Properties

Label 525.2.a.j.1.1
Level $525$
Weight $2$
Character 525.1
Self dual yes
Analytic conductor $4.192$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} +1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} -2.70928 q^{6} +1.00000 q^{7} -9.04945 q^{8} +1.00000 q^{9} +2.00000 q^{11} +5.34017 q^{12} +0.921622 q^{13} -2.70928 q^{14} +13.8371 q^{16} +1.07838 q^{17} -2.70928 q^{18} +3.07838 q^{19} +1.00000 q^{21} -5.41855 q^{22} +2.34017 q^{23} -9.04945 q^{24} -2.49693 q^{26} +1.00000 q^{27} +5.34017 q^{28} -6.68035 q^{29} -7.75872 q^{31} -19.3896 q^{32} +2.00000 q^{33} -2.92162 q^{34} +5.34017 q^{36} +10.8371 q^{37} -8.34017 q^{38} +0.921622 q^{39} +6.49693 q^{41} -2.70928 q^{42} -6.52359 q^{43} +10.6803 q^{44} -6.34017 q^{46} +4.68035 q^{47} +13.8371 q^{48} +1.00000 q^{49} +1.07838 q^{51} +4.92162 q^{52} +3.75872 q^{53} -2.70928 q^{54} -9.04945 q^{56} +3.07838 q^{57} +18.0989 q^{58} +10.5236 q^{59} -4.15676 q^{61} +21.0205 q^{62} +1.00000 q^{63} +24.8576 q^{64} -5.41855 q^{66} +4.68035 q^{67} +5.75872 q^{68} +2.34017 q^{69} +2.00000 q^{71} -9.04945 q^{72} +7.07838 q^{73} -29.3607 q^{74} +16.4391 q^{76} +2.00000 q^{77} -2.49693 q^{78} +6.15676 q^{79} +1.00000 q^{81} -17.6020 q^{82} +6.83710 q^{83} +5.34017 q^{84} +17.6742 q^{86} -6.68035 q^{87} -18.0989 q^{88} +8.34017 q^{89} +0.921622 q^{91} +12.4969 q^{92} -7.75872 q^{93} -12.6803 q^{94} -19.3896 q^{96} -8.43907 q^{97} -2.70928 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} - q^{14} + 13 q^{16} - q^{18} + 6 q^{19} + 3 q^{21} - 2 q^{22} - 4 q^{23} - 9 q^{24} + 10 q^{26} + 3 q^{27} + 5 q^{28} + 2 q^{29} + 2 q^{31} - 29 q^{32} + 6 q^{33} - 12 q^{34} + 5 q^{36} + 4 q^{37} - 14 q^{38} + 6 q^{39} + 2 q^{41} - q^{42} - 4 q^{43} + 10 q^{44} - 8 q^{46} - 8 q^{47} + 13 q^{48} + 3 q^{49} + 18 q^{52} - 14 q^{53} - q^{54} - 9 q^{56} + 6 q^{57} + 18 q^{58} + 16 q^{59} - 6 q^{61} + 30 q^{62} + 3 q^{63} + 13 q^{64} - 2 q^{66} - 8 q^{67} - 8 q^{68} - 4 q^{69} + 6 q^{71} - 9 q^{72} + 18 q^{73} - 44 q^{74} + 2 q^{76} + 6 q^{77} + 10 q^{78} + 12 q^{79} + 3 q^{81} - 34 q^{82} - 8 q^{83} + 5 q^{84} - 4 q^{86} + 2 q^{87} - 18 q^{88} + 14 q^{89} + 6 q^{91} + 20 q^{92} + 2 q^{93} - 16 q^{94} - 29 q^{96} + 22 q^{97} - q^{98} + 6 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70928 −1.91575 −0.957873 0.287190i \(-0.907279\pi\)
−0.957873 + 0.287190i \(0.907279\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.34017 2.67009
\(5\) 0 0
\(6\) −2.70928 −1.10606
\(7\) 1.00000 0.377964
\(8\) −9.04945 −3.19946
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 5.34017 1.54158
\(13\) 0.921622 0.255612 0.127806 0.991799i \(-0.459207\pi\)
0.127806 + 0.991799i \(0.459207\pi\)
\(14\) −2.70928 −0.724084
\(15\) 0 0
\(16\) 13.8371 3.45928
\(17\) 1.07838 0.261545 0.130773 0.991412i \(-0.458254\pi\)
0.130773 + 0.991412i \(0.458254\pi\)
\(18\) −2.70928 −0.638582
\(19\) 3.07838 0.706228 0.353114 0.935580i \(-0.385123\pi\)
0.353114 + 0.935580i \(0.385123\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −5.41855 −1.15524
\(23\) 2.34017 0.487960 0.243980 0.969780i \(-0.421547\pi\)
0.243980 + 0.969780i \(0.421547\pi\)
\(24\) −9.04945 −1.84721
\(25\) 0 0
\(26\) −2.49693 −0.489688
\(27\) 1.00000 0.192450
\(28\) 5.34017 1.00920
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) −7.75872 −1.39351 −0.696754 0.717310i \(-0.745373\pi\)
−0.696754 + 0.717310i \(0.745373\pi\)
\(32\) −19.3896 −3.42763
\(33\) 2.00000 0.348155
\(34\) −2.92162 −0.501054
\(35\) 0 0
\(36\) 5.34017 0.890029
\(37\) 10.8371 1.78161 0.890804 0.454387i \(-0.150142\pi\)
0.890804 + 0.454387i \(0.150142\pi\)
\(38\) −8.34017 −1.35295
\(39\) 0.921622 0.147578
\(40\) 0 0
\(41\) 6.49693 1.01465 0.507325 0.861755i \(-0.330634\pi\)
0.507325 + 0.861755i \(0.330634\pi\)
\(42\) −2.70928 −0.418050
\(43\) −6.52359 −0.994838 −0.497419 0.867510i \(-0.665719\pi\)
−0.497419 + 0.867510i \(0.665719\pi\)
\(44\) 10.6803 1.61012
\(45\) 0 0
\(46\) −6.34017 −0.934808
\(47\) 4.68035 0.682699 0.341349 0.939937i \(-0.389116\pi\)
0.341349 + 0.939937i \(0.389116\pi\)
\(48\) 13.8371 1.99721
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.07838 0.151003
\(52\) 4.92162 0.682506
\(53\) 3.75872 0.516300 0.258150 0.966105i \(-0.416887\pi\)
0.258150 + 0.966105i \(0.416887\pi\)
\(54\) −2.70928 −0.368686
\(55\) 0 0
\(56\) −9.04945 −1.20928
\(57\) 3.07838 0.407741
\(58\) 18.0989 2.37650
\(59\) 10.5236 1.37005 0.685027 0.728517i \(-0.259790\pi\)
0.685027 + 0.728517i \(0.259790\pi\)
\(60\) 0 0
\(61\) −4.15676 −0.532218 −0.266109 0.963943i \(-0.585738\pi\)
−0.266109 + 0.963943i \(0.585738\pi\)
\(62\) 21.0205 2.66961
\(63\) 1.00000 0.125988
\(64\) 24.8576 3.10720
\(65\) 0 0
\(66\) −5.41855 −0.666977
\(67\) 4.68035 0.571795 0.285898 0.958260i \(-0.407708\pi\)
0.285898 + 0.958260i \(0.407708\pi\)
\(68\) 5.75872 0.698348
\(69\) 2.34017 0.281724
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −9.04945 −1.06649
\(73\) 7.07838 0.828461 0.414231 0.910172i \(-0.364051\pi\)
0.414231 + 0.910172i \(0.364051\pi\)
\(74\) −29.3607 −3.41311
\(75\) 0 0
\(76\) 16.4391 1.88569
\(77\) 2.00000 0.227921
\(78\) −2.49693 −0.282721
\(79\) 6.15676 0.692689 0.346345 0.938107i \(-0.387423\pi\)
0.346345 + 0.938107i \(0.387423\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −17.6020 −1.94381
\(83\) 6.83710 0.750469 0.375235 0.926930i \(-0.377562\pi\)
0.375235 + 0.926930i \(0.377562\pi\)
\(84\) 5.34017 0.582661
\(85\) 0 0
\(86\) 17.6742 1.90586
\(87\) −6.68035 −0.716208
\(88\) −18.0989 −1.92935
\(89\) 8.34017 0.884057 0.442028 0.897001i \(-0.354259\pi\)
0.442028 + 0.897001i \(0.354259\pi\)
\(90\) 0 0
\(91\) 0.921622 0.0966123
\(92\) 12.4969 1.30289
\(93\) −7.75872 −0.804542
\(94\) −12.6803 −1.30788
\(95\) 0 0
\(96\) −19.3896 −1.97894
\(97\) −8.43907 −0.856858 −0.428429 0.903576i \(-0.640933\pi\)
−0.428429 + 0.903576i \(0.640933\pi\)
\(98\) −2.70928 −0.273678
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −5.81658 −0.578772 −0.289386 0.957213i \(-0.593451\pi\)
−0.289386 + 0.957213i \(0.593451\pi\)
\(102\) −2.92162 −0.289284
\(103\) −2.15676 −0.212511 −0.106256 0.994339i \(-0.533886\pi\)
−0.106256 + 0.994339i \(0.533886\pi\)
\(104\) −8.34017 −0.817821
\(105\) 0 0
\(106\) −10.1834 −0.989101
\(107\) −16.4969 −1.59482 −0.797409 0.603439i \(-0.793797\pi\)
−0.797409 + 0.603439i \(0.793797\pi\)
\(108\) 5.34017 0.513858
\(109\) −12.8371 −1.22957 −0.614786 0.788694i \(-0.710757\pi\)
−0.614786 + 0.788694i \(0.710757\pi\)
\(110\) 0 0
\(111\) 10.8371 1.02861
\(112\) 13.8371 1.30748
\(113\) −5.23513 −0.492480 −0.246240 0.969209i \(-0.579195\pi\)
−0.246240 + 0.969209i \(0.579195\pi\)
\(114\) −8.34017 −0.781129
\(115\) 0 0
\(116\) −35.6742 −3.31227
\(117\) 0.921622 0.0852040
\(118\) −28.5113 −2.62468
\(119\) 1.07838 0.0988547
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 11.2618 1.01960
\(123\) 6.49693 0.585808
\(124\) −41.4329 −3.72079
\(125\) 0 0
\(126\) −2.70928 −0.241361
\(127\) 1.84324 0.163562 0.0817808 0.996650i \(-0.473939\pi\)
0.0817808 + 0.996650i \(0.473939\pi\)
\(128\) −28.5669 −2.52498
\(129\) −6.52359 −0.574370
\(130\) 0 0
\(131\) 1.47641 0.128995 0.0644973 0.997918i \(-0.479456\pi\)
0.0644973 + 0.997918i \(0.479456\pi\)
\(132\) 10.6803 0.929605
\(133\) 3.07838 0.266929
\(134\) −12.6803 −1.09542
\(135\) 0 0
\(136\) −9.75872 −0.836804
\(137\) 4.43907 0.379255 0.189628 0.981856i \(-0.439272\pi\)
0.189628 + 0.981856i \(0.439272\pi\)
\(138\) −6.34017 −0.539711
\(139\) 13.6020 1.15370 0.576852 0.816849i \(-0.304281\pi\)
0.576852 + 0.816849i \(0.304281\pi\)
\(140\) 0 0
\(141\) 4.68035 0.394156
\(142\) −5.41855 −0.454715
\(143\) 1.84324 0.154140
\(144\) 13.8371 1.15309
\(145\) 0 0
\(146\) −19.1773 −1.58712
\(147\) 1.00000 0.0824786
\(148\) 57.8720 4.75705
\(149\) 15.6742 1.28408 0.642040 0.766671i \(-0.278088\pi\)
0.642040 + 0.766671i \(0.278088\pi\)
\(150\) 0 0
\(151\) 5.84324 0.475516 0.237758 0.971324i \(-0.423587\pi\)
0.237758 + 0.971324i \(0.423587\pi\)
\(152\) −27.8576 −2.25955
\(153\) 1.07838 0.0871817
\(154\) −5.41855 −0.436639
\(155\) 0 0
\(156\) 4.92162 0.394045
\(157\) 4.92162 0.392788 0.196394 0.980525i \(-0.437077\pi\)
0.196394 + 0.980525i \(0.437077\pi\)
\(158\) −16.6803 −1.32702
\(159\) 3.75872 0.298086
\(160\) 0 0
\(161\) 2.34017 0.184431
\(162\) −2.70928 −0.212861
\(163\) −9.84324 −0.770982 −0.385491 0.922712i \(-0.625968\pi\)
−0.385491 + 0.922712i \(0.625968\pi\)
\(164\) 34.6947 2.70920
\(165\) 0 0
\(166\) −18.5236 −1.43771
\(167\) −19.2039 −1.48605 −0.743023 0.669266i \(-0.766609\pi\)
−0.743023 + 0.669266i \(0.766609\pi\)
\(168\) −9.04945 −0.698180
\(169\) −12.1506 −0.934662
\(170\) 0 0
\(171\) 3.07838 0.235409
\(172\) −34.8371 −2.65630
\(173\) −22.4391 −1.70601 −0.853005 0.521902i \(-0.825223\pi\)
−0.853005 + 0.521902i \(0.825223\pi\)
\(174\) 18.0989 1.37207
\(175\) 0 0
\(176\) 27.6742 2.08602
\(177\) 10.5236 0.791001
\(178\) −22.5958 −1.69363
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −8.52359 −0.633553 −0.316777 0.948500i \(-0.602601\pi\)
−0.316777 + 0.948500i \(0.602601\pi\)
\(182\) −2.49693 −0.185085
\(183\) −4.15676 −0.307276
\(184\) −21.1773 −1.56121
\(185\) 0 0
\(186\) 21.0205 1.54130
\(187\) 2.15676 0.157718
\(188\) 24.9939 1.82286
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 15.3607 1.11146 0.555730 0.831363i \(-0.312439\pi\)
0.555730 + 0.831363i \(0.312439\pi\)
\(192\) 24.8576 1.79394
\(193\) 8.36683 0.602258 0.301129 0.953583i \(-0.402637\pi\)
0.301129 + 0.953583i \(0.402637\pi\)
\(194\) 22.8638 1.64152
\(195\) 0 0
\(196\) 5.34017 0.381441
\(197\) −11.7587 −0.837774 −0.418887 0.908038i \(-0.637580\pi\)
−0.418887 + 0.908038i \(0.637580\pi\)
\(198\) −5.41855 −0.385080
\(199\) −22.5958 −1.60178 −0.800888 0.598814i \(-0.795639\pi\)
−0.800888 + 0.598814i \(0.795639\pi\)
\(200\) 0 0
\(201\) 4.68035 0.330126
\(202\) 15.7587 1.10878
\(203\) −6.68035 −0.468868
\(204\) 5.75872 0.403191
\(205\) 0 0
\(206\) 5.84324 0.407118
\(207\) 2.34017 0.162653
\(208\) 12.7526 0.884232
\(209\) 6.15676 0.425872
\(210\) 0 0
\(211\) −13.6742 −0.941371 −0.470685 0.882301i \(-0.655993\pi\)
−0.470685 + 0.882301i \(0.655993\pi\)
\(212\) 20.0722 1.37857
\(213\) 2.00000 0.137038
\(214\) 44.6947 3.05527
\(215\) 0 0
\(216\) −9.04945 −0.615737
\(217\) −7.75872 −0.526696
\(218\) 34.7792 2.35555
\(219\) 7.07838 0.478312
\(220\) 0 0
\(221\) 0.993857 0.0668541
\(222\) −29.3607 −1.97056
\(223\) −21.6742 −1.45141 −0.725706 0.688005i \(-0.758487\pi\)
−0.725706 + 0.688005i \(0.758487\pi\)
\(224\) −19.3896 −1.29552
\(225\) 0 0
\(226\) 14.1834 0.943467
\(227\) −11.5174 −0.764440 −0.382220 0.924071i \(-0.624840\pi\)
−0.382220 + 0.924071i \(0.624840\pi\)
\(228\) 16.4391 1.08870
\(229\) 12.8371 0.848300 0.424150 0.905592i \(-0.360573\pi\)
0.424150 + 0.905592i \(0.360573\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 60.4534 3.96896
\(233\) −6.76487 −0.443181 −0.221591 0.975140i \(-0.571125\pi\)
−0.221591 + 0.975140i \(0.571125\pi\)
\(234\) −2.49693 −0.163229
\(235\) 0 0
\(236\) 56.1978 3.65816
\(237\) 6.15676 0.399924
\(238\) −2.92162 −0.189381
\(239\) −23.3607 −1.51108 −0.755539 0.655104i \(-0.772625\pi\)
−0.755539 + 0.655104i \(0.772625\pi\)
\(240\) 0 0
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) 18.9649 1.21911
\(243\) 1.00000 0.0641500
\(244\) −22.1978 −1.42107
\(245\) 0 0
\(246\) −17.6020 −1.12226
\(247\) 2.83710 0.180520
\(248\) 70.2122 4.45848
\(249\) 6.83710 0.433284
\(250\) 0 0
\(251\) 9.16290 0.578357 0.289179 0.957275i \(-0.406618\pi\)
0.289179 + 0.957275i \(0.406618\pi\)
\(252\) 5.34017 0.336399
\(253\) 4.68035 0.294251
\(254\) −4.99386 −0.313342
\(255\) 0 0
\(256\) 27.6803 1.73002
\(257\) −5.07838 −0.316781 −0.158390 0.987377i \(-0.550630\pi\)
−0.158390 + 0.987377i \(0.550630\pi\)
\(258\) 17.6742 1.10035
\(259\) 10.8371 0.673385
\(260\) 0 0
\(261\) −6.68035 −0.413503
\(262\) −4.00000 −0.247121
\(263\) 5.65983 0.349000 0.174500 0.984657i \(-0.444169\pi\)
0.174500 + 0.984657i \(0.444169\pi\)
\(264\) −18.0989 −1.11391
\(265\) 0 0
\(266\) −8.34017 −0.511369
\(267\) 8.34017 0.510410
\(268\) 24.9939 1.52674
\(269\) −27.8576 −1.69851 −0.849255 0.527984i \(-0.822948\pi\)
−0.849255 + 0.527984i \(0.822948\pi\)
\(270\) 0 0
\(271\) 25.1194 1.52590 0.762948 0.646460i \(-0.223751\pi\)
0.762948 + 0.646460i \(0.223751\pi\)
\(272\) 14.9216 0.904756
\(273\) 0.921622 0.0557791
\(274\) −12.0267 −0.726557
\(275\) 0 0
\(276\) 12.4969 0.752227
\(277\) −28.1978 −1.69424 −0.847121 0.531401i \(-0.821666\pi\)
−0.847121 + 0.531401i \(0.821666\pi\)
\(278\) −36.8515 −2.21020
\(279\) −7.75872 −0.464503
\(280\) 0 0
\(281\) −20.3545 −1.21425 −0.607125 0.794606i \(-0.707677\pi\)
−0.607125 + 0.794606i \(0.707677\pi\)
\(282\) −12.6803 −0.755104
\(283\) 23.5174 1.39797 0.698984 0.715138i \(-0.253636\pi\)
0.698984 + 0.715138i \(0.253636\pi\)
\(284\) 10.6803 0.633762
\(285\) 0 0
\(286\) −4.99386 −0.295293
\(287\) 6.49693 0.383502
\(288\) −19.3896 −1.14254
\(289\) −15.8371 −0.931594
\(290\) 0 0
\(291\) −8.43907 −0.494707
\(292\) 37.7998 2.21206
\(293\) −2.92162 −0.170683 −0.0853415 0.996352i \(-0.527198\pi\)
−0.0853415 + 0.996352i \(0.527198\pi\)
\(294\) −2.70928 −0.158008
\(295\) 0 0
\(296\) −98.0698 −5.70019
\(297\) 2.00000 0.116052
\(298\) −42.4657 −2.45997
\(299\) 2.15676 0.124728
\(300\) 0 0
\(301\) −6.52359 −0.376014
\(302\) −15.8310 −0.910969
\(303\) −5.81658 −0.334154
\(304\) 42.5958 2.44304
\(305\) 0 0
\(306\) −2.92162 −0.167018
\(307\) −10.4703 −0.597570 −0.298785 0.954321i \(-0.596581\pi\)
−0.298785 + 0.954321i \(0.596581\pi\)
\(308\) 10.6803 0.608569
\(309\) −2.15676 −0.122694
\(310\) 0 0
\(311\) 23.8310 1.35133 0.675665 0.737209i \(-0.263857\pi\)
0.675665 + 0.737209i \(0.263857\pi\)
\(312\) −8.34017 −0.472169
\(313\) 32.7526 1.85129 0.925643 0.378399i \(-0.123525\pi\)
0.925643 + 0.378399i \(0.123525\pi\)
\(314\) −13.3340 −0.752483
\(315\) 0 0
\(316\) 32.8781 1.84954
\(317\) −17.9155 −1.00623 −0.503117 0.864218i \(-0.667813\pi\)
−0.503117 + 0.864218i \(0.667813\pi\)
\(318\) −10.1834 −0.571058
\(319\) −13.3607 −0.748055
\(320\) 0 0
\(321\) −16.4969 −0.920769
\(322\) −6.34017 −0.353324
\(323\) 3.31965 0.184710
\(324\) 5.34017 0.296676
\(325\) 0 0
\(326\) 26.6681 1.47701
\(327\) −12.8371 −0.709893
\(328\) −58.7936 −3.24633
\(329\) 4.68035 0.258036
\(330\) 0 0
\(331\) −1.36069 −0.0747904 −0.0373952 0.999301i \(-0.511906\pi\)
−0.0373952 + 0.999301i \(0.511906\pi\)
\(332\) 36.5113 2.00382
\(333\) 10.8371 0.593870
\(334\) 52.0288 2.84689
\(335\) 0 0
\(336\) 13.8371 0.754876
\(337\) −25.3607 −1.38148 −0.690742 0.723101i \(-0.742716\pi\)
−0.690742 + 0.723101i \(0.742716\pi\)
\(338\) 32.9194 1.79058
\(339\) −5.23513 −0.284333
\(340\) 0 0
\(341\) −15.5174 −0.840317
\(342\) −8.34017 −0.450985
\(343\) 1.00000 0.0539949
\(344\) 59.0349 3.18295
\(345\) 0 0
\(346\) 60.7936 3.26829
\(347\) 16.8638 0.905294 0.452647 0.891690i \(-0.350480\pi\)
0.452647 + 0.891690i \(0.350480\pi\)
\(348\) −35.6742 −1.91234
\(349\) 9.51745 0.509457 0.254729 0.967013i \(-0.418014\pi\)
0.254729 + 0.967013i \(0.418014\pi\)
\(350\) 0 0
\(351\) 0.921622 0.0491926
\(352\) −38.7792 −2.06694
\(353\) −35.7998 −1.90543 −0.952715 0.303867i \(-0.901722\pi\)
−0.952715 + 0.303867i \(0.901722\pi\)
\(354\) −28.5113 −1.51536
\(355\) 0 0
\(356\) 44.5380 2.36051
\(357\) 1.07838 0.0570738
\(358\) −27.0928 −1.43190
\(359\) −22.3135 −1.17766 −0.588831 0.808256i \(-0.700412\pi\)
−0.588831 + 0.808256i \(0.700412\pi\)
\(360\) 0 0
\(361\) −9.52359 −0.501242
\(362\) 23.0928 1.21373
\(363\) −7.00000 −0.367405
\(364\) 4.92162 0.257963
\(365\) 0 0
\(366\) 11.2618 0.588663
\(367\) 20.3135 1.06036 0.530178 0.847886i \(-0.322125\pi\)
0.530178 + 0.847886i \(0.322125\pi\)
\(368\) 32.3812 1.68799
\(369\) 6.49693 0.338217
\(370\) 0 0
\(371\) 3.75872 0.195143
\(372\) −41.4329 −2.14820
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −5.84324 −0.302147
\(375\) 0 0
\(376\) −42.3545 −2.18427
\(377\) −6.15676 −0.317089
\(378\) −2.70928 −0.139350
\(379\) 6.15676 0.316251 0.158126 0.987419i \(-0.449455\pi\)
0.158126 + 0.987419i \(0.449455\pi\)
\(380\) 0 0
\(381\) 1.84324 0.0944323
\(382\) −41.6163 −2.12928
\(383\) 26.8371 1.37131 0.685656 0.727926i \(-0.259515\pi\)
0.685656 + 0.727926i \(0.259515\pi\)
\(384\) −28.5669 −1.45780
\(385\) 0 0
\(386\) −22.6681 −1.15377
\(387\) −6.52359 −0.331613
\(388\) −45.0661 −2.28788
\(389\) −5.63317 −0.285613 −0.142806 0.989751i \(-0.545613\pi\)
−0.142806 + 0.989751i \(0.545613\pi\)
\(390\) 0 0
\(391\) 2.52359 0.127623
\(392\) −9.04945 −0.457066
\(393\) 1.47641 0.0744750
\(394\) 31.8576 1.60496
\(395\) 0 0
\(396\) 10.6803 0.536708
\(397\) 37.7998 1.89712 0.948558 0.316604i \(-0.102543\pi\)
0.948558 + 0.316604i \(0.102543\pi\)
\(398\) 61.2183 3.06860
\(399\) 3.07838 0.154112
\(400\) 0 0
\(401\) −13.6332 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(402\) −12.6803 −0.632438
\(403\) −7.15061 −0.356197
\(404\) −31.0616 −1.54537
\(405\) 0 0
\(406\) 18.0989 0.898233
\(407\) 21.6742 1.07435
\(408\) −9.75872 −0.483129
\(409\) 12.3545 0.610893 0.305447 0.952209i \(-0.401194\pi\)
0.305447 + 0.952209i \(0.401194\pi\)
\(410\) 0 0
\(411\) 4.43907 0.218963
\(412\) −11.5174 −0.567424
\(413\) 10.5236 0.517832
\(414\) −6.34017 −0.311603
\(415\) 0 0
\(416\) −17.8699 −0.876144
\(417\) 13.6020 0.666091
\(418\) −16.6803 −0.815862
\(419\) 28.9939 1.41644 0.708221 0.705991i \(-0.249498\pi\)
0.708221 + 0.705991i \(0.249498\pi\)
\(420\) 0 0
\(421\) −15.1629 −0.738994 −0.369497 0.929232i \(-0.620470\pi\)
−0.369497 + 0.929232i \(0.620470\pi\)
\(422\) 37.0472 1.80343
\(423\) 4.68035 0.227566
\(424\) −34.0144 −1.65188
\(425\) 0 0
\(426\) −5.41855 −0.262530
\(427\) −4.15676 −0.201159
\(428\) −88.0965 −4.25830
\(429\) 1.84324 0.0889927
\(430\) 0 0
\(431\) −10.3135 −0.496784 −0.248392 0.968660i \(-0.579902\pi\)
−0.248392 + 0.968660i \(0.579902\pi\)
\(432\) 13.8371 0.665738
\(433\) 20.4391 0.982239 0.491120 0.871092i \(-0.336588\pi\)
0.491120 + 0.871092i \(0.336588\pi\)
\(434\) 21.0205 1.00902
\(435\) 0 0
\(436\) −68.5523 −3.28306
\(437\) 7.20394 0.344611
\(438\) −19.1773 −0.916326
\(439\) −16.9216 −0.807625 −0.403812 0.914842i \(-0.632315\pi\)
−0.403812 + 0.914842i \(0.632315\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −2.69263 −0.128075
\(443\) −12.8104 −0.608642 −0.304321 0.952569i \(-0.598430\pi\)
−0.304321 + 0.952569i \(0.598430\pi\)
\(444\) 57.8720 2.74648
\(445\) 0 0
\(446\) 58.7214 2.78054
\(447\) 15.6742 0.741364
\(448\) 24.8576 1.17441
\(449\) −14.6270 −0.690292 −0.345146 0.938549i \(-0.612171\pi\)
−0.345146 + 0.938549i \(0.612171\pi\)
\(450\) 0 0
\(451\) 12.9939 0.611857
\(452\) −27.9565 −1.31496
\(453\) 5.84324 0.274540
\(454\) 31.2039 1.46447
\(455\) 0 0
\(456\) −27.8576 −1.30455
\(457\) 14.1568 0.662225 0.331113 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(458\) −34.7792 −1.62513
\(459\) 1.07838 0.0503344
\(460\) 0 0
\(461\) 0.340173 0.0158434 0.00792172 0.999969i \(-0.497478\pi\)
0.00792172 + 0.999969i \(0.497478\pi\)
\(462\) −5.41855 −0.252094
\(463\) −9.84324 −0.457454 −0.228727 0.973491i \(-0.573456\pi\)
−0.228727 + 0.973491i \(0.573456\pi\)
\(464\) −92.4366 −4.29126
\(465\) 0 0
\(466\) 18.3279 0.849023
\(467\) −11.5174 −0.532964 −0.266482 0.963840i \(-0.585861\pi\)
−0.266482 + 0.963840i \(0.585861\pi\)
\(468\) 4.92162 0.227502
\(469\) 4.68035 0.216118
\(470\) 0 0
\(471\) 4.92162 0.226776
\(472\) −95.2327 −4.38344
\(473\) −13.0472 −0.599910
\(474\) −16.6803 −0.766154
\(475\) 0 0
\(476\) 5.75872 0.263951
\(477\) 3.75872 0.172100
\(478\) 63.2905 2.89484
\(479\) −19.5174 −0.891775 −0.445887 0.895089i \(-0.647112\pi\)
−0.445887 + 0.895089i \(0.647112\pi\)
\(480\) 0 0
\(481\) 9.98771 0.455401
\(482\) 39.7731 1.81162
\(483\) 2.34017 0.106482
\(484\) −37.3812 −1.69915
\(485\) 0 0
\(486\) −2.70928 −0.122895
\(487\) 23.1506 1.04905 0.524527 0.851394i \(-0.324242\pi\)
0.524527 + 0.851394i \(0.324242\pi\)
\(488\) 37.6163 1.70281
\(489\) −9.84324 −0.445127
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 34.6947 1.56416
\(493\) −7.20394 −0.324449
\(494\) −7.68649 −0.345831
\(495\) 0 0
\(496\) −107.358 −4.82053
\(497\) 2.00000 0.0897123
\(498\) −18.5236 −0.830062
\(499\) 27.2039 1.21782 0.608908 0.793241i \(-0.291608\pi\)
0.608908 + 0.793241i \(0.291608\pi\)
\(500\) 0 0
\(501\) −19.2039 −0.857969
\(502\) −24.8248 −1.10799
\(503\) −18.8371 −0.839905 −0.419952 0.907546i \(-0.637953\pi\)
−0.419952 + 0.907546i \(0.637953\pi\)
\(504\) −9.04945 −0.403094
\(505\) 0 0
\(506\) −12.6803 −0.563710
\(507\) −12.1506 −0.539628
\(508\) 9.84324 0.436723
\(509\) 6.81044 0.301867 0.150934 0.988544i \(-0.451772\pi\)
0.150934 + 0.988544i \(0.451772\pi\)
\(510\) 0 0
\(511\) 7.07838 0.313129
\(512\) −17.8599 −0.789303
\(513\) 3.07838 0.135914
\(514\) 13.7587 0.606871
\(515\) 0 0
\(516\) −34.8371 −1.53362
\(517\) 9.36069 0.411683
\(518\) −29.3607 −1.29003
\(519\) −22.4391 −0.984966
\(520\) 0 0
\(521\) −25.8166 −1.13105 −0.565523 0.824733i \(-0.691325\pi\)
−0.565523 + 0.824733i \(0.691325\pi\)
\(522\) 18.0989 0.792167
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 7.88428 0.344426
\(525\) 0 0
\(526\) −15.3340 −0.668595
\(527\) −8.36683 −0.364465
\(528\) 27.6742 1.20437
\(529\) −17.5236 −0.761895
\(530\) 0 0
\(531\) 10.5236 0.456685
\(532\) 16.4391 0.712724
\(533\) 5.98771 0.259357
\(534\) −22.5958 −0.977817
\(535\) 0 0
\(536\) −42.3545 −1.82944
\(537\) 10.0000 0.431532
\(538\) 75.4740 3.25391
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 25.8843 1.11285 0.556426 0.830897i \(-0.312172\pi\)
0.556426 + 0.830897i \(0.312172\pi\)
\(542\) −68.0554 −2.92323
\(543\) −8.52359 −0.365782
\(544\) −20.9093 −0.896480
\(545\) 0 0
\(546\) −2.49693 −0.106859
\(547\) 11.3197 0.483993 0.241997 0.970277i \(-0.422198\pi\)
0.241997 + 0.970277i \(0.422198\pi\)
\(548\) 23.7054 1.01264
\(549\) −4.15676 −0.177406
\(550\) 0 0
\(551\) −20.5646 −0.876083
\(552\) −21.1773 −0.901365
\(553\) 6.15676 0.261812
\(554\) 76.3956 3.24574
\(555\) 0 0
\(556\) 72.6369 3.08049
\(557\) −26.6491 −1.12916 −0.564580 0.825378i \(-0.690962\pi\)
−0.564580 + 0.825378i \(0.690962\pi\)
\(558\) 21.0205 0.889870
\(559\) −6.01229 −0.254293
\(560\) 0 0
\(561\) 2.15676 0.0910583
\(562\) 55.1461 2.32620
\(563\) 46.3545 1.95361 0.976806 0.214128i \(-0.0686909\pi\)
0.976806 + 0.214128i \(0.0686909\pi\)
\(564\) 24.9939 1.05243
\(565\) 0 0
\(566\) −63.7152 −2.67815
\(567\) 1.00000 0.0419961
\(568\) −18.0989 −0.759413
\(569\) −14.3668 −0.602289 −0.301145 0.953579i \(-0.597369\pi\)
−0.301145 + 0.953579i \(0.597369\pi\)
\(570\) 0 0
\(571\) 38.7214 1.62044 0.810220 0.586126i \(-0.199348\pi\)
0.810220 + 0.586126i \(0.199348\pi\)
\(572\) 9.84324 0.411567
\(573\) 15.3607 0.641702
\(574\) −17.6020 −0.734692
\(575\) 0 0
\(576\) 24.8576 1.03573
\(577\) 43.4740 1.80984 0.904922 0.425577i \(-0.139929\pi\)
0.904922 + 0.425577i \(0.139929\pi\)
\(578\) 42.9071 1.78470
\(579\) 8.36683 0.347714
\(580\) 0 0
\(581\) 6.83710 0.283651
\(582\) 22.8638 0.947733
\(583\) 7.51745 0.311341
\(584\) −64.0554 −2.65063
\(585\) 0 0
\(586\) 7.91548 0.326985
\(587\) 36.0288 1.48707 0.743533 0.668699i \(-0.233149\pi\)
0.743533 + 0.668699i \(0.233149\pi\)
\(588\) 5.34017 0.220225
\(589\) −23.8843 −0.984135
\(590\) 0 0
\(591\) −11.7587 −0.483689
\(592\) 149.954 6.16307
\(593\) 31.4863 1.29299 0.646493 0.762920i \(-0.276235\pi\)
0.646493 + 0.762920i \(0.276235\pi\)
\(594\) −5.41855 −0.222326
\(595\) 0 0
\(596\) 83.7030 3.42861
\(597\) −22.5958 −0.924786
\(598\) −5.84324 −0.238948
\(599\) 29.0349 1.18633 0.593167 0.805080i \(-0.297877\pi\)
0.593167 + 0.805080i \(0.297877\pi\)
\(600\) 0 0
\(601\) 15.3607 0.626576 0.313288 0.949658i \(-0.398570\pi\)
0.313288 + 0.949658i \(0.398570\pi\)
\(602\) 17.6742 0.720347
\(603\) 4.68035 0.190598
\(604\) 31.2039 1.26967
\(605\) 0 0
\(606\) 15.7587 0.640154
\(607\) −13.0472 −0.529569 −0.264784 0.964308i \(-0.585301\pi\)
−0.264784 + 0.964308i \(0.585301\pi\)
\(608\) −59.6886 −2.42069
\(609\) −6.68035 −0.270701
\(610\) 0 0
\(611\) 4.31351 0.174506
\(612\) 5.75872 0.232783
\(613\) −15.5174 −0.626744 −0.313372 0.949630i \(-0.601459\pi\)
−0.313372 + 0.949630i \(0.601459\pi\)
\(614\) 28.3668 1.14479
\(615\) 0 0
\(616\) −18.0989 −0.729225
\(617\) −22.7649 −0.916479 −0.458240 0.888829i \(-0.651520\pi\)
−0.458240 + 0.888829i \(0.651520\pi\)
\(618\) 5.84324 0.235050
\(619\) 7.92777 0.318644 0.159322 0.987227i \(-0.449069\pi\)
0.159322 + 0.987227i \(0.449069\pi\)
\(620\) 0 0
\(621\) 2.34017 0.0939079
\(622\) −64.5646 −2.58881
\(623\) 8.34017 0.334142
\(624\) 12.7526 0.510512
\(625\) 0 0
\(626\) −88.7358 −3.54659
\(627\) 6.15676 0.245877
\(628\) 26.2823 1.04878
\(629\) 11.6865 0.465971
\(630\) 0 0
\(631\) 19.2039 0.764497 0.382248 0.924060i \(-0.375150\pi\)
0.382248 + 0.924060i \(0.375150\pi\)
\(632\) −55.7152 −2.21623
\(633\) −13.6742 −0.543501
\(634\) 48.5380 1.92769
\(635\) 0 0
\(636\) 20.0722 0.795916
\(637\) 0.921622 0.0365160
\(638\) 36.1978 1.43308
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −5.94668 −0.234880 −0.117440 0.993080i \(-0.537469\pi\)
−0.117440 + 0.993080i \(0.537469\pi\)
\(642\) 44.6947 1.76396
\(643\) −30.8904 −1.21820 −0.609100 0.793094i \(-0.708469\pi\)
−0.609100 + 0.793094i \(0.708469\pi\)
\(644\) 12.4969 0.492448
\(645\) 0 0
\(646\) −8.99386 −0.353859
\(647\) −19.2039 −0.754985 −0.377492 0.926013i \(-0.623214\pi\)
−0.377492 + 0.926013i \(0.623214\pi\)
\(648\) −9.04945 −0.355496
\(649\) 21.0472 0.826174
\(650\) 0 0
\(651\) −7.75872 −0.304088
\(652\) −52.5646 −2.05859
\(653\) −28.5548 −1.11744 −0.558718 0.829358i \(-0.688706\pi\)
−0.558718 + 0.829358i \(0.688706\pi\)
\(654\) 34.7792 1.35998
\(655\) 0 0
\(656\) 89.8987 3.50995
\(657\) 7.07838 0.276154
\(658\) −12.6803 −0.494331
\(659\) −27.9877 −1.09025 −0.545123 0.838356i \(-0.683517\pi\)
−0.545123 + 0.838356i \(0.683517\pi\)
\(660\) 0 0
\(661\) −22.1445 −0.861320 −0.430660 0.902514i \(-0.641719\pi\)
−0.430660 + 0.902514i \(0.641719\pi\)
\(662\) 3.68649 0.143279
\(663\) 0.993857 0.0385982
\(664\) −61.8720 −2.40110
\(665\) 0 0
\(666\) −29.3607 −1.13770
\(667\) −15.6332 −0.605319
\(668\) −102.552 −3.96787
\(669\) −21.6742 −0.837973
\(670\) 0 0
\(671\) −8.31351 −0.320940
\(672\) −19.3896 −0.747971
\(673\) 2.21008 0.0851923 0.0425962 0.999092i \(-0.486437\pi\)
0.0425962 + 0.999092i \(0.486437\pi\)
\(674\) 68.7091 2.64658
\(675\) 0 0
\(676\) −64.8864 −2.49563
\(677\) 19.5486 0.751315 0.375658 0.926758i \(-0.377417\pi\)
0.375658 + 0.926758i \(0.377417\pi\)
\(678\) 14.1834 0.544711
\(679\) −8.43907 −0.323862
\(680\) 0 0
\(681\) −11.5174 −0.441350
\(682\) 42.0410 1.60983
\(683\) 11.8166 0.452149 0.226074 0.974110i \(-0.427411\pi\)
0.226074 + 0.974110i \(0.427411\pi\)
\(684\) 16.4391 0.628564
\(685\) 0 0
\(686\) −2.70928 −0.103441
\(687\) 12.8371 0.489766
\(688\) −90.2676 −3.44142
\(689\) 3.46412 0.131973
\(690\) 0 0
\(691\) 11.7587 0.447323 0.223661 0.974667i \(-0.428199\pi\)
0.223661 + 0.974667i \(0.428199\pi\)
\(692\) −119.829 −4.55520
\(693\) 2.00000 0.0759737
\(694\) −45.6886 −1.73431
\(695\) 0 0
\(696\) 60.4534 2.29148
\(697\) 7.00614 0.265377
\(698\) −25.7854 −0.975991
\(699\) −6.76487 −0.255871
\(700\) 0 0
\(701\) 9.94668 0.375681 0.187840 0.982200i \(-0.439851\pi\)
0.187840 + 0.982200i \(0.439851\pi\)
\(702\) −2.49693 −0.0942405
\(703\) 33.3607 1.25822
\(704\) 49.7152 1.87371
\(705\) 0 0
\(706\) 96.9914 3.65032
\(707\) −5.81658 −0.218755
\(708\) 56.1978 2.11204
\(709\) −11.0472 −0.414886 −0.207443 0.978247i \(-0.566514\pi\)
−0.207443 + 0.978247i \(0.566514\pi\)
\(710\) 0 0
\(711\) 6.15676 0.230896
\(712\) −75.4740 −2.82851
\(713\) −18.1568 −0.679976
\(714\) −2.92162 −0.109339
\(715\) 0 0
\(716\) 53.4017 1.99572
\(717\) −23.3607 −0.872421
\(718\) 60.4534 2.25610
\(719\) −6.15676 −0.229608 −0.114804 0.993388i \(-0.536624\pi\)
−0.114804 + 0.993388i \(0.536624\pi\)
\(720\) 0 0
\(721\) −2.15676 −0.0803218
\(722\) 25.8020 0.960252
\(723\) −14.6803 −0.545968
\(724\) −45.5174 −1.69164
\(725\) 0 0
\(726\) 18.9649 0.703854
\(727\) 2.89043 0.107200 0.0536000 0.998562i \(-0.482930\pi\)
0.0536000 + 0.998562i \(0.482930\pi\)
\(728\) −8.34017 −0.309107
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −7.03489 −0.260195
\(732\) −22.1978 −0.820454
\(733\) −25.7998 −0.952936 −0.476468 0.879192i \(-0.658083\pi\)
−0.476468 + 0.879192i \(0.658083\pi\)
\(734\) −55.0349 −2.03138
\(735\) 0 0
\(736\) −45.3751 −1.67255
\(737\) 9.36069 0.344806
\(738\) −17.6020 −0.647937
\(739\) −1.04718 −0.0385212 −0.0192606 0.999814i \(-0.506131\pi\)
−0.0192606 + 0.999814i \(0.506131\pi\)
\(740\) 0 0
\(741\) 2.83710 0.104224
\(742\) −10.1834 −0.373845
\(743\) −9.97334 −0.365886 −0.182943 0.983123i \(-0.558562\pi\)
−0.182943 + 0.983123i \(0.558562\pi\)
\(744\) 70.2122 2.57410
\(745\) 0 0
\(746\) 43.3484 1.58710
\(747\) 6.83710 0.250156
\(748\) 11.5174 0.421120
\(749\) −16.4969 −0.602785
\(750\) 0 0
\(751\) 3.26633 0.119190 0.0595950 0.998223i \(-0.481019\pi\)
0.0595950 + 0.998223i \(0.481019\pi\)
\(752\) 64.7624 2.36164
\(753\) 9.16290 0.333915
\(754\) 16.6803 0.607462
\(755\) 0 0
\(756\) 5.34017 0.194220
\(757\) −49.9877 −1.81683 −0.908417 0.418065i \(-0.862708\pi\)
−0.908417 + 0.418065i \(0.862708\pi\)
\(758\) −16.6803 −0.605857
\(759\) 4.68035 0.169886
\(760\) 0 0
\(761\) 2.61265 0.0947083 0.0473542 0.998878i \(-0.484921\pi\)
0.0473542 + 0.998878i \(0.484921\pi\)
\(762\) −4.99386 −0.180908
\(763\) −12.8371 −0.464734
\(764\) 82.0288 2.96770
\(765\) 0 0
\(766\) −72.7091 −2.62709
\(767\) 9.69878 0.350202
\(768\) 27.6803 0.998828
\(769\) −15.6742 −0.565226 −0.282613 0.959234i \(-0.591201\pi\)
−0.282613 + 0.959234i \(0.591201\pi\)
\(770\) 0 0
\(771\) −5.07838 −0.182893
\(772\) 44.6803 1.60808
\(773\) 5.81205 0.209045 0.104522 0.994523i \(-0.466669\pi\)
0.104522 + 0.994523i \(0.466669\pi\)
\(774\) 17.6742 0.635286
\(775\) 0 0
\(776\) 76.3689 2.74148
\(777\) 10.8371 0.388779
\(778\) 15.2618 0.547162
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −6.83710 −0.244494
\(783\) −6.68035 −0.238736
\(784\) 13.8371 0.494182
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 39.3484 1.40262 0.701310 0.712857i \(-0.252599\pi\)
0.701310 + 0.712857i \(0.252599\pi\)
\(788\) −62.7936 −2.23693
\(789\) 5.65983 0.201495
\(790\) 0 0
\(791\) −5.23513 −0.186140
\(792\) −18.0989 −0.643116
\(793\) −3.83096 −0.136041
\(794\) −102.410 −3.63439
\(795\) 0 0
\(796\) −120.666 −4.27688
\(797\) 28.2823 1.00181 0.500905 0.865502i \(-0.333000\pi\)
0.500905 + 0.865502i \(0.333000\pi\)
\(798\) −8.34017 −0.295239
\(799\) 5.04718 0.178556
\(800\) 0 0
\(801\) 8.34017 0.294686
\(802\) 36.9360 1.30426
\(803\) 14.1568 0.499581
\(804\) 24.9939 0.881465
\(805\) 0 0
\(806\) 19.3730 0.682384
\(807\) −27.8576 −0.980635
\(808\) 52.6369 1.85176
\(809\) −15.6742 −0.551076 −0.275538 0.961290i \(-0.588856\pi\)
−0.275538 + 0.961290i \(0.588856\pi\)
\(810\) 0 0
\(811\) −42.1666 −1.48067 −0.740335 0.672238i \(-0.765333\pi\)
−0.740335 + 0.672238i \(0.765333\pi\)
\(812\) −35.6742 −1.25192
\(813\) 25.1194 0.880976
\(814\) −58.7214 −2.05818
\(815\) 0 0
\(816\) 14.9216 0.522361
\(817\) −20.0821 −0.702583
\(818\) −33.4719 −1.17032
\(819\) 0.921622 0.0322041
\(820\) 0 0
\(821\) −39.0472 −1.36276 −0.681378 0.731932i \(-0.738619\pi\)
−0.681378 + 0.731932i \(0.738619\pi\)
\(822\) −12.0267 −0.419478
\(823\) −36.5646 −1.27456 −0.637281 0.770631i \(-0.719941\pi\)
−0.637281 + 0.770631i \(0.719941\pi\)
\(824\) 19.5174 0.679922
\(825\) 0 0
\(826\) −28.5113 −0.992035
\(827\) 50.2245 1.74648 0.873238 0.487294i \(-0.162016\pi\)
0.873238 + 0.487294i \(0.162016\pi\)
\(828\) 12.4969 0.434298
\(829\) 32.8371 1.14048 0.570240 0.821478i \(-0.306850\pi\)
0.570240 + 0.821478i \(0.306850\pi\)
\(830\) 0 0
\(831\) −28.1978 −0.978171
\(832\) 22.9093 0.794238
\(833\) 1.07838 0.0373636
\(834\) −36.8515 −1.27606
\(835\) 0 0
\(836\) 32.8781 1.13711
\(837\) −7.75872 −0.268181
\(838\) −78.5523 −2.71355
\(839\) −13.3607 −0.461262 −0.230631 0.973041i \(-0.574079\pi\)
−0.230631 + 0.973041i \(0.574079\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) 41.0805 1.41573
\(843\) −20.3545 −0.701048
\(844\) −73.0226 −2.51354
\(845\) 0 0
\(846\) −12.6803 −0.435959
\(847\) −7.00000 −0.240523
\(848\) 52.0098 1.78603
\(849\) 23.5174 0.807117
\(850\) 0 0
\(851\) 25.3607 0.869353
\(852\) 10.6803 0.365903
\(853\) −39.6430 −1.35735 −0.678675 0.734438i \(-0.737446\pi\)
−0.678675 + 0.734438i \(0.737446\pi\)
\(854\) 11.2618 0.385371
\(855\) 0 0
\(856\) 149.288 5.10256
\(857\) −29.7054 −1.01472 −0.507359 0.861735i \(-0.669378\pi\)
−0.507359 + 0.861735i \(0.669378\pi\)
\(858\) −4.99386 −0.170487
\(859\) −3.07838 −0.105033 −0.0525164 0.998620i \(-0.516724\pi\)
−0.0525164 + 0.998620i \(0.516724\pi\)
\(860\) 0 0
\(861\) 6.49693 0.221415
\(862\) 27.9421 0.951713
\(863\) −6.39350 −0.217637 −0.108819 0.994062i \(-0.534707\pi\)
−0.108819 + 0.994062i \(0.534707\pi\)
\(864\) −19.3896 −0.659648
\(865\) 0 0
\(866\) −55.3751 −1.88172
\(867\) −15.8371 −0.537856
\(868\) −41.4329 −1.40633
\(869\) 12.3135 0.417707
\(870\) 0 0
\(871\) 4.31351 0.146158
\(872\) 116.169 3.93397
\(873\) −8.43907 −0.285619
\(874\) −19.5174 −0.660188
\(875\) 0 0
\(876\) 37.7998 1.27714
\(877\) −1.21622 −0.0410689 −0.0205345 0.999789i \(-0.506537\pi\)
−0.0205345 + 0.999789i \(0.506537\pi\)
\(878\) 45.8453 1.54721
\(879\) −2.92162 −0.0985439
\(880\) 0 0
\(881\) 15.9733 0.538155 0.269078 0.963118i \(-0.413281\pi\)
0.269078 + 0.963118i \(0.413281\pi\)
\(882\) −2.70928 −0.0912260
\(883\) 11.6865 0.393282 0.196641 0.980476i \(-0.436997\pi\)
0.196641 + 0.980476i \(0.436997\pi\)
\(884\) 5.30737 0.178506
\(885\) 0 0
\(886\) 34.7070 1.16600
\(887\) −25.6209 −0.860265 −0.430132 0.902766i \(-0.641533\pi\)
−0.430132 + 0.902766i \(0.641533\pi\)
\(888\) −98.0698 −3.29101
\(889\) 1.84324 0.0618204
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −115.744 −3.87540
\(893\) 14.4079 0.482141
\(894\) −42.4657 −1.42027
\(895\) 0 0
\(896\) −28.5669 −0.954353
\(897\) 2.15676 0.0720120
\(898\) 39.6286 1.32242
\(899\) 51.8310 1.72866
\(900\) 0 0
\(901\) 4.05332 0.135036
\(902\) −35.2039 −1.17216
\(903\) −6.52359 −0.217091
\(904\) 47.3751 1.57567
\(905\) 0 0
\(906\) −15.8310 −0.525948
\(907\) −57.7563 −1.91777 −0.958883 0.283802i \(-0.908404\pi\)
−0.958883 + 0.283802i \(0.908404\pi\)
\(908\) −61.5052 −2.04112
\(909\) −5.81658 −0.192924
\(910\) 0 0
\(911\) −35.9877 −1.19233 −0.596163 0.802863i \(-0.703309\pi\)
−0.596163 + 0.802863i \(0.703309\pi\)
\(912\) 42.5958 1.41049
\(913\) 13.6742 0.452550
\(914\) −38.3545 −1.26866
\(915\) 0 0
\(916\) 68.5523 2.26503
\(917\) 1.47641 0.0487553
\(918\) −2.92162 −0.0964279
\(919\) 46.7214 1.54120 0.770598 0.637321i \(-0.219958\pi\)
0.770598 + 0.637321i \(0.219958\pi\)
\(920\) 0 0
\(921\) −10.4703 −0.345007
\(922\) −0.921622 −0.0303520
\(923\) 1.84324 0.0606711
\(924\) 10.6803 0.351358
\(925\) 0 0
\(926\) 26.6681 0.876367
\(927\) −2.15676 −0.0708371
\(928\) 129.529 4.25201
\(929\) −53.0493 −1.74049 −0.870245 0.492619i \(-0.836040\pi\)
−0.870245 + 0.492619i \(0.836040\pi\)
\(930\) 0 0
\(931\) 3.07838 0.100890
\(932\) −36.1256 −1.18333
\(933\) 23.8310 0.780191
\(934\) 31.2039 1.02102
\(935\) 0 0
\(936\) −8.34017 −0.272607
\(937\) −16.1256 −0.526799 −0.263400 0.964687i \(-0.584844\pi\)
−0.263400 + 0.964687i \(0.584844\pi\)
\(938\) −12.6803 −0.414028
\(939\) 32.7526 1.06884
\(940\) 0 0
\(941\) 24.7070 0.805425 0.402713 0.915326i \(-0.368067\pi\)
0.402713 + 0.915326i \(0.368067\pi\)
\(942\) −13.3340 −0.434446
\(943\) 15.2039 0.495108
\(944\) 145.616 4.73940
\(945\) 0 0
\(946\) 35.3484 1.14928
\(947\) −6.53797 −0.212455 −0.106228 0.994342i \(-0.533877\pi\)
−0.106228 + 0.994342i \(0.533877\pi\)
\(948\) 32.8781 1.06783
\(949\) 6.52359 0.211765
\(950\) 0 0
\(951\) −17.9155 −0.580949
\(952\) −9.75872 −0.316282
\(953\) 6.11327 0.198028 0.0990142 0.995086i \(-0.468431\pi\)
0.0990142 + 0.995086i \(0.468431\pi\)
\(954\) −10.1834 −0.329700
\(955\) 0 0
\(956\) −124.750 −4.03471
\(957\) −13.3607 −0.431890
\(958\) 52.8781 1.70842
\(959\) 4.43907 0.143345
\(960\) 0 0
\(961\) 29.1978 0.941864
\(962\) −27.0595 −0.872432
\(963\) −16.4969 −0.531606
\(964\) −78.3956 −2.52495
\(965\) 0 0
\(966\) −6.34017 −0.203992
\(967\) −25.6209 −0.823912 −0.411956 0.911204i \(-0.635154\pi\)
−0.411956 + 0.911204i \(0.635154\pi\)
\(968\) 63.3461 2.03602
\(969\) 3.31965 0.106643
\(970\) 0 0
\(971\) 4.05332 0.130077 0.0650387 0.997883i \(-0.479283\pi\)
0.0650387 + 0.997883i \(0.479283\pi\)
\(972\) 5.34017 0.171286
\(973\) 13.6020 0.436059
\(974\) −62.7214 −2.00972
\(975\) 0 0
\(976\) −57.5174 −1.84109
\(977\) −3.81205 −0.121958 −0.0609791 0.998139i \(-0.519422\pi\)
−0.0609791 + 0.998139i \(0.519422\pi\)
\(978\) 26.6681 0.852751
\(979\) 16.6803 0.533106
\(980\) 0 0
\(981\) −12.8371 −0.409857
\(982\) −5.41855 −0.172913
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −58.7936 −1.87427
\(985\) 0 0
\(986\) 19.5174 0.621562
\(987\) 4.68035 0.148977
\(988\) 15.1506 0.482005
\(989\) −15.2663 −0.485441
\(990\) 0 0
\(991\) −42.4079 −1.34713 −0.673565 0.739128i \(-0.735238\pi\)
−0.673565 + 0.739128i \(0.735238\pi\)
\(992\) 150.439 4.77643
\(993\) −1.36069 −0.0431803
\(994\) −5.41855 −0.171866
\(995\) 0 0
\(996\) 36.5113 1.15690
\(997\) 43.4740 1.37683 0.688417 0.725315i \(-0.258306\pi\)
0.688417 + 0.725315i \(0.258306\pi\)
\(998\) −73.7030 −2.33303
\(999\) 10.8371 0.342871
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 525.2.a.j.1.1 3
3.2 odd 2 1575.2.a.x.1.3 3
4.3 odd 2 8400.2.a.dg.1.2 3
5.2 odd 4 105.2.d.b.64.1 6
5.3 odd 4 105.2.d.b.64.6 yes 6
5.4 even 2 525.2.a.k.1.3 3
7.6 odd 2 3675.2.a.bi.1.1 3
15.2 even 4 315.2.d.e.64.6 6
15.8 even 4 315.2.d.e.64.1 6
15.14 odd 2 1575.2.a.w.1.1 3
20.3 even 4 1680.2.t.k.1009.3 6
20.7 even 4 1680.2.t.k.1009.6 6
20.19 odd 2 8400.2.a.dj.1.2 3
35.2 odd 12 735.2.q.e.214.1 12
35.3 even 12 735.2.q.f.79.1 12
35.12 even 12 735.2.q.f.214.1 12
35.13 even 4 735.2.d.b.589.6 6
35.17 even 12 735.2.q.f.79.6 12
35.18 odd 12 735.2.q.e.79.1 12
35.23 odd 12 735.2.q.e.214.6 12
35.27 even 4 735.2.d.b.589.1 6
35.32 odd 12 735.2.q.e.79.6 12
35.33 even 12 735.2.q.f.214.6 12
35.34 odd 2 3675.2.a.bj.1.3 3
60.23 odd 4 5040.2.t.v.1009.1 6
60.47 odd 4 5040.2.t.v.1009.2 6
105.62 odd 4 2205.2.d.l.1324.6 6
105.83 odd 4 2205.2.d.l.1324.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.1 6 5.2 odd 4
105.2.d.b.64.6 yes 6 5.3 odd 4
315.2.d.e.64.1 6 15.8 even 4
315.2.d.e.64.6 6 15.2 even 4
525.2.a.j.1.1 3 1.1 even 1 trivial
525.2.a.k.1.3 3 5.4 even 2
735.2.d.b.589.1 6 35.27 even 4
735.2.d.b.589.6 6 35.13 even 4
735.2.q.e.79.1 12 35.18 odd 12
735.2.q.e.79.6 12 35.32 odd 12
735.2.q.e.214.1 12 35.2 odd 12
735.2.q.e.214.6 12 35.23 odd 12
735.2.q.f.79.1 12 35.3 even 12
735.2.q.f.79.6 12 35.17 even 12
735.2.q.f.214.1 12 35.12 even 12
735.2.q.f.214.6 12 35.33 even 12
1575.2.a.w.1.1 3 15.14 odd 2
1575.2.a.x.1.3 3 3.2 odd 2
1680.2.t.k.1009.3 6 20.3 even 4
1680.2.t.k.1009.6 6 20.7 even 4
2205.2.d.l.1324.1 6 105.83 odd 4
2205.2.d.l.1324.6 6 105.62 odd 4
3675.2.a.bi.1.1 3 7.6 odd 2
3675.2.a.bj.1.3 3 35.34 odd 2
5040.2.t.v.1009.1 6 60.23 odd 4
5040.2.t.v.1009.2 6 60.47 odd 4
8400.2.a.dg.1.2 3 4.3 odd 2
8400.2.a.dj.1.2 3 20.19 odd 2