Properties

Label 6001.2.a.d.1.17
Level 6001
Weight 2
Character 6001.1
Self dual yes
Analytic conductor 47.918
Analytic rank 0
Dimension 121
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6001 = 17 \cdot 353 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9182262530\)
Analytic rank: \(0\)
Dimension: \(121\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) = 6001.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.15090 q^{2} +0.00192410 q^{3} +2.62636 q^{4} -2.18392 q^{5} -0.00413853 q^{6} -0.0660171 q^{7} -1.34723 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-2.15090 q^{2} +0.00192410 q^{3} +2.62636 q^{4} -2.18392 q^{5} -0.00413853 q^{6} -0.0660171 q^{7} -1.34723 q^{8} -3.00000 q^{9} +4.69738 q^{10} -1.22409 q^{11} +0.00505337 q^{12} +4.00583 q^{13} +0.141996 q^{14} -0.00420207 q^{15} -2.35496 q^{16} +1.00000 q^{17} +6.45268 q^{18} +3.51056 q^{19} -5.73574 q^{20} -0.000127023 q^{21} +2.63290 q^{22} +6.94081 q^{23} -0.00259220 q^{24} -0.230514 q^{25} -8.61613 q^{26} -0.0115446 q^{27} -0.173384 q^{28} +2.17980 q^{29} +0.00903821 q^{30} -4.00292 q^{31} +7.75974 q^{32} -0.00235528 q^{33} -2.15090 q^{34} +0.144176 q^{35} -7.87906 q^{36} +1.73156 q^{37} -7.55085 q^{38} +0.00770761 q^{39} +2.94224 q^{40} -0.957973 q^{41} +0.000273214 q^{42} +0.963143 q^{43} -3.21491 q^{44} +6.55174 q^{45} -14.9290 q^{46} +7.86225 q^{47} -0.00453118 q^{48} -6.99564 q^{49} +0.495812 q^{50} +0.00192410 q^{51} +10.5207 q^{52} +11.0535 q^{53} +0.0248312 q^{54} +2.67332 q^{55} +0.0889402 q^{56} +0.00675466 q^{57} -4.68852 q^{58} -8.58175 q^{59} -0.0110361 q^{60} -5.45989 q^{61} +8.60987 q^{62} +0.198051 q^{63} -11.9805 q^{64} -8.74839 q^{65} +0.00506596 q^{66} +2.72843 q^{67} +2.62636 q^{68} +0.0133548 q^{69} -0.310107 q^{70} -3.68059 q^{71} +4.04169 q^{72} -6.56518 q^{73} -3.72441 q^{74} -0.000443531 q^{75} +9.21999 q^{76} +0.0808111 q^{77} -0.0165783 q^{78} -15.3460 q^{79} +5.14304 q^{80} +8.99997 q^{81} +2.06050 q^{82} -1.95438 q^{83} -0.000333608 q^{84} -2.18392 q^{85} -2.07162 q^{86} +0.00419414 q^{87} +1.64914 q^{88} +2.65682 q^{89} -14.0921 q^{90} -0.264453 q^{91} +18.2290 q^{92} -0.00770201 q^{93} -16.9109 q^{94} -7.66677 q^{95} +0.0149305 q^{96} -14.0058 q^{97} +15.0469 q^{98} +3.67228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 121q + 9q^{2} + 21q^{3} + 127q^{4} + 27q^{5} + 17q^{6} + 39q^{7} + 24q^{8} + 134q^{9} + O(q^{10}) \) \( 121q + 9q^{2} + 21q^{3} + 127q^{4} + 27q^{5} + 17q^{6} + 39q^{7} + 24q^{8} + 134q^{9} + 19q^{10} + 48q^{11} + 43q^{12} + 6q^{13} + 40q^{14} + 49q^{15} + 135q^{16} + 121q^{17} + 30q^{19} + 50q^{20} + 18q^{21} + 24q^{22} + 75q^{23} + 24q^{24} + 128q^{25} + 59q^{26} + 75q^{27} + 52q^{28} + 49q^{29} - 34q^{30} + 101q^{31} + 47q^{32} + 20q^{33} + 9q^{34} + 47q^{35} + 138q^{36} + 32q^{37} + 30q^{38} + 101q^{39} + 36q^{40} + 83q^{41} - 11q^{42} + 8q^{43} + 98q^{44} + 49q^{45} + 45q^{46} + 135q^{47} + 54q^{48} + 116q^{49} + 3q^{50} + 21q^{51} - 5q^{52} + 28q^{53} + 10q^{54} + 37q^{55} + 75q^{56} + 31q^{58} + 150q^{59} + 50q^{60} + 36q^{61} + 34q^{62} + 118q^{63} + 110q^{64} + 18q^{65} - 28q^{66} - 6q^{67} + 127q^{68} + 25q^{69} - 22q^{70} + 223q^{71} + q^{72} + 38q^{73} - 10q^{74} + 88q^{75} - 4q^{76} + 38q^{77} + 42q^{78} + 74q^{79} + 106q^{80} + 133q^{81} + 28q^{82} + 55q^{83} + 10q^{84} + 27q^{85} + 64q^{86} + 14q^{87} + 56q^{88} + 118q^{89} + 51q^{90} + 73q^{91} + 82q^{92} + 31q^{93} + 33q^{94} + 106q^{95} + 38q^{96} + 37q^{97} + 88q^{98} + 81q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15090 −1.52091 −0.760457 0.649388i \(-0.775025\pi\)
−0.760457 + 0.649388i \(0.775025\pi\)
\(3\) 0.00192410 0.00111088 0.000555439 1.00000i \(-0.499823\pi\)
0.000555439 1.00000i \(0.499823\pi\)
\(4\) 2.62636 1.31318
\(5\) −2.18392 −0.976677 −0.488338 0.872654i \(-0.662397\pi\)
−0.488338 + 0.872654i \(0.662397\pi\)
\(6\) −0.00413853 −0.00168955
\(7\) −0.0660171 −0.0249521 −0.0124761 0.999922i \(-0.503971\pi\)
−0.0124761 + 0.999922i \(0.503971\pi\)
\(8\) −1.34723 −0.476318
\(9\) −3.00000 −0.999999
\(10\) 4.69738 1.48544
\(11\) −1.22409 −0.369078 −0.184539 0.982825i \(-0.559079\pi\)
−0.184539 + 0.982825i \(0.559079\pi\)
\(12\) 0.00505337 0.00145878
\(13\) 4.00583 1.11102 0.555509 0.831511i \(-0.312524\pi\)
0.555509 + 0.831511i \(0.312524\pi\)
\(14\) 0.141996 0.0379500
\(15\) −0.00420207 −0.00108497
\(16\) −2.35496 −0.588740
\(17\) 1.00000 0.242536
\(18\) 6.45268 1.52091
\(19\) 3.51056 0.805378 0.402689 0.915337i \(-0.368076\pi\)
0.402689 + 0.915337i \(0.368076\pi\)
\(20\) −5.73574 −1.28255
\(21\) −0.000127023 0 −2.77187e−5 0
\(22\) 2.63290 0.561336
\(23\) 6.94081 1.44726 0.723629 0.690189i \(-0.242472\pi\)
0.723629 + 0.690189i \(0.242472\pi\)
\(24\) −0.00259220 −0.000529131 0
\(25\) −0.230514 −0.0461028
\(26\) −8.61613 −1.68976
\(27\) −0.0115446 −0.00222175
\(28\) −0.173384 −0.0327666
\(29\) 2.17980 0.404778 0.202389 0.979305i \(-0.435129\pi\)
0.202389 + 0.979305i \(0.435129\pi\)
\(30\) 0.00903821 0.00165014
\(31\) −4.00292 −0.718946 −0.359473 0.933156i \(-0.617043\pi\)
−0.359473 + 0.933156i \(0.617043\pi\)
\(32\) 7.75974 1.37174
\(33\) −0.00235528 −0.000410001 0
\(34\) −2.15090 −0.368876
\(35\) 0.144176 0.0243701
\(36\) −7.87906 −1.31318
\(37\) 1.73156 0.284667 0.142333 0.989819i \(-0.454539\pi\)
0.142333 + 0.989819i \(0.454539\pi\)
\(38\) −7.55085 −1.22491
\(39\) 0.00770761 0.00123420
\(40\) 2.94224 0.465208
\(41\) −0.957973 −0.149610 −0.0748051 0.997198i \(-0.523833\pi\)
−0.0748051 + 0.997198i \(0.523833\pi\)
\(42\) 0.000273214 0 4.21578e−5 0
\(43\) 0.963143 0.146878 0.0734390 0.997300i \(-0.476603\pi\)
0.0734390 + 0.997300i \(0.476603\pi\)
\(44\) −3.21491 −0.484666
\(45\) 6.55174 0.976675
\(46\) −14.9290 −2.20116
\(47\) 7.86225 1.14683 0.573413 0.819266i \(-0.305619\pi\)
0.573413 + 0.819266i \(0.305619\pi\)
\(48\) −0.00453118 −0.000654019 0
\(49\) −6.99564 −0.999377
\(50\) 0.495812 0.0701184
\(51\) 0.00192410 0.000269428 0
\(52\) 10.5207 1.45896
\(53\) 11.0535 1.51831 0.759155 0.650910i \(-0.225613\pi\)
0.759155 + 0.650910i \(0.225613\pi\)
\(54\) 0.0248312 0.00337910
\(55\) 2.67332 0.360470
\(56\) 0.0889402 0.0118851
\(57\) 0.00675466 0.000894677 0
\(58\) −4.68852 −0.615633
\(59\) −8.58175 −1.11725 −0.558625 0.829421i \(-0.688671\pi\)
−0.558625 + 0.829421i \(0.688671\pi\)
\(60\) −0.0110361 −0.00142476
\(61\) −5.45989 −0.699067 −0.349533 0.936924i \(-0.613660\pi\)
−0.349533 + 0.936924i \(0.613660\pi\)
\(62\) 8.60987 1.09345
\(63\) 0.198051 0.0249521
\(64\) −11.9805 −1.49756
\(65\) −8.74839 −1.08510
\(66\) 0.00506596 0.000623576 0
\(67\) 2.72843 0.333331 0.166666 0.986013i \(-0.446700\pi\)
0.166666 + 0.986013i \(0.446700\pi\)
\(68\) 2.62636 0.318493
\(69\) 0.0133548 0.00160773
\(70\) −0.310107 −0.0370649
\(71\) −3.68059 −0.436805 −0.218403 0.975859i \(-0.570085\pi\)
−0.218403 + 0.975859i \(0.570085\pi\)
\(72\) 4.04169 0.476317
\(73\) −6.56518 −0.768397 −0.384198 0.923251i \(-0.625522\pi\)
−0.384198 + 0.923251i \(0.625522\pi\)
\(74\) −3.72441 −0.432954
\(75\) −0.000443531 0 −5.12146e−5 0
\(76\) 9.21999 1.05761
\(77\) 0.0808111 0.00920928
\(78\) −0.0165783 −0.00187712
\(79\) −15.3460 −1.72656 −0.863280 0.504726i \(-0.831594\pi\)
−0.863280 + 0.504726i \(0.831594\pi\)
\(80\) 5.14304 0.575009
\(81\) 8.99997 0.999996
\(82\) 2.06050 0.227544
\(83\) −1.95438 −0.214522 −0.107261 0.994231i \(-0.534208\pi\)
−0.107261 + 0.994231i \(0.534208\pi\)
\(84\) −0.000333608 0 −3.63997e−5 0
\(85\) −2.18392 −0.236879
\(86\) −2.07162 −0.223389
\(87\) 0.00419414 0.000449659 0
\(88\) 1.64914 0.175799
\(89\) 2.65682 0.281623 0.140811 0.990036i \(-0.455029\pi\)
0.140811 + 0.990036i \(0.455029\pi\)
\(90\) −14.0921 −1.48544
\(91\) −0.264453 −0.0277222
\(92\) 18.2290 1.90051
\(93\) −0.00770201 −0.000798661 0
\(94\) −16.9109 −1.74422
\(95\) −7.66677 −0.786594
\(96\) 0.0149305 0.00152384
\(97\) −14.0058 −1.42207 −0.711036 0.703155i \(-0.751774\pi\)
−0.711036 + 0.703155i \(0.751774\pi\)
\(98\) 15.0469 1.51997
\(99\) 3.67228 0.369078
\(100\) −0.605412 −0.0605412
\(101\) −1.04744 −0.104224 −0.0521120 0.998641i \(-0.516595\pi\)
−0.0521120 + 0.998641i \(0.516595\pi\)
\(102\) −0.00413853 −0.000409776 0
\(103\) −1.96651 −0.193766 −0.0968828 0.995296i \(-0.530887\pi\)
−0.0968828 + 0.995296i \(0.530887\pi\)
\(104\) −5.39678 −0.529197
\(105\) 0.000277408 0 2.70723e−5 0
\(106\) −23.7748 −2.30922
\(107\) 13.8145 1.33550 0.667751 0.744385i \(-0.267257\pi\)
0.667751 + 0.744385i \(0.267257\pi\)
\(108\) −0.0303202 −0.00291756
\(109\) 8.54786 0.818737 0.409368 0.912369i \(-0.365749\pi\)
0.409368 + 0.912369i \(0.365749\pi\)
\(110\) −5.75003 −0.548244
\(111\) 0.00333169 0.000316230 0
\(112\) 0.155468 0.0146903
\(113\) −2.66162 −0.250384 −0.125192 0.992133i \(-0.539955\pi\)
−0.125192 + 0.992133i \(0.539955\pi\)
\(114\) −0.0145286 −0.00136073
\(115\) −15.1581 −1.41350
\(116\) 5.72493 0.531546
\(117\) −12.0175 −1.11102
\(118\) 18.4585 1.69924
\(119\) −0.0660171 −0.00605178
\(120\) 0.00566115 0.000516790 0
\(121\) −9.50159 −0.863781
\(122\) 11.7437 1.06322
\(123\) −0.00184323 −0.000166199 0
\(124\) −10.5131 −0.944104
\(125\) 11.4230 1.02170
\(126\) −0.425987 −0.0379500
\(127\) −10.2995 −0.913934 −0.456967 0.889484i \(-0.651064\pi\)
−0.456967 + 0.889484i \(0.651064\pi\)
\(128\) 10.2493 0.905918
\(129\) 0.00185318 0.000163163 0
\(130\) 18.8169 1.65035
\(131\) 2.97846 0.260229 0.130115 0.991499i \(-0.458465\pi\)
0.130115 + 0.991499i \(0.458465\pi\)
\(132\) −0.00618580 −0.000538405 0
\(133\) −0.231757 −0.0200959
\(134\) −5.86858 −0.506968
\(135\) 0.0252124 0.00216994
\(136\) −1.34723 −0.115524
\(137\) 18.0870 1.54528 0.772639 0.634846i \(-0.218936\pi\)
0.772639 + 0.634846i \(0.218936\pi\)
\(138\) −0.0287248 −0.00244522
\(139\) 22.5177 1.90993 0.954964 0.296722i \(-0.0958936\pi\)
0.954964 + 0.296722i \(0.0958936\pi\)
\(140\) 0.378657 0.0320023
\(141\) 0.0151277 0.00127398
\(142\) 7.91656 0.664343
\(143\) −4.90352 −0.410053
\(144\) 7.06488 0.588740
\(145\) −4.76050 −0.395338
\(146\) 14.1210 1.16867
\(147\) −0.0134603 −0.00111019
\(148\) 4.54770 0.373819
\(149\) −11.5854 −0.949110 −0.474555 0.880226i \(-0.657391\pi\)
−0.474555 + 0.880226i \(0.657391\pi\)
\(150\) 0.000953990 0 7.78929e−5 0
\(151\) −12.4322 −1.01172 −0.505859 0.862616i \(-0.668824\pi\)
−0.505859 + 0.862616i \(0.668824\pi\)
\(152\) −4.72953 −0.383616
\(153\) −3.00000 −0.242535
\(154\) −0.173816 −0.0140065
\(155\) 8.74204 0.702177
\(156\) 0.0202429 0.00162073
\(157\) 2.80771 0.224080 0.112040 0.993704i \(-0.464262\pi\)
0.112040 + 0.993704i \(0.464262\pi\)
\(158\) 33.0077 2.62595
\(159\) 0.0212679 0.00168666
\(160\) −16.9466 −1.33975
\(161\) −0.458212 −0.0361122
\(162\) −19.3580 −1.52091
\(163\) −11.3378 −0.888041 −0.444021 0.896017i \(-0.646448\pi\)
−0.444021 + 0.896017i \(0.646448\pi\)
\(164\) −2.51598 −0.196465
\(165\) 0.00514373 0.000400438 0
\(166\) 4.20368 0.326269
\(167\) 3.19854 0.247511 0.123755 0.992313i \(-0.460506\pi\)
0.123755 + 0.992313i \(0.460506\pi\)
\(168\) 0.000171130 0 1.32029e−5 0
\(169\) 3.04668 0.234360
\(170\) 4.69738 0.360272
\(171\) −10.5317 −0.805377
\(172\) 2.52956 0.192877
\(173\) −4.75516 −0.361528 −0.180764 0.983526i \(-0.557857\pi\)
−0.180764 + 0.983526i \(0.557857\pi\)
\(174\) −0.00902117 −0.000683893 0
\(175\) 0.0152179 0.00115036
\(176\) 2.88270 0.217291
\(177\) −0.0165121 −0.00124113
\(178\) −5.71455 −0.428324
\(179\) −14.0034 −1.04667 −0.523333 0.852128i \(-0.675312\pi\)
−0.523333 + 0.852128i \(0.675312\pi\)
\(180\) 17.2072 1.28255
\(181\) 1.84346 0.137023 0.0685115 0.997650i \(-0.478175\pi\)
0.0685115 + 0.997650i \(0.478175\pi\)
\(182\) 0.568812 0.0421631
\(183\) −0.0105054 −0.000776578 0
\(184\) −9.35087 −0.689355
\(185\) −3.78158 −0.278028
\(186\) 0.0165662 0.00121469
\(187\) −1.22409 −0.0895147
\(188\) 20.6491 1.50599
\(189\) 0.000762139 0 5.54375e−5 0
\(190\) 16.4904 1.19634
\(191\) −1.02605 −0.0742421 −0.0371211 0.999311i \(-0.511819\pi\)
−0.0371211 + 0.999311i \(0.511819\pi\)
\(192\) −0.0230516 −0.00166361
\(193\) 18.0690 1.30064 0.650319 0.759662i \(-0.274635\pi\)
0.650319 + 0.759662i \(0.274635\pi\)
\(194\) 30.1250 2.16285
\(195\) −0.0168328 −0.00120542
\(196\) −18.3731 −1.31236
\(197\) 22.7681 1.62216 0.811080 0.584935i \(-0.198880\pi\)
0.811080 + 0.584935i \(0.198880\pi\)
\(198\) −7.89869 −0.561336
\(199\) −0.940724 −0.0666862 −0.0333431 0.999444i \(-0.510615\pi\)
−0.0333431 + 0.999444i \(0.510615\pi\)
\(200\) 0.310555 0.0219596
\(201\) 0.00524977 0.000370290 0
\(202\) 2.25293 0.158516
\(203\) −0.143904 −0.0101001
\(204\) 0.00505337 0.000353806 0
\(205\) 2.09213 0.146121
\(206\) 4.22975 0.294701
\(207\) −20.8224 −1.44726
\(208\) −9.43358 −0.654101
\(209\) −4.29726 −0.297248
\(210\) −0.000596676 0 −4.11746e−5 0
\(211\) 27.0996 1.86561 0.932805 0.360381i \(-0.117353\pi\)
0.932805 + 0.360381i \(0.117353\pi\)
\(212\) 29.0303 1.99381
\(213\) −0.00708181 −0.000485237 0
\(214\) −29.7136 −2.03118
\(215\) −2.10342 −0.143452
\(216\) 0.0155532 0.00105826
\(217\) 0.264261 0.0179392
\(218\) −18.3856 −1.24523
\(219\) −0.0126321 −0.000853595 0
\(220\) 7.02109 0.473362
\(221\) 4.00583 0.269461
\(222\) −0.00716613 −0.000480959 0
\(223\) −3.89565 −0.260872 −0.130436 0.991457i \(-0.541638\pi\)
−0.130436 + 0.991457i \(0.541638\pi\)
\(224\) −0.512275 −0.0342278
\(225\) 0.691541 0.0461027
\(226\) 5.72488 0.380813
\(227\) 19.7182 1.30874 0.654370 0.756174i \(-0.272934\pi\)
0.654370 + 0.756174i \(0.272934\pi\)
\(228\) 0.0177402 0.00117487
\(229\) −16.5887 −1.09621 −0.548106 0.836409i \(-0.684651\pi\)
−0.548106 + 0.836409i \(0.684651\pi\)
\(230\) 32.6036 2.14982
\(231\) 0.000155488 0 1.02304e−5 0
\(232\) −2.93669 −0.192803
\(233\) −21.3086 −1.39597 −0.697986 0.716111i \(-0.745920\pi\)
−0.697986 + 0.716111i \(0.745920\pi\)
\(234\) 25.8484 1.68976
\(235\) −17.1705 −1.12008
\(236\) −22.5388 −1.46715
\(237\) −0.0295272 −0.00191800
\(238\) 0.141996 0.00920423
\(239\) −25.1268 −1.62532 −0.812660 0.582738i \(-0.801981\pi\)
−0.812660 + 0.582738i \(0.801981\pi\)
\(240\) 0.00989570 0.000638765 0
\(241\) 26.6567 1.71711 0.858554 0.512723i \(-0.171363\pi\)
0.858554 + 0.512723i \(0.171363\pi\)
\(242\) 20.4369 1.31374
\(243\) 0.0519505 0.00333263
\(244\) −14.3396 −0.918000
\(245\) 15.2779 0.976069
\(246\) 0.00396460 0.000252774 0
\(247\) 14.0627 0.894789
\(248\) 5.39285 0.342447
\(249\) −0.00376043 −0.000238307 0
\(250\) −24.5697 −1.55392
\(251\) 6.07268 0.383304 0.191652 0.981463i \(-0.438615\pi\)
0.191652 + 0.981463i \(0.438615\pi\)
\(252\) 0.520153 0.0327665
\(253\) −8.49621 −0.534152
\(254\) 22.1532 1.39001
\(255\) −0.00420207 −0.000263144 0
\(256\) 1.91579 0.119737
\(257\) 26.4315 1.64875 0.824374 0.566045i \(-0.191527\pi\)
0.824374 + 0.566045i \(0.191527\pi\)
\(258\) −0.00398600 −0.000248158 0
\(259\) −0.114313 −0.00710304
\(260\) −22.9764 −1.42494
\(261\) −6.53939 −0.404778
\(262\) −6.40636 −0.395786
\(263\) −9.51496 −0.586717 −0.293359 0.956002i \(-0.594773\pi\)
−0.293359 + 0.956002i \(0.594773\pi\)
\(264\) 0.00317310 0.000195291 0
\(265\) −24.1398 −1.48290
\(266\) 0.498485 0.0305641
\(267\) 0.00511199 0.000312849 0
\(268\) 7.16584 0.437723
\(269\) 3.92078 0.239054 0.119527 0.992831i \(-0.461862\pi\)
0.119527 + 0.992831i \(0.461862\pi\)
\(270\) −0.0542292 −0.00330029
\(271\) −0.360394 −0.0218924 −0.0109462 0.999940i \(-0.503484\pi\)
−0.0109462 + 0.999940i \(0.503484\pi\)
\(272\) −2.35496 −0.142791
\(273\) −0.000508834 0 −3.07960e−5 0
\(274\) −38.9033 −2.35023
\(275\) 0.282171 0.0170155
\(276\) 0.0350745 0.00211123
\(277\) −24.2096 −1.45461 −0.727306 0.686314i \(-0.759228\pi\)
−0.727306 + 0.686314i \(0.759228\pi\)
\(278\) −48.4333 −2.90484
\(279\) 12.0087 0.718945
\(280\) −0.194238 −0.0116079
\(281\) −9.77843 −0.583332 −0.291666 0.956520i \(-0.594210\pi\)
−0.291666 + 0.956520i \(0.594210\pi\)
\(282\) −0.0325382 −0.00193762
\(283\) 19.3972 1.15305 0.576523 0.817081i \(-0.304409\pi\)
0.576523 + 0.817081i \(0.304409\pi\)
\(284\) −9.66653 −0.573603
\(285\) −0.0147516 −0.000873810 0
\(286\) 10.5470 0.623655
\(287\) 0.0632425 0.00373309
\(288\) −23.2792 −1.37174
\(289\) 1.00000 0.0588235
\(290\) 10.2393 0.601274
\(291\) −0.0269485 −0.00157975
\(292\) −17.2425 −1.00904
\(293\) −5.92616 −0.346210 −0.173105 0.984903i \(-0.555380\pi\)
−0.173105 + 0.984903i \(0.555380\pi\)
\(294\) 0.0289517 0.00168850
\(295\) 18.7418 1.09119
\(296\) −2.33281 −0.135592
\(297\) 0.0141317 0.000820002 0
\(298\) 24.9189 1.44351
\(299\) 27.8037 1.60793
\(300\) −0.00116487 −6.72539e−5 0
\(301\) −0.0635839 −0.00366491
\(302\) 26.7404 1.53874
\(303\) −0.00201537 −0.000115780 0
\(304\) −8.26724 −0.474158
\(305\) 11.9239 0.682762
\(306\) 6.45268 0.368875
\(307\) 28.0133 1.59880 0.799402 0.600797i \(-0.205150\pi\)
0.799402 + 0.600797i \(0.205150\pi\)
\(308\) 0.212239 0.0120934
\(309\) −0.00378375 −0.000215250 0
\(310\) −18.8032 −1.06795
\(311\) −17.1756 −0.973939 −0.486970 0.873419i \(-0.661898\pi\)
−0.486970 + 0.873419i \(0.661898\pi\)
\(312\) −0.0103839 −0.000587874 0
\(313\) 0.972274 0.0549562 0.0274781 0.999622i \(-0.491252\pi\)
0.0274781 + 0.999622i \(0.491252\pi\)
\(314\) −6.03910 −0.340806
\(315\) −0.432527 −0.0243701
\(316\) −40.3041 −2.26728
\(317\) 2.10893 0.118449 0.0592247 0.998245i \(-0.481137\pi\)
0.0592247 + 0.998245i \(0.481137\pi\)
\(318\) −0.0457451 −0.00256526
\(319\) −2.66828 −0.149395
\(320\) 26.1643 1.46263
\(321\) 0.0265805 0.00148358
\(322\) 0.985567 0.0549235
\(323\) 3.51056 0.195333
\(324\) 23.6371 1.31317
\(325\) −0.923400 −0.0512210
\(326\) 24.3863 1.35063
\(327\) 0.0164469 0.000909516 0
\(328\) 1.29061 0.0712620
\(329\) −0.519043 −0.0286157
\(330\) −0.0110636 −0.000609032 0
\(331\) −18.8176 −1.03431 −0.517154 0.855892i \(-0.673009\pi\)
−0.517154 + 0.855892i \(0.673009\pi\)
\(332\) −5.13291 −0.281705
\(333\) −5.19468 −0.284667
\(334\) −6.87973 −0.376442
\(335\) −5.95867 −0.325557
\(336\) 0.000299135 0 1.63191e−5 0
\(337\) 27.1204 1.47734 0.738670 0.674067i \(-0.235454\pi\)
0.738670 + 0.674067i \(0.235454\pi\)
\(338\) −6.55308 −0.356441
\(339\) −0.00512122 −0.000278147 0
\(340\) −5.73574 −0.311064
\(341\) 4.89995 0.265347
\(342\) 22.6525 1.22491
\(343\) 0.923951 0.0498887
\(344\) −1.29758 −0.0699606
\(345\) −0.0291657 −0.00157023
\(346\) 10.2279 0.549853
\(347\) 3.04343 0.163380 0.0816900 0.996658i \(-0.473968\pi\)
0.0816900 + 0.996658i \(0.473968\pi\)
\(348\) 0.0110153 0.000590483 0
\(349\) 17.9538 0.961045 0.480523 0.876982i \(-0.340447\pi\)
0.480523 + 0.876982i \(0.340447\pi\)
\(350\) −0.0327320 −0.00174960
\(351\) −0.0462456 −0.00246841
\(352\) −9.49866 −0.506280
\(353\) −1.00000 −0.0532246
\(354\) 0.0355159 0.00188765
\(355\) 8.03809 0.426617
\(356\) 6.97777 0.369821
\(357\) −0.000127023 0 −6.72278e−6 0
\(358\) 30.1199 1.59189
\(359\) 28.0790 1.48195 0.740977 0.671531i \(-0.234363\pi\)
0.740977 + 0.671531i \(0.234363\pi\)
\(360\) −8.82670 −0.465208
\(361\) −6.67596 −0.351367
\(362\) −3.96509 −0.208400
\(363\) −0.0182820 −0.000959555 0
\(364\) −0.694549 −0.0364042
\(365\) 14.3378 0.750475
\(366\) 0.0225959 0.00118111
\(367\) −14.8703 −0.776221 −0.388111 0.921613i \(-0.626872\pi\)
−0.388111 + 0.921613i \(0.626872\pi\)
\(368\) −16.3453 −0.852060
\(369\) 2.87391 0.149610
\(370\) 8.13380 0.422856
\(371\) −0.729717 −0.0378850
\(372\) −0.0202282 −0.00104878
\(373\) −10.0753 −0.521680 −0.260840 0.965382i \(-0.583999\pi\)
−0.260840 + 0.965382i \(0.583999\pi\)
\(374\) 2.63290 0.136144
\(375\) 0.0219790 0.00113499
\(376\) −10.5923 −0.546254
\(377\) 8.73190 0.449716
\(378\) −0.00163928 −8.43156e−5 0
\(379\) 7.24287 0.372041 0.186021 0.982546i \(-0.440441\pi\)
0.186021 + 0.982546i \(0.440441\pi\)
\(380\) −20.1357 −1.03294
\(381\) −0.0198173 −0.00101527
\(382\) 2.20692 0.112916
\(383\) 8.85020 0.452224 0.226112 0.974101i \(-0.427398\pi\)
0.226112 + 0.974101i \(0.427398\pi\)
\(384\) 0.0197206 0.00100636
\(385\) −0.176485 −0.00899449
\(386\) −38.8646 −1.97816
\(387\) −2.88943 −0.146878
\(388\) −36.7842 −1.86744
\(389\) 5.22915 0.265128 0.132564 0.991174i \(-0.457679\pi\)
0.132564 + 0.991174i \(0.457679\pi\)
\(390\) 0.0362055 0.00183334
\(391\) 6.94081 0.351012
\(392\) 9.42474 0.476021
\(393\) 0.00573084 0.000289083 0
\(394\) −48.9719 −2.46717
\(395\) 33.5144 1.68629
\(396\) 9.64472 0.484665
\(397\) −2.35073 −0.117980 −0.0589899 0.998259i \(-0.518788\pi\)
−0.0589899 + 0.998259i \(0.518788\pi\)
\(398\) 2.02340 0.101424
\(399\) −0.000445923 0 −2.23241e−5 0
\(400\) 0.542851 0.0271426
\(401\) 3.02122 0.150872 0.0754362 0.997151i \(-0.475965\pi\)
0.0754362 + 0.997151i \(0.475965\pi\)
\(402\) −0.0112917 −0.000563180 0
\(403\) −16.0350 −0.798761
\(404\) −2.75095 −0.136865
\(405\) −19.6552 −0.976673
\(406\) 0.309522 0.0153613
\(407\) −2.11960 −0.105064
\(408\) −0.00259220 −0.000128333 0
\(409\) −25.0321 −1.23776 −0.618879 0.785486i \(-0.712413\pi\)
−0.618879 + 0.785486i \(0.712413\pi\)
\(410\) −4.49996 −0.222237
\(411\) 0.0348012 0.00171662
\(412\) −5.16475 −0.254449
\(413\) 0.566542 0.0278777
\(414\) 44.7868 2.20115
\(415\) 4.26821 0.209518
\(416\) 31.0842 1.52403
\(417\) 0.0433263 0.00212170
\(418\) 9.24296 0.452088
\(419\) −20.4178 −0.997474 −0.498737 0.866753i \(-0.666203\pi\)
−0.498737 + 0.866753i \(0.666203\pi\)
\(420\) 0.000728573 0 3.55507e−5 0
\(421\) 27.6799 1.34904 0.674518 0.738258i \(-0.264351\pi\)
0.674518 + 0.738258i \(0.264351\pi\)
\(422\) −58.2884 −2.83743
\(423\) −23.5867 −1.14683
\(424\) −14.8916 −0.723198
\(425\) −0.230514 −0.0111816
\(426\) 0.0152322 0.000738004 0
\(427\) 0.360446 0.0174432
\(428\) 36.2819 1.75375
\(429\) −0.00943484 −0.000455518 0
\(430\) 4.52425 0.218178
\(431\) 34.2033 1.64751 0.823757 0.566943i \(-0.191874\pi\)
0.823757 + 0.566943i \(0.191874\pi\)
\(432\) 0.0271870 0.00130804
\(433\) 8.54142 0.410474 0.205237 0.978712i \(-0.434203\pi\)
0.205237 + 0.978712i \(0.434203\pi\)
\(434\) −0.568398 −0.0272840
\(435\) −0.00915966 −0.000439172 0
\(436\) 22.4497 1.07515
\(437\) 24.3661 1.16559
\(438\) 0.0271702 0.00129824
\(439\) −14.0159 −0.668941 −0.334470 0.942406i \(-0.608557\pi\)
−0.334470 + 0.942406i \(0.608557\pi\)
\(440\) −3.60158 −0.171698
\(441\) 20.9869 0.999376
\(442\) −8.61613 −0.409827
\(443\) −16.7385 −0.795268 −0.397634 0.917544i \(-0.630169\pi\)
−0.397634 + 0.917544i \(0.630169\pi\)
\(444\) 0.00875022 0.000415267 0
\(445\) −5.80228 −0.275054
\(446\) 8.37914 0.396764
\(447\) −0.0222914 −0.00105434
\(448\) 0.790916 0.0373673
\(449\) 24.5082 1.15661 0.578306 0.815820i \(-0.303714\pi\)
0.578306 + 0.815820i \(0.303714\pi\)
\(450\) −1.48743 −0.0701183
\(451\) 1.17265 0.0552179
\(452\) −6.99037 −0.328800
\(453\) −0.0239208 −0.00112390
\(454\) −42.4117 −1.99048
\(455\) 0.577543 0.0270756
\(456\) −0.00910008 −0.000426150 0
\(457\) 29.3567 1.37325 0.686625 0.727012i \(-0.259092\pi\)
0.686625 + 0.727012i \(0.259092\pi\)
\(458\) 35.6806 1.66724
\(459\) −0.0115446 −0.000538855 0
\(460\) −39.8107 −1.85618
\(461\) 7.82734 0.364556 0.182278 0.983247i \(-0.441653\pi\)
0.182278 + 0.983247i \(0.441653\pi\)
\(462\) −0.000334440 0 −1.55595e−5 0
\(463\) −20.1106 −0.934618 −0.467309 0.884094i \(-0.654776\pi\)
−0.467309 + 0.884094i \(0.654776\pi\)
\(464\) −5.13334 −0.238309
\(465\) 0.0168205 0.000780034 0
\(466\) 45.8326 2.12315
\(467\) −39.4279 −1.82451 −0.912253 0.409627i \(-0.865659\pi\)
−0.912253 + 0.409627i \(0.865659\pi\)
\(468\) −31.5622 −1.45896
\(469\) −0.180123 −0.00831732
\(470\) 36.9319 1.70354
\(471\) 0.00540231 0.000248925 0
\(472\) 11.5616 0.532166
\(473\) −1.17898 −0.0542095
\(474\) 0.0635099 0.00291711
\(475\) −0.809233 −0.0371302
\(476\) −0.173384 −0.00794706
\(477\) −33.1603 −1.51831
\(478\) 54.0452 2.47197
\(479\) −10.2907 −0.470196 −0.235098 0.971972i \(-0.575541\pi\)
−0.235098 + 0.971972i \(0.575541\pi\)
\(480\) −0.0326069 −0.00148830
\(481\) 6.93634 0.316270
\(482\) −57.3358 −2.61157
\(483\) −0.000881644 0 −4.01162e−5 0
\(484\) −24.9546 −1.13430
\(485\) 30.5875 1.38890
\(486\) −0.111740 −0.00506864
\(487\) −5.09851 −0.231036 −0.115518 0.993305i \(-0.536853\pi\)
−0.115518 + 0.993305i \(0.536853\pi\)
\(488\) 7.35572 0.332978
\(489\) −0.0218149 −0.000986506 0
\(490\) −32.8612 −1.48452
\(491\) 28.4709 1.28488 0.642438 0.766338i \(-0.277923\pi\)
0.642438 + 0.766338i \(0.277923\pi\)
\(492\) −0.00484099 −0.000218249 0
\(493\) 2.17980 0.0981732
\(494\) −30.2474 −1.36090
\(495\) −8.01995 −0.360470
\(496\) 9.42672 0.423272
\(497\) 0.242982 0.0108992
\(498\) 0.00808829 0.000362445 0
\(499\) −16.0159 −0.716972 −0.358486 0.933535i \(-0.616707\pi\)
−0.358486 + 0.933535i \(0.616707\pi\)
\(500\) 30.0009 1.34168
\(501\) 0.00615430 0.000274954 0
\(502\) −13.0617 −0.582972
\(503\) 33.9449 1.51353 0.756764 0.653689i \(-0.226779\pi\)
0.756764 + 0.653689i \(0.226779\pi\)
\(504\) −0.266820 −0.0118851
\(505\) 2.28752 0.101793
\(506\) 18.2745 0.812399
\(507\) 0.00586210 0.000260345 0
\(508\) −27.0502 −1.20016
\(509\) −36.3553 −1.61142 −0.805709 0.592311i \(-0.798216\pi\)
−0.805709 + 0.592311i \(0.798216\pi\)
\(510\) 0.00903821 0.000400219 0
\(511\) 0.433414 0.0191731
\(512\) −24.6192 −1.08803
\(513\) −0.0405279 −0.00178935
\(514\) −56.8513 −2.50760
\(515\) 4.29468 0.189246
\(516\) 0.00486712 0.000214263 0
\(517\) −9.62414 −0.423269
\(518\) 0.245875 0.0108031
\(519\) −0.00914939 −0.000401614 0
\(520\) 11.7861 0.516855
\(521\) 20.0091 0.876614 0.438307 0.898825i \(-0.355578\pi\)
0.438307 + 0.898825i \(0.355578\pi\)
\(522\) 14.0655 0.615632
\(523\) 12.9235 0.565103 0.282552 0.959252i \(-0.408819\pi\)
0.282552 + 0.959252i \(0.408819\pi\)
\(524\) 7.82250 0.341727
\(525\) 2.92806e−5 0 1.27791e−6 0
\(526\) 20.4657 0.892347
\(527\) −4.00292 −0.174370
\(528\) 0.00554659 0.000241384 0
\(529\) 25.1748 1.09456
\(530\) 51.9223 2.25536
\(531\) 25.7452 1.11725
\(532\) −0.608677 −0.0263895
\(533\) −3.83748 −0.166220
\(534\) −0.0109954 −0.000475816 0
\(535\) −30.1698 −1.30435
\(536\) −3.67583 −0.158772
\(537\) −0.0269440 −0.00116272
\(538\) −8.43319 −0.363581
\(539\) 8.56333 0.368849
\(540\) 0.0662167 0.00284951
\(541\) 15.7378 0.676622 0.338311 0.941034i \(-0.390144\pi\)
0.338311 + 0.941034i \(0.390144\pi\)
\(542\) 0.775170 0.0332964
\(543\) 0.00354699 0.000152216 0
\(544\) 7.75974 0.332696
\(545\) −18.6678 −0.799641
\(546\) 0.00109445 4.68381e−5 0
\(547\) −18.9470 −0.810115 −0.405057 0.914291i \(-0.632748\pi\)
−0.405057 + 0.914291i \(0.632748\pi\)
\(548\) 47.5030 2.02923
\(549\) 16.3796 0.699066
\(550\) −0.606920 −0.0258792
\(551\) 7.65231 0.326000
\(552\) −0.0179920 −0.000765789 0
\(553\) 1.01310 0.0430813
\(554\) 52.0723 2.21234
\(555\) −0.00727613 −0.000308855 0
\(556\) 59.1396 2.50808
\(557\) −2.18886 −0.0927451 −0.0463725 0.998924i \(-0.514766\pi\)
−0.0463725 + 0.998924i \(0.514766\pi\)
\(558\) −25.8296 −1.09345
\(559\) 3.85819 0.163184
\(560\) −0.339528 −0.0143477
\(561\) −0.00235528 −9.94399e−5 0
\(562\) 21.0324 0.887198
\(563\) 4.44828 0.187473 0.0937363 0.995597i \(-0.470119\pi\)
0.0937363 + 0.995597i \(0.470119\pi\)
\(564\) 0.0397308 0.00167297
\(565\) 5.81276 0.244545
\(566\) −41.7215 −1.75368
\(567\) −0.594151 −0.0249520
\(568\) 4.95860 0.208058
\(569\) 30.2155 1.26670 0.633351 0.773865i \(-0.281679\pi\)
0.633351 + 0.773865i \(0.281679\pi\)
\(570\) 0.0317292 0.00132899
\(571\) 21.0723 0.881848 0.440924 0.897544i \(-0.354651\pi\)
0.440924 + 0.897544i \(0.354651\pi\)
\(572\) −12.8784 −0.538472
\(573\) −0.00197421 −8.24739e−5 0
\(574\) −0.136028 −0.00567771
\(575\) −1.59995 −0.0667227
\(576\) 35.9414 1.49756
\(577\) 18.2837 0.761161 0.380581 0.924748i \(-0.375724\pi\)
0.380581 + 0.924748i \(0.375724\pi\)
\(578\) −2.15090 −0.0894655
\(579\) 0.0347666 0.00144485
\(580\) −12.5028 −0.519149
\(581\) 0.129023 0.00535276
\(582\) 0.0579634 0.00240266
\(583\) −13.5305 −0.560375
\(584\) 8.84481 0.366001
\(585\) 26.2451 1.08510
\(586\) 12.7466 0.526555
\(587\) 11.7756 0.486030 0.243015 0.970022i \(-0.421864\pi\)
0.243015 + 0.970022i \(0.421864\pi\)
\(588\) −0.0353515 −0.00145787
\(589\) −14.0525 −0.579023
\(590\) −40.3117 −1.65961
\(591\) 0.0438081 0.00180202
\(592\) −4.07776 −0.167595
\(593\) −5.47969 −0.225024 −0.112512 0.993650i \(-0.535890\pi\)
−0.112512 + 0.993650i \(0.535890\pi\)
\(594\) −0.0303957 −0.00124715
\(595\) 0.144176 0.00591063
\(596\) −30.4273 −1.24635
\(597\) −0.00181004 −7.40802e−5 0
\(598\) −59.8029 −2.44552
\(599\) 34.6462 1.41560 0.707802 0.706411i \(-0.249687\pi\)
0.707802 + 0.706411i \(0.249687\pi\)
\(600\) 0.000597539 0 2.43944e−5 0
\(601\) −28.4667 −1.16118 −0.580591 0.814195i \(-0.697179\pi\)
−0.580591 + 0.814195i \(0.697179\pi\)
\(602\) 0.136762 0.00557402
\(603\) −8.18529 −0.333331
\(604\) −32.6514 −1.32857
\(605\) 20.7507 0.843635
\(606\) 0.00433486 0.000176092 0
\(607\) 29.8214 1.21041 0.605206 0.796069i \(-0.293091\pi\)
0.605206 + 0.796069i \(0.293091\pi\)
\(608\) 27.2410 1.10477
\(609\) −0.000276885 0 −1.12200e−5 0
\(610\) −25.6471 −1.03842
\(611\) 31.4948 1.27414
\(612\) −7.87906 −0.318492
\(613\) 17.6244 0.711842 0.355921 0.934516i \(-0.384167\pi\)
0.355921 + 0.934516i \(0.384167\pi\)
\(614\) −60.2537 −2.43164
\(615\) 0.00402546 0.000162322 0
\(616\) −0.108871 −0.00438655
\(617\) 20.8462 0.839238 0.419619 0.907700i \(-0.362164\pi\)
0.419619 + 0.907700i \(0.362164\pi\)
\(618\) 0.00813845 0.000327377 0
\(619\) 32.5000 1.30629 0.653144 0.757234i \(-0.273450\pi\)
0.653144 + 0.757234i \(0.273450\pi\)
\(620\) 22.9597 0.922085
\(621\) −0.0801287 −0.00321545
\(622\) 36.9429 1.48128
\(623\) −0.175396 −0.00702708
\(624\) −0.0181511 −0.000726626 0
\(625\) −23.7943 −0.951772
\(626\) −2.09126 −0.0835836
\(627\) −0.00826834 −0.000330206 0
\(628\) 7.37406 0.294257
\(629\) 1.73156 0.0690419
\(630\) 0.930320 0.0370648
\(631\) −16.0987 −0.640878 −0.320439 0.947269i \(-0.603830\pi\)
−0.320439 + 0.947269i \(0.603830\pi\)
\(632\) 20.6746 0.822391
\(633\) 0.0521422 0.00207247
\(634\) −4.53609 −0.180151
\(635\) 22.4933 0.892618
\(636\) 0.0558572 0.00221488
\(637\) −28.0234 −1.11033
\(638\) 5.73919 0.227217
\(639\) 11.0417 0.436805
\(640\) −22.3836 −0.884789
\(641\) −12.9469 −0.511373 −0.255686 0.966760i \(-0.582301\pi\)
−0.255686 + 0.966760i \(0.582301\pi\)
\(642\) −0.0571719 −0.00225640
\(643\) 44.1599 1.74150 0.870748 0.491730i \(-0.163635\pi\)
0.870748 + 0.491730i \(0.163635\pi\)
\(644\) −1.20343 −0.0474217
\(645\) −0.00404719 −0.000159358 0
\(646\) −7.55085 −0.297084
\(647\) 16.1624 0.635409 0.317705 0.948190i \(-0.397088\pi\)
0.317705 + 0.948190i \(0.397088\pi\)
\(648\) −12.1250 −0.476316
\(649\) 10.5049 0.412353
\(650\) 1.98614 0.0779027
\(651\) 0.000508464 0 1.99283e−5 0
\(652\) −29.7770 −1.16616
\(653\) −38.6017 −1.51060 −0.755301 0.655378i \(-0.772509\pi\)
−0.755301 + 0.655378i \(0.772509\pi\)
\(654\) −0.0353756 −0.00138330
\(655\) −6.50470 −0.254160
\(656\) 2.25599 0.0880816
\(657\) 19.6955 0.768396
\(658\) 1.11641 0.0435221
\(659\) 33.4201 1.30186 0.650931 0.759137i \(-0.274379\pi\)
0.650931 + 0.759137i \(0.274379\pi\)
\(660\) 0.0135093 0.000525847 0
\(661\) 4.36077 0.169614 0.0848071 0.996397i \(-0.472973\pi\)
0.0848071 + 0.996397i \(0.472973\pi\)
\(662\) 40.4747 1.57309
\(663\) 0.00770761 0.000299339 0
\(664\) 2.63301 0.102180
\(665\) 0.506138 0.0196272
\(666\) 11.1732 0.432953
\(667\) 15.1296 0.585819
\(668\) 8.40051 0.325026
\(669\) −0.00749561 −0.000289797 0
\(670\) 12.8165 0.495144
\(671\) 6.68342 0.258010
\(672\) −0.000985668 0 −3.80229e−5 0
\(673\) 47.4623 1.82954 0.914769 0.403977i \(-0.132372\pi\)
0.914769 + 0.403977i \(0.132372\pi\)
\(674\) −58.3331 −2.24691
\(675\) 0.00266119 0.000102429 0
\(676\) 8.00166 0.307756
\(677\) −6.69512 −0.257314 −0.128657 0.991689i \(-0.541067\pi\)
−0.128657 + 0.991689i \(0.541067\pi\)
\(678\) 0.0110152 0.000423037 0
\(679\) 0.924621 0.0354837
\(680\) 2.94224 0.112830
\(681\) 0.0379397 0.00145385
\(682\) −10.5393 −0.403570
\(683\) −34.2662 −1.31116 −0.655579 0.755127i \(-0.727575\pi\)
−0.655579 + 0.755127i \(0.727575\pi\)
\(684\) −27.6599 −1.05760
\(685\) −39.5005 −1.50924
\(686\) −1.98732 −0.0758764
\(687\) −0.0319183 −0.00121776
\(688\) −2.26817 −0.0864730
\(689\) 44.2783 1.68687
\(690\) 0.0627325 0.00238819
\(691\) 37.6204 1.43115 0.715574 0.698537i \(-0.246165\pi\)
0.715574 + 0.698537i \(0.246165\pi\)
\(692\) −12.4887 −0.474751
\(693\) −0.242433 −0.00920927
\(694\) −6.54611 −0.248487
\(695\) −49.1768 −1.86538
\(696\) −0.00565048 −0.000214181 0
\(697\) −0.957973 −0.0362858
\(698\) −38.6168 −1.46167
\(699\) −0.0409998 −0.00155075
\(700\) 0.0399675 0.00151063
\(701\) 22.8514 0.863084 0.431542 0.902093i \(-0.357970\pi\)
0.431542 + 0.902093i \(0.357970\pi\)
\(702\) 0.0994695 0.00375424
\(703\) 6.07875 0.229264
\(704\) 14.6652 0.552717
\(705\) −0.0330377 −0.00124427
\(706\) 2.15090 0.0809501
\(707\) 0.0691488 0.00260061
\(708\) −0.0433667 −0.00162982
\(709\) 28.5678 1.07289 0.536444 0.843936i \(-0.319767\pi\)
0.536444 + 0.843936i \(0.319767\pi\)
\(710\) −17.2891 −0.648848
\(711\) 46.0379 1.72656
\(712\) −3.57935 −0.134142
\(713\) −27.7835 −1.04050
\(714\) 0.000273214 0 1.02248e−5 0
\(715\) 10.7089 0.400489
\(716\) −36.7780 −1.37446
\(717\) −0.0483465 −0.00180553
\(718\) −60.3950 −2.25392
\(719\) −17.5267 −0.653634 −0.326817 0.945088i \(-0.605976\pi\)
−0.326817 + 0.945088i \(0.605976\pi\)
\(720\) −15.4291 −0.575008
\(721\) 0.129823 0.00483486
\(722\) 14.3593 0.534398
\(723\) 0.0512901 0.00190750
\(724\) 4.84158 0.179936
\(725\) −0.502474 −0.0186614
\(726\) 0.0393227 0.00145940
\(727\) 34.3346 1.27340 0.636699 0.771112i \(-0.280299\pi\)
0.636699 + 0.771112i \(0.280299\pi\)
\(728\) 0.356279 0.0132046
\(729\) −26.9998 −0.999993
\(730\) −30.8391 −1.14141
\(731\) 0.963143 0.0356231
\(732\) −0.0275908 −0.00101979
\(733\) 6.02979 0.222715 0.111358 0.993780i \(-0.464480\pi\)
0.111358 + 0.993780i \(0.464480\pi\)
\(734\) 31.9844 1.18057
\(735\) 0.0293961 0.00108429
\(736\) 53.8589 1.98526
\(737\) −3.33986 −0.123025
\(738\) −6.18149 −0.227544
\(739\) −49.2832 −1.81291 −0.906457 0.422299i \(-0.861223\pi\)
−0.906457 + 0.422299i \(0.861223\pi\)
\(740\) −9.93179 −0.365100
\(741\) 0.0270580 0.000994001 0
\(742\) 1.56955 0.0576198
\(743\) 41.4348 1.52010 0.760048 0.649867i \(-0.225175\pi\)
0.760048 + 0.649867i \(0.225175\pi\)
\(744\) 0.0103764 0.000380416 0
\(745\) 25.3014 0.926973
\(746\) 21.6710 0.793430
\(747\) 5.86315 0.214521
\(748\) −3.21491 −0.117549
\(749\) −0.911995 −0.0333236
\(750\) −0.0472745 −0.00172622
\(751\) −7.13645 −0.260413 −0.130206 0.991487i \(-0.541564\pi\)
−0.130206 + 0.991487i \(0.541564\pi\)
\(752\) −18.5153 −0.675183
\(753\) 0.0116844 0.000425804 0
\(754\) −18.7814 −0.683979
\(755\) 27.1509 0.988122
\(756\) 0.00200165 7.27993e−5 0
\(757\) 39.6273 1.44028 0.720140 0.693829i \(-0.244078\pi\)
0.720140 + 0.693829i \(0.244078\pi\)
\(758\) −15.5787 −0.565843
\(759\) −0.0163475 −0.000593378 0
\(760\) 10.3289 0.374669
\(761\) 34.1137 1.23662 0.618310 0.785935i \(-0.287818\pi\)
0.618310 + 0.785935i \(0.287818\pi\)
\(762\) 0.0426249 0.00154414
\(763\) −0.564305 −0.0204292
\(764\) −2.69476 −0.0974931
\(765\) 6.55174 0.236879
\(766\) −19.0359 −0.687794
\(767\) −34.3770 −1.24128
\(768\) 0.00368616 0.000133013 0
\(769\) 14.9781 0.540122 0.270061 0.962843i \(-0.412956\pi\)
0.270061 + 0.962843i \(0.412956\pi\)
\(770\) 0.379600 0.0136798
\(771\) 0.0508567 0.00183156
\(772\) 47.4557 1.70797
\(773\) 24.0200 0.863940 0.431970 0.901888i \(-0.357819\pi\)
0.431970 + 0.901888i \(0.357819\pi\)
\(774\) 6.21486 0.223388
\(775\) 0.922729 0.0331454
\(776\) 18.8690 0.677358
\(777\) −0.000219949 0 −7.89061e−6 0
\(778\) −11.2474 −0.403237
\(779\) −3.36302 −0.120493
\(780\) −0.0442088 −0.00158293
\(781\) 4.50539 0.161215
\(782\) −14.9290 −0.533859
\(783\) −0.0251649 −0.000899318 0
\(784\) 16.4745 0.588374
\(785\) −6.13181 −0.218854
\(786\) −0.0123265 −0.000439670 0
\(787\) 24.4061 0.869982 0.434991 0.900435i \(-0.356751\pi\)
0.434991 + 0.900435i \(0.356751\pi\)
\(788\) 59.7972 2.13019
\(789\) −0.0183077 −0.000651772 0
\(790\) −72.0859 −2.56470
\(791\) 0.175713 0.00624762
\(792\) −4.94741 −0.175798
\(793\) −21.8714 −0.776675
\(794\) 5.05618 0.179437
\(795\) −0.0464474 −0.00164732
\(796\) −2.47068 −0.0875709
\(797\) −20.5525 −0.728008 −0.364004 0.931397i \(-0.618591\pi\)
−0.364004 + 0.931397i \(0.618591\pi\)
\(798\) 0.000959134 0 3.39530e−5 0
\(799\) 7.86225 0.278146
\(800\) −1.78873 −0.0632411
\(801\) −7.97046 −0.281622
\(802\) −6.49833 −0.229464
\(803\) 8.03641 0.283599
\(804\) 0.0137878 0.000486257 0
\(805\) 1.00070 0.0352699
\(806\) 34.4897 1.21485
\(807\) 0.00754396 0.000265560 0
\(808\) 1.41114 0.0496437
\(809\) 23.9840 0.843232 0.421616 0.906774i \(-0.361463\pi\)
0.421616 + 0.906774i \(0.361463\pi\)
\(810\) 42.2762 1.48544
\(811\) −27.4895 −0.965288 −0.482644 0.875817i \(-0.660324\pi\)
−0.482644 + 0.875817i \(0.660324\pi\)
\(812\) −0.377943 −0.0132632
\(813\) −0.000693433 0 −2.43198e−5 0
\(814\) 4.55903 0.159794
\(815\) 24.7607 0.867329
\(816\) −0.00453118 −0.000158623 0
\(817\) 3.38117 0.118292
\(818\) 53.8415 1.88252
\(819\) 0.793359 0.0277222
\(820\) 5.49468 0.191883
\(821\) 37.1874 1.29785 0.648925 0.760852i \(-0.275219\pi\)
0.648925 + 0.760852i \(0.275219\pi\)
\(822\) −0.0748537 −0.00261082
\(823\) 17.9888 0.627050 0.313525 0.949580i \(-0.398490\pi\)
0.313525 + 0.949580i \(0.398490\pi\)
\(824\) 2.64934 0.0922940
\(825\) 0.000542924 0 1.89022e−5 0
\(826\) −1.21857 −0.0423996
\(827\) 1.31666 0.0457848 0.0228924 0.999738i \(-0.492712\pi\)
0.0228924 + 0.999738i \(0.492712\pi\)
\(828\) −54.6871 −1.90051
\(829\) 20.6743 0.718049 0.359025 0.933328i \(-0.383109\pi\)
0.359025 + 0.933328i \(0.383109\pi\)
\(830\) −9.18048 −0.318659
\(831\) −0.0465815 −0.00161590
\(832\) −47.9918 −1.66381
\(833\) −6.99564 −0.242385
\(834\) −0.0931903 −0.00322692
\(835\) −6.98534 −0.241738
\(836\) −11.2861 −0.390339
\(837\) 0.0462120 0.00159732
\(838\) 43.9166 1.51707
\(839\) −4.26481 −0.147238 −0.0736189 0.997286i \(-0.523455\pi\)
−0.0736189 + 0.997286i \(0.523455\pi\)
\(840\) −0.000373732 0 −1.28950e−5 0
\(841\) −24.2485 −0.836154
\(842\) −59.5367 −2.05177
\(843\) −0.0188146 −0.000648011 0
\(844\) 71.1731 2.44988
\(845\) −6.65368 −0.228894
\(846\) 50.7326 1.74422
\(847\) 0.627267 0.0215532
\(848\) −26.0305 −0.893890
\(849\) 0.0373222 0.00128089
\(850\) 0.495812 0.0170062
\(851\) 12.0184 0.411987
\(852\) −0.0185994 −0.000637203 0
\(853\) 13.4613 0.460908 0.230454 0.973083i \(-0.425979\pi\)
0.230454 + 0.973083i \(0.425979\pi\)
\(854\) −0.775282 −0.0265296
\(855\) 23.0003 0.786593
\(856\) −18.6114 −0.636123
\(857\) −32.8656 −1.12267 −0.561335 0.827589i \(-0.689712\pi\)
−0.561335 + 0.827589i \(0.689712\pi\)
\(858\) 0.0202934 0.000692804 0
\(859\) 48.3685 1.65031 0.825157 0.564904i \(-0.191087\pi\)
0.825157 + 0.564904i \(0.191087\pi\)
\(860\) −5.52434 −0.188378
\(861\) 0.000121685 0 4.14701e−6 0
\(862\) −73.5678 −2.50573
\(863\) 30.6660 1.04388 0.521942 0.852981i \(-0.325208\pi\)
0.521942 + 0.852981i \(0.325208\pi\)
\(864\) −0.0895829 −0.00304767
\(865\) 10.3849 0.353096
\(866\) −18.3717 −0.624296
\(867\) 0.00192410 6.53458e−5 0
\(868\) 0.694044 0.0235574
\(869\) 18.7850 0.637236
\(870\) 0.0197015 0.000667943 0
\(871\) 10.9296 0.370337
\(872\) −11.5159 −0.389979
\(873\) 42.0173 1.42207
\(874\) −52.4090 −1.77276
\(875\) −0.754113 −0.0254937
\(876\) −0.0331763 −0.00112092
\(877\) 30.8966 1.04330 0.521652 0.853159i \(-0.325316\pi\)
0.521652 + 0.853159i \(0.325316\pi\)
\(878\) 30.1467 1.01740
\(879\) −0.0114025 −0.000384597 0
\(880\) −6.29556 −0.212223
\(881\) 21.6270 0.728631 0.364315 0.931276i \(-0.381303\pi\)
0.364315 + 0.931276i \(0.381303\pi\)
\(882\) −45.1407 −1.51996
\(883\) 12.8697 0.433098 0.216549 0.976272i \(-0.430520\pi\)
0.216549 + 0.976272i \(0.430520\pi\)
\(884\) 10.5207 0.353851
\(885\) 0.0360611 0.00121218
\(886\) 36.0027 1.20953
\(887\) 2.89962 0.0973598 0.0486799 0.998814i \(-0.484499\pi\)
0.0486799 + 0.998814i \(0.484499\pi\)
\(888\) −0.00448856 −0.000150626 0
\(889\) 0.679944 0.0228046
\(890\) 12.4801 0.418334
\(891\) −11.0168 −0.369077
\(892\) −10.2314 −0.342571
\(893\) 27.6009 0.923629
\(894\) 0.0479464 0.00160357
\(895\) 30.5823 1.02225
\(896\) −0.676628 −0.0226046
\(897\) 0.0534970 0.00178621
\(898\) −52.7145 −1.75911
\(899\) −8.72556 −0.291014
\(900\) 1.81623 0.0605411
\(901\) 11.0535 0.368244
\(902\) −2.52225 −0.0839817
\(903\) −0.000122342 0 −4.07127e−6 0
\(904\) 3.58582 0.119263
\(905\) −4.02595 −0.133827
\(906\) 0.0514511 0.00170935
\(907\) −12.8277 −0.425937 −0.212969 0.977059i \(-0.568313\pi\)
−0.212969 + 0.977059i \(0.568313\pi\)
\(908\) 51.7870 1.71861
\(909\) 3.14231 0.104224
\(910\) −1.24224 −0.0411797
\(911\) 33.5315 1.11095 0.555474 0.831534i \(-0.312537\pi\)
0.555474 + 0.831534i \(0.312537\pi\)
\(912\) −0.0159070 −0.000526732 0
\(913\) 2.39235 0.0791753
\(914\) −63.1433 −2.08859
\(915\) 0.0229428 0.000758466 0
\(916\) −43.5678 −1.43952
\(917\) −0.196629 −0.00649327
\(918\) 0.0248312 0.000819551 0
\(919\) 15.3195 0.505344 0.252672 0.967552i \(-0.418691\pi\)
0.252672 + 0.967552i \(0.418691\pi\)
\(920\) 20.4215 0.673277
\(921\) 0.0539003 0.00177608
\(922\) −16.8358 −0.554458
\(923\) −14.7438 −0.485298
\(924\) 0.000408368 0 1.34343e−5 0
\(925\) −0.399149 −0.0131239
\(926\) 43.2558 1.42147
\(927\) 5.89951 0.193765
\(928\) 16.9147 0.555251
\(929\) −41.9901 −1.37765 −0.688825 0.724928i \(-0.741873\pi\)
−0.688825 + 0.724928i \(0.741873\pi\)
\(930\) −0.0361792 −0.00118636
\(931\) −24.5586 −0.804876
\(932\) −55.9640 −1.83316
\(933\) −0.0330475 −0.00108193
\(934\) 84.8054 2.77492
\(935\) 2.67332 0.0874269
\(936\) 16.1903 0.529197
\(937\) 47.7920 1.56130 0.780648 0.624971i \(-0.214889\pi\)
0.780648 + 0.624971i \(0.214889\pi\)
\(938\) 0.387426 0.0126499
\(939\) 0.00187075 6.10496e−5 0
\(940\) −45.0958 −1.47086
\(941\) −11.4752 −0.374082 −0.187041 0.982352i \(-0.559890\pi\)
−0.187041 + 0.982352i \(0.559890\pi\)
\(942\) −0.0116198 −0.000378594 0
\(943\) −6.64911 −0.216525
\(944\) 20.2097 0.657770
\(945\) −0.00166445 −5.41445e−5 0
\(946\) 2.53586 0.0824479
\(947\) −1.57913 −0.0513148 −0.0256574 0.999671i \(-0.508168\pi\)
−0.0256574 + 0.999671i \(0.508168\pi\)
\(948\) −0.0775489 −0.00251867
\(949\) −26.2990 −0.853702
\(950\) 1.74058 0.0564718
\(951\) 0.00405779 0.000131583 0
\(952\) 0.0889402 0.00288257
\(953\) 6.81875 0.220881 0.110440 0.993883i \(-0.464774\pi\)
0.110440 + 0.993883i \(0.464774\pi\)
\(954\) 71.3245 2.30921
\(955\) 2.24080 0.0725105
\(956\) −65.9921 −2.13434
\(957\) −0.00513403 −0.000165960 0
\(958\) 22.1343 0.715127
\(959\) −1.19405 −0.0385579
\(960\) 0.0503427 0.00162481
\(961\) −14.9766 −0.483117
\(962\) −14.9194 −0.481019
\(963\) −41.4435 −1.33550
\(964\) 70.0100 2.25487
\(965\) −39.4612 −1.27030
\(966\) 0.00189633 6.10133e−5 0
\(967\) −44.4238 −1.42857 −0.714286 0.699854i \(-0.753249\pi\)
−0.714286 + 0.699854i \(0.753249\pi\)
\(968\) 12.8008 0.411434
\(969\) 0.00675466 0.000216991 0
\(970\) −65.7905 −2.11240
\(971\) 15.3517 0.492660 0.246330 0.969186i \(-0.420775\pi\)
0.246330 + 0.969186i \(0.420775\pi\)
\(972\) 0.136441 0.00437634
\(973\) −1.48655 −0.0476567
\(974\) 10.9664 0.351385
\(975\) −0.00177671 −5.69003e−5 0
\(976\) 12.8578 0.411569
\(977\) −46.0827 −1.47432 −0.737158 0.675720i \(-0.763833\pi\)
−0.737158 + 0.675720i \(0.763833\pi\)
\(978\) 0.0469217 0.00150039
\(979\) −3.25220 −0.103941
\(980\) 40.1252 1.28175
\(981\) −25.6435 −0.818736
\(982\) −61.2380 −1.95418
\(983\) −34.8750 −1.11234 −0.556170 0.831069i \(-0.687730\pi\)
−0.556170 + 0.831069i \(0.687730\pi\)
\(984\) 0.00248326 7.91634e−5 0
\(985\) −49.7236 −1.58433
\(986\) −4.68852 −0.149313
\(987\) −0.000998688 0 −3.17886e−5 0
\(988\) 36.9337 1.17502
\(989\) 6.68499 0.212570
\(990\) 17.2501 0.548244
\(991\) −7.12912 −0.226464 −0.113232 0.993569i \(-0.536120\pi\)
−0.113232 + 0.993569i \(0.536120\pi\)
\(992\) −31.0616 −0.986207
\(993\) −0.0362069 −0.00114899
\(994\) −0.522628 −0.0165768
\(995\) 2.05446 0.0651308
\(996\) −0.00987622 −0.000312940 0
\(997\) 48.2534 1.52820 0.764101 0.645097i \(-0.223183\pi\)
0.764101 + 0.645097i \(0.223183\pi\)
\(998\) 34.4486 1.09045
\(999\) −0.0199901 −0.000632460 0
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 6001.2.a.d.1.17 121
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6001.2.a.d.1.17 121 1.1 even 1 trivial