Properties

Label 4650.2.d.y.3349.2
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
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 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.y.3349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} +3.00000 q^{19} +3.00000 q^{21} +3.00000i q^{22} +5.00000i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} -4.00000 q^{34} +1.00000 q^{36} +3.00000i q^{38} -2.00000 q^{39} +4.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +4.00000 q^{51} +2.00000i q^{52} +3.00000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -3.00000i q^{57} -4.00000i q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -2.00000i q^{67} -4.00000i q^{68} +5.00000 q^{69} +7.00000 q^{71} +1.00000i q^{72} +5.00000i q^{73} -3.00000 q^{76} +9.00000i q^{77} -2.00000i q^{78} +1.00000 q^{79} +1.00000 q^{81} +4.00000i q^{82} +12.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +4.00000i q^{87} -3.00000i q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000i q^{92} -1.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} -2.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 6q^{11} - 6q^{14} + 2q^{16} + 6q^{19} + 6q^{21} - 2q^{24} + 4q^{26} - 8q^{29} + 2q^{31} - 8q^{34} + 2q^{36} - 4q^{39} + 8q^{41} - 6q^{44} - 10q^{46} - 4q^{49} + 8q^{51} - 2q^{54} + 6q^{56} - 12q^{59} - 4q^{61} - 2q^{64} + 6q^{66} + 10q^{69} + 14q^{71} - 6q^{76} + 2q^{79} + 2q^{81} - 6q^{84} - 2q^{86} - 2q^{89} + 12q^{91} + 20q^{94} + 2q^{96} - 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 3.00000i 0.639602i
\(23\) 5.00000i 1.04257i 0.853382 + 0.521286i \(0.174548\pi\)
−0.853382 + 0.521286i \(0.825452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 3.00000i − 0.566947i
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 3.00000i 0.486664i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 3.00000i 0.462910i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) − 10.0000i − 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 2.00000i 0.277350i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) − 3.00000i − 0.397360i
\(58\) − 4.00000i − 0.525226i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 5.00000i 0.585206i 0.956234 + 0.292603i \(0.0945214\pi\)
−0.956234 + 0.292603i \(0.905479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 9.00000i 1.02565i
\(78\) − 2.00000i − 0.226455i
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 4.00000i 0.441726i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 4.00000i 0.428845i
\(88\) − 3.00000i − 0.319801i
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) − 5.00000i − 0.521286i
\(93\) − 1.00000i − 0.103695i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 4.00000i 0.396059i
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.00000i 0.283473i
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 2.00000i 0.184900i
\(118\) − 6.00000i − 0.552345i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 2.00000i − 0.181071i
\(123\) − 4.00000i − 0.360668i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 3.00000i 0.261116i
\(133\) 9.00000i 0.780399i
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 10.0000i 0.854358i 0.904167 + 0.427179i \(0.140493\pi\)
−0.904167 + 0.427179i \(0.859507\pi\)
\(138\) 5.00000i 0.425628i
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 7.00000i 0.587427i
\(143\) − 6.00000i − 0.501745i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −5.00000 −0.413803
\(147\) 2.00000i 0.164957i
\(148\) 0 0
\(149\) 11.0000 0.901155 0.450578 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) − 4.00000i − 0.323381i
\(154\) −9.00000 −0.725241
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 5.00000i 0.399043i 0.979893 + 0.199522i \(0.0639388\pi\)
−0.979893 + 0.199522i \(0.936061\pi\)
\(158\) 1.00000i 0.0795557i
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) 1.00000i 0.0785674i
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 19.0000i 1.47026i 0.677924 + 0.735132i \(0.262880\pi\)
−0.677924 + 0.735132i \(0.737120\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) − 1.00000i − 0.0762493i
\(173\) − 22.0000i − 1.67263i −0.548250 0.836315i \(-0.684706\pi\)
0.548250 0.836315i \(-0.315294\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 6.00000i 0.450988i
\(178\) − 1.00000i − 0.0749532i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 2.00000i 0.147844i
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 12.0000i 0.877527i
\(188\) 10.0000i 0.729325i
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) − 1.00000i − 0.0703598i
\(203\) − 12.0000i − 0.842235i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) − 5.00000i − 0.347524i
\(208\) − 2.00000i − 0.138675i
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) − 3.00000i − 0.206041i
\(213\) − 7.00000i − 0.479632i
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 3.00000i 0.203653i
\(218\) − 20.0000i − 1.35457i
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000i 1.79205i 0.444001 + 0.896026i \(0.353559\pi\)
−0.444001 + 0.896026i \(0.646441\pi\)
\(228\) 3.00000i 0.198680i
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) 4.00000i 0.262613i
\(233\) 15.0000i 0.982683i 0.870967 + 0.491341i \(0.163493\pi\)
−0.870967 + 0.491341i \(0.836507\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) − 1.00000i − 0.0649570i
\(238\) − 12.0000i − 0.777844i
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) − 2.00000i − 0.128565i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 4.00000 0.255031
\(247\) − 6.00000i − 0.381771i
\(248\) − 1.00000i − 0.0635001i
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 15.0000i 0.943042i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.0000i 1.43470i 0.696713 + 0.717350i \(0.254645\pi\)
−0.696713 + 0.717350i \(0.745355\pi\)
\(258\) 1.00000i 0.0622573i
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 6.00000i 0.370681i
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −9.00000 −0.551825
\(267\) 1.00000i 0.0611990i
\(268\) 2.00000i 0.122169i
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 4.00000i 0.242536i
\(273\) − 6.00000i − 0.363137i
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) − 12.0000i − 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) 14.0000i 0.839664i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) − 10.0000i − 0.595491i
\(283\) − 24.0000i − 1.42665i −0.700832 0.713326i \(-0.747188\pi\)
0.700832 0.713326i \(-0.252812\pi\)
\(284\) −7.00000 −0.415374
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 12.0000i 0.708338i
\(288\) − 1.00000i − 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 5.00000i − 0.292603i
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000i 0.174078i
\(298\) 11.0000i 0.637213i
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 1.00000i 0.0574485i
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) − 9.00000i − 0.512823i
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000i 0.113228i
\(313\) 30.0000i 1.69570i 0.530236 + 0.847850i \(0.322103\pi\)
−0.530236 + 0.847850i \(0.677897\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 8.00000i 0.449325i 0.974437 + 0.224662i \(0.0721279\pi\)
−0.974437 + 0.224662i \(0.927872\pi\)
\(318\) 3.00000i 0.168232i
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) − 15.0000i − 0.835917i
\(323\) 12.0000i 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 20.0000i 1.10600i
\(328\) − 4.00000i − 0.220863i
\(329\) 30.0000 1.65395
\(330\) 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
\(333\) 0 0
\(334\) −19.0000 −1.03963
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) − 3.00000i − 0.162221i
\(343\) 15.0000i 0.809924i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 3.00000i 0.159901i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) 1.00000 0.0529999
\(357\) 12.0000i 0.635107i
\(358\) − 4.00000i − 0.211407i
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 5.00000i 0.262794i
\(363\) 2.00000i 0.104973i
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) − 2.00000i − 0.104399i −0.998637 0.0521996i \(-0.983377\pi\)
0.998637 0.0521996i \(-0.0166232\pi\)
\(368\) 5.00000i 0.260643i
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 1.00000i 0.0518476i
\(373\) − 9.00000i − 0.466002i −0.972476 0.233001i \(-0.925145\pi\)
0.972476 0.233001i \(-0.0748546\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 8.00000i 0.412021i
\(378\) − 3.00000i − 0.154303i
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 16.0000i 0.818631i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) − 1.00000i − 0.0508329i
\(388\) − 10.0000i − 0.507673i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 2.00000i 0.101015i
\(393\) − 6.00000i − 0.302660i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) − 35.0000i − 1.75660i −0.478110 0.878300i \(-0.658678\pi\)
0.478110 0.878300i \(-0.341322\pi\)
\(398\) − 7.00000i − 0.350878i
\(399\) 9.00000 0.450564
\(400\) 0 0
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) − 2.00000i − 0.0997509i
\(403\) − 2.00000i − 0.0996271i
\(404\) 1.00000 0.0497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) − 4.00000i − 0.198030i
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) − 16.0000i − 0.788263i
\(413\) − 18.0000i − 0.885722i
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) − 14.0000i − 0.685583i
\(418\) 9.00000i 0.440204i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) − 19.0000i − 0.924906i
\(423\) 10.0000i 0.486217i
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 7.00000 0.339151
\(427\) − 6.00000i − 0.290360i
\(428\) 9.00000i 0.435031i
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 15.0000i 0.717547i
\(438\) 5.00000i 0.238909i
\(439\) −18.0000 −0.859093 −0.429547 0.903045i \(-0.641327\pi\)
−0.429547 + 0.903045i \(0.641327\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 8.00000i 0.380521i
\(443\) 15.0000i 0.712672i 0.934358 + 0.356336i \(0.115974\pi\)
−0.934358 + 0.356336i \(0.884026\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.00000 0.284108
\(447\) − 11.0000i − 0.520282i
\(448\) − 3.00000i − 0.141737i
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) − 9.00000i − 0.423324i
\(453\) 0 0
\(454\) −27.0000 −1.26717
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 13.0000i − 0.607450i
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 9.00000i 0.418718i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −15.0000 −0.694862
\(467\) − 24.0000i − 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 5.00000 0.230388
\(472\) 6.00000i 0.276172i
\(473\) 3.00000i 0.137940i
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) − 3.00000i − 0.137361i
\(478\) − 12.0000i − 0.548867i
\(479\) −3.00000 −0.137073 −0.0685367 0.997649i \(-0.521833\pi\)
−0.0685367 + 0.997649i \(0.521833\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 18.0000i − 0.819878i
\(483\) 15.0000i 0.682524i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 32.0000i − 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 2.00000i 0.0905357i
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −37.0000 −1.66979 −0.834893 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) 4.00000i 0.180334i
\(493\) − 16.0000i − 0.720604i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 21.0000i 0.941979i
\(498\) 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
\(501\) 19.0000 0.848857
\(502\) − 12.0000i − 0.535586i
\(503\) 20.0000i 0.891756i 0.895094 + 0.445878i \(0.147108\pi\)
−0.895094 + 0.445878i \(0.852892\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −15.0000 −0.666831
\(507\) − 9.00000i − 0.399704i
\(508\) 8.00000i 0.354943i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) 1.00000i 0.0441942i
\(513\) 3.00000i 0.132453i
\(514\) −23.0000 −1.01449
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) − 30.0000i − 1.31940i
\(518\) 0 0
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 4.00000i 0.175075i
\(523\) − 43.0000i − 1.88026i −0.340818 0.940129i \(-0.610704\pi\)
0.340818 0.940129i \(-0.389296\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 4.00000i 0.174243i
\(528\) − 3.00000i − 0.130558i
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) − 9.00000i − 0.390199i
\(533\) − 8.00000i − 0.346518i
\(534\) −1.00000 −0.0432742
\(535\) 0 0
\(536\) −2.00000 −0.0863868
\(537\) 4.00000i 0.172613i
\(538\) − 2.00000i − 0.0862261i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 13.0000i 0.558398i
\(543\) − 5.00000i − 0.214571i
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 6.00000 0.256776
\(547\) − 22.0000i − 0.940652i −0.882493 0.470326i \(-0.844136\pi\)
0.882493 0.470326i \(-0.155864\pi\)
\(548\) − 10.0000i − 0.427179i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) − 5.00000i − 0.212814i
\(553\) 3.00000i 0.127573i
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 1.00000i 0.0423714i 0.999776 + 0.0211857i \(0.00674412\pi\)
−0.999776 + 0.0211857i \(0.993256\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 16.0000i 0.674919i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 10.0000 0.421076
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) 3.00000i 0.125988i
\(568\) − 7.00000i − 0.293713i
\(569\) 23.0000 0.964210 0.482105 0.876113i \(-0.339872\pi\)
0.482105 + 0.876113i \(0.339872\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 6.00000i 0.250873i
\(573\) − 16.0000i − 0.668410i
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 10.0000i 0.414513i
\(583\) 9.00000i 0.372742i
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) − 44.0000i − 1.81607i −0.418890 0.908037i \(-0.637581\pi\)
0.418890 0.908037i \(-0.362419\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 22.0000i 0.903432i 0.892162 + 0.451716i \(0.149188\pi\)
−0.892162 + 0.451716i \(0.850812\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) −11.0000 −0.450578
\(597\) 7.00000i 0.286491i
\(598\) 10.0000i 0.408930i
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) − 3.00000i − 0.122271i
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) 0 0
\(606\) −1.00000 −0.0406222
\(607\) − 29.0000i − 1.17707i −0.808470 0.588537i \(-0.799704\pi\)
0.808470 0.588537i \(-0.200296\pi\)
\(608\) 3.00000i 0.121666i
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) −20.0000 −0.809113
\(612\) 4.00000i 0.161690i
\(613\) − 22.0000i − 0.888572i −0.895885 0.444286i \(-0.853457\pi\)
0.895885 0.444286i \(-0.146543\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 9.00000 0.362620
\(617\) 27.0000i 1.08698i 0.839416 + 0.543490i \(0.182897\pi\)
−0.839416 + 0.543490i \(0.817103\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) − 3.00000i − 0.120192i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) − 9.00000i − 0.359425i
\(628\) − 5.00000i − 0.199522i
\(629\) 0 0
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) − 1.00000i − 0.0397779i
\(633\) 19.0000i 0.755182i
\(634\) −8.00000 −0.317721
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 4.00000i 0.158486i
\(638\) − 12.0000i − 0.475085i
\(639\) −7.00000 −0.276916
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) − 9.00000i − 0.355202i
\(643\) 19.0000i 0.749287i 0.927169 + 0.374643i \(0.122235\pi\)
−0.927169 + 0.374643i \(0.877765\pi\)
\(644\) 15.0000 0.591083
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) − 3.00000i − 0.117942i −0.998260 0.0589711i \(-0.981218\pi\)
0.998260 0.0589711i \(-0.0187820\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) − 14.0000i − 0.548282i
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) − 5.00000i − 0.195069i
\(658\) 30.0000i 1.16952i
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 12.0000i 0.466393i
\(663\) − 8.00000i − 0.310694i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) − 20.0000i − 0.774403i
\(668\) − 19.0000i − 0.735132i
\(669\) −6.00000 −0.231973
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 3.00000i 0.115728i
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 27.0000i − 1.03769i −0.854867 0.518847i \(-0.826361\pi\)
0.854867 0.518847i \(-0.173639\pi\)
\(678\) 9.00000i 0.345643i
\(679\) −30.0000 −1.15129
\(680\) 0 0
\(681\) 27.0000 1.03464
\(682\) 3.00000i 0.114876i
\(683\) − 19.0000i − 0.727015i −0.931591 0.363507i \(-0.881579\pi\)
0.931591 0.363507i \(-0.118421\pi\)
\(684\) 3.00000 0.114708
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 13.0000i 0.495981i
\(688\) 1.00000i 0.0381246i
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) 22.0000i 0.836315i
\(693\) − 9.00000i − 0.341882i
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 16.0000i 0.606043i
\(698\) 24.0000i 0.908413i
\(699\) 15.0000 0.567352
\(700\) 0 0
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) − 3.00000i − 0.112827i
\(708\) − 6.00000i − 0.225494i
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 1.00000i 0.0374766i
\(713\) 5.00000i 0.187251i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 12.0000i 0.448148i
\(718\) − 15.0000i − 0.559795i
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) − 10.0000i − 0.372161i
\(723\) 18.0000i 0.669427i
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) − 45.0000i − 1.66896i −0.551040 0.834479i \(-0.685769\pi\)
0.551040 0.834479i \(-0.314231\pi\)
\(728\) − 6.00000i − 0.222375i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) − 2.00000i − 0.0739221i
\(733\) − 18.0000i − 0.664845i −0.943131 0.332423i \(-0.892134\pi\)
0.943131 0.332423i \(-0.107866\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) − 6.00000i − 0.221013i
\(738\) − 4.00000i − 0.147242i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) − 9.00000i − 0.330400i
\(743\) − 37.0000i − 1.35740i −0.734416 0.678699i \(-0.762544\pi\)
0.734416 0.678699i \(-0.237456\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 9.00000 0.329513
\(747\) − 12.0000i − 0.439057i
\(748\) − 12.0000i − 0.438763i
\(749\) 27.0000 0.986559
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) − 10.0000i − 0.364662i
\(753\) 12.0000i 0.437304i
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) − 50.0000i − 1.81728i −0.417579 0.908640i \(-0.637121\pi\)
0.417579 0.908640i \(-0.362879\pi\)
\(758\) − 15.0000i − 0.544825i
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) − 8.00000i − 0.289809i
\(763\) − 60.0000i − 2.17215i
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 12.0000i 0.433295i
\(768\) − 1.00000i − 0.0360844i
\(769\) −9.00000 −0.324548 −0.162274 0.986746i \(-0.551883\pi\)
−0.162274 + 0.986746i \(0.551883\pi\)
\(770\) 0 0
\(771\) 23.0000 0.828325
\(772\) 6.00000i 0.215945i
\(773\) − 31.0000i − 1.11499i −0.830179 0.557496i \(-0.811762\pi\)
0.830179 0.557496i \(-0.188238\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 26.0000i 0.932145i
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 21.0000 0.751439
\(782\) − 20.0000i − 0.715199i
\(783\) − 4.00000i − 0.142948i
\(784\) −2.00000 −0.0714286
\(785\) 0 0
\(786\) 6.00000 0.214013
\(787\) 35.0000i 1.24762i 0.781578 + 0.623808i \(0.214415\pi\)
−0.781578 + 0.623808i \(0.785585\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −27.0000 −0.960009
\(792\) 3.00000i 0.106600i
\(793\) 4.00000i 0.142044i
\(794\) 35.0000 1.24210
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) − 2.00000i − 0.0708436i −0.999372 0.0354218i \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) 9.00000i 0.318597i
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) 1.00000 0.0353333
\(802\) 25.0000i 0.882781i
\(803\) 15.0000i 0.529339i
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 2.00000i 0.0704033i
\(808\) 1.00000i 0.0351799i
\(809\) −17.0000 −0.597688 −0.298844 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 12.0000i 0.421117i
\(813\) − 13.0000i − 0.455930i
\(814\) 0 0
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 3.00000i 0.104957i
\(818\) − 8.00000i − 0.279713i
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 40.0000 1.39601 0.698005 0.716093i \(-0.254071\pi\)
0.698005 + 0.716093i \(0.254071\pi\)
\(822\) 10.0000i 0.348790i
\(823\) − 18.0000i − 0.627441i −0.949515 0.313720i \(-0.898425\pi\)
0.949515 0.313720i \(-0.101575\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 5.00000i 0.173762i
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 2.00000i 0.0693375i
\(833\) − 8.00000i − 0.277184i
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) −9.00000 −0.311272
\(837\) 1.00000i 0.0345651i
\(838\) − 12.0000i − 0.414533i
\(839\) −51.0000 −1.76072 −0.880358 0.474310i \(-0.842698\pi\)
−0.880358 + 0.474310i \(0.842698\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) − 22.0000i − 0.758170i
\(843\) − 16.0000i − 0.551069i
\(844\) 19.0000 0.654007
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) − 6.00000i − 0.206162i
\(848\) 3.00000i 0.103020i
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 7.00000i 0.239816i
\(853\) − 41.0000i − 1.40381i −0.712269 0.701907i \(-0.752332\pi\)
0.712269 0.701907i \(-0.247668\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) − 6.00000i − 0.204837i
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 16.0000i 0.544962i
\(863\) 43.0000i 1.46374i 0.681446 + 0.731869i \(0.261351\pi\)
−0.681446 + 0.731869i \(0.738649\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 1.00000 0.0339814
\(867\) − 1.00000i − 0.0339618i
\(868\) − 3.00000i − 0.101827i
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 20.0000i 0.677285i
\(873\) − 10.0000i − 0.338449i
\(874\) −15.0000 −0.507383
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) − 18.0000i − 0.607471i
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 2.00000i 0.0673435i
\(883\) 11.0000i 0.370179i 0.982722 + 0.185090i \(0.0592576\pi\)
−0.982722 + 0.185090i \(0.940742\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −15.0000 −0.503935
\(887\) 46.0000i 1.54453i 0.635301 + 0.772264i \(0.280876\pi\)
−0.635301 + 0.772264i \(0.719124\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 6.00000i 0.200895i
\(893\) − 30.0000i − 1.00391i
\(894\) 11.0000 0.367895
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) − 10.0000i − 0.333890i
\(898\) − 2.00000i − 0.0667409i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 12.0000i 0.399556i
\(903\) 3.00000i 0.0998337i
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000i 1.72663i 0.504664 + 0.863316i \(0.331616\pi\)
−0.504664 + 0.863316i \(0.668384\pi\)
\(908\) − 27.0000i − 0.896026i
\(909\) 1.00000 0.0331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) 36.0000i 1.19143i
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 13.0000 0.429532
\(917\) 18.0000i 0.594412i
\(918\) − 4.00000i − 0.132020i
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) − 14.0000i − 0.460816i
\(924\) −9.00000 −0.296078
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) − 16.0000i − 0.525509i
\(928\) − 4.00000i − 0.131306i
\(929\) 41.0000 1.34517 0.672583 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) − 15.0000i − 0.491341i
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 52.0000i 1.69877i 0.527777 + 0.849383i \(0.323026\pi\)
−0.527777 + 0.849383i \(0.676974\pi\)
\(938\) 6.00000i 0.195907i
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 5.00000i 0.162909i
\(943\) 20.0000i 0.651290i
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) − 30.0000i − 0.974869i −0.873160 0.487435i \(-0.837933\pi\)
0.873160 0.487435i \(-0.162067\pi\)
\(948\) 1.00000i 0.0324785i
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 8.00000 0.259418
\(952\) 12.0000i 0.388922i
\(953\) − 52.0000i − 1.68445i −0.539130 0.842223i \(-0.681247\pi\)
0.539130 0.842223i \(-0.318753\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 12.0000i 0.387905i
\(958\) − 3.00000i − 0.0969256i
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 9.00000i 0.290021i
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) −15.0000 −0.482617
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 2.00000i 0.0642824i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 42.0000i 1.34646i
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 38.0000i − 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 14.0000i 0.447671i
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) − 37.0000i − 1.18072i
\(983\) − 56.0000i − 1.78612i −0.449935 0.893061i \(-0.648553\pi\)
0.449935 0.893061i \(-0.351447\pi\)
\(984\) −4.00000 −0.127515
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) − 30.0000i − 0.954911i
\(988\) 6.00000i 0.190885i
\(989\) −5.00000 −0.158991
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) − 12.0000i − 0.380808i
\(994\) −21.0000 −0.666080
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) 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 4650.2.d.y.3349.2 2
5.2 odd 4 930.2.a.a.1.1 1
5.3 odd 4 4650.2.a.bv.1.1 1
5.4 even 2 inner 4650.2.d.y.3349.1 2
15.2 even 4 2790.2.a.y.1.1 1
20.7 even 4 7440.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.a.1.1 1 5.2 odd 4
2790.2.a.y.1.1 1 15.2 even 4
4650.2.a.bv.1.1 1 5.3 odd 4
4650.2.d.y.3349.1 2 5.4 even 2 inner
4650.2.d.y.3349.2 2 1.1 even 1 trivial
7440.2.a.v.1.1 1 20.7 even 4