Properties

Label 2001.2.a.i.1.6
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.21072\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.21072 q^{2} +1.00000 q^{3} -0.534159 q^{4} +3.03795 q^{5} +1.21072 q^{6} -3.40847 q^{7} -3.06816 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.21072 q^{2} +1.00000 q^{3} -0.534159 q^{4} +3.03795 q^{5} +1.21072 q^{6} -3.40847 q^{7} -3.06816 q^{8} +1.00000 q^{9} +3.67810 q^{10} -3.85707 q^{11} -0.534159 q^{12} -5.15691 q^{13} -4.12670 q^{14} +3.03795 q^{15} -2.64635 q^{16} -2.07488 q^{17} +1.21072 q^{18} -0.0953170 q^{19} -1.62275 q^{20} -3.40847 q^{21} -4.66983 q^{22} +1.00000 q^{23} -3.06816 q^{24} +4.22914 q^{25} -6.24357 q^{26} +1.00000 q^{27} +1.82067 q^{28} -1.00000 q^{29} +3.67810 q^{30} -3.70527 q^{31} +2.93232 q^{32} -3.85707 q^{33} -2.51210 q^{34} -10.3548 q^{35} -0.534159 q^{36} -8.24097 q^{37} -0.115402 q^{38} -5.15691 q^{39} -9.32090 q^{40} +4.31059 q^{41} -4.12670 q^{42} +2.81904 q^{43} +2.06029 q^{44} +3.03795 q^{45} +1.21072 q^{46} -2.48043 q^{47} -2.64635 q^{48} +4.61770 q^{49} +5.12030 q^{50} -2.07488 q^{51} +2.75461 q^{52} -6.40448 q^{53} +1.21072 q^{54} -11.7176 q^{55} +10.4577 q^{56} -0.0953170 q^{57} -1.21072 q^{58} -10.8564 q^{59} -1.62275 q^{60} +5.67822 q^{61} -4.48604 q^{62} -3.40847 q^{63} +8.84292 q^{64} -15.6664 q^{65} -4.66983 q^{66} +12.8177 q^{67} +1.10832 q^{68} +1.00000 q^{69} -12.5367 q^{70} -2.25971 q^{71} -3.06816 q^{72} +14.0533 q^{73} -9.97750 q^{74} +4.22914 q^{75} +0.0509145 q^{76} +13.1467 q^{77} -6.24357 q^{78} +0.679730 q^{79} -8.03950 q^{80} +1.00000 q^{81} +5.21891 q^{82} +4.99221 q^{83} +1.82067 q^{84} -6.30339 q^{85} +3.41307 q^{86} -1.00000 q^{87} +11.8341 q^{88} +8.87404 q^{89} +3.67810 q^{90} +17.5772 q^{91} -0.534159 q^{92} -3.70527 q^{93} -3.00310 q^{94} -0.289568 q^{95} +2.93232 q^{96} +12.0257 q^{97} +5.59073 q^{98} -3.85707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + O(q^{10}) \) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + 3q^{10} - 4q^{11} + 5q^{12} - 18q^{13} - 2q^{14} - 3q^{15} - 7q^{16} - 3q^{17} - 3q^{18} - 4q^{19} - 2q^{20} - 5q^{21} - 26q^{22} + 7q^{23} - 6q^{24} - 8q^{25} - 7q^{26} + 7q^{27} - 6q^{28} - 7q^{29} + 3q^{30} - 22q^{31} + 5q^{32} - 4q^{33} + 9q^{34} + 3q^{35} + 5q^{36} - 25q^{37} + 14q^{38} - 18q^{39} - 10q^{40} - 13q^{41} - 2q^{42} - 2q^{43} + 4q^{44} - 3q^{45} - 3q^{46} - 25q^{47} - 7q^{48} - 8q^{49} + 19q^{50} - 3q^{51} - 12q^{52} - 5q^{53} - 3q^{54} - 15q^{55} + 18q^{56} - 4q^{57} + 3q^{58} + 11q^{59} - 2q^{60} - 33q^{61} + 28q^{62} - 5q^{63} - 14q^{64} - 2q^{65} - 26q^{66} + 8q^{67} + 12q^{68} + 7q^{69} - 22q^{70} - 6q^{71} - 6q^{72} + 15q^{73} + 34q^{74} - 8q^{75} - 28q^{76} - q^{77} - 7q^{78} - 15q^{79} - 12q^{80} + 7q^{81} - 14q^{82} + 21q^{83} - 6q^{84} - 28q^{85} - 12q^{86} - 7q^{87} - 13q^{88} + 8q^{89} + 3q^{90} + 6q^{91} + 5q^{92} - 22q^{93} - 35q^{94} - 25q^{95} + 5q^{96} + 13q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21072 0.856108 0.428054 0.903753i \(-0.359199\pi\)
0.428054 + 0.903753i \(0.359199\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.534159 −0.267080
\(5\) 3.03795 1.35861 0.679306 0.733855i \(-0.262281\pi\)
0.679306 + 0.733855i \(0.262281\pi\)
\(6\) 1.21072 0.494274
\(7\) −3.40847 −1.28828 −0.644141 0.764907i \(-0.722785\pi\)
−0.644141 + 0.764907i \(0.722785\pi\)
\(8\) −3.06816 −1.08476
\(9\) 1.00000 0.333333
\(10\) 3.67810 1.16312
\(11\) −3.85707 −1.16295 −0.581476 0.813564i \(-0.697524\pi\)
−0.581476 + 0.813564i \(0.697524\pi\)
\(12\) −0.534159 −0.154199
\(13\) −5.15691 −1.43027 −0.715135 0.698987i \(-0.753635\pi\)
−0.715135 + 0.698987i \(0.753635\pi\)
\(14\) −4.12670 −1.10291
\(15\) 3.03795 0.784395
\(16\) −2.64635 −0.661589
\(17\) −2.07488 −0.503233 −0.251616 0.967827i \(-0.580962\pi\)
−0.251616 + 0.967827i \(0.580962\pi\)
\(18\) 1.21072 0.285369
\(19\) −0.0953170 −0.0218672 −0.0109336 0.999940i \(-0.503480\pi\)
−0.0109336 + 0.999940i \(0.503480\pi\)
\(20\) −1.62275 −0.362858
\(21\) −3.40847 −0.743790
\(22\) −4.66983 −0.995612
\(23\) 1.00000 0.208514
\(24\) −3.06816 −0.626285
\(25\) 4.22914 0.845828
\(26\) −6.24357 −1.22446
\(27\) 1.00000 0.192450
\(28\) 1.82067 0.344074
\(29\) −1.00000 −0.185695
\(30\) 3.67810 0.671527
\(31\) −3.70527 −0.665485 −0.332743 0.943018i \(-0.607974\pi\)
−0.332743 + 0.943018i \(0.607974\pi\)
\(32\) 2.93232 0.518365
\(33\) −3.85707 −0.671430
\(34\) −2.51210 −0.430821
\(35\) −10.3548 −1.75028
\(36\) −0.534159 −0.0890266
\(37\) −8.24097 −1.35481 −0.677404 0.735612i \(-0.736895\pi\)
−0.677404 + 0.735612i \(0.736895\pi\)
\(38\) −0.115402 −0.0187207
\(39\) −5.15691 −0.825766
\(40\) −9.32090 −1.47376
\(41\) 4.31059 0.673201 0.336600 0.941648i \(-0.390723\pi\)
0.336600 + 0.941648i \(0.390723\pi\)
\(42\) −4.12670 −0.636764
\(43\) 2.81904 0.429900 0.214950 0.976625i \(-0.431041\pi\)
0.214950 + 0.976625i \(0.431041\pi\)
\(44\) 2.06029 0.310601
\(45\) 3.03795 0.452871
\(46\) 1.21072 0.178511
\(47\) −2.48043 −0.361808 −0.180904 0.983501i \(-0.557902\pi\)
−0.180904 + 0.983501i \(0.557902\pi\)
\(48\) −2.64635 −0.381968
\(49\) 4.61770 0.659671
\(50\) 5.12030 0.724120
\(51\) −2.07488 −0.290542
\(52\) 2.75461 0.381996
\(53\) −6.40448 −0.879723 −0.439861 0.898066i \(-0.644972\pi\)
−0.439861 + 0.898066i \(0.644972\pi\)
\(54\) 1.21072 0.164758
\(55\) −11.7176 −1.58000
\(56\) 10.4577 1.39747
\(57\) −0.0953170 −0.0126250
\(58\) −1.21072 −0.158975
\(59\) −10.8564 −1.41338 −0.706692 0.707521i \(-0.749813\pi\)
−0.706692 + 0.707521i \(0.749813\pi\)
\(60\) −1.62275 −0.209496
\(61\) 5.67822 0.727021 0.363510 0.931590i \(-0.381578\pi\)
0.363510 + 0.931590i \(0.381578\pi\)
\(62\) −4.48604 −0.569727
\(63\) −3.40847 −0.429427
\(64\) 8.84292 1.10537
\(65\) −15.6664 −1.94318
\(66\) −4.66983 −0.574817
\(67\) 12.8177 1.56593 0.782963 0.622068i \(-0.213707\pi\)
0.782963 + 0.622068i \(0.213707\pi\)
\(68\) 1.10832 0.134403
\(69\) 1.00000 0.120386
\(70\) −12.5367 −1.49843
\(71\) −2.25971 −0.268178 −0.134089 0.990969i \(-0.542811\pi\)
−0.134089 + 0.990969i \(0.542811\pi\)
\(72\) −3.06816 −0.361586
\(73\) 14.0533 1.64482 0.822408 0.568898i \(-0.192630\pi\)
0.822408 + 0.568898i \(0.192630\pi\)
\(74\) −9.97750 −1.15986
\(75\) 4.22914 0.488339
\(76\) 0.0509145 0.00584029
\(77\) 13.1467 1.49821
\(78\) −6.24357 −0.706945
\(79\) 0.679730 0.0764756 0.0382378 0.999269i \(-0.487826\pi\)
0.0382378 + 0.999269i \(0.487826\pi\)
\(80\) −8.03950 −0.898843
\(81\) 1.00000 0.111111
\(82\) 5.21891 0.576332
\(83\) 4.99221 0.547967 0.273983 0.961734i \(-0.411659\pi\)
0.273983 + 0.961734i \(0.411659\pi\)
\(84\) 1.82067 0.198651
\(85\) −6.30339 −0.683698
\(86\) 3.41307 0.368041
\(87\) −1.00000 −0.107211
\(88\) 11.8341 1.26152
\(89\) 8.87404 0.940646 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(90\) 3.67810 0.387706
\(91\) 17.5772 1.84259
\(92\) −0.534159 −0.0556900
\(93\) −3.70527 −0.384218
\(94\) −3.00310 −0.309747
\(95\) −0.289568 −0.0297091
\(96\) 2.93232 0.299278
\(97\) 12.0257 1.22102 0.610511 0.792007i \(-0.290964\pi\)
0.610511 + 0.792007i \(0.290964\pi\)
\(98\) 5.59073 0.564749
\(99\) −3.85707 −0.387651
\(100\) −2.25904 −0.225904
\(101\) −12.8874 −1.28234 −0.641172 0.767397i \(-0.721552\pi\)
−0.641172 + 0.767397i \(0.721552\pi\)
\(102\) −2.51210 −0.248735
\(103\) −16.6608 −1.64164 −0.820818 0.571190i \(-0.806482\pi\)
−0.820818 + 0.571190i \(0.806482\pi\)
\(104\) 15.8222 1.55149
\(105\) −10.3548 −1.01052
\(106\) −7.75402 −0.753137
\(107\) −3.34894 −0.323754 −0.161877 0.986811i \(-0.551755\pi\)
−0.161877 + 0.986811i \(0.551755\pi\)
\(108\) −0.534159 −0.0513995
\(109\) 1.99607 0.191189 0.0955944 0.995420i \(-0.469525\pi\)
0.0955944 + 0.995420i \(0.469525\pi\)
\(110\) −14.1867 −1.35265
\(111\) −8.24097 −0.782198
\(112\) 9.02003 0.852313
\(113\) −7.18948 −0.676330 −0.338165 0.941087i \(-0.609806\pi\)
−0.338165 + 0.941087i \(0.609806\pi\)
\(114\) −0.115402 −0.0108084
\(115\) 3.03795 0.283290
\(116\) 0.534159 0.0495955
\(117\) −5.15691 −0.476756
\(118\) −13.1441 −1.21001
\(119\) 7.07218 0.648306
\(120\) −9.32090 −0.850878
\(121\) 3.87702 0.352456
\(122\) 6.87472 0.622408
\(123\) 4.31059 0.388673
\(124\) 1.97920 0.177738
\(125\) −2.34183 −0.209459
\(126\) −4.12670 −0.367636
\(127\) 3.54176 0.314280 0.157140 0.987576i \(-0.449773\pi\)
0.157140 + 0.987576i \(0.449773\pi\)
\(128\) 4.84166 0.427946
\(129\) 2.81904 0.248203
\(130\) −18.9676 −1.66357
\(131\) −4.63906 −0.405316 −0.202658 0.979250i \(-0.564958\pi\)
−0.202658 + 0.979250i \(0.564958\pi\)
\(132\) 2.06029 0.179325
\(133\) 0.324885 0.0281711
\(134\) 15.5186 1.34060
\(135\) 3.03795 0.261465
\(136\) 6.36606 0.545885
\(137\) −16.9239 −1.44591 −0.722954 0.690897i \(-0.757216\pi\)
−0.722954 + 0.690897i \(0.757216\pi\)
\(138\) 1.21072 0.103063
\(139\) 6.59644 0.559503 0.279751 0.960072i \(-0.409748\pi\)
0.279751 + 0.960072i \(0.409748\pi\)
\(140\) 5.53110 0.467463
\(141\) −2.48043 −0.208890
\(142\) −2.73587 −0.229589
\(143\) 19.8906 1.66333
\(144\) −2.64635 −0.220530
\(145\) −3.03795 −0.252288
\(146\) 17.0146 1.40814
\(147\) 4.61770 0.380861
\(148\) 4.40199 0.361841
\(149\) 3.08255 0.252532 0.126266 0.991996i \(-0.459701\pi\)
0.126266 + 0.991996i \(0.459701\pi\)
\(150\) 5.12030 0.418071
\(151\) −11.3917 −0.927046 −0.463523 0.886085i \(-0.653415\pi\)
−0.463523 + 0.886085i \(0.653415\pi\)
\(152\) 0.292447 0.0237206
\(153\) −2.07488 −0.167744
\(154\) 15.9170 1.28263
\(155\) −11.2564 −0.904137
\(156\) 2.75461 0.220545
\(157\) −7.45670 −0.595109 −0.297555 0.954705i \(-0.596171\pi\)
−0.297555 + 0.954705i \(0.596171\pi\)
\(158\) 0.822962 0.0654713
\(159\) −6.40448 −0.507908
\(160\) 8.90824 0.704258
\(161\) −3.40847 −0.268625
\(162\) 1.21072 0.0951231
\(163\) −16.9258 −1.32573 −0.662865 0.748739i \(-0.730660\pi\)
−0.662865 + 0.748739i \(0.730660\pi\)
\(164\) −2.30254 −0.179798
\(165\) −11.7176 −0.912214
\(166\) 6.04417 0.469118
\(167\) 5.32428 0.412005 0.206002 0.978551i \(-0.433955\pi\)
0.206002 + 0.978551i \(0.433955\pi\)
\(168\) 10.4577 0.806831
\(169\) 13.5937 1.04567
\(170\) −7.63163 −0.585319
\(171\) −0.0953170 −0.00728907
\(172\) −1.50582 −0.114818
\(173\) 24.7597 1.88244 0.941222 0.337789i \(-0.109679\pi\)
0.941222 + 0.337789i \(0.109679\pi\)
\(174\) −1.21072 −0.0917844
\(175\) −14.4149 −1.08967
\(176\) 10.2072 0.769396
\(177\) −10.8564 −0.816017
\(178\) 10.7440 0.805294
\(179\) −6.67712 −0.499071 −0.249536 0.968366i \(-0.580278\pi\)
−0.249536 + 0.968366i \(0.580278\pi\)
\(180\) −1.62275 −0.120953
\(181\) 10.7524 0.799219 0.399609 0.916686i \(-0.369146\pi\)
0.399609 + 0.916686i \(0.369146\pi\)
\(182\) 21.2810 1.57746
\(183\) 5.67822 0.419746
\(184\) −3.06816 −0.226187
\(185\) −25.0357 −1.84066
\(186\) −4.48604 −0.328932
\(187\) 8.00297 0.585235
\(188\) 1.32494 0.0966315
\(189\) −3.40847 −0.247930
\(190\) −0.350586 −0.0254342
\(191\) −9.96489 −0.721034 −0.360517 0.932753i \(-0.617400\pi\)
−0.360517 + 0.932753i \(0.617400\pi\)
\(192\) 8.84292 0.638183
\(193\) 5.72373 0.412003 0.206002 0.978552i \(-0.433955\pi\)
0.206002 + 0.978552i \(0.433955\pi\)
\(194\) 14.5597 1.04533
\(195\) −15.6664 −1.12190
\(196\) −2.46659 −0.176185
\(197\) −7.92245 −0.564451 −0.282226 0.959348i \(-0.591073\pi\)
−0.282226 + 0.959348i \(0.591073\pi\)
\(198\) −4.66983 −0.331871
\(199\) −16.2857 −1.15446 −0.577230 0.816581i \(-0.695866\pi\)
−0.577230 + 0.816581i \(0.695866\pi\)
\(200\) −12.9757 −0.917518
\(201\) 12.8177 0.904088
\(202\) −15.6030 −1.09782
\(203\) 3.40847 0.239228
\(204\) 1.10832 0.0775977
\(205\) 13.0953 0.914619
\(206\) −20.1715 −1.40542
\(207\) 1.00000 0.0695048
\(208\) 13.6470 0.946250
\(209\) 0.367645 0.0254305
\(210\) −12.5367 −0.865116
\(211\) −8.49660 −0.584930 −0.292465 0.956276i \(-0.594476\pi\)
−0.292465 + 0.956276i \(0.594476\pi\)
\(212\) 3.42101 0.234956
\(213\) −2.25971 −0.154833
\(214\) −4.05462 −0.277168
\(215\) 8.56411 0.584068
\(216\) −3.06816 −0.208762
\(217\) 12.6293 0.857333
\(218\) 2.41668 0.163678
\(219\) 14.0533 0.949635
\(220\) 6.25907 0.421986
\(221\) 10.7000 0.719758
\(222\) −9.97750 −0.669646
\(223\) −22.8578 −1.53067 −0.765334 0.643633i \(-0.777426\pi\)
−0.765334 + 0.643633i \(0.777426\pi\)
\(224\) −9.99473 −0.667801
\(225\) 4.22914 0.281943
\(226\) −8.70444 −0.579011
\(227\) −7.74967 −0.514363 −0.257182 0.966363i \(-0.582794\pi\)
−0.257182 + 0.966363i \(0.582794\pi\)
\(228\) 0.0509145 0.00337189
\(229\) −22.2253 −1.46869 −0.734344 0.678778i \(-0.762510\pi\)
−0.734344 + 0.678778i \(0.762510\pi\)
\(230\) 3.67810 0.242527
\(231\) 13.1467 0.864992
\(232\) 3.06816 0.201434
\(233\) 10.7454 0.703954 0.351977 0.936009i \(-0.385510\pi\)
0.351977 + 0.936009i \(0.385510\pi\)
\(234\) −6.24357 −0.408155
\(235\) −7.53542 −0.491557
\(236\) 5.79905 0.377486
\(237\) 0.679730 0.0441532
\(238\) 8.56242 0.555019
\(239\) −13.8541 −0.896146 −0.448073 0.893997i \(-0.647890\pi\)
−0.448073 + 0.893997i \(0.647890\pi\)
\(240\) −8.03950 −0.518947
\(241\) 10.4272 0.671676 0.335838 0.941920i \(-0.390980\pi\)
0.335838 + 0.941920i \(0.390980\pi\)
\(242\) 4.69398 0.301741
\(243\) 1.00000 0.0641500
\(244\) −3.03307 −0.194173
\(245\) 14.0283 0.896237
\(246\) 5.21891 0.332746
\(247\) 0.491541 0.0312760
\(248\) 11.3683 0.721890
\(249\) 4.99221 0.316369
\(250\) −2.83529 −0.179320
\(251\) 1.35842 0.0857428 0.0428714 0.999081i \(-0.486349\pi\)
0.0428714 + 0.999081i \(0.486349\pi\)
\(252\) 1.82067 0.114691
\(253\) −3.85707 −0.242492
\(254\) 4.28807 0.269058
\(255\) −6.30339 −0.394733
\(256\) −11.8240 −0.738997
\(257\) 5.57472 0.347741 0.173871 0.984768i \(-0.444373\pi\)
0.173871 + 0.984768i \(0.444373\pi\)
\(258\) 3.41307 0.212488
\(259\) 28.0891 1.74537
\(260\) 8.36837 0.518984
\(261\) −1.00000 −0.0618984
\(262\) −5.61660 −0.346994
\(263\) 31.2075 1.92433 0.962167 0.272460i \(-0.0878374\pi\)
0.962167 + 0.272460i \(0.0878374\pi\)
\(264\) 11.8341 0.728339
\(265\) −19.4565 −1.19520
\(266\) 0.393345 0.0241175
\(267\) 8.87404 0.543082
\(268\) −6.84667 −0.418227
\(269\) 15.0114 0.915259 0.457629 0.889143i \(-0.348699\pi\)
0.457629 + 0.889143i \(0.348699\pi\)
\(270\) 3.67810 0.223842
\(271\) −16.3564 −0.993584 −0.496792 0.867870i \(-0.665489\pi\)
−0.496792 + 0.867870i \(0.665489\pi\)
\(272\) 5.49087 0.332933
\(273\) 17.5772 1.06382
\(274\) −20.4901 −1.23785
\(275\) −16.3121 −0.983658
\(276\) −0.534159 −0.0321526
\(277\) 21.8350 1.31194 0.655969 0.754788i \(-0.272260\pi\)
0.655969 + 0.754788i \(0.272260\pi\)
\(278\) 7.98644 0.478995
\(279\) −3.70527 −0.221828
\(280\) 31.7701 1.89862
\(281\) 24.1432 1.44026 0.720130 0.693839i \(-0.244082\pi\)
0.720130 + 0.693839i \(0.244082\pi\)
\(282\) −3.00310 −0.178832
\(283\) −26.8035 −1.59330 −0.796651 0.604439i \(-0.793397\pi\)
−0.796651 + 0.604439i \(0.793397\pi\)
\(284\) 1.20704 0.0716249
\(285\) −0.289568 −0.0171525
\(286\) 24.0819 1.42399
\(287\) −14.6925 −0.867272
\(288\) 2.93232 0.172788
\(289\) −12.6949 −0.746757
\(290\) −3.67810 −0.215986
\(291\) 12.0257 0.704958
\(292\) −7.50671 −0.439297
\(293\) 13.5767 0.793156 0.396578 0.918001i \(-0.370198\pi\)
0.396578 + 0.918001i \(0.370198\pi\)
\(294\) 5.59073 0.326058
\(295\) −32.9812 −1.92024
\(296\) 25.2846 1.46964
\(297\) −3.85707 −0.223810
\(298\) 3.73210 0.216195
\(299\) −5.15691 −0.298232
\(300\) −2.25904 −0.130426
\(301\) −9.60863 −0.553833
\(302\) −13.7922 −0.793651
\(303\) −12.8874 −0.740362
\(304\) 0.252243 0.0144671
\(305\) 17.2501 0.987740
\(306\) −2.51210 −0.143607
\(307\) −19.6964 −1.12414 −0.562068 0.827091i \(-0.689994\pi\)
−0.562068 + 0.827091i \(0.689994\pi\)
\(308\) −7.02245 −0.400141
\(309\) −16.6608 −0.947799
\(310\) −13.6284 −0.774039
\(311\) −29.6322 −1.68029 −0.840143 0.542364i \(-0.817529\pi\)
−0.840143 + 0.542364i \(0.817529\pi\)
\(312\) 15.8222 0.895755
\(313\) −23.8630 −1.34882 −0.674408 0.738359i \(-0.735601\pi\)
−0.674408 + 0.738359i \(0.735601\pi\)
\(314\) −9.02797 −0.509478
\(315\) −10.3548 −0.583425
\(316\) −0.363084 −0.0204251
\(317\) 27.3388 1.53550 0.767751 0.640748i \(-0.221376\pi\)
0.767751 + 0.640748i \(0.221376\pi\)
\(318\) −7.75402 −0.434824
\(319\) 3.85707 0.215955
\(320\) 26.8644 1.50176
\(321\) −3.34894 −0.186919
\(322\) −4.12670 −0.229972
\(323\) 0.197771 0.0110043
\(324\) −0.534159 −0.0296755
\(325\) −21.8093 −1.20976
\(326\) −20.4924 −1.13497
\(327\) 1.99607 0.110383
\(328\) −13.2255 −0.730259
\(329\) 8.45448 0.466111
\(330\) −14.1867 −0.780953
\(331\) 33.9104 1.86388 0.931942 0.362607i \(-0.118113\pi\)
0.931942 + 0.362607i \(0.118113\pi\)
\(332\) −2.66664 −0.146351
\(333\) −8.24097 −0.451602
\(334\) 6.44620 0.352721
\(335\) 38.9394 2.12749
\(336\) 9.02003 0.492083
\(337\) 7.35469 0.400636 0.200318 0.979731i \(-0.435803\pi\)
0.200318 + 0.979731i \(0.435803\pi\)
\(338\) 16.4582 0.895206
\(339\) −7.18948 −0.390479
\(340\) 3.36701 0.182602
\(341\) 14.2915 0.773927
\(342\) −0.115402 −0.00624023
\(343\) 8.12002 0.438440
\(344\) −8.64926 −0.466337
\(345\) 3.03795 0.163558
\(346\) 29.9770 1.61157
\(347\) 11.9665 0.642394 0.321197 0.947012i \(-0.395915\pi\)
0.321197 + 0.947012i \(0.395915\pi\)
\(348\) 0.534159 0.0286339
\(349\) 21.1613 1.13274 0.566368 0.824152i \(-0.308348\pi\)
0.566368 + 0.824152i \(0.308348\pi\)
\(350\) −17.4524 −0.932871
\(351\) −5.15691 −0.275255
\(352\) −11.3102 −0.602834
\(353\) −29.9515 −1.59416 −0.797080 0.603874i \(-0.793623\pi\)
−0.797080 + 0.603874i \(0.793623\pi\)
\(354\) −13.1441 −0.698599
\(355\) −6.86488 −0.364350
\(356\) −4.74015 −0.251227
\(357\) 7.07218 0.374299
\(358\) −8.08411 −0.427259
\(359\) −33.8970 −1.78901 −0.894507 0.447053i \(-0.852473\pi\)
−0.894507 + 0.447053i \(0.852473\pi\)
\(360\) −9.32090 −0.491255
\(361\) −18.9909 −0.999522
\(362\) 13.0181 0.684217
\(363\) 3.87702 0.203491
\(364\) −9.38902 −0.492118
\(365\) 42.6933 2.23467
\(366\) 6.87472 0.359348
\(367\) −1.12298 −0.0586189 −0.0293094 0.999570i \(-0.509331\pi\)
−0.0293094 + 0.999570i \(0.509331\pi\)
\(368\) −2.64635 −0.137951
\(369\) 4.31059 0.224400
\(370\) −30.3111 −1.57580
\(371\) 21.8295 1.13333
\(372\) 1.97920 0.102617
\(373\) 11.0715 0.573261 0.286630 0.958041i \(-0.407465\pi\)
0.286630 + 0.958041i \(0.407465\pi\)
\(374\) 9.68935 0.501024
\(375\) −2.34183 −0.120931
\(376\) 7.61034 0.392474
\(377\) 5.15691 0.265594
\(378\) −4.12670 −0.212255
\(379\) 13.9430 0.716203 0.358101 0.933683i \(-0.383424\pi\)
0.358101 + 0.933683i \(0.383424\pi\)
\(380\) 0.154676 0.00793469
\(381\) 3.54176 0.181450
\(382\) −12.0647 −0.617283
\(383\) 15.0622 0.769644 0.384822 0.922991i \(-0.374263\pi\)
0.384822 + 0.922991i \(0.374263\pi\)
\(384\) 4.84166 0.247075
\(385\) 39.9391 2.03549
\(386\) 6.92983 0.352719
\(387\) 2.81904 0.143300
\(388\) −6.42363 −0.326110
\(389\) 10.4235 0.528491 0.264245 0.964456i \(-0.414877\pi\)
0.264245 + 0.964456i \(0.414877\pi\)
\(390\) −18.9676 −0.960464
\(391\) −2.07488 −0.104931
\(392\) −14.1678 −0.715582
\(393\) −4.63906 −0.234010
\(394\) −9.59186 −0.483231
\(395\) 2.06499 0.103901
\(396\) 2.06029 0.103534
\(397\) 26.3518 1.32256 0.661280 0.750139i \(-0.270013\pi\)
0.661280 + 0.750139i \(0.270013\pi\)
\(398\) −19.7174 −0.988342
\(399\) 0.324885 0.0162646
\(400\) −11.1918 −0.559591
\(401\) 16.3597 0.816964 0.408482 0.912766i \(-0.366058\pi\)
0.408482 + 0.912766i \(0.366058\pi\)
\(402\) 15.5186 0.773996
\(403\) 19.1077 0.951823
\(404\) 6.88393 0.342488
\(405\) 3.03795 0.150957
\(406\) 4.12670 0.204805
\(407\) 31.7860 1.57558
\(408\) 6.36606 0.315167
\(409\) −16.8579 −0.833568 −0.416784 0.909005i \(-0.636843\pi\)
−0.416784 + 0.909005i \(0.636843\pi\)
\(410\) 15.8548 0.783012
\(411\) −16.9239 −0.834795
\(412\) 8.89951 0.438447
\(413\) 37.0038 1.82084
\(414\) 1.21072 0.0595036
\(415\) 15.1661 0.744474
\(416\) −15.1217 −0.741402
\(417\) 6.59644 0.323029
\(418\) 0.445114 0.0217713
\(419\) −0.578582 −0.0282656 −0.0141328 0.999900i \(-0.504499\pi\)
−0.0141328 + 0.999900i \(0.504499\pi\)
\(420\) 5.53110 0.269890
\(421\) 21.3261 1.03937 0.519686 0.854357i \(-0.326049\pi\)
0.519686 + 0.854357i \(0.326049\pi\)
\(422\) −10.2870 −0.500763
\(423\) −2.48043 −0.120603
\(424\) 19.6499 0.954285
\(425\) −8.77497 −0.425649
\(426\) −2.73587 −0.132553
\(427\) −19.3540 −0.936608
\(428\) 1.78887 0.0864681
\(429\) 19.8906 0.960326
\(430\) 10.3687 0.500025
\(431\) −1.62870 −0.0784518 −0.0392259 0.999230i \(-0.512489\pi\)
−0.0392259 + 0.999230i \(0.512489\pi\)
\(432\) −2.64635 −0.127323
\(433\) 6.74749 0.324264 0.162132 0.986769i \(-0.448163\pi\)
0.162132 + 0.986769i \(0.448163\pi\)
\(434\) 15.2905 0.733969
\(435\) −3.03795 −0.145659
\(436\) −1.06622 −0.0510626
\(437\) −0.0953170 −0.00455963
\(438\) 17.0146 0.812990
\(439\) −7.43797 −0.354995 −0.177498 0.984121i \(-0.556800\pi\)
−0.177498 + 0.984121i \(0.556800\pi\)
\(440\) 35.9514 1.71392
\(441\) 4.61770 0.219890
\(442\) 12.9547 0.616191
\(443\) 2.20485 0.104755 0.0523777 0.998627i \(-0.483320\pi\)
0.0523777 + 0.998627i \(0.483320\pi\)
\(444\) 4.40199 0.208909
\(445\) 26.9589 1.27797
\(446\) −27.6743 −1.31042
\(447\) 3.08255 0.145799
\(448\) −30.1409 −1.42402
\(449\) −37.0431 −1.74817 −0.874087 0.485770i \(-0.838539\pi\)
−0.874087 + 0.485770i \(0.838539\pi\)
\(450\) 5.12030 0.241373
\(451\) −16.6263 −0.782900
\(452\) 3.84033 0.180634
\(453\) −11.3917 −0.535230
\(454\) −9.38267 −0.440351
\(455\) 53.3986 2.50337
\(456\) 0.292447 0.0136951
\(457\) 28.8088 1.34762 0.673810 0.738904i \(-0.264657\pi\)
0.673810 + 0.738904i \(0.264657\pi\)
\(458\) −26.9086 −1.25735
\(459\) −2.07488 −0.0968472
\(460\) −1.62275 −0.0756611
\(461\) −22.4332 −1.04482 −0.522409 0.852695i \(-0.674966\pi\)
−0.522409 + 0.852695i \(0.674966\pi\)
\(462\) 15.9170 0.740526
\(463\) −0.887793 −0.0412593 −0.0206296 0.999787i \(-0.506567\pi\)
−0.0206296 + 0.999787i \(0.506567\pi\)
\(464\) 2.64635 0.122854
\(465\) −11.2564 −0.522004
\(466\) 13.0096 0.602660
\(467\) 21.3965 0.990113 0.495056 0.868861i \(-0.335147\pi\)
0.495056 + 0.868861i \(0.335147\pi\)
\(468\) 2.75461 0.127332
\(469\) −43.6886 −2.01735
\(470\) −9.12328 −0.420826
\(471\) −7.45670 −0.343586
\(472\) 33.3091 1.53318
\(473\) −10.8733 −0.499953
\(474\) 0.822962 0.0377999
\(475\) −0.403109 −0.0184959
\(476\) −3.77767 −0.173149
\(477\) −6.40448 −0.293241
\(478\) −16.7734 −0.767198
\(479\) −27.8301 −1.27159 −0.635795 0.771858i \(-0.719327\pi\)
−0.635795 + 0.771858i \(0.719327\pi\)
\(480\) 8.90824 0.406603
\(481\) 42.4979 1.93774
\(482\) 12.6244 0.575027
\(483\) −3.40847 −0.155091
\(484\) −2.07095 −0.0941339
\(485\) 36.5334 1.65890
\(486\) 1.21072 0.0549193
\(487\) −6.27093 −0.284163 −0.142081 0.989855i \(-0.545379\pi\)
−0.142081 + 0.989855i \(0.545379\pi\)
\(488\) −17.4216 −0.788641
\(489\) −16.9258 −0.765410
\(490\) 16.9844 0.767275
\(491\) −23.6758 −1.06847 −0.534236 0.845335i \(-0.679401\pi\)
−0.534236 + 0.845335i \(0.679401\pi\)
\(492\) −2.30254 −0.103807
\(493\) 2.07488 0.0934480
\(494\) 0.595118 0.0267756
\(495\) −11.7176 −0.526667
\(496\) 9.80545 0.440278
\(497\) 7.70216 0.345489
\(498\) 6.04417 0.270846
\(499\) −16.9094 −0.756969 −0.378484 0.925608i \(-0.623555\pi\)
−0.378484 + 0.925608i \(0.623555\pi\)
\(500\) 1.25091 0.0559424
\(501\) 5.32428 0.237871
\(502\) 1.64467 0.0734050
\(503\) 4.36684 0.194708 0.0973539 0.995250i \(-0.468962\pi\)
0.0973539 + 0.995250i \(0.468962\pi\)
\(504\) 10.4577 0.465824
\(505\) −39.1513 −1.74221
\(506\) −4.66983 −0.207599
\(507\) 13.5937 0.603718
\(508\) −1.89186 −0.0839378
\(509\) −23.8848 −1.05868 −0.529338 0.848411i \(-0.677559\pi\)
−0.529338 + 0.848411i \(0.677559\pi\)
\(510\) −7.63163 −0.337934
\(511\) −47.9004 −2.11899
\(512\) −23.9988 −1.06061
\(513\) −0.0953170 −0.00420835
\(514\) 6.74942 0.297704
\(515\) −50.6146 −2.23035
\(516\) −1.50582 −0.0662900
\(517\) 9.56720 0.420765
\(518\) 34.0080 1.49423
\(519\) 24.7597 1.08683
\(520\) 48.0670 2.10788
\(521\) 11.2081 0.491037 0.245519 0.969392i \(-0.421042\pi\)
0.245519 + 0.969392i \(0.421042\pi\)
\(522\) −1.21072 −0.0529917
\(523\) 2.67524 0.116980 0.0584901 0.998288i \(-0.481371\pi\)
0.0584901 + 0.998288i \(0.481371\pi\)
\(524\) 2.47800 0.108252
\(525\) −14.4149 −0.629119
\(526\) 37.7835 1.64744
\(527\) 7.68799 0.334894
\(528\) 10.2072 0.444211
\(529\) 1.00000 0.0434783
\(530\) −23.5563 −1.02322
\(531\) −10.8564 −0.471128
\(532\) −0.173541 −0.00752394
\(533\) −22.2293 −0.962858
\(534\) 10.7440 0.464937
\(535\) −10.1739 −0.439856
\(536\) −39.3266 −1.69865
\(537\) −6.67712 −0.288139
\(538\) 18.1745 0.783560
\(539\) −17.8108 −0.767165
\(540\) −1.62275 −0.0698320
\(541\) −4.28908 −0.184402 −0.0922010 0.995740i \(-0.529390\pi\)
−0.0922010 + 0.995740i \(0.529390\pi\)
\(542\) −19.8031 −0.850614
\(543\) 10.7524 0.461429
\(544\) −6.08421 −0.260858
\(545\) 6.06396 0.259752
\(546\) 21.2810 0.910744
\(547\) −35.7338 −1.52787 −0.763933 0.645296i \(-0.776734\pi\)
−0.763933 + 0.645296i \(0.776734\pi\)
\(548\) 9.04006 0.386172
\(549\) 5.67822 0.242340
\(550\) −19.7494 −0.842117
\(551\) 0.0953170 0.00406064
\(552\) −3.06816 −0.130589
\(553\) −2.31684 −0.0985221
\(554\) 26.4361 1.12316
\(555\) −25.0357 −1.06270
\(556\) −3.52355 −0.149432
\(557\) −3.81954 −0.161839 −0.0809195 0.996721i \(-0.525786\pi\)
−0.0809195 + 0.996721i \(0.525786\pi\)
\(558\) −4.48604 −0.189909
\(559\) −14.5375 −0.614873
\(560\) 27.4024 1.15796
\(561\) 8.00297 0.337886
\(562\) 29.2306 1.23302
\(563\) −20.4663 −0.862554 −0.431277 0.902220i \(-0.641937\pi\)
−0.431277 + 0.902220i \(0.641937\pi\)
\(564\) 1.32494 0.0557902
\(565\) −21.8413 −0.918870
\(566\) −32.4515 −1.36404
\(567\) −3.40847 −0.143142
\(568\) 6.93314 0.290908
\(569\) 10.7702 0.451510 0.225755 0.974184i \(-0.427515\pi\)
0.225755 + 0.974184i \(0.427515\pi\)
\(570\) −0.350586 −0.0146844
\(571\) −19.2253 −0.804556 −0.402278 0.915518i \(-0.631781\pi\)
−0.402278 + 0.915518i \(0.631781\pi\)
\(572\) −10.6247 −0.444243
\(573\) −9.96489 −0.416289
\(574\) −17.7885 −0.742478
\(575\) 4.22914 0.176367
\(576\) 8.84292 0.368455
\(577\) −16.6266 −0.692174 −0.346087 0.938203i \(-0.612490\pi\)
−0.346087 + 0.938203i \(0.612490\pi\)
\(578\) −15.3699 −0.639304
\(579\) 5.72373 0.237870
\(580\) 1.62275 0.0673810
\(581\) −17.0158 −0.705936
\(582\) 14.5597 0.603520
\(583\) 24.7025 1.02308
\(584\) −43.1177 −1.78423
\(585\) −15.6664 −0.647727
\(586\) 16.4375 0.679027
\(587\) 38.1799 1.57585 0.787926 0.615769i \(-0.211155\pi\)
0.787926 + 0.615769i \(0.211155\pi\)
\(588\) −2.46659 −0.101720
\(589\) 0.353175 0.0145523
\(590\) −39.9310 −1.64393
\(591\) −7.92245 −0.325886
\(592\) 21.8085 0.896325
\(593\) 24.9127 1.02304 0.511522 0.859270i \(-0.329082\pi\)
0.511522 + 0.859270i \(0.329082\pi\)
\(594\) −4.66983 −0.191606
\(595\) 21.4849 0.880796
\(596\) −1.64657 −0.0674462
\(597\) −16.2857 −0.666528
\(598\) −6.24357 −0.255318
\(599\) −41.2275 −1.68451 −0.842255 0.539079i \(-0.818772\pi\)
−0.842255 + 0.539079i \(0.818772\pi\)
\(600\) −12.9757 −0.529729
\(601\) 20.1084 0.820241 0.410120 0.912031i \(-0.365487\pi\)
0.410120 + 0.912031i \(0.365487\pi\)
\(602\) −11.6334 −0.474140
\(603\) 12.8177 0.521975
\(604\) 6.08500 0.247595
\(605\) 11.7782 0.478852
\(606\) −15.6030 −0.633829
\(607\) −15.0972 −0.612776 −0.306388 0.951907i \(-0.599121\pi\)
−0.306388 + 0.951907i \(0.599121\pi\)
\(608\) −0.279500 −0.0113352
\(609\) 3.40847 0.138118
\(610\) 20.8851 0.845612
\(611\) 12.7914 0.517483
\(612\) 1.10832 0.0448011
\(613\) −26.9957 −1.09035 −0.545174 0.838323i \(-0.683536\pi\)
−0.545174 + 0.838323i \(0.683536\pi\)
\(614\) −23.8469 −0.962381
\(615\) 13.0953 0.528055
\(616\) −40.3362 −1.62519
\(617\) 8.06126 0.324534 0.162267 0.986747i \(-0.448119\pi\)
0.162267 + 0.986747i \(0.448119\pi\)
\(618\) −20.1715 −0.811418
\(619\) −38.6945 −1.55526 −0.777631 0.628721i \(-0.783579\pi\)
−0.777631 + 0.628721i \(0.783579\pi\)
\(620\) 6.01272 0.241477
\(621\) 1.00000 0.0401286
\(622\) −35.8762 −1.43851
\(623\) −30.2469 −1.21182
\(624\) 13.6470 0.546318
\(625\) −28.2601 −1.13040
\(626\) −28.8914 −1.15473
\(627\) 0.367645 0.0146823
\(628\) 3.98306 0.158942
\(629\) 17.0990 0.681783
\(630\) −12.5367 −0.499475
\(631\) 18.5647 0.739048 0.369524 0.929221i \(-0.379521\pi\)
0.369524 + 0.929221i \(0.379521\pi\)
\(632\) −2.08552 −0.0829574
\(633\) −8.49660 −0.337710
\(634\) 33.0997 1.31456
\(635\) 10.7597 0.426985
\(636\) 3.42101 0.135652
\(637\) −23.8130 −0.943507
\(638\) 4.66983 0.184880
\(639\) −2.25971 −0.0893927
\(640\) 14.7087 0.581413
\(641\) −3.84832 −0.151999 −0.0759997 0.997108i \(-0.524215\pi\)
−0.0759997 + 0.997108i \(0.524215\pi\)
\(642\) −4.05462 −0.160023
\(643\) 19.0776 0.752348 0.376174 0.926549i \(-0.377239\pi\)
0.376174 + 0.926549i \(0.377239\pi\)
\(644\) 1.82067 0.0717444
\(645\) 8.56411 0.337212
\(646\) 0.239446 0.00942086
\(647\) 35.1122 1.38040 0.690202 0.723617i \(-0.257522\pi\)
0.690202 + 0.723617i \(0.257522\pi\)
\(648\) −3.06816 −0.120529
\(649\) 41.8740 1.64370
\(650\) −26.4049 −1.03569
\(651\) 12.6293 0.494981
\(652\) 9.04106 0.354075
\(653\) −12.2338 −0.478745 −0.239372 0.970928i \(-0.576942\pi\)
−0.239372 + 0.970928i \(0.576942\pi\)
\(654\) 2.41668 0.0944996
\(655\) −14.0932 −0.550668
\(656\) −11.4073 −0.445382
\(657\) 14.0533 0.548272
\(658\) 10.2360 0.399041
\(659\) −2.80788 −0.109380 −0.0546898 0.998503i \(-0.517417\pi\)
−0.0546898 + 0.998503i \(0.517417\pi\)
\(660\) 6.25907 0.243634
\(661\) −37.6385 −1.46397 −0.731984 0.681322i \(-0.761405\pi\)
−0.731984 + 0.681322i \(0.761405\pi\)
\(662\) 41.0560 1.59569
\(663\) 10.7000 0.415553
\(664\) −15.3169 −0.594410
\(665\) 0.986986 0.0382737
\(666\) −9.97750 −0.386620
\(667\) −1.00000 −0.0387202
\(668\) −2.84401 −0.110038
\(669\) −22.8578 −0.883732
\(670\) 47.1447 1.82136
\(671\) −21.9013 −0.845490
\(672\) −9.99473 −0.385555
\(673\) −50.4621 −1.94517 −0.972586 0.232543i \(-0.925295\pi\)
−0.972586 + 0.232543i \(0.925295\pi\)
\(674\) 8.90447 0.342987
\(675\) 4.22914 0.162780
\(676\) −7.26121 −0.279277
\(677\) 18.5792 0.714055 0.357028 0.934094i \(-0.383790\pi\)
0.357028 + 0.934094i \(0.383790\pi\)
\(678\) −8.70444 −0.334292
\(679\) −40.9892 −1.57302
\(680\) 19.3398 0.741646
\(681\) −7.74967 −0.296968
\(682\) 17.3030 0.662565
\(683\) 49.7804 1.90480 0.952398 0.304858i \(-0.0986089\pi\)
0.952398 + 0.304858i \(0.0986089\pi\)
\(684\) 0.0509145 0.00194676
\(685\) −51.4140 −1.96443
\(686\) 9.83107 0.375352
\(687\) −22.2253 −0.847947
\(688\) −7.46019 −0.284417
\(689\) 33.0273 1.25824
\(690\) 3.67810 0.140023
\(691\) 45.4174 1.72776 0.863880 0.503698i \(-0.168027\pi\)
0.863880 + 0.503698i \(0.168027\pi\)
\(692\) −13.2256 −0.502763
\(693\) 13.1467 0.499403
\(694\) 14.4880 0.549959
\(695\) 20.0397 0.760148
\(696\) 3.06816 0.116298
\(697\) −8.94396 −0.338777
\(698\) 25.6203 0.969744
\(699\) 10.7454 0.406428
\(700\) 7.69987 0.291028
\(701\) 16.6480 0.628788 0.314394 0.949293i \(-0.398199\pi\)
0.314394 + 0.949293i \(0.398199\pi\)
\(702\) −6.24357 −0.235648
\(703\) 0.785504 0.0296259
\(704\) −34.1078 −1.28549
\(705\) −7.53542 −0.283800
\(706\) −36.2629 −1.36477
\(707\) 43.9264 1.65202
\(708\) 5.79905 0.217942
\(709\) −8.65867 −0.325183 −0.162592 0.986693i \(-0.551985\pi\)
−0.162592 + 0.986693i \(0.551985\pi\)
\(710\) −8.31145 −0.311923
\(711\) 0.679730 0.0254919
\(712\) −27.2269 −1.02037
\(713\) −3.70527 −0.138763
\(714\) 8.56242 0.320441
\(715\) 60.4266 2.25983
\(716\) 3.56664 0.133292
\(717\) −13.8541 −0.517390
\(718\) −41.0397 −1.53159
\(719\) −19.1222 −0.713138 −0.356569 0.934269i \(-0.616054\pi\)
−0.356569 + 0.934269i \(0.616054\pi\)
\(720\) −8.03950 −0.299614
\(721\) 56.7878 2.11489
\(722\) −22.9927 −0.855698
\(723\) 10.4272 0.387792
\(724\) −5.74349 −0.213455
\(725\) −4.22914 −0.157066
\(726\) 4.69398 0.174210
\(727\) −9.85298 −0.365427 −0.182713 0.983166i \(-0.558488\pi\)
−0.182713 + 0.983166i \(0.558488\pi\)
\(728\) −53.9295 −1.99876
\(729\) 1.00000 0.0370370
\(730\) 51.6896 1.91312
\(731\) −5.84918 −0.216340
\(732\) −3.03307 −0.112106
\(733\) 29.0623 1.07344 0.536721 0.843760i \(-0.319663\pi\)
0.536721 + 0.843760i \(0.319663\pi\)
\(734\) −1.35961 −0.0501841
\(735\) 14.0283 0.517443
\(736\) 2.93232 0.108087
\(737\) −49.4386 −1.82110
\(738\) 5.21891 0.192111
\(739\) −18.5216 −0.681327 −0.340663 0.940185i \(-0.610652\pi\)
−0.340663 + 0.940185i \(0.610652\pi\)
\(740\) 13.3730 0.491602
\(741\) 0.491541 0.0180572
\(742\) 26.4294 0.970253
\(743\) 33.6795 1.23558 0.617790 0.786343i \(-0.288028\pi\)
0.617790 + 0.786343i \(0.288028\pi\)
\(744\) 11.3683 0.416783
\(745\) 9.36462 0.343093
\(746\) 13.4045 0.490773
\(747\) 4.99221 0.182656
\(748\) −4.27486 −0.156304
\(749\) 11.4148 0.417086
\(750\) −2.83529 −0.103530
\(751\) 51.9526 1.89578 0.947889 0.318600i \(-0.103213\pi\)
0.947889 + 0.318600i \(0.103213\pi\)
\(752\) 6.56410 0.239368
\(753\) 1.35842 0.0495036
\(754\) 6.24357 0.227377
\(755\) −34.6075 −1.25950
\(756\) 1.82067 0.0662171
\(757\) 33.8994 1.23210 0.616048 0.787709i \(-0.288733\pi\)
0.616048 + 0.787709i \(0.288733\pi\)
\(758\) 16.8810 0.613147
\(759\) −3.85707 −0.140003
\(760\) 0.888440 0.0322271
\(761\) 47.1104 1.70775 0.853875 0.520478i \(-0.174246\pi\)
0.853875 + 0.520478i \(0.174246\pi\)
\(762\) 4.28807 0.155340
\(763\) −6.80355 −0.246305
\(764\) 5.32284 0.192574
\(765\) −6.30339 −0.227899
\(766\) 18.2361 0.658898
\(767\) 55.9855 2.02152
\(768\) −11.8240 −0.426660
\(769\) −23.6926 −0.854377 −0.427188 0.904163i \(-0.640496\pi\)
−0.427188 + 0.904163i \(0.640496\pi\)
\(770\) 48.3551 1.74260
\(771\) 5.57472 0.200769
\(772\) −3.05738 −0.110038
\(773\) −20.8577 −0.750201 −0.375100 0.926984i \(-0.622392\pi\)
−0.375100 + 0.926984i \(0.622392\pi\)
\(774\) 3.41307 0.122680
\(775\) −15.6701 −0.562887
\(776\) −36.8967 −1.32451
\(777\) 28.0891 1.00769
\(778\) 12.6199 0.452445
\(779\) −0.410872 −0.0147210
\(780\) 8.36837 0.299636
\(781\) 8.71586 0.311878
\(782\) −2.51210 −0.0898325
\(783\) −1.00000 −0.0357371
\(784\) −12.2201 −0.436431
\(785\) −22.6531 −0.808523
\(786\) −5.61660 −0.200337
\(787\) 15.5046 0.552680 0.276340 0.961060i \(-0.410878\pi\)
0.276340 + 0.961060i \(0.410878\pi\)
\(788\) 4.23185 0.150753
\(789\) 31.2075 1.11101
\(790\) 2.50012 0.0889502
\(791\) 24.5052 0.871303
\(792\) 11.8341 0.420506
\(793\) −29.2820 −1.03984
\(794\) 31.9047 1.13225
\(795\) −19.4565 −0.690051
\(796\) 8.69914 0.308333
\(797\) −18.8889 −0.669080 −0.334540 0.942382i \(-0.608581\pi\)
−0.334540 + 0.942382i \(0.608581\pi\)
\(798\) 0.393345 0.0139243
\(799\) 5.14660 0.182074
\(800\) 12.4012 0.438448
\(801\) 8.87404 0.313549
\(802\) 19.8070 0.699409
\(803\) −54.2047 −1.91284
\(804\) −6.84667 −0.241463
\(805\) −10.3548 −0.364958
\(806\) 23.1341 0.814863
\(807\) 15.0114 0.528425
\(808\) 39.5405 1.39103
\(809\) 10.3408 0.363563 0.181781 0.983339i \(-0.441814\pi\)
0.181781 + 0.983339i \(0.441814\pi\)
\(810\) 3.67810 0.129235
\(811\) −31.1199 −1.09277 −0.546383 0.837535i \(-0.683996\pi\)
−0.546383 + 0.837535i \(0.683996\pi\)
\(812\) −1.82067 −0.0638929
\(813\) −16.3564 −0.573646
\(814\) 38.4840 1.34886
\(815\) −51.4197 −1.80115
\(816\) 5.49087 0.192219
\(817\) −0.268703 −0.00940072
\(818\) −20.4102 −0.713624
\(819\) 17.5772 0.614197
\(820\) −6.99500 −0.244276
\(821\) −20.8286 −0.726922 −0.363461 0.931609i \(-0.618405\pi\)
−0.363461 + 0.931609i \(0.618405\pi\)
\(822\) −20.4901 −0.714674
\(823\) 23.2091 0.809018 0.404509 0.914534i \(-0.367443\pi\)
0.404509 + 0.914534i \(0.367443\pi\)
\(824\) 51.1179 1.78077
\(825\) −16.3121 −0.567915
\(826\) 44.8012 1.55883
\(827\) −18.4758 −0.642468 −0.321234 0.947000i \(-0.604098\pi\)
−0.321234 + 0.947000i \(0.604098\pi\)
\(828\) −0.534159 −0.0185633
\(829\) 20.8312 0.723497 0.361749 0.932276i \(-0.382180\pi\)
0.361749 + 0.932276i \(0.382180\pi\)
\(830\) 18.3619 0.637350
\(831\) 21.8350 0.757448
\(832\) −45.6021 −1.58097
\(833\) −9.58117 −0.331968
\(834\) 7.98644 0.276548
\(835\) 16.1749 0.559755
\(836\) −0.196381 −0.00679197
\(837\) −3.70527 −0.128073
\(838\) −0.700501 −0.0241984
\(839\) −7.97701 −0.275397 −0.137698 0.990474i \(-0.543970\pi\)
−0.137698 + 0.990474i \(0.543970\pi\)
\(840\) 31.7701 1.09617
\(841\) 1.00000 0.0344828
\(842\) 25.8200 0.889815
\(843\) 24.1432 0.831534
\(844\) 4.53854 0.156223
\(845\) 41.2970 1.42066
\(846\) −3.00310 −0.103249
\(847\) −13.2147 −0.454063
\(848\) 16.9485 0.582015
\(849\) −26.8035 −0.919894
\(850\) −10.6240 −0.364401
\(851\) −8.24097 −0.282497
\(852\) 1.20704 0.0413527
\(853\) −28.3893 −0.972030 −0.486015 0.873950i \(-0.661550\pi\)
−0.486015 + 0.873950i \(0.661550\pi\)
\(854\) −23.4323 −0.801837
\(855\) −0.289568 −0.00990302
\(856\) 10.2751 0.351194
\(857\) 1.06086 0.0362384 0.0181192 0.999836i \(-0.494232\pi\)
0.0181192 + 0.999836i \(0.494232\pi\)
\(858\) 24.0819 0.822143
\(859\) −34.9191 −1.19142 −0.595711 0.803199i \(-0.703130\pi\)
−0.595711 + 0.803199i \(0.703130\pi\)
\(860\) −4.57460 −0.155993
\(861\) −14.6925 −0.500720
\(862\) −1.97190 −0.0671632
\(863\) −31.8477 −1.08411 −0.542054 0.840343i \(-0.682353\pi\)
−0.542054 + 0.840343i \(0.682353\pi\)
\(864\) 2.93232 0.0997595
\(865\) 75.2187 2.55751
\(866\) 8.16932 0.277605
\(867\) −12.6949 −0.431140
\(868\) −6.74606 −0.228976
\(869\) −2.62177 −0.0889374
\(870\) −3.67810 −0.124699
\(871\) −66.0995 −2.23970
\(872\) −6.12425 −0.207393
\(873\) 12.0257 0.407008
\(874\) −0.115402 −0.00390353
\(875\) 7.98206 0.269843
\(876\) −7.50671 −0.253628
\(877\) −28.7892 −0.972141 −0.486071 0.873919i \(-0.661570\pi\)
−0.486071 + 0.873919i \(0.661570\pi\)
\(878\) −9.00530 −0.303914
\(879\) 13.5767 0.457929
\(880\) 31.0089 1.04531
\(881\) 6.20651 0.209103 0.104551 0.994519i \(-0.466659\pi\)
0.104551 + 0.994519i \(0.466659\pi\)
\(882\) 5.59073 0.188250
\(883\) −38.2460 −1.28708 −0.643540 0.765413i \(-0.722535\pi\)
−0.643540 + 0.765413i \(0.722535\pi\)
\(884\) −5.71549 −0.192233
\(885\) −32.9812 −1.10865
\(886\) 2.66945 0.0896819
\(887\) −13.5713 −0.455681 −0.227841 0.973698i \(-0.573166\pi\)
−0.227841 + 0.973698i \(0.573166\pi\)
\(888\) 25.2846 0.848495
\(889\) −12.0720 −0.404881
\(890\) 32.6396 1.09408
\(891\) −3.85707 −0.129217
\(892\) 12.2097 0.408810
\(893\) 0.236427 0.00791173
\(894\) 3.73210 0.124820
\(895\) −20.2847 −0.678044
\(896\) −16.5027 −0.551316
\(897\) −5.15691 −0.172184
\(898\) −44.8488 −1.49662
\(899\) 3.70527 0.123578
\(900\) −2.25904 −0.0753012
\(901\) 13.2885 0.442705
\(902\) −20.1297 −0.670246
\(903\) −9.60863 −0.319755
\(904\) 22.0584 0.733653
\(905\) 32.6652 1.08583
\(906\) −13.7922 −0.458215
\(907\) −8.32618 −0.276466 −0.138233 0.990400i \(-0.544142\pi\)
−0.138233 + 0.990400i \(0.544142\pi\)
\(908\) 4.13956 0.137376
\(909\) −12.8874 −0.427448
\(910\) 64.6507 2.14315
\(911\) 35.8689 1.18839 0.594194 0.804321i \(-0.297471\pi\)
0.594194 + 0.804321i \(0.297471\pi\)
\(912\) 0.252243 0.00835259
\(913\) −19.2553 −0.637259
\(914\) 34.8794 1.15371
\(915\) 17.2501 0.570272
\(916\) 11.8718 0.392256
\(917\) 15.8121 0.522162
\(918\) −2.51210 −0.0829116
\(919\) 38.8879 1.28279 0.641397 0.767209i \(-0.278355\pi\)
0.641397 + 0.767209i \(0.278355\pi\)
\(920\) −9.32090 −0.307301
\(921\) −19.6964 −0.649020
\(922\) −27.1603 −0.894476
\(923\) 11.6531 0.383567
\(924\) −7.02245 −0.231022
\(925\) −34.8522 −1.14593
\(926\) −1.07487 −0.0353224
\(927\) −16.6608 −0.547212
\(928\) −2.93232 −0.0962580
\(929\) −6.58929 −0.216188 −0.108094 0.994141i \(-0.534475\pi\)
−0.108094 + 0.994141i \(0.534475\pi\)
\(930\) −13.6284 −0.446891
\(931\) −0.440145 −0.0144252
\(932\) −5.73975 −0.188012
\(933\) −29.6322 −0.970114
\(934\) 25.9052 0.847643
\(935\) 24.3126 0.795108
\(936\) 15.8222 0.517165
\(937\) 4.77562 0.156013 0.0780063 0.996953i \(-0.475145\pi\)
0.0780063 + 0.996953i \(0.475145\pi\)
\(938\) −52.8947 −1.72707
\(939\) −23.8630 −0.778739
\(940\) 4.02512 0.131285
\(941\) −48.1961 −1.57115 −0.785573 0.618769i \(-0.787632\pi\)
−0.785573 + 0.618769i \(0.787632\pi\)
\(942\) −9.02797 −0.294147
\(943\) 4.31059 0.140372
\(944\) 28.7299 0.935079
\(945\) −10.3548 −0.336841
\(946\) −13.1645 −0.428014
\(947\) 35.1573 1.14246 0.571229 0.820790i \(-0.306467\pi\)
0.571229 + 0.820790i \(0.306467\pi\)
\(948\) −0.363084 −0.0117924
\(949\) −72.4717 −2.35253
\(950\) −0.488052 −0.0158345
\(951\) 27.3388 0.886523
\(952\) −21.6985 −0.703254
\(953\) −24.9046 −0.806740 −0.403370 0.915037i \(-0.632161\pi\)
−0.403370 + 0.915037i \(0.632161\pi\)
\(954\) −7.75402 −0.251046
\(955\) −30.2728 −0.979606
\(956\) 7.40029 0.239342
\(957\) 3.85707 0.124681
\(958\) −33.6944 −1.08862
\(959\) 57.6847 1.86274
\(960\) 26.8644 0.867044
\(961\) −17.2710 −0.557129
\(962\) 51.4531 1.65891
\(963\) −3.34894 −0.107918
\(964\) −5.56980 −0.179391
\(965\) 17.3884 0.559753
\(966\) −4.12670 −0.132775
\(967\) −10.6116 −0.341245 −0.170622 0.985337i \(-0.554578\pi\)
−0.170622 + 0.985337i \(0.554578\pi\)
\(968\) −11.8953 −0.382329
\(969\) 0.197771 0.00635333
\(970\) 44.2317 1.42019
\(971\) 25.5705 0.820595 0.410298 0.911952i \(-0.365425\pi\)
0.410298 + 0.911952i \(0.365425\pi\)
\(972\) −0.534159 −0.0171332
\(973\) −22.4838 −0.720798
\(974\) −7.59233 −0.243274
\(975\) −21.8093 −0.698457
\(976\) −15.0266 −0.480989
\(977\) 48.7768 1.56051 0.780255 0.625462i \(-0.215089\pi\)
0.780255 + 0.625462i \(0.215089\pi\)
\(978\) −20.4924 −0.655273
\(979\) −34.2278 −1.09393
\(980\) −7.49336 −0.239367
\(981\) 1.99607 0.0637296
\(982\) −28.6647 −0.914727
\(983\) −42.2885 −1.34879 −0.674397 0.738369i \(-0.735596\pi\)
−0.674397 + 0.738369i \(0.735596\pi\)
\(984\) −13.2255 −0.421615
\(985\) −24.0680 −0.766871
\(986\) 2.51210 0.0800015
\(987\) 8.45448 0.269109
\(988\) −0.262561 −0.00835319
\(989\) 2.81904 0.0896404
\(990\) −14.1867 −0.450884
\(991\) −2.03129 −0.0645259 −0.0322630 0.999479i \(-0.510271\pi\)
−0.0322630 + 0.999479i \(0.510271\pi\)
\(992\) −10.8650 −0.344965
\(993\) 33.9104 1.07611
\(994\) 9.32515 0.295776
\(995\) −49.4750 −1.56846
\(996\) −2.66664 −0.0844957
\(997\) 4.13490 0.130954 0.0654768 0.997854i \(-0.479143\pi\)
0.0654768 + 0.997854i \(0.479143\pi\)
\(998\) −20.4725 −0.648047
\(999\) −8.24097 −0.260733
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 2001.2.a.i.1.6 7
3.2 odd 2 6003.2.a.j.1.2 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.6 7 1.1 even 1 trivial
6003.2.a.j.1.2 7 3.2 odd 2