Properties

Label 105.2.a.b.1.2
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 1.2
Root \(-0.618034\)
Character \(\chi\) = 105.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -1.00000 q^{5} -2.23607 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{10} -2.47214 q^{11} -3.00000 q^{12} -4.47214 q^{13} +2.23607 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +2.23607 q^{18} +6.47214 q^{19} -3.00000 q^{20} -1.00000 q^{21} -5.52786 q^{22} +4.00000 q^{23} -2.23607 q^{24} +1.00000 q^{25} -10.0000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -2.00000 q^{29} +2.23607 q^{30} +10.4721 q^{31} -6.70820 q^{32} +2.47214 q^{33} -4.47214 q^{34} -1.00000 q^{35} +3.00000 q^{36} +10.9443 q^{37} +14.4721 q^{38} +4.47214 q^{39} -2.23607 q^{40} -2.00000 q^{41} -2.23607 q^{42} -8.94427 q^{43} -7.41641 q^{44} -1.00000 q^{45} +8.94427 q^{46} -4.94427 q^{47} +1.00000 q^{48} +1.00000 q^{49} +2.23607 q^{50} +2.00000 q^{51} -13.4164 q^{52} -12.4721 q^{53} -2.23607 q^{54} +2.47214 q^{55} +2.23607 q^{56} -6.47214 q^{57} -4.47214 q^{58} +8.94427 q^{59} +3.00000 q^{60} -2.00000 q^{61} +23.4164 q^{62} +1.00000 q^{63} -13.0000 q^{64} +4.47214 q^{65} +5.52786 q^{66} -4.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} -2.23607 q^{70} +14.4721 q^{71} +2.23607 q^{72} -3.52786 q^{73} +24.4721 q^{74} -1.00000 q^{75} +19.4164 q^{76} -2.47214 q^{77} +10.0000 q^{78} -4.94427 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.47214 q^{82} +0.944272 q^{83} -3.00000 q^{84} +2.00000 q^{85} -20.0000 q^{86} +2.00000 q^{87} -5.52786 q^{88} -2.00000 q^{89} -2.23607 q^{90} -4.47214 q^{91} +12.0000 q^{92} -10.4721 q^{93} -11.0557 q^{94} -6.47214 q^{95} +6.70820 q^{96} -0.472136 q^{97} +2.23607 q^{98} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 6q^{4} - 2q^{5} + 2q^{7} + 2q^{9} + 4q^{11} - 6q^{12} + 2q^{15} - 2q^{16} - 4q^{17} + 4q^{19} - 6q^{20} - 2q^{21} - 20q^{22} + 8q^{23} + 2q^{25} - 20q^{26} - 2q^{27} + 6q^{28} - 4q^{29} + 12q^{31} - 4q^{33} - 2q^{35} + 6q^{36} + 4q^{37} + 20q^{38} - 4q^{41} + 12q^{44} - 2q^{45} + 8q^{47} + 2q^{48} + 2q^{49} + 4q^{51} - 16q^{53} - 4q^{55} - 4q^{57} + 6q^{60} - 4q^{61} + 20q^{62} + 2q^{63} - 26q^{64} + 20q^{66} - 8q^{67} - 12q^{68} - 8q^{69} + 20q^{71} - 16q^{73} + 40q^{74} - 2q^{75} + 12q^{76} + 4q^{77} + 20q^{78} + 8q^{79} + 2q^{80} + 2q^{81} - 16q^{83} - 6q^{84} + 4q^{85} - 40q^{86} + 4q^{87} - 20q^{88} - 4q^{89} + 24q^{92} - 12q^{93} - 40q^{94} - 4q^{95} + 8q^{97} + 4q^{99} + O(q^{100}) \)

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 \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −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
\(11\) −2.47214 −0.745377 −0.372689 0.927957i \(-0.621564\pi\)
−0.372689 + 0.927957i \(0.621564\pi\)
\(12\) −3.00000 −0.866025
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 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
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.00000 −0.218218
\(22\) −5.52786 −1.17854
\(23\) 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
\(31\) 10.4721 1.88085 0.940426 0.340000i \(-0.110427\pi\)
0.940426 + 0.340000i \(0.110427\pi\)
\(32\) −6.70820 −1.18585
\(33\) 2.47214 0.430344
\(34\) −4.47214 −0.766965
\(35\) −1.00000 −0.169031
\(36\) 3.00000 0.500000
\(37\) 10.9443 1.79923 0.899614 0.436687i \(-0.143848\pi\)
0.899614 + 0.436687i \(0.143848\pi\)
\(38\) 14.4721 2.34769
\(39\) 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 \(-0.738887\pi\)
−0.681994 + 0.731357i \(0.738887\pi\)
\(44\) −7.41641 −1.11807
\(45\) −1.00000 −0.149071
\(46\) 8.94427 1.31876
\(47\) −4.94427 −0.721196 −0.360598 0.932721i \(-0.617427\pi\)
−0.360598 + 0.932721i \(0.617427\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 2.23607 0.316228
\(51\) 2.00000 0.280056
\(52\) −13.4164 −1.86052
\(53\) −12.4721 −1.71318 −0.856590 0.515998i \(-0.827421\pi\)
−0.856590 + 0.515998i \(0.827421\pi\)
\(54\) −2.23607 −0.304290
\(55\) 2.47214 0.333343
\(56\) 2.23607 0.298807
\(57\) −6.47214 −0.857255
\(58\) −4.47214 −0.587220
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 3.00000 0.387298
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 23.4164 2.97389
\(63\) 1.00000 0.125988
\(64\) −13.0000 −1.62500
\(65\) 4.47214 0.554700
\(66\) 5.52786 0.680433
\(67\) −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
\(71\) 14.4721 1.71753 0.858763 0.512373i \(-0.171233\pi\)
0.858763 + 0.512373i \(0.171233\pi\)
\(72\) 2.23607 0.263523
\(73\) −3.52786 −0.412905 −0.206453 0.978457i \(-0.566192\pi\)
−0.206453 + 0.978457i \(0.566192\pi\)
\(74\) 24.4721 2.84483
\(75\) −1.00000 −0.115470
\(76\) 19.4164 2.22721
\(77\) −2.47214 −0.281726
\(78\) 10.0000 1.13228
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −4.47214 −0.493865
\(83\) 0.944272 0.103647 0.0518237 0.998656i \(-0.483497\pi\)
0.0518237 + 0.998656i \(0.483497\pi\)
\(84\) −3.00000 −0.327327
\(85\) 2.00000 0.216930
\(86\) −20.0000 −2.15666
\(87\) 2.00000 0.214423
\(88\) −5.52786 −0.589272
\(89\) −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
\(93\) −10.4721 −1.08591
\(94\) −11.0557 −1.14031
\(95\) −6.47214 −0.664027
\(96\) 6.70820 0.684653
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 2.23607 0.225877
\(99\) −2.47214 −0.248459
\(100\) 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
\(106\) −27.8885 −2.70877
\(107\) 4.94427 0.477981 0.238990 0.971022i \(-0.423184\pi\)
0.238990 + 0.971022i \(0.423184\pi\)
\(108\) −3.00000 −0.288675
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 5.52786 0.527061
\(111\) −10.9443 −1.03878
\(112\) −1.00000 −0.0944911
\(113\) −8.47214 −0.796992 −0.398496 0.917170i \(-0.630468\pi\)
−0.398496 + 0.917170i \(0.630468\pi\)
\(114\) −14.4721 −1.35544
\(115\) −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
\(121\) −4.88854 −0.444413
\(122\) −4.47214 −0.404888
\(123\) 2.00000 0.180334
\(124\) 31.4164 2.82128
\(125\) −1.00000 −0.0894427
\(126\) 2.23607 0.199205
\(127\) 12.9443 1.14862 0.574309 0.818638i \(-0.305271\pi\)
0.574309 + 0.818638i \(0.305271\pi\)
\(128\) −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 \(0.444091\pi\)
\(132\) 7.41641 0.645515
\(133\) 6.47214 0.561205
\(134\) −8.94427 −0.772667
\(135\) 1.00000 0.0860663
\(136\) −4.47214 −0.383482
\(137\) 12.4721 1.06557 0.532783 0.846252i \(-0.321146\pi\)
0.532783 + 0.846252i \(0.321146\pi\)
\(138\) −8.94427 −0.761387
\(139\) −19.4164 −1.64688 −0.823439 0.567405i \(-0.807948\pi\)
−0.823439 + 0.567405i \(0.807948\pi\)
\(140\) −3.00000 −0.253546
\(141\) 4.94427 0.416383
\(142\) 32.3607 2.71565
\(143\) 11.0557 0.924526
\(144\) −1.00000 −0.0833333
\(145\) 2.00000 0.166091
\(146\) −7.88854 −0.652861
\(147\) −1.00000 −0.0824786
\(148\) 32.8328 2.69884
\(149\) 2.94427 0.241204 0.120602 0.992701i \(-0.461517\pi\)
0.120602 + 0.992701i \(0.461517\pi\)
\(150\) −2.23607 −0.182574
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 14.4721 1.17385
\(153\) −2.00000 −0.161690
\(154\) −5.52786 −0.445448
\(155\) −10.4721 −0.841142
\(156\) 13.4164 1.07417
\(157\) 8.47214 0.676150 0.338075 0.941119i \(-0.390224\pi\)
0.338075 + 0.941119i \(0.390224\pi\)
\(158\) −11.0557 −0.879547
\(159\) 12.4721 0.989105
\(160\) 6.70820 0.530330
\(161\) 4.00000 0.315244
\(162\) 2.23607 0.175682
\(163\) 0.944272 0.0739611 0.0369805 0.999316i \(-0.488226\pi\)
0.0369805 + 0.999316i \(0.488226\pi\)
\(164\) −6.00000 −0.468521
\(165\) −2.47214 −0.192456
\(166\) 2.11146 0.163881
\(167\) −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
\(171\) 6.47214 0.494937
\(172\) −26.8328 −2.04598
\(173\) 14.9443 1.13619 0.568096 0.822962i \(-0.307680\pi\)
0.568096 + 0.822962i \(0.307680\pi\)
\(174\) 4.47214 0.339032
\(175\) 1.00000 0.0755929
\(176\) 2.47214 0.186344
\(177\) −8.94427 −0.672293
\(178\) −4.47214 −0.335201
\(179\) −2.47214 −0.184776 −0.0923881 0.995723i \(-0.529450\pi\)
−0.0923881 + 0.995723i \(0.529450\pi\)
\(180\) −3.00000 −0.223607
\(181\) 18.9443 1.40812 0.704058 0.710142i \(-0.251369\pi\)
0.704058 + 0.710142i \(0.251369\pi\)
\(182\) −10.0000 −0.741249
\(183\) 2.00000 0.147844
\(184\) 8.94427 0.659380
\(185\) −10.9443 −0.804639
\(186\) −23.4164 −1.71697
\(187\) 4.94427 0.361561
\(188\) −14.8328 −1.08179
\(189\) −1.00000 −0.0727393
\(190\) −14.4721 −1.04992
\(191\) 27.4164 1.98378 0.991891 0.127093i \(-0.0405646\pi\)
0.991891 + 0.127093i \(0.0405646\pi\)
\(192\) 13.0000 0.938194
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −1.05573 −0.0757969
\(195\) −4.47214 −0.320256
\(196\) 3.00000 0.214286
\(197\) 24.4721 1.74357 0.871784 0.489891i \(-0.162963\pi\)
0.871784 + 0.489891i \(0.162963\pi\)
\(198\) −5.52786 −0.392848
\(199\) 0.583592 0.0413697 0.0206849 0.999786i \(-0.493415\pi\)
0.0206849 + 0.999786i \(0.493415\pi\)
\(200\) 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
\(211\) 0.944272 0.0650064 0.0325032 0.999472i \(-0.489652\pi\)
0.0325032 + 0.999472i \(0.489652\pi\)
\(212\) −37.4164 −2.56977
\(213\) −14.4721 −0.991614
\(214\) 11.0557 0.755754
\(215\) 8.94427 0.609994
\(216\) −2.23607 −0.152145
\(217\) 10.4721 0.710895
\(218\) −4.47214 −0.302891
\(219\) 3.52786 0.238391
\(220\) 7.41641 0.500014
\(221\) 8.94427 0.601657
\(222\) −24.4721 −1.64246
\(223\) 4.94427 0.331093 0.165546 0.986202i \(-0.447061\pi\)
0.165546 + 0.986202i \(0.447061\pi\)
\(224\) −6.70820 −0.448211
\(225\) 1.00000 0.0666667
\(226\) −18.9443 −1.26015
\(227\) −16.9443 −1.12463 −0.562315 0.826923i \(-0.690089\pi\)
−0.562315 + 0.826923i \(0.690089\pi\)
\(228\) −19.4164 −1.28588
\(229\) −11.8885 −0.785617 −0.392809 0.919620i \(-0.628496\pi\)
−0.392809 + 0.919620i \(0.628496\pi\)
\(230\) −8.94427 −0.589768
\(231\) 2.47214 0.162655
\(232\) −4.47214 −0.293610
\(233\) 17.4164 1.14099 0.570493 0.821302i \(-0.306752\pi\)
0.570493 + 0.821302i \(0.306752\pi\)
\(234\) −10.0000 −0.653720
\(235\) 4.94427 0.322529
\(236\) 26.8328 1.74667
\(237\) 4.94427 0.321165
\(238\) −4.47214 −0.289886
\(239\) −1.52786 −0.0988293 −0.0494147 0.998778i \(-0.515736\pi\)
−0.0494147 + 0.998778i \(0.515736\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −1.05573 −0.0680054 −0.0340027 0.999422i \(-0.510825\pi\)
−0.0340027 + 0.999422i \(0.510825\pi\)
\(242\) −10.9311 −0.702679
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −1.00000 −0.0638877
\(246\) 4.47214 0.285133
\(247\) −28.9443 −1.84168
\(248\) 23.4164 1.48694
\(249\) −0.944272 −0.0598408
\(250\) −2.23607 −0.141421
\(251\) −0.944272 −0.0596019 −0.0298010 0.999556i \(-0.509487\pi\)
−0.0298010 + 0.999556i \(0.509487\pi\)
\(252\) 3.00000 0.188982
\(253\) −9.88854 −0.621687
\(254\) 28.9443 1.81613
\(255\) −2.00000 −0.125245
\(256\) −9.00000 −0.562500
\(257\) 1.05573 0.0658545 0.0329273 0.999458i \(-0.489517\pi\)
0.0329273 + 0.999458i \(0.489517\pi\)
\(258\) 20.0000 1.24515
\(259\) 10.9443 0.680044
\(260\) 13.4164 0.832050
\(261\) −2.00000 −0.123797
\(262\) 8.94427 0.552579
\(263\) 24.9443 1.53813 0.769065 0.639171i \(-0.220722\pi\)
0.769065 + 0.639171i \(0.220722\pi\)
\(264\) 5.52786 0.340217
\(265\) 12.4721 0.766157
\(266\) 14.4721 0.887344
\(267\) 2.00000 0.122398
\(268\) −12.0000 −0.733017
\(269\) −23.8885 −1.45651 −0.728255 0.685306i \(-0.759668\pi\)
−0.728255 + 0.685306i \(0.759668\pi\)
\(270\) 2.23607 0.136083
\(271\) −10.4721 −0.636137 −0.318068 0.948068i \(-0.603034\pi\)
−0.318068 + 0.948068i \(0.603034\pi\)
\(272\) 2.00000 0.121268
\(273\) 4.47214 0.270666
\(274\) 27.8885 1.68481
\(275\) −2.47214 −0.149075
\(276\) −12.0000 −0.722315
\(277\) 1.05573 0.0634326 0.0317163 0.999497i \(-0.489903\pi\)
0.0317163 + 0.999497i \(0.489903\pi\)
\(278\) −43.4164 −2.60394
\(279\) 10.4721 0.626950
\(280\) −2.23607 −0.133631
\(281\) 6.94427 0.414261 0.207130 0.978313i \(-0.433588\pi\)
0.207130 + 0.978313i \(0.433588\pi\)
\(282\) 11.0557 0.658359
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 43.4164 2.57629
\(285\) 6.47214 0.383376
\(286\) 24.7214 1.46180
\(287\) −2.00000 −0.118056
\(288\) −6.70820 −0.395285
\(289\) −13.0000 −0.764706
\(290\) 4.47214 0.262613
\(291\) 0.472136 0.0276771
\(292\) −10.5836 −0.619358
\(293\) 22.9443 1.34042 0.670209 0.742172i \(-0.266204\pi\)
0.670209 + 0.742172i \(0.266204\pi\)
\(294\) −2.23607 −0.130410
\(295\) −8.94427 −0.520756
\(296\) 24.4721 1.42241
\(297\) 2.47214 0.143448
\(298\) 6.58359 0.381377
\(299\) −17.8885 −1.03452
\(300\) −3.00000 −0.173205
\(301\) −8.94427 −0.515539
\(302\) −35.7771 −2.05874
\(303\) 14.0000 0.804279
\(304\) −6.47214 −0.371202
\(305\) 2.00000 0.114520
\(306\) −4.47214 −0.255655
\(307\) −32.9443 −1.88023 −0.940114 0.340859i \(-0.889282\pi\)
−0.940114 + 0.340859i \(0.889282\pi\)
\(308\) −7.41641 −0.422589
\(309\) 0 0
\(310\) −23.4164 −1.32996
\(311\) −9.88854 −0.560728 −0.280364 0.959894i \(-0.590455\pi\)
−0.280364 + 0.959894i \(0.590455\pi\)
\(312\) 10.0000 0.566139
\(313\) 9.41641 0.532247 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(314\) 18.9443 1.06909
\(315\) −1.00000 −0.0563436
\(316\) −14.8328 −0.834411
\(317\) −30.3607 −1.70523 −0.852613 0.522543i \(-0.824983\pi\)
−0.852613 + 0.522543i \(0.824983\pi\)
\(318\) 27.8885 1.56391
\(319\) 4.94427 0.276826
\(320\) 13.0000 0.726722
\(321\) −4.94427 −0.275962
\(322\) 8.94427 0.498445
\(323\) −12.9443 −0.720239
\(324\) 3.00000 0.166667
\(325\) −4.47214 −0.248069
\(326\) 2.11146 0.116943
\(327\) 2.00000 0.110600
\(328\) −4.47214 −0.246932
\(329\) −4.94427 −0.272587
\(330\) −5.52786 −0.304299
\(331\) −16.9443 −0.931341 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(332\) 2.83282 0.155471
\(333\) 10.9443 0.599742
\(334\) −17.8885 −0.978818
\(335\) 4.00000 0.218543
\(336\) 1.00000 0.0545545
\(337\) 11.8885 0.647610 0.323805 0.946124i \(-0.395038\pi\)
0.323805 + 0.946124i \(0.395038\pi\)
\(338\) 15.6525 0.851382
\(339\) 8.47214 0.460143
\(340\) 6.00000 0.325396
\(341\) −25.8885 −1.40194
\(342\) 14.4721 0.782563
\(343\) 1.00000 0.0539949
\(344\) −20.0000 −1.07833
\(345\) 4.00000 0.215353
\(346\) 33.4164 1.79648
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 6.00000 0.321634
\(349\) 23.8885 1.27872 0.639362 0.768906i \(-0.279198\pi\)
0.639362 + 0.768906i \(0.279198\pi\)
\(350\) 2.23607 0.119523
\(351\) 4.47214 0.238705
\(352\) 16.5836 0.883908
\(353\) −27.8885 −1.48436 −0.742179 0.670202i \(-0.766207\pi\)
−0.742179 + 0.670202i \(0.766207\pi\)
\(354\) −20.0000 −1.06299
\(355\) −14.4721 −0.768101
\(356\) −6.00000 −0.317999
\(357\) 2.00000 0.105851
\(358\) −5.52786 −0.292157
\(359\) 9.52786 0.502861 0.251431 0.967875i \(-0.419099\pi\)
0.251431 + 0.967875i \(0.419099\pi\)
\(360\) −2.23607 −0.117851
\(361\) 22.8885 1.20466
\(362\) 42.3607 2.22643
\(363\) 4.88854 0.256582
\(364\) −13.4164 −0.703211
\(365\) 3.52786 0.184657
\(366\) 4.47214 0.233762
\(367\) 20.9443 1.09328 0.546641 0.837367i \(-0.315906\pi\)
0.546641 + 0.837367i \(0.315906\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) −24.4721 −1.27225
\(371\) −12.4721 −0.647521
\(372\) −31.4164 −1.62886
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 11.0557 0.571678
\(375\) 1.00000 0.0516398
\(376\) −11.0557 −0.570156
\(377\) 8.94427 0.460653
\(378\) −2.23607 −0.115011
\(379\) −2.11146 −0.108458 −0.0542291 0.998529i \(-0.517270\pi\)
−0.0542291 + 0.998529i \(0.517270\pi\)
\(380\) −19.4164 −0.996041
\(381\) −12.9443 −0.663155
\(382\) 61.3050 3.13663
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 15.6525 0.798762
\(385\) 2.47214 0.125992
\(386\) −31.3050 −1.59338
\(387\) −8.94427 −0.454663
\(388\) −1.41641 −0.0719072
\(389\) 10.9443 0.554897 0.277448 0.960741i \(-0.410511\pi\)
0.277448 + 0.960741i \(0.410511\pi\)
\(390\) −10.0000 −0.506370
\(391\) −8.00000 −0.404577
\(392\) 2.23607 0.112938
\(393\) −4.00000 −0.201773
\(394\) 54.7214 2.75682
\(395\) 4.94427 0.248773
\(396\) −7.41641 −0.372689
\(397\) 13.4164 0.673350 0.336675 0.941621i \(-0.390698\pi\)
0.336675 + 0.941621i \(0.390698\pi\)
\(398\) 1.30495 0.0654113
\(399\) −6.47214 −0.324012
\(400\) −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
\(403\) −46.8328 −2.33291
\(404\) −42.0000 −2.08958
\(405\) −1.00000 −0.0496904
\(406\) −4.47214 −0.221948
\(407\) −27.0557 −1.34110
\(408\) 4.47214 0.221404
\(409\) −23.8885 −1.18121 −0.590606 0.806960i \(-0.701111\pi\)
−0.590606 + 0.806960i \(0.701111\pi\)
\(410\) 4.47214 0.220863
\(411\) −12.4721 −0.615205
\(412\) 0 0
\(413\) 8.94427 0.440119
\(414\) 8.94427 0.439587
\(415\) −0.944272 −0.0463525
\(416\) 30.0000 1.47087
\(417\) 19.4164 0.950826
\(418\) −35.7771 −1.74991
\(419\) 5.88854 0.287674 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(420\) 3.00000 0.146385
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 2.11146 0.102784
\(423\) −4.94427 −0.240399
\(424\) −27.8885 −1.35439
\(425\) −2.00000 −0.0970143
\(426\) −32.3607 −1.56788
\(427\) −2.00000 −0.0967868
\(428\) 14.8328 0.716971
\(429\) −11.0557 −0.533776
\(430\) 20.0000 0.964486
\(431\) −9.52786 −0.458941 −0.229471 0.973316i \(-0.573699\pi\)
−0.229471 + 0.973316i \(0.573699\pi\)
\(432\) 1.00000 0.0481125
\(433\) 7.52786 0.361766 0.180883 0.983505i \(-0.442104\pi\)
0.180883 + 0.983505i \(0.442104\pi\)
\(434\) 23.4164 1.12402
\(435\) −2.00000 −0.0958927
\(436\) −6.00000 −0.287348
\(437\) 25.8885 1.23842
\(438\) 7.88854 0.376929
\(439\) 10.4721 0.499808 0.249904 0.968271i \(-0.419601\pi\)
0.249904 + 0.968271i \(0.419601\pi\)
\(440\) 5.52786 0.263531
\(441\) 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 \(0.560864\pi\)
\(444\) −32.8328 −1.55818
\(445\) 2.00000 0.0948091
\(446\) 11.0557 0.523504
\(447\) −2.94427 −0.139259
\(448\) −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
\(451\) 4.94427 0.232817
\(452\) −25.4164 −1.19549
\(453\) 16.0000 0.751746
\(454\) −37.8885 −1.77820
\(455\) 4.47214 0.209657
\(456\) −14.4721 −0.677720
\(457\) −10.9443 −0.511951 −0.255976 0.966683i \(-0.582397\pi\)
−0.255976 + 0.966683i \(0.582397\pi\)
\(458\) −26.5836 −1.24217
\(459\) 2.00000 0.0933520
\(460\) −12.0000 −0.559503
\(461\) −31.8885 −1.48520 −0.742599 0.669737i \(-0.766407\pi\)
−0.742599 + 0.669737i \(0.766407\pi\)
\(462\) 5.52786 0.257180
\(463\) 3.05573 0.142012 0.0710059 0.997476i \(-0.477379\pi\)
0.0710059 + 0.997476i \(0.477379\pi\)
\(464\) 2.00000 0.0928477
\(465\) 10.4721 0.485634
\(466\) 38.9443 1.80406
\(467\) 8.94427 0.413892 0.206946 0.978352i \(-0.433648\pi\)
0.206946 + 0.978352i \(0.433648\pi\)
\(468\) −13.4164 −0.620174
\(469\) −4.00000 −0.184703
\(470\) 11.0557 0.509963
\(471\) −8.47214 −0.390375
\(472\) 20.0000 0.920575
\(473\) 22.1115 1.01669
\(474\) 11.0557 0.507806
\(475\) 6.47214 0.296962
\(476\) −6.00000 −0.275010
\(477\) −12.4721 −0.571060
\(478\) −3.41641 −0.156263
\(479\) 17.8885 0.817348 0.408674 0.912680i \(-0.365991\pi\)
0.408674 + 0.912680i \(0.365991\pi\)
\(480\) −6.70820 −0.306186
\(481\) −48.9443 −2.23167
\(482\) −2.36068 −0.107526
\(483\) −4.00000 −0.182006
\(484\) −14.6656 −0.666620
\(485\) 0.472136 0.0214386
\(486\) −2.23607 −0.101430
\(487\) −3.05573 −0.138468 −0.0692341 0.997600i \(-0.522056\pi\)
−0.0692341 + 0.997600i \(0.522056\pi\)
\(488\) −4.47214 −0.202444
\(489\) −0.944272 −0.0427015
\(490\) −2.23607 −0.101015
\(491\) 41.3050 1.86407 0.932033 0.362373i \(-0.118033\pi\)
0.932033 + 0.362373i \(0.118033\pi\)
\(492\) 6.00000 0.270501
\(493\) 4.00000 0.180151
\(494\) −64.7214 −2.91195
\(495\) 2.47214 0.111114
\(496\) −10.4721 −0.470213
\(497\) 14.4721 0.649164
\(498\) −2.11146 −0.0946166
\(499\) 21.8885 0.979866 0.489933 0.871760i \(-0.337021\pi\)
0.489933 + 0.871760i \(0.337021\pi\)
\(500\) −3.00000 −0.134164
\(501\) 8.00000 0.357414
\(502\) −2.11146 −0.0942389
\(503\) 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
\(506\) −22.1115 −0.982974
\(507\) −7.00000 −0.310881
\(508\) 38.8328 1.72293
\(509\) 11.8885 0.526950 0.263475 0.964666i \(-0.415131\pi\)
0.263475 + 0.964666i \(0.415131\pi\)
\(510\) −4.47214 −0.198030
\(511\) −3.52786 −0.156064
\(512\) 11.1803 0.494106
\(513\) −6.47214 −0.285752
\(514\) 2.36068 0.104125
\(515\) 0 0
\(516\) 26.8328 1.18125
\(517\) 12.2229 0.537563
\(518\) 24.4721 1.07524
\(519\) −14.9443 −0.655981
\(520\) 10.0000 0.438529
\(521\) 15.8885 0.696090 0.348045 0.937478i \(-0.386846\pi\)
0.348045 + 0.937478i \(0.386846\pi\)
\(522\) −4.47214 −0.195740
\(523\) 8.94427 0.391106 0.195553 0.980693i \(-0.437350\pi\)
0.195553 + 0.980693i \(0.437350\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) 55.7771 2.43200
\(527\) −20.9443 −0.912347
\(528\) −2.47214 −0.107586
\(529\) −7.00000 −0.304348
\(530\) 27.8885 1.21140
\(531\) 8.94427 0.388148
\(532\) 19.4164 0.841808
\(533\) 8.94427 0.387419
\(534\) 4.47214 0.193528
\(535\) −4.94427 −0.213760
\(536\) −8.94427 −0.386334
\(537\) 2.47214 0.106681
\(538\) −53.4164 −2.30294
\(539\) −2.47214 −0.106482
\(540\) 3.00000 0.129099
\(541\) 23.8885 1.02705 0.513524 0.858075i \(-0.328340\pi\)
0.513524 + 0.858075i \(0.328340\pi\)
\(542\) −23.4164 −1.00582
\(543\) −18.9443 −0.812977
\(544\) 13.4164 0.575224
\(545\) 2.00000 0.0856706
\(546\) 10.0000 0.427960
\(547\) −29.8885 −1.27794 −0.638971 0.769231i \(-0.720640\pi\)
−0.638971 + 0.769231i \(0.720640\pi\)
\(548\) 37.4164 1.59835
\(549\) −2.00000 −0.0853579
\(550\) −5.52786 −0.235709
\(551\) −12.9443 −0.551445
\(552\) −8.94427 −0.380693
\(553\) −4.94427 −0.210252
\(554\) 2.36068 0.100296
\(555\) 10.9443 0.464558
\(556\) −58.2492 −2.47032
\(557\) 11.5279 0.488451 0.244226 0.969718i \(-0.421466\pi\)
0.244226 + 0.969718i \(0.421466\pi\)
\(558\) 23.4164 0.991296
\(559\) 40.0000 1.69182
\(560\) 1.00000 0.0422577
\(561\) −4.94427 −0.208747
\(562\) 15.5279 0.655003
\(563\) −21.8885 −0.922492 −0.461246 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(564\) 14.8328 0.624574
\(565\) 8.47214 0.356425
\(566\) 26.8328 1.12787
\(567\) 1.00000 0.0419961
\(568\) 32.3607 1.35782
\(569\) −4.11146 −0.172361 −0.0861806 0.996280i \(-0.527466\pi\)
−0.0861806 + 0.996280i \(0.527466\pi\)
\(570\) 14.4721 0.606171
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 33.1672 1.38679
\(573\) −27.4164 −1.14534
\(574\) −4.47214 −0.186663
\(575\) 4.00000 0.166812
\(576\) −13.0000 −0.541667
\(577\) −34.3607 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(578\) −29.0689 −1.20911
\(579\) 14.0000 0.581820
\(580\) 6.00000 0.249136
\(581\) 0.944272 0.0391750
\(582\) 1.05573 0.0437613
\(583\) 30.8328 1.27696
\(584\) −7.88854 −0.326430
\(585\) 4.47214 0.184900
\(586\) 51.3050 2.11939
\(587\) −4.00000 −0.165098 −0.0825488 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) −3.00000 −0.123718
\(589\) 67.7771 2.79271
\(590\) −20.0000 −0.823387
\(591\) −24.4721 −1.00665
\(592\) −10.9443 −0.449807
\(593\) −11.8885 −0.488204 −0.244102 0.969750i \(-0.578493\pi\)
−0.244102 + 0.969750i \(0.578493\pi\)
\(594\) 5.52786 0.226811
\(595\) 2.00000 0.0819920
\(596\) 8.83282 0.361806
\(597\) −0.583592 −0.0238848
\(598\) −40.0000 −1.63572
\(599\) −32.3607 −1.32222 −0.661111 0.750288i \(-0.729915\pi\)
−0.661111 + 0.750288i \(0.729915\pi\)
\(600\) −2.23607 −0.0912871
\(601\) 21.0557 0.858881 0.429441 0.903095i \(-0.358711\pi\)
0.429441 + 0.903095i \(0.358711\pi\)
\(602\) −20.0000 −0.815139
\(603\) −4.00000 −0.162893
\(604\) −48.0000 −1.95309
\(605\) 4.88854 0.198748
\(606\) 31.3050 1.27168
\(607\) −14.8328 −0.602045 −0.301023 0.953617i \(-0.597328\pi\)
−0.301023 + 0.953617i \(0.597328\pi\)
\(608\) −43.4164 −1.76077
\(609\) 2.00000 0.0810441
\(610\) 4.47214 0.181071
\(611\) 22.1115 0.894534
\(612\) −6.00000 −0.242536
\(613\) 10.9443 0.442035 0.221017 0.975270i \(-0.429062\pi\)
0.221017 + 0.975270i \(0.429062\pi\)
\(614\) −73.6656 −2.97290
\(615\) −2.00000 −0.0806478
\(616\) −5.52786 −0.222724
\(617\) 7.52786 0.303060 0.151530 0.988453i \(-0.451580\pi\)
0.151530 + 0.988453i \(0.451580\pi\)
\(618\) 0 0
\(619\) 12.5836 0.505777 0.252889 0.967495i \(-0.418619\pi\)
0.252889 + 0.967495i \(0.418619\pi\)
\(620\) −31.4164 −1.26171
\(621\) −4.00000 −0.160514
\(622\) −22.1115 −0.886589
\(623\) −2.00000 −0.0801283
\(624\) −4.47214 −0.179029
\(625\) 1.00000 0.0400000
\(626\) 21.0557 0.841556
\(627\) 16.0000 0.638978
\(628\) 25.4164 1.01423
\(629\) −21.8885 −0.872753
\(630\) −2.23607 −0.0890871
\(631\) −22.8328 −0.908960 −0.454480 0.890757i \(-0.650175\pi\)
−0.454480 + 0.890757i \(0.650175\pi\)
\(632\) −11.0557 −0.439773
\(633\) −0.944272 −0.0375314
\(634\) −67.8885 −2.69620
\(635\) −12.9443 −0.513678
\(636\) 37.4164 1.48366
\(637\) −4.47214 −0.177192
\(638\) 11.0557 0.437700
\(639\) 14.4721 0.572509
\(640\) 15.6525 0.618718
\(641\) −36.8328 −1.45481 −0.727404 0.686209i \(-0.759274\pi\)
−0.727404 + 0.686209i \(0.759274\pi\)
\(642\) −11.0557 −0.436335
\(643\) −32.9443 −1.29920 −0.649598 0.760278i \(-0.725063\pi\)
−0.649598 + 0.760278i \(0.725063\pi\)
\(644\) 12.0000 0.472866
\(645\) −8.94427 −0.352180
\(646\) −28.9443 −1.13880
\(647\) 33.8885 1.33230 0.666148 0.745820i \(-0.267942\pi\)
0.666148 + 0.745820i \(0.267942\pi\)
\(648\) 2.23607 0.0878410
\(649\) −22.1115 −0.867951
\(650\) −10.0000 −0.392232
\(651\) −10.4721 −0.410435
\(652\) 2.83282 0.110942
\(653\) −49.4164 −1.93381 −0.966907 0.255130i \(-0.917882\pi\)
−0.966907 + 0.255130i \(0.917882\pi\)
\(654\) 4.47214 0.174874
\(655\) −4.00000 −0.156293
\(656\) 2.00000 0.0780869
\(657\) −3.52786 −0.137635
\(658\) −11.0557 −0.430997
\(659\) −41.3050 −1.60901 −0.804506 0.593944i \(-0.797570\pi\)
−0.804506 + 0.593944i \(0.797570\pi\)
\(660\) −7.41641 −0.288683
\(661\) −0.111456 −0.00433514 −0.00216757 0.999998i \(-0.500690\pi\)
−0.00216757 + 0.999998i \(0.500690\pi\)
\(662\) −37.8885 −1.47258
\(663\) −8.94427 −0.347367
\(664\) 2.11146 0.0819404
\(665\) −6.47214 −0.250979
\(666\) 24.4721 0.948276
\(667\) −8.00000 −0.309761
\(668\) −24.0000 −0.928588
\(669\) −4.94427 −0.191157
\(670\) 8.94427 0.345547
\(671\) 4.94427 0.190872
\(672\) 6.70820 0.258775
\(673\) −44.8328 −1.72818 −0.864089 0.503339i \(-0.832105\pi\)
−0.864089 + 0.503339i \(0.832105\pi\)
\(674\) 26.5836 1.02396
\(675\) −1.00000 −0.0384900
\(676\) 21.0000 0.807692
\(677\) 38.9443 1.49675 0.748375 0.663276i \(-0.230834\pi\)
0.748375 + 0.663276i \(0.230834\pi\)
\(678\) 18.9443 0.727550
\(679\) −0.472136 −0.0181189
\(680\) 4.47214 0.171499
\(681\) 16.9443 0.649306
\(682\) −57.8885 −2.21667
\(683\) 33.8885 1.29671 0.648355 0.761339i \(-0.275457\pi\)
0.648355 + 0.761339i \(0.275457\pi\)
\(684\) 19.4164 0.742405
\(685\) −12.4721 −0.476536
\(686\) 2.23607 0.0853735
\(687\) 11.8885 0.453576
\(688\) 8.94427 0.340997
\(689\) 55.7771 2.12494
\(690\) 8.94427 0.340503
\(691\) −0.360680 −0.0137209 −0.00686045 0.999976i \(-0.502184\pi\)
−0.00686045 + 0.999976i \(0.502184\pi\)
\(692\) 44.8328 1.70429
\(693\) −2.47214 −0.0939087
\(694\) −17.8885 −0.679040
\(695\) 19.4164 0.736506
\(696\) 4.47214 0.169516
\(697\) 4.00000 0.151511
\(698\) 53.4164 2.02184
\(699\) −17.4164 −0.658749
\(700\) 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
\(703\) 70.8328 2.67151
\(704\) 32.1378 1.21124
\(705\) −4.94427 −0.186212
\(706\) −62.3607 −2.34698
\(707\) −14.0000 −0.526524
\(708\) −26.8328 −1.00844
\(709\) −45.7771 −1.71919 −0.859597 0.510972i \(-0.829286\pi\)
−0.859597 + 0.510972i \(0.829286\pi\)
\(710\) −32.3607 −1.21447
\(711\) −4.94427 −0.185425
\(712\) −4.47214 −0.167600
\(713\) 41.8885 1.56874
\(714\) 4.47214 0.167365
\(715\) −11.0557 −0.413461
\(716\) −7.41641 −0.277164
\(717\) 1.52786 0.0570591
\(718\) 21.3050 0.795094
\(719\) 46.8328 1.74657 0.873285 0.487210i \(-0.161985\pi\)
0.873285 + 0.487210i \(0.161985\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 51.1803 1.90474
\(723\) 1.05573 0.0392630
\(724\) 56.8328 2.11217
\(725\) −2.00000 −0.0742781
\(726\) 10.9311 0.405692
\(727\) 14.8328 0.550119 0.275059 0.961427i \(-0.411302\pi\)
0.275059 + 0.961427i \(0.411302\pi\)
\(728\) −10.0000 −0.370625
\(729\) 1.00000 0.0370370
\(730\) 7.88854 0.291968
\(731\) 17.8885 0.661632
\(732\) 6.00000 0.221766
\(733\) 37.4164 1.38201 0.691003 0.722852i \(-0.257169\pi\)
0.691003 + 0.722852i \(0.257169\pi\)
\(734\) 46.8328 1.72863
\(735\) 1.00000 0.0368856
\(736\) −26.8328 −0.989071
\(737\) 9.88854 0.364249
\(738\) −4.47214 −0.164622
\(739\) 29.8885 1.09947 0.549734 0.835340i \(-0.314729\pi\)
0.549734 + 0.835340i \(0.314729\pi\)
\(740\) −32.8328 −1.20696
\(741\) 28.9443 1.06329
\(742\) −27.8885 −1.02382
\(743\) −18.8328 −0.690909 −0.345455 0.938436i \(-0.612275\pi\)
−0.345455 + 0.938436i \(0.612275\pi\)
\(744\) −23.4164 −0.858487
\(745\) −2.94427 −0.107870
\(746\) 13.4164 0.491210
\(747\) 0.944272 0.0345491
\(748\) 14.8328 0.542341
\(749\) 4.94427 0.180660
\(750\) 2.23607 0.0816497
\(751\) −3.05573 −0.111505 −0.0557526 0.998445i \(-0.517756\pi\)
−0.0557526 + 0.998445i \(0.517756\pi\)
\(752\) 4.94427 0.180299
\(753\) 0.944272 0.0344112
\(754\) 20.0000 0.728357
\(755\) 16.0000 0.582300
\(756\) −3.00000 −0.109109
\(757\) −3.88854 −0.141332 −0.0706658 0.997500i \(-0.522512\pi\)
−0.0706658 + 0.997500i \(0.522512\pi\)
\(758\) −4.72136 −0.171488
\(759\) 9.88854 0.358931
\(760\) −14.4721 −0.524960
\(761\) 7.88854 0.285959 0.142980 0.989726i \(-0.454332\pi\)
0.142980 + 0.989726i \(0.454332\pi\)
\(762\) −28.9443 −1.04854
\(763\) −2.00000 −0.0724049
\(764\) 82.2492 2.97567
\(765\) 2.00000 0.0723102
\(766\) −17.8885 −0.646339
\(767\) −40.0000 −1.44432
\(768\) 9.00000 0.324760
\(769\) 0.832816 0.0300321 0.0150161 0.999887i \(-0.495220\pi\)
0.0150161 + 0.999887i \(0.495220\pi\)
\(770\) 5.52786 0.199210
\(771\) −1.05573 −0.0380211
\(772\) −42.0000 −1.51161
\(773\) −25.0557 −0.901192 −0.450596 0.892728i \(-0.648788\pi\)
−0.450596 + 0.892728i \(0.648788\pi\)
\(774\) −20.0000 −0.718885
\(775\) 10.4721 0.376170
\(776\) −1.05573 −0.0378984
\(777\) −10.9443 −0.392624
\(778\) 24.4721 0.877369
\(779\) −12.9443 −0.463777
\(780\) −13.4164 −0.480384
\(781\) −35.7771 −1.28020
\(782\) −17.8885 −0.639693
\(783\) 2.00000 0.0714742
\(784\) −1.00000 −0.0357143
\(785\) −8.47214 −0.302383
\(786\) −8.94427 −0.319032
\(787\) 48.9443 1.74467 0.872337 0.488904i \(-0.162603\pi\)
0.872337 + 0.488904i \(0.162603\pi\)
\(788\) 73.4164 2.61535
\(789\) −24.9443 −0.888040
\(790\) 11.0557 0.393345
\(791\) −8.47214 −0.301234
\(792\) −5.52786 −0.196424
\(793\) 8.94427 0.317620
\(794\) 30.0000 1.06466
\(795\) −12.4721 −0.442341
\(796\) 1.75078 0.0620546
\(797\) −1.05573 −0.0373958 −0.0186979 0.999825i \(-0.505952\pi\)
−0.0186979 + 0.999825i \(0.505952\pi\)
\(798\) −14.4721 −0.512308
\(799\) 9.88854 0.349832
\(800\) −6.70820 −0.237171
\(801\) −2.00000 −0.0706665
\(802\) 22.3607 0.789583
\(803\) 8.72136 0.307770
\(804\) 12.0000 0.423207
\(805\) −4.00000 −0.140981
\(806\) −104.721 −3.68865
\(807\) 23.8885 0.840917
\(808\) −31.3050 −1.10130
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) −2.23607 −0.0785674
\(811\) 28.5836 1.00371 0.501853 0.864953i \(-0.332652\pi\)
0.501853 + 0.864953i \(0.332652\pi\)
\(812\) −6.00000 −0.210559
\(813\) 10.4721 0.367274
\(814\) −60.4984 −2.12047
\(815\) −0.944272 −0.0330764
\(816\) −2.00000 −0.0700140
\(817\) −57.8885 −2.02526
\(818\) −53.4164 −1.86766
\(819\) −4.47214 −0.156269
\(820\) 6.00000 0.209529
\(821\) −37.7771 −1.31843 −0.659215 0.751955i \(-0.729111\pi\)
−0.659215 + 0.751955i \(0.729111\pi\)
\(822\) −27.8885 −0.972725
\(823\) 27.0557 0.943103 0.471552 0.881838i \(-0.343694\pi\)
0.471552 + 0.881838i \(0.343694\pi\)
\(824\) 0 0
\(825\) 2.47214 0.0860687
\(826\) 20.0000 0.695889
\(827\) −4.94427 −0.171929 −0.0859646 0.996298i \(-0.527397\pi\)
−0.0859646 + 0.996298i \(0.527397\pi\)
\(828\) 12.0000 0.417029
\(829\) −30.9443 −1.07474 −0.537369 0.843347i \(-0.680582\pi\)
−0.537369 + 0.843347i \(0.680582\pi\)
\(830\) −2.11146 −0.0732897
\(831\) −1.05573 −0.0366228
\(832\) 58.1378 2.01556
\(833\) −2.00000 −0.0692959
\(834\) 43.4164 1.50339
\(835\) 8.00000 0.276851
\(836\) −48.0000 −1.66011
\(837\) −10.4721 −0.361970
\(838\) 13.1672 0.454853
\(839\) 1.16718 0.0402957 0.0201478 0.999797i \(-0.493586\pi\)
0.0201478 + 0.999797i \(0.493586\pi\)
\(840\) 2.23607 0.0771517
\(841\) −25.0000 −0.862069
\(842\) 49.1935 1.69532
\(843\) −6.94427 −0.239173
\(844\) 2.83282 0.0975095
\(845\) −7.00000 −0.240807
\(846\) −11.0557 −0.380104
\(847\) −4.88854 −0.167972
\(848\) 12.4721 0.428295
\(849\) −12.0000 −0.411839
\(850\) −4.47214 −0.153393
\(851\) 43.7771 1.50066
\(852\) −43.4164 −1.48742
\(853\) 31.3050 1.07186 0.535931 0.844262i \(-0.319961\pi\)
0.535931 + 0.844262i \(0.319961\pi\)
\(854\) −4.47214 −0.153033
\(855\) −6.47214 −0.221342
\(856\) 11.0557 0.377877
\(857\) −16.8328 −0.574998 −0.287499 0.957781i \(-0.592824\pi\)
−0.287499 + 0.957781i \(0.592824\pi\)
\(858\) −24.7214 −0.843973
\(859\) 41.5279 1.41691 0.708456 0.705755i \(-0.249392\pi\)
0.708456 + 0.705755i \(0.249392\pi\)
\(860\) 26.8328 0.914991
\(861\) 2.00000 0.0681598
\(862\) −21.3050 −0.725650
\(863\) −13.8885 −0.472772 −0.236386 0.971659i \(-0.575963\pi\)
−0.236386 + 0.971659i \(0.575963\pi\)
\(864\) 6.70820 0.228218
\(865\) −14.9443 −0.508120
\(866\) 16.8328 0.572002
\(867\) 13.0000 0.441503
\(868\) 31.4164 1.06634
\(869\) 12.2229 0.414634
\(870\) −4.47214 −0.151620
\(871\) 17.8885 0.606130
\(872\) −4.47214 −0.151446
\(873\) −0.472136 −0.0159794
\(874\) 57.8885 1.95811
\(875\) −1.00000 −0.0338062
\(876\) 10.5836 0.357586
\(877\) −3.16718 −0.106948 −0.0534741 0.998569i \(-0.517029\pi\)
−0.0534741 + 0.998569i \(0.517029\pi\)
\(878\) 23.4164 0.790265
\(879\) −22.9443 −0.773891
\(880\) −2.47214 −0.0833357
\(881\) 7.88854 0.265772 0.132886 0.991131i \(-0.457576\pi\)
0.132886 + 0.991131i \(0.457576\pi\)
\(882\) 2.23607 0.0752923
\(883\) −2.11146 −0.0710562 −0.0355281 0.999369i \(-0.511311\pi\)
−0.0355281 + 0.999369i \(0.511311\pi\)
\(884\) 26.8328 0.902485
\(885\) 8.94427 0.300658
\(886\) −17.8885 −0.600977
\(887\) 22.8328 0.766651 0.383325 0.923613i \(-0.374779\pi\)
0.383325 + 0.923613i \(0.374779\pi\)
\(888\) −24.4721 −0.821231
\(889\) 12.9443 0.434137
\(890\) 4.47214 0.149906
\(891\) −2.47214 −0.0828197
\(892\) 14.8328 0.496639
\(893\) −32.0000 −1.07084
\(894\) −6.58359 −0.220188
\(895\) 2.47214 0.0826344
\(896\) −15.6525 −0.522913
\(897\) 17.8885 0.597281
\(898\) −31.3050 −1.04466
\(899\) −20.9443 −0.698531
\(900\) 3.00000 0.100000
\(901\) 24.9443 0.831014
\(902\) 11.0557 0.368115
\(903\) 8.94427 0.297647
\(904\) −18.9443 −0.630077
\(905\) −18.9443 −0.629729
\(906\) 35.7771 1.18861
\(907\) 18.1115 0.601381 0.300691 0.953722i \(-0.402783\pi\)
0.300691 + 0.953722i \(0.402783\pi\)
\(908\) −50.8328 −1.68695
\(909\) −14.0000 −0.464351
\(910\) 10.0000 0.331497
\(911\) −34.2492 −1.13473 −0.567364 0.823467i \(-0.692037\pi\)
−0.567364 + 0.823467i \(0.692037\pi\)
\(912\) 6.47214 0.214314
\(913\) −2.33437 −0.0772563
\(914\) −24.4721 −0.809466
\(915\) −2.00000 −0.0661180
\(916\) −35.6656 −1.17843
\(917\) 4.00000 0.132092
\(918\) 4.47214 0.147602
\(919\) −52.9443 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(920\) −8.94427 −0.294884
\(921\) 32.9443 1.08555
\(922\) −71.3050 −2.34830
\(923\) −64.7214 −2.13033
\(924\) 7.41641 0.243982
\(925\) 10.9443 0.359845
\(926\) 6.83282 0.224540
\(927\) 0 0
\(928\) 13.4164 0.440415
\(929\) −51.8885 −1.70241 −0.851204 0.524835i \(-0.824127\pi\)
−0.851204 + 0.524835i \(0.824127\pi\)
\(930\) 23.4164 0.767854
\(931\) 6.47214 0.212116
\(932\) 52.2492 1.71148
\(933\) 9.88854 0.323736
\(934\) 20.0000 0.654420
\(935\) −4.94427 −0.161695
\(936\) −10.0000 −0.326860
\(937\) −43.5279 −1.42199 −0.710997 0.703195i \(-0.751756\pi\)
−0.710997 + 0.703195i \(0.751756\pi\)
\(938\) −8.94427 −0.292041
\(939\) −9.41641 −0.307293
\(940\) 14.8328 0.483793
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −18.9443 −0.617238
\(943\) −8.00000 −0.260516
\(944\) −8.94427 −0.291111
\(945\) 1.00000 0.0325300
\(946\) 49.4427 1.60752
\(947\) 17.8885 0.581300 0.290650 0.956830i \(-0.406129\pi\)
0.290650 + 0.956830i \(0.406129\pi\)
\(948\) 14.8328 0.481747
\(949\) 15.7771 0.512146
\(950\) 14.4721 0.469538
\(951\) 30.3607 0.984512
\(952\) −4.47214 −0.144943
\(953\) −6.58359 −0.213263 −0.106632 0.994299i \(-0.534007\pi\)
−0.106632 + 0.994299i \(0.534007\pi\)
\(954\) −27.8885 −0.902925
\(955\) −27.4164 −0.887174
\(956\) −4.58359 −0.148244
\(957\) −4.94427 −0.159826
\(958\) 40.0000 1.29234
\(959\) 12.4721 0.402746
\(960\) −13.0000 −0.419573
\(961\) 78.6656 2.53760
\(962\) −109.443 −3.52857
\(963\) 4.94427 0.159327
\(964\) −3.16718 −0.102008
\(965\) 14.0000 0.450676
\(966\) −8.94427 −0.287777
\(967\) −9.88854 −0.317994 −0.158997 0.987279i \(-0.550826\pi\)
−0.158997 + 0.987279i \(0.550826\pi\)
\(968\) −10.9311 −0.351339
\(969\) 12.9443 0.415830
\(970\) 1.05573 0.0338974
\(971\) 23.0557 0.739894 0.369947 0.929053i \(-0.379376\pi\)
0.369947 + 0.929053i \(0.379376\pi\)
\(972\) −3.00000 −0.0962250
\(973\) −19.4164 −0.622461
\(974\) −6.83282 −0.218938
\(975\) 4.47214 0.143223
\(976\) 2.00000 0.0640184
\(977\) 57.4164 1.83691 0.918457 0.395521i \(-0.129436\pi\)
0.918457 + 0.395521i \(0.129436\pi\)
\(978\) −2.11146 −0.0675169
\(979\) 4.94427 0.158020
\(980\) −3.00000 −0.0958315
\(981\) −2.00000 −0.0638551
\(982\) 92.3607 2.94735
\(983\) 30.8328 0.983414 0.491707 0.870761i \(-0.336373\pi\)
0.491707 + 0.870761i \(0.336373\pi\)
\(984\) 4.47214 0.142566
\(985\) −24.4721 −0.779747
\(986\) 8.94427 0.284844
\(987\) 4.94427 0.157378
\(988\) −86.8328 −2.76252
\(989\) −35.7771 −1.13765
\(990\) 5.52786 0.175687
\(991\) −12.9443 −0.411188 −0.205594 0.978637i \(-0.565913\pi\)
−0.205594 + 0.978637i \(0.565913\pi\)
\(992\) −70.2492 −2.23042
\(993\) 16.9443 0.537710
\(994\) 32.3607 1.02642
\(995\) −0.583592 −0.0185011
\(996\) −2.83282 −0.0897612
\(997\) 21.4164 0.678264 0.339132 0.940739i \(-0.389867\pi\)
0.339132 + 0.940739i \(0.389867\pi\)
\(998\) 48.9443 1.54930
\(999\) −10.9443 −0.346261
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.1.2 2
3.2 odd 2 315.2.a.d.1.1 2
4.3 odd 2 1680.2.a.v.1.2 2
5.2 odd 4 525.2.d.c.274.4 4
5.3 odd 4 525.2.d.c.274.1 4
5.4 even 2 525.2.a.g.1.1 2
7.2 even 3 735.2.i.k.361.1 4
7.3 odd 6 735.2.i.i.226.1 4
7.4 even 3 735.2.i.k.226.1 4
7.5 odd 6 735.2.i.i.361.1 4
7.6 odd 2 735.2.a.k.1.2 2
8.3 odd 2 6720.2.a.cs.1.1 2
8.5 even 2 6720.2.a.cx.1.2 2
12.11 even 2 5040.2.a.bw.1.1 2
15.2 even 4 1575.2.d.d.1324.2 4
15.8 even 4 1575.2.d.d.1324.3 4
15.14 odd 2 1575.2.a.r.1.2 2
20.19 odd 2 8400.2.a.cx.1.2 2
21.20 even 2 2205.2.a.w.1.1 2
35.34 odd 2 3675.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.b.1.2 2 1.1 even 1 trivial
315.2.a.d.1.1 2 3.2 odd 2
525.2.a.g.1.1 2 5.4 even 2
525.2.d.c.274.1 4 5.3 odd 4
525.2.d.c.274.4 4 5.2 odd 4
735.2.a.k.1.2 2 7.6 odd 2
735.2.i.i.226.1 4 7.3 odd 6
735.2.i.i.361.1 4 7.5 odd 6
735.2.i.k.226.1 4 7.4 even 3
735.2.i.k.361.1 4 7.2 even 3
1575.2.a.r.1.2 2 15.14 odd 2
1575.2.d.d.1324.2 4 15.2 even 4
1575.2.d.d.1324.3 4 15.8 even 4
1680.2.a.v.1.2 2 4.3 odd 2
2205.2.a.w.1.1 2 21.20 even 2
3675.2.a.y.1.1 2 35.34 odd 2
5040.2.a.bw.1.1 2 12.11 even 2
6720.2.a.cs.1.1 2 8.3 odd 2
6720.2.a.cx.1.2 2 8.5 even 2
8400.2.a.cx.1.2 2 20.19 odd 2