Properties

Label 2842.2.a.z.1.1
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1019601.1
Defining polynomial: \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.298978\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.44245 q^{3} +1.00000 q^{4} +4.14347 q^{5} -2.44245 q^{6} +1.00000 q^{8} +2.96555 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.44245 q^{3} +1.00000 q^{4} +4.14347 q^{5} -2.44245 q^{6} +1.00000 q^{8} +2.96555 q^{9} +4.14347 q^{10} -4.70697 q^{11} -2.44245 q^{12} -1.91061 q^{13} -10.1202 q^{15} +1.00000 q^{16} +2.76714 q^{17} +2.96555 q^{18} +4.04040 q^{19} +4.14347 q^{20} -4.70697 q^{22} -2.17268 q^{23} -2.44245 q^{24} +12.1683 q^{25} -1.91061 q^{26} +0.0841506 q^{27} -1.00000 q^{29} -10.1202 q^{30} -0.969044 q^{31} +1.00000 q^{32} +11.4965 q^{33} +2.76714 q^{34} +2.96555 q^{36} -1.47340 q^{37} +4.04040 q^{38} +4.66657 q^{39} +4.14347 q^{40} +10.4741 q^{41} +7.84520 q^{43} -4.70697 q^{44} +12.2876 q^{45} -2.17268 q^{46} +11.7294 q^{47} -2.44245 q^{48} +12.1683 q^{50} -6.75860 q^{51} -1.91061 q^{52} -2.78082 q^{53} +0.0841506 q^{54} -19.5032 q^{55} -9.86847 q^{57} -1.00000 q^{58} +3.41673 q^{59} -10.1202 q^{60} +11.4192 q^{61} -0.969044 q^{62} +1.00000 q^{64} -7.91656 q^{65} +11.4965 q^{66} -2.27575 q^{67} +2.76714 q^{68} +5.30667 q^{69} -6.24653 q^{71} +2.96555 q^{72} -13.5230 q^{73} -1.47340 q^{74} -29.7205 q^{75} +4.04040 q^{76} +4.66657 q^{78} -4.46293 q^{79} +4.14347 q^{80} -9.10217 q^{81} +10.4741 q^{82} -6.06003 q^{83} +11.4656 q^{85} +7.84520 q^{86} +2.44245 q^{87} -4.70697 q^{88} +13.2662 q^{89} +12.2876 q^{90} -2.17268 q^{92} +2.36684 q^{93} +11.7294 q^{94} +16.7413 q^{95} -2.44245 q^{96} +16.8452 q^{97} -13.9587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 3 q^{3} + 5 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} + 3 q^{3} + 5 q^{4} + 7 q^{5} + 3 q^{6} + 5 q^{8} + 8 q^{9} + 7 q^{10} + 3 q^{12} + 10 q^{13} - 10 q^{15} + 5 q^{16} + 8 q^{17} + 8 q^{18} + 2 q^{19} + 7 q^{20} + q^{23} + 3 q^{24} + 12 q^{25} + 10 q^{26} + 15 q^{27} - 5 q^{29} - 10 q^{30} + 11 q^{31} + 5 q^{32} + 9 q^{33} + 8 q^{34} + 8 q^{36} - 8 q^{37} + 2 q^{38} + 18 q^{39} + 7 q^{40} + 23 q^{41} - 3 q^{43} + 4 q^{45} + q^{46} + 16 q^{47} + 3 q^{48} + 12 q^{50} + 7 q^{51} + 10 q^{52} + 7 q^{53} + 15 q^{54} + 6 q^{55} - 34 q^{57} - 5 q^{58} - 9 q^{59} - 10 q^{60} + 15 q^{61} + 11 q^{62} + 5 q^{64} + 5 q^{65} + 9 q^{66} - 4 q^{67} + 8 q^{68} + 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} - 34 q^{75} + 2 q^{76} + 18 q^{78} - 13 q^{79} + 7 q^{80} + 17 q^{81} + 23 q^{82} + 28 q^{83} - 7 q^{85} - 3 q^{86} - 3 q^{87} + 17 q^{89} + 4 q^{90} + q^{92} + 17 q^{93} + 16 q^{94} - 9 q^{95} + 3 q^{96} + 42 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.44245 −1.41015 −0.705074 0.709134i \(-0.749086\pi\)
−0.705074 + 0.709134i \(0.749086\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.14347 1.85302 0.926508 0.376276i \(-0.122795\pi\)
0.926508 + 0.376276i \(0.122795\pi\)
\(6\) −2.44245 −0.997125
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 2.96555 0.988516
\(10\) 4.14347 1.31028
\(11\) −4.70697 −1.41921 −0.709603 0.704602i \(-0.751126\pi\)
−0.709603 + 0.704602i \(0.751126\pi\)
\(12\) −2.44245 −0.705074
\(13\) −1.91061 −0.529908 −0.264954 0.964261i \(-0.585357\pi\)
−0.264954 + 0.964261i \(0.585357\pi\)
\(14\) 0 0
\(15\) −10.1202 −2.61302
\(16\) 1.00000 0.250000
\(17\) 2.76714 0.671131 0.335565 0.942017i \(-0.391073\pi\)
0.335565 + 0.942017i \(0.391073\pi\)
\(18\) 2.96555 0.698986
\(19\) 4.04040 0.926932 0.463466 0.886115i \(-0.346606\pi\)
0.463466 + 0.886115i \(0.346606\pi\)
\(20\) 4.14347 0.926508
\(21\) 0 0
\(22\) −4.70697 −1.00353
\(23\) −2.17268 −0.453036 −0.226518 0.974007i \(-0.572734\pi\)
−0.226518 + 0.974007i \(0.572734\pi\)
\(24\) −2.44245 −0.498562
\(25\) 12.1683 2.43367
\(26\) −1.91061 −0.374702
\(27\) 0.0841506 0.0161948
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) −10.1202 −1.84769
\(31\) −0.969044 −0.174045 −0.0870227 0.996206i \(-0.527735\pi\)
−0.0870227 + 0.996206i \(0.527735\pi\)
\(32\) 1.00000 0.176777
\(33\) 11.4965 2.00129
\(34\) 2.76714 0.474561
\(35\) 0 0
\(36\) 2.96555 0.494258
\(37\) −1.47340 −0.242226 −0.121113 0.992639i \(-0.538646\pi\)
−0.121113 + 0.992639i \(0.538646\pi\)
\(38\) 4.04040 0.655440
\(39\) 4.66657 0.747249
\(40\) 4.14347 0.655140
\(41\) 10.4741 1.63578 0.817891 0.575373i \(-0.195143\pi\)
0.817891 + 0.575373i \(0.195143\pi\)
\(42\) 0 0
\(43\) 7.84520 1.19638 0.598191 0.801353i \(-0.295886\pi\)
0.598191 + 0.801353i \(0.295886\pi\)
\(44\) −4.70697 −0.709603
\(45\) 12.2876 1.83173
\(46\) −2.17268 −0.320345
\(47\) 11.7294 1.71091 0.855453 0.517880i \(-0.173279\pi\)
0.855453 + 0.517880i \(0.173279\pi\)
\(48\) −2.44245 −0.352537
\(49\) 0 0
\(50\) 12.1683 1.72086
\(51\) −6.75860 −0.946393
\(52\) −1.91061 −0.264954
\(53\) −2.78082 −0.381975 −0.190988 0.981592i \(-0.561169\pi\)
−0.190988 + 0.981592i \(0.561169\pi\)
\(54\) 0.0841506 0.0114514
\(55\) −19.5032 −2.62981
\(56\) 0 0
\(57\) −9.86847 −1.30711
\(58\) −1.00000 −0.131306
\(59\) 3.41673 0.444820 0.222410 0.974953i \(-0.428608\pi\)
0.222410 + 0.974953i \(0.428608\pi\)
\(60\) −10.1202 −1.30651
\(61\) 11.4192 1.46208 0.731038 0.682337i \(-0.239036\pi\)
0.731038 + 0.682337i \(0.239036\pi\)
\(62\) −0.969044 −0.123069
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.91656 −0.981929
\(66\) 11.4965 1.41512
\(67\) −2.27575 −0.278027 −0.139014 0.990290i \(-0.544393\pi\)
−0.139014 + 0.990290i \(0.544393\pi\)
\(68\) 2.76714 0.335565
\(69\) 5.30667 0.638848
\(70\) 0 0
\(71\) −6.24653 −0.741327 −0.370664 0.928767i \(-0.620870\pi\)
−0.370664 + 0.928767i \(0.620870\pi\)
\(72\) 2.96555 0.349493
\(73\) −13.5230 −1.58274 −0.791371 0.611336i \(-0.790632\pi\)
−0.791371 + 0.611336i \(0.790632\pi\)
\(74\) −1.47340 −0.171280
\(75\) −29.7205 −3.43183
\(76\) 4.04040 0.463466
\(77\) 0 0
\(78\) 4.66657 0.528385
\(79\) −4.46293 −0.502119 −0.251059 0.967972i \(-0.580779\pi\)
−0.251059 + 0.967972i \(0.580779\pi\)
\(80\) 4.14347 0.463254
\(81\) −9.10217 −1.01135
\(82\) 10.4741 1.15667
\(83\) −6.06003 −0.665175 −0.332587 0.943072i \(-0.607922\pi\)
−0.332587 + 0.943072i \(0.607922\pi\)
\(84\) 0 0
\(85\) 11.4656 1.24362
\(86\) 7.84520 0.845970
\(87\) 2.44245 0.261858
\(88\) −4.70697 −0.501765
\(89\) 13.2662 1.40621 0.703105 0.711086i \(-0.251796\pi\)
0.703105 + 0.711086i \(0.251796\pi\)
\(90\) 12.2876 1.29523
\(91\) 0 0
\(92\) −2.17268 −0.226518
\(93\) 2.36684 0.245430
\(94\) 11.7294 1.20979
\(95\) 16.7413 1.71762
\(96\) −2.44245 −0.249281
\(97\) 16.8452 1.71037 0.855186 0.518322i \(-0.173443\pi\)
0.855186 + 0.518322i \(0.173443\pi\)
\(98\) 0 0
\(99\) −13.9587 −1.40291
\(100\) 12.1683 1.21683
\(101\) −12.4261 −1.23644 −0.618219 0.786006i \(-0.712146\pi\)
−0.618219 + 0.786006i \(0.712146\pi\)
\(102\) −6.75860 −0.669201
\(103\) −16.1718 −1.59346 −0.796729 0.604337i \(-0.793438\pi\)
−0.796729 + 0.604337i \(0.793438\pi\)
\(104\) −1.91061 −0.187351
\(105\) 0 0
\(106\) −2.78082 −0.270097
\(107\) 10.6246 1.02712 0.513558 0.858055i \(-0.328327\pi\)
0.513558 + 0.858055i \(0.328327\pi\)
\(108\) 0.0841506 0.00809739
\(109\) 6.09884 0.584162 0.292081 0.956394i \(-0.405652\pi\)
0.292081 + 0.956394i \(0.405652\pi\)
\(110\) −19.5032 −1.85956
\(111\) 3.59871 0.341574
\(112\) 0 0
\(113\) 14.5875 1.37228 0.686139 0.727471i \(-0.259304\pi\)
0.686139 + 0.727471i \(0.259304\pi\)
\(114\) −9.86847 −0.924267
\(115\) −9.00245 −0.839483
\(116\) −1.00000 −0.0928477
\(117\) −5.66601 −0.523823
\(118\) 3.41673 0.314535
\(119\) 0 0
\(120\) −10.1202 −0.923844
\(121\) 11.1556 1.01414
\(122\) 11.4192 1.03384
\(123\) −25.5825 −2.30669
\(124\) −0.969044 −0.0870227
\(125\) 29.7018 2.65661
\(126\) 0 0
\(127\) 5.46381 0.484835 0.242418 0.970172i \(-0.422060\pi\)
0.242418 + 0.970172i \(0.422060\pi\)
\(128\) 1.00000 0.0883883
\(129\) −19.1615 −1.68708
\(130\) −7.91656 −0.694328
\(131\) −13.4182 −1.17235 −0.586176 0.810184i \(-0.699367\pi\)
−0.586176 + 0.810184i \(0.699367\pi\)
\(132\) 11.4965 1.00064
\(133\) 0 0
\(134\) −2.27575 −0.196595
\(135\) 0.348675 0.0300092
\(136\) 2.76714 0.237281
\(137\) 17.1032 1.46123 0.730613 0.682791i \(-0.239234\pi\)
0.730613 + 0.682791i \(0.239234\pi\)
\(138\) 5.30667 0.451734
\(139\) −3.90711 −0.331397 −0.165699 0.986176i \(-0.552988\pi\)
−0.165699 + 0.986176i \(0.552988\pi\)
\(140\) 0 0
\(141\) −28.6484 −2.41263
\(142\) −6.24653 −0.524198
\(143\) 8.99320 0.752049
\(144\) 2.96555 0.247129
\(145\) −4.14347 −0.344096
\(146\) −13.5230 −1.11917
\(147\) 0 0
\(148\) −1.47340 −0.121113
\(149\) −7.49827 −0.614282 −0.307141 0.951664i \(-0.599372\pi\)
−0.307141 + 0.951664i \(0.599372\pi\)
\(150\) −29.7205 −2.42667
\(151\) 9.41817 0.766440 0.383220 0.923657i \(-0.374815\pi\)
0.383220 + 0.923657i \(0.374815\pi\)
\(152\) 4.04040 0.327720
\(153\) 8.20609 0.663423
\(154\) 0 0
\(155\) −4.01520 −0.322509
\(156\) 4.66657 0.373625
\(157\) 17.8286 1.42288 0.711439 0.702747i \(-0.248044\pi\)
0.711439 + 0.702747i \(0.248044\pi\)
\(158\) −4.46293 −0.355051
\(159\) 6.79201 0.538641
\(160\) 4.14347 0.327570
\(161\) 0 0
\(162\) −9.10217 −0.715134
\(163\) −10.8789 −0.852102 −0.426051 0.904699i \(-0.640096\pi\)
−0.426051 + 0.904699i \(0.640096\pi\)
\(164\) 10.4741 0.817891
\(165\) 47.6355 3.70842
\(166\) −6.06003 −0.470350
\(167\) 16.0326 1.24064 0.620318 0.784350i \(-0.287003\pi\)
0.620318 + 0.784350i \(0.287003\pi\)
\(168\) 0 0
\(169\) −9.34956 −0.719197
\(170\) 11.4656 0.879369
\(171\) 11.9820 0.916287
\(172\) 7.84520 0.598191
\(173\) −6.18878 −0.470524 −0.235262 0.971932i \(-0.575595\pi\)
−0.235262 + 0.971932i \(0.575595\pi\)
\(174\) 2.44245 0.185161
\(175\) 0 0
\(176\) −4.70697 −0.354801
\(177\) −8.34518 −0.627262
\(178\) 13.2662 0.994341
\(179\) 1.20540 0.0900957 0.0450479 0.998985i \(-0.485656\pi\)
0.0450479 + 0.998985i \(0.485656\pi\)
\(180\) 12.2876 0.915867
\(181\) 6.95105 0.516668 0.258334 0.966056i \(-0.416827\pi\)
0.258334 + 0.966056i \(0.416827\pi\)
\(182\) 0 0
\(183\) −27.8907 −2.06174
\(184\) −2.17268 −0.160172
\(185\) −6.10500 −0.448848
\(186\) 2.36684 0.173545
\(187\) −13.0249 −0.952472
\(188\) 11.7294 0.855453
\(189\) 0 0
\(190\) 16.7413 1.21454
\(191\) −6.80847 −0.492644 −0.246322 0.969188i \(-0.579222\pi\)
−0.246322 + 0.969188i \(0.579222\pi\)
\(192\) −2.44245 −0.176268
\(193\) −3.89664 −0.280486 −0.140243 0.990117i \(-0.544788\pi\)
−0.140243 + 0.990117i \(0.544788\pi\)
\(194\) 16.8452 1.20942
\(195\) 19.3358 1.38466
\(196\) 0 0
\(197\) 15.5624 1.10877 0.554386 0.832260i \(-0.312953\pi\)
0.554386 + 0.832260i \(0.312953\pi\)
\(198\) −13.9587 −0.992005
\(199\) 24.5735 1.74197 0.870986 0.491308i \(-0.163481\pi\)
0.870986 + 0.491308i \(0.163481\pi\)
\(200\) 12.1683 0.860431
\(201\) 5.55840 0.392059
\(202\) −12.4261 −0.874294
\(203\) 0 0
\(204\) −6.75860 −0.473197
\(205\) 43.3992 3.03113
\(206\) −16.1718 −1.12674
\(207\) −6.44320 −0.447833
\(208\) −1.91061 −0.132477
\(209\) −19.0181 −1.31551
\(210\) 0 0
\(211\) 8.65646 0.595935 0.297968 0.954576i \(-0.403691\pi\)
0.297968 + 0.954576i \(0.403691\pi\)
\(212\) −2.78082 −0.190988
\(213\) 15.2568 1.04538
\(214\) 10.6246 0.726281
\(215\) 32.5064 2.21691
\(216\) 0.0841506 0.00572572
\(217\) 0 0
\(218\) 6.09884 0.413065
\(219\) 33.0291 2.23190
\(220\) −19.5032 −1.31490
\(221\) −5.28694 −0.355638
\(222\) 3.59871 0.241529
\(223\) −10.9191 −0.731194 −0.365597 0.930773i \(-0.619135\pi\)
−0.365597 + 0.930773i \(0.619135\pi\)
\(224\) 0 0
\(225\) 36.0858 2.40572
\(226\) 14.5875 0.970347
\(227\) −10.4951 −0.696583 −0.348292 0.937386i \(-0.613238\pi\)
−0.348292 + 0.937386i \(0.613238\pi\)
\(228\) −9.86847 −0.653555
\(229\) −7.51525 −0.496622 −0.248311 0.968680i \(-0.579875\pi\)
−0.248311 + 0.968680i \(0.579875\pi\)
\(230\) −9.00245 −0.593604
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) 5.14262 0.336904 0.168452 0.985710i \(-0.446123\pi\)
0.168452 + 0.985710i \(0.446123\pi\)
\(234\) −5.66601 −0.370399
\(235\) 48.6003 3.17034
\(236\) 3.41673 0.222410
\(237\) 10.9005 0.708061
\(238\) 0 0
\(239\) 0.749673 0.0484923 0.0242462 0.999706i \(-0.492281\pi\)
0.0242462 + 0.999706i \(0.492281\pi\)
\(240\) −10.1202 −0.653256
\(241\) −9.01959 −0.581003 −0.290501 0.956875i \(-0.593822\pi\)
−0.290501 + 0.956875i \(0.593822\pi\)
\(242\) 11.1556 0.717108
\(243\) 21.9791 1.40996
\(244\) 11.4192 0.731038
\(245\) 0 0
\(246\) −25.5825 −1.63108
\(247\) −7.71964 −0.491189
\(248\) −0.969044 −0.0615344
\(249\) 14.8013 0.937994
\(250\) 29.7018 1.87850
\(251\) −29.1169 −1.83784 −0.918921 0.394441i \(-0.870938\pi\)
−0.918921 + 0.394441i \(0.870938\pi\)
\(252\) 0 0
\(253\) 10.2268 0.642951
\(254\) 5.46381 0.342830
\(255\) −28.0041 −1.75368
\(256\) 1.00000 0.0625000
\(257\) −23.2275 −1.44889 −0.724447 0.689330i \(-0.757905\pi\)
−0.724447 + 0.689330i \(0.757905\pi\)
\(258\) −19.1615 −1.19294
\(259\) 0 0
\(260\) −7.91656 −0.490964
\(261\) −2.96555 −0.183563
\(262\) −13.4182 −0.828978
\(263\) −20.5033 −1.26429 −0.632145 0.774850i \(-0.717825\pi\)
−0.632145 + 0.774850i \(0.717825\pi\)
\(264\) 11.4965 0.707562
\(265\) −11.5222 −0.707806
\(266\) 0 0
\(267\) −32.4019 −1.98296
\(268\) −2.27575 −0.139014
\(269\) −11.3952 −0.694778 −0.347389 0.937721i \(-0.612932\pi\)
−0.347389 + 0.937721i \(0.612932\pi\)
\(270\) 0.348675 0.0212197
\(271\) 17.9751 1.09191 0.545954 0.837815i \(-0.316167\pi\)
0.545954 + 0.837815i \(0.316167\pi\)
\(272\) 2.76714 0.167783
\(273\) 0 0
\(274\) 17.1032 1.03324
\(275\) −57.2760 −3.45387
\(276\) 5.30667 0.319424
\(277\) −2.08032 −0.124994 −0.0624971 0.998045i \(-0.519906\pi\)
−0.0624971 + 0.998045i \(0.519906\pi\)
\(278\) −3.90711 −0.234333
\(279\) −2.87375 −0.172047
\(280\) 0 0
\(281\) 7.35741 0.438906 0.219453 0.975623i \(-0.429573\pi\)
0.219453 + 0.975623i \(0.429573\pi\)
\(282\) −28.6484 −1.70599
\(283\) −6.08053 −0.361450 −0.180725 0.983534i \(-0.557844\pi\)
−0.180725 + 0.983534i \(0.557844\pi\)
\(284\) −6.24653 −0.370664
\(285\) −40.8897 −2.42210
\(286\) 8.99320 0.531779
\(287\) 0 0
\(288\) 2.96555 0.174747
\(289\) −9.34292 −0.549583
\(290\) −4.14347 −0.243313
\(291\) −41.1435 −2.41188
\(292\) −13.5230 −0.791371
\(293\) 13.7159 0.801291 0.400646 0.916233i \(-0.368786\pi\)
0.400646 + 0.916233i \(0.368786\pi\)
\(294\) 0 0
\(295\) 14.1571 0.824259
\(296\) −1.47340 −0.0856398
\(297\) −0.396094 −0.0229837
\(298\) −7.49827 −0.434363
\(299\) 4.15116 0.240068
\(300\) −29.7205 −1.71591
\(301\) 0 0
\(302\) 9.41817 0.541955
\(303\) 30.3500 1.74356
\(304\) 4.04040 0.231733
\(305\) 47.3150 2.70925
\(306\) 8.20609 0.469111
\(307\) 7.64494 0.436320 0.218160 0.975913i \(-0.429995\pi\)
0.218160 + 0.975913i \(0.429995\pi\)
\(308\) 0 0
\(309\) 39.4988 2.24701
\(310\) −4.01520 −0.228048
\(311\) 24.4252 1.38502 0.692512 0.721406i \(-0.256504\pi\)
0.692512 + 0.721406i \(0.256504\pi\)
\(312\) 4.66657 0.264192
\(313\) −18.2645 −1.03237 −0.516185 0.856477i \(-0.672648\pi\)
−0.516185 + 0.856477i \(0.672648\pi\)
\(314\) 17.8286 1.00613
\(315\) 0 0
\(316\) −4.46293 −0.251059
\(317\) −10.4217 −0.585342 −0.292671 0.956213i \(-0.594544\pi\)
−0.292671 + 0.956213i \(0.594544\pi\)
\(318\) 6.79201 0.380877
\(319\) 4.70697 0.263540
\(320\) 4.14347 0.231627
\(321\) −25.9500 −1.44839
\(322\) 0 0
\(323\) 11.1804 0.622093
\(324\) −9.10217 −0.505676
\(325\) −23.2490 −1.28962
\(326\) −10.8789 −0.602527
\(327\) −14.8961 −0.823755
\(328\) 10.4741 0.578336
\(329\) 0 0
\(330\) 47.6355 2.62225
\(331\) 4.68351 0.257429 0.128715 0.991682i \(-0.458915\pi\)
0.128715 + 0.991682i \(0.458915\pi\)
\(332\) −6.06003 −0.332587
\(333\) −4.36944 −0.239444
\(334\) 16.0326 0.877262
\(335\) −9.42950 −0.515189
\(336\) 0 0
\(337\) −25.3719 −1.38210 −0.691048 0.722809i \(-0.742851\pi\)
−0.691048 + 0.722809i \(0.742851\pi\)
\(338\) −9.34956 −0.508549
\(339\) −35.6292 −1.93511
\(340\) 11.4656 0.621808
\(341\) 4.56126 0.247006
\(342\) 11.9820 0.647913
\(343\) 0 0
\(344\) 7.84520 0.422985
\(345\) 21.9880 1.18379
\(346\) −6.18878 −0.332711
\(347\) −1.93057 −0.103639 −0.0518193 0.998656i \(-0.516502\pi\)
−0.0518193 + 0.998656i \(0.516502\pi\)
\(348\) 2.44245 0.130929
\(349\) 1.95782 0.104800 0.0523998 0.998626i \(-0.483313\pi\)
0.0523998 + 0.998626i \(0.483313\pi\)
\(350\) 0 0
\(351\) −0.160779 −0.00858175
\(352\) −4.70697 −0.250882
\(353\) −16.3330 −0.869317 −0.434658 0.900595i \(-0.643131\pi\)
−0.434658 + 0.900595i \(0.643131\pi\)
\(354\) −8.34518 −0.443541
\(355\) −25.8823 −1.37369
\(356\) 13.2662 0.703105
\(357\) 0 0
\(358\) 1.20540 0.0637073
\(359\) −11.7801 −0.621729 −0.310865 0.950454i \(-0.600619\pi\)
−0.310865 + 0.950454i \(0.600619\pi\)
\(360\) 12.2876 0.647616
\(361\) −2.67514 −0.140797
\(362\) 6.95105 0.365339
\(363\) −27.2469 −1.43009
\(364\) 0 0
\(365\) −56.0320 −2.93285
\(366\) −27.8907 −1.45787
\(367\) 35.6438 1.86059 0.930295 0.366813i \(-0.119551\pi\)
0.930295 + 0.366813i \(0.119551\pi\)
\(368\) −2.17268 −0.113259
\(369\) 31.0615 1.61700
\(370\) −6.10500 −0.317384
\(371\) 0 0
\(372\) 2.36684 0.122715
\(373\) 17.7483 0.918971 0.459486 0.888185i \(-0.348034\pi\)
0.459486 + 0.888185i \(0.348034\pi\)
\(374\) −13.0249 −0.673500
\(375\) −72.5450 −3.74621
\(376\) 11.7294 0.604897
\(377\) 1.91061 0.0984015
\(378\) 0 0
\(379\) −15.2263 −0.782124 −0.391062 0.920364i \(-0.627892\pi\)
−0.391062 + 0.920364i \(0.627892\pi\)
\(380\) 16.7413 0.858810
\(381\) −13.3451 −0.683689
\(382\) −6.80847 −0.348352
\(383\) −15.8118 −0.807943 −0.403971 0.914772i \(-0.632371\pi\)
−0.403971 + 0.914772i \(0.632371\pi\)
\(384\) −2.44245 −0.124641
\(385\) 0 0
\(386\) −3.89664 −0.198334
\(387\) 23.2653 1.18264
\(388\) 16.8452 0.855186
\(389\) −21.7172 −1.10110 −0.550552 0.834801i \(-0.685583\pi\)
−0.550552 + 0.834801i \(0.685583\pi\)
\(390\) 19.3358 0.979105
\(391\) −6.01213 −0.304047
\(392\) 0 0
\(393\) 32.7732 1.65319
\(394\) 15.5624 0.784020
\(395\) −18.4920 −0.930434
\(396\) −13.9587 −0.701453
\(397\) 1.26033 0.0632543 0.0316272 0.999500i \(-0.489931\pi\)
0.0316272 + 0.999500i \(0.489931\pi\)
\(398\) 24.5735 1.23176
\(399\) 0 0
\(400\) 12.1683 0.608417
\(401\) −1.91181 −0.0954713 −0.0477356 0.998860i \(-0.515201\pi\)
−0.0477356 + 0.998860i \(0.515201\pi\)
\(402\) 5.55840 0.277228
\(403\) 1.85147 0.0922282
\(404\) −12.4261 −0.618219
\(405\) −37.7146 −1.87405
\(406\) 0 0
\(407\) 6.93526 0.343768
\(408\) −6.75860 −0.334601
\(409\) −3.01815 −0.149238 −0.0746189 0.997212i \(-0.523774\pi\)
−0.0746189 + 0.997212i \(0.523774\pi\)
\(410\) 43.3992 2.14333
\(411\) −41.7737 −2.06054
\(412\) −16.1718 −0.796729
\(413\) 0 0
\(414\) −6.44320 −0.316666
\(415\) −25.1095 −1.23258
\(416\) −1.91061 −0.0936755
\(417\) 9.54292 0.467319
\(418\) −19.0181 −0.930204
\(419\) 4.64504 0.226925 0.113463 0.993542i \(-0.463806\pi\)
0.113463 + 0.993542i \(0.463806\pi\)
\(420\) 0 0
\(421\) −26.6202 −1.29739 −0.648695 0.761048i \(-0.724685\pi\)
−0.648695 + 0.761048i \(0.724685\pi\)
\(422\) 8.65646 0.421390
\(423\) 34.7840 1.69126
\(424\) −2.78082 −0.135049
\(425\) 33.6715 1.63331
\(426\) 15.2568 0.739196
\(427\) 0 0
\(428\) 10.6246 0.513558
\(429\) −21.9654 −1.06050
\(430\) 32.5064 1.56760
\(431\) −9.63869 −0.464279 −0.232140 0.972682i \(-0.574573\pi\)
−0.232140 + 0.972682i \(0.574573\pi\)
\(432\) 0.0841506 0.00404870
\(433\) −3.16466 −0.152084 −0.0760420 0.997105i \(-0.524228\pi\)
−0.0760420 + 0.997105i \(0.524228\pi\)
\(434\) 0 0
\(435\) 10.1202 0.485227
\(436\) 6.09884 0.292081
\(437\) −8.77852 −0.419934
\(438\) 33.0291 1.57819
\(439\) 13.8900 0.662933 0.331467 0.943467i \(-0.392457\pi\)
0.331467 + 0.943467i \(0.392457\pi\)
\(440\) −19.5032 −0.929778
\(441\) 0 0
\(442\) −5.28694 −0.251474
\(443\) 20.3708 0.967845 0.483923 0.875111i \(-0.339212\pi\)
0.483923 + 0.875111i \(0.339212\pi\)
\(444\) 3.59871 0.170787
\(445\) 54.9679 2.60573
\(446\) −10.9191 −0.517032
\(447\) 18.3141 0.866228
\(448\) 0 0
\(449\) 33.1442 1.56417 0.782086 0.623170i \(-0.214156\pi\)
0.782086 + 0.623170i \(0.214156\pi\)
\(450\) 36.0858 1.70110
\(451\) −49.3014 −2.32151
\(452\) 14.5875 0.686139
\(453\) −23.0034 −1.08079
\(454\) −10.4951 −0.492559
\(455\) 0 0
\(456\) −9.86847 −0.462133
\(457\) −18.2087 −0.851767 −0.425883 0.904778i \(-0.640037\pi\)
−0.425883 + 0.904778i \(0.640037\pi\)
\(458\) −7.51525 −0.351165
\(459\) 0.232857 0.0108688
\(460\) −9.00245 −0.419741
\(461\) −3.52484 −0.164168 −0.0820841 0.996625i \(-0.526158\pi\)
−0.0820841 + 0.996625i \(0.526158\pi\)
\(462\) 0 0
\(463\) 10.7653 0.500304 0.250152 0.968207i \(-0.419519\pi\)
0.250152 + 0.968207i \(0.419519\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 9.80692 0.454785
\(466\) 5.14262 0.238227
\(467\) 12.0199 0.556214 0.278107 0.960550i \(-0.410293\pi\)
0.278107 + 0.960550i \(0.410293\pi\)
\(468\) −5.66601 −0.261911
\(469\) 0 0
\(470\) 48.6003 2.24177
\(471\) −43.5455 −2.00647
\(472\) 3.41673 0.157268
\(473\) −36.9271 −1.69791
\(474\) 10.9005 0.500675
\(475\) 49.1650 2.25584
\(476\) 0 0
\(477\) −8.24666 −0.377588
\(478\) 0.749673 0.0342893
\(479\) 8.29202 0.378872 0.189436 0.981893i \(-0.439334\pi\)
0.189436 + 0.981893i \(0.439334\pi\)
\(480\) −10.1202 −0.461922
\(481\) 2.81510 0.128358
\(482\) −9.01959 −0.410831
\(483\) 0 0
\(484\) 11.1556 0.507072
\(485\) 69.7976 3.16934
\(486\) 21.9791 0.996993
\(487\) 7.89345 0.357687 0.178843 0.983878i \(-0.442764\pi\)
0.178843 + 0.983878i \(0.442764\pi\)
\(488\) 11.4192 0.516922
\(489\) 26.5711 1.20159
\(490\) 0 0
\(491\) −37.7250 −1.70250 −0.851252 0.524757i \(-0.824156\pi\)
−0.851252 + 0.524757i \(0.824156\pi\)
\(492\) −25.5825 −1.15335
\(493\) −2.76714 −0.124626
\(494\) −7.71964 −0.347323
\(495\) −57.8376 −2.59961
\(496\) −0.969044 −0.0435114
\(497\) 0 0
\(498\) 14.8013 0.663262
\(499\) −8.47942 −0.379591 −0.189796 0.981824i \(-0.560783\pi\)
−0.189796 + 0.981824i \(0.560783\pi\)
\(500\) 29.7018 1.32830
\(501\) −39.1587 −1.74948
\(502\) −29.1169 −1.29955
\(503\) −25.2796 −1.12716 −0.563581 0.826061i \(-0.690577\pi\)
−0.563581 + 0.826061i \(0.690577\pi\)
\(504\) 0 0
\(505\) −51.4870 −2.29114
\(506\) 10.2268 0.454635
\(507\) 22.8358 1.01417
\(508\) 5.46381 0.242418
\(509\) 4.96964 0.220275 0.110138 0.993916i \(-0.464871\pi\)
0.110138 + 0.993916i \(0.464871\pi\)
\(510\) −28.0041 −1.24004
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0.340002 0.0150115
\(514\) −23.2275 −1.02452
\(515\) −67.0075 −2.95270
\(516\) −19.1615 −0.843538
\(517\) −55.2099 −2.42813
\(518\) 0 0
\(519\) 15.1158 0.663508
\(520\) −7.91656 −0.347164
\(521\) 26.9628 1.18126 0.590631 0.806942i \(-0.298879\pi\)
0.590631 + 0.806942i \(0.298879\pi\)
\(522\) −2.96555 −0.129798
\(523\) −10.4899 −0.458691 −0.229345 0.973345i \(-0.573659\pi\)
−0.229345 + 0.973345i \(0.573659\pi\)
\(524\) −13.4182 −0.586176
\(525\) 0 0
\(526\) −20.5033 −0.893988
\(527\) −2.68148 −0.116807
\(528\) 11.4965 0.500322
\(529\) −18.2794 −0.794758
\(530\) −11.5222 −0.500494
\(531\) 10.1325 0.439712
\(532\) 0 0
\(533\) −20.0120 −0.866815
\(534\) −32.4019 −1.40217
\(535\) 44.0226 1.90326
\(536\) −2.27575 −0.0982975
\(537\) −2.94412 −0.127048
\(538\) −11.3952 −0.491282
\(539\) 0 0
\(540\) 0.348675 0.0150046
\(541\) −44.5694 −1.91619 −0.958093 0.286456i \(-0.907523\pi\)
−0.958093 + 0.286456i \(0.907523\pi\)
\(542\) 17.9751 0.772095
\(543\) −16.9776 −0.728578
\(544\) 2.76714 0.118640
\(545\) 25.2703 1.08246
\(546\) 0 0
\(547\) 34.8005 1.48796 0.743980 0.668202i \(-0.232936\pi\)
0.743980 + 0.668202i \(0.232936\pi\)
\(548\) 17.1032 0.730613
\(549\) 33.8641 1.44529
\(550\) −57.2760 −2.44226
\(551\) −4.04040 −0.172127
\(552\) 5.30667 0.225867
\(553\) 0 0
\(554\) −2.08032 −0.0883843
\(555\) 14.9111 0.632942
\(556\) −3.90711 −0.165699
\(557\) −33.7665 −1.43073 −0.715365 0.698751i \(-0.753740\pi\)
−0.715365 + 0.698751i \(0.753740\pi\)
\(558\) −2.87375 −0.121655
\(559\) −14.9891 −0.633973
\(560\) 0 0
\(561\) 31.8125 1.34313
\(562\) 7.35741 0.310354
\(563\) 41.2408 1.73809 0.869047 0.494730i \(-0.164733\pi\)
0.869047 + 0.494730i \(0.164733\pi\)
\(564\) −28.6484 −1.20632
\(565\) 60.4429 2.54285
\(566\) −6.08053 −0.255584
\(567\) 0 0
\(568\) −6.24653 −0.262099
\(569\) −25.9185 −1.08656 −0.543281 0.839551i \(-0.682818\pi\)
−0.543281 + 0.839551i \(0.682818\pi\)
\(570\) −40.8897 −1.71268
\(571\) 15.8953 0.665196 0.332598 0.943069i \(-0.392075\pi\)
0.332598 + 0.943069i \(0.392075\pi\)
\(572\) 8.99320 0.376024
\(573\) 16.6293 0.694700
\(574\) 0 0
\(575\) −26.4380 −1.10254
\(576\) 2.96555 0.123564
\(577\) −10.2723 −0.427639 −0.213820 0.976873i \(-0.568591\pi\)
−0.213820 + 0.976873i \(0.568591\pi\)
\(578\) −9.34292 −0.388614
\(579\) 9.51734 0.395527
\(580\) −4.14347 −0.172048
\(581\) 0 0
\(582\) −41.1435 −1.70545
\(583\) 13.0892 0.542101
\(584\) −13.5230 −0.559584
\(585\) −23.4769 −0.970652
\(586\) 13.7159 0.566599
\(587\) −23.9902 −0.990183 −0.495091 0.868841i \(-0.664865\pi\)
−0.495091 + 0.868841i \(0.664865\pi\)
\(588\) 0 0
\(589\) −3.91533 −0.161328
\(590\) 14.1571 0.582839
\(591\) −38.0102 −1.56353
\(592\) −1.47340 −0.0605565
\(593\) −15.5367 −0.638014 −0.319007 0.947752i \(-0.603349\pi\)
−0.319007 + 0.947752i \(0.603349\pi\)
\(594\) −0.396094 −0.0162519
\(595\) 0 0
\(596\) −7.49827 −0.307141
\(597\) −60.0196 −2.45644
\(598\) 4.15116 0.169753
\(599\) −20.8308 −0.851122 −0.425561 0.904930i \(-0.639923\pi\)
−0.425561 + 0.904930i \(0.639923\pi\)
\(600\) −29.7205 −1.21333
\(601\) 17.7149 0.722606 0.361303 0.932448i \(-0.382332\pi\)
0.361303 + 0.932448i \(0.382332\pi\)
\(602\) 0 0
\(603\) −6.74884 −0.274834
\(604\) 9.41817 0.383220
\(605\) 46.2228 1.87922
\(606\) 30.3500 1.23288
\(607\) −2.44517 −0.0992466 −0.0496233 0.998768i \(-0.515802\pi\)
−0.0496233 + 0.998768i \(0.515802\pi\)
\(608\) 4.04040 0.163860
\(609\) 0 0
\(610\) 47.3150 1.91573
\(611\) −22.4103 −0.906624
\(612\) 8.20609 0.331712
\(613\) −1.80610 −0.0729477 −0.0364739 0.999335i \(-0.511613\pi\)
−0.0364739 + 0.999335i \(0.511613\pi\)
\(614\) 7.64494 0.308525
\(615\) −106.000 −4.27434
\(616\) 0 0
\(617\) −15.0917 −0.607567 −0.303784 0.952741i \(-0.598250\pi\)
−0.303784 + 0.952741i \(0.598250\pi\)
\(618\) 39.4988 1.58888
\(619\) −44.5910 −1.79227 −0.896133 0.443786i \(-0.853635\pi\)
−0.896133 + 0.443786i \(0.853635\pi\)
\(620\) −4.01520 −0.161254
\(621\) −0.182833 −0.00733682
\(622\) 24.4252 0.979360
\(623\) 0 0
\(624\) 4.66657 0.186812
\(625\) 62.2267 2.48907
\(626\) −18.2645 −0.729996
\(627\) 46.4506 1.85506
\(628\) 17.8286 0.711439
\(629\) −4.07712 −0.162565
\(630\) 0 0
\(631\) −14.3265 −0.570328 −0.285164 0.958479i \(-0.592048\pi\)
−0.285164 + 0.958479i \(0.592048\pi\)
\(632\) −4.46293 −0.177526
\(633\) −21.1429 −0.840357
\(634\) −10.4217 −0.413899
\(635\) 22.6391 0.898407
\(636\) 6.79201 0.269321
\(637\) 0 0
\(638\) 4.70697 0.186351
\(639\) −18.5244 −0.732814
\(640\) 4.14347 0.163785
\(641\) −17.4354 −0.688658 −0.344329 0.938849i \(-0.611894\pi\)
−0.344329 + 0.938849i \(0.611894\pi\)
\(642\) −25.9500 −1.02416
\(643\) −6.99067 −0.275685 −0.137843 0.990454i \(-0.544017\pi\)
−0.137843 + 0.990454i \(0.544017\pi\)
\(644\) 0 0
\(645\) −79.3950 −3.12618
\(646\) 11.1804 0.439886
\(647\) −16.2341 −0.638227 −0.319113 0.947717i \(-0.603385\pi\)
−0.319113 + 0.947717i \(0.603385\pi\)
\(648\) −9.10217 −0.357567
\(649\) −16.0824 −0.631291
\(650\) −23.2490 −0.911899
\(651\) 0 0
\(652\) −10.8789 −0.426051
\(653\) 22.5201 0.881281 0.440641 0.897684i \(-0.354751\pi\)
0.440641 + 0.897684i \(0.354751\pi\)
\(654\) −14.8961 −0.582483
\(655\) −55.5978 −2.17239
\(656\) 10.4741 0.408945
\(657\) −40.1030 −1.56457
\(658\) 0 0
\(659\) −6.71682 −0.261650 −0.130825 0.991405i \(-0.541763\pi\)
−0.130825 + 0.991405i \(0.541763\pi\)
\(660\) 47.6355 1.85421
\(661\) −35.3608 −1.37538 −0.687689 0.726006i \(-0.741375\pi\)
−0.687689 + 0.726006i \(0.741375\pi\)
\(662\) 4.68351 0.182030
\(663\) 12.9131 0.501502
\(664\) −6.06003 −0.235175
\(665\) 0 0
\(666\) −4.36944 −0.169313
\(667\) 2.17268 0.0841267
\(668\) 16.0326 0.620318
\(669\) 26.6692 1.03109
\(670\) −9.42950 −0.364293
\(671\) −53.7498 −2.07499
\(672\) 0 0
\(673\) −23.1388 −0.891935 −0.445967 0.895049i \(-0.647140\pi\)
−0.445967 + 0.895049i \(0.647140\pi\)
\(674\) −25.3719 −0.977290
\(675\) 1.02397 0.0394127
\(676\) −9.34956 −0.359599
\(677\) 22.7344 0.873755 0.436878 0.899521i \(-0.356084\pi\)
0.436878 + 0.899521i \(0.356084\pi\)
\(678\) −35.6292 −1.36833
\(679\) 0 0
\(680\) 11.4656 0.439685
\(681\) 25.6337 0.982285
\(682\) 4.56126 0.174660
\(683\) 20.2928 0.776483 0.388242 0.921558i \(-0.373083\pi\)
0.388242 + 0.921558i \(0.373083\pi\)
\(684\) 11.9820 0.458143
\(685\) 70.8667 2.70768
\(686\) 0 0
\(687\) 18.3556 0.700310
\(688\) 7.84520 0.299096
\(689\) 5.31307 0.202412
\(690\) 21.9880 0.837069
\(691\) 2.01180 0.0765324 0.0382662 0.999268i \(-0.487817\pi\)
0.0382662 + 0.999268i \(0.487817\pi\)
\(692\) −6.18878 −0.235262
\(693\) 0 0
\(694\) −1.93057 −0.0732836
\(695\) −16.1890 −0.614084
\(696\) 2.44245 0.0925807
\(697\) 28.9834 1.09782
\(698\) 1.95782 0.0741045
\(699\) −12.5606 −0.475084
\(700\) 0 0
\(701\) −3.02432 −0.114227 −0.0571136 0.998368i \(-0.518190\pi\)
−0.0571136 + 0.998368i \(0.518190\pi\)
\(702\) −0.160779 −0.00606822
\(703\) −5.95314 −0.224527
\(704\) −4.70697 −0.177401
\(705\) −118.704 −4.47064
\(706\) −16.3330 −0.614700
\(707\) 0 0
\(708\) −8.34518 −0.313631
\(709\) −45.4669 −1.70754 −0.853772 0.520647i \(-0.825691\pi\)
−0.853772 + 0.520647i \(0.825691\pi\)
\(710\) −25.8823 −0.971346
\(711\) −13.2350 −0.496352
\(712\) 13.2662 0.497170
\(713\) 2.10543 0.0788489
\(714\) 0 0
\(715\) 37.2630 1.39356
\(716\) 1.20540 0.0450479
\(717\) −1.83104 −0.0683813
\(718\) −11.7801 −0.439629
\(719\) 30.1441 1.12418 0.562092 0.827075i \(-0.309997\pi\)
0.562092 + 0.827075i \(0.309997\pi\)
\(720\) 12.2876 0.457934
\(721\) 0 0
\(722\) −2.67514 −0.0995585
\(723\) 22.0299 0.819300
\(724\) 6.95105 0.258334
\(725\) −12.1683 −0.451921
\(726\) −27.2469 −1.01123
\(727\) 45.6599 1.69343 0.846717 0.532044i \(-0.178576\pi\)
0.846717 + 0.532044i \(0.178576\pi\)
\(728\) 0 0
\(729\) −26.3763 −0.976901
\(730\) −56.0320 −2.07384
\(731\) 21.7088 0.802929
\(732\) −27.8907 −1.03087
\(733\) −27.3458 −1.01004 −0.505020 0.863107i \(-0.668515\pi\)
−0.505020 + 0.863107i \(0.668515\pi\)
\(734\) 35.6438 1.31564
\(735\) 0 0
\(736\) −2.17268 −0.0800862
\(737\) 10.7119 0.394578
\(738\) 31.0615 1.14339
\(739\) 49.9166 1.83621 0.918106 0.396335i \(-0.129718\pi\)
0.918106 + 0.396335i \(0.129718\pi\)
\(740\) −6.10500 −0.224424
\(741\) 18.8548 0.692649
\(742\) 0 0
\(743\) 28.6655 1.05164 0.525818 0.850597i \(-0.323759\pi\)
0.525818 + 0.850597i \(0.323759\pi\)
\(744\) 2.36684 0.0867725
\(745\) −31.0688 −1.13827
\(746\) 17.7483 0.649811
\(747\) −17.9713 −0.657536
\(748\) −13.0249 −0.476236
\(749\) 0 0
\(750\) −72.5450 −2.64897
\(751\) −19.0316 −0.694473 −0.347236 0.937778i \(-0.612880\pi\)
−0.347236 + 0.937778i \(0.612880\pi\)
\(752\) 11.7294 0.427727
\(753\) 71.1165 2.59163
\(754\) 1.91061 0.0695804
\(755\) 39.0239 1.42023
\(756\) 0 0
\(757\) −9.39694 −0.341538 −0.170769 0.985311i \(-0.554625\pi\)
−0.170769 + 0.985311i \(0.554625\pi\)
\(758\) −15.2263 −0.553045
\(759\) −24.9783 −0.906656
\(760\) 16.7413 0.607270
\(761\) −10.7493 −0.389661 −0.194831 0.980837i \(-0.562416\pi\)
−0.194831 + 0.980837i \(0.562416\pi\)
\(762\) −13.3451 −0.483441
\(763\) 0 0
\(764\) −6.80847 −0.246322
\(765\) 34.0017 1.22933
\(766\) −15.8118 −0.571302
\(767\) −6.52804 −0.235714
\(768\) −2.44245 −0.0881342
\(769\) 39.3774 1.41998 0.709992 0.704209i \(-0.248698\pi\)
0.709992 + 0.704209i \(0.248698\pi\)
\(770\) 0 0
\(771\) 56.7321 2.04316
\(772\) −3.89664 −0.140243
\(773\) −19.0471 −0.685078 −0.342539 0.939504i \(-0.611287\pi\)
−0.342539 + 0.939504i \(0.611287\pi\)
\(774\) 23.2653 0.836254
\(775\) −11.7917 −0.423569
\(776\) 16.8452 0.604708
\(777\) 0 0
\(778\) −21.7172 −0.778598
\(779\) 42.3196 1.51626
\(780\) 19.3358 0.692332
\(781\) 29.4023 1.05210
\(782\) −6.01213 −0.214993
\(783\) −0.0841506 −0.00300730
\(784\) 0 0
\(785\) 73.8723 2.63662
\(786\) 32.7732 1.16898
\(787\) 26.4446 0.942650 0.471325 0.881960i \(-0.343776\pi\)
0.471325 + 0.881960i \(0.343776\pi\)
\(788\) 15.5624 0.554386
\(789\) 50.0783 1.78283
\(790\) −18.4920 −0.657916
\(791\) 0 0
\(792\) −13.9587 −0.496002
\(793\) −21.8176 −0.774767
\(794\) 1.26033 0.0447275
\(795\) 28.1425 0.998111
\(796\) 24.5735 0.870986
\(797\) −42.8772 −1.51879 −0.759395 0.650630i \(-0.774505\pi\)
−0.759395 + 0.650630i \(0.774505\pi\)
\(798\) 0 0
\(799\) 32.4569 1.14824
\(800\) 12.1683 0.430216
\(801\) 39.3414 1.39006
\(802\) −1.91181 −0.0675084
\(803\) 63.6522 2.24624
\(804\) 5.55840 0.196030
\(805\) 0 0
\(806\) 1.85147 0.0652152
\(807\) 27.8322 0.979739
\(808\) −12.4261 −0.437147
\(809\) −1.72392 −0.0606097 −0.0303048 0.999541i \(-0.509648\pi\)
−0.0303048 + 0.999541i \(0.509648\pi\)
\(810\) −37.7146 −1.32515
\(811\) 7.23311 0.253989 0.126995 0.991903i \(-0.459467\pi\)
0.126995 + 0.991903i \(0.459467\pi\)
\(812\) 0 0
\(813\) −43.9031 −1.53975
\(814\) 6.93526 0.243081
\(815\) −45.0764 −1.57896
\(816\) −6.75860 −0.236598
\(817\) 31.6978 1.10896
\(818\) −3.01815 −0.105527
\(819\) 0 0
\(820\) 43.3992 1.51556
\(821\) 30.7040 1.07158 0.535789 0.844352i \(-0.320014\pi\)
0.535789 + 0.844352i \(0.320014\pi\)
\(822\) −41.7737 −1.45703
\(823\) −9.58578 −0.334139 −0.167070 0.985945i \(-0.553430\pi\)
−0.167070 + 0.985945i \(0.553430\pi\)
\(824\) −16.1718 −0.563372
\(825\) 139.894 4.87047
\(826\) 0 0
\(827\) −6.26986 −0.218025 −0.109012 0.994040i \(-0.534769\pi\)
−0.109012 + 0.994040i \(0.534769\pi\)
\(828\) −6.44320 −0.223917
\(829\) 11.2898 0.392110 0.196055 0.980593i \(-0.437187\pi\)
0.196055 + 0.980593i \(0.437187\pi\)
\(830\) −25.1095 −0.871565
\(831\) 5.08107 0.176260
\(832\) −1.91061 −0.0662386
\(833\) 0 0
\(834\) 9.54292 0.330444
\(835\) 66.4304 2.29892
\(836\) −19.0181 −0.657753
\(837\) −0.0815456 −0.00281863
\(838\) 4.64504 0.160460
\(839\) −44.0101 −1.51940 −0.759698 0.650276i \(-0.774653\pi\)
−0.759698 + 0.650276i \(0.774653\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −26.6202 −0.917394
\(843\) −17.9701 −0.618923
\(844\) 8.65646 0.297968
\(845\) −38.7396 −1.33268
\(846\) 34.7840 1.19590
\(847\) 0 0
\(848\) −2.78082 −0.0954938
\(849\) 14.8514 0.509698
\(850\) 33.6715 1.15492
\(851\) 3.20124 0.109737
\(852\) 15.2568 0.522690
\(853\) 10.9272 0.374141 0.187070 0.982347i \(-0.440101\pi\)
0.187070 + 0.982347i \(0.440101\pi\)
\(854\) 0 0
\(855\) 49.6471 1.69789
\(856\) 10.6246 0.363141
\(857\) 29.2754 1.00003 0.500015 0.866017i \(-0.333328\pi\)
0.500015 + 0.866017i \(0.333328\pi\)
\(858\) −21.9654 −0.749887
\(859\) −43.3222 −1.47813 −0.739067 0.673632i \(-0.764733\pi\)
−0.739067 + 0.673632i \(0.764733\pi\)
\(860\) 32.5064 1.10846
\(861\) 0 0
\(862\) −9.63869 −0.328295
\(863\) −43.8381 −1.49227 −0.746133 0.665797i \(-0.768092\pi\)
−0.746133 + 0.665797i \(0.768092\pi\)
\(864\) 0.0841506 0.00286286
\(865\) −25.6430 −0.871888
\(866\) −3.16466 −0.107540
\(867\) 22.8196 0.774994
\(868\) 0 0
\(869\) 21.0069 0.712609
\(870\) 10.1202 0.343107
\(871\) 4.34808 0.147329
\(872\) 6.09884 0.206533
\(873\) 49.9552 1.69073
\(874\) −8.77852 −0.296938
\(875\) 0 0
\(876\) 33.0291 1.11595
\(877\) 29.5072 0.996388 0.498194 0.867066i \(-0.333997\pi\)
0.498194 + 0.867066i \(0.333997\pi\)
\(878\) 13.8900 0.468764
\(879\) −33.5004 −1.12994
\(880\) −19.5032 −0.657452
\(881\) −13.1060 −0.441554 −0.220777 0.975324i \(-0.570859\pi\)
−0.220777 + 0.975324i \(0.570859\pi\)
\(882\) 0 0
\(883\) −7.44435 −0.250522 −0.125261 0.992124i \(-0.539977\pi\)
−0.125261 + 0.992124i \(0.539977\pi\)
\(884\) −5.28694 −0.177819
\(885\) −34.5780 −1.16233
\(886\) 20.3708 0.684370
\(887\) 38.6833 1.29886 0.649430 0.760422i \(-0.275008\pi\)
0.649430 + 0.760422i \(0.275008\pi\)
\(888\) 3.59871 0.120765
\(889\) 0 0
\(890\) 54.9679 1.84253
\(891\) 42.8437 1.43532
\(892\) −10.9191 −0.365597
\(893\) 47.3914 1.58589
\(894\) 18.3141 0.612516
\(895\) 4.99453 0.166949
\(896\) 0 0
\(897\) −10.1390 −0.338531
\(898\) 33.1442 1.10604
\(899\) 0.969044 0.0323194
\(900\) 36.0858 1.20286
\(901\) −7.69493 −0.256355
\(902\) −49.3014 −1.64156
\(903\) 0 0
\(904\) 14.5875 0.485173
\(905\) 28.8015 0.957393
\(906\) −23.0034 −0.764236
\(907\) 2.42625 0.0805624 0.0402812 0.999188i \(-0.487175\pi\)
0.0402812 + 0.999188i \(0.487175\pi\)
\(908\) −10.4951 −0.348292
\(909\) −36.8500 −1.22224
\(910\) 0 0
\(911\) −1.86661 −0.0618435 −0.0309218 0.999522i \(-0.509844\pi\)
−0.0309218 + 0.999522i \(0.509844\pi\)
\(912\) −9.86847 −0.326778
\(913\) 28.5244 0.944020
\(914\) −18.2087 −0.602290
\(915\) −115.564 −3.82044
\(916\) −7.51525 −0.248311
\(917\) 0 0
\(918\) 0.232857 0.00768542
\(919\) 27.4057 0.904031 0.452015 0.892010i \(-0.350705\pi\)
0.452015 + 0.892010i \(0.350705\pi\)
\(920\) −9.00245 −0.296802
\(921\) −18.6724 −0.615275
\(922\) −3.52484 −0.116084
\(923\) 11.9347 0.392836
\(924\) 0 0
\(925\) −17.9289 −0.589497
\(926\) 10.7653 0.353768
\(927\) −47.9583 −1.57516
\(928\) −1.00000 −0.0328266
\(929\) 43.9595 1.44227 0.721133 0.692797i \(-0.243622\pi\)
0.721133 + 0.692797i \(0.243622\pi\)
\(930\) 9.80692 0.321582
\(931\) 0 0
\(932\) 5.14262 0.168452
\(933\) −59.6572 −1.95309
\(934\) 12.0199 0.393303
\(935\) −53.9681 −1.76495
\(936\) −5.66601 −0.185199
\(937\) −29.9446 −0.978246 −0.489123 0.872215i \(-0.662683\pi\)
−0.489123 + 0.872215i \(0.662683\pi\)
\(938\) 0 0
\(939\) 44.6101 1.45579
\(940\) 48.6003 1.58517
\(941\) 21.5723 0.703236 0.351618 0.936144i \(-0.385632\pi\)
0.351618 + 0.936144i \(0.385632\pi\)
\(942\) −43.5455 −1.41879
\(943\) −22.7570 −0.741068
\(944\) 3.41673 0.111205
\(945\) 0 0
\(946\) −36.9271 −1.20061
\(947\) 4.39212 0.142725 0.0713624 0.997450i \(-0.477265\pi\)
0.0713624 + 0.997450i \(0.477265\pi\)
\(948\) 10.9005 0.354031
\(949\) 25.8371 0.838709
\(950\) 49.1650 1.59512
\(951\) 25.4545 0.825418
\(952\) 0 0
\(953\) 32.9052 1.06591 0.532953 0.846145i \(-0.321082\pi\)
0.532953 + 0.846145i \(0.321082\pi\)
\(954\) −8.24666 −0.266995
\(955\) −28.2107 −0.912876
\(956\) 0.749673 0.0242462
\(957\) −11.4965 −0.371630
\(958\) 8.29202 0.267903
\(959\) 0 0
\(960\) −10.1202 −0.326628
\(961\) −30.0610 −0.969708
\(962\) 2.81510 0.0907625
\(963\) 31.5077 1.01532
\(964\) −9.01959 −0.290501
\(965\) −16.1456 −0.519745
\(966\) 0 0
\(967\) 4.50365 0.144827 0.0724137 0.997375i \(-0.476930\pi\)
0.0724137 + 0.997375i \(0.476930\pi\)
\(968\) 11.1556 0.358554
\(969\) −27.3075 −0.877242
\(970\) 69.7976 2.24107
\(971\) 30.4189 0.976187 0.488094 0.872791i \(-0.337692\pi\)
0.488094 + 0.872791i \(0.337692\pi\)
\(972\) 21.9791 0.704981
\(973\) 0 0
\(974\) 7.89345 0.252923
\(975\) 56.7844 1.81856
\(976\) 11.4192 0.365519
\(977\) 37.5202 1.20038 0.600189 0.799858i \(-0.295092\pi\)
0.600189 + 0.799858i \(0.295092\pi\)
\(978\) 26.5711 0.849652
\(979\) −62.4434 −1.99570
\(980\) 0 0
\(981\) 18.0864 0.577454
\(982\) −37.7250 −1.20385
\(983\) 43.4878 1.38705 0.693523 0.720435i \(-0.256058\pi\)
0.693523 + 0.720435i \(0.256058\pi\)
\(984\) −25.5825 −0.815539
\(985\) 64.4821 2.05457
\(986\) −2.76714 −0.0881238
\(987\) 0 0
\(988\) −7.71964 −0.245595
\(989\) −17.0452 −0.542004
\(990\) −57.8376 −1.83820
\(991\) −55.4198 −1.76047 −0.880233 0.474541i \(-0.842614\pi\)
−0.880233 + 0.474541i \(0.842614\pi\)
\(992\) −0.969044 −0.0307672
\(993\) −11.4392 −0.363013
\(994\) 0 0
\(995\) 101.820 3.22790
\(996\) 14.8013 0.468997
\(997\) −23.8889 −0.756570 −0.378285 0.925689i \(-0.623486\pi\)
−0.378285 + 0.925689i \(0.623486\pi\)
\(998\) −8.47942 −0.268411
\(999\) −0.123988 −0.00392280
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 2842.2.a.z.1.1 5
7.2 even 3 406.2.e.a.291.5 yes 10
7.4 even 3 406.2.e.a.233.5 10
7.6 odd 2 2842.2.a.x.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.5 10 7.4 even 3
406.2.e.a.291.5 yes 10 7.2 even 3
2842.2.a.x.1.5 5 7.6 odd 2
2842.2.a.z.1.1 5 1.1 even 1 trivial