Properties

Label 5070.2.b.h.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.h.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -5.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +5.00000 q^{22} +1.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +5.00000 q^{29} -1.00000 q^{30} +11.0000i q^{31} +1.00000i q^{32} +5.00000i q^{33} +2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +3.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.0000 q^{43} +5.00000i q^{44} -1.00000i q^{45} +1.00000i q^{46} +9.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000i q^{54} -5.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} +5.00000i q^{58} -15.0000i q^{59} -1.00000i q^{60} +10.0000 q^{61} -11.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} -16.0000i q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -6.00000i q^{73} -3.00000 q^{74} +1.00000 q^{75} -2.00000i q^{76} -10.0000 q^{77} -11.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -2.00000 q^{82} -6.00000i q^{83} -2.00000i q^{84} -2.00000i q^{85} +11.0000i q^{86} -5.00000 q^{87} -5.00000 q^{88} +2.00000i q^{89} +1.00000 q^{90} -1.00000 q^{92} -11.0000i q^{93} -9.00000 q^{94} +2.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +3.00000i q^{98} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} + 4q^{17} + 10q^{22} + 2q^{23} - 2q^{25} - 2q^{27} + 10q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} - 4q^{42} + 22q^{43} - 2q^{48} + 6q^{49} - 4q^{51} + 12q^{53} - 10q^{55} - 4q^{56} + 20q^{61} - 22q^{62} - 2q^{64} - 10q^{66} - 4q^{68} - 2q^{69} - 6q^{74} + 2q^{75} - 20q^{77} - 22q^{79} + 2q^{81} - 4q^{82} - 10q^{87} - 10q^{88} + 2q^{90} - 2q^{92} - 18q^{94} + 4q^{95} + O(q^{100}) \)

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 \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 5.00000i − 1.50756i −0.657129 0.753778i \(-0.728229\pi\)
0.657129 0.753778i \(-0.271771\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 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
\(22\) 5.00000 1.06600
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000i 0.377964i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −1.00000 −0.182574
\(31\) 11.0000i 1.97566i 0.155543 + 0.987829i \(0.450287\pi\)
−0.155543 + 0.987829i \(0.549713\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.00000i 0.870388i
\(34\) 2.00000i 0.342997i
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) −2.00000 −0.308607
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 5.00000i 0.753778i
\(45\) − 1.00000i − 0.149071i
\(46\) 1.00000i 0.147442i
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) − 1.00000i − 0.141421i
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −5.00000 −0.674200
\(56\) −2.00000 −0.267261
\(57\) − 2.00000i − 0.264906i
\(58\) 5.00000i 0.656532i
\(59\) − 15.0000i − 1.95283i −0.215894 0.976417i \(-0.569267\pi\)
0.215894 0.976417i \(-0.430733\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −11.0000 −1.39700
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) − 16.0000i − 1.95471i −0.211604 0.977356i \(-0.567869\pi\)
0.211604 0.977356i \(-0.432131\pi\)
\(68\) −2.00000 −0.242536
\(69\) −1.00000 −0.120386
\(70\) − 2.00000i − 0.239046i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −3.00000 −0.348743
\(75\) 1.00000 0.115470
\(76\) − 2.00000i − 0.229416i
\(77\) −10.0000 −1.13961
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) − 2.00000i − 0.218218i
\(85\) − 2.00000i − 0.216930i
\(86\) 11.0000i 1.18616i
\(87\) −5.00000 −0.536056
\(88\) −5.00000 −0.533002
\(89\) 2.00000i 0.212000i 0.994366 + 0.106000i \(0.0338043\pi\)
−0.994366 + 0.106000i \(0.966196\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) − 11.0000i − 1.14065i
\(94\) −9.00000 −0.928279
\(95\) 2.00000 0.205196
\(96\) − 1.00000i − 0.102062i
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 3.00000i 0.303046i
\(99\) − 5.00000i − 0.502519i
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 6.00000i 0.582772i
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) − 5.00000i − 0.476731i
\(111\) − 3.00000i − 0.284747i
\(112\) − 2.00000i − 0.188982i
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) 2.00000 0.187317
\(115\) − 1.00000i − 0.0932505i
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) 15.0000 1.38086
\(119\) − 4.00000i − 0.366679i
\(120\) 1.00000 0.0912871
\(121\) −14.0000 −1.27273
\(122\) 10.0000i 0.905357i
\(123\) − 2.00000i − 0.180334i
\(124\) − 11.0000i − 0.987829i
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000 0.178174
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −11.0000 −0.968496
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) − 5.00000i − 0.435194i
\(133\) 4.00000 0.346844
\(134\) 16.0000 1.38219
\(135\) 1.00000i 0.0860663i
\(136\) − 2.00000i − 0.171499i
\(137\) − 11.0000i − 0.939793i −0.882721 0.469897i \(-0.844291\pi\)
0.882721 0.469897i \(-0.155709\pi\)
\(138\) − 1.00000i − 0.0851257i
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 2.00000 0.169031
\(141\) − 9.00000i − 0.757937i
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 5.00000i − 0.415227i
\(146\) 6.00000 0.496564
\(147\) −3.00000 −0.247436
\(148\) − 3.00000i − 0.246598i
\(149\) − 17.0000i − 1.39269i −0.717705 0.696347i \(-0.754807\pi\)
0.717705 0.696347i \(-0.245193\pi\)
\(150\) 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
\(153\) 2.00000 0.161690
\(154\) − 10.0000i − 0.805823i
\(155\) 11.0000 0.883541
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) − 11.0000i − 0.875113i
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) − 2.00000i − 0.157622i
\(162\) 1.00000i 0.0785674i
\(163\) − 15.0000i − 1.17489i −0.809264 0.587445i \(-0.800134\pi\)
0.809264 0.587445i \(-0.199866\pi\)
\(164\) − 2.00000i − 0.156174i
\(165\) 5.00000 0.389249
\(166\) 6.00000 0.465690
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 2.00000 0.153393
\(171\) 2.00000i 0.152944i
\(172\) −11.0000 −0.838742
\(173\) −20.0000 −1.52057 −0.760286 0.649589i \(-0.774941\pi\)
−0.760286 + 0.649589i \(0.774941\pi\)
\(174\) − 5.00000i − 0.379049i
\(175\) 2.00000i 0.151186i
\(176\) − 5.00000i − 0.376889i
\(177\) 15.0000i 1.12747i
\(178\) −2.00000 −0.149906
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 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
\(183\) −10.0000 −0.739221
\(184\) − 1.00000i − 0.0737210i
\(185\) 3.00000 0.220564
\(186\) 11.0000 0.806559
\(187\) − 10.0000i − 0.731272i
\(188\) − 9.00000i − 0.656392i
\(189\) 2.00000i 0.145479i
\(190\) 2.00000i 0.145095i
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 24.0000i − 1.72756i −0.503871 0.863779i \(-0.668091\pi\)
0.503871 0.863779i \(-0.331909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 8.00000i 0.569976i 0.958531 + 0.284988i \(0.0919897\pi\)
−0.958531 + 0.284988i \(0.908010\pi\)
\(198\) 5.00000 0.355335
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 16.0000i 1.12855i
\(202\) − 2.00000i − 0.140720i
\(203\) − 10.0000i − 0.701862i
\(204\) 2.00000 0.140028
\(205\) 2.00000 0.139686
\(206\) − 10.0000i − 0.696733i
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 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
\(214\) 10.0000i 0.683586i
\(215\) − 11.0000i − 0.750194i
\(216\) 1.00000i 0.0680414i
\(217\) 22.0000 1.49346
\(218\) −2.00000 −0.135457
\(219\) 6.00000i 0.405442i
\(220\) 5.00000 0.337100
\(221\) 0 0
\(222\) 3.00000 0.201347
\(223\) − 26.0000i − 1.74109i −0.492090 0.870544i \(-0.663767\pi\)
0.492090 0.870544i \(-0.336233\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 11.0000i 0.731709i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 2.00000i 0.132453i
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 1.00000 0.0659380
\(231\) 10.0000 0.657952
\(232\) − 5.00000i − 0.328266i
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 15.0000i 0.976417i
\(237\) 11.0000 0.714527
\(238\) 4.00000 0.259281
\(239\) 20.0000i 1.29369i 0.762620 + 0.646846i \(0.223912\pi\)
−0.762620 + 0.646846i \(0.776088\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) − 7.00000i − 0.450910i −0.974254 0.225455i \(-0.927613\pi\)
0.974254 0.225455i \(-0.0723868\pi\)
\(242\) − 14.0000i − 0.899954i
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) − 3.00000i − 0.191663i
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 11.0000 0.698501
\(249\) 6.00000i 0.380235i
\(250\) −1.00000 −0.0632456
\(251\) −25.0000 −1.57799 −0.788993 0.614402i \(-0.789397\pi\)
−0.788993 + 0.614402i \(0.789397\pi\)
\(252\) 2.00000i 0.125988i
\(253\) − 5.00000i − 0.314347i
\(254\) − 2.00000i − 0.125491i
\(255\) 2.00000i 0.125245i
\(256\) 1.00000 0.0625000
\(257\) 17.0000 1.06043 0.530215 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(258\) − 11.0000i − 0.684830i
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) − 1.00000i − 0.0617802i
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 5.00000 0.307729
\(265\) − 6.00000i − 0.368577i
\(266\) 4.00000i 0.245256i
\(267\) − 2.00000i − 0.122398i
\(268\) 16.0000i 0.977356i
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 13.0000i − 0.789694i −0.918747 0.394847i \(-0.870798\pi\)
0.918747 0.394847i \(-0.129202\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 11.0000 0.664534
\(275\) 5.00000i 0.301511i
\(276\) 1.00000 0.0601929
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 2.00000i 0.119952i
\(279\) 11.0000i 0.658553i
\(280\) 2.00000i 0.119523i
\(281\) − 10.0000i − 0.596550i −0.954480 0.298275i \(-0.903589\pi\)
0.954480 0.298275i \(-0.0964112\pi\)
\(282\) 9.00000 0.535942
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 1.00000i 0.0589256i
\(289\) −13.0000 −0.764706
\(290\) 5.00000 0.293610
\(291\) − 2.00000i − 0.117242i
\(292\) 6.00000i 0.351123i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) −15.0000 −0.873334
\(296\) 3.00000 0.174371
\(297\) 5.00000i 0.290129i
\(298\) 17.0000 0.984784
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 22.0000i − 1.26806i
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) 2.00000i 0.114708i
\(305\) − 10.0000i − 0.572598i
\(306\) 2.00000i 0.114332i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 10.0000 0.569803
\(309\) 10.0000 0.568880
\(310\) 11.0000i 0.624758i
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) − 7.00000i − 0.395033i
\(315\) −2.00000 −0.112687
\(316\) 11.0000 0.618798
\(317\) − 16.0000i − 0.898650i −0.893368 0.449325i \(-0.851665\pi\)
0.893368 0.449325i \(-0.148335\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) − 25.0000i − 1.39973i
\(320\) 1.00000i 0.0559017i
\(321\) −10.0000 −0.558146
\(322\) 2.00000 0.111456
\(323\) 4.00000i 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 15.0000 0.830773
\(327\) − 2.00000i − 0.110600i
\(328\) 2.00000 0.110432
\(329\) 18.0000 0.992372
\(330\) 5.00000i 0.275241i
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 3.00000i 0.164399i
\(334\) −3.00000 −0.164153
\(335\) −16.0000 −0.874173
\(336\) 2.00000i 0.109109i
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −11.0000 −0.597438
\(340\) 2.00000i 0.108465i
\(341\) 55.0000 2.97842
\(342\) −2.00000 −0.108148
\(343\) − 20.0000i − 1.07990i
\(344\) − 11.0000i − 0.593080i
\(345\) 1.00000i 0.0538382i
\(346\) − 20.0000i − 1.07521i
\(347\) 34.0000 1.82522 0.912608 0.408836i \(-0.134065\pi\)
0.912608 + 0.408836i \(0.134065\pi\)
\(348\) 5.00000 0.268028
\(349\) 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
\(352\) 5.00000 0.266501
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) −15.0000 −0.797241
\(355\) 0 0
\(356\) − 2.00000i − 0.106000i
\(357\) 4.00000i 0.211702i
\(358\) 13.0000i 0.687071i
\(359\) 12.0000i 0.633336i 0.948536 + 0.316668i \(0.102564\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 15.0000 0.789474
\(362\) 16.0000i 0.840941i
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) − 10.0000i − 0.522708i
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000i 0.104116i
\(370\) 3.00000i 0.155963i
\(371\) − 12.0000i − 0.623009i
\(372\) 11.0000i 0.570323i
\(373\) −19.0000 −0.983783 −0.491891 0.870657i \(-0.663694\pi\)
−0.491891 + 0.870657i \(0.663694\pi\)
\(374\) 10.0000 0.517088
\(375\) − 1.00000i − 0.0516398i
\(376\) 9.00000 0.464140
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) −2.00000 −0.102598
\(381\) 2.00000 0.102463
\(382\) 4.00000i 0.204658i
\(383\) − 31.0000i − 1.58403i −0.610504 0.792013i \(-0.709033\pi\)
0.610504 0.792013i \(-0.290967\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 10.0000i 0.509647i
\(386\) 24.0000 1.22157
\(387\) 11.0000 0.559161
\(388\) − 2.00000i − 0.101535i
\(389\) −5.00000 −0.253510 −0.126755 0.991934i \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) − 3.00000i − 0.151523i
\(393\) 1.00000 0.0504433
\(394\) −8.00000 −0.403034
\(395\) 11.0000i 0.553470i
\(396\) 5.00000i 0.251259i
\(397\) 3.00000i 0.150566i 0.997162 + 0.0752828i \(0.0239860\pi\)
−0.997162 + 0.0752828i \(0.976014\pi\)
\(398\) − 24.0000i − 1.20301i
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 36.0000i 1.79775i 0.438201 + 0.898877i \(0.355616\pi\)
−0.438201 + 0.898877i \(0.644384\pi\)
\(402\) −16.0000 −0.798007
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) − 1.00000i − 0.0496904i
\(406\) 10.0000 0.496292
\(407\) 15.0000 0.743522
\(408\) 2.00000i 0.0990148i
\(409\) − 30.0000i − 1.48340i −0.670729 0.741702i \(-0.734019\pi\)
0.670729 0.741702i \(-0.265981\pi\)
\(410\) 2.00000i 0.0987730i
\(411\) 11.0000i 0.542590i
\(412\) 10.0000 0.492665
\(413\) −30.0000 −1.47620
\(414\) 1.00000i 0.0491473i
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −2.00000 −0.0979404
\(418\) 10.0000i 0.489116i
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 16.0000i 0.779792i 0.920859 + 0.389896i \(0.127489\pi\)
−0.920859 + 0.389896i \(0.872511\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 9.00000i 0.437595i
\(424\) − 6.00000i − 0.291386i
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) − 20.0000i − 0.967868i
\(428\) −10.0000 −0.483368
\(429\) 0 0
\(430\) 11.0000 0.530467
\(431\) − 40.0000i − 1.92673i −0.268190 0.963366i \(-0.586425\pi\)
0.268190 0.963366i \(-0.413575\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 22.0000i 1.05603i
\(435\) 5.00000i 0.239732i
\(436\) − 2.00000i − 0.0957826i
\(437\) 2.00000i 0.0956730i
\(438\) −6.00000 −0.286691
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 5.00000i 0.238366i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 3.00000i 0.142374i
\(445\) 2.00000 0.0948091
\(446\) 26.0000 1.23114
\(447\) 17.0000i 0.804072i
\(448\) 2.00000i 0.0944911i
\(449\) 12.0000i 0.566315i 0.959073 + 0.283158i \(0.0913819\pi\)
−0.959073 + 0.283158i \(0.908618\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 10.0000 0.470882
\(452\) −11.0000 −0.517396
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 1.00000i 0.0466252i
\(461\) 39.0000i 1.81641i 0.418524 + 0.908206i \(0.362547\pi\)
−0.418524 + 0.908206i \(0.637453\pi\)
\(462\) 10.0000i 0.465242i
\(463\) 14.0000i 0.650635i 0.945605 + 0.325318i \(0.105471\pi\)
−0.945605 + 0.325318i \(0.894529\pi\)
\(464\) 5.00000 0.232119
\(465\) −11.0000 −0.510113
\(466\) − 1.00000i − 0.0463241i
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 9.00000i 0.415139i
\(471\) 7.00000 0.322543
\(472\) −15.0000 −0.690431
\(473\) − 55.0000i − 2.52890i
\(474\) 11.0000i 0.505247i
\(475\) − 2.00000i − 0.0917663i
\(476\) 4.00000i 0.183340i
\(477\) 6.00000 0.274721
\(478\) −20.0000 −0.914779
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 7.00000 0.318841
\(483\) 2.00000i 0.0910032i
\(484\) 14.0000 0.636364
\(485\) 2.00000 0.0908153
\(486\) − 1.00000i − 0.0453609i
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 15.0000i 0.678323i
\(490\) 3.00000 0.135526
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 10.0000 0.450377
\(494\) 0 0
\(495\) −5.00000 −0.224733
\(496\) 11.0000i 0.493915i
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 3.00000i − 0.134030i
\(502\) − 25.0000i − 1.11580i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 2.00000i 0.0889988i
\(506\) 5.00000 0.222277
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) − 29.0000i − 1.28540i −0.766117 0.642701i \(-0.777814\pi\)
0.766117 0.642701i \(-0.222186\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −12.0000 −0.530849
\(512\) 1.00000i 0.0441942i
\(513\) − 2.00000i − 0.0883022i
\(514\) 17.0000i 0.749838i
\(515\) 10.0000i 0.440653i
\(516\) 11.0000 0.484248
\(517\) 45.0000 1.97910
\(518\) 6.00000i 0.263625i
\(519\) 20.0000 0.877903
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 5.00000i 0.218844i
\(523\) −31.0000 −1.35554 −0.677768 0.735276i \(-0.737052\pi\)
−0.677768 + 0.735276i \(0.737052\pi\)
\(524\) 1.00000 0.0436852
\(525\) − 2.00000i − 0.0872872i
\(526\) 21.0000i 0.915644i
\(527\) 22.0000i 0.958335i
\(528\) 5.00000i 0.217597i
\(529\) −22.0000 −0.956522
\(530\) 6.00000 0.260623
\(531\) − 15.0000i − 0.650945i
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 2.00000 0.0865485
\(535\) − 10.0000i − 0.432338i
\(536\) −16.0000 −0.691095
\(537\) −13.0000 −0.560991
\(538\) 14.0000i 0.603583i
\(539\) − 15.0000i − 0.646096i
\(540\) − 1.00000i − 0.0430331i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 13.0000 0.558398
\(543\) −16.0000 −0.686626
\(544\) 2.00000i 0.0857493i
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 11.0000i 0.469897i
\(549\) 10.0000 0.426790
\(550\) −5.00000 −0.213201
\(551\) 10.0000i 0.426014i
\(552\) 1.00000i 0.0425628i
\(553\) 22.0000i 0.935535i
\(554\) 11.0000i 0.467345i
\(555\) −3.00000 −0.127343
\(556\) −2.00000 −0.0848189
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) −11.0000 −0.465667
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 10.0000i 0.422200i
\(562\) 10.0000 0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 9.00000i 0.378968i
\(565\) − 11.0000i − 0.462773i
\(566\) − 19.0000i − 0.798630i
\(567\) − 2.00000i − 0.0839921i
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) − 2.00000i − 0.0837708i
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 4.00000i 0.166957i
\(575\) −1.00000 −0.0417029
\(576\) −1.00000 −0.0416667
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 24.0000i 0.997406i
\(580\) 5.00000i 0.207614i
\(581\) −12.0000 −0.497844
\(582\) 2.00000 0.0829027
\(583\) − 30.0000i − 1.24247i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 3.00000 0.123718
\(589\) −22.0000 −0.906494
\(590\) − 15.0000i − 0.617540i
\(591\) − 8.00000i − 0.329076i
\(592\) 3.00000i 0.123299i
\(593\) 31.0000i 1.27302i 0.771270 + 0.636509i \(0.219622\pi\)
−0.771270 + 0.636509i \(0.780378\pi\)
\(594\) −5.00000 −0.205152
\(595\) −4.00000 −0.163984
\(596\) 17.0000i 0.696347i
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 22.0000 0.896653
\(603\) − 16.0000i − 0.651570i
\(604\) − 8.00000i − 0.325515i
\(605\) 14.0000i 0.569181i
\(606\) 2.00000i 0.0812444i
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 10.0000i 0.405220i
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 29.0000i 1.17130i 0.810564 + 0.585649i \(0.199160\pi\)
−0.810564 + 0.585649i \(0.800840\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 10.0000i 0.402911i
\(617\) − 41.0000i − 1.65060i −0.564696 0.825299i \(-0.691007\pi\)
0.564696 0.825299i \(-0.308993\pi\)
\(618\) 10.0000i 0.402259i
\(619\) − 10.0000i − 0.401934i −0.979598 0.200967i \(-0.935592\pi\)
0.979598 0.200967i \(-0.0644084\pi\)
\(620\) −11.0000 −0.441771
\(621\) −1.00000 −0.0401286
\(622\) − 20.0000i − 0.801927i
\(623\) 4.00000 0.160257
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 20.0000i 0.799361i
\(627\) −10.0000 −0.399362
\(628\) 7.00000 0.279330
\(629\) 6.00000i 0.239236i
\(630\) − 2.00000i − 0.0796819i
\(631\) 40.0000i 1.59237i 0.605050 + 0.796187i \(0.293153\pi\)
−0.605050 + 0.796187i \(0.706847\pi\)
\(632\) 11.0000i 0.437557i
\(633\) 4.00000 0.158986
\(634\) 16.0000 0.635441
\(635\) 2.00000i 0.0793676i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 25.0000 0.989759
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) − 10.0000i − 0.394669i
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) 2.00000i 0.0788110i
\(645\) 11.0000i 0.433125i
\(646\) −4.00000 −0.157378
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −75.0000 −2.94401
\(650\) 0 0
\(651\) −22.0000 −0.862248
\(652\) 15.0000i 0.587445i
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 2.00000 0.0782062
\(655\) 1.00000i 0.0390732i
\(656\) 2.00000i 0.0780869i
\(657\) − 6.00000i − 0.234082i
\(658\) 18.0000i 0.701713i
\(659\) −39.0000 −1.51922 −0.759612 0.650376i \(-0.774611\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(660\) −5.00000 −0.194625
\(661\) 32.0000i 1.24466i 0.782757 + 0.622328i \(0.213813\pi\)
−0.782757 + 0.622328i \(0.786187\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) − 4.00000i − 0.155113i
\(666\) −3.00000 −0.116248
\(667\) 5.00000 0.193601
\(668\) − 3.00000i − 0.116073i
\(669\) 26.0000i 1.00522i
\(670\) − 16.0000i − 0.618134i
\(671\) − 50.0000i − 1.93023i
\(672\) −2.00000 −0.0771517
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 2.00000i 0.0770371i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.00000 −0.153732 −0.0768662 0.997041i \(-0.524491\pi\)
−0.0768662 + 0.997041i \(0.524491\pi\)
\(678\) − 11.0000i − 0.422452i
\(679\) 4.00000 0.153506
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 55.0000i 2.10606i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) − 2.00000i − 0.0764719i
\(685\) −11.0000 −0.420288
\(686\) 20.0000 0.763604
\(687\) − 10.0000i − 0.381524i
\(688\) 11.0000 0.419371
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) 30.0000i 1.14125i 0.821209 + 0.570627i \(0.193300\pi\)
−0.821209 + 0.570627i \(0.806700\pi\)
\(692\) 20.0000 0.760286
\(693\) −10.0000 −0.379869
\(694\) 34.0000i 1.29062i
\(695\) − 2.00000i − 0.0758643i
\(696\) 5.00000i 0.189525i
\(697\) 4.00000i 0.151511i
\(698\) −20.0000 −0.757011
\(699\) 1.00000 0.0378235
\(700\) − 2.00000i − 0.0755929i
\(701\) −13.0000 −0.491003 −0.245502 0.969396i \(-0.578953\pi\)
−0.245502 + 0.969396i \(0.578953\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) 5.00000i 0.188445i
\(705\) −9.00000 −0.338960
\(706\) −18.0000 −0.677439
\(707\) 4.00000i 0.150435i
\(708\) − 15.0000i − 0.563735i
\(709\) − 8.00000i − 0.300446i −0.988652 0.150223i \(-0.952001\pi\)
0.988652 0.150223i \(-0.0479992\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 2.00000 0.0749532
\(713\) 11.0000i 0.411953i
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −13.0000 −0.485833
\(717\) − 20.0000i − 0.746914i
\(718\) −12.0000 −0.447836
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 20.0000i 0.744839i
\(722\) 15.0000i 0.558242i
\(723\) 7.00000i 0.260333i
\(724\) −16.0000 −0.594635
\(725\) −5.00000 −0.185695
\(726\) 14.0000i 0.519589i
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 6.00000i − 0.222070i
\(731\) 22.0000 0.813699
\(732\) 10.0000 0.369611
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) − 16.0000i − 0.590571i
\(735\) 3.00000i 0.110657i
\(736\) 1.00000i 0.0368605i
\(737\) −80.0000 −2.94684
\(738\) −2.00000 −0.0736210
\(739\) 44.0000i 1.61857i 0.587419 + 0.809283i \(0.300144\pi\)
−0.587419 + 0.809283i \(0.699856\pi\)
\(740\) −3.00000 −0.110282
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) − 51.0000i − 1.87101i −0.353315 0.935504i \(-0.614946\pi\)
0.353315 0.935504i \(-0.385054\pi\)
\(744\) −11.0000 −0.403280
\(745\) −17.0000 −0.622832
\(746\) − 19.0000i − 0.695639i
\(747\) − 6.00000i − 0.219529i
\(748\) 10.0000i 0.365636i
\(749\) − 20.0000i − 0.730784i
\(750\) 1.00000 0.0365148
\(751\) 23.0000 0.839282 0.419641 0.907690i \(-0.362156\pi\)
0.419641 + 0.907690i \(0.362156\pi\)
\(752\) 9.00000i 0.328196i
\(753\) 25.0000 0.911051
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) − 2.00000i − 0.0727393i
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 5.00000i 0.181489i
\(760\) − 2.00000i − 0.0725476i
\(761\) 20.0000i 0.724999i 0.931984 + 0.362500i \(0.118077\pi\)
−0.931984 + 0.362500i \(0.881923\pi\)
\(762\) 2.00000i 0.0724524i
\(763\) 4.00000 0.144810
\(764\) −4.00000 −0.144715
\(765\) − 2.00000i − 0.0723102i
\(766\) 31.0000 1.12008
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 43.0000i − 1.55062i −0.631581 0.775310i \(-0.717594\pi\)
0.631581 0.775310i \(-0.282406\pi\)
\(770\) −10.0000 −0.360375
\(771\) −17.0000 −0.612240
\(772\) 24.0000i 0.863779i
\(773\) − 32.0000i − 1.15096i −0.817816 0.575480i \(-0.804815\pi\)
0.817816 0.575480i \(-0.195185\pi\)
\(774\) 11.0000i 0.395387i
\(775\) − 11.0000i − 0.395132i
\(776\) 2.00000 0.0717958
\(777\) −6.00000 −0.215249
\(778\) − 5.00000i − 0.179259i
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) 2.00000i 0.0715199i
\(783\) −5.00000 −0.178685
\(784\) 3.00000 0.107143
\(785\) 7.00000i 0.249841i
\(786\) 1.00000i 0.0356688i
\(787\) − 31.0000i − 1.10503i −0.833503 0.552515i \(-0.813668\pi\)
0.833503 0.552515i \(-0.186332\pi\)
\(788\) − 8.00000i − 0.284988i
\(789\) −21.0000 −0.747620
\(790\) −11.0000 −0.391362
\(791\) − 22.0000i − 0.782230i
\(792\) −5.00000 −0.177667
\(793\) 0 0
\(794\) −3.00000 −0.106466
\(795\) 6.00000i 0.212798i
\(796\) 24.0000 0.850657
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 18.0000i 0.636794i
\(800\) − 1.00000i − 0.0353553i
\(801\) 2.00000i 0.0706665i
\(802\) −36.0000 −1.27120
\(803\) −30.0000 −1.05868
\(804\) − 16.0000i − 0.564276i
\(805\) −2.00000 −0.0704907
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 2.00000i 0.0703598i
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 20.0000i − 0.702295i −0.936320 0.351147i \(-0.885792\pi\)
0.936320 0.351147i \(-0.114208\pi\)
\(812\) 10.0000i 0.350931i
\(813\) 13.0000i 0.455930i
\(814\) 15.0000i 0.525750i
\(815\) −15.0000 −0.525427
\(816\) −2.00000 −0.0700140
\(817\) 22.0000i 0.769683i
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 51.0000i 1.77991i 0.456046 + 0.889956i \(0.349265\pi\)
−0.456046 + 0.889956i \(0.650735\pi\)
\(822\) −11.0000 −0.383669
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) 10.0000i 0.348367i
\(825\) − 5.00000i − 0.174078i
\(826\) − 30.0000i − 1.04383i
\(827\) − 50.0000i − 1.73867i −0.494223 0.869335i \(-0.664547\pi\)
0.494223 0.869335i \(-0.335453\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) − 6.00000i − 0.208263i
\(831\) −11.0000 −0.381586
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) − 2.00000i − 0.0692543i
\(835\) 3.00000 0.103819
\(836\) −10.0000 −0.345857
\(837\) − 11.0000i − 0.380216i
\(838\) − 4.00000i − 0.138178i
\(839\) − 54.0000i − 1.86429i −0.362089 0.932144i \(-0.617936\pi\)
0.362089 0.932144i \(-0.382064\pi\)
\(840\) − 2.00000i − 0.0690066i
\(841\) −4.00000 −0.137931
\(842\) −16.0000 −0.551396
\(843\) 10.0000i 0.344418i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) −9.00000 −0.309426
\(847\) 28.0000i 0.962091i
\(848\) 6.00000 0.206041
\(849\) 19.0000 0.652078
\(850\) − 2.00000i − 0.0685994i
\(851\) 3.00000i 0.102839i
\(852\) 0 0
\(853\) − 49.0000i − 1.67773i −0.544341 0.838864i \(-0.683220\pi\)
0.544341 0.838864i \(-0.316780\pi\)
\(854\) 20.0000 0.684386
\(855\) 2.00000 0.0683986
\(856\) − 10.0000i − 0.341793i
\(857\) 25.0000 0.853984 0.426992 0.904255i \(-0.359573\pi\)
0.426992 + 0.904255i \(0.359573\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 11.0000i 0.375097i
\(861\) −4.00000 −0.136320
\(862\) 40.0000 1.36241
\(863\) − 39.0000i − 1.32758i −0.747921 0.663788i \(-0.768948\pi\)
0.747921 0.663788i \(-0.231052\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 20.0000i 0.680020i
\(866\) − 28.0000i − 0.951479i
\(867\) 13.0000 0.441503
\(868\) −22.0000 −0.746729
\(869\) 55.0000i 1.86575i
\(870\) −5.00000 −0.169516
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 2.00000i 0.0676897i
\(874\) −2.00000 −0.0676510
\(875\) 2.00000 0.0676123
\(876\) − 6.00000i − 0.202721i
\(877\) 19.0000i 0.641584i 0.947150 + 0.320792i \(0.103949\pi\)
−0.947150 + 0.320792i \(0.896051\pi\)
\(878\) 4.00000i 0.134993i
\(879\) − 6.00000i − 0.202375i
\(880\) −5.00000 −0.168550
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) 15.0000 0.504219
\(886\) 20.0000i 0.671913i
\(887\) 3.00000 0.100730 0.0503651 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) −3.00000 −0.100673
\(889\) 4.00000i 0.134156i
\(890\) 2.00000i 0.0670402i
\(891\) − 5.00000i − 0.167506i
\(892\) 26.0000i 0.870544i
\(893\) −18.0000 −0.602347
\(894\) −17.0000 −0.568565
\(895\) − 13.0000i − 0.434542i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) 55.0000i 1.83435i
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 10.0000i 0.332964i
\(903\) 22.0000i 0.732114i
\(904\) − 11.0000i − 0.365855i
\(905\) − 16.0000i − 0.531858i
\(906\) 8.00000 0.265782
\(907\) −19.0000 −0.630885 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) − 2.00000i − 0.0662266i
\(913\) −30.0000 −0.992855
\(914\) 38.0000 1.25693
\(915\) 10.0000i 0.330590i
\(916\) − 10.0000i − 0.330409i
\(917\) 2.00000i 0.0660458i
\(918\) − 2.00000i − 0.0660098i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −39.0000 −1.28440
\(923\) 0 0
\(924\) −10.0000 −0.328976
\(925\) − 3.00000i − 0.0986394i
\(926\) −14.0000 −0.460069
\(927\) −10.0000 −0.328443
\(928\) 5.00000i 0.164133i
\(929\) 16.0000i 0.524943i 0.964940 + 0.262471i \(0.0845376\pi\)
−0.964940 + 0.262471i \(0.915462\pi\)
\(930\) − 11.0000i − 0.360704i
\(931\) 6.00000i 0.196642i
\(932\) 1.00000 0.0327561
\(933\) 20.0000 0.654771
\(934\) − 6.00000i − 0.196326i
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) −50.0000 −1.63343 −0.816714 0.577042i \(-0.804207\pi\)
−0.816714 + 0.577042i \(0.804207\pi\)
\(938\) − 32.0000i − 1.04484i
\(939\) −20.0000 −0.652675
\(940\) −9.00000 −0.293548
\(941\) 50.0000i 1.62995i 0.579494 + 0.814977i \(0.303250\pi\)
−0.579494 + 0.814977i \(0.696750\pi\)
\(942\) 7.00000i 0.228072i
\(943\) 2.00000i 0.0651290i
\(944\) − 15.0000i − 0.488208i
\(945\) 2.00000 0.0650600
\(946\) 55.0000 1.78820
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) −11.0000 −0.357263
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) 16.0000i 0.518836i
\(952\) −4.00000 −0.129641
\(953\) 51.0000 1.65205 0.826026 0.563632i \(-0.190596\pi\)
0.826026 + 0.563632i \(0.190596\pi\)
\(954\) 6.00000i 0.194257i
\(955\) − 4.00000i − 0.129437i
\(956\) − 20.0000i − 0.646846i
\(957\) 25.0000i 0.808135i
\(958\) 24.0000 0.775405
\(959\) −22.0000 −0.710417
\(960\) − 1.00000i − 0.0322749i
\(961\) −90.0000 −2.90323
\(962\) 0 0
\(963\) 10.0000 0.322245
\(964\) 7.00000i 0.225455i
\(965\) −24.0000 −0.772587
\(966\) −2.00000 −0.0643489
\(967\) − 20.0000i − 0.643157i −0.946883 0.321578i \(-0.895787\pi\)
0.946883 0.321578i \(-0.104213\pi\)
\(968\) 14.0000i 0.449977i
\(969\) − 4.00000i − 0.128499i
\(970\) 2.00000i 0.0642161i
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 4.00000i − 0.128234i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 21.0000i 0.671850i 0.941889 + 0.335925i \(0.109049\pi\)
−0.941889 + 0.335925i \(0.890951\pi\)
\(978\) −15.0000 −0.479647
\(979\) 10.0000 0.319601
\(980\) 3.00000i 0.0958315i
\(981\) 2.00000i 0.0638551i
\(982\) 16.0000i 0.510581i
\(983\) − 3.00000i − 0.0956851i −0.998855 0.0478426i \(-0.984765\pi\)
0.998855 0.0478426i \(-0.0152346\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 8.00000 0.254901
\(986\) 10.0000i 0.318465i
\(987\) −18.0000 −0.572946
\(988\) 0 0
\(989\) 11.0000 0.349780
\(990\) − 5.00000i − 0.158910i
\(991\) −17.0000 −0.540023 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) −11.0000 −0.349250
\(993\) − 28.0000i − 0.888553i
\(994\) 0 0
\(995\) 24.0000i 0.760851i
\(996\) − 6.00000i − 0.190117i
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 28.0000 0.886325
\(999\) − 3.00000i − 0.0949158i
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.h.1351.2 2
13.5 odd 4 5070.2.a.p.1.1 1
13.7 odd 12 390.2.i.f.211.1 yes 2
13.8 odd 4 5070.2.a.d.1.1 1
13.11 odd 12 390.2.i.f.61.1 2
13.12 even 2 inner 5070.2.b.h.1351.1 2
39.11 even 12 1170.2.i.a.451.1 2
39.20 even 12 1170.2.i.a.991.1 2
65.7 even 12 1950.2.z.e.1849.1 4
65.24 odd 12 1950.2.i.d.451.1 2
65.33 even 12 1950.2.z.e.1849.2 4
65.37 even 12 1950.2.z.e.1699.2 4
65.59 odd 12 1950.2.i.d.601.1 2
65.63 even 12 1950.2.z.e.1699.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.f.61.1 2 13.11 odd 12
390.2.i.f.211.1 yes 2 13.7 odd 12
1170.2.i.a.451.1 2 39.11 even 12
1170.2.i.a.991.1 2 39.20 even 12
1950.2.i.d.451.1 2 65.24 odd 12
1950.2.i.d.601.1 2 65.59 odd 12
1950.2.z.e.1699.1 4 65.63 even 12
1950.2.z.e.1699.2 4 65.37 even 12
1950.2.z.e.1849.1 4 65.7 even 12
1950.2.z.e.1849.2 4 65.33 even 12
5070.2.a.d.1.1 1 13.8 odd 4
5070.2.a.p.1.1 1 13.5 odd 4
5070.2.b.h.1351.1 2 13.12 even 2 inner
5070.2.b.h.1351.2 2 1.1 even 1 trivial