Properties

Label 2310.2.a.e.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -8.00000 q^{37} +1.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -2.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} +14.0000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{56} -6.00000 q^{58} -1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} +2.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +1.00000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} -12.0000 q^{89} +1.00000 q^{90} +2.00000 q^{92} -8.00000 q^{93} -12.0000 q^{94} -1.00000 q^{96} +6.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 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
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 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
\(46\) −2.00000 −0.294884
\(47\) 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
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 8.00000 1.01600
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) 2.00000 0.240772
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 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
\(82\) −2.00000 −0.220863
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.00000 −0.433861
\(86\) 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
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −8.00000 −0.829561
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −4.00000 −0.396059
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) −14.0000 −1.35980
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −8.00000 −0.759326
\(112\) −1.00000 −0.0944911
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 2.00000 0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) −4.00000 −0.342997
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −2.00000 −0.170251
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 1.00000 0.0845154
\(141\) 12.0000 1.01058
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −6.00000 −0.496564
\(147\) 1.00000 0.0824786
\(148\) −8.00000 −0.657596
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −8.00000 −0.636446
\(159\) 14.0000 1.11027
\(160\) 1.00000 0.0790569
\(161\) −2.00000 −0.157622
\(162\) −1.00000 −0.0785674
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 2.00000 0.156174
\(165\) 1.00000 0.0778499
\(166\) 8.00000 0.620920
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 1.00000 0.0771517
\(169\) −13.0000 −1.00000
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) −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
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −2.00000 −0.147442
\(185\) 8.00000 0.588172
\(186\) 8.00000 0.586588
\(187\) −4.00000 −0.292509
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.00000 0.141069
\(202\) −10.0000 −0.703598
\(203\) −6.00000 −0.421117
\(204\) 4.00000 0.280056
\(205\) −2.00000 −0.139686
\(206\) −16.0000 −1.11477
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) 14.0000 0.961524
\(213\) −8.00000 −0.548151
\(214\) −20.0000 −1.36717
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) −4.00000 −0.270914
\(219\) 6.00000 0.405442
\(220\) 1.00000 0.0674200
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −4.00000 −0.266076
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 2.00000 0.131876
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 4.00000 0.259281
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −8.00000 −0.506979
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) −8.00000 −0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 4.00000 0.249029
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −20.0000 −1.23560
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 1.00000 0.0615457
\(265\) −14.0000 −0.860013
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 2.00000 0.122169
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 1.00000 0.0608581
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) −1.00000 −0.0603023
\(276\) 2.00000 0.120386
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 20.0000 1.19952
\(279\) −8.00000 −0.478947
\(280\) −1.00000 −0.0597614
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) −12.0000 −0.714590
\(283\) 6.00000 0.356663 0.178331 0.983970i \(-0.442930\pi\)
0.178331 + 0.983970i \(0.442930\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 6.00000 0.352332
\(291\) 6.00000 0.351726
\(292\) 6.00000 0.351123
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) −4.00000 −0.228665
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 1.00000 0.0569803
\(309\) 16.0000 0.910208
\(310\) −8.00000 −0.454369
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) 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 \(0.446109\pi\)
\(318\) −14.0000 −0.785081
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) 20.0000 1.11629
\(322\) 2.00000 0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) 4.00000 0.221201
\(328\) −2.00000 −0.110432
\(329\) −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 \(0.535063\pi\)
\(332\) −8.00000 −0.439057
\(333\) −8.00000 −0.438397
\(334\) 6.00000 0.328305
\(335\) −2.00000 −0.109272
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 13.0000 0.707107
\(339\) 4.00000 0.217250
\(340\) −4.00000 −0.216930
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) −2.00000 −0.107676
\(346\) 6.00000 0.322562
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −12.0000 −0.635999
\(357\) −4.00000 −0.211702
\(358\) 20.0000 1.05703
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) 16.0000 0.840941
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) −10.0000 −0.522708
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 2.00000 0.104257
\(369\) 2.00000 0.104116
\(370\) −8.00000 −0.415900
\(371\) −14.0000 −0.726844
\(372\) −8.00000 −0.414781
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 4.00000 0.206835
\(375\) −1.00000 −0.0516398
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 12.0000 0.613973
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.00000 −0.0509647
\(386\) 18.0000 0.916176
\(387\) −4.00000 −0.203331
\(388\) 6.00000 0.304604
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −1.00000 −0.0505076
\(393\) 20.0000 1.00887
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) −1.00000 −0.0502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) −4.00000 −0.198030
\(409\) −2.00000 −0.0988936 −0.0494468 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) 2.00000 0.0987730
\(411\) 4.00000 0.197305
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 1.00000 0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −26.0000 −1.26566
\(423\) 12.0000 0.583460
\(424\) −14.0000 −0.679900
\(425\) 4.00000 0.194029
\(426\) 8.00000 0.387601
\(427\) −10.0000 −0.483934
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 1.00000 0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −8.00000 −0.384012
\(435\) −6.00000 −0.287678
\(436\) 4.00000 0.191565
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) −8.00000 −0.379663
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) 18.0000 0.851371
\(448\) −1.00000 −0.0472456
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −20.0000 −0.934539
\(459\) 4.00000 0.186704
\(460\) −2.00000 −0.0932505
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) 2.00000 0.0926482
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 12.0000 0.553519
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 14.0000 0.641016
\(478\) −2.00000 −0.0914779
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 14.0000 0.637683
\(483\) −2.00000 −0.0910032
\(484\) 1.00000 0.0454545
\(485\) −6.00000 −0.272446
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −10.0000 −0.452679
\(489\) −22.0000 −0.994874
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) −8.00000 −0.359211
\(497\) 8.00000 0.358849
\(498\) 8.00000 0.358489
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 1.00000 0.0445435
\(505\) −10.0000 −0.444994
\(506\) 2.00000 0.0889108
\(507\) −13.0000 −0.577350
\(508\) 8.00000 0.354943
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 4.00000 0.177123
\(511\) −6.00000 −0.265424
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.0000 −0.970378
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) −12.0000 −0.527759
\(518\) −8.00000 −0.351500
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) −6.00000 −0.262613
\(523\) −6.00000 −0.262362 −0.131181 0.991358i \(-0.541877\pi\)
−0.131181 + 0.991358i \(0.541877\pi\)
\(524\) 20.0000 0.873704
\(525\) −1.00000 −0.0436436
\(526\) −12.0000 −0.523225
\(527\) −32.0000 −1.39394
\(528\) −1.00000 −0.0435194
\(529\) −19.0000 −0.826087
\(530\) 14.0000 0.608121
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) −20.0000 −0.864675
\(536\) −2.00000 −0.0863868
\(537\) −20.0000 −0.863064
\(538\) 18.0000 0.776035
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) −24.0000 −1.03184 −0.515920 0.856637i \(-0.672550\pi\)
−0.515920 + 0.856637i \(0.672550\pi\)
\(542\) −24.0000 −1.03089
\(543\) −16.0000 −0.686626
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 4.00000 0.170872
\(549\) 10.0000 0.426790
\(550\) 1.00000 0.0426401
\(551\) 0 0
\(552\) −2.00000 −0.0851257
\(553\) −8.00000 −0.340195
\(554\) −6.00000 −0.254916
\(555\) 8.00000 0.339581
\(556\) −20.0000 −0.848189
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) −4.00000 −0.168880
\(562\) −8.00000 −0.337460
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 12.0000 0.505291
\(565\) −4.00000 −0.168281
\(566\) −6.00000 −0.252199
\(567\) −1.00000 −0.0419961
\(568\) 8.00000 0.335673
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 2.00000 0.0834784
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 1.00000 0.0415945
\(579\) −18.0000 −0.748054
\(580\) −6.00000 −0.249136
\(581\) 8.00000 0.331896
\(582\) −6.00000 −0.248708
\(583\) −14.0000 −0.579821
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 10.0000 0.413096
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −8.00000 −0.328798
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 1.00000 0.0410305
\(595\) 4.00000 0.163984
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) −4.00000 −0.163028
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) −10.0000 −0.406222
\(607\) 48.0000 1.94826 0.974130 0.225989i \(-0.0725612\pi\)
0.974130 + 0.225989i \(0.0725612\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −6.00000 −0.242140
\(615\) −2.00000 −0.0806478
\(616\) −1.00000 −0.0402911
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −16.0000 −0.643614
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 8.00000 0.321288
\(621\) 2.00000 0.0802572
\(622\) −18.0000 −0.721734
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −32.0000 −1.27592
\(630\) −1.00000 −0.0398410
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −8.00000 −0.318223
\(633\) 26.0000 1.03341
\(634\) −6.00000 −0.238290
\(635\) −8.00000 −0.317470
\(636\) 14.0000 0.555136
\(637\) 0 0
\(638\) 6.00000 0.237542
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −20.0000 −0.789337
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) −22.0000 −0.861586
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −4.00000 −0.156412
\(655\) −20.0000 −0.781465
\(656\) 2.00000 0.0780869
\(657\) 6.00000 0.234082
\(658\) 12.0000 0.467809
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 1.00000 0.0389249
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 12.0000 0.464642
\(668\) −6.00000 −0.232147
\(669\) 16.0000 0.618596
\(670\) 2.00000 0.0772667
\(671\) −10.0000 −0.386046
\(672\) 1.00000 0.0385758
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 14.0000 0.539260
\(675\) 1.00000 0.0384900
\(676\) −13.0000 −0.500000
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −4.00000 −0.153619
\(679\) −6.00000 −0.230259
\(680\) 4.00000 0.153393
\(681\) 24.0000 0.919682
\(682\) −8.00000 −0.306336
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 1.00000 0.0381802
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 2.00000 0.0761387
\(691\) 18.0000 0.684752 0.342376 0.939563i \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) −6.00000 −0.228086
\(693\) 1.00000 0.0379869
\(694\) 20.0000 0.759190
\(695\) 20.0000 0.758643
\(696\) −6.00000 −0.227429
\(697\) 8.00000 0.303022
\(698\) −26.0000 −0.984115
\(699\) −2.00000 −0.0756469
\(700\) −1.00000 −0.0377964
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) −12.0000 −0.451946
\(706\) −6.00000 −0.225813
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) 8.00000 0.300023
\(712\) 12.0000 0.449719
\(713\) −16.0000 −0.599205
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 2.00000 0.0746914
\(718\) 34.0000 1.26887
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −16.0000 −0.595871
\(722\) 19.0000 0.707107
\(723\) −14.0000 −0.520666
\(724\) −16.0000 −0.594635
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −16.0000 −0.591781
\(732\) 10.0000 0.369611
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 32.0000 1.18114
\(735\) −1.00000 −0.0368856
\(736\) −2.00000 −0.0737210
\(737\) −2.00000 −0.0736709
\(738\) −2.00000 −0.0736210
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 14.0000 0.513956
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 8.00000 0.293294
\(745\) −18.0000 −0.659469
\(746\) −26.0000 −0.951928
\(747\) −8.00000 −0.292705
\(748\) −4.00000 −0.146254
\(749\) −20.0000 −0.730784
\(750\) 1.00000 0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 20.0000 0.726433
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) 50.0000 1.81250 0.906249 0.422744i \(-0.138933\pi\)
0.906249 + 0.422744i \(0.138933\pi\)
\(762\) −8.00000 −0.289809
\(763\) −4.00000 −0.144810
\(764\) −12.0000 −0.434145
\(765\) −4.00000 −0.144620
\(766\) −20.0000 −0.722629
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 1.00000 0.0360375
\(771\) 22.0000 0.792311
\(772\) −18.0000 −0.647834
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) −6.00000 −0.215387
\(777\) 8.00000 0.286998
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −8.00000 −0.286079
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) −18.0000 −0.642448
\(786\) −20.0000 −0.713376
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 6.00000 0.213741
\(789\) 12.0000 0.427211
\(790\) 8.00000 0.284627
\(791\) −4.00000 −0.142224
\(792\) 1.00000 0.0355335
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) −14.0000 −0.496529
\(796\) −8.00000 −0.283552
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) −12.0000 −0.423999
\(802\) −18.0000 −0.635602
\(803\) −6.00000 −0.211735
\(804\) 2.00000 0.0705346
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −10.0000 −0.351799
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 1.00000 0.0351364
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) −6.00000 −0.210559
\(813\) 24.0000 0.841717
\(814\) −8.00000 −0.280400
\(815\) 22.0000 0.770626
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) 2.00000 0.0699284
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −4.00000 −0.139516
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −16.0000 −0.557386
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 2.00000 0.0695048
\(829\) 24.0000 0.833554 0.416777 0.909009i \(-0.363160\pi\)
0.416777 + 0.909009i \(0.363160\pi\)
\(830\) −8.00000 −0.277684
\(831\) 6.00000 0.208138
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 20.0000 0.692543
\(835\) 6.00000 0.207639
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 20.0000 0.690889
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) 8.00000 0.275535
\(844\) 26.0000 0.894957
\(845\) 13.0000 0.447214
\(846\) −12.0000 −0.412568
\(847\) −1.00000 −0.0343604
\(848\) 14.0000 0.480762
\(849\) 6.00000 0.205919
\(850\) −4.00000 −0.137199
\(851\) −16.0000 −0.548473
\(852\) −8.00000 −0.274075
\(853\) 24.0000 0.821744 0.410872 0.911693i \(-0.365224\pi\)
0.410872 + 0.911693i \(0.365224\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 4.00000 0.136399
\(861\) −2.00000 −0.0681598
\(862\) −10.0000 −0.340601
\(863\) −54.0000 −1.83818 −0.919091 0.394046i \(-0.871075\pi\)
−0.919091 + 0.394046i \(0.871075\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.00000 0.204006
\(866\) 34.0000 1.15537
\(867\) −1.00000 −0.0339618
\(868\) 8.00000 0.271538
\(869\) −8.00000 −0.271381
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) −4.00000 −0.135457
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 6.00000 0.202721
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 32.0000 1.07995
\(879\) −10.0000 −0.337292
\(880\) 1.00000 0.0337100
\(881\) 28.0000 0.943344 0.471672 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 8.00000 0.268462
\(889\) −8.00000 −0.268311
\(890\) −12.0000 −0.402241
\(891\) −1.00000 −0.0335013
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 20.0000 0.668526
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −26.0000 −0.867631
\(899\) −48.0000 −1.60089
\(900\) 1.00000 0.0333333
\(901\) 56.0000 1.86563
\(902\) 2.00000 0.0665927
\(903\) 4.00000 0.133112
\(904\) −4.00000 −0.133038
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) −58.0000 −1.92586 −0.962929 0.269754i \(-0.913058\pi\)
−0.962929 + 0.269754i \(0.913058\pi\)
\(908\) 24.0000 0.796468
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 2.00000 0.0661541
\(915\) −10.0000 −0.330590
\(916\) 20.0000 0.660819
\(917\) −20.0000 −0.660458
\(918\) −4.00000 −0.132020
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 2.00000 0.0659380
\(921\) 6.00000 0.197707
\(922\) 34.0000 1.11973
\(923\) 0 0
\(924\) 1.00000 0.0328976
\(925\) −8.00000 −0.263038
\(926\) 32.0000 1.05159
\(927\) 16.0000 0.525509
\(928\) −6.00000 −0.196960
\(929\) −32.0000 −1.04989 −0.524943 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) −2.00000 −0.0655122
\(933\) 18.0000 0.589294
\(934\) −4.00000 −0.130884
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 2.00000 0.0653023
\(939\) −6.00000 −0.195803
\(940\) −12.0000 −0.391397
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −18.0000 −0.586472
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) −4.00000 −0.130051
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 4.00000 0.129641
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) −14.0000 −0.453267
\(955\) 12.0000 0.388311
\(956\) 2.00000 0.0646846
\(957\) −6.00000 −0.193952
\(958\) 24.0000 0.775405
\(959\) −4.00000 −0.129167
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) −14.0000 −0.450910
\(965\) 18.0000 0.579441
\(966\) 2.00000 0.0643489
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) −4.00000 −0.128366 −0.0641831 0.997938i \(-0.520444\pi\)
−0.0641831 + 0.997938i \(0.520444\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 32.0000 1.02377 0.511885 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 22.0000 0.703482
\(979\) 12.0000 0.383522
\(980\) −1.00000 −0.0319438
\(981\) 4.00000 0.127710
\(982\) 12.0000 0.382935
\(983\) 52.0000 1.65854 0.829271 0.558846i \(-0.188756\pi\)
0.829271 + 0.558846i \(0.188756\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −6.00000 −0.191176
\(986\) −24.0000 −0.764316
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) −1.00000 −0.0317821
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) −4.00000 −0.126936
\(994\) −8.00000 −0.253745
\(995\) 8.00000 0.253617
\(996\) −8.00000 −0.253490
\(997\) −4.00000 −0.126681 −0.0633406 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) 0 0
\(999\) −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.e.1.1 1
3.2 odd 2 6930.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.e.1.1 1 1.1 even 1 trivial
6930.2.a.be.1.1 1 3.2 odd 2