Properties

Label 7623.2.a.cg.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.43091\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} +O(q^{10})\) \(q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} -0.210456 q^{10} +4.43091 q^{13} +0.301143 q^{14} +3.46410 q^{16} +6.80432 q^{17} +1.78954 q^{19} -1.33434 q^{20} +9.20204 q^{23} -4.51160 q^{25} -1.33434 q^{26} +1.90931 q^{28} -7.89932 q^{29} +10.4600 q^{31} -3.39771 q^{32} -2.04907 q^{34} -0.698857 q^{35} +6.16884 q^{37} -0.538909 q^{38} +0.822738 q^{40} -1.33434 q^{41} +6.33434 q^{43} -2.77113 q^{46} -8.20634 q^{47} +1.00000 q^{49} +1.35864 q^{50} -8.45999 q^{52} +2.35452 q^{53} -1.17726 q^{56} +2.37882 q^{58} +4.76524 q^{59} -1.69455 q^{61} -3.14995 q^{62} -5.90501 q^{64} +3.09657 q^{65} -12.5423 q^{67} -12.9916 q^{68} +0.210456 q^{70} -7.58387 q^{71} +6.96681 q^{73} -1.85770 q^{74} -3.41680 q^{76} +8.85751 q^{79} +2.42091 q^{80} +0.401826 q^{82} -5.71775 q^{83} +4.75525 q^{85} -1.90754 q^{86} +2.22935 q^{89} -4.43091 q^{91} -17.5696 q^{92} +2.47128 q^{94} +1.25064 q^{95} -0.171378 q^{97} -0.301143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + 10q^{10} + 10q^{13} + 2q^{14} - 6q^{17} + 18q^{19} + 2q^{23} - 8q^{25} - 4q^{28} - 6q^{29} - 12q^{32} - 2q^{34} - 2q^{35} - 4q^{37} + 8q^{40} + 20q^{43} + 16q^{46} + 6q^{47} + 4q^{49} + 24q^{50} + 8q^{52} + 24q^{58} + 6q^{59} - 10q^{61} - 16q^{64} + 10q^{65} + 4q^{67} - 28q^{68} - 10q^{70} + 6q^{71} + 34q^{73} + 36q^{74} + 36q^{76} + 24q^{79} - 12q^{80} + 28q^{82} - 6q^{83} - 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} - 6q^{94} + 18q^{95} - 10q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.301143 −0.212940 −0.106470 0.994316i \(-0.533955\pi\)
−0.106470 + 0.994316i \(0.533955\pi\)
\(3\) 0 0
\(4\) −1.90931 −0.954656
\(5\) 0.698857 0.312538 0.156269 0.987715i \(-0.450053\pi\)
0.156269 + 0.987715i \(0.450053\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.17726 0.416225
\(9\) 0 0
\(10\) −0.210456 −0.0665520
\(11\) 0 0
\(12\) 0 0
\(13\) 4.43091 1.22891 0.614456 0.788951i \(-0.289375\pi\)
0.614456 + 0.788951i \(0.289375\pi\)
\(14\) 0.301143 0.0804838
\(15\) 0 0
\(16\) 3.46410 0.866025
\(17\) 6.80432 1.65029 0.825145 0.564921i \(-0.191093\pi\)
0.825145 + 0.564921i \(0.191093\pi\)
\(18\) 0 0
\(19\) 1.78954 0.410550 0.205275 0.978704i \(-0.434191\pi\)
0.205275 + 0.978704i \(0.434191\pi\)
\(20\) −1.33434 −0.298367
\(21\) 0 0
\(22\) 0 0
\(23\) 9.20204 1.91876 0.959379 0.282122i \(-0.0910382\pi\)
0.959379 + 0.282122i \(0.0910382\pi\)
\(24\) 0 0
\(25\) −4.51160 −0.902320
\(26\) −1.33434 −0.261685
\(27\) 0 0
\(28\) 1.90931 0.360826
\(29\) −7.89932 −1.46687 −0.733433 0.679762i \(-0.762083\pi\)
−0.733433 + 0.679762i \(0.762083\pi\)
\(30\) 0 0
\(31\) 10.4600 1.87867 0.939335 0.343002i \(-0.111444\pi\)
0.939335 + 0.343002i \(0.111444\pi\)
\(32\) −3.39771 −0.600637
\(33\) 0 0
\(34\) −2.04907 −0.351413
\(35\) −0.698857 −0.118128
\(36\) 0 0
\(37\) 6.16884 1.01415 0.507076 0.861901i \(-0.330726\pi\)
0.507076 + 0.861901i \(0.330726\pi\)
\(38\) −0.538909 −0.0874225
\(39\) 0 0
\(40\) 0.822738 0.130086
\(41\) −1.33434 −0.208388 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(42\) 0 0
\(43\) 6.33434 0.965977 0.482989 0.875627i \(-0.339551\pi\)
0.482989 + 0.875627i \(0.339551\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.77113 −0.408581
\(47\) −8.20634 −1.19702 −0.598509 0.801116i \(-0.704240\pi\)
−0.598509 + 0.801116i \(0.704240\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.35864 0.192140
\(51\) 0 0
\(52\) −8.45999 −1.17319
\(53\) 2.35452 0.323419 0.161709 0.986838i \(-0.448299\pi\)
0.161709 + 0.986838i \(0.448299\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.17726 −0.157318
\(57\) 0 0
\(58\) 2.37882 0.312355
\(59\) 4.76524 0.620382 0.310191 0.950674i \(-0.399607\pi\)
0.310191 + 0.950674i \(0.399607\pi\)
\(60\) 0 0
\(61\) −1.69455 −0.216965 −0.108482 0.994098i \(-0.534599\pi\)
−0.108482 + 0.994098i \(0.534599\pi\)
\(62\) −3.14995 −0.400044
\(63\) 0 0
\(64\) −5.90501 −0.738126
\(65\) 3.09657 0.384082
\(66\) 0 0
\(67\) −12.5423 −1.53228 −0.766140 0.642673i \(-0.777825\pi\)
−0.766140 + 0.642673i \(0.777825\pi\)
\(68\) −12.9916 −1.57546
\(69\) 0 0
\(70\) 0.210456 0.0251543
\(71\) −7.58387 −0.900040 −0.450020 0.893019i \(-0.648583\pi\)
−0.450020 + 0.893019i \(0.648583\pi\)
\(72\) 0 0
\(73\) 6.96681 0.815403 0.407701 0.913115i \(-0.366330\pi\)
0.407701 + 0.913115i \(0.366330\pi\)
\(74\) −1.85770 −0.215954
\(75\) 0 0
\(76\) −3.41680 −0.391934
\(77\) 0 0
\(78\) 0 0
\(79\) 8.85751 0.996548 0.498274 0.867020i \(-0.333967\pi\)
0.498274 + 0.867020i \(0.333967\pi\)
\(80\) 2.42091 0.270666
\(81\) 0 0
\(82\) 0.401826 0.0443743
\(83\) −5.71775 −0.627604 −0.313802 0.949488i \(-0.601603\pi\)
−0.313802 + 0.949488i \(0.601603\pi\)
\(84\) 0 0
\(85\) 4.75525 0.515779
\(86\) −1.90754 −0.205695
\(87\) 0 0
\(88\) 0 0
\(89\) 2.22935 0.236310 0.118155 0.992995i \(-0.462302\pi\)
0.118155 + 0.992995i \(0.462302\pi\)
\(90\) 0 0
\(91\) −4.43091 −0.464485
\(92\) −17.5696 −1.83175
\(93\) 0 0
\(94\) 2.47128 0.254893
\(95\) 1.25064 0.128312
\(96\) 0 0
\(97\) −0.171378 −0.0174008 −0.00870041 0.999962i \(-0.502769\pi\)
−0.00870041 + 0.999962i \(0.502769\pi\)
\(98\) −0.301143 −0.0304200
\(99\) 0 0
\(100\) 8.61405 0.861405
\(101\) −18.0723 −1.79826 −0.899129 0.437683i \(-0.855799\pi\)
−0.899129 + 0.437683i \(0.855799\pi\)
\(102\) 0 0
\(103\) −1.73047 −0.170509 −0.0852543 0.996359i \(-0.527170\pi\)
−0.0852543 + 0.996359i \(0.527170\pi\)
\(104\) 5.21634 0.511504
\(105\) 0 0
\(106\) −0.709048 −0.0688689
\(107\) 6.96982 0.673798 0.336899 0.941541i \(-0.390622\pi\)
0.336899 + 0.941541i \(0.390622\pi\)
\(108\) 0 0
\(109\) 17.6531 1.69086 0.845432 0.534084i \(-0.179343\pi\)
0.845432 + 0.534084i \(0.179343\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.46410 −0.327327
\(113\) −11.7236 −1.10287 −0.551433 0.834219i \(-0.685919\pi\)
−0.551433 + 0.834219i \(0.685919\pi\)
\(114\) 0 0
\(115\) 6.43091 0.599685
\(116\) 15.0823 1.40035
\(117\) 0 0
\(118\) −1.43502 −0.132104
\(119\) −6.80432 −0.623751
\(120\) 0 0
\(121\) 0 0
\(122\) 0.510302 0.0462005
\(123\) 0 0
\(124\) −19.9714 −1.79348
\(125\) −6.64725 −0.594548
\(126\) 0 0
\(127\) 12.1530 1.07840 0.539201 0.842177i \(-0.318726\pi\)
0.539201 + 0.842177i \(0.318726\pi\)
\(128\) 8.57368 0.757813
\(129\) 0 0
\(130\) −0.932511 −0.0817866
\(131\) −18.9111 −1.65227 −0.826135 0.563473i \(-0.809465\pi\)
−0.826135 + 0.563473i \(0.809465\pi\)
\(132\) 0 0
\(133\) −1.78954 −0.155173
\(134\) 3.77701 0.326284
\(135\) 0 0
\(136\) 8.01047 0.686892
\(137\) 8.37930 0.715892 0.357946 0.933742i \(-0.383477\pi\)
0.357946 + 0.933742i \(0.383477\pi\)
\(138\) 0 0
\(139\) 9.84039 0.834651 0.417325 0.908757i \(-0.362968\pi\)
0.417325 + 0.908757i \(0.362968\pi\)
\(140\) 1.33434 0.112772
\(141\) 0 0
\(142\) 2.28383 0.191655
\(143\) 0 0
\(144\) 0 0
\(145\) −5.52049 −0.458452
\(146\) −2.09800 −0.173632
\(147\) 0 0
\(148\) −11.7783 −0.968166
\(149\) −7.13435 −0.584469 −0.292234 0.956347i \(-0.594399\pi\)
−0.292234 + 0.956347i \(0.594399\pi\)
\(150\) 0 0
\(151\) 7.83881 0.637914 0.318957 0.947769i \(-0.396668\pi\)
0.318957 + 0.947769i \(0.396668\pi\)
\(152\) 2.10676 0.170881
\(153\) 0 0
\(154\) 0 0
\(155\) 7.31004 0.587156
\(156\) 0 0
\(157\) −18.9959 −1.51604 −0.758018 0.652233i \(-0.773832\pi\)
−0.758018 + 0.652233i \(0.773832\pi\)
\(158\) −2.66738 −0.212205
\(159\) 0 0
\(160\) −2.37452 −0.187722
\(161\) −9.20204 −0.725222
\(162\) 0 0
\(163\) −12.8779 −1.00867 −0.504337 0.863507i \(-0.668263\pi\)
−0.504337 + 0.863507i \(0.668263\pi\)
\(164\) 2.54767 0.198939
\(165\) 0 0
\(166\) 1.72186 0.133642
\(167\) −0.00588400 −0.000455318 0 −0.000227659 1.00000i \(-0.500072\pi\)
−0.000227659 1.00000i \(0.500072\pi\)
\(168\) 0 0
\(169\) 6.63294 0.510226
\(170\) −1.43201 −0.109830
\(171\) 0 0
\(172\) −12.0942 −0.922176
\(173\) 12.9193 0.982237 0.491118 0.871093i \(-0.336588\pi\)
0.491118 + 0.871093i \(0.336588\pi\)
\(174\) 0 0
\(175\) 4.51160 0.341045
\(176\) 0 0
\(177\) 0 0
\(178\) −0.671352 −0.0503200
\(179\) 1.31133 0.0980137 0.0490069 0.998798i \(-0.484394\pi\)
0.0490069 + 0.998798i \(0.484394\pi\)
\(180\) 0 0
\(181\) 4.43138 0.329382 0.164691 0.986345i \(-0.447337\pi\)
0.164691 + 0.986345i \(0.447337\pi\)
\(182\) 1.33434 0.0989076
\(183\) 0 0
\(184\) 10.8332 0.798635
\(185\) 4.31114 0.316961
\(186\) 0 0
\(187\) 0 0
\(188\) 15.6685 1.14274
\(189\) 0 0
\(190\) −0.376620 −0.0273229
\(191\) −5.34563 −0.386796 −0.193398 0.981120i \(-0.561951\pi\)
−0.193398 + 0.981120i \(0.561951\pi\)
\(192\) 0 0
\(193\) 0.803848 0.0578622 0.0289311 0.999581i \(-0.490790\pi\)
0.0289311 + 0.999581i \(0.490790\pi\)
\(194\) 0.0516093 0.00370533
\(195\) 0 0
\(196\) −1.90931 −0.136379
\(197\) 23.4082 1.66776 0.833882 0.551943i \(-0.186113\pi\)
0.833882 + 0.551943i \(0.186113\pi\)
\(198\) 0 0
\(199\) 23.3118 1.65253 0.826265 0.563281i \(-0.190461\pi\)
0.826265 + 0.563281i \(0.190461\pi\)
\(200\) −5.31133 −0.375568
\(201\) 0 0
\(202\) 5.44234 0.382922
\(203\) 7.89932 0.554423
\(204\) 0 0
\(205\) −0.932511 −0.0651294
\(206\) 0.521120 0.0363082
\(207\) 0 0
\(208\) 15.3491 1.06427
\(209\) 0 0
\(210\) 0 0
\(211\) 10.1455 0.698445 0.349223 0.937040i \(-0.386446\pi\)
0.349223 + 0.937040i \(0.386446\pi\)
\(212\) −4.49552 −0.308754
\(213\) 0 0
\(214\) −2.09891 −0.143479
\(215\) 4.42680 0.301905
\(216\) 0 0
\(217\) −10.4600 −0.710070
\(218\) −5.31612 −0.360053
\(219\) 0 0
\(220\) 0 0
\(221\) 30.1493 2.02806
\(222\) 0 0
\(223\) −6.86052 −0.459414 −0.229707 0.973260i \(-0.573777\pi\)
−0.229707 + 0.973260i \(0.573777\pi\)
\(224\) 3.39771 0.227019
\(225\) 0 0
\(226\) 3.53049 0.234845
\(227\) −9.65136 −0.640583 −0.320292 0.947319i \(-0.603781\pi\)
−0.320292 + 0.947319i \(0.603781\pi\)
\(228\) 0 0
\(229\) 2.37452 0.156912 0.0784562 0.996918i \(-0.475001\pi\)
0.0784562 + 0.996918i \(0.475001\pi\)
\(230\) −1.93662 −0.127697
\(231\) 0 0
\(232\) −9.29957 −0.610546
\(233\) −16.2643 −1.06551 −0.532755 0.846269i \(-0.678843\pi\)
−0.532755 + 0.846269i \(0.678843\pi\)
\(234\) 0 0
\(235\) −5.73506 −0.374114
\(236\) −9.09834 −0.592252
\(237\) 0 0
\(238\) 2.04907 0.132822
\(239\) 17.2386 1.11507 0.557535 0.830153i \(-0.311747\pi\)
0.557535 + 0.830153i \(0.311747\pi\)
\(240\) 0 0
\(241\) 5.41818 0.349016 0.174508 0.984656i \(-0.444167\pi\)
0.174508 + 0.984656i \(0.444167\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 3.23543 0.207127
\(245\) 0.698857 0.0446483
\(246\) 0 0
\(247\) 7.92931 0.504530
\(248\) 12.3141 0.781949
\(249\) 0 0
\(250\) 2.00177 0.126603
\(251\) −15.1995 −0.959384 −0.479692 0.877437i \(-0.659252\pi\)
−0.479692 + 0.877437i \(0.659252\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3.65978 −0.229635
\(255\) 0 0
\(256\) 9.22811 0.576757
\(257\) 26.6304 1.66116 0.830580 0.556900i \(-0.188009\pi\)
0.830580 + 0.556900i \(0.188009\pi\)
\(258\) 0 0
\(259\) −6.16884 −0.383313
\(260\) −5.91232 −0.366667
\(261\) 0 0
\(262\) 5.69494 0.351835
\(263\) 5.63979 0.347764 0.173882 0.984766i \(-0.444369\pi\)
0.173882 + 0.984766i \(0.444369\pi\)
\(264\) 0 0
\(265\) 1.64548 0.101081
\(266\) 0.538909 0.0330426
\(267\) 0 0
\(268\) 23.9471 1.46280
\(269\) 1.10958 0.0676521 0.0338261 0.999428i \(-0.489231\pi\)
0.0338261 + 0.999428i \(0.489231\pi\)
\(270\) 0 0
\(271\) −13.7713 −0.836548 −0.418274 0.908321i \(-0.637365\pi\)
−0.418274 + 0.908321i \(0.637365\pi\)
\(272\) 23.5709 1.42919
\(273\) 0 0
\(274\) −2.52337 −0.152442
\(275\) 0 0
\(276\) 0 0
\(277\) −7.90089 −0.474719 −0.237359 0.971422i \(-0.576282\pi\)
−0.237359 + 0.971422i \(0.576282\pi\)
\(278\) −2.96336 −0.177731
\(279\) 0 0
\(280\) −0.822738 −0.0491680
\(281\) 13.7596 0.820826 0.410413 0.911900i \(-0.365384\pi\)
0.410413 + 0.911900i \(0.365384\pi\)
\(282\) 0 0
\(283\) 17.8650 1.06197 0.530983 0.847383i \(-0.321823\pi\)
0.530983 + 0.847383i \(0.321823\pi\)
\(284\) 14.4800 0.859229
\(285\) 0 0
\(286\) 0 0
\(287\) 1.33434 0.0787634
\(288\) 0 0
\(289\) 29.2988 1.72346
\(290\) 1.66246 0.0976229
\(291\) 0 0
\(292\) −13.3018 −0.778430
\(293\) 23.9721 1.40046 0.700231 0.713916i \(-0.253080\pi\)
0.700231 + 0.713916i \(0.253080\pi\)
\(294\) 0 0
\(295\) 3.33022 0.193893
\(296\) 7.26234 0.422115
\(297\) 0 0
\(298\) 2.14846 0.124457
\(299\) 40.7734 2.35799
\(300\) 0 0
\(301\) −6.33434 −0.365105
\(302\) −2.36060 −0.135837
\(303\) 0 0
\(304\) 6.19916 0.355546
\(305\) −1.18425 −0.0678098
\(306\) 0 0
\(307\) −24.2718 −1.38526 −0.692632 0.721292i \(-0.743549\pi\)
−0.692632 + 0.721292i \(0.743549\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.20137 −0.125029
\(311\) 1.97542 0.112016 0.0560079 0.998430i \(-0.482163\pi\)
0.0560079 + 0.998430i \(0.482163\pi\)
\(312\) 0 0
\(313\) 3.04189 0.171938 0.0859690 0.996298i \(-0.472601\pi\)
0.0859690 + 0.996298i \(0.472601\pi\)
\(314\) 5.72048 0.322825
\(315\) 0 0
\(316\) −16.9118 −0.951361
\(317\) −27.7329 −1.55763 −0.778816 0.627252i \(-0.784180\pi\)
−0.778816 + 0.627252i \(0.784180\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.12675 −0.230693
\(321\) 0 0
\(322\) 2.77113 0.154429
\(323\) 12.1766 0.677526
\(324\) 0 0
\(325\) −19.9905 −1.10887
\(326\) 3.87809 0.214787
\(327\) 0 0
\(328\) −1.57086 −0.0867365
\(329\) 8.20634 0.452430
\(330\) 0 0
\(331\) −13.3502 −0.733794 −0.366897 0.930262i \(-0.619580\pi\)
−0.366897 + 0.930262i \(0.619580\pi\)
\(332\) 10.9170 0.599147
\(333\) 0 0
\(334\) 0.00177192 9.69554e−5 0
\(335\) −8.76524 −0.478896
\(336\) 0 0
\(337\) 6.41345 0.349363 0.174681 0.984625i \(-0.444110\pi\)
0.174681 + 0.984625i \(0.444110\pi\)
\(338\) −1.99746 −0.108648
\(339\) 0 0
\(340\) −9.07926 −0.492392
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 7.45717 0.402064
\(345\) 0 0
\(346\) −3.89056 −0.209158
\(347\) 9.97432 0.535450 0.267725 0.963495i \(-0.413728\pi\)
0.267725 + 0.963495i \(0.413728\pi\)
\(348\) 0 0
\(349\) 8.18267 0.438008 0.219004 0.975724i \(-0.429719\pi\)
0.219004 + 0.975724i \(0.429719\pi\)
\(350\) −1.35864 −0.0726222
\(351\) 0 0
\(352\) 0 0
\(353\) 0.173150 0.00921585 0.00460792 0.999989i \(-0.498533\pi\)
0.00460792 + 0.999989i \(0.498533\pi\)
\(354\) 0 0
\(355\) −5.30004 −0.281297
\(356\) −4.25652 −0.225595
\(357\) 0 0
\(358\) −0.394899 −0.0208711
\(359\) 24.9257 1.31553 0.657763 0.753225i \(-0.271503\pi\)
0.657763 + 0.753225i \(0.271503\pi\)
\(360\) 0 0
\(361\) −15.7975 −0.831449
\(362\) −1.33448 −0.0701387
\(363\) 0 0
\(364\) 8.45999 0.443424
\(365\) 4.86880 0.254845
\(366\) 0 0
\(367\) −34.1736 −1.78385 −0.891924 0.452185i \(-0.850645\pi\)
−0.891924 + 0.452185i \(0.850645\pi\)
\(368\) 31.8768 1.66169
\(369\) 0 0
\(370\) −1.29827 −0.0674938
\(371\) −2.35452 −0.122241
\(372\) 0 0
\(373\) −1.51030 −0.0782005 −0.0391002 0.999235i \(-0.512449\pi\)
−0.0391002 + 0.999235i \(0.512449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.66102 −0.498229
\(377\) −35.0011 −1.80265
\(378\) 0 0
\(379\) −7.46013 −0.383201 −0.191601 0.981473i \(-0.561368\pi\)
−0.191601 + 0.981473i \(0.561368\pi\)
\(380\) −2.38785 −0.122494
\(381\) 0 0
\(382\) 1.60980 0.0823645
\(383\) −25.7482 −1.31567 −0.657836 0.753161i \(-0.728528\pi\)
−0.657836 + 0.753161i \(0.728528\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.242073 −0.0123212
\(387\) 0 0
\(388\) 0.327214 0.0166118
\(389\) 33.6464 1.70594 0.852971 0.521959i \(-0.174799\pi\)
0.852971 + 0.521959i \(0.174799\pi\)
\(390\) 0 0
\(391\) 62.6136 3.16651
\(392\) 1.17726 0.0594607
\(393\) 0 0
\(394\) −7.04921 −0.355134
\(395\) 6.19013 0.311459
\(396\) 0 0
\(397\) −19.6950 −0.988465 −0.494232 0.869330i \(-0.664551\pi\)
−0.494232 + 0.869330i \(0.664551\pi\)
\(398\) −7.02019 −0.351890
\(399\) 0 0
\(400\) −15.6286 −0.781432
\(401\) −17.4368 −0.870752 −0.435376 0.900249i \(-0.643385\pi\)
−0.435376 + 0.900249i \(0.643385\pi\)
\(402\) 0 0
\(403\) 46.3472 2.30872
\(404\) 34.5056 1.71672
\(405\) 0 0
\(406\) −2.37882 −0.118059
\(407\) 0 0
\(408\) 0 0
\(409\) 7.92547 0.391889 0.195945 0.980615i \(-0.437223\pi\)
0.195945 + 0.980615i \(0.437223\pi\)
\(410\) 0.280819 0.0138687
\(411\) 0 0
\(412\) 3.30402 0.162777
\(413\) −4.76524 −0.234482
\(414\) 0 0
\(415\) −3.99589 −0.196150
\(416\) −15.0550 −0.738130
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0302 0.587713 0.293856 0.955850i \(-0.405061\pi\)
0.293856 + 0.955850i \(0.405061\pi\)
\(420\) 0 0
\(421\) −1.36314 −0.0664353 −0.0332177 0.999448i \(-0.510575\pi\)
−0.0332177 + 0.999448i \(0.510575\pi\)
\(422\) −3.05525 −0.148727
\(423\) 0 0
\(424\) 2.77189 0.134615
\(425\) −30.6984 −1.48909
\(426\) 0 0
\(427\) 1.69455 0.0820050
\(428\) −13.3076 −0.643245
\(429\) 0 0
\(430\) −1.33310 −0.0642877
\(431\) 8.23954 0.396885 0.198442 0.980113i \(-0.436412\pi\)
0.198442 + 0.980113i \(0.436412\pi\)
\(432\) 0 0
\(433\) 13.7708 0.661785 0.330892 0.943668i \(-0.392650\pi\)
0.330892 + 0.943668i \(0.392650\pi\)
\(434\) 3.14995 0.151203
\(435\) 0 0
\(436\) −33.7054 −1.61419
\(437\) 16.4675 0.787745
\(438\) 0 0
\(439\) −18.7090 −0.892934 −0.446467 0.894800i \(-0.647318\pi\)
−0.446467 + 0.894800i \(0.647318\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9.07926 −0.431856
\(443\) −33.3148 −1.58283 −0.791417 0.611276i \(-0.790656\pi\)
−0.791417 + 0.611276i \(0.790656\pi\)
\(444\) 0 0
\(445\) 1.55799 0.0738560
\(446\) 2.06600 0.0978278
\(447\) 0 0
\(448\) 5.90501 0.278985
\(449\) 3.74233 0.176611 0.0883057 0.996093i \(-0.471855\pi\)
0.0883057 + 0.996093i \(0.471855\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 22.3841 1.05286
\(453\) 0 0
\(454\) 2.90644 0.136406
\(455\) −3.09657 −0.145169
\(456\) 0 0
\(457\) 19.9747 0.934379 0.467190 0.884157i \(-0.345267\pi\)
0.467190 + 0.884157i \(0.345267\pi\)
\(458\) −0.715069 −0.0334130
\(459\) 0 0
\(460\) −12.2786 −0.572493
\(461\) 0.904388 0.0421215 0.0210608 0.999778i \(-0.493296\pi\)
0.0210608 + 0.999778i \(0.493296\pi\)
\(462\) 0 0
\(463\) 23.2784 1.08184 0.540920 0.841074i \(-0.318076\pi\)
0.540920 + 0.841074i \(0.318076\pi\)
\(464\) −27.3640 −1.27034
\(465\) 0 0
\(466\) 4.89788 0.226890
\(467\) −3.75018 −0.173538 −0.0867688 0.996228i \(-0.527654\pi\)
−0.0867688 + 0.996228i \(0.527654\pi\)
\(468\) 0 0
\(469\) 12.5423 0.579148
\(470\) 1.72707 0.0796640
\(471\) 0 0
\(472\) 5.60994 0.258219
\(473\) 0 0
\(474\) 0 0
\(475\) −8.07371 −0.370447
\(476\) 12.9916 0.595468
\(477\) 0 0
\(478\) −5.19128 −0.237443
\(479\) 29.0418 1.32695 0.663477 0.748197i \(-0.269080\pi\)
0.663477 + 0.748197i \(0.269080\pi\)
\(480\) 0 0
\(481\) 27.3336 1.24630
\(482\) −1.63165 −0.0743195
\(483\) 0 0
\(484\) 0 0
\(485\) −0.119769 −0.00543842
\(486\) 0 0
\(487\) −12.4381 −0.563624 −0.281812 0.959470i \(-0.590935\pi\)
−0.281812 + 0.959470i \(0.590935\pi\)
\(488\) −1.99493 −0.0903062
\(489\) 0 0
\(490\) −0.210456 −0.00950743
\(491\) −6.46343 −0.291691 −0.145845 0.989307i \(-0.546590\pi\)
−0.145845 + 0.989307i \(0.546590\pi\)
\(492\) 0 0
\(493\) −53.7495 −2.42076
\(494\) −2.38785 −0.107435
\(495\) 0 0
\(496\) 36.2345 1.62698
\(497\) 7.58387 0.340183
\(498\) 0 0
\(499\) −13.2848 −0.594708 −0.297354 0.954767i \(-0.596104\pi\)
−0.297354 + 0.954767i \(0.596104\pi\)
\(500\) 12.6917 0.567589
\(501\) 0 0
\(502\) 4.57722 0.204291
\(503\) −36.5805 −1.63104 −0.815522 0.578726i \(-0.803550\pi\)
−0.815522 + 0.578726i \(0.803550\pi\)
\(504\) 0 0
\(505\) −12.6299 −0.562025
\(506\) 0 0
\(507\) 0 0
\(508\) −23.2038 −1.02950
\(509\) 4.36993 0.193694 0.0968469 0.995299i \(-0.469124\pi\)
0.0968469 + 0.995299i \(0.469124\pi\)
\(510\) 0 0
\(511\) −6.96681 −0.308193
\(512\) −19.9263 −0.880628
\(513\) 0 0
\(514\) −8.01956 −0.353728
\(515\) −1.20935 −0.0532905
\(516\) 0 0
\(517\) 0 0
\(518\) 1.85770 0.0816228
\(519\) 0 0
\(520\) 3.64548 0.159865
\(521\) 36.1430 1.58345 0.791726 0.610876i \(-0.209183\pi\)
0.791726 + 0.610876i \(0.209183\pi\)
\(522\) 0 0
\(523\) −28.5269 −1.24739 −0.623696 0.781667i \(-0.714370\pi\)
−0.623696 + 0.781667i \(0.714370\pi\)
\(524\) 36.1072 1.57735
\(525\) 0 0
\(526\) −1.69838 −0.0740530
\(527\) 71.1731 3.10035
\(528\) 0 0
\(529\) 61.6775 2.68163
\(530\) −0.495523 −0.0215242
\(531\) 0 0
\(532\) 3.41680 0.148137
\(533\) −5.91232 −0.256091
\(534\) 0 0
\(535\) 4.87091 0.210588
\(536\) −14.7655 −0.637774
\(537\) 0 0
\(538\) −0.334141 −0.0144059
\(539\) 0 0
\(540\) 0 0
\(541\) −26.3725 −1.13384 −0.566921 0.823772i \(-0.691866\pi\)
−0.566921 + 0.823772i \(0.691866\pi\)
\(542\) 4.14714 0.178135
\(543\) 0 0
\(544\) −23.1191 −0.991225
\(545\) 12.3370 0.528460
\(546\) 0 0
\(547\) 32.2848 1.38040 0.690199 0.723620i \(-0.257523\pi\)
0.690199 + 0.723620i \(0.257523\pi\)
\(548\) −15.9987 −0.683431
\(549\) 0 0
\(550\) 0 0
\(551\) −14.1362 −0.602221
\(552\) 0 0
\(553\) −8.85751 −0.376660
\(554\) 2.37930 0.101087
\(555\) 0 0
\(556\) −18.7884 −0.796805
\(557\) 37.4768 1.58794 0.793972 0.607954i \(-0.208010\pi\)
0.793972 + 0.607954i \(0.208010\pi\)
\(558\) 0 0
\(559\) 28.0669 1.18710
\(560\) −2.42091 −0.102302
\(561\) 0 0
\(562\) −4.14359 −0.174787
\(563\) 14.3834 0.606188 0.303094 0.952961i \(-0.401980\pi\)
0.303094 + 0.952961i \(0.401980\pi\)
\(564\) 0 0
\(565\) −8.19314 −0.344688
\(566\) −5.37993 −0.226135
\(567\) 0 0
\(568\) −8.92820 −0.374619
\(569\) −29.2888 −1.22785 −0.613925 0.789364i \(-0.710410\pi\)
−0.613925 + 0.789364i \(0.710410\pi\)
\(570\) 0 0
\(571\) 18.5737 0.777284 0.388642 0.921389i \(-0.372944\pi\)
0.388642 + 0.921389i \(0.372944\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.401826 −0.0167719
\(575\) −41.5159 −1.73133
\(576\) 0 0
\(577\) 20.3524 0.847282 0.423641 0.905830i \(-0.360752\pi\)
0.423641 + 0.905830i \(0.360752\pi\)
\(578\) −8.82313 −0.366994
\(579\) 0 0
\(580\) 10.5403 0.437664
\(581\) 5.71775 0.237212
\(582\) 0 0
\(583\) 0 0
\(584\) 8.20176 0.339391
\(585\) 0 0
\(586\) −7.21902 −0.298215
\(587\) 30.8643 1.27391 0.636954 0.770902i \(-0.280194\pi\)
0.636954 + 0.770902i \(0.280194\pi\)
\(588\) 0 0
\(589\) 18.7186 0.771287
\(590\) −1.00287 −0.0412877
\(591\) 0 0
\(592\) 21.3695 0.878281
\(593\) −4.76749 −0.195777 −0.0978887 0.995197i \(-0.531209\pi\)
−0.0978887 + 0.995197i \(0.531209\pi\)
\(594\) 0 0
\(595\) −4.75525 −0.194946
\(596\) 13.6217 0.557967
\(597\) 0 0
\(598\) −12.2786 −0.502110
\(599\) −31.5423 −1.28878 −0.644391 0.764696i \(-0.722889\pi\)
−0.644391 + 0.764696i \(0.722889\pi\)
\(600\) 0 0
\(601\) −2.59544 −0.105870 −0.0529352 0.998598i \(-0.516858\pi\)
−0.0529352 + 0.998598i \(0.516858\pi\)
\(602\) 1.90754 0.0777456
\(603\) 0 0
\(604\) −14.9667 −0.608988
\(605\) 0 0
\(606\) 0 0
\(607\) 24.2991 0.986269 0.493135 0.869953i \(-0.335851\pi\)
0.493135 + 0.869953i \(0.335851\pi\)
\(608\) −6.08036 −0.246591
\(609\) 0 0
\(610\) 0.356628 0.0144394
\(611\) −36.3616 −1.47103
\(612\) 0 0
\(613\) −4.32611 −0.174730 −0.0873650 0.996176i \(-0.527845\pi\)
−0.0873650 + 0.996176i \(0.527845\pi\)
\(614\) 7.30927 0.294978
\(615\) 0 0
\(616\) 0 0
\(617\) −27.3809 −1.10231 −0.551156 0.834402i \(-0.685813\pi\)
−0.551156 + 0.834402i \(0.685813\pi\)
\(618\) 0 0
\(619\) 9.13728 0.367258 0.183629 0.982996i \(-0.441215\pi\)
0.183629 + 0.982996i \(0.441215\pi\)
\(620\) −13.9571 −0.560533
\(621\) 0 0
\(622\) −0.594884 −0.0238527
\(623\) −2.22935 −0.0893169
\(624\) 0 0
\(625\) 17.9125 0.716501
\(626\) −0.916045 −0.0366125
\(627\) 0 0
\(628\) 36.2691 1.44729
\(629\) 41.9748 1.67364
\(630\) 0 0
\(631\) 29.7629 1.18484 0.592421 0.805628i \(-0.298172\pi\)
0.592421 + 0.805628i \(0.298172\pi\)
\(632\) 10.4276 0.414788
\(633\) 0 0
\(634\) 8.35156 0.331683
\(635\) 8.49318 0.337042
\(636\) 0 0
\(637\) 4.43091 0.175559
\(638\) 0 0
\(639\) 0 0
\(640\) 5.99178 0.236846
\(641\) 17.6506 0.697159 0.348579 0.937279i \(-0.386664\pi\)
0.348579 + 0.937279i \(0.386664\pi\)
\(642\) 0 0
\(643\) 6.56451 0.258879 0.129439 0.991587i \(-0.458682\pi\)
0.129439 + 0.991587i \(0.458682\pi\)
\(644\) 17.5696 0.692338
\(645\) 0 0
\(646\) −3.66691 −0.144273
\(647\) −28.8178 −1.13294 −0.566472 0.824081i \(-0.691692\pi\)
−0.566472 + 0.824081i \(0.691692\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.01999 0.236124
\(651\) 0 0
\(652\) 24.5879 0.962937
\(653\) −32.3666 −1.26660 −0.633302 0.773905i \(-0.718301\pi\)
−0.633302 + 0.773905i \(0.718301\pi\)
\(654\) 0 0
\(655\) −13.2161 −0.516397
\(656\) −4.62228 −0.180470
\(657\) 0 0
\(658\) −2.47128 −0.0963406
\(659\) 3.30435 0.128719 0.0643596 0.997927i \(-0.479500\pi\)
0.0643596 + 0.997927i \(0.479500\pi\)
\(660\) 0 0
\(661\) −7.07404 −0.275148 −0.137574 0.990491i \(-0.543931\pi\)
−0.137574 + 0.990491i \(0.543931\pi\)
\(662\) 4.02032 0.156254
\(663\) 0 0
\(664\) −6.73129 −0.261225
\(665\) −1.25064 −0.0484976
\(666\) 0 0
\(667\) −72.6898 −2.81456
\(668\) 0.0112344 0.000434672 0
\(669\) 0 0
\(670\) 2.63959 0.101976
\(671\) 0 0
\(672\) 0 0
\(673\) 21.2582 0.819445 0.409722 0.912210i \(-0.365626\pi\)
0.409722 + 0.912210i \(0.365626\pi\)
\(674\) −1.93137 −0.0743934
\(675\) 0 0
\(676\) −12.6644 −0.487091
\(677\) 43.2282 1.66140 0.830698 0.556723i \(-0.187942\pi\)
0.830698 + 0.556723i \(0.187942\pi\)
\(678\) 0 0
\(679\) 0.171378 0.00657689
\(680\) 5.59817 0.214680
\(681\) 0 0
\(682\) 0 0
\(683\) −30.9708 −1.18507 −0.592533 0.805546i \(-0.701872\pi\)
−0.592533 + 0.805546i \(0.701872\pi\)
\(684\) 0 0
\(685\) 5.85593 0.223744
\(686\) 0.301143 0.0114977
\(687\) 0 0
\(688\) 21.9428 0.836561
\(689\) 10.4327 0.397453
\(690\) 0 0
\(691\) −6.50729 −0.247549 −0.123775 0.992310i \(-0.539500\pi\)
−0.123775 + 0.992310i \(0.539500\pi\)
\(692\) −24.6670 −0.937699
\(693\) 0 0
\(694\) −3.00370 −0.114019
\(695\) 6.87703 0.260860
\(696\) 0 0
\(697\) −9.07926 −0.343901
\(698\) −2.46415 −0.0932696
\(699\) 0 0
\(700\) −8.61405 −0.325581
\(701\) 13.3174 0.502992 0.251496 0.967858i \(-0.419078\pi\)
0.251496 + 0.967858i \(0.419078\pi\)
\(702\) 0 0
\(703\) 11.0394 0.416359
\(704\) 0 0
\(705\) 0 0
\(706\) −0.0521429 −0.00196242
\(707\) 18.0723 0.679678
\(708\) 0 0
\(709\) 48.2055 1.81040 0.905198 0.424990i \(-0.139722\pi\)
0.905198 + 0.424990i \(0.139722\pi\)
\(710\) 1.59607 0.0598994
\(711\) 0 0
\(712\) 2.62452 0.0983582
\(713\) 96.2532 3.60471
\(714\) 0 0
\(715\) 0 0
\(716\) −2.50375 −0.0935694
\(717\) 0 0
\(718\) −7.50619 −0.280129
\(719\) 11.4733 0.427881 0.213940 0.976847i \(-0.431370\pi\)
0.213940 + 0.976847i \(0.431370\pi\)
\(720\) 0 0
\(721\) 1.73047 0.0644462
\(722\) 4.75732 0.177049
\(723\) 0 0
\(724\) −8.46090 −0.314447
\(725\) 35.6385 1.32358
\(726\) 0 0
\(727\) −29.5904 −1.09745 −0.548724 0.836004i \(-0.684886\pi\)
−0.548724 + 0.836004i \(0.684886\pi\)
\(728\) −5.21634 −0.193330
\(729\) 0 0
\(730\) −1.46621 −0.0542667
\(731\) 43.1009 1.59414
\(732\) 0 0
\(733\) 39.1482 1.44597 0.722986 0.690862i \(-0.242769\pi\)
0.722986 + 0.690862i \(0.242769\pi\)
\(734\) 10.2911 0.379853
\(735\) 0 0
\(736\) −31.2659 −1.15248
\(737\) 0 0
\(738\) 0 0
\(739\) 15.6126 0.574317 0.287159 0.957883i \(-0.407289\pi\)
0.287159 + 0.957883i \(0.407289\pi\)
\(740\) −8.23131 −0.302589
\(741\) 0 0
\(742\) 0.709048 0.0260300
\(743\) −29.8447 −1.09490 −0.547448 0.836840i \(-0.684401\pi\)
−0.547448 + 0.836840i \(0.684401\pi\)
\(744\) 0 0
\(745\) −4.98589 −0.182669
\(746\) 0.454817 0.0166520
\(747\) 0 0
\(748\) 0 0
\(749\) −6.96982 −0.254672
\(750\) 0 0
\(751\) 1.52452 0.0556306 0.0278153 0.999613i \(-0.491145\pi\)
0.0278153 + 0.999613i \(0.491145\pi\)
\(752\) −28.4276 −1.03665
\(753\) 0 0
\(754\) 10.5403 0.383857
\(755\) 5.47821 0.199372
\(756\) 0 0
\(757\) −13.3841 −0.486453 −0.243226 0.969970i \(-0.578206\pi\)
−0.243226 + 0.969970i \(0.578206\pi\)
\(758\) 2.24657 0.0815990
\(759\) 0 0
\(760\) 1.47233 0.0534069
\(761\) 17.1185 0.620544 0.310272 0.950648i \(-0.399580\pi\)
0.310272 + 0.950648i \(0.399580\pi\)
\(762\) 0 0
\(763\) −17.6531 −0.639086
\(764\) 10.2065 0.369257
\(765\) 0 0
\(766\) 7.75389 0.280160
\(767\) 21.1144 0.762395
\(768\) 0 0
\(769\) −15.9428 −0.574912 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.53480 −0.0552385
\(773\) 11.9350 0.429274 0.214637 0.976694i \(-0.431143\pi\)
0.214637 + 0.976694i \(0.431143\pi\)
\(774\) 0 0
\(775\) −47.1913 −1.69516
\(776\) −0.201757 −0.00724265
\(777\) 0 0
\(778\) −10.1324 −0.363264
\(779\) −2.38785 −0.0855538
\(780\) 0 0
\(781\) 0 0
\(782\) −18.8557 −0.674277
\(783\) 0 0
\(784\) 3.46410 0.123718
\(785\) −13.2754 −0.473820
\(786\) 0 0
\(787\) −2.80275 −0.0999071 −0.0499535 0.998752i \(-0.515907\pi\)
−0.0499535 + 0.998752i \(0.515907\pi\)
\(788\) −44.6935 −1.59214
\(789\) 0 0
\(790\) −1.86411 −0.0663222
\(791\) 11.7236 0.416844
\(792\) 0 0
\(793\) −7.50839 −0.266631
\(794\) 5.93102 0.210484
\(795\) 0 0
\(796\) −44.5095 −1.57760
\(797\) 13.6873 0.484828 0.242414 0.970173i \(-0.422061\pi\)
0.242414 + 0.970173i \(0.422061\pi\)
\(798\) 0 0
\(799\) −55.8386 −1.97543
\(800\) 15.3291 0.541966
\(801\) 0 0
\(802\) 5.25097 0.185418
\(803\) 0 0
\(804\) 0 0
\(805\) −6.43091 −0.226660
\(806\) −13.9571 −0.491620
\(807\) 0 0
\(808\) −21.2758 −0.748480
\(809\) 25.4980 0.896464 0.448232 0.893917i \(-0.352054\pi\)
0.448232 + 0.893917i \(0.352054\pi\)
\(810\) 0 0
\(811\) −36.7404 −1.29013 −0.645065 0.764128i \(-0.723170\pi\)
−0.645065 + 0.764128i \(0.723170\pi\)
\(812\) −15.0823 −0.529284
\(813\) 0 0
\(814\) 0 0
\(815\) −8.99980 −0.315249
\(816\) 0 0
\(817\) 11.3356 0.396582
\(818\) −2.38670 −0.0834490
\(819\) 0 0
\(820\) 1.78045 0.0621762
\(821\) −13.0737 −0.456274 −0.228137 0.973629i \(-0.573263\pi\)
−0.228137 + 0.973629i \(0.573263\pi\)
\(822\) 0 0
\(823\) 48.7149 1.69809 0.849047 0.528318i \(-0.177177\pi\)
0.849047 + 0.528318i \(0.177177\pi\)
\(824\) −2.03722 −0.0709700
\(825\) 0 0
\(826\) 1.43502 0.0499307
\(827\) −53.8443 −1.87235 −0.936176 0.351532i \(-0.885661\pi\)
−0.936176 + 0.351532i \(0.885661\pi\)
\(828\) 0 0
\(829\) 11.1183 0.386154 0.193077 0.981184i \(-0.438153\pi\)
0.193077 + 0.981184i \(0.438153\pi\)
\(830\) 1.20333 0.0417683
\(831\) 0 0
\(832\) −26.1645 −0.907092
\(833\) 6.80432 0.235756
\(834\) 0 0
\(835\) −0.00411207 −0.000142304 0
\(836\) 0 0
\(837\) 0 0
\(838\) −3.62281 −0.125148
\(839\) −26.8372 −0.926524 −0.463262 0.886221i \(-0.653321\pi\)
−0.463262 + 0.886221i \(0.653321\pi\)
\(840\) 0 0
\(841\) 33.3992 1.15170
\(842\) 0.410500 0.0141468
\(843\) 0 0
\(844\) −19.3709 −0.666775
\(845\) 4.63548 0.159465
\(846\) 0 0
\(847\) 0 0
\(848\) 8.15631 0.280089
\(849\) 0 0
\(850\) 9.24460 0.317087
\(851\) 56.7659 1.94591
\(852\) 0 0
\(853\) −34.8276 −1.19247 −0.596237 0.802809i \(-0.703338\pi\)
−0.596237 + 0.802809i \(0.703338\pi\)
\(854\) −0.510302 −0.0174622
\(855\) 0 0
\(856\) 8.20530 0.280451
\(857\) −38.4741 −1.31425 −0.657126 0.753781i \(-0.728228\pi\)
−0.657126 + 0.753781i \(0.728228\pi\)
\(858\) 0 0
\(859\) −7.58039 −0.258639 −0.129320 0.991603i \(-0.541279\pi\)
−0.129320 + 0.991603i \(0.541279\pi\)
\(860\) −8.45214 −0.288216
\(861\) 0 0
\(862\) −2.48128 −0.0845127
\(863\) 32.4054 1.10309 0.551546 0.834145i \(-0.314038\pi\)
0.551546 + 0.834145i \(0.314038\pi\)
\(864\) 0 0
\(865\) 9.02875 0.306987
\(866\) −4.14699 −0.140921
\(867\) 0 0
\(868\) 19.9714 0.677873
\(869\) 0 0
\(870\) 0 0
\(871\) −55.5736 −1.88304
\(872\) 20.7824 0.703780
\(873\) 0 0
\(874\) −4.95906 −0.167743
\(875\) 6.64725 0.224718
\(876\) 0 0
\(877\) 11.2438 0.379676 0.189838 0.981815i \(-0.439204\pi\)
0.189838 + 0.981815i \(0.439204\pi\)
\(878\) 5.63410 0.190142
\(879\) 0 0
\(880\) 0 0
\(881\) 35.8854 1.20901 0.604505 0.796601i \(-0.293371\pi\)
0.604505 + 0.796601i \(0.293371\pi\)
\(882\) 0 0
\(883\) −12.8564 −0.432653 −0.216326 0.976321i \(-0.569407\pi\)
−0.216326 + 0.976321i \(0.569407\pi\)
\(884\) −57.5645 −1.93610
\(885\) 0 0
\(886\) 10.0325 0.337049
\(887\) −44.5324 −1.49525 −0.747627 0.664119i \(-0.768807\pi\)
−0.747627 + 0.664119i \(0.768807\pi\)
\(888\) 0 0
\(889\) −12.1530 −0.407597
\(890\) −0.469179 −0.0157269
\(891\) 0 0
\(892\) 13.0989 0.438583
\(893\) −14.6856 −0.491435
\(894\) 0 0
\(895\) 0.916435 0.0306331
\(896\) −8.57368 −0.286427
\(897\) 0 0
\(898\) −1.12698 −0.0376077
\(899\) −82.6268 −2.75576
\(900\) 0 0
\(901\) 16.0209 0.533735
\(902\) 0 0
\(903\) 0 0
\(904\) −13.8018 −0.459041
\(905\) 3.09690 0.102945
\(906\) 0 0
\(907\) −0.321610 −0.0106789 −0.00533944 0.999986i \(-0.501700\pi\)
−0.00533944 + 0.999986i \(0.501700\pi\)
\(908\) 18.4275 0.611537
\(909\) 0 0
\(910\) 0.932511 0.0309124
\(911\) −42.8345 −1.41917 −0.709585 0.704620i \(-0.751118\pi\)
−0.709585 + 0.704620i \(0.751118\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.01525 −0.198967
\(915\) 0 0
\(916\) −4.53369 −0.149797
\(917\) 18.9111 0.624499
\(918\) 0 0
\(919\) 41.3657 1.36453 0.682265 0.731105i \(-0.260995\pi\)
0.682265 + 0.731105i \(0.260995\pi\)
\(920\) 7.57086 0.249604
\(921\) 0 0
\(922\) −0.272350 −0.00896937
\(923\) −33.6034 −1.10607
\(924\) 0 0
\(925\) −27.8313 −0.915089
\(926\) −7.01013 −0.230367
\(927\) 0 0
\(928\) 26.8396 0.881054
\(929\) −39.4522 −1.29439 −0.647193 0.762326i \(-0.724057\pi\)
−0.647193 + 0.762326i \(0.724057\pi\)
\(930\) 0 0
\(931\) 1.78954 0.0586499
\(932\) 31.0537 1.01720
\(933\) 0 0
\(934\) 1.12934 0.0369531
\(935\) 0 0
\(936\) 0 0
\(937\) 17.9805 0.587397 0.293698 0.955898i \(-0.405114\pi\)
0.293698 + 0.955898i \(0.405114\pi\)
\(938\) −3.77701 −0.123324
\(939\) 0 0
\(940\) 10.9500 0.357150
\(941\) 30.5804 0.996892 0.498446 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(942\) 0 0
\(943\) −12.2786 −0.399847
\(944\) 16.5073 0.537267
\(945\) 0 0
\(946\) 0 0
\(947\) 8.17056 0.265507 0.132754 0.991149i \(-0.457618\pi\)
0.132754 + 0.991149i \(0.457618\pi\)
\(948\) 0 0
\(949\) 30.8693 1.00206
\(950\) 2.43134 0.0788831
\(951\) 0 0
\(952\) −8.01047 −0.259621
\(953\) −8.38538 −0.271629 −0.135815 0.990734i \(-0.543365\pi\)
−0.135815 + 0.990734i \(0.543365\pi\)
\(954\) 0 0
\(955\) −3.73583 −0.120889
\(956\) −32.9138 −1.06451
\(957\) 0 0
\(958\) −8.74573 −0.282562
\(959\) −8.37930 −0.270582
\(960\) 0 0
\(961\) 78.4114 2.52940
\(962\) −8.23131 −0.265388
\(963\) 0 0
\(964\) −10.3450 −0.333190
\(965\) 0.561775 0.0180842
\(966\) 0 0
\(967\) 51.4778 1.65541 0.827707 0.561161i \(-0.189645\pi\)
0.827707 + 0.561161i \(0.189645\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.0360675 0.00115806
\(971\) 13.5974 0.436360 0.218180 0.975909i \(-0.429988\pi\)
0.218180 + 0.975909i \(0.429988\pi\)
\(972\) 0 0
\(973\) −9.84039 −0.315468
\(974\) 3.74564 0.120018
\(975\) 0 0
\(976\) −5.87009 −0.187897
\(977\) −19.6433 −0.628446 −0.314223 0.949349i \(-0.601744\pi\)
−0.314223 + 0.949349i \(0.601744\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.33434 −0.0426238
\(981\) 0 0
\(982\) 1.94642 0.0621127
\(983\) 57.0855 1.82074 0.910372 0.413791i \(-0.135796\pi\)
0.910372 + 0.413791i \(0.135796\pi\)
\(984\) 0 0
\(985\) 16.3590 0.521240
\(986\) 16.1863 0.515476
\(987\) 0 0
\(988\) −15.1395 −0.481652
\(989\) 58.2888 1.85348
\(990\) 0 0
\(991\) −32.7803 −1.04130 −0.520650 0.853770i \(-0.674310\pi\)
−0.520650 + 0.853770i \(0.674310\pi\)
\(992\) −35.5401 −1.12840
\(993\) 0 0
\(994\) −2.28383 −0.0724387
\(995\) 16.2916 0.516479
\(996\) 0 0
\(997\) 33.3145 1.05508 0.527540 0.849530i \(-0.323114\pi\)
0.527540 + 0.849530i \(0.323114\pi\)
\(998\) 4.00062 0.126637
\(999\) 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.cg.1.3 4
3.2 odd 2 2541.2.a.bp.1.2 yes 4
11.10 odd 2 7623.2.a.cn.1.2 4
33.32 even 2 2541.2.a.bl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bl.1.3 4 33.32 even 2
2541.2.a.bp.1.2 yes 4 3.2 odd 2
7623.2.a.cg.1.3 4 1.1 even 1 trivial
7623.2.a.cn.1.2 4 11.10 odd 2