Properties

Label 105.2.a.b.1.1
Level 105
Weight 2
Character 105.1
Self dual yes
Analytic conductor 0.838
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.838429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\)
Character \(\chi\) = 105.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -1.00000 q^{5} +2.23607 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{10} +6.47214 q^{11} -3.00000 q^{12} +4.47214 q^{13} -2.23607 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -2.23607 q^{18} -2.47214 q^{19} -3.00000 q^{20} -1.00000 q^{21} -14.4721 q^{22} +4.00000 q^{23} +2.23607 q^{24} +1.00000 q^{25} -10.0000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -2.00000 q^{29} -2.23607 q^{30} +1.52786 q^{31} +6.70820 q^{32} -6.47214 q^{33} +4.47214 q^{34} -1.00000 q^{35} +3.00000 q^{36} -6.94427 q^{37} +5.52786 q^{38} -4.47214 q^{39} +2.23607 q^{40} -2.00000 q^{41} +2.23607 q^{42} +8.94427 q^{43} +19.4164 q^{44} -1.00000 q^{45} -8.94427 q^{46} +12.9443 q^{47} +1.00000 q^{48} +1.00000 q^{49} -2.23607 q^{50} +2.00000 q^{51} +13.4164 q^{52} -3.52786 q^{53} +2.23607 q^{54} -6.47214 q^{55} -2.23607 q^{56} +2.47214 q^{57} +4.47214 q^{58} -8.94427 q^{59} +3.00000 q^{60} -2.00000 q^{61} -3.41641 q^{62} +1.00000 q^{63} -13.0000 q^{64} -4.47214 q^{65} +14.4721 q^{66} -4.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} +2.23607 q^{70} +5.52786 q^{71} -2.23607 q^{72} -12.4721 q^{73} +15.5279 q^{74} -1.00000 q^{75} -7.41641 q^{76} +6.47214 q^{77} +10.0000 q^{78} +12.9443 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.47214 q^{82} -16.9443 q^{83} -3.00000 q^{84} +2.00000 q^{85} -20.0000 q^{86} +2.00000 q^{87} -14.4721 q^{88} -2.00000 q^{89} +2.23607 q^{90} +4.47214 q^{91} +12.0000 q^{92} -1.52786 q^{93} -28.9443 q^{94} +2.47214 q^{95} -6.70820 q^{96} +8.47214 q^{97} -2.23607 q^{98} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{11} - 6q^{12} + 2q^{15} - 2q^{16} - 4q^{17} + 4q^{19} - 6q^{20} - 2q^{21} - 20q^{22} + 8q^{23} + 2q^{25} - 20q^{26} - 2q^{27} + 6q^{28} - 4q^{29} + 12q^{31} - 4q^{33} - 2q^{35} + 6q^{36} + 4q^{37} + 20q^{38} - 4q^{41} + 12q^{44} - 2q^{45} + 8q^{47} + 2q^{48} + 2q^{49} + 4q^{51} - 16q^{53} - 4q^{55} - 4q^{57} + 6q^{60} - 4q^{61} + 20q^{62} + 2q^{63} - 26q^{64} + 20q^{66} - 8q^{67} - 12q^{68} - 8q^{69} + 20q^{71} - 16q^{73} + 40q^{74} - 2q^{75} + 12q^{76} + 4q^{77} + 20q^{78} + 8q^{79} + 2q^{80} + 2q^{81} - 16q^{83} - 6q^{84} + 4q^{85} - 40q^{86} + 4q^{87} - 20q^{88} - 4q^{89} + 24q^{92} - 12q^{93} - 40q^{94} - 4q^{95} + 8q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) −1.00000 −0.447214
\(6\) 2.23607 0.912871
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 2.23607 0.707107
\(11\) 6.47214 1.95142 0.975711 0.219061i \(-0.0702993\pi\)
0.975711 + 0.219061i \(0.0702993\pi\)
\(12\) −3.00000 −0.866025
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) −2.23607 −0.597614
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −2.23607 −0.527046
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.00000 −0.218218
\(22\) −14.4721 −3.08547
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.00000 0.200000
\(26\) −10.0000 −1.96116
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −2.23607 −0.408248
\(31\) 1.52786 0.274412 0.137206 0.990543i \(-0.456188\pi\)
0.137206 + 0.990543i \(0.456188\pi\)
\(32\) 6.70820 1.18585
\(33\) −6.47214 −1.12665
\(34\) 4.47214 0.766965
\(35\) −1.00000 −0.169031
\(36\) 3.00000 0.500000
\(37\) −6.94427 −1.14163 −0.570816 0.821078i \(-0.693373\pi\)
−0.570816 + 0.821078i \(0.693373\pi\)
\(38\) 5.52786 0.896738
\(39\) −4.47214 −0.716115
\(40\) 2.23607 0.353553
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.23607 0.345033
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 19.4164 2.92713
\(45\) −1.00000 −0.149071
\(46\) −8.94427 −1.31876
\(47\) 12.9443 1.88812 0.944058 0.329779i \(-0.106974\pi\)
0.944058 + 0.329779i \(0.106974\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −2.23607 −0.316228
\(51\) 2.00000 0.280056
\(52\) 13.4164 1.86052
\(53\) −3.52786 −0.484589 −0.242295 0.970203i \(-0.577900\pi\)
−0.242295 + 0.970203i \(0.577900\pi\)
\(54\) 2.23607 0.304290
\(55\) −6.47214 −0.872703
\(56\) −2.23607 −0.298807
\(57\) 2.47214 0.327442
\(58\) 4.47214 0.587220
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 3.00000 0.387298
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −3.41641 −0.433884
\(63\) 1.00000 0.125988
\(64\) −13.0000 −1.62500
\(65\) −4.47214 −0.554700
\(66\) 14.4721 1.78140
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) 2.23607 0.267261
\(71\) 5.52786 0.656037 0.328018 0.944671i \(-0.393619\pi\)
0.328018 + 0.944671i \(0.393619\pi\)
\(72\) −2.23607 −0.263523
\(73\) −12.4721 −1.45975 −0.729877 0.683579i \(-0.760422\pi\)
−0.729877 + 0.683579i \(0.760422\pi\)
\(74\) 15.5279 1.80508
\(75\) −1.00000 −0.115470
\(76\) −7.41641 −0.850720
\(77\) 6.47214 0.737568
\(78\) 10.0000 1.13228
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.47214 0.493865
\(83\) −16.9443 −1.85988 −0.929938 0.367717i \(-0.880140\pi\)
−0.929938 + 0.367717i \(0.880140\pi\)
\(84\) −3.00000 −0.327327
\(85\) 2.00000 0.216930
\(86\) −20.0000 −2.15666
\(87\) 2.00000 0.214423
\(88\) −14.4721 −1.54273
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.23607 0.235702
\(91\) 4.47214 0.468807
\(92\) 12.0000 1.25109
\(93\) −1.52786 −0.158432
\(94\) −28.9443 −2.98537
\(95\) 2.47214 0.253636
\(96\) −6.70820 −0.684653
\(97\) 8.47214 0.860215 0.430108 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) −2.23607 −0.225877
\(99\) 6.47214 0.650474
\(100\) 3.00000 0.300000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −4.47214 −0.442807
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −10.0000 −0.980581
\(105\) 1.00000 0.0975900
\(106\) 7.88854 0.766203
\(107\) −12.9443 −1.25137 −0.625685 0.780076i \(-0.715180\pi\)
−0.625685 + 0.780076i \(0.715180\pi\)
\(108\) −3.00000 −0.288675
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 14.4721 1.37986
\(111\) 6.94427 0.659121
\(112\) −1.00000 −0.0944911
\(113\) 0.472136 0.0444148 0.0222074 0.999753i \(-0.492931\pi\)
0.0222074 + 0.999753i \(0.492931\pi\)
\(114\) −5.52786 −0.517732
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 4.47214 0.413449
\(118\) 20.0000 1.84115
\(119\) −2.00000 −0.183340
\(120\) −2.23607 −0.204124
\(121\) 30.8885 2.80805
\(122\) 4.47214 0.404888
\(123\) 2.00000 0.180334
\(124\) 4.58359 0.411619
\(125\) −1.00000 −0.0894427
\(126\) −2.23607 −0.199205
\(127\) −4.94427 −0.438733 −0.219367 0.975643i \(-0.570399\pi\)
−0.219367 + 0.975643i \(0.570399\pi\)
\(128\) 15.6525 1.38350
\(129\) −8.94427 −0.787499
\(130\) 10.0000 0.877058
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −19.4164 −1.68998
\(133\) −2.47214 −0.214361
\(134\) 8.94427 0.772667
\(135\) 1.00000 0.0860663
\(136\) 4.47214 0.383482
\(137\) 3.52786 0.301406 0.150703 0.988579i \(-0.451846\pi\)
0.150703 + 0.988579i \(0.451846\pi\)
\(138\) 8.94427 0.761387
\(139\) 7.41641 0.629052 0.314526 0.949249i \(-0.398155\pi\)
0.314526 + 0.949249i \(0.398155\pi\)
\(140\) −3.00000 −0.253546
\(141\) −12.9443 −1.09010
\(142\) −12.3607 −1.03729
\(143\) 28.9443 2.42044
\(144\) −1.00000 −0.0833333
\(145\) 2.00000 0.166091
\(146\) 27.8885 2.30807
\(147\) −1.00000 −0.0824786
\(148\) −20.8328 −1.71245
\(149\) −14.9443 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(150\) 2.23607 0.182574
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 5.52786 0.448369
\(153\) −2.00000 −0.161690
\(154\) −14.4721 −1.16620
\(155\) −1.52786 −0.122721
\(156\) −13.4164 −1.07417
\(157\) −0.472136 −0.0376806 −0.0188403 0.999823i \(-0.505997\pi\)
−0.0188403 + 0.999823i \(0.505997\pi\)
\(158\) −28.9443 −2.30268
\(159\) 3.52786 0.279778
\(160\) −6.70820 −0.530330
\(161\) 4.00000 0.315244
\(162\) −2.23607 −0.175682
\(163\) −16.9443 −1.32718 −0.663589 0.748097i \(-0.730968\pi\)
−0.663589 + 0.748097i \(0.730968\pi\)
\(164\) −6.00000 −0.468521
\(165\) 6.47214 0.503855
\(166\) 37.8885 2.94072
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 2.23607 0.172516
\(169\) 7.00000 0.538462
\(170\) −4.47214 −0.342997
\(171\) −2.47214 −0.189049
\(172\) 26.8328 2.04598
\(173\) −2.94427 −0.223849 −0.111924 0.993717i \(-0.535701\pi\)
−0.111924 + 0.993717i \(0.535701\pi\)
\(174\) −4.47214 −0.339032
\(175\) 1.00000 0.0755929
\(176\) −6.47214 −0.487856
\(177\) 8.94427 0.672293
\(178\) 4.47214 0.335201
\(179\) 6.47214 0.483750 0.241875 0.970307i \(-0.422238\pi\)
0.241875 + 0.970307i \(0.422238\pi\)
\(180\) −3.00000 −0.223607
\(181\) 1.05573 0.0784717 0.0392358 0.999230i \(-0.487508\pi\)
0.0392358 + 0.999230i \(0.487508\pi\)
\(182\) −10.0000 −0.741249
\(183\) 2.00000 0.147844
\(184\) −8.94427 −0.659380
\(185\) 6.94427 0.510553
\(186\) 3.41641 0.250503
\(187\) −12.9443 −0.946579
\(188\) 38.8328 2.83217
\(189\) −1.00000 −0.0727393
\(190\) −5.52786 −0.401033
\(191\) 0.583592 0.0422272 0.0211136 0.999777i \(-0.493279\pi\)
0.0211136 + 0.999777i \(0.493279\pi\)
\(192\) 13.0000 0.938194
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −18.9443 −1.36012
\(195\) 4.47214 0.320256
\(196\) 3.00000 0.214286
\(197\) 15.5279 1.10631 0.553157 0.833077i \(-0.313423\pi\)
0.553157 + 0.833077i \(0.313423\pi\)
\(198\) −14.4721 −1.02849
\(199\) 27.4164 1.94350 0.971749 0.236017i \(-0.0758423\pi\)
0.971749 + 0.236017i \(0.0758423\pi\)
\(200\) −2.23607 −0.158114
\(201\) 4.00000 0.282138
\(202\) 31.3050 2.20261
\(203\) −2.00000 −0.140372
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) −4.47214 −0.310087
\(209\) −16.0000 −1.10674
\(210\) −2.23607 −0.154303
\(211\) −16.9443 −1.16649 −0.583246 0.812296i \(-0.698218\pi\)
−0.583246 + 0.812296i \(0.698218\pi\)
\(212\) −10.5836 −0.726884
\(213\) −5.52786 −0.378763
\(214\) 28.9443 1.97859
\(215\) −8.94427 −0.609994
\(216\) 2.23607 0.152145
\(217\) 1.52786 0.103718
\(218\) 4.47214 0.302891
\(219\) 12.4721 0.842789
\(220\) −19.4164 −1.30905
\(221\) −8.94427 −0.601657
\(222\) −15.5279 −1.04216
\(223\) −12.9443 −0.866813 −0.433406 0.901199i \(-0.642688\pi\)
−0.433406 + 0.901199i \(0.642688\pi\)
\(224\) 6.70820 0.448211
\(225\) 1.00000 0.0666667
\(226\) −1.05573 −0.0702260
\(227\) 0.944272 0.0626735 0.0313368 0.999509i \(-0.490024\pi\)
0.0313368 + 0.999509i \(0.490024\pi\)
\(228\) 7.41641 0.491164
\(229\) 23.8885 1.57860 0.789300 0.614008i \(-0.210444\pi\)
0.789300 + 0.614008i \(0.210444\pi\)
\(230\) 8.94427 0.589768
\(231\) −6.47214 −0.425835
\(232\) 4.47214 0.293610
\(233\) −9.41641 −0.616889 −0.308445 0.951242i \(-0.599808\pi\)
−0.308445 + 0.951242i \(0.599808\pi\)
\(234\) −10.0000 −0.653720
\(235\) −12.9443 −0.844391
\(236\) −26.8328 −1.74667
\(237\) −12.9443 −0.840821
\(238\) 4.47214 0.289886
\(239\) −10.4721 −0.677386 −0.338693 0.940897i \(-0.609985\pi\)
−0.338693 + 0.940897i \(0.609985\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −18.9443 −1.22031 −0.610154 0.792283i \(-0.708892\pi\)
−0.610154 + 0.792283i \(0.708892\pi\)
\(242\) −69.0689 −4.43992
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −1.00000 −0.0638877
\(246\) −4.47214 −0.285133
\(247\) −11.0557 −0.703459
\(248\) −3.41641 −0.216942
\(249\) 16.9443 1.07380
\(250\) 2.23607 0.141421
\(251\) 16.9443 1.06951 0.534756 0.845006i \(-0.320403\pi\)
0.534756 + 0.845006i \(0.320403\pi\)
\(252\) 3.00000 0.188982
\(253\) 25.8885 1.62760
\(254\) 11.0557 0.693698
\(255\) −2.00000 −0.125245
\(256\) −9.00000 −0.562500
\(257\) 18.9443 1.18171 0.590856 0.806777i \(-0.298790\pi\)
0.590856 + 0.806777i \(0.298790\pi\)
\(258\) 20.0000 1.24515
\(259\) −6.94427 −0.431496
\(260\) −13.4164 −0.832050
\(261\) −2.00000 −0.123797
\(262\) −8.94427 −0.552579
\(263\) 7.05573 0.435075 0.217537 0.976052i \(-0.430198\pi\)
0.217537 + 0.976052i \(0.430198\pi\)
\(264\) 14.4721 0.890698
\(265\) 3.52786 0.216715
\(266\) 5.52786 0.338935
\(267\) 2.00000 0.122398
\(268\) −12.0000 −0.733017
\(269\) 11.8885 0.724857 0.362429 0.932012i \(-0.381948\pi\)
0.362429 + 0.932012i \(0.381948\pi\)
\(270\) −2.23607 −0.136083
\(271\) −1.52786 −0.0928111 −0.0464056 0.998923i \(-0.514777\pi\)
−0.0464056 + 0.998923i \(0.514777\pi\)
\(272\) 2.00000 0.121268
\(273\) −4.47214 −0.270666
\(274\) −7.88854 −0.476564
\(275\) 6.47214 0.390284
\(276\) −12.0000 −0.722315
\(277\) 18.9443 1.13825 0.569125 0.822251i \(-0.307282\pi\)
0.569125 + 0.822251i \(0.307282\pi\)
\(278\) −16.5836 −0.994618
\(279\) 1.52786 0.0914708
\(280\) 2.23607 0.133631
\(281\) −10.9443 −0.652881 −0.326440 0.945218i \(-0.605849\pi\)
−0.326440 + 0.945218i \(0.605849\pi\)
\(282\) 28.9443 1.72361
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 16.5836 0.984055
\(285\) −2.47214 −0.146437
\(286\) −64.7214 −3.82705
\(287\) −2.00000 −0.118056
\(288\) 6.70820 0.395285
\(289\) −13.0000 −0.764706
\(290\) −4.47214 −0.262613
\(291\) −8.47214 −0.496645
\(292\) −37.4164 −2.18963
\(293\) 5.05573 0.295359 0.147679 0.989035i \(-0.452820\pi\)
0.147679 + 0.989035i \(0.452820\pi\)
\(294\) 2.23607 0.130410
\(295\) 8.94427 0.520756
\(296\) 15.5279 0.902539
\(297\) −6.47214 −0.375551
\(298\) 33.4164 1.93576
\(299\) 17.8885 1.03452
\(300\) −3.00000 −0.173205
\(301\) 8.94427 0.515539
\(302\) 35.7771 2.05874
\(303\) 14.0000 0.804279
\(304\) 2.47214 0.141787
\(305\) 2.00000 0.114520
\(306\) 4.47214 0.255655
\(307\) −15.0557 −0.859276 −0.429638 0.903001i \(-0.641359\pi\)
−0.429638 + 0.903001i \(0.641359\pi\)
\(308\) 19.4164 1.10635
\(309\) 0 0
\(310\) 3.41641 0.194039
\(311\) 25.8885 1.46800 0.734002 0.679147i \(-0.237650\pi\)
0.734002 + 0.679147i \(0.237650\pi\)
\(312\) 10.0000 0.566139
\(313\) −17.4164 −0.984434 −0.492217 0.870473i \(-0.663813\pi\)
−0.492217 + 0.870473i \(0.663813\pi\)
\(314\) 1.05573 0.0595782
\(315\) −1.00000 −0.0563436
\(316\) 38.8328 2.18452
\(317\) 14.3607 0.806576 0.403288 0.915073i \(-0.367867\pi\)
0.403288 + 0.915073i \(0.367867\pi\)
\(318\) −7.88854 −0.442368
\(319\) −12.9443 −0.724740
\(320\) 13.0000 0.726722
\(321\) 12.9443 0.722479
\(322\) −8.94427 −0.498445
\(323\) 4.94427 0.275107
\(324\) 3.00000 0.166667
\(325\) 4.47214 0.248069
\(326\) 37.8885 2.09845
\(327\) 2.00000 0.110600
\(328\) 4.47214 0.246932
\(329\) 12.9443 0.713641
\(330\) −14.4721 −0.796665
\(331\) 0.944272 0.0519019 0.0259509 0.999663i \(-0.491739\pi\)
0.0259509 + 0.999663i \(0.491739\pi\)
\(332\) −50.8328 −2.78981
\(333\) −6.94427 −0.380544
\(334\) 17.8885 0.978818
\(335\) 4.00000 0.218543
\(336\) 1.00000 0.0545545
\(337\) −23.8885 −1.30129 −0.650646 0.759381i \(-0.725502\pi\)
−0.650646 + 0.759381i \(0.725502\pi\)
\(338\) −15.6525 −0.851382
\(339\) −0.472136 −0.0256429
\(340\) 6.00000 0.325396
\(341\) 9.88854 0.535495
\(342\) 5.52786 0.298913
\(343\) 1.00000 0.0539949
\(344\) −20.0000 −1.07833
\(345\) 4.00000 0.215353
\(346\) 6.58359 0.353936
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 6.00000 0.321634
\(349\) −11.8885 −0.636379 −0.318190 0.948027i \(-0.603075\pi\)
−0.318190 + 0.948027i \(0.603075\pi\)
\(350\) −2.23607 −0.119523
\(351\) −4.47214 −0.238705
\(352\) 43.4164 2.31410
\(353\) 7.88854 0.419865 0.209932 0.977716i \(-0.432676\pi\)
0.209932 + 0.977716i \(0.432676\pi\)
\(354\) −20.0000 −1.06299
\(355\) −5.52786 −0.293389
\(356\) −6.00000 −0.317999
\(357\) 2.00000 0.105851
\(358\) −14.4721 −0.764876
\(359\) 18.4721 0.974922 0.487461 0.873145i \(-0.337923\pi\)
0.487461 + 0.873145i \(0.337923\pi\)
\(360\) 2.23607 0.117851
\(361\) −12.8885 −0.678344
\(362\) −2.36068 −0.124075
\(363\) −30.8885 −1.62123
\(364\) 13.4164 0.703211
\(365\) 12.4721 0.652821
\(366\) −4.47214 −0.233762
\(367\) 3.05573 0.159508 0.0797539 0.996815i \(-0.474587\pi\)
0.0797539 + 0.996815i \(0.474587\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) −15.5279 −0.807255
\(371\) −3.52786 −0.183158
\(372\) −4.58359 −0.237648
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 28.9443 1.49667
\(375\) 1.00000 0.0516398
\(376\) −28.9443 −1.49269
\(377\) −8.94427 −0.460653
\(378\) 2.23607 0.115011
\(379\) −37.8885 −1.94620 −0.973102 0.230375i \(-0.926005\pi\)
−0.973102 + 0.230375i \(0.926005\pi\)
\(380\) 7.41641 0.380454
\(381\) 4.94427 0.253303
\(382\) −1.30495 −0.0667671
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −15.6525 −0.798762
\(385\) −6.47214 −0.329851
\(386\) 31.3050 1.59338
\(387\) 8.94427 0.454663
\(388\) 25.4164 1.29032
\(389\) −6.94427 −0.352089 −0.176044 0.984382i \(-0.556330\pi\)
−0.176044 + 0.984382i \(0.556330\pi\)
\(390\) −10.0000 −0.506370
\(391\) −8.00000 −0.404577
\(392\) −2.23607 −0.112938
\(393\) −4.00000 −0.201773
\(394\) −34.7214 −1.74924
\(395\) −12.9443 −0.651297
\(396\) 19.4164 0.975711
\(397\) −13.4164 −0.673350 −0.336675 0.941621i \(-0.609302\pi\)
−0.336675 + 0.941621i \(0.609302\pi\)
\(398\) −61.3050 −3.07294
\(399\) 2.47214 0.123762
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −8.94427 −0.446100
\(403\) 6.83282 0.340367
\(404\) −42.0000 −2.08958
\(405\) −1.00000 −0.0496904
\(406\) 4.47214 0.221948
\(407\) −44.9443 −2.22780
\(408\) −4.47214 −0.221404
\(409\) 11.8885 0.587851 0.293925 0.955828i \(-0.405038\pi\)
0.293925 + 0.955828i \(0.405038\pi\)
\(410\) −4.47214 −0.220863
\(411\) −3.52786 −0.174017
\(412\) 0 0
\(413\) −8.94427 −0.440119
\(414\) −8.94427 −0.439587
\(415\) 16.9443 0.831762
\(416\) 30.0000 1.47087
\(417\) −7.41641 −0.363183
\(418\) 35.7771 1.74991
\(419\) −29.8885 −1.46015 −0.730075 0.683367i \(-0.760515\pi\)
−0.730075 + 0.683367i \(0.760515\pi\)
\(420\) 3.00000 0.146385
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 37.8885 1.84439
\(423\) 12.9443 0.629372
\(424\) 7.88854 0.383102
\(425\) −2.00000 −0.0970143
\(426\) 12.3607 0.598877
\(427\) −2.00000 −0.0967868
\(428\) −38.8328 −1.87705
\(429\) −28.9443 −1.39744
\(430\) 20.0000 0.964486
\(431\) −18.4721 −0.889771 −0.444886 0.895587i \(-0.646756\pi\)
−0.444886 + 0.895587i \(0.646756\pi\)
\(432\) 1.00000 0.0481125
\(433\) 16.4721 0.791600 0.395800 0.918337i \(-0.370467\pi\)
0.395800 + 0.918337i \(0.370467\pi\)
\(434\) −3.41641 −0.163993
\(435\) −2.00000 −0.0958927
\(436\) −6.00000 −0.287348
\(437\) −9.88854 −0.473033
\(438\) −27.8885 −1.33257
\(439\) 1.52786 0.0729210 0.0364605 0.999335i \(-0.488392\pi\)
0.0364605 + 0.999335i \(0.488392\pi\)
\(440\) 14.4721 0.689932
\(441\) 1.00000 0.0476190
\(442\) 20.0000 0.951303
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) 20.8328 0.988682
\(445\) 2.00000 0.0948091
\(446\) 28.9443 1.37055
\(447\) 14.9443 0.706840
\(448\) −13.0000 −0.614192
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −2.23607 −0.105409
\(451\) −12.9443 −0.609522
\(452\) 1.41641 0.0666222
\(453\) 16.0000 0.751746
\(454\) −2.11146 −0.0990955
\(455\) −4.47214 −0.209657
\(456\) −5.52786 −0.258866
\(457\) 6.94427 0.324839 0.162420 0.986722i \(-0.448070\pi\)
0.162420 + 0.986722i \(0.448070\pi\)
\(458\) −53.4164 −2.49598
\(459\) 2.00000 0.0933520
\(460\) −12.0000 −0.559503
\(461\) 3.88854 0.181108 0.0905538 0.995892i \(-0.471136\pi\)
0.0905538 + 0.995892i \(0.471136\pi\)
\(462\) 14.4721 0.673305
\(463\) 20.9443 0.973363 0.486681 0.873580i \(-0.338207\pi\)
0.486681 + 0.873580i \(0.338207\pi\)
\(464\) 2.00000 0.0928477
\(465\) 1.52786 0.0708530
\(466\) 21.0557 0.975388
\(467\) −8.94427 −0.413892 −0.206946 0.978352i \(-0.566352\pi\)
−0.206946 + 0.978352i \(0.566352\pi\)
\(468\) 13.4164 0.620174
\(469\) −4.00000 −0.184703
\(470\) 28.9443 1.33510
\(471\) 0.472136 0.0217549
\(472\) 20.0000 0.920575
\(473\) 57.8885 2.66172
\(474\) 28.9443 1.32945
\(475\) −2.47214 −0.113429
\(476\) −6.00000 −0.275010
\(477\) −3.52786 −0.161530
\(478\) 23.4164 1.07104
\(479\) −17.8885 −0.817348 −0.408674 0.912680i \(-0.634009\pi\)
−0.408674 + 0.912680i \(0.634009\pi\)
\(480\) 6.70820 0.306186
\(481\) −31.0557 −1.41602
\(482\) 42.3607 1.92948
\(483\) −4.00000 −0.182006
\(484\) 92.6656 4.21207
\(485\) −8.47214 −0.384700
\(486\) 2.23607 0.101430
\(487\) −20.9443 −0.949076 −0.474538 0.880235i \(-0.657385\pi\)
−0.474538 + 0.880235i \(0.657385\pi\)
\(488\) 4.47214 0.202444
\(489\) 16.9443 0.766246
\(490\) 2.23607 0.101015
\(491\) −21.3050 −0.961479 −0.480740 0.876863i \(-0.659632\pi\)
−0.480740 + 0.876863i \(0.659632\pi\)
\(492\) 6.00000 0.270501
\(493\) 4.00000 0.180151
\(494\) 24.7214 1.11227
\(495\) −6.47214 −0.290901
\(496\) −1.52786 −0.0686031
\(497\) 5.52786 0.247959
\(498\) −37.8885 −1.69783
\(499\) −13.8885 −0.621737 −0.310868 0.950453i \(-0.600620\pi\)
−0.310868 + 0.950453i \(0.600620\pi\)
\(500\) −3.00000 −0.134164
\(501\) 8.00000 0.357414
\(502\) −37.8885 −1.69105
\(503\) 32.0000 1.42681 0.713405 0.700752i \(-0.247152\pi\)
0.713405 + 0.700752i \(0.247152\pi\)
\(504\) −2.23607 −0.0996024
\(505\) 14.0000 0.622992
\(506\) −57.8885 −2.57346
\(507\) −7.00000 −0.310881
\(508\) −14.8328 −0.658100
\(509\) −23.8885 −1.05884 −0.529421 0.848360i \(-0.677591\pi\)
−0.529421 + 0.848360i \(0.677591\pi\)
\(510\) 4.47214 0.198030
\(511\) −12.4721 −0.551735
\(512\) −11.1803 −0.494106
\(513\) 2.47214 0.109147
\(514\) −42.3607 −1.86845
\(515\) 0 0
\(516\) −26.8328 −1.18125
\(517\) 83.7771 3.68451
\(518\) 15.5279 0.682255
\(519\) 2.94427 0.129239
\(520\) 10.0000 0.438529
\(521\) −19.8885 −0.871333 −0.435666 0.900108i \(-0.643487\pi\)
−0.435666 + 0.900108i \(0.643487\pi\)
\(522\) 4.47214 0.195740
\(523\) −8.94427 −0.391106 −0.195553 0.980693i \(-0.562650\pi\)
−0.195553 + 0.980693i \(0.562650\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) −15.7771 −0.687914
\(527\) −3.05573 −0.133110
\(528\) 6.47214 0.281664
\(529\) −7.00000 −0.304348
\(530\) −7.88854 −0.342656
\(531\) −8.94427 −0.388148
\(532\) −7.41641 −0.321542
\(533\) −8.94427 −0.387419
\(534\) −4.47214 −0.193528
\(535\) 12.9443 0.559630
\(536\) 8.94427 0.386334
\(537\) −6.47214 −0.279293
\(538\) −26.5836 −1.14610
\(539\) 6.47214 0.278775
\(540\) 3.00000 0.129099
\(541\) −11.8885 −0.511128 −0.255564 0.966792i \(-0.582261\pi\)
−0.255564 + 0.966792i \(0.582261\pi\)
\(542\) 3.41641 0.146747
\(543\) −1.05573 −0.0453056
\(544\) −13.4164 −0.575224
\(545\) 2.00000 0.0856706
\(546\) 10.0000 0.427960
\(547\) 5.88854 0.251776 0.125888 0.992044i \(-0.459822\pi\)
0.125888 + 0.992044i \(0.459822\pi\)
\(548\) 10.5836 0.452109
\(549\) −2.00000 −0.0853579
\(550\) −14.4721 −0.617094
\(551\) 4.94427 0.210633
\(552\) 8.94427 0.380693
\(553\) 12.9443 0.550446
\(554\) −42.3607 −1.79973
\(555\) −6.94427 −0.294768
\(556\) 22.2492 0.943577
\(557\) 20.4721 0.867432 0.433716 0.901050i \(-0.357202\pi\)
0.433716 + 0.901050i \(0.357202\pi\)
\(558\) −3.41641 −0.144628
\(559\) 40.0000 1.69182
\(560\) 1.00000 0.0422577
\(561\) 12.9443 0.546508
\(562\) 24.4721 1.03229
\(563\) 13.8885 0.585332 0.292666 0.956215i \(-0.405458\pi\)
0.292666 + 0.956215i \(0.405458\pi\)
\(564\) −38.8328 −1.63516
\(565\) −0.472136 −0.0198629
\(566\) −26.8328 −1.12787
\(567\) 1.00000 0.0419961
\(568\) −12.3607 −0.518643
\(569\) −39.8885 −1.67221 −0.836107 0.548566i \(-0.815174\pi\)
−0.836107 + 0.548566i \(0.815174\pi\)
\(570\) 5.52786 0.231537
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 86.8328 3.63066
\(573\) −0.583592 −0.0243799
\(574\) 4.47214 0.186663
\(575\) 4.00000 0.166812
\(576\) −13.0000 −0.541667
\(577\) 10.3607 0.431321 0.215660 0.976468i \(-0.430810\pi\)
0.215660 + 0.976468i \(0.430810\pi\)
\(578\) 29.0689 1.20911
\(579\) 14.0000 0.581820
\(580\) 6.00000 0.249136
\(581\) −16.9443 −0.702967
\(582\) 18.9443 0.785265
\(583\) −22.8328 −0.945639
\(584\) 27.8885 1.15404
\(585\) −4.47214 −0.184900
\(586\) −11.3050 −0.467003
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −3.00000 −0.123718
\(589\) −3.77709 −0.155632
\(590\) −20.0000 −0.823387
\(591\) −15.5279 −0.638731
\(592\) 6.94427 0.285408
\(593\) 23.8885 0.980985 0.490492 0.871445i \(-0.336817\pi\)
0.490492 + 0.871445i \(0.336817\pi\)
\(594\) 14.4721 0.593799
\(595\) 2.00000 0.0819920
\(596\) −44.8328 −1.83642
\(597\) −27.4164 −1.12208
\(598\) −40.0000 −1.63572
\(599\) 12.3607 0.505044 0.252522 0.967591i \(-0.418740\pi\)
0.252522 + 0.967591i \(0.418740\pi\)
\(600\) 2.23607 0.0912871
\(601\) 38.9443 1.58857 0.794285 0.607545i \(-0.207846\pi\)
0.794285 + 0.607545i \(0.207846\pi\)
\(602\) −20.0000 −0.815139
\(603\) −4.00000 −0.162893
\(604\) −48.0000 −1.95309
\(605\) −30.8885 −1.25580
\(606\) −31.3050 −1.27168
\(607\) 38.8328 1.57618 0.788088 0.615563i \(-0.211071\pi\)
0.788088 + 0.615563i \(0.211071\pi\)
\(608\) −16.5836 −0.672553
\(609\) 2.00000 0.0810441
\(610\) −4.47214 −0.181071
\(611\) 57.8885 2.34192
\(612\) −6.00000 −0.242536
\(613\) −6.94427 −0.280477 −0.140238 0.990118i \(-0.544787\pi\)
−0.140238 + 0.990118i \(0.544787\pi\)
\(614\) 33.6656 1.35863
\(615\) −2.00000 −0.0806478
\(616\) −14.4721 −0.583099
\(617\) 16.4721 0.663143 0.331572 0.943430i \(-0.392421\pi\)
0.331572 + 0.943430i \(0.392421\pi\)
\(618\) 0 0
\(619\) 39.4164 1.58428 0.792140 0.610340i \(-0.208967\pi\)
0.792140 + 0.610340i \(0.208967\pi\)
\(620\) −4.58359 −0.184081
\(621\) −4.00000 −0.160514
\(622\) −57.8885 −2.32112
\(623\) −2.00000 −0.0801283
\(624\) 4.47214 0.179029
\(625\) 1.00000 0.0400000
\(626\) 38.9443 1.55653
\(627\) 16.0000 0.638978
\(628\) −1.41641 −0.0565208
\(629\) 13.8885 0.553773
\(630\) 2.23607 0.0890871
\(631\) 30.8328 1.22744 0.613718 0.789526i \(-0.289673\pi\)
0.613718 + 0.789526i \(0.289673\pi\)
\(632\) −28.9443 −1.15134
\(633\) 16.9443 0.673474
\(634\) −32.1115 −1.27531
\(635\) 4.94427 0.196207
\(636\) 10.5836 0.419667
\(637\) 4.47214 0.177192
\(638\) 28.9443 1.14591
\(639\) 5.52786 0.218679
\(640\) −15.6525 −0.618718
\(641\) 16.8328 0.664856 0.332428 0.943129i \(-0.392132\pi\)
0.332428 + 0.943129i \(0.392132\pi\)
\(642\) −28.9443 −1.14234
\(643\) −15.0557 −0.593740 −0.296870 0.954918i \(-0.595943\pi\)
−0.296870 + 0.954918i \(0.595943\pi\)
\(644\) 12.0000 0.472866
\(645\) 8.94427 0.352180
\(646\) −11.0557 −0.434982
\(647\) −1.88854 −0.0742463 −0.0371232 0.999311i \(-0.511819\pi\)
−0.0371232 + 0.999311i \(0.511819\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −57.8885 −2.27232
\(650\) −10.0000 −0.392232
\(651\) −1.52786 −0.0598817
\(652\) −50.8328 −1.99077
\(653\) −22.5836 −0.883764 −0.441882 0.897073i \(-0.645689\pi\)
−0.441882 + 0.897073i \(0.645689\pi\)
\(654\) −4.47214 −0.174874
\(655\) −4.00000 −0.156293
\(656\) 2.00000 0.0780869
\(657\) −12.4721 −0.486584
\(658\) −28.9443 −1.12837
\(659\) 21.3050 0.829923 0.414962 0.909839i \(-0.363795\pi\)
0.414962 + 0.909839i \(0.363795\pi\)
\(660\) 19.4164 0.755783
\(661\) −35.8885 −1.39590 −0.697951 0.716145i \(-0.745905\pi\)
−0.697951 + 0.716145i \(0.745905\pi\)
\(662\) −2.11146 −0.0820641
\(663\) 8.94427 0.347367
\(664\) 37.8885 1.47036
\(665\) 2.47214 0.0958653
\(666\) 15.5279 0.601693
\(667\) −8.00000 −0.309761
\(668\) −24.0000 −0.928588
\(669\) 12.9443 0.500454
\(670\) −8.94427 −0.345547
\(671\) −12.9443 −0.499708
\(672\) −6.70820 −0.258775
\(673\) 8.83282 0.340480 0.170240 0.985403i \(-0.445546\pi\)
0.170240 + 0.985403i \(0.445546\pi\)
\(674\) 53.4164 2.05752
\(675\) −1.00000 −0.0384900
\(676\) 21.0000 0.807692
\(677\) 21.0557 0.809237 0.404619 0.914485i \(-0.367404\pi\)
0.404619 + 0.914485i \(0.367404\pi\)
\(678\) 1.05573 0.0405450
\(679\) 8.47214 0.325131
\(680\) −4.47214 −0.171499
\(681\) −0.944272 −0.0361846
\(682\) −22.1115 −0.846691
\(683\) −1.88854 −0.0722631 −0.0361316 0.999347i \(-0.511504\pi\)
−0.0361316 + 0.999347i \(0.511504\pi\)
\(684\) −7.41641 −0.283573
\(685\) −3.52786 −0.134793
\(686\) −2.23607 −0.0853735
\(687\) −23.8885 −0.911405
\(688\) −8.94427 −0.340997
\(689\) −15.7771 −0.601059
\(690\) −8.94427 −0.340503
\(691\) 44.3607 1.68756 0.843780 0.536689i \(-0.180325\pi\)
0.843780 + 0.536689i \(0.180325\pi\)
\(692\) −8.83282 −0.335773
\(693\) 6.47214 0.245856
\(694\) 17.8885 0.679040
\(695\) −7.41641 −0.281320
\(696\) −4.47214 −0.169516
\(697\) 4.00000 0.151511
\(698\) 26.5836 1.00620
\(699\) 9.41641 0.356161
\(700\) 3.00000 0.113389
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 10.0000 0.377426
\(703\) 17.1672 0.647473
\(704\) −84.1378 −3.17106
\(705\) 12.9443 0.487509
\(706\) −17.6393 −0.663865
\(707\) −14.0000 −0.526524
\(708\) 26.8328 1.00844
\(709\) 25.7771 0.968079 0.484039 0.875046i \(-0.339169\pi\)
0.484039 + 0.875046i \(0.339169\pi\)
\(710\) 12.3607 0.463888
\(711\) 12.9443 0.485448
\(712\) 4.47214 0.167600
\(713\) 6.11146 0.228876
\(714\) −4.47214 −0.167365
\(715\) −28.9443 −1.08245
\(716\) 19.4164 0.725625
\(717\) 10.4721 0.391089
\(718\) −41.3050 −1.54149
\(719\) −6.83282 −0.254821 −0.127411 0.991850i \(-0.540667\pi\)
−0.127411 + 0.991850i \(0.540667\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 28.8197 1.07256
\(723\) 18.9443 0.704545
\(724\) 3.16718 0.117707
\(725\) −2.00000 −0.0742781
\(726\) 69.0689 2.56339
\(727\) −38.8328 −1.44023 −0.720115 0.693855i \(-0.755911\pi\)
−0.720115 + 0.693855i \(0.755911\pi\)
\(728\) −10.0000 −0.370625
\(729\) 1.00000 0.0370370
\(730\) −27.8885 −1.03220
\(731\) −17.8885 −0.661632
\(732\) 6.00000 0.221766
\(733\) 10.5836 0.390914 0.195457 0.980712i \(-0.437381\pi\)
0.195457 + 0.980712i \(0.437381\pi\)
\(734\) −6.83282 −0.252204
\(735\) 1.00000 0.0368856
\(736\) 26.8328 0.989071
\(737\) −25.8885 −0.953617
\(738\) 4.47214 0.164622
\(739\) −5.88854 −0.216614 −0.108307 0.994118i \(-0.534543\pi\)
−0.108307 + 0.994118i \(0.534543\pi\)
\(740\) 20.8328 0.765830
\(741\) 11.0557 0.406142
\(742\) 7.88854 0.289598
\(743\) 34.8328 1.27789 0.638946 0.769252i \(-0.279371\pi\)
0.638946 + 0.769252i \(0.279371\pi\)
\(744\) 3.41641 0.125252
\(745\) 14.9443 0.547516
\(746\) −13.4164 −0.491210
\(747\) −16.9443 −0.619958
\(748\) −38.8328 −1.41987
\(749\) −12.9443 −0.472973
\(750\) −2.23607 −0.0816497
\(751\) −20.9443 −0.764267 −0.382134 0.924107i \(-0.624811\pi\)
−0.382134 + 0.924107i \(0.624811\pi\)
\(752\) −12.9443 −0.472029
\(753\) −16.9443 −0.617484
\(754\) 20.0000 0.728357
\(755\) 16.0000 0.582300
\(756\) −3.00000 −0.109109
\(757\) 31.8885 1.15901 0.579504 0.814969i \(-0.303246\pi\)
0.579504 + 0.814969i \(0.303246\pi\)
\(758\) 84.7214 3.07722
\(759\) −25.8885 −0.939695
\(760\) −5.52786 −0.200517
\(761\) −27.8885 −1.01096 −0.505479 0.862839i \(-0.668684\pi\)
−0.505479 + 0.862839i \(0.668684\pi\)
\(762\) −11.0557 −0.400507
\(763\) −2.00000 −0.0724049
\(764\) 1.75078 0.0633409
\(765\) 2.00000 0.0723102
\(766\) 17.8885 0.646339
\(767\) −40.0000 −1.44432
\(768\) 9.00000 0.324760
\(769\) −52.8328 −1.90520 −0.952600 0.304226i \(-0.901602\pi\)
−0.952600 + 0.304226i \(0.901602\pi\)
\(770\) 14.4721 0.521540
\(771\) −18.9443 −0.682261
\(772\) −42.0000 −1.51161
\(773\) −42.9443 −1.54460 −0.772299 0.635259i \(-0.780893\pi\)
−0.772299 + 0.635259i \(0.780893\pi\)
\(774\) −20.0000 −0.718885
\(775\) 1.52786 0.0548825
\(776\) −18.9443 −0.680060
\(777\) 6.94427 0.249124
\(778\) 15.5279 0.556701
\(779\) 4.94427 0.177147
\(780\) 13.4164 0.480384
\(781\) 35.7771 1.28020
\(782\) 17.8885 0.639693
\(783\) 2.00000 0.0714742
\(784\) −1.00000 −0.0357143
\(785\) 0.472136 0.0168513
\(786\) 8.94427 0.319032
\(787\) 31.0557 1.10702 0.553509 0.832843i \(-0.313289\pi\)
0.553509 + 0.832843i \(0.313289\pi\)
\(788\) 46.5836 1.65947
\(789\) −7.05573 −0.251191
\(790\) 28.9443 1.02979
\(791\) 0.472136 0.0167872
\(792\) −14.4721 −0.514245
\(793\) −8.94427 −0.317620
\(794\) 30.0000 1.06466
\(795\) −3.52786 −0.125120
\(796\) 82.2492 2.91525
\(797\) −18.9443 −0.671041 −0.335520 0.942033i \(-0.608912\pi\)
−0.335520 + 0.942033i \(0.608912\pi\)
\(798\) −5.52786 −0.195684
\(799\) −25.8885 −0.915871
\(800\) 6.70820 0.237171
\(801\) −2.00000 −0.0706665
\(802\) −22.3607 −0.789583
\(803\) −80.7214 −2.84859
\(804\) 12.0000 0.423207
\(805\) −4.00000 −0.140981
\(806\) −15.2786 −0.538167
\(807\) −11.8885 −0.418497
\(808\) 31.3050 1.10130
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 2.23607 0.0785674
\(811\) 55.4164 1.94593 0.972967 0.230946i \(-0.0741820\pi\)
0.972967 + 0.230946i \(0.0741820\pi\)
\(812\) −6.00000 −0.210559
\(813\) 1.52786 0.0535845
\(814\) 100.498 3.52247
\(815\) 16.9443 0.593532
\(816\) −2.00000 −0.0700140
\(817\) −22.1115 −0.773582
\(818\) −26.5836 −0.929474
\(819\) 4.47214 0.156269
\(820\) 6.00000 0.209529
\(821\) 33.7771 1.17883 0.589414 0.807831i \(-0.299359\pi\)
0.589414 + 0.807831i \(0.299359\pi\)
\(822\) 7.88854 0.275145
\(823\) 44.9443 1.56666 0.783329 0.621607i \(-0.213520\pi\)
0.783329 + 0.621607i \(0.213520\pi\)
\(824\) 0 0
\(825\) −6.47214 −0.225331
\(826\) 20.0000 0.695889
\(827\) 12.9443 0.450116 0.225058 0.974345i \(-0.427743\pi\)
0.225058 + 0.974345i \(0.427743\pi\)
\(828\) 12.0000 0.417029
\(829\) −13.0557 −0.453444 −0.226722 0.973959i \(-0.572801\pi\)
−0.226722 + 0.973959i \(0.572801\pi\)
\(830\) −37.8885 −1.31513
\(831\) −18.9443 −0.657170
\(832\) −58.1378 −2.01556
\(833\) −2.00000 −0.0692959
\(834\) 16.5836 0.574243
\(835\) 8.00000 0.276851
\(836\) −48.0000 −1.66011
\(837\) −1.52786 −0.0528107
\(838\) 66.8328 2.30870
\(839\) 54.8328 1.89304 0.946520 0.322647i \(-0.104573\pi\)
0.946520 + 0.322647i \(0.104573\pi\)
\(840\) −2.23607 −0.0771517
\(841\) −25.0000 −0.862069
\(842\) −49.1935 −1.69532
\(843\) 10.9443 0.376941
\(844\) −50.8328 −1.74974
\(845\) −7.00000 −0.240807
\(846\) −28.9443 −0.995125
\(847\) 30.8885 1.06134
\(848\) 3.52786 0.121147
\(849\) −12.0000 −0.411839
\(850\) 4.47214 0.153393
\(851\) −27.7771 −0.952186
\(852\) −16.5836 −0.568145
\(853\) −31.3050 −1.07186 −0.535931 0.844262i \(-0.680039\pi\)
−0.535931 + 0.844262i \(0.680039\pi\)
\(854\) 4.47214 0.153033
\(855\) 2.47214 0.0845453
\(856\) 28.9443 0.989295
\(857\) 36.8328 1.25819 0.629093 0.777330i \(-0.283427\pi\)
0.629093 + 0.777330i \(0.283427\pi\)
\(858\) 64.7214 2.20955
\(859\) 50.4721 1.72209 0.861044 0.508531i \(-0.169811\pi\)
0.861044 + 0.508531i \(0.169811\pi\)
\(860\) −26.8328 −0.914991
\(861\) 2.00000 0.0681598
\(862\) 41.3050 1.40685
\(863\) 21.8885 0.745095 0.372547 0.928013i \(-0.378484\pi\)
0.372547 + 0.928013i \(0.378484\pi\)
\(864\) −6.70820 −0.228218
\(865\) 2.94427 0.100108
\(866\) −36.8328 −1.25163
\(867\) 13.0000 0.441503
\(868\) 4.58359 0.155577
\(869\) 83.7771 2.84194
\(870\) 4.47214 0.151620
\(871\) −17.8885 −0.606130
\(872\) 4.47214 0.151446
\(873\) 8.47214 0.286738
\(874\) 22.1115 0.747931
\(875\) −1.00000 −0.0338062
\(876\) 37.4164 1.26418
\(877\) −56.8328 −1.91911 −0.959554 0.281525i \(-0.909160\pi\)
−0.959554 + 0.281525i \(0.909160\pi\)
\(878\) −3.41641 −0.115298
\(879\) −5.05573 −0.170525
\(880\) 6.47214 0.218176
\(881\) −27.8885 −0.939589 −0.469794 0.882776i \(-0.655672\pi\)
−0.469794 + 0.882776i \(0.655672\pi\)
\(882\) −2.23607 −0.0752923
\(883\) −37.8885 −1.27505 −0.637526 0.770429i \(-0.720042\pi\)
−0.637526 + 0.770429i \(0.720042\pi\)
\(884\) −26.8328 −0.902485
\(885\) −8.94427 −0.300658
\(886\) 17.8885 0.600977
\(887\) −30.8328 −1.03526 −0.517632 0.855603i \(-0.673186\pi\)
−0.517632 + 0.855603i \(0.673186\pi\)
\(888\) −15.5279 −0.521081
\(889\) −4.94427 −0.165826
\(890\) −4.47214 −0.149906
\(891\) 6.47214 0.216825
\(892\) −38.8328 −1.30022
\(893\) −32.0000 −1.07084
\(894\) −33.4164 −1.11761
\(895\) −6.47214 −0.216340
\(896\) 15.6525 0.522913
\(897\) −17.8885 −0.597281
\(898\) 31.3050 1.04466
\(899\) −3.05573 −0.101914
\(900\) 3.00000 0.100000
\(901\) 7.05573 0.235060
\(902\) 28.9443 0.963739
\(903\) −8.94427 −0.297647
\(904\) −1.05573 −0.0351130
\(905\) −1.05573 −0.0350936
\(906\) −35.7771 −1.18861
\(907\) 53.8885 1.78934 0.894670 0.446728i \(-0.147411\pi\)
0.894670 + 0.446728i \(0.147411\pi\)
\(908\) 2.83282 0.0940103
\(909\) −14.0000 −0.464351
\(910\) 10.0000 0.331497
\(911\) 46.2492 1.53231 0.766153 0.642659i \(-0.222169\pi\)
0.766153 + 0.642659i \(0.222169\pi\)
\(912\) −2.47214 −0.0818606
\(913\) −109.666 −3.62940
\(914\) −15.5279 −0.513616
\(915\) −2.00000 −0.0661180
\(916\) 71.6656 2.36790
\(917\) 4.00000 0.132092
\(918\) −4.47214 −0.147602
\(919\) −35.0557 −1.15638 −0.578191 0.815902i \(-0.696241\pi\)
−0.578191 + 0.815902i \(0.696241\pi\)
\(920\) 8.94427 0.294884
\(921\) 15.0557 0.496103
\(922\) −8.69505 −0.286356
\(923\) 24.7214 0.813713
\(924\) −19.4164 −0.638753
\(925\) −6.94427 −0.228326
\(926\) −46.8328 −1.53902
\(927\) 0 0
\(928\) −13.4164 −0.440415
\(929\) −16.1115 −0.528600 −0.264300 0.964441i \(-0.585141\pi\)
−0.264300 + 0.964441i \(0.585141\pi\)
\(930\) −3.41641 −0.112028
\(931\) −2.47214 −0.0810210
\(932\) −28.2492 −0.925334
\(933\) −25.8885 −0.847553
\(934\) 20.0000 0.654420
\(935\) 12.9443 0.423323
\(936\) −10.0000 −0.326860
\(937\) −52.4721 −1.71419 −0.857095 0.515158i \(-0.827733\pi\)
−0.857095 + 0.515158i \(0.827733\pi\)
\(938\) 8.94427 0.292041
\(939\) 17.4164 0.568363
\(940\) −38.8328 −1.26659
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −1.05573 −0.0343975
\(943\) −8.00000 −0.260516
\(944\) 8.94427 0.291111
\(945\) 1.00000 0.0325300
\(946\) −129.443 −4.20855
\(947\) −17.8885 −0.581300 −0.290650 0.956830i \(-0.593871\pi\)
−0.290650 + 0.956830i \(0.593871\pi\)
\(948\) −38.8328 −1.26123
\(949\) −55.7771 −1.81060
\(950\) 5.52786 0.179348
\(951\) −14.3607 −0.465677
\(952\) 4.47214 0.144943
\(953\) −33.4164 −1.08246 −0.541232 0.840873i \(-0.682042\pi\)
−0.541232 + 0.840873i \(0.682042\pi\)
\(954\) 7.88854 0.255401
\(955\) −0.583592 −0.0188846
\(956\) −31.4164 −1.01608
\(957\) 12.9443 0.418429
\(958\) 40.0000 1.29234
\(959\) 3.52786 0.113921
\(960\) −13.0000 −0.419573
\(961\) −28.6656 −0.924698
\(962\) 69.4427 2.23892
\(963\) −12.9443 −0.417123
\(964\) −56.8328 −1.83046
\(965\) 14.0000 0.450676
\(966\) 8.94427 0.287777
\(967\) 25.8885 0.832519 0.416260 0.909246i \(-0.363341\pi\)
0.416260 + 0.909246i \(0.363341\pi\)
\(968\) −69.0689 −2.21996
\(969\) −4.94427 −0.158833
\(970\) 18.9443 0.608264
\(971\) 40.9443 1.31396 0.656982 0.753906i \(-0.271833\pi\)
0.656982 + 0.753906i \(0.271833\pi\)
\(972\) −3.00000 −0.0962250
\(973\) 7.41641 0.237759
\(974\) 46.8328 1.50062
\(975\) −4.47214 −0.143223
\(976\) 2.00000 0.0640184
\(977\) 30.5836 0.978456 0.489228 0.872156i \(-0.337279\pi\)
0.489228 + 0.872156i \(0.337279\pi\)
\(978\) −37.8885 −1.21154
\(979\) −12.9443 −0.413701
\(980\) −3.00000 −0.0958315
\(981\) −2.00000 −0.0638551
\(982\) 47.6393 1.52023
\(983\) −22.8328 −0.728254 −0.364127 0.931349i \(-0.618633\pi\)
−0.364127 + 0.931349i \(0.618633\pi\)
\(984\) −4.47214 −0.142566
\(985\) −15.5279 −0.494759
\(986\) −8.94427 −0.284844
\(987\) −12.9443 −0.412021
\(988\) −33.1672 −1.05519
\(989\) 35.7771 1.13765
\(990\) 14.4721 0.459955
\(991\) 4.94427 0.157060 0.0785300 0.996912i \(-0.474977\pi\)
0.0785300 + 0.996912i \(0.474977\pi\)
\(992\) 10.2492 0.325413
\(993\) −0.944272 −0.0299656
\(994\) −12.3607 −0.392057
\(995\) −27.4164 −0.869159
\(996\) 50.8328 1.61070
\(997\) −5.41641 −0.171539 −0.0857697 0.996315i \(-0.527335\pi\)
−0.0857697 + 0.996315i \(0.527335\pi\)
\(998\) 31.0557 0.983052
\(999\) 6.94427 0.219707
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 105.2.a.b.1.1 2
3.2 odd 2 315.2.a.d.1.2 2
4.3 odd 2 1680.2.a.v.1.1 2
5.2 odd 4 525.2.d.c.274.2 4
5.3 odd 4 525.2.d.c.274.3 4
5.4 even 2 525.2.a.g.1.2 2
7.2 even 3 735.2.i.k.361.2 4
7.3 odd 6 735.2.i.i.226.2 4
7.4 even 3 735.2.i.k.226.2 4
7.5 odd 6 735.2.i.i.361.2 4
7.6 odd 2 735.2.a.k.1.1 2
8.3 odd 2 6720.2.a.cs.1.2 2
8.5 even 2 6720.2.a.cx.1.1 2
12.11 even 2 5040.2.a.bw.1.2 2
15.2 even 4 1575.2.d.d.1324.4 4
15.8 even 4 1575.2.d.d.1324.1 4
15.14 odd 2 1575.2.a.r.1.1 2
20.19 odd 2 8400.2.a.cx.1.1 2
21.20 even 2 2205.2.a.w.1.2 2
35.34 odd 2 3675.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.b.1.1 2 1.1 even 1 trivial
315.2.a.d.1.2 2 3.2 odd 2
525.2.a.g.1.2 2 5.4 even 2
525.2.d.c.274.2 4 5.2 odd 4
525.2.d.c.274.3 4 5.3 odd 4
735.2.a.k.1.1 2 7.6 odd 2
735.2.i.i.226.2 4 7.3 odd 6
735.2.i.i.361.2 4 7.5 odd 6
735.2.i.k.226.2 4 7.4 even 3
735.2.i.k.361.2 4 7.2 even 3
1575.2.a.r.1.1 2 15.14 odd 2
1575.2.d.d.1324.1 4 15.8 even 4
1575.2.d.d.1324.4 4 15.2 even 4
1680.2.a.v.1.1 2 4.3 odd 2
2205.2.a.w.1.2 2 21.20 even 2
3675.2.a.y.1.2 2 35.34 odd 2
5040.2.a.bw.1.2 2 12.11 even 2
6720.2.a.cs.1.2 2 8.3 odd 2
6720.2.a.cx.1.1 2 8.5 even 2
8400.2.a.cx.1.1 2 20.19 odd 2