Properties

Label 1216.4.a.r.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7057.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 22x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.86867\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.28632 q^{3} -2.38427 q^{5} +5.83529 q^{7} +41.6631 q^{9} +O(q^{10})\) \(q-8.28632 q^{3} -2.38427 q^{5} +5.83529 q^{7} +41.6631 q^{9} +7.33364 q^{11} -55.6319 q^{13} +19.7568 q^{15} -10.0356 q^{17} +19.0000 q^{19} -48.3531 q^{21} -9.26357 q^{23} -119.315 q^{25} -121.503 q^{27} +83.9613 q^{29} +202.424 q^{31} -60.7689 q^{33} -13.9129 q^{35} -95.2017 q^{37} +460.983 q^{39} -25.9916 q^{41} +119.579 q^{43} -99.3358 q^{45} +467.824 q^{47} -308.949 q^{49} +83.1579 q^{51} +764.970 q^{53} -17.4853 q^{55} -157.440 q^{57} -69.1344 q^{59} +398.549 q^{61} +243.116 q^{63} +132.641 q^{65} +243.650 q^{67} +76.7609 q^{69} -781.354 q^{71} -711.911 q^{73} +988.684 q^{75} +42.7939 q^{77} +723.944 q^{79} -118.092 q^{81} +1227.94 q^{83} +23.9275 q^{85} -695.730 q^{87} -653.687 q^{89} -324.628 q^{91} -1677.35 q^{93} -45.3010 q^{95} +1692.40 q^{97} +305.542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −8.28632 −1.59470 −0.797351 0.603516i \(-0.793766\pi\)
−0.797351 + 0.603516i \(0.793766\pi\)
\(4\) 0 0
\(5\) −2.38427 −0.213255 −0.106628 0.994299i \(-0.534005\pi\)
−0.106628 + 0.994299i \(0.534005\pi\)
\(6\) 0 0
\(7\) 5.83529 0.315076 0.157538 0.987513i \(-0.449644\pi\)
0.157538 + 0.987513i \(0.449644\pi\)
\(8\) 0 0
\(9\) 41.6631 1.54308
\(10\) 0 0
\(11\) 7.33364 0.201016 0.100508 0.994936i \(-0.467953\pi\)
0.100508 + 0.994936i \(0.467953\pi\)
\(12\) 0 0
\(13\) −55.6319 −1.18688 −0.593442 0.804876i \(-0.702232\pi\)
−0.593442 + 0.804876i \(0.702232\pi\)
\(14\) 0 0
\(15\) 19.7568 0.340079
\(16\) 0 0
\(17\) −10.0356 −0.143176 −0.0715878 0.997434i \(-0.522807\pi\)
−0.0715878 + 0.997434i \(0.522807\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) −48.3531 −0.502453
\(22\) 0 0
\(23\) −9.26357 −0.0839821 −0.0419911 0.999118i \(-0.513370\pi\)
−0.0419911 + 0.999118i \(0.513370\pi\)
\(24\) 0 0
\(25\) −119.315 −0.954522
\(26\) 0 0
\(27\) −121.503 −0.866045
\(28\) 0 0
\(29\) 83.9613 0.537628 0.268814 0.963192i \(-0.413368\pi\)
0.268814 + 0.963192i \(0.413368\pi\)
\(30\) 0 0
\(31\) 202.424 1.17279 0.586394 0.810026i \(-0.300547\pi\)
0.586394 + 0.810026i \(0.300547\pi\)
\(32\) 0 0
\(33\) −60.7689 −0.320561
\(34\) 0 0
\(35\) −13.9129 −0.0671916
\(36\) 0 0
\(37\) −95.2017 −0.423002 −0.211501 0.977378i \(-0.567835\pi\)
−0.211501 + 0.977378i \(0.567835\pi\)
\(38\) 0 0
\(39\) 460.983 1.89273
\(40\) 0 0
\(41\) −25.9916 −0.0990049 −0.0495024 0.998774i \(-0.515764\pi\)
−0.0495024 + 0.998774i \(0.515764\pi\)
\(42\) 0 0
\(43\) 119.579 0.424084 0.212042 0.977261i \(-0.431989\pi\)
0.212042 + 0.977261i \(0.431989\pi\)
\(44\) 0 0
\(45\) −99.3358 −0.329069
\(46\) 0 0
\(47\) 467.824 1.45190 0.725948 0.687749i \(-0.241401\pi\)
0.725948 + 0.687749i \(0.241401\pi\)
\(48\) 0 0
\(49\) −308.949 −0.900727
\(50\) 0 0
\(51\) 83.1579 0.228322
\(52\) 0 0
\(53\) 764.970 1.98258 0.991290 0.131695i \(-0.0420420\pi\)
0.991290 + 0.131695i \(0.0420420\pi\)
\(54\) 0 0
\(55\) −17.4853 −0.0428677
\(56\) 0 0
\(57\) −157.440 −0.365850
\(58\) 0 0
\(59\) −69.1344 −0.152551 −0.0762757 0.997087i \(-0.524303\pi\)
−0.0762757 + 0.997087i \(0.524303\pi\)
\(60\) 0 0
\(61\) 398.549 0.836541 0.418271 0.908322i \(-0.362636\pi\)
0.418271 + 0.908322i \(0.362636\pi\)
\(62\) 0 0
\(63\) 243.116 0.486186
\(64\) 0 0
\(65\) 132.641 0.253109
\(66\) 0 0
\(67\) 243.650 0.444278 0.222139 0.975015i \(-0.428696\pi\)
0.222139 + 0.975015i \(0.428696\pi\)
\(68\) 0 0
\(69\) 76.7609 0.133926
\(70\) 0 0
\(71\) −781.354 −1.30605 −0.653026 0.757336i \(-0.726501\pi\)
−0.653026 + 0.757336i \(0.726501\pi\)
\(72\) 0 0
\(73\) −711.911 −1.14141 −0.570705 0.821155i \(-0.693330\pi\)
−0.570705 + 0.821155i \(0.693330\pi\)
\(74\) 0 0
\(75\) 988.684 1.52218
\(76\) 0 0
\(77\) 42.7939 0.0633353
\(78\) 0 0
\(79\) 723.944 1.03101 0.515507 0.856885i \(-0.327604\pi\)
0.515507 + 0.856885i \(0.327604\pi\)
\(80\) 0 0
\(81\) −118.092 −0.161992
\(82\) 0 0
\(83\) 1227.94 1.62391 0.811953 0.583723i \(-0.198405\pi\)
0.811953 + 0.583723i \(0.198405\pi\)
\(84\) 0 0
\(85\) 23.9275 0.0305329
\(86\) 0 0
\(87\) −695.730 −0.857357
\(88\) 0 0
\(89\) −653.687 −0.778548 −0.389274 0.921122i \(-0.627274\pi\)
−0.389274 + 0.921122i \(0.627274\pi\)
\(90\) 0 0
\(91\) −324.628 −0.373959
\(92\) 0 0
\(93\) −1677.35 −1.87025
\(94\) 0 0
\(95\) −45.3010 −0.0489241
\(96\) 0 0
\(97\) 1692.40 1.77152 0.885758 0.464148i \(-0.153639\pi\)
0.885758 + 0.464148i \(0.153639\pi\)
\(98\) 0 0
\(99\) 305.542 0.310183
\(100\) 0 0
\(101\) −839.619 −0.827180 −0.413590 0.910463i \(-0.635725\pi\)
−0.413590 + 0.910463i \(0.635725\pi\)
\(102\) 0 0
\(103\) 352.945 0.337638 0.168819 0.985647i \(-0.446005\pi\)
0.168819 + 0.985647i \(0.446005\pi\)
\(104\) 0 0
\(105\) 115.287 0.107151
\(106\) 0 0
\(107\) −375.247 −0.339033 −0.169516 0.985527i \(-0.554221\pi\)
−0.169516 + 0.985527i \(0.554221\pi\)
\(108\) 0 0
\(109\) −1297.69 −1.14033 −0.570164 0.821531i \(-0.693120\pi\)
−0.570164 + 0.821531i \(0.693120\pi\)
\(110\) 0 0
\(111\) 788.871 0.674562
\(112\) 0 0
\(113\) −83.3803 −0.0694138 −0.0347069 0.999398i \(-0.511050\pi\)
−0.0347069 + 0.999398i \(0.511050\pi\)
\(114\) 0 0
\(115\) 22.0868 0.0179096
\(116\) 0 0
\(117\) −2317.79 −1.83145
\(118\) 0 0
\(119\) −58.5605 −0.0451112
\(120\) 0 0
\(121\) −1277.22 −0.959593
\(122\) 0 0
\(123\) 215.374 0.157883
\(124\) 0 0
\(125\) 582.513 0.416812
\(126\) 0 0
\(127\) −1083.10 −0.756772 −0.378386 0.925648i \(-0.623521\pi\)
−0.378386 + 0.925648i \(0.623521\pi\)
\(128\) 0 0
\(129\) −990.870 −0.676288
\(130\) 0 0
\(131\) −2400.48 −1.60100 −0.800500 0.599333i \(-0.795433\pi\)
−0.800500 + 0.599333i \(0.795433\pi\)
\(132\) 0 0
\(133\) 110.871 0.0722834
\(134\) 0 0
\(135\) 289.695 0.184689
\(136\) 0 0
\(137\) −2349.46 −1.46517 −0.732584 0.680677i \(-0.761686\pi\)
−0.732584 + 0.680677i \(0.761686\pi\)
\(138\) 0 0
\(139\) −2828.01 −1.72568 −0.862838 0.505481i \(-0.831315\pi\)
−0.862838 + 0.505481i \(0.831315\pi\)
\(140\) 0 0
\(141\) −3876.54 −2.31534
\(142\) 0 0
\(143\) −407.984 −0.238583
\(144\) 0 0
\(145\) −200.186 −0.114652
\(146\) 0 0
\(147\) 2560.05 1.43639
\(148\) 0 0
\(149\) 1609.68 0.885033 0.442517 0.896760i \(-0.354086\pi\)
0.442517 + 0.896760i \(0.354086\pi\)
\(150\) 0 0
\(151\) 1625.10 0.875819 0.437910 0.899019i \(-0.355719\pi\)
0.437910 + 0.899019i \(0.355719\pi\)
\(152\) 0 0
\(153\) −418.113 −0.220931
\(154\) 0 0
\(155\) −482.633 −0.250103
\(156\) 0 0
\(157\) −1747.90 −0.888521 −0.444261 0.895898i \(-0.646534\pi\)
−0.444261 + 0.895898i \(0.646534\pi\)
\(158\) 0 0
\(159\) −6338.79 −3.16163
\(160\) 0 0
\(161\) −54.0556 −0.0264608
\(162\) 0 0
\(163\) −805.324 −0.386981 −0.193490 0.981102i \(-0.561981\pi\)
−0.193490 + 0.981102i \(0.561981\pi\)
\(164\) 0 0
\(165\) 144.889 0.0683612
\(166\) 0 0
\(167\) 1297.79 0.601354 0.300677 0.953726i \(-0.402787\pi\)
0.300677 + 0.953726i \(0.402787\pi\)
\(168\) 0 0
\(169\) 897.905 0.408696
\(170\) 0 0
\(171\) 791.598 0.354006
\(172\) 0 0
\(173\) 1244.83 0.547068 0.273534 0.961862i \(-0.411807\pi\)
0.273534 + 0.961862i \(0.411807\pi\)
\(174\) 0 0
\(175\) −696.239 −0.300747
\(176\) 0 0
\(177\) 572.870 0.243274
\(178\) 0 0
\(179\) −1715.51 −0.716331 −0.358166 0.933658i \(-0.616598\pi\)
−0.358166 + 0.933658i \(0.616598\pi\)
\(180\) 0 0
\(181\) −2382.52 −0.978406 −0.489203 0.872170i \(-0.662712\pi\)
−0.489203 + 0.872170i \(0.662712\pi\)
\(182\) 0 0
\(183\) −3302.51 −1.33403
\(184\) 0 0
\(185\) 226.986 0.0902073
\(186\) 0 0
\(187\) −73.5973 −0.0287806
\(188\) 0 0
\(189\) −709.004 −0.272870
\(190\) 0 0
\(191\) 1209.12 0.458056 0.229028 0.973420i \(-0.426445\pi\)
0.229028 + 0.973420i \(0.426445\pi\)
\(192\) 0 0
\(193\) 3304.21 1.23234 0.616172 0.787612i \(-0.288683\pi\)
0.616172 + 0.787612i \(0.288683\pi\)
\(194\) 0 0
\(195\) −1099.11 −0.403634
\(196\) 0 0
\(197\) 263.227 0.0951989 0.0475994 0.998867i \(-0.484843\pi\)
0.0475994 + 0.998867i \(0.484843\pi\)
\(198\) 0 0
\(199\) 1172.29 0.417593 0.208797 0.977959i \(-0.433045\pi\)
0.208797 + 0.977959i \(0.433045\pi\)
\(200\) 0 0
\(201\) −2018.96 −0.708491
\(202\) 0 0
\(203\) 489.939 0.169394
\(204\) 0 0
\(205\) 61.9708 0.0211133
\(206\) 0 0
\(207\) −385.949 −0.129591
\(208\) 0 0
\(209\) 139.339 0.0461162
\(210\) 0 0
\(211\) −4970.45 −1.62170 −0.810852 0.585251i \(-0.800996\pi\)
−0.810852 + 0.585251i \(0.800996\pi\)
\(212\) 0 0
\(213\) 6474.55 2.08276
\(214\) 0 0
\(215\) −285.108 −0.0904382
\(216\) 0 0
\(217\) 1181.20 0.369518
\(218\) 0 0
\(219\) 5899.12 1.82021
\(220\) 0 0
\(221\) 558.298 0.169933
\(222\) 0 0
\(223\) −4348.86 −1.30592 −0.652962 0.757391i \(-0.726474\pi\)
−0.652962 + 0.757391i \(0.726474\pi\)
\(224\) 0 0
\(225\) −4971.04 −1.47290
\(226\) 0 0
\(227\) 735.340 0.215006 0.107503 0.994205i \(-0.465715\pi\)
0.107503 + 0.994205i \(0.465715\pi\)
\(228\) 0 0
\(229\) −392.347 −0.113218 −0.0566092 0.998396i \(-0.518029\pi\)
−0.0566092 + 0.998396i \(0.518029\pi\)
\(230\) 0 0
\(231\) −354.604 −0.101001
\(232\) 0 0
\(233\) −4007.01 −1.12664 −0.563322 0.826237i \(-0.690477\pi\)
−0.563322 + 0.826237i \(0.690477\pi\)
\(234\) 0 0
\(235\) −1115.42 −0.309625
\(236\) 0 0
\(237\) −5998.83 −1.64416
\(238\) 0 0
\(239\) −5551.22 −1.50242 −0.751211 0.660062i \(-0.770530\pi\)
−0.751211 + 0.660062i \(0.770530\pi\)
\(240\) 0 0
\(241\) 3185.68 0.851486 0.425743 0.904844i \(-0.360013\pi\)
0.425743 + 0.904844i \(0.360013\pi\)
\(242\) 0 0
\(243\) 4259.12 1.12437
\(244\) 0 0
\(245\) 736.617 0.192085
\(246\) 0 0
\(247\) −1057.01 −0.272290
\(248\) 0 0
\(249\) −10175.1 −2.58965
\(250\) 0 0
\(251\) −5916.38 −1.48780 −0.743902 0.668289i \(-0.767027\pi\)
−0.743902 + 0.668289i \(0.767027\pi\)
\(252\) 0 0
\(253\) −67.9357 −0.0168817
\(254\) 0 0
\(255\) −198.271 −0.0486909
\(256\) 0 0
\(257\) −1094.22 −0.265586 −0.132793 0.991144i \(-0.542395\pi\)
−0.132793 + 0.991144i \(0.542395\pi\)
\(258\) 0 0
\(259\) −555.530 −0.133278
\(260\) 0 0
\(261\) 3498.08 0.829601
\(262\) 0 0
\(263\) 5474.53 1.28355 0.641775 0.766893i \(-0.278198\pi\)
0.641775 + 0.766893i \(0.278198\pi\)
\(264\) 0 0
\(265\) −1823.89 −0.422796
\(266\) 0 0
\(267\) 5416.66 1.24155
\(268\) 0 0
\(269\) −5031.11 −1.14034 −0.570172 0.821525i \(-0.693123\pi\)
−0.570172 + 0.821525i \(0.693123\pi\)
\(270\) 0 0
\(271\) 7531.38 1.68819 0.844093 0.536196i \(-0.180139\pi\)
0.844093 + 0.536196i \(0.180139\pi\)
\(272\) 0 0
\(273\) 2689.97 0.596354
\(274\) 0 0
\(275\) −875.015 −0.191874
\(276\) 0 0
\(277\) −4082.11 −0.885452 −0.442726 0.896657i \(-0.645989\pi\)
−0.442726 + 0.896657i \(0.645989\pi\)
\(278\) 0 0
\(279\) 8433.60 1.80970
\(280\) 0 0
\(281\) −8485.56 −1.80145 −0.900723 0.434395i \(-0.856962\pi\)
−0.900723 + 0.434395i \(0.856962\pi\)
\(282\) 0 0
\(283\) 966.510 0.203014 0.101507 0.994835i \(-0.467634\pi\)
0.101507 + 0.994835i \(0.467634\pi\)
\(284\) 0 0
\(285\) 375.379 0.0780194
\(286\) 0 0
\(287\) −151.668 −0.0311941
\(288\) 0 0
\(289\) −4812.29 −0.979501
\(290\) 0 0
\(291\) −14023.7 −2.82504
\(292\) 0 0
\(293\) 1361.87 0.271540 0.135770 0.990740i \(-0.456649\pi\)
0.135770 + 0.990740i \(0.456649\pi\)
\(294\) 0 0
\(295\) 164.835 0.0325324
\(296\) 0 0
\(297\) −891.057 −0.174089
\(298\) 0 0
\(299\) 515.350 0.0996771
\(300\) 0 0
\(301\) 697.778 0.133619
\(302\) 0 0
\(303\) 6957.35 1.31911
\(304\) 0 0
\(305\) −950.248 −0.178397
\(306\) 0 0
\(307\) −1526.15 −0.283720 −0.141860 0.989887i \(-0.545308\pi\)
−0.141860 + 0.989887i \(0.545308\pi\)
\(308\) 0 0
\(309\) −2924.61 −0.538432
\(310\) 0 0
\(311\) 1893.60 0.345261 0.172630 0.984987i \(-0.444773\pi\)
0.172630 + 0.984987i \(0.444773\pi\)
\(312\) 0 0
\(313\) 9049.39 1.63419 0.817096 0.576502i \(-0.195583\pi\)
0.817096 + 0.576502i \(0.195583\pi\)
\(314\) 0 0
\(315\) −579.653 −0.103682
\(316\) 0 0
\(317\) −1317.33 −0.233403 −0.116702 0.993167i \(-0.537232\pi\)
−0.116702 + 0.993167i \(0.537232\pi\)
\(318\) 0 0
\(319\) 615.742 0.108072
\(320\) 0 0
\(321\) 3109.42 0.540656
\(322\) 0 0
\(323\) −190.676 −0.0328467
\(324\) 0 0
\(325\) 6637.73 1.13291
\(326\) 0 0
\(327\) 10753.0 1.81848
\(328\) 0 0
\(329\) 2729.89 0.457458
\(330\) 0 0
\(331\) 9896.94 1.64346 0.821730 0.569877i \(-0.193009\pi\)
0.821730 + 0.569877i \(0.193009\pi\)
\(332\) 0 0
\(333\) −3966.39 −0.652724
\(334\) 0 0
\(335\) −580.927 −0.0947446
\(336\) 0 0
\(337\) 1263.75 0.204276 0.102138 0.994770i \(-0.467432\pi\)
0.102138 + 0.994770i \(0.467432\pi\)
\(338\) 0 0
\(339\) 690.915 0.110694
\(340\) 0 0
\(341\) 1484.50 0.235749
\(342\) 0 0
\(343\) −3804.31 −0.598874
\(344\) 0 0
\(345\) −183.018 −0.0285605
\(346\) 0 0
\(347\) −7582.42 −1.17304 −0.586521 0.809934i \(-0.699503\pi\)
−0.586521 + 0.809934i \(0.699503\pi\)
\(348\) 0 0
\(349\) −10227.9 −1.56872 −0.784362 0.620303i \(-0.787009\pi\)
−0.784362 + 0.620303i \(0.787009\pi\)
\(350\) 0 0
\(351\) 6759.42 1.02790
\(352\) 0 0
\(353\) 7056.98 1.06404 0.532019 0.846732i \(-0.321434\pi\)
0.532019 + 0.846732i \(0.321434\pi\)
\(354\) 0 0
\(355\) 1862.96 0.278522
\(356\) 0 0
\(357\) 485.251 0.0719389
\(358\) 0 0
\(359\) −8459.22 −1.24362 −0.621811 0.783167i \(-0.713603\pi\)
−0.621811 + 0.783167i \(0.713603\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 10583.4 1.53026
\(364\) 0 0
\(365\) 1697.39 0.243412
\(366\) 0 0
\(367\) −1344.85 −0.191282 −0.0956410 0.995416i \(-0.530490\pi\)
−0.0956410 + 0.995416i \(0.530490\pi\)
\(368\) 0 0
\(369\) −1082.89 −0.152772
\(370\) 0 0
\(371\) 4463.83 0.624664
\(372\) 0 0
\(373\) 3928.85 0.545384 0.272692 0.962101i \(-0.412086\pi\)
0.272692 + 0.962101i \(0.412086\pi\)
\(374\) 0 0
\(375\) −4826.88 −0.664691
\(376\) 0 0
\(377\) −4670.92 −0.638103
\(378\) 0 0
\(379\) 10874.9 1.47390 0.736951 0.675947i \(-0.236265\pi\)
0.736951 + 0.675947i \(0.236265\pi\)
\(380\) 0 0
\(381\) 8974.95 1.20683
\(382\) 0 0
\(383\) −8209.99 −1.09533 −0.547664 0.836698i \(-0.684483\pi\)
−0.547664 + 0.836698i \(0.684483\pi\)
\(384\) 0 0
\(385\) −102.032 −0.0135066
\(386\) 0 0
\(387\) 4982.03 0.654394
\(388\) 0 0
\(389\) 6353.98 0.828174 0.414087 0.910237i \(-0.364101\pi\)
0.414087 + 0.910237i \(0.364101\pi\)
\(390\) 0 0
\(391\) 92.9652 0.0120242
\(392\) 0 0
\(393\) 19891.2 2.55312
\(394\) 0 0
\(395\) −1726.08 −0.219869
\(396\) 0 0
\(397\) −4184.09 −0.528951 −0.264476 0.964392i \(-0.585199\pi\)
−0.264476 + 0.964392i \(0.585199\pi\)
\(398\) 0 0
\(399\) −918.708 −0.115271
\(400\) 0 0
\(401\) −12492.3 −1.55569 −0.777847 0.628453i \(-0.783688\pi\)
−0.777847 + 0.628453i \(0.783688\pi\)
\(402\) 0 0
\(403\) −11261.2 −1.39196
\(404\) 0 0
\(405\) 281.563 0.0345457
\(406\) 0 0
\(407\) −698.175 −0.0850301
\(408\) 0 0
\(409\) 7559.01 0.913861 0.456931 0.889502i \(-0.348949\pi\)
0.456931 + 0.889502i \(0.348949\pi\)
\(410\) 0 0
\(411\) 19468.4 2.33651
\(412\) 0 0
\(413\) −403.420 −0.0480653
\(414\) 0 0
\(415\) −2927.74 −0.346306
\(416\) 0 0
\(417\) 23433.8 2.75194
\(418\) 0 0
\(419\) 12805.0 1.49300 0.746498 0.665388i \(-0.231734\pi\)
0.746498 + 0.665388i \(0.231734\pi\)
\(420\) 0 0
\(421\) 6126.65 0.709251 0.354626 0.935008i \(-0.384608\pi\)
0.354626 + 0.935008i \(0.384608\pi\)
\(422\) 0 0
\(423\) 19491.0 2.24039
\(424\) 0 0
\(425\) 1197.40 0.136664
\(426\) 0 0
\(427\) 2325.65 0.263574
\(428\) 0 0
\(429\) 3380.68 0.380469
\(430\) 0 0
\(431\) 9784.87 1.09355 0.546776 0.837279i \(-0.315855\pi\)
0.546776 + 0.837279i \(0.315855\pi\)
\(432\) 0 0
\(433\) 9483.80 1.05257 0.526285 0.850308i \(-0.323585\pi\)
0.526285 + 0.850308i \(0.323585\pi\)
\(434\) 0 0
\(435\) 1658.80 0.182836
\(436\) 0 0
\(437\) −176.008 −0.0192668
\(438\) 0 0
\(439\) −16627.0 −1.80766 −0.903828 0.427895i \(-0.859255\pi\)
−0.903828 + 0.427895i \(0.859255\pi\)
\(440\) 0 0
\(441\) −12871.8 −1.38989
\(442\) 0 0
\(443\) −12294.8 −1.31861 −0.659304 0.751877i \(-0.729149\pi\)
−0.659304 + 0.751877i \(0.729149\pi\)
\(444\) 0 0
\(445\) 1558.56 0.166029
\(446\) 0 0
\(447\) −13338.3 −1.41136
\(448\) 0 0
\(449\) −4523.75 −0.475476 −0.237738 0.971329i \(-0.576406\pi\)
−0.237738 + 0.971329i \(0.576406\pi\)
\(450\) 0 0
\(451\) −190.613 −0.0199016
\(452\) 0 0
\(453\) −13466.1 −1.39667
\(454\) 0 0
\(455\) 774.000 0.0797487
\(456\) 0 0
\(457\) −14738.0 −1.50856 −0.754281 0.656551i \(-0.772015\pi\)
−0.754281 + 0.656551i \(0.772015\pi\)
\(458\) 0 0
\(459\) 1219.35 0.123996
\(460\) 0 0
\(461\) −10875.2 −1.09872 −0.549360 0.835586i \(-0.685128\pi\)
−0.549360 + 0.835586i \(0.685128\pi\)
\(462\) 0 0
\(463\) −12721.9 −1.27697 −0.638484 0.769635i \(-0.720438\pi\)
−0.638484 + 0.769635i \(0.720438\pi\)
\(464\) 0 0
\(465\) 3999.25 0.398840
\(466\) 0 0
\(467\) −5255.10 −0.520721 −0.260361 0.965511i \(-0.583841\pi\)
−0.260361 + 0.965511i \(0.583841\pi\)
\(468\) 0 0
\(469\) 1421.77 0.139981
\(470\) 0 0
\(471\) 14483.7 1.41693
\(472\) 0 0
\(473\) 876.949 0.0852477
\(474\) 0 0
\(475\) −2266.99 −0.218982
\(476\) 0 0
\(477\) 31871.0 3.05927
\(478\) 0 0
\(479\) −6261.62 −0.597288 −0.298644 0.954365i \(-0.596534\pi\)
−0.298644 + 0.954365i \(0.596534\pi\)
\(480\) 0 0
\(481\) 5296.25 0.502054
\(482\) 0 0
\(483\) 447.922 0.0421970
\(484\) 0 0
\(485\) −4035.13 −0.377785
\(486\) 0 0
\(487\) −9957.10 −0.926487 −0.463243 0.886231i \(-0.653314\pi\)
−0.463243 + 0.886231i \(0.653314\pi\)
\(488\) 0 0
\(489\) 6673.17 0.617119
\(490\) 0 0
\(491\) −3112.32 −0.286064 −0.143032 0.989718i \(-0.545685\pi\)
−0.143032 + 0.989718i \(0.545685\pi\)
\(492\) 0 0
\(493\) −842.600 −0.0769752
\(494\) 0 0
\(495\) −728.493 −0.0661481
\(496\) 0 0
\(497\) −4559.43 −0.411506
\(498\) 0 0
\(499\) −1278.95 −0.114737 −0.0573685 0.998353i \(-0.518271\pi\)
−0.0573685 + 0.998353i \(0.518271\pi\)
\(500\) 0 0
\(501\) −10753.9 −0.958981
\(502\) 0 0
\(503\) 10223.8 0.906272 0.453136 0.891441i \(-0.350305\pi\)
0.453136 + 0.891441i \(0.350305\pi\)
\(504\) 0 0
\(505\) 2001.88 0.176401
\(506\) 0 0
\(507\) −7440.32 −0.651748
\(508\) 0 0
\(509\) 8872.97 0.772667 0.386333 0.922359i \(-0.373741\pi\)
0.386333 + 0.922359i \(0.373741\pi\)
\(510\) 0 0
\(511\) −4154.21 −0.359631
\(512\) 0 0
\(513\) −2308.55 −0.198684
\(514\) 0 0
\(515\) −841.515 −0.0720031
\(516\) 0 0
\(517\) 3430.85 0.291854
\(518\) 0 0
\(519\) −10315.1 −0.872411
\(520\) 0 0
\(521\) −9651.60 −0.811601 −0.405801 0.913962i \(-0.633007\pi\)
−0.405801 + 0.913962i \(0.633007\pi\)
\(522\) 0 0
\(523\) −2707.90 −0.226402 −0.113201 0.993572i \(-0.536110\pi\)
−0.113201 + 0.993572i \(0.536110\pi\)
\(524\) 0 0
\(525\) 5769.26 0.479602
\(526\) 0 0
\(527\) −2031.44 −0.167915
\(528\) 0 0
\(529\) −12081.2 −0.992947
\(530\) 0 0
\(531\) −2880.35 −0.235399
\(532\) 0 0
\(533\) 1445.96 0.117507
\(534\) 0 0
\(535\) 894.689 0.0723005
\(536\) 0 0
\(537\) 14215.3 1.14234
\(538\) 0 0
\(539\) −2265.72 −0.181060
\(540\) 0 0
\(541\) 14478.5 1.15061 0.575304 0.817940i \(-0.304884\pi\)
0.575304 + 0.817940i \(0.304884\pi\)
\(542\) 0 0
\(543\) 19742.3 1.56027
\(544\) 0 0
\(545\) 3094.03 0.243181
\(546\) 0 0
\(547\) 10220.5 0.798895 0.399448 0.916756i \(-0.369202\pi\)
0.399448 + 0.916756i \(0.369202\pi\)
\(548\) 0 0
\(549\) 16604.8 1.29085
\(550\) 0 0
\(551\) 1595.26 0.123340
\(552\) 0 0
\(553\) 4224.43 0.324848
\(554\) 0 0
\(555\) −1880.88 −0.143854
\(556\) 0 0
\(557\) −11008.2 −0.837400 −0.418700 0.908125i \(-0.637514\pi\)
−0.418700 + 0.908125i \(0.637514\pi\)
\(558\) 0 0
\(559\) −6652.40 −0.503339
\(560\) 0 0
\(561\) 609.850 0.0458964
\(562\) 0 0
\(563\) −5553.05 −0.415689 −0.207845 0.978162i \(-0.566645\pi\)
−0.207845 + 0.978162i \(0.566645\pi\)
\(564\) 0 0
\(565\) 198.801 0.0148028
\(566\) 0 0
\(567\) −689.103 −0.0510399
\(568\) 0 0
\(569\) 14737.2 1.08580 0.542898 0.839799i \(-0.317327\pi\)
0.542898 + 0.839799i \(0.317327\pi\)
\(570\) 0 0
\(571\) −3141.77 −0.230261 −0.115131 0.993350i \(-0.536729\pi\)
−0.115131 + 0.993350i \(0.536729\pi\)
\(572\) 0 0
\(573\) −10019.1 −0.730464
\(574\) 0 0
\(575\) 1105.29 0.0801628
\(576\) 0 0
\(577\) 3639.45 0.262586 0.131293 0.991344i \(-0.458087\pi\)
0.131293 + 0.991344i \(0.458087\pi\)
\(578\) 0 0
\(579\) −27379.8 −1.96522
\(580\) 0 0
\(581\) 7165.40 0.511654
\(582\) 0 0
\(583\) 5610.02 0.398530
\(584\) 0 0
\(585\) 5526.24 0.390567
\(586\) 0 0
\(587\) −14028.9 −0.986433 −0.493216 0.869907i \(-0.664179\pi\)
−0.493216 + 0.869907i \(0.664179\pi\)
\(588\) 0 0
\(589\) 3846.06 0.269056
\(590\) 0 0
\(591\) −2181.19 −0.151814
\(592\) 0 0
\(593\) 20191.0 1.39822 0.699110 0.715014i \(-0.253580\pi\)
0.699110 + 0.715014i \(0.253580\pi\)
\(594\) 0 0
\(595\) 139.624 0.00962020
\(596\) 0 0
\(597\) −9713.93 −0.665937
\(598\) 0 0
\(599\) 3131.35 0.213595 0.106798 0.994281i \(-0.465940\pi\)
0.106798 + 0.994281i \(0.465940\pi\)
\(600\) 0 0
\(601\) 18210.1 1.23595 0.617976 0.786197i \(-0.287953\pi\)
0.617976 + 0.786197i \(0.287953\pi\)
\(602\) 0 0
\(603\) 10151.2 0.685555
\(604\) 0 0
\(605\) 3045.23 0.204638
\(606\) 0 0
\(607\) −12605.1 −0.842875 −0.421437 0.906858i \(-0.638474\pi\)
−0.421437 + 0.906858i \(0.638474\pi\)
\(608\) 0 0
\(609\) −4059.79 −0.270133
\(610\) 0 0
\(611\) −26025.9 −1.72323
\(612\) 0 0
\(613\) −10096.7 −0.665256 −0.332628 0.943058i \(-0.607935\pi\)
−0.332628 + 0.943058i \(0.607935\pi\)
\(614\) 0 0
\(615\) −513.510 −0.0336694
\(616\) 0 0
\(617\) 5008.97 0.326829 0.163415 0.986557i \(-0.447749\pi\)
0.163415 + 0.986557i \(0.447749\pi\)
\(618\) 0 0
\(619\) −6083.17 −0.394997 −0.197499 0.980303i \(-0.563282\pi\)
−0.197499 + 0.980303i \(0.563282\pi\)
\(620\) 0 0
\(621\) 1125.55 0.0727323
\(622\) 0 0
\(623\) −3814.46 −0.245302
\(624\) 0 0
\(625\) 13525.5 0.865635
\(626\) 0 0
\(627\) −1154.61 −0.0735416
\(628\) 0 0
\(629\) 955.403 0.0605635
\(630\) 0 0
\(631\) 3719.15 0.234639 0.117319 0.993094i \(-0.462570\pi\)
0.117319 + 0.993094i \(0.462570\pi\)
\(632\) 0 0
\(633\) 41186.7 2.58614
\(634\) 0 0
\(635\) 2582.41 0.161386
\(636\) 0 0
\(637\) 17187.4 1.06906
\(638\) 0 0
\(639\) −32553.6 −2.01534
\(640\) 0 0
\(641\) −10626.9 −0.654816 −0.327408 0.944883i \(-0.606175\pi\)
−0.327408 + 0.944883i \(0.606175\pi\)
\(642\) 0 0
\(643\) −1914.95 −0.117447 −0.0587234 0.998274i \(-0.518703\pi\)
−0.0587234 + 0.998274i \(0.518703\pi\)
\(644\) 0 0
\(645\) 2362.50 0.144222
\(646\) 0 0
\(647\) 3804.40 0.231169 0.115584 0.993298i \(-0.463126\pi\)
0.115584 + 0.993298i \(0.463126\pi\)
\(648\) 0 0
\(649\) −507.007 −0.0306653
\(650\) 0 0
\(651\) −9787.83 −0.589271
\(652\) 0 0
\(653\) −15110.4 −0.905534 −0.452767 0.891629i \(-0.649563\pi\)
−0.452767 + 0.891629i \(0.649563\pi\)
\(654\) 0 0
\(655\) 5723.39 0.341422
\(656\) 0 0
\(657\) −29660.4 −1.76128
\(658\) 0 0
\(659\) −25627.7 −1.51489 −0.757445 0.652899i \(-0.773553\pi\)
−0.757445 + 0.652899i \(0.773553\pi\)
\(660\) 0 0
\(661\) 22255.1 1.30956 0.654782 0.755818i \(-0.272761\pi\)
0.654782 + 0.755818i \(0.272761\pi\)
\(662\) 0 0
\(663\) −4626.23 −0.270992
\(664\) 0 0
\(665\) −264.345 −0.0154148
\(666\) 0 0
\(667\) −777.781 −0.0451511
\(668\) 0 0
\(669\) 36036.0 2.08256
\(670\) 0 0
\(671\) 2922.82 0.168158
\(672\) 0 0
\(673\) −20867.0 −1.19519 −0.597595 0.801798i \(-0.703877\pi\)
−0.597595 + 0.801798i \(0.703877\pi\)
\(674\) 0 0
\(675\) 14497.1 0.826659
\(676\) 0 0
\(677\) −13538.0 −0.768551 −0.384275 0.923219i \(-0.625549\pi\)
−0.384275 + 0.923219i \(0.625549\pi\)
\(678\) 0 0
\(679\) 9875.64 0.558162
\(680\) 0 0
\(681\) −6093.26 −0.342870
\(682\) 0 0
\(683\) 19790.9 1.10875 0.554377 0.832266i \(-0.312957\pi\)
0.554377 + 0.832266i \(0.312957\pi\)
\(684\) 0 0
\(685\) 5601.74 0.312455
\(686\) 0 0
\(687\) 3251.11 0.180550
\(688\) 0 0
\(689\) −42556.7 −2.35309
\(690\) 0 0
\(691\) 4170.34 0.229591 0.114795 0.993389i \(-0.463379\pi\)
0.114795 + 0.993389i \(0.463379\pi\)
\(692\) 0 0
\(693\) 1782.93 0.0977312
\(694\) 0 0
\(695\) 6742.73 0.368009
\(696\) 0 0
\(697\) 260.840 0.0141751
\(698\) 0 0
\(699\) 33203.4 1.79666
\(700\) 0 0
\(701\) −1836.49 −0.0989491 −0.0494746 0.998775i \(-0.515755\pi\)
−0.0494746 + 0.998775i \(0.515755\pi\)
\(702\) 0 0
\(703\) −1808.83 −0.0970432
\(704\) 0 0
\(705\) 9242.70 0.493759
\(706\) 0 0
\(707\) −4899.42 −0.260625
\(708\) 0 0
\(709\) −9458.42 −0.501013 −0.250507 0.968115i \(-0.580597\pi\)
−0.250507 + 0.968115i \(0.580597\pi\)
\(710\) 0 0
\(711\) 30161.7 1.59093
\(712\) 0 0
\(713\) −1875.17 −0.0984932
\(714\) 0 0
\(715\) 972.742 0.0508790
\(716\) 0 0
\(717\) 45999.2 2.39592
\(718\) 0 0
\(719\) −28024.8 −1.45362 −0.726808 0.686840i \(-0.758997\pi\)
−0.726808 + 0.686840i \(0.758997\pi\)
\(720\) 0 0
\(721\) 2059.54 0.106382
\(722\) 0 0
\(723\) −26397.6 −1.35787
\(724\) 0 0
\(725\) −10017.9 −0.513178
\(726\) 0 0
\(727\) −729.786 −0.0372301 −0.0186150 0.999827i \(-0.505926\pi\)
−0.0186150 + 0.999827i \(0.505926\pi\)
\(728\) 0 0
\(729\) −32104.0 −1.63105
\(730\) 0 0
\(731\) −1200.04 −0.0607185
\(732\) 0 0
\(733\) 28958.9 1.45924 0.729618 0.683855i \(-0.239698\pi\)
0.729618 + 0.683855i \(0.239698\pi\)
\(734\) 0 0
\(735\) −6103.85 −0.306318
\(736\) 0 0
\(737\) 1786.84 0.0893070
\(738\) 0 0
\(739\) −4906.85 −0.244251 −0.122125 0.992515i \(-0.538971\pi\)
−0.122125 + 0.992515i \(0.538971\pi\)
\(740\) 0 0
\(741\) 8758.68 0.434222
\(742\) 0 0
\(743\) 29014.2 1.43261 0.716304 0.697788i \(-0.245832\pi\)
0.716304 + 0.697788i \(0.245832\pi\)
\(744\) 0 0
\(745\) −3837.90 −0.188738
\(746\) 0 0
\(747\) 51159.8 2.50581
\(748\) 0 0
\(749\) −2189.68 −0.106821
\(750\) 0 0
\(751\) −14611.4 −0.709958 −0.354979 0.934874i \(-0.615512\pi\)
−0.354979 + 0.934874i \(0.615512\pi\)
\(752\) 0 0
\(753\) 49025.0 2.37261
\(754\) 0 0
\(755\) −3874.67 −0.186773
\(756\) 0 0
\(757\) −34632.6 −1.66281 −0.831403 0.555670i \(-0.812462\pi\)
−0.831403 + 0.555670i \(0.812462\pi\)
\(758\) 0 0
\(759\) 562.937 0.0269214
\(760\) 0 0
\(761\) −1887.07 −0.0898899 −0.0449449 0.998989i \(-0.514311\pi\)
−0.0449449 + 0.998989i \(0.514311\pi\)
\(762\) 0 0
\(763\) −7572.37 −0.359290
\(764\) 0 0
\(765\) 996.892 0.0471146
\(766\) 0 0
\(767\) 3846.08 0.181061
\(768\) 0 0
\(769\) −16854.3 −0.790351 −0.395176 0.918606i \(-0.629316\pi\)
−0.395176 + 0.918606i \(0.629316\pi\)
\(770\) 0 0
\(771\) 9067.05 0.423530
\(772\) 0 0
\(773\) 37853.8 1.76133 0.880665 0.473740i \(-0.157096\pi\)
0.880665 + 0.473740i \(0.157096\pi\)
\(774\) 0 0
\(775\) −24152.3 −1.11945
\(776\) 0 0
\(777\) 4603.29 0.212538
\(778\) 0 0
\(779\) −493.840 −0.0227133
\(780\) 0 0
\(781\) −5730.17 −0.262537
\(782\) 0 0
\(783\) −10201.5 −0.465610
\(784\) 0 0
\(785\) 4167.47 0.189482
\(786\) 0 0
\(787\) −40230.0 −1.82216 −0.911082 0.412225i \(-0.864752\pi\)
−0.911082 + 0.412225i \(0.864752\pi\)
\(788\) 0 0
\(789\) −45363.7 −2.04688
\(790\) 0 0
\(791\) −486.548 −0.0218706
\(792\) 0 0
\(793\) −22172.0 −0.992878
\(794\) 0 0
\(795\) 15113.4 0.674233
\(796\) 0 0
\(797\) 14861.6 0.660507 0.330253 0.943892i \(-0.392866\pi\)
0.330253 + 0.943892i \(0.392866\pi\)
\(798\) 0 0
\(799\) −4694.88 −0.207876
\(800\) 0 0
\(801\) −27234.6 −1.20136
\(802\) 0 0
\(803\) −5220.90 −0.229442
\(804\) 0 0
\(805\) 128.883 0.00564289
\(806\) 0 0
\(807\) 41689.4 1.81851
\(808\) 0 0
\(809\) 17346.5 0.753857 0.376928 0.926242i \(-0.376980\pi\)
0.376928 + 0.926242i \(0.376980\pi\)
\(810\) 0 0
\(811\) 19405.3 0.840210 0.420105 0.907475i \(-0.361993\pi\)
0.420105 + 0.907475i \(0.361993\pi\)
\(812\) 0 0
\(813\) −62407.4 −2.69216
\(814\) 0 0
\(815\) 1920.11 0.0825256
\(816\) 0 0
\(817\) 2272.00 0.0972916
\(818\) 0 0
\(819\) −13525.0 −0.577047
\(820\) 0 0
\(821\) −9008.08 −0.382928 −0.191464 0.981500i \(-0.561324\pi\)
−0.191464 + 0.981500i \(0.561324\pi\)
\(822\) 0 0
\(823\) 22387.4 0.948207 0.474103 0.880469i \(-0.342772\pi\)
0.474103 + 0.880469i \(0.342772\pi\)
\(824\) 0 0
\(825\) 7250.65 0.305982
\(826\) 0 0
\(827\) 8244.50 0.346662 0.173331 0.984864i \(-0.444547\pi\)
0.173331 + 0.984864i \(0.444547\pi\)
\(828\) 0 0
\(829\) 19708.0 0.825680 0.412840 0.910804i \(-0.364537\pi\)
0.412840 + 0.910804i \(0.364537\pi\)
\(830\) 0 0
\(831\) 33825.6 1.41203
\(832\) 0 0
\(833\) 3100.48 0.128962
\(834\) 0 0
\(835\) −3094.28 −0.128242
\(836\) 0 0
\(837\) −24595.1 −1.01569
\(838\) 0 0
\(839\) 24561.7 1.01069 0.505343 0.862919i \(-0.331366\pi\)
0.505343 + 0.862919i \(0.331366\pi\)
\(840\) 0 0
\(841\) −17339.5 −0.710956
\(842\) 0 0
\(843\) 70314.1 2.87277
\(844\) 0 0
\(845\) −2140.84 −0.0871565
\(846\) 0 0
\(847\) −7452.94 −0.302345
\(848\) 0 0
\(849\) −8008.81 −0.323747
\(850\) 0 0
\(851\) 881.908 0.0355246
\(852\) 0 0
\(853\) −13832.7 −0.555243 −0.277622 0.960690i \(-0.589546\pi\)
−0.277622 + 0.960690i \(0.589546\pi\)
\(854\) 0 0
\(855\) −1887.38 −0.0754936
\(856\) 0 0
\(857\) 22597.5 0.900720 0.450360 0.892847i \(-0.351296\pi\)
0.450360 + 0.892847i \(0.351296\pi\)
\(858\) 0 0
\(859\) 33731.1 1.33980 0.669902 0.742449i \(-0.266336\pi\)
0.669902 + 0.742449i \(0.266336\pi\)
\(860\) 0 0
\(861\) 1256.77 0.0497453
\(862\) 0 0
\(863\) 15230.6 0.600759 0.300379 0.953820i \(-0.402887\pi\)
0.300379 + 0.953820i \(0.402887\pi\)
\(864\) 0 0
\(865\) −2968.01 −0.116665
\(866\) 0 0
\(867\) 39876.1 1.56201
\(868\) 0 0
\(869\) 5309.15 0.207250
\(870\) 0 0
\(871\) −13554.7 −0.527307
\(872\) 0 0
\(873\) 70510.5 2.73358
\(874\) 0 0
\(875\) 3399.13 0.131328
\(876\) 0 0
\(877\) −44693.6 −1.72086 −0.860432 0.509566i \(-0.829806\pi\)
−0.860432 + 0.509566i \(0.829806\pi\)
\(878\) 0 0
\(879\) −11284.9 −0.433025
\(880\) 0 0
\(881\) −45125.9 −1.72569 −0.862844 0.505471i \(-0.831319\pi\)
−0.862844 + 0.505471i \(0.831319\pi\)
\(882\) 0 0
\(883\) 35570.8 1.35567 0.677833 0.735216i \(-0.262919\pi\)
0.677833 + 0.735216i \(0.262919\pi\)
\(884\) 0 0
\(885\) −1365.87 −0.0518795
\(886\) 0 0
\(887\) −34224.2 −1.29553 −0.647766 0.761840i \(-0.724296\pi\)
−0.647766 + 0.761840i \(0.724296\pi\)
\(888\) 0 0
\(889\) −6320.23 −0.238441
\(890\) 0 0
\(891\) −866.046 −0.0325630
\(892\) 0 0
\(893\) 8888.66 0.333088
\(894\) 0 0
\(895\) 4090.23 0.152761
\(896\) 0 0
\(897\) −4270.35 −0.158955
\(898\) 0 0
\(899\) 16995.8 0.630524
\(900\) 0 0
\(901\) −7676.92 −0.283857
\(902\) 0 0
\(903\) −5782.01 −0.213082
\(904\) 0 0
\(905\) 5680.57 0.208650
\(906\) 0 0
\(907\) −35101.3 −1.28503 −0.642514 0.766274i \(-0.722109\pi\)
−0.642514 + 0.766274i \(0.722109\pi\)
\(908\) 0 0
\(909\) −34981.1 −1.27640
\(910\) 0 0
\(911\) −35322.9 −1.28463 −0.642317 0.766439i \(-0.722027\pi\)
−0.642317 + 0.766439i \(0.722027\pi\)
\(912\) 0 0
\(913\) 9005.28 0.326431
\(914\) 0 0
\(915\) 7874.05 0.284490
\(916\) 0 0
\(917\) −14007.5 −0.504437
\(918\) 0 0
\(919\) 21222.7 0.761778 0.380889 0.924621i \(-0.375618\pi\)
0.380889 + 0.924621i \(0.375618\pi\)
\(920\) 0 0
\(921\) 12646.2 0.452450
\(922\) 0 0
\(923\) 43468.2 1.55013
\(924\) 0 0
\(925\) 11359.0 0.403764
\(926\) 0 0
\(927\) 14704.8 0.521001
\(928\) 0 0
\(929\) 49566.4 1.75051 0.875253 0.483666i \(-0.160695\pi\)
0.875253 + 0.483666i \(0.160695\pi\)
\(930\) 0 0
\(931\) −5870.04 −0.206641
\(932\) 0 0
\(933\) −15691.0 −0.550589
\(934\) 0 0
\(935\) 175.475 0.00613760
\(936\) 0 0
\(937\) 23500.7 0.819354 0.409677 0.912231i \(-0.365641\pi\)
0.409677 + 0.912231i \(0.365641\pi\)
\(938\) 0 0
\(939\) −74986.1 −2.60605
\(940\) 0 0
\(941\) −31868.2 −1.10401 −0.552005 0.833841i \(-0.686137\pi\)
−0.552005 + 0.833841i \(0.686137\pi\)
\(942\) 0 0
\(943\) 240.775 0.00831464
\(944\) 0 0
\(945\) 1690.45 0.0581910
\(946\) 0 0
\(947\) 19262.6 0.660983 0.330491 0.943809i \(-0.392786\pi\)
0.330491 + 0.943809i \(0.392786\pi\)
\(948\) 0 0
\(949\) 39605.0 1.35472
\(950\) 0 0
\(951\) 10915.8 0.372209
\(952\) 0 0
\(953\) −32426.9 −1.10221 −0.551106 0.834435i \(-0.685794\pi\)
−0.551106 + 0.834435i \(0.685794\pi\)
\(954\) 0 0
\(955\) −2882.86 −0.0976829
\(956\) 0 0
\(957\) −5102.23 −0.172342
\(958\) 0 0
\(959\) −13709.8 −0.461639
\(960\) 0 0
\(961\) 11184.5 0.375432
\(962\) 0 0
\(963\) −15633.9 −0.523153
\(964\) 0 0
\(965\) −7878.12 −0.262804
\(966\) 0 0
\(967\) 25418.9 0.845313 0.422656 0.906290i \(-0.361098\pi\)
0.422656 + 0.906290i \(0.361098\pi\)
\(968\) 0 0
\(969\) 1580.00 0.0523807
\(970\) 0 0
\(971\) −11477.7 −0.379339 −0.189670 0.981848i \(-0.560742\pi\)
−0.189670 + 0.981848i \(0.560742\pi\)
\(972\) 0 0
\(973\) −16502.3 −0.543719
\(974\) 0 0
\(975\) −55002.4 −1.80665
\(976\) 0 0
\(977\) −31621.3 −1.03547 −0.517735 0.855541i \(-0.673225\pi\)
−0.517735 + 0.855541i \(0.673225\pi\)
\(978\) 0 0
\(979\) −4793.91 −0.156500
\(980\) 0 0
\(981\) −54065.6 −1.75961
\(982\) 0 0
\(983\) 22000.0 0.713827 0.356913 0.934137i \(-0.383829\pi\)
0.356913 + 0.934137i \(0.383829\pi\)
\(984\) 0 0
\(985\) −627.604 −0.0203017
\(986\) 0 0
\(987\) −22620.7 −0.729509
\(988\) 0 0
\(989\) −1107.73 −0.0356155
\(990\) 0 0
\(991\) 31554.0 1.01145 0.505724 0.862695i \(-0.331225\pi\)
0.505724 + 0.862695i \(0.331225\pi\)
\(992\) 0 0
\(993\) −82009.2 −2.62083
\(994\) 0 0
\(995\) −2795.04 −0.0890540
\(996\) 0 0
\(997\) 57792.3 1.83581 0.917904 0.396802i \(-0.129880\pi\)
0.917904 + 0.396802i \(0.129880\pi\)
\(998\) 0 0
\(999\) 11567.3 0.366338
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 1216.4.a.r.1.1 3
4.3 odd 2 1216.4.a.w.1.3 3
8.3 odd 2 304.4.a.g.1.1 3
8.5 even 2 152.4.a.c.1.3 3
24.5 odd 2 1368.4.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.4.a.c.1.3 3 8.5 even 2
304.4.a.g.1.1 3 8.3 odd 2
1216.4.a.r.1.1 3 1.1 even 1 trivial
1216.4.a.w.1.3 3 4.3 odd 2
1368.4.a.d.1.2 3 24.5 odd 2