Properties

Label 2842.2.a.x.1.5
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
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: \(1\)
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 1.5
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 \(-0.250914\pi\)
0.705074 + 0.709134i \(0.250914\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.14347 −1.85302 −0.926508 0.376276i \(-0.877205\pi\)
−0.926508 + 0.376276i \(0.877205\pi\)
\(6\) 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 \(-0.414643\pi\)
0.264954 + 0.964261i \(0.414643\pi\)
\(14\) 0 0
\(15\) −10.1202 −2.61302
\(16\) 1.00000 0.250000
\(17\) −2.76714 −0.671131 −0.335565 0.942017i \(-0.608927\pi\)
−0.335565 + 0.942017i \(0.608927\pi\)
\(18\) 2.96555 0.698986
\(19\) −4.04040 −0.926932 −0.463466 0.886115i \(-0.653394\pi\)
−0.463466 + 0.886115i \(0.653394\pi\)
\(20\) −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 \(-0.472265\pi\)
0.0870227 + 0.996206i \(0.472265\pi\)
\(32\) 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 \(-0.804857\pi\)
−0.817891 + 0.575373i \(0.804857\pi\)
\(42\) 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 \(-0.826721\pi\)
−0.855453 + 0.517880i \(0.826721\pi\)
\(48\) 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 \(-0.571392\pi\)
−0.222410 + 0.974953i \(0.571392\pi\)
\(60\) −10.1202 −1.30651
\(61\) −11.4192 −1.46208 −0.731038 0.682337i \(-0.760964\pi\)
−0.731038 + 0.682337i \(0.760964\pi\)
\(62\) 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 \(-0.209368\pi\)
0.791371 + 0.611336i \(0.209368\pi\)
\(74\) −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 \(-0.392078\pi\)
0.332587 + 0.943072i \(0.392078\pi\)
\(84\) 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 \(-0.748204\pi\)
−0.703105 + 0.711086i \(0.748204\pi\)
\(90\) −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 \(-0.826557\pi\)
−0.855186 + 0.518322i \(0.826557\pi\)
\(98\) 0 0
\(99\) −13.9587 −1.40291
\(100\) 12.1683 1.21683
\(101\) 12.4261 1.23644 0.618219 0.786006i \(-0.287854\pi\)
0.618219 + 0.786006i \(0.287854\pi\)
\(102\) −6.75860 −0.669201
\(103\) 16.1718 1.59346 0.796729 0.604337i \(-0.206562\pi\)
0.796729 + 0.604337i \(0.206562\pi\)
\(104\) 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 \(-0.300633\pi\)
0.586176 + 0.810184i \(0.300633\pi\)
\(132\) −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 \(-0.447012\pi\)
0.165699 + 0.986176i \(0.447012\pi\)
\(140\) 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 \(-0.751956\pi\)
−0.711439 + 0.702747i \(0.751956\pi\)
\(158\) −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 \(-0.712997\pi\)
−0.620318 + 0.784350i \(0.712997\pi\)
\(168\) 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 \(-0.424405\pi\)
0.235262 + 0.971932i \(0.424405\pi\)
\(174\) −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 \(-0.583173\pi\)
−0.258334 + 0.966056i \(0.583173\pi\)
\(182\) 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 \(-0.836519\pi\)
−0.870986 + 0.491308i \(0.836519\pi\)
\(200\) 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 \(-0.380865\pi\)
0.365597 + 0.930773i \(0.380865\pi\)
\(224\) 0 0
\(225\) 36.0858 2.40572
\(226\) 14.5875 0.970347
\(227\) 10.4951 0.696583 0.348292 0.937386i \(-0.386762\pi\)
0.348292 + 0.937386i \(0.386762\pi\)
\(228\) −9.86847 −0.653555
\(229\) 7.51525 0.496622 0.248311 0.968680i \(-0.420125\pi\)
0.248311 + 0.968680i \(0.420125\pi\)
\(230\) 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 \(-0.406178\pi\)
0.290501 + 0.956875i \(0.406178\pi\)
\(242\) 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 \(-0.129062\pi\)
0.918921 + 0.394441i \(0.129062\pi\)
\(252\) 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 \(-0.242095\pi\)
0.724447 + 0.689330i \(0.242095\pi\)
\(258\) 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 \(-0.387068\pi\)
0.347389 + 0.937721i \(0.387068\pi\)
\(270\) 0.348675 0.0212197
\(271\) −17.9751 −1.09191 −0.545954 0.837815i \(-0.683833\pi\)
−0.545954 + 0.837815i \(0.683833\pi\)
\(272\) −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 \(-0.442156\pi\)
0.180725 + 0.983534i \(0.442156\pi\)
\(284\) −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 \(-0.631214\pi\)
−0.400646 + 0.916233i \(0.631214\pi\)
\(294\) 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 \(-0.570005\pi\)
−0.218160 + 0.975913i \(0.570005\pi\)
\(308\) 0 0
\(309\) 39.4988 2.24701
\(310\) −4.01520 −0.228048
\(311\) −24.4252 −1.38502 −0.692512 0.721406i \(-0.743496\pi\)
−0.692512 + 0.721406i \(0.743496\pi\)
\(312\) 4.66657 0.264192
\(313\) 18.2645 1.03237 0.516185 0.856477i \(-0.327352\pi\)
0.516185 + 0.856477i \(0.327352\pi\)
\(314\) −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 \(-0.516687\pi\)
−0.0523998 + 0.998626i \(0.516687\pi\)
\(350\) 0 0
\(351\) −0.160779 −0.00858175
\(352\) −4.70697 −0.250882
\(353\) 16.3330 0.869317 0.434658 0.900595i \(-0.356869\pi\)
0.434658 + 0.900595i \(0.356869\pi\)
\(354\) −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 \(-0.880449\pi\)
−0.930295 + 0.366813i \(0.880449\pi\)
\(368\) −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 \(-0.367629\pi\)
0.403971 + 0.914772i \(0.367629\pi\)
\(384\) 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 \(-0.510069\pi\)
−0.0316272 + 0.999500i \(0.510069\pi\)
\(398\) −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 \(-0.476226\pi\)
0.0746189 + 0.997212i \(0.476226\pi\)
\(410\) 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 \(-0.536194\pi\)
−0.113463 + 0.993542i \(0.536194\pi\)
\(420\) 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 \(-0.475772\pi\)
0.0760420 + 0.997105i \(0.475772\pi\)
\(434\) 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 \(-0.607543\pi\)
−0.331467 + 0.943467i \(0.607543\pi\)
\(440\) 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 \(-0.473842\pi\)
0.0820841 + 0.996625i \(0.473842\pi\)
\(462\) 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 \(-0.589707\pi\)
−0.278107 + 0.960550i \(0.589707\pi\)
\(468\) 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 \(-0.560666\pi\)
−0.189436 + 0.981893i \(0.560666\pi\)
\(480\) −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 \(-0.309423\pi\)
0.563581 + 0.826061i \(0.309423\pi\)
\(504\) 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 \(-0.535129\pi\)
−0.110138 + 0.993916i \(0.535129\pi\)
\(510\) 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 \(-0.701121\pi\)
−0.590631 + 0.806942i \(0.701121\pi\)
\(522\) −2.96555 −0.129798
\(523\) 10.4899 0.458691 0.229345 0.973345i \(-0.426341\pi\)
0.229345 + 0.973345i \(0.426341\pi\)
\(524\) 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 \(-0.835267\pi\)
−0.869047 + 0.494730i \(0.835267\pi\)
\(564\) −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 \(-0.431409\pi\)
0.213820 + 0.976873i \(0.431409\pi\)
\(578\) −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 \(-0.335135\pi\)
0.495091 + 0.868841i \(0.335135\pi\)
\(588\) 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 \(-0.396651\pi\)
0.319007 + 0.947752i \(0.396651\pi\)
\(594\) 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 \(-0.617668\pi\)
−0.361303 + 0.932448i \(0.617668\pi\)
\(602\) 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 \(-0.484198\pi\)
0.0496233 + 0.998768i \(0.484198\pi\)
\(608\) −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 \(-0.146365\pi\)
0.896133 + 0.443786i \(0.146365\pi\)
\(620\) −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 \(-0.455983\pi\)
0.137843 + 0.990454i \(0.455983\pi\)
\(644\) 0 0
\(645\) −79.3950 −3.12618
\(646\) 11.1804 0.439886
\(647\) 16.2341 0.638227 0.319113 0.947717i \(-0.396615\pi\)
0.319113 + 0.947717i \(0.396615\pi\)
\(648\) −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 \(-0.258625\pi\)
0.687689 + 0.726006i \(0.258625\pi\)
\(662\) 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 \(-0.643916\pi\)
−0.436878 + 0.899521i \(0.643916\pi\)
\(678\) 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 \(-0.512183\pi\)
−0.0382662 + 0.999268i \(0.512183\pi\)
\(692\) 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 \(-0.690003\pi\)
−0.562092 + 0.827075i \(0.690003\pi\)
\(720\) −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 \(-0.821424\pi\)
−0.846717 + 0.532044i \(0.821424\pi\)
\(728\) 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 \(-0.331485\pi\)
0.505020 + 0.863107i \(0.331485\pi\)
\(734\) −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 \(-0.437584\pi\)
0.194831 + 0.980837i \(0.437584\pi\)
\(762\) 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 \(-0.751302\pi\)
−0.709992 + 0.704209i \(0.751302\pi\)
\(770\) 0 0
\(771\) 56.7321 2.04316
\(772\) −3.89664 −0.140243
\(773\) 19.0471 0.685078 0.342539 0.939504i \(-0.388713\pi\)
0.342539 + 0.939504i \(0.388713\pi\)
\(774\) 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 \(-0.656224\pi\)
−0.471325 + 0.881960i \(0.656224\pi\)
\(788\) 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 \(-0.225495\pi\)
0.759395 + 0.650630i \(0.225495\pi\)
\(798\) 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 \(-0.540533\pi\)
−0.126995 + 0.991903i \(0.540533\pi\)
\(812\) 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 \(-0.562813\pi\)
−0.196055 + 0.980593i \(0.562813\pi\)
\(830\) −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 \(-0.225347\pi\)
0.759698 + 0.650276i \(0.225347\pi\)
\(840\) 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 \(-0.559899\pi\)
−0.187070 + 0.982347i \(0.559899\pi\)
\(854\) 0 0
\(855\) 49.6471 1.69789
\(856\) 10.6246 0.363141
\(857\) −29.2754 −1.00003 −0.500015 0.866017i \(-0.666672\pi\)
−0.500015 + 0.866017i \(0.666672\pi\)
\(858\) −21.9654 −0.749887
\(859\) 43.3222 1.47813 0.739067 0.673632i \(-0.235267\pi\)
0.739067 + 0.673632i \(0.235267\pi\)
\(860\) −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 \(-0.429141\pi\)
0.220777 + 0.975324i \(0.429141\pi\)
\(882\) 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 \(-0.724992\pi\)
−0.649430 + 0.760422i \(0.724992\pi\)
\(888\) −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 \(-0.756378\pi\)
−0.721133 + 0.692797i \(0.756378\pi\)
\(930\) −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 \(-0.337317\pi\)
0.489123 + 0.872215i \(0.337317\pi\)
\(938\) 0 0
\(939\) 44.6101 1.45579
\(940\) 48.6003 1.58517
\(941\) −21.5723 −0.703236 −0.351618 0.936144i \(-0.614368\pi\)
−0.351618 + 0.936144i \(0.614368\pi\)
\(942\) −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 \(-0.662308\pi\)
−0.488094 + 0.872791i \(0.662308\pi\)
\(972\) −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 \(-0.743942\pi\)
−0.693523 + 0.720435i \(0.743942\pi\)
\(984\) −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 \(-0.376514\pi\)
0.378285 + 0.925689i \(0.376514\pi\)
\(998\) −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.x.1.5 5
7.3 odd 6 406.2.e.a.233.5 10
7.5 odd 6 406.2.e.a.291.5 yes 10
7.6 odd 2 2842.2.a.z.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.5 10 7.3 odd 6
406.2.e.a.291.5 yes 10 7.5 odd 6
2842.2.a.x.1.5 5 1.1 even 1 trivial
2842.2.a.z.1.1 5 7.6 odd 2