Properties

Label 3549.2.a.bg.1.1
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.80363\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.80363 q^{2} +1.00000 q^{3} +5.86034 q^{4} +3.14248 q^{5} -2.80363 q^{6} +1.00000 q^{7} -10.8230 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.80363 q^{2} +1.00000 q^{3} +5.86034 q^{4} +3.14248 q^{5} -2.80363 q^{6} +1.00000 q^{7} -10.8230 q^{8} +1.00000 q^{9} -8.81036 q^{10} -3.00418 q^{11} +5.86034 q^{12} -2.80363 q^{14} +3.14248 q^{15} +18.6229 q^{16} +3.41447 q^{17} -2.80363 q^{18} +4.86056 q^{19} +18.4160 q^{20} +1.00000 q^{21} +8.42260 q^{22} +5.37767 q^{23} -10.8230 q^{24} +4.87520 q^{25} +1.00000 q^{27} +5.86034 q^{28} +0.848910 q^{29} -8.81036 q^{30} -0.451052 q^{31} -30.5658 q^{32} -3.00418 q^{33} -9.57292 q^{34} +3.14248 q^{35} +5.86034 q^{36} +3.88581 q^{37} -13.6272 q^{38} -34.0110 q^{40} -10.5244 q^{41} -2.80363 q^{42} +7.20767 q^{43} -17.6055 q^{44} +3.14248 q^{45} -15.0770 q^{46} +9.33384 q^{47} +18.6229 q^{48} +1.00000 q^{49} -13.6683 q^{50} +3.41447 q^{51} -13.1147 q^{53} -2.80363 q^{54} -9.44058 q^{55} -10.8230 q^{56} +4.86056 q^{57} -2.38003 q^{58} -1.45785 q^{59} +18.4160 q^{60} +4.95821 q^{61} +1.26458 q^{62} +1.00000 q^{63} +48.4493 q^{64} +8.42260 q^{66} -4.57857 q^{67} +20.0100 q^{68} +5.37767 q^{69} -8.81036 q^{70} +2.36205 q^{71} -10.8230 q^{72} -1.24210 q^{73} -10.8944 q^{74} +4.87520 q^{75} +28.4845 q^{76} -3.00418 q^{77} -7.10832 q^{79} +58.5222 q^{80} +1.00000 q^{81} +29.5065 q^{82} -5.04895 q^{83} +5.86034 q^{84} +10.7299 q^{85} -20.2076 q^{86} +0.848910 q^{87} +32.5141 q^{88} +12.6846 q^{89} -8.81036 q^{90} +31.5150 q^{92} -0.451052 q^{93} -26.1686 q^{94} +15.2742 q^{95} -30.5658 q^{96} +4.29846 q^{97} -2.80363 q^{98} -3.00418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + 21q^{10} - 5q^{11} + 28q^{12} - 2q^{14} + 9q^{15} + 50q^{16} - q^{17} - 2q^{18} - 3q^{19} + 23q^{20} + 15q^{21} + 21q^{22} + 4q^{23} - 9q^{24} + 50q^{25} + 15q^{27} + 28q^{28} + 9q^{29} + 21q^{30} - 7q^{31} - 35q^{32} - 5q^{33} + 2q^{34} + 9q^{35} + 28q^{36} - 17q^{37} - 12q^{38} + 46q^{40} + 22q^{41} - 2q^{42} + 36q^{43} - 29q^{44} + 9q^{45} + q^{46} + 12q^{47} + 50q^{48} + 15q^{49} - 53q^{50} - q^{51} - 5q^{53} - 2q^{54} + 43q^{55} - 9q^{56} - 3q^{57} - 29q^{58} + 29q^{59} + 23q^{60} + 12q^{61} + 14q^{62} + 15q^{63} + 95q^{64} + 21q^{66} + 12q^{67} - 16q^{68} + 4q^{69} + 21q^{70} - 36q^{71} - 9q^{72} + 29q^{73} - 5q^{74} + 50q^{75} - 25q^{76} - 5q^{77} + 35q^{79} + 89q^{80} + 15q^{81} + 51q^{82} + 10q^{83} + 28q^{84} - 23q^{85} - 19q^{86} + 9q^{87} + 73q^{88} - 25q^{89} + 21q^{90} - 31q^{92} - 7q^{93} - 19q^{94} - 7q^{95} - 35q^{96} + 26q^{97} - 2q^{98} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.80363 −1.98247 −0.991233 0.132127i \(-0.957819\pi\)
−0.991233 + 0.132127i \(0.957819\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.86034 2.93017
\(5\) 3.14248 1.40536 0.702681 0.711505i \(-0.251986\pi\)
0.702681 + 0.711505i \(0.251986\pi\)
\(6\) −2.80363 −1.14458
\(7\) 1.00000 0.377964
\(8\) −10.8230 −3.82649
\(9\) 1.00000 0.333333
\(10\) −8.81036 −2.78608
\(11\) −3.00418 −0.905793 −0.452897 0.891563i \(-0.649609\pi\)
−0.452897 + 0.891563i \(0.649609\pi\)
\(12\) 5.86034 1.69173
\(13\) 0 0
\(14\) −2.80363 −0.749302
\(15\) 3.14248 0.811386
\(16\) 18.6229 4.65572
\(17\) 3.41447 0.828132 0.414066 0.910247i \(-0.364108\pi\)
0.414066 + 0.910247i \(0.364108\pi\)
\(18\) −2.80363 −0.660822
\(19\) 4.86056 1.11509 0.557544 0.830147i \(-0.311744\pi\)
0.557544 + 0.830147i \(0.311744\pi\)
\(20\) 18.4160 4.11795
\(21\) 1.00000 0.218218
\(22\) 8.42260 1.79570
\(23\) 5.37767 1.12132 0.560661 0.828045i \(-0.310547\pi\)
0.560661 + 0.828045i \(0.310547\pi\)
\(24\) −10.8230 −2.20923
\(25\) 4.87520 0.975041
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 5.86034 1.10750
\(29\) 0.848910 0.157639 0.0788193 0.996889i \(-0.474885\pi\)
0.0788193 + 0.996889i \(0.474885\pi\)
\(30\) −8.81036 −1.60854
\(31\) −0.451052 −0.0810113 −0.0405057 0.999179i \(-0.512897\pi\)
−0.0405057 + 0.999179i \(0.512897\pi\)
\(32\) −30.5658 −5.40332
\(33\) −3.00418 −0.522960
\(34\) −9.57292 −1.64174
\(35\) 3.14248 0.531177
\(36\) 5.86034 0.976723
\(37\) 3.88581 0.638824 0.319412 0.947616i \(-0.396515\pi\)
0.319412 + 0.947616i \(0.396515\pi\)
\(38\) −13.6272 −2.21063
\(39\) 0 0
\(40\) −34.0110 −5.37761
\(41\) −10.5244 −1.64363 −0.821817 0.569752i \(-0.807039\pi\)
−0.821817 + 0.569752i \(0.807039\pi\)
\(42\) −2.80363 −0.432609
\(43\) 7.20767 1.09916 0.549579 0.835441i \(-0.314788\pi\)
0.549579 + 0.835441i \(0.314788\pi\)
\(44\) −17.6055 −2.65413
\(45\) 3.14248 0.468454
\(46\) −15.0770 −2.22298
\(47\) 9.33384 1.36148 0.680741 0.732525i \(-0.261658\pi\)
0.680741 + 0.732525i \(0.261658\pi\)
\(48\) 18.6229 2.68798
\(49\) 1.00000 0.142857
\(50\) −13.6683 −1.93298
\(51\) 3.41447 0.478122
\(52\) 0 0
\(53\) −13.1147 −1.80144 −0.900722 0.434395i \(-0.856962\pi\)
−0.900722 + 0.434395i \(0.856962\pi\)
\(54\) −2.80363 −0.381526
\(55\) −9.44058 −1.27297
\(56\) −10.8230 −1.44628
\(57\) 4.86056 0.643797
\(58\) −2.38003 −0.312513
\(59\) −1.45785 −0.189797 −0.0948983 0.995487i \(-0.530253\pi\)
−0.0948983 + 0.995487i \(0.530253\pi\)
\(60\) 18.4160 2.37750
\(61\) 4.95821 0.634833 0.317417 0.948286i \(-0.397185\pi\)
0.317417 + 0.948286i \(0.397185\pi\)
\(62\) 1.26458 0.160602
\(63\) 1.00000 0.125988
\(64\) 48.4493 6.05617
\(65\) 0 0
\(66\) 8.42260 1.03675
\(67\) −4.57857 −0.559361 −0.279680 0.960093i \(-0.590229\pi\)
−0.279680 + 0.960093i \(0.590229\pi\)
\(68\) 20.0100 2.42657
\(69\) 5.37767 0.647396
\(70\) −8.81036 −1.05304
\(71\) 2.36205 0.280324 0.140162 0.990129i \(-0.455238\pi\)
0.140162 + 0.990129i \(0.455238\pi\)
\(72\) −10.8230 −1.27550
\(73\) −1.24210 −0.145377 −0.0726887 0.997355i \(-0.523158\pi\)
−0.0726887 + 0.997355i \(0.523158\pi\)
\(74\) −10.8944 −1.26645
\(75\) 4.87520 0.562940
\(76\) 28.4845 3.26740
\(77\) −3.00418 −0.342358
\(78\) 0 0
\(79\) −7.10832 −0.799748 −0.399874 0.916570i \(-0.630946\pi\)
−0.399874 + 0.916570i \(0.630946\pi\)
\(80\) 58.5222 6.54298
\(81\) 1.00000 0.111111
\(82\) 29.5065 3.25845
\(83\) −5.04895 −0.554195 −0.277097 0.960842i \(-0.589372\pi\)
−0.277097 + 0.960842i \(0.589372\pi\)
\(84\) 5.86034 0.639415
\(85\) 10.7299 1.16382
\(86\) −20.2076 −2.17904
\(87\) 0.848910 0.0910127
\(88\) 32.5141 3.46601
\(89\) 12.6846 1.34456 0.672282 0.740295i \(-0.265314\pi\)
0.672282 + 0.740295i \(0.265314\pi\)
\(90\) −8.81036 −0.928694
\(91\) 0 0
\(92\) 31.5150 3.28567
\(93\) −0.451052 −0.0467719
\(94\) −26.1686 −2.69909
\(95\) 15.2742 1.56710
\(96\) −30.5658 −3.11961
\(97\) 4.29846 0.436442 0.218221 0.975899i \(-0.429975\pi\)
0.218221 + 0.975899i \(0.429975\pi\)
\(98\) −2.80363 −0.283209
\(99\) −3.00418 −0.301931
\(100\) 28.5703 2.85703
\(101\) 12.0756 1.20157 0.600784 0.799411i \(-0.294855\pi\)
0.600784 + 0.799411i \(0.294855\pi\)
\(102\) −9.57292 −0.947861
\(103\) −1.57289 −0.154981 −0.0774907 0.996993i \(-0.524691\pi\)
−0.0774907 + 0.996993i \(0.524691\pi\)
\(104\) 0 0
\(105\) 3.14248 0.306675
\(106\) 36.7688 3.57130
\(107\) −12.2770 −1.18687 −0.593433 0.804884i \(-0.702228\pi\)
−0.593433 + 0.804884i \(0.702228\pi\)
\(108\) 5.86034 0.563911
\(109\) 17.7042 1.69576 0.847879 0.530190i \(-0.177880\pi\)
0.847879 + 0.530190i \(0.177880\pi\)
\(110\) 26.4679 2.52361
\(111\) 3.88581 0.368825
\(112\) 18.6229 1.75970
\(113\) −7.67290 −0.721806 −0.360903 0.932603i \(-0.617531\pi\)
−0.360903 + 0.932603i \(0.617531\pi\)
\(114\) −13.6272 −1.27631
\(115\) 16.8993 1.57586
\(116\) 4.97490 0.461908
\(117\) 0 0
\(118\) 4.08728 0.376265
\(119\) 3.41447 0.313004
\(120\) −34.0110 −3.10476
\(121\) −1.97492 −0.179538
\(122\) −13.9010 −1.25854
\(123\) −10.5244 −0.948952
\(124\) −2.64332 −0.237377
\(125\) −0.392171 −0.0350768
\(126\) −2.80363 −0.249767
\(127\) −15.0680 −1.33707 −0.668534 0.743682i \(-0.733078\pi\)
−0.668534 + 0.743682i \(0.733078\pi\)
\(128\) −74.7024 −6.60283
\(129\) 7.20767 0.634600
\(130\) 0 0
\(131\) −7.00728 −0.612229 −0.306114 0.951995i \(-0.599029\pi\)
−0.306114 + 0.951995i \(0.599029\pi\)
\(132\) −17.6055 −1.53236
\(133\) 4.86056 0.421464
\(134\) 12.8366 1.10891
\(135\) 3.14248 0.270462
\(136\) −36.9547 −3.16884
\(137\) 12.3949 1.05897 0.529486 0.848319i \(-0.322385\pi\)
0.529486 + 0.848319i \(0.322385\pi\)
\(138\) −15.0770 −1.28344
\(139\) −2.68693 −0.227903 −0.113951 0.993486i \(-0.536351\pi\)
−0.113951 + 0.993486i \(0.536351\pi\)
\(140\) 18.4160 1.55644
\(141\) 9.33384 0.786051
\(142\) −6.62231 −0.555732
\(143\) 0 0
\(144\) 18.6229 1.55191
\(145\) 2.66769 0.221539
\(146\) 3.48240 0.288206
\(147\) 1.00000 0.0824786
\(148\) 22.7722 1.87186
\(149\) −2.60806 −0.213660 −0.106830 0.994277i \(-0.534070\pi\)
−0.106830 + 0.994277i \(0.534070\pi\)
\(150\) −13.6683 −1.11601
\(151\) 5.13680 0.418027 0.209013 0.977913i \(-0.432975\pi\)
0.209013 + 0.977913i \(0.432975\pi\)
\(152\) −52.6056 −4.26688
\(153\) 3.41447 0.276044
\(154\) 8.42260 0.678712
\(155\) −1.41742 −0.113850
\(156\) 0 0
\(157\) 10.8870 0.868874 0.434437 0.900702i \(-0.356947\pi\)
0.434437 + 0.900702i \(0.356947\pi\)
\(158\) 19.9291 1.58547
\(159\) −13.1147 −1.04006
\(160\) −96.0525 −7.59362
\(161\) 5.37767 0.423820
\(162\) −2.80363 −0.220274
\(163\) 2.53323 0.198418 0.0992088 0.995067i \(-0.468369\pi\)
0.0992088 + 0.995067i \(0.468369\pi\)
\(164\) −61.6765 −4.81612
\(165\) −9.44058 −0.734948
\(166\) 14.1554 1.09867
\(167\) −1.10194 −0.0852703 −0.0426352 0.999091i \(-0.513575\pi\)
−0.0426352 + 0.999091i \(0.513575\pi\)
\(168\) −10.8230 −0.835010
\(169\) 0 0
\(170\) −30.0828 −2.30724
\(171\) 4.86056 0.371696
\(172\) 42.2394 3.22072
\(173\) −22.0711 −1.67803 −0.839016 0.544107i \(-0.816868\pi\)
−0.839016 + 0.544107i \(0.816868\pi\)
\(174\) −2.38003 −0.180430
\(175\) 4.87520 0.368531
\(176\) −55.9465 −4.21712
\(177\) −1.45785 −0.109579
\(178\) −35.5629 −2.66555
\(179\) 7.20834 0.538777 0.269388 0.963032i \(-0.413178\pi\)
0.269388 + 0.963032i \(0.413178\pi\)
\(180\) 18.4160 1.37265
\(181\) 13.0966 0.973466 0.486733 0.873551i \(-0.338188\pi\)
0.486733 + 0.873551i \(0.338188\pi\)
\(182\) 0 0
\(183\) 4.95821 0.366521
\(184\) −58.2024 −4.29073
\(185\) 12.2111 0.897778
\(186\) 1.26458 0.0927237
\(187\) −10.2577 −0.750116
\(188\) 54.6995 3.98937
\(189\) 1.00000 0.0727393
\(190\) −42.8233 −3.10673
\(191\) 24.5517 1.77650 0.888249 0.459363i \(-0.151922\pi\)
0.888249 + 0.459363i \(0.151922\pi\)
\(192\) 48.4493 3.49653
\(193\) −5.78789 −0.416621 −0.208311 0.978063i \(-0.566797\pi\)
−0.208311 + 0.978063i \(0.566797\pi\)
\(194\) −12.0513 −0.865232
\(195\) 0 0
\(196\) 5.86034 0.418596
\(197\) −0.0857381 −0.00610859 −0.00305429 0.999995i \(-0.500972\pi\)
−0.00305429 + 0.999995i \(0.500972\pi\)
\(198\) 8.42260 0.598568
\(199\) 5.62060 0.398434 0.199217 0.979955i \(-0.436160\pi\)
0.199217 + 0.979955i \(0.436160\pi\)
\(200\) −52.7641 −3.73099
\(201\) −4.57857 −0.322947
\(202\) −33.8555 −2.38207
\(203\) 0.848910 0.0595818
\(204\) 20.0100 1.40098
\(205\) −33.0727 −2.30990
\(206\) 4.40980 0.307245
\(207\) 5.37767 0.373774
\(208\) 0 0
\(209\) −14.6020 −1.01004
\(210\) −8.81036 −0.607973
\(211\) 18.1162 1.24717 0.623587 0.781754i \(-0.285675\pi\)
0.623587 + 0.781754i \(0.285675\pi\)
\(212\) −76.8567 −5.27854
\(213\) 2.36205 0.161845
\(214\) 34.4203 2.35292
\(215\) 22.6500 1.54472
\(216\) −10.8230 −0.736409
\(217\) −0.451052 −0.0306194
\(218\) −49.6361 −3.36178
\(219\) −1.24210 −0.0839336
\(220\) −55.3250 −3.73001
\(221\) 0 0
\(222\) −10.8944 −0.731183
\(223\) −5.32513 −0.356597 −0.178298 0.983976i \(-0.557059\pi\)
−0.178298 + 0.983976i \(0.557059\pi\)
\(224\) −30.5658 −2.04226
\(225\) 4.87520 0.325014
\(226\) 21.5120 1.43096
\(227\) 22.2569 1.47725 0.738623 0.674119i \(-0.235477\pi\)
0.738623 + 0.674119i \(0.235477\pi\)
\(228\) 28.4845 1.88643
\(229\) 8.26095 0.545899 0.272949 0.962028i \(-0.412001\pi\)
0.272949 + 0.962028i \(0.412001\pi\)
\(230\) −47.3792 −3.12409
\(231\) −3.00418 −0.197660
\(232\) −9.18772 −0.603203
\(233\) 1.14151 0.0747825 0.0373913 0.999301i \(-0.488095\pi\)
0.0373913 + 0.999301i \(0.488095\pi\)
\(234\) 0 0
\(235\) 29.3315 1.91337
\(236\) −8.54352 −0.556136
\(237\) −7.10832 −0.461735
\(238\) −9.57292 −0.620520
\(239\) 11.3967 0.737190 0.368595 0.929590i \(-0.379839\pi\)
0.368595 + 0.929590i \(0.379839\pi\)
\(240\) 58.5222 3.77759
\(241\) −14.7693 −0.951374 −0.475687 0.879615i \(-0.657800\pi\)
−0.475687 + 0.879615i \(0.657800\pi\)
\(242\) 5.53695 0.355929
\(243\) 1.00000 0.0641500
\(244\) 29.0568 1.86017
\(245\) 3.14248 0.200766
\(246\) 29.5065 1.88126
\(247\) 0 0
\(248\) 4.88172 0.309989
\(249\) −5.04895 −0.319964
\(250\) 1.09950 0.0695386
\(251\) 13.8604 0.874858 0.437429 0.899253i \(-0.355889\pi\)
0.437429 + 0.899253i \(0.355889\pi\)
\(252\) 5.86034 0.369167
\(253\) −16.1555 −1.01569
\(254\) 42.2451 2.65069
\(255\) 10.7299 0.671934
\(256\) 112.539 7.03371
\(257\) −13.1717 −0.821630 −0.410815 0.911719i \(-0.634756\pi\)
−0.410815 + 0.911719i \(0.634756\pi\)
\(258\) −20.2076 −1.25807
\(259\) 3.88581 0.241453
\(260\) 0 0
\(261\) 0.848910 0.0525462
\(262\) 19.6458 1.21372
\(263\) −30.2616 −1.86601 −0.933006 0.359860i \(-0.882824\pi\)
−0.933006 + 0.359860i \(0.882824\pi\)
\(264\) 32.5141 2.00110
\(265\) −41.2128 −2.53168
\(266\) −13.6272 −0.835538
\(267\) 12.6846 0.776284
\(268\) −26.8320 −1.63902
\(269\) −24.4467 −1.49054 −0.745272 0.666760i \(-0.767680\pi\)
−0.745272 + 0.666760i \(0.767680\pi\)
\(270\) −8.81036 −0.536181
\(271\) 19.0695 1.15839 0.579193 0.815190i \(-0.303368\pi\)
0.579193 + 0.815190i \(0.303368\pi\)
\(272\) 63.5874 3.85555
\(273\) 0 0
\(274\) −34.7508 −2.09938
\(275\) −14.6460 −0.883185
\(276\) 31.5150 1.89698
\(277\) 3.66058 0.219943 0.109971 0.993935i \(-0.464924\pi\)
0.109971 + 0.993935i \(0.464924\pi\)
\(278\) 7.53316 0.451809
\(279\) −0.451052 −0.0270038
\(280\) −34.0110 −2.03254
\(281\) 8.89390 0.530565 0.265283 0.964171i \(-0.414535\pi\)
0.265283 + 0.964171i \(0.414535\pi\)
\(282\) −26.1686 −1.55832
\(283\) −12.7960 −0.760646 −0.380323 0.924854i \(-0.624187\pi\)
−0.380323 + 0.924854i \(0.624187\pi\)
\(284\) 13.8424 0.821396
\(285\) 15.2742 0.904767
\(286\) 0 0
\(287\) −10.5244 −0.621235
\(288\) −30.5658 −1.80111
\(289\) −5.34136 −0.314198
\(290\) −7.47920 −0.439194
\(291\) 4.29846 0.251980
\(292\) −7.27915 −0.425980
\(293\) −7.11839 −0.415861 −0.207930 0.978144i \(-0.566673\pi\)
−0.207930 + 0.978144i \(0.566673\pi\)
\(294\) −2.80363 −0.163511
\(295\) −4.58128 −0.266733
\(296\) −42.0560 −2.44446
\(297\) −3.00418 −0.174320
\(298\) 7.31203 0.423574
\(299\) 0 0
\(300\) 28.5703 1.64951
\(301\) 7.20767 0.415443
\(302\) −14.4017 −0.828724
\(303\) 12.0756 0.693726
\(304\) 90.5177 5.19155
\(305\) 15.5811 0.892170
\(306\) −9.57292 −0.547248
\(307\) −26.1182 −1.49065 −0.745323 0.666704i \(-0.767705\pi\)
−0.745323 + 0.666704i \(0.767705\pi\)
\(308\) −17.6055 −1.00317
\(309\) −1.57289 −0.0894786
\(310\) 3.97393 0.225704
\(311\) 23.4754 1.33117 0.665584 0.746323i \(-0.268183\pi\)
0.665584 + 0.746323i \(0.268183\pi\)
\(312\) 0 0
\(313\) −17.0234 −0.962220 −0.481110 0.876660i \(-0.659766\pi\)
−0.481110 + 0.876660i \(0.659766\pi\)
\(314\) −30.5230 −1.72251
\(315\) 3.14248 0.177059
\(316\) −41.6572 −2.34340
\(317\) 9.59342 0.538820 0.269410 0.963026i \(-0.413171\pi\)
0.269410 + 0.963026i \(0.413171\pi\)
\(318\) 36.7688 2.06189
\(319\) −2.55028 −0.142788
\(320\) 152.251 8.51110
\(321\) −12.2770 −0.685237
\(322\) −15.0770 −0.840209
\(323\) 16.5963 0.923440
\(324\) 5.86034 0.325574
\(325\) 0 0
\(326\) −7.10223 −0.393356
\(327\) 17.7042 0.979046
\(328\) 113.905 6.28935
\(329\) 9.33384 0.514591
\(330\) 26.4679 1.45701
\(331\) −1.52112 −0.0836084 −0.0418042 0.999126i \(-0.513311\pi\)
−0.0418042 + 0.999126i \(0.513311\pi\)
\(332\) −29.5886 −1.62388
\(333\) 3.88581 0.212941
\(334\) 3.08942 0.169045
\(335\) −14.3881 −0.786104
\(336\) 18.6229 1.01596
\(337\) 28.2047 1.53641 0.768204 0.640206i \(-0.221151\pi\)
0.768204 + 0.640206i \(0.221151\pi\)
\(338\) 0 0
\(339\) −7.67290 −0.416735
\(340\) 62.8810 3.41020
\(341\) 1.35504 0.0733795
\(342\) −13.6272 −0.736875
\(343\) 1.00000 0.0539949
\(344\) −78.0083 −4.20593
\(345\) 16.8993 0.909825
\(346\) 61.8791 3.32664
\(347\) 7.59551 0.407748 0.203874 0.978997i \(-0.434647\pi\)
0.203874 + 0.978997i \(0.434647\pi\)
\(348\) 4.97490 0.266683
\(349\) −33.9613 −1.81791 −0.908954 0.416896i \(-0.863118\pi\)
−0.908954 + 0.416896i \(0.863118\pi\)
\(350\) −13.6683 −0.730600
\(351\) 0 0
\(352\) 91.8250 4.89429
\(353\) −14.4675 −0.770027 −0.385013 0.922911i \(-0.625803\pi\)
−0.385013 + 0.922911i \(0.625803\pi\)
\(354\) 4.08728 0.217237
\(355\) 7.42270 0.393956
\(356\) 74.3360 3.93980
\(357\) 3.41447 0.180713
\(358\) −20.2095 −1.06811
\(359\) 9.99212 0.527364 0.263682 0.964610i \(-0.415063\pi\)
0.263682 + 0.964610i \(0.415063\pi\)
\(360\) −34.0110 −1.79254
\(361\) 4.62504 0.243423
\(362\) −36.7181 −1.92986
\(363\) −1.97492 −0.103657
\(364\) 0 0
\(365\) −3.90329 −0.204308
\(366\) −13.9010 −0.726616
\(367\) −19.5834 −1.02224 −0.511122 0.859508i \(-0.670770\pi\)
−0.511122 + 0.859508i \(0.670770\pi\)
\(368\) 100.148 5.22057
\(369\) −10.5244 −0.547878
\(370\) −34.2354 −1.77981
\(371\) −13.1147 −0.680882
\(372\) −2.64332 −0.137050
\(373\) −3.57778 −0.185250 −0.0926252 0.995701i \(-0.529526\pi\)
−0.0926252 + 0.995701i \(0.529526\pi\)
\(374\) 28.7587 1.48708
\(375\) −0.392171 −0.0202516
\(376\) −101.020 −5.20970
\(377\) 0 0
\(378\) −2.80363 −0.144203
\(379\) 9.43289 0.484535 0.242268 0.970209i \(-0.422109\pi\)
0.242268 + 0.970209i \(0.422109\pi\)
\(380\) 89.5122 4.59188
\(381\) −15.0680 −0.771956
\(382\) −68.8338 −3.52184
\(383\) 18.2451 0.932281 0.466141 0.884711i \(-0.345644\pi\)
0.466141 + 0.884711i \(0.345644\pi\)
\(384\) −74.7024 −3.81214
\(385\) −9.44058 −0.481136
\(386\) 16.2271 0.825938
\(387\) 7.20767 0.366386
\(388\) 25.1904 1.27885
\(389\) 7.06982 0.358454 0.179227 0.983808i \(-0.442640\pi\)
0.179227 + 0.983808i \(0.442640\pi\)
\(390\) 0 0
\(391\) 18.3619 0.928603
\(392\) −10.8230 −0.546642
\(393\) −7.00728 −0.353470
\(394\) 0.240378 0.0121101
\(395\) −22.3378 −1.12394
\(396\) −17.6055 −0.884709
\(397\) 4.76261 0.239029 0.119514 0.992832i \(-0.461866\pi\)
0.119514 + 0.992832i \(0.461866\pi\)
\(398\) −15.7581 −0.789881
\(399\) 4.86056 0.243332
\(400\) 90.7904 4.53952
\(401\) 32.3631 1.61614 0.808068 0.589090i \(-0.200513\pi\)
0.808068 + 0.589090i \(0.200513\pi\)
\(402\) 12.8366 0.640232
\(403\) 0 0
\(404\) 70.7672 3.52080
\(405\) 3.14248 0.156151
\(406\) −2.38003 −0.118119
\(407\) −11.6737 −0.578642
\(408\) −36.9547 −1.82953
\(409\) 2.00033 0.0989101 0.0494550 0.998776i \(-0.484252\pi\)
0.0494550 + 0.998776i \(0.484252\pi\)
\(410\) 92.7236 4.57929
\(411\) 12.3949 0.611398
\(412\) −9.21767 −0.454122
\(413\) −1.45785 −0.0717363
\(414\) −15.0770 −0.740994
\(415\) −15.8662 −0.778844
\(416\) 0 0
\(417\) −2.68693 −0.131580
\(418\) 40.9385 2.00237
\(419\) −32.9805 −1.61120 −0.805601 0.592458i \(-0.798158\pi\)
−0.805601 + 0.592458i \(0.798158\pi\)
\(420\) 18.4160 0.898610
\(421\) −14.2283 −0.693445 −0.346723 0.937968i \(-0.612705\pi\)
−0.346723 + 0.937968i \(0.612705\pi\)
\(422\) −50.7912 −2.47248
\(423\) 9.33384 0.453827
\(424\) 141.940 6.89322
\(425\) 16.6463 0.807462
\(426\) −6.62231 −0.320852
\(427\) 4.95821 0.239945
\(428\) −71.9476 −3.47772
\(429\) 0 0
\(430\) −63.5021 −3.06235
\(431\) 23.9213 1.15225 0.576124 0.817362i \(-0.304564\pi\)
0.576124 + 0.817362i \(0.304564\pi\)
\(432\) 18.6229 0.895995
\(433\) 30.3234 1.45725 0.728624 0.684914i \(-0.240160\pi\)
0.728624 + 0.684914i \(0.240160\pi\)
\(434\) 1.26458 0.0607019
\(435\) 2.66769 0.127906
\(436\) 103.753 4.96886
\(437\) 26.1385 1.25037
\(438\) 3.48240 0.166396
\(439\) −11.7009 −0.558452 −0.279226 0.960225i \(-0.590078\pi\)
−0.279226 + 0.960225i \(0.590078\pi\)
\(440\) 102.175 4.87100
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 5.05620 0.240227 0.120114 0.992760i \(-0.461674\pi\)
0.120114 + 0.992760i \(0.461674\pi\)
\(444\) 22.7722 1.08072
\(445\) 39.8611 1.88960
\(446\) 14.9297 0.706941
\(447\) −2.60806 −0.123357
\(448\) 48.4493 2.28902
\(449\) −17.3705 −0.819763 −0.409882 0.912139i \(-0.634430\pi\)
−0.409882 + 0.912139i \(0.634430\pi\)
\(450\) −13.6683 −0.644328
\(451\) 31.6171 1.48879
\(452\) −44.9658 −2.11501
\(453\) 5.13680 0.241348
\(454\) −62.4002 −2.92859
\(455\) 0 0
\(456\) −52.6056 −2.46349
\(457\) −34.5376 −1.61560 −0.807800 0.589456i \(-0.799342\pi\)
−0.807800 + 0.589456i \(0.799342\pi\)
\(458\) −23.1606 −1.08223
\(459\) 3.41447 0.159374
\(460\) 99.0354 4.61755
\(461\) 11.1375 0.518723 0.259361 0.965780i \(-0.416488\pi\)
0.259361 + 0.965780i \(0.416488\pi\)
\(462\) 8.42260 0.391855
\(463\) 8.14843 0.378690 0.189345 0.981911i \(-0.439364\pi\)
0.189345 + 0.981911i \(0.439364\pi\)
\(464\) 15.8092 0.733922
\(465\) −1.41742 −0.0657314
\(466\) −3.20036 −0.148254
\(467\) 14.8950 0.689258 0.344629 0.938739i \(-0.388005\pi\)
0.344629 + 0.938739i \(0.388005\pi\)
\(468\) 0 0
\(469\) −4.57857 −0.211419
\(470\) −82.2345 −3.79320
\(471\) 10.8870 0.501645
\(472\) 15.7783 0.726255
\(473\) −21.6531 −0.995611
\(474\) 19.9291 0.915374
\(475\) 23.6962 1.08726
\(476\) 20.0100 0.917156
\(477\) −13.1147 −0.600482
\(478\) −31.9521 −1.46145
\(479\) 13.3435 0.609679 0.304840 0.952404i \(-0.401397\pi\)
0.304840 + 0.952404i \(0.401397\pi\)
\(480\) −96.0525 −4.38418
\(481\) 0 0
\(482\) 41.4076 1.88607
\(483\) 5.37767 0.244693
\(484\) −11.5737 −0.526078
\(485\) 13.5078 0.613359
\(486\) −2.80363 −0.127175
\(487\) 15.0870 0.683657 0.341828 0.939762i \(-0.388954\pi\)
0.341828 + 0.939762i \(0.388954\pi\)
\(488\) −53.6625 −2.42919
\(489\) 2.53323 0.114556
\(490\) −8.81036 −0.398012
\(491\) −18.6136 −0.840022 −0.420011 0.907519i \(-0.637974\pi\)
−0.420011 + 0.907519i \(0.637974\pi\)
\(492\) −61.6765 −2.78059
\(493\) 2.89858 0.130546
\(494\) 0 0
\(495\) −9.44058 −0.424322
\(496\) −8.39990 −0.377166
\(497\) 2.36205 0.105952
\(498\) 14.1554 0.634318
\(499\) 18.3295 0.820543 0.410272 0.911963i \(-0.365434\pi\)
0.410272 + 0.911963i \(0.365434\pi\)
\(500\) −2.29825 −0.102781
\(501\) −1.10194 −0.0492308
\(502\) −38.8593 −1.73438
\(503\) −13.0416 −0.581495 −0.290748 0.956800i \(-0.593904\pi\)
−0.290748 + 0.956800i \(0.593904\pi\)
\(504\) −10.8230 −0.482093
\(505\) 37.9474 1.68864
\(506\) 45.2940 2.01356
\(507\) 0 0
\(508\) −88.3035 −3.91783
\(509\) 17.0920 0.757590 0.378795 0.925481i \(-0.376339\pi\)
0.378795 + 0.925481i \(0.376339\pi\)
\(510\) −30.0828 −1.33209
\(511\) −1.24210 −0.0549475
\(512\) −166.114 −7.34126
\(513\) 4.86056 0.214599
\(514\) 36.9287 1.62885
\(515\) −4.94278 −0.217805
\(516\) 42.2394 1.85948
\(517\) −28.0405 −1.23322
\(518\) −10.8944 −0.478672
\(519\) −22.0711 −0.968812
\(520\) 0 0
\(521\) −33.7225 −1.47741 −0.738705 0.674029i \(-0.764562\pi\)
−0.738705 + 0.674029i \(0.764562\pi\)
\(522\) −2.38003 −0.104171
\(523\) 35.8559 1.56787 0.783934 0.620844i \(-0.213210\pi\)
0.783934 + 0.620844i \(0.213210\pi\)
\(524\) −41.0650 −1.79393
\(525\) 4.87520 0.212771
\(526\) 84.8424 3.69930
\(527\) −1.54011 −0.0670881
\(528\) −55.9465 −2.43476
\(529\) 5.91938 0.257364
\(530\) 115.545 5.01897
\(531\) −1.45785 −0.0632655
\(532\) 28.4845 1.23496
\(533\) 0 0
\(534\) −35.5629 −1.53896
\(535\) −38.5804 −1.66797
\(536\) 49.5537 2.14039
\(537\) 7.20834 0.311063
\(538\) 68.5396 2.95495
\(539\) −3.00418 −0.129399
\(540\) 18.4160 0.792499
\(541\) −17.9589 −0.772113 −0.386056 0.922475i \(-0.626163\pi\)
−0.386056 + 0.922475i \(0.626163\pi\)
\(542\) −53.4637 −2.29646
\(543\) 13.0966 0.562031
\(544\) −104.366 −4.47466
\(545\) 55.6352 2.38315
\(546\) 0 0
\(547\) 30.1484 1.28905 0.644526 0.764583i \(-0.277055\pi\)
0.644526 + 0.764583i \(0.277055\pi\)
\(548\) 72.6386 3.10297
\(549\) 4.95821 0.211611
\(550\) 41.0619 1.75088
\(551\) 4.12618 0.175781
\(552\) −58.2024 −2.47726
\(553\) −7.10832 −0.302276
\(554\) −10.2629 −0.436029
\(555\) 12.2111 0.518333
\(556\) −15.7463 −0.667793
\(557\) −13.5998 −0.576243 −0.288122 0.957594i \(-0.593031\pi\)
−0.288122 + 0.957594i \(0.593031\pi\)
\(558\) 1.26458 0.0535341
\(559\) 0 0
\(560\) 58.5222 2.47301
\(561\) −10.2577 −0.433080
\(562\) −24.9352 −1.05183
\(563\) 11.4717 0.483475 0.241737 0.970342i \(-0.422283\pi\)
0.241737 + 0.970342i \(0.422283\pi\)
\(564\) 54.6995 2.30326
\(565\) −24.1120 −1.01440
\(566\) 35.8754 1.50795
\(567\) 1.00000 0.0419961
\(568\) −25.5644 −1.07266
\(569\) 10.7718 0.451578 0.225789 0.974176i \(-0.427504\pi\)
0.225789 + 0.974176i \(0.427504\pi\)
\(570\) −42.8233 −1.79367
\(571\) 13.6895 0.572887 0.286443 0.958097i \(-0.407527\pi\)
0.286443 + 0.958097i \(0.407527\pi\)
\(572\) 0 0
\(573\) 24.5517 1.02566
\(574\) 29.5065 1.23158
\(575\) 26.2173 1.09334
\(576\) 48.4493 2.01872
\(577\) −31.9864 −1.33161 −0.665805 0.746126i \(-0.731912\pi\)
−0.665805 + 0.746126i \(0.731912\pi\)
\(578\) 14.9752 0.622886
\(579\) −5.78789 −0.240536
\(580\) 15.6335 0.649148
\(581\) −5.04895 −0.209466
\(582\) −12.0513 −0.499542
\(583\) 39.3989 1.63174
\(584\) 13.4432 0.556286
\(585\) 0 0
\(586\) 19.9573 0.824429
\(587\) −36.5126 −1.50704 −0.753519 0.657427i \(-0.771645\pi\)
−0.753519 + 0.657427i \(0.771645\pi\)
\(588\) 5.86034 0.241676
\(589\) −2.19237 −0.0903348
\(590\) 12.8442 0.528788
\(591\) −0.0857381 −0.00352679
\(592\) 72.3651 2.97419
\(593\) −7.28387 −0.299112 −0.149556 0.988753i \(-0.547785\pi\)
−0.149556 + 0.988753i \(0.547785\pi\)
\(594\) 8.42260 0.345583
\(595\) 10.7299 0.439884
\(596\) −15.2841 −0.626061
\(597\) 5.62060 0.230036
\(598\) 0 0
\(599\) 46.1528 1.88575 0.942876 0.333144i \(-0.108110\pi\)
0.942876 + 0.333144i \(0.108110\pi\)
\(600\) −52.7641 −2.15409
\(601\) −24.4355 −0.996746 −0.498373 0.866963i \(-0.666069\pi\)
−0.498373 + 0.866963i \(0.666069\pi\)
\(602\) −20.2076 −0.823602
\(603\) −4.57857 −0.186454
\(604\) 30.1034 1.22489
\(605\) −6.20616 −0.252316
\(606\) −33.8555 −1.37529
\(607\) −26.2738 −1.06642 −0.533210 0.845983i \(-0.679014\pi\)
−0.533210 + 0.845983i \(0.679014\pi\)
\(608\) −148.567 −6.02518
\(609\) 0.848910 0.0343996
\(610\) −43.6836 −1.76870
\(611\) 0 0
\(612\) 20.0100 0.808855
\(613\) −39.0382 −1.57674 −0.788370 0.615202i \(-0.789074\pi\)
−0.788370 + 0.615202i \(0.789074\pi\)
\(614\) 73.2258 2.95515
\(615\) −33.0727 −1.33362
\(616\) 32.5141 1.31003
\(617\) 9.26856 0.373138 0.186569 0.982442i \(-0.440263\pi\)
0.186569 + 0.982442i \(0.440263\pi\)
\(618\) 4.40980 0.177388
\(619\) −45.4890 −1.82836 −0.914179 0.405310i \(-0.867164\pi\)
−0.914179 + 0.405310i \(0.867164\pi\)
\(620\) −8.30658 −0.333600
\(621\) 5.37767 0.215799
\(622\) −65.8163 −2.63899
\(623\) 12.6846 0.508197
\(624\) 0 0
\(625\) −25.6084 −1.02434
\(626\) 47.7273 1.90757
\(627\) −14.6020 −0.583147
\(628\) 63.8013 2.54595
\(629\) 13.2680 0.529030
\(630\) −8.81036 −0.351013
\(631\) −20.9773 −0.835092 −0.417546 0.908656i \(-0.637110\pi\)
−0.417546 + 0.908656i \(0.637110\pi\)
\(632\) 76.9331 3.06023
\(633\) 18.1162 0.720056
\(634\) −26.8964 −1.06819
\(635\) −47.3509 −1.87906
\(636\) −76.8567 −3.04757
\(637\) 0 0
\(638\) 7.15003 0.283072
\(639\) 2.36205 0.0934412
\(640\) −234.751 −9.27936
\(641\) −11.4475 −0.452148 −0.226074 0.974110i \(-0.572589\pi\)
−0.226074 + 0.974110i \(0.572589\pi\)
\(642\) 34.4203 1.35846
\(643\) 27.7242 1.09334 0.546668 0.837350i \(-0.315896\pi\)
0.546668 + 0.837350i \(0.315896\pi\)
\(644\) 31.5150 1.24186
\(645\) 22.6500 0.891842
\(646\) −46.5298 −1.83069
\(647\) 24.7236 0.971984 0.485992 0.873963i \(-0.338458\pi\)
0.485992 + 0.873963i \(0.338458\pi\)
\(648\) −10.8230 −0.425166
\(649\) 4.37965 0.171916
\(650\) 0 0
\(651\) −0.451052 −0.0176781
\(652\) 14.8456 0.581397
\(653\) −12.2921 −0.481027 −0.240513 0.970646i \(-0.577316\pi\)
−0.240513 + 0.970646i \(0.577316\pi\)
\(654\) −49.6361 −1.94092
\(655\) −22.0203 −0.860402
\(656\) −195.995 −7.65230
\(657\) −1.24210 −0.0484591
\(658\) −26.1686 −1.02016
\(659\) 43.4003 1.69063 0.845317 0.534265i \(-0.179411\pi\)
0.845317 + 0.534265i \(0.179411\pi\)
\(660\) −55.3250 −2.15352
\(661\) −0.585779 −0.0227841 −0.0113921 0.999935i \(-0.503626\pi\)
−0.0113921 + 0.999935i \(0.503626\pi\)
\(662\) 4.26466 0.165751
\(663\) 0 0
\(664\) 54.6446 2.12062
\(665\) 15.2742 0.592309
\(666\) −10.8944 −0.422149
\(667\) 4.56516 0.176764
\(668\) −6.45772 −0.249857
\(669\) −5.32513 −0.205881
\(670\) 40.3388 1.55842
\(671\) −14.8953 −0.575028
\(672\) −30.5658 −1.17910
\(673\) −11.6662 −0.449698 −0.224849 0.974394i \(-0.572189\pi\)
−0.224849 + 0.974394i \(0.572189\pi\)
\(674\) −79.0755 −3.04587
\(675\) 4.87520 0.187647
\(676\) 0 0
\(677\) 9.21779 0.354268 0.177134 0.984187i \(-0.443317\pi\)
0.177134 + 0.984187i \(0.443317\pi\)
\(678\) 21.5120 0.826163
\(679\) 4.29846 0.164960
\(680\) −116.130 −4.45337
\(681\) 22.2569 0.852888
\(682\) −3.79903 −0.145472
\(683\) −23.5144 −0.899753 −0.449876 0.893091i \(-0.648532\pi\)
−0.449876 + 0.893091i \(0.648532\pi\)
\(684\) 28.4845 1.08913
\(685\) 38.9509 1.48824
\(686\) −2.80363 −0.107043
\(687\) 8.26095 0.315175
\(688\) 134.228 5.11738
\(689\) 0 0
\(690\) −47.3792 −1.80370
\(691\) 6.30097 0.239700 0.119850 0.992792i \(-0.461759\pi\)
0.119850 + 0.992792i \(0.461759\pi\)
\(692\) −129.344 −4.91692
\(693\) −3.00418 −0.114119
\(694\) −21.2950 −0.808347
\(695\) −8.44364 −0.320286
\(696\) −9.18772 −0.348260
\(697\) −35.9353 −1.36114
\(698\) 95.2150 3.60394
\(699\) 1.14151 0.0431757
\(700\) 28.5703 1.07986
\(701\) 25.9334 0.979492 0.489746 0.871865i \(-0.337090\pi\)
0.489746 + 0.871865i \(0.337090\pi\)
\(702\) 0 0
\(703\) 18.8872 0.712345
\(704\) −145.550 −5.48564
\(705\) 29.3315 1.10469
\(706\) 40.5615 1.52655
\(707\) 12.0756 0.454150
\(708\) −8.54352 −0.321085
\(709\) −44.7279 −1.67979 −0.839895 0.542748i \(-0.817384\pi\)
−0.839895 + 0.542748i \(0.817384\pi\)
\(710\) −20.8105 −0.781004
\(711\) −7.10832 −0.266583
\(712\) −137.285 −5.14496
\(713\) −2.42561 −0.0908398
\(714\) −9.57292 −0.358258
\(715\) 0 0
\(716\) 42.2433 1.57871
\(717\) 11.3967 0.425617
\(718\) −28.0142 −1.04548
\(719\) 1.13604 0.0423672 0.0211836 0.999776i \(-0.493257\pi\)
0.0211836 + 0.999776i \(0.493257\pi\)
\(720\) 58.5222 2.18099
\(721\) −1.57289 −0.0585775
\(722\) −12.9669 −0.482578
\(723\) −14.7693 −0.549276
\(724\) 76.7508 2.85242
\(725\) 4.13861 0.153704
\(726\) 5.53695 0.205496
\(727\) −41.7837 −1.54967 −0.774836 0.632162i \(-0.782168\pi\)
−0.774836 + 0.632162i \(0.782168\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.9434 0.405033
\(731\) 24.6104 0.910248
\(732\) 29.0568 1.07397
\(733\) −46.4593 −1.71601 −0.858006 0.513639i \(-0.828297\pi\)
−0.858006 + 0.513639i \(0.828297\pi\)
\(734\) 54.9046 2.02656
\(735\) 3.14248 0.115912
\(736\) −164.373 −6.05886
\(737\) 13.7548 0.506665
\(738\) 29.5065 1.08615
\(739\) 5.69151 0.209365 0.104683 0.994506i \(-0.466617\pi\)
0.104683 + 0.994506i \(0.466617\pi\)
\(740\) 71.5612 2.63064
\(741\) 0 0
\(742\) 36.7688 1.34983
\(743\) −3.88256 −0.142437 −0.0712186 0.997461i \(-0.522689\pi\)
−0.0712186 + 0.997461i \(0.522689\pi\)
\(744\) 4.88172 0.178972
\(745\) −8.19578 −0.300270
\(746\) 10.0308 0.367253
\(747\) −5.04895 −0.184732
\(748\) −60.1135 −2.19797
\(749\) −12.2770 −0.448593
\(750\) 1.09950 0.0401481
\(751\) −40.3197 −1.47129 −0.735644 0.677368i \(-0.763120\pi\)
−0.735644 + 0.677368i \(0.763120\pi\)
\(752\) 173.823 6.33868
\(753\) 13.8604 0.505100
\(754\) 0 0
\(755\) 16.1423 0.587479
\(756\) 5.86034 0.213138
\(757\) 9.61102 0.349319 0.174659 0.984629i \(-0.444118\pi\)
0.174659 + 0.984629i \(0.444118\pi\)
\(758\) −26.4463 −0.960574
\(759\) −16.1555 −0.586407
\(760\) −165.312 −5.99651
\(761\) −30.1840 −1.09417 −0.547085 0.837077i \(-0.684262\pi\)
−0.547085 + 0.837077i \(0.684262\pi\)
\(762\) 42.2451 1.53038
\(763\) 17.7042 0.640936
\(764\) 143.881 5.20544
\(765\) 10.7299 0.387941
\(766\) −51.1525 −1.84822
\(767\) 0 0
\(768\) 112.539 4.06091
\(769\) 10.8144 0.389976 0.194988 0.980806i \(-0.437533\pi\)
0.194988 + 0.980806i \(0.437533\pi\)
\(770\) 26.4679 0.953836
\(771\) −13.1717 −0.474368
\(772\) −33.9190 −1.22077
\(773\) 45.5987 1.64007 0.820036 0.572311i \(-0.193953\pi\)
0.820036 + 0.572311i \(0.193953\pi\)
\(774\) −20.2076 −0.726348
\(775\) −2.19897 −0.0789894
\(776\) −46.5220 −1.67004
\(777\) 3.88581 0.139403
\(778\) −19.8212 −0.710623
\(779\) −51.1544 −1.83280
\(780\) 0 0
\(781\) −7.09601 −0.253915
\(782\) −51.4801 −1.84092
\(783\) 0.848910 0.0303376
\(784\) 18.6229 0.665103
\(785\) 34.2121 1.22108
\(786\) 19.6458 0.700743
\(787\) 32.4330 1.15611 0.578055 0.815997i \(-0.303812\pi\)
0.578055 + 0.815997i \(0.303812\pi\)
\(788\) −0.502454 −0.0178992
\(789\) −30.2616 −1.07734
\(790\) 62.6268 2.22816
\(791\) −7.67290 −0.272817
\(792\) 32.5141 1.15534
\(793\) 0 0
\(794\) −13.3526 −0.473866
\(795\) −41.2128 −1.46167
\(796\) 32.9386 1.16748
\(797\) 41.0287 1.45331 0.726656 0.687001i \(-0.241073\pi\)
0.726656 + 0.687001i \(0.241073\pi\)
\(798\) −13.6272 −0.482398
\(799\) 31.8702 1.12749
\(800\) −149.014 −5.26846
\(801\) 12.6846 0.448188
\(802\) −90.7341 −3.20393
\(803\) 3.73150 0.131682
\(804\) −26.8320 −0.946290
\(805\) 16.8993 0.595620
\(806\) 0 0
\(807\) −24.4467 −0.860566
\(808\) −130.694 −4.59779
\(809\) −10.6632 −0.374898 −0.187449 0.982274i \(-0.560022\pi\)
−0.187449 + 0.982274i \(0.560022\pi\)
\(810\) −8.81036 −0.309565
\(811\) 5.60921 0.196966 0.0984831 0.995139i \(-0.468601\pi\)
0.0984831 + 0.995139i \(0.468601\pi\)
\(812\) 4.97490 0.174585
\(813\) 19.0695 0.668795
\(814\) 32.7286 1.14714
\(815\) 7.96062 0.278848
\(816\) 63.5874 2.22600
\(817\) 35.0333 1.22566
\(818\) −5.60819 −0.196086
\(819\) 0 0
\(820\) −193.817 −6.76840
\(821\) −12.4008 −0.432790 −0.216395 0.976306i \(-0.569430\pi\)
−0.216395 + 0.976306i \(0.569430\pi\)
\(822\) −34.7508 −1.21207
\(823\) 23.9208 0.833827 0.416913 0.908946i \(-0.363112\pi\)
0.416913 + 0.908946i \(0.363112\pi\)
\(824\) 17.0233 0.593036
\(825\) −14.6460 −0.509907
\(826\) 4.08728 0.142215
\(827\) 2.10966 0.0733600 0.0366800 0.999327i \(-0.488322\pi\)
0.0366800 + 0.999327i \(0.488322\pi\)
\(828\) 31.5150 1.09522
\(829\) 47.4770 1.64895 0.824473 0.565901i \(-0.191472\pi\)
0.824473 + 0.565901i \(0.191472\pi\)
\(830\) 44.4831 1.54403
\(831\) 3.66058 0.126984
\(832\) 0 0
\(833\) 3.41447 0.118305
\(834\) 7.53316 0.260852
\(835\) −3.46281 −0.119836
\(836\) −85.5726 −2.95959
\(837\) −0.451052 −0.0155906
\(838\) 92.4651 3.19415
\(839\) 8.30337 0.286664 0.143332 0.989675i \(-0.454218\pi\)
0.143332 + 0.989675i \(0.454218\pi\)
\(840\) −34.0110 −1.17349
\(841\) −28.2794 −0.975150
\(842\) 39.8909 1.37473
\(843\) 8.89390 0.306322
\(844\) 106.167 3.65443
\(845\) 0 0
\(846\) −26.1686 −0.899696
\(847\) −1.97492 −0.0678592
\(848\) −244.234 −8.38703
\(849\) −12.7960 −0.439159
\(850\) −46.6699 −1.60077
\(851\) 20.8966 0.716327
\(852\) 13.8424 0.474233
\(853\) −5.14634 −0.176207 −0.0881036 0.996111i \(-0.528081\pi\)
−0.0881036 + 0.996111i \(0.528081\pi\)
\(854\) −13.9010 −0.475682
\(855\) 15.2742 0.522368
\(856\) 132.874 4.54153
\(857\) 6.25760 0.213756 0.106878 0.994272i \(-0.465915\pi\)
0.106878 + 0.994272i \(0.465915\pi\)
\(858\) 0 0
\(859\) −4.10387 −0.140022 −0.0700112 0.997546i \(-0.522304\pi\)
−0.0700112 + 0.997546i \(0.522304\pi\)
\(860\) 132.737 4.52628
\(861\) −10.5244 −0.358670
\(862\) −67.0665 −2.28429
\(863\) 37.8902 1.28980 0.644899 0.764268i \(-0.276899\pi\)
0.644899 + 0.764268i \(0.276899\pi\)
\(864\) −30.5658 −1.03987
\(865\) −69.3579 −2.35824
\(866\) −85.0155 −2.88895
\(867\) −5.34136 −0.181402
\(868\) −2.64332 −0.0897201
\(869\) 21.3546 0.724407
\(870\) −7.47920 −0.253569
\(871\) 0 0
\(872\) −191.612 −6.48881
\(873\) 4.29846 0.145481
\(874\) −73.2827 −2.47882
\(875\) −0.392171 −0.0132578
\(876\) −7.27915 −0.245940
\(877\) −55.7199 −1.88153 −0.940764 0.339063i \(-0.889890\pi\)
−0.940764 + 0.339063i \(0.889890\pi\)
\(878\) 32.8049 1.10711
\(879\) −7.11839 −0.240097
\(880\) −175.811 −5.92658
\(881\) −34.4048 −1.15913 −0.579564 0.814927i \(-0.696777\pi\)
−0.579564 + 0.814927i \(0.696777\pi\)
\(882\) −2.80363 −0.0944031
\(883\) −36.9973 −1.24506 −0.622529 0.782597i \(-0.713895\pi\)
−0.622529 + 0.782597i \(0.713895\pi\)
\(884\) 0 0
\(885\) −4.58128 −0.153998
\(886\) −14.1757 −0.476242
\(887\) −55.3301 −1.85780 −0.928902 0.370325i \(-0.879246\pi\)
−0.928902 + 0.370325i \(0.879246\pi\)
\(888\) −42.0560 −1.41131
\(889\) −15.0680 −0.505364
\(890\) −111.756 −3.74606
\(891\) −3.00418 −0.100644
\(892\) −31.2070 −1.04489
\(893\) 45.3677 1.51817
\(894\) 7.31203 0.244551
\(895\) 22.6521 0.757176
\(896\) −74.7024 −2.49563
\(897\) 0 0
\(898\) 48.7004 1.62515
\(899\) −0.382903 −0.0127705
\(900\) 28.5703 0.952345
\(901\) −44.7799 −1.49183
\(902\) −88.6427 −2.95148
\(903\) 7.20767 0.239856
\(904\) 83.0435 2.76199
\(905\) 41.1560 1.36807
\(906\) −14.4017 −0.478464
\(907\) 12.5397 0.416375 0.208188 0.978089i \(-0.433243\pi\)
0.208188 + 0.978089i \(0.433243\pi\)
\(908\) 130.433 4.32858
\(909\) 12.0756 0.400523
\(910\) 0 0
\(911\) −11.4861 −0.380552 −0.190276 0.981731i \(-0.560938\pi\)
−0.190276 + 0.981731i \(0.560938\pi\)
\(912\) 90.5177 2.99734
\(913\) 15.1679 0.501986
\(914\) 96.8306 3.20287
\(915\) 15.5811 0.515095
\(916\) 48.4120 1.59958
\(917\) −7.00728 −0.231401
\(918\) −9.57292 −0.315954
\(919\) 33.9296 1.11924 0.559618 0.828751i \(-0.310948\pi\)
0.559618 + 0.828751i \(0.310948\pi\)
\(920\) −182.900 −6.03003
\(921\) −26.1182 −0.860624
\(922\) −31.2253 −1.02835
\(923\) 0 0
\(924\) −17.6055 −0.579178
\(925\) 18.9441 0.622879
\(926\) −22.8452 −0.750739
\(927\) −1.57289 −0.0516605
\(928\) −25.9476 −0.851772
\(929\) −50.4939 −1.65665 −0.828326 0.560247i \(-0.810706\pi\)
−0.828326 + 0.560247i \(0.810706\pi\)
\(930\) 3.97393 0.130310
\(931\) 4.86056 0.159298
\(932\) 6.68961 0.219125
\(933\) 23.4754 0.768550
\(934\) −41.7601 −1.36643
\(935\) −32.2346 −1.05418
\(936\) 0 0
\(937\) 14.7689 0.482478 0.241239 0.970466i \(-0.422446\pi\)
0.241239 + 0.970466i \(0.422446\pi\)
\(938\) 12.8366 0.419130
\(939\) −17.0234 −0.555538
\(940\) 171.892 5.60651
\(941\) 30.1402 0.982544 0.491272 0.871006i \(-0.336532\pi\)
0.491272 + 0.871006i \(0.336532\pi\)
\(942\) −30.5230 −0.994493
\(943\) −56.5967 −1.84304
\(944\) −27.1495 −0.883640
\(945\) 3.14248 0.102225
\(946\) 60.7073 1.97376
\(947\) −27.1523 −0.882331 −0.441165 0.897426i \(-0.645435\pi\)
−0.441165 + 0.897426i \(0.645435\pi\)
\(948\) −41.6572 −1.35296
\(949\) 0 0
\(950\) −66.4354 −2.15545
\(951\) 9.59342 0.311088
\(952\) −36.9547 −1.19771
\(953\) −19.0337 −0.616560 −0.308280 0.951296i \(-0.599753\pi\)
−0.308280 + 0.951296i \(0.599753\pi\)
\(954\) 36.7688 1.19043
\(955\) 77.1533 2.49662
\(956\) 66.7884 2.16009
\(957\) −2.55028 −0.0824387
\(958\) −37.4102 −1.20867
\(959\) 12.3949 0.400254
\(960\) 152.251 4.91389
\(961\) −30.7966 −0.993437
\(962\) 0 0
\(963\) −12.2770 −0.395622
\(964\) −86.5530 −2.78769
\(965\) −18.1883 −0.585504
\(966\) −15.0770 −0.485095
\(967\) −2.60639 −0.0838158 −0.0419079 0.999121i \(-0.513344\pi\)
−0.0419079 + 0.999121i \(0.513344\pi\)
\(968\) 21.3745 0.687003
\(969\) 16.5963 0.533149
\(970\) −37.8710 −1.21596
\(971\) 38.7695 1.24417 0.622086 0.782949i \(-0.286286\pi\)
0.622086 + 0.782949i \(0.286286\pi\)
\(972\) 5.86034 0.187970
\(973\) −2.68693 −0.0861391
\(974\) −42.2983 −1.35533
\(975\) 0 0
\(976\) 92.3362 2.95561
\(977\) 41.1599 1.31682 0.658411 0.752658i \(-0.271229\pi\)
0.658411 + 0.752658i \(0.271229\pi\)
\(978\) −7.10223 −0.227104
\(979\) −38.1067 −1.21790
\(980\) 18.4160 0.588278
\(981\) 17.7042 0.565252
\(982\) 52.1857 1.66531
\(983\) 8.61935 0.274914 0.137457 0.990508i \(-0.456107\pi\)
0.137457 + 0.990508i \(0.456107\pi\)
\(984\) 113.905 3.63116
\(985\) −0.269431 −0.00858477
\(986\) −8.12655 −0.258802
\(987\) 9.33384 0.297100
\(988\) 0 0
\(989\) 38.7605 1.23251
\(990\) 26.4679 0.841204
\(991\) 32.1624 1.02167 0.510835 0.859679i \(-0.329336\pi\)
0.510835 + 0.859679i \(0.329336\pi\)
\(992\) 13.7868 0.437730
\(993\) −1.52112 −0.0482714
\(994\) −6.62231 −0.210047
\(995\) 17.6626 0.559943
\(996\) −29.5886 −0.937550
\(997\) −28.8875 −0.914876 −0.457438 0.889242i \(-0.651233\pi\)
−0.457438 + 0.889242i \(0.651233\pi\)
\(998\) −51.3893 −1.62670
\(999\) 3.88581 0.122942
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 3549.2.a.bg.1.1 15
13.12 even 2 3549.2.a.bh.1.15 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.1 15 1.1 even 1 trivial
3549.2.a.bh.1.15 yes 15 13.12 even 2