Properties

Label 1950.2.a.t.1.1
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
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
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} +5.00000 q^{19} -4.00000 q^{21} -1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +3.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{36} +7.00000 q^{37} +5.00000 q^{38} +1.00000 q^{39} +3.00000 q^{41} -4.00000 q^{42} -2.00000 q^{43} -9.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -5.00000 q^{57} +3.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -5.00000 q^{67} -3.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +7.00000 q^{74} +5.00000 q^{76} +1.00000 q^{78} +11.0000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -2.00000 q^{86} -3.00000 q^{87} +6.00000 q^{89} -4.00000 q^{91} +4.00000 q^{93} -9.00000 q^{94} -1.00000 q^{96} -8.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 5.00000 0.811107
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −4.00000 −0.617213
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −5.00000 −0.662266
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −7.00000 −0.664411
\(112\) 4.00000 0.377964
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −5.00000 −0.468293
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −1.00000 −0.0924500
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 8.00000 0.724286
\(123\) −3.00000 −0.270501
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 20.0000 1.73422
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) −3.00000 −0.251754
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −9.00000 −0.742307
\(148\) 7.00000 0.575396
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 5.00000 0.405554
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 11.0000 0.875113
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) −2.00000 −0.152499
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) −4.00000 −0.296500
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −9.00000 −0.656392
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 18.0000 1.26648
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −9.00000 −0.618123
\(213\) 3.00000 0.205557
\(214\) −9.00000 −0.615227
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −16.0000 −1.08615
\(218\) 11.0000 0.745014
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) −7.00000 −0.469809
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) −5.00000 −0.331133
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) −5.00000 −0.318142
\(248\) −4.00000 −0.254000
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 9.00000 0.568075 0.284037 0.958813i \(-0.408326\pi\)
0.284037 + 0.958813i \(0.408326\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 7.00000 0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 2.00000 0.124515
\(259\) 28.0000 1.73984
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 3.00000 0.185341
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 20.0000 1.22628
\(267\) −6.00000 −0.367194
\(268\) −5.00000 −0.305424
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 8.00000 0.479808
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 9.00000 0.535942
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 4.00000 0.234082
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) −12.0000 −0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −16.0000 −0.920697
\(303\) −18.0000 −1.03407
\(304\) 5.00000 0.286770
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 1.00000 0.0566139
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 11.0000 0.618798
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 9.00000 0.504695
\(319\) 0 0
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −11.0000 −0.608301
\(328\) 3.00000 0.165647
\(329\) −36.0000 −1.98474
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 6.00000 0.329293
\(333\) 7.00000 0.383598
\(334\) 21.0000 1.14907
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 1.00000 0.0543928
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 5.00000 0.270369
\(343\) 8.00000 0.431959
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) 9.00000 0.483145 0.241573 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) −3.00000 −0.160817
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −3.00000 −0.158334 −0.0791670 0.996861i \(-0.525226\pi\)
−0.0791670 + 0.996861i \(0.525226\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −16.0000 −0.840941
\(363\) 11.0000 0.577350
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 4.00000 0.207390
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −3.00000 −0.154508
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 18.0000 0.920960
\(383\) −15.0000 −0.766464 −0.383232 0.923652i \(-0.625189\pi\)
−0.383232 + 0.923652i \(0.625189\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) −2.00000 −0.101666
\(388\) −8.00000 −0.406138
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.00000 0.454569
\(393\) −3.00000 −0.151330
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) 31.0000 1.55585 0.777923 0.628360i \(-0.216273\pi\)
0.777923 + 0.628360i \(0.216273\pi\)
\(398\) −7.00000 −0.350878
\(399\) −20.0000 −1.00125
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 5.00000 0.249377
\(403\) 4.00000 0.199254
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 16.0000 0.788263
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −4.00000 −0.194717
\(423\) −9.00000 −0.437595
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 3.00000 0.145350
\(427\) 32.0000 1.54859
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 29.0000 1.38409 0.692047 0.721852i \(-0.256709\pi\)
0.692047 + 0.721852i \(0.256709\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 3.00000 0.142534 0.0712672 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) −7.00000 −0.332205
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 12.0000 0.567581
\(448\) 4.00000 0.188982
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 16.0000 0.751746
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −7.00000 −0.327089
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) −11.0000 −0.505247
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) −24.0000 −1.09773
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 8.00000 0.362143
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −3.00000 −0.135250
\(493\) 0 0
\(494\) −5.00000 −0.224961
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −12.0000 −0.538274
\(498\) −6.00000 −0.268866
\(499\) 11.0000 0.492428 0.246214 0.969216i \(-0.420813\pi\)
0.246214 + 0.969216i \(0.420813\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) 9.00000 0.401690
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 7.00000 0.310575
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 1.00000 0.0441942
\(513\) −5.00000 −0.220755
\(514\) 12.0000 0.529297
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 28.0000 1.23025
\(519\) 21.0000 0.921798
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 3.00000 0.131306
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 20.0000 0.867110
\(533\) −3.00000 −0.129944
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) −12.0000 −0.517838
\(538\) −21.0000 −0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −28.0000 −1.20270
\(543\) 16.0000 0.686626
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −3.00000 −0.128154
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 15.0000 0.639021
\(552\) 0 0
\(553\) 44.0000 1.87107
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −4.00000 −0.169334
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) −15.0000 −0.632737
\(563\) 21.0000 0.885044 0.442522 0.896758i \(-0.354084\pi\)
0.442522 + 0.896758i \(0.354084\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) −32.0000 −1.34506
\(567\) 4.00000 0.167984
\(568\) −3.00000 −0.125877
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) −17.0000 −0.707107
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −9.00000 −0.371154
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 7.00000 0.287698
\(593\) −3.00000 −0.123195 −0.0615976 0.998101i \(-0.519620\pi\)
−0.0615976 + 0.998101i \(0.519620\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 7.00000 0.286491
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) −8.00000 −0.326056
\(603\) −5.00000 −0.203616
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) −23.0000 −0.933541 −0.466771 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) 5.00000 0.202777
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 7.00000 0.282497
\(615\) 0 0
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) −16.0000 −0.643614
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) 24.0000 0.961540
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 0 0
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 11.0000 0.437557
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) −9.00000 −0.356593
\(638\) 0 0
\(639\) −3.00000 −0.118678
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 9.00000 0.355202
\(643\) −23.0000 −0.907031 −0.453516 0.891248i \(-0.649830\pi\)
−0.453516 + 0.891248i \(0.649830\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 16.0000 0.626608
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −11.0000 −0.430134
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 4.00000 0.156055
\(658\) −36.0000 −1.40343
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) −25.0000 −0.972387 −0.486194 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) 0 0
\(668\) 21.0000 0.812514
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 18.0000 0.691286
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 5.00000 0.191180
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 7.00000 0.267067
\(688\) −2.00000 −0.0762493
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) 9.00000 0.341635
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 2.00000 0.0757011
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 1.00000 0.0377426
\(703\) 35.0000 1.32005
\(704\) 0 0
\(705\) 0 0
\(706\) 3.00000 0.112906
\(707\) 72.0000 2.70784
\(708\) −6.00000 −0.225494
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 11.0000 0.412532
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) −3.00000 −0.111959
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) 6.00000 0.223297
\(723\) −8.00000 −0.297523
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 52.0000 1.92857 0.964287 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −8.00000 −0.295689
\(733\) −5.00000 −0.184679 −0.0923396 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 3.00000 0.110432
\(739\) −7.00000 −0.257499 −0.128750 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) 5.00000 0.183680
\(742\) −36.0000 −1.32160
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) 5.00000 0.182453 0.0912263 0.995830i \(-0.470921\pi\)
0.0912263 + 0.995830i \(0.470921\pi\)
\(752\) −9.00000 −0.328196
\(753\) −9.00000 −0.327978
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) −7.00000 −0.253583
\(763\) 44.0000 1.59291
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) −6.00000 −0.216647
\(768\) −1.00000 −0.0360844
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) −20.0000 −0.719816
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) −28.0000 −1.00449
\(778\) 27.0000 0.967997
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −3.00000 −0.107006
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) −24.0000 −0.854965
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −72.0000 −2.56003
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) 31.0000 1.10015
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −20.0000 −0.707992
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 21.0000 0.739235
\(808\) 18.0000 0.633238
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 12.0000 0.421117
\(813\) 28.0000 0.982003
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −10.0000 −0.349856
\(818\) −4.00000 −0.139857
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 3.00000 0.104637
\(823\) −23.0000 −0.801730 −0.400865 0.916137i \(-0.631290\pi\)
−0.400865 + 0.916137i \(0.631290\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −27.0000 −0.932700
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −22.0000 −0.758170
\(843\) 15.0000 0.516627
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) −44.0000 −1.51186
\(848\) −9.00000 −0.309061
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 0 0
\(852\) 3.00000 0.102778
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 32.0000 1.09502
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −33.0000 −1.12398
\(863\) 27.0000 0.919091 0.459545 0.888154i \(-0.348012\pi\)
0.459545 + 0.888154i \(0.348012\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −11.0000 −0.373795
\(867\) 17.0000 0.577350
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 5.00000 0.169419
\(872\) 11.0000 0.372507
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −35.0000 −1.18187 −0.590933 0.806721i \(-0.701240\pi\)
−0.590933 + 0.806721i \(0.701240\pi\)
\(878\) 29.0000 0.978703
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 9.00000 0.303046
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) −7.00000 −0.234905
\(889\) 28.0000 0.939090
\(890\) 0 0
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −45.0000 −1.50587
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 21.0000 0.700779
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −6.00000 −0.199117
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) −5.00000 −0.165567
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −7.00000 −0.231287
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 24.0000 0.790398
\(923\) 3.00000 0.0987462
\(924\) 0 0
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 16.0000 0.525509
\(928\) 3.00000 0.0984798
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 45.0000 1.47482
\(932\) −24.0000 −0.786146
\(933\) 18.0000 0.589294
\(934\) 33.0000 1.07979
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −20.0000 −0.653023
\(939\) −1.00000 −0.0326338
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) −10.0000 −0.325818
\(943\) 0 0
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −11.0000 −0.357263
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −21.0000 −0.678479
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −7.00000 −0.225689
\(963\) −9.00000 −0.290021
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −11.0000 −0.353553
\(969\) 0 0
\(970\) 0 0
\(971\) −39.0000 −1.25157 −0.625785 0.779996i \(-0.715221\pi\)
−0.625785 + 0.779996i \(0.715221\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 32.0000 1.02587
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −16.0000 −0.511624
\(979\) 0 0
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) 0 0
\(987\) 36.0000 1.14589
\(988\) −5.00000 −0.159071
\(989\) 0 0
\(990\) 0 0
\(991\) 41.0000 1.30241 0.651204 0.758903i \(-0.274264\pi\)
0.651204 + 0.758903i \(0.274264\pi\)
\(992\) −4.00000 −0.127000
\(993\) −20.0000 −0.634681
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 11.0000 0.348199
\(999\) −7.00000 −0.221470
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 1950.2.a.t.1.1 yes 1
3.2 odd 2 5850.2.a.y.1.1 1
5.2 odd 4 1950.2.e.c.1249.2 2
5.3 odd 4 1950.2.e.c.1249.1 2
5.4 even 2 1950.2.a.f.1.1 1
15.2 even 4 5850.2.e.s.5149.1 2
15.8 even 4 5850.2.e.s.5149.2 2
15.14 odd 2 5850.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.f.1.1 1 5.4 even 2
1950.2.a.t.1.1 yes 1 1.1 even 1 trivial
1950.2.e.c.1249.1 2 5.3 odd 4
1950.2.e.c.1249.2 2 5.2 odd 4
5850.2.a.y.1.1 1 3.2 odd 2
5850.2.a.be.1.1 1 15.14 odd 2
5850.2.e.s.5149.1 2 15.2 even 4
5850.2.e.s.5149.2 2 15.8 even 4