Properties

Label 3381.2.a.bi.1.8
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.27502\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.27502 q^{2} -1.00000 q^{3} +3.17573 q^{4} -1.23869 q^{5} -2.27502 q^{6} +2.67481 q^{8} +1.00000 q^{9} -2.81806 q^{10} +6.12669 q^{11} -3.17573 q^{12} +0.910400 q^{13} +1.23869 q^{15} -0.266205 q^{16} +4.90873 q^{17} +2.27502 q^{18} -6.90370 q^{19} -3.93376 q^{20} +13.9384 q^{22} +1.00000 q^{23} -2.67481 q^{24} -3.46564 q^{25} +2.07118 q^{26} -1.00000 q^{27} +3.36004 q^{29} +2.81806 q^{30} +5.43735 q^{31} -5.95524 q^{32} -6.12669 q^{33} +11.1675 q^{34} +3.17573 q^{36} +6.54665 q^{37} -15.7061 q^{38} -0.910400 q^{39} -3.31327 q^{40} -2.52207 q^{41} -0.231369 q^{43} +19.4567 q^{44} -1.23869 q^{45} +2.27502 q^{46} +13.3578 q^{47} +0.266205 q^{48} -7.88440 q^{50} -4.90873 q^{51} +2.89118 q^{52} +4.22681 q^{53} -2.27502 q^{54} -7.58909 q^{55} +6.90370 q^{57} +7.64417 q^{58} -0.754201 q^{59} +3.93376 q^{60} +3.57664 q^{61} +12.3701 q^{62} -13.0159 q^{64} -1.12771 q^{65} -13.9384 q^{66} -4.49406 q^{67} +15.5888 q^{68} -1.00000 q^{69} +15.8575 q^{71} +2.67481 q^{72} -4.27776 q^{73} +14.8938 q^{74} +3.46564 q^{75} -21.9243 q^{76} -2.07118 q^{78} -0.222015 q^{79} +0.329747 q^{80} +1.00000 q^{81} -5.73777 q^{82} +15.5197 q^{83} -6.08041 q^{85} -0.526369 q^{86} -3.36004 q^{87} +16.3877 q^{88} +0.559363 q^{89} -2.81806 q^{90} +3.17573 q^{92} -5.43735 q^{93} +30.3893 q^{94} +8.55157 q^{95} +5.95524 q^{96} +17.6476 q^{97} +6.12669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27502 1.60868 0.804342 0.594167i \(-0.202518\pi\)
0.804342 + 0.594167i \(0.202518\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.17573 1.58786
\(5\) −1.23869 −0.553961 −0.276980 0.960876i \(-0.589334\pi\)
−0.276980 + 0.960876i \(0.589334\pi\)
\(6\) −2.27502 −0.928774
\(7\) 0 0
\(8\) 2.67481 0.945688
\(9\) 1.00000 0.333333
\(10\) −2.81806 −0.891148
\(11\) 6.12669 1.84727 0.923633 0.383277i \(-0.125205\pi\)
0.923633 + 0.383277i \(0.125205\pi\)
\(12\) −3.17573 −0.916754
\(13\) 0.910400 0.252500 0.126250 0.991998i \(-0.459706\pi\)
0.126250 + 0.991998i \(0.459706\pi\)
\(14\) 0 0
\(15\) 1.23869 0.319829
\(16\) −0.266205 −0.0665513
\(17\) 4.90873 1.19054 0.595271 0.803525i \(-0.297045\pi\)
0.595271 + 0.803525i \(0.297045\pi\)
\(18\) 2.27502 0.536228
\(19\) −6.90370 −1.58382 −0.791909 0.610639i \(-0.790913\pi\)
−0.791909 + 0.610639i \(0.790913\pi\)
\(20\) −3.93376 −0.879614
\(21\) 0 0
\(22\) 13.9384 2.97167
\(23\) 1.00000 0.208514
\(24\) −2.67481 −0.545993
\(25\) −3.46564 −0.693128
\(26\) 2.07118 0.406192
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.36004 0.623944 0.311972 0.950091i \(-0.399010\pi\)
0.311972 + 0.950091i \(0.399010\pi\)
\(30\) 2.81806 0.514504
\(31\) 5.43735 0.976577 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(32\) −5.95524 −1.05275
\(33\) −6.12669 −1.06652
\(34\) 11.1675 1.91521
\(35\) 0 0
\(36\) 3.17573 0.529288
\(37\) 6.54665 1.07626 0.538132 0.842861i \(-0.319130\pi\)
0.538132 + 0.842861i \(0.319130\pi\)
\(38\) −15.7061 −2.54786
\(39\) −0.910400 −0.145781
\(40\) −3.31327 −0.523874
\(41\) −2.52207 −0.393882 −0.196941 0.980415i \(-0.563101\pi\)
−0.196941 + 0.980415i \(0.563101\pi\)
\(42\) 0 0
\(43\) −0.231369 −0.0352834 −0.0176417 0.999844i \(-0.505616\pi\)
−0.0176417 + 0.999844i \(0.505616\pi\)
\(44\) 19.4567 2.93321
\(45\) −1.23869 −0.184654
\(46\) 2.27502 0.335434
\(47\) 13.3578 1.94843 0.974217 0.225614i \(-0.0724388\pi\)
0.974217 + 0.225614i \(0.0724388\pi\)
\(48\) 0.266205 0.0384234
\(49\) 0 0
\(50\) −7.88440 −1.11502
\(51\) −4.90873 −0.687360
\(52\) 2.89118 0.400935
\(53\) 4.22681 0.580597 0.290298 0.956936i \(-0.406245\pi\)
0.290298 + 0.956936i \(0.406245\pi\)
\(54\) −2.27502 −0.309591
\(55\) −7.58909 −1.02331
\(56\) 0 0
\(57\) 6.90370 0.914418
\(58\) 7.64417 1.00373
\(59\) −0.754201 −0.0981886 −0.0490943 0.998794i \(-0.515633\pi\)
−0.0490943 + 0.998794i \(0.515633\pi\)
\(60\) 3.93376 0.507846
\(61\) 3.57664 0.457942 0.228971 0.973433i \(-0.426464\pi\)
0.228971 + 0.973433i \(0.426464\pi\)
\(62\) 12.3701 1.57100
\(63\) 0 0
\(64\) −13.0159 −1.62699
\(65\) −1.12771 −0.139875
\(66\) −13.9384 −1.71569
\(67\) −4.49406 −0.549036 −0.274518 0.961582i \(-0.588518\pi\)
−0.274518 + 0.961582i \(0.588518\pi\)
\(68\) 15.5888 1.89042
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 15.8575 1.88194 0.940968 0.338495i \(-0.109918\pi\)
0.940968 + 0.338495i \(0.109918\pi\)
\(72\) 2.67481 0.315229
\(73\) −4.27776 −0.500674 −0.250337 0.968159i \(-0.580541\pi\)
−0.250337 + 0.968159i \(0.580541\pi\)
\(74\) 14.8938 1.73137
\(75\) 3.46564 0.400177
\(76\) −21.9243 −2.51489
\(77\) 0 0
\(78\) −2.07118 −0.234515
\(79\) −0.222015 −0.0249787 −0.0124893 0.999922i \(-0.503976\pi\)
−0.0124893 + 0.999922i \(0.503976\pi\)
\(80\) 0.329747 0.0368668
\(81\) 1.00000 0.111111
\(82\) −5.73777 −0.633631
\(83\) 15.5197 1.70351 0.851753 0.523944i \(-0.175540\pi\)
0.851753 + 0.523944i \(0.175540\pi\)
\(84\) 0 0
\(85\) −6.08041 −0.659513
\(86\) −0.526369 −0.0567598
\(87\) −3.36004 −0.360234
\(88\) 16.3877 1.74694
\(89\) 0.559363 0.0592924 0.0296462 0.999560i \(-0.490562\pi\)
0.0296462 + 0.999560i \(0.490562\pi\)
\(90\) −2.81806 −0.297049
\(91\) 0 0
\(92\) 3.17573 0.331093
\(93\) −5.43735 −0.563827
\(94\) 30.3893 3.13441
\(95\) 8.55157 0.877373
\(96\) 5.95524 0.607804
\(97\) 17.6476 1.79185 0.895923 0.444209i \(-0.146515\pi\)
0.895923 + 0.444209i \(0.146515\pi\)
\(98\) 0 0
\(99\) 6.12669 0.615756
\(100\) −11.0059 −1.10059
\(101\) −8.13075 −0.809040 −0.404520 0.914529i \(-0.632561\pi\)
−0.404520 + 0.914529i \(0.632561\pi\)
\(102\) −11.1675 −1.10574
\(103\) −14.2795 −1.40700 −0.703500 0.710695i \(-0.748381\pi\)
−0.703500 + 0.710695i \(0.748381\pi\)
\(104\) 2.43515 0.238786
\(105\) 0 0
\(106\) 9.61609 0.933997
\(107\) −6.55764 −0.633951 −0.316975 0.948434i \(-0.602667\pi\)
−0.316975 + 0.948434i \(0.602667\pi\)
\(108\) −3.17573 −0.305585
\(109\) 18.2358 1.74668 0.873338 0.487114i \(-0.161950\pi\)
0.873338 + 0.487114i \(0.161950\pi\)
\(110\) −17.2654 −1.64619
\(111\) −6.54665 −0.621381
\(112\) 0 0
\(113\) −18.5945 −1.74922 −0.874610 0.484828i \(-0.838882\pi\)
−0.874610 + 0.484828i \(0.838882\pi\)
\(114\) 15.7061 1.47101
\(115\) −1.23869 −0.115509
\(116\) 10.6706 0.990738
\(117\) 0.910400 0.0841665
\(118\) −1.71582 −0.157954
\(119\) 0 0
\(120\) 3.31327 0.302459
\(121\) 26.5363 2.41239
\(122\) 8.13695 0.736685
\(123\) 2.52207 0.227408
\(124\) 17.2676 1.55067
\(125\) 10.4863 0.937926
\(126\) 0 0
\(127\) −4.19807 −0.372518 −0.186259 0.982501i \(-0.559636\pi\)
−0.186259 + 0.982501i \(0.559636\pi\)
\(128\) −17.7010 −1.56456
\(129\) 0.231369 0.0203709
\(130\) −2.56556 −0.225014
\(131\) 3.83542 0.335102 0.167551 0.985863i \(-0.446414\pi\)
0.167551 + 0.985863i \(0.446414\pi\)
\(132\) −19.4567 −1.69349
\(133\) 0 0
\(134\) −10.2241 −0.883226
\(135\) 1.23869 0.106610
\(136\) 13.1299 1.12588
\(137\) 7.25647 0.619962 0.309981 0.950743i \(-0.399677\pi\)
0.309981 + 0.950743i \(0.399677\pi\)
\(138\) −2.27502 −0.193663
\(139\) 3.52170 0.298707 0.149353 0.988784i \(-0.452281\pi\)
0.149353 + 0.988784i \(0.452281\pi\)
\(140\) 0 0
\(141\) −13.3578 −1.12493
\(142\) 36.0761 3.02744
\(143\) 5.57774 0.466434
\(144\) −0.266205 −0.0221838
\(145\) −4.16206 −0.345640
\(146\) −9.73201 −0.805427
\(147\) 0 0
\(148\) 20.7904 1.70896
\(149\) −10.9243 −0.894951 −0.447475 0.894296i \(-0.647677\pi\)
−0.447475 + 0.894296i \(0.647677\pi\)
\(150\) 7.88440 0.643759
\(151\) −15.8535 −1.29014 −0.645071 0.764122i \(-0.723172\pi\)
−0.645071 + 0.764122i \(0.723172\pi\)
\(152\) −18.4661 −1.49780
\(153\) 4.90873 0.396847
\(154\) 0 0
\(155\) −6.73521 −0.540985
\(156\) −2.89118 −0.231480
\(157\) −9.19996 −0.734237 −0.367118 0.930174i \(-0.619656\pi\)
−0.367118 + 0.930174i \(0.619656\pi\)
\(158\) −0.505090 −0.0401828
\(159\) −4.22681 −0.335208
\(160\) 7.37672 0.583181
\(161\) 0 0
\(162\) 2.27502 0.178743
\(163\) −1.64877 −0.129142 −0.0645709 0.997913i \(-0.520568\pi\)
−0.0645709 + 0.997913i \(0.520568\pi\)
\(164\) −8.00942 −0.625431
\(165\) 7.58909 0.590810
\(166\) 35.3076 2.74040
\(167\) 5.46655 0.423015 0.211507 0.977376i \(-0.432163\pi\)
0.211507 + 0.977376i \(0.432163\pi\)
\(168\) 0 0
\(169\) −12.1712 −0.936244
\(170\) −13.8331 −1.06095
\(171\) −6.90370 −0.527939
\(172\) −0.734764 −0.0560252
\(173\) −0.509172 −0.0387117 −0.0193558 0.999813i \(-0.506162\pi\)
−0.0193558 + 0.999813i \(0.506162\pi\)
\(174\) −7.64417 −0.579503
\(175\) 0 0
\(176\) −1.63096 −0.122938
\(177\) 0.754201 0.0566892
\(178\) 1.27256 0.0953827
\(179\) 8.35087 0.624173 0.312087 0.950054i \(-0.398972\pi\)
0.312087 + 0.950054i \(0.398972\pi\)
\(180\) −3.93376 −0.293205
\(181\) −2.42861 −0.180517 −0.0902584 0.995918i \(-0.528769\pi\)
−0.0902584 + 0.995918i \(0.528769\pi\)
\(182\) 0 0
\(183\) −3.57664 −0.264393
\(184\) 2.67481 0.197190
\(185\) −8.10930 −0.596208
\(186\) −12.3701 −0.907020
\(187\) 30.0743 2.19925
\(188\) 42.4207 3.09385
\(189\) 0 0
\(190\) 19.4550 1.41142
\(191\) −14.2931 −1.03421 −0.517107 0.855921i \(-0.672991\pi\)
−0.517107 + 0.855921i \(0.672991\pi\)
\(192\) 13.0159 0.939342
\(193\) 2.09005 0.150445 0.0752225 0.997167i \(-0.476033\pi\)
0.0752225 + 0.997167i \(0.476033\pi\)
\(194\) 40.1488 2.88251
\(195\) 1.12771 0.0807568
\(196\) 0 0
\(197\) 10.8863 0.775614 0.387807 0.921741i \(-0.373233\pi\)
0.387807 + 0.921741i \(0.373233\pi\)
\(198\) 13.9384 0.990556
\(199\) 3.44053 0.243893 0.121946 0.992537i \(-0.461086\pi\)
0.121946 + 0.992537i \(0.461086\pi\)
\(200\) −9.26992 −0.655482
\(201\) 4.49406 0.316986
\(202\) −18.4976 −1.30149
\(203\) 0 0
\(204\) −15.5888 −1.09143
\(205\) 3.12408 0.218195
\(206\) −32.4862 −2.26342
\(207\) 1.00000 0.0695048
\(208\) −0.242353 −0.0168042
\(209\) −42.2968 −2.92573
\(210\) 0 0
\(211\) 11.6690 0.803324 0.401662 0.915788i \(-0.368433\pi\)
0.401662 + 0.915788i \(0.368433\pi\)
\(212\) 13.4232 0.921909
\(213\) −15.8575 −1.08654
\(214\) −14.9188 −1.01983
\(215\) 0.286595 0.0195456
\(216\) −2.67481 −0.181998
\(217\) 0 0
\(218\) 41.4869 2.80985
\(219\) 4.27776 0.289065
\(220\) −24.1009 −1.62488
\(221\) 4.46891 0.300611
\(222\) −14.8938 −0.999605
\(223\) 2.48168 0.166185 0.0830926 0.996542i \(-0.473520\pi\)
0.0830926 + 0.996542i \(0.473520\pi\)
\(224\) 0 0
\(225\) −3.46564 −0.231043
\(226\) −42.3028 −2.81394
\(227\) −22.0495 −1.46348 −0.731738 0.681586i \(-0.761290\pi\)
−0.731738 + 0.681586i \(0.761290\pi\)
\(228\) 21.9243 1.45197
\(229\) 14.7467 0.974488 0.487244 0.873266i \(-0.338002\pi\)
0.487244 + 0.873266i \(0.338002\pi\)
\(230\) −2.81806 −0.185817
\(231\) 0 0
\(232\) 8.98747 0.590056
\(233\) −3.41615 −0.223799 −0.111900 0.993720i \(-0.535694\pi\)
−0.111900 + 0.993720i \(0.535694\pi\)
\(234\) 2.07118 0.135397
\(235\) −16.5462 −1.07936
\(236\) −2.39514 −0.155910
\(237\) 0.222015 0.0144215
\(238\) 0 0
\(239\) −10.8751 −0.703453 −0.351727 0.936103i \(-0.614405\pi\)
−0.351727 + 0.936103i \(0.614405\pi\)
\(240\) −0.329747 −0.0212851
\(241\) −21.3965 −1.37827 −0.689134 0.724634i \(-0.742009\pi\)
−0.689134 + 0.724634i \(0.742009\pi\)
\(242\) 60.3708 3.88078
\(243\) −1.00000 −0.0641500
\(244\) 11.3585 0.727150
\(245\) 0 0
\(246\) 5.73777 0.365827
\(247\) −6.28513 −0.399913
\(248\) 14.5439 0.923537
\(249\) −15.5197 −0.983520
\(250\) 23.8566 1.50883
\(251\) 4.68560 0.295752 0.147876 0.989006i \(-0.452756\pi\)
0.147876 + 0.989006i \(0.452756\pi\)
\(252\) 0 0
\(253\) 6.12669 0.385182
\(254\) −9.55070 −0.599264
\(255\) 6.08041 0.380770
\(256\) −14.2383 −0.889897
\(257\) −27.0174 −1.68530 −0.842650 0.538462i \(-0.819006\pi\)
−0.842650 + 0.538462i \(0.819006\pi\)
\(258\) 0.526369 0.0327703
\(259\) 0 0
\(260\) −3.58129 −0.222102
\(261\) 3.36004 0.207981
\(262\) 8.72566 0.539073
\(263\) −20.0063 −1.23364 −0.616822 0.787103i \(-0.711580\pi\)
−0.616822 + 0.787103i \(0.711580\pi\)
\(264\) −16.3877 −1.00859
\(265\) −5.23572 −0.321628
\(266\) 0 0
\(267\) −0.559363 −0.0342325
\(268\) −14.2719 −0.871795
\(269\) −6.28072 −0.382942 −0.191471 0.981498i \(-0.561326\pi\)
−0.191471 + 0.981498i \(0.561326\pi\)
\(270\) 2.81806 0.171501
\(271\) 25.4203 1.54417 0.772085 0.635519i \(-0.219214\pi\)
0.772085 + 0.635519i \(0.219214\pi\)
\(272\) −1.30673 −0.0792321
\(273\) 0 0
\(274\) 16.5086 0.997323
\(275\) −21.2329 −1.28039
\(276\) −3.17573 −0.191156
\(277\) −15.1411 −0.909739 −0.454869 0.890558i \(-0.650314\pi\)
−0.454869 + 0.890558i \(0.650314\pi\)
\(278\) 8.01195 0.480525
\(279\) 5.43735 0.325526
\(280\) 0 0
\(281\) −27.3149 −1.62947 −0.814736 0.579833i \(-0.803118\pi\)
−0.814736 + 0.579833i \(0.803118\pi\)
\(282\) −30.3893 −1.80965
\(283\) 8.54261 0.507805 0.253903 0.967230i \(-0.418286\pi\)
0.253903 + 0.967230i \(0.418286\pi\)
\(284\) 50.3590 2.98826
\(285\) −8.55157 −0.506551
\(286\) 12.6895 0.750345
\(287\) 0 0
\(288\) −5.95524 −0.350916
\(289\) 7.09562 0.417390
\(290\) −9.46879 −0.556026
\(291\) −17.6476 −1.03452
\(292\) −13.5850 −0.795003
\(293\) 13.6153 0.795413 0.397707 0.917513i \(-0.369806\pi\)
0.397707 + 0.917513i \(0.369806\pi\)
\(294\) 0 0
\(295\) 0.934224 0.0543926
\(296\) 17.5110 1.01781
\(297\) −6.12669 −0.355507
\(298\) −24.8530 −1.43969
\(299\) 0.910400 0.0526498
\(300\) 11.0059 0.635427
\(301\) 0 0
\(302\) −36.0672 −2.07543
\(303\) 8.13075 0.467100
\(304\) 1.83780 0.105405
\(305\) −4.43037 −0.253682
\(306\) 11.1675 0.638402
\(307\) −1.94494 −0.111003 −0.0555017 0.998459i \(-0.517676\pi\)
−0.0555017 + 0.998459i \(0.517676\pi\)
\(308\) 0 0
\(309\) 14.2795 0.812332
\(310\) −15.3228 −0.870275
\(311\) −14.2020 −0.805319 −0.402660 0.915350i \(-0.631914\pi\)
−0.402660 + 0.915350i \(0.631914\pi\)
\(312\) −2.43515 −0.137863
\(313\) −4.20457 −0.237656 −0.118828 0.992915i \(-0.537914\pi\)
−0.118828 + 0.992915i \(0.537914\pi\)
\(314\) −20.9301 −1.18115
\(315\) 0 0
\(316\) −0.705061 −0.0396628
\(317\) 0.738433 0.0414745 0.0207373 0.999785i \(-0.493399\pi\)
0.0207373 + 0.999785i \(0.493399\pi\)
\(318\) −9.61609 −0.539243
\(319\) 20.5859 1.15259
\(320\) 16.1227 0.901287
\(321\) 6.55764 0.366012
\(322\) 0 0
\(323\) −33.8884 −1.88560
\(324\) 3.17573 0.176429
\(325\) −3.15512 −0.175014
\(326\) −3.75100 −0.207748
\(327\) −18.2358 −1.00844
\(328\) −6.74607 −0.372489
\(329\) 0 0
\(330\) 17.2654 0.950427
\(331\) −20.5989 −1.13222 −0.566108 0.824331i \(-0.691552\pi\)
−0.566108 + 0.824331i \(0.691552\pi\)
\(332\) 49.2863 2.70494
\(333\) 6.54665 0.358754
\(334\) 12.4365 0.680497
\(335\) 5.56676 0.304145
\(336\) 0 0
\(337\) −27.9032 −1.51999 −0.759993 0.649932i \(-0.774797\pi\)
−0.759993 + 0.649932i \(0.774797\pi\)
\(338\) −27.6897 −1.50612
\(339\) 18.5945 1.00991
\(340\) −19.3097 −1.04722
\(341\) 33.3130 1.80400
\(342\) −15.7061 −0.849287
\(343\) 0 0
\(344\) −0.618867 −0.0333671
\(345\) 1.23869 0.0666890
\(346\) −1.15838 −0.0622748
\(347\) −2.08533 −0.111946 −0.0559732 0.998432i \(-0.517826\pi\)
−0.0559732 + 0.998432i \(0.517826\pi\)
\(348\) −10.6706 −0.572003
\(349\) −4.04322 −0.216429 −0.108214 0.994128i \(-0.534513\pi\)
−0.108214 + 0.994128i \(0.534513\pi\)
\(350\) 0 0
\(351\) −0.910400 −0.0485936
\(352\) −36.4859 −1.94471
\(353\) −4.50793 −0.239933 −0.119966 0.992778i \(-0.538279\pi\)
−0.119966 + 0.992778i \(0.538279\pi\)
\(354\) 1.71582 0.0911950
\(355\) −19.6426 −1.04252
\(356\) 1.77639 0.0941483
\(357\) 0 0
\(358\) 18.9984 1.00410
\(359\) −20.8138 −1.09851 −0.549254 0.835656i \(-0.685088\pi\)
−0.549254 + 0.835656i \(0.685088\pi\)
\(360\) −3.31327 −0.174625
\(361\) 28.6611 1.50848
\(362\) −5.52513 −0.290394
\(363\) −26.5363 −1.39280
\(364\) 0 0
\(365\) 5.29884 0.277354
\(366\) −8.13695 −0.425325
\(367\) 8.57218 0.447464 0.223732 0.974651i \(-0.428176\pi\)
0.223732 + 0.974651i \(0.428176\pi\)
\(368\) −0.266205 −0.0138769
\(369\) −2.52207 −0.131294
\(370\) −18.4488 −0.959109
\(371\) 0 0
\(372\) −17.2676 −0.895281
\(373\) −1.45507 −0.0753407 −0.0376703 0.999290i \(-0.511994\pi\)
−0.0376703 + 0.999290i \(0.511994\pi\)
\(374\) 68.4196 3.53790
\(375\) −10.4863 −0.541512
\(376\) 35.7295 1.84261
\(377\) 3.05898 0.157546
\(378\) 0 0
\(379\) −2.61370 −0.134257 −0.0671283 0.997744i \(-0.521384\pi\)
−0.0671283 + 0.997744i \(0.521384\pi\)
\(380\) 27.1575 1.39315
\(381\) 4.19807 0.215074
\(382\) −32.5172 −1.66372
\(383\) −11.3203 −0.578438 −0.289219 0.957263i \(-0.593396\pi\)
−0.289219 + 0.957263i \(0.593396\pi\)
\(384\) 17.7010 0.903300
\(385\) 0 0
\(386\) 4.75491 0.242018
\(387\) −0.231369 −0.0117611
\(388\) 56.0441 2.84521
\(389\) −30.9710 −1.57029 −0.785147 0.619310i \(-0.787412\pi\)
−0.785147 + 0.619310i \(0.787412\pi\)
\(390\) 2.56556 0.129912
\(391\) 4.90873 0.248245
\(392\) 0 0
\(393\) −3.83542 −0.193471
\(394\) 24.7665 1.24772
\(395\) 0.275009 0.0138372
\(396\) 19.4567 0.977736
\(397\) 25.0396 1.25670 0.628350 0.777931i \(-0.283731\pi\)
0.628350 + 0.777931i \(0.283731\pi\)
\(398\) 7.82728 0.392346
\(399\) 0 0
\(400\) 0.922571 0.0461285
\(401\) 34.9745 1.74654 0.873272 0.487234i \(-0.161994\pi\)
0.873272 + 0.487234i \(0.161994\pi\)
\(402\) 10.2241 0.509931
\(403\) 4.95017 0.246585
\(404\) −25.8211 −1.28465
\(405\) −1.23869 −0.0615512
\(406\) 0 0
\(407\) 40.1093 1.98815
\(408\) −13.1299 −0.650028
\(409\) −24.1704 −1.19515 −0.597576 0.801812i \(-0.703869\pi\)
−0.597576 + 0.801812i \(0.703869\pi\)
\(410\) 7.10735 0.351007
\(411\) −7.25647 −0.357935
\(412\) −45.3478 −2.23413
\(413\) 0 0
\(414\) 2.27502 0.111811
\(415\) −19.2241 −0.943675
\(416\) −5.42165 −0.265818
\(417\) −3.52170 −0.172458
\(418\) −96.2263 −4.70658
\(419\) −19.9385 −0.974059 −0.487029 0.873386i \(-0.661920\pi\)
−0.487029 + 0.873386i \(0.661920\pi\)
\(420\) 0 0
\(421\) −36.1003 −1.75942 −0.879711 0.475508i \(-0.842264\pi\)
−0.879711 + 0.475508i \(0.842264\pi\)
\(422\) 26.5471 1.29229
\(423\) 13.3578 0.649478
\(424\) 11.3059 0.549063
\(425\) −17.0119 −0.825197
\(426\) −36.0761 −1.74789
\(427\) 0 0
\(428\) −20.8253 −1.00663
\(429\) −5.57774 −0.269296
\(430\) 0.652010 0.0314427
\(431\) 10.8677 0.523479 0.261739 0.965139i \(-0.415704\pi\)
0.261739 + 0.965139i \(0.415704\pi\)
\(432\) 0.266205 0.0128078
\(433\) −26.1898 −1.25860 −0.629301 0.777161i \(-0.716659\pi\)
−0.629301 + 0.777161i \(0.716659\pi\)
\(434\) 0 0
\(435\) 4.16206 0.199556
\(436\) 57.9121 2.77349
\(437\) −6.90370 −0.330249
\(438\) 9.73201 0.465013
\(439\) −31.4062 −1.49894 −0.749468 0.662041i \(-0.769690\pi\)
−0.749468 + 0.662041i \(0.769690\pi\)
\(440\) −20.2994 −0.967735
\(441\) 0 0
\(442\) 10.1669 0.483589
\(443\) −9.07886 −0.431350 −0.215675 0.976465i \(-0.569195\pi\)
−0.215675 + 0.976465i \(0.569195\pi\)
\(444\) −20.7904 −0.986668
\(445\) −0.692880 −0.0328457
\(446\) 5.64587 0.267340
\(447\) 10.9243 0.516700
\(448\) 0 0
\(449\) 23.9369 1.12965 0.564825 0.825211i \(-0.308944\pi\)
0.564825 + 0.825211i \(0.308944\pi\)
\(450\) −7.88440 −0.371674
\(451\) −15.4520 −0.727604
\(452\) −59.0510 −2.77752
\(453\) 15.8535 0.744864
\(454\) −50.1631 −2.35427
\(455\) 0 0
\(456\) 18.4661 0.864754
\(457\) −4.81036 −0.225019 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(458\) 33.5490 1.56764
\(459\) −4.90873 −0.229120
\(460\) −3.93376 −0.183412
\(461\) 23.4694 1.09308 0.546540 0.837433i \(-0.315945\pi\)
0.546540 + 0.837433i \(0.315945\pi\)
\(462\) 0 0
\(463\) 23.4343 1.08908 0.544542 0.838733i \(-0.316703\pi\)
0.544542 + 0.838733i \(0.316703\pi\)
\(464\) −0.894460 −0.0415243
\(465\) 6.73521 0.312338
\(466\) −7.77182 −0.360022
\(467\) 13.4506 0.622417 0.311209 0.950342i \(-0.399266\pi\)
0.311209 + 0.950342i \(0.399266\pi\)
\(468\) 2.89118 0.133645
\(469\) 0 0
\(470\) −37.6430 −1.73634
\(471\) 9.19996 0.423912
\(472\) −2.01734 −0.0928557
\(473\) −1.41752 −0.0651778
\(474\) 0.505090 0.0231996
\(475\) 23.9257 1.09779
\(476\) 0 0
\(477\) 4.22681 0.193532
\(478\) −24.7411 −1.13163
\(479\) 6.76164 0.308947 0.154474 0.987997i \(-0.450632\pi\)
0.154474 + 0.987997i \(0.450632\pi\)
\(480\) −7.37672 −0.336700
\(481\) 5.96007 0.271756
\(482\) −48.6774 −2.21720
\(483\) 0 0
\(484\) 84.2722 3.83055
\(485\) −21.8600 −0.992612
\(486\) −2.27502 −0.103197
\(487\) 19.8666 0.900241 0.450121 0.892968i \(-0.351381\pi\)
0.450121 + 0.892968i \(0.351381\pi\)
\(488\) 9.56684 0.433071
\(489\) 1.64877 0.0745601
\(490\) 0 0
\(491\) 4.20340 0.189697 0.0948483 0.995492i \(-0.469763\pi\)
0.0948483 + 0.995492i \(0.469763\pi\)
\(492\) 8.00942 0.361093
\(493\) 16.4935 0.742831
\(494\) −14.2988 −0.643334
\(495\) −7.58909 −0.341104
\(496\) −1.44745 −0.0649925
\(497\) 0 0
\(498\) −35.3076 −1.58217
\(499\) 6.94840 0.311053 0.155527 0.987832i \(-0.450293\pi\)
0.155527 + 0.987832i \(0.450293\pi\)
\(500\) 33.3017 1.48930
\(501\) −5.46655 −0.244228
\(502\) 10.6598 0.475772
\(503\) 5.90346 0.263222 0.131611 0.991301i \(-0.457985\pi\)
0.131611 + 0.991301i \(0.457985\pi\)
\(504\) 0 0
\(505\) 10.0715 0.448176
\(506\) 13.9384 0.619636
\(507\) 12.1712 0.540541
\(508\) −13.3319 −0.591508
\(509\) −26.6647 −1.18189 −0.590947 0.806711i \(-0.701246\pi\)
−0.590947 + 0.806711i \(0.701246\pi\)
\(510\) 13.8331 0.612539
\(511\) 0 0
\(512\) 3.00941 0.132998
\(513\) 6.90370 0.304806
\(514\) −61.4652 −2.71111
\(515\) 17.6879 0.779423
\(516\) 0.734764 0.0323462
\(517\) 81.8390 3.59928
\(518\) 0 0
\(519\) 0.509172 0.0223502
\(520\) −3.01640 −0.132278
\(521\) 5.31311 0.232771 0.116386 0.993204i \(-0.462869\pi\)
0.116386 + 0.993204i \(0.462869\pi\)
\(522\) 7.64417 0.334576
\(523\) 17.0386 0.745047 0.372524 0.928023i \(-0.378493\pi\)
0.372524 + 0.928023i \(0.378493\pi\)
\(524\) 12.1802 0.532096
\(525\) 0 0
\(526\) −45.5149 −1.98454
\(527\) 26.6905 1.16266
\(528\) 1.63096 0.0709783
\(529\) 1.00000 0.0434783
\(530\) −11.9114 −0.517398
\(531\) −0.754201 −0.0327295
\(532\) 0 0
\(533\) −2.29610 −0.0994550
\(534\) −1.27256 −0.0550692
\(535\) 8.12290 0.351184
\(536\) −12.0207 −0.519217
\(537\) −8.35087 −0.360367
\(538\) −14.2888 −0.616033
\(539\) 0 0
\(540\) 3.93376 0.169282
\(541\) 38.7960 1.66797 0.833986 0.551786i \(-0.186053\pi\)
0.833986 + 0.551786i \(0.186053\pi\)
\(542\) 57.8317 2.48408
\(543\) 2.42861 0.104221
\(544\) −29.2327 −1.25334
\(545\) −22.5886 −0.967590
\(546\) 0 0
\(547\) 29.7356 1.27140 0.635701 0.771935i \(-0.280711\pi\)
0.635701 + 0.771935i \(0.280711\pi\)
\(548\) 23.0446 0.984415
\(549\) 3.57664 0.152647
\(550\) −48.3053 −2.05975
\(551\) −23.1967 −0.988214
\(552\) −2.67481 −0.113847
\(553\) 0 0
\(554\) −34.4463 −1.46348
\(555\) 8.10930 0.344221
\(556\) 11.1840 0.474306
\(557\) −33.5645 −1.42217 −0.711086 0.703105i \(-0.751796\pi\)
−0.711086 + 0.703105i \(0.751796\pi\)
\(558\) 12.3701 0.523668
\(559\) −0.210638 −0.00890904
\(560\) 0 0
\(561\) −30.0743 −1.26974
\(562\) −62.1421 −2.62130
\(563\) −30.5532 −1.28766 −0.643831 0.765167i \(-0.722656\pi\)
−0.643831 + 0.765167i \(0.722656\pi\)
\(564\) −42.4207 −1.78623
\(565\) 23.0328 0.968999
\(566\) 19.4346 0.816898
\(567\) 0 0
\(568\) 42.4157 1.77972
\(569\) 42.1160 1.76559 0.882797 0.469754i \(-0.155657\pi\)
0.882797 + 0.469754i \(0.155657\pi\)
\(570\) −19.4550 −0.814881
\(571\) 6.51703 0.272729 0.136365 0.990659i \(-0.456458\pi\)
0.136365 + 0.990659i \(0.456458\pi\)
\(572\) 17.7134 0.740634
\(573\) 14.2931 0.597104
\(574\) 0 0
\(575\) −3.46564 −0.144527
\(576\) −13.0159 −0.542329
\(577\) 41.5356 1.72915 0.864575 0.502503i \(-0.167588\pi\)
0.864575 + 0.502503i \(0.167588\pi\)
\(578\) 16.1427 0.671448
\(579\) −2.09005 −0.0868594
\(580\) −13.2176 −0.548830
\(581\) 0 0
\(582\) −40.1488 −1.66422
\(583\) 25.8963 1.07252
\(584\) −11.4422 −0.473482
\(585\) −1.12771 −0.0466249
\(586\) 30.9751 1.27957
\(587\) 12.6109 0.520506 0.260253 0.965540i \(-0.416194\pi\)
0.260253 + 0.965540i \(0.416194\pi\)
\(588\) 0 0
\(589\) −37.5379 −1.54672
\(590\) 2.12538 0.0875005
\(591\) −10.8863 −0.447801
\(592\) −1.74275 −0.0716267
\(593\) 10.3859 0.426498 0.213249 0.976998i \(-0.431596\pi\)
0.213249 + 0.976998i \(0.431596\pi\)
\(594\) −13.9384 −0.571898
\(595\) 0 0
\(596\) −34.6925 −1.42106
\(597\) −3.44053 −0.140812
\(598\) 2.07118 0.0846969
\(599\) 28.0358 1.14551 0.572757 0.819725i \(-0.305874\pi\)
0.572757 + 0.819725i \(0.305874\pi\)
\(600\) 9.26992 0.378443
\(601\) 3.81190 0.155491 0.0777454 0.996973i \(-0.475228\pi\)
0.0777454 + 0.996973i \(0.475228\pi\)
\(602\) 0 0
\(603\) −4.49406 −0.183012
\(604\) −50.3465 −2.04857
\(605\) −32.8704 −1.33637
\(606\) 18.4976 0.751416
\(607\) 0.159930 0.00649136 0.00324568 0.999995i \(-0.498967\pi\)
0.00324568 + 0.999995i \(0.498967\pi\)
\(608\) 41.1132 1.66736
\(609\) 0 0
\(610\) −10.0792 −0.408094
\(611\) 12.1609 0.491979
\(612\) 15.5888 0.630140
\(613\) −16.5339 −0.667797 −0.333899 0.942609i \(-0.608364\pi\)
−0.333899 + 0.942609i \(0.608364\pi\)
\(614\) −4.42477 −0.178569
\(615\) −3.12408 −0.125975
\(616\) 0 0
\(617\) 35.8727 1.44418 0.722091 0.691798i \(-0.243181\pi\)
0.722091 + 0.691798i \(0.243181\pi\)
\(618\) 32.4862 1.30679
\(619\) −36.6950 −1.47490 −0.737448 0.675404i \(-0.763969\pi\)
−0.737448 + 0.675404i \(0.763969\pi\)
\(620\) −21.3892 −0.859011
\(621\) −1.00000 −0.0401286
\(622\) −32.3098 −1.29550
\(623\) 0 0
\(624\) 0.242353 0.00970189
\(625\) 4.33883 0.173553
\(626\) −9.56548 −0.382314
\(627\) 42.2968 1.68917
\(628\) −29.2166 −1.16587
\(629\) 32.1358 1.28134
\(630\) 0 0
\(631\) 12.3865 0.493098 0.246549 0.969130i \(-0.420703\pi\)
0.246549 + 0.969130i \(0.420703\pi\)
\(632\) −0.593849 −0.0236220
\(633\) −11.6690 −0.463799
\(634\) 1.67995 0.0667194
\(635\) 5.20012 0.206360
\(636\) −13.4232 −0.532264
\(637\) 0 0
\(638\) 46.8335 1.85415
\(639\) 15.8575 0.627312
\(640\) 21.9261 0.866705
\(641\) 27.0383 1.06795 0.533974 0.845501i \(-0.320698\pi\)
0.533974 + 0.845501i \(0.320698\pi\)
\(642\) 14.9188 0.588797
\(643\) −27.4371 −1.08201 −0.541006 0.841019i \(-0.681956\pi\)
−0.541006 + 0.841019i \(0.681956\pi\)
\(644\) 0 0
\(645\) −0.286595 −0.0112847
\(646\) −77.0969 −3.03334
\(647\) 0.875536 0.0344209 0.0172104 0.999852i \(-0.494521\pi\)
0.0172104 + 0.999852i \(0.494521\pi\)
\(648\) 2.67481 0.105076
\(649\) −4.62075 −0.181380
\(650\) −7.17796 −0.281543
\(651\) 0 0
\(652\) −5.23606 −0.205060
\(653\) −15.5487 −0.608466 −0.304233 0.952598i \(-0.598400\pi\)
−0.304233 + 0.952598i \(0.598400\pi\)
\(654\) −41.4869 −1.62227
\(655\) −4.75091 −0.185633
\(656\) 0.671389 0.0262133
\(657\) −4.27776 −0.166891
\(658\) 0 0
\(659\) 21.1989 0.825794 0.412897 0.910778i \(-0.364517\pi\)
0.412897 + 0.910778i \(0.364517\pi\)
\(660\) 24.1009 0.938126
\(661\) −8.65093 −0.336482 −0.168241 0.985746i \(-0.553809\pi\)
−0.168241 + 0.985746i \(0.553809\pi\)
\(662\) −46.8629 −1.82138
\(663\) −4.46891 −0.173558
\(664\) 41.5122 1.61099
\(665\) 0 0
\(666\) 14.8938 0.577122
\(667\) 3.36004 0.130101
\(668\) 17.3603 0.671690
\(669\) −2.48168 −0.0959471
\(670\) 12.6645 0.489273
\(671\) 21.9130 0.845942
\(672\) 0 0
\(673\) −9.05137 −0.348905 −0.174452 0.984666i \(-0.555816\pi\)
−0.174452 + 0.984666i \(0.555816\pi\)
\(674\) −63.4805 −2.44518
\(675\) 3.46564 0.133392
\(676\) −38.6523 −1.48663
\(677\) −16.8996 −0.649503 −0.324751 0.945799i \(-0.605281\pi\)
−0.324751 + 0.945799i \(0.605281\pi\)
\(678\) 42.3028 1.62463
\(679\) 0 0
\(680\) −16.2639 −0.623694
\(681\) 22.0495 0.844938
\(682\) 75.7878 2.90206
\(683\) 22.7419 0.870194 0.435097 0.900384i \(-0.356714\pi\)
0.435097 + 0.900384i \(0.356714\pi\)
\(684\) −21.9243 −0.838296
\(685\) −8.98854 −0.343434
\(686\) 0 0
\(687\) −14.7467 −0.562621
\(688\) 0.0615915 0.00234815
\(689\) 3.84809 0.146600
\(690\) 2.81806 0.107282
\(691\) 35.1619 1.33762 0.668810 0.743433i \(-0.266804\pi\)
0.668810 + 0.743433i \(0.266804\pi\)
\(692\) −1.61699 −0.0614689
\(693\) 0 0
\(694\) −4.74418 −0.180086
\(695\) −4.36231 −0.165472
\(696\) −8.98747 −0.340669
\(697\) −12.3802 −0.468933
\(698\) −9.19842 −0.348165
\(699\) 3.41615 0.129211
\(700\) 0 0
\(701\) −35.0402 −1.32345 −0.661725 0.749746i \(-0.730175\pi\)
−0.661725 + 0.749746i \(0.730175\pi\)
\(702\) −2.07118 −0.0781717
\(703\) −45.1961 −1.70460
\(704\) −79.7444 −3.00548
\(705\) 16.5462 0.623166
\(706\) −10.2556 −0.385976
\(707\) 0 0
\(708\) 2.39514 0.0900147
\(709\) −49.6644 −1.86519 −0.932593 0.360930i \(-0.882459\pi\)
−0.932593 + 0.360930i \(0.882459\pi\)
\(710\) −44.6873 −1.67708
\(711\) −0.222015 −0.00832623
\(712\) 1.49619 0.0560721
\(713\) 5.43735 0.203630
\(714\) 0 0
\(715\) −6.90911 −0.258386
\(716\) 26.5201 0.991103
\(717\) 10.8751 0.406139
\(718\) −47.3518 −1.76715
\(719\) 3.54087 0.132052 0.0660261 0.997818i \(-0.478968\pi\)
0.0660261 + 0.997818i \(0.478968\pi\)
\(720\) 0.329747 0.0122889
\(721\) 0 0
\(722\) 65.2046 2.42667
\(723\) 21.3965 0.795743
\(724\) −7.71259 −0.286636
\(725\) −11.6447 −0.432473
\(726\) −60.3708 −2.24057
\(727\) 7.64097 0.283388 0.141694 0.989911i \(-0.454745\pi\)
0.141694 + 0.989911i \(0.454745\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 12.0550 0.446175
\(731\) −1.13573 −0.0420063
\(732\) −11.3585 −0.419820
\(733\) 20.1456 0.744094 0.372047 0.928214i \(-0.378656\pi\)
0.372047 + 0.928214i \(0.378656\pi\)
\(734\) 19.5019 0.719829
\(735\) 0 0
\(736\) −5.95524 −0.219513
\(737\) −27.5337 −1.01422
\(738\) −5.73777 −0.211210
\(739\) 38.9079 1.43125 0.715626 0.698484i \(-0.246142\pi\)
0.715626 + 0.698484i \(0.246142\pi\)
\(740\) −25.7529 −0.946697
\(741\) 6.28513 0.230890
\(742\) 0 0
\(743\) −12.5828 −0.461617 −0.230809 0.972999i \(-0.574137\pi\)
−0.230809 + 0.972999i \(0.574137\pi\)
\(744\) −14.5439 −0.533205
\(745\) 13.5318 0.495768
\(746\) −3.31032 −0.121199
\(747\) 15.5197 0.567835
\(748\) 95.5077 3.49211
\(749\) 0 0
\(750\) −23.8566 −0.871122
\(751\) −23.2255 −0.847511 −0.423756 0.905777i \(-0.639288\pi\)
−0.423756 + 0.905777i \(0.639288\pi\)
\(752\) −3.55591 −0.129671
\(753\) −4.68560 −0.170753
\(754\) 6.95925 0.253441
\(755\) 19.6377 0.714688
\(756\) 0 0
\(757\) −31.3210 −1.13838 −0.569191 0.822205i \(-0.692744\pi\)
−0.569191 + 0.822205i \(0.692744\pi\)
\(758\) −5.94622 −0.215976
\(759\) −6.12669 −0.222385
\(760\) 22.8738 0.829721
\(761\) −25.8770 −0.938040 −0.469020 0.883188i \(-0.655393\pi\)
−0.469020 + 0.883188i \(0.655393\pi\)
\(762\) 9.55070 0.345985
\(763\) 0 0
\(764\) −45.3911 −1.64219
\(765\) −6.08041 −0.219838
\(766\) −25.7538 −0.930524
\(767\) −0.686624 −0.0247926
\(768\) 14.2383 0.513782
\(769\) 2.66864 0.0962338 0.0481169 0.998842i \(-0.484678\pi\)
0.0481169 + 0.998842i \(0.484678\pi\)
\(770\) 0 0
\(771\) 27.0174 0.973008
\(772\) 6.63742 0.238886
\(773\) −5.78286 −0.207995 −0.103998 0.994578i \(-0.533163\pi\)
−0.103998 + 0.994578i \(0.533163\pi\)
\(774\) −0.526369 −0.0189199
\(775\) −18.8439 −0.676893
\(776\) 47.2041 1.69453
\(777\) 0 0
\(778\) −70.4598 −2.52611
\(779\) 17.4116 0.623837
\(780\) 3.58129 0.128231
\(781\) 97.1539 3.47644
\(782\) 11.1675 0.399348
\(783\) −3.36004 −0.120078
\(784\) 0 0
\(785\) 11.3959 0.406738
\(786\) −8.72566 −0.311234
\(787\) −7.55455 −0.269291 −0.134645 0.990894i \(-0.542990\pi\)
−0.134645 + 0.990894i \(0.542990\pi\)
\(788\) 34.5718 1.23157
\(789\) 20.0063 0.712244
\(790\) 0.625652 0.0222597
\(791\) 0 0
\(792\) 16.3877 0.582313
\(793\) 3.25618 0.115630
\(794\) 56.9656 2.02163
\(795\) 5.23572 0.185692
\(796\) 10.9262 0.387269
\(797\) −17.6173 −0.624035 −0.312018 0.950076i \(-0.601005\pi\)
−0.312018 + 0.950076i \(0.601005\pi\)
\(798\) 0 0
\(799\) 65.5698 2.31969
\(800\) 20.6387 0.729689
\(801\) 0.559363 0.0197641
\(802\) 79.5678 2.80964
\(803\) −26.2085 −0.924879
\(804\) 14.2719 0.503331
\(805\) 0 0
\(806\) 11.2617 0.396678
\(807\) 6.28072 0.221092
\(808\) −21.7482 −0.765099
\(809\) 23.2841 0.818624 0.409312 0.912394i \(-0.365769\pi\)
0.409312 + 0.912394i \(0.365769\pi\)
\(810\) −2.81806 −0.0990164
\(811\) 35.2946 1.23936 0.619680 0.784855i \(-0.287262\pi\)
0.619680 + 0.784855i \(0.287262\pi\)
\(812\) 0 0
\(813\) −25.4203 −0.891527
\(814\) 91.2496 3.19830
\(815\) 2.04232 0.0715395
\(816\) 1.30673 0.0457447
\(817\) 1.59730 0.0558824
\(818\) −54.9883 −1.92262
\(819\) 0 0
\(820\) 9.92122 0.346464
\(821\) 1.86839 0.0652072 0.0326036 0.999468i \(-0.489620\pi\)
0.0326036 + 0.999468i \(0.489620\pi\)
\(822\) −16.5086 −0.575804
\(823\) −19.4504 −0.677997 −0.338999 0.940787i \(-0.610088\pi\)
−0.338999 + 0.940787i \(0.610088\pi\)
\(824\) −38.1949 −1.33058
\(825\) 21.2329 0.739234
\(826\) 0 0
\(827\) 11.1883 0.389055 0.194527 0.980897i \(-0.437683\pi\)
0.194527 + 0.980897i \(0.437683\pi\)
\(828\) 3.17573 0.110364
\(829\) 0.598968 0.0208030 0.0104015 0.999946i \(-0.496689\pi\)
0.0104015 + 0.999946i \(0.496689\pi\)
\(830\) −43.7353 −1.51808
\(831\) 15.1411 0.525238
\(832\) −11.8497 −0.410814
\(833\) 0 0
\(834\) −8.01195 −0.277431
\(835\) −6.77139 −0.234333
\(836\) −134.323 −4.64567
\(837\) −5.43735 −0.187942
\(838\) −45.3605 −1.56695
\(839\) −43.0684 −1.48689 −0.743443 0.668799i \(-0.766809\pi\)
−0.743443 + 0.668799i \(0.766809\pi\)
\(840\) 0 0
\(841\) −17.7101 −0.610694
\(842\) −82.1291 −2.83036
\(843\) 27.3149 0.940776
\(844\) 37.0574 1.27557
\(845\) 15.0764 0.518642
\(846\) 30.3893 1.04480
\(847\) 0 0
\(848\) −1.12520 −0.0386395
\(849\) −8.54261 −0.293182
\(850\) −38.7024 −1.32748
\(851\) 6.54665 0.224416
\(852\) −50.3590 −1.72527
\(853\) 32.0125 1.09609 0.548043 0.836450i \(-0.315373\pi\)
0.548043 + 0.836450i \(0.315373\pi\)
\(854\) 0 0
\(855\) 8.55157 0.292458
\(856\) −17.5404 −0.599520
\(857\) −45.7684 −1.56342 −0.781710 0.623642i \(-0.785652\pi\)
−0.781710 + 0.623642i \(0.785652\pi\)
\(858\) −12.6895 −0.433212
\(859\) −43.2491 −1.47564 −0.737820 0.674998i \(-0.764145\pi\)
−0.737820 + 0.674998i \(0.764145\pi\)
\(860\) 0.910147 0.0310358
\(861\) 0 0
\(862\) 24.7243 0.842112
\(863\) −36.0384 −1.22676 −0.613380 0.789788i \(-0.710191\pi\)
−0.613380 + 0.789788i \(0.710191\pi\)
\(864\) 5.95524 0.202601
\(865\) 0.630709 0.0214447
\(866\) −59.5825 −2.02469
\(867\) −7.09562 −0.240980
\(868\) 0 0
\(869\) −1.36022 −0.0461423
\(870\) 9.46879 0.321022
\(871\) −4.09139 −0.138631
\(872\) 48.7774 1.65181
\(873\) 17.6476 0.597282
\(874\) −15.7061 −0.531266
\(875\) 0 0
\(876\) 13.5850 0.458995
\(877\) 26.9432 0.909806 0.454903 0.890541i \(-0.349674\pi\)
0.454903 + 0.890541i \(0.349674\pi\)
\(878\) −71.4498 −2.41131
\(879\) −13.6153 −0.459232
\(880\) 2.02026 0.0681028
\(881\) −36.5231 −1.23050 −0.615248 0.788334i \(-0.710944\pi\)
−0.615248 + 0.788334i \(0.710944\pi\)
\(882\) 0 0
\(883\) −54.7823 −1.84357 −0.921785 0.387702i \(-0.873269\pi\)
−0.921785 + 0.387702i \(0.873269\pi\)
\(884\) 14.1920 0.477330
\(885\) −0.934224 −0.0314036
\(886\) −20.6546 −0.693906
\(887\) −13.7438 −0.461470 −0.230735 0.973017i \(-0.574113\pi\)
−0.230735 + 0.973017i \(0.574113\pi\)
\(888\) −17.5110 −0.587632
\(889\) 0 0
\(890\) −1.57632 −0.0528383
\(891\) 6.12669 0.205252
\(892\) 7.88113 0.263880
\(893\) −92.2182 −3.08596
\(894\) 24.8530 0.831207
\(895\) −10.3442 −0.345767
\(896\) 0 0
\(897\) −0.910400 −0.0303974
\(898\) 54.4569 1.81725
\(899\) 18.2697 0.609330
\(900\) −11.0059 −0.366864
\(901\) 20.7483 0.691225
\(902\) −35.1536 −1.17049
\(903\) 0 0
\(904\) −49.7366 −1.65422
\(905\) 3.00830 0.0999992
\(906\) 36.0672 1.19825
\(907\) −21.9841 −0.729970 −0.364985 0.931013i \(-0.618926\pi\)
−0.364985 + 0.931013i \(0.618926\pi\)
\(908\) −70.0232 −2.32380
\(909\) −8.13075 −0.269680
\(910\) 0 0
\(911\) −17.3602 −0.575170 −0.287585 0.957755i \(-0.592852\pi\)
−0.287585 + 0.957755i \(0.592852\pi\)
\(912\) −1.83780 −0.0608557
\(913\) 95.0843 3.14683
\(914\) −10.9437 −0.361984
\(915\) 4.43037 0.146463
\(916\) 46.8315 1.54735
\(917\) 0 0
\(918\) −11.1675 −0.368581
\(919\) 5.14845 0.169832 0.0849159 0.996388i \(-0.472938\pi\)
0.0849159 + 0.996388i \(0.472938\pi\)
\(920\) −3.31327 −0.109235
\(921\) 1.94494 0.0640878
\(922\) 53.3935 1.75842
\(923\) 14.4367 0.475188
\(924\) 0 0
\(925\) −22.6883 −0.745988
\(926\) 53.3136 1.75199
\(927\) −14.2795 −0.469000
\(928\) −20.0099 −0.656856
\(929\) 12.4649 0.408961 0.204480 0.978871i \(-0.434450\pi\)
0.204480 + 0.978871i \(0.434450\pi\)
\(930\) 15.3228 0.502453
\(931\) 0 0
\(932\) −10.8488 −0.355363
\(933\) 14.2020 0.464951
\(934\) 30.6003 1.00127
\(935\) −37.2528 −1.21830
\(936\) 2.43515 0.0795953
\(937\) −1.66532 −0.0544038 −0.0272019 0.999630i \(-0.508660\pi\)
−0.0272019 + 0.999630i \(0.508660\pi\)
\(938\) 0 0
\(939\) 4.20457 0.137211
\(940\) −52.5463 −1.71387
\(941\) −34.1661 −1.11378 −0.556892 0.830585i \(-0.688006\pi\)
−0.556892 + 0.830585i \(0.688006\pi\)
\(942\) 20.9301 0.681940
\(943\) −2.52207 −0.0821300
\(944\) 0.200772 0.00653457
\(945\) 0 0
\(946\) −3.22490 −0.104851
\(947\) 52.3162 1.70005 0.850024 0.526744i \(-0.176587\pi\)
0.850024 + 0.526744i \(0.176587\pi\)
\(948\) 0.705061 0.0228993
\(949\) −3.89448 −0.126420
\(950\) 54.4316 1.76599
\(951\) −0.738433 −0.0239453
\(952\) 0 0
\(953\) −35.3873 −1.14631 −0.573154 0.819448i \(-0.694280\pi\)
−0.573154 + 0.819448i \(0.694280\pi\)
\(954\) 9.61609 0.311332
\(955\) 17.7048 0.572914
\(956\) −34.5364 −1.11699
\(957\) −20.5859 −0.665449
\(958\) 15.3829 0.496998
\(959\) 0 0
\(960\) −16.1227 −0.520358
\(961\) −1.43520 −0.0462967
\(962\) 13.5593 0.437170
\(963\) −6.55764 −0.211317
\(964\) −67.9494 −2.18850
\(965\) −2.58893 −0.0833406
\(966\) 0 0
\(967\) −1.97543 −0.0635254 −0.0317627 0.999495i \(-0.510112\pi\)
−0.0317627 + 0.999495i \(0.510112\pi\)
\(968\) 70.9796 2.28137
\(969\) 33.8884 1.08865
\(970\) −49.7320 −1.59680
\(971\) −28.7536 −0.922746 −0.461373 0.887206i \(-0.652643\pi\)
−0.461373 + 0.887206i \(0.652643\pi\)
\(972\) −3.17573 −0.101862
\(973\) 0 0
\(974\) 45.1969 1.44820
\(975\) 3.15512 0.101045
\(976\) −0.952121 −0.0304767
\(977\) −10.7374 −0.343520 −0.171760 0.985139i \(-0.554945\pi\)
−0.171760 + 0.985139i \(0.554945\pi\)
\(978\) 3.75100 0.119944
\(979\) 3.42705 0.109529
\(980\) 0 0
\(981\) 18.2358 0.582226
\(982\) 9.56282 0.305162
\(983\) −46.2766 −1.47599 −0.737997 0.674804i \(-0.764228\pi\)
−0.737997 + 0.674804i \(0.764228\pi\)
\(984\) 6.74607 0.215057
\(985\) −13.4848 −0.429660
\(986\) 37.5232 1.19498
\(987\) 0 0
\(988\) −19.9599 −0.635008
\(989\) −0.231369 −0.00735709
\(990\) −17.2654 −0.548729
\(991\) −42.9752 −1.36515 −0.682576 0.730814i \(-0.739141\pi\)
−0.682576 + 0.730814i \(0.739141\pi\)
\(992\) −32.3807 −1.02809
\(993\) 20.5989 0.653686
\(994\) 0 0
\(995\) −4.26176 −0.135107
\(996\) −49.2863 −1.56170
\(997\) −12.0816 −0.382628 −0.191314 0.981529i \(-0.561275\pi\)
−0.191314 + 0.981529i \(0.561275\pi\)
\(998\) 15.8078 0.500386
\(999\) −6.54665 −0.207127
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 3381.2.a.bi.1.8 10
7.2 even 3 483.2.i.h.277.3 20
7.4 even 3 483.2.i.h.415.3 yes 20
7.6 odd 2 3381.2.a.bj.1.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.3 20 7.2 even 3
483.2.i.h.415.3 yes 20 7.4 even 3
3381.2.a.bi.1.8 10 1.1 even 1 trivial
3381.2.a.bj.1.8 10 7.6 odd 2