Properties

Label 1805.2.a.n.1.3
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.00000 q^{5} +1.93974 q^{6} -4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +O(q^{10})\) \(q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.00000 q^{5} +1.93974 q^{6} -4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +1.13856 q^{10} +3.46027 q^{11} -1.19882 q^{12} -1.17127 q^{13} -5.41569 q^{14} +1.70367 q^{15} -2.09750 q^{16} -6.20500 q^{17} -0.111014 q^{18} -0.703671 q^{20} -8.10368 q^{21} +3.93974 q^{22} +5.05910 q^{23} -5.24442 q^{24} +1.00000 q^{25} -1.33357 q^{26} -5.27713 q^{27} +3.34708 q^{28} +1.61070 q^{29} +1.93974 q^{30} -7.49400 q^{31} +3.76846 q^{32} +5.89516 q^{33} -7.06479 q^{34} -4.75660 q^{35} +0.0686106 q^{36} -5.98080 q^{37} -1.99547 q^{39} -3.07830 q^{40} -5.43374 q^{41} -9.22656 q^{42} -10.0664 q^{43} -2.43489 q^{44} -0.0975037 q^{45} +5.76011 q^{46} -8.06479 q^{47} -3.57346 q^{48} +15.6252 q^{49} +1.13856 q^{50} -10.5713 q^{51} +0.824193 q^{52} -6.68283 q^{53} -6.00835 q^{54} +3.46027 q^{55} +14.6423 q^{56} +1.83389 q^{58} +2.17229 q^{59} -1.19882 q^{60} +6.20852 q^{61} -8.53240 q^{62} +0.463786 q^{63} +8.48565 q^{64} -1.17127 q^{65} +6.71202 q^{66} -5.62257 q^{67} +4.36628 q^{68} +8.61905 q^{69} -5.41569 q^{70} +2.72287 q^{71} +0.300146 q^{72} +3.15661 q^{73} -6.80953 q^{74} +1.70367 q^{75} -16.4591 q^{77} -2.27197 q^{78} +12.0783 q^{79} -2.09750 q^{80} -8.69798 q^{81} -6.18666 q^{82} -8.24958 q^{83} +5.70233 q^{84} -6.20500 q^{85} -11.4613 q^{86} +2.74410 q^{87} -10.6518 q^{88} +8.83490 q^{89} -0.111014 q^{90} +5.57128 q^{91} -3.55995 q^{92} -12.7673 q^{93} -9.18229 q^{94} +6.42023 q^{96} +0.707489 q^{97} +17.7903 q^{98} -0.337389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} + 3 q^{4} + 4 q^{5} - 7 q^{6} - 11 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} - 16 q^{12} - 2 q^{13} - 11 q^{14} + q^{15} - 3 q^{16} - 7 q^{17} + 17 q^{18} + 3 q^{20} + 2 q^{21} + q^{22} - 11 q^{23} - 13 q^{24} + 4 q^{25} + 9 q^{26} - 14 q^{27} - 13 q^{28} - 15 q^{29} - 7 q^{30} - q^{31} + 3 q^{32} + 12 q^{33} - 22 q^{34} - 11 q^{35} + 16 q^{36} - 11 q^{37} - 29 q^{39} + 6 q^{40} + 22 q^{41} + 19 q^{42} - 26 q^{43} - 12 q^{44} + 5 q^{45} + 10 q^{46} - 26 q^{47} - 13 q^{48} + 13 q^{49} + q^{50} - 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} - 8 q^{56} - 3 q^{58} - 10 q^{59} - 16 q^{60} + 2 q^{61} - 31 q^{62} - 17 q^{63} + 4 q^{64} - 2 q^{65} + 22 q^{66} + 3 q^{67} + 4 q^{68} + 14 q^{69} - 11 q^{70} + 18 q^{71} + 29 q^{72} - 24 q^{73} - 17 q^{74} + q^{75} - 6 q^{77} - 15 q^{78} + 30 q^{79} - 3 q^{80} - 4 q^{81} - 13 q^{82} - 12 q^{83} + 52 q^{84} - 7 q^{85} - 16 q^{86} - q^{87} - 23 q^{88} + 9 q^{89} + 17 q^{90} - 9 q^{91} - 25 q^{92} + 7 q^{93} - 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13856 0.805087 0.402543 0.915401i \(-0.368126\pi\)
0.402543 + 0.915401i \(0.368126\pi\)
\(3\) 1.70367 0.983615 0.491808 0.870704i \(-0.336336\pi\)
0.491808 + 0.870704i \(0.336336\pi\)
\(4\) −0.703671 −0.351836
\(5\) 1.00000 0.447214
\(6\) 1.93974 0.791895
\(7\) −4.75660 −1.79783 −0.898913 0.438128i \(-0.855642\pi\)
−0.898913 + 0.438128i \(0.855642\pi\)
\(8\) −3.07830 −1.08834
\(9\) −0.0975037 −0.0325012
\(10\) 1.13856 0.360046
\(11\) 3.46027 1.04331 0.521655 0.853156i \(-0.325315\pi\)
0.521655 + 0.853156i \(0.325315\pi\)
\(12\) −1.19882 −0.346071
\(13\) −1.17127 −0.324853 −0.162427 0.986721i \(-0.551932\pi\)
−0.162427 + 0.986721i \(0.551932\pi\)
\(14\) −5.41569 −1.44740
\(15\) 1.70367 0.439886
\(16\) −2.09750 −0.524376
\(17\) −6.20500 −1.50493 −0.752467 0.658630i \(-0.771136\pi\)
−0.752467 + 0.658630i \(0.771136\pi\)
\(18\) −0.111014 −0.0261663
\(19\) 0 0
\(20\) −0.703671 −0.157346
\(21\) −8.10368 −1.76837
\(22\) 3.93974 0.839955
\(23\) 5.05910 1.05490 0.527448 0.849587i \(-0.323149\pi\)
0.527448 + 0.849587i \(0.323149\pi\)
\(24\) −5.24442 −1.07051
\(25\) 1.00000 0.200000
\(26\) −1.33357 −0.261535
\(27\) −5.27713 −1.01558
\(28\) 3.34708 0.632539
\(29\) 1.61070 0.299100 0.149550 0.988754i \(-0.452218\pi\)
0.149550 + 0.988754i \(0.452218\pi\)
\(30\) 1.93974 0.354146
\(31\) −7.49400 −1.34596 −0.672981 0.739660i \(-0.734986\pi\)
−0.672981 + 0.739660i \(0.734986\pi\)
\(32\) 3.76846 0.666177
\(33\) 5.89516 1.02622
\(34\) −7.06479 −1.21160
\(35\) −4.75660 −0.804012
\(36\) 0.0686106 0.0114351
\(37\) −5.98080 −0.983237 −0.491619 0.870811i \(-0.663595\pi\)
−0.491619 + 0.870811i \(0.663595\pi\)
\(38\) 0 0
\(39\) −1.99547 −0.319531
\(40\) −3.07830 −0.486723
\(41\) −5.43374 −0.848607 −0.424303 0.905520i \(-0.639481\pi\)
−0.424303 + 0.905520i \(0.639481\pi\)
\(42\) −9.22656 −1.42369
\(43\) −10.0664 −1.53512 −0.767559 0.640979i \(-0.778529\pi\)
−0.767559 + 0.640979i \(0.778529\pi\)
\(44\) −2.43489 −0.367074
\(45\) −0.0975037 −0.0145350
\(46\) 5.76011 0.849283
\(47\) −8.06479 −1.17637 −0.588185 0.808726i \(-0.700158\pi\)
−0.588185 + 0.808726i \(0.700158\pi\)
\(48\) −3.57346 −0.515784
\(49\) 15.6252 2.23218
\(50\) 1.13856 0.161017
\(51\) −10.5713 −1.48028
\(52\) 0.824193 0.114295
\(53\) −6.68283 −0.917957 −0.458978 0.888447i \(-0.651784\pi\)
−0.458978 + 0.888447i \(0.651784\pi\)
\(54\) −6.00835 −0.817633
\(55\) 3.46027 0.466583
\(56\) 14.6423 1.95665
\(57\) 0 0
\(58\) 1.83389 0.240801
\(59\) 2.17229 0.282808 0.141404 0.989952i \(-0.454838\pi\)
0.141404 + 0.989952i \(0.454838\pi\)
\(60\) −1.19882 −0.154768
\(61\) 6.20852 0.794919 0.397460 0.917620i \(-0.369892\pi\)
0.397460 + 0.917620i \(0.369892\pi\)
\(62\) −8.53240 −1.08362
\(63\) 0.463786 0.0584315
\(64\) 8.48565 1.06071
\(65\) −1.17127 −0.145279
\(66\) 6.71202 0.826193
\(67\) −5.62257 −0.686906 −0.343453 0.939170i \(-0.611597\pi\)
−0.343453 + 0.939170i \(0.611597\pi\)
\(68\) 4.36628 0.529490
\(69\) 8.61905 1.03761
\(70\) −5.41569 −0.647299
\(71\) 2.72287 0.323145 0.161573 0.986861i \(-0.448343\pi\)
0.161573 + 0.986861i \(0.448343\pi\)
\(72\) 0.300146 0.0353725
\(73\) 3.15661 0.369453 0.184726 0.982790i \(-0.440860\pi\)
0.184726 + 0.982790i \(0.440860\pi\)
\(74\) −6.80953 −0.791591
\(75\) 1.70367 0.196723
\(76\) 0 0
\(77\) −16.4591 −1.87569
\(78\) −2.27197 −0.257250
\(79\) 12.0783 1.35892 0.679458 0.733715i \(-0.262215\pi\)
0.679458 + 0.733715i \(0.262215\pi\)
\(80\) −2.09750 −0.234508
\(81\) −8.69798 −0.966442
\(82\) −6.18666 −0.683202
\(83\) −8.24958 −0.905509 −0.452754 0.891635i \(-0.649559\pi\)
−0.452754 + 0.891635i \(0.649559\pi\)
\(84\) 5.70233 0.622175
\(85\) −6.20500 −0.673027
\(86\) −11.4613 −1.23590
\(87\) 2.74410 0.294199
\(88\) −10.6518 −1.13548
\(89\) 8.83490 0.936498 0.468249 0.883597i \(-0.344885\pi\)
0.468249 + 0.883597i \(0.344885\pi\)
\(90\) −0.111014 −0.0117019
\(91\) 5.57128 0.584029
\(92\) −3.55995 −0.371150
\(93\) −12.7673 −1.32391
\(94\) −9.18229 −0.947080
\(95\) 0 0
\(96\) 6.42023 0.655262
\(97\) 0.707489 0.0718346 0.0359173 0.999355i \(-0.488565\pi\)
0.0359173 + 0.999355i \(0.488565\pi\)
\(98\) 17.7903 1.79709
\(99\) −0.337389 −0.0339089
\(100\) −0.703671 −0.0703671
\(101\) −0.112658 −0.0112099 −0.00560497 0.999984i \(-0.501784\pi\)
−0.00560497 + 0.999984i \(0.501784\pi\)
\(102\) −12.0361 −1.19175
\(103\) 11.3233 1.11572 0.557861 0.829934i \(-0.311622\pi\)
0.557861 + 0.829934i \(0.311622\pi\)
\(104\) 3.60554 0.353552
\(105\) −8.10368 −0.790838
\(106\) −7.60883 −0.739035
\(107\) −2.17861 −0.210614 −0.105307 0.994440i \(-0.533583\pi\)
−0.105307 + 0.994440i \(0.533583\pi\)
\(108\) 3.71336 0.357319
\(109\) −13.3663 −1.28026 −0.640129 0.768268i \(-0.721119\pi\)
−0.640129 + 0.768268i \(0.721119\pi\)
\(110\) 3.93974 0.375639
\(111\) −10.1893 −0.967127
\(112\) 9.97698 0.942736
\(113\) 11.8586 1.11557 0.557783 0.829987i \(-0.311652\pi\)
0.557783 + 0.829987i \(0.311652\pi\)
\(114\) 0 0
\(115\) 5.05910 0.471764
\(116\) −1.13340 −0.105234
\(117\) 0.114204 0.0105581
\(118\) 2.47329 0.227685
\(119\) 29.5147 2.70561
\(120\) −5.24442 −0.478748
\(121\) 0.973466 0.0884969
\(122\) 7.06880 0.639979
\(123\) −9.25730 −0.834703
\(124\) 5.27331 0.473557
\(125\) 1.00000 0.0894427
\(126\) 0.528050 0.0470424
\(127\) 20.0596 1.78000 0.890000 0.455960i \(-0.150704\pi\)
0.890000 + 0.455960i \(0.150704\pi\)
\(128\) 2.12452 0.187783
\(129\) −17.1499 −1.50996
\(130\) −1.33357 −0.116962
\(131\) 9.21953 0.805514 0.402757 0.915307i \(-0.368052\pi\)
0.402757 + 0.915307i \(0.368052\pi\)
\(132\) −4.14826 −0.361059
\(133\) 0 0
\(134\) −6.40165 −0.553019
\(135\) −5.27713 −0.454183
\(136\) 19.1009 1.63789
\(137\) −3.46596 −0.296117 −0.148058 0.988979i \(-0.547302\pi\)
−0.148058 + 0.988979i \(0.547302\pi\)
\(138\) 9.81334 0.835367
\(139\) −1.92271 −0.163082 −0.0815412 0.996670i \(-0.525984\pi\)
−0.0815412 + 0.996670i \(0.525984\pi\)
\(140\) 3.34708 0.282880
\(141\) −13.7398 −1.15710
\(142\) 3.10016 0.260160
\(143\) −4.05293 −0.338923
\(144\) 0.204514 0.0170429
\(145\) 1.61070 0.133761
\(146\) 3.59400 0.297442
\(147\) 26.6203 2.19560
\(148\) 4.20852 0.345938
\(149\) 3.37579 0.276555 0.138278 0.990393i \(-0.455843\pi\)
0.138278 + 0.990393i \(0.455843\pi\)
\(150\) 1.93974 0.158379
\(151\) 19.6888 1.60225 0.801125 0.598497i \(-0.204235\pi\)
0.801125 + 0.598497i \(0.204235\pi\)
\(152\) 0 0
\(153\) 0.605011 0.0489122
\(154\) −18.7398 −1.51009
\(155\) −7.49400 −0.601932
\(156\) 1.40415 0.112422
\(157\) 10.8049 0.862321 0.431161 0.902275i \(-0.358104\pi\)
0.431161 + 0.902275i \(0.358104\pi\)
\(158\) 13.7519 1.09404
\(159\) −11.3853 −0.902916
\(160\) 3.76846 0.297923
\(161\) −24.0641 −1.89652
\(162\) −9.90321 −0.778070
\(163\) −2.40117 −0.188074 −0.0940369 0.995569i \(-0.529977\pi\)
−0.0940369 + 0.995569i \(0.529977\pi\)
\(164\) 3.82356 0.298570
\(165\) 5.89516 0.458938
\(166\) −9.39268 −0.729013
\(167\) 18.8862 1.46146 0.730728 0.682668i \(-0.239181\pi\)
0.730728 + 0.682668i \(0.239181\pi\)
\(168\) 24.9456 1.92459
\(169\) −11.6281 −0.894470
\(170\) −7.06479 −0.541845
\(171\) 0 0
\(172\) 7.08346 0.540109
\(173\) 7.01969 0.533697 0.266848 0.963738i \(-0.414018\pi\)
0.266848 + 0.963738i \(0.414018\pi\)
\(174\) 3.12434 0.236856
\(175\) −4.75660 −0.359565
\(176\) −7.25793 −0.547087
\(177\) 3.70087 0.278174
\(178\) 10.0591 0.753962
\(179\) −9.61639 −0.718763 −0.359381 0.933191i \(-0.617012\pi\)
−0.359381 + 0.933191i \(0.617012\pi\)
\(180\) 0.0686106 0.00511393
\(181\) 18.4974 1.37490 0.687449 0.726232i \(-0.258730\pi\)
0.687449 + 0.726232i \(0.258730\pi\)
\(182\) 6.34326 0.470194
\(183\) 10.5773 0.781895
\(184\) −15.5735 −1.14809
\(185\) −5.98080 −0.439717
\(186\) −14.5364 −1.06586
\(187\) −21.4710 −1.57011
\(188\) 5.67496 0.413889
\(189\) 25.1012 1.82584
\(190\) 0 0
\(191\) −10.8586 −0.785703 −0.392852 0.919602i \(-0.628511\pi\)
−0.392852 + 0.919602i \(0.628511\pi\)
\(192\) 14.4568 1.04333
\(193\) −12.5059 −0.900192 −0.450096 0.892980i \(-0.648610\pi\)
−0.450096 + 0.892980i \(0.648610\pi\)
\(194\) 0.805522 0.0578331
\(195\) −1.99547 −0.142898
\(196\) −10.9950 −0.785359
\(197\) −10.0643 −0.717049 −0.358525 0.933520i \(-0.616720\pi\)
−0.358525 + 0.933520i \(0.616720\pi\)
\(198\) −0.384139 −0.0272996
\(199\) 4.81054 0.341010 0.170505 0.985357i \(-0.445460\pi\)
0.170505 + 0.985357i \(0.445460\pi\)
\(200\) −3.07830 −0.217669
\(201\) −9.57901 −0.675651
\(202\) −0.128269 −0.00902497
\(203\) −7.66145 −0.537729
\(204\) 7.43871 0.520814
\(205\) −5.43374 −0.379509
\(206\) 12.8924 0.898253
\(207\) −0.493281 −0.0342854
\(208\) 2.45675 0.170345
\(209\) 0 0
\(210\) −9.22656 −0.636693
\(211\) −1.42250 −0.0979288 −0.0489644 0.998801i \(-0.515592\pi\)
−0.0489644 + 0.998801i \(0.515592\pi\)
\(212\) 4.70251 0.322970
\(213\) 4.63888 0.317851
\(214\) −2.48049 −0.169563
\(215\) −10.0664 −0.686525
\(216\) 16.2446 1.10531
\(217\) 35.6459 2.41980
\(218\) −15.2184 −1.03072
\(219\) 5.37782 0.363400
\(220\) −2.43489 −0.164160
\(221\) 7.26776 0.488883
\(222\) −11.6012 −0.778621
\(223\) −0.280645 −0.0187934 −0.00939668 0.999956i \(-0.502991\pi\)
−0.00939668 + 0.999956i \(0.502991\pi\)
\(224\) −17.9251 −1.19767
\(225\) −0.0975037 −0.00650025
\(226\) 13.5018 0.898128
\(227\) −19.0252 −1.26275 −0.631375 0.775478i \(-0.717509\pi\)
−0.631375 + 0.775478i \(0.717509\pi\)
\(228\) 0 0
\(229\) 11.6753 0.771523 0.385762 0.922598i \(-0.373939\pi\)
0.385762 + 0.922598i \(0.373939\pi\)
\(230\) 5.76011 0.379811
\(231\) −28.0409 −1.84496
\(232\) −4.95822 −0.325523
\(233\) −18.1431 −1.18859 −0.594297 0.804246i \(-0.702570\pi\)
−0.594297 + 0.804246i \(0.702570\pi\)
\(234\) 0.130028 0.00850021
\(235\) −8.06479 −0.526089
\(236\) −1.52858 −0.0995020
\(237\) 20.5775 1.33665
\(238\) 33.6044 2.17825
\(239\) −27.2177 −1.76057 −0.880285 0.474446i \(-0.842648\pi\)
−0.880285 + 0.474446i \(0.842648\pi\)
\(240\) −3.57346 −0.230666
\(241\) 9.53341 0.614101 0.307051 0.951693i \(-0.400658\pi\)
0.307051 + 0.951693i \(0.400658\pi\)
\(242\) 1.10835 0.0712477
\(243\) 1.01288 0.0649764
\(244\) −4.36876 −0.279681
\(245\) 15.6252 0.998259
\(246\) −10.5400 −0.672008
\(247\) 0 0
\(248\) 23.0688 1.46487
\(249\) −14.0546 −0.890672
\(250\) 1.13856 0.0720091
\(251\) −10.1985 −0.643725 −0.321863 0.946786i \(-0.604309\pi\)
−0.321863 + 0.946786i \(0.604309\pi\)
\(252\) −0.326353 −0.0205583
\(253\) 17.5059 1.10058
\(254\) 22.8391 1.43305
\(255\) −10.5713 −0.661999
\(256\) −14.5524 −0.909524
\(257\) −20.9818 −1.30881 −0.654405 0.756144i \(-0.727081\pi\)
−0.654405 + 0.756144i \(0.727081\pi\)
\(258\) −19.5263 −1.21565
\(259\) 28.4483 1.76769
\(260\) 0.824193 0.0511143
\(261\) −0.157049 −0.00972110
\(262\) 10.4970 0.648508
\(263\) −24.4437 −1.50726 −0.753632 0.657296i \(-0.771700\pi\)
−0.753632 + 0.657296i \(0.771700\pi\)
\(264\) −18.1471 −1.11688
\(265\) −6.68283 −0.410523
\(266\) 0 0
\(267\) 15.0518 0.921153
\(268\) 3.95644 0.241678
\(269\) −28.7189 −1.75102 −0.875512 0.483197i \(-0.839475\pi\)
−0.875512 + 0.483197i \(0.839475\pi\)
\(270\) −6.00835 −0.365657
\(271\) −20.9374 −1.27186 −0.635929 0.771748i \(-0.719383\pi\)
−0.635929 + 0.771748i \(0.719383\pi\)
\(272\) 13.0150 0.789151
\(273\) 9.49164 0.574460
\(274\) −3.94622 −0.238400
\(275\) 3.46027 0.208662
\(276\) −6.06498 −0.365069
\(277\) −18.7217 −1.12488 −0.562439 0.826839i \(-0.690137\pi\)
−0.562439 + 0.826839i \(0.690137\pi\)
\(278\) −2.18913 −0.131295
\(279\) 0.730692 0.0437454
\(280\) 14.6423 0.875042
\(281\) −19.1468 −1.14220 −0.571100 0.820880i \(-0.693483\pi\)
−0.571100 + 0.820880i \(0.693483\pi\)
\(282\) −15.6436 −0.931563
\(283\) −17.9013 −1.06412 −0.532062 0.846705i \(-0.678583\pi\)
−0.532062 + 0.846705i \(0.678583\pi\)
\(284\) −1.91601 −0.113694
\(285\) 0 0
\(286\) −4.61452 −0.272862
\(287\) 25.8461 1.52565
\(288\) −0.367439 −0.0216516
\(289\) 21.5020 1.26483
\(290\) 1.83389 0.107689
\(291\) 1.20533 0.0706576
\(292\) −2.22121 −0.129987
\(293\) −6.42187 −0.375170 −0.187585 0.982248i \(-0.560066\pi\)
−0.187585 + 0.982248i \(0.560066\pi\)
\(294\) 30.3089 1.76765
\(295\) 2.17229 0.126476
\(296\) 18.4107 1.07010
\(297\) −18.2603 −1.05957
\(298\) 3.84355 0.222651
\(299\) −5.92560 −0.342686
\(300\) −1.19882 −0.0692142
\(301\) 47.8820 2.75987
\(302\) 22.4169 1.28995
\(303\) −0.191933 −0.0110263
\(304\) 0 0
\(305\) 6.20852 0.355499
\(306\) 0.688844 0.0393786
\(307\) −6.15675 −0.351384 −0.175692 0.984445i \(-0.556216\pi\)
−0.175692 + 0.984445i \(0.556216\pi\)
\(308\) 11.5818 0.659935
\(309\) 19.2913 1.09744
\(310\) −8.53240 −0.484608
\(311\) 3.34988 0.189954 0.0949772 0.995479i \(-0.469722\pi\)
0.0949772 + 0.995479i \(0.469722\pi\)
\(312\) 6.14265 0.347759
\(313\) 24.2386 1.37005 0.685023 0.728521i \(-0.259792\pi\)
0.685023 + 0.728521i \(0.259792\pi\)
\(314\) 12.3020 0.694243
\(315\) 0.463786 0.0261314
\(316\) −8.49916 −0.478115
\(317\) −10.9842 −0.616933 −0.308466 0.951235i \(-0.599816\pi\)
−0.308466 + 0.951235i \(0.599816\pi\)
\(318\) −12.9629 −0.726926
\(319\) 5.57346 0.312054
\(320\) 8.48565 0.474362
\(321\) −3.71163 −0.207163
\(322\) −27.3986 −1.52686
\(323\) 0 0
\(324\) 6.12052 0.340029
\(325\) −1.17127 −0.0649706
\(326\) −2.73388 −0.151416
\(327\) −22.7718 −1.25928
\(328\) 16.7267 0.923577
\(329\) 38.3610 2.11491
\(330\) 6.71202 0.369485
\(331\) −25.2522 −1.38799 −0.693994 0.719981i \(-0.744151\pi\)
−0.693994 + 0.719981i \(0.744151\pi\)
\(332\) 5.80499 0.318590
\(333\) 0.583150 0.0319564
\(334\) 21.5031 1.17660
\(335\) −5.62257 −0.307194
\(336\) 16.9975 0.927290
\(337\) −8.26425 −0.450182 −0.225091 0.974338i \(-0.572268\pi\)
−0.225091 + 0.974338i \(0.572268\pi\)
\(338\) −13.2394 −0.720126
\(339\) 20.2032 1.09729
\(340\) 4.36628 0.236795
\(341\) −25.9312 −1.40426
\(342\) 0 0
\(343\) −41.0267 −2.21524
\(344\) 30.9876 1.67074
\(345\) 8.61905 0.464034
\(346\) 7.99237 0.429672
\(347\) 21.3088 1.14391 0.571957 0.820283i \(-0.306184\pi\)
0.571957 + 0.820283i \(0.306184\pi\)
\(348\) −1.93095 −0.103510
\(349\) −23.2574 −1.24494 −0.622470 0.782644i \(-0.713871\pi\)
−0.622470 + 0.782644i \(0.713871\pi\)
\(350\) −5.41569 −0.289481
\(351\) 6.18097 0.329916
\(352\) 13.0399 0.695029
\(353\) 24.7741 1.31859 0.659296 0.751883i \(-0.270854\pi\)
0.659296 + 0.751883i \(0.270854\pi\)
\(354\) 4.21368 0.223954
\(355\) 2.72287 0.144515
\(356\) −6.21687 −0.329493
\(357\) 50.2834 2.66128
\(358\) −10.9489 −0.578666
\(359\) −3.85028 −0.203210 −0.101605 0.994825i \(-0.532398\pi\)
−0.101605 + 0.994825i \(0.532398\pi\)
\(360\) 0.300146 0.0158191
\(361\) 0 0
\(362\) 21.0604 1.10691
\(363\) 1.65847 0.0870469
\(364\) −3.92035 −0.205482
\(365\) 3.15661 0.165224
\(366\) 12.0429 0.629493
\(367\) −28.6314 −1.49455 −0.747274 0.664517i \(-0.768638\pi\)
−0.747274 + 0.664517i \(0.768638\pi\)
\(368\) −10.6115 −0.553162
\(369\) 0.529809 0.0275808
\(370\) −6.80953 −0.354010
\(371\) 31.7875 1.65033
\(372\) 8.98399 0.465798
\(373\) 22.9648 1.18907 0.594537 0.804069i \(-0.297336\pi\)
0.594537 + 0.804069i \(0.297336\pi\)
\(374\) −24.4461 −1.26408
\(375\) 1.70367 0.0879772
\(376\) 24.8259 1.28030
\(377\) −1.88657 −0.0971634
\(378\) 28.5793 1.46996
\(379\) −19.3472 −0.993801 −0.496901 0.867807i \(-0.665529\pi\)
−0.496901 + 0.867807i \(0.665529\pi\)
\(380\) 0 0
\(381\) 34.1750 1.75084
\(382\) −12.3633 −0.632559
\(383\) 6.13972 0.313725 0.156863 0.987620i \(-0.449862\pi\)
0.156863 + 0.987620i \(0.449862\pi\)
\(384\) 3.61949 0.184706
\(385\) −16.4591 −0.838834
\(386\) −14.2387 −0.724732
\(387\) 0.981515 0.0498932
\(388\) −0.497840 −0.0252740
\(389\) 3.66728 0.185939 0.0929693 0.995669i \(-0.470364\pi\)
0.0929693 + 0.995669i \(0.470364\pi\)
\(390\) −2.27197 −0.115046
\(391\) −31.3917 −1.58755
\(392\) −48.0992 −2.42938
\(393\) 15.7070 0.792316
\(394\) −11.4588 −0.577287
\(395\) 12.0783 0.607725
\(396\) 0.237411 0.0119304
\(397\) 24.5839 1.23383 0.616914 0.787030i \(-0.288382\pi\)
0.616914 + 0.787030i \(0.288382\pi\)
\(398\) 5.47711 0.274543
\(399\) 0 0
\(400\) −2.09750 −0.104875
\(401\) 6.84605 0.341876 0.170938 0.985282i \(-0.445320\pi\)
0.170938 + 0.985282i \(0.445320\pi\)
\(402\) −10.9063 −0.543957
\(403\) 8.77753 0.437240
\(404\) 0.0792746 0.00394406
\(405\) −8.69798 −0.432206
\(406\) −8.72306 −0.432918
\(407\) −20.6952 −1.02582
\(408\) 32.5416 1.61105
\(409\) 7.92406 0.391819 0.195910 0.980622i \(-0.437234\pi\)
0.195910 + 0.980622i \(0.437234\pi\)
\(410\) −6.18666 −0.305537
\(411\) −5.90486 −0.291265
\(412\) −7.96792 −0.392551
\(413\) −10.3327 −0.508440
\(414\) −0.561633 −0.0276027
\(415\) −8.24958 −0.404956
\(416\) −4.41391 −0.216410
\(417\) −3.27567 −0.160410
\(418\) 0 0
\(419\) 1.66830 0.0815018 0.0407509 0.999169i \(-0.487025\pi\)
0.0407509 + 0.999169i \(0.487025\pi\)
\(420\) 5.70233 0.278245
\(421\) −19.0982 −0.930790 −0.465395 0.885103i \(-0.654088\pi\)
−0.465395 + 0.885103i \(0.654088\pi\)
\(422\) −1.61960 −0.0788411
\(423\) 0.786347 0.0382335
\(424\) 20.5718 0.999054
\(425\) −6.20500 −0.300987
\(426\) 5.28166 0.255897
\(427\) −29.5314 −1.42913
\(428\) 1.53302 0.0741015
\(429\) −6.90486 −0.333370
\(430\) −11.4613 −0.552712
\(431\) −18.5276 −0.892442 −0.446221 0.894923i \(-0.647231\pi\)
−0.446221 + 0.894923i \(0.647231\pi\)
\(432\) 11.0688 0.532548
\(433\) 21.8081 1.04803 0.524016 0.851708i \(-0.324433\pi\)
0.524016 + 0.851708i \(0.324433\pi\)
\(434\) 40.5852 1.94815
\(435\) 2.74410 0.131570
\(436\) 9.40547 0.450440
\(437\) 0 0
\(438\) 6.12300 0.292568
\(439\) −19.6692 −0.938759 −0.469379 0.882997i \(-0.655522\pi\)
−0.469379 + 0.882997i \(0.655522\pi\)
\(440\) −10.6518 −0.507803
\(441\) −1.52352 −0.0725485
\(442\) 8.27481 0.393593
\(443\) 3.48680 0.165663 0.0828315 0.996564i \(-0.473604\pi\)
0.0828315 + 0.996564i \(0.473604\pi\)
\(444\) 7.16993 0.340270
\(445\) 8.83490 0.418815
\(446\) −0.319532 −0.0151303
\(447\) 5.75124 0.272024
\(448\) −40.3628 −1.90696
\(449\) 8.06276 0.380505 0.190253 0.981735i \(-0.439069\pi\)
0.190253 + 0.981735i \(0.439069\pi\)
\(450\) −0.111014 −0.00523326
\(451\) −18.8022 −0.885361
\(452\) −8.34458 −0.392496
\(453\) 33.5432 1.57600
\(454\) −21.6615 −1.01662
\(455\) 5.57128 0.261186
\(456\) 0 0
\(457\) −25.7296 −1.20358 −0.601790 0.798654i \(-0.705546\pi\)
−0.601790 + 0.798654i \(0.705546\pi\)
\(458\) 13.2930 0.621143
\(459\) 32.7446 1.52839
\(460\) −3.55995 −0.165983
\(461\) 7.48361 0.348547 0.174273 0.984697i \(-0.444242\pi\)
0.174273 + 0.984697i \(0.444242\pi\)
\(462\) −31.9264 −1.48535
\(463\) 24.4776 1.13757 0.568786 0.822485i \(-0.307413\pi\)
0.568786 + 0.822485i \(0.307413\pi\)
\(464\) −3.37845 −0.156841
\(465\) −12.7673 −0.592070
\(466\) −20.6571 −0.956921
\(467\) 6.03350 0.279197 0.139599 0.990208i \(-0.455419\pi\)
0.139599 + 0.990208i \(0.455419\pi\)
\(468\) −0.0803618 −0.00371473
\(469\) 26.7443 1.23494
\(470\) −9.18229 −0.423547
\(471\) 18.4079 0.848192
\(472\) −6.68697 −0.307793
\(473\) −34.8326 −1.60160
\(474\) 23.4288 1.07612
\(475\) 0 0
\(476\) −20.7687 −0.951930
\(477\) 0.651600 0.0298347
\(478\) −30.9891 −1.41741
\(479\) 22.2776 1.01789 0.508945 0.860799i \(-0.330036\pi\)
0.508945 + 0.860799i \(0.330036\pi\)
\(480\) 6.42023 0.293042
\(481\) 7.00516 0.319408
\(482\) 10.8544 0.494405
\(483\) −40.9974 −1.86544
\(484\) −0.685000 −0.0311364
\(485\) 0.707489 0.0321254
\(486\) 1.15323 0.0523117
\(487\) 21.2868 0.964598 0.482299 0.876007i \(-0.339802\pi\)
0.482299 + 0.876007i \(0.339802\pi\)
\(488\) −19.1117 −0.865146
\(489\) −4.09080 −0.184992
\(490\) 17.7903 0.803685
\(491\) −33.6313 −1.51776 −0.758879 0.651232i \(-0.774253\pi\)
−0.758879 + 0.651232i \(0.774253\pi\)
\(492\) 6.51410 0.293678
\(493\) −9.99440 −0.450125
\(494\) 0 0
\(495\) −0.337389 −0.0151645
\(496\) 15.7187 0.705790
\(497\) −12.9516 −0.580959
\(498\) −16.0020 −0.717068
\(499\) −35.2646 −1.57866 −0.789330 0.613969i \(-0.789572\pi\)
−0.789330 + 0.613969i \(0.789572\pi\)
\(500\) −0.703671 −0.0314691
\(501\) 32.1759 1.43751
\(502\) −11.6117 −0.518254
\(503\) 1.00821 0.0449538 0.0224769 0.999747i \(-0.492845\pi\)
0.0224769 + 0.999747i \(0.492845\pi\)
\(504\) −1.42767 −0.0635937
\(505\) −0.112658 −0.00501324
\(506\) 19.9315 0.886065
\(507\) −19.8105 −0.879815
\(508\) −14.1154 −0.626268
\(509\) 22.0570 0.977660 0.488830 0.872379i \(-0.337424\pi\)
0.488830 + 0.872379i \(0.337424\pi\)
\(510\) −12.0361 −0.532967
\(511\) −15.0147 −0.664212
\(512\) −20.8179 −0.920029
\(513\) 0 0
\(514\) −23.8891 −1.05371
\(515\) 11.3233 0.498966
\(516\) 12.0679 0.531259
\(517\) −27.9064 −1.22732
\(518\) 32.3902 1.42314
\(519\) 11.9592 0.524952
\(520\) 3.60554 0.158113
\(521\) −16.4008 −0.718532 −0.359266 0.933235i \(-0.616973\pi\)
−0.359266 + 0.933235i \(0.616973\pi\)
\(522\) −0.178811 −0.00782633
\(523\) −17.8476 −0.780419 −0.390210 0.920726i \(-0.627597\pi\)
−0.390210 + 0.920726i \(0.627597\pi\)
\(524\) −6.48752 −0.283409
\(525\) −8.10368 −0.353674
\(526\) −27.8308 −1.21348
\(527\) 46.5003 2.02558
\(528\) −12.3651 −0.538123
\(529\) 2.59453 0.112806
\(530\) −7.60883 −0.330506
\(531\) −0.211806 −0.00919162
\(532\) 0 0
\(533\) 6.36440 0.275673
\(534\) 17.1374 0.741608
\(535\) −2.17861 −0.0941895
\(536\) 17.3080 0.747590
\(537\) −16.3832 −0.706986
\(538\) −32.6983 −1.40973
\(539\) 54.0675 2.32885
\(540\) 3.71336 0.159798
\(541\) 22.1825 0.953699 0.476849 0.878985i \(-0.341779\pi\)
0.476849 + 0.878985i \(0.341779\pi\)
\(542\) −23.8386 −1.02396
\(543\) 31.5134 1.35237
\(544\) −23.3833 −1.00255
\(545\) −13.3663 −0.572549
\(546\) 10.8068 0.462490
\(547\) 24.9289 1.06588 0.532941 0.846152i \(-0.321087\pi\)
0.532941 + 0.846152i \(0.321087\pi\)
\(548\) 2.43890 0.104184
\(549\) −0.605354 −0.0258359
\(550\) 3.93974 0.167991
\(551\) 0 0
\(552\) −26.5321 −1.12928
\(553\) −57.4516 −2.44309
\(554\) −21.3159 −0.905625
\(555\) −10.1893 −0.432512
\(556\) 1.35296 0.0573782
\(557\) −7.31171 −0.309807 −0.154904 0.987930i \(-0.549507\pi\)
−0.154904 + 0.987930i \(0.549507\pi\)
\(558\) 0.831940 0.0352188
\(559\) 11.7906 0.498688
\(560\) 9.97698 0.421604
\(561\) −36.5795 −1.54439
\(562\) −21.7998 −0.919570
\(563\) −16.5614 −0.697978 −0.348989 0.937127i \(-0.613475\pi\)
−0.348989 + 0.937127i \(0.613475\pi\)
\(564\) 9.66827 0.407108
\(565\) 11.8586 0.498897
\(566\) −20.3818 −0.856712
\(567\) 41.3728 1.73749
\(568\) −8.38183 −0.351694
\(569\) 31.2306 1.30926 0.654628 0.755951i \(-0.272825\pi\)
0.654628 + 0.755951i \(0.272825\pi\)
\(570\) 0 0
\(571\) −1.27375 −0.0533049 −0.0266525 0.999645i \(-0.508485\pi\)
−0.0266525 + 0.999645i \(0.508485\pi\)
\(572\) 2.85193 0.119245
\(573\) −18.4995 −0.772830
\(574\) 29.4274 1.22828
\(575\) 5.05910 0.210979
\(576\) −0.827382 −0.0344743
\(577\) −1.77096 −0.0737262 −0.0368631 0.999320i \(-0.511737\pi\)
−0.0368631 + 0.999320i \(0.511737\pi\)
\(578\) 24.4815 1.01829
\(579\) −21.3059 −0.885442
\(580\) −1.13340 −0.0470620
\(581\) 39.2399 1.62795
\(582\) 1.37234 0.0568855
\(583\) −23.1244 −0.957714
\(584\) −9.71700 −0.402092
\(585\) 0.114204 0.00472174
\(586\) −7.31171 −0.302044
\(587\) 8.25987 0.340921 0.170461 0.985364i \(-0.445474\pi\)
0.170461 + 0.985364i \(0.445474\pi\)
\(588\) −18.7319 −0.772491
\(589\) 0 0
\(590\) 2.47329 0.101824
\(591\) −17.1462 −0.705300
\(592\) 12.5448 0.515586
\(593\) −20.8241 −0.855141 −0.427571 0.903982i \(-0.640630\pi\)
−0.427571 + 0.903982i \(0.640630\pi\)
\(594\) −20.7905 −0.853045
\(595\) 29.5147 1.20998
\(596\) −2.37545 −0.0973021
\(597\) 8.19558 0.335423
\(598\) −6.74668 −0.275892
\(599\) 40.0833 1.63776 0.818880 0.573965i \(-0.194595\pi\)
0.818880 + 0.573965i \(0.194595\pi\)
\(600\) −5.24442 −0.214102
\(601\) 48.3109 1.97064 0.985321 0.170710i \(-0.0546060\pi\)
0.985321 + 0.170710i \(0.0546060\pi\)
\(602\) 54.5167 2.22194
\(603\) 0.548221 0.0223253
\(604\) −13.8544 −0.563729
\(605\) 0.973466 0.0395770
\(606\) −0.218528 −0.00887710
\(607\) −23.7640 −0.964552 −0.482276 0.876019i \(-0.660190\pi\)
−0.482276 + 0.876019i \(0.660190\pi\)
\(608\) 0 0
\(609\) −13.0526 −0.528918
\(610\) 7.06880 0.286207
\(611\) 9.44609 0.382148
\(612\) −0.425729 −0.0172091
\(613\) 4.70479 0.190025 0.0950123 0.995476i \(-0.469711\pi\)
0.0950123 + 0.995476i \(0.469711\pi\)
\(614\) −7.00985 −0.282895
\(615\) −9.25730 −0.373290
\(616\) 50.6661 2.04140
\(617\) 2.11358 0.0850893 0.0425447 0.999095i \(-0.486454\pi\)
0.0425447 + 0.999095i \(0.486454\pi\)
\(618\) 21.9643 0.883536
\(619\) −34.0091 −1.36694 −0.683470 0.729978i \(-0.739530\pi\)
−0.683470 + 0.729978i \(0.739530\pi\)
\(620\) 5.27331 0.211781
\(621\) −26.6975 −1.07134
\(622\) 3.81406 0.152930
\(623\) −42.0241 −1.68366
\(624\) 4.18550 0.167554
\(625\) 1.00000 0.0400000
\(626\) 27.5972 1.10301
\(627\) 0 0
\(628\) −7.60307 −0.303395
\(629\) 37.1109 1.47971
\(630\) 0.528050 0.0210380
\(631\) 9.57620 0.381223 0.190611 0.981666i \(-0.438953\pi\)
0.190611 + 0.981666i \(0.438953\pi\)
\(632\) −37.1807 −1.47897
\(633\) −2.42347 −0.0963242
\(634\) −12.5062 −0.496684
\(635\) 20.0596 0.796041
\(636\) 8.01154 0.317678
\(637\) −18.3014 −0.725129
\(638\) 6.34574 0.251230
\(639\) −0.265490 −0.0105026
\(640\) 2.12452 0.0839792
\(641\) 23.8257 0.941058 0.470529 0.882384i \(-0.344063\pi\)
0.470529 + 0.882384i \(0.344063\pi\)
\(642\) −4.22593 −0.166784
\(643\) 27.3687 1.07931 0.539657 0.841885i \(-0.318554\pi\)
0.539657 + 0.841885i \(0.318554\pi\)
\(644\) 16.9332 0.667263
\(645\) −17.1499 −0.675277
\(646\) 0 0
\(647\) −38.5660 −1.51619 −0.758093 0.652147i \(-0.773869\pi\)
−0.758093 + 0.652147i \(0.773869\pi\)
\(648\) 26.7750 1.05182
\(649\) 7.51671 0.295057
\(650\) −1.33357 −0.0523070
\(651\) 60.7290 2.38016
\(652\) 1.68963 0.0661711
\(653\) −14.6990 −0.575216 −0.287608 0.957748i \(-0.592860\pi\)
−0.287608 + 0.957748i \(0.592860\pi\)
\(654\) −25.9271 −1.01383
\(655\) 9.21953 0.360237
\(656\) 11.3973 0.444989
\(657\) −0.307781 −0.0120077
\(658\) 43.6764 1.70268
\(659\) 7.91649 0.308383 0.154191 0.988041i \(-0.450723\pi\)
0.154191 + 0.988041i \(0.450723\pi\)
\(660\) −4.14826 −0.161471
\(661\) 33.2446 1.29307 0.646533 0.762886i \(-0.276218\pi\)
0.646533 + 0.762886i \(0.276218\pi\)
\(662\) −28.7513 −1.11745
\(663\) 12.3819 0.480872
\(664\) 25.3947 0.985506
\(665\) 0 0
\(666\) 0.663954 0.0257277
\(667\) 8.14870 0.315519
\(668\) −13.2897 −0.514193
\(669\) −0.478127 −0.0184854
\(670\) −6.40165 −0.247317
\(671\) 21.4831 0.829348
\(672\) −30.5384 −1.17805
\(673\) 34.7182 1.33829 0.669144 0.743133i \(-0.266661\pi\)
0.669144 + 0.743133i \(0.266661\pi\)
\(674\) −9.40938 −0.362436
\(675\) −5.27713 −0.203117
\(676\) 8.18237 0.314707
\(677\) 13.6369 0.524107 0.262054 0.965053i \(-0.415600\pi\)
0.262054 + 0.965053i \(0.415600\pi\)
\(678\) 23.0027 0.883412
\(679\) −3.36524 −0.129146
\(680\) 19.1009 0.732485
\(681\) −32.4128 −1.24206
\(682\) −29.5244 −1.13055
\(683\) 32.0188 1.22516 0.612582 0.790407i \(-0.290131\pi\)
0.612582 + 0.790407i \(0.290131\pi\)
\(684\) 0 0
\(685\) −3.46596 −0.132427
\(686\) −46.7116 −1.78346
\(687\) 19.8908 0.758882
\(688\) 21.1144 0.804979
\(689\) 7.82743 0.298201
\(690\) 9.81334 0.373588
\(691\) −42.5000 −1.61678 −0.808389 0.588649i \(-0.799660\pi\)
−0.808389 + 0.588649i \(0.799660\pi\)
\(692\) −4.93955 −0.187774
\(693\) 1.60482 0.0609622
\(694\) 24.2614 0.920950
\(695\) −1.92271 −0.0729326
\(696\) −8.44719 −0.320190
\(697\) 33.7163 1.27710
\(698\) −26.4800 −1.00228
\(699\) −30.9099 −1.16912
\(700\) 3.34708 0.126508
\(701\) 17.0007 0.642108 0.321054 0.947061i \(-0.395963\pi\)
0.321054 + 0.947061i \(0.395963\pi\)
\(702\) 7.03743 0.265611
\(703\) 0 0
\(704\) 29.3626 1.10665
\(705\) −13.7398 −0.517469
\(706\) 28.2069 1.06158
\(707\) 0.535871 0.0201535
\(708\) −2.60420 −0.0978717
\(709\) −2.22434 −0.0835369 −0.0417685 0.999127i \(-0.513299\pi\)
−0.0417685 + 0.999127i \(0.513299\pi\)
\(710\) 3.10016 0.116347
\(711\) −1.17768 −0.0441664
\(712\) −27.1965 −1.01923
\(713\) −37.9129 −1.41985
\(714\) 57.2508 2.14256
\(715\) −4.05293 −0.151571
\(716\) 6.76678 0.252886
\(717\) −46.3701 −1.73172
\(718\) −4.38380 −0.163602
\(719\) 29.4500 1.09830 0.549150 0.835724i \(-0.314952\pi\)
0.549150 + 0.835724i \(0.314952\pi\)
\(720\) 0.204514 0.00762180
\(721\) −53.8606 −2.00587
\(722\) 0 0
\(723\) 16.2418 0.604039
\(724\) −13.0161 −0.483739
\(725\) 1.61070 0.0598199
\(726\) 1.88827 0.0700803
\(727\) −5.16809 −0.191674 −0.0958368 0.995397i \(-0.530553\pi\)
−0.0958368 + 0.995397i \(0.530553\pi\)
\(728\) −17.1501 −0.635625
\(729\) 27.8196 1.03035
\(730\) 3.59400 0.133020
\(731\) 62.4623 2.31025
\(732\) −7.44293 −0.275098
\(733\) 6.04955 0.223445 0.111723 0.993739i \(-0.464363\pi\)
0.111723 + 0.993739i \(0.464363\pi\)
\(734\) −32.5987 −1.20324
\(735\) 26.6203 0.981903
\(736\) 19.0651 0.702747
\(737\) −19.4556 −0.716656
\(738\) 0.603222 0.0222049
\(739\) −27.1443 −0.998518 −0.499259 0.866453i \(-0.666394\pi\)
−0.499259 + 0.866453i \(0.666394\pi\)
\(740\) 4.20852 0.154708
\(741\) 0 0
\(742\) 36.1921 1.32866
\(743\) −36.3675 −1.33419 −0.667097 0.744971i \(-0.732463\pi\)
−0.667097 + 0.744971i \(0.732463\pi\)
\(744\) 39.3016 1.44087
\(745\) 3.37579 0.123679
\(746\) 26.1469 0.957307
\(747\) 0.804365 0.0294302
\(748\) 15.1085 0.552422
\(749\) 10.3628 0.378647
\(750\) 1.93974 0.0708293
\(751\) 30.6467 1.11831 0.559157 0.829062i \(-0.311125\pi\)
0.559157 + 0.829062i \(0.311125\pi\)
\(752\) 16.9159 0.616861
\(753\) −17.3749 −0.633178
\(754\) −2.14798 −0.0782250
\(755\) 19.6888 0.716548
\(756\) −17.6630 −0.642396
\(757\) −8.24452 −0.299652 −0.149826 0.988712i \(-0.547871\pi\)
−0.149826 + 0.988712i \(0.547871\pi\)
\(758\) −22.0281 −0.800096
\(759\) 29.8242 1.08255
\(760\) 0 0
\(761\) 11.2064 0.406231 0.203116 0.979155i \(-0.434893\pi\)
0.203116 + 0.979155i \(0.434893\pi\)
\(762\) 38.9104 1.40957
\(763\) 63.5780 2.30168
\(764\) 7.64091 0.276438
\(765\) 0.605011 0.0218742
\(766\) 6.99047 0.252576
\(767\) −2.54435 −0.0918711
\(768\) −24.7925 −0.894622
\(769\) −26.1536 −0.943122 −0.471561 0.881833i \(-0.656309\pi\)
−0.471561 + 0.881833i \(0.656309\pi\)
\(770\) −18.7398 −0.675334
\(771\) −35.7461 −1.28737
\(772\) 8.80002 0.316720
\(773\) 14.5134 0.522010 0.261005 0.965337i \(-0.415946\pi\)
0.261005 + 0.965337i \(0.415946\pi\)
\(774\) 1.11752 0.0401684
\(775\) −7.49400 −0.269192
\(776\) −2.17787 −0.0781808
\(777\) 48.4665 1.73873
\(778\) 4.17544 0.149697
\(779\) 0 0
\(780\) 1.40415 0.0502768
\(781\) 9.42187 0.337141
\(782\) −35.7415 −1.27811
\(783\) −8.49987 −0.303761
\(784\) −32.7740 −1.17050
\(785\) 10.8049 0.385642
\(786\) 17.8835 0.637883
\(787\) −48.3569 −1.72374 −0.861869 0.507130i \(-0.830706\pi\)
−0.861869 + 0.507130i \(0.830706\pi\)
\(788\) 7.08193 0.252283
\(789\) −41.6441 −1.48257
\(790\) 13.7519 0.489272
\(791\) −56.4068 −2.00559
\(792\) 1.03859 0.0369046
\(793\) −7.27188 −0.258232
\(794\) 27.9903 0.993339
\(795\) −11.3853 −0.403796
\(796\) −3.38504 −0.119980
\(797\) 28.0793 0.994619 0.497310 0.867573i \(-0.334321\pi\)
0.497310 + 0.867573i \(0.334321\pi\)
\(798\) 0 0
\(799\) 50.0421 1.77036
\(800\) 3.76846 0.133235
\(801\) −0.861436 −0.0304373
\(802\) 7.79467 0.275239
\(803\) 10.9227 0.385454
\(804\) 6.74047 0.237718
\(805\) −24.0641 −0.848149
\(806\) 9.99378 0.352016
\(807\) −48.9276 −1.72233
\(808\) 0.346797 0.0122003
\(809\) −33.4598 −1.17638 −0.588191 0.808722i \(-0.700160\pi\)
−0.588191 + 0.808722i \(0.700160\pi\)
\(810\) −9.90321 −0.347963
\(811\) 10.8723 0.381778 0.190889 0.981612i \(-0.438863\pi\)
0.190889 + 0.981612i \(0.438863\pi\)
\(812\) 5.39115 0.189192
\(813\) −35.6705 −1.25102
\(814\) −23.5628 −0.825875
\(815\) −2.40117 −0.0841092
\(816\) 22.1733 0.776221
\(817\) 0 0
\(818\) 9.02205 0.315448
\(819\) −0.543221 −0.0189817
\(820\) 3.82356 0.133525
\(821\) −26.8644 −0.937576 −0.468788 0.883311i \(-0.655309\pi\)
−0.468788 + 0.883311i \(0.655309\pi\)
\(822\) −6.72306 −0.234494
\(823\) −26.6654 −0.929496 −0.464748 0.885443i \(-0.653855\pi\)
−0.464748 + 0.885443i \(0.653855\pi\)
\(824\) −34.8567 −1.21429
\(825\) 5.89516 0.205243
\(826\) −11.7645 −0.409338
\(827\) −20.5863 −0.715856 −0.357928 0.933749i \(-0.616517\pi\)
−0.357928 + 0.933749i \(0.616517\pi\)
\(828\) 0.347108 0.0120628
\(829\) −23.1471 −0.803932 −0.401966 0.915655i \(-0.631673\pi\)
−0.401966 + 0.915655i \(0.631673\pi\)
\(830\) −9.39268 −0.326025
\(831\) −31.8956 −1.10645
\(832\) −9.93902 −0.344574
\(833\) −96.9546 −3.35928
\(834\) −3.72956 −0.129144
\(835\) 18.8862 0.653583
\(836\) 0 0
\(837\) 39.5468 1.36694
\(838\) 1.89947 0.0656160
\(839\) 41.4744 1.43186 0.715928 0.698174i \(-0.246004\pi\)
0.715928 + 0.698174i \(0.246004\pi\)
\(840\) 24.9456 0.860705
\(841\) −26.4056 −0.910539
\(842\) −21.7445 −0.749367
\(843\) −32.6198 −1.12349
\(844\) 1.00097 0.0344548
\(845\) −11.6281 −0.400019
\(846\) 0.895307 0.0307813
\(847\) −4.63039 −0.159102
\(848\) 14.0173 0.481355
\(849\) −30.4980 −1.04669
\(850\) −7.06479 −0.242320
\(851\) −30.2575 −1.03721
\(852\) −3.26425 −0.111831
\(853\) 49.7956 1.70497 0.852484 0.522753i \(-0.175095\pi\)
0.852484 + 0.522753i \(0.175095\pi\)
\(854\) −33.6234 −1.15057
\(855\) 0 0
\(856\) 6.70642 0.229221
\(857\) −37.3465 −1.27573 −0.637866 0.770148i \(-0.720182\pi\)
−0.637866 + 0.770148i \(0.720182\pi\)
\(858\) −7.86162 −0.268391
\(859\) −23.7810 −0.811397 −0.405698 0.914007i \(-0.632972\pi\)
−0.405698 + 0.914007i \(0.632972\pi\)
\(860\) 7.08346 0.241544
\(861\) 44.0333 1.50065
\(862\) −21.0948 −0.718493
\(863\) −28.6969 −0.976855 −0.488427 0.872605i \(-0.662429\pi\)
−0.488427 + 0.872605i \(0.662429\pi\)
\(864\) −19.8867 −0.676558
\(865\) 7.01969 0.238677
\(866\) 24.8300 0.843757
\(867\) 36.6324 1.24410
\(868\) −25.0830 −0.851373
\(869\) 41.7942 1.41777
\(870\) 3.12434 0.105925
\(871\) 6.58557 0.223144
\(872\) 41.1455 1.39336
\(873\) −0.0689828 −0.00233471
\(874\) 0 0
\(875\) −4.75660 −0.160802
\(876\) −3.78422 −0.127857
\(877\) −33.4018 −1.12790 −0.563950 0.825809i \(-0.690719\pi\)
−0.563950 + 0.825809i \(0.690719\pi\)
\(878\) −22.3946 −0.755782
\(879\) −10.9408 −0.369023
\(880\) −7.25793 −0.244665
\(881\) −33.0377 −1.11307 −0.556535 0.830824i \(-0.687869\pi\)
−0.556535 + 0.830824i \(0.687869\pi\)
\(882\) −1.73462 −0.0584078
\(883\) −32.1095 −1.08057 −0.540286 0.841482i \(-0.681684\pi\)
−0.540286 + 0.841482i \(0.681684\pi\)
\(884\) −5.11412 −0.172006
\(885\) 3.70087 0.124403
\(886\) 3.96995 0.133373
\(887\) −13.6924 −0.459747 −0.229873 0.973221i \(-0.573831\pi\)
−0.229873 + 0.973221i \(0.573831\pi\)
\(888\) 31.3658 1.05257
\(889\) −95.4154 −3.20013
\(890\) 10.0591 0.337182
\(891\) −30.0974 −1.00830
\(892\) 0.197482 0.00661218
\(893\) 0 0
\(894\) 6.54815 0.219003
\(895\) −9.61639 −0.321440
\(896\) −10.1055 −0.337601
\(897\) −10.0953 −0.337071
\(898\) 9.17997 0.306340
\(899\) −12.0706 −0.402576
\(900\) 0.0686106 0.00228702
\(901\) 41.4670 1.38146
\(902\) −21.4075 −0.712792
\(903\) 81.5752 2.71465
\(904\) −36.5045 −1.21412
\(905\) 18.4974 0.614873
\(906\) 38.1911 1.26881
\(907\) −32.4144 −1.07630 −0.538151 0.842848i \(-0.680877\pi\)
−0.538151 + 0.842848i \(0.680877\pi\)
\(908\) 13.3875 0.444280
\(909\) 0.0109846 0.000364337 0
\(910\) 6.34326 0.210277
\(911\) −7.16536 −0.237399 −0.118699 0.992930i \(-0.537872\pi\)
−0.118699 + 0.992930i \(0.537872\pi\)
\(912\) 0 0
\(913\) −28.5458 −0.944727
\(914\) −29.2948 −0.968987
\(915\) 10.5773 0.349674
\(916\) −8.21555 −0.271449
\(917\) −43.8536 −1.44817
\(918\) 37.2818 1.23048
\(919\) −13.3921 −0.441765 −0.220883 0.975300i \(-0.570894\pi\)
−0.220883 + 0.975300i \(0.570894\pi\)
\(920\) −15.5735 −0.513442
\(921\) −10.4891 −0.345627
\(922\) 8.52058 0.280610
\(923\) −3.18923 −0.104975
\(924\) 19.7316 0.649122
\(925\) −5.98080 −0.196647
\(926\) 27.8694 0.915844
\(927\) −1.10407 −0.0362624
\(928\) 6.06987 0.199253
\(929\) 8.49231 0.278624 0.139312 0.990249i \(-0.455511\pi\)
0.139312 + 0.990249i \(0.455511\pi\)
\(930\) −14.5364 −0.476667
\(931\) 0 0
\(932\) 12.7668 0.418190
\(933\) 5.70710 0.186842
\(934\) 6.86953 0.224778
\(935\) −21.4710 −0.702176
\(936\) −0.351554 −0.0114909
\(937\) −50.7981 −1.65950 −0.829751 0.558134i \(-0.811517\pi\)
−0.829751 + 0.558134i \(0.811517\pi\)
\(938\) 30.4501 0.994231
\(939\) 41.2946 1.34760
\(940\) 5.67496 0.185097
\(941\) −3.94534 −0.128614 −0.0643072 0.997930i \(-0.520484\pi\)
−0.0643072 + 0.997930i \(0.520484\pi\)
\(942\) 20.9586 0.682868
\(943\) −27.4898 −0.895192
\(944\) −4.55639 −0.148298
\(945\) 25.1012 0.816541
\(946\) −39.6591 −1.28943
\(947\) 1.12607 0.0365924 0.0182962 0.999833i \(-0.494176\pi\)
0.0182962 + 0.999833i \(0.494176\pi\)
\(948\) −14.4798 −0.470281
\(949\) −3.69725 −0.120018
\(950\) 0 0
\(951\) −18.7134 −0.606824
\(952\) −90.8552 −2.94463
\(953\) 51.9998 1.68444 0.842220 0.539135i \(-0.181249\pi\)
0.842220 + 0.539135i \(0.181249\pi\)
\(954\) 0.741889 0.0240195
\(955\) −10.8586 −0.351377
\(956\) 19.1523 0.619431
\(957\) 9.49534 0.306941
\(958\) 25.3645 0.819490
\(959\) 16.4862 0.532366
\(960\) 14.4568 0.466590
\(961\) 25.1600 0.811612
\(962\) 7.97583 0.257151
\(963\) 0.212422 0.00684522
\(964\) −6.70839 −0.216063
\(965\) −12.5059 −0.402578
\(966\) −46.6781 −1.50184
\(967\) −24.6191 −0.791696 −0.395848 0.918316i \(-0.629549\pi\)
−0.395848 + 0.918316i \(0.629549\pi\)
\(968\) −2.99662 −0.0963152
\(969\) 0 0
\(970\) 0.805522 0.0258637
\(971\) 53.3884 1.71332 0.856658 0.515884i \(-0.172537\pi\)
0.856658 + 0.515884i \(0.172537\pi\)
\(972\) −0.712736 −0.0228610
\(973\) 9.14557 0.293194
\(974\) 24.2364 0.776585
\(975\) −1.99547 −0.0639061
\(976\) −13.0224 −0.416837
\(977\) 21.4513 0.686288 0.343144 0.939283i \(-0.388508\pi\)
0.343144 + 0.939283i \(0.388508\pi\)
\(978\) −4.65764 −0.148935
\(979\) 30.5711 0.977058
\(980\) −10.9950 −0.351223
\(981\) 1.30326 0.0416100
\(982\) −38.2914 −1.22193
\(983\) −14.5902 −0.465355 −0.232677 0.972554i \(-0.574749\pi\)
−0.232677 + 0.972554i \(0.574749\pi\)
\(984\) 28.4968 0.908444
\(985\) −10.0643 −0.320674
\(986\) −11.3793 −0.362390
\(987\) 65.3545 2.08026
\(988\) 0 0
\(989\) −50.9271 −1.61939
\(990\) −0.384139 −0.0122087
\(991\) 5.62007 0.178527 0.0892636 0.996008i \(-0.471549\pi\)
0.0892636 + 0.996008i \(0.471549\pi\)
\(992\) −28.2409 −0.896648
\(993\) −43.0215 −1.36525
\(994\) −14.7462 −0.467722
\(995\) 4.81054 0.152504
\(996\) 9.88980 0.313370
\(997\) 31.1721 0.987230 0.493615 0.869681i \(-0.335675\pi\)
0.493615 + 0.869681i \(0.335675\pi\)
\(998\) −40.1510 −1.27096
\(999\) 31.5615 0.998560
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 1805.2.a.n.1.3 yes 4
5.4 even 2 9025.2.a.bh.1.2 4
19.18 odd 2 1805.2.a.j.1.2 4
95.94 odd 2 9025.2.a.bo.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.j.1.2 4 19.18 odd 2
1805.2.a.n.1.3 yes 4 1.1 even 1 trivial
9025.2.a.bh.1.2 4 5.4 even 2
9025.2.a.bo.1.3 4 95.94 odd 2