Properties

Label 405.2.a.j.1.1
Level $405$
Weight $2$
Character 405.1
Self dual yes
Analytic conductor $3.234$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.08613 q^{2} +2.35194 q^{4} +1.00000 q^{5} +4.08613 q^{7} -0.734191 q^{8} +O(q^{10})\) \(q-2.08613 q^{2} +2.35194 q^{4} +1.00000 q^{5} +4.08613 q^{7} -0.734191 q^{8} -2.08613 q^{10} -1.35194 q^{11} +0.648061 q^{13} -8.52420 q^{14} -3.17226 q^{16} -1.35194 q^{17} +0.648061 q^{19} +2.35194 q^{20} +2.82032 q^{22} +4.79001 q^{23} +1.00000 q^{25} -1.35194 q^{26} +9.61033 q^{28} +3.87614 q^{29} -7.69646 q^{31} +8.08613 q^{32} +2.82032 q^{34} +4.08613 q^{35} +7.52420 q^{37} -1.35194 q^{38} -0.734191 q^{40} -0.179679 q^{41} -0.820321 q^{43} -3.17968 q^{44} -9.99258 q^{46} +10.9065 q^{47} +9.69646 q^{49} -2.08613 q^{50} +1.52420 q^{52} +4.17226 q^{53} -1.35194 q^{55} -3.00000 q^{56} -8.08613 q^{58} +4.17226 q^{59} -3.82032 q^{61} +16.0558 q^{62} -10.5242 q^{64} +0.648061 q^{65} +8.14195 q^{67} -3.17968 q^{68} -8.52420 q^{70} -6.11644 q^{71} -12.3445 q^{73} -15.6965 q^{74} +1.52420 q^{76} -5.52420 q^{77} +10.3445 q^{79} -3.17226 q^{80} +0.374833 q^{82} -12.2584 q^{83} -1.35194 q^{85} +1.71130 q^{86} +0.992582 q^{88} -3.00000 q^{89} +2.64806 q^{91} +11.2658 q^{92} -22.7523 q^{94} +0.648061 q^{95} -13.5800 q^{97} -20.2281 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.08613 −1.47512 −0.737558 0.675283i \(-0.764021\pi\)
−0.737558 + 0.675283i \(0.764021\pi\)
\(3\) 0 0
\(4\) 2.35194 1.17597
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.08613 1.54441 0.772206 0.635372i \(-0.219153\pi\)
0.772206 + 0.635372i \(0.219153\pi\)
\(8\) −0.734191 −0.259576
\(9\) 0 0
\(10\) −2.08613 −0.659692
\(11\) −1.35194 −0.407625 −0.203813 0.979010i \(-0.565333\pi\)
−0.203813 + 0.979010i \(0.565333\pi\)
\(12\) 0 0
\(13\) 0.648061 0.179740 0.0898699 0.995954i \(-0.471355\pi\)
0.0898699 + 0.995954i \(0.471355\pi\)
\(14\) −8.52420 −2.27819
\(15\) 0 0
\(16\) −3.17226 −0.793065
\(17\) −1.35194 −0.327893 −0.163947 0.986469i \(-0.552423\pi\)
−0.163947 + 0.986469i \(0.552423\pi\)
\(18\) 0 0
\(19\) 0.648061 0.148675 0.0743377 0.997233i \(-0.476316\pi\)
0.0743377 + 0.997233i \(0.476316\pi\)
\(20\) 2.35194 0.525910
\(21\) 0 0
\(22\) 2.82032 0.601294
\(23\) 4.79001 0.998786 0.499393 0.866376i \(-0.333556\pi\)
0.499393 + 0.866376i \(0.333556\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.35194 −0.265137
\(27\) 0 0
\(28\) 9.61033 1.81618
\(29\) 3.87614 0.719781 0.359890 0.932995i \(-0.382814\pi\)
0.359890 + 0.932995i \(0.382814\pi\)
\(30\) 0 0
\(31\) −7.69646 −1.38233 −0.691163 0.722699i \(-0.742901\pi\)
−0.691163 + 0.722699i \(0.742901\pi\)
\(32\) 8.08613 1.42944
\(33\) 0 0
\(34\) 2.82032 0.483681
\(35\) 4.08613 0.690682
\(36\) 0 0
\(37\) 7.52420 1.23697 0.618485 0.785796i \(-0.287747\pi\)
0.618485 + 0.785796i \(0.287747\pi\)
\(38\) −1.35194 −0.219313
\(39\) 0 0
\(40\) −0.734191 −0.116086
\(41\) −0.179679 −0.0280611 −0.0140306 0.999902i \(-0.504466\pi\)
−0.0140306 + 0.999902i \(0.504466\pi\)
\(42\) 0 0
\(43\) −0.820321 −0.125098 −0.0625489 0.998042i \(-0.519923\pi\)
−0.0625489 + 0.998042i \(0.519923\pi\)
\(44\) −3.17968 −0.479355
\(45\) 0 0
\(46\) −9.99258 −1.47333
\(47\) 10.9065 1.59087 0.795435 0.606039i \(-0.207243\pi\)
0.795435 + 0.606039i \(0.207243\pi\)
\(48\) 0 0
\(49\) 9.69646 1.38521
\(50\) −2.08613 −0.295023
\(51\) 0 0
\(52\) 1.52420 0.211368
\(53\) 4.17226 0.573104 0.286552 0.958065i \(-0.407491\pi\)
0.286552 + 0.958065i \(0.407491\pi\)
\(54\) 0 0
\(55\) −1.35194 −0.182295
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −8.08613 −1.06176
\(59\) 4.17226 0.543182 0.271591 0.962413i \(-0.412450\pi\)
0.271591 + 0.962413i \(0.412450\pi\)
\(60\) 0 0
\(61\) −3.82032 −0.489142 −0.244571 0.969631i \(-0.578647\pi\)
−0.244571 + 0.969631i \(0.578647\pi\)
\(62\) 16.0558 2.03909
\(63\) 0 0
\(64\) −10.5242 −1.31552
\(65\) 0.648061 0.0803820
\(66\) 0 0
\(67\) 8.14195 0.994697 0.497349 0.867551i \(-0.334307\pi\)
0.497349 + 0.867551i \(0.334307\pi\)
\(68\) −3.17968 −0.385593
\(69\) 0 0
\(70\) −8.52420 −1.01884
\(71\) −6.11644 −0.725888 −0.362944 0.931811i \(-0.618228\pi\)
−0.362944 + 0.931811i \(0.618228\pi\)
\(72\) 0 0
\(73\) −12.3445 −1.44482 −0.722408 0.691467i \(-0.756965\pi\)
−0.722408 + 0.691467i \(0.756965\pi\)
\(74\) −15.6965 −1.82468
\(75\) 0 0
\(76\) 1.52420 0.174838
\(77\) −5.52420 −0.629541
\(78\) 0 0
\(79\) 10.3445 1.16385 0.581925 0.813243i \(-0.302300\pi\)
0.581925 + 0.813243i \(0.302300\pi\)
\(80\) −3.17226 −0.354669
\(81\) 0 0
\(82\) 0.374833 0.0413934
\(83\) −12.2584 −1.34553 −0.672767 0.739855i \(-0.734894\pi\)
−0.672767 + 0.739855i \(0.734894\pi\)
\(84\) 0 0
\(85\) −1.35194 −0.146638
\(86\) 1.71130 0.184534
\(87\) 0 0
\(88\) 0.992582 0.105810
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 2.64806 0.277592
\(92\) 11.2658 1.17454
\(93\) 0 0
\(94\) −22.7523 −2.34672
\(95\) 0.648061 0.0664896
\(96\) 0 0
\(97\) −13.5800 −1.37884 −0.689421 0.724361i \(-0.742135\pi\)
−0.689421 + 0.724361i \(0.742135\pi\)
\(98\) −20.2281 −2.04334
\(99\) 0 0
\(100\) 2.35194 0.235194
\(101\) −1.46838 −0.146109 −0.0730547 0.997328i \(-0.523275\pi\)
−0.0730547 + 0.997328i \(0.523275\pi\)
\(102\) 0 0
\(103\) 7.52420 0.741381 0.370691 0.928756i \(-0.379121\pi\)
0.370691 + 0.928756i \(0.379121\pi\)
\(104\) −0.475800 −0.0466561
\(105\) 0 0
\(106\) −8.70388 −0.845395
\(107\) 1.20999 0.116974 0.0584871 0.998288i \(-0.481372\pi\)
0.0584871 + 0.998288i \(0.481372\pi\)
\(108\) 0 0
\(109\) 14.1042 1.35094 0.675469 0.737388i \(-0.263941\pi\)
0.675469 + 0.737388i \(0.263941\pi\)
\(110\) 2.82032 0.268907
\(111\) 0 0
\(112\) −12.9623 −1.22482
\(113\) −11.9245 −1.12177 −0.560883 0.827895i \(-0.689538\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(114\) 0 0
\(115\) 4.79001 0.446671
\(116\) 9.11644 0.846440
\(117\) 0 0
\(118\) −8.70388 −0.801257
\(119\) −5.52420 −0.506403
\(120\) 0 0
\(121\) −9.17226 −0.833842
\(122\) 7.96969 0.721542
\(123\) 0 0
\(124\) −18.1016 −1.62557
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.07871 −0.628134 −0.314067 0.949401i \(-0.601692\pi\)
−0.314067 + 0.949401i \(0.601692\pi\)
\(128\) 5.78259 0.511114
\(129\) 0 0
\(130\) −1.35194 −0.118573
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 2.64806 0.229616
\(134\) −16.9852 −1.46729
\(135\) 0 0
\(136\) 0.992582 0.0851132
\(137\) −7.46838 −0.638067 −0.319033 0.947743i \(-0.603358\pi\)
−0.319033 + 0.947743i \(0.603358\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 9.61033 0.812221
\(141\) 0 0
\(142\) 12.7597 1.07077
\(143\) −0.876139 −0.0732664
\(144\) 0 0
\(145\) 3.87614 0.321896
\(146\) 25.7523 2.13127
\(147\) 0 0
\(148\) 17.6965 1.45464
\(149\) −10.5848 −0.867143 −0.433571 0.901119i \(-0.642747\pi\)
−0.433571 + 0.901119i \(0.642747\pi\)
\(150\) 0 0
\(151\) 17.6965 1.44012 0.720059 0.693913i \(-0.244115\pi\)
0.720059 + 0.693913i \(0.244115\pi\)
\(152\) −0.475800 −0.0385925
\(153\) 0 0
\(154\) 11.5242 0.928646
\(155\) −7.69646 −0.618195
\(156\) 0 0
\(157\) −2.53162 −0.202045 −0.101023 0.994884i \(-0.532211\pi\)
−0.101023 + 0.994884i \(0.532211\pi\)
\(158\) −21.5800 −1.71681
\(159\) 0 0
\(160\) 8.08613 0.639265
\(161\) 19.5726 1.54254
\(162\) 0 0
\(163\) 8.47580 0.663876 0.331938 0.943301i \(-0.392298\pi\)
0.331938 + 0.943301i \(0.392298\pi\)
\(164\) −0.422594 −0.0329990
\(165\) 0 0
\(166\) 25.5726 1.98482
\(167\) 12.7342 0.985401 0.492701 0.870199i \(-0.336010\pi\)
0.492701 + 0.870199i \(0.336010\pi\)
\(168\) 0 0
\(169\) −12.5800 −0.967694
\(170\) 2.82032 0.216309
\(171\) 0 0
\(172\) −1.92935 −0.147111
\(173\) 23.0484 1.75234 0.876169 0.482005i \(-0.160091\pi\)
0.876169 + 0.482005i \(0.160091\pi\)
\(174\) 0 0
\(175\) 4.08613 0.308882
\(176\) 4.28870 0.323273
\(177\) 0 0
\(178\) 6.25839 0.469086
\(179\) 2.22808 0.166534 0.0832672 0.996527i \(-0.473465\pi\)
0.0832672 + 0.996527i \(0.473465\pi\)
\(180\) 0 0
\(181\) 0.468382 0.0348146 0.0174073 0.999848i \(-0.494459\pi\)
0.0174073 + 0.999848i \(0.494459\pi\)
\(182\) −5.52420 −0.409481
\(183\) 0 0
\(184\) −3.51678 −0.259261
\(185\) 7.52420 0.553190
\(186\) 0 0
\(187\) 1.82774 0.133658
\(188\) 25.6513 1.87081
\(189\) 0 0
\(190\) −1.35194 −0.0980800
\(191\) −20.2281 −1.46365 −0.731826 0.681491i \(-0.761332\pi\)
−0.731826 + 0.681491i \(0.761332\pi\)
\(192\) 0 0
\(193\) −19.9293 −1.43455 −0.717273 0.696792i \(-0.754610\pi\)
−0.717273 + 0.696792i \(0.754610\pi\)
\(194\) 28.3297 2.03395
\(195\) 0 0
\(196\) 22.8055 1.62896
\(197\) −15.5800 −1.11003 −0.555015 0.831840i \(-0.687288\pi\)
−0.555015 + 0.831840i \(0.687288\pi\)
\(198\) 0 0
\(199\) 3.58482 0.254121 0.127061 0.991895i \(-0.459446\pi\)
0.127061 + 0.991895i \(0.459446\pi\)
\(200\) −0.734191 −0.0519151
\(201\) 0 0
\(202\) 3.06324 0.215529
\(203\) 15.8384 1.11164
\(204\) 0 0
\(205\) −0.179679 −0.0125493
\(206\) −15.6965 −1.09362
\(207\) 0 0
\(208\) −2.05582 −0.142545
\(209\) −0.876139 −0.0606038
\(210\) 0 0
\(211\) 14.9926 1.03213 0.516066 0.856549i \(-0.327396\pi\)
0.516066 + 0.856549i \(0.327396\pi\)
\(212\) 9.81290 0.673953
\(213\) 0 0
\(214\) −2.52420 −0.172551
\(215\) −0.820321 −0.0559454
\(216\) 0 0
\(217\) −31.4487 −2.13488
\(218\) −29.4232 −1.99279
\(219\) 0 0
\(220\) −3.17968 −0.214374
\(221\) −0.876139 −0.0589355
\(222\) 0 0
\(223\) −26.8310 −1.79674 −0.898368 0.439244i \(-0.855246\pi\)
−0.898368 + 0.439244i \(0.855246\pi\)
\(224\) 33.0410 2.20764
\(225\) 0 0
\(226\) 24.8761 1.65474
\(227\) 1.35194 0.0897314 0.0448657 0.998993i \(-0.485714\pi\)
0.0448657 + 0.998993i \(0.485714\pi\)
\(228\) 0 0
\(229\) −8.23550 −0.544217 −0.272108 0.962267i \(-0.587721\pi\)
−0.272108 + 0.962267i \(0.587721\pi\)
\(230\) −9.99258 −0.658891
\(231\) 0 0
\(232\) −2.84583 −0.186838
\(233\) 8.58744 0.562582 0.281291 0.959623i \(-0.409237\pi\)
0.281291 + 0.959623i \(0.409237\pi\)
\(234\) 0 0
\(235\) 10.9065 0.711458
\(236\) 9.81290 0.638766
\(237\) 0 0
\(238\) 11.5242 0.747003
\(239\) −23.9245 −1.54755 −0.773775 0.633461i \(-0.781634\pi\)
−0.773775 + 0.633461i \(0.781634\pi\)
\(240\) 0 0
\(241\) −6.24030 −0.401973 −0.200987 0.979594i \(-0.564415\pi\)
−0.200987 + 0.979594i \(0.564415\pi\)
\(242\) 19.1345 1.23001
\(243\) 0 0
\(244\) −8.98516 −0.575216
\(245\) 9.69646 0.619484
\(246\) 0 0
\(247\) 0.419983 0.0267229
\(248\) 5.65067 0.358818
\(249\) 0 0
\(250\) −2.08613 −0.131938
\(251\) −28.5726 −1.80349 −0.901743 0.432272i \(-0.857712\pi\)
−0.901743 + 0.432272i \(0.857712\pi\)
\(252\) 0 0
\(253\) −6.47580 −0.407130
\(254\) 14.7671 0.926571
\(255\) 0 0
\(256\) 8.98516 0.561573
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 30.7449 1.91039
\(260\) 1.52420 0.0945268
\(261\) 0 0
\(262\) 12.5168 0.773289
\(263\) 31.8687 1.96511 0.982555 0.185974i \(-0.0595439\pi\)
0.982555 + 0.185974i \(0.0595439\pi\)
\(264\) 0 0
\(265\) 4.17226 0.256300
\(266\) −5.52420 −0.338710
\(267\) 0 0
\(268\) 19.1494 1.16973
\(269\) −31.4971 −1.92041 −0.960207 0.279289i \(-0.909901\pi\)
−0.960207 + 0.279289i \(0.909901\pi\)
\(270\) 0 0
\(271\) −3.24030 −0.196834 −0.0984172 0.995145i \(-0.531378\pi\)
−0.0984172 + 0.995145i \(0.531378\pi\)
\(272\) 4.28870 0.260041
\(273\) 0 0
\(274\) 15.5800 0.941223
\(275\) −1.35194 −0.0815250
\(276\) 0 0
\(277\) −5.58482 −0.335560 −0.167780 0.985824i \(-0.553660\pi\)
−0.167780 + 0.985824i \(0.553660\pi\)
\(278\) −16.6890 −1.00094
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −24.1042 −1.43794 −0.718969 0.695043i \(-0.755385\pi\)
−0.718969 + 0.695043i \(0.755385\pi\)
\(282\) 0 0
\(283\) −10.5423 −0.626674 −0.313337 0.949642i \(-0.601447\pi\)
−0.313337 + 0.949642i \(0.601447\pi\)
\(284\) −14.3855 −0.853622
\(285\) 0 0
\(286\) 1.82774 0.108077
\(287\) −0.734191 −0.0433379
\(288\) 0 0
\(289\) −15.1723 −0.892486
\(290\) −8.08613 −0.474834
\(291\) 0 0
\(292\) −29.0336 −1.69906
\(293\) 18.9926 1.10956 0.554779 0.831998i \(-0.312803\pi\)
0.554779 + 0.831998i \(0.312803\pi\)
\(294\) 0 0
\(295\) 4.17226 0.242918
\(296\) −5.52420 −0.321088
\(297\) 0 0
\(298\) 22.0813 1.27914
\(299\) 3.10422 0.179521
\(300\) 0 0
\(301\) −3.35194 −0.193203
\(302\) −36.9171 −2.12434
\(303\) 0 0
\(304\) −2.05582 −0.117909
\(305\) −3.82032 −0.218751
\(306\) 0 0
\(307\) 29.4791 1.68246 0.841229 0.540679i \(-0.181833\pi\)
0.841229 + 0.540679i \(0.181833\pi\)
\(308\) −12.9926 −0.740321
\(309\) 0 0
\(310\) 16.0558 0.911909
\(311\) −9.41256 −0.533738 −0.266869 0.963733i \(-0.585989\pi\)
−0.266869 + 0.963733i \(0.585989\pi\)
\(312\) 0 0
\(313\) 11.6210 0.656858 0.328429 0.944529i \(-0.393481\pi\)
0.328429 + 0.944529i \(0.393481\pi\)
\(314\) 5.28128 0.298040
\(315\) 0 0
\(316\) 24.3297 1.36865
\(317\) −9.17968 −0.515582 −0.257791 0.966201i \(-0.582995\pi\)
−0.257791 + 0.966201i \(0.582995\pi\)
\(318\) 0 0
\(319\) −5.24030 −0.293401
\(320\) −10.5242 −0.588321
\(321\) 0 0
\(322\) −40.8310 −2.27542
\(323\) −0.876139 −0.0487497
\(324\) 0 0
\(325\) 0.648061 0.0359479
\(326\) −17.6816 −0.979295
\(327\) 0 0
\(328\) 0.131919 0.00728398
\(329\) 44.5652 2.45696
\(330\) 0 0
\(331\) −7.22066 −0.396883 −0.198442 0.980113i \(-0.563588\pi\)
−0.198442 + 0.980113i \(0.563588\pi\)
\(332\) −28.8310 −1.58231
\(333\) 0 0
\(334\) −26.5652 −1.45358
\(335\) 8.14195 0.444842
\(336\) 0 0
\(337\) 2.28390 0.124412 0.0622059 0.998063i \(-0.480186\pi\)
0.0622059 + 0.998063i \(0.480186\pi\)
\(338\) 26.2436 1.42746
\(339\) 0 0
\(340\) −3.17968 −0.172442
\(341\) 10.4051 0.563470
\(342\) 0 0
\(343\) 11.0181 0.594921
\(344\) 0.602272 0.0324724
\(345\) 0 0
\(346\) −48.0820 −2.58490
\(347\) −0.708686 −0.0380443 −0.0190221 0.999819i \(-0.506055\pi\)
−0.0190221 + 0.999819i \(0.506055\pi\)
\(348\) 0 0
\(349\) −21.3445 −1.14255 −0.571273 0.820760i \(-0.693550\pi\)
−0.571273 + 0.820760i \(0.693550\pi\)
\(350\) −8.52420 −0.455638
\(351\) 0 0
\(352\) −10.9320 −0.582675
\(353\) −10.0968 −0.537398 −0.268699 0.963224i \(-0.586594\pi\)
−0.268699 + 0.963224i \(0.586594\pi\)
\(354\) 0 0
\(355\) −6.11644 −0.324627
\(356\) −7.05582 −0.373958
\(357\) 0 0
\(358\) −4.64806 −0.245658
\(359\) 30.5578 1.61278 0.806388 0.591386i \(-0.201419\pi\)
0.806388 + 0.591386i \(0.201419\pi\)
\(360\) 0 0
\(361\) −18.5800 −0.977896
\(362\) −0.977106 −0.0513555
\(363\) 0 0
\(364\) 6.22808 0.326440
\(365\) −12.3445 −0.646142
\(366\) 0 0
\(367\) −7.17968 −0.374776 −0.187388 0.982286i \(-0.560002\pi\)
−0.187388 + 0.982286i \(0.560002\pi\)
\(368\) −15.1952 −0.792102
\(369\) 0 0
\(370\) −15.6965 −0.816020
\(371\) 17.0484 0.885109
\(372\) 0 0
\(373\) −21.9245 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(374\) −3.81290 −0.197161
\(375\) 0 0
\(376\) −8.00742 −0.412951
\(377\) 2.51197 0.129373
\(378\) 0 0
\(379\) −17.3929 −0.893414 −0.446707 0.894680i \(-0.647403\pi\)
−0.446707 + 0.894680i \(0.647403\pi\)
\(380\) 1.52420 0.0781898
\(381\) 0 0
\(382\) 42.1984 2.15906
\(383\) 0.475800 0.0243123 0.0121561 0.999926i \(-0.496130\pi\)
0.0121561 + 0.999926i \(0.496130\pi\)
\(384\) 0 0
\(385\) −5.52420 −0.281539
\(386\) 41.5752 2.11612
\(387\) 0 0
\(388\) −31.9394 −1.62148
\(389\) −5.58744 −0.283294 −0.141647 0.989917i \(-0.545240\pi\)
−0.141647 + 0.989917i \(0.545240\pi\)
\(390\) 0 0
\(391\) −6.47580 −0.327495
\(392\) −7.11905 −0.359567
\(393\) 0 0
\(394\) 32.5019 1.63742
\(395\) 10.3445 0.520489
\(396\) 0 0
\(397\) 3.75228 0.188321 0.0941607 0.995557i \(-0.469983\pi\)
0.0941607 + 0.995557i \(0.469983\pi\)
\(398\) −7.47841 −0.374859
\(399\) 0 0
\(400\) −3.17226 −0.158613
\(401\) 23.5652 1.17679 0.588394 0.808574i \(-0.299760\pi\)
0.588394 + 0.808574i \(0.299760\pi\)
\(402\) 0 0
\(403\) −4.98777 −0.248459
\(404\) −3.45355 −0.171820
\(405\) 0 0
\(406\) −33.0410 −1.63980
\(407\) −10.1723 −0.504220
\(408\) 0 0
\(409\) 1.04840 0.0518400 0.0259200 0.999664i \(-0.491748\pi\)
0.0259200 + 0.999664i \(0.491748\pi\)
\(410\) 0.374833 0.0185117
\(411\) 0 0
\(412\) 17.6965 0.871842
\(413\) 17.0484 0.838897
\(414\) 0 0
\(415\) −12.2584 −0.601741
\(416\) 5.24030 0.256927
\(417\) 0 0
\(418\) 1.82774 0.0893977
\(419\) −25.9197 −1.26626 −0.633131 0.774045i \(-0.718231\pi\)
−0.633131 + 0.774045i \(0.718231\pi\)
\(420\) 0 0
\(421\) 7.64064 0.372382 0.186191 0.982514i \(-0.440386\pi\)
0.186191 + 0.982514i \(0.440386\pi\)
\(422\) −31.2765 −1.52252
\(423\) 0 0
\(424\) −3.06324 −0.148764
\(425\) −1.35194 −0.0655787
\(426\) 0 0
\(427\) −15.6103 −0.755437
\(428\) 2.84583 0.137558
\(429\) 0 0
\(430\) 1.71130 0.0825261
\(431\) 7.98516 0.384632 0.192316 0.981333i \(-0.438400\pi\)
0.192316 + 0.981333i \(0.438400\pi\)
\(432\) 0 0
\(433\) −12.5120 −0.601287 −0.300644 0.953737i \(-0.597201\pi\)
−0.300644 + 0.953737i \(0.597201\pi\)
\(434\) 65.6062 3.14920
\(435\) 0 0
\(436\) 33.1723 1.58866
\(437\) 3.10422 0.148495
\(438\) 0 0
\(439\) −8.76450 −0.418307 −0.209153 0.977883i \(-0.567071\pi\)
−0.209153 + 0.977883i \(0.567071\pi\)
\(440\) 0.992582 0.0473195
\(441\) 0 0
\(442\) 1.82774 0.0869367
\(443\) 3.67095 0.174412 0.0872062 0.996190i \(-0.472206\pi\)
0.0872062 + 0.996190i \(0.472206\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 55.9729 2.65040
\(447\) 0 0
\(448\) −43.0032 −2.03171
\(449\) 28.1723 1.32953 0.664766 0.747052i \(-0.268531\pi\)
0.664766 + 0.747052i \(0.268531\pi\)
\(450\) 0 0
\(451\) 0.242915 0.0114384
\(452\) −28.0458 −1.31916
\(453\) 0 0
\(454\) −2.82032 −0.132364
\(455\) 2.64806 0.124143
\(456\) 0 0
\(457\) 35.2616 1.64947 0.824735 0.565519i \(-0.191324\pi\)
0.824735 + 0.565519i \(0.191324\pi\)
\(458\) 17.1803 0.802784
\(459\) 0 0
\(460\) 11.2658 0.525271
\(461\) 34.6768 1.61506 0.807530 0.589826i \(-0.200804\pi\)
0.807530 + 0.589826i \(0.200804\pi\)
\(462\) 0 0
\(463\) 7.44874 0.346172 0.173086 0.984907i \(-0.444626\pi\)
0.173086 + 0.984907i \(0.444626\pi\)
\(464\) −12.2961 −0.570833
\(465\) 0 0
\(466\) −17.9145 −0.829874
\(467\) 29.9655 1.38664 0.693319 0.720630i \(-0.256148\pi\)
0.693319 + 0.720630i \(0.256148\pi\)
\(468\) 0 0
\(469\) 33.2691 1.53622
\(470\) −22.7523 −1.04948
\(471\) 0 0
\(472\) −3.06324 −0.140997
\(473\) 1.10902 0.0509930
\(474\) 0 0
\(475\) 0.648061 0.0297351
\(476\) −12.9926 −0.595514
\(477\) 0 0
\(478\) 49.9097 2.28282
\(479\) −7.98516 −0.364851 −0.182426 0.983220i \(-0.558395\pi\)
−0.182426 + 0.983220i \(0.558395\pi\)
\(480\) 0 0
\(481\) 4.87614 0.222333
\(482\) 13.0181 0.592958
\(483\) 0 0
\(484\) −21.5726 −0.980573
\(485\) −13.5800 −0.616637
\(486\) 0 0
\(487\) −11.9442 −0.541243 −0.270621 0.962686i \(-0.587229\pi\)
−0.270621 + 0.962686i \(0.587229\pi\)
\(488\) 2.80485 0.126969
\(489\) 0 0
\(490\) −20.2281 −0.913811
\(491\) −9.22066 −0.416123 −0.208061 0.978116i \(-0.566715\pi\)
−0.208061 + 0.978116i \(0.566715\pi\)
\(492\) 0 0
\(493\) −5.24030 −0.236011
\(494\) −0.876139 −0.0394193
\(495\) 0 0
\(496\) 24.4152 1.09627
\(497\) −24.9926 −1.12107
\(498\) 0 0
\(499\) 30.1723 1.35070 0.675348 0.737499i \(-0.263993\pi\)
0.675348 + 0.737499i \(0.263993\pi\)
\(500\) 2.35194 0.105182
\(501\) 0 0
\(502\) 59.6062 2.66035
\(503\) −10.5981 −0.472546 −0.236273 0.971687i \(-0.575926\pi\)
−0.236273 + 0.971687i \(0.575926\pi\)
\(504\) 0 0
\(505\) −1.46838 −0.0653421
\(506\) 13.5094 0.600564
\(507\) 0 0
\(508\) −16.6487 −0.738667
\(509\) 28.7523 1.27442 0.637211 0.770689i \(-0.280088\pi\)
0.637211 + 0.770689i \(0.280088\pi\)
\(510\) 0 0
\(511\) −50.4413 −2.23139
\(512\) −30.3094 −1.33950
\(513\) 0 0
\(514\) 37.5503 1.65627
\(515\) 7.52420 0.331556
\(516\) 0 0
\(517\) −14.7449 −0.648478
\(518\) −64.1378 −2.81805
\(519\) 0 0
\(520\) −0.475800 −0.0208652
\(521\) −36.0942 −1.58132 −0.790658 0.612259i \(-0.790261\pi\)
−0.790658 + 0.612259i \(0.790261\pi\)
\(522\) 0 0
\(523\) 11.1297 0.486669 0.243334 0.969942i \(-0.421759\pi\)
0.243334 + 0.969942i \(0.421759\pi\)
\(524\) −14.1116 −0.616470
\(525\) 0 0
\(526\) −66.4823 −2.89877
\(527\) 10.4051 0.453255
\(528\) 0 0
\(529\) −0.0558176 −0.00242685
\(530\) −8.70388 −0.378072
\(531\) 0 0
\(532\) 6.22808 0.270021
\(533\) −0.116443 −0.00504370
\(534\) 0 0
\(535\) 1.20999 0.0523125
\(536\) −5.97774 −0.258199
\(537\) 0 0
\(538\) 65.7071 2.83284
\(539\) −13.1090 −0.564646
\(540\) 0 0
\(541\) −34.7374 −1.49348 −0.746740 0.665116i \(-0.768382\pi\)
−0.746740 + 0.665116i \(0.768382\pi\)
\(542\) 6.75970 0.290354
\(543\) 0 0
\(544\) −10.9320 −0.468704
\(545\) 14.1042 0.604158
\(546\) 0 0
\(547\) −2.71455 −0.116066 −0.0580328 0.998315i \(-0.518483\pi\)
−0.0580328 + 0.998315i \(0.518483\pi\)
\(548\) −17.5652 −0.750347
\(549\) 0 0
\(550\) 2.82032 0.120259
\(551\) 2.51197 0.107014
\(552\) 0 0
\(553\) 42.2691 1.79746
\(554\) 11.6507 0.494990
\(555\) 0 0
\(556\) 18.8155 0.797956
\(557\) −8.93676 −0.378663 −0.189331 0.981913i \(-0.560632\pi\)
−0.189331 + 0.981913i \(0.560632\pi\)
\(558\) 0 0
\(559\) −0.531618 −0.0224850
\(560\) −12.9623 −0.547756
\(561\) 0 0
\(562\) 50.2845 2.12113
\(563\) −9.36261 −0.394587 −0.197293 0.980344i \(-0.563215\pi\)
−0.197293 + 0.980344i \(0.563215\pi\)
\(564\) 0 0
\(565\) −11.9245 −0.501669
\(566\) 21.9926 0.924417
\(567\) 0 0
\(568\) 4.49064 0.188423
\(569\) 35.8735 1.50390 0.751948 0.659222i \(-0.229114\pi\)
0.751948 + 0.659222i \(0.229114\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −2.06063 −0.0861591
\(573\) 0 0
\(574\) 1.53162 0.0639285
\(575\) 4.79001 0.199757
\(576\) 0 0
\(577\) 1.35675 0.0564821 0.0282411 0.999601i \(-0.491009\pi\)
0.0282411 + 0.999601i \(0.491009\pi\)
\(578\) 31.6513 1.31652
\(579\) 0 0
\(580\) 9.11644 0.378540
\(581\) −50.0894 −2.07806
\(582\) 0 0
\(583\) −5.64064 −0.233612
\(584\) 9.06324 0.375039
\(585\) 0 0
\(586\) −39.6210 −1.63673
\(587\) 28.7900 1.18829 0.594145 0.804358i \(-0.297490\pi\)
0.594145 + 0.804358i \(0.297490\pi\)
\(588\) 0 0
\(589\) −4.98777 −0.205518
\(590\) −8.70388 −0.358333
\(591\) 0 0
\(592\) −23.8687 −0.980998
\(593\) 30.9171 1.26961 0.634807 0.772671i \(-0.281080\pi\)
0.634807 + 0.772671i \(0.281080\pi\)
\(594\) 0 0
\(595\) −5.52420 −0.226470
\(596\) −24.8949 −1.01973
\(597\) 0 0
\(598\) −6.47580 −0.264815
\(599\) 1.39292 0.0569132 0.0284566 0.999595i \(-0.490941\pi\)
0.0284566 + 0.999595i \(0.490941\pi\)
\(600\) 0 0
\(601\) 8.82513 0.359985 0.179992 0.983668i \(-0.442393\pi\)
0.179992 + 0.983668i \(0.442393\pi\)
\(602\) 6.99258 0.284996
\(603\) 0 0
\(604\) 41.6210 1.69353
\(605\) −9.17226 −0.372905
\(606\) 0 0
\(607\) −2.15678 −0.0875412 −0.0437706 0.999042i \(-0.513937\pi\)
−0.0437706 + 0.999042i \(0.513937\pi\)
\(608\) 5.24030 0.212522
\(609\) 0 0
\(610\) 7.96969 0.322683
\(611\) 7.06804 0.285942
\(612\) 0 0
\(613\) −9.57521 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(614\) −61.4971 −2.48182
\(615\) 0 0
\(616\) 4.05582 0.163414
\(617\) −37.6768 −1.51681 −0.758406 0.651783i \(-0.774021\pi\)
−0.758406 + 0.651783i \(0.774021\pi\)
\(618\) 0 0
\(619\) 17.1042 0.687477 0.343738 0.939065i \(-0.388307\pi\)
0.343738 + 0.939065i \(0.388307\pi\)
\(620\) −18.1016 −0.726978
\(621\) 0 0
\(622\) 19.6358 0.787325
\(623\) −12.2584 −0.491122
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −24.2429 −0.968942
\(627\) 0 0
\(628\) −5.95421 −0.237599
\(629\) −10.1723 −0.405595
\(630\) 0 0
\(631\) 33.1090 1.31805 0.659025 0.752121i \(-0.270969\pi\)
0.659025 + 0.752121i \(0.270969\pi\)
\(632\) −7.59485 −0.302107
\(633\) 0 0
\(634\) 19.1500 0.760544
\(635\) −7.07871 −0.280910
\(636\) 0 0
\(637\) 6.28390 0.248977
\(638\) 10.9320 0.432800
\(639\) 0 0
\(640\) 5.78259 0.228577
\(641\) −23.1526 −0.914473 −0.457237 0.889345i \(-0.651161\pi\)
−0.457237 + 0.889345i \(0.651161\pi\)
\(642\) 0 0
\(643\) 43.0639 1.69827 0.849137 0.528173i \(-0.177123\pi\)
0.849137 + 0.528173i \(0.177123\pi\)
\(644\) 46.0336 1.81398
\(645\) 0 0
\(646\) 1.82774 0.0719115
\(647\) −20.6439 −0.811595 −0.405798 0.913963i \(-0.633006\pi\)
−0.405798 + 0.913963i \(0.633006\pi\)
\(648\) 0 0
\(649\) −5.64064 −0.221415
\(650\) −1.35194 −0.0530274
\(651\) 0 0
\(652\) 19.9346 0.780698
\(653\) 6.83516 0.267480 0.133740 0.991016i \(-0.457301\pi\)
0.133740 + 0.991016i \(0.457301\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 0.569988 0.0222543
\(657\) 0 0
\(658\) −92.9688 −3.62430
\(659\) −26.8613 −1.04637 −0.523184 0.852220i \(-0.675256\pi\)
−0.523184 + 0.852220i \(0.675256\pi\)
\(660\) 0 0
\(661\) −2.12125 −0.0825071 −0.0412535 0.999149i \(-0.513135\pi\)
−0.0412535 + 0.999149i \(0.513135\pi\)
\(662\) 15.0632 0.585449
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 2.64806 0.102687
\(666\) 0 0
\(667\) 18.5667 0.718907
\(668\) 29.9500 1.15880
\(669\) 0 0
\(670\) −16.9852 −0.656194
\(671\) 5.16484 0.199387
\(672\) 0 0
\(673\) 34.8203 1.34222 0.671112 0.741356i \(-0.265817\pi\)
0.671112 + 0.741356i \(0.265817\pi\)
\(674\) −4.76450 −0.183522
\(675\) 0 0
\(676\) −29.5874 −1.13798
\(677\) −24.6842 −0.948692 −0.474346 0.880338i \(-0.657315\pi\)
−0.474346 + 0.880338i \(0.657315\pi\)
\(678\) 0 0
\(679\) −55.4897 −2.12950
\(680\) 0.992582 0.0380638
\(681\) 0 0
\(682\) −21.7065 −0.831184
\(683\) −38.4610 −1.47167 −0.735834 0.677162i \(-0.763210\pi\)
−0.735834 + 0.677162i \(0.763210\pi\)
\(684\) 0 0
\(685\) −7.46838 −0.285352
\(686\) −22.9852 −0.877578
\(687\) 0 0
\(688\) 2.60227 0.0992107
\(689\) 2.70388 0.103010
\(690\) 0 0
\(691\) 0.480608 0.0182832 0.00914159 0.999958i \(-0.497090\pi\)
0.00914159 + 0.999958i \(0.497090\pi\)
\(692\) 54.2084 2.06070
\(693\) 0 0
\(694\) 1.47841 0.0561197
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 0.242915 0.00920105
\(698\) 44.5274 1.68539
\(699\) 0 0
\(700\) 9.61033 0.363236
\(701\) −18.1797 −0.686637 −0.343318 0.939219i \(-0.611551\pi\)
−0.343318 + 0.939219i \(0.611551\pi\)
\(702\) 0 0
\(703\) 4.87614 0.183907
\(704\) 14.2281 0.536241
\(705\) 0 0
\(706\) 21.0632 0.792725
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 7.18710 0.269917 0.134959 0.990851i \(-0.456910\pi\)
0.134959 + 0.990851i \(0.456910\pi\)
\(710\) 12.7597 0.478863
\(711\) 0 0
\(712\) 2.20257 0.0825449
\(713\) −36.8661 −1.38065
\(714\) 0 0
\(715\) −0.876139 −0.0327657
\(716\) 5.24030 0.195839
\(717\) 0 0
\(718\) −63.7475 −2.37903
\(719\) 12.5168 0.466797 0.233399 0.972381i \(-0.425015\pi\)
0.233399 + 0.972381i \(0.425015\pi\)
\(720\) 0 0
\(721\) 30.7449 1.14500
\(722\) 38.7603 1.44251
\(723\) 0 0
\(724\) 1.10161 0.0409409
\(725\) 3.87614 0.143956
\(726\) 0 0
\(727\) 8.42584 0.312497 0.156249 0.987718i \(-0.450060\pi\)
0.156249 + 0.987718i \(0.450060\pi\)
\(728\) −1.94418 −0.0720562
\(729\) 0 0
\(730\) 25.7523 0.953135
\(731\) 1.10902 0.0410187
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 14.9777 0.552839
\(735\) 0 0
\(736\) 38.7326 1.42770
\(737\) −11.0074 −0.405463
\(738\) 0 0
\(739\) −1.81290 −0.0666887 −0.0333444 0.999444i \(-0.510616\pi\)
−0.0333444 + 0.999444i \(0.510616\pi\)
\(740\) 17.6965 0.650535
\(741\) 0 0
\(742\) −35.5652 −1.30564
\(743\) 20.1371 0.738760 0.369380 0.929278i \(-0.379570\pi\)
0.369380 + 0.929278i \(0.379570\pi\)
\(744\) 0 0
\(745\) −10.5848 −0.387798
\(746\) 45.7374 1.67457
\(747\) 0 0
\(748\) 4.29873 0.157177
\(749\) 4.94418 0.180656
\(750\) 0 0
\(751\) 12.2132 0.445668 0.222834 0.974856i \(-0.428469\pi\)
0.222834 + 0.974856i \(0.428469\pi\)
\(752\) −34.5981 −1.26166
\(753\) 0 0
\(754\) −5.24030 −0.190841
\(755\) 17.6965 0.644040
\(756\) 0 0
\(757\) 52.9533 1.92462 0.962310 0.271955i \(-0.0876701\pi\)
0.962310 + 0.271955i \(0.0876701\pi\)
\(758\) 36.2839 1.31789
\(759\) 0 0
\(760\) −0.475800 −0.0172591
\(761\) −18.4535 −0.668940 −0.334470 0.942406i \(-0.608557\pi\)
−0.334470 + 0.942406i \(0.608557\pi\)
\(762\) 0 0
\(763\) 57.6317 2.08641
\(764\) −47.5752 −1.72121
\(765\) 0 0
\(766\) −0.992582 −0.0358634
\(767\) 2.70388 0.0976314
\(768\) 0 0
\(769\) −4.45355 −0.160599 −0.0802995 0.996771i \(-0.525588\pi\)
−0.0802995 + 0.996771i \(0.525588\pi\)
\(770\) 11.5242 0.415303
\(771\) 0 0
\(772\) −46.8726 −1.68698
\(773\) 38.9368 1.40046 0.700229 0.713918i \(-0.253081\pi\)
0.700229 + 0.713918i \(0.253081\pi\)
\(774\) 0 0
\(775\) −7.69646 −0.276465
\(776\) 9.97033 0.357914
\(777\) 0 0
\(778\) 11.6561 0.417892
\(779\) −0.116443 −0.00417200
\(780\) 0 0
\(781\) 8.26906 0.295890
\(782\) 13.5094 0.483094
\(783\) 0 0
\(784\) −30.7597 −1.09856
\(785\) −2.53162 −0.0903573
\(786\) 0 0
\(787\) 34.2281 1.22010 0.610050 0.792363i \(-0.291150\pi\)
0.610050 + 0.792363i \(0.291150\pi\)
\(788\) −36.6433 −1.30536
\(789\) 0 0
\(790\) −21.5800 −0.767783
\(791\) −48.7252 −1.73247
\(792\) 0 0
\(793\) −2.47580 −0.0879183
\(794\) −7.82774 −0.277796
\(795\) 0 0
\(796\) 8.43129 0.298839
\(797\) 23.9655 0.848902 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(798\) 0 0
\(799\) −14.7449 −0.521636
\(800\) 8.08613 0.285888
\(801\) 0 0
\(802\) −49.1600 −1.73590
\(803\) 16.6890 0.588943
\(804\) 0 0
\(805\) 19.5726 0.689843
\(806\) 10.4051 0.366506
\(807\) 0 0
\(808\) 1.07807 0.0379265
\(809\) −0.283896 −0.00998124 −0.00499062 0.999988i \(-0.501589\pi\)
−0.00499062 + 0.999988i \(0.501589\pi\)
\(810\) 0 0
\(811\) 32.4413 1.13917 0.569584 0.821933i \(-0.307104\pi\)
0.569584 + 0.821933i \(0.307104\pi\)
\(812\) 37.2510 1.30725
\(813\) 0 0
\(814\) 21.2207 0.743784
\(815\) 8.47580 0.296894
\(816\) 0 0
\(817\) −0.531618 −0.0185990
\(818\) −2.18710 −0.0764701
\(819\) 0 0
\(820\) −0.422594 −0.0147576
\(821\) 41.6694 1.45427 0.727136 0.686493i \(-0.240851\pi\)
0.727136 + 0.686493i \(0.240851\pi\)
\(822\) 0 0
\(823\) −19.3626 −0.674938 −0.337469 0.941337i \(-0.609571\pi\)
−0.337469 + 0.941337i \(0.609571\pi\)
\(824\) −5.52420 −0.192445
\(825\) 0 0
\(826\) −35.5652 −1.23747
\(827\) 18.8097 0.654076 0.327038 0.945011i \(-0.393950\pi\)
0.327038 + 0.945011i \(0.393950\pi\)
\(828\) 0 0
\(829\) −33.1016 −1.14967 −0.574833 0.818271i \(-0.694933\pi\)
−0.574833 + 0.818271i \(0.694933\pi\)
\(830\) 25.5726 0.887638
\(831\) 0 0
\(832\) −6.82032 −0.236452
\(833\) −13.1090 −0.454201
\(834\) 0 0
\(835\) 12.7342 0.440685
\(836\) −2.06063 −0.0712682
\(837\) 0 0
\(838\) 54.0719 1.86788
\(839\) −39.6965 −1.37047 −0.685237 0.728320i \(-0.740301\pi\)
−0.685237 + 0.728320i \(0.740301\pi\)
\(840\) 0 0
\(841\) −13.9755 −0.481915
\(842\) −15.9394 −0.549307
\(843\) 0 0
\(844\) 35.2616 1.21376
\(845\) −12.5800 −0.432766
\(846\) 0 0
\(847\) −37.4791 −1.28780
\(848\) −13.2355 −0.454509
\(849\) 0 0
\(850\) 2.82032 0.0967362
\(851\) 36.0410 1.23547
\(852\) 0 0
\(853\) −4.11644 −0.140944 −0.0704722 0.997514i \(-0.522451\pi\)
−0.0704722 + 0.997514i \(0.522451\pi\)
\(854\) 32.5652 1.11436
\(855\) 0 0
\(856\) −0.888365 −0.0303637
\(857\) 14.7449 0.503675 0.251837 0.967770i \(-0.418965\pi\)
0.251837 + 0.967770i \(0.418965\pi\)
\(858\) 0 0
\(859\) −37.8539 −1.29156 −0.645779 0.763524i \(-0.723467\pi\)
−0.645779 + 0.763524i \(0.723467\pi\)
\(860\) −1.92935 −0.0657901
\(861\) 0 0
\(862\) −16.6581 −0.567377
\(863\) 26.7704 0.911274 0.455637 0.890166i \(-0.349412\pi\)
0.455637 + 0.890166i \(0.349412\pi\)
\(864\) 0 0
\(865\) 23.0484 0.783669
\(866\) 26.1016 0.886969
\(867\) 0 0
\(868\) −73.9655 −2.51055
\(869\) −13.9852 −0.474414
\(870\) 0 0
\(871\) 5.27648 0.178787
\(872\) −10.3552 −0.350671
\(873\) 0 0
\(874\) −6.47580 −0.219047
\(875\) 4.08613 0.138136
\(876\) 0 0
\(877\) −46.9681 −1.58600 −0.793001 0.609221i \(-0.791482\pi\)
−0.793001 + 0.609221i \(0.791482\pi\)
\(878\) 18.2839 0.617052
\(879\) 0 0
\(880\) 4.28870 0.144572
\(881\) −19.8055 −0.667264 −0.333632 0.942703i \(-0.608274\pi\)
−0.333632 + 0.942703i \(0.608274\pi\)
\(882\) 0 0
\(883\) −6.20257 −0.208733 −0.104367 0.994539i \(-0.533282\pi\)
−0.104367 + 0.994539i \(0.533282\pi\)
\(884\) −2.06063 −0.0693063
\(885\) 0 0
\(886\) −7.65809 −0.257279
\(887\) 13.4274 0.450848 0.225424 0.974261i \(-0.427623\pi\)
0.225424 + 0.974261i \(0.427623\pi\)
\(888\) 0 0
\(889\) −28.9245 −0.970098
\(890\) 6.25839 0.209782
\(891\) 0 0
\(892\) −63.1049 −2.11291
\(893\) 7.06804 0.236523
\(894\) 0 0
\(895\) 2.22808 0.0744764
\(896\) 23.6284 0.789370
\(897\) 0 0
\(898\) −58.7710 −1.96121
\(899\) −29.8325 −0.994971
\(900\) 0 0
\(901\) −5.64064 −0.187917
\(902\) −0.506752 −0.0168730
\(903\) 0 0
\(904\) 8.75489 0.291183
\(905\) 0.468382 0.0155695
\(906\) 0 0
\(907\) 0.673566 0.0223654 0.0111827 0.999937i \(-0.496440\pi\)
0.0111827 + 0.999937i \(0.496440\pi\)
\(908\) 3.17968 0.105521
\(909\) 0 0
\(910\) −5.52420 −0.183125
\(911\) −7.90970 −0.262060 −0.131030 0.991378i \(-0.541828\pi\)
−0.131030 + 0.991378i \(0.541828\pi\)
\(912\) 0 0
\(913\) 16.5726 0.548473
\(914\) −73.5604 −2.43316
\(915\) 0 0
\(916\) −19.3694 −0.639983
\(917\) −24.5168 −0.809615
\(918\) 0 0
\(919\) −8.58263 −0.283115 −0.141557 0.989930i \(-0.545211\pi\)
−0.141557 + 0.989930i \(0.545211\pi\)
\(920\) −3.51678 −0.115945
\(921\) 0 0
\(922\) −72.3404 −2.38240
\(923\) −3.96383 −0.130471
\(924\) 0 0
\(925\) 7.52420 0.247394
\(926\) −15.5390 −0.510644
\(927\) 0 0
\(928\) 31.3430 1.02888
\(929\) 29.6162 0.971676 0.485838 0.874049i \(-0.338515\pi\)
0.485838 + 0.874049i \(0.338515\pi\)
\(930\) 0 0
\(931\) 6.28390 0.205946
\(932\) 20.1971 0.661579
\(933\) 0 0
\(934\) −62.5120 −2.04545
\(935\) 1.82774 0.0597735
\(936\) 0 0
\(937\) −15.2058 −0.496753 −0.248376 0.968664i \(-0.579897\pi\)
−0.248376 + 0.968664i \(0.579897\pi\)
\(938\) −69.4036 −2.26611
\(939\) 0 0
\(940\) 25.6513 0.836654
\(941\) 5.65287 0.184278 0.0921391 0.995746i \(-0.470630\pi\)
0.0921391 + 0.995746i \(0.470630\pi\)
\(942\) 0 0
\(943\) −0.860663 −0.0280270
\(944\) −13.2355 −0.430779
\(945\) 0 0
\(946\) −2.31357 −0.0752206
\(947\) −40.3962 −1.31270 −0.656350 0.754457i \(-0.727900\pi\)
−0.656350 + 0.754457i \(0.727900\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) −1.35194 −0.0438627
\(951\) 0 0
\(952\) 4.05582 0.131450
\(953\) −22.9320 −0.742839 −0.371419 0.928465i \(-0.621129\pi\)
−0.371419 + 0.928465i \(0.621129\pi\)
\(954\) 0 0
\(955\) −20.2281 −0.654565
\(956\) −56.2691 −1.81987
\(957\) 0 0
\(958\) 16.6581 0.538198
\(959\) −30.5168 −0.985438
\(960\) 0 0
\(961\) 28.2355 0.910822
\(962\) −10.1723 −0.327967
\(963\) 0 0
\(964\) −14.6768 −0.472708
\(965\) −19.9293 −0.641548
\(966\) 0 0
\(967\) 10.3700 0.333478 0.166739 0.986001i \(-0.446676\pi\)
0.166739 + 0.986001i \(0.446676\pi\)
\(968\) 6.73419 0.216445
\(969\) 0 0
\(970\) 28.3297 0.909611
\(971\) 48.0410 1.54171 0.770854 0.637012i \(-0.219830\pi\)
0.770854 + 0.637012i \(0.219830\pi\)
\(972\) 0 0
\(973\) 32.6890 1.04796
\(974\) 24.9171 0.798396
\(975\) 0 0
\(976\) 12.1191 0.387921
\(977\) 27.0532 0.865509 0.432754 0.901512i \(-0.357542\pi\)
0.432754 + 0.901512i \(0.357542\pi\)
\(978\) 0 0
\(979\) 4.05582 0.129624
\(980\) 22.8055 0.728494
\(981\) 0 0
\(982\) 19.2355 0.613829
\(983\) −22.4817 −0.717054 −0.358527 0.933519i \(-0.616721\pi\)
−0.358527 + 0.933519i \(0.616721\pi\)
\(984\) 0 0
\(985\) −15.5800 −0.496421
\(986\) 10.9320 0.348144
\(987\) 0 0
\(988\) 0.987774 0.0314253
\(989\) −3.92935 −0.124946
\(990\) 0 0
\(991\) −26.5316 −0.842805 −0.421402 0.906874i \(-0.638462\pi\)
−0.421402 + 0.906874i \(0.638462\pi\)
\(992\) −62.2346 −1.97595
\(993\) 0 0
\(994\) 52.1378 1.65371
\(995\) 3.58482 0.113647
\(996\) 0 0
\(997\) −28.3659 −0.898356 −0.449178 0.893442i \(-0.648283\pi\)
−0.449178 + 0.893442i \(0.648283\pi\)
\(998\) −62.9433 −1.99243
\(999\) 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 405.2.a.j.1.1 3
3.2 odd 2 405.2.a.i.1.3 3
4.3 odd 2 6480.2.a.bv.1.1 3
5.2 odd 4 2025.2.b.l.649.2 6
5.3 odd 4 2025.2.b.l.649.5 6
5.4 even 2 2025.2.a.n.1.3 3
9.2 odd 6 135.2.e.b.91.1 6
9.4 even 3 45.2.e.b.16.3 6
9.5 odd 6 135.2.e.b.46.1 6
9.7 even 3 45.2.e.b.31.3 yes 6
12.11 even 2 6480.2.a.bs.1.1 3
15.2 even 4 2025.2.b.m.649.5 6
15.8 even 4 2025.2.b.m.649.2 6
15.14 odd 2 2025.2.a.o.1.1 3
36.7 odd 6 720.2.q.i.481.3 6
36.11 even 6 2160.2.q.k.1441.3 6
36.23 even 6 2160.2.q.k.721.3 6
36.31 odd 6 720.2.q.i.241.3 6
45.2 even 12 675.2.k.b.199.5 12
45.4 even 6 225.2.e.b.151.1 6
45.7 odd 12 225.2.k.b.49.2 12
45.13 odd 12 225.2.k.b.124.2 12
45.14 odd 6 675.2.e.b.451.3 6
45.22 odd 12 225.2.k.b.124.5 12
45.23 even 12 675.2.k.b.424.5 12
45.29 odd 6 675.2.e.b.226.3 6
45.32 even 12 675.2.k.b.424.2 12
45.34 even 6 225.2.e.b.76.1 6
45.38 even 12 675.2.k.b.199.2 12
45.43 odd 12 225.2.k.b.49.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.3 6 9.4 even 3
45.2.e.b.31.3 yes 6 9.7 even 3
135.2.e.b.46.1 6 9.5 odd 6
135.2.e.b.91.1 6 9.2 odd 6
225.2.e.b.76.1 6 45.34 even 6
225.2.e.b.151.1 6 45.4 even 6
225.2.k.b.49.2 12 45.7 odd 12
225.2.k.b.49.5 12 45.43 odd 12
225.2.k.b.124.2 12 45.13 odd 12
225.2.k.b.124.5 12 45.22 odd 12
405.2.a.i.1.3 3 3.2 odd 2
405.2.a.j.1.1 3 1.1 even 1 trivial
675.2.e.b.226.3 6 45.29 odd 6
675.2.e.b.451.3 6 45.14 odd 6
675.2.k.b.199.2 12 45.38 even 12
675.2.k.b.199.5 12 45.2 even 12
675.2.k.b.424.2 12 45.32 even 12
675.2.k.b.424.5 12 45.23 even 12
720.2.q.i.241.3 6 36.31 odd 6
720.2.q.i.481.3 6 36.7 odd 6
2025.2.a.n.1.3 3 5.4 even 2
2025.2.a.o.1.1 3 15.14 odd 2
2025.2.b.l.649.2 6 5.2 odd 4
2025.2.b.l.649.5 6 5.3 odd 4
2025.2.b.m.649.2 6 15.8 even 4
2025.2.b.m.649.5 6 15.2 even 4
2160.2.q.k.721.3 6 36.23 even 6
2160.2.q.k.1441.3 6 36.11 even 6
6480.2.a.bs.1.1 3 12.11 even 2
6480.2.a.bv.1.1 3 4.3 odd 2