Properties

Label 5070.2.b.m.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
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 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.m.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.00000i q^{28} +1.00000 q^{30} -8.00000i q^{31} +1.00000i q^{32} +2.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} +2.00000 q^{38} -1.00000 q^{40} -6.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000i q^{45} +6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -6.00000 q^{53} +1.00000i q^{54} +2.00000 q^{56} -2.00000i q^{57} +1.00000i q^{60} +14.0000 q^{61} +8.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000 q^{69} +2.00000i q^{70} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} +2.00000i q^{76} -16.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +4.00000i q^{86} -6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} -8.00000i q^{93} -2.00000 q^{95} +1.00000i q^{96} +4.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 12 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{30} + 4 q^{35} - 2 q^{36} + 4 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{53} + 4 q^{56} + 28 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{69} - 4 q^{74} - 2 q^{75} - 32 q^{79} + 2 q^{81} + 12 q^{82} + 2 q^{90} - 12 q^{92} - 4 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 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
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.00000i − 0.377964i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) − 8.00000i − 1.43684i −0.695608 0.718421i \(-0.744865\pi\)
0.695608 0.718421i \(-0.255135\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) −1.00000 −0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 6.00000i − 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 6.00000i 0.884652i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 2.00000i 0.239046i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) 2.00000i 0.229416i
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) 0 0
\(86\) 4.00000i 0.431331i
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 1.00000i 0.102062i
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) − 6.00000i − 0.582772i
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 0 0
\(111\) 2.00000i 0.189832i
\(112\) 2.00000i 0.188982i
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 2.00000 0.187317
\(115\) − 6.00000i − 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 11.0000 1.00000
\(122\) 14.0000i 1.26750i
\(123\) − 6.00000i − 0.541002i
\(124\) 8.00000i 0.718421i
\(125\) 1.00000i 0.0894427i
\(126\) −2.00000 −0.178174
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) − 1.00000i − 0.0860663i
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 6.00000i 0.510754i
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) − 2.00000i − 0.164399i
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 8.00000i 0.651031i 0.945537 + 0.325515i \(0.105538\pi\)
−0.945537 + 0.325515i \(0.894462\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) − 16.0000i − 1.27289i
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 12.0000i 0.945732i
\(162\) 1.00000i 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) − 2.00000i − 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) − 6.00000i − 0.442326i
\(185\) 2.00000 0.147043
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) − 2.00000i − 0.145095i
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.00000i 0.575853i 0.957653 + 0.287926i \(0.0929658\pi\)
−0.957653 + 0.287926i \(0.907034\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 4.00000i 0.282138i
\(202\) 12.0000i 0.844317i
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 16.0000i 1.11477i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 2.00000i 0.138013i
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) − 12.0000i − 0.820303i
\(215\) − 4.00000i − 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) 16.0000 1.08615
\(218\) −4.00000 −0.270914
\(219\) − 4.00000i − 0.270295i
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 10.0000i 0.669650i 0.942280 + 0.334825i \(0.108677\pi\)
−0.942280 + 0.334825i \(0.891323\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) 12.0000i 0.798228i
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 2.00000i 0.132453i
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 26.0000i 1.67481i 0.546585 + 0.837404i \(0.315928\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) − 3.00000i − 0.191663i
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) 12.0000i 0.760469i
\(250\) −1.00000 −0.0632456
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 0 0
\(254\) 16.0000i 1.00393i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000i 0.370681i
\(263\) 30.0000 1.84988 0.924940 0.380114i \(-0.124115\pi\)
0.924940 + 0.380114i \(0.124115\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 4.00000i 0.245256i
\(267\) − 6.00000i − 0.367194i
\(268\) − 4.00000i − 0.244339i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 1.00000 0.0608581
\(271\) 8.00000i 0.485965i 0.970031 + 0.242983i \(0.0781258\pi\)
−0.970031 + 0.242983i \(0.921874\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 8.00000i 0.479808i
\(279\) − 8.00000i − 0.478947i
\(280\) − 2.00000i − 0.119523i
\(281\) − 30.0000i − 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 1.00000i 0.0589256i
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 4.00000i 0.234484i
\(292\) 4.00000i 0.234082i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 8.00000i 0.461112i
\(302\) −8.00000 −0.460348
\(303\) 12.0000 0.689382
\(304\) − 2.00000i − 0.114708i
\(305\) − 14.0000i − 0.801638i
\(306\) 0 0
\(307\) − 4.00000i − 0.228292i −0.993464 0.114146i \(-0.963587\pi\)
0.993464 0.114146i \(-0.0364132\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) − 8.00000i − 0.454369i
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 2.00000i 0.112867i
\(315\) 2.00000 0.112687
\(316\) 16.0000 0.900070
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 1.00000i 0.0559017i
\(321\) −12.0000 −0.669775
\(322\) −12.0000 −0.668734
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 4.00000i 0.221201i
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) − 2.00000i − 0.109930i −0.998488 0.0549650i \(-0.982495\pi\)
0.998488 0.0549650i \(-0.0175047\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) 2.00000i 0.109599i
\(334\) −12.0000 −0.656611
\(335\) 4.00000 0.218543
\(336\) 2.00000i 0.109109i
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 20.0000i 1.07990i
\(344\) − 4.00000i − 0.215666i
\(345\) − 6.00000i − 0.323029i
\(346\) − 18.0000i − 0.967686i
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 20.0000i 1.07058i 0.844670 + 0.535288i \(0.179797\pi\)
−0.844670 + 0.535288i \(0.820203\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000i 0.317999i
\(357\) 0 0
\(358\) 6.00000i 0.317110i
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 15.0000 0.789474
\(362\) − 2.00000i − 0.105118i
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 14.0000i 0.731792i
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 6.00000 0.312772
\(369\) − 6.00000i − 0.312348i
\(370\) 2.00000i 0.103975i
\(371\) − 12.0000i − 0.623009i
\(372\) 8.00000i 0.414781i
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 34.0000i 1.74646i 0.487306 + 0.873231i \(0.337980\pi\)
−0.487306 + 0.873231i \(0.662020\pi\)
\(380\) 2.00000 0.102598
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 4.00000 0.203331
\(388\) − 4.00000i − 0.203069i
\(389\) −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
\(393\) 6.00000 0.302660
\(394\) 18.0000 0.906827
\(395\) 16.0000i 0.805047i
\(396\) 0 0
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) − 18.0000i − 0.898877i −0.893311 0.449439i \(-0.851624\pi\)
0.893311 0.449439i \(-0.148376\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 14.0000i − 0.692255i −0.938187 0.346128i \(-0.887496\pi\)
0.938187 0.346128i \(-0.112504\pi\)
\(410\) − 6.00000i − 0.296319i
\(411\) 6.00000i 0.295958i
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 6.00000i 0.294884i
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) −2.00000 −0.0975900
\(421\) − 32.0000i − 1.55958i −0.626038 0.779792i \(-0.715325\pi\)
0.626038 0.779792i \(-0.284675\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 0 0
\(426\) 0 0
\(427\) 28.0000i 1.35501i
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) − 24.0000i − 1.15604i −0.816023 0.578020i \(-0.803826\pi\)
0.816023 0.578020i \(-0.196174\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 16.0000i 0.768025i
\(435\) 0 0
\(436\) − 4.00000i − 0.191565i
\(437\) − 12.0000i − 0.574038i
\(438\) 4.00000 0.191127
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) − 2.00000i − 0.0949158i
\(445\) −6.00000 −0.284427
\(446\) −10.0000 −0.473514
\(447\) − 18.0000i − 0.851371i
\(448\) − 2.00000i − 0.0944911i
\(449\) − 30.0000i − 1.41579i −0.706319 0.707894i \(-0.749646\pi\)
0.706319 0.707894i \(-0.250354\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 0 0
\(452\) −12.0000 −0.564433
\(453\) 8.00000i 0.375873i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) 6.00000i 0.279751i
\(461\) 42.0000i 1.95614i 0.208288 + 0.978068i \(0.433211\pi\)
−0.208288 + 0.978068i \(0.566789\pi\)
\(462\) 0 0
\(463\) − 10.0000i − 0.464739i −0.972628 0.232370i \(-0.925352\pi\)
0.972628 0.232370i \(-0.0746479\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) − 16.0000i − 0.734904i
\(475\) 2.00000i 0.0917663i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) 12.0000i 0.546019i
\(484\) −11.0000 −0.500000
\(485\) 4.00000 0.181631
\(486\) 1.00000i 0.0453609i
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) − 16.0000i − 0.723545i
\(490\) 3.00000 0.135526
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) − 8.00000i − 0.359211i
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) − 14.0000i − 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 12.0000i 0.536120i
\(502\) 30.0000i 1.33897i
\(503\) −18.0000 −0.802580 −0.401290 0.915951i \(-0.631438\pi\)
−0.401290 + 0.915951i \(0.631438\pi\)
\(504\) 2.00000 0.0890871
\(505\) − 12.0000i − 0.533993i
\(506\) 0 0
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000i 0.0441942i
\(513\) − 2.00000i − 0.0883022i
\(514\) 12.0000i 0.529297i
\(515\) − 16.0000i − 0.705044i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 4.00000i − 0.175750i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −6.00000 −0.262111
\(525\) − 2.00000i − 0.0872872i
\(526\) 30.0000i 1.30806i
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) 12.0000i 0.518805i
\(536\) 4.00000 0.172774
\(537\) 6.00000 0.258919
\(538\) 12.0000i 0.517357i
\(539\) 0 0
\(540\) 1.00000i 0.0430331i
\(541\) 8.00000i 0.343947i 0.985102 + 0.171973i \(0.0550143\pi\)
−0.985102 + 0.171973i \(0.944986\pi\)
\(542\) −8.00000 −0.343629
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 0 0
\(552\) − 6.00000i − 0.255377i
\(553\) − 32.0000i − 1.36078i
\(554\) − 26.0000i − 1.10463i
\(555\) 2.00000 0.0848953
\(556\) −8.00000 −0.339276
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) − 12.0000i − 0.504844i
\(566\) 28.0000i 1.17693i
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) − 2.00000i − 0.0837708i
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000i 0.500870i
\(575\) −6.00000 −0.250217
\(576\) −1.00000 −0.0416667
\(577\) 16.0000i 0.666089i 0.942911 + 0.333044i \(0.108076\pi\)
−0.942911 + 0.333044i \(0.891924\pi\)
\(578\) − 17.0000i − 0.707107i
\(579\) 8.00000i 0.332469i
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) −4.00000 −0.165805
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) −3.00000 −0.123718
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) − 18.0000i − 0.740421i
\(592\) 2.00000i 0.0821995i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000i 0.737309i
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −8.00000 −0.326056
\(603\) 4.00000i 0.162893i
\(604\) − 8.00000i − 0.325515i
\(605\) − 11.0000i − 0.447214i
\(606\) 12.0000i 0.487467i
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 14.0000 0.566843
\(611\) 0 0
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 38.0000i 1.52735i 0.645601 + 0.763674i \(0.276607\pi\)
−0.645601 + 0.763674i \(0.723393\pi\)
\(620\) 8.00000 0.321288
\(621\) 6.00000 0.240772
\(622\) − 24.0000i − 0.962312i
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 10.0000i − 0.399680i
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) 2.00000i 0.0796819i
\(631\) 20.0000i 0.796187i 0.917345 + 0.398094i \(0.130328\pi\)
−0.917345 + 0.398094i \(0.869672\pi\)
\(632\) 16.0000i 0.636446i
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) − 16.0000i − 0.634941i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) − 12.0000i − 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) − 12.0000i − 0.472866i
\(645\) − 4.00000i − 0.157500i
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 16.0000i 0.626608i
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −4.00000 −0.156412
\(655\) − 6.00000i − 0.234439i
\(656\) − 6.00000i − 0.234261i
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) − 16.0000i − 0.622328i −0.950356 0.311164i \(-0.899281\pi\)
0.950356 0.311164i \(-0.100719\pi\)
\(662\) 2.00000 0.0777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) − 4.00000i − 0.155113i
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 10.0000i 0.386622i
\(670\) 4.00000i 0.154533i
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) − 14.0000i − 0.539260i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 12.0000i 0.460857i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) − 12.0000i − 0.459841i
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 2.00000i 0.0764719i
\(685\) 6.00000 0.229248
\(686\) −20.0000 −0.763604
\(687\) − 4.00000i − 0.152610i
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 6.00000 0.228416
\(691\) − 26.0000i − 0.989087i −0.869153 0.494543i \(-0.835335\pi\)
0.869153 0.494543i \(-0.164665\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 24.0000i 0.911028i
\(695\) − 8.00000i − 0.303457i
\(696\) 0 0
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 0 0
\(700\) 2.00000i 0.0755929i
\(701\) −36.0000 −1.35970 −0.679851 0.733351i \(-0.737955\pi\)
−0.679851 + 0.733351i \(0.737955\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) 20.0000i 0.751116i 0.926799 + 0.375558i \(0.122549\pi\)
−0.926799 + 0.375558i \(0.877451\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) −6.00000 −0.224860
\(713\) − 48.0000i − 1.79761i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 32.0000i 1.19174i
\(722\) 15.0000i 0.558242i
\(723\) 26.0000i 0.966950i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 11.0000i 0.408248i
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 4.00000i − 0.148047i
\(731\) 0 0
\(732\) −14.0000 −0.517455
\(733\) − 50.0000i − 1.84679i −0.383849 0.923396i \(-0.625402\pi\)
0.383849 0.923396i \(-0.374598\pi\)
\(734\) 8.00000i 0.295285i
\(735\) − 3.00000i − 0.110657i
\(736\) 6.00000i 0.221163i
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) 38.0000i 1.39785i 0.715194 + 0.698926i \(0.246338\pi\)
−0.715194 + 0.698926i \(0.753662\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) −8.00000 −0.293294
\(745\) −18.0000 −0.659469
\(746\) 26.0000i 0.951928i
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) − 24.0000i − 0.876941i
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) − 2.00000i − 0.0727393i
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −34.0000 −1.23494
\(759\) 0 0
\(760\) 2.00000i 0.0725476i
\(761\) 6.00000i 0.217500i 0.994069 + 0.108750i \(0.0346848\pi\)
−0.994069 + 0.108750i \(0.965315\pi\)
\(762\) 16.0000i 0.579619i
\(763\) −8.00000 −0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 22.0000i 0.793340i 0.917961 + 0.396670i \(0.129834\pi\)
−0.917961 + 0.396670i \(0.870166\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) − 8.00000i − 0.287926i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 8.00000i 0.287368i
\(776\) 4.00000 0.143592
\(777\) −4.00000 −0.143499
\(778\) − 36.0000i − 1.29066i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) − 2.00000i − 0.0713831i
\(786\) 6.00000i 0.214013i
\(787\) − 16.0000i − 0.570338i −0.958477 0.285169i \(-0.907950\pi\)
0.958477 0.285169i \(-0.0920498\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 30.0000 1.06803
\(790\) −16.0000 −0.569254
\(791\) 24.0000i 0.853342i
\(792\) 0 0
\(793\) 0 0
\(794\) −26.0000 −0.922705
\(795\) 6.00000i 0.212798i
\(796\) −16.0000 −0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) − 1.00000i − 0.0353553i
\(801\) − 6.00000i − 0.212000i
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) − 4.00000i − 0.141069i
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) − 12.0000i − 0.422159i
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 26.0000i − 0.912983i −0.889728 0.456492i \(-0.849106\pi\)
0.889728 0.456492i \(-0.150894\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) − 8.00000i − 0.279885i
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) −6.00000 −0.209274
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) − 16.0000i − 0.557386i
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) −6.00000 −0.208514
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 12.0000i 0.416526i
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 0 0
\(834\) 8.00000i 0.277017i
\(835\) 12.0000 0.415277
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) 6.00000i 0.207267i
\(839\) − 48.0000i − 1.65714i −0.559883 0.828572i \(-0.689154\pi\)
0.559883 0.828572i \(-0.310846\pi\)
\(840\) − 2.00000i − 0.0690066i
\(841\) −29.0000 −1.00000
\(842\) 32.0000 1.10279
\(843\) − 30.0000i − 1.03325i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) −6.00000 −0.206041
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) −28.0000 −0.958140
\(855\) −2.00000 −0.0683986
\(856\) 12.0000i 0.410152i
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000i 0.136399i
\(861\) 12.0000 0.408959
\(862\) 24.0000 0.817443
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 18.0000i 0.612018i
\(866\) − 14.0000i − 0.475739i
\(867\) −17.0000 −0.577350
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) 4.00000i 0.135379i
\(874\) 12.0000 0.405906
\(875\) −2.00000 −0.0676123
\(876\) 4.00000i 0.135147i
\(877\) 58.0000i 1.95852i 0.202606 + 0.979260i \(0.435059\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) − 32.0000i − 1.07995i
\(879\) − 6.00000i − 0.202375i
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −30.0000 −1.00730 −0.503651 0.863907i \(-0.668010\pi\)
−0.503651 + 0.863907i \(0.668010\pi\)
\(888\) 2.00000 0.0671156
\(889\) 32.0000i 1.07325i
\(890\) − 6.00000i − 0.201120i
\(891\) 0 0
\(892\) − 10.0000i − 0.334825i
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) − 6.00000i − 0.200558i
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) − 12.0000i − 0.399114i
\(905\) 2.00000i 0.0664822i
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 12.0000i 0.398234i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) 0 0
\(914\) −28.0000 −0.926158
\(915\) − 14.0000i − 0.462826i
\(916\) 4.00000i 0.132164i
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −6.00000 −0.197814
\(921\) − 4.00000i − 0.131804i
\(922\) −42.0000 −1.38320
\(923\) 0 0
\(924\) 0 0
\(925\) − 2.00000i − 0.0657596i
\(926\) 10.0000 0.328620
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 54.0000i 1.77168i 0.463988 + 0.885841i \(0.346418\pi\)
−0.463988 + 0.885841i \(0.653582\pi\)
\(930\) − 8.00000i − 0.262330i
\(931\) − 6.00000i − 0.196642i
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) − 36.0000i − 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) − 36.0000i − 1.17232i
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) 18.0000i 0.583690i
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 1.00000i 0.0322749i
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) − 26.0000i − 0.837404i
\(965\) 8.00000 0.257529
\(966\) −12.0000 −0.386094
\(967\) − 50.0000i − 1.60789i −0.594703 0.803946i \(-0.702730\pi\)
0.594703 0.803946i \(-0.297270\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 0 0
\(970\) 4.00000i 0.128432i
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.0000i 0.512936i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 16.0000 0.511624
\(979\) 0 0
\(980\) 3.00000i 0.0958315i
\(981\) 4.00000i 0.127710i
\(982\) − 6.00000i − 0.191468i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) −6.00000 −0.191273
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000 0.254000
\(993\) − 2.00000i − 0.0634681i
\(994\) 0 0
\(995\) − 16.0000i − 0.507234i
\(996\) − 12.0000i − 0.380235i
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 14.0000 0.443162
\(999\) 2.00000i 0.0632772i
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 5070.2.b.m.1351.2 2
13.5 odd 4 5070.2.a.t.1.1 1
13.8 odd 4 390.2.a.d.1.1 1
13.12 even 2 inner 5070.2.b.m.1351.1 2
39.8 even 4 1170.2.a.k.1.1 1
52.47 even 4 3120.2.a.j.1.1 1
65.8 even 4 1950.2.e.d.1249.2 2
65.34 odd 4 1950.2.a.o.1.1 1
65.47 even 4 1950.2.e.d.1249.1 2
156.47 odd 4 9360.2.a.g.1.1 1
195.8 odd 4 5850.2.e.o.5149.1 2
195.47 odd 4 5850.2.e.o.5149.2 2
195.164 even 4 5850.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 13.8 odd 4
1170.2.a.k.1.1 1 39.8 even 4
1950.2.a.o.1.1 1 65.34 odd 4
1950.2.e.d.1249.1 2 65.47 even 4
1950.2.e.d.1249.2 2 65.8 even 4
3120.2.a.j.1.1 1 52.47 even 4
5070.2.a.t.1.1 1 13.5 odd 4
5070.2.b.m.1351.1 2 13.12 even 2 inner
5070.2.b.m.1351.2 2 1.1 even 1 trivial
5850.2.a.g.1.1 1 195.164 even 4
5850.2.e.o.5149.1 2 195.8 odd 4
5850.2.e.o.5149.2 2 195.47 odd 4
9360.2.a.g.1.1 1 156.47 odd 4