Properties

Label 4730.2.a.bf.1.2
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.26258\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.26258 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.26258 q^{6} +2.73152 q^{7} +1.00000 q^{8} +7.64446 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.26258 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.26258 q^{6} +2.73152 q^{7} +1.00000 q^{8} +7.64446 q^{9} -1.00000 q^{10} +1.00000 q^{11} -3.26258 q^{12} +6.11752 q^{13} +2.73152 q^{14} +3.26258 q^{15} +1.00000 q^{16} +7.48258 q^{17} +7.64446 q^{18} -7.78028 q^{19} -1.00000 q^{20} -8.91182 q^{21} +1.00000 q^{22} +8.88984 q^{23} -3.26258 q^{24} +1.00000 q^{25} +6.11752 q^{26} -15.1529 q^{27} +2.73152 q^{28} +6.36025 q^{29} +3.26258 q^{30} +6.01389 q^{31} +1.00000 q^{32} -3.26258 q^{33} +7.48258 q^{34} -2.73152 q^{35} +7.64446 q^{36} -0.607804 q^{37} -7.78028 q^{38} -19.9589 q^{39} -1.00000 q^{40} +2.66769 q^{41} -8.91182 q^{42} +1.00000 q^{43} +1.00000 q^{44} -7.64446 q^{45} +8.88984 q^{46} +0.852378 q^{47} -3.26258 q^{48} +0.461206 q^{49} +1.00000 q^{50} -24.4126 q^{51} +6.11752 q^{52} -9.54066 q^{53} -15.1529 q^{54} -1.00000 q^{55} +2.73152 q^{56} +25.3838 q^{57} +6.36025 q^{58} -1.12856 q^{59} +3.26258 q^{60} +9.95844 q^{61} +6.01389 q^{62} +20.8810 q^{63} +1.00000 q^{64} -6.11752 q^{65} -3.26258 q^{66} -11.0418 q^{67} +7.48258 q^{68} -29.0039 q^{69} -2.73152 q^{70} -4.13401 q^{71} +7.64446 q^{72} -13.2485 q^{73} -0.607804 q^{74} -3.26258 q^{75} -7.78028 q^{76} +2.73152 q^{77} -19.9589 q^{78} -2.52486 q^{79} -1.00000 q^{80} +26.5044 q^{81} +2.66769 q^{82} -12.7935 q^{83} -8.91182 q^{84} -7.48258 q^{85} +1.00000 q^{86} -20.7509 q^{87} +1.00000 q^{88} +1.91784 q^{89} -7.64446 q^{90} +16.7101 q^{91} +8.88984 q^{92} -19.6208 q^{93} +0.852378 q^{94} +7.78028 q^{95} -3.26258 q^{96} +12.9532 q^{97} +0.461206 q^{98} +7.64446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.26258 −1.88365 −0.941827 0.336098i \(-0.890893\pi\)
−0.941827 + 0.336098i \(0.890893\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.26258 −1.33194
\(7\) 2.73152 1.03242 0.516209 0.856463i \(-0.327343\pi\)
0.516209 + 0.856463i \(0.327343\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.64446 2.54815
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −3.26258 −0.941827
\(13\) 6.11752 1.69670 0.848348 0.529439i \(-0.177598\pi\)
0.848348 + 0.529439i \(0.177598\pi\)
\(14\) 2.73152 0.730030
\(15\) 3.26258 0.842396
\(16\) 1.00000 0.250000
\(17\) 7.48258 1.81479 0.907396 0.420276i \(-0.138067\pi\)
0.907396 + 0.420276i \(0.138067\pi\)
\(18\) 7.64446 1.80182
\(19\) −7.78028 −1.78492 −0.892459 0.451128i \(-0.851022\pi\)
−0.892459 + 0.451128i \(0.851022\pi\)
\(20\) −1.00000 −0.223607
\(21\) −8.91182 −1.94472
\(22\) 1.00000 0.213201
\(23\) 8.88984 1.85366 0.926830 0.375481i \(-0.122523\pi\)
0.926830 + 0.375481i \(0.122523\pi\)
\(24\) −3.26258 −0.665972
\(25\) 1.00000 0.200000
\(26\) 6.11752 1.19974
\(27\) −15.1529 −2.91618
\(28\) 2.73152 0.516209
\(29\) 6.36025 1.18107 0.590534 0.807012i \(-0.298917\pi\)
0.590534 + 0.807012i \(0.298917\pi\)
\(30\) 3.26258 0.595664
\(31\) 6.01389 1.08013 0.540064 0.841624i \(-0.318400\pi\)
0.540064 + 0.841624i \(0.318400\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.26258 −0.567943
\(34\) 7.48258 1.28325
\(35\) −2.73152 −0.461711
\(36\) 7.64446 1.27408
\(37\) −0.607804 −0.0999224 −0.0499612 0.998751i \(-0.515910\pi\)
−0.0499612 + 0.998751i \(0.515910\pi\)
\(38\) −7.78028 −1.26213
\(39\) −19.9589 −3.19599
\(40\) −1.00000 −0.158114
\(41\) 2.66769 0.416624 0.208312 0.978062i \(-0.433203\pi\)
0.208312 + 0.978062i \(0.433203\pi\)
\(42\) −8.91182 −1.37512
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) −7.64446 −1.13957
\(46\) 8.88984 1.31074
\(47\) 0.852378 0.124332 0.0621660 0.998066i \(-0.480199\pi\)
0.0621660 + 0.998066i \(0.480199\pi\)
\(48\) −3.26258 −0.470914
\(49\) 0.461206 0.0658865
\(50\) 1.00000 0.141421
\(51\) −24.4126 −3.41844
\(52\) 6.11752 0.848348
\(53\) −9.54066 −1.31051 −0.655255 0.755408i \(-0.727439\pi\)
−0.655255 + 0.755408i \(0.727439\pi\)
\(54\) −15.1529 −2.06205
\(55\) −1.00000 −0.134840
\(56\) 2.73152 0.365015
\(57\) 25.3838 3.36217
\(58\) 6.36025 0.835142
\(59\) −1.12856 −0.146926 −0.0734632 0.997298i \(-0.523405\pi\)
−0.0734632 + 0.997298i \(0.523405\pi\)
\(60\) 3.26258 0.421198
\(61\) 9.95844 1.27505 0.637524 0.770431i \(-0.279959\pi\)
0.637524 + 0.770431i \(0.279959\pi\)
\(62\) 6.01389 0.763765
\(63\) 20.8810 2.63076
\(64\) 1.00000 0.125000
\(65\) −6.11752 −0.758785
\(66\) −3.26258 −0.401596
\(67\) −11.0418 −1.34897 −0.674487 0.738286i \(-0.735635\pi\)
−0.674487 + 0.738286i \(0.735635\pi\)
\(68\) 7.48258 0.907396
\(69\) −29.0039 −3.49165
\(70\) −2.73152 −0.326479
\(71\) −4.13401 −0.490616 −0.245308 0.969445i \(-0.578889\pi\)
−0.245308 + 0.969445i \(0.578889\pi\)
\(72\) 7.64446 0.900908
\(73\) −13.2485 −1.55062 −0.775309 0.631582i \(-0.782406\pi\)
−0.775309 + 0.631582i \(0.782406\pi\)
\(74\) −0.607804 −0.0706558
\(75\) −3.26258 −0.376731
\(76\) −7.78028 −0.892459
\(77\) 2.73152 0.311286
\(78\) −19.9589 −2.25990
\(79\) −2.52486 −0.284069 −0.142034 0.989862i \(-0.545364\pi\)
−0.142034 + 0.989862i \(0.545364\pi\)
\(80\) −1.00000 −0.111803
\(81\) 26.5044 2.94493
\(82\) 2.66769 0.294597
\(83\) −12.7935 −1.40427 −0.702135 0.712043i \(-0.747770\pi\)
−0.702135 + 0.712043i \(0.747770\pi\)
\(84\) −8.91182 −0.972359
\(85\) −7.48258 −0.811600
\(86\) 1.00000 0.107833
\(87\) −20.7509 −2.22473
\(88\) 1.00000 0.106600
\(89\) 1.91784 0.203290 0.101645 0.994821i \(-0.467589\pi\)
0.101645 + 0.994821i \(0.467589\pi\)
\(90\) −7.64446 −0.805797
\(91\) 16.7101 1.75170
\(92\) 8.88984 0.926830
\(93\) −19.6208 −2.03459
\(94\) 0.852378 0.0879160
\(95\) 7.78028 0.798240
\(96\) −3.26258 −0.332986
\(97\) 12.9532 1.31520 0.657601 0.753366i \(-0.271571\pi\)
0.657601 + 0.753366i \(0.271571\pi\)
\(98\) 0.461206 0.0465888
\(99\) 7.64446 0.768297
\(100\) 1.00000 0.100000
\(101\) 3.60944 0.359152 0.179576 0.983744i \(-0.442527\pi\)
0.179576 + 0.983744i \(0.442527\pi\)
\(102\) −24.4126 −2.41720
\(103\) −1.89824 −0.187039 −0.0935197 0.995617i \(-0.529812\pi\)
−0.0935197 + 0.995617i \(0.529812\pi\)
\(104\) 6.11752 0.599872
\(105\) 8.91182 0.869704
\(106\) −9.54066 −0.926670
\(107\) 3.55840 0.344003 0.172002 0.985097i \(-0.444977\pi\)
0.172002 + 0.985097i \(0.444977\pi\)
\(108\) −15.1529 −1.45809
\(109\) 6.28847 0.602326 0.301163 0.953573i \(-0.402625\pi\)
0.301163 + 0.953573i \(0.402625\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 1.98301 0.188219
\(112\) 2.73152 0.258104
\(113\) −4.24313 −0.399160 −0.199580 0.979882i \(-0.563958\pi\)
−0.199580 + 0.979882i \(0.563958\pi\)
\(114\) 25.3838 2.37741
\(115\) −8.88984 −0.828982
\(116\) 6.36025 0.590534
\(117\) 46.7652 4.32344
\(118\) −1.12856 −0.103893
\(119\) 20.4388 1.87362
\(120\) 3.26258 0.297832
\(121\) 1.00000 0.0909091
\(122\) 9.95844 0.901595
\(123\) −8.70358 −0.784775
\(124\) 6.01389 0.540064
\(125\) −1.00000 −0.0894427
\(126\) 20.8810 1.86023
\(127\) −1.72903 −0.153426 −0.0767132 0.997053i \(-0.524443\pi\)
−0.0767132 + 0.997053i \(0.524443\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.26258 −0.287255
\(130\) −6.11752 −0.536542
\(131\) −17.4710 −1.52645 −0.763223 0.646135i \(-0.776384\pi\)
−0.763223 + 0.646135i \(0.776384\pi\)
\(132\) −3.26258 −0.283972
\(133\) −21.2520 −1.84278
\(134\) −11.0418 −0.953869
\(135\) 15.1529 1.30416
\(136\) 7.48258 0.641626
\(137\) −0.759375 −0.0648778 −0.0324389 0.999474i \(-0.510327\pi\)
−0.0324389 + 0.999474i \(0.510327\pi\)
\(138\) −29.0039 −2.46897
\(139\) −4.51146 −0.382657 −0.191328 0.981526i \(-0.561280\pi\)
−0.191328 + 0.981526i \(0.561280\pi\)
\(140\) −2.73152 −0.230856
\(141\) −2.78095 −0.234199
\(142\) −4.13401 −0.346918
\(143\) 6.11752 0.511573
\(144\) 7.64446 0.637038
\(145\) −6.36025 −0.528190
\(146\) −13.2485 −1.09645
\(147\) −1.50472 −0.124107
\(148\) −0.607804 −0.0499612
\(149\) 6.08570 0.498560 0.249280 0.968431i \(-0.419806\pi\)
0.249280 + 0.968431i \(0.419806\pi\)
\(150\) −3.26258 −0.266389
\(151\) 8.36366 0.680625 0.340312 0.940312i \(-0.389467\pi\)
0.340312 + 0.940312i \(0.389467\pi\)
\(152\) −7.78028 −0.631064
\(153\) 57.2003 4.62437
\(154\) 2.73152 0.220112
\(155\) −6.01389 −0.483048
\(156\) −19.9589 −1.59799
\(157\) −6.68607 −0.533606 −0.266803 0.963751i \(-0.585967\pi\)
−0.266803 + 0.963751i \(0.585967\pi\)
\(158\) −2.52486 −0.200867
\(159\) 31.1272 2.46855
\(160\) −1.00000 −0.0790569
\(161\) 24.2828 1.91375
\(162\) 26.5044 2.08238
\(163\) 22.5617 1.76717 0.883585 0.468270i \(-0.155123\pi\)
0.883585 + 0.468270i \(0.155123\pi\)
\(164\) 2.66769 0.208312
\(165\) 3.26258 0.253992
\(166\) −12.7935 −0.992969
\(167\) −14.7906 −1.14453 −0.572267 0.820067i \(-0.693936\pi\)
−0.572267 + 0.820067i \(0.693936\pi\)
\(168\) −8.91182 −0.687562
\(169\) 24.4241 1.87878
\(170\) −7.48258 −0.573888
\(171\) −59.4760 −4.54825
\(172\) 1.00000 0.0762493
\(173\) −5.80673 −0.441478 −0.220739 0.975333i \(-0.570847\pi\)
−0.220739 + 0.975333i \(0.570847\pi\)
\(174\) −20.7509 −1.57312
\(175\) 2.73152 0.206484
\(176\) 1.00000 0.0753778
\(177\) 3.68203 0.276759
\(178\) 1.91784 0.143748
\(179\) 9.00912 0.673373 0.336687 0.941617i \(-0.390694\pi\)
0.336687 + 0.941617i \(0.390694\pi\)
\(180\) −7.64446 −0.569784
\(181\) −20.2234 −1.50319 −0.751597 0.659623i \(-0.770716\pi\)
−0.751597 + 0.659623i \(0.770716\pi\)
\(182\) 16.7101 1.23864
\(183\) −32.4903 −2.40175
\(184\) 8.88984 0.655368
\(185\) 0.607804 0.0446866
\(186\) −19.6208 −1.43867
\(187\) 7.48258 0.547181
\(188\) 0.852378 0.0621660
\(189\) −41.3906 −3.01072
\(190\) 7.78028 0.564441
\(191\) 16.1486 1.16847 0.584237 0.811583i \(-0.301394\pi\)
0.584237 + 0.811583i \(0.301394\pi\)
\(192\) −3.26258 −0.235457
\(193\) 11.6496 0.838558 0.419279 0.907857i \(-0.362283\pi\)
0.419279 + 0.907857i \(0.362283\pi\)
\(194\) 12.9532 0.929988
\(195\) 19.9589 1.42929
\(196\) 0.461206 0.0329433
\(197\) 15.2043 1.08326 0.541630 0.840617i \(-0.317808\pi\)
0.541630 + 0.840617i \(0.317808\pi\)
\(198\) 7.64446 0.543268
\(199\) −2.21588 −0.157079 −0.0785396 0.996911i \(-0.525026\pi\)
−0.0785396 + 0.996911i \(0.525026\pi\)
\(200\) 1.00000 0.0707107
\(201\) 36.0249 2.54100
\(202\) 3.60944 0.253959
\(203\) 17.3732 1.21936
\(204\) −24.4126 −1.70922
\(205\) −2.66769 −0.186320
\(206\) −1.89824 −0.132257
\(207\) 67.9580 4.72341
\(208\) 6.11752 0.424174
\(209\) −7.78028 −0.538173
\(210\) 8.91182 0.614974
\(211\) 12.3897 0.852939 0.426470 0.904502i \(-0.359757\pi\)
0.426470 + 0.904502i \(0.359757\pi\)
\(212\) −9.54066 −0.655255
\(213\) 13.4875 0.924152
\(214\) 3.55840 0.243247
\(215\) −1.00000 −0.0681994
\(216\) −15.1529 −1.03103
\(217\) 16.4271 1.11514
\(218\) 6.28847 0.425909
\(219\) 43.2243 2.92083
\(220\) −1.00000 −0.0674200
\(221\) 45.7749 3.07915
\(222\) 1.98301 0.133091
\(223\) −3.05848 −0.204811 −0.102405 0.994743i \(-0.532654\pi\)
−0.102405 + 0.994743i \(0.532654\pi\)
\(224\) 2.73152 0.182507
\(225\) 7.64446 0.509631
\(226\) −4.24313 −0.282249
\(227\) 14.6877 0.974860 0.487430 0.873162i \(-0.337934\pi\)
0.487430 + 0.873162i \(0.337934\pi\)
\(228\) 25.3838 1.68108
\(229\) −13.4531 −0.889008 −0.444504 0.895777i \(-0.646620\pi\)
−0.444504 + 0.895777i \(0.646620\pi\)
\(230\) −8.88984 −0.586179
\(231\) −8.91182 −0.586355
\(232\) 6.36025 0.417571
\(233\) −17.1819 −1.12562 −0.562811 0.826586i \(-0.690280\pi\)
−0.562811 + 0.826586i \(0.690280\pi\)
\(234\) 46.7652 3.05713
\(235\) −0.852378 −0.0556030
\(236\) −1.12856 −0.0734632
\(237\) 8.23757 0.535088
\(238\) 20.4388 1.32485
\(239\) 17.8779 1.15643 0.578213 0.815886i \(-0.303750\pi\)
0.578213 + 0.815886i \(0.303750\pi\)
\(240\) 3.26258 0.210599
\(241\) −23.9789 −1.54462 −0.772308 0.635249i \(-0.780898\pi\)
−0.772308 + 0.635249i \(0.780898\pi\)
\(242\) 1.00000 0.0642824
\(243\) −41.0139 −2.63105
\(244\) 9.95844 0.637524
\(245\) −0.461206 −0.0294653
\(246\) −8.70358 −0.554920
\(247\) −47.5960 −3.02846
\(248\) 6.01389 0.381883
\(249\) 41.7399 2.64516
\(250\) −1.00000 −0.0632456
\(251\) 12.1141 0.764633 0.382316 0.924031i \(-0.375126\pi\)
0.382316 + 0.924031i \(0.375126\pi\)
\(252\) 20.8810 1.31538
\(253\) 8.88984 0.558899
\(254\) −1.72903 −0.108489
\(255\) 24.4126 1.52877
\(256\) 1.00000 0.0625000
\(257\) 12.6375 0.788305 0.394152 0.919045i \(-0.371038\pi\)
0.394152 + 0.919045i \(0.371038\pi\)
\(258\) −3.26258 −0.203120
\(259\) −1.66023 −0.103162
\(260\) −6.11752 −0.379393
\(261\) 48.6207 3.00954
\(262\) −17.4710 −1.07936
\(263\) −30.8809 −1.90420 −0.952099 0.305790i \(-0.901079\pi\)
−0.952099 + 0.305790i \(0.901079\pi\)
\(264\) −3.26258 −0.200798
\(265\) 9.54066 0.586078
\(266\) −21.2520 −1.30304
\(267\) −6.25711 −0.382929
\(268\) −11.0418 −0.674487
\(269\) −18.0663 −1.10152 −0.550762 0.834662i \(-0.685663\pi\)
−0.550762 + 0.834662i \(0.685663\pi\)
\(270\) 15.1529 0.922179
\(271\) 21.2955 1.29361 0.646805 0.762656i \(-0.276105\pi\)
0.646805 + 0.762656i \(0.276105\pi\)
\(272\) 7.48258 0.453698
\(273\) −54.5182 −3.29959
\(274\) −0.759375 −0.0458755
\(275\) 1.00000 0.0603023
\(276\) −29.0039 −1.74583
\(277\) 8.04461 0.483354 0.241677 0.970357i \(-0.422303\pi\)
0.241677 + 0.970357i \(0.422303\pi\)
\(278\) −4.51146 −0.270579
\(279\) 45.9730 2.75233
\(280\) −2.73152 −0.163240
\(281\) −29.1658 −1.73988 −0.869942 0.493154i \(-0.835844\pi\)
−0.869942 + 0.493154i \(0.835844\pi\)
\(282\) −2.78095 −0.165603
\(283\) −5.12601 −0.304710 −0.152355 0.988326i \(-0.548686\pi\)
−0.152355 + 0.988326i \(0.548686\pi\)
\(284\) −4.13401 −0.245308
\(285\) −25.3838 −1.50361
\(286\) 6.11752 0.361737
\(287\) 7.28686 0.430130
\(288\) 7.64446 0.450454
\(289\) 38.9891 2.29347
\(290\) −6.36025 −0.373487
\(291\) −42.2610 −2.47739
\(292\) −13.2485 −0.775309
\(293\) −17.4294 −1.01824 −0.509118 0.860697i \(-0.670028\pi\)
−0.509118 + 0.860697i \(0.670028\pi\)
\(294\) −1.50472 −0.0877572
\(295\) 1.12856 0.0657075
\(296\) −0.607804 −0.0353279
\(297\) −15.1529 −0.879263
\(298\) 6.08570 0.352535
\(299\) 54.3838 3.14510
\(300\) −3.26258 −0.188365
\(301\) 2.73152 0.157442
\(302\) 8.36366 0.481274
\(303\) −11.7761 −0.676519
\(304\) −7.78028 −0.446230
\(305\) −9.95844 −0.570219
\(306\) 57.2003 3.26992
\(307\) 11.6324 0.663898 0.331949 0.943297i \(-0.392294\pi\)
0.331949 + 0.943297i \(0.392294\pi\)
\(308\) 2.73152 0.155643
\(309\) 6.19317 0.352317
\(310\) −6.01389 −0.341566
\(311\) −6.14313 −0.348345 −0.174172 0.984715i \(-0.555725\pi\)
−0.174172 + 0.984715i \(0.555725\pi\)
\(312\) −19.9589 −1.12995
\(313\) −24.6979 −1.39601 −0.698003 0.716095i \(-0.745928\pi\)
−0.698003 + 0.716095i \(0.745928\pi\)
\(314\) −6.68607 −0.377316
\(315\) −20.8810 −1.17651
\(316\) −2.52486 −0.142034
\(317\) 8.66426 0.486633 0.243317 0.969947i \(-0.421765\pi\)
0.243317 + 0.969947i \(0.421765\pi\)
\(318\) 31.1272 1.74553
\(319\) 6.36025 0.356106
\(320\) −1.00000 −0.0559017
\(321\) −11.6096 −0.647983
\(322\) 24.2828 1.35323
\(323\) −58.2166 −3.23926
\(324\) 26.5044 1.47247
\(325\) 6.11752 0.339339
\(326\) 22.5617 1.24958
\(327\) −20.5167 −1.13457
\(328\) 2.66769 0.147299
\(329\) 2.32829 0.128363
\(330\) 3.26258 0.179599
\(331\) 35.9138 1.97400 0.986999 0.160724i \(-0.0513828\pi\)
0.986999 + 0.160724i \(0.0513828\pi\)
\(332\) −12.7935 −0.702135
\(333\) −4.64633 −0.254617
\(334\) −14.7906 −0.809307
\(335\) 11.0418 0.603280
\(336\) −8.91182 −0.486180
\(337\) 0.835503 0.0455128 0.0227564 0.999741i \(-0.492756\pi\)
0.0227564 + 0.999741i \(0.492756\pi\)
\(338\) 24.4241 1.32850
\(339\) 13.8436 0.751880
\(340\) −7.48258 −0.405800
\(341\) 6.01389 0.325671
\(342\) −59.4760 −3.21610
\(343\) −17.8609 −0.964395
\(344\) 1.00000 0.0539164
\(345\) 29.0039 1.56152
\(346\) −5.80673 −0.312172
\(347\) −1.30007 −0.0697913 −0.0348956 0.999391i \(-0.511110\pi\)
−0.0348956 + 0.999391i \(0.511110\pi\)
\(348\) −20.7509 −1.11236
\(349\) 26.2403 1.40461 0.702306 0.711875i \(-0.252154\pi\)
0.702306 + 0.711875i \(0.252154\pi\)
\(350\) 2.73152 0.146006
\(351\) −92.6985 −4.94788
\(352\) 1.00000 0.0533002
\(353\) 9.93627 0.528854 0.264427 0.964406i \(-0.414817\pi\)
0.264427 + 0.964406i \(0.414817\pi\)
\(354\) 3.68203 0.195698
\(355\) 4.13401 0.219410
\(356\) 1.91784 0.101645
\(357\) −66.6834 −3.52926
\(358\) 9.00912 0.476147
\(359\) −28.8363 −1.52192 −0.760962 0.648797i \(-0.775272\pi\)
−0.760962 + 0.648797i \(0.775272\pi\)
\(360\) −7.64446 −0.402898
\(361\) 41.5328 2.18594
\(362\) −20.2234 −1.06292
\(363\) −3.26258 −0.171241
\(364\) 16.7101 0.875849
\(365\) 13.2485 0.693458
\(366\) −32.4903 −1.69829
\(367\) −36.6213 −1.91162 −0.955808 0.293992i \(-0.905016\pi\)
−0.955808 + 0.293992i \(0.905016\pi\)
\(368\) 8.88984 0.463415
\(369\) 20.3931 1.06162
\(370\) 0.607804 0.0315982
\(371\) −26.0605 −1.35299
\(372\) −19.6208 −1.01729
\(373\) 13.8817 0.718766 0.359383 0.933190i \(-0.382987\pi\)
0.359383 + 0.933190i \(0.382987\pi\)
\(374\) 7.48258 0.386915
\(375\) 3.26258 0.168479
\(376\) 0.852378 0.0439580
\(377\) 38.9090 2.00391
\(378\) −41.3906 −2.12890
\(379\) −7.18540 −0.369089 −0.184545 0.982824i \(-0.559081\pi\)
−0.184545 + 0.982824i \(0.559081\pi\)
\(380\) 7.78028 0.399120
\(381\) 5.64110 0.289002
\(382\) 16.1486 0.826236
\(383\) −28.7845 −1.47082 −0.735410 0.677622i \(-0.763011\pi\)
−0.735410 + 0.677622i \(0.763011\pi\)
\(384\) −3.26258 −0.166493
\(385\) −2.73152 −0.139211
\(386\) 11.6496 0.592950
\(387\) 7.64446 0.388590
\(388\) 12.9532 0.657601
\(389\) 0.309694 0.0157021 0.00785106 0.999969i \(-0.497501\pi\)
0.00785106 + 0.999969i \(0.497501\pi\)
\(390\) 19.9589 1.01066
\(391\) 66.5190 3.36401
\(392\) 0.461206 0.0232944
\(393\) 57.0005 2.87530
\(394\) 15.2043 0.765980
\(395\) 2.52486 0.127039
\(396\) 7.64446 0.384149
\(397\) −29.8800 −1.49964 −0.749818 0.661644i \(-0.769859\pi\)
−0.749818 + 0.661644i \(0.769859\pi\)
\(398\) −2.21588 −0.111072
\(399\) 69.3364 3.47116
\(400\) 1.00000 0.0500000
\(401\) −7.23252 −0.361175 −0.180587 0.983559i \(-0.557800\pi\)
−0.180587 + 0.983559i \(0.557800\pi\)
\(402\) 36.0249 1.79676
\(403\) 36.7901 1.83265
\(404\) 3.60944 0.179576
\(405\) −26.5044 −1.31701
\(406\) 17.3732 0.862215
\(407\) −0.607804 −0.0301277
\(408\) −24.4126 −1.20860
\(409\) 12.8063 0.633229 0.316615 0.948554i \(-0.397454\pi\)
0.316615 + 0.948554i \(0.397454\pi\)
\(410\) −2.66769 −0.131748
\(411\) 2.47752 0.122207
\(412\) −1.89824 −0.0935197
\(413\) −3.08269 −0.151689
\(414\) 67.9580 3.33995
\(415\) 12.7935 0.628009
\(416\) 6.11752 0.299936
\(417\) 14.7190 0.720793
\(418\) −7.78028 −0.380546
\(419\) 2.39313 0.116912 0.0584561 0.998290i \(-0.481382\pi\)
0.0584561 + 0.998290i \(0.481382\pi\)
\(420\) 8.91182 0.434852
\(421\) 6.58937 0.321146 0.160573 0.987024i \(-0.448666\pi\)
0.160573 + 0.987024i \(0.448666\pi\)
\(422\) 12.3897 0.603119
\(423\) 6.51597 0.316817
\(424\) −9.54066 −0.463335
\(425\) 7.48258 0.362959
\(426\) 13.4875 0.653474
\(427\) 27.2017 1.31638
\(428\) 3.55840 0.172002
\(429\) −19.9589 −0.963627
\(430\) −1.00000 −0.0482243
\(431\) −38.2689 −1.84335 −0.921674 0.387965i \(-0.873178\pi\)
−0.921674 + 0.387965i \(0.873178\pi\)
\(432\) −15.1529 −0.729046
\(433\) −26.5851 −1.27760 −0.638799 0.769374i \(-0.720568\pi\)
−0.638799 + 0.769374i \(0.720568\pi\)
\(434\) 16.4271 0.788525
\(435\) 20.7509 0.994927
\(436\) 6.28847 0.301163
\(437\) −69.1655 −3.30863
\(438\) 43.2243 2.06534
\(439\) 5.69352 0.271737 0.135868 0.990727i \(-0.456618\pi\)
0.135868 + 0.990727i \(0.456618\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 3.52567 0.167889
\(442\) 45.7749 2.17729
\(443\) −21.1563 −1.00516 −0.502582 0.864529i \(-0.667617\pi\)
−0.502582 + 0.864529i \(0.667617\pi\)
\(444\) 1.98301 0.0941096
\(445\) −1.91784 −0.0909143
\(446\) −3.05848 −0.144823
\(447\) −19.8551 −0.939114
\(448\) 2.73152 0.129052
\(449\) −8.72132 −0.411585 −0.205792 0.978596i \(-0.565977\pi\)
−0.205792 + 0.978596i \(0.565977\pi\)
\(450\) 7.64446 0.360363
\(451\) 2.66769 0.125617
\(452\) −4.24313 −0.199580
\(453\) −27.2871 −1.28206
\(454\) 14.6877 0.689330
\(455\) −16.7101 −0.783383
\(456\) 25.3838 1.18871
\(457\) 0.125694 0.00587973 0.00293987 0.999996i \(-0.499064\pi\)
0.00293987 + 0.999996i \(0.499064\pi\)
\(458\) −13.4531 −0.628623
\(459\) −113.383 −5.29227
\(460\) −8.88984 −0.414491
\(461\) 3.91909 0.182530 0.0912651 0.995827i \(-0.470909\pi\)
0.0912651 + 0.995827i \(0.470909\pi\)
\(462\) −8.91182 −0.414615
\(463\) −18.4445 −0.857187 −0.428594 0.903497i \(-0.640991\pi\)
−0.428594 + 0.903497i \(0.640991\pi\)
\(464\) 6.36025 0.295267
\(465\) 19.6208 0.909894
\(466\) −17.1819 −0.795935
\(467\) −19.7612 −0.914440 −0.457220 0.889354i \(-0.651155\pi\)
−0.457220 + 0.889354i \(0.651155\pi\)
\(468\) 46.7652 2.16172
\(469\) −30.1610 −1.39271
\(470\) −0.852378 −0.0393172
\(471\) 21.8139 1.00513
\(472\) −1.12856 −0.0519463
\(473\) 1.00000 0.0459800
\(474\) 8.23757 0.378364
\(475\) −7.78028 −0.356984
\(476\) 20.4388 0.936812
\(477\) −72.9331 −3.33938
\(478\) 17.8779 0.817716
\(479\) 39.0414 1.78385 0.891924 0.452185i \(-0.149355\pi\)
0.891924 + 0.452185i \(0.149355\pi\)
\(480\) 3.26258 0.148916
\(481\) −3.71825 −0.169538
\(482\) −23.9789 −1.09221
\(483\) −79.2246 −3.60485
\(484\) 1.00000 0.0454545
\(485\) −12.9532 −0.588176
\(486\) −41.0139 −1.86043
\(487\) 23.8887 1.08250 0.541251 0.840861i \(-0.317951\pi\)
0.541251 + 0.840861i \(0.317951\pi\)
\(488\) 9.95844 0.450797
\(489\) −73.6095 −3.32874
\(490\) −0.461206 −0.0208351
\(491\) −23.3319 −1.05295 −0.526477 0.850189i \(-0.676487\pi\)
−0.526477 + 0.850189i \(0.676487\pi\)
\(492\) −8.70358 −0.392388
\(493\) 47.5911 2.14340
\(494\) −47.5960 −2.14145
\(495\) −7.64446 −0.343593
\(496\) 6.01389 0.270032
\(497\) −11.2921 −0.506521
\(498\) 41.7399 1.87041
\(499\) −28.8772 −1.29272 −0.646361 0.763032i \(-0.723710\pi\)
−0.646361 + 0.763032i \(0.723710\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 48.2557 2.15591
\(502\) 12.1141 0.540677
\(503\) −38.5921 −1.72074 −0.860369 0.509671i \(-0.829767\pi\)
−0.860369 + 0.509671i \(0.829767\pi\)
\(504\) 20.8810 0.930114
\(505\) −3.60944 −0.160618
\(506\) 8.88984 0.395202
\(507\) −79.6857 −3.53896
\(508\) −1.72903 −0.0767132
\(509\) −4.40494 −0.195245 −0.0976227 0.995223i \(-0.531124\pi\)
−0.0976227 + 0.995223i \(0.531124\pi\)
\(510\) 24.4126 1.08101
\(511\) −36.1885 −1.60089
\(512\) 1.00000 0.0441942
\(513\) 117.894 5.20515
\(514\) 12.6375 0.557416
\(515\) 1.89824 0.0836465
\(516\) −3.26258 −0.143627
\(517\) 0.852378 0.0374875
\(518\) −1.66023 −0.0729463
\(519\) 18.9450 0.831591
\(520\) −6.11752 −0.268271
\(521\) −22.1098 −0.968646 −0.484323 0.874889i \(-0.660934\pi\)
−0.484323 + 0.874889i \(0.660934\pi\)
\(522\) 48.6207 2.12807
\(523\) −4.11099 −0.179761 −0.0898805 0.995953i \(-0.528649\pi\)
−0.0898805 + 0.995953i \(0.528649\pi\)
\(524\) −17.4710 −0.763223
\(525\) −8.91182 −0.388944
\(526\) −30.8809 −1.34647
\(527\) 44.9995 1.96021
\(528\) −3.26258 −0.141986
\(529\) 56.0293 2.43606
\(530\) 9.54066 0.414420
\(531\) −8.62726 −0.374391
\(532\) −21.2520 −0.921391
\(533\) 16.3197 0.706884
\(534\) −6.25711 −0.270772
\(535\) −3.55840 −0.153843
\(536\) −11.0418 −0.476935
\(537\) −29.3930 −1.26840
\(538\) −18.0663 −0.778895
\(539\) 0.461206 0.0198655
\(540\) 15.1529 0.652079
\(541\) 10.5245 0.452482 0.226241 0.974071i \(-0.427356\pi\)
0.226241 + 0.974071i \(0.427356\pi\)
\(542\) 21.2955 0.914720
\(543\) 65.9806 2.83150
\(544\) 7.48258 0.320813
\(545\) −6.28847 −0.269369
\(546\) −54.5182 −2.33317
\(547\) 37.3005 1.59486 0.797428 0.603414i \(-0.206193\pi\)
0.797428 + 0.603414i \(0.206193\pi\)
\(548\) −0.759375 −0.0324389
\(549\) 76.1269 3.24902
\(550\) 1.00000 0.0426401
\(551\) −49.4845 −2.10811
\(552\) −29.0039 −1.23449
\(553\) −6.89671 −0.293278
\(554\) 8.04461 0.341783
\(555\) −1.98301 −0.0841742
\(556\) −4.51146 −0.191328
\(557\) −14.8754 −0.630293 −0.315146 0.949043i \(-0.602054\pi\)
−0.315146 + 0.949043i \(0.602054\pi\)
\(558\) 45.9730 1.94619
\(559\) 6.11752 0.258744
\(560\) −2.73152 −0.115428
\(561\) −24.4126 −1.03070
\(562\) −29.1658 −1.23028
\(563\) 18.4913 0.779315 0.389658 0.920960i \(-0.372593\pi\)
0.389658 + 0.920960i \(0.372593\pi\)
\(564\) −2.78095 −0.117099
\(565\) 4.24313 0.178510
\(566\) −5.12601 −0.215462
\(567\) 72.3972 3.04040
\(568\) −4.13401 −0.173459
\(569\) 8.73588 0.366227 0.183114 0.983092i \(-0.441382\pi\)
0.183114 + 0.983092i \(0.441382\pi\)
\(570\) −25.3838 −1.06321
\(571\) −0.811948 −0.0339790 −0.0169895 0.999856i \(-0.505408\pi\)
−0.0169895 + 0.999856i \(0.505408\pi\)
\(572\) 6.11752 0.255786
\(573\) −52.6863 −2.20100
\(574\) 7.28686 0.304148
\(575\) 8.88984 0.370732
\(576\) 7.64446 0.318519
\(577\) 20.7394 0.863393 0.431696 0.902019i \(-0.357915\pi\)
0.431696 + 0.902019i \(0.357915\pi\)
\(578\) 38.9891 1.62173
\(579\) −38.0079 −1.57955
\(580\) −6.36025 −0.264095
\(581\) −34.9458 −1.44979
\(582\) −42.2610 −1.75178
\(583\) −9.54066 −0.395134
\(584\) −13.2485 −0.548226
\(585\) −46.7652 −1.93350
\(586\) −17.4294 −0.720001
\(587\) 22.0637 0.910668 0.455334 0.890321i \(-0.349520\pi\)
0.455334 + 0.890321i \(0.349520\pi\)
\(588\) −1.50472 −0.0620537
\(589\) −46.7898 −1.92794
\(590\) 1.12856 0.0464622
\(591\) −49.6052 −2.04049
\(592\) −0.607804 −0.0249806
\(593\) 42.7997 1.75757 0.878786 0.477216i \(-0.158354\pi\)
0.878786 + 0.477216i \(0.158354\pi\)
\(594\) −15.1529 −0.621733
\(595\) −20.4388 −0.837910
\(596\) 6.08570 0.249280
\(597\) 7.22948 0.295883
\(598\) 54.3838 2.22392
\(599\) −44.3238 −1.81102 −0.905511 0.424323i \(-0.860512\pi\)
−0.905511 + 0.424323i \(0.860512\pi\)
\(600\) −3.26258 −0.133194
\(601\) 34.5402 1.40893 0.704463 0.709741i \(-0.251188\pi\)
0.704463 + 0.709741i \(0.251188\pi\)
\(602\) 2.73152 0.111328
\(603\) −84.4089 −3.43739
\(604\) 8.36366 0.340312
\(605\) −1.00000 −0.0406558
\(606\) −11.7761 −0.478371
\(607\) −21.4937 −0.872402 −0.436201 0.899849i \(-0.643676\pi\)
−0.436201 + 0.899849i \(0.643676\pi\)
\(608\) −7.78028 −0.315532
\(609\) −56.6814 −2.29685
\(610\) −9.95844 −0.403205
\(611\) 5.21444 0.210954
\(612\) 57.2003 2.31219
\(613\) −2.16333 −0.0873761 −0.0436881 0.999045i \(-0.513911\pi\)
−0.0436881 + 0.999045i \(0.513911\pi\)
\(614\) 11.6324 0.469446
\(615\) 8.70358 0.350962
\(616\) 2.73152 0.110056
\(617\) 24.5899 0.989950 0.494975 0.868907i \(-0.335177\pi\)
0.494975 + 0.868907i \(0.335177\pi\)
\(618\) 6.19317 0.249126
\(619\) 15.5361 0.624447 0.312223 0.950009i \(-0.398926\pi\)
0.312223 + 0.950009i \(0.398926\pi\)
\(620\) −6.01389 −0.241524
\(621\) −134.707 −5.40561
\(622\) −6.14313 −0.246317
\(623\) 5.23862 0.209881
\(624\) −19.9589 −0.798997
\(625\) 1.00000 0.0400000
\(626\) −24.6979 −0.987125
\(627\) 25.3838 1.01373
\(628\) −6.68607 −0.266803
\(629\) −4.54794 −0.181338
\(630\) −20.8810 −0.831919
\(631\) 35.2931 1.40500 0.702498 0.711686i \(-0.252068\pi\)
0.702498 + 0.711686i \(0.252068\pi\)
\(632\) −2.52486 −0.100434
\(633\) −40.4223 −1.60664
\(634\) 8.66426 0.344102
\(635\) 1.72903 0.0686144
\(636\) 31.1272 1.23427
\(637\) 2.82144 0.111789
\(638\) 6.36025 0.251805
\(639\) −31.6022 −1.25017
\(640\) −1.00000 −0.0395285
\(641\) −8.95945 −0.353877 −0.176939 0.984222i \(-0.556619\pi\)
−0.176939 + 0.984222i \(0.556619\pi\)
\(642\) −11.6096 −0.458193
\(643\) 11.3438 0.447356 0.223678 0.974663i \(-0.428194\pi\)
0.223678 + 0.974663i \(0.428194\pi\)
\(644\) 24.2828 0.956876
\(645\) 3.26258 0.128464
\(646\) −58.2166 −2.29050
\(647\) −26.8885 −1.05710 −0.528548 0.848903i \(-0.677263\pi\)
−0.528548 + 0.848903i \(0.677263\pi\)
\(648\) 26.5044 1.04119
\(649\) −1.12856 −0.0443000
\(650\) 6.11752 0.239949
\(651\) −53.5947 −2.10054
\(652\) 22.5617 0.883585
\(653\) 17.2333 0.674393 0.337196 0.941434i \(-0.390521\pi\)
0.337196 + 0.941434i \(0.390521\pi\)
\(654\) −20.5167 −0.802265
\(655\) 17.4710 0.682647
\(656\) 2.66769 0.104156
\(657\) −101.278 −3.95121
\(658\) 2.32829 0.0907661
\(659\) 2.81633 0.109709 0.0548543 0.998494i \(-0.482531\pi\)
0.0548543 + 0.998494i \(0.482531\pi\)
\(660\) 3.26258 0.126996
\(661\) 13.2691 0.516110 0.258055 0.966130i \(-0.416918\pi\)
0.258055 + 0.966130i \(0.416918\pi\)
\(662\) 35.9138 1.39583
\(663\) −149.344 −5.80006
\(664\) −12.7935 −0.496485
\(665\) 21.2520 0.824117
\(666\) −4.64633 −0.180042
\(667\) 56.5416 2.18930
\(668\) −14.7906 −0.572267
\(669\) 9.97854 0.385793
\(670\) 11.0418 0.426583
\(671\) 9.95844 0.384441
\(672\) −8.91182 −0.343781
\(673\) −7.65905 −0.295235 −0.147617 0.989045i \(-0.547160\pi\)
−0.147617 + 0.989045i \(0.547160\pi\)
\(674\) 0.835503 0.0321824
\(675\) −15.1529 −0.583237
\(676\) 24.4241 0.939388
\(677\) 24.9266 0.958007 0.479004 0.877813i \(-0.340998\pi\)
0.479004 + 0.877813i \(0.340998\pi\)
\(678\) 13.8436 0.531659
\(679\) 35.3820 1.35784
\(680\) −7.48258 −0.286944
\(681\) −47.9200 −1.83630
\(682\) 6.01389 0.230284
\(683\) −0.841780 −0.0322098 −0.0161049 0.999870i \(-0.505127\pi\)
−0.0161049 + 0.999870i \(0.505127\pi\)
\(684\) −59.4760 −2.27412
\(685\) 0.759375 0.0290142
\(686\) −17.8609 −0.681931
\(687\) 43.8920 1.67458
\(688\) 1.00000 0.0381246
\(689\) −58.3652 −2.22354
\(690\) 29.0039 1.10416
\(691\) 26.1079 0.993191 0.496596 0.867982i \(-0.334583\pi\)
0.496596 + 0.867982i \(0.334583\pi\)
\(692\) −5.80673 −0.220739
\(693\) 20.8810 0.793204
\(694\) −1.30007 −0.0493499
\(695\) 4.51146 0.171129
\(696\) −20.7509 −0.786559
\(697\) 19.9612 0.756086
\(698\) 26.2403 0.993211
\(699\) 56.0573 2.12028
\(700\) 2.73152 0.103242
\(701\) 18.2442 0.689074 0.344537 0.938773i \(-0.388036\pi\)
0.344537 + 0.938773i \(0.388036\pi\)
\(702\) −92.6985 −3.49868
\(703\) 4.72889 0.178353
\(704\) 1.00000 0.0376889
\(705\) 2.78095 0.104737
\(706\) 9.93627 0.373956
\(707\) 9.85925 0.370795
\(708\) 3.68203 0.138379
\(709\) 9.38743 0.352552 0.176276 0.984341i \(-0.443595\pi\)
0.176276 + 0.984341i \(0.443595\pi\)
\(710\) 4.13401 0.155147
\(711\) −19.3012 −0.723851
\(712\) 1.91784 0.0718740
\(713\) 53.4625 2.00219
\(714\) −66.6834 −2.49556
\(715\) −6.11752 −0.228782
\(716\) 9.00912 0.336687
\(717\) −58.3282 −2.17831
\(718\) −28.8363 −1.07616
\(719\) −34.7065 −1.29433 −0.647166 0.762349i \(-0.724046\pi\)
−0.647166 + 0.762349i \(0.724046\pi\)
\(720\) −7.64446 −0.284892
\(721\) −5.18509 −0.193103
\(722\) 41.5328 1.54569
\(723\) 78.2331 2.90952
\(724\) −20.2234 −0.751597
\(725\) 6.36025 0.236214
\(726\) −3.26258 −0.121086
\(727\) −2.61803 −0.0970975 −0.0485487 0.998821i \(-0.515460\pi\)
−0.0485487 + 0.998821i \(0.515460\pi\)
\(728\) 16.7101 0.619319
\(729\) 54.2983 2.01105
\(730\) 13.2485 0.490349
\(731\) 7.48258 0.276753
\(732\) −32.4903 −1.20087
\(733\) −25.9663 −0.959087 −0.479544 0.877518i \(-0.659198\pi\)
−0.479544 + 0.877518i \(0.659198\pi\)
\(734\) −36.6213 −1.35172
\(735\) 1.50472 0.0555025
\(736\) 8.88984 0.327684
\(737\) −11.0418 −0.406731
\(738\) 20.3931 0.750679
\(739\) 1.38279 0.0508668 0.0254334 0.999677i \(-0.491903\pi\)
0.0254334 + 0.999677i \(0.491903\pi\)
\(740\) 0.607804 0.0223433
\(741\) 155.286 5.70458
\(742\) −26.0605 −0.956711
\(743\) 1.68351 0.0617621 0.0308811 0.999523i \(-0.490169\pi\)
0.0308811 + 0.999523i \(0.490169\pi\)
\(744\) −19.6208 −0.719335
\(745\) −6.08570 −0.222963
\(746\) 13.8817 0.508244
\(747\) −97.7995 −3.57830
\(748\) 7.48258 0.273590
\(749\) 9.71983 0.355155
\(750\) 3.26258 0.119133
\(751\) 7.26875 0.265241 0.132620 0.991167i \(-0.457661\pi\)
0.132620 + 0.991167i \(0.457661\pi\)
\(752\) 0.852378 0.0310830
\(753\) −39.5232 −1.44030
\(754\) 38.9090 1.41698
\(755\) −8.36366 −0.304385
\(756\) −41.3906 −1.50536
\(757\) 38.3526 1.39395 0.696975 0.717096i \(-0.254529\pi\)
0.696975 + 0.717096i \(0.254529\pi\)
\(758\) −7.18540 −0.260985
\(759\) −29.0039 −1.05277
\(760\) 7.78028 0.282220
\(761\) −17.3124 −0.627574 −0.313787 0.949493i \(-0.601598\pi\)
−0.313787 + 0.949493i \(0.601598\pi\)
\(762\) 5.64110 0.204356
\(763\) 17.1771 0.621852
\(764\) 16.1486 0.584237
\(765\) −57.2003 −2.06808
\(766\) −28.7845 −1.04003
\(767\) −6.90401 −0.249289
\(768\) −3.26258 −0.117728
\(769\) 5.07471 0.182999 0.0914994 0.995805i \(-0.470834\pi\)
0.0914994 + 0.995805i \(0.470834\pi\)
\(770\) −2.73152 −0.0984372
\(771\) −41.2309 −1.48489
\(772\) 11.6496 0.419279
\(773\) −22.6272 −0.813845 −0.406922 0.913463i \(-0.633398\pi\)
−0.406922 + 0.913463i \(0.633398\pi\)
\(774\) 7.64446 0.274774
\(775\) 6.01389 0.216025
\(776\) 12.9532 0.464994
\(777\) 5.41664 0.194321
\(778\) 0.309694 0.0111031
\(779\) −20.7554 −0.743640
\(780\) 19.9589 0.714645
\(781\) −4.13401 −0.147926
\(782\) 66.5190 2.37871
\(783\) −96.3765 −3.44421
\(784\) 0.461206 0.0164716
\(785\) 6.68607 0.238636
\(786\) 57.0005 2.03314
\(787\) −9.94530 −0.354512 −0.177256 0.984165i \(-0.556722\pi\)
−0.177256 + 0.984165i \(0.556722\pi\)
\(788\) 15.2043 0.541630
\(789\) 100.752 3.58685
\(790\) 2.52486 0.0898305
\(791\) −11.5902 −0.412100
\(792\) 7.64446 0.271634
\(793\) 60.9210 2.16337
\(794\) −29.8800 −1.06040
\(795\) −31.1272 −1.10397
\(796\) −2.21588 −0.0785396
\(797\) −7.89423 −0.279628 −0.139814 0.990178i \(-0.544650\pi\)
−0.139814 + 0.990178i \(0.544650\pi\)
\(798\) 69.3364 2.45448
\(799\) 6.37799 0.225637
\(800\) 1.00000 0.0353553
\(801\) 14.6608 0.518015
\(802\) −7.23252 −0.255389
\(803\) −13.2485 −0.467529
\(804\) 36.0249 1.27050
\(805\) −24.2828 −0.855856
\(806\) 36.7901 1.29588
\(807\) 58.9429 2.07489
\(808\) 3.60944 0.126980
\(809\) 15.9642 0.561271 0.280635 0.959814i \(-0.409455\pi\)
0.280635 + 0.959814i \(0.409455\pi\)
\(810\) −26.5044 −0.931269
\(811\) −9.47545 −0.332728 −0.166364 0.986064i \(-0.553203\pi\)
−0.166364 + 0.986064i \(0.553203\pi\)
\(812\) 17.3732 0.609678
\(813\) −69.4783 −2.43671
\(814\) −0.607804 −0.0213035
\(815\) −22.5617 −0.790303
\(816\) −24.4126 −0.854611
\(817\) −7.78028 −0.272198
\(818\) 12.8063 0.447761
\(819\) 127.740 4.46360
\(820\) −2.66769 −0.0931599
\(821\) 38.8017 1.35419 0.677094 0.735896i \(-0.263239\pi\)
0.677094 + 0.735896i \(0.263239\pi\)
\(822\) 2.47752 0.0864136
\(823\) −4.72647 −0.164754 −0.0823772 0.996601i \(-0.526251\pi\)
−0.0823772 + 0.996601i \(0.526251\pi\)
\(824\) −1.89824 −0.0661284
\(825\) −3.26258 −0.113589
\(826\) −3.08269 −0.107261
\(827\) −49.4963 −1.72116 −0.860578 0.509319i \(-0.829897\pi\)
−0.860578 + 0.509319i \(0.829897\pi\)
\(828\) 67.9580 2.36170
\(829\) 18.0650 0.627423 0.313712 0.949518i \(-0.398428\pi\)
0.313712 + 0.949518i \(0.398428\pi\)
\(830\) 12.7935 0.444069
\(831\) −26.2462 −0.910471
\(832\) 6.11752 0.212087
\(833\) 3.45101 0.119570
\(834\) 14.7190 0.509678
\(835\) 14.7906 0.511851
\(836\) −7.78028 −0.269087
\(837\) −91.1282 −3.14985
\(838\) 2.39313 0.0826693
\(839\) 13.5439 0.467587 0.233793 0.972286i \(-0.424886\pi\)
0.233793 + 0.972286i \(0.424886\pi\)
\(840\) 8.91182 0.307487
\(841\) 11.4528 0.394923
\(842\) 6.58937 0.227085
\(843\) 95.1558 3.27734
\(844\) 12.3897 0.426470
\(845\) −24.4241 −0.840214
\(846\) 6.51597 0.224023
\(847\) 2.73152 0.0938562
\(848\) −9.54066 −0.327627
\(849\) 16.7240 0.573968
\(850\) 7.48258 0.256650
\(851\) −5.40328 −0.185222
\(852\) 13.4875 0.462076
\(853\) 13.4081 0.459085 0.229543 0.973299i \(-0.426277\pi\)
0.229543 + 0.973299i \(0.426277\pi\)
\(854\) 27.2017 0.930822
\(855\) 59.4760 2.03404
\(856\) 3.55840 0.121623
\(857\) −2.10280 −0.0718302 −0.0359151 0.999355i \(-0.511435\pi\)
−0.0359151 + 0.999355i \(0.511435\pi\)
\(858\) −19.9589 −0.681387
\(859\) −7.01321 −0.239288 −0.119644 0.992817i \(-0.538175\pi\)
−0.119644 + 0.992817i \(0.538175\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −23.7740 −0.810216
\(862\) −38.2689 −1.30344
\(863\) 32.2177 1.09670 0.548352 0.836247i \(-0.315255\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(864\) −15.1529 −0.515513
\(865\) 5.80673 0.197435
\(866\) −26.5851 −0.903398
\(867\) −127.205 −4.32011
\(868\) 16.4271 0.557571
\(869\) −2.52486 −0.0856500
\(870\) 20.7509 0.703520
\(871\) −67.5487 −2.28880
\(872\) 6.28847 0.212955
\(873\) 99.0205 3.35134
\(874\) −69.1655 −2.33956
\(875\) −2.73152 −0.0923423
\(876\) 43.2243 1.46041
\(877\) −30.7262 −1.03755 −0.518776 0.854910i \(-0.673612\pi\)
−0.518776 + 0.854910i \(0.673612\pi\)
\(878\) 5.69352 0.192147
\(879\) 56.8648 1.91800
\(880\) −1.00000 −0.0337100
\(881\) 41.3315 1.39249 0.696247 0.717803i \(-0.254852\pi\)
0.696247 + 0.717803i \(0.254852\pi\)
\(882\) 3.52567 0.118715
\(883\) 5.38708 0.181290 0.0906448 0.995883i \(-0.471107\pi\)
0.0906448 + 0.995883i \(0.471107\pi\)
\(884\) 45.7749 1.53958
\(885\) −3.68203 −0.123770
\(886\) −21.1563 −0.710759
\(887\) 40.0017 1.34313 0.671563 0.740947i \(-0.265623\pi\)
0.671563 + 0.740947i \(0.265623\pi\)
\(888\) 1.98301 0.0665455
\(889\) −4.72288 −0.158400
\(890\) −1.91784 −0.0642861
\(891\) 26.5044 0.887930
\(892\) −3.05848 −0.102405
\(893\) −6.63174 −0.221923
\(894\) −19.8551 −0.664054
\(895\) −9.00912 −0.301142
\(896\) 2.73152 0.0912537
\(897\) −177.432 −5.92427
\(898\) −8.72132 −0.291034
\(899\) 38.2499 1.27570
\(900\) 7.64446 0.254815
\(901\) −71.3887 −2.37830
\(902\) 2.66769 0.0888245
\(903\) −8.91182 −0.296567
\(904\) −4.24313 −0.141124
\(905\) 20.2234 0.672249
\(906\) −27.2871 −0.906555
\(907\) −22.5610 −0.749125 −0.374562 0.927202i \(-0.622207\pi\)
−0.374562 + 0.927202i \(0.622207\pi\)
\(908\) 14.6877 0.487430
\(909\) 27.5922 0.915175
\(910\) −16.7101 −0.553936
\(911\) 11.6888 0.387266 0.193633 0.981074i \(-0.437973\pi\)
0.193633 + 0.981074i \(0.437973\pi\)
\(912\) 25.3838 0.840542
\(913\) −12.7935 −0.423404
\(914\) 0.125694 0.00415760
\(915\) 32.4903 1.07409
\(916\) −13.4531 −0.444504
\(917\) −47.7223 −1.57593
\(918\) −113.383 −3.74220
\(919\) 36.6817 1.21002 0.605009 0.796218i \(-0.293169\pi\)
0.605009 + 0.796218i \(0.293169\pi\)
\(920\) −8.88984 −0.293089
\(921\) −37.9518 −1.25055
\(922\) 3.91909 0.129068
\(923\) −25.2899 −0.832427
\(924\) −8.91182 −0.293177
\(925\) −0.607804 −0.0199845
\(926\) −18.4445 −0.606123
\(927\) −14.5110 −0.476605
\(928\) 6.36025 0.208785
\(929\) −17.6418 −0.578808 −0.289404 0.957207i \(-0.593457\pi\)
−0.289404 + 0.957207i \(0.593457\pi\)
\(930\) 19.6208 0.643393
\(931\) −3.58831 −0.117602
\(932\) −17.1819 −0.562811
\(933\) 20.0425 0.656161
\(934\) −19.7612 −0.646607
\(935\) −7.48258 −0.244707
\(936\) 46.7652 1.52857
\(937\) 6.40942 0.209387 0.104693 0.994505i \(-0.466614\pi\)
0.104693 + 0.994505i \(0.466614\pi\)
\(938\) −30.1610 −0.984792
\(939\) 80.5788 2.62959
\(940\) −0.852378 −0.0278015
\(941\) −6.93849 −0.226188 −0.113094 0.993584i \(-0.536076\pi\)
−0.113094 + 0.993584i \(0.536076\pi\)
\(942\) 21.8139 0.710734
\(943\) 23.7154 0.772279
\(944\) −1.12856 −0.0367316
\(945\) 41.3906 1.34644
\(946\) 1.00000 0.0325128
\(947\) −10.1517 −0.329887 −0.164944 0.986303i \(-0.552744\pi\)
−0.164944 + 0.986303i \(0.552744\pi\)
\(948\) 8.23757 0.267544
\(949\) −81.0479 −2.63093
\(950\) −7.78028 −0.252426
\(951\) −28.2679 −0.916649
\(952\) 20.4388 0.662426
\(953\) −16.4986 −0.534443 −0.267221 0.963635i \(-0.586105\pi\)
−0.267221 + 0.963635i \(0.586105\pi\)
\(954\) −72.9331 −2.36130
\(955\) −16.1486 −0.522557
\(956\) 17.8779 0.578213
\(957\) −20.7509 −0.670780
\(958\) 39.0414 1.26137
\(959\) −2.07425 −0.0669810
\(960\) 3.26258 0.105299
\(961\) 5.16691 0.166674
\(962\) −3.71825 −0.119881
\(963\) 27.2020 0.876573
\(964\) −23.9789 −0.772308
\(965\) −11.6496 −0.375015
\(966\) −79.2246 −2.54901
\(967\) 48.6597 1.56479 0.782396 0.622781i \(-0.213997\pi\)
0.782396 + 0.622781i \(0.213997\pi\)
\(968\) 1.00000 0.0321412
\(969\) 189.937 6.10164
\(970\) −12.9532 −0.415903
\(971\) 2.38795 0.0766330 0.0383165 0.999266i \(-0.487800\pi\)
0.0383165 + 0.999266i \(0.487800\pi\)
\(972\) −41.0139 −1.31552
\(973\) −12.3231 −0.395062
\(974\) 23.8887 0.765444
\(975\) −19.9589 −0.639198
\(976\) 9.95844 0.318762
\(977\) −38.4820 −1.23115 −0.615575 0.788079i \(-0.711076\pi\)
−0.615575 + 0.788079i \(0.711076\pi\)
\(978\) −73.6095 −2.35377
\(979\) 1.91784 0.0612944
\(980\) −0.461206 −0.0147327
\(981\) 48.0720 1.53482
\(982\) −23.3319 −0.744551
\(983\) −49.1853 −1.56877 −0.784384 0.620275i \(-0.787021\pi\)
−0.784384 + 0.620275i \(0.787021\pi\)
\(984\) −8.70358 −0.277460
\(985\) −15.2043 −0.484448
\(986\) 47.5911 1.51561
\(987\) −7.59623 −0.241791
\(988\) −47.5960 −1.51423
\(989\) 8.88984 0.282680
\(990\) −7.64446 −0.242957
\(991\) −19.9159 −0.632649 −0.316325 0.948651i \(-0.602449\pi\)
−0.316325 + 0.948651i \(0.602449\pi\)
\(992\) 6.01389 0.190941
\(993\) −117.172 −3.71833
\(994\) −11.2921 −0.358165
\(995\) 2.21588 0.0702480
\(996\) 41.7399 1.32258
\(997\) 1.91558 0.0606670 0.0303335 0.999540i \(-0.490343\pi\)
0.0303335 + 0.999540i \(0.490343\pi\)
\(998\) −28.8772 −0.914093
\(999\) 9.21002 0.291392
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 4730.2.a.bf.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.2 13 1.1 even 1 trivial