Properties

Label 3450.2.d.a.2899.1
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 690)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} +10.0000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} +14.0000i q^{53} +1.00000 q^{54} +4.00000i q^{57} -6.00000i q^{58} +10.0000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} +6.00000i q^{68} -1.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} -2.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -6.00000i q^{78} +12.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +16.0000i q^{83} -4.00000 q^{86} -6.00000i q^{87} -4.00000i q^{88} +2.00000 q^{89} +1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -14.0000i q^{97} -7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} + 2q^{24} + 12q^{26} + 12q^{29} - 16q^{31} - 12q^{34} + 2q^{36} + 12q^{39} + 20q^{41} + 8q^{44} - 2q^{46} + 14q^{49} - 12q^{51} + 2q^{54} + 20q^{61} - 2q^{64} + 8q^{66} - 2q^{69} + 16q^{71} + 12q^{74} + 8q^{76} + 24q^{79} + 2q^{81} - 8q^{86} + 4q^{89} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100}) \)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) − 1.00000i − 0.208514i
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 6.00000i − 0.832050i
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) − 6.00000i − 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) − 6.00000i − 0.679366i
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 6.00000i − 0.643268i
\(88\) − 4.00000i − 0.426401i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000i 0.104257i
\(93\) 8.00000i 0.829561i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 6.00000i 0.594089i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 10.0000i − 0.905357i
\(123\) − 10.0000i − 0.901670i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000i 1.06483i 0.846484 + 0.532414i \(0.178715\pi\)
−0.846484 + 0.532414i \(0.821285\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 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
\(142\) − 8.00000i − 0.671345i
\(143\) − 24.0000i − 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 7.00000i − 0.577350i
\(148\) − 6.00000i − 0.493197i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) − 12.0000i − 0.954669i
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000i 0.304997i
\(173\) − 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
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) − 2.00000i − 0.149906i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 24.0000i 1.75505i
\(188\) 8.00000i 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 26.0000i 1.85242i 0.377004 + 0.926212i \(0.376954\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) − 18.0000i − 1.26648i
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 1.00000i 0.0695048i
\(208\) 6.00000i 0.416025i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) − 14.0000i − 0.961524i
\(213\) − 8.00000i − 0.548151i
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) − 6.00000i − 0.402694i
\(223\) 20.0000i 1.33930i 0.742677 + 0.669650i \(0.233556\pi\)
−0.742677 + 0.669650i \(0.766444\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 22.0000i − 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) − 12.0000i − 0.779484i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) − 24.0000i − 1.52708i
\(248\) − 8.00000i − 0.508001i
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 4.00000i 0.251478i
\(254\) 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 \(0.940080\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) − 8.00000i − 0.494242i
\(263\) 32.0000i 1.97320i 0.163144 + 0.986602i \(0.447836\pi\)
−0.163144 + 0.986602i \(0.552164\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.00000i − 0.122398i
\(268\) − 4.00000i − 0.244339i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) − 30.0000i − 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 8.00000i 0.476393i
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 2.00000i 0.117041i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) − 4.00000i − 0.232104i
\(298\) 10.0000i 0.579284i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) − 16.0000i − 0.920697i
\(303\) − 18.0000i − 1.03407i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 6.00000i 0.339683i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 14.0000i − 0.785081i
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 2.00000i 0.110600i
\(328\) 10.0000i 0.552158i
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) − 6.00000i − 0.328798i
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 6.00000i 0.321634i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 4.00000i 0.213201i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 6.00000i 0.315353i
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) − 32.0000i − 1.67039i −0.549957 0.835193i \(-0.685356\pi\)
0.549957 0.835193i \(-0.314644\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) − 8.00000i − 0.414781i
\(373\) 18.0000i 0.932005i 0.884783 + 0.466002i \(0.154306\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 8.00000i 0.409316i
\(383\) − 16.0000i − 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 4.00000i 0.203331i
\(388\) 14.0000i 0.710742i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 7.00000i 0.353553i
\(393\) − 8.00000i − 0.403547i
\(394\) 26.0000 1.30986
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 12.0000i − 0.601506i
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) − 48.0000i − 2.39105i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) − 6.00000i − 0.297044i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 12.0000i 0.587643i
\(418\) − 16.0000i − 0.782586i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 8.00000i 0.388973i
\(424\) −14.0000 −0.679900
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) − 8.00000i − 0.386695i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 4.00000i 0.191346i
\(438\) 2.00000i 0.0955637i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 36.0000i − 1.71235i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 20.0000 0.947027
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) − 14.0000i − 0.658505i
\(453\) − 16.0000i − 0.751746i
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) 0 0
\(477\) − 14.0000i − 0.641016i
\(478\) − 8.00000i − 0.365911i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) − 2.00000i − 0.0910975i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 10.0000i 0.452679i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 10.0000i 0.450835i
\(493\) − 36.0000i − 1.62136i
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) − 16.0000i − 0.716977i
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 12.0000i 0.535586i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 23.0000i 1.02147i
\(508\) − 12.0000i − 0.532414i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 6.00000i 0.262613i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 48.0000i 2.09091i
\(528\) 4.00000i 0.174078i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 60.0000i 2.59889i
\(534\) −2.00000 −0.0865485
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 10.0000i 0.431131i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 8.00000i 0.343629i
\(543\) 6.00000i 0.257485i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 2.00000i − 0.0854358i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) − 1.00000i − 0.0425628i
\(553\) 0 0
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) − 30.0000i − 1.26547i
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 8.00000i 0.335673i
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 24.0000i 1.00349i
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000i 1.41544i 0.706494 + 0.707719i \(0.250276\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000i 0.580319i
\(583\) − 56.0000i − 2.31928i
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) − 4.00000i − 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) 7.00000i 0.288675i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) 26.0000 1.06950
\(592\) 6.00000i 0.246598i
\(593\) − 6.00000i − 0.246390i −0.992382 0.123195i \(-0.960686\pi\)
0.992382 0.123195i \(-0.0393141\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) − 12.0000i − 0.491127i
\(598\) − 6.00000i − 0.245358i
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) − 6.00000i − 0.242536i
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 8.00000i 0.320771i
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) − 16.0000i − 0.638978i
\(628\) − 6.00000i − 0.239426i
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 12.0000i 0.477334i
\(633\) 20.0000i 0.794929i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −14.0000 −0.555136
\(637\) 42.0000i 1.66410i
\(638\) 24.0000i 0.950169i
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) − 8.00000i − 0.315735i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 34.0000 1.32245 0.661223 0.750189i \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) 20.0000i 0.777322i
\(663\) − 36.0000i − 1.39812i
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) − 6.00000i − 0.232321i
\(668\) 16.0000i 0.619059i
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) − 32.0000i − 1.22534i
\(683\) − 44.0000i − 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.0000i − 0.534133i
\(688\) − 4.00000i − 0.152499i
\(689\) −84.0000 −3.20015
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 60.0000i − 2.27266i
\(698\) − 34.0000i − 1.28692i
\(699\) −22.0000 −0.832116
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 6.00000i 0.226455i
\(703\) − 24.0000i − 0.905177i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 2.00000i 0.0749532i
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 8.00000i − 0.298765i
\(718\) − 8.00000i − 0.298557i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000i 0.111648i
\(723\) − 2.00000i − 0.0743808i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 10.0000i 0.369611i
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) − 16.0000i − 0.589368i
\(738\) 10.0000i 0.368105i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) − 16.0000i − 0.585409i
\(748\) − 24.0000i − 0.877527i
\(749\) 0 0
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 12.0000i 0.437304i
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 54.0000i 1.96266i 0.192323 + 0.981332i \(0.438398\pi\)
−0.192323 + 0.981332i \(0.561602\pi\)
\(758\) 4.00000i 0.145287i
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) − 12.0000i − 0.434714i
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 10.0000i 0.359908i
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 6.00000i 0.214560i
\(783\) 6.00000i 0.214423i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 36.0000i 1.28326i 0.767014 + 0.641631i \(0.221742\pi\)
−0.767014 + 0.641631i \(0.778258\pi\)
\(788\) − 26.0000i − 0.926212i
\(789\) 32.0000 1.13923
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000i 0.142134i
\(793\) 60.0000i 2.13066i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) − 6.00000i − 0.211867i
\(803\) 8.00000i 0.282314i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 10.0000i 0.352017i
\(808\) 18.0000i 0.633238i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 16.0000i 0.559769i
\(818\) 10.0000i 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) − 2.00000i − 0.0697580i
\(823\) − 20.0000i − 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) − 1.00000i − 0.0347524i
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) − 6.00000i − 0.208013i
\(833\) − 42.0000i − 1.45521i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) − 8.00000i − 0.276520i
\(838\) − 12.0000i − 0.414533i
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 38.0000i 1.30957i
\(843\) − 30.0000i − 1.03325i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) 14.0000i 0.480762i
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 8.00000i 0.274075i
\(853\) 22.0000i 0.753266i 0.926363 + 0.376633i \(0.122918\pi\)
−0.926363 + 0.376633i \(0.877082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 24.0000i 0.819346i
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 8.00000i − 0.272481i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 19.0000i 0.645274i
\(868\) 0 0
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) − 2.00000i − 0.0677285i
\(873\) 14.0000i 0.473828i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 7.00000i 0.235702i
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 20.0000i − 0.669650i
\(893\) 32.0000i 1.07084i
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.00000i − 0.200334i
\(898\) 10.0000i 0.333704i
\(899\) −48.0000 −1.60089
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) 40.0000i 1.33185i
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) − 4.00000i − 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000i 0.132453i
\(913\) − 64.0000i − 2.11809i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) − 6.00000i − 0.198030i
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 2.00000i − 0.0658665i
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) 20.0000 0.657241
\(927\) − 16.0000i − 0.525509i
\(928\) − 6.00000i − 0.196960i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 22.0000i 0.720634i
\(933\) 8.00000i 0.261908i
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 10.0000i 0.326686i 0.986569 + 0.163343i \(0.0522277\pi\)
−0.986569 + 0.163343i \(0.947772\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) − 6.00000i − 0.195491i
\(943\) − 10.0000i − 0.325645i
\(944\) 0 0
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 20.0000i − 0.649913i −0.945729 0.324956i \(-0.894650\pi\)
0.945729 0.324956i \(-0.105350\pi\)
\(948\) 12.0000i 0.389742i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) −14.0000 −0.453267
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 24.0000i 0.775810i
\(958\) 24.0000i 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 36.0000i 1.16069i
\(963\) − 8.00000i − 0.257796i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) 28.0000i 0.900419i 0.892923 + 0.450210i \(0.148651\pi\)
−0.892923 + 0.450210i \(0.851349\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 4.00000i 0.127906i
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 24.0000i 0.763542i
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) −16.0000 −0.506979
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 12.0000i 0.379853i
\(999\) −6.00000 −0.189832
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.a.2899.1 2
5.2 odd 4 3450.2.a.p.1.1 1
5.3 odd 4 690.2.a.e.1.1 1
5.4 even 2 inner 3450.2.d.a.2899.2 2
15.8 even 4 2070.2.a.r.1.1 1
20.3 even 4 5520.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.e.1.1 1 5.3 odd 4
2070.2.a.r.1.1 1 15.8 even 4
3450.2.a.p.1.1 1 5.2 odd 4
3450.2.d.a.2899.1 2 1.1 even 1 trivial
3450.2.d.a.2899.2 2 5.4 even 2 inner
5520.2.a.f.1.1 1 20.3 even 4