Properties

Label 198.2.a.e.1.1
Level $198$
Weight $2$
Character 198.1
Self dual yes
Analytic conductor $1.581$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 198.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.58103796002\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 198.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{11} -4.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} +1.00000 q^{22} -6.00000 q^{23} -5.00000 q^{25} -4.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -10.0000 q^{37} -4.00000 q^{38} -6.00000 q^{41} +8.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -5.00000 q^{50} -4.00000 q^{52} +2.00000 q^{56} -6.00000 q^{58} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{67} +6.00000 q^{68} -6.00000 q^{71} +2.00000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +2.00000 q^{77} +14.0000 q^{79} -6.00000 q^{82} +12.0000 q^{83} +8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -8.00000 q^{91} -6.00000 q^{92} +6.00000 q^{94} +14.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 14.0000 1.11378
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −10.0000 −0.755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −8.00000 −0.592999
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) −10.0000 −0.534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −22.0000 −1.15629
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −22.0000 −1.02799
\(459\) 0 0
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 30.0000 1.32324
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) −20.0000 −0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 24.0000 1.03471
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) −5.00000 −0.213201
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 28.0000 1.19068
\(554\) −16.0000 −0.679775
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 30.0000 1.25109
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 16.0000 0.652111
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 2.00000 0.0805823
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −60.0000 −2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 14.0000 0.556890
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 20.0000 0.784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) 28.0000 1.07454
\(680\) 0 0
\(681\) 0 0
\(682\) 8.00000 0.306336
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) −4.00000 −0.151402
\(699\) 0 0
\(700\) −10.0000 −0.377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 40.0000 1.50863
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) 30.0000 1.11417
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −8.00000 −0.296500
\(729\) 0 0
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 20.0000 0.732252
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −8.00000 −0.289619
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −36.0000 −1.28736
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −34.0000 −1.18878
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) −24.0000 −0.829066
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) −30.0000 −1.02899
\(851\) 60.0000 2.05677
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −42.0000 −1.42970 −0.714848 0.699280i \(-0.753504\pi\)
−0.714848 + 0.699280i \(0.753504\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) −52.0000 −1.75592 −0.877958 0.478738i \(-0.841094\pi\)
−0.877958 + 0.478738i \(0.841094\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 42.0000 1.39152 0.695761 0.718273i \(-0.255067\pi\)
0.695761 + 0.718273i \(0.255067\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 2.00000 0.0659739 0.0329870 0.999456i \(-0.489498\pi\)
0.0329870 + 0.999456i \(0.489498\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −42.0000 −1.38320
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 50.0000 1.64399
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 20.0000 0.648886
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 40.0000 1.28965
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) 44.0000 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 198.2.a.e.1.1 1
3.2 odd 2 66.2.a.a.1.1 1
4.3 odd 2 1584.2.a.h.1.1 1
5.2 odd 4 4950.2.c.r.199.2 2
5.3 odd 4 4950.2.c.r.199.1 2
5.4 even 2 4950.2.a.g.1.1 1
7.6 odd 2 9702.2.a.bu.1.1 1
8.3 odd 2 6336.2.a.bf.1.1 1
8.5 even 2 6336.2.a.bj.1.1 1
9.2 odd 6 1782.2.e.s.595.1 2
9.4 even 3 1782.2.e.f.1189.1 2
9.5 odd 6 1782.2.e.s.1189.1 2
9.7 even 3 1782.2.e.f.595.1 2
11.10 odd 2 2178.2.a.b.1.1 1
12.11 even 2 528.2.a.d.1.1 1
15.2 even 4 1650.2.c.d.199.1 2
15.8 even 4 1650.2.c.d.199.2 2
15.14 odd 2 1650.2.a.m.1.1 1
21.20 even 2 3234.2.a.d.1.1 1
24.5 odd 2 2112.2.a.i.1.1 1
24.11 even 2 2112.2.a.v.1.1 1
33.2 even 10 726.2.e.b.565.1 4
33.5 odd 10 726.2.e.k.487.1 4
33.8 even 10 726.2.e.b.493.1 4
33.14 odd 10 726.2.e.k.493.1 4
33.17 even 10 726.2.e.b.487.1 4
33.20 odd 10 726.2.e.k.565.1 4
33.26 odd 10 726.2.e.k.511.1 4
33.29 even 10 726.2.e.b.511.1 4
33.32 even 2 726.2.a.i.1.1 1
132.131 odd 2 5808.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.a.a.1.1 1 3.2 odd 2
198.2.a.e.1.1 1 1.1 even 1 trivial
528.2.a.d.1.1 1 12.11 even 2
726.2.a.i.1.1 1 33.32 even 2
726.2.e.b.487.1 4 33.17 even 10
726.2.e.b.493.1 4 33.8 even 10
726.2.e.b.511.1 4 33.29 even 10
726.2.e.b.565.1 4 33.2 even 10
726.2.e.k.487.1 4 33.5 odd 10
726.2.e.k.493.1 4 33.14 odd 10
726.2.e.k.511.1 4 33.26 odd 10
726.2.e.k.565.1 4 33.20 odd 10
1584.2.a.h.1.1 1 4.3 odd 2
1650.2.a.m.1.1 1 15.14 odd 2
1650.2.c.d.199.1 2 15.2 even 4
1650.2.c.d.199.2 2 15.8 even 4
1782.2.e.f.595.1 2 9.7 even 3
1782.2.e.f.1189.1 2 9.4 even 3
1782.2.e.s.595.1 2 9.2 odd 6
1782.2.e.s.1189.1 2 9.5 odd 6
2112.2.a.i.1.1 1 24.5 odd 2
2112.2.a.v.1.1 1 24.11 even 2
2178.2.a.b.1.1 1 11.10 odd 2
3234.2.a.d.1.1 1 21.20 even 2
4950.2.a.g.1.1 1 5.4 even 2
4950.2.c.r.199.1 2 5.3 odd 4
4950.2.c.r.199.2 2 5.2 odd 4
5808.2.a.l.1.1 1 132.131 odd 2
6336.2.a.bf.1.1 1 8.3 odd 2
6336.2.a.bj.1.1 1 8.5 even 2
9702.2.a.bu.1.1 1 7.6 odd 2