Properties

Label 1150.4.b.o.599.2
Level $1150$
Weight $4$
Character 1150.599
Analytic conductor $67.852$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
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: \( x^{8} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(-0.0119250i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.o.599.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +2.98808i q^{3} -4.00000 q^{4} +5.97615 q^{6} -2.98517i q^{7} +8.00000i q^{8} +18.0714 q^{9} +68.6262 q^{11} -11.9523i q^{12} -12.0028i q^{13} -5.97035 q^{14} +16.0000 q^{16} -106.208i q^{17} -36.1428i q^{18} +48.4891 q^{19} +8.91992 q^{21} -137.252i q^{22} -23.0000i q^{23} -23.9046 q^{24} -24.0055 q^{26} +134.677i q^{27} +11.9407i q^{28} -135.226 q^{29} -230.782 q^{31} -32.0000i q^{32} +205.060i q^{33} -212.416 q^{34} -72.2856 q^{36} +107.956i q^{37} -96.9782i q^{38} +35.8651 q^{39} +394.637 q^{41} -17.8398i q^{42} +136.063i q^{43} -274.505 q^{44} -46.0000 q^{46} +50.4512i q^{47} +47.8092i q^{48} +334.089 q^{49} +317.357 q^{51} +48.0110i q^{52} -414.707i q^{53} +269.353 q^{54} +23.8814 q^{56} +144.889i q^{57} +270.452i q^{58} +183.586 q^{59} -98.4751 q^{61} +461.565i q^{62} -53.9463i q^{63} -64.0000 q^{64} +410.121 q^{66} -136.925i q^{67} +424.831i q^{68} +68.7257 q^{69} -708.800 q^{71} +144.571i q^{72} -689.605i q^{73} +215.912 q^{74} -193.956 q^{76} -204.861i q^{77} -71.7303i q^{78} -546.583 q^{79} +85.5038 q^{81} -789.275i q^{82} -20.2622i q^{83} -35.6797 q^{84} +272.125 q^{86} -404.065i q^{87} +549.010i q^{88} +1087.31 q^{89} -35.8303 q^{91} +92.0000i q^{92} -689.595i q^{93} +100.902 q^{94} +95.6184 q^{96} +1115.90i q^{97} -668.177i q^{98} +1240.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 2.98808i 0.575055i 0.957772 + 0.287528i \(0.0928333\pi\)
−0.957772 + 0.287528i \(0.907167\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 5.97615 0.406626
\(7\) − 2.98517i − 0.161184i −0.996747 0.0805921i \(-0.974319\pi\)
0.996747 0.0805921i \(-0.0256811\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 18.0714 0.669311
\(10\) 0 0
\(11\) 68.6262 1.88105 0.940527 0.339720i \(-0.110332\pi\)
0.940527 + 0.339720i \(0.110332\pi\)
\(12\) − 11.9523i − 0.287528i
\(13\) − 12.0028i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408677\pi\)
\(14\) −5.97035 −0.113974
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 106.208i − 1.51525i −0.652693 0.757623i \(-0.726361\pi\)
0.652693 0.757623i \(-0.273639\pi\)
\(18\) − 36.1428i − 0.473275i
\(19\) 48.4891 0.585482 0.292741 0.956192i \(-0.405433\pi\)
0.292741 + 0.956192i \(0.405433\pi\)
\(20\) 0 0
\(21\) 8.91992 0.0926898
\(22\) − 137.252i − 1.33011i
\(23\) − 23.0000i − 0.208514i
\(24\) −23.9046 −0.203313
\(25\) 0 0
\(26\) −24.0055 −0.181072
\(27\) 134.677i 0.959946i
\(28\) 11.9407i 0.0805921i
\(29\) −135.226 −0.865890 −0.432945 0.901420i \(-0.642526\pi\)
−0.432945 + 0.901420i \(0.642526\pi\)
\(30\) 0 0
\(31\) −230.782 −1.33709 −0.668544 0.743673i \(-0.733082\pi\)
−0.668544 + 0.743673i \(0.733082\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 205.060i 1.08171i
\(34\) −212.416 −1.07144
\(35\) 0 0
\(36\) −72.2856 −0.334656
\(37\) 107.956i 0.479672i 0.970813 + 0.239836i \(0.0770937\pi\)
−0.970813 + 0.239836i \(0.922906\pi\)
\(38\) − 96.9782i − 0.413998i
\(39\) 35.8651 0.147257
\(40\) 0 0
\(41\) 394.637 1.50322 0.751610 0.659608i \(-0.229278\pi\)
0.751610 + 0.659608i \(0.229278\pi\)
\(42\) − 17.8398i − 0.0655416i
\(43\) 136.063i 0.482543i 0.970458 + 0.241272i \(0.0775645\pi\)
−0.970458 + 0.241272i \(0.922436\pi\)
\(44\) −274.505 −0.940527
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 50.4512i 0.156576i 0.996931 + 0.0782880i \(0.0249454\pi\)
−0.996931 + 0.0782880i \(0.975055\pi\)
\(48\) 47.8092i 0.143764i
\(49\) 334.089 0.974020
\(50\) 0 0
\(51\) 317.357 0.871350
\(52\) 48.0110i 0.128037i
\(53\) − 414.707i − 1.07480i −0.843327 0.537400i \(-0.819406\pi\)
0.843327 0.537400i \(-0.180594\pi\)
\(54\) 269.353 0.678785
\(55\) 0 0
\(56\) 23.8814 0.0569872
\(57\) 144.889i 0.336685i
\(58\) 270.452i 0.612277i
\(59\) 183.586 0.405100 0.202550 0.979272i \(-0.435077\pi\)
0.202550 + 0.979272i \(0.435077\pi\)
\(60\) 0 0
\(61\) −98.4751 −0.206696 −0.103348 0.994645i \(-0.532956\pi\)
−0.103348 + 0.994645i \(0.532956\pi\)
\(62\) 461.565i 0.945464i
\(63\) − 53.9463i − 0.107882i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 410.121 0.764884
\(67\) − 136.925i − 0.249672i −0.992177 0.124836i \(-0.960160\pi\)
0.992177 0.124836i \(-0.0398404\pi\)
\(68\) 424.831i 0.757623i
\(69\) 68.7257 0.119907
\(70\) 0 0
\(71\) −708.800 −1.18478 −0.592388 0.805653i \(-0.701815\pi\)
−0.592388 + 0.805653i \(0.701815\pi\)
\(72\) 144.571i 0.236637i
\(73\) − 689.605i − 1.10565i −0.833299 0.552823i \(-0.813551\pi\)
0.833299 0.552823i \(-0.186449\pi\)
\(74\) 215.912 0.339179
\(75\) 0 0
\(76\) −193.956 −0.292741
\(77\) − 204.861i − 0.303196i
\(78\) − 71.7303i − 0.104126i
\(79\) −546.583 −0.778423 −0.389212 0.921148i \(-0.627252\pi\)
−0.389212 + 0.921148i \(0.627252\pi\)
\(80\) 0 0
\(81\) 85.5038 0.117289
\(82\) − 789.275i − 1.06294i
\(83\) − 20.2622i − 0.0267959i −0.999910 0.0133980i \(-0.995735\pi\)
0.999910 0.0133980i \(-0.00426483\pi\)
\(84\) −35.6797 −0.0463449
\(85\) 0 0
\(86\) 272.125 0.341210
\(87\) − 404.065i − 0.497935i
\(88\) 549.010i 0.665053i
\(89\) 1087.31 1.29499 0.647495 0.762069i \(-0.275817\pi\)
0.647495 + 0.762069i \(0.275817\pi\)
\(90\) 0 0
\(91\) −35.8303 −0.0412751
\(92\) 92.0000i 0.104257i
\(93\) − 689.595i − 0.768900i
\(94\) 100.902 0.110716
\(95\) 0 0
\(96\) 95.6184 0.101656
\(97\) 1115.90i 1.16807i 0.811728 + 0.584036i \(0.198527\pi\)
−0.811728 + 0.584036i \(0.801473\pi\)
\(98\) − 668.177i − 0.688736i
\(99\) 1240.17 1.25901
\(100\) 0 0
\(101\) −134.565 −0.132572 −0.0662859 0.997801i \(-0.521115\pi\)
−0.0662859 + 0.997801i \(0.521115\pi\)
\(102\) − 634.714i − 0.616137i
\(103\) − 925.953i − 0.885795i −0.896572 0.442898i \(-0.853951\pi\)
0.896572 0.442898i \(-0.146049\pi\)
\(104\) 96.0221 0.0905360
\(105\) 0 0
\(106\) −829.415 −0.759999
\(107\) − 1871.13i − 1.69055i −0.534334 0.845273i \(-0.679438\pi\)
0.534334 0.845273i \(-0.320562\pi\)
\(108\) − 538.707i − 0.479973i
\(109\) 978.242 0.859620 0.429810 0.902919i \(-0.358581\pi\)
0.429810 + 0.902919i \(0.358581\pi\)
\(110\) 0 0
\(111\) −322.581 −0.275838
\(112\) − 47.7628i − 0.0402961i
\(113\) − 1760.45i − 1.46557i −0.680459 0.732786i \(-0.738220\pi\)
0.680459 0.732786i \(-0.261780\pi\)
\(114\) 289.778 0.238072
\(115\) 0 0
\(116\) 540.903 0.432945
\(117\) − 216.907i − 0.171393i
\(118\) − 367.172i − 0.286449i
\(119\) −317.049 −0.244234
\(120\) 0 0
\(121\) 3378.56 2.53836
\(122\) 196.950i 0.146156i
\(123\) 1179.21i 0.864434i
\(124\) 923.129 0.668544
\(125\) 0 0
\(126\) −107.893 −0.0762844
\(127\) 1260.87i 0.880980i 0.897758 + 0.440490i \(0.145195\pi\)
−0.897758 + 0.440490i \(0.854805\pi\)
\(128\) 128.000i 0.0883883i
\(129\) −406.566 −0.277489
\(130\) 0 0
\(131\) 444.283 0.296314 0.148157 0.988964i \(-0.452666\pi\)
0.148157 + 0.988964i \(0.452666\pi\)
\(132\) − 820.241i − 0.540855i
\(133\) − 144.748i − 0.0943705i
\(134\) −273.849 −0.176545
\(135\) 0 0
\(136\) 849.662 0.535720
\(137\) − 429.072i − 0.267577i −0.991010 0.133789i \(-0.957286\pi\)
0.991010 0.133789i \(-0.0427143\pi\)
\(138\) − 137.451i − 0.0847873i
\(139\) 249.755 0.152402 0.0762012 0.997092i \(-0.475721\pi\)
0.0762012 + 0.997092i \(0.475721\pi\)
\(140\) 0 0
\(141\) −150.752 −0.0900398
\(142\) 1417.60i 0.837763i
\(143\) − 823.704i − 0.481689i
\(144\) 289.143 0.167328
\(145\) 0 0
\(146\) −1379.21 −0.781809
\(147\) 998.282i 0.560115i
\(148\) − 431.824i − 0.239836i
\(149\) 1564.18 0.860020 0.430010 0.902824i \(-0.358510\pi\)
0.430010 + 0.902824i \(0.358510\pi\)
\(150\) 0 0
\(151\) −1860.04 −1.00244 −0.501219 0.865321i \(-0.667115\pi\)
−0.501219 + 0.865321i \(0.667115\pi\)
\(152\) 387.913i 0.206999i
\(153\) − 1919.32i − 1.01417i
\(154\) −409.722 −0.214392
\(155\) 0 0
\(156\) −143.461 −0.0736285
\(157\) − 2834.52i − 1.44089i −0.693512 0.720445i \(-0.743938\pi\)
0.693512 0.720445i \(-0.256062\pi\)
\(158\) 1093.17i 0.550428i
\(159\) 1239.18 0.618070
\(160\) 0 0
\(161\) −68.6590 −0.0336092
\(162\) − 171.008i − 0.0829359i
\(163\) 2114.51i 1.01608i 0.861333 + 0.508041i \(0.169630\pi\)
−0.861333 + 0.508041i \(0.830370\pi\)
\(164\) −1578.55 −0.751610
\(165\) 0 0
\(166\) −40.5244 −0.0189476
\(167\) − 2487.41i − 1.15258i −0.817244 0.576291i \(-0.804499\pi\)
0.817244 0.576291i \(-0.195501\pi\)
\(168\) 71.3594i 0.0327708i
\(169\) 2052.93 0.934426
\(170\) 0 0
\(171\) 876.266 0.391870
\(172\) − 544.251i − 0.241272i
\(173\) 3672.32i 1.61388i 0.590634 + 0.806939i \(0.298878\pi\)
−0.590634 + 0.806939i \(0.701122\pi\)
\(174\) −808.130 −0.352093
\(175\) 0 0
\(176\) 1098.02 0.470263
\(177\) 548.569i 0.232955i
\(178\) − 2174.61i − 0.915697i
\(179\) 3224.63 1.34648 0.673240 0.739424i \(-0.264902\pi\)
0.673240 + 0.739424i \(0.264902\pi\)
\(180\) 0 0
\(181\) −2224.23 −0.913401 −0.456700 0.889621i \(-0.650969\pi\)
−0.456700 + 0.889621i \(0.650969\pi\)
\(182\) 71.6606i 0.0291859i
\(183\) − 294.251i − 0.118862i
\(184\) 184.000 0.0737210
\(185\) 0 0
\(186\) −1379.19 −0.543694
\(187\) − 7288.64i − 2.85026i
\(188\) − 201.805i − 0.0782880i
\(189\) 402.033 0.154728
\(190\) 0 0
\(191\) 3273.53 1.24013 0.620063 0.784552i \(-0.287107\pi\)
0.620063 + 0.784552i \(0.287107\pi\)
\(192\) − 191.237i − 0.0718819i
\(193\) 5126.86i 1.91212i 0.293166 + 0.956062i \(0.405291\pi\)
−0.293166 + 0.956062i \(0.594709\pi\)
\(194\) 2231.81 0.825951
\(195\) 0 0
\(196\) −1336.35 −0.487010
\(197\) − 3262.57i − 1.17994i −0.807425 0.589971i \(-0.799139\pi\)
0.807425 0.589971i \(-0.200861\pi\)
\(198\) − 2480.34i − 0.890255i
\(199\) 168.282 0.0599457 0.0299728 0.999551i \(-0.490458\pi\)
0.0299728 + 0.999551i \(0.490458\pi\)
\(200\) 0 0
\(201\) 409.141 0.143575
\(202\) 269.131i 0.0937424i
\(203\) 403.673i 0.139568i
\(204\) −1269.43 −0.435675
\(205\) 0 0
\(206\) −1851.91 −0.626352
\(207\) − 415.642i − 0.139561i
\(208\) − 192.044i − 0.0640186i
\(209\) 3327.62 1.10132
\(210\) 0 0
\(211\) 4197.01 1.36936 0.684678 0.728846i \(-0.259943\pi\)
0.684678 + 0.728846i \(0.259943\pi\)
\(212\) 1658.83i 0.537400i
\(213\) − 2117.95i − 0.681312i
\(214\) −3742.25 −1.19540
\(215\) 0 0
\(216\) −1077.41 −0.339392
\(217\) 688.925i 0.215518i
\(218\) − 1956.48i − 0.607843i
\(219\) 2060.59 0.635807
\(220\) 0 0
\(221\) −1274.79 −0.388015
\(222\) 645.162i 0.195047i
\(223\) 5192.66i 1.55931i 0.626208 + 0.779656i \(0.284606\pi\)
−0.626208 + 0.779656i \(0.715394\pi\)
\(224\) −95.5256 −0.0284936
\(225\) 0 0
\(226\) −3520.91 −1.03632
\(227\) − 4701.24i − 1.37459i −0.726378 0.687295i \(-0.758798\pi\)
0.726378 0.687295i \(-0.241202\pi\)
\(228\) − 579.556i − 0.168342i
\(229\) 2125.17 0.613255 0.306628 0.951830i \(-0.400799\pi\)
0.306628 + 0.951830i \(0.400799\pi\)
\(230\) 0 0
\(231\) 612.141 0.174355
\(232\) − 1081.81i − 0.306138i
\(233\) − 1646.67i − 0.462992i −0.972836 0.231496i \(-0.925638\pi\)
0.972836 0.231496i \(-0.0743620\pi\)
\(234\) −433.814 −0.121193
\(235\) 0 0
\(236\) −734.345 −0.202550
\(237\) − 1633.23i − 0.447636i
\(238\) 634.097i 0.172699i
\(239\) −2877.88 −0.778888 −0.389444 0.921050i \(-0.627333\pi\)
−0.389444 + 0.921050i \(0.627333\pi\)
\(240\) 0 0
\(241\) −4244.54 −1.13450 −0.567251 0.823545i \(-0.691993\pi\)
−0.567251 + 0.823545i \(0.691993\pi\)
\(242\) − 6757.12i − 1.79489i
\(243\) 3891.76i 1.02739i
\(244\) 393.900 0.103348
\(245\) 0 0
\(246\) 2358.41 0.611247
\(247\) − 582.003i − 0.149927i
\(248\) − 1846.26i − 0.472732i
\(249\) 60.5449 0.0154092
\(250\) 0 0
\(251\) 4401.99 1.10698 0.553488 0.832857i \(-0.313296\pi\)
0.553488 + 0.832857i \(0.313296\pi\)
\(252\) 215.785i 0.0539412i
\(253\) − 1578.40i − 0.392227i
\(254\) 2521.75 0.622947
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 1694.78i − 0.411352i −0.978620 0.205676i \(-0.934061\pi\)
0.978620 0.205676i \(-0.0659393\pi\)
\(258\) 813.131i 0.196214i
\(259\) 322.268 0.0773156
\(260\) 0 0
\(261\) −2443.72 −0.579550
\(262\) − 888.566i − 0.209526i
\(263\) − 727.462i − 0.170560i −0.996357 0.0852799i \(-0.972822\pi\)
0.996357 0.0852799i \(-0.0271784\pi\)
\(264\) −1640.48 −0.382442
\(265\) 0 0
\(266\) −289.497 −0.0667300
\(267\) 3248.95i 0.744691i
\(268\) 547.698i 0.124836i
\(269\) −94.9937 −0.0215311 −0.0107656 0.999942i \(-0.503427\pi\)
−0.0107656 + 0.999942i \(0.503427\pi\)
\(270\) 0 0
\(271\) 5321.54 1.19284 0.596422 0.802671i \(-0.296589\pi\)
0.596422 + 0.802671i \(0.296589\pi\)
\(272\) − 1699.32i − 0.378811i
\(273\) − 107.064i − 0.0237355i
\(274\) −858.143 −0.189206
\(275\) 0 0
\(276\) −274.903 −0.0599537
\(277\) − 2561.25i − 0.555562i −0.960644 0.277781i \(-0.910401\pi\)
0.960644 0.277781i \(-0.0895989\pi\)
\(278\) − 499.510i − 0.107765i
\(279\) −4170.56 −0.894928
\(280\) 0 0
\(281\) −8753.73 −1.85838 −0.929188 0.369606i \(-0.879493\pi\)
−0.929188 + 0.369606i \(0.879493\pi\)
\(282\) 301.504i 0.0636678i
\(283\) − 2400.14i − 0.504147i −0.967708 0.252073i \(-0.918888\pi\)
0.967708 0.252073i \(-0.0811125\pi\)
\(284\) 2835.20 0.592388
\(285\) 0 0
\(286\) −1647.41 −0.340606
\(287\) − 1178.06i − 0.242295i
\(288\) − 578.285i − 0.118319i
\(289\) −6367.09 −1.29597
\(290\) 0 0
\(291\) −3334.41 −0.671706
\(292\) 2758.42i 0.552823i
\(293\) 1045.56i 0.208472i 0.994553 + 0.104236i \(0.0332397\pi\)
−0.994553 + 0.104236i \(0.966760\pi\)
\(294\) 1996.56 0.396061
\(295\) 0 0
\(296\) −863.649 −0.169590
\(297\) 9242.36i 1.80571i
\(298\) − 3128.37i − 0.608126i
\(299\) −276.063 −0.0533952
\(300\) 0 0
\(301\) 406.171 0.0777784
\(302\) 3720.08i 0.708830i
\(303\) − 402.091i − 0.0762361i
\(304\) 775.826 0.146371
\(305\) 0 0
\(306\) −3838.65 −0.717127
\(307\) 7905.84i 1.46974i 0.678208 + 0.734870i \(0.262757\pi\)
−0.678208 + 0.734870i \(0.737243\pi\)
\(308\) 819.445i 0.151598i
\(309\) 2766.82 0.509381
\(310\) 0 0
\(311\) −9227.63 −1.68248 −0.841240 0.540662i \(-0.818174\pi\)
−0.841240 + 0.540662i \(0.818174\pi\)
\(312\) 286.921i 0.0520632i
\(313\) 2712.57i 0.489852i 0.969542 + 0.244926i \(0.0787637\pi\)
−0.969542 + 0.244926i \(0.921236\pi\)
\(314\) −5669.05 −1.01886
\(315\) 0 0
\(316\) 2186.33 0.389212
\(317\) 3440.29i 0.609546i 0.952425 + 0.304773i \(0.0985805\pi\)
−0.952425 + 0.304773i \(0.901419\pi\)
\(318\) − 2478.35i − 0.437041i
\(319\) −9280.04 −1.62879
\(320\) 0 0
\(321\) 5591.06 0.972158
\(322\) 137.318i 0.0237653i
\(323\) − 5149.92i − 0.887149i
\(324\) −342.015 −0.0586446
\(325\) 0 0
\(326\) 4229.03 0.718479
\(327\) 2923.06i 0.494329i
\(328\) 3157.10i 0.531468i
\(329\) 150.606 0.0252376
\(330\) 0 0
\(331\) 3259.75 0.541305 0.270652 0.962677i \(-0.412761\pi\)
0.270652 + 0.962677i \(0.412761\pi\)
\(332\) 81.0487i 0.0133980i
\(333\) 1950.92i 0.321050i
\(334\) −4974.81 −0.814999
\(335\) 0 0
\(336\) 142.719 0.0231725
\(337\) − 7163.38i − 1.15790i −0.815361 0.578952i \(-0.803462\pi\)
0.815361 0.578952i \(-0.196538\pi\)
\(338\) − 4105.87i − 0.660739i
\(339\) 5260.37 0.842785
\(340\) 0 0
\(341\) −15837.7 −2.51513
\(342\) − 1752.53i − 0.277094i
\(343\) − 2021.23i − 0.318181i
\(344\) −1088.50 −0.170605
\(345\) 0 0
\(346\) 7344.63 1.14118
\(347\) 1833.45i 0.283645i 0.989892 + 0.141822i \(0.0452962\pi\)
−0.989892 + 0.141822i \(0.954704\pi\)
\(348\) 1616.26i 0.248967i
\(349\) 10400.3 1.59517 0.797583 0.603209i \(-0.206112\pi\)
0.797583 + 0.603209i \(0.206112\pi\)
\(350\) 0 0
\(351\) 1616.49 0.245818
\(352\) − 2196.04i − 0.332526i
\(353\) − 3156.48i − 0.475928i −0.971274 0.237964i \(-0.923520\pi\)
0.971274 0.237964i \(-0.0764800\pi\)
\(354\) 1097.14 0.164724
\(355\) 0 0
\(356\) −4349.22 −0.647495
\(357\) − 947.365i − 0.140448i
\(358\) − 6449.25i − 0.952105i
\(359\) 9324.16 1.37078 0.685390 0.728176i \(-0.259632\pi\)
0.685390 + 0.728176i \(0.259632\pi\)
\(360\) 0 0
\(361\) −4507.81 −0.657211
\(362\) 4448.45i 0.645872i
\(363\) 10095.4i 1.45970i
\(364\) 143.321 0.0206376
\(365\) 0 0
\(366\) −588.502 −0.0840478
\(367\) − 3802.68i − 0.540867i −0.962739 0.270434i \(-0.912833\pi\)
0.962739 0.270434i \(-0.0871671\pi\)
\(368\) − 368.000i − 0.0521286i
\(369\) 7131.65 1.00612
\(370\) 0 0
\(371\) −1237.97 −0.173241
\(372\) 2758.38i 0.384450i
\(373\) − 8928.03i − 1.23935i −0.784860 0.619673i \(-0.787265\pi\)
0.784860 0.619673i \(-0.212735\pi\)
\(374\) −14577.3 −2.01544
\(375\) 0 0
\(376\) −403.610 −0.0553580
\(377\) 1623.08i 0.221732i
\(378\) − 804.067i − 0.109409i
\(379\) −3798.13 −0.514768 −0.257384 0.966309i \(-0.582860\pi\)
−0.257384 + 0.966309i \(0.582860\pi\)
\(380\) 0 0
\(381\) −3767.58 −0.506612
\(382\) − 6547.05i − 0.876901i
\(383\) 1456.41i 0.194306i 0.995269 + 0.0971530i \(0.0309736\pi\)
−0.995269 + 0.0971530i \(0.969026\pi\)
\(384\) −382.474 −0.0508282
\(385\) 0 0
\(386\) 10253.7 1.35208
\(387\) 2458.84i 0.322972i
\(388\) − 4463.62i − 0.584036i
\(389\) 8211.34 1.07026 0.535130 0.844769i \(-0.320262\pi\)
0.535130 + 0.844769i \(0.320262\pi\)
\(390\) 0 0
\(391\) −2442.78 −0.315950
\(392\) 2672.71i 0.344368i
\(393\) 1327.55i 0.170397i
\(394\) −6525.14 −0.834345
\(395\) 0 0
\(396\) −4960.69 −0.629505
\(397\) − 914.365i − 0.115594i −0.998328 0.0577968i \(-0.981592\pi\)
0.998328 0.0577968i \(-0.0184076\pi\)
\(398\) − 336.564i − 0.0423880i
\(399\) 432.519 0.0542683
\(400\) 0 0
\(401\) 9414.97 1.17247 0.586236 0.810140i \(-0.300609\pi\)
0.586236 + 0.810140i \(0.300609\pi\)
\(402\) − 818.282i − 0.101523i
\(403\) 2770.02i 0.342394i
\(404\) 538.261 0.0662859
\(405\) 0 0
\(406\) 807.345 0.0986894
\(407\) 7408.62i 0.902289i
\(408\) 2538.85i 0.308069i
\(409\) −3299.41 −0.398889 −0.199444 0.979909i \(-0.563914\pi\)
−0.199444 + 0.979909i \(0.563914\pi\)
\(410\) 0 0
\(411\) 1282.10 0.153872
\(412\) 3703.81i 0.442898i
\(413\) − 548.037i − 0.0652957i
\(414\) −831.285 −0.0986846
\(415\) 0 0
\(416\) −384.088 −0.0452680
\(417\) 746.286i 0.0876398i
\(418\) − 6655.25i − 0.778753i
\(419\) 3554.64 0.414452 0.207226 0.978293i \(-0.433556\pi\)
0.207226 + 0.978293i \(0.433556\pi\)
\(420\) 0 0
\(421\) 11655.5 1.34930 0.674649 0.738139i \(-0.264295\pi\)
0.674649 + 0.738139i \(0.264295\pi\)
\(422\) − 8394.02i − 0.968281i
\(423\) 911.725i 0.104798i
\(424\) 3317.66 0.380000
\(425\) 0 0
\(426\) −4235.90 −0.481760
\(427\) 293.965i 0.0333161i
\(428\) 7484.50i 0.845273i
\(429\) 2461.29 0.276998
\(430\) 0 0
\(431\) 4727.38 0.528330 0.264165 0.964478i \(-0.414904\pi\)
0.264165 + 0.964478i \(0.414904\pi\)
\(432\) 2154.83i 0.239987i
\(433\) 3936.24i 0.436868i 0.975852 + 0.218434i \(0.0700948\pi\)
−0.975852 + 0.218434i \(0.929905\pi\)
\(434\) 1377.85 0.152394
\(435\) 0 0
\(436\) −3912.97 −0.429810
\(437\) − 1115.25i − 0.122081i
\(438\) − 4121.18i − 0.449584i
\(439\) −6859.77 −0.745784 −0.372892 0.927875i \(-0.621634\pi\)
−0.372892 + 0.927875i \(0.621634\pi\)
\(440\) 0 0
\(441\) 6037.45 0.651922
\(442\) 2549.57i 0.274368i
\(443\) 16814.4i 1.80334i 0.432430 + 0.901668i \(0.357656\pi\)
−0.432430 + 0.901668i \(0.642344\pi\)
\(444\) 1290.32 0.137919
\(445\) 0 0
\(446\) 10385.3 1.10260
\(447\) 4673.90i 0.494559i
\(448\) 191.051i 0.0201480i
\(449\) −10673.8 −1.12189 −0.560943 0.827855i \(-0.689561\pi\)
−0.560943 + 0.827855i \(0.689561\pi\)
\(450\) 0 0
\(451\) 27082.5 2.82764
\(452\) 7041.82i 0.732786i
\(453\) − 5557.94i − 0.576457i
\(454\) −9402.47 −0.971982
\(455\) 0 0
\(456\) −1159.11 −0.119036
\(457\) − 15493.0i − 1.58585i −0.609318 0.792926i \(-0.708557\pi\)
0.609318 0.792926i \(-0.291443\pi\)
\(458\) − 4250.35i − 0.433637i
\(459\) 14303.7 1.45455
\(460\) 0 0
\(461\) −10793.0 −1.09041 −0.545207 0.838302i \(-0.683549\pi\)
−0.545207 + 0.838302i \(0.683549\pi\)
\(462\) − 1224.28i − 0.123287i
\(463\) − 6165.87i − 0.618904i −0.950915 0.309452i \(-0.899854\pi\)
0.950915 0.309452i \(-0.100146\pi\)
\(464\) −2163.61 −0.216473
\(465\) 0 0
\(466\) −3293.34 −0.327385
\(467\) 10736.4i 1.06386i 0.846789 + 0.531929i \(0.178533\pi\)
−0.846789 + 0.531929i \(0.821467\pi\)
\(468\) 867.627i 0.0856967i
\(469\) −408.744 −0.0402431
\(470\) 0 0
\(471\) 8469.77 0.828591
\(472\) 1468.69i 0.143224i
\(473\) 9337.47i 0.907690i
\(474\) −3266.46 −0.316527
\(475\) 0 0
\(476\) 1268.19 0.122117
\(477\) − 7494.35i − 0.719377i
\(478\) 5755.75i 0.550757i
\(479\) −6333.30 −0.604125 −0.302062 0.953288i \(-0.597675\pi\)
−0.302062 + 0.953288i \(0.597675\pi\)
\(480\) 0 0
\(481\) 1295.77 0.122832
\(482\) 8489.08i 0.802213i
\(483\) − 205.158i − 0.0193272i
\(484\) −13514.2 −1.26918
\(485\) 0 0
\(486\) 7783.53 0.726477
\(487\) 16833.3i 1.56630i 0.621834 + 0.783149i \(0.286388\pi\)
−0.621834 + 0.783149i \(0.713612\pi\)
\(488\) − 787.801i − 0.0730780i
\(489\) −6318.33 −0.584304
\(490\) 0 0
\(491\) −4.80428 −0.000441577 0 −0.000220788 1.00000i \(-0.500070\pi\)
−0.000220788 1.00000i \(0.500070\pi\)
\(492\) − 4716.82i − 0.432217i
\(493\) 14362.0i 1.31204i
\(494\) −1164.01 −0.106014
\(495\) 0 0
\(496\) −3692.52 −0.334272
\(497\) 2115.89i 0.190967i
\(498\) − 121.090i − 0.0108959i
\(499\) −21206.9 −1.90251 −0.951253 0.308412i \(-0.900203\pi\)
−0.951253 + 0.308412i \(0.900203\pi\)
\(500\) 0 0
\(501\) 7432.56 0.662799
\(502\) − 8803.98i − 0.782750i
\(503\) 14513.7i 1.28655i 0.765635 + 0.643275i \(0.222425\pi\)
−0.765635 + 0.643275i \(0.777575\pi\)
\(504\) 431.570 0.0381422
\(505\) 0 0
\(506\) −3156.81 −0.277346
\(507\) 6134.32i 0.537347i
\(508\) − 5043.49i − 0.440490i
\(509\) 2545.77 0.221689 0.110844 0.993838i \(-0.464645\pi\)
0.110844 + 0.993838i \(0.464645\pi\)
\(510\) 0 0
\(511\) −2058.59 −0.178213
\(512\) − 512.000i − 0.0441942i
\(513\) 6530.35i 0.562032i
\(514\) −3389.56 −0.290870
\(515\) 0 0
\(516\) 1626.26 0.138745
\(517\) 3462.28i 0.294528i
\(518\) − 644.535i − 0.0546704i
\(519\) −10973.2 −0.928070
\(520\) 0 0
\(521\) −21343.1 −1.79473 −0.897367 0.441285i \(-0.854523\pi\)
−0.897367 + 0.441285i \(0.854523\pi\)
\(522\) 4887.44i 0.409804i
\(523\) − 789.191i − 0.0659827i −0.999456 0.0329913i \(-0.989497\pi\)
0.999456 0.0329913i \(-0.0105034\pi\)
\(524\) −1777.13 −0.148157
\(525\) 0 0
\(526\) −1454.92 −0.120604
\(527\) 24510.9i 2.02602i
\(528\) 3280.96i 0.270427i
\(529\) −529.000 −0.0434783
\(530\) 0 0
\(531\) 3317.66 0.271138
\(532\) 578.994i 0.0471852i
\(533\) − 4736.74i − 0.384936i
\(534\) 6497.90 0.526576
\(535\) 0 0
\(536\) 1095.40 0.0882723
\(537\) 9635.42i 0.774300i
\(538\) 189.987i 0.0152248i
\(539\) 22927.2 1.83218
\(540\) 0 0
\(541\) 13405.1 1.06531 0.532653 0.846334i \(-0.321195\pi\)
0.532653 + 0.846334i \(0.321195\pi\)
\(542\) − 10643.1i − 0.843468i
\(543\) − 6646.16i − 0.525256i
\(544\) −3398.65 −0.267860
\(545\) 0 0
\(546\) −214.127 −0.0167835
\(547\) − 10088.8i − 0.788605i −0.918981 0.394303i \(-0.870986\pi\)
0.918981 0.394303i \(-0.129014\pi\)
\(548\) 1716.29i 0.133789i
\(549\) −1779.58 −0.138344
\(550\) 0 0
\(551\) −6556.98 −0.506963
\(552\) 549.806i 0.0423936i
\(553\) 1631.65i 0.125470i
\(554\) −5122.50 −0.392842
\(555\) 0 0
\(556\) −999.019 −0.0762012
\(557\) 11657.7i 0.886809i 0.896322 + 0.443405i \(0.146230\pi\)
−0.896322 + 0.443405i \(0.853770\pi\)
\(558\) 8341.12i 0.632810i
\(559\) 1633.13 0.123567
\(560\) 0 0
\(561\) 21779.0 1.63906
\(562\) 17507.5i 1.31407i
\(563\) 4839.43i 0.362270i 0.983458 + 0.181135i \(0.0579770\pi\)
−0.983458 + 0.181135i \(0.942023\pi\)
\(564\) 603.008 0.0450199
\(565\) 0 0
\(566\) −4800.28 −0.356486
\(567\) − 255.244i − 0.0189052i
\(568\) − 5670.40i − 0.418882i
\(569\) 646.680 0.0476454 0.0238227 0.999716i \(-0.492416\pi\)
0.0238227 + 0.999716i \(0.492416\pi\)
\(570\) 0 0
\(571\) 3263.66 0.239194 0.119597 0.992822i \(-0.461840\pi\)
0.119597 + 0.992822i \(0.461840\pi\)
\(572\) 3294.82i 0.240845i
\(573\) 9781.54i 0.713141i
\(574\) −2356.12 −0.171329
\(575\) 0 0
\(576\) −1156.57 −0.0836639
\(577\) 8980.46i 0.647940i 0.946067 + 0.323970i \(0.105018\pi\)
−0.946067 + 0.323970i \(0.894982\pi\)
\(578\) 12734.2i 0.916388i
\(579\) −15319.5 −1.09958
\(580\) 0 0
\(581\) −60.4861 −0.00431908
\(582\) 6668.81i 0.474968i
\(583\) − 28459.8i − 2.02176i
\(584\) 5516.84 0.390905
\(585\) 0 0
\(586\) 2091.12 0.147412
\(587\) 6538.05i 0.459718i 0.973224 + 0.229859i \(0.0738265\pi\)
−0.973224 + 0.229859i \(0.926174\pi\)
\(588\) − 3993.13i − 0.280058i
\(589\) −11190.4 −0.782841
\(590\) 0 0
\(591\) 9748.80 0.678532
\(592\) 1727.30i 0.119918i
\(593\) 7803.41i 0.540384i 0.962806 + 0.270192i \(0.0870872\pi\)
−0.962806 + 0.270192i \(0.912913\pi\)
\(594\) 18484.7 1.27683
\(595\) 0 0
\(596\) −6256.73 −0.430010
\(597\) 502.839i 0.0344721i
\(598\) 552.127i 0.0377561i
\(599\) 27812.0 1.89711 0.948555 0.316613i \(-0.102546\pi\)
0.948555 + 0.316613i \(0.102546\pi\)
\(600\) 0 0
\(601\) 483.216 0.0327966 0.0163983 0.999866i \(-0.494780\pi\)
0.0163983 + 0.999866i \(0.494780\pi\)
\(602\) − 812.342i − 0.0549976i
\(603\) − 2474.42i − 0.167108i
\(604\) 7440.17 0.501219
\(605\) 0 0
\(606\) −804.183 −0.0539071
\(607\) 18104.0i 1.21058i 0.796006 + 0.605289i \(0.206942\pi\)
−0.796006 + 0.605289i \(0.793058\pi\)
\(608\) − 1551.65i − 0.103500i
\(609\) −1206.20 −0.0802592
\(610\) 0 0
\(611\) 605.554 0.0400951
\(612\) 7677.30i 0.507085i
\(613\) 2191.19i 0.144374i 0.997391 + 0.0721870i \(0.0229978\pi\)
−0.997391 + 0.0721870i \(0.977002\pi\)
\(614\) 15811.7 1.03926
\(615\) 0 0
\(616\) 1638.89 0.107196
\(617\) 8243.06i 0.537849i 0.963161 + 0.268925i \(0.0866683\pi\)
−0.963161 + 0.268925i \(0.913332\pi\)
\(618\) − 5533.64i − 0.360187i
\(619\) −18298.0 −1.18814 −0.594069 0.804414i \(-0.702479\pi\)
−0.594069 + 0.804414i \(0.702479\pi\)
\(620\) 0 0
\(621\) 3097.57 0.200163
\(622\) 18455.3i 1.18969i
\(623\) − 3245.80i − 0.208732i
\(624\) 573.842 0.0368142
\(625\) 0 0
\(626\) 5425.15 0.346378
\(627\) 9943.19i 0.633322i
\(628\) 11338.1i 0.720445i
\(629\) 11465.8 0.726821
\(630\) 0 0
\(631\) −26472.0 −1.67010 −0.835051 0.550172i \(-0.814562\pi\)
−0.835051 + 0.550172i \(0.814562\pi\)
\(632\) − 4372.67i − 0.275214i
\(633\) 12541.0i 0.787455i
\(634\) 6880.58 0.431014
\(635\) 0 0
\(636\) −4956.71 −0.309035
\(637\) − 4009.99i − 0.249421i
\(638\) 18560.1i 1.15173i
\(639\) −12809.0 −0.792984
\(640\) 0 0
\(641\) 7411.93 0.456714 0.228357 0.973577i \(-0.426665\pi\)
0.228357 + 0.973577i \(0.426665\pi\)
\(642\) − 11182.1i − 0.687419i
\(643\) 8405.96i 0.515550i 0.966205 + 0.257775i \(0.0829894\pi\)
−0.966205 + 0.257775i \(0.917011\pi\)
\(644\) 274.636 0.0168046
\(645\) 0 0
\(646\) −10299.8 −0.627309
\(647\) − 27878.3i − 1.69398i −0.531605 0.846992i \(-0.678411\pi\)
0.531605 0.846992i \(-0.321589\pi\)
\(648\) 684.030i 0.0414680i
\(649\) 12598.8 0.762014
\(650\) 0 0
\(651\) −2058.56 −0.123934
\(652\) − 8458.06i − 0.508041i
\(653\) − 18871.1i − 1.13091i −0.824779 0.565455i \(-0.808701\pi\)
0.824779 0.565455i \(-0.191299\pi\)
\(654\) 5846.12 0.349543
\(655\) 0 0
\(656\) 6314.20 0.375805
\(657\) − 12462.1i − 0.740021i
\(658\) − 301.211i − 0.0178457i
\(659\) 12144.4 0.717872 0.358936 0.933362i \(-0.383140\pi\)
0.358936 + 0.933362i \(0.383140\pi\)
\(660\) 0 0
\(661\) −32504.2 −1.91266 −0.956328 0.292295i \(-0.905581\pi\)
−0.956328 + 0.292295i \(0.905581\pi\)
\(662\) − 6519.49i − 0.382760i
\(663\) − 3809.16i − 0.223130i
\(664\) 162.097 0.00947380
\(665\) 0 0
\(666\) 3901.84 0.227017
\(667\) 3110.19i 0.180551i
\(668\) 9949.62i 0.576291i
\(669\) −15516.1 −0.896690
\(670\) 0 0
\(671\) −6757.98 −0.388806
\(672\) − 285.438i − 0.0163854i
\(673\) − 21387.8i − 1.22502i −0.790462 0.612511i \(-0.790160\pi\)
0.790462 0.612511i \(-0.209840\pi\)
\(674\) −14326.8 −0.818762
\(675\) 0 0
\(676\) −8211.74 −0.467213
\(677\) − 10494.6i − 0.595776i −0.954601 0.297888i \(-0.903718\pi\)
0.954601 0.297888i \(-0.0962821\pi\)
\(678\) − 10520.7i − 0.595939i
\(679\) 3331.17 0.188275
\(680\) 0 0
\(681\) 14047.6 0.790466
\(682\) 31675.4i 1.77847i
\(683\) 24014.2i 1.34535i 0.739937 + 0.672677i \(0.234855\pi\)
−0.739937 + 0.672677i \(0.765145\pi\)
\(684\) −3505.07 −0.195935
\(685\) 0 0
\(686\) −4042.45 −0.224988
\(687\) 6350.18i 0.352656i
\(688\) 2177.00i 0.120636i
\(689\) −4977.63 −0.275229
\(690\) 0 0
\(691\) −10825.8 −0.595995 −0.297997 0.954567i \(-0.596319\pi\)
−0.297997 + 0.954567i \(0.596319\pi\)
\(692\) − 14689.3i − 0.806939i
\(693\) − 3702.13i − 0.202933i
\(694\) 3666.90 0.200567
\(695\) 0 0
\(696\) 3232.52 0.176046
\(697\) − 41913.6i − 2.27775i
\(698\) − 20800.5i − 1.12795i
\(699\) 4920.38 0.266246
\(700\) 0 0
\(701\) −171.021 −0.00921451 −0.00460726 0.999989i \(-0.501467\pi\)
−0.00460726 + 0.999989i \(0.501467\pi\)
\(702\) − 3232.99i − 0.173819i
\(703\) 5234.69i 0.280840i
\(704\) −4392.08 −0.235132
\(705\) 0 0
\(706\) −6312.96 −0.336532
\(707\) 401.701i 0.0213685i
\(708\) − 2194.28i − 0.116477i
\(709\) −28732.7 −1.52197 −0.760987 0.648767i \(-0.775285\pi\)
−0.760987 + 0.648767i \(0.775285\pi\)
\(710\) 0 0
\(711\) −9877.53 −0.521007
\(712\) 8698.45i 0.457848i
\(713\) 5307.99i 0.278802i
\(714\) −1894.73 −0.0993116
\(715\) 0 0
\(716\) −12898.5 −0.673240
\(717\) − 8599.31i − 0.447904i
\(718\) − 18648.3i − 0.969288i
\(719\) −6976.22 −0.361849 −0.180924 0.983497i \(-0.557909\pi\)
−0.180924 + 0.983497i \(0.557909\pi\)
\(720\) 0 0
\(721\) −2764.13 −0.142776
\(722\) 9015.61i 0.464718i
\(723\) − 12683.0i − 0.652401i
\(724\) 8896.91 0.456700
\(725\) 0 0
\(726\) 20190.8 1.03216
\(727\) − 13114.6i − 0.669042i −0.942388 0.334521i \(-0.891426\pi\)
0.942388 0.334521i \(-0.108574\pi\)
\(728\) − 286.643i − 0.0145930i
\(729\) −9320.28 −0.473519
\(730\) 0 0
\(731\) 14450.9 0.731172
\(732\) 1177.00i 0.0594308i
\(733\) − 22681.7i − 1.14293i −0.820626 0.571466i \(-0.806375\pi\)
0.820626 0.571466i \(-0.193625\pi\)
\(734\) −7605.36 −0.382451
\(735\) 0 0
\(736\) −736.000 −0.0368605
\(737\) − 9396.62i − 0.469646i
\(738\) − 14263.3i − 0.711436i
\(739\) −24237.8 −1.20650 −0.603249 0.797553i \(-0.706127\pi\)
−0.603249 + 0.797553i \(0.706127\pi\)
\(740\) 0 0
\(741\) 1739.07 0.0862163
\(742\) 2475.95i 0.122500i
\(743\) 27356.7i 1.35077i 0.737466 + 0.675384i \(0.236022\pi\)
−0.737466 + 0.675384i \(0.763978\pi\)
\(744\) 5516.76 0.271847
\(745\) 0 0
\(746\) −17856.1 −0.876350
\(747\) − 366.166i − 0.0179348i
\(748\) 29154.6i 1.42513i
\(749\) −5585.63 −0.272489
\(750\) 0 0
\(751\) 22297.8 1.08343 0.541716 0.840561i \(-0.317775\pi\)
0.541716 + 0.840561i \(0.317775\pi\)
\(752\) 807.220i 0.0391440i
\(753\) 13153.5i 0.636573i
\(754\) 3246.17 0.156788
\(755\) 0 0
\(756\) −1608.13 −0.0773641
\(757\) 16088.4i 0.772447i 0.922405 + 0.386224i \(0.126221\pi\)
−0.922405 + 0.386224i \(0.873779\pi\)
\(758\) 7596.26i 0.363996i
\(759\) 4716.39 0.225552
\(760\) 0 0
\(761\) 22500.2 1.07179 0.535894 0.844285i \(-0.319975\pi\)
0.535894 + 0.844285i \(0.319975\pi\)
\(762\) 7535.17i 0.358229i
\(763\) − 2920.22i − 0.138557i
\(764\) −13094.1 −0.620063
\(765\) 0 0
\(766\) 2912.82 0.137395
\(767\) − 2203.54i − 0.103736i
\(768\) 764.947i 0.0359410i
\(769\) 22677.7 1.06343 0.531716 0.846923i \(-0.321547\pi\)
0.531716 + 0.846923i \(0.321547\pi\)
\(770\) 0 0
\(771\) 5064.13 0.236550
\(772\) − 20507.5i − 0.956062i
\(773\) 20465.1i 0.952235i 0.879382 + 0.476118i \(0.157956\pi\)
−0.879382 + 0.476118i \(0.842044\pi\)
\(774\) 4917.69 0.228376
\(775\) 0 0
\(776\) −8927.24 −0.412976
\(777\) 962.960i 0.0444607i
\(778\) − 16422.7i − 0.756789i
\(779\) 19135.6 0.880108
\(780\) 0 0
\(781\) −48642.3 −2.22863
\(782\) 4885.56i 0.223411i
\(783\) − 18211.8i − 0.831208i
\(784\) 5345.42 0.243505
\(785\) 0 0
\(786\) 2655.10 0.120489
\(787\) 28850.8i 1.30676i 0.757031 + 0.653379i \(0.226649\pi\)
−0.757031 + 0.653379i \(0.773351\pi\)
\(788\) 13050.3i 0.589971i
\(789\) 2173.71 0.0980813
\(790\) 0 0
\(791\) −5255.26 −0.236227
\(792\) 9921.38i 0.445127i
\(793\) 1181.97i 0.0529295i
\(794\) −1828.73 −0.0817370
\(795\) 0 0
\(796\) −673.128 −0.0299728
\(797\) 39332.9i 1.74811i 0.485828 + 0.874054i \(0.338518\pi\)
−0.485828 + 0.874054i \(0.661482\pi\)
\(798\) − 865.038i − 0.0383735i
\(799\) 5358.31 0.237251
\(800\) 0 0
\(801\) 19649.1 0.866752
\(802\) − 18829.9i − 0.829063i
\(803\) − 47325.0i − 2.07978i
\(804\) −1636.56 −0.0717875
\(805\) 0 0
\(806\) 5540.05 0.242109
\(807\) − 283.848i − 0.0123816i
\(808\) − 1076.52i − 0.0468712i
\(809\) −33946.5 −1.47527 −0.737637 0.675198i \(-0.764058\pi\)
−0.737637 + 0.675198i \(0.764058\pi\)
\(810\) 0 0
\(811\) −20772.2 −0.899398 −0.449699 0.893180i \(-0.648469\pi\)
−0.449699 + 0.893180i \(0.648469\pi\)
\(812\) − 1614.69i − 0.0697839i
\(813\) 15901.2i 0.685951i
\(814\) 14817.2 0.638015
\(815\) 0 0
\(816\) 5077.71 0.217837
\(817\) 6597.56i 0.282521i
\(818\) 6598.83i 0.282057i
\(819\) −647.504 −0.0276259
\(820\) 0 0
\(821\) 6466.37 0.274882 0.137441 0.990510i \(-0.456112\pi\)
0.137441 + 0.990510i \(0.456112\pi\)
\(822\) − 2564.20i − 0.108804i
\(823\) − 16503.7i − 0.699008i −0.936935 0.349504i \(-0.886350\pi\)
0.936935 0.349504i \(-0.113650\pi\)
\(824\) 7407.63 0.313176
\(825\) 0 0
\(826\) −1096.07 −0.0461710
\(827\) − 36540.8i − 1.53645i −0.640177 0.768227i \(-0.721139\pi\)
0.640177 0.768227i \(-0.278861\pi\)
\(828\) 1662.57i 0.0697805i
\(829\) 24874.5 1.04213 0.521066 0.853516i \(-0.325535\pi\)
0.521066 + 0.853516i \(0.325535\pi\)
\(830\) 0 0
\(831\) 7653.21 0.319479
\(832\) 768.177i 0.0320093i
\(833\) − 35482.8i − 1.47588i
\(834\) 1492.57 0.0619707
\(835\) 0 0
\(836\) −13310.5 −0.550662
\(837\) − 31081.0i − 1.28353i
\(838\) − 7109.28i − 0.293062i
\(839\) −15814.3 −0.650741 −0.325370 0.945587i \(-0.605489\pi\)
−0.325370 + 0.945587i \(0.605489\pi\)
\(840\) 0 0
\(841\) −6102.97 −0.250234
\(842\) − 23311.0i − 0.954097i
\(843\) − 26156.8i − 1.06867i
\(844\) −16788.0 −0.684678
\(845\) 0 0
\(846\) 1823.45 0.0741034
\(847\) − 10085.6i − 0.409144i
\(848\) − 6635.32i − 0.268700i
\(849\) 7171.80 0.289912
\(850\) 0 0
\(851\) 2482.99 0.100019
\(852\) 8471.79i 0.340656i
\(853\) − 31093.2i − 1.24808i −0.781394 0.624039i \(-0.785491\pi\)
0.781394 0.624039i \(-0.214509\pi\)
\(854\) 587.931 0.0235580
\(855\) 0 0
\(856\) 14969.0 0.597699
\(857\) − 34212.4i − 1.36368i −0.731501 0.681840i \(-0.761180\pi\)
0.731501 0.681840i \(-0.238820\pi\)
\(858\) − 4922.58i − 0.195867i
\(859\) −4821.57 −0.191513 −0.0957567 0.995405i \(-0.530527\pi\)
−0.0957567 + 0.995405i \(0.530527\pi\)
\(860\) 0 0
\(861\) 3520.13 0.139333
\(862\) − 9454.77i − 0.373585i
\(863\) 19145.6i 0.755185i 0.925972 + 0.377593i \(0.123248\pi\)
−0.925972 + 0.377593i \(0.876752\pi\)
\(864\) 4309.66 0.169696
\(865\) 0 0
\(866\) 7872.49 0.308912
\(867\) − 19025.4i − 0.745254i
\(868\) − 2755.70i − 0.107759i
\(869\) −37509.9 −1.46426
\(870\) 0 0
\(871\) −1643.47 −0.0639345
\(872\) 7825.93i 0.303921i
\(873\) 20166.0i 0.781804i
\(874\) −2230.50 −0.0863246
\(875\) 0 0
\(876\) −8242.36 −0.317904
\(877\) − 16890.8i − 0.650354i −0.945653 0.325177i \(-0.894576\pi\)
0.945653 0.325177i \(-0.105424\pi\)
\(878\) 13719.5i 0.527349i
\(879\) −3124.21 −0.119883
\(880\) 0 0
\(881\) −27802.3 −1.06321 −0.531603 0.846994i \(-0.678410\pi\)
−0.531603 + 0.846994i \(0.678410\pi\)
\(882\) − 12074.9i − 0.460979i
\(883\) 9794.15i 0.373272i 0.982429 + 0.186636i \(0.0597585\pi\)
−0.982429 + 0.186636i \(0.940241\pi\)
\(884\) 5099.15 0.194008
\(885\) 0 0
\(886\) 33628.9 1.27515
\(887\) 41652.4i 1.57672i 0.615214 + 0.788360i \(0.289070\pi\)
−0.615214 + 0.788360i \(0.710930\pi\)
\(888\) − 2580.65i − 0.0975235i
\(889\) 3763.93 0.142000
\(890\) 0 0
\(891\) 5867.80 0.220627
\(892\) − 20770.7i − 0.779656i
\(893\) 2446.34i 0.0916724i
\(894\) 9347.80 0.349706
\(895\) 0 0
\(896\) 382.102 0.0142468
\(897\) − 824.898i − 0.0307052i
\(898\) 21347.5i 0.793293i
\(899\) 31207.7 1.15777
\(900\) 0 0
\(901\) −44045.2 −1.62859
\(902\) − 54164.9i − 1.99944i
\(903\) 1213.67i 0.0447269i
\(904\) 14083.6 0.518158
\(905\) 0 0
\(906\) −11115.9 −0.407617
\(907\) − 5467.21i − 0.200150i −0.994980 0.100075i \(-0.968092\pi\)
0.994980 0.100075i \(-0.0319082\pi\)
\(908\) 18804.9i 0.687295i
\(909\) −2431.79 −0.0887318
\(910\) 0 0
\(911\) 33520.9 1.21910 0.609549 0.792749i \(-0.291351\pi\)
0.609549 + 0.792749i \(0.291351\pi\)
\(912\) 2318.23i 0.0841712i
\(913\) − 1390.52i − 0.0504046i
\(914\) −30986.1 −1.12137
\(915\) 0 0
\(916\) −8500.70 −0.306628
\(917\) − 1326.26i − 0.0477612i
\(918\) − 28607.4i − 1.02853i
\(919\) −21423.9 −0.768996 −0.384498 0.923126i \(-0.625625\pi\)
−0.384498 + 0.923126i \(0.625625\pi\)
\(920\) 0 0
\(921\) −23623.2 −0.845182
\(922\) 21586.0i 0.771039i
\(923\) 8507.56i 0.303391i
\(924\) −2448.56 −0.0871773
\(925\) 0 0
\(926\) −12331.7 −0.437631
\(927\) − 16733.3i − 0.592873i
\(928\) 4327.23i 0.153069i
\(929\) −25838.9 −0.912538 −0.456269 0.889842i \(-0.650814\pi\)
−0.456269 + 0.889842i \(0.650814\pi\)
\(930\) 0 0
\(931\) 16199.7 0.570271
\(932\) 6586.69i 0.231496i
\(933\) − 27572.9i − 0.967519i
\(934\) 21472.8 0.752261
\(935\) 0 0
\(936\) 1735.25 0.0605967
\(937\) − 48095.9i − 1.67687i −0.545003 0.838434i \(-0.683472\pi\)
0.545003 0.838434i \(-0.316528\pi\)
\(938\) 817.487i 0.0284562i
\(939\) −8105.38 −0.281692
\(940\) 0 0
\(941\) −32450.4 −1.12418 −0.562089 0.827077i \(-0.690002\pi\)
−0.562089 + 0.827077i \(0.690002\pi\)
\(942\) − 16939.5i − 0.585903i
\(943\) − 9076.66i − 0.313443i
\(944\) 2937.38 0.101275
\(945\) 0 0
\(946\) 18674.9 0.641834
\(947\) − 39067.9i − 1.34059i −0.742096 0.670294i \(-0.766168\pi\)
0.742096 0.670294i \(-0.233832\pi\)
\(948\) 6532.93i 0.223818i
\(949\) −8277.16 −0.283127
\(950\) 0 0
\(951\) −10279.8 −0.350522
\(952\) − 2536.39i − 0.0863496i
\(953\) 17756.2i 0.603546i 0.953380 + 0.301773i \(0.0975784\pi\)
−0.953380 + 0.301773i \(0.902422\pi\)
\(954\) −14988.7 −0.508676
\(955\) 0 0
\(956\) 11511.5 0.389444
\(957\) − 27729.5i − 0.936642i
\(958\) 12666.6i 0.427181i
\(959\) −1280.85 −0.0431292
\(960\) 0 0
\(961\) 23469.5 0.787805
\(962\) − 2591.54i − 0.0868552i
\(963\) − 33813.9i − 1.13150i
\(964\) 16978.2 0.567251
\(965\) 0 0
\(966\) −410.316 −0.0136664
\(967\) 10696.2i 0.355705i 0.984057 + 0.177853i \(0.0569151\pi\)
−0.984057 + 0.177853i \(0.943085\pi\)
\(968\) 27028.5i 0.897446i
\(969\) 15388.3 0.510160
\(970\) 0 0
\(971\) −53635.1 −1.77264 −0.886320 0.463074i \(-0.846746\pi\)
−0.886320 + 0.463074i \(0.846746\pi\)
\(972\) − 15567.1i − 0.513697i
\(973\) − 745.561i − 0.0245649i
\(974\) 33666.5 1.10754
\(975\) 0 0
\(976\) −1575.60 −0.0516740
\(977\) 21627.7i 0.708220i 0.935204 + 0.354110i \(0.115216\pi\)
−0.935204 + 0.354110i \(0.884784\pi\)
\(978\) 12636.7i 0.413165i
\(979\) 74617.7 2.43595
\(980\) 0 0
\(981\) 17678.2 0.575353
\(982\) 9.60856i 0 0.000312242i
\(983\) 12031.4i 0.390378i 0.980766 + 0.195189i \(0.0625321\pi\)
−0.980766 + 0.195189i \(0.937468\pi\)
\(984\) −9433.65 −0.305624
\(985\) 0 0
\(986\) 28724.1 0.927749
\(987\) 450.021i 0.0145130i
\(988\) 2328.01i 0.0749635i
\(989\) 3129.44 0.100617
\(990\) 0 0
\(991\) −24573.0 −0.787675 −0.393838 0.919180i \(-0.628853\pi\)
−0.393838 + 0.919180i \(0.628853\pi\)
\(992\) 7385.03i 0.236366i
\(993\) 9740.36i 0.311280i
\(994\) 4231.78 0.135034
\(995\) 0 0
\(996\) −242.180 −0.00770458
\(997\) 14209.7i 0.451379i 0.974199 + 0.225690i \(0.0724635\pi\)
−0.974199 + 0.225690i \(0.927536\pi\)
\(998\) 42413.8i 1.34527i
\(999\) −14539.2 −0.460460
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 1150.4.b.o.599.2 8
5.2 odd 4 230.4.a.j.1.2 4
5.3 odd 4 1150.4.a.n.1.3 4
5.4 even 2 inner 1150.4.b.o.599.7 8
15.2 even 4 2070.4.a.bg.1.2 4
20.7 even 4 1840.4.a.k.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.j.1.2 4 5.2 odd 4
1150.4.a.n.1.3 4 5.3 odd 4
1150.4.b.o.599.2 8 1.1 even 1 trivial
1150.4.b.o.599.7 8 5.4 even 2 inner
1840.4.a.k.1.3 4 20.7 even 4
2070.4.a.bg.1.2 4 15.2 even 4