Properties

Label 5070.2.b.j.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.j.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000i q^{18} -2.00000i q^{19} -1.00000i q^{20} +2.00000i q^{21} +3.00000 q^{22} -3.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.00000i q^{28} +3.00000 q^{29} -1.00000 q^{30} -5.00000i q^{31} +1.00000i q^{32} -3.00000i q^{33} -6.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -7.00000i q^{37} +2.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -2.00000 q^{42} +1.00000 q^{43} +3.00000i q^{44} +1.00000i q^{45} -3.00000i q^{46} -3.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} +1.00000i q^{54} +3.00000 q^{55} +2.00000 q^{56} -2.00000i q^{57} +3.00000i q^{58} -9.00000i q^{59} -1.00000i q^{60} +2.00000 q^{61} +5.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -8.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} -2.00000i q^{70} +12.0000i q^{71} -1.00000i q^{72} +14.0000i q^{73} +7.00000 q^{74} -1.00000 q^{75} +2.00000i q^{76} +6.00000 q^{77} +5.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +6.00000i q^{83} -2.00000i q^{84} -6.00000i q^{85} +1.00000i q^{86} +3.00000 q^{87} -3.00000 q^{88} -18.0000i q^{89} -1.00000 q^{90} +3.00000 q^{92} -5.00000i q^{93} +3.00000 q^{94} +2.00000 q^{95} +1.00000i q^{96} -14.0000i q^{97} +3.00000i q^{98} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} - 2q^{12} - 4q^{14} + 2q^{16} - 12q^{17} + 6q^{22} - 6q^{23} - 2q^{25} + 2q^{27} + 6q^{29} - 2q^{30} - 4q^{35} - 2q^{36} + 4q^{38} + 2q^{40} - 4q^{42} + 2q^{43} + 2q^{48} + 6q^{49} - 12q^{51} - 12q^{53} + 6q^{55} + 4q^{56} + 4q^{61} + 10q^{62} - 2q^{64} + 6q^{66} + 12q^{68} - 6q^{69} + 14q^{74} - 2q^{75} + 12q^{77} + 10q^{79} + 2q^{81} + 12q^{82} + 6q^{87} - 6q^{88} - 2q^{90} + 6q^{92} + 6q^{94} + 4q^{95} + O(q^{100}) \)

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
\(11\) − 3.00000i − 0.904534i −0.891883 0.452267i \(-0.850615\pi\)
0.891883 0.452267i \(-0.149385\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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
\(22\) 3.00000 0.639602
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 2.00000i − 0.377964i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 5.00000i − 0.898027i −0.893525 0.449013i \(-0.851776\pi\)
0.893525 0.449013i \(-0.148224\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) − 6.00000i − 1.02899i
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) 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
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 3.00000i 0.452267i
\(45\) 1.00000i 0.149071i
\(46\) − 3.00000i − 0.442326i
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) −6.00000 −0.840168
\(52\) 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
\(55\) 3.00000 0.404520
\(56\) 2.00000 0.267261
\(57\) − 2.00000i − 0.264906i
\(58\) 3.00000i 0.393919i
\(59\) − 9.00000i − 1.17170i −0.810419 0.585850i \(-0.800761\pi\)
0.810419 0.585850i \(-0.199239\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 5.00000 0.635001
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 6.00000 0.727607
\(69\) −3.00000 −0.361158
\(70\) − 2.00000i − 0.239046i
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 7.00000 0.813733
\(75\) −1.00000 −0.115470
\(76\) 2.00000i 0.229416i
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) − 6.00000i − 0.650791i
\(86\) 1.00000i 0.107833i
\(87\) 3.00000 0.321634
\(88\) −3.00000 −0.319801
\(89\) − 18.0000i − 1.90800i −0.299813 0.953998i \(-0.596924\pi\)
0.299813 0.953998i \(-0.403076\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) − 5.00000i − 0.518476i
\(94\) 3.00000 0.309426
\(95\) 2.00000 0.205196
\(96\) 1.00000i 0.102062i
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 3.00000i 0.303046i
\(99\) − 3.00000i − 0.301511i
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) − 6.00000i − 0.582772i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 14.0000i − 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 3.00000i 0.286039i
\(111\) − 7.00000i − 0.664411i
\(112\) 2.00000i 0.188982i
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 2.00000 0.187317
\(115\) − 3.00000i − 0.279751i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 9.00000 0.828517
\(119\) − 12.0000i − 1.10004i
\(120\) 1.00000 0.0912871
\(121\) 2.00000 0.181818
\(122\) 2.00000i 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) 5.00000i 0.449013i
\(125\) − 1.00000i − 0.0894427i
\(126\) −2.00000 −0.178174
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 9.00000 0.786334 0.393167 0.919467i \(-0.371379\pi\)
0.393167 + 0.919467i \(0.371379\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 4.00000 0.346844
\(134\) 8.00000 0.691095
\(135\) 1.00000i 0.0860663i
\(136\) 6.00000i 0.514496i
\(137\) 9.00000i 0.768922i 0.923141 + 0.384461i \(0.125613\pi\)
−0.923141 + 0.384461i \(0.874387\pi\)
\(138\) − 3.00000i − 0.255377i
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 2.00000 0.169031
\(141\) − 3.00000i − 0.252646i
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.00000i 0.249136i
\(146\) −14.0000 −1.15865
\(147\) 3.00000 0.247436
\(148\) 7.00000i 0.575396i
\(149\) 9.00000i 0.737309i 0.929567 + 0.368654i \(0.120181\pi\)
−0.929567 + 0.368654i \(0.879819\pi\)
\(150\) − 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
\(153\) −6.00000 −0.485071
\(154\) 6.00000i 0.483494i
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 5.00000i 0.397779i
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) − 6.00000i − 0.472866i
\(162\) 1.00000i 0.0785674i
\(163\) − 13.0000i − 1.01824i −0.860696 0.509119i \(-0.829971\pi\)
0.860696 0.509119i \(-0.170029\pi\)
\(164\) 6.00000i 0.468521i
\(165\) 3.00000 0.233550
\(166\) −6.00000 −0.465690
\(167\) − 9.00000i − 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 6.00000 0.460179
\(171\) − 2.00000i − 0.152944i
\(172\) −1.00000 −0.0762493
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 3.00000i 0.227429i
\(175\) − 2.00000i − 0.151186i
\(176\) − 3.00000i − 0.226134i
\(177\) − 9.00000i − 0.676481i
\(178\) 18.0000 1.34916
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 3.00000i 0.221163i
\(185\) 7.00000 0.514650
\(186\) 5.00000 0.366618
\(187\) 18.0000i 1.31629i
\(188\) 3.00000i 0.218797i
\(189\) 2.00000i 0.145479i
\(190\) 2.00000i 0.145095i
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 24.0000i − 1.70993i −0.518686 0.854965i \(-0.673579\pi\)
0.518686 0.854965i \(-0.326421\pi\)
\(198\) 3.00000 0.213201
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) − 8.00000i − 0.564276i
\(202\) − 6.00000i − 0.422159i
\(203\) 6.00000i 0.421117i
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) − 14.0000i − 0.975426i
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) − 2.00000i − 0.138013i
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000i 0.822226i
\(214\) − 6.00000i − 0.410152i
\(215\) 1.00000i 0.0681994i
\(216\) − 1.00000i − 0.0680414i
\(217\) 10.0000 0.678844
\(218\) 14.0000 0.948200
\(219\) 14.0000i 0.946032i
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) 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
\(226\) 15.0000i 0.997785i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 2.00000i 0.132453i
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 3.00000 0.197814
\(231\) 6.00000 0.394771
\(232\) − 3.00000i − 0.196960i
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 9.00000i 0.585850i
\(237\) 5.00000 0.324785
\(238\) 12.0000 0.777844
\(239\) − 24.0000i − 1.55243i −0.630468 0.776215i \(-0.717137\pi\)
0.630468 0.776215i \(-0.282863\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 17.0000i 1.09507i 0.836784 + 0.547533i \(0.184433\pi\)
−0.836784 + 0.547533i \(0.815567\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 3.00000i 0.191663i
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) −5.00000 −0.317500
\(249\) 6.00000i 0.380235i
\(250\) 1.00000 0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 9.00000i 0.565825i
\(254\) − 14.0000i − 0.878438i
\(255\) − 6.00000i − 0.375735i
\(256\) 1.00000 0.0625000
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 1.00000i 0.0622573i
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 9.00000i 0.556022i
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) −3.00000 −0.184637
\(265\) − 6.00000i − 0.368577i
\(266\) 4.00000i 0.245256i
\(267\) − 18.0000i − 1.10158i
\(268\) 8.00000i 0.488678i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 11.0000i 0.668202i 0.942537 + 0.334101i \(0.108433\pi\)
−0.942537 + 0.334101i \(0.891567\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 3.00000i 0.180907i
\(276\) 3.00000 0.180579
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 14.0000i 0.839664i
\(279\) − 5.00000i − 0.299342i
\(280\) 2.00000i 0.119523i
\(281\) 18.0000i 1.07379i 0.843649 + 0.536895i \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(282\) 3.00000 0.178647
\(283\) 31.0000 1.84276 0.921379 0.388664i \(-0.127063\pi\)
0.921379 + 0.388664i \(0.127063\pi\)
\(284\) − 12.0000i − 0.712069i
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 1.00000i 0.0589256i
\(289\) 19.0000 1.11765
\(290\) −3.00000 −0.176166
\(291\) − 14.0000i − 0.820695i
\(292\) − 14.0000i − 0.819288i
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 9.00000 0.524000
\(296\) −7.00000 −0.406867
\(297\) − 3.00000i − 0.174078i
\(298\) −9.00000 −0.521356
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 2.00000i 0.115278i
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) − 2.00000i − 0.114708i
\(305\) 2.00000i 0.114520i
\(306\) − 6.00000i − 0.342997i
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) −6.00000 −0.341882
\(309\) −14.0000 −0.796432
\(310\) 5.00000i 0.283981i
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) − 13.0000i − 0.733632i
\(315\) −2.00000 −0.112687
\(316\) −5.00000 −0.281272
\(317\) − 12.0000i − 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) − 9.00000i − 0.503903i
\(320\) − 1.00000i − 0.0559017i
\(321\) −6.00000 −0.334887
\(322\) 6.00000 0.334367
\(323\) 12.0000i 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 13.0000 0.720003
\(327\) − 14.0000i − 0.774202i
\(328\) −6.00000 −0.331295
\(329\) 6.00000 0.330791
\(330\) 3.00000i 0.165145i
\(331\) − 32.0000i − 1.75888i −0.476011 0.879440i \(-0.657918\pi\)
0.476011 0.879440i \(-0.342082\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) − 7.00000i − 0.383598i
\(334\) 9.00000 0.492458
\(335\) 8.00000 0.437087
\(336\) 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
\(339\) 15.0000 0.814688
\(340\) 6.00000i 0.325396i
\(341\) −15.0000 −0.812296
\(342\) 2.00000 0.108148
\(343\) 20.0000i 1.07990i
\(344\) − 1.00000i − 0.0539164i
\(345\) − 3.00000i − 0.161515i
\(346\) − 12.0000i − 0.645124i
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −3.00000 −0.160817
\(349\) 8.00000i 0.428230i 0.976808 + 0.214115i \(0.0686868\pi\)
−0.976808 + 0.214115i \(0.931313\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 9.00000 0.478345
\(355\) −12.0000 −0.636894
\(356\) 18.0000i 0.953998i
\(357\) − 12.0000i − 0.635107i
\(358\) 3.00000i 0.158555i
\(359\) 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
\(362\) 16.0000i 0.840941i
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) 2.00000i 0.104542i
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) −3.00000 −0.156386
\(369\) − 6.00000i − 0.312348i
\(370\) 7.00000i 0.363913i
\(371\) − 12.0000i − 0.623009i
\(372\) 5.00000i 0.259238i
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) −18.0000 −0.930758
\(375\) − 1.00000i − 0.0516398i
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 38.0000i − 1.95193i −0.217930 0.975964i \(-0.569930\pi\)
0.217930 0.975964i \(-0.430070\pi\)
\(380\) −2.00000 −0.102598
\(381\) −14.0000 −0.717242
\(382\) − 12.0000i − 0.613973i
\(383\) 21.0000i 1.07305i 0.843884 + 0.536525i \(0.180263\pi\)
−0.843884 + 0.536525i \(0.819737\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 6.00000i 0.305788i
\(386\) 4.00000 0.203595
\(387\) 1.00000 0.0508329
\(388\) 14.0000i 0.710742i
\(389\) −3.00000 −0.152106 −0.0760530 0.997104i \(-0.524232\pi\)
−0.0760530 + 0.997104i \(0.524232\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) − 3.00000i − 0.151523i
\(393\) 9.00000 0.453990
\(394\) 24.0000 1.20910
\(395\) 5.00000i 0.251577i
\(396\) 3.00000i 0.150756i
\(397\) − 31.0000i − 1.55585i −0.628360 0.777923i \(-0.716273\pi\)
0.628360 0.777923i \(-0.283727\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) − 12.0000i − 0.599251i −0.954057 0.299626i \(-0.903138\pi\)
0.954057 0.299626i \(-0.0968618\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000i 0.0496904i
\(406\) −6.00000 −0.297775
\(407\) −21.0000 −1.04093
\(408\) 6.00000i 0.297044i
\(409\) 10.0000i 0.494468i 0.968956 + 0.247234i \(0.0795217\pi\)
−0.968956 + 0.247234i \(0.920478\pi\)
\(410\) 6.00000i 0.296319i
\(411\) 9.00000i 0.443937i
\(412\) 14.0000 0.689730
\(413\) 18.0000 0.885722
\(414\) − 3.00000i − 0.147442i
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) − 6.00000i − 0.293470i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) 16.0000i 0.779792i 0.920859 + 0.389896i \(0.127489\pi\)
−0.920859 + 0.389896i \(0.872511\pi\)
\(422\) 20.0000i 0.973585i
\(423\) − 3.00000i − 0.145865i
\(424\) 6.00000i 0.291386i
\(425\) 6.00000 0.291043
\(426\) −12.0000 −0.581402
\(427\) 4.00000i 0.193574i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) − 12.0000i − 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 1.00000 0.0481125
\(433\) 40.0000 1.92228 0.961139 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 10.0000i 0.480015i
\(435\) 3.00000i 0.143839i
\(436\) 14.0000i 0.670478i
\(437\) 6.00000i 0.287019i
\(438\) −14.0000 −0.668946
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) − 3.00000i − 0.143019i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 7.00000i 0.332205i
\(445\) 18.0000 0.853282
\(446\) −10.0000 −0.473514
\(447\) 9.00000i 0.425685i
\(448\) − 2.00000i − 0.0944911i
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −18.0000 −0.847587
\(452\) −15.0000 −0.705541
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) −14.0000 −0.654177
\(459\) −6.00000 −0.280056
\(460\) 3.00000i 0.139876i
\(461\) − 15.0000i − 0.698620i −0.937007 0.349310i \(-0.886416\pi\)
0.937007 0.349310i \(-0.113584\pi\)
\(462\) 6.00000i 0.279145i
\(463\) − 34.0000i − 1.58011i −0.613033 0.790057i \(-0.710051\pi\)
0.613033 0.790057i \(-0.289949\pi\)
\(464\) 3.00000 0.139272
\(465\) 5.00000 0.231869
\(466\) − 21.0000i − 0.972806i
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 3.00000i 0.138380i
\(471\) −13.0000 −0.599008
\(472\) −9.00000 −0.414259
\(473\) − 3.00000i − 0.137940i
\(474\) 5.00000i 0.229658i
\(475\) 2.00000i 0.0917663i
\(476\) 12.0000i 0.550019i
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −17.0000 −0.774329
\(483\) − 6.00000i − 0.273009i
\(484\) −2.00000 −0.0909091
\(485\) 14.0000 0.635707
\(486\) 1.00000i 0.0453609i
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) − 13.0000i − 0.587880i
\(490\) −3.00000 −0.135526
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000i 0.270501i
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) − 5.00000i − 0.224507i
\(497\) −24.0000 −1.07655
\(498\) −6.00000 −0.268866
\(499\) − 32.0000i − 1.43252i −0.697835 0.716258i \(-0.745853\pi\)
0.697835 0.716258i \(-0.254147\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 9.00000i − 0.402090i
\(502\) − 15.0000i − 0.669483i
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 2.00000 0.0890871
\(505\) − 6.00000i − 0.266996i
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) 14.0000 0.621150
\(509\) − 3.00000i − 0.132973i −0.997787 0.0664863i \(-0.978821\pi\)
0.997787 0.0664863i \(-0.0211789\pi\)
\(510\) 6.00000 0.265684
\(511\) −28.0000 −1.23865
\(512\) 1.00000i 0.0441942i
\(513\) − 2.00000i − 0.0883022i
\(514\) 21.0000i 0.926270i
\(515\) − 14.0000i − 0.616914i
\(516\) −1.00000 −0.0440225
\(517\) −9.00000 −0.395820
\(518\) 14.0000i 0.615125i
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 3.00000i 0.131306i
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −9.00000 −0.393167
\(525\) − 2.00000i − 0.0872872i
\(526\) − 15.0000i − 0.654031i
\(527\) 30.0000i 1.30682i
\(528\) − 3.00000i − 0.130558i
\(529\) −14.0000 −0.608696
\(530\) 6.00000 0.260623
\(531\) − 9.00000i − 0.390567i
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) − 6.00000i − 0.259403i
\(536\) −8.00000 −0.345547
\(537\) 3.00000 0.129460
\(538\) 18.0000i 0.776035i
\(539\) − 9.00000i − 0.387657i
\(540\) − 1.00000i − 0.0430331i
\(541\) 20.0000i 0.859867i 0.902861 + 0.429934i \(0.141463\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) −11.0000 −0.472490
\(543\) 16.0000 0.686626
\(544\) − 6.00000i − 0.257248i
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) − 9.00000i − 0.384461i
\(549\) 2.00000 0.0853579
\(550\) −3.00000 −0.127920
\(551\) − 6.00000i − 0.255609i
\(552\) 3.00000i 0.127688i
\(553\) 10.0000i 0.425243i
\(554\) 1.00000i 0.0424859i
\(555\) 7.00000 0.297133
\(556\) −14.0000 −0.593732
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) 5.00000 0.211667
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 18.0000i 0.759961i
\(562\) −18.0000 −0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 3.00000i 0.126323i
\(565\) 15.0000i 0.631055i
\(566\) 31.0000i 1.30303i
\(567\) 2.00000i 0.0839921i
\(568\) 12.0000 0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 2.00000i 0.0837708i
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 12.0000i 0.500870i
\(575\) 3.00000 0.125109
\(576\) −1.00000 −0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 19.0000i 0.790296i
\(579\) − 4.00000i − 0.166234i
\(580\) − 3.00000i − 0.124568i
\(581\) −12.0000 −0.497844
\(582\) 14.0000 0.580319
\(583\) 18.0000i 0.745484i
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 30.0000 1.23929
\(587\) − 18.0000i − 0.742940i −0.928445 0.371470i \(-0.878854\pi\)
0.928445 0.371470i \(-0.121146\pi\)
\(588\) −3.00000 −0.123718
\(589\) −10.0000 −0.412043
\(590\) 9.00000i 0.370524i
\(591\) − 24.0000i − 0.987228i
\(592\) − 7.00000i − 0.287698i
\(593\) 27.0000i 1.10876i 0.832265 + 0.554379i \(0.187044\pi\)
−0.832265 + 0.554379i \(0.812956\pi\)
\(594\) 3.00000 0.123091
\(595\) 12.0000 0.491952
\(596\) − 9.00000i − 0.368654i
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) −2.00000 −0.0815139
\(603\) − 8.00000i − 0.325785i
\(604\) − 8.00000i − 0.325515i
\(605\) 2.00000i 0.0813116i
\(606\) − 6.00000i − 0.243733i
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 2.00000 0.0811107
\(609\) 6.00000i 0.243132i
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 31.0000i 1.25208i 0.779792 + 0.626039i \(0.215325\pi\)
−0.779792 + 0.626039i \(0.784675\pi\)
\(614\) −8.00000 −0.322854
\(615\) 6.00000 0.241943
\(616\) − 6.00000i − 0.241747i
\(617\) − 21.0000i − 0.845428i −0.906263 0.422714i \(-0.861077\pi\)
0.906263 0.422714i \(-0.138923\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) − 46.0000i − 1.84890i −0.381308 0.924448i \(-0.624526\pi\)
0.381308 0.924448i \(-0.375474\pi\)
\(620\) −5.00000 −0.200805
\(621\) −3.00000 −0.120386
\(622\) 12.0000i 0.481156i
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000i 0.319744i
\(627\) −6.00000 −0.239617
\(628\) 13.0000 0.518756
\(629\) 42.0000i 1.67465i
\(630\) − 2.00000i − 0.0796819i
\(631\) − 16.0000i − 0.636950i −0.947931 0.318475i \(-0.896829\pi\)
0.947931 0.318475i \(-0.103171\pi\)
\(632\) − 5.00000i − 0.198889i
\(633\) 20.0000 0.794929
\(634\) 12.0000 0.476581
\(635\) − 14.0000i − 0.555573i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) 12.0000i 0.474713i
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) − 6.00000i − 0.236801i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 6.00000i 0.236433i
\(645\) 1.00000i 0.0393750i
\(646\) −12.0000 −0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −27.0000 −1.05984
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 13.0000i 0.509119i
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 14.0000 0.547443
\(655\) 9.00000i 0.351659i
\(656\) − 6.00000i − 0.234261i
\(657\) 14.0000i 0.546192i
\(658\) 6.00000i 0.233904i
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) −3.00000 −0.116775
\(661\) 32.0000i 1.24466i 0.782757 + 0.622328i \(0.213813\pi\)
−0.782757 + 0.622328i \(0.786187\pi\)
\(662\) 32.0000 1.24372
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 4.00000i 0.155113i
\(666\) 7.00000 0.271244
\(667\) −9.00000 −0.348481
\(668\) 9.00000i 0.348220i
\(669\) 10.0000i 0.386622i
\(670\) 8.00000i 0.309067i
\(671\) − 6.00000i − 0.231627i
\(672\) −2.00000 −0.0771517
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) − 14.0000i − 0.539260i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 15.0000i 0.576072i
\(679\) 28.0000 1.07454
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) − 15.0000i − 0.574380i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) 2.00000i 0.0764719i
\(685\) −9.00000 −0.343872
\(686\) −20.0000 −0.763604
\(687\) 14.0000i 0.534133i
\(688\) 1.00000 0.0381246
\(689\) 0 0
\(690\) 3.00000 0.114208
\(691\) 46.0000i 1.74992i 0.484193 + 0.874961i \(0.339113\pi\)
−0.484193 + 0.874961i \(0.660887\pi\)
\(692\) 12.0000 0.456172
\(693\) 6.00000 0.227921
\(694\) − 30.0000i − 1.13878i
\(695\) 14.0000i 0.531050i
\(696\) − 3.00000i − 0.113715i
\(697\) 36.0000i 1.36360i
\(698\) −8.00000 −0.302804
\(699\) −21.0000 −0.794293
\(700\) 2.00000i 0.0755929i
\(701\) 21.0000 0.793159 0.396580 0.918000i \(-0.370197\pi\)
0.396580 + 0.918000i \(0.370197\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 3.00000i 0.113067i
\(705\) 3.00000 0.112987
\(706\) 30.0000 1.12906
\(707\) − 12.0000i − 0.451306i
\(708\) 9.00000i 0.338241i
\(709\) 32.0000i 1.20179i 0.799330 + 0.600893i \(0.205188\pi\)
−0.799330 + 0.600893i \(0.794812\pi\)
\(710\) − 12.0000i − 0.450352i
\(711\) 5.00000 0.187515
\(712\) −18.0000 −0.674579
\(713\) 15.0000i 0.561754i
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −3.00000 −0.112115
\(717\) − 24.0000i − 0.896296i
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 28.0000i − 1.04277i
\(722\) 15.0000i 0.558242i
\(723\) 17.0000i 0.632237i
\(724\) −16.0000 −0.594635
\(725\) −3.00000 −0.111417
\(726\) 2.00000i 0.0742270i
\(727\) 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
\(730\) − 14.0000i − 0.518163i
\(731\) −6.00000 −0.221918
\(732\) −2.00000 −0.0739221
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 20.0000i 0.738213i
\(735\) 3.00000i 0.110657i
\(736\) − 3.00000i − 0.110581i
\(737\) −24.0000 −0.884051
\(738\) 6.00000 0.220863
\(739\) − 16.0000i − 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) −7.00000 −0.257325
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) −5.00000 −0.183309
\(745\) −9.00000 −0.329734
\(746\) − 25.0000i − 0.915315i
\(747\) 6.00000i 0.219529i
\(748\) − 18.0000i − 0.658145i
\(749\) − 12.0000i − 0.438470i
\(750\) 1.00000 0.0365148
\(751\) −41.0000 −1.49611 −0.748056 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) −15.0000 −0.546630
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) − 2.00000i − 0.0727393i
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 38.0000 1.38022
\(759\) 9.00000i 0.326679i
\(760\) − 2.00000i − 0.0725476i
\(761\) − 12.0000i − 0.435000i −0.976060 0.217500i \(-0.930210\pi\)
0.976060 0.217500i \(-0.0697902\pi\)
\(762\) − 14.0000i − 0.507166i
\(763\) 28.0000 1.01367
\(764\) 12.0000 0.434145
\(765\) − 6.00000i − 0.216930i
\(766\) −21.0000 −0.758761
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 13.0000i 0.468792i 0.972141 + 0.234396i \(0.0753112\pi\)
−0.972141 + 0.234396i \(0.924689\pi\)
\(770\) −6.00000 −0.216225
\(771\) 21.0000 0.756297
\(772\) 4.00000i 0.143963i
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 1.00000i 0.0359443i
\(775\) 5.00000i 0.179605i
\(776\) −14.0000 −0.502571
\(777\) 14.0000 0.502247
\(778\) − 3.00000i − 0.107555i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 18.0000i 0.643679i
\(783\) 3.00000 0.107211
\(784\) 3.00000 0.107143
\(785\) − 13.0000i − 0.463990i
\(786\) 9.00000i 0.321019i
\(787\) 35.0000i 1.24762i 0.781578 + 0.623808i \(0.214415\pi\)
−0.781578 + 0.623808i \(0.785585\pi\)
\(788\) 24.0000i 0.854965i
\(789\) −15.0000 −0.534014
\(790\) −5.00000 −0.177892
\(791\) 30.0000i 1.06668i
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) 31.0000 1.10015
\(795\) − 6.00000i − 0.212798i
\(796\) 8.00000 0.283552
\(797\) 24.0000 0.850124 0.425062 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 18.0000i 0.636794i
\(800\) − 1.00000i − 0.0353553i
\(801\) − 18.0000i − 0.635999i
\(802\) 12.0000 0.423735
\(803\) 42.0000 1.48215
\(804\) 8.00000i 0.282138i
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 6.00000i 0.211079i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 32.0000i − 1.12367i −0.827249 0.561836i \(-0.810095\pi\)
0.827249 0.561836i \(-0.189905\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 11.0000i 0.385787i
\(814\) − 21.0000i − 0.736050i
\(815\) 13.0000 0.455370
\(816\) −6.00000 −0.210042
\(817\) − 2.00000i − 0.0699711i
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) − 27.0000i − 0.942306i −0.882051 0.471153i \(-0.843838\pi\)
0.882051 0.471153i \(-0.156162\pi\)
\(822\) −9.00000 −0.313911
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) 14.0000i 0.487713i
\(825\) 3.00000i 0.104447i
\(826\) 18.0000i 0.626300i
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) 3.00000 0.104257
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) − 6.00000i − 0.208263i
\(831\) 1.00000 0.0346896
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 14.0000i 0.484780i
\(835\) 9.00000 0.311458
\(836\) 6.00000 0.207514
\(837\) − 5.00000i − 0.172825i
\(838\) − 12.0000i − 0.414533i
\(839\) 42.0000i 1.45000i 0.688748 + 0.725001i \(0.258161\pi\)
−0.688748 + 0.725001i \(0.741839\pi\)
\(840\) 2.00000i 0.0690066i
\(841\) −20.0000 −0.689655
\(842\) −16.0000 −0.551396
\(843\) 18.0000i 0.619953i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 4.00000i 0.137442i
\(848\) −6.00000 −0.206041
\(849\) 31.0000 1.06392
\(850\) 6.00000i 0.205798i
\(851\) 21.0000i 0.719871i
\(852\) − 12.0000i − 0.411113i
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) −4.00000 −0.136877
\(855\) 2.00000 0.0683986
\(856\) 6.00000i 0.205076i
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) 0 0
\(859\) −46.0000 −1.56950 −0.784750 0.619813i \(-0.787209\pi\)
−0.784750 + 0.619813i \(0.787209\pi\)
\(860\) − 1.00000i − 0.0340997i
\(861\) 12.0000 0.408959
\(862\) 12.0000 0.408722
\(863\) 45.0000i 1.53182i 0.642949 + 0.765909i \(0.277711\pi\)
−0.642949 + 0.765909i \(0.722289\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 12.0000i − 0.408012i
\(866\) 40.0000i 1.35926i
\(867\) 19.0000 0.645274
\(868\) −10.0000 −0.339422
\(869\) − 15.0000i − 0.508840i
\(870\) −3.00000 −0.101710
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) − 14.0000i − 0.473828i
\(874\) −6.00000 −0.202953
\(875\) 2.00000 0.0676123
\(876\) − 14.0000i − 0.473016i
\(877\) − 23.0000i − 0.776655i −0.921521 0.388327i \(-0.873053\pi\)
0.921521 0.388327i \(-0.126947\pi\)
\(878\) 4.00000i 0.134993i
\(879\) − 30.0000i − 1.01187i
\(880\) 3.00000 0.101130
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 19.0000 0.639401 0.319700 0.947519i \(-0.396418\pi\)
0.319700 + 0.947519i \(0.396418\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) − 36.0000i − 1.20944i
\(887\) −57.0000 −1.91387 −0.956936 0.290298i \(-0.906246\pi\)
−0.956936 + 0.290298i \(0.906246\pi\)
\(888\) −7.00000 −0.234905
\(889\) − 28.0000i − 0.939090i
\(890\) 18.0000i 0.603361i
\(891\) − 3.00000i − 0.100504i
\(892\) − 10.0000i − 0.334825i
\(893\) −6.00000 −0.200782
\(894\) −9.00000 −0.301005
\(895\) 3.00000i 0.100279i
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) − 15.0000i − 0.500278i
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) − 18.0000i − 0.599334i
\(903\) 2.00000i 0.0665558i
\(904\) − 15.0000i − 0.498893i
\(905\) 16.0000i 0.531858i
\(906\) −8.00000 −0.265782
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) 18.0000 0.595713
\(914\) 2.00000 0.0661541
\(915\) 2.00000i 0.0661180i
\(916\) − 14.0000i − 0.462573i
\(917\) 18.0000i 0.594412i
\(918\) − 6.00000i − 0.198030i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −3.00000 −0.0989071
\(921\) 8.00000i 0.263609i
\(922\) 15.0000 0.493999
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 7.00000i 0.230159i
\(926\) 34.0000 1.11731
\(927\) −14.0000 −0.459820
\(928\) 3.00000i 0.0984798i
\(929\) 12.0000i 0.393707i 0.980433 + 0.196854i \(0.0630724\pi\)
−0.980433 + 0.196854i \(0.936928\pi\)
\(930\) 5.00000i 0.163956i
\(931\) − 6.00000i − 0.196642i
\(932\) 21.0000 0.687878
\(933\) 12.0000 0.392862
\(934\) − 18.0000i − 0.588978i
\(935\) −18.0000 −0.588663
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 16.0000i 0.522419i
\(939\) 8.00000 0.261070
\(940\) −3.00000 −0.0978492
\(941\) − 42.0000i − 1.36916i −0.728937 0.684580i \(-0.759985\pi\)
0.728937 0.684580i \(-0.240015\pi\)
\(942\) − 13.0000i − 0.423563i
\(943\) 18.0000i 0.586161i
\(944\) − 9.00000i − 0.292925i
\(945\) −2.00000 −0.0650600
\(946\) 3.00000 0.0975384
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) −5.00000 −0.162392
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) − 12.0000i − 0.389127i
\(952\) −12.0000 −0.388922
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) − 12.0000i − 0.388311i
\(956\) 24.0000i 0.776215i
\(957\) − 9.00000i − 0.290929i
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) − 1.00000i − 0.0322749i
\(961\) 6.00000 0.193548
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) − 17.0000i − 0.547533i
\(965\) 4.00000 0.128765
\(966\) 6.00000 0.193047
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 12.0000i 0.385496i
\(970\) 14.0000i 0.449513i
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 28.0000i 0.897639i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 57.0000i 1.82359i 0.410644 + 0.911796i \(0.365304\pi\)
−0.410644 + 0.911796i \(0.634696\pi\)
\(978\) 13.0000 0.415694
\(979\) −54.0000 −1.72585
\(980\) − 3.00000i − 0.0958315i
\(981\) − 14.0000i − 0.446986i
\(982\) 0 0
\(983\) 9.00000i 0.287055i 0.989646 + 0.143528i \(0.0458446\pi\)
−0.989646 + 0.143528i \(0.954155\pi\)
\(984\) −6.00000 −0.191273
\(985\) 24.0000 0.764704
\(986\) − 18.0000i − 0.573237i
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 3.00000i 0.0953463i
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 5.00000 0.158750
\(993\) − 32.0000i − 1.01549i
\(994\) − 24.0000i − 0.761234i
\(995\) − 8.00000i − 0.253617i
\(996\) − 6.00000i − 0.190117i
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 32.0000 1.01294
\(999\) − 7.00000i − 0.221470i
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.j.1351.2 2
13.5 odd 4 5070.2.a.v.1.1 1
13.7 odd 12 390.2.i.c.211.1 yes 2
13.8 odd 4 5070.2.a.j.1.1 1
13.11 odd 12 390.2.i.c.61.1 2
13.12 even 2 inner 5070.2.b.j.1351.1 2
39.11 even 12 1170.2.i.d.451.1 2
39.20 even 12 1170.2.i.d.991.1 2
65.7 even 12 1950.2.z.k.1849.1 4
65.24 odd 12 1950.2.i.n.451.1 2
65.33 even 12 1950.2.z.k.1849.2 4
65.37 even 12 1950.2.z.k.1699.2 4
65.59 odd 12 1950.2.i.n.601.1 2
65.63 even 12 1950.2.z.k.1699.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.c.61.1 2 13.11 odd 12
390.2.i.c.211.1 yes 2 13.7 odd 12
1170.2.i.d.451.1 2 39.11 even 12
1170.2.i.d.991.1 2 39.20 even 12
1950.2.i.n.451.1 2 65.24 odd 12
1950.2.i.n.601.1 2 65.59 odd 12
1950.2.z.k.1699.1 4 65.63 even 12
1950.2.z.k.1699.2 4 65.37 even 12
1950.2.z.k.1849.1 4 65.7 even 12
1950.2.z.k.1849.2 4 65.33 even 12
5070.2.a.j.1.1 1 13.8 odd 4
5070.2.a.v.1.1 1 13.5 odd 4
5070.2.b.j.1351.1 2 13.12 even 2 inner
5070.2.b.j.1351.2 2 1.1 even 1 trivial