Properties

Label 2169.2.a.e.1.3
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.27758\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} +O(q^{10})\) \(q-0.277577 q^{2} -1.92295 q^{4} +1.23324 q^{5} +1.36627 q^{7} +1.08892 q^{8} -0.342320 q^{10} +4.69806 q^{11} -0.0431968 q^{13} -0.379244 q^{14} +3.54364 q^{16} +7.31430 q^{17} -0.697489 q^{19} -2.37146 q^{20} -1.30407 q^{22} -1.41195 q^{23} -3.47911 q^{25} +0.0119904 q^{26} -2.62726 q^{28} -8.30334 q^{29} +3.39655 q^{31} -3.16148 q^{32} -2.03028 q^{34} +1.68494 q^{35} +7.15948 q^{37} +0.193607 q^{38} +1.34290 q^{40} -5.45541 q^{41} -11.7568 q^{43} -9.03414 q^{44} +0.391925 q^{46} +5.24836 q^{47} -5.13332 q^{49} +0.965723 q^{50} +0.0830653 q^{52} +8.57769 q^{53} +5.79385 q^{55} +1.48776 q^{56} +2.30482 q^{58} +12.9925 q^{59} +10.1636 q^{61} -0.942804 q^{62} -6.20973 q^{64} -0.0532721 q^{65} +10.1259 q^{67} -14.0650 q^{68} -0.467700 q^{70} -1.86703 q^{71} +6.47826 q^{73} -1.98731 q^{74} +1.34124 q^{76} +6.41880 q^{77} -12.9436 q^{79} +4.37017 q^{80} +1.51430 q^{82} +2.32915 q^{83} +9.02030 q^{85} +3.26342 q^{86} +5.11582 q^{88} +14.5180 q^{89} -0.0590183 q^{91} +2.71511 q^{92} -1.45682 q^{94} -0.860173 q^{95} -2.23725 q^{97} +1.42489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.277577 −0.196277 −0.0981383 0.995173i \(-0.531289\pi\)
−0.0981383 + 0.995173i \(0.531289\pi\)
\(3\) 0 0
\(4\) −1.92295 −0.961475
\(5\) 1.23324 0.551523 0.275761 0.961226i \(-0.411070\pi\)
0.275761 + 0.961226i \(0.411070\pi\)
\(6\) 0 0
\(7\) 1.36627 0.516400 0.258200 0.966092i \(-0.416871\pi\)
0.258200 + 0.966092i \(0.416871\pi\)
\(8\) 1.08892 0.384992
\(9\) 0 0
\(10\) −0.342320 −0.108251
\(11\) 4.69806 1.41652 0.708259 0.705952i \(-0.249481\pi\)
0.708259 + 0.705952i \(0.249481\pi\)
\(12\) 0 0
\(13\) −0.0431968 −0.0119806 −0.00599032 0.999982i \(-0.501907\pi\)
−0.00599032 + 0.999982i \(0.501907\pi\)
\(14\) −0.379244 −0.101357
\(15\) 0 0
\(16\) 3.54364 0.885911
\(17\) 7.31430 1.77398 0.886989 0.461790i \(-0.152793\pi\)
0.886989 + 0.461790i \(0.152793\pi\)
\(18\) 0 0
\(19\) −0.697489 −0.160015 −0.0800075 0.996794i \(-0.525494\pi\)
−0.0800075 + 0.996794i \(0.525494\pi\)
\(20\) −2.37146 −0.530275
\(21\) 0 0
\(22\) −1.30407 −0.278030
\(23\) −1.41195 −0.294412 −0.147206 0.989106i \(-0.547028\pi\)
−0.147206 + 0.989106i \(0.547028\pi\)
\(24\) 0 0
\(25\) −3.47911 −0.695823
\(26\) 0.0119904 0.00235152
\(27\) 0 0
\(28\) −2.62726 −0.496506
\(29\) −8.30334 −1.54189 −0.770946 0.636901i \(-0.780216\pi\)
−0.770946 + 0.636901i \(0.780216\pi\)
\(30\) 0 0
\(31\) 3.39655 0.610038 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(32\) −3.16148 −0.558875
\(33\) 0 0
\(34\) −2.03028 −0.348191
\(35\) 1.68494 0.284806
\(36\) 0 0
\(37\) 7.15948 1.17701 0.588506 0.808493i \(-0.299716\pi\)
0.588506 + 0.808493i \(0.299716\pi\)
\(38\) 0.193607 0.0314072
\(39\) 0 0
\(40\) 1.34290 0.212332
\(41\) −5.45541 −0.851992 −0.425996 0.904725i \(-0.640076\pi\)
−0.425996 + 0.904725i \(0.640076\pi\)
\(42\) 0 0
\(43\) −11.7568 −1.79290 −0.896448 0.443149i \(-0.853861\pi\)
−0.896448 + 0.443149i \(0.853861\pi\)
\(44\) −9.03414 −1.36195
\(45\) 0 0
\(46\) 0.391925 0.0577862
\(47\) 5.24836 0.765551 0.382776 0.923841i \(-0.374968\pi\)
0.382776 + 0.923841i \(0.374968\pi\)
\(48\) 0 0
\(49\) −5.13332 −0.733331
\(50\) 0.965723 0.136574
\(51\) 0 0
\(52\) 0.0830653 0.0115191
\(53\) 8.57769 1.17824 0.589118 0.808047i \(-0.299475\pi\)
0.589118 + 0.808047i \(0.299475\pi\)
\(54\) 0 0
\(55\) 5.79385 0.781242
\(56\) 1.48776 0.198810
\(57\) 0 0
\(58\) 2.30482 0.302637
\(59\) 12.9925 1.69148 0.845738 0.533598i \(-0.179160\pi\)
0.845738 + 0.533598i \(0.179160\pi\)
\(60\) 0 0
\(61\) 10.1636 1.30132 0.650658 0.759371i \(-0.274493\pi\)
0.650658 + 0.759371i \(0.274493\pi\)
\(62\) −0.942804 −0.119736
\(63\) 0 0
\(64\) −6.20973 −0.776216
\(65\) −0.0532721 −0.00660759
\(66\) 0 0
\(67\) 10.1259 1.23707 0.618536 0.785757i \(-0.287726\pi\)
0.618536 + 0.785757i \(0.287726\pi\)
\(68\) −14.0650 −1.70564
\(69\) 0 0
\(70\) −0.467700 −0.0559008
\(71\) −1.86703 −0.221576 −0.110788 0.993844i \(-0.535337\pi\)
−0.110788 + 0.993844i \(0.535337\pi\)
\(72\) 0 0
\(73\) 6.47826 0.758223 0.379111 0.925351i \(-0.376230\pi\)
0.379111 + 0.925351i \(0.376230\pi\)
\(74\) −1.98731 −0.231020
\(75\) 0 0
\(76\) 1.34124 0.153850
\(77\) 6.41880 0.731490
\(78\) 0 0
\(79\) −12.9436 −1.45627 −0.728136 0.685432i \(-0.759613\pi\)
−0.728136 + 0.685432i \(0.759613\pi\)
\(80\) 4.37017 0.488600
\(81\) 0 0
\(82\) 1.51430 0.167226
\(83\) 2.32915 0.255657 0.127829 0.991796i \(-0.459199\pi\)
0.127829 + 0.991796i \(0.459199\pi\)
\(84\) 0 0
\(85\) 9.02030 0.978389
\(86\) 3.26342 0.351904
\(87\) 0 0
\(88\) 5.11582 0.545348
\(89\) 14.5180 1.53890 0.769450 0.638707i \(-0.220530\pi\)
0.769450 + 0.638707i \(0.220530\pi\)
\(90\) 0 0
\(91\) −0.0590183 −0.00618680
\(92\) 2.71511 0.283070
\(93\) 0 0
\(94\) −1.45682 −0.150260
\(95\) −0.860173 −0.0882519
\(96\) 0 0
\(97\) −2.23725 −0.227158 −0.113579 0.993529i \(-0.536231\pi\)
−0.113579 + 0.993529i \(0.536231\pi\)
\(98\) 1.42489 0.143936
\(99\) 0 0
\(100\) 6.69017 0.669017
\(101\) 4.01607 0.399614 0.199807 0.979835i \(-0.435969\pi\)
0.199807 + 0.979835i \(0.435969\pi\)
\(102\) 0 0
\(103\) −5.25187 −0.517482 −0.258741 0.965947i \(-0.583308\pi\)
−0.258741 + 0.965947i \(0.583308\pi\)
\(104\) −0.0470379 −0.00461245
\(105\) 0 0
\(106\) −2.38097 −0.231260
\(107\) −5.60385 −0.541745 −0.270873 0.962615i \(-0.587312\pi\)
−0.270873 + 0.962615i \(0.587312\pi\)
\(108\) 0 0
\(109\) −4.77557 −0.457416 −0.228708 0.973495i \(-0.573450\pi\)
−0.228708 + 0.973495i \(0.573450\pi\)
\(110\) −1.60824 −0.153340
\(111\) 0 0
\(112\) 4.84155 0.457484
\(113\) 12.6186 1.18706 0.593531 0.804811i \(-0.297733\pi\)
0.593531 + 0.804811i \(0.297733\pi\)
\(114\) 0 0
\(115\) −1.74127 −0.162375
\(116\) 15.9669 1.48249
\(117\) 0 0
\(118\) −3.60642 −0.331997
\(119\) 9.99327 0.916082
\(120\) 0 0
\(121\) 11.0718 1.00653
\(122\) −2.82118 −0.255418
\(123\) 0 0
\(124\) −6.53139 −0.586536
\(125\) −10.4568 −0.935285
\(126\) 0 0
\(127\) 0.427651 0.0379479 0.0189740 0.999820i \(-0.493960\pi\)
0.0189740 + 0.999820i \(0.493960\pi\)
\(128\) 8.04663 0.711229
\(129\) 0 0
\(130\) 0.0147871 0.00129692
\(131\) 13.3633 1.16756 0.583780 0.811912i \(-0.301573\pi\)
0.583780 + 0.811912i \(0.301573\pi\)
\(132\) 0 0
\(133\) −0.952955 −0.0826317
\(134\) −2.81071 −0.242808
\(135\) 0 0
\(136\) 7.96470 0.682967
\(137\) −11.2702 −0.962874 −0.481437 0.876481i \(-0.659885\pi\)
−0.481437 + 0.876481i \(0.659885\pi\)
\(138\) 0 0
\(139\) −0.927184 −0.0786427 −0.0393214 0.999227i \(-0.512520\pi\)
−0.0393214 + 0.999227i \(0.512520\pi\)
\(140\) −3.24005 −0.273834
\(141\) 0 0
\(142\) 0.518245 0.0434902
\(143\) −0.202941 −0.0169708
\(144\) 0 0
\(145\) −10.2400 −0.850388
\(146\) −1.79822 −0.148821
\(147\) 0 0
\(148\) −13.7673 −1.13167
\(149\) −12.8757 −1.05481 −0.527407 0.849612i \(-0.676836\pi\)
−0.527407 + 0.849612i \(0.676836\pi\)
\(150\) 0 0
\(151\) −21.2436 −1.72878 −0.864390 0.502823i \(-0.832295\pi\)
−0.864390 + 0.502823i \(0.832295\pi\)
\(152\) −0.759511 −0.0616045
\(153\) 0 0
\(154\) −1.78171 −0.143574
\(155\) 4.18876 0.336450
\(156\) 0 0
\(157\) 17.1798 1.37110 0.685550 0.728026i \(-0.259562\pi\)
0.685550 + 0.728026i \(0.259562\pi\)
\(158\) 3.59286 0.285832
\(159\) 0 0
\(160\) −3.89887 −0.308232
\(161\) −1.92910 −0.152034
\(162\) 0 0
\(163\) 11.1081 0.870055 0.435028 0.900417i \(-0.356739\pi\)
0.435028 + 0.900417i \(0.356739\pi\)
\(164\) 10.4905 0.819170
\(165\) 0 0
\(166\) −0.646519 −0.0501796
\(167\) 22.3791 1.73174 0.865872 0.500266i \(-0.166764\pi\)
0.865872 + 0.500266i \(0.166764\pi\)
\(168\) 0 0
\(169\) −12.9981 −0.999856
\(170\) −2.50383 −0.192035
\(171\) 0 0
\(172\) 22.6078 1.72383
\(173\) −4.16100 −0.316355 −0.158178 0.987411i \(-0.550562\pi\)
−0.158178 + 0.987411i \(0.550562\pi\)
\(174\) 0 0
\(175\) −4.75339 −0.359323
\(176\) 16.6482 1.25491
\(177\) 0 0
\(178\) −4.02985 −0.302050
\(179\) 5.79009 0.432772 0.216386 0.976308i \(-0.430573\pi\)
0.216386 + 0.976308i \(0.430573\pi\)
\(180\) 0 0
\(181\) 17.4917 1.30015 0.650076 0.759869i \(-0.274737\pi\)
0.650076 + 0.759869i \(0.274737\pi\)
\(182\) 0.0163821 0.00121432
\(183\) 0 0
\(184\) −1.53750 −0.113346
\(185\) 8.82937 0.649148
\(186\) 0 0
\(187\) 34.3630 2.51287
\(188\) −10.0923 −0.736059
\(189\) 0 0
\(190\) 0.238764 0.0173218
\(191\) 21.6074 1.56346 0.781728 0.623620i \(-0.214339\pi\)
0.781728 + 0.623620i \(0.214339\pi\)
\(192\) 0 0
\(193\) 2.30886 0.166195 0.0830977 0.996541i \(-0.473519\pi\)
0.0830977 + 0.996541i \(0.473519\pi\)
\(194\) 0.621008 0.0445858
\(195\) 0 0
\(196\) 9.87112 0.705080
\(197\) −1.91876 −0.136706 −0.0683531 0.997661i \(-0.521774\pi\)
−0.0683531 + 0.997661i \(0.521774\pi\)
\(198\) 0 0
\(199\) 21.2430 1.50588 0.752938 0.658091i \(-0.228636\pi\)
0.752938 + 0.658091i \(0.228636\pi\)
\(200\) −3.78848 −0.267886
\(201\) 0 0
\(202\) −1.11477 −0.0784349
\(203\) −11.3446 −0.796232
\(204\) 0 0
\(205\) −6.72784 −0.469893
\(206\) 1.45780 0.101570
\(207\) 0 0
\(208\) −0.153074 −0.0106138
\(209\) −3.27685 −0.226664
\(210\) 0 0
\(211\) −19.0961 −1.31463 −0.657314 0.753616i \(-0.728308\pi\)
−0.657314 + 0.753616i \(0.728308\pi\)
\(212\) −16.4945 −1.13285
\(213\) 0 0
\(214\) 1.55550 0.106332
\(215\) −14.4990 −0.988823
\(216\) 0 0
\(217\) 4.64058 0.315023
\(218\) 1.32559 0.0897802
\(219\) 0 0
\(220\) −11.1413 −0.751145
\(221\) −0.315954 −0.0212534
\(222\) 0 0
\(223\) 8.41781 0.563698 0.281849 0.959459i \(-0.409052\pi\)
0.281849 + 0.959459i \(0.409052\pi\)
\(224\) −4.31942 −0.288603
\(225\) 0 0
\(226\) −3.50265 −0.232993
\(227\) −1.82854 −0.121364 −0.0606821 0.998157i \(-0.519328\pi\)
−0.0606821 + 0.998157i \(0.519328\pi\)
\(228\) 0 0
\(229\) 2.52038 0.166551 0.0832756 0.996527i \(-0.473462\pi\)
0.0832756 + 0.996527i \(0.473462\pi\)
\(230\) 0.483338 0.0318704
\(231\) 0 0
\(232\) −9.04169 −0.593616
\(233\) −16.3812 −1.07317 −0.536585 0.843847i \(-0.680286\pi\)
−0.536585 + 0.843847i \(0.680286\pi\)
\(234\) 0 0
\(235\) 6.47249 0.422219
\(236\) −24.9839 −1.62631
\(237\) 0 0
\(238\) −2.77390 −0.179805
\(239\) 10.7430 0.694909 0.347455 0.937697i \(-0.387046\pi\)
0.347455 + 0.937697i \(0.387046\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) −3.07327 −0.197557
\(243\) 0 0
\(244\) −19.5441 −1.25118
\(245\) −6.33063 −0.404449
\(246\) 0 0
\(247\) 0.0301293 0.00191708
\(248\) 3.69857 0.234860
\(249\) 0 0
\(250\) 2.90257 0.183575
\(251\) −23.8819 −1.50741 −0.753706 0.657212i \(-0.771736\pi\)
−0.753706 + 0.657212i \(0.771736\pi\)
\(252\) 0 0
\(253\) −6.63342 −0.417040
\(254\) −0.118706 −0.00744829
\(255\) 0 0
\(256\) 10.1859 0.636619
\(257\) −5.25407 −0.327740 −0.163870 0.986482i \(-0.552398\pi\)
−0.163870 + 0.986482i \(0.552398\pi\)
\(258\) 0 0
\(259\) 9.78175 0.607808
\(260\) 0.102440 0.00635304
\(261\) 0 0
\(262\) −3.70936 −0.229165
\(263\) 1.10964 0.0684235 0.0342118 0.999415i \(-0.489108\pi\)
0.0342118 + 0.999415i \(0.489108\pi\)
\(264\) 0 0
\(265\) 10.5784 0.649824
\(266\) 0.264519 0.0162187
\(267\) 0 0
\(268\) −19.4715 −1.18941
\(269\) 2.84634 0.173544 0.0867722 0.996228i \(-0.472345\pi\)
0.0867722 + 0.996228i \(0.472345\pi\)
\(270\) 0 0
\(271\) 13.1029 0.795946 0.397973 0.917397i \(-0.369714\pi\)
0.397973 + 0.917397i \(0.369714\pi\)
\(272\) 25.9193 1.57159
\(273\) 0 0
\(274\) 3.12834 0.188990
\(275\) −16.3451 −0.985646
\(276\) 0 0
\(277\) −23.4634 −1.40978 −0.704888 0.709318i \(-0.749003\pi\)
−0.704888 + 0.709318i \(0.749003\pi\)
\(278\) 0.257365 0.0154357
\(279\) 0 0
\(280\) 1.83476 0.109648
\(281\) −8.91165 −0.531625 −0.265812 0.964025i \(-0.585640\pi\)
−0.265812 + 0.964025i \(0.585640\pi\)
\(282\) 0 0
\(283\) −15.8040 −0.939453 −0.469726 0.882812i \(-0.655647\pi\)
−0.469726 + 0.882812i \(0.655647\pi\)
\(284\) 3.59021 0.213040
\(285\) 0 0
\(286\) 0.0563318 0.00333097
\(287\) −7.45354 −0.439968
\(288\) 0 0
\(289\) 36.4990 2.14700
\(290\) 2.84240 0.166911
\(291\) 0 0
\(292\) −12.4574 −0.729013
\(293\) −32.9425 −1.92452 −0.962261 0.272127i \(-0.912273\pi\)
−0.962261 + 0.272127i \(0.912273\pi\)
\(294\) 0 0
\(295\) 16.0229 0.932888
\(296\) 7.79611 0.453140
\(297\) 0 0
\(298\) 3.57399 0.207036
\(299\) 0.0609917 0.00352724
\(300\) 0 0
\(301\) −16.0629 −0.925851
\(302\) 5.89674 0.339319
\(303\) 0 0
\(304\) −2.47165 −0.141759
\(305\) 12.5342 0.717705
\(306\) 0 0
\(307\) 20.2343 1.15483 0.577416 0.816450i \(-0.304061\pi\)
0.577416 + 0.816450i \(0.304061\pi\)
\(308\) −12.3430 −0.703309
\(309\) 0 0
\(310\) −1.16271 −0.0660372
\(311\) −2.27289 −0.128884 −0.0644420 0.997921i \(-0.520527\pi\)
−0.0644420 + 0.997921i \(0.520527\pi\)
\(312\) 0 0
\(313\) 13.1142 0.741257 0.370629 0.928781i \(-0.379142\pi\)
0.370629 + 0.928781i \(0.379142\pi\)
\(314\) −4.76873 −0.269115
\(315\) 0 0
\(316\) 24.8900 1.40017
\(317\) 7.86525 0.441757 0.220878 0.975301i \(-0.429108\pi\)
0.220878 + 0.975301i \(0.429108\pi\)
\(318\) 0 0
\(319\) −39.0096 −2.18412
\(320\) −7.65810 −0.428101
\(321\) 0 0
\(322\) 0.535473 0.0298407
\(323\) −5.10164 −0.283863
\(324\) 0 0
\(325\) 0.150287 0.00833640
\(326\) −3.08336 −0.170772
\(327\) 0 0
\(328\) −5.94051 −0.328010
\(329\) 7.17065 0.395331
\(330\) 0 0
\(331\) −19.3752 −1.06496 −0.532480 0.846443i \(-0.678740\pi\)
−0.532480 + 0.846443i \(0.678740\pi\)
\(332\) −4.47884 −0.245808
\(333\) 0 0
\(334\) −6.21192 −0.339901
\(335\) 12.4876 0.682273
\(336\) 0 0
\(337\) −26.2459 −1.42970 −0.714852 0.699276i \(-0.753506\pi\)
−0.714852 + 0.699276i \(0.753506\pi\)
\(338\) 3.60798 0.196249
\(339\) 0 0
\(340\) −17.3456 −0.940697
\(341\) 15.9572 0.864130
\(342\) 0 0
\(343\) −16.5773 −0.895092
\(344\) −12.8022 −0.690250
\(345\) 0 0
\(346\) 1.15500 0.0620931
\(347\) 3.65189 0.196044 0.0980219 0.995184i \(-0.468748\pi\)
0.0980219 + 0.995184i \(0.468748\pi\)
\(348\) 0 0
\(349\) −0.818652 −0.0438214 −0.0219107 0.999760i \(-0.506975\pi\)
−0.0219107 + 0.999760i \(0.506975\pi\)
\(350\) 1.31943 0.0705267
\(351\) 0 0
\(352\) −14.8528 −0.791658
\(353\) 28.5367 1.51886 0.759428 0.650591i \(-0.225479\pi\)
0.759428 + 0.650591i \(0.225479\pi\)
\(354\) 0 0
\(355\) −2.30250 −0.122204
\(356\) −27.9173 −1.47961
\(357\) 0 0
\(358\) −1.60720 −0.0849430
\(359\) 15.8623 0.837180 0.418590 0.908175i \(-0.362524\pi\)
0.418590 + 0.908175i \(0.362524\pi\)
\(360\) 0 0
\(361\) −18.5135 −0.974395
\(362\) −4.85531 −0.255189
\(363\) 0 0
\(364\) 0.113489 0.00594845
\(365\) 7.98926 0.418177
\(366\) 0 0
\(367\) 13.9937 0.730463 0.365232 0.930917i \(-0.380990\pi\)
0.365232 + 0.930917i \(0.380990\pi\)
\(368\) −5.00344 −0.260822
\(369\) 0 0
\(370\) −2.45083 −0.127413
\(371\) 11.7194 0.608441
\(372\) 0 0
\(373\) 12.2336 0.633434 0.316717 0.948520i \(-0.397419\pi\)
0.316717 + 0.948520i \(0.397419\pi\)
\(374\) −9.53839 −0.493218
\(375\) 0 0
\(376\) 5.71505 0.294731
\(377\) 0.358678 0.0184728
\(378\) 0 0
\(379\) 0.284829 0.0146307 0.00731535 0.999973i \(-0.497671\pi\)
0.00731535 + 0.999973i \(0.497671\pi\)
\(380\) 1.65407 0.0848520
\(381\) 0 0
\(382\) −5.99772 −0.306870
\(383\) −29.8268 −1.52408 −0.762038 0.647532i \(-0.775801\pi\)
−0.762038 + 0.647532i \(0.775801\pi\)
\(384\) 0 0
\(385\) 7.91593 0.403433
\(386\) −0.640887 −0.0326203
\(387\) 0 0
\(388\) 4.30211 0.218407
\(389\) 8.88544 0.450510 0.225255 0.974300i \(-0.427679\pi\)
0.225255 + 0.974300i \(0.427679\pi\)
\(390\) 0 0
\(391\) −10.3274 −0.522280
\(392\) −5.58978 −0.282327
\(393\) 0 0
\(394\) 0.532605 0.0268322
\(395\) −15.9626 −0.803167
\(396\) 0 0
\(397\) −29.5831 −1.48473 −0.742367 0.669993i \(-0.766297\pi\)
−0.742367 + 0.669993i \(0.766297\pi\)
\(398\) −5.89657 −0.295569
\(399\) 0 0
\(400\) −12.3287 −0.616437
\(401\) −6.30306 −0.314760 −0.157380 0.987538i \(-0.550305\pi\)
−0.157380 + 0.987538i \(0.550305\pi\)
\(402\) 0 0
\(403\) −0.146720 −0.00730864
\(404\) −7.72270 −0.384219
\(405\) 0 0
\(406\) 3.14899 0.156282
\(407\) 33.6357 1.66726
\(408\) 0 0
\(409\) −7.63163 −0.377360 −0.188680 0.982039i \(-0.560421\pi\)
−0.188680 + 0.982039i \(0.560421\pi\)
\(410\) 1.86750 0.0922290
\(411\) 0 0
\(412\) 10.0991 0.497546
\(413\) 17.7512 0.873478
\(414\) 0 0
\(415\) 2.87240 0.141001
\(416\) 0.136566 0.00669568
\(417\) 0 0
\(418\) 0.909578 0.0444889
\(419\) −2.34760 −0.114688 −0.0573439 0.998354i \(-0.518263\pi\)
−0.0573439 + 0.998354i \(0.518263\pi\)
\(420\) 0 0
\(421\) 8.61693 0.419963 0.209982 0.977705i \(-0.432660\pi\)
0.209982 + 0.977705i \(0.432660\pi\)
\(422\) 5.30064 0.258031
\(423\) 0 0
\(424\) 9.34043 0.453611
\(425\) −25.4473 −1.23437
\(426\) 0 0
\(427\) 13.8862 0.671999
\(428\) 10.7759 0.520875
\(429\) 0 0
\(430\) 4.02459 0.194083
\(431\) 2.73295 0.131641 0.0658207 0.997831i \(-0.479033\pi\)
0.0658207 + 0.997831i \(0.479033\pi\)
\(432\) 0 0
\(433\) 14.8768 0.714934 0.357467 0.933926i \(-0.383640\pi\)
0.357467 + 0.933926i \(0.383640\pi\)
\(434\) −1.28812 −0.0618317
\(435\) 0 0
\(436\) 9.18318 0.439795
\(437\) 0.984819 0.0471103
\(438\) 0 0
\(439\) −19.8115 −0.945552 −0.472776 0.881183i \(-0.656748\pi\)
−0.472776 + 0.881183i \(0.656748\pi\)
\(440\) 6.30904 0.300772
\(441\) 0 0
\(442\) 0.0877017 0.00417154
\(443\) 39.5818 1.88059 0.940293 0.340366i \(-0.110551\pi\)
0.940293 + 0.340366i \(0.110551\pi\)
\(444\) 0 0
\(445\) 17.9041 0.848738
\(446\) −2.33659 −0.110641
\(447\) 0 0
\(448\) −8.48414 −0.400838
\(449\) 22.9205 1.08169 0.540843 0.841124i \(-0.318105\pi\)
0.540843 + 0.841124i \(0.318105\pi\)
\(450\) 0 0
\(451\) −25.6299 −1.20686
\(452\) −24.2650 −1.14133
\(453\) 0 0
\(454\) 0.507560 0.0238210
\(455\) −0.0727838 −0.00341216
\(456\) 0 0
\(457\) −9.66494 −0.452107 −0.226053 0.974115i \(-0.572582\pi\)
−0.226053 + 0.974115i \(0.572582\pi\)
\(458\) −0.699599 −0.0326901
\(459\) 0 0
\(460\) 3.34839 0.156119
\(461\) −6.28046 −0.292510 −0.146255 0.989247i \(-0.546722\pi\)
−0.146255 + 0.989247i \(0.546722\pi\)
\(462\) 0 0
\(463\) −7.64337 −0.355218 −0.177609 0.984101i \(-0.556836\pi\)
−0.177609 + 0.984101i \(0.556836\pi\)
\(464\) −29.4241 −1.36598
\(465\) 0 0
\(466\) 4.54705 0.210638
\(467\) −25.8943 −1.19824 −0.599122 0.800658i \(-0.704484\pi\)
−0.599122 + 0.800658i \(0.704484\pi\)
\(468\) 0 0
\(469\) 13.8346 0.638823
\(470\) −1.79662 −0.0828717
\(471\) 0 0
\(472\) 14.1478 0.651205
\(473\) −55.2342 −2.53967
\(474\) 0 0
\(475\) 2.42664 0.111342
\(476\) −19.2166 −0.880790
\(477\) 0 0
\(478\) −2.98202 −0.136394
\(479\) −12.6131 −0.576309 −0.288155 0.957584i \(-0.593042\pi\)
−0.288155 + 0.957584i \(0.593042\pi\)
\(480\) 0 0
\(481\) −0.309267 −0.0141013
\(482\) 0.277577 0.0126433
\(483\) 0 0
\(484\) −21.2905 −0.967749
\(485\) −2.75907 −0.125283
\(486\) 0 0
\(487\) 31.7807 1.44012 0.720061 0.693910i \(-0.244114\pi\)
0.720061 + 0.693910i \(0.244114\pi\)
\(488\) 11.0674 0.500996
\(489\) 0 0
\(490\) 1.75724 0.0793839
\(491\) 10.2151 0.461001 0.230500 0.973072i \(-0.425964\pi\)
0.230500 + 0.973072i \(0.425964\pi\)
\(492\) 0 0
\(493\) −60.7331 −2.73528
\(494\) −0.00836321 −0.000376278 0
\(495\) 0 0
\(496\) 12.0361 0.540439
\(497\) −2.55086 −0.114422
\(498\) 0 0
\(499\) −9.35390 −0.418738 −0.209369 0.977837i \(-0.567141\pi\)
−0.209369 + 0.977837i \(0.567141\pi\)
\(500\) 20.1079 0.899253
\(501\) 0 0
\(502\) 6.62907 0.295870
\(503\) 13.7659 0.613792 0.306896 0.951743i \(-0.400710\pi\)
0.306896 + 0.951743i \(0.400710\pi\)
\(504\) 0 0
\(505\) 4.95278 0.220396
\(506\) 1.84129 0.0818552
\(507\) 0 0
\(508\) −0.822353 −0.0364860
\(509\) −22.6367 −1.00335 −0.501677 0.865055i \(-0.667283\pi\)
−0.501677 + 0.865055i \(0.667283\pi\)
\(510\) 0 0
\(511\) 8.85102 0.391546
\(512\) −18.9206 −0.836182
\(513\) 0 0
\(514\) 1.45841 0.0643277
\(515\) −6.47682 −0.285403
\(516\) 0 0
\(517\) 24.6571 1.08442
\(518\) −2.71519 −0.119299
\(519\) 0 0
\(520\) −0.0580091 −0.00254387
\(521\) 10.8738 0.476389 0.238194 0.971218i \(-0.423445\pi\)
0.238194 + 0.971218i \(0.423445\pi\)
\(522\) 0 0
\(523\) −3.43232 −0.150085 −0.0750424 0.997180i \(-0.523909\pi\)
−0.0750424 + 0.997180i \(0.523909\pi\)
\(524\) −25.6970 −1.12258
\(525\) 0 0
\(526\) −0.308011 −0.0134299
\(527\) 24.8434 1.08219
\(528\) 0 0
\(529\) −21.0064 −0.913322
\(530\) −2.93631 −0.127545
\(531\) 0 0
\(532\) 1.83249 0.0794483
\(533\) 0.235656 0.0102074
\(534\) 0 0
\(535\) −6.91091 −0.298785
\(536\) 11.0263 0.476263
\(537\) 0 0
\(538\) −0.790079 −0.0340627
\(539\) −24.1167 −1.03878
\(540\) 0 0
\(541\) −1.00758 −0.0433193 −0.0216597 0.999765i \(-0.506895\pi\)
−0.0216597 + 0.999765i \(0.506895\pi\)
\(542\) −3.63707 −0.156226
\(543\) 0 0
\(544\) −23.1240 −0.991433
\(545\) −5.88943 −0.252275
\(546\) 0 0
\(547\) 10.6391 0.454894 0.227447 0.973791i \(-0.426962\pi\)
0.227447 + 0.973791i \(0.426962\pi\)
\(548\) 21.6720 0.925780
\(549\) 0 0
\(550\) 4.53702 0.193459
\(551\) 5.79149 0.246726
\(552\) 0 0
\(553\) −17.6844 −0.752019
\(554\) 6.51289 0.276706
\(555\) 0 0
\(556\) 1.78293 0.0756130
\(557\) −18.9092 −0.801207 −0.400603 0.916252i \(-0.631199\pi\)
−0.400603 + 0.916252i \(0.631199\pi\)
\(558\) 0 0
\(559\) 0.507856 0.0214800
\(560\) 5.97081 0.252313
\(561\) 0 0
\(562\) 2.47367 0.104346
\(563\) −28.1768 −1.18751 −0.593756 0.804645i \(-0.702356\pi\)
−0.593756 + 0.804645i \(0.702356\pi\)
\(564\) 0 0
\(565\) 15.5618 0.654692
\(566\) 4.38684 0.184393
\(567\) 0 0
\(568\) −2.03305 −0.0853049
\(569\) 0.499818 0.0209534 0.0104767 0.999945i \(-0.496665\pi\)
0.0104767 + 0.999945i \(0.496665\pi\)
\(570\) 0 0
\(571\) −28.2303 −1.18140 −0.590702 0.806890i \(-0.701149\pi\)
−0.590702 + 0.806890i \(0.701149\pi\)
\(572\) 0.390246 0.0163170
\(573\) 0 0
\(574\) 2.06893 0.0863555
\(575\) 4.91233 0.204858
\(576\) 0 0
\(577\) −36.3728 −1.51422 −0.757111 0.653287i \(-0.773390\pi\)
−0.757111 + 0.653287i \(0.773390\pi\)
\(578\) −10.1313 −0.421406
\(579\) 0 0
\(580\) 19.6911 0.817627
\(581\) 3.18224 0.132021
\(582\) 0 0
\(583\) 40.2985 1.66899
\(584\) 7.05432 0.291910
\(585\) 0 0
\(586\) 9.14409 0.377739
\(587\) −16.1223 −0.665437 −0.332718 0.943026i \(-0.607966\pi\)
−0.332718 + 0.943026i \(0.607966\pi\)
\(588\) 0 0
\(589\) −2.36905 −0.0976152
\(590\) −4.44758 −0.183104
\(591\) 0 0
\(592\) 25.3706 1.04273
\(593\) −24.8487 −1.02041 −0.510207 0.860052i \(-0.670431\pi\)
−0.510207 + 0.860052i \(0.670431\pi\)
\(594\) 0 0
\(595\) 12.3241 0.505240
\(596\) 24.7593 1.01418
\(597\) 0 0
\(598\) −0.0169299 −0.000692315 0
\(599\) 1.50155 0.0613517 0.0306759 0.999529i \(-0.490234\pi\)
0.0306759 + 0.999529i \(0.490234\pi\)
\(600\) 0 0
\(601\) −24.0828 −0.982356 −0.491178 0.871059i \(-0.663434\pi\)
−0.491178 + 0.871059i \(0.663434\pi\)
\(602\) 4.45870 0.181723
\(603\) 0 0
\(604\) 40.8504 1.66218
\(605\) 13.6542 0.555121
\(606\) 0 0
\(607\) 16.0760 0.652506 0.326253 0.945283i \(-0.394214\pi\)
0.326253 + 0.945283i \(0.394214\pi\)
\(608\) 2.20510 0.0894284
\(609\) 0 0
\(610\) −3.47920 −0.140869
\(611\) −0.226712 −0.00917179
\(612\) 0 0
\(613\) 43.5233 1.75789 0.878945 0.476923i \(-0.158248\pi\)
0.878945 + 0.476923i \(0.158248\pi\)
\(614\) −5.61658 −0.226667
\(615\) 0 0
\(616\) 6.98957 0.281618
\(617\) 3.82544 0.154006 0.0770032 0.997031i \(-0.475465\pi\)
0.0770032 + 0.997031i \(0.475465\pi\)
\(618\) 0 0
\(619\) 8.69881 0.349635 0.174817 0.984601i \(-0.444066\pi\)
0.174817 + 0.984601i \(0.444066\pi\)
\(620\) −8.05479 −0.323488
\(621\) 0 0
\(622\) 0.630903 0.0252969
\(623\) 19.8354 0.794687
\(624\) 0 0
\(625\) 4.49981 0.179992
\(626\) −3.64020 −0.145492
\(627\) 0 0
\(628\) −33.0360 −1.31828
\(629\) 52.3666 2.08799
\(630\) 0 0
\(631\) −28.0264 −1.11571 −0.557856 0.829938i \(-0.688376\pi\)
−0.557856 + 0.829938i \(0.688376\pi\)
\(632\) −14.0946 −0.560653
\(633\) 0 0
\(634\) −2.18321 −0.0867065
\(635\) 0.527398 0.0209291
\(636\) 0 0
\(637\) 0.221743 0.00878578
\(638\) 10.8282 0.428692
\(639\) 0 0
\(640\) 9.92345 0.392259
\(641\) −22.7621 −0.899048 −0.449524 0.893268i \(-0.648406\pi\)
−0.449524 + 0.893268i \(0.648406\pi\)
\(642\) 0 0
\(643\) 35.9559 1.41796 0.708981 0.705228i \(-0.249155\pi\)
0.708981 + 0.705228i \(0.249155\pi\)
\(644\) 3.70956 0.146177
\(645\) 0 0
\(646\) 1.41610 0.0557157
\(647\) 1.94619 0.0765127 0.0382564 0.999268i \(-0.487820\pi\)
0.0382564 + 0.999268i \(0.487820\pi\)
\(648\) 0 0
\(649\) 61.0395 2.39601
\(650\) −0.0417161 −0.00163624
\(651\) 0 0
\(652\) −21.3604 −0.836537
\(653\) −24.3392 −0.952467 −0.476233 0.879319i \(-0.657998\pi\)
−0.476233 + 0.879319i \(0.657998\pi\)
\(654\) 0 0
\(655\) 16.4802 0.643936
\(656\) −19.3320 −0.754789
\(657\) 0 0
\(658\) −1.99041 −0.0775942
\(659\) 6.16907 0.240313 0.120156 0.992755i \(-0.461660\pi\)
0.120156 + 0.992755i \(0.461660\pi\)
\(660\) 0 0
\(661\) −48.1704 −1.87361 −0.936807 0.349848i \(-0.886233\pi\)
−0.936807 + 0.349848i \(0.886233\pi\)
\(662\) 5.37812 0.209027
\(663\) 0 0
\(664\) 2.53626 0.0984260
\(665\) −1.17522 −0.0455732
\(666\) 0 0
\(667\) 11.7239 0.453951
\(668\) −43.0338 −1.66503
\(669\) 0 0
\(670\) −3.46628 −0.133914
\(671\) 47.7492 1.84334
\(672\) 0 0
\(673\) −19.2071 −0.740381 −0.370190 0.928956i \(-0.620708\pi\)
−0.370190 + 0.928956i \(0.620708\pi\)
\(674\) 7.28525 0.280617
\(675\) 0 0
\(676\) 24.9948 0.961337
\(677\) 14.9094 0.573014 0.286507 0.958078i \(-0.407506\pi\)
0.286507 + 0.958078i \(0.407506\pi\)
\(678\) 0 0
\(679\) −3.05667 −0.117304
\(680\) 9.82240 0.376672
\(681\) 0 0
\(682\) −4.42935 −0.169609
\(683\) 22.8682 0.875029 0.437515 0.899211i \(-0.355859\pi\)
0.437515 + 0.899211i \(0.355859\pi\)
\(684\) 0 0
\(685\) −13.8988 −0.531047
\(686\) 4.60149 0.175686
\(687\) 0 0
\(688\) −41.6619 −1.58835
\(689\) −0.370529 −0.0141160
\(690\) 0 0
\(691\) −35.5210 −1.35128 −0.675641 0.737231i \(-0.736133\pi\)
−0.675641 + 0.737231i \(0.736133\pi\)
\(692\) 8.00140 0.304168
\(693\) 0 0
\(694\) −1.01368 −0.0384788
\(695\) −1.14344 −0.0433732
\(696\) 0 0
\(697\) −39.9025 −1.51142
\(698\) 0.227239 0.00860112
\(699\) 0 0
\(700\) 9.14054 0.345480
\(701\) −9.22303 −0.348349 −0.174175 0.984715i \(-0.555726\pi\)
−0.174175 + 0.984715i \(0.555726\pi\)
\(702\) 0 0
\(703\) −4.99366 −0.188339
\(704\) −29.1737 −1.09952
\(705\) 0 0
\(706\) −7.92114 −0.298116
\(707\) 5.48701 0.206360
\(708\) 0 0
\(709\) 28.7763 1.08072 0.540358 0.841435i \(-0.318289\pi\)
0.540358 + 0.841435i \(0.318289\pi\)
\(710\) 0.639122 0.0239858
\(711\) 0 0
\(712\) 15.8089 0.592464
\(713\) −4.79575 −0.179602
\(714\) 0 0
\(715\) −0.250276 −0.00935978
\(716\) −11.1341 −0.416100
\(717\) 0 0
\(718\) −4.40301 −0.164319
\(719\) 24.1878 0.902055 0.451027 0.892510i \(-0.351058\pi\)
0.451027 + 0.892510i \(0.351058\pi\)
\(720\) 0 0
\(721\) −7.17544 −0.267227
\(722\) 5.13893 0.191251
\(723\) 0 0
\(724\) −33.6358 −1.25006
\(725\) 28.8883 1.07288
\(726\) 0 0
\(727\) 9.73994 0.361235 0.180617 0.983553i \(-0.442190\pi\)
0.180617 + 0.983553i \(0.442190\pi\)
\(728\) −0.0642663 −0.00238187
\(729\) 0 0
\(730\) −2.21764 −0.0820784
\(731\) −85.9928 −3.18056
\(732\) 0 0
\(733\) −4.38060 −0.161801 −0.0809006 0.996722i \(-0.525780\pi\)
−0.0809006 + 0.996722i \(0.525780\pi\)
\(734\) −3.88432 −0.143373
\(735\) 0 0
\(736\) 4.46384 0.164539
\(737\) 47.5719 1.75234
\(738\) 0 0
\(739\) −6.21520 −0.228630 −0.114315 0.993445i \(-0.536467\pi\)
−0.114315 + 0.993445i \(0.536467\pi\)
\(740\) −16.9784 −0.624140
\(741\) 0 0
\(742\) −3.25304 −0.119423
\(743\) −26.7097 −0.979885 −0.489942 0.871755i \(-0.662982\pi\)
−0.489942 + 0.871755i \(0.662982\pi\)
\(744\) 0 0
\(745\) −15.8788 −0.581754
\(746\) −3.39578 −0.124328
\(747\) 0 0
\(748\) −66.0784 −2.41607
\(749\) −7.65635 −0.279757
\(750\) 0 0
\(751\) 46.9924 1.71478 0.857388 0.514670i \(-0.172085\pi\)
0.857388 + 0.514670i \(0.172085\pi\)
\(752\) 18.5983 0.678210
\(753\) 0 0
\(754\) −0.0995608 −0.00362579
\(755\) −26.1985 −0.953461
\(756\) 0 0
\(757\) −35.2425 −1.28091 −0.640456 0.767995i \(-0.721255\pi\)
−0.640456 + 0.767995i \(0.721255\pi\)
\(758\) −0.0790621 −0.00287166
\(759\) 0 0
\(760\) −0.936661 −0.0339763
\(761\) 47.1109 1.70777 0.853885 0.520461i \(-0.174240\pi\)
0.853885 + 0.520461i \(0.174240\pi\)
\(762\) 0 0
\(763\) −6.52469 −0.236210
\(764\) −41.5499 −1.50322
\(765\) 0 0
\(766\) 8.27923 0.299141
\(767\) −0.561234 −0.0202650
\(768\) 0 0
\(769\) −53.8827 −1.94306 −0.971530 0.236915i \(-0.923864\pi\)
−0.971530 + 0.236915i \(0.923864\pi\)
\(770\) −2.19728 −0.0791845
\(771\) 0 0
\(772\) −4.43983 −0.159793
\(773\) 7.33241 0.263728 0.131864 0.991268i \(-0.457904\pi\)
0.131864 + 0.991268i \(0.457904\pi\)
\(774\) 0 0
\(775\) −11.8170 −0.424478
\(776\) −2.43618 −0.0874539
\(777\) 0 0
\(778\) −2.46639 −0.0884245
\(779\) 3.80509 0.136332
\(780\) 0 0
\(781\) −8.77143 −0.313866
\(782\) 2.86666 0.102511
\(783\) 0 0
\(784\) −18.1906 −0.649666
\(785\) 21.1869 0.756193
\(786\) 0 0
\(787\) −0.274880 −0.00979841 −0.00489921 0.999988i \(-0.501559\pi\)
−0.00489921 + 0.999988i \(0.501559\pi\)
\(788\) 3.68969 0.131440
\(789\) 0 0
\(790\) 4.43086 0.157643
\(791\) 17.2404 0.612998
\(792\) 0 0
\(793\) −0.439035 −0.0155906
\(794\) 8.21160 0.291419
\(795\) 0 0
\(796\) −40.8493 −1.44786
\(797\) 7.42858 0.263134 0.131567 0.991307i \(-0.457999\pi\)
0.131567 + 0.991307i \(0.457999\pi\)
\(798\) 0 0
\(799\) 38.3881 1.35807
\(800\) 10.9991 0.388878
\(801\) 0 0
\(802\) 1.74958 0.0617800
\(803\) 30.4353 1.07404
\(804\) 0 0
\(805\) −2.37904 −0.0838502
\(806\) 0.0407261 0.00143452
\(807\) 0 0
\(808\) 4.37318 0.153848
\(809\) −42.6491 −1.49946 −0.749730 0.661743i \(-0.769817\pi\)
−0.749730 + 0.661743i \(0.769817\pi\)
\(810\) 0 0
\(811\) 25.6139 0.899425 0.449713 0.893173i \(-0.351526\pi\)
0.449713 + 0.893173i \(0.351526\pi\)
\(812\) 21.8150 0.765558
\(813\) 0 0
\(814\) −9.33649 −0.327244
\(815\) 13.6990 0.479855
\(816\) 0 0
\(817\) 8.20024 0.286890
\(818\) 2.11837 0.0740669
\(819\) 0 0
\(820\) 12.9373 0.451790
\(821\) −21.8573 −0.762826 −0.381413 0.924405i \(-0.624562\pi\)
−0.381413 + 0.924405i \(0.624562\pi\)
\(822\) 0 0
\(823\) 1.58270 0.0551693 0.0275847 0.999619i \(-0.491218\pi\)
0.0275847 + 0.999619i \(0.491218\pi\)
\(824\) −5.71887 −0.199226
\(825\) 0 0
\(826\) −4.92732 −0.171443
\(827\) −36.2198 −1.25949 −0.629743 0.776803i \(-0.716840\pi\)
−0.629743 + 0.776803i \(0.716840\pi\)
\(828\) 0 0
\(829\) 30.5957 1.06263 0.531316 0.847174i \(-0.321698\pi\)
0.531316 + 0.847174i \(0.321698\pi\)
\(830\) −0.797314 −0.0276752
\(831\) 0 0
\(832\) 0.268241 0.00929957
\(833\) −37.5466 −1.30091
\(834\) 0 0
\(835\) 27.5988 0.955096
\(836\) 6.30122 0.217932
\(837\) 0 0
\(838\) 0.651640 0.0225105
\(839\) 9.85906 0.340373 0.170186 0.985412i \(-0.445563\pi\)
0.170186 + 0.985412i \(0.445563\pi\)
\(840\) 0 0
\(841\) 39.9455 1.37743
\(842\) −2.39186 −0.0824290
\(843\) 0 0
\(844\) 36.7208 1.26398
\(845\) −16.0298 −0.551443
\(846\) 0 0
\(847\) 15.1270 0.519769
\(848\) 30.3963 1.04381
\(849\) 0 0
\(850\) 7.06358 0.242279
\(851\) −10.1088 −0.346526
\(852\) 0 0
\(853\) 20.6701 0.707730 0.353865 0.935296i \(-0.384867\pi\)
0.353865 + 0.935296i \(0.384867\pi\)
\(854\) −3.85448 −0.131898
\(855\) 0 0
\(856\) −6.10216 −0.208567
\(857\) −36.0858 −1.23267 −0.616334 0.787485i \(-0.711383\pi\)
−0.616334 + 0.787485i \(0.711383\pi\)
\(858\) 0 0
\(859\) 40.7566 1.39060 0.695299 0.718720i \(-0.255272\pi\)
0.695299 + 0.718720i \(0.255272\pi\)
\(860\) 27.8808 0.950729
\(861\) 0 0
\(862\) −0.758604 −0.0258381
\(863\) −21.5125 −0.732295 −0.366148 0.930557i \(-0.619323\pi\)
−0.366148 + 0.930557i \(0.619323\pi\)
\(864\) 0 0
\(865\) −5.13152 −0.174477
\(866\) −4.12947 −0.140325
\(867\) 0 0
\(868\) −8.92362 −0.302887
\(869\) −60.8100 −2.06284
\(870\) 0 0
\(871\) −0.437405 −0.0148209
\(872\) −5.20022 −0.176102
\(873\) 0 0
\(874\) −0.273363 −0.00924665
\(875\) −14.2868 −0.482981
\(876\) 0 0
\(877\) −47.8912 −1.61717 −0.808586 0.588378i \(-0.799767\pi\)
−0.808586 + 0.588378i \(0.799767\pi\)
\(878\) 5.49922 0.185590
\(879\) 0 0
\(880\) 20.5313 0.692111
\(881\) 18.2689 0.615496 0.307748 0.951468i \(-0.400425\pi\)
0.307748 + 0.951468i \(0.400425\pi\)
\(882\) 0 0
\(883\) −37.0354 −1.24634 −0.623170 0.782087i \(-0.714155\pi\)
−0.623170 + 0.782087i \(0.714155\pi\)
\(884\) 0.607565 0.0204346
\(885\) 0 0
\(886\) −10.9870 −0.369115
\(887\) 13.7619 0.462080 0.231040 0.972944i \(-0.425787\pi\)
0.231040 + 0.972944i \(0.425787\pi\)
\(888\) 0 0
\(889\) 0.584285 0.0195963
\(890\) −4.96978 −0.166587
\(891\) 0 0
\(892\) −16.1870 −0.541982
\(893\) −3.66067 −0.122500
\(894\) 0 0
\(895\) 7.14059 0.238684
\(896\) 10.9938 0.367278
\(897\) 0 0
\(898\) −6.36221 −0.212310
\(899\) −28.2027 −0.940613
\(900\) 0 0
\(901\) 62.7398 2.09017
\(902\) 7.11426 0.236879
\(903\) 0 0
\(904\) 13.7407 0.457009
\(905\) 21.5716 0.717063
\(906\) 0 0
\(907\) −27.1842 −0.902637 −0.451319 0.892363i \(-0.649046\pi\)
−0.451319 + 0.892363i \(0.649046\pi\)
\(908\) 3.51619 0.116689
\(909\) 0 0
\(910\) 0.0202031 0.000669727 0
\(911\) −29.4894 −0.977027 −0.488513 0.872556i \(-0.662461\pi\)
−0.488513 + 0.872556i \(0.662461\pi\)
\(912\) 0 0
\(913\) 10.9425 0.362143
\(914\) 2.68277 0.0887380
\(915\) 0 0
\(916\) −4.84656 −0.160135
\(917\) 18.2579 0.602928
\(918\) 0 0
\(919\) −44.2354 −1.45919 −0.729596 0.683878i \(-0.760292\pi\)
−0.729596 + 0.683878i \(0.760292\pi\)
\(920\) −1.89611 −0.0625129
\(921\) 0 0
\(922\) 1.74331 0.0574130
\(923\) 0.0806498 0.00265462
\(924\) 0 0
\(925\) −24.9086 −0.818991
\(926\) 2.12163 0.0697210
\(927\) 0 0
\(928\) 26.2508 0.861726
\(929\) −17.7425 −0.582112 −0.291056 0.956706i \(-0.594007\pi\)
−0.291056 + 0.956706i \(0.594007\pi\)
\(930\) 0 0
\(931\) 3.58043 0.117344
\(932\) 31.5003 1.03183
\(933\) 0 0
\(934\) 7.18766 0.235187
\(935\) 42.3779 1.38591
\(936\) 0 0
\(937\) 46.1680 1.50824 0.754121 0.656735i \(-0.228063\pi\)
0.754121 + 0.656735i \(0.228063\pi\)
\(938\) −3.84017 −0.125386
\(939\) 0 0
\(940\) −12.4463 −0.405953
\(941\) 22.0383 0.718428 0.359214 0.933255i \(-0.383045\pi\)
0.359214 + 0.933255i \(0.383045\pi\)
\(942\) 0 0
\(943\) 7.70276 0.250836
\(944\) 46.0407 1.49850
\(945\) 0 0
\(946\) 15.3317 0.498478
\(947\) 34.5940 1.12416 0.562078 0.827085i \(-0.310002\pi\)
0.562078 + 0.827085i \(0.310002\pi\)
\(948\) 0 0
\(949\) −0.279840 −0.00908399
\(950\) −0.673581 −0.0218539
\(951\) 0 0
\(952\) 10.8819 0.352684
\(953\) 20.9799 0.679607 0.339803 0.940496i \(-0.389639\pi\)
0.339803 + 0.940496i \(0.389639\pi\)
\(954\) 0 0
\(955\) 26.6471 0.862281
\(956\) −20.6583 −0.668138
\(957\) 0 0
\(958\) 3.50112 0.113116
\(959\) −15.3980 −0.497228
\(960\) 0 0
\(961\) −19.4635 −0.627854
\(962\) 0.0858453 0.00276777
\(963\) 0 0
\(964\) 1.92295 0.0619341
\(965\) 2.84738 0.0916605
\(966\) 0 0
\(967\) −0.0716187 −0.00230310 −0.00115155 0.999999i \(-0.500367\pi\)
−0.00115155 + 0.999999i \(0.500367\pi\)
\(968\) 12.0563 0.387504
\(969\) 0 0
\(970\) 0.765853 0.0245901
\(971\) 8.18205 0.262575 0.131287 0.991344i \(-0.458089\pi\)
0.131287 + 0.991344i \(0.458089\pi\)
\(972\) 0 0
\(973\) −1.26678 −0.0406111
\(974\) −8.82161 −0.282663
\(975\) 0 0
\(976\) 36.0162 1.15285
\(977\) −22.6886 −0.725872 −0.362936 0.931814i \(-0.618226\pi\)
−0.362936 + 0.931814i \(0.618226\pi\)
\(978\) 0 0
\(979\) 68.2062 2.17988
\(980\) 12.1735 0.388868
\(981\) 0 0
\(982\) −2.83548 −0.0904837
\(983\) 18.8463 0.601103 0.300551 0.953766i \(-0.402829\pi\)
0.300551 + 0.953766i \(0.402829\pi\)
\(984\) 0 0
\(985\) −2.36630 −0.0753966
\(986\) 16.8581 0.536872
\(987\) 0 0
\(988\) −0.0579372 −0.00184323
\(989\) 16.6000 0.527850
\(990\) 0 0
\(991\) −16.8310 −0.534655 −0.267328 0.963606i \(-0.586141\pi\)
−0.267328 + 0.963606i \(0.586141\pi\)
\(992\) −10.7381 −0.340935
\(993\) 0 0
\(994\) 0.708060 0.0224583
\(995\) 26.1978 0.830525
\(996\) 0 0
\(997\) 60.1343 1.90447 0.952236 0.305363i \(-0.0987779\pi\)
0.952236 + 0.305363i \(0.0987779\pi\)
\(998\) 2.59643 0.0821885
\(999\) 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 2169.2.a.e.1.3 7
3.2 odd 2 241.2.a.a.1.5 7
12.11 even 2 3856.2.a.j.1.4 7
15.14 odd 2 6025.2.a.f.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.5 7 3.2 odd 2
2169.2.a.e.1.3 7 1.1 even 1 trivial
3856.2.a.j.1.4 7 12.11 even 2
6025.2.a.f.1.3 7 15.14 odd 2