Properties

Label 3450.2.d.j.2899.2
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.j.2899.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +2.00000 q^{21} -6.00000i q^{22} -1.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} +6.00000i q^{33} +1.00000 q^{36} -2.00000 q^{39} +10.0000 q^{41} +2.00000i q^{42} -12.0000i q^{43} +6.00000 q^{44} +1.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +2.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{58} +12.0000 q^{59} +4.00000 q^{61} +8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} +12.0000i q^{67} -1.00000 q^{69} +1.00000i q^{72} -10.0000i q^{73} -12.0000i q^{77} -2.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +14.0000i q^{83} -2.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +6.00000i q^{88} +4.00000 q^{91} +1.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} +1.00000 q^{96} +6.00000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{21} - 2 q^{24} + 4 q^{26} - 12 q^{29} + 16 q^{31} + 2 q^{36} - 4 q^{39} + 20 q^{41} + 12 q^{44} + 2 q^{46} + 6 q^{49} - 2 q^{54} + 4 q^{56} + 24 q^{59} + 8 q^{61} - 2 q^{64} - 12 q^{66} - 2 q^{69} + 12 q^{79} + 2 q^{81} - 4 q^{84} + 24 q^{86} + 8 q^{91} - 16 q^{94} + 2 q^{96} + 12 q^{99} + O(q^{100}) \)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 6.00000i − 1.27920i
\(23\) − 1.00000i − 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 6.00000i 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000i 1.01600i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 12.0000i − 1.36753i
\(78\) − 2.00000i − 0.226455i
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000i 0.643268i
\(88\) 6.00000i 0.639602i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 1.00000i 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 14.0000i − 1.35343i −0.736245 0.676716i \(-0.763403\pi\)
0.736245 0.676716i \(-0.236597\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) − 8.00000i − 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 2.00000i 0.184900i
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 4.00000i 0.362143i
\(123\) − 10.0000i − 0.901670i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) − 6.00000i − 0.522233i
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 1.00000i − 0.0851257i
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) − 16.0000i − 1.27694i −0.769647 0.638470i \(-0.779568\pi\)
0.769647 0.638470i \(-0.220432\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000i 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −14.0000 −1.08661
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000i 0.914991i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 4.00000i 0.296500i
\(183\) − 4.00000i − 0.295689i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) − 8.00000i − 0.583460i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 6.00000i 0.426401i
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) − 6.00000i − 0.422159i
\(203\) − 12.0000i − 0.842235i
\(204\) 0 0
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) 1.00000i 0.0695048i
\(208\) − 2.00000i − 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 0 0
\(214\) 14.0000 0.957020
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000i 1.08615i
\(218\) 16.0000i 1.08366i
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0000i 0.803579i 0.915732 + 0.401790i \(0.131612\pi\)
−0.915732 + 0.401790i \(0.868388\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 8.00000 0.532152
\(227\) 10.0000i 0.663723i 0.943328 + 0.331862i \(0.107677\pi\)
−0.943328 + 0.331862i \(0.892323\pi\)
\(228\) 0 0
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 6.00000i 0.393919i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) − 6.00000i − 0.389742i
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 25.0000i 1.60706i
\(243\) − 1.00000i − 0.0641500i
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) − 8.00000i − 0.508001i
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 6.00000i 0.377217i
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 8.00000i − 0.494242i
\(263\) 4.00000i 0.246651i 0.992366 + 0.123325i \(0.0393559\pi\)
−0.992366 + 0.123325i \(0.960644\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) − 4.00000i − 0.242091i
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) − 12.0000i − 0.719712i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 8.00000i 0.476393i
\(283\) 24.0000i 1.42665i 0.700832 + 0.713326i \(0.252812\pi\)
−0.700832 + 0.713326i \(0.747188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 20.0000i 1.18056i
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 10.0000i 0.585206i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 0 0
\(297\) − 6.00000i − 0.348155i
\(298\) 18.0000i 1.04271i
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) − 12.0000i − 0.690522i
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 12.0000i 0.683763i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 2.00000i 0.113228i
\(313\) − 26.0000i − 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) − 22.0000i − 1.23564i −0.786318 0.617822i \(-0.788015\pi\)
0.786318 0.617822i \(-0.211985\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 36.0000 2.01561
\(320\) 0 0
\(321\) −14.0000 −0.781404
\(322\) 2.00000i 0.111456i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) − 16.0000i − 0.884802i
\(328\) − 10.0000i − 0.552158i
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) − 14.0000i − 0.768350i
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 10.0000i 0.544735i 0.962193 + 0.272367i \(0.0878066\pi\)
−0.962193 + 0.272367i \(0.912193\pi\)
\(338\) 9.00000i 0.489535i
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) − 6.00000i − 0.319801i
\(353\) − 2.00000i − 0.106449i −0.998583 0.0532246i \(-0.983050\pi\)
0.998583 0.0532246i \(-0.0169499\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 16.0000i 0.845626i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 8.00000i 0.420471i
\(363\) − 25.0000i − 1.31216i
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 14.0000i 0.730794i 0.930852 + 0.365397i \(0.119067\pi\)
−0.930852 + 0.365397i \(0.880933\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 8.00000i 0.414781i
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 12.0000i 0.618031i
\(378\) − 2.00000i − 0.102869i
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000i 0.609994i
\(388\) − 6.00000i − 0.304604i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) 8.00000i 0.403547i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) − 2.00000i − 0.100251i
\(399\) 0 0
\(400\) 0 0
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 16.0000i − 0.797017i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) − 14.0000i − 0.689730i
\(413\) 24.0000i 1.18096i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) 20.0000i 0.973585i
\(423\) − 8.00000i − 0.388973i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 14.0000i 0.676716i
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) − 10.0000i − 0.477818i
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) − 18.0000i − 0.851371i
\(448\) − 2.00000i − 0.0944911i
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −60.0000 −2.82529
\(452\) 8.00000i 0.376288i
\(453\) 12.0000i 0.563809i
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 24.0000i 1.12145i
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) − 12.0000i − 0.558291i
\(463\) − 40.0000i − 1.85896i −0.368875 0.929479i \(-0.620257\pi\)
0.368875 0.929479i \(-0.379743\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) − 18.0000i − 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) − 12.0000i − 0.552345i
\(473\) 72.0000i 3.31056i
\(474\) 6.00000 0.275589
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 16.0000i 0.731823i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 6.00000i − 0.273293i
\(483\) − 2.00000i − 0.0910032i
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 8.00000i − 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) − 4.00000i − 0.181071i
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 10.0000i 0.450835i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 14.0000i 0.627355i
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) − 6.00000i − 0.267793i
\(503\) − 12.0000i − 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) − 9.00000i − 0.399704i
\(508\) 12.0000i 0.532414i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) − 48.0000i − 2.11104i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 6.00000i 0.261116i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 16.0000i − 0.690451i
\(538\) − 18.0000i − 0.776035i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 8.00000i − 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 1.00000i 0.0425628i
\(553\) 12.0000i 0.510292i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 16.0000i 0.674919i
\(563\) 14.0000i 0.590030i 0.955493 + 0.295015i \(0.0953246\pi\)
−0.955493 + 0.295015i \(0.904675\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −24.0000 −1.00880
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −16.0000 −0.670755 −0.335377 0.942084i \(-0.608864\pi\)
−0.335377 + 0.942084i \(0.608864\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) − 12.0000i − 0.501745i
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 6.00000i 0.248708i
\(583\) − 12.0000i − 0.496989i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 16.0000i − 0.660391i −0.943913 0.330195i \(-0.892885\pi\)
0.943913 0.330195i \(-0.107115\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 2.00000i 0.0818546i
\(598\) − 2.00000i − 0.0817861i
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 24.0000i 0.978167i
\(603\) − 12.0000i − 0.488678i
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 4.00000i 0.162355i 0.996700 + 0.0811775i \(0.0258681\pi\)
−0.996700 + 0.0811775i \(0.974132\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) − 4.00000i − 0.161558i −0.996732 0.0807792i \(-0.974259\pi\)
0.996732 0.0807792i \(-0.0257409\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 32.0000i 1.28827i 0.764911 + 0.644136i \(0.222783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 14.0000i 0.563163i
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 16.0000i 0.641542i
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 16.0000i 0.638470i
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) − 6.00000i − 0.238667i
\(633\) − 20.0000i − 0.794929i
\(634\) 22.0000 0.873732
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) − 6.00000i − 0.237729i
\(638\) 36.0000i 1.42525i
\(639\) 0 0
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) − 14.0000i − 0.552536i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 40.0000i 1.57256i 0.617869 + 0.786281i \(0.287996\pi\)
−0.617869 + 0.786281i \(0.712004\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −72.0000 −2.82625
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 12.0000i 0.469956i
\(653\) − 22.0000i − 0.860927i −0.902608 0.430463i \(-0.858350\pi\)
0.902608 0.430463i \(-0.141650\pi\)
\(654\) 16.0000 0.625650
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 10.0000i 0.390137i
\(658\) − 16.0000i − 0.623745i
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000i 0.232321i
\(668\) 8.00000i 0.309529i
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 2.00000i 0.0771517i
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) − 8.00000i − 0.307238i
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) − 48.0000i − 1.83801i
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 24.0000i − 0.915657i
\(688\) − 12.0000i − 0.457496i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 12.0000i 0.455842i
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) − 26.0000i − 0.984115i
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 0 0
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) − 12.0000i − 0.451306i
\(708\) 12.0000i 0.450988i
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) − 8.00000i − 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) − 16.0000i − 0.597531i
\(718\) − 24.0000i − 0.895672i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) − 19.0000i − 0.707107i
\(723\) 6.00000i 0.223142i
\(724\) −8.00000 −0.297318
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 18.0000i 0.667583i 0.942647 + 0.333792i \(0.108328\pi\)
−0.942647 + 0.333792i \(0.891672\pi\)
\(728\) − 4.00000i − 0.148250i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 4.00000i 0.147844i
\(733\) 36.0000i 1.32969i 0.746981 + 0.664845i \(0.231502\pi\)
−0.746981 + 0.664845i \(0.768498\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 72.0000i − 2.65215i
\(738\) − 10.0000i − 0.368105i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 4.00000i − 0.146845i
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) − 14.0000i − 0.512233i
\(748\) 0 0
\(749\) 28.0000 1.02310
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 6.00000i 0.218652i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 32.0000i 1.16306i 0.813525 + 0.581530i \(0.197546\pi\)
−0.813525 + 0.581530i \(0.802454\pi\)
\(758\) 0 0
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 32.0000i 1.15848i
\(764\) 0 0
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) − 24.0000i − 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 14.0000i − 0.503871i
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 6.00000i − 0.214423i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) − 6.00000i − 0.213201i
\(793\) − 8.00000i − 0.284088i
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 32.0000i − 1.12996i
\(803\) 60.0000i 2.11735i
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 18.0000i 0.633630i
\(808\) 6.00000i 0.211079i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 12.0000i 0.421117i
\(813\) − 28.0000i − 0.982003i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000i 0.349642i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 12.0000i 0.418548i
\(823\) − 24.0000i − 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 2.00000i 0.0693375i
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 26.0000i 0.898155i
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 4.00000i 0.137849i
\(843\) − 16.0000i − 0.551069i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 50.0000i 1.71802i
\(848\) 2.00000i 0.0686803i
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) − 2.00000i − 0.0683187i −0.999416 0.0341593i \(-0.989125\pi\)
0.999416 0.0341593i \(-0.0108754\pi\)
\(858\) 12.0000i 0.409673i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) − 12.0000i − 0.408722i
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) − 17.0000i − 0.577350i
\(868\) − 16.0000i − 0.543075i
\(869\) −36.0000 −1.22122
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) − 16.0000i − 0.541828i
\(873\) − 6.00000i − 0.203069i
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 16.0000i 0.539974i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 16.0000i 0.537227i 0.963248 + 0.268614i \(0.0865655\pi\)
−0.963248 + 0.268614i \(0.913434\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) − 12.0000i − 0.401790i
\(893\) 0 0
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 2.00000i 0.0667781i
\(898\) − 2.00000i − 0.0667409i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) − 60.0000i − 1.99778i
\(903\) − 24.0000i − 0.798670i
\(904\) −8.00000 −0.266076
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) − 16.0000i − 0.531271i −0.964073 0.265636i \(-0.914418\pi\)
0.964073 0.265636i \(-0.0855818\pi\)
\(908\) − 10.0000i − 0.331862i
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) − 84.0000i − 2.77999i
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −24.0000 −0.792982
\(917\) − 16.0000i − 0.528367i
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) − 30.0000i − 0.987997i
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) − 14.0000i − 0.459820i
\(928\) − 6.00000i − 0.196960i
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 10.0000i − 0.327561i
\(933\) − 16.0000i − 0.523816i
\(934\) 18.0000 0.588978
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) − 24.0000i − 0.783628i
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) − 16.0000i − 0.521308i
\(943\) − 10.0000i − 0.325645i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −72.0000 −2.34092
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 6.00000i 0.194871i
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) − 36.0000i − 1.16371i
\(958\) − 24.0000i − 0.775405i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 14.0000i 0.451144i
\(964\) 6.00000 0.193247
\(965\) 0 0
\(966\) 2.00000 0.0643489
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) − 25.0000i − 0.803530i
\(969\) 0 0
\(970\) 0 0
\(971\) −10.0000 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 24.0000i − 0.769405i
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) 0 0
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) − 32.0000i − 1.02116i
\(983\) 52.0000i 1.65854i 0.558846 + 0.829271i \(0.311244\pi\)
−0.558846 + 0.829271i \(0.688756\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000i 0.509286i
\(988\) 0 0
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 4.00000i − 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 20.0000i 0.633089i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.d.j.2899.2 2
5.2 odd 4 138.2.a.a.1.1 1
5.3 odd 4 3450.2.a.y.1.1 1
5.4 even 2 inner 3450.2.d.j.2899.1 2
15.2 even 4 414.2.a.d.1.1 1
20.7 even 4 1104.2.a.e.1.1 1
35.27 even 4 6762.2.a.q.1.1 1
40.27 even 4 4416.2.a.m.1.1 1
40.37 odd 4 4416.2.a.z.1.1 1
60.47 odd 4 3312.2.a.n.1.1 1
115.22 even 4 3174.2.a.b.1.1 1
345.137 odd 4 9522.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 1 5.2 odd 4
414.2.a.d.1.1 1 15.2 even 4
1104.2.a.e.1.1 1 20.7 even 4
3174.2.a.b.1.1 1 115.22 even 4
3312.2.a.n.1.1 1 60.47 odd 4
3450.2.a.y.1.1 1 5.3 odd 4
3450.2.d.j.2899.1 2 5.4 even 2 inner
3450.2.d.j.2899.2 2 1.1 even 1 trivial
4416.2.a.m.1.1 1 40.27 even 4
4416.2.a.z.1.1 1 40.37 odd 4
6762.2.a.q.1.1 1 35.27 even 4
9522.2.a.i.1.1 1 345.137 odd 4