Properties

Label 770.2.a.k.1.1
Level $770$
Weight $2$
Character 770.1
Self dual yes
Analytic conductor $6.148$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 770.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} -4.74456 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +4.74456 q^{17} +1.00000 q^{18} +4.74456 q^{19} +1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} +4.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} -4.74456 q^{26} -4.00000 q^{27} +1.00000 q^{28} -2.74456 q^{29} +2.00000 q^{30} -6.74456 q^{31} +1.00000 q^{32} -2.00000 q^{33} +4.74456 q^{34} +1.00000 q^{35} +1.00000 q^{36} -10.7446 q^{37} +4.74456 q^{38} -9.48913 q^{39} +1.00000 q^{40} -4.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +4.74456 q^{46} -6.74456 q^{47} +2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +9.48913 q^{51} -4.74456 q^{52} -1.25544 q^{53} -4.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +9.48913 q^{57} -2.74456 q^{58} +2.74456 q^{59} +2.00000 q^{60} +12.7446 q^{61} -6.74456 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.74456 q^{65} -2.00000 q^{66} -4.00000 q^{67} +4.74456 q^{68} +9.48913 q^{69} +1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -0.744563 q^{73} -10.7446 q^{74} +2.00000 q^{75} +4.74456 q^{76} -1.00000 q^{77} -9.48913 q^{78} -4.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} -4.00000 q^{82} +8.00000 q^{83} +2.00000 q^{84} +4.74456 q^{85} +4.00000 q^{86} -5.48913 q^{87} -1.00000 q^{88} -7.48913 q^{89} +1.00000 q^{90} -4.74456 q^{91} +4.74456 q^{92} -13.4891 q^{93} -6.74456 q^{94} +4.74456 q^{95} +2.00000 q^{96} -5.25544 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + 2 q^{13} + 2 q^{14} + 4 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + 4 q^{21} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{30} - 2 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} - 10 q^{37} - 2 q^{38} + 4 q^{39} + 2 q^{40} - 8 q^{41} + 4 q^{42} + 8 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 2 q^{47} + 4 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 2 q^{52} - 14 q^{53} - 8 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} + 6 q^{58} - 6 q^{59} + 4 q^{60} + 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 4 q^{66} - 8 q^{67} - 2 q^{68} - 4 q^{69} + 2 q^{70} - 8 q^{71} + 2 q^{72} + 10 q^{73} - 10 q^{74} + 4 q^{75} - 2 q^{76} - 2 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} - 8 q^{82} + 16 q^{83} + 4 q^{84} - 2 q^{85} + 8 q^{86} + 12 q^{87} - 2 q^{88} + 8 q^{89} + 2 q^{90} + 2 q^{91} - 2 q^{92} - 4 q^{93} - 2 q^{94} - 2 q^{95} + 4 q^{96} - 22 q^{97} + 2 q^{98} - 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 4.74456 1.15073 0.575363 0.817898i \(-0.304861\pi\)
0.575363 + 0.817898i \(0.304861\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.74456 1.08848 0.544239 0.838930i \(-0.316819\pi\)
0.544239 + 0.838930i \(0.316819\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −1.00000 −0.213201
\(23\) 4.74456 0.989310 0.494655 0.869090i \(-0.335294\pi\)
0.494655 + 0.869090i \(0.335294\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −4.74456 −0.930485
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −2.74456 −0.509652 −0.254826 0.966987i \(-0.582018\pi\)
−0.254826 + 0.966987i \(0.582018\pi\)
\(30\) 2.00000 0.365148
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 4.74456 0.813686
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −10.7446 −1.76640 −0.883198 0.469001i \(-0.844614\pi\)
−0.883198 + 0.469001i \(0.844614\pi\)
\(38\) 4.74456 0.769670
\(39\) −9.48913 −1.51948
\(40\) 1.00000 0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 2.00000 0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 4.74456 0.699548
\(47\) −6.74456 −0.983796 −0.491898 0.870653i \(-0.663697\pi\)
−0.491898 + 0.870653i \(0.663697\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 9.48913 1.32874
\(52\) −4.74456 −0.657952
\(53\) −1.25544 −0.172448 −0.0862238 0.996276i \(-0.527480\pi\)
−0.0862238 + 0.996276i \(0.527480\pi\)
\(54\) −4.00000 −0.544331
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 9.48913 1.25687
\(58\) −2.74456 −0.360379
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 2.00000 0.258199
\(61\) 12.7446 1.63177 0.815887 0.578211i \(-0.196249\pi\)
0.815887 + 0.578211i \(0.196249\pi\)
\(62\) −6.74456 −0.856560
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −4.74456 −0.588491
\(66\) −2.00000 −0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 4.74456 0.575363
\(69\) 9.48913 1.14236
\(70\) 1.00000 0.119523
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.744563 −0.0871445 −0.0435722 0.999050i \(-0.513874\pi\)
−0.0435722 + 0.999050i \(0.513874\pi\)
\(74\) −10.7446 −1.24903
\(75\) 2.00000 0.230940
\(76\) 4.74456 0.544239
\(77\) −1.00000 −0.113961
\(78\) −9.48913 −1.07443
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −4.00000 −0.441726
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 2.00000 0.218218
\(85\) 4.74456 0.514620
\(86\) 4.00000 0.431331
\(87\) −5.48913 −0.588496
\(88\) −1.00000 −0.106600
\(89\) −7.48913 −0.793846 −0.396923 0.917852i \(-0.629922\pi\)
−0.396923 + 0.917852i \(0.629922\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.74456 −0.497365
\(92\) 4.74456 0.494655
\(93\) −13.4891 −1.39876
\(94\) −6.74456 −0.695649
\(95\) 4.74456 0.486782
\(96\) 2.00000 0.204124
\(97\) −5.25544 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −8.74456 −0.870117 −0.435058 0.900402i \(-0.643272\pi\)
−0.435058 + 0.900402i \(0.643272\pi\)
\(102\) 9.48913 0.939563
\(103\) 10.7446 1.05869 0.529347 0.848406i \(-0.322437\pi\)
0.529347 + 0.848406i \(0.322437\pi\)
\(104\) −4.74456 −0.465243
\(105\) 2.00000 0.195180
\(106\) −1.25544 −0.121939
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) 16.2337 1.55491 0.777453 0.628941i \(-0.216511\pi\)
0.777453 + 0.628941i \(0.216511\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −21.4891 −2.03966
\(112\) 1.00000 0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 9.48913 0.888738
\(115\) 4.74456 0.442433
\(116\) −2.74456 −0.254826
\(117\) −4.74456 −0.438635
\(118\) 2.74456 0.252657
\(119\) 4.74456 0.434933
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 12.7446 1.15384
\(123\) −8.00000 −0.721336
\(124\) −6.74456 −0.605680
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) −4.74456 −0.416126
\(131\) 8.74456 0.764016 0.382008 0.924159i \(-0.375233\pi\)
0.382008 + 0.924159i \(0.375233\pi\)
\(132\) −2.00000 −0.174078
\(133\) 4.74456 0.411406
\(134\) −4.00000 −0.345547
\(135\) −4.00000 −0.344265
\(136\) 4.74456 0.406843
\(137\) −19.4891 −1.66507 −0.832534 0.553974i \(-0.813111\pi\)
−0.832534 + 0.553974i \(0.813111\pi\)
\(138\) 9.48913 0.807768
\(139\) 3.25544 0.276123 0.138061 0.990424i \(-0.455913\pi\)
0.138061 + 0.990424i \(0.455913\pi\)
\(140\) 1.00000 0.0845154
\(141\) −13.4891 −1.13599
\(142\) −4.00000 −0.335673
\(143\) 4.74456 0.396760
\(144\) 1.00000 0.0833333
\(145\) −2.74456 −0.227924
\(146\) −0.744563 −0.0616204
\(147\) 2.00000 0.164957
\(148\) −10.7446 −0.883198
\(149\) 10.7446 0.880229 0.440114 0.897942i \(-0.354938\pi\)
0.440114 + 0.897942i \(0.354938\pi\)
\(150\) 2.00000 0.163299
\(151\) −20.7446 −1.68817 −0.844084 0.536211i \(-0.819855\pi\)
−0.844084 + 0.536211i \(0.819855\pi\)
\(152\) 4.74456 0.384835
\(153\) 4.74456 0.383575
\(154\) −1.00000 −0.0805823
\(155\) −6.74456 −0.541736
\(156\) −9.48913 −0.759738
\(157\) 23.4891 1.87464 0.937318 0.348475i \(-0.113300\pi\)
0.937318 + 0.348475i \(0.113300\pi\)
\(158\) −4.74456 −0.377457
\(159\) −2.51087 −0.199125
\(160\) 1.00000 0.0790569
\(161\) 4.74456 0.373924
\(162\) −11.0000 −0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −4.00000 −0.312348
\(165\) −2.00000 −0.155700
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) 9.51087 0.731606
\(170\) 4.74456 0.363891
\(171\) 4.74456 0.362826
\(172\) 4.00000 0.304997
\(173\) −14.2337 −1.08217 −0.541084 0.840969i \(-0.681986\pi\)
−0.541084 + 0.840969i \(0.681986\pi\)
\(174\) −5.48913 −0.416130
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 5.48913 0.412588
\(178\) −7.48913 −0.561334
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) 3.48913 0.259345 0.129672 0.991557i \(-0.458607\pi\)
0.129672 + 0.991557i \(0.458607\pi\)
\(182\) −4.74456 −0.351690
\(183\) 25.4891 1.88421
\(184\) 4.74456 0.349774
\(185\) −10.7446 −0.789956
\(186\) −13.4891 −0.989071
\(187\) −4.74456 −0.346957
\(188\) −6.74456 −0.491898
\(189\) −4.00000 −0.290957
\(190\) 4.74456 0.344207
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 2.00000 0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −5.25544 −0.377318
\(195\) −9.48913 −0.679530
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 14.7446 1.04521 0.522607 0.852574i \(-0.324959\pi\)
0.522607 + 0.852574i \(0.324959\pi\)
\(200\) 1.00000 0.0707107
\(201\) −8.00000 −0.564276
\(202\) −8.74456 −0.615265
\(203\) −2.74456 −0.192631
\(204\) 9.48913 0.664372
\(205\) −4.00000 −0.279372
\(206\) 10.7446 0.748609
\(207\) 4.74456 0.329770
\(208\) −4.74456 −0.328976
\(209\) −4.74456 −0.328188
\(210\) 2.00000 0.138013
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −1.25544 −0.0862238
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) −4.00000 −0.272166
\(217\) −6.74456 −0.457851
\(218\) 16.2337 1.09948
\(219\) −1.48913 −0.100626
\(220\) −1.00000 −0.0674200
\(221\) −22.5109 −1.51425
\(222\) −21.4891 −1.44226
\(223\) 26.7446 1.79095 0.895474 0.445113i \(-0.146837\pi\)
0.895474 + 0.445113i \(0.146837\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 9.48913 0.628433
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 4.74456 0.312847
\(231\) −2.00000 −0.131590
\(232\) −2.74456 −0.180189
\(233\) 20.9783 1.37433 0.687165 0.726501i \(-0.258855\pi\)
0.687165 + 0.726501i \(0.258855\pi\)
\(234\) −4.74456 −0.310162
\(235\) −6.74456 −0.439967
\(236\) 2.74456 0.178656
\(237\) −9.48913 −0.616385
\(238\) 4.74456 0.307544
\(239\) −3.25544 −0.210577 −0.105288 0.994442i \(-0.533577\pi\)
−0.105288 + 0.994442i \(0.533577\pi\)
\(240\) 2.00000 0.129099
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 12.7446 0.815887
\(245\) 1.00000 0.0638877
\(246\) −8.00000 −0.510061
\(247\) −22.5109 −1.43233
\(248\) −6.74456 −0.428280
\(249\) 16.0000 1.01396
\(250\) 1.00000 0.0632456
\(251\) 8.23369 0.519706 0.259853 0.965648i \(-0.416326\pi\)
0.259853 + 0.965648i \(0.416326\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.74456 −0.298288
\(254\) 0 0
\(255\) 9.48913 0.594232
\(256\) 1.00000 0.0625000
\(257\) 24.2337 1.51166 0.755828 0.654770i \(-0.227235\pi\)
0.755828 + 0.654770i \(0.227235\pi\)
\(258\) 8.00000 0.498058
\(259\) −10.7446 −0.667635
\(260\) −4.74456 −0.294245
\(261\) −2.74456 −0.169884
\(262\) 8.74456 0.540241
\(263\) −18.9783 −1.17025 −0.585125 0.810943i \(-0.698954\pi\)
−0.585125 + 0.810943i \(0.698954\pi\)
\(264\) −2.00000 −0.123091
\(265\) −1.25544 −0.0771209
\(266\) 4.74456 0.290908
\(267\) −14.9783 −0.916654
\(268\) −4.00000 −0.244339
\(269\) −24.9783 −1.52295 −0.761475 0.648194i \(-0.775525\pi\)
−0.761475 + 0.648194i \(0.775525\pi\)
\(270\) −4.00000 −0.243432
\(271\) −30.9783 −1.88179 −0.940897 0.338692i \(-0.890016\pi\)
−0.940897 + 0.338692i \(0.890016\pi\)
\(272\) 4.74456 0.287681
\(273\) −9.48913 −0.574308
\(274\) −19.4891 −1.17738
\(275\) −1.00000 −0.0603023
\(276\) 9.48913 0.571178
\(277\) −7.48913 −0.449978 −0.224989 0.974361i \(-0.572235\pi\)
−0.224989 + 0.974361i \(0.572235\pi\)
\(278\) 3.25544 0.195248
\(279\) −6.74456 −0.403786
\(280\) 1.00000 0.0597614
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) −13.4891 −0.803266
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −4.00000 −0.237356
\(285\) 9.48913 0.562087
\(286\) 4.74456 0.280552
\(287\) −4.00000 −0.236113
\(288\) 1.00000 0.0589256
\(289\) 5.51087 0.324169
\(290\) −2.74456 −0.161166
\(291\) −10.5109 −0.616158
\(292\) −0.744563 −0.0435722
\(293\) 24.7446 1.44559 0.722796 0.691061i \(-0.242856\pi\)
0.722796 + 0.691061i \(0.242856\pi\)
\(294\) 2.00000 0.116642
\(295\) 2.74456 0.159795
\(296\) −10.7446 −0.624515
\(297\) 4.00000 0.232104
\(298\) 10.7446 0.622416
\(299\) −22.5109 −1.30184
\(300\) 2.00000 0.115470
\(301\) 4.00000 0.230556
\(302\) −20.7446 −1.19372
\(303\) −17.4891 −1.00472
\(304\) 4.74456 0.272119
\(305\) 12.7446 0.729752
\(306\) 4.74456 0.271229
\(307\) 21.4891 1.22645 0.613225 0.789909i \(-0.289872\pi\)
0.613225 + 0.789909i \(0.289872\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 21.4891 1.22247
\(310\) −6.74456 −0.383065
\(311\) −9.25544 −0.524828 −0.262414 0.964955i \(-0.584519\pi\)
−0.262414 + 0.964955i \(0.584519\pi\)
\(312\) −9.48913 −0.537216
\(313\) 32.2337 1.82196 0.910978 0.412455i \(-0.135329\pi\)
0.910978 + 0.412455i \(0.135329\pi\)
\(314\) 23.4891 1.32557
\(315\) 1.00000 0.0563436
\(316\) −4.74456 −0.266903
\(317\) −32.2337 −1.81042 −0.905212 0.424960i \(-0.860288\pi\)
−0.905212 + 0.424960i \(0.860288\pi\)
\(318\) −2.51087 −0.140803
\(319\) 2.74456 0.153666
\(320\) 1.00000 0.0559017
\(321\) −24.0000 −1.33955
\(322\) 4.74456 0.264404
\(323\) 22.5109 1.25254
\(324\) −11.0000 −0.611111
\(325\) −4.74456 −0.263181
\(326\) −4.00000 −0.221540
\(327\) 32.4674 1.79545
\(328\) −4.00000 −0.220863
\(329\) −6.74456 −0.371840
\(330\) −2.00000 −0.110096
\(331\) −30.9783 −1.70272 −0.851359 0.524583i \(-0.824221\pi\)
−0.851359 + 0.524583i \(0.824221\pi\)
\(332\) 8.00000 0.439057
\(333\) −10.7446 −0.588798
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 2.00000 0.109109
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 9.51087 0.517323
\(339\) 20.0000 1.08625
\(340\) 4.74456 0.257310
\(341\) 6.74456 0.365239
\(342\) 4.74456 0.256557
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 9.48913 0.510877
\(346\) −14.2337 −0.765208
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) −5.48913 −0.294248
\(349\) 19.2554 1.03072 0.515360 0.856974i \(-0.327658\pi\)
0.515360 + 0.856974i \(0.327658\pi\)
\(350\) 1.00000 0.0534522
\(351\) 18.9783 1.01298
\(352\) −1.00000 −0.0533002
\(353\) −2.74456 −0.146078 −0.0730392 0.997329i \(-0.523270\pi\)
−0.0730392 + 0.997329i \(0.523270\pi\)
\(354\) 5.48913 0.291744
\(355\) −4.00000 −0.212298
\(356\) −7.48913 −0.396923
\(357\) 9.48913 0.502218
\(358\) −4.00000 −0.211407
\(359\) −12.7446 −0.672632 −0.336316 0.941749i \(-0.609181\pi\)
−0.336316 + 0.941749i \(0.609181\pi\)
\(360\) 1.00000 0.0527046
\(361\) 3.51087 0.184783
\(362\) 3.48913 0.183384
\(363\) 2.00000 0.104973
\(364\) −4.74456 −0.248683
\(365\) −0.744563 −0.0389722
\(366\) 25.4891 1.33234
\(367\) 2.74456 0.143265 0.0716325 0.997431i \(-0.477179\pi\)
0.0716325 + 0.997431i \(0.477179\pi\)
\(368\) 4.74456 0.247327
\(369\) −4.00000 −0.208232
\(370\) −10.7446 −0.558583
\(371\) −1.25544 −0.0651791
\(372\) −13.4891 −0.699379
\(373\) 28.9783 1.50044 0.750218 0.661190i \(-0.229948\pi\)
0.750218 + 0.661190i \(0.229948\pi\)
\(374\) −4.74456 −0.245335
\(375\) 2.00000 0.103280
\(376\) −6.74456 −0.347824
\(377\) 13.0217 0.670654
\(378\) −4.00000 −0.205738
\(379\) −14.5109 −0.745374 −0.372687 0.927957i \(-0.621563\pi\)
−0.372687 + 0.927957i \(0.621563\pi\)
\(380\) 4.74456 0.243391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −37.7228 −1.92755 −0.963773 0.266724i \(-0.914059\pi\)
−0.963773 + 0.266724i \(0.914059\pi\)
\(384\) 2.00000 0.102062
\(385\) −1.00000 −0.0509647
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) −5.25544 −0.266804
\(389\) 15.4891 0.785330 0.392665 0.919682i \(-0.371553\pi\)
0.392665 + 0.919682i \(0.371553\pi\)
\(390\) −9.48913 −0.480501
\(391\) 22.5109 1.13842
\(392\) 1.00000 0.0505076
\(393\) 17.4891 0.882210
\(394\) 10.0000 0.503793
\(395\) −4.74456 −0.238725
\(396\) −1.00000 −0.0502519
\(397\) 12.5109 0.627903 0.313951 0.949439i \(-0.398347\pi\)
0.313951 + 0.949439i \(0.398347\pi\)
\(398\) 14.7446 0.739078
\(399\) 9.48913 0.475050
\(400\) 1.00000 0.0500000
\(401\) 0.510875 0.0255119 0.0127559 0.999919i \(-0.495940\pi\)
0.0127559 + 0.999919i \(0.495940\pi\)
\(402\) −8.00000 −0.399004
\(403\) 32.0000 1.59403
\(404\) −8.74456 −0.435058
\(405\) −11.0000 −0.546594
\(406\) −2.74456 −0.136210
\(407\) 10.7446 0.532588
\(408\) 9.48913 0.469782
\(409\) 1.48913 0.0736325 0.0368163 0.999322i \(-0.488278\pi\)
0.0368163 + 0.999322i \(0.488278\pi\)
\(410\) −4.00000 −0.197546
\(411\) −38.9783 −1.92266
\(412\) 10.7446 0.529347
\(413\) 2.74456 0.135051
\(414\) 4.74456 0.233183
\(415\) 8.00000 0.392705
\(416\) −4.74456 −0.232621
\(417\) 6.51087 0.318839
\(418\) −4.74456 −0.232064
\(419\) 10.7446 0.524906 0.262453 0.964945i \(-0.415468\pi\)
0.262453 + 0.964945i \(0.415468\pi\)
\(420\) 2.00000 0.0975900
\(421\) 27.4891 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(422\) −12.0000 −0.584151
\(423\) −6.74456 −0.327932
\(424\) −1.25544 −0.0609694
\(425\) 4.74456 0.230145
\(426\) −8.00000 −0.387601
\(427\) 12.7446 0.616753
\(428\) −12.0000 −0.580042
\(429\) 9.48913 0.458139
\(430\) 4.00000 0.192897
\(431\) −28.7446 −1.38458 −0.692288 0.721621i \(-0.743397\pi\)
−0.692288 + 0.721621i \(0.743397\pi\)
\(432\) −4.00000 −0.192450
\(433\) 25.2554 1.21370 0.606849 0.794817i \(-0.292433\pi\)
0.606849 + 0.794817i \(0.292433\pi\)
\(434\) −6.74456 −0.323749
\(435\) −5.48913 −0.263183
\(436\) 16.2337 0.777453
\(437\) 22.5109 1.07684
\(438\) −1.48913 −0.0711532
\(439\) −30.9783 −1.47851 −0.739256 0.673425i \(-0.764822\pi\)
−0.739256 + 0.673425i \(0.764822\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −22.5109 −1.07073
\(443\) −6.51087 −0.309341 −0.154670 0.987966i \(-0.549432\pi\)
−0.154670 + 0.987966i \(0.549432\pi\)
\(444\) −21.4891 −1.01983
\(445\) −7.48913 −0.355019
\(446\) 26.7446 1.26639
\(447\) 21.4891 1.01640
\(448\) 1.00000 0.0472456
\(449\) 40.9783 1.93388 0.966942 0.254998i \(-0.0820748\pi\)
0.966942 + 0.254998i \(0.0820748\pi\)
\(450\) 1.00000 0.0471405
\(451\) 4.00000 0.188353
\(452\) 10.0000 0.470360
\(453\) −41.4891 −1.94933
\(454\) 20.0000 0.938647
\(455\) −4.74456 −0.222429
\(456\) 9.48913 0.444369
\(457\) −19.4891 −0.911663 −0.455831 0.890066i \(-0.650658\pi\)
−0.455831 + 0.890066i \(0.650658\pi\)
\(458\) −6.00000 −0.280362
\(459\) −18.9783 −0.885829
\(460\) 4.74456 0.221216
\(461\) −23.7228 −1.10488 −0.552441 0.833552i \(-0.686303\pi\)
−0.552441 + 0.833552i \(0.686303\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 12.7446 0.592290 0.296145 0.955143i \(-0.404299\pi\)
0.296145 + 0.955143i \(0.404299\pi\)
\(464\) −2.74456 −0.127413
\(465\) −13.4891 −0.625543
\(466\) 20.9783 0.971799
\(467\) −28.9783 −1.34095 −0.670477 0.741931i \(-0.733910\pi\)
−0.670477 + 0.741931i \(0.733910\pi\)
\(468\) −4.74456 −0.219317
\(469\) −4.00000 −0.184703
\(470\) −6.74456 −0.311103
\(471\) 46.9783 2.16464
\(472\) 2.74456 0.126329
\(473\) −4.00000 −0.183920
\(474\) −9.48913 −0.435850
\(475\) 4.74456 0.217695
\(476\) 4.74456 0.217467
\(477\) −1.25544 −0.0574825
\(478\) −3.25544 −0.148900
\(479\) −18.5109 −0.845783 −0.422892 0.906180i \(-0.638985\pi\)
−0.422892 + 0.906180i \(0.638985\pi\)
\(480\) 2.00000 0.0912871
\(481\) 50.9783 2.32441
\(482\) 20.0000 0.910975
\(483\) 9.48913 0.431770
\(484\) 1.00000 0.0454545
\(485\) −5.25544 −0.238637
\(486\) −10.0000 −0.453609
\(487\) 20.7446 0.940026 0.470013 0.882660i \(-0.344249\pi\)
0.470013 + 0.882660i \(0.344249\pi\)
\(488\) 12.7446 0.576919
\(489\) −8.00000 −0.361773
\(490\) 1.00000 0.0451754
\(491\) −14.9783 −0.675959 −0.337979 0.941153i \(-0.609743\pi\)
−0.337979 + 0.941153i \(0.609743\pi\)
\(492\) −8.00000 −0.360668
\(493\) −13.0217 −0.586470
\(494\) −22.5109 −1.01281
\(495\) −1.00000 −0.0449467
\(496\) −6.74456 −0.302840
\(497\) −4.00000 −0.179425
\(498\) 16.0000 0.716977
\(499\) 9.48913 0.424792 0.212396 0.977184i \(-0.431873\pi\)
0.212396 + 0.977184i \(0.431873\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 8.23369 0.367487
\(503\) 29.4891 1.31486 0.657428 0.753518i \(-0.271645\pi\)
0.657428 + 0.753518i \(0.271645\pi\)
\(504\) 1.00000 0.0445435
\(505\) −8.74456 −0.389128
\(506\) −4.74456 −0.210922
\(507\) 19.0217 0.844786
\(508\) 0 0
\(509\) −12.5109 −0.554535 −0.277267 0.960793i \(-0.589429\pi\)
−0.277267 + 0.960793i \(0.589429\pi\)
\(510\) 9.48913 0.420186
\(511\) −0.744563 −0.0329375
\(512\) 1.00000 0.0441942
\(513\) −18.9783 −0.837910
\(514\) 24.2337 1.06890
\(515\) 10.7446 0.473462
\(516\) 8.00000 0.352180
\(517\) 6.74456 0.296626
\(518\) −10.7446 −0.472089
\(519\) −28.4674 −1.24958
\(520\) −4.74456 −0.208063
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) −2.74456 −0.120126
\(523\) 5.48913 0.240023 0.120011 0.992773i \(-0.461707\pi\)
0.120011 + 0.992773i \(0.461707\pi\)
\(524\) 8.74456 0.382008
\(525\) 2.00000 0.0872872
\(526\) −18.9783 −0.827491
\(527\) −32.0000 −1.39394
\(528\) −2.00000 −0.0870388
\(529\) −0.489125 −0.0212663
\(530\) −1.25544 −0.0545327
\(531\) 2.74456 0.119104
\(532\) 4.74456 0.205703
\(533\) 18.9783 0.822039
\(534\) −14.9783 −0.648172
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) −8.00000 −0.345225
\(538\) −24.9783 −1.07689
\(539\) −1.00000 −0.0430730
\(540\) −4.00000 −0.172133
\(541\) 36.2337 1.55781 0.778904 0.627143i \(-0.215776\pi\)
0.778904 + 0.627143i \(0.215776\pi\)
\(542\) −30.9783 −1.33063
\(543\) 6.97825 0.299465
\(544\) 4.74456 0.203421
\(545\) 16.2337 0.695375
\(546\) −9.48913 −0.406097
\(547\) −30.9783 −1.32453 −0.662267 0.749268i \(-0.730406\pi\)
−0.662267 + 0.749268i \(0.730406\pi\)
\(548\) −19.4891 −0.832534
\(549\) 12.7446 0.543925
\(550\) −1.00000 −0.0426401
\(551\) −13.0217 −0.554745
\(552\) 9.48913 0.403884
\(553\) −4.74456 −0.201759
\(554\) −7.48913 −0.318182
\(555\) −21.4891 −0.912163
\(556\) 3.25544 0.138061
\(557\) 44.9783 1.90579 0.952895 0.303301i \(-0.0980887\pi\)
0.952895 + 0.303301i \(0.0980887\pi\)
\(558\) −6.74456 −0.285520
\(559\) −18.9783 −0.802694
\(560\) 1.00000 0.0422577
\(561\) −9.48913 −0.400631
\(562\) 14.0000 0.590554
\(563\) −17.4891 −0.737079 −0.368539 0.929612i \(-0.620142\pi\)
−0.368539 + 0.929612i \(0.620142\pi\)
\(564\) −13.4891 −0.567995
\(565\) 10.0000 0.420703
\(566\) 28.0000 1.17693
\(567\) −11.0000 −0.461957
\(568\) −4.00000 −0.167836
\(569\) 39.4891 1.65547 0.827735 0.561119i \(-0.189629\pi\)
0.827735 + 0.561119i \(0.189629\pi\)
\(570\) 9.48913 0.397456
\(571\) 5.48913 0.229713 0.114856 0.993382i \(-0.463359\pi\)
0.114856 + 0.993382i \(0.463359\pi\)
\(572\) 4.74456 0.198380
\(573\) −32.0000 −1.33682
\(574\) −4.00000 −0.166957
\(575\) 4.74456 0.197862
\(576\) 1.00000 0.0416667
\(577\) 2.74456 0.114258 0.0571288 0.998367i \(-0.481805\pi\)
0.0571288 + 0.998367i \(0.481805\pi\)
\(578\) 5.51087 0.229222
\(579\) −4.00000 −0.166234
\(580\) −2.74456 −0.113962
\(581\) 8.00000 0.331896
\(582\) −10.5109 −0.435690
\(583\) 1.25544 0.0519949
\(584\) −0.744563 −0.0308102
\(585\) −4.74456 −0.196164
\(586\) 24.7446 1.02219
\(587\) 40.9783 1.69135 0.845677 0.533696i \(-0.179197\pi\)
0.845677 + 0.533696i \(0.179197\pi\)
\(588\) 2.00000 0.0824786
\(589\) −32.0000 −1.31854
\(590\) 2.74456 0.112992
\(591\) 20.0000 0.822690
\(592\) −10.7446 −0.441599
\(593\) −5.76631 −0.236794 −0.118397 0.992966i \(-0.537776\pi\)
−0.118397 + 0.992966i \(0.537776\pi\)
\(594\) 4.00000 0.164122
\(595\) 4.74456 0.194508
\(596\) 10.7446 0.440114
\(597\) 29.4891 1.20691
\(598\) −22.5109 −0.920538
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 2.00000 0.0816497
\(601\) −10.5109 −0.428748 −0.214374 0.976752i \(-0.568771\pi\)
−0.214374 + 0.976752i \(0.568771\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) −20.7446 −0.844084
\(605\) 1.00000 0.0406558
\(606\) −17.4891 −0.710447
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.74456 0.192417
\(609\) −5.48913 −0.222431
\(610\) 12.7446 0.516012
\(611\) 32.0000 1.29458
\(612\) 4.74456 0.191788
\(613\) −11.4891 −0.464041 −0.232021 0.972711i \(-0.574534\pi\)
−0.232021 + 0.972711i \(0.574534\pi\)
\(614\) 21.4891 0.867231
\(615\) −8.00000 −0.322591
\(616\) −1.00000 −0.0402911
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 21.4891 0.864419
\(619\) −10.7446 −0.431860 −0.215930 0.976409i \(-0.569278\pi\)
−0.215930 + 0.976409i \(0.569278\pi\)
\(620\) −6.74456 −0.270868
\(621\) −18.9783 −0.761571
\(622\) −9.25544 −0.371109
\(623\) −7.48913 −0.300045
\(624\) −9.48913 −0.379869
\(625\) 1.00000 0.0400000
\(626\) 32.2337 1.28832
\(627\) −9.48913 −0.378959
\(628\) 23.4891 0.937318
\(629\) −50.9783 −2.03264
\(630\) 1.00000 0.0398410
\(631\) 42.9783 1.71094 0.855469 0.517855i \(-0.173269\pi\)
0.855469 + 0.517855i \(0.173269\pi\)
\(632\) −4.74456 −0.188729
\(633\) −24.0000 −0.953914
\(634\) −32.2337 −1.28016
\(635\) 0 0
\(636\) −2.51087 −0.0995627
\(637\) −4.74456 −0.187986
\(638\) 2.74456 0.108658
\(639\) −4.00000 −0.158238
\(640\) 1.00000 0.0395285
\(641\) −11.4891 −0.453793 −0.226897 0.973919i \(-0.572858\pi\)
−0.226897 + 0.973919i \(0.572858\pi\)
\(642\) −24.0000 −0.947204
\(643\) −22.4674 −0.886027 −0.443013 0.896515i \(-0.646091\pi\)
−0.443013 + 0.896515i \(0.646091\pi\)
\(644\) 4.74456 0.186962
\(645\) 8.00000 0.315000
\(646\) 22.5109 0.885679
\(647\) −2.74456 −0.107900 −0.0539499 0.998544i \(-0.517181\pi\)
−0.0539499 + 0.998544i \(0.517181\pi\)
\(648\) −11.0000 −0.432121
\(649\) −2.74456 −0.107734
\(650\) −4.74456 −0.186097
\(651\) −13.4891 −0.528681
\(652\) −4.00000 −0.156652
\(653\) −41.7228 −1.63274 −0.816370 0.577529i \(-0.804017\pi\)
−0.816370 + 0.577529i \(0.804017\pi\)
\(654\) 32.4674 1.26957
\(655\) 8.74456 0.341678
\(656\) −4.00000 −0.156174
\(657\) −0.744563 −0.0290482
\(658\) −6.74456 −0.262930
\(659\) −18.5109 −0.721081 −0.360541 0.932744i \(-0.617408\pi\)
−0.360541 + 0.932744i \(0.617408\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −30.9783 −1.20400
\(663\) −45.0217 −1.74850
\(664\) 8.00000 0.310460
\(665\) 4.74456 0.183986
\(666\) −10.7446 −0.416343
\(667\) −13.0217 −0.504204
\(668\) 0 0
\(669\) 53.4891 2.06801
\(670\) −4.00000 −0.154533
\(671\) −12.7446 −0.491998
\(672\) 2.00000 0.0771517
\(673\) −24.9783 −0.962841 −0.481420 0.876490i \(-0.659879\pi\)
−0.481420 + 0.876490i \(0.659879\pi\)
\(674\) 26.0000 1.00148
\(675\) −4.00000 −0.153960
\(676\) 9.51087 0.365803
\(677\) 1.76631 0.0678849 0.0339424 0.999424i \(-0.489194\pi\)
0.0339424 + 0.999424i \(0.489194\pi\)
\(678\) 20.0000 0.768095
\(679\) −5.25544 −0.201685
\(680\) 4.74456 0.181946
\(681\) 40.0000 1.53280
\(682\) 6.74456 0.258263
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 4.74456 0.181413
\(685\) −19.4891 −0.744641
\(686\) 1.00000 0.0381802
\(687\) −12.0000 −0.457829
\(688\) 4.00000 0.152499
\(689\) 5.95650 0.226925
\(690\) 9.48913 0.361245
\(691\) 36.2337 1.37839 0.689197 0.724574i \(-0.257963\pi\)
0.689197 + 0.724574i \(0.257963\pi\)
\(692\) −14.2337 −0.541084
\(693\) −1.00000 −0.0379869
\(694\) 22.9783 0.872242
\(695\) 3.25544 0.123486
\(696\) −5.48913 −0.208065
\(697\) −18.9783 −0.718853
\(698\) 19.2554 0.728829
\(699\) 41.9565 1.58694
\(700\) 1.00000 0.0377964
\(701\) −12.2337 −0.462060 −0.231030 0.972947i \(-0.574210\pi\)
−0.231030 + 0.972947i \(0.574210\pi\)
\(702\) 18.9783 0.716288
\(703\) −50.9783 −1.92268
\(704\) −1.00000 −0.0376889
\(705\) −13.4891 −0.508030
\(706\) −2.74456 −0.103293
\(707\) −8.74456 −0.328873
\(708\) 5.48913 0.206294
\(709\) 23.4891 0.882153 0.441076 0.897470i \(-0.354597\pi\)
0.441076 + 0.897470i \(0.354597\pi\)
\(710\) −4.00000 −0.150117
\(711\) −4.74456 −0.177935
\(712\) −7.48913 −0.280667
\(713\) −32.0000 −1.19841
\(714\) 9.48913 0.355122
\(715\) 4.74456 0.177437
\(716\) −4.00000 −0.149487
\(717\) −6.51087 −0.243153
\(718\) −12.7446 −0.475623
\(719\) −49.7228 −1.85435 −0.927174 0.374631i \(-0.877769\pi\)
−0.927174 + 0.374631i \(0.877769\pi\)
\(720\) 1.00000 0.0372678
\(721\) 10.7446 0.400148
\(722\) 3.51087 0.130661
\(723\) 40.0000 1.48762
\(724\) 3.48913 0.129672
\(725\) −2.74456 −0.101930
\(726\) 2.00000 0.0742270
\(727\) −20.2337 −0.750426 −0.375213 0.926939i \(-0.622430\pi\)
−0.375213 + 0.926939i \(0.622430\pi\)
\(728\) −4.74456 −0.175845
\(729\) 13.0000 0.481481
\(730\) −0.744563 −0.0275575
\(731\) 18.9783 0.701936
\(732\) 25.4891 0.942105
\(733\) 18.2337 0.673477 0.336738 0.941598i \(-0.390676\pi\)
0.336738 + 0.941598i \(0.390676\pi\)
\(734\) 2.74456 0.101304
\(735\) 2.00000 0.0737711
\(736\) 4.74456 0.174887
\(737\) 4.00000 0.147342
\(738\) −4.00000 −0.147242
\(739\) 14.9783 0.550984 0.275492 0.961303i \(-0.411159\pi\)
0.275492 + 0.961303i \(0.411159\pi\)
\(740\) −10.7446 −0.394978
\(741\) −45.0217 −1.65392
\(742\) −1.25544 −0.0460886
\(743\) −18.9783 −0.696244 −0.348122 0.937449i \(-0.613181\pi\)
−0.348122 + 0.937449i \(0.613181\pi\)
\(744\) −13.4891 −0.494535
\(745\) 10.7446 0.393650
\(746\) 28.9783 1.06097
\(747\) 8.00000 0.292705
\(748\) −4.74456 −0.173478
\(749\) −12.0000 −0.438470
\(750\) 2.00000 0.0730297
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −6.74456 −0.245949
\(753\) 16.4674 0.600105
\(754\) 13.0217 0.474224
\(755\) −20.7446 −0.754972
\(756\) −4.00000 −0.145479
\(757\) −30.7446 −1.11743 −0.558715 0.829360i \(-0.688705\pi\)
−0.558715 + 0.829360i \(0.688705\pi\)
\(758\) −14.5109 −0.527059
\(759\) −9.48913 −0.344433
\(760\) 4.74456 0.172103
\(761\) −6.51087 −0.236019 −0.118010 0.993012i \(-0.537651\pi\)
−0.118010 + 0.993012i \(0.537651\pi\)
\(762\) 0 0
\(763\) 16.2337 0.587699
\(764\) −16.0000 −0.578860
\(765\) 4.74456 0.171540
\(766\) −37.7228 −1.36298
\(767\) −13.0217 −0.470188
\(768\) 2.00000 0.0721688
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 48.4674 1.74551
\(772\) −2.00000 −0.0719816
\(773\) −28.5109 −1.02546 −0.512732 0.858548i \(-0.671367\pi\)
−0.512732 + 0.858548i \(0.671367\pi\)
\(774\) 4.00000 0.143777
\(775\) −6.74456 −0.242272
\(776\) −5.25544 −0.188659
\(777\) −21.4891 −0.770918
\(778\) 15.4891 0.555312
\(779\) −18.9783 −0.679966
\(780\) −9.48913 −0.339765
\(781\) 4.00000 0.143131
\(782\) 22.5109 0.804987
\(783\) 10.9783 0.392331
\(784\) 1.00000 0.0357143
\(785\) 23.4891 0.838363
\(786\) 17.4891 0.623816
\(787\) −17.4891 −0.623420 −0.311710 0.950177i \(-0.600902\pi\)
−0.311710 + 0.950177i \(0.600902\pi\)
\(788\) 10.0000 0.356235
\(789\) −37.9565 −1.35129
\(790\) −4.74456 −0.168804
\(791\) 10.0000 0.355559
\(792\) −1.00000 −0.0355335
\(793\) −60.4674 −2.14726
\(794\) 12.5109 0.443994
\(795\) −2.51087 −0.0890515
\(796\) 14.7446 0.522607
\(797\) 26.4674 0.937523 0.468761 0.883325i \(-0.344700\pi\)
0.468761 + 0.883325i \(0.344700\pi\)
\(798\) 9.48913 0.335911
\(799\) −32.0000 −1.13208
\(800\) 1.00000 0.0353553
\(801\) −7.48913 −0.264615
\(802\) 0.510875 0.0180396
\(803\) 0.744563 0.0262750
\(804\) −8.00000 −0.282138
\(805\) 4.74456 0.167224
\(806\) 32.0000 1.12715
\(807\) −49.9565 −1.75855
\(808\) −8.74456 −0.307633
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) −11.0000 −0.386501
\(811\) −7.25544 −0.254773 −0.127386 0.991853i \(-0.540659\pi\)
−0.127386 + 0.991853i \(0.540659\pi\)
\(812\) −2.74456 −0.0963153
\(813\) −61.9565 −2.17291
\(814\) 10.7446 0.376597
\(815\) −4.00000 −0.140114
\(816\) 9.48913 0.332186
\(817\) 18.9783 0.663965
\(818\) 1.48913 0.0520660
\(819\) −4.74456 −0.165788
\(820\) −4.00000 −0.139686
\(821\) −49.7228 −1.73534 −0.867669 0.497142i \(-0.834383\pi\)
−0.867669 + 0.497142i \(0.834383\pi\)
\(822\) −38.9783 −1.35952
\(823\) 42.2337 1.47217 0.736087 0.676887i \(-0.236671\pi\)
0.736087 + 0.676887i \(0.236671\pi\)
\(824\) 10.7446 0.374305
\(825\) −2.00000 −0.0696311
\(826\) 2.74456 0.0954955
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 4.74456 0.164885
\(829\) −54.4674 −1.89173 −0.945865 0.324560i \(-0.894784\pi\)
−0.945865 + 0.324560i \(0.894784\pi\)
\(830\) 8.00000 0.277684
\(831\) −14.9783 −0.519590
\(832\) −4.74456 −0.164488
\(833\) 4.74456 0.164389
\(834\) 6.51087 0.225453
\(835\) 0 0
\(836\) −4.74456 −0.164094
\(837\) 26.9783 0.932505
\(838\) 10.7446 0.371165
\(839\) −14.7446 −0.509039 −0.254519 0.967068i \(-0.581917\pi\)
−0.254519 + 0.967068i \(0.581917\pi\)
\(840\) 2.00000 0.0690066
\(841\) −21.4674 −0.740254
\(842\) 27.4891 0.947338
\(843\) 28.0000 0.964371
\(844\) −12.0000 −0.413057
\(845\) 9.51087 0.327184
\(846\) −6.74456 −0.231883
\(847\) 1.00000 0.0343604
\(848\) −1.25544 −0.0431119
\(849\) 56.0000 1.92192
\(850\) 4.74456 0.162737
\(851\) −50.9783 −1.74751
\(852\) −8.00000 −0.274075
\(853\) −11.2554 −0.385379 −0.192689 0.981260i \(-0.561721\pi\)
−0.192689 + 0.981260i \(0.561721\pi\)
\(854\) 12.7446 0.436110
\(855\) 4.74456 0.162261
\(856\) −12.0000 −0.410152
\(857\) 37.2119 1.27114 0.635568 0.772045i \(-0.280766\pi\)
0.635568 + 0.772045i \(0.280766\pi\)
\(858\) 9.48913 0.323953
\(859\) 51.2119 1.74733 0.873664 0.486529i \(-0.161737\pi\)
0.873664 + 0.486529i \(0.161737\pi\)
\(860\) 4.00000 0.136399
\(861\) −8.00000 −0.272639
\(862\) −28.7446 −0.979044
\(863\) −5.76631 −0.196288 −0.0981438 0.995172i \(-0.531291\pi\)
−0.0981438 + 0.995172i \(0.531291\pi\)
\(864\) −4.00000 −0.136083
\(865\) −14.2337 −0.483960
\(866\) 25.2554 0.858215
\(867\) 11.0217 0.374318
\(868\) −6.74456 −0.228925
\(869\) 4.74456 0.160948
\(870\) −5.48913 −0.186099
\(871\) 18.9783 0.643053
\(872\) 16.2337 0.549742
\(873\) −5.25544 −0.177870
\(874\) 22.5109 0.761442
\(875\) 1.00000 0.0338062
\(876\) −1.48913 −0.0503129
\(877\) 42.4674 1.43402 0.717011 0.697062i \(-0.245510\pi\)
0.717011 + 0.697062i \(0.245510\pi\)
\(878\) −30.9783 −1.04547
\(879\) 49.4891 1.66923
\(880\) −1.00000 −0.0337100
\(881\) −32.5109 −1.09532 −0.547660 0.836701i \(-0.684481\pi\)
−0.547660 + 0.836701i \(0.684481\pi\)
\(882\) 1.00000 0.0336718
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) −22.5109 −0.757123
\(885\) 5.48913 0.184515
\(886\) −6.51087 −0.218737
\(887\) −18.5109 −0.621534 −0.310767 0.950486i \(-0.600586\pi\)
−0.310767 + 0.950486i \(0.600586\pi\)
\(888\) −21.4891 −0.721128
\(889\) 0 0
\(890\) −7.48913 −0.251036
\(891\) 11.0000 0.368514
\(892\) 26.7446 0.895474
\(893\) −32.0000 −1.07084
\(894\) 21.4891 0.718704
\(895\) −4.00000 −0.133705
\(896\) 1.00000 0.0334077
\(897\) −45.0217 −1.50323
\(898\) 40.9783 1.36746
\(899\) 18.5109 0.617372
\(900\) 1.00000 0.0333333
\(901\) −5.95650 −0.198440
\(902\) 4.00000 0.133185
\(903\) 8.00000 0.266223
\(904\) 10.0000 0.332595
\(905\) 3.48913 0.115982
\(906\) −41.4891 −1.37838
\(907\) −37.4891 −1.24481 −0.622403 0.782697i \(-0.713843\pi\)
−0.622403 + 0.782697i \(0.713843\pi\)
\(908\) 20.0000 0.663723
\(909\) −8.74456 −0.290039
\(910\) −4.74456 −0.157281
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 9.48913 0.314216
\(913\) −8.00000 −0.264761
\(914\) −19.4891 −0.644643
\(915\) 25.4891 0.842644
\(916\) −6.00000 −0.198246
\(917\) 8.74456 0.288771
\(918\) −18.9783 −0.626376
\(919\) 25.2119 0.831665 0.415833 0.909441i \(-0.363490\pi\)
0.415833 + 0.909441i \(0.363490\pi\)
\(920\) 4.74456 0.156424
\(921\) 42.9783 1.41618
\(922\) −23.7228 −0.781269
\(923\) 18.9783 0.624677
\(924\) −2.00000 −0.0657952
\(925\) −10.7446 −0.353279
\(926\) 12.7446 0.418812
\(927\) 10.7446 0.352898
\(928\) −2.74456 −0.0900947
\(929\) 16.9783 0.557038 0.278519 0.960431i \(-0.410156\pi\)
0.278519 + 0.960431i \(0.410156\pi\)
\(930\) −13.4891 −0.442326
\(931\) 4.74456 0.155497
\(932\) 20.9783 0.687165
\(933\) −18.5109 −0.606019
\(934\) −28.9783 −0.948197
\(935\) −4.74456 −0.155164
\(936\) −4.74456 −0.155081
\(937\) −7.25544 −0.237025 −0.118512 0.992953i \(-0.537813\pi\)
−0.118512 + 0.992953i \(0.537813\pi\)
\(938\) −4.00000 −0.130605
\(939\) 64.4674 2.10381
\(940\) −6.74456 −0.219983
\(941\) −22.2337 −0.724798 −0.362399 0.932023i \(-0.618042\pi\)
−0.362399 + 0.932023i \(0.618042\pi\)
\(942\) 46.9783 1.53063
\(943\) −18.9783 −0.618017
\(944\) 2.74456 0.0893279
\(945\) −4.00000 −0.130120
\(946\) −4.00000 −0.130051
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −9.48913 −0.308192
\(949\) 3.53262 0.114674
\(950\) 4.74456 0.153934
\(951\) −64.4674 −2.09050
\(952\) 4.74456 0.153772
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −1.25544 −0.0406463
\(955\) −16.0000 −0.517748
\(956\) −3.25544 −0.105288
\(957\) 5.48913 0.177438
\(958\) −18.5109 −0.598059
\(959\) −19.4891 −0.629337
\(960\) 2.00000 0.0645497
\(961\) 14.4891 0.467391
\(962\) 50.9783 1.64360
\(963\) −12.0000 −0.386695
\(964\) 20.0000 0.644157
\(965\) −2.00000 −0.0643823
\(966\) 9.48913 0.305308
\(967\) −5.02175 −0.161489 −0.0807443 0.996735i \(-0.525730\pi\)
−0.0807443 + 0.996735i \(0.525730\pi\)
\(968\) 1.00000 0.0321412
\(969\) 45.0217 1.44631
\(970\) −5.25544 −0.168742
\(971\) −37.7228 −1.21058 −0.605291 0.796004i \(-0.706943\pi\)
−0.605291 + 0.796004i \(0.706943\pi\)
\(972\) −10.0000 −0.320750
\(973\) 3.25544 0.104365
\(974\) 20.7446 0.664699
\(975\) −9.48913 −0.303895
\(976\) 12.7446 0.407944
\(977\) −32.9783 −1.05507 −0.527534 0.849534i \(-0.676883\pi\)
−0.527534 + 0.849534i \(0.676883\pi\)
\(978\) −8.00000 −0.255812
\(979\) 7.48913 0.239353
\(980\) 1.00000 0.0319438
\(981\) 16.2337 0.518302
\(982\) −14.9783 −0.477975
\(983\) −32.2337 −1.02809 −0.514047 0.857762i \(-0.671854\pi\)
−0.514047 + 0.857762i \(0.671854\pi\)
\(984\) −8.00000 −0.255031
\(985\) 10.0000 0.318626
\(986\) −13.0217 −0.414697
\(987\) −13.4891 −0.429364
\(988\) −22.5109 −0.716166
\(989\) 18.9783 0.603473
\(990\) −1.00000 −0.0317821
\(991\) −21.4891 −0.682625 −0.341312 0.939950i \(-0.610871\pi\)
−0.341312 + 0.939950i \(0.610871\pi\)
\(992\) −6.74456 −0.214140
\(993\) −61.9565 −1.96613
\(994\) −4.00000 −0.126872
\(995\) 14.7446 0.467434
\(996\) 16.0000 0.506979
\(997\) −41.2119 −1.30520 −0.652598 0.757705i \(-0.726321\pi\)
−0.652598 + 0.757705i \(0.726321\pi\)
\(998\) 9.48913 0.300373
\(999\) 42.9783 1.35977
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 770.2.a.k.1.1 2
3.2 odd 2 6930.2.a.bo.1.1 2
4.3 odd 2 6160.2.a.r.1.1 2
5.2 odd 4 3850.2.c.y.1849.4 4
5.3 odd 4 3850.2.c.y.1849.1 4
5.4 even 2 3850.2.a.bc.1.2 2
7.6 odd 2 5390.2.a.bq.1.2 2
11.10 odd 2 8470.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.1 2 1.1 even 1 trivial
3850.2.a.bc.1.2 2 5.4 even 2
3850.2.c.y.1849.1 4 5.3 odd 4
3850.2.c.y.1849.4 4 5.2 odd 4
5390.2.a.bq.1.2 2 7.6 odd 2
6160.2.a.r.1.1 2 4.3 odd 2
6930.2.a.bo.1.1 2 3.2 odd 2
8470.2.a.bu.1.2 2 11.10 odd 2