Properties

Label 5445.2.a.bv.1.3
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.737640\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +O(q^{10})\) \(q+2.45589 q^{2} +4.03138 q^{4} +1.00000 q^{5} -3.28684 q^{7} +4.98884 q^{8} +2.45589 q^{10} +0.313133 q^{13} -8.07211 q^{14} +4.18926 q^{16} +5.00000 q^{17} +7.45408 q^{19} +4.03138 q^{20} -1.07392 q^{23} +1.00000 q^{25} +0.769020 q^{26} -13.2505 q^{28} -5.03647 q^{29} +3.44899 q^{31} +0.310680 q^{32} +12.2794 q^{34} -3.28684 q^{35} +2.63428 q^{37} +18.3064 q^{38} +4.98884 q^{40} +10.8472 q^{41} +5.51468 q^{43} -2.63743 q^{46} +11.9982 q^{47} +3.80333 q^{49} +2.45589 q^{50} +1.26236 q^{52} -4.93543 q^{53} -16.3975 q^{56} -12.3690 q^{58} +9.16409 q^{59} -9.18431 q^{61} +8.47033 q^{62} -7.61553 q^{64} +0.313133 q^{65} -15.2739 q^{67} +20.1569 q^{68} -8.07211 q^{70} -3.07211 q^{71} +8.65269 q^{73} +6.46950 q^{74} +30.0502 q^{76} -5.41446 q^{79} +4.18926 q^{80} +26.6395 q^{82} +16.2454 q^{83} +5.00000 q^{85} +13.5434 q^{86} -1.62118 q^{89} -1.02922 q^{91} -4.32938 q^{92} +29.4662 q^{94} +7.45408 q^{95} +0.224082 q^{97} +9.34054 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + O(q^{10}) \) \( 4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.45589 1.73657 0.868287 0.496062i \(-0.165221\pi\)
0.868287 + 0.496062i \(0.165221\pi\)
\(3\) 0 0
\(4\) 4.03138 2.01569
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.28684 −1.24231 −0.621155 0.783688i \(-0.713336\pi\)
−0.621155 + 0.783688i \(0.713336\pi\)
\(8\) 4.98884 1.76382
\(9\) 0 0
\(10\) 2.45589 0.776620
\(11\) 0 0
\(12\) 0 0
\(13\) 0.313133 0.0868476 0.0434238 0.999057i \(-0.486173\pi\)
0.0434238 + 0.999057i \(0.486173\pi\)
\(14\) −8.07211 −2.15736
\(15\) 0 0
\(16\) 4.18926 1.04732
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 7.45408 1.71008 0.855041 0.518560i \(-0.173532\pi\)
0.855041 + 0.518560i \(0.173532\pi\)
\(20\) 4.03138 0.901444
\(21\) 0 0
\(22\) 0 0
\(23\) −1.07392 −0.223928 −0.111964 0.993712i \(-0.535714\pi\)
−0.111964 + 0.993712i \(0.535714\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.769020 0.150817
\(27\) 0 0
\(28\) −13.2505 −2.50411
\(29\) −5.03647 −0.935249 −0.467624 0.883927i \(-0.654890\pi\)
−0.467624 + 0.883927i \(0.654890\pi\)
\(30\) 0 0
\(31\) 3.44899 0.619457 0.309728 0.950825i \(-0.399762\pi\)
0.309728 + 0.950825i \(0.399762\pi\)
\(32\) 0.310680 0.0549210
\(33\) 0 0
\(34\) 12.2794 2.10591
\(35\) −3.28684 −0.555578
\(36\) 0 0
\(37\) 2.63428 0.433073 0.216537 0.976274i \(-0.430524\pi\)
0.216537 + 0.976274i \(0.430524\pi\)
\(38\) 18.3064 2.96969
\(39\) 0 0
\(40\) 4.98884 0.788805
\(41\) 10.8472 1.69405 0.847024 0.531554i \(-0.178392\pi\)
0.847024 + 0.531554i \(0.178392\pi\)
\(42\) 0 0
\(43\) 5.51468 0.840980 0.420490 0.907297i \(-0.361858\pi\)
0.420490 + 0.907297i \(0.361858\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.63743 −0.388868
\(47\) 11.9982 1.75012 0.875058 0.484018i \(-0.160823\pi\)
0.875058 + 0.484018i \(0.160823\pi\)
\(48\) 0 0
\(49\) 3.80333 0.543333
\(50\) 2.45589 0.347315
\(51\) 0 0
\(52\) 1.26236 0.175058
\(53\) −4.93543 −0.677934 −0.338967 0.940798i \(-0.610077\pi\)
−0.338967 + 0.940798i \(0.610077\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −16.3975 −2.19121
\(57\) 0 0
\(58\) −12.3690 −1.62413
\(59\) 9.16409 1.19306 0.596531 0.802590i \(-0.296545\pi\)
0.596531 + 0.802590i \(0.296545\pi\)
\(60\) 0 0
\(61\) −9.18431 −1.17593 −0.587965 0.808886i \(-0.700071\pi\)
−0.587965 + 0.808886i \(0.700071\pi\)
\(62\) 8.47033 1.07573
\(63\) 0 0
\(64\) −7.61553 −0.951942
\(65\) 0.313133 0.0388394
\(66\) 0 0
\(67\) −15.2739 −1.86600 −0.933000 0.359876i \(-0.882819\pi\)
−0.933000 + 0.359876i \(0.882819\pi\)
\(68\) 20.1569 2.44438
\(69\) 0 0
\(70\) −8.07211 −0.964802
\(71\) −3.07211 −0.364593 −0.182296 0.983244i \(-0.558353\pi\)
−0.182296 + 0.983244i \(0.558353\pi\)
\(72\) 0 0
\(73\) 8.65269 1.01272 0.506361 0.862322i \(-0.330991\pi\)
0.506361 + 0.862322i \(0.330991\pi\)
\(74\) 6.46950 0.752064
\(75\) 0 0
\(76\) 30.0502 3.44700
\(77\) 0 0
\(78\) 0 0
\(79\) −5.41446 −0.609175 −0.304587 0.952484i \(-0.598519\pi\)
−0.304587 + 0.952484i \(0.598519\pi\)
\(80\) 4.18926 0.468374
\(81\) 0 0
\(82\) 26.6395 2.94184
\(83\) 16.2454 1.78317 0.891583 0.452857i \(-0.149595\pi\)
0.891583 + 0.452857i \(0.149595\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 13.5434 1.46042
\(87\) 0 0
\(88\) 0 0
\(89\) −1.62118 −0.171845 −0.0859223 0.996302i \(-0.527384\pi\)
−0.0859223 + 0.996302i \(0.527384\pi\)
\(90\) 0 0
\(91\) −1.02922 −0.107892
\(92\) −4.32938 −0.451369
\(93\) 0 0
\(94\) 29.4662 3.03921
\(95\) 7.45408 0.764772
\(96\) 0 0
\(97\) 0.224082 0.0227521 0.0113760 0.999935i \(-0.496379\pi\)
0.0113760 + 0.999935i \(0.496379\pi\)
\(98\) 9.34054 0.943537
\(99\) 0 0
\(100\) 4.03138 0.403138
\(101\) −0.505326 −0.0502818 −0.0251409 0.999684i \(-0.508003\pi\)
−0.0251409 + 0.999684i \(0.508003\pi\)
\(102\) 0 0
\(103\) −6.40197 −0.630805 −0.315402 0.948958i \(-0.602139\pi\)
−0.315402 + 0.948958i \(0.602139\pi\)
\(104\) 1.56217 0.153184
\(105\) 0 0
\(106\) −12.1209 −1.17728
\(107\) 2.09249 0.202289 0.101144 0.994872i \(-0.467750\pi\)
0.101144 + 0.994872i \(0.467750\pi\)
\(108\) 0 0
\(109\) 6.69278 0.641052 0.320526 0.947240i \(-0.396140\pi\)
0.320526 + 0.947240i \(0.396140\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −13.7694 −1.30109
\(113\) 10.7941 1.01542 0.507712 0.861527i \(-0.330491\pi\)
0.507712 + 0.861527i \(0.330491\pi\)
\(114\) 0 0
\(115\) −1.07392 −0.100144
\(116\) −20.3039 −1.88517
\(117\) 0 0
\(118\) 22.5060 2.07184
\(119\) −16.4342 −1.50652
\(120\) 0 0
\(121\) 0 0
\(122\) −22.5556 −2.04209
\(123\) 0 0
\(124\) 13.9042 1.24863
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.0033 1.50880 0.754398 0.656417i \(-0.227929\pi\)
0.754398 + 0.656417i \(0.227929\pi\)
\(128\) −19.3242 −1.70804
\(129\) 0 0
\(130\) 0.769020 0.0674475
\(131\) −0.0430508 −0.00376136 −0.00188068 0.999998i \(-0.500599\pi\)
−0.00188068 + 0.999998i \(0.500599\pi\)
\(132\) 0 0
\(133\) −24.5004 −2.12445
\(134\) −37.5109 −3.24045
\(135\) 0 0
\(136\) 24.9442 2.13895
\(137\) −7.36257 −0.629027 −0.314513 0.949253i \(-0.601841\pi\)
−0.314513 + 0.949253i \(0.601841\pi\)
\(138\) 0 0
\(139\) −13.2393 −1.12295 −0.561473 0.827495i \(-0.689765\pi\)
−0.561473 + 0.827495i \(0.689765\pi\)
\(140\) −13.2505 −1.11987
\(141\) 0 0
\(142\) −7.54476 −0.633142
\(143\) 0 0
\(144\) 0 0
\(145\) −5.03647 −0.418256
\(146\) 21.2500 1.75867
\(147\) 0 0
\(148\) 10.6198 0.872942
\(149\) −5.87858 −0.481592 −0.240796 0.970576i \(-0.577409\pi\)
−0.240796 + 0.970576i \(0.577409\pi\)
\(150\) 0 0
\(151\) −7.62821 −0.620775 −0.310387 0.950610i \(-0.600459\pi\)
−0.310387 + 0.950610i \(0.600459\pi\)
\(152\) 37.1872 3.01628
\(153\) 0 0
\(154\) 0 0
\(155\) 3.44899 0.277029
\(156\) 0 0
\(157\) −10.1332 −0.808719 −0.404360 0.914600i \(-0.632506\pi\)
−0.404360 + 0.914600i \(0.632506\pi\)
\(158\) −13.2973 −1.05788
\(159\) 0 0
\(160\) 0.310680 0.0245614
\(161\) 3.52981 0.278188
\(162\) 0 0
\(163\) −5.02906 −0.393906 −0.196953 0.980413i \(-0.563105\pi\)
−0.196953 + 0.980413i \(0.563105\pi\)
\(164\) 43.7292 3.41468
\(165\) 0 0
\(166\) 39.8969 3.09660
\(167\) 5.79105 0.448125 0.224062 0.974575i \(-0.428068\pi\)
0.224062 + 0.974575i \(0.428068\pi\)
\(168\) 0 0
\(169\) −12.9019 −0.992457
\(170\) 12.2794 0.941790
\(171\) 0 0
\(172\) 22.2318 1.69516
\(173\) 16.0652 1.22142 0.610708 0.791856i \(-0.290885\pi\)
0.610708 + 0.791856i \(0.290885\pi\)
\(174\) 0 0
\(175\) −3.28684 −0.248462
\(176\) 0 0
\(177\) 0 0
\(178\) −3.98143 −0.298421
\(179\) 8.30309 0.620602 0.310301 0.950638i \(-0.399570\pi\)
0.310301 + 0.950638i \(0.399570\pi\)
\(180\) 0 0
\(181\) −6.46425 −0.480484 −0.240242 0.970713i \(-0.577227\pi\)
−0.240242 + 0.970713i \(0.577227\pi\)
\(182\) −2.52765 −0.187362
\(183\) 0 0
\(184\) −5.35762 −0.394969
\(185\) 2.63428 0.193676
\(186\) 0 0
\(187\) 0 0
\(188\) 48.3693 3.52769
\(189\) 0 0
\(190\) 18.3064 1.32808
\(191\) −15.3693 −1.11208 −0.556041 0.831155i \(-0.687680\pi\)
−0.556041 + 0.831155i \(0.687680\pi\)
\(192\) 0 0
\(193\) −15.7518 −1.13384 −0.566919 0.823773i \(-0.691865\pi\)
−0.566919 + 0.823773i \(0.691865\pi\)
\(194\) 0.550320 0.0395107
\(195\) 0 0
\(196\) 15.3327 1.09519
\(197\) 16.3940 1.16802 0.584010 0.811746i \(-0.301483\pi\)
0.584010 + 0.811746i \(0.301483\pi\)
\(198\) 0 0
\(199\) 6.96500 0.493736 0.246868 0.969049i \(-0.420599\pi\)
0.246868 + 0.969049i \(0.420599\pi\)
\(200\) 4.98884 0.352764
\(201\) 0 0
\(202\) −1.24102 −0.0873180
\(203\) 16.5541 1.16187
\(204\) 0 0
\(205\) 10.8472 0.757602
\(206\) −15.7225 −1.09544
\(207\) 0 0
\(208\) 1.31180 0.0909569
\(209\) 0 0
\(210\) 0 0
\(211\) 19.9531 1.37363 0.686814 0.726833i \(-0.259008\pi\)
0.686814 + 0.726833i \(0.259008\pi\)
\(212\) −19.8966 −1.36650
\(213\) 0 0
\(214\) 5.13892 0.351289
\(215\) 5.51468 0.376098
\(216\) 0 0
\(217\) −11.3363 −0.769557
\(218\) 16.4367 1.11323
\(219\) 0 0
\(220\) 0 0
\(221\) 1.56567 0.105318
\(222\) 0 0
\(223\) −20.1466 −1.34912 −0.674559 0.738221i \(-0.735666\pi\)
−0.674559 + 0.738221i \(0.735666\pi\)
\(224\) −1.02116 −0.0682289
\(225\) 0 0
\(226\) 26.5091 1.76336
\(227\) 0.533937 0.0354386 0.0177193 0.999843i \(-0.494359\pi\)
0.0177193 + 0.999843i \(0.494359\pi\)
\(228\) 0 0
\(229\) −22.1931 −1.46656 −0.733279 0.679928i \(-0.762011\pi\)
−0.733279 + 0.679928i \(0.762011\pi\)
\(230\) −2.63743 −0.173907
\(231\) 0 0
\(232\) −25.1261 −1.64961
\(233\) −4.56567 −0.299107 −0.149553 0.988754i \(-0.547784\pi\)
−0.149553 + 0.988754i \(0.547784\pi\)
\(234\) 0 0
\(235\) 11.9982 0.782676
\(236\) 36.9439 2.40485
\(237\) 0 0
\(238\) −40.3606 −2.61619
\(239\) −5.86053 −0.379086 −0.189543 0.981872i \(-0.560701\pi\)
−0.189543 + 0.981872i \(0.560701\pi\)
\(240\) 0 0
\(241\) −9.96074 −0.641628 −0.320814 0.947142i \(-0.603956\pi\)
−0.320814 + 0.947142i \(0.603956\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −37.0254 −2.37031
\(245\) 3.80333 0.242986
\(246\) 0 0
\(247\) 2.33412 0.148517
\(248\) 17.2065 1.09261
\(249\) 0 0
\(250\) 2.45589 0.155324
\(251\) 16.8788 1.06538 0.532690 0.846310i \(-0.321181\pi\)
0.532690 + 0.846310i \(0.321181\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 41.7581 2.62014
\(255\) 0 0
\(256\) −32.2271 −2.01419
\(257\) −11.1436 −0.695117 −0.347559 0.937658i \(-0.612989\pi\)
−0.347559 + 0.937658i \(0.612989\pi\)
\(258\) 0 0
\(259\) −8.65847 −0.538011
\(260\) 1.26236 0.0782882
\(261\) 0 0
\(262\) −0.105728 −0.00653189
\(263\) −26.8726 −1.65704 −0.828519 0.559961i \(-0.810816\pi\)
−0.828519 + 0.559961i \(0.810816\pi\)
\(264\) 0 0
\(265\) −4.93543 −0.303181
\(266\) −60.1701 −3.68927
\(267\) 0 0
\(268\) −61.5748 −3.76128
\(269\) 10.0629 0.613545 0.306773 0.951783i \(-0.400751\pi\)
0.306773 + 0.951783i \(0.400751\pi\)
\(270\) 0 0
\(271\) −10.5441 −0.640509 −0.320255 0.947331i \(-0.603768\pi\)
−0.320255 + 0.947331i \(0.603768\pi\)
\(272\) 20.9463 1.27006
\(273\) 0 0
\(274\) −18.0816 −1.09235
\(275\) 0 0
\(276\) 0 0
\(277\) 17.9376 1.07777 0.538883 0.842381i \(-0.318847\pi\)
0.538883 + 0.842381i \(0.318847\pi\)
\(278\) −32.5143 −1.95008
\(279\) 0 0
\(280\) −16.3975 −0.979939
\(281\) 7.41103 0.442105 0.221052 0.975262i \(-0.429051\pi\)
0.221052 + 0.975262i \(0.429051\pi\)
\(282\) 0 0
\(283\) −5.38684 −0.320214 −0.160107 0.987100i \(-0.551184\pi\)
−0.160107 + 0.987100i \(0.551184\pi\)
\(284\) −12.3848 −0.734905
\(285\) 0 0
\(286\) 0 0
\(287\) −35.6530 −2.10453
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) −12.3690 −0.726332
\(291\) 0 0
\(292\) 34.8823 2.04133
\(293\) −1.74006 −0.101655 −0.0508277 0.998707i \(-0.516186\pi\)
−0.0508277 + 0.998707i \(0.516186\pi\)
\(294\) 0 0
\(295\) 9.16409 0.533554
\(296\) 13.1420 0.763864
\(297\) 0 0
\(298\) −14.4371 −0.836321
\(299\) −0.336280 −0.0194476
\(300\) 0 0
\(301\) −18.1259 −1.04476
\(302\) −18.7340 −1.07802
\(303\) 0 0
\(304\) 31.2271 1.79100
\(305\) −9.18431 −0.525892
\(306\) 0 0
\(307\) −21.3566 −1.21889 −0.609444 0.792829i \(-0.708607\pi\)
−0.609444 + 0.792829i \(0.708607\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.47033 0.481082
\(311\) −32.8096 −1.86046 −0.930231 0.366975i \(-0.880393\pi\)
−0.930231 + 0.366975i \(0.880393\pi\)
\(312\) 0 0
\(313\) −3.45852 −0.195487 −0.0977436 0.995212i \(-0.531163\pi\)
−0.0977436 + 0.995212i \(0.531163\pi\)
\(314\) −24.8860 −1.40440
\(315\) 0 0
\(316\) −21.8278 −1.22791
\(317\) 2.87566 0.161513 0.0807565 0.996734i \(-0.474266\pi\)
0.0807565 + 0.996734i \(0.474266\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.61553 −0.425721
\(321\) 0 0
\(322\) 8.66881 0.483094
\(323\) 37.2704 2.07378
\(324\) 0 0
\(325\) 0.313133 0.0173695
\(326\) −12.3508 −0.684048
\(327\) 0 0
\(328\) 54.1150 2.98800
\(329\) −39.4362 −2.17419
\(330\) 0 0
\(331\) −14.1221 −0.776219 −0.388109 0.921613i \(-0.626872\pi\)
−0.388109 + 0.921613i \(0.626872\pi\)
\(332\) 65.4915 3.59431
\(333\) 0 0
\(334\) 14.2222 0.778202
\(335\) −15.2739 −0.834501
\(336\) 0 0
\(337\) −15.9490 −0.868796 −0.434398 0.900721i \(-0.643039\pi\)
−0.434398 + 0.900721i \(0.643039\pi\)
\(338\) −31.6857 −1.72348
\(339\) 0 0
\(340\) 20.1569 1.09316
\(341\) 0 0
\(342\) 0 0
\(343\) 10.5070 0.567322
\(344\) 27.5118 1.48334
\(345\) 0 0
\(346\) 39.4543 2.12108
\(347\) 29.6801 1.59331 0.796656 0.604433i \(-0.206600\pi\)
0.796656 + 0.604433i \(0.206600\pi\)
\(348\) 0 0
\(349\) −31.6937 −1.69653 −0.848263 0.529574i \(-0.822352\pi\)
−0.848263 + 0.529574i \(0.822352\pi\)
\(350\) −8.07211 −0.431472
\(351\) 0 0
\(352\) 0 0
\(353\) −1.20189 −0.0639703 −0.0319852 0.999488i \(-0.510183\pi\)
−0.0319852 + 0.999488i \(0.510183\pi\)
\(354\) 0 0
\(355\) −3.07211 −0.163051
\(356\) −6.53559 −0.346385
\(357\) 0 0
\(358\) 20.3915 1.07772
\(359\) −11.6591 −0.615343 −0.307671 0.951493i \(-0.599550\pi\)
−0.307671 + 0.951493i \(0.599550\pi\)
\(360\) 0 0
\(361\) 36.5633 1.92438
\(362\) −15.8755 −0.834396
\(363\) 0 0
\(364\) −4.14918 −0.217476
\(365\) 8.65269 0.452903
\(366\) 0 0
\(367\) 15.9860 0.834465 0.417232 0.908800i \(-0.363000\pi\)
0.417232 + 0.908800i \(0.363000\pi\)
\(368\) −4.49894 −0.234523
\(369\) 0 0
\(370\) 6.46950 0.336333
\(371\) 16.2220 0.842203
\(372\) 0 0
\(373\) −0.321975 −0.0166712 −0.00833561 0.999965i \(-0.502653\pi\)
−0.00833561 + 0.999965i \(0.502653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 59.8570 3.08689
\(377\) −1.57709 −0.0812241
\(378\) 0 0
\(379\) 11.4174 0.586475 0.293237 0.956040i \(-0.405267\pi\)
0.293237 + 0.956040i \(0.405267\pi\)
\(380\) 30.0502 1.54154
\(381\) 0 0
\(382\) −37.7452 −1.93121
\(383\) −28.3673 −1.44950 −0.724750 0.689012i \(-0.758045\pi\)
−0.724750 + 0.689012i \(0.758045\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −38.6846 −1.96899
\(387\) 0 0
\(388\) 0.903359 0.0458611
\(389\) −15.1802 −0.769666 −0.384833 0.922986i \(-0.625741\pi\)
−0.384833 + 0.922986i \(0.625741\pi\)
\(390\) 0 0
\(391\) −5.36960 −0.271553
\(392\) 18.9742 0.958341
\(393\) 0 0
\(394\) 40.2617 2.02835
\(395\) −5.41446 −0.272431
\(396\) 0 0
\(397\) 5.22461 0.262216 0.131108 0.991368i \(-0.458147\pi\)
0.131108 + 0.991368i \(0.458147\pi\)
\(398\) 17.1053 0.857409
\(399\) 0 0
\(400\) 4.18926 0.209463
\(401\) −14.0007 −0.699160 −0.349580 0.936907i \(-0.613676\pi\)
−0.349580 + 0.936907i \(0.613676\pi\)
\(402\) 0 0
\(403\) 1.07999 0.0537983
\(404\) −2.03716 −0.101352
\(405\) 0 0
\(406\) 40.6549 2.01767
\(407\) 0 0
\(408\) 0 0
\(409\) −33.3112 −1.64713 −0.823567 0.567218i \(-0.808020\pi\)
−0.823567 + 0.567218i \(0.808020\pi\)
\(410\) 26.6395 1.31563
\(411\) 0 0
\(412\) −25.8088 −1.27151
\(413\) −30.1209 −1.48215
\(414\) 0 0
\(415\) 16.2454 0.797456
\(416\) 0.0972843 0.00476975
\(417\) 0 0
\(418\) 0 0
\(419\) 5.28460 0.258170 0.129085 0.991634i \(-0.458796\pi\)
0.129085 + 0.991634i \(0.458796\pi\)
\(420\) 0 0
\(421\) −30.7810 −1.50017 −0.750087 0.661340i \(-0.769988\pi\)
−0.750087 + 0.661340i \(0.769988\pi\)
\(422\) 49.0026 2.38541
\(423\) 0 0
\(424\) −24.6221 −1.19575
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 30.1874 1.46087
\(428\) 8.43562 0.407751
\(429\) 0 0
\(430\) 13.5434 0.653122
\(431\) 12.3506 0.594910 0.297455 0.954736i \(-0.403862\pi\)
0.297455 + 0.954736i \(0.403862\pi\)
\(432\) 0 0
\(433\) 1.41287 0.0678983 0.0339491 0.999424i \(-0.489192\pi\)
0.0339491 + 0.999424i \(0.489192\pi\)
\(434\) −27.8406 −1.33639
\(435\) 0 0
\(436\) 26.9811 1.29216
\(437\) −8.00509 −0.382935
\(438\) 0 0
\(439\) 7.58532 0.362028 0.181014 0.983481i \(-0.442062\pi\)
0.181014 + 0.983481i \(0.442062\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.84510 0.182893
\(443\) −11.0662 −0.525771 −0.262885 0.964827i \(-0.584674\pi\)
−0.262885 + 0.964827i \(0.584674\pi\)
\(444\) 0 0
\(445\) −1.62118 −0.0768512
\(446\) −49.4779 −2.34284
\(447\) 0 0
\(448\) 25.0311 1.18261
\(449\) −6.32856 −0.298663 −0.149332 0.988787i \(-0.547712\pi\)
−0.149332 + 0.988787i \(0.547712\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 43.5152 2.04678
\(453\) 0 0
\(454\) 1.31129 0.0615418
\(455\) −1.02922 −0.0482506
\(456\) 0 0
\(457\) 0.189579 0.00886814 0.00443407 0.999990i \(-0.498589\pi\)
0.00443407 + 0.999990i \(0.498589\pi\)
\(458\) −54.5036 −2.54679
\(459\) 0 0
\(460\) −4.32938 −0.201859
\(461\) −26.6198 −1.23981 −0.619904 0.784678i \(-0.712828\pi\)
−0.619904 + 0.784678i \(0.712828\pi\)
\(462\) 0 0
\(463\) 20.9935 0.975652 0.487826 0.872941i \(-0.337790\pi\)
0.487826 + 0.872941i \(0.337790\pi\)
\(464\) −21.0991 −0.979501
\(465\) 0 0
\(466\) −11.2128 −0.519421
\(467\) −7.89989 −0.365563 −0.182782 0.983154i \(-0.558510\pi\)
−0.182782 + 0.983154i \(0.558510\pi\)
\(468\) 0 0
\(469\) 50.2028 2.31815
\(470\) 29.4662 1.35917
\(471\) 0 0
\(472\) 45.7182 2.10435
\(473\) 0 0
\(474\) 0 0
\(475\) 7.45408 0.342017
\(476\) −66.2525 −3.03668
\(477\) 0 0
\(478\) −14.3928 −0.658311
\(479\) 39.9728 1.82640 0.913201 0.407509i \(-0.133602\pi\)
0.913201 + 0.407509i \(0.133602\pi\)
\(480\) 0 0
\(481\) 0.824882 0.0376114
\(482\) −24.4624 −1.11423
\(483\) 0 0
\(484\) 0 0
\(485\) 0.224082 0.0101750
\(486\) 0 0
\(487\) −9.93556 −0.450223 −0.225112 0.974333i \(-0.572275\pi\)
−0.225112 + 0.974333i \(0.572275\pi\)
\(488\) −45.8190 −2.07413
\(489\) 0 0
\(490\) 9.34054 0.421963
\(491\) 4.97349 0.224451 0.112225 0.993683i \(-0.464202\pi\)
0.112225 + 0.993683i \(0.464202\pi\)
\(492\) 0 0
\(493\) −25.1823 −1.13416
\(494\) 5.73234 0.257910
\(495\) 0 0
\(496\) 14.4487 0.648767
\(497\) 10.0975 0.452937
\(498\) 0 0
\(499\) 43.7757 1.95967 0.979834 0.199812i \(-0.0640332\pi\)
0.979834 + 0.199812i \(0.0640332\pi\)
\(500\) 4.03138 0.180289
\(501\) 0 0
\(502\) 41.4524 1.85011
\(503\) 6.28236 0.280117 0.140058 0.990143i \(-0.455271\pi\)
0.140058 + 0.990143i \(0.455271\pi\)
\(504\) 0 0
\(505\) −0.505326 −0.0224867
\(506\) 0 0
\(507\) 0 0
\(508\) 68.5467 3.04127
\(509\) −24.8381 −1.10093 −0.550465 0.834858i \(-0.685550\pi\)
−0.550465 + 0.834858i \(0.685550\pi\)
\(510\) 0 0
\(511\) −28.4400 −1.25811
\(512\) −40.4976 −1.78976
\(513\) 0 0
\(514\) −27.3674 −1.20712
\(515\) −6.40197 −0.282104
\(516\) 0 0
\(517\) 0 0
\(518\) −21.2642 −0.934296
\(519\) 0 0
\(520\) 1.56217 0.0685058
\(521\) 6.94869 0.304428 0.152214 0.988348i \(-0.451360\pi\)
0.152214 + 0.988348i \(0.451360\pi\)
\(522\) 0 0
\(523\) 26.7510 1.16974 0.584869 0.811128i \(-0.301146\pi\)
0.584869 + 0.811128i \(0.301146\pi\)
\(524\) −0.173554 −0.00758175
\(525\) 0 0
\(526\) −65.9962 −2.87757
\(527\) 17.2449 0.751202
\(528\) 0 0
\(529\) −21.8467 −0.949856
\(530\) −12.1209 −0.526496
\(531\) 0 0
\(532\) −98.7703 −4.28224
\(533\) 3.39662 0.147124
\(534\) 0 0
\(535\) 2.09249 0.0904662
\(536\) −76.1989 −3.29129
\(537\) 0 0
\(538\) 24.7133 1.06547
\(539\) 0 0
\(540\) 0 0
\(541\) −14.5084 −0.623767 −0.311883 0.950120i \(-0.600960\pi\)
−0.311883 + 0.950120i \(0.600960\pi\)
\(542\) −25.8951 −1.11229
\(543\) 0 0
\(544\) 1.55340 0.0666015
\(545\) 6.69278 0.286687
\(546\) 0 0
\(547\) 26.7346 1.14309 0.571543 0.820572i \(-0.306345\pi\)
0.571543 + 0.820572i \(0.306345\pi\)
\(548\) −29.6813 −1.26792
\(549\) 0 0
\(550\) 0 0
\(551\) −37.5422 −1.59935
\(552\) 0 0
\(553\) 17.7965 0.756784
\(554\) 44.0527 1.87162
\(555\) 0 0
\(556\) −53.3728 −2.26351
\(557\) 17.2444 0.730670 0.365335 0.930876i \(-0.380954\pi\)
0.365335 + 0.930876i \(0.380954\pi\)
\(558\) 0 0
\(559\) 1.72683 0.0730371
\(560\) −13.7694 −0.581865
\(561\) 0 0
\(562\) 18.2006 0.767748
\(563\) −0.831914 −0.0350610 −0.0175305 0.999846i \(-0.505580\pi\)
−0.0175305 + 0.999846i \(0.505580\pi\)
\(564\) 0 0
\(565\) 10.7941 0.454112
\(566\) −13.2295 −0.556076
\(567\) 0 0
\(568\) −15.3263 −0.643076
\(569\) 11.5961 0.486132 0.243066 0.970010i \(-0.421847\pi\)
0.243066 + 0.970010i \(0.421847\pi\)
\(570\) 0 0
\(571\) −21.8414 −0.914034 −0.457017 0.889458i \(-0.651082\pi\)
−0.457017 + 0.889458i \(0.651082\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −87.5598 −3.65468
\(575\) −1.07392 −0.0447856
\(576\) 0 0
\(577\) 9.74587 0.405726 0.202863 0.979207i \(-0.434975\pi\)
0.202863 + 0.979207i \(0.434975\pi\)
\(578\) 19.6471 0.817211
\(579\) 0 0
\(580\) −20.3039 −0.843074
\(581\) −53.3961 −2.21524
\(582\) 0 0
\(583\) 0 0
\(584\) 43.1669 1.78626
\(585\) 0 0
\(586\) −4.27339 −0.176532
\(587\) −22.9441 −0.947005 −0.473502 0.880793i \(-0.657010\pi\)
−0.473502 + 0.880793i \(0.657010\pi\)
\(588\) 0 0
\(589\) 25.7090 1.05932
\(590\) 22.5060 0.926556
\(591\) 0 0
\(592\) 11.0357 0.453565
\(593\) 28.7819 1.18193 0.590965 0.806697i \(-0.298747\pi\)
0.590965 + 0.806697i \(0.298747\pi\)
\(594\) 0 0
\(595\) −16.4342 −0.673737
\(596\) −23.6988 −0.970741
\(597\) 0 0
\(598\) −0.825867 −0.0337722
\(599\) 29.1951 1.19288 0.596440 0.802657i \(-0.296581\pi\)
0.596440 + 0.802657i \(0.296581\pi\)
\(600\) 0 0
\(601\) 6.68087 0.272518 0.136259 0.990673i \(-0.456492\pi\)
0.136259 + 0.990673i \(0.456492\pi\)
\(602\) −44.5151 −1.81430
\(603\) 0 0
\(604\) −30.7522 −1.25129
\(605\) 0 0
\(606\) 0 0
\(607\) 42.6108 1.72952 0.864759 0.502188i \(-0.167471\pi\)
0.864759 + 0.502188i \(0.167471\pi\)
\(608\) 2.31583 0.0939194
\(609\) 0 0
\(610\) −22.5556 −0.913251
\(611\) 3.75703 0.151993
\(612\) 0 0
\(613\) 5.83156 0.235535 0.117767 0.993041i \(-0.462426\pi\)
0.117767 + 0.993041i \(0.462426\pi\)
\(614\) −52.4495 −2.11669
\(615\) 0 0
\(616\) 0 0
\(617\) 33.6386 1.35424 0.677119 0.735874i \(-0.263228\pi\)
0.677119 + 0.735874i \(0.263228\pi\)
\(618\) 0 0
\(619\) 42.5616 1.71070 0.855349 0.518053i \(-0.173343\pi\)
0.855349 + 0.518053i \(0.173343\pi\)
\(620\) 13.9042 0.558405
\(621\) 0 0
\(622\) −80.5766 −3.23083
\(623\) 5.32856 0.213484
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.49374 −0.339478
\(627\) 0 0
\(628\) −40.8509 −1.63013
\(629\) 13.1714 0.525179
\(630\) 0 0
\(631\) −8.89989 −0.354299 −0.177149 0.984184i \(-0.556688\pi\)
−0.177149 + 0.984184i \(0.556688\pi\)
\(632\) −27.0119 −1.07448
\(633\) 0 0
\(634\) 7.06228 0.280479
\(635\) 17.0033 0.674755
\(636\) 0 0
\(637\) 1.19095 0.0471871
\(638\) 0 0
\(639\) 0 0
\(640\) −19.3242 −0.763858
\(641\) 6.16806 0.243624 0.121812 0.992553i \(-0.461130\pi\)
0.121812 + 0.992553i \(0.461130\pi\)
\(642\) 0 0
\(643\) 4.35335 0.171680 0.0858398 0.996309i \(-0.472643\pi\)
0.0858398 + 0.996309i \(0.472643\pi\)
\(644\) 14.2300 0.560740
\(645\) 0 0
\(646\) 91.5318 3.60127
\(647\) −13.4933 −0.530478 −0.265239 0.964183i \(-0.585451\pi\)
−0.265239 + 0.964183i \(0.585451\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.769020 0.0301635
\(651\) 0 0
\(652\) −20.2741 −0.793993
\(653\) −27.4481 −1.07413 −0.537064 0.843541i \(-0.680467\pi\)
−0.537064 + 0.843541i \(0.680467\pi\)
\(654\) 0 0
\(655\) −0.0430508 −0.00168213
\(656\) 45.4418 1.77420
\(657\) 0 0
\(658\) −96.8507 −3.77563
\(659\) −18.7768 −0.731441 −0.365721 0.930725i \(-0.619177\pi\)
−0.365721 + 0.930725i \(0.619177\pi\)
\(660\) 0 0
\(661\) −21.6525 −0.842184 −0.421092 0.907018i \(-0.638353\pi\)
−0.421092 + 0.907018i \(0.638353\pi\)
\(662\) −34.6822 −1.34796
\(663\) 0 0
\(664\) 81.0458 3.14519
\(665\) −24.5004 −0.950084
\(666\) 0 0
\(667\) 5.40877 0.209428
\(668\) 23.3459 0.903281
\(669\) 0 0
\(670\) −37.5109 −1.44917
\(671\) 0 0
\(672\) 0 0
\(673\) −23.3021 −0.898232 −0.449116 0.893474i \(-0.648261\pi\)
−0.449116 + 0.893474i \(0.648261\pi\)
\(674\) −39.1689 −1.50873
\(675\) 0 0
\(676\) −52.0127 −2.00049
\(677\) 33.2808 1.27909 0.639543 0.768756i \(-0.279124\pi\)
0.639543 + 0.768756i \(0.279124\pi\)
\(678\) 0 0
\(679\) −0.736522 −0.0282651
\(680\) 24.9442 0.956566
\(681\) 0 0
\(682\) 0 0
\(683\) −16.9244 −0.647593 −0.323796 0.946127i \(-0.604959\pi\)
−0.323796 + 0.946127i \(0.604959\pi\)
\(684\) 0 0
\(685\) −7.36257 −0.281309
\(686\) 25.8039 0.985197
\(687\) 0 0
\(688\) 23.1024 0.880772
\(689\) −1.54545 −0.0588769
\(690\) 0 0
\(691\) −48.4335 −1.84250 −0.921249 0.388973i \(-0.872830\pi\)
−0.921249 + 0.388973i \(0.872830\pi\)
\(692\) 64.7650 2.46200
\(693\) 0 0
\(694\) 72.8910 2.76690
\(695\) −13.2393 −0.502197
\(696\) 0 0
\(697\) 54.2360 2.05434
\(698\) −77.8362 −2.94614
\(699\) 0 0
\(700\) −13.2505 −0.500822
\(701\) −45.4161 −1.71534 −0.857672 0.514197i \(-0.828090\pi\)
−0.857672 + 0.514197i \(0.828090\pi\)
\(702\) 0 0
\(703\) 19.6361 0.740591
\(704\) 0 0
\(705\) 0 0
\(706\) −2.95171 −0.111089
\(707\) 1.66093 0.0624655
\(708\) 0 0
\(709\) −1.76497 −0.0662848 −0.0331424 0.999451i \(-0.510551\pi\)
−0.0331424 + 0.999451i \(0.510551\pi\)
\(710\) −7.54476 −0.283150
\(711\) 0 0
\(712\) −8.08780 −0.303103
\(713\) −3.70394 −0.138714
\(714\) 0 0
\(715\) 0 0
\(716\) 33.4729 1.25094
\(717\) 0 0
\(718\) −28.6334 −1.06859
\(719\) 3.29998 0.123069 0.0615343 0.998105i \(-0.480401\pi\)
0.0615343 + 0.998105i \(0.480401\pi\)
\(720\) 0 0
\(721\) 21.0423 0.783655
\(722\) 89.7952 3.34183
\(723\) 0 0
\(724\) −26.0599 −0.968507
\(725\) −5.03647 −0.187050
\(726\) 0 0
\(727\) 11.7838 0.437037 0.218519 0.975833i \(-0.429878\pi\)
0.218519 + 0.975833i \(0.429878\pi\)
\(728\) −5.13461 −0.190301
\(729\) 0 0
\(730\) 21.2500 0.786499
\(731\) 27.5734 1.01984
\(732\) 0 0
\(733\) −5.73108 −0.211682 −0.105841 0.994383i \(-0.533754\pi\)
−0.105841 + 0.994383i \(0.533754\pi\)
\(734\) 39.2599 1.44911
\(735\) 0 0
\(736\) −0.333646 −0.0122983
\(737\) 0 0
\(738\) 0 0
\(739\) 21.0551 0.774524 0.387262 0.921970i \(-0.373421\pi\)
0.387262 + 0.921970i \(0.373421\pi\)
\(740\) 10.6198 0.390391
\(741\) 0 0
\(742\) 39.8393 1.46255
\(743\) 13.1283 0.481630 0.240815 0.970571i \(-0.422585\pi\)
0.240815 + 0.970571i \(0.422585\pi\)
\(744\) 0 0
\(745\) −5.87858 −0.215375
\(746\) −0.790734 −0.0289508
\(747\) 0 0
\(748\) 0 0
\(749\) −6.87768 −0.251305
\(750\) 0 0
\(751\) 25.6251 0.935073 0.467537 0.883974i \(-0.345142\pi\)
0.467537 + 0.883974i \(0.345142\pi\)
\(752\) 50.2636 1.83292
\(753\) 0 0
\(754\) −3.87315 −0.141052
\(755\) −7.62821 −0.277619
\(756\) 0 0
\(757\) 31.1970 1.13387 0.566936 0.823762i \(-0.308129\pi\)
0.566936 + 0.823762i \(0.308129\pi\)
\(758\) 28.0400 1.01846
\(759\) 0 0
\(760\) 37.1872 1.34892
\(761\) 11.3761 0.412382 0.206191 0.978512i \(-0.433893\pi\)
0.206191 + 0.978512i \(0.433893\pi\)
\(762\) 0 0
\(763\) −21.9981 −0.796385
\(764\) −61.9594 −2.24161
\(765\) 0 0
\(766\) −69.6668 −2.51717
\(767\) 2.86958 0.103615
\(768\) 0 0
\(769\) −10.3938 −0.374811 −0.187405 0.982283i \(-0.560008\pi\)
−0.187405 + 0.982283i \(0.560008\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −63.5014 −2.28547
\(773\) −14.0348 −0.504796 −0.252398 0.967623i \(-0.581219\pi\)
−0.252398 + 0.967623i \(0.581219\pi\)
\(774\) 0 0
\(775\) 3.44899 0.123891
\(776\) 1.11791 0.0401306
\(777\) 0 0
\(778\) −37.2808 −1.33658
\(779\) 80.8559 2.89696
\(780\) 0 0
\(781\) 0 0
\(782\) −13.1871 −0.471571
\(783\) 0 0
\(784\) 15.9331 0.569041
\(785\) −10.1332 −0.361670
\(786\) 0 0
\(787\) 13.8176 0.492545 0.246273 0.969201i \(-0.420794\pi\)
0.246273 + 0.969201i \(0.420794\pi\)
\(788\) 66.0902 2.35437
\(789\) 0 0
\(790\) −13.2973 −0.473097
\(791\) −35.4785 −1.26147
\(792\) 0 0
\(793\) −2.87591 −0.102127
\(794\) 12.8311 0.455357
\(795\) 0 0
\(796\) 28.0786 0.995218
\(797\) 5.38594 0.190780 0.0953898 0.995440i \(-0.469590\pi\)
0.0953898 + 0.995440i \(0.469590\pi\)
\(798\) 0 0
\(799\) 59.9910 2.12233
\(800\) 0.310680 0.0109842
\(801\) 0 0
\(802\) −34.3840 −1.21414
\(803\) 0 0
\(804\) 0 0
\(805\) 3.52981 0.124409
\(806\) 2.65234 0.0934248
\(807\) 0 0
\(808\) −2.52099 −0.0886880
\(809\) 23.7748 0.835876 0.417938 0.908476i \(-0.362753\pi\)
0.417938 + 0.908476i \(0.362753\pi\)
\(810\) 0 0
\(811\) −9.46335 −0.332303 −0.166152 0.986100i \(-0.553134\pi\)
−0.166152 + 0.986100i \(0.553134\pi\)
\(812\) 66.7358 2.34197
\(813\) 0 0
\(814\) 0 0
\(815\) −5.02906 −0.176160
\(816\) 0 0
\(817\) 41.1068 1.43815
\(818\) −81.8086 −2.86037
\(819\) 0 0
\(820\) 43.7292 1.52709
\(821\) −10.4189 −0.363622 −0.181811 0.983334i \(-0.558196\pi\)
−0.181811 + 0.983334i \(0.558196\pi\)
\(822\) 0 0
\(823\) 24.3540 0.848928 0.424464 0.905445i \(-0.360463\pi\)
0.424464 + 0.905445i \(0.360463\pi\)
\(824\) −31.9384 −1.11263
\(825\) 0 0
\(826\) −73.9736 −2.57387
\(827\) −22.0006 −0.765037 −0.382519 0.923948i \(-0.624943\pi\)
−0.382519 + 0.923948i \(0.624943\pi\)
\(828\) 0 0
\(829\) 3.03289 0.105337 0.0526683 0.998612i \(-0.483227\pi\)
0.0526683 + 0.998612i \(0.483227\pi\)
\(830\) 39.8969 1.38484
\(831\) 0 0
\(832\) −2.38468 −0.0826738
\(833\) 19.0166 0.658888
\(834\) 0 0
\(835\) 5.79105 0.200408
\(836\) 0 0
\(837\) 0 0
\(838\) 12.9784 0.448331
\(839\) 48.5383 1.67573 0.837864 0.545879i \(-0.183804\pi\)
0.837864 + 0.545879i \(0.183804\pi\)
\(840\) 0 0
\(841\) −3.63399 −0.125310
\(842\) −75.5946 −2.60516
\(843\) 0 0
\(844\) 80.4386 2.76881
\(845\) −12.9019 −0.443840
\(846\) 0 0
\(847\) 0 0
\(848\) −20.6758 −0.710011
\(849\) 0 0
\(850\) 12.2794 0.421181
\(851\) −2.82901 −0.0969773
\(852\) 0 0
\(853\) 11.7632 0.402766 0.201383 0.979513i \(-0.435456\pi\)
0.201383 + 0.979513i \(0.435456\pi\)
\(854\) 74.1368 2.53691
\(855\) 0 0
\(856\) 10.4391 0.356801
\(857\) 1.61311 0.0551026 0.0275513 0.999620i \(-0.491229\pi\)
0.0275513 + 0.999620i \(0.491229\pi\)
\(858\) 0 0
\(859\) 47.3263 1.61475 0.807376 0.590038i \(-0.200887\pi\)
0.807376 + 0.590038i \(0.200887\pi\)
\(860\) 22.2318 0.758097
\(861\) 0 0
\(862\) 30.3318 1.03310
\(863\) 9.97233 0.339462 0.169731 0.985490i \(-0.445710\pi\)
0.169731 + 0.985490i \(0.445710\pi\)
\(864\) 0 0
\(865\) 16.0652 0.546234
\(866\) 3.46985 0.117910
\(867\) 0 0
\(868\) −45.7009 −1.55119
\(869\) 0 0
\(870\) 0 0
\(871\) −4.78276 −0.162058
\(872\) 33.3892 1.13070
\(873\) 0 0
\(874\) −19.6596 −0.664996
\(875\) −3.28684 −0.111116
\(876\) 0 0
\(877\) −26.0057 −0.878151 −0.439076 0.898450i \(-0.644694\pi\)
−0.439076 + 0.898450i \(0.644694\pi\)
\(878\) 18.6287 0.628688
\(879\) 0 0
\(880\) 0 0
\(881\) −10.4081 −0.350657 −0.175329 0.984510i \(-0.556099\pi\)
−0.175329 + 0.984510i \(0.556099\pi\)
\(882\) 0 0
\(883\) −53.7283 −1.80810 −0.904051 0.427424i \(-0.859421\pi\)
−0.904051 + 0.427424i \(0.859421\pi\)
\(884\) 6.31180 0.212289
\(885\) 0 0
\(886\) −27.1773 −0.913040
\(887\) −40.6246 −1.36404 −0.682021 0.731333i \(-0.738899\pi\)
−0.682021 + 0.731333i \(0.738899\pi\)
\(888\) 0 0
\(889\) −55.8871 −1.87439
\(890\) −3.98143 −0.133458
\(891\) 0 0
\(892\) −81.2188 −2.71941
\(893\) 89.4354 2.99284
\(894\) 0 0
\(895\) 8.30309 0.277542
\(896\) 63.5157 2.12191
\(897\) 0 0
\(898\) −15.5422 −0.518651
\(899\) −17.3707 −0.579346
\(900\) 0 0
\(901\) −24.6772 −0.822115
\(902\) 0 0
\(903\) 0 0
\(904\) 53.8501 1.79103
\(905\) −6.46425 −0.214879
\(906\) 0 0
\(907\) −19.4070 −0.644398 −0.322199 0.946672i \(-0.604422\pi\)
−0.322199 + 0.946672i \(0.604422\pi\)
\(908\) 2.15250 0.0714333
\(909\) 0 0
\(910\) −2.52765 −0.0837907
\(911\) −10.7208 −0.355195 −0.177597 0.984103i \(-0.556832\pi\)
−0.177597 + 0.984103i \(0.556832\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.465585 0.0154002
\(915\) 0 0
\(916\) −89.4686 −2.95613
\(917\) 0.141501 0.00467278
\(918\) 0 0
\(919\) 23.1310 0.763021 0.381511 0.924364i \(-0.375404\pi\)
0.381511 + 0.924364i \(0.375404\pi\)
\(920\) −5.35762 −0.176635
\(921\) 0 0
\(922\) −65.3752 −2.15302
\(923\) −0.961981 −0.0316640
\(924\) 0 0
\(925\) 2.63428 0.0866147
\(926\) 51.5577 1.69429
\(927\) 0 0
\(928\) −1.56473 −0.0513648
\(929\) −26.2273 −0.860489 −0.430245 0.902712i \(-0.641573\pi\)
−0.430245 + 0.902712i \(0.641573\pi\)
\(930\) 0 0
\(931\) 28.3503 0.929144
\(932\) −18.4059 −0.602907
\(933\) 0 0
\(934\) −19.4012 −0.634828
\(935\) 0 0
\(936\) 0 0
\(937\) −23.4011 −0.764480 −0.382240 0.924063i \(-0.624847\pi\)
−0.382240 + 0.924063i \(0.624847\pi\)
\(938\) 123.292 4.02564
\(939\) 0 0
\(940\) 48.3693 1.57763
\(941\) −10.9687 −0.357570 −0.178785 0.983888i \(-0.557217\pi\)
−0.178785 + 0.983888i \(0.557217\pi\)
\(942\) 0 0
\(943\) −11.6490 −0.379345
\(944\) 38.3908 1.24951
\(945\) 0 0
\(946\) 0 0
\(947\) 13.3652 0.434310 0.217155 0.976137i \(-0.430322\pi\)
0.217155 + 0.976137i \(0.430322\pi\)
\(948\) 0 0
\(949\) 2.70945 0.0879524
\(950\) 18.3064 0.593937
\(951\) 0 0
\(952\) −81.9876 −2.65723
\(953\) −21.8242 −0.706956 −0.353478 0.935443i \(-0.615001\pi\)
−0.353478 + 0.935443i \(0.615001\pi\)
\(954\) 0 0
\(955\) −15.3693 −0.497339
\(956\) −23.6260 −0.764120
\(957\) 0 0
\(958\) 98.1686 3.17168
\(959\) 24.1996 0.781446
\(960\) 0 0
\(961\) −19.1045 −0.616273
\(962\) 2.02582 0.0653150
\(963\) 0 0
\(964\) −40.1555 −1.29332
\(965\) −15.7518 −0.507068
\(966\) 0 0
\(967\) −16.6600 −0.535750 −0.267875 0.963454i \(-0.586321\pi\)
−0.267875 + 0.963454i \(0.586321\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.550320 0.0176697
\(971\) −11.1032 −0.356320 −0.178160 0.984002i \(-0.557014\pi\)
−0.178160 + 0.984002i \(0.557014\pi\)
\(972\) 0 0
\(973\) 43.5156 1.39505
\(974\) −24.4006 −0.781846
\(975\) 0 0
\(976\) −38.4755 −1.23157
\(977\) −18.8144 −0.601926 −0.300963 0.953636i \(-0.597308\pi\)
−0.300963 + 0.953636i \(0.597308\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 15.3327 0.489784
\(981\) 0 0
\(982\) 12.2143 0.389775
\(983\) 1.37848 0.0439667 0.0219833 0.999758i \(-0.493002\pi\)
0.0219833 + 0.999758i \(0.493002\pi\)
\(984\) 0 0
\(985\) 16.3940 0.522355
\(986\) −61.8450 −1.96955
\(987\) 0 0
\(988\) 9.40973 0.299363
\(989\) −5.92233 −0.188319
\(990\) 0 0
\(991\) −46.3186 −1.47136 −0.735680 0.677329i \(-0.763137\pi\)
−0.735680 + 0.677329i \(0.763137\pi\)
\(992\) 1.07153 0.0340212
\(993\) 0 0
\(994\) 24.7984 0.786558
\(995\) 6.96500 0.220805
\(996\) 0 0
\(997\) −14.5470 −0.460709 −0.230355 0.973107i \(-0.573989\pi\)
−0.230355 + 0.973107i \(0.573989\pi\)
\(998\) 107.508 3.40311
\(999\) 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 5445.2.a.bv.1.3 4
3.2 odd 2 1815.2.a.o.1.2 4
11.7 odd 10 495.2.n.d.181.1 8
11.8 odd 10 495.2.n.d.361.1 8
11.10 odd 2 5445.2.a.be.1.2 4
15.14 odd 2 9075.2.a.dj.1.3 4
33.8 even 10 165.2.m.a.31.2 yes 8
33.29 even 10 165.2.m.a.16.2 8
33.32 even 2 1815.2.a.x.1.3 4
165.8 odd 20 825.2.bx.h.724.1 16
165.29 even 10 825.2.n.k.676.1 8
165.62 odd 20 825.2.bx.h.49.1 16
165.74 even 10 825.2.n.k.526.1 8
165.107 odd 20 825.2.bx.h.724.4 16
165.128 odd 20 825.2.bx.h.49.4 16
165.164 even 2 9075.2.a.cl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.2 8 33.29 even 10
165.2.m.a.31.2 yes 8 33.8 even 10
495.2.n.d.181.1 8 11.7 odd 10
495.2.n.d.361.1 8 11.8 odd 10
825.2.n.k.526.1 8 165.74 even 10
825.2.n.k.676.1 8 165.29 even 10
825.2.bx.h.49.1 16 165.62 odd 20
825.2.bx.h.49.4 16 165.128 odd 20
825.2.bx.h.724.1 16 165.8 odd 20
825.2.bx.h.724.4 16 165.107 odd 20
1815.2.a.o.1.2 4 3.2 odd 2
1815.2.a.x.1.3 4 33.32 even 2
5445.2.a.be.1.2 4 11.10 odd 2
5445.2.a.bv.1.3 4 1.1 even 1 trivial
9075.2.a.cl.1.2 4 165.164 even 2
9075.2.a.dj.1.3 4 15.14 odd 2