Properties

Label 2240.2.e.g.2239.9
Level $2240$
Weight $2$
Character 2240.2239
Analytic conductor $17.886$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(2239,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 1120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2239.9
Character \(\chi\) \(=\) 2240.2239
Dual form 2240.2.e.g.2239.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.86733i q^{3} +(-2.07366 + 0.836614i) q^{5} +(-2.64040 + 0.168201i) q^{7} -0.486908 q^{9} +O(q^{10})\) \(q-1.86733i q^{3} +(-2.07366 + 0.836614i) q^{5} +(-2.64040 + 0.168201i) q^{7} -0.486908 q^{9} -4.26303i q^{11} +5.57238 q^{13} +(1.56223 + 3.87221i) q^{15} -3.01761 q^{17} -0.0464832 q^{19} +(0.314085 + 4.93049i) q^{21} -5.58465 q^{23} +(3.60015 - 3.46971i) q^{25} -4.69276i q^{27} -1.21067 q^{29} -4.62124 q^{31} -7.96047 q^{33} +(5.33458 - 2.55779i) q^{35} -9.87892i q^{37} -10.4055i q^{39} +10.5213i q^{41} -12.4259 q^{43} +(1.00968 - 0.407354i) q^{45} +6.78840i q^{47} +(6.94342 - 0.888233i) q^{49} +5.63486i q^{51} +4.99936i q^{53} +(3.56651 + 8.84009i) q^{55} +0.0867992i q^{57} +7.06269 q^{59} +11.5761i q^{61} +(1.28563 - 0.0818982i) q^{63} +(-11.5552 + 4.66193i) q^{65} +6.65798 q^{67} +10.4284i q^{69} +5.11624i q^{71} -9.67546 q^{73} +(-6.47908 - 6.72266i) q^{75} +(0.717044 + 11.2561i) q^{77} -5.90273i q^{79} -10.2236 q^{81} +11.1201i q^{83} +(6.25750 - 2.52457i) q^{85} +2.26072i q^{87} +1.51888i q^{89} +(-14.7133 + 0.937278i) q^{91} +8.62936i q^{93} +(0.0963904 - 0.0388885i) q^{95} -3.01761 q^{97} +2.07570i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.86733i 1.07810i −0.842273 0.539051i \(-0.818783\pi\)
0.842273 0.539051i \(-0.181217\pi\)
\(4\) 0 0
\(5\) −2.07366 + 0.836614i −0.927370 + 0.374145i
\(6\) 0 0
\(7\) −2.64040 + 0.168201i −0.997977 + 0.0635738i
\(8\) 0 0
\(9\) −0.486908 −0.162303
\(10\) 0 0
\(11\) 4.26303i 1.28535i −0.766138 0.642676i \(-0.777824\pi\)
0.766138 0.642676i \(-0.222176\pi\)
\(12\) 0 0
\(13\) 5.57238 1.54550 0.772750 0.634710i \(-0.218881\pi\)
0.772750 + 0.634710i \(0.218881\pi\)
\(14\) 0 0
\(15\) 1.56223 + 3.87221i 0.403366 + 0.999799i
\(16\) 0 0
\(17\) −3.01761 −0.731878 −0.365939 0.930639i \(-0.619252\pi\)
−0.365939 + 0.930639i \(0.619252\pi\)
\(18\) 0 0
\(19\) −0.0464832 −0.0106640 −0.00533198 0.999986i \(-0.501697\pi\)
−0.00533198 + 0.999986i \(0.501697\pi\)
\(20\) 0 0
\(21\) 0.314085 + 4.93049i 0.0685390 + 1.07592i
\(22\) 0 0
\(23\) −5.58465 −1.16448 −0.582240 0.813017i \(-0.697824\pi\)
−0.582240 + 0.813017i \(0.697824\pi\)
\(24\) 0 0
\(25\) 3.60015 3.46971i 0.720031 0.693942i
\(26\) 0 0
\(27\) 4.69276i 0.903123i
\(28\) 0 0
\(29\) −1.21067 −0.224817 −0.112408 0.993662i \(-0.535856\pi\)
−0.112408 + 0.993662i \(0.535856\pi\)
\(30\) 0 0
\(31\) −4.62124 −0.829999 −0.414999 0.909822i \(-0.636218\pi\)
−0.414999 + 0.909822i \(0.636218\pi\)
\(32\) 0 0
\(33\) −7.96047 −1.38574
\(34\) 0 0
\(35\) 5.33458 2.55779i 0.901708 0.432345i
\(36\) 0 0
\(37\) 9.87892i 1.62408i −0.583599 0.812042i \(-0.698356\pi\)
0.583599 0.812042i \(-0.301644\pi\)
\(38\) 0 0
\(39\) 10.4055i 1.66621i
\(40\) 0 0
\(41\) 10.5213i 1.64316i 0.570096 + 0.821578i \(0.306906\pi\)
−0.570096 + 0.821578i \(0.693094\pi\)
\(42\) 0 0
\(43\) −12.4259 −1.89493 −0.947467 0.319854i \(-0.896366\pi\)
−0.947467 + 0.319854i \(0.896366\pi\)
\(44\) 0 0
\(45\) 1.00968 0.407354i 0.150515 0.0607247i
\(46\) 0 0
\(47\) 6.78840i 0.990190i 0.868839 + 0.495095i \(0.164867\pi\)
−0.868839 + 0.495095i \(0.835133\pi\)
\(48\) 0 0
\(49\) 6.94342 0.888233i 0.991917 0.126890i
\(50\) 0 0
\(51\) 5.63486i 0.789038i
\(52\) 0 0
\(53\) 4.99936i 0.686715i 0.939205 + 0.343358i \(0.111564\pi\)
−0.939205 + 0.343358i \(0.888436\pi\)
\(54\) 0 0
\(55\) 3.56651 + 8.84009i 0.480908 + 1.19200i
\(56\) 0 0
\(57\) 0.0867992i 0.0114968i
\(58\) 0 0
\(59\) 7.06269 0.919484 0.459742 0.888053i \(-0.347942\pi\)
0.459742 + 0.888053i \(0.347942\pi\)
\(60\) 0 0
\(61\) 11.5761i 1.48216i 0.671415 + 0.741081i \(0.265687\pi\)
−0.671415 + 0.741081i \(0.734313\pi\)
\(62\) 0 0
\(63\) 1.28563 0.0818982i 0.161974 0.0103182i
\(64\) 0 0
\(65\) −11.5552 + 4.66193i −1.43325 + 0.578241i
\(66\) 0 0
\(67\) 6.65798 0.813402 0.406701 0.913561i \(-0.366679\pi\)
0.406701 + 0.913561i \(0.366679\pi\)
\(68\) 0 0
\(69\) 10.4284i 1.25543i
\(70\) 0 0
\(71\) 5.11624i 0.607186i 0.952802 + 0.303593i \(0.0981863\pi\)
−0.952802 + 0.303593i \(0.901814\pi\)
\(72\) 0 0
\(73\) −9.67546 −1.13243 −0.566214 0.824258i \(-0.691592\pi\)
−0.566214 + 0.824258i \(0.691592\pi\)
\(74\) 0 0
\(75\) −6.47908 6.72266i −0.748140 0.776266i
\(76\) 0 0
\(77\) 0.717044 + 11.2561i 0.0817148 + 1.28275i
\(78\) 0 0
\(79\) 5.90273i 0.664109i −0.943260 0.332054i \(-0.892258\pi\)
0.943260 0.332054i \(-0.107742\pi\)
\(80\) 0 0
\(81\) −10.2236 −1.13596
\(82\) 0 0
\(83\) 11.1201i 1.22059i 0.792175 + 0.610294i \(0.208949\pi\)
−0.792175 + 0.610294i \(0.791051\pi\)
\(84\) 0 0
\(85\) 6.25750 2.52457i 0.678721 0.273828i
\(86\) 0 0
\(87\) 2.26072i 0.242375i
\(88\) 0 0
\(89\) 1.51888i 0.161001i 0.996755 + 0.0805005i \(0.0256519\pi\)
−0.996755 + 0.0805005i \(0.974348\pi\)
\(90\) 0 0
\(91\) −14.7133 + 0.937278i −1.54237 + 0.0982534i
\(92\) 0 0
\(93\) 8.62936i 0.894823i
\(94\) 0 0
\(95\) 0.0963904 0.0388885i 0.00988945 0.00398987i
\(96\) 0 0
\(97\) −3.01761 −0.306392 −0.153196 0.988196i \(-0.548957\pi\)
−0.153196 + 0.988196i \(0.548957\pi\)
\(98\) 0 0
\(99\) 2.07570i 0.208616i
\(100\) 0 0
\(101\) 0.464155i 0.0461852i 0.999733 + 0.0230926i \(0.00735125\pi\)
−0.999733 + 0.0230926i \(0.992649\pi\)
\(102\) 0 0
\(103\) 1.67886i 0.165423i 0.996574 + 0.0827113i \(0.0263579\pi\)
−0.996574 + 0.0827113i \(0.973642\pi\)
\(104\) 0 0
\(105\) −4.77622 9.96140i −0.466111 0.972133i
\(106\) 0 0
\(107\) −6.65798 −0.643651 −0.321826 0.946799i \(-0.604297\pi\)
−0.321826 + 0.946799i \(0.604297\pi\)
\(108\) 0 0
\(109\) −13.0975 −1.25451 −0.627257 0.778812i \(-0.715822\pi\)
−0.627257 + 0.778812i \(0.715822\pi\)
\(110\) 0 0
\(111\) −18.4472 −1.75093
\(112\) 0 0
\(113\) 16.1608i 1.52028i 0.649757 + 0.760142i \(0.274871\pi\)
−0.649757 + 0.760142i \(0.725129\pi\)
\(114\) 0 0
\(115\) 11.5807 4.67220i 1.07990 0.435685i
\(116\) 0 0
\(117\) −2.71324 −0.250839
\(118\) 0 0
\(119\) 7.96769 0.507563i 0.730397 0.0465283i
\(120\) 0 0
\(121\) −7.17343 −0.652130
\(122\) 0 0
\(123\) 19.6468 1.77149
\(124\) 0 0
\(125\) −4.56270 + 10.2069i −0.408100 + 0.912937i
\(126\) 0 0
\(127\) 8.40526 0.745846 0.372923 0.927862i \(-0.378356\pi\)
0.372923 + 0.927862i \(0.378356\pi\)
\(128\) 0 0
\(129\) 23.2032i 2.04293i
\(130\) 0 0
\(131\) −1.12073 −0.0979187 −0.0489594 0.998801i \(-0.515590\pi\)
−0.0489594 + 0.998801i \(0.515590\pi\)
\(132\) 0 0
\(133\) 0.122734 0.00781849i 0.0106424 0.000677949i
\(134\) 0 0
\(135\) 3.92603 + 9.73121i 0.337899 + 0.837529i
\(136\) 0 0
\(137\) 4.27185i 0.364969i −0.983209 0.182484i \(-0.941586\pi\)
0.983209 0.182484i \(-0.0584139\pi\)
\(138\) 0 0
\(139\) 0.369778 0.0313641 0.0156821 0.999877i \(-0.495008\pi\)
0.0156821 + 0.999877i \(0.495008\pi\)
\(140\) 0 0
\(141\) 12.6762 1.06753
\(142\) 0 0
\(143\) 23.7552i 1.98651i
\(144\) 0 0
\(145\) 2.51053 1.01287i 0.208488 0.0841140i
\(146\) 0 0
\(147\) −1.65862 12.9656i −0.136801 1.06939i
\(148\) 0 0
\(149\) −16.5383 −1.35487 −0.677437 0.735581i \(-0.736909\pi\)
−0.677437 + 0.735581i \(0.736909\pi\)
\(150\) 0 0
\(151\) 1.84168i 0.149874i −0.997188 0.0749370i \(-0.976124\pi\)
0.997188 0.0749370i \(-0.0238756\pi\)
\(152\) 0 0
\(153\) 1.46930 0.118786
\(154\) 0 0
\(155\) 9.58289 3.86619i 0.769716 0.310540i
\(156\) 0 0
\(157\) −6.88403 −0.549405 −0.274703 0.961529i \(-0.588579\pi\)
−0.274703 + 0.961529i \(0.588579\pi\)
\(158\) 0 0
\(159\) 9.33544 0.740348
\(160\) 0 0
\(161\) 14.7457 0.939342i 1.16212 0.0740305i
\(162\) 0 0
\(163\) 3.94845 0.309266 0.154633 0.987972i \(-0.450580\pi\)
0.154633 + 0.987972i \(0.450580\pi\)
\(164\) 0 0
\(165\) 16.5073 6.65984i 1.28509 0.518468i
\(166\) 0 0
\(167\) 7.87928i 0.609717i −0.952398 0.304858i \(-0.901391\pi\)
0.952398 0.304858i \(-0.0986091\pi\)
\(168\) 0 0
\(169\) 18.0514 1.38857
\(170\) 0 0
\(171\) 0.0226330 0.00173079
\(172\) 0 0
\(173\) −12.9532 −0.984814 −0.492407 0.870365i \(-0.663883\pi\)
−0.492407 + 0.870365i \(0.663883\pi\)
\(174\) 0 0
\(175\) −8.92224 + 9.76697i −0.674458 + 0.738313i
\(176\) 0 0
\(177\) 13.1883i 0.991297i
\(178\) 0 0
\(179\) 6.98609i 0.522165i 0.965316 + 0.261083i \(0.0840795\pi\)
−0.965316 + 0.261083i \(0.915920\pi\)
\(180\) 0 0
\(181\) 21.1459i 1.57176i −0.618377 0.785882i \(-0.712210\pi\)
0.618377 0.785882i \(-0.287790\pi\)
\(182\) 0 0
\(183\) 21.6163 1.59792
\(184\) 0 0
\(185\) 8.26484 + 20.4855i 0.607643 + 1.50613i
\(186\) 0 0
\(187\) 12.8642i 0.940720i
\(188\) 0 0
\(189\) 0.789325 + 12.3908i 0.0574150 + 0.901296i
\(190\) 0 0
\(191\) 15.7773i 1.14160i 0.821088 + 0.570802i \(0.193368\pi\)
−0.821088 + 0.570802i \(0.806632\pi\)
\(192\) 0 0
\(193\) 4.96235i 0.357198i 0.983922 + 0.178599i \(0.0571564\pi\)
−0.983922 + 0.178599i \(0.942844\pi\)
\(194\) 0 0
\(195\) 8.70535 + 21.5774i 0.623403 + 1.54519i
\(196\) 0 0
\(197\) 17.5088i 1.24745i −0.781643 0.623726i \(-0.785618\pi\)
0.781643 0.623726i \(-0.214382\pi\)
\(198\) 0 0
\(199\) −7.43247 −0.526874 −0.263437 0.964677i \(-0.584856\pi\)
−0.263437 + 0.964677i \(0.584856\pi\)
\(200\) 0 0
\(201\) 12.4326i 0.876930i
\(202\) 0 0
\(203\) 3.19666 0.203636i 0.224362 0.0142925i
\(204\) 0 0
\(205\) −8.80230 21.8177i −0.614779 1.52381i
\(206\) 0 0
\(207\) 2.71921 0.188998
\(208\) 0 0
\(209\) 0.198159i 0.0137070i
\(210\) 0 0
\(211\) 21.8042i 1.50106i −0.660834 0.750532i \(-0.729797\pi\)
0.660834 0.750532i \(-0.270203\pi\)
\(212\) 0 0
\(213\) 9.55369 0.654608
\(214\) 0 0
\(215\) 25.7671 10.3957i 1.75730 0.708980i
\(216\) 0 0
\(217\) 12.2019 0.777295i 0.828320 0.0527662i
\(218\) 0 0
\(219\) 18.0673i 1.22087i
\(220\) 0 0
\(221\) −16.8153 −1.13112
\(222\) 0 0
\(223\) 18.6451i 1.24857i 0.781197 + 0.624284i \(0.214609\pi\)
−0.781197 + 0.624284i \(0.785391\pi\)
\(224\) 0 0
\(225\) −1.75294 + 1.68943i −0.116863 + 0.112629i
\(226\) 0 0
\(227\) 12.4234i 0.824571i 0.911055 + 0.412285i \(0.135269\pi\)
−0.911055 + 0.412285i \(0.864731\pi\)
\(228\) 0 0
\(229\) 9.06384i 0.598955i 0.954103 + 0.299478i \(0.0968124\pi\)
−0.954103 + 0.299478i \(0.903188\pi\)
\(230\) 0 0
\(231\) 21.0188 1.33896i 1.38294 0.0880968i
\(232\) 0 0
\(233\) 1.36535i 0.0894473i 0.998999 + 0.0447236i \(0.0142407\pi\)
−0.998999 + 0.0447236i \(0.985759\pi\)
\(234\) 0 0
\(235\) −5.67927 14.0769i −0.370475 0.918273i
\(236\) 0 0
\(237\) −11.0223 −0.715977
\(238\) 0 0
\(239\) 20.1208i 1.30151i −0.759288 0.650754i \(-0.774453\pi\)
0.759288 0.650754i \(-0.225547\pi\)
\(240\) 0 0
\(241\) 3.60404i 0.232157i 0.993240 + 0.116078i \(0.0370324\pi\)
−0.993240 + 0.116078i \(0.962968\pi\)
\(242\) 0 0
\(243\) 5.01259i 0.321558i
\(244\) 0 0
\(245\) −13.6552 + 7.65086i −0.872399 + 0.488795i
\(246\) 0 0
\(247\) −0.259022 −0.0164812
\(248\) 0 0
\(249\) 20.7648 1.31592
\(250\) 0 0
\(251\) −21.8309 −1.37795 −0.688976 0.724784i \(-0.741939\pi\)
−0.688976 + 0.724784i \(0.741939\pi\)
\(252\) 0 0
\(253\) 23.8075i 1.49677i
\(254\) 0 0
\(255\) −4.71420 11.6848i −0.295215 0.731731i
\(256\) 0 0
\(257\) 21.0520 1.31319 0.656595 0.754244i \(-0.271996\pi\)
0.656595 + 0.754244i \(0.271996\pi\)
\(258\) 0 0
\(259\) 1.66164 + 26.0843i 0.103249 + 1.62080i
\(260\) 0 0
\(261\) 0.589487 0.0364883
\(262\) 0 0
\(263\) −2.71567 −0.167455 −0.0837276 0.996489i \(-0.526683\pi\)
−0.0837276 + 0.996489i \(0.526683\pi\)
\(264\) 0 0
\(265\) −4.18253 10.3670i −0.256931 0.636839i
\(266\) 0 0
\(267\) 2.83625 0.173575
\(268\) 0 0
\(269\) 25.9926i 1.58479i −0.610005 0.792397i \(-0.708833\pi\)
0.610005 0.792397i \(-0.291167\pi\)
\(270\) 0 0
\(271\) −25.4482 −1.54587 −0.772933 0.634487i \(-0.781211\pi\)
−0.772933 + 0.634487i \(0.781211\pi\)
\(272\) 0 0
\(273\) 1.75020 + 27.4746i 0.105927 + 1.66284i
\(274\) 0 0
\(275\) −14.7915 15.3476i −0.891960 0.925493i
\(276\) 0 0
\(277\) 8.50956i 0.511290i 0.966771 + 0.255645i \(0.0822878\pi\)
−0.966771 + 0.255645i \(0.917712\pi\)
\(278\) 0 0
\(279\) 2.25012 0.134711
\(280\) 0 0
\(281\) −10.1347 −0.604588 −0.302294 0.953215i \(-0.597753\pi\)
−0.302294 + 0.953215i \(0.597753\pi\)
\(282\) 0 0
\(283\) 13.5926i 0.807993i −0.914760 0.403997i \(-0.867621\pi\)
0.914760 0.403997i \(-0.132379\pi\)
\(284\) 0 0
\(285\) −0.0726174 0.179992i −0.00430149 0.0106618i
\(286\) 0 0
\(287\) −1.76969 27.7805i −0.104462 1.63983i
\(288\) 0 0
\(289\) −7.89404 −0.464355
\(290\) 0 0
\(291\) 5.63486i 0.330321i
\(292\) 0 0
\(293\) −18.0627 −1.05524 −0.527619 0.849481i \(-0.676915\pi\)
−0.527619 + 0.849481i \(0.676915\pi\)
\(294\) 0 0
\(295\) −14.6456 + 5.90874i −0.852702 + 0.344020i
\(296\) 0 0
\(297\) −20.0054 −1.16083
\(298\) 0 0
\(299\) −31.1198 −1.79971
\(300\) 0 0
\(301\) 32.8094 2.09005i 1.89110 0.120468i
\(302\) 0 0
\(303\) 0.866730 0.0497923
\(304\) 0 0
\(305\) −9.68469 24.0048i −0.554544 1.37451i
\(306\) 0 0
\(307\) 18.5799i 1.06041i 0.847869 + 0.530205i \(0.177885\pi\)
−0.847869 + 0.530205i \(0.822115\pi\)
\(308\) 0 0
\(309\) 3.13497 0.178342
\(310\) 0 0
\(311\) −12.2396 −0.694046 −0.347023 0.937857i \(-0.612807\pi\)
−0.347023 + 0.937857i \(0.612807\pi\)
\(312\) 0 0
\(313\) 23.1509 1.30857 0.654283 0.756250i \(-0.272971\pi\)
0.654283 + 0.756250i \(0.272971\pi\)
\(314\) 0 0
\(315\) −2.59745 + 1.24541i −0.146350 + 0.0701707i
\(316\) 0 0
\(317\) 2.61492i 0.146868i −0.997300 0.0734342i \(-0.976604\pi\)
0.997300 0.0734342i \(-0.0233959\pi\)
\(318\) 0 0
\(319\) 5.16114i 0.288969i
\(320\) 0 0
\(321\) 12.4326i 0.693922i
\(322\) 0 0
\(323\) 0.140268 0.00780472
\(324\) 0 0
\(325\) 20.0614 19.3345i 1.11281 1.07249i
\(326\) 0 0
\(327\) 24.4573i 1.35249i
\(328\) 0 0
\(329\) −1.14181 17.9241i −0.0629502 0.988187i
\(330\) 0 0
\(331\) 12.8342i 0.705434i −0.935730 0.352717i \(-0.885258\pi\)
0.935730 0.352717i \(-0.114742\pi\)
\(332\) 0 0
\(333\) 4.81013i 0.263593i
\(334\) 0 0
\(335\) −13.8064 + 5.57016i −0.754325 + 0.304330i
\(336\) 0 0
\(337\) 3.05930i 0.166651i −0.996522 0.0833254i \(-0.973446\pi\)
0.996522 0.0833254i \(-0.0265541\pi\)
\(338\) 0 0
\(339\) 30.1776 1.63902
\(340\) 0 0
\(341\) 19.7005i 1.06684i
\(342\) 0 0
\(343\) −18.1840 + 3.51318i −0.981843 + 0.189694i
\(344\) 0 0
\(345\) −8.72452 21.6249i −0.469712 1.16425i
\(346\) 0 0
\(347\) −16.7386 −0.898574 −0.449287 0.893387i \(-0.648322\pi\)
−0.449287 + 0.893387i \(0.648322\pi\)
\(348\) 0 0
\(349\) 32.8680i 1.75938i −0.475544 0.879692i \(-0.657749\pi\)
0.475544 0.879692i \(-0.342251\pi\)
\(350\) 0 0
\(351\) 26.1499i 1.39578i
\(352\) 0 0
\(353\) 33.7338 1.79547 0.897736 0.440535i \(-0.145211\pi\)
0.897736 + 0.440535i \(0.145211\pi\)
\(354\) 0 0
\(355\) −4.28032 10.6094i −0.227176 0.563086i
\(356\) 0 0
\(357\) −0.947787 14.8783i −0.0501622 0.787442i
\(358\) 0 0
\(359\) 11.1542i 0.588698i −0.955698 0.294349i \(-0.904897\pi\)
0.955698 0.294349i \(-0.0951028\pi\)
\(360\) 0 0
\(361\) −18.9978 −0.999886
\(362\) 0 0
\(363\) 13.3951i 0.703063i
\(364\) 0 0
\(365\) 20.0637 8.09463i 1.05018 0.423692i
\(366\) 0 0
\(367\) 28.3476i 1.47973i −0.672755 0.739865i \(-0.734889\pi\)
0.672755 0.739865i \(-0.265111\pi\)
\(368\) 0 0
\(369\) 5.12292i 0.266689i
\(370\) 0 0
\(371\) −0.840895 13.2003i −0.0436571 0.685326i
\(372\) 0 0
\(373\) 28.6333i 1.48258i 0.671187 + 0.741288i \(0.265785\pi\)
−0.671187 + 0.741288i \(0.734215\pi\)
\(374\) 0 0
\(375\) 19.0597 + 8.52005i 0.984239 + 0.439973i
\(376\) 0 0
\(377\) −6.74634 −0.347454
\(378\) 0 0
\(379\) 19.9281i 1.02364i 0.859093 + 0.511820i \(0.171029\pi\)
−0.859093 + 0.511820i \(0.828971\pi\)
\(380\) 0 0
\(381\) 15.6954i 0.804098i
\(382\) 0 0
\(383\) 5.22155i 0.266809i 0.991062 + 0.133404i \(0.0425909\pi\)
−0.991062 + 0.133404i \(0.957409\pi\)
\(384\) 0 0
\(385\) −10.9039 22.7415i −0.555715 1.15901i
\(386\) 0 0
\(387\) 6.05027 0.307553
\(388\) 0 0
\(389\) 10.9831 0.556866 0.278433 0.960456i \(-0.410185\pi\)
0.278433 + 0.960456i \(0.410185\pi\)
\(390\) 0 0
\(391\) 16.8523 0.852257
\(392\) 0 0
\(393\) 2.09277i 0.105566i
\(394\) 0 0
\(395\) 4.93830 + 12.2403i 0.248473 + 0.615875i
\(396\) 0 0
\(397\) 33.4503 1.67882 0.839412 0.543495i \(-0.182899\pi\)
0.839412 + 0.543495i \(0.182899\pi\)
\(398\) 0 0
\(399\) −0.0145997 0.229185i −0.000730898 0.0114736i
\(400\) 0 0
\(401\) 3.51024 0.175293 0.0876466 0.996152i \(-0.472065\pi\)
0.0876466 + 0.996152i \(0.472065\pi\)
\(402\) 0 0
\(403\) −25.7513 −1.28276
\(404\) 0 0
\(405\) 21.2004 8.55324i 1.05346 0.425014i
\(406\) 0 0
\(407\) −42.1141 −2.08752
\(408\) 0 0
\(409\) 22.4267i 1.10893i −0.832207 0.554465i \(-0.812923\pi\)
0.832207 0.554465i \(-0.187077\pi\)
\(410\) 0 0
\(411\) −7.97694 −0.393474
\(412\) 0 0
\(413\) −18.6483 + 1.18795i −0.917624 + 0.0584551i
\(414\) 0 0
\(415\) −9.30322 23.0593i −0.456677 1.13194i
\(416\) 0 0
\(417\) 0.690495i 0.0338137i
\(418\) 0 0
\(419\) −0.797437 −0.0389573 −0.0194787 0.999810i \(-0.506201\pi\)
−0.0194787 + 0.999810i \(0.506201\pi\)
\(420\) 0 0
\(421\) −11.1666 −0.544227 −0.272114 0.962265i \(-0.587723\pi\)
−0.272114 + 0.962265i \(0.587723\pi\)
\(422\) 0 0
\(423\) 3.30533i 0.160710i
\(424\) 0 0
\(425\) −10.8639 + 10.4702i −0.526974 + 0.507881i
\(426\) 0 0
\(427\) −1.94710 30.5654i −0.0942268 1.47916i
\(428\) 0 0
\(429\) −44.3588 −2.14166
\(430\) 0 0
\(431\) 10.8568i 0.522954i 0.965210 + 0.261477i \(0.0842096\pi\)
−0.965210 + 0.261477i \(0.915790\pi\)
\(432\) 0 0
\(433\) −10.5291 −0.505996 −0.252998 0.967467i \(-0.581417\pi\)
−0.252998 + 0.967467i \(0.581417\pi\)
\(434\) 0 0
\(435\) −1.89135 4.68798i −0.0906835 0.224771i
\(436\) 0 0
\(437\) 0.259592 0.0124180
\(438\) 0 0
\(439\) 34.9523 1.66818 0.834092 0.551626i \(-0.185992\pi\)
0.834092 + 0.551626i \(0.185992\pi\)
\(440\) 0 0
\(441\) −3.38081 + 0.432488i −0.160991 + 0.0205947i
\(442\) 0 0
\(443\) −13.0336 −0.619246 −0.309623 0.950859i \(-0.600203\pi\)
−0.309623 + 0.950859i \(0.600203\pi\)
\(444\) 0 0
\(445\) −1.27072 3.14965i −0.0602377 0.149308i
\(446\) 0 0
\(447\) 30.8825i 1.46069i
\(448\) 0 0
\(449\) 13.9459 0.658147 0.329073 0.944304i \(-0.393264\pi\)
0.329073 + 0.944304i \(0.393264\pi\)
\(450\) 0 0
\(451\) 44.8528 2.11203
\(452\) 0 0
\(453\) −3.43902 −0.161579
\(454\) 0 0
\(455\) 29.7263 14.2530i 1.39359 0.668189i
\(456\) 0 0
\(457\) 17.4758i 0.817485i 0.912650 + 0.408743i \(0.134033\pi\)
−0.912650 + 0.408743i \(0.865967\pi\)
\(458\) 0 0
\(459\) 14.1609i 0.660975i
\(460\) 0 0
\(461\) 0.0195231i 0.000909280i 1.00000 0.000454640i \(0.000144716\pi\)
−1.00000 0.000454640i \(0.999855\pi\)
\(462\) 0 0
\(463\) −0.582327 −0.0270630 −0.0135315 0.999908i \(-0.504307\pi\)
−0.0135315 + 0.999908i \(0.504307\pi\)
\(464\) 0 0
\(465\) −7.21944 17.8944i −0.334793 0.829832i
\(466\) 0 0
\(467\) 16.3130i 0.754878i 0.926034 + 0.377439i \(0.123195\pi\)
−0.926034 + 0.377439i \(0.876805\pi\)
\(468\) 0 0
\(469\) −17.5797 + 1.11988i −0.811757 + 0.0517111i
\(470\) 0 0
\(471\) 12.8547i 0.592315i
\(472\) 0 0
\(473\) 52.9720i 2.43566i
\(474\) 0 0
\(475\) −0.167347 + 0.161283i −0.00767839 + 0.00740018i
\(476\) 0 0
\(477\) 2.43423i 0.111456i
\(478\) 0 0
\(479\) −12.1467 −0.554995 −0.277498 0.960726i \(-0.589505\pi\)
−0.277498 + 0.960726i \(0.589505\pi\)
\(480\) 0 0
\(481\) 55.0491i 2.51002i
\(482\) 0 0
\(483\) −1.75406 27.5351i −0.0798124 1.25289i
\(484\) 0 0
\(485\) 6.25750 2.52457i 0.284139 0.114635i
\(486\) 0 0
\(487\) 0.964581 0.0437093 0.0218547 0.999761i \(-0.493043\pi\)
0.0218547 + 0.999761i \(0.493043\pi\)
\(488\) 0 0
\(489\) 7.37304i 0.333420i
\(490\) 0 0
\(491\) 29.1152i 1.31395i 0.753912 + 0.656976i \(0.228165\pi\)
−0.753912 + 0.656976i \(0.771835\pi\)
\(492\) 0 0
\(493\) 3.65334 0.164538
\(494\) 0 0
\(495\) −1.73656 4.30431i −0.0780527 0.193464i
\(496\) 0 0
\(497\) −0.860554 13.5089i −0.0386011 0.605958i
\(498\) 0 0
\(499\) 24.4725i 1.09554i −0.836630 0.547769i \(-0.815477\pi\)
0.836630 0.547769i \(-0.184523\pi\)
\(500\) 0 0
\(501\) −14.7132 −0.657336
\(502\) 0 0
\(503\) 24.7444i 1.10330i 0.834077 + 0.551649i \(0.186001\pi\)
−0.834077 + 0.551649i \(0.813999\pi\)
\(504\) 0 0
\(505\) −0.388319 0.962502i −0.0172800 0.0428308i
\(506\) 0 0
\(507\) 33.7079i 1.49702i
\(508\) 0 0
\(509\) 14.2465i 0.631466i −0.948848 0.315733i \(-0.897750\pi\)
0.948848 0.315733i \(-0.102250\pi\)
\(510\) 0 0
\(511\) 25.5471 1.62742i 1.13014 0.0719928i
\(512\) 0 0
\(513\) 0.218134i 0.00963087i
\(514\) 0 0
\(515\) −1.40455 3.48138i −0.0618920 0.153408i
\(516\) 0 0
\(517\) 28.9392 1.27274
\(518\) 0 0
\(519\) 24.1879i 1.06173i
\(520\) 0 0
\(521\) 24.1921i 1.05987i 0.848037 + 0.529937i \(0.177784\pi\)
−0.848037 + 0.529937i \(0.822216\pi\)
\(522\) 0 0
\(523\) 36.4422i 1.59351i −0.604306 0.796753i \(-0.706549\pi\)
0.604306 0.796753i \(-0.293451\pi\)
\(524\) 0 0
\(525\) 18.2381 + 16.6607i 0.795977 + 0.727134i
\(526\) 0 0
\(527\) 13.9451 0.607457
\(528\) 0 0
\(529\) 8.18834 0.356015
\(530\) 0 0
\(531\) −3.43888 −0.149235
\(532\) 0 0
\(533\) 58.6289i 2.53950i
\(534\) 0 0
\(535\) 13.8064 5.57016i 0.596903 0.240819i
\(536\) 0 0
\(537\) 13.0453 0.562947
\(538\) 0 0
\(539\) −3.78657 29.6000i −0.163099 1.27496i
\(540\) 0 0
\(541\) 17.0685 0.733831 0.366916 0.930254i \(-0.380414\pi\)
0.366916 + 0.930254i \(0.380414\pi\)
\(542\) 0 0
\(543\) −39.4863 −1.69452
\(544\) 0 0
\(545\) 27.1598 10.9576i 1.16340 0.469370i
\(546\) 0 0
\(547\) 31.9460 1.36591 0.682955 0.730460i \(-0.260694\pi\)
0.682955 + 0.730460i \(0.260694\pi\)
\(548\) 0 0
\(549\) 5.63648i 0.240559i
\(550\) 0 0
\(551\) 0.0562760 0.00239744
\(552\) 0 0
\(553\) 0.992842 + 15.5856i 0.0422200 + 0.662765i
\(554\) 0 0
\(555\) 38.2532 15.4332i 1.62376 0.655101i
\(556\) 0 0
\(557\) 40.8312i 1.73008i −0.501707 0.865038i \(-0.667294\pi\)
0.501707 0.865038i \(-0.332706\pi\)
\(558\) 0 0
\(559\) −69.2419 −2.92862
\(560\) 0 0
\(561\) 24.0216 1.01419
\(562\) 0 0
\(563\) 19.7917i 0.834120i −0.908879 0.417060i \(-0.863061\pi\)
0.908879 0.417060i \(-0.136939\pi\)
\(564\) 0 0
\(565\) −13.5204 33.5121i −0.568807 1.40987i
\(566\) 0 0
\(567\) 26.9945 1.71962i 1.13366 0.0722174i
\(568\) 0 0
\(569\) 10.8023 0.452856 0.226428 0.974028i \(-0.427295\pi\)
0.226428 + 0.974028i \(0.427295\pi\)
\(570\) 0 0
\(571\) 17.2186i 0.720575i −0.932841 0.360287i \(-0.882679\pi\)
0.932841 0.360287i \(-0.117321\pi\)
\(572\) 0 0
\(573\) 29.4613 1.23077
\(574\) 0 0
\(575\) −20.1056 + 19.3771i −0.838462 + 0.808082i
\(576\) 0 0
\(577\) −31.2572 −1.30126 −0.650628 0.759397i \(-0.725494\pi\)
−0.650628 + 0.759397i \(0.725494\pi\)
\(578\) 0 0
\(579\) 9.26632 0.385095
\(580\) 0 0
\(581\) −1.87041 29.3615i −0.0775975 1.21812i
\(582\) 0 0
\(583\) 21.3124 0.882671
\(584\) 0 0
\(585\) 5.62634 2.26993i 0.232620 0.0938501i
\(586\) 0 0
\(587\) 27.9549i 1.15382i 0.816808 + 0.576910i \(0.195742\pi\)
−0.816808 + 0.576910i \(0.804258\pi\)
\(588\) 0 0
\(589\) 0.214810 0.00885108
\(590\) 0 0
\(591\) −32.6947 −1.34488
\(592\) 0 0
\(593\) −23.6892 −0.972800 −0.486400 0.873736i \(-0.661690\pi\)
−0.486400 + 0.873736i \(0.661690\pi\)
\(594\) 0 0
\(595\) −16.0977 + 7.71840i −0.659940 + 0.316423i
\(596\) 0 0
\(597\) 13.8788i 0.568023i
\(598\) 0 0
\(599\) 9.76391i 0.398942i −0.979904 0.199471i \(-0.936078\pi\)
0.979904 0.199471i \(-0.0639224\pi\)
\(600\) 0 0
\(601\) 15.3030i 0.624221i −0.950046 0.312110i \(-0.898964\pi\)
0.950046 0.312110i \(-0.101036\pi\)
\(602\) 0 0
\(603\) −3.24182 −0.132017
\(604\) 0 0
\(605\) 14.8753 6.00139i 0.604766 0.243991i
\(606\) 0 0
\(607\) 15.4718i 0.627982i 0.949426 + 0.313991i \(0.101666\pi\)
−0.949426 + 0.313991i \(0.898334\pi\)
\(608\) 0 0
\(609\) −0.380255 5.96922i −0.0154087 0.241885i
\(610\) 0 0
\(611\) 37.8276i 1.53034i
\(612\) 0 0
\(613\) 8.87239i 0.358352i −0.983817 0.179176i \(-0.942657\pi\)
0.983817 0.179176i \(-0.0573432\pi\)
\(614\) 0 0
\(615\) −40.7408 + 16.4368i −1.64283 + 0.662794i
\(616\) 0 0
\(617\) 16.9225i 0.681274i −0.940195 0.340637i \(-0.889357\pi\)
0.940195 0.340637i \(-0.110643\pi\)
\(618\) 0 0
\(619\) −22.3878 −0.899841 −0.449921 0.893069i \(-0.648548\pi\)
−0.449921 + 0.893069i \(0.648548\pi\)
\(620\) 0 0
\(621\) 26.2075i 1.05167i
\(622\) 0 0
\(623\) −0.255477 4.01045i −0.0102355 0.160675i
\(624\) 0 0
\(625\) 0.922223 24.9830i 0.0368889 0.999319i
\(626\) 0 0
\(627\) 0.370028 0.0147775
\(628\) 0 0
\(629\) 29.8107i 1.18863i
\(630\) 0 0
\(631\) 20.5691i 0.818844i 0.912345 + 0.409422i \(0.134270\pi\)
−0.912345 + 0.409422i \(0.865730\pi\)
\(632\) 0 0
\(633\) −40.7156 −1.61830
\(634\) 0 0
\(635\) −17.4297 + 7.03196i −0.691676 + 0.279055i
\(636\) 0 0
\(637\) 38.6914 4.94957i 1.53301 0.196109i
\(638\) 0 0
\(639\) 2.49114i 0.0985479i
\(640\) 0 0
\(641\) −11.4936 −0.453969 −0.226985 0.973898i \(-0.572887\pi\)
−0.226985 + 0.973898i \(0.572887\pi\)
\(642\) 0 0
\(643\) 23.7105i 0.935051i 0.883980 + 0.467525i \(0.154854\pi\)
−0.883980 + 0.467525i \(0.845146\pi\)
\(644\) 0 0
\(645\) −19.4121 48.1157i −0.764352 1.89455i
\(646\) 0 0
\(647\) 22.3311i 0.877926i −0.898505 0.438963i \(-0.855346\pi\)
0.898505 0.438963i \(-0.144654\pi\)
\(648\) 0 0
\(649\) 30.1085i 1.18186i
\(650\) 0 0
\(651\) −1.45146 22.7849i −0.0568873 0.893012i
\(652\) 0 0
\(653\) 14.8441i 0.580896i −0.956891 0.290448i \(-0.906196\pi\)
0.956891 0.290448i \(-0.0938043\pi\)
\(654\) 0 0
\(655\) 2.32402 0.937619i 0.0908069 0.0366358i
\(656\) 0 0
\(657\) 4.71106 0.183796
\(658\) 0 0
\(659\) 9.04938i 0.352514i 0.984344 + 0.176257i \(0.0563990\pi\)
−0.984344 + 0.176257i \(0.943601\pi\)
\(660\) 0 0
\(661\) 26.4095i 1.02721i −0.858027 0.513605i \(-0.828310\pi\)
0.858027 0.513605i \(-0.171690\pi\)
\(662\) 0 0
\(663\) 31.3996i 1.21946i
\(664\) 0 0
\(665\) −0.247968 + 0.118894i −0.00961579 + 0.00461051i
\(666\) 0 0
\(667\) 6.76120 0.261795
\(668\) 0 0
\(669\) 34.8165 1.34608
\(670\) 0 0
\(671\) 49.3491 1.90510
\(672\) 0 0
\(673\) 46.7468i 1.80196i −0.433865 0.900978i \(-0.642851\pi\)
0.433865 0.900978i \(-0.357149\pi\)
\(674\) 0 0
\(675\) −16.2825 16.8947i −0.626715 0.650276i
\(676\) 0 0
\(677\) −13.1268 −0.504504 −0.252252 0.967662i \(-0.581171\pi\)
−0.252252 + 0.967662i \(0.581171\pi\)
\(678\) 0 0
\(679\) 7.96769 0.507563i 0.305772 0.0194785i
\(680\) 0 0
\(681\) 23.1986 0.888971
\(682\) 0 0
\(683\) 24.9102 0.953162 0.476581 0.879130i \(-0.341876\pi\)
0.476581 + 0.879130i \(0.341876\pi\)
\(684\) 0 0
\(685\) 3.57389 + 8.85838i 0.136551 + 0.338461i
\(686\) 0 0
\(687\) 16.9251 0.645734
\(688\) 0 0
\(689\) 27.8583i 1.06132i
\(690\) 0 0
\(691\) 37.2251 1.41611 0.708054 0.706158i \(-0.249573\pi\)
0.708054 + 0.706158i \(0.249573\pi\)
\(692\) 0 0
\(693\) −0.349135 5.48069i −0.0132625 0.208194i
\(694\) 0 0
\(695\) −0.766794 + 0.309361i −0.0290861 + 0.0117347i
\(696\) 0 0
\(697\) 31.7493i 1.20259i
\(698\) 0 0
\(699\) 2.54956 0.0964333
\(700\) 0 0
\(701\) −10.1857 −0.384709 −0.192355 0.981326i \(-0.561612\pi\)
−0.192355 + 0.981326i \(0.561612\pi\)
\(702\) 0 0
\(703\) 0.459203i 0.0173192i
\(704\) 0 0
\(705\) −26.2861 + 10.6051i −0.989991 + 0.399409i
\(706\) 0 0
\(707\) −0.0780712 1.22556i −0.00293617 0.0460918i
\(708\) 0 0
\(709\) 49.7016 1.86658 0.933291 0.359121i \(-0.116924\pi\)
0.933291 + 0.359121i \(0.116924\pi\)
\(710\) 0 0
\(711\) 2.87409i 0.107787i
\(712\) 0 0
\(713\) 25.8080 0.966517
\(714\) 0 0
\(715\) 19.8740 + 49.2603i 0.743244 + 1.84223i
\(716\) 0 0
\(717\) −37.5722 −1.40316
\(718\) 0 0
\(719\) −18.2407 −0.680263 −0.340131 0.940378i \(-0.610472\pi\)
−0.340131 + 0.940378i \(0.610472\pi\)
\(720\) 0 0
\(721\) −0.282384 4.43285i −0.0105165 0.165088i
\(722\) 0 0
\(723\) 6.72992 0.250289
\(724\) 0 0
\(725\) −4.35862 + 4.20069i −0.161875 + 0.156010i
\(726\) 0 0
\(727\) 39.5072i 1.46524i 0.680637 + 0.732621i \(0.261703\pi\)
−0.680637 + 0.732621i \(0.738297\pi\)
\(728\) 0 0
\(729\) −21.3108 −0.789288
\(730\) 0 0
\(731\) 37.4965 1.38686
\(732\) 0 0
\(733\) 19.6704 0.726545 0.363272 0.931683i \(-0.381659\pi\)
0.363272 + 0.931683i \(0.381659\pi\)
\(734\) 0 0
\(735\) 14.2866 + 25.4987i 0.526971 + 0.940534i
\(736\) 0 0
\(737\) 28.3832i 1.04551i
\(738\) 0 0
\(739\) 3.40360i 0.125204i −0.998039 0.0626018i \(-0.980060\pi\)
0.998039 0.0626018i \(-0.0199398\pi\)
\(740\) 0 0
\(741\) 0.483678i 0.0177684i
\(742\) 0 0
\(743\) 5.21886 0.191462 0.0957308 0.995407i \(-0.469481\pi\)
0.0957308 + 0.995407i \(0.469481\pi\)
\(744\) 0 0
\(745\) 34.2949 13.8362i 1.25647 0.506919i
\(746\) 0 0
\(747\) 5.41446i 0.198105i
\(748\) 0 0
\(749\) 17.5797 1.11988i 0.642349 0.0409194i
\(750\) 0 0
\(751\) 3.17145i 0.115728i −0.998324 0.0578640i \(-0.981571\pi\)
0.998324 0.0578640i \(-0.0184290\pi\)
\(752\) 0 0
\(753\) 40.7653i 1.48557i
\(754\) 0 0
\(755\) 1.54078 + 3.81903i 0.0560746 + 0.138989i
\(756\) 0 0
\(757\) 27.3380i 0.993615i 0.867861 + 0.496808i \(0.165495\pi\)
−0.867861 + 0.496808i \(0.834505\pi\)
\(758\) 0 0
\(759\) 44.4565 1.61367
\(760\) 0 0
\(761\) 19.9547i 0.723356i −0.932303 0.361678i \(-0.882204\pi\)
0.932303 0.361678i \(-0.117796\pi\)
\(762\) 0 0
\(763\) 34.5827 2.20301i 1.25198 0.0797543i
\(764\) 0 0
\(765\) −3.04683 + 1.22923i −0.110158 + 0.0444431i
\(766\) 0 0
\(767\) 39.3560 1.42106
\(768\) 0 0
\(769\) 10.8811i 0.392384i 0.980566 + 0.196192i \(0.0628576\pi\)
−0.980566 + 0.196192i \(0.937142\pi\)
\(770\) 0 0
\(771\) 39.3110i 1.41575i
\(772\) 0 0
\(773\) 45.5968 1.64000 0.820002 0.572360i \(-0.193972\pi\)
0.820002 + 0.572360i \(0.193972\pi\)
\(774\) 0 0
\(775\) −16.6372 + 16.0343i −0.597625 + 0.575971i
\(776\) 0 0
\(777\) 48.7079 3.10282i 1.74739 0.111313i
\(778\) 0 0
\(779\) 0.489065i 0.0175226i
\(780\) 0 0
\(781\) 21.8107 0.780448
\(782\) 0 0
\(783\) 5.68141i 0.203037i
\(784\) 0 0
\(785\) 14.2752 5.75927i 0.509502 0.205557i
\(786\) 0 0
\(787\) 25.0781i 0.893937i −0.894550 0.446969i \(-0.852504\pi\)
0.894550 0.446969i \(-0.147496\pi\)
\(788\) 0 0
\(789\) 5.07104i 0.180534i
\(790\) 0 0
\(791\) −2.71826 42.6711i −0.0966503 1.51721i
\(792\) 0 0
\(793\) 64.5062i 2.29068i
\(794\) 0 0
\(795\) −19.3586 + 7.81016i −0.686577 + 0.276998i
\(796\) 0 0
\(797\) −27.1965 −0.963350 −0.481675 0.876350i \(-0.659971\pi\)
−0.481675 + 0.876350i \(0.659971\pi\)
\(798\) 0 0
\(799\) 20.4847i 0.724698i
\(800\) 0 0
\(801\) 0.739555i 0.0261309i
\(802\) 0 0
\(803\) 41.2468i 1.45557i
\(804\) 0 0
\(805\) −29.7918 + 14.2843i −1.05002 + 0.503457i
\(806\) 0 0
\(807\) −48.5366 −1.70857
\(808\) 0 0
\(809\) −36.4489 −1.28148 −0.640738 0.767760i \(-0.721371\pi\)
−0.640738 + 0.767760i \(0.721371\pi\)
\(810\) 0 0
\(811\) −25.2471 −0.886546 −0.443273 0.896387i \(-0.646183\pi\)
−0.443273 + 0.896387i \(0.646183\pi\)
\(812\) 0 0
\(813\) 47.5201i 1.66660i
\(814\) 0 0
\(815\) −8.18775 + 3.30333i −0.286804 + 0.115710i
\(816\) 0 0
\(817\) 0.577596 0.0202075
\(818\) 0 0
\(819\) 7.16403 0.456368i 0.250331 0.0159468i
\(820\) 0 0
\(821\) 13.2810 0.463510 0.231755 0.972774i \(-0.425553\pi\)
0.231755 + 0.972774i \(0.425553\pi\)
\(822\) 0 0
\(823\) 18.7856 0.654824 0.327412 0.944882i \(-0.393823\pi\)
0.327412 + 0.944882i \(0.393823\pi\)
\(824\) 0 0
\(825\) −28.6589 + 27.6205i −0.997776 + 0.961623i
\(826\) 0 0
\(827\) −45.0550 −1.56672 −0.783358 0.621571i \(-0.786495\pi\)
−0.783358 + 0.621571i \(0.786495\pi\)
\(828\) 0 0
\(829\) 3.20136i 0.111188i −0.998453 0.0555939i \(-0.982295\pi\)
0.998453 0.0555939i \(-0.0177052\pi\)
\(830\) 0 0
\(831\) 15.8901 0.551222
\(832\) 0 0
\(833\) −20.9525 + 2.68034i −0.725962 + 0.0928683i
\(834\) 0 0
\(835\) 6.59191 + 16.3390i 0.228123 + 0.565433i
\(836\) 0 0
\(837\) 21.6864i 0.749590i
\(838\) 0 0
\(839\) 3.11067 0.107392 0.0536962 0.998557i \(-0.482900\pi\)
0.0536962 + 0.998557i \(0.482900\pi\)
\(840\) 0 0
\(841\) −27.5343 −0.949457
\(842\) 0 0
\(843\) 18.9249i 0.651808i
\(844\) 0 0
\(845\) −37.4326 + 15.1021i −1.28772 + 0.519527i
\(846\) 0 0
\(847\) 18.9407 1.20658i 0.650811 0.0414584i
\(848\) 0 0
\(849\) −25.3817 −0.871099
\(850\) 0 0
\(851\) 55.1703i 1.89121i
\(852\) 0 0
\(853\) −17.1854 −0.588416 −0.294208 0.955741i \(-0.595056\pi\)
−0.294208 + 0.955741i \(0.595056\pi\)
\(854\) 0 0
\(855\) −0.0469333 + 0.0189351i −0.00160508 + 0.000647567i
\(856\) 0 0
\(857\) −32.3754 −1.10592 −0.552961 0.833207i \(-0.686502\pi\)
−0.552961 + 0.833207i \(0.686502\pi\)
\(858\) 0 0
\(859\) −29.4040 −1.00325 −0.501626 0.865085i \(-0.667264\pi\)
−0.501626 + 0.865085i \(0.667264\pi\)
\(860\) 0 0
\(861\) −51.8753 + 3.30460i −1.76791 + 0.112620i
\(862\) 0 0
\(863\) 30.0664 1.02347 0.511736 0.859143i \(-0.329003\pi\)
0.511736 + 0.859143i \(0.329003\pi\)
\(864\) 0 0
\(865\) 26.8606 10.8368i 0.913287 0.368463i
\(866\) 0 0
\(867\) 14.7407i 0.500622i
\(868\) 0 0
\(869\) −25.1635 −0.853614
\(870\) 0 0
\(871\) 37.1008 1.25711
\(872\) 0 0
\(873\) 1.46930 0.0497282
\(874\) 0 0
\(875\) 10.3305 27.7179i 0.349236 0.937035i
\(876\) 0 0
\(877\) 33.9614i 1.14679i −0.819278 0.573397i \(-0.805625\pi\)
0.819278 0.573397i \(-0.194375\pi\)
\(878\) 0 0
\(879\) 33.7290i 1.13765i
\(880\) 0 0
\(881\) 1.07536i 0.0362299i 0.999836 + 0.0181149i \(0.00576648\pi\)
−0.999836 + 0.0181149i \(0.994234\pi\)
\(882\) 0 0
\(883\) −26.0500 −0.876653 −0.438327 0.898816i \(-0.644429\pi\)
−0.438327 + 0.898816i \(0.644429\pi\)
\(884\) 0 0
\(885\) 11.0336 + 27.3482i 0.370889 + 0.919299i
\(886\) 0 0
\(887\) 28.1243i 0.944320i −0.881513 0.472160i \(-0.843474\pi\)
0.881513 0.472160i \(-0.156526\pi\)
\(888\) 0 0
\(889\) −22.1932 + 1.41377i −0.744338 + 0.0474163i
\(890\) 0 0
\(891\) 43.5837i 1.46011i
\(892\) 0 0
\(893\) 0.315546i 0.0105594i
\(894\) 0 0
\(895\) −5.84466 14.4868i −0.195366 0.484240i
\(896\) 0 0
\(897\) 58.1108i 1.94026i
\(898\) 0 0
\(899\) 5.59481 0.186597
\(900\) 0 0
\(901\) 15.0861i 0.502591i
\(902\) 0 0
\(903\) −3.90280 61.2658i −0.129877 2.03880i
\(904\) 0 0
\(905\) 17.6910 + 43.8495i 0.588067 + 1.45761i
\(906\) 0 0
\(907\) −33.9347 −1.12678 −0.563391 0.826190i \(-0.690504\pi\)
−0.563391 + 0.826190i \(0.690504\pi\)
\(908\) 0 0
\(909\) 0.226001i 0.00749598i
\(910\) 0 0
\(911\) 20.8703i 0.691465i −0.938333 0.345732i \(-0.887631\pi\)
0.938333 0.345732i \(-0.112369\pi\)
\(912\) 0 0
\(913\) 47.4053 1.56889
\(914\) 0 0
\(915\) −44.8249 + 18.0845i −1.48186 + 0.597855i
\(916\) 0 0
\(917\) 2.95918 0.188508i 0.0977207 0.00622507i
\(918\) 0 0
\(919\) 35.6755i 1.17682i −0.808561 0.588412i \(-0.799753\pi\)
0.808561 0.588412i \(-0.200247\pi\)
\(920\) 0 0
\(921\) 34.6947 1.14323
\(922\) 0 0
\(923\) 28.5096i 0.938406i
\(924\) 0 0
\(925\) −34.2770 35.5656i −1.12702 1.16939i
\(926\) 0 0
\(927\) 0.817448i 0.0268485i
\(928\) 0 0
\(929\) 49.6649i 1.62945i 0.579847 + 0.814726i \(0.303112\pi\)
−0.579847 + 0.814726i \(0.696888\pi\)
\(930\) 0 0
\(931\) −0.322752 + 0.0412879i −0.0105778 + 0.00135316i
\(932\) 0 0
\(933\) 22.8554i 0.748252i
\(934\) 0 0
\(935\) −10.7623 26.6759i −0.351966 0.872396i
\(936\) 0 0
\(937\) −8.12715 −0.265503 −0.132751 0.991149i \(-0.542381\pi\)
−0.132751 + 0.991149i \(0.542381\pi\)
\(938\) 0 0
\(939\) 43.2303i 1.41077i
\(940\) 0 0
\(941\) 23.1049i 0.753198i 0.926376 + 0.376599i \(0.122907\pi\)
−0.926376 + 0.376599i \(0.877093\pi\)
\(942\) 0 0
\(943\) 58.7580i 1.91342i
\(944\) 0 0
\(945\) −12.0031 25.0339i −0.390460 0.814353i
\(946\) 0 0
\(947\) 48.2578 1.56817 0.784084 0.620655i \(-0.213133\pi\)
0.784084 + 0.620655i \(0.213133\pi\)
\(948\) 0 0
\(949\) −53.9154 −1.75017
\(950\) 0 0
\(951\) −4.88291 −0.158339
\(952\) 0 0
\(953\) 52.9217i 1.71430i −0.515066 0.857151i \(-0.672233\pi\)
0.515066 0.857151i \(-0.327767\pi\)
\(954\) 0 0
\(955\) −13.1995 32.7168i −0.427126 1.05869i
\(956\) 0 0
\(957\) 9.63754 0.311537
\(958\) 0 0
\(959\) 0.718528 + 11.2794i 0.0232025 + 0.364231i
\(960\) 0 0
\(961\) −9.64418 −0.311102
\(962\) 0 0
\(963\) 3.24182 0.104466
\(964\) 0 0
\(965\) −4.15157 10.2902i −0.133644 0.331254i
\(966\) 0 0
\(967\) −8.36043 −0.268853 −0.134427 0.990924i \(-0.542919\pi\)
−0.134427 + 0.990924i \(0.542919\pi\)
\(968\) 0 0
\(969\) 0.261926i 0.00841428i
\(970\) 0 0
\(971\) 44.4955 1.42793 0.713964 0.700183i \(-0.246898\pi\)
0.713964 + 0.700183i \(0.246898\pi\)
\(972\) 0 0
\(973\) −0.976360 + 0.0621968i −0.0313007 + 0.00199394i
\(974\) 0 0
\(975\) −36.1039 37.4612i −1.15625 1.19972i
\(976\) 0 0
\(977\) 15.9555i 0.510461i −0.966880 0.255231i \(-0.917849\pi\)
0.966880 0.255231i \(-0.0821514\pi\)
\(978\) 0 0
\(979\) 6.47503 0.206943
\(980\) 0 0
\(981\) 6.37728 0.203611
\(982\) 0 0
\(983\) 48.5017i 1.54696i −0.633818 0.773482i \(-0.718513\pi\)
0.633818 0.773482i \(-0.281487\pi\)
\(984\) 0 0
\(985\) 14.6481 + 36.3074i 0.466728 + 1.15685i
\(986\) 0 0
\(987\) −33.4701 + 2.13214i −1.06537 + 0.0678667i
\(988\) 0 0
\(989\) 69.3944 2.20661
\(990\) 0 0
\(991\) 50.4999i 1.60418i 0.597202 + 0.802091i \(0.296279\pi\)
−0.597202 + 0.802091i \(0.703721\pi\)
\(992\) 0 0
\(993\) −23.9657 −0.760529
\(994\) 0 0
\(995\) 15.4124 6.21810i 0.488607 0.197127i
\(996\) 0 0
\(997\) −36.0833 −1.14277 −0.571384 0.820683i \(-0.693593\pi\)
−0.571384 + 0.820683i \(0.693593\pi\)
\(998\) 0 0
\(999\) −46.3594 −1.46675
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 2240.2.e.g.2239.9 48
4.3 odd 2 inner 2240.2.e.g.2239.40 48
5.4 even 2 inner 2240.2.e.g.2239.6 48
7.6 odd 2 inner 2240.2.e.g.2239.44 48
8.3 odd 2 1120.2.e.a.1119.43 yes 48
8.5 even 2 1120.2.e.a.1119.6 yes 48
20.19 odd 2 inner 2240.2.e.g.2239.43 48
28.27 even 2 inner 2240.2.e.g.2239.5 48
35.34 odd 2 inner 2240.2.e.g.2239.39 48
40.19 odd 2 1120.2.e.a.1119.40 yes 48
40.29 even 2 1120.2.e.a.1119.9 yes 48
56.13 odd 2 1120.2.e.a.1119.39 yes 48
56.27 even 2 1120.2.e.a.1119.10 yes 48
140.139 even 2 inner 2240.2.e.g.2239.10 48
280.69 odd 2 1120.2.e.a.1119.44 yes 48
280.139 even 2 1120.2.e.a.1119.5 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1120.2.e.a.1119.5 48 280.139 even 2
1120.2.e.a.1119.6 yes 48 8.5 even 2
1120.2.e.a.1119.9 yes 48 40.29 even 2
1120.2.e.a.1119.10 yes 48 56.27 even 2
1120.2.e.a.1119.39 yes 48 56.13 odd 2
1120.2.e.a.1119.40 yes 48 40.19 odd 2
1120.2.e.a.1119.43 yes 48 8.3 odd 2
1120.2.e.a.1119.44 yes 48 280.69 odd 2
2240.2.e.g.2239.5 48 28.27 even 2 inner
2240.2.e.g.2239.6 48 5.4 even 2 inner
2240.2.e.g.2239.9 48 1.1 even 1 trivial
2240.2.e.g.2239.10 48 140.139 even 2 inner
2240.2.e.g.2239.39 48 35.34 odd 2 inner
2240.2.e.g.2239.40 48 4.3 odd 2 inner
2240.2.e.g.2239.43 48 20.19 odd 2 inner
2240.2.e.g.2239.44 48 7.6 odd 2 inner