Properties

Label 896.2.m.g.225.3
Level $896$
Weight $2$
Character 896.225
Analytic conductor $7.155$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(225,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.m (of order \(4\), degree \(2\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.20138089353117696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 225.3
Root \(0.402577 + 1.35570i\) of defining polynomial
Character \(\chi\) \(=\) 896.225
Dual form 896.2.m.g.673.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.631188 + 0.631188i) q^{3} +(2.34259 + 2.34259i) q^{5} +1.00000i q^{7} +2.20320i q^{9} +O(q^{10})\) \(q+(-0.631188 + 0.631188i) q^{3} +(2.34259 + 2.34259i) q^{5} +1.00000i q^{7} +2.20320i q^{9} +(2.18310 + 2.18310i) q^{11} +(4.03390 - 4.03390i) q^{13} -2.95723 q^{15} +0.347931 q^{17} +(-4.26332 + 4.26332i) q^{19} +(-0.631188 - 0.631188i) q^{21} -6.23788i q^{23} +5.97550i q^{25} +(-3.28420 - 3.28420i) q^{27} +(-1.21961 + 1.21961i) q^{29} -1.26238 q^{31} -2.75589 q^{33} +(-2.34259 + 2.34259i) q^{35} +(6.42281 + 6.42281i) q^{37} +5.09229i q^{39} +2.68519i q^{41} +(-4.05478 - 4.05478i) q^{43} +(-5.16121 + 5.16121i) q^{45} +4.64498 q^{47} -1.00000 q^{49} +(-0.219610 + 0.219610i) q^{51} +(-8.44108 - 8.44108i) q^{53} +10.2282i q^{55} -5.38191i q^{57} +(5.17776 + 5.17776i) q^{59} +(0.00533660 - 0.00533660i) q^{61} -2.20320 q^{63} +18.8996 q^{65} +(-3.02011 + 3.02011i) q^{67} +(3.93727 + 3.93727i) q^{69} -0.828913i q^{71} +6.25173i q^{73} +(-3.77166 - 3.77166i) q^{75} +(-2.18310 + 2.18310i) q^{77} -0.755891 q^{79} -2.46372 q^{81} +(3.66586 - 3.66586i) q^{83} +(0.815062 + 0.815062i) q^{85} -1.53961i q^{87} -6.24461i q^{89} +(4.03390 + 4.03390i) q^{91} +(0.796796 - 0.796796i) q^{93} -19.9745 q^{95} +2.18393 q^{97} +(-4.80981 + 4.80981i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 4 q^{5} - 24 q^{15} - 8 q^{17} - 4 q^{21} - 4 q^{27} + 4 q^{29} - 8 q^{31} + 4 q^{35} + 20 q^{37} - 16 q^{43} - 40 q^{45} + 16 q^{47} - 12 q^{49} + 16 q^{51} - 4 q^{53} + 16 q^{59} + 20 q^{61} + 12 q^{63} + 32 q^{65} - 24 q^{67} + 4 q^{69} + 40 q^{75} + 24 q^{79} - 44 q^{81} + 20 q^{83} + 8 q^{85} + 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −0.631188 + 0.631188i −0.364416 + 0.364416i −0.865436 0.501020i \(-0.832958\pi\)
0.501020 + 0.865436i \(0.332958\pi\)
\(4\) 0 0
\(5\) 2.34259 + 2.34259i 1.04764 + 1.04764i 0.998807 + 0.0488333i \(0.0155503\pi\)
0.0488333 + 0.998807i \(0.484450\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.20320i 0.734401i
\(10\) 0 0
\(11\) 2.18310 + 2.18310i 0.658229 + 0.658229i 0.954961 0.296732i \(-0.0958967\pi\)
−0.296732 + 0.954961i \(0.595897\pi\)
\(12\) 0 0
\(13\) 4.03390 4.03390i 1.11880 1.11880i 0.126884 0.991918i \(-0.459502\pi\)
0.991918 0.126884i \(-0.0404976\pi\)
\(14\) 0 0
\(15\) −2.95723 −0.763555
\(16\) 0 0
\(17\) 0.347931 0.0843858 0.0421929 0.999109i \(-0.486566\pi\)
0.0421929 + 0.999109i \(0.486566\pi\)
\(18\) 0 0
\(19\) −4.26332 + 4.26332i −0.978072 + 0.978072i −0.999765 0.0216925i \(-0.993095\pi\)
0.0216925 + 0.999765i \(0.493095\pi\)
\(20\) 0 0
\(21\) −0.631188 0.631188i −0.137736 0.137736i
\(22\) 0 0
\(23\) 6.23788i 1.30069i −0.759640 0.650344i \(-0.774625\pi\)
0.759640 0.650344i \(-0.225375\pi\)
\(24\) 0 0
\(25\) 5.97550i 1.19510i
\(26\) 0 0
\(27\) −3.28420 3.28420i −0.632044 0.632044i
\(28\) 0 0
\(29\) −1.21961 + 1.21961i −0.226476 + 0.226476i −0.811219 0.584743i \(-0.801195\pi\)
0.584743 + 0.811219i \(0.301195\pi\)
\(30\) 0 0
\(31\) −1.26238 −0.226729 −0.113365 0.993553i \(-0.536163\pi\)
−0.113365 + 0.993553i \(0.536163\pi\)
\(32\) 0 0
\(33\) −2.75589 −0.479739
\(34\) 0 0
\(35\) −2.34259 + 2.34259i −0.395971 + 0.395971i
\(36\) 0 0
\(37\) 6.42281 + 6.42281i 1.05590 + 1.05590i 0.998342 + 0.0575622i \(0.0183327\pi\)
0.0575622 + 0.998342i \(0.481667\pi\)
\(38\) 0 0
\(39\) 5.09229i 0.815419i
\(40\) 0 0
\(41\) 2.68519i 0.419356i 0.977770 + 0.209678i \(0.0672416\pi\)
−0.977770 + 0.209678i \(0.932758\pi\)
\(42\) 0 0
\(43\) −4.05478 4.05478i −0.618348 0.618348i 0.326760 0.945107i \(-0.394043\pi\)
−0.945107 + 0.326760i \(0.894043\pi\)
\(44\) 0 0
\(45\) −5.16121 + 5.16121i −0.769388 + 0.769388i
\(46\) 0 0
\(47\) 4.64498 0.677540 0.338770 0.940869i \(-0.389989\pi\)
0.338770 + 0.940869i \(0.389989\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −0.219610 + 0.219610i −0.0307516 + 0.0307516i
\(52\) 0 0
\(53\) −8.44108 8.44108i −1.15947 1.15947i −0.984589 0.174882i \(-0.944046\pi\)
−0.174882 0.984589i \(-0.555954\pi\)
\(54\) 0 0
\(55\) 10.2282i 1.37917i
\(56\) 0 0
\(57\) 5.38191i 0.712851i
\(58\) 0 0
\(59\) 5.17776 + 5.17776i 0.674087 + 0.674087i 0.958656 0.284568i \(-0.0918503\pi\)
−0.284568 + 0.958656i \(0.591850\pi\)
\(60\) 0 0
\(61\) 0.00533660 0.00533660i 0.000683281 0.000683281i −0.706765 0.707448i \(-0.749846\pi\)
0.707448 + 0.706765i \(0.249846\pi\)
\(62\) 0 0
\(63\) −2.20320 −0.277578
\(64\) 0 0
\(65\) 18.8996 2.34420
\(66\) 0 0
\(67\) −3.02011 + 3.02011i −0.368965 + 0.368965i −0.867100 0.498135i \(-0.834018\pi\)
0.498135 + 0.867100i \(0.334018\pi\)
\(68\) 0 0
\(69\) 3.93727 + 3.93727i 0.473992 + 0.473992i
\(70\) 0 0
\(71\) 0.828913i 0.0983739i −0.998790 0.0491869i \(-0.984337\pi\)
0.998790 0.0491869i \(-0.0156630\pi\)
\(72\) 0 0
\(73\) 6.25173i 0.731709i 0.930672 + 0.365855i \(0.119223\pi\)
−0.930672 + 0.365855i \(0.880777\pi\)
\(74\) 0 0
\(75\) −3.77166 3.77166i −0.435514 0.435514i
\(76\) 0 0
\(77\) −2.18310 + 2.18310i −0.248787 + 0.248787i
\(78\) 0 0
\(79\) −0.755891 −0.0850443 −0.0425222 0.999096i \(-0.513539\pi\)
−0.0425222 + 0.999096i \(0.513539\pi\)
\(80\) 0 0
\(81\) −2.46372 −0.273747
\(82\) 0 0
\(83\) 3.66586 3.66586i 0.402380 0.402380i −0.476691 0.879071i \(-0.658164\pi\)
0.879071 + 0.476691i \(0.158164\pi\)
\(84\) 0 0
\(85\) 0.815062 + 0.815062i 0.0884059 + 0.0884059i
\(86\) 0 0
\(87\) 1.53961i 0.165063i
\(88\) 0 0
\(89\) 6.24461i 0.661928i −0.943643 0.330964i \(-0.892626\pi\)
0.943643 0.330964i \(-0.107374\pi\)
\(90\) 0 0
\(91\) 4.03390 + 4.03390i 0.422867 + 0.422867i
\(92\) 0 0
\(93\) 0.796796 0.796796i 0.0826239 0.0826239i
\(94\) 0 0
\(95\) −19.9745 −2.04934
\(96\) 0 0
\(97\) 2.18393 0.221745 0.110872 0.993835i \(-0.464635\pi\)
0.110872 + 0.993835i \(0.464635\pi\)
\(98\) 0 0
\(99\) −4.80981 + 4.80981i −0.483404 + 0.483404i
\(100\) 0 0
\(101\) 4.75325 + 4.75325i 0.472967 + 0.472967i 0.902873 0.429907i \(-0.141454\pi\)
−0.429907 + 0.902873i \(0.641454\pi\)
\(102\) 0 0
\(103\) 15.7259i 1.54952i 0.632256 + 0.774760i \(0.282129\pi\)
−0.632256 + 0.774760i \(0.717871\pi\)
\(104\) 0 0
\(105\) 2.95723i 0.288597i
\(106\) 0 0
\(107\) 6.68080 + 6.68080i 0.645857 + 0.645857i 0.951989 0.306132i \(-0.0990349\pi\)
−0.306132 + 0.951989i \(0.599035\pi\)
\(108\) 0 0
\(109\) 0.812507 0.812507i 0.0778241 0.0778241i −0.667123 0.744947i \(-0.732475\pi\)
0.744947 + 0.667123i \(0.232475\pi\)
\(110\) 0 0
\(111\) −8.10800 −0.769578
\(112\) 0 0
\(113\) −18.5170 −1.74193 −0.870966 0.491343i \(-0.836506\pi\)
−0.870966 + 0.491343i \(0.836506\pi\)
\(114\) 0 0
\(115\) 14.6128 14.6128i 1.36265 1.36265i
\(116\) 0 0
\(117\) 8.88750 + 8.88750i 0.821649 + 0.821649i
\(118\) 0 0
\(119\) 0.347931i 0.0318948i
\(120\) 0 0
\(121\) 1.46816i 0.133469i
\(122\) 0 0
\(123\) −1.69486 1.69486i −0.152820 0.152820i
\(124\) 0 0
\(125\) −2.28520 + 2.28520i −0.204395 + 0.204395i
\(126\) 0 0
\(127\) 13.7063 1.21624 0.608121 0.793845i \(-0.291924\pi\)
0.608121 + 0.793845i \(0.291924\pi\)
\(128\) 0 0
\(129\) 5.11865 0.450672
\(130\) 0 0
\(131\) 3.52925 3.52925i 0.308352 0.308352i −0.535918 0.844270i \(-0.680034\pi\)
0.844270 + 0.535918i \(0.180034\pi\)
\(132\) 0 0
\(133\) −4.26332 4.26332i −0.369677 0.369677i
\(134\) 0 0
\(135\) 15.3871i 1.32431i
\(136\) 0 0
\(137\) 15.6540i 1.33741i −0.743529 0.668704i \(-0.766849\pi\)
0.743529 0.668704i \(-0.233151\pi\)
\(138\) 0 0
\(139\) −1.52569 1.52569i −0.129408 0.129408i 0.639436 0.768844i \(-0.279168\pi\)
−0.768844 + 0.639436i \(0.779168\pi\)
\(140\) 0 0
\(141\) −2.93185 + 2.93185i −0.246907 + 0.246907i
\(142\) 0 0
\(143\) 17.6128 1.47286
\(144\) 0 0
\(145\) −5.71410 −0.474531
\(146\) 0 0
\(147\) 0.631188 0.631188i 0.0520595 0.0520595i
\(148\) 0 0
\(149\) −9.23788 9.23788i −0.756796 0.756796i 0.218942 0.975738i \(-0.429740\pi\)
−0.975738 + 0.218942i \(0.929740\pi\)
\(150\) 0 0
\(151\) 11.7266i 0.954297i −0.878823 0.477149i \(-0.841670\pi\)
0.878823 0.477149i \(-0.158330\pi\)
\(152\) 0 0
\(153\) 0.766564i 0.0619730i
\(154\) 0 0
\(155\) −2.95723 2.95723i −0.237531 0.237531i
\(156\) 0 0
\(157\) 10.3818 10.3818i 0.828560 0.828560i −0.158758 0.987318i \(-0.550749\pi\)
0.987318 + 0.158758i \(0.0507488\pi\)
\(158\) 0 0
\(159\) 10.6558 0.845061
\(160\) 0 0
\(161\) 6.23788 0.491614
\(162\) 0 0
\(163\) 15.6554 15.6554i 1.22623 1.22623i 0.260849 0.965380i \(-0.415997\pi\)
0.965380 0.260849i \(-0.0840025\pi\)
\(164\) 0 0
\(165\) −6.45593 6.45593i −0.502594 0.502594i
\(166\) 0 0
\(167\) 2.12023i 0.164068i −0.996630 0.0820341i \(-0.973858\pi\)
0.996630 0.0820341i \(-0.0261416\pi\)
\(168\) 0 0
\(169\) 19.5446i 1.50343i
\(170\) 0 0
\(171\) −9.39296 9.39296i −0.718298 0.718298i
\(172\) 0 0
\(173\) −2.12654 + 2.12654i −0.161678 + 0.161678i −0.783310 0.621632i \(-0.786470\pi\)
0.621632 + 0.783310i \(0.286470\pi\)
\(174\) 0 0
\(175\) −5.97550 −0.451705
\(176\) 0 0
\(177\) −6.53628 −0.491297
\(178\) 0 0
\(179\) −1.41911 + 1.41911i −0.106070 + 0.106070i −0.758150 0.652080i \(-0.773896\pi\)
0.652080 + 0.758150i \(0.273896\pi\)
\(180\) 0 0
\(181\) −14.3191 14.3191i −1.06433 1.06433i −0.997783 0.0665470i \(-0.978802\pi\)
−0.0665470 0.997783i \(-0.521198\pi\)
\(182\) 0 0
\(183\) 0.00673679i 0.000497998i
\(184\) 0 0
\(185\) 30.0921i 2.21242i
\(186\) 0 0
\(187\) 0.759569 + 0.759569i 0.0555452 + 0.0555452i
\(188\) 0 0
\(189\) 3.28420 3.28420i 0.238890 0.238890i
\(190\) 0 0
\(191\) 6.22279 0.450265 0.225133 0.974328i \(-0.427718\pi\)
0.225133 + 0.974328i \(0.427718\pi\)
\(192\) 0 0
\(193\) 1.57618 0.113456 0.0567280 0.998390i \(-0.481933\pi\)
0.0567280 + 0.998390i \(0.481933\pi\)
\(194\) 0 0
\(195\) −11.9292 + 11.9292i −0.854266 + 0.854266i
\(196\) 0 0
\(197\) −10.7183 10.7183i −0.763648 0.763648i 0.213332 0.976980i \(-0.431568\pi\)
−0.976980 + 0.213332i \(0.931568\pi\)
\(198\) 0 0
\(199\) 25.5363i 1.81022i −0.425180 0.905109i \(-0.639789\pi\)
0.425180 0.905109i \(-0.360211\pi\)
\(200\) 0 0
\(201\) 3.81251i 0.268914i
\(202\) 0 0
\(203\) −1.21961 1.21961i −0.0855998 0.0855998i
\(204\) 0 0
\(205\) −6.29031 + 6.29031i −0.439334 + 0.439334i
\(206\) 0 0
\(207\) 13.7433 0.955226
\(208\) 0 0
\(209\) −18.6145 −1.28759
\(210\) 0 0
\(211\) −9.19881 + 9.19881i −0.633272 + 0.633272i −0.948887 0.315615i \(-0.897789\pi\)
0.315615 + 0.948887i \(0.397789\pi\)
\(212\) 0 0
\(213\) 0.523200 + 0.523200i 0.0358491 + 0.0358491i
\(214\) 0 0
\(215\) 18.9974i 1.29561i
\(216\) 0 0
\(217\) 1.26238i 0.0856956i
\(218\) 0 0
\(219\) −3.94601 3.94601i −0.266647 0.266647i
\(220\) 0 0
\(221\) 1.40352 1.40352i 0.0944109 0.0944109i
\(222\) 0 0
\(223\) 12.7530 0.854003 0.427001 0.904251i \(-0.359570\pi\)
0.427001 + 0.904251i \(0.359570\pi\)
\(224\) 0 0
\(225\) −13.1652 −0.877683
\(226\) 0 0
\(227\) 12.2451 12.2451i 0.812733 0.812733i −0.172310 0.985043i \(-0.555123\pi\)
0.985043 + 0.172310i \(0.0551231\pi\)
\(228\) 0 0
\(229\) 9.93193 + 9.93193i 0.656321 + 0.656321i 0.954508 0.298187i \(-0.0963818\pi\)
−0.298187 + 0.954508i \(0.596382\pi\)
\(230\) 0 0
\(231\) 2.75589i 0.181324i
\(232\) 0 0
\(233\) 18.3143i 1.19981i 0.800072 + 0.599905i \(0.204795\pi\)
−0.800072 + 0.599905i \(0.795205\pi\)
\(234\) 0 0
\(235\) 10.8813 + 10.8813i 0.709818 + 0.709818i
\(236\) 0 0
\(237\) 0.477109 0.477109i 0.0309916 0.0309916i
\(238\) 0 0
\(239\) −18.3443 −1.18660 −0.593298 0.804983i \(-0.702174\pi\)
−0.593298 + 0.804983i \(0.702174\pi\)
\(240\) 0 0
\(241\) 25.4147 1.63710 0.818552 0.574433i \(-0.194777\pi\)
0.818552 + 0.574433i \(0.194777\pi\)
\(242\) 0 0
\(243\) 11.4077 11.4077i 0.731802 0.731802i
\(244\) 0 0
\(245\) −2.34259 2.34259i −0.149663 0.149663i
\(246\) 0 0
\(247\) 34.3956i 2.18854i
\(248\) 0 0
\(249\) 4.62769i 0.293268i
\(250\) 0 0
\(251\) 10.9301 + 10.9301i 0.689903 + 0.689903i 0.962210 0.272307i \(-0.0877867\pi\)
−0.272307 + 0.962210i \(0.587787\pi\)
\(252\) 0 0
\(253\) 13.6179 13.6179i 0.856150 0.856150i
\(254\) 0 0
\(255\) −1.02891 −0.0644331
\(256\) 0 0
\(257\) 2.29652 0.143253 0.0716265 0.997432i \(-0.477181\pi\)
0.0716265 + 0.997432i \(0.477181\pi\)
\(258\) 0 0
\(259\) −6.42281 + 6.42281i −0.399094 + 0.399094i
\(260\) 0 0
\(261\) −2.68705 2.68705i −0.166324 0.166324i
\(262\) 0 0
\(263\) 11.8508i 0.730749i 0.930861 + 0.365375i \(0.119059\pi\)
−0.930861 + 0.365375i \(0.880941\pi\)
\(264\) 0 0
\(265\) 39.5481i 2.42942i
\(266\) 0 0
\(267\) 3.94152 + 3.94152i 0.241217 + 0.241217i
\(268\) 0 0
\(269\) −5.70791 + 5.70791i −0.348017 + 0.348017i −0.859371 0.511353i \(-0.829144\pi\)
0.511353 + 0.859371i \(0.329144\pi\)
\(270\) 0 0
\(271\) 26.1234 1.58688 0.793441 0.608648i \(-0.208288\pi\)
0.793441 + 0.608648i \(0.208288\pi\)
\(272\) 0 0
\(273\) −5.09229 −0.308200
\(274\) 0 0
\(275\) −13.0451 + 13.0451i −0.786650 + 0.786650i
\(276\) 0 0
\(277\) −1.98274 1.98274i −0.119131 0.119131i 0.645028 0.764159i \(-0.276846\pi\)
−0.764159 + 0.645028i \(0.776846\pi\)
\(278\) 0 0
\(279\) 2.78127i 0.166510i
\(280\) 0 0
\(281\) 6.52475i 0.389234i 0.980879 + 0.194617i \(0.0623464\pi\)
−0.980879 + 0.194617i \(0.937654\pi\)
\(282\) 0 0
\(283\) 13.9578 + 13.9578i 0.829705 + 0.829705i 0.987476 0.157771i \(-0.0504307\pi\)
−0.157771 + 0.987476i \(0.550431\pi\)
\(284\) 0 0
\(285\) 12.6076 12.6076i 0.746812 0.746812i
\(286\) 0 0
\(287\) −2.68519 −0.158502
\(288\) 0 0
\(289\) −16.8789 −0.992879
\(290\) 0 0
\(291\) −1.37847 + 1.37847i −0.0808075 + 0.0808075i
\(292\) 0 0
\(293\) 2.57087 + 2.57087i 0.150192 + 0.150192i 0.778204 0.628012i \(-0.216131\pi\)
−0.628012 + 0.778204i \(0.716131\pi\)
\(294\) 0 0
\(295\) 24.2588i 1.41240i
\(296\) 0 0
\(297\) 14.3395i 0.832060i
\(298\) 0 0
\(299\) −25.1629 25.1629i −1.45521 1.45521i
\(300\) 0 0
\(301\) 4.05478 4.05478i 0.233713 0.233713i
\(302\) 0 0
\(303\) −6.00039 −0.344714
\(304\) 0 0
\(305\) 0.0250030 0.00143167
\(306\) 0 0
\(307\) −18.8253 + 18.8253i −1.07441 + 1.07441i −0.0774148 + 0.996999i \(0.524667\pi\)
−0.996999 + 0.0774148i \(0.975333\pi\)
\(308\) 0 0
\(309\) −9.92600 9.92600i −0.564670 0.564670i
\(310\) 0 0
\(311\) 23.8918i 1.35478i −0.735625 0.677389i \(-0.763111\pi\)
0.735625 0.677389i \(-0.236889\pi\)
\(312\) 0 0
\(313\) 18.0884i 1.02242i 0.859457 + 0.511208i \(0.170802\pi\)
−0.859457 + 0.511208i \(0.829198\pi\)
\(314\) 0 0
\(315\) −5.16121 5.16121i −0.290801 0.290801i
\(316\) 0 0
\(317\) 5.36465 5.36465i 0.301309 0.301309i −0.540217 0.841526i \(-0.681658\pi\)
0.841526 + 0.540217i \(0.181658\pi\)
\(318\) 0 0
\(319\) −5.32506 −0.298146
\(320\) 0 0
\(321\) −8.43367 −0.470722
\(322\) 0 0
\(323\) −1.48334 + 1.48334i −0.0825354 + 0.0825354i
\(324\) 0 0
\(325\) 24.1046 + 24.1046i 1.33708 + 1.33708i
\(326\) 0 0
\(327\) 1.02569i 0.0567207i
\(328\) 0 0
\(329\) 4.64498i 0.256086i
\(330\) 0 0
\(331\) 22.3563 + 22.3563i 1.22881 + 1.22881i 0.964413 + 0.264402i \(0.0851744\pi\)
0.264402 + 0.964413i \(0.414826\pi\)
\(332\) 0 0
\(333\) −14.1508 + 14.1508i −0.775457 + 0.775457i
\(334\) 0 0
\(335\) −14.1498 −0.773084
\(336\) 0 0
\(337\) 11.5086 0.626914 0.313457 0.949602i \(-0.398513\pi\)
0.313457 + 0.949602i \(0.398513\pi\)
\(338\) 0 0
\(339\) 11.6877 11.6877i 0.634789 0.634789i
\(340\) 0 0
\(341\) −2.75589 2.75589i −0.149240 0.149240i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 18.4469i 0.993146i
\(346\) 0 0
\(347\) −12.3190 12.3190i −0.661319 0.661319i 0.294372 0.955691i \(-0.404890\pi\)
−0.955691 + 0.294372i \(0.904890\pi\)
\(348\) 0 0
\(349\) −0.251501 + 0.251501i −0.0134625 + 0.0134625i −0.713806 0.700343i \(-0.753030\pi\)
0.700343 + 0.713806i \(0.253030\pi\)
\(350\) 0 0
\(351\) −26.4962 −1.41426
\(352\) 0 0
\(353\) 16.6888 0.888255 0.444128 0.895964i \(-0.353514\pi\)
0.444128 + 0.895964i \(0.353514\pi\)
\(354\) 0 0
\(355\) 1.94181 1.94181i 0.103060 0.103060i
\(356\) 0 0
\(357\) −0.219610 0.219610i −0.0116230 0.0116230i
\(358\) 0 0
\(359\) 17.9781i 0.948849i −0.880296 0.474425i \(-0.842656\pi\)
0.880296 0.474425i \(-0.157344\pi\)
\(360\) 0 0
\(361\) 17.3518i 0.913251i
\(362\) 0 0
\(363\) 0.926684 + 0.926684i 0.0486383 + 0.0486383i
\(364\) 0 0
\(365\) −14.6453 + 14.6453i −0.766568 + 0.766568i
\(366\) 0 0
\(367\) 21.6270 1.12892 0.564461 0.825460i \(-0.309084\pi\)
0.564461 + 0.825460i \(0.309084\pi\)
\(368\) 0 0
\(369\) −5.91602 −0.307976
\(370\) 0 0
\(371\) 8.44108 8.44108i 0.438239 0.438239i
\(372\) 0 0
\(373\) 12.1124 + 12.1124i 0.627159 + 0.627159i 0.947352 0.320194i \(-0.103748\pi\)
−0.320194 + 0.947352i \(0.603748\pi\)
\(374\) 0 0
\(375\) 2.88478i 0.148970i
\(376\) 0 0
\(377\) 9.83956i 0.506763i
\(378\) 0 0
\(379\) −4.34251 4.34251i −0.223060 0.223060i 0.586726 0.809786i \(-0.300417\pi\)
−0.809786 + 0.586726i \(0.800417\pi\)
\(380\) 0 0
\(381\) −8.65128 + 8.65128i −0.443218 + 0.443218i
\(382\) 0 0
\(383\) −32.8242 −1.67724 −0.838619 0.544718i \(-0.816637\pi\)
−0.838619 + 0.544718i \(0.816637\pi\)
\(384\) 0 0
\(385\) −10.2282 −0.521279
\(386\) 0 0
\(387\) 8.93350 8.93350i 0.454115 0.454115i
\(388\) 0 0
\(389\) −0.234988 0.234988i −0.0119143 0.0119143i 0.701125 0.713039i \(-0.252682\pi\)
−0.713039 + 0.701125i \(0.752682\pi\)
\(390\) 0 0
\(391\) 2.17035i 0.109759i
\(392\) 0 0
\(393\) 4.45524i 0.224737i
\(394\) 0 0
\(395\) −1.77075 1.77075i −0.0890959 0.0890959i
\(396\) 0 0
\(397\) −1.97983 + 1.97983i −0.0993650 + 0.0993650i −0.755042 0.655677i \(-0.772383\pi\)
0.655677 + 0.755042i \(0.272383\pi\)
\(398\) 0 0
\(399\) 5.38191 0.269432
\(400\) 0 0
\(401\) −5.15044 −0.257200 −0.128600 0.991697i \(-0.541048\pi\)
−0.128600 + 0.991697i \(0.541048\pi\)
\(402\) 0 0
\(403\) −5.09229 + 5.09229i −0.253665 + 0.253665i
\(404\) 0 0
\(405\) −5.77150 5.77150i −0.286788 0.286788i
\(406\) 0 0
\(407\) 28.0433i 1.39005i
\(408\) 0 0
\(409\) 0.0729036i 0.00360485i −0.999998 0.00180243i \(-0.999426\pi\)
0.999998 0.00180243i \(-0.000573731\pi\)
\(410\) 0 0
\(411\) 9.88058 + 9.88058i 0.487373 + 0.487373i
\(412\) 0 0
\(413\) −5.17776 + 5.17776i −0.254781 + 0.254781i
\(414\) 0 0
\(415\) 17.1752 0.843100
\(416\) 0 0
\(417\) 1.92600 0.0943165
\(418\) 0 0
\(419\) 13.7426 13.7426i 0.671370 0.671370i −0.286662 0.958032i \(-0.592546\pi\)
0.958032 + 0.286662i \(0.0925456\pi\)
\(420\) 0 0
\(421\) −3.88097 3.88097i −0.189147 0.189147i 0.606180 0.795327i \(-0.292701\pi\)
−0.795327 + 0.606180i \(0.792701\pi\)
\(422\) 0 0
\(423\) 10.2338i 0.497586i
\(424\) 0 0
\(425\) 2.07906i 0.100849i
\(426\) 0 0
\(427\) 0.00533660 + 0.00533660i 0.000258256 + 0.000258256i
\(428\) 0 0
\(429\) −11.1170 + 11.1170i −0.536733 + 0.536733i
\(430\) 0 0
\(431\) −33.8033 −1.62825 −0.814125 0.580690i \(-0.802783\pi\)
−0.814125 + 0.580690i \(0.802783\pi\)
\(432\) 0 0
\(433\) 18.6931 0.898334 0.449167 0.893448i \(-0.351721\pi\)
0.449167 + 0.893448i \(0.351721\pi\)
\(434\) 0 0
\(435\) 3.60667 3.60667i 0.172927 0.172927i
\(436\) 0 0
\(437\) 26.5940 + 26.5940i 1.27217 + 1.27217i
\(438\) 0 0
\(439\) 6.03142i 0.287864i −0.989588 0.143932i \(-0.954025\pi\)
0.989588 0.143932i \(-0.0459747\pi\)
\(440\) 0 0
\(441\) 2.20320i 0.104914i
\(442\) 0 0
\(443\) −24.9158 24.9158i −1.18378 1.18378i −0.978756 0.205027i \(-0.934272\pi\)
−0.205027 0.978756i \(-0.565728\pi\)
\(444\) 0 0
\(445\) 14.6286 14.6286i 0.693462 0.693462i
\(446\) 0 0
\(447\) 11.6617 0.551578
\(448\) 0 0
\(449\) 21.9883 1.03769 0.518846 0.854868i \(-0.326362\pi\)
0.518846 + 0.854868i \(0.326362\pi\)
\(450\) 0 0
\(451\) −5.86203 + 5.86203i −0.276032 + 0.276032i
\(452\) 0 0
\(453\) 7.40169 + 7.40169i 0.347762 + 0.347762i
\(454\) 0 0
\(455\) 18.8996i 0.886026i
\(456\) 0 0
\(457\) 26.7381i 1.25076i −0.780322 0.625378i \(-0.784945\pi\)
0.780322 0.625378i \(-0.215055\pi\)
\(458\) 0 0
\(459\) −1.14268 1.14268i −0.0533355 0.0533355i
\(460\) 0 0
\(461\) −23.5563 + 23.5563i −1.09713 + 1.09713i −0.102381 + 0.994745i \(0.532646\pi\)
−0.994745 + 0.102381i \(0.967354\pi\)
\(462\) 0 0
\(463\) −9.83629 −0.457131 −0.228566 0.973529i \(-0.573404\pi\)
−0.228566 + 0.973529i \(0.573404\pi\)
\(464\) 0 0
\(465\) 3.73314 0.173120
\(466\) 0 0
\(467\) −1.96594 + 1.96594i −0.0909727 + 0.0909727i −0.751129 0.660156i \(-0.770490\pi\)
0.660156 + 0.751129i \(0.270490\pi\)
\(468\) 0 0
\(469\) −3.02011 3.02011i −0.139455 0.139455i
\(470\) 0 0
\(471\) 13.1058i 0.603882i
\(472\) 0 0
\(473\) 17.7040i 0.814029i
\(474\) 0 0
\(475\) −25.4755 25.4755i −1.16889 1.16889i
\(476\) 0 0
\(477\) 18.5974 18.5974i 0.851517 0.851517i
\(478\) 0 0
\(479\) 3.78688 0.173027 0.0865136 0.996251i \(-0.472427\pi\)
0.0865136 + 0.996251i \(0.472427\pi\)
\(480\) 0 0
\(481\) 51.8179 2.36269
\(482\) 0 0
\(483\) −3.93727 + 3.93727i −0.179152 + 0.179152i
\(484\) 0 0
\(485\) 5.11607 + 5.11607i 0.232309 + 0.232309i
\(486\) 0 0
\(487\) 16.8200i 0.762186i 0.924537 + 0.381093i \(0.124452\pi\)
−0.924537 + 0.381093i \(0.875548\pi\)
\(488\) 0 0
\(489\) 19.7630i 0.893716i
\(490\) 0 0
\(491\) 6.41618 + 6.41618i 0.289558 + 0.289558i 0.836906 0.547347i \(-0.184362\pi\)
−0.547347 + 0.836906i \(0.684362\pi\)
\(492\) 0 0
\(493\) −0.424341 + 0.424341i −0.0191113 + 0.0191113i
\(494\) 0 0
\(495\) −22.5349 −1.01287
\(496\) 0 0
\(497\) 0.828913 0.0371818
\(498\) 0 0
\(499\) 4.69302 4.69302i 0.210088 0.210088i −0.594217 0.804305i \(-0.702538\pi\)
0.804305 + 0.594217i \(0.202538\pi\)
\(500\) 0 0
\(501\) 1.33826 + 1.33826i 0.0597891 + 0.0597891i
\(502\) 0 0
\(503\) 4.37360i 0.195009i 0.995235 + 0.0975045i \(0.0310860\pi\)
−0.995235 + 0.0975045i \(0.968914\pi\)
\(504\) 0 0
\(505\) 22.2699i 0.990998i
\(506\) 0 0
\(507\) 12.3363 + 12.3363i 0.547876 + 0.547876i
\(508\) 0 0
\(509\) −1.93619 + 1.93619i −0.0858200 + 0.0858200i −0.748714 0.662894i \(-0.769328\pi\)
0.662894 + 0.748714i \(0.269328\pi\)
\(510\) 0 0
\(511\) −6.25173 −0.276560
\(512\) 0 0
\(513\) 28.0032 1.23637
\(514\) 0 0
\(515\) −36.8394 + 36.8394i −1.62334 + 1.62334i
\(516\) 0 0
\(517\) 10.1404 + 10.1404i 0.445976 + 0.445976i
\(518\) 0 0
\(519\) 2.68449i 0.117836i
\(520\) 0 0
\(521\) 35.3300i 1.54783i 0.633287 + 0.773917i \(0.281705\pi\)
−0.633287 + 0.773917i \(0.718295\pi\)
\(522\) 0 0
\(523\) 23.3854 + 23.3854i 1.02257 + 1.02257i 0.999739 + 0.0228331i \(0.00726864\pi\)
0.0228331 + 0.999739i \(0.492731\pi\)
\(524\) 0 0
\(525\) 3.77166 3.77166i 0.164609 0.164609i
\(526\) 0 0
\(527\) −0.439220 −0.0191327
\(528\) 0 0
\(529\) −15.9111 −0.691787
\(530\) 0 0
\(531\) −11.4077 + 11.4077i −0.495051 + 0.495051i
\(532\) 0 0
\(533\) 10.8318 + 10.8318i 0.469176 + 0.469176i
\(534\) 0 0
\(535\) 31.3008i 1.35325i
\(536\) 0 0
\(537\) 1.79146i 0.0773070i
\(538\) 0 0
\(539\) −2.18310 2.18310i −0.0940327 0.0940327i
\(540\) 0 0
\(541\) 5.13176 5.13176i 0.220631 0.220631i −0.588133 0.808764i \(-0.700137\pi\)
0.808764 + 0.588133i \(0.200137\pi\)
\(542\) 0 0
\(543\) 18.0761 0.775719
\(544\) 0 0
\(545\) 3.80675 0.163063
\(546\) 0 0
\(547\) −16.8157 + 16.8157i −0.718989 + 0.718989i −0.968398 0.249409i \(-0.919764\pi\)
0.249409 + 0.968398i \(0.419764\pi\)
\(548\) 0 0
\(549\) 0.0117576 + 0.0117576i 0.000501803 + 0.000501803i
\(550\) 0 0
\(551\) 10.3992i 0.443020i
\(552\) 0 0
\(553\) 0.755891i 0.0321437i
\(554\) 0 0
\(555\) −18.9938 18.9938i −0.806240 0.806240i
\(556\) 0 0
\(557\) 11.0874 11.0874i 0.469789 0.469789i −0.432057 0.901846i \(-0.642212\pi\)
0.901846 + 0.432057i \(0.142212\pi\)
\(558\) 0 0
\(559\) −32.7131 −1.38362
\(560\) 0 0
\(561\) −0.958861 −0.0404831
\(562\) 0 0
\(563\) −22.2057 + 22.2057i −0.935857 + 0.935857i −0.998063 0.0622066i \(-0.980186\pi\)
0.0622066 + 0.998063i \(0.480186\pi\)
\(564\) 0 0
\(565\) −43.3778 43.3778i −1.82492 1.82492i
\(566\) 0 0
\(567\) 2.46372i 0.103466i
\(568\) 0 0
\(569\) 0.317171i 0.0132965i 0.999978 + 0.00664825i \(0.00211622\pi\)
−0.999978 + 0.00664825i \(0.997884\pi\)
\(570\) 0 0
\(571\) 7.80358 + 7.80358i 0.326570 + 0.326570i 0.851280 0.524711i \(-0.175827\pi\)
−0.524711 + 0.851280i \(0.675827\pi\)
\(572\) 0 0
\(573\) −3.92775 + 3.92775i −0.164084 + 0.164084i
\(574\) 0 0
\(575\) 37.2744 1.55445
\(576\) 0 0
\(577\) −10.2699 −0.427540 −0.213770 0.976884i \(-0.568574\pi\)
−0.213770 + 0.976884i \(0.568574\pi\)
\(578\) 0 0
\(579\) −0.994867 + 0.994867i −0.0413453 + 0.0413453i
\(580\) 0 0
\(581\) 3.66586 + 3.66586i 0.152086 + 0.152086i
\(582\) 0 0
\(583\) 36.8554i 1.52640i
\(584\) 0 0
\(585\) 41.6396i 1.72159i
\(586\) 0 0
\(587\) 27.9965 + 27.9965i 1.15554 + 1.15554i 0.985425 + 0.170113i \(0.0544133\pi\)
0.170113 + 0.985425i \(0.445587\pi\)
\(588\) 0 0
\(589\) 5.38191 5.38191i 0.221758 0.221758i
\(590\) 0 0
\(591\) 13.5305 0.556572
\(592\) 0 0
\(593\) −34.5902 −1.42045 −0.710225 0.703974i \(-0.751407\pi\)
−0.710225 + 0.703974i \(0.751407\pi\)
\(594\) 0 0
\(595\) −0.815062 + 0.815062i −0.0334143 + 0.0334143i
\(596\) 0 0
\(597\) 16.1182 + 16.1182i 0.659673 + 0.659673i
\(598\) 0 0
\(599\) 14.0866i 0.575562i 0.957696 + 0.287781i \(0.0929176\pi\)
−0.957696 + 0.287781i \(0.907082\pi\)
\(600\) 0 0
\(601\) 7.07501i 0.288595i −0.989534 0.144298i \(-0.953908\pi\)
0.989534 0.144298i \(-0.0460923\pi\)
\(602\) 0 0
\(603\) −6.65391 6.65391i −0.270968 0.270968i
\(604\) 0 0
\(605\) 3.43930 3.43930i 0.139828 0.139828i
\(606\) 0 0
\(607\) 5.99294 0.243246 0.121623 0.992576i \(-0.461190\pi\)
0.121623 + 0.992576i \(0.461190\pi\)
\(608\) 0 0
\(609\) 1.53961 0.0623880
\(610\) 0 0
\(611\) 18.7374 18.7374i 0.758033 0.758033i
\(612\) 0 0
\(613\) 12.3166 + 12.3166i 0.497464 + 0.497464i 0.910648 0.413184i \(-0.135583\pi\)
−0.413184 + 0.910648i \(0.635583\pi\)
\(614\) 0 0
\(615\) 7.94074i 0.320201i
\(616\) 0 0
\(617\) 40.3690i 1.62519i −0.582827 0.812596i \(-0.698053\pi\)
0.582827 0.812596i \(-0.301947\pi\)
\(618\) 0 0
\(619\) −30.9375 30.9375i −1.24348 1.24348i −0.958546 0.284938i \(-0.908027\pi\)
−0.284938 0.958546i \(-0.591973\pi\)
\(620\) 0 0
\(621\) −20.4864 + 20.4864i −0.822092 + 0.822092i
\(622\) 0 0
\(623\) 6.24461 0.250185
\(624\) 0 0
\(625\) 19.1709 0.766836
\(626\) 0 0
\(627\) 11.7492 11.7492i 0.469219 0.469219i
\(628\) 0 0
\(629\) 2.23470 + 2.23470i 0.0891033 + 0.0891033i
\(630\) 0 0
\(631\) 23.7329i 0.944792i −0.881386 0.472396i \(-0.843389\pi\)
0.881386 0.472396i \(-0.156611\pi\)
\(632\) 0 0
\(633\) 11.6124i 0.461550i
\(634\) 0 0
\(635\) 32.1084 + 32.1084i 1.27418 + 1.27418i
\(636\) 0 0
\(637\) −4.03390 + 4.03390i −0.159829 + 0.159829i
\(638\) 0 0
\(639\) 1.82626 0.0722459
\(640\) 0 0
\(641\) −14.9883 −0.592000 −0.296000 0.955188i \(-0.595653\pi\)
−0.296000 + 0.955188i \(0.595653\pi\)
\(642\) 0 0
\(643\) 11.8452 11.8452i 0.467130 0.467130i −0.433854 0.900983i \(-0.642847\pi\)
0.900983 + 0.433854i \(0.142847\pi\)
\(644\) 0 0
\(645\) 11.9909 + 11.9909i 0.472142 + 0.472142i
\(646\) 0 0
\(647\) 4.12471i 0.162159i −0.996708 0.0810795i \(-0.974163\pi\)
0.996708 0.0810795i \(-0.0258368\pi\)
\(648\) 0 0
\(649\) 22.6071i 0.887408i
\(650\) 0 0
\(651\) 0.796796 + 0.796796i 0.0312289 + 0.0312289i
\(652\) 0 0
\(653\) 25.4516 25.4516i 0.995997 0.995997i −0.00399544 0.999992i \(-0.501272\pi\)
0.999992 + 0.00399544i \(0.00127179\pi\)
\(654\) 0 0
\(655\) 16.5352 0.646084
\(656\) 0 0
\(657\) −13.7738 −0.537368
\(658\) 0 0
\(659\) −7.38409 + 7.38409i −0.287643 + 0.287643i −0.836148 0.548504i \(-0.815197\pi\)
0.548504 + 0.836148i \(0.315197\pi\)
\(660\) 0 0
\(661\) −14.0924 14.0924i −0.548130 0.548130i 0.377770 0.925900i \(-0.376691\pi\)
−0.925900 + 0.377770i \(0.876691\pi\)
\(662\) 0 0
\(663\) 1.77177i 0.0688098i
\(664\) 0 0
\(665\) 19.9745i 0.774576i
\(666\) 0 0
\(667\) 7.60778 + 7.60778i 0.294574 + 0.294574i
\(668\) 0 0
\(669\) −8.04953 + 8.04953i −0.311213 + 0.311213i
\(670\) 0 0
\(671\) 0.0233006 0.000899511
\(672\) 0 0
\(673\) −35.0089 −1.34949 −0.674746 0.738050i \(-0.735747\pi\)
−0.674746 + 0.738050i \(0.735747\pi\)
\(674\) 0 0
\(675\) 19.6247 19.6247i 0.755356 0.755356i
\(676\) 0 0
\(677\) −11.2860 11.2860i −0.433755 0.433755i 0.456148 0.889904i \(-0.349229\pi\)
−0.889904 + 0.456148i \(0.849229\pi\)
\(678\) 0 0
\(679\) 2.18393i 0.0838117i
\(680\) 0 0
\(681\) 15.4579i 0.592346i
\(682\) 0 0
\(683\) −33.4270 33.4270i −1.27905 1.27905i −0.941201 0.337848i \(-0.890301\pi\)
−0.337848 0.941201i \(-0.609699\pi\)
\(684\) 0 0
\(685\) 36.6709 36.6709i 1.40112 1.40112i
\(686\) 0 0
\(687\) −12.5378 −0.478348
\(688\) 0 0
\(689\) −68.1009 −2.59444
\(690\) 0 0
\(691\) 5.03700 5.03700i 0.191616 0.191616i −0.604778 0.796394i \(-0.706738\pi\)
0.796394 + 0.604778i \(0.206738\pi\)
\(692\) 0 0
\(693\) −4.80981 4.80981i −0.182710 0.182710i
\(694\) 0 0
\(695\) 7.14816i 0.271145i
\(696\) 0 0
\(697\) 0.934262i 0.0353877i
\(698\) 0 0
\(699\) −11.5598 11.5598i −0.437230 0.437230i
\(700\) 0 0
\(701\) −1.77330 + 1.77330i −0.0669766 + 0.0669766i −0.739802 0.672825i \(-0.765081\pi\)
0.672825 + 0.739802i \(0.265081\pi\)
\(702\) 0 0
\(703\) −54.7650 −2.06550
\(704\) 0 0
\(705\) −13.7363 −0.517339
\(706\) 0 0
\(707\) −4.75325 + 4.75325i −0.178765 + 0.178765i
\(708\) 0 0
\(709\) 8.84781 + 8.84781i 0.332286 + 0.332286i 0.853454 0.521168i \(-0.174503\pi\)
−0.521168 + 0.853454i \(0.674503\pi\)
\(710\) 0 0
\(711\) 1.66538i 0.0624567i
\(712\) 0 0
\(713\) 7.87454i 0.294904i
\(714\) 0 0
\(715\) 41.2596 + 41.2596i 1.54302 + 1.54302i
\(716\) 0 0
\(717\) 11.5787 11.5787i 0.432415 0.432415i
\(718\) 0 0
\(719\) −11.8257 −0.441025 −0.220512 0.975384i \(-0.570773\pi\)
−0.220512 + 0.975384i \(0.570773\pi\)
\(720\) 0 0
\(721\) −15.7259 −0.585663
\(722\) 0 0
\(723\) −16.0414 + 16.0414i −0.596587 + 0.596587i
\(724\) 0 0
\(725\) −7.28778 7.28778i −0.270661 0.270661i
\(726\) 0 0
\(727\) 27.7703i 1.02994i −0.857207 0.514972i \(-0.827802\pi\)
0.857207 0.514972i \(-0.172198\pi\)
\(728\) 0 0
\(729\) 7.00960i 0.259615i
\(730\) 0 0
\(731\) −1.41078 1.41078i −0.0521797 0.0521797i
\(732\) 0 0
\(733\) −35.8466 + 35.8466i −1.32402 + 1.32402i −0.413535 + 0.910488i \(0.635706\pi\)
−0.910488 + 0.413535i \(0.864294\pi\)
\(734\) 0 0
\(735\) 2.95723 0.109079
\(736\) 0 0
\(737\) −13.1864 −0.485726
\(738\) 0 0
\(739\) 10.1088 10.1088i 0.371860 0.371860i −0.496295 0.868154i \(-0.665306\pi\)
0.868154 + 0.496295i \(0.165306\pi\)
\(740\) 0 0
\(741\) −21.7101 21.7101i −0.797539 0.797539i
\(742\) 0 0
\(743\) 31.4037i 1.15209i −0.817418 0.576045i \(-0.804595\pi\)
0.817418 0.576045i \(-0.195405\pi\)
\(744\) 0 0
\(745\) 43.2812i 1.58570i
\(746\) 0 0
\(747\) 8.07664 + 8.07664i 0.295509 + 0.295509i
\(748\) 0 0
\(749\) −6.68080 + 6.68080i −0.244111 + 0.244111i
\(750\) 0 0
\(751\) 12.6531 0.461717 0.230859 0.972987i \(-0.425847\pi\)
0.230859 + 0.972987i \(0.425847\pi\)
\(752\) 0 0
\(753\) −13.7979 −0.502824
\(754\) 0 0
\(755\) 27.4707 27.4707i 0.999760 0.999760i
\(756\) 0 0
\(757\) −19.6555 19.6555i −0.714391 0.714391i 0.253060 0.967451i \(-0.418563\pi\)
−0.967451 + 0.253060i \(0.918563\pi\)
\(758\) 0 0
\(759\) 17.1909i 0.623990i
\(760\) 0 0
\(761\) 12.2754i 0.444984i 0.974934 + 0.222492i \(0.0714192\pi\)
−0.974934 + 0.222492i \(0.928581\pi\)
\(762\) 0 0
\(763\) 0.812507 + 0.812507i 0.0294147 + 0.0294147i
\(764\) 0 0
\(765\) −1.79575 + 1.79575i −0.0649254 + 0.0649254i
\(766\) 0 0
\(767\) 41.7731 1.50834
\(768\) 0 0
\(769\) 44.0633 1.58896 0.794481 0.607289i \(-0.207743\pi\)
0.794481 + 0.607289i \(0.207743\pi\)
\(770\) 0 0
\(771\) −1.44954 + 1.44954i −0.0522037 + 0.0522037i
\(772\) 0 0
\(773\) −30.9188 30.9188i −1.11207 1.11207i −0.992870 0.119200i \(-0.961967\pi\)
−0.119200 0.992870i \(-0.538033\pi\)
\(774\) 0 0
\(775\) 7.54333i 0.270964i
\(776\) 0 0
\(777\) 8.10800i 0.290873i
\(778\) 0 0
\(779\) −11.4478 11.4478i −0.410161 0.410161i
\(780\) 0 0
\(781\) 1.80960 1.80960i 0.0647526 0.0647526i
\(782\) 0 0
\(783\) 8.01088 0.286286
\(784\) 0 0
\(785\) 48.6408 1.73607
\(786\) 0 0
\(787\) −13.3994 + 13.3994i −0.477638 + 0.477638i −0.904375 0.426738i \(-0.859663\pi\)
0.426738 + 0.904375i \(0.359663\pi\)
\(788\) 0 0
\(789\) −7.48006 7.48006i −0.266297 0.266297i
\(790\) 0 0
\(791\) 18.5170i 0.658388i
\(792\) 0 0
\(793\) 0.0430546i 0.00152891i
\(794\) 0 0
\(795\) 24.9623 + 24.9623i 0.885320 + 0.885320i
\(796\) 0 0
\(797\) 16.1055 16.1055i 0.570485 0.570485i −0.361779 0.932264i \(-0.617831\pi\)
0.932264 + 0.361779i \(0.117831\pi\)
\(798\) 0 0
\(799\) 1.61613 0.0571747
\(800\) 0 0
\(801\) 13.7582 0.486121
\(802\) 0 0
\(803\) −13.6481 + 13.6481i −0.481632 + 0.481632i
\(804\) 0 0
\(805\) 14.6128 + 14.6128i 0.515034 + 0.515034i
\(806\) 0 0
\(807\) 7.20553i 0.253647i
\(808\) 0 0
\(809\) 37.6047i 1.32211i −0.750336 0.661056i \(-0.770109\pi\)
0.750336 0.661056i \(-0.229891\pi\)
\(810\) 0 0
\(811\) −7.00106 7.00106i −0.245841 0.245841i 0.573421 0.819261i \(-0.305616\pi\)
−0.819261 + 0.573421i \(0.805616\pi\)
\(812\) 0 0
\(813\) −16.4887 + 16.4887i −0.578286 + 0.578286i
\(814\) 0 0
\(815\) 73.3487 2.56929
\(816\) 0 0
\(817\) 34.5736 1.20958
\(818\) 0 0
\(819\) −8.88750 + 8.88750i −0.310554 + 0.310554i
\(820\) 0 0
\(821\) −0.711892 0.711892i −0.0248452 0.0248452i 0.694575 0.719420i \(-0.255592\pi\)
−0.719420 + 0.694575i \(0.755592\pi\)
\(822\) 0 0
\(823\) 12.0278i 0.419262i −0.977781 0.209631i \(-0.932774\pi\)
0.977781 0.209631i \(-0.0672263\pi\)
\(824\) 0 0
\(825\) 16.4678i 0.573336i
\(826\) 0 0
\(827\) −16.1838 16.1838i −0.562764 0.562764i 0.367328 0.930092i \(-0.380273\pi\)
−0.930092 + 0.367328i \(0.880273\pi\)
\(828\) 0 0
\(829\) −3.31569 + 3.31569i −0.115159 + 0.115159i −0.762338 0.647179i \(-0.775949\pi\)
0.647179 + 0.762338i \(0.275949\pi\)
\(830\) 0 0
\(831\) 2.50296 0.0868267
\(832\) 0 0
\(833\) −0.347931 −0.0120551
\(834\) 0 0
\(835\) 4.96684 4.96684i 0.171884 0.171884i
\(836\) 0 0
\(837\) 4.14589 + 4.14589i 0.143303 + 0.143303i
\(838\) 0 0
\(839\) 47.4324i 1.63755i 0.574116 + 0.818774i \(0.305346\pi\)
−0.574116 + 0.818774i \(0.694654\pi\)
\(840\) 0 0
\(841\) 26.0251i 0.897417i
\(842\) 0 0
\(843\) −4.11834 4.11834i −0.141843 0.141843i
\(844\) 0 0
\(845\) 45.7852 45.7852i 1.57506 1.57506i
\(846\) 0 0
\(847\) 1.46816 0.0504466
\(848\) 0 0
\(849\) −17.6200 −0.604716
\(850\) 0 0
\(851\) 40.0647 40.0647i 1.37340 1.37340i
\(852\) 0 0
\(853\) −3.99601 3.99601i −0.136821 0.136821i 0.635379 0.772200i \(-0.280844\pi\)
−0.772200 + 0.635379i \(0.780844\pi\)
\(854\) 0 0
\(855\) 44.0078i 1.50503i
\(856\) 0 0
\(857\) 11.1854i 0.382086i 0.981582 + 0.191043i \(0.0611869\pi\)
−0.981582 + 0.191043i \(0.938813\pi\)
\(858\) 0 0
\(859\) −15.0707 15.0707i −0.514207 0.514207i 0.401606 0.915813i \(-0.368452\pi\)
−0.915813 + 0.401606i \(0.868452\pi\)
\(860\) 0 0
\(861\) 1.69486 1.69486i 0.0577606 0.0577606i
\(862\) 0 0
\(863\) 29.6751 1.01015 0.505076 0.863075i \(-0.331464\pi\)
0.505076 + 0.863075i \(0.331464\pi\)
\(864\) 0 0
\(865\) −9.96325 −0.338761
\(866\) 0 0
\(867\) 10.6538 10.6538i 0.361821 0.361821i
\(868\) 0 0
\(869\) −1.65018 1.65018i −0.0559787 0.0559787i
\(870\) 0 0
\(871\) 24.3656i 0.825596i
\(872\) 0 0
\(873\) 4.81165i 0.162850i
\(874\) 0 0
\(875\) −2.28520 2.28520i −0.0772540 0.0772540i
\(876\) 0 0
\(877\) −2.85105 + 2.85105i −0.0962730 + 0.0962730i −0.753603 0.657330i \(-0.771686\pi\)
0.657330 + 0.753603i \(0.271686\pi\)
\(878\) 0 0
\(879\) −3.24541 −0.109465
\(880\) 0 0
\(881\) −23.4719 −0.790789 −0.395394 0.918511i \(-0.629392\pi\)
−0.395394 + 0.918511i \(0.629392\pi\)
\(882\) 0 0
\(883\) 22.0401 22.0401i 0.741710 0.741710i −0.231197 0.972907i \(-0.574264\pi\)
0.972907 + 0.231197i \(0.0742642\pi\)
\(884\) 0 0
\(885\) −15.3119 15.3119i −0.514702 0.514702i
\(886\) 0 0
\(887\) 18.6629i 0.626640i 0.949648 + 0.313320i \(0.101441\pi\)
−0.949648 + 0.313320i \(0.898559\pi\)
\(888\) 0 0
\(889\) 13.7063i 0.459696i
\(890\) 0 0
\(891\) −5.37854 5.37854i −0.180188 0.180188i
\(892\) 0 0
\(893\) −19.8030 + 19.8030i −0.662683 + 0.662683i
\(894\) 0 0
\(895\) −6.64882 −0.222246
\(896\) 0 0
\(897\) 31.7651 1.06061
\(898\) 0 0
\(899\) 1.53961 1.53961i 0.0513487 0.0513487i
\(900\) 0 0
\(901\) −2.93692 2.93692i −0.0978429 0.0978429i
\(902\) 0 0
\(903\) 5.11865i 0.170338i
\(904\) 0 0
\(905\) 67.0877i 2.23007i
\(906\) 0 0
\(907\) 2.95043 + 2.95043i 0.0979673 + 0.0979673i 0.754392 0.656424i \(-0.227932\pi\)
−0.656424 + 0.754392i \(0.727932\pi\)
\(908\) 0 0
\(909\) −10.4724 + 10.4724i −0.347347 + 0.347347i
\(910\) 0 0
\(911\) −26.5648 −0.880132 −0.440066 0.897965i \(-0.645045\pi\)
−0.440066 + 0.897965i \(0.645045\pi\)
\(912\) 0 0
\(913\) 16.0059 0.529717
\(914\) 0 0
\(915\) −0.0157816 + 0.0157816i −0.000521723 + 0.000521723i
\(916\) 0 0
\(917\) 3.52925 + 3.52925i 0.116546 + 0.116546i
\(918\) 0 0
\(919\) 21.9819i 0.725116i 0.931961 + 0.362558i \(0.118096\pi\)
−0.931961 + 0.362558i \(0.881904\pi\)
\(920\) 0 0
\(921\) 23.7645i 0.783068i
\(922\) 0 0
\(923\) −3.34375 3.34375i −0.110061 0.110061i
\(924\) 0 0
\(925\) −38.3795 + 38.3795i −1.26191 + 1.26191i
\(926\) 0 0
\(927\) −34.6474 −1.13797
\(928\) 0 0
\(929\) 14.0560 0.461163 0.230581 0.973053i \(-0.425937\pi\)
0.230581 + 0.973053i \(0.425937\pi\)
\(930\) 0 0
\(931\) 4.26332 4.26332i 0.139725 0.139725i
\(932\) 0 0
\(933\) 15.0802 + 15.0802i 0.493704 + 0.493704i
\(934\) 0 0
\(935\) 3.55872i 0.116383i
\(936\) 0 0
\(937\) 57.9965i 1.89466i 0.320256 + 0.947331i \(0.396231\pi\)
−0.320256 + 0.947331i \(0.603769\pi\)
\(938\) 0 0
\(939\) −11.4172 11.4172i −0.372585 0.372585i
\(940\) 0 0
\(941\) 15.4672 15.4672i 0.504217 0.504217i −0.408529 0.912745i \(-0.633958\pi\)
0.912745 + 0.408529i \(0.133958\pi\)
\(942\) 0 0
\(943\) 16.7499 0.545451
\(944\) 0 0
\(945\) 15.3871 0.500542
\(946\) 0 0
\(947\) −18.9435 + 18.9435i −0.615580 + 0.615580i −0.944395 0.328814i \(-0.893351\pi\)
0.328814 + 0.944395i \(0.393351\pi\)
\(948\) 0 0
\(949\) 25.2188 + 25.2188i 0.818638 + 0.818638i
\(950\) 0 0
\(951\) 6.77220i 0.219604i
\(952\) 0 0
\(953\) 20.2198i 0.654983i −0.944854 0.327491i \(-0.893797\pi\)
0.944854 0.327491i \(-0.106203\pi\)
\(954\) 0 0
\(955\) 14.5775 + 14.5775i 0.471716 + 0.471716i
\(956\) 0 0
\(957\) 3.36111 3.36111i 0.108649 0.108649i
\(958\) 0 0
\(959\) 15.6540 0.505493
\(960\) 0 0
\(961\) −29.4064 −0.948594
\(962\) 0 0
\(963\) −14.7192 + 14.7192i −0.474318 + 0.474318i
\(964\) 0 0
\(965\) 3.69236 + 3.69236i 0.118861 + 0.118861i
\(966\) 0 0
\(967\) 33.3933i 1.07385i 0.843628 + 0.536927i \(0.180415\pi\)
−0.843628 + 0.536927i \(0.819585\pi\)
\(968\) 0 0
\(969\) 1.87253i 0.0601545i
\(970\) 0 0
\(971\) −7.52202 7.52202i −0.241393 0.241393i 0.576033 0.817426i \(-0.304600\pi\)
−0.817426 + 0.576033i \(0.804600\pi\)
\(972\) 0 0
\(973\) 1.52569 1.52569i 0.0489115 0.0489115i
\(974\) 0 0
\(975\) −30.4290 −0.974508
\(976\) 0 0
\(977\) 28.2764 0.904641 0.452321 0.891855i \(-0.350596\pi\)
0.452321 + 0.891855i \(0.350596\pi\)
\(978\) 0 0
\(979\) 13.6326 13.6326i 0.435700 0.435700i
\(980\) 0 0
\(981\) 1.79012 + 1.79012i 0.0571541 + 0.0571541i
\(982\) 0 0
\(983\) 1.94376i 0.0619964i −0.999519 0.0309982i \(-0.990131\pi\)
0.999519 0.0309982i \(-0.00986862\pi\)
\(984\) 0 0
\(985\) 50.2173i 1.60006i
\(986\) 0 0
\(987\) −2.93185 2.93185i −0.0933219 0.0933219i
\(988\) 0 0
\(989\) −25.2932 + 25.2932i −0.804277 + 0.804277i
\(990\) 0 0
\(991\) −8.60542 −0.273360 −0.136680 0.990615i \(-0.543643\pi\)
−0.136680 + 0.990615i \(0.543643\pi\)
\(992\) 0 0
\(993\) −28.2221 −0.895600
\(994\) 0 0
\(995\) 59.8211 59.8211i 1.89646 1.89646i
\(996\) 0 0
\(997\) −7.78348 7.78348i −0.246505 0.246505i 0.573029 0.819535i \(-0.305768\pi\)
−0.819535 + 0.573029i \(0.805768\pi\)
\(998\) 0 0
\(999\) 42.1876i 1.33476i
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 896.2.m.g.225.3 12
4.3 odd 2 896.2.m.h.225.4 12
8.3 odd 2 448.2.m.d.113.3 12
8.5 even 2 112.2.m.d.85.6 yes 12
16.3 odd 4 896.2.m.h.673.4 12
16.5 even 4 112.2.m.d.29.6 12
16.11 odd 4 448.2.m.d.337.3 12
16.13 even 4 inner 896.2.m.g.673.3 12
32.3 odd 8 7168.2.a.bi.1.9 12
32.13 even 8 7168.2.a.bj.1.9 12
32.19 odd 8 7168.2.a.bi.1.4 12
32.29 even 8 7168.2.a.bj.1.4 12
56.5 odd 6 784.2.x.m.165.3 24
56.13 odd 2 784.2.m.h.197.6 12
56.37 even 6 784.2.x.l.165.3 24
56.45 odd 6 784.2.x.m.373.2 24
56.53 even 6 784.2.x.l.373.2 24
112.5 odd 12 784.2.x.m.557.2 24
112.37 even 12 784.2.x.l.557.2 24
112.53 even 12 784.2.x.l.765.3 24
112.69 odd 4 784.2.m.h.589.6 12
112.101 odd 12 784.2.x.m.765.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.d.29.6 12 16.5 even 4
112.2.m.d.85.6 yes 12 8.5 even 2
448.2.m.d.113.3 12 8.3 odd 2
448.2.m.d.337.3 12 16.11 odd 4
784.2.m.h.197.6 12 56.13 odd 2
784.2.m.h.589.6 12 112.69 odd 4
784.2.x.l.165.3 24 56.37 even 6
784.2.x.l.373.2 24 56.53 even 6
784.2.x.l.557.2 24 112.37 even 12
784.2.x.l.765.3 24 112.53 even 12
784.2.x.m.165.3 24 56.5 odd 6
784.2.x.m.373.2 24 56.45 odd 6
784.2.x.m.557.2 24 112.5 odd 12
784.2.x.m.765.3 24 112.101 odd 12
896.2.m.g.225.3 12 1.1 even 1 trivial
896.2.m.g.673.3 12 16.13 even 4 inner
896.2.m.h.225.4 12 4.3 odd 2
896.2.m.h.673.4 12 16.3 odd 4
7168.2.a.bi.1.4 12 32.19 odd 8
7168.2.a.bi.1.9 12 32.3 odd 8
7168.2.a.bj.1.4 12 32.29 even 8
7168.2.a.bj.1.9 12 32.13 even 8