Properties

Label 3450.2.d.ba.2899.6
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.181494784.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 12 x^{3} + 49 x^{2} - 14 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.6
Root \(2.38150 - 2.38150i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.ba.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.76300i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.76300i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.18291 q^{11} -1.00000i q^{12} -6.34310i q^{13} -3.76300 q^{14} +1.00000 q^{16} +2.58010i q^{17} -1.00000i q^{18} +6.34310 q^{19} -3.76300 q^{21} +3.18291i q^{22} -1.00000i q^{23} +1.00000 q^{24} +6.34310 q^{26} -1.00000i q^{27} -3.76300i q^{28} +1.00000 q^{29} +6.00000 q^{31} +1.00000i q^{32} +3.18291i q^{33} -2.58010 q^{34} +1.00000 q^{36} -2.58010i q^{37} +6.34310i q^{38} +6.34310 q^{39} +12.3431 q^{41} -3.76300i q^{42} +10.3431i q^{43} -3.18291 q^{44} +1.00000 q^{46} -3.34310i q^{47} +1.00000i q^{48} -7.16019 q^{49} -2.58010 q^{51} +6.34310i q^{52} +7.16019i q^{53} +1.00000 q^{54} +3.76300 q^{56} +6.34310i q^{57} +1.00000i q^{58} +4.36581 q^{59} -1.52601 q^{61} +6.00000i q^{62} -3.76300i q^{63} -1.00000 q^{64} -3.18291 q^{66} -10.6862i q^{67} -2.58010i q^{68} +1.00000 q^{69} +1.81709 q^{71} +1.00000i q^{72} +8.52601i q^{73} +2.58010 q^{74} -6.34310 q^{76} +11.9773i q^{77} +6.34310i q^{78} +3.18291 q^{79} +1.00000 q^{81} +12.3431i q^{82} -7.28901i q^{83} +3.76300 q^{84} -10.3431 q^{86} +1.00000i q^{87} -3.18291i q^{88} -4.58010 q^{89} +23.8691 q^{91} +1.00000i q^{92} +6.00000i q^{93} +3.34310 q^{94} -1.00000 q^{96} -11.5260i q^{97} -7.16019i q^{98} -3.18291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10}) \) \( 6q - 6q^{4} - 6q^{6} - 6q^{9} + 6q^{11} + 2q^{14} + 6q^{16} + 2q^{19} + 2q^{21} + 6q^{24} + 2q^{26} + 6q^{29} + 36q^{31} - 4q^{34} + 6q^{36} + 2q^{39} + 38q^{41} - 6q^{44} + 6q^{46} - 20q^{49} - 4q^{51} + 6q^{54} - 2q^{56} + 40q^{61} - 6q^{64} - 6q^{66} + 6q^{69} + 24q^{71} + 4q^{74} - 2q^{76} + 6q^{79} + 6q^{81} - 2q^{84} - 26q^{86} - 16q^{89} + 58q^{91} - 16q^{94} - 6q^{96} - 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 3.76300i 1.42228i 0.703050 + 0.711141i \(0.251821\pi\)
−0.703050 + 0.711141i \(0.748179\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.18291 0.959682 0.479841 0.877355i \(-0.340694\pi\)
0.479841 + 0.877355i \(0.340694\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.34310i − 1.75926i −0.475659 0.879630i \(-0.657790\pi\)
0.475659 0.879630i \(-0.342210\pi\)
\(14\) −3.76300 −1.00570
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.58010i 0.625765i 0.949792 + 0.312883i \(0.101295\pi\)
−0.949792 + 0.312883i \(0.898705\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 6.34310 1.45521 0.727603 0.685998i \(-0.240634\pi\)
0.727603 + 0.685998i \(0.240634\pi\)
\(20\) 0 0
\(21\) −3.76300 −0.821155
\(22\) 3.18291i 0.678598i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.34310 1.24398
\(27\) − 1.00000i − 0.192450i
\(28\) − 3.76300i − 0.711141i
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 3.18291i 0.554073i
\(34\) −2.58010 −0.442483
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 2.58010i − 0.424165i −0.977252 0.212083i \(-0.931975\pi\)
0.977252 0.212083i \(-0.0680246\pi\)
\(38\) 6.34310i 1.02899i
\(39\) 6.34310 1.01571
\(40\) 0 0
\(41\) 12.3431 1.92767 0.963834 0.266503i \(-0.0858681\pi\)
0.963834 + 0.266503i \(0.0858681\pi\)
\(42\) − 3.76300i − 0.580644i
\(43\) 10.3431i 1.57731i 0.614837 + 0.788654i \(0.289222\pi\)
−0.614837 + 0.788654i \(0.710778\pi\)
\(44\) −3.18291 −0.479841
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 3.34310i − 0.487641i −0.969820 0.243821i \(-0.921599\pi\)
0.969820 0.243821i \(-0.0784008\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −7.16019 −1.02288
\(50\) 0 0
\(51\) −2.58010 −0.361286
\(52\) 6.34310i 0.879630i
\(53\) 7.16019i 0.983528i 0.870728 + 0.491764i \(0.163648\pi\)
−0.870728 + 0.491764i \(0.836352\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.76300 0.502852
\(57\) 6.34310i 0.840164i
\(58\) 1.00000i 0.131306i
\(59\) 4.36581 0.568380 0.284190 0.958768i \(-0.408275\pi\)
0.284190 + 0.958768i \(0.408275\pi\)
\(60\) 0 0
\(61\) −1.52601 −0.195385 −0.0976926 0.995217i \(-0.531146\pi\)
−0.0976926 + 0.995217i \(0.531146\pi\)
\(62\) 6.00000i 0.762001i
\(63\) − 3.76300i − 0.474094i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.18291 −0.391789
\(67\) − 10.6862i − 1.30553i −0.757562 0.652764i \(-0.773610\pi\)
0.757562 0.652764i \(-0.226390\pi\)
\(68\) − 2.58010i − 0.312883i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.81709 0.215649 0.107825 0.994170i \(-0.465611\pi\)
0.107825 + 0.994170i \(0.465611\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 8.52601i 0.997894i 0.866633 + 0.498947i \(0.166280\pi\)
−0.866633 + 0.498947i \(0.833720\pi\)
\(74\) 2.58010 0.299930
\(75\) 0 0
\(76\) −6.34310 −0.727603
\(77\) 11.9773i 1.36494i
\(78\) 6.34310i 0.718215i
\(79\) 3.18291 0.358105 0.179052 0.983840i \(-0.442697\pi\)
0.179052 + 0.983840i \(0.442697\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.3431i 1.36307i
\(83\) − 7.28901i − 0.800073i −0.916499 0.400036i \(-0.868997\pi\)
0.916499 0.400036i \(-0.131003\pi\)
\(84\) 3.76300 0.410577
\(85\) 0 0
\(86\) −10.3431 −1.11533
\(87\) 1.00000i 0.107211i
\(88\) − 3.18291i − 0.339299i
\(89\) −4.58010 −0.485489 −0.242745 0.970090i \(-0.578048\pi\)
−0.242745 + 0.970090i \(0.578048\pi\)
\(90\) 0 0
\(91\) 23.8691 2.50216
\(92\) 1.00000i 0.104257i
\(93\) 6.00000i 0.622171i
\(94\) 3.34310 0.344814
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 11.5260i − 1.17029i −0.810929 0.585144i \(-0.801038\pi\)
0.810929 0.585144i \(-0.198962\pi\)
\(98\) − 7.16019i − 0.723289i
\(99\) −3.18291 −0.319894
\(100\) 0 0
\(101\) −7.70891 −0.767066 −0.383533 0.923527i \(-0.625293\pi\)
−0.383533 + 0.923527i \(0.625293\pi\)
\(102\) − 2.58010i − 0.255468i
\(103\) 3.39719i 0.334735i 0.985895 + 0.167368i \(0.0535267\pi\)
−0.985895 + 0.167368i \(0.946473\pi\)
\(104\) −6.34310 −0.621992
\(105\) 0 0
\(106\) −7.16019 −0.695459
\(107\) − 6.68620i − 0.646379i −0.946334 0.323190i \(-0.895245\pi\)
0.946334 0.323190i \(-0.104755\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −5.78572 −0.554171 −0.277086 0.960845i \(-0.589369\pi\)
−0.277086 + 0.960845i \(0.589369\pi\)
\(110\) 0 0
\(111\) 2.58010 0.244892
\(112\) 3.76300i 0.355570i
\(113\) − 0.580097i − 0.0545709i −0.999628 0.0272855i \(-0.991314\pi\)
0.999628 0.0272855i \(-0.00868631\pi\)
\(114\) −6.34310 −0.594086
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 6.34310i 0.586420i
\(118\) 4.36581i 0.401906i
\(119\) −9.70891 −0.890015
\(120\) 0 0
\(121\) −0.869107 −0.0790097
\(122\) − 1.52601i − 0.138158i
\(123\) 12.3431i 1.11294i
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 3.76300 0.335235
\(127\) 14.3204i 1.27073i 0.772212 + 0.635364i \(0.219150\pi\)
−0.772212 + 0.635364i \(0.780850\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −10.3431 −0.910659
\(130\) 0 0
\(131\) 15.8918 1.38847 0.694237 0.719746i \(-0.255742\pi\)
0.694237 + 0.719746i \(0.255742\pi\)
\(132\) − 3.18291i − 0.277036i
\(133\) 23.8691i 2.06971i
\(134\) 10.6862 0.923147
\(135\) 0 0
\(136\) 2.58010 0.221241
\(137\) 15.4199i 1.31741i 0.752401 + 0.658706i \(0.228896\pi\)
−0.752401 + 0.658706i \(0.771104\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −3.02271 −0.256383 −0.128192 0.991749i \(-0.540917\pi\)
−0.128192 + 0.991749i \(0.540917\pi\)
\(140\) 0 0
\(141\) 3.34310 0.281540
\(142\) 1.81709i 0.152487i
\(143\) − 20.1895i − 1.68833i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −8.52601 −0.705617
\(147\) − 7.16019i − 0.590563i
\(148\) 2.58010i 0.212083i
\(149\) −16.6862 −1.36699 −0.683493 0.729957i \(-0.739540\pi\)
−0.683493 + 0.729957i \(0.739540\pi\)
\(150\) 0 0
\(151\) −7.16019 −0.582688 −0.291344 0.956618i \(-0.594102\pi\)
−0.291344 + 0.956618i \(0.594102\pi\)
\(152\) − 6.34310i − 0.514493i
\(153\) − 2.58010i − 0.208588i
\(154\) −11.9773 −0.965157
\(155\) 0 0
\(156\) −6.34310 −0.507854
\(157\) − 22.2122i − 1.77273i −0.462990 0.886364i \(-0.653223\pi\)
0.462990 0.886364i \(-0.346777\pi\)
\(158\) 3.18291i 0.253218i
\(159\) −7.16019 −0.567840
\(160\) 0 0
\(161\) 3.76300 0.296566
\(162\) 1.00000i 0.0785674i
\(163\) − 8.68620i − 0.680356i −0.940361 0.340178i \(-0.889513\pi\)
0.940361 0.340178i \(-0.110487\pi\)
\(164\) −12.3431 −0.963834
\(165\) 0 0
\(166\) 7.28901 0.565737
\(167\) 25.2349i 1.95274i 0.216113 + 0.976368i \(0.430662\pi\)
−0.216113 + 0.976368i \(0.569338\pi\)
\(168\) 3.76300i 0.290322i
\(169\) −27.2349 −2.09499
\(170\) 0 0
\(171\) −6.34310 −0.485069
\(172\) − 10.3431i − 0.788654i
\(173\) 0.817094i 0.0621225i 0.999517 + 0.0310612i \(0.00988869\pi\)
−0.999517 + 0.0310612i \(0.990111\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 3.18291 0.239921
\(177\) 4.36581i 0.328155i
\(178\) − 4.58010i − 0.343293i
\(179\) 3.16019 0.236204 0.118102 0.993001i \(-0.462319\pi\)
0.118102 + 0.993001i \(0.462319\pi\)
\(180\) 0 0
\(181\) 3.37448 0.250823 0.125411 0.992105i \(-0.459975\pi\)
0.125411 + 0.992105i \(0.459975\pi\)
\(182\) 23.8691i 1.76930i
\(183\) − 1.52601i − 0.112806i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 8.21221i 0.600536i
\(188\) 3.34310i 0.243821i
\(189\) 3.76300 0.273718
\(190\) 0 0
\(191\) −11.8691 −0.858818 −0.429409 0.903110i \(-0.641278\pi\)
−0.429409 + 0.903110i \(0.641278\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 2.97729i − 0.214310i −0.994242 0.107155i \(-0.965826\pi\)
0.994242 0.107155i \(-0.0341741\pi\)
\(194\) 11.5260 0.827519
\(195\) 0 0
\(196\) 7.16019 0.511442
\(197\) 24.8918i 1.77347i 0.462279 + 0.886734i \(0.347032\pi\)
−0.462279 + 0.886734i \(0.652968\pi\)
\(198\) − 3.18291i − 0.226199i
\(199\) −22.4492 −1.59138 −0.795691 0.605703i \(-0.792892\pi\)
−0.795691 + 0.605703i \(0.792892\pi\)
\(200\) 0 0
\(201\) 10.6862 0.753746
\(202\) − 7.70891i − 0.542397i
\(203\) 3.76300i 0.264111i
\(204\) 2.58010 0.180643
\(205\) 0 0
\(206\) −3.39719 −0.236693
\(207\) 1.00000i 0.0695048i
\(208\) − 6.34310i − 0.439815i
\(209\) 20.1895 1.39654
\(210\) 0 0
\(211\) 19.5553 1.34624 0.673121 0.739532i \(-0.264953\pi\)
0.673121 + 0.739532i \(0.264953\pi\)
\(212\) − 7.16019i − 0.491764i
\(213\) 1.81709i 0.124505i
\(214\) 6.68620 0.457059
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 22.5780i 1.53270i
\(218\) − 5.78572i − 0.391858i
\(219\) −8.52601 −0.576134
\(220\) 0 0
\(221\) 16.3658 1.10088
\(222\) 2.58010i 0.173165i
\(223\) 28.8984i 1.93518i 0.252525 + 0.967590i \(0.418739\pi\)
−0.252525 + 0.967590i \(0.581261\pi\)
\(224\) −3.76300 −0.251426
\(225\) 0 0
\(226\) 0.580097 0.0385875
\(227\) − 17.6321i − 1.17028i −0.810931 0.585142i \(-0.801039\pi\)
0.810931 0.585142i \(-0.198961\pi\)
\(228\) − 6.34310i − 0.420082i
\(229\) 11.8918 0.785834 0.392917 0.919574i \(-0.371466\pi\)
0.392917 + 0.919574i \(0.371466\pi\)
\(230\) 0 0
\(231\) −11.9773 −0.788048
\(232\) − 1.00000i − 0.0656532i
\(233\) 9.02930i 0.591529i 0.955261 + 0.295765i \(0.0955744\pi\)
−0.955261 + 0.295765i \(0.904426\pi\)
\(234\) −6.34310 −0.414661
\(235\) 0 0
\(236\) −4.36581 −0.284190
\(237\) 3.18291i 0.206752i
\(238\) − 9.70891i − 0.629335i
\(239\) 20.0747 1.29853 0.649263 0.760564i \(-0.275077\pi\)
0.649263 + 0.760564i \(0.275077\pi\)
\(240\) 0 0
\(241\) −23.0066 −1.48198 −0.740992 0.671514i \(-0.765645\pi\)
−0.740992 + 0.671514i \(0.765645\pi\)
\(242\) − 0.869107i − 0.0558683i
\(243\) 1.00000i 0.0641500i
\(244\) 1.52601 0.0976926
\(245\) 0 0
\(246\) −12.3431 −0.786967
\(247\) − 40.2349i − 2.56009i
\(248\) − 6.00000i − 0.381000i
\(249\) 7.28901 0.461922
\(250\) 0 0
\(251\) −10.9459 −0.690900 −0.345450 0.938437i \(-0.612274\pi\)
−0.345450 + 0.938437i \(0.612274\pi\)
\(252\) 3.76300i 0.237047i
\(253\) − 3.18291i − 0.200108i
\(254\) −14.3204 −0.898541
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.3204i 1.01804i 0.860755 + 0.509019i \(0.169992\pi\)
−0.860755 + 0.509019i \(0.830008\pi\)
\(258\) − 10.3431i − 0.643933i
\(259\) 9.70891 0.603282
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 15.8918i 0.981800i
\(263\) 29.0066i 1.78862i 0.447445 + 0.894311i \(0.352334\pi\)
−0.447445 + 0.894311i \(0.647666\pi\)
\(264\) 3.18291 0.195894
\(265\) 0 0
\(266\) −23.8691 −1.46351
\(267\) − 4.58010i − 0.280297i
\(268\) 10.6862i 0.652764i
\(269\) 26.1895 1.59680 0.798401 0.602126i \(-0.205680\pi\)
0.798401 + 0.602126i \(0.205680\pi\)
\(270\) 0 0
\(271\) 6.36581 0.386696 0.193348 0.981130i \(-0.438065\pi\)
0.193348 + 0.981130i \(0.438065\pi\)
\(272\) 2.58010i 0.156441i
\(273\) 23.8691i 1.44462i
\(274\) −15.4199 −0.931550
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 9.50329i 0.570998i 0.958379 + 0.285499i \(0.0921593\pi\)
−0.958379 + 0.285499i \(0.907841\pi\)
\(278\) − 3.02271i − 0.181290i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 27.2663 1.62657 0.813285 0.581865i \(-0.197677\pi\)
0.813285 + 0.581865i \(0.197677\pi\)
\(282\) 3.34310i 0.199079i
\(283\) − 10.3204i − 0.613483i −0.951793 0.306742i \(-0.900761\pi\)
0.951793 0.306742i \(-0.0992388\pi\)
\(284\) −1.81709 −0.107825
\(285\) 0 0
\(286\) 20.1895 1.19383
\(287\) 46.4471i 2.74169i
\(288\) − 1.00000i − 0.0589256i
\(289\) 10.3431 0.608418
\(290\) 0 0
\(291\) 11.5260 0.675666
\(292\) − 8.52601i − 0.498947i
\(293\) 8.36581i 0.488736i 0.969683 + 0.244368i \(0.0785805\pi\)
−0.969683 + 0.244368i \(0.921420\pi\)
\(294\) 7.16019 0.417591
\(295\) 0 0
\(296\) −2.58010 −0.149965
\(297\) − 3.18291i − 0.184691i
\(298\) − 16.6862i − 0.966606i
\(299\) −6.34310 −0.366831
\(300\) 0 0
\(301\) −38.9211 −2.24338
\(302\) − 7.16019i − 0.412023i
\(303\) − 7.70891i − 0.442865i
\(304\) 6.34310 0.363802
\(305\) 0 0
\(306\) 2.58010 0.147494
\(307\) − 33.7089i − 1.92387i −0.273280 0.961935i \(-0.588108\pi\)
0.273280 0.961935i \(-0.411892\pi\)
\(308\) − 11.9773i − 0.682469i
\(309\) −3.39719 −0.193259
\(310\) 0 0
\(311\) −1.86252 −0.105614 −0.0528069 0.998605i \(-0.516817\pi\)
−0.0528069 + 0.998605i \(0.516817\pi\)
\(312\) − 6.34310i − 0.359107i
\(313\) − 26.8984i − 1.52039i −0.649696 0.760194i \(-0.725104\pi\)
0.649696 0.760194i \(-0.274896\pi\)
\(314\) 22.2122 1.25351
\(315\) 0 0
\(316\) −3.18291 −0.179052
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) − 7.16019i − 0.401524i
\(319\) 3.18291 0.178209
\(320\) 0 0
\(321\) 6.68620 0.373187
\(322\) 3.76300i 0.209704i
\(323\) 16.3658i 0.910618i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 8.68620 0.481084
\(327\) − 5.78572i − 0.319951i
\(328\) − 12.3431i − 0.681534i
\(329\) 12.5801 0.693563
\(330\) 0 0
\(331\) −6.18291 −0.339843 −0.169922 0.985458i \(-0.554351\pi\)
−0.169922 + 0.985458i \(0.554351\pi\)
\(332\) 7.28901i 0.400036i
\(333\) 2.58010i 0.141388i
\(334\) −25.2349 −1.38079
\(335\) 0 0
\(336\) −3.76300 −0.205289
\(337\) 15.1602i 0.825828i 0.910770 + 0.412914i \(0.135489\pi\)
−0.910770 + 0.412914i \(0.864511\pi\)
\(338\) − 27.2349i − 1.48138i
\(339\) 0.580097 0.0315065
\(340\) 0 0
\(341\) 19.0974 1.03418
\(342\) − 6.34310i − 0.342996i
\(343\) − 0.602810i − 0.0325487i
\(344\) 10.3431 0.557663
\(345\) 0 0
\(346\) −0.817094 −0.0439272
\(347\) − 29.8918i − 1.60468i −0.596869 0.802338i \(-0.703589\pi\)
0.596869 0.802338i \(-0.296411\pi\)
\(348\) − 1.00000i − 0.0536056i
\(349\) −7.02930 −0.376270 −0.188135 0.982143i \(-0.560244\pi\)
−0.188135 + 0.982143i \(0.560244\pi\)
\(350\) 0 0
\(351\) −6.34310 −0.338570
\(352\) 3.18291i 0.169649i
\(353\) 26.2349i 1.39634i 0.715930 + 0.698172i \(0.246003\pi\)
−0.715930 + 0.698172i \(0.753997\pi\)
\(354\) −4.36581 −0.232040
\(355\) 0 0
\(356\) 4.58010 0.242745
\(357\) − 9.70891i − 0.513850i
\(358\) 3.16019i 0.167021i
\(359\) −19.5033 −1.02934 −0.514672 0.857387i \(-0.672086\pi\)
−0.514672 + 0.857387i \(0.672086\pi\)
\(360\) 0 0
\(361\) 21.2349 1.11763
\(362\) 3.37448i 0.177359i
\(363\) − 0.869107i − 0.0456163i
\(364\) −23.8691 −1.25108
\(365\) 0 0
\(366\) 1.52601 0.0797656
\(367\) − 33.5033i − 1.74886i −0.485154 0.874429i \(-0.661236\pi\)
0.485154 0.874429i \(-0.338764\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −12.3431 −0.642556
\(370\) 0 0
\(371\) −26.9438 −1.39885
\(372\) − 6.00000i − 0.311086i
\(373\) − 10.1515i − 0.525626i −0.964847 0.262813i \(-0.915350\pi\)
0.964847 0.262813i \(-0.0846503\pi\)
\(374\) −8.21221 −0.424643
\(375\) 0 0
\(376\) −3.34310 −0.172407
\(377\) − 6.34310i − 0.326686i
\(378\) 3.76300i 0.193548i
\(379\) −20.5780 −1.05702 −0.528511 0.848926i \(-0.677249\pi\)
−0.528511 + 0.848926i \(0.677249\pi\)
\(380\) 0 0
\(381\) −14.3204 −0.733656
\(382\) − 11.8691i − 0.607276i
\(383\) 1.22833i 0.0627648i 0.999507 + 0.0313824i \(0.00999097\pi\)
−0.999507 + 0.0313824i \(0.990009\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.97729 0.151540
\(387\) − 10.3431i − 0.525769i
\(388\) 11.5260i 0.585144i
\(389\) −35.3724 −1.79345 −0.896726 0.442586i \(-0.854061\pi\)
−0.896726 + 0.442586i \(0.854061\pi\)
\(390\) 0 0
\(391\) 2.58010 0.130481
\(392\) 7.16019i 0.361644i
\(393\) 15.8918i 0.801636i
\(394\) −24.8918 −1.25403
\(395\) 0 0
\(396\) 3.18291 0.159947
\(397\) − 31.0066i − 1.55618i −0.628155 0.778088i \(-0.716190\pi\)
0.628155 0.778088i \(-0.283810\pi\)
\(398\) − 22.4492i − 1.12528i
\(399\) −23.8691 −1.19495
\(400\) 0 0
\(401\) 15.2056 0.759332 0.379666 0.925124i \(-0.376039\pi\)
0.379666 + 0.925124i \(0.376039\pi\)
\(402\) 10.6862i 0.532979i
\(403\) − 38.0586i − 1.89583i
\(404\) 7.70891 0.383533
\(405\) 0 0
\(406\) −3.76300 −0.186755
\(407\) − 8.21221i − 0.407064i
\(408\) 2.58010i 0.127734i
\(409\) 2.57143 0.127149 0.0635746 0.997977i \(-0.479750\pi\)
0.0635746 + 0.997977i \(0.479750\pi\)
\(410\) 0 0
\(411\) −15.4199 −0.760608
\(412\) − 3.39719i − 0.167368i
\(413\) 16.4286i 0.808397i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 6.34310 0.310996
\(417\) − 3.02271i − 0.148023i
\(418\) 20.1895i 0.987500i
\(419\) 19.0768 0.931963 0.465981 0.884795i \(-0.345701\pi\)
0.465981 + 0.884795i \(0.345701\pi\)
\(420\) 0 0
\(421\) −15.8918 −0.774520 −0.387260 0.921971i \(-0.626578\pi\)
−0.387260 + 0.921971i \(0.626578\pi\)
\(422\) 19.5553i 0.951937i
\(423\) 3.34310i 0.162547i
\(424\) 7.16019 0.347730
\(425\) 0 0
\(426\) −1.81709 −0.0880385
\(427\) − 5.74237i − 0.277893i
\(428\) 6.68620i 0.323190i
\(429\) 20.1895 0.974758
\(430\) 0 0
\(431\) −24.2122 −1.16626 −0.583130 0.812379i \(-0.698172\pi\)
−0.583130 + 0.812379i \(0.698172\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 12.2122i − 0.586881i −0.955977 0.293441i \(-0.905200\pi\)
0.955977 0.293441i \(-0.0948003\pi\)
\(434\) −22.5780 −1.08378
\(435\) 0 0
\(436\) 5.78572 0.277086
\(437\) − 6.34310i − 0.303432i
\(438\) − 8.52601i − 0.407388i
\(439\) 0.794381 0.0379137 0.0189569 0.999820i \(-0.493965\pi\)
0.0189569 + 0.999820i \(0.493965\pi\)
\(440\) 0 0
\(441\) 7.16019 0.340962
\(442\) 16.3658i 0.778442i
\(443\) 14.2122i 0.675242i 0.941282 + 0.337621i \(0.109622\pi\)
−0.941282 + 0.337621i \(0.890378\pi\)
\(444\) −2.58010 −0.122446
\(445\) 0 0
\(446\) −28.8984 −1.36838
\(447\) − 16.6862i − 0.789230i
\(448\) − 3.76300i − 0.177785i
\(449\) −1.05201 −0.0496476 −0.0248238 0.999692i \(-0.507902\pi\)
−0.0248238 + 0.999692i \(0.507902\pi\)
\(450\) 0 0
\(451\) 39.2869 1.84995
\(452\) 0.580097i 0.0272855i
\(453\) − 7.16019i − 0.336415i
\(454\) 17.6321 0.827516
\(455\) 0 0
\(456\) 6.34310 0.297043
\(457\) 0.686200i 0.0320991i 0.999871 + 0.0160495i \(0.00510895\pi\)
−0.999871 + 0.0160495i \(0.994891\pi\)
\(458\) 11.8918i 0.555668i
\(459\) 2.58010 0.120429
\(460\) 0 0
\(461\) −29.5780 −1.37759 −0.688793 0.724958i \(-0.741859\pi\)
−0.688793 + 0.724958i \(0.741859\pi\)
\(462\) − 11.9773i − 0.557234i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −9.02930 −0.418274
\(467\) 29.2890i 1.35533i 0.735369 + 0.677667i \(0.237009\pi\)
−0.735369 + 0.677667i \(0.762991\pi\)
\(468\) − 6.34310i − 0.293210i
\(469\) 40.2122 1.85683
\(470\) 0 0
\(471\) 22.2122 1.02348
\(472\) − 4.36581i − 0.200953i
\(473\) 32.9211i 1.51371i
\(474\) −3.18291 −0.146196
\(475\) 0 0
\(476\) 9.70891 0.445007
\(477\) − 7.16019i − 0.327843i
\(478\) 20.0747i 0.918197i
\(479\) −15.9145 −0.727154 −0.363577 0.931564i \(-0.618445\pi\)
−0.363577 + 0.931564i \(0.618445\pi\)
\(480\) 0 0
\(481\) −16.3658 −0.746217
\(482\) − 23.0066i − 1.04792i
\(483\) 3.76300i 0.171223i
\(484\) 0.869107 0.0395049
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 38.9438i − 1.76471i −0.470581 0.882357i \(-0.655956\pi\)
0.470581 0.882357i \(-0.344044\pi\)
\(488\) 1.52601i 0.0690791i
\(489\) 8.68620 0.392804
\(490\) 0 0
\(491\) 27.4806 1.24018 0.620091 0.784530i \(-0.287096\pi\)
0.620091 + 0.784530i \(0.287096\pi\)
\(492\) − 12.3431i − 0.556470i
\(493\) 2.58010i 0.116202i
\(494\) 40.2349 1.81025
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) 6.83773i 0.306714i
\(498\) 7.28901i 0.326628i
\(499\) −6.97729 −0.312346 −0.156173 0.987730i \(-0.549916\pi\)
−0.156173 + 0.987730i \(0.549916\pi\)
\(500\) 0 0
\(501\) −25.2349 −1.12741
\(502\) − 10.9459i − 0.488540i
\(503\) 7.65690i 0.341404i 0.985323 + 0.170702i \(0.0546036\pi\)
−0.985323 + 0.170702i \(0.945396\pi\)
\(504\) −3.76300 −0.167617
\(505\) 0 0
\(506\) 3.18291 0.141497
\(507\) − 27.2349i − 1.20955i
\(508\) − 14.3204i − 0.635364i
\(509\) 34.6073 1.53394 0.766971 0.641681i \(-0.221763\pi\)
0.766971 + 0.641681i \(0.221763\pi\)
\(510\) 0 0
\(511\) −32.0834 −1.41929
\(512\) 1.00000i 0.0441942i
\(513\) − 6.34310i − 0.280055i
\(514\) −16.3204 −0.719861
\(515\) 0 0
\(516\) 10.3431 0.455330
\(517\) − 10.6408i − 0.467981i
\(518\) 9.70891i 0.426585i
\(519\) −0.817094 −0.0358664
\(520\) 0 0
\(521\) 25.4199 1.11367 0.556833 0.830624i \(-0.312016\pi\)
0.556833 + 0.830624i \(0.312016\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 25.8691i 1.13118i 0.824688 + 0.565589i \(0.191351\pi\)
−0.824688 + 0.565589i \(0.808649\pi\)
\(524\) −15.8918 −0.694237
\(525\) 0 0
\(526\) −29.0066 −1.26475
\(527\) 15.4806i 0.674345i
\(528\) 3.18291i 0.138518i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −4.36581 −0.189460
\(532\) − 23.8691i − 1.03486i
\(533\) − 78.2935i − 3.39127i
\(534\) 4.58010 0.198200
\(535\) 0 0
\(536\) −10.6862 −0.461574
\(537\) 3.16019i 0.136372i
\(538\) 26.1895i 1.12911i
\(539\) −22.7902 −0.981645
\(540\) 0 0
\(541\) 34.6007 1.48760 0.743801 0.668401i \(-0.233021\pi\)
0.743801 + 0.668401i \(0.233021\pi\)
\(542\) 6.36581i 0.273435i
\(543\) 3.37448i 0.144813i
\(544\) −2.58010 −0.110621
\(545\) 0 0
\(546\) −23.8691 −1.02150
\(547\) 3.55531i 0.152014i 0.997107 + 0.0760070i \(0.0242171\pi\)
−0.997107 + 0.0760070i \(0.975783\pi\)
\(548\) − 15.4199i − 0.658706i
\(549\) 1.52601 0.0651284
\(550\) 0 0
\(551\) 6.34310 0.270225
\(552\) − 1.00000i − 0.0425628i
\(553\) 11.9773i 0.509326i
\(554\) −9.50329 −0.403756
\(555\) 0 0
\(556\) 3.02271 0.128192
\(557\) − 15.2642i − 0.646766i −0.946268 0.323383i \(-0.895180\pi\)
0.946268 0.323383i \(-0.104820\pi\)
\(558\) − 6.00000i − 0.254000i
\(559\) 65.6073 2.77489
\(560\) 0 0
\(561\) −8.21221 −0.346720
\(562\) 27.2663i 1.15016i
\(563\) 13.5487i 0.571010i 0.958377 + 0.285505i \(0.0921614\pi\)
−0.958377 + 0.285505i \(0.907839\pi\)
\(564\) −3.34310 −0.140770
\(565\) 0 0
\(566\) 10.3204 0.433798
\(567\) 3.76300i 0.158031i
\(568\) − 1.81709i − 0.0762436i
\(569\) 18.4740 0.774470 0.387235 0.921981i \(-0.373430\pi\)
0.387235 + 0.921981i \(0.373430\pi\)
\(570\) 0 0
\(571\) −9.89182 −0.413960 −0.206980 0.978345i \(-0.566363\pi\)
−0.206980 + 0.978345i \(0.566363\pi\)
\(572\) 20.1895i 0.844165i
\(573\) − 11.8691i − 0.495839i
\(574\) −46.4471 −1.93867
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 29.5033i − 1.22824i −0.789213 0.614119i \(-0.789511\pi\)
0.789213 0.614119i \(-0.210489\pi\)
\(578\) 10.3431i 0.430216i
\(579\) 2.97729 0.123732
\(580\) 0 0
\(581\) 27.4286 1.13793
\(582\) 11.5260i 0.477768i
\(583\) 22.7902i 0.943875i
\(584\) 8.52601 0.352809
\(585\) 0 0
\(586\) −8.36581 −0.345589
\(587\) 14.4740i 0.597406i 0.954346 + 0.298703i \(0.0965539\pi\)
−0.954346 + 0.298703i \(0.903446\pi\)
\(588\) 7.16019i 0.295281i
\(589\) 38.0586 1.56818
\(590\) 0 0
\(591\) −24.8918 −1.02391
\(592\) − 2.58010i − 0.106041i
\(593\) − 5.86911i − 0.241015i −0.992712 0.120508i \(-0.961548\pi\)
0.992712 0.120508i \(-0.0384522\pi\)
\(594\) 3.18291 0.130596
\(595\) 0 0
\(596\) 16.6862 0.683493
\(597\) − 22.4492i − 0.918785i
\(598\) − 6.34310i − 0.259389i
\(599\) 3.95457 0.161580 0.0807898 0.996731i \(-0.474256\pi\)
0.0807898 + 0.996731i \(0.474256\pi\)
\(600\) 0 0
\(601\) 9.23492 0.376700 0.188350 0.982102i \(-0.439686\pi\)
0.188350 + 0.982102i \(0.439686\pi\)
\(602\) − 38.9211i − 1.58631i
\(603\) 10.6862i 0.435176i
\(604\) 7.16019 0.291344
\(605\) 0 0
\(606\) 7.70891 0.313153
\(607\) 48.7448i 1.97849i 0.146266 + 0.989245i \(0.453274\pi\)
−0.146266 + 0.989245i \(0.546726\pi\)
\(608\) 6.34310i 0.257247i
\(609\) −3.76300 −0.152485
\(610\) 0 0
\(611\) −21.2056 −0.857888
\(612\) 2.58010i 0.104294i
\(613\) − 3.67754i − 0.148534i −0.997238 0.0742671i \(-0.976338\pi\)
0.997238 0.0742671i \(-0.0236617\pi\)
\(614\) 33.7089 1.36038
\(615\) 0 0
\(616\) 11.9773 0.482579
\(617\) 0.794381i 0.0319806i 0.999872 + 0.0159903i \(0.00509008\pi\)
−0.999872 + 0.0159903i \(0.994910\pi\)
\(618\) − 3.39719i − 0.136655i
\(619\) 4.57802 0.184006 0.0920031 0.995759i \(-0.470673\pi\)
0.0920031 + 0.995759i \(0.470673\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) − 1.86252i − 0.0746802i
\(623\) − 17.2349i − 0.690502i
\(624\) 6.34310 0.253927
\(625\) 0 0
\(626\) 26.8984 1.07508
\(627\) 20.1895i 0.806291i
\(628\) 22.2122i 0.886364i
\(629\) 6.65690 0.265428
\(630\) 0 0
\(631\) −31.1808 −1.24129 −0.620645 0.784092i \(-0.713129\pi\)
−0.620645 + 0.784092i \(0.713129\pi\)
\(632\) − 3.18291i − 0.126609i
\(633\) 19.5553i 0.777254i
\(634\) −11.0000 −0.436866
\(635\) 0 0
\(636\) 7.16019 0.283920
\(637\) 45.4178i 1.79952i
\(638\) 3.18291i 0.126012i
\(639\) −1.81709 −0.0718831
\(640\) 0 0
\(641\) 46.7295 1.84571 0.922853 0.385152i \(-0.125851\pi\)
0.922853 + 0.385152i \(0.125851\pi\)
\(642\) 6.68620i 0.263883i
\(643\) 4.81709i 0.189968i 0.995479 + 0.0949838i \(0.0302799\pi\)
−0.995479 + 0.0949838i \(0.969720\pi\)
\(644\) −3.76300 −0.148283
\(645\) 0 0
\(646\) −16.3658 −0.643904
\(647\) − 19.4967i − 0.766495i −0.923646 0.383247i \(-0.874806\pi\)
0.923646 0.383247i \(-0.125194\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 13.8960 0.545465
\(650\) 0 0
\(651\) −22.5780 −0.884902
\(652\) 8.68620i 0.340178i
\(653\) − 30.5260i − 1.19457i −0.802027 0.597287i \(-0.796245\pi\)
0.802027 0.597287i \(-0.203755\pi\)
\(654\) 5.78572 0.226239
\(655\) 0 0
\(656\) 12.3431 0.481917
\(657\) − 8.52601i − 0.332631i
\(658\) 12.5801i 0.490423i
\(659\) 2.49463 0.0971769 0.0485885 0.998819i \(-0.484528\pi\)
0.0485885 + 0.998819i \(0.484528\pi\)
\(660\) 0 0
\(661\) −16.7337 −0.650866 −0.325433 0.945565i \(-0.605510\pi\)
−0.325433 + 0.945565i \(0.605510\pi\)
\(662\) − 6.18291i − 0.240305i
\(663\) 16.3658i 0.635596i
\(664\) −7.28901 −0.282868
\(665\) 0 0
\(666\) −2.58010 −0.0999767
\(667\) − 1.00000i − 0.0387202i
\(668\) − 25.2349i − 0.976368i
\(669\) −28.8984 −1.11728
\(670\) 0 0
\(671\) −4.85714 −0.187508
\(672\) − 3.76300i − 0.145161i
\(673\) 2.73821i 0.105550i 0.998606 + 0.0527752i \(0.0168067\pi\)
−0.998606 + 0.0527752i \(0.983193\pi\)
\(674\) −15.1602 −0.583949
\(675\) 0 0
\(676\) 27.2349 1.04750
\(677\) − 38.1040i − 1.46446i −0.681059 0.732228i \(-0.738480\pi\)
0.681059 0.732228i \(-0.261520\pi\)
\(678\) 0.580097i 0.0222785i
\(679\) 43.3724 1.66448
\(680\) 0 0
\(681\) 17.6321 0.675664
\(682\) 19.0974i 0.731279i
\(683\) − 24.6234i − 0.942190i −0.882082 0.471095i \(-0.843859\pi\)
0.882082 0.471095i \(-0.156141\pi\)
\(684\) 6.34310 0.242534
\(685\) 0 0
\(686\) 0.602810 0.0230154
\(687\) 11.8918i 0.453701i
\(688\) 10.3431i 0.394327i
\(689\) 45.4178 1.73028
\(690\) 0 0
\(691\) 43.2935 1.64696 0.823482 0.567343i \(-0.192029\pi\)
0.823482 + 0.567343i \(0.192029\pi\)
\(692\) − 0.817094i − 0.0310612i
\(693\) − 11.9773i − 0.454980i
\(694\) 29.8918 1.13468
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 31.8464i 1.20627i
\(698\) − 7.02930i − 0.266063i
\(699\) −9.02930 −0.341520
\(700\) 0 0
\(701\) 9.00659 0.340174 0.170087 0.985429i \(-0.445595\pi\)
0.170087 + 0.985429i \(0.445595\pi\)
\(702\) − 6.34310i − 0.239405i
\(703\) − 16.3658i − 0.617248i
\(704\) −3.18291 −0.119960
\(705\) 0 0
\(706\) −26.2349 −0.987364
\(707\) − 29.0087i − 1.09098i
\(708\) − 4.36581i − 0.164077i
\(709\) −22.3183 −0.838182 −0.419091 0.907944i \(-0.637651\pi\)
−0.419091 + 0.907944i \(0.637651\pi\)
\(710\) 0 0
\(711\) −3.18291 −0.119368
\(712\) 4.58010i 0.171646i
\(713\) − 6.00000i − 0.224702i
\(714\) 9.70891 0.363347
\(715\) 0 0
\(716\) −3.16019 −0.118102
\(717\) 20.0747i 0.749704i
\(718\) − 19.5033i − 0.727856i
\(719\) 32.1829 1.20022 0.600110 0.799918i \(-0.295123\pi\)
0.600110 + 0.799918i \(0.295123\pi\)
\(720\) 0 0
\(721\) −12.7836 −0.476088
\(722\) 21.2349i 0.790282i
\(723\) − 23.0066i − 0.855624i
\(724\) −3.37448 −0.125411
\(725\) 0 0
\(726\) 0.869107 0.0322556
\(727\) − 32.3658i − 1.20038i −0.799857 0.600191i \(-0.795091\pi\)
0.799857 0.600191i \(-0.204909\pi\)
\(728\) − 23.8691i − 0.884648i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −26.6862 −0.987025
\(732\) 1.52601i 0.0564028i
\(733\) − 34.6841i − 1.28109i −0.767922 0.640544i \(-0.778709\pi\)
0.767922 0.640544i \(-0.221291\pi\)
\(734\) 33.5033 1.23663
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 34.0132i − 1.25289i
\(738\) − 12.3431i − 0.454356i
\(739\) −45.1441 −1.66065 −0.830326 0.557278i \(-0.811846\pi\)
−0.830326 + 0.557278i \(0.811846\pi\)
\(740\) 0 0
\(741\) 40.2349 1.47807
\(742\) − 26.9438i − 0.989139i
\(743\) 3.02930i 0.111134i 0.998455 + 0.0555671i \(0.0176967\pi\)
−0.998455 + 0.0555671i \(0.982303\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) 10.1515 0.371674
\(747\) 7.28901i 0.266691i
\(748\) − 8.21221i − 0.300268i
\(749\) 25.1602 0.919333
\(750\) 0 0
\(751\) −12.3410 −0.450330 −0.225165 0.974321i \(-0.572292\pi\)
−0.225165 + 0.974321i \(0.572292\pi\)
\(752\) − 3.34310i − 0.121910i
\(753\) − 10.9459i − 0.398891i
\(754\) 6.34310 0.231002
\(755\) 0 0
\(756\) −3.76300 −0.136859
\(757\) 6.00208i 0.218149i 0.994034 + 0.109075i \(0.0347887\pi\)
−0.994034 + 0.109075i \(0.965211\pi\)
\(758\) − 20.5780i − 0.747427i
\(759\) 3.18291 0.115532
\(760\) 0 0
\(761\) −26.2977 −0.953290 −0.476645 0.879096i \(-0.658147\pi\)
−0.476645 + 0.879096i \(0.658147\pi\)
\(762\) − 14.3204i − 0.518773i
\(763\) − 21.7717i − 0.788187i
\(764\) 11.8691 0.429409
\(765\) 0 0
\(766\) −1.22833 −0.0443814
\(767\) − 27.6928i − 0.999929i
\(768\) 1.00000i 0.0360844i
\(769\) 45.8918 1.65490 0.827451 0.561538i \(-0.189790\pi\)
0.827451 + 0.561538i \(0.189790\pi\)
\(770\) 0 0
\(771\) −16.3204 −0.587764
\(772\) 2.97729i 0.107155i
\(773\) − 37.5846i − 1.35182i −0.736982 0.675912i \(-0.763750\pi\)
0.736982 0.675912i \(-0.236250\pi\)
\(774\) 10.3431 0.371775
\(775\) 0 0
\(776\) −11.5260 −0.413760
\(777\) 9.70891i 0.348305i
\(778\) − 35.3724i − 1.26816i
\(779\) 78.2935 2.80516
\(780\) 0 0
\(781\) 5.78364 0.206955
\(782\) 2.58010i 0.0922641i
\(783\) − 1.00000i − 0.0357371i
\(784\) −7.16019 −0.255721
\(785\) 0 0
\(786\) −15.8918 −0.566842
\(787\) − 20.9211i − 0.745757i −0.927880 0.372879i \(-0.878371\pi\)
0.927880 0.372879i \(-0.121629\pi\)
\(788\) − 24.8918i − 0.886734i
\(789\) −29.0066 −1.03266
\(790\) 0 0
\(791\) 2.18291 0.0776152
\(792\) 3.18291i 0.113100i
\(793\) 9.67961i 0.343733i
\(794\) 31.0066 1.10038
\(795\) 0 0
\(796\) 22.4492 0.795691
\(797\) 46.5326i 1.64827i 0.566394 + 0.824135i \(0.308338\pi\)
−0.566394 + 0.824135i \(0.691662\pi\)
\(798\) − 23.8691i − 0.844957i
\(799\) 8.62552 0.305149
\(800\) 0 0
\(801\) 4.58010 0.161830
\(802\) 15.2056i 0.536929i
\(803\) 27.1375i 0.957661i
\(804\) −10.6862 −0.376873
\(805\) 0 0
\(806\) 38.0586 1.34056
\(807\) 26.1895i 0.921914i
\(808\) 7.70891i 0.271199i
\(809\) −26.1895 −0.920774 −0.460387 0.887718i \(-0.652289\pi\)
−0.460387 + 0.887718i \(0.652289\pi\)
\(810\) 0 0
\(811\) 10.4112 0.365588 0.182794 0.983151i \(-0.441486\pi\)
0.182794 + 0.983151i \(0.441486\pi\)
\(812\) − 3.76300i − 0.132056i
\(813\) 6.36581i 0.223259i
\(814\) 8.21221 0.287838
\(815\) 0 0
\(816\) −2.58010 −0.0903215
\(817\) 65.6073i 2.29531i
\(818\) 2.57143i 0.0899080i
\(819\) −23.8691 −0.834054
\(820\) 0 0
\(821\) 32.8171 1.14532 0.572662 0.819791i \(-0.305911\pi\)
0.572662 + 0.819791i \(0.305911\pi\)
\(822\) − 15.4199i − 0.537831i
\(823\) 20.5780i 0.717305i 0.933471 + 0.358652i \(0.116764\pi\)
−0.933471 + 0.358652i \(0.883236\pi\)
\(824\) 3.39719 0.118347
\(825\) 0 0
\(826\) −16.4286 −0.571623
\(827\) 21.1982i 0.737132i 0.929602 + 0.368566i \(0.120151\pi\)
−0.929602 + 0.368566i \(0.879849\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 3.29109 0.114304 0.0571521 0.998365i \(-0.481798\pi\)
0.0571521 + 0.998365i \(0.481798\pi\)
\(830\) 0 0
\(831\) −9.50329 −0.329666
\(832\) 6.34310i 0.219907i
\(833\) − 18.4740i − 0.640086i
\(834\) 3.02271 0.104668
\(835\) 0 0
\(836\) −20.1895 −0.698268
\(837\) − 6.00000i − 0.207390i
\(838\) 19.0768i 0.658997i
\(839\) 20.0813 0.693284 0.346642 0.937998i \(-0.387322\pi\)
0.346642 + 0.937998i \(0.387322\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) − 15.8918i − 0.547668i
\(843\) 27.2663i 0.939101i
\(844\) −19.5553 −0.673121
\(845\) 0 0
\(846\) −3.34310 −0.114938
\(847\) − 3.27045i − 0.112374i
\(848\) 7.16019i 0.245882i
\(849\) 10.3204 0.354195
\(850\) 0 0
\(851\) −2.58010 −0.0884446
\(852\) − 1.81709i − 0.0622526i
\(853\) 32.9211i 1.12720i 0.826049 + 0.563599i \(0.190584\pi\)
−0.826049 + 0.563599i \(0.809416\pi\)
\(854\) 5.74237 0.196500
\(855\) 0 0
\(856\) −6.68620 −0.228530
\(857\) 46.5780i 1.59107i 0.605904 + 0.795537i \(0.292811\pi\)
−0.605904 + 0.795537i \(0.707189\pi\)
\(858\) 20.1895i 0.689258i
\(859\) 42.0132 1.43347 0.716736 0.697345i \(-0.245635\pi\)
0.716736 + 0.697345i \(0.245635\pi\)
\(860\) 0 0
\(861\) −46.4471 −1.58291
\(862\) − 24.2122i − 0.824671i
\(863\) 15.5553i 0.529509i 0.964316 + 0.264754i \(0.0852909\pi\)
−0.964316 + 0.264754i \(0.914709\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 12.2122 0.414988
\(867\) 10.3431i 0.351270i
\(868\) − 22.5780i − 0.766348i
\(869\) 10.1309 0.343667
\(870\) 0 0
\(871\) −67.7836 −2.29676
\(872\) 5.78572i 0.195929i
\(873\) 11.5260i 0.390096i
\(874\) 6.34310 0.214559
\(875\) 0 0
\(876\) 8.52601 0.288067
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) 0.794381i 0.0268091i
\(879\) −8.36581 −0.282172
\(880\) 0 0
\(881\) 10.4740 0.352878 0.176439 0.984312i \(-0.443542\pi\)
0.176439 + 0.984312i \(0.443542\pi\)
\(882\) 7.16019i 0.241096i
\(883\) 35.8757i 1.20731i 0.797244 + 0.603657i \(0.206290\pi\)
−0.797244 + 0.603657i \(0.793710\pi\)
\(884\) −16.3658 −0.550442
\(885\) 0 0
\(886\) −14.2122 −0.477468
\(887\) − 7.29767i − 0.245032i −0.992467 0.122516i \(-0.960904\pi\)
0.992467 0.122516i \(-0.0390963\pi\)
\(888\) − 2.58010i − 0.0865824i
\(889\) −53.8877 −1.80733
\(890\) 0 0
\(891\) 3.18291 0.106631
\(892\) − 28.8984i − 0.967590i
\(893\) − 21.2056i − 0.709619i
\(894\) 16.6862 0.558070
\(895\) 0 0
\(896\) 3.76300 0.125713
\(897\) − 6.34310i − 0.211790i
\(898\) − 1.05201i − 0.0351061i
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) −18.4740 −0.615458
\(902\) 39.2869i 1.30811i
\(903\) − 38.9211i − 1.29521i
\(904\) −0.580097 −0.0192937
\(905\) 0 0
\(906\) 7.16019 0.237882
\(907\) − 40.7089i − 1.35172i −0.737031 0.675859i \(-0.763773\pi\)
0.737031 0.675859i \(-0.236227\pi\)
\(908\) 17.6321i 0.585142i
\(909\) 7.70891 0.255689
\(910\) 0 0
\(911\) 22.7089 0.752380 0.376190 0.926543i \(-0.377234\pi\)
0.376190 + 0.926543i \(0.377234\pi\)
\(912\) 6.34310i 0.210041i
\(913\) − 23.2002i − 0.767816i
\(914\) −0.686200 −0.0226975
\(915\) 0 0
\(916\) −11.8918 −0.392917
\(917\) 59.8010i 1.97480i
\(918\) 2.58010i 0.0851559i
\(919\) 26.5801 0.876796 0.438398 0.898781i \(-0.355546\pi\)
0.438398 + 0.898781i \(0.355546\pi\)
\(920\) 0 0
\(921\) 33.7089 1.11075
\(922\) − 29.5780i − 0.974100i
\(923\) − 11.5260i − 0.379383i
\(924\) 11.9773 0.394024
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) − 3.39719i − 0.111578i
\(928\) 1.00000i 0.0328266i
\(929\) −36.5553 −1.19934 −0.599670 0.800247i \(-0.704702\pi\)
−0.599670 + 0.800247i \(0.704702\pi\)
\(930\) 0 0
\(931\) −45.4178 −1.48851
\(932\) − 9.02930i − 0.295765i
\(933\) − 1.86252i − 0.0609761i
\(934\) −29.2890 −0.958366
\(935\) 0 0
\(936\) 6.34310 0.207331
\(937\) 19.3724i 0.632869i 0.948614 + 0.316434i \(0.102486\pi\)
−0.948614 + 0.316434i \(0.897514\pi\)
\(938\) 40.2122i 1.31298i
\(939\) 26.8984 0.877796
\(940\) 0 0
\(941\) −36.3204 −1.18401 −0.592005 0.805934i \(-0.701664\pi\)
−0.592005 + 0.805934i \(0.701664\pi\)
\(942\) 22.2122i 0.723713i
\(943\) − 12.3431i − 0.401947i
\(944\) 4.36581 0.142095
\(945\) 0 0
\(946\) −32.9211 −1.07036
\(947\) − 5.63419i − 0.183086i −0.995801 0.0915432i \(-0.970820\pi\)
0.995801 0.0915432i \(-0.0291800\pi\)
\(948\) − 3.18291i − 0.103376i
\(949\) 54.0813 1.75555
\(950\) 0 0
\(951\) −11.0000 −0.356699
\(952\) 9.70891i 0.314668i
\(953\) − 28.9005i − 0.936179i −0.883681 0.468089i \(-0.844943\pi\)
0.883681 0.468089i \(-0.155057\pi\)
\(954\) 7.16019 0.231820
\(955\) 0 0
\(956\) −20.0747 −0.649263
\(957\) 3.18291i 0.102889i
\(958\) − 15.9145i − 0.514175i
\(959\) −58.0251 −1.87373
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) − 16.3658i − 0.527655i
\(963\) 6.68620i 0.215460i
\(964\) 23.0066 0.740992
\(965\) 0 0
\(966\) −3.76300 −0.121073
\(967\) − 12.3658i − 0.397658i −0.980034 0.198829i \(-0.936286\pi\)
0.980034 0.198829i \(-0.0637138\pi\)
\(968\) 0.869107i 0.0279341i
\(969\) −16.3658 −0.525746
\(970\) 0 0
\(971\) −33.9125 −1.08830 −0.544151 0.838987i \(-0.683148\pi\)
−0.544151 + 0.838987i \(0.683148\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 11.3745i − 0.364649i
\(974\) 38.9438 1.24784
\(975\) 0 0
\(976\) −1.52601 −0.0488463
\(977\) − 41.3703i − 1.32355i −0.749701 0.661777i \(-0.769803\pi\)
0.749701 0.661777i \(-0.230197\pi\)
\(978\) 8.68620i 0.277754i
\(979\) −14.5780 −0.465916
\(980\) 0 0
\(981\) 5.78572 0.184724
\(982\) 27.4806i 0.876941i
\(983\) 1.91453i 0.0610641i 0.999534 + 0.0305320i \(0.00972016\pi\)
−0.999534 + 0.0305320i \(0.990280\pi\)
\(984\) 12.3431 0.393484
\(985\) 0 0
\(986\) −2.58010 −0.0821670
\(987\) 12.5801i 0.400429i
\(988\) 40.2349i 1.28004i
\(989\) 10.3431 0.328891
\(990\) 0 0
\(991\) 3.31380 0.105266 0.0526332 0.998614i \(-0.483239\pi\)
0.0526332 + 0.998614i \(0.483239\pi\)
\(992\) 6.00000i 0.190500i
\(993\) − 6.18291i − 0.196209i
\(994\) −6.83773 −0.216880
\(995\) 0 0
\(996\) −7.28901 −0.230961
\(997\) 8.70891i 0.275814i 0.990445 + 0.137907i \(0.0440375\pi\)
−0.990445 + 0.137907i \(0.955962\pi\)
\(998\) − 6.97729i − 0.220862i
\(999\) −2.58010 −0.0816307
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 3450.2.d.ba.2899.6 6
5.2 odd 4 3450.2.a.bp.1.1 3
5.3 odd 4 3450.2.a.bs.1.3 yes 3
5.4 even 2 inner 3450.2.d.ba.2899.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bp.1.1 3 5.2 odd 4
3450.2.a.bs.1.3 yes 3 5.3 odd 4
3450.2.d.ba.2899.1 6 5.4 even 2 inner
3450.2.d.ba.2899.6 6 1.1 even 1 trivial