Properties

Label 528.4.a.p.1.1
Level $528$
Weight $4$
Character 528.1
Self dual yes
Analytic conductor $31.153$
Analytic rank $1$
Dimension $2$
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] = [528,4,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("528.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.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: \(31.1530084830\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 528.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -16.8489 q^{5} +7.69772 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -16.8489 q^{5} +7.69772 q^{7} +9.00000 q^{9} +11.0000 q^{11} +24.8489 q^{13} -50.5466 q^{15} -15.9420 q^{17} -15.1511 q^{19} +23.0931 q^{21} -17.7557 q^{23} +158.884 q^{25} +27.0000 q^{27} -128.547 q^{29} -219.395 q^{31} +33.0000 q^{33} -129.698 q^{35} +92.0703 q^{37} +74.5466 q^{39} -459.942 q^{41} -64.9648 q^{43} -151.640 q^{45} -497.408 q^{47} -283.745 q^{49} -47.8260 q^{51} -526.919 q^{53} -185.337 q^{55} -45.4534 q^{57} +578.443 q^{59} -221.569 q^{61} +69.2794 q^{63} -418.675 q^{65} +860.745 q^{67} -53.2671 q^{69} -580.919 q^{71} +510.116 q^{73} +476.652 q^{75} +84.6749 q^{77} -1035.12 q^{79} +81.0000 q^{81} -606.211 q^{83} +268.605 q^{85} -385.640 q^{87} -23.4411 q^{89} +191.279 q^{91} -658.186 q^{93} +255.279 q^{95} +719.490 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9} + 22 q^{11} + 30 q^{13} - 42 q^{15} + 106 q^{17} - 50 q^{19} - 72 q^{21} - 134 q^{23} + 42 q^{25} + 54 q^{27} - 198 q^{29} - 360 q^{31} + 66 q^{33} - 220 q^{35} - 328 q^{37} + 90 q^{39} - 782 q^{41} - 386 q^{43} - 126 q^{45} - 266 q^{47} + 378 q^{49} + 318 q^{51} - 522 q^{53} - 154 q^{55} - 150 q^{57} + 172 q^{59} - 778 q^{61} - 216 q^{63} - 404 q^{65} + 776 q^{67} - 402 q^{69} - 630 q^{71} + 1296 q^{73} + 126 q^{75} - 264 q^{77} - 652 q^{79} + 162 q^{81} + 324 q^{83} + 616 q^{85} - 594 q^{87} - 756 q^{89} + 28 q^{91} - 1080 q^{93} + 156 q^{95} - 452 q^{97} + 198 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −16.8489 −1.50701 −0.753504 0.657444i \(-0.771638\pi\)
−0.753504 + 0.657444i \(0.771638\pi\)
\(6\) 0 0
\(7\) 7.69772 0.415638 0.207819 0.978167i \(-0.433364\pi\)
0.207819 + 0.978167i \(0.433364\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 24.8489 0.530141 0.265071 0.964229i \(-0.414605\pi\)
0.265071 + 0.964229i \(0.414605\pi\)
\(14\) 0 0
\(15\) −50.5466 −0.870071
\(16\) 0 0
\(17\) −15.9420 −0.227441 −0.113721 0.993513i \(-0.536277\pi\)
−0.113721 + 0.993513i \(0.536277\pi\)
\(18\) 0 0
\(19\) −15.1511 −0.182943 −0.0914713 0.995808i \(-0.529157\pi\)
−0.0914713 + 0.995808i \(0.529157\pi\)
\(20\) 0 0
\(21\) 23.0931 0.239968
\(22\) 0 0
\(23\) −17.7557 −0.160971 −0.0804853 0.996756i \(-0.525647\pi\)
−0.0804853 + 0.996756i \(0.525647\pi\)
\(24\) 0 0
\(25\) 158.884 1.27107
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −128.547 −0.823121 −0.411560 0.911383i \(-0.635016\pi\)
−0.411560 + 0.911383i \(0.635016\pi\)
\(30\) 0 0
\(31\) −219.395 −1.27112 −0.635558 0.772053i \(-0.719230\pi\)
−0.635558 + 0.772053i \(0.719230\pi\)
\(32\) 0 0
\(33\) 33.0000 0.174078
\(34\) 0 0
\(35\) −129.698 −0.626369
\(36\) 0 0
\(37\) 92.0703 0.409088 0.204544 0.978857i \(-0.434429\pi\)
0.204544 + 0.978857i \(0.434429\pi\)
\(38\) 0 0
\(39\) 74.5466 0.306077
\(40\) 0 0
\(41\) −459.942 −1.75197 −0.875986 0.482336i \(-0.839788\pi\)
−0.875986 + 0.482336i \(0.839788\pi\)
\(42\) 0 0
\(43\) −64.9648 −0.230396 −0.115198 0.993343i \(-0.536750\pi\)
−0.115198 + 0.993343i \(0.536750\pi\)
\(44\) 0 0
\(45\) −151.640 −0.502336
\(46\) 0 0
\(47\) −497.408 −1.54371 −0.771855 0.635799i \(-0.780671\pi\)
−0.771855 + 0.635799i \(0.780671\pi\)
\(48\) 0 0
\(49\) −283.745 −0.827245
\(50\) 0 0
\(51\) −47.8260 −0.131313
\(52\) 0 0
\(53\) −526.919 −1.36562 −0.682811 0.730596i \(-0.739243\pi\)
−0.682811 + 0.730596i \(0.739243\pi\)
\(54\) 0 0
\(55\) −185.337 −0.454380
\(56\) 0 0
\(57\) −45.4534 −0.105622
\(58\) 0 0
\(59\) 578.443 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(60\) 0 0
\(61\) −221.569 −0.465067 −0.232533 0.972588i \(-0.574701\pi\)
−0.232533 + 0.972588i \(0.574701\pi\)
\(62\) 0 0
\(63\) 69.2794 0.138546
\(64\) 0 0
\(65\) −418.675 −0.798927
\(66\) 0 0
\(67\) 860.745 1.56950 0.784752 0.619810i \(-0.212790\pi\)
0.784752 + 0.619810i \(0.212790\pi\)
\(68\) 0 0
\(69\) −53.2671 −0.0929364
\(70\) 0 0
\(71\) −580.919 −0.971020 −0.485510 0.874231i \(-0.661366\pi\)
−0.485510 + 0.874231i \(0.661366\pi\)
\(72\) 0 0
\(73\) 510.116 0.817871 0.408935 0.912563i \(-0.365900\pi\)
0.408935 + 0.912563i \(0.365900\pi\)
\(74\) 0 0
\(75\) 476.652 0.733854
\(76\) 0 0
\(77\) 84.6749 0.125319
\(78\) 0 0
\(79\) −1035.12 −1.47418 −0.737088 0.675797i \(-0.763800\pi\)
−0.737088 + 0.675797i \(0.763800\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −606.211 −0.801690 −0.400845 0.916146i \(-0.631283\pi\)
−0.400845 + 0.916146i \(0.631283\pi\)
\(84\) 0 0
\(85\) 268.605 0.342756
\(86\) 0 0
\(87\) −385.640 −0.475229
\(88\) 0 0
\(89\) −23.4411 −0.0279186 −0.0139593 0.999903i \(-0.504444\pi\)
−0.0139593 + 0.999903i \(0.504444\pi\)
\(90\) 0 0
\(91\) 191.279 0.220347
\(92\) 0 0
\(93\) −658.186 −0.733879
\(94\) 0 0
\(95\) 255.279 0.275696
\(96\) 0 0
\(97\) 719.490 0.753126 0.376563 0.926391i \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 1871.27 1.84355 0.921774 0.387727i \(-0.126740\pi\)
0.921774 + 0.387727i \(0.126740\pi\)
\(102\) 0 0
\(103\) 428.745 0.410151 0.205075 0.978746i \(-0.434256\pi\)
0.205075 + 0.978746i \(0.434256\pi\)
\(104\) 0 0
\(105\) −389.093 −0.361634
\(106\) 0 0
\(107\) −1148.02 −1.03723 −0.518616 0.855008i \(-0.673552\pi\)
−0.518616 + 0.855008i \(0.673552\pi\)
\(108\) 0 0
\(109\) −1828.32 −1.60662 −0.803308 0.595564i \(-0.796929\pi\)
−0.803308 + 0.595564i \(0.796929\pi\)
\(110\) 0 0
\(111\) 276.211 0.236187
\(112\) 0 0
\(113\) 1126.40 0.937722 0.468861 0.883272i \(-0.344665\pi\)
0.468861 + 0.883272i \(0.344665\pi\)
\(114\) 0 0
\(115\) 299.163 0.242584
\(116\) 0 0
\(117\) 223.640 0.176714
\(118\) 0 0
\(119\) −122.717 −0.0945332
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) −1379.83 −1.01150
\(124\) 0 0
\(125\) −570.907 −0.408508
\(126\) 0 0
\(127\) −661.304 −0.462057 −0.231029 0.972947i \(-0.574209\pi\)
−0.231029 + 0.972947i \(0.574209\pi\)
\(128\) 0 0
\(129\) −194.895 −0.133019
\(130\) 0 0
\(131\) −622.186 −0.414967 −0.207483 0.978239i \(-0.566527\pi\)
−0.207483 + 0.978239i \(0.566527\pi\)
\(132\) 0 0
\(133\) −116.629 −0.0760378
\(134\) 0 0
\(135\) −454.919 −0.290024
\(136\) 0 0
\(137\) −1872.84 −1.16794 −0.583969 0.811776i \(-0.698501\pi\)
−0.583969 + 0.811776i \(0.698501\pi\)
\(138\) 0 0
\(139\) 954.058 0.582174 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(140\) 0 0
\(141\) −1492.22 −0.891261
\(142\) 0 0
\(143\) 273.337 0.159844
\(144\) 0 0
\(145\) 2165.86 1.24045
\(146\) 0 0
\(147\) −851.236 −0.477610
\(148\) 0 0
\(149\) −2047.01 −1.12549 −0.562745 0.826631i \(-0.690255\pi\)
−0.562745 + 0.826631i \(0.690255\pi\)
\(150\) 0 0
\(151\) −475.863 −0.256458 −0.128229 0.991745i \(-0.540929\pi\)
−0.128229 + 0.991745i \(0.540929\pi\)
\(152\) 0 0
\(153\) −143.478 −0.0758138
\(154\) 0 0
\(155\) 3696.56 1.91558
\(156\) 0 0
\(157\) −647.466 −0.329130 −0.164565 0.986366i \(-0.552622\pi\)
−0.164565 + 0.986366i \(0.552622\pi\)
\(158\) 0 0
\(159\) −1580.76 −0.788442
\(160\) 0 0
\(161\) −136.678 −0.0669054
\(162\) 0 0
\(163\) −1093.23 −0.525329 −0.262665 0.964887i \(-0.584601\pi\)
−0.262665 + 0.964887i \(0.584601\pi\)
\(164\) 0 0
\(165\) −556.012 −0.262336
\(166\) 0 0
\(167\) −1123.25 −0.520479 −0.260240 0.965544i \(-0.583802\pi\)
−0.260240 + 0.965544i \(0.583802\pi\)
\(168\) 0 0
\(169\) −1579.53 −0.718951
\(170\) 0 0
\(171\) −136.360 −0.0609809
\(172\) 0 0
\(173\) 46.0123 0.0202211 0.0101106 0.999949i \(-0.496782\pi\)
0.0101106 + 0.999949i \(0.496782\pi\)
\(174\) 0 0
\(175\) 1223.04 0.528305
\(176\) 0 0
\(177\) 1735.33 0.736923
\(178\) 0 0
\(179\) 831.975 0.347401 0.173700 0.984799i \(-0.444428\pi\)
0.173700 + 0.984799i \(0.444428\pi\)
\(180\) 0 0
\(181\) −1810.63 −0.743553 −0.371776 0.928322i \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) 0 0
\(183\) −664.708 −0.268506
\(184\) 0 0
\(185\) −1551.28 −0.616499
\(186\) 0 0
\(187\) −175.362 −0.0685762
\(188\) 0 0
\(189\) 207.838 0.0799895
\(190\) 0 0
\(191\) −458.898 −0.173847 −0.0869233 0.996215i \(-0.527704\pi\)
−0.0869233 + 0.996215i \(0.527704\pi\)
\(192\) 0 0
\(193\) 1778.91 0.663465 0.331733 0.943373i \(-0.392367\pi\)
0.331733 + 0.943373i \(0.392367\pi\)
\(194\) 0 0
\(195\) −1256.02 −0.461260
\(196\) 0 0
\(197\) 5304.53 1.91844 0.959218 0.282666i \(-0.0912188\pi\)
0.959218 + 0.282666i \(0.0912188\pi\)
\(198\) 0 0
\(199\) 5138.40 1.83041 0.915205 0.402989i \(-0.132029\pi\)
0.915205 + 0.402989i \(0.132029\pi\)
\(200\) 0 0
\(201\) 2582.24 0.906153
\(202\) 0 0
\(203\) −989.515 −0.342120
\(204\) 0 0
\(205\) 7749.50 2.64024
\(206\) 0 0
\(207\) −159.801 −0.0536568
\(208\) 0 0
\(209\) −166.663 −0.0551593
\(210\) 0 0
\(211\) 4262.36 1.39068 0.695339 0.718682i \(-0.255254\pi\)
0.695339 + 0.718682i \(0.255254\pi\)
\(212\) 0 0
\(213\) −1742.76 −0.560619
\(214\) 0 0
\(215\) 1094.58 0.347209
\(216\) 0 0
\(217\) −1688.84 −0.528323
\(218\) 0 0
\(219\) 1530.35 0.472198
\(220\) 0 0
\(221\) −396.141 −0.120576
\(222\) 0 0
\(223\) 1377.80 0.413740 0.206870 0.978368i \(-0.433672\pi\)
0.206870 + 0.978368i \(0.433672\pi\)
\(224\) 0 0
\(225\) 1429.96 0.423691
\(226\) 0 0
\(227\) 1227.28 0.358843 0.179422 0.983772i \(-0.442577\pi\)
0.179422 + 0.983772i \(0.442577\pi\)
\(228\) 0 0
\(229\) 3890.28 1.12261 0.561304 0.827610i \(-0.310300\pi\)
0.561304 + 0.827610i \(0.310300\pi\)
\(230\) 0 0
\(231\) 254.025 0.0723532
\(232\) 0 0
\(233\) −3218.14 −0.904837 −0.452419 0.891806i \(-0.649439\pi\)
−0.452419 + 0.891806i \(0.649439\pi\)
\(234\) 0 0
\(235\) 8380.75 2.32638
\(236\) 0 0
\(237\) −3105.35 −0.851115
\(238\) 0 0
\(239\) 428.098 0.115864 0.0579318 0.998321i \(-0.481549\pi\)
0.0579318 + 0.998321i \(0.481549\pi\)
\(240\) 0 0
\(241\) 1231.16 0.329070 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 4780.78 1.24667
\(246\) 0 0
\(247\) −376.489 −0.0969854
\(248\) 0 0
\(249\) −1818.63 −0.462856
\(250\) 0 0
\(251\) −2838.22 −0.713732 −0.356866 0.934156i \(-0.616155\pi\)
−0.356866 + 0.934156i \(0.616155\pi\)
\(252\) 0 0
\(253\) −195.313 −0.0485344
\(254\) 0 0
\(255\) 805.814 0.197890
\(256\) 0 0
\(257\) 342.007 0.0830110 0.0415055 0.999138i \(-0.486785\pi\)
0.0415055 + 0.999138i \(0.486785\pi\)
\(258\) 0 0
\(259\) 708.731 0.170032
\(260\) 0 0
\(261\) −1156.92 −0.274374
\(262\) 0 0
\(263\) −5895.00 −1.38213 −0.691067 0.722791i \(-0.742859\pi\)
−0.691067 + 0.722791i \(0.742859\pi\)
\(264\) 0 0
\(265\) 8877.99 2.05800
\(266\) 0 0
\(267\) −70.3234 −0.0161188
\(268\) 0 0
\(269\) −2496.18 −0.565779 −0.282890 0.959152i \(-0.591293\pi\)
−0.282890 + 0.959152i \(0.591293\pi\)
\(270\) 0 0
\(271\) 2249.68 0.504274 0.252137 0.967692i \(-0.418867\pi\)
0.252137 + 0.967692i \(0.418867\pi\)
\(272\) 0 0
\(273\) 573.838 0.127217
\(274\) 0 0
\(275\) 1747.72 0.383243
\(276\) 0 0
\(277\) 4082.59 0.885556 0.442778 0.896631i \(-0.353993\pi\)
0.442778 + 0.896631i \(0.353993\pi\)
\(278\) 0 0
\(279\) −1974.56 −0.423705
\(280\) 0 0
\(281\) −1033.79 −0.219468 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(282\) 0 0
\(283\) 7809.14 1.64030 0.820150 0.572148i \(-0.193890\pi\)
0.820150 + 0.572148i \(0.193890\pi\)
\(284\) 0 0
\(285\) 765.838 0.159173
\(286\) 0 0
\(287\) −3540.50 −0.728186
\(288\) 0 0
\(289\) −4658.85 −0.948270
\(290\) 0 0
\(291\) 2158.47 0.434817
\(292\) 0 0
\(293\) −1949.19 −0.388645 −0.194323 0.980938i \(-0.562251\pi\)
−0.194323 + 0.980938i \(0.562251\pi\)
\(294\) 0 0
\(295\) −9746.10 −1.92353
\(296\) 0 0
\(297\) 297.000 0.0580259
\(298\) 0 0
\(299\) −441.209 −0.0853371
\(300\) 0 0
\(301\) −500.081 −0.0957614
\(302\) 0 0
\(303\) 5613.81 1.06437
\(304\) 0 0
\(305\) 3733.19 0.700859
\(306\) 0 0
\(307\) −2364.09 −0.439497 −0.219748 0.975557i \(-0.570524\pi\)
−0.219748 + 0.975557i \(0.570524\pi\)
\(308\) 0 0
\(309\) 1286.24 0.236801
\(310\) 0 0
\(311\) 1989.17 0.362686 0.181343 0.983420i \(-0.441956\pi\)
0.181343 + 0.983420i \(0.441956\pi\)
\(312\) 0 0
\(313\) −3878.67 −0.700433 −0.350216 0.936669i \(-0.613892\pi\)
−0.350216 + 0.936669i \(0.613892\pi\)
\(314\) 0 0
\(315\) −1167.28 −0.208790
\(316\) 0 0
\(317\) 2913.73 0.516251 0.258126 0.966111i \(-0.416895\pi\)
0.258126 + 0.966111i \(0.416895\pi\)
\(318\) 0 0
\(319\) −1414.01 −0.248180
\(320\) 0 0
\(321\) −3444.07 −0.598846
\(322\) 0 0
\(323\) 241.540 0.0416087
\(324\) 0 0
\(325\) 3948.09 0.673847
\(326\) 0 0
\(327\) −5484.95 −0.927580
\(328\) 0 0
\(329\) −3828.90 −0.641624
\(330\) 0 0
\(331\) −8104.46 −1.34580 −0.672902 0.739731i \(-0.734953\pi\)
−0.672902 + 0.739731i \(0.734953\pi\)
\(332\) 0 0
\(333\) 828.633 0.136363
\(334\) 0 0
\(335\) −14502.6 −2.36525
\(336\) 0 0
\(337\) 5919.19 0.956792 0.478396 0.878144i \(-0.341218\pi\)
0.478396 + 0.878144i \(0.341218\pi\)
\(338\) 0 0
\(339\) 3379.19 0.541394
\(340\) 0 0
\(341\) −2413.35 −0.383256
\(342\) 0 0
\(343\) −4824.51 −0.759472
\(344\) 0 0
\(345\) 897.490 0.140056
\(346\) 0 0
\(347\) 8540.59 1.32128 0.660638 0.750705i \(-0.270286\pi\)
0.660638 + 0.750705i \(0.270286\pi\)
\(348\) 0 0
\(349\) 937.337 0.143767 0.0718833 0.997413i \(-0.477099\pi\)
0.0718833 + 0.997413i \(0.477099\pi\)
\(350\) 0 0
\(351\) 670.919 0.102026
\(352\) 0 0
\(353\) −211.118 −0.0318319 −0.0159160 0.999873i \(-0.505066\pi\)
−0.0159160 + 0.999873i \(0.505066\pi\)
\(354\) 0 0
\(355\) 9787.82 1.46333
\(356\) 0 0
\(357\) −368.151 −0.0545788
\(358\) 0 0
\(359\) 1376.31 0.202337 0.101169 0.994869i \(-0.467742\pi\)
0.101169 + 0.994869i \(0.467742\pi\)
\(360\) 0 0
\(361\) −6629.44 −0.966532
\(362\) 0 0
\(363\) 363.000 0.0524864
\(364\) 0 0
\(365\) −8594.87 −1.23254
\(366\) 0 0
\(367\) −1030.45 −0.146564 −0.0732821 0.997311i \(-0.523347\pi\)
−0.0732821 + 0.997311i \(0.523347\pi\)
\(368\) 0 0
\(369\) −4139.48 −0.583991
\(370\) 0 0
\(371\) −4056.07 −0.567603
\(372\) 0 0
\(373\) 9365.39 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(374\) 0 0
\(375\) −1712.72 −0.235852
\(376\) 0 0
\(377\) −3194.24 −0.436370
\(378\) 0 0
\(379\) 7120.23 0.965017 0.482509 0.875891i \(-0.339726\pi\)
0.482509 + 0.875891i \(0.339726\pi\)
\(380\) 0 0
\(381\) −1983.91 −0.266769
\(382\) 0 0
\(383\) 1163.56 0.155235 0.0776176 0.996983i \(-0.475269\pi\)
0.0776176 + 0.996983i \(0.475269\pi\)
\(384\) 0 0
\(385\) −1426.67 −0.188857
\(386\) 0 0
\(387\) −584.684 −0.0767988
\(388\) 0 0
\(389\) −10958.9 −1.42838 −0.714188 0.699954i \(-0.753204\pi\)
−0.714188 + 0.699954i \(0.753204\pi\)
\(390\) 0 0
\(391\) 283.062 0.0366114
\(392\) 0 0
\(393\) −1866.56 −0.239581
\(394\) 0 0
\(395\) 17440.6 2.22159
\(396\) 0 0
\(397\) −2172.09 −0.274595 −0.137298 0.990530i \(-0.543842\pi\)
−0.137298 + 0.990530i \(0.543842\pi\)
\(398\) 0 0
\(399\) −349.888 −0.0439005
\(400\) 0 0
\(401\) 7830.71 0.975180 0.487590 0.873073i \(-0.337876\pi\)
0.487590 + 0.873073i \(0.337876\pi\)
\(402\) 0 0
\(403\) −5451.73 −0.673870
\(404\) 0 0
\(405\) −1364.76 −0.167445
\(406\) 0 0
\(407\) 1012.77 0.123345
\(408\) 0 0
\(409\) −10731.2 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(410\) 0 0
\(411\) −5618.52 −0.674309
\(412\) 0 0
\(413\) 4452.69 0.530515
\(414\) 0 0
\(415\) 10214.0 1.20815
\(416\) 0 0
\(417\) 2862.17 0.336118
\(418\) 0 0
\(419\) 7315.88 0.852994 0.426497 0.904489i \(-0.359748\pi\)
0.426497 + 0.904489i \(0.359748\pi\)
\(420\) 0 0
\(421\) −12495.7 −1.44657 −0.723284 0.690551i \(-0.757368\pi\)
−0.723284 + 0.690551i \(0.757368\pi\)
\(422\) 0 0
\(423\) −4476.67 −0.514570
\(424\) 0 0
\(425\) −2532.93 −0.289094
\(426\) 0 0
\(427\) −1705.58 −0.193299
\(428\) 0 0
\(429\) 820.012 0.0922857
\(430\) 0 0
\(431\) −6075.01 −0.678939 −0.339470 0.940617i \(-0.610248\pi\)
−0.339470 + 0.940617i \(0.610248\pi\)
\(432\) 0 0
\(433\) 5641.79 0.626160 0.313080 0.949727i \(-0.398639\pi\)
0.313080 + 0.949727i \(0.398639\pi\)
\(434\) 0 0
\(435\) 6497.59 0.716174
\(436\) 0 0
\(437\) 269.019 0.0294484
\(438\) 0 0
\(439\) −10897.0 −1.18470 −0.592351 0.805680i \(-0.701800\pi\)
−0.592351 + 0.805680i \(0.701800\pi\)
\(440\) 0 0
\(441\) −2553.71 −0.275748
\(442\) 0 0
\(443\) 7720.83 0.828054 0.414027 0.910265i \(-0.364122\pi\)
0.414027 + 0.910265i \(0.364122\pi\)
\(444\) 0 0
\(445\) 394.956 0.0420735
\(446\) 0 0
\(447\) −6141.04 −0.649802
\(448\) 0 0
\(449\) −7473.86 −0.785553 −0.392776 0.919634i \(-0.628485\pi\)
−0.392776 + 0.919634i \(0.628485\pi\)
\(450\) 0 0
\(451\) −5059.36 −0.528240
\(452\) 0 0
\(453\) −1427.59 −0.148066
\(454\) 0 0
\(455\) −3222.84 −0.332064
\(456\) 0 0
\(457\) 11140.5 1.14033 0.570167 0.821529i \(-0.306879\pi\)
0.570167 + 0.821529i \(0.306879\pi\)
\(458\) 0 0
\(459\) −430.434 −0.0437711
\(460\) 0 0
\(461\) 14328.8 1.44763 0.723817 0.689992i \(-0.242386\pi\)
0.723817 + 0.689992i \(0.242386\pi\)
\(462\) 0 0
\(463\) −11760.7 −1.18049 −0.590246 0.807223i \(-0.700969\pi\)
−0.590246 + 0.807223i \(0.700969\pi\)
\(464\) 0 0
\(465\) 11089.7 1.10596
\(466\) 0 0
\(467\) −11854.9 −1.17469 −0.587343 0.809338i \(-0.699826\pi\)
−0.587343 + 0.809338i \(0.699826\pi\)
\(468\) 0 0
\(469\) 6625.77 0.652345
\(470\) 0 0
\(471\) −1942.40 −0.190023
\(472\) 0 0
\(473\) −714.613 −0.0694671
\(474\) 0 0
\(475\) −2407.27 −0.232533
\(476\) 0 0
\(477\) −4742.27 −0.455207
\(478\) 0 0
\(479\) 1324.68 0.126359 0.0631796 0.998002i \(-0.479876\pi\)
0.0631796 + 0.998002i \(0.479876\pi\)
\(480\) 0 0
\(481\) 2287.84 0.216874
\(482\) 0 0
\(483\) −410.035 −0.0386278
\(484\) 0 0
\(485\) −12122.6 −1.13497
\(486\) 0 0
\(487\) −18636.4 −1.73408 −0.867040 0.498239i \(-0.833980\pi\)
−0.867040 + 0.498239i \(0.833980\pi\)
\(488\) 0 0
\(489\) −3279.70 −0.303299
\(490\) 0 0
\(491\) −124.552 −0.0114480 −0.00572398 0.999984i \(-0.501822\pi\)
−0.00572398 + 0.999984i \(0.501822\pi\)
\(492\) 0 0
\(493\) 2049.29 0.187212
\(494\) 0 0
\(495\) −1668.04 −0.151460
\(496\) 0 0
\(497\) −4471.75 −0.403592
\(498\) 0 0
\(499\) −10230.2 −0.917768 −0.458884 0.888496i \(-0.651751\pi\)
−0.458884 + 0.888496i \(0.651751\pi\)
\(500\) 0 0
\(501\) −3369.76 −0.300499
\(502\) 0 0
\(503\) −5150.81 −0.456587 −0.228294 0.973592i \(-0.573315\pi\)
−0.228294 + 0.973592i \(0.573315\pi\)
\(504\) 0 0
\(505\) −31528.8 −2.77824
\(506\) 0 0
\(507\) −4738.60 −0.415086
\(508\) 0 0
\(509\) −22.7715 −0.00198296 −0.000991481 1.00000i \(-0.500316\pi\)
−0.000991481 1.00000i \(0.500316\pi\)
\(510\) 0 0
\(511\) 3926.73 0.339938
\(512\) 0 0
\(513\) −409.081 −0.0352073
\(514\) 0 0
\(515\) −7223.87 −0.618100
\(516\) 0 0
\(517\) −5471.49 −0.465446
\(518\) 0 0
\(519\) 138.037 0.0116747
\(520\) 0 0
\(521\) 21521.7 1.80976 0.904879 0.425669i \(-0.139961\pi\)
0.904879 + 0.425669i \(0.139961\pi\)
\(522\) 0 0
\(523\) −2923.36 −0.244416 −0.122208 0.992504i \(-0.538998\pi\)
−0.122208 + 0.992504i \(0.538998\pi\)
\(524\) 0 0
\(525\) 3669.13 0.305017
\(526\) 0 0
\(527\) 3497.60 0.289104
\(528\) 0 0
\(529\) −11851.7 −0.974088
\(530\) 0 0
\(531\) 5205.99 0.425462
\(532\) 0 0
\(533\) −11429.0 −0.928792
\(534\) 0 0
\(535\) 19342.9 1.56312
\(536\) 0 0
\(537\) 2495.93 0.200572
\(538\) 0 0
\(539\) −3121.20 −0.249424
\(540\) 0 0
\(541\) 21272.8 1.69056 0.845278 0.534327i \(-0.179435\pi\)
0.845278 + 0.534327i \(0.179435\pi\)
\(542\) 0 0
\(543\) −5431.89 −0.429290
\(544\) 0 0
\(545\) 30805.1 2.42118
\(546\) 0 0
\(547\) 18730.5 1.46409 0.732046 0.681256i \(-0.238566\pi\)
0.732046 + 0.681256i \(0.238566\pi\)
\(548\) 0 0
\(549\) −1994.12 −0.155022
\(550\) 0 0
\(551\) 1947.63 0.150584
\(552\) 0 0
\(553\) −7968.04 −0.612723
\(554\) 0 0
\(555\) −4653.84 −0.355936
\(556\) 0 0
\(557\) −18885.0 −1.43659 −0.718297 0.695736i \(-0.755078\pi\)
−0.718297 + 0.695736i \(0.755078\pi\)
\(558\) 0 0
\(559\) −1614.30 −0.122143
\(560\) 0 0
\(561\) −526.086 −0.0395925
\(562\) 0 0
\(563\) −10285.1 −0.769922 −0.384961 0.922933i \(-0.625785\pi\)
−0.384961 + 0.922933i \(0.625785\pi\)
\(564\) 0 0
\(565\) −18978.5 −1.41315
\(566\) 0 0
\(567\) 623.515 0.0461820
\(568\) 0 0
\(569\) 18008.8 1.32683 0.663415 0.748251i \(-0.269106\pi\)
0.663415 + 0.748251i \(0.269106\pi\)
\(570\) 0 0
\(571\) 7010.79 0.513822 0.256911 0.966435i \(-0.417295\pi\)
0.256911 + 0.966435i \(0.417295\pi\)
\(572\) 0 0
\(573\) −1376.69 −0.100370
\(574\) 0 0
\(575\) −2821.10 −0.204605
\(576\) 0 0
\(577\) −16398.9 −1.18318 −0.591589 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(578\) 0 0
\(579\) 5336.73 0.383052
\(580\) 0 0
\(581\) −4666.44 −0.333213
\(582\) 0 0
\(583\) −5796.11 −0.411750
\(584\) 0 0
\(585\) −3768.07 −0.266309
\(586\) 0 0
\(587\) −12823.5 −0.901671 −0.450836 0.892607i \(-0.648874\pi\)
−0.450836 + 0.892607i \(0.648874\pi\)
\(588\) 0 0
\(589\) 3324.09 0.232541
\(590\) 0 0
\(591\) 15913.6 1.10761
\(592\) 0 0
\(593\) 16899.5 1.17029 0.585144 0.810929i \(-0.301038\pi\)
0.585144 + 0.810929i \(0.301038\pi\)
\(594\) 0 0
\(595\) 2067.64 0.142462
\(596\) 0 0
\(597\) 15415.2 1.05679
\(598\) 0 0
\(599\) 15074.9 1.02829 0.514143 0.857704i \(-0.328110\pi\)
0.514143 + 0.857704i \(0.328110\pi\)
\(600\) 0 0
\(601\) −11418.8 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(602\) 0 0
\(603\) 7746.71 0.523168
\(604\) 0 0
\(605\) −2038.71 −0.137001
\(606\) 0 0
\(607\) 17952.8 1.20046 0.600232 0.799826i \(-0.295075\pi\)
0.600232 + 0.799826i \(0.295075\pi\)
\(608\) 0 0
\(609\) −2968.54 −0.197523
\(610\) 0 0
\(611\) −12360.0 −0.818384
\(612\) 0 0
\(613\) 12528.9 0.825507 0.412753 0.910843i \(-0.364567\pi\)
0.412753 + 0.910843i \(0.364567\pi\)
\(614\) 0 0
\(615\) 23248.5 1.52434
\(616\) 0 0
\(617\) −8586.10 −0.560232 −0.280116 0.959966i \(-0.590373\pi\)
−0.280116 + 0.959966i \(0.590373\pi\)
\(618\) 0 0
\(619\) −18415.4 −1.19576 −0.597882 0.801584i \(-0.703991\pi\)
−0.597882 + 0.801584i \(0.703991\pi\)
\(620\) 0 0
\(621\) −479.404 −0.0309788
\(622\) 0 0
\(623\) −180.443 −0.0116040
\(624\) 0 0
\(625\) −10241.4 −0.655448
\(626\) 0 0
\(627\) −499.988 −0.0318462
\(628\) 0 0
\(629\) −1467.79 −0.0930436
\(630\) 0 0
\(631\) −2374.38 −0.149798 −0.0748989 0.997191i \(-0.523863\pi\)
−0.0748989 + 0.997191i \(0.523863\pi\)
\(632\) 0 0
\(633\) 12787.1 0.802909
\(634\) 0 0
\(635\) 11142.2 0.696324
\(636\) 0 0
\(637\) −7050.74 −0.438557
\(638\) 0 0
\(639\) −5228.27 −0.323673
\(640\) 0 0
\(641\) 11086.0 0.683104 0.341552 0.939863i \(-0.389047\pi\)
0.341552 + 0.939863i \(0.389047\pi\)
\(642\) 0 0
\(643\) −19934.1 −1.22259 −0.611294 0.791403i \(-0.709351\pi\)
−0.611294 + 0.791403i \(0.709351\pi\)
\(644\) 0 0
\(645\) 3283.75 0.200461
\(646\) 0 0
\(647\) 30634.8 1.86148 0.930739 0.365684i \(-0.119165\pi\)
0.930739 + 0.365684i \(0.119165\pi\)
\(648\) 0 0
\(649\) 6362.87 0.384845
\(650\) 0 0
\(651\) −5066.53 −0.305028
\(652\) 0 0
\(653\) −9818.07 −0.588378 −0.294189 0.955747i \(-0.595049\pi\)
−0.294189 + 0.955747i \(0.595049\pi\)
\(654\) 0 0
\(655\) 10483.1 0.625358
\(656\) 0 0
\(657\) 4591.04 0.272624
\(658\) 0 0
\(659\) −16478.5 −0.974070 −0.487035 0.873383i \(-0.661922\pi\)
−0.487035 + 0.873383i \(0.661922\pi\)
\(660\) 0 0
\(661\) 2958.12 0.174066 0.0870328 0.996205i \(-0.472262\pi\)
0.0870328 + 0.996205i \(0.472262\pi\)
\(662\) 0 0
\(663\) −1188.42 −0.0696146
\(664\) 0 0
\(665\) 1965.07 0.114590
\(666\) 0 0
\(667\) 2282.44 0.132498
\(668\) 0 0
\(669\) 4133.39 0.238873
\(670\) 0 0
\(671\) −2437.26 −0.140223
\(672\) 0 0
\(673\) −29960.3 −1.71602 −0.858012 0.513630i \(-0.828300\pi\)
−0.858012 + 0.513630i \(0.828300\pi\)
\(674\) 0 0
\(675\) 4289.87 0.244618
\(676\) 0 0
\(677\) −4514.73 −0.256300 −0.128150 0.991755i \(-0.540904\pi\)
−0.128150 + 0.991755i \(0.540904\pi\)
\(678\) 0 0
\(679\) 5538.43 0.313027
\(680\) 0 0
\(681\) 3681.84 0.207178
\(682\) 0 0
\(683\) 13555.7 0.759438 0.379719 0.925102i \(-0.376021\pi\)
0.379719 + 0.925102i \(0.376021\pi\)
\(684\) 0 0
\(685\) 31555.2 1.76009
\(686\) 0 0
\(687\) 11670.8 0.648137
\(688\) 0 0
\(689\) −13093.3 −0.723972
\(690\) 0 0
\(691\) 11471.3 0.631535 0.315768 0.948837i \(-0.397738\pi\)
0.315768 + 0.948837i \(0.397738\pi\)
\(692\) 0 0
\(693\) 762.074 0.0417731
\(694\) 0 0
\(695\) −16074.8 −0.877340
\(696\) 0 0
\(697\) 7332.40 0.398471
\(698\) 0 0
\(699\) −9654.41 −0.522408
\(700\) 0 0
\(701\) 22229.0 1.19769 0.598843 0.800866i \(-0.295627\pi\)
0.598843 + 0.800866i \(0.295627\pi\)
\(702\) 0 0
\(703\) −1394.97 −0.0748397
\(704\) 0 0
\(705\) 25142.3 1.34314
\(706\) 0 0
\(707\) 14404.5 0.766248
\(708\) 0 0
\(709\) −15081.2 −0.798851 −0.399426 0.916766i \(-0.630790\pi\)
−0.399426 + 0.916766i \(0.630790\pi\)
\(710\) 0 0
\(711\) −9316.06 −0.491392
\(712\) 0 0
\(713\) 3895.52 0.204612
\(714\) 0 0
\(715\) −4605.42 −0.240885
\(716\) 0 0
\(717\) 1284.30 0.0668938
\(718\) 0 0
\(719\) 7399.80 0.383819 0.191910 0.981413i \(-0.438532\pi\)
0.191910 + 0.981413i \(0.438532\pi\)
\(720\) 0 0
\(721\) 3300.36 0.170474
\(722\) 0 0
\(723\) 3693.48 0.189989
\(724\) 0 0
\(725\) −20424.0 −1.04625
\(726\) 0 0
\(727\) −1705.77 −0.0870202 −0.0435101 0.999053i \(-0.513854\pi\)
−0.0435101 + 0.999053i \(0.513854\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1035.67 0.0524017
\(732\) 0 0
\(733\) 37122.6 1.87061 0.935303 0.353847i \(-0.115127\pi\)
0.935303 + 0.353847i \(0.115127\pi\)
\(734\) 0 0
\(735\) 14342.3 0.719762
\(736\) 0 0
\(737\) 9468.20 0.473223
\(738\) 0 0
\(739\) −34256.3 −1.70520 −0.852598 0.522568i \(-0.824974\pi\)
−0.852598 + 0.522568i \(0.824974\pi\)
\(740\) 0 0
\(741\) −1129.47 −0.0559945
\(742\) 0 0
\(743\) −1567.88 −0.0774160 −0.0387080 0.999251i \(-0.512324\pi\)
−0.0387080 + 0.999251i \(0.512324\pi\)
\(744\) 0 0
\(745\) 34489.8 1.69612
\(746\) 0 0
\(747\) −5455.90 −0.267230
\(748\) 0 0
\(749\) −8837.17 −0.431112
\(750\) 0 0
\(751\) 955.613 0.0464325 0.0232163 0.999730i \(-0.492609\pi\)
0.0232163 + 0.999730i \(0.492609\pi\)
\(752\) 0 0
\(753\) −8514.65 −0.412073
\(754\) 0 0
\(755\) 8017.75 0.386484
\(756\) 0 0
\(757\) −14015.4 −0.672918 −0.336459 0.941698i \(-0.609229\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(758\) 0 0
\(759\) −585.938 −0.0280214
\(760\) 0 0
\(761\) −36271.0 −1.72776 −0.863879 0.503699i \(-0.831972\pi\)
−0.863879 + 0.503699i \(0.831972\pi\)
\(762\) 0 0
\(763\) −14073.9 −0.667770
\(764\) 0 0
\(765\) 2417.44 0.114252
\(766\) 0 0
\(767\) 14373.6 0.676665
\(768\) 0 0
\(769\) −18163.6 −0.851749 −0.425874 0.904782i \(-0.640033\pi\)
−0.425874 + 0.904782i \(0.640033\pi\)
\(770\) 0 0
\(771\) 1026.02 0.0479264
\(772\) 0 0
\(773\) −8345.65 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(774\) 0 0
\(775\) −34858.4 −1.61568
\(776\) 0 0
\(777\) 2126.19 0.0981683
\(778\) 0 0
\(779\) 6968.65 0.320510
\(780\) 0 0
\(781\) −6390.11 −0.292774
\(782\) 0 0
\(783\) −3470.76 −0.158410
\(784\) 0 0
\(785\) 10909.1 0.496001
\(786\) 0 0
\(787\) 22996.2 1.04158 0.520791 0.853684i \(-0.325637\pi\)
0.520791 + 0.853684i \(0.325637\pi\)
\(788\) 0 0
\(789\) −17685.0 −0.797975
\(790\) 0 0
\(791\) 8670.69 0.389752
\(792\) 0 0
\(793\) −5505.75 −0.246551
\(794\) 0 0
\(795\) 26634.0 1.18819
\(796\) 0 0
\(797\) −2743.82 −0.121946 −0.0609730 0.998139i \(-0.519420\pi\)
−0.0609730 + 0.998139i \(0.519420\pi\)
\(798\) 0 0
\(799\) 7929.68 0.351104
\(800\) 0 0
\(801\) −210.970 −0.00930619
\(802\) 0 0
\(803\) 5611.28 0.246597
\(804\) 0 0
\(805\) 2302.88 0.100827
\(806\) 0 0
\(807\) −7488.53 −0.326653
\(808\) 0 0
\(809\) 41241.7 1.79231 0.896156 0.443738i \(-0.146348\pi\)
0.896156 + 0.443738i \(0.146348\pi\)
\(810\) 0 0
\(811\) −12832.9 −0.555641 −0.277820 0.960633i \(-0.589612\pi\)
−0.277820 + 0.960633i \(0.589612\pi\)
\(812\) 0 0
\(813\) 6749.03 0.291142
\(814\) 0 0
\(815\) 18419.7 0.791675
\(816\) 0 0
\(817\) 984.292 0.0421493
\(818\) 0 0
\(819\) 1721.51 0.0734488
\(820\) 0 0
\(821\) 16368.5 0.695817 0.347908 0.937529i \(-0.386892\pi\)
0.347908 + 0.937529i \(0.386892\pi\)
\(822\) 0 0
\(823\) −3869.53 −0.163892 −0.0819461 0.996637i \(-0.526114\pi\)
−0.0819461 + 0.996637i \(0.526114\pi\)
\(824\) 0 0
\(825\) 5243.17 0.221265
\(826\) 0 0
\(827\) −7388.69 −0.310677 −0.155339 0.987861i \(-0.549647\pi\)
−0.155339 + 0.987861i \(0.549647\pi\)
\(828\) 0 0
\(829\) 23990.1 1.00508 0.502539 0.864554i \(-0.332399\pi\)
0.502539 + 0.864554i \(0.332399\pi\)
\(830\) 0 0
\(831\) 12247.8 0.511276
\(832\) 0 0
\(833\) 4523.47 0.188150
\(834\) 0 0
\(835\) 18925.6 0.784367
\(836\) 0 0
\(837\) −5923.68 −0.244626
\(838\) 0 0
\(839\) 18228.3 0.750074 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(840\) 0 0
\(841\) −7864.78 −0.322472
\(842\) 0 0
\(843\) −3101.36 −0.126710
\(844\) 0 0
\(845\) 26613.3 1.08346
\(846\) 0 0
\(847\) 931.424 0.0377852
\(848\) 0 0
\(849\) 23427.4 0.947028
\(850\) 0 0
\(851\) −1634.77 −0.0658511
\(852\) 0 0
\(853\) −21737.3 −0.872534 −0.436267 0.899817i \(-0.643700\pi\)
−0.436267 + 0.899817i \(0.643700\pi\)
\(854\) 0 0
\(855\) 2297.51 0.0918987
\(856\) 0 0
\(857\) −18712.2 −0.745852 −0.372926 0.927861i \(-0.621645\pi\)
−0.372926 + 0.927861i \(0.621645\pi\)
\(858\) 0 0
\(859\) −30527.6 −1.21256 −0.606279 0.795252i \(-0.707339\pi\)
−0.606279 + 0.795252i \(0.707339\pi\)
\(860\) 0 0
\(861\) −10621.5 −0.420418
\(862\) 0 0
\(863\) −10906.4 −0.430196 −0.215098 0.976592i \(-0.569007\pi\)
−0.215098 + 0.976592i \(0.569007\pi\)
\(864\) 0 0
\(865\) −775.255 −0.0304734
\(866\) 0 0
\(867\) −13976.6 −0.547484
\(868\) 0 0
\(869\) −11386.3 −0.444481
\(870\) 0 0
\(871\) 21388.5 0.832058
\(872\) 0 0
\(873\) 6475.41 0.251042
\(874\) 0 0
\(875\) −4394.68 −0.169791
\(876\) 0 0
\(877\) 21770.9 0.838256 0.419128 0.907927i \(-0.362336\pi\)
0.419128 + 0.907927i \(0.362336\pi\)
\(878\) 0 0
\(879\) −5847.58 −0.224384
\(880\) 0 0
\(881\) 47206.9 1.80527 0.902634 0.430409i \(-0.141631\pi\)
0.902634 + 0.430409i \(0.141631\pi\)
\(882\) 0 0
\(883\) 6059.68 0.230945 0.115473 0.993311i \(-0.463162\pi\)
0.115473 + 0.993311i \(0.463162\pi\)
\(884\) 0 0
\(885\) −29238.3 −1.11055
\(886\) 0 0
\(887\) 37130.2 1.40553 0.702767 0.711420i \(-0.251948\pi\)
0.702767 + 0.711420i \(0.251948\pi\)
\(888\) 0 0
\(889\) −5090.53 −0.192048
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 0 0
\(893\) 7536.30 0.282410
\(894\) 0 0
\(895\) −14017.8 −0.523536
\(896\) 0 0
\(897\) −1323.63 −0.0492694
\(898\) 0 0
\(899\) 28202.5 1.04628
\(900\) 0 0
\(901\) 8400.15 0.310599
\(902\) 0 0
\(903\) −1500.24 −0.0552879
\(904\) 0 0
\(905\) 30507.0 1.12054
\(906\) 0 0
\(907\) −1182.94 −0.0433064 −0.0216532 0.999766i \(-0.506893\pi\)
−0.0216532 + 0.999766i \(0.506893\pi\)
\(908\) 0 0
\(909\) 16841.4 0.614516
\(910\) 0 0
\(911\) −37676.4 −1.37022 −0.685112 0.728438i \(-0.740247\pi\)
−0.685112 + 0.728438i \(0.740247\pi\)
\(912\) 0 0
\(913\) −6668.32 −0.241719
\(914\) 0 0
\(915\) 11199.6 0.404641
\(916\) 0 0
\(917\) −4789.41 −0.172476
\(918\) 0 0
\(919\) −8697.82 −0.312203 −0.156101 0.987741i \(-0.549893\pi\)
−0.156101 + 0.987741i \(0.549893\pi\)
\(920\) 0 0
\(921\) −7092.26 −0.253744
\(922\) 0 0
\(923\) −14435.2 −0.514778
\(924\) 0 0
\(925\) 14628.5 0.519981
\(926\) 0 0
\(927\) 3858.71 0.136717
\(928\) 0 0
\(929\) −17247.5 −0.609119 −0.304559 0.952493i \(-0.598509\pi\)
−0.304559 + 0.952493i \(0.598509\pi\)
\(930\) 0 0
\(931\) 4299.06 0.151338
\(932\) 0 0
\(933\) 5967.51 0.209397
\(934\) 0 0
\(935\) 2954.65 0.103345
\(936\) 0 0
\(937\) −41812.4 −1.45779 −0.728896 0.684624i \(-0.759966\pi\)
−0.728896 + 0.684624i \(0.759966\pi\)
\(938\) 0 0
\(939\) −11636.0 −0.404395
\(940\) 0 0
\(941\) 37655.9 1.30451 0.652257 0.757998i \(-0.273822\pi\)
0.652257 + 0.757998i \(0.273822\pi\)
\(942\) 0 0
\(943\) 8166.60 0.282016
\(944\) 0 0
\(945\) −3501.84 −0.120545
\(946\) 0 0
\(947\) 21244.4 0.728986 0.364493 0.931206i \(-0.381242\pi\)
0.364493 + 0.931206i \(0.381242\pi\)
\(948\) 0 0
\(949\) 12675.8 0.433587
\(950\) 0 0
\(951\) 8741.20 0.298058
\(952\) 0 0
\(953\) 1324.27 0.0450130 0.0225065 0.999747i \(-0.492835\pi\)
0.0225065 + 0.999747i \(0.492835\pi\)
\(954\) 0 0
\(955\) 7731.91 0.261988
\(956\) 0 0
\(957\) −4242.04 −0.143287
\(958\) 0 0
\(959\) −14416.6 −0.485439
\(960\) 0 0
\(961\) 18343.4 0.615735
\(962\) 0 0
\(963\) −10332.2 −0.345744
\(964\) 0 0
\(965\) −29972.6 −0.999847
\(966\) 0 0
\(967\) −52267.1 −1.73815 −0.869077 0.494676i \(-0.835287\pi\)
−0.869077 + 0.494676i \(0.835287\pi\)
\(968\) 0 0
\(969\) 724.619 0.0240228
\(970\) 0 0
\(971\) 52489.8 1.73479 0.867394 0.497622i \(-0.165793\pi\)
0.867394 + 0.497622i \(0.165793\pi\)
\(972\) 0 0
\(973\) 7344.07 0.241973
\(974\) 0 0
\(975\) 11844.3 0.389046
\(976\) 0 0
\(977\) 8324.11 0.272581 0.136291 0.990669i \(-0.456482\pi\)
0.136291 + 0.990669i \(0.456482\pi\)
\(978\) 0 0
\(979\) −257.852 −0.00841777
\(980\) 0 0
\(981\) −16454.9 −0.535539
\(982\) 0 0
\(983\) 44407.1 1.44086 0.720431 0.693527i \(-0.243944\pi\)
0.720431 + 0.693527i \(0.243944\pi\)
\(984\) 0 0
\(985\) −89375.3 −2.89110
\(986\) 0 0
\(987\) −11486.7 −0.370442
\(988\) 0 0
\(989\) 1153.50 0.0370870
\(990\) 0 0
\(991\) 45124.7 1.44645 0.723226 0.690612i \(-0.242659\pi\)
0.723226 + 0.690612i \(0.242659\pi\)
\(992\) 0 0
\(993\) −24313.4 −0.777001
\(994\) 0 0
\(995\) −86576.2 −2.75844
\(996\) 0 0
\(997\) 5480.61 0.174095 0.0870474 0.996204i \(-0.472257\pi\)
0.0870474 + 0.996204i \(0.472257\pi\)
\(998\) 0 0
\(999\) 2485.90 0.0787291
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 528.4.a.p.1.1 2
3.2 odd 2 1584.4.a.bj.1.2 2
4.3 odd 2 33.4.a.c.1.2 2
8.3 odd 2 2112.4.a.bn.1.2 2
8.5 even 2 2112.4.a.bg.1.2 2
12.11 even 2 99.4.a.f.1.1 2
20.3 even 4 825.4.c.h.199.1 4
20.7 even 4 825.4.c.h.199.4 4
20.19 odd 2 825.4.a.l.1.1 2
28.27 even 2 1617.4.a.k.1.2 2
44.43 even 2 363.4.a.i.1.1 2
60.59 even 2 2475.4.a.p.1.2 2
132.131 odd 2 1089.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 4.3 odd 2
99.4.a.f.1.1 2 12.11 even 2
363.4.a.i.1.1 2 44.43 even 2
528.4.a.p.1.1 2 1.1 even 1 trivial
825.4.a.l.1.1 2 20.19 odd 2
825.4.c.h.199.1 4 20.3 even 4
825.4.c.h.199.4 4 20.7 even 4
1089.4.a.u.1.2 2 132.131 odd 2
1584.4.a.bj.1.2 2 3.2 odd 2
1617.4.a.k.1.2 2 28.27 even 2
2112.4.a.bg.1.2 2 8.5 even 2
2112.4.a.bn.1.2 2 8.3 odd 2
2475.4.a.p.1.2 2 60.59 even 2