Properties

Label 7623.2.a.bl.1.2
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 1.2
Root \(-0.618034\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +4.47214 q^{10} -3.23607 q^{13} -2.23607 q^{14} -1.00000 q^{16} -3.23607 q^{17} -6.47214 q^{19} +6.00000 q^{20} -2.47214 q^{23} -1.00000 q^{25} -7.23607 q^{26} -3.00000 q^{28} +8.47214 q^{29} -2.76393 q^{31} -6.70820 q^{32} -7.23607 q^{34} -2.00000 q^{35} -8.47214 q^{37} -14.4721 q^{38} +4.47214 q^{40} -11.2361 q^{41} -8.00000 q^{43} -5.52786 q^{46} -2.76393 q^{47} +1.00000 q^{49} -2.23607 q^{50} -9.70820 q^{52} +0.472136 q^{53} -2.23607 q^{56} +18.9443 q^{58} +1.23607 q^{59} +7.23607 q^{61} -6.18034 q^{62} -13.0000 q^{64} -6.47214 q^{65} +14.4721 q^{67} -9.70820 q^{68} -4.47214 q^{70} +10.4721 q^{71} +0.763932 q^{73} -18.9443 q^{74} -19.4164 q^{76} +8.94427 q^{79} -2.00000 q^{80} -25.1246 q^{82} -11.4164 q^{83} -6.47214 q^{85} -17.8885 q^{86} -2.00000 q^{89} +3.23607 q^{91} -7.41641 q^{92} -6.18034 q^{94} -12.9443 q^{95} +17.4164 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{4} + 4q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + 6q^{4} + 4q^{5} - 2q^{7} - 2q^{13} - 2q^{16} - 2q^{17} - 4q^{19} + 12q^{20} + 4q^{23} - 2q^{25} - 10q^{26} - 6q^{28} + 8q^{29} - 10q^{31} - 10q^{34} - 4q^{35} - 8q^{37} - 20q^{38} - 18q^{41} - 16q^{43} - 20q^{46} - 10q^{47} + 2q^{49} - 6q^{52} - 8q^{53} + 20q^{58} - 2q^{59} + 10q^{61} + 10q^{62} - 26q^{64} - 4q^{65} + 20q^{67} - 6q^{68} + 12q^{71} + 6q^{73} - 20q^{74} - 12q^{76} - 4q^{80} - 10q^{82} + 4q^{83} - 4q^{85} - 4q^{89} + 2q^{91} + 12q^{92} + 10q^{94} - 8q^{95} + 8q^{97} + O(q^{100}) \)

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 \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) 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
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) −2.23607 −0.597614
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −3.23607 −0.784862 −0.392431 0.919781i \(-0.628366\pi\)
−0.392431 + 0.919781i \(0.628366\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −7.23607 −1.41911
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) −2.76393 −0.496417 −0.248208 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) −7.23607 −1.24098
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) −14.4721 −2.34769
\(39\) 0 0
\(40\) 4.47214 0.707107
\(41\) −11.2361 −1.75478 −0.877390 0.479779i \(-0.840717\pi\)
−0.877390 + 0.479779i \(0.840717\pi\)
\(42\) 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
\(46\) −5.52786 −0.815039
\(47\) −2.76393 −0.403161 −0.201580 0.979472i \(-0.564608\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.23607 −0.316228
\(51\) 0 0
\(52\) −9.70820 −1.34629
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) 18.9443 2.48750
\(59\) 1.23607 0.160922 0.0804612 0.996758i \(-0.474361\pi\)
0.0804612 + 0.996758i \(0.474361\pi\)
\(60\) 0 0
\(61\) 7.23607 0.926484 0.463242 0.886232i \(-0.346686\pi\)
0.463242 + 0.886232i \(0.346686\pi\)
\(62\) −6.18034 −0.784904
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) 14.4721 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) −9.70820 −1.17729
\(69\) 0 0
\(70\) −4.47214 −0.534522
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 0 0
\(73\) 0.763932 0.0894115 0.0447057 0.999000i \(-0.485765\pi\)
0.0447057 + 0.999000i \(0.485765\pi\)
\(74\) −18.9443 −2.20223
\(75\) 0 0
\(76\) −19.4164 −2.22721
\(77\) 0 0
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) −25.1246 −2.77455
\(83\) −11.4164 −1.25311 −0.626557 0.779376i \(-0.715536\pi\)
−0.626557 + 0.779376i \(0.715536\pi\)
\(84\) 0 0
\(85\) −6.47214 −0.702002
\(86\) −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
\(91\) 3.23607 0.339232
\(92\) −7.41641 −0.773214
\(93\) 0 0
\(94\) −6.18034 −0.637453
\(95\) −12.9443 −1.32805
\(96\) 0 0
\(97\) 17.4164 1.76837 0.884184 0.467139i \(-0.154715\pi\)
0.884184 + 0.467139i \(0.154715\pi\)
\(98\) 2.23607 0.225877
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) −4.76393 −0.474029 −0.237014 0.971506i \(-0.576169\pi\)
−0.237014 + 0.971506i \(0.576169\pi\)
\(102\) 0 0
\(103\) 7.70820 0.759512 0.379756 0.925087i \(-0.376008\pi\)
0.379756 + 0.925087i \(0.376008\pi\)
\(104\) −7.23607 −0.709555
\(105\) 0 0
\(106\) 1.05573 0.102541
\(107\) −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 \(-0.431293\pi\)
0.214176 + 0.976795i \(0.431293\pi\)
\(110\) 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
\(115\) −4.94427 −0.461056
\(116\) 25.4164 2.35985
\(117\) 0 0
\(118\) 2.76393 0.254441
\(119\) 3.23607 0.296650
\(120\) 0 0
\(121\) 0 0
\(122\) 16.1803 1.46490
\(123\) 0 0
\(124\) −8.29180 −0.744625
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −3.05573 −0.271152 −0.135576 0.990767i \(-0.543288\pi\)
−0.135576 + 0.990767i \(0.543288\pi\)
\(128\) −15.6525 −1.38350
\(129\) 0 0
\(130\) −14.4721 −1.26929
\(131\) 21.8885 1.91241 0.956205 0.292696i \(-0.0945525\pi\)
0.956205 + 0.292696i \(0.0945525\pi\)
\(132\) 0 0
\(133\) 6.47214 0.561205
\(134\) 32.3607 2.79554
\(135\) 0 0
\(136\) −7.23607 −0.620488
\(137\) −16.4721 −1.40731 −0.703655 0.710542i \(-0.748450\pi\)
−0.703655 + 0.710542i \(0.748450\pi\)
\(138\) 0 0
\(139\) 1.52786 0.129592 0.0647959 0.997899i \(-0.479360\pi\)
0.0647959 + 0.997899i \(0.479360\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) 23.4164 1.96506
\(143\) 0 0
\(144\) 0 0
\(145\) 16.9443 1.40715
\(146\) 1.70820 0.141372
\(147\) 0 0
\(148\) −25.4164 −2.08922
\(149\) 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 \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) −14.4721 −1.17385
\(153\) 0 0
\(154\) 0 0
\(155\) −5.52786 −0.444009
\(156\) 0 0
\(157\) 10.9443 0.873448 0.436724 0.899596i \(-0.356139\pi\)
0.436724 + 0.899596i \(0.356139\pi\)
\(158\) 20.0000 1.59111
\(159\) 0 0
\(160\) −13.4164 −1.06066
\(161\) 2.47214 0.194832
\(162\) 0 0
\(163\) −3.41641 −0.267594 −0.133797 0.991009i \(-0.542717\pi\)
−0.133797 + 0.991009i \(0.542717\pi\)
\(164\) −33.7082 −2.63217
\(165\) 0 0
\(166\) −25.5279 −1.98135
\(167\) 4.94427 0.382599 0.191300 0.981532i \(-0.438730\pi\)
0.191300 + 0.981532i \(0.438730\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) −14.4721 −1.10996
\(171\) 0 0
\(172\) −24.0000 −1.82998
\(173\) −12.7639 −0.970424 −0.485212 0.874397i \(-0.661258\pi\)
−0.485212 + 0.874397i \(0.661258\pi\)
\(174\) 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 \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) −25.4164 −1.88919 −0.944593 0.328243i \(-0.893544\pi\)
−0.944593 + 0.328243i \(0.893544\pi\)
\(182\) 7.23607 0.536373
\(183\) 0 0
\(184\) −5.52786 −0.407520
\(185\) −16.9443 −1.24577
\(186\) 0 0
\(187\) 0 0
\(188\) −8.29180 −0.604741
\(189\) 0 0
\(190\) −28.9443 −2.09984
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) 0 0
\(193\) −11.8885 −0.855756 −0.427878 0.903836i \(-0.640739\pi\)
−0.427878 + 0.903836i \(0.640739\pi\)
\(194\) 38.9443 2.79604
\(195\) 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
\(199\) −2.18034 −0.154560 −0.0772801 0.997009i \(-0.524624\pi\)
−0.0772801 + 0.997009i \(0.524624\pi\)
\(200\) −2.23607 −0.158114
\(201\) 0 0
\(202\) −10.6525 −0.749506
\(203\) −8.47214 −0.594627
\(204\) 0 0
\(205\) −22.4721 −1.56952
\(206\) 17.2361 1.20089
\(207\) 0 0
\(208\) 3.23607 0.224381
\(209\) 0 0
\(210\) 0 0
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) 1.41641 0.0972793
\(213\) 0 0
\(214\) −8.94427 −0.611418
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 2.76393 0.187628
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 0 0
\(221\) 10.4721 0.704432
\(222\) 0 0
\(223\) −10.1803 −0.681726 −0.340863 0.940113i \(-0.610719\pi\)
−0.340863 + 0.940113i \(0.610719\pi\)
\(224\) 6.70820 0.448211
\(225\) 0 0
\(226\) −4.47214 −0.297482
\(227\) 5.88854 0.390836 0.195418 0.980720i \(-0.437394\pi\)
0.195418 + 0.980720i \(0.437394\pi\)
\(228\) 0 0
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) −11.0557 −0.728993
\(231\) 0 0
\(232\) 18.9443 1.24375
\(233\) 9.41641 0.616889 0.308445 0.951242i \(-0.400192\pi\)
0.308445 + 0.951242i \(0.400192\pi\)
\(234\) 0 0
\(235\) −5.52786 −0.360598
\(236\) 3.70820 0.241384
\(237\) 0 0
\(238\) 7.23607 0.469045
\(239\) −9.88854 −0.639637 −0.319818 0.947479i \(-0.603622\pi\)
−0.319818 + 0.947479i \(0.603622\pi\)
\(240\) 0 0
\(241\) 13.1246 0.845431 0.422715 0.906263i \(-0.361077\pi\)
0.422715 + 0.906263i \(0.361077\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 21.7082 1.38973
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 20.9443 1.33265
\(248\) −6.18034 −0.392452
\(249\) 0 0
\(250\) −26.8328 −1.69706
\(251\) −4.29180 −0.270896 −0.135448 0.990784i \(-0.543247\pi\)
−0.135448 + 0.990784i \(0.543247\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −6.83282 −0.428729
\(255\) 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
\(259\) 8.47214 0.526433
\(260\) −19.4164 −1.20415
\(261\) 0 0
\(262\) 48.9443 3.02379
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0.944272 0.0580062
\(266\) 14.4721 0.887344
\(267\) 0 0
\(268\) 43.4164 2.65208
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) 10.4721 0.636137 0.318068 0.948068i \(-0.396966\pi\)
0.318068 + 0.948068i \(0.396966\pi\)
\(272\) 3.23607 0.196215
\(273\) 0 0
\(274\) −36.8328 −2.22515
\(275\) 0 0
\(276\) 0 0
\(277\) 19.8885 1.19499 0.597493 0.801874i \(-0.296163\pi\)
0.597493 + 0.801874i \(0.296163\pi\)
\(278\) 3.41641 0.204903
\(279\) 0 0
\(280\) −4.47214 −0.267261
\(281\) −3.52786 −0.210455 −0.105227 0.994448i \(-0.533557\pi\)
−0.105227 + 0.994448i \(0.533557\pi\)
\(282\) 0 0
\(283\) −29.8885 −1.77669 −0.888345 0.459177i \(-0.848144\pi\)
−0.888345 + 0.459177i \(0.848144\pi\)
\(284\) 31.4164 1.86422
\(285\) 0 0
\(286\) 0 0
\(287\) 11.2361 0.663244
\(288\) 0 0
\(289\) −6.52786 −0.383992
\(290\) 37.8885 2.22489
\(291\) 0 0
\(292\) 2.29180 0.134117
\(293\) −25.1246 −1.46780 −0.733898 0.679260i \(-0.762301\pi\)
−0.733898 + 0.679260i \(0.762301\pi\)
\(294\) 0 0
\(295\) 2.47214 0.143933
\(296\) −18.9443 −1.10111
\(297\) 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
\(304\) 6.47214 0.371202
\(305\) 14.4721 0.828672
\(306\) 0 0
\(307\) 8.94427 0.510477 0.255238 0.966878i \(-0.417846\pi\)
0.255238 + 0.966878i \(0.417846\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.3607 −0.702039
\(311\) 8.29180 0.470185 0.235092 0.971973i \(-0.424461\pi\)
0.235092 + 0.971973i \(0.424461\pi\)
\(312\) 0 0
\(313\) 14.9443 0.844700 0.422350 0.906433i \(-0.361205\pi\)
0.422350 + 0.906433i \(0.361205\pi\)
\(314\) 24.4721 1.38104
\(315\) 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
\(322\) 5.52786 0.308056
\(323\) 20.9443 1.16537
\(324\) 0 0
\(325\) 3.23607 0.179505
\(326\) −7.63932 −0.423103
\(327\) 0 0
\(328\) −25.1246 −1.38727
\(329\) 2.76393 0.152381
\(330\) 0 0
\(331\) −13.8885 −0.763383 −0.381692 0.924290i \(-0.624658\pi\)
−0.381692 + 0.924290i \(0.624658\pi\)
\(332\) −34.2492 −1.87967
\(333\) 0 0
\(334\) 11.0557 0.604943
\(335\) 28.9443 1.58139
\(336\) 0 0
\(337\) 11.5279 0.627963 0.313981 0.949429i \(-0.398337\pi\)
0.313981 + 0.949429i \(0.398337\pi\)
\(338\) −5.65248 −0.307454
\(339\) 0 0
\(340\) −19.4164 −1.05300
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −17.8885 −0.964486
\(345\) 0 0
\(346\) −28.5410 −1.53437
\(347\) 20.9443 1.12435 0.562174 0.827019i \(-0.309965\pi\)
0.562174 + 0.827019i \(0.309965\pi\)
\(348\) 0 0
\(349\) 7.23607 0.387338 0.193669 0.981067i \(-0.437961\pi\)
0.193669 + 0.981067i \(0.437961\pi\)
\(350\) 2.23607 0.119523
\(351\) 0 0
\(352\) 0 0
\(353\) −19.8885 −1.05856 −0.529280 0.848447i \(-0.677538\pi\)
−0.529280 + 0.848447i \(0.677538\pi\)
\(354\) 0 0
\(355\) 20.9443 1.11161
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −24.9443 −1.31651 −0.658254 0.752796i \(-0.728705\pi\)
−0.658254 + 0.752796i \(0.728705\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) −56.8328 −2.98707
\(363\) 0 0
\(364\) 9.70820 0.508848
\(365\) 1.52786 0.0799721
\(366\) 0 0
\(367\) −23.1246 −1.20709 −0.603547 0.797327i \(-0.706247\pi\)
−0.603547 + 0.797327i \(0.706247\pi\)
\(368\) 2.47214 0.128869
\(369\) 0 0
\(370\) −37.8885 −1.96973
\(371\) −0.472136 −0.0245121
\(372\) 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
\(376\) −6.18034 −0.318727
\(377\) −27.4164 −1.41202
\(378\) 0 0
\(379\) 37.3050 1.91623 0.958113 0.286389i \(-0.0924551\pi\)
0.958113 + 0.286389i \(0.0924551\pi\)
\(380\) −38.8328 −1.99208
\(381\) 0 0
\(382\) 6.83282 0.349597
\(383\) 4.65248 0.237730 0.118865 0.992910i \(-0.462074\pi\)
0.118865 + 0.992910i \(0.462074\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.5836 −1.35307
\(387\) 0 0
\(388\) 52.2492 2.65255
\(389\) −15.8885 −0.805581 −0.402791 0.915292i \(-0.631960\pi\)
−0.402791 + 0.915292i \(0.631960\pi\)
\(390\) 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
\(397\) −35.8885 −1.80119 −0.900597 0.434655i \(-0.856870\pi\)
−0.900597 + 0.434655i \(0.856870\pi\)
\(398\) −4.87539 −0.244381
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.9443 −1.14578 −0.572891 0.819631i \(-0.694178\pi\)
−0.572891 + 0.819631i \(0.694178\pi\)
\(402\) 0 0
\(403\) 8.94427 0.445546
\(404\) −14.2918 −0.711043
\(405\) 0 0
\(406\) −18.9443 −0.940188
\(407\) 0 0
\(408\) 0 0
\(409\) −9.12461 −0.451183 −0.225592 0.974222i \(-0.572431\pi\)
−0.225592 + 0.974222i \(0.572431\pi\)
\(410\) −50.2492 −2.48163
\(411\) 0 0
\(412\) 23.1246 1.13927
\(413\) −1.23607 −0.0608229
\(414\) 0 0
\(415\) −22.8328 −1.12082
\(416\) 21.7082 1.06433
\(417\) 0 0
\(418\) 0 0
\(419\) −24.6525 −1.20435 −0.602176 0.798363i \(-0.705700\pi\)
−0.602176 + 0.798363i \(0.705700\pi\)
\(420\) 0 0
\(421\) 22.3607 1.08979 0.544896 0.838503i \(-0.316569\pi\)
0.544896 + 0.838503i \(0.316569\pi\)
\(422\) 31.0557 1.51177
\(423\) 0 0
\(424\) 1.05573 0.0512707
\(425\) 3.23607 0.156972
\(426\) 0 0
\(427\) −7.23607 −0.350178
\(428\) −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
\(433\) −8.47214 −0.407145 −0.203572 0.979060i \(-0.565255\pi\)
−0.203572 + 0.979060i \(0.565255\pi\)
\(434\) 6.18034 0.296666
\(435\) 0 0
\(436\) 13.4164 0.642529
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) −10.4721 −0.499808 −0.249904 0.968271i \(-0.580399\pi\)
−0.249904 + 0.968271i \(0.580399\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 23.4164 1.11380
\(443\) 24.9443 1.18514 0.592569 0.805520i \(-0.298114\pi\)
0.592569 + 0.805520i \(0.298114\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) −22.7639 −1.07790
\(447\) 0 0
\(448\) 13.0000 0.614192
\(449\) 28.4721 1.34368 0.671842 0.740695i \(-0.265504\pi\)
0.671842 + 0.740695i \(0.265504\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 13.1672 0.617967
\(455\) 6.47214 0.303418
\(456\) 0 0
\(457\) 28.8328 1.34874 0.674371 0.738393i \(-0.264415\pi\)
0.674371 + 0.738393i \(0.264415\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) −14.8328 −0.691584
\(461\) −12.1803 −0.567295 −0.283647 0.958929i \(-0.591545\pi\)
−0.283647 + 0.958929i \(0.591545\pi\)
\(462\) 0 0
\(463\) −5.52786 −0.256902 −0.128451 0.991716i \(-0.541000\pi\)
−0.128451 + 0.991716i \(0.541000\pi\)
\(464\) −8.47214 −0.393309
\(465\) 0 0
\(466\) 21.0557 0.975388
\(467\) −24.0689 −1.11378 −0.556888 0.830588i \(-0.688005\pi\)
−0.556888 + 0.830588i \(0.688005\pi\)
\(468\) 0 0
\(469\) −14.4721 −0.668261
\(470\) −12.3607 −0.570156
\(471\) 0 0
\(472\) 2.76393 0.127220
\(473\) 0 0
\(474\) 0 0
\(475\) 6.47214 0.296962
\(476\) 9.70820 0.444975
\(477\) 0 0
\(478\) −22.1115 −1.01135
\(479\) 13.5279 0.618104 0.309052 0.951045i \(-0.399988\pi\)
0.309052 + 0.951045i \(0.399988\pi\)
\(480\) 0 0
\(481\) 27.4164 1.25008
\(482\) 29.3475 1.33674
\(483\) 0 0
\(484\) 0 0
\(485\) 34.8328 1.58168
\(486\) 0 0
\(487\) −36.3607 −1.64766 −0.823830 0.566837i \(-0.808167\pi\)
−0.823830 + 0.566837i \(0.808167\pi\)
\(488\) 16.1803 0.732450
\(489\) 0 0
\(490\) 4.47214 0.202031
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −27.4164 −1.23477
\(494\) 46.8328 2.10711
\(495\) 0 0
\(496\) 2.76393 0.124104
\(497\) −10.4721 −0.469739
\(498\) 0 0
\(499\) 1.52786 0.0683966 0.0341983 0.999415i \(-0.489112\pi\)
0.0341983 + 0.999415i \(0.489112\pi\)
\(500\) −36.0000 −1.60997
\(501\) 0 0
\(502\) −9.59675 −0.428324
\(503\) 23.4164 1.04409 0.522043 0.852919i \(-0.325170\pi\)
0.522043 + 0.852919i \(0.325170\pi\)
\(504\) 0 0
\(505\) −9.52786 −0.423984
\(506\) 0 0
\(507\) 0 0
\(508\) −9.16718 −0.406728
\(509\) −40.4721 −1.79390 −0.896948 0.442136i \(-0.854221\pi\)
−0.896948 + 0.442136i \(0.854221\pi\)
\(510\) 0 0
\(511\) −0.763932 −0.0337944
\(512\) 11.1803 0.494106
\(513\) 0 0
\(514\) 13.4164 0.591772
\(515\) 15.4164 0.679328
\(516\) 0 0
\(517\) 0 0
\(518\) 18.9443 0.832364
\(519\) 0 0
\(520\) −14.4721 −0.634645
\(521\) −30.3607 −1.33013 −0.665063 0.746787i \(-0.731595\pi\)
−0.665063 + 0.746787i \(0.731595\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 65.6656 2.86862
\(525\) 0 0
\(526\) 0 0
\(527\) 8.94427 0.389619
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 2.11146 0.0917158
\(531\) 0 0
\(532\) 19.4164 0.841808
\(533\) 36.3607 1.57496
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 32.3607 1.39777
\(537\) 0 0
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) 0 0
\(541\) −20.8328 −0.895673 −0.447836 0.894116i \(-0.647805\pi\)
−0.447836 + 0.894116i \(0.647805\pi\)
\(542\) 23.4164 1.00582
\(543\) 0 0
\(544\) 21.7082 0.930732
\(545\) 8.94427 0.383131
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −49.4164 −2.11096
\(549\) 0 0
\(550\) 0 0
\(551\) −54.8328 −2.33596
\(552\) 0 0
\(553\) −8.94427 −0.380349
\(554\) 44.4721 1.88944
\(555\) 0 0
\(556\) 4.58359 0.194388
\(557\) −38.9443 −1.65012 −0.825061 0.565044i \(-0.808859\pi\)
−0.825061 + 0.565044i \(0.808859\pi\)
\(558\) 0 0
\(559\) 25.8885 1.09497
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −7.88854 −0.332758
\(563\) 12.5836 0.530335 0.265168 0.964202i \(-0.414573\pi\)
0.265168 + 0.964202i \(0.414573\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) −66.8328 −2.80919
\(567\) 0 0
\(568\) 23.4164 0.982531
\(569\) 7.52786 0.315584 0.157792 0.987472i \(-0.449562\pi\)
0.157792 + 0.987472i \(0.449562\pi\)
\(570\) 0 0
\(571\) 15.0557 0.630063 0.315031 0.949081i \(-0.397985\pi\)
0.315031 + 0.949081i \(0.397985\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 25.1246 1.04868
\(575\) 2.47214 0.103095
\(576\) 0 0
\(577\) −19.5279 −0.812956 −0.406478 0.913661i \(-0.633243\pi\)
−0.406478 + 0.913661i \(0.633243\pi\)
\(578\) −14.5967 −0.607145
\(579\) 0 0
\(580\) 50.8328 2.11072
\(581\) 11.4164 0.473632
\(582\) 0 0
\(583\) 0 0
\(584\) 1.70820 0.0706860
\(585\) 0 0
\(586\) −56.1803 −2.32079
\(587\) −27.1246 −1.11955 −0.559776 0.828644i \(-0.689113\pi\)
−0.559776 + 0.828644i \(0.689113\pi\)
\(588\) 0 0
\(589\) 17.8885 0.737085
\(590\) 5.52786 0.227579
\(591\) 0 0
\(592\) 8.47214 0.348203
\(593\) −45.7082 −1.87701 −0.938505 0.345264i \(-0.887789\pi\)
−0.938505 + 0.345264i \(0.887789\pi\)
\(594\) 0 0
\(595\) 6.47214 0.265332
\(596\) 42.0000 1.72039
\(597\) 0 0
\(598\) 17.8885 0.731517
\(599\) 23.4164 0.956768 0.478384 0.878151i \(-0.341223\pi\)
0.478384 + 0.878151i \(0.341223\pi\)
\(600\) 0 0
\(601\) 37.1246 1.51434 0.757172 0.653215i \(-0.226580\pi\)
0.757172 + 0.653215i \(0.226580\pi\)
\(602\) 17.8885 0.729083
\(603\) 0 0
\(604\) −26.8328 −1.09181
\(605\) 0 0
\(606\) 0 0
\(607\) −12.9443 −0.525392 −0.262696 0.964879i \(-0.584612\pi\)
−0.262696 + 0.964879i \(0.584612\pi\)
\(608\) 43.4164 1.76077
\(609\) 0 0
\(610\) 32.3607 1.31025
\(611\) 8.94427 0.361847
\(612\) 0 0
\(613\) −15.3050 −0.618161 −0.309081 0.951036i \(-0.600021\pi\)
−0.309081 + 0.951036i \(0.600021\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) −6.58359 −0.265045 −0.132523 0.991180i \(-0.542308\pi\)
−0.132523 + 0.991180i \(0.542308\pi\)
\(618\) 0 0
\(619\) 11.1246 0.447136 0.223568 0.974688i \(-0.428230\pi\)
0.223568 + 0.974688i \(0.428230\pi\)
\(620\) −16.5836 −0.666013
\(621\) 0 0
\(622\) 18.5410 0.743427
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 33.4164 1.33559
\(627\) 0 0
\(628\) 32.8328 1.31017
\(629\) 27.4164 1.09316
\(630\) 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
\(635\) −6.11146 −0.242526
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 0 0
\(640\) −31.3050 −1.23744
\(641\) 15.5279 0.613314 0.306657 0.951820i \(-0.400790\pi\)
0.306657 + 0.951820i \(0.400790\pi\)
\(642\) 0 0
\(643\) 11.1246 0.438712 0.219356 0.975645i \(-0.429604\pi\)
0.219356 + 0.975645i \(0.429604\pi\)
\(644\) 7.41641 0.292247
\(645\) 0 0
\(646\) 46.8328 1.84261
\(647\) −36.0689 −1.41801 −0.709007 0.705201i \(-0.750857\pi\)
−0.709007 + 0.705201i \(0.750857\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 7.23607 0.283822
\(651\) 0 0
\(652\) −10.2492 −0.401391
\(653\) −25.0557 −0.980506 −0.490253 0.871580i \(-0.663096\pi\)
−0.490253 + 0.871580i \(0.663096\pi\)
\(654\) 0 0
\(655\) 43.7771 1.71051
\(656\) 11.2361 0.438695
\(657\) 0 0
\(658\) 6.18034 0.240935
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −40.8328 −1.58821 −0.794106 0.607779i \(-0.792061\pi\)
−0.794106 + 0.607779i \(0.792061\pi\)
\(662\) −31.0557 −1.20702
\(663\) 0 0
\(664\) −25.5279 −0.990673
\(665\) 12.9443 0.501957
\(666\) 0 0
\(667\) −20.9443 −0.810965
\(668\) 14.8328 0.573899
\(669\) 0 0
\(670\) 64.7214 2.50040
\(671\) 0 0
\(672\) 0 0
\(673\) 21.4164 0.825542 0.412771 0.910835i \(-0.364561\pi\)
0.412771 + 0.910835i \(0.364561\pi\)
\(674\) 25.7771 0.992896
\(675\) 0 0
\(676\) −7.58359 −0.291677
\(677\) −9.70820 −0.373117 −0.186558 0.982444i \(-0.559733\pi\)
−0.186558 + 0.982444i \(0.559733\pi\)
\(678\) 0 0
\(679\) −17.4164 −0.668380
\(680\) −14.4721 −0.554981
\(681\) 0 0
\(682\) 0 0
\(683\) 5.88854 0.225319 0.112659 0.993634i \(-0.464063\pi\)
0.112659 + 0.993634i \(0.464063\pi\)
\(684\) 0 0
\(685\) −32.9443 −1.25874
\(686\) −2.23607 −0.0853735
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −1.52786 −0.0582070
\(690\) 0 0
\(691\) 18.5410 0.705334 0.352667 0.935749i \(-0.385275\pi\)
0.352667 + 0.935749i \(0.385275\pi\)
\(692\) −38.2918 −1.45564
\(693\) 0 0
\(694\) 46.8328 1.77775
\(695\) 3.05573 0.115910
\(696\) 0 0
\(697\) 36.3607 1.37726
\(698\) 16.1803 0.612435
\(699\) 0 0
\(700\) 3.00000 0.113389
\(701\) −15.5279 −0.586479 −0.293240 0.956039i \(-0.594733\pi\)
−0.293240 + 0.956039i \(0.594733\pi\)
\(702\) 0 0
\(703\) 54.8328 2.06806
\(704\) 0 0
\(705\) 0 0
\(706\) −44.4721 −1.67373
\(707\) 4.76393 0.179166
\(708\) 0 0
\(709\) −14.9443 −0.561244 −0.280622 0.959818i \(-0.590541\pi\)
−0.280622 + 0.959818i \(0.590541\pi\)
\(710\) 46.8328 1.75760
\(711\) 0 0
\(712\) −4.47214 −0.167600
\(713\) 6.83282 0.255891
\(714\) 0 0
\(715\) 0 0
\(716\) 26.8328 1.00279
\(717\) 0 0
\(718\) −55.7771 −2.08158
\(719\) 51.4853 1.92008 0.960039 0.279867i \(-0.0902905\pi\)
0.960039 + 0.279867i \(0.0902905\pi\)
\(720\) 0 0
\(721\) −7.70820 −0.287069
\(722\) 51.1803 1.90474
\(723\) 0 0
\(724\) −76.2492 −2.83378
\(725\) −8.47214 −0.314647
\(726\) 0 0
\(727\) 25.0132 0.927687 0.463843 0.885917i \(-0.346470\pi\)
0.463843 + 0.885917i \(0.346470\pi\)
\(728\) 7.23607 0.268187
\(729\) 0 0
\(730\) 3.41641 0.126447
\(731\) 25.8885 0.957522
\(732\) 0 0
\(733\) −8.76393 −0.323703 −0.161852 0.986815i \(-0.551747\pi\)
−0.161852 + 0.986815i \(0.551747\pi\)
\(734\) −51.7082 −1.90858
\(735\) 0 0
\(736\) 16.5836 0.611279
\(737\) 0 0
\(738\) 0 0
\(739\) −24.9443 −0.917590 −0.458795 0.888542i \(-0.651719\pi\)
−0.458795 + 0.888542i \(0.651719\pi\)
\(740\) −50.8328 −1.86865
\(741\) 0 0
\(742\) −1.05573 −0.0387570
\(743\) 1.88854 0.0692840 0.0346420 0.999400i \(-0.488971\pi\)
0.0346420 + 0.999400i \(0.488971\pi\)
\(744\) 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
\(751\) 29.5279 1.07749 0.538744 0.842470i \(-0.318899\pi\)
0.538744 + 0.842470i \(0.318899\pi\)
\(752\) 2.76393 0.100790
\(753\) 0 0
\(754\) −61.3050 −2.23259
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) 15.8885 0.577479 0.288739 0.957408i \(-0.406764\pi\)
0.288739 + 0.957408i \(0.406764\pi\)
\(758\) 83.4164 3.02982
\(759\) 0 0
\(760\) −28.9443 −1.04992
\(761\) −31.5967 −1.14538 −0.572691 0.819772i \(-0.694100\pi\)
−0.572691 + 0.819772i \(0.694100\pi\)
\(762\) 0 0
\(763\) −4.47214 −0.161902
\(764\) 9.16718 0.331657
\(765\) 0 0
\(766\) 10.4033 0.375885
\(767\) −4.00000 −0.144432
\(768\) 0 0
\(769\) −18.2918 −0.659619 −0.329810 0.944047i \(-0.606985\pi\)
−0.329810 + 0.944047i \(0.606985\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35.6656 −1.28363
\(773\) 38.3607 1.37974 0.689869 0.723934i \(-0.257668\pi\)
0.689869 + 0.723934i \(0.257668\pi\)
\(774\) 0 0
\(775\) 2.76393 0.0992834
\(776\) 38.9443 1.39802
\(777\) 0 0
\(778\) −35.5279 −1.27374
\(779\) 72.7214 2.60551
\(780\) 0 0
\(781\) 0 0
\(782\) 17.8885 0.639693
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 21.8885 0.781236
\(786\) 0 0
\(787\) 43.4164 1.54763 0.773814 0.633413i \(-0.218347\pi\)
0.773814 + 0.633413i \(0.218347\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 40.0000 1.42314
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) −23.4164 −0.831541
\(794\) −80.2492 −2.84794
\(795\) 0 0
\(796\) −6.54102 −0.231840
\(797\) 14.9443 0.529353 0.264677 0.964337i \(-0.414735\pi\)
0.264677 + 0.964337i \(0.414735\pi\)
\(798\) 0 0
\(799\) 8.94427 0.316426
\(800\) 6.70820 0.237171
\(801\) 0 0
\(802\) −51.3050 −1.81164
\(803\) 0 0
\(804\) 0 0
\(805\) 4.94427 0.174263
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) −10.6525 −0.374753
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) 34.8328 1.22315 0.611573 0.791188i \(-0.290537\pi\)
0.611573 + 0.791188i \(0.290537\pi\)
\(812\) −25.4164 −0.891941
\(813\) 0 0
\(814\) 0 0
\(815\) −6.83282 −0.239343
\(816\) 0 0
\(817\) 51.7771 1.81145
\(818\) −20.4033 −0.713383
\(819\) 0 0
\(820\) −67.4164 −2.35428
\(821\) 44.8328 1.56468 0.782338 0.622854i \(-0.214027\pi\)
0.782338 + 0.622854i \(0.214027\pi\)
\(822\) 0 0
\(823\) −14.1115 −0.491894 −0.245947 0.969283i \(-0.579099\pi\)
−0.245947 + 0.969283i \(0.579099\pi\)
\(824\) 17.2361 0.600447
\(825\) 0 0
\(826\) −2.76393 −0.0961695
\(827\) −12.9443 −0.450116 −0.225058 0.974345i \(-0.572257\pi\)
−0.225058 + 0.974345i \(0.572257\pi\)
\(828\) 0 0
\(829\) 36.8328 1.27926 0.639628 0.768684i \(-0.279088\pi\)
0.639628 + 0.768684i \(0.279088\pi\)
\(830\) −51.0557 −1.77217
\(831\) 0 0
\(832\) 42.0689 1.45848
\(833\) −3.23607 −0.112123
\(834\) 0 0
\(835\) 9.88854 0.342207
\(836\) 0 0
\(837\) 0 0
\(838\) −55.1246 −1.90425
\(839\) −44.0689 −1.52143 −0.760713 0.649088i \(-0.775151\pi\)
−0.760713 + 0.649088i \(0.775151\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 50.0000 1.72311
\(843\) 0 0
\(844\) 41.6656 1.43419
\(845\) −5.05573 −0.173922
\(846\) 0 0
\(847\) 0 0
\(848\) −0.472136 −0.0162132
\(849\) 0 0
\(850\) 7.23607 0.248195
\(851\) 20.9443 0.717960
\(852\) 0 0
\(853\) 30.6525 1.04952 0.524760 0.851250i \(-0.324155\pi\)
0.524760 + 0.851250i \(0.324155\pi\)
\(854\) −16.1803 −0.553680
\(855\) 0 0
\(856\) −8.94427 −0.305709
\(857\) 15.2361 0.520454 0.260227 0.965547i \(-0.416203\pi\)
0.260227 + 0.965547i \(0.416203\pi\)
\(858\) 0 0
\(859\) −26.5410 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(860\) −48.0000 −1.63679
\(861\) 0 0
\(862\) 26.8328 0.913929
\(863\) 3.05573 0.104018 0.0520091 0.998647i \(-0.483438\pi\)
0.0520091 + 0.998647i \(0.483438\pi\)
\(864\) 0 0
\(865\) −25.5279 −0.867973
\(866\) −18.9443 −0.643753
\(867\) 0 0
\(868\) 8.29180 0.281442
\(869\) 0 0
\(870\) 0 0
\(871\) −46.8328 −1.58687
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 35.7771 1.21018
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −14.5836 −0.492453 −0.246226 0.969212i \(-0.579191\pi\)
−0.246226 + 0.969212i \(0.579191\pi\)
\(878\) −23.4164 −0.790265
\(879\) 0 0
\(880\) 0 0
\(881\) −2.58359 −0.0870434 −0.0435217 0.999052i \(-0.513858\pi\)
−0.0435217 + 0.999052i \(0.513858\pi\)
\(882\) 0 0
\(883\) −8.94427 −0.300999 −0.150499 0.988610i \(-0.548088\pi\)
−0.150499 + 0.988610i \(0.548088\pi\)
\(884\) 31.4164 1.05665
\(885\) 0 0
\(886\) 55.7771 1.87387
\(887\) −4.36068 −0.146417 −0.0732086 0.997317i \(-0.523324\pi\)
−0.0732086 + 0.997317i \(0.523324\pi\)
\(888\) 0 0
\(889\) 3.05573 0.102486
\(890\) −8.94427 −0.299813
\(891\) 0 0
\(892\) −30.5410 −1.02259
\(893\) 17.8885 0.598617
\(894\) 0 0
\(895\) 17.8885 0.597948
\(896\) 15.6525 0.522913
\(897\) 0 0
\(898\) 63.6656 2.12455
\(899\) −23.4164 −0.780981
\(900\) 0 0
\(901\) −1.52786 −0.0509005
\(902\) 0 0
\(903\) 0 0
\(904\) −4.47214 −0.148741
\(905\) −50.8328 −1.68974
\(906\) 0 0
\(907\) 22.4721 0.746175 0.373088 0.927796i \(-0.378299\pi\)
0.373088 + 0.927796i \(0.378299\pi\)
\(908\) 17.6656 0.586255
\(909\) 0 0
\(910\) 14.4721 0.479747
\(911\) −42.4721 −1.40716 −0.703582 0.710614i \(-0.748417\pi\)
−0.703582 + 0.710614i \(0.748417\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 64.4721 2.13255
\(915\) 0 0
\(916\) −13.4164 −0.443291
\(917\) −21.8885 −0.722823
\(918\) 0 0
\(919\) −41.8885 −1.38178 −0.690888 0.722962i \(-0.742780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(920\) −11.0557 −0.364497
\(921\) 0 0
\(922\) −27.2361 −0.896972
\(923\) −33.8885 −1.11546
\(924\) 0 0
\(925\) 8.47214 0.278562
\(926\) −12.3607 −0.406197
\(927\) 0 0
\(928\) −56.8328 −1.86563
\(929\) 52.2492 1.71424 0.857121 0.515116i \(-0.172251\pi\)
0.857121 + 0.515116i \(0.172251\pi\)
\(930\) 0 0
\(931\) −6.47214 −0.212116
\(932\) 28.2492 0.925334
\(933\) 0 0
\(934\) −53.8197 −1.76103
\(935\) 0 0
\(936\) 0 0
\(937\) 10.6525 0.348001 0.174001 0.984746i \(-0.444331\pi\)
0.174001 + 0.984746i \(0.444331\pi\)
\(938\) −32.3607 −1.05661
\(939\) 0 0
\(940\) −16.5836 −0.540897
\(941\) 7.59675 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(942\) 0 0
\(943\) 27.7771 0.904546
\(944\) −1.23607 −0.0402306
\(945\) 0 0
\(946\) 0 0
\(947\) 5.16718 0.167911 0.0839555 0.996470i \(-0.473245\pi\)
0.0839555 + 0.996470i \(0.473245\pi\)
\(948\) 0 0
\(949\) −2.47214 −0.0802489
\(950\) 14.4721 0.469538
\(951\) 0 0
\(952\) 7.23607 0.234522
\(953\) 22.9443 0.743238 0.371619 0.928385i \(-0.378803\pi\)
0.371619 + 0.928385i \(0.378803\pi\)
\(954\) 0 0
\(955\) 6.11146 0.197762
\(956\) −29.6656 −0.959455
\(957\) 0 0
\(958\) 30.2492 0.977308
\(959\) 16.4721 0.531913
\(960\) 0 0
\(961\) −23.3607 −0.753570
\(962\) 61.3050 1.97655
\(963\) 0 0
\(964\) 39.3738 1.26815
\(965\) −23.7771 −0.765412
\(966\) 0 0
\(967\) 13.8885 0.446625 0.223313 0.974747i \(-0.428313\pi\)
0.223313 + 0.974747i \(0.428313\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 77.8885 2.50085
\(971\) −11.1246 −0.357006 −0.178503 0.983939i \(-0.557125\pi\)
−0.178503 + 0.983939i \(0.557125\pi\)
\(972\) 0 0
\(973\) −1.52786 −0.0489811
\(974\) −81.3050 −2.60518
\(975\) 0 0
\(976\) −7.23607 −0.231621
\(977\) −22.9443 −0.734052 −0.367026 0.930211i \(-0.619624\pi\)
−0.367026 + 0.930211i \(0.619624\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 0 0
\(983\) −21.8197 −0.695939 −0.347970 0.937506i \(-0.613129\pi\)
−0.347970 + 0.937506i \(0.613129\pi\)
\(984\) 0 0
\(985\) −4.00000 −0.127451
\(986\) −61.3050 −1.95235
\(987\) 0 0
\(988\) 62.8328 1.99898
\(989\) 19.7771 0.628875
\(990\) 0 0
\(991\) 54.2492 1.72328 0.861642 0.507517i \(-0.169437\pi\)
0.861642 + 0.507517i \(0.169437\pi\)
\(992\) 18.5410 0.588678
\(993\) 0 0
\(994\) −23.4164 −0.742723
\(995\) −4.36068 −0.138243
\(996\) 0 0
\(997\) −1.34752 −0.0426765 −0.0213383 0.999772i \(-0.506793\pi\)
−0.0213383 + 0.999772i \(0.506793\pi\)
\(998\) 3.41641 0.108145
\(999\) 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.1.2 2
3.2 odd 2 847.2.a.f.1.1 2
11.10 odd 2 693.2.a.h.1.1 2
21.20 even 2 5929.2.a.m.1.1 2
33.2 even 10 847.2.f.a.323.1 4
33.5 odd 10 847.2.f.m.729.1 4
33.8 even 10 847.2.f.n.372.1 4
33.14 odd 10 847.2.f.b.372.1 4
33.17 even 10 847.2.f.a.729.1 4
33.20 odd 10 847.2.f.m.323.1 4
33.26 odd 10 847.2.f.b.148.1 4
33.29 even 10 847.2.f.n.148.1 4
33.32 even 2 77.2.a.d.1.2 2
77.76 even 2 4851.2.a.y.1.1 2
132.131 odd 2 1232.2.a.m.1.2 2
165.32 odd 4 1925.2.b.h.1849.3 4
165.98 odd 4 1925.2.b.h.1849.2 4
165.164 even 2 1925.2.a.r.1.1 2
231.32 even 6 539.2.e.i.177.1 4
231.65 even 6 539.2.e.i.67.1 4
231.131 odd 6 539.2.e.j.67.1 4
231.164 odd 6 539.2.e.j.177.1 4
231.230 odd 2 539.2.a.f.1.2 2
264.131 odd 2 4928.2.a.bv.1.1 2
264.197 even 2 4928.2.a.bm.1.2 2
924.923 even 2 8624.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.d.1.2 2 33.32 even 2
539.2.a.f.1.2 2 231.230 odd 2
539.2.e.i.67.1 4 231.65 even 6
539.2.e.i.177.1 4 231.32 even 6
539.2.e.j.67.1 4 231.131 odd 6
539.2.e.j.177.1 4 231.164 odd 6
693.2.a.h.1.1 2 11.10 odd 2
847.2.a.f.1.1 2 3.2 odd 2
847.2.f.a.323.1 4 33.2 even 10
847.2.f.a.729.1 4 33.17 even 10
847.2.f.b.148.1 4 33.26 odd 10
847.2.f.b.372.1 4 33.14 odd 10
847.2.f.m.323.1 4 33.20 odd 10
847.2.f.m.729.1 4 33.5 odd 10
847.2.f.n.148.1 4 33.29 even 10
847.2.f.n.372.1 4 33.8 even 10
1232.2.a.m.1.2 2 132.131 odd 2
1925.2.a.r.1.1 2 165.164 even 2
1925.2.b.h.1849.2 4 165.98 odd 4
1925.2.b.h.1849.3 4 165.32 odd 4
4851.2.a.y.1.1 2 77.76 even 2
4928.2.a.bm.1.2 2 264.197 even 2
4928.2.a.bv.1.1 2 264.131 odd 2
5929.2.a.m.1.1 2 21.20 even 2
7623.2.a.bl.1.2 2 1.1 even 1 trivial
8624.2.a.ce.1.1 2 924.923 even 2