Properties

Label 75.2.b.b.49.2
Level $75$
Weight $2$
Character 75.49
Analytic conductor $0.599$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
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 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.2.b.b.49.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} -4.00000i q^{22} +3.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.00000 q^{29} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +10.0000i q^{37} -4.00000i q^{38} -2.00000 q^{39} +10.0000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -2.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -7.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -2.00000i q^{68} -8.00000 q^{71} -3.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} -2.00000i q^{78} +1.00000 q^{81} +10.0000i q^{82} +12.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} -12.0000i q^{88} +6.00000 q^{89} +8.00000 q^{94} +5.00000 q^{96} -2.00000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} - 2q^{16} - 8q^{19} + 6q^{24} + 4q^{26} + 4q^{29} + 4q^{34} - 2q^{36} - 4q^{39} + 20q^{41} - 8q^{44} + 14q^{49} - 4q^{51} - 2q^{54} + 8q^{59} - 4q^{61} - 14q^{64} - 8q^{66} - 16q^{71} - 20q^{74} - 8q^{76} + 2q^{81} - 8q^{86} + 12q^{89} + 16q^{94} + 10q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) − 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 \(\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −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
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 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
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 0 0
\(47\) − 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
\(51\) −2.00000 −0.280056
\(52\) − 2.00000i − 0.277350i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 2.00000i 0.262613i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 2.00000i − 0.214423i
\(88\) − 12.0000i − 1.27920i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000i 0.184900i
\(118\) 4.00000i 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) − 10.0000i − 0.901670i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 8.00000i − 0.671345i
\(143\) 8.00000i 0.668994i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) − 7.00000i − 0.577350i
\(148\) 10.0000i 0.821995i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 12.0000i − 0.973329i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 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
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 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 \(-0.239873\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) 6.00000i 0.449719i
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) − 8.00000i − 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 7.00000i 0.505181i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 2.00000i 0.138675i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) 8.00000i 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) − 14.0000i − 0.948200i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 10.0000i 0.671156i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000i 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 12.0000i − 0.741362i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) − 12.0000i − 0.733017i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 2.00000i 0.121268i
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) − 5.00000i − 0.294628i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 10.0000i 0.585206i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −30.0000 −1.74371
\(297\) − 4.00000i − 0.232104i
\(298\) − 22.0000i − 1.27443i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.00000i − 0.460348i
\(303\) − 6.00000i − 0.344691i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 8.00000i 0.445132i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 14.0000i 0.774202i
\(328\) 30.0000i 1.65647i
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 10.0000i − 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000i 0.216295i
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) − 20.0000i − 1.06600i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) − 20.0000i − 1.05703i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 10.0000i − 0.525588i
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 24.0000i 1.25279i 0.779506 + 0.626395i \(0.215470\pi\)
−0.779506 + 0.626395i \(0.784530\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) − 4.00000i − 0.206010i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 16.0000i 0.818631i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) − 4.00000i − 0.203331i
\(388\) − 2.00000i − 0.101535i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 21.0000i 1.06066i
\(393\) 12.0000i 0.605320i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) − 40.0000i − 1.98273i
\(408\) − 6.00000i − 0.297044i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) − 16.0000i − 0.788263i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) − 4.00000i − 0.195881i
\(418\) 16.0000i 0.782586i
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 8.00000i 0.388973i
\(424\) 30.0000 1.45693
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 12.0000i 0.580042i
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) 10.0000i 0.477818i
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 4.00000i − 0.190261i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 22.0000i 1.04056i
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 2.00000i 0.0940721i
\(453\) 8.00000i 0.375873i
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) 0 0
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 12.0000i 0.552345i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 16.0000i 0.731823i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) − 14.0000i − 0.637683i
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) − 4.00000i − 0.180151i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) − 32.0000i − 1.42681i −0.700752 0.713405i \(-0.747152\pi\)
0.700752 0.713405i \(-0.252848\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 8.00000i 0.354943i
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11.0000i − 0.486136i
\(513\) − 4.00000i − 0.176604i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) − 4.00000i − 0.174078i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 36.0000 1.55496
\(537\) 20.0000i 0.863064i
\(538\) − 14.0000i − 0.603583i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 16.0000i 0.687259i
\(543\) 10.0000i 0.429141i
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) 0 0
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) − 6.00000i − 0.253095i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 0 0
\(568\) − 24.0000i − 1.00702i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 8.00000i 0.334497i
\(573\) − 16.0000i − 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) − 2.00000i − 0.0829027i
\(583\) 40.0000i 1.65663i
\(584\) −30.0000 −1.24141
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 10.0000i − 0.410997i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) − 8.00000i − 0.327418i
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 20.0000i − 0.811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 2.00000i 0.0808452i
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) − 16.0000i − 0.643614i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) − 16.0000i − 0.638978i
\(628\) − 14.0000i − 0.558661i
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) − 20.0000i − 0.794929i
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −10.0000 −0.396526
\(637\) − 14.0000i − 0.554700i
\(638\) − 8.00000i − 0.316723i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 12.0000i 0.473602i
\(643\) − 36.0000i − 1.41970i −0.704352 0.709851i \(-0.748762\pi\)
0.704352 0.709851i \(-0.251238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) − 32.0000i − 1.25805i −0.777385 0.629025i \(-0.783454\pi\)
0.777385 0.629025i \(-0.216546\pi\)
\(648\) 3.00000i 0.117851i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) − 4.00000i − 0.156652i
\(653\) 46.0000i 1.80012i 0.435767 + 0.900060i \(0.356477\pi\)
−0.435767 + 0.900060i \(0.643523\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000i 0.466393i
\(663\) 4.00000i 0.155347i
\(664\) −36.0000 −1.39707
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) − 30.0000i − 1.15642i −0.815890 0.578208i \(-0.803752\pi\)
0.815890 0.578208i \(-0.196248\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 2.00000i 0.0768095i
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 36.0000i 1.37750i 0.724998 + 0.688751i \(0.241841\pi\)
−0.724998 + 0.688751i \(0.758159\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 6.00000i 0.228914i
\(688\) − 4.00000i − 0.152499i
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) − 18.0000i − 0.684257i
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 20.0000i − 0.757554i
\(698\) 2.00000i 0.0757011i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) − 40.0000i − 1.50863i
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) − 4.00000i − 0.150329i
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 18.0000i 0.674579i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) − 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
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) 14.0000i 0.520666i
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 16.0000i 0.593407i 0.954970 + 0.296704i \(0.0958873\pi\)
−0.954970 + 0.296704i \(0.904113\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 2.00000i 0.0739221i
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000i 1.76810i
\(738\) − 10.0000i − 0.368105i
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) − 12.0000i − 0.439057i
\(748\) 8.00000i 0.292509i
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 8.00000i 0.291730i
\(753\) − 12.0000i − 0.437304i
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 8.00000i − 0.288863i
\(768\) 17.0000i 0.613435i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000i 0.0719816i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 2.00000i 0.0714742i
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 12.0000i 0.426401i
\(793\) 4.00000i 0.142044i
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 2.00000i 0.0708436i 0.999372 + 0.0354218i \(0.0112775\pi\)
−0.999372 + 0.0354218i \(0.988723\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 18.0000i 0.635602i
\(803\) − 40.0000i − 1.41157i
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000i 0.492823i
\(808\) 18.0000i 0.633238i
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) − 16.0000i − 0.561144i
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) − 16.0000i − 0.559769i
\(818\) − 26.0000i − 0.909069i
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) 48.0000 1.67216
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000i 0.973655i 0.873498 + 0.486828i \(0.161846\pi\)
−0.873498 + 0.486828i \(0.838154\pi\)
\(828\) 0 0
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 14.0000i 0.485363i
\(833\) − 14.0000i − 0.485071i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) − 4.00000i − 0.138178i
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 26.0000i − 0.896019i
\(843\) 6.00000i 0.206651i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 10.0000i 0.343401i
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 8.00000i 0.274075i
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) 8.00000i 0.273115i
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 56.0000i − 1.90626i −0.302558 0.953131i \(-0.597840\pi\)
0.302558 0.953131i \(-0.402160\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 14.0000 0.475739
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) − 42.0000i − 1.42230i
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) − 30.0000i − 1.01303i −0.862232 0.506514i \(-0.830934\pi\)
0.862232 0.506514i \(-0.169066\pi\)
\(878\) − 40.0000i − 1.34993i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 30.0000i 1.00673i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000i 0.267860i
\(893\) 32.0000i 1.07084i
\(894\) −22.0000 −0.735790
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) − 2.00000i − 0.0667409i
\(899\) 0 0
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) − 40.0000i − 1.33185i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 48.0000i − 1.58857i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) − 18.0000i − 0.592798i
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 16.0000i 0.525509i
\(928\) 10.0000i 0.328266i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) − 6.00000i − 0.196537i
\(933\) 24.0000i 0.785725i
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 54.0000i 1.76410i 0.471153 + 0.882052i \(0.343838\pi\)
−0.471153 + 0.882052i \(0.656162\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) − 22.0000i − 0.712650i −0.934362 0.356325i \(-0.884030\pi\)
0.934362 0.356325i \(-0.115970\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 8.00000i 0.258603i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 20.0000i 0.644826i
\(963\) − 12.0000i − 0.386695i
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 15.0000i 0.482118i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 2.00000i − 0.0639857i −0.999488 0.0319928i \(-0.989815\pi\)
0.999488 0.0319928i \(-0.0101854\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 28.0000i 0.893516i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 30.0000 0.956365
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) − 12.0000i − 0.380808i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 54.0000i − 1.71020i −0.518465 0.855099i \(-0.673497\pi\)
0.518465 0.855099i \(-0.326503\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) −10.0000 −0.316386
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 75.2.b.b.49.2 2
3.2 odd 2 225.2.b.b.199.1 2
4.3 odd 2 1200.2.f.h.49.2 2
5.2 odd 4 15.2.a.a.1.1 1
5.3 odd 4 75.2.a.b.1.1 1
5.4 even 2 inner 75.2.b.b.49.1 2
8.3 odd 2 4800.2.f.c.3649.1 2
8.5 even 2 4800.2.f.bf.3649.2 2
12.11 even 2 3600.2.f.e.2449.1 2
15.2 even 4 45.2.a.a.1.1 1
15.8 even 4 225.2.a.b.1.1 1
15.14 odd 2 225.2.b.b.199.2 2
20.3 even 4 1200.2.a.e.1.1 1
20.7 even 4 240.2.a.d.1.1 1
20.19 odd 2 1200.2.f.h.49.1 2
35.2 odd 12 735.2.i.e.361.1 2
35.12 even 12 735.2.i.d.361.1 2
35.13 even 4 3675.2.a.j.1.1 1
35.17 even 12 735.2.i.d.226.1 2
35.27 even 4 735.2.a.c.1.1 1
35.32 odd 12 735.2.i.e.226.1 2
40.3 even 4 4800.2.a.bz.1.1 1
40.13 odd 4 4800.2.a.t.1.1 1
40.19 odd 2 4800.2.f.c.3649.2 2
40.27 even 4 960.2.a.a.1.1 1
40.29 even 2 4800.2.f.bf.3649.1 2
40.37 odd 4 960.2.a.l.1.1 1
45.2 even 12 405.2.e.c.271.1 2
45.7 odd 12 405.2.e.f.271.1 2
45.22 odd 12 405.2.e.f.136.1 2
45.32 even 12 405.2.e.c.136.1 2
55.32 even 4 1815.2.a.d.1.1 1
55.43 even 4 9075.2.a.g.1.1 1
60.23 odd 4 3600.2.a.u.1.1 1
60.47 odd 4 720.2.a.c.1.1 1
60.59 even 2 3600.2.f.e.2449.2 2
65.12 odd 4 2535.2.a.j.1.1 1
80.27 even 4 3840.2.k.r.1921.1 2
80.37 odd 4 3840.2.k.m.1921.2 2
80.67 even 4 3840.2.k.r.1921.2 2
80.77 odd 4 3840.2.k.m.1921.1 2
85.67 odd 4 4335.2.a.c.1.1 1
95.37 even 4 5415.2.a.j.1.1 1
105.62 odd 4 2205.2.a.i.1.1 1
115.22 even 4 7935.2.a.d.1.1 1
120.77 even 4 2880.2.a.y.1.1 1
120.107 odd 4 2880.2.a.bc.1.1 1
165.32 odd 4 5445.2.a.c.1.1 1
195.77 even 4 7605.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.2.a.a.1.1 1 5.2 odd 4
45.2.a.a.1.1 1 15.2 even 4
75.2.a.b.1.1 1 5.3 odd 4
75.2.b.b.49.1 2 5.4 even 2 inner
75.2.b.b.49.2 2 1.1 even 1 trivial
225.2.a.b.1.1 1 15.8 even 4
225.2.b.b.199.1 2 3.2 odd 2
225.2.b.b.199.2 2 15.14 odd 2
240.2.a.d.1.1 1 20.7 even 4
405.2.e.c.136.1 2 45.32 even 12
405.2.e.c.271.1 2 45.2 even 12
405.2.e.f.136.1 2 45.22 odd 12
405.2.e.f.271.1 2 45.7 odd 12
720.2.a.c.1.1 1 60.47 odd 4
735.2.a.c.1.1 1 35.27 even 4
735.2.i.d.226.1 2 35.17 even 12
735.2.i.d.361.1 2 35.12 even 12
735.2.i.e.226.1 2 35.32 odd 12
735.2.i.e.361.1 2 35.2 odd 12
960.2.a.a.1.1 1 40.27 even 4
960.2.a.l.1.1 1 40.37 odd 4
1200.2.a.e.1.1 1 20.3 even 4
1200.2.f.h.49.1 2 20.19 odd 2
1200.2.f.h.49.2 2 4.3 odd 2
1815.2.a.d.1.1 1 55.32 even 4
2205.2.a.i.1.1 1 105.62 odd 4
2535.2.a.j.1.1 1 65.12 odd 4
2880.2.a.y.1.1 1 120.77 even 4
2880.2.a.bc.1.1 1 120.107 odd 4
3600.2.a.u.1.1 1 60.23 odd 4
3600.2.f.e.2449.1 2 12.11 even 2
3600.2.f.e.2449.2 2 60.59 even 2
3675.2.a.j.1.1 1 35.13 even 4
3840.2.k.m.1921.1 2 80.77 odd 4
3840.2.k.m.1921.2 2 80.37 odd 4
3840.2.k.r.1921.1 2 80.27 even 4
3840.2.k.r.1921.2 2 80.67 even 4
4335.2.a.c.1.1 1 85.67 odd 4
4800.2.a.t.1.1 1 40.13 odd 4
4800.2.a.bz.1.1 1 40.3 even 4
4800.2.f.c.3649.1 2 8.3 odd 2
4800.2.f.c.3649.2 2 40.19 odd 2
4800.2.f.bf.3649.1 2 40.29 even 2
4800.2.f.bf.3649.2 2 8.5 even 2
5415.2.a.j.1.1 1 95.37 even 4
5445.2.a.c.1.1 1 165.32 odd 4
7605.2.a.g.1.1 1 195.77 even 4
7935.2.a.d.1.1 1 115.22 even 4
9075.2.a.g.1.1 1 55.43 even 4