Properties

Label 4650.2.d.p.3349.1
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.p.3349.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} +4.00000i q^{22} -1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000 q^{39} -6.00000 q^{41} +2.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +6.00000 q^{51} -4.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -2.00000 q^{56} +4.00000i q^{58} -6.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -10.0000i q^{67} +6.00000i q^{68} -14.0000 q^{71} -1.00000i q^{72} -4.00000i q^{73} +4.00000 q^{74} -8.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} +2.00000 q^{84} +8.00000 q^{86} -4.00000i q^{87} -4.00000i q^{88} +8.00000 q^{89} -8.00000 q^{91} +1.00000i q^{93} -12.0000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{21} - 2 q^{24} + 8 q^{26} - 8 q^{29} + 2 q^{31} - 12 q^{34} + 2 q^{36} - 8 q^{39} - 12 q^{41} + 8 q^{44} + 6 q^{49} + 12 q^{51} - 2 q^{54} - 4 q^{56} - 12 q^{59} + 20 q^{61} - 2 q^{64} - 8 q^{66} - 28 q^{71} + 8 q^{74} + 16 q^{79} + 2 q^{81} + 4 q^{84} + 16 q^{86} + 16 q^{89} - 16 q^{91} - 24 q^{94} + 2 q^{96} + 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.0000i − 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 4.00000i − 0.554700i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 4.00000i 0.525226i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.00000i − 0.911685i
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) − 4.00000i − 0.428845i
\(88\) − 4.00000i − 0.426401i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 1.00000i 0.103695i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 2.00000i 0.188982i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) − 4.00000i − 0.369800i
\(118\) 6.00000i 0.552345i
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 10.0000i − 0.905357i
\(123\) − 6.00000i − 0.541002i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 14.0000i 1.17485i
\(143\) − 16.0000i − 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 3.00000i 0.247436i
\(148\) − 4.00000i − 0.328798i
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) − 8.00000i − 0.609994i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 6.00000i − 0.450988i
\(178\) − 8.00000i − 0.599625i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 8.00000i 0.592999i
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 24.0000i 1.75505i
\(188\) 12.0000i 0.875190i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) − 14.0000i − 0.985037i
\(203\) − 8.00000i − 0.561490i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) − 14.0000i − 0.959264i
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 2.00000i 0.135769i
\(218\) 18.0000i 1.21911i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 4.00000i 0.268462i
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) − 4.00000i − 0.262613i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 8.00000i 0.519656i
\(238\) − 12.0000i − 0.777844i
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 1.00000i 0.0635001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 8.00000i 0.498058i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) − 14.0000i − 0.864923i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) 8.00000i 0.489592i
\(268\) 10.0000i 0.610847i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) − 8.00000i − 0.484182i
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) − 14.0000i − 0.832214i −0.909316 0.416107i \(-0.863394\pi\)
0.909316 0.416107i \(-0.136606\pi\)
\(284\) 14.0000 0.830747
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) − 12.0000i − 0.708338i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 4.00000i 0.234082i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 4.00000i 0.232104i
\(298\) 14.0000i 0.810998i
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 16.0000i 0.920697i
\(303\) 14.0000i 0.804279i
\(304\) 0 0
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 8.00000i 0.455842i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 26.0000i 1.46031i 0.683284 + 0.730153i \(0.260551\pi\)
−0.683284 + 0.730153i \(0.739449\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) − 18.0000i − 0.995402i
\(328\) − 6.00000i − 0.331295i
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 4.00000i − 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 4.00000i 0.214423i
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 4.00000i 0.213201i
\(353\) − 10.0000i − 0.532246i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857428\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 12.0000i 0.635107i
\(358\) − 4.00000i − 0.211407i
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 18.0000i − 0.946059i
\(363\) 5.00000i 0.262432i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) − 1.00000i − 0.0518476i
\(373\) − 18.0000i − 0.932005i −0.884783 0.466002i \(-0.845694\pi\)
0.884783 0.466002i \(-0.154306\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) − 16.0000i − 0.824042i
\(378\) − 2.00000i − 0.102869i
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 10.0000i 0.511645i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 8.00000i − 0.406663i
\(388\) 2.00000i 0.101535i
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 3.00000i 0.151523i
\(393\) 14.0000i 0.706207i
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.00000 −0.199750 −0.0998752 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) − 10.0000i − 0.498755i
\(403\) 4.00000i 0.199254i
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) − 16.0000i − 0.793091i
\(408\) 6.00000i 0.297044i
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 14.0000i 0.689730i
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) 12.0000i 0.583460i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −14.0000 −0.678302
\(427\) 20.0000i 0.967868i
\(428\) 16.0000i 0.773389i
\(429\) 16.0000 0.772487
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) − 4.00000i − 0.191127i
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 24.0000i − 1.14156i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) − 14.0000i − 0.662177i
\(448\) − 2.00000i − 0.0944911i
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 6.00000i − 0.282216i
\(453\) − 16.0000i − 0.751746i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.00000i − 0.187112i −0.995614 0.0935561i \(-0.970177\pi\)
0.995614 0.0935561i \(-0.0298234\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) − 8.00000i − 0.372194i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) − 24.0000i − 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) − 6.00000i − 0.276172i
\(473\) − 32.0000i − 1.47136i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) − 2.00000i − 0.0915737i
\(478\) 20.0000i 0.914779i
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) − 10.0000i − 0.455488i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 24.0000i 1.08091i
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 28.0000i − 1.25597i
\(498\) 4.00000i 0.179244i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 28.0000i 1.24970i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) 0 0
\(507\) − 3.00000i − 0.133235i
\(508\) 12.0000i 0.532414i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 48.0000i 2.11104i
\(518\) 8.00000i 0.351500i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) − 4.00000i − 0.175075i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) − 6.00000i − 0.261364i
\(528\) − 4.00000i − 0.174078i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) − 24.0000i − 1.03956i
\(534\) 8.00000 0.346194
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) 4.00000i 0.172613i
\(538\) − 12.0000i − 0.517357i
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 24.0000i 1.03089i
\(543\) 18.0000i 0.772454i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) 46.0000i 1.96682i 0.181402 + 0.983409i \(0.441936\pi\)
−0.181402 + 0.983409i \(0.558064\pi\)
\(548\) 18.0000i 0.768922i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 1.00000i 0.0423334i
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) − 6.00000i − 0.253095i
\(563\) − 8.00000i − 0.337160i −0.985688 0.168580i \(-0.946082\pi\)
0.985688 0.168580i \(-0.0539181\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 2.00000i 0.0839921i
\(568\) − 14.0000i − 0.587427i
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 16.0000i 0.668994i
\(573\) − 10.0000i − 0.417756i
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 42.0000i 1.74848i 0.485491 + 0.874241i \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) − 2.00000i − 0.0829027i
\(583\) − 8.00000i − 0.331326i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 4.00000i 0.164399i
\(593\) 46.0000i 1.88899i 0.328521 + 0.944497i \(0.393450\pi\)
−0.328521 + 0.944497i \(0.606550\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 −0.245153 −0.122577 0.992459i \(-0.539116\pi\)
−0.122577 + 0.992459i \(0.539116\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 16.0000i 0.652111i
\(603\) 10.0000i 0.407231i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 14.0000 0.568711
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) − 6.00000i − 0.242536i
\(613\) 24.0000i 0.969351i 0.874694 + 0.484675i \(0.161062\pi\)
−0.874694 + 0.484675i \(0.838938\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 26.0000i 1.04672i 0.852111 + 0.523360i \(0.175322\pi\)
−0.852111 + 0.523360i \(0.824678\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 6.00000i − 0.240578i
\(623\) 16.0000i 0.641026i
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) 6.00000i 0.239426i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 16.0000i 0.635943i
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 12.0000i 0.475457i
\(638\) − 16.0000i − 0.633446i
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) − 16.0000i − 0.631470i
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 6.00000i 0.234978i
\(653\) 22.0000i 0.860927i 0.902608 + 0.430463i \(0.141650\pi\)
−0.902608 + 0.430463i \(0.858350\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 4.00000i 0.156055i
\(658\) − 24.0000i − 0.935617i
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 24.0000i 0.932083i
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 2.00000i 0.0771517i
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 34.0000i − 1.30673i −0.757045 0.653363i \(-0.773358\pi\)
0.757045 0.653363i \(-0.226642\pi\)
\(678\) 6.00000i 0.230429i
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 4.00000i 0.153168i
\(683\) − 4.00000i − 0.153056i −0.997067 0.0765279i \(-0.975617\pi\)
0.997067 0.0765279i \(-0.0243834\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 26.0000i 0.991962i
\(688\) 8.00000i 0.304997i
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 8.00000i 0.303895i
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 36.0000i 1.36360i
\(698\) 2.00000i 0.0757011i
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) 28.0000i 1.05305i
\(708\) 6.00000i 0.225494i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 8.00000i 0.299813i
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) − 20.0000i − 0.746914i
\(718\) 30.0000i 1.11959i
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 19.0000i 0.707107i
\(723\) 10.0000i 0.371904i
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 30.0000i − 1.11264i −0.830969 0.556319i \(-0.812213\pi\)
0.830969 0.556319i \(-0.187787\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) − 10.0000i − 0.369611i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.0000i 1.47342i
\(738\) − 6.00000i − 0.220863i
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000i 0.146845i
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) − 4.00000i − 0.146352i
\(748\) − 24.0000i − 0.877527i
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) − 28.0000i − 1.02038i
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) − 12.0000i − 0.436147i −0.975932 0.218074i \(-0.930023\pi\)
0.975932 0.218074i \(-0.0699773\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) − 36.0000i − 1.30329i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) − 24.0000i − 0.866590i
\(768\) 1.00000i 0.0360844i
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) − 2.00000i − 0.0719816i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) − 8.00000i − 0.286998i
\(778\) 36.0000i 1.29066i
\(779\) 0 0
\(780\) 0 0
\(781\) 56.0000 2.00384
\(782\) 0 0
\(783\) 4.00000i 0.142948i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 14.0000 0.499363
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 4.00000i 0.142134i
\(793\) 40.0000i 1.42044i
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 0 0
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 4.00000i 0.141245i
\(803\) 16.0000i 0.564628i
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 12.0000i 0.422420i
\(808\) 14.0000i 0.492518i
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 8.00000i 0.280745i
\(813\) − 24.0000i − 0.841717i
\(814\) −16.0000 −0.560800
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 0 0
\(818\) 6.00000i 0.209785i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) − 18.0000i − 0.627822i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 4.00000i − 0.139094i −0.997579 0.0695468i \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) − 4.00000i − 0.138675i
\(833\) − 18.0000i − 0.623663i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) − 1.00000i − 0.0345651i
\(838\) − 30.0000i − 1.03633i
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 30.0000i 1.03387i
\(843\) 6.00000i 0.206651i
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 10.0000i 0.343604i
\(848\) 2.00000i 0.0686803i
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 0 0
\(852\) 14.0000i 0.479632i
\(853\) − 30.0000i − 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) − 16.0000i − 0.546231i
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 10.0000i 0.340601i
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 4.00000 0.135926
\(867\) − 19.0000i − 0.645274i
\(868\) − 2.00000i − 0.0678844i
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) − 18.0000i − 0.609557i
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) − 18.0000i − 0.607817i −0.952701 0.303908i \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) 24.0000i 0.809961i
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −56.0000 −1.88669 −0.943344 0.331816i \(-0.892339\pi\)
−0.943344 + 0.331816i \(0.892339\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) − 4.00000i − 0.134231i
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 16.0000i 0.535720i
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 20.0000i 0.667409i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 24.0000i − 0.799113i
\(903\) − 16.0000i − 0.532447i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 58.0000i 1.92586i 0.269754 + 0.962929i \(0.413058\pi\)
−0.269754 + 0.962929i \(0.586942\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) − 16.0000i − 0.529523i
\(914\) −4.00000 −0.132308
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 28.0000i 0.924641i
\(918\) 6.00000i 0.198030i
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) − 24.0000i − 0.790398i
\(923\) − 56.0000i − 1.84326i
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 14.0000i 0.459820i
\(928\) 4.00000i 0.131306i
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000i 0.196537i
\(933\) 6.00000i 0.196431i
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) − 20.0000i − 0.653023i
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) − 6.00000i − 0.195491i
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −26.0000 −0.843108
\(952\) 12.0000i 0.388922i
\(953\) 50.0000i 1.61966i 0.586665 + 0.809829i \(0.300440\pi\)
−0.586665 + 0.809829i \(0.699560\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 16.0000i 0.517207i
\(958\) − 14.0000i − 0.452319i
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 16.0000i 0.515861i
\(963\) 16.0000i 0.515593i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) − 40.0000i − 1.28631i −0.765735 0.643157i \(-0.777624\pi\)
0.765735 0.643157i \(-0.222376\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 24.0000i 0.769405i
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) − 6.00000i − 0.191859i
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) 0 0
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) − 1.00000i − 0.0317500i
\(993\) − 28.0000i − 0.888553i
\(994\) −28.0000 −0.888106
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 4.00000 0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4650.2.d.p.3349.1 2
5.2 odd 4 4650.2.a.bl.1.1 1
5.3 odd 4 930.2.a.d.1.1 1
5.4 even 2 inner 4650.2.d.p.3349.2 2
15.8 even 4 2790.2.a.t.1.1 1
20.3 even 4 7440.2.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.d.1.1 1 5.3 odd 4
2790.2.a.t.1.1 1 15.8 even 4
4650.2.a.bl.1.1 1 5.2 odd 4
4650.2.d.p.3349.1 2 1.1 even 1 trivial
4650.2.d.p.3349.2 2 5.4 even 2 inner
7440.2.a.y.1.1 1 20.3 even 4