Properties

Label 6039.2.a.i.1.7
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} - 760 x^{4} - 742 x^{3} + 366 x^{2} + 236 x - 47\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.171582\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} +O(q^{10})\) \(q-0.171582 q^{2} -1.97056 q^{4} +0.133072 q^{5} +0.615329 q^{7} +0.681275 q^{8} -0.0228327 q^{10} -1.00000 q^{11} -4.70459 q^{13} -0.105579 q^{14} +3.82423 q^{16} -2.34281 q^{17} -7.85906 q^{19} -0.262226 q^{20} +0.171582 q^{22} -1.86581 q^{23} -4.98229 q^{25} +0.807221 q^{26} -1.21254 q^{28} -4.53951 q^{29} +3.06900 q^{31} -2.01872 q^{32} +0.401984 q^{34} +0.0818830 q^{35} +9.24562 q^{37} +1.34847 q^{38} +0.0906587 q^{40} -1.52969 q^{41} +1.17504 q^{43} +1.97056 q^{44} +0.320139 q^{46} +7.78342 q^{47} -6.62137 q^{49} +0.854870 q^{50} +9.27067 q^{52} +11.1113 q^{53} -0.133072 q^{55} +0.419208 q^{56} +0.778896 q^{58} -11.0751 q^{59} -1.00000 q^{61} -0.526584 q^{62} -7.30207 q^{64} -0.626049 q^{65} +11.2239 q^{67} +4.61665 q^{68} -0.0140496 q^{70} +6.59487 q^{71} +4.36946 q^{73} -1.58638 q^{74} +15.4867 q^{76} -0.615329 q^{77} +11.7882 q^{79} +0.508897 q^{80} +0.262467 q^{82} -10.8873 q^{83} -0.311763 q^{85} -0.201616 q^{86} -0.681275 q^{88} -15.1533 q^{89} -2.89487 q^{91} +3.67669 q^{92} -1.33549 q^{94} -1.04582 q^{95} +15.1739 q^{97} +1.13611 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + O(q^{10}) \) \( 13q - 2q^{2} + 16q^{4} - 3q^{5} + 11q^{7} - 9q^{8} + 6q^{10} - 13q^{11} + 13q^{13} - q^{14} + 18q^{16} - 17q^{17} + 14q^{19} + 7q^{20} + 2q^{22} - 7q^{23} + 18q^{25} + 10q^{26} + 19q^{28} + 6q^{29} + 27q^{31} - 5q^{32} + 6q^{34} - 14q^{35} + 10q^{37} - 2q^{38} + 8q^{40} - 3q^{41} + 29q^{43} - 16q^{44} - 24q^{46} - 8q^{47} + 8q^{49} + 27q^{50} + 37q^{52} + 24q^{53} + 3q^{55} - 24q^{56} - 5q^{58} - 13q^{59} - 13q^{61} - 39q^{62} + 47q^{64} + 11q^{65} + 44q^{67} + 8q^{68} - 12q^{70} - 3q^{71} + 48q^{73} + 22q^{74} + 47q^{76} - 11q^{77} - 17q^{79} + 26q^{80} + 56q^{82} - 50q^{83} + 8q^{85} - 18q^{86} + 9q^{88} + 15q^{89} + 47q^{91} - 14q^{92} + 45q^{94} + q^{95} + 27q^{97} - 47q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.171582 −0.121327 −0.0606633 0.998158i \(-0.519322\pi\)
−0.0606633 + 0.998158i \(0.519322\pi\)
\(3\) 0 0
\(4\) −1.97056 −0.985280
\(5\) 0.133072 0.0595116 0.0297558 0.999557i \(-0.490527\pi\)
0.0297558 + 0.999557i \(0.490527\pi\)
\(6\) 0 0
\(7\) 0.615329 0.232572 0.116286 0.993216i \(-0.462901\pi\)
0.116286 + 0.993216i \(0.462901\pi\)
\(8\) 0.681275 0.240867
\(9\) 0 0
\(10\) −0.0228327 −0.00722034
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.70459 −1.30482 −0.652409 0.757867i \(-0.726241\pi\)
−0.652409 + 0.757867i \(0.726241\pi\)
\(14\) −0.105579 −0.0282172
\(15\) 0 0
\(16\) 3.82423 0.956056
\(17\) −2.34281 −0.568215 −0.284108 0.958792i \(-0.591697\pi\)
−0.284108 + 0.958792i \(0.591697\pi\)
\(18\) 0 0
\(19\) −7.85906 −1.80299 −0.901496 0.432788i \(-0.857530\pi\)
−0.901496 + 0.432788i \(0.857530\pi\)
\(20\) −0.262226 −0.0586356
\(21\) 0 0
\(22\) 0.171582 0.0365813
\(23\) −1.86581 −0.389048 −0.194524 0.980898i \(-0.562316\pi\)
−0.194524 + 0.980898i \(0.562316\pi\)
\(24\) 0 0
\(25\) −4.98229 −0.996458
\(26\) 0.807221 0.158309
\(27\) 0 0
\(28\) −1.21254 −0.229149
\(29\) −4.53951 −0.842965 −0.421482 0.906837i \(-0.638490\pi\)
−0.421482 + 0.906837i \(0.638490\pi\)
\(30\) 0 0
\(31\) 3.06900 0.551208 0.275604 0.961271i \(-0.411122\pi\)
0.275604 + 0.961271i \(0.411122\pi\)
\(32\) −2.01872 −0.356862
\(33\) 0 0
\(34\) 0.401984 0.0689396
\(35\) 0.0818830 0.0138408
\(36\) 0 0
\(37\) 9.24562 1.51997 0.759986 0.649940i \(-0.225206\pi\)
0.759986 + 0.649940i \(0.225206\pi\)
\(38\) 1.34847 0.218751
\(39\) 0 0
\(40\) 0.0906587 0.0143344
\(41\) −1.52969 −0.238897 −0.119449 0.992840i \(-0.538113\pi\)
−0.119449 + 0.992840i \(0.538113\pi\)
\(42\) 0 0
\(43\) 1.17504 0.179193 0.0895963 0.995978i \(-0.471442\pi\)
0.0895963 + 0.995978i \(0.471442\pi\)
\(44\) 1.97056 0.297073
\(45\) 0 0
\(46\) 0.320139 0.0472019
\(47\) 7.78342 1.13533 0.567664 0.823260i \(-0.307847\pi\)
0.567664 + 0.823260i \(0.307847\pi\)
\(48\) 0 0
\(49\) −6.62137 −0.945910
\(50\) 0.854870 0.120897
\(51\) 0 0
\(52\) 9.27067 1.28561
\(53\) 11.1113 1.52626 0.763130 0.646246i \(-0.223662\pi\)
0.763130 + 0.646246i \(0.223662\pi\)
\(54\) 0 0
\(55\) −0.133072 −0.0179434
\(56\) 0.419208 0.0560191
\(57\) 0 0
\(58\) 0.778896 0.102274
\(59\) −11.0751 −1.44185 −0.720925 0.693013i \(-0.756283\pi\)
−0.720925 + 0.693013i \(0.756283\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) −0.526584 −0.0668762
\(63\) 0 0
\(64\) −7.30207 −0.912759
\(65\) −0.626049 −0.0776518
\(66\) 0 0
\(67\) 11.2239 1.37121 0.685607 0.727972i \(-0.259537\pi\)
0.685607 + 0.727972i \(0.259537\pi\)
\(68\) 4.61665 0.559851
\(69\) 0 0
\(70\) −0.0140496 −0.00167925
\(71\) 6.59487 0.782668 0.391334 0.920249i \(-0.372014\pi\)
0.391334 + 0.920249i \(0.372014\pi\)
\(72\) 0 0
\(73\) 4.36946 0.511406 0.255703 0.966755i \(-0.417693\pi\)
0.255703 + 0.966755i \(0.417693\pi\)
\(74\) −1.58638 −0.184413
\(75\) 0 0
\(76\) 15.4867 1.77645
\(77\) −0.615329 −0.0701232
\(78\) 0 0
\(79\) 11.7882 1.32628 0.663141 0.748495i \(-0.269223\pi\)
0.663141 + 0.748495i \(0.269223\pi\)
\(80\) 0.508897 0.0568964
\(81\) 0 0
\(82\) 0.262467 0.0289846
\(83\) −10.8873 −1.19503 −0.597517 0.801856i \(-0.703846\pi\)
−0.597517 + 0.801856i \(0.703846\pi\)
\(84\) 0 0
\(85\) −0.311763 −0.0338154
\(86\) −0.201616 −0.0217408
\(87\) 0 0
\(88\) −0.681275 −0.0726242
\(89\) −15.1533 −1.60625 −0.803123 0.595813i \(-0.796830\pi\)
−0.803123 + 0.595813i \(0.796830\pi\)
\(90\) 0 0
\(91\) −2.89487 −0.303465
\(92\) 3.67669 0.383321
\(93\) 0 0
\(94\) −1.33549 −0.137745
\(95\) −1.04582 −0.107299
\(96\) 0 0
\(97\) 15.1739 1.54068 0.770340 0.637633i \(-0.220086\pi\)
0.770340 + 0.637633i \(0.220086\pi\)
\(98\) 1.13611 0.114764
\(99\) 0 0
\(100\) 9.81790 0.981790
\(101\) −14.9729 −1.48986 −0.744931 0.667141i \(-0.767518\pi\)
−0.744931 + 0.667141i \(0.767518\pi\)
\(102\) 0 0
\(103\) 2.65906 0.262005 0.131003 0.991382i \(-0.458180\pi\)
0.131003 + 0.991382i \(0.458180\pi\)
\(104\) −3.20512 −0.314288
\(105\) 0 0
\(106\) −1.90650 −0.185176
\(107\) −9.97635 −0.964450 −0.482225 0.876047i \(-0.660171\pi\)
−0.482225 + 0.876047i \(0.660171\pi\)
\(108\) 0 0
\(109\) −8.83359 −0.846104 −0.423052 0.906105i \(-0.639041\pi\)
−0.423052 + 0.906105i \(0.639041\pi\)
\(110\) 0.0228327 0.00217701
\(111\) 0 0
\(112\) 2.35316 0.222352
\(113\) 4.24584 0.399415 0.199707 0.979856i \(-0.436001\pi\)
0.199707 + 0.979856i \(0.436001\pi\)
\(114\) 0 0
\(115\) −0.248287 −0.0231529
\(116\) 8.94537 0.830556
\(117\) 0 0
\(118\) 1.90028 0.174935
\(119\) −1.44160 −0.132151
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.171582 0.0155343
\(123\) 0 0
\(124\) −6.04764 −0.543094
\(125\) −1.32836 −0.118812
\(126\) 0 0
\(127\) 19.3085 1.71335 0.856675 0.515856i \(-0.172526\pi\)
0.856675 + 0.515856i \(0.172526\pi\)
\(128\) 5.29034 0.467604
\(129\) 0 0
\(130\) 0.107419 0.00942123
\(131\) −16.1882 −1.41437 −0.707187 0.707027i \(-0.750036\pi\)
−0.707187 + 0.707027i \(0.750036\pi\)
\(132\) 0 0
\(133\) −4.83590 −0.419326
\(134\) −1.92581 −0.166365
\(135\) 0 0
\(136\) −1.59610 −0.136864
\(137\) 16.9876 1.45135 0.725674 0.688038i \(-0.241528\pi\)
0.725674 + 0.688038i \(0.241528\pi\)
\(138\) 0 0
\(139\) 0.981625 0.0832604 0.0416302 0.999133i \(-0.486745\pi\)
0.0416302 + 0.999133i \(0.486745\pi\)
\(140\) −0.161355 −0.0136370
\(141\) 0 0
\(142\) −1.13156 −0.0949584
\(143\) 4.70459 0.393417
\(144\) 0 0
\(145\) −0.604081 −0.0501662
\(146\) −0.749719 −0.0620472
\(147\) 0 0
\(148\) −18.2191 −1.49760
\(149\) −19.9181 −1.63175 −0.815876 0.578227i \(-0.803745\pi\)
−0.815876 + 0.578227i \(0.803745\pi\)
\(150\) 0 0
\(151\) −9.27179 −0.754528 −0.377264 0.926106i \(-0.623135\pi\)
−0.377264 + 0.926106i \(0.623135\pi\)
\(152\) −5.35418 −0.434282
\(153\) 0 0
\(154\) 0.105579 0.00850781
\(155\) 0.408397 0.0328033
\(156\) 0 0
\(157\) 9.62702 0.768320 0.384160 0.923267i \(-0.374491\pi\)
0.384160 + 0.923267i \(0.374491\pi\)
\(158\) −2.02265 −0.160913
\(159\) 0 0
\(160\) −0.268635 −0.0212374
\(161\) −1.14809 −0.0904818
\(162\) 0 0
\(163\) 17.0593 1.33619 0.668093 0.744077i \(-0.267111\pi\)
0.668093 + 0.744077i \(0.267111\pi\)
\(164\) 3.01434 0.235381
\(165\) 0 0
\(166\) 1.86806 0.144989
\(167\) −8.92176 −0.690387 −0.345193 0.938532i \(-0.612187\pi\)
−0.345193 + 0.938532i \(0.612187\pi\)
\(168\) 0 0
\(169\) 9.13314 0.702549
\(170\) 0.0534928 0.00410271
\(171\) 0 0
\(172\) −2.31549 −0.176555
\(173\) −3.85738 −0.293271 −0.146636 0.989191i \(-0.546844\pi\)
−0.146636 + 0.989191i \(0.546844\pi\)
\(174\) 0 0
\(175\) −3.06575 −0.231749
\(176\) −3.82423 −0.288262
\(177\) 0 0
\(178\) 2.60003 0.194880
\(179\) 10.4383 0.780194 0.390097 0.920774i \(-0.372442\pi\)
0.390097 + 0.920774i \(0.372442\pi\)
\(180\) 0 0
\(181\) 13.9705 1.03842 0.519208 0.854648i \(-0.326227\pi\)
0.519208 + 0.854648i \(0.326227\pi\)
\(182\) 0.496706 0.0368183
\(183\) 0 0
\(184\) −1.27113 −0.0937089
\(185\) 1.23033 0.0904559
\(186\) 0 0
\(187\) 2.34281 0.171323
\(188\) −15.3377 −1.11862
\(189\) 0 0
\(190\) 0.179444 0.0130182
\(191\) 24.4448 1.76876 0.884382 0.466763i \(-0.154580\pi\)
0.884382 + 0.466763i \(0.154580\pi\)
\(192\) 0 0
\(193\) −6.16239 −0.443578 −0.221789 0.975095i \(-0.571190\pi\)
−0.221789 + 0.975095i \(0.571190\pi\)
\(194\) −2.60357 −0.186926
\(195\) 0 0
\(196\) 13.0478 0.931986
\(197\) 16.1438 1.15020 0.575100 0.818083i \(-0.304963\pi\)
0.575100 + 0.818083i \(0.304963\pi\)
\(198\) 0 0
\(199\) −11.7030 −0.829600 −0.414800 0.909913i \(-0.636149\pi\)
−0.414800 + 0.909913i \(0.636149\pi\)
\(200\) −3.39431 −0.240014
\(201\) 0 0
\(202\) 2.56908 0.180760
\(203\) −2.79329 −0.196050
\(204\) 0 0
\(205\) −0.203559 −0.0142172
\(206\) −0.456247 −0.0317882
\(207\) 0 0
\(208\) −17.9914 −1.24748
\(209\) 7.85906 0.543622
\(210\) 0 0
\(211\) −13.0881 −0.901019 −0.450509 0.892772i \(-0.648758\pi\)
−0.450509 + 0.892772i \(0.648758\pi\)
\(212\) −21.8955 −1.50379
\(213\) 0 0
\(214\) 1.71176 0.117013
\(215\) 0.156365 0.0106640
\(216\) 0 0
\(217\) 1.88844 0.128196
\(218\) 1.51568 0.102655
\(219\) 0 0
\(220\) 0.262226 0.0176793
\(221\) 11.0220 0.741417
\(222\) 0 0
\(223\) 2.06775 0.138467 0.0692334 0.997600i \(-0.477945\pi\)
0.0692334 + 0.997600i \(0.477945\pi\)
\(224\) −1.24218 −0.0829963
\(225\) 0 0
\(226\) −0.728508 −0.0484596
\(227\) 6.15200 0.408322 0.204161 0.978937i \(-0.434553\pi\)
0.204161 + 0.978937i \(0.434553\pi\)
\(228\) 0 0
\(229\) 19.5963 1.29496 0.647479 0.762083i \(-0.275823\pi\)
0.647479 + 0.762083i \(0.275823\pi\)
\(230\) 0.0426015 0.00280906
\(231\) 0 0
\(232\) −3.09265 −0.203043
\(233\) 20.5752 1.34792 0.673962 0.738766i \(-0.264591\pi\)
0.673962 + 0.738766i \(0.264591\pi\)
\(234\) 0 0
\(235\) 1.03575 0.0675652
\(236\) 21.8241 1.42063
\(237\) 0 0
\(238\) 0.247352 0.0160335
\(239\) 23.1378 1.49666 0.748330 0.663327i \(-0.230856\pi\)
0.748330 + 0.663327i \(0.230856\pi\)
\(240\) 0 0
\(241\) 0.0983366 0.00633442 0.00316721 0.999995i \(-0.498992\pi\)
0.00316721 + 0.999995i \(0.498992\pi\)
\(242\) −0.171582 −0.0110297
\(243\) 0 0
\(244\) 1.97056 0.126152
\(245\) −0.881119 −0.0562926
\(246\) 0 0
\(247\) 36.9736 2.35257
\(248\) 2.09083 0.132768
\(249\) 0 0
\(250\) 0.227923 0.0144151
\(251\) 3.43655 0.216913 0.108457 0.994101i \(-0.465409\pi\)
0.108457 + 0.994101i \(0.465409\pi\)
\(252\) 0 0
\(253\) 1.86581 0.117302
\(254\) −3.31298 −0.207875
\(255\) 0 0
\(256\) 13.6964 0.856027
\(257\) 12.4152 0.774439 0.387220 0.921988i \(-0.373436\pi\)
0.387220 + 0.921988i \(0.373436\pi\)
\(258\) 0 0
\(259\) 5.68910 0.353503
\(260\) 1.23367 0.0765087
\(261\) 0 0
\(262\) 2.77761 0.171601
\(263\) −4.10959 −0.253408 −0.126704 0.991941i \(-0.540440\pi\)
−0.126704 + 0.991941i \(0.540440\pi\)
\(264\) 0 0
\(265\) 1.47861 0.0908301
\(266\) 0.829753 0.0508754
\(267\) 0 0
\(268\) −22.1173 −1.35103
\(269\) −17.7222 −1.08054 −0.540270 0.841492i \(-0.681678\pi\)
−0.540270 + 0.841492i \(0.681678\pi\)
\(270\) 0 0
\(271\) 6.97502 0.423702 0.211851 0.977302i \(-0.432051\pi\)
0.211851 + 0.977302i \(0.432051\pi\)
\(272\) −8.95944 −0.543246
\(273\) 0 0
\(274\) −2.91476 −0.176087
\(275\) 4.98229 0.300444
\(276\) 0 0
\(277\) −2.67748 −0.160874 −0.0804371 0.996760i \(-0.525632\pi\)
−0.0804371 + 0.996760i \(0.525632\pi\)
\(278\) −0.168429 −0.0101017
\(279\) 0 0
\(280\) 0.0557849 0.00333379
\(281\) 16.1747 0.964905 0.482452 0.875922i \(-0.339746\pi\)
0.482452 + 0.875922i \(0.339746\pi\)
\(282\) 0 0
\(283\) 14.1430 0.840715 0.420357 0.907359i \(-0.361905\pi\)
0.420357 + 0.907359i \(0.361905\pi\)
\(284\) −12.9956 −0.771147
\(285\) 0 0
\(286\) −0.807221 −0.0477320
\(287\) −0.941262 −0.0555609
\(288\) 0 0
\(289\) −11.5112 −0.677131
\(290\) 0.103649 0.00608649
\(291\) 0 0
\(292\) −8.61028 −0.503878
\(293\) 8.25000 0.481970 0.240985 0.970529i \(-0.422529\pi\)
0.240985 + 0.970529i \(0.422529\pi\)
\(294\) 0 0
\(295\) −1.47378 −0.0858068
\(296\) 6.29882 0.366111
\(297\) 0 0
\(298\) 3.41758 0.197975
\(299\) 8.77786 0.507637
\(300\) 0 0
\(301\) 0.723038 0.0416752
\(302\) 1.59087 0.0915443
\(303\) 0 0
\(304\) −30.0548 −1.72376
\(305\) −0.133072 −0.00761968
\(306\) 0 0
\(307\) −5.17139 −0.295147 −0.147573 0.989051i \(-0.547146\pi\)
−0.147573 + 0.989051i \(0.547146\pi\)
\(308\) 1.21254 0.0690910
\(309\) 0 0
\(310\) −0.0700735 −0.00397991
\(311\) −5.56505 −0.315565 −0.157783 0.987474i \(-0.550435\pi\)
−0.157783 + 0.987474i \(0.550435\pi\)
\(312\) 0 0
\(313\) −26.1339 −1.47718 −0.738589 0.674156i \(-0.764507\pi\)
−0.738589 + 0.674156i \(0.764507\pi\)
\(314\) −1.65182 −0.0932176
\(315\) 0 0
\(316\) −23.2294 −1.30676
\(317\) 9.26111 0.520156 0.260078 0.965588i \(-0.416252\pi\)
0.260078 + 0.965588i \(0.416252\pi\)
\(318\) 0 0
\(319\) 4.53951 0.254163
\(320\) −0.971702 −0.0543198
\(321\) 0 0
\(322\) 0.196991 0.0109779
\(323\) 18.4123 1.02449
\(324\) 0 0
\(325\) 23.4396 1.30020
\(326\) −2.92706 −0.162115
\(327\) 0 0
\(328\) −1.04214 −0.0575425
\(329\) 4.78936 0.264046
\(330\) 0 0
\(331\) −2.06884 −0.113714 −0.0568569 0.998382i \(-0.518108\pi\)
−0.0568569 + 0.998382i \(0.518108\pi\)
\(332\) 21.4540 1.17744
\(333\) 0 0
\(334\) 1.53081 0.0837622
\(335\) 1.49358 0.0816031
\(336\) 0 0
\(337\) −0.238290 −0.0129805 −0.00649024 0.999979i \(-0.502066\pi\)
−0.00649024 + 0.999979i \(0.502066\pi\)
\(338\) −1.56708 −0.0852379
\(339\) 0 0
\(340\) 0.614347 0.0333176
\(341\) −3.06900 −0.166195
\(342\) 0 0
\(343\) −8.38162 −0.452565
\(344\) 0.800529 0.0431616
\(345\) 0 0
\(346\) 0.661856 0.0355816
\(347\) −27.9023 −1.49787 −0.748937 0.662641i \(-0.769436\pi\)
−0.748937 + 0.662641i \(0.769436\pi\)
\(348\) 0 0
\(349\) 30.6969 1.64317 0.821585 0.570086i \(-0.193090\pi\)
0.821585 + 0.570086i \(0.193090\pi\)
\(350\) 0.526026 0.0281173
\(351\) 0 0
\(352\) 2.01872 0.107598
\(353\) −32.2120 −1.71447 −0.857236 0.514924i \(-0.827820\pi\)
−0.857236 + 0.514924i \(0.827820\pi\)
\(354\) 0 0
\(355\) 0.877593 0.0465778
\(356\) 29.8605 1.58260
\(357\) 0 0
\(358\) −1.79102 −0.0946582
\(359\) 25.4721 1.34437 0.672184 0.740384i \(-0.265356\pi\)
0.672184 + 0.740384i \(0.265356\pi\)
\(360\) 0 0
\(361\) 42.7648 2.25078
\(362\) −2.39708 −0.125987
\(363\) 0 0
\(364\) 5.70451 0.298998
\(365\) 0.581452 0.0304346
\(366\) 0 0
\(367\) 35.8891 1.87340 0.936698 0.350138i \(-0.113865\pi\)
0.936698 + 0.350138i \(0.113865\pi\)
\(368\) −7.13527 −0.371952
\(369\) 0 0
\(370\) −0.211103 −0.0109747
\(371\) 6.83713 0.354966
\(372\) 0 0
\(373\) 26.2640 1.35990 0.679950 0.733259i \(-0.262002\pi\)
0.679950 + 0.733259i \(0.262002\pi\)
\(374\) −0.401984 −0.0207861
\(375\) 0 0
\(376\) 5.30265 0.273463
\(377\) 21.3565 1.09992
\(378\) 0 0
\(379\) −22.5811 −1.15991 −0.579957 0.814647i \(-0.696931\pi\)
−0.579957 + 0.814647i \(0.696931\pi\)
\(380\) 2.06085 0.105719
\(381\) 0 0
\(382\) −4.19428 −0.214598
\(383\) −1.52359 −0.0778520 −0.0389260 0.999242i \(-0.512394\pi\)
−0.0389260 + 0.999242i \(0.512394\pi\)
\(384\) 0 0
\(385\) −0.0818830 −0.00417315
\(386\) 1.05735 0.0538179
\(387\) 0 0
\(388\) −29.9012 −1.51800
\(389\) 21.4415 1.08713 0.543563 0.839369i \(-0.317075\pi\)
0.543563 + 0.839369i \(0.317075\pi\)
\(390\) 0 0
\(391\) 4.37124 0.221063
\(392\) −4.51098 −0.227839
\(393\) 0 0
\(394\) −2.76999 −0.139550
\(395\) 1.56869 0.0789291
\(396\) 0 0
\(397\) −7.89768 −0.396373 −0.198187 0.980164i \(-0.563505\pi\)
−0.198187 + 0.980164i \(0.563505\pi\)
\(398\) 2.00801 0.100653
\(399\) 0 0
\(400\) −19.0534 −0.952670
\(401\) −11.0900 −0.553807 −0.276903 0.960898i \(-0.589308\pi\)
−0.276903 + 0.960898i \(0.589308\pi\)
\(402\) 0 0
\(403\) −14.4384 −0.719226
\(404\) 29.5051 1.46793
\(405\) 0 0
\(406\) 0.479277 0.0237861
\(407\) −9.24562 −0.458289
\(408\) 0 0
\(409\) 1.82984 0.0904797 0.0452399 0.998976i \(-0.485595\pi\)
0.0452399 + 0.998976i \(0.485595\pi\)
\(410\) 0.0349270 0.00172492
\(411\) 0 0
\(412\) −5.23985 −0.258149
\(413\) −6.81481 −0.335335
\(414\) 0 0
\(415\) −1.44879 −0.0711183
\(416\) 9.49723 0.465640
\(417\) 0 0
\(418\) −1.34847 −0.0659558
\(419\) −29.0901 −1.42115 −0.710573 0.703623i \(-0.751564\pi\)
−0.710573 + 0.703623i \(0.751564\pi\)
\(420\) 0 0
\(421\) −0.0560143 −0.00272997 −0.00136499 0.999999i \(-0.500434\pi\)
−0.00136499 + 0.999999i \(0.500434\pi\)
\(422\) 2.24567 0.109318
\(423\) 0 0
\(424\) 7.56988 0.367626
\(425\) 11.6726 0.566203
\(426\) 0 0
\(427\) −0.615329 −0.0297779
\(428\) 19.6590 0.950253
\(429\) 0 0
\(430\) −0.0268294 −0.00129383
\(431\) 2.07838 0.100112 0.0500561 0.998746i \(-0.484060\pi\)
0.0500561 + 0.998746i \(0.484060\pi\)
\(432\) 0 0
\(433\) −5.73899 −0.275798 −0.137899 0.990446i \(-0.544035\pi\)
−0.137899 + 0.990446i \(0.544035\pi\)
\(434\) −0.324022 −0.0155536
\(435\) 0 0
\(436\) 17.4071 0.833650
\(437\) 14.6635 0.701450
\(438\) 0 0
\(439\) 32.6721 1.55935 0.779677 0.626182i \(-0.215383\pi\)
0.779677 + 0.626182i \(0.215383\pi\)
\(440\) −0.0906587 −0.00432198
\(441\) 0 0
\(442\) −1.89117 −0.0899537
\(443\) 25.8218 1.22683 0.613415 0.789761i \(-0.289795\pi\)
0.613415 + 0.789761i \(0.289795\pi\)
\(444\) 0 0
\(445\) −2.01648 −0.0955903
\(446\) −0.354788 −0.0167997
\(447\) 0 0
\(448\) −4.49318 −0.212283
\(449\) 18.6591 0.880579 0.440290 0.897856i \(-0.354876\pi\)
0.440290 + 0.897856i \(0.354876\pi\)
\(450\) 0 0
\(451\) 1.52969 0.0720303
\(452\) −8.36667 −0.393535
\(453\) 0 0
\(454\) −1.05557 −0.0495404
\(455\) −0.385226 −0.0180597
\(456\) 0 0
\(457\) −4.37660 −0.204729 −0.102364 0.994747i \(-0.532641\pi\)
−0.102364 + 0.994747i \(0.532641\pi\)
\(458\) −3.36236 −0.157113
\(459\) 0 0
\(460\) 0.489264 0.0228120
\(461\) 36.4443 1.69738 0.848689 0.528891i \(-0.177392\pi\)
0.848689 + 0.528891i \(0.177392\pi\)
\(462\) 0 0
\(463\) 14.5793 0.677557 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(464\) −17.3601 −0.805922
\(465\) 0 0
\(466\) −3.53032 −0.163539
\(467\) −26.5701 −1.22952 −0.614760 0.788714i \(-0.710747\pi\)
−0.614760 + 0.788714i \(0.710747\pi\)
\(468\) 0 0
\(469\) 6.90637 0.318906
\(470\) −0.177717 −0.00819745
\(471\) 0 0
\(472\) −7.54517 −0.347295
\(473\) −1.17504 −0.0540286
\(474\) 0 0
\(475\) 39.1561 1.79661
\(476\) 2.84076 0.130206
\(477\) 0 0
\(478\) −3.97002 −0.181585
\(479\) −4.66442 −0.213123 −0.106561 0.994306i \(-0.533984\pi\)
−0.106561 + 0.994306i \(0.533984\pi\)
\(480\) 0 0
\(481\) −43.4968 −1.98329
\(482\) −0.0168728 −0.000768533 0
\(483\) 0 0
\(484\) −1.97056 −0.0895709
\(485\) 2.01923 0.0916884
\(486\) 0 0
\(487\) 16.2253 0.735240 0.367620 0.929976i \(-0.380173\pi\)
0.367620 + 0.929976i \(0.380173\pi\)
\(488\) −0.681275 −0.0308399
\(489\) 0 0
\(490\) 0.151184 0.00682979
\(491\) −37.8484 −1.70808 −0.854038 0.520210i \(-0.825854\pi\)
−0.854038 + 0.520210i \(0.825854\pi\)
\(492\) 0 0
\(493\) 10.6352 0.478986
\(494\) −6.34400 −0.285430
\(495\) 0 0
\(496\) 11.7365 0.526986
\(497\) 4.05802 0.182027
\(498\) 0 0
\(499\) −12.9470 −0.579586 −0.289793 0.957089i \(-0.593587\pi\)
−0.289793 + 0.957089i \(0.593587\pi\)
\(500\) 2.61762 0.117063
\(501\) 0 0
\(502\) −0.589650 −0.0263174
\(503\) 22.8525 1.01894 0.509471 0.860488i \(-0.329841\pi\)
0.509471 + 0.860488i \(0.329841\pi\)
\(504\) 0 0
\(505\) −1.99248 −0.0886641
\(506\) −0.320139 −0.0142319
\(507\) 0 0
\(508\) −38.0485 −1.68813
\(509\) −1.26053 −0.0558720 −0.0279360 0.999610i \(-0.508893\pi\)
−0.0279360 + 0.999610i \(0.508893\pi\)
\(510\) 0 0
\(511\) 2.68865 0.118939
\(512\) −12.9307 −0.571463
\(513\) 0 0
\(514\) −2.13022 −0.0939601
\(515\) 0.353847 0.0155924
\(516\) 0 0
\(517\) −7.78342 −0.342314
\(518\) −0.976146 −0.0428894
\(519\) 0 0
\(520\) −0.426512 −0.0187038
\(521\) −33.2078 −1.45486 −0.727430 0.686182i \(-0.759285\pi\)
−0.727430 + 0.686182i \(0.759285\pi\)
\(522\) 0 0
\(523\) 22.8888 1.00086 0.500428 0.865778i \(-0.333176\pi\)
0.500428 + 0.865778i \(0.333176\pi\)
\(524\) 31.8999 1.39355
\(525\) 0 0
\(526\) 0.705131 0.0307452
\(527\) −7.19008 −0.313205
\(528\) 0 0
\(529\) −19.5188 −0.848642
\(530\) −0.253702 −0.0110201
\(531\) 0 0
\(532\) 9.52944 0.413154
\(533\) 7.19656 0.311717
\(534\) 0 0
\(535\) −1.32757 −0.0573959
\(536\) 7.64654 0.330280
\(537\) 0 0
\(538\) 3.04080 0.131098
\(539\) 6.62137 0.285203
\(540\) 0 0
\(541\) −7.47250 −0.321268 −0.160634 0.987014i \(-0.551354\pi\)
−0.160634 + 0.987014i \(0.551354\pi\)
\(542\) −1.19679 −0.0514064
\(543\) 0 0
\(544\) 4.72948 0.202775
\(545\) −1.17550 −0.0503530
\(546\) 0 0
\(547\) 42.4596 1.81544 0.907721 0.419574i \(-0.137820\pi\)
0.907721 + 0.419574i \(0.137820\pi\)
\(548\) −33.4751 −1.42998
\(549\) 0 0
\(550\) −0.854870 −0.0364518
\(551\) 35.6762 1.51986
\(552\) 0 0
\(553\) 7.25365 0.308457
\(554\) 0.459407 0.0195183
\(555\) 0 0
\(556\) −1.93435 −0.0820348
\(557\) −33.5690 −1.42236 −0.711182 0.703008i \(-0.751840\pi\)
−0.711182 + 0.703008i \(0.751840\pi\)
\(558\) 0 0
\(559\) −5.52810 −0.233814
\(560\) 0.313139 0.0132325
\(561\) 0 0
\(562\) −2.77529 −0.117069
\(563\) 6.90978 0.291212 0.145606 0.989343i \(-0.453487\pi\)
0.145606 + 0.989343i \(0.453487\pi\)
\(564\) 0 0
\(565\) 0.565002 0.0237698
\(566\) −2.42668 −0.102001
\(567\) 0 0
\(568\) 4.49293 0.188519
\(569\) 43.2342 1.81247 0.906236 0.422772i \(-0.138943\pi\)
0.906236 + 0.422772i \(0.138943\pi\)
\(570\) 0 0
\(571\) 36.9917 1.54805 0.774027 0.633153i \(-0.218240\pi\)
0.774027 + 0.633153i \(0.218240\pi\)
\(572\) −9.27067 −0.387626
\(573\) 0 0
\(574\) 0.161503 0.00674102
\(575\) 9.29600 0.387670
\(576\) 0 0
\(577\) −43.0272 −1.79125 −0.895623 0.444815i \(-0.853270\pi\)
−0.895623 + 0.444815i \(0.853270\pi\)
\(578\) 1.97512 0.0821540
\(579\) 0 0
\(580\) 1.19038 0.0494277
\(581\) −6.69925 −0.277932
\(582\) 0 0
\(583\) −11.1113 −0.460184
\(584\) 2.97680 0.123181
\(585\) 0 0
\(586\) −1.41555 −0.0584758
\(587\) −2.31390 −0.0955048 −0.0477524 0.998859i \(-0.515206\pi\)
−0.0477524 + 0.998859i \(0.515206\pi\)
\(588\) 0 0
\(589\) −24.1194 −0.993823
\(590\) 0.252874 0.0104107
\(591\) 0 0
\(592\) 35.3573 1.45318
\(593\) −10.2698 −0.421731 −0.210865 0.977515i \(-0.567628\pi\)
−0.210865 + 0.977515i \(0.567628\pi\)
\(594\) 0 0
\(595\) −0.191837 −0.00786453
\(596\) 39.2498 1.60773
\(597\) 0 0
\(598\) −1.50612 −0.0615898
\(599\) −21.7905 −0.890335 −0.445167 0.895447i \(-0.646856\pi\)
−0.445167 + 0.895447i \(0.646856\pi\)
\(600\) 0 0
\(601\) −20.7713 −0.847279 −0.423640 0.905831i \(-0.639248\pi\)
−0.423640 + 0.905831i \(0.639248\pi\)
\(602\) −0.124060 −0.00505632
\(603\) 0 0
\(604\) 18.2706 0.743421
\(605\) 0.133072 0.00541015
\(606\) 0 0
\(607\) 1.14578 0.0465057 0.0232528 0.999730i \(-0.492598\pi\)
0.0232528 + 0.999730i \(0.492598\pi\)
\(608\) 15.8652 0.643420
\(609\) 0 0
\(610\) 0.0228327 0.000924470 0
\(611\) −36.6178 −1.48140
\(612\) 0 0
\(613\) 13.2402 0.534766 0.267383 0.963590i \(-0.413841\pi\)
0.267383 + 0.963590i \(0.413841\pi\)
\(614\) 0.887316 0.0358092
\(615\) 0 0
\(616\) −0.419208 −0.0168904
\(617\) 19.1708 0.771787 0.385894 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(618\) 0 0
\(619\) 3.63574 0.146133 0.0730663 0.997327i \(-0.476722\pi\)
0.0730663 + 0.997327i \(0.476722\pi\)
\(620\) −0.804771 −0.0323204
\(621\) 0 0
\(622\) 0.954861 0.0382864
\(623\) −9.32426 −0.373569
\(624\) 0 0
\(625\) 24.7347 0.989388
\(626\) 4.48411 0.179221
\(627\) 0 0
\(628\) −18.9706 −0.757010
\(629\) −21.6608 −0.863671
\(630\) 0 0
\(631\) 34.0488 1.35546 0.677732 0.735309i \(-0.262963\pi\)
0.677732 + 0.735309i \(0.262963\pi\)
\(632\) 8.03105 0.319458
\(633\) 0 0
\(634\) −1.58904 −0.0631087
\(635\) 2.56942 0.101964
\(636\) 0 0
\(637\) 31.1508 1.23424
\(638\) −0.778896 −0.0308368
\(639\) 0 0
\(640\) 0.703996 0.0278279
\(641\) −1.72492 −0.0681302 −0.0340651 0.999420i \(-0.510845\pi\)
−0.0340651 + 0.999420i \(0.510845\pi\)
\(642\) 0 0
\(643\) −37.6517 −1.48484 −0.742420 0.669935i \(-0.766322\pi\)
−0.742420 + 0.669935i \(0.766322\pi\)
\(644\) 2.26237 0.0891499
\(645\) 0 0
\(646\) −3.15921 −0.124298
\(647\) −46.6191 −1.83279 −0.916393 0.400279i \(-0.868913\pi\)
−0.916393 + 0.400279i \(0.868913\pi\)
\(648\) 0 0
\(649\) 11.0751 0.434734
\(650\) −4.02181 −0.157748
\(651\) 0 0
\(652\) −33.6164 −1.31652
\(653\) 32.6575 1.27799 0.638993 0.769212i \(-0.279351\pi\)
0.638993 + 0.769212i \(0.279351\pi\)
\(654\) 0 0
\(655\) −2.15420 −0.0841716
\(656\) −5.84988 −0.228399
\(657\) 0 0
\(658\) −0.821767 −0.0320358
\(659\) −36.3322 −1.41530 −0.707651 0.706562i \(-0.750245\pi\)
−0.707651 + 0.706562i \(0.750245\pi\)
\(660\) 0 0
\(661\) 20.6126 0.801736 0.400868 0.916136i \(-0.368709\pi\)
0.400868 + 0.916136i \(0.368709\pi\)
\(662\) 0.354975 0.0137965
\(663\) 0 0
\(664\) −7.41723 −0.287844
\(665\) −0.643523 −0.0249548
\(666\) 0 0
\(667\) 8.46985 0.327954
\(668\) 17.5809 0.680224
\(669\) 0 0
\(670\) −0.256271 −0.00990063
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −5.41924 −0.208896 −0.104448 0.994530i \(-0.533308\pi\)
−0.104448 + 0.994530i \(0.533308\pi\)
\(674\) 0.0408862 0.00157488
\(675\) 0 0
\(676\) −17.9974 −0.692208
\(677\) 15.1357 0.581713 0.290857 0.956767i \(-0.406060\pi\)
0.290857 + 0.956767i \(0.406060\pi\)
\(678\) 0 0
\(679\) 9.33697 0.358320
\(680\) −0.212396 −0.00814502
\(681\) 0 0
\(682\) 0.526584 0.0201639
\(683\) −5.25700 −0.201154 −0.100577 0.994929i \(-0.532069\pi\)
−0.100577 + 0.994929i \(0.532069\pi\)
\(684\) 0 0
\(685\) 2.26057 0.0863721
\(686\) 1.43813 0.0549082
\(687\) 0 0
\(688\) 4.49363 0.171318
\(689\) −52.2742 −1.99149
\(690\) 0 0
\(691\) −37.8441 −1.43966 −0.719828 0.694152i \(-0.755780\pi\)
−0.719828 + 0.694152i \(0.755780\pi\)
\(692\) 7.60120 0.288954
\(693\) 0 0
\(694\) 4.78753 0.181732
\(695\) 0.130627 0.00495496
\(696\) 0 0
\(697\) 3.58377 0.135745
\(698\) −5.26703 −0.199360
\(699\) 0 0
\(700\) 6.04124 0.228337
\(701\) 26.2157 0.990154 0.495077 0.868849i \(-0.335140\pi\)
0.495077 + 0.868849i \(0.335140\pi\)
\(702\) 0 0
\(703\) −72.6619 −2.74049
\(704\) 7.30207 0.275207
\(705\) 0 0
\(706\) 5.52699 0.208011
\(707\) −9.21328 −0.346501
\(708\) 0 0
\(709\) −22.3870 −0.840762 −0.420381 0.907348i \(-0.638104\pi\)
−0.420381 + 0.907348i \(0.638104\pi\)
\(710\) −0.150579 −0.00565113
\(711\) 0 0
\(712\) −10.3236 −0.386892
\(713\) −5.72616 −0.214446
\(714\) 0 0
\(715\) 0.626049 0.0234129
\(716\) −20.5693 −0.768709
\(717\) 0 0
\(718\) −4.37055 −0.163108
\(719\) 21.7396 0.810751 0.405376 0.914150i \(-0.367141\pi\)
0.405376 + 0.914150i \(0.367141\pi\)
\(720\) 0 0
\(721\) 1.63620 0.0609352
\(722\) −7.33765 −0.273079
\(723\) 0 0
\(724\) −27.5296 −1.02313
\(725\) 22.6171 0.839979
\(726\) 0 0
\(727\) −24.4460 −0.906653 −0.453327 0.891344i \(-0.649763\pi\)
−0.453327 + 0.891344i \(0.649763\pi\)
\(728\) −1.97220 −0.0730947
\(729\) 0 0
\(730\) −0.0997666 −0.00369253
\(731\) −2.75291 −0.101820
\(732\) 0 0
\(733\) 13.6034 0.502452 0.251226 0.967928i \(-0.419166\pi\)
0.251226 + 0.967928i \(0.419166\pi\)
\(734\) −6.15792 −0.227293
\(735\) 0 0
\(736\) 3.76654 0.138837
\(737\) −11.2239 −0.413436
\(738\) 0 0
\(739\) 1.68561 0.0620061 0.0310031 0.999519i \(-0.490130\pi\)
0.0310031 + 0.999519i \(0.490130\pi\)
\(740\) −2.42445 −0.0891244
\(741\) 0 0
\(742\) −1.17313 −0.0430668
\(743\) −45.8835 −1.68330 −0.841652 0.540020i \(-0.818417\pi\)
−0.841652 + 0.540020i \(0.818417\pi\)
\(744\) 0 0
\(745\) −2.65054 −0.0971082
\(746\) −4.50643 −0.164992
\(747\) 0 0
\(748\) −4.61665 −0.168801
\(749\) −6.13873 −0.224304
\(750\) 0 0
\(751\) −47.2695 −1.72489 −0.862445 0.506151i \(-0.831068\pi\)
−0.862445 + 0.506151i \(0.831068\pi\)
\(752\) 29.7655 1.08544
\(753\) 0 0
\(754\) −3.66438 −0.133449
\(755\) −1.23382 −0.0449032
\(756\) 0 0
\(757\) −33.4045 −1.21411 −0.607054 0.794661i \(-0.707649\pi\)
−0.607054 + 0.794661i \(0.707649\pi\)
\(758\) 3.87451 0.140728
\(759\) 0 0
\(760\) −0.712492 −0.0258448
\(761\) −19.2821 −0.698974 −0.349487 0.936941i \(-0.613644\pi\)
−0.349487 + 0.936941i \(0.613644\pi\)
\(762\) 0 0
\(763\) −5.43556 −0.196781
\(764\) −48.1700 −1.74273
\(765\) 0 0
\(766\) 0.261421 0.00944551
\(767\) 52.1036 1.88135
\(768\) 0 0
\(769\) −27.7997 −1.00248 −0.501242 0.865307i \(-0.667123\pi\)
−0.501242 + 0.865307i \(0.667123\pi\)
\(770\) 0.0140496 0.000506314 0
\(771\) 0 0
\(772\) 12.1434 0.437049
\(773\) −15.4428 −0.555438 −0.277719 0.960662i \(-0.589578\pi\)
−0.277719 + 0.960662i \(0.589578\pi\)
\(774\) 0 0
\(775\) −15.2906 −0.549256
\(776\) 10.3376 0.371100
\(777\) 0 0
\(778\) −3.67896 −0.131897
\(779\) 12.0219 0.430730
\(780\) 0 0
\(781\) −6.59487 −0.235983
\(782\) −0.750024 −0.0268208
\(783\) 0 0
\(784\) −25.3216 −0.904343
\(785\) 1.28109 0.0457239
\(786\) 0 0
\(787\) −39.8594 −1.42083 −0.710417 0.703781i \(-0.751494\pi\)
−0.710417 + 0.703781i \(0.751494\pi\)
\(788\) −31.8124 −1.13327
\(789\) 0 0
\(790\) −0.269158 −0.00957620
\(791\) 2.61259 0.0928929
\(792\) 0 0
\(793\) 4.70459 0.167065
\(794\) 1.35510 0.0480906
\(795\) 0 0
\(796\) 23.0614 0.817388
\(797\) 17.9931 0.637348 0.318674 0.947864i \(-0.396762\pi\)
0.318674 + 0.947864i \(0.396762\pi\)
\(798\) 0 0
\(799\) −18.2351 −0.645111
\(800\) 10.0578 0.355598
\(801\) 0 0
\(802\) 1.90284 0.0671915
\(803\) −4.36946 −0.154195
\(804\) 0 0
\(805\) −0.152778 −0.00538472
\(806\) 2.47736 0.0872612
\(807\) 0 0
\(808\) −10.2007 −0.358859
\(809\) 36.0517 1.26751 0.633755 0.773534i \(-0.281513\pi\)
0.633755 + 0.773534i \(0.281513\pi\)
\(810\) 0 0
\(811\) 17.6858 0.621032 0.310516 0.950568i \(-0.399498\pi\)
0.310516 + 0.950568i \(0.399498\pi\)
\(812\) 5.50434 0.193165
\(813\) 0 0
\(814\) 1.58638 0.0556026
\(815\) 2.27011 0.0795186
\(816\) 0 0
\(817\) −9.23474 −0.323082
\(818\) −0.313967 −0.0109776
\(819\) 0 0
\(820\) 0.401125 0.0140079
\(821\) 14.5517 0.507858 0.253929 0.967223i \(-0.418277\pi\)
0.253929 + 0.967223i \(0.418277\pi\)
\(822\) 0 0
\(823\) 48.1384 1.67800 0.838999 0.544133i \(-0.183141\pi\)
0.838999 + 0.544133i \(0.183141\pi\)
\(824\) 1.81156 0.0631085
\(825\) 0 0
\(826\) 1.16930 0.0406850
\(827\) −27.5769 −0.958943 −0.479472 0.877557i \(-0.659172\pi\)
−0.479472 + 0.877557i \(0.659172\pi\)
\(828\) 0 0
\(829\) −10.6222 −0.368926 −0.184463 0.982839i \(-0.559055\pi\)
−0.184463 + 0.982839i \(0.559055\pi\)
\(830\) 0.248586 0.00862855
\(831\) 0 0
\(832\) 34.3532 1.19098
\(833\) 15.5126 0.537481
\(834\) 0 0
\(835\) −1.18724 −0.0410860
\(836\) −15.4867 −0.535620
\(837\) 0 0
\(838\) 4.99134 0.172423
\(839\) 48.1933 1.66382 0.831909 0.554912i \(-0.187248\pi\)
0.831909 + 0.554912i \(0.187248\pi\)
\(840\) 0 0
\(841\) −8.39289 −0.289410
\(842\) 0.00961104 0.000331218 0
\(843\) 0 0
\(844\) 25.7908 0.887756
\(845\) 1.21536 0.0418098
\(846\) 0 0
\(847\) 0.615329 0.0211430
\(848\) 42.4922 1.45919
\(849\) 0 0
\(850\) −2.00280 −0.0686955
\(851\) −17.2506 −0.591342
\(852\) 0 0
\(853\) 46.5258 1.59301 0.796507 0.604629i \(-0.206679\pi\)
0.796507 + 0.604629i \(0.206679\pi\)
\(854\) 0.105579 0.00361285
\(855\) 0 0
\(856\) −6.79664 −0.232304
\(857\) 10.3918 0.354979 0.177489 0.984123i \(-0.443202\pi\)
0.177489 + 0.984123i \(0.443202\pi\)
\(858\) 0 0
\(859\) 57.1315 1.94930 0.974651 0.223732i \(-0.0718241\pi\)
0.974651 + 0.223732i \(0.0718241\pi\)
\(860\) −0.308127 −0.0105071
\(861\) 0 0
\(862\) −0.356613 −0.0121463
\(863\) −21.5752 −0.734429 −0.367214 0.930136i \(-0.619688\pi\)
−0.367214 + 0.930136i \(0.619688\pi\)
\(864\) 0 0
\(865\) −0.513309 −0.0174530
\(866\) 0.984706 0.0334617
\(867\) 0 0
\(868\) −3.72129 −0.126309
\(869\) −11.7882 −0.399889
\(870\) 0 0
\(871\) −52.8036 −1.78918
\(872\) −6.01811 −0.203799
\(873\) 0 0
\(874\) −2.51599 −0.0851045
\(875\) −0.817380 −0.0276325
\(876\) 0 0
\(877\) −19.6961 −0.665092 −0.332546 0.943087i \(-0.607908\pi\)
−0.332546 + 0.943087i \(0.607908\pi\)
\(878\) −5.60594 −0.189191
\(879\) 0 0
\(880\) −0.508897 −0.0171549
\(881\) 13.2882 0.447691 0.223845 0.974625i \(-0.428139\pi\)
0.223845 + 0.974625i \(0.428139\pi\)
\(882\) 0 0
\(883\) 12.8872 0.433688 0.216844 0.976206i \(-0.430424\pi\)
0.216844 + 0.976206i \(0.430424\pi\)
\(884\) −21.7194 −0.730504
\(885\) 0 0
\(886\) −4.43055 −0.148847
\(887\) −1.36425 −0.0458071 −0.0229036 0.999738i \(-0.507291\pi\)
−0.0229036 + 0.999738i \(0.507291\pi\)
\(888\) 0 0
\(889\) 11.8811 0.398478
\(890\) 0.345991 0.0115976
\(891\) 0 0
\(892\) −4.07463 −0.136429
\(893\) −61.1703 −2.04699
\(894\) 0 0
\(895\) 1.38904 0.0464306
\(896\) 3.25530 0.108752
\(897\) 0 0
\(898\) −3.20157 −0.106838
\(899\) −13.9317 −0.464649
\(900\) 0 0
\(901\) −26.0318 −0.867244
\(902\) −0.262467 −0.00873919
\(903\) 0 0
\(904\) 2.89258 0.0962059
\(905\) 1.85908 0.0617978
\(906\) 0 0
\(907\) 52.3304 1.73760 0.868802 0.495160i \(-0.164890\pi\)
0.868802 + 0.495160i \(0.164890\pi\)
\(908\) −12.1229 −0.402312
\(909\) 0 0
\(910\) 0.0660977 0.00219112
\(911\) −4.50180 −0.149151 −0.0745756 0.997215i \(-0.523760\pi\)
−0.0745756 + 0.997215i \(0.523760\pi\)
\(912\) 0 0
\(913\) 10.8873 0.360316
\(914\) 0.750944 0.0248390
\(915\) 0 0
\(916\) −38.6156 −1.27590
\(917\) −9.96110 −0.328944
\(918\) 0 0
\(919\) 25.3477 0.836143 0.418072 0.908414i \(-0.362706\pi\)
0.418072 + 0.908414i \(0.362706\pi\)
\(920\) −0.169152 −0.00557677
\(921\) 0 0
\(922\) −6.25317 −0.205937
\(923\) −31.0262 −1.02124
\(924\) 0 0
\(925\) −46.0644 −1.51459
\(926\) −2.50154 −0.0822057
\(927\) 0 0
\(928\) 9.16398 0.300822
\(929\) −20.9203 −0.686372 −0.343186 0.939267i \(-0.611506\pi\)
−0.343186 + 0.939267i \(0.611506\pi\)
\(930\) 0 0
\(931\) 52.0377 1.70547
\(932\) −40.5446 −1.32808
\(933\) 0 0
\(934\) 4.55895 0.149173
\(935\) 0.311763 0.0101957
\(936\) 0 0
\(937\) 33.7225 1.10167 0.550833 0.834615i \(-0.314310\pi\)
0.550833 + 0.834615i \(0.314310\pi\)
\(938\) −1.18501 −0.0386918
\(939\) 0 0
\(940\) −2.04102 −0.0665706
\(941\) 45.8406 1.49436 0.747180 0.664621i \(-0.231407\pi\)
0.747180 + 0.664621i \(0.231407\pi\)
\(942\) 0 0
\(943\) 2.85411 0.0929425
\(944\) −42.3535 −1.37849
\(945\) 0 0
\(946\) 0.201616 0.00655510
\(947\) 31.2864 1.01667 0.508336 0.861159i \(-0.330261\pi\)
0.508336 + 0.861159i \(0.330261\pi\)
\(948\) 0 0
\(949\) −20.5565 −0.667292
\(950\) −6.71847 −0.217976
\(951\) 0 0
\(952\) −0.982127 −0.0318309
\(953\) −58.6406 −1.89955 −0.949777 0.312926i \(-0.898691\pi\)
−0.949777 + 0.312926i \(0.898691\pi\)
\(954\) 0 0
\(955\) 3.25292 0.105262
\(956\) −45.5944 −1.47463
\(957\) 0 0
\(958\) 0.800329 0.0258574
\(959\) 10.4530 0.337544
\(960\) 0 0
\(961\) −21.5813 −0.696170
\(962\) 7.46326 0.240625
\(963\) 0 0
\(964\) −0.193778 −0.00624117
\(965\) −0.820041 −0.0263981
\(966\) 0 0
\(967\) −35.3660 −1.13729 −0.568647 0.822582i \(-0.692533\pi\)
−0.568647 + 0.822582i \(0.692533\pi\)
\(968\) 0.681275 0.0218970
\(969\) 0 0
\(970\) −0.346462 −0.0111242
\(971\) −21.4189 −0.687366 −0.343683 0.939086i \(-0.611675\pi\)
−0.343683 + 0.939086i \(0.611675\pi\)
\(972\) 0 0
\(973\) 0.604022 0.0193641
\(974\) −2.78397 −0.0892042
\(975\) 0 0
\(976\) −3.82423 −0.122410
\(977\) 50.4860 1.61519 0.807595 0.589737i \(-0.200769\pi\)
0.807595 + 0.589737i \(0.200769\pi\)
\(978\) 0 0
\(979\) 15.1533 0.484301
\(980\) 1.73630 0.0554640
\(981\) 0 0
\(982\) 6.49410 0.207235
\(983\) 4.20465 0.134107 0.0670537 0.997749i \(-0.478640\pi\)
0.0670537 + 0.997749i \(0.478640\pi\)
\(984\) 0 0
\(985\) 2.14829 0.0684503
\(986\) −1.82481 −0.0581137
\(987\) 0 0
\(988\) −72.8587 −2.31794
\(989\) −2.19241 −0.0697145
\(990\) 0 0
\(991\) 14.4063 0.457631 0.228816 0.973470i \(-0.426515\pi\)
0.228816 + 0.973470i \(0.426515\pi\)
\(992\) −6.19544 −0.196705
\(993\) 0 0
\(994\) −0.696282 −0.0220847
\(995\) −1.55734 −0.0493708
\(996\) 0 0
\(997\) 25.4235 0.805169 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(998\) 2.22147 0.0703193
\(999\) 0 0
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 6039.2.a.i.1.7 13
3.2 odd 2 2013.2.a.e.1.7 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.e.1.7 13 3.2 odd 2
6039.2.a.i.1.7 13 1.1 even 1 trivial