Properties

Label 405.2.a.i.1.3
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 1.3
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 \(-0.235979\pi\)
0.737558 + 0.675283i \(0.235979\pi\)
\(3\) 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 \(-0.434667\pi\)
0.203813 + 0.979010i \(0.434667\pi\)
\(12\) 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 \(-0.447577\pi\)
0.163947 + 0.986469i \(0.447577\pi\)
\(18\) 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 \(-0.666444\pi\)
−0.499393 + 0.866376i \(0.666444\pi\)
\(24\) 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 \(-0.617186\pi\)
−0.359890 + 0.932995i \(0.617186\pi\)
\(30\) 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 \(-0.495534\pi\)
0.0140306 + 0.999902i \(0.495534\pi\)
\(42\) 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 \(-0.792757\pi\)
−0.795435 + 0.606039i \(0.792757\pi\)
\(48\) 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 \(-0.592509\pi\)
−0.286552 + 0.958065i \(0.592509\pi\)
\(54\) 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 \(-0.587550\pi\)
−0.271591 + 0.962413i \(0.587550\pi\)
\(60\) 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 \(-0.381772\pi\)
0.362944 + 0.931811i \(0.381772\pi\)
\(72\) 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 \(-0.265106\pi\)
0.672767 + 0.739855i \(0.265106\pi\)
\(84\) 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 \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 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 \(-0.476725\pi\)
0.0730547 + 0.997328i \(0.476725\pi\)
\(102\) 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 \(-0.518628\pi\)
−0.0584871 + 0.998288i \(0.518628\pi\)
\(108\) 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 \(-0.310462\pi\)
0.560883 + 0.827895i \(0.310462\pi\)
\(114\) 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 \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 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 \(-0.396642\pi\)
0.319033 + 0.947743i \(0.396642\pi\)
\(138\) 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 \(-0.357253\pi\)
0.433571 + 0.901119i \(0.357253\pi\)
\(150\) 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 \(-0.663990\pi\)
−0.492701 + 0.870199i \(0.663990\pi\)
\(168\) 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 \(-0.839909\pi\)
−0.876169 + 0.482005i \(0.839909\pi\)
\(174\) 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 \(-0.526535\pi\)
−0.0832672 + 0.996527i \(0.526535\pi\)
\(180\) 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 \(-0.238668\pi\)
0.731826 + 0.681491i \(0.238668\pi\)
\(192\) 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 \(-0.312712\pi\)
0.555015 + 0.831840i \(0.312712\pi\)
\(198\) 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 \(-0.514286\pi\)
−0.0448657 + 0.998993i \(0.514286\pi\)
\(228\) 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 \(-0.590763\pi\)
−0.281291 + 0.959623i \(0.590763\pi\)
\(234\) 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 \(-0.218366\pi\)
0.773775 + 0.633461i \(0.218366\pi\)
\(240\) 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 \(-0.142288\pi\)
0.901743 + 0.432272i \(0.142288\pi\)
\(252\) 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 \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 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 \(-0.940456\pi\)
−0.982555 + 0.185974i \(0.940456\pi\)
\(264\) 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 \(-0.0900987\pi\)
0.960207 + 0.279289i \(0.0900987\pi\)
\(270\) 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 \(-0.244615\pi\)
0.718969 + 0.695043i \(0.244615\pi\)
\(282\) 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 \(-0.687197\pi\)
−0.554779 + 0.831998i \(0.687197\pi\)
\(294\) 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 \(-0.414011\pi\)
0.266869 + 0.963733i \(0.414011\pi\)
\(312\) 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 \(-0.417005\pi\)
0.257791 + 0.966201i \(0.417005\pi\)
\(318\) 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 \(-0.493945\pi\)
0.0190221 + 0.999819i \(0.493945\pi\)
\(348\) 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 \(-0.413406\pi\)
0.268699 + 0.963224i \(0.413406\pi\)
\(354\) 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 \(-0.798581\pi\)
−0.806388 + 0.591386i \(0.798581\pi\)
\(360\) 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 \(-0.503870\pi\)
−0.0121561 + 0.999926i \(0.503870\pi\)
\(384\) 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 \(-0.454760\pi\)
0.141647 + 0.989917i \(0.454760\pi\)
\(390\) 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 \(-0.700240\pi\)
−0.588394 + 0.808574i \(0.700240\pi\)
\(402\) 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 \(-0.281769\pi\)
0.633131 + 0.774045i \(0.281769\pi\)
\(420\) 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 \(-0.561600\pi\)
−0.192316 + 0.981333i \(0.561600\pi\)
\(432\) 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 \(-0.527794\pi\)
−0.0872062 + 0.996190i \(0.527794\pi\)
\(444\) 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 \(-0.731469\pi\)
−0.664766 + 0.747052i \(0.731469\pi\)
\(450\) 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 \(-0.799196\pi\)
−0.807530 + 0.589826i \(0.799196\pi\)
\(462\) 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 \(-0.743852\pi\)
−0.693319 + 0.720630i \(0.743852\pi\)
\(468\) 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 \(-0.441605\pi\)
0.182426 + 0.983220i \(0.441605\pi\)
\(480\) 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 \(-0.433285\pi\)
0.208061 + 0.978116i \(0.433285\pi\)
\(492\) 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 \(-0.424074\pi\)
0.236273 + 0.971687i \(0.424074\pi\)
\(504\) 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 \(-0.719912\pi\)
−0.637211 + 0.770689i \(0.719912\pi\)
\(510\) 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 \(-0.209739\pi\)
0.790658 + 0.612259i \(0.209739\pi\)
\(522\) 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 \(-0.439368\pi\)
0.189331 + 0.981913i \(0.439368\pi\)
\(558\) 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 \(-0.436785\pi\)
0.197293 + 0.980344i \(0.436785\pi\)
\(564\) 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 \(-0.770886\pi\)
−0.751948 + 0.659222i \(0.770886\pi\)
\(570\) 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 \(-0.702510\pi\)
−0.594145 + 0.804358i \(0.702510\pi\)
\(588\) 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 \(-0.718920\pi\)
−0.634807 + 0.772671i \(0.718920\pi\)
\(594\) 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 \(-0.509059\pi\)
−0.0284566 + 0.999595i \(0.509059\pi\)
\(600\) 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 \(-0.225979\pi\)
0.758406 + 0.651783i \(0.225979\pi\)
\(618\) 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 \(-0.348839\pi\)
0.457237 + 0.889345i \(0.348839\pi\)
\(642\) 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 \(-0.366994\pi\)
0.405798 + 0.913963i \(0.366994\pi\)
\(648\) 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 \(-0.542699\pi\)
−0.133740 + 0.991016i \(0.542699\pi\)
\(654\) 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 \(-0.324744\pi\)
0.523184 + 0.852220i \(0.324744\pi\)
\(660\) 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 \(-0.342685\pi\)
0.474346 + 0.880338i \(0.342685\pi\)
\(678\) 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 \(-0.236790\pi\)
0.735834 + 0.677162i \(0.236790\pi\)
\(684\) 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 \(-0.388449\pi\)
0.343318 + 0.939219i \(0.388449\pi\)
\(702\) 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 \(-0.574985\pi\)
−0.233399 + 0.972381i \(0.574985\pi\)
\(720\) 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 \(-0.620430\pi\)
−0.369380 + 0.929278i \(0.620430\pi\)
\(744\) 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 \(-0.391443\pi\)
0.334470 + 0.942406i \(0.391443\pi\)
\(762\) 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 \(-0.746919\pi\)
−0.700229 + 0.713918i \(0.746919\pi\)
\(774\) 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 \(-0.639533\pi\)
−0.424451 + 0.905451i \(0.639533\pi\)
\(798\) 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 \(-0.498411\pi\)
0.00499062 + 0.999988i \(0.498411\pi\)
\(810\) 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 \(-0.759149\pi\)
−0.727136 + 0.686493i \(0.759149\pi\)
\(822\) 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 \(-0.606050\pi\)
−0.327038 + 0.945011i \(0.606050\pi\)
\(828\) 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 \(-0.259699\pi\)
0.685237 + 0.728320i \(0.259699\pi\)
\(840\) 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 \(-0.581035\pi\)
−0.251837 + 0.967770i \(0.581035\pi\)
\(858\) 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 \(-0.650588\pi\)
−0.455637 + 0.890166i \(0.650588\pi\)
\(864\) 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 \(-0.391726\pi\)
0.333632 + 0.942703i \(0.391726\pi\)
\(882\) 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 \(-0.572377\pi\)
−0.225424 + 0.974261i \(0.572377\pi\)
\(888\) 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 \(-0.458172\pi\)
0.131030 + 0.991378i \(0.458172\pi\)
\(912\) 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 \(-0.661485\pi\)
−0.485838 + 0.874049i \(0.661485\pi\)
\(930\) 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 \(-0.529370\pi\)
−0.0921391 + 0.995746i \(0.529370\pi\)
\(942\) 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 \(-0.272100\pi\)
0.656350 + 0.754457i \(0.272100\pi\)
\(948\) 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 \(-0.378871\pi\)
0.371419 + 0.928465i \(0.378871\pi\)
\(954\) 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 \(-0.780170\pi\)
−0.770854 + 0.637012i \(0.780170\pi\)
\(972\) 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 \(-0.642458\pi\)
−0.432754 + 0.901512i \(0.642458\pi\)
\(978\) 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 \(-0.383279\pi\)
0.358527 + 0.933519i \(0.383279\pi\)
\(984\) 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.i.1.3 3
3.2 odd 2 405.2.a.j.1.1 3
4.3 odd 2 6480.2.a.bs.1.1 3
5.2 odd 4 2025.2.b.m.649.5 6
5.3 odd 4 2025.2.b.m.649.2 6
5.4 even 2 2025.2.a.o.1.1 3
9.2 odd 6 45.2.e.b.31.3 yes 6
9.4 even 3 135.2.e.b.46.1 6
9.5 odd 6 45.2.e.b.16.3 6
9.7 even 3 135.2.e.b.91.1 6
12.11 even 2 6480.2.a.bv.1.1 3
15.2 even 4 2025.2.b.l.649.2 6
15.8 even 4 2025.2.b.l.649.5 6
15.14 odd 2 2025.2.a.n.1.3 3
36.7 odd 6 2160.2.q.k.1441.3 6
36.11 even 6 720.2.q.i.481.3 6
36.23 even 6 720.2.q.i.241.3 6
36.31 odd 6 2160.2.q.k.721.3 6
45.2 even 12 225.2.k.b.49.2 12
45.4 even 6 675.2.e.b.451.3 6
45.7 odd 12 675.2.k.b.199.5 12
45.13 odd 12 675.2.k.b.424.5 12
45.14 odd 6 225.2.e.b.151.1 6
45.22 odd 12 675.2.k.b.424.2 12
45.23 even 12 225.2.k.b.124.2 12
45.29 odd 6 225.2.e.b.76.1 6
45.32 even 12 225.2.k.b.124.5 12
45.34 even 6 675.2.e.b.226.3 6
45.38 even 12 225.2.k.b.49.5 12
45.43 odd 12 675.2.k.b.199.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.3 6 9.5 odd 6
45.2.e.b.31.3 yes 6 9.2 odd 6
135.2.e.b.46.1 6 9.4 even 3
135.2.e.b.91.1 6 9.7 even 3
225.2.e.b.76.1 6 45.29 odd 6
225.2.e.b.151.1 6 45.14 odd 6
225.2.k.b.49.2 12 45.2 even 12
225.2.k.b.49.5 12 45.38 even 12
225.2.k.b.124.2 12 45.23 even 12
225.2.k.b.124.5 12 45.32 even 12
405.2.a.i.1.3 3 1.1 even 1 trivial
405.2.a.j.1.1 3 3.2 odd 2
675.2.e.b.226.3 6 45.34 even 6
675.2.e.b.451.3 6 45.4 even 6
675.2.k.b.199.2 12 45.43 odd 12
675.2.k.b.199.5 12 45.7 odd 12
675.2.k.b.424.2 12 45.22 odd 12
675.2.k.b.424.5 12 45.13 odd 12
720.2.q.i.241.3 6 36.23 even 6
720.2.q.i.481.3 6 36.11 even 6
2025.2.a.n.1.3 3 15.14 odd 2
2025.2.a.o.1.1 3 5.4 even 2
2025.2.b.l.649.2 6 15.2 even 4
2025.2.b.l.649.5 6 15.8 even 4
2025.2.b.m.649.2 6 5.3 odd 4
2025.2.b.m.649.5 6 5.2 odd 4
2160.2.q.k.721.3 6 36.31 odd 6
2160.2.q.k.1441.3 6 36.7 odd 6
6480.2.a.bs.1.1 3 4.3 odd 2
6480.2.a.bv.1.1 3 12.11 even 2