Properties

Label 2310.2.a.s.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2310.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} -6.00000 q^{67} +6.00000 q^{69} +1.00000 q^{70} +4.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +4.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -4.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} +6.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −4.00000 −0.648886
\(39\) 4.00000 0.640513
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 1.00000 0.123091
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 1.00000 0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 4.00000 0.464991
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 4.00000 0.452911
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 4.00000 0.392232
\(105\) 1.00000 0.0975900
\(106\) −10.0000 −0.971286
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 4.00000 0.379663
\(112\) −1.00000 −0.0944911
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −4.00000 −0.374634
\(115\) −6.00000 −0.559503
\(116\) −2.00000 −0.185695
\(117\) 4.00000 0.369800
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) −4.00000 −0.350823
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 1.00000 0.0870388
\(133\) 4.00000 0.346844
\(134\) −6.00000 −0.518321
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 6.00000 0.510754
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.00000 0.673722
\(142\) 4.00000 0.335673
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) 4.00000 0.328798
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 1.00000 0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −8.00000 −0.642575
\(156\) 4.00000 0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 4.00000 0.318223
\(159\) −10.0000 −0.793052
\(160\) −1.00000 −0.0790569
\(161\) −6.00000 −0.472866
\(162\) 1.00000 0.0785674
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 10.0000 0.780869
\(165\) −1.00000 −0.0778499
\(166\) 4.00000 0.310460
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −8.00000 −0.609994
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 4.00000 0.300658
\(178\) −4.00000 −0.299813
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000 0.147844
\(184\) 6.00000 0.442326
\(185\) −4.00000 −0.294086
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 1.00000 0.0710669
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.00000 −0.423207
\(202\) −6.00000 −0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 8.00000 0.557386
\(207\) 6.00000 0.417029
\(208\) 4.00000 0.277350
\(209\) −4.00000 −0.276686
\(210\) 1.00000 0.0690066
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) −10.0000 −0.686803
\(213\) 4.00000 0.274075
\(214\) 4.00000 0.273434
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) −8.00000 −0.543075
\(218\) −12.0000 −0.812743
\(219\) −2.00000 −0.135147
\(220\) −1.00000 −0.0674200
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 16.0000 1.06430
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −4.00000 −0.264906
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −6.00000 −0.395628
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 4.00000 0.261488
\(235\) −8.00000 −0.521862
\(236\) 4.00000 0.260378
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) 10.0000 0.637577
\(247\) −16.0000 −1.01806
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −8.00000 −0.498058
\(259\) −4.00000 −0.248548
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) 4.00000 0.247121
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 1.00000 0.0615457
\(265\) 10.0000 0.614295
\(266\) 4.00000 0.245256
\(267\) −4.00000 −0.244796
\(268\) −6.00000 −0.366508
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) −8.00000 −0.483298
\(275\) 1.00000 0.0603023
\(276\) 6.00000 0.361158
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −8.00000 −0.479808
\(279\) 8.00000 0.478947
\(280\) 1.00000 0.0597614
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 8.00000 0.476393
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 4.00000 0.237356
\(285\) 4.00000 0.236940
\(286\) 4.00000 0.236525
\(287\) −10.0000 −0.590281
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 2.00000 0.117444
\(291\) 2.00000 0.117242
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) 4.00000 0.232495
\(297\) 1.00000 0.0580259
\(298\) 2.00000 0.115857
\(299\) 24.0000 1.38796
\(300\) 1.00000 0.0577350
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000 0.226455
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) −10.0000 −0.564333
\(315\) 1.00000 0.0563436
\(316\) 4.00000 0.225018
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) −10.0000 −0.560772
\(319\) −2.00000 −0.111979
\(320\) −1.00000 −0.0559017
\(321\) 4.00000 0.223258
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) −6.00000 −0.332309
\(327\) −12.0000 −0.663602
\(328\) 10.0000 0.552158
\(329\) −8.00000 −0.441054
\(330\) −1.00000 −0.0550482
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 4.00000 0.219529
\(333\) 4.00000 0.219199
\(334\) 10.0000 0.547176
\(335\) 6.00000 0.327815
\(336\) −1.00000 −0.0545545
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 3.00000 0.163178
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) −6.00000 −0.323029
\(346\) −14.0000 −0.752645
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 4.00000 0.213504
\(352\) 1.00000 0.0533002
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 4.00000 0.212598
\(355\) −4.00000 −0.212298
\(356\) −4.00000 −0.212000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) −8.00000 −0.420471
\(363\) 1.00000 0.0524864
\(364\) −4.00000 −0.209657
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 6.00000 0.312772
\(369\) 10.0000 0.520579
\(370\) −4.00000 −0.207950
\(371\) 10.0000 0.519174
\(372\) 8.00000 0.414781
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) −8.00000 −0.412021
\(378\) −1.00000 −0.0514344
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 4.00000 0.205196
\(381\) −8.00000 −0.409852
\(382\) −24.0000 −1.22795
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.00000 0.0509647
\(386\) −2.00000 −0.101797
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 4.00000 0.201773
\(394\) −14.0000 −0.705310
\(395\) −4.00000 −0.201262
\(396\) 1.00000 0.0502519
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −16.0000 −0.802008
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −6.00000 −0.299253
\(403\) 32.0000 1.59403
\(404\) −6.00000 −0.298511
\(405\) −1.00000 −0.0496904
\(406\) 2.00000 0.0992583
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −10.0000 −0.493865
\(411\) −8.00000 −0.394611
\(412\) 8.00000 0.394132
\(413\) −4.00000 −0.196827
\(414\) 6.00000 0.294884
\(415\) −4.00000 −0.196352
\(416\) 4.00000 0.196116
\(417\) −8.00000 −0.391762
\(418\) −4.00000 −0.195646
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 1.00000 0.0487950
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 8.00000 0.388973
\(424\) −10.0000 −0.485643
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) −2.00000 −0.0967868
\(428\) 4.00000 0.193347
\(429\) 4.00000 0.193122
\(430\) 8.00000 0.385794
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −8.00000 −0.384012
\(435\) 2.00000 0.0958927
\(436\) −12.0000 −0.574696
\(437\) −24.0000 −1.14808
\(438\) −2.00000 −0.0955637
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 4.00000 0.189832
\(445\) 4.00000 0.189618
\(446\) −8.00000 −0.378811
\(447\) 2.00000 0.0945968
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) 10.0000 0.470882
\(452\) 16.0000 0.752577
\(453\) 4.00000 0.187936
\(454\) 12.0000 0.563188
\(455\) 4.00000 0.187523
\(456\) −4.00000 −0.187317
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −8.00000 −0.370991
\(466\) 18.0000 0.833834
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 4.00000 0.184900
\(469\) 6.00000 0.277054
\(470\) −8.00000 −0.369012
\(471\) −10.0000 −0.460776
\(472\) 4.00000 0.184115
\(473\) −8.00000 −0.367840
\(474\) 4.00000 0.183726
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −6.00000 −0.274434
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 16.0000 0.729537
\(482\) −10.0000 −0.455488
\(483\) −6.00000 −0.273009
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 2.00000 0.0905357
\(489\) −6.00000 −0.271329
\(490\) −1.00000 −0.0451754
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) −1.00000 −0.0449467
\(496\) 8.00000 0.359211
\(497\) −4.00000 −0.179425
\(498\) 4.00000 0.179244
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 10.0000 0.446767
\(502\) 12.0000 0.535586
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 6.00000 0.266996
\(506\) 6.00000 0.266733
\(507\) 3.00000 0.133235
\(508\) −8.00000 −0.354943
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) −8.00000 −0.352180
\(517\) 8.00000 0.351840
\(518\) −4.00000 −0.175750
\(519\) −14.0000 −0.614532
\(520\) −4.00000 −0.175412
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 4.00000 0.174741
\(525\) −1.00000 −0.0436436
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 1.00000 0.0435194
\(529\) 13.0000 0.565217
\(530\) 10.0000 0.434372
\(531\) 4.00000 0.173585
\(532\) 4.00000 0.173422
\(533\) 40.0000 1.73259
\(534\) −4.00000 −0.173097
\(535\) −4.00000 −0.172935
\(536\) −6.00000 −0.259161
\(537\) −4.00000 −0.172613
\(538\) 6.00000 0.258678
\(539\) 1.00000 0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 24.0000 1.03089
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) 12.0000 0.514024
\(546\) −4.00000 −0.171184
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −8.00000 −0.341743
\(549\) 2.00000 0.0853579
\(550\) 1.00000 0.0426401
\(551\) 8.00000 0.340811
\(552\) 6.00000 0.255377
\(553\) −4.00000 −0.170097
\(554\) 2.00000 0.0849719
\(555\) −4.00000 −0.169791
\(556\) −8.00000 −0.339276
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 8.00000 0.338667
\(559\) −32.0000 −1.35346
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) 8.00000 0.337160 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(564\) 8.00000 0.336861
\(565\) −16.0000 −0.673125
\(566\) −14.0000 −0.588464
\(567\) −1.00000 −0.0419961
\(568\) 4.00000 0.167836
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 4.00000 0.167542
\(571\) −6.00000 −0.251092 −0.125546 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(572\) 4.00000 0.167248
\(573\) −24.0000 −1.00261
\(574\) −10.0000 −0.417392
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −17.0000 −0.707107
\(579\) −2.00000 −0.0831172
\(580\) 2.00000 0.0830455
\(581\) −4.00000 −0.165948
\(582\) 2.00000 0.0829027
\(583\) −10.0000 −0.414158
\(584\) −2.00000 −0.0827606
\(585\) −4.00000 −0.165380
\(586\) 6.00000 0.247858
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.0000 −1.31854
\(590\) −4.00000 −0.164677
\(591\) −14.0000 −0.575883
\(592\) 4.00000 0.164399
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) −16.0000 −0.654836
\(598\) 24.0000 0.981433
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 8.00000 0.326056
\(603\) −6.00000 −0.244339
\(604\) 4.00000 0.162758
\(605\) −1.00000 −0.0406558
\(606\) −6.00000 −0.243733
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −4.00000 −0.162221
\(609\) 2.00000 0.0810441
\(610\) −2.00000 −0.0809776
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) −22.0000 −0.887848
\(615\) −10.0000 −0.403239
\(616\) −1.00000 −0.0402911
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 8.00000 0.321807
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) −8.00000 −0.321288
\(621\) 6.00000 0.240772
\(622\) −18.0000 −0.721734
\(623\) 4.00000 0.160257
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 22.0000 0.879297
\(627\) −4.00000 −0.159745
\(628\) −10.0000 −0.399043
\(629\) 0 0
\(630\) 1.00000 0.0398410
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 4.00000 0.159111
\(633\) −2.00000 −0.0794929
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) −10.0000 −0.396526
\(637\) 4.00000 0.158486
\(638\) −2.00000 −0.0791808
\(639\) 4.00000 0.158238
\(640\) −1.00000 −0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 4.00000 0.157867
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −6.00000 −0.236433
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000 0.0392837
\(649\) 4.00000 0.157014
\(650\) 4.00000 0.156893
\(651\) −8.00000 −0.313545
\(652\) −6.00000 −0.234978
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) −12.0000 −0.469237
\(655\) −4.00000 −0.156293
\(656\) 10.0000 0.390434
\(657\) −2.00000 −0.0780274
\(658\) −8.00000 −0.311872
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −48.0000 −1.86698 −0.933492 0.358599i \(-0.883255\pi\)
−0.933492 + 0.358599i \(0.883255\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −4.00000 −0.155113
\(666\) 4.00000 0.154997
\(667\) −12.0000 −0.464642
\(668\) 10.0000 0.386912
\(669\) −8.00000 −0.309298
\(670\) 6.00000 0.231800
\(671\) 2.00000 0.0772091
\(672\) −1.00000 −0.0385758
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 16.0000 0.614476
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 8.00000 0.306336
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) −4.00000 −0.152944
\(685\) 8.00000 0.305664
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) −8.00000 −0.304997
\(689\) −40.0000 −1.52388
\(690\) −6.00000 −0.228416
\(691\) −30.0000 −1.14125 −0.570627 0.821209i \(-0.693300\pi\)
−0.570627 + 0.821209i \(0.693300\pi\)
\(692\) −14.0000 −0.532200
\(693\) −1.00000 −0.0379869
\(694\) 28.0000 1.06287
\(695\) 8.00000 0.303457
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) 18.0000 0.680823
\(700\) −1.00000 −0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 4.00000 0.150970
\(703\) −16.0000 −0.603451
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) 18.0000 0.677439
\(707\) 6.00000 0.225653
\(708\) 4.00000 0.150329
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) −4.00000 −0.150117
\(711\) 4.00000 0.150012
\(712\) −4.00000 −0.149906
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) −4.00000 −0.149487
\(717\) −6.00000 −0.224074
\(718\) −18.0000 −0.671754
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) −3.00000 −0.111648
\(723\) −10.0000 −0.371904
\(724\) −8.00000 −0.297318
\(725\) −2.00000 −0.0742781
\(726\) 1.00000 0.0371135
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −24.0000 −0.885856
\(735\) −1.00000 −0.0368856
\(736\) 6.00000 0.221163
\(737\) −6.00000 −0.221013
\(738\) 10.0000 0.368105
\(739\) 26.0000 0.956425 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) −4.00000 −0.147043
\(741\) −16.0000 −0.587775
\(742\) 10.0000 0.367112
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 8.00000 0.293294
\(745\) −2.00000 −0.0732743
\(746\) −34.0000 −1.24483
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) −8.00000 −0.291343
\(755\) −4.00000 −0.145575
\(756\) −1.00000 −0.0363696
\(757\) −36.0000 −1.30844 −0.654221 0.756303i \(-0.727003\pi\)
−0.654221 + 0.756303i \(0.727003\pi\)
\(758\) −12.0000 −0.435860
\(759\) 6.00000 0.217786
\(760\) 4.00000 0.145095
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −8.00000 −0.289809
\(763\) 12.0000 0.434429
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 1.00000 0.0360844
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 1.00000 0.0360375
\(771\) 18.0000 0.648254
\(772\) −2.00000 −0.0719816
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) −8.00000 −0.287554
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) −4.00000 −0.143499
\(778\) 6.00000 0.215110
\(779\) −40.0000 −1.43315
\(780\) −4.00000 −0.143223
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 4.00000 0.142675
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) −14.0000 −0.498729
\(789\) −4.00000 −0.142404
\(790\) −4.00000 −0.142314
\(791\) −16.0000 −0.568895
\(792\) 1.00000 0.0355335
\(793\) 8.00000 0.284088
\(794\) 18.0000 0.638796
\(795\) 10.0000 0.354663
\(796\) −16.0000 −0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −4.00000 −0.141333
\(802\) 10.0000 0.353112
\(803\) −2.00000 −0.0705785
\(804\) −6.00000 −0.211604
\(805\) 6.00000 0.211472
\(806\) 32.0000 1.12715
\(807\) 6.00000 0.211210
\(808\) −6.00000 −0.211079
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 2.00000 0.0701862
\(813\) 24.0000 0.841717
\(814\) 4.00000 0.140200
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) −22.0000 −0.769212
\(819\) −4.00000 −0.139771
\(820\) −10.0000 −0.349215
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −8.00000 −0.279032
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) −4.00000 −0.139178
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 6.00000 0.208514
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) −4.00000 −0.138842
\(831\) 2.00000 0.0693792
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) −10.0000 −0.346064
\(836\) −4.00000 −0.138343
\(837\) 8.00000 0.276520
\(838\) −16.0000 −0.552711
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 1.00000 0.0345033
\(841\) −25.0000 −0.862069
\(842\) 30.0000 1.03387
\(843\) −16.0000 −0.551069
\(844\) −2.00000 −0.0688428
\(845\) −3.00000 −0.103203
\(846\) 8.00000 0.275046
\(847\) −1.00000 −0.0343604
\(848\) −10.0000 −0.343401
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 4.00000 0.137038
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 4.00000 0.136797
\(856\) 4.00000 0.136717
\(857\) 44.0000 1.50301 0.751506 0.659727i \(-0.229328\pi\)
0.751506 + 0.659727i \(0.229328\pi\)
\(858\) 4.00000 0.136558
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 8.00000 0.272798
\(861\) −10.0000 −0.340799
\(862\) −30.0000 −1.02180
\(863\) −34.0000 −1.15737 −0.578687 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 34.0000 1.15537
\(867\) −17.0000 −0.577350
\(868\) −8.00000 −0.271538
\(869\) 4.00000 0.135691
\(870\) 2.00000 0.0678064
\(871\) −24.0000 −0.813209
\(872\) −12.0000 −0.406371
\(873\) 2.00000 0.0676897
\(874\) −24.0000 −0.811812
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) −1.00000 −0.0337100
\(881\) −44.0000 −1.48240 −0.741199 0.671286i \(-0.765742\pi\)
−0.741199 + 0.671286i \(0.765742\pi\)
\(882\) 1.00000 0.0336718
\(883\) 50.0000 1.68263 0.841317 0.540542i \(-0.181781\pi\)
0.841317 + 0.540542i \(0.181781\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 24.0000 0.806296
\(887\) −34.0000 −1.14161 −0.570804 0.821086i \(-0.693368\pi\)
−0.570804 + 0.821086i \(0.693368\pi\)
\(888\) 4.00000 0.134231
\(889\) 8.00000 0.268311
\(890\) 4.00000 0.134080
\(891\) 1.00000 0.0335013
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) 2.00000 0.0668900
\(895\) 4.00000 0.133705
\(896\) −1.00000 −0.0334077
\(897\) 24.0000 0.801337
\(898\) −6.00000 −0.200223
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 10.0000 0.332964
\(903\) 8.00000 0.266223
\(904\) 16.0000 0.532152
\(905\) 8.00000 0.265929
\(906\) 4.00000 0.132891
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) 12.0000 0.398234
\(909\) −6.00000 −0.199007
\(910\) 4.00000 0.132599
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −4.00000 −0.132453
\(913\) 4.00000 0.132381
\(914\) −10.0000 −0.330771
\(915\) −2.00000 −0.0661180
\(916\) −12.0000 −0.396491
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −6.00000 −0.197814
\(921\) −22.0000 −0.724925
\(922\) −2.00000 −0.0658665
\(923\) 16.0000 0.526646
\(924\) −1.00000 −0.0328976
\(925\) 4.00000 0.131519
\(926\) −20.0000 −0.657241
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −8.00000 −0.262330
\(931\) −4.00000 −0.131095
\(932\) 18.0000 0.589610
\(933\) −18.0000 −0.589294
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) 6.00000 0.195907
\(939\) 22.0000 0.717943
\(940\) −8.00000 −0.260931
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) −10.0000 −0.325818
\(943\) 60.0000 1.95387
\(944\) 4.00000 0.130189
\(945\) 1.00000 0.0325300
\(946\) −8.00000 −0.260102
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 4.00000 0.129914
\(949\) −8.00000 −0.259691
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 46.0000 1.49009 0.745043 0.667016i \(-0.232429\pi\)
0.745043 + 0.667016i \(0.232429\pi\)
\(954\) −10.0000 −0.323762
\(955\) 24.0000 0.776622
\(956\) −6.00000 −0.194054
\(957\) −2.00000 −0.0646508
\(958\) −16.0000 −0.516937
\(959\) 8.00000 0.258333
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 16.0000 0.515861
\(963\) 4.00000 0.128898
\(964\) −10.0000 −0.322078
\(965\) 2.00000 0.0643823
\(966\) −6.00000 −0.193047
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000 0.256468
\(974\) 20.0000 0.640841
\(975\) 4.00000 0.128103
\(976\) 2.00000 0.0640184
\(977\) 44.0000 1.40768 0.703842 0.710356i \(-0.251466\pi\)
0.703842 + 0.710356i \(0.251466\pi\)
\(978\) −6.00000 −0.191859
\(979\) −4.00000 −0.127841
\(980\) −1.00000 −0.0319438
\(981\) −12.0000 −0.383131
\(982\) 4.00000 0.127645
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 10.0000 0.318788
\(985\) 14.0000 0.446077
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) −16.0000 −0.509028
\(989\) −48.0000 −1.52631
\(990\) −1.00000 −0.0317821
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 8.00000 0.254000
\(993\) 4.00000 0.126936
\(994\) −4.00000 −0.126872
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) 16.0000 0.506471
\(999\) 4.00000 0.126554
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 2310.2.a.s.1.1 1
3.2 odd 2 6930.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.s.1.1 1 1.1 even 1 trivial
6930.2.a.j.1.1 1 3.2 odd 2