Properties

Label 5445.2.a.d.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -4.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} -4.00000 q^{19} +1.00000 q^{20} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -3.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -3.00000 q^{35} -8.00000 q^{37} +4.00000 q^{38} -3.00000 q^{40} -5.00000 q^{41} -5.00000 q^{43} -8.00000 q^{46} +3.00000 q^{47} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -4.00000 q^{53} +9.00000 q^{56} -6.00000 q^{58} +2.00000 q^{59} +11.0000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} -13.0000 q^{67} +3.00000 q^{70} -2.00000 q^{71} +8.00000 q^{73} +8.00000 q^{74} +4.00000 q^{76} -10.0000 q^{79} +1.00000 q^{80} +5.00000 q^{82} +4.00000 q^{83} +5.00000 q^{86} -1.00000 q^{89} -12.0000 q^{91} -8.00000 q^{92} -3.00000 q^{94} +4.00000 q^{95} -8.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.00000 1.20268
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 5.00000 0.552158
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.00000 0.539164
\(87\) 0 0
\(88\) 0 0
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −12.0000 −1.17670
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) −11.0000 −0.995893
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 13.0000 1.12303
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −8.00000 −0.662085
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −17.0000 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) 24.0000 1.89146
\(162\) 0 0
\(163\) −17.0000 −1.33154 −0.665771 0.746156i \(-0.731897\pi\)
−0.665771 + 0.746156i \(0.731897\pi\)
\(164\) 5.00000 0.390434
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −7.00000 −0.541676 −0.270838 0.962625i \(-0.587301\pi\)
−0.270838 + 0.962625i \(0.587301\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 5.00000 0.381246
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) 0 0
\(178\) 1.00000 0.0749532
\(179\) 26.0000 1.94333 0.971666 0.236360i \(-0.0759544\pi\)
0.971666 + 0.236360i \(0.0759544\pi\)
\(180\) 0 0
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 12.0000 0.889499
\(183\) 0 0
\(184\) 24.0000 1.76930
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) 5.00000 0.351799
\(203\) 18.0000 1.26335
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 5.00000 0.340997
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) −9.00000 −0.609557
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5.00000 0.334825 0.167412 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −1.00000 −0.0663723 −0.0331862 0.999449i \(-0.510565\pi\)
−0.0331862 + 0.999449i \(0.510565\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −11.0000 −0.704203
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 11.0000 0.690201
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 12.0000 0.735767
\(267\) 0 0
\(268\) 13.0000 0.794101
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) −18.0000 −1.07957
\(279\) 0 0
\(280\) −9.00000 −0.537853
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −15.0000 −0.885422
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) −34.0000 −1.98630 −0.993151 0.116841i \(-0.962723\pi\)
−0.993151 + 0.116841i \(0.962723\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) −24.0000 −1.39497
\(297\) 0 0
\(298\) 17.0000 0.984784
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) −15.0000 −0.864586
\(302\) −14.0000 −0.805609
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −11.0000 −0.629858
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 8.00000 0.451466
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 17.0000 0.941543
\(327\) 0 0
\(328\) −15.0000 −0.828236
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 7.00000 0.383023
\(335\) 13.0000 0.710266
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) −15.0000 −0.808746
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) 7.00000 0.375780 0.187890 0.982190i \(-0.439835\pi\)
0.187890 + 0.982190i \(0.439835\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 1.00000 0.0529999
\(357\) 0 0
\(358\) −26.0000 −1.37414
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 19.0000 0.998618
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −22.0000 −1.13006 −0.565032 0.825069i \(-0.691136\pi\)
−0.565032 + 0.825069i \(0.691136\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) −12.0000 −0.602263 −0.301131 0.953583i \(-0.597364\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 6.00000 0.300753
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 37.0000 1.84769 0.923846 0.382765i \(-0.125028\pi\)
0.923846 + 0.382765i \(0.125028\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 5.00000 0.248759
\(405\) 0 0
\(406\) −18.0000 −0.893325
\(407\) 0 0
\(408\) 0 0
\(409\) −21.0000 −1.03838 −0.519192 0.854658i \(-0.673767\pi\)
−0.519192 + 0.854658i \(0.673767\pi\)
\(410\) −5.00000 −0.246932
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 20.0000 0.980581
\(417\) 0 0
\(418\) 0 0
\(419\) −32.0000 −1.56330 −0.781651 0.623716i \(-0.785622\pi\)
−0.781651 + 0.623716i \(0.785622\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) −22.0000 −1.07094
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) 33.0000 1.59698
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) −5.00000 −0.241121
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) −9.00000 −0.431022
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 25.0000 1.18779 0.593893 0.804544i \(-0.297590\pi\)
0.593893 + 0.804544i \(0.297590\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −5.00000 −0.236757
\(447\) 0 0
\(448\) 21.0000 0.992157
\(449\) 13.0000 0.613508 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 1.00000 0.0469323
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 1.00000 0.0467269
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 7.00000 0.326023 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) −39.0000 −1.80085
\(470\) 3.00000 0.138380
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) −4.00000 −0.182956
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 23.0000 1.04762
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 33.0000 1.49384
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −42.0000 −1.88018 −0.940089 0.340929i \(-0.889258\pi\)
−0.940089 + 0.340929i \(0.889258\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 13.0000 0.579641 0.289821 0.957081i \(-0.406404\pi\)
0.289821 + 0.957081i \(0.406404\pi\)
\(504\) 0 0
\(505\) 5.00000 0.222497
\(506\) 0 0
\(507\) 0 0
\(508\) 11.0000 0.488046
\(509\) −23.0000 −1.01946 −0.509729 0.860335i \(-0.670254\pi\)
−0.509729 + 0.860335i \(0.670254\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 24.0000 1.05450
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −4.00000 −0.173749
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) −39.0000 −1.68454
\(537\) 0 0
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) −31.0000 −1.33279 −0.666397 0.745597i \(-0.732164\pi\)
−0.666397 + 0.745597i \(0.732164\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.00000 −0.385518
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −18.0000 −0.763370
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) −13.0000 −0.546431
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 15.0000 0.626088
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 17.0000 0.707107
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 24.0000 0.993127
\(585\) 0 0
\(586\) 34.0000 1.40453
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 2.00000 0.0823387
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0000 0.696347
\(597\) 0 0
\(598\) 32.0000 1.30858
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 15.0000 0.611354
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) 11.0000 0.445377
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) −3.00000 −0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) 0 0
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) −30.0000 −1.19334
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 11.0000 0.436522
\(636\) 0 0
\(637\) −8.00000 −0.316972
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 35.0000 1.38027 0.690133 0.723683i \(-0.257552\pi\)
0.690133 + 0.723683i \(0.257552\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 0 0
\(647\) 25.0000 0.982851 0.491426 0.870919i \(-0.336476\pi\)
0.491426 + 0.870919i \(0.336476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 17.0000 0.665771
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) −9.00000 −0.350857
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −37.0000 −1.43913 −0.719567 0.694423i \(-0.755660\pi\)
−0.719567 + 0.694423i \(0.755660\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 7.00000 0.270838
\(669\) 0 0
\(670\) −13.0000 −0.502234
\(671\) 0 0
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −12.0000 −0.462223
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) −28.0000 −1.07613 −0.538064 0.842904i \(-0.680844\pi\)
−0.538064 + 0.842904i \(0.680844\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −47.0000 −1.79841 −0.899203 0.437533i \(-0.855852\pi\)
−0.899203 + 0.437533i \(0.855852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) 5.00000 0.190623
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) −24.0000 −0.912343
\(693\) 0 0
\(694\) −7.00000 −0.265716
\(695\) −18.0000 −0.682779
\(696\) 0 0
\(697\) 0 0
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 0 0
\(712\) −3.00000 −0.112430
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −26.0000 −0.971666
\(717\) 0 0
\(718\) 28.0000 1.04495
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 3.00000 0.111648
\(723\) 0 0
\(724\) 19.0000 0.706129
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) −36.0000 −1.33425
\(729\) 0 0
\(730\) 8.00000 0.296093
\(731\) 0 0
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) −1.00000 −0.0369107
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 0 0
\(738\) 0 0
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −7.00000 −0.256805 −0.128403 0.991722i \(-0.540985\pi\)
−0.128403 + 0.991722i \(0.540985\pi\)
\(744\) 0 0
\(745\) 17.0000 0.622832
\(746\) 18.0000 0.659027
\(747\) 0 0
\(748\) 0 0
\(749\) 27.0000 0.986559
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) −3.00000 −0.109399
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) −14.0000 −0.509512
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 22.0000 0.799076
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 0 0
\(763\) 27.0000 0.977466
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 22.0000 0.791285 0.395643 0.918405i \(-0.370522\pi\)
0.395643 + 0.918405i \(0.370522\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) −24.0000 −0.861550
\(777\) 0 0
\(778\) −3.00000 −0.107555
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −2.00000 −0.0714286
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) 43.0000 1.53278 0.766392 0.642373i \(-0.222050\pi\)
0.766392 + 0.642373i \(0.222050\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −44.0000 −1.56249
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) 50.0000 1.77109 0.885545 0.464553i \(-0.153785\pi\)
0.885545 + 0.464553i \(0.153785\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −37.0000 −1.30652
\(803\) 0 0
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) −15.0000 −0.527698
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) −18.0000 −0.631676
\(813\) 0 0
\(814\) 0 0
\(815\) 17.0000 0.595484
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 21.0000 0.734248
\(819\) 0 0
\(820\) −5.00000 −0.174608
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 0 0
\(823\) 51.0000 1.77775 0.888874 0.458151i \(-0.151488\pi\)
0.888874 + 0.458151i \(0.151488\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) 0 0
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −28.0000 −0.970725
\(833\) 0 0
\(834\) 0 0
\(835\) 7.00000 0.242245
\(836\) 0 0
\(837\) 0 0
\(838\) 32.0000 1.10542
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −3.00000 −0.103387
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) 0 0
\(851\) −64.0000 −2.19389
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) −33.0000 −1.12924
\(855\) 0 0
\(856\) 27.0000 0.922841
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −5.00000 −0.170499
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −31.0000 −1.05525 −0.527626 0.849477i \(-0.676918\pi\)
−0.527626 + 0.849477i \(0.676918\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 0 0
\(870\) 0 0
\(871\) 52.0000 1.76195
\(872\) 27.0000 0.914335
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) 0 0
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −25.0000 −0.839891
\(887\) 1.00000 0.0335767 0.0167884 0.999859i \(-0.494656\pi\)
0.0167884 + 0.999859i \(0.494656\pi\)
\(888\) 0 0
\(889\) −33.0000 −1.10678
\(890\) −1.00000 −0.0335201
\(891\) 0 0
\(892\) −5.00000 −0.167412
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −26.0000 −0.869084
\(896\) 9.00000 0.300669
\(897\) 0 0
\(898\) −13.0000 −0.433816
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) 19.0000 0.631581
\(906\) 0 0
\(907\) −43.0000 −1.42779 −0.713896 0.700252i \(-0.753071\pi\)
−0.713896 + 0.700252i \(0.753071\pi\)
\(908\) 1.00000 0.0331862
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 34.0000 1.12647 0.563235 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) 1.00000 0.0330409
\(917\) 0 0
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) −7.00000 −0.230533
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 15.0000 0.492931
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 24.0000 0.786146
\(933\) 0 0
\(934\) 27.0000 0.883467
\(935\) 0 0
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 39.0000 1.27340
\(939\) 0 0
\(940\) 3.00000 0.0978492
\(941\) −13.0000 −0.423788 −0.211894 0.977293i \(-0.567963\pi\)
−0.211894 + 0.977293i \(0.567963\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −32.0000 −1.03876
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) −4.00000 −0.129369
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −32.0000 −1.03172
\(963\) 0 0
\(964\) 23.0000 0.740780
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 54.0000 1.73116
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −11.0000 −0.352101
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −22.0000 −0.702048
\(983\) 61.0000 1.94560 0.972799 0.231651i \(-0.0744128\pi\)
0.972799 + 0.231651i \(0.0744128\pi\)
\(984\) 0 0
\(985\) 14.0000 0.446077
\(986\) 0 0
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) 6.00000 0.190308
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −52.0000 −1.64686 −0.823428 0.567420i \(-0.807941\pi\)
−0.823428 + 0.567420i \(0.807941\pi\)
\(998\) 42.0000 1.32949
\(999\) 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 5445.2.a.d.1.1 1
3.2 odd 2 605.2.a.c.1.1 yes 1
11.10 odd 2 5445.2.a.h.1.1 1
12.11 even 2 9680.2.a.be.1.1 1
15.14 odd 2 3025.2.a.c.1.1 1
33.2 even 10 605.2.g.d.81.1 4
33.5 odd 10 605.2.g.b.366.1 4
33.8 even 10 605.2.g.d.251.1 4
33.14 odd 10 605.2.g.b.251.1 4
33.17 even 10 605.2.g.d.366.1 4
33.20 odd 10 605.2.g.b.81.1 4
33.26 odd 10 605.2.g.b.511.1 4
33.29 even 10 605.2.g.d.511.1 4
33.32 even 2 605.2.a.a.1.1 1
132.131 odd 2 9680.2.a.bf.1.1 1
165.164 even 2 3025.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.a.1.1 1 33.32 even 2
605.2.a.c.1.1 yes 1 3.2 odd 2
605.2.g.b.81.1 4 33.20 odd 10
605.2.g.b.251.1 4 33.14 odd 10
605.2.g.b.366.1 4 33.5 odd 10
605.2.g.b.511.1 4 33.26 odd 10
605.2.g.d.81.1 4 33.2 even 10
605.2.g.d.251.1 4 33.8 even 10
605.2.g.d.366.1 4 33.17 even 10
605.2.g.d.511.1 4 33.29 even 10
3025.2.a.c.1.1 1 15.14 odd 2
3025.2.a.g.1.1 1 165.164 even 2
5445.2.a.d.1.1 1 1.1 even 1 trivial
5445.2.a.h.1.1 1 11.10 odd 2
9680.2.a.be.1.1 1 12.11 even 2
9680.2.a.bf.1.1 1 132.131 odd 2