Properties

Label 5550.2.a.bx.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
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: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.81507\) of defining polynomial
Character \(\chi\) \(=\) 5550.1

$q$-expansion

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

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(-0.894017\pi\)
−0.945081 + 0.326837i \(0.894017\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.81507 −0.645822 −0.322911 0.946429i \(-0.604661\pi\)
−0.322911 + 0.946429i \(0.604661\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) −1.00000 −0.213201
\(23\) 4.81507 1.00401 0.502006 0.864864i \(-0.332596\pi\)
0.502006 + 0.864864i \(0.332596\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −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 \(0.384746\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −2.81507 −0.456665
\(39\) 6.81507 1.09129
\(40\) 0 0
\(41\) 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 \(-0.646226\pi\)
−0.443395 + 0.896326i \(0.646226\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.81507 0.709944
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) −6.81507 −0.945081
\(53\) 10.6301 1.46016 0.730081 0.683360i \(-0.239482\pi\)
0.730081 + 0.683360i \(0.239482\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −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 \(0.458442\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 13.6301 1.66519 0.832594 0.553884i \(-0.186855\pi\)
0.832594 + 0.553884i \(0.186855\pi\)
\(68\) −1.00000 −0.121268
\(69\) −4.81507 −0.579667
\(70\) 0 0
\(71\) 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 \(-0.590925\pi\)
−0.281781 + 0.959479i \(0.590925\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −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 \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 5.81507 0.642167
\(83\) −6.81507 −0.748051 −0.374026 0.927418i \(-0.622023\pi\)
−0.374026 + 0.927418i \(0.622023\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −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 \(0.479997\pi\)
\(90\) 0 0
\(91\) 20.4452 2.14324
\(92\) 4.81507 0.502006
\(93\) 3.81507 0.395605
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −5.81507 −0.590431 −0.295216 0.955431i \(-0.595391\pi\)
−0.295216 + 0.955431i \(0.595391\pi\)
\(98\) 2.00000 0.202031
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −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
\(105\) 0 0
\(106\) 10.6301 1.03249
\(107\) 18.8151 1.81892 0.909461 0.415790i \(-0.136495\pi\)
0.909461 + 0.415790i \(0.136495\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) −3.00000 −0.283473
\(113\) −3.81507 −0.358892 −0.179446 0.983768i \(-0.557430\pi\)
−0.179446 + 0.983768i \(0.557430\pi\)
\(114\) 2.81507 0.263656
\(115\) 0 0
\(116\) 3.81507 0.354221
\(117\) −6.81507 −0.630054
\(118\) 2.00000 0.184115
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −3.81507 −0.345400
\(123\) −5.81507 −0.524327
\(124\) −3.81507 −0.342604
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) −10.4452 −0.926863 −0.463432 0.886133i \(-0.653382\pi\)
−0.463432 + 0.886133i \(0.653382\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.81507 0.511989
\(130\) 0 0
\(131\) 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
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −4.81507 −0.409886
\(139\) 12.1849 1.03351 0.516756 0.856133i \(-0.327139\pi\)
0.516756 + 0.856133i \(0.327139\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 9.63015 0.808144
\(143\) 6.81507 0.569905
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −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 \(0.473894\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) 6.81507 0.545643
\(157\) 2.18493 0.174376 0.0871881 0.996192i \(-0.472212\pi\)
0.0871881 + 0.996192i \(0.472212\pi\)
\(158\) 12.0000 0.954669
\(159\) −10.6301 −0.843025
\(160\) 0 0
\(161\) −14.4452 −1.13844
\(162\) 1.00000 0.0785674
\(163\) 4.63015 0.362661 0.181331 0.983422i \(-0.441960\pi\)
0.181331 + 0.983422i \(0.441960\pi\)
\(164\) 5.81507 0.454081
\(165\) 0 0
\(166\) −6.81507 −0.528952
\(167\) 11.1849 0.865516 0.432758 0.901510i \(-0.357541\pi\)
0.432758 + 0.901510i \(0.357541\pi\)
\(168\) 3.00000 0.231455
\(169\) 33.4452 2.57271
\(170\) 0 0
\(171\) −2.81507 −0.215274
\(172\) −5.81507 −0.443395
\(173\) 1.00000 0.0760286 0.0380143 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) −3.81507 −0.289220
\(175\) 0 0
\(176\) −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 \(0.295983\pi\)
\(180\) 0 0
\(181\) −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
\(185\) 0 0
\(186\) 3.81507 0.279735
\(187\) 1.00000 0.0731272
\(188\) 8.00000 0.583460
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) −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 \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −5.81507 −0.417498
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 14.8151 1.05553 0.527765 0.849390i \(-0.323030\pi\)
0.527765 + 0.849390i \(0.323030\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −13.6301 −0.961396
\(202\) −9.63015 −0.677575
\(203\) −11.4452 −0.803297
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 0 0
\(207\) 4.81507 0.334671
\(208\) −6.81507 −0.472540
\(209\) 2.81507 0.194723
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 11.4452 0.776952
\(218\) 13.0000 0.880471
\(219\) 4.81507 0.325372
\(220\) 0 0
\(221\) 6.81507 0.458431
\(222\) 1.00000 0.0671156
\(223\) 6.18493 0.414173 0.207087 0.978323i \(-0.433602\pi\)
0.207087 + 0.978323i \(0.433602\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −3.81507 −0.253775
\(227\) 14.1849 0.941487 0.470743 0.882270i \(-0.343986\pi\)
0.470743 + 0.882270i \(0.343986\pi\)
\(228\) 2.81507 0.186433
\(229\) 21.6301 1.42936 0.714680 0.699451i \(-0.246572\pi\)
0.714680 + 0.699451i \(0.246572\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 3.81507 0.250472
\(233\) −13.6301 −0.892941 −0.446470 0.894798i \(-0.647319\pi\)
−0.446470 + 0.894798i \(0.647319\pi\)
\(234\) −6.81507 −0.445515
\(235\) 0 0
\(236\) 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 \(0.807677\pi\)
\(240\) 0 0
\(241\) −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
\(245\) 0 0
\(246\) −5.81507 −0.370756
\(247\) 19.1849 1.22071
\(248\) −3.81507 −0.242257
\(249\) 6.81507 0.431888
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.81507 −0.425113 −0.212556 0.977149i \(-0.568179\pi\)
−0.212556 + 0.977149i \(0.568179\pi\)
\(258\) 5.81507 0.362031
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 3.81507 0.236147
\(262\) 11.6301 0.718513
\(263\) −1.44522 −0.0891160 −0.0445580 0.999007i \(-0.514188\pi\)
−0.0445580 + 0.999007i \(0.514188\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 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 \(0.507910\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −4.81507 −0.289833
\(277\) 10.4452 0.627592 0.313796 0.949490i \(-0.398399\pi\)
0.313796 + 0.949490i \(0.398399\pi\)
\(278\) 12.1849 0.730803
\(279\) −3.81507 −0.228402
\(280\) 0 0
\(281\) 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 \(-0.392129\pi\)
0.332437 + 0.943125i \(0.392129\pi\)
\(284\) 9.63015 0.571444
\(285\) 0 0
\(286\) 6.81507 0.402984
\(287\) −17.4452 −1.02976
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 5.81507 0.340886
\(292\) −4.81507 −0.281781
\(293\) 14.2603 0.833095 0.416548 0.909114i \(-0.363240\pi\)
0.416548 + 0.909114i \(0.363240\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 1.00000 0.0580259
\(298\) 2.00000 0.115857
\(299\) −32.8151 −1.89774
\(300\) 0 0
\(301\) 17.4452 1.00553
\(302\) −6.81507 −0.392163
\(303\) 9.63015 0.553237
\(304\) −2.81507 −0.161456
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 18.0000 1.02731 0.513657 0.857996i \(-0.328290\pi\)
0.513657 + 0.857996i \(0.328290\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 0 0
\(311\) −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 \(-0.162161\pi\)
0.873015 + 0.487693i \(0.162161\pi\)
\(314\) 2.18493 0.123303
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −7.81507 −0.438938 −0.219469 0.975619i \(-0.570433\pi\)
−0.219469 + 0.975619i \(0.570433\pi\)
\(318\) −10.6301 −0.596109
\(319\) −3.81507 −0.213603
\(320\) 0 0
\(321\) −18.8151 −1.05015
\(322\) −14.4452 −0.805001
\(323\) 2.81507 0.156635
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 4.63015 0.256440
\(327\) −13.0000 −0.718902
\(328\) 5.81507 0.321084
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) 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
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 14.8151 0.807028 0.403514 0.914973i \(-0.367789\pi\)
0.403514 + 0.914973i \(0.367789\pi\)
\(338\) 33.4452 1.81918
\(339\) 3.81507 0.207206
\(340\) 0 0
\(341\) 3.81507 0.206598
\(342\) −2.81507 −0.152222
\(343\) 15.0000 0.809924
\(344\) −5.81507 −0.313528
\(345\) 0 0
\(346\) 1.00000 0.0537603
\(347\) 31.2603 1.67814 0.839070 0.544023i \(-0.183100\pi\)
0.839070 + 0.544023i \(0.183100\pi\)
\(348\) −3.81507 −0.204509
\(349\) −35.2603 −1.88744 −0.943720 0.330745i \(-0.892700\pi\)
−0.943720 + 0.330745i \(0.892700\pi\)
\(350\) 0 0
\(351\) 6.81507 0.363762
\(352\) −1.00000 −0.0533002
\(353\) 23.8151 1.26755 0.633774 0.773518i \(-0.281505\pi\)
0.633774 + 0.773518i \(0.281505\pi\)
\(354\) −2.00000 −0.106299
\(355\) 0 0
\(356\) 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 \(0.239924\pi\)
\(360\) 0 0
\(361\) −11.0754 −0.582914
\(362\) −10.0000 −0.525588
\(363\) 10.0000 0.524864
\(364\) 20.4452 1.07162
\(365\) 0 0
\(366\) 3.81507 0.199417
\(367\) 21.0000 1.09619 0.548096 0.836416i \(-0.315353\pi\)
0.548096 + 0.836416i \(0.315353\pi\)
\(368\) 4.81507 0.251003
\(369\) 5.81507 0.302721
\(370\) 0 0
\(371\) −31.8904 −1.65567
\(372\) 3.81507 0.197802
\(373\) −25.2603 −1.30793 −0.653964 0.756526i \(-0.726895\pi\)
−0.653964 + 0.756526i \(0.726895\pi\)
\(374\) 1.00000 0.0517088
\(375\) 0 0
\(376\) 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 \(0.486669\pi\)
\(380\) 0 0
\(381\) 10.4452 0.535125
\(382\) −16.6301 −0.850872
\(383\) −1.55478 −0.0794456 −0.0397228 0.999211i \(-0.512647\pi\)
−0.0397228 + 0.999211i \(0.512647\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 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 \(0.0617807\pi\)
\(390\) 0 0
\(391\) −4.81507 −0.243509
\(392\) 2.00000 0.101015
\(393\) −11.6301 −0.586663
\(394\) 14.8151 0.746373
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −15.6301 −0.784455 −0.392227 0.919868i \(-0.628295\pi\)
−0.392227 + 0.919868i \(0.628295\pi\)
\(398\) −4.00000 −0.200502
\(399\) −8.44522 −0.422790
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) −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 \(0.260626\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 4.81507 0.236648
\(415\) 0 0
\(416\) −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 \(0.449678\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) −9.63015 −0.466582
\(427\) 11.4452 0.553873
\(428\) 18.8151 0.909461
\(429\) −6.81507 −0.329035
\(430\) 0 0
\(431\) 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 \(-0.432063\pi\)
0.211813 + 0.977310i \(0.432063\pi\)
\(434\) 11.4452 0.549388
\(435\) 0 0
\(436\) 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 \(0.396031\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 6.81507 0.324160
\(443\) 7.63015 0.362519 0.181260 0.983435i \(-0.441983\pi\)
0.181260 + 0.983435i \(0.441983\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 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 \(0.782804\pi\)
\(450\) 0 0
\(451\) −5.81507 −0.273821
\(452\) −3.81507 −0.179446
\(453\) 6.81507 0.320200
\(454\) 14.1849 0.665732
\(455\) 0 0
\(456\) 2.81507 0.131828
\(457\) −25.0754 −1.17298 −0.586488 0.809958i \(-0.699490\pi\)
−0.586488 + 0.809958i \(0.699490\pi\)
\(458\) 21.6301 1.01071
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 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 \(-0.526883\pi\)
−0.0843536 + 0.996436i \(0.526883\pi\)
\(464\) 3.81507 0.177110
\(465\) 0 0
\(466\) −13.6301 −0.631404
\(467\) −18.1849 −0.841498 −0.420749 0.907177i \(-0.638233\pi\)
−0.420749 + 0.907177i \(0.638233\pi\)
\(468\) −6.81507 −0.315027
\(469\) −40.8904 −1.88814
\(470\) 0 0
\(471\) −2.18493 −0.100676
\(472\) 2.00000 0.0920575
\(473\) 5.81507 0.267377
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) 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 \(0.505928\pi\)
\(480\) 0 0
\(481\) 6.81507 0.310741
\(482\) −26.0000 −1.18427
\(483\) 14.4452 0.657280
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −3.81507 −0.172700
\(489\) −4.63015 −0.209382
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −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 \(0.395198\pi\)
\(500\) 0 0
\(501\) −11.1849 −0.499706
\(502\) 9.63015 0.429814
\(503\) −43.2603 −1.92888 −0.964441 0.264300i \(-0.914859\pi\)
−0.964441 + 0.264300i \(0.914859\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) −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 \(0.603729\pi\)
\(510\) 0 0
\(511\) 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
\(520\) 0 0
\(521\) −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 \(-0.105260\pi\)
0.945820 + 0.324691i \(0.105260\pi\)
\(524\) 11.6301 0.508065
\(525\) 0 0
\(526\) −1.44522 −0.0630145
\(527\) 3.81507 0.166187
\(528\) 1.00000 0.0435194
\(529\) 0.184927 0.00804031
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 8.44522 0.366147
\(533\) −39.6301 −1.71657
\(534\) −1.18493 −0.0512768
\(535\) 0 0
\(536\) 13.6301 0.588733
\(537\) −16.0000 −0.690451
\(538\) −0.815073 −0.0351403
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) −20.4452 −0.874975
\(547\) 39.8904 1.70559 0.852796 0.522244i \(-0.174905\pi\)
0.852796 + 0.522244i \(0.174905\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −3.81507 −0.162823
\(550\) 0 0
\(551\) −10.7397 −0.457527
\(552\) −4.81507 −0.204943
\(553\) −36.0000 −1.53088
\(554\) 10.4452 0.443775
\(555\) 0 0
\(556\) 12.1849 0.516756
\(557\) 0.369854 0.0156712 0.00783561 0.999969i \(-0.497506\pi\)
0.00783561 + 0.999969i \(0.497506\pi\)
\(558\) −3.81507 −0.161505
\(559\) 39.6301 1.67618
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 12.4452 0.524970
\(563\) −23.4452 −0.988098 −0.494049 0.869434i \(-0.664484\pi\)
−0.494049 + 0.869434i \(0.664484\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 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 \(0.677022\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −16.0000 −0.665512
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 20.4452 0.848211
\(582\) 5.81507 0.241043
\(583\) −10.6301 −0.440256
\(584\) −4.81507 −0.199249
\(585\) 0 0
\(586\) 14.2603 0.589087
\(587\) −10.1849 −0.420377 −0.210188 0.977661i \(-0.567408\pi\)
−0.210188 + 0.977661i \(0.567408\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 10.7397 0.442522
\(590\) 0 0
\(591\) −14.8151 −0.609411
\(592\) −1.00000 −0.0410997
\(593\) −7.63015 −0.313333 −0.156666 0.987652i \(-0.550075\pi\)
−0.156666 + 0.987652i \(0.550075\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 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 \(0.394006\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) 9.63015 0.391198
\(607\) −5.26029 −0.213509 −0.106754 0.994285i \(-0.534046\pi\)
−0.106754 + 0.994285i \(0.534046\pi\)
\(608\) −2.81507 −0.114166
\(609\) 11.4452 0.463784
\(610\) 0 0
\(611\) −54.5206 −2.20567
\(612\) −1.00000 −0.0404226
\(613\) −47.0754 −1.90136 −0.950678 0.310179i \(-0.899611\pi\)
−0.950678 + 0.310179i \(0.899611\pi\)
\(614\) 18.0000 0.726421
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −45.4452 −1.82660 −0.913299 0.407290i \(-0.866474\pi\)
−0.913299 + 0.407290i \(0.866474\pi\)
\(620\) 0 0
\(621\) −4.81507 −0.193222
\(622\) −21.4452 −0.859875
\(623\) −3.55478 −0.142419
\(624\) 6.81507 0.272821
\(625\) 0 0
\(626\) 30.8904 1.23463
\(627\) −2.81507 −0.112423
\(628\) 2.18493 0.0871881
\(629\) 1.00000 0.0398726
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) −10.6301 −0.421513
\(637\) −13.6301 −0.540046
\(638\) −3.81507 −0.151040
\(639\) 9.63015 0.380963
\(640\) 0 0
\(641\) −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 \(-0.608806\pi\)
−0.335207 + 0.942144i \(0.608806\pi\)
\(644\) −14.4452 −0.569221
\(645\) 0 0
\(646\) 2.81507 0.110758
\(647\) 28.0754 1.10376 0.551878 0.833925i \(-0.313911\pi\)
0.551878 + 0.833925i \(0.313911\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.00000 −0.0785069
\(650\) 0 0
\(651\) −11.4452 −0.448573
\(652\) 4.63015 0.181331
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) −13.0000 −0.508340
\(655\) 0 0
\(656\) 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 \(0.429616\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 18.3699 0.711284
\(668\) 11.1849 0.432758
\(669\) −6.18493 −0.239123
\(670\) 0 0
\(671\) 3.81507 0.147279
\(672\) 3.00000 0.115728
\(673\) −16.8151 −0.648173 −0.324087 0.946027i \(-0.605057\pi\)
−0.324087 + 0.946027i \(0.605057\pi\)
\(674\) 14.8151 0.570655
\(675\) 0 0
\(676\) 33.4452 1.28635
\(677\) 6.07536 0.233495 0.116748 0.993162i \(-0.462753\pi\)
0.116748 + 0.993162i \(0.462753\pi\)
\(678\) 3.81507 0.146517
\(679\) 17.4452 0.669486
\(680\) 0 0
\(681\) −14.1849 −0.543568
\(682\) 3.81507 0.146087
\(683\) −48.3357 −1.84951 −0.924756 0.380560i \(-0.875731\pi\)
−0.924756 + 0.380560i \(0.875731\pi\)
\(684\) −2.81507 −0.107637
\(685\) 0 0
\(686\) 15.0000 0.572703
\(687\) −21.6301 −0.825242
\(688\) −5.81507 −0.221698
\(689\) −72.4452 −2.75994
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) −3.81507 −0.144610
\(697\) −5.81507 −0.220262
\(698\) −35.2603 −1.33462
\(699\) 13.6301 0.515539
\(700\) 0 0
\(701\) 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
\(705\) 0 0
\(706\) 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 \(0.503767\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 1.18493 0.0444070
\(713\) −18.3699 −0.687956
\(714\) −3.00000 −0.112272
\(715\) 0 0
\(716\) 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 \(0.281415\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −11.0754 −0.412182
\(723\) 26.0000 0.966950
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 10.0000 0.371135
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 20.4452 0.757750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.81507 0.215078
\(732\) 3.81507 0.141009
\(733\) −8.55478 −0.315978 −0.157989 0.987441i \(-0.550501\pi\)
−0.157989 + 0.987441i \(0.550501\pi\)
\(734\) 21.0000 0.775124
\(735\) 0 0
\(736\) 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 \(0.614107\pi\)
\(740\) 0 0
\(741\) −19.1849 −0.704776
\(742\) −31.8904 −1.17073
\(743\) 27.4452 1.00687 0.503434 0.864034i \(-0.332070\pi\)
0.503434 + 0.864034i \(0.332070\pi\)
\(744\) 3.81507 0.139867
\(745\) 0 0
\(746\) −25.2603 −0.924845
\(747\) −6.81507 −0.249350
\(748\) 1.00000 0.0365636
\(749\) −56.4452 −2.06246
\(750\) 0 0
\(751\) 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
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) −13.1849 −0.479214 −0.239607 0.970870i \(-0.577019\pi\)
−0.239607 + 0.970870i \(0.577019\pi\)
\(758\) 1.63015 0.0592096
\(759\) 4.81507 0.174776
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) −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 \(0.461223\pi\)
\(770\) 0 0
\(771\) 6.81507 0.245439
\(772\) 2.00000 0.0719816
\(773\) 28.6301 1.02975 0.514877 0.857264i \(-0.327837\pi\)
0.514877 + 0.857264i \(0.327837\pi\)
\(774\) −5.81507 −0.209018
\(775\) 0 0
\(776\) −5.81507 −0.208749
\(777\) −3.00000 −0.107624
\(778\) 38.7055 1.38766
\(779\) −16.3699 −0.586511
\(780\) 0 0
\(781\) −9.63015 −0.344594
\(782\) −4.81507 −0.172187
\(783\) −3.81507 −0.136340
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) −11.6301 −0.414834
\(787\) 18.3699 0.654815 0.327407 0.944883i \(-0.393825\pi\)
0.327407 + 0.944883i \(0.393825\pi\)
\(788\) 14.8151 0.527765
\(789\) 1.44522 0.0514511
\(790\) 0 0
\(791\) 11.4452 0.406945
\(792\) −1.00000 −0.0355335
\(793\) 26.0000 0.923287
\(794\) −15.6301 −0.554693
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −28.8904 −1.02335 −0.511676 0.859179i \(-0.670975\pi\)
−0.511676 + 0.859179i \(0.670975\pi\)
\(798\) −8.44522 −0.298958
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 1.18493 0.0418673
\(802\) 32.4452 1.14568
\(803\) 4.81507 0.169920
\(804\) −13.6301 −0.480698
\(805\) 0 0
\(806\) 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 \(0.251177\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) 16.3699 0.572709
\(818\) 27.6301 0.966065
\(819\) 20.4452 0.714414
\(820\) 0 0
\(821\) −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 \(-0.432504\pi\)
0.210460 + 0.977602i \(0.432504\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −25.0754 −0.871956 −0.435978 0.899957i \(-0.643597\pi\)
−0.435978 + 0.899957i \(0.643597\pi\)
\(828\) 4.81507 0.167335
\(829\) 31.5206 1.09476 0.547378 0.836886i \(-0.315626\pi\)
0.547378 + 0.836886i \(0.315626\pi\)
\(830\) 0 0
\(831\) −10.4452 −0.362341
\(832\) −6.81507 −0.236270
\(833\) −2.00000 −0.0692959
\(834\) −12.1849 −0.421930
\(835\) 0 0
\(836\) 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 \(0.399432\pi\)
\(840\) 0 0
\(841\) −14.4452 −0.498111
\(842\) 13.2603 0.456980
\(843\) −12.4452 −0.428636
\(844\) −23.4452 −0.807018
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 30.0000 1.03081
\(848\) 10.6301 0.365041
\(849\) −11.1849 −0.383866
\(850\) 0 0
\(851\) −4.81507 −0.165059
\(852\) −9.63015 −0.329923
\(853\) −42.0754 −1.44063 −0.720317 0.693646i \(-0.756003\pi\)
−0.720317 + 0.693646i \(0.756003\pi\)
\(854\) 11.4452 0.391647
\(855\) 0 0
\(856\) 18.8151 0.643086
\(857\) 12.2603 0.418804 0.209402 0.977830i \(-0.432848\pi\)
0.209402 + 0.977830i \(0.432848\pi\)
\(858\) −6.81507 −0.232663
\(859\) −0.445219 −0.0151907 −0.00759533 0.999971i \(-0.502418\pi\)
−0.00759533 + 0.999971i \(0.502418\pi\)
\(860\) 0 0
\(861\) 17.4452 0.594531
\(862\) 21.0000 0.715263
\(863\) 33.4452 1.13849 0.569244 0.822168i \(-0.307236\pi\)
0.569244 + 0.822168i \(0.307236\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 8.81507 0.299548
\(867\) 16.0000 0.543388
\(868\) 11.4452 0.388476
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −92.8904 −3.14747
\(872\) 13.0000 0.440236
\(873\) −5.81507 −0.196810
\(874\) −13.5548 −0.458497
\(875\) 0 0
\(876\) 4.81507 0.162686
\(877\) 24.7055 0.834246 0.417123 0.908850i \(-0.363038\pi\)
0.417123 + 0.908850i \(0.363038\pi\)
\(878\) 13.4452 0.453754
\(879\) −14.2603 −0.480988
\(880\) 0 0
\(881\) 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 \(-0.650115\pi\)
−0.454311 + 0.890843i \(0.650115\pi\)
\(884\) 6.81507 0.229216
\(885\) 0 0
\(886\) 7.63015 0.256340
\(887\) −39.0754 −1.31202 −0.656011 0.754751i \(-0.727758\pi\)
−0.656011 + 0.754751i \(0.727758\pi\)
\(888\) 1.00000 0.0335578
\(889\) 31.3357 1.05096
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 6.18493 0.207087
\(893\) −22.5206 −0.753623
\(894\) −2.00000 −0.0668900
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 32.8151 1.09566
\(898\) −32.8904 −1.09757
\(899\) −14.5548 −0.485429
\(900\) 0 0
\(901\) −10.6301 −0.354142
\(902\) −5.81507 −0.193621
\(903\) −17.4452 −0.580541
\(904\) −3.81507 −0.126887
\(905\) 0 0
\(906\) 6.81507 0.226416
\(907\) 35.1849 1.16830 0.584148 0.811647i \(-0.301429\pi\)
0.584148 + 0.811647i \(0.301429\pi\)
\(908\) 14.1849 0.470743
\(909\) −9.63015 −0.319412
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) 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 \(0.370452\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 24.1849 0.796488
\(923\) −65.6301 −2.16024
\(924\) −3.00000 −0.0986928
\(925\) 0 0
\(926\) −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 \(0.538974\pi\)
\(930\) 0 0
\(931\) −5.63015 −0.184521
\(932\) −13.6301 −0.446470
\(933\) 21.4452 0.702085
\(934\) −18.1849 −0.595029
\(935\) 0 0
\(936\) −6.81507 −0.222758
\(937\) −9.63015 −0.314603 −0.157302 0.987551i \(-0.550279\pi\)
−0.157302 + 0.987551i \(0.550279\pi\)
\(938\) −40.8904 −1.33512
\(939\) −30.8904 −1.00807
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) 5.81507 0.189064
\(947\) 57.4452 1.86672 0.933359 0.358943i \(-0.116863\pi\)
0.933359 + 0.358943i \(0.116863\pi\)
\(948\) −12.0000 −0.389742
\(949\) 32.8151 1.06522
\(950\) 0 0
\(951\) 7.81507 0.253421
\(952\) 3.00000 0.0972306
\(953\) 48.8904 1.58372 0.791858 0.610705i \(-0.209114\pi\)
0.791858 + 0.610705i \(0.209114\pi\)
\(954\) 10.6301 0.344164
\(955\) 0 0
\(956\) −25.4452 −0.822957
\(957\) 3.81507 0.123324
\(958\) −0.815073 −0.0263338
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −16.4452 −0.530491
\(962\) 6.81507 0.219727
\(963\) 18.8151 0.606307
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 14.4452 0.464767
\(967\) 13.6301 0.438316 0.219158 0.975689i \(-0.429669\pi\)
0.219158 + 0.975689i \(0.429669\pi\)
\(968\) −10.0000 −0.321412
\(969\) −2.81507 −0.0904332
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −3.81507 −0.122118
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −4.63015 −0.148056
\(979\) −1.18493 −0.0378704
\(980\) 0 0
\(981\) 13.0000 0.415058
\(982\) 12.8151 0.408945
\(983\) 38.3357 1.22272 0.611359 0.791354i \(-0.290623\pi\)
0.611359 + 0.791354i \(0.290623\pi\)
\(984\) −5.81507 −0.185378
\(985\) 0 0
\(986\) −3.81507 −0.121497
\(987\) 24.0000 0.763928
\(988\) 19.1849 0.610354
\(989\) −28.0000 −0.890348
\(990\) 0 0
\(991\) −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
\(995\) 0 0
\(996\) 6.81507 0.215944
\(997\) −1.92464 −0.0609538 −0.0304769 0.999535i \(-0.509703\pi\)
−0.0304769 + 0.999535i \(0.509703\pi\)
\(998\) 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 5550.2.a.bx.1.1 2
5.4 even 2 1110.2.a.q.1.2 2
15.14 odd 2 3330.2.a.bf.1.2 2
20.19 odd 2 8880.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.2 2 5.4 even 2
3330.2.a.bf.1.2 2 15.14 odd 2
5550.2.a.bx.1.1 2 1.1 even 1 trivial
8880.2.a.bh.1.2 2 20.19 odd 2