Properties

Label 9408.2.a.cr.1.1
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{15} +4.00000 q^{17} -8.00000 q^{19} +4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +5.00000 q^{29} +3.00000 q^{31} +5.00000 q^{33} +4.00000 q^{37} +2.00000 q^{43} +1.00000 q^{45} -6.00000 q^{47} +4.00000 q^{51} +9.00000 q^{53} +5.00000 q^{55} -8.00000 q^{57} +11.0000 q^{59} -6.00000 q^{61} -2.00000 q^{67} +4.00000 q^{69} -2.00000 q^{71} -10.0000 q^{73} -4.00000 q^{75} -3.00000 q^{79} +1.00000 q^{81} +7.00000 q^{83} +4.00000 q^{85} +5.00000 q^{87} +6.00000 q^{89} +3.00000 q^{93} -8.00000 q^{95} -7.00000 q^{97} +5.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.00000 0.768350 0.384175 0.923260i \(-0.374486\pi\)
0.384175 + 0.923260i \(0.374486\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 5.00000 0.389249
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.0000 0.826811
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 20.0000 1.46254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −40.0000 −2.76686
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 25.0000 1.61039 0.805196 0.593009i \(-0.202060\pi\)
0.805196 + 0.593009i \(0.202060\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 7.00000 0.443607
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 20.0000 1.25739
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 31.0000 1.89010 0.945052 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) 15.0000 0.911185 0.455593 0.890188i \(-0.349427\pi\)
0.455593 + 0.890188i \(0.349427\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −7.00000 −0.410347
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 −0.168497 −0.0842484 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) 0 0
\(319\) 25.0000 1.39973
\(320\) 0 0
\(321\) 3.00000 0.167444
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 9.00000 0.490261 0.245131 0.969490i \(-0.421169\pi\)
0.245131 + 0.969490i \(0.421169\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −9.00000 −0.461084
\(382\) 0 0
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −1.00000 −0.0504433
\(394\) 0 0
\(395\) −3.00000 −0.150946
\(396\) 0 0
\(397\) 36.0000 1.80679 0.903394 0.428811i \(-0.141067\pi\)
0.903394 + 0.428811i \(0.141067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 7.00000 0.343616
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −16.0000 −0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 5.00000 0.239732
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 15.0000 0.715911 0.357955 0.933739i \(-0.383474\pi\)
0.357955 + 0.933739i \(0.383474\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.0000 0.807694 0.403847 0.914826i \(-0.367673\pi\)
0.403847 + 0.914826i \(0.367673\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −19.0000 −0.892698
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) 32.0000 1.46826
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) 38.0000 1.73626 0.868132 0.496333i \(-0.165321\pi\)
0.868132 + 0.496333i \(0.165321\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.00000 −0.317854
\(486\) 0 0
\(487\) −5.00000 −0.226572 −0.113286 0.993562i \(-0.536138\pi\)
−0.113286 + 0.993562i \(0.536138\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 5.00000 0.224733
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) −14.0000 −0.625474
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) 23.0000 0.974541 0.487271 0.873251i \(-0.337993\pi\)
0.487271 + 0.873251i \(0.337993\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) 0 0
\(563\) −17.0000 −0.716465 −0.358232 0.933632i \(-0.616620\pi\)
−0.358232 + 0.933632i \(0.616620\pi\)
\(564\) 0 0
\(565\) 16.0000 0.673125
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 0 0
\(579\) 5.00000 0.207793
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 45.0000 1.86371
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −35.0000 −1.44460 −0.722302 0.691577i \(-0.756916\pi\)
−0.722302 + 0.691577i \(0.756916\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −27.0000 −1.09590 −0.547948 0.836512i \(-0.684591\pi\)
−0.547948 + 0.836512i \(0.684591\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −12.0000 −0.484675 −0.242338 0.970192i \(-0.577914\pi\)
−0.242338 + 0.970192i \(0.577914\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −40.0000 −1.59745
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 19.0000 0.756378 0.378189 0.925728i \(-0.376547\pi\)
0.378189 + 0.925728i \(0.376547\pi\)
\(632\) 0 0
\(633\) 2.00000 0.0794929
\(634\) 0 0
\(635\) −9.00000 −0.357154
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 55.0000 2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) −1.00000 −0.0390732
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) 0 0
\(669\) −7.00000 −0.270636
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) −20.0000 −0.763048
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.0000 0.531050
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −3.00000 −0.112509
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 25.0000 0.929760
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 7.00000 0.256117
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −45.0000 −1.64207 −0.821037 0.570875i \(-0.806604\pi\)
−0.821037 + 0.570875i \(0.806604\pi\)
\(752\) 0 0
\(753\) −21.0000 −0.765283
\(754\) 0 0
\(755\) −19.0000 −0.691481
\(756\) 0 0
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) 0 0
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) −8.00000 −0.290000 −0.145000 0.989432i \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.00000 0.144620
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 0 0
\(789\) 30.0000 1.06803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 9.00000 0.319197
\(796\) 0 0
\(797\) 21.0000 0.743858 0.371929 0.928261i \(-0.378696\pi\)
0.371929 + 0.928261i \(0.378696\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −50.0000 −1.76446
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 31.0000 1.09125
\(808\) 0 0
\(809\) 40.0000 1.40633 0.703163 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 15.0000 0.526073
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −32.0000 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) 16.0000 0.555034
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −14.0000 −0.484490
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 2.00000 0.0688837
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.00000 −0.236914
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) 0 0
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 0 0
\(885\) 11.0000 0.369761
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 35.0000 1.15833
\(914\) 0 0
\(915\) −6.00000 −0.198354
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −16.0000 −0.526077
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −32.0000 −1.04763
\(934\) 0 0
\(935\) 20.0000 0.654070
\(936\) 0 0
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) −1.00000 −0.0326338
\(940\) 0 0
\(941\) −11.0000 −0.358590 −0.179295 0.983795i \(-0.557382\pi\)
−0.179295 + 0.983795i \(0.557382\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 0 0
\(957\) 25.0000 0.808135
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 3.00000 0.0966736
\(964\) 0 0
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) 61.0000 1.96163 0.980814 0.194946i \(-0.0624533\pi\)
0.980814 + 0.194946i \(0.0624533\pi\)
\(968\) 0 0
\(969\) −32.0000 −1.02799
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −47.0000 −1.49300 −0.746502 0.665383i \(-0.768268\pi\)
−0.746502 + 0.665383i \(0.768268\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 9408.2.a.cr.1.1 1
4.3 odd 2 9408.2.a.z.1.1 1
7.3 odd 6 1344.2.q.s.961.1 2
7.5 odd 6 1344.2.q.s.193.1 2
7.6 odd 2 9408.2.a.q.1.1 1
8.3 odd 2 294.2.a.f.1.1 1
8.5 even 2 2352.2.a.f.1.1 1
24.5 odd 2 7056.2.a.bl.1.1 1
24.11 even 2 882.2.a.d.1.1 1
28.3 even 6 1344.2.q.g.961.1 2
28.19 even 6 1344.2.q.g.193.1 2
28.27 even 2 9408.2.a.ce.1.1 1
40.19 odd 2 7350.2.a.q.1.1 1
56.3 even 6 42.2.e.a.37.1 yes 2
56.5 odd 6 336.2.q.b.193.1 2
56.11 odd 6 294.2.e.b.79.1 2
56.13 odd 2 2352.2.a.t.1.1 1
56.19 even 6 42.2.e.a.25.1 2
56.27 even 2 294.2.a.e.1.1 1
56.37 even 6 2352.2.q.u.1537.1 2
56.45 odd 6 336.2.q.b.289.1 2
56.51 odd 6 294.2.e.b.67.1 2
56.53 even 6 2352.2.q.u.961.1 2
168.5 even 6 1008.2.s.k.865.1 2
168.11 even 6 882.2.g.i.667.1 2
168.59 odd 6 126.2.g.c.37.1 2
168.83 odd 2 882.2.a.c.1.1 1
168.101 even 6 1008.2.s.k.289.1 2
168.107 even 6 882.2.g.i.361.1 2
168.125 even 2 7056.2.a.w.1.1 1
168.131 odd 6 126.2.g.c.109.1 2
280.3 odd 12 1050.2.o.a.499.2 4
280.19 even 6 1050.2.i.l.151.1 2
280.59 even 6 1050.2.i.l.751.1 2
280.139 even 2 7350.2.a.bl.1.1 1
280.187 odd 12 1050.2.o.a.949.2 4
280.227 odd 12 1050.2.o.a.499.1 4
280.243 odd 12 1050.2.o.a.949.1 4
504.59 odd 6 1134.2.h.l.541.1 2
504.115 even 6 1134.2.e.l.919.1 2
504.131 odd 6 1134.2.e.e.865.1 2
504.187 even 6 1134.2.h.e.109.1 2
504.227 odd 6 1134.2.e.e.919.1 2
504.283 even 6 1134.2.h.e.541.1 2
504.299 odd 6 1134.2.h.l.109.1 2
504.355 even 6 1134.2.e.l.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.a.25.1 2 56.19 even 6
42.2.e.a.37.1 yes 2 56.3 even 6
126.2.g.c.37.1 2 168.59 odd 6
126.2.g.c.109.1 2 168.131 odd 6
294.2.a.e.1.1 1 56.27 even 2
294.2.a.f.1.1 1 8.3 odd 2
294.2.e.b.67.1 2 56.51 odd 6
294.2.e.b.79.1 2 56.11 odd 6
336.2.q.b.193.1 2 56.5 odd 6
336.2.q.b.289.1 2 56.45 odd 6
882.2.a.c.1.1 1 168.83 odd 2
882.2.a.d.1.1 1 24.11 even 2
882.2.g.i.361.1 2 168.107 even 6
882.2.g.i.667.1 2 168.11 even 6
1008.2.s.k.289.1 2 168.101 even 6
1008.2.s.k.865.1 2 168.5 even 6
1050.2.i.l.151.1 2 280.19 even 6
1050.2.i.l.751.1 2 280.59 even 6
1050.2.o.a.499.1 4 280.227 odd 12
1050.2.o.a.499.2 4 280.3 odd 12
1050.2.o.a.949.1 4 280.243 odd 12
1050.2.o.a.949.2 4 280.187 odd 12
1134.2.e.e.865.1 2 504.131 odd 6
1134.2.e.e.919.1 2 504.227 odd 6
1134.2.e.l.865.1 2 504.355 even 6
1134.2.e.l.919.1 2 504.115 even 6
1134.2.h.e.109.1 2 504.187 even 6
1134.2.h.e.541.1 2 504.283 even 6
1134.2.h.l.109.1 2 504.299 odd 6
1134.2.h.l.541.1 2 504.59 odd 6
1344.2.q.g.193.1 2 28.19 even 6
1344.2.q.g.961.1 2 28.3 even 6
1344.2.q.s.193.1 2 7.5 odd 6
1344.2.q.s.961.1 2 7.3 odd 6
2352.2.a.f.1.1 1 8.5 even 2
2352.2.a.t.1.1 1 56.13 odd 2
2352.2.q.u.961.1 2 56.53 even 6
2352.2.q.u.1537.1 2 56.37 even 6
7056.2.a.w.1.1 1 168.125 even 2
7056.2.a.bl.1.1 1 24.5 odd 2
7350.2.a.q.1.1 1 40.19 odd 2
7350.2.a.bl.1.1 1 280.139 even 2
9408.2.a.q.1.1 1 7.6 odd 2
9408.2.a.z.1.1 1 4.3 odd 2
9408.2.a.ce.1.1 1 28.27 even 2
9408.2.a.cr.1.1 1 1.1 even 1 trivial