Properties

Label 1110.2.a.q.1.2
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 1.2
Root \(-4.81507\) of defining polynomial
Character \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.81507 q^{13} -3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -2.81507 q^{19} +1.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} -4.81507 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.81507 q^{26} +1.00000 q^{27} +3.00000 q^{28} +3.81507 q^{29} -1.00000 q^{30} -3.81507 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.81507 q^{38} +6.81507 q^{39} -1.00000 q^{40} +5.81507 q^{41} -3.00000 q^{42} +5.81507 q^{43} -1.00000 q^{44} +1.00000 q^{45} +4.81507 q^{46} -8.00000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +6.81507 q^{52} -10.6301 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} -2.81507 q^{57} -3.81507 q^{58} +2.00000 q^{59} +1.00000 q^{60} -3.81507 q^{61} +3.81507 q^{62} +3.00000 q^{63} +1.00000 q^{64} +6.81507 q^{65} +1.00000 q^{66} -13.6301 q^{67} +1.00000 q^{68} -4.81507 q^{69} -3.00000 q^{70} +9.63015 q^{71} -1.00000 q^{72} +4.81507 q^{73} -1.00000 q^{74} +1.00000 q^{75} -2.81507 q^{76} -3.00000 q^{77} -6.81507 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -5.81507 q^{82} +6.81507 q^{83} +3.00000 q^{84} +1.00000 q^{85} -5.81507 q^{86} +3.81507 q^{87} +1.00000 q^{88} +1.18493 q^{89} -1.00000 q^{90} +20.4452 q^{91} -4.81507 q^{92} -3.81507 q^{93} +8.00000 q^{94} -2.81507 q^{95} -1.00000 q^{96} +5.81507 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 3q^{13} - 6q^{14} + 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 5q^{19} + 2q^{20} + 6q^{21} + 2q^{22} + q^{23} - 2q^{24} + 2q^{25} - 3q^{26} + 2q^{27} + 6q^{28} - 3q^{29} - 2q^{30} + 3q^{31} - 2q^{32} - 2q^{33} - 2q^{34} + 6q^{35} + 2q^{36} + 2q^{37} - 5q^{38} + 3q^{39} - 2q^{40} + q^{41} - 6q^{42} + q^{43} - 2q^{44} + 2q^{45} - q^{46} - 16q^{47} + 2q^{48} + 4q^{49} - 2q^{50} + 2q^{51} + 3q^{52} - 2q^{54} - 2q^{55} - 6q^{56} + 5q^{57} + 3q^{58} + 4q^{59} + 2q^{60} + 3q^{61} - 3q^{62} + 6q^{63} + 2q^{64} + 3q^{65} + 2q^{66} - 6q^{67} + 2q^{68} + q^{69} - 6q^{70} - 2q^{71} - 2q^{72} - q^{73} - 2q^{74} + 2q^{75} + 5q^{76} - 6q^{77} - 3q^{78} + 24q^{79} + 2q^{80} + 2q^{81} - q^{82} + 3q^{83} + 6q^{84} + 2q^{85} - q^{86} - 3q^{87} + 2q^{88} + 13q^{89} - 2q^{90} + 9q^{91} + q^{92} + 3q^{93} + 16q^{94} + 5q^{95} - 2q^{96} + q^{97} - 4q^{98} - 2q^{99} + O(q^{100}) \)

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
\(13\) 6.81507 1.89016 0.945081 0.326837i \(-0.105983\pi\)
0.945081 + 0.326837i \(0.105983\pi\)
\(14\) −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
\(19\) −2.81507 −0.645822 −0.322911 0.946429i \(-0.604661\pi\)
−0.322911 + 0.946429i \(0.604661\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) 1.00000 0.213201
\(23\) −4.81507 −1.00401 −0.502006 0.864864i \(-0.667404\pi\)
−0.502006 + 0.864864i \(0.667404\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −6.81507 −1.33655
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) 3.81507 0.708441 0.354221 0.935162i \(-0.384746\pi\)
0.354221 + 0.935162i \(0.384746\pi\)
\(30\) −1.00000 −0.182574
\(31\) −3.81507 −0.685207 −0.342604 0.939480i \(-0.611309\pi\)
−0.342604 + 0.939480i \(0.611309\pi\)
\(32\) −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
\(38\) 2.81507 0.456665
\(39\) 6.81507 1.09129
\(40\) −1.00000 −0.158114
\(41\) 5.81507 0.908162 0.454081 0.890960i \(-0.349968\pi\)
0.454081 + 0.890960i \(0.349968\pi\)
\(42\) −3.00000 −0.462910
\(43\) 5.81507 0.886790 0.443395 0.896326i \(-0.353774\pi\)
0.443395 + 0.896326i \(0.353774\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 4.81507 0.709944
\(47\) −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
\(52\) 6.81507 0.945081
\(53\) −10.6301 −1.46016 −0.730081 0.683360i \(-0.760518\pi\)
−0.730081 + 0.683360i \(0.760518\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −3.00000 −0.400892
\(57\) −2.81507 −0.372866
\(58\) −3.81507 −0.500944
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 1.00000 0.129099
\(61\) −3.81507 −0.488470 −0.244235 0.969716i \(-0.578537\pi\)
−0.244235 + 0.969716i \(0.578537\pi\)
\(62\) 3.81507 0.484515
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 6.81507 0.845306
\(66\) 1.00000 0.123091
\(67\) −13.6301 −1.66519 −0.832594 0.553884i \(-0.813145\pi\)
−0.832594 + 0.553884i \(0.813145\pi\)
\(68\) 1.00000 0.121268
\(69\) −4.81507 −0.579667
\(70\) −3.00000 −0.358569
\(71\) 9.63015 1.14289 0.571444 0.820641i \(-0.306383\pi\)
0.571444 + 0.820641i \(0.306383\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.81507 0.563562 0.281781 0.959479i \(-0.409075\pi\)
0.281781 + 0.959479i \(0.409075\pi\)
\(74\) −1.00000 −0.116248
\(75\) 1.00000 0.115470
\(76\) −2.81507 −0.322911
\(77\) −3.00000 −0.341882
\(78\) −6.81507 −0.771655
\(79\) 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
\(82\) −5.81507 −0.642167
\(83\) 6.81507 0.748051 0.374026 0.927418i \(-0.377977\pi\)
0.374026 + 0.927418i \(0.377977\pi\)
\(84\) 3.00000 0.327327
\(85\) 1.00000 0.108465
\(86\) −5.81507 −0.627055
\(87\) 3.81507 0.409019
\(88\) 1.00000 0.106600
\(89\) 1.18493 0.125602 0.0628010 0.998026i \(-0.479997\pi\)
0.0628010 + 0.998026i \(0.479997\pi\)
\(90\) −1.00000 −0.105409
\(91\) 20.4452 2.14324
\(92\) −4.81507 −0.502006
\(93\) −3.81507 −0.395605
\(94\) 8.00000 0.825137
\(95\) −2.81507 −0.288820
\(96\) −1.00000 −0.102062
\(97\) 5.81507 0.590431 0.295216 0.955431i \(-0.404609\pi\)
0.295216 + 0.955431i \(0.404609\pi\)
\(98\) −2.00000 −0.202031
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −9.63015 −0.958235 −0.479118 0.877751i \(-0.659043\pi\)
−0.479118 + 0.877751i \(0.659043\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.81507 −0.668273
\(105\) 3.00000 0.292770
\(106\) 10.6301 1.03249
\(107\) −18.8151 −1.81892 −0.909461 0.415790i \(-0.863505\pi\)
−0.909461 + 0.415790i \(0.863505\pi\)
\(108\) 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
\(113\) 3.81507 0.358892 0.179446 0.983768i \(-0.442570\pi\)
0.179446 + 0.983768i \(0.442570\pi\)
\(114\) 2.81507 0.263656
\(115\) −4.81507 −0.449008
\(116\) 3.81507 0.354221
\(117\) 6.81507 0.630054
\(118\) −2.00000 −0.184115
\(119\) 3.00000 0.275010
\(120\) −1.00000 −0.0912871
\(121\) −10.0000 −0.909091
\(122\) 3.81507 0.345400
\(123\) 5.81507 0.524327
\(124\) −3.81507 −0.342604
\(125\) 1.00000 0.0894427
\(126\) −3.00000 −0.267261
\(127\) 10.4452 0.926863 0.463432 0.886133i \(-0.346618\pi\)
0.463432 + 0.886133i \(0.346618\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.81507 0.511989
\(130\) −6.81507 −0.597721
\(131\) 11.6301 1.01613 0.508065 0.861319i \(-0.330361\pi\)
0.508065 + 0.861319i \(0.330361\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −8.44522 −0.732293
\(134\) 13.6301 1.17747
\(135\) 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 \(0.472772\pi\)
\(138\) 4.81507 0.409886
\(139\) 12.1849 1.03351 0.516756 0.856133i \(-0.327139\pi\)
0.516756 + 0.856133i \(0.327139\pi\)
\(140\) 3.00000 0.253546
\(141\) −8.00000 −0.673722
\(142\) −9.63015 −0.808144
\(143\) −6.81507 −0.569905
\(144\) 1.00000 0.0833333
\(145\) 3.81507 0.316825
\(146\) −4.81507 −0.398498
\(147\) 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
\(151\) −6.81507 −0.554603 −0.277301 0.960783i \(-0.589440\pi\)
−0.277301 + 0.960783i \(0.589440\pi\)
\(152\) 2.81507 0.228333
\(153\) 1.00000 0.0808452
\(154\) 3.00000 0.241747
\(155\) −3.81507 −0.306434
\(156\) 6.81507 0.545643
\(157\) −2.18493 −0.174376 −0.0871881 0.996192i \(-0.527788\pi\)
−0.0871881 + 0.996192i \(0.527788\pi\)
\(158\) −12.0000 −0.954669
\(159\) −10.6301 −0.843025
\(160\) −1.00000 −0.0790569
\(161\) −14.4452 −1.13844
\(162\) −1.00000 −0.0785674
\(163\) −4.63015 −0.362661 −0.181331 0.983422i \(-0.558040\pi\)
−0.181331 + 0.983422i \(0.558040\pi\)
\(164\) 5.81507 0.454081
\(165\) −1.00000 −0.0778499
\(166\) −6.81507 −0.528952
\(167\) −11.1849 −0.865516 −0.432758 0.901510i \(-0.642459\pi\)
−0.432758 + 0.901510i \(0.642459\pi\)
\(168\) −3.00000 −0.231455
\(169\) 33.4452 2.57271
\(170\) −1.00000 −0.0766965
\(171\) −2.81507 −0.215274
\(172\) 5.81507 0.443395
\(173\) −1.00000 −0.0760286 −0.0380143 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) −3.81507 −0.289220
\(175\) 3.00000 0.226779
\(176\) −1.00000 −0.0753778
\(177\) 2.00000 0.150329
\(178\) −1.18493 −0.0888140
\(179\) 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 \(0.621207\pi\)
\(182\) −20.4452 −1.51550
\(183\) −3.81507 −0.282018
\(184\) 4.81507 0.354972
\(185\) 1.00000 0.0735215
\(186\) 3.81507 0.279735
\(187\) −1.00000 −0.0731272
\(188\) −8.00000 −0.583460
\(189\) 3.00000 0.218218
\(190\) 2.81507 0.204227
\(191\) −16.6301 −1.20332 −0.601658 0.798754i \(-0.705493\pi\)
−0.601658 + 0.798754i \(0.705493\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −5.81507 −0.417498
\(195\) 6.81507 0.488038
\(196\) 2.00000 0.142857
\(197\) −14.8151 −1.05553 −0.527765 0.849390i \(-0.676970\pi\)
−0.527765 + 0.849390i \(0.676970\pi\)
\(198\) 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
\(201\) −13.6301 −0.961396
\(202\) 9.63015 0.677575
\(203\) 11.4452 0.803297
\(204\) 1.00000 0.0700140
\(205\) 5.81507 0.406142
\(206\) 0 0
\(207\) −4.81507 −0.334671
\(208\) 6.81507 0.472540
\(209\) 2.81507 0.194723
\(210\) −3.00000 −0.207020
\(211\) −23.4452 −1.61404 −0.807018 0.590527i \(-0.798920\pi\)
−0.807018 + 0.590527i \(0.798920\pi\)
\(212\) −10.6301 −0.730081
\(213\) 9.63015 0.659847
\(214\) 18.8151 1.28617
\(215\) 5.81507 0.396585
\(216\) −1.00000 −0.0680414
\(217\) −11.4452 −0.776952
\(218\) −13.0000 −0.880471
\(219\) 4.81507 0.325372
\(220\) −1.00000 −0.0674200
\(221\) 6.81507 0.458431
\(222\) −1.00000 −0.0671156
\(223\) −6.18493 −0.414173 −0.207087 0.978323i \(-0.566398\pi\)
−0.207087 + 0.978323i \(0.566398\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) −3.81507 −0.253775
\(227\) −14.1849 −0.941487 −0.470743 0.882270i \(-0.656014\pi\)
−0.470743 + 0.882270i \(0.656014\pi\)
\(228\) −2.81507 −0.186433
\(229\) 21.6301 1.42936 0.714680 0.699451i \(-0.246572\pi\)
0.714680 + 0.699451i \(0.246572\pi\)
\(230\) 4.81507 0.317497
\(231\) −3.00000 −0.197386
\(232\) −3.81507 −0.250472
\(233\) 13.6301 0.892941 0.446470 0.894798i \(-0.352681\pi\)
0.446470 + 0.894798i \(0.352681\pi\)
\(234\) −6.81507 −0.445515
\(235\) −8.00000 −0.521862
\(236\) 2.00000 0.130189
\(237\) 12.0000 0.779484
\(238\) −3.00000 −0.194461
\(239\) −25.4452 −1.64591 −0.822957 0.568103i \(-0.807677\pi\)
−0.822957 + 0.568103i \(0.807677\pi\)
\(240\) 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
\(244\) −3.81507 −0.244235
\(245\) 2.00000 0.127775
\(246\) −5.81507 −0.370756
\(247\) −19.1849 −1.22071
\(248\) 3.81507 0.242257
\(249\) 6.81507 0.431888
\(250\) −1.00000 −0.0632456
\(251\) 9.63015 0.607849 0.303925 0.952696i \(-0.401703\pi\)
0.303925 + 0.952696i \(0.401703\pi\)
\(252\) 3.00000 0.188982
\(253\) 4.81507 0.302721
\(254\) −10.4452 −0.655391
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 6.81507 0.425113 0.212556 0.977149i \(-0.431821\pi\)
0.212556 + 0.977149i \(0.431821\pi\)
\(258\) −5.81507 −0.362031
\(259\) 3.00000 0.186411
\(260\) 6.81507 0.422653
\(261\) 3.81507 0.236147
\(262\) −11.6301 −0.718513
\(263\) 1.44522 0.0891160 0.0445580 0.999007i \(-0.485812\pi\)
0.0445580 + 0.999007i \(0.485812\pi\)
\(264\) 1.00000 0.0615457
\(265\) −10.6301 −0.653005
\(266\) 8.44522 0.517810
\(267\) 1.18493 0.0725164
\(268\) −13.6301 −0.832594
\(269\) −0.815073 −0.0496959 −0.0248479 0.999691i \(-0.507910\pi\)
−0.0248479 + 0.999691i \(0.507910\pi\)
\(270\) −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
\(273\) 20.4452 1.23740
\(274\) −2.00000 −0.120824
\(275\) −1.00000 −0.0603023
\(276\) −4.81507 −0.289833
\(277\) −10.4452 −0.627592 −0.313796 0.949490i \(-0.601601\pi\)
−0.313796 + 0.949490i \(0.601601\pi\)
\(278\) −12.1849 −0.730803
\(279\) −3.81507 −0.228402
\(280\) −3.00000 −0.179284
\(281\) 12.4452 0.742420 0.371210 0.928549i \(-0.378943\pi\)
0.371210 + 0.928549i \(0.378943\pi\)
\(282\) 8.00000 0.476393
\(283\) −11.1849 −0.664875 −0.332437 0.943125i \(-0.607871\pi\)
−0.332437 + 0.943125i \(0.607871\pi\)
\(284\) 9.63015 0.571444
\(285\) −2.81507 −0.166751
\(286\) 6.81507 0.402984
\(287\) 17.4452 1.02976
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) −3.81507 −0.224029
\(291\) 5.81507 0.340886
\(292\) 4.81507 0.281781
\(293\) −14.2603 −0.833095 −0.416548 0.909114i \(-0.636760\pi\)
−0.416548 + 0.909114i \(0.636760\pi\)
\(294\) −2.00000 −0.116642
\(295\) 2.00000 0.116445
\(296\) −1.00000 −0.0581238
\(297\) −1.00000 −0.0580259
\(298\) −2.00000 −0.115857
\(299\) −32.8151 −1.89774
\(300\) 1.00000 0.0577350
\(301\) 17.4452 1.00553
\(302\) 6.81507 0.392163
\(303\) −9.63015 −0.553237
\(304\) −2.81507 −0.161456
\(305\) −3.81507 −0.218450
\(306\) −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
\(310\) 3.81507 0.216682
\(311\) −21.4452 −1.21605 −0.608023 0.793919i \(-0.708037\pi\)
−0.608023 + 0.793919i \(0.708037\pi\)
\(312\) −6.81507 −0.385828
\(313\) −30.8904 −1.74603 −0.873015 0.487693i \(-0.837839\pi\)
−0.873015 + 0.487693i \(0.837839\pi\)
\(314\) 2.18493 0.123303
\(315\) 3.00000 0.169031
\(316\) 12.0000 0.675053
\(317\) 7.81507 0.438938 0.219469 0.975619i \(-0.429567\pi\)
0.219469 + 0.975619i \(0.429567\pi\)
\(318\) 10.6301 0.596109
\(319\) −3.81507 −0.213603
\(320\) 1.00000 0.0559017
\(321\) −18.8151 −1.05015
\(322\) 14.4452 0.805001
\(323\) −2.81507 −0.156635
\(324\) 1.00000 0.0555556
\(325\) 6.81507 0.378032
\(326\) 4.63015 0.256440
\(327\) 13.0000 0.718902
\(328\) −5.81507 −0.321084
\(329\) −24.0000 −1.32316
\(330\) 1.00000 0.0550482
\(331\) 19.2603 1.05864 0.529321 0.848422i \(-0.322447\pi\)
0.529321 + 0.848422i \(0.322447\pi\)
\(332\) 6.81507 0.374026
\(333\) 1.00000 0.0547997
\(334\) 11.1849 0.612012
\(335\) −13.6301 −0.744694
\(336\) 3.00000 0.163663
\(337\) −14.8151 −0.807028 −0.403514 0.914973i \(-0.632211\pi\)
−0.403514 + 0.914973i \(0.632211\pi\)
\(338\) −33.4452 −1.81918
\(339\) 3.81507 0.207206
\(340\) 1.00000 0.0542326
\(341\) 3.81507 0.206598
\(342\) 2.81507 0.152222
\(343\) −15.0000 −0.809924
\(344\) −5.81507 −0.313528
\(345\) −4.81507 −0.259235
\(346\) 1.00000 0.0537603
\(347\) −31.2603 −1.67814 −0.839070 0.544023i \(-0.816900\pi\)
−0.839070 + 0.544023i \(0.816900\pi\)
\(348\) 3.81507 0.204509
\(349\) −35.2603 −1.88744 −0.943720 0.330745i \(-0.892700\pi\)
−0.943720 + 0.330745i \(0.892700\pi\)
\(350\) −3.00000 −0.160357
\(351\) 6.81507 0.363762
\(352\) 1.00000 0.0533002
\(353\) −23.8151 −1.26755 −0.633774 0.773518i \(-0.718495\pi\)
−0.633774 + 0.773518i \(0.718495\pi\)
\(354\) −2.00000 −0.106299
\(355\) 9.63015 0.511115
\(356\) 1.18493 0.0628010
\(357\) 3.00000 0.158777
\(358\) −16.0000 −0.845626
\(359\) 27.6301 1.45826 0.729132 0.684373i \(-0.239924\pi\)
0.729132 + 0.684373i \(0.239924\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −11.0754 −0.582914
\(362\) 10.0000 0.525588
\(363\) −10.0000 −0.524864
\(364\) 20.4452 1.07162
\(365\) 4.81507 0.252032
\(366\) 3.81507 0.199417
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) −4.81507 −0.251003
\(369\) 5.81507 0.302721
\(370\) −1.00000 −0.0519875
\(371\) −31.8904 −1.65567
\(372\) −3.81507 −0.197802
\(373\) 25.2603 1.30793 0.653964 0.756526i \(-0.273105\pi\)
0.653964 + 0.756526i \(0.273105\pi\)
\(374\) 1.00000 0.0517088
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) 26.0000 1.33907
\(378\) −3.00000 −0.154303
\(379\) 1.63015 0.0837350 0.0418675 0.999123i \(-0.486669\pi\)
0.0418675 + 0.999123i \(0.486669\pi\)
\(380\) −2.81507 −0.144410
\(381\) 10.4452 0.535125
\(382\) 16.6301 0.850872
\(383\) 1.55478 0.0794456 0.0397228 0.999211i \(-0.487353\pi\)
0.0397228 + 0.999211i \(0.487353\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.00000 −0.152894
\(386\) 2.00000 0.101797
\(387\) 5.81507 0.295597
\(388\) 5.81507 0.295216
\(389\) 38.7055 1.96245 0.981224 0.192873i \(-0.0617807\pi\)
0.981224 + 0.192873i \(0.0617807\pi\)
\(390\) −6.81507 −0.345095
\(391\) −4.81507 −0.243509
\(392\) −2.00000 −0.101015
\(393\) 11.6301 0.586663
\(394\) 14.8151 0.746373
\(395\) 12.0000 0.603786
\(396\) −1.00000 −0.0502519
\(397\) 15.6301 0.784455 0.392227 0.919868i \(-0.371705\pi\)
0.392227 + 0.919868i \(0.371705\pi\)
\(398\) 4.00000 0.200502
\(399\) −8.44522 −0.422790
\(400\) 1.00000 0.0500000
\(401\) 32.4452 1.62024 0.810118 0.586266i \(-0.199403\pi\)
0.810118 + 0.586266i \(0.199403\pi\)
\(402\) 13.6301 0.679810
\(403\) −26.0000 −1.29515
\(404\) −9.63015 −0.479118
\(405\) 1.00000 0.0496904
\(406\) −11.4452 −0.568017
\(407\) −1.00000 −0.0495682
\(408\) −1.00000 −0.0495074
\(409\) 27.6301 1.36622 0.683111 0.730314i \(-0.260626\pi\)
0.683111 + 0.730314i \(0.260626\pi\)
\(410\) −5.81507 −0.287186
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 4.81507 0.236648
\(415\) 6.81507 0.334539
\(416\) −6.81507 −0.334136
\(417\) 12.1849 0.596698
\(418\) −2.81507 −0.137690
\(419\) 6.44522 0.314870 0.157435 0.987529i \(-0.449678\pi\)
0.157435 + 0.987529i \(0.449678\pi\)
\(420\) 3.00000 0.146385
\(421\) 13.2603 0.646267 0.323134 0.946353i \(-0.395264\pi\)
0.323134 + 0.946353i \(0.395264\pi\)
\(422\) 23.4452 1.14130
\(423\) −8.00000 −0.388973
\(424\) 10.6301 0.516246
\(425\) 1.00000 0.0485071
\(426\) −9.63015 −0.466582
\(427\) −11.4452 −0.553873
\(428\) −18.8151 −0.909461
\(429\) −6.81507 −0.329035
\(430\) −5.81507 −0.280428
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 1.00000 0.0481125
\(433\) −8.81507 −0.423625 −0.211813 0.977310i \(-0.567937\pi\)
−0.211813 + 0.977310i \(0.567937\pi\)
\(434\) 11.4452 0.549388
\(435\) 3.81507 0.182919
\(436\) 13.0000 0.622587
\(437\) 13.5548 0.648413
\(438\) −4.81507 −0.230073
\(439\) 13.4452 0.641705 0.320853 0.947129i \(-0.396031\pi\)
0.320853 + 0.947129i \(0.396031\pi\)
\(440\) 1.00000 0.0476731
\(441\) 2.00000 0.0952381
\(442\) −6.81507 −0.324160
\(443\) −7.63015 −0.362519 −0.181260 0.983435i \(-0.558017\pi\)
−0.181260 + 0.983435i \(0.558017\pi\)
\(444\) 1.00000 0.0474579
\(445\) 1.18493 0.0561709
\(446\) 6.18493 0.292865
\(447\) 2.00000 0.0945968
\(448\) 3.00000 0.141737
\(449\) −32.8904 −1.55220 −0.776098 0.630612i \(-0.782804\pi\)
−0.776098 + 0.630612i \(0.782804\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −5.81507 −0.273821
\(452\) 3.81507 0.179446
\(453\) −6.81507 −0.320200
\(454\) 14.1849 0.665732
\(455\) 20.4452 0.958487
\(456\) 2.81507 0.131828
\(457\) 25.0754 1.17298 0.586488 0.809958i \(-0.300510\pi\)
0.586488 + 0.809958i \(0.300510\pi\)
\(458\) −21.6301 −1.01071
\(459\) 1.00000 0.0466760
\(460\) −4.81507 −0.224504
\(461\) 24.1849 1.12640 0.563202 0.826319i \(-0.309569\pi\)
0.563202 + 0.826319i \(0.309569\pi\)
\(462\) 3.00000 0.139573
\(463\) 3.63015 0.168707 0.0843536 0.996436i \(-0.473117\pi\)
0.0843536 + 0.996436i \(0.473117\pi\)
\(464\) 3.81507 0.177110
\(465\) −3.81507 −0.176920
\(466\) −13.6301 −0.631404
\(467\) 18.1849 0.841498 0.420749 0.907177i \(-0.361767\pi\)
0.420749 + 0.907177i \(0.361767\pi\)
\(468\) 6.81507 0.315027
\(469\) −40.8904 −1.88814
\(470\) 8.00000 0.369012
\(471\) −2.18493 −0.100676
\(472\) −2.00000 −0.0920575
\(473\) −5.81507 −0.267377
\(474\) −12.0000 −0.551178
\(475\) −2.81507 −0.129164
\(476\) 3.00000 0.137505
\(477\) −10.6301 −0.486721
\(478\) 25.4452 1.16384
\(479\) −0.815073 −0.0372416 −0.0186208 0.999827i \(-0.505928\pi\)
−0.0186208 + 0.999827i \(0.505928\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 6.81507 0.310741
\(482\) 26.0000 1.18427
\(483\) −14.4452 −0.657280
\(484\) −10.0000 −0.454545
\(485\) 5.81507 0.264049
\(486\) −1.00000 −0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 3.81507 0.172700
\(489\) −4.63015 −0.209382
\(490\) −2.00000 −0.0903508
\(491\) 12.8151 0.578336 0.289168 0.957278i \(-0.406621\pi\)
0.289168 + 0.957278i \(0.406621\pi\)
\(492\) 5.81507 0.262164
\(493\) 3.81507 0.171822
\(494\) 19.1849 0.863171
\(495\) −1.00000 −0.0449467
\(496\) −3.81507 −0.171302
\(497\) 28.8904 1.29591
\(498\) −6.81507 −0.305391
\(499\) 14.4452 0.646657 0.323328 0.946287i \(-0.395198\pi\)
0.323328 + 0.946287i \(0.395198\pi\)
\(500\) 1.00000 0.0447214
\(501\) −11.1849 −0.499706
\(502\) −9.63015 −0.429814
\(503\) 43.2603 1.92888 0.964441 0.264300i \(-0.0851409\pi\)
0.964441 + 0.264300i \(0.0851409\pi\)
\(504\) −3.00000 −0.133631
\(505\) −9.63015 −0.428536
\(506\) −4.81507 −0.214056
\(507\) 33.4452 1.48535
\(508\) 10.4452 0.463432
\(509\) −14.4452 −0.640273 −0.320137 0.947371i \(-0.603729\pi\)
−0.320137 + 0.947371i \(0.603729\pi\)
\(510\) −1.00000 −0.0442807
\(511\) 14.4452 0.639019
\(512\) −1.00000 −0.0441942
\(513\) −2.81507 −0.124289
\(514\) −6.81507 −0.300600
\(515\) 0 0
\(516\) 5.81507 0.255994
\(517\) 8.00000 0.351840
\(518\) −3.00000 −0.131812
\(519\) −1.00000 −0.0438951
\(520\) −6.81507 −0.298861
\(521\) −9.81507 −0.430006 −0.215003 0.976613i \(-0.568976\pi\)
−0.215003 + 0.976613i \(0.568976\pi\)
\(522\) −3.81507 −0.166981
\(523\) −43.2603 −1.89164 −0.945820 0.324691i \(-0.894740\pi\)
−0.945820 + 0.324691i \(0.894740\pi\)
\(524\) 11.6301 0.508065
\(525\) 3.00000 0.130931
\(526\) −1.44522 −0.0630145
\(527\) −3.81507 −0.166187
\(528\) −1.00000 −0.0435194
\(529\) 0.184927 0.00804031
\(530\) 10.6301 0.461744
\(531\) 2.00000 0.0867926
\(532\) −8.44522 −0.366147
\(533\) 39.6301 1.71657
\(534\) −1.18493 −0.0512768
\(535\) −18.8151 −0.813447
\(536\) 13.6301 0.588733
\(537\) 16.0000 0.690451
\(538\) 0.815073 0.0351403
\(539\) −2.00000 −0.0861461
\(540\) 1.00000 0.0430331
\(541\) 17.1849 0.738838 0.369419 0.929263i \(-0.379557\pi\)
0.369419 + 0.929263i \(0.379557\pi\)
\(542\) 20.0000 0.859074
\(543\) −10.0000 −0.429141
\(544\) −1.00000 −0.0428746
\(545\) 13.0000 0.556859
\(546\) −20.4452 −0.874975
\(547\) −39.8904 −1.70559 −0.852796 0.522244i \(-0.825095\pi\)
−0.852796 + 0.522244i \(0.825095\pi\)
\(548\) 2.00000 0.0854358
\(549\) −3.81507 −0.162823
\(550\) 1.00000 0.0426401
\(551\) −10.7397 −0.457527
\(552\) 4.81507 0.204943
\(553\) 36.0000 1.53088
\(554\) 10.4452 0.443775
\(555\) 1.00000 0.0424476
\(556\) 12.1849 0.516756
\(557\) −0.369854 −0.0156712 −0.00783561 0.999969i \(-0.502494\pi\)
−0.00783561 + 0.999969i \(0.502494\pi\)
\(558\) 3.81507 0.161505
\(559\) 39.6301 1.67618
\(560\) 3.00000 0.126773
\(561\) −1.00000 −0.0422200
\(562\) −12.4452 −0.524970
\(563\) 23.4452 0.988098 0.494049 0.869434i \(-0.335516\pi\)
0.494049 + 0.869434i \(0.335516\pi\)
\(564\) −8.00000 −0.336861
\(565\) 3.81507 0.160501
\(566\) 11.1849 0.470138
\(567\) 3.00000 0.125988
\(568\) −9.63015 −0.404072
\(569\) −25.1849 −1.05581 −0.527904 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(570\) 2.81507 0.117910
\(571\) −4.18493 −0.175134 −0.0875669 0.996159i \(-0.527909\pi\)
−0.0875669 + 0.996159i \(0.527909\pi\)
\(572\) −6.81507 −0.284953
\(573\) −16.6301 −0.694734
\(574\) −17.4452 −0.728149
\(575\) −4.81507 −0.200802
\(576\) 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
\(580\) 3.81507 0.158412
\(581\) 20.4452 0.848211
\(582\) −5.81507 −0.241043
\(583\) 10.6301 0.440256
\(584\) −4.81507 −0.199249
\(585\) 6.81507 0.281769
\(586\) 14.2603 0.589087
\(587\) 10.1849 0.420377 0.210188 0.977661i \(-0.432592\pi\)
0.210188 + 0.977661i \(0.432592\pi\)
\(588\) 2.00000 0.0824786
\(589\) 10.7397 0.442522
\(590\) −2.00000 −0.0823387
\(591\) −14.8151 −0.609411
\(592\) 1.00000 0.0410997
\(593\) 7.63015 0.313333 0.156666 0.987652i \(-0.449925\pi\)
0.156666 + 0.987652i \(0.449925\pi\)
\(594\) 1.00000 0.0410305
\(595\) 3.00000 0.122988
\(596\) 2.00000 0.0819232
\(597\) −4.00000 −0.163709
\(598\) 32.8151 1.34191
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −40.6301 −1.65734 −0.828669 0.559739i \(-0.810901\pi\)
−0.828669 + 0.559739i \(0.810901\pi\)
\(602\) −17.4452 −0.711014
\(603\) −13.6301 −0.555062
\(604\) −6.81507 −0.277301
\(605\) −10.0000 −0.406558
\(606\) 9.63015 0.391198
\(607\) 5.26029 0.213509 0.106754 0.994285i \(-0.465954\pi\)
0.106754 + 0.994285i \(0.465954\pi\)
\(608\) 2.81507 0.114166
\(609\) 11.4452 0.463784
\(610\) 3.81507 0.154468
\(611\) −54.5206 −2.20567
\(612\) 1.00000 0.0404226
\(613\) 47.0754 1.90136 0.950678 0.310179i \(-0.100389\pi\)
0.950678 + 0.310179i \(0.100389\pi\)
\(614\) 18.0000 0.726421
\(615\) 5.81507 0.234486
\(616\) 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
\(619\) −45.4452 −1.82660 −0.913299 0.407290i \(-0.866474\pi\)
−0.913299 + 0.407290i \(0.866474\pi\)
\(620\) −3.81507 −0.153217
\(621\) −4.81507 −0.193222
\(622\) 21.4452 0.859875
\(623\) 3.55478 0.142419
\(624\) 6.81507 0.272821
\(625\) 1.00000 0.0400000
\(626\) 30.8904 1.23463
\(627\) 2.81507 0.112423
\(628\) −2.18493 −0.0871881
\(629\) 1.00000 0.0398726
\(630\) −3.00000 −0.119523
\(631\) −40.7055 −1.62046 −0.810230 0.586112i \(-0.800658\pi\)
−0.810230 + 0.586112i \(0.800658\pi\)
\(632\) −12.0000 −0.477334
\(633\) −23.4452 −0.931864
\(634\) −7.81507 −0.310376
\(635\) 10.4452 0.414506
\(636\) −10.6301 −0.421513
\(637\) 13.6301 0.540046
\(638\) 3.81507 0.151040
\(639\) 9.63015 0.380963
\(640\) −1.00000 −0.0395285
\(641\) −45.4452 −1.79498 −0.897489 0.441037i \(-0.854611\pi\)
−0.897489 + 0.441037i \(0.854611\pi\)
\(642\) 18.8151 0.742572
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) −14.4452 −0.569221
\(645\) 5.81507 0.228968
\(646\) 2.81507 0.110758
\(647\) −28.0754 −1.10376 −0.551878 0.833925i \(-0.686089\pi\)
−0.551878 + 0.833925i \(0.686089\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.00000 −0.0785069
\(650\) −6.81507 −0.267309
\(651\) −11.4452 −0.448573
\(652\) −4.63015 −0.181331
\(653\) 4.00000 0.156532 0.0782660 0.996933i \(-0.475062\pi\)
0.0782660 + 0.996933i \(0.475062\pi\)
\(654\) −13.0000 −0.508340
\(655\) 11.6301 0.454427
\(656\) 5.81507 0.227040
\(657\) 4.81507 0.187854
\(658\) 24.0000 0.935617
\(659\) 11.2603 0.438639 0.219319 0.975653i \(-0.429616\pi\)
0.219319 + 0.975653i \(0.429616\pi\)
\(660\) −1.00000 −0.0389249
\(661\) 46.2603 1.79932 0.899658 0.436595i \(-0.143816\pi\)
0.899658 + 0.436595i \(0.143816\pi\)
\(662\) −19.2603 −0.748572
\(663\) 6.81507 0.264675
\(664\) −6.81507 −0.264476
\(665\) −8.44522 −0.327492
\(666\) −1.00000 −0.0387492
\(667\) −18.3699 −0.711284
\(668\) −11.1849 −0.432758
\(669\) −6.18493 −0.239123
\(670\) 13.6301 0.526578
\(671\) 3.81507 0.147279
\(672\) −3.00000 −0.115728
\(673\) 16.8151 0.648173 0.324087 0.946027i \(-0.394943\pi\)
0.324087 + 0.946027i \(0.394943\pi\)
\(674\) 14.8151 0.570655
\(675\) 1.00000 0.0384900
\(676\) 33.4452 1.28635
\(677\) −6.07536 −0.233495 −0.116748 0.993162i \(-0.537247\pi\)
−0.116748 + 0.993162i \(0.537247\pi\)
\(678\) −3.81507 −0.146517
\(679\) 17.4452 0.669486
\(680\) −1.00000 −0.0383482
\(681\) −14.1849 −0.543568
\(682\) −3.81507 −0.146087
\(683\) 48.3357 1.84951 0.924756 0.380560i \(-0.124269\pi\)
0.924756 + 0.380560i \(0.124269\pi\)
\(684\) −2.81507 −0.107637
\(685\) 2.00000 0.0764161
\(686\) 15.0000 0.572703
\(687\) 21.6301 0.825242
\(688\) 5.81507 0.221698
\(689\) −72.4452 −2.75994
\(690\) 4.81507 0.183307
\(691\) 19.0754 0.725661 0.362831 0.931855i \(-0.381810\pi\)
0.362831 + 0.931855i \(0.381810\pi\)
\(692\) −1.00000 −0.0380143
\(693\) −3.00000 −0.113961
\(694\) 31.2603 1.18662
\(695\) 12.1849 0.462201
\(696\) −3.81507 −0.144610
\(697\) 5.81507 0.220262
\(698\) 35.2603 1.33462
\(699\) 13.6301 0.515539
\(700\) 3.00000 0.113389
\(701\) 12.3699 0.467203 0.233601 0.972332i \(-0.424949\pi\)
0.233601 + 0.972332i \(0.424949\pi\)
\(702\) −6.81507 −0.257218
\(703\) −2.81507 −0.106172
\(704\) −1.00000 −0.0376889
\(705\) −8.00000 −0.301297
\(706\) 23.8151 0.896292
\(707\) −28.8904 −1.08654
\(708\) 2.00000 0.0751646
\(709\) −0.630146 −0.0236656 −0.0118328 0.999930i \(-0.503767\pi\)
−0.0118328 + 0.999930i \(0.503767\pi\)
\(710\) −9.63015 −0.361413
\(711\) 12.0000 0.450035
\(712\) −1.18493 −0.0444070
\(713\) 18.3699 0.687956
\(714\) −3.00000 −0.112272
\(715\) −6.81507 −0.254869
\(716\) 16.0000 0.597948
\(717\) −25.4452 −0.950269
\(718\) −27.6301 −1.03115
\(719\) 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
\(722\) 11.0754 0.412182
\(723\) −26.0000 −0.966950
\(724\) −10.0000 −0.371647
\(725\) 3.81507 0.141688
\(726\) 10.0000 0.371135
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) −20.4452 −0.757750
\(729\) 1.00000 0.0370370
\(730\) −4.81507 −0.178214
\(731\) 5.81507 0.215078
\(732\) −3.81507 −0.141009
\(733\) 8.55478 0.315978 0.157989 0.987441i \(-0.449499\pi\)
0.157989 + 0.987441i \(0.449499\pi\)
\(734\) 21.0000 0.775124
\(735\) 2.00000 0.0737711
\(736\) 4.81507 0.177486
\(737\) 13.6301 0.502073
\(738\) −5.81507 −0.214056
\(739\) −19.0754 −0.701699 −0.350849 0.936432i \(-0.614107\pi\)
−0.350849 + 0.936432i \(0.614107\pi\)
\(740\) 1.00000 0.0367607
\(741\) −19.1849 −0.704776
\(742\) 31.8904 1.17073
\(743\) −27.4452 −1.00687 −0.503434 0.864034i \(-0.667930\pi\)
−0.503434 + 0.864034i \(0.667930\pi\)
\(744\) 3.81507 0.139867
\(745\) 2.00000 0.0732743
\(746\) −25.2603 −0.924845
\(747\) 6.81507 0.249350
\(748\) −1.00000 −0.0365636
\(749\) −56.4452 −2.06246
\(750\) −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
\(753\) 9.63015 0.350942
\(754\) −26.0000 −0.946864
\(755\) −6.81507 −0.248026
\(756\) 3.00000 0.109109
\(757\) 13.1849 0.479214 0.239607 0.970870i \(-0.422981\pi\)
0.239607 + 0.970870i \(0.422981\pi\)
\(758\) −1.63015 −0.0592096
\(759\) 4.81507 0.174776
\(760\) 2.81507 0.102113
\(761\) −19.8151 −0.718296 −0.359148 0.933281i \(-0.616933\pi\)
−0.359148 + 0.933281i \(0.616933\pi\)
\(762\) −10.4452 −0.378390
\(763\) 39.0000 1.41189
\(764\) −16.6301 −0.601658
\(765\) 1.00000 0.0361551
\(766\) −1.55478 −0.0561765
\(767\) 13.6301 0.492156
\(768\) 1.00000 0.0360844
\(769\) 6.73971 0.243040 0.121520 0.992589i \(-0.461223\pi\)
0.121520 + 0.992589i \(0.461223\pi\)
\(770\) 3.00000 0.108112
\(771\) 6.81507 0.245439
\(772\) −2.00000 −0.0719816
\(773\) −28.6301 −1.02975 −0.514877 0.857264i \(-0.672163\pi\)
−0.514877 + 0.857264i \(0.672163\pi\)
\(774\) −5.81507 −0.209018
\(775\) −3.81507 −0.137041
\(776\) −5.81507 −0.208749
\(777\) 3.00000 0.107624
\(778\) −38.7055 −1.38766
\(779\) −16.3699 −0.586511
\(780\) 6.81507 0.244019
\(781\) −9.63015 −0.344594
\(782\) 4.81507 0.172187
\(783\) 3.81507 0.136340
\(784\) 2.00000 0.0714286
\(785\) −2.18493 −0.0779834
\(786\) −11.6301 −0.414834
\(787\) −18.3699 −0.654815 −0.327407 0.944883i \(-0.606175\pi\)
−0.327407 + 0.944883i \(0.606175\pi\)
\(788\) −14.8151 −0.527765
\(789\) 1.44522 0.0514511
\(790\) −12.0000 −0.426941
\(791\) 11.4452 0.406945
\(792\) 1.00000 0.0355335
\(793\) −26.0000 −0.923287
\(794\) −15.6301 −0.554693
\(795\) −10.6301 −0.377012
\(796\) −4.00000 −0.141776
\(797\) 28.8904 1.02335 0.511676 0.859179i \(-0.329025\pi\)
0.511676 + 0.859179i \(0.329025\pi\)
\(798\) 8.44522 0.298958
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) 1.18493 0.0418673
\(802\) −32.4452 −1.14568
\(803\) −4.81507 −0.169920
\(804\) −13.6301 −0.480698
\(805\) −14.4452 −0.509127
\(806\) 26.0000 0.915811
\(807\) −0.815073 −0.0286919
\(808\) 9.63015 0.338787
\(809\) 40.0754 1.40897 0.704487 0.709717i \(-0.251177\pi\)
0.704487 + 0.709717i \(0.251177\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 11.4452 0.401648
\(813\) −20.0000 −0.701431
\(814\) 1.00000 0.0350500
\(815\) −4.63015 −0.162187
\(816\) 1.00000 0.0350070
\(817\) −16.3699 −0.572709
\(818\) −27.6301 −0.966065
\(819\) 20.4452 0.714414
\(820\) 5.81507 0.203071
\(821\) −42.0754 −1.46844 −0.734220 0.678911i \(-0.762452\pi\)
−0.734220 + 0.678911i \(0.762452\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −12.0754 −0.420921 −0.210460 0.977602i \(-0.567496\pi\)
−0.210460 + 0.977602i \(0.567496\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) −6.00000 −0.208767
\(827\) 25.0754 0.871956 0.435978 0.899957i \(-0.356403\pi\)
0.435978 + 0.899957i \(0.356403\pi\)
\(828\) −4.81507 −0.167335
\(829\) 31.5206 1.09476 0.547378 0.836886i \(-0.315626\pi\)
0.547378 + 0.836886i \(0.315626\pi\)
\(830\) −6.81507 −0.236555
\(831\) −10.4452 −0.362341
\(832\) 6.81507 0.236270
\(833\) 2.00000 0.0692959
\(834\) −12.1849 −0.421930
\(835\) −11.1849 −0.387070
\(836\) 2.81507 0.0973613
\(837\) −3.81507 −0.131868
\(838\) −6.44522 −0.222646
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) −3.00000 −0.103510
\(841\) −14.4452 −0.498111
\(842\) −13.2603 −0.456980
\(843\) 12.4452 0.428636
\(844\) −23.4452 −0.807018
\(845\) 33.4452 1.15055
\(846\) 8.00000 0.275046
\(847\) −30.0000 −1.03081
\(848\) −10.6301 −0.365041
\(849\) −11.1849 −0.383866
\(850\) −1.00000 −0.0342997
\(851\) −4.81507 −0.165059
\(852\) 9.63015 0.329923
\(853\) 42.0754 1.44063 0.720317 0.693646i \(-0.243997\pi\)
0.720317 + 0.693646i \(0.243997\pi\)
\(854\) 11.4452 0.391647
\(855\) −2.81507 −0.0962735
\(856\) 18.8151 0.643086
\(857\) −12.2603 −0.418804 −0.209402 0.977830i \(-0.567152\pi\)
−0.209402 + 0.977830i \(0.567152\pi\)
\(858\) 6.81507 0.232663
\(859\) −0.445219 −0.0151907 −0.00759533 0.999971i \(-0.502418\pi\)
−0.00759533 + 0.999971i \(0.502418\pi\)
\(860\) 5.81507 0.198292
\(861\) 17.4452 0.594531
\(862\) −21.0000 −0.715263
\(863\) −33.4452 −1.13849 −0.569244 0.822168i \(-0.692764\pi\)
−0.569244 + 0.822168i \(0.692764\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.00000 −0.0340010
\(866\) 8.81507 0.299548
\(867\) −16.0000 −0.543388
\(868\) −11.4452 −0.388476
\(869\) −12.0000 −0.407072
\(870\) −3.81507 −0.129343
\(871\) −92.8904 −3.14747
\(872\) −13.0000 −0.440236
\(873\) 5.81507 0.196810
\(874\) −13.5548 −0.458497
\(875\) 3.00000 0.101419
\(876\) 4.81507 0.162686
\(877\) −24.7055 −0.834246 −0.417123 0.908850i \(-0.636962\pi\)
−0.417123 + 0.908850i \(0.636962\pi\)
\(878\) −13.4452 −0.453754
\(879\) −14.2603 −0.480988
\(880\) −1.00000 −0.0337100
\(881\) 53.8151 1.81308 0.906538 0.422124i \(-0.138715\pi\)
0.906538 + 0.422124i \(0.138715\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 27.0000 0.908622 0.454311 0.890843i \(-0.349885\pi\)
0.454311 + 0.890843i \(0.349885\pi\)
\(884\) 6.81507 0.229216
\(885\) 2.00000 0.0672293
\(886\) 7.63015 0.256340
\(887\) 39.0754 1.31202 0.656011 0.754751i \(-0.272242\pi\)
0.656011 + 0.754751i \(0.272242\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 31.3357 1.05096
\(890\) −1.18493 −0.0397188
\(891\) −1.00000 −0.0335013
\(892\) −6.18493 −0.207087
\(893\) 22.5206 0.753623
\(894\) −2.00000 −0.0668900
\(895\) 16.0000 0.534821
\(896\) −3.00000 −0.100223
\(897\) −32.8151 −1.09566
\(898\) 32.8904 1.09757
\(899\) −14.5548 −0.485429
\(900\) 1.00000 0.0333333
\(901\) −10.6301 −0.354142
\(902\) 5.81507 0.193621
\(903\) 17.4452 0.580541
\(904\) −3.81507 −0.126887
\(905\) −10.0000 −0.332411
\(906\) 6.81507 0.226416
\(907\) −35.1849 −1.16830 −0.584148 0.811647i \(-0.698571\pi\)
−0.584148 + 0.811647i \(0.698571\pi\)
\(908\) −14.1849 −0.470743
\(909\) −9.63015 −0.319412
\(910\) −20.4452 −0.677752
\(911\) 30.5206 1.01119 0.505596 0.862770i \(-0.331273\pi\)
0.505596 + 0.862770i \(0.331273\pi\)
\(912\) −2.81507 −0.0932164
\(913\) −6.81507 −0.225546
\(914\) −25.0754 −0.829419
\(915\) −3.81507 −0.126122
\(916\) 21.6301 0.714680
\(917\) 34.8904 1.15218
\(918\) −1.00000 −0.0330049
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 4.81507 0.158748
\(921\) −18.0000 −0.593120
\(922\) −24.1849 −0.796488
\(923\) 65.6301 2.16024
\(924\) −3.00000 −0.0986928
\(925\) 1.00000 0.0328798
\(926\) −3.63015 −0.119294
\(927\) 0 0
\(928\) −3.81507 −0.125236
\(929\) −7.44522 −0.244270 −0.122135 0.992514i \(-0.538974\pi\)
−0.122135 + 0.992514i \(0.538974\pi\)
\(930\) 3.81507 0.125101
\(931\) −5.63015 −0.184521
\(932\) 13.6301 0.446470
\(933\) −21.4452 −0.702085
\(934\) −18.1849 −0.595029
\(935\) −1.00000 −0.0327035
\(936\) −6.81507 −0.222758
\(937\) 9.63015 0.314603 0.157302 0.987551i \(-0.449721\pi\)
0.157302 + 0.987551i \(0.449721\pi\)
\(938\) 40.8904 1.33512
\(939\) −30.8904 −1.00807
\(940\) −8.00000 −0.260931
\(941\) 41.2603 1.34505 0.672524 0.740076i \(-0.265210\pi\)
0.672524 + 0.740076i \(0.265210\pi\)
\(942\) 2.18493 0.0711888
\(943\) −28.0000 −0.911805
\(944\) 2.00000 0.0650945
\(945\) 3.00000 0.0975900
\(946\) 5.81507 0.189064
\(947\) −57.4452 −1.86672 −0.933359 0.358943i \(-0.883137\pi\)
−0.933359 + 0.358943i \(0.883137\pi\)
\(948\) 12.0000 0.389742
\(949\) 32.8151 1.06522
\(950\) 2.81507 0.0913330
\(951\) 7.81507 0.253421
\(952\) −3.00000 −0.0972306
\(953\) −48.8904 −1.58372 −0.791858 0.610705i \(-0.790886\pi\)
−0.791858 + 0.610705i \(0.790886\pi\)
\(954\) 10.6301 0.344164
\(955\) −16.6301 −0.538139
\(956\) −25.4452 −0.822957
\(957\) −3.81507 −0.123324
\(958\) 0.815073 0.0263338
\(959\) 6.00000 0.193750
\(960\) 1.00000 0.0322749
\(961\) −16.4452 −0.530491
\(962\) −6.81507 −0.219727
\(963\) −18.8151 −0.606307
\(964\) −26.0000 −0.837404
\(965\) −2.00000 −0.0643823
\(966\) 14.4452 0.464767
\(967\) −13.6301 −0.438316 −0.219158 0.975689i \(-0.570331\pi\)
−0.219158 + 0.975689i \(0.570331\pi\)
\(968\) 10.0000 0.321412
\(969\) −2.81507 −0.0904332
\(970\) −5.81507 −0.186711
\(971\) 5.07536 0.162876 0.0814381 0.996678i \(-0.474049\pi\)
0.0814381 + 0.996678i \(0.474049\pi\)
\(972\) 1.00000 0.0320750
\(973\) 36.5548 1.17189
\(974\) −2.00000 −0.0640841
\(975\) 6.81507 0.218257
\(976\) −3.81507 −0.122118
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) 4.63015 0.148056
\(979\) −1.18493 −0.0378704
\(980\) 2.00000 0.0638877
\(981\) 13.0000 0.415058
\(982\) −12.8151 −0.408945
\(983\) −38.3357 −1.22272 −0.611359 0.791354i \(-0.709377\pi\)
−0.611359 + 0.791354i \(0.709377\pi\)
\(984\) −5.81507 −0.185378
\(985\) −14.8151 −0.472047
\(986\) −3.81507 −0.121497
\(987\) −24.0000 −0.763928
\(988\) −19.1849 −0.610354
\(989\) −28.0000 −0.890348
\(990\) 1.00000 0.0317821
\(991\) −6.55478 −0.208219 −0.104110 0.994566i \(-0.533199\pi\)
−0.104110 + 0.994566i \(0.533199\pi\)
\(992\) 3.81507 0.121129
\(993\) 19.2603 0.611207
\(994\) −28.8904 −0.916349
\(995\) −4.00000 −0.126809
\(996\) 6.81507 0.215944
\(997\) 1.92464 0.0609538 0.0304769 0.999535i \(-0.490297\pi\)
0.0304769 + 0.999535i \(0.490297\pi\)
\(998\) −14.4452 −0.457255
\(999\) 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.1.2 2
3.2 odd 2 3330.2.a.bf.1.2 2
4.3 odd 2 8880.2.a.bh.1.2 2
5.4 even 2 5550.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.2 2 1.1 even 1 trivial
3330.2.a.bf.1.2 2 3.2 odd 2
5550.2.a.bx.1.1 2 5.4 even 2
8880.2.a.bh.1.2 2 4.3 odd 2