0.165333 0.986238i \(-0.552870\pi\)
−0.165333 + 0.986238i \(0.552870\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 5.19615 0.247156
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 11.0000 0.522037
\(445\) 0 0
\(446\) 34.6410 1.64030
\(447\) −12.1244 −0.573462
\(448\) 3.46410 0.163663
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 21.0000 0.987757
\(453\) 13.8564 0.651031
\(454\) −42.0000 −1.97116
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 29.4449 1.37737 0.688686 0.725059i \(-0.258188\pi\)
0.688686 + 0.725059i \(0.258188\pi\)
\(458\) 39.8372 1.86147
\(459\) −1.73205 −0.0808452
\(460\) 0 0
\(461\) −15.5885 −0.726027 −0.363013 0.931784i \(-0.618252\pi\)
−0.363013 + 0.931784i \(0.618252\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 8.66025 0.402042
\(465\) 0 0
\(466\) 51.0000 2.36253
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 1.73205 0.0800641
\(469\) 6.92820 0.319915
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) −10.3923 −0.478345
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 9.00000 0.412082
\(478\) 12.0000 0.548867
\(479\) 24.2487 1.10795 0.553976 0.832533i \(-0.313110\pi\)
0.553976 + 0.832533i \(0.313110\pi\)
\(480\) 0 0
\(481\) 19.0526 0.868722
\(482\) 36.0000 1.63976
\(483\) 20.7846 0.945732
\(484\) 0 0
\(485\) 0 0
\(486\) −1.73205 −0.0785674
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) −20.7846 −0.937996 −0.468998 0.883199i \(-0.655385\pi\)
−0.468998 + 0.883199i \(0.655385\pi\)
\(492\) −1.73205 −0.0780869
\(493\) 3.00000 0.135113
\(494\) −20.7846 −0.935144
\(495\) 0 0
\(496\) −20.0000 −0.898027
\(497\) −20.7846 −0.932317
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 3.46410 0.154765
\(502\) 10.3923 0.463831
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) 0 0
\(507\) −10.0000 −0.444116
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −24.0000 −1.06170
\(512\) −8.66025 −0.382733
\(513\) 6.92820 0.305888
\(514\) 15.5885 0.687577
\(515\) 0 0
\(516\) −3.46410 −0.152499
\(517\) 0 0
\(518\) −66.0000 −2.89987
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 3.00000 0.131306
\(523\) 41.5692 1.81770 0.908848 0.417128i \(-0.136963\pi\)
0.908848 + 0.417128i \(0.136963\pi\)
\(524\) −17.3205 −0.756650
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −6.92820 −0.301797
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 24.0000 1.04053
\(533\) −3.00000 −0.129944
\(534\) −15.5885 −0.674579
\(535\) 0 0
\(536\) 3.46410 0.149626
\(537\) 12.0000 0.517838
\(538\) 36.3731 1.56815
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 6.00000 0.257722
\(543\) −7.00000 −0.300399
\(544\) −9.00000 −0.385872
\(545\) 0 0
\(546\) −10.3923 −0.444750
\(547\) 34.6410 1.48114 0.740571 0.671978i \(-0.234555\pi\)
0.740571 + 0.671978i \(0.234555\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 10.3923 0.442326
\(553\) 0 0
\(554\) −9.00000 −0.382373
\(555\) 0 0
\(556\) 10.3923 0.440732
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −6.92820 −0.293294
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −17.3205 −0.729972 −0.364986 0.931013i \(-0.618926\pi\)
−0.364986 + 0.931013i \(0.618926\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −54.0000 −2.26979
\(567\) 3.46410 0.145479
\(568\) −10.3923 −0.436051
\(569\) −27.7128 −1.16178 −0.580891 0.813982i \(-0.697296\pi\)
−0.580891 + 0.813982i \(0.697296\pi\)
\(570\) 0 0
\(571\) 3.46410 0.144968 0.0724841 0.997370i \(-0.476907\pi\)
0.0724841 + 0.997370i \(0.476907\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 10.3923 0.433766
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 24.2487 1.00861
\(579\) 5.19615 0.215945
\(580\) 0 0
\(581\) 0 0
\(582\) −12.1244 −0.502571
\(583\) 0 0
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 33.0000 1.36322
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 5.00000 0.206197
\(589\) 27.7128 1.14189
\(590\) 0 0
\(591\) 19.0526 0.783718
\(592\) −55.0000 −2.26049
\(593\) 22.5167 0.924648 0.462324 0.886711i \(-0.347016\pi\)
0.462324 + 0.886711i \(0.347016\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.1244 −0.496633
\(597\) 10.0000 0.409273
\(598\) −18.0000 −0.736075
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −22.5167 −0.918474 −0.459237 0.888314i \(-0.651877\pi\)
−0.459237 + 0.888314i \(0.651877\pi\)
\(602\) 20.7846 0.847117
\(603\) 2.00000 0.0814463
\(604\) 13.8564 0.563809
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) −20.7846 −0.843621 −0.421811 0.906684i \(-0.638605\pi\)
−0.421811 + 0.906684i \(0.638605\pi\)
\(608\) 36.0000 1.45999
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) −1.73205 −0.0700140
\(613\) −19.0526 −0.769526 −0.384763 0.923015i \(-0.625717\pi\)
−0.384763 + 0.923015i \(0.625717\pi\)
\(614\) 6.00000 0.242140
\(615\) 0 0
\(616\) 0 0
\(617\) 9.00000 0.362326 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(618\) 24.2487 0.975426
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −20.7846 −0.833387
\(623\) 31.1769 1.24908
\(624\) −8.66025 −0.346688
\(625\) 0 0
\(626\) 12.1244 0.484587
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) −19.0526 −0.759675
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 0 0
\(633\) −17.3205 −0.688428
\(634\) 10.3923 0.412731
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 8.66025 0.343132
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 6.00000 0.236801
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 20.7846 0.819028
\(645\) 0 0
\(646\) 20.7846 0.817760
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.73205 0.0680414
\(649\) 0 0
\(650\) 0 0
\(651\) 13.8564 0.543075
\(652\) −2.00000 −0.0783260
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −27.0000 −1.05578
\(655\) 0 0
\(656\) 8.66025 0.338126
\(657\) −6.92820 −0.270295
\(658\) 0 0
\(659\) 13.8564 0.539769 0.269884 0.962893i \(-0.413014\pi\)
0.269884 + 0.962893i \(0.413014\pi\)
\(660\) 0 0
\(661\) −41.0000 −1.59472 −0.797358 0.603507i \(-0.793769\pi\)
−0.797358 + 0.603507i \(0.793769\pi\)
\(662\) −34.6410 −1.34636
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) 0 0
\(666\) −19.0526 −0.738272
\(667\) −10.3923 −0.402392
\(668\) 3.46410 0.134030
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) 0 0
\(672\) 18.0000 0.694365
\(673\) 20.7846 0.801188 0.400594 0.916256i \(-0.368804\pi\)
0.400594 + 0.916256i \(0.368804\pi\)
\(674\) −21.0000 −0.808890
\(675\) 0 0
\(676\) −10.0000 −0.384615
\(677\) −32.9090 −1.26479 −0.632397 0.774644i \(-0.717929\pi\)
−0.632397 + 0.774644i \(0.717929\pi\)
\(678\) −36.3731 −1.39690
\(679\) 24.2487 0.930580
\(680\) 0 0
\(681\) 24.2487 0.929213
\(682\) 0 0
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 6.92820 0.264906
\(685\) 0 0
\(686\) 12.0000 0.458162
\(687\) −23.0000 −0.877505
\(688\) 17.3205 0.660338
\(689\) 15.5885 0.593873
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −20.7846 −0.790112
\(693\) 0 0
\(694\) 48.0000 1.82206
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 3.00000 0.113633
\(698\) 3.00000 0.113552
\(699\) −29.4449 −1.11371
\(700\) 0 0
\(701\) −39.8372 −1.50463 −0.752315 0.658804i \(-0.771063\pi\)
−0.752315 + 0.658804i \(0.771063\pi\)
\(702\) −3.00000 −0.113228
\(703\) 76.2102 2.87432
\(704\) 0 0
\(705\) 0 0
\(706\) 36.3731 1.36892
\(707\) −48.0000 −1.80523
\(708\) −6.00000 −0.225494
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.5885 0.584202
\(713\) 24.0000 0.898807
\(714\) 10.3923 0.388922
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −6.92820 −0.258738
\(718\) −54.0000 −2.01526
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −48.4974 −1.80614
\(722\) −50.2295 −1.86935
\(723\) −20.7846 −0.772988
\(724\) −7.00000 −0.260153
\(725\) 0 0
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 10.3923 0.385164
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 0 0
\(733\) −8.66025 −0.319874 −0.159937 0.987127i \(-0.551129\pi\)
−0.159937 + 0.987127i \(0.551129\pi\)
\(734\) 45.0333 1.66221
\(735\) 0 0
\(736\) 31.1769 1.14920
\(737\) 0 0
\(738\) 3.00000 0.110432
\(739\) −27.7128 −1.01943 −0.509716 0.860343i \(-0.670250\pi\)
−0.509716 + 0.860343i \(0.670250\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) −54.0000 −1.98240
\(743\) 51.9615 1.90628 0.953142 0.302524i \(-0.0978293\pi\)
0.953142 + 0.302524i \(0.0978293\pi\)
\(744\) 6.92820 0.254000
\(745\) 0 0
\(746\) 36.0000 1.31805
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 5.19615 0.189233
\(755\) 0 0
\(756\) 3.46410 0.125988
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) −24.2487 −0.880753
\(759\) 0 0
\(760\) 0 0
\(761\) 32.9090 1.19295 0.596475 0.802632i \(-0.296568\pi\)
0.596475 + 0.802632i \(0.296568\pi\)
\(762\) 0 0
\(763\) 54.0000 1.95493
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 10.3923 0.375489
\(767\) −10.3923 −0.375244
\(768\) 19.0000 0.685603
\(769\) 25.9808 0.936890 0.468445 0.883493i \(-0.344814\pi\)
0.468445 + 0.883493i \(0.344814\pi\)
\(770\) 0 0
\(771\) −9.00000 −0.324127
\(772\) 5.19615 0.187014
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) 12.1244 0.435239
\(777\) 38.1051 1.36701
\(778\) 46.7654 1.67662
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) −1.73205 −0.0618984
\(784\) −25.0000 −0.892857
\(785\) 0 0
\(786\) 30.0000 1.07006
\(787\) 6.92820 0.246964 0.123482 0.992347i \(-0.460594\pi\)
0.123482 + 0.992347i \(0.460594\pi\)
\(788\) 19.0526 0.678719
\(789\) −13.8564 −0.493301
\(790\) 0 0
\(791\) 72.7461 2.58655
\(792\) 0 0
\(793\) 0 0
\(794\) 19.0526 0.676150
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −41.5692 −1.47153
\(799\) 0 0
\(800\) 0 0
\(801\) 9.00000 0.317999
\(802\) −5.19615 −0.183483
\(803\) 0 0
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −21.0000 −0.739235
\(808\) −24.0000 −0.844317
\(809\) 20.7846 0.730748 0.365374 0.930861i \(-0.380941\pi\)
0.365374 + 0.930861i \(0.380941\pi\)
\(810\) 0 0
\(811\) −27.7128 −0.973128 −0.486564 0.873645i \(-0.661750\pi\)
−0.486564 + 0.873645i \(0.661750\pi\)
\(812\) −6.00000 −0.210559
\(813\) −3.46410 −0.121491
\(814\) 0 0
\(815\) 0 0
\(816\) 8.66025 0.303170
\(817\) −24.0000 −0.839654
\(818\) −3.00000 −0.104893
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 55.4256 1.93437 0.967184 0.254078i \(-0.0817719\pi\)
0.967184 + 0.254078i \(0.0817719\pi\)
\(822\) −10.3923 −0.362473
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) −24.2487 −0.844744
\(825\) 0 0
\(826\) 36.0000 1.25260
\(827\) 38.1051 1.32504 0.662522 0.749042i \(-0.269486\pi\)
0.662522 + 0.749042i \(0.269486\pi\)
\(828\) 6.00000 0.208514
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) 5.19615 0.180253
\(832\) 1.73205 0.0600481
\(833\) −8.66025 −0.300060
\(834\) −18.0000 −0.623289
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 51.9615 1.79498
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) −12.1244 −0.417833
\(843\) −6.92820 −0.238620
\(844\) −17.3205 −0.596196
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −45.0000 −1.54531
\(849\) 31.1769 1.06999
\(850\) 0 0
\(851\) 66.0000 2.26245
\(852\) −6.00000 −0.205557
\(853\) −8.66025 −0.296521 −0.148261 0.988948i \(-0.547367\pi\)
−0.148261 + 0.988948i \(0.547367\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.00000 −0.205076
\(857\) −41.5692 −1.41998 −0.709989 0.704213i \(-0.751300\pi\)
−0.709989 + 0.704213i \(0.751300\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 30.0000 1.02180
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 5.19615 0.176777
\(865\) 0 0
\(866\) −32.9090 −1.11829
\(867\) −14.0000 −0.475465
\(868\) 13.8564 0.470317
\(869\) 0 0
\(870\) 0 0
\(871\) 3.46410 0.117377
\(872\) 27.0000 0.914335
\(873\) 7.00000 0.236914
\(874\) −72.0000 −2.43544
\(875\) 0 0
\(876\) −6.92820 −0.234082
\(877\) 19.0526 0.643359 0.321680 0.946849i \(-0.395753\pi\)
0.321680 + 0.946849i \(0.395753\pi\)
\(878\) 12.0000 0.404980
\(879\) −19.0526 −0.642627
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) −8.66025 −0.291606
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) −3.00000 −0.100901
\(885\) 0 0
\(886\) −31.1769 −1.04741
\(887\) −31.1769 −1.04682 −0.523409 0.852081i \(-0.675340\pi\)
−0.523409 + 0.852081i \(0.675340\pi\)
\(888\) 19.0526 0.639362
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −20.0000 −0.669650
\(893\) 0 0
\(894\) 21.0000 0.702345
\(895\) 0 0
\(896\) −42.0000 −1.40312
\(897\) 10.3923 0.346989
\(898\) −36.3731 −1.21378
\(899\) −6.92820 −0.231069
\(900\) 0 0
\(901\) −15.5885 −0.519327
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 36.3731 1.20975
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 24.2487 0.804722
\(909\) −13.8564 −0.459588
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −34.6410 −1.14708
\(913\) 0 0
\(914\) −51.0000 −1.68693
\(915\) 0 0
\(916\) −23.0000 −0.759941
\(917\) −60.0000 −1.98137
\(918\) 3.00000 0.0990148
\(919\) −24.2487 −0.799891 −0.399946 0.916539i \(-0.630971\pi\)
−0.399946 + 0.916539i \(0.630971\pi\)
\(920\) 0 0
\(921\) −3.46410 −0.114146
\(922\) 27.0000 0.889198
\(923\) −10.3923 −0.342067
\(924\) 0 0
\(925\) 0 0
\(926\) −58.8897 −1.93524
\(927\) −14.0000 −0.459820
\(928\) −9.00000 −0.295439
\(929\) 39.0000 1.27955 0.639774 0.768563i \(-0.279028\pi\)
0.639774 + 0.768563i \(0.279028\pi\)
\(930\) 0 0
\(931\) 34.6410 1.13531
\(932\) −29.4449 −0.964499
\(933\) 12.0000 0.392862
\(934\) −31.1769 −1.02014
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) −50.2295 −1.64093 −0.820463 0.571700i \(-0.806284\pi\)
−0.820463 + 0.571700i \(0.806284\pi\)
\(938\) −12.0000 −0.391814
\(939\) −7.00000 −0.228436
\(940\) 0 0
\(941\) 29.4449 0.959875 0.479938 0.877303i \(-0.340659\pi\)
0.479938 + 0.877303i \(0.340659\pi\)
\(942\) −24.2487 −0.790066
\(943\) −10.3923 −0.338420
\(944\) 30.0000 0.976417
\(945\) 0 0
\(946\) 0 0
\(947\) −54.0000 −1.75476 −0.877382 0.479792i \(-0.840712\pi\)
−0.877382 + 0.479792i \(0.840712\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) −10.3923 −0.336817
\(953\) 57.1577 1.85152 0.925759 0.378113i \(-0.123427\pi\)
0.925759 + 0.378113i \(0.123427\pi\)
\(954\) −15.5885 −0.504695
\(955\) 0 0
\(956\) −6.92820 −0.224074
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) 20.7846 0.671170
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −33.0000 −1.06396
\(963\) −3.46410 −0.111629
\(964\) −20.7846 −0.669427
\(965\) 0 0
\(966\) −36.0000 −1.15828
\(967\) −10.3923 −0.334194 −0.167097 0.985940i \(-0.553439\pi\)
−0.167097 + 0.985940i \(0.553439\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 1.00000 0.0320750
\(973\) 36.0000 1.15411
\(974\) −65.8179 −2.10894
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 0.287936 0.143968 0.989582i \(-0.454014\pi\)
0.143968 + 0.989582i \(0.454014\pi\)
\(978\) 3.46410 0.110770
\(979\) 0 0
\(980\) 0 0
\(981\) 15.5885 0.497701
\(982\) 36.0000 1.14881
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) −5.19615 −0.165479
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −20.7846 −0.660912
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 20.7846 0.659912
\(993\) 20.0000 0.634681
\(994\) 36.0000 1.14185
\(995\) 0 0
\(996\) 0 0
\(997\) 1.73205 0.0548546 0.0274273 0.999624i \(-0.491269\pi\)
0.0274273 + 0.999624i \(0.491269\pi\)
\(998\) −38.1051 −1.20620
\(999\) 11.0000 0.348025
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 9075.2.a.bo.1.1 2
5.4 even 2 363.2.a.f.1.2 yes 2
11.10 odd 2 inner 9075.2.a.bo.1.2 2
15.14 odd 2 1089.2.a.o.1.1 2
20.19 odd 2 5808.2.a.ca.1.2 2
55.4 even 10 363.2.e.m.148.2 8
55.9 even 10 363.2.e.m.202.1 8
55.14 even 10 363.2.e.m.130.2 8
55.19 odd 10 363.2.e.m.130.1 8
55.24 odd 10 363.2.e.m.202.2 8
55.29 odd 10 363.2.e.m.148.1 8
55.39 odd 10 363.2.e.m.124.2 8
55.49 even 10 363.2.e.m.124.1 8
55.54 odd 2 363.2.a.f.1.1 2
165.164 even 2 1089.2.a.o.1.2 2
220.219 even 2 5808.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.f.1.1 2 55.54 odd 2
363.2.a.f.1.2 yes 2 5.4 even 2
363.2.e.m.124.1 8 55.49 even 10
363.2.e.m.124.2 8 55.39 odd 10
363.2.e.m.130.1 8 55.19 odd 10
363.2.e.m.130.2 8 55.14 even 10
363.2.e.m.148.1 8 55.29 odd 10
363.2.e.m.148.2 8 55.4 even 10
363.2.e.m.202.1 8 55.9 even 10
363.2.e.m.202.2 8 55.24 odd 10
1089.2.a.o.1.1 2 15.14 odd 2
1089.2.a.o.1.2 2 165.164 even 2
5808.2.a.ca.1.1 2 220.219 even 2
5808.2.a.ca.1.2 2 20.19 odd 2
9075.2.a.bo.1.1 2 1.1 even 1 trivial
9075.2.a.bo.1.2 2 11.10 odd 2 inner