Properties

Label 7623.2.a.ct.1.4
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 1.4
Root \(0.669744\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +O(q^{10})\) \(q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +1.43578 q^{10} +2.52826 q^{13} -0.669744 q^{14} +1.50986 q^{16} +1.79092 q^{17} -6.72740 q^{19} +3.32595 q^{20} +3.16429 q^{23} -0.404214 q^{25} -1.69329 q^{26} -1.55144 q^{28} +0.924170 q^{29} +3.00178 q^{31} -5.76834 q^{32} -1.19946 q^{34} -2.14378 q^{35} -1.50718 q^{37} +4.50564 q^{38} -5.09910 q^{40} -5.56104 q^{41} -8.42985 q^{43} -2.11926 q^{46} -4.39771 q^{47} +1.00000 q^{49} +0.270720 q^{50} -3.92246 q^{52} -0.667421 q^{53} +2.37856 q^{56} -0.618957 q^{58} +0.368360 q^{59} +5.01149 q^{61} -2.01043 q^{62} +0.843584 q^{64} -5.42003 q^{65} -0.902129 q^{67} -2.77851 q^{68} +1.43578 q^{70} +14.8694 q^{71} +8.03816 q^{73} +1.00942 q^{74} +10.4372 q^{76} +4.05668 q^{79} -3.23681 q^{80} +3.72447 q^{82} +4.05134 q^{83} -3.83934 q^{85} +5.64584 q^{86} +8.30727 q^{89} +2.52826 q^{91} -4.90921 q^{92} +2.94534 q^{94} +14.4221 q^{95} +8.51583 q^{97} -0.669744 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + O(q^{10}) \) \( 8q - q^{2} + 7q^{4} - 10q^{5} + 8q^{7} + 6q^{10} - 6q^{13} - q^{14} + q^{16} + 5q^{17} - 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} + 7q^{28} - 9q^{29} + 9q^{31} - 16q^{32} - 12q^{34} - 10q^{35} + 7q^{37} + 10q^{38} + 5q^{40} + 10q^{41} - 4q^{43} + 4q^{46} - 16q^{47} + 8q^{49} - 6q^{50} - 41q^{52} - 37q^{53} - 15q^{58} - q^{59} + 19q^{61} + 18q^{62} - 4q^{64} + 4q^{65} - 19q^{67} - 9q^{68} + 6q^{70} - 13q^{71} - 25q^{73} - 33q^{74} + 26q^{76} - 4q^{80} - 13q^{82} + 25q^{83} + 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} - 42q^{94} - 21q^{95} + 15q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.669744 −0.473580 −0.236790 0.971561i \(-0.576095\pi\)
−0.236790 + 0.971561i \(0.576095\pi\)
\(3\) 0 0
\(4\) −1.55144 −0.775722
\(5\) −2.14378 −0.958727 −0.479363 0.877616i \(-0.659132\pi\)
−0.479363 + 0.877616i \(0.659132\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.37856 0.840947
\(9\) 0 0
\(10\) 1.43578 0.454034
\(11\) 0 0
\(12\) 0 0
\(13\) 2.52826 0.701214 0.350607 0.936523i \(-0.385975\pi\)
0.350607 + 0.936523i \(0.385975\pi\)
\(14\) −0.669744 −0.178997
\(15\) 0 0
\(16\) 1.50986 0.377465
\(17\) 1.79092 0.434362 0.217181 0.976131i \(-0.430314\pi\)
0.217181 + 0.976131i \(0.430314\pi\)
\(18\) 0 0
\(19\) −6.72740 −1.54337 −0.771686 0.636004i \(-0.780586\pi\)
−0.771686 + 0.636004i \(0.780586\pi\)
\(20\) 3.32595 0.743705
\(21\) 0 0
\(22\) 0 0
\(23\) 3.16429 0.659799 0.329900 0.944016i \(-0.392985\pi\)
0.329900 + 0.944016i \(0.392985\pi\)
\(24\) 0 0
\(25\) −0.404214 −0.0808427
\(26\) −1.69329 −0.332081
\(27\) 0 0
\(28\) −1.55144 −0.293195
\(29\) 0.924170 0.171614 0.0858070 0.996312i \(-0.472653\pi\)
0.0858070 + 0.996312i \(0.472653\pi\)
\(30\) 0 0
\(31\) 3.00178 0.539136 0.269568 0.962981i \(-0.413119\pi\)
0.269568 + 0.962981i \(0.413119\pi\)
\(32\) −5.76834 −1.01971
\(33\) 0 0
\(34\) −1.19946 −0.205705
\(35\) −2.14378 −0.362365
\(36\) 0 0
\(37\) −1.50718 −0.247779 −0.123889 0.992296i \(-0.539537\pi\)
−0.123889 + 0.992296i \(0.539537\pi\)
\(38\) 4.50564 0.730911
\(39\) 0 0
\(40\) −5.09910 −0.806239
\(41\) −5.56104 −0.868488 −0.434244 0.900795i \(-0.642984\pi\)
−0.434244 + 0.900795i \(0.642984\pi\)
\(42\) 0 0
\(43\) −8.42985 −1.28554 −0.642770 0.766059i \(-0.722215\pi\)
−0.642770 + 0.766059i \(0.722215\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.11926 −0.312468
\(47\) −4.39771 −0.641471 −0.320736 0.947169i \(-0.603930\pi\)
−0.320736 + 0.947169i \(0.603930\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.270720 0.0382855
\(51\) 0 0
\(52\) −3.92246 −0.543947
\(53\) −0.667421 −0.0916773 −0.0458387 0.998949i \(-0.514596\pi\)
−0.0458387 + 0.998949i \(0.514596\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.37856 0.317848
\(57\) 0 0
\(58\) −0.618957 −0.0812731
\(59\) 0.368360 0.0479564 0.0239782 0.999712i \(-0.492367\pi\)
0.0239782 + 0.999712i \(0.492367\pi\)
\(60\) 0 0
\(61\) 5.01149 0.641655 0.320828 0.947138i \(-0.396039\pi\)
0.320828 + 0.947138i \(0.396039\pi\)
\(62\) −2.01043 −0.255324
\(63\) 0 0
\(64\) 0.843584 0.105448
\(65\) −5.42003 −0.672273
\(66\) 0 0
\(67\) −0.902129 −0.110213 −0.0551063 0.998480i \(-0.517550\pi\)
−0.0551063 + 0.998480i \(0.517550\pi\)
\(68\) −2.77851 −0.336944
\(69\) 0 0
\(70\) 1.43578 0.171609
\(71\) 14.8694 1.76467 0.882335 0.470623i \(-0.155971\pi\)
0.882335 + 0.470623i \(0.155971\pi\)
\(72\) 0 0
\(73\) 8.03816 0.940795 0.470398 0.882455i \(-0.344110\pi\)
0.470398 + 0.882455i \(0.344110\pi\)
\(74\) 1.00942 0.117343
\(75\) 0 0
\(76\) 10.4372 1.19723
\(77\) 0 0
\(78\) 0 0
\(79\) 4.05668 0.456412 0.228206 0.973613i \(-0.426714\pi\)
0.228206 + 0.973613i \(0.426714\pi\)
\(80\) −3.23681 −0.361886
\(81\) 0 0
\(82\) 3.72447 0.411299
\(83\) 4.05134 0.444692 0.222346 0.974968i \(-0.428628\pi\)
0.222346 + 0.974968i \(0.428628\pi\)
\(84\) 0 0
\(85\) −3.83934 −0.416434
\(86\) 5.64584 0.608807
\(87\) 0 0
\(88\) 0 0
\(89\) 8.30727 0.880569 0.440284 0.897858i \(-0.354878\pi\)
0.440284 + 0.897858i \(0.354878\pi\)
\(90\) 0 0
\(91\) 2.52826 0.265034
\(92\) −4.90921 −0.511820
\(93\) 0 0
\(94\) 2.94534 0.303788
\(95\) 14.4221 1.47967
\(96\) 0 0
\(97\) 8.51583 0.864652 0.432326 0.901717i \(-0.357693\pi\)
0.432326 + 0.901717i \(0.357693\pi\)
\(98\) −0.669744 −0.0676544
\(99\) 0 0
\(100\) 0.627115 0.0627115
\(101\) 4.02707 0.400708 0.200354 0.979724i \(-0.435791\pi\)
0.200354 + 0.979724i \(0.435791\pi\)
\(102\) 0 0
\(103\) −17.6258 −1.73672 −0.868362 0.495930i \(-0.834827\pi\)
−0.868362 + 0.495930i \(0.834827\pi\)
\(104\) 6.01362 0.589684
\(105\) 0 0
\(106\) 0.447001 0.0434166
\(107\) 15.4037 1.48913 0.744567 0.667548i \(-0.232656\pi\)
0.744567 + 0.667548i \(0.232656\pi\)
\(108\) 0 0
\(109\) −18.9265 −1.81283 −0.906416 0.422386i \(-0.861193\pi\)
−0.906416 + 0.422386i \(0.861193\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.50986 0.142669
\(113\) −1.68062 −0.158100 −0.0790498 0.996871i \(-0.525189\pi\)
−0.0790498 + 0.996871i \(0.525189\pi\)
\(114\) 0 0
\(115\) −6.78353 −0.632567
\(116\) −1.43380 −0.133125
\(117\) 0 0
\(118\) −0.246707 −0.0227112
\(119\) 1.79092 0.164173
\(120\) 0 0
\(121\) 0 0
\(122\) −3.35641 −0.303875
\(123\) 0 0
\(124\) −4.65710 −0.418220
\(125\) 11.5854 1.03623
\(126\) 0 0
\(127\) 17.5669 1.55881 0.779406 0.626519i \(-0.215521\pi\)
0.779406 + 0.626519i \(0.215521\pi\)
\(128\) 10.9717 0.969769
\(129\) 0 0
\(130\) 3.63003 0.318375
\(131\) −6.72557 −0.587616 −0.293808 0.955865i \(-0.594923\pi\)
−0.293808 + 0.955865i \(0.594923\pi\)
\(132\) 0 0
\(133\) −6.72740 −0.583340
\(134\) 0.604195 0.0521945
\(135\) 0 0
\(136\) 4.25981 0.365275
\(137\) −13.8676 −1.18479 −0.592396 0.805647i \(-0.701818\pi\)
−0.592396 + 0.805647i \(0.701818\pi\)
\(138\) 0 0
\(139\) 14.2707 1.21043 0.605214 0.796063i \(-0.293088\pi\)
0.605214 + 0.796063i \(0.293088\pi\)
\(140\) 3.32595 0.281094
\(141\) 0 0
\(142\) −9.95867 −0.835713
\(143\) 0 0
\(144\) 0 0
\(145\) −1.98122 −0.164531
\(146\) −5.38351 −0.445542
\(147\) 0 0
\(148\) 2.33830 0.192207
\(149\) −2.62292 −0.214878 −0.107439 0.994212i \(-0.534265\pi\)
−0.107439 + 0.994212i \(0.534265\pi\)
\(150\) 0 0
\(151\) −2.98960 −0.243290 −0.121645 0.992574i \(-0.538817\pi\)
−0.121645 + 0.992574i \(0.538817\pi\)
\(152\) −16.0015 −1.29789
\(153\) 0 0
\(154\) 0 0
\(155\) −6.43516 −0.516884
\(156\) 0 0
\(157\) 11.2547 0.898222 0.449111 0.893476i \(-0.351741\pi\)
0.449111 + 0.893476i \(0.351741\pi\)
\(158\) −2.71694 −0.216148
\(159\) 0 0
\(160\) 12.3660 0.977621
\(161\) 3.16429 0.249381
\(162\) 0 0
\(163\) −18.2030 −1.42577 −0.712883 0.701283i \(-0.752611\pi\)
−0.712883 + 0.701283i \(0.752611\pi\)
\(164\) 8.62763 0.673705
\(165\) 0 0
\(166\) −2.71336 −0.210598
\(167\) −19.9312 −1.54233 −0.771163 0.636638i \(-0.780325\pi\)
−0.771163 + 0.636638i \(0.780325\pi\)
\(168\) 0 0
\(169\) −6.60789 −0.508299
\(170\) 2.57137 0.197215
\(171\) 0 0
\(172\) 13.0784 0.997221
\(173\) −5.99008 −0.455417 −0.227709 0.973729i \(-0.573123\pi\)
−0.227709 + 0.973729i \(0.573123\pi\)
\(174\) 0 0
\(175\) −0.404214 −0.0305557
\(176\) 0 0
\(177\) 0 0
\(178\) −5.56374 −0.417020
\(179\) 1.61894 0.121005 0.0605026 0.998168i \(-0.480730\pi\)
0.0605026 + 0.998168i \(0.480730\pi\)
\(180\) 0 0
\(181\) 2.42682 0.180384 0.0901921 0.995924i \(-0.471252\pi\)
0.0901921 + 0.995924i \(0.471252\pi\)
\(182\) −1.69329 −0.125515
\(183\) 0 0
\(184\) 7.52643 0.554856
\(185\) 3.23106 0.237552
\(186\) 0 0
\(187\) 0 0
\(188\) 6.82279 0.497603
\(189\) 0 0
\(190\) −9.65909 −0.700744
\(191\) −12.7009 −0.919006 −0.459503 0.888176i \(-0.651973\pi\)
−0.459503 + 0.888176i \(0.651973\pi\)
\(192\) 0 0
\(193\) −1.75931 −0.126638 −0.0633190 0.997993i \(-0.520169\pi\)
−0.0633190 + 0.997993i \(0.520169\pi\)
\(194\) −5.70343 −0.409482
\(195\) 0 0
\(196\) −1.55144 −0.110817
\(197\) 0.903053 0.0643399 0.0321699 0.999482i \(-0.489758\pi\)
0.0321699 + 0.999482i \(0.489758\pi\)
\(198\) 0 0
\(199\) −15.6296 −1.10795 −0.553976 0.832533i \(-0.686890\pi\)
−0.553976 + 0.832533i \(0.686890\pi\)
\(200\) −0.961445 −0.0679845
\(201\) 0 0
\(202\) −2.69710 −0.189768
\(203\) 0.924170 0.0648640
\(204\) 0 0
\(205\) 11.9216 0.832643
\(206\) 11.8048 0.822479
\(207\) 0 0
\(208\) 3.81733 0.264684
\(209\) 0 0
\(210\) 0 0
\(211\) 14.7687 1.01672 0.508360 0.861145i \(-0.330252\pi\)
0.508360 + 0.861145i \(0.330252\pi\)
\(212\) 1.03547 0.0711161
\(213\) 0 0
\(214\) −10.3165 −0.705225
\(215\) 18.0717 1.23248
\(216\) 0 0
\(217\) 3.00178 0.203774
\(218\) 12.6759 0.858522
\(219\) 0 0
\(220\) 0 0
\(221\) 4.52791 0.304581
\(222\) 0 0
\(223\) −1.84537 −0.123575 −0.0617875 0.998089i \(-0.519680\pi\)
−0.0617875 + 0.998089i \(0.519680\pi\)
\(224\) −5.76834 −0.385413
\(225\) 0 0
\(226\) 1.12559 0.0748728
\(227\) 21.9399 1.45620 0.728102 0.685468i \(-0.240403\pi\)
0.728102 + 0.685468i \(0.240403\pi\)
\(228\) 0 0
\(229\) −20.2518 −1.33828 −0.669138 0.743138i \(-0.733337\pi\)
−0.669138 + 0.743138i \(0.733337\pi\)
\(230\) 4.54323 0.299571
\(231\) 0 0
\(232\) 2.19819 0.144318
\(233\) −21.0626 −1.37986 −0.689928 0.723878i \(-0.742358\pi\)
−0.689928 + 0.723878i \(0.742358\pi\)
\(234\) 0 0
\(235\) 9.42771 0.614996
\(236\) −0.571490 −0.0372008
\(237\) 0 0
\(238\) −1.19946 −0.0777493
\(239\) −15.4965 −1.00239 −0.501194 0.865335i \(-0.667106\pi\)
−0.501194 + 0.865335i \(0.667106\pi\)
\(240\) 0 0
\(241\) 14.0848 0.907283 0.453641 0.891184i \(-0.350125\pi\)
0.453641 + 0.891184i \(0.350125\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −7.77504 −0.497746
\(245\) −2.14378 −0.136961
\(246\) 0 0
\(247\) −17.0086 −1.08223
\(248\) 7.13991 0.453385
\(249\) 0 0
\(250\) −7.75928 −0.490740
\(251\) 1.07727 0.0679966 0.0339983 0.999422i \(-0.489176\pi\)
0.0339983 + 0.999422i \(0.489176\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −11.7653 −0.738223
\(255\) 0 0
\(256\) −9.03539 −0.564712
\(257\) −13.0810 −0.815971 −0.407986 0.912988i \(-0.633769\pi\)
−0.407986 + 0.912988i \(0.633769\pi\)
\(258\) 0 0
\(259\) −1.50718 −0.0936516
\(260\) 8.40887 0.521496
\(261\) 0 0
\(262\) 4.50441 0.278283
\(263\) 9.57216 0.590245 0.295122 0.955459i \(-0.404640\pi\)
0.295122 + 0.955459i \(0.404640\pi\)
\(264\) 0 0
\(265\) 1.43080 0.0878935
\(266\) 4.50564 0.276258
\(267\) 0 0
\(268\) 1.39960 0.0854943
\(269\) 4.76853 0.290743 0.145371 0.989377i \(-0.453562\pi\)
0.145371 + 0.989377i \(0.453562\pi\)
\(270\) 0 0
\(271\) −20.0523 −1.21809 −0.609045 0.793136i \(-0.708447\pi\)
−0.609045 + 0.793136i \(0.708447\pi\)
\(272\) 2.70404 0.163957
\(273\) 0 0
\(274\) 9.28776 0.561094
\(275\) 0 0
\(276\) 0 0
\(277\) −11.6024 −0.697124 −0.348562 0.937286i \(-0.613330\pi\)
−0.348562 + 0.937286i \(0.613330\pi\)
\(278\) −9.55773 −0.573235
\(279\) 0 0
\(280\) −5.09910 −0.304730
\(281\) −12.0788 −0.720562 −0.360281 0.932844i \(-0.617319\pi\)
−0.360281 + 0.932844i \(0.617319\pi\)
\(282\) 0 0
\(283\) 21.9932 1.30736 0.653681 0.756771i \(-0.273224\pi\)
0.653681 + 0.756771i \(0.273224\pi\)
\(284\) −23.0690 −1.36889
\(285\) 0 0
\(286\) 0 0
\(287\) −5.56104 −0.328258
\(288\) 0 0
\(289\) −13.7926 −0.811330
\(290\) 1.32691 0.0779187
\(291\) 0 0
\(292\) −12.4707 −0.729795
\(293\) −1.41721 −0.0827941 −0.0413971 0.999143i \(-0.513181\pi\)
−0.0413971 + 0.999143i \(0.513181\pi\)
\(294\) 0 0
\(295\) −0.789683 −0.0459771
\(296\) −3.58491 −0.208369
\(297\) 0 0
\(298\) 1.75669 0.101762
\(299\) 8.00014 0.462660
\(300\) 0 0
\(301\) −8.42985 −0.485888
\(302\) 2.00227 0.115217
\(303\) 0 0
\(304\) −10.1574 −0.582570
\(305\) −10.7435 −0.615172
\(306\) 0 0
\(307\) 29.4646 1.68163 0.840817 0.541319i \(-0.182075\pi\)
0.840817 + 0.541319i \(0.182075\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.30991 0.244786
\(311\) −26.8787 −1.52415 −0.762075 0.647489i \(-0.775819\pi\)
−0.762075 + 0.647489i \(0.775819\pi\)
\(312\) 0 0
\(313\) −4.49270 −0.253942 −0.126971 0.991906i \(-0.540526\pi\)
−0.126971 + 0.991906i \(0.540526\pi\)
\(314\) −7.53776 −0.425381
\(315\) 0 0
\(316\) −6.29371 −0.354049
\(317\) 10.9501 0.615017 0.307508 0.951545i \(-0.400505\pi\)
0.307508 + 0.951545i \(0.400505\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.80846 −0.101096
\(321\) 0 0
\(322\) −2.11926 −0.118102
\(323\) −12.0482 −0.670382
\(324\) 0 0
\(325\) −1.02196 −0.0566880
\(326\) 12.1913 0.675215
\(327\) 0 0
\(328\) −13.2272 −0.730353
\(329\) −4.39771 −0.242453
\(330\) 0 0
\(331\) 16.5226 0.908166 0.454083 0.890959i \(-0.349967\pi\)
0.454083 + 0.890959i \(0.349967\pi\)
\(332\) −6.28543 −0.344958
\(333\) 0 0
\(334\) 13.3488 0.730415
\(335\) 1.93397 0.105664
\(336\) 0 0
\(337\) −12.1207 −0.660258 −0.330129 0.943936i \(-0.607092\pi\)
−0.330129 + 0.943936i \(0.607092\pi\)
\(338\) 4.42559 0.240721
\(339\) 0 0
\(340\) 5.95651 0.323037
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −20.0509 −1.08107
\(345\) 0 0
\(346\) 4.01182 0.215677
\(347\) 24.2851 1.30369 0.651847 0.758350i \(-0.273994\pi\)
0.651847 + 0.758350i \(0.273994\pi\)
\(348\) 0 0
\(349\) −3.31880 −0.177651 −0.0888257 0.996047i \(-0.528311\pi\)
−0.0888257 + 0.996047i \(0.528311\pi\)
\(350\) 0.270720 0.0144706
\(351\) 0 0
\(352\) 0 0
\(353\) −20.3272 −1.08191 −0.540955 0.841051i \(-0.681937\pi\)
−0.540955 + 0.841051i \(0.681937\pi\)
\(354\) 0 0
\(355\) −31.8766 −1.69184
\(356\) −12.8883 −0.683076
\(357\) 0 0
\(358\) −1.08427 −0.0573057
\(359\) −29.0412 −1.53274 −0.766369 0.642401i \(-0.777938\pi\)
−0.766369 + 0.642401i \(0.777938\pi\)
\(360\) 0 0
\(361\) 26.2579 1.38200
\(362\) −1.62535 −0.0854264
\(363\) 0 0
\(364\) −3.92246 −0.205593
\(365\) −17.2320 −0.901966
\(366\) 0 0
\(367\) −22.7588 −1.18800 −0.593999 0.804465i \(-0.702452\pi\)
−0.593999 + 0.804465i \(0.702452\pi\)
\(368\) 4.77763 0.249051
\(369\) 0 0
\(370\) −2.16398 −0.112500
\(371\) −0.667421 −0.0346508
\(372\) 0 0
\(373\) −22.2412 −1.15160 −0.575802 0.817589i \(-0.695310\pi\)
−0.575802 + 0.817589i \(0.695310\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −10.4602 −0.539444
\(377\) 2.33654 0.120338
\(378\) 0 0
\(379\) 33.4707 1.71927 0.859637 0.510906i \(-0.170690\pi\)
0.859637 + 0.510906i \(0.170690\pi\)
\(380\) −22.3750 −1.14781
\(381\) 0 0
\(382\) 8.50637 0.435224
\(383\) 6.14592 0.314042 0.157021 0.987595i \(-0.449811\pi\)
0.157021 + 0.987595i \(0.449811\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.17829 0.0599733
\(387\) 0 0
\(388\) −13.2118 −0.670729
\(389\) −7.40364 −0.375380 −0.187690 0.982228i \(-0.560100\pi\)
−0.187690 + 0.982228i \(0.560100\pi\)
\(390\) 0 0
\(391\) 5.66698 0.286592
\(392\) 2.37856 0.120135
\(393\) 0 0
\(394\) −0.604814 −0.0304701
\(395\) −8.69662 −0.437575
\(396\) 0 0
\(397\) 17.8079 0.893752 0.446876 0.894596i \(-0.352537\pi\)
0.446876 + 0.894596i \(0.352537\pi\)
\(398\) 10.4678 0.524704
\(399\) 0 0
\(400\) −0.610307 −0.0305153
\(401\) −25.6789 −1.28234 −0.641170 0.767398i \(-0.721551\pi\)
−0.641170 + 0.767398i \(0.721551\pi\)
\(402\) 0 0
\(403\) 7.58929 0.378050
\(404\) −6.24777 −0.310838
\(405\) 0 0
\(406\) −0.618957 −0.0307183
\(407\) 0 0
\(408\) 0 0
\(409\) 1.33754 0.0661373 0.0330687 0.999453i \(-0.489472\pi\)
0.0330687 + 0.999453i \(0.489472\pi\)
\(410\) −7.98444 −0.394323
\(411\) 0 0
\(412\) 27.3455 1.34722
\(413\) 0.368360 0.0181258
\(414\) 0 0
\(415\) −8.68518 −0.426339
\(416\) −14.5839 −0.715033
\(417\) 0 0
\(418\) 0 0
\(419\) −37.4618 −1.83013 −0.915064 0.403310i \(-0.867860\pi\)
−0.915064 + 0.403310i \(0.867860\pi\)
\(420\) 0 0
\(421\) 8.26156 0.402644 0.201322 0.979525i \(-0.435476\pi\)
0.201322 + 0.979525i \(0.435476\pi\)
\(422\) −9.89125 −0.481499
\(423\) 0 0
\(424\) −1.58750 −0.0770958
\(425\) −0.723914 −0.0351150
\(426\) 0 0
\(427\) 5.01149 0.242523
\(428\) −23.8980 −1.15515
\(429\) 0 0
\(430\) −12.1034 −0.583679
\(431\) −32.6564 −1.57301 −0.786503 0.617587i \(-0.788110\pi\)
−0.786503 + 0.617587i \(0.788110\pi\)
\(432\) 0 0
\(433\) 15.7953 0.759072 0.379536 0.925177i \(-0.376084\pi\)
0.379536 + 0.925177i \(0.376084\pi\)
\(434\) −2.01043 −0.0965035
\(435\) 0 0
\(436\) 29.3634 1.40625
\(437\) −21.2874 −1.01832
\(438\) 0 0
\(439\) −20.6942 −0.987678 −0.493839 0.869553i \(-0.664407\pi\)
−0.493839 + 0.869553i \(0.664407\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.03254 −0.144243
\(443\) −30.0955 −1.42988 −0.714939 0.699186i \(-0.753546\pi\)
−0.714939 + 0.699186i \(0.753546\pi\)
\(444\) 0 0
\(445\) −17.8089 −0.844225
\(446\) 1.23592 0.0585227
\(447\) 0 0
\(448\) 0.843584 0.0398556
\(449\) −36.3944 −1.71756 −0.858779 0.512345i \(-0.828777\pi\)
−0.858779 + 0.512345i \(0.828777\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.60739 0.122641
\(453\) 0 0
\(454\) −14.6941 −0.689630
\(455\) −5.42003 −0.254095
\(456\) 0 0
\(457\) −9.79401 −0.458145 −0.229072 0.973409i \(-0.573569\pi\)
−0.229072 + 0.973409i \(0.573569\pi\)
\(458\) 13.5635 0.633781
\(459\) 0 0
\(460\) 10.5243 0.490696
\(461\) 21.8596 1.01810 0.509052 0.860736i \(-0.329996\pi\)
0.509052 + 0.860736i \(0.329996\pi\)
\(462\) 0 0
\(463\) 6.75889 0.314112 0.157056 0.987590i \(-0.449800\pi\)
0.157056 + 0.987590i \(0.449800\pi\)
\(464\) 1.39537 0.0647784
\(465\) 0 0
\(466\) 14.1065 0.653473
\(467\) 30.0232 1.38931 0.694654 0.719344i \(-0.255558\pi\)
0.694654 + 0.719344i \(0.255558\pi\)
\(468\) 0 0
\(469\) −0.902129 −0.0416565
\(470\) −6.31415 −0.291250
\(471\) 0 0
\(472\) 0.876166 0.0403288
\(473\) 0 0
\(474\) 0 0
\(475\) 2.71931 0.124770
\(476\) −2.77851 −0.127353
\(477\) 0 0
\(478\) 10.3787 0.474711
\(479\) 6.00353 0.274308 0.137154 0.990550i \(-0.456204\pi\)
0.137154 + 0.990550i \(0.456204\pi\)
\(480\) 0 0
\(481\) −3.81054 −0.173746
\(482\) −9.43322 −0.429671
\(483\) 0 0
\(484\) 0 0
\(485\) −18.2561 −0.828965
\(486\) 0 0
\(487\) 6.36723 0.288527 0.144263 0.989539i \(-0.453919\pi\)
0.144263 + 0.989539i \(0.453919\pi\)
\(488\) 11.9201 0.539598
\(489\) 0 0
\(490\) 1.43578 0.0648620
\(491\) −12.3769 −0.558561 −0.279281 0.960210i \(-0.590096\pi\)
−0.279281 + 0.960210i \(0.590096\pi\)
\(492\) 0 0
\(493\) 1.65511 0.0745426
\(494\) 11.3914 0.512525
\(495\) 0 0
\(496\) 4.53228 0.203505
\(497\) 14.8694 0.666982
\(498\) 0 0
\(499\) −14.2797 −0.639245 −0.319623 0.947545i \(-0.603556\pi\)
−0.319623 + 0.947545i \(0.603556\pi\)
\(500\) −17.9741 −0.803828
\(501\) 0 0
\(502\) −0.721494 −0.0322019
\(503\) −5.92573 −0.264215 −0.132108 0.991235i \(-0.542174\pi\)
−0.132108 + 0.991235i \(0.542174\pi\)
\(504\) 0 0
\(505\) −8.63314 −0.384170
\(506\) 0 0
\(507\) 0 0
\(508\) −27.2541 −1.20920
\(509\) −30.8735 −1.36844 −0.684222 0.729274i \(-0.739858\pi\)
−0.684222 + 0.729274i \(0.739858\pi\)
\(510\) 0 0
\(511\) 8.03816 0.355587
\(512\) −15.8920 −0.702333
\(513\) 0 0
\(514\) 8.76093 0.386428
\(515\) 37.7859 1.66504
\(516\) 0 0
\(517\) 0 0
\(518\) 1.00942 0.0443516
\(519\) 0 0
\(520\) −12.8919 −0.565346
\(521\) −18.8870 −0.827453 −0.413726 0.910401i \(-0.635773\pi\)
−0.413726 + 0.910401i \(0.635773\pi\)
\(522\) 0 0
\(523\) −7.94209 −0.347283 −0.173642 0.984809i \(-0.555553\pi\)
−0.173642 + 0.984809i \(0.555553\pi\)
\(524\) 10.4343 0.455826
\(525\) 0 0
\(526\) −6.41090 −0.279528
\(527\) 5.37595 0.234180
\(528\) 0 0
\(529\) −12.9873 −0.564665
\(530\) −0.958271 −0.0416246
\(531\) 0 0
\(532\) 10.4372 0.452509
\(533\) −14.0598 −0.608996
\(534\) 0 0
\(535\) −33.0222 −1.42767
\(536\) −2.14577 −0.0926830
\(537\) 0 0
\(538\) −3.19370 −0.137690
\(539\) 0 0
\(540\) 0 0
\(541\) 7.60165 0.326820 0.163410 0.986558i \(-0.447751\pi\)
0.163410 + 0.986558i \(0.447751\pi\)
\(542\) 13.4299 0.576863
\(543\) 0 0
\(544\) −10.3306 −0.442922
\(545\) 40.5743 1.73801
\(546\) 0 0
\(547\) 21.6989 0.927780 0.463890 0.885893i \(-0.346453\pi\)
0.463890 + 0.885893i \(0.346453\pi\)
\(548\) 21.5148 0.919068
\(549\) 0 0
\(550\) 0 0
\(551\) −6.21726 −0.264864
\(552\) 0 0
\(553\) 4.05668 0.172508
\(554\) 7.77067 0.330144
\(555\) 0 0
\(556\) −22.1402 −0.938955
\(557\) 40.6134 1.72084 0.860422 0.509582i \(-0.170200\pi\)
0.860422 + 0.509582i \(0.170200\pi\)
\(558\) 0 0
\(559\) −21.3129 −0.901438
\(560\) −3.23681 −0.136780
\(561\) 0 0
\(562\) 8.08972 0.341244
\(563\) 23.4293 0.987425 0.493713 0.869625i \(-0.335639\pi\)
0.493713 + 0.869625i \(0.335639\pi\)
\(564\) 0 0
\(565\) 3.60288 0.151574
\(566\) −14.7298 −0.619141
\(567\) 0 0
\(568\) 35.3676 1.48399
\(569\) 11.1453 0.467237 0.233619 0.972328i \(-0.424943\pi\)
0.233619 + 0.972328i \(0.424943\pi\)
\(570\) 0 0
\(571\) −6.15846 −0.257724 −0.128862 0.991663i \(-0.541132\pi\)
−0.128862 + 0.991663i \(0.541132\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 3.72447 0.155456
\(575\) −1.27905 −0.0533400
\(576\) 0 0
\(577\) −14.4842 −0.602983 −0.301492 0.953469i \(-0.597485\pi\)
−0.301492 + 0.953469i \(0.597485\pi\)
\(578\) 9.23751 0.384230
\(579\) 0 0
\(580\) 3.07374 0.127630
\(581\) 4.05134 0.168078
\(582\) 0 0
\(583\) 0 0
\(584\) 19.1192 0.791159
\(585\) 0 0
\(586\) 0.949166 0.0392097
\(587\) −15.8570 −0.654487 −0.327243 0.944940i \(-0.606120\pi\)
−0.327243 + 0.944940i \(0.606120\pi\)
\(588\) 0 0
\(589\) −20.1942 −0.832087
\(590\) 0.528885 0.0217739
\(591\) 0 0
\(592\) −2.27563 −0.0935279
\(593\) −22.9285 −0.941560 −0.470780 0.882251i \(-0.656027\pi\)
−0.470780 + 0.882251i \(0.656027\pi\)
\(594\) 0 0
\(595\) −3.83934 −0.157397
\(596\) 4.06931 0.166686
\(597\) 0 0
\(598\) −5.35805 −0.219107
\(599\) −13.1512 −0.537344 −0.268672 0.963232i \(-0.586585\pi\)
−0.268672 + 0.963232i \(0.586585\pi\)
\(600\) 0 0
\(601\) 27.3525 1.11573 0.557865 0.829932i \(-0.311621\pi\)
0.557865 + 0.829932i \(0.311621\pi\)
\(602\) 5.64584 0.230107
\(603\) 0 0
\(604\) 4.63819 0.188725
\(605\) 0 0
\(606\) 0 0
\(607\) −46.8743 −1.90257 −0.951285 0.308313i \(-0.900236\pi\)
−0.951285 + 0.308313i \(0.900236\pi\)
\(608\) 38.8059 1.57379
\(609\) 0 0
\(610\) 7.19540 0.291333
\(611\) −11.1186 −0.449809
\(612\) 0 0
\(613\) −20.4694 −0.826752 −0.413376 0.910560i \(-0.635651\pi\)
−0.413376 + 0.910560i \(0.635651\pi\)
\(614\) −19.7337 −0.796389
\(615\) 0 0
\(616\) 0 0
\(617\) −44.1691 −1.77818 −0.889090 0.457733i \(-0.848662\pi\)
−0.889090 + 0.457733i \(0.848662\pi\)
\(618\) 0 0
\(619\) 0.681584 0.0273952 0.0136976 0.999906i \(-0.495640\pi\)
0.0136976 + 0.999906i \(0.495640\pi\)
\(620\) 9.98378 0.400958
\(621\) 0 0
\(622\) 18.0018 0.721808
\(623\) 8.30727 0.332824
\(624\) 0 0
\(625\) −22.8155 −0.912622
\(626\) 3.00896 0.120262
\(627\) 0 0
\(628\) −17.4610 −0.696770
\(629\) −2.69924 −0.107626
\(630\) 0 0
\(631\) −36.8718 −1.46784 −0.733921 0.679234i \(-0.762312\pi\)
−0.733921 + 0.679234i \(0.762312\pi\)
\(632\) 9.64904 0.383818
\(633\) 0 0
\(634\) −7.33374 −0.291260
\(635\) −37.6596 −1.49447
\(636\) 0 0
\(637\) 2.52826 0.100173
\(638\) 0 0
\(639\) 0 0
\(640\) −23.5209 −0.929744
\(641\) 20.5126 0.810201 0.405100 0.914272i \(-0.367237\pi\)
0.405100 + 0.914272i \(0.367237\pi\)
\(642\) 0 0
\(643\) −7.63660 −0.301158 −0.150579 0.988598i \(-0.548114\pi\)
−0.150579 + 0.988598i \(0.548114\pi\)
\(644\) −4.90921 −0.193450
\(645\) 0 0
\(646\) 8.06923 0.317480
\(647\) −14.6828 −0.577241 −0.288621 0.957444i \(-0.593197\pi\)
−0.288621 + 0.957444i \(0.593197\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.684450 0.0268463
\(651\) 0 0
\(652\) 28.2408 1.10600
\(653\) −1.12703 −0.0441043 −0.0220521 0.999757i \(-0.507020\pi\)
−0.0220521 + 0.999757i \(0.507020\pi\)
\(654\) 0 0
\(655\) 14.4181 0.563363
\(656\) −8.39640 −0.327824
\(657\) 0 0
\(658\) 2.94534 0.114821
\(659\) −10.0215 −0.390384 −0.195192 0.980765i \(-0.562533\pi\)
−0.195192 + 0.980765i \(0.562533\pi\)
\(660\) 0 0
\(661\) 15.7371 0.612101 0.306050 0.952015i \(-0.400992\pi\)
0.306050 + 0.952015i \(0.400992\pi\)
\(662\) −11.0659 −0.430090
\(663\) 0 0
\(664\) 9.63635 0.373963
\(665\) 14.4221 0.559263
\(666\) 0 0
\(667\) 2.92434 0.113231
\(668\) 30.9222 1.19642
\(669\) 0 0
\(670\) −1.29526 −0.0500403
\(671\) 0 0
\(672\) 0 0
\(673\) 32.0120 1.23397 0.616986 0.786974i \(-0.288353\pi\)
0.616986 + 0.786974i \(0.288353\pi\)
\(674\) 8.11779 0.312686
\(675\) 0 0
\(676\) 10.2518 0.394299
\(677\) 15.3400 0.589566 0.294783 0.955564i \(-0.404753\pi\)
0.294783 + 0.955564i \(0.404753\pi\)
\(678\) 0 0
\(679\) 8.51583 0.326808
\(680\) −9.13208 −0.350199
\(681\) 0 0
\(682\) 0 0
\(683\) −1.04764 −0.0400868 −0.0200434 0.999799i \(-0.506380\pi\)
−0.0200434 + 0.999799i \(0.506380\pi\)
\(684\) 0 0
\(685\) 29.7291 1.13589
\(686\) −0.669744 −0.0255709
\(687\) 0 0
\(688\) −12.7279 −0.485247
\(689\) −1.68742 −0.0642854
\(690\) 0 0
\(691\) 29.1883 1.11038 0.555188 0.831725i \(-0.312646\pi\)
0.555188 + 0.831725i \(0.312646\pi\)
\(692\) 9.29326 0.353277
\(693\) 0 0
\(694\) −16.2648 −0.617404
\(695\) −30.5933 −1.16047
\(696\) 0 0
\(697\) −9.95937 −0.377238
\(698\) 2.22275 0.0841322
\(699\) 0 0
\(700\) 0.627115 0.0237027
\(701\) −12.7785 −0.482637 −0.241318 0.970446i \(-0.577580\pi\)
−0.241318 + 0.970446i \(0.577580\pi\)
\(702\) 0 0
\(703\) 10.1394 0.382415
\(704\) 0 0
\(705\) 0 0
\(706\) 13.6141 0.512372
\(707\) 4.02707 0.151453
\(708\) 0 0
\(709\) 38.0710 1.42978 0.714892 0.699234i \(-0.246476\pi\)
0.714892 + 0.699234i \(0.246476\pi\)
\(710\) 21.3492 0.801220
\(711\) 0 0
\(712\) 19.7593 0.740512
\(713\) 9.49850 0.355722
\(714\) 0 0
\(715\) 0 0
\(716\) −2.51169 −0.0938663
\(717\) 0 0
\(718\) 19.4502 0.725875
\(719\) 39.2694 1.46450 0.732250 0.681036i \(-0.238470\pi\)
0.732250 + 0.681036i \(0.238470\pi\)
\(720\) 0 0
\(721\) −17.6258 −0.656420
\(722\) −17.5861 −0.654486
\(723\) 0 0
\(724\) −3.76507 −0.139928
\(725\) −0.373562 −0.0138737
\(726\) 0 0
\(727\) −28.4699 −1.05589 −0.527946 0.849278i \(-0.677037\pi\)
−0.527946 + 0.849278i \(0.677037\pi\)
\(728\) 6.01362 0.222879
\(729\) 0 0
\(730\) 11.5410 0.427153
\(731\) −15.0972 −0.558389
\(732\) 0 0
\(733\) −2.99337 −0.110563 −0.0552813 0.998471i \(-0.517606\pi\)
−0.0552813 + 0.998471i \(0.517606\pi\)
\(734\) 15.2426 0.562613
\(735\) 0 0
\(736\) −18.2527 −0.672802
\(737\) 0 0
\(738\) 0 0
\(739\) −50.7329 −1.86624 −0.933120 0.359566i \(-0.882925\pi\)
−0.933120 + 0.359566i \(0.882925\pi\)
\(740\) −5.01280 −0.184274
\(741\) 0 0
\(742\) 0.447001 0.0164099
\(743\) 0.383877 0.0140831 0.00704154 0.999975i \(-0.497759\pi\)
0.00704154 + 0.999975i \(0.497759\pi\)
\(744\) 0 0
\(745\) 5.62296 0.206009
\(746\) 14.8959 0.545378
\(747\) 0 0
\(748\) 0 0
\(749\) 15.4037 0.562840
\(750\) 0 0
\(751\) −39.3570 −1.43616 −0.718078 0.695963i \(-0.754978\pi\)
−0.718078 + 0.695963i \(0.754978\pi\)
\(752\) −6.63993 −0.242133
\(753\) 0 0
\(754\) −1.56489 −0.0569898
\(755\) 6.40904 0.233249
\(756\) 0 0
\(757\) 11.3867 0.413856 0.206928 0.978356i \(-0.433653\pi\)
0.206928 + 0.978356i \(0.433653\pi\)
\(758\) −22.4168 −0.814214
\(759\) 0 0
\(760\) 34.3037 1.24433
\(761\) 7.48165 0.271209 0.135605 0.990763i \(-0.456702\pi\)
0.135605 + 0.990763i \(0.456702\pi\)
\(762\) 0 0
\(763\) −18.9265 −0.685186
\(764\) 19.7048 0.712893
\(765\) 0 0
\(766\) −4.11619 −0.148724
\(767\) 0.931311 0.0336277
\(768\) 0 0
\(769\) −26.8378 −0.967798 −0.483899 0.875124i \(-0.660780\pi\)
−0.483899 + 0.875124i \(0.660780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.72947 0.0982358
\(773\) −4.46781 −0.160696 −0.0803479 0.996767i \(-0.525603\pi\)
−0.0803479 + 0.996767i \(0.525603\pi\)
\(774\) 0 0
\(775\) −1.21336 −0.0435852
\(776\) 20.2554 0.727126
\(777\) 0 0
\(778\) 4.95855 0.177772
\(779\) 37.4113 1.34040
\(780\) 0 0
\(781\) 0 0
\(782\) −3.79543 −0.135724
\(783\) 0 0
\(784\) 1.50986 0.0539236
\(785\) −24.1276 −0.861150
\(786\) 0 0
\(787\) −13.4859 −0.480721 −0.240361 0.970684i \(-0.577266\pi\)
−0.240361 + 0.970684i \(0.577266\pi\)
\(788\) −1.40104 −0.0499098
\(789\) 0 0
\(790\) 5.82451 0.207227
\(791\) −1.68062 −0.0597560
\(792\) 0 0
\(793\) 12.6704 0.449937
\(794\) −11.9267 −0.423263
\(795\) 0 0
\(796\) 24.2484 0.859462
\(797\) −52.5052 −1.85983 −0.929915 0.367775i \(-0.880120\pi\)
−0.929915 + 0.367775i \(0.880120\pi\)
\(798\) 0 0
\(799\) −7.87594 −0.278631
\(800\) 2.33164 0.0824359
\(801\) 0 0
\(802\) 17.1983 0.607292
\(803\) 0 0
\(804\) 0 0
\(805\) −6.78353 −0.239088
\(806\) −5.08288 −0.179037
\(807\) 0 0
\(808\) 9.57861 0.336974
\(809\) 1.28984 0.0453484 0.0226742 0.999743i \(-0.492782\pi\)
0.0226742 + 0.999743i \(0.492782\pi\)
\(810\) 0 0
\(811\) −34.1878 −1.20049 −0.600247 0.799815i \(-0.704931\pi\)
−0.600247 + 0.799815i \(0.704931\pi\)
\(812\) −1.43380 −0.0503164
\(813\) 0 0
\(814\) 0 0
\(815\) 39.0231 1.36692
\(816\) 0 0
\(817\) 56.7110 1.98407
\(818\) −0.895813 −0.0313214
\(819\) 0 0
\(820\) −18.4957 −0.645899
\(821\) −39.6599 −1.38414 −0.692071 0.721830i \(-0.743301\pi\)
−0.692071 + 0.721830i \(0.743301\pi\)
\(822\) 0 0
\(823\) −9.22714 −0.321638 −0.160819 0.986984i \(-0.551414\pi\)
−0.160819 + 0.986984i \(0.551414\pi\)
\(824\) −41.9241 −1.46049
\(825\) 0 0
\(826\) −0.246707 −0.00858403
\(827\) 25.6958 0.893530 0.446765 0.894651i \(-0.352576\pi\)
0.446765 + 0.894651i \(0.352576\pi\)
\(828\) 0 0
\(829\) 10.9385 0.379911 0.189955 0.981793i \(-0.439166\pi\)
0.189955 + 0.981793i \(0.439166\pi\)
\(830\) 5.81685 0.201906
\(831\) 0 0
\(832\) 2.13280 0.0739416
\(833\) 1.79092 0.0620517
\(834\) 0 0
\(835\) 42.7282 1.47867
\(836\) 0 0
\(837\) 0 0
\(838\) 25.0898 0.866712
\(839\) 34.7374 1.19927 0.599634 0.800275i \(-0.295313\pi\)
0.599634 + 0.800275i \(0.295313\pi\)
\(840\) 0 0
\(841\) −28.1459 −0.970549
\(842\) −5.53313 −0.190684
\(843\) 0 0
\(844\) −22.9128 −0.788691
\(845\) 14.1659 0.487320
\(846\) 0 0
\(847\) 0 0
\(848\) −1.00771 −0.0346050
\(849\) 0 0
\(850\) 0.484837 0.0166298
\(851\) −4.76915 −0.163484
\(852\) 0 0
\(853\) −20.6422 −0.706777 −0.353389 0.935477i \(-0.614971\pi\)
−0.353389 + 0.935477i \(0.614971\pi\)
\(854\) −3.35641 −0.114854
\(855\) 0 0
\(856\) 36.6386 1.25228
\(857\) 34.1512 1.16658 0.583291 0.812263i \(-0.301765\pi\)
0.583291 + 0.812263i \(0.301765\pi\)
\(858\) 0 0
\(859\) −33.4493 −1.14127 −0.570637 0.821202i \(-0.693304\pi\)
−0.570637 + 0.821202i \(0.693304\pi\)
\(860\) −28.0373 −0.956063
\(861\) 0 0
\(862\) 21.8715 0.744945
\(863\) 34.0340 1.15853 0.579265 0.815139i \(-0.303340\pi\)
0.579265 + 0.815139i \(0.303340\pi\)
\(864\) 0 0
\(865\) 12.8414 0.436621
\(866\) −10.5788 −0.359482
\(867\) 0 0
\(868\) −4.65710 −0.158072
\(869\) 0 0
\(870\) 0 0
\(871\) −2.28082 −0.0772826
\(872\) −45.0178 −1.52450
\(873\) 0 0
\(874\) 14.2571 0.482254
\(875\) 11.5854 0.391659
\(876\) 0 0
\(877\) 7.86706 0.265652 0.132826 0.991139i \(-0.457595\pi\)
0.132826 + 0.991139i \(0.457595\pi\)
\(878\) 13.8598 0.467745
\(879\) 0 0
\(880\) 0 0
\(881\) −13.3289 −0.449063 −0.224531 0.974467i \(-0.572085\pi\)
−0.224531 + 0.974467i \(0.572085\pi\)
\(882\) 0 0
\(883\) −17.0998 −0.575454 −0.287727 0.957712i \(-0.592900\pi\)
−0.287727 + 0.957712i \(0.592900\pi\)
\(884\) −7.02480 −0.236270
\(885\) 0 0
\(886\) 20.1563 0.677163
\(887\) 26.5185 0.890403 0.445201 0.895430i \(-0.353132\pi\)
0.445201 + 0.895430i \(0.353132\pi\)
\(888\) 0 0
\(889\) 17.5669 0.589176
\(890\) 11.9274 0.399808
\(891\) 0 0
\(892\) 2.86298 0.0958598
\(893\) 29.5851 0.990029
\(894\) 0 0
\(895\) −3.47065 −0.116011
\(896\) 10.9717 0.366538
\(897\) 0 0
\(898\) 24.3749 0.813402
\(899\) 2.77416 0.0925233
\(900\) 0 0
\(901\) −1.19530 −0.0398211
\(902\) 0 0
\(903\) 0 0
\(904\) −3.99745 −0.132953
\(905\) −5.20257 −0.172939
\(906\) 0 0
\(907\) 8.85256 0.293944 0.146972 0.989141i \(-0.453047\pi\)
0.146972 + 0.989141i \(0.453047\pi\)
\(908\) −34.0386 −1.12961
\(909\) 0 0
\(910\) 3.63003 0.120334
\(911\) 55.8998 1.85204 0.926022 0.377469i \(-0.123206\pi\)
0.926022 + 0.377469i \(0.123206\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 6.55948 0.216968
\(915\) 0 0
\(916\) 31.4195 1.03813
\(917\) −6.72557 −0.222098
\(918\) 0 0
\(919\) 48.3776 1.59583 0.797915 0.602770i \(-0.205936\pi\)
0.797915 + 0.602770i \(0.205936\pi\)
\(920\) −16.1350 −0.531955
\(921\) 0 0
\(922\) −14.6404 −0.482155
\(923\) 37.5937 1.23741
\(924\) 0 0
\(925\) 0.609223 0.0200311
\(926\) −4.52673 −0.148757
\(927\) 0 0
\(928\) −5.33092 −0.174996
\(929\) −0.864295 −0.0283566 −0.0141783 0.999899i \(-0.504513\pi\)
−0.0141783 + 0.999899i \(0.504513\pi\)
\(930\) 0 0
\(931\) −6.72740 −0.220482
\(932\) 32.6774 1.07038
\(933\) 0 0
\(934\) −20.1079 −0.657949
\(935\) 0 0
\(936\) 0 0
\(937\) 53.4580 1.74640 0.873199 0.487364i \(-0.162042\pi\)
0.873199 + 0.487364i \(0.162042\pi\)
\(938\) 0.604195 0.0197277
\(939\) 0 0
\(940\) −14.6266 −0.477066
\(941\) −13.7000 −0.446607 −0.223304 0.974749i \(-0.571684\pi\)
−0.223304 + 0.974749i \(0.571684\pi\)
\(942\) 0 0
\(943\) −17.5967 −0.573028
\(944\) 0.556173 0.0181019
\(945\) 0 0
\(946\) 0 0
\(947\) −31.6444 −1.02830 −0.514152 0.857699i \(-0.671893\pi\)
−0.514152 + 0.857699i \(0.671893\pi\)
\(948\) 0 0
\(949\) 20.3226 0.659699
\(950\) −1.82124 −0.0590888
\(951\) 0 0
\(952\) 4.25981 0.138061
\(953\) −32.7379 −1.06049 −0.530243 0.847846i \(-0.677899\pi\)
−0.530243 + 0.847846i \(0.677899\pi\)
\(954\) 0 0
\(955\) 27.2280 0.881076
\(956\) 24.0420 0.777573
\(957\) 0 0
\(958\) −4.02083 −0.129907
\(959\) −13.8676 −0.447809
\(960\) 0 0
\(961\) −21.9893 −0.709332
\(962\) 2.55209 0.0822827
\(963\) 0 0
\(964\) −21.8518 −0.703799
\(965\) 3.77157 0.121411
\(966\) 0 0
\(967\) −32.3487 −1.04026 −0.520132 0.854086i \(-0.674117\pi\)
−0.520132 + 0.854086i \(0.674117\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 12.2269 0.392582
\(971\) 29.4270 0.944358 0.472179 0.881503i \(-0.343468\pi\)
0.472179 + 0.881503i \(0.343468\pi\)
\(972\) 0 0
\(973\) 14.2707 0.457498
\(974\) −4.26441 −0.136641
\(975\) 0 0
\(976\) 7.56665 0.242203
\(977\) −13.6944 −0.438124 −0.219062 0.975711i \(-0.570300\pi\)
−0.219062 + 0.975711i \(0.570300\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.32595 0.106244
\(981\) 0 0
\(982\) 8.28935 0.264524
\(983\) −17.0319 −0.543234 −0.271617 0.962405i \(-0.587558\pi\)
−0.271617 + 0.962405i \(0.587558\pi\)
\(984\) 0 0
\(985\) −1.93595 −0.0616844
\(986\) −1.10850 −0.0353019
\(987\) 0 0
\(988\) 26.3879 0.839512
\(989\) −26.6744 −0.848198
\(990\) 0 0
\(991\) 23.2202 0.737614 0.368807 0.929506i \(-0.379766\pi\)
0.368807 + 0.929506i \(0.379766\pi\)
\(992\) −17.3153 −0.549761
\(993\) 0 0
\(994\) −9.95867 −0.315870
\(995\) 33.5064 1.06222
\(996\) 0 0
\(997\) −18.3776 −0.582025 −0.291012 0.956719i \(-0.593992\pi\)
−0.291012 + 0.956719i \(0.593992\pi\)
\(998\) 9.56371 0.302734
\(999\) 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.1.4 8
3.2 odd 2 847.2.a.p.1.5 8
11.3 even 5 693.2.m.i.64.2 16
11.4 even 5 693.2.m.i.379.2 16
11.10 odd 2 7623.2.a.cw.1.5 8
21.20 even 2 5929.2.a.bt.1.5 8
33.2 even 10 847.2.f.v.323.3 16
33.5 odd 10 847.2.f.w.729.2 16
33.8 even 10 847.2.f.x.372.2 16
33.14 odd 10 77.2.f.b.64.3 16
33.17 even 10 847.2.f.v.729.3 16
33.20 odd 10 847.2.f.w.323.2 16
33.26 odd 10 77.2.f.b.71.3 yes 16
33.29 even 10 847.2.f.x.148.2 16
33.32 even 2 847.2.a.o.1.4 8
231.26 even 30 539.2.q.f.214.3 32
231.47 even 30 539.2.q.f.361.2 32
231.59 even 30 539.2.q.f.324.2 32
231.80 even 30 539.2.q.f.471.3 32
231.125 even 10 539.2.f.e.148.3 16
231.146 even 10 539.2.f.e.295.3 16
231.158 odd 30 539.2.q.g.324.2 32
231.179 odd 30 539.2.q.g.471.3 32
231.191 odd 30 539.2.q.g.214.3 32
231.212 odd 30 539.2.q.g.361.2 32
231.230 odd 2 5929.2.a.bs.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.64.3 16 33.14 odd 10
77.2.f.b.71.3 yes 16 33.26 odd 10
539.2.f.e.148.3 16 231.125 even 10
539.2.f.e.295.3 16 231.146 even 10
539.2.q.f.214.3 32 231.26 even 30
539.2.q.f.324.2 32 231.59 even 30
539.2.q.f.361.2 32 231.47 even 30
539.2.q.f.471.3 32 231.80 even 30
539.2.q.g.214.3 32 231.191 odd 30
539.2.q.g.324.2 32 231.158 odd 30
539.2.q.g.361.2 32 231.212 odd 30
539.2.q.g.471.3 32 231.179 odd 30
693.2.m.i.64.2 16 11.3 even 5
693.2.m.i.379.2 16 11.4 even 5
847.2.a.o.1.4 8 33.32 even 2
847.2.a.p.1.5 8 3.2 odd 2
847.2.f.v.323.3 16 33.2 even 10
847.2.f.v.729.3 16 33.17 even 10
847.2.f.w.323.2 16 33.20 odd 10
847.2.f.w.729.2 16 33.5 odd 10
847.2.f.x.148.2 16 33.29 even 10
847.2.f.x.372.2 16 33.8 even 10
5929.2.a.bs.1.4 8 231.230 odd 2
5929.2.a.bt.1.5 8 21.20 even 2
7623.2.a.ct.1.4 8 1.1 even 1 trivial
7623.2.a.cw.1.5 8 11.10 odd 2