Properties

Label 7623.2.a.ct.1.6
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.40927\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +O(q^{10})\) \(q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +2.58091 q^{10} -4.64706 q^{13} +1.40927 q^{14} -3.97188 q^{16} +5.47021 q^{17} -5.80118 q^{19} -0.0255744 q^{20} +0.719682 q^{23} -1.64602 q^{25} -6.54895 q^{26} -0.0139645 q^{28} +1.17247 q^{29} -1.30787 q^{31} +0.0789938 q^{32} +7.70900 q^{34} +1.83139 q^{35} +2.09474 q^{37} -8.17541 q^{38} -5.19787 q^{40} +0.916645 q^{41} +8.02379 q^{43} +1.01423 q^{46} -5.97584 q^{47} +1.00000 q^{49} -2.31969 q^{50} +0.0648939 q^{52} -10.1449 q^{53} -2.83822 q^{56} +1.65233 q^{58} +7.68081 q^{59} -6.27612 q^{61} -1.84313 q^{62} +8.05508 q^{64} -8.51056 q^{65} -15.4673 q^{67} -0.0763889 q^{68} +2.58091 q^{70} -13.9019 q^{71} +6.01462 q^{73} +2.95205 q^{74} +0.0810106 q^{76} -15.6409 q^{79} -7.27404 q^{80} +1.29180 q^{82} +4.37573 q^{83} +10.0181 q^{85} +11.3077 q^{86} -15.3437 q^{89} -4.64706 q^{91} -0.0100500 q^{92} -8.42155 q^{94} -10.6242 q^{95} +2.41124 q^{97} +1.40927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40927 0.996503 0.498251 0.867033i \(-0.333976\pi\)
0.498251 + 0.867033i \(0.333976\pi\)
\(3\) 0 0
\(4\) −0.0139645 −0.00698226
\(5\) 1.83139 0.819021 0.409510 0.912305i \(-0.365700\pi\)
0.409510 + 0.912305i \(0.365700\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.83822 −1.00346
\(9\) 0 0
\(10\) 2.58091 0.816157
\(11\) 0 0
\(12\) 0 0
\(13\) −4.64706 −1.28886 −0.644431 0.764663i \(-0.722906\pi\)
−0.644431 + 0.764663i \(0.722906\pi\)
\(14\) 1.40927 0.376643
\(15\) 0 0
\(16\) −3.97188 −0.992969
\(17\) 5.47021 1.32672 0.663361 0.748300i \(-0.269129\pi\)
0.663361 + 0.748300i \(0.269129\pi\)
\(18\) 0 0
\(19\) −5.80118 −1.33088 −0.665441 0.746451i \(-0.731756\pi\)
−0.665441 + 0.746451i \(0.731756\pi\)
\(20\) −0.0255744 −0.00571861
\(21\) 0 0
\(22\) 0 0
\(23\) 0.719682 0.150064 0.0750321 0.997181i \(-0.476094\pi\)
0.0750321 + 0.997181i \(0.476094\pi\)
\(24\) 0 0
\(25\) −1.64602 −0.329205
\(26\) −6.54895 −1.28435
\(27\) 0 0
\(28\) −0.0139645 −0.00263905
\(29\) 1.17247 0.217723 0.108861 0.994057i \(-0.465280\pi\)
0.108861 + 0.994057i \(0.465280\pi\)
\(30\) 0 0
\(31\) −1.30787 −0.234900 −0.117450 0.993079i \(-0.537472\pi\)
−0.117450 + 0.993079i \(0.537472\pi\)
\(32\) 0.0789938 0.0139643
\(33\) 0 0
\(34\) 7.70900 1.32208
\(35\) 1.83139 0.309561
\(36\) 0 0
\(37\) 2.09474 0.344373 0.172186 0.985064i \(-0.444917\pi\)
0.172186 + 0.985064i \(0.444917\pi\)
\(38\) −8.17541 −1.32623
\(39\) 0 0
\(40\) −5.19787 −0.821855
\(41\) 0.916645 0.143156 0.0715780 0.997435i \(-0.477197\pi\)
0.0715780 + 0.997435i \(0.477197\pi\)
\(42\) 0 0
\(43\) 8.02379 1.22362 0.611808 0.791006i \(-0.290442\pi\)
0.611808 + 0.791006i \(0.290442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.01423 0.149539
\(47\) −5.97584 −0.871665 −0.435833 0.900028i \(-0.643546\pi\)
−0.435833 + 0.900028i \(0.643546\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.31969 −0.328053
\(51\) 0 0
\(52\) 0.0648939 0.00899916
\(53\) −10.1449 −1.39351 −0.696757 0.717307i \(-0.745374\pi\)
−0.696757 + 0.717307i \(0.745374\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.83822 −0.379272
\(57\) 0 0
\(58\) 1.65233 0.216962
\(59\) 7.68081 0.999956 0.499978 0.866038i \(-0.333341\pi\)
0.499978 + 0.866038i \(0.333341\pi\)
\(60\) 0 0
\(61\) −6.27612 −0.803575 −0.401788 0.915733i \(-0.631611\pi\)
−0.401788 + 0.915733i \(0.631611\pi\)
\(62\) −1.84313 −0.234078
\(63\) 0 0
\(64\) 8.05508 1.00688
\(65\) −8.51056 −1.05560
\(66\) 0 0
\(67\) −15.4673 −1.88963 −0.944814 0.327608i \(-0.893758\pi\)
−0.944814 + 0.327608i \(0.893758\pi\)
\(68\) −0.0763889 −0.00926351
\(69\) 0 0
\(70\) 2.58091 0.308478
\(71\) −13.9019 −1.64985 −0.824927 0.565240i \(-0.808784\pi\)
−0.824927 + 0.565240i \(0.808784\pi\)
\(72\) 0 0
\(73\) 6.01462 0.703959 0.351979 0.936008i \(-0.385509\pi\)
0.351979 + 0.936008i \(0.385509\pi\)
\(74\) 2.95205 0.343169
\(75\) 0 0
\(76\) 0.0810106 0.00929255
\(77\) 0 0
\(78\) 0 0
\(79\) −15.6409 −1.75974 −0.879872 0.475211i \(-0.842372\pi\)
−0.879872 + 0.475211i \(0.842372\pi\)
\(80\) −7.27404 −0.813262
\(81\) 0 0
\(82\) 1.29180 0.142655
\(83\) 4.37573 0.480299 0.240149 0.970736i \(-0.422804\pi\)
0.240149 + 0.970736i \(0.422804\pi\)
\(84\) 0 0
\(85\) 10.0181 1.08661
\(86\) 11.3077 1.21934
\(87\) 0 0
\(88\) 0 0
\(89\) −15.3437 −1.62643 −0.813215 0.581963i \(-0.802285\pi\)
−0.813215 + 0.581963i \(0.802285\pi\)
\(90\) 0 0
\(91\) −4.64706 −0.487144
\(92\) −0.0100500 −0.00104779
\(93\) 0 0
\(94\) −8.42155 −0.868617
\(95\) −10.6242 −1.09002
\(96\) 0 0
\(97\) 2.41124 0.244824 0.122412 0.992479i \(-0.460937\pi\)
0.122412 + 0.992479i \(0.460937\pi\)
\(98\) 1.40927 0.142358
\(99\) 0 0
\(100\) 0.0229859 0.00229859
\(101\) −11.8959 −1.18368 −0.591842 0.806054i \(-0.701599\pi\)
−0.591842 + 0.806054i \(0.701599\pi\)
\(102\) 0 0
\(103\) 0.396314 0.0390500 0.0195250 0.999809i \(-0.493785\pi\)
0.0195250 + 0.999809i \(0.493785\pi\)
\(104\) 13.1893 1.29332
\(105\) 0 0
\(106\) −14.2969 −1.38864
\(107\) 3.26935 0.316060 0.158030 0.987434i \(-0.449486\pi\)
0.158030 + 0.987434i \(0.449486\pi\)
\(108\) 0 0
\(109\) −2.84638 −0.272634 −0.136317 0.990665i \(-0.543527\pi\)
−0.136317 + 0.990665i \(0.543527\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.97188 −0.375307
\(113\) −14.5445 −1.36823 −0.684117 0.729372i \(-0.739812\pi\)
−0.684117 + 0.729372i \(0.739812\pi\)
\(114\) 0 0
\(115\) 1.31802 0.122906
\(116\) −0.0163730 −0.00152020
\(117\) 0 0
\(118\) 10.8243 0.996459
\(119\) 5.47021 0.501454
\(120\) 0 0
\(121\) 0 0
\(122\) −8.84474 −0.800765
\(123\) 0 0
\(124\) 0.0182637 0.00164013
\(125\) −12.1714 −1.08865
\(126\) 0 0
\(127\) 5.03287 0.446595 0.223298 0.974750i \(-0.428318\pi\)
0.223298 + 0.974750i \(0.428318\pi\)
\(128\) 11.1938 0.989399
\(129\) 0 0
\(130\) −11.9937 −1.05191
\(131\) 0.180053 0.0157313 0.00786565 0.999969i \(-0.497496\pi\)
0.00786565 + 0.999969i \(0.497496\pi\)
\(132\) 0 0
\(133\) −5.80118 −0.503026
\(134\) −21.7975 −1.88302
\(135\) 0 0
\(136\) −15.5256 −1.33131
\(137\) −8.32395 −0.711163 −0.355582 0.934645i \(-0.615717\pi\)
−0.355582 + 0.934645i \(0.615717\pi\)
\(138\) 0 0
\(139\) −6.96119 −0.590441 −0.295220 0.955429i \(-0.595393\pi\)
−0.295220 + 0.955429i \(0.595393\pi\)
\(140\) −0.0255744 −0.00216143
\(141\) 0 0
\(142\) −19.5915 −1.64408
\(143\) 0 0
\(144\) 0 0
\(145\) 2.14725 0.178320
\(146\) 8.47622 0.701497
\(147\) 0 0
\(148\) −0.0292520 −0.00240450
\(149\) 3.21431 0.263327 0.131663 0.991294i \(-0.457968\pi\)
0.131663 + 0.991294i \(0.457968\pi\)
\(150\) 0 0
\(151\) 22.2670 1.81206 0.906032 0.423210i \(-0.139097\pi\)
0.906032 + 0.423210i \(0.139097\pi\)
\(152\) 16.4650 1.33549
\(153\) 0 0
\(154\) 0 0
\(155\) −2.39521 −0.192388
\(156\) 0 0
\(157\) 13.2548 1.05785 0.528923 0.848670i \(-0.322596\pi\)
0.528923 + 0.848670i \(0.322596\pi\)
\(158\) −22.0423 −1.75359
\(159\) 0 0
\(160\) 0.144668 0.0114370
\(161\) 0.719682 0.0567189
\(162\) 0 0
\(163\) 13.7183 1.07450 0.537252 0.843422i \(-0.319462\pi\)
0.537252 + 0.843422i \(0.319462\pi\)
\(164\) −0.0128005 −0.000999552 0
\(165\) 0 0
\(166\) 6.16657 0.478619
\(167\) −9.30860 −0.720321 −0.360160 0.932890i \(-0.617278\pi\)
−0.360160 + 0.932890i \(0.617278\pi\)
\(168\) 0 0
\(169\) 8.59513 0.661164
\(170\) 14.1181 1.08281
\(171\) 0 0
\(172\) −0.112048 −0.00854361
\(173\) −10.5057 −0.798732 −0.399366 0.916792i \(-0.630770\pi\)
−0.399366 + 0.916792i \(0.630770\pi\)
\(174\) 0 0
\(175\) −1.64602 −0.124428
\(176\) 0 0
\(177\) 0 0
\(178\) −21.6234 −1.62074
\(179\) 8.32331 0.622113 0.311057 0.950391i \(-0.399317\pi\)
0.311057 + 0.950391i \(0.399317\pi\)
\(180\) 0 0
\(181\) −14.8030 −1.10030 −0.550148 0.835067i \(-0.685429\pi\)
−0.550148 + 0.835067i \(0.685429\pi\)
\(182\) −6.54895 −0.485440
\(183\) 0 0
\(184\) −2.04261 −0.150583
\(185\) 3.83628 0.282049
\(186\) 0 0
\(187\) 0 0
\(188\) 0.0834496 0.00608619
\(189\) 0 0
\(190\) −14.9723 −1.08621
\(191\) 9.60676 0.695121 0.347560 0.937658i \(-0.387010\pi\)
0.347560 + 0.937658i \(0.387010\pi\)
\(192\) 0 0
\(193\) −1.48781 −0.107095 −0.0535474 0.998565i \(-0.517053\pi\)
−0.0535474 + 0.998565i \(0.517053\pi\)
\(194\) 3.39808 0.243968
\(195\) 0 0
\(196\) −0.0139645 −0.000997465 0
\(197\) 14.0434 1.00055 0.500274 0.865867i \(-0.333233\pi\)
0.500274 + 0.865867i \(0.333233\pi\)
\(198\) 0 0
\(199\) −4.28729 −0.303918 −0.151959 0.988387i \(-0.548558\pi\)
−0.151959 + 0.988387i \(0.548558\pi\)
\(200\) 4.67177 0.330344
\(201\) 0 0
\(202\) −16.7645 −1.17954
\(203\) 1.17247 0.0822915
\(204\) 0 0
\(205\) 1.67873 0.117248
\(206\) 0.558512 0.0389134
\(207\) 0 0
\(208\) 18.4575 1.27980
\(209\) 0 0
\(210\) 0 0
\(211\) −1.45527 −0.100185 −0.0500925 0.998745i \(-0.515952\pi\)
−0.0500925 + 0.998745i \(0.515952\pi\)
\(212\) 0.141669 0.00972987
\(213\) 0 0
\(214\) 4.60739 0.314955
\(215\) 14.6947 1.00217
\(216\) 0 0
\(217\) −1.30787 −0.0887837
\(218\) −4.01132 −0.271681
\(219\) 0 0
\(220\) 0 0
\(221\) −25.4204 −1.70996
\(222\) 0 0
\(223\) −4.85642 −0.325210 −0.162605 0.986691i \(-0.551990\pi\)
−0.162605 + 0.986691i \(0.551990\pi\)
\(224\) 0.0789938 0.00527799
\(225\) 0 0
\(226\) −20.4971 −1.36345
\(227\) −0.397144 −0.0263594 −0.0131797 0.999913i \(-0.504195\pi\)
−0.0131797 + 0.999913i \(0.504195\pi\)
\(228\) 0 0
\(229\) 2.18963 0.144695 0.0723475 0.997379i \(-0.476951\pi\)
0.0723475 + 0.997379i \(0.476951\pi\)
\(230\) 1.85744 0.122476
\(231\) 0 0
\(232\) −3.32773 −0.218476
\(233\) 1.26016 0.0825555 0.0412778 0.999148i \(-0.486857\pi\)
0.0412778 + 0.999148i \(0.486857\pi\)
\(234\) 0 0
\(235\) −10.9441 −0.713912
\(236\) −0.107259 −0.00698195
\(237\) 0 0
\(238\) 7.70900 0.499700
\(239\) 11.1617 0.721987 0.360994 0.932568i \(-0.382438\pi\)
0.360994 + 0.932568i \(0.382438\pi\)
\(240\) 0 0
\(241\) −21.4843 −1.38392 −0.691962 0.721934i \(-0.743254\pi\)
−0.691962 + 0.721934i \(0.743254\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0.0876430 0.00561077
\(245\) 1.83139 0.117003
\(246\) 0 0
\(247\) 26.9584 1.71532
\(248\) 3.71200 0.235712
\(249\) 0 0
\(250\) −17.1528 −1.08484
\(251\) 0.423820 0.0267513 0.0133756 0.999911i \(-0.495742\pi\)
0.0133756 + 0.999911i \(0.495742\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 7.09267 0.445033
\(255\) 0 0
\(256\) −0.335132 −0.0209457
\(257\) −17.4401 −1.08788 −0.543941 0.839124i \(-0.683068\pi\)
−0.543941 + 0.839124i \(0.683068\pi\)
\(258\) 0 0
\(259\) 2.09474 0.130161
\(260\) 0.118846 0.00737050
\(261\) 0 0
\(262\) 0.253743 0.0156763
\(263\) −1.51519 −0.0934307 −0.0467153 0.998908i \(-0.514875\pi\)
−0.0467153 + 0.998908i \(0.514875\pi\)
\(264\) 0 0
\(265\) −18.5793 −1.14132
\(266\) −8.17541 −0.501267
\(267\) 0 0
\(268\) 0.215993 0.0131939
\(269\) 2.03103 0.123834 0.0619170 0.998081i \(-0.480279\pi\)
0.0619170 + 0.998081i \(0.480279\pi\)
\(270\) 0 0
\(271\) 7.60444 0.461937 0.230968 0.972961i \(-0.425811\pi\)
0.230968 + 0.972961i \(0.425811\pi\)
\(272\) −21.7270 −1.31739
\(273\) 0 0
\(274\) −11.7307 −0.708676
\(275\) 0 0
\(276\) 0 0
\(277\) 14.4268 0.866823 0.433411 0.901196i \(-0.357310\pi\)
0.433411 + 0.901196i \(0.357310\pi\)
\(278\) −9.81019 −0.588376
\(279\) 0 0
\(280\) −5.19787 −0.310632
\(281\) −17.7496 −1.05886 −0.529428 0.848355i \(-0.677593\pi\)
−0.529428 + 0.848355i \(0.677593\pi\)
\(282\) 0 0
\(283\) −31.1361 −1.85085 −0.925426 0.378929i \(-0.876292\pi\)
−0.925426 + 0.378929i \(0.876292\pi\)
\(284\) 0.194133 0.0115197
\(285\) 0 0
\(286\) 0 0
\(287\) 0.916645 0.0541079
\(288\) 0 0
\(289\) 12.9232 0.760190
\(290\) 3.02605 0.177696
\(291\) 0 0
\(292\) −0.0839913 −0.00491522
\(293\) 24.0303 1.40386 0.701932 0.712244i \(-0.252321\pi\)
0.701932 + 0.712244i \(0.252321\pi\)
\(294\) 0 0
\(295\) 14.0665 0.818985
\(296\) −5.94532 −0.345565
\(297\) 0 0
\(298\) 4.52983 0.262406
\(299\) −3.34441 −0.193412
\(300\) 0 0
\(301\) 8.02379 0.462484
\(302\) 31.3802 1.80573
\(303\) 0 0
\(304\) 23.0416 1.32152
\(305\) −11.4940 −0.658145
\(306\) 0 0
\(307\) −5.46298 −0.311789 −0.155894 0.987774i \(-0.549826\pi\)
−0.155894 + 0.987774i \(0.549826\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.37549 −0.191715
\(311\) −13.8884 −0.787541 −0.393770 0.919209i \(-0.628830\pi\)
−0.393770 + 0.919209i \(0.628830\pi\)
\(312\) 0 0
\(313\) 27.4486 1.55148 0.775742 0.631050i \(-0.217376\pi\)
0.775742 + 0.631050i \(0.217376\pi\)
\(314\) 18.6795 1.05415
\(315\) 0 0
\(316\) 0.218418 0.0122870
\(317\) −7.82570 −0.439535 −0.219768 0.975552i \(-0.570530\pi\)
−0.219768 + 0.975552i \(0.570530\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 14.7520 0.824659
\(321\) 0 0
\(322\) 1.01423 0.0565206
\(323\) −31.7337 −1.76571
\(324\) 0 0
\(325\) 7.64917 0.424299
\(326\) 19.3328 1.07075
\(327\) 0 0
\(328\) −2.60164 −0.143651
\(329\) −5.97584 −0.329458
\(330\) 0 0
\(331\) −28.1462 −1.54705 −0.773527 0.633764i \(-0.781509\pi\)
−0.773527 + 0.633764i \(0.781509\pi\)
\(332\) −0.0611049 −0.00335357
\(333\) 0 0
\(334\) −13.1183 −0.717802
\(335\) −28.3265 −1.54764
\(336\) 0 0
\(337\) −24.9789 −1.36069 −0.680345 0.732892i \(-0.738170\pi\)
−0.680345 + 0.732892i \(0.738170\pi\)
\(338\) 12.1128 0.658852
\(339\) 0 0
\(340\) −0.139898 −0.00758701
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −22.7732 −1.22785
\(345\) 0 0
\(346\) −14.8053 −0.795939
\(347\) −20.6492 −1.10851 −0.554254 0.832347i \(-0.686997\pi\)
−0.554254 + 0.832347i \(0.686997\pi\)
\(348\) 0 0
\(349\) −5.99721 −0.321023 −0.160512 0.987034i \(-0.551314\pi\)
−0.160512 + 0.987034i \(0.551314\pi\)
\(350\) −2.31969 −0.123993
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0382 −1.27942 −0.639712 0.768615i \(-0.720946\pi\)
−0.639712 + 0.768615i \(0.720946\pi\)
\(354\) 0 0
\(355\) −25.4598 −1.35126
\(356\) 0.214267 0.0113562
\(357\) 0 0
\(358\) 11.7298 0.619938
\(359\) 10.9501 0.577925 0.288963 0.957340i \(-0.406690\pi\)
0.288963 + 0.957340i \(0.406690\pi\)
\(360\) 0 0
\(361\) 14.6536 0.771244
\(362\) −20.8613 −1.09645
\(363\) 0 0
\(364\) 0.0648939 0.00340136
\(365\) 11.0151 0.576557
\(366\) 0 0
\(367\) 10.6178 0.554243 0.277121 0.960835i \(-0.410620\pi\)
0.277121 + 0.960835i \(0.410620\pi\)
\(368\) −2.85849 −0.149009
\(369\) 0 0
\(370\) 5.40634 0.281062
\(371\) −10.1449 −0.526699
\(372\) 0 0
\(373\) 36.6036 1.89526 0.947631 0.319367i \(-0.103470\pi\)
0.947631 + 0.319367i \(0.103470\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 16.9607 0.874682
\(377\) −5.44855 −0.280615
\(378\) 0 0
\(379\) −12.6578 −0.650186 −0.325093 0.945682i \(-0.605396\pi\)
−0.325093 + 0.945682i \(0.605396\pi\)
\(380\) 0.148362 0.00761080
\(381\) 0 0
\(382\) 13.5385 0.692690
\(383\) 15.4679 0.790372 0.395186 0.918601i \(-0.370680\pi\)
0.395186 + 0.918601i \(0.370680\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.09672 −0.106720
\(387\) 0 0
\(388\) −0.0336718 −0.00170943
\(389\) 12.7130 0.644572 0.322286 0.946642i \(-0.395549\pi\)
0.322286 + 0.946642i \(0.395549\pi\)
\(390\) 0 0
\(391\) 3.93682 0.199093
\(392\) −2.83822 −0.143352
\(393\) 0 0
\(394\) 19.7909 0.997049
\(395\) −28.6446 −1.44127
\(396\) 0 0
\(397\) −18.9574 −0.951445 −0.475722 0.879596i \(-0.657813\pi\)
−0.475722 + 0.879596i \(0.657813\pi\)
\(398\) −6.04193 −0.302855
\(399\) 0 0
\(400\) 6.53780 0.326890
\(401\) 8.68208 0.433563 0.216781 0.976220i \(-0.430444\pi\)
0.216781 + 0.976220i \(0.430444\pi\)
\(402\) 0 0
\(403\) 6.07772 0.302753
\(404\) 0.166120 0.00826478
\(405\) 0 0
\(406\) 1.65233 0.0820037
\(407\) 0 0
\(408\) 0 0
\(409\) 5.71406 0.282542 0.141271 0.989971i \(-0.454881\pi\)
0.141271 + 0.989971i \(0.454881\pi\)
\(410\) 2.36578 0.116838
\(411\) 0 0
\(412\) −0.00553433 −0.000272657 0
\(413\) 7.68081 0.377948
\(414\) 0 0
\(415\) 8.01365 0.393375
\(416\) −0.367089 −0.0179980
\(417\) 0 0
\(418\) 0 0
\(419\) 27.1909 1.32836 0.664181 0.747571i \(-0.268780\pi\)
0.664181 + 0.747571i \(0.268780\pi\)
\(420\) 0 0
\(421\) −23.9651 −1.16799 −0.583993 0.811759i \(-0.698510\pi\)
−0.583993 + 0.811759i \(0.698510\pi\)
\(422\) −2.05087 −0.0998347
\(423\) 0 0
\(424\) 28.7935 1.39834
\(425\) −9.00410 −0.436763
\(426\) 0 0
\(427\) −6.27612 −0.303723
\(428\) −0.0456549 −0.00220681
\(429\) 0 0
\(430\) 20.7087 0.998663
\(431\) −16.4732 −0.793484 −0.396742 0.917930i \(-0.629859\pi\)
−0.396742 + 0.917930i \(0.629859\pi\)
\(432\) 0 0
\(433\) 20.0909 0.965509 0.482754 0.875756i \(-0.339636\pi\)
0.482754 + 0.875756i \(0.339636\pi\)
\(434\) −1.84313 −0.0884732
\(435\) 0 0
\(436\) 0.0397484 0.00190360
\(437\) −4.17500 −0.199718
\(438\) 0 0
\(439\) 26.7682 1.27758 0.638788 0.769383i \(-0.279436\pi\)
0.638788 + 0.769383i \(0.279436\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −35.8241 −1.70398
\(443\) −26.2153 −1.24553 −0.622764 0.782410i \(-0.713990\pi\)
−0.622764 + 0.782410i \(0.713990\pi\)
\(444\) 0 0
\(445\) −28.1003 −1.33208
\(446\) −6.84400 −0.324073
\(447\) 0 0
\(448\) 8.05508 0.380567
\(449\) −9.74740 −0.460008 −0.230004 0.973190i \(-0.573874\pi\)
−0.230004 + 0.973190i \(0.573874\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0.203107 0.00955336
\(453\) 0 0
\(454\) −0.559682 −0.0262672
\(455\) −8.51056 −0.398981
\(456\) 0 0
\(457\) 11.8853 0.555971 0.277986 0.960585i \(-0.410333\pi\)
0.277986 + 0.960585i \(0.410333\pi\)
\(458\) 3.08578 0.144189
\(459\) 0 0
\(460\) −0.0184055 −0.000858159 0
\(461\) −9.14737 −0.426035 −0.213018 0.977048i \(-0.568329\pi\)
−0.213018 + 0.977048i \(0.568329\pi\)
\(462\) 0 0
\(463\) 38.9342 1.80943 0.904713 0.426021i \(-0.140085\pi\)
0.904713 + 0.426021i \(0.140085\pi\)
\(464\) −4.65692 −0.216192
\(465\) 0 0
\(466\) 1.77590 0.0822668
\(467\) 20.7834 0.961742 0.480871 0.876791i \(-0.340320\pi\)
0.480871 + 0.876791i \(0.340320\pi\)
\(468\) 0 0
\(469\) −15.4673 −0.714212
\(470\) −15.4231 −0.711415
\(471\) 0 0
\(472\) −21.7998 −1.00342
\(473\) 0 0
\(474\) 0 0
\(475\) 9.54887 0.438132
\(476\) −0.0763889 −0.00350128
\(477\) 0 0
\(478\) 15.7298 0.719462
\(479\) 24.5363 1.12109 0.560546 0.828123i \(-0.310591\pi\)
0.560546 + 0.828123i \(0.310591\pi\)
\(480\) 0 0
\(481\) −9.73437 −0.443849
\(482\) −30.2771 −1.37908
\(483\) 0 0
\(484\) 0 0
\(485\) 4.41591 0.200516
\(486\) 0 0
\(487\) −12.3713 −0.560598 −0.280299 0.959913i \(-0.590434\pi\)
−0.280299 + 0.959913i \(0.590434\pi\)
\(488\) 17.8130 0.806356
\(489\) 0 0
\(490\) 2.58091 0.116594
\(491\) −16.5251 −0.745766 −0.372883 0.927878i \(-0.621631\pi\)
−0.372883 + 0.927878i \(0.621631\pi\)
\(492\) 0 0
\(493\) 6.41368 0.288858
\(494\) 37.9916 1.70932
\(495\) 0 0
\(496\) 5.19468 0.233248
\(497\) −13.9019 −0.623586
\(498\) 0 0
\(499\) −13.7410 −0.615129 −0.307565 0.951527i \(-0.599514\pi\)
−0.307565 + 0.951527i \(0.599514\pi\)
\(500\) 0.169968 0.00760121
\(501\) 0 0
\(502\) 0.597276 0.0266577
\(503\) 22.5968 1.00754 0.503770 0.863838i \(-0.331946\pi\)
0.503770 + 0.863838i \(0.331946\pi\)
\(504\) 0 0
\(505\) −21.7859 −0.969461
\(506\) 0 0
\(507\) 0 0
\(508\) −0.0702816 −0.00311824
\(509\) 21.4636 0.951358 0.475679 0.879619i \(-0.342202\pi\)
0.475679 + 0.879619i \(0.342202\pi\)
\(510\) 0 0
\(511\) 6.01462 0.266071
\(512\) −22.8598 −1.01027
\(513\) 0 0
\(514\) −24.5777 −1.08408
\(515\) 0.725804 0.0319827
\(516\) 0 0
\(517\) 0 0
\(518\) 2.95205 0.129706
\(519\) 0 0
\(520\) 24.1548 1.05926
\(521\) −34.7116 −1.52074 −0.760371 0.649489i \(-0.774983\pi\)
−0.760371 + 0.649489i \(0.774983\pi\)
\(522\) 0 0
\(523\) 19.7281 0.862651 0.431326 0.902196i \(-0.358046\pi\)
0.431326 + 0.902196i \(0.358046\pi\)
\(524\) −0.00251435 −0.000109840 0
\(525\) 0 0
\(526\) −2.13531 −0.0931039
\(527\) −7.15430 −0.311646
\(528\) 0 0
\(529\) −22.4821 −0.977481
\(530\) −26.1832 −1.13733
\(531\) 0 0
\(532\) 0.0810106 0.00351226
\(533\) −4.25970 −0.184508
\(534\) 0 0
\(535\) 5.98745 0.258860
\(536\) 43.8994 1.89617
\(537\) 0 0
\(538\) 2.86226 0.123401
\(539\) 0 0
\(540\) 0 0
\(541\) −5.54942 −0.238588 −0.119294 0.992859i \(-0.538063\pi\)
−0.119294 + 0.992859i \(0.538063\pi\)
\(542\) 10.7167 0.460321
\(543\) 0 0
\(544\) 0.432113 0.0185267
\(545\) −5.21283 −0.223293
\(546\) 0 0
\(547\) −8.44671 −0.361155 −0.180578 0.983561i \(-0.557797\pi\)
−0.180578 + 0.983561i \(0.557797\pi\)
\(548\) 0.116240 0.00496553
\(549\) 0 0
\(550\) 0 0
\(551\) −6.80173 −0.289763
\(552\) 0 0
\(553\) −15.6409 −0.665120
\(554\) 20.3312 0.863791
\(555\) 0 0
\(556\) 0.0972097 0.00412261
\(557\) 12.1835 0.516231 0.258115 0.966114i \(-0.416898\pi\)
0.258115 + 0.966114i \(0.416898\pi\)
\(558\) 0 0
\(559\) −37.2870 −1.57707
\(560\) −7.27404 −0.307384
\(561\) 0 0
\(562\) −25.0140 −1.05515
\(563\) 27.3535 1.15281 0.576407 0.817163i \(-0.304454\pi\)
0.576407 + 0.817163i \(0.304454\pi\)
\(564\) 0 0
\(565\) −26.6366 −1.12061
\(566\) −43.8792 −1.84438
\(567\) 0 0
\(568\) 39.4566 1.65556
\(569\) −7.13524 −0.299125 −0.149562 0.988752i \(-0.547786\pi\)
−0.149562 + 0.988752i \(0.547786\pi\)
\(570\) 0 0
\(571\) 32.4839 1.35941 0.679705 0.733486i \(-0.262108\pi\)
0.679705 + 0.733486i \(0.262108\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.29180 0.0539186
\(575\) −1.18461 −0.0494018
\(576\) 0 0
\(577\) 34.7819 1.44799 0.723995 0.689806i \(-0.242304\pi\)
0.723995 + 0.689806i \(0.242304\pi\)
\(578\) 18.2123 0.757532
\(579\) 0 0
\(580\) −0.0299853 −0.00124507
\(581\) 4.37573 0.181536
\(582\) 0 0
\(583\) 0 0
\(584\) −17.0708 −0.706395
\(585\) 0 0
\(586\) 33.8651 1.39896
\(587\) 14.4749 0.597445 0.298722 0.954340i \(-0.403440\pi\)
0.298722 + 0.954340i \(0.403440\pi\)
\(588\) 0 0
\(589\) 7.58716 0.312623
\(590\) 19.8235 0.816120
\(591\) 0 0
\(592\) −8.32004 −0.341952
\(593\) 15.0291 0.617169 0.308585 0.951197i \(-0.400145\pi\)
0.308585 + 0.951197i \(0.400145\pi\)
\(594\) 0 0
\(595\) 10.0181 0.410701
\(596\) −0.0448863 −0.00183862
\(597\) 0 0
\(598\) −4.71316 −0.192736
\(599\) −1.76045 −0.0719302 −0.0359651 0.999353i \(-0.511451\pi\)
−0.0359651 + 0.999353i \(0.511451\pi\)
\(600\) 0 0
\(601\) 23.4365 0.955993 0.477997 0.878362i \(-0.341363\pi\)
0.477997 + 0.878362i \(0.341363\pi\)
\(602\) 11.3077 0.460866
\(603\) 0 0
\(604\) −0.310948 −0.0126523
\(605\) 0 0
\(606\) 0 0
\(607\) −25.2785 −1.02602 −0.513012 0.858382i \(-0.671470\pi\)
−0.513012 + 0.858382i \(0.671470\pi\)
\(608\) −0.458257 −0.0185848
\(609\) 0 0
\(610\) −16.1981 −0.655843
\(611\) 27.7700 1.12346
\(612\) 0 0
\(613\) 1.16094 0.0468900 0.0234450 0.999725i \(-0.492537\pi\)
0.0234450 + 0.999725i \(0.492537\pi\)
\(614\) −7.69880 −0.310698
\(615\) 0 0
\(616\) 0 0
\(617\) −12.9711 −0.522197 −0.261098 0.965312i \(-0.584085\pi\)
−0.261098 + 0.965312i \(0.584085\pi\)
\(618\) 0 0
\(619\) 45.7920 1.84053 0.920267 0.391291i \(-0.127971\pi\)
0.920267 + 0.391291i \(0.127971\pi\)
\(620\) 0.0334479 0.00134330
\(621\) 0 0
\(622\) −19.5725 −0.784787
\(623\) −15.3437 −0.614733
\(624\) 0 0
\(625\) −14.0605 −0.562419
\(626\) 38.6824 1.54606
\(627\) 0 0
\(628\) −0.185097 −0.00738615
\(629\) 11.4587 0.456887
\(630\) 0 0
\(631\) −12.6207 −0.502421 −0.251211 0.967932i \(-0.580829\pi\)
−0.251211 + 0.967932i \(0.580829\pi\)
\(632\) 44.3924 1.76583
\(633\) 0 0
\(634\) −11.0285 −0.437998
\(635\) 9.21713 0.365771
\(636\) 0 0
\(637\) −4.64706 −0.184123
\(638\) 0 0
\(639\) 0 0
\(640\) 20.5001 0.810338
\(641\) −27.9567 −1.10422 −0.552112 0.833770i \(-0.686178\pi\)
−0.552112 + 0.833770i \(0.686178\pi\)
\(642\) 0 0
\(643\) −49.6981 −1.95990 −0.979950 0.199243i \(-0.936152\pi\)
−0.979950 + 0.199243i \(0.936152\pi\)
\(644\) −0.0100500 −0.000396026 0
\(645\) 0 0
\(646\) −44.7212 −1.75953
\(647\) 11.0067 0.432718 0.216359 0.976314i \(-0.430582\pi\)
0.216359 + 0.976314i \(0.430582\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 10.7797 0.422816
\(651\) 0 0
\(652\) −0.191570 −0.00750246
\(653\) −28.9364 −1.13237 −0.566185 0.824278i \(-0.691581\pi\)
−0.566185 + 0.824278i \(0.691581\pi\)
\(654\) 0 0
\(655\) 0.329747 0.0128843
\(656\) −3.64080 −0.142149
\(657\) 0 0
\(658\) −8.42155 −0.328306
\(659\) −10.8405 −0.422288 −0.211144 0.977455i \(-0.567719\pi\)
−0.211144 + 0.977455i \(0.567719\pi\)
\(660\) 0 0
\(661\) 20.3444 0.791305 0.395652 0.918400i \(-0.370519\pi\)
0.395652 + 0.918400i \(0.370519\pi\)
\(662\) −39.6655 −1.54164
\(663\) 0 0
\(664\) −12.4193 −0.481961
\(665\) −10.6242 −0.411989
\(666\) 0 0
\(667\) 0.843809 0.0326724
\(668\) 0.129990 0.00502946
\(669\) 0 0
\(670\) −39.9197 −1.54223
\(671\) 0 0
\(672\) 0 0
\(673\) −12.0788 −0.465604 −0.232802 0.972524i \(-0.574789\pi\)
−0.232802 + 0.972524i \(0.574789\pi\)
\(674\) −35.2020 −1.35593
\(675\) 0 0
\(676\) −0.120027 −0.00461642
\(677\) −3.39630 −0.130530 −0.0652651 0.997868i \(-0.520789\pi\)
−0.0652651 + 0.997868i \(0.520789\pi\)
\(678\) 0 0
\(679\) 2.41124 0.0925349
\(680\) −28.4335 −1.09037
\(681\) 0 0
\(682\) 0 0
\(683\) 4.75643 0.182000 0.0909999 0.995851i \(-0.470994\pi\)
0.0909999 + 0.995851i \(0.470994\pi\)
\(684\) 0 0
\(685\) −15.2444 −0.582458
\(686\) 1.40927 0.0538061
\(687\) 0 0
\(688\) −31.8695 −1.21501
\(689\) 47.1441 1.79605
\(690\) 0 0
\(691\) 6.63388 0.252365 0.126182 0.992007i \(-0.459728\pi\)
0.126182 + 0.992007i \(0.459728\pi\)
\(692\) 0.146707 0.00557695
\(693\) 0 0
\(694\) −29.1003 −1.10463
\(695\) −12.7486 −0.483583
\(696\) 0 0
\(697\) 5.01425 0.189928
\(698\) −8.45168 −0.319901
\(699\) 0 0
\(700\) 0.0229859 0.000868786 0
\(701\) −3.03003 −0.114443 −0.0572213 0.998362i \(-0.518224\pi\)
−0.0572213 + 0.998362i \(0.518224\pi\)
\(702\) 0 0
\(703\) −12.1519 −0.458319
\(704\) 0 0
\(705\) 0 0
\(706\) −33.8762 −1.27495
\(707\) −11.8959 −0.447390
\(708\) 0 0
\(709\) −13.6570 −0.512901 −0.256451 0.966557i \(-0.582553\pi\)
−0.256451 + 0.966557i \(0.582553\pi\)
\(710\) −35.8796 −1.34654
\(711\) 0 0
\(712\) 43.5488 1.63206
\(713\) −0.941248 −0.0352500
\(714\) 0 0
\(715\) 0 0
\(716\) −0.116231 −0.00434376
\(717\) 0 0
\(718\) 15.4317 0.575904
\(719\) 1.90029 0.0708689 0.0354344 0.999372i \(-0.488719\pi\)
0.0354344 + 0.999372i \(0.488719\pi\)
\(720\) 0 0
\(721\) 0.396314 0.0147595
\(722\) 20.6509 0.768547
\(723\) 0 0
\(724\) 0.206716 0.00768255
\(725\) −1.92992 −0.0716754
\(726\) 0 0
\(727\) −13.8211 −0.512595 −0.256298 0.966598i \(-0.582503\pi\)
−0.256298 + 0.966598i \(0.582503\pi\)
\(728\) 13.1893 0.488830
\(729\) 0 0
\(730\) 15.5232 0.574540
\(731\) 43.8918 1.62340
\(732\) 0 0
\(733\) 48.3744 1.78675 0.893374 0.449314i \(-0.148331\pi\)
0.893374 + 0.449314i \(0.148331\pi\)
\(734\) 14.9633 0.552304
\(735\) 0 0
\(736\) 0.0568504 0.00209553
\(737\) 0 0
\(738\) 0 0
\(739\) 23.2081 0.853724 0.426862 0.904317i \(-0.359619\pi\)
0.426862 + 0.904317i \(0.359619\pi\)
\(740\) −0.0535717 −0.00196934
\(741\) 0 0
\(742\) −14.2969 −0.524857
\(743\) 44.8311 1.64469 0.822347 0.568986i \(-0.192664\pi\)
0.822347 + 0.568986i \(0.192664\pi\)
\(744\) 0 0
\(745\) 5.88665 0.215670
\(746\) 51.5843 1.88863
\(747\) 0 0
\(748\) 0 0
\(749\) 3.26935 0.119460
\(750\) 0 0
\(751\) −41.1856 −1.50288 −0.751442 0.659799i \(-0.770641\pi\)
−0.751442 + 0.659799i \(0.770641\pi\)
\(752\) 23.7353 0.865537
\(753\) 0 0
\(754\) −7.67847 −0.279633
\(755\) 40.7795 1.48412
\(756\) 0 0
\(757\) −21.8888 −0.795561 −0.397781 0.917481i \(-0.630219\pi\)
−0.397781 + 0.917481i \(0.630219\pi\)
\(758\) −17.8382 −0.647912
\(759\) 0 0
\(760\) 30.1537 1.09379
\(761\) 35.7154 1.29468 0.647340 0.762201i \(-0.275881\pi\)
0.647340 + 0.762201i \(0.275881\pi\)
\(762\) 0 0
\(763\) −2.84638 −0.103046
\(764\) −0.134154 −0.00485351
\(765\) 0 0
\(766\) 21.7984 0.787608
\(767\) −35.6931 −1.28880
\(768\) 0 0
\(769\) 5.30246 0.191212 0.0956058 0.995419i \(-0.469521\pi\)
0.0956058 + 0.995419i \(0.469521\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.0207765 0.000747763 0
\(773\) 49.8912 1.79446 0.897231 0.441561i \(-0.145575\pi\)
0.897231 + 0.441561i \(0.145575\pi\)
\(774\) 0 0
\(775\) 2.15278 0.0773300
\(776\) −6.84362 −0.245672
\(777\) 0 0
\(778\) 17.9160 0.642318
\(779\) −5.31762 −0.190524
\(780\) 0 0
\(781\) 0 0
\(782\) 5.54803 0.198397
\(783\) 0 0
\(784\) −3.97188 −0.141853
\(785\) 24.2746 0.866398
\(786\) 0 0
\(787\) 35.8570 1.27816 0.639081 0.769139i \(-0.279315\pi\)
0.639081 + 0.769139i \(0.279315\pi\)
\(788\) −0.196109 −0.00698609
\(789\) 0 0
\(790\) −40.3679 −1.43623
\(791\) −14.5445 −0.517144
\(792\) 0 0
\(793\) 29.1655 1.03570
\(794\) −26.7161 −0.948117
\(795\) 0 0
\(796\) 0.0598699 0.00212203
\(797\) −14.7413 −0.522163 −0.261081 0.965317i \(-0.584079\pi\)
−0.261081 + 0.965317i \(0.584079\pi\)
\(798\) 0 0
\(799\) −32.6891 −1.15646
\(800\) −0.130026 −0.00459710
\(801\) 0 0
\(802\) 12.2354 0.432046
\(803\) 0 0
\(804\) 0 0
\(805\) 1.31802 0.0464540
\(806\) 8.56514 0.301694
\(807\) 0 0
\(808\) 33.7630 1.18778
\(809\) 26.9758 0.948420 0.474210 0.880412i \(-0.342734\pi\)
0.474210 + 0.880412i \(0.342734\pi\)
\(810\) 0 0
\(811\) 1.46440 0.0514219 0.0257109 0.999669i \(-0.491815\pi\)
0.0257109 + 0.999669i \(0.491815\pi\)
\(812\) −0.0163730 −0.000574581 0
\(813\) 0 0
\(814\) 0 0
\(815\) 25.1236 0.880041
\(816\) 0 0
\(817\) −46.5474 −1.62849
\(818\) 8.05264 0.281554
\(819\) 0 0
\(820\) −0.0234427 −0.000818654 0
\(821\) −30.6287 −1.06895 −0.534474 0.845185i \(-0.679490\pi\)
−0.534474 + 0.845185i \(0.679490\pi\)
\(822\) 0 0
\(823\) −24.5338 −0.855195 −0.427598 0.903969i \(-0.640640\pi\)
−0.427598 + 0.903969i \(0.640640\pi\)
\(824\) −1.12482 −0.0391851
\(825\) 0 0
\(826\) 10.8243 0.376626
\(827\) 6.56686 0.228352 0.114176 0.993461i \(-0.463577\pi\)
0.114176 + 0.993461i \(0.463577\pi\)
\(828\) 0 0
\(829\) −19.8442 −0.689219 −0.344610 0.938746i \(-0.611989\pi\)
−0.344610 + 0.938746i \(0.611989\pi\)
\(830\) 11.2934 0.391999
\(831\) 0 0
\(832\) −37.4324 −1.29773
\(833\) 5.47021 0.189532
\(834\) 0 0
\(835\) −17.0476 −0.589958
\(836\) 0 0
\(837\) 0 0
\(838\) 38.3193 1.32372
\(839\) 7.96051 0.274827 0.137414 0.990514i \(-0.456121\pi\)
0.137414 + 0.990514i \(0.456121\pi\)
\(840\) 0 0
\(841\) −27.6253 −0.952597
\(842\) −33.7732 −1.16390
\(843\) 0 0
\(844\) 0.0203222 0.000699518 0
\(845\) 15.7410 0.541507
\(846\) 0 0
\(847\) 0 0
\(848\) 40.2944 1.38372
\(849\) 0 0
\(850\) −12.6892 −0.435236
\(851\) 1.50755 0.0516780
\(852\) 0 0
\(853\) 34.0732 1.16664 0.583322 0.812241i \(-0.301753\pi\)
0.583322 + 0.812241i \(0.301753\pi\)
\(854\) −8.84474 −0.302661
\(855\) 0 0
\(856\) −9.27913 −0.317154
\(857\) −24.8539 −0.848992 −0.424496 0.905430i \(-0.639549\pi\)
−0.424496 + 0.905430i \(0.639549\pi\)
\(858\) 0 0
\(859\) 2.05654 0.0701683 0.0350841 0.999384i \(-0.488830\pi\)
0.0350841 + 0.999384i \(0.488830\pi\)
\(860\) −0.205204 −0.00699739
\(861\) 0 0
\(862\) −23.2151 −0.790709
\(863\) 0.259476 0.00883265 0.00441633 0.999990i \(-0.498594\pi\)
0.00441633 + 0.999990i \(0.498594\pi\)
\(864\) 0 0
\(865\) −19.2400 −0.654178
\(866\) 28.3135 0.962132
\(867\) 0 0
\(868\) 0.0182637 0.000619910 0
\(869\) 0 0
\(870\) 0 0
\(871\) 71.8773 2.43547
\(872\) 8.07865 0.273578
\(873\) 0 0
\(874\) −5.88370 −0.199019
\(875\) −12.1714 −0.411470
\(876\) 0 0
\(877\) 18.5930 0.627840 0.313920 0.949449i \(-0.398358\pi\)
0.313920 + 0.949449i \(0.398358\pi\)
\(878\) 37.7235 1.27311
\(879\) 0 0
\(880\) 0 0
\(881\) −6.45292 −0.217404 −0.108702 0.994074i \(-0.534670\pi\)
−0.108702 + 0.994074i \(0.534670\pi\)
\(882\) 0 0
\(883\) −0.278487 −0.00937185 −0.00468592 0.999989i \(-0.501492\pi\)
−0.00468592 + 0.999989i \(0.501492\pi\)
\(884\) 0.354983 0.0119394
\(885\) 0 0
\(886\) −36.9444 −1.24117
\(887\) −30.5570 −1.02600 −0.513001 0.858388i \(-0.671466\pi\)
−0.513001 + 0.858388i \(0.671466\pi\)
\(888\) 0 0
\(889\) 5.03287 0.168797
\(890\) −39.6008 −1.32742
\(891\) 0 0
\(892\) 0.0678176 0.00227070
\(893\) 34.6669 1.16008
\(894\) 0 0
\(895\) 15.2432 0.509524
\(896\) 11.1938 0.373958
\(897\) 0 0
\(898\) −13.7367 −0.458400
\(899\) −1.53344 −0.0511430
\(900\) 0 0
\(901\) −55.4949 −1.84880
\(902\) 0 0
\(903\) 0 0
\(904\) 41.2805 1.37297
\(905\) −27.1099 −0.901165
\(906\) 0 0
\(907\) 31.9347 1.06037 0.530187 0.847881i \(-0.322122\pi\)
0.530187 + 0.847881i \(0.322122\pi\)
\(908\) 0.00554592 0.000184048 0
\(909\) 0 0
\(910\) −11.9937 −0.397586
\(911\) −2.80042 −0.0927821 −0.0463911 0.998923i \(-0.514772\pi\)
−0.0463911 + 0.998923i \(0.514772\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 16.7496 0.554027
\(915\) 0 0
\(916\) −0.0305772 −0.00101030
\(917\) 0.180053 0.00594587
\(918\) 0 0
\(919\) −15.8339 −0.522311 −0.261156 0.965297i \(-0.584104\pi\)
−0.261156 + 0.965297i \(0.584104\pi\)
\(920\) −3.74081 −0.123331
\(921\) 0 0
\(922\) −12.8911 −0.424545
\(923\) 64.6030 2.12643
\(924\) 0 0
\(925\) −3.44799 −0.113369
\(926\) 54.8687 1.80310
\(927\) 0 0
\(928\) 0.0926181 0.00304034
\(929\) 27.4834 0.901699 0.450850 0.892600i \(-0.351121\pi\)
0.450850 + 0.892600i \(0.351121\pi\)
\(930\) 0 0
\(931\) −5.80118 −0.190126
\(932\) −0.0175975 −0.000576424 0
\(933\) 0 0
\(934\) 29.2894 0.958379
\(935\) 0 0
\(936\) 0 0
\(937\) −12.3086 −0.402105 −0.201053 0.979580i \(-0.564436\pi\)
−0.201053 + 0.979580i \(0.564436\pi\)
\(938\) −21.7975 −0.711714
\(939\) 0 0
\(940\) 0.152829 0.00498472
\(941\) −1.46014 −0.0475991 −0.0237996 0.999717i \(-0.507576\pi\)
−0.0237996 + 0.999717i \(0.507576\pi\)
\(942\) 0 0
\(943\) 0.659693 0.0214826
\(944\) −30.5072 −0.992925
\(945\) 0 0
\(946\) 0 0
\(947\) −11.0714 −0.359771 −0.179885 0.983688i \(-0.557573\pi\)
−0.179885 + 0.983688i \(0.557573\pi\)
\(948\) 0 0
\(949\) −27.9503 −0.907305
\(950\) 13.4569 0.436600
\(951\) 0 0
\(952\) −15.5256 −0.503189
\(953\) 14.8234 0.480176 0.240088 0.970751i \(-0.422824\pi\)
0.240088 + 0.970751i \(0.422824\pi\)
\(954\) 0 0
\(955\) 17.5937 0.569318
\(956\) −0.155867 −0.00504110
\(957\) 0 0
\(958\) 34.5782 1.11717
\(959\) −8.32395 −0.268794
\(960\) 0 0
\(961\) −29.2895 −0.944822
\(962\) −13.7183 −0.442297
\(963\) 0 0
\(964\) 0.300018 0.00966292
\(965\) −2.72475 −0.0877129
\(966\) 0 0
\(967\) 16.5193 0.531224 0.265612 0.964080i \(-0.414426\pi\)
0.265612 + 0.964080i \(0.414426\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.22320 0.199815
\(971\) −40.7993 −1.30931 −0.654656 0.755927i \(-0.727187\pi\)
−0.654656 + 0.755927i \(0.727187\pi\)
\(972\) 0 0
\(973\) −6.96119 −0.223166
\(974\) −17.4345 −0.558637
\(975\) 0 0
\(976\) 24.9280 0.797925
\(977\) −49.1145 −1.57131 −0.785656 0.618664i \(-0.787674\pi\)
−0.785656 + 0.618664i \(0.787674\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.0255744 −0.000816945 0
\(981\) 0 0
\(982\) −23.2883 −0.743158
\(983\) −1.10679 −0.0353011 −0.0176505 0.999844i \(-0.505619\pi\)
−0.0176505 + 0.999844i \(0.505619\pi\)
\(984\) 0 0
\(985\) 25.7188 0.819470
\(986\) 9.03860 0.287848
\(987\) 0 0
\(988\) −0.376461 −0.0119768
\(989\) 5.77458 0.183621
\(990\) 0 0
\(991\) 41.7851 1.32735 0.663674 0.748022i \(-0.268996\pi\)
0.663674 + 0.748022i \(0.268996\pi\)
\(992\) −0.103313 −0.00328020
\(993\) 0 0
\(994\) −19.5915 −0.621405
\(995\) −7.85168 −0.248915
\(996\) 0 0
\(997\) 44.1945 1.39965 0.699827 0.714313i \(-0.253261\pi\)
0.699827 + 0.714313i \(0.253261\pi\)
\(998\) −19.3647 −0.612978
\(999\) 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.ct.1.6 8
3.2 odd 2 847.2.a.p.1.3 8
11.3 even 5 693.2.m.i.64.3 16
11.4 even 5 693.2.m.i.379.3 16
11.10 odd 2 7623.2.a.cw.1.3 8
21.20 even 2 5929.2.a.bt.1.3 8
33.2 even 10 847.2.f.v.323.2 16
33.5 odd 10 847.2.f.w.729.3 16
33.8 even 10 847.2.f.x.372.3 16
33.14 odd 10 77.2.f.b.64.2 16
33.17 even 10 847.2.f.v.729.2 16
33.20 odd 10 847.2.f.w.323.3 16
33.26 odd 10 77.2.f.b.71.2 yes 16
33.29 even 10 847.2.f.x.148.3 16
33.32 even 2 847.2.a.o.1.6 8
231.26 even 30 539.2.q.f.214.2 32
231.47 even 30 539.2.q.f.361.3 32
231.59 even 30 539.2.q.f.324.3 32
231.80 even 30 539.2.q.f.471.2 32
231.125 even 10 539.2.f.e.148.2 16
231.146 even 10 539.2.f.e.295.2 16
231.158 odd 30 539.2.q.g.324.3 32
231.179 odd 30 539.2.q.g.471.2 32
231.191 odd 30 539.2.q.g.214.2 32
231.212 odd 30 539.2.q.g.361.3 32
231.230 odd 2 5929.2.a.bs.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.64.2 16 33.14 odd 10
77.2.f.b.71.2 yes 16 33.26 odd 10
539.2.f.e.148.2 16 231.125 even 10
539.2.f.e.295.2 16 231.146 even 10
539.2.q.f.214.2 32 231.26 even 30
539.2.q.f.324.3 32 231.59 even 30
539.2.q.f.361.3 32 231.47 even 30
539.2.q.f.471.2 32 231.80 even 30
539.2.q.g.214.2 32 231.191 odd 30
539.2.q.g.324.3 32 231.158 odd 30
539.2.q.g.361.3 32 231.212 odd 30
539.2.q.g.471.2 32 231.179 odd 30
693.2.m.i.64.3 16 11.3 even 5
693.2.m.i.379.3 16 11.4 even 5
847.2.a.o.1.6 8 33.32 even 2
847.2.a.p.1.3 8 3.2 odd 2
847.2.f.v.323.2 16 33.2 even 10
847.2.f.v.729.2 16 33.17 even 10
847.2.f.w.323.3 16 33.20 odd 10
847.2.f.w.729.3 16 33.5 odd 10
847.2.f.x.148.3 16 33.29 even 10
847.2.f.x.372.3 16 33.8 even 10
5929.2.a.bs.1.6 8 231.230 odd 2
5929.2.a.bt.1.3 8 21.20 even 2
7623.2.a.ct.1.6 8 1.1 even 1 trivial
7623.2.a.cw.1.3 8 11.10 odd 2