Properties

Label 2541.2.a.a.1.1
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +3.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +4.00000 q^{19} -6.00000 q^{20} -1.00000 q^{21} +2.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -8.00000 q^{29} -6.00000 q^{30} +2.00000 q^{31} +8.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} +2.00000 q^{36} +10.0000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +2.00000 q^{42} -9.00000 q^{43} -3.00000 q^{45} -4.00000 q^{46} +9.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -8.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} -8.00000 q^{53} +2.00000 q^{54} -4.00000 q^{57} +16.0000 q^{58} -3.00000 q^{59} +6.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +13.0000 q^{67} -6.00000 q^{68} -2.00000 q^{69} +6.00000 q^{70} +14.0000 q^{71} +10.0000 q^{73} -20.0000 q^{74} -4.00000 q^{75} +8.00000 q^{76} -4.00000 q^{78} +4.00000 q^{79} +12.0000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -1.00000 q^{83} -2.00000 q^{84} +9.00000 q^{85} +18.0000 q^{86} +8.00000 q^{87} -9.00000 q^{89} +6.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} -2.00000 q^{93} -18.0000 q^{94} -12.0000 q^{95} -8.00000 q^{96} -16.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 6.00000 1.89737
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 3.00000 0.774597
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −2.00000 −0.471405
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −6.00000 −1.34164
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −6.00000 −1.09545
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.00000 −0.507093
\(36\) 2.00000 0.333333
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) −4.00000 −0.589768
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 4.00000 0.577350
\(49\) 1.00000 0.142857
\(50\) −8.00000 −1.13137
\(51\) 3.00000 0.420084
\(52\) −4.00000 −0.554700
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 16.0000 2.10090
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 6.00000 0.774597
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) −6.00000 −0.727607
\(69\) −2.00000 −0.240772
\(70\) 6.00000 0.717137
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −20.0000 −2.32495
\(75\) −4.00000 −0.461880
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 12.0000 1.34164
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) −2.00000 −0.218218
\(85\) 9.00000 0.976187
\(86\) 18.0000 1.94099
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 6.00000 0.632456
\(91\) −2.00000 −0.209657
\(92\) 4.00000 0.417029
\(93\) −2.00000 −0.207390
\(94\) −18.0000 −1.85656
\(95\) −12.0000 −1.23117
\(96\) −8.00000 −0.816497
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −17.0000 −1.69156 −0.845782 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(102\) −6.00000 −0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 16.0000 1.55406
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) −2.00000 −0.192450
\(109\) −13.0000 −1.24517 −0.622587 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) −4.00000 −0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 8.00000 0.749269
\(115\) −6.00000 −0.559503
\(116\) −16.0000 −1.48556
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 0 0
\(122\) −20.0000 −1.81071
\(123\) 2.00000 0.180334
\(124\) 4.00000 0.359211
\(125\) 3.00000 0.268328
\(126\) −2.00000 −0.178174
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) 9.00000 0.792406
\(130\) −12.0000 −1.05247
\(131\) −13.0000 −1.13582 −0.567908 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −26.0000 −2.24606
\(135\) 3.00000 0.258199
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 4.00000 0.340503
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −6.00000 −0.507093
\(141\) −9.00000 −0.757937
\(142\) −28.0000 −2.34971
\(143\) 0 0
\(144\) −4.00000 −0.333333
\(145\) 24.0000 1.99309
\(146\) −20.0000 −1.65521
\(147\) −1.00000 −0.0824786
\(148\) 20.0000 1.64399
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 8.00000 0.653197
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 4.00000 0.320256
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −8.00000 −0.636446
\(159\) 8.00000 0.634441
\(160\) −24.0000 −1.89737
\(161\) 2.00000 0.157622
\(162\) −2.00000 −0.157135
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −18.0000 −1.38054
\(171\) 4.00000 0.305888
\(172\) −18.0000 −1.37249
\(173\) −25.0000 −1.90071 −0.950357 0.311160i \(-0.899282\pi\)
−0.950357 + 0.311160i \(0.899282\pi\)
\(174\) −16.0000 −1.21296
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 18.0000 1.34916
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −6.00000 −0.447214
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 4.00000 0.296500
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −30.0000 −2.20564
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) 18.0000 1.31278
\(189\) −1.00000 −0.0727393
\(190\) 24.0000 1.74114
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 8.00000 0.577350
\(193\) 23.0000 1.65558 0.827788 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 32.0000 2.29747
\(195\) −6.00000 −0.429669
\(196\) 2.00000 0.142857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 34.0000 2.39223
\(203\) −8.00000 −0.561490
\(204\) 6.00000 0.420084
\(205\) 6.00000 0.419058
\(206\) −28.0000 −1.95085
\(207\) 2.00000 0.139010
\(208\) 8.00000 0.554700
\(209\) 0 0
\(210\) −6.00000 −0.414039
\(211\) −7.00000 −0.481900 −0.240950 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) −16.0000 −1.09888
\(213\) −14.0000 −0.959264
\(214\) −20.0000 −1.36717
\(215\) 27.0000 1.84138
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 26.0000 1.76094
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 20.0000 1.34231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 8.00000 0.534522
\(225\) 4.00000 0.266667
\(226\) −12.0000 −0.798228
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) −8.00000 −0.529813
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 4.00000 0.261488
\(235\) −27.0000 −1.76129
\(236\) −6.00000 −0.390567
\(237\) −4.00000 −0.259828
\(238\) 6.00000 0.388922
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −12.0000 −0.774597
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 20.0000 1.28037
\(245\) −3.00000 −0.191663
\(246\) −4.00000 −0.255031
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 1.00000 0.0633724
\(250\) −6.00000 −0.379473
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 26.0000 1.63139
\(255\) −9.00000 −0.563602
\(256\) 16.0000 1.00000
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) −18.0000 −1.12063
\(259\) 10.0000 0.621370
\(260\) 12.0000 0.744208
\(261\) −8.00000 −0.495188
\(262\) 26.0000 1.60629
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) −8.00000 −0.490511
\(267\) 9.00000 0.550791
\(268\) 26.0000 1.58820
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −6.00000 −0.365148
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 12.0000 0.727607
\(273\) 2.00000 0.121046
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) −4.00000 −0.239904
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 18.0000 1.07188
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 28.0000 1.66149
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 8.00000 0.471405
\(289\) −8.00000 −0.470588
\(290\) −48.0000 −2.81866
\(291\) 16.0000 0.937937
\(292\) 20.0000 1.17041
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 2.00000 0.116642
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) −4.00000 −0.231326
\(300\) −8.00000 −0.461880
\(301\) −9.00000 −0.518751
\(302\) −10.0000 −0.575435
\(303\) 17.0000 0.976624
\(304\) −16.0000 −0.917663
\(305\) −30.0000 −1.71780
\(306\) 6.00000 0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 12.0000 0.681554
\(311\) −25.0000 −1.41762 −0.708810 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 16.0000 0.902932
\(315\) −3.00000 −0.169031
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −16.0000 −0.897235
\(319\) 0 0
\(320\) 24.0000 1.34164
\(321\) −10.0000 −0.558146
\(322\) −4.00000 −0.222911
\(323\) −12.0000 −0.667698
\(324\) 2.00000 0.111111
\(325\) −8.00000 −0.443760
\(326\) −8.00000 −0.443079
\(327\) 13.0000 0.718902
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) −2.00000 −0.109764
\(333\) 10.0000 0.547997
\(334\) −6.00000 −0.328305
\(335\) −39.0000 −2.13080
\(336\) 4.00000 0.218218
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 18.0000 0.979071
\(339\) −6.00000 −0.325875
\(340\) 18.0000 0.976187
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 50.0000 2.68802
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 16.0000 0.857690
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −8.00000 −0.427618
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −6.00000 −0.318896
\(355\) −42.0000 −2.22913
\(356\) −18.0000 −0.953998
\(357\) 3.00000 0.158777
\(358\) −40.0000 −2.11407
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 32.0000 1.68188
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −30.0000 −1.57027
\(366\) 20.0000 1.04542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −8.00000 −0.417029
\(369\) −2.00000 −0.104116
\(370\) 60.0000 3.11925
\(371\) −8.00000 −0.415339
\(372\) −4.00000 −0.207390
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 2.00000 0.102869
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) −24.0000 −1.23117
\(381\) 13.0000 0.666010
\(382\) 40.0000 2.04658
\(383\) 19.0000 0.970855 0.485427 0.874277i \(-0.338664\pi\)
0.485427 + 0.874277i \(0.338664\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −46.0000 −2.34134
\(387\) −9.00000 −0.457496
\(388\) −32.0000 −1.62455
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 12.0000 0.607644
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 13.0000 0.655763
\(394\) 44.0000 2.21669
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 28.0000 1.40351
\(399\) −4.00000 −0.200250
\(400\) −16.0000 −0.800000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 26.0000 1.29676
\(403\) −4.00000 −0.199254
\(404\) −34.0000 −1.69156
\(405\) −3.00000 −0.149071
\(406\) 16.0000 0.794067
\(407\) 0 0
\(408\) 0 0
\(409\) −28.0000 −1.38451 −0.692255 0.721653i \(-0.743383\pi\)
−0.692255 + 0.721653i \(0.743383\pi\)
\(410\) −12.0000 −0.592638
\(411\) −8.00000 −0.394611
\(412\) 28.0000 1.37946
\(413\) −3.00000 −0.147620
\(414\) −4.00000 −0.196589
\(415\) 3.00000 0.147264
\(416\) −16.0000 −0.784465
\(417\) −2.00000 −0.0979404
\(418\) 0 0
\(419\) 19.0000 0.928211 0.464105 0.885780i \(-0.346376\pi\)
0.464105 + 0.885780i \(0.346376\pi\)
\(420\) 6.00000 0.292770
\(421\) 33.0000 1.60832 0.804161 0.594412i \(-0.202615\pi\)
0.804161 + 0.594412i \(0.202615\pi\)
\(422\) 14.0000 0.681509
\(423\) 9.00000 0.437595
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 28.0000 1.35660
\(427\) 10.0000 0.483934
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) −54.0000 −2.60411
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 4.00000 0.192450
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −4.00000 −0.192006
\(435\) −24.0000 −1.15071
\(436\) −26.0000 −1.24517
\(437\) 8.00000 0.382692
\(438\) 20.0000 0.955637
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −38.0000 −1.80543 −0.902717 0.430234i \(-0.858431\pi\)
−0.902717 + 0.430234i \(0.858431\pi\)
\(444\) −20.0000 −0.949158
\(445\) 27.0000 1.27992
\(446\) 48.0000 2.27287
\(447\) −12.0000 −0.567581
\(448\) −8.00000 −0.377964
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −8.00000 −0.377124
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) −5.00000 −0.234920
\(454\) 14.0000 0.657053
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −3.00000 −0.140334 −0.0701670 0.997535i \(-0.522353\pi\)
−0.0701670 + 0.997535i \(0.522353\pi\)
\(458\) 8.00000 0.373815
\(459\) 3.00000 0.140028
\(460\) −12.0000 −0.559503
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 32.0000 1.48556
\(465\) 6.00000 0.278243
\(466\) −8.00000 −0.370593
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) −4.00000 −0.184900
\(469\) 13.0000 0.600284
\(470\) 54.0000 2.49083
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 16.0000 0.734130
\(476\) −6.00000 −0.275010
\(477\) −8.00000 −0.366295
\(478\) −12.0000 −0.548867
\(479\) −29.0000 −1.32504 −0.662522 0.749043i \(-0.730514\pi\)
−0.662522 + 0.749043i \(0.730514\pi\)
\(480\) 24.0000 1.09545
\(481\) −20.0000 −0.911922
\(482\) 56.0000 2.55073
\(483\) −2.00000 −0.0910032
\(484\) 0 0
\(485\) 48.0000 2.17957
\(486\) 2.00000 0.0907218
\(487\) 37.0000 1.67663 0.838315 0.545186i \(-0.183541\pi\)
0.838315 + 0.545186i \(0.183541\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 6.00000 0.271052
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 4.00000 0.180334
\(493\) 24.0000 1.08091
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 14.0000 0.627986
\(498\) −2.00000 −0.0896221
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 6.00000 0.268328
\(501\) −3.00000 −0.134030
\(502\) −24.0000 −1.07117
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) 0 0
\(505\) 51.0000 2.26947
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −26.0000 −1.15356
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 18.0000 0.797053
\(511\) 10.0000 0.442374
\(512\) −32.0000 −1.41421
\(513\) −4.00000 −0.176604
\(514\) −14.0000 −0.617514
\(515\) −42.0000 −1.85074
\(516\) 18.0000 0.792406
\(517\) 0 0
\(518\) −20.0000 −0.878750
\(519\) 25.0000 1.09738
\(520\) 0 0
\(521\) 1.00000 0.0438108 0.0219054 0.999760i \(-0.493027\pi\)
0.0219054 + 0.999760i \(0.493027\pi\)
\(522\) 16.0000 0.700301
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) −26.0000 −1.13582
\(525\) −4.00000 −0.174574
\(526\) −36.0000 −1.56967
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −48.0000 −2.08499
\(531\) −3.00000 −0.130189
\(532\) 8.00000 0.346844
\(533\) 4.00000 0.173259
\(534\) −18.0000 −0.778936
\(535\) −30.0000 −1.29701
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 28.0000 1.20717
\(539\) 0 0
\(540\) 6.00000 0.258199
\(541\) 33.0000 1.41878 0.709390 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) 40.0000 1.71815
\(543\) 16.0000 0.686626
\(544\) −24.0000 −1.02899
\(545\) 39.0000 1.67058
\(546\) −4.00000 −0.171184
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 16.0000 0.683486
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −34.0000 −1.44452
\(555\) 30.0000 1.27343
\(556\) 4.00000 0.169638
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −4.00000 −0.169334
\(559\) 18.0000 0.761319
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) −18.0000 −0.757937
\(565\) −18.0000 −0.757266
\(566\) 8.00000 0.336265
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) −24.0000 −1.00525
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 20.0000 0.835512
\(574\) 4.00000 0.166957
\(575\) 8.00000 0.333623
\(576\) −8.00000 −0.333333
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 16.0000 0.665512
\(579\) −23.0000 −0.955847
\(580\) 48.0000 1.99309
\(581\) −1.00000 −0.0414870
\(582\) −32.0000 −1.32644
\(583\) 0 0
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 18.0000 0.743573
\(587\) 39.0000 1.60970 0.804851 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 8.00000 0.329634
\(590\) −18.0000 −0.741048
\(591\) 22.0000 0.904959
\(592\) −40.0000 −1.64399
\(593\) −11.0000 −0.451716 −0.225858 0.974160i \(-0.572519\pi\)
−0.225858 + 0.974160i \(0.572519\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) 24.0000 0.983078
\(597\) 14.0000 0.572982
\(598\) 8.00000 0.327144
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 18.0000 0.733625
\(603\) 13.0000 0.529401
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −34.0000 −1.38116
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 32.0000 1.29777
\(609\) 8.00000 0.324176
\(610\) 60.0000 2.42933
\(611\) −18.0000 −0.728202
\(612\) −6.00000 −0.242536
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) 24.0000 0.968561
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 28.0000 1.12633
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) −12.0000 −0.481932
\(621\) −2.00000 −0.0802572
\(622\) 50.0000 2.00482
\(623\) −9.00000 −0.360577
\(624\) −8.00000 −0.320256
\(625\) −29.0000 −1.16000
\(626\) 48.0000 1.91847
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −30.0000 −1.19618
\(630\) 6.00000 0.239046
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 7.00000 0.278225
\(634\) 12.0000 0.476581
\(635\) 39.0000 1.54767
\(636\) 16.0000 0.634441
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 14.0000 0.553831
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 20.0000 0.789337
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 4.00000 0.157622
\(645\) −27.0000 −1.06312
\(646\) 24.0000 0.944267
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.0000 0.627572
\(651\) −2.00000 −0.0783862
\(652\) 8.00000 0.313304
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −26.0000 −1.01668
\(655\) 39.0000 1.52386
\(656\) 8.00000 0.312348
\(657\) 10.0000 0.390137
\(658\) −18.0000 −0.701713
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) −6.00000 −0.233197
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) −20.0000 −0.774984
\(667\) −16.0000 −0.619522
\(668\) 6.00000 0.232147
\(669\) 24.0000 0.927894
\(670\) 78.0000 3.01340
\(671\) 0 0
\(672\) −8.00000 −0.308607
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 4.00000 0.154074
\(675\) −4.00000 −0.153960
\(676\) −18.0000 −0.692308
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 12.0000 0.460857
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) 0 0
\(683\) −34.0000 −1.30097 −0.650487 0.759517i \(-0.725435\pi\)
−0.650487 + 0.759517i \(0.725435\pi\)
\(684\) 8.00000 0.305888
\(685\) −24.0000 −0.916993
\(686\) −2.00000 −0.0763604
\(687\) 4.00000 0.152610
\(688\) 36.0000 1.37249
\(689\) 16.0000 0.609551
\(690\) −12.0000 −0.456832
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −50.0000 −1.90071
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 12.0000 0.454207
\(699\) −4.00000 −0.151294
\(700\) 8.00000 0.302372
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) −4.00000 −0.150970
\(703\) 40.0000 1.50863
\(704\) 0 0
\(705\) 27.0000 1.01688
\(706\) −60.0000 −2.25813
\(707\) −17.0000 −0.639351
\(708\) 6.00000 0.225494
\(709\) −9.00000 −0.338002 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(710\) 84.0000 3.15246
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 40.0000 1.49487
\(717\) −6.00000 −0.224074
\(718\) −36.0000 −1.34351
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 12.0000 0.447214
\(721\) 14.0000 0.521387
\(722\) 6.00000 0.223297
\(723\) 28.0000 1.04133
\(724\) −32.0000 −1.18927
\(725\) −32.0000 −1.18845
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 60.0000 2.22070
\(731\) 27.0000 0.998631
\(732\) −20.0000 −0.739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 16.0000 0.590571
\(735\) 3.00000 0.110657
\(736\) 16.0000 0.589768
\(737\) 0 0
\(738\) 4.00000 0.147242
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) −60.0000 −2.20564
\(741\) 8.00000 0.293887
\(742\) 16.0000 0.587378
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 42.0000 1.53773
\(747\) −1.00000 −0.0365881
\(748\) 0 0
\(749\) 10.0000 0.365392
\(750\) 6.00000 0.219089
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) −36.0000 −1.31278
\(753\) −12.0000 −0.437304
\(754\) −32.0000 −1.16537
\(755\) −15.0000 −0.545906
\(756\) −2.00000 −0.0727393
\(757\) −9.00000 −0.327111 −0.163555 0.986534i \(-0.552296\pi\)
−0.163555 + 0.986534i \(0.552296\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) −26.0000 −0.941881
\(763\) −13.0000 −0.470632
\(764\) −40.0000 −1.44715
\(765\) 9.00000 0.325396
\(766\) −38.0000 −1.37300
\(767\) 6.00000 0.216647
\(768\) −16.0000 −0.577350
\(769\) −24.0000 −0.865462 −0.432731 0.901523i \(-0.642450\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) −7.00000 −0.252099
\(772\) 46.0000 1.65558
\(773\) 15.0000 0.539513 0.269756 0.962929i \(-0.413057\pi\)
0.269756 + 0.962929i \(0.413057\pi\)
\(774\) 18.0000 0.646997
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) −8.00000 −0.286814
\(779\) −8.00000 −0.286630
\(780\) −12.0000 −0.429669
\(781\) 0 0
\(782\) 12.0000 0.429119
\(783\) 8.00000 0.285897
\(784\) −4.00000 −0.142857
\(785\) 24.0000 0.856597
\(786\) −26.0000 −0.927389
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) −44.0000 −1.56744
\(789\) −18.0000 −0.640817
\(790\) 24.0000 0.853882
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 36.0000 1.27759
\(795\) −24.0000 −0.851192
\(796\) −28.0000 −0.992434
\(797\) 11.0000 0.389640 0.194820 0.980839i \(-0.437588\pi\)
0.194820 + 0.980839i \(0.437588\pi\)
\(798\) 8.00000 0.283197
\(799\) −27.0000 −0.955191
\(800\) 32.0000 1.13137
\(801\) −9.00000 −0.317999
\(802\) 0 0
\(803\) 0 0
\(804\) −26.0000 −0.916949
\(805\) −6.00000 −0.211472
\(806\) 8.00000 0.281788
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 6.00000 0.210819
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) −16.0000 −0.561490
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −12.0000 −0.420342
\(816\) −12.0000 −0.420084
\(817\) −36.0000 −1.25948
\(818\) 56.0000 1.95799
\(819\) −2.00000 −0.0698857
\(820\) 12.0000 0.419058
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 16.0000 0.558064
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −38.0000 −1.32139 −0.660695 0.750655i \(-0.729738\pi\)
−0.660695 + 0.750655i \(0.729738\pi\)
\(828\) 4.00000 0.139010
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −6.00000 −0.208263
\(831\) −17.0000 −0.589723
\(832\) 16.0000 0.554700
\(833\) −3.00000 −0.103944
\(834\) 4.00000 0.138509
\(835\) −9.00000 −0.311458
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) −38.0000 −1.31269
\(839\) 51.0000 1.76072 0.880358 0.474310i \(-0.157302\pi\)
0.880358 + 0.474310i \(0.157302\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −66.0000 −2.27451
\(843\) 12.0000 0.413302
\(844\) −14.0000 −0.481900
\(845\) 27.0000 0.928828
\(846\) −18.0000 −0.618853
\(847\) 0 0
\(848\) 32.0000 1.09888
\(849\) 4.00000 0.137280
\(850\) 24.0000 0.823193
\(851\) 20.0000 0.685591
\(852\) −28.0000 −0.959264
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) −20.0000 −0.684386
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 54.0000 1.84138
\(861\) 2.00000 0.0681598
\(862\) −8.00000 −0.272481
\(863\) −26.0000 −0.885050 −0.442525 0.896756i \(-0.645917\pi\)
−0.442525 + 0.896756i \(0.645917\pi\)
\(864\) −8.00000 −0.272166
\(865\) 75.0000 2.55008
\(866\) −8.00000 −0.271851
\(867\) 8.00000 0.271694
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 48.0000 1.62735
\(871\) −26.0000 −0.880976
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) −16.0000 −0.541208
\(875\) 3.00000 0.101419
\(876\) −20.0000 −0.675737
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 52.0000 1.75491
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) 12.0000 0.403604
\(885\) −9.00000 −0.302532
\(886\) 76.0000 2.55327
\(887\) 55.0000 1.84672 0.923360 0.383936i \(-0.125432\pi\)
0.923360 + 0.383936i \(0.125432\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) −54.0000 −1.81008
\(891\) 0 0
\(892\) −48.0000 −1.60716
\(893\) 36.0000 1.20469
\(894\) 24.0000 0.802680
\(895\) −60.0000 −2.00558
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −12.0000 −0.400445
\(899\) −16.0000 −0.533630
\(900\) 8.00000 0.266667
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 9.00000 0.299501
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 10.0000 0.332228
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −14.0000 −0.464606
\(909\) −17.0000 −0.563854
\(910\) −12.0000 −0.397796
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 16.0000 0.529813
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) 30.0000 0.991769
\(916\) −8.00000 −0.264327
\(917\) −13.0000 −0.429298
\(918\) −6.00000 −0.198030
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) −30.0000 −0.987997
\(923\) −28.0000 −0.921631
\(924\) 0 0
\(925\) 40.0000 1.31519
\(926\) −32.0000 −1.05159
\(927\) 14.0000 0.459820
\(928\) −64.0000 −2.10090
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) −12.0000 −0.393496
\(931\) 4.00000 0.131095
\(932\) 8.00000 0.262049
\(933\) 25.0000 0.818463
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 16.0000 0.522697 0.261349 0.965244i \(-0.415833\pi\)
0.261349 + 0.965244i \(0.415833\pi\)
\(938\) −26.0000 −0.848930
\(939\) 24.0000 0.783210
\(940\) −54.0000 −1.76129
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −16.0000 −0.521308
\(943\) −4.00000 −0.130258
\(944\) 12.0000 0.390567
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) −8.00000 −0.259828
\(949\) −20.0000 −0.649227
\(950\) −32.0000 −1.03822
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 16.0000 0.518019
\(955\) 60.0000 1.94155
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 58.0000 1.87389
\(959\) 8.00000 0.258333
\(960\) −24.0000 −0.774597
\(961\) −27.0000 −0.870968
\(962\) 40.0000 1.28965
\(963\) 10.0000 0.322245
\(964\) −56.0000 −1.80364
\(965\) −69.0000 −2.22119
\(966\) 4.00000 0.128698
\(967\) −1.00000 −0.0321578 −0.0160789 0.999871i \(-0.505118\pi\)
−0.0160789 + 0.999871i \(0.505118\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) −96.0000 −3.08237
\(971\) 37.0000 1.18739 0.593693 0.804691i \(-0.297669\pi\)
0.593693 + 0.804691i \(0.297669\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 2.00000 0.0641171
\(974\) −74.0000 −2.37111
\(975\) 8.00000 0.256205
\(976\) −40.0000 −1.28037
\(977\) −14.0000 −0.447900 −0.223950 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 8.00000 0.255812
\(979\) 0 0
\(980\) −6.00000 −0.191663
\(981\) −13.0000 −0.415058
\(982\) 84.0000 2.68055
\(983\) −8.00000 −0.255160 −0.127580 0.991828i \(-0.540721\pi\)
−0.127580 + 0.991828i \(0.540721\pi\)
\(984\) 0 0
\(985\) 66.0000 2.10293
\(986\) −48.0000 −1.52863
\(987\) −9.00000 −0.286473
\(988\) −16.0000 −0.509028
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 16.0000 0.508001
\(993\) −3.00000 −0.0952021
\(994\) −28.0000 −0.888106
\(995\) 42.0000 1.33149
\(996\) 2.00000 0.0633724
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 10.0000 0.316544
\(999\) −10.0000 −0.316386
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 2541.2.a.a.1.1 1
3.2 odd 2 7623.2.a.s.1.1 1
11.10 odd 2 2541.2.a.k.1.1 yes 1
33.32 even 2 7623.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.a.1.1 1 1.1 even 1 trivial
2541.2.a.k.1.1 yes 1 11.10 odd 2
7623.2.a.d.1.1 1 33.32 even 2
7623.2.a.s.1.1 1 3.2 odd 2