Properties

Label 224.3.n.a.145.10
Level $224$
Weight $3$
Character 224.145
Analytic conductor $6.104$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(17,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.n (of order \(6\), degree \(2\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 145.10
Character \(\chi\) \(=\) 224.145
Dual form 224.3.n.a.17.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.16781 + 2.02271i) q^{3} +(1.55055 - 2.68563i) q^{5} +(6.89374 + 1.21502i) q^{7} +(1.77242 - 3.06992i) q^{9} +O(q^{10})\) \(q+(1.16781 + 2.02271i) q^{3} +(1.55055 - 2.68563i) q^{5} +(6.89374 + 1.21502i) q^{7} +(1.77242 - 3.06992i) q^{9} +(-4.06604 + 2.34753i) q^{11} +6.88097 q^{13} +7.24301 q^{15} +(14.7184 - 8.49765i) q^{17} +(-13.1099 + 22.7070i) q^{19} +(5.59297 + 15.3630i) q^{21} +(12.9403 - 22.4132i) q^{23} +(7.69160 + 13.3222i) q^{25} +29.3001 q^{27} +42.2701i q^{29} +(-15.9024 + 9.18126i) q^{31} +(-9.49676 - 5.48296i) q^{33} +(13.9522 - 16.6301i) q^{35} +(-43.1997 - 24.9413i) q^{37} +(8.03569 + 13.9182i) q^{39} -10.7844i q^{41} -24.1791i q^{43} +(-5.49645 - 9.52013i) q^{45} +(-11.8480 - 6.84046i) q^{47} +(46.0474 + 16.7521i) q^{49} +(34.3766 + 19.8474i) q^{51} +(-6.03948 + 3.48690i) q^{53} +14.5598i q^{55} -61.2396 q^{57} +(-53.0922 - 91.9584i) q^{59} +(46.7304 - 80.9395i) q^{61} +(15.9486 - 19.0097i) q^{63} +(10.6693 - 18.4797i) q^{65} +(-77.2753 + 44.6149i) q^{67} +60.4473 q^{69} -81.7898 q^{71} +(-119.473 + 68.9780i) q^{73} +(-17.9647 + 31.1158i) q^{75} +(-30.8826 + 11.2429i) q^{77} +(-6.55090 + 11.3465i) q^{79} +(18.2653 + 31.6364i) q^{81} +2.15689 q^{83} -52.7041i q^{85} +(-85.5003 + 49.3636i) q^{87} +(-87.8261 - 50.7064i) q^{89} +(47.4356 + 8.36055i) q^{91} +(-37.1421 - 21.4440i) q^{93} +(40.6550 + 70.4166i) q^{95} -88.9318i q^{97} +16.6432i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 4 q^{7} - 32 q^{9} - 28 q^{15} - 6 q^{17} - 30 q^{23} - 32 q^{25} + 6 q^{31} - 6 q^{33} + 20 q^{39} + 294 q^{47} - 20 q^{49} + 124 q^{57} - 432 q^{63} - 52 q^{65} + 136 q^{71} + 234 q^{73} + 162 q^{79} - 18 q^{81} - 48 q^{87} - 150 q^{89} - 290 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.16781 + 2.02271i 0.389271 + 0.674238i 0.992352 0.123443i \(-0.0393935\pi\)
−0.603080 + 0.797680i \(0.706060\pi\)
\(4\) 0 0
\(5\) 1.55055 2.68563i 0.310110 0.537126i −0.668276 0.743913i \(-0.732968\pi\)
0.978386 + 0.206787i \(0.0663009\pi\)
\(6\) 0 0
\(7\) 6.89374 + 1.21502i 0.984821 + 0.173575i
\(8\) 0 0
\(9\) 1.77242 3.06992i 0.196936 0.341102i
\(10\) 0 0
\(11\) −4.06604 + 2.34753i −0.369640 + 0.213412i −0.673301 0.739368i \(-0.735124\pi\)
0.303661 + 0.952780i \(0.401791\pi\)
\(12\) 0 0
\(13\) 6.88097 0.529305 0.264653 0.964344i \(-0.414743\pi\)
0.264653 + 0.964344i \(0.414743\pi\)
\(14\) 0 0
\(15\) 7.24301 0.482868
\(16\) 0 0
\(17\) 14.7184 8.49765i 0.865786 0.499862i −0.000159428 1.00000i \(-0.500051\pi\)
0.865946 + 0.500138i \(0.166717\pi\)
\(18\) 0 0
\(19\) −13.1099 + 22.7070i −0.689994 + 1.19510i 0.281845 + 0.959460i \(0.409053\pi\)
−0.971839 + 0.235645i \(0.924280\pi\)
\(20\) 0 0
\(21\) 5.59297 + 15.3630i 0.266332 + 0.731571i
\(22\) 0 0
\(23\) 12.9403 22.4132i 0.562620 0.974486i −0.434647 0.900601i \(-0.643127\pi\)
0.997267 0.0738851i \(-0.0235398\pi\)
\(24\) 0 0
\(25\) 7.69160 + 13.3222i 0.307664 + 0.532889i
\(26\) 0 0
\(27\) 29.3001 1.08519
\(28\) 0 0
\(29\) 42.2701i 1.45759i 0.684732 + 0.728795i \(0.259919\pi\)
−0.684732 + 0.728795i \(0.740081\pi\)
\(30\) 0 0
\(31\) −15.9024 + 9.18126i −0.512981 + 0.296170i −0.734058 0.679087i \(-0.762376\pi\)
0.221077 + 0.975256i \(0.429043\pi\)
\(32\) 0 0
\(33\) −9.49676 5.48296i −0.287781 0.166150i
\(34\) 0 0
\(35\) 13.9522 16.6301i 0.398634 0.475145i
\(36\) 0 0
\(37\) −43.1997 24.9413i −1.16756 0.674090i −0.214455 0.976734i \(-0.568797\pi\)
−0.953104 + 0.302644i \(0.902131\pi\)
\(38\) 0 0
\(39\) 8.03569 + 13.9182i 0.206043 + 0.356878i
\(40\) 0 0
\(41\) 10.7844i 0.263035i −0.991314 0.131517i \(-0.958015\pi\)
0.991314 0.131517i \(-0.0419849\pi\)
\(42\) 0 0
\(43\) 24.1791i 0.562304i −0.959663 0.281152i \(-0.909284\pi\)
0.959663 0.281152i \(-0.0907165\pi\)
\(44\) 0 0
\(45\) −5.49645 9.52013i −0.122143 0.211558i
\(46\) 0 0
\(47\) −11.8480 6.84046i −0.252086 0.145542i 0.368633 0.929575i \(-0.379826\pi\)
−0.620719 + 0.784033i \(0.713159\pi\)
\(48\) 0 0
\(49\) 46.0474 + 16.7521i 0.939743 + 0.341880i
\(50\) 0 0
\(51\) 34.3766 + 19.8474i 0.674052 + 0.389164i
\(52\) 0 0
\(53\) −6.03948 + 3.48690i −0.113952 + 0.0657905i −0.555893 0.831254i \(-0.687624\pi\)
0.441940 + 0.897044i \(0.354290\pi\)
\(54\) 0 0
\(55\) 14.5598i 0.264724i
\(56\) 0 0
\(57\) −61.2396 −1.07438
\(58\) 0 0
\(59\) −53.0922 91.9584i −0.899868 1.55862i −0.827662 0.561227i \(-0.810329\pi\)
−0.0722059 0.997390i \(-0.523004\pi\)
\(60\) 0 0
\(61\) 46.7304 80.9395i 0.766073 1.32688i −0.173605 0.984815i \(-0.555542\pi\)
0.939678 0.342062i \(-0.111125\pi\)
\(62\) 0 0
\(63\) 15.9486 19.0097i 0.253153 0.301742i
\(64\) 0 0
\(65\) 10.6693 18.4797i 0.164143 0.284304i
\(66\) 0 0
\(67\) −77.2753 + 44.6149i −1.15336 + 0.665894i −0.949705 0.313147i \(-0.898617\pi\)
−0.203659 + 0.979042i \(0.565283\pi\)
\(68\) 0 0
\(69\) 60.4473 0.876047
\(70\) 0 0
\(71\) −81.7898 −1.15197 −0.575984 0.817461i \(-0.695381\pi\)
−0.575984 + 0.817461i \(0.695381\pi\)
\(72\) 0 0
\(73\) −119.473 + 68.9780i −1.63662 + 0.944904i −0.654637 + 0.755943i \(0.727179\pi\)
−0.981985 + 0.188961i \(0.939488\pi\)
\(74\) 0 0
\(75\) −17.9647 + 31.1158i −0.239529 + 0.414877i
\(76\) 0 0
\(77\) −30.8826 + 11.2429i −0.401072 + 0.146012i
\(78\) 0 0
\(79\) −6.55090 + 11.3465i −0.0829228 + 0.143627i −0.904504 0.426465i \(-0.859759\pi\)
0.821581 + 0.570091i \(0.193092\pi\)
\(80\) 0 0
\(81\) 18.2653 + 31.6364i 0.225497 + 0.390573i
\(82\) 0 0
\(83\) 2.15689 0.0259867 0.0129933 0.999916i \(-0.495864\pi\)
0.0129933 + 0.999916i \(0.495864\pi\)
\(84\) 0 0
\(85\) 52.7041i 0.620048i
\(86\) 0 0
\(87\) −85.5003 + 49.3636i −0.982762 + 0.567398i
\(88\) 0 0
\(89\) −87.8261 50.7064i −0.986810 0.569735i −0.0824908 0.996592i \(-0.526288\pi\)
−0.904319 + 0.426857i \(0.859621\pi\)
\(90\) 0 0
\(91\) 47.4356 + 8.36055i 0.521271 + 0.0918742i
\(92\) 0 0
\(93\) −37.1421 21.4440i −0.399378 0.230581i
\(94\) 0 0
\(95\) 40.6550 + 70.4166i 0.427948 + 0.741227i
\(96\) 0 0
\(97\) 88.9318i 0.916823i −0.888740 0.458412i \(-0.848419\pi\)
0.888740 0.458412i \(-0.151581\pi\)
\(98\) 0 0
\(99\) 16.6432i 0.168114i
\(100\) 0 0
\(101\) 10.4239 + 18.0546i 0.103206 + 0.178759i 0.913004 0.407951i \(-0.133756\pi\)
−0.809798 + 0.586709i \(0.800423\pi\)
\(102\) 0 0
\(103\) −2.97469 1.71744i −0.0288805 0.0166741i 0.485490 0.874242i \(-0.338641\pi\)
−0.514371 + 0.857568i \(0.671974\pi\)
\(104\) 0 0
\(105\) 49.9315 + 8.80044i 0.475538 + 0.0838137i
\(106\) 0 0
\(107\) 58.4603 + 33.7521i 0.546358 + 0.315440i 0.747652 0.664091i \(-0.231181\pi\)
−0.201294 + 0.979531i \(0.564515\pi\)
\(108\) 0 0
\(109\) −116.961 + 67.5273i −1.07303 + 0.619516i −0.929009 0.370058i \(-0.879338\pi\)
−0.144025 + 0.989574i \(0.546005\pi\)
\(110\) 0 0
\(111\) 116.507i 1.04962i
\(112\) 0 0
\(113\) 136.328 1.20645 0.603223 0.797573i \(-0.293883\pi\)
0.603223 + 0.797573i \(0.293883\pi\)
\(114\) 0 0
\(115\) −40.1290 69.5055i −0.348948 0.604395i
\(116\) 0 0
\(117\) 12.1960 21.1240i 0.104239 0.180547i
\(118\) 0 0
\(119\) 111.790 40.6975i 0.939408 0.341996i
\(120\) 0 0
\(121\) −49.4782 + 85.6988i −0.408911 + 0.708254i
\(122\) 0 0
\(123\) 21.8138 12.5942i 0.177348 0.102392i
\(124\) 0 0
\(125\) 125.232 1.00186
\(126\) 0 0
\(127\) 6.39702 0.0503702 0.0251851 0.999683i \(-0.491982\pi\)
0.0251851 + 0.999683i \(0.491982\pi\)
\(128\) 0 0
\(129\) 48.9073 28.2366i 0.379126 0.218889i
\(130\) 0 0
\(131\) 86.8472 150.424i 0.662956 1.14827i −0.316879 0.948466i \(-0.602635\pi\)
0.979835 0.199807i \(-0.0640316\pi\)
\(132\) 0 0
\(133\) −117.966 + 140.607i −0.886960 + 1.05720i
\(134\) 0 0
\(135\) 45.4312 78.6892i 0.336528 0.582883i
\(136\) 0 0
\(137\) 38.2926 + 66.3247i 0.279508 + 0.484122i 0.971262 0.238011i \(-0.0764954\pi\)
−0.691755 + 0.722133i \(0.743162\pi\)
\(138\) 0 0
\(139\) −72.4724 −0.521384 −0.260692 0.965422i \(-0.583951\pi\)
−0.260692 + 0.965422i \(0.583951\pi\)
\(140\) 0 0
\(141\) 31.9535i 0.226621i
\(142\) 0 0
\(143\) −27.9783 + 16.1533i −0.195653 + 0.112960i
\(144\) 0 0
\(145\) 113.522 + 65.5419i 0.782909 + 0.452013i
\(146\) 0 0
\(147\) 19.8901 + 112.704i 0.135307 + 0.766695i
\(148\) 0 0
\(149\) 211.542 + 122.134i 1.41974 + 0.819690i 0.996276 0.0862205i \(-0.0274790\pi\)
0.423469 + 0.905911i \(0.360812\pi\)
\(150\) 0 0
\(151\) 103.109 + 178.590i 0.682841 + 1.18272i 0.974110 + 0.226074i \(0.0725893\pi\)
−0.291269 + 0.956641i \(0.594077\pi\)
\(152\) 0 0
\(153\) 60.2456i 0.393762i
\(154\) 0 0
\(155\) 56.9440i 0.367380i
\(156\) 0 0
\(157\) −37.2714 64.5559i −0.237397 0.411184i 0.722569 0.691298i \(-0.242961\pi\)
−0.959967 + 0.280114i \(0.909628\pi\)
\(158\) 0 0
\(159\) −14.1060 8.14409i −0.0887169 0.0512207i
\(160\) 0 0
\(161\) 116.439 138.788i 0.723226 0.862037i
\(162\) 0 0
\(163\) 85.1169 + 49.1422i 0.522189 + 0.301486i 0.737830 0.674987i \(-0.235851\pi\)
−0.215641 + 0.976473i \(0.569184\pi\)
\(164\) 0 0
\(165\) −29.4504 + 17.0032i −0.178487 + 0.103050i
\(166\) 0 0
\(167\) 252.539i 1.51221i −0.654449 0.756106i \(-0.727099\pi\)
0.654449 0.756106i \(-0.272901\pi\)
\(168\) 0 0
\(169\) −121.652 −0.719836
\(170\) 0 0
\(171\) 46.4724 + 80.4926i 0.271769 + 0.470717i
\(172\) 0 0
\(173\) −75.1889 + 130.231i −0.434618 + 0.752781i −0.997264 0.0739171i \(-0.976450\pi\)
0.562646 + 0.826698i \(0.309783\pi\)
\(174\) 0 0
\(175\) 36.8371 + 101.186i 0.210497 + 0.578203i
\(176\) 0 0
\(177\) 124.004 214.781i 0.700586 1.21345i
\(178\) 0 0
\(179\) −89.6246 + 51.7448i −0.500696 + 0.289077i −0.729001 0.684513i \(-0.760015\pi\)
0.228305 + 0.973590i \(0.426682\pi\)
\(180\) 0 0
\(181\) −95.1121 −0.525481 −0.262741 0.964867i \(-0.584626\pi\)
−0.262741 + 0.964867i \(0.584626\pi\)
\(182\) 0 0
\(183\) 218.290 1.19284
\(184\) 0 0
\(185\) −133.966 + 77.3455i −0.724143 + 0.418084i
\(186\) 0 0
\(187\) −39.8970 + 69.1036i −0.213353 + 0.369538i
\(188\) 0 0
\(189\) 201.987 + 35.6003i 1.06872 + 0.188362i
\(190\) 0 0
\(191\) 1.97252 3.41650i 0.0103273 0.0178874i −0.860816 0.508917i \(-0.830046\pi\)
0.871143 + 0.491030i \(0.163379\pi\)
\(192\) 0 0
\(193\) 146.091 + 253.037i 0.756949 + 1.31107i 0.944399 + 0.328800i \(0.106644\pi\)
−0.187450 + 0.982274i \(0.560022\pi\)
\(194\) 0 0
\(195\) 49.8390 0.255584
\(196\) 0 0
\(197\) 160.503i 0.814735i −0.913264 0.407367i \(-0.866447\pi\)
0.913264 0.407367i \(-0.133553\pi\)
\(198\) 0 0
\(199\) −174.461 + 100.725i −0.876688 + 0.506156i −0.869565 0.493819i \(-0.835601\pi\)
−0.00712311 + 0.999975i \(0.502267\pi\)
\(200\) 0 0
\(201\) −180.486 104.204i −0.897943 0.518427i
\(202\) 0 0
\(203\) −51.3592 + 291.399i −0.253001 + 1.43546i
\(204\) 0 0
\(205\) −28.9630 16.7218i −0.141283 0.0815697i
\(206\) 0 0
\(207\) −45.8711 79.4511i −0.221600 0.383822i
\(208\) 0 0
\(209\) 123.103i 0.589012i
\(210\) 0 0
\(211\) 170.542i 0.808256i −0.914702 0.404128i \(-0.867575\pi\)
0.914702 0.404128i \(-0.132425\pi\)
\(212\) 0 0
\(213\) −95.5152 165.437i −0.448428 0.776701i
\(214\) 0 0
\(215\) −64.9360 37.4908i −0.302028 0.174376i
\(216\) 0 0
\(217\) −120.783 + 43.9714i −0.556602 + 0.202633i
\(218\) 0 0
\(219\) −279.045 161.107i −1.27418 0.735648i
\(220\) 0 0
\(221\) 101.277 58.4721i 0.458265 0.264580i
\(222\) 0 0
\(223\) 143.446i 0.643255i 0.946866 + 0.321628i \(0.104230\pi\)
−0.946866 + 0.321628i \(0.895770\pi\)
\(224\) 0 0
\(225\) 54.5309 0.242360
\(226\) 0 0
\(227\) −11.7597 20.3683i −0.0518047 0.0897284i 0.838960 0.544193i \(-0.183164\pi\)
−0.890765 + 0.454464i \(0.849831\pi\)
\(228\) 0 0
\(229\) 30.7040 53.1809i 0.134079 0.232231i −0.791167 0.611601i \(-0.790526\pi\)
0.925245 + 0.379370i \(0.123859\pi\)
\(230\) 0 0
\(231\) −58.8063 49.3369i −0.254573 0.213580i
\(232\) 0 0
\(233\) −52.0991 + 90.2384i −0.223601 + 0.387289i −0.955899 0.293696i \(-0.905115\pi\)
0.732297 + 0.680985i \(0.238448\pi\)
\(234\) 0 0
\(235\) −36.7419 + 21.2129i −0.156348 + 0.0902678i
\(236\) 0 0
\(237\) −30.6010 −0.129118
\(238\) 0 0
\(239\) −104.695 −0.438056 −0.219028 0.975719i \(-0.570289\pi\)
−0.219028 + 0.975719i \(0.570289\pi\)
\(240\) 0 0
\(241\) −142.650 + 82.3591i −0.591910 + 0.341739i −0.765852 0.643017i \(-0.777683\pi\)
0.173943 + 0.984756i \(0.444349\pi\)
\(242\) 0 0
\(243\) 89.1895 154.481i 0.367035 0.635723i
\(244\) 0 0
\(245\) 116.389 97.6913i 0.475056 0.398740i
\(246\) 0 0
\(247\) −90.2087 + 156.246i −0.365217 + 0.632575i
\(248\) 0 0
\(249\) 2.51885 + 4.36278i 0.0101159 + 0.0175212i
\(250\) 0 0
\(251\) 399.066 1.58990 0.794952 0.606672i \(-0.207496\pi\)
0.794952 + 0.606672i \(0.207496\pi\)
\(252\) 0 0
\(253\) 121.511i 0.480279i
\(254\) 0 0
\(255\) 106.605 61.5486i 0.418060 0.241367i
\(256\) 0 0
\(257\) 2.12341 + 1.22595i 0.00826231 + 0.00477025i 0.504125 0.863630i \(-0.331815\pi\)
−0.495863 + 0.868401i \(0.665148\pi\)
\(258\) 0 0
\(259\) −267.503 224.428i −1.03283 0.866517i
\(260\) 0 0
\(261\) 129.766 + 74.9204i 0.497187 + 0.287051i
\(262\) 0 0
\(263\) 28.7798 + 49.8481i 0.109429 + 0.189536i 0.915539 0.402229i \(-0.131764\pi\)
−0.806110 + 0.591766i \(0.798431\pi\)
\(264\) 0 0
\(265\) 21.6264i 0.0816091i
\(266\) 0 0
\(267\) 236.863i 0.887126i
\(268\) 0 0
\(269\) −120.201 208.195i −0.446845 0.773958i 0.551334 0.834285i \(-0.314119\pi\)
−0.998179 + 0.0603267i \(0.980786\pi\)
\(270\) 0 0
\(271\) −116.507 67.2655i −0.429916 0.248212i 0.269395 0.963030i \(-0.413176\pi\)
−0.699311 + 0.714818i \(0.746510\pi\)
\(272\) 0 0
\(273\) 38.4850 + 105.712i 0.140971 + 0.387225i
\(274\) 0 0
\(275\) −62.5487 36.1125i −0.227450 0.131318i
\(276\) 0 0
\(277\) −95.6097 + 55.2003i −0.345161 + 0.199279i −0.662552 0.749016i \(-0.730527\pi\)
0.317391 + 0.948295i \(0.397193\pi\)
\(278\) 0 0
\(279\) 65.0922i 0.233305i
\(280\) 0 0
\(281\) −154.087 −0.548351 −0.274175 0.961680i \(-0.588405\pi\)
−0.274175 + 0.961680i \(0.588405\pi\)
\(282\) 0 0
\(283\) 15.4714 + 26.7972i 0.0546692 + 0.0946899i 0.892065 0.451907i \(-0.149256\pi\)
−0.837396 + 0.546597i \(0.815923\pi\)
\(284\) 0 0
\(285\) −94.9551 + 164.467i −0.333176 + 0.577077i
\(286\) 0 0
\(287\) 13.1034 74.3451i 0.0456563 0.259042i
\(288\) 0 0
\(289\) −0.0797964 + 0.138211i −0.000276112 + 0.000478240i
\(290\) 0 0
\(291\) 179.884 103.856i 0.618157 0.356893i
\(292\) 0 0
\(293\) −511.686 −1.74637 −0.873184 0.487390i \(-0.837949\pi\)
−0.873184 + 0.487390i \(0.837949\pi\)
\(294\) 0 0
\(295\) −329.288 −1.11623
\(296\) 0 0
\(297\) −119.135 + 68.7828i −0.401129 + 0.231592i
\(298\) 0 0
\(299\) 89.0415 154.224i 0.297798 0.515801i
\(300\) 0 0
\(301\) 29.3782 166.684i 0.0976018 0.553768i
\(302\) 0 0
\(303\) −24.3463 + 42.1689i −0.0803507 + 0.139171i
\(304\) 0 0
\(305\) −144.916 251.001i −0.475133 0.822955i
\(306\) 0 0
\(307\) 51.2670 0.166993 0.0834967 0.996508i \(-0.473391\pi\)
0.0834967 + 0.996508i \(0.473391\pi\)
\(308\) 0 0
\(309\) 8.02259i 0.0259631i
\(310\) 0 0
\(311\) 17.7940 10.2734i 0.0572153 0.0330333i −0.471119 0.882069i \(-0.656150\pi\)
0.528335 + 0.849036i \(0.322817\pi\)
\(312\) 0 0
\(313\) 291.960 + 168.563i 0.932780 + 0.538541i 0.887690 0.460442i \(-0.152309\pi\)
0.0450905 + 0.998983i \(0.485642\pi\)
\(314\) 0 0
\(315\) −26.3239 72.3076i −0.0835680 0.229548i
\(316\) 0 0
\(317\) −80.7634 46.6288i −0.254774 0.147094i 0.367174 0.930152i \(-0.380325\pi\)
−0.621948 + 0.783058i \(0.713659\pi\)
\(318\) 0 0
\(319\) −99.2304 171.872i −0.311067 0.538784i
\(320\) 0 0
\(321\) 157.665i 0.491167i
\(322\) 0 0
\(323\) 445.613i 1.37961i
\(324\) 0 0
\(325\) 52.9256 + 91.6699i 0.162848 + 0.282061i
\(326\) 0 0
\(327\) −273.177 157.719i −0.835403 0.482320i
\(328\) 0 0
\(329\) −73.3659 61.5520i −0.222997 0.187088i
\(330\) 0 0
\(331\) −64.9939 37.5242i −0.196356 0.113366i 0.398599 0.917125i \(-0.369497\pi\)
−0.594955 + 0.803759i \(0.702830\pi\)
\(332\) 0 0
\(333\) −153.136 + 88.4130i −0.459867 + 0.265505i
\(334\) 0 0
\(335\) 276.711i 0.826002i
\(336\) 0 0
\(337\) −140.105 −0.415743 −0.207872 0.978156i \(-0.566654\pi\)
−0.207872 + 0.978156i \(0.566654\pi\)
\(338\) 0 0
\(339\) 159.206 + 275.753i 0.469635 + 0.813431i
\(340\) 0 0
\(341\) 43.1066 74.6628i 0.126412 0.218952i
\(342\) 0 0
\(343\) 297.085 + 171.434i 0.866137 + 0.499807i
\(344\) 0 0
\(345\) 93.7264 162.339i 0.271671 0.470548i
\(346\) 0 0
\(347\) −64.9715 + 37.5113i −0.187238 + 0.108102i −0.590689 0.806899i \(-0.701144\pi\)
0.403451 + 0.915001i \(0.367811\pi\)
\(348\) 0 0
\(349\) 603.618 1.72956 0.864782 0.502148i \(-0.167457\pi\)
0.864782 + 0.502148i \(0.167457\pi\)
\(350\) 0 0
\(351\) 201.613 0.574396
\(352\) 0 0
\(353\) 337.515 194.864i 0.956132 0.552023i 0.0611514 0.998129i \(-0.480523\pi\)
0.894980 + 0.446106i \(0.147189\pi\)
\(354\) 0 0
\(355\) −126.819 + 219.657i −0.357237 + 0.618752i
\(356\) 0 0
\(357\) 212.869 + 178.591i 0.596271 + 0.500255i
\(358\) 0 0
\(359\) 69.2214 119.895i 0.192817 0.333969i −0.753366 0.657602i \(-0.771571\pi\)
0.946183 + 0.323633i \(0.104904\pi\)
\(360\) 0 0
\(361\) −163.238 282.737i −0.452183 0.783204i
\(362\) 0 0
\(363\) −231.125 −0.636709
\(364\) 0 0
\(365\) 427.815i 1.17210i
\(366\) 0 0
\(367\) 408.823 236.034i 1.11396 0.643145i 0.174108 0.984727i \(-0.444296\pi\)
0.939852 + 0.341581i \(0.110963\pi\)
\(368\) 0 0
\(369\) −33.1073 19.1145i −0.0897218 0.0518009i
\(370\) 0 0
\(371\) −45.8713 + 16.6997i −0.123642 + 0.0450125i
\(372\) 0 0
\(373\) 30.5419 + 17.6334i 0.0818818 + 0.0472745i 0.540382 0.841420i \(-0.318280\pi\)
−0.458500 + 0.888694i \(0.651613\pi\)
\(374\) 0 0
\(375\) 146.248 + 253.309i 0.389995 + 0.675491i
\(376\) 0 0
\(377\) 290.859i 0.771510i
\(378\) 0 0
\(379\) 230.447i 0.608039i 0.952666 + 0.304019i \(0.0983287\pi\)
−0.952666 + 0.304019i \(0.901671\pi\)
\(380\) 0 0
\(381\) 7.47053 + 12.9393i 0.0196077 + 0.0339615i
\(382\) 0 0
\(383\) 480.020 + 277.140i 1.25332 + 0.723602i 0.971766 0.235945i \(-0.0758184\pi\)
0.281549 + 0.959547i \(0.409152\pi\)
\(384\) 0 0
\(385\) −17.6906 + 100.372i −0.0459495 + 0.260706i
\(386\) 0 0
\(387\) −74.2278 42.8554i −0.191803 0.110738i
\(388\) 0 0
\(389\) 344.401 198.840i 0.885349 0.511157i 0.0129310 0.999916i \(-0.495884\pi\)
0.872418 + 0.488760i \(0.162550\pi\)
\(390\) 0 0
\(391\) 439.847i 1.12493i
\(392\) 0 0
\(393\) 405.686 1.03228
\(394\) 0 0
\(395\) 20.3150 + 35.1866i 0.0514304 + 0.0890800i
\(396\) 0 0
\(397\) 60.9545 105.576i 0.153538 0.265935i −0.778988 0.627039i \(-0.784267\pi\)
0.932526 + 0.361104i \(0.117600\pi\)
\(398\) 0 0
\(399\) −422.170 74.4077i −1.05807 0.186485i
\(400\) 0 0
\(401\) −124.337 + 215.358i −0.310067 + 0.537051i −0.978377 0.206832i \(-0.933685\pi\)
0.668310 + 0.743883i \(0.267018\pi\)
\(402\) 0 0
\(403\) −109.424 + 63.1760i −0.271524 + 0.156764i
\(404\) 0 0
\(405\) 113.285 0.279716
\(406\) 0 0
\(407\) 234.202 0.575435
\(408\) 0 0
\(409\) 582.721 336.434i 1.42475 0.822578i 0.428047 0.903757i \(-0.359202\pi\)
0.996700 + 0.0811790i \(0.0258685\pi\)
\(410\) 0 0
\(411\) −89.4372 + 154.910i −0.217609 + 0.376909i
\(412\) 0 0
\(413\) −254.272 698.446i −0.615672 1.69115i
\(414\) 0 0
\(415\) 3.34437 5.79262i 0.00805872 0.0139581i
\(416\) 0 0
\(417\) −84.6343 146.591i −0.202960 0.351537i
\(418\) 0 0
\(419\) −178.795 −0.426718 −0.213359 0.976974i \(-0.568440\pi\)
−0.213359 + 0.976974i \(0.568440\pi\)
\(420\) 0 0
\(421\) 212.470i 0.504679i −0.967639 0.252340i \(-0.918800\pi\)
0.967639 0.252340i \(-0.0812000\pi\)
\(422\) 0 0
\(423\) −41.9993 + 24.2483i −0.0992892 + 0.0573247i
\(424\) 0 0
\(425\) 226.415 + 130.721i 0.532742 + 0.307579i
\(426\) 0 0
\(427\) 420.491 501.198i 0.984757 1.17376i
\(428\) 0 0
\(429\) −65.3470 37.7281i −0.152324 0.0879442i
\(430\) 0 0
\(431\) −345.732 598.826i −0.802163 1.38939i −0.918190 0.396141i \(-0.870349\pi\)
0.116027 0.993246i \(-0.462984\pi\)
\(432\) 0 0
\(433\) 99.8389i 0.230575i 0.993332 + 0.115287i \(0.0367789\pi\)
−0.993332 + 0.115287i \(0.963221\pi\)
\(434\) 0 0
\(435\) 306.163i 0.703823i
\(436\) 0 0
\(437\) 339.290 + 587.668i 0.776408 + 1.34478i
\(438\) 0 0
\(439\) 599.369 + 346.046i 1.36530 + 0.788259i 0.990324 0.138774i \(-0.0443161\pi\)
0.374980 + 0.927033i \(0.377649\pi\)
\(440\) 0 0
\(441\) 133.043 111.670i 0.301685 0.253220i
\(442\) 0 0
\(443\) 233.569 + 134.851i 0.527244 + 0.304405i 0.739893 0.672724i \(-0.234876\pi\)
−0.212649 + 0.977129i \(0.568209\pi\)
\(444\) 0 0
\(445\) −272.357 + 157.246i −0.612039 + 0.353361i
\(446\) 0 0
\(447\) 570.519i 1.27633i
\(448\) 0 0
\(449\) −76.6510 −0.170715 −0.0853575 0.996350i \(-0.527203\pi\)
−0.0853575 + 0.996350i \(0.527203\pi\)
\(450\) 0 0
\(451\) 25.3168 + 43.8500i 0.0561348 + 0.0972283i
\(452\) 0 0
\(453\) −240.824 + 417.120i −0.531621 + 0.920795i
\(454\) 0 0
\(455\) 96.0046 114.431i 0.210999 0.251497i
\(456\) 0 0
\(457\) 175.616 304.177i 0.384281 0.665594i −0.607388 0.794405i \(-0.707783\pi\)
0.991669 + 0.128811i \(0.0411160\pi\)
\(458\) 0 0
\(459\) 431.249 248.982i 0.939541 0.542444i
\(460\) 0 0
\(461\) −296.940 −0.644122 −0.322061 0.946719i \(-0.604376\pi\)
−0.322061 + 0.946719i \(0.604376\pi\)
\(462\) 0 0
\(463\) −25.5350 −0.0551513 −0.0275756 0.999620i \(-0.508779\pi\)
−0.0275756 + 0.999620i \(0.508779\pi\)
\(464\) 0 0
\(465\) −115.181 + 66.5000i −0.247702 + 0.143011i
\(466\) 0 0
\(467\) 62.7601 108.704i 0.134390 0.232770i −0.790974 0.611849i \(-0.790426\pi\)
0.925364 + 0.379079i \(0.123759\pi\)
\(468\) 0 0
\(469\) −586.925 + 213.673i −1.25144 + 0.455592i
\(470\) 0 0
\(471\) 87.0521 150.779i 0.184824 0.320125i
\(472\) 0 0
\(473\) 56.7611 + 98.3131i 0.120002 + 0.207850i
\(474\) 0 0
\(475\) −403.344 −0.849145
\(476\) 0 0
\(477\) 24.7210i 0.0518259i
\(478\) 0 0
\(479\) 150.188 86.7108i 0.313544 0.181025i −0.334967 0.942230i \(-0.608725\pi\)
0.648511 + 0.761205i \(0.275392\pi\)
\(480\) 0 0
\(481\) −297.256 171.621i −0.617995 0.356800i
\(482\) 0 0
\(483\) 416.708 + 73.4449i 0.862749 + 0.152060i
\(484\) 0 0
\(485\) −238.838 137.893i −0.492449 0.284316i
\(486\) 0 0
\(487\) −340.756 590.206i −0.699704 1.21192i −0.968569 0.248744i \(-0.919982\pi\)
0.268866 0.963178i \(-0.413351\pi\)
\(488\) 0 0
\(489\) 229.556i 0.469440i
\(490\) 0 0
\(491\) 278.104i 0.566404i −0.959060 0.283202i \(-0.908603\pi\)
0.959060 0.283202i \(-0.0913966\pi\)
\(492\) 0 0
\(493\) 359.197 + 622.147i 0.728594 + 1.26196i
\(494\) 0 0
\(495\) 44.6976 + 25.8062i 0.0902981 + 0.0521336i
\(496\) 0 0
\(497\) −563.838 99.3766i −1.13448 0.199953i
\(498\) 0 0
\(499\) 355.447 + 205.217i 0.712319 + 0.411258i 0.811919 0.583770i \(-0.198423\pi\)
−0.0996002 + 0.995028i \(0.531756\pi\)
\(500\) 0 0
\(501\) 510.815 294.919i 1.01959 0.588661i
\(502\) 0 0
\(503\) 554.042i 1.10148i 0.834678 + 0.550738i \(0.185654\pi\)
−0.834678 + 0.550738i \(0.814346\pi\)
\(504\) 0 0
\(505\) 64.6508 0.128021
\(506\) 0 0
\(507\) −142.067 246.068i −0.280212 0.485341i
\(508\) 0 0
\(509\) −22.0971 + 38.2733i −0.0434127 + 0.0751931i −0.886915 0.461932i \(-0.847156\pi\)
0.843503 + 0.537125i \(0.180490\pi\)
\(510\) 0 0
\(511\) −907.429 + 330.354i −1.77579 + 0.646485i
\(512\) 0 0
\(513\) −384.121 + 665.317i −0.748773 + 1.29691i
\(514\) 0 0
\(515\) −9.22480 + 5.32594i −0.0179122 + 0.0103416i
\(516\) 0 0
\(517\) 64.2327 0.124241
\(518\) 0 0
\(519\) −351.227 −0.676738
\(520\) 0 0
\(521\) 363.862 210.076i 0.698392 0.403217i −0.108356 0.994112i \(-0.534559\pi\)
0.806748 + 0.590895i \(0.201225\pi\)
\(522\) 0 0
\(523\) −137.447 + 238.065i −0.262805 + 0.455191i −0.966986 0.254829i \(-0.917981\pi\)
0.704181 + 0.710020i \(0.251314\pi\)
\(524\) 0 0
\(525\) −161.651 + 192.677i −0.307906 + 0.367003i
\(526\) 0 0
\(527\) −156.038 + 270.266i −0.296088 + 0.512839i
\(528\) 0 0
\(529\) −70.4004 121.937i −0.133082 0.230505i
\(530\) 0 0
\(531\) −376.407 −0.708864
\(532\) 0 0
\(533\) 74.2073i 0.139226i
\(534\) 0 0
\(535\) 181.291 104.668i 0.338862 0.195642i
\(536\) 0 0
\(537\) −209.330 120.857i −0.389813 0.225059i
\(538\) 0 0
\(539\) −226.557 + 39.9828i −0.420328 + 0.0741797i
\(540\) 0 0
\(541\) −485.358 280.221i −0.897149 0.517969i −0.0208748 0.999782i \(-0.506645\pi\)
−0.876274 + 0.481813i \(0.839978\pi\)
\(542\) 0 0
\(543\) −111.073 192.385i −0.204555 0.354299i
\(544\) 0 0
\(545\) 418.818i 0.768473i
\(546\) 0 0
\(547\) 655.564i 1.19847i −0.800573 0.599235i \(-0.795471\pi\)
0.800573 0.599235i \(-0.204529\pi\)
\(548\) 0 0
\(549\) −165.652 286.917i −0.301734 0.522618i
\(550\) 0 0
\(551\) −959.826 554.156i −1.74197 1.00573i
\(552\) 0 0
\(553\) −58.9465 + 70.2604i −0.106594 + 0.127053i
\(554\) 0 0
\(555\) −312.896 180.650i −0.563776 0.325496i
\(556\) 0 0
\(557\) 650.172 375.377i 1.16728 0.673927i 0.214239 0.976781i \(-0.431273\pi\)
0.953037 + 0.302855i \(0.0979397\pi\)
\(558\) 0 0
\(559\) 166.375i 0.297630i
\(560\) 0 0
\(561\) −186.369 −0.332209
\(562\) 0 0
\(563\) −317.191 549.391i −0.563394 0.975827i −0.997197 0.0748195i \(-0.976162\pi\)
0.433803 0.901008i \(-0.357171\pi\)
\(564\) 0 0
\(565\) 211.384 366.127i 0.374130 0.648013i
\(566\) 0 0
\(567\) 87.4772 + 240.286i 0.154281 + 0.423785i
\(568\) 0 0
\(569\) −196.482 + 340.317i −0.345312 + 0.598097i −0.985410 0.170195i \(-0.945560\pi\)
0.640099 + 0.768293i \(0.278893\pi\)
\(570\) 0 0
\(571\) 466.776 269.493i 0.817471 0.471967i −0.0320728 0.999486i \(-0.510211\pi\)
0.849543 + 0.527519i \(0.176878\pi\)
\(572\) 0 0
\(573\) 9.21414 0.0160805
\(574\) 0 0
\(575\) 398.125 0.692391
\(576\) 0 0
\(577\) −301.353 + 173.986i −0.522276 + 0.301536i −0.737865 0.674948i \(-0.764166\pi\)
0.215589 + 0.976484i \(0.430833\pi\)
\(578\) 0 0
\(579\) −341.215 + 591.001i −0.589317 + 1.02073i
\(580\) 0 0
\(581\) 14.8691 + 2.62068i 0.0255922 + 0.00451064i
\(582\) 0 0
\(583\) 16.3712 28.3557i 0.0280809 0.0486376i
\(584\) 0 0
\(585\) −37.8209 65.5077i −0.0646511 0.111979i
\(586\) 0 0
\(587\) −258.936 −0.441118 −0.220559 0.975374i \(-0.570788\pi\)
−0.220559 + 0.975374i \(0.570788\pi\)
\(588\) 0 0
\(589\) 481.461i 0.817421i
\(590\) 0 0
\(591\) 324.651 187.437i 0.549325 0.317153i
\(592\) 0 0
\(593\) 66.6525 + 38.4819i 0.112399 + 0.0648935i 0.555146 0.831753i \(-0.312663\pi\)
−0.442747 + 0.896647i \(0.645996\pi\)
\(594\) 0 0
\(595\) 64.0368 363.329i 0.107625 0.610636i
\(596\) 0 0
\(597\) −407.476 235.256i −0.682539 0.394064i
\(598\) 0 0
\(599\) 579.488 + 1003.70i 0.967426 + 1.67563i 0.702951 + 0.711239i \(0.251865\pi\)
0.264475 + 0.964392i \(0.414801\pi\)
\(600\) 0 0
\(601\) 976.895i 1.62545i −0.582648 0.812724i \(-0.697983\pi\)
0.582648 0.812724i \(-0.302017\pi\)
\(602\) 0 0
\(603\) 316.306i 0.524553i
\(604\) 0 0
\(605\) 153.437 + 265.760i 0.253614 + 0.439273i
\(606\) 0 0
\(607\) 684.187 + 395.015i 1.12716 + 0.650767i 0.943219 0.332170i \(-0.107781\pi\)
0.183942 + 0.982937i \(0.441114\pi\)
\(608\) 0 0
\(609\) −649.395 + 236.415i −1.06633 + 0.388202i
\(610\) 0 0
\(611\) −81.5259 47.0690i −0.133430 0.0770360i
\(612\) 0 0
\(613\) 233.690 134.921i 0.381223 0.220099i −0.297127 0.954838i \(-0.596029\pi\)
0.678350 + 0.734739i \(0.262695\pi\)
\(614\) 0 0
\(615\) 78.1118i 0.127011i
\(616\) 0 0
\(617\) 701.515 1.13698 0.568489 0.822691i \(-0.307528\pi\)
0.568489 + 0.822691i \(0.307528\pi\)
\(618\) 0 0
\(619\) 434.760 + 753.026i 0.702358 + 1.21652i 0.967636 + 0.252349i \(0.0812029\pi\)
−0.265278 + 0.964172i \(0.585464\pi\)
\(620\) 0 0
\(621\) 379.151 656.708i 0.610548 1.05750i
\(622\) 0 0
\(623\) −543.841 456.268i −0.872939 0.732372i
\(624\) 0 0
\(625\) 1.88880 3.27149i 0.00302208 0.00523439i
\(626\) 0 0
\(627\) 249.003 143.762i 0.397134 0.229285i
\(628\) 0 0
\(629\) −847.771 −1.34781
\(630\) 0 0
\(631\) −100.362 −0.159052 −0.0795258 0.996833i \(-0.525341\pi\)
−0.0795258 + 0.996833i \(0.525341\pi\)
\(632\) 0 0
\(633\) 344.958 199.162i 0.544957 0.314631i
\(634\) 0 0
\(635\) 9.91889 17.1800i 0.0156203 0.0270552i
\(636\) 0 0
\(637\) 316.851 + 115.271i 0.497411 + 0.180959i
\(638\) 0 0
\(639\) −144.966 + 251.088i −0.226863 + 0.392939i
\(640\) 0 0
\(641\) −530.571 918.977i −0.827724 1.43366i −0.899819 0.436263i \(-0.856302\pi\)
0.0720947 0.997398i \(-0.477032\pi\)
\(642\) 0 0
\(643\) −132.853 −0.206615 −0.103308 0.994649i \(-0.532943\pi\)
−0.103308 + 0.994649i \(0.532943\pi\)
\(644\) 0 0
\(645\) 175.129i 0.271518i
\(646\) 0 0
\(647\) −998.259 + 576.345i −1.54290 + 0.890796i −0.544250 + 0.838923i \(0.683186\pi\)
−0.998654 + 0.0518730i \(0.983481\pi\)
\(648\) 0 0
\(649\) 431.750 + 249.271i 0.665255 + 0.384085i
\(650\) 0 0
\(651\) −229.993 192.958i −0.353292 0.296403i
\(652\) 0 0
\(653\) −25.1758 14.5352i −0.0385540 0.0222592i 0.480599 0.876940i \(-0.340419\pi\)
−0.519153 + 0.854681i \(0.673753\pi\)
\(654\) 0 0
\(655\) −269.322 466.479i −0.411178 0.712181i
\(656\) 0 0
\(657\) 489.032i 0.744341i
\(658\) 0 0
\(659\) 705.504i 1.07057i −0.844672 0.535283i \(-0.820205\pi\)
0.844672 0.535283i \(-0.179795\pi\)
\(660\) 0 0
\(661\) 63.6678 + 110.276i 0.0963204 + 0.166832i 0.910159 0.414259i \(-0.135959\pi\)
−0.813838 + 0.581091i \(0.802626\pi\)
\(662\) 0 0
\(663\) 236.545 + 136.569i 0.356779 + 0.205987i
\(664\) 0 0
\(665\) 194.708 + 534.831i 0.292793 + 0.804257i
\(666\) 0 0
\(667\) 947.407 + 546.986i 1.42040 + 0.820069i
\(668\) 0 0
\(669\) −290.150 + 167.518i −0.433707 + 0.250401i
\(670\) 0 0
\(671\) 438.805i 0.653956i
\(672\) 0 0
\(673\) −463.380 −0.688528 −0.344264 0.938873i \(-0.611872\pi\)
−0.344264 + 0.938873i \(0.611872\pi\)
\(674\) 0 0
\(675\) 225.364 + 390.343i 0.333873 + 0.578285i
\(676\) 0 0
\(677\) 188.138 325.864i 0.277899 0.481335i −0.692963 0.720973i \(-0.743695\pi\)
0.970862 + 0.239637i \(0.0770286\pi\)
\(678\) 0 0
\(679\) 108.054 613.073i 0.159138 0.902906i
\(680\) 0 0
\(681\) 27.4662 47.5729i 0.0403322 0.0698574i
\(682\) 0 0
\(683\) −897.932 + 518.421i −1.31469 + 0.759035i −0.982869 0.184308i \(-0.940996\pi\)
−0.331819 + 0.943343i \(0.607662\pi\)
\(684\) 0 0
\(685\) 237.498 0.346712
\(686\) 0 0
\(687\) 143.426 0.208772
\(688\) 0 0
\(689\) −41.5575 + 23.9932i −0.0603157 + 0.0348233i
\(690\) 0 0
\(691\) −177.535 + 307.499i −0.256924 + 0.445006i −0.965416 0.260713i \(-0.916042\pi\)
0.708492 + 0.705719i \(0.249376\pi\)
\(692\) 0 0
\(693\) −20.2219 + 114.734i −0.0291803 + 0.165562i
\(694\) 0 0
\(695\) −112.372 + 194.634i −0.161686 + 0.280049i
\(696\) 0 0
\(697\) −91.6423 158.729i −0.131481 0.227732i
\(698\) 0 0
\(699\) −243.368 −0.348167
\(700\) 0 0
\(701\) 1278.63i 1.82400i 0.410185 + 0.912002i \(0.365464\pi\)
−0.410185 + 0.912002i \(0.634536\pi\)
\(702\) 0 0
\(703\) 1132.69 653.956i 1.61122 0.930236i
\(704\) 0 0
\(705\) −85.8154 49.5455i −0.121724 0.0702774i
\(706\) 0 0
\(707\) 49.9226 + 137.129i 0.0706118 + 0.193959i
\(708\) 0 0
\(709\) 1040.03 + 600.464i 1.46690 + 0.846917i 0.999314 0.0370292i \(-0.0117895\pi\)
0.467589 + 0.883946i \(0.345123\pi\)
\(710\) 0 0
\(711\) 23.2219 + 40.2215i 0.0326609 + 0.0565704i
\(712\) 0 0
\(713\) 475.231i 0.666524i
\(714\) 0 0
\(715\) 100.186i 0.140120i
\(716\) 0 0
\(717\) −122.265 211.769i −0.170523 0.295354i
\(718\) 0 0
\(719\) 6.39954 + 3.69478i 0.00890061 + 0.00513877i 0.504444 0.863445i \(-0.331698\pi\)
−0.495543 + 0.868583i \(0.665031\pi\)
\(720\) 0 0
\(721\) −18.4200 15.4539i −0.0255479 0.0214340i
\(722\) 0 0
\(723\) −333.178 192.360i −0.460827 0.266059i
\(724\) 0 0
\(725\) −563.132 + 325.125i −0.776734 + 0.448448i
\(726\) 0 0
\(727\) 1184.67i 1.62953i −0.579788 0.814767i \(-0.696865\pi\)
0.579788 0.814767i \(-0.303135\pi\)
\(728\) 0 0
\(729\) 745.402 1.02250
\(730\) 0 0
\(731\) −205.465 355.876i −0.281074 0.486835i
\(732\) 0 0
\(733\) −469.714 + 813.569i −0.640810 + 1.10992i 0.344442 + 0.938808i \(0.388068\pi\)
−0.985252 + 0.171108i \(0.945265\pi\)
\(734\) 0 0
\(735\) 333.522 + 121.336i 0.453772 + 0.165083i
\(736\) 0 0
\(737\) 209.470 362.812i 0.284220 0.492283i
\(738\) 0 0
\(739\) 984.945 568.658i 1.33281 0.769497i 0.347079 0.937836i \(-0.387174\pi\)
0.985729 + 0.168339i \(0.0538404\pi\)
\(740\) 0 0
\(741\) −421.388 −0.568675
\(742\) 0 0
\(743\) 455.212 0.612667 0.306333 0.951924i \(-0.400898\pi\)
0.306333 + 0.951924i \(0.400898\pi\)
\(744\) 0 0
\(745\) 656.012 378.749i 0.880554 0.508388i
\(746\) 0 0
\(747\) 3.82292 6.62150i 0.00511770 0.00886412i
\(748\) 0 0
\(749\) 362.001 + 303.709i 0.483312 + 0.405486i
\(750\) 0 0
\(751\) 94.2623 163.267i 0.125516 0.217400i −0.796419 0.604746i \(-0.793275\pi\)
0.921934 + 0.387346i \(0.126608\pi\)
\(752\) 0 0
\(753\) 466.035 + 807.196i 0.618905 + 1.07197i
\(754\) 0 0
\(755\) 639.502 0.847023
\(756\) 0 0
\(757\) 199.539i 0.263591i 0.991277 + 0.131796i \(0.0420743\pi\)
−0.991277 + 0.131796i \(0.957926\pi\)
\(758\) 0 0
\(759\) −245.781 + 141.902i −0.323822 + 0.186959i
\(760\) 0 0
\(761\) 292.734 + 169.010i 0.384670 + 0.222089i 0.679848 0.733353i \(-0.262046\pi\)
−0.295178 + 0.955442i \(0.595379\pi\)
\(762\) 0 0
\(763\) −888.345 + 323.406i −1.16428 + 0.423861i
\(764\) 0 0
\(765\) −161.797 93.4138i −0.211500 0.122110i
\(766\) 0 0
\(767\) −365.326 632.763i −0.476305 0.824984i
\(768\) 0 0
\(769\) 604.446i 0.786015i −0.919535 0.393008i \(-0.871435\pi\)
0.919535 0.393008i \(-0.128565\pi\)
\(770\) 0 0
\(771\) 5.72674i 0.00742768i
\(772\) 0 0
\(773\) 390.213 + 675.869i 0.504804 + 0.874346i 0.999985 + 0.00555593i \(0.00176852\pi\)
−0.495181 + 0.868790i \(0.664898\pi\)
\(774\) 0 0
\(775\) −244.630 141.237i −0.315651 0.182241i
\(776\) 0 0
\(777\) 141.559 803.172i 0.182187 1.03368i
\(778\) 0 0
\(779\) 244.882 + 141.383i 0.314354 + 0.181492i
\(780\) 0 0
\(781\) 332.561 192.004i 0.425814 0.245844i
\(782\) 0 0
\(783\) 1238.52i 1.58176i
\(784\) 0 0
\(785\) −231.164 −0.294477
\(786\) 0 0
\(787\) 481.905 + 834.684i 0.612332 + 1.06059i 0.990846 + 0.134994i \(0.0431016\pi\)
−0.378515 + 0.925595i \(0.623565\pi\)
\(788\) 0 0
\(789\) −67.2189 + 116.427i −0.0851951 + 0.147562i
\(790\) 0 0
\(791\) 939.813 + 165.642i 1.18813 + 0.209409i
\(792\) 0 0
\(793\) 321.551 556.942i 0.405486 0.702323i
\(794\) 0 0
\(795\) −43.7440 + 25.2556i −0.0550239 + 0.0317681i
\(796\) 0 0
\(797\) −677.191 −0.849675 −0.424837 0.905270i \(-0.639669\pi\)
−0.424837 + 0.905270i \(0.639669\pi\)
\(798\) 0 0
\(799\) −232.511 −0.291003
\(800\) 0 0
\(801\) −311.329 + 179.746i −0.388676 + 0.224402i
\(802\) 0 0
\(803\) 323.856 560.935i 0.403308 0.698549i
\(804\) 0 0
\(805\) −192.188 527.911i −0.238743 0.655790i
\(806\) 0 0
\(807\) 280.746 486.265i 0.347888 0.602559i
\(808\) 0 0
\(809\) 208.617 + 361.335i 0.257870 + 0.446645i 0.965671 0.259767i \(-0.0836459\pi\)
−0.707801 + 0.706412i \(0.750313\pi\)
\(810\) 0 0
\(811\) 1431.94 1.76564 0.882821 0.469709i \(-0.155641\pi\)
0.882821 + 0.469709i \(0.155641\pi\)
\(812\) 0 0
\(813\) 314.215i 0.386488i
\(814\) 0 0
\(815\) 263.956 152.395i 0.323872 0.186988i
\(816\) 0 0
\(817\) 549.033 + 316.985i 0.672012 + 0.387986i
\(818\) 0 0
\(819\) 109.742 130.805i 0.133995 0.159713i
\(820\) 0 0
\(821\) 15.8963 + 9.17774i 0.0193621 + 0.0111787i 0.509650 0.860382i \(-0.329775\pi\)
−0.490288 + 0.871561i \(0.663108\pi\)
\(822\) 0 0
\(823\) 720.293 + 1247.58i 0.875204 + 1.51590i 0.856545 + 0.516073i \(0.172607\pi\)
0.0186596 + 0.999826i \(0.494060\pi\)
\(824\) 0 0
\(825\) 168.691i 0.204474i
\(826\) 0 0
\(827\) 240.040i 0.290255i 0.989413 + 0.145127i \(0.0463592\pi\)
−0.989413 + 0.145127i \(0.953641\pi\)
\(828\) 0 0
\(829\) −732.065 1267.97i −0.883070 1.52952i −0.847910 0.530140i \(-0.822139\pi\)
−0.0351599 0.999382i \(-0.511194\pi\)
\(830\) 0 0
\(831\) −223.309 128.927i −0.268723 0.155147i
\(832\) 0 0
\(833\) 820.097 144.731i 0.984510 0.173747i
\(834\) 0 0
\(835\) −678.227 391.575i −0.812248 0.468952i
\(836\) 0 0
\(837\) −465.942 + 269.012i −0.556681 + 0.321400i
\(838\) 0 0
\(839\) 896.568i 1.06861i −0.845290 0.534307i \(-0.820573\pi\)
0.845290 0.534307i \(-0.179427\pi\)
\(840\) 0 0
\(841\) −945.761 −1.12457
\(842\) 0 0
\(843\) −179.944 311.673i −0.213457 0.369719i
\(844\) 0 0
\(845\) −188.628 + 326.713i −0.223228 + 0.386642i
\(846\) 0 0
\(847\) −445.216 + 530.668i −0.525639 + 0.626527i
\(848\) 0 0
\(849\) −36.1354 + 62.5884i −0.0425623 + 0.0737201i
\(850\) 0 0
\(851\) −1118.03 + 645.495i −1.31378 + 0.758513i
\(852\) 0 0
\(853\) 1376.70 1.61395 0.806973 0.590588i \(-0.201104\pi\)
0.806973 + 0.590588i \(0.201104\pi\)
\(854\) 0 0
\(855\) 288.231 0.337112
\(856\) 0 0
\(857\) −1048.58 + 605.400i −1.22355 + 0.706418i −0.965673 0.259760i \(-0.916357\pi\)
−0.257878 + 0.966177i \(0.583023\pi\)
\(858\) 0 0
\(859\) 287.326 497.663i 0.334488 0.579351i −0.648898 0.760875i \(-0.724770\pi\)
0.983386 + 0.181524i \(0.0581031\pi\)
\(860\) 0 0
\(861\) 165.681 60.3170i 0.192429 0.0700545i
\(862\) 0 0
\(863\) −243.335 + 421.468i −0.281964 + 0.488375i −0.971868 0.235525i \(-0.924319\pi\)
0.689905 + 0.723900i \(0.257652\pi\)
\(864\) 0 0
\(865\) 233.168 + 403.859i 0.269559 + 0.466889i
\(866\) 0 0
\(867\) −0.372749 −0.000429930
\(868\) 0 0
\(869\) 61.5138i 0.0707869i
\(870\) 0 0
\(871\) −531.729 + 306.994i −0.610481 + 0.352462i
\(872\) 0 0
\(873\) −273.014 157.625i −0.312730 0.180555i
\(874\) 0 0
\(875\) 863.319 + 152.160i 0.986650 + 0.173897i
\(876\) 0 0
\(877\) 1018.25 + 587.886i 1.16106 + 0.670337i 0.951557 0.307472i \(-0.0994829\pi\)
0.209500 + 0.977809i \(0.432816\pi\)
\(878\) 0 0
\(879\) −597.554 1034.99i −0.679811 1.17747i
\(880\) 0 0
\(881\) 197.506i 0.224183i −0.993698 0.112092i \(-0.964245\pi\)
0.993698 0.112092i \(-0.0357550\pi\)
\(882\) 0 0
\(883\) 739.290i 0.837248i 0.908160 + 0.418624i \(0.137487\pi\)
−0.908160 + 0.418624i \(0.862513\pi\)
\(884\) 0 0
\(885\) −384.547 666.056i −0.434517 0.752605i
\(886\) 0 0
\(887\) 379.322 + 219.002i 0.427646 + 0.246902i 0.698343 0.715763i \(-0.253921\pi\)
−0.270697 + 0.962665i \(0.587254\pi\)
\(888\) 0 0
\(889\) 44.0994 + 7.77254i 0.0496056 + 0.00874301i
\(890\) 0 0
\(891\) −148.535 85.7566i −0.166706 0.0962476i
\(892\) 0 0
\(893\) 310.652 179.355i 0.347875 0.200846i
\(894\) 0 0
\(895\) 320.931i 0.358582i
\(896\) 0 0
\(897\) 415.936 0.463696
\(898\) 0 0
\(899\) −388.093 672.196i −0.431694 0.747716i
\(900\) 0 0
\(901\) −59.2609 + 102.643i −0.0657723 + 0.113921i
\(902\) 0 0
\(903\) 371.463 135.233i 0.411365 0.149759i
\(904\) 0 0
\(905\) −147.476 + 255.436i −0.162957 + 0.282250i
\(906\) 0 0
\(907\) −1142.72 + 659.752i −1.25989 + 0.727400i −0.973054 0.230579i \(-0.925938\pi\)
−0.286840 + 0.957979i \(0.592605\pi\)
\(908\) 0 0
\(909\) 73.9018 0.0813001
\(910\) 0 0
\(911\) −206.561 −0.226741 −0.113371 0.993553i \(-0.536165\pi\)
−0.113371 + 0.993553i \(0.536165\pi\)
\(912\) 0 0
\(913\) −8.77002 + 5.06338i −0.00960572 + 0.00554587i
\(914\) 0 0
\(915\) 338.469 586.246i 0.369912 0.640706i
\(916\) 0 0
\(917\) 781.471 931.462i 0.852204 1.01577i
\(918\) 0 0
\(919\) 281.693 487.906i 0.306521 0.530910i −0.671078 0.741387i \(-0.734168\pi\)
0.977599 + 0.210477i \(0.0675018\pi\)
\(920\) 0 0
\(921\) 59.8703 + 103.698i 0.0650058 + 0.112593i
\(922\) 0 0
\(923\) −562.793 −0.609743
\(924\) 0 0
\(925\) 767.355i 0.829573i
\(926\) 0 0
\(927\) −10.5448 + 6.08804i −0.0113752 + 0.00656746i
\(928\) 0 0
\(929\) −1050.66 606.600i −1.13096 0.652960i −0.186784 0.982401i \(-0.559806\pi\)
−0.944176 + 0.329441i \(0.893140\pi\)
\(930\) 0 0
\(931\) −984.067 + 825.980i −1.05700 + 0.887196i
\(932\) 0 0
\(933\) 41.5601 + 23.9947i 0.0445446 + 0.0257178i
\(934\) 0 0
\(935\) 123.724 + 214.297i 0.132326 + 0.229195i
\(936\) 0 0
\(937\) 237.201i 0.253149i −0.991957 0.126575i \(-0.959602\pi\)
0.991957 0.126575i \(-0.0403983\pi\)
\(938\) 0 0
\(939\) 787.403i 0.838554i
\(940\) 0 0
\(941\) −59.6021 103.234i −0.0633391 0.109707i 0.832617 0.553849i \(-0.186842\pi\)
−0.895956 + 0.444143i \(0.853508\pi\)
\(942\) 0 0
\(943\) −241.713 139.553i −0.256324 0.147989i
\(944\) 0 0
\(945\) 408.800 487.263i 0.432593 0.515622i
\(946\) 0 0
\(947\) −842.482 486.407i −0.889633 0.513630i −0.0158103 0.999875i \(-0.505033\pi\)
−0.873822 + 0.486245i \(0.838366\pi\)
\(948\) 0 0
\(949\) −822.093 + 474.636i −0.866273 + 0.500143i
\(950\) 0 0
\(951\) 217.815i 0.229038i
\(952\) 0 0
\(953\) −840.555 −0.882010 −0.441005 0.897505i \(-0.645378\pi\)
−0.441005 + 0.897505i \(0.645378\pi\)
\(954\) 0 0
\(955\) −6.11697 10.5949i −0.00640520 0.0110941i
\(956\) 0 0
\(957\) 231.765 401.429i 0.242179 0.419466i
\(958\) 0 0
\(959\) 183.393 + 503.752i 0.191234 + 0.525289i
\(960\) 0 0
\(961\) −311.909 + 540.242i −0.324567 + 0.562167i
\(962\) 0 0
\(963\) 207.232 119.646i 0.215195 0.124243i
\(964\) 0 0
\(965\) 906.086 0.938949
\(966\) 0 0
\(967\) −1696.40 −1.75429 −0.877147 0.480222i \(-0.840556\pi\)
−0.877147 + 0.480222i \(0.840556\pi\)
\(968\) 0 0
\(969\) −901.347 + 520.393i −0.930183 + 0.537041i
\(970\) 0 0
\(971\) 644.056 1115.54i 0.663291 1.14885i −0.316454 0.948608i \(-0.602492\pi\)
0.979746 0.200247i \(-0.0641743\pi\)
\(972\) 0 0
\(973\) −499.606 88.0558i −0.513470 0.0904993i
\(974\) 0 0
\(975\) −123.615 + 214.107i −0.126784 + 0.219597i
\(976\) 0 0
\(977\) 61.9545 + 107.308i 0.0634130 + 0.109835i 0.895989 0.444076i \(-0.146468\pi\)
−0.832576 + 0.553911i \(0.813135\pi\)
\(978\) 0 0
\(979\) 476.139 0.486353
\(980\) 0 0
\(981\) 478.747i 0.488019i
\(982\) 0 0
\(983\) −1425.29 + 822.891i −1.44994 + 0.837122i −0.998477 0.0551686i \(-0.982430\pi\)
−0.451461 + 0.892291i \(0.649097\pi\)
\(984\) 0 0
\(985\) −431.051 248.867i −0.437615 0.252657i
\(986\) 0 0
\(987\) 38.8243 220.280i 0.0393357 0.223181i
\(988\) 0 0
\(989\) −541.930 312.883i −0.547957 0.316363i
\(990\) 0 0
\(991\) 226.918 + 393.034i 0.228979 + 0.396603i 0.957506 0.288414i \(-0.0931280\pi\)
−0.728527 + 0.685017i \(0.759795\pi\)
\(992\) 0 0
\(993\) 175.285i 0.176521i
\(994\) 0 0
\(995\) 624.717i 0.627856i
\(996\) 0 0
\(997\) −453.413 785.334i −0.454777 0.787697i 0.543898 0.839151i \(-0.316948\pi\)
−0.998675 + 0.0514540i \(0.983614\pi\)
\(998\) 0 0
\(999\) −1265.75 730.783i −1.26702 0.731515i
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 224.3.n.a.145.10 28
4.3 odd 2 56.3.j.a.5.7 28
7.2 even 3 1568.3.h.a.881.10 28
7.3 odd 6 inner 224.3.n.a.17.5 28
7.5 odd 6 1568.3.h.a.881.20 28
8.3 odd 2 56.3.j.a.5.13 yes 28
8.5 even 2 inner 224.3.n.a.145.5 28
28.3 even 6 56.3.j.a.45.13 yes 28
28.11 odd 6 392.3.j.e.325.13 28
28.19 even 6 392.3.h.a.293.7 28
28.23 odd 6 392.3.h.a.293.8 28
28.27 even 2 392.3.j.e.117.7 28
56.3 even 6 56.3.j.a.45.7 yes 28
56.5 odd 6 1568.3.h.a.881.9 28
56.11 odd 6 392.3.j.e.325.7 28
56.19 even 6 392.3.h.a.293.6 28
56.27 even 2 392.3.j.e.117.13 28
56.37 even 6 1568.3.h.a.881.19 28
56.45 odd 6 inner 224.3.n.a.17.10 28
56.51 odd 6 392.3.h.a.293.5 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.7 28 4.3 odd 2
56.3.j.a.5.13 yes 28 8.3 odd 2
56.3.j.a.45.7 yes 28 56.3 even 6
56.3.j.a.45.13 yes 28 28.3 even 6
224.3.n.a.17.5 28 7.3 odd 6 inner
224.3.n.a.17.10 28 56.45 odd 6 inner
224.3.n.a.145.5 28 8.5 even 2 inner
224.3.n.a.145.10 28 1.1 even 1 trivial
392.3.h.a.293.5 28 56.51 odd 6
392.3.h.a.293.6 28 56.19 even 6
392.3.h.a.293.7 28 28.19 even 6
392.3.h.a.293.8 28 28.23 odd 6
392.3.j.e.117.7 28 28.27 even 2
392.3.j.e.117.13 28 56.27 even 2
392.3.j.e.325.7 28 56.11 odd 6
392.3.j.e.325.13 28 28.11 odd 6
1568.3.h.a.881.9 28 56.5 odd 6
1568.3.h.a.881.10 28 7.2 even 3
1568.3.h.a.881.19 28 56.37 even 6
1568.3.h.a.881.20 28 7.5 odd 6