Properties

Label 1110.2.a.q.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
Defining polynomial: \(x^{2} - x - 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.81507\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -3.81507 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +7.81507 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} +5.81507 q^{23} -1.00000 q^{24} +1.00000 q^{25} +3.81507 q^{26} +1.00000 q^{27} +3.00000 q^{28} -6.81507 q^{29} -1.00000 q^{30} +6.81507 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} -7.81507 q^{38} -3.81507 q^{39} -1.00000 q^{40} -4.81507 q^{41} -3.00000 q^{42} -4.81507 q^{43} -1.00000 q^{44} +1.00000 q^{45} -5.81507 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -3.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} +7.81507 q^{57} +6.81507 q^{58} +2.00000 q^{59} +1.00000 q^{60} +6.81507 q^{61} -6.81507 q^{62} +3.00000 q^{63} +1.00000 q^{64} -3.81507 q^{65} +1.00000 q^{66} +7.63015 q^{67} +1.00000 q^{68} +5.81507 q^{69} -3.00000 q^{70} -11.6301 q^{71} -1.00000 q^{72} -5.81507 q^{73} -1.00000 q^{74} +1.00000 q^{75} +7.81507 q^{76} -3.00000 q^{77} +3.81507 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.81507 q^{82} -3.81507 q^{83} +3.00000 q^{84} +1.00000 q^{85} +4.81507 q^{86} -6.81507 q^{87} +1.00000 q^{88} +11.8151 q^{89} -1.00000 q^{90} -11.4452 q^{91} +5.81507 q^{92} +6.81507 q^{93} +8.00000 q^{94} +7.81507 q^{95} -1.00000 q^{96} -4.81507 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 3q^{13} - 6q^{14} + 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 5q^{19} + 2q^{20} + 6q^{21} + 2q^{22} + q^{23} - 2q^{24} + 2q^{25} - 3q^{26} + 2q^{27} + 6q^{28} - 3q^{29} - 2q^{30} + 3q^{31} - 2q^{32} - 2q^{33} - 2q^{34} + 6q^{35} + 2q^{36} + 2q^{37} - 5q^{38} + 3q^{39} - 2q^{40} + q^{41} - 6q^{42} + q^{43} - 2q^{44} + 2q^{45} - q^{46} - 16q^{47} + 2q^{48} + 4q^{49} - 2q^{50} + 2q^{51} + 3q^{52} - 2q^{54} - 2q^{55} - 6q^{56} + 5q^{57} + 3q^{58} + 4q^{59} + 2q^{60} + 3q^{61} - 3q^{62} + 6q^{63} + 2q^{64} + 3q^{65} + 2q^{66} - 6q^{67} + 2q^{68} + q^{69} - 6q^{70} - 2q^{71} - 2q^{72} - q^{73} - 2q^{74} + 2q^{75} + 5q^{76} - 6q^{77} - 3q^{78} + 24q^{79} + 2q^{80} + 2q^{81} - q^{82} + 3q^{83} + 6q^{84} + 2q^{85} - q^{86} - 3q^{87} + 2q^{88} + 13q^{89} - 2q^{90} + 9q^{91} + q^{92} + 3q^{93} + 16q^{94} + 5q^{95} - 2q^{96} + q^{97} - 4q^{98} - 2q^{99} + O(q^{100}) \)

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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.81507 −1.05811 −0.529055 0.848587i \(-0.677454\pi\)
−0.529055 + 0.848587i \(0.677454\pi\)
\(14\) −3.00000 −0.801784
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.81507 1.79290 0.896450 0.443144i \(-0.146137\pi\)
0.896450 + 0.443144i \(0.146137\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) 1.00000 0.213201
\(23\) 5.81507 1.21253 0.606263 0.795264i \(-0.292668\pi\)
0.606263 + 0.795264i \(0.292668\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 3.81507 0.748197
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) −6.81507 −1.26553 −0.632764 0.774345i \(-0.718080\pi\)
−0.632764 + 0.774345i \(0.718080\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.81507 1.22402 0.612012 0.790849i \(-0.290361\pi\)
0.612012 + 0.790849i \(0.290361\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −7.81507 −1.26777
\(39\) −3.81507 −0.610901
\(40\) −1.00000 −0.158114
\(41\) −4.81507 −0.751988 −0.375994 0.926622i \(-0.622699\pi\)
−0.375994 + 0.926622i \(0.622699\pi\)
\(42\) −3.00000 −0.462910
\(43\) −4.81507 −0.734292 −0.367146 0.930163i \(-0.619665\pi\)
−0.367146 + 0.930163i \(0.619665\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) −5.81507 −0.857386
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −3.81507 −0.529055
\(53\) 10.6301 1.46016 0.730081 0.683360i \(-0.239482\pi\)
0.730081 + 0.683360i \(0.239482\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −3.00000 −0.400892
\(57\) 7.81507 1.03513
\(58\) 6.81507 0.894863
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.81507 0.872581 0.436290 0.899806i \(-0.356292\pi\)
0.436290 + 0.899806i \(0.356292\pi\)
\(62\) −6.81507 −0.865515
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) −3.81507 −0.473202
\(66\) 1.00000 0.123091
\(67\) 7.63015 0.932171 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(68\) 1.00000 0.121268
\(69\) 5.81507 0.700053
\(70\) −3.00000 −0.358569
\(71\) −11.6301 −1.38024 −0.690122 0.723693i \(-0.742443\pi\)
−0.690122 + 0.723693i \(0.742443\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.81507 −0.680603 −0.340301 0.940316i \(-0.610529\pi\)
−0.340301 + 0.940316i \(0.610529\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) 7.81507 0.896450
\(77\) −3.00000 −0.341882
\(78\) 3.81507 0.431972
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.81507 0.531736
\(83\) −3.81507 −0.418759 −0.209379 0.977834i \(-0.567144\pi\)
−0.209379 + 0.977834i \(0.567144\pi\)
\(84\) 3.00000 0.327327
\(85\) 1.00000 0.108465
\(86\) 4.81507 0.519223
\(87\) −6.81507 −0.730653
\(88\) 1.00000 0.106600
\(89\) 11.8151 1.25240 0.626198 0.779664i \(-0.284610\pi\)
0.626198 + 0.779664i \(0.284610\pi\)
\(90\) −1.00000 −0.105409
\(91\) −11.4452 −1.19978
\(92\) 5.81507 0.606263
\(93\) 6.81507 0.706690
\(94\) 8.00000 0.825137
\(95\) 7.81507 0.801810
\(96\) −1.00000 −0.102062
\(97\) −4.81507 −0.488897 −0.244448 0.969662i \(-0.578607\pi\)
−0.244448 + 0.969662i \(0.578607\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 11.6301 1.15724 0.578621 0.815596i \(-0.303591\pi\)
0.578621 + 0.815596i \(0.303591\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 3.81507 0.374099
\(105\) 3.00000 0.292770
\(106\) −10.6301 −1.03249
\(107\) −8.18493 −0.791267 −0.395633 0.918409i \(-0.629475\pi\)
−0.395633 + 0.918409i \(0.629475\pi\)
\(108\) 1.00000 0.0962250
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 1.00000 0.0953463
\(111\) 1.00000 0.0949158
\(112\) 3.00000 0.283473
\(113\) −6.81507 −0.641108 −0.320554 0.947230i \(-0.603869\pi\)
−0.320554 + 0.947230i \(0.603869\pi\)
\(114\) −7.81507 −0.731949
\(115\) 5.81507 0.542258
\(116\) −6.81507 −0.632764
\(117\) −3.81507 −0.352704
\(118\) −2.00000 −0.184115
\(119\) 3.00000 0.275010
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) −6.81507 −0.617008
\(123\) −4.81507 −0.434161
\(124\) 6.81507 0.612012
\(125\) 1.00000 0.0894427
\(126\) −3.00000 −0.267261
\(127\) −21.4452 −1.90296 −0.951478 0.307718i \(-0.900435\pi\)
−0.951478 + 0.307718i \(0.900435\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.81507 −0.423944
\(130\) 3.81507 0.334604
\(131\) −9.63015 −0.841390 −0.420695 0.907202i \(-0.638214\pi\)
−0.420695 + 0.907202i \(0.638214\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 23.4452 2.03296
\(134\) −7.63015 −0.659144
\(135\) 1.00000 0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −5.81507 −0.495012
\(139\) 22.8151 1.93515 0.967575 0.252585i \(-0.0812809\pi\)
0.967575 + 0.252585i \(0.0812809\pi\)
\(140\) 3.00000 0.253546
\(141\) −8.00000 −0.673722
\(142\) 11.6301 0.975980
\(143\) 3.81507 0.319032
\(144\) 1.00000 0.0833333
\(145\) −6.81507 −0.565961
\(146\) 5.81507 0.481259
\(147\) 2.00000 0.164957
\(148\) 1.00000 0.0821995
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 3.81507 0.310466 0.155233 0.987878i \(-0.450387\pi\)
0.155233 + 0.987878i \(0.450387\pi\)
\(152\) −7.81507 −0.633886
\(153\) 1.00000 0.0808452
\(154\) 3.00000 0.241747
\(155\) 6.81507 0.547400
\(156\) −3.81507 −0.305450
\(157\) −12.8151 −1.02275 −0.511377 0.859356i \(-0.670864\pi\)
−0.511377 + 0.859356i \(0.670864\pi\)
\(158\) −12.0000 −0.954669
\(159\) 10.6301 0.843025
\(160\) −1.00000 −0.0790569
\(161\) 17.4452 1.37488
\(162\) −1.00000 −0.0785674
\(163\) 16.6301 1.30257 0.651287 0.758832i \(-0.274230\pi\)
0.651287 + 0.758832i \(0.274230\pi\)
\(164\) −4.81507 −0.375994
\(165\) −1.00000 −0.0778499
\(166\) 3.81507 0.296107
\(167\) −21.8151 −1.68810 −0.844051 0.536264i \(-0.819835\pi\)
−0.844051 + 0.536264i \(0.819835\pi\)
\(168\) −3.00000 −0.231455
\(169\) 1.55478 0.119599
\(170\) −1.00000 −0.0766965
\(171\) 7.81507 0.597634
\(172\) −4.81507 −0.367146
\(173\) −1.00000 −0.0760286 −0.0380143 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) 6.81507 0.516649
\(175\) 3.00000 0.226779
\(176\) −1.00000 −0.0753778
\(177\) 2.00000 0.150329
\(178\) −11.8151 −0.885577
\(179\) 16.0000 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 1.00000 0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 11.4452 0.848376
\(183\) 6.81507 0.503785
\(184\) −5.81507 −0.428693
\(185\) 1.00000 0.0735215
\(186\) −6.81507 −0.499705
\(187\) −1.00000 −0.0731272
\(188\) −8.00000 −0.583460
\(189\) 3.00000 0.218218
\(190\) −7.81507 −0.566965
\(191\) 4.63015 0.335026 0.167513 0.985870i \(-0.446426\pi\)
0.167513 + 0.985870i \(0.446426\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 4.81507 0.345702
\(195\) −3.81507 −0.273203
\(196\) 2.00000 0.142857
\(197\) −4.18493 −0.298164 −0.149082 0.988825i \(-0.547632\pi\)
−0.149082 + 0.988825i \(0.547632\pi\)
\(198\) 1.00000 0.0710669
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 7.63015 0.538189
\(202\) −11.6301 −0.818294
\(203\) −20.4452 −1.43497
\(204\) 1.00000 0.0700140
\(205\) −4.81507 −0.336299
\(206\) 0 0
\(207\) 5.81507 0.404176
\(208\) −3.81507 −0.264528
\(209\) −7.81507 −0.540580
\(210\) −3.00000 −0.207020
\(211\) 8.44522 0.581393 0.290696 0.956815i \(-0.406113\pi\)
0.290696 + 0.956815i \(0.406113\pi\)
\(212\) 10.6301 0.730081
\(213\) −11.6301 −0.796884
\(214\) 8.18493 0.559510
\(215\) −4.81507 −0.328385
\(216\) −1.00000 −0.0680414
\(217\) 20.4452 1.38791
\(218\) −13.0000 −0.880471
\(219\) −5.81507 −0.392946
\(220\) −1.00000 −0.0674200
\(221\) −3.81507 −0.256630
\(222\) −1.00000 −0.0671156
\(223\) −16.8151 −1.12602 −0.563010 0.826450i \(-0.690357\pi\)
−0.563010 + 0.826450i \(0.690357\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) 6.81507 0.453332
\(227\) −24.8151 −1.64703 −0.823517 0.567291i \(-0.807991\pi\)
−0.823517 + 0.567291i \(0.807991\pi\)
\(228\) 7.81507 0.517566
\(229\) 0.369854 0.0244407 0.0122203 0.999925i \(-0.496110\pi\)
0.0122203 + 0.999925i \(0.496110\pi\)
\(230\) −5.81507 −0.383435
\(231\) −3.00000 −0.197386
\(232\) 6.81507 0.447431
\(233\) −7.63015 −0.499867 −0.249934 0.968263i \(-0.580409\pi\)
−0.249934 + 0.968263i \(0.580409\pi\)
\(234\) 3.81507 0.249399
\(235\) −8.00000 −0.521862
\(236\) 2.00000 0.130189
\(237\) 12.0000 0.779484
\(238\) −3.00000 −0.194461
\(239\) 6.44522 0.416907 0.208453 0.978032i \(-0.433157\pi\)
0.208453 + 0.978032i \(0.433157\pi\)
\(240\) 1.00000 0.0645497
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 10.0000 0.642824
\(243\) 1.00000 0.0641500
\(244\) 6.81507 0.436290
\(245\) 2.00000 0.127775
\(246\) 4.81507 0.306998
\(247\) −29.8151 −1.89709
\(248\) −6.81507 −0.432758
\(249\) −3.81507 −0.241770
\(250\) −1.00000 −0.0632456
\(251\) −11.6301 −0.734088 −0.367044 0.930204i \(-0.619630\pi\)
−0.367044 + 0.930204i \(0.619630\pi\)
\(252\) 3.00000 0.188982
\(253\) −5.81507 −0.365591
\(254\) 21.4452 1.34559
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) −3.81507 −0.237978 −0.118989 0.992896i \(-0.537965\pi\)
−0.118989 + 0.992896i \(0.537965\pi\)
\(258\) 4.81507 0.299773
\(259\) 3.00000 0.186411
\(260\) −3.81507 −0.236601
\(261\) −6.81507 −0.421842
\(262\) 9.63015 0.594952
\(263\) −30.4452 −1.87733 −0.938666 0.344827i \(-0.887938\pi\)
−0.938666 + 0.344827i \(0.887938\pi\)
\(264\) 1.00000 0.0615457
\(265\) 10.6301 0.653005
\(266\) −23.4452 −1.43752
\(267\) 11.8151 0.723071
\(268\) 7.63015 0.466085
\(269\) 9.81507 0.598436 0.299218 0.954185i \(-0.403274\pi\)
0.299218 + 0.954185i \(0.403274\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 1.00000 0.0606339
\(273\) −11.4452 −0.692696
\(274\) −2.00000 −0.120824
\(275\) −1.00000 −0.0603023
\(276\) 5.81507 0.350026
\(277\) 21.4452 1.28852 0.644259 0.764807i \(-0.277166\pi\)
0.644259 + 0.764807i \(0.277166\pi\)
\(278\) −22.8151 −1.36836
\(279\) 6.81507 0.408008
\(280\) −3.00000 −0.179284
\(281\) −19.4452 −1.16000 −0.580002 0.814615i \(-0.696948\pi\)
−0.580002 + 0.814615i \(0.696948\pi\)
\(282\) 8.00000 0.476393
\(283\) −21.8151 −1.29677 −0.648386 0.761312i \(-0.724556\pi\)
−0.648386 + 0.761312i \(0.724556\pi\)
\(284\) −11.6301 −0.690122
\(285\) 7.81507 0.462925
\(286\) −3.81507 −0.225590
\(287\) −14.4452 −0.852674
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) 6.81507 0.400195
\(291\) −4.81507 −0.282265
\(292\) −5.81507 −0.340301
\(293\) 28.2603 1.65098 0.825492 0.564414i \(-0.190898\pi\)
0.825492 + 0.564414i \(0.190898\pi\)
\(294\) −2.00000 −0.116642
\(295\) 2.00000 0.116445
\(296\) −1.00000 −0.0581238
\(297\) −1.00000 −0.0580259
\(298\) −2.00000 −0.115857
\(299\) −22.1849 −1.28299
\(300\) 1.00000 0.0577350
\(301\) −14.4452 −0.832609
\(302\) −3.81507 −0.219533
\(303\) 11.6301 0.668134
\(304\) 7.81507 0.448225
\(305\) 6.81507 0.390230
\(306\) −1.00000 −0.0571662
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) −6.81507 −0.387070
\(311\) 10.4452 0.592294 0.296147 0.955142i \(-0.404298\pi\)
0.296147 + 0.955142i \(0.404298\pi\)
\(312\) 3.81507 0.215986
\(313\) 32.8904 1.85908 0.929539 0.368725i \(-0.120205\pi\)
0.929539 + 0.368725i \(0.120205\pi\)
\(314\) 12.8151 0.723196
\(315\) 3.00000 0.169031
\(316\) 12.0000 0.675053
\(317\) −2.81507 −0.158110 −0.0790551 0.996870i \(-0.525190\pi\)
−0.0790551 + 0.996870i \(0.525190\pi\)
\(318\) −10.6301 −0.596109
\(319\) 6.81507 0.381571
\(320\) 1.00000 0.0559017
\(321\) −8.18493 −0.456838
\(322\) −17.4452 −0.972184
\(323\) 7.81507 0.434842
\(324\) 1.00000 0.0555556
\(325\) −3.81507 −0.211622
\(326\) −16.6301 −0.921059
\(327\) 13.0000 0.718902
\(328\) 4.81507 0.265868
\(329\) −24.0000 −1.32316
\(330\) 1.00000 0.0550482
\(331\) −23.2603 −1.27850 −0.639251 0.768998i \(-0.720755\pi\)
−0.639251 + 0.768998i \(0.720755\pi\)
\(332\) −3.81507 −0.209379
\(333\) 1.00000 0.0547997
\(334\) 21.8151 1.19367
\(335\) 7.63015 0.416879
\(336\) 3.00000 0.163663
\(337\) −4.18493 −0.227968 −0.113984 0.993483i \(-0.536361\pi\)
−0.113984 + 0.993483i \(0.536361\pi\)
\(338\) −1.55478 −0.0845690
\(339\) −6.81507 −0.370144
\(340\) 1.00000 0.0542326
\(341\) −6.81507 −0.369057
\(342\) −7.81507 −0.422591
\(343\) −15.0000 −0.809924
\(344\) 4.81507 0.259611
\(345\) 5.81507 0.313073
\(346\) 1.00000 0.0537603
\(347\) 11.2603 0.604484 0.302242 0.953231i \(-0.402265\pi\)
0.302242 + 0.953231i \(0.402265\pi\)
\(348\) −6.81507 −0.365326
\(349\) 7.26029 0.388635 0.194317 0.980939i \(-0.437751\pi\)
0.194317 + 0.980939i \(0.437751\pi\)
\(350\) −3.00000 −0.160357
\(351\) −3.81507 −0.203634
\(352\) 1.00000 0.0533002
\(353\) −13.1849 −0.701763 −0.350881 0.936420i \(-0.614118\pi\)
−0.350881 + 0.936420i \(0.614118\pi\)
\(354\) −2.00000 −0.106299
\(355\) −11.6301 −0.617264
\(356\) 11.8151 0.626198
\(357\) 3.00000 0.158777
\(358\) −16.0000 −0.845626
\(359\) 6.36985 0.336188 0.168094 0.985771i \(-0.446239\pi\)
0.168094 + 0.985771i \(0.446239\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 42.0754 2.21449
\(362\) 10.0000 0.525588
\(363\) −10.0000 −0.524864
\(364\) −11.4452 −0.599892
\(365\) −5.81507 −0.304375
\(366\) −6.81507 −0.356230
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) 5.81507 0.303132
\(369\) −4.81507 −0.250663
\(370\) −1.00000 −0.0519875
\(371\) 31.8904 1.65567
\(372\) 6.81507 0.353345
\(373\) −17.2603 −0.893704 −0.446852 0.894608i \(-0.647455\pi\)
−0.446852 + 0.894608i \(0.647455\pi\)
\(374\) 1.00000 0.0517088
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) 26.0000 1.33907
\(378\) −3.00000 −0.154303
\(379\) −19.6301 −1.00833 −0.504166 0.863607i \(-0.668200\pi\)
−0.504166 + 0.863607i \(0.668200\pi\)
\(380\) 7.81507 0.400905
\(381\) −21.4452 −1.09867
\(382\) −4.63015 −0.236899
\(383\) 33.4452 1.70897 0.854485 0.519475i \(-0.173873\pi\)
0.854485 + 0.519475i \(0.173873\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.00000 −0.152894
\(386\) 2.00000 0.101797
\(387\) −4.81507 −0.244764
\(388\) −4.81507 −0.244448
\(389\) −35.7055 −1.81034 −0.905171 0.425048i \(-0.860257\pi\)
−0.905171 + 0.425048i \(0.860257\pi\)
\(390\) 3.81507 0.193184
\(391\) 5.81507 0.294081
\(392\) −2.00000 −0.101015
\(393\) −9.63015 −0.485777
\(394\) 4.18493 0.210834
\(395\) 12.0000 0.603786
\(396\) −1.00000 −0.0502519
\(397\) −5.63015 −0.282569 −0.141284 0.989969i \(-0.545123\pi\)
−0.141284 + 0.989969i \(0.545123\pi\)
\(398\) 4.00000 0.200502
\(399\) 23.4452 1.17373
\(400\) 1.00000 0.0500000
\(401\) 0.554781 0.0277045 0.0138522 0.999904i \(-0.495591\pi\)
0.0138522 + 0.999904i \(0.495591\pi\)
\(402\) −7.63015 −0.380557
\(403\) −26.0000 −1.29515
\(404\) 11.6301 0.578621
\(405\) 1.00000 0.0496904
\(406\) 20.4452 1.01468
\(407\) −1.00000 −0.0495682
\(408\) −1.00000 −0.0495074
\(409\) 6.36985 0.314969 0.157485 0.987521i \(-0.449662\pi\)
0.157485 + 0.987521i \(0.449662\pi\)
\(410\) 4.81507 0.237800
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) −5.81507 −0.285795
\(415\) −3.81507 −0.187275
\(416\) 3.81507 0.187049
\(417\) 22.8151 1.11726
\(418\) 7.81507 0.382248
\(419\) −25.4452 −1.24308 −0.621540 0.783382i \(-0.713493\pi\)
−0.621540 + 0.783382i \(0.713493\pi\)
\(420\) 3.00000 0.146385
\(421\) −29.2603 −1.42606 −0.713030 0.701134i \(-0.752678\pi\)
−0.713030 + 0.701134i \(0.752678\pi\)
\(422\) −8.44522 −0.411107
\(423\) −8.00000 −0.388973
\(424\) −10.6301 −0.516246
\(425\) 1.00000 0.0485071
\(426\) 11.6301 0.563482
\(427\) 20.4452 0.989413
\(428\) −8.18493 −0.395633
\(429\) 3.81507 0.184193
\(430\) 4.81507 0.232203
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 1.00000 0.0481125
\(433\) 1.81507 0.0872268 0.0436134 0.999048i \(-0.486113\pi\)
0.0436134 + 0.999048i \(0.486113\pi\)
\(434\) −20.4452 −0.981402
\(435\) −6.81507 −0.326758
\(436\) 13.0000 0.622587
\(437\) 45.4452 2.17394
\(438\) 5.81507 0.277855
\(439\) −18.4452 −0.880342 −0.440171 0.897914i \(-0.645082\pi\)
−0.440171 + 0.897914i \(0.645082\pi\)
\(440\) 1.00000 0.0476731
\(441\) 2.00000 0.0952381
\(442\) 3.81507 0.181465
\(443\) 13.6301 0.647588 0.323794 0.946128i \(-0.395042\pi\)
0.323794 + 0.946128i \(0.395042\pi\)
\(444\) 1.00000 0.0474579
\(445\) 11.8151 0.560088
\(446\) 16.8151 0.796217
\(447\) 2.00000 0.0945968
\(448\) 3.00000 0.141737
\(449\) 30.8904 1.45781 0.728905 0.684615i \(-0.240030\pi\)
0.728905 + 0.684615i \(0.240030\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 4.81507 0.226733
\(452\) −6.81507 −0.320554
\(453\) 3.81507 0.179248
\(454\) 24.8151 1.16463
\(455\) −11.4452 −0.536560
\(456\) −7.81507 −0.365974
\(457\) −28.0754 −1.31331 −0.656655 0.754191i \(-0.728029\pi\)
−0.656655 + 0.754191i \(0.728029\pi\)
\(458\) −0.369854 −0.0172822
\(459\) 1.00000 0.0466760
\(460\) 5.81507 0.271129
\(461\) 34.8151 1.62150 0.810750 0.585393i \(-0.199060\pi\)
0.810750 + 0.585393i \(0.199060\pi\)
\(462\) 3.00000 0.139573
\(463\) −17.6301 −0.819342 −0.409671 0.912233i \(-0.634357\pi\)
−0.409671 + 0.912233i \(0.634357\pi\)
\(464\) −6.81507 −0.316382
\(465\) 6.81507 0.316041
\(466\) 7.63015 0.353460
\(467\) 28.8151 1.33340 0.666701 0.745325i \(-0.267706\pi\)
0.666701 + 0.745325i \(0.267706\pi\)
\(468\) −3.81507 −0.176352
\(469\) 22.8904 1.05698
\(470\) 8.00000 0.369012
\(471\) −12.8151 −0.590487
\(472\) −2.00000 −0.0920575
\(473\) 4.81507 0.221397
\(474\) −12.0000 −0.551178
\(475\) 7.81507 0.358580
\(476\) 3.00000 0.137505
\(477\) 10.6301 0.486721
\(478\) −6.44522 −0.294797
\(479\) 9.81507 0.448462 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −3.81507 −0.173952
\(482\) 26.0000 1.18427
\(483\) 17.4452 0.793785
\(484\) −10.0000 −0.454545
\(485\) −4.81507 −0.218641
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −6.81507 −0.308504
\(489\) 16.6301 0.752041
\(490\) −2.00000 −0.0903508
\(491\) 2.18493 0.0986044 0.0493022 0.998784i \(-0.484300\pi\)
0.0493022 + 0.998784i \(0.484300\pi\)
\(492\) −4.81507 −0.217080
\(493\) −6.81507 −0.306935
\(494\) 29.8151 1.34144
\(495\) −1.00000 −0.0449467
\(496\) 6.81507 0.306006
\(497\) −34.8904 −1.56505
\(498\) 3.81507 0.170958
\(499\) −17.4452 −0.780955 −0.390478 0.920612i \(-0.627690\pi\)
−0.390478 + 0.920612i \(0.627690\pi\)
\(500\) 1.00000 0.0447214
\(501\) −21.8151 −0.974626
\(502\) 11.6301 0.519079
\(503\) 0.739708 0.0329820 0.0164910 0.999864i \(-0.494751\pi\)
0.0164910 + 0.999864i \(0.494751\pi\)
\(504\) −3.00000 −0.133631
\(505\) 11.6301 0.517535
\(506\) 5.81507 0.258512
\(507\) 1.55478 0.0690503
\(508\) −21.4452 −0.951478
\(509\) 17.4452 0.773246 0.386623 0.922238i \(-0.373642\pi\)
0.386623 + 0.922238i \(0.373642\pi\)
\(510\) −1.00000 −0.0442807
\(511\) −17.4452 −0.771731
\(512\) −1.00000 −0.0441942
\(513\) 7.81507 0.345044
\(514\) 3.81507 0.168276
\(515\) 0 0
\(516\) −4.81507 −0.211972
\(517\) 8.00000 0.351840
\(518\) −3.00000 −0.131812
\(519\) −1.00000 −0.0438951
\(520\) 3.81507 0.167302
\(521\) 0.815073 0.0357090 0.0178545 0.999841i \(-0.494316\pi\)
0.0178545 + 0.999841i \(0.494316\pi\)
\(522\) 6.81507 0.298288
\(523\) −0.739708 −0.0323452 −0.0161726 0.999869i \(-0.505148\pi\)
−0.0161726 + 0.999869i \(0.505148\pi\)
\(524\) −9.63015 −0.420695
\(525\) 3.00000 0.130931
\(526\) 30.4452 1.32747
\(527\) 6.81507 0.296869
\(528\) −1.00000 −0.0435194
\(529\) 10.8151 0.470221
\(530\) −10.6301 −0.461744
\(531\) 2.00000 0.0867926
\(532\) 23.4452 1.01648
\(533\) 18.3699 0.795687
\(534\) −11.8151 −0.511288
\(535\) −8.18493 −0.353865
\(536\) −7.63015 −0.329572
\(537\) 16.0000 0.690451
\(538\) −9.81507 −0.423158
\(539\) −2.00000 −0.0861461
\(540\) 1.00000 0.0430331
\(541\) 27.8151 1.19586 0.597932 0.801547i \(-0.295989\pi\)
0.597932 + 0.801547i \(0.295989\pi\)
\(542\) 20.0000 0.859074
\(543\) −10.0000 −0.429141
\(544\) −1.00000 −0.0428746
\(545\) 13.0000 0.556859
\(546\) 11.4452 0.489810
\(547\) 23.8904 1.02148 0.510741 0.859735i \(-0.329371\pi\)
0.510741 + 0.859735i \(0.329371\pi\)
\(548\) 2.00000 0.0854358
\(549\) 6.81507 0.290860
\(550\) 1.00000 0.0426401
\(551\) −53.2603 −2.26896
\(552\) −5.81507 −0.247506
\(553\) 36.0000 1.53088
\(554\) −21.4452 −0.911120
\(555\) 1.00000 0.0424476
\(556\) 22.8151 0.967575
\(557\) −21.6301 −0.916499 −0.458249 0.888824i \(-0.651523\pi\)
−0.458249 + 0.888824i \(0.651523\pi\)
\(558\) −6.81507 −0.288505
\(559\) 18.3699 0.776962
\(560\) 3.00000 0.126773
\(561\) −1.00000 −0.0422200
\(562\) 19.4452 0.820247
\(563\) −8.44522 −0.355924 −0.177962 0.984037i \(-0.556950\pi\)
−0.177962 + 0.984037i \(0.556950\pi\)
\(564\) −8.00000 −0.336861
\(565\) −6.81507 −0.286712
\(566\) 21.8151 0.916956
\(567\) 3.00000 0.125988
\(568\) 11.6301 0.487990
\(569\) −35.8151 −1.50145 −0.750723 0.660617i \(-0.770295\pi\)
−0.750723 + 0.660617i \(0.770295\pi\)
\(570\) −7.81507 −0.327337
\(571\) −14.8151 −0.619992 −0.309996 0.950738i \(-0.600328\pi\)
−0.309996 + 0.950738i \(0.600328\pi\)
\(572\) 3.81507 0.159516
\(573\) 4.63015 0.193427
\(574\) 14.4452 0.602932
\(575\) 5.81507 0.242505
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 16.0000 0.665512
\(579\) −2.00000 −0.0831172
\(580\) −6.81507 −0.282980
\(581\) −11.4452 −0.474828
\(582\) 4.81507 0.199591
\(583\) −10.6301 −0.440256
\(584\) 5.81507 0.240629
\(585\) −3.81507 −0.157734
\(586\) −28.2603 −1.16742
\(587\) 20.8151 0.859130 0.429565 0.903036i \(-0.358667\pi\)
0.429565 + 0.903036i \(0.358667\pi\)
\(588\) 2.00000 0.0824786
\(589\) 53.2603 2.19455
\(590\) −2.00000 −0.0823387
\(591\) −4.18493 −0.172145
\(592\) 1.00000 0.0410997
\(593\) −13.6301 −0.559723 −0.279862 0.960040i \(-0.590289\pi\)
−0.279862 + 0.960040i \(0.590289\pi\)
\(594\) 1.00000 0.0410305
\(595\) 3.00000 0.122988
\(596\) 2.00000 0.0819232
\(597\) −4.00000 −0.163709
\(598\) 22.1849 0.907209
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −19.3699 −0.790113 −0.395056 0.918657i \(-0.629275\pi\)
−0.395056 + 0.918657i \(0.629275\pi\)
\(602\) 14.4452 0.588743
\(603\) 7.63015 0.310724
\(604\) 3.81507 0.155233
\(605\) −10.0000 −0.406558
\(606\) −11.6301 −0.472442
\(607\) −37.2603 −1.51235 −0.756174 0.654370i \(-0.772934\pi\)
−0.756174 + 0.654370i \(0.772934\pi\)
\(608\) −7.81507 −0.316943
\(609\) −20.4452 −0.828482
\(610\) −6.81507 −0.275934
\(611\) 30.5206 1.23473
\(612\) 1.00000 0.0404226
\(613\) −6.07536 −0.245382 −0.122691 0.992445i \(-0.539152\pi\)
−0.122691 + 0.992445i \(0.539152\pi\)
\(614\) 18.0000 0.726421
\(615\) −4.81507 −0.194162
\(616\) 3.00000 0.120873
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −13.5548 −0.544813 −0.272406 0.962182i \(-0.587819\pi\)
−0.272406 + 0.962182i \(0.587819\pi\)
\(620\) 6.81507 0.273700
\(621\) 5.81507 0.233351
\(622\) −10.4452 −0.418815
\(623\) 35.4452 1.42008
\(624\) −3.81507 −0.152725
\(625\) 1.00000 0.0400000
\(626\) −32.8904 −1.31457
\(627\) −7.81507 −0.312104
\(628\) −12.8151 −0.511377
\(629\) 1.00000 0.0398726
\(630\) −3.00000 −0.119523
\(631\) 33.7055 1.34180 0.670898 0.741550i \(-0.265909\pi\)
0.670898 + 0.741550i \(0.265909\pi\)
\(632\) −12.0000 −0.477334
\(633\) 8.44522 0.335667
\(634\) 2.81507 0.111801
\(635\) −21.4452 −0.851028
\(636\) 10.6301 0.421513
\(637\) −7.63015 −0.302317
\(638\) −6.81507 −0.269811
\(639\) −11.6301 −0.460081
\(640\) −1.00000 −0.0395285
\(641\) −13.5548 −0.535382 −0.267691 0.963505i \(-0.586261\pi\)
−0.267691 + 0.963505i \(0.586261\pi\)
\(642\) 8.18493 0.323033
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 17.4452 0.687438
\(645\) −4.81507 −0.189593
\(646\) −7.81507 −0.307480
\(647\) 25.0754 0.985814 0.492907 0.870082i \(-0.335934\pi\)
0.492907 + 0.870082i \(0.335934\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.00000 −0.0785069
\(650\) 3.81507 0.149639
\(651\) 20.4452 0.801311
\(652\) 16.6301 0.651287
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) −13.0000 −0.508340
\(655\) −9.63015 −0.376281
\(656\) −4.81507 −0.187997
\(657\) −5.81507 −0.226868
\(658\) 24.0000 0.935617
\(659\) −31.2603 −1.21773 −0.608864 0.793275i \(-0.708375\pi\)
−0.608864 + 0.793275i \(0.708375\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 3.73971 0.145458 0.0727289 0.997352i \(-0.476829\pi\)
0.0727289 + 0.997352i \(0.476829\pi\)
\(662\) 23.2603 0.904037
\(663\) −3.81507 −0.148165
\(664\) 3.81507 0.148054
\(665\) 23.4452 0.909167
\(666\) −1.00000 −0.0387492
\(667\) −39.6301 −1.53449
\(668\) −21.8151 −0.844051
\(669\) −16.8151 −0.650108
\(670\) −7.63015 −0.294778
\(671\) −6.81507 −0.263093
\(672\) −3.00000 −0.115728
\(673\) 6.18493 0.238411 0.119206 0.992870i \(-0.461965\pi\)
0.119206 + 0.992870i \(0.461965\pi\)
\(674\) 4.18493 0.161197
\(675\) 1.00000 0.0384900
\(676\) 1.55478 0.0597993
\(677\) 47.0754 1.80925 0.904627 0.426205i \(-0.140150\pi\)
0.904627 + 0.426205i \(0.140150\pi\)
\(678\) 6.81507 0.261731
\(679\) −14.4452 −0.554357
\(680\) −1.00000 −0.0383482
\(681\) −24.8151 −0.950916
\(682\) 6.81507 0.260963
\(683\) −47.3357 −1.81125 −0.905624 0.424081i \(-0.860597\pi\)
−0.905624 + 0.424081i \(0.860597\pi\)
\(684\) 7.81507 0.298817
\(685\) 2.00000 0.0764161
\(686\) 15.0000 0.572703
\(687\) 0.369854 0.0141108
\(688\) −4.81507 −0.183573
\(689\) −40.5548 −1.54501
\(690\) −5.81507 −0.221376
\(691\) −34.0754 −1.29629 −0.648144 0.761518i \(-0.724455\pi\)
−0.648144 + 0.761518i \(0.724455\pi\)
\(692\) −1.00000 −0.0380143
\(693\) −3.00000 −0.113961
\(694\) −11.2603 −0.427435
\(695\) 22.8151 0.865425
\(696\) 6.81507 0.258325
\(697\) −4.81507 −0.182384
\(698\) −7.26029 −0.274806
\(699\) −7.63015 −0.288599
\(700\) 3.00000 0.113389
\(701\) 33.6301 1.27019 0.635097 0.772433i \(-0.280960\pi\)
0.635097 + 0.772433i \(0.280960\pi\)
\(702\) 3.81507 0.143991
\(703\) 7.81507 0.294751
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) 13.1849 0.496221
\(707\) 34.8904 1.31219
\(708\) 2.00000 0.0751646
\(709\) 20.6301 0.774781 0.387391 0.921916i \(-0.373376\pi\)
0.387391 + 0.921916i \(0.373376\pi\)
\(710\) 11.6301 0.436472
\(711\) 12.0000 0.450035
\(712\) −11.8151 −0.442789
\(713\) 39.6301 1.48416
\(714\) −3.00000 −0.112272
\(715\) 3.81507 0.142676
\(716\) 16.0000 0.597948
\(717\) 6.44522 0.240701
\(718\) −6.36985 −0.237721
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −42.0754 −1.56588
\(723\) −26.0000 −0.966950
\(724\) −10.0000 −0.371647
\(725\) −6.81507 −0.253105
\(726\) 10.0000 0.371135
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 11.4452 0.424188
\(729\) 1.00000 0.0370370
\(730\) 5.81507 0.215226
\(731\) −4.81507 −0.178092
\(732\) 6.81507 0.251892
\(733\) 40.4452 1.49388 0.746939 0.664892i \(-0.231523\pi\)
0.746939 + 0.664892i \(0.231523\pi\)
\(734\) 21.0000 0.775124
\(735\) 2.00000 0.0737711
\(736\) −5.81507 −0.214346
\(737\) −7.63015 −0.281060
\(738\) 4.81507 0.177245
\(739\) 34.0754 1.25348 0.626741 0.779227i \(-0.284388\pi\)
0.626741 + 0.779227i \(0.284388\pi\)
\(740\) 1.00000 0.0367607
\(741\) −29.8151 −1.09528
\(742\) −31.8904 −1.17073
\(743\) 4.44522 0.163079 0.0815396 0.996670i \(-0.474016\pi\)
0.0815396 + 0.996670i \(0.474016\pi\)
\(744\) −6.81507 −0.249853
\(745\) 2.00000 0.0732743
\(746\) 17.2603 0.631944
\(747\) −3.81507 −0.139586
\(748\) −1.00000 −0.0365636
\(749\) −24.5548 −0.897212
\(750\) −1.00000 −0.0365148
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −8.00000 −0.291730
\(753\) −11.6301 −0.423826
\(754\) −26.0000 −0.946864
\(755\) 3.81507 0.138845
\(756\) 3.00000 0.109109
\(757\) 23.8151 0.865574 0.432787 0.901496i \(-0.357530\pi\)
0.432787 + 0.901496i \(0.357530\pi\)
\(758\) 19.6301 0.712999
\(759\) −5.81507 −0.211074
\(760\) −7.81507 −0.283482
\(761\) −9.18493 −0.332953 −0.166477 0.986045i \(-0.553239\pi\)
−0.166477 + 0.986045i \(0.553239\pi\)
\(762\) 21.4452 0.776878
\(763\) 39.0000 1.41189
\(764\) 4.63015 0.167513
\(765\) 1.00000 0.0361551
\(766\) −33.4452 −1.20842
\(767\) −7.63015 −0.275509
\(768\) 1.00000 0.0360844
\(769\) 49.2603 1.77637 0.888186 0.459485i \(-0.151966\pi\)
0.888186 + 0.459485i \(0.151966\pi\)
\(770\) 3.00000 0.108112
\(771\) −3.81507 −0.137396
\(772\) −2.00000 −0.0719816
\(773\) −7.36985 −0.265075 −0.132538 0.991178i \(-0.542313\pi\)
−0.132538 + 0.991178i \(0.542313\pi\)
\(774\) 4.81507 0.173074
\(775\) 6.81507 0.244805
\(776\) 4.81507 0.172851
\(777\) 3.00000 0.107624
\(778\) 35.7055 1.28010
\(779\) −37.6301 −1.34824
\(780\) −3.81507 −0.136602
\(781\) 11.6301 0.416159
\(782\) −5.81507 −0.207947
\(783\) −6.81507 −0.243551
\(784\) 2.00000 0.0714286
\(785\) −12.8151 −0.457390
\(786\) 9.63015 0.343496
\(787\) −39.6301 −1.41266 −0.706331 0.707882i \(-0.749651\pi\)
−0.706331 + 0.707882i \(0.749651\pi\)
\(788\) −4.18493 −0.149082
\(789\) −30.4452 −1.08388
\(790\) −12.0000 −0.426941
\(791\) −20.4452 −0.726948
\(792\) 1.00000 0.0355335
\(793\) −26.0000 −0.923287
\(794\) 5.63015 0.199806
\(795\) 10.6301 0.377012
\(796\) −4.00000 −0.141776
\(797\) −34.8904 −1.23588 −0.617941 0.786224i \(-0.712033\pi\)
−0.617941 + 0.786224i \(0.712033\pi\)
\(798\) −23.4452 −0.829952
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) 11.8151 0.417465
\(802\) −0.554781 −0.0195900
\(803\) 5.81507 0.205209
\(804\) 7.63015 0.269094
\(805\) 17.4452 0.614863
\(806\) 26.0000 0.915811
\(807\) 9.81507 0.345507
\(808\) −11.6301 −0.409147
\(809\) −13.0754 −0.459705 −0.229853 0.973225i \(-0.573824\pi\)
−0.229853 + 0.973225i \(0.573824\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −20.4452 −0.717487
\(813\) −20.0000 −0.701431
\(814\) 1.00000 0.0350500
\(815\) 16.6301 0.582529
\(816\) 1.00000 0.0350070
\(817\) −37.6301 −1.31651
\(818\) −6.36985 −0.222717
\(819\) −11.4452 −0.399928
\(820\) −4.81507 −0.168150
\(821\) 11.0754 0.386533 0.193266 0.981146i \(-0.438092\pi\)
0.193266 + 0.981146i \(0.438092\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 41.0754 1.43180 0.715899 0.698204i \(-0.246017\pi\)
0.715899 + 0.698204i \(0.246017\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) −6.00000 −0.208767
\(827\) −28.0754 −0.976276 −0.488138 0.872766i \(-0.662324\pi\)
−0.488138 + 0.872766i \(0.662324\pi\)
\(828\) 5.81507 0.202088
\(829\) −53.5206 −1.85885 −0.929423 0.369015i \(-0.879695\pi\)
−0.929423 + 0.369015i \(0.879695\pi\)
\(830\) 3.81507 0.132423
\(831\) 21.4452 0.743926
\(832\) −3.81507 −0.132264
\(833\) 2.00000 0.0692959
\(834\) −22.8151 −0.790021
\(835\) −21.8151 −0.754942
\(836\) −7.81507 −0.270290
\(837\) 6.81507 0.235563
\(838\) 25.4452 0.878990
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) −3.00000 −0.103510
\(841\) 17.4452 0.601559
\(842\) 29.2603 1.00838
\(843\) −19.4452 −0.669729
\(844\) 8.44522 0.290696
\(845\) 1.55478 0.0534861
\(846\) 8.00000 0.275046
\(847\) −30.0000 −1.03081
\(848\) 10.6301 0.365041
\(849\) −21.8151 −0.748691
\(850\) −1.00000 −0.0342997
\(851\) 5.81507 0.199338
\(852\) −11.6301 −0.398442
\(853\) −11.0754 −0.379213 −0.189607 0.981860i \(-0.560721\pi\)
−0.189607 + 0.981860i \(0.560721\pi\)
\(854\) −20.4452 −0.699621
\(855\) 7.81507 0.267270
\(856\) 8.18493 0.279755
\(857\) 30.2603 1.03367 0.516836 0.856084i \(-0.327110\pi\)
0.516836 + 0.856084i \(0.327110\pi\)
\(858\) −3.81507 −0.130244
\(859\) 31.4452 1.07290 0.536449 0.843933i \(-0.319766\pi\)
0.536449 + 0.843933i \(0.319766\pi\)
\(860\) −4.81507 −0.164193
\(861\) −14.4452 −0.492292
\(862\) −21.0000 −0.715263
\(863\) −1.55478 −0.0529254 −0.0264627 0.999650i \(-0.508424\pi\)
−0.0264627 + 0.999650i \(0.508424\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.00000 −0.0340010
\(866\) −1.81507 −0.0616787
\(867\) −16.0000 −0.543388
\(868\) 20.4452 0.693956
\(869\) −12.0000 −0.407072
\(870\) 6.81507 0.231053
\(871\) −29.1096 −0.986340
\(872\) −13.0000 −0.440236
\(873\) −4.81507 −0.162966
\(874\) −45.4452 −1.53721
\(875\) 3.00000 0.101419
\(876\) −5.81507 −0.196473
\(877\) 49.7055 1.67844 0.839218 0.543795i \(-0.183013\pi\)
0.839218 + 0.543795i \(0.183013\pi\)
\(878\) 18.4452 0.622496
\(879\) 28.2603 0.953196
\(880\) −1.00000 −0.0337100
\(881\) 43.1849 1.45494 0.727469 0.686141i \(-0.240697\pi\)
0.727469 + 0.686141i \(0.240697\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 27.0000 0.908622 0.454311 0.890843i \(-0.349885\pi\)
0.454311 + 0.890843i \(0.349885\pi\)
\(884\) −3.81507 −0.128315
\(885\) 2.00000 0.0672293
\(886\) −13.6301 −0.457914
\(887\) −14.0754 −0.472604 −0.236302 0.971680i \(-0.575936\pi\)
−0.236302 + 0.971680i \(0.575936\pi\)
\(888\) −1.00000 −0.0335578
\(889\) −64.3357 −2.15775
\(890\) −11.8151 −0.396042
\(891\) −1.00000 −0.0335013
\(892\) −16.8151 −0.563010
\(893\) −62.5206 −2.09217
\(894\) −2.00000 −0.0668900
\(895\) 16.0000 0.534821
\(896\) −3.00000 −0.100223
\(897\) −22.1849 −0.740733
\(898\) −30.8904 −1.03083
\(899\) −46.4452 −1.54903
\(900\) 1.00000 0.0333333
\(901\) 10.6301 0.354142
\(902\) −4.81507 −0.160324
\(903\) −14.4452 −0.480707
\(904\) 6.81507 0.226666
\(905\) −10.0000 −0.332411
\(906\) −3.81507 −0.126747
\(907\) −45.8151 −1.52126 −0.760632 0.649183i \(-0.775111\pi\)
−0.760632 + 0.649183i \(0.775111\pi\)
\(908\) −24.8151 −0.823517
\(909\) 11.6301 0.385748
\(910\) 11.4452 0.379405
\(911\) −54.5206 −1.80635 −0.903174 0.429275i \(-0.858769\pi\)
−0.903174 + 0.429275i \(0.858769\pi\)
\(912\) 7.81507 0.258783
\(913\) 3.81507 0.126260
\(914\) 28.0754 0.928651
\(915\) 6.81507 0.225299
\(916\) 0.369854 0.0122203
\(917\) −28.8904 −0.954046
\(918\) −1.00000 −0.0330049
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −5.81507 −0.191717
\(921\) −18.0000 −0.593120
\(922\) −34.8151 −1.14657
\(923\) 44.3699 1.46045
\(924\) −3.00000 −0.0986928
\(925\) 1.00000 0.0328798
\(926\) 17.6301 0.579363
\(927\) 0 0
\(928\) 6.81507 0.223716
\(929\) 24.4452 0.802022 0.401011 0.916073i \(-0.368659\pi\)
0.401011 + 0.916073i \(0.368659\pi\)
\(930\) −6.81507 −0.223475
\(931\) 15.6301 0.512257
\(932\) −7.63015 −0.249934
\(933\) 10.4452 0.341961
\(934\) −28.8151 −0.942858
\(935\) −1.00000 −0.0327035
\(936\) 3.81507 0.124700
\(937\) −11.6301 −0.379940 −0.189970 0.981790i \(-0.560839\pi\)
−0.189970 + 0.981790i \(0.560839\pi\)
\(938\) −22.8904 −0.747399
\(939\) 32.8904 1.07334
\(940\) −8.00000 −0.260931
\(941\) −1.26029 −0.0410843 −0.0205422 0.999789i \(-0.506539\pi\)
−0.0205422 + 0.999789i \(0.506539\pi\)
\(942\) 12.8151 0.417538
\(943\) −28.0000 −0.911805
\(944\) 2.00000 0.0650945
\(945\) 3.00000 0.0975900
\(946\) −4.81507 −0.156552
\(947\) −25.5548 −0.830419 −0.415209 0.909726i \(-0.636292\pi\)
−0.415209 + 0.909726i \(0.636292\pi\)
\(948\) 12.0000 0.389742
\(949\) 22.1849 0.720153
\(950\) −7.81507 −0.253554
\(951\) −2.81507 −0.0912850
\(952\) −3.00000 −0.0972306
\(953\) 14.8904 0.482349 0.241174 0.970482i \(-0.422467\pi\)
0.241174 + 0.970482i \(0.422467\pi\)
\(954\) −10.6301 −0.344164
\(955\) 4.63015 0.149828
\(956\) 6.44522 0.208453
\(957\) 6.81507 0.220300
\(958\) −9.81507 −0.317111
\(959\) 6.00000 0.193750
\(960\) 1.00000 0.0322749
\(961\) 15.4452 0.498233
\(962\) 3.81507 0.123003
\(963\) −8.18493 −0.263756
\(964\) −26.0000 −0.837404
\(965\) −2.00000 −0.0643823
\(966\) −17.4452 −0.561291
\(967\) 7.63015 0.245369 0.122684 0.992446i \(-0.460850\pi\)
0.122684 + 0.992446i \(0.460850\pi\)
\(968\) 10.0000 0.321412
\(969\) 7.81507 0.251056
\(970\) 4.81507 0.154603
\(971\) −48.0754 −1.54281 −0.771406 0.636343i \(-0.780446\pi\)
−0.771406 + 0.636343i \(0.780446\pi\)
\(972\) 1.00000 0.0320750
\(973\) 68.4452 2.19425
\(974\) −2.00000 −0.0640841
\(975\) −3.81507 −0.122180
\(976\) 6.81507 0.218145
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) −16.6301 −0.531773
\(979\) −11.8151 −0.377611
\(980\) 2.00000 0.0638877
\(981\) 13.0000 0.415058
\(982\) −2.18493 −0.0697238
\(983\) 57.3357 1.82872 0.914362 0.404898i \(-0.132693\pi\)
0.914362 + 0.404898i \(0.132693\pi\)
\(984\) 4.81507 0.153499
\(985\) −4.18493 −0.133343
\(986\) 6.81507 0.217036
\(987\) −24.0000 −0.763928
\(988\) −29.8151 −0.948544
\(989\) −28.0000 −0.890348
\(990\) 1.00000 0.0317821
\(991\) −38.4452 −1.22125 −0.610626 0.791919i \(-0.709082\pi\)
−0.610626 + 0.791919i \(0.709082\pi\)
\(992\) −6.81507 −0.216379
\(993\) −23.2603 −0.738143
\(994\) 34.8904 1.10666
\(995\) −4.00000 −0.126809
\(996\) −3.81507 −0.120885
\(997\) 55.0754 1.74425 0.872127 0.489279i \(-0.162740\pi\)
0.872127 + 0.489279i \(0.162740\pi\)
\(998\) 17.4452 0.552219
\(999\) 1.00000 0.0316386
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 1110.2.a.q.1.1 2
3.2 odd 2 3330.2.a.bf.1.1 2
4.3 odd 2 8880.2.a.bh.1.1 2
5.4 even 2 5550.2.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.1 2 1.1 even 1 trivial
3330.2.a.bf.1.1 2 3.2 odd 2
5550.2.a.bx.1.2 2 5.4 even 2
8880.2.a.bh.1.1 2 4.3 odd 2