Properties

Label 1805.2.a.j.1.2
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 1.2
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 \(-0.631874\pi\)
−0.402543 + 0.915401i \(0.631874\pi\)
\(3\) −1.70367 −0.983615 −0.491808 0.870704i \(-0.663664\pi\)
−0.491808 + 0.870704i \(0.663664\pi\)
\(4\) −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 \(-0.448068\pi\)
0.162427 + 0.986721i \(0.448068\pi\)
\(14\) 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 \(-0.547782\pi\)
−0.149550 + 0.988754i \(0.547782\pi\)
\(30\) 1.93974 0.354146
\(31\) 7.49400 1.34596 0.672981 0.739660i \(-0.265014\pi\)
0.672981 + 0.739660i \(0.265014\pi\)
\(32\) −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 \(-0.336405\pi\)
0.491619 + 0.870811i \(0.336405\pi\)
\(38\) 0 0
\(39\) −1.99547 −0.319531
\(40\) 3.07830 0.486723
\(41\) 5.43374 0.848607 0.424303 0.905520i \(-0.360519\pi\)
0.424303 + 0.905520i \(0.360519\pi\)
\(42\) −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 \(-0.348216\pi\)
0.458978 + 0.888447i \(0.348216\pi\)
\(54\) −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 \(-0.545162\pi\)
−0.141404 + 0.989952i \(0.545162\pi\)
\(60\) 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 \(-0.388403\pi\)
0.343453 + 0.939170i \(0.388403\pi\)
\(68\) 4.36628 0.529490
\(69\) −8.61905 −1.03761
\(70\) 5.41569 0.647299
\(71\) −2.72287 −0.323145 −0.161573 0.986861i \(-0.551657\pi\)
−0.161573 + 0.986861i \(0.551657\pi\)
\(72\) −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 \(-0.737785\pi\)
−0.679458 + 0.733715i \(0.737785\pi\)
\(80\) −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 \(-0.655115\pi\)
−0.468249 + 0.883597i \(0.655115\pi\)
\(90\) 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 \(-0.511435\pi\)
−0.0359173 + 0.999355i \(0.511435\pi\)
\(98\) −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 \(-0.688378\pi\)
−0.557861 + 0.829934i \(0.688378\pi\)
\(104\) 3.60554 0.353552
\(105\) 8.10368 0.790838
\(106\) −7.60883 −0.739035
\(107\) 2.17861 0.210614 0.105307 0.994440i \(-0.466417\pi\)
0.105307 + 0.994440i \(0.466417\pi\)
\(108\) −3.71336 −0.357319
\(109\) 13.3663 1.28026 0.640129 0.768268i \(-0.278881\pi\)
0.640129 + 0.768268i \(0.278881\pi\)
\(110\) −3.93974 −0.375639
\(111\) −10.1893 −0.967127
\(112\) 9.97698 0.942736
\(113\) −11.8586 −1.11557 −0.557783 0.829987i \(-0.688348\pi\)
−0.557783 + 0.829987i \(0.688348\pi\)
\(114\) 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 \(-0.849296\pi\)
−0.890000 + 0.455960i \(0.849296\pi\)
\(128\) −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 \(-0.795765\pi\)
−0.801125 + 0.598497i \(0.795765\pi\)
\(152\) 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 \(-0.760819\pi\)
−0.730728 + 0.682668i \(0.760819\pi\)
\(168\) 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 \(-0.585982\pi\)
−0.266848 + 0.963738i \(0.585982\pi\)
\(174\) −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 \(-0.382988\pi\)
0.359381 + 0.933191i \(0.382988\pi\)
\(180\) 0.0686106 0.00511393
\(181\) −18.4974 −1.37490 −0.687449 0.726232i \(-0.741270\pi\)
−0.687449 + 0.726232i \(0.741270\pi\)
\(182\) 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 \(-0.351390\pi\)
0.450096 + 0.892980i \(0.351390\pi\)
\(194\) 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 \(-0.484408\pi\)
0.0489644 + 0.998801i \(0.484408\pi\)
\(212\) −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 \(-0.497009\pi\)
0.00939668 + 0.999956i \(0.497009\pi\)
\(224\) 17.9251 1.19767
\(225\) −0.0975037 −0.00650025
\(226\) 13.5018 0.898128
\(227\) 19.0252 1.26275 0.631375 0.775478i \(-0.282491\pi\)
0.631375 + 0.775478i \(0.282491\pi\)
\(228\) 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 \(-0.599342\pi\)
−0.307051 + 0.951693i \(0.599342\pi\)
\(242\) −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 \(-0.272919\pi\)
0.654405 + 0.756144i \(0.272919\pi\)
\(258\) −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 \(-0.160525\pi\)
0.875512 + 0.483197i \(0.160525\pi\)
\(270\) −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 \(-0.306517\pi\)
0.571100 + 0.820880i \(0.306517\pi\)
\(282\) −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 \(-0.439934\pi\)
0.187585 + 0.982248i \(0.439934\pi\)
\(294\) 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 \(-0.443784\pi\)
0.175692 + 0.984445i \(0.443784\pi\)
\(308\) 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 \(-0.400184\pi\)
0.308466 + 0.951235i \(0.400184\pi\)
\(318\) 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 \(-0.255849\pi\)
0.693994 + 0.719981i \(0.255849\pi\)
\(332\) 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 \(-0.427732\pi\)
0.225091 + 0.974338i \(0.427732\pi\)
\(338\) 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 \(-0.702664\pi\)
−0.594537 + 0.804069i \(0.702664\pi\)
\(374\) 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 \(-0.334471\pi\)
0.496901 + 0.867807i \(0.334471\pi\)
\(380\) 0 0
\(381\) 34.1750 1.75084
\(382\) 12.3633 0.632559
\(383\) −6.13972 −0.313725 −0.156863 0.987620i \(-0.550138\pi\)
−0.156863 + 0.987620i \(0.550138\pi\)
\(384\) 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 \(-0.554680\pi\)
−0.170938 + 0.985282i \(0.554680\pi\)
\(402\) 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 \(-0.562766\pi\)
−0.195910 + 0.980622i \(0.562766\pi\)
\(410\) −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 \(-0.345912\pi\)
0.465395 + 0.885103i \(0.345912\pi\)
\(422\) −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 \(-0.352769\pi\)
0.446221 + 0.894923i \(0.352769\pi\)
\(432\) −11.0688 −0.532548
\(433\) −21.8081 −1.04803 −0.524016 0.851708i \(-0.675567\pi\)
−0.524016 + 0.851708i \(0.675567\pi\)
\(434\) 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 \(-0.344478\pi\)
0.469379 + 0.882997i \(0.344478\pi\)
\(440\) 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 \(-0.560931\pi\)
−0.190253 + 0.981735i \(0.560931\pi\)
\(450\) 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 \(-0.660198\pi\)
−0.482299 + 0.876007i \(0.660198\pi\)
\(488\) 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 \(-0.662576\pi\)
−0.488830 + 0.872379i \(0.662576\pi\)
\(510\) −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 \(-0.383027\pi\)
0.359266 + 0.933235i \(0.383027\pi\)
\(522\) −0.178811 −0.00782633
\(523\) 17.8476 0.780419 0.390210 0.920726i \(-0.372403\pi\)
0.390210 + 0.920726i \(0.372403\pi\)
\(524\) −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 \(-0.678913\pi\)
−0.532941 + 0.846152i \(0.678913\pi\)
\(548\) 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 \(-0.386525\pi\)
0.348989 + 0.937127i \(0.386525\pi\)
\(564\) −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 \(-0.727175\pi\)
−0.654628 + 0.755951i \(0.727175\pi\)
\(570\) 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 \(-0.805405\pi\)
−0.818880 + 0.573965i \(0.805405\pi\)
\(600\) −5.24442 −0.214102
\(601\) −48.3109 −1.97064 −0.985321 0.170710i \(-0.945394\pi\)
−0.985321 + 0.170710i \(0.945394\pi\)
\(602\) −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 \(-0.339810\pi\)
0.482276 + 0.876019i \(0.339810\pi\)
\(608\) 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 \(-0.655937\pi\)
−0.470529 + 0.882384i \(0.655937\pi\)
\(642\) 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 \(-0.549277\pi\)
−0.154191 + 0.988041i \(0.549277\pi\)
\(660\) 4.14826 0.161471
\(661\) −33.2446 −1.29307 −0.646533 0.762886i \(-0.723782\pi\)
−0.646533 + 0.762886i \(0.723782\pi\)
\(662\) −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 \(-0.733339\pi\)
−0.669144 + 0.743133i \(0.733339\pi\)
\(674\) −9.40938 −0.362436
\(675\) 5.27713 0.203117
\(676\) 8.18237 0.314707
\(677\) −13.6369 −0.524107 −0.262054 0.965053i \(-0.584400\pi\)
−0.262054 + 0.965053i \(0.584400\pi\)
\(678\) −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 \(-0.709869\pi\)
−0.612582 + 0.790407i \(0.709869\pi\)
\(684\) 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 \(-0.267537\pi\)
0.667097 + 0.744971i \(0.267537\pi\)
\(744\) −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 \(-0.688875\pi\)
−0.559157 + 0.829062i \(0.688875\pi\)
\(752\) 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 \(-0.584054\pi\)
−0.261005 + 0.965337i \(0.584054\pi\)
\(774\) −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 \(-0.169294\pi\)
0.861869 + 0.507130i \(0.169294\pi\)
\(788\) 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 \(-0.665679\pi\)
−0.497310 + 0.867573i \(0.665679\pi\)
\(798\) 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 \(-0.561137\pi\)
−0.190889 + 0.981612i \(0.561137\pi\)
\(812\) −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 \(-0.383483\pi\)
0.357928 + 0.933749i \(0.383483\pi\)
\(828\) 0.347108 0.0120628
\(829\) 23.1471 0.803932 0.401966 0.915655i \(-0.368327\pi\)
0.401966 + 0.915655i \(0.368327\pi\)
\(830\) 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 \(-0.753996\pi\)
−0.715928 + 0.698174i \(0.753996\pi\)
\(840\) 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 \(-0.279818\pi\)
0.637866 + 0.770148i \(0.279818\pi\)
\(858\) 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 \(-0.337571\pi\)
0.488427 + 0.872605i \(0.337571\pi\)
\(864\) −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 \(-0.309281\pi\)
0.563950 + 0.825809i \(0.309281\pi\)
\(878\) −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 \(-0.426169\pi\)
0.229873 + 0.973221i \(0.426169\pi\)
\(888\) −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 \(-0.319123\pi\)
0.538151 + 0.842848i \(0.319123\pi\)
\(908\) −13.3875 −0.444280
\(909\) 0.0109846 0.000364337 0
\(910\) 6.34326 0.210277
\(911\) 7.16536 0.237399 0.118699 0.992930i \(-0.462128\pi\)
0.118699 + 0.992930i \(0.462128\pi\)
\(912\) 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 \(-0.479516\pi\)
0.0643072 + 0.997930i \(0.479516\pi\)
\(942\) 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 \(-0.818751\pi\)
−0.842220 + 0.539135i \(0.818751\pi\)
\(954\) 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 \(-0.827463\pi\)
−0.856658 + 0.515884i \(0.827463\pi\)
\(972\) 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 \(-0.611492\pi\)
−0.343144 + 0.939283i \(0.611492\pi\)
\(978\) −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 \(-0.425251\pi\)
0.232677 + 0.972554i \(0.425251\pi\)
\(984\) −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 \(-0.528451\pi\)
−0.0892636 + 0.996008i \(0.528451\pi\)
\(992\) −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.j.1.2 4
5.4 even 2 9025.2.a.bo.1.3 4
19.18 odd 2 1805.2.a.n.1.3 yes 4
95.94 odd 2 9025.2.a.bh.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.j.1.2 4 1.1 even 1 trivial
1805.2.a.n.1.3 yes 4 19.18 odd 2
9025.2.a.bh.1.2 4 95.94 odd 2
9025.2.a.bo.1.3 4 5.4 even 2