Properties

Label 2151.2.a.e.1.3
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.94590\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} +O(q^{10})\) \(q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} -0.0449852 q^{10} +1.08824 q^{11} -2.81084 q^{13} -0.288090 q^{14} +3.93506 q^{16} -5.82744 q^{17} -7.25266 q^{19} -0.859312 q^{20} -0.113322 q^{22} +3.48861 q^{23} -4.81338 q^{25} +0.292701 q^{26} -5.50313 q^{28} +5.03919 q^{29} -2.02388 q^{31} -1.24057 q^{32} +0.606828 q^{34} +1.19515 q^{35} +2.08347 q^{37} +0.755240 q^{38} +0.179453 q^{40} +5.40397 q^{41} -5.91283 q^{43} -2.16468 q^{44} -0.363279 q^{46} +0.795216 q^{47} +0.653884 q^{49} +0.501230 q^{50} +5.59121 q^{52} +3.47735 q^{53} +0.470119 q^{55} +1.14924 q^{56} -0.524745 q^{58} +9.03190 q^{59} -10.4206 q^{61} +0.210753 q^{62} -7.74093 q^{64} -1.21428 q^{65} -3.98418 q^{67} +11.5917 q^{68} -0.124454 q^{70} -10.9852 q^{71} +1.10684 q^{73} -0.216957 q^{74} +14.4267 q^{76} +3.01069 q^{77} -0.409081 q^{79} +1.69994 q^{80} -0.562731 q^{82} -15.0992 q^{83} -2.51744 q^{85} +0.615719 q^{86} +0.452058 q^{88} -14.2188 q^{89} -7.77639 q^{91} -6.93940 q^{92} -0.0828081 q^{94} -3.13314 q^{95} +12.5418 q^{97} -0.0680907 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.104133 −0.0736330 −0.0368165 0.999322i \(-0.511722\pi\)
−0.0368165 + 0.999322i \(0.511722\pi\)
\(3\) 0 0
\(4\) −1.98916 −0.994578
\(5\) 0.431998 0.193195 0.0965977 0.995324i \(-0.469204\pi\)
0.0965977 + 0.995324i \(0.469204\pi\)
\(6\) 0 0
\(7\) 2.76657 1.04566 0.522832 0.852436i \(-0.324876\pi\)
0.522832 + 0.852436i \(0.324876\pi\)
\(8\) 0.415402 0.146867
\(9\) 0 0
\(10\) −0.0449852 −0.0142256
\(11\) 1.08824 0.328117 0.164059 0.986451i \(-0.447541\pi\)
0.164059 + 0.986451i \(0.447541\pi\)
\(12\) 0 0
\(13\) −2.81084 −0.779588 −0.389794 0.920902i \(-0.627454\pi\)
−0.389794 + 0.920902i \(0.627454\pi\)
\(14\) −0.288090 −0.0769953
\(15\) 0 0
\(16\) 3.93506 0.983764
\(17\) −5.82744 −1.41336 −0.706681 0.707532i \(-0.749808\pi\)
−0.706681 + 0.707532i \(0.749808\pi\)
\(18\) 0 0
\(19\) −7.25266 −1.66388 −0.831938 0.554869i \(-0.812768\pi\)
−0.831938 + 0.554869i \(0.812768\pi\)
\(20\) −0.859312 −0.192148
\(21\) 0 0
\(22\) −0.113322 −0.0241603
\(23\) 3.48861 0.727426 0.363713 0.931511i \(-0.381509\pi\)
0.363713 + 0.931511i \(0.381509\pi\)
\(24\) 0 0
\(25\) −4.81338 −0.962676
\(26\) 0.292701 0.0574034
\(27\) 0 0
\(28\) −5.50313 −1.03999
\(29\) 5.03919 0.935754 0.467877 0.883794i \(-0.345019\pi\)
0.467877 + 0.883794i \(0.345019\pi\)
\(30\) 0 0
\(31\) −2.02388 −0.363500 −0.181750 0.983345i \(-0.558176\pi\)
−0.181750 + 0.983345i \(0.558176\pi\)
\(32\) −1.24057 −0.219304
\(33\) 0 0
\(34\) 0.606828 0.104070
\(35\) 1.19515 0.202017
\(36\) 0 0
\(37\) 2.08347 0.342520 0.171260 0.985226i \(-0.445216\pi\)
0.171260 + 0.985226i \(0.445216\pi\)
\(38\) 0.755240 0.122516
\(39\) 0 0
\(40\) 0.179453 0.0283740
\(41\) 5.40397 0.843959 0.421979 0.906605i \(-0.361335\pi\)
0.421979 + 0.906605i \(0.361335\pi\)
\(42\) 0 0
\(43\) −5.91283 −0.901698 −0.450849 0.892600i \(-0.648879\pi\)
−0.450849 + 0.892600i \(0.648879\pi\)
\(44\) −2.16468 −0.326338
\(45\) 0 0
\(46\) −0.363279 −0.0535626
\(47\) 0.795216 0.115994 0.0579971 0.998317i \(-0.481529\pi\)
0.0579971 + 0.998317i \(0.481529\pi\)
\(48\) 0 0
\(49\) 0.653884 0.0934120
\(50\) 0.501230 0.0708847
\(51\) 0 0
\(52\) 5.59121 0.775361
\(53\) 3.47735 0.477650 0.238825 0.971063i \(-0.423238\pi\)
0.238825 + 0.971063i \(0.423238\pi\)
\(54\) 0 0
\(55\) 0.470119 0.0633908
\(56\) 1.14924 0.153573
\(57\) 0 0
\(58\) −0.524745 −0.0689024
\(59\) 9.03190 1.17585 0.587927 0.808914i \(-0.299944\pi\)
0.587927 + 0.808914i \(0.299944\pi\)
\(60\) 0 0
\(61\) −10.4206 −1.33422 −0.667109 0.744961i \(-0.732468\pi\)
−0.667109 + 0.744961i \(0.732468\pi\)
\(62\) 0.210753 0.0267656
\(63\) 0 0
\(64\) −7.74093 −0.967616
\(65\) −1.21428 −0.150613
\(66\) 0 0
\(67\) −3.98418 −0.486745 −0.243373 0.969933i \(-0.578254\pi\)
−0.243373 + 0.969933i \(0.578254\pi\)
\(68\) 11.5917 1.40570
\(69\) 0 0
\(70\) −0.124454 −0.0148752
\(71\) −10.9852 −1.30370 −0.651851 0.758347i \(-0.726007\pi\)
−0.651851 + 0.758347i \(0.726007\pi\)
\(72\) 0 0
\(73\) 1.10684 0.129546 0.0647729 0.997900i \(-0.479368\pi\)
0.0647729 + 0.997900i \(0.479368\pi\)
\(74\) −0.216957 −0.0252208
\(75\) 0 0
\(76\) 14.4267 1.65485
\(77\) 3.01069 0.343100
\(78\) 0 0
\(79\) −0.409081 −0.0460252 −0.0230126 0.999735i \(-0.507326\pi\)
−0.0230126 + 0.999735i \(0.507326\pi\)
\(80\) 1.69994 0.190059
\(81\) 0 0
\(82\) −0.562731 −0.0621432
\(83\) −15.0992 −1.65736 −0.828679 0.559725i \(-0.810907\pi\)
−0.828679 + 0.559725i \(0.810907\pi\)
\(84\) 0 0
\(85\) −2.51744 −0.273055
\(86\) 0.615719 0.0663947
\(87\) 0 0
\(88\) 0.452058 0.0481895
\(89\) −14.2188 −1.50719 −0.753593 0.657342i \(-0.771681\pi\)
−0.753593 + 0.657342i \(0.771681\pi\)
\(90\) 0 0
\(91\) −7.77639 −0.815187
\(92\) −6.93940 −0.723482
\(93\) 0 0
\(94\) −0.0828081 −0.00854100
\(95\) −3.13314 −0.321453
\(96\) 0 0
\(97\) 12.5418 1.27342 0.636712 0.771101i \(-0.280294\pi\)
0.636712 + 0.771101i \(0.280294\pi\)
\(98\) −0.0680907 −0.00687820
\(99\) 0 0
\(100\) 9.57456 0.957456
\(101\) −4.71052 −0.468714 −0.234357 0.972151i \(-0.575298\pi\)
−0.234357 + 0.972151i \(0.575298\pi\)
\(102\) 0 0
\(103\) −19.1362 −1.88555 −0.942775 0.333431i \(-0.891794\pi\)
−0.942775 + 0.333431i \(0.891794\pi\)
\(104\) −1.16763 −0.114496
\(105\) 0 0
\(106\) −0.362106 −0.0351708
\(107\) −6.40124 −0.618831 −0.309416 0.950927i \(-0.600133\pi\)
−0.309416 + 0.950927i \(0.600133\pi\)
\(108\) 0 0
\(109\) −2.95365 −0.282908 −0.141454 0.989945i \(-0.545178\pi\)
−0.141454 + 0.989945i \(0.545178\pi\)
\(110\) −0.0489548 −0.00466765
\(111\) 0 0
\(112\) 10.8866 1.02869
\(113\) −0.888386 −0.0835724 −0.0417862 0.999127i \(-0.513305\pi\)
−0.0417862 + 0.999127i \(0.513305\pi\)
\(114\) 0 0
\(115\) 1.50707 0.140535
\(116\) −10.0237 −0.930680
\(117\) 0 0
\(118\) −0.940517 −0.0865816
\(119\) −16.1220 −1.47790
\(120\) 0 0
\(121\) −9.81573 −0.892339
\(122\) 1.08512 0.0982424
\(123\) 0 0
\(124\) 4.02582 0.361530
\(125\) −4.23936 −0.379180
\(126\) 0 0
\(127\) −2.46764 −0.218968 −0.109484 0.993989i \(-0.534920\pi\)
−0.109484 + 0.993989i \(0.534920\pi\)
\(128\) 3.28723 0.290553
\(129\) 0 0
\(130\) 0.126446 0.0110901
\(131\) 6.69156 0.584644 0.292322 0.956320i \(-0.405572\pi\)
0.292322 + 0.956320i \(0.405572\pi\)
\(132\) 0 0
\(133\) −20.0650 −1.73985
\(134\) 0.414884 0.0358405
\(135\) 0 0
\(136\) −2.42073 −0.207576
\(137\) 6.91690 0.590951 0.295475 0.955350i \(-0.404522\pi\)
0.295475 + 0.955350i \(0.404522\pi\)
\(138\) 0 0
\(139\) −9.48581 −0.804575 −0.402288 0.915513i \(-0.631785\pi\)
−0.402288 + 0.915513i \(0.631785\pi\)
\(140\) −2.37734 −0.200922
\(141\) 0 0
\(142\) 1.14392 0.0959955
\(143\) −3.05888 −0.255796
\(144\) 0 0
\(145\) 2.17692 0.180783
\(146\) −0.115258 −0.00953885
\(147\) 0 0
\(148\) −4.14434 −0.340663
\(149\) −7.32345 −0.599961 −0.299980 0.953945i \(-0.596980\pi\)
−0.299980 + 0.953945i \(0.596980\pi\)
\(150\) 0 0
\(151\) −1.83931 −0.149681 −0.0748404 0.997196i \(-0.523845\pi\)
−0.0748404 + 0.997196i \(0.523845\pi\)
\(152\) −3.01277 −0.244368
\(153\) 0 0
\(154\) −0.313512 −0.0252635
\(155\) −0.874314 −0.0702266
\(156\) 0 0
\(157\) −1.48477 −0.118498 −0.0592488 0.998243i \(-0.518871\pi\)
−0.0592488 + 0.998243i \(0.518871\pi\)
\(158\) 0.0425988 0.00338897
\(159\) 0 0
\(160\) −0.535925 −0.0423686
\(161\) 9.65148 0.760643
\(162\) 0 0
\(163\) 9.95721 0.779909 0.389954 0.920834i \(-0.372491\pi\)
0.389954 + 0.920834i \(0.372491\pi\)
\(164\) −10.7493 −0.839383
\(165\) 0 0
\(166\) 1.57233 0.122036
\(167\) 1.51314 0.117090 0.0585452 0.998285i \(-0.481354\pi\)
0.0585452 + 0.998285i \(0.481354\pi\)
\(168\) 0 0
\(169\) −5.09915 −0.392242
\(170\) 0.262149 0.0201059
\(171\) 0 0
\(172\) 11.7615 0.896809
\(173\) −21.1660 −1.60922 −0.804612 0.593801i \(-0.797627\pi\)
−0.804612 + 0.593801i \(0.797627\pi\)
\(174\) 0 0
\(175\) −13.3165 −1.00663
\(176\) 4.28229 0.322790
\(177\) 0 0
\(178\) 1.48064 0.110979
\(179\) 23.4094 1.74970 0.874850 0.484394i \(-0.160960\pi\)
0.874850 + 0.484394i \(0.160960\pi\)
\(180\) 0 0
\(181\) 14.5477 1.08132 0.540660 0.841241i \(-0.318175\pi\)
0.540660 + 0.841241i \(0.318175\pi\)
\(182\) 0.809777 0.0600247
\(183\) 0 0
\(184\) 1.44918 0.106835
\(185\) 0.900055 0.0661733
\(186\) 0 0
\(187\) −6.34167 −0.463749
\(188\) −1.58181 −0.115365
\(189\) 0 0
\(190\) 0.326262 0.0236696
\(191\) −24.9050 −1.80206 −0.901031 0.433755i \(-0.857188\pi\)
−0.901031 + 0.433755i \(0.857188\pi\)
\(192\) 0 0
\(193\) 5.84285 0.420577 0.210289 0.977639i \(-0.432560\pi\)
0.210289 + 0.977639i \(0.432560\pi\)
\(194\) −1.30601 −0.0937661
\(195\) 0 0
\(196\) −1.30068 −0.0929055
\(197\) 16.2436 1.15731 0.578655 0.815572i \(-0.303578\pi\)
0.578655 + 0.815572i \(0.303578\pi\)
\(198\) 0 0
\(199\) −22.7909 −1.61560 −0.807801 0.589455i \(-0.799343\pi\)
−0.807801 + 0.589455i \(0.799343\pi\)
\(200\) −1.99949 −0.141385
\(201\) 0 0
\(202\) 0.490519 0.0345128
\(203\) 13.9412 0.978484
\(204\) 0 0
\(205\) 2.33451 0.163049
\(206\) 1.99271 0.138839
\(207\) 0 0
\(208\) −11.0608 −0.766931
\(209\) −7.89265 −0.545946
\(210\) 0 0
\(211\) 18.5913 1.27988 0.639938 0.768427i \(-0.278960\pi\)
0.639938 + 0.768427i \(0.278960\pi\)
\(212\) −6.91698 −0.475060
\(213\) 0 0
\(214\) 0.666579 0.0455664
\(215\) −2.55433 −0.174204
\(216\) 0 0
\(217\) −5.59921 −0.380099
\(218\) 0.307572 0.0208314
\(219\) 0 0
\(220\) −0.935140 −0.0630471
\(221\) 16.3800 1.10184
\(222\) 0 0
\(223\) −12.7025 −0.850622 −0.425311 0.905047i \(-0.639835\pi\)
−0.425311 + 0.905047i \(0.639835\pi\)
\(224\) −3.43212 −0.229318
\(225\) 0 0
\(226\) 0.0925102 0.00615369
\(227\) 11.9490 0.793086 0.396543 0.918016i \(-0.370210\pi\)
0.396543 + 0.918016i \(0.370210\pi\)
\(228\) 0 0
\(229\) 2.84730 0.188155 0.0940773 0.995565i \(-0.470010\pi\)
0.0940773 + 0.995565i \(0.470010\pi\)
\(230\) −0.156936 −0.0103480
\(231\) 0 0
\(232\) 2.09329 0.137431
\(233\) 4.49249 0.294313 0.147156 0.989113i \(-0.452988\pi\)
0.147156 + 0.989113i \(0.452988\pi\)
\(234\) 0 0
\(235\) 0.343532 0.0224096
\(236\) −17.9659 −1.16948
\(237\) 0 0
\(238\) 1.67883 0.108822
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 13.6633 0.880133 0.440067 0.897965i \(-0.354955\pi\)
0.440067 + 0.897965i \(0.354955\pi\)
\(242\) 1.02214 0.0657056
\(243\) 0 0
\(244\) 20.7281 1.32698
\(245\) 0.282477 0.0180468
\(246\) 0 0
\(247\) 20.3861 1.29714
\(248\) −0.840726 −0.0533861
\(249\) 0 0
\(250\) 0.441457 0.0279202
\(251\) 18.2147 1.14970 0.574850 0.818259i \(-0.305060\pi\)
0.574850 + 0.818259i \(0.305060\pi\)
\(252\) 0 0
\(253\) 3.79646 0.238681
\(254\) 0.256963 0.0161233
\(255\) 0 0
\(256\) 15.1395 0.946222
\(257\) −4.82747 −0.301129 −0.150565 0.988600i \(-0.548109\pi\)
−0.150565 + 0.988600i \(0.548109\pi\)
\(258\) 0 0
\(259\) 5.76405 0.358161
\(260\) 2.41539 0.149796
\(261\) 0 0
\(262\) −0.696811 −0.0430491
\(263\) 0.934556 0.0576272 0.0288136 0.999585i \(-0.490827\pi\)
0.0288136 + 0.999585i \(0.490827\pi\)
\(264\) 0 0
\(265\) 1.50221 0.0922799
\(266\) 2.08942 0.128111
\(267\) 0 0
\(268\) 7.92516 0.484106
\(269\) −22.8799 −1.39501 −0.697506 0.716579i \(-0.745707\pi\)
−0.697506 + 0.716579i \(0.745707\pi\)
\(270\) 0 0
\(271\) 14.5729 0.885243 0.442621 0.896709i \(-0.354049\pi\)
0.442621 + 0.896709i \(0.354049\pi\)
\(272\) −22.9313 −1.39041
\(273\) 0 0
\(274\) −0.720276 −0.0435135
\(275\) −5.23812 −0.315871
\(276\) 0 0
\(277\) −6.42790 −0.386215 −0.193108 0.981178i \(-0.561857\pi\)
−0.193108 + 0.981178i \(0.561857\pi\)
\(278\) 0.987783 0.0592433
\(279\) 0 0
\(280\) 0.496468 0.0296697
\(281\) 11.9766 0.714462 0.357231 0.934016i \(-0.383721\pi\)
0.357231 + 0.934016i \(0.383721\pi\)
\(282\) 0 0
\(283\) −18.4031 −1.09395 −0.546975 0.837149i \(-0.684221\pi\)
−0.546975 + 0.837149i \(0.684221\pi\)
\(284\) 21.8513 1.29663
\(285\) 0 0
\(286\) 0.318530 0.0188351
\(287\) 14.9504 0.882497
\(288\) 0 0
\(289\) 16.9591 0.997592
\(290\) −0.226689 −0.0133116
\(291\) 0 0
\(292\) −2.20168 −0.128843
\(293\) −24.8968 −1.45449 −0.727244 0.686379i \(-0.759199\pi\)
−0.727244 + 0.686379i \(0.759199\pi\)
\(294\) 0 0
\(295\) 3.90176 0.227170
\(296\) 0.865477 0.0503048
\(297\) 0 0
\(298\) 0.762612 0.0441769
\(299\) −9.80595 −0.567093
\(300\) 0 0
\(301\) −16.3582 −0.942872
\(302\) 0.191532 0.0110214
\(303\) 0 0
\(304\) −28.5396 −1.63686
\(305\) −4.50167 −0.257765
\(306\) 0 0
\(307\) −3.67995 −0.210026 −0.105013 0.994471i \(-0.533488\pi\)
−0.105013 + 0.994471i \(0.533488\pi\)
\(308\) −5.98874 −0.341240
\(309\) 0 0
\(310\) 0.0910448 0.00517100
\(311\) 28.6157 1.62265 0.811324 0.584596i \(-0.198747\pi\)
0.811324 + 0.584596i \(0.198747\pi\)
\(312\) 0 0
\(313\) 14.3414 0.810622 0.405311 0.914179i \(-0.367163\pi\)
0.405311 + 0.914179i \(0.367163\pi\)
\(314\) 0.154613 0.00872534
\(315\) 0 0
\(316\) 0.813726 0.0457757
\(317\) 24.7113 1.38793 0.693964 0.720010i \(-0.255863\pi\)
0.693964 + 0.720010i \(0.255863\pi\)
\(318\) 0 0
\(319\) 5.48386 0.307037
\(320\) −3.34407 −0.186939
\(321\) 0 0
\(322\) −1.00504 −0.0560084
\(323\) 42.2645 2.35166
\(324\) 0 0
\(325\) 13.5297 0.750490
\(326\) −1.03687 −0.0574270
\(327\) 0 0
\(328\) 2.24482 0.123949
\(329\) 2.20002 0.121291
\(330\) 0 0
\(331\) −30.5404 −1.67865 −0.839327 0.543627i \(-0.817051\pi\)
−0.839327 + 0.543627i \(0.817051\pi\)
\(332\) 30.0348 1.64837
\(333\) 0 0
\(334\) −0.157568 −0.00862171
\(335\) −1.72116 −0.0940370
\(336\) 0 0
\(337\) 26.8195 1.46095 0.730474 0.682940i \(-0.239299\pi\)
0.730474 + 0.682940i \(0.239299\pi\)
\(338\) 0.530989 0.0288820
\(339\) 0 0
\(340\) 5.00759 0.271575
\(341\) −2.20248 −0.119271
\(342\) 0 0
\(343\) −17.5569 −0.947986
\(344\) −2.45620 −0.132429
\(345\) 0 0
\(346\) 2.20408 0.118492
\(347\) 9.11072 0.489089 0.244544 0.969638i \(-0.421362\pi\)
0.244544 + 0.969638i \(0.421362\pi\)
\(348\) 0 0
\(349\) 31.8626 1.70557 0.852783 0.522265i \(-0.174913\pi\)
0.852783 + 0.522265i \(0.174913\pi\)
\(350\) 1.38669 0.0741215
\(351\) 0 0
\(352\) −1.35004 −0.0719575
\(353\) −13.4752 −0.717214 −0.358607 0.933489i \(-0.616748\pi\)
−0.358607 + 0.933489i \(0.616748\pi\)
\(354\) 0 0
\(355\) −4.74558 −0.251869
\(356\) 28.2833 1.49901
\(357\) 0 0
\(358\) −2.43769 −0.128836
\(359\) −2.38673 −0.125967 −0.0629835 0.998015i \(-0.520062\pi\)
−0.0629835 + 0.998015i \(0.520062\pi\)
\(360\) 0 0
\(361\) 33.6011 1.76848
\(362\) −1.51489 −0.0796208
\(363\) 0 0
\(364\) 15.4684 0.810767
\(365\) 0.478153 0.0250277
\(366\) 0 0
\(367\) 1.65472 0.0863755 0.0431877 0.999067i \(-0.486249\pi\)
0.0431877 + 0.999067i \(0.486249\pi\)
\(368\) 13.7279 0.715616
\(369\) 0 0
\(370\) −0.0937252 −0.00487254
\(371\) 9.62030 0.499461
\(372\) 0 0
\(373\) −6.13773 −0.317800 −0.158900 0.987295i \(-0.550795\pi\)
−0.158900 + 0.987295i \(0.550795\pi\)
\(374\) 0.660376 0.0341472
\(375\) 0 0
\(376\) 0.330334 0.0170357
\(377\) −14.1644 −0.729503
\(378\) 0 0
\(379\) 14.3037 0.734731 0.367365 0.930077i \(-0.380260\pi\)
0.367365 + 0.930077i \(0.380260\pi\)
\(380\) 6.23230 0.319710
\(381\) 0 0
\(382\) 2.59343 0.132691
\(383\) −18.2784 −0.933981 −0.466990 0.884262i \(-0.654662\pi\)
−0.466990 + 0.884262i \(0.654662\pi\)
\(384\) 0 0
\(385\) 1.30061 0.0662854
\(386\) −0.608432 −0.0309684
\(387\) 0 0
\(388\) −24.9476 −1.26652
\(389\) −12.9086 −0.654492 −0.327246 0.944939i \(-0.606120\pi\)
−0.327246 + 0.944939i \(0.606120\pi\)
\(390\) 0 0
\(391\) −20.3297 −1.02812
\(392\) 0.271625 0.0137191
\(393\) 0 0
\(394\) −1.69149 −0.0852163
\(395\) −0.176722 −0.00889186
\(396\) 0 0
\(397\) −17.5435 −0.880484 −0.440242 0.897879i \(-0.645107\pi\)
−0.440242 + 0.897879i \(0.645107\pi\)
\(398\) 2.37328 0.118962
\(399\) 0 0
\(400\) −18.9409 −0.947045
\(401\) −35.1950 −1.75755 −0.878777 0.477233i \(-0.841640\pi\)
−0.878777 + 0.477233i \(0.841640\pi\)
\(402\) 0 0
\(403\) 5.68883 0.283381
\(404\) 9.36995 0.466173
\(405\) 0 0
\(406\) −1.45174 −0.0720487
\(407\) 2.26732 0.112387
\(408\) 0 0
\(409\) 6.31266 0.312141 0.156070 0.987746i \(-0.450117\pi\)
0.156070 + 0.987746i \(0.450117\pi\)
\(410\) −0.243099 −0.0120058
\(411\) 0 0
\(412\) 38.0650 1.87533
\(413\) 24.9873 1.22955
\(414\) 0 0
\(415\) −6.52284 −0.320194
\(416\) 3.48706 0.170967
\(417\) 0 0
\(418\) 0.821884 0.0401997
\(419\) −24.6530 −1.20438 −0.602190 0.798353i \(-0.705705\pi\)
−0.602190 + 0.798353i \(0.705705\pi\)
\(420\) 0 0
\(421\) −25.4233 −1.23906 −0.619529 0.784974i \(-0.712676\pi\)
−0.619529 + 0.784974i \(0.712676\pi\)
\(422\) −1.93596 −0.0942411
\(423\) 0 0
\(424\) 1.44450 0.0701510
\(425\) 28.0497 1.36061
\(426\) 0 0
\(427\) −28.8292 −1.39514
\(428\) 12.7331 0.615476
\(429\) 0 0
\(430\) 0.265990 0.0128272
\(431\) 35.1475 1.69299 0.846497 0.532393i \(-0.178707\pi\)
0.846497 + 0.532393i \(0.178707\pi\)
\(432\) 0 0
\(433\) −28.5125 −1.37022 −0.685111 0.728439i \(-0.740246\pi\)
−0.685111 + 0.728439i \(0.740246\pi\)
\(434\) 0.583061 0.0279878
\(435\) 0 0
\(436\) 5.87527 0.281374
\(437\) −25.3017 −1.21035
\(438\) 0 0
\(439\) 19.6250 0.936648 0.468324 0.883557i \(-0.344858\pi\)
0.468324 + 0.883557i \(0.344858\pi\)
\(440\) 0.195288 0.00931000
\(441\) 0 0
\(442\) −1.70570 −0.0811318
\(443\) 13.5449 0.643539 0.321770 0.946818i \(-0.395722\pi\)
0.321770 + 0.946818i \(0.395722\pi\)
\(444\) 0 0
\(445\) −6.14248 −0.291181
\(446\) 1.32275 0.0626338
\(447\) 0 0
\(448\) −21.4158 −1.01180
\(449\) −31.3156 −1.47787 −0.738936 0.673775i \(-0.764672\pi\)
−0.738936 + 0.673775i \(0.764672\pi\)
\(450\) 0 0
\(451\) 5.88083 0.276917
\(452\) 1.76714 0.0831193
\(453\) 0 0
\(454\) −1.24429 −0.0583973
\(455\) −3.35938 −0.157490
\(456\) 0 0
\(457\) −11.7404 −0.549193 −0.274597 0.961560i \(-0.588544\pi\)
−0.274597 + 0.961560i \(0.588544\pi\)
\(458\) −0.296497 −0.0138544
\(459\) 0 0
\(460\) −2.99781 −0.139773
\(461\) 0.0379172 0.00176598 0.000882990 1.00000i \(-0.499719\pi\)
0.000882990 1.00000i \(0.499719\pi\)
\(462\) 0 0
\(463\) 3.78515 0.175911 0.0879554 0.996124i \(-0.471967\pi\)
0.0879554 + 0.996124i \(0.471967\pi\)
\(464\) 19.8295 0.920561
\(465\) 0 0
\(466\) −0.467816 −0.0216711
\(467\) −32.5797 −1.50761 −0.753804 0.657099i \(-0.771783\pi\)
−0.753804 + 0.657099i \(0.771783\pi\)
\(468\) 0 0
\(469\) −11.0225 −0.508972
\(470\) −0.0357729 −0.00165008
\(471\) 0 0
\(472\) 3.75187 0.172694
\(473\) −6.43459 −0.295863
\(474\) 0 0
\(475\) 34.9098 1.60177
\(476\) 32.0692 1.46989
\(477\) 0 0
\(478\) −0.104133 −0.00476292
\(479\) 19.3874 0.885833 0.442917 0.896563i \(-0.353944\pi\)
0.442917 + 0.896563i \(0.353944\pi\)
\(480\) 0 0
\(481\) −5.85631 −0.267025
\(482\) −1.42280 −0.0648069
\(483\) 0 0
\(484\) 19.5250 0.887501
\(485\) 5.41803 0.246020
\(486\) 0 0
\(487\) 4.84919 0.219738 0.109869 0.993946i \(-0.464957\pi\)
0.109869 + 0.993946i \(0.464957\pi\)
\(488\) −4.32873 −0.195952
\(489\) 0 0
\(490\) −0.0294151 −0.00132884
\(491\) 24.7326 1.11617 0.558084 0.829784i \(-0.311537\pi\)
0.558084 + 0.829784i \(0.311537\pi\)
\(492\) 0 0
\(493\) −29.3656 −1.32256
\(494\) −2.12286 −0.0955121
\(495\) 0 0
\(496\) −7.96410 −0.357599
\(497\) −30.3912 −1.36323
\(498\) 0 0
\(499\) −13.0377 −0.583647 −0.291823 0.956472i \(-0.594262\pi\)
−0.291823 + 0.956472i \(0.594262\pi\)
\(500\) 8.43275 0.377124
\(501\) 0 0
\(502\) −1.89675 −0.0846559
\(503\) 3.27894 0.146201 0.0731004 0.997325i \(-0.476711\pi\)
0.0731004 + 0.997325i \(0.476711\pi\)
\(504\) 0 0
\(505\) −2.03493 −0.0905534
\(506\) −0.395336 −0.0175748
\(507\) 0 0
\(508\) 4.90853 0.217781
\(509\) −19.1552 −0.849040 −0.424520 0.905419i \(-0.639557\pi\)
−0.424520 + 0.905419i \(0.639557\pi\)
\(510\) 0 0
\(511\) 3.06214 0.135461
\(512\) −8.15098 −0.360226
\(513\) 0 0
\(514\) 0.502698 0.0221731
\(515\) −8.26682 −0.364280
\(516\) 0 0
\(517\) 0.865388 0.0380597
\(518\) −0.600227 −0.0263725
\(519\) 0 0
\(520\) −0.504414 −0.0221200
\(521\) 9.95719 0.436232 0.218116 0.975923i \(-0.430009\pi\)
0.218116 + 0.975923i \(0.430009\pi\)
\(522\) 0 0
\(523\) 23.4154 1.02389 0.511943 0.859020i \(-0.328926\pi\)
0.511943 + 0.859020i \(0.328926\pi\)
\(524\) −13.3106 −0.581474
\(525\) 0 0
\(526\) −0.0973180 −0.00424326
\(527\) 11.7941 0.513758
\(528\) 0 0
\(529\) −10.8296 −0.470851
\(530\) −0.156429 −0.00679484
\(531\) 0 0
\(532\) 39.9124 1.73042
\(533\) −15.1897 −0.657940
\(534\) 0 0
\(535\) −2.76532 −0.119555
\(536\) −1.65504 −0.0714867
\(537\) 0 0
\(538\) 2.38255 0.102719
\(539\) 0.711584 0.0306501
\(540\) 0 0
\(541\) 15.9305 0.684904 0.342452 0.939535i \(-0.388743\pi\)
0.342452 + 0.939535i \(0.388743\pi\)
\(542\) −1.51752 −0.0651831
\(543\) 0 0
\(544\) 7.22936 0.309956
\(545\) −1.27597 −0.0546566
\(546\) 0 0
\(547\) 33.0838 1.41456 0.707280 0.706934i \(-0.249922\pi\)
0.707280 + 0.706934i \(0.249922\pi\)
\(548\) −13.7588 −0.587746
\(549\) 0 0
\(550\) 0.545460 0.0232585
\(551\) −36.5475 −1.55698
\(552\) 0 0
\(553\) −1.13175 −0.0481269
\(554\) 0.669355 0.0284382
\(555\) 0 0
\(556\) 18.8687 0.800213
\(557\) 31.7597 1.34570 0.672851 0.739778i \(-0.265069\pi\)
0.672851 + 0.739778i \(0.265069\pi\)
\(558\) 0 0
\(559\) 16.6200 0.702953
\(560\) 4.70299 0.198737
\(561\) 0 0
\(562\) −1.24715 −0.0526080
\(563\) 33.8109 1.42496 0.712479 0.701694i \(-0.247572\pi\)
0.712479 + 0.701694i \(0.247572\pi\)
\(564\) 0 0
\(565\) −0.383781 −0.0161458
\(566\) 1.91636 0.0805508
\(567\) 0 0
\(568\) −4.56327 −0.191471
\(569\) −13.6032 −0.570274 −0.285137 0.958487i \(-0.592039\pi\)
−0.285137 + 0.958487i \(0.592039\pi\)
\(570\) 0 0
\(571\) 19.2244 0.804517 0.402259 0.915526i \(-0.368225\pi\)
0.402259 + 0.915526i \(0.368225\pi\)
\(572\) 6.08459 0.254410
\(573\) 0 0
\(574\) −1.55683 −0.0649809
\(575\) −16.7920 −0.700275
\(576\) 0 0
\(577\) −30.1474 −1.25505 −0.627525 0.778596i \(-0.715932\pi\)
−0.627525 + 0.778596i \(0.715932\pi\)
\(578\) −1.76600 −0.0734557
\(579\) 0 0
\(580\) −4.33023 −0.179803
\(581\) −41.7730 −1.73304
\(582\) 0 0
\(583\) 3.78419 0.156725
\(584\) 0.459783 0.0190260
\(585\) 0 0
\(586\) 2.59258 0.107098
\(587\) 34.4451 1.42170 0.710850 0.703344i \(-0.248311\pi\)
0.710850 + 0.703344i \(0.248311\pi\)
\(588\) 0 0
\(589\) 14.6786 0.604819
\(590\) −0.406302 −0.0167272
\(591\) 0 0
\(592\) 8.19856 0.336959
\(593\) 18.4600 0.758061 0.379030 0.925384i \(-0.376258\pi\)
0.379030 + 0.925384i \(0.376258\pi\)
\(594\) 0 0
\(595\) −6.96467 −0.285524
\(596\) 14.5675 0.596708
\(597\) 0 0
\(598\) 1.02112 0.0417567
\(599\) 44.8094 1.83086 0.915432 0.402473i \(-0.131849\pi\)
0.915432 + 0.402473i \(0.131849\pi\)
\(600\) 0 0
\(601\) −1.25830 −0.0513271 −0.0256636 0.999671i \(-0.508170\pi\)
−0.0256636 + 0.999671i \(0.508170\pi\)
\(602\) 1.70343 0.0694265
\(603\) 0 0
\(604\) 3.65867 0.148869
\(605\) −4.24038 −0.172396
\(606\) 0 0
\(607\) −24.6395 −1.00009 −0.500044 0.866000i \(-0.666683\pi\)
−0.500044 + 0.866000i \(0.666683\pi\)
\(608\) 8.99746 0.364895
\(609\) 0 0
\(610\) 0.468771 0.0189800
\(611\) −2.23523 −0.0904277
\(612\) 0 0
\(613\) 1.92675 0.0778205 0.0389103 0.999243i \(-0.487611\pi\)
0.0389103 + 0.999243i \(0.487611\pi\)
\(614\) 0.383204 0.0154648
\(615\) 0 0
\(616\) 1.25065 0.0503900
\(617\) −25.5030 −1.02671 −0.513357 0.858175i \(-0.671598\pi\)
−0.513357 + 0.858175i \(0.671598\pi\)
\(618\) 0 0
\(619\) −18.7143 −0.752190 −0.376095 0.926581i \(-0.622733\pi\)
−0.376095 + 0.926581i \(0.622733\pi\)
\(620\) 1.73915 0.0698459
\(621\) 0 0
\(622\) −2.97984 −0.119481
\(623\) −39.3371 −1.57601
\(624\) 0 0
\(625\) 22.2355 0.889420
\(626\) −1.49341 −0.0596886
\(627\) 0 0
\(628\) 2.95344 0.117855
\(629\) −12.1413 −0.484105
\(630\) 0 0
\(631\) −2.67027 −0.106302 −0.0531510 0.998586i \(-0.516926\pi\)
−0.0531510 + 0.998586i \(0.516926\pi\)
\(632\) −0.169933 −0.00675957
\(633\) 0 0
\(634\) −2.57326 −0.102197
\(635\) −1.06602 −0.0423036
\(636\) 0 0
\(637\) −1.83797 −0.0728229
\(638\) −0.571049 −0.0226081
\(639\) 0 0
\(640\) 1.42008 0.0561335
\(641\) 12.0848 0.477320 0.238660 0.971103i \(-0.423292\pi\)
0.238660 + 0.971103i \(0.423292\pi\)
\(642\) 0 0
\(643\) 43.4945 1.71525 0.857627 0.514272i \(-0.171938\pi\)
0.857627 + 0.514272i \(0.171938\pi\)
\(644\) −19.1983 −0.756519
\(645\) 0 0
\(646\) −4.40112 −0.173160
\(647\) −0.330370 −0.0129882 −0.00649409 0.999979i \(-0.502067\pi\)
−0.00649409 + 0.999979i \(0.502067\pi\)
\(648\) 0 0
\(649\) 9.82889 0.385818
\(650\) −1.40888 −0.0552609
\(651\) 0 0
\(652\) −19.8064 −0.775680
\(653\) 29.4599 1.15285 0.576427 0.817148i \(-0.304446\pi\)
0.576427 + 0.817148i \(0.304446\pi\)
\(654\) 0 0
\(655\) 2.89074 0.112951
\(656\) 21.2649 0.830256
\(657\) 0 0
\(658\) −0.229094 −0.00893101
\(659\) −42.6534 −1.66154 −0.830770 0.556616i \(-0.812100\pi\)
−0.830770 + 0.556616i \(0.812100\pi\)
\(660\) 0 0
\(661\) 44.7939 1.74228 0.871140 0.491034i \(-0.163381\pi\)
0.871140 + 0.491034i \(0.163381\pi\)
\(662\) 3.18026 0.123604
\(663\) 0 0
\(664\) −6.27226 −0.243411
\(665\) −8.66803 −0.336132
\(666\) 0 0
\(667\) 17.5798 0.680692
\(668\) −3.00987 −0.116455
\(669\) 0 0
\(670\) 0.179229 0.00692423
\(671\) −11.3401 −0.437780
\(672\) 0 0
\(673\) 0.183092 0.00705769 0.00352885 0.999994i \(-0.498877\pi\)
0.00352885 + 0.999994i \(0.498877\pi\)
\(674\) −2.79279 −0.107574
\(675\) 0 0
\(676\) 10.1430 0.390116
\(677\) 2.12245 0.0815725 0.0407862 0.999168i \(-0.487014\pi\)
0.0407862 + 0.999168i \(0.487014\pi\)
\(678\) 0 0
\(679\) 34.6976 1.33157
\(680\) −1.04575 −0.0401027
\(681\) 0 0
\(682\) 0.229350 0.00878227
\(683\) −6.23555 −0.238597 −0.119298 0.992858i \(-0.538064\pi\)
−0.119298 + 0.992858i \(0.538064\pi\)
\(684\) 0 0
\(685\) 2.98809 0.114169
\(686\) 1.82825 0.0698031
\(687\) 0 0
\(688\) −23.2673 −0.887057
\(689\) −9.77428 −0.372370
\(690\) 0 0
\(691\) 29.0740 1.10603 0.553014 0.833172i \(-0.313478\pi\)
0.553014 + 0.833172i \(0.313478\pi\)
\(692\) 42.1025 1.60050
\(693\) 0 0
\(694\) −0.948725 −0.0360131
\(695\) −4.09785 −0.155440
\(696\) 0 0
\(697\) −31.4913 −1.19282
\(698\) −3.31794 −0.125586
\(699\) 0 0
\(700\) 26.4886 1.00118
\(701\) −4.01876 −0.151786 −0.0758932 0.997116i \(-0.524181\pi\)
−0.0758932 + 0.997116i \(0.524181\pi\)
\(702\) 0 0
\(703\) −15.1107 −0.569911
\(704\) −8.42400 −0.317492
\(705\) 0 0
\(706\) 1.40321 0.0528106
\(707\) −13.0320 −0.490117
\(708\) 0 0
\(709\) 38.7677 1.45595 0.727975 0.685603i \(-0.240462\pi\)
0.727975 + 0.685603i \(0.240462\pi\)
\(710\) 0.494171 0.0185459
\(711\) 0 0
\(712\) −5.90650 −0.221356
\(713\) −7.06055 −0.264420
\(714\) 0 0
\(715\) −1.32143 −0.0494187
\(716\) −46.5649 −1.74021
\(717\) 0 0
\(718\) 0.248537 0.00927533
\(719\) −36.1988 −1.34999 −0.674993 0.737824i \(-0.735854\pi\)
−0.674993 + 0.737824i \(0.735854\pi\)
\(720\) 0 0
\(721\) −52.9416 −1.97165
\(722\) −3.49898 −0.130219
\(723\) 0 0
\(724\) −28.9376 −1.07546
\(725\) −24.2555 −0.900827
\(726\) 0 0
\(727\) −9.24566 −0.342902 −0.171451 0.985193i \(-0.554846\pi\)
−0.171451 + 0.985193i \(0.554846\pi\)
\(728\) −3.23033 −0.119724
\(729\) 0 0
\(730\) −0.0497914 −0.00184286
\(731\) 34.4566 1.27443
\(732\) 0 0
\(733\) 8.57554 0.316745 0.158372 0.987379i \(-0.449375\pi\)
0.158372 + 0.987379i \(0.449375\pi\)
\(734\) −0.172310 −0.00636009
\(735\) 0 0
\(736\) −4.32788 −0.159528
\(737\) −4.33576 −0.159710
\(738\) 0 0
\(739\) 14.0883 0.518246 0.259123 0.965844i \(-0.416566\pi\)
0.259123 + 0.965844i \(0.416566\pi\)
\(740\) −1.79035 −0.0658145
\(741\) 0 0
\(742\) −1.00179 −0.0367768
\(743\) 47.7553 1.75197 0.875986 0.482337i \(-0.160212\pi\)
0.875986 + 0.482337i \(0.160212\pi\)
\(744\) 0 0
\(745\) −3.16372 −0.115910
\(746\) 0.639139 0.0234005
\(747\) 0 0
\(748\) 12.6146 0.461234
\(749\) −17.7095 −0.647089
\(750\) 0 0
\(751\) 24.4902 0.893662 0.446831 0.894618i \(-0.352553\pi\)
0.446831 + 0.894618i \(0.352553\pi\)
\(752\) 3.12922 0.114111
\(753\) 0 0
\(754\) 1.47498 0.0537155
\(755\) −0.794578 −0.0289176
\(756\) 0 0
\(757\) −18.3094 −0.665465 −0.332733 0.943021i \(-0.607971\pi\)
−0.332733 + 0.943021i \(0.607971\pi\)
\(758\) −1.48948 −0.0541004
\(759\) 0 0
\(760\) −1.30151 −0.0472108
\(761\) 37.5505 1.36120 0.680602 0.732653i \(-0.261718\pi\)
0.680602 + 0.732653i \(0.261718\pi\)
\(762\) 0 0
\(763\) −8.17146 −0.295827
\(764\) 49.5399 1.79229
\(765\) 0 0
\(766\) 1.90338 0.0687718
\(767\) −25.3873 −0.916681
\(768\) 0 0
\(769\) −35.3202 −1.27368 −0.636839 0.770996i \(-0.719759\pi\)
−0.636839 + 0.770996i \(0.719759\pi\)
\(770\) −0.135437 −0.00488080
\(771\) 0 0
\(772\) −11.6223 −0.418297
\(773\) 7.48572 0.269243 0.134621 0.990897i \(-0.457018\pi\)
0.134621 + 0.990897i \(0.457018\pi\)
\(774\) 0 0
\(775\) 9.74172 0.349933
\(776\) 5.20988 0.187024
\(777\) 0 0
\(778\) 1.34421 0.0481922
\(779\) −39.1932 −1.40424
\(780\) 0 0
\(781\) −11.9545 −0.427767
\(782\) 2.11699 0.0757033
\(783\) 0 0
\(784\) 2.57307 0.0918953
\(785\) −0.641419 −0.0228932
\(786\) 0 0
\(787\) −4.60088 −0.164004 −0.0820018 0.996632i \(-0.526131\pi\)
−0.0820018 + 0.996632i \(0.526131\pi\)
\(788\) −32.3111 −1.15104
\(789\) 0 0
\(790\) 0.0184026 0.000654735 0
\(791\) −2.45778 −0.0873886
\(792\) 0 0
\(793\) 29.2906 1.04014
\(794\) 1.82686 0.0648327
\(795\) 0 0
\(796\) 45.3346 1.60684
\(797\) 4.86780 0.172426 0.0862132 0.996277i \(-0.472523\pi\)
0.0862132 + 0.996277i \(0.472523\pi\)
\(798\) 0 0
\(799\) −4.63408 −0.163942
\(800\) 5.97134 0.211119
\(801\) 0 0
\(802\) 3.66495 0.129414
\(803\) 1.20451 0.0425062
\(804\) 0 0
\(805\) 4.16942 0.146953
\(806\) −0.592393 −0.0208662
\(807\) 0 0
\(808\) −1.95676 −0.0688385
\(809\) 29.9137 1.05171 0.525855 0.850574i \(-0.323745\pi\)
0.525855 + 0.850574i \(0.323745\pi\)
\(810\) 0 0
\(811\) 32.8629 1.15397 0.576986 0.816754i \(-0.304229\pi\)
0.576986 + 0.816754i \(0.304229\pi\)
\(812\) −27.7313 −0.973178
\(813\) 0 0
\(814\) −0.236102 −0.00827538
\(815\) 4.30150 0.150675
\(816\) 0 0
\(817\) 42.8837 1.50031
\(818\) −0.657355 −0.0229839
\(819\) 0 0
\(820\) −4.64370 −0.162165
\(821\) 3.75765 0.131143 0.0655715 0.997848i \(-0.479113\pi\)
0.0655715 + 0.997848i \(0.479113\pi\)
\(822\) 0 0
\(823\) −28.8183 −1.00454 −0.502271 0.864710i \(-0.667502\pi\)
−0.502271 + 0.864710i \(0.667502\pi\)
\(824\) −7.94923 −0.276925
\(825\) 0 0
\(826\) −2.60200 −0.0905352
\(827\) 55.3088 1.92328 0.961638 0.274322i \(-0.0884533\pi\)
0.961638 + 0.274322i \(0.0884533\pi\)
\(828\) 0 0
\(829\) 35.2389 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(830\) 0.679242 0.0235768
\(831\) 0 0
\(832\) 21.7585 0.754342
\(833\) −3.81047 −0.132025
\(834\) 0 0
\(835\) 0.653674 0.0226213
\(836\) 15.6997 0.542986
\(837\) 0 0
\(838\) 2.56719 0.0886821
\(839\) −18.9566 −0.654454 −0.327227 0.944946i \(-0.606114\pi\)
−0.327227 + 0.944946i \(0.606114\pi\)
\(840\) 0 0
\(841\) −3.60658 −0.124365
\(842\) 2.64740 0.0912355
\(843\) 0 0
\(844\) −36.9809 −1.27294
\(845\) −2.20282 −0.0757794
\(846\) 0 0
\(847\) −27.1559 −0.933086
\(848\) 13.6836 0.469895
\(849\) 0 0
\(850\) −2.92089 −0.100186
\(851\) 7.26841 0.249158
\(852\) 0 0
\(853\) −5.80817 −0.198868 −0.0994339 0.995044i \(-0.531703\pi\)
−0.0994339 + 0.995044i \(0.531703\pi\)
\(854\) 3.00206 0.102728
\(855\) 0 0
\(856\) −2.65909 −0.0908858
\(857\) −20.8413 −0.711924 −0.355962 0.934500i \(-0.615847\pi\)
−0.355962 + 0.934500i \(0.615847\pi\)
\(858\) 0 0
\(859\) 20.6840 0.705728 0.352864 0.935675i \(-0.385208\pi\)
0.352864 + 0.935675i \(0.385208\pi\)
\(860\) 5.08096 0.173259
\(861\) 0 0
\(862\) −3.66000 −0.124660
\(863\) 32.6930 1.11288 0.556441 0.830887i \(-0.312166\pi\)
0.556441 + 0.830887i \(0.312166\pi\)
\(864\) 0 0
\(865\) −9.14369 −0.310895
\(866\) 2.96908 0.100894
\(867\) 0 0
\(868\) 11.1377 0.378038
\(869\) −0.445179 −0.0151017
\(870\) 0 0
\(871\) 11.1989 0.379461
\(872\) −1.22695 −0.0415498
\(873\) 0 0
\(874\) 2.63474 0.0891214
\(875\) −11.7285 −0.396495
\(876\) 0 0
\(877\) 31.3932 1.06007 0.530036 0.847975i \(-0.322178\pi\)
0.530036 + 0.847975i \(0.322178\pi\)
\(878\) −2.04360 −0.0689682
\(879\) 0 0
\(880\) 1.84994 0.0623616
\(881\) −54.7572 −1.84482 −0.922409 0.386214i \(-0.873783\pi\)
−0.922409 + 0.386214i \(0.873783\pi\)
\(882\) 0 0
\(883\) −44.9140 −1.51148 −0.755738 0.654874i \(-0.772722\pi\)
−0.755738 + 0.654874i \(0.772722\pi\)
\(884\) −32.5824 −1.09587
\(885\) 0 0
\(886\) −1.41047 −0.0473857
\(887\) 29.8624 1.00268 0.501340 0.865250i \(-0.332841\pi\)
0.501340 + 0.865250i \(0.332841\pi\)
\(888\) 0 0
\(889\) −6.82690 −0.228967
\(890\) 0.639634 0.0214406
\(891\) 0 0
\(892\) 25.2672 0.846010
\(893\) −5.76744 −0.193000
\(894\) 0 0
\(895\) 10.1128 0.338034
\(896\) 9.09433 0.303820
\(897\) 0 0
\(898\) 3.26098 0.108820
\(899\) −10.1987 −0.340147
\(900\) 0 0
\(901\) −20.2640 −0.675093
\(902\) −0.612387 −0.0203903
\(903\) 0 0
\(904\) −0.369038 −0.0122740
\(905\) 6.28456 0.208906
\(906\) 0 0
\(907\) −5.00636 −0.166234 −0.0831168 0.996540i \(-0.526487\pi\)
−0.0831168 + 0.996540i \(0.526487\pi\)
\(908\) −23.7685 −0.788786
\(909\) 0 0
\(910\) 0.349822 0.0115965
\(911\) −32.0748 −1.06269 −0.531343 0.847157i \(-0.678312\pi\)
−0.531343 + 0.847157i \(0.678312\pi\)
\(912\) 0 0
\(913\) −16.4316 −0.543808
\(914\) 1.22256 0.0404388
\(915\) 0 0
\(916\) −5.66371 −0.187134
\(917\) 18.5126 0.611341
\(918\) 0 0
\(919\) −24.0569 −0.793565 −0.396783 0.917913i \(-0.629873\pi\)
−0.396783 + 0.917913i \(0.629873\pi\)
\(920\) 0.626042 0.0206400
\(921\) 0 0
\(922\) −0.00394842 −0.000130034 0
\(923\) 30.8777 1.01635
\(924\) 0 0
\(925\) −10.0285 −0.329736
\(926\) −0.394158 −0.0129528
\(927\) 0 0
\(928\) −6.25148 −0.205215
\(929\) 39.0700 1.28184 0.640922 0.767606i \(-0.278552\pi\)
0.640922 + 0.767606i \(0.278552\pi\)
\(930\) 0 0
\(931\) −4.74240 −0.155426
\(932\) −8.93627 −0.292717
\(933\) 0 0
\(934\) 3.39262 0.111010
\(935\) −2.73959 −0.0895941
\(936\) 0 0
\(937\) −4.37686 −0.142986 −0.0714929 0.997441i \(-0.522776\pi\)
−0.0714929 + 0.997441i \(0.522776\pi\)
\(938\) 1.14780 0.0374771
\(939\) 0 0
\(940\) −0.683339 −0.0222881
\(941\) −42.0947 −1.37225 −0.686123 0.727485i \(-0.740689\pi\)
−0.686123 + 0.727485i \(0.740689\pi\)
\(942\) 0 0
\(943\) 18.8524 0.613918
\(944\) 35.5410 1.15676
\(945\) 0 0
\(946\) 0.670052 0.0217853
\(947\) −28.4050 −0.923037 −0.461519 0.887131i \(-0.652695\pi\)
−0.461519 + 0.887131i \(0.652695\pi\)
\(948\) 0 0
\(949\) −3.11115 −0.100992
\(950\) −3.63526 −0.117943
\(951\) 0 0
\(952\) −6.69711 −0.217055
\(953\) 17.0187 0.551289 0.275645 0.961260i \(-0.411109\pi\)
0.275645 + 0.961260i \(0.411109\pi\)
\(954\) 0 0
\(955\) −10.7589 −0.348150
\(956\) −1.98916 −0.0643339
\(957\) 0 0
\(958\) −2.01886 −0.0652266
\(959\) 19.1361 0.617935
\(960\) 0 0
\(961\) −26.9039 −0.867867
\(962\) 0.609834 0.0196618
\(963\) 0 0
\(964\) −27.1785 −0.875361
\(965\) 2.52410 0.0812536
\(966\) 0 0
\(967\) 9.67491 0.311124 0.155562 0.987826i \(-0.450281\pi\)
0.155562 + 0.987826i \(0.450281\pi\)
\(968\) −4.07747 −0.131055
\(969\) 0 0
\(970\) −0.564194 −0.0181152
\(971\) 34.6807 1.11296 0.556478 0.830862i \(-0.312152\pi\)
0.556478 + 0.830862i \(0.312152\pi\)
\(972\) 0 0
\(973\) −26.2431 −0.841315
\(974\) −0.504960 −0.0161800
\(975\) 0 0
\(976\) −41.0055 −1.31255
\(977\) 24.2830 0.776882 0.388441 0.921474i \(-0.373014\pi\)
0.388441 + 0.921474i \(0.373014\pi\)
\(978\) 0 0
\(979\) −15.4735 −0.494534
\(980\) −0.561890 −0.0179489
\(981\) 0 0
\(982\) −2.57548 −0.0821869
\(983\) −28.0091 −0.893351 −0.446675 0.894696i \(-0.647392\pi\)
−0.446675 + 0.894696i \(0.647392\pi\)
\(984\) 0 0
\(985\) 7.01722 0.223587
\(986\) 3.05792 0.0973840
\(987\) 0 0
\(988\) −40.5512 −1.29010
\(989\) −20.6276 −0.655918
\(990\) 0 0
\(991\) −0.266294 −0.00845912 −0.00422956 0.999991i \(-0.501346\pi\)
−0.00422956 + 0.999991i \(0.501346\pi\)
\(992\) 2.51078 0.0797172
\(993\) 0 0
\(994\) 3.16473 0.100379
\(995\) −9.84562 −0.312127
\(996\) 0 0
\(997\) −45.0714 −1.42743 −0.713713 0.700439i \(-0.752988\pi\)
−0.713713 + 0.700439i \(0.752988\pi\)
\(998\) 1.35765 0.0429757
\(999\) 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 2151.2.a.e.1.3 6
3.2 odd 2 717.2.a.d.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.4 6 3.2 odd 2
2151.2.a.e.1.3 6 1.1 even 1 trivial