Properties

Label 4851.2.a.bg.1.2
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1617)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 4851.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} +O(q^{10})\) \(q+2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} +7.47214 q^{8} -5.85410 q^{10} +1.00000 q^{11} -3.47214 q^{13} +9.85410 q^{16} +6.00000 q^{17} +4.23607 q^{19} -10.8541 q^{20} +2.61803 q^{22} +2.47214 q^{23} -9.09017 q^{26} +10.2361 q^{29} -8.47214 q^{31} +10.8541 q^{32} +15.7082 q^{34} -0.527864 q^{37} +11.0902 q^{38} -16.7082 q^{40} +6.00000 q^{41} -0.472136 q^{43} +4.85410 q^{44} +6.47214 q^{46} +11.4721 q^{47} -16.8541 q^{52} +12.9443 q^{53} -2.23607 q^{55} +26.7984 q^{58} -11.9443 q^{59} +6.94427 q^{61} -22.1803 q^{62} +8.70820 q^{64} +7.76393 q^{65} +0.708204 q^{67} +29.1246 q^{68} -4.47214 q^{71} +1.00000 q^{73} -1.38197 q^{74} +20.5623 q^{76} -2.47214 q^{79} -22.0344 q^{80} +15.7082 q^{82} +12.4721 q^{83} -13.4164 q^{85} -1.23607 q^{86} +7.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} +30.0344 q^{94} -9.47214 q^{95} +6.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 2 q^{11} + 2 q^{13} + 13 q^{16} + 12 q^{17} + 4 q^{19} - 15 q^{20} + 3 q^{22} - 4 q^{23} - 7 q^{26} + 16 q^{29} - 8 q^{31} + 15 q^{32} + 18 q^{34} - 10 q^{37} + 11 q^{38} - 20 q^{40} + 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} + 14 q^{47} - 27 q^{52} + 8 q^{53} + 29 q^{58} - 6 q^{59} - 4 q^{61} - 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} + 18 q^{68} + 2 q^{73} - 5 q^{74} + 21 q^{76} + 4 q^{79} - 15 q^{80} + 18 q^{82} + 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} + 31 q^{94} - 10 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61803 1.85123 0.925615 0.378467i \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) 0 0
\(4\) 4.85410 2.42705
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 7.47214 2.64180
\(9\) 0 0
\(10\) −5.85410 −1.85123
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.47214 −0.962997 −0.481499 0.876447i \(-0.659907\pi\)
−0.481499 + 0.876447i \(0.659907\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.23607 0.971821 0.485910 0.874009i \(-0.338488\pi\)
0.485910 + 0.874009i \(0.338488\pi\)
\(20\) −10.8541 −2.42705
\(21\) 0 0
\(22\) 2.61803 0.558167
\(23\) 2.47214 0.515476 0.257738 0.966215i \(-0.417023\pi\)
0.257738 + 0.966215i \(0.417023\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −9.09017 −1.78273
\(27\) 0 0
\(28\) 0 0
\(29\) 10.2361 1.90079 0.950395 0.311045i \(-0.100679\pi\)
0.950395 + 0.311045i \(0.100679\pi\)
\(30\) 0 0
\(31\) −8.47214 −1.52164 −0.760820 0.648963i \(-0.775203\pi\)
−0.760820 + 0.648963i \(0.775203\pi\)
\(32\) 10.8541 1.91875
\(33\) 0 0
\(34\) 15.7082 2.69393
\(35\) 0 0
\(36\) 0 0
\(37\) −0.527864 −0.0867803 −0.0433902 0.999058i \(-0.513816\pi\)
−0.0433902 + 0.999058i \(0.513816\pi\)
\(38\) 11.0902 1.79906
\(39\) 0 0
\(40\) −16.7082 −2.64180
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −0.472136 −0.0720001 −0.0360000 0.999352i \(-0.511462\pi\)
−0.0360000 + 0.999352i \(0.511462\pi\)
\(44\) 4.85410 0.731783
\(45\) 0 0
\(46\) 6.47214 0.954264
\(47\) 11.4721 1.67338 0.836692 0.547674i \(-0.184487\pi\)
0.836692 + 0.547674i \(0.184487\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −16.8541 −2.33724
\(53\) 12.9443 1.77803 0.889016 0.457876i \(-0.151390\pi\)
0.889016 + 0.457876i \(0.151390\pi\)
\(54\) 0 0
\(55\) −2.23607 −0.301511
\(56\) 0 0
\(57\) 0 0
\(58\) 26.7984 3.51880
\(59\) −11.9443 −1.55501 −0.777506 0.628876i \(-0.783515\pi\)
−0.777506 + 0.628876i \(0.783515\pi\)
\(60\) 0 0
\(61\) 6.94427 0.889123 0.444561 0.895748i \(-0.353360\pi\)
0.444561 + 0.895748i \(0.353360\pi\)
\(62\) −22.1803 −2.81691
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 7.76393 0.962997
\(66\) 0 0
\(67\) 0.708204 0.0865209 0.0432604 0.999064i \(-0.486225\pi\)
0.0432604 + 0.999064i \(0.486225\pi\)
\(68\) 29.1246 3.53188
\(69\) 0 0
\(70\) 0 0
\(71\) −4.47214 −0.530745 −0.265372 0.964146i \(-0.585495\pi\)
−0.265372 + 0.964146i \(0.585495\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −1.38197 −0.160650
\(75\) 0 0
\(76\) 20.5623 2.35866
\(77\) 0 0
\(78\) 0 0
\(79\) −2.47214 −0.278137 −0.139069 0.990283i \(-0.544411\pi\)
−0.139069 + 0.990283i \(0.544411\pi\)
\(80\) −22.0344 −2.46353
\(81\) 0 0
\(82\) 15.7082 1.73468
\(83\) 12.4721 1.36899 0.684497 0.729015i \(-0.260022\pi\)
0.684497 + 0.729015i \(0.260022\pi\)
\(84\) 0 0
\(85\) −13.4164 −1.45521
\(86\) −1.23607 −0.133289
\(87\) 0 0
\(88\) 7.47214 0.796532
\(89\) −13.4164 −1.42214 −0.711068 0.703123i \(-0.751788\pi\)
−0.711068 + 0.703123i \(0.751788\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 30.0344 3.09782
\(95\) −9.47214 −0.971821
\(96\) 0 0
\(97\) 6.94427 0.705084 0.352542 0.935796i \(-0.385317\pi\)
0.352542 + 0.935796i \(0.385317\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.52786 0.351036 0.175518 0.984476i \(-0.443840\pi\)
0.175518 + 0.984476i \(0.443840\pi\)
\(102\) 0 0
\(103\) 7.41641 0.730760 0.365380 0.930858i \(-0.380939\pi\)
0.365380 + 0.930858i \(0.380939\pi\)
\(104\) −25.9443 −2.54405
\(105\) 0 0
\(106\) 33.8885 3.29155
\(107\) −19.9443 −1.92809 −0.964043 0.265747i \(-0.914381\pi\)
−0.964043 + 0.265747i \(0.914381\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −5.85410 −0.558167
\(111\) 0 0
\(112\) 0 0
\(113\) −6.47214 −0.608847 −0.304424 0.952537i \(-0.598464\pi\)
−0.304424 + 0.952537i \(0.598464\pi\)
\(114\) 0 0
\(115\) −5.52786 −0.515476
\(116\) 49.6869 4.61331
\(117\) 0 0
\(118\) −31.2705 −2.87868
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 18.1803 1.64597
\(123\) 0 0
\(124\) −41.1246 −3.69310
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.09017 0.0963583
\(129\) 0 0
\(130\) 20.3262 1.78273
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.85410 0.160170
\(135\) 0 0
\(136\) 44.8328 3.84438
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.7082 −0.982531
\(143\) −3.47214 −0.290355
\(144\) 0 0
\(145\) −22.8885 −1.90079
\(146\) 2.61803 0.216670
\(147\) 0 0
\(148\) −2.56231 −0.210620
\(149\) −15.6525 −1.28230 −0.641150 0.767415i \(-0.721543\pi\)
−0.641150 + 0.767415i \(0.721543\pi\)
\(150\) 0 0
\(151\) −6.94427 −0.565117 −0.282558 0.959250i \(-0.591183\pi\)
−0.282558 + 0.959250i \(0.591183\pi\)
\(152\) 31.6525 2.56735
\(153\) 0 0
\(154\) 0 0
\(155\) 18.9443 1.52164
\(156\) 0 0
\(157\) −3.41641 −0.272659 −0.136330 0.990664i \(-0.543531\pi\)
−0.136330 + 0.990664i \(0.543531\pi\)
\(158\) −6.47214 −0.514895
\(159\) 0 0
\(160\) −24.2705 −1.91875
\(161\) 0 0
\(162\) 0 0
\(163\) −18.1246 −1.41963 −0.709815 0.704389i \(-0.751221\pi\)
−0.709815 + 0.704389i \(0.751221\pi\)
\(164\) 29.1246 2.27425
\(165\) 0 0
\(166\) 32.6525 2.53432
\(167\) 18.4721 1.42942 0.714708 0.699423i \(-0.246559\pi\)
0.714708 + 0.699423i \(0.246559\pi\)
\(168\) 0 0
\(169\) −0.944272 −0.0726363
\(170\) −35.1246 −2.69393
\(171\) 0 0
\(172\) −2.29180 −0.174748
\(173\) 5.52786 0.420276 0.210138 0.977672i \(-0.432609\pi\)
0.210138 + 0.977672i \(0.432609\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 9.85410 0.742781
\(177\) 0 0
\(178\) −35.1246 −2.63270
\(179\) 5.52786 0.413172 0.206586 0.978428i \(-0.433765\pi\)
0.206586 + 0.978428i \(0.433765\pi\)
\(180\) 0 0
\(181\) −0.472136 −0.0350936 −0.0175468 0.999846i \(-0.505586\pi\)
−0.0175468 + 0.999846i \(0.505586\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 18.4721 1.36178
\(185\) 1.18034 0.0867803
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 55.6869 4.06139
\(189\) 0 0
\(190\) −24.7984 −1.79906
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 25.8885 1.86350 0.931749 0.363103i \(-0.118283\pi\)
0.931749 + 0.363103i \(0.118283\pi\)
\(194\) 18.1803 1.30527
\(195\) 0 0
\(196\) 0 0
\(197\) 8.47214 0.603615 0.301807 0.953369i \(-0.402410\pi\)
0.301807 + 0.953369i \(0.402410\pi\)
\(198\) 0 0
\(199\) −24.4721 −1.73478 −0.867392 0.497626i \(-0.834205\pi\)
−0.867392 + 0.497626i \(0.834205\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 9.23607 0.649847
\(203\) 0 0
\(204\) 0 0
\(205\) −13.4164 −0.937043
\(206\) 19.4164 1.35281
\(207\) 0 0
\(208\) −34.2148 −2.37237
\(209\) 4.23607 0.293015
\(210\) 0 0
\(211\) 18.9443 1.30418 0.652089 0.758143i \(-0.273893\pi\)
0.652089 + 0.758143i \(0.273893\pi\)
\(212\) 62.8328 4.31538
\(213\) 0 0
\(214\) −52.2148 −3.56933
\(215\) 1.05573 0.0720001
\(216\) 0 0
\(217\) 0 0
\(218\) 15.7082 1.06389
\(219\) 0 0
\(220\) −10.8541 −0.731783
\(221\) −20.8328 −1.40137
\(222\) 0 0
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.9443 −1.12712
\(227\) 26.8328 1.78096 0.890478 0.455026i \(-0.150370\pi\)
0.890478 + 0.455026i \(0.150370\pi\)
\(228\) 0 0
\(229\) −15.4164 −1.01874 −0.509372 0.860546i \(-0.670122\pi\)
−0.509372 + 0.860546i \(0.670122\pi\)
\(230\) −14.4721 −0.954264
\(231\) 0 0
\(232\) 76.4853 5.02151
\(233\) −13.4164 −0.878938 −0.439469 0.898258i \(-0.644833\pi\)
−0.439469 + 0.898258i \(0.644833\pi\)
\(234\) 0 0
\(235\) −25.6525 −1.67338
\(236\) −57.9787 −3.77409
\(237\) 0 0
\(238\) 0 0
\(239\) −17.3607 −1.12297 −0.561485 0.827487i \(-0.689770\pi\)
−0.561485 + 0.827487i \(0.689770\pi\)
\(240\) 0 0
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) 2.61803 0.168294
\(243\) 0 0
\(244\) 33.7082 2.15795
\(245\) 0 0
\(246\) 0 0
\(247\) −14.7082 −0.935861
\(248\) −63.3050 −4.01987
\(249\) 0 0
\(250\) 29.2705 1.85123
\(251\) 3.94427 0.248960 0.124480 0.992222i \(-0.460274\pi\)
0.124480 + 0.992222i \(0.460274\pi\)
\(252\) 0 0
\(253\) 2.47214 0.155422
\(254\) 41.8885 2.62832
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) −7.76393 −0.484301 −0.242150 0.970239i \(-0.577853\pi\)
−0.242150 + 0.970239i \(0.577853\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 37.6869 2.33724
\(261\) 0 0
\(262\) −11.7082 −0.723335
\(263\) 4.41641 0.272327 0.136164 0.990686i \(-0.456523\pi\)
0.136164 + 0.990686i \(0.456523\pi\)
\(264\) 0 0
\(265\) −28.9443 −1.77803
\(266\) 0 0
\(267\) 0 0
\(268\) 3.43769 0.209991
\(269\) −16.4721 −1.00432 −0.502162 0.864774i \(-0.667462\pi\)
−0.502162 + 0.864774i \(0.667462\pi\)
\(270\) 0 0
\(271\) −5.29180 −0.321454 −0.160727 0.986999i \(-0.551384\pi\)
−0.160727 + 0.986999i \(0.551384\pi\)
\(272\) 59.1246 3.58496
\(273\) 0 0
\(274\) −15.7082 −0.948967
\(275\) 0 0
\(276\) 0 0
\(277\) −18.9443 −1.13825 −0.569125 0.822251i \(-0.692718\pi\)
−0.569125 + 0.822251i \(0.692718\pi\)
\(278\) −20.9443 −1.25615
\(279\) 0 0
\(280\) 0 0
\(281\) −5.18034 −0.309033 −0.154517 0.987990i \(-0.549382\pi\)
−0.154517 + 0.987990i \(0.549382\pi\)
\(282\) 0 0
\(283\) −4.70820 −0.279874 −0.139937 0.990160i \(-0.544690\pi\)
−0.139937 + 0.990160i \(0.544690\pi\)
\(284\) −21.7082 −1.28814
\(285\) 0 0
\(286\) −9.09017 −0.537513
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −59.9230 −3.51880
\(291\) 0 0
\(292\) 4.85410 0.284065
\(293\) −9.88854 −0.577695 −0.288847 0.957375i \(-0.593272\pi\)
−0.288847 + 0.957375i \(0.593272\pi\)
\(294\) 0 0
\(295\) 26.7082 1.55501
\(296\) −3.94427 −0.229256
\(297\) 0 0
\(298\) −40.9787 −2.37383
\(299\) −8.58359 −0.496402
\(300\) 0 0
\(301\) 0 0
\(302\) −18.1803 −1.04616
\(303\) 0 0
\(304\) 41.7426 2.39410
\(305\) −15.5279 −0.889123
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 49.5967 2.81691
\(311\) −12.9443 −0.734002 −0.367001 0.930220i \(-0.619615\pi\)
−0.367001 + 0.930220i \(0.619615\pi\)
\(312\) 0 0
\(313\) −26.3607 −1.48999 −0.744997 0.667068i \(-0.767549\pi\)
−0.744997 + 0.667068i \(0.767549\pi\)
\(314\) −8.94427 −0.504754
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −26.9443 −1.51334 −0.756671 0.653796i \(-0.773175\pi\)
−0.756671 + 0.653796i \(0.773175\pi\)
\(318\) 0 0
\(319\) 10.2361 0.573110
\(320\) −19.4721 −1.08853
\(321\) 0 0
\(322\) 0 0
\(323\) 25.4164 1.41421
\(324\) 0 0
\(325\) 0 0
\(326\) −47.4508 −2.62806
\(327\) 0 0
\(328\) 44.8328 2.47548
\(329\) 0 0
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 60.5410 3.32262
\(333\) 0 0
\(334\) 48.3607 2.64618
\(335\) −1.58359 −0.0865209
\(336\) 0 0
\(337\) 11.4164 0.621891 0.310946 0.950428i \(-0.399354\pi\)
0.310946 + 0.950428i \(0.399354\pi\)
\(338\) −2.47214 −0.134466
\(339\) 0 0
\(340\) −65.1246 −3.53188
\(341\) −8.47214 −0.458792
\(342\) 0 0
\(343\) 0 0
\(344\) −3.52786 −0.190210
\(345\) 0 0
\(346\) 14.4721 0.778027
\(347\) −0.944272 −0.0506912 −0.0253456 0.999679i \(-0.508069\pi\)
−0.0253456 + 0.999679i \(0.508069\pi\)
\(348\) 0 0
\(349\) 2.41641 0.129347 0.0646737 0.997906i \(-0.479399\pi\)
0.0646737 + 0.997906i \(0.479399\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 10.8541 0.578526
\(353\) −4.23607 −0.225463 −0.112732 0.993625i \(-0.535960\pi\)
−0.112732 + 0.993625i \(0.535960\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) −65.1246 −3.45160
\(357\) 0 0
\(358\) 14.4721 0.764876
\(359\) 20.9443 1.10540 0.552698 0.833381i \(-0.313598\pi\)
0.552698 + 0.833381i \(0.313598\pi\)
\(360\) 0 0
\(361\) −1.05573 −0.0555646
\(362\) −1.23607 −0.0649663
\(363\) 0 0
\(364\) 0 0
\(365\) −2.23607 −0.117041
\(366\) 0 0
\(367\) 25.4164 1.32673 0.663363 0.748298i \(-0.269129\pi\)
0.663363 + 0.748298i \(0.269129\pi\)
\(368\) 24.3607 1.26989
\(369\) 0 0
\(370\) 3.09017 0.160650
\(371\) 0 0
\(372\) 0 0
\(373\) −16.9443 −0.877341 −0.438671 0.898648i \(-0.644550\pi\)
−0.438671 + 0.898648i \(0.644550\pi\)
\(374\) 15.7082 0.812252
\(375\) 0 0
\(376\) 85.7214 4.42074
\(377\) −35.5410 −1.83046
\(378\) 0 0
\(379\) −25.6525 −1.31768 −0.658840 0.752283i \(-0.728952\pi\)
−0.658840 + 0.752283i \(0.728952\pi\)
\(380\) −45.9787 −2.35866
\(381\) 0 0
\(382\) −31.4164 −1.60740
\(383\) 8.94427 0.457031 0.228515 0.973540i \(-0.426613\pi\)
0.228515 + 0.973540i \(0.426613\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 67.7771 3.44976
\(387\) 0 0
\(388\) 33.7082 1.71127
\(389\) −20.8328 −1.05627 −0.528133 0.849162i \(-0.677108\pi\)
−0.528133 + 0.849162i \(0.677108\pi\)
\(390\) 0 0
\(391\) 14.8328 0.750128
\(392\) 0 0
\(393\) 0 0
\(394\) 22.1803 1.11743
\(395\) 5.52786 0.278137
\(396\) 0 0
\(397\) −9.41641 −0.472596 −0.236298 0.971681i \(-0.575934\pi\)
−0.236298 + 0.971681i \(0.575934\pi\)
\(398\) −64.0689 −3.21148
\(399\) 0 0
\(400\) 0 0
\(401\) 8.47214 0.423078 0.211539 0.977370i \(-0.432152\pi\)
0.211539 + 0.977370i \(0.432152\pi\)
\(402\) 0 0
\(403\) 29.4164 1.46534
\(404\) 17.1246 0.851981
\(405\) 0 0
\(406\) 0 0
\(407\) −0.527864 −0.0261652
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −35.1246 −1.73468
\(411\) 0 0
\(412\) 36.0000 1.77359
\(413\) 0 0
\(414\) 0 0
\(415\) −27.8885 −1.36899
\(416\) −37.6869 −1.84775
\(417\) 0 0
\(418\) 11.0902 0.542438
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 29.4721 1.43638 0.718192 0.695845i \(-0.244970\pi\)
0.718192 + 0.695845i \(0.244970\pi\)
\(422\) 49.5967 2.41433
\(423\) 0 0
\(424\) 96.7214 4.69720
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −96.8115 −4.67956
\(429\) 0 0
\(430\) 2.76393 0.133289
\(431\) 17.3607 0.836235 0.418117 0.908393i \(-0.362690\pi\)
0.418117 + 0.908393i \(0.362690\pi\)
\(432\) 0 0
\(433\) 12.5836 0.604729 0.302364 0.953192i \(-0.402224\pi\)
0.302364 + 0.953192i \(0.402224\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 29.1246 1.39482
\(437\) 10.4721 0.500950
\(438\) 0 0
\(439\) −8.12461 −0.387767 −0.193883 0.981025i \(-0.562108\pi\)
−0.193883 + 0.981025i \(0.562108\pi\)
\(440\) −16.7082 −0.796532
\(441\) 0 0
\(442\) −54.5410 −2.59425
\(443\) −9.05573 −0.430251 −0.215125 0.976586i \(-0.569016\pi\)
−0.215125 + 0.976586i \(0.569016\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) −47.1246 −2.23142
\(447\) 0 0
\(448\) 0 0
\(449\) 25.4164 1.19947 0.599737 0.800197i \(-0.295272\pi\)
0.599737 + 0.800197i \(0.295272\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −31.4164 −1.47770
\(453\) 0 0
\(454\) 70.2492 3.29696
\(455\) 0 0
\(456\) 0 0
\(457\) −11.4164 −0.534037 −0.267019 0.963691i \(-0.586038\pi\)
−0.267019 + 0.963691i \(0.586038\pi\)
\(458\) −40.3607 −1.88593
\(459\) 0 0
\(460\) −26.8328 −1.25109
\(461\) 16.4721 0.767184 0.383592 0.923503i \(-0.374687\pi\)
0.383592 + 0.923503i \(0.374687\pi\)
\(462\) 0 0
\(463\) −2.70820 −0.125861 −0.0629305 0.998018i \(-0.520045\pi\)
−0.0629305 + 0.998018i \(0.520045\pi\)
\(464\) 100.867 4.68264
\(465\) 0 0
\(466\) −35.1246 −1.62712
\(467\) 23.8328 1.10285 0.551426 0.834224i \(-0.314084\pi\)
0.551426 + 0.834224i \(0.314084\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −67.1591 −3.09782
\(471\) 0 0
\(472\) −89.2492 −4.10803
\(473\) −0.472136 −0.0217088
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −45.4508 −2.07887
\(479\) 21.8885 1.00011 0.500057 0.865993i \(-0.333313\pi\)
0.500057 + 0.865993i \(0.333313\pi\)
\(480\) 0 0
\(481\) 1.83282 0.0835692
\(482\) 49.7426 2.26572
\(483\) 0 0
\(484\) 4.85410 0.220641
\(485\) −15.5279 −0.705084
\(486\) 0 0
\(487\) −21.8885 −0.991865 −0.495932 0.868361i \(-0.665174\pi\)
−0.495932 + 0.868361i \(0.665174\pi\)
\(488\) 51.8885 2.34888
\(489\) 0 0
\(490\) 0 0
\(491\) 10.8885 0.491393 0.245697 0.969347i \(-0.420983\pi\)
0.245697 + 0.969347i \(0.420983\pi\)
\(492\) 0 0
\(493\) 61.4164 2.76606
\(494\) −38.5066 −1.73249
\(495\) 0 0
\(496\) −83.4853 −3.74860
\(497\) 0 0
\(498\) 0 0
\(499\) 22.1246 0.990434 0.495217 0.868769i \(-0.335089\pi\)
0.495217 + 0.868769i \(0.335089\pi\)
\(500\) 54.2705 2.42705
\(501\) 0 0
\(502\) 10.3262 0.460883
\(503\) 10.5836 0.471899 0.235950 0.971765i \(-0.424180\pi\)
0.235950 + 0.971765i \(0.424180\pi\)
\(504\) 0 0
\(505\) −7.88854 −0.351036
\(506\) 6.47214 0.287722
\(507\) 0 0
\(508\) 77.6656 3.44586
\(509\) 35.5279 1.57474 0.787372 0.616478i \(-0.211441\pi\)
0.787372 + 0.616478i \(0.211441\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −40.3050 −1.78124
\(513\) 0 0
\(514\) −20.3262 −0.896552
\(515\) −16.5836 −0.730760
\(516\) 0 0
\(517\) 11.4721 0.504544
\(518\) 0 0
\(519\) 0 0
\(520\) 58.0132 2.54405
\(521\) −23.7639 −1.04112 −0.520558 0.853826i \(-0.674276\pi\)
−0.520558 + 0.853826i \(0.674276\pi\)
\(522\) 0 0
\(523\) −3.29180 −0.143940 −0.0719701 0.997407i \(-0.522929\pi\)
−0.0719701 + 0.997407i \(0.522929\pi\)
\(524\) −21.7082 −0.948327
\(525\) 0 0
\(526\) 11.5623 0.504140
\(527\) −50.8328 −2.21431
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) −75.7771 −3.29155
\(531\) 0 0
\(532\) 0 0
\(533\) −20.8328 −0.902369
\(534\) 0 0
\(535\) 44.5967 1.92809
\(536\) 5.29180 0.228571
\(537\) 0 0
\(538\) −43.1246 −1.85923
\(539\) 0 0
\(540\) 0 0
\(541\) 21.4164 0.920763 0.460382 0.887721i \(-0.347713\pi\)
0.460382 + 0.887721i \(0.347713\pi\)
\(542\) −13.8541 −0.595085
\(543\) 0 0
\(544\) 65.1246 2.79219
\(545\) −13.4164 −0.574696
\(546\) 0 0
\(547\) −13.4164 −0.573644 −0.286822 0.957984i \(-0.592599\pi\)
−0.286822 + 0.957984i \(0.592599\pi\)
\(548\) −29.1246 −1.24414
\(549\) 0 0
\(550\) 0 0
\(551\) 43.3607 1.84723
\(552\) 0 0
\(553\) 0 0
\(554\) −49.5967 −2.10716
\(555\) 0 0
\(556\) −38.8328 −1.64688
\(557\) 8.12461 0.344251 0.172125 0.985075i \(-0.444937\pi\)
0.172125 + 0.985075i \(0.444937\pi\)
\(558\) 0 0
\(559\) 1.63932 0.0693359
\(560\) 0 0
\(561\) 0 0
\(562\) −13.5623 −0.572091
\(563\) −4.47214 −0.188478 −0.0942390 0.995550i \(-0.530042\pi\)
−0.0942390 + 0.995550i \(0.530042\pi\)
\(564\) 0 0
\(565\) 14.4721 0.608847
\(566\) −12.3262 −0.518110
\(567\) 0 0
\(568\) −33.4164 −1.40212
\(569\) 12.4721 0.522859 0.261430 0.965223i \(-0.415806\pi\)
0.261430 + 0.965223i \(0.415806\pi\)
\(570\) 0 0
\(571\) −10.5836 −0.442910 −0.221455 0.975171i \(-0.571081\pi\)
−0.221455 + 0.975171i \(0.571081\pi\)
\(572\) −16.8541 −0.704705
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.9443 −1.45475 −0.727375 0.686241i \(-0.759260\pi\)
−0.727375 + 0.686241i \(0.759260\pi\)
\(578\) 49.7426 2.06902
\(579\) 0 0
\(580\) −111.103 −4.61331
\(581\) 0 0
\(582\) 0 0
\(583\) 12.9443 0.536097
\(584\) 7.47214 0.309199
\(585\) 0 0
\(586\) −25.8885 −1.06945
\(587\) 14.0557 0.580142 0.290071 0.957005i \(-0.406321\pi\)
0.290071 + 0.957005i \(0.406321\pi\)
\(588\) 0 0
\(589\) −35.8885 −1.47876
\(590\) 69.9230 2.87868
\(591\) 0 0
\(592\) −5.20163 −0.213786
\(593\) −37.4164 −1.53651 −0.768254 0.640145i \(-0.778874\pi\)
−0.768254 + 0.640145i \(0.778874\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −75.9787 −3.11221
\(597\) 0 0
\(598\) −22.4721 −0.918954
\(599\) −25.3050 −1.03393 −0.516966 0.856006i \(-0.672939\pi\)
−0.516966 + 0.856006i \(0.672939\pi\)
\(600\) 0 0
\(601\) −5.83282 −0.237926 −0.118963 0.992899i \(-0.537957\pi\)
−0.118963 + 0.992899i \(0.537957\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −33.7082 −1.37157
\(605\) −2.23607 −0.0909091
\(606\) 0 0
\(607\) 22.7082 0.921698 0.460849 0.887479i \(-0.347545\pi\)
0.460849 + 0.887479i \(0.347545\pi\)
\(608\) 45.9787 1.86468
\(609\) 0 0
\(610\) −40.6525 −1.64597
\(611\) −39.8328 −1.61146
\(612\) 0 0
\(613\) 19.4164 0.784221 0.392111 0.919918i \(-0.371745\pi\)
0.392111 + 0.919918i \(0.371745\pi\)
\(614\) −41.8885 −1.69048
\(615\) 0 0
\(616\) 0 0
\(617\) 40.2492 1.62037 0.810186 0.586172i \(-0.199366\pi\)
0.810186 + 0.586172i \(0.199366\pi\)
\(618\) 0 0
\(619\) −27.4164 −1.10196 −0.550979 0.834519i \(-0.685746\pi\)
−0.550979 + 0.834519i \(0.685746\pi\)
\(620\) 91.9574 3.69310
\(621\) 0 0
\(622\) −33.8885 −1.35881
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) −69.0132 −2.75832
\(627\) 0 0
\(628\) −16.5836 −0.661757
\(629\) −3.16718 −0.126284
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) −18.4721 −0.734782
\(633\) 0 0
\(634\) −70.5410 −2.80154
\(635\) −35.7771 −1.41977
\(636\) 0 0
\(637\) 0 0
\(638\) 26.7984 1.06096
\(639\) 0 0
\(640\) −2.43769 −0.0963583
\(641\) −31.5279 −1.24528 −0.622638 0.782510i \(-0.713939\pi\)
−0.622638 + 0.782510i \(0.713939\pi\)
\(642\) 0 0
\(643\) −5.41641 −0.213602 −0.106801 0.994280i \(-0.534061\pi\)
−0.106801 + 0.994280i \(0.534061\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 66.5410 2.61802
\(647\) −4.41641 −0.173627 −0.0868135 0.996225i \(-0.527668\pi\)
−0.0868135 + 0.996225i \(0.527668\pi\)
\(648\) 0 0
\(649\) −11.9443 −0.468854
\(650\) 0 0
\(651\) 0 0
\(652\) −87.9787 −3.44551
\(653\) −5.88854 −0.230437 −0.115218 0.993340i \(-0.536757\pi\)
−0.115218 + 0.993340i \(0.536757\pi\)
\(654\) 0 0
\(655\) 10.0000 0.390732
\(656\) 59.1246 2.30843
\(657\) 0 0
\(658\) 0 0
\(659\) −32.8885 −1.28116 −0.640578 0.767893i \(-0.721305\pi\)
−0.640578 + 0.767893i \(0.721305\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −41.8885 −1.62804
\(663\) 0 0
\(664\) 93.1935 3.61661
\(665\) 0 0
\(666\) 0 0
\(667\) 25.3050 0.979812
\(668\) 89.6656 3.46927
\(669\) 0 0
\(670\) −4.14590 −0.160170
\(671\) 6.94427 0.268081
\(672\) 0 0
\(673\) −18.5836 −0.716345 −0.358172 0.933655i \(-0.616600\pi\)
−0.358172 + 0.933655i \(0.616600\pi\)
\(674\) 29.8885 1.15126
\(675\) 0 0
\(676\) −4.58359 −0.176292
\(677\) 43.4164 1.66863 0.834314 0.551289i \(-0.185864\pi\)
0.834314 + 0.551289i \(0.185864\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −100.249 −3.84438
\(681\) 0 0
\(682\) −22.1803 −0.849329
\(683\) −9.52786 −0.364574 −0.182287 0.983245i \(-0.558350\pi\)
−0.182287 + 0.983245i \(0.558350\pi\)
\(684\) 0 0
\(685\) 13.4164 0.512615
\(686\) 0 0
\(687\) 0 0
\(688\) −4.65248 −0.177374
\(689\) −44.9443 −1.71224
\(690\) 0 0
\(691\) −6.58359 −0.250452 −0.125226 0.992128i \(-0.539966\pi\)
−0.125226 + 0.992128i \(0.539966\pi\)
\(692\) 26.8328 1.02003
\(693\) 0 0
\(694\) −2.47214 −0.0938410
\(695\) 17.8885 0.678551
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 6.32624 0.239452
\(699\) 0 0
\(700\) 0 0
\(701\) 13.4164 0.506731 0.253365 0.967371i \(-0.418463\pi\)
0.253365 + 0.967371i \(0.418463\pi\)
\(702\) 0 0
\(703\) −2.23607 −0.0843349
\(704\) 8.70820 0.328203
\(705\) 0 0
\(706\) −11.0902 −0.417384
\(707\) 0 0
\(708\) 0 0
\(709\) 22.4164 0.841866 0.420933 0.907092i \(-0.361703\pi\)
0.420933 + 0.907092i \(0.361703\pi\)
\(710\) 26.1803 0.982531
\(711\) 0 0
\(712\) −100.249 −3.75700
\(713\) −20.9443 −0.784369
\(714\) 0 0
\(715\) 7.76393 0.290355
\(716\) 26.8328 1.00279
\(717\) 0 0
\(718\) 54.8328 2.04634
\(719\) −4.63932 −0.173017 −0.0865087 0.996251i \(-0.527571\pi\)
−0.0865087 + 0.996251i \(0.527571\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.76393 −0.102863
\(723\) 0 0
\(724\) −2.29180 −0.0851739
\(725\) 0 0
\(726\) 0 0
\(727\) −4.58359 −0.169996 −0.0849980 0.996381i \(-0.527088\pi\)
−0.0849980 + 0.996381i \(0.527088\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −5.85410 −0.216670
\(731\) −2.83282 −0.104775
\(732\) 0 0
\(733\) −29.0557 −1.07320 −0.536599 0.843837i \(-0.680291\pi\)
−0.536599 + 0.843837i \(0.680291\pi\)
\(734\) 66.5410 2.45607
\(735\) 0 0
\(736\) 26.8328 0.989071
\(737\) 0.708204 0.0260870
\(738\) 0 0
\(739\) 43.4164 1.59710 0.798549 0.601930i \(-0.205601\pi\)
0.798549 + 0.601930i \(0.205601\pi\)
\(740\) 5.72949 0.210620
\(741\) 0 0
\(742\) 0 0
\(743\) −12.3050 −0.451425 −0.225712 0.974194i \(-0.572471\pi\)
−0.225712 + 0.974194i \(0.572471\pi\)
\(744\) 0 0
\(745\) 35.0000 1.28230
\(746\) −44.3607 −1.62416
\(747\) 0 0
\(748\) 29.1246 1.06490
\(749\) 0 0
\(750\) 0 0
\(751\) 12.1246 0.442433 0.221217 0.975225i \(-0.428997\pi\)
0.221217 + 0.975225i \(0.428997\pi\)
\(752\) 113.048 4.12242
\(753\) 0 0
\(754\) −93.0476 −3.38859
\(755\) 15.5279 0.565117
\(756\) 0 0
\(757\) −11.5836 −0.421013 −0.210506 0.977592i \(-0.567511\pi\)
−0.210506 + 0.977592i \(0.567511\pi\)
\(758\) −67.1591 −2.43933
\(759\) 0 0
\(760\) −70.7771 −2.56735
\(761\) 13.5279 0.490385 0.245192 0.969474i \(-0.421149\pi\)
0.245192 + 0.969474i \(0.421149\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −58.2492 −2.10738
\(765\) 0 0
\(766\) 23.4164 0.846069
\(767\) 41.4721 1.49747
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 125.666 4.52281
\(773\) −20.3475 −0.731850 −0.365925 0.930644i \(-0.619247\pi\)
−0.365925 + 0.930644i \(0.619247\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 51.8885 1.86269
\(777\) 0 0
\(778\) −54.5410 −1.95539
\(779\) 25.4164 0.910637
\(780\) 0 0
\(781\) −4.47214 −0.160026
\(782\) 38.8328 1.38866
\(783\) 0 0
\(784\) 0 0
\(785\) 7.63932 0.272659
\(786\) 0 0
\(787\) −38.1246 −1.35900 −0.679498 0.733678i \(-0.737802\pi\)
−0.679498 + 0.733678i \(0.737802\pi\)
\(788\) 41.1246 1.46500
\(789\) 0 0
\(790\) 14.4721 0.514895
\(791\) 0 0
\(792\) 0 0
\(793\) −24.1115 −0.856223
\(794\) −24.6525 −0.874884
\(795\) 0 0
\(796\) −118.790 −4.21041
\(797\) −45.7639 −1.62104 −0.810521 0.585710i \(-0.800816\pi\)
−0.810521 + 0.585710i \(0.800816\pi\)
\(798\) 0 0
\(799\) 68.8328 2.43513
\(800\) 0 0
\(801\) 0 0
\(802\) 22.1803 0.783215
\(803\) 1.00000 0.0352892
\(804\) 0 0
\(805\) 0 0
\(806\) 77.0132 2.71267
\(807\) 0 0
\(808\) 26.3607 0.927365
\(809\) 14.8197 0.521032 0.260516 0.965470i \(-0.416107\pi\)
0.260516 + 0.965470i \(0.416107\pi\)
\(810\) 0 0
\(811\) 27.5410 0.967096 0.483548 0.875318i \(-0.339348\pi\)
0.483548 + 0.875318i \(0.339348\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.38197 −0.0484379
\(815\) 40.5279 1.41963
\(816\) 0 0
\(817\) −2.00000 −0.0699711
\(818\) −36.6525 −1.28152
\(819\) 0 0
\(820\) −65.1246 −2.27425
\(821\) 33.5410 1.17059 0.585295 0.810821i \(-0.300979\pi\)
0.585295 + 0.810821i \(0.300979\pi\)
\(822\) 0 0
\(823\) −45.5410 −1.58746 −0.793730 0.608270i \(-0.791864\pi\)
−0.793730 + 0.608270i \(0.791864\pi\)
\(824\) 55.4164 1.93052
\(825\) 0 0
\(826\) 0 0
\(827\) −0.167184 −0.00581357 −0.00290678 0.999996i \(-0.500925\pi\)
−0.00290678 + 0.999996i \(0.500925\pi\)
\(828\) 0 0
\(829\) 34.8328 1.20979 0.604897 0.796304i \(-0.293214\pi\)
0.604897 + 0.796304i \(0.293214\pi\)
\(830\) −73.0132 −2.53432
\(831\) 0 0
\(832\) −30.2361 −1.04825
\(833\) 0 0
\(834\) 0 0
\(835\) −41.3050 −1.42942
\(836\) 20.5623 0.711162
\(837\) 0 0
\(838\) 7.85410 0.271315
\(839\) −20.5279 −0.708701 −0.354350 0.935113i \(-0.615298\pi\)
−0.354350 + 0.935113i \(0.615298\pi\)
\(840\) 0 0
\(841\) 75.7771 2.61300
\(842\) 77.1591 2.65908
\(843\) 0 0
\(844\) 91.9574 3.16531
\(845\) 2.11146 0.0726363
\(846\) 0 0
\(847\) 0 0
\(848\) 127.554 4.38023
\(849\) 0 0
\(850\) 0 0
\(851\) −1.30495 −0.0447332
\(852\) 0 0
\(853\) 54.7214 1.87362 0.936812 0.349834i \(-0.113762\pi\)
0.936812 + 0.349834i \(0.113762\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −149.026 −5.09361
\(857\) −50.7214 −1.73261 −0.866304 0.499517i \(-0.833511\pi\)
−0.866304 + 0.499517i \(0.833511\pi\)
\(858\) 0 0
\(859\) −15.4164 −0.526001 −0.263001 0.964796i \(-0.584712\pi\)
−0.263001 + 0.964796i \(0.584712\pi\)
\(860\) 5.12461 0.174748
\(861\) 0 0
\(862\) 45.4508 1.54806
\(863\) −35.3050 −1.20179 −0.600897 0.799326i \(-0.705190\pi\)
−0.600897 + 0.799326i \(0.705190\pi\)
\(864\) 0 0
\(865\) −12.3607 −0.420276
\(866\) 32.9443 1.11949
\(867\) 0 0
\(868\) 0 0
\(869\) −2.47214 −0.0838615
\(870\) 0 0
\(871\) −2.45898 −0.0833194
\(872\) 44.8328 1.51823
\(873\) 0 0
\(874\) 27.4164 0.927374
\(875\) 0 0
\(876\) 0 0
\(877\) −21.8885 −0.739124 −0.369562 0.929206i \(-0.620492\pi\)
−0.369562 + 0.929206i \(0.620492\pi\)
\(878\) −21.2705 −0.717845
\(879\) 0 0
\(880\) −22.0344 −0.742781
\(881\) −22.5967 −0.761304 −0.380652 0.924718i \(-0.624300\pi\)
−0.380652 + 0.924718i \(0.624300\pi\)
\(882\) 0 0
\(883\) 4.23607 0.142555 0.0712775 0.997457i \(-0.477292\pi\)
0.0712775 + 0.997457i \(0.477292\pi\)
\(884\) −101.125 −3.40119
\(885\) 0 0
\(886\) −23.7082 −0.796493
\(887\) −28.3607 −0.952258 −0.476129 0.879375i \(-0.657961\pi\)
−0.476129 + 0.879375i \(0.657961\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 78.5410 2.63270
\(891\) 0 0
\(892\) −87.3738 −2.92549
\(893\) 48.5967 1.62623
\(894\) 0 0
\(895\) −12.3607 −0.413172
\(896\) 0 0
\(897\) 0 0
\(898\) 66.5410 2.22050
\(899\) −86.7214 −2.89232
\(900\) 0 0
\(901\) 77.6656 2.58742
\(902\) 15.7082 0.523026
\(903\) 0 0
\(904\) −48.3607 −1.60845
\(905\) 1.05573 0.0350936
\(906\) 0 0
\(907\) −14.1115 −0.468563 −0.234282 0.972169i \(-0.575274\pi\)
−0.234282 + 0.972169i \(0.575274\pi\)
\(908\) 130.249 4.32247
\(909\) 0 0
\(910\) 0 0
\(911\) 31.4164 1.04087 0.520436 0.853901i \(-0.325769\pi\)
0.520436 + 0.853901i \(0.325769\pi\)
\(912\) 0 0
\(913\) 12.4721 0.412767
\(914\) −29.8885 −0.988625
\(915\) 0 0
\(916\) −74.8328 −2.47255
\(917\) 0 0
\(918\) 0 0
\(919\) 1.88854 0.0622973 0.0311487 0.999515i \(-0.490083\pi\)
0.0311487 + 0.999515i \(0.490083\pi\)
\(920\) −41.3050 −1.36178
\(921\) 0 0
\(922\) 43.1246 1.42023
\(923\) 15.5279 0.511106
\(924\) 0 0
\(925\) 0 0
\(926\) −7.09017 −0.232997
\(927\) 0 0
\(928\) 111.103 3.64715
\(929\) 52.4853 1.72199 0.860993 0.508616i \(-0.169843\pi\)
0.860993 + 0.508616i \(0.169843\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −65.1246 −2.13323
\(933\) 0 0
\(934\) 62.3951 2.04163
\(935\) −13.4164 −0.438763
\(936\) 0 0
\(937\) −42.7214 −1.39565 −0.697823 0.716270i \(-0.745848\pi\)
−0.697823 + 0.716270i \(0.745848\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −124.520 −4.06139
\(941\) 51.1935 1.66886 0.834430 0.551114i \(-0.185797\pi\)
0.834430 + 0.551114i \(0.185797\pi\)
\(942\) 0 0
\(943\) 14.8328 0.483023
\(944\) −117.700 −3.83081
\(945\) 0 0
\(946\) −1.23607 −0.0401880
\(947\) −0.111456 −0.00362184 −0.00181092 0.999998i \(-0.500576\pi\)
−0.00181092 + 0.999998i \(0.500576\pi\)
\(948\) 0 0
\(949\) −3.47214 −0.112710
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 28.4853 0.922729 0.461365 0.887211i \(-0.347360\pi\)
0.461365 + 0.887211i \(0.347360\pi\)
\(954\) 0 0
\(955\) 26.8328 0.868290
\(956\) −84.2705 −2.72550
\(957\) 0 0
\(958\) 57.3050 1.85144
\(959\) 0 0
\(960\) 0 0
\(961\) 40.7771 1.31539
\(962\) 4.79837 0.154706
\(963\) 0 0
\(964\) 92.2279 2.97046
\(965\) −57.8885 −1.86350
\(966\) 0 0
\(967\) 46.2492 1.48727 0.743637 0.668583i \(-0.233099\pi\)
0.743637 + 0.668583i \(0.233099\pi\)
\(968\) 7.47214 0.240164
\(969\) 0 0
\(970\) −40.6525 −1.30527
\(971\) 24.8885 0.798711 0.399356 0.916796i \(-0.369234\pi\)
0.399356 + 0.916796i \(0.369234\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −57.3050 −1.83617
\(975\) 0 0
\(976\) 68.4296 2.19038
\(977\) 13.5279 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(978\) 0 0
\(979\) −13.4164 −0.428790
\(980\) 0 0
\(981\) 0 0
\(982\) 28.5066 0.909681
\(983\) 38.8328 1.23857 0.619287 0.785165i \(-0.287422\pi\)
0.619287 + 0.785165i \(0.287422\pi\)
\(984\) 0 0
\(985\) −18.9443 −0.603615
\(986\) 160.790 5.12060
\(987\) 0 0
\(988\) −71.3951 −2.27138
\(989\) −1.16718 −0.0371143
\(990\) 0 0
\(991\) 13.2918 0.422228 0.211114 0.977461i \(-0.432291\pi\)
0.211114 + 0.977461i \(0.432291\pi\)
\(992\) −91.9574 −2.91965
\(993\) 0 0
\(994\) 0 0
\(995\) 54.7214 1.73478
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 57.9230 1.83352
\(999\) 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 4851.2.a.bg.1.2 2
3.2 odd 2 1617.2.a.l.1.1 yes 2
7.6 odd 2 4851.2.a.bh.1.2 2
21.20 even 2 1617.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.k.1.1 2 21.20 even 2
1617.2.a.l.1.1 yes 2 3.2 odd 2
4851.2.a.bg.1.2 2 1.1 even 1 trivial
4851.2.a.bh.1.2 2 7.6 odd 2