Properties

Label 2310.2.a.f.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -1.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} -1.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 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
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) −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
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −1.00000 −0.113961
\(78\) 2.00000 0.226455
\(79\) 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
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) −1.00000 −0.106600
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) −4.00000 −0.417029
\(93\) −4.00000 −0.414781
\(94\) 8.00000 0.825137
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 1.00000 0.0953463
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 4.00000 0.373002
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 2.00000 0.183340
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 10.0000 0.901670
\(124\) −4.00000 −0.359211
\(125\) −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
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) −4.00000 −0.346844
\(134\) 12.0000 1.03664
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 4.00000 0.340503
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 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 \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −4.00000 −0.324443
\(153\) −2.00000 −0.161690
\(154\) 1.00000 0.0805823
\(155\) 4.00000 0.321288
\(156\) −2.00000 −0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 4.00000 0.315244
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −2.00000 −0.148250
\(183\) −2.00000 −0.147844
\(184\) 4.00000 0.294884
\(185\) 2.00000 0.147043
\(186\) 4.00000 0.293294
\(187\) −2.00000 −0.146254
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) 10.0000 0.703598
\(203\) 6.00000 0.421117
\(204\) −2.00000 −0.140028
\(205\) −10.0000 −0.698430
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) −1.00000 −0.0690066
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) 4.00000 0.273434
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 4.00000 0.271538
\(218\) −6.00000 −0.406371
\(219\) 2.00000 0.135147
\(220\) −1.00000 −0.0674200
\(221\) 4.00000 0.269069
\(222\) 2.00000 0.134231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 4.00000 0.264906
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −4.00000 −0.263752
\(231\) −1.00000 −0.0657952
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) 4.00000 0.259828
\(238\) −2.00000 −0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −4.00000 −0.251478
\(254\) −8.00000 −0.501965
\(255\) 2.00000 0.125245
\(256\) 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
\(259\) 2.00000 0.124274
\(260\) 2.00000 0.124035
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 1.00000 0.0608581
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) −2.00000 −0.121268
\(273\) 2.00000 0.121046
\(274\) 18.0000 1.08742
\(275\) 1.00000 0.0603023
\(276\) −4.00000 −0.240772
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 4.00000 0.239904
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 8.00000 0.476393
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −8.00000 −0.474713
\(285\) −4.00000 −0.236940
\(286\) 2.00000 0.118262
\(287\) −10.0000 −0.590281
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −2.00000 −0.117242
\(292\) 2.00000 0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 4.00000 0.232889
\(296\) 2.00000 0.116248
\(297\) 1.00000 0.0580259
\(298\) −2.00000 −0.115857
\(299\) 8.00000 0.462652
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 4.00000 0.230174
\(303\) −10.0000 −0.574485
\(304\) 4.00000 0.229416
\(305\) 2.00000 0.114520
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −2.00000 −0.112867
\(315\) 1.00000 0.0563436
\(316\) 4.00000 0.225018
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 6.00000 0.336463
\(319\) −6.00000 −0.335936
\(320\) −1.00000 −0.0559017
\(321\) −4.00000 −0.223258
\(322\) −4.00000 −0.222911
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −12.0000 −0.664619
\(327\) 6.00000 0.331801
\(328\) −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 \(0.464937\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 2.00000 0.108465
\(341\) −4.00000 −0.216612
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 4.00000 0.215353
\(346\) 22.0000 1.18273
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −6.00000 −0.321634
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 1.00000 0.0534522
\(351\) −2.00000 −0.106752
\(352\) −1.00000 −0.0533002
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 4.00000 0.212598
\(355\) 8.00000 0.424596
\(356\) −10.0000 −0.529999
\(357\) 2.00000 0.105851
\(358\) −20.0000 −1.05703
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) −2.00000 −0.104685
\(366\) 2.00000 0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) −2.00000 −0.103975
\(371\) 6.00000 0.311504
\(372\) −4.00000 −0.207390
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 2.00000 0.103418
\(375\) −1.00000 −0.0516398
\(376\) 8.00000 0.412568
\(377\) 12.0000 0.618031
\(378\) 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
\(382\) 16.0000 0.818631
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 1.00000 0.0509647
\(386\) −22.0000 −1.11977
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) −2.00000 −0.101274
\(391\) 8.00000 0.404577
\(392\) −1.00000 −0.0505076
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 1.00000 0.0502519
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −4.00000 −0.200502
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 12.0000 0.598506
\(403\) 8.00000 0.398508
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) −2.00000 −0.0991363
\(408\) 2.00000 0.0990148
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 10.0000 0.493865
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) −4.00000 −0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 1.00000 0.0487950
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −8.00000 −0.389434
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 8.00000 0.387601
\(427\) 2.00000 0.0967868
\(428\) −4.00000 −0.193347
\(429\) −2.00000 −0.0965609
\(430\) 4.00000 0.192897
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −4.00000 −0.192006
\(435\) 6.00000 0.287678
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) −2.00000 −0.0955637
\(439\) 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
\(442\) −4.00000 −0.190261
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 10.0000 0.474045
\(446\) −8.00000 −0.378811
\(447\) 2.00000 0.0945968
\(448\) −1.00000 −0.0472456
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 10.0000 0.470882
\(452\) 6.00000 0.282216
\(453\) −4.00000 −0.187936
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) −4.00000 −0.187317
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) −6.00000 −0.280362
\(459\) −2.00000 −0.0933520
\(460\) 4.00000 0.186501
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 1.00000 0.0465242
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) −6.00000 −0.277945
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 12.0000 0.554109
\(470\) −8.00000 −0.369012
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) 4.00000 0.183920
\(474\) −4.00000 −0.183726
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) −2.00000 −0.0910975
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 2.00000 0.0905357
\(489\) 12.0000 0.542659
\(490\) 1.00000 0.0451754
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 10.0000 0.450835
\(493\) 12.0000 0.540453
\(494\) 8.00000 0.359937
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) −20.0000 −0.892644
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 1.00000 0.0445435
\(505\) 10.0000 0.444994
\(506\) 4.00000 0.177822
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) 18.0000 0.793946
\(515\) 8.00000 0.352522
\(516\) 4.00000 0.176090
\(517\) −8.00000 −0.351840
\(518\) −2.00000 −0.0878750
\(519\) −22.0000 −0.965693
\(520\) −2.00000 −0.0877058
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 6.00000 0.262613
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) −16.0000 −0.697633
\(527\) 8.00000 0.348485
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) −4.00000 −0.173585
\(532\) −4.00000 −0.173422
\(533\) −20.0000 −0.866296
\(534\) 10.0000 0.432742
\(535\) 4.00000 0.172935
\(536\) 12.0000 0.518321
\(537\) 20.0000 0.863064
\(538\) 6.00000 0.258678
\(539\) 1.00000 0.0430730
\(540\) −1.00000 −0.0430331
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 24.0000 1.03089
\(543\) −2.00000 −0.0858282
\(544\) 2.00000 0.0857493
\(545\) −6.00000 −0.257012
\(546\) −2.00000 −0.0855921
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −18.0000 −0.768922
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) 4.00000 0.170251
\(553\) −4.00000 −0.170097
\(554\) −2.00000 −0.0849719
\(555\) 2.00000 0.0848953
\(556\) −4.00000 −0.169638
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) −2.00000 −0.0844401
\(562\) 18.0000 0.759284
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −8.00000 −0.336861
\(565\) −6.00000 −0.252422
\(566\) 28.0000 1.17693
\(567\) −1.00000 −0.0419961
\(568\) 8.00000 0.335673
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 4.00000 0.167542
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −16.0000 −0.668410
\(574\) 10.0000 0.417392
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 13.0000 0.540729
\(579\) 22.0000 0.914289
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −6.00000 −0.248495
\(584\) −2.00000 −0.0827606
\(585\) 2.00000 0.0826898
\(586\) −18.0000 −0.743573
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 1.00000 0.0412393
\(589\) −16.0000 −0.659269
\(590\) −4.00000 −0.164677
\(591\) −6.00000 −0.246807
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −2.00000 −0.0819920
\(596\) 2.00000 0.0819232
\(597\) 4.00000 0.163709
\(598\) −8.00000 −0.327144
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) −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
\(603\) −12.0000 −0.488678
\(604\) −4.00000 −0.162758
\(605\) −1.00000 −0.0406558
\(606\) 10.0000 0.406222
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) −4.00000 −0.162221
\(609\) 6.00000 0.243132
\(610\) −2.00000 −0.0809776
\(611\) 16.0000 0.647291
\(612\) −2.00000 −0.0808452
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 20.0000 0.807134
\(615\) −10.0000 −0.403239
\(616\) 1.00000 0.0402911
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 8.00000 0.321807
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 4.00000 0.160644
\(621\) −4.00000 −0.160514
\(622\) −16.0000 −0.641542
\(623\) 10.0000 0.400642
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) 4.00000 0.159490
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) 8.00000 0.317971
\(634\) −2.00000 −0.0794301
\(635\) −8.00000 −0.317470
\(636\) −6.00000 −0.237915
\(637\) −2.00000 −0.0792429
\(638\) 6.00000 0.237542
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 4.00000 0.157867
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 4.00000 0.157622
\(645\) −4.00000 −0.157500
\(646\) 8.00000 0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 4.00000 0.156772
\(652\) 12.0000 0.469956
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −6.00000 −0.234619
\(655\) 12.0000 0.468879
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) −8.00000 −0.311872
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −4.00000 −0.155464
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 2.00000 0.0774984
\(667\) 24.0000 0.929284
\(668\) −12.0000 −0.464294
\(669\) 8.00000 0.309298
\(670\) −12.0000 −0.463600
\(671\) −2.00000 −0.0772091
\(672\) 1.00000 0.0385758
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) 2.00000 0.0767530
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 4.00000 0.152944
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) 6.00000 0.228914
\(688\) 4.00000 0.152499
\(689\) 12.0000 0.457164
\(690\) −4.00000 −0.152277
\(691\) −48.0000 −1.82601 −0.913003 0.407953i \(-0.866243\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) −22.0000 −0.836315
\(693\) −1.00000 −0.0379869
\(694\) 20.0000 0.759190
\(695\) 4.00000 0.151729
\(696\) 6.00000 0.227429
\(697\) −20.0000 −0.757554
\(698\) −30.0000 −1.13552
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) 1.00000 0.0376889
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) 10.0000 0.376089
\(708\) −4.00000 −0.150329
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) 16.0000 0.599205
\(714\) −2.00000 −0.0748481
\(715\) 2.00000 0.0747958
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −8.00000 −0.295891
\(732\) −2.00000 −0.0739221
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 8.00000 0.295285
\(735\) −1.00000 −0.0368856
\(736\) 4.00000 0.147442
\(737\) −12.0000 −0.442026
\(738\) −10.0000 −0.368105
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 2.00000 0.0735215
\(741\) −8.00000 −0.293887
\(742\) −6.00000 −0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 4.00000 0.146647
\(745\) −2.00000 −0.0732743
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 4.00000 0.146157
\(750\) 1.00000 0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −8.00000 −0.291730
\(753\) 20.0000 0.728841
\(754\) −12.0000 −0.437014
\(755\) 4.00000 0.145575
\(756\) −1.00000 −0.0363696
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 12.0000 0.435860
\(759\) −4.00000 −0.145191
\(760\) 4.00000 0.145095
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −8.00000 −0.289809
\(763\) −6.00000 −0.217215
\(764\) −16.0000 −0.578860
\(765\) 2.00000 0.0723102
\(766\) 24.0000 0.867155
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) −1.00000 −0.0360375
\(771\) −18.0000 −0.648254
\(772\) 22.0000 0.791797
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −4.00000 −0.143777
\(775\) −4.00000 −0.143684
\(776\) 2.00000 0.0717958
\(777\) 2.00000 0.0717496
\(778\) −14.0000 −0.501924
\(779\) 40.0000 1.43315
\(780\) 2.00000 0.0716115
\(781\) −8.00000 −0.286263
\(782\) −8.00000 −0.286079
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 12.0000 0.428026
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) −6.00000 −0.213741
\(789\) 16.0000 0.569615
\(790\) 4.00000 0.142314
\(791\) −6.00000 −0.213335
\(792\) −1.00000 −0.0355335
\(793\) 4.00000 0.142044
\(794\) 38.0000 1.34857
\(795\) 6.00000 0.212798
\(796\) 4.00000 0.141776
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 4.00000 0.141598
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) −10.0000 −0.353333
\(802\) 30.0000 1.05934
\(803\) 2.00000 0.0705785
\(804\) −12.0000 −0.423207
\(805\) −4.00000 −0.140981
\(806\) −8.00000 −0.281788
\(807\) −6.00000 −0.211210
\(808\) 10.0000 0.351799
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 1.00000 0.0351364
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 6.00000 0.210559
\(813\) −24.0000 −0.841717
\(814\) 2.00000 0.0701000
\(815\) −12.0000 −0.420342
\(816\) −2.00000 −0.0700140
\(817\) 16.0000 0.559769
\(818\) −10.0000 −0.349642
\(819\) 2.00000 0.0698857
\(820\) −10.0000 −0.349215
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 18.0000 0.627822
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) −4.00000 −0.139178
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) −42.0000 −1.45872 −0.729360 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 4.00000 0.138509
\(835\) 12.0000 0.415277
\(836\) 4.00000 0.138343
\(837\) −4.00000 −0.138260
\(838\) −12.0000 −0.414533
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −6.00000 −0.206774
\(843\) −18.0000 −0.619953
\(844\) 8.00000 0.275371
\(845\) 9.00000 0.309609
\(846\) 8.00000 0.275046
\(847\) −1.00000 −0.0343604
\(848\) −6.00000 −0.206041
\(849\) −28.0000 −0.960958
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) −8.00000 −0.274075
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −4.00000 −0.136797
\(856\) 4.00000 0.136717
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 2.00000 0.0682789
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) −4.00000 −0.136399
\(861\) −10.0000 −0.340799
\(862\) −16.0000 −0.544962
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 22.0000 0.748022
\(866\) −6.00000 −0.203888
\(867\) −13.0000 −0.441503
\(868\) 4.00000 0.135769
\(869\) 4.00000 0.135691
\(870\) −6.00000 −0.203419
\(871\) 24.0000 0.813209
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 16.0000 0.541208
\(875\) 1.00000 0.0338062
\(876\) 2.00000 0.0675737
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −8.00000 −0.269987
\(879\) 18.0000 0.607125
\(880\) −1.00000 −0.0337100
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 4.00000 0.134535
\(885\) 4.00000 0.134459
\(886\) −16.0000 −0.537531
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 2.00000 0.0671156
\(889\) −8.00000 −0.268311
\(890\) −10.0000 −0.335201
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) −32.0000 −1.07084
\(894\) −2.00000 −0.0668900
\(895\) −20.0000 −0.668526
\(896\) 1.00000 0.0334077
\(897\) 8.00000 0.267112
\(898\) −34.0000 −1.13459
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −10.0000 −0.332964
\(903\) −4.00000 −0.133112
\(904\) −6.00000 −0.199557
\(905\) 2.00000 0.0664822
\(906\) 4.00000 0.132891
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 2.00000 0.0662994
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 2.00000 0.0661180
\(916\) 6.00000 0.198246
\(917\) 12.0000 0.396275
\(918\) 2.00000 0.0660098
\(919\) −28.0000 −0.923635 −0.461817 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) −4.00000 −0.131876
\(921\) −20.0000 −0.659022
\(922\) −22.0000 −0.724531
\(923\) 16.0000 0.526646
\(924\) −1.00000 −0.0328976
\(925\) −2.00000 −0.0657596
\(926\) −16.0000 −0.525793
\(927\) −8.00000 −0.262754
\(928\) 6.00000 0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) −4.00000 −0.131165
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) 16.0000 0.523816
\(934\) −20.0000 −0.654420
\(935\) 2.00000 0.0654070
\(936\) 2.00000 0.0653720
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −12.0000 −0.391814
\(939\) −10.0000 −0.326338
\(940\) 8.00000 0.260931
\(941\) −58.0000 −1.89075 −0.945373 0.325991i \(-0.894302\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −40.0000 −1.30258
\(944\) −4.00000 −0.130189
\(945\) 1.00000 0.0325300
\(946\) −4.00000 −0.130051
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 4.00000 0.129914
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) −2.00000 −0.0648204
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) −6.00000 −0.193952
\(958\) −40.0000 −1.29234
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −4.00000 −0.128898
\(964\) 2.00000 0.0644157
\(965\) −22.0000 −0.708205
\(966\) −4.00000 −0.128698
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −8.00000 −0.256997
\(970\) −2.00000 −0.0642161
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 1.00000 0.0320750
\(973\) 4.00000 0.128234
\(974\) −40.0000 −1.28168
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −12.0000 −0.383718
\(979\) −10.0000 −0.319601
\(980\) −1.00000 −0.0319438
\(981\) 6.00000 0.191565
\(982\) 4.00000 0.127645
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −10.0000 −0.318788
\(985\) 6.00000 0.191176
\(986\) −12.0000 −0.382158
\(987\) 8.00000 0.254643
\(988\) −8.00000 −0.254514
\(989\) −16.0000 −0.508770
\(990\) 1.00000 0.0317821
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 4.00000 0.127000
\(993\) 4.00000 0.126936
\(994\) −8.00000 −0.253745
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −20.0000 −0.633089
\(999\) −2.00000 −0.0632772
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.f.1.1 1
3.2 odd 2 6930.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.f.1.1 1 1.1 even 1 trivial
6930.2.a.bb.1.1 1 3.2 odd 2