Properties

Label 7623.2.a.c.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.00000 q^{10} +4.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{20} -4.00000 q^{23} -4.00000 q^{25} -8.00000 q^{26} -2.00000 q^{28} -2.00000 q^{31} +8.00000 q^{32} -2.00000 q^{34} -1.00000 q^{35} +6.00000 q^{37} -2.00000 q^{41} +3.00000 q^{43} +8.00000 q^{46} -7.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} +8.00000 q^{52} -12.0000 q^{53} +5.00000 q^{59} +12.0000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +4.00000 q^{65} +5.00000 q^{67} +2.00000 q^{68} +2.00000 q^{70} +6.00000 q^{71} -2.00000 q^{73} -12.0000 q^{74} +8.00000 q^{79} -4.00000 q^{80} +4.00000 q^{82} +15.0000 q^{83} +1.00000 q^{85} -6.00000 q^{86} -9.00000 q^{89} -4.00000 q^{91} -8.00000 q^{92} +14.0000 q^{94} +2.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 0 0
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −8.00000 −1.56893
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 3.00000 0.457496 0.228748 0.973486i \(-0.426537\pi\)
0.228748 + 0.973486i \(0.426537\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 8.00000 1.13137
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 14.0000 1.44399
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −13.0000 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(102\) 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
\(106\) 24.0000 2.33109
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 0 0
\(122\) −24.0000 −2.17286
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 12.0000 0.986394
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −16.0000 −1.27289
\(159\) 0 0
\(160\) 8.00000 0.632456
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −30.0000 −2.32845
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 6.00000 0.457496
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) −14.0000 −1.02105
\(189\) 0 0
\(190\) 0 0
\(191\) 2.00000 0.144715 0.0723575 0.997379i \(-0.476948\pi\)
0.0723575 + 0.997379i \(0.476948\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 26.0000 1.82935
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −16.0000 −1.10940
\(209\) 0 0
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) 24.0000 1.64061
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) −8.00000 −0.534522
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) −27.0000 −1.79205 −0.896026 0.444001i \(-0.853559\pi\)
−0.896026 + 0.444001i \(0.853559\pi\)
\(228\) 0 0
\(229\) 30.0000 1.98246 0.991228 0.132164i \(-0.0421925\pi\)
0.991228 + 0.132164i \(0.0421925\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −7.00000 −0.456630
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 24.0000 1.53644
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 18.0000 1.13842
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −22.0000 −1.38040
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 19.0000 1.18519 0.592594 0.805502i \(-0.298104\pi\)
0.592594 + 0.805502i \(0.298104\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) −30.0000 −1.85341
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 0 0
\(277\) 25.0000 1.50210 0.751052 0.660243i \(-0.229547\pi\)
0.751052 + 0.660243i \(0.229547\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 0.687118
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.00000 0.227185
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 28.0000 1.57264 0.786318 0.617822i \(-0.211985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 40.0000 2.21540
\(327\) 0 0
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 30.0000 1.64646
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 5.00000 0.273179
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −6.00000 −0.326357
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 10.0000 0.536828 0.268414 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 0 0
\(359\) −26.0000 −1.37223 −0.686114 0.727494i \(-0.740685\pi\)
−0.686114 + 0.727494i \(0.740685\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −32.0000 −1.68188
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 16.0000 0.834058
\(369\) 0 0
\(370\) −12.0000 −0.623850
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 19.0000 0.975964 0.487982 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.00000 −0.204658
\(383\) −1.00000 −0.0510976 −0.0255488 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) −44.0000 −2.21669
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −26.0000 −1.29355
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) −5.00000 −0.246034
\(414\) 0 0
\(415\) 15.0000 0.736321
\(416\) 32.0000 1.56893
\(417\) 0 0
\(418\) 0 0
\(419\) 23.0000 1.12362 0.561812 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) 0 0
\(421\) 25.0000 1.21843 0.609213 0.793007i \(-0.291486\pi\)
0.609213 + 0.793007i \(0.291486\pi\)
\(422\) 6.00000 0.292075
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) −24.0000 −1.16008
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −32.0000 −1.53782 −0.768911 0.639356i \(-0.779201\pi\)
−0.768911 + 0.639356i \(0.779201\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) −20.0000 −0.947027
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 54.0000 2.53435
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −27.0000 −1.26301 −0.631503 0.775373i \(-0.717562\pi\)
−0.631503 + 0.775373i \(0.717562\pi\)
\(458\) −60.0000 −2.80362
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 3.00000 0.139724 0.0698620 0.997557i \(-0.477744\pi\)
0.0698620 + 0.997557i \(0.477744\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −5.00000 −0.230879
\(470\) 14.0000 0.645772
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −37.0000 −1.69057 −0.845287 0.534313i \(-0.820570\pi\)
−0.845287 + 0.534313i \(0.820570\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −31.0000 −1.40474 −0.702372 0.711810i \(-0.747876\pi\)
−0.702372 + 0.711810i \(0.747876\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 34.0000 1.53440 0.767199 0.641409i \(-0.221650\pi\)
0.767199 + 0.641409i \(0.221650\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −1.00000 −0.0447661 −0.0223831 0.999749i \(-0.507125\pi\)
−0.0223831 + 0.999749i \(0.507125\pi\)
\(500\) −18.0000 −0.804984
\(501\) 0 0
\(502\) 40.0000 1.78529
\(503\) 13.0000 0.579641 0.289821 0.957081i \(-0.406404\pi\)
0.289821 + 0.957081i \(0.406404\pi\)
\(504\) 0 0
\(505\) −13.0000 −0.578492
\(506\) 0 0
\(507\) 0 0
\(508\) 22.0000 0.976092
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) −38.0000 −1.67611
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) 0 0
\(521\) 45.0000 1.97149 0.985743 0.168259i \(-0.0538144\pi\)
0.985743 + 0.168259i \(0.0538144\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 30.0000 1.31056
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 24.0000 1.04249
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) 0 0
\(538\) 4.00000 0.172452
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 0.0429934 0.0214967 0.999769i \(-0.493157\pi\)
0.0214967 + 0.999769i \(0.493157\pi\)
\(542\) −4.00000 −0.171815
\(543\) 0 0
\(544\) 8.00000 0.342997
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 16.0000 0.683486
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −50.0000 −2.12430
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −48.0000 −2.02476
\(563\) 31.0000 1.30649 0.653247 0.757145i \(-0.273406\pi\)
0.653247 + 0.757145i \(0.273406\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) −44.0000 −1.84946
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 31.0000 1.27951 0.639753 0.768580i \(-0.279036\pi\)
0.639753 + 0.768580i \(0.279036\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) −24.0000 −0.986394
\(593\) 17.0000 0.698106 0.349053 0.937103i \(-0.386503\pi\)
0.349053 + 0.937103i \(0.386503\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) 32.0000 1.30858
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 6.00000 0.244542
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) −28.0000 −1.13276
\(612\) 0 0
\(613\) −7.00000 −0.282727 −0.141364 0.989958i \(-0.545149\pi\)
−0.141364 + 0.989958i \(0.545149\pi\)
\(614\) 44.0000 1.77570
\(615\) 0 0
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) 42.0000 1.68405
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −40.0000 −1.59872
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −56.0000 −2.22404
\(635\) 11.0000 0.436522
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 0 0
\(647\) 11.0000 0.432455 0.216227 0.976343i \(-0.430625\pi\)
0.216227 + 0.976343i \(0.430625\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 32.0000 1.25514
\(651\) 0 0
\(652\) −40.0000 −1.56652
\(653\) 40.0000 1.56532 0.782660 0.622449i \(-0.213862\pi\)
0.782660 + 0.622449i \(0.213862\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) −14.0000 −0.545777
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) 0 0
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) 15.0000 0.576497 0.288248 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −2.00000 −0.0758643
\(696\) 0 0
\(697\) −2.00000 −0.0757554
\(698\) 12.0000 0.454207
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 60.0000 2.25813
\(707\) 13.0000 0.488915
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) −12.0000 −0.450352
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 52.0000 1.94062
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 38.0000 1.41421
\(723\) 0 0
\(724\) 32.0000 1.18927
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 64.0000 2.36228
\(735\) 0 0
\(736\) −32.0000 −1.17954
\(737\) 0 0
\(738\) 0 0
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) −38.0000 −1.39128
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 28.0000 1.02105
\(753\) 0 0
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 35.0000 1.27210 0.636048 0.771649i \(-0.280568\pi\)
0.636048 + 0.771649i \(0.280568\pi\)
\(758\) −38.0000 −1.38022
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) −7.00000 −0.253417
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) 44.0000 1.56744
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) 0 0
\(797\) 23.0000 0.814702 0.407351 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) 0 0
\(799\) −7.00000 −0.247642
\(800\) −32.0000 −1.13137
\(801\) 0 0
\(802\) −28.0000 −0.988714
\(803\) 0 0
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 0 0
\(818\) −48.0000 −1.67828
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) −30.0000 −1.04132
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) 0 0
\(837\) 0 0
\(838\) −46.0000 −1.58904
\(839\) 43.0000 1.48452 0.742262 0.670109i \(-0.233753\pi\)
0.742262 + 0.670109i \(0.233753\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −50.0000 −1.72311
\(843\) 0 0
\(844\) −6.00000 −0.206529
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) 48.0000 1.64833
\(849\) 0 0
\(850\) 8.00000 0.274398
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 56.0000 1.91740 0.958702 0.284413i \(-0.0917988\pi\)
0.958702 + 0.284413i \(0.0917988\pi\)
\(854\) 24.0000 0.821263
\(855\) 0 0
\(856\) 0 0
\(857\) −49.0000 −1.67381 −0.836904 0.547350i \(-0.815637\pi\)
−0.836904 + 0.547350i \(0.815637\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) 60.0000 2.04361
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −9.00000 −0.306009
\(866\) 64.0000 2.17481
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 64.0000 2.15990
\(879\) 0 0
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −56.0000 −1.88136
\(887\) −37.0000 −1.24234 −0.621169 0.783676i \(-0.713342\pi\)
−0.621169 + 0.783676i \(0.713342\pi\)
\(888\) 0 0
\(889\) −11.0000 −0.368928
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) 20.0000 0.669650
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 68.0000 2.26919
\(899\) 0 0
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −54.0000 −1.79205
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 54.0000 1.78616
\(915\) 0 0
\(916\) 60.0000 1.98246
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −6.00000 −0.197599
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −24.0000 −0.789115
\(926\) 64.0000 2.10317
\(927\) 0 0
\(928\) 0 0
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 10.0000 0.326512
\(939\) 0 0
\(940\) −14.0000 −0.456630
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) −20.0000 −0.650945
\(945\) 0 0
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) 2.00000 0.0647185
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 74.0000 2.39083
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −48.0000 −1.54758
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) −5.00000 −0.160956
\(966\) 0 0
\(967\) 31.0000 0.996893 0.498446 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) −55.0000 −1.76503 −0.882517 0.470281i \(-0.844153\pi\)
−0.882517 + 0.470281i \(0.844153\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 62.0000 1.98661
\(975\) 0 0
\(976\) −48.0000 −1.53644
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −68.0000 −2.16997
\(983\) 48.0000 1.53096 0.765481 0.643458i \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −16.0000 −0.508001
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) 2.00000 0.0633089
\(999\) 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 7623.2.a.c.1.1 yes 1
3.2 odd 2 7623.2.a.p.1.1 yes 1
11.10 odd 2 7623.2.a.r.1.1 yes 1
33.32 even 2 7623.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.b.1.1 1 33.32 even 2
7623.2.a.c.1.1 yes 1 1.1 even 1 trivial
7623.2.a.p.1.1 yes 1 3.2 odd 2
7623.2.a.r.1.1 yes 1 11.10 odd 2