Properties

Label 7623.2.a.bl.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q-2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -2.23607 q^{8} -4.47214 q^{10} +1.23607 q^{13} +2.23607 q^{14} -1.00000 q^{16} +1.23607 q^{17} +2.47214 q^{19} +6.00000 q^{20} +6.47214 q^{23} -1.00000 q^{25} -2.76393 q^{26} -3.00000 q^{28} -0.472136 q^{29} -7.23607 q^{31} +6.70820 q^{32} -2.76393 q^{34} -2.00000 q^{35} +0.472136 q^{37} -5.52786 q^{38} -4.47214 q^{40} -6.76393 q^{41} -8.00000 q^{43} -14.4721 q^{46} -7.23607 q^{47} +1.00000 q^{49} +2.23607 q^{50} +3.70820 q^{52} -8.47214 q^{53} +2.23607 q^{56} +1.05573 q^{58} -3.23607 q^{59} +2.76393 q^{61} +16.1803 q^{62} -13.0000 q^{64} +2.47214 q^{65} +5.52786 q^{67} +3.70820 q^{68} +4.47214 q^{70} +1.52786 q^{71} +5.23607 q^{73} -1.05573 q^{74} +7.41641 q^{76} -8.94427 q^{79} -2.00000 q^{80} +15.1246 q^{82} +15.4164 q^{83} +2.47214 q^{85} +17.8885 q^{86} -2.00000 q^{89} -1.23607 q^{91} +19.4164 q^{92} +16.1803 q^{94} +4.94427 q^{95} -9.41641 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{4} + 4q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 6q^{4} + 4q^{5} - 2q^{7} - 2q^{13} - 2q^{16} - 2q^{17} - 4q^{19} + 12q^{20} + 4q^{23} - 2q^{25} - 10q^{26} - 6q^{28} + 8q^{29} - 10q^{31} - 10q^{34} - 4q^{35} - 8q^{37} - 20q^{38} - 18q^{41} - 16q^{43} - 20q^{46} - 10q^{47} + 2q^{49} - 6q^{52} - 8q^{53} + 20q^{58} - 2q^{59} + 10q^{61} + 10q^{62} - 26q^{64} - 4q^{65} + 20q^{67} - 6q^{68} + 12q^{71} + 6q^{73} - 20q^{74} - 12q^{76} - 4q^{80} - 10q^{82} + 4q^{83} - 4q^{85} - 4q^{89} + 2q^{91} + 12q^{92} + 10q^{94} - 8q^{95} + 8q^{97} + O(q^{100}) \)

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 \(0.790215\pi\)
\(3\) 0 0
\(4\) 3.00000 1.50000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −4.47214 −1.41421
\(11\) 0 0
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 2.23607 0.597614
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.23607 0.299791 0.149895 0.988702i \(-0.452106\pi\)
0.149895 + 0.988702i \(0.452106\pi\)
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) 6.47214 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.76393 −0.542052
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) −0.472136 −0.0876734 −0.0438367 0.999039i \(-0.513958\pi\)
−0.0438367 + 0.999039i \(0.513958\pi\)
\(30\) 0 0
\(31\) −7.23607 −1.29964 −0.649818 0.760090i \(-0.725155\pi\)
−0.649818 + 0.760090i \(0.725155\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) −2.76393 −0.474010
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 0.472136 0.0776187 0.0388093 0.999247i \(-0.487644\pi\)
0.0388093 + 0.999247i \(0.487644\pi\)
\(38\) −5.52786 −0.896738
\(39\) 0 0
\(40\) −4.47214 −0.707107
\(41\) −6.76393 −1.05635 −0.528174 0.849136i \(-0.677123\pi\)
−0.528174 + 0.849136i \(0.677123\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −14.4721 −2.13380
\(47\) −7.23607 −1.05549 −0.527744 0.849403i \(-0.676962\pi\)
−0.527744 + 0.849403i \(0.676962\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.23607 0.316228
\(51\) 0 0
\(52\) 3.70820 0.514235
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) 1.05573 0.138624
\(59\) −3.23607 −0.421300 −0.210650 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(60\) 0 0
\(61\) 2.76393 0.353885 0.176943 0.984221i \(-0.443379\pi\)
0.176943 + 0.984221i \(0.443379\pi\)
\(62\) 16.1803 2.05491
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) 2.47214 0.306631
\(66\) 0 0
\(67\) 5.52786 0.675336 0.337668 0.941265i \(-0.390362\pi\)
0.337668 + 0.941265i \(0.390362\pi\)
\(68\) 3.70820 0.449686
\(69\) 0 0
\(70\) 4.47214 0.534522
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) 0 0
\(73\) 5.23607 0.612835 0.306418 0.951897i \(-0.400870\pi\)
0.306418 + 0.951897i \(0.400870\pi\)
\(74\) −1.05573 −0.122726
\(75\) 0 0
\(76\) 7.41641 0.850720
\(77\) 0 0
\(78\) 0 0
\(79\) −8.94427 −1.00631 −0.503155 0.864196i \(-0.667827\pi\)
−0.503155 + 0.864196i \(0.667827\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 15.1246 1.67023
\(83\) 15.4164 1.69217 0.846085 0.533048i \(-0.178953\pi\)
0.846085 + 0.533048i \(0.178953\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) 17.8885 1.92897
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 19.4164 2.02430
\(93\) 0 0
\(94\) 16.1803 1.66887
\(95\) 4.94427 0.507272
\(96\) 0 0
\(97\) −9.41641 −0.956091 −0.478046 0.878335i \(-0.658655\pi\)
−0.478046 + 0.878335i \(0.658655\pi\)
\(98\) −2.23607 −0.225877
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −9.23607 −0.919023 −0.459512 0.888172i \(-0.651976\pi\)
−0.459512 + 0.888172i \(0.651976\pi\)
\(102\) 0 0
\(103\) −5.70820 −0.562446 −0.281223 0.959642i \(-0.590740\pi\)
−0.281223 + 0.959642i \(0.590740\pi\)
\(104\) −2.76393 −0.271026
\(105\) 0 0
\(106\) 18.9443 1.84003
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −4.47214 −0.428353 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 12.9443 1.20706
\(116\) −1.41641 −0.131510
\(117\) 0 0
\(118\) 7.23607 0.666134
\(119\) −1.23607 −0.113310
\(120\) 0 0
\(121\) 0 0
\(122\) −6.18034 −0.559542
\(123\) 0 0
\(124\) −21.7082 −1.94945
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −20.9443 −1.85850 −0.929252 0.369447i \(-0.879547\pi\)
−0.929252 + 0.369447i \(0.879547\pi\)
\(128\) 15.6525 1.38350
\(129\) 0 0
\(130\) −5.52786 −0.484826
\(131\) −13.8885 −1.21345 −0.606724 0.794913i \(-0.707517\pi\)
−0.606724 + 0.794913i \(0.707517\pi\)
\(132\) 0 0
\(133\) −2.47214 −0.214361
\(134\) −12.3607 −1.06780
\(135\) 0 0
\(136\) −2.76393 −0.237005
\(137\) −7.52786 −0.643149 −0.321574 0.946884i \(-0.604212\pi\)
−0.321574 + 0.946884i \(0.604212\pi\)
\(138\) 0 0
\(139\) 10.4721 0.888235 0.444117 0.895969i \(-0.353517\pi\)
0.444117 + 0.895969i \(0.353517\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −3.41641 −0.286699
\(143\) 0 0
\(144\) 0 0
\(145\) −0.944272 −0.0784175
\(146\) −11.7082 −0.968978
\(147\) 0 0
\(148\) 1.41641 0.116428
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 8.94427 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(152\) −5.52786 −0.448369
\(153\) 0 0
\(154\) 0 0
\(155\) −14.4721 −1.16243
\(156\) 0 0
\(157\) −6.94427 −0.554213 −0.277107 0.960839i \(-0.589376\pi\)
−0.277107 + 0.960839i \(0.589376\pi\)
\(158\) 20.0000 1.59111
\(159\) 0 0
\(160\) 13.4164 1.06066
\(161\) −6.47214 −0.510076
\(162\) 0 0
\(163\) 23.4164 1.83411 0.917057 0.398755i \(-0.130558\pi\)
0.917057 + 0.398755i \(0.130558\pi\)
\(164\) −20.2918 −1.58452
\(165\) 0 0
\(166\) −34.4721 −2.67556
\(167\) −12.9443 −1.00166 −0.500829 0.865546i \(-0.666971\pi\)
−0.500829 + 0.865546i \(0.666971\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) −5.52786 −0.423968
\(171\) 0 0
\(172\) −24.0000 −1.82998
\(173\) −17.2361 −1.31043 −0.655217 0.755441i \(-0.727423\pi\)
−0.655217 + 0.755441i \(0.727423\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 4.47214 0.335201
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 0 0
\(181\) 1.41641 0.105281 0.0526404 0.998614i \(-0.483236\pi\)
0.0526404 + 0.998614i \(0.483236\pi\)
\(182\) 2.76393 0.204876
\(183\) 0 0
\(184\) −14.4721 −1.06690
\(185\) 0.944272 0.0694243
\(186\) 0 0
\(187\) 0 0
\(188\) −21.7082 −1.58323
\(189\) 0 0
\(190\) −11.0557 −0.802067
\(191\) 20.9443 1.51547 0.757737 0.652560i \(-0.226305\pi\)
0.757737 + 0.652560i \(0.226305\pi\)
\(192\) 0 0
\(193\) 23.8885 1.71954 0.859768 0.510686i \(-0.170608\pi\)
0.859768 + 0.510686i \(0.170608\pi\)
\(194\) 21.0557 1.51171
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 20.1803 1.43055 0.715273 0.698845i \(-0.246302\pi\)
0.715273 + 0.698845i \(0.246302\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 20.6525 1.45310
\(203\) 0.472136 0.0331374
\(204\) 0 0
\(205\) −13.5279 −0.944827
\(206\) 12.7639 0.889305
\(207\) 0 0
\(208\) −1.23607 −0.0857059
\(209\) 0 0
\(210\) 0 0
\(211\) −21.8885 −1.50687 −0.753435 0.657523i \(-0.771604\pi\)
−0.753435 + 0.657523i \(0.771604\pi\)
\(212\) −25.4164 −1.74561
\(213\) 0 0
\(214\) 8.94427 0.611418
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 7.23607 0.491216
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) 1.52786 0.102775
\(222\) 0 0
\(223\) 12.1803 0.815656 0.407828 0.913059i \(-0.366286\pi\)
0.407828 + 0.913059i \(0.366286\pi\)
\(224\) −6.70820 −0.448211
\(225\) 0 0
\(226\) 4.47214 0.297482
\(227\) −29.8885 −1.98377 −0.991886 0.127129i \(-0.959424\pi\)
−0.991886 + 0.127129i \(0.959424\pi\)
\(228\) 0 0
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) −28.9443 −1.90853
\(231\) 0 0
\(232\) 1.05573 0.0693119
\(233\) −17.4164 −1.14099 −0.570493 0.821302i \(-0.693248\pi\)
−0.570493 + 0.821302i \(0.693248\pi\)
\(234\) 0 0
\(235\) −14.4721 −0.944058
\(236\) −9.70820 −0.631950
\(237\) 0 0
\(238\) 2.76393 0.179159
\(239\) 25.8885 1.67459 0.837295 0.546751i \(-0.184136\pi\)
0.837295 + 0.546751i \(0.184136\pi\)
\(240\) 0 0
\(241\) −27.1246 −1.74725 −0.873625 0.486600i \(-0.838237\pi\)
−0.873625 + 0.486600i \(0.838237\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 8.29180 0.530828
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 3.05573 0.194431
\(248\) 16.1803 1.02745
\(249\) 0 0
\(250\) 26.8328 1.69706
\(251\) −17.7082 −1.11773 −0.558866 0.829258i \(-0.688763\pi\)
−0.558866 + 0.829258i \(0.688763\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 46.8328 2.93855
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −0.472136 −0.0293371
\(260\) 7.41641 0.459946
\(261\) 0 0
\(262\) 31.0557 1.91863
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −16.9443 −1.04088
\(266\) 5.52786 0.338935
\(267\) 0 0
\(268\) 16.5836 1.01300
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) 0 0
\(271\) 1.52786 0.0928111 0.0464056 0.998923i \(-0.485223\pi\)
0.0464056 + 0.998923i \(0.485223\pi\)
\(272\) −1.23607 −0.0749476
\(273\) 0 0
\(274\) 16.8328 1.01691
\(275\) 0 0
\(276\) 0 0
\(277\) −15.8885 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(278\) −23.4164 −1.40442
\(279\) 0 0
\(280\) 4.47214 0.267261
\(281\) −12.4721 −0.744025 −0.372013 0.928228i \(-0.621332\pi\)
−0.372013 + 0.928228i \(0.621332\pi\)
\(282\) 0 0
\(283\) 5.88854 0.350038 0.175019 0.984565i \(-0.444001\pi\)
0.175019 + 0.984565i \(0.444001\pi\)
\(284\) 4.58359 0.271986
\(285\) 0 0
\(286\) 0 0
\(287\) 6.76393 0.399262
\(288\) 0 0
\(289\) −15.4721 −0.910126
\(290\) 2.11146 0.123989
\(291\) 0 0
\(292\) 15.7082 0.919253
\(293\) 15.1246 0.883589 0.441795 0.897116i \(-0.354342\pi\)
0.441795 + 0.897116i \(0.354342\pi\)
\(294\) 0 0
\(295\) −6.47214 −0.376822
\(296\) −1.05573 −0.0613629
\(297\) 0 0
\(298\) −31.3050 −1.81345
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) −2.47214 −0.141787
\(305\) 5.52786 0.316525
\(306\) 0 0
\(307\) −8.94427 −0.510477 −0.255238 0.966878i \(-0.582154\pi\)
−0.255238 + 0.966878i \(0.582154\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 32.3607 1.83796
\(311\) 21.7082 1.23096 0.615480 0.788153i \(-0.288962\pi\)
0.615480 + 0.788153i \(0.288962\pi\)
\(312\) 0 0
\(313\) −2.94427 −0.166420 −0.0832100 0.996532i \(-0.526517\pi\)
−0.0832100 + 0.996532i \(0.526517\pi\)
\(314\) 15.5279 0.876288
\(315\) 0 0
\(316\) −26.8328 −1.50946
\(317\) −14.0000 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −26.0000 −1.45344
\(321\) 0 0
\(322\) 14.4721 0.806501
\(323\) 3.05573 0.170025
\(324\) 0 0
\(325\) −1.23607 −0.0685647
\(326\) −52.3607 −2.89999
\(327\) 0 0
\(328\) 15.1246 0.835117
\(329\) 7.23607 0.398937
\(330\) 0 0
\(331\) 21.8885 1.20310 0.601552 0.798834i \(-0.294549\pi\)
0.601552 + 0.798834i \(0.294549\pi\)
\(332\) 46.2492 2.53826
\(333\) 0 0
\(334\) 28.9443 1.58376
\(335\) 11.0557 0.604039
\(336\) 0 0
\(337\) 20.4721 1.11519 0.557594 0.830114i \(-0.311725\pi\)
0.557594 + 0.830114i \(0.311725\pi\)
\(338\) 25.6525 1.39531
\(339\) 0 0
\(340\) 7.41641 0.402211
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 17.8885 0.964486
\(345\) 0 0
\(346\) 38.5410 2.07198
\(347\) 3.05573 0.164040 0.0820200 0.996631i \(-0.473863\pi\)
0.0820200 + 0.996631i \(0.473863\pi\)
\(348\) 0 0
\(349\) 2.76393 0.147950 0.0739749 0.997260i \(-0.476432\pi\)
0.0739749 + 0.997260i \(0.476432\pi\)
\(350\) −2.23607 −0.119523
\(351\) 0 0
\(352\) 0 0
\(353\) 15.8885 0.845662 0.422831 0.906209i \(-0.361036\pi\)
0.422831 + 0.906209i \(0.361036\pi\)
\(354\) 0 0
\(355\) 3.05573 0.162181
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −7.05573 −0.372387 −0.186194 0.982513i \(-0.559615\pi\)
−0.186194 + 0.982513i \(0.559615\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) −3.16718 −0.166464
\(363\) 0 0
\(364\) −3.70820 −0.194363
\(365\) 10.4721 0.548137
\(366\) 0 0
\(367\) 17.1246 0.893897 0.446949 0.894560i \(-0.352511\pi\)
0.446949 + 0.894560i \(0.352511\pi\)
\(368\) −6.47214 −0.337383
\(369\) 0 0
\(370\) −2.11146 −0.109769
\(371\) 8.47214 0.439851
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 16.1803 0.834437
\(377\) −0.583592 −0.0300565
\(378\) 0 0
\(379\) −25.3050 −1.29983 −0.649914 0.760008i \(-0.725195\pi\)
−0.649914 + 0.760008i \(0.725195\pi\)
\(380\) 14.8328 0.760907
\(381\) 0 0
\(382\) −46.8328 −2.39618
\(383\) −26.6525 −1.36188 −0.680939 0.732340i \(-0.738428\pi\)
−0.680939 + 0.732340i \(0.738428\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −53.4164 −2.71882
\(387\) 0 0
\(388\) −28.2492 −1.43414
\(389\) 19.8885 1.00839 0.504195 0.863590i \(-0.331789\pi\)
0.504195 + 0.863590i \(0.331789\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −2.23607 −0.112938
\(393\) 0 0
\(394\) 4.47214 0.225303
\(395\) −17.8885 −0.900070
\(396\) 0 0
\(397\) −0.111456 −0.00559383 −0.00279691 0.999996i \(-0.500890\pi\)
−0.00279691 + 0.999996i \(0.500890\pi\)
\(398\) −45.1246 −2.26189
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −5.05573 −0.252471 −0.126236 0.992000i \(-0.540290\pi\)
−0.126236 + 0.992000i \(0.540290\pi\)
\(402\) 0 0
\(403\) −8.94427 −0.445546
\(404\) −27.7082 −1.37853
\(405\) 0 0
\(406\) −1.05573 −0.0523949
\(407\) 0 0
\(408\) 0 0
\(409\) 31.1246 1.53901 0.769507 0.638639i \(-0.220502\pi\)
0.769507 + 0.638639i \(0.220502\pi\)
\(410\) 30.2492 1.49390
\(411\) 0 0
\(412\) −17.1246 −0.843669
\(413\) 3.23607 0.159236
\(414\) 0 0
\(415\) 30.8328 1.51352
\(416\) 8.29180 0.406539
\(417\) 0 0
\(418\) 0 0
\(419\) 6.65248 0.324995 0.162497 0.986709i \(-0.448045\pi\)
0.162497 + 0.986709i \(0.448045\pi\)
\(420\) 0 0
\(421\) −22.3607 −1.08979 −0.544896 0.838503i \(-0.683431\pi\)
−0.544896 + 0.838503i \(0.683431\pi\)
\(422\) 48.9443 2.38257
\(423\) 0 0
\(424\) 18.9443 0.920015
\(425\) −1.23607 −0.0599581
\(426\) 0 0
\(427\) −2.76393 −0.133756
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 35.7771 1.72532
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 0.472136 0.0226894 0.0113447 0.999936i \(-0.496389\pi\)
0.0113447 + 0.999936i \(0.496389\pi\)
\(434\) −16.1803 −0.776681
\(435\) 0 0
\(436\) −13.4164 −0.642529
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −1.52786 −0.0729210 −0.0364605 0.999335i \(-0.511608\pi\)
−0.0364605 + 0.999335i \(0.511608\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.41641 −0.162502
\(443\) 7.05573 0.335228 0.167614 0.985853i \(-0.446394\pi\)
0.167614 + 0.985853i \(0.446394\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) −27.2361 −1.28967
\(447\) 0 0
\(448\) 13.0000 0.614192
\(449\) 19.5279 0.921577 0.460788 0.887510i \(-0.347567\pi\)
0.460788 + 0.887510i \(0.347567\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 66.8328 3.13662
\(455\) −2.47214 −0.115896
\(456\) 0 0
\(457\) −24.8328 −1.16163 −0.580815 0.814036i \(-0.697266\pi\)
−0.580815 + 0.814036i \(0.697266\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 38.8328 1.81059
\(461\) 10.1803 0.474146 0.237073 0.971492i \(-0.423812\pi\)
0.237073 + 0.971492i \(0.423812\pi\)
\(462\) 0 0
\(463\) −14.4721 −0.672577 −0.336289 0.941759i \(-0.609172\pi\)
−0.336289 + 0.941759i \(0.609172\pi\)
\(464\) 0.472136 0.0219184
\(465\) 0 0
\(466\) 38.9443 1.80406
\(467\) 34.0689 1.57652 0.788260 0.615342i \(-0.210982\pi\)
0.788260 + 0.615342i \(0.210982\pi\)
\(468\) 0 0
\(469\) −5.52786 −0.255253
\(470\) 32.3607 1.49269
\(471\) 0 0
\(472\) 7.23607 0.333067
\(473\) 0 0
\(474\) 0 0
\(475\) −2.47214 −0.113429
\(476\) −3.70820 −0.169965
\(477\) 0 0
\(478\) −57.8885 −2.64776
\(479\) 22.4721 1.02678 0.513389 0.858156i \(-0.328390\pi\)
0.513389 + 0.858156i \(0.328390\pi\)
\(480\) 0 0
\(481\) 0.583592 0.0266095
\(482\) 60.6525 2.76264
\(483\) 0 0
\(484\) 0 0
\(485\) −18.8328 −0.855154
\(486\) 0 0
\(487\) 8.36068 0.378859 0.189429 0.981894i \(-0.439336\pi\)
0.189429 + 0.981894i \(0.439336\pi\)
\(488\) −6.18034 −0.279771
\(489\) 0 0
\(490\) −4.47214 −0.202031
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −0.583592 −0.0262837
\(494\) −6.83282 −0.307423
\(495\) 0 0
\(496\) 7.23607 0.324909
\(497\) −1.52786 −0.0685341
\(498\) 0 0
\(499\) 10.4721 0.468797 0.234399 0.972141i \(-0.424688\pi\)
0.234399 + 0.972141i \(0.424688\pi\)
\(500\) −36.0000 −1.60997
\(501\) 0 0
\(502\) 39.5967 1.76729
\(503\) −3.41641 −0.152330 −0.0761650 0.997095i \(-0.524268\pi\)
−0.0761650 + 0.997095i \(0.524268\pi\)
\(504\) 0 0
\(505\) −18.4721 −0.821999
\(506\) 0 0
\(507\) 0 0
\(508\) −62.8328 −2.78776
\(509\) −31.5279 −1.39745 −0.698724 0.715391i \(-0.746248\pi\)
−0.698724 + 0.715391i \(0.746248\pi\)
\(510\) 0 0
\(511\) −5.23607 −0.231630
\(512\) −11.1803 −0.494106
\(513\) 0 0
\(514\) −13.4164 −0.591772
\(515\) −11.4164 −0.503067
\(516\) 0 0
\(517\) 0 0
\(518\) 1.05573 0.0463860
\(519\) 0 0
\(520\) −5.52786 −0.242413
\(521\) 14.3607 0.629153 0.314576 0.949232i \(-0.398138\pi\)
0.314576 + 0.949232i \(0.398138\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −41.6656 −1.82017
\(525\) 0 0
\(526\) 0 0
\(527\) −8.94427 −0.389619
\(528\) 0 0
\(529\) 18.8885 0.821241
\(530\) 37.8885 1.64577
\(531\) 0 0
\(532\) −7.41641 −0.321542
\(533\) −8.36068 −0.362141
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) −12.3607 −0.533900
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) 32.8328 1.41159 0.705797 0.708415i \(-0.250589\pi\)
0.705797 + 0.708415i \(0.250589\pi\)
\(542\) −3.41641 −0.146747
\(543\) 0 0
\(544\) 8.29180 0.355508
\(545\) −8.94427 −0.383131
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −22.5836 −0.964723
\(549\) 0 0
\(550\) 0 0
\(551\) −1.16718 −0.0497237
\(552\) 0 0
\(553\) 8.94427 0.380349
\(554\) 35.5279 1.50943
\(555\) 0 0
\(556\) 31.4164 1.33235
\(557\) −21.0557 −0.892160 −0.446080 0.894993i \(-0.647180\pi\)
−0.446080 + 0.894993i \(0.647180\pi\)
\(558\) 0 0
\(559\) −9.88854 −0.418241
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 27.8885 1.17641
\(563\) 39.4164 1.66120 0.830602 0.556867i \(-0.187997\pi\)
0.830602 + 0.556867i \(0.187997\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) −13.1672 −0.553458
\(567\) 0 0
\(568\) −3.41641 −0.143349
\(569\) 16.4721 0.690548 0.345274 0.938502i \(-0.387786\pi\)
0.345274 + 0.938502i \(0.387786\pi\)
\(570\) 0 0
\(571\) 32.9443 1.37867 0.689337 0.724440i \(-0.257902\pi\)
0.689337 + 0.724440i \(0.257902\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −15.1246 −0.631289
\(575\) −6.47214 −0.269907
\(576\) 0 0
\(577\) −28.4721 −1.18531 −0.592655 0.805456i \(-0.701920\pi\)
−0.592655 + 0.805456i \(0.701920\pi\)
\(578\) 34.5967 1.43903
\(579\) 0 0
\(580\) −2.83282 −0.117626
\(581\) −15.4164 −0.639580
\(582\) 0 0
\(583\) 0 0
\(584\) −11.7082 −0.484489
\(585\) 0 0
\(586\) −33.8197 −1.39708
\(587\) 13.1246 0.541711 0.270855 0.962620i \(-0.412693\pi\)
0.270855 + 0.962620i \(0.412693\pi\)
\(588\) 0 0
\(589\) −17.8885 −0.737085
\(590\) 14.4721 0.595808
\(591\) 0 0
\(592\) −0.472136 −0.0194047
\(593\) −32.2918 −1.32607 −0.663033 0.748591i \(-0.730731\pi\)
−0.663033 + 0.748591i \(0.730731\pi\)
\(594\) 0 0
\(595\) −2.47214 −0.101348
\(596\) 42.0000 1.72039
\(597\) 0 0
\(598\) −17.8885 −0.731517
\(599\) −3.41641 −0.139591 −0.0697953 0.997561i \(-0.522235\pi\)
−0.0697953 + 0.997561i \(0.522235\pi\)
\(600\) 0 0
\(601\) −3.12461 −0.127456 −0.0637278 0.997967i \(-0.520299\pi\)
−0.0637278 + 0.997967i \(0.520299\pi\)
\(602\) −17.8885 −0.729083
\(603\) 0 0
\(604\) 26.8328 1.09181
\(605\) 0 0
\(606\) 0 0
\(607\) 4.94427 0.200682 0.100341 0.994953i \(-0.468007\pi\)
0.100341 + 0.994953i \(0.468007\pi\)
\(608\) 16.5836 0.672553
\(609\) 0 0
\(610\) −12.3607 −0.500469
\(611\) −8.94427 −0.361847
\(612\) 0 0
\(613\) 47.3050 1.91063 0.955315 0.295591i \(-0.0955166\pi\)
0.955315 + 0.295591i \(0.0955166\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −33.4164 −1.34529 −0.672647 0.739964i \(-0.734843\pi\)
−0.672647 + 0.739964i \(0.734843\pi\)
\(618\) 0 0
\(619\) −29.1246 −1.17062 −0.585308 0.810811i \(-0.699027\pi\)
−0.585308 + 0.810811i \(0.699027\pi\)
\(620\) −43.4164 −1.74364
\(621\) 0 0
\(622\) −48.5410 −1.94632
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 6.58359 0.263133
\(627\) 0 0
\(628\) −20.8328 −0.831320
\(629\) 0.583592 0.0232693
\(630\) 0 0
\(631\) −24.0000 −0.955425 −0.477712 0.878516i \(-0.658534\pi\)
−0.477712 + 0.878516i \(0.658534\pi\)
\(632\) 20.0000 0.795557
\(633\) 0 0
\(634\) 31.3050 1.24328
\(635\) −41.8885 −1.66230
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) 0 0
\(639\) 0 0
\(640\) 31.3050 1.23744
\(641\) 24.4721 0.966591 0.483296 0.875457i \(-0.339440\pi\)
0.483296 + 0.875457i \(0.339440\pi\)
\(642\) 0 0
\(643\) −29.1246 −1.14856 −0.574281 0.818658i \(-0.694718\pi\)
−0.574281 + 0.818658i \(0.694718\pi\)
\(644\) −19.4164 −0.765114
\(645\) 0 0
\(646\) −6.83282 −0.268834
\(647\) 22.0689 0.867617 0.433809 0.901005i \(-0.357169\pi\)
0.433809 + 0.901005i \(0.357169\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.76393 0.108410
\(651\) 0 0
\(652\) 70.2492 2.75117
\(653\) −42.9443 −1.68054 −0.840270 0.542169i \(-0.817603\pi\)
−0.840270 + 0.542169i \(0.817603\pi\)
\(654\) 0 0
\(655\) −27.7771 −1.08534
\(656\) 6.76393 0.264087
\(657\) 0 0
\(658\) −16.1803 −0.630775
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) 12.8328 0.499139 0.249569 0.968357i \(-0.419711\pi\)
0.249569 + 0.968357i \(0.419711\pi\)
\(662\) −48.9443 −1.90227
\(663\) 0 0
\(664\) −34.4721 −1.33778
\(665\) −4.94427 −0.191731
\(666\) 0 0
\(667\) −3.05573 −0.118318
\(668\) −38.8328 −1.50249
\(669\) 0 0
\(670\) −24.7214 −0.955069
\(671\) 0 0
\(672\) 0 0
\(673\) −5.41641 −0.208787 −0.104394 0.994536i \(-0.533290\pi\)
−0.104394 + 0.994536i \(0.533290\pi\)
\(674\) −45.7771 −1.76327
\(675\) 0 0
\(676\) −34.4164 −1.32371
\(677\) 3.70820 0.142518 0.0712589 0.997458i \(-0.477298\pi\)
0.0712589 + 0.997458i \(0.477298\pi\)
\(678\) 0 0
\(679\) 9.41641 0.361369
\(680\) −5.52786 −0.211984
\(681\) 0 0
\(682\) 0 0
\(683\) −29.8885 −1.14365 −0.571827 0.820374i \(-0.693765\pi\)
−0.571827 + 0.820374i \(0.693765\pi\)
\(684\) 0 0
\(685\) −15.0557 −0.575250
\(686\) 2.23607 0.0853735
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −10.4721 −0.398957
\(690\) 0 0
\(691\) −48.5410 −1.84659 −0.923294 0.384095i \(-0.874514\pi\)
−0.923294 + 0.384095i \(0.874514\pi\)
\(692\) −51.7082 −1.96565
\(693\) 0 0
\(694\) −6.83282 −0.259370
\(695\) 20.9443 0.794462
\(696\) 0 0
\(697\) −8.36068 −0.316683
\(698\) −6.18034 −0.233929
\(699\) 0 0
\(700\) 3.00000 0.113389
\(701\) −24.4721 −0.924300 −0.462150 0.886802i \(-0.652922\pi\)
−0.462150 + 0.886802i \(0.652922\pi\)
\(702\) 0 0
\(703\) 1.16718 0.0440212
\(704\) 0 0
\(705\) 0 0
\(706\) −35.5279 −1.33711
\(707\) 9.23607 0.347358
\(708\) 0 0
\(709\) 2.94427 0.110574 0.0552872 0.998470i \(-0.482393\pi\)
0.0552872 + 0.998470i \(0.482393\pi\)
\(710\) −6.83282 −0.256431
\(711\) 0 0
\(712\) 4.47214 0.167600
\(713\) −46.8328 −1.75390
\(714\) 0 0
\(715\) 0 0
\(716\) −26.8328 −1.00279
\(717\) 0 0
\(718\) 15.7771 0.588796
\(719\) −33.4853 −1.24879 −0.624395 0.781108i \(-0.714655\pi\)
−0.624395 + 0.781108i \(0.714655\pi\)
\(720\) 0 0
\(721\) 5.70820 0.212585
\(722\) 28.8197 1.07256
\(723\) 0 0
\(724\) 4.24922 0.157921
\(725\) 0.472136 0.0175347
\(726\) 0 0
\(727\) −51.0132 −1.89197 −0.945987 0.324206i \(-0.894903\pi\)
−0.945987 + 0.324206i \(0.894903\pi\)
\(728\) 2.76393 0.102438
\(729\) 0 0
\(730\) −23.4164 −0.866680
\(731\) −9.88854 −0.365741
\(732\) 0 0
\(733\) −13.2361 −0.488885 −0.244443 0.969664i \(-0.578605\pi\)
−0.244443 + 0.969664i \(0.578605\pi\)
\(734\) −38.2918 −1.41338
\(735\) 0 0
\(736\) 43.4164 1.60035
\(737\) 0 0
\(738\) 0 0
\(739\) −7.05573 −0.259549 −0.129775 0.991544i \(-0.541425\pi\)
−0.129775 + 0.991544i \(0.541425\pi\)
\(740\) 2.83282 0.104136
\(741\) 0 0
\(742\) −18.9443 −0.695466
\(743\) −33.8885 −1.24325 −0.621625 0.783315i \(-0.713527\pi\)
−0.621625 + 0.783315i \(0.713527\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) 13.4164 0.491210
\(747\) 0 0
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 38.4721 1.40387 0.701934 0.712242i \(-0.252320\pi\)
0.701934 + 0.712242i \(0.252320\pi\)
\(752\) 7.23607 0.263872
\(753\) 0 0
\(754\) 1.30495 0.0475235
\(755\) 17.8885 0.651031
\(756\) 0 0
\(757\) −19.8885 −0.722861 −0.361431 0.932399i \(-0.617712\pi\)
−0.361431 + 0.932399i \(0.617712\pi\)
\(758\) 56.5836 2.05521
\(759\) 0 0
\(760\) −11.0557 −0.401033
\(761\) 17.5967 0.637882 0.318941 0.947775i \(-0.396673\pi\)
0.318941 + 0.947775i \(0.396673\pi\)
\(762\) 0 0
\(763\) 4.47214 0.161902
\(764\) 62.8328 2.27321
\(765\) 0 0
\(766\) 59.5967 2.15332
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) −31.7082 −1.14343 −0.571714 0.820453i \(-0.693721\pi\)
−0.571714 + 0.820453i \(0.693721\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 71.6656 2.57930
\(773\) −6.36068 −0.228778 −0.114389 0.993436i \(-0.536491\pi\)
−0.114389 + 0.993436i \(0.536491\pi\)
\(774\) 0 0
\(775\) 7.23607 0.259927
\(776\) 21.0557 0.755857
\(777\) 0 0
\(778\) −44.4721 −1.59440
\(779\) −16.7214 −0.599105
\(780\) 0 0
\(781\) 0 0
\(782\) −17.8885 −0.639693
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −13.8885 −0.495703
\(786\) 0 0
\(787\) 16.5836 0.591141 0.295571 0.955321i \(-0.404490\pi\)
0.295571 + 0.955321i \(0.404490\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 40.0000 1.42314
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 3.41641 0.121320
\(794\) 0.249224 0.00884461
\(795\) 0 0
\(796\) 60.5410 2.14582
\(797\) −2.94427 −0.104291 −0.0521457 0.998639i \(-0.516606\pi\)
−0.0521457 + 0.998639i \(0.516606\pi\)
\(798\) 0 0
\(799\) −8.94427 −0.316426
\(800\) −6.70820 −0.237171
\(801\) 0 0
\(802\) 11.3050 0.399192
\(803\) 0 0
\(804\) 0 0
\(805\) −12.9443 −0.456226
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 20.6525 0.726552
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 0 0
\(811\) −18.8328 −0.661310 −0.330655 0.943752i \(-0.607270\pi\)
−0.330655 + 0.943752i \(0.607270\pi\)
\(812\) 1.41641 0.0497062
\(813\) 0 0
\(814\) 0 0
\(815\) 46.8328 1.64048
\(816\) 0 0
\(817\) −19.7771 −0.691913
\(818\) −69.5967 −2.43339
\(819\) 0 0
\(820\) −40.5836 −1.41724
\(821\) −8.83282 −0.308267 −0.154134 0.988050i \(-0.549259\pi\)
−0.154134 + 0.988050i \(0.549259\pi\)
\(822\) 0 0
\(823\) −49.8885 −1.73901 −0.869503 0.493928i \(-0.835561\pi\)
−0.869503 + 0.493928i \(0.835561\pi\)
\(824\) 12.7639 0.444653
\(825\) 0 0
\(826\) −7.23607 −0.251775
\(827\) 4.94427 0.171929 0.0859646 0.996298i \(-0.472603\pi\)
0.0859646 + 0.996298i \(0.472603\pi\)
\(828\) 0 0
\(829\) −16.8328 −0.584628 −0.292314 0.956322i \(-0.594425\pi\)
−0.292314 + 0.956322i \(0.594425\pi\)
\(830\) −68.9443 −2.39309
\(831\) 0 0
\(832\) −16.0689 −0.557088
\(833\) 1.23607 0.0428272
\(834\) 0 0
\(835\) −25.8885 −0.895910
\(836\) 0 0
\(837\) 0 0
\(838\) −14.8754 −0.513862
\(839\) 14.0689 0.485712 0.242856 0.970062i \(-0.421916\pi\)
0.242856 + 0.970062i \(0.421916\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) 50.0000 1.72311
\(843\) 0 0
\(844\) −65.6656 −2.26030
\(845\) −22.9443 −0.789307
\(846\) 0 0
\(847\) 0 0
\(848\) 8.47214 0.290934
\(849\) 0 0
\(850\) 2.76393 0.0948021
\(851\) 3.05573 0.104749
\(852\) 0 0
\(853\) −0.652476 −0.0223403 −0.0111702 0.999938i \(-0.503556\pi\)
−0.0111702 + 0.999938i \(0.503556\pi\)
\(854\) 6.18034 0.211487
\(855\) 0 0
\(856\) 8.94427 0.305709
\(857\) 10.7639 0.367689 0.183844 0.982955i \(-0.441146\pi\)
0.183844 + 0.982955i \(0.441146\pi\)
\(858\) 0 0
\(859\) 40.5410 1.38324 0.691621 0.722261i \(-0.256897\pi\)
0.691621 + 0.722261i \(0.256897\pi\)
\(860\) −48.0000 −1.63679
\(861\) 0 0
\(862\) −26.8328 −0.913929
\(863\) 20.9443 0.712951 0.356476 0.934305i \(-0.383978\pi\)
0.356476 + 0.934305i \(0.383978\pi\)
\(864\) 0 0
\(865\) −34.4721 −1.17209
\(866\) −1.05573 −0.0358751
\(867\) 0 0
\(868\) 21.7082 0.736824
\(869\) 0 0
\(870\) 0 0
\(871\) 6.83282 0.231521
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −35.7771 −1.21018
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −41.4164 −1.39853 −0.699266 0.714861i \(-0.746490\pi\)
−0.699266 + 0.714861i \(0.746490\pi\)
\(878\) 3.41641 0.115298
\(879\) 0 0
\(880\) 0 0
\(881\) −29.4164 −0.991064 −0.495532 0.868590i \(-0.665027\pi\)
−0.495532 + 0.868590i \(0.665027\pi\)
\(882\) 0 0
\(883\) 8.94427 0.300999 0.150499 0.988610i \(-0.451912\pi\)
0.150499 + 0.988610i \(0.451912\pi\)
\(884\) 4.58359 0.154163
\(885\) 0 0
\(886\) −15.7771 −0.530042
\(887\) 40.3607 1.35518 0.677589 0.735440i \(-0.263025\pi\)
0.677589 + 0.735440i \(0.263025\pi\)
\(888\) 0 0
\(889\) 20.9443 0.702448
\(890\) 8.94427 0.299813
\(891\) 0 0
\(892\) 36.5410 1.22348
\(893\) −17.8885 −0.598617
\(894\) 0 0
\(895\) −17.8885 −0.597948
\(896\) −15.6525 −0.522913
\(897\) 0 0
\(898\) −43.6656 −1.45714
\(899\) 3.41641 0.113944
\(900\) 0 0
\(901\) −10.4721 −0.348877
\(902\) 0 0
\(903\) 0 0
\(904\) 4.47214 0.148741
\(905\) 2.83282 0.0941660
\(906\) 0 0
\(907\) 13.5279 0.449185 0.224593 0.974453i \(-0.427895\pi\)
0.224593 + 0.974453i \(0.427895\pi\)
\(908\) −89.6656 −2.97566
\(909\) 0 0
\(910\) 5.52786 0.183247
\(911\) −33.5279 −1.11083 −0.555414 0.831574i \(-0.687440\pi\)
−0.555414 + 0.831574i \(0.687440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 55.5279 1.83670
\(915\) 0 0
\(916\) 13.4164 0.443291
\(917\) 13.8885 0.458640
\(918\) 0 0
\(919\) −6.11146 −0.201598 −0.100799 0.994907i \(-0.532140\pi\)
−0.100799 + 0.994907i \(0.532140\pi\)
\(920\) −28.9443 −0.954264
\(921\) 0 0
\(922\) −22.7639 −0.749690
\(923\) 1.88854 0.0621622
\(924\) 0 0
\(925\) −0.472136 −0.0155237
\(926\) 32.3607 1.06344
\(927\) 0 0
\(928\) −3.16718 −0.103968
\(929\) −28.2492 −0.926827 −0.463413 0.886142i \(-0.653376\pi\)
−0.463413 + 0.886142i \(0.653376\pi\)
\(930\) 0 0
\(931\) 2.47214 0.0810210
\(932\) −52.2492 −1.71148
\(933\) 0 0
\(934\) −76.1803 −2.49270
\(935\) 0 0
\(936\) 0 0
\(937\) −20.6525 −0.674687 −0.337343 0.941382i \(-0.609528\pi\)
−0.337343 + 0.941382i \(0.609528\pi\)
\(938\) 12.3607 0.403591
\(939\) 0 0
\(940\) −43.4164 −1.41609
\(941\) −41.5967 −1.35602 −0.678008 0.735055i \(-0.737156\pi\)
−0.678008 + 0.735055i \(0.737156\pi\)
\(942\) 0 0
\(943\) −43.7771 −1.42558
\(944\) 3.23607 0.105325
\(945\) 0 0
\(946\) 0 0
\(947\) 58.8328 1.91181 0.955905 0.293677i \(-0.0948789\pi\)
0.955905 + 0.293677i \(0.0948789\pi\)
\(948\) 0 0
\(949\) 6.47214 0.210094
\(950\) 5.52786 0.179348
\(951\) 0 0
\(952\) 2.76393 0.0895796
\(953\) 5.05573 0.163771 0.0818855 0.996642i \(-0.473906\pi\)
0.0818855 + 0.996642i \(0.473906\pi\)
\(954\) 0 0
\(955\) 41.8885 1.35548
\(956\) 77.6656 2.51189
\(957\) 0 0
\(958\) −50.2492 −1.62348
\(959\) 7.52786 0.243087
\(960\) 0 0
\(961\) 21.3607 0.689054
\(962\) −1.30495 −0.0420733
\(963\) 0 0
\(964\) −81.3738 −2.62087
\(965\) 47.7771 1.53800
\(966\) 0 0
\(967\) −21.8885 −0.703888 −0.351944 0.936021i \(-0.614479\pi\)
−0.351944 + 0.936021i \(0.614479\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 42.1115 1.35212
\(971\) 29.1246 0.934653 0.467327 0.884085i \(-0.345217\pi\)
0.467327 + 0.884085i \(0.345217\pi\)
\(972\) 0 0
\(973\) −10.4721 −0.335721
\(974\) −18.6950 −0.599028
\(975\) 0 0
\(976\) −2.76393 −0.0884713
\(977\) −5.05573 −0.161747 −0.0808735 0.996724i \(-0.525771\pi\)
−0.0808735 + 0.996724i \(0.525771\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 0 0
\(983\) −44.1803 −1.40913 −0.704567 0.709637i \(-0.748859\pi\)
−0.704567 + 0.709637i \(0.748859\pi\)
\(984\) 0 0
\(985\) −4.00000 −0.127451
\(986\) 1.30495 0.0415581
\(987\) 0 0
\(988\) 9.16718 0.291647
\(989\) −51.7771 −1.64642
\(990\) 0 0
\(991\) −26.2492 −0.833834 −0.416917 0.908945i \(-0.636889\pi\)
−0.416917 + 0.908945i \(0.636889\pi\)
\(992\) −48.5410 −1.54118
\(993\) 0 0
\(994\) 3.41641 0.108362
\(995\) 40.3607 1.27952
\(996\) 0 0
\(997\) −32.6525 −1.03411 −0.517057 0.855951i \(-0.672973\pi\)
−0.517057 + 0.855951i \(0.672973\pi\)
\(998\) −23.4164 −0.741233
\(999\) 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 7623.2.a.bl.1.1 2
3.2 odd 2 847.2.a.f.1.2 2
11.10 odd 2 693.2.a.h.1.2 2
21.20 even 2 5929.2.a.m.1.2 2
33.2 even 10 847.2.f.n.323.1 4
33.5 odd 10 847.2.f.b.729.1 4
33.8 even 10 847.2.f.a.372.1 4
33.14 odd 10 847.2.f.m.372.1 4
33.17 even 10 847.2.f.n.729.1 4
33.20 odd 10 847.2.f.b.323.1 4
33.26 odd 10 847.2.f.m.148.1 4
33.29 even 10 847.2.f.a.148.1 4
33.32 even 2 77.2.a.d.1.1 2
77.76 even 2 4851.2.a.y.1.2 2
132.131 odd 2 1232.2.a.m.1.1 2
165.32 odd 4 1925.2.b.h.1849.1 4
165.98 odd 4 1925.2.b.h.1849.4 4
165.164 even 2 1925.2.a.r.1.2 2
231.32 even 6 539.2.e.i.177.2 4
231.65 even 6 539.2.e.i.67.2 4
231.131 odd 6 539.2.e.j.67.2 4
231.164 odd 6 539.2.e.j.177.2 4
231.230 odd 2 539.2.a.f.1.1 2
264.131 odd 2 4928.2.a.bv.1.2 2
264.197 even 2 4928.2.a.bm.1.1 2
924.923 even 2 8624.2.a.ce.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.d.1.1 2 33.32 even 2
539.2.a.f.1.1 2 231.230 odd 2
539.2.e.i.67.2 4 231.65 even 6
539.2.e.i.177.2 4 231.32 even 6
539.2.e.j.67.2 4 231.131 odd 6
539.2.e.j.177.2 4 231.164 odd 6
693.2.a.h.1.2 2 11.10 odd 2
847.2.a.f.1.2 2 3.2 odd 2
847.2.f.a.148.1 4 33.29 even 10
847.2.f.a.372.1 4 33.8 even 10
847.2.f.b.323.1 4 33.20 odd 10
847.2.f.b.729.1 4 33.5 odd 10
847.2.f.m.148.1 4 33.26 odd 10
847.2.f.m.372.1 4 33.14 odd 10
847.2.f.n.323.1 4 33.2 even 10
847.2.f.n.729.1 4 33.17 even 10
1232.2.a.m.1.1 2 132.131 odd 2
1925.2.a.r.1.2 2 165.164 even 2
1925.2.b.h.1849.1 4 165.32 odd 4
1925.2.b.h.1849.4 4 165.98 odd 4
4851.2.a.y.1.2 2 77.76 even 2
4928.2.a.bm.1.1 2 264.197 even 2
4928.2.a.bv.1.2 2 264.131 odd 2
5929.2.a.m.1.2 2 21.20 even 2
7623.2.a.bl.1.1 2 1.1 even 1 trivial
8624.2.a.ce.1.2 2 924.923 even 2