Properties

Label 736.4.a.e.1.4
Level $736$
Weight $4$
Character 736.1
Self dual yes
Analytic conductor $43.425$
Analytic rank $1$
Dimension $8$
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] = [736,4,Mod(1,736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(736, 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("736.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 736.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: \(43.4254057642\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 137x^{6} + 344x^{5} + 6175x^{4} - 7924x^{3} - 89643x^{2} + 45072x + 51084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.557812\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.55781 q^{3} -15.7437 q^{5} +25.1863 q^{7} -20.4576 q^{9} -15.8480 q^{11} +15.4997 q^{13} +40.2693 q^{15} +56.7767 q^{17} +107.137 q^{19} -64.4219 q^{21} +23.0000 q^{23} +122.863 q^{25} +121.388 q^{27} -267.565 q^{29} -35.0813 q^{31} +40.5361 q^{33} -396.525 q^{35} +84.9213 q^{37} -39.6452 q^{39} +296.807 q^{41} -353.475 q^{43} +322.078 q^{45} +86.6482 q^{47} +291.352 q^{49} -145.224 q^{51} -126.747 q^{53} +249.505 q^{55} -274.036 q^{57} -853.472 q^{59} +647.571 q^{61} -515.252 q^{63} -244.021 q^{65} -603.269 q^{67} -58.8297 q^{69} +467.385 q^{71} +301.578 q^{73} -314.260 q^{75} -399.152 q^{77} -766.955 q^{79} +241.868 q^{81} -660.843 q^{83} -893.873 q^{85} +684.381 q^{87} -1101.08 q^{89} +390.380 q^{91} +89.7314 q^{93} -1686.73 q^{95} -1494.52 q^{97} +324.211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} + 14 q^{7} + 90 q^{9} - 88 q^{11} - 30 q^{13} - 30 q^{15} + 58 q^{17} - 190 q^{19} - 66 q^{21} + 184 q^{23} + 28 q^{25} - 432 q^{27} + 190 q^{29} + 60 q^{31} + 346 q^{33} - 192 q^{35}+ \cdots - 5986 q^{99}+O(q^{100}) \) 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
\(3\) −2.55781 −0.492251 −0.246126 0.969238i \(-0.579158\pi\)
−0.246126 + 0.969238i \(0.579158\pi\)
\(4\) 0 0
\(5\) −15.7437 −1.40816 −0.704078 0.710123i \(-0.748640\pi\)
−0.704078 + 0.710123i \(0.748640\pi\)
\(6\) 0 0
\(7\) 25.1863 1.35993 0.679967 0.733242i \(-0.261994\pi\)
0.679967 + 0.733242i \(0.261994\pi\)
\(8\) 0 0
\(9\) −20.4576 −0.757689
\(10\) 0 0
\(11\) −15.8480 −0.434394 −0.217197 0.976128i \(-0.569691\pi\)
−0.217197 + 0.976128i \(0.569691\pi\)
\(12\) 0 0
\(13\) 15.4997 0.330679 0.165340 0.986237i \(-0.447128\pi\)
0.165340 + 0.986237i \(0.447128\pi\)
\(14\) 0 0
\(15\) 40.2693 0.693167
\(16\) 0 0
\(17\) 56.7767 0.810021 0.405011 0.914312i \(-0.367268\pi\)
0.405011 + 0.914312i \(0.367268\pi\)
\(18\) 0 0
\(19\) 107.137 1.29362 0.646812 0.762649i \(-0.276102\pi\)
0.646812 + 0.762649i \(0.276102\pi\)
\(20\) 0 0
\(21\) −64.4219 −0.669430
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 122.863 0.982904
\(26\) 0 0
\(27\) 121.388 0.865224
\(28\) 0 0
\(29\) −267.565 −1.71330 −0.856648 0.515901i \(-0.827457\pi\)
−0.856648 + 0.515901i \(0.827457\pi\)
\(30\) 0 0
\(31\) −35.0813 −0.203251 −0.101626 0.994823i \(-0.532404\pi\)
−0.101626 + 0.994823i \(0.532404\pi\)
\(32\) 0 0
\(33\) 40.5361 0.213831
\(34\) 0 0
\(35\) −396.525 −1.91500
\(36\) 0 0
\(37\) 84.9213 0.377324 0.188662 0.982042i \(-0.439585\pi\)
0.188662 + 0.982042i \(0.439585\pi\)
\(38\) 0 0
\(39\) −39.6452 −0.162777
\(40\) 0 0
\(41\) 296.807 1.13057 0.565286 0.824895i \(-0.308766\pi\)
0.565286 + 0.824895i \(0.308766\pi\)
\(42\) 0 0
\(43\) −353.475 −1.25359 −0.626795 0.779184i \(-0.715634\pi\)
−0.626795 + 0.779184i \(0.715634\pi\)
\(44\) 0 0
\(45\) 322.078 1.06694
\(46\) 0 0
\(47\) 86.6482 0.268913 0.134457 0.990919i \(-0.457071\pi\)
0.134457 + 0.990919i \(0.457071\pi\)
\(48\) 0 0
\(49\) 291.352 0.849423
\(50\) 0 0
\(51\) −145.224 −0.398734
\(52\) 0 0
\(53\) −126.747 −0.328491 −0.164246 0.986419i \(-0.552519\pi\)
−0.164246 + 0.986419i \(0.552519\pi\)
\(54\) 0 0
\(55\) 249.505 0.611695
\(56\) 0 0
\(57\) −274.036 −0.636788
\(58\) 0 0
\(59\) −853.472 −1.88326 −0.941632 0.336644i \(-0.890708\pi\)
−0.941632 + 0.336644i \(0.890708\pi\)
\(60\) 0 0
\(61\) 647.571 1.35923 0.679614 0.733570i \(-0.262147\pi\)
0.679614 + 0.733570i \(0.262147\pi\)
\(62\) 0 0
\(63\) −515.252 −1.03041
\(64\) 0 0
\(65\) −244.021 −0.465648
\(66\) 0 0
\(67\) −603.269 −1.10002 −0.550008 0.835160i \(-0.685375\pi\)
−0.550008 + 0.835160i \(0.685375\pi\)
\(68\) 0 0
\(69\) −58.8297 −0.102641
\(70\) 0 0
\(71\) 467.385 0.781245 0.390623 0.920551i \(-0.372260\pi\)
0.390623 + 0.920551i \(0.372260\pi\)
\(72\) 0 0
\(73\) 301.578 0.483522 0.241761 0.970336i \(-0.422275\pi\)
0.241761 + 0.970336i \(0.422275\pi\)
\(74\) 0 0
\(75\) −314.260 −0.483836
\(76\) 0 0
\(77\) −399.152 −0.590748
\(78\) 0 0
\(79\) −766.955 −1.09227 −0.546134 0.837698i \(-0.683901\pi\)
−0.546134 + 0.837698i \(0.683901\pi\)
\(80\) 0 0
\(81\) 241.868 0.331781
\(82\) 0 0
\(83\) −660.843 −0.873939 −0.436969 0.899476i \(-0.643948\pi\)
−0.436969 + 0.899476i \(0.643948\pi\)
\(84\) 0 0
\(85\) −893.873 −1.14064
\(86\) 0 0
\(87\) 684.381 0.843372
\(88\) 0 0
\(89\) −1101.08 −1.31140 −0.655700 0.755021i \(-0.727626\pi\)
−0.655700 + 0.755021i \(0.727626\pi\)
\(90\) 0 0
\(91\) 390.380 0.449702
\(92\) 0 0
\(93\) 89.7314 0.100051
\(94\) 0 0
\(95\) −1686.73 −1.82163
\(96\) 0 0
\(97\) −1494.52 −1.56438 −0.782192 0.623037i \(-0.785899\pi\)
−0.782192 + 0.623037i \(0.785899\pi\)
\(98\) 0 0
\(99\) 324.211 0.329136
\(100\) 0 0
\(101\) −1184.71 −1.16716 −0.583578 0.812057i \(-0.698348\pi\)
−0.583578 + 0.812057i \(0.698348\pi\)
\(102\) 0 0
\(103\) −374.047 −0.357825 −0.178912 0.983865i \(-0.557258\pi\)
−0.178912 + 0.983865i \(0.557258\pi\)
\(104\) 0 0
\(105\) 1014.24 0.942661
\(106\) 0 0
\(107\) −2052.88 −1.85476 −0.927380 0.374122i \(-0.877944\pi\)
−0.927380 + 0.374122i \(0.877944\pi\)
\(108\) 0 0
\(109\) 547.325 0.480956 0.240478 0.970655i \(-0.422696\pi\)
0.240478 + 0.970655i \(0.422696\pi\)
\(110\) 0 0
\(111\) −217.213 −0.185738
\(112\) 0 0
\(113\) 250.013 0.208135 0.104067 0.994570i \(-0.466814\pi\)
0.104067 + 0.994570i \(0.466814\pi\)
\(114\) 0 0
\(115\) −362.104 −0.293621
\(116\) 0 0
\(117\) −317.086 −0.250552
\(118\) 0 0
\(119\) 1430.00 1.10158
\(120\) 0 0
\(121\) −1079.84 −0.811302
\(122\) 0 0
\(123\) −759.176 −0.556525
\(124\) 0 0
\(125\) 33.6445 0.0240741
\(126\) 0 0
\(127\) −654.293 −0.457158 −0.228579 0.973525i \(-0.573408\pi\)
−0.228579 + 0.973525i \(0.573408\pi\)
\(128\) 0 0
\(129\) 904.122 0.617081
\(130\) 0 0
\(131\) 1679.16 1.11992 0.559958 0.828521i \(-0.310817\pi\)
0.559958 + 0.828521i \(0.310817\pi\)
\(132\) 0 0
\(133\) 2698.38 1.75924
\(134\) 0 0
\(135\) −1911.09 −1.21837
\(136\) 0 0
\(137\) −1344.51 −0.838463 −0.419232 0.907879i \(-0.637700\pi\)
−0.419232 + 0.907879i \(0.637700\pi\)
\(138\) 0 0
\(139\) 1729.97 1.05564 0.527821 0.849356i \(-0.323009\pi\)
0.527821 + 0.849356i \(0.323009\pi\)
\(140\) 0 0
\(141\) −221.630 −0.132373
\(142\) 0 0
\(143\) −245.638 −0.143645
\(144\) 0 0
\(145\) 4212.46 2.41259
\(146\) 0 0
\(147\) −745.224 −0.418129
\(148\) 0 0
\(149\) 272.695 0.149933 0.0749667 0.997186i \(-0.476115\pi\)
0.0749667 + 0.997186i \(0.476115\pi\)
\(150\) 0 0
\(151\) 1696.91 0.914519 0.457259 0.889333i \(-0.348831\pi\)
0.457259 + 0.889333i \(0.348831\pi\)
\(152\) 0 0
\(153\) −1161.51 −0.613744
\(154\) 0 0
\(155\) 552.308 0.286209
\(156\) 0 0
\(157\) −2111.93 −1.07357 −0.536784 0.843720i \(-0.680361\pi\)
−0.536784 + 0.843720i \(0.680361\pi\)
\(158\) 0 0
\(159\) 324.195 0.161700
\(160\) 0 0
\(161\) 579.286 0.283566
\(162\) 0 0
\(163\) −726.275 −0.348995 −0.174498 0.984658i \(-0.555830\pi\)
−0.174498 + 0.984658i \(0.555830\pi\)
\(164\) 0 0
\(165\) −638.187 −0.301108
\(166\) 0 0
\(167\) −1282.83 −0.594423 −0.297212 0.954812i \(-0.596057\pi\)
−0.297212 + 0.954812i \(0.596057\pi\)
\(168\) 0 0
\(169\) −1956.76 −0.890651
\(170\) 0 0
\(171\) −2191.76 −0.980165
\(172\) 0 0
\(173\) 1437.63 0.631797 0.315899 0.948793i \(-0.397694\pi\)
0.315899 + 0.948793i \(0.397694\pi\)
\(174\) 0 0
\(175\) 3094.47 1.33669
\(176\) 0 0
\(177\) 2183.02 0.927039
\(178\) 0 0
\(179\) −4243.83 −1.77206 −0.886031 0.463627i \(-0.846548\pi\)
−0.886031 + 0.463627i \(0.846548\pi\)
\(180\) 0 0
\(181\) 693.537 0.284808 0.142404 0.989809i \(-0.454517\pi\)
0.142404 + 0.989809i \(0.454517\pi\)
\(182\) 0 0
\(183\) −1656.36 −0.669082
\(184\) 0 0
\(185\) −1336.97 −0.531331
\(186\) 0 0
\(187\) −899.794 −0.351869
\(188\) 0 0
\(189\) 3057.31 1.17665
\(190\) 0 0
\(191\) −1692.66 −0.641239 −0.320620 0.947208i \(-0.603891\pi\)
−0.320620 + 0.947208i \(0.603891\pi\)
\(192\) 0 0
\(193\) 1028.12 0.383447 0.191724 0.981449i \(-0.438592\pi\)
0.191724 + 0.981449i \(0.438592\pi\)
\(194\) 0 0
\(195\) 624.161 0.229216
\(196\) 0 0
\(197\) −548.484 −0.198365 −0.0991823 0.995069i \(-0.531623\pi\)
−0.0991823 + 0.995069i \(0.531623\pi\)
\(198\) 0 0
\(199\) 2784.57 0.991925 0.495962 0.868344i \(-0.334815\pi\)
0.495962 + 0.868344i \(0.334815\pi\)
\(200\) 0 0
\(201\) 1543.05 0.541484
\(202\) 0 0
\(203\) −6738.99 −2.32997
\(204\) 0 0
\(205\) −4672.83 −1.59202
\(206\) 0 0
\(207\) −470.525 −0.157989
\(208\) 0 0
\(209\) −1697.90 −0.561943
\(210\) 0 0
\(211\) 3344.97 1.09136 0.545681 0.837993i \(-0.316271\pi\)
0.545681 + 0.837993i \(0.316271\pi\)
\(212\) 0 0
\(213\) −1195.48 −0.384569
\(214\) 0 0
\(215\) 5564.98 1.76525
\(216\) 0 0
\(217\) −883.569 −0.276408
\(218\) 0 0
\(219\) −771.381 −0.238014
\(220\) 0 0
\(221\) 880.019 0.267857
\(222\) 0 0
\(223\) 3449.19 1.03576 0.517881 0.855453i \(-0.326721\pi\)
0.517881 + 0.855453i \(0.326721\pi\)
\(224\) 0 0
\(225\) −2513.48 −0.744735
\(226\) 0 0
\(227\) −2897.08 −0.847075 −0.423538 0.905879i \(-0.639212\pi\)
−0.423538 + 0.905879i \(0.639212\pi\)
\(228\) 0 0
\(229\) −461.258 −0.133104 −0.0665519 0.997783i \(-0.521200\pi\)
−0.0665519 + 0.997783i \(0.521200\pi\)
\(230\) 0 0
\(231\) 1020.96 0.290796
\(232\) 0 0
\(233\) −889.861 −0.250200 −0.125100 0.992144i \(-0.539925\pi\)
−0.125100 + 0.992144i \(0.539925\pi\)
\(234\) 0 0
\(235\) −1364.16 −0.378672
\(236\) 0 0
\(237\) 1961.73 0.537670
\(238\) 0 0
\(239\) −4239.79 −1.14749 −0.573744 0.819035i \(-0.694509\pi\)
−0.573744 + 0.819035i \(0.694509\pi\)
\(240\) 0 0
\(241\) 4493.26 1.20098 0.600491 0.799632i \(-0.294972\pi\)
0.600491 + 0.799632i \(0.294972\pi\)
\(242\) 0 0
\(243\) −3896.12 −1.02854
\(244\) 0 0
\(245\) −4586.95 −1.19612
\(246\) 0 0
\(247\) 1660.58 0.427775
\(248\) 0 0
\(249\) 1690.31 0.430197
\(250\) 0 0
\(251\) −5012.13 −1.26041 −0.630205 0.776429i \(-0.717029\pi\)
−0.630205 + 0.776429i \(0.717029\pi\)
\(252\) 0 0
\(253\) −364.503 −0.0905775
\(254\) 0 0
\(255\) 2286.36 0.561480
\(256\) 0 0
\(257\) −3557.61 −0.863493 −0.431746 0.901995i \(-0.642102\pi\)
−0.431746 + 0.901995i \(0.642102\pi\)
\(258\) 0 0
\(259\) 2138.86 0.513136
\(260\) 0 0
\(261\) 5473.74 1.29815
\(262\) 0 0
\(263\) 6499.08 1.52377 0.761883 0.647715i \(-0.224275\pi\)
0.761883 + 0.647715i \(0.224275\pi\)
\(264\) 0 0
\(265\) 1995.46 0.462567
\(266\) 0 0
\(267\) 2816.37 0.645538
\(268\) 0 0
\(269\) 1314.12 0.297856 0.148928 0.988848i \(-0.452418\pi\)
0.148928 + 0.988848i \(0.452418\pi\)
\(270\) 0 0
\(271\) −2987.30 −0.669614 −0.334807 0.942287i \(-0.608671\pi\)
−0.334807 + 0.942287i \(0.608671\pi\)
\(272\) 0 0
\(273\) −998.518 −0.221367
\(274\) 0 0
\(275\) −1947.13 −0.426968
\(276\) 0 0
\(277\) 3036.46 0.658641 0.329320 0.944218i \(-0.393180\pi\)
0.329320 + 0.944218i \(0.393180\pi\)
\(278\) 0 0
\(279\) 717.679 0.154001
\(280\) 0 0
\(281\) 2686.34 0.570298 0.285149 0.958483i \(-0.407957\pi\)
0.285149 + 0.958483i \(0.407957\pi\)
\(282\) 0 0
\(283\) 2167.83 0.455349 0.227675 0.973737i \(-0.426888\pi\)
0.227675 + 0.973737i \(0.426888\pi\)
\(284\) 0 0
\(285\) 4314.33 0.896697
\(286\) 0 0
\(287\) 7475.48 1.53750
\(288\) 0 0
\(289\) −1689.41 −0.343865
\(290\) 0 0
\(291\) 3822.70 0.770070
\(292\) 0 0
\(293\) 7312.10 1.45794 0.728972 0.684543i \(-0.239998\pi\)
0.728972 + 0.684543i \(0.239998\pi\)
\(294\) 0 0
\(295\) 13436.8 2.65193
\(296\) 0 0
\(297\) −1923.75 −0.375849
\(298\) 0 0
\(299\) 356.492 0.0689514
\(300\) 0 0
\(301\) −8902.73 −1.70480
\(302\) 0 0
\(303\) 3030.26 0.574534
\(304\) 0 0
\(305\) −10195.1 −1.91401
\(306\) 0 0
\(307\) 4204.98 0.781729 0.390864 0.920448i \(-0.372176\pi\)
0.390864 + 0.920448i \(0.372176\pi\)
\(308\) 0 0
\(309\) 956.742 0.176140
\(310\) 0 0
\(311\) 6334.77 1.15502 0.577511 0.816383i \(-0.304024\pi\)
0.577511 + 0.816383i \(0.304024\pi\)
\(312\) 0 0
\(313\) −3328.27 −0.601038 −0.300519 0.953776i \(-0.597160\pi\)
−0.300519 + 0.953776i \(0.597160\pi\)
\(314\) 0 0
\(315\) 8111.96 1.45097
\(316\) 0 0
\(317\) 8132.70 1.44094 0.720470 0.693487i \(-0.243926\pi\)
0.720470 + 0.693487i \(0.243926\pi\)
\(318\) 0 0
\(319\) 4240.36 0.744246
\(320\) 0 0
\(321\) 5250.88 0.913007
\(322\) 0 0
\(323\) 6082.87 1.04786
\(324\) 0 0
\(325\) 1904.33 0.325026
\(326\) 0 0
\(327\) −1399.95 −0.236751
\(328\) 0 0
\(329\) 2182.35 0.365705
\(330\) 0 0
\(331\) −6300.52 −1.04625 −0.523124 0.852257i \(-0.675234\pi\)
−0.523124 + 0.852257i \(0.675234\pi\)
\(332\) 0 0
\(333\) −1737.29 −0.285894
\(334\) 0 0
\(335\) 9497.66 1.54899
\(336\) 0 0
\(337\) −8473.63 −1.36970 −0.684848 0.728686i \(-0.740132\pi\)
−0.684848 + 0.728686i \(0.740132\pi\)
\(338\) 0 0
\(339\) −639.486 −0.102455
\(340\) 0 0
\(341\) 555.967 0.0882911
\(342\) 0 0
\(343\) −1300.82 −0.204775
\(344\) 0 0
\(345\) 926.195 0.144535
\(346\) 0 0
\(347\) 2859.61 0.442398 0.221199 0.975229i \(-0.429003\pi\)
0.221199 + 0.975229i \(0.429003\pi\)
\(348\) 0 0
\(349\) −10651.8 −1.63375 −0.816876 0.576814i \(-0.804296\pi\)
−0.816876 + 0.576814i \(0.804296\pi\)
\(350\) 0 0
\(351\) 1881.47 0.286112
\(352\) 0 0
\(353\) 7723.04 1.16446 0.582232 0.813022i \(-0.302179\pi\)
0.582232 + 0.813022i \(0.302179\pi\)
\(354\) 0 0
\(355\) −7358.36 −1.10012
\(356\) 0 0
\(357\) −3657.66 −0.542252
\(358\) 0 0
\(359\) −5468.59 −0.803959 −0.401979 0.915649i \(-0.631678\pi\)
−0.401979 + 0.915649i \(0.631678\pi\)
\(360\) 0 0
\(361\) 4619.29 0.673464
\(362\) 0 0
\(363\) 2762.03 0.399364
\(364\) 0 0
\(365\) −4747.95 −0.680874
\(366\) 0 0
\(367\) −13353.5 −1.89931 −0.949656 0.313295i \(-0.898567\pi\)
−0.949656 + 0.313295i \(0.898567\pi\)
\(368\) 0 0
\(369\) −6071.95 −0.856621
\(370\) 0 0
\(371\) −3192.29 −0.446727
\(372\) 0 0
\(373\) 4189.05 0.581503 0.290752 0.956799i \(-0.406095\pi\)
0.290752 + 0.956799i \(0.406095\pi\)
\(374\) 0 0
\(375\) −86.0564 −0.0118505
\(376\) 0 0
\(377\) −4147.17 −0.566552
\(378\) 0 0
\(379\) 10371.6 1.40568 0.702839 0.711349i \(-0.251915\pi\)
0.702839 + 0.711349i \(0.251915\pi\)
\(380\) 0 0
\(381\) 1673.56 0.225037
\(382\) 0 0
\(383\) −6828.72 −0.911047 −0.455524 0.890224i \(-0.650548\pi\)
−0.455524 + 0.890224i \(0.650548\pi\)
\(384\) 0 0
\(385\) 6284.12 0.831865
\(386\) 0 0
\(387\) 7231.24 0.949831
\(388\) 0 0
\(389\) −1413.57 −0.184244 −0.0921221 0.995748i \(-0.529365\pi\)
−0.0921221 + 0.995748i \(0.529365\pi\)
\(390\) 0 0
\(391\) 1305.86 0.168901
\(392\) 0 0
\(393\) −4294.98 −0.551280
\(394\) 0 0
\(395\) 12074.7 1.53808
\(396\) 0 0
\(397\) −12730.3 −1.60936 −0.804679 0.593711i \(-0.797662\pi\)
−0.804679 + 0.593711i \(0.797662\pi\)
\(398\) 0 0
\(399\) −6901.96 −0.865990
\(400\) 0 0
\(401\) 11371.4 1.41611 0.708056 0.706157i \(-0.249573\pi\)
0.708056 + 0.706157i \(0.249573\pi\)
\(402\) 0 0
\(403\) −543.748 −0.0672110
\(404\) 0 0
\(405\) −3807.89 −0.467199
\(406\) 0 0
\(407\) −1345.83 −0.163907
\(408\) 0 0
\(409\) 9102.21 1.10043 0.550215 0.835023i \(-0.314546\pi\)
0.550215 + 0.835023i \(0.314546\pi\)
\(410\) 0 0
\(411\) 3439.01 0.412735
\(412\) 0 0
\(413\) −21495.8 −2.56112
\(414\) 0 0
\(415\) 10404.1 1.23064
\(416\) 0 0
\(417\) −4424.94 −0.519641
\(418\) 0 0
\(419\) −15235.3 −1.77635 −0.888176 0.459504i \(-0.848027\pi\)
−0.888176 + 0.459504i \(0.848027\pi\)
\(420\) 0 0
\(421\) 7711.38 0.892707 0.446354 0.894857i \(-0.352722\pi\)
0.446354 + 0.894857i \(0.352722\pi\)
\(422\) 0 0
\(423\) −1772.61 −0.203753
\(424\) 0 0
\(425\) 6975.75 0.796173
\(426\) 0 0
\(427\) 16309.9 1.84846
\(428\) 0 0
\(429\) 628.296 0.0707096
\(430\) 0 0
\(431\) −2248.78 −0.251322 −0.125661 0.992073i \(-0.540105\pi\)
−0.125661 + 0.992073i \(0.540105\pi\)
\(432\) 0 0
\(433\) 7405.23 0.821877 0.410938 0.911663i \(-0.365201\pi\)
0.410938 + 0.911663i \(0.365201\pi\)
\(434\) 0 0
\(435\) −10774.7 −1.18760
\(436\) 0 0
\(437\) 2464.15 0.269739
\(438\) 0 0
\(439\) −6226.42 −0.676927 −0.338463 0.940980i \(-0.609907\pi\)
−0.338463 + 0.940980i \(0.609907\pi\)
\(440\) 0 0
\(441\) −5960.36 −0.643598
\(442\) 0 0
\(443\) −15367.3 −1.64813 −0.824064 0.566497i \(-0.808298\pi\)
−0.824064 + 0.566497i \(0.808298\pi\)
\(444\) 0 0
\(445\) 17335.1 1.84666
\(446\) 0 0
\(447\) −697.503 −0.0738049
\(448\) 0 0
\(449\) −16205.8 −1.70334 −0.851668 0.524081i \(-0.824409\pi\)
−0.851668 + 0.524081i \(0.824409\pi\)
\(450\) 0 0
\(451\) −4703.78 −0.491114
\(452\) 0 0
\(453\) −4340.37 −0.450173
\(454\) 0 0
\(455\) −6146.01 −0.633251
\(456\) 0 0
\(457\) −15393.8 −1.57569 −0.787846 0.615872i \(-0.788804\pi\)
−0.787846 + 0.615872i \(0.788804\pi\)
\(458\) 0 0
\(459\) 6891.98 0.700850
\(460\) 0 0
\(461\) 3078.57 0.311026 0.155513 0.987834i \(-0.450297\pi\)
0.155513 + 0.987834i \(0.450297\pi\)
\(462\) 0 0
\(463\) −12389.8 −1.24364 −0.621820 0.783160i \(-0.713606\pi\)
−0.621820 + 0.783160i \(0.713606\pi\)
\(464\) 0 0
\(465\) −1412.70 −0.140887
\(466\) 0 0
\(467\) 9711.19 0.962271 0.481135 0.876646i \(-0.340225\pi\)
0.481135 + 0.876646i \(0.340225\pi\)
\(468\) 0 0
\(469\) −15194.1 −1.49595
\(470\) 0 0
\(471\) 5401.91 0.528465
\(472\) 0 0
\(473\) 5601.85 0.544552
\(474\) 0 0
\(475\) 13163.1 1.27151
\(476\) 0 0
\(477\) 2592.94 0.248894
\(478\) 0 0
\(479\) 11418.2 1.08917 0.544583 0.838707i \(-0.316688\pi\)
0.544583 + 0.838707i \(0.316688\pi\)
\(480\) 0 0
\(481\) 1316.25 0.124773
\(482\) 0 0
\(483\) −1481.70 −0.139586
\(484\) 0 0
\(485\) 23529.2 2.20290
\(486\) 0 0
\(487\) −8515.00 −0.792303 −0.396151 0.918185i \(-0.629654\pi\)
−0.396151 + 0.918185i \(0.629654\pi\)
\(488\) 0 0
\(489\) 1857.68 0.171793
\(490\) 0 0
\(491\) −5868.20 −0.539365 −0.269683 0.962949i \(-0.586919\pi\)
−0.269683 + 0.962949i \(0.586919\pi\)
\(492\) 0 0
\(493\) −15191.5 −1.38781
\(494\) 0 0
\(495\) −5104.27 −0.463474
\(496\) 0 0
\(497\) 11771.7 1.06244
\(498\) 0 0
\(499\) −8500.38 −0.762584 −0.381292 0.924455i \(-0.624521\pi\)
−0.381292 + 0.924455i \(0.624521\pi\)
\(500\) 0 0
\(501\) 3281.25 0.292606
\(502\) 0 0
\(503\) −12383.7 −1.09774 −0.548869 0.835909i \(-0.684941\pi\)
−0.548869 + 0.835909i \(0.684941\pi\)
\(504\) 0 0
\(505\) 18651.6 1.64354
\(506\) 0 0
\(507\) 5005.03 0.438424
\(508\) 0 0
\(509\) 7608.92 0.662593 0.331296 0.943527i \(-0.392514\pi\)
0.331296 + 0.943527i \(0.392514\pi\)
\(510\) 0 0
\(511\) 7595.66 0.657558
\(512\) 0 0
\(513\) 13005.1 1.11928
\(514\) 0 0
\(515\) 5888.87 0.503873
\(516\) 0 0
\(517\) −1373.20 −0.116814
\(518\) 0 0
\(519\) −3677.19 −0.311003
\(520\) 0 0
\(521\) 19155.0 1.61074 0.805372 0.592770i \(-0.201966\pi\)
0.805372 + 0.592770i \(0.201966\pi\)
\(522\) 0 0
\(523\) 2705.89 0.226234 0.113117 0.993582i \(-0.463917\pi\)
0.113117 + 0.993582i \(0.463917\pi\)
\(524\) 0 0
\(525\) −7915.07 −0.657985
\(526\) 0 0
\(527\) −1991.80 −0.164638
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) 17460.0 1.42693
\(532\) 0 0
\(533\) 4600.40 0.373857
\(534\) 0 0
\(535\) 32319.8 2.61179
\(536\) 0 0
\(537\) 10854.9 0.872299
\(538\) 0 0
\(539\) −4617.33 −0.368984
\(540\) 0 0
\(541\) 14435.0 1.14715 0.573575 0.819153i \(-0.305556\pi\)
0.573575 + 0.819153i \(0.305556\pi\)
\(542\) 0 0
\(543\) −1773.94 −0.140197
\(544\) 0 0
\(545\) −8616.90 −0.677262
\(546\) 0 0
\(547\) −6800.62 −0.531578 −0.265789 0.964031i \(-0.585633\pi\)
−0.265789 + 0.964031i \(0.585633\pi\)
\(548\) 0 0
\(549\) −13247.7 −1.02987
\(550\) 0 0
\(551\) −28666.1 −2.21636
\(552\) 0 0
\(553\) −19316.8 −1.48541
\(554\) 0 0
\(555\) 3419.73 0.261548
\(556\) 0 0
\(557\) 19804.9 1.50657 0.753285 0.657694i \(-0.228468\pi\)
0.753285 + 0.657694i \(0.228468\pi\)
\(558\) 0 0
\(559\) −5478.74 −0.414536
\(560\) 0 0
\(561\) 2301.50 0.173208
\(562\) 0 0
\(563\) −13125.5 −0.982546 −0.491273 0.871006i \(-0.663468\pi\)
−0.491273 + 0.871006i \(0.663468\pi\)
\(564\) 0 0
\(565\) −3936.12 −0.293087
\(566\) 0 0
\(567\) 6091.78 0.451200
\(568\) 0 0
\(569\) −14961.4 −1.10231 −0.551155 0.834403i \(-0.685813\pi\)
−0.551155 + 0.834403i \(0.685813\pi\)
\(570\) 0 0
\(571\) −6809.52 −0.499071 −0.249536 0.968366i \(-0.580278\pi\)
−0.249536 + 0.968366i \(0.580278\pi\)
\(572\) 0 0
\(573\) 4329.51 0.315651
\(574\) 0 0
\(575\) 2825.85 0.204950
\(576\) 0 0
\(577\) −21373.0 −1.54206 −0.771032 0.636796i \(-0.780259\pi\)
−0.771032 + 0.636796i \(0.780259\pi\)
\(578\) 0 0
\(579\) −2629.73 −0.188752
\(580\) 0 0
\(581\) −16644.2 −1.18850
\(582\) 0 0
\(583\) 2008.68 0.142695
\(584\) 0 0
\(585\) 4992.09 0.352816
\(586\) 0 0
\(587\) 1796.18 0.126297 0.0631483 0.998004i \(-0.479886\pi\)
0.0631483 + 0.998004i \(0.479886\pi\)
\(588\) 0 0
\(589\) −3758.50 −0.262931
\(590\) 0 0
\(591\) 1402.92 0.0976452
\(592\) 0 0
\(593\) 14393.9 0.996773 0.498386 0.866955i \(-0.333926\pi\)
0.498386 + 0.866955i \(0.333926\pi\)
\(594\) 0 0
\(595\) −22513.4 −1.55119
\(596\) 0 0
\(597\) −7122.41 −0.488276
\(598\) 0 0
\(599\) 6674.39 0.455272 0.227636 0.973746i \(-0.426900\pi\)
0.227636 + 0.973746i \(0.426900\pi\)
\(600\) 0 0
\(601\) −10008.1 −0.679263 −0.339631 0.940559i \(-0.610302\pi\)
−0.339631 + 0.940559i \(0.610302\pi\)
\(602\) 0 0
\(603\) 12341.4 0.833469
\(604\) 0 0
\(605\) 17000.7 1.14244
\(606\) 0 0
\(607\) −25442.5 −1.70128 −0.850642 0.525745i \(-0.823787\pi\)
−0.850642 + 0.525745i \(0.823787\pi\)
\(608\) 0 0
\(609\) 17237.1 1.14693
\(610\) 0 0
\(611\) 1343.02 0.0889241
\(612\) 0 0
\(613\) 7269.39 0.478969 0.239484 0.970900i \(-0.423022\pi\)
0.239484 + 0.970900i \(0.423022\pi\)
\(614\) 0 0
\(615\) 11952.2 0.783674
\(616\) 0 0
\(617\) −13634.2 −0.889614 −0.444807 0.895626i \(-0.646728\pi\)
−0.444807 + 0.895626i \(0.646728\pi\)
\(618\) 0 0
\(619\) 16180.2 1.05063 0.525313 0.850909i \(-0.323948\pi\)
0.525313 + 0.850909i \(0.323948\pi\)
\(620\) 0 0
\(621\) 2791.92 0.180412
\(622\) 0 0
\(623\) −27732.3 −1.78342
\(624\) 0 0
\(625\) −15887.6 −1.01680
\(626\) 0 0
\(627\) 4342.91 0.276617
\(628\) 0 0
\(629\) 4821.55 0.305640
\(630\) 0 0
\(631\) −4454.90 −0.281057 −0.140528 0.990077i \(-0.544880\pi\)
−0.140528 + 0.990077i \(0.544880\pi\)
\(632\) 0 0
\(633\) −8555.81 −0.537224
\(634\) 0 0
\(635\) 10301.0 0.643750
\(636\) 0 0
\(637\) 4515.86 0.280887
\(638\) 0 0
\(639\) −9561.58 −0.591941
\(640\) 0 0
\(641\) −14638.1 −0.901980 −0.450990 0.892529i \(-0.648929\pi\)
−0.450990 + 0.892529i \(0.648929\pi\)
\(642\) 0 0
\(643\) −18574.8 −1.13922 −0.569610 0.821915i \(-0.692906\pi\)
−0.569610 + 0.821915i \(0.692906\pi\)
\(644\) 0 0
\(645\) −14234.2 −0.868947
\(646\) 0 0
\(647\) 9846.84 0.598330 0.299165 0.954201i \(-0.403292\pi\)
0.299165 + 0.954201i \(0.403292\pi\)
\(648\) 0 0
\(649\) 13525.8 0.818079
\(650\) 0 0
\(651\) 2260.00 0.136062
\(652\) 0 0
\(653\) −23223.5 −1.39174 −0.695869 0.718169i \(-0.744980\pi\)
−0.695869 + 0.718169i \(0.744980\pi\)
\(654\) 0 0
\(655\) −26436.1 −1.57702
\(656\) 0 0
\(657\) −6169.57 −0.366359
\(658\) 0 0
\(659\) −28368.7 −1.67692 −0.838459 0.544965i \(-0.816543\pi\)
−0.838459 + 0.544965i \(0.816543\pi\)
\(660\) 0 0
\(661\) 15263.8 0.898173 0.449087 0.893488i \(-0.351749\pi\)
0.449087 + 0.893488i \(0.351749\pi\)
\(662\) 0 0
\(663\) −2250.92 −0.131853
\(664\) 0 0
\(665\) −42482.5 −2.47729
\(666\) 0 0
\(667\) −6154.00 −0.357247
\(668\) 0 0
\(669\) −8822.38 −0.509855
\(670\) 0 0
\(671\) −10262.7 −0.590441
\(672\) 0 0
\(673\) −11576.8 −0.663082 −0.331541 0.943441i \(-0.607569\pi\)
−0.331541 + 0.943441i \(0.607569\pi\)
\(674\) 0 0
\(675\) 14914.0 0.850432
\(676\) 0 0
\(677\) −8245.18 −0.468077 −0.234039 0.972227i \(-0.575194\pi\)
−0.234039 + 0.972227i \(0.575194\pi\)
\(678\) 0 0
\(679\) −37641.5 −2.12746
\(680\) 0 0
\(681\) 7410.19 0.416974
\(682\) 0 0
\(683\) 34421.5 1.92841 0.964203 0.265167i \(-0.0854269\pi\)
0.964203 + 0.265167i \(0.0854269\pi\)
\(684\) 0 0
\(685\) 21167.6 1.18069
\(686\) 0 0
\(687\) 1179.81 0.0655205
\(688\) 0 0
\(689\) −1964.54 −0.108625
\(690\) 0 0
\(691\) −28258.4 −1.55571 −0.777857 0.628441i \(-0.783693\pi\)
−0.777857 + 0.628441i \(0.783693\pi\)
\(692\) 0 0
\(693\) 8165.69 0.447603
\(694\) 0 0
\(695\) −27236.1 −1.48651
\(696\) 0 0
\(697\) 16851.7 0.915787
\(698\) 0 0
\(699\) 2276.10 0.123161
\(700\) 0 0
\(701\) 19513.9 1.05140 0.525700 0.850670i \(-0.323804\pi\)
0.525700 + 0.850670i \(0.323804\pi\)
\(702\) 0 0
\(703\) 9098.20 0.488115
\(704\) 0 0
\(705\) 3489.26 0.186402
\(706\) 0 0
\(707\) −29838.4 −1.58725
\(708\) 0 0
\(709\) 12679.3 0.671624 0.335812 0.941929i \(-0.390989\pi\)
0.335812 + 0.941929i \(0.390989\pi\)
\(710\) 0 0
\(711\) 15690.0 0.827599
\(712\) 0 0
\(713\) −806.870 −0.0423808
\(714\) 0 0
\(715\) 3867.24 0.202275
\(716\) 0 0
\(717\) 10844.6 0.564852
\(718\) 0 0
\(719\) −25606.3 −1.32817 −0.664085 0.747657i \(-0.731179\pi\)
−0.664085 + 0.747657i \(0.731179\pi\)
\(720\) 0 0
\(721\) −9420.87 −0.486618
\(722\) 0 0
\(723\) −11492.9 −0.591184
\(724\) 0 0
\(725\) −32873.8 −1.68401
\(726\) 0 0
\(727\) −5687.29 −0.290138 −0.145069 0.989422i \(-0.546340\pi\)
−0.145069 + 0.989422i \(0.546340\pi\)
\(728\) 0 0
\(729\) 3435.10 0.174521
\(730\) 0 0
\(731\) −20069.1 −1.01543
\(732\) 0 0
\(733\) 26753.0 1.34808 0.674040 0.738694i \(-0.264557\pi\)
0.674040 + 0.738694i \(0.264557\pi\)
\(734\) 0 0
\(735\) 11732.6 0.588791
\(736\) 0 0
\(737\) 9560.58 0.477840
\(738\) 0 0
\(739\) 18859.2 0.938763 0.469381 0.882996i \(-0.344477\pi\)
0.469381 + 0.882996i \(0.344477\pi\)
\(740\) 0 0
\(741\) −4247.46 −0.210573
\(742\) 0 0
\(743\) 19561.0 0.965848 0.482924 0.875662i \(-0.339575\pi\)
0.482924 + 0.875662i \(0.339575\pi\)
\(744\) 0 0
\(745\) −4293.22 −0.211130
\(746\) 0 0
\(747\) 13519.3 0.662173
\(748\) 0 0
\(749\) −51704.5 −2.52235
\(750\) 0 0
\(751\) −14946.6 −0.726242 −0.363121 0.931742i \(-0.618289\pi\)
−0.363121 + 0.931742i \(0.618289\pi\)
\(752\) 0 0
\(753\) 12820.1 0.620438
\(754\) 0 0
\(755\) −26715.5 −1.28778
\(756\) 0 0
\(757\) 13741.0 0.659743 0.329871 0.944026i \(-0.392995\pi\)
0.329871 + 0.944026i \(0.392995\pi\)
\(758\) 0 0
\(759\) 932.330 0.0445869
\(760\) 0 0
\(761\) −25846.0 −1.23116 −0.615582 0.788073i \(-0.711079\pi\)
−0.615582 + 0.788073i \(0.711079\pi\)
\(762\) 0 0
\(763\) 13785.1 0.654069
\(764\) 0 0
\(765\) 18286.5 0.864247
\(766\) 0 0
\(767\) −13228.5 −0.622757
\(768\) 0 0
\(769\) 16125.9 0.756195 0.378097 0.925766i \(-0.376578\pi\)
0.378097 + 0.925766i \(0.376578\pi\)
\(770\) 0 0
\(771\) 9099.70 0.425055
\(772\) 0 0
\(773\) 8845.77 0.411592 0.205796 0.978595i \(-0.434022\pi\)
0.205796 + 0.978595i \(0.434022\pi\)
\(774\) 0 0
\(775\) −4310.19 −0.199776
\(776\) 0 0
\(777\) −5470.80 −0.252592
\(778\) 0 0
\(779\) 31798.9 1.46253
\(780\) 0 0
\(781\) −7407.10 −0.339369
\(782\) 0 0
\(783\) −32479.1 −1.48239
\(784\) 0 0
\(785\) 33249.5 1.51175
\(786\) 0 0
\(787\) 15915.8 0.720885 0.360443 0.932781i \(-0.382626\pi\)
0.360443 + 0.932781i \(0.382626\pi\)
\(788\) 0 0
\(789\) −16623.4 −0.750076
\(790\) 0 0
\(791\) 6296.91 0.283050
\(792\) 0 0
\(793\) 10037.1 0.449469
\(794\) 0 0
\(795\) −5104.02 −0.227699
\(796\) 0 0
\(797\) −16815.4 −0.747342 −0.373671 0.927561i \(-0.621901\pi\)
−0.373671 + 0.927561i \(0.621901\pi\)
\(798\) 0 0
\(799\) 4919.59 0.217826
\(800\) 0 0
\(801\) 22525.5 0.993633
\(802\) 0 0
\(803\) −4779.40 −0.210039
\(804\) 0 0
\(805\) −9120.08 −0.399305
\(806\) 0 0
\(807\) −3361.27 −0.146620
\(808\) 0 0
\(809\) 1908.47 0.0829395 0.0414698 0.999140i \(-0.486796\pi\)
0.0414698 + 0.999140i \(0.486796\pi\)
\(810\) 0 0
\(811\) −18224.0 −0.789066 −0.394533 0.918882i \(-0.629094\pi\)
−0.394533 + 0.918882i \(0.629094\pi\)
\(812\) 0 0
\(813\) 7640.95 0.329618
\(814\) 0 0
\(815\) 11434.2 0.491440
\(816\) 0 0
\(817\) −37870.1 −1.62167
\(818\) 0 0
\(819\) −7986.23 −0.340734
\(820\) 0 0
\(821\) −16782.4 −0.713410 −0.356705 0.934217i \(-0.616100\pi\)
−0.356705 + 0.934217i \(0.616100\pi\)
\(822\) 0 0
\(823\) 35573.1 1.50668 0.753341 0.657630i \(-0.228441\pi\)
0.753341 + 0.657630i \(0.228441\pi\)
\(824\) 0 0
\(825\) 4980.38 0.210175
\(826\) 0 0
\(827\) 23590.3 0.991918 0.495959 0.868346i \(-0.334817\pi\)
0.495959 + 0.868346i \(0.334817\pi\)
\(828\) 0 0
\(829\) −20356.1 −0.852829 −0.426414 0.904528i \(-0.640223\pi\)
−0.426414 + 0.904528i \(0.640223\pi\)
\(830\) 0 0
\(831\) −7766.70 −0.324217
\(832\) 0 0
\(833\) 16542.0 0.688050
\(834\) 0 0
\(835\) 20196.5 0.837041
\(836\) 0 0
\(837\) −4258.43 −0.175858
\(838\) 0 0
\(839\) −43791.0 −1.80195 −0.900974 0.433874i \(-0.857146\pi\)
−0.900974 + 0.433874i \(0.857146\pi\)
\(840\) 0 0
\(841\) 47202.1 1.93538
\(842\) 0 0
\(843\) −6871.16 −0.280730
\(844\) 0 0
\(845\) 30806.6 1.25418
\(846\) 0 0
\(847\) −27197.3 −1.10332
\(848\) 0 0
\(849\) −5544.89 −0.224146
\(850\) 0 0
\(851\) 1953.19 0.0786775
\(852\) 0 0
\(853\) 4492.42 0.180325 0.0901627 0.995927i \(-0.471261\pi\)
0.0901627 + 0.995927i \(0.471261\pi\)
\(854\) 0 0
\(855\) 34506.3 1.38022
\(856\) 0 0
\(857\) 33204.5 1.32351 0.661753 0.749722i \(-0.269813\pi\)
0.661753 + 0.749722i \(0.269813\pi\)
\(858\) 0 0
\(859\) 7947.14 0.315661 0.157831 0.987466i \(-0.449550\pi\)
0.157831 + 0.987466i \(0.449550\pi\)
\(860\) 0 0
\(861\) −19120.9 −0.756838
\(862\) 0 0
\(863\) −10840.6 −0.427599 −0.213799 0.976878i \(-0.568584\pi\)
−0.213799 + 0.976878i \(0.568584\pi\)
\(864\) 0 0
\(865\) −22633.6 −0.889669
\(866\) 0 0
\(867\) 4321.20 0.169268
\(868\) 0 0
\(869\) 12154.7 0.474475
\(870\) 0 0
\(871\) −9350.46 −0.363752
\(872\) 0 0
\(873\) 30574.3 1.18532
\(874\) 0 0
\(875\) 847.383 0.0327392
\(876\) 0 0
\(877\) 11412.4 0.439419 0.219709 0.975565i \(-0.429489\pi\)
0.219709 + 0.975565i \(0.429489\pi\)
\(878\) 0 0
\(879\) −18703.0 −0.717675
\(880\) 0 0
\(881\) 2502.70 0.0957073 0.0478537 0.998854i \(-0.484762\pi\)
0.0478537 + 0.998854i \(0.484762\pi\)
\(882\) 0 0
\(883\) 52137.0 1.98703 0.993517 0.113684i \(-0.0362651\pi\)
0.993517 + 0.113684i \(0.0362651\pi\)
\(884\) 0 0
\(885\) −34368.7 −1.30542
\(886\) 0 0
\(887\) −14793.6 −0.560002 −0.280001 0.960000i \(-0.590335\pi\)
−0.280001 + 0.960000i \(0.590335\pi\)
\(888\) 0 0
\(889\) −16479.2 −0.621705
\(890\) 0 0
\(891\) −3833.12 −0.144124
\(892\) 0 0
\(893\) 9283.20 0.347873
\(894\) 0 0
\(895\) 66813.5 2.49534
\(896\) 0 0
\(897\) −911.840 −0.0339414
\(898\) 0 0
\(899\) 9386.53 0.348229
\(900\) 0 0
\(901\) −7196.27 −0.266085
\(902\) 0 0
\(903\) 22771.5 0.839190
\(904\) 0 0
\(905\) −10918.8 −0.401054
\(906\) 0 0
\(907\) −13661.8 −0.500146 −0.250073 0.968227i \(-0.580455\pi\)
−0.250073 + 0.968227i \(0.580455\pi\)
\(908\) 0 0
\(909\) 24236.2 0.884340
\(910\) 0 0
\(911\) 6644.55 0.241651 0.120825 0.992674i \(-0.461446\pi\)
0.120825 + 0.992674i \(0.461446\pi\)
\(912\) 0 0
\(913\) 10473.0 0.379634
\(914\) 0 0
\(915\) 26077.3 0.942172
\(916\) 0 0
\(917\) 42291.9 1.52301
\(918\) 0 0
\(919\) 9464.13 0.339709 0.169855 0.985469i \(-0.445670\pi\)
0.169855 + 0.985469i \(0.445670\pi\)
\(920\) 0 0
\(921\) −10755.5 −0.384807
\(922\) 0 0
\(923\) 7244.31 0.258342
\(924\) 0 0
\(925\) 10433.7 0.370873
\(926\) 0 0
\(927\) 7652.10 0.271120
\(928\) 0 0
\(929\) −16562.5 −0.584927 −0.292464 0.956277i \(-0.594475\pi\)
−0.292464 + 0.956277i \(0.594475\pi\)
\(930\) 0 0
\(931\) 31214.5 1.09883
\(932\) 0 0
\(933\) −16203.2 −0.568561
\(934\) 0 0
\(935\) 14166.1 0.495486
\(936\) 0 0
\(937\) −38227.3 −1.33280 −0.666398 0.745596i \(-0.732165\pi\)
−0.666398 + 0.745596i \(0.732165\pi\)
\(938\) 0 0
\(939\) 8513.09 0.295862
\(940\) 0 0
\(941\) −47320.8 −1.63933 −0.819667 0.572840i \(-0.805842\pi\)
−0.819667 + 0.572840i \(0.805842\pi\)
\(942\) 0 0
\(943\) 6826.56 0.235740
\(944\) 0 0
\(945\) −48133.3 −1.65691
\(946\) 0 0
\(947\) −7613.32 −0.261246 −0.130623 0.991432i \(-0.541698\pi\)
−0.130623 + 0.991432i \(0.541698\pi\)
\(948\) 0 0
\(949\) 4674.36 0.159891
\(950\) 0 0
\(951\) −20801.9 −0.709304
\(952\) 0 0
\(953\) −44717.8 −1.51999 −0.759995 0.649928i \(-0.774799\pi\)
−0.759995 + 0.649928i \(0.774799\pi\)
\(954\) 0 0
\(955\) 26648.7 0.902965
\(956\) 0 0
\(957\) −10846.0 −0.366356
\(958\) 0 0
\(959\) −33863.4 −1.14026
\(960\) 0 0
\(961\) −28560.3 −0.958689
\(962\) 0 0
\(963\) 41997.0 1.40533
\(964\) 0 0
\(965\) −16186.3 −0.539954
\(966\) 0 0
\(967\) 47052.4 1.56474 0.782370 0.622813i \(-0.214010\pi\)
0.782370 + 0.622813i \(0.214010\pi\)
\(968\) 0 0
\(969\) −15558.8 −0.515812
\(970\) 0 0
\(971\) 40080.3 1.32465 0.662326 0.749215i \(-0.269569\pi\)
0.662326 + 0.749215i \(0.269569\pi\)
\(972\) 0 0
\(973\) 43571.6 1.43560
\(974\) 0 0
\(975\) −4870.93 −0.159994
\(976\) 0 0
\(977\) 56773.8 1.85911 0.929557 0.368679i \(-0.120190\pi\)
0.929557 + 0.368679i \(0.120190\pi\)
\(978\) 0 0
\(979\) 17449.9 0.569665
\(980\) 0 0
\(981\) −11197.0 −0.364415
\(982\) 0 0
\(983\) −33919.1 −1.10056 −0.550281 0.834980i \(-0.685479\pi\)
−0.550281 + 0.834980i \(0.685479\pi\)
\(984\) 0 0
\(985\) 8635.14 0.279328
\(986\) 0 0
\(987\) −5582.04 −0.180019
\(988\) 0 0
\(989\) −8129.91 −0.261392
\(990\) 0 0
\(991\) 5183.79 0.166164 0.0830820 0.996543i \(-0.473524\pi\)
0.0830820 + 0.996543i \(0.473524\pi\)
\(992\) 0 0
\(993\) 16115.6 0.515017
\(994\) 0 0
\(995\) −43839.3 −1.39678
\(996\) 0 0
\(997\) 41570.9 1.32053 0.660263 0.751034i \(-0.270445\pi\)
0.660263 + 0.751034i \(0.270445\pi\)
\(998\) 0 0
\(999\) 10308.4 0.326470
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 736.4.a.e.1.4 8
4.3 odd 2 736.4.a.f.1.5 yes 8
8.3 odd 2 1472.4.a.bg.1.4 8
8.5 even 2 1472.4.a.bh.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.4.a.e.1.4 8 1.1 even 1 trivial
736.4.a.f.1.5 yes 8 4.3 odd 2
1472.4.a.bg.1.4 8 8.3 odd 2
1472.4.a.bh.1.5 8 8.5 even 2