Properties

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

$q$-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 \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 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 \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) − 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 \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) − 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 \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) −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 \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 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 \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −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 \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 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 \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 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 \(-0.934908\pi\)
0.979164 0.203069i \(-0.0650917\pi\)
\(98\) − 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 \(-0.938644\pi\)
0.981480 0.191565i \(-0.0613564\pi\)
\(110\) 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 \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 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 \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 8.00000i − 0.651031i −0.945537 0.325515i \(-0.894462\pi\)
0.945537 0.325515i \(-0.105538\pi\)
\(152\) −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 \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) − 6.00000i − 0.468521i
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 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 \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 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 \(-0.891323\pi\)
0.942280 0.334825i \(-0.108677\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) − 12.0000i − 0.798228i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 4.00000i 0.264327i 0.991228 + 0.132164i \(0.0421925\pi\)
−0.991228 + 0.132164i \(0.957808\pi\)
\(230\) 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 \(-0.684072\pi\)
0.546585 0.837404i \(-0.315928\pi\)
\(242\) − 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 \(-0.921874\pi\)
0.970031 0.242983i \(-0.0781258\pi\)
\(272\) 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 \(0.352700\pi\)
−0.446417 + 0.894825i \(0.647300\pi\)
\(282\) 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 \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) − 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 \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 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 \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 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 \(0.0175047\pi\)
−0.998488 + 0.0549650i \(0.982495\pi\)
\(332\) 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 \(-0.820203\pi\)
0.844670 0.535288i \(-0.179797\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 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 \(0.218315\pi\)
−0.773877 + 0.633336i \(0.781685\pi\)
\(360\) −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 \(-0.662020\pi\)
0.487306 0.873231i \(-0.337980\pi\)
\(380\) 2.00000 0.102598
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 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 \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) −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 \(0.112504\pi\)
−0.938187 + 0.346128i \(0.887496\pi\)
\(410\) 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 \(0.284675\pi\)
−0.626038 + 0.779792i \(0.715325\pi\)
\(422\) 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 \(0.196174\pi\)
−0.816023 + 0.578020i \(0.803826\pi\)
\(432\) 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 \(0.250354\pi\)
−0.706319 + 0.707894i \(0.749646\pi\)
\(450\) 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 \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) − 6.00000i − 0.279751i
\(461\) − 42.0000i − 1.95614i −0.208288 0.978068i \(-0.566789\pi\)
0.208288 0.978068i \(-0.433211\pi\)
\(462\) 0 0
\(463\) 10.0000i 0.464739i 0.972628 + 0.232370i \(0.0746479\pi\)
−0.972628 + 0.232370i \(0.925352\pi\)
\(464\) 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 \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 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 \(0.101456\pi\)
−0.949633 + 0.313363i \(0.898544\pi\)
\(500\) 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 \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) 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 \(-0.944986\pi\)
0.985102 0.171973i \(-0.0550143\pi\)
\(542\) −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 \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 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 \(-0.891924\pi\)
0.942911 0.333044i \(-0.108076\pi\)
\(578\) 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 \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) −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 \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 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 \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) − 38.0000i − 1.52735i −0.645601 0.763674i \(-0.723393\pi\)
0.645601 0.763674i \(-0.276607\pi\)
\(620\) 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 \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) − 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 \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 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 \(0.100719\pi\)
−0.950356 + 0.311164i \(0.899281\pi\)
\(662\) 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 \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) − 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 \(0.164665\pi\)
−0.869153 + 0.494543i \(0.835335\pi\)
\(692\) 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 \(-0.877451\pi\)
0.926799 0.375558i \(-0.122549\pi\)
\(710\) 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 \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) − 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 \(-0.753662\pi\)
0.715194 0.698926i \(-0.246338\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) −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 \(-0.965315\pi\)
0.994069 0.108750i \(-0.0346848\pi\)
\(762\) − 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 \(-0.870166\pi\)
0.917961 0.396670i \(-0.129834\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 8.00000i 0.287926i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) − 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 \(0.0920498\pi\)
−0.958477 + 0.285169i \(0.907950\pi\)
\(788\) − 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 \(0.150894\pi\)
−0.889728 + 0.456492i \(0.849106\pi\)
\(812\) 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 \(-0.966612\pi\)
0.994504 0.104701i \(-0.0333885\pi\)
\(822\) −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 \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) −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 \(0.310846\pi\)
−0.559883 + 0.828572i \(0.689154\pi\)
\(840\) 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 \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) −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 \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) − 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 \(-0.564941\pi\)
0.202606 0.979260i \(-0.435059\pi\)
\(878\) 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 \(-0.653582\pi\)
0.463988 0.885841i \(-0.346418\pi\)
\(930\) 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 \(0.240015\pi\)
−0.728937 + 0.684580i \(0.759985\pi\)
\(942\) − 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 \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 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 \(0.297270\pi\)
−0.594703 + 0.803946i \(0.702730\pi\)
\(968\) 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 \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 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 \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) −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.1351.1 2
13.5 odd 4 390.2.a.d.1.1 1
13.8 odd 4 5070.2.a.t.1.1 1
13.12 even 2 inner 5070.2.b.m.1351.2 2
39.5 even 4 1170.2.a.k.1.1 1
52.31 even 4 3120.2.a.j.1.1 1
65.18 even 4 1950.2.e.d.1249.2 2
65.44 odd 4 1950.2.a.o.1.1 1
65.57 even 4 1950.2.e.d.1249.1 2
156.83 odd 4 9360.2.a.g.1.1 1
195.44 even 4 5850.2.a.g.1.1 1
195.83 odd 4 5850.2.e.o.5149.1 2
195.122 odd 4 5850.2.e.o.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 13.5 odd 4
1170.2.a.k.1.1 1 39.5 even 4
1950.2.a.o.1.1 1 65.44 odd 4
1950.2.e.d.1249.1 2 65.57 even 4
1950.2.e.d.1249.2 2 65.18 even 4
3120.2.a.j.1.1 1 52.31 even 4
5070.2.a.t.1.1 1 13.8 odd 4
5070.2.b.m.1351.1 2 1.1 even 1 trivial
5070.2.b.m.1351.2 2 13.12 even 2 inner
5850.2.a.g.1.1 1 195.44 even 4
5850.2.e.o.5149.1 2 195.83 odd 4
5850.2.e.o.5149.2 2 195.122 odd 4
9360.2.a.g.1.1 1 156.83 odd 4