Properties

Label 2310.2.a.t.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 $1$

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} +8.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -1.00000 q^{66} -10.0000 q^{67} +6.00000 q^{69} -1.00000 q^{70} +1.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -1.00000 q^{77} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +1.00000 q^{84} -4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} -4.00000 q^{91} +6.00000 q^{92} -4.00000 q^{93} +12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 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
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −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
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 8.00000 1.29777
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 8.00000 1.05963
\(58\) 6.00000 0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −1.00000 −0.123091
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) −1.00000 −0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 8.00000 0.917663
\(77\) −1.00000 −0.113961
\(78\) −4.00000 −0.452911
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) −4.00000 −0.414781
\(94\) 12.0000 1.23771
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −4.00000 −0.392232
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 1.00000 0.0953463
\(111\) 8.00000 0.759326
\(112\) 1.00000 0.0944911
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 8.00000 0.749269
\(115\) −6.00000 −0.559503
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 6.00000 0.541002
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 8.00000 0.693688
\(134\) −10.0000 −0.863868
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 6.00000 0.510754
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) 14.0000 1.15865
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.00000 0.648886
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 8.00000 0.636446
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 6.00000 0.468521
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 1.00000 0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −12.0000 −0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) −4.00000 −0.296500
\(183\) −10.0000 −0.739221
\(184\) 6.00000 0.442326
\(185\) −8.00000 −0.588172
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 1.00000 0.0727393
\(190\) −8.00000 −0.580381
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −10.0000 −0.717958
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.0000 −0.705346
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −16.0000 −1.11477
\(207\) 6.00000 0.417029
\(208\) −4.00000 −0.277350
\(209\) −8.00000 −0.553372
\(210\) −1.00000 −0.0690066
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −16.0000 −1.08366
\(219\) 14.0000 0.946032
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 8.00000 0.529813
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) −6.00000 −0.395628
\(231\) −1.00000 −0.0657952
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) −12.0000 −0.782794
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) −32.0000 −2.03611
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) −6.00000 −0.377217
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) 8.00000 0.497096
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −6.00000 −0.368577
\(266\) 8.00000 0.490511
\(267\) −12.0000 −0.734388
\(268\) −10.0000 −0.610847
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 12.0000 0.724947
\(275\) −1.00000 −0.0603023
\(276\) 6.00000 0.361158
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −4.00000 −0.239904
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 12.0000 0.714590
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 4.00000 0.236525
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) −6.00000 −0.352332
\(291\) −10.0000 −0.586210
\(292\) 14.0000 0.819288
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 8.00000 0.464991
\(297\) −1.00000 −0.0580259
\(298\) −6.00000 −0.347571
\(299\) −24.0000 −1.38796
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) 8.00000 0.458831
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −16.0000 −0.910208
\(310\) 4.00000 0.227185
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 2.00000 0.112867
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 6.00000 0.334367
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 14.0000 0.775388
\(327\) −16.0000 −0.884802
\(328\) 6.00000 0.331295
\(329\) 12.0000 0.661581
\(330\) 1.00000 0.0550482
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) −18.0000 −0.984916
\(335\) 10.0000 0.546358
\(336\) 1.00000 0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 3.00000 0.163178
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 8.00000 0.432590
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) −6.00000 −0.323029
\(346\) −6.00000 −0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 1.00000 0.0534522
\(351\) −4.00000 −0.213504
\(352\) −1.00000 −0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.0000 2.36842
\(362\) −16.0000 −0.840941
\(363\) 1.00000 0.0524864
\(364\) −4.00000 −0.209657
\(365\) −14.0000 −0.732793
\(366\) −10.0000 −0.522708
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 6.00000 0.312772
\(369\) 6.00000 0.312348
\(370\) −8.00000 −0.415900
\(371\) 6.00000 0.311504
\(372\) −4.00000 −0.207390
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) −24.0000 −1.23606
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) −12.0000 −0.613973
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.00000 0.0509647
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 18.0000 0.906827
\(395\) −8.00000 −0.402524
\(396\) −1.00000 −0.0502519
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 20.0000 1.00251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −10.0000 −0.498755
\(403\) 16.0000 0.797017
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) 12.0000 0.591916
\(412\) −16.0000 −0.788263
\(413\) −12.0000 −0.590481
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −4.00000 −0.195881
\(418\) −8.00000 −0.391293
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 14.0000 0.681509
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 4.00000 0.193122
\(430\) 4.00000 0.192897
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −4.00000 −0.192006
\(435\) −6.00000 −0.287678
\(436\) −16.0000 −0.766261
\(437\) 48.0000 2.29615
\(438\) 14.0000 0.668946
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 8.00000 0.379663
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) −6.00000 −0.282529
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) 24.0000 1.12638
\(455\) 4.00000 0.187523
\(456\) 8.00000 0.374634
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −4.00000 −0.184900
\(469\) −10.0000 −0.461757
\(470\) −12.0000 −0.553519
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) 8.00000 0.367452
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 6.00000 0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −32.0000 −1.45907
\(482\) 14.0000 0.637683
\(483\) 6.00000 0.273009
\(484\) 1.00000 0.0454545
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) −10.0000 −0.452679
\(489\) 14.0000 0.633102
\(490\) −1.00000 −0.0451754
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 6.00000 0.270501
\(493\) 0 0
\(494\) −32.0000 −1.43975
\(495\) 1.00000 0.0449467
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −18.0000 −0.804181
\(502\) −12.0000 −0.535586
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 1.00000 0.0445435
\(505\) −6.00000 −0.266996
\(506\) −6.00000 −0.266733
\(507\) 3.00000 0.133235
\(508\) −16.0000 −0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −18.0000 −0.793946
\(515\) 16.0000 0.705044
\(516\) −4.00000 −0.176090
\(517\) −12.0000 −0.527759
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) 4.00000 0.175412
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 6.00000 0.262613
\(523\) −10.0000 −0.437269 −0.218635 0.975807i \(-0.570160\pi\)
−0.218635 + 0.975807i \(0.570160\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) −12.0000 −0.523225
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 8.00000 0.346844
\(533\) −24.0000 −1.03956
\(534\) −12.0000 −0.519291
\(535\) 12.0000 0.518805
\(536\) −10.0000 −0.431934
\(537\) 12.0000 0.517838
\(538\) 6.00000 0.258678
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −16.0000 −0.687259
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) −4.00000 −0.171184
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 12.0000 0.512615
\(549\) −10.0000 −0.426790
\(550\) −1.00000 −0.0426401
\(551\) 48.0000 2.04487
\(552\) 6.00000 0.255377
\(553\) 8.00000 0.340195
\(554\) −22.0000 −0.934690
\(555\) −8.00000 −0.339581
\(556\) −4.00000 −0.169638
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −4.00000 −0.169334
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 12.0000 0.505291
\(565\) −12.0000 −0.504844
\(566\) 2.00000 0.0840663
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) −8.00000 −0.335083
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 4.00000 0.167248
\(573\) −12.0000 −0.501307
\(574\) 6.00000 0.250435
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) −17.0000 −0.707107
\(579\) 2.00000 0.0831172
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) −6.00000 −0.248495
\(584\) 14.0000 0.579324
\(585\) 4.00000 0.165380
\(586\) −18.0000 −0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) −32.0000 −1.31854
\(590\) 12.0000 0.494032
\(591\) 18.0000 0.740421
\(592\) 8.00000 0.328798
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 20.0000 0.818546
\(598\) −24.0000 −0.981433
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 1.00000 0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −4.00000 −0.163028
\(603\) −10.0000 −0.407231
\(604\) 8.00000 0.325515
\(605\) −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 \(0.801498\pi\)
\(608\) 8.00000 0.324443
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 26.0000 1.04927
\(615\) −6.00000 −0.241943
\(616\) −1.00000 −0.0402911
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) −16.0000 −0.643614
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) −18.0000 −0.721734
\(623\) −12.0000 −0.480770
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) −8.00000 −0.319489
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000 0.318223
\(633\) 14.0000 0.556450
\(634\) 6.00000 0.238290
\(635\) 16.0000 0.634941
\(636\) 6.00000 0.237915
\(637\) −4.00000 −0.158486
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −12.0000 −0.473602
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 6.00000 0.236433
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.0000 0.471041
\(650\) −4.00000 −0.156893
\(651\) −4.00000 −0.156772
\(652\) 14.0000 0.548282
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −16.0000 −0.625650
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) 14.0000 0.546192
\(658\) 12.0000 0.467809
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 1.00000 0.0389249
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 8.00000 0.309994
\(667\) 36.0000 1.39393
\(668\) −18.0000 −0.696441
\(669\) −16.0000 −0.618596
\(670\) 10.0000 0.386334
\(671\) 10.0000 0.386046
\(672\) 1.00000 0.0385758
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 12.0000 0.460857
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 4.00000 0.153168
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 8.00000 0.305888
\(685\) −12.0000 −0.458496
\(686\) 1.00000 0.0381802
\(687\) −28.0000 −1.06827
\(688\) −4.00000 −0.152499
\(689\) −24.0000 −0.914327
\(690\) −6.00000 −0.228416
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) −6.00000 −0.228086
\(693\) −1.00000 −0.0379869
\(694\) −12.0000 −0.455514
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −34.0000 −1.28692
\(699\) −6.00000 −0.226941
\(700\) 1.00000 0.0377964
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −4.00000 −0.150970
\(703\) 64.0000 2.41381
\(704\) −1.00000 −0.0376889
\(705\) −12.0000 −0.451946
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) −12.0000 −0.450988
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −12.0000 −0.449719
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 12.0000 0.448461
\(717\) 6.00000 0.224074
\(718\) 18.0000 0.671754
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −16.0000 −0.595871
\(722\) 45.0000 1.67473
\(723\) 14.0000 0.520666
\(724\) −16.0000 −0.594635
\(725\) 6.00000 0.222834
\(726\) 1.00000 0.0371135
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −16.0000 −0.590571
\(735\) −1.00000 −0.0368856
\(736\) 6.00000 0.221163
\(737\) 10.0000 0.368355
\(738\) 6.00000 0.220863
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) −8.00000 −0.294086
\(741\) −32.0000 −1.17555
\(742\) 6.00000 0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −4.00000 −0.146647
\(745\) 6.00000 0.219823
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000 0.437595
\(753\) −12.0000 −0.437304
\(754\) −24.0000 −0.874028
\(755\) −8.00000 −0.291150
\(756\) 1.00000 0.0363696
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) −4.00000 −0.145287
\(759\) −6.00000 −0.217786
\(760\) −8.00000 −0.290191
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −16.0000 −0.579619
\(763\) −16.0000 −0.579239
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 1.00000 0.0360375
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −4.00000 −0.143777
\(775\) −4.00000 −0.143684
\(776\) −10.0000 −0.358979
\(777\) 8.00000 0.286998
\(778\) −6.00000 −0.215110
\(779\) 48.0000 1.71978
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 12.0000 0.428026
\(787\) 50.0000 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(788\) 18.0000 0.641223
\(789\) −12.0000 −0.427211
\(790\) −8.00000 −0.284627
\(791\) 12.0000 0.426671
\(792\) −1.00000 −0.0355335
\(793\) 40.0000 1.42044
\(794\) −34.0000 −1.20661
\(795\) −6.00000 −0.212798
\(796\) 20.0000 0.708881
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) 18.0000 0.635602
\(803\) −14.0000 −0.494049
\(804\) −10.0000 −0.352673
\(805\) −6.00000 −0.211472
\(806\) 16.0000 0.563576
\(807\) 6.00000 0.211210
\(808\) 6.00000 0.211079
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) 6.00000 0.210559
\(813\) −16.0000 −0.561144
\(814\) −8.00000 −0.280400
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −22.0000 −0.769212
\(819\) −4.00000 −0.139771
\(820\) −6.00000 −0.209529
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 12.0000 0.418548
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) −16.0000 −0.557386
\(825\) −1.00000 −0.0348155
\(826\) −12.0000 −0.417533
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 6.00000 0.208514
\(829\) 32.0000 1.11141 0.555703 0.831381i \(-0.312449\pi\)
0.555703 + 0.831381i \(0.312449\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 18.0000 0.622916
\(836\) −8.00000 −0.276686
\(837\) −4.00000 −0.138260
\(838\) −24.0000 −0.829066
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −12.0000 −0.413302
\(844\) 14.0000 0.481900
\(845\) −3.00000 −0.103203
\(846\) 12.0000 0.412568
\(847\) 1.00000 0.0343604
\(848\) 6.00000 0.206041
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) −10.0000 −0.342193
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 4.00000 0.136558
\(859\) −46.0000 −1.56950 −0.784750 0.619813i \(-0.787209\pi\)
−0.784750 + 0.619813i \(0.787209\pi\)
\(860\) 4.00000 0.136399
\(861\) 6.00000 0.204479
\(862\) 6.00000 0.204361
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) 1.00000 0.0340207
\(865\) 6.00000 0.204006
\(866\) 14.0000 0.475739
\(867\) −17.0000 −0.577350
\(868\) −4.00000 −0.135769
\(869\) −8.00000 −0.271381
\(870\) −6.00000 −0.203419
\(871\) 40.0000 1.35535
\(872\) −16.0000 −0.541828
\(873\) −10.0000 −0.338449
\(874\) 48.0000 1.62362
\(875\) −1.00000 −0.0338062
\(876\) 14.0000 0.473016
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 8.00000 0.269987
\(879\) −18.0000 −0.607125
\(880\) 1.00000 0.0337100
\(881\) −12.0000 −0.404290 −0.202145 0.979356i \(-0.564791\pi\)
−0.202145 + 0.979356i \(0.564791\pi\)
\(882\) 1.00000 0.0336718
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 36.0000 1.20944
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 8.00000 0.268462
\(889\) −16.0000 −0.536623
\(890\) 12.0000 0.402241
\(891\) −1.00000 −0.0335013
\(892\) −16.0000 −0.535720
\(893\) 96.0000 3.21252
\(894\) −6.00000 −0.200670
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) −24.0000 −0.801337
\(898\) −30.0000 −1.00111
\(899\) −24.0000 −0.800445
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −6.00000 −0.199778
\(903\) −4.00000 −0.133112
\(904\) 12.0000 0.399114
\(905\) 16.0000 0.531858
\(906\) 8.00000 0.265782
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 24.0000 0.796468
\(909\) 6.00000 0.199007
\(910\) 4.00000 0.132599
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 10.0000 0.330590
\(916\) −28.0000 −0.925146
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) −6.00000 −0.197814
\(921\) 26.0000 0.856729
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) 8.00000 0.263038
\(926\) −4.00000 −0.131448
\(927\) −16.0000 −0.525509
\(928\) 6.00000 0.196960
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 4.00000 0.131165
\(931\) 8.00000 0.262189
\(932\) −6.00000 −0.196537
\(933\) −18.0000 −0.589294
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −10.0000 −0.326512
\(939\) 26.0000 0.848478
\(940\) −12.0000 −0.391397
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 2.00000 0.0651635
\(943\) 36.0000 1.17232
\(944\) −12.0000 −0.390567
\(945\) −1.00000 −0.0325300
\(946\) 4.00000 0.130051
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 8.00000 0.259828
\(949\) −56.0000 −1.81784
\(950\) 8.00000 0.259554
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) 12.0000 0.388311
\(956\) 6.00000 0.194054
\(957\) −6.00000 −0.193952
\(958\) −24.0000 −0.775405
\(959\) 12.0000 0.387500
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −32.0000 −1.03172
\(963\) −12.0000 −0.386695
\(964\) 14.0000 0.450910
\(965\) −2.00000 −0.0643823
\(966\) 6.00000 0.193047
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) −28.0000 −0.897178
\(975\) −4.00000 −0.128103
\(976\) −10.0000 −0.320092
\(977\) 24.0000 0.767828 0.383914 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) 14.0000 0.447671
\(979\) 12.0000 0.383522
\(980\) −1.00000 −0.0319438
\(981\) −16.0000 −0.510841
\(982\) 12.0000 0.382935
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 6.00000 0.191273
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) −32.0000 −1.01806
\(989\) −24.0000 −0.763156
\(990\) 1.00000 0.0317821
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −4.00000 −0.127000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −40.0000 −1.26681 −0.633406 0.773819i \(-0.718344\pi\)
−0.633406 + 0.773819i \(0.718344\pi\)
\(998\) −40.0000 −1.26618
\(999\) 8.00000 0.253109
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.t.1.1 1
3.2 odd 2 6930.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.t.1.1 1 1.1 even 1 trivial
6930.2.a.p.1.1 1 3.2 odd 2