Properties

Label 547.2.a.b.1.14
Level $547$
Weight $2$
Character 547.1
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36781699056\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(-1.15793\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.15793 q^{2} -0.220625 q^{3} -0.659200 q^{4} -0.421419 q^{5} -0.255468 q^{6} -0.645304 q^{7} -3.07917 q^{8} -2.95132 q^{9} +O(q^{10})\) \(q+1.15793 q^{2} -0.220625 q^{3} -0.659200 q^{4} -0.421419 q^{5} -0.255468 q^{6} -0.645304 q^{7} -3.07917 q^{8} -2.95132 q^{9} -0.487974 q^{10} -0.640037 q^{11} +0.145436 q^{12} -4.07156 q^{13} -0.747216 q^{14} +0.0929756 q^{15} -2.24706 q^{16} +6.47629 q^{17} -3.41743 q^{18} -4.15152 q^{19} +0.277799 q^{20} +0.142370 q^{21} -0.741118 q^{22} -7.48578 q^{23} +0.679341 q^{24} -4.82241 q^{25} -4.71457 q^{26} +1.31301 q^{27} +0.425384 q^{28} -0.298162 q^{29} +0.107659 q^{30} +10.2445 q^{31} +3.55640 q^{32} +0.141208 q^{33} +7.49908 q^{34} +0.271944 q^{35} +1.94551 q^{36} -6.30257 q^{37} -4.80717 q^{38} +0.898287 q^{39} +1.29762 q^{40} +3.79643 q^{41} +0.164855 q^{42} +0.987983 q^{43} +0.421912 q^{44} +1.24375 q^{45} -8.66801 q^{46} +0.803927 q^{47} +0.495757 q^{48} -6.58358 q^{49} -5.58401 q^{50} -1.42883 q^{51} +2.68397 q^{52} -0.867236 q^{53} +1.52037 q^{54} +0.269724 q^{55} +1.98700 q^{56} +0.915929 q^{57} -0.345251 q^{58} +0.841036 q^{59} -0.0612895 q^{60} -5.88765 q^{61} +11.8624 q^{62} +1.90450 q^{63} +8.61217 q^{64} +1.71583 q^{65} +0.163509 q^{66} +3.61378 q^{67} -4.26917 q^{68} +1.65155 q^{69} +0.314891 q^{70} -4.80121 q^{71} +9.08762 q^{72} +9.79568 q^{73} -7.29793 q^{74} +1.06394 q^{75} +2.73668 q^{76} +0.413018 q^{77} +1.04015 q^{78} +5.23123 q^{79} +0.946953 q^{80} +8.56429 q^{81} +4.39600 q^{82} -17.0050 q^{83} -0.0938503 q^{84} -2.72923 q^{85} +1.14401 q^{86} +0.0657820 q^{87} +1.97078 q^{88} -9.06607 q^{89} +1.44017 q^{90} +2.62739 q^{91} +4.93463 q^{92} -2.26019 q^{93} +0.930890 q^{94} +1.74953 q^{95} -0.784630 q^{96} +5.13040 q^{97} -7.62332 q^{98} +1.88896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.15793 0.818780 0.409390 0.912360i \(-0.365742\pi\)
0.409390 + 0.912360i \(0.365742\pi\)
\(3\) −0.220625 −0.127378 −0.0636889 0.997970i \(-0.520287\pi\)
−0.0636889 + 0.997970i \(0.520287\pi\)
\(4\) −0.659200 −0.329600
\(5\) −0.421419 −0.188464 −0.0942322 0.995550i \(-0.530040\pi\)
−0.0942322 + 0.995550i \(0.530040\pi\)
\(6\) −0.255468 −0.104294
\(7\) −0.645304 −0.243902 −0.121951 0.992536i \(-0.538915\pi\)
−0.121951 + 0.992536i \(0.538915\pi\)
\(8\) −3.07917 −1.08865
\(9\) −2.95132 −0.983775
\(10\) −0.487974 −0.154311
\(11\) −0.640037 −0.192978 −0.0964892 0.995334i \(-0.530761\pi\)
−0.0964892 + 0.995334i \(0.530761\pi\)
\(12\) 0.145436 0.0419837
\(13\) −4.07156 −1.12925 −0.564623 0.825349i \(-0.690979\pi\)
−0.564623 + 0.825349i \(0.690979\pi\)
\(14\) −0.747216 −0.199702
\(15\) 0.0929756 0.0240062
\(16\) −2.24706 −0.561764
\(17\) 6.47629 1.57073 0.785365 0.619033i \(-0.212475\pi\)
0.785365 + 0.619033i \(0.212475\pi\)
\(18\) −3.41743 −0.805495
\(19\) −4.15152 −0.952425 −0.476212 0.879330i \(-0.657991\pi\)
−0.476212 + 0.879330i \(0.657991\pi\)
\(20\) 0.277799 0.0621179
\(21\) 0.142370 0.0310677
\(22\) −0.741118 −0.158007
\(23\) −7.48578 −1.56089 −0.780447 0.625222i \(-0.785008\pi\)
−0.780447 + 0.625222i \(0.785008\pi\)
\(24\) 0.679341 0.138670
\(25\) −4.82241 −0.964481
\(26\) −4.71457 −0.924604
\(27\) 1.31301 0.252689
\(28\) 0.425384 0.0803900
\(29\) −0.298162 −0.0553673 −0.0276837 0.999617i \(-0.508813\pi\)
−0.0276837 + 0.999617i \(0.508813\pi\)
\(30\) 0.107659 0.0196558
\(31\) 10.2445 1.83996 0.919981 0.391962i \(-0.128203\pi\)
0.919981 + 0.391962i \(0.128203\pi\)
\(32\) 3.55640 0.628688
\(33\) 0.141208 0.0245812
\(34\) 7.49908 1.28608
\(35\) 0.271944 0.0459668
\(36\) 1.94551 0.324252
\(37\) −6.30257 −1.03614 −0.518068 0.855339i \(-0.673349\pi\)
−0.518068 + 0.855339i \(0.673349\pi\)
\(38\) −4.80717 −0.779826
\(39\) 0.898287 0.143841
\(40\) 1.29762 0.205172
\(41\) 3.79643 0.592903 0.296452 0.955048i \(-0.404197\pi\)
0.296452 + 0.955048i \(0.404197\pi\)
\(42\) 0.164855 0.0254376
\(43\) 0.987983 0.150666 0.0753330 0.997158i \(-0.475998\pi\)
0.0753330 + 0.997158i \(0.475998\pi\)
\(44\) 0.421912 0.0636057
\(45\) 1.24375 0.185407
\(46\) −8.66801 −1.27803
\(47\) 0.803927 0.117265 0.0586324 0.998280i \(-0.481326\pi\)
0.0586324 + 0.998280i \(0.481326\pi\)
\(48\) 0.495757 0.0715563
\(49\) −6.58358 −0.940512
\(50\) −5.58401 −0.789698
\(51\) −1.42883 −0.200076
\(52\) 2.68397 0.372199
\(53\) −0.867236 −0.119124 −0.0595620 0.998225i \(-0.518970\pi\)
−0.0595620 + 0.998225i \(0.518970\pi\)
\(54\) 1.52037 0.206897
\(55\) 0.269724 0.0363696
\(56\) 1.98700 0.265524
\(57\) 0.915929 0.121318
\(58\) −0.345251 −0.0453337
\(59\) 0.841036 0.109494 0.0547468 0.998500i \(-0.482565\pi\)
0.0547468 + 0.998500i \(0.482565\pi\)
\(60\) −0.0612895 −0.00791244
\(61\) −5.88765 −0.753836 −0.376918 0.926247i \(-0.623016\pi\)
−0.376918 + 0.926247i \(0.623016\pi\)
\(62\) 11.8624 1.50652
\(63\) 1.90450 0.239945
\(64\) 8.61217 1.07652
\(65\) 1.71583 0.212823
\(66\) 0.163509 0.0201266
\(67\) 3.61378 0.441493 0.220747 0.975331i \(-0.429151\pi\)
0.220747 + 0.975331i \(0.429151\pi\)
\(68\) −4.26917 −0.517712
\(69\) 1.65155 0.198823
\(70\) 0.314891 0.0376367
\(71\) −4.80121 −0.569799 −0.284900 0.958557i \(-0.591960\pi\)
−0.284900 + 0.958557i \(0.591960\pi\)
\(72\) 9.08762 1.07099
\(73\) 9.79568 1.14650 0.573249 0.819382i \(-0.305683\pi\)
0.573249 + 0.819382i \(0.305683\pi\)
\(74\) −7.29793 −0.848367
\(75\) 1.06394 0.122854
\(76\) 2.73668 0.313919
\(77\) 0.413018 0.0470678
\(78\) 1.04015 0.117774
\(79\) 5.23123 0.588559 0.294280 0.955719i \(-0.404920\pi\)
0.294280 + 0.955719i \(0.404920\pi\)
\(80\) 0.946953 0.105873
\(81\) 8.56429 0.951588
\(82\) 4.39600 0.485457
\(83\) −17.0050 −1.86654 −0.933272 0.359170i \(-0.883060\pi\)
−0.933272 + 0.359170i \(0.883060\pi\)
\(84\) −0.0938503 −0.0102399
\(85\) −2.72923 −0.296027
\(86\) 1.14401 0.123362
\(87\) 0.0657820 0.00705257
\(88\) 1.97078 0.210086
\(89\) −9.06607 −0.961001 −0.480501 0.876994i \(-0.659545\pi\)
−0.480501 + 0.876994i \(0.659545\pi\)
\(90\) 1.44017 0.151807
\(91\) 2.62739 0.275425
\(92\) 4.93463 0.514470
\(93\) −2.26019 −0.234371
\(94\) 0.930890 0.0960140
\(95\) 1.74953 0.179498
\(96\) −0.784630 −0.0800810
\(97\) 5.13040 0.520913 0.260457 0.965486i \(-0.416127\pi\)
0.260457 + 0.965486i \(0.416127\pi\)
\(98\) −7.62332 −0.770072
\(99\) 1.88896 0.189847
\(100\) 3.17893 0.317893
\(101\) −0.429975 −0.0427841 −0.0213921 0.999771i \(-0.506810\pi\)
−0.0213921 + 0.999771i \(0.506810\pi\)
\(102\) −1.65448 −0.163818
\(103\) 6.28129 0.618914 0.309457 0.950913i \(-0.399853\pi\)
0.309457 + 0.950913i \(0.399853\pi\)
\(104\) 12.5370 1.22935
\(105\) −0.0599975 −0.00585516
\(106\) −1.00420 −0.0975363
\(107\) 8.04485 0.777725 0.388863 0.921296i \(-0.372868\pi\)
0.388863 + 0.921296i \(0.372868\pi\)
\(108\) −0.865536 −0.0832862
\(109\) −1.54427 −0.147914 −0.0739571 0.997261i \(-0.523563\pi\)
−0.0739571 + 0.997261i \(0.523563\pi\)
\(110\) 0.312321 0.0297787
\(111\) 1.39050 0.131981
\(112\) 1.45003 0.137015
\(113\) 12.8195 1.20596 0.602979 0.797757i \(-0.293980\pi\)
0.602979 + 0.797757i \(0.293980\pi\)
\(114\) 1.06058 0.0993326
\(115\) 3.15465 0.294173
\(116\) 0.196548 0.0182491
\(117\) 12.0165 1.11092
\(118\) 0.973860 0.0896511
\(119\) −4.17917 −0.383104
\(120\) −0.286287 −0.0261343
\(121\) −10.5904 −0.962759
\(122\) −6.81748 −0.617226
\(123\) −0.837588 −0.0755228
\(124\) −6.75316 −0.606451
\(125\) 4.13935 0.370235
\(126\) 2.20528 0.196462
\(127\) −20.4407 −1.81381 −0.906907 0.421330i \(-0.861563\pi\)
−0.906907 + 0.421330i \(0.861563\pi\)
\(128\) 2.85949 0.252745
\(129\) −0.217974 −0.0191915
\(130\) 1.98681 0.174255
\(131\) 0.581501 0.0508060 0.0254030 0.999677i \(-0.491913\pi\)
0.0254030 + 0.999677i \(0.491913\pi\)
\(132\) −0.0930844 −0.00810195
\(133\) 2.67899 0.232298
\(134\) 4.18450 0.361486
\(135\) −0.553328 −0.0476229
\(136\) −19.9416 −1.70997
\(137\) −12.2771 −1.04890 −0.524452 0.851440i \(-0.675730\pi\)
−0.524452 + 0.851440i \(0.675730\pi\)
\(138\) 1.91238 0.162792
\(139\) −1.35565 −0.114985 −0.0574923 0.998346i \(-0.518310\pi\)
−0.0574923 + 0.998346i \(0.518310\pi\)
\(140\) −0.179265 −0.0151507
\(141\) −0.177366 −0.0149369
\(142\) −5.55947 −0.466540
\(143\) 2.60595 0.217920
\(144\) 6.63179 0.552649
\(145\) 0.125651 0.0104348
\(146\) 11.3427 0.938729
\(147\) 1.45250 0.119800
\(148\) 4.15465 0.341510
\(149\) 3.61357 0.296035 0.148017 0.988985i \(-0.452711\pi\)
0.148017 + 0.988985i \(0.452711\pi\)
\(150\) 1.23197 0.100590
\(151\) −4.04873 −0.329481 −0.164741 0.986337i \(-0.552679\pi\)
−0.164741 + 0.986337i \(0.552679\pi\)
\(152\) 12.7832 1.03686
\(153\) −19.1136 −1.54524
\(154\) 0.478246 0.0385382
\(155\) −4.31722 −0.346768
\(156\) −0.592150 −0.0474100
\(157\) −16.8338 −1.34348 −0.671742 0.740786i \(-0.734453\pi\)
−0.671742 + 0.740786i \(0.734453\pi\)
\(158\) 6.05739 0.481900
\(159\) 0.191334 0.0151738
\(160\) −1.49873 −0.118485
\(161\) 4.83061 0.380705
\(162\) 9.91684 0.779141
\(163\) 23.5478 1.84441 0.922203 0.386706i \(-0.126387\pi\)
0.922203 + 0.386706i \(0.126387\pi\)
\(164\) −2.50261 −0.195421
\(165\) −0.0595078 −0.00463268
\(166\) −19.6906 −1.52829
\(167\) −11.2055 −0.867108 −0.433554 0.901128i \(-0.642741\pi\)
−0.433554 + 0.901128i \(0.642741\pi\)
\(168\) −0.438381 −0.0338218
\(169\) 3.57757 0.275198
\(170\) −3.16026 −0.242381
\(171\) 12.2525 0.936972
\(172\) −0.651278 −0.0496595
\(173\) −6.09402 −0.463320 −0.231660 0.972797i \(-0.574416\pi\)
−0.231660 + 0.972797i \(0.574416\pi\)
\(174\) 0.0761709 0.00577450
\(175\) 3.11192 0.235239
\(176\) 1.43820 0.108408
\(177\) −0.185553 −0.0139470
\(178\) −10.4979 −0.786848
\(179\) −11.8489 −0.885629 −0.442814 0.896613i \(-0.646020\pi\)
−0.442814 + 0.896613i \(0.646020\pi\)
\(180\) −0.819876 −0.0611100
\(181\) −11.3326 −0.842346 −0.421173 0.906980i \(-0.638381\pi\)
−0.421173 + 0.906980i \(0.638381\pi\)
\(182\) 3.04233 0.225513
\(183\) 1.29896 0.0960220
\(184\) 23.0500 1.69927
\(185\) 2.65602 0.195275
\(186\) −2.61714 −0.191898
\(187\) −4.14506 −0.303117
\(188\) −0.529948 −0.0386504
\(189\) −0.847291 −0.0616313
\(190\) 2.02583 0.146969
\(191\) 6.48714 0.469393 0.234697 0.972069i \(-0.424590\pi\)
0.234697 + 0.972069i \(0.424590\pi\)
\(192\) −1.90006 −0.137125
\(193\) −11.6701 −0.840033 −0.420017 0.907516i \(-0.637976\pi\)
−0.420017 + 0.907516i \(0.637976\pi\)
\(194\) 5.94064 0.426513
\(195\) −0.378555 −0.0271089
\(196\) 4.33990 0.309993
\(197\) −19.8657 −1.41537 −0.707685 0.706528i \(-0.750260\pi\)
−0.707685 + 0.706528i \(0.750260\pi\)
\(198\) 2.18728 0.155443
\(199\) 21.3669 1.51466 0.757331 0.653031i \(-0.226503\pi\)
0.757331 + 0.653031i \(0.226503\pi\)
\(200\) 14.8490 1.04998
\(201\) −0.797289 −0.0562364
\(202\) −0.497881 −0.0350308
\(203\) 0.192405 0.0135042
\(204\) 0.941884 0.0659451
\(205\) −1.59989 −0.111741
\(206\) 7.27330 0.506754
\(207\) 22.0930 1.53557
\(208\) 9.14902 0.634370
\(209\) 2.65713 0.183797
\(210\) −0.0694729 −0.00479408
\(211\) 17.2121 1.18493 0.592465 0.805596i \(-0.298155\pi\)
0.592465 + 0.805596i \(0.298155\pi\)
\(212\) 0.571682 0.0392633
\(213\) 1.05927 0.0725798
\(214\) 9.31537 0.636786
\(215\) −0.416355 −0.0283952
\(216\) −4.04298 −0.275090
\(217\) −6.61080 −0.448771
\(218\) −1.78816 −0.121109
\(219\) −2.16117 −0.146038
\(220\) −0.177802 −0.0119874
\(221\) −26.3686 −1.77374
\(222\) 1.61010 0.108063
\(223\) −19.3469 −1.29557 −0.647783 0.761825i \(-0.724304\pi\)
−0.647783 + 0.761825i \(0.724304\pi\)
\(224\) −2.29496 −0.153338
\(225\) 14.2325 0.948832
\(226\) 14.8441 0.987413
\(227\) −25.0121 −1.66011 −0.830057 0.557679i \(-0.811692\pi\)
−0.830057 + 0.557679i \(0.811692\pi\)
\(228\) −0.603780 −0.0399863
\(229\) −14.3939 −0.951178 −0.475589 0.879668i \(-0.657765\pi\)
−0.475589 + 0.879668i \(0.657765\pi\)
\(230\) 3.65287 0.240863
\(231\) −0.0911222 −0.00599540
\(232\) 0.918091 0.0602756
\(233\) −19.0426 −1.24752 −0.623762 0.781614i \(-0.714397\pi\)
−0.623762 + 0.781614i \(0.714397\pi\)
\(234\) 13.9142 0.909602
\(235\) −0.338790 −0.0221002
\(236\) −0.554410 −0.0360890
\(237\) −1.15414 −0.0749694
\(238\) −4.83919 −0.313678
\(239\) 11.6405 0.752963 0.376481 0.926424i \(-0.377134\pi\)
0.376481 + 0.926424i \(0.377134\pi\)
\(240\) −0.208921 −0.0134858
\(241\) −19.3895 −1.24899 −0.624494 0.781029i \(-0.714695\pi\)
−0.624494 + 0.781029i \(0.714695\pi\)
\(242\) −12.2629 −0.788288
\(243\) −5.82853 −0.373900
\(244\) 3.88114 0.248464
\(245\) 2.77445 0.177253
\(246\) −0.969868 −0.0618365
\(247\) 16.9032 1.07552
\(248\) −31.5444 −2.00307
\(249\) 3.75173 0.237756
\(250\) 4.79308 0.303141
\(251\) −2.07993 −0.131284 −0.0656419 0.997843i \(-0.520910\pi\)
−0.0656419 + 0.997843i \(0.520910\pi\)
\(252\) −1.25545 −0.0790857
\(253\) 4.79118 0.301219
\(254\) −23.6688 −1.48511
\(255\) 0.602137 0.0377073
\(256\) −13.9133 −0.869578
\(257\) 23.6693 1.47645 0.738225 0.674555i \(-0.235664\pi\)
0.738225 + 0.674555i \(0.235664\pi\)
\(258\) −0.252398 −0.0157136
\(259\) 4.06707 0.252716
\(260\) −1.13108 −0.0701464
\(261\) 0.879974 0.0544690
\(262\) 0.673337 0.0415989
\(263\) 13.5122 0.833198 0.416599 0.909090i \(-0.363222\pi\)
0.416599 + 0.909090i \(0.363222\pi\)
\(264\) −0.434803 −0.0267603
\(265\) 0.365470 0.0224506
\(266\) 3.10209 0.190201
\(267\) 2.00020 0.122410
\(268\) −2.38220 −0.145516
\(269\) −13.0912 −0.798186 −0.399093 0.916910i \(-0.630675\pi\)
−0.399093 + 0.916910i \(0.630675\pi\)
\(270\) −0.640715 −0.0389927
\(271\) 22.6734 1.37731 0.688656 0.725089i \(-0.258201\pi\)
0.688656 + 0.725089i \(0.258201\pi\)
\(272\) −14.5526 −0.882380
\(273\) −0.579668 −0.0350831
\(274\) −14.2160 −0.858821
\(275\) 3.08652 0.186124
\(276\) −1.08870 −0.0655321
\(277\) 11.2897 0.678335 0.339167 0.940726i \(-0.389855\pi\)
0.339167 + 0.940726i \(0.389855\pi\)
\(278\) −1.56975 −0.0941471
\(279\) −30.2348 −1.81011
\(280\) −0.837359 −0.0500418
\(281\) −31.5464 −1.88190 −0.940951 0.338543i \(-0.890066\pi\)
−0.940951 + 0.338543i \(0.890066\pi\)
\(282\) −0.205378 −0.0122301
\(283\) 12.2749 0.729666 0.364833 0.931073i \(-0.381126\pi\)
0.364833 + 0.931073i \(0.381126\pi\)
\(284\) 3.16496 0.187806
\(285\) −0.385990 −0.0228641
\(286\) 3.01750 0.178429
\(287\) −2.44985 −0.144610
\(288\) −10.4961 −0.618488
\(289\) 24.9423 1.46719
\(290\) 0.145495 0.00854378
\(291\) −1.13189 −0.0663528
\(292\) −6.45731 −0.377885
\(293\) −17.1994 −1.00480 −0.502400 0.864635i \(-0.667550\pi\)
−0.502400 + 0.864635i \(0.667550\pi\)
\(294\) 1.68189 0.0980901
\(295\) −0.354429 −0.0206356
\(296\) 19.4067 1.12799
\(297\) −0.840375 −0.0487635
\(298\) 4.18425 0.242387
\(299\) 30.4788 1.76263
\(300\) −0.701351 −0.0404925
\(301\) −0.637549 −0.0367477
\(302\) −4.68815 −0.269773
\(303\) 0.0948632 0.00544975
\(304\) 9.32871 0.535038
\(305\) 2.48117 0.142071
\(306\) −22.1322 −1.26522
\(307\) 31.1189 1.77605 0.888025 0.459795i \(-0.152077\pi\)
0.888025 + 0.459795i \(0.152077\pi\)
\(308\) −0.272262 −0.0155135
\(309\) −1.38581 −0.0788360
\(310\) −4.99904 −0.283926
\(311\) 3.57300 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(312\) −2.76597 −0.156592
\(313\) 25.7114 1.45330 0.726648 0.687010i \(-0.241077\pi\)
0.726648 + 0.687010i \(0.241077\pi\)
\(314\) −19.4923 −1.10002
\(315\) −0.802594 −0.0452210
\(316\) −3.44842 −0.193989
\(317\) −18.3410 −1.03013 −0.515067 0.857150i \(-0.672233\pi\)
−0.515067 + 0.857150i \(0.672233\pi\)
\(318\) 0.221551 0.0124240
\(319\) 0.190835 0.0106847
\(320\) −3.62933 −0.202886
\(321\) −1.77489 −0.0990650
\(322\) 5.59350 0.311714
\(323\) −26.8865 −1.49600
\(324\) −5.64558 −0.313643
\(325\) 19.6347 1.08914
\(326\) 27.2667 1.51016
\(327\) 0.340704 0.0188410
\(328\) −11.6898 −0.645464
\(329\) −0.518777 −0.0286011
\(330\) −0.0689059 −0.00379314
\(331\) −6.93883 −0.381393 −0.190696 0.981649i \(-0.561075\pi\)
−0.190696 + 0.981649i \(0.561075\pi\)
\(332\) 11.2097 0.615213
\(333\) 18.6009 1.01932
\(334\) −12.9752 −0.709971
\(335\) −1.52292 −0.0832057
\(336\) −0.319914 −0.0174527
\(337\) −17.6976 −0.964047 −0.482024 0.876158i \(-0.660098\pi\)
−0.482024 + 0.876158i \(0.660098\pi\)
\(338\) 4.14257 0.225326
\(339\) −2.82830 −0.153612
\(340\) 1.79911 0.0975704
\(341\) −6.55685 −0.355073
\(342\) 14.1875 0.767173
\(343\) 8.76554 0.473295
\(344\) −3.04216 −0.164022
\(345\) −0.695995 −0.0374711
\(346\) −7.05645 −0.379357
\(347\) 8.98600 0.482393 0.241197 0.970476i \(-0.422460\pi\)
0.241197 + 0.970476i \(0.422460\pi\)
\(348\) −0.0433635 −0.00232453
\(349\) −29.3335 −1.57018 −0.785092 0.619379i \(-0.787385\pi\)
−0.785092 + 0.619379i \(0.787385\pi\)
\(350\) 3.60338 0.192609
\(351\) −5.34600 −0.285348
\(352\) −2.27623 −0.121323
\(353\) −31.5641 −1.67999 −0.839995 0.542594i \(-0.817442\pi\)
−0.839995 + 0.542594i \(0.817442\pi\)
\(354\) −0.214858 −0.0114196
\(355\) 2.02332 0.107387
\(356\) 5.97635 0.316746
\(357\) 0.922030 0.0487990
\(358\) −13.7202 −0.725135
\(359\) −27.6250 −1.45799 −0.728997 0.684517i \(-0.760013\pi\)
−0.728997 + 0.684517i \(0.760013\pi\)
\(360\) −3.82970 −0.201843
\(361\) −1.76485 −0.0928870
\(362\) −13.1224 −0.689696
\(363\) 2.33650 0.122634
\(364\) −1.73198 −0.0907802
\(365\) −4.12809 −0.216074
\(366\) 1.50411 0.0786209
\(367\) 8.43787 0.440453 0.220227 0.975449i \(-0.429320\pi\)
0.220227 + 0.975449i \(0.429320\pi\)
\(368\) 16.8210 0.876854
\(369\) −11.2045 −0.583284
\(370\) 3.07549 0.159887
\(371\) 0.559631 0.0290546
\(372\) 1.48991 0.0772485
\(373\) 10.8965 0.564201 0.282101 0.959385i \(-0.408969\pi\)
0.282101 + 0.959385i \(0.408969\pi\)
\(374\) −4.79969 −0.248186
\(375\) −0.913244 −0.0471597
\(376\) −2.47542 −0.127660
\(377\) 1.21398 0.0625234
\(378\) −0.981103 −0.0504625
\(379\) 20.0041 1.02754 0.513770 0.857928i \(-0.328248\pi\)
0.513770 + 0.857928i \(0.328248\pi\)
\(380\) −1.15329 −0.0591626
\(381\) 4.50972 0.231040
\(382\) 7.51165 0.384330
\(383\) 2.92786 0.149607 0.0748033 0.997198i \(-0.476167\pi\)
0.0748033 + 0.997198i \(0.476167\pi\)
\(384\) −0.630874 −0.0321942
\(385\) −0.174054 −0.00887061
\(386\) −13.5132 −0.687802
\(387\) −2.91586 −0.148221
\(388\) −3.38196 −0.171693
\(389\) −8.39847 −0.425819 −0.212910 0.977072i \(-0.568294\pi\)
−0.212910 + 0.977072i \(0.568294\pi\)
\(390\) −0.438340 −0.0221962
\(391\) −48.4801 −2.45174
\(392\) 20.2719 1.02389
\(393\) −0.128294 −0.00647156
\(394\) −23.0030 −1.15888
\(395\) −2.20454 −0.110922
\(396\) −1.24520 −0.0625737
\(397\) 32.4143 1.62683 0.813413 0.581686i \(-0.197607\pi\)
0.813413 + 0.581686i \(0.197607\pi\)
\(398\) 24.7414 1.24017
\(399\) −0.591053 −0.0295897
\(400\) 10.8362 0.541811
\(401\) 5.01237 0.250306 0.125153 0.992137i \(-0.460058\pi\)
0.125153 + 0.992137i \(0.460058\pi\)
\(402\) −0.923204 −0.0460452
\(403\) −41.7110 −2.07777
\(404\) 0.283439 0.0141016
\(405\) −3.60916 −0.179340
\(406\) 0.222792 0.0110570
\(407\) 4.03388 0.199952
\(408\) 4.39960 0.217813
\(409\) 25.6745 1.26952 0.634760 0.772709i \(-0.281099\pi\)
0.634760 + 0.772709i \(0.281099\pi\)
\(410\) −1.85256 −0.0914914
\(411\) 2.70863 0.133607
\(412\) −4.14063 −0.203994
\(413\) −0.542724 −0.0267057
\(414\) 25.5821 1.25729
\(415\) 7.16625 0.351777
\(416\) −14.4801 −0.709944
\(417\) 0.299090 0.0146465
\(418\) 3.07677 0.150490
\(419\) 2.84015 0.138751 0.0693753 0.997591i \(-0.477899\pi\)
0.0693753 + 0.997591i \(0.477899\pi\)
\(420\) 0.0395503 0.00192986
\(421\) 32.1637 1.56757 0.783783 0.621035i \(-0.213288\pi\)
0.783783 + 0.621035i \(0.213288\pi\)
\(422\) 19.9304 0.970196
\(423\) −2.37265 −0.115362
\(424\) 2.67036 0.129684
\(425\) −31.2313 −1.51494
\(426\) 1.22656 0.0594269
\(427\) 3.79932 0.183862
\(428\) −5.30316 −0.256338
\(429\) −0.574937 −0.0277582
\(430\) −0.482110 −0.0232494
\(431\) 7.34330 0.353714 0.176857 0.984237i \(-0.443407\pi\)
0.176857 + 0.984237i \(0.443407\pi\)
\(432\) −2.95041 −0.141952
\(433\) −10.5912 −0.508983 −0.254491 0.967075i \(-0.581908\pi\)
−0.254491 + 0.967075i \(0.581908\pi\)
\(434\) −7.65484 −0.367444
\(435\) −0.0277218 −0.00132916
\(436\) 1.01798 0.0487525
\(437\) 31.0774 1.48663
\(438\) −2.50248 −0.119573
\(439\) −10.6116 −0.506464 −0.253232 0.967406i \(-0.581494\pi\)
−0.253232 + 0.967406i \(0.581494\pi\)
\(440\) −0.830525 −0.0395937
\(441\) 19.4303 0.925252
\(442\) −30.5329 −1.45230
\(443\) −31.0558 −1.47551 −0.737753 0.675071i \(-0.764113\pi\)
−0.737753 + 0.675071i \(0.764113\pi\)
\(444\) −0.916620 −0.0435008
\(445\) 3.82062 0.181115
\(446\) −22.4024 −1.06078
\(447\) −0.797243 −0.0377083
\(448\) −5.55747 −0.262566
\(449\) 27.9951 1.32117 0.660586 0.750751i \(-0.270308\pi\)
0.660586 + 0.750751i \(0.270308\pi\)
\(450\) 16.4802 0.776885
\(451\) −2.42986 −0.114418
\(452\) −8.45061 −0.397483
\(453\) 0.893251 0.0419686
\(454\) −28.9623 −1.35927
\(455\) −1.10723 −0.0519079
\(456\) −2.82030 −0.132073
\(457\) 26.8641 1.25665 0.628326 0.777950i \(-0.283741\pi\)
0.628326 + 0.777950i \(0.283741\pi\)
\(458\) −16.6672 −0.778805
\(459\) 8.50343 0.396906
\(460\) −2.07955 −0.0969594
\(461\) 30.6776 1.42880 0.714399 0.699738i \(-0.246700\pi\)
0.714399 + 0.699738i \(0.246700\pi\)
\(462\) −0.105513 −0.00490891
\(463\) 15.2132 0.707015 0.353508 0.935432i \(-0.384989\pi\)
0.353508 + 0.935432i \(0.384989\pi\)
\(464\) 0.669987 0.0311034
\(465\) 0.952487 0.0441705
\(466\) −22.0500 −1.02145
\(467\) −17.9194 −0.829209 −0.414604 0.910002i \(-0.636080\pi\)
−0.414604 + 0.910002i \(0.636080\pi\)
\(468\) −7.92126 −0.366160
\(469\) −2.33198 −0.107681
\(470\) −0.392295 −0.0180952
\(471\) 3.71395 0.171130
\(472\) −2.58969 −0.119200
\(473\) −0.632346 −0.0290753
\(474\) −1.33641 −0.0613834
\(475\) 20.0203 0.918596
\(476\) 2.75491 0.126271
\(477\) 2.55949 0.117191
\(478\) 13.4789 0.616510
\(479\) 34.8507 1.59237 0.796186 0.605052i \(-0.206848\pi\)
0.796186 + 0.605052i \(0.206848\pi\)
\(480\) 0.330658 0.0150924
\(481\) 25.6613 1.17005
\(482\) −22.4517 −1.02265
\(483\) −1.06575 −0.0484934
\(484\) 6.98116 0.317325
\(485\) −2.16205 −0.0981737
\(486\) −6.74902 −0.306142
\(487\) 3.51576 0.159314 0.0796571 0.996822i \(-0.474617\pi\)
0.0796571 + 0.996822i \(0.474617\pi\)
\(488\) 18.1290 0.820663
\(489\) −5.19523 −0.234936
\(490\) 3.21262 0.145131
\(491\) 17.8253 0.804446 0.402223 0.915542i \(-0.368238\pi\)
0.402223 + 0.915542i \(0.368238\pi\)
\(492\) 0.552138 0.0248923
\(493\) −1.93098 −0.0869672
\(494\) 19.5727 0.880616
\(495\) −0.796043 −0.0357795
\(496\) −23.0199 −1.03363
\(497\) 3.09824 0.138975
\(498\) 4.34424 0.194670
\(499\) 32.2263 1.44265 0.721323 0.692599i \(-0.243534\pi\)
0.721323 + 0.692599i \(0.243534\pi\)
\(500\) −2.72866 −0.122029
\(501\) 2.47221 0.110450
\(502\) −2.40841 −0.107493
\(503\) 11.2441 0.501350 0.250675 0.968071i \(-0.419348\pi\)
0.250675 + 0.968071i \(0.419348\pi\)
\(504\) −5.86427 −0.261216
\(505\) 0.181200 0.00806329
\(506\) 5.54785 0.246632
\(507\) −0.789301 −0.0350541
\(508\) 13.4745 0.597833
\(509\) −27.6441 −1.22530 −0.612651 0.790353i \(-0.709897\pi\)
−0.612651 + 0.790353i \(0.709897\pi\)
\(510\) 0.697232 0.0308739
\(511\) −6.32119 −0.279633
\(512\) −21.8295 −0.964739
\(513\) −5.45099 −0.240667
\(514\) 27.4074 1.20889
\(515\) −2.64706 −0.116643
\(516\) 0.143688 0.00632552
\(517\) −0.514543 −0.0226296
\(518\) 4.70938 0.206918
\(519\) 1.34449 0.0590167
\(520\) −5.28333 −0.231689
\(521\) −24.0447 −1.05342 −0.526709 0.850046i \(-0.676574\pi\)
−0.526709 + 0.850046i \(0.676574\pi\)
\(522\) 1.01895 0.0445981
\(523\) −6.79138 −0.296966 −0.148483 0.988915i \(-0.547439\pi\)
−0.148483 + 0.988915i \(0.547439\pi\)
\(524\) −0.383325 −0.0167456
\(525\) −0.686566 −0.0299642
\(526\) 15.6462 0.682205
\(527\) 66.3462 2.89009
\(528\) −0.317303 −0.0138088
\(529\) 33.0370 1.43639
\(530\) 0.423188 0.0183821
\(531\) −2.48217 −0.107717
\(532\) −1.76599 −0.0765655
\(533\) −15.4574 −0.669534
\(534\) 2.31609 0.100227
\(535\) −3.39026 −0.146574
\(536\) −11.1274 −0.480631
\(537\) 2.61416 0.112809
\(538\) −15.1587 −0.653538
\(539\) 4.21374 0.181499
\(540\) 0.364754 0.0156965
\(541\) −3.22089 −0.138477 −0.0692385 0.997600i \(-0.522057\pi\)
−0.0692385 + 0.997600i \(0.522057\pi\)
\(542\) 26.2542 1.12771
\(543\) 2.50025 0.107296
\(544\) 23.0323 0.987500
\(545\) 0.650785 0.0278766
\(546\) −0.671214 −0.0287253
\(547\) −1.00000 −0.0427569
\(548\) 8.09306 0.345718
\(549\) 17.3764 0.741605
\(550\) 3.57397 0.152395
\(551\) 1.23783 0.0527332
\(552\) −5.08540 −0.216449
\(553\) −3.37573 −0.143551
\(554\) 13.0727 0.555407
\(555\) −0.585985 −0.0248737
\(556\) 0.893644 0.0378989
\(557\) 2.24964 0.0953202 0.0476601 0.998864i \(-0.484824\pi\)
0.0476601 + 0.998864i \(0.484824\pi\)
\(558\) −35.0097 −1.48208
\(559\) −4.02263 −0.170139
\(560\) −0.611072 −0.0258225
\(561\) 0.914504 0.0386104
\(562\) −36.5285 −1.54086
\(563\) 13.2181 0.557077 0.278539 0.960425i \(-0.410150\pi\)
0.278539 + 0.960425i \(0.410150\pi\)
\(564\) 0.116920 0.00492321
\(565\) −5.40238 −0.227280
\(566\) 14.2135 0.597436
\(567\) −5.52657 −0.232094
\(568\) 14.7837 0.620312
\(569\) 9.91716 0.415749 0.207875 0.978156i \(-0.433345\pi\)
0.207875 + 0.978156i \(0.433345\pi\)
\(570\) −0.446950 −0.0187207
\(571\) 37.4262 1.56624 0.783119 0.621872i \(-0.213627\pi\)
0.783119 + 0.621872i \(0.213627\pi\)
\(572\) −1.71784 −0.0718265
\(573\) −1.43123 −0.0597903
\(574\) −2.83676 −0.118404
\(575\) 36.0995 1.50545
\(576\) −25.4173 −1.05905
\(577\) −3.61492 −0.150491 −0.0752455 0.997165i \(-0.523974\pi\)
−0.0752455 + 0.997165i \(0.523974\pi\)
\(578\) 28.8814 1.20131
\(579\) 2.57472 0.107002
\(580\) −0.0828293 −0.00343930
\(581\) 10.9734 0.455254
\(582\) −1.31065 −0.0543283
\(583\) 0.555063 0.0229884
\(584\) −30.1625 −1.24813
\(585\) −5.06398 −0.209370
\(586\) −19.9157 −0.822710
\(587\) −22.0003 −0.908048 −0.454024 0.890989i \(-0.650012\pi\)
−0.454024 + 0.890989i \(0.650012\pi\)
\(588\) −0.957489 −0.0394862
\(589\) −42.5302 −1.75243
\(590\) −0.410403 −0.0168960
\(591\) 4.38286 0.180287
\(592\) 14.1622 0.582064
\(593\) −23.0882 −0.948117 −0.474059 0.880493i \(-0.657212\pi\)
−0.474059 + 0.880493i \(0.657212\pi\)
\(594\) −0.973095 −0.0399266
\(595\) 1.76118 0.0722015
\(596\) −2.38206 −0.0975730
\(597\) −4.71408 −0.192934
\(598\) 35.2923 1.44321
\(599\) 0.624248 0.0255061 0.0127530 0.999919i \(-0.495940\pi\)
0.0127530 + 0.999919i \(0.495940\pi\)
\(600\) −3.27606 −0.133744
\(601\) 26.0763 1.06368 0.531838 0.846846i \(-0.321501\pi\)
0.531838 + 0.846846i \(0.321501\pi\)
\(602\) −0.738237 −0.0300883
\(603\) −10.6654 −0.434330
\(604\) 2.66892 0.108597
\(605\) 4.46298 0.181446
\(606\) 0.109845 0.00446214
\(607\) −4.24424 −0.172268 −0.0861341 0.996284i \(-0.527451\pi\)
−0.0861341 + 0.996284i \(0.527451\pi\)
\(608\) −14.7645 −0.598778
\(609\) −0.0424494 −0.00172014
\(610\) 2.87302 0.116325
\(611\) −3.27323 −0.132421
\(612\) 12.5997 0.509312
\(613\) −43.4402 −1.75453 −0.877267 0.480003i \(-0.840636\pi\)
−0.877267 + 0.480003i \(0.840636\pi\)
\(614\) 36.0335 1.45419
\(615\) 0.352976 0.0142334
\(616\) −1.27175 −0.0512404
\(617\) 31.9260 1.28529 0.642646 0.766163i \(-0.277836\pi\)
0.642646 + 0.766163i \(0.277836\pi\)
\(618\) −1.60467 −0.0645493
\(619\) 41.8091 1.68045 0.840225 0.542237i \(-0.182423\pi\)
0.840225 + 0.542237i \(0.182423\pi\)
\(620\) 2.84591 0.114295
\(621\) −9.82891 −0.394421
\(622\) 4.13728 0.165890
\(623\) 5.85037 0.234390
\(624\) −2.01850 −0.0808047
\(625\) 22.3676 0.894705
\(626\) 29.7720 1.18993
\(627\) −0.586229 −0.0234117
\(628\) 11.0968 0.442812
\(629\) −40.8172 −1.62749
\(630\) −0.929347 −0.0370261
\(631\) 10.6280 0.423094 0.211547 0.977368i \(-0.432150\pi\)
0.211547 + 0.977368i \(0.432150\pi\)
\(632\) −16.1078 −0.640734
\(633\) −3.79742 −0.150934
\(634\) −21.2376 −0.843452
\(635\) 8.61409 0.341840
\(636\) −0.126127 −0.00500127
\(637\) 26.8054 1.06207
\(638\) 0.220973 0.00874842
\(639\) 14.1699 0.560554
\(640\) −1.20504 −0.0476335
\(641\) 3.46304 0.136782 0.0683909 0.997659i \(-0.478213\pi\)
0.0683909 + 0.997659i \(0.478213\pi\)
\(642\) −2.05520 −0.0811124
\(643\) 12.0057 0.473459 0.236730 0.971576i \(-0.423924\pi\)
0.236730 + 0.971576i \(0.423924\pi\)
\(644\) −3.18433 −0.125480
\(645\) 0.0918583 0.00361692
\(646\) −31.1326 −1.22490
\(647\) −17.6257 −0.692937 −0.346468 0.938062i \(-0.612619\pi\)
−0.346468 + 0.938062i \(0.612619\pi\)
\(648\) −26.3709 −1.03595
\(649\) −0.538294 −0.0211299
\(650\) 22.7356 0.891763
\(651\) 1.45851 0.0571634
\(652\) −15.5227 −0.607916
\(653\) 47.1062 1.84341 0.921704 0.387895i \(-0.126798\pi\)
0.921704 + 0.387895i \(0.126798\pi\)
\(654\) 0.394512 0.0154266
\(655\) −0.245056 −0.00957512
\(656\) −8.53080 −0.333072
\(657\) −28.9102 −1.12790
\(658\) −0.600707 −0.0234180
\(659\) 11.8304 0.460848 0.230424 0.973090i \(-0.425989\pi\)
0.230424 + 0.973090i \(0.425989\pi\)
\(660\) 0.0392275 0.00152693
\(661\) 11.7551 0.457219 0.228610 0.973518i \(-0.426582\pi\)
0.228610 + 0.973518i \(0.426582\pi\)
\(662\) −8.03468 −0.312277
\(663\) 5.81756 0.225935
\(664\) 52.3613 2.03201
\(665\) −1.12898 −0.0437800
\(666\) 21.5386 0.834602
\(667\) 2.23198 0.0864226
\(668\) 7.38667 0.285799
\(669\) 4.26842 0.165027
\(670\) −1.76343 −0.0681272
\(671\) 3.76831 0.145474
\(672\) 0.506325 0.0195319
\(673\) −24.5574 −0.946618 −0.473309 0.880897i \(-0.656941\pi\)
−0.473309 + 0.880897i \(0.656941\pi\)
\(674\) −20.4925 −0.789342
\(675\) −6.33187 −0.243714
\(676\) −2.35833 −0.0907051
\(677\) −39.3171 −1.51108 −0.755540 0.655103i \(-0.772625\pi\)
−0.755540 + 0.655103i \(0.772625\pi\)
\(678\) −3.27497 −0.125775
\(679\) −3.31067 −0.127052
\(680\) 8.40376 0.322269
\(681\) 5.51830 0.211462
\(682\) −7.59237 −0.290727
\(683\) −6.62542 −0.253515 −0.126757 0.991934i \(-0.540457\pi\)
−0.126757 + 0.991934i \(0.540457\pi\)
\(684\) −8.07684 −0.308826
\(685\) 5.17381 0.197681
\(686\) 10.1499 0.387524
\(687\) 3.17566 0.121159
\(688\) −2.22005 −0.0846387
\(689\) 3.53100 0.134520
\(690\) −0.805913 −0.0306806
\(691\) −21.9346 −0.834430 −0.417215 0.908808i \(-0.636994\pi\)
−0.417215 + 0.908808i \(0.636994\pi\)
\(692\) 4.01718 0.152710
\(693\) −1.21895 −0.0463041
\(694\) 10.4051 0.394974
\(695\) 0.571297 0.0216705
\(696\) −0.202554 −0.00767778
\(697\) 24.5868 0.931291
\(698\) −33.9661 −1.28564
\(699\) 4.20128 0.158907
\(700\) −2.05137 −0.0775347
\(701\) −18.0547 −0.681917 −0.340958 0.940078i \(-0.610752\pi\)
−0.340958 + 0.940078i \(0.610752\pi\)
\(702\) −6.19028 −0.233637
\(703\) 26.1653 0.986842
\(704\) −5.51211 −0.207745
\(705\) 0.0747456 0.00281508
\(706\) −36.5490 −1.37554
\(707\) 0.277465 0.0104351
\(708\) 0.122317 0.00459694
\(709\) 14.5503 0.546448 0.273224 0.961950i \(-0.411910\pi\)
0.273224 + 0.961950i \(0.411910\pi\)
\(710\) 2.34287 0.0879262
\(711\) −15.4390 −0.579010
\(712\) 27.9159 1.04619
\(713\) −76.6880 −2.87199
\(714\) 1.06765 0.0399556
\(715\) −1.09820 −0.0410702
\(716\) 7.81079 0.291903
\(717\) −2.56819 −0.0959107
\(718\) −31.9878 −1.19378
\(719\) 2.03755 0.0759879 0.0379939 0.999278i \(-0.487903\pi\)
0.0379939 + 0.999278i \(0.487903\pi\)
\(720\) −2.79477 −0.104155
\(721\) −4.05334 −0.150954
\(722\) −2.04357 −0.0760540
\(723\) 4.27781 0.159093
\(724\) 7.47045 0.277637
\(725\) 1.43786 0.0534008
\(726\) 2.70550 0.100410
\(727\) −36.1230 −1.33973 −0.669865 0.742483i \(-0.733648\pi\)
−0.669865 + 0.742483i \(0.733648\pi\)
\(728\) −8.09017 −0.299842
\(729\) −24.4070 −0.903961
\(730\) −4.78003 −0.176917
\(731\) 6.39846 0.236656
\(732\) −0.856275 −0.0316488
\(733\) −19.5342 −0.721512 −0.360756 0.932660i \(-0.617481\pi\)
−0.360756 + 0.932660i \(0.617481\pi\)
\(734\) 9.77046 0.360634
\(735\) −0.612112 −0.0225781
\(736\) −26.6224 −0.981316
\(737\) −2.31295 −0.0851986
\(738\) −12.9740 −0.477581
\(739\) −18.9474 −0.696990 −0.348495 0.937311i \(-0.613307\pi\)
−0.348495 + 0.937311i \(0.613307\pi\)
\(740\) −1.75085 −0.0643625
\(741\) −3.72926 −0.136998
\(742\) 0.648013 0.0237893
\(743\) 19.7628 0.725028 0.362514 0.931978i \(-0.381918\pi\)
0.362514 + 0.931978i \(0.381918\pi\)
\(744\) 6.95949 0.255147
\(745\) −1.52283 −0.0557920
\(746\) 12.6174 0.461957
\(747\) 50.1874 1.83626
\(748\) 2.73242 0.0999073
\(749\) −5.19138 −0.189689
\(750\) −1.05747 −0.0386134
\(751\) 28.5750 1.04272 0.521359 0.853337i \(-0.325425\pi\)
0.521359 + 0.853337i \(0.325425\pi\)
\(752\) −1.80647 −0.0658751
\(753\) 0.458884 0.0167227
\(754\) 1.40571 0.0511929
\(755\) 1.70621 0.0620955
\(756\) 0.558534 0.0203137
\(757\) −27.6434 −1.00472 −0.502358 0.864660i \(-0.667534\pi\)
−0.502358 + 0.864660i \(0.667534\pi\)
\(758\) 23.1633 0.841329
\(759\) −1.05705 −0.0383686
\(760\) −5.38710 −0.195411
\(761\) 20.4285 0.740531 0.370266 0.928926i \(-0.379267\pi\)
0.370266 + 0.928926i \(0.379267\pi\)
\(762\) 5.22193 0.189171
\(763\) 0.996524 0.0360766
\(764\) −4.27632 −0.154712
\(765\) 8.05485 0.291224
\(766\) 3.39025 0.122495
\(767\) −3.42432 −0.123645
\(768\) 3.06961 0.110765
\(769\) 14.6134 0.526974 0.263487 0.964663i \(-0.415127\pi\)
0.263487 + 0.964663i \(0.415127\pi\)
\(770\) −0.201542 −0.00726308
\(771\) −5.22203 −0.188067
\(772\) 7.69294 0.276875
\(773\) −8.68704 −0.312451 −0.156225 0.987721i \(-0.549933\pi\)
−0.156225 + 0.987721i \(0.549933\pi\)
\(774\) −3.37636 −0.121361
\(775\) −49.4030 −1.77461
\(776\) −15.7974 −0.567092
\(777\) −0.897297 −0.0321904
\(778\) −9.72483 −0.348652
\(779\) −15.7610 −0.564696
\(780\) 0.249544 0.00893509
\(781\) 3.07296 0.109959
\(782\) −56.1365 −2.00744
\(783\) −0.391490 −0.0139907
\(784\) 14.7937 0.528346
\(785\) 7.09409 0.253199
\(786\) −0.148555 −0.00529878
\(787\) −50.1106 −1.78625 −0.893125 0.449808i \(-0.851493\pi\)
−0.893125 + 0.449808i \(0.851493\pi\)
\(788\) 13.0954 0.466506
\(789\) −2.98113 −0.106131
\(790\) −2.55270 −0.0908211
\(791\) −8.27247 −0.294135
\(792\) −5.81641 −0.206677
\(793\) 23.9719 0.851267
\(794\) 37.5335 1.33201
\(795\) −0.0806318 −0.00285971
\(796\) −14.0851 −0.499232
\(797\) −23.7864 −0.842557 −0.421279 0.906931i \(-0.638419\pi\)
−0.421279 + 0.906931i \(0.638419\pi\)
\(798\) −0.684397 −0.0242274
\(799\) 5.20646 0.184191
\(800\) −17.1504 −0.606358
\(801\) 26.7569 0.945409
\(802\) 5.80397 0.204945
\(803\) −6.26960 −0.221249
\(804\) 0.525573 0.0185355
\(805\) −2.03571 −0.0717494
\(806\) −48.2984 −1.70124
\(807\) 2.88825 0.101671
\(808\) 1.32396 0.0465769
\(809\) 37.8000 1.32898 0.664489 0.747298i \(-0.268649\pi\)
0.664489 + 0.747298i \(0.268649\pi\)
\(810\) −4.17915 −0.146840
\(811\) −16.7465 −0.588049 −0.294024 0.955798i \(-0.594995\pi\)
−0.294024 + 0.955798i \(0.594995\pi\)
\(812\) −0.126834 −0.00445098
\(813\) −5.00232 −0.175439
\(814\) 4.67095 0.163717
\(815\) −9.92350 −0.347605
\(816\) 3.21066 0.112396
\(817\) −4.10163 −0.143498
\(818\) 29.7292 1.03946
\(819\) −7.75428 −0.270957
\(820\) 1.05465 0.0368299
\(821\) −6.67989 −0.233130 −0.116565 0.993183i \(-0.537188\pi\)
−0.116565 + 0.993183i \(0.537188\pi\)
\(822\) 3.13641 0.109395
\(823\) −42.3883 −1.47756 −0.738782 0.673944i \(-0.764599\pi\)
−0.738782 + 0.673944i \(0.764599\pi\)
\(824\) −19.3411 −0.673781
\(825\) −0.680963 −0.0237081
\(826\) −0.628436 −0.0218661
\(827\) 36.9785 1.28587 0.642934 0.765922i \(-0.277717\pi\)
0.642934 + 0.765922i \(0.277717\pi\)
\(828\) −14.5637 −0.506123
\(829\) −10.0901 −0.350442 −0.175221 0.984529i \(-0.556064\pi\)
−0.175221 + 0.984529i \(0.556064\pi\)
\(830\) 8.29801 0.288028
\(831\) −2.49080 −0.0864048
\(832\) −35.0649 −1.21566
\(833\) −42.6372 −1.47729
\(834\) 0.346325 0.0119923
\(835\) 4.72222 0.163419
\(836\) −1.75158 −0.0605796
\(837\) 13.4511 0.464938
\(838\) 3.28870 0.113606
\(839\) 1.93149 0.0666825 0.0333412 0.999444i \(-0.489385\pi\)
0.0333412 + 0.999444i \(0.489385\pi\)
\(840\) 0.184742 0.00637421
\(841\) −28.9111 −0.996934
\(842\) 37.2433 1.28349
\(843\) 6.95993 0.239713
\(844\) −11.3462 −0.390553
\(845\) −1.50766 −0.0518650
\(846\) −2.74736 −0.0944562
\(847\) 6.83400 0.234819
\(848\) 1.94873 0.0669196
\(849\) −2.70815 −0.0929433
\(850\) −36.1636 −1.24040
\(851\) 47.1797 1.61730
\(852\) −0.698269 −0.0239223
\(853\) −33.0856 −1.13283 −0.566414 0.824121i \(-0.691670\pi\)
−0.566414 + 0.824121i \(0.691670\pi\)
\(854\) 4.39935 0.150543
\(855\) −5.16344 −0.176586
\(856\) −24.7714 −0.846670
\(857\) 2.23398 0.0763113 0.0381557 0.999272i \(-0.487852\pi\)
0.0381557 + 0.999272i \(0.487852\pi\)
\(858\) −0.665736 −0.0227279
\(859\) −53.8019 −1.83570 −0.917848 0.396932i \(-0.870075\pi\)
−0.917848 + 0.396932i \(0.870075\pi\)
\(860\) 0.274461 0.00935905
\(861\) 0.540499 0.0184202
\(862\) 8.50302 0.289614
\(863\) 22.7366 0.773962 0.386981 0.922088i \(-0.373518\pi\)
0.386981 + 0.922088i \(0.373518\pi\)
\(864\) 4.66959 0.158863
\(865\) 2.56814 0.0873193
\(866\) −12.2639 −0.416745
\(867\) −5.50289 −0.186888
\(868\) 4.35784 0.147915
\(869\) −3.34818 −0.113579
\(870\) −0.0320999 −0.00108829
\(871\) −14.7137 −0.498554
\(872\) 4.75506 0.161027
\(873\) −15.1415 −0.512462
\(874\) 35.9854 1.21723
\(875\) −2.67114 −0.0903010
\(876\) 1.42464 0.0481342
\(877\) −19.3780 −0.654349 −0.327175 0.944964i \(-0.606097\pi\)
−0.327175 + 0.944964i \(0.606097\pi\)
\(878\) −12.2875 −0.414682
\(879\) 3.79462 0.127989
\(880\) −0.606085 −0.0204311
\(881\) −33.8051 −1.13892 −0.569461 0.822018i \(-0.692848\pi\)
−0.569461 + 0.822018i \(0.692848\pi\)
\(882\) 22.4989 0.757577
\(883\) −0.646989 −0.0217729 −0.0108864 0.999941i \(-0.503465\pi\)
−0.0108864 + 0.999941i \(0.503465\pi\)
\(884\) 17.3821 0.584625
\(885\) 0.0781958 0.00262852
\(886\) −35.9604 −1.20811
\(887\) −52.9615 −1.77827 −0.889136 0.457643i \(-0.848694\pi\)
−0.889136 + 0.457643i \(0.848694\pi\)
\(888\) −4.28159 −0.143681
\(889\) 13.1904 0.442393
\(890\) 4.42400 0.148293
\(891\) −5.48146 −0.183636
\(892\) 12.7535 0.427019
\(893\) −3.33752 −0.111686
\(894\) −0.923151 −0.0308748
\(895\) 4.99336 0.166910
\(896\) −1.84524 −0.0616451
\(897\) −6.72438 −0.224521
\(898\) 32.4164 1.08175
\(899\) −3.05452 −0.101874
\(900\) −9.38205 −0.312735
\(901\) −5.61647 −0.187112
\(902\) −2.81361 −0.0936828
\(903\) 0.140659 0.00468085
\(904\) −39.4734 −1.31286
\(905\) 4.77578 0.158752
\(906\) 1.03432 0.0343630
\(907\) −17.7548 −0.589539 −0.294770 0.955568i \(-0.595243\pi\)
−0.294770 + 0.955568i \(0.595243\pi\)
\(908\) 16.4880 0.547173
\(909\) 1.26900 0.0420899
\(910\) −1.28210 −0.0425011
\(911\) −49.6346 −1.64447 −0.822233 0.569150i \(-0.807272\pi\)
−0.822233 + 0.569150i \(0.807272\pi\)
\(912\) −2.05815 −0.0681520
\(913\) 10.8839 0.360203
\(914\) 31.1068 1.02892
\(915\) −0.547408 −0.0180967
\(916\) 9.48848 0.313508
\(917\) −0.375245 −0.0123917
\(918\) 9.84637 0.324979
\(919\) 34.5625 1.14011 0.570055 0.821606i \(-0.306922\pi\)
0.570055 + 0.821606i \(0.306922\pi\)
\(920\) −9.71370 −0.320251
\(921\) −6.86561 −0.226229
\(922\) 35.5225 1.16987
\(923\) 19.5484 0.643444
\(924\) 0.0600677 0.00197608
\(925\) 30.3935 0.999334
\(926\) 17.6158 0.578890
\(927\) −18.5381 −0.608872
\(928\) −1.06038 −0.0348088
\(929\) 37.0060 1.21413 0.607063 0.794654i \(-0.292348\pi\)
0.607063 + 0.794654i \(0.292348\pi\)
\(930\) 1.10291 0.0361659
\(931\) 27.3319 0.895767
\(932\) 12.5529 0.411184
\(933\) −0.788293 −0.0258075
\(934\) −20.7493 −0.678939
\(935\) 1.74681 0.0571268
\(936\) −37.0007 −1.20941
\(937\) 13.3304 0.435487 0.217743 0.976006i \(-0.430130\pi\)
0.217743 + 0.976006i \(0.430130\pi\)
\(938\) −2.70027 −0.0881670
\(939\) −5.67258 −0.185118
\(940\) 0.223330 0.00728423
\(941\) −2.02721 −0.0660850 −0.0330425 0.999454i \(-0.510520\pi\)
−0.0330425 + 0.999454i \(0.510520\pi\)
\(942\) 4.30050 0.140118
\(943\) −28.4193 −0.925459
\(944\) −1.88985 −0.0615095
\(945\) 0.357065 0.0116153
\(946\) −0.732212 −0.0238063
\(947\) −40.3563 −1.31140 −0.655701 0.755020i \(-0.727627\pi\)
−0.655701 + 0.755020i \(0.727627\pi\)
\(948\) 0.760808 0.0247099
\(949\) −39.8836 −1.29468
\(950\) 23.1821 0.752128
\(951\) 4.04648 0.131216
\(952\) 12.8684 0.417066
\(953\) 22.7099 0.735646 0.367823 0.929896i \(-0.380103\pi\)
0.367823 + 0.929896i \(0.380103\pi\)
\(954\) 2.96371 0.0959538
\(955\) −2.73381 −0.0884639
\(956\) −7.67343 −0.248176
\(957\) −0.0421029 −0.00136099
\(958\) 40.3547 1.30380
\(959\) 7.92246 0.255830
\(960\) 0.800722 0.0258432
\(961\) 73.9494 2.38546
\(962\) 29.7139 0.958016
\(963\) −23.7430 −0.765107
\(964\) 12.7816 0.411666
\(965\) 4.91801 0.158316
\(966\) −1.23407 −0.0397054
\(967\) −14.9197 −0.479785 −0.239892 0.970799i \(-0.577112\pi\)
−0.239892 + 0.970799i \(0.577112\pi\)
\(968\) 32.6094 1.04811
\(969\) 5.93182 0.190558
\(970\) −2.50350 −0.0803826
\(971\) 49.3577 1.58396 0.791982 0.610544i \(-0.209049\pi\)
0.791982 + 0.610544i \(0.209049\pi\)
\(972\) 3.84216 0.123237
\(973\) 0.874806 0.0280450
\(974\) 4.07100 0.130443
\(975\) −4.33190 −0.138732
\(976\) 13.2299 0.423478
\(977\) 4.29541 0.137422 0.0687112 0.997637i \(-0.478111\pi\)
0.0687112 + 0.997637i \(0.478111\pi\)
\(978\) −6.01571 −0.192361
\(979\) 5.80262 0.185453
\(980\) −1.82892 −0.0584226
\(981\) 4.55764 0.145514
\(982\) 20.6405 0.658664
\(983\) −44.2697 −1.41198 −0.705992 0.708220i \(-0.749499\pi\)
−0.705992 + 0.708220i \(0.749499\pi\)
\(984\) 2.57907 0.0822178
\(985\) 8.37177 0.266747
\(986\) −2.23594 −0.0712069
\(987\) 0.114455 0.00364315
\(988\) −11.1426 −0.354492
\(989\) −7.39583 −0.235174
\(990\) −0.921762 −0.0292955
\(991\) −4.60210 −0.146190 −0.0730952 0.997325i \(-0.523288\pi\)
−0.0730952 + 0.997325i \(0.523288\pi\)
\(992\) 36.4334 1.15676
\(993\) 1.53088 0.0485810
\(994\) 3.58755 0.113790
\(995\) −9.00444 −0.285460
\(996\) −2.47314 −0.0783645
\(997\) −4.30801 −0.136436 −0.0682180 0.997670i \(-0.521731\pi\)
−0.0682180 + 0.997670i \(0.521731\pi\)
\(998\) 37.3157 1.18121
\(999\) −8.27534 −0.261820
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 547.2.a.b.1.14 18
3.2 odd 2 4923.2.a.l.1.5 18
4.3 odd 2 8752.2.a.s.1.9 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.14 18 1.1 even 1 trivial
4923.2.a.l.1.5 18 3.2 odd 2
8752.2.a.s.1.9 18 4.3 odd 2