Properties

Label 8281.2.a.cd.1.5
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.35100\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.17619 q^{2} +3.35100 q^{3} -0.616586 q^{4} -3.14862 q^{5} +3.94140 q^{6} -3.07759 q^{8} +8.22917 q^{9} +O(q^{10})\) \(q+1.17619 q^{2} +3.35100 q^{3} -0.616586 q^{4} -3.14862 q^{5} +3.94140 q^{6} -3.07759 q^{8} +8.22917 q^{9} -3.70337 q^{10} +0.773390 q^{11} -2.06618 q^{12} -10.5510 q^{15} -2.38665 q^{16} -5.75340 q^{17} +9.67904 q^{18} +1.22298 q^{19} +1.94140 q^{20} +0.909650 q^{22} -2.99182 q^{23} -10.3130 q^{24} +4.91383 q^{25} +17.5229 q^{27} -2.46882 q^{29} -12.4100 q^{30} +6.13487 q^{31} +3.34804 q^{32} +2.59163 q^{33} -6.76707 q^{34} -5.07399 q^{36} -4.99933 q^{37} +1.43845 q^{38} +9.69018 q^{40} +2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} -25.9106 q^{45} -3.51894 q^{46} -5.37169 q^{47} -7.99766 q^{48} +5.77958 q^{50} -19.2796 q^{51} -9.79015 q^{53} +20.6102 q^{54} -2.43511 q^{55} +4.09820 q^{57} -2.90379 q^{58} -2.50456 q^{59} +6.50561 q^{60} -10.9167 q^{61} +7.21575 q^{62} +8.71122 q^{64} +3.04823 q^{66} -4.32518 q^{67} +3.54746 q^{68} -10.0256 q^{69} -10.6649 q^{71} -25.3260 q^{72} +5.17450 q^{73} -5.88014 q^{74} +16.4662 q^{75} -0.754072 q^{76} -0.542038 q^{79} +7.51467 q^{80} +34.0318 q^{81} +3.01081 q^{82} -15.2259 q^{83} +18.1153 q^{85} -3.21275 q^{86} -8.27301 q^{87} -2.38018 q^{88} -9.23208 q^{89} -30.4757 q^{90} +1.84471 q^{92} +20.5579 q^{93} -6.31811 q^{94} -3.85070 q^{95} +11.2193 q^{96} +1.26291 q^{97} +6.36436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + 4q^{10} - 4q^{11} - 4q^{12} - 12q^{15} + 16q^{17} + 4q^{18} - 2q^{19} - 16q^{20} - 12q^{22} - 6q^{23} - 12q^{24} - 4q^{25} + 20q^{27} - 6q^{29} - 6q^{31} + 20q^{32} - 4q^{33} - 24q^{36} + 8q^{38} + 4q^{40} + 8q^{41} + 2q^{43} + 4q^{44} - 14q^{45} - 8q^{46} - 30q^{47} - 8q^{48} - 8q^{50} - 4q^{51} - 14q^{53} + 48q^{54} - 8q^{55} - 4q^{57} + 8q^{58} - 24q^{59} - 12q^{60} + 28q^{62} - 20q^{64} - 4q^{66} - 16q^{67} + 28q^{68} - 20q^{69} - 8q^{71} - 28q^{72} + 6q^{73} - 12q^{74} + 12q^{75} + 16q^{76} - 22q^{79} + 28q^{80} + 46q^{81} - 40q^{82} - 50q^{83} + 8q^{85} + 16q^{86} - 16q^{87} - 44q^{88} - 26q^{89} - 40q^{90} + 20q^{92} - 16q^{93} - 32q^{94} - 6q^{95} + 20q^{96} + 14q^{97} - 12q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17619 0.831689 0.415845 0.909436i \(-0.363486\pi\)
0.415845 + 0.909436i \(0.363486\pi\)
\(3\) 3.35100 1.93470 0.967349 0.253447i \(-0.0815644\pi\)
0.967349 + 0.253447i \(0.0815644\pi\)
\(4\) −0.616586 −0.308293
\(5\) −3.14862 −1.40811 −0.704054 0.710147i \(-0.748629\pi\)
−0.704054 + 0.710147i \(0.748629\pi\)
\(6\) 3.94140 1.60907
\(7\) 0 0
\(8\) −3.07759 −1.08809
\(9\) 8.22917 2.74306
\(10\) −3.70337 −1.17111
\(11\) 0.773390 0.233186 0.116593 0.993180i \(-0.462803\pi\)
0.116593 + 0.993180i \(0.462803\pi\)
\(12\) −2.06618 −0.596454
\(13\) 0 0
\(14\) 0 0
\(15\) −10.5510 −2.72426
\(16\) −2.38665 −0.596663
\(17\) −5.75340 −1.39540 −0.697702 0.716388i \(-0.745794\pi\)
−0.697702 + 0.716388i \(0.745794\pi\)
\(18\) 9.67904 2.28137
\(19\) 1.22298 0.280571 0.140285 0.990111i \(-0.455198\pi\)
0.140285 + 0.990111i \(0.455198\pi\)
\(20\) 1.94140 0.434109
\(21\) 0 0
\(22\) 0.909650 0.193938
\(23\) −2.99182 −0.623838 −0.311919 0.950109i \(-0.600972\pi\)
−0.311919 + 0.950109i \(0.600972\pi\)
\(24\) −10.3130 −2.10513
\(25\) 4.91383 0.982767
\(26\) 0 0
\(27\) 17.5229 3.37229
\(28\) 0 0
\(29\) −2.46882 −0.458449 −0.229224 0.973374i \(-0.573619\pi\)
−0.229224 + 0.973374i \(0.573619\pi\)
\(30\) −12.4100 −2.26574
\(31\) 6.13487 1.10185 0.550927 0.834553i \(-0.314274\pi\)
0.550927 + 0.834553i \(0.314274\pi\)
\(32\) 3.34804 0.591855
\(33\) 2.59163 0.451144
\(34\) −6.76707 −1.16054
\(35\) 0 0
\(36\) −5.07399 −0.845665
\(37\) −4.99933 −0.821884 −0.410942 0.911661i \(-0.634800\pi\)
−0.410942 + 0.911661i \(0.634800\pi\)
\(38\) 1.43845 0.233348
\(39\) 0 0
\(40\) 9.69018 1.53215
\(41\) 2.55981 0.399774 0.199887 0.979819i \(-0.435942\pi\)
0.199887 + 0.979819i \(0.435942\pi\)
\(42\) 0 0
\(43\) −2.73150 −0.416550 −0.208275 0.978070i \(-0.566785\pi\)
−0.208275 + 0.978070i \(0.566785\pi\)
\(44\) −0.476861 −0.0718895
\(45\) −25.9106 −3.86252
\(46\) −3.51894 −0.518839
\(47\) −5.37169 −0.783542 −0.391771 0.920063i \(-0.628137\pi\)
−0.391771 + 0.920063i \(0.628137\pi\)
\(48\) −7.99766 −1.15436
\(49\) 0 0
\(50\) 5.77958 0.817357
\(51\) −19.2796 −2.69969
\(52\) 0 0
\(53\) −9.79015 −1.34478 −0.672390 0.740197i \(-0.734732\pi\)
−0.672390 + 0.740197i \(0.734732\pi\)
\(54\) 20.6102 2.80470
\(55\) −2.43511 −0.328351
\(56\) 0 0
\(57\) 4.09820 0.542820
\(58\) −2.90379 −0.381287
\(59\) −2.50456 −0.326066 −0.163033 0.986621i \(-0.552128\pi\)
−0.163033 + 0.986621i \(0.552128\pi\)
\(60\) 6.50561 0.839871
\(61\) −10.9167 −1.39774 −0.698870 0.715248i \(-0.746314\pi\)
−0.698870 + 0.715248i \(0.746314\pi\)
\(62\) 7.21575 0.916401
\(63\) 0 0
\(64\) 8.71122 1.08890
\(65\) 0 0
\(66\) 3.04823 0.375212
\(67\) −4.32518 −0.528404 −0.264202 0.964467i \(-0.585109\pi\)
−0.264202 + 0.964467i \(0.585109\pi\)
\(68\) 3.54746 0.430193
\(69\) −10.0256 −1.20694
\(70\) 0 0
\(71\) −10.6649 −1.26570 −0.632848 0.774276i \(-0.718114\pi\)
−0.632848 + 0.774276i \(0.718114\pi\)
\(72\) −25.3260 −2.98470
\(73\) 5.17450 0.605630 0.302815 0.953049i \(-0.402074\pi\)
0.302815 + 0.953049i \(0.402074\pi\)
\(74\) −5.88014 −0.683552
\(75\) 16.4662 1.90136
\(76\) −0.754072 −0.0864979
\(77\) 0 0
\(78\) 0 0
\(79\) −0.542038 −0.0609841 −0.0304920 0.999535i \(-0.509707\pi\)
−0.0304920 + 0.999535i \(0.509707\pi\)
\(80\) 7.51467 0.840165
\(81\) 34.0318 3.78131
\(82\) 3.01081 0.332488
\(83\) −15.2259 −1.67125 −0.835627 0.549297i \(-0.814896\pi\)
−0.835627 + 0.549297i \(0.814896\pi\)
\(84\) 0 0
\(85\) 18.1153 1.96488
\(86\) −3.21275 −0.346440
\(87\) −8.27301 −0.886960
\(88\) −2.38018 −0.253728
\(89\) −9.23208 −0.978599 −0.489299 0.872116i \(-0.662747\pi\)
−0.489299 + 0.872116i \(0.662747\pi\)
\(90\) −30.4757 −3.21242
\(91\) 0 0
\(92\) 1.84471 0.192325
\(93\) 20.5579 2.13176
\(94\) −6.31811 −0.651663
\(95\) −3.85070 −0.395074
\(96\) 11.2193 1.14506
\(97\) 1.26291 0.128229 0.0641145 0.997943i \(-0.479578\pi\)
0.0641145 + 0.997943i \(0.479578\pi\)
\(98\) 0 0
\(99\) 6.36436 0.639642
\(100\) −3.02980 −0.302980
\(101\) 0.605447 0.0602443 0.0301221 0.999546i \(-0.490410\pi\)
0.0301221 + 0.999546i \(0.490410\pi\)
\(102\) −22.6764 −2.24530
\(103\) 7.64804 0.753584 0.376792 0.926298i \(-0.377027\pi\)
0.376792 + 0.926298i \(0.377027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −11.5150 −1.11844
\(107\) 4.82965 0.466900 0.233450 0.972369i \(-0.424999\pi\)
0.233450 + 0.972369i \(0.424999\pi\)
\(108\) −10.8044 −1.03965
\(109\) −7.23092 −0.692596 −0.346298 0.938125i \(-0.612561\pi\)
−0.346298 + 0.938125i \(0.612561\pi\)
\(110\) −2.86415 −0.273086
\(111\) −16.7527 −1.59010
\(112\) 0 0
\(113\) −9.19375 −0.864875 −0.432438 0.901664i \(-0.642346\pi\)
−0.432438 + 0.901664i \(0.642346\pi\)
\(114\) 4.82025 0.451458
\(115\) 9.42012 0.878430
\(116\) 1.52224 0.141336
\(117\) 0 0
\(118\) −2.94583 −0.271186
\(119\) 0 0
\(120\) 32.4718 2.96425
\(121\) −10.4019 −0.945624
\(122\) −12.8401 −1.16249
\(123\) 8.57790 0.773443
\(124\) −3.78267 −0.339694
\(125\) 0.271305 0.0242663
\(126\) 0 0
\(127\) −11.2118 −0.994888 −0.497444 0.867496i \(-0.665728\pi\)
−0.497444 + 0.867496i \(0.665728\pi\)
\(128\) 3.54994 0.313773
\(129\) −9.15325 −0.805899
\(130\) 0 0
\(131\) 15.7380 1.37503 0.687517 0.726169i \(-0.258701\pi\)
0.687517 + 0.726169i \(0.258701\pi\)
\(132\) −1.59796 −0.139084
\(133\) 0 0
\(134\) −5.08721 −0.439468
\(135\) −55.1732 −4.74855
\(136\) 17.7066 1.51833
\(137\) 18.6210 1.59090 0.795449 0.606020i \(-0.207235\pi\)
0.795449 + 0.606020i \(0.207235\pi\)
\(138\) −11.7919 −1.00380
\(139\) −11.9137 −1.01050 −0.505252 0.862972i \(-0.668601\pi\)
−0.505252 + 0.862972i \(0.668601\pi\)
\(140\) 0 0
\(141\) −18.0005 −1.51592
\(142\) −12.5440 −1.05267
\(143\) 0 0
\(144\) −19.6402 −1.63668
\(145\) 7.77339 0.645545
\(146\) 6.08618 0.503696
\(147\) 0 0
\(148\) 3.08251 0.253381
\(149\) −11.3753 −0.931904 −0.465952 0.884810i \(-0.654288\pi\)
−0.465952 + 0.884810i \(0.654288\pi\)
\(150\) 19.3674 1.58134
\(151\) 3.69250 0.300492 0.150246 0.988649i \(-0.451993\pi\)
0.150246 + 0.988649i \(0.451993\pi\)
\(152\) −3.76383 −0.305287
\(153\) −47.3457 −3.82767
\(154\) 0 0
\(155\) −19.3164 −1.55153
\(156\) 0 0
\(157\) 1.12065 0.0894374 0.0447187 0.999000i \(-0.485761\pi\)
0.0447187 + 0.999000i \(0.485761\pi\)
\(158\) −0.637538 −0.0507198
\(159\) −32.8068 −2.60175
\(160\) −10.5417 −0.833396
\(161\) 0 0
\(162\) 40.0277 3.14488
\(163\) 20.3435 1.59342 0.796712 0.604359i \(-0.206571\pi\)
0.796712 + 0.604359i \(0.206571\pi\)
\(164\) −1.57834 −0.123248
\(165\) −8.16005 −0.635259
\(166\) −17.9084 −1.38996
\(167\) −13.0063 −1.00646 −0.503228 0.864154i \(-0.667854\pi\)
−0.503228 + 0.864154i \(0.667854\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 21.3070 1.63417
\(171\) 10.0641 0.769622
\(172\) 1.68420 0.128419
\(173\) −25.9110 −1.96998 −0.984989 0.172616i \(-0.944778\pi\)
−0.984989 + 0.172616i \(0.944778\pi\)
\(174\) −9.73060 −0.737675
\(175\) 0 0
\(176\) −1.84581 −0.139133
\(177\) −8.39278 −0.630840
\(178\) −10.8587 −0.813890
\(179\) 9.64163 0.720649 0.360325 0.932827i \(-0.382666\pi\)
0.360325 + 0.932827i \(0.382666\pi\)
\(180\) 15.9761 1.19079
\(181\) −2.92683 −0.217550 −0.108775 0.994066i \(-0.534693\pi\)
−0.108775 + 0.994066i \(0.534693\pi\)
\(182\) 0 0
\(183\) −36.5818 −2.70421
\(184\) 9.20760 0.678793
\(185\) 15.7410 1.15730
\(186\) 24.1799 1.77296
\(187\) −4.44962 −0.325388
\(188\) 3.31211 0.241560
\(189\) 0 0
\(190\) −4.52914 −0.328579
\(191\) −16.1364 −1.16759 −0.583796 0.811901i \(-0.698433\pi\)
−0.583796 + 0.811901i \(0.698433\pi\)
\(192\) 29.1913 2.10670
\(193\) 0.0863817 0.00621789 0.00310894 0.999995i \(-0.499010\pi\)
0.00310894 + 0.999995i \(0.499010\pi\)
\(194\) 1.48542 0.106647
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0589 −1.07290 −0.536451 0.843932i \(-0.680235\pi\)
−0.536451 + 0.843932i \(0.680235\pi\)
\(198\) 7.48567 0.531983
\(199\) 15.4466 1.09498 0.547492 0.836811i \(-0.315583\pi\)
0.547492 + 0.836811i \(0.315583\pi\)
\(200\) −15.1228 −1.06934
\(201\) −14.4936 −1.02230
\(202\) 0.712119 0.0501045
\(203\) 0 0
\(204\) 11.8875 0.832294
\(205\) −8.05987 −0.562925
\(206\) 8.99552 0.626748
\(207\) −24.6202 −1.71122
\(208\) 0 0
\(209\) 0.945840 0.0654251
\(210\) 0 0
\(211\) −25.0561 −1.72493 −0.862466 0.506115i \(-0.831081\pi\)
−0.862466 + 0.506115i \(0.831081\pi\)
\(212\) 6.03647 0.414586
\(213\) −35.7382 −2.44874
\(214\) 5.68057 0.388316
\(215\) 8.60047 0.586547
\(216\) −53.9285 −3.66937
\(217\) 0 0
\(218\) −8.50491 −0.576025
\(219\) 17.3397 1.17171
\(220\) 1.50146 0.101228
\(221\) 0 0
\(222\) −19.7043 −1.32247
\(223\) 20.6640 1.38376 0.691881 0.722012i \(-0.256782\pi\)
0.691881 + 0.722012i \(0.256782\pi\)
\(224\) 0 0
\(225\) 40.4368 2.69579
\(226\) −10.8136 −0.719307
\(227\) −0.982378 −0.0652027 −0.0326013 0.999468i \(-0.510379\pi\)
−0.0326013 + 0.999468i \(0.510379\pi\)
\(228\) −2.52689 −0.167347
\(229\) 14.2742 0.943269 0.471634 0.881794i \(-0.343664\pi\)
0.471634 + 0.881794i \(0.343664\pi\)
\(230\) 11.0798 0.730581
\(231\) 0 0
\(232\) 7.59802 0.498835
\(233\) 1.39534 0.0914119 0.0457060 0.998955i \(-0.485446\pi\)
0.0457060 + 0.998955i \(0.485446\pi\)
\(234\) 0 0
\(235\) 16.9134 1.10331
\(236\) 1.54428 0.100524
\(237\) −1.81637 −0.117986
\(238\) 0 0
\(239\) −2.54875 −0.164865 −0.0824326 0.996597i \(-0.526269\pi\)
−0.0824326 + 0.996597i \(0.526269\pi\)
\(240\) 25.1816 1.62547
\(241\) −29.6746 −1.91151 −0.955755 0.294164i \(-0.904959\pi\)
−0.955755 + 0.294164i \(0.904959\pi\)
\(242\) −12.2345 −0.786466
\(243\) 61.4716 3.94340
\(244\) 6.73108 0.430913
\(245\) 0 0
\(246\) 10.0892 0.643264
\(247\) 0 0
\(248\) −18.8806 −1.19892
\(249\) −51.0218 −3.23337
\(250\) 0.319106 0.0201820
\(251\) 18.0858 1.14157 0.570784 0.821100i \(-0.306639\pi\)
0.570784 + 0.821100i \(0.306639\pi\)
\(252\) 0 0
\(253\) −2.31384 −0.145470
\(254\) −13.1872 −0.827437
\(255\) 60.7043 3.80145
\(256\) −13.2470 −0.827940
\(257\) 14.7403 0.919473 0.459737 0.888055i \(-0.347944\pi\)
0.459737 + 0.888055i \(0.347944\pi\)
\(258\) −10.7659 −0.670257
\(259\) 0 0
\(260\) 0 0
\(261\) −20.3164 −1.25755
\(262\) 18.5108 1.14360
\(263\) −8.15922 −0.503119 −0.251560 0.967842i \(-0.580943\pi\)
−0.251560 + 0.967842i \(0.580943\pi\)
\(264\) −7.97597 −0.490887
\(265\) 30.8255 1.89360
\(266\) 0 0
\(267\) −30.9367 −1.89329
\(268\) 2.66684 0.162903
\(269\) 0.442582 0.0269847 0.0134924 0.999909i \(-0.495705\pi\)
0.0134924 + 0.999909i \(0.495705\pi\)
\(270\) −64.8939 −3.94932
\(271\) 21.0831 1.28071 0.640354 0.768080i \(-0.278788\pi\)
0.640354 + 0.768080i \(0.278788\pi\)
\(272\) 13.7314 0.832586
\(273\) 0 0
\(274\) 21.9018 1.32313
\(275\) 3.80031 0.229167
\(276\) 6.18163 0.372090
\(277\) 3.59421 0.215955 0.107977 0.994153i \(-0.465563\pi\)
0.107977 + 0.994153i \(0.465563\pi\)
\(278\) −14.0127 −0.840426
\(279\) 50.4849 3.02245
\(280\) 0 0
\(281\) 9.01252 0.537642 0.268821 0.963190i \(-0.413366\pi\)
0.268821 + 0.963190i \(0.413366\pi\)
\(282\) −21.1720 −1.26077
\(283\) 27.7411 1.64904 0.824520 0.565833i \(-0.191445\pi\)
0.824520 + 0.565833i \(0.191445\pi\)
\(284\) 6.57585 0.390205
\(285\) −12.9037 −0.764349
\(286\) 0 0
\(287\) 0 0
\(288\) 27.5516 1.62349
\(289\) 16.1016 0.947153
\(290\) 9.14296 0.536893
\(291\) 4.23201 0.248085
\(292\) −3.19052 −0.186711
\(293\) −12.7409 −0.744333 −0.372167 0.928166i \(-0.621385\pi\)
−0.372167 + 0.928166i \(0.621385\pi\)
\(294\) 0 0
\(295\) 7.88593 0.459137
\(296\) 15.3859 0.894287
\(297\) 13.5521 0.786370
\(298\) −13.3795 −0.775055
\(299\) 0 0
\(300\) −10.1528 −0.586175
\(301\) 0 0
\(302\) 4.34307 0.249916
\(303\) 2.02885 0.116555
\(304\) −2.91883 −0.167406
\(305\) 34.3726 1.96817
\(306\) −55.6874 −3.18344
\(307\) −10.1384 −0.578626 −0.289313 0.957235i \(-0.593427\pi\)
−0.289313 + 0.957235i \(0.593427\pi\)
\(308\) 0 0
\(309\) 25.6286 1.45796
\(310\) −22.7197 −1.29039
\(311\) 20.6013 1.16819 0.584096 0.811685i \(-0.301449\pi\)
0.584096 + 0.811685i \(0.301449\pi\)
\(312\) 0 0
\(313\) 16.5456 0.935213 0.467606 0.883937i \(-0.345117\pi\)
0.467606 + 0.883937i \(0.345117\pi\)
\(314\) 1.31809 0.0743841
\(315\) 0 0
\(316\) 0.334213 0.0188010
\(317\) 21.2340 1.19262 0.596310 0.802754i \(-0.296633\pi\)
0.596310 + 0.802754i \(0.296633\pi\)
\(318\) −38.5869 −2.16384
\(319\) −1.90936 −0.106904
\(320\) −27.4284 −1.53329
\(321\) 16.1841 0.903310
\(322\) 0 0
\(323\) −7.03629 −0.391510
\(324\) −20.9835 −1.16575
\(325\) 0 0
\(326\) 23.9277 1.32523
\(327\) −24.2308 −1.33996
\(328\) −7.87804 −0.434992
\(329\) 0 0
\(330\) −9.59774 −0.528338
\(331\) −16.2415 −0.892712 −0.446356 0.894855i \(-0.647279\pi\)
−0.446356 + 0.894855i \(0.647279\pi\)
\(332\) 9.38804 0.515236
\(333\) −41.1403 −2.25448
\(334\) −15.2978 −0.837058
\(335\) 13.6184 0.744050
\(336\) 0 0
\(337\) 6.42141 0.349797 0.174898 0.984586i \(-0.444040\pi\)
0.174898 + 0.984586i \(0.444040\pi\)
\(338\) 0 0
\(339\) −30.8082 −1.67327
\(340\) −11.1696 −0.605758
\(341\) 4.74464 0.256937
\(342\) 11.8373 0.640086
\(343\) 0 0
\(344\) 8.40645 0.453245
\(345\) 31.5668 1.69950
\(346\) −30.4762 −1.63841
\(347\) 14.4809 0.777376 0.388688 0.921369i \(-0.372928\pi\)
0.388688 + 0.921369i \(0.372928\pi\)
\(348\) 5.10102 0.273443
\(349\) −1.74500 −0.0934079 −0.0467040 0.998909i \(-0.514872\pi\)
−0.0467040 + 0.998909i \(0.514872\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.58934 0.138012
\(353\) −15.9580 −0.849358 −0.424679 0.905344i \(-0.639613\pi\)
−0.424679 + 0.905344i \(0.639613\pi\)
\(354\) −9.87148 −0.524663
\(355\) 33.5799 1.78224
\(356\) 5.69237 0.301695
\(357\) 0 0
\(358\) 11.3404 0.599356
\(359\) −8.81400 −0.465185 −0.232593 0.972574i \(-0.574721\pi\)
−0.232593 + 0.972574i \(0.574721\pi\)
\(360\) 79.7422 4.20278
\(361\) −17.5043 −0.921280
\(362\) −3.44250 −0.180934
\(363\) −34.8566 −1.82950
\(364\) 0 0
\(365\) −16.2926 −0.852792
\(366\) −43.0271 −2.24906
\(367\) −19.7080 −1.02875 −0.514374 0.857566i \(-0.671976\pi\)
−0.514374 + 0.857566i \(0.671976\pi\)
\(368\) 7.14043 0.372221
\(369\) 21.0651 1.09660
\(370\) 18.5144 0.962515
\(371\) 0 0
\(372\) −12.6757 −0.657205
\(373\) −0.365792 −0.0189400 −0.00947000 0.999955i \(-0.503014\pi\)
−0.00947000 + 0.999955i \(0.503014\pi\)
\(374\) −5.23358 −0.270622
\(375\) 0.909143 0.0469479
\(376\) 16.5319 0.852566
\(377\) 0 0
\(378\) 0 0
\(379\) −7.39215 −0.379709 −0.189855 0.981812i \(-0.560802\pi\)
−0.189855 + 0.981812i \(0.560802\pi\)
\(380\) 2.37429 0.121798
\(381\) −37.5707 −1.92481
\(382\) −18.9795 −0.971073
\(383\) 3.35364 0.171363 0.0856814 0.996323i \(-0.472693\pi\)
0.0856814 + 0.996323i \(0.472693\pi\)
\(384\) 11.8958 0.607057
\(385\) 0 0
\(386\) 0.101601 0.00517135
\(387\) −22.4780 −1.14262
\(388\) −0.778692 −0.0395321
\(389\) −2.59344 −0.131492 −0.0657462 0.997836i \(-0.520943\pi\)
−0.0657462 + 0.997836i \(0.520943\pi\)
\(390\) 0 0
\(391\) 17.2131 0.870506
\(392\) 0 0
\(393\) 52.7379 2.66027
\(394\) −17.7121 −0.892321
\(395\) 1.70668 0.0858721
\(396\) −3.92417 −0.197197
\(397\) 33.4914 1.68088 0.840441 0.541902i \(-0.182296\pi\)
0.840441 + 0.541902i \(0.182296\pi\)
\(398\) 18.1681 0.910686
\(399\) 0 0
\(400\) −11.7276 −0.586380
\(401\) −39.5156 −1.97331 −0.986657 0.162811i \(-0.947944\pi\)
−0.986657 + 0.162811i \(0.947944\pi\)
\(402\) −17.0472 −0.850239
\(403\) 0 0
\(404\) −0.373310 −0.0185729
\(405\) −107.153 −5.32449
\(406\) 0 0
\(407\) −3.86643 −0.191652
\(408\) 59.3348 2.93751
\(409\) 10.9799 0.542922 0.271461 0.962449i \(-0.412493\pi\)
0.271461 + 0.962449i \(0.412493\pi\)
\(410\) −9.47990 −0.468179
\(411\) 62.3989 3.07791
\(412\) −4.71567 −0.232324
\(413\) 0 0
\(414\) −28.9580 −1.42321
\(415\) 47.9405 2.35331
\(416\) 0 0
\(417\) −39.9227 −1.95502
\(418\) 1.11248 0.0544134
\(419\) 31.5621 1.54191 0.770954 0.636891i \(-0.219780\pi\)
0.770954 + 0.636891i \(0.219780\pi\)
\(420\) 0 0
\(421\) 17.7055 0.862914 0.431457 0.902134i \(-0.358000\pi\)
0.431457 + 0.902134i \(0.358000\pi\)
\(422\) −29.4706 −1.43461
\(423\) −44.2046 −2.14930
\(424\) 30.1301 1.46325
\(425\) −28.2712 −1.37136
\(426\) −42.0348 −2.03659
\(427\) 0 0
\(428\) −2.97789 −0.143942
\(429\) 0 0
\(430\) 10.1158 0.487825
\(431\) 21.4816 1.03473 0.517367 0.855764i \(-0.326912\pi\)
0.517367 + 0.855764i \(0.326912\pi\)
\(432\) −41.8212 −2.01212
\(433\) 34.7200 1.66854 0.834269 0.551358i \(-0.185890\pi\)
0.834269 + 0.551358i \(0.185890\pi\)
\(434\) 0 0
\(435\) 26.0486 1.24893
\(436\) 4.45848 0.213522
\(437\) −3.65894 −0.175031
\(438\) 20.3948 0.974500
\(439\) 0.728505 0.0347697 0.0173848 0.999849i \(-0.494466\pi\)
0.0173848 + 0.999849i \(0.494466\pi\)
\(440\) 7.49428 0.357276
\(441\) 0 0
\(442\) 0 0
\(443\) 0.837291 0.0397809 0.0198904 0.999802i \(-0.493668\pi\)
0.0198904 + 0.999802i \(0.493668\pi\)
\(444\) 10.3295 0.490216
\(445\) 29.0684 1.37797
\(446\) 24.3047 1.15086
\(447\) −38.1187 −1.80295
\(448\) 0 0
\(449\) −26.4312 −1.24737 −0.623683 0.781677i \(-0.714365\pi\)
−0.623683 + 0.781677i \(0.714365\pi\)
\(450\) 47.5612 2.24206
\(451\) 1.97973 0.0932217
\(452\) 5.66873 0.266635
\(453\) 12.3736 0.581361
\(454\) −1.15546 −0.0542284
\(455\) 0 0
\(456\) −12.6126 −0.590639
\(457\) 2.40475 0.112489 0.0562447 0.998417i \(-0.482087\pi\)
0.0562447 + 0.998417i \(0.482087\pi\)
\(458\) 16.7892 0.784507
\(459\) −100.816 −4.70571
\(460\) −5.80831 −0.270814
\(461\) 22.2702 1.03722 0.518612 0.855010i \(-0.326449\pi\)
0.518612 + 0.855010i \(0.326449\pi\)
\(462\) 0 0
\(463\) −32.3085 −1.50151 −0.750753 0.660583i \(-0.770309\pi\)
−0.750753 + 0.660583i \(0.770309\pi\)
\(464\) 5.89221 0.273539
\(465\) −64.7291 −3.00174
\(466\) 1.64118 0.0760263
\(467\) −12.8744 −0.595756 −0.297878 0.954604i \(-0.596279\pi\)
−0.297878 + 0.954604i \(0.596279\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 19.8934 0.917612
\(471\) 3.75528 0.173034
\(472\) 7.70802 0.354791
\(473\) −2.11251 −0.0971335
\(474\) −2.13639 −0.0981275
\(475\) 6.00952 0.275736
\(476\) 0 0
\(477\) −80.5649 −3.68881
\(478\) −2.99781 −0.137117
\(479\) −10.5419 −0.481670 −0.240835 0.970566i \(-0.577421\pi\)
−0.240835 + 0.970566i \(0.577421\pi\)
\(480\) −35.3252 −1.61237
\(481\) 0 0
\(482\) −34.9029 −1.58978
\(483\) 0 0
\(484\) 6.41364 0.291529
\(485\) −3.97643 −0.180560
\(486\) 72.3020 3.27969
\(487\) −38.2416 −1.73289 −0.866446 0.499271i \(-0.833601\pi\)
−0.866446 + 0.499271i \(0.833601\pi\)
\(488\) 33.5972 1.52087
\(489\) 68.1709 3.08280
\(490\) 0 0
\(491\) −15.4291 −0.696306 −0.348153 0.937438i \(-0.613191\pi\)
−0.348153 + 0.937438i \(0.613191\pi\)
\(492\) −5.28901 −0.238447
\(493\) 14.2041 0.639721
\(494\) 0 0
\(495\) −20.0390 −0.900685
\(496\) −14.6418 −0.657436
\(497\) 0 0
\(498\) −60.0111 −2.68916
\(499\) 38.8212 1.73788 0.868938 0.494921i \(-0.164803\pi\)
0.868938 + 0.494921i \(0.164803\pi\)
\(500\) −0.167283 −0.00748112
\(501\) −43.5840 −1.94719
\(502\) 21.2723 0.949430
\(503\) −27.0935 −1.20804 −0.604020 0.796969i \(-0.706435\pi\)
−0.604020 + 0.796969i \(0.706435\pi\)
\(504\) 0 0
\(505\) −1.90633 −0.0848304
\(506\) −2.72151 −0.120986
\(507\) 0 0
\(508\) 6.91304 0.306717
\(509\) 15.7693 0.698961 0.349480 0.936944i \(-0.386358\pi\)
0.349480 + 0.936944i \(0.386358\pi\)
\(510\) 71.3995 3.16162
\(511\) 0 0
\(512\) −22.6809 −1.00236
\(513\) 21.4302 0.946167
\(514\) 17.3373 0.764716
\(515\) −24.0808 −1.06113
\(516\) 5.64376 0.248453
\(517\) −4.15441 −0.182711
\(518\) 0 0
\(519\) −86.8277 −3.81131
\(520\) 0 0
\(521\) 7.30737 0.320142 0.160071 0.987106i \(-0.448828\pi\)
0.160071 + 0.987106i \(0.448828\pi\)
\(522\) −23.8958 −1.04589
\(523\) 14.0826 0.615788 0.307894 0.951421i \(-0.400376\pi\)
0.307894 + 0.951421i \(0.400376\pi\)
\(524\) −9.70381 −0.423913
\(525\) 0 0
\(526\) −9.59677 −0.418439
\(527\) −35.2963 −1.53753
\(528\) −6.18531 −0.269181
\(529\) −14.0490 −0.610827
\(530\) 36.2565 1.57488
\(531\) −20.6105 −0.894419
\(532\) 0 0
\(533\) 0 0
\(534\) −36.3873 −1.57463
\(535\) −15.2067 −0.657445
\(536\) 13.3111 0.574953
\(537\) 32.3091 1.39424
\(538\) 0.520559 0.0224429
\(539\) 0 0
\(540\) 34.0190 1.46394
\(541\) 39.1750 1.68426 0.842132 0.539271i \(-0.181300\pi\)
0.842132 + 0.539271i \(0.181300\pi\)
\(542\) 24.7977 1.06515
\(543\) −9.80781 −0.420893
\(544\) −19.2626 −0.825877
\(545\) 22.7674 0.975250
\(546\) 0 0
\(547\) 38.7917 1.65862 0.829308 0.558792i \(-0.188735\pi\)
0.829308 + 0.558792i \(0.188735\pi\)
\(548\) −11.4814 −0.490463
\(549\) −89.8355 −3.83408
\(550\) 4.46987 0.190596
\(551\) −3.01932 −0.128627
\(552\) 30.8546 1.31326
\(553\) 0 0
\(554\) 4.22746 0.179607
\(555\) 52.7480 2.23903
\(556\) 7.34580 0.311531
\(557\) 27.4693 1.16391 0.581956 0.813220i \(-0.302287\pi\)
0.581956 + 0.813220i \(0.302287\pi\)
\(558\) 59.3796 2.51374
\(559\) 0 0
\(560\) 0 0
\(561\) −14.9107 −0.629528
\(562\) 10.6004 0.447151
\(563\) 14.1210 0.595129 0.297565 0.954702i \(-0.403826\pi\)
0.297565 + 0.954702i \(0.403826\pi\)
\(564\) 11.0989 0.467346
\(565\) 28.9477 1.21784
\(566\) 32.6288 1.37149
\(567\) 0 0
\(568\) 32.8223 1.37720
\(569\) −8.28649 −0.347388 −0.173694 0.984800i \(-0.555570\pi\)
−0.173694 + 0.984800i \(0.555570\pi\)
\(570\) −15.1771 −0.635701
\(571\) −23.5274 −0.984591 −0.492295 0.870428i \(-0.663842\pi\)
−0.492295 + 0.870428i \(0.663842\pi\)
\(572\) 0 0
\(573\) −54.0731 −2.25894
\(574\) 0 0
\(575\) −14.7013 −0.613087
\(576\) 71.6861 2.98692
\(577\) 17.7732 0.739906 0.369953 0.929050i \(-0.379374\pi\)
0.369953 + 0.929050i \(0.379374\pi\)
\(578\) 18.9385 0.787737
\(579\) 0.289465 0.0120297
\(580\) −4.79296 −0.199017
\(581\) 0 0
\(582\) 4.97763 0.206329
\(583\) −7.57160 −0.313584
\(584\) −15.9250 −0.658982
\(585\) 0 0
\(586\) −14.9857 −0.619054
\(587\) 6.64096 0.274102 0.137051 0.990564i \(-0.456238\pi\)
0.137051 + 0.990564i \(0.456238\pi\)
\(588\) 0 0
\(589\) 7.50282 0.309148
\(590\) 9.27532 0.381859
\(591\) −50.4623 −2.07574
\(592\) 11.9317 0.490388
\(593\) −34.0504 −1.39828 −0.699141 0.714984i \(-0.746434\pi\)
−0.699141 + 0.714984i \(0.746434\pi\)
\(594\) 15.9398 0.654016
\(595\) 0 0
\(596\) 7.01387 0.287299
\(597\) 51.7616 2.11846
\(598\) 0 0
\(599\) 9.38441 0.383437 0.191718 0.981450i \(-0.438594\pi\)
0.191718 + 0.981450i \(0.438594\pi\)
\(600\) −50.6764 −2.06885
\(601\) 8.80294 0.359079 0.179540 0.983751i \(-0.442539\pi\)
0.179540 + 0.983751i \(0.442539\pi\)
\(602\) 0 0
\(603\) −35.5926 −1.44944
\(604\) −2.27675 −0.0926394
\(605\) 32.7516 1.33154
\(606\) 2.38631 0.0969371
\(607\) 17.7720 0.721342 0.360671 0.932693i \(-0.382548\pi\)
0.360671 + 0.932693i \(0.382548\pi\)
\(608\) 4.09458 0.166057
\(609\) 0 0
\(610\) 40.4286 1.63691
\(611\) 0 0
\(612\) 29.1927 1.18004
\(613\) −10.6081 −0.428457 −0.214228 0.976784i \(-0.568724\pi\)
−0.214228 + 0.976784i \(0.568724\pi\)
\(614\) −11.9246 −0.481237
\(615\) −27.0086 −1.08909
\(616\) 0 0
\(617\) −49.3483 −1.98669 −0.993344 0.115188i \(-0.963253\pi\)
−0.993344 + 0.115188i \(0.963253\pi\)
\(618\) 30.1440 1.21257
\(619\) −7.42203 −0.298317 −0.149158 0.988813i \(-0.547656\pi\)
−0.149158 + 0.988813i \(0.547656\pi\)
\(620\) 11.9102 0.478325
\(621\) −52.4255 −2.10376
\(622\) 24.2309 0.971573
\(623\) 0 0
\(624\) 0 0
\(625\) −25.4234 −1.01694
\(626\) 19.4607 0.777806
\(627\) 3.16951 0.126578
\(628\) −0.690975 −0.0275729
\(629\) 28.7631 1.14686
\(630\) 0 0
\(631\) −35.5184 −1.41396 −0.706982 0.707231i \(-0.749944\pi\)
−0.706982 + 0.707231i \(0.749944\pi\)
\(632\) 1.66817 0.0663564
\(633\) −83.9628 −3.33722
\(634\) 24.9752 0.991890
\(635\) 35.3018 1.40091
\(636\) 20.2282 0.802099
\(637\) 0 0
\(638\) −2.24576 −0.0889106
\(639\) −87.7637 −3.47188
\(640\) −11.1774 −0.441827
\(641\) 37.1554 1.46755 0.733775 0.679393i \(-0.237757\pi\)
0.733775 + 0.679393i \(0.237757\pi\)
\(642\) 19.0356 0.751274
\(643\) −39.7694 −1.56835 −0.784176 0.620538i \(-0.786914\pi\)
−0.784176 + 0.620538i \(0.786914\pi\)
\(644\) 0 0
\(645\) 28.8201 1.13479
\(646\) −8.27599 −0.325614
\(647\) 22.9754 0.903256 0.451628 0.892206i \(-0.350843\pi\)
0.451628 + 0.892206i \(0.350843\pi\)
\(648\) −104.736 −4.11442
\(649\) −1.93700 −0.0760340
\(650\) 0 0
\(651\) 0 0
\(652\) −12.5435 −0.491241
\(653\) 18.1757 0.711268 0.355634 0.934625i \(-0.384265\pi\)
0.355634 + 0.934625i \(0.384265\pi\)
\(654\) −28.4999 −1.11443
\(655\) −49.5530 −1.93619
\(656\) −6.10936 −0.238531
\(657\) 42.5819 1.66128
\(658\) 0 0
\(659\) 14.1044 0.549431 0.274716 0.961526i \(-0.411416\pi\)
0.274716 + 0.961526i \(0.411416\pi\)
\(660\) 5.03137 0.195846
\(661\) 12.8557 0.500027 0.250013 0.968242i \(-0.419565\pi\)
0.250013 + 0.968242i \(0.419565\pi\)
\(662\) −19.1030 −0.742459
\(663\) 0 0
\(664\) 46.8590 1.81848
\(665\) 0 0
\(666\) −48.3887 −1.87502
\(667\) 7.38627 0.285997
\(668\) 8.01948 0.310283
\(669\) 69.2449 2.67716
\(670\) 16.0177 0.618819
\(671\) −8.44287 −0.325933
\(672\) 0 0
\(673\) −45.6138 −1.75828 −0.879141 0.476561i \(-0.841883\pi\)
−0.879141 + 0.476561i \(0.841883\pi\)
\(674\) 7.55278 0.290922
\(675\) 86.1048 3.31418
\(676\) 0 0
\(677\) 10.2469 0.393821 0.196910 0.980421i \(-0.436909\pi\)
0.196910 + 0.980421i \(0.436909\pi\)
\(678\) −36.2362 −1.39164
\(679\) 0 0
\(680\) −55.7515 −2.13797
\(681\) −3.29194 −0.126148
\(682\) 5.58058 0.213692
\(683\) −2.90040 −0.110981 −0.0554904 0.998459i \(-0.517672\pi\)
−0.0554904 + 0.998459i \(0.517672\pi\)
\(684\) −6.20539 −0.237269
\(685\) −58.6305 −2.24016
\(686\) 0 0
\(687\) 47.8329 1.82494
\(688\) 6.51914 0.248540
\(689\) 0 0
\(690\) 37.1284 1.41345
\(691\) −6.91470 −0.263048 −0.131524 0.991313i \(-0.541987\pi\)
−0.131524 + 0.991313i \(0.541987\pi\)
\(692\) 15.9764 0.607330
\(693\) 0 0
\(694\) 17.0323 0.646536
\(695\) 37.5117 1.42290
\(696\) 25.4610 0.965095
\(697\) −14.7276 −0.557847
\(698\) −2.05245 −0.0776864
\(699\) 4.67579 0.176855
\(700\) 0 0
\(701\) −26.0973 −0.985682 −0.492841 0.870119i \(-0.664042\pi\)
−0.492841 + 0.870119i \(0.664042\pi\)
\(702\) 0 0
\(703\) −6.11408 −0.230597
\(704\) 6.73717 0.253916
\(705\) 56.6769 2.13457
\(706\) −18.7696 −0.706402
\(707\) 0 0
\(708\) 5.17487 0.194483
\(709\) 3.38472 0.127116 0.0635580 0.997978i \(-0.479755\pi\)
0.0635580 + 0.997978i \(0.479755\pi\)
\(710\) 39.4962 1.48227
\(711\) −4.46053 −0.167283
\(712\) 28.4126 1.06481
\(713\) −18.3544 −0.687378
\(714\) 0 0
\(715\) 0 0
\(716\) −5.94489 −0.222171
\(717\) −8.54087 −0.318964
\(718\) −10.3669 −0.386890
\(719\) −2.41574 −0.0900918 −0.0450459 0.998985i \(-0.514343\pi\)
−0.0450459 + 0.998985i \(0.514343\pi\)
\(720\) 61.8395 2.30462
\(721\) 0 0
\(722\) −20.5883 −0.766219
\(723\) −99.4395 −3.69819
\(724\) 1.80464 0.0670691
\(725\) −12.1314 −0.450548
\(726\) −40.9979 −1.52157
\(727\) 17.0150 0.631050 0.315525 0.948917i \(-0.397819\pi\)
0.315525 + 0.948917i \(0.397819\pi\)
\(728\) 0 0
\(729\) 103.896 3.84799
\(730\) −19.1631 −0.709258
\(731\) 15.7154 0.581256
\(732\) 22.5558 0.833687
\(733\) −4.18453 −0.154559 −0.0772795 0.997009i \(-0.524623\pi\)
−0.0772795 + 0.997009i \(0.524623\pi\)
\(734\) −23.1803 −0.855599
\(735\) 0 0
\(736\) −10.0167 −0.369221
\(737\) −3.34505 −0.123216
\(738\) 24.7765 0.912034
\(739\) 28.0794 1.03292 0.516458 0.856312i \(-0.327250\pi\)
0.516458 + 0.856312i \(0.327250\pi\)
\(740\) −9.70567 −0.356788
\(741\) 0 0
\(742\) 0 0
\(743\) 26.5210 0.972961 0.486481 0.873691i \(-0.338280\pi\)
0.486481 + 0.873691i \(0.338280\pi\)
\(744\) −63.2689 −2.31955
\(745\) 35.8167 1.31222
\(746\) −0.430240 −0.0157522
\(747\) −125.296 −4.58435
\(748\) 2.74357 0.100315
\(749\) 0 0
\(750\) 1.06932 0.0390461
\(751\) −23.7162 −0.865418 −0.432709 0.901534i \(-0.642442\pi\)
−0.432709 + 0.901534i \(0.642442\pi\)
\(752\) 12.8204 0.467510
\(753\) 60.6056 2.20859
\(754\) 0 0
\(755\) −11.6263 −0.423125
\(756\) 0 0
\(757\) −52.3661 −1.90328 −0.951639 0.307219i \(-0.900602\pi\)
−0.951639 + 0.307219i \(0.900602\pi\)
\(758\) −8.69454 −0.315800
\(759\) −7.75368 −0.281441
\(760\) 11.8509 0.429877
\(761\) −23.8695 −0.865268 −0.432634 0.901570i \(-0.642416\pi\)
−0.432634 + 0.901570i \(0.642416\pi\)
\(762\) −44.1902 −1.60084
\(763\) 0 0
\(764\) 9.94949 0.359960
\(765\) 149.074 5.38978
\(766\) 3.94450 0.142521
\(767\) 0 0
\(768\) −44.3908 −1.60181
\(769\) −41.7599 −1.50590 −0.752950 0.658077i \(-0.771370\pi\)
−0.752950 + 0.658077i \(0.771370\pi\)
\(770\) 0 0
\(771\) 49.3946 1.77890
\(772\) −0.0532617 −0.00191693
\(773\) −2.97178 −0.106888 −0.0534438 0.998571i \(-0.517020\pi\)
−0.0534438 + 0.998571i \(0.517020\pi\)
\(774\) −26.4383 −0.950306
\(775\) 30.1457 1.08287
\(776\) −3.88672 −0.139525
\(777\) 0 0
\(778\) −3.05037 −0.109361
\(779\) 3.13059 0.112165
\(780\) 0 0
\(781\) −8.24815 −0.295142
\(782\) 20.2459 0.723990
\(783\) −43.2610 −1.54602
\(784\) 0 0
\(785\) −3.52850 −0.125937
\(786\) 62.0296 2.21252
\(787\) 48.6142 1.73291 0.866454 0.499256i \(-0.166393\pi\)
0.866454 + 0.499256i \(0.166393\pi\)
\(788\) 9.28509 0.330768
\(789\) −27.3415 −0.973384
\(790\) 2.00737 0.0714189
\(791\) 0 0
\(792\) −19.5869 −0.695990
\(793\) 0 0
\(794\) 39.3921 1.39797
\(795\) 103.296 3.66354
\(796\) −9.52418 −0.337576
\(797\) 25.6470 0.908463 0.454232 0.890884i \(-0.349914\pi\)
0.454232 + 0.890884i \(0.349914\pi\)
\(798\) 0 0
\(799\) 30.9055 1.09336
\(800\) 16.4517 0.581655
\(801\) −75.9724 −2.68435
\(802\) −46.4777 −1.64118
\(803\) 4.00191 0.141224
\(804\) 8.93657 0.315169
\(805\) 0 0
\(806\) 0 0
\(807\) 1.48309 0.0522073
\(808\) −1.86332 −0.0655514
\(809\) −11.9205 −0.419101 −0.209550 0.977798i \(-0.567200\pi\)
−0.209550 + 0.977798i \(0.567200\pi\)
\(810\) −126.032 −4.42832
\(811\) 10.5564 0.370685 0.185343 0.982674i \(-0.440661\pi\)
0.185343 + 0.982674i \(0.440661\pi\)
\(812\) 0 0
\(813\) 70.6494 2.47778
\(814\) −4.54764 −0.159395
\(815\) −64.0540 −2.24371
\(816\) 46.0137 1.61080
\(817\) −3.34057 −0.116872
\(818\) 12.9144 0.451543
\(819\) 0 0
\(820\) 4.96960 0.173546
\(821\) 46.2139 1.61288 0.806438 0.591319i \(-0.201393\pi\)
0.806438 + 0.591319i \(0.201393\pi\)
\(822\) 73.3927 2.55986
\(823\) 41.2171 1.43674 0.718368 0.695663i \(-0.244889\pi\)
0.718368 + 0.695663i \(0.244889\pi\)
\(824\) −23.5375 −0.819969
\(825\) 12.7348 0.443369
\(826\) 0 0
\(827\) −5.94430 −0.206704 −0.103352 0.994645i \(-0.532957\pi\)
−0.103352 + 0.994645i \(0.532957\pi\)
\(828\) 15.1805 0.527558
\(829\) 34.8106 1.20902 0.604511 0.796596i \(-0.293368\pi\)
0.604511 + 0.796596i \(0.293368\pi\)
\(830\) 56.3869 1.95722
\(831\) 12.0442 0.417808
\(832\) 0 0
\(833\) 0 0
\(834\) −46.9565 −1.62597
\(835\) 40.9519 1.41720
\(836\) −0.583191 −0.0201701
\(837\) 107.501 3.71578
\(838\) 37.1229 1.28239
\(839\) −10.7896 −0.372500 −0.186250 0.982502i \(-0.559633\pi\)
−0.186250 + 0.982502i \(0.559633\pi\)
\(840\) 0 0
\(841\) −22.9049 −0.789825
\(842\) 20.8250 0.717676
\(843\) 30.2009 1.04018
\(844\) 15.4492 0.531784
\(845\) 0 0
\(846\) −51.9928 −1.78755
\(847\) 0 0
\(848\) 23.3657 0.802381
\(849\) 92.9605 3.19039
\(850\) −33.2523 −1.14054
\(851\) 14.9571 0.512722
\(852\) 22.0356 0.754929
\(853\) −39.1053 −1.33894 −0.669470 0.742839i \(-0.733479\pi\)
−0.669470 + 0.742839i \(0.733479\pi\)
\(854\) 0 0
\(855\) −31.6881 −1.08371
\(856\) −14.8637 −0.508030
\(857\) −33.3654 −1.13974 −0.569870 0.821735i \(-0.693006\pi\)
−0.569870 + 0.821735i \(0.693006\pi\)
\(858\) 0 0
\(859\) −29.4914 −1.00623 −0.503116 0.864219i \(-0.667813\pi\)
−0.503116 + 0.864219i \(0.667813\pi\)
\(860\) −5.30293 −0.180828
\(861\) 0 0
\(862\) 25.2664 0.860578
\(863\) 24.6224 0.838157 0.419079 0.907950i \(-0.362353\pi\)
0.419079 + 0.907950i \(0.362353\pi\)
\(864\) 58.6675 1.99591
\(865\) 81.5841 2.77394
\(866\) 40.8372 1.38771
\(867\) 53.9564 1.83246
\(868\) 0 0
\(869\) −0.419207 −0.0142206
\(870\) 30.6380 1.03873
\(871\) 0 0
\(872\) 22.2538 0.753609
\(873\) 10.3927 0.351740
\(874\) −4.30359 −0.145571
\(875\) 0 0
\(876\) −10.6914 −0.361230
\(877\) 4.62556 0.156194 0.0780971 0.996946i \(-0.475116\pi\)
0.0780971 + 0.996946i \(0.475116\pi\)
\(878\) 0.856858 0.0289176
\(879\) −42.6948 −1.44006
\(880\) 5.81177 0.195915
\(881\) −42.1720 −1.42081 −0.710405 0.703793i \(-0.751488\pi\)
−0.710405 + 0.703793i \(0.751488\pi\)
\(882\) 0 0
\(883\) 20.7992 0.699950 0.349975 0.936759i \(-0.386190\pi\)
0.349975 + 0.936759i \(0.386190\pi\)
\(884\) 0 0
\(885\) 26.4257 0.888291
\(886\) 0.984810 0.0330853
\(887\) 34.4889 1.15802 0.579012 0.815319i \(-0.303438\pi\)
0.579012 + 0.815319i \(0.303438\pi\)
\(888\) 51.5581 1.73018
\(889\) 0 0
\(890\) 34.1898 1.14605
\(891\) 26.3198 0.881748
\(892\) −12.7411 −0.426604
\(893\) −6.56947 −0.219839
\(894\) −44.8347 −1.49950
\(895\) −30.3579 −1.01475
\(896\) 0 0
\(897\) 0 0
\(898\) −31.0880 −1.03742
\(899\) −15.1459 −0.505144
\(900\) −24.9327 −0.831091
\(901\) 56.3267 1.87651
\(902\) 2.32853 0.0775315
\(903\) 0 0
\(904\) 28.2946 0.941065
\(905\) 9.21550 0.306334
\(906\) 14.5536 0.483512
\(907\) 13.3619 0.443675 0.221838 0.975084i \(-0.428795\pi\)
0.221838 + 0.975084i \(0.428795\pi\)
\(908\) 0.605720 0.0201015
\(909\) 4.98233 0.165254
\(910\) 0 0
\(911\) 5.29058 0.175285 0.0876424 0.996152i \(-0.472067\pi\)
0.0876424 + 0.996152i \(0.472067\pi\)
\(912\) −9.78097 −0.323880
\(913\) −11.7755 −0.389713
\(914\) 2.82843 0.0935562
\(915\) 115.182 3.80781
\(916\) −8.80129 −0.290803
\(917\) 0 0
\(918\) −118.579 −3.91369
\(919\) −18.8306 −0.621164 −0.310582 0.950547i \(-0.600524\pi\)
−0.310582 + 0.950547i \(0.600524\pi\)
\(920\) −28.9913 −0.955814
\(921\) −33.9736 −1.11947
\(922\) 26.1939 0.862649
\(923\) 0 0
\(924\) 0 0
\(925\) −24.5659 −0.807721
\(926\) −38.0009 −1.24879
\(927\) 62.9371 2.06712
\(928\) −8.26571 −0.271335
\(929\) 44.9537 1.47488 0.737442 0.675411i \(-0.236034\pi\)
0.737442 + 0.675411i \(0.236034\pi\)
\(930\) −76.1335 −2.49652
\(931\) 0 0
\(932\) −0.860348 −0.0281816
\(933\) 69.0348 2.26010
\(934\) −15.1427 −0.495484
\(935\) 14.0102 0.458182
\(936\) 0 0
\(937\) 37.9272 1.23903 0.619514 0.784986i \(-0.287330\pi\)
0.619514 + 0.784986i \(0.287330\pi\)
\(938\) 0 0
\(939\) 55.4442 1.80935
\(940\) −10.4286 −0.340143
\(941\) −35.7869 −1.16662 −0.583310 0.812250i \(-0.698243\pi\)
−0.583310 + 0.812250i \(0.698243\pi\)
\(942\) 4.41691 0.143911
\(943\) −7.65848 −0.249394
\(944\) 5.97752 0.194552
\(945\) 0 0
\(946\) −2.48471 −0.0807849
\(947\) 14.7443 0.479125 0.239563 0.970881i \(-0.422996\pi\)
0.239563 + 0.970881i \(0.422996\pi\)
\(948\) 1.11995 0.0363742
\(949\) 0 0
\(950\) 7.06831 0.229326
\(951\) 71.1551 2.30736
\(952\) 0 0
\(953\) 0.649669 0.0210448 0.0105224 0.999945i \(-0.496651\pi\)
0.0105224 + 0.999945i \(0.496651\pi\)
\(954\) −94.7593 −3.06795
\(955\) 50.8076 1.64409
\(956\) 1.57153 0.0508268
\(957\) −6.39826 −0.206826
\(958\) −12.3992 −0.400600
\(959\) 0 0
\(960\) −91.9123 −2.96646
\(961\) 6.63659 0.214083
\(962\) 0 0
\(963\) 39.7440 1.28073
\(964\) 18.2969 0.589305
\(965\) −0.271983 −0.00875545
\(966\) 0 0
\(967\) 13.0802 0.420632 0.210316 0.977633i \(-0.432551\pi\)
0.210316 + 0.977633i \(0.432551\pi\)
\(968\) 32.0127 1.02893
\(969\) −23.5786 −0.757453
\(970\) −4.67702 −0.150170
\(971\) −29.1203 −0.934515 −0.467258 0.884121i \(-0.654758\pi\)
−0.467258 + 0.884121i \(0.654758\pi\)
\(972\) −37.9025 −1.21572
\(973\) 0 0
\(974\) −44.9792 −1.44123
\(975\) 0 0
\(976\) 26.0544 0.833980
\(977\) 18.5000 0.591868 0.295934 0.955208i \(-0.404369\pi\)
0.295934 + 0.955208i \(0.404369\pi\)
\(978\) 80.1817 2.56393
\(979\) −7.14000 −0.228195
\(980\) 0 0
\(981\) −59.5045 −1.89983
\(982\) −18.1475 −0.579111
\(983\) 11.0462 0.352318 0.176159 0.984362i \(-0.443633\pi\)
0.176159 + 0.984362i \(0.443633\pi\)
\(984\) −26.3993 −0.841578
\(985\) 47.4148 1.51076
\(986\) 16.7067 0.532049
\(987\) 0 0
\(988\) 0 0
\(989\) 8.17216 0.259860
\(990\) −23.5696 −0.749090
\(991\) −40.4757 −1.28575 −0.642877 0.765969i \(-0.722259\pi\)
−0.642877 + 0.765969i \(0.722259\pi\)
\(992\) 20.5398 0.652138
\(993\) −54.4251 −1.72713
\(994\) 0 0
\(995\) −48.6357 −1.54185
\(996\) 31.4593 0.996826
\(997\) −12.0623 −0.382018 −0.191009 0.981588i \(-0.561176\pi\)
−0.191009 + 0.981588i \(0.561176\pi\)
\(998\) 45.6610 1.44537
\(999\) −87.6029 −2.77163
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 8281.2.a.cd.1.5 6
7.6 odd 2 8281.2.a.cc.1.5 6
13.12 even 2 637.2.a.n.1.2 yes 6
39.38 odd 2 5733.2.a.br.1.5 6
91.12 odd 6 637.2.e.o.508.5 12
91.25 even 6 637.2.e.n.79.5 12
91.38 odd 6 637.2.e.o.79.5 12
91.51 even 6 637.2.e.n.508.5 12
91.90 odd 2 637.2.a.m.1.2 6
273.272 even 2 5733.2.a.bu.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.2 6 91.90 odd 2
637.2.a.n.1.2 yes 6 13.12 even 2
637.2.e.n.79.5 12 91.25 even 6
637.2.e.n.508.5 12 91.51 even 6
637.2.e.o.79.5 12 91.38 odd 6
637.2.e.o.508.5 12 91.12 odd 6
5733.2.a.br.1.5 6 39.38 odd 2
5733.2.a.bu.1.5 6 273.272 even 2
8281.2.a.cc.1.5 6 7.6 odd 2
8281.2.a.cd.1.5 6 1.1 even 1 trivial