Properties

Label 1815.2.a.y.1.3
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.437199552.1
Defining polynomial: \(x^{6} - 13 x^{4} + 49 x^{2} - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.23396\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.23396 q^{2} -1.00000 q^{3} -0.477352 q^{4} +1.00000 q^{5} +1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.23396 q^{2} -1.00000 q^{3} -0.477352 q^{4} +1.00000 q^{5} +1.23396 q^{6} -3.79281 q^{7} +3.05694 q^{8} +1.00000 q^{9} -1.23396 q^{10} +0.477352 q^{12} -2.46791 q^{13} +4.68016 q^{14} -1.00000 q^{15} -2.81743 q^{16} +6.52105 q^{17} -1.23396 q^{18} -8.25310 q^{19} -0.477352 q^{20} +3.79281 q^{21} -2.34008 q^{23} -3.05694 q^{24} +1.00000 q^{25} +3.04530 q^{26} -1.00000 q^{27} +1.81050 q^{28} -2.46791 q^{29} +1.23396 q^{30} +5.27455 q^{31} -2.63730 q^{32} -8.04668 q^{34} -3.79281 q^{35} -0.477352 q^{36} +3.34008 q^{37} +10.1840 q^{38} +2.46791 q^{39} +3.05694 q^{40} +3.46410 q^{41} -4.68016 q^{42} -12.0459 q^{43} +1.00000 q^{45} +2.88755 q^{46} -9.56933 q^{47} +2.81743 q^{48} +7.38537 q^{49} -1.23396 q^{50} -6.52105 q^{51} +1.17806 q^{52} -6.61463 q^{53} +1.23396 q^{54} -11.5944 q^{56} +8.25310 q^{57} +3.04530 q^{58} +10.1840 q^{59} +0.477352 q^{60} +6.66788 q^{61} -6.50856 q^{62} -3.79281 q^{63} +8.88918 q^{64} -2.46791 q^{65} +3.34008 q^{67} -3.11284 q^{68} +2.34008 q^{69} +4.68016 q^{70} +7.22925 q^{71} +3.05694 q^{72} -9.72482 q^{73} -4.12151 q^{74} -1.00000 q^{75} +3.93963 q^{76} -3.04530 q^{78} +9.65644 q^{79} -2.81743 q^{80} +1.00000 q^{81} -4.27455 q^{82} -11.0497 q^{83} -1.81050 q^{84} +6.52105 q^{85} +14.8641 q^{86} +2.46791 q^{87} +10.2745 q^{89} -1.23396 q^{90} +9.36031 q^{91} +1.11704 q^{92} -5.27455 q^{93} +11.8081 q^{94} -8.25310 q^{95} +2.63730 q^{96} +13.2495 q^{97} -9.11323 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} - 14q^{12} + 8q^{14} - 6q^{15} + 10q^{16} + 14q^{20} - 4q^{23} + 6q^{25} + 52q^{26} - 6q^{27} + 18q^{31} + 26q^{34} + 14q^{36} + 10q^{37} - 20q^{38} - 8q^{42} + 6q^{45} - 10q^{48} + 68q^{49} - 16q^{53} - 76q^{56} + 52q^{58} - 20q^{59} - 14q^{60} + 16q^{64} + 10q^{67} + 4q^{69} + 8q^{70} - 4q^{71} - 6q^{75} - 52q^{78} + 10q^{80} + 6q^{81} - 12q^{82} - 12q^{86} + 48q^{89} + 16q^{91} + 30q^{92} - 18q^{93} + 2q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23396 −0.872539 −0.436269 0.899816i \(-0.643701\pi\)
−0.436269 + 0.899816i \(0.643701\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.477352 −0.238676
\(5\) 1.00000 0.447214
\(6\) 1.23396 0.503760
\(7\) −3.79281 −1.43355 −0.716773 0.697307i \(-0.754382\pi\)
−0.716773 + 0.697307i \(0.754382\pi\)
\(8\) 3.05694 1.08079
\(9\) 1.00000 0.333333
\(10\) −1.23396 −0.390211
\(11\) 0 0
\(12\) 0.477352 0.137800
\(13\) −2.46791 −0.684476 −0.342238 0.939613i \(-0.611185\pi\)
−0.342238 + 0.939613i \(0.611185\pi\)
\(14\) 4.68016 1.25082
\(15\) −1.00000 −0.258199
\(16\) −2.81743 −0.704358
\(17\) 6.52105 1.58159 0.790793 0.612084i \(-0.209668\pi\)
0.790793 + 0.612084i \(0.209668\pi\)
\(18\) −1.23396 −0.290846
\(19\) −8.25310 −1.89339 −0.946695 0.322131i \(-0.895601\pi\)
−0.946695 + 0.322131i \(0.895601\pi\)
\(20\) −0.477352 −0.106739
\(21\) 3.79281 0.827658
\(22\) 0 0
\(23\) −2.34008 −0.487940 −0.243970 0.969783i \(-0.578450\pi\)
−0.243970 + 0.969783i \(0.578450\pi\)
\(24\) −3.05694 −0.623996
\(25\) 1.00000 0.200000
\(26\) 3.04530 0.597232
\(27\) −1.00000 −0.192450
\(28\) 1.81050 0.342153
\(29\) −2.46791 −0.458280 −0.229140 0.973394i \(-0.573591\pi\)
−0.229140 + 0.973394i \(0.573591\pi\)
\(30\) 1.23396 0.225289
\(31\) 5.27455 0.947337 0.473669 0.880703i \(-0.342929\pi\)
0.473669 + 0.880703i \(0.342929\pi\)
\(32\) −2.63730 −0.466214
\(33\) 0 0
\(34\) −8.04668 −1.37999
\(35\) −3.79281 −0.641101
\(36\) −0.477352 −0.0795587
\(37\) 3.34008 0.549105 0.274553 0.961572i \(-0.411470\pi\)
0.274553 + 0.961572i \(0.411470\pi\)
\(38\) 10.1840 1.65206
\(39\) 2.46791 0.395182
\(40\) 3.05694 0.483345
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) −4.68016 −0.722164
\(43\) −12.0459 −1.83698 −0.918491 0.395441i \(-0.870592\pi\)
−0.918491 + 0.395441i \(0.870592\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 2.88755 0.425746
\(47\) −9.56933 −1.39583 −0.697915 0.716180i \(-0.745889\pi\)
−0.697915 + 0.716180i \(0.745889\pi\)
\(48\) 2.81743 0.406661
\(49\) 7.38537 1.05505
\(50\) −1.23396 −0.174508
\(51\) −6.52105 −0.913129
\(52\) 1.17806 0.163368
\(53\) −6.61463 −0.908589 −0.454294 0.890852i \(-0.650109\pi\)
−0.454294 + 0.890852i \(0.650109\pi\)
\(54\) 1.23396 0.167920
\(55\) 0 0
\(56\) −11.5944 −1.54937
\(57\) 8.25310 1.09315
\(58\) 3.04530 0.399867
\(59\) 10.1840 1.32584 0.662919 0.748691i \(-0.269317\pi\)
0.662919 + 0.748691i \(0.269317\pi\)
\(60\) 0.477352 0.0616259
\(61\) 6.66788 0.853734 0.426867 0.904314i \(-0.359617\pi\)
0.426867 + 0.904314i \(0.359617\pi\)
\(62\) −6.50856 −0.826588
\(63\) −3.79281 −0.477849
\(64\) 8.88918 1.11115
\(65\) −2.46791 −0.306107
\(66\) 0 0
\(67\) 3.34008 0.408055 0.204028 0.978965i \(-0.434597\pi\)
0.204028 + 0.978965i \(0.434597\pi\)
\(68\) −3.11284 −0.377487
\(69\) 2.34008 0.281712
\(70\) 4.68016 0.559386
\(71\) 7.22925 0.857955 0.428977 0.903315i \(-0.358874\pi\)
0.428977 + 0.903315i \(0.358874\pi\)
\(72\) 3.05694 0.360264
\(73\) −9.72482 −1.13820 −0.569102 0.822267i \(-0.692709\pi\)
−0.569102 + 0.822267i \(0.692709\pi\)
\(74\) −4.12151 −0.479116
\(75\) −1.00000 −0.115470
\(76\) 3.93963 0.451907
\(77\) 0 0
\(78\) −3.04530 −0.344812
\(79\) 9.65644 1.08643 0.543217 0.839592i \(-0.317206\pi\)
0.543217 + 0.839592i \(0.317206\pi\)
\(80\) −2.81743 −0.314998
\(81\) 1.00000 0.111111
\(82\) −4.27455 −0.472045
\(83\) −11.0497 −1.21286 −0.606432 0.795136i \(-0.707400\pi\)
−0.606432 + 0.795136i \(0.707400\pi\)
\(84\) −1.81050 −0.197542
\(85\) 6.52105 0.707307
\(86\) 14.8641 1.60284
\(87\) 2.46791 0.264588
\(88\) 0 0
\(89\) 10.2745 1.08910 0.544550 0.838728i \(-0.316700\pi\)
0.544550 + 0.838728i \(0.316700\pi\)
\(90\) −1.23396 −0.130070
\(91\) 9.36031 0.981227
\(92\) 1.11704 0.116460
\(93\) −5.27455 −0.546945
\(94\) 11.8081 1.21792
\(95\) −8.25310 −0.846750
\(96\) 2.63730 0.269169
\(97\) 13.2495 1.34528 0.672641 0.739969i \(-0.265160\pi\)
0.672641 + 0.739969i \(0.265160\pi\)
\(98\) −9.11323 −0.920575
\(99\) 0 0
\(100\) −0.477352 −0.0477352
\(101\) −1.99238 −0.198249 −0.0991246 0.995075i \(-0.531604\pi\)
−0.0991246 + 0.995075i \(0.531604\pi\)
\(102\) 8.04668 0.796740
\(103\) −1.61463 −0.159094 −0.0795470 0.996831i \(-0.525347\pi\)
−0.0795470 + 0.996831i \(0.525347\pi\)
\(104\) −7.54427 −0.739776
\(105\) 3.79281 0.370140
\(106\) 8.16216 0.792779
\(107\) −0.589032 −0.0569439 −0.0284719 0.999595i \(-0.509064\pi\)
−0.0284719 + 0.999595i \(0.509064\pi\)
\(108\) 0.477352 0.0459333
\(109\) −4.78899 −0.458703 −0.229351 0.973344i \(-0.573660\pi\)
−0.229351 + 0.973344i \(0.573660\pi\)
\(110\) 0 0
\(111\) −3.34008 −0.317026
\(112\) 10.6860 1.00973
\(113\) 18.5240 1.74259 0.871297 0.490755i \(-0.163279\pi\)
0.871297 + 0.490755i \(0.163279\pi\)
\(114\) −10.1840 −0.953815
\(115\) −2.34008 −0.218213
\(116\) 1.17806 0.109380
\(117\) −2.46791 −0.228159
\(118\) −12.5666 −1.15685
\(119\) −24.7331 −2.26728
\(120\) −3.05694 −0.279060
\(121\) 0 0
\(122\) −8.22787 −0.744916
\(123\) −3.46410 −0.312348
\(124\) −2.51782 −0.226107
\(125\) 1.00000 0.0894427
\(126\) 4.68016 0.416941
\(127\) 12.1927 1.08193 0.540965 0.841045i \(-0.318059\pi\)
0.540965 + 0.841045i \(0.318059\pi\)
\(128\) −5.69425 −0.503305
\(129\) 12.0459 1.06058
\(130\) 3.04530 0.267090
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) 31.3024 2.71426
\(134\) −4.12151 −0.356044
\(135\) −1.00000 −0.0860663
\(136\) 19.9345 1.70937
\(137\) 7.93447 0.677888 0.338944 0.940807i \(-0.389930\pi\)
0.338944 + 0.940807i \(0.389930\pi\)
\(138\) −2.88755 −0.245805
\(139\) 6.18226 0.524373 0.262186 0.965017i \(-0.415556\pi\)
0.262186 + 0.965017i \(0.415556\pi\)
\(140\) 1.81050 0.153016
\(141\) 9.56933 0.805883
\(142\) −8.92058 −0.748599
\(143\) 0 0
\(144\) −2.81743 −0.234786
\(145\) −2.46791 −0.204949
\(146\) 12.0000 0.993127
\(147\) −7.38537 −0.609135
\(148\) −1.59439 −0.131058
\(149\) −1.81050 −0.148322 −0.0741612 0.997246i \(-0.523628\pi\)
−0.0741612 + 0.997246i \(0.523628\pi\)
\(150\) 1.23396 0.100752
\(151\) 20.0387 1.63072 0.815362 0.578952i \(-0.196538\pi\)
0.815362 + 0.578952i \(0.196538\pi\)
\(152\) −25.2293 −2.04636
\(153\) 6.52105 0.527195
\(154\) 0 0
\(155\) 5.27455 0.423662
\(156\) −1.17806 −0.0943206
\(157\) −14.5693 −1.16276 −0.581380 0.813632i \(-0.697487\pi\)
−0.581380 + 0.813632i \(0.697487\pi\)
\(158\) −11.9156 −0.947956
\(159\) 6.61463 0.524574
\(160\) −2.63730 −0.208497
\(161\) 8.87546 0.699484
\(162\) −1.23396 −0.0969487
\(163\) −5.88918 −0.461276 −0.230638 0.973040i \(-0.574081\pi\)
−0.230638 + 0.973040i \(0.574081\pi\)
\(164\) −1.65360 −0.129124
\(165\) 0 0
\(166\) 13.6349 1.05827
\(167\) 22.8454 1.76783 0.883914 0.467650i \(-0.154899\pi\)
0.883914 + 0.467650i \(0.154899\pi\)
\(168\) 11.5944 0.894527
\(169\) −6.90941 −0.531493
\(170\) −8.04668 −0.617152
\(171\) −8.25310 −0.631130
\(172\) 5.75014 0.438444
\(173\) 12.5214 0.951987 0.475994 0.879449i \(-0.342089\pi\)
0.475994 + 0.879449i \(0.342089\pi\)
\(174\) −3.04530 −0.230863
\(175\) −3.79281 −0.286709
\(176\) 0 0
\(177\) −10.1840 −0.765473
\(178\) −12.6783 −0.950282
\(179\) −3.04530 −0.227616 −0.113808 0.993503i \(-0.536305\pi\)
−0.113808 + 0.993503i \(0.536305\pi\)
\(180\) −0.477352 −0.0355797
\(181\) 4.38537 0.325962 0.162981 0.986629i \(-0.447889\pi\)
0.162981 + 0.986629i \(0.447889\pi\)
\(182\) −11.5502 −0.856159
\(183\) −6.66788 −0.492904
\(184\) −7.15349 −0.527362
\(185\) 3.34008 0.245567
\(186\) 6.50856 0.477231
\(187\) 0 0
\(188\) 4.56794 0.333151
\(189\) 3.79281 0.275886
\(190\) 10.1840 0.738822
\(191\) 23.4132 1.69412 0.847060 0.531497i \(-0.178370\pi\)
0.847060 + 0.531497i \(0.178370\pi\)
\(192\) −8.88918 −0.641521
\(193\) 23.2424 1.67303 0.836514 0.547946i \(-0.184590\pi\)
0.836514 + 0.547946i \(0.184590\pi\)
\(194\) −16.3493 −1.17381
\(195\) 2.46791 0.176731
\(196\) −3.52543 −0.251816
\(197\) −1.99238 −0.141951 −0.0709756 0.997478i \(-0.522611\pi\)
−0.0709756 + 0.997478i \(0.522611\pi\)
\(198\) 0 0
\(199\) 8.72545 0.618531 0.309265 0.950976i \(-0.399917\pi\)
0.309265 + 0.950976i \(0.399917\pi\)
\(200\) 3.05694 0.216159
\(201\) −3.34008 −0.235591
\(202\) 2.45851 0.172980
\(203\) 9.36031 0.656965
\(204\) 3.11284 0.217942
\(205\) 3.46410 0.241943
\(206\) 1.99238 0.138816
\(207\) −2.34008 −0.162647
\(208\) 6.95317 0.482116
\(209\) 0 0
\(210\) −4.68016 −0.322961
\(211\) 16.0640 1.10589 0.552945 0.833218i \(-0.313504\pi\)
0.552945 + 0.833218i \(0.313504\pi\)
\(212\) 3.15751 0.216859
\(213\) −7.22925 −0.495340
\(214\) 0.726839 0.0496857
\(215\) −12.0459 −0.821524
\(216\) −3.05694 −0.207999
\(217\) −20.0053 −1.35805
\(218\) 5.90941 0.400236
\(219\) 9.72482 0.657142
\(220\) 0 0
\(221\) −16.0934 −1.08256
\(222\) 4.12151 0.276618
\(223\) −11.0655 −0.741003 −0.370501 0.928832i \(-0.620814\pi\)
−0.370501 + 0.928832i \(0.620814\pi\)
\(224\) 10.0028 0.668339
\(225\) 1.00000 0.0666667
\(226\) −22.8578 −1.52048
\(227\) 15.3965 1.02190 0.510951 0.859610i \(-0.329293\pi\)
0.510951 + 0.859610i \(0.329293\pi\)
\(228\) −3.93963 −0.260909
\(229\) 4.70522 0.310930 0.155465 0.987841i \(-0.450312\pi\)
0.155465 + 0.987841i \(0.450312\pi\)
\(230\) 2.88755 0.190400
\(231\) 0 0
\(232\) −7.54427 −0.495306
\(233\) 14.9210 0.977505 0.488753 0.872422i \(-0.337452\pi\)
0.488753 + 0.872422i \(0.337452\pi\)
\(234\) 3.04530 0.199077
\(235\) −9.56933 −0.624234
\(236\) −4.86134 −0.316446
\(237\) −9.65644 −0.627253
\(238\) 30.5195 1.97829
\(239\) −11.3885 −0.736660 −0.368330 0.929695i \(-0.620070\pi\)
−0.368330 + 0.929695i \(0.620070\pi\)
\(240\) 2.81743 0.181864
\(241\) 12.1143 0.780349 0.390175 0.920741i \(-0.372415\pi\)
0.390175 + 0.920741i \(0.372415\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −3.18293 −0.203766
\(245\) 7.38537 0.471834
\(246\) 4.27455 0.272535
\(247\) 20.3679 1.29598
\(248\) 16.1240 1.02388
\(249\) 11.0497 0.700247
\(250\) −1.23396 −0.0780422
\(251\) −17.4132 −1.09911 −0.549556 0.835457i \(-0.685203\pi\)
−0.549556 + 0.835457i \(0.685203\pi\)
\(252\) 1.81050 0.114051
\(253\) 0 0
\(254\) −15.0453 −0.944026
\(255\) −6.52105 −0.408364
\(256\) −10.7519 −0.671994
\(257\) −18.4334 −1.14985 −0.574923 0.818207i \(-0.694968\pi\)
−0.574923 + 0.818207i \(0.694968\pi\)
\(258\) −14.8641 −0.925399
\(259\) −12.6683 −0.787168
\(260\) 1.17806 0.0730604
\(261\) −2.46791 −0.152760
\(262\) −4.27455 −0.264083
\(263\) −2.06075 −0.127072 −0.0635358 0.997980i \(-0.520238\pi\)
−0.0635358 + 0.997980i \(0.520238\pi\)
\(264\) 0 0
\(265\) −6.61463 −0.406333
\(266\) −38.6258 −2.36830
\(267\) −10.2745 −0.628792
\(268\) −1.59439 −0.0973931
\(269\) 3.45090 0.210405 0.105203 0.994451i \(-0.466451\pi\)
0.105203 + 0.994451i \(0.466451\pi\)
\(270\) 1.23396 0.0750962
\(271\) 21.2167 1.28882 0.644412 0.764678i \(-0.277102\pi\)
0.644412 + 0.764678i \(0.277102\pi\)
\(272\) −18.3726 −1.11400
\(273\) −9.36031 −0.566512
\(274\) −9.79079 −0.591483
\(275\) 0 0
\(276\) −1.11704 −0.0672380
\(277\) 4.13159 0.248243 0.124122 0.992267i \(-0.460389\pi\)
0.124122 + 0.992267i \(0.460389\pi\)
\(278\) −7.62864 −0.457536
\(279\) 5.27455 0.315779
\(280\) −11.5944 −0.692898
\(281\) −23.0956 −1.37777 −0.688884 0.724871i \(-0.741899\pi\)
−0.688884 + 0.724871i \(0.741899\pi\)
\(282\) −11.8081 −0.703164
\(283\) 24.7593 1.47179 0.735893 0.677097i \(-0.236762\pi\)
0.735893 + 0.677097i \(0.236762\pi\)
\(284\) −3.45090 −0.204773
\(285\) 8.25310 0.488871
\(286\) 0 0
\(287\) −13.1387 −0.775551
\(288\) −2.63730 −0.155405
\(289\) 25.5240 1.50141
\(290\) 3.04530 0.178826
\(291\) −13.2495 −0.776699
\(292\) 4.64217 0.271662
\(293\) −7.33536 −0.428536 −0.214268 0.976775i \(-0.568737\pi\)
−0.214268 + 0.976775i \(0.568737\pi\)
\(294\) 9.11323 0.531494
\(295\) 10.1840 0.592933
\(296\) 10.2104 0.593469
\(297\) 0 0
\(298\) 2.23408 0.129417
\(299\) 5.77511 0.333983
\(300\) 0.477352 0.0275600
\(301\) 45.6878 2.63340
\(302\) −24.7268 −1.42287
\(303\) 1.99238 0.114459
\(304\) 23.2525 1.33362
\(305\) 6.66788 0.381801
\(306\) −8.04668 −0.459998
\(307\) −22.7669 −1.29938 −0.649688 0.760201i \(-0.725101\pi\)
−0.649688 + 0.760201i \(0.725101\pi\)
\(308\) 0 0
\(309\) 1.61463 0.0918529
\(310\) −6.50856 −0.369661
\(311\) −25.5443 −1.44848 −0.724241 0.689547i \(-0.757810\pi\)
−0.724241 + 0.689547i \(0.757810\pi\)
\(312\) 7.54427 0.427110
\(313\) −34.7735 −1.96552 −0.982758 0.184897i \(-0.940805\pi\)
−0.982758 + 0.184897i \(0.940805\pi\)
\(314\) 17.9779 1.01455
\(315\) −3.79281 −0.213700
\(316\) −4.60953 −0.259306
\(317\) −5.38537 −0.302473 −0.151236 0.988498i \(-0.548325\pi\)
−0.151236 + 0.988498i \(0.548325\pi\)
\(318\) −8.16216 −0.457711
\(319\) 0 0
\(320\) 8.88918 0.496920
\(321\) 0.589032 0.0328766
\(322\) −10.9519 −0.610327
\(323\) −53.8188 −2.99456
\(324\) −0.477352 −0.0265196
\(325\) −2.46791 −0.136895
\(326\) 7.26699 0.402481
\(327\) 4.78899 0.264832
\(328\) 10.5896 0.584711
\(329\) 36.2946 2.00099
\(330\) 0 0
\(331\) 19.8641 1.09183 0.545915 0.837840i \(-0.316182\pi\)
0.545915 + 0.837840i \(0.316182\pi\)
\(332\) 5.27461 0.289482
\(333\) 3.34008 0.183035
\(334\) −28.1902 −1.54250
\(335\) 3.34008 0.182488
\(336\) −10.6860 −0.582967
\(337\) −10.9029 −0.593918 −0.296959 0.954890i \(-0.595972\pi\)
−0.296959 + 0.954890i \(0.595972\pi\)
\(338\) 8.52591 0.463748
\(339\) −18.5240 −1.00609
\(340\) −3.11284 −0.168817
\(341\) 0 0
\(342\) 10.1840 0.550685
\(343\) −1.46164 −0.0789214
\(344\) −36.8236 −1.98540
\(345\) 2.34008 0.125986
\(346\) −15.4509 −0.830646
\(347\) 5.00420 0.268640 0.134320 0.990938i \(-0.457115\pi\)
0.134320 + 0.990938i \(0.457115\pi\)
\(348\) −1.17806 −0.0631508
\(349\) −26.7749 −1.43323 −0.716614 0.697470i \(-0.754309\pi\)
−0.716614 + 0.697470i \(0.754309\pi\)
\(350\) 4.68016 0.250165
\(351\) 2.46791 0.131727
\(352\) 0 0
\(353\) 17.2042 0.915687 0.457843 0.889033i \(-0.348622\pi\)
0.457843 + 0.889033i \(0.348622\pi\)
\(354\) 12.5666 0.667905
\(355\) 7.22925 0.383689
\(356\) −4.90458 −0.259942
\(357\) 24.7331 1.30901
\(358\) 3.75776 0.198604
\(359\) −0.657408 −0.0346966 −0.0173483 0.999850i \(-0.505522\pi\)
−0.0173483 + 0.999850i \(0.505522\pi\)
\(360\) 3.05694 0.161115
\(361\) 49.1136 2.58493
\(362\) −5.41136 −0.284415
\(363\) 0 0
\(364\) −4.46817 −0.234196
\(365\) −9.72482 −0.509020
\(366\) 8.22787 0.430077
\(367\) −33.5038 −1.74888 −0.874442 0.485130i \(-0.838772\pi\)
−0.874442 + 0.485130i \(0.838772\pi\)
\(368\) 6.59301 0.343684
\(369\) 3.46410 0.180334
\(370\) −4.12151 −0.214267
\(371\) 25.0880 1.30250
\(372\) 2.51782 0.130543
\(373\) 7.73244 0.400371 0.200185 0.979758i \(-0.435846\pi\)
0.200185 + 0.979758i \(0.435846\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −29.2529 −1.50860
\(377\) 6.09059 0.313681
\(378\) −4.68016 −0.240721
\(379\) −0.340078 −0.0174686 −0.00873431 0.999962i \(-0.502780\pi\)
−0.00873431 + 0.999962i \(0.502780\pi\)
\(380\) 3.93963 0.202099
\(381\) −12.1927 −0.624653
\(382\) −28.8909 −1.47819
\(383\) 31.0481 1.58648 0.793241 0.608908i \(-0.208392\pi\)
0.793241 + 0.608908i \(0.208392\pi\)
\(384\) 5.69425 0.290583
\(385\) 0 0
\(386\) −28.6802 −1.45978
\(387\) −12.0459 −0.612328
\(388\) −6.32467 −0.321087
\(389\) −24.3150 −1.23282 −0.616410 0.787425i \(-0.711414\pi\)
−0.616410 + 0.787425i \(0.711414\pi\)
\(390\) −3.04530 −0.154205
\(391\) −15.2598 −0.771719
\(392\) 22.5767 1.14029
\(393\) −3.46410 −0.174741
\(394\) 2.45851 0.123858
\(395\) 9.65644 0.485868
\(396\) 0 0
\(397\) 10.2948 0.516680 0.258340 0.966054i \(-0.416824\pi\)
0.258340 + 0.966054i \(0.416824\pi\)
\(398\) −10.7668 −0.539692
\(399\) −31.3024 −1.56708
\(400\) −2.81743 −0.140872
\(401\) −19.2293 −0.960263 −0.480132 0.877196i \(-0.659411\pi\)
−0.480132 + 0.877196i \(0.659411\pi\)
\(402\) 4.12151 0.205562
\(403\) −13.0171 −0.648429
\(404\) 0.951067 0.0473173
\(405\) 1.00000 0.0496904
\(406\) −11.5502 −0.573227
\(407\) 0 0
\(408\) −19.9345 −0.986903
\(409\) −6.53112 −0.322943 −0.161472 0.986877i \(-0.551624\pi\)
−0.161472 + 0.986877i \(0.551624\pi\)
\(410\) −4.27455 −0.211105
\(411\) −7.93447 −0.391379
\(412\) 0.770746 0.0379719
\(413\) −38.6258 −1.90065
\(414\) 2.88755 0.141915
\(415\) −11.0497 −0.542409
\(416\) 6.50863 0.319112
\(417\) −6.18226 −0.302747
\(418\) 0 0
\(419\) 27.6878 1.35264 0.676318 0.736610i \(-0.263575\pi\)
0.676318 + 0.736610i \(0.263575\pi\)
\(420\) −1.81050 −0.0883436
\(421\) −11.1637 −0.544087 −0.272043 0.962285i \(-0.587699\pi\)
−0.272043 + 0.962285i \(0.587699\pi\)
\(422\) −19.8223 −0.964932
\(423\) −9.56933 −0.465277
\(424\) −20.2205 −0.981996
\(425\) 6.52105 0.316317
\(426\) 8.92058 0.432204
\(427\) −25.2900 −1.22387
\(428\) 0.281176 0.0135911
\(429\) 0 0
\(430\) 14.8641 0.716811
\(431\) −13.3607 −0.643563 −0.321782 0.946814i \(-0.604282\pi\)
−0.321782 + 0.946814i \(0.604282\pi\)
\(432\) 2.81743 0.135554
\(433\) 19.2495 0.925071 0.462536 0.886601i \(-0.346940\pi\)
0.462536 + 0.886601i \(0.346940\pi\)
\(434\) 24.6857 1.18495
\(435\) 2.46791 0.118327
\(436\) 2.28604 0.109481
\(437\) 19.3129 0.923861
\(438\) −12.0000 −0.573382
\(439\) −21.6572 −1.03364 −0.516821 0.856093i \(-0.672885\pi\)
−0.516821 + 0.856093i \(0.672885\pi\)
\(440\) 0 0
\(441\) 7.38537 0.351684
\(442\) 19.8585 0.944573
\(443\) 1.13866 0.0540995 0.0270498 0.999634i \(-0.491389\pi\)
0.0270498 + 0.999634i \(0.491389\pi\)
\(444\) 1.59439 0.0756666
\(445\) 10.2745 0.487060
\(446\) 13.6544 0.646553
\(447\) 1.81050 0.0856339
\(448\) −33.7149 −1.59288
\(449\) −0.0905906 −0.00427524 −0.00213762 0.999998i \(-0.500680\pi\)
−0.00213762 + 0.999998i \(0.500680\pi\)
\(450\) −1.23396 −0.0581692
\(451\) 0 0
\(452\) −8.84249 −0.415916
\(453\) −20.0387 −0.941499
\(454\) −18.9986 −0.891649
\(455\) 9.36031 0.438818
\(456\) 25.2293 1.18147
\(457\) −4.12151 −0.192796 −0.0963980 0.995343i \(-0.530732\pi\)
−0.0963980 + 0.995343i \(0.530732\pi\)
\(458\) −5.80603 −0.271298
\(459\) −6.52105 −0.304376
\(460\) 1.11704 0.0520823
\(461\) 6.74633 0.314208 0.157104 0.987582i \(-0.449784\pi\)
0.157104 + 0.987582i \(0.449784\pi\)
\(462\) 0 0
\(463\) −20.4990 −0.952668 −0.476334 0.879264i \(-0.658035\pi\)
−0.476334 + 0.879264i \(0.658035\pi\)
\(464\) 6.95317 0.322793
\(465\) −5.27455 −0.244601
\(466\) −18.4118 −0.852911
\(467\) −4.79859 −0.222052 −0.111026 0.993817i \(-0.535414\pi\)
−0.111026 + 0.993817i \(0.535414\pi\)
\(468\) 1.17806 0.0544560
\(469\) −12.6683 −0.584966
\(470\) 11.8081 0.544669
\(471\) 14.5693 0.671319
\(472\) 31.1318 1.43296
\(473\) 0 0
\(474\) 11.9156 0.547303
\(475\) −8.25310 −0.378678
\(476\) 11.8064 0.541145
\(477\) −6.61463 −0.302863
\(478\) 14.0529 0.642765
\(479\) 9.39612 0.429319 0.214660 0.976689i \(-0.431136\pi\)
0.214660 + 0.976689i \(0.431136\pi\)
\(480\) 2.63730 0.120376
\(481\) −8.24302 −0.375849
\(482\) −14.9485 −0.680885
\(483\) −8.87546 −0.403847
\(484\) 0 0
\(485\) 13.2495 0.601628
\(486\) 1.23396 0.0559734
\(487\) −24.0934 −1.09177 −0.545887 0.837859i \(-0.683807\pi\)
−0.545887 + 0.837859i \(0.683807\pi\)
\(488\) 20.3833 0.922710
\(489\) 5.88918 0.266318
\(490\) −9.11323 −0.411694
\(491\) 6.63454 0.299413 0.149706 0.988730i \(-0.452167\pi\)
0.149706 + 0.988730i \(0.452167\pi\)
\(492\) 1.65360 0.0745499
\(493\) −16.0934 −0.724809
\(494\) −25.1331 −1.13079
\(495\) 0 0
\(496\) −14.8607 −0.667264
\(497\) −27.4192 −1.22992
\(498\) −13.6349 −0.610993
\(499\) 22.1637 0.992185 0.496092 0.868270i \(-0.334768\pi\)
0.496092 + 0.868270i \(0.334768\pi\)
\(500\) −0.477352 −0.0213478
\(501\) −22.8454 −1.02066
\(502\) 21.4871 0.959018
\(503\) −37.8597 −1.68808 −0.844040 0.536281i \(-0.819829\pi\)
−0.844040 + 0.536281i \(0.819829\pi\)
\(504\) −11.5944 −0.516455
\(505\) −1.99238 −0.0886597
\(506\) 0 0
\(507\) 6.90941 0.306858
\(508\) −5.82023 −0.258231
\(509\) −0.405606 −0.0179782 −0.00898909 0.999960i \(-0.502861\pi\)
−0.00898909 + 0.999960i \(0.502861\pi\)
\(510\) 8.04668 0.356313
\(511\) 36.8843 1.63167
\(512\) 24.6559 1.08965
\(513\) 8.25310 0.364383
\(514\) 22.7461 1.00329
\(515\) −1.61463 −0.0711490
\(516\) −5.75014 −0.253136
\(517\) 0 0
\(518\) 15.6321 0.686834
\(519\) −12.5214 −0.549630
\(520\) −7.54427 −0.330838
\(521\) 22.9952 1.00744 0.503718 0.863868i \(-0.331965\pi\)
0.503718 + 0.863868i \(0.331965\pi\)
\(522\) 3.04530 0.133289
\(523\) 1.14302 0.0499807 0.0249904 0.999688i \(-0.492044\pi\)
0.0249904 + 0.999688i \(0.492044\pi\)
\(524\) −1.65360 −0.0722377
\(525\) 3.79281 0.165532
\(526\) 2.54288 0.110875
\(527\) 34.3956 1.49829
\(528\) 0 0
\(529\) −17.5240 −0.761915
\(530\) 8.16216 0.354542
\(531\) 10.1840 0.441946
\(532\) −14.9423 −0.647830
\(533\) −8.54910 −0.370303
\(534\) 12.6783 0.548646
\(535\) −0.589032 −0.0254661
\(536\) 10.2104 0.441023
\(537\) 3.04530 0.131414
\(538\) −4.25826 −0.183587
\(539\) 0 0
\(540\) 0.477352 0.0205420
\(541\) −8.92058 −0.383526 −0.191763 0.981441i \(-0.561420\pi\)
−0.191763 + 0.981441i \(0.561420\pi\)
\(542\) −26.1805 −1.12455
\(543\) −4.38537 −0.188194
\(544\) −17.1980 −0.737357
\(545\) −4.78899 −0.205138
\(546\) 11.5502 0.494303
\(547\) −8.73871 −0.373640 −0.186820 0.982394i \(-0.559818\pi\)
−0.186820 + 0.982394i \(0.559818\pi\)
\(548\) −3.78754 −0.161796
\(549\) 6.66788 0.284578
\(550\) 0 0
\(551\) 20.3679 0.867702
\(552\) 7.15349 0.304473
\(553\) −36.6250 −1.55745
\(554\) −5.09820 −0.216602
\(555\) −3.34008 −0.141778
\(556\) −2.95112 −0.125155
\(557\) −16.6197 −0.704199 −0.352099 0.935963i \(-0.614532\pi\)
−0.352099 + 0.935963i \(0.614532\pi\)
\(558\) −6.50856 −0.275529
\(559\) 29.7282 1.25737
\(560\) 10.6860 0.451564
\(561\) 0 0
\(562\) 28.4990 1.20216
\(563\) 41.7060 1.75770 0.878849 0.477101i \(-0.158312\pi\)
0.878849 + 0.477101i \(0.158312\pi\)
\(564\) −4.56794 −0.192345
\(565\) 18.5240 0.779312
\(566\) −30.5519 −1.28419
\(567\) −3.79281 −0.159283
\(568\) 22.0994 0.927271
\(569\) 7.76749 0.325630 0.162815 0.986657i \(-0.447943\pi\)
0.162815 + 0.986657i \(0.447943\pi\)
\(570\) −10.1840 −0.426559
\(571\) 26.9568 1.12811 0.564053 0.825738i \(-0.309241\pi\)
0.564053 + 0.825738i \(0.309241\pi\)
\(572\) 0 0
\(573\) −23.4132 −0.978101
\(574\) 16.2125 0.676698
\(575\) −2.34008 −0.0975880
\(576\) 8.88918 0.370382
\(577\) 26.9345 1.12130 0.560648 0.828054i \(-0.310552\pi\)
0.560648 + 0.828054i \(0.310552\pi\)
\(578\) −31.4955 −1.31004
\(579\) −23.2424 −0.965923
\(580\) 1.17806 0.0489164
\(581\) 41.9094 1.73870
\(582\) 16.3493 0.677700
\(583\) 0 0
\(584\) −29.7282 −1.23016
\(585\) −2.46791 −0.102036
\(586\) 9.05151 0.373915
\(587\) −13.4383 −0.554657 −0.277328 0.960775i \(-0.589449\pi\)
−0.277328 + 0.960775i \(0.589449\pi\)
\(588\) 3.52543 0.145386
\(589\) −43.5314 −1.79368
\(590\) −12.5666 −0.517357
\(591\) 1.99238 0.0819555
\(592\) −9.41044 −0.386767
\(593\) 23.4344 0.962335 0.481168 0.876629i \(-0.340213\pi\)
0.481168 + 0.876629i \(0.340213\pi\)
\(594\) 0 0
\(595\) −24.7331 −1.01396
\(596\) 0.864249 0.0354010
\(597\) −8.72545 −0.357109
\(598\) −7.12623 −0.291413
\(599\) −40.0028 −1.63447 −0.817235 0.576305i \(-0.804494\pi\)
−0.817235 + 0.576305i \(0.804494\pi\)
\(600\) −3.05694 −0.124799
\(601\) −15.3181 −0.624836 −0.312418 0.949945i \(-0.601139\pi\)
−0.312418 + 0.949945i \(0.601139\pi\)
\(602\) −56.3767 −2.29774
\(603\) 3.34008 0.136018
\(604\) −9.56551 −0.389215
\(605\) 0 0
\(606\) −2.45851 −0.0998701
\(607\) 34.0086 1.38037 0.690183 0.723635i \(-0.257530\pi\)
0.690183 + 0.723635i \(0.257530\pi\)
\(608\) 21.7659 0.882724
\(609\) −9.36031 −0.379299
\(610\) −8.22787 −0.333137
\(611\) 23.6163 0.955412
\(612\) −3.11284 −0.125829
\(613\) 40.0874 1.61912 0.809558 0.587040i \(-0.199707\pi\)
0.809558 + 0.587040i \(0.199707\pi\)
\(614\) 28.0934 1.13376
\(615\) −3.46410 −0.139686
\(616\) 0 0
\(617\) 3.36031 0.135281 0.0676405 0.997710i \(-0.478453\pi\)
0.0676405 + 0.997710i \(0.478453\pi\)
\(618\) −1.99238 −0.0801452
\(619\) 28.5491 1.14749 0.573743 0.819036i \(-0.305491\pi\)
0.573743 + 0.819036i \(0.305491\pi\)
\(620\) −2.51782 −0.101118
\(621\) 2.34008 0.0939041
\(622\) 31.5205 1.26386
\(623\) −38.9694 −1.56127
\(624\) −6.95317 −0.278350
\(625\) 1.00000 0.0400000
\(626\) 42.9090 1.71499
\(627\) 0 0
\(628\) 6.95470 0.277523
\(629\) 21.7808 0.868457
\(630\) 4.68016 0.186462
\(631\) −18.7986 −0.748360 −0.374180 0.927356i \(-0.622076\pi\)
−0.374180 + 0.927356i \(0.622076\pi\)
\(632\) 29.5192 1.17421
\(633\) −16.0640 −0.638486
\(634\) 6.64531 0.263919
\(635\) 12.1927 0.483854
\(636\) −3.15751 −0.125203
\(637\) −18.2265 −0.722158
\(638\) 0 0
\(639\) 7.22925 0.285985
\(640\) −5.69425 −0.225085
\(641\) 45.1010 1.78138 0.890691 0.454610i \(-0.150221\pi\)
0.890691 + 0.454610i \(0.150221\pi\)
\(642\) −0.726839 −0.0286861
\(643\) 30.8439 1.21636 0.608182 0.793798i \(-0.291899\pi\)
0.608182 + 0.793798i \(0.291899\pi\)
\(644\) −4.23672 −0.166950
\(645\) 12.0459 0.474307
\(646\) 66.4101 2.61287
\(647\) 31.3477 1.23240 0.616202 0.787588i \(-0.288670\pi\)
0.616202 + 0.787588i \(0.288670\pi\)
\(648\) 3.05694 0.120088
\(649\) 0 0
\(650\) 3.04530 0.119446
\(651\) 20.0053 0.784071
\(652\) 2.81121 0.110096
\(653\) 32.4585 1.27020 0.635100 0.772430i \(-0.280959\pi\)
0.635100 + 0.772430i \(0.280959\pi\)
\(654\) −5.90941 −0.231076
\(655\) 3.46410 0.135354
\(656\) −9.75986 −0.381059
\(657\) −9.72482 −0.379401
\(658\) −44.7860 −1.74594
\(659\) −21.7808 −0.848459 −0.424230 0.905555i \(-0.639455\pi\)
−0.424230 + 0.905555i \(0.639455\pi\)
\(660\) 0 0
\(661\) 3.83627 0.149214 0.0746069 0.997213i \(-0.476230\pi\)
0.0746069 + 0.997213i \(0.476230\pi\)
\(662\) −24.5114 −0.952664
\(663\) 16.0934 0.625015
\(664\) −33.7784 −1.31085
\(665\) 31.3024 1.21385
\(666\) −4.12151 −0.159705
\(667\) 5.77511 0.223613
\(668\) −10.9053 −0.421938
\(669\) 11.0655 0.427818
\(670\) −4.12151 −0.159228
\(671\) 0 0
\(672\) −10.0028 −0.385866
\(673\) −24.8711 −0.958709 −0.479355 0.877621i \(-0.659129\pi\)
−0.479355 + 0.877621i \(0.659129\pi\)
\(674\) 13.4537 0.518216
\(675\) −1.00000 −0.0384900
\(676\) 3.29822 0.126855
\(677\) 14.2202 0.546525 0.273262 0.961940i \(-0.411897\pi\)
0.273262 + 0.961940i \(0.411897\pi\)
\(678\) 22.8578 0.877850
\(679\) −50.2527 −1.92852
\(680\) 19.9345 0.764452
\(681\) −15.3965 −0.589995
\(682\) 0 0
\(683\) −9.68776 −0.370692 −0.185346 0.982673i \(-0.559341\pi\)
−0.185346 + 0.982673i \(0.559341\pi\)
\(684\) 3.93963 0.150636
\(685\) 7.93447 0.303161
\(686\) 1.80361 0.0688620
\(687\) −4.70522 −0.179515
\(688\) 33.9385 1.29389
\(689\) 16.3243 0.621907
\(690\) −2.88755 −0.109927
\(691\) −0.594394 −0.0226118 −0.0113059 0.999936i \(-0.503599\pi\)
−0.0113059 + 0.999936i \(0.503599\pi\)
\(692\) −5.97714 −0.227217
\(693\) 0 0
\(694\) −6.17496 −0.234398
\(695\) 6.18226 0.234507
\(696\) 7.54427 0.285965
\(697\) 22.5896 0.855641
\(698\) 33.0391 1.25055
\(699\) −14.9210 −0.564363
\(700\) 1.81050 0.0684306
\(701\) −18.1598 −0.685886 −0.342943 0.939356i \(-0.611424\pi\)
−0.342943 + 0.939356i \(0.611424\pi\)
\(702\) −3.04530 −0.114937
\(703\) −27.5660 −1.03967
\(704\) 0 0
\(705\) 9.56933 0.360402
\(706\) −21.2292 −0.798972
\(707\) 7.55671 0.284199
\(708\) 4.86134 0.182700
\(709\) −1.80341 −0.0677287 −0.0338643 0.999426i \(-0.510781\pi\)
−0.0338643 + 0.999426i \(0.510781\pi\)
\(710\) −8.92058 −0.334783
\(711\) 9.65644 0.362145
\(712\) 31.4087 1.17709
\(713\) −12.3429 −0.462244
\(714\) −30.5195 −1.14216
\(715\) 0 0
\(716\) 1.45368 0.0543265
\(717\) 11.3885 0.425311
\(718\) 0.811212 0.0302742
\(719\) −37.0481 −1.38166 −0.690830 0.723017i \(-0.742755\pi\)
−0.690830 + 0.723017i \(0.742755\pi\)
\(720\) −2.81743 −0.104999
\(721\) 6.12397 0.228068
\(722\) −60.6040 −2.25545
\(723\) −12.1143 −0.450535
\(724\) −2.09337 −0.0777994
\(725\) −2.46791 −0.0916560
\(726\) 0 0
\(727\) 23.3198 0.864885 0.432443 0.901661i \(-0.357652\pi\)
0.432443 + 0.901661i \(0.357652\pi\)
\(728\) 28.6139 1.06050
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −78.5519 −2.90535
\(732\) 3.18293 0.117644
\(733\) 15.9855 0.590439 0.295220 0.955429i \(-0.404607\pi\)
0.295220 + 0.955429i \(0.404607\pi\)
\(734\) 41.3422 1.52597
\(735\) −7.38537 −0.272414
\(736\) 6.17149 0.227484
\(737\) 0 0
\(738\) −4.27455 −0.157348
\(739\) 32.9339 1.21149 0.605747 0.795657i \(-0.292874\pi\)
0.605747 + 0.795657i \(0.292874\pi\)
\(740\) −1.59439 −0.0586111
\(741\) −20.3679 −0.748234
\(742\) −30.9575 −1.13648
\(743\) −23.2758 −0.853905 −0.426953 0.904274i \(-0.640413\pi\)
−0.426953 + 0.904274i \(0.640413\pi\)
\(744\) −16.1240 −0.591135
\(745\) −1.81050 −0.0663318
\(746\) −9.54149 −0.349339
\(747\) −11.0497 −0.404288
\(748\) 0 0
\(749\) 2.23408 0.0816316
\(750\) 1.23396 0.0450577
\(751\) −9.18396 −0.335127 −0.167564 0.985861i \(-0.553590\pi\)
−0.167564 + 0.985861i \(0.553590\pi\)
\(752\) 26.9609 0.983164
\(753\) 17.4132 0.634573
\(754\) −7.51552 −0.273699
\(755\) 20.0387 0.729282
\(756\) −1.81050 −0.0658474
\(757\) −47.2523 −1.71741 −0.858706 0.512468i \(-0.828731\pi\)
−0.858706 + 0.512468i \(0.828731\pi\)
\(758\) 0.419641 0.0152420
\(759\) 0 0
\(760\) −25.2293 −0.915161
\(761\) −49.9941 −1.81229 −0.906143 0.422972i \(-0.860987\pi\)
−0.906143 + 0.422972i \(0.860987\pi\)
\(762\) 15.0453 0.545034
\(763\) 18.1637 0.657571
\(764\) −11.1764 −0.404346
\(765\) 6.52105 0.235769
\(766\) −38.3120 −1.38427
\(767\) −25.1331 −0.907504
\(768\) 10.7519 0.387976
\(769\) 4.53874 0.163671 0.0818357 0.996646i \(-0.473922\pi\)
0.0818357 + 0.996646i \(0.473922\pi\)
\(770\) 0 0
\(771\) 18.4334 0.663864
\(772\) −11.0948 −0.399312
\(773\) −31.3352 −1.12705 −0.563525 0.826099i \(-0.690555\pi\)
−0.563525 + 0.826099i \(0.690555\pi\)
\(774\) 14.8641 0.534280
\(775\) 5.27455 0.189467
\(776\) 40.5029 1.45397
\(777\) 12.6683 0.454471
\(778\) 30.0037 1.07568
\(779\) −28.5896 −1.02433
\(780\) −1.17806 −0.0421814
\(781\) 0 0
\(782\) 18.8299 0.673355
\(783\) 2.46791 0.0881960
\(784\) −20.8078 −0.743135
\(785\) −14.5693 −0.520002
\(786\) 4.27455 0.152468
\(787\) 5.59323 0.199377 0.0996886 0.995019i \(-0.468215\pi\)
0.0996886 + 0.995019i \(0.468215\pi\)
\(788\) 0.951067 0.0338803
\(789\) 2.06075 0.0733648
\(790\) −11.9156 −0.423939
\(791\) −70.2581 −2.49809
\(792\) 0 0
\(793\) −16.4557 −0.584360
\(794\) −12.7033 −0.450824
\(795\) 6.61463 0.234597
\(796\) −4.16511 −0.147629
\(797\) −32.4585 −1.14974 −0.574870 0.818245i \(-0.694947\pi\)
−0.574870 + 0.818245i \(0.694947\pi\)
\(798\) 38.6258 1.36734
\(799\) −62.4020 −2.20763
\(800\) −2.63730 −0.0932427
\(801\) 10.2745 0.363033
\(802\) 23.7281 0.837867
\(803\) 0 0
\(804\) 1.59439 0.0562299
\(805\) 8.87546 0.312819
\(806\) 16.0626 0.565780
\(807\) −3.45090 −0.121477
\(808\) −6.09059 −0.214266
\(809\) 41.9128 1.47358 0.736788 0.676124i \(-0.236342\pi\)
0.736788 + 0.676124i \(0.236342\pi\)
\(810\) −1.23396 −0.0433568
\(811\) 28.4285 0.998260 0.499130 0.866527i \(-0.333653\pi\)
0.499130 + 0.866527i \(0.333653\pi\)
\(812\) −4.46817 −0.156802
\(813\) −21.2167 −0.744103
\(814\) 0 0
\(815\) −5.88918 −0.206289
\(816\) 18.3726 0.643169
\(817\) 99.4160 3.47813
\(818\) 8.05912 0.281781
\(819\) 9.36031 0.327076
\(820\) −1.65360 −0.0577461
\(821\) 53.6401 1.87205 0.936026 0.351931i \(-0.114475\pi\)
0.936026 + 0.351931i \(0.114475\pi\)
\(822\) 9.79079 0.341493
\(823\) −11.1561 −0.388878 −0.194439 0.980915i \(-0.562289\pi\)
−0.194439 + 0.980915i \(0.562289\pi\)
\(824\) −4.93582 −0.171948
\(825\) 0 0
\(826\) 47.6625 1.65839
\(827\) 27.8496 0.968424 0.484212 0.874951i \(-0.339106\pi\)
0.484212 + 0.874951i \(0.339106\pi\)
\(828\) 1.11704 0.0388199
\(829\) 39.4084 1.36871 0.684355 0.729149i \(-0.260084\pi\)
0.684355 + 0.729149i \(0.260084\pi\)
\(830\) 13.6349 0.473273
\(831\) −4.13159 −0.143323
\(832\) −21.9377 −0.760553
\(833\) 48.1604 1.66866
\(834\) 7.62864 0.264158
\(835\) 22.8454 0.790596
\(836\) 0 0
\(837\) −5.27455 −0.182315
\(838\) −34.1655 −1.18023
\(839\) 35.8188 1.23660 0.618301 0.785941i \(-0.287821\pi\)
0.618301 + 0.785941i \(0.287821\pi\)
\(840\) 11.5944 0.400045
\(841\) −22.9094 −0.789980
\(842\) 13.7755 0.474737
\(843\) 23.0956 0.795455
\(844\) −7.66818 −0.263950
\(845\) −6.90941 −0.237691
\(846\) 11.8081 0.405972
\(847\) 0 0
\(848\) 18.6362 0.639971
\(849\) −24.7593 −0.849737
\(850\) −8.04668 −0.275999
\(851\) −7.81604 −0.267930
\(852\) 3.45090 0.118226
\(853\) 27.4091 0.938469 0.469234 0.883074i \(-0.344530\pi\)
0.469234 + 0.883074i \(0.344530\pi\)
\(854\) 31.2067 1.06787
\(855\) −8.25310 −0.282250
\(856\) −1.80064 −0.0615445
\(857\) −9.82824 −0.335726 −0.167863 0.985810i \(-0.553687\pi\)
−0.167863 + 0.985810i \(0.553687\pi\)
\(858\) 0 0
\(859\) −33.9825 −1.15947 −0.579735 0.814805i \(-0.696844\pi\)
−0.579735 + 0.814805i \(0.696844\pi\)
\(860\) 5.75014 0.196078
\(861\) 13.1387 0.447764
\(862\) 16.4865 0.561534
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 2.63730 0.0897229
\(865\) 12.5214 0.425742
\(866\) −23.7530 −0.807161
\(867\) −25.5240 −0.866842
\(868\) 9.54960 0.324134
\(869\) 0 0
\(870\) −3.04530 −0.103245
\(871\) −8.24302 −0.279304
\(872\) −14.6397 −0.495762
\(873\) 13.2495 0.448427
\(874\) −23.8313 −0.806104
\(875\) −3.79281 −0.128220
\(876\) −4.64217 −0.156844
\(877\) 42.8941 1.44843 0.724216 0.689574i \(-0.242202\pi\)
0.724216 + 0.689574i \(0.242202\pi\)
\(878\) 26.7241 0.901893
\(879\) 7.33536 0.247416
\(880\) 0 0
\(881\) −17.7282 −0.597279 −0.298640 0.954366i \(-0.596533\pi\)
−0.298640 + 0.954366i \(0.596533\pi\)
\(882\) −9.11323 −0.306858
\(883\) −0.201414 −0.00677813 −0.00338907 0.999994i \(-0.501079\pi\)
−0.00338907 + 0.999994i \(0.501079\pi\)
\(884\) 7.68221 0.258381
\(885\) −10.1840 −0.342330
\(886\) −1.40506 −0.0472039
\(887\) 43.0409 1.44517 0.722587 0.691280i \(-0.242953\pi\)
0.722587 + 0.691280i \(0.242953\pi\)
\(888\) −10.2104 −0.342640
\(889\) −46.2447 −1.55100
\(890\) −12.6783 −0.424979
\(891\) 0 0
\(892\) 5.28216 0.176860
\(893\) 78.9766 2.64285
\(894\) −2.23408 −0.0747189
\(895\) −3.04530 −0.101793
\(896\) 21.5972 0.721511
\(897\) −5.77511 −0.192825
\(898\) 0.111785 0.00373031
\(899\) −13.0171 −0.434145
\(900\) −0.477352 −0.0159117
\(901\) −43.1343 −1.43701
\(902\) 0 0
\(903\) −45.6878 −1.52039
\(904\) 56.6269 1.88338
\(905\) 4.38537 0.145775
\(906\) 24.7268 0.821494
\(907\) −0.204192 −0.00678008 −0.00339004 0.999994i \(-0.501079\pi\)
−0.00339004 + 0.999994i \(0.501079\pi\)
\(908\) −7.34956 −0.243904
\(909\) −1.99238 −0.0660830
\(910\) −11.5502 −0.382886
\(911\) −32.9547 −1.09184 −0.545919 0.837838i \(-0.683819\pi\)
−0.545919 + 0.837838i \(0.683819\pi\)
\(912\) −23.2525 −0.769968
\(913\) 0 0
\(914\) 5.08576 0.168222
\(915\) −6.66788 −0.220433
\(916\) −2.24605 −0.0742115
\(917\) −13.1387 −0.433877
\(918\) 8.04668 0.265580
\(919\) 50.8435 1.67717 0.838586 0.544770i \(-0.183383\pi\)
0.838586 + 0.544770i \(0.183383\pi\)
\(920\) −7.15349 −0.235843
\(921\) 22.7669 0.750195
\(922\) −8.32467 −0.274159
\(923\) −17.8412 −0.587249
\(924\) 0 0
\(925\) 3.34008 0.109821
\(926\) 25.2948 0.831240
\(927\) −1.61463 −0.0530313
\(928\) 6.50863 0.213656
\(929\) 48.5394 1.59253 0.796264 0.604950i \(-0.206807\pi\)
0.796264 + 0.604950i \(0.206807\pi\)
\(930\) 6.50856 0.213424
\(931\) −60.9522 −1.99763
\(932\) −7.12256 −0.233307
\(933\) 25.5443 0.836282
\(934\) 5.92124 0.193749
\(935\) 0 0
\(936\) −7.54427 −0.246592
\(937\) −10.3573 −0.338357 −0.169178 0.985585i \(-0.554111\pi\)
−0.169178 + 0.985585i \(0.554111\pi\)
\(938\) 15.6321 0.510406
\(939\) 34.7735 1.13479
\(940\) 4.56794 0.148990
\(941\) −48.9528 −1.59582 −0.797908 0.602779i \(-0.794060\pi\)
−0.797908 + 0.602779i \(0.794060\pi\)
\(942\) −17.9779 −0.585752
\(943\) −8.10627 −0.263976
\(944\) −28.6926 −0.933864
\(945\) 3.79281 0.123380
\(946\) 0 0
\(947\) −20.5213 −0.666851 −0.333426 0.942776i \(-0.608205\pi\)
−0.333426 + 0.942776i \(0.608205\pi\)
\(948\) 4.60953 0.149710
\(949\) 24.0000 0.779073
\(950\) 10.1840 0.330411
\(951\) 5.38537 0.174633
\(952\) −75.6076 −2.45046
\(953\) −11.2066 −0.363018 −0.181509 0.983389i \(-0.558098\pi\)
−0.181509 + 0.983389i \(0.558098\pi\)
\(954\) 8.16216 0.264260
\(955\) 23.4132 0.757634
\(956\) 5.43632 0.175823
\(957\) 0 0
\(958\) −11.5944 −0.374598
\(959\) −30.0939 −0.971783
\(960\) −8.88918 −0.286897
\(961\) −3.17913 −0.102553
\(962\) 10.1715 0.327943
\(963\) −0.589032 −0.0189813
\(964\) −5.78278 −0.186251
\(965\) 23.2424 0.748201
\(966\) 10.9519 0.352372
\(967\) −5.76022 −0.185236 −0.0926181 0.995702i \(-0.529524\pi\)
−0.0926181 + 0.995702i \(0.529524\pi\)
\(968\) 0 0
\(969\) 53.8188 1.72891
\(970\) −16.3493 −0.524944
\(971\) 14.9547 0.479919 0.239960 0.970783i \(-0.422866\pi\)
0.239960 + 0.970783i \(0.422866\pi\)
\(972\) 0.477352 0.0153111
\(973\) −23.4481 −0.751712
\(974\) 29.7302 0.952616
\(975\) 2.46791 0.0790364
\(976\) −18.7863 −0.601334
\(977\) 14.7457 0.471756 0.235878 0.971783i \(-0.424203\pi\)
0.235878 + 0.971783i \(0.424203\pi\)
\(978\) −7.26699 −0.232373
\(979\) 0 0
\(980\) −3.52543 −0.112616
\(981\) −4.78899 −0.152901
\(982\) −8.18674 −0.261249
\(983\) 34.8892 1.11279 0.556396 0.830917i \(-0.312184\pi\)
0.556396 + 0.830917i \(0.312184\pi\)
\(984\) −10.5896 −0.337583
\(985\) −1.99238 −0.0634825
\(986\) 19.8585 0.632424
\(987\) −36.2946 −1.15527
\(988\) −9.72267 −0.309319
\(989\) 28.1883 0.896337
\(990\) 0 0
\(991\) −15.5289 −0.493291 −0.246645 0.969106i \(-0.579328\pi\)
−0.246645 + 0.969106i \(0.579328\pi\)
\(992\) −13.9106 −0.441662
\(993\) −19.8641 −0.630369
\(994\) 33.8340 1.07315
\(995\) 8.72545 0.276615
\(996\) −5.27461 −0.167132
\(997\) 3.63590 0.115150 0.0575750 0.998341i \(-0.481663\pi\)
0.0575750 + 0.998341i \(0.481663\pi\)
\(998\) −27.3491 −0.865720
\(999\) −3.34008 −0.105675
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 1815.2.a.y.1.3 6
3.2 odd 2 5445.2.a.bz.1.4 6
5.4 even 2 9075.2.a.dq.1.4 6
11.10 odd 2 inner 1815.2.a.y.1.4 yes 6
33.32 even 2 5445.2.a.bz.1.3 6
55.54 odd 2 9075.2.a.dq.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.y.1.3 6 1.1 even 1 trivial
1815.2.a.y.1.4 yes 6 11.10 odd 2 inner
5445.2.a.bz.1.3 6 33.32 even 2
5445.2.a.bz.1.4 6 3.2 odd 2
9075.2.a.dq.1.3 6 55.54 odd 2
9075.2.a.dq.1.4 6 5.4 even 2