Properties

Label 1815.2.a.y.1.4
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 1.4
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 \(-0.356299\pi\)
0.436269 + 0.899816i \(0.356299\pi\)
\(3\) −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 \(-0.245618\pi\)
0.716773 + 0.697307i \(0.245618\pi\)
\(8\) −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 \(-0.388815\pi\)
0.342238 + 0.939613i \(0.388815\pi\)
\(14\) 4.68016 1.25082
\(15\) −1.00000 −0.258199
\(16\) −2.81743 −0.704358
\(17\) −6.52105 −1.58159 −0.790793 0.612084i \(-0.790332\pi\)
−0.790793 + 0.612084i \(0.790332\pi\)
\(18\) 1.23396 0.290846
\(19\) 8.25310 1.89339 0.946695 0.322131i \(-0.104399\pi\)
0.946695 + 0.322131i \(0.104399\pi\)
\(20\) −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 \(-0.426409\pi\)
0.229140 + 0.973394i \(0.426409\pi\)
\(30\) −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 \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) −4.68016 −0.722164
\(43\) 12.0459 1.83698 0.918491 0.395441i \(-0.129408\pi\)
0.918491 + 0.395441i \(0.129408\pi\)
\(44\) 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 \(-0.640383\pi\)
−0.426867 + 0.904314i \(0.640383\pi\)
\(62\) 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 \(-0.307291\pi\)
0.569102 + 0.822267i \(0.307291\pi\)
\(74\) 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 \(-0.682794\pi\)
−0.543217 + 0.839592i \(0.682794\pi\)
\(80\) −2.81743 −0.314998
\(81\) 1.00000 0.111111
\(82\) −4.27455 −0.472045
\(83\) 11.0497 1.21286 0.606432 0.795136i \(-0.292600\pi\)
0.606432 + 0.795136i \(0.292600\pi\)
\(84\) 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 \(-0.468396\pi\)
0.0991246 + 0.995075i \(0.468396\pi\)
\(102\) 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 \(-0.490936\pi\)
0.0284719 + 0.999595i \(0.490936\pi\)
\(108\) 0.477352 0.0459333
\(109\) 4.78899 0.458703 0.229351 0.973344i \(-0.426340\pi\)
0.229351 + 0.973344i \(0.426340\pi\)
\(110\) 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 \(-0.681941\pi\)
−0.540965 + 0.841045i \(0.681941\pi\)
\(128\) 5.69425 0.503305
\(129\) −12.0459 −1.06058
\(130\) 3.04530 0.267090
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 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 \(-0.584444\pi\)
−0.262186 + 0.965017i \(0.584444\pi\)
\(140\) −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 \(-0.476372\pi\)
0.0741612 + 0.997246i \(0.476372\pi\)
\(150\) −1.23396 −0.100752
\(151\) −20.0387 −1.63072 −0.815362 0.578952i \(-0.803462\pi\)
−0.815362 + 0.578952i \(0.803462\pi\)
\(152\) −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 \(-0.845101\pi\)
−0.883914 + 0.467650i \(0.845101\pi\)
\(168\) 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 \(-0.657911\pi\)
−0.475994 + 0.879449i \(0.657911\pi\)
\(174\) −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 \(-0.815410\pi\)
−0.836514 + 0.547946i \(0.815410\pi\)
\(194\) 16.3493 1.17381
\(195\) −2.46791 −0.176731
\(196\) −3.52543 −0.251816
\(197\) 1.99238 0.141951 0.0709756 0.997478i \(-0.477389\pi\)
0.0709756 + 0.997478i \(0.477389\pi\)
\(198\) 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 \(-0.686496\pi\)
−0.552945 + 0.833218i \(0.686496\pi\)
\(212\) 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 \(-0.670707\pi\)
−0.510951 + 0.859610i \(0.670707\pi\)
\(228\) 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 \(-0.662548\pi\)
−0.488753 + 0.872422i \(0.662548\pi\)
\(234\) 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 \(-0.379930\pi\)
0.368330 + 0.929695i \(0.379930\pi\)
\(240\) 2.81743 0.181864
\(241\) −12.1143 −0.780349 −0.390175 0.920741i \(-0.627585\pi\)
−0.390175 + 0.920741i \(0.627585\pi\)
\(242\) 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 \(-0.479762\pi\)
0.0635358 + 0.997980i \(0.479762\pi\)
\(264\) 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 \(-0.722898\pi\)
−0.644412 + 0.764678i \(0.722898\pi\)
\(272\) 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 \(-0.539611\pi\)
−0.124122 + 0.992267i \(0.539611\pi\)
\(278\) −7.62864 −0.457536
\(279\) 5.27455 0.315779
\(280\) −11.5944 −0.692898
\(281\) 23.0956 1.37777 0.688884 0.724871i \(-0.258101\pi\)
0.688884 + 0.724871i \(0.258101\pi\)
\(282\) 11.8081 0.703164
\(283\) −24.7593 −1.47179 −0.735893 0.677097i \(-0.763238\pi\)
−0.735893 + 0.677097i \(0.763238\pi\)
\(284\) −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 \(-0.431263\pi\)
0.214268 + 0.976775i \(0.431263\pi\)
\(294\) −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 \(-0.274899\pi\)
0.649688 + 0.760201i \(0.274899\pi\)
\(308\) 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 \(-0.404028\pi\)
0.296959 + 0.954890i \(0.404028\pi\)
\(338\) −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 \(-0.542885\pi\)
−0.134320 + 0.990938i \(0.542885\pi\)
\(348\) 1.17806 0.0631508
\(349\) 26.7749 1.43323 0.716614 0.697470i \(-0.245691\pi\)
0.716614 + 0.697470i \(0.245691\pi\)
\(350\) 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 \(-0.494478\pi\)
0.0173483 + 0.999850i \(0.494478\pi\)
\(360\) −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 \(-0.564154\pi\)
−0.200185 + 0.979758i \(0.564154\pi\)
\(374\) 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 \(-0.448376\pi\)
0.161472 + 0.986877i \(0.448376\pi\)
\(410\) −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 \(-0.395718\pi\)
0.321782 + 0.946814i \(0.395718\pi\)
\(432\) 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 \(-0.327115\pi\)
0.516821 + 0.856093i \(0.327115\pi\)
\(440\) 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 \(-0.469268\pi\)
0.0963980 + 0.995343i \(0.469268\pi\)
\(458\) 5.80603 0.271298
\(459\) 6.52105 0.304376
\(460\) 1.11704 0.0520823
\(461\) −6.74633 −0.314208 −0.157104 0.987582i \(-0.550216\pi\)
−0.157104 + 0.987582i \(0.550216\pi\)
\(462\) 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 \(-0.568864\pi\)
−0.214660 + 0.976689i \(0.568864\pi\)
\(480\) −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 \(-0.547833\pi\)
−0.149706 + 0.988730i \(0.547833\pi\)
\(492\) −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 \(-0.180171\pi\)
0.844040 + 0.536281i \(0.180171\pi\)
\(504\) −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 \(-0.507956\pi\)
−0.0249904 + 0.999688i \(0.507956\pi\)
\(524\) 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 \(-0.438580\pi\)
0.191763 + 0.981441i \(0.438580\pi\)
\(542\) −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 \(-0.440182\pi\)
0.186820 + 0.982394i \(0.440182\pi\)
\(548\) −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 \(-0.385468\pi\)
0.352099 + 0.935963i \(0.385468\pi\)
\(558\) 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 \(-0.841688\pi\)
−0.878849 + 0.477101i \(0.841688\pi\)
\(564\) −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 \(-0.552057\pi\)
−0.162815 + 0.986657i \(0.552057\pi\)
\(570\) −10.1840 −0.426559
\(571\) −26.9568 −1.12811 −0.564053 0.825738i \(-0.690759\pi\)
−0.564053 + 0.825738i \(0.690759\pi\)
\(572\) 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 \(-0.659787\pi\)
−0.481168 + 0.876629i \(0.659787\pi\)
\(594\) 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 \(-0.398861\pi\)
0.312418 + 0.949945i \(0.398861\pi\)
\(602\) 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 \(-0.742470\pi\)
−0.690183 + 0.723635i \(0.742470\pi\)
\(608\) 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 \(-0.800293\pi\)
−0.809558 + 0.587040i \(0.800293\pi\)
\(614\) 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 \(-0.360545\pi\)
0.424230 + 0.905555i \(0.360545\pi\)
\(660\) 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 \(-0.340871\pi\)
0.479355 + 0.877621i \(0.340871\pi\)
\(674\) 13.4537 0.518216
\(675\) −1.00000 −0.0384900
\(676\) 3.29822 0.126855
\(677\) −14.2202 −0.546525 −0.273262 0.961940i \(-0.588103\pi\)
−0.273262 + 0.961940i \(0.588103\pi\)
\(678\) −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 \(-0.388576\pi\)
0.342943 + 0.939356i \(0.388576\pi\)
\(702\) −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 \(-0.595393\pi\)
−0.295220 + 0.955429i \(0.595393\pi\)
\(734\) −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 \(-0.707126\pi\)
−0.605747 + 0.795657i \(0.707126\pi\)
\(740\) −1.59439 −0.0586111
\(741\) −20.3679 −0.748234
\(742\) −30.9575 −1.13648
\(743\) 23.2758 0.853905 0.426953 0.904274i \(-0.359587\pi\)
0.426953 + 0.904274i \(0.359587\pi\)
\(744\) 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 \(-0.139013\pi\)
0.906143 + 0.422972i \(0.139013\pi\)
\(762\) 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 \(-0.526078\pi\)
−0.0818357 + 0.996646i \(0.526078\pi\)
\(770\) 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 \(-0.531785\pi\)
−0.0996886 + 0.995019i \(0.531785\pi\)
\(788\) −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 \(-0.763658\pi\)
−0.736788 + 0.676124i \(0.763658\pi\)
\(810\) 1.23396 0.0433568
\(811\) −28.4285 −0.998260 −0.499130 0.866527i \(-0.666347\pi\)
−0.499130 + 0.866527i \(0.666347\pi\)
\(812\) −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 \(-0.885525\pi\)
−0.936026 + 0.351931i \(0.885525\pi\)
\(822\) −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 \(-0.660894\pi\)
−0.484212 + 0.874951i \(0.660894\pi\)
\(828\) 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 \(-0.655470\pi\)
−0.469234 + 0.883074i \(0.655470\pi\)
\(854\) −31.2067 −1.06787
\(855\) 8.25310 0.282250
\(856\) −1.80064 −0.0615445
\(857\) 9.82824 0.335726 0.167863 0.985810i \(-0.446313\pi\)
0.167863 + 0.985810i \(0.446313\pi\)
\(858\) 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 \(-0.757798\pi\)
−0.724216 + 0.689574i \(0.757798\pi\)
\(878\) 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 \(-0.757047\pi\)
−0.722587 + 0.691280i \(0.757047\pi\)
\(888\) 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 \(-0.816617\pi\)
−0.838586 + 0.544770i \(0.816617\pi\)
\(920\) 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 \(-0.445889\pi\)
0.169178 + 0.985585i \(0.445889\pi\)
\(938\) 15.6321 0.510406
\(939\) 34.7735 1.13479
\(940\) 4.56794 0.148990
\(941\) 48.9528 1.59582 0.797908 0.602779i \(-0.205940\pi\)
0.797908 + 0.602779i \(0.205940\pi\)
\(942\) 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 \(-0.441902\pi\)
0.181509 + 0.983389i \(0.441902\pi\)
\(954\) −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 \(-0.470476\pi\)
0.0926181 + 0.995702i \(0.470476\pi\)
\(968\) 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 \(-0.518337\pi\)
−0.0575750 + 0.998341i \(0.518337\pi\)
\(998\) 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.1.4 yes 6
3.2 odd 2 5445.2.a.bz.1.3 6
5.4 even 2 9075.2.a.dq.1.3 6
11.10 odd 2 inner 1815.2.a.y.1.3 6
33.32 even 2 5445.2.a.bz.1.4 6
55.54 odd 2 9075.2.a.dq.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.y.1.3 6 11.10 odd 2 inner
1815.2.a.y.1.4 yes 6 1.1 even 1 trivial
5445.2.a.bz.1.3 6 3.2 odd 2
5445.2.a.bz.1.4 6 33.32 even 2
9075.2.a.dq.1.3 6 5.4 even 2
9075.2.a.dq.1.4 6 55.54 odd 2