Properties

Label 4650.2.d.b.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.b.3349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +8.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} -1.00000i q^{27} -2.00000i q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +12.0000i q^{37} +8.00000i q^{38} +4.00000 q^{39} +10.0000 q^{41} -2.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +4.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000i q^{57} -4.00000i q^{58} -2.00000 q^{59} +10.0000 q^{61} -1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +6.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +6.00000 q^{71} +1.00000i q^{72} -4.00000i q^{73} -12.0000 q^{74} -8.00000 q^{76} -8.00000i q^{77} +4.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} +10.0000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -8.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +8.00000 q^{91} +8.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} +18.0000i q^{97} +3.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{11} - 4 q^{14} + 2 q^{16} + 16 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 8 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 8 q^{39} + 20 q^{41} + 8 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} + 2 q^{54} + 4 q^{56} - 4 q^{59} + 20 q^{61} - 2 q^{64} + 8 q^{66} + 16 q^{69} + 12 q^{71} - 24 q^{74} - 16 q^{76} + 16 q^{79} + 2 q^{81} + 4 q^{84} - 16 q^{86} + 16 q^{91} - 8 q^{94} - 2 q^{96} + 8 q^{99} + O(q^{100}) \)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) − 4.00000i − 0.852803i
\(23\) − 8.00000i − 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −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
\(33\) − 4.00000i − 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 12.0000i 1.97279i 0.164399 + 0.986394i \(0.447432\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 4.00000i 0.554700i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000i 1.05963i
\(58\) − 4.00000i − 0.525226i
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 6.00000i 0.733017i 0.930415 + 0.366508i \(0.119447\pi\)
−0.930415 + 0.366508i \(0.880553\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) − 8.00000i − 0.911685i
\(78\) 4.00000i 0.452911i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) − 4.00000i − 0.428845i
\(88\) 4.00000i 0.426401i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 8.00000i 0.834058i
\(93\) − 1.00000i − 0.103695i
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 2.00000i 0.198030i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 2.00000i 0.188982i
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 4.00000i 0.369800i
\(118\) − 2.00000i − 0.184115i
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) 10.0000i 0.901670i
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 16.0000i 1.38738i
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 8.00000i 0.681005i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 6.00000i 0.503509i
\(143\) 16.0000i 1.33799i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 3.00000i 0.247436i
\(148\) − 12.0000i − 0.986394i
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 8.00000i − 0.648886i
\(153\) 2.00000i 0.161690i
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000i 0.0785674i
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 2.00000i 0.154303i
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) − 8.00000i − 0.609994i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 2.00000i − 0.150329i
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 8.00000i 0.592999i
\(183\) 10.0000i 0.739221i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 8.00000i 0.585018i
\(188\) − 4.00000i − 0.291730i
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 10.0000i − 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 4.00000i 0.284268i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) − 18.0000i − 1.26648i
\(203\) − 8.00000i − 0.561490i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 8.00000i 0.556038i
\(208\) − 4.00000i − 0.277350i
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 6.00000i 0.411113i
\(214\) 8.00000 0.546869
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 2.00000i − 0.135769i
\(218\) − 18.0000i − 1.21911i
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 12.0000i − 0.805387i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 4.00000i 0.262613i
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 2.00000 0.130189
\(237\) 8.00000i 0.519656i
\(238\) 4.00000i 0.259281i
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) − 32.0000i − 2.03611i
\(248\) 1.00000i 0.0635001i
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 32.0000i 2.01182i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 10.0000i 0.617802i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) − 6.00000i − 0.366508i
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 8.00000i 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) 2.00000i 0.118888i 0.998232 + 0.0594438i \(0.0189327\pi\)
−0.998232 + 0.0594438i \(0.981067\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 20.0000i 1.18056i
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 4.00000i 0.234082i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 4.00000i 0.232104i
\(298\) 18.0000i 1.04271i
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) − 16.0000i − 0.920697i
\(303\) − 18.0000i − 1.03407i
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) 8.00000i 0.455842i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000i 1.23564i 0.786318 + 0.617822i \(0.211985\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 16.0000i 0.891645i
\(323\) − 16.0000i − 0.890264i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) − 18.0000i − 0.995402i
\(328\) − 10.0000i − 0.552158i
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) − 12.0000i − 0.657596i
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 32.0000i 1.74315i 0.490261 + 0.871576i \(0.336901\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) − 8.00000i − 0.432590i
\(343\) 20.0000i 1.07990i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 4.00000i 0.214423i
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) − 4.00000i − 0.213201i
\(353\) 34.0000i 1.80964i 0.425797 + 0.904819i \(0.359994\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 2.00000 0.106299
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 20.0000i 1.05703i
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 18.0000i 0.946059i
\(363\) 5.00000i 0.262432i
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 1.00000i 0.0518476i
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 16.0000i 0.824042i
\(378\) 2.00000i 0.102869i
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 18.0000i 0.920960i
\(383\) − 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
\(386\) 2.00000 0.101797
\(387\) − 8.00000i − 0.406663i
\(388\) − 18.0000i − 0.913812i
\(389\) 28.0000 1.41966 0.709828 0.704375i \(-0.248773\pi\)
0.709828 + 0.704375i \(0.248773\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 3.00000i − 0.151523i
\(393\) 10.0000i 0.504433i
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) − 6.00000i − 0.299253i
\(403\) 4.00000i 0.199254i
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) − 48.0000i − 2.37927i
\(408\) − 2.00000i − 0.0990148i
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 14.0000i 0.689730i
\(413\) − 4.00000i − 0.196827i
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) − 4.00000i − 0.195881i
\(418\) − 32.0000i − 1.56517i
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 24.0000i 1.16830i
\(423\) − 4.00000i − 0.194487i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 20.0000i 0.967868i
\(428\) 8.00000i 0.386695i
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) − 64.0000i − 3.06154i
\(438\) 4.00000i 0.191127i
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) − 8.00000i − 0.380521i
\(443\) 4.00000i 0.190046i 0.995475 + 0.0950229i \(0.0302924\pi\)
−0.995475 + 0.0950229i \(0.969708\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 18.0000i 0.851371i
\(448\) − 2.00000i − 0.0944911i
\(449\) −28.0000 −1.32140 −0.660701 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 6.00000i 0.282216i
\(453\) − 16.0000i − 0.751746i
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) − 36.0000i − 1.68401i −0.539471 0.842004i \(-0.681376\pi\)
0.539471 0.842004i \(-0.318624\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 8.00000i 0.372194i
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 2.00000i 0.0920575i
\(473\) − 32.0000i − 1.47136i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) − 6.00000i − 0.274721i
\(478\) 4.00000i 0.182956i
\(479\) 10.0000 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 10.0000i 0.455488i
\(483\) 16.0000i 0.728025i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) 8.00000i 0.360302i
\(494\) 32.0000 1.43975
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 12.0000i 0.538274i
\(498\) − 4.00000i − 0.179244i
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 20.0000i 0.892644i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) − 3.00000i − 0.133235i
\(508\) 4.00000i 0.177471i
\(509\) −16.0000 −0.709188 −0.354594 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000i 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) − 16.0000i − 0.703679i
\(518\) − 24.0000i − 1.05450i
\(519\) −22.0000 −0.965693
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000i 0.0871214i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) − 16.0000i − 0.693688i
\(533\) − 40.0000i − 1.73259i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 20.0000i 0.863064i
\(538\) 12.0000i 0.517357i
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 8.00000i 0.343629i
\(543\) 18.0000i 0.772454i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 2.00000i − 0.0855138i −0.999086 0.0427569i \(-0.986386\pi\)
0.999086 0.0427569i \(-0.0136141\pi\)
\(548\) 6.00000i 0.256307i
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) − 8.00000i − 0.340503i
\(553\) 16.0000i 0.680389i
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) − 6.00000i − 0.254228i −0.991888 0.127114i \(-0.959429\pi\)
0.991888 0.127114i \(-0.0405714\pi\)
\(558\) 1.00000i 0.0423334i
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 6.00000i 0.253095i
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 2.00000i 0.0839921i
\(568\) − 6.00000i − 0.251754i
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) − 16.0000i − 0.668994i
\(573\) 18.0000i 0.751961i
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 6.00000i 0.249783i 0.992170 + 0.124892i \(0.0398583\pi\)
−0.992170 + 0.124892i \(0.960142\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) − 18.0000i − 0.746124i
\(583\) − 24.0000i − 0.993978i
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 28.0000i − 1.15568i −0.816149 0.577842i \(-0.803895\pi\)
0.816149 0.577842i \(-0.196105\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 12.0000i 0.493197i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) − 16.0000i − 0.654836i
\(598\) − 32.0000i − 1.30858i
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) − 6.00000i − 0.244339i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 14.0000i 0.568242i 0.958788 + 0.284121i \(0.0917018\pi\)
−0.958788 + 0.284121i \(0.908298\pi\)
\(608\) 8.00000i 0.324443i
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) − 2.00000i − 0.0808452i
\(613\) 8.00000i 0.323117i 0.986863 + 0.161558i \(0.0516520\pi\)
−0.986863 + 0.161558i \(0.948348\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 14.0000i 0.563163i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) − 30.0000i − 1.20289i
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −20.0000 −0.799361
\(627\) − 32.0000i − 1.27796i
\(628\) − 22.0000i − 0.877896i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 24.0000i 0.953914i
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 12.0000i − 0.475457i
\(638\) 16.0000i 0.633446i
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 8.00000i 0.315735i
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) − 8.00000i − 0.314512i −0.987558 0.157256i \(-0.949735\pi\)
0.987558 0.157256i \(-0.0502649\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 6.00000i 0.234978i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 18.0000 0.703856
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) 4.00000i 0.156055i
\(658\) − 8.00000i − 0.311872i
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) − 8.00000i − 0.310694i
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 32.0000i 1.23904i
\(668\) − 8.00000i − 0.309529i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) − 2.00000i − 0.0771517i
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) 6.00000i 0.230429i
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 4.00000i 0.153168i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) − 22.0000i − 0.839352i
\(688\) 8.00000i 0.304997i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) − 22.0000i − 0.836315i
\(693\) 8.00000i 0.303895i
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) − 20.0000i − 0.757554i
\(698\) 30.0000i 1.13552i
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 96.0000i 3.62071i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) − 36.0000i − 1.35392i
\(708\) 2.00000i 0.0751646i
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 4.00000i 0.149383i
\(718\) − 10.0000i − 0.373197i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 45.0000i 1.67473i
\(723\) 10.0000i 0.371904i
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 2.00000i 0.0741759i 0.999312 + 0.0370879i \(0.0118082\pi\)
−0.999312 + 0.0370879i \(0.988192\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) − 10.0000i − 0.369611i
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) − 24.0000i − 0.884051i
\(738\) − 10.0000i − 0.368105i
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 32.0000 1.17555
\(742\) − 12.0000i − 0.440534i
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) − 4.00000i − 0.146352i
\(748\) − 8.00000i − 0.292509i
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 20.0000i 0.728841i
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) − 36.0000i − 1.30844i −0.756303 0.654221i \(-0.772997\pi\)
0.756303 0.654221i \(-0.227003\pi\)
\(758\) − 16.0000i − 0.581146i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) −20.0000 −0.724999 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 4.00000i 0.144905i
\(763\) − 36.0000i − 1.30329i
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 8.00000i 0.288863i
\(768\) 1.00000i 0.0360844i
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000i 0.0719816i
\(773\) 14.0000i 0.503545i 0.967786 + 0.251773i \(0.0810135\pi\)
−0.967786 + 0.251773i \(0.918987\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) − 24.0000i − 0.860995i
\(778\) 28.0000i 1.00385i
\(779\) 80.0000 2.86630
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) − 16.0000i − 0.572159i
\(783\) 4.00000i 0.142948i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 10.0000i 0.356235i
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) − 4.00000i − 0.142134i
\(793\) − 40.0000i − 1.42044i
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 4.00000i 0.141245i
\(803\) 16.0000i 0.564628i
\(804\) 6.00000 0.211604
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 12.0000i 0.422420i
\(808\) 18.0000i 0.633238i
\(809\) −28.0000 −0.984428 −0.492214 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 8.00000i 0.280745i
\(813\) 8.00000i 0.280572i
\(814\) 48.0000 1.68240
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 64.0000i 2.23908i
\(818\) − 6.00000i − 0.209785i
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) 6.00000i 0.209274i
\(823\) 28.0000i 0.976019i 0.872838 + 0.488009i \(0.162277\pi\)
−0.872838 + 0.488009i \(0.837723\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) − 36.0000i − 1.25184i −0.779886 0.625921i \(-0.784723\pi\)
0.779886 0.625921i \(-0.215277\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 4.00000i 0.138675i
\(833\) − 6.00000i − 0.207888i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 32.0000 1.10674
\(837\) 1.00000i 0.0345651i
\(838\) 10.0000i 0.345444i
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 2.00000i 0.0689246i
\(843\) 6.00000i 0.206651i
\(844\) −24.0000 −0.826114
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 10.0000i 0.343604i
\(848\) 6.00000i 0.206041i
\(849\) −2.00000 −0.0686398
\(850\) 0 0
\(851\) 96.0000 3.29084
\(852\) − 6.00000i − 0.205557i
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) − 16.0000i − 0.546231i
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −20.0000 −0.681598
\(862\) − 30.0000i − 1.02180i
\(863\) 16.0000i 0.544646i 0.962206 + 0.272323i \(0.0877920\pi\)
−0.962206 + 0.272323i \(0.912208\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −4.00000 −0.135926
\(867\) 13.0000i 0.441503i
\(868\) 2.00000i 0.0678844i
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 18.0000i 0.609557i
\(873\) − 18.0000i − 0.609208i
\(874\) 64.0000 2.16483
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 40.0000i 1.34993i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) 20.0000i 0.673054i 0.941674 + 0.336527i \(0.109252\pi\)
−0.941674 + 0.336527i \(0.890748\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 56.0000i 1.88030i 0.340766 + 0.940148i \(0.389313\pi\)
−0.340766 + 0.940148i \(0.610687\pi\)
\(888\) 12.0000i 0.402694i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000i 0.267860i
\(893\) 32.0000i 1.07084i
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) − 32.0000i − 1.06845i
\(898\) − 28.0000i − 0.934372i
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 40.0000i − 1.33185i
\(903\) − 16.0000i − 0.532447i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 10.0000i 0.332045i 0.986122 + 0.166022i \(0.0530924\pi\)
−0.986122 + 0.166022i \(0.946908\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 8.00000i 0.264906i
\(913\) − 16.0000i − 0.529523i
\(914\) 36.0000 1.19077
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 20.0000i 0.660458i
\(918\) − 2.00000i − 0.0660098i
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) − 40.0000i − 1.31733i
\(923\) − 24.0000i − 0.789970i
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 14.0000i 0.459820i
\(928\) − 4.00000i − 0.131306i
\(929\) −32.0000 −1.04989 −0.524943 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 10.0000i 0.327561i
\(933\) − 30.0000i − 0.982156i
\(934\) 0 0
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) − 22.0000i − 0.716799i
\(943\) − 80.0000i − 2.60516i
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −16.0000 −0.519382
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) − 4.00000i − 0.129641i
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) 16.0000i 0.517207i
\(958\) 10.0000i 0.323085i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 48.0000i 1.54758i
\(963\) 8.00000i 0.257796i
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) −2.00000 −0.0641831 −0.0320915 0.999485i \(-0.510217\pi\)
−0.0320915 + 0.999485i \(0.510217\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 8.00000i − 0.256468i
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 46.0000i 1.47167i 0.677161 + 0.735835i \(0.263210\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(978\) 6.00000i 0.191859i
\(979\) 0 0
\(980\) 0 0
\(981\) 18.0000 0.574696
\(982\) − 8.00000i − 0.255290i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) − 8.00000i − 0.254643i
\(988\) 32.0000i 1.01806i
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) − 1.00000i − 0.0317500i
\(993\) − 12.0000i − 0.380808i
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 36.0000i 1.13956i
\(999\) 12.0000 0.379663
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.b.3349.2 2
5.2 odd 4 930.2.a.h.1.1 1
5.3 odd 4 4650.2.a.bg.1.1 1
5.4 even 2 inner 4650.2.d.b.3349.1 2
15.2 even 4 2790.2.a.p.1.1 1
20.7 even 4 7440.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 1 5.2 odd 4
2790.2.a.p.1.1 1 15.2 even 4
4650.2.a.bg.1.1 1 5.3 odd 4
4650.2.d.b.3349.1 2 5.4 even 2 inner
4650.2.d.b.3349.2 2 1.1 even 1 trivial
7440.2.a.m.1.1 1 20.7 even 4