Properties

Label 1280.3.g.e.1151.4
Level $1280$
Weight $3$
Character 1280.1151
Analytic conductor $34.877$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,3,Mod(1151,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1280.g (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: \(34.8774738381\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.4
Root \(0.587785 + 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1280.1151
Dual form 1280.3.g.e.1151.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.35114 q^{3} +2.23607i q^{5} -5.25731i q^{7} -3.47214 q^{9} +O(q^{10})\) \(q-2.35114 q^{3} +2.23607i q^{5} -5.25731i q^{7} -3.47214 q^{9} -19.9192 q^{11} -8.47214i q^{13} -5.25731i q^{15} +11.8885 q^{17} -15.2169 q^{19} +12.3607i q^{21} -0.555029i q^{23} -5.00000 q^{25} +29.3238 q^{27} -10.9443i q^{29} +8.29451i q^{31} +46.8328 q^{33} +11.7557 q^{35} +18.3607i q^{37} +19.9192i q^{39} +14.5836 q^{41} -22.2703 q^{43} -7.76393i q^{45} -53.3902i q^{47} +21.3607 q^{49} -27.9516 q^{51} +66.3607i q^{53} -44.5407i q^{55} +35.7771 q^{57} -17.4370 q^{59} +90.1378i q^{61} +18.2541i q^{63} +18.9443 q^{65} +50.2220 q^{67} +1.30495i q^{69} -80.7868i q^{71} +5.55418 q^{73} +11.7557 q^{75} +104.721i q^{77} +13.8448i q^{79} -37.6950 q^{81} -76.2155 q^{83} +26.5836i q^{85} +25.7315i q^{87} +111.443 q^{89} -44.5407 q^{91} -19.5016i q^{93} -34.0260i q^{95} -92.8328 q^{97} +69.1621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 48 q^{17} - 40 q^{25} + 160 q^{33} + 224 q^{41} - 8 q^{49} + 80 q^{65} - 528 q^{73} - 552 q^{81} + 176 q^{89} - 528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −2.35114 −0.783714 −0.391857 0.920026i \(-0.628167\pi\)
−0.391857 + 0.920026i \(0.628167\pi\)
\(4\) 0 0
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) − 5.25731i − 0.751044i −0.926813 0.375522i \(-0.877463\pi\)
0.926813 0.375522i \(-0.122537\pi\)
\(8\) 0 0
\(9\) −3.47214 −0.385793
\(10\) 0 0
\(11\) −19.9192 −1.81084 −0.905418 0.424522i \(-0.860442\pi\)
−0.905418 + 0.424522i \(0.860442\pi\)
\(12\) 0 0
\(13\) − 8.47214i − 0.651703i −0.945421 0.325851i \(-0.894349\pi\)
0.945421 0.325851i \(-0.105651\pi\)
\(14\) 0 0
\(15\) − 5.25731i − 0.350487i
\(16\) 0 0
\(17\) 11.8885 0.699326 0.349663 0.936876i \(-0.386296\pi\)
0.349663 + 0.936876i \(0.386296\pi\)
\(18\) 0 0
\(19\) −15.2169 −0.800890 −0.400445 0.916321i \(-0.631144\pi\)
−0.400445 + 0.916321i \(0.631144\pi\)
\(20\) 0 0
\(21\) 12.3607i 0.588604i
\(22\) 0 0
\(23\) − 0.555029i − 0.0241317i −0.999927 0.0120659i \(-0.996159\pi\)
0.999927 0.0120659i \(-0.00384077\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) 29.3238 1.08606
\(28\) 0 0
\(29\) − 10.9443i − 0.377389i −0.982036 0.188694i \(-0.939574\pi\)
0.982036 0.188694i \(-0.0604255\pi\)
\(30\) 0 0
\(31\) 8.29451i 0.267565i 0.991011 + 0.133782i \(0.0427123\pi\)
−0.991011 + 0.133782i \(0.957288\pi\)
\(32\) 0 0
\(33\) 46.8328 1.41918
\(34\) 0 0
\(35\) 11.7557 0.335877
\(36\) 0 0
\(37\) 18.3607i 0.496235i 0.968730 + 0.248117i \(0.0798118\pi\)
−0.968730 + 0.248117i \(0.920188\pi\)
\(38\) 0 0
\(39\) 19.9192i 0.510748i
\(40\) 0 0
\(41\) 14.5836 0.355697 0.177849 0.984058i \(-0.443086\pi\)
0.177849 + 0.984058i \(0.443086\pi\)
\(42\) 0 0
\(43\) −22.2703 −0.517915 −0.258957 0.965889i \(-0.583379\pi\)
−0.258957 + 0.965889i \(0.583379\pi\)
\(44\) 0 0
\(45\) − 7.76393i − 0.172532i
\(46\) 0 0
\(47\) − 53.3902i − 1.13596i −0.823042 0.567981i \(-0.807725\pi\)
0.823042 0.567981i \(-0.192275\pi\)
\(48\) 0 0
\(49\) 21.3607 0.435932
\(50\) 0 0
\(51\) −27.9516 −0.548071
\(52\) 0 0
\(53\) 66.3607i 1.25209i 0.779788 + 0.626044i \(0.215327\pi\)
−0.779788 + 0.626044i \(0.784673\pi\)
\(54\) 0 0
\(55\) − 44.5407i − 0.809830i
\(56\) 0 0
\(57\) 35.7771 0.627668
\(58\) 0 0
\(59\) −17.4370 −0.295543 −0.147771 0.989022i \(-0.547210\pi\)
−0.147771 + 0.989022i \(0.547210\pi\)
\(60\) 0 0
\(61\) 90.1378i 1.47767i 0.673887 + 0.738834i \(0.264623\pi\)
−0.673887 + 0.738834i \(0.735377\pi\)
\(62\) 0 0
\(63\) 18.2541i 0.289748i
\(64\) 0 0
\(65\) 18.9443 0.291450
\(66\) 0 0
\(67\) 50.2220 0.749582 0.374791 0.927109i \(-0.377715\pi\)
0.374791 + 0.927109i \(0.377715\pi\)
\(68\) 0 0
\(69\) 1.30495i 0.0189123i
\(70\) 0 0
\(71\) − 80.7868i − 1.13784i −0.822392 0.568921i \(-0.807361\pi\)
0.822392 0.568921i \(-0.192639\pi\)
\(72\) 0 0
\(73\) 5.55418 0.0760846 0.0380423 0.999276i \(-0.487888\pi\)
0.0380423 + 0.999276i \(0.487888\pi\)
\(74\) 0 0
\(75\) 11.7557 0.156743
\(76\) 0 0
\(77\) 104.721i 1.36002i
\(78\) 0 0
\(79\) 13.8448i 0.175251i 0.996154 + 0.0876253i \(0.0279278\pi\)
−0.996154 + 0.0876253i \(0.972072\pi\)
\(80\) 0 0
\(81\) −37.6950 −0.465371
\(82\) 0 0
\(83\) −76.2155 −0.918260 −0.459130 0.888369i \(-0.651839\pi\)
−0.459130 + 0.888369i \(0.651839\pi\)
\(84\) 0 0
\(85\) 26.5836i 0.312748i
\(86\) 0 0
\(87\) 25.7315i 0.295765i
\(88\) 0 0
\(89\) 111.443 1.25217 0.626083 0.779757i \(-0.284657\pi\)
0.626083 + 0.779757i \(0.284657\pi\)
\(90\) 0 0
\(91\) −44.5407 −0.489458
\(92\) 0 0
\(93\) − 19.5016i − 0.209694i
\(94\) 0 0
\(95\) − 34.0260i − 0.358169i
\(96\) 0 0
\(97\) −92.8328 −0.957039 −0.478520 0.878077i \(-0.658826\pi\)
−0.478520 + 0.878077i \(0.658826\pi\)
\(98\) 0 0
\(99\) 69.1621 0.698607
\(100\) 0 0
\(101\) − 64.1115i − 0.634767i −0.948297 0.317383i \(-0.897196\pi\)
0.948297 0.317383i \(-0.102804\pi\)
\(102\) 0 0
\(103\) 137.769i 1.33757i 0.743458 + 0.668783i \(0.233184\pi\)
−0.743458 + 0.668783i \(0.766816\pi\)
\(104\) 0 0
\(105\) −27.6393 −0.263232
\(106\) 0 0
\(107\) 51.3320 0.479739 0.239869 0.970805i \(-0.422895\pi\)
0.239869 + 0.970805i \(0.422895\pi\)
\(108\) 0 0
\(109\) 133.469i 1.22449i 0.790669 + 0.612243i \(0.209733\pi\)
−0.790669 + 0.612243i \(0.790267\pi\)
\(110\) 0 0
\(111\) − 43.1685i − 0.388906i
\(112\) 0 0
\(113\) 170.721 1.51081 0.755404 0.655259i \(-0.227441\pi\)
0.755404 + 0.655259i \(0.227441\pi\)
\(114\) 0 0
\(115\) 1.24108 0.0107920
\(116\) 0 0
\(117\) 29.4164i 0.251422i
\(118\) 0 0
\(119\) − 62.5018i − 0.525225i
\(120\) 0 0
\(121\) 275.774 2.27912
\(122\) 0 0
\(123\) −34.2881 −0.278765
\(124\) 0 0
\(125\) − 11.1803i − 0.0894427i
\(126\) 0 0
\(127\) 198.637i 1.56407i 0.623235 + 0.782035i \(0.285818\pi\)
−0.623235 + 0.782035i \(0.714182\pi\)
\(128\) 0 0
\(129\) 52.3607 0.405897
\(130\) 0 0
\(131\) 7.77041 0.0593161 0.0296580 0.999560i \(-0.490558\pi\)
0.0296580 + 0.999560i \(0.490558\pi\)
\(132\) 0 0
\(133\) 80.0000i 0.601504i
\(134\) 0 0
\(135\) 65.5699i 0.485703i
\(136\) 0 0
\(137\) −0.832816 −0.00607895 −0.00303947 0.999995i \(-0.500967\pi\)
−0.00303947 + 0.999995i \(0.500967\pi\)
\(138\) 0 0
\(139\) 237.658 1.70977 0.854885 0.518817i \(-0.173627\pi\)
0.854885 + 0.518817i \(0.173627\pi\)
\(140\) 0 0
\(141\) 125.528i 0.890269i
\(142\) 0 0
\(143\) 168.758i 1.18013i
\(144\) 0 0
\(145\) 24.4721 0.168773
\(146\) 0 0
\(147\) −50.2220 −0.341646
\(148\) 0 0
\(149\) 36.9706i 0.248125i 0.992274 + 0.124062i \(0.0395923\pi\)
−0.992274 + 0.124062i \(0.960408\pi\)
\(150\) 0 0
\(151\) 282.723i 1.87234i 0.351552 + 0.936168i \(0.385654\pi\)
−0.351552 + 0.936168i \(0.614346\pi\)
\(152\) 0 0
\(153\) −41.2786 −0.269795
\(154\) 0 0
\(155\) −18.5471 −0.119659
\(156\) 0 0
\(157\) 204.748i 1.30413i 0.758165 + 0.652063i \(0.226096\pi\)
−0.758165 + 0.652063i \(0.773904\pi\)
\(158\) 0 0
\(159\) − 156.023i − 0.981279i
\(160\) 0 0
\(161\) −2.91796 −0.0181240
\(162\) 0 0
\(163\) 107.235 0.657885 0.328943 0.944350i \(-0.393308\pi\)
0.328943 + 0.944350i \(0.393308\pi\)
\(164\) 0 0
\(165\) 104.721i 0.634675i
\(166\) 0 0
\(167\) 33.2090i 0.198856i 0.995045 + 0.0994280i \(0.0317013\pi\)
−0.995045 + 0.0994280i \(0.968299\pi\)
\(168\) 0 0
\(169\) 97.2229 0.575284
\(170\) 0 0
\(171\) 52.8352 0.308978
\(172\) 0 0
\(173\) − 226.361i − 1.30844i −0.756303 0.654222i \(-0.772996\pi\)
0.756303 0.654222i \(-0.227004\pi\)
\(174\) 0 0
\(175\) 26.2866i 0.150209i
\(176\) 0 0
\(177\) 40.9969 0.231621
\(178\) 0 0
\(179\) 224.337 1.25328 0.626641 0.779308i \(-0.284429\pi\)
0.626641 + 0.779308i \(0.284429\pi\)
\(180\) 0 0
\(181\) − 86.2229i − 0.476370i −0.971220 0.238185i \(-0.923448\pi\)
0.971220 0.238185i \(-0.0765524\pi\)
\(182\) 0 0
\(183\) − 211.927i − 1.15807i
\(184\) 0 0
\(185\) −41.0557 −0.221923
\(186\) 0 0
\(187\) −236.810 −1.26636
\(188\) 0 0
\(189\) − 154.164i − 0.815683i
\(190\) 0 0
\(191\) 31.0198i 0.162407i 0.996698 + 0.0812036i \(0.0258764\pi\)
−0.996698 + 0.0812036i \(0.974124\pi\)
\(192\) 0 0
\(193\) 110.223 0.571103 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(194\) 0 0
\(195\) −44.5407 −0.228414
\(196\) 0 0
\(197\) 172.525i 0.875760i 0.899033 + 0.437880i \(0.144271\pi\)
−0.899033 + 0.437880i \(0.855729\pi\)
\(198\) 0 0
\(199\) − 272.208i − 1.36788i −0.729538 0.683940i \(-0.760265\pi\)
0.729538 0.683940i \(-0.239735\pi\)
\(200\) 0 0
\(201\) −118.079 −0.587457
\(202\) 0 0
\(203\) −57.5374 −0.283436
\(204\) 0 0
\(205\) 32.6099i 0.159073i
\(206\) 0 0
\(207\) 1.92714i 0.00930984i
\(208\) 0 0
\(209\) 303.108 1.45028
\(210\) 0 0
\(211\) −205.266 −0.972826 −0.486413 0.873729i \(-0.661695\pi\)
−0.486413 + 0.873729i \(0.661695\pi\)
\(212\) 0 0
\(213\) 189.941i 0.891743i
\(214\) 0 0
\(215\) − 49.7980i − 0.231618i
\(216\) 0 0
\(217\) 43.6068 0.200953
\(218\) 0 0
\(219\) −13.0586 −0.0596285
\(220\) 0 0
\(221\) − 100.721i − 0.455753i
\(222\) 0 0
\(223\) − 235.731i − 1.05709i −0.848905 0.528545i \(-0.822738\pi\)
0.848905 0.528545i \(-0.177262\pi\)
\(224\) 0 0
\(225\) 17.3607 0.0771586
\(226\) 0 0
\(227\) −58.5165 −0.257782 −0.128891 0.991659i \(-0.541142\pi\)
−0.128891 + 0.991659i \(0.541142\pi\)
\(228\) 0 0
\(229\) − 162.721i − 0.710574i −0.934757 0.355287i \(-0.884383\pi\)
0.934757 0.355287i \(-0.115617\pi\)
\(230\) 0 0
\(231\) − 246.215i − 1.06586i
\(232\) 0 0
\(233\) −319.050 −1.36931 −0.684656 0.728867i \(-0.740047\pi\)
−0.684656 + 0.728867i \(0.740047\pi\)
\(234\) 0 0
\(235\) 119.384 0.508017
\(236\) 0 0
\(237\) − 32.5511i − 0.137346i
\(238\) 0 0
\(239\) − 236.810i − 0.990837i −0.868654 0.495419i \(-0.835015\pi\)
0.868654 0.495419i \(-0.164985\pi\)
\(240\) 0 0
\(241\) −0.917961 −0.00380897 −0.00190448 0.999998i \(-0.500606\pi\)
−0.00190448 + 0.999998i \(0.500606\pi\)
\(242\) 0 0
\(243\) −175.287 −0.721347
\(244\) 0 0
\(245\) 47.7639i 0.194955i
\(246\) 0 0
\(247\) 128.920i 0.521942i
\(248\) 0 0
\(249\) 179.193 0.719653
\(250\) 0 0
\(251\) 136.690 0.544582 0.272291 0.962215i \(-0.412219\pi\)
0.272291 + 0.962215i \(0.412219\pi\)
\(252\) 0 0
\(253\) 11.0557i 0.0436985i
\(254\) 0 0
\(255\) − 62.5018i − 0.245105i
\(256\) 0 0
\(257\) −274.944 −1.06982 −0.534911 0.844908i \(-0.679655\pi\)
−0.534911 + 0.844908i \(0.679655\pi\)
\(258\) 0 0
\(259\) 96.5278 0.372694
\(260\) 0 0
\(261\) 38.0000i 0.145594i
\(262\) 0 0
\(263\) 406.385i 1.54519i 0.634899 + 0.772596i \(0.281042\pi\)
−0.634899 + 0.772596i \(0.718958\pi\)
\(264\) 0 0
\(265\) −148.387 −0.559951
\(266\) 0 0
\(267\) −262.018 −0.981339
\(268\) 0 0
\(269\) − 348.525i − 1.29563i −0.761797 0.647816i \(-0.775683\pi\)
0.761797 0.647816i \(-0.224317\pi\)
\(270\) 0 0
\(271\) 247.849i 0.914571i 0.889320 + 0.457286i \(0.151178\pi\)
−0.889320 + 0.457286i \(0.848822\pi\)
\(272\) 0 0
\(273\) 104.721 0.383595
\(274\) 0 0
\(275\) 99.5959 0.362167
\(276\) 0 0
\(277\) 54.7539i 0.197667i 0.995104 + 0.0988337i \(0.0315112\pi\)
−0.995104 + 0.0988337i \(0.968489\pi\)
\(278\) 0 0
\(279\) − 28.7997i − 0.103225i
\(280\) 0 0
\(281\) 50.3607 0.179220 0.0896098 0.995977i \(-0.471438\pi\)
0.0896098 + 0.995977i \(0.471438\pi\)
\(282\) 0 0
\(283\) 147.336 0.520621 0.260310 0.965525i \(-0.416175\pi\)
0.260310 + 0.965525i \(0.416175\pi\)
\(284\) 0 0
\(285\) 80.0000i 0.280702i
\(286\) 0 0
\(287\) − 76.6705i − 0.267145i
\(288\) 0 0
\(289\) −147.663 −0.510943
\(290\) 0 0
\(291\) 218.263 0.750045
\(292\) 0 0
\(293\) − 178.859i − 0.610441i −0.952282 0.305220i \(-0.901270\pi\)
0.952282 0.305220i \(-0.0987301\pi\)
\(294\) 0 0
\(295\) − 38.9904i − 0.132171i
\(296\) 0 0
\(297\) −584.105 −1.96668
\(298\) 0 0
\(299\) −4.70228 −0.0157267
\(300\) 0 0
\(301\) 117.082i 0.388977i
\(302\) 0 0
\(303\) 150.735i 0.497475i
\(304\) 0 0
\(305\) −201.554 −0.660833
\(306\) 0 0
\(307\) −284.550 −0.926873 −0.463436 0.886130i \(-0.653384\pi\)
−0.463436 + 0.886130i \(0.653384\pi\)
\(308\) 0 0
\(309\) − 323.915i − 1.04827i
\(310\) 0 0
\(311\) 282.199i 0.907392i 0.891157 + 0.453696i \(0.149895\pi\)
−0.891157 + 0.453696i \(0.850105\pi\)
\(312\) 0 0
\(313\) 567.548 1.81325 0.906626 0.421935i \(-0.138649\pi\)
0.906626 + 0.421935i \(0.138649\pi\)
\(314\) 0 0
\(315\) −40.8174 −0.129579
\(316\) 0 0
\(317\) 161.141i 0.508331i 0.967161 + 0.254165i \(0.0818008\pi\)
−0.967161 + 0.254165i \(0.918199\pi\)
\(318\) 0 0
\(319\) 218.001i 0.683389i
\(320\) 0 0
\(321\) −120.689 −0.375978
\(322\) 0 0
\(323\) −180.907 −0.560083
\(324\) 0 0
\(325\) 42.3607i 0.130341i
\(326\) 0 0
\(327\) − 313.805i − 0.959647i
\(328\) 0 0
\(329\) −280.689 −0.853158
\(330\) 0 0
\(331\) −331.966 −1.00292 −0.501459 0.865181i \(-0.667203\pi\)
−0.501459 + 0.865181i \(0.667203\pi\)
\(332\) 0 0
\(333\) − 63.7508i − 0.191444i
\(334\) 0 0
\(335\) 112.300i 0.335223i
\(336\) 0 0
\(337\) −269.108 −0.798541 −0.399271 0.916833i \(-0.630737\pi\)
−0.399271 + 0.916833i \(0.630737\pi\)
\(338\) 0 0
\(339\) −401.390 −1.18404
\(340\) 0 0
\(341\) − 165.220i − 0.484516i
\(342\) 0 0
\(343\) − 369.908i − 1.07845i
\(344\) 0 0
\(345\) −2.91796 −0.00845786
\(346\) 0 0
\(347\) 503.075 1.44978 0.724892 0.688863i \(-0.241890\pi\)
0.724892 + 0.688863i \(0.241890\pi\)
\(348\) 0 0
\(349\) − 0.504658i − 0.00144601i −1.00000 0.000723006i \(-0.999770\pi\)
1.00000 0.000723006i \(-0.000230140\pi\)
\(350\) 0 0
\(351\) − 248.435i − 0.707791i
\(352\) 0 0
\(353\) −335.994 −0.951824 −0.475912 0.879493i \(-0.657882\pi\)
−0.475912 + 0.879493i \(0.657882\pi\)
\(354\) 0 0
\(355\) 180.645 0.508859
\(356\) 0 0
\(357\) 146.950i 0.411626i
\(358\) 0 0
\(359\) 98.4859i 0.274334i 0.990548 + 0.137167i \(0.0437997\pi\)
−0.990548 + 0.137167i \(0.956200\pi\)
\(360\) 0 0
\(361\) −129.446 −0.358576
\(362\) 0 0
\(363\) −648.384 −1.78618
\(364\) 0 0
\(365\) 12.4195i 0.0340261i
\(366\) 0 0
\(367\) − 498.473i − 1.35824i −0.734029 0.679118i \(-0.762362\pi\)
0.734029 0.679118i \(-0.237638\pi\)
\(368\) 0 0
\(369\) −50.6362 −0.137226
\(370\) 0 0
\(371\) 348.879 0.940374
\(372\) 0 0
\(373\) − 600.354i − 1.60953i −0.593594 0.804765i \(-0.702291\pi\)
0.593594 0.804765i \(-0.297709\pi\)
\(374\) 0 0
\(375\) 26.2866i 0.0700975i
\(376\) 0 0
\(377\) −92.7214 −0.245945
\(378\) 0 0
\(379\) −303.490 −0.800765 −0.400383 0.916348i \(-0.631123\pi\)
−0.400383 + 0.916348i \(0.631123\pi\)
\(380\) 0 0
\(381\) − 467.023i − 1.22578i
\(382\) 0 0
\(383\) 332.583i 0.868362i 0.900826 + 0.434181i \(0.142962\pi\)
−0.900826 + 0.434181i \(0.857038\pi\)
\(384\) 0 0
\(385\) −234.164 −0.608218
\(386\) 0 0
\(387\) 77.3256 0.199808
\(388\) 0 0
\(389\) − 392.354i − 1.00862i −0.863522 0.504312i \(-0.831746\pi\)
0.863522 0.504312i \(-0.168254\pi\)
\(390\) 0 0
\(391\) − 6.59849i − 0.0168759i
\(392\) 0 0
\(393\) −18.2693 −0.0464868
\(394\) 0 0
\(395\) −30.9579 −0.0783744
\(396\) 0 0
\(397\) 334.190i 0.841789i 0.907110 + 0.420895i \(0.138284\pi\)
−0.907110 + 0.420895i \(0.861716\pi\)
\(398\) 0 0
\(399\) − 188.091i − 0.471407i
\(400\) 0 0
\(401\) 121.003 0.301753 0.150877 0.988553i \(-0.451790\pi\)
0.150877 + 0.988553i \(0.451790\pi\)
\(402\) 0 0
\(403\) 70.2722 0.174373
\(404\) 0 0
\(405\) − 84.2887i − 0.208120i
\(406\) 0 0
\(407\) − 365.730i − 0.898599i
\(408\) 0 0
\(409\) 607.410 1.48511 0.742555 0.669785i \(-0.233614\pi\)
0.742555 + 0.669785i \(0.233614\pi\)
\(410\) 0 0
\(411\) 1.95807 0.00476415
\(412\) 0 0
\(413\) 91.6718i 0.221966i
\(414\) 0 0
\(415\) − 170.423i − 0.410658i
\(416\) 0 0
\(417\) −558.768 −1.33997
\(418\) 0 0
\(419\) −466.760 −1.11398 −0.556992 0.830518i \(-0.688045\pi\)
−0.556992 + 0.830518i \(0.688045\pi\)
\(420\) 0 0
\(421\) 73.0883i 0.173606i 0.996225 + 0.0868031i \(0.0276651\pi\)
−0.996225 + 0.0868031i \(0.972335\pi\)
\(422\) 0 0
\(423\) 185.378i 0.438246i
\(424\) 0 0
\(425\) −59.4427 −0.139865
\(426\) 0 0
\(427\) 473.882 1.10979
\(428\) 0 0
\(429\) − 396.774i − 0.924881i
\(430\) 0 0
\(431\) 463.630i 1.07571i 0.843038 + 0.537853i \(0.180765\pi\)
−0.843038 + 0.537853i \(0.819235\pi\)
\(432\) 0 0
\(433\) −99.8359 −0.230568 −0.115284 0.993333i \(-0.536778\pi\)
−0.115284 + 0.993333i \(0.536778\pi\)
\(434\) 0 0
\(435\) −57.5374 −0.132270
\(436\) 0 0
\(437\) 8.44582i 0.0193268i
\(438\) 0 0
\(439\) 374.086i 0.852133i 0.904692 + 0.426066i \(0.140101\pi\)
−0.904692 + 0.426066i \(0.859899\pi\)
\(440\) 0 0
\(441\) −74.1672 −0.168180
\(442\) 0 0
\(443\) 290.100 0.654854 0.327427 0.944877i \(-0.393818\pi\)
0.327427 + 0.944877i \(0.393818\pi\)
\(444\) 0 0
\(445\) 249.193i 0.559985i
\(446\) 0 0
\(447\) − 86.9231i − 0.194459i
\(448\) 0 0
\(449\) 299.921 0.667976 0.333988 0.942577i \(-0.391606\pi\)
0.333988 + 0.942577i \(0.391606\pi\)
\(450\) 0 0
\(451\) −290.493 −0.644109
\(452\) 0 0
\(453\) − 664.721i − 1.46738i
\(454\) 0 0
\(455\) − 99.5959i − 0.218892i
\(456\) 0 0
\(457\) −822.328 −1.79941 −0.899703 0.436503i \(-0.856217\pi\)
−0.899703 + 0.436503i \(0.856217\pi\)
\(458\) 0 0
\(459\) 348.617 0.759513
\(460\) 0 0
\(461\) − 456.885i − 0.991075i −0.868587 0.495537i \(-0.834971\pi\)
0.868587 0.495537i \(-0.165029\pi\)
\(462\) 0 0
\(463\) − 400.249i − 0.864469i −0.901761 0.432234i \(-0.857725\pi\)
0.901761 0.432234i \(-0.142275\pi\)
\(464\) 0 0
\(465\) 43.6068 0.0937781
\(466\) 0 0
\(467\) 913.145 1.95534 0.977672 0.210139i \(-0.0673915\pi\)
0.977672 + 0.210139i \(0.0673915\pi\)
\(468\) 0 0
\(469\) − 264.033i − 0.562969i
\(470\) 0 0
\(471\) − 481.391i − 1.02206i
\(472\) 0 0
\(473\) 443.607 0.937858
\(474\) 0 0
\(475\) 76.0845 0.160178
\(476\) 0 0
\(477\) − 230.413i − 0.483047i
\(478\) 0 0
\(479\) 526.131i 1.09840i 0.835692 + 0.549198i \(0.185067\pi\)
−0.835692 + 0.549198i \(0.814933\pi\)
\(480\) 0 0
\(481\) 155.554 0.323397
\(482\) 0 0
\(483\) 6.86054 0.0142040
\(484\) 0 0
\(485\) − 207.580i − 0.428001i
\(486\) 0 0
\(487\) 443.541i 0.910762i 0.890297 + 0.455381i \(0.150497\pi\)
−0.890297 + 0.455381i \(0.849503\pi\)
\(488\) 0 0
\(489\) −252.125 −0.515594
\(490\) 0 0
\(491\) 287.163 0.584854 0.292427 0.956288i \(-0.405537\pi\)
0.292427 + 0.956288i \(0.405537\pi\)
\(492\) 0 0
\(493\) − 130.111i − 0.263918i
\(494\) 0 0
\(495\) 154.651i 0.312427i
\(496\) 0 0
\(497\) −424.721 −0.854570
\(498\) 0 0
\(499\) 810.936 1.62512 0.812561 0.582876i \(-0.198073\pi\)
0.812561 + 0.582876i \(0.198073\pi\)
\(500\) 0 0
\(501\) − 78.0789i − 0.155846i
\(502\) 0 0
\(503\) 642.471i 1.27728i 0.769506 + 0.638639i \(0.220502\pi\)
−0.769506 + 0.638639i \(0.779498\pi\)
\(504\) 0 0
\(505\) 143.358 0.283876
\(506\) 0 0
\(507\) −228.585 −0.450858
\(508\) 0 0
\(509\) − 915.050i − 1.79774i −0.438216 0.898870i \(-0.644389\pi\)
0.438216 0.898870i \(-0.355611\pi\)
\(510\) 0 0
\(511\) − 29.2000i − 0.0571429i
\(512\) 0 0
\(513\) −446.217 −0.869818
\(514\) 0 0
\(515\) −308.061 −0.598177
\(516\) 0 0
\(517\) 1063.49i 2.05704i
\(518\) 0 0
\(519\) 532.206i 1.02544i
\(520\) 0 0
\(521\) −1006.98 −1.93279 −0.966396 0.257058i \(-0.917247\pi\)
−0.966396 + 0.257058i \(0.917247\pi\)
\(522\) 0 0
\(523\) 774.173 1.48025 0.740127 0.672467i \(-0.234765\pi\)
0.740127 + 0.672467i \(0.234765\pi\)
\(524\) 0 0
\(525\) − 61.8034i − 0.117721i
\(526\) 0 0
\(527\) 98.6096i 0.187115i
\(528\) 0 0
\(529\) 528.692 0.999418
\(530\) 0 0
\(531\) 60.5437 0.114018
\(532\) 0 0
\(533\) − 123.554i − 0.231809i
\(534\) 0 0
\(535\) 114.782i 0.214546i
\(536\) 0 0
\(537\) −527.449 −0.982214
\(538\) 0 0
\(539\) −425.487 −0.789401
\(540\) 0 0
\(541\) − 259.115i − 0.478955i −0.970902 0.239477i \(-0.923024\pi\)
0.970902 0.239477i \(-0.0769761\pi\)
\(542\) 0 0
\(543\) 202.722i 0.373337i
\(544\) 0 0
\(545\) −298.446 −0.547607
\(546\) 0 0
\(547\) −149.818 −0.273890 −0.136945 0.990579i \(-0.543728\pi\)
−0.136945 + 0.990579i \(0.543728\pi\)
\(548\) 0 0
\(549\) − 312.971i − 0.570074i
\(550\) 0 0
\(551\) 166.538i 0.302247i
\(552\) 0 0
\(553\) 72.7864 0.131621
\(554\) 0 0
\(555\) 96.5278 0.173924
\(556\) 0 0
\(557\) 511.698i 0.918668i 0.888264 + 0.459334i \(0.151912\pi\)
−0.888264 + 0.459334i \(0.848088\pi\)
\(558\) 0 0
\(559\) 188.677i 0.337526i
\(560\) 0 0
\(561\) 556.774 0.992467
\(562\) 0 0
\(563\) 490.726 0.871627 0.435814 0.900037i \(-0.356461\pi\)
0.435814 + 0.900037i \(0.356461\pi\)
\(564\) 0 0
\(565\) 381.745i 0.675654i
\(566\) 0 0
\(567\) 198.175i 0.349514i
\(568\) 0 0
\(569\) 232.748 0.409047 0.204523 0.978862i \(-0.434436\pi\)
0.204523 + 0.978862i \(0.434436\pi\)
\(570\) 0 0
\(571\) 210.755 0.369098 0.184549 0.982823i \(-0.440918\pi\)
0.184549 + 0.982823i \(0.440918\pi\)
\(572\) 0 0
\(573\) − 72.9318i − 0.127281i
\(574\) 0 0
\(575\) 2.77515i 0.00482634i
\(576\) 0 0
\(577\) 341.712 0.592222 0.296111 0.955154i \(-0.404310\pi\)
0.296111 + 0.955154i \(0.404310\pi\)
\(578\) 0 0
\(579\) −259.150 −0.447581
\(580\) 0 0
\(581\) 400.689i 0.689654i
\(582\) 0 0
\(583\) − 1321.85i − 2.26733i
\(584\) 0 0
\(585\) −65.7771 −0.112439
\(586\) 0 0
\(587\) −618.412 −1.05351 −0.526756 0.850016i \(-0.676592\pi\)
−0.526756 + 0.850016i \(0.676592\pi\)
\(588\) 0 0
\(589\) − 126.217i − 0.214290i
\(590\) 0 0
\(591\) − 405.630i − 0.686345i
\(592\) 0 0
\(593\) 120.663 0.203478 0.101739 0.994811i \(-0.467559\pi\)
0.101739 + 0.994811i \(0.467559\pi\)
\(594\) 0 0
\(595\) 139.758 0.234888
\(596\) 0 0
\(597\) 640.000i 1.07203i
\(598\) 0 0
\(599\) − 849.927i − 1.41891i −0.704751 0.709455i \(-0.748941\pi\)
0.704751 0.709455i \(-0.251059\pi\)
\(600\) 0 0
\(601\) 11.3576 0.0188978 0.00944890 0.999955i \(-0.496992\pi\)
0.00944890 + 0.999955i \(0.496992\pi\)
\(602\) 0 0
\(603\) −174.378 −0.289183
\(604\) 0 0
\(605\) 616.649i 1.01926i
\(606\) 0 0
\(607\) 1115.12i 1.83710i 0.395305 + 0.918550i \(0.370639\pi\)
−0.395305 + 0.918550i \(0.629361\pi\)
\(608\) 0 0
\(609\) 135.279 0.222132
\(610\) 0 0
\(611\) −452.329 −0.740309
\(612\) 0 0
\(613\) − 499.475i − 0.814805i −0.913249 0.407402i \(-0.866435\pi\)
0.913249 0.407402i \(-0.133565\pi\)
\(614\) 0 0
\(615\) − 76.6705i − 0.124667i
\(616\) 0 0
\(617\) −545.935 −0.884822 −0.442411 0.896813i \(-0.645877\pi\)
−0.442411 + 0.896813i \(0.645877\pi\)
\(618\) 0 0
\(619\) −455.011 −0.735075 −0.367537 0.930009i \(-0.619799\pi\)
−0.367537 + 0.930009i \(0.619799\pi\)
\(620\) 0 0
\(621\) − 16.2755i − 0.0262086i
\(622\) 0 0
\(623\) − 585.889i − 0.940432i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −712.650 −1.13660
\(628\) 0 0
\(629\) 218.282i 0.347030i
\(630\) 0 0
\(631\) 267.706i 0.424257i 0.977242 + 0.212128i \(0.0680395\pi\)
−0.977242 + 0.212128i \(0.931960\pi\)
\(632\) 0 0
\(633\) 482.610 0.762417
\(634\) 0 0
\(635\) −444.165 −0.699473
\(636\) 0 0
\(637\) − 180.971i − 0.284098i
\(638\) 0 0
\(639\) 280.503i 0.438971i
\(640\) 0 0
\(641\) −418.571 −0.652997 −0.326499 0.945198i \(-0.605869\pi\)
−0.326499 + 0.945198i \(0.605869\pi\)
\(642\) 0 0
\(643\) 439.339 0.683265 0.341633 0.939834i \(-0.389020\pi\)
0.341633 + 0.939834i \(0.389020\pi\)
\(644\) 0 0
\(645\) 117.082i 0.181523i
\(646\) 0 0
\(647\) 419.644i 0.648600i 0.945954 + 0.324300i \(0.105129\pi\)
−0.945954 + 0.324300i \(0.894871\pi\)
\(648\) 0 0
\(649\) 347.331 0.535179
\(650\) 0 0
\(651\) −102.526 −0.157490
\(652\) 0 0
\(653\) − 370.085i − 0.566746i −0.959010 0.283373i \(-0.908547\pi\)
0.959010 0.283373i \(-0.0914534\pi\)
\(654\) 0 0
\(655\) 17.3752i 0.0265270i
\(656\) 0 0
\(657\) −19.2849 −0.0293529
\(658\) 0 0
\(659\) −322.823 −0.489868 −0.244934 0.969540i \(-0.578766\pi\)
−0.244934 + 0.969540i \(0.578766\pi\)
\(660\) 0 0
\(661\) 812.735i 1.22955i 0.788701 + 0.614777i \(0.210754\pi\)
−0.788701 + 0.614777i \(0.789246\pi\)
\(662\) 0 0
\(663\) 236.810i 0.357180i
\(664\) 0 0
\(665\) −178.885 −0.269001
\(666\) 0 0
\(667\) −6.07439 −0.00910703
\(668\) 0 0
\(669\) 554.237i 0.828456i
\(670\) 0 0
\(671\) − 1795.47i − 2.67581i
\(672\) 0 0
\(673\) 467.378 0.694469 0.347235 0.937778i \(-0.387121\pi\)
0.347235 + 0.937778i \(0.387121\pi\)
\(674\) 0 0
\(675\) −146.619 −0.217213
\(676\) 0 0
\(677\) − 548.237i − 0.809803i −0.914360 0.404902i \(-0.867306\pi\)
0.914360 0.404902i \(-0.132694\pi\)
\(678\) 0 0
\(679\) 488.051i 0.718779i
\(680\) 0 0
\(681\) 137.580 0.202027
\(682\) 0 0
\(683\) −23.9663 −0.0350898 −0.0175449 0.999846i \(-0.505585\pi\)
−0.0175449 + 0.999846i \(0.505585\pi\)
\(684\) 0 0
\(685\) − 1.86223i − 0.00271859i
\(686\) 0 0
\(687\) 382.581i 0.556886i
\(688\) 0 0
\(689\) 562.217 0.815989
\(690\) 0 0
\(691\) −186.981 −0.270595 −0.135298 0.990805i \(-0.543199\pi\)
−0.135298 + 0.990805i \(0.543199\pi\)
\(692\) 0 0
\(693\) − 363.607i − 0.524685i
\(694\) 0 0
\(695\) 531.420i 0.764633i
\(696\) 0 0
\(697\) 173.378 0.248748
\(698\) 0 0
\(699\) 750.130 1.07315
\(700\) 0 0
\(701\) − 706.636i − 1.00804i −0.863692 0.504020i \(-0.831854\pi\)
0.863692 0.504020i \(-0.168146\pi\)
\(702\) 0 0
\(703\) − 279.393i − 0.397429i
\(704\) 0 0
\(705\) −280.689 −0.398140
\(706\) 0 0
\(707\) −337.054 −0.476738
\(708\) 0 0
\(709\) 188.597i 0.266005i 0.991116 + 0.133002i \(0.0424618\pi\)
−0.991116 + 0.133002i \(0.957538\pi\)
\(710\) 0 0
\(711\) − 48.0710i − 0.0676104i
\(712\) 0 0
\(713\) 4.60369 0.00645679
\(714\) 0 0
\(715\) −377.354 −0.527769
\(716\) 0 0
\(717\) 556.774i 0.776533i
\(718\) 0 0
\(719\) − 156.085i − 0.217086i −0.994092 0.108543i \(-0.965381\pi\)
0.994092 0.108543i \(-0.0346186\pi\)
\(720\) 0 0
\(721\) 724.296 1.00457
\(722\) 0 0
\(723\) 2.15825 0.00298514
\(724\) 0 0
\(725\) 54.7214i 0.0754777i
\(726\) 0 0
\(727\) − 715.164i − 0.983719i −0.870675 0.491859i \(-0.836317\pi\)
0.870675 0.491859i \(-0.163683\pi\)
\(728\) 0 0
\(729\) 751.381 1.03070
\(730\) 0 0
\(731\) −264.762 −0.362191
\(732\) 0 0
\(733\) 1233.29i 1.68252i 0.540632 + 0.841259i \(0.318185\pi\)
−0.540632 + 0.841259i \(0.681815\pi\)
\(734\) 0 0
\(735\) − 112.300i − 0.152789i
\(736\) 0 0
\(737\) −1000.38 −1.35737
\(738\) 0 0
\(739\) 8.55656 0.0115786 0.00578928 0.999983i \(-0.498157\pi\)
0.00578928 + 0.999983i \(0.498157\pi\)
\(740\) 0 0
\(741\) − 303.108i − 0.409053i
\(742\) 0 0
\(743\) − 1010.56i − 1.36011i −0.733163 0.680053i \(-0.761957\pi\)
0.733163 0.680053i \(-0.238043\pi\)
\(744\) 0 0
\(745\) −82.6687 −0.110965
\(746\) 0 0
\(747\) 264.631 0.354258
\(748\) 0 0
\(749\) − 269.868i − 0.360305i
\(750\) 0 0
\(751\) 1104.31i 1.47046i 0.677820 + 0.735228i \(0.262925\pi\)
−0.677820 + 0.735228i \(0.737075\pi\)
\(752\) 0 0
\(753\) −321.378 −0.426796
\(754\) 0 0
\(755\) −632.188 −0.837335
\(756\) 0 0
\(757\) 875.633i 1.15671i 0.815783 + 0.578357i \(0.196306\pi\)
−0.815783 + 0.578357i \(0.803694\pi\)
\(758\) 0 0
\(759\) − 25.9936i − 0.0342471i
\(760\) 0 0
\(761\) 647.207 0.850470 0.425235 0.905083i \(-0.360192\pi\)
0.425235 + 0.905083i \(0.360192\pi\)
\(762\) 0 0
\(763\) 701.688 0.919644
\(764\) 0 0
\(765\) − 92.3018i − 0.120656i
\(766\) 0 0
\(767\) 147.729i 0.192606i
\(768\) 0 0
\(769\) 631.430 0.821106 0.410553 0.911837i \(-0.365336\pi\)
0.410553 + 0.911837i \(0.365336\pi\)
\(770\) 0 0
\(771\) 646.433 0.838434
\(772\) 0 0
\(773\) 421.522i 0.545306i 0.962112 + 0.272653i \(0.0879011\pi\)
−0.962112 + 0.272653i \(0.912099\pi\)
\(774\) 0 0
\(775\) − 41.4725i − 0.0535129i
\(776\) 0 0
\(777\) −226.950 −0.292086
\(778\) 0 0
\(779\) −221.917 −0.284874
\(780\) 0 0
\(781\) 1609.21i 2.06044i
\(782\) 0 0
\(783\) − 320.927i − 0.409869i
\(784\) 0 0
\(785\) −457.830 −0.583223
\(786\) 0 0
\(787\) 838.633 1.06561 0.532804 0.846239i \(-0.321138\pi\)
0.532804 + 0.846239i \(0.321138\pi\)
\(788\) 0 0
\(789\) − 955.469i − 1.21099i
\(790\) 0 0
\(791\) − 897.535i − 1.13468i
\(792\) 0 0
\(793\) 763.659 0.963001
\(794\) 0 0
\(795\) 348.879 0.438841
\(796\) 0 0
\(797\) 1213.57i 1.52268i 0.648354 + 0.761339i \(0.275458\pi\)
−0.648354 + 0.761339i \(0.724542\pi\)
\(798\) 0 0
\(799\) − 634.732i − 0.794408i
\(800\) 0 0
\(801\) −386.944 −0.483076
\(802\) 0 0
\(803\) −110.635 −0.137777
\(804\) 0 0
\(805\) − 6.52476i − 0.00810529i
\(806\) 0 0
\(807\) 819.431i 1.01540i
\(808\) 0 0
\(809\) 229.214 0.283330 0.141665 0.989915i \(-0.454754\pi\)
0.141665 + 0.989915i \(0.454754\pi\)
\(810\) 0 0
\(811\) −454.225 −0.560080 −0.280040 0.959988i \(-0.590348\pi\)
−0.280040 + 0.959988i \(0.590348\pi\)
\(812\) 0 0
\(813\) − 582.728i − 0.716762i
\(814\) 0 0
\(815\) 239.785i 0.294215i
\(816\) 0 0
\(817\) 338.885 0.414792
\(818\) 0 0
\(819\) 154.651 0.188829
\(820\) 0 0
\(821\) − 1130.90i − 1.37747i −0.725015 0.688733i \(-0.758167\pi\)
0.725015 0.688733i \(-0.241833\pi\)
\(822\) 0 0
\(823\) 780.148i 0.947931i 0.880543 + 0.473966i \(0.157178\pi\)
−0.880543 + 0.473966i \(0.842822\pi\)
\(824\) 0 0
\(825\) −234.164 −0.283835
\(826\) 0 0
\(827\) −209.175 −0.252932 −0.126466 0.991971i \(-0.540364\pi\)
−0.126466 + 0.991971i \(0.540364\pi\)
\(828\) 0 0
\(829\) − 508.525i − 0.613419i −0.951803 0.306710i \(-0.900772\pi\)
0.951803 0.306710i \(-0.0992281\pi\)
\(830\) 0 0
\(831\) − 128.734i − 0.154915i
\(832\) 0 0
\(833\) 253.947 0.304859
\(834\) 0 0
\(835\) −74.2575 −0.0889311
\(836\) 0 0
\(837\) 243.226i 0.290593i
\(838\) 0 0
\(839\) − 274.028i − 0.326613i −0.986575 0.163306i \(-0.947784\pi\)
0.986575 0.163306i \(-0.0522159\pi\)
\(840\) 0 0
\(841\) 721.223 0.857578
\(842\) 0 0
\(843\) −118.405 −0.140457
\(844\) 0 0
\(845\) 217.397i 0.257275i
\(846\) 0 0
\(847\) − 1449.83i − 1.71172i
\(848\) 0 0
\(849\) −346.407 −0.408018
\(850\) 0 0
\(851\) 10.1907 0.0119750
\(852\) 0 0
\(853\) 1583.28i 1.85613i 0.372416 + 0.928066i \(0.378529\pi\)
−0.372416 + 0.928066i \(0.621471\pi\)
\(854\) 0 0
\(855\) 118.143i 0.138179i
\(856\) 0 0
\(857\) 1007.38 1.17547 0.587735 0.809054i \(-0.300020\pi\)
0.587735 + 0.809054i \(0.300020\pi\)
\(858\) 0 0
\(859\) 76.6086 0.0891835 0.0445917 0.999005i \(-0.485801\pi\)
0.0445917 + 0.999005i \(0.485801\pi\)
\(860\) 0 0
\(861\) 180.263i 0.209365i
\(862\) 0 0
\(863\) 255.450i 0.296002i 0.988987 + 0.148001i \(0.0472839\pi\)
−0.988987 + 0.148001i \(0.952716\pi\)
\(864\) 0 0
\(865\) 506.158 0.585154
\(866\) 0 0
\(867\) 347.175 0.400433
\(868\) 0 0
\(869\) − 275.777i − 0.317350i
\(870\) 0 0
\(871\) − 425.487i − 0.488504i
\(872\) 0 0
\(873\) 322.328 0.369219
\(874\) 0 0
\(875\) −58.7785 −0.0671755
\(876\) 0 0
\(877\) 601.522i 0.685886i 0.939356 + 0.342943i \(0.111424\pi\)
−0.939356 + 0.342943i \(0.888576\pi\)
\(878\) 0 0
\(879\) 420.523i 0.478411i
\(880\) 0 0
\(881\) 237.850 0.269977 0.134989 0.990847i \(-0.456900\pi\)
0.134989 + 0.990847i \(0.456900\pi\)
\(882\) 0 0
\(883\) −1.30294 −0.00147559 −0.000737794 1.00000i \(-0.500235\pi\)
−0.000737794 1.00000i \(0.500235\pi\)
\(884\) 0 0
\(885\) 91.6718i 0.103584i
\(886\) 0 0
\(887\) − 536.353i − 0.604682i −0.953200 0.302341i \(-0.902232\pi\)
0.953200 0.302341i \(-0.0977682\pi\)
\(888\) 0 0
\(889\) 1044.30 1.17469
\(890\) 0 0
\(891\) 750.855 0.842710
\(892\) 0 0
\(893\) 812.433i 0.909780i
\(894\) 0 0
\(895\) 501.634i 0.560485i
\(896\) 0 0
\(897\) 11.0557 0.0123252
\(898\) 0 0
\(899\) 90.7773 0.100976
\(900\) 0 0
\(901\) 788.932i 0.875618i
\(902\) 0 0
\(903\) − 275.276i − 0.304846i
\(904\) 0 0
\(905\) 192.800 0.213039
\(906\) 0 0
\(907\) −332.159 −0.366217 −0.183108 0.983093i \(-0.558616\pi\)
−0.183108 + 0.983093i \(0.558616\pi\)
\(908\) 0 0
\(909\) 222.604i 0.244889i
\(910\) 0 0
\(911\) 1450.06i 1.59172i 0.605478 + 0.795862i \(0.292982\pi\)
−0.605478 + 0.795862i \(0.707018\pi\)
\(912\) 0 0
\(913\) 1518.15 1.66282
\(914\) 0 0
\(915\) 473.882 0.517904
\(916\) 0 0
\(917\) − 40.8514i − 0.0445490i
\(918\) 0 0
\(919\) 814.405i 0.886186i 0.896476 + 0.443093i \(0.146119\pi\)
−0.896476 + 0.443093i \(0.853881\pi\)
\(920\) 0 0
\(921\) 669.017 0.726403
\(922\) 0 0
\(923\) −684.437 −0.741535
\(924\) 0 0
\(925\) − 91.8034i − 0.0992469i
\(926\) 0 0
\(927\) − 478.353i − 0.516023i
\(928\) 0 0
\(929\) 400.039 0.430612 0.215306 0.976547i \(-0.430925\pi\)
0.215306 + 0.976547i \(0.430925\pi\)
\(930\) 0 0
\(931\) −325.043 −0.349134
\(932\) 0 0
\(933\) − 663.489i − 0.711135i
\(934\) 0 0
\(935\) − 529.524i − 0.566335i
\(936\) 0 0
\(937\) −249.279 −0.266039 −0.133020 0.991113i \(-0.542467\pi\)
−0.133020 + 0.991113i \(0.542467\pi\)
\(938\) 0 0
\(939\) −1334.39 −1.42107
\(940\) 0 0
\(941\) 724.229i 0.769638i 0.922992 + 0.384819i \(0.125736\pi\)
−0.922992 + 0.384819i \(0.874264\pi\)
\(942\) 0 0
\(943\) − 8.09432i − 0.00858358i
\(944\) 0 0
\(945\) 344.721 0.364785
\(946\) 0 0
\(947\) −1141.54 −1.20542 −0.602712 0.797959i \(-0.705913\pi\)
−0.602712 + 0.797959i \(0.705913\pi\)
\(948\) 0 0
\(949\) − 47.0557i − 0.0495845i
\(950\) 0 0
\(951\) − 378.865i − 0.398386i
\(952\) 0 0
\(953\) −1295.33 −1.35921 −0.679604 0.733579i \(-0.737848\pi\)
−0.679604 + 0.733579i \(0.737848\pi\)
\(954\) 0 0
\(955\) −69.3623 −0.0726307
\(956\) 0 0
\(957\) − 512.551i − 0.535581i
\(958\) 0 0
\(959\) 4.37837i 0.00456556i
\(960\) 0 0
\(961\) 892.201 0.928409
\(962\) 0 0
\(963\) −178.232 −0.185080
\(964\) 0 0
\(965\) 246.466i 0.255405i
\(966\) 0 0
\(967\) − 398.477i − 0.412075i −0.978544 0.206037i \(-0.933943\pi\)
0.978544 0.206037i \(-0.0660569\pi\)
\(968\) 0 0
\(969\) 425.337 0.438945
\(970\) 0 0
\(971\) 928.093 0.955811 0.477906 0.878411i \(-0.341396\pi\)
0.477906 + 0.878411i \(0.341396\pi\)
\(972\) 0 0
\(973\) − 1249.44i − 1.28411i
\(974\) 0 0
\(975\) − 99.5959i − 0.102150i
\(976\) 0 0
\(977\) −1378.05 −1.41049 −0.705247 0.708962i \(-0.749164\pi\)
−0.705247 + 0.708962i \(0.749164\pi\)
\(978\) 0 0
\(979\) −2219.85 −2.26747
\(980\) 0 0
\(981\) − 463.423i − 0.472398i
\(982\) 0 0
\(983\) 311.291i 0.316675i 0.987385 + 0.158337i \(0.0506134\pi\)
−0.987385 + 0.158337i \(0.949387\pi\)
\(984\) 0 0
\(985\) −385.777 −0.391652
\(986\) 0 0
\(987\) 659.939 0.668631
\(988\) 0 0
\(989\) 12.3607i 0.0124982i
\(990\) 0 0
\(991\) − 961.147i − 0.969876i −0.874549 0.484938i \(-0.838842\pi\)
0.874549 0.484938i \(-0.161158\pi\)
\(992\) 0 0
\(993\) 780.498 0.786000
\(994\) 0 0
\(995\) 608.676 0.611735
\(996\) 0 0
\(997\) 1089.68i 1.09296i 0.837473 + 0.546479i \(0.184032\pi\)
−0.837473 + 0.546479i \(0.815968\pi\)
\(998\) 0 0
\(999\) 538.404i 0.538943i
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 1280.3.g.e.1151.4 8
4.3 odd 2 inner 1280.3.g.e.1151.6 8
8.3 odd 2 inner 1280.3.g.e.1151.3 8
8.5 even 2 inner 1280.3.g.e.1151.5 8
16.3 odd 4 320.3.b.c.191.3 4
16.5 even 4 20.3.b.a.11.3 4
16.11 odd 4 20.3.b.a.11.4 yes 4
16.13 even 4 320.3.b.c.191.2 4
48.5 odd 4 180.3.c.a.91.2 4
48.11 even 4 180.3.c.a.91.1 4
48.29 odd 4 2880.3.e.e.2431.2 4
48.35 even 4 2880.3.e.e.2431.1 4
80.3 even 4 1600.3.h.n.1599.3 8
80.13 odd 4 1600.3.h.n.1599.6 8
80.19 odd 4 1600.3.b.s.1151.2 4
80.27 even 4 100.3.d.b.99.2 8
80.29 even 4 1600.3.b.s.1151.3 4
80.37 odd 4 100.3.d.b.99.8 8
80.43 even 4 100.3.d.b.99.7 8
80.53 odd 4 100.3.d.b.99.1 8
80.59 odd 4 100.3.b.f.51.1 4
80.67 even 4 1600.3.h.n.1599.5 8
80.69 even 4 100.3.b.f.51.2 4
80.77 odd 4 1600.3.h.n.1599.4 8
240.53 even 4 900.3.f.e.199.8 8
240.59 even 4 900.3.c.k.451.4 4
240.107 odd 4 900.3.f.e.199.7 8
240.149 odd 4 900.3.c.k.451.3 4
240.197 even 4 900.3.f.e.199.1 8
240.203 odd 4 900.3.f.e.199.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.3 4 16.5 even 4
20.3.b.a.11.4 yes 4 16.11 odd 4
100.3.b.f.51.1 4 80.59 odd 4
100.3.b.f.51.2 4 80.69 even 4
100.3.d.b.99.1 8 80.53 odd 4
100.3.d.b.99.2 8 80.27 even 4
100.3.d.b.99.7 8 80.43 even 4
100.3.d.b.99.8 8 80.37 odd 4
180.3.c.a.91.1 4 48.11 even 4
180.3.c.a.91.2 4 48.5 odd 4
320.3.b.c.191.2 4 16.13 even 4
320.3.b.c.191.3 4 16.3 odd 4
900.3.c.k.451.3 4 240.149 odd 4
900.3.c.k.451.4 4 240.59 even 4
900.3.f.e.199.1 8 240.197 even 4
900.3.f.e.199.2 8 240.203 odd 4
900.3.f.e.199.7 8 240.107 odd 4
900.3.f.e.199.8 8 240.53 even 4
1280.3.g.e.1151.3 8 8.3 odd 2 inner
1280.3.g.e.1151.4 8 1.1 even 1 trivial
1280.3.g.e.1151.5 8 8.5 even 2 inner
1280.3.g.e.1151.6 8 4.3 odd 2 inner
1600.3.b.s.1151.2 4 80.19 odd 4
1600.3.b.s.1151.3 4 80.29 even 4
1600.3.h.n.1599.3 8 80.3 even 4
1600.3.h.n.1599.4 8 80.77 odd 4
1600.3.h.n.1599.5 8 80.67 even 4
1600.3.h.n.1599.6 8 80.13 odd 4
2880.3.e.e.2431.1 4 48.35 even 4
2880.3.e.e.2431.2 4 48.29 odd 4