Properties

Label 136.3.t.a.65.1
Level $136$
Weight $3$
Character 136.65
Analytic conductor $3.706$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,3,Mod(41,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.t (of order \(16\), degree \(8\), 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: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 65.1
Character \(\chi\) \(=\) 136.65
Dual form 136.3.t.a.113.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.912174 + 4.58581i) q^{3} +(7.13034 - 4.76434i) q^{5} +(7.38570 + 4.93497i) q^{7} +(-11.8827 - 4.92196i) q^{9} +O(q^{10})\) \(q+(-0.912174 + 4.58581i) q^{3} +(7.13034 - 4.76434i) q^{5} +(7.38570 + 4.93497i) q^{7} +(-11.8827 - 4.92196i) q^{9} +(-10.3779 + 2.06429i) q^{11} +(7.09039 + 7.09039i) q^{13} +(15.3442 + 37.0443i) q^{15} +(-9.06777 + 14.3797i) q^{17} +(0.265133 - 0.109822i) q^{19} +(-29.3679 + 29.3679i) q^{21} +(-7.43010 - 37.3536i) q^{23} +(18.5757 - 44.8458i) q^{25} +(10.0313 - 15.0130i) q^{27} +(11.2269 + 16.8022i) q^{29} +(13.1773 + 2.62114i) q^{31} -49.4740i q^{33} +76.1745 q^{35} +(8.07571 - 40.5993i) q^{37} +(-38.9828 + 26.0475i) q^{39} +(-41.2112 - 27.5364i) q^{41} +(33.4063 + 13.8374i) q^{43} +(-108.177 + 21.5178i) q^{45} +(-28.3642 - 28.3642i) q^{47} +(11.4432 + 27.6264i) q^{49} +(-57.6711 - 54.6998i) q^{51} +(77.8996 - 32.2671i) q^{53} +(-64.1630 + 64.1630i) q^{55} +(0.261774 + 1.31602i) q^{57} +(-27.7385 + 66.9668i) q^{59} +(-26.1745 + 39.1729i) q^{61} +(-63.4721 - 94.9927i) q^{63} +(84.3379 + 16.7759i) q^{65} -46.9366i q^{67} +178.074 q^{69} +(24.0034 - 120.673i) q^{71} +(-31.3109 + 20.9213i) q^{73} +(188.710 + 126.092i) q^{75} +(-86.8353 - 35.9683i) q^{77} +(-44.2960 + 8.81103i) q^{79} +(-22.1551 - 22.1551i) q^{81} +(-16.6189 - 40.1215i) q^{83} +(3.85346 + 145.734i) q^{85} +(-87.2924 + 36.1577i) q^{87} +(74.9321 - 74.9321i) q^{89} +(17.3767 + 87.3583i) q^{91} +(-24.0401 + 58.0378i) q^{93} +(1.36726 - 2.04625i) q^{95} +(78.3135 + 117.204i) q^{97} +(133.477 + 26.5503i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 8 q^{3} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 8 q^{3} + 8 q^{7} + 16 q^{9} - 24 q^{11} + 48 q^{13} + 96 q^{15} + 40 q^{19} - 80 q^{21} - 48 q^{23} + 112 q^{25} - 80 q^{27} + 56 q^{29} - 24 q^{31} - 96 q^{35} + 48 q^{37} - 72 q^{39} - 160 q^{41} + 112 q^{43} - 504 q^{45} + 48 q^{47} + 208 q^{49} - 400 q^{51} + 304 q^{53} - 368 q^{55} - 264 q^{57} + 192 q^{59} - 288 q^{61} + 56 q^{63} + 8 q^{65} + 32 q^{69} + 352 q^{71} - 184 q^{73} + 24 q^{75} + 688 q^{77} - 424 q^{79} + 312 q^{81} + 600 q^{83} - 512 q^{85} + 1336 q^{87} - 144 q^{89} - 24 q^{91} + 944 q^{93} - 256 q^{95} + 416 q^{97} + 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{9}{16}\right)\)

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
\(3\) −0.912174 + 4.58581i −0.304058 + 1.52860i 0.462612 + 0.886561i \(0.346912\pi\)
−0.766670 + 0.642042i \(0.778088\pi\)
\(4\) 0 0
\(5\) 7.13034 4.76434i 1.42607 0.952869i 0.427257 0.904130i \(-0.359480\pi\)
0.998812 0.0487385i \(-0.0155201\pi\)
\(6\) 0 0
\(7\) 7.38570 + 4.93497i 1.05510 + 0.704996i 0.956973 0.290178i \(-0.0937145\pi\)
0.0981280 + 0.995174i \(0.468715\pi\)
\(8\) 0 0
\(9\) −11.8827 4.92196i −1.32030 0.546884i
\(10\) 0 0
\(11\) −10.3779 + 2.06429i −0.943445 + 0.187663i −0.642766 0.766063i \(-0.722213\pi\)
−0.300679 + 0.953725i \(0.597213\pi\)
\(12\) 0 0
\(13\) 7.09039 + 7.09039i 0.545414 + 0.545414i 0.925111 0.379697i \(-0.123972\pi\)
−0.379697 + 0.925111i \(0.623972\pi\)
\(14\) 0 0
\(15\) 15.3442 + 37.0443i 1.02295 + 2.46962i
\(16\) 0 0
\(17\) −9.06777 + 14.3797i −0.533398 + 0.845864i
\(18\) 0 0
\(19\) 0.265133 0.109822i 0.0139544 0.00578009i −0.375695 0.926743i \(-0.622596\pi\)
0.389650 + 0.920963i \(0.372596\pi\)
\(20\) 0 0
\(21\) −29.3679 + 29.3679i −1.39847 + 1.39847i
\(22\) 0 0
\(23\) −7.43010 37.3536i −0.323048 1.62407i −0.711558 0.702627i \(-0.752010\pi\)
0.388511 0.921444i \(-0.372990\pi\)
\(24\) 0 0
\(25\) 18.5757 44.8458i 0.743030 1.79383i
\(26\) 0 0
\(27\) 10.0313 15.0130i 0.371531 0.556035i
\(28\) 0 0
\(29\) 11.2269 + 16.8022i 0.387133 + 0.579385i 0.972938 0.231067i \(-0.0742216\pi\)
−0.585805 + 0.810452i \(0.699222\pi\)
\(30\) 0 0
\(31\) 13.1773 + 2.62114i 0.425076 + 0.0845528i 0.402992 0.915203i \(-0.367970\pi\)
0.0220832 + 0.999756i \(0.492970\pi\)
\(32\) 0 0
\(33\) 49.4740i 1.49921i
\(34\) 0 0
\(35\) 76.1745 2.17641
\(36\) 0 0
\(37\) 8.07571 40.5993i 0.218262 1.09728i −0.703847 0.710351i \(-0.748536\pi\)
0.922110 0.386928i \(-0.126464\pi\)
\(38\) 0 0
\(39\) −38.9828 + 26.0475i −0.999559 + 0.667884i
\(40\) 0 0
\(41\) −41.2112 27.5364i −1.00515 0.671621i −0.0599814 0.998199i \(-0.519104\pi\)
−0.945170 + 0.326579i \(0.894104\pi\)
\(42\) 0 0
\(43\) 33.4063 + 13.8374i 0.776892 + 0.321799i 0.735660 0.677351i \(-0.236872\pi\)
0.0412313 + 0.999150i \(0.486872\pi\)
\(44\) 0 0
\(45\) −108.177 + 21.5178i −2.40394 + 0.478173i
\(46\) 0 0
\(47\) −28.3642 28.3642i −0.603495 0.603495i 0.337744 0.941238i \(-0.390336\pi\)
−0.941238 + 0.337744i \(0.890336\pi\)
\(48\) 0 0
\(49\) 11.4432 + 27.6264i 0.233535 + 0.563803i
\(50\) 0 0
\(51\) −57.6711 54.6998i −1.13081 1.07255i
\(52\) 0 0
\(53\) 77.8996 32.2671i 1.46980 0.608812i 0.502989 0.864293i \(-0.332234\pi\)
0.966814 + 0.255480i \(0.0822336\pi\)
\(54\) 0 0
\(55\) −64.1630 + 64.1630i −1.16660 + 1.16660i
\(56\) 0 0
\(57\) 0.261774 + 1.31602i 0.00459252 + 0.0230882i
\(58\) 0 0
\(59\) −27.7385 + 66.9668i −0.470145 + 1.13503i 0.493954 + 0.869488i \(0.335551\pi\)
−0.964099 + 0.265542i \(0.914449\pi\)
\(60\) 0 0
\(61\) −26.1745 + 39.1729i −0.429090 + 0.642179i −0.981515 0.191384i \(-0.938702\pi\)
0.552425 + 0.833562i \(0.313702\pi\)
\(62\) 0 0
\(63\) −63.4721 94.9927i −1.00749 1.50782i
\(64\) 0 0
\(65\) 84.3379 + 16.7759i 1.29751 + 0.258090i
\(66\) 0 0
\(67\) 46.9366i 0.700547i −0.936648 0.350273i \(-0.886089\pi\)
0.936648 0.350273i \(-0.113911\pi\)
\(68\) 0 0
\(69\) 178.074 2.58078
\(70\) 0 0
\(71\) 24.0034 120.673i 0.338076 1.69962i −0.320617 0.947209i \(-0.603890\pi\)
0.658693 0.752412i \(-0.271110\pi\)
\(72\) 0 0
\(73\) −31.3109 + 20.9213i −0.428917 + 0.286593i −0.751231 0.660040i \(-0.770539\pi\)
0.322314 + 0.946633i \(0.395539\pi\)
\(74\) 0 0
\(75\) 188.710 + 126.092i 2.51613 + 1.68123i
\(76\) 0 0
\(77\) −86.8353 35.9683i −1.12773 0.467121i
\(78\) 0 0
\(79\) −44.2960 + 8.81103i −0.560709 + 0.111532i −0.467305 0.884096i \(-0.654775\pi\)
−0.0934045 + 0.995628i \(0.529775\pi\)
\(80\) 0 0
\(81\) −22.1551 22.1551i −0.273520 0.273520i
\(82\) 0 0
\(83\) −16.6189 40.1215i −0.200227 0.483392i 0.791591 0.611052i \(-0.209253\pi\)
−0.991818 + 0.127660i \(0.959253\pi\)
\(84\) 0 0
\(85\) 3.85346 + 145.734i 0.0453348 + 1.71452i
\(86\) 0 0
\(87\) −87.2924 + 36.1577i −1.00336 + 0.415605i
\(88\) 0 0
\(89\) 74.9321 74.9321i 0.841933 0.841933i −0.147177 0.989110i \(-0.547019\pi\)
0.989110 + 0.147177i \(0.0470187\pi\)
\(90\) 0 0
\(91\) 17.3767 + 87.3583i 0.190952 + 0.959982i
\(92\) 0 0
\(93\) −24.0401 + 58.0378i −0.258495 + 0.624063i
\(94\) 0 0
\(95\) 1.36726 2.04625i 0.0143922 0.0215395i
\(96\) 0 0
\(97\) 78.3135 + 117.204i 0.807356 + 1.20829i 0.974947 + 0.222437i \(0.0714010\pi\)
−0.167592 + 0.985857i \(0.553599\pi\)
\(98\) 0 0
\(99\) 133.477 + 26.5503i 1.34826 + 0.268185i
\(100\) 0 0
\(101\) 140.520i 1.39129i −0.718386 0.695645i \(-0.755119\pi\)
0.718386 0.695645i \(-0.244881\pi\)
\(102\) 0 0
\(103\) −101.755 −0.987912 −0.493956 0.869487i \(-0.664450\pi\)
−0.493956 + 0.869487i \(0.664450\pi\)
\(104\) 0 0
\(105\) −69.4844 + 349.322i −0.661756 + 3.32687i
\(106\) 0 0
\(107\) −6.94287 + 4.63908i −0.0648867 + 0.0433559i −0.587590 0.809158i \(-0.699923\pi\)
0.522704 + 0.852514i \(0.324923\pi\)
\(108\) 0 0
\(109\) 69.1148 + 46.1810i 0.634081 + 0.423679i 0.830638 0.556813i \(-0.187976\pi\)
−0.196557 + 0.980492i \(0.562976\pi\)
\(110\) 0 0
\(111\) 178.814 + 74.0673i 1.61094 + 0.667273i
\(112\) 0 0
\(113\) −89.7284 + 17.8481i −0.794057 + 0.157948i −0.575420 0.817858i \(-0.695161\pi\)
−0.218637 + 0.975806i \(0.570161\pi\)
\(114\) 0 0
\(115\) −230.945 230.945i −2.00821 2.00821i
\(116\) 0 0
\(117\) −49.3540 119.151i −0.421829 1.01839i
\(118\) 0 0
\(119\) −137.935 + 61.4550i −1.15912 + 0.516428i
\(120\) 0 0
\(121\) −8.35004 + 3.45870i −0.0690086 + 0.0285843i
\(122\) 0 0
\(123\) 163.869 163.869i 1.33227 1.33227i
\(124\) 0 0
\(125\) −39.3841 197.997i −0.315072 1.58398i
\(126\) 0 0
\(127\) −4.44386 + 10.7284i −0.0349911 + 0.0844759i −0.940409 0.340045i \(-0.889558\pi\)
0.905418 + 0.424521i \(0.139558\pi\)
\(128\) 0 0
\(129\) −93.9279 + 140.573i −0.728123 + 1.08971i
\(130\) 0 0
\(131\) 70.3003 + 105.212i 0.536644 + 0.803144i 0.996389 0.0849068i \(-0.0270592\pi\)
−0.459745 + 0.888051i \(0.652059\pi\)
\(132\) 0 0
\(133\) 2.50016 + 0.497313i 0.0187982 + 0.00373919i
\(134\) 0 0
\(135\) 154.840i 1.14696i
\(136\) 0 0
\(137\) −92.3943 −0.674411 −0.337206 0.941431i \(-0.609482\pi\)
−0.337206 + 0.941431i \(0.609482\pi\)
\(138\) 0 0
\(139\) −0.130138 + 0.654249i −0.000936246 + 0.00470683i −0.981251 0.192736i \(-0.938264\pi\)
0.980314 + 0.197443i \(0.0632638\pi\)
\(140\) 0 0
\(141\) 155.946 104.200i 1.10600 0.739006i
\(142\) 0 0
\(143\) −88.2199 58.9467i −0.616922 0.412214i
\(144\) 0 0
\(145\) 160.103 + 66.3167i 1.10416 + 0.457356i
\(146\) 0 0
\(147\) −137.127 + 27.2763i −0.932840 + 0.185553i
\(148\) 0 0
\(149\) 44.4637 + 44.4637i 0.298414 + 0.298414i 0.840392 0.541978i \(-0.182325\pi\)
−0.541978 + 0.840392i \(0.682325\pi\)
\(150\) 0 0
\(151\) 2.76538 + 6.67621i 0.0183137 + 0.0442133i 0.932772 0.360467i \(-0.117383\pi\)
−0.914458 + 0.404681i \(0.867383\pi\)
\(152\) 0 0
\(153\) 178.525 126.238i 1.16683 0.825083i
\(154\) 0 0
\(155\) 106.447 44.0918i 0.686755 0.284463i
\(156\) 0 0
\(157\) −52.5857 + 52.5857i −0.334941 + 0.334941i −0.854459 0.519518i \(-0.826111\pi\)
0.519518 + 0.854459i \(0.326111\pi\)
\(158\) 0 0
\(159\) 76.9126 + 386.666i 0.483727 + 2.43186i
\(160\) 0 0
\(161\) 129.463 312.550i 0.804115 1.94131i
\(162\) 0 0
\(163\) −18.7682 + 28.0886i −0.115142 + 0.172322i −0.884564 0.466418i \(-0.845544\pi\)
0.769422 + 0.638740i \(0.220544\pi\)
\(164\) 0 0
\(165\) −235.711 352.767i −1.42855 2.13798i
\(166\) 0 0
\(167\) −92.6964 18.4385i −0.555068 0.110410i −0.0904173 0.995904i \(-0.528820\pi\)
−0.464651 + 0.885494i \(0.653820\pi\)
\(168\) 0 0
\(169\) 68.4528i 0.405046i
\(170\) 0 0
\(171\) −3.69102 −0.0215849
\(172\) 0 0
\(173\) −44.0902 + 221.657i −0.254857 + 1.28125i 0.615228 + 0.788349i \(0.289064\pi\)
−0.870085 + 0.492902i \(0.835936\pi\)
\(174\) 0 0
\(175\) 358.508 239.547i 2.04862 1.36884i
\(176\) 0 0
\(177\) −281.794 188.289i −1.59206 1.06378i
\(178\) 0 0
\(179\) 91.7076 + 37.9865i 0.512333 + 0.212215i 0.623845 0.781548i \(-0.285569\pi\)
−0.111512 + 0.993763i \(0.535569\pi\)
\(180\) 0 0
\(181\) −83.0467 + 16.5190i −0.458822 + 0.0912653i −0.419089 0.907945i \(-0.637651\pi\)
−0.0397325 + 0.999210i \(0.512651\pi\)
\(182\) 0 0
\(183\) −155.764 155.764i −0.851168 0.851168i
\(184\) 0 0
\(185\) −135.847 327.963i −0.734306 1.77277i
\(186\) 0 0
\(187\) 64.4205 167.949i 0.344495 0.898125i
\(188\) 0 0
\(189\) 148.177 61.3769i 0.784005 0.324745i
\(190\) 0 0
\(191\) −203.313 + 203.313i −1.06447 + 1.06447i −0.0666915 + 0.997774i \(0.521244\pi\)
−0.997774 + 0.0666915i \(0.978756\pi\)
\(192\) 0 0
\(193\) −19.1117 96.0809i −0.0990242 0.497829i −0.998186 0.0602114i \(-0.980823\pi\)
0.899161 0.437617i \(-0.144177\pi\)
\(194\) 0 0
\(195\) −153.862 + 371.455i −0.789034 + 1.90490i
\(196\) 0 0
\(197\) −115.812 + 173.325i −0.587879 + 0.879823i −0.999503 0.0315311i \(-0.989962\pi\)
0.411624 + 0.911354i \(0.364962\pi\)
\(198\) 0 0
\(199\) 37.4762 + 56.0871i 0.188323 + 0.281845i 0.913602 0.406610i \(-0.133289\pi\)
−0.725279 + 0.688455i \(0.758289\pi\)
\(200\) 0 0
\(201\) 215.242 + 42.8144i 1.07086 + 0.213007i
\(202\) 0 0
\(203\) 179.500i 0.884237i
\(204\) 0 0
\(205\) −425.043 −2.07338
\(206\) 0 0
\(207\) −95.5637 + 480.431i −0.461660 + 2.32092i
\(208\) 0 0
\(209\) −2.52482 + 1.68703i −0.0120805 + 0.00807191i
\(210\) 0 0
\(211\) 26.3783 + 17.6254i 0.125016 + 0.0835328i 0.616506 0.787350i \(-0.288548\pi\)
−0.491490 + 0.870883i \(0.663548\pi\)
\(212\) 0 0
\(213\) 531.488 + 220.150i 2.49525 + 1.03357i
\(214\) 0 0
\(215\) 304.125 60.4942i 1.41453 0.281368i
\(216\) 0 0
\(217\) 84.3887 + 84.3887i 0.388888 + 0.388888i
\(218\) 0 0
\(219\) −67.3801 162.670i −0.307672 0.742785i
\(220\) 0 0
\(221\) −166.252 + 37.6636i −0.752270 + 0.170423i
\(222\) 0 0
\(223\) 184.159 76.2809i 0.825823 0.342067i 0.0705751 0.997506i \(-0.477517\pi\)
0.755248 + 0.655439i \(0.227517\pi\)
\(224\) 0 0
\(225\) −441.458 + 441.458i −1.96204 + 1.96204i
\(226\) 0 0
\(227\) −30.2374 152.014i −0.133205 0.669664i −0.988463 0.151463i \(-0.951602\pi\)
0.855258 0.518202i \(-0.173398\pi\)
\(228\) 0 0
\(229\) −99.1387 + 239.342i −0.432920 + 1.04516i 0.545421 + 0.838162i \(0.316370\pi\)
−0.978341 + 0.206999i \(0.933630\pi\)
\(230\) 0 0
\(231\) 244.153 365.400i 1.05694 1.58182i
\(232\) 0 0
\(233\) 52.4419 + 78.4849i 0.225073 + 0.336845i 0.926769 0.375631i \(-0.122574\pi\)
−0.701697 + 0.712476i \(0.747574\pi\)
\(234\) 0 0
\(235\) −337.384 67.1098i −1.43568 0.285574i
\(236\) 0 0
\(237\) 211.170i 0.891014i
\(238\) 0 0
\(239\) 243.237 1.01773 0.508864 0.860847i \(-0.330065\pi\)
0.508864 + 0.860847i \(0.330065\pi\)
\(240\) 0 0
\(241\) −57.1441 + 287.283i −0.237113 + 1.19205i 0.660351 + 0.750957i \(0.270407\pi\)
−0.897464 + 0.441088i \(0.854593\pi\)
\(242\) 0 0
\(243\) 256.925 171.672i 1.05730 0.706468i
\(244\) 0 0
\(245\) 213.216 + 142.466i 0.870268 + 0.581494i
\(246\) 0 0
\(247\) 2.65857 + 1.10122i 0.0107635 + 0.00445837i
\(248\) 0 0
\(249\) 199.149 39.6132i 0.799794 0.159089i
\(250\) 0 0
\(251\) 279.162 + 279.162i 1.11220 + 1.11220i 0.992853 + 0.119347i \(0.0380802\pi\)
0.119347 + 0.992853i \(0.461920\pi\)
\(252\) 0 0
\(253\) 154.218 + 372.314i 0.609556 + 1.47160i
\(254\) 0 0
\(255\) −671.824 115.264i −2.63460 0.452014i
\(256\) 0 0
\(257\) 203.234 84.1822i 0.790793 0.327557i 0.0495308 0.998773i \(-0.484227\pi\)
0.741262 + 0.671215i \(0.234227\pi\)
\(258\) 0 0
\(259\) 260.001 260.001i 1.00387 1.00387i
\(260\) 0 0
\(261\) −50.7052 254.912i −0.194273 0.976676i
\(262\) 0 0
\(263\) 100.491 242.606i 0.382093 0.922455i −0.609467 0.792811i \(-0.708617\pi\)
0.991561 0.129644i \(-0.0413834\pi\)
\(264\) 0 0
\(265\) 401.719 601.216i 1.51592 2.26874i
\(266\) 0 0
\(267\) 275.273 + 411.975i 1.03098 + 1.54298i
\(268\) 0 0
\(269\) 345.233 + 68.6712i 1.28340 + 0.255283i 0.789237 0.614089i \(-0.210476\pi\)
0.494158 + 0.869372i \(0.335476\pi\)
\(270\) 0 0
\(271\) 69.2042i 0.255366i 0.991815 + 0.127683i \(0.0407540\pi\)
−0.991815 + 0.127683i \(0.959246\pi\)
\(272\) 0 0
\(273\) −416.459 −1.52549
\(274\) 0 0
\(275\) −100.202 + 503.751i −0.364372 + 1.83182i
\(276\) 0 0
\(277\) −206.761 + 138.153i −0.746430 + 0.498749i −0.869673 0.493628i \(-0.835671\pi\)
0.123243 + 0.992377i \(0.460671\pi\)
\(278\) 0 0
\(279\) −143.681 96.0044i −0.514985 0.344102i
\(280\) 0 0
\(281\) −279.217 115.656i −0.993656 0.411586i −0.174189 0.984712i \(-0.555730\pi\)
−0.819467 + 0.573126i \(0.805730\pi\)
\(282\) 0 0
\(283\) −210.437 + 41.8584i −0.743592 + 0.147910i −0.552326 0.833628i \(-0.686259\pi\)
−0.191267 + 0.981538i \(0.561259\pi\)
\(284\) 0 0
\(285\) 8.13653 + 8.13653i 0.0285492 + 0.0285492i
\(286\) 0 0
\(287\) −168.482 406.752i −0.587046 1.41725i
\(288\) 0 0
\(289\) −124.551 260.784i −0.430972 0.902365i
\(290\) 0 0
\(291\) −608.912 + 252.220i −2.09248 + 0.866735i
\(292\) 0 0
\(293\) −1.68954 + 1.68954i −0.00576636 + 0.00576636i −0.709984 0.704218i \(-0.751298\pi\)
0.704218 + 0.709984i \(0.251298\pi\)
\(294\) 0 0
\(295\) 121.267 + 609.652i 0.411076 + 2.06662i
\(296\) 0 0
\(297\) −73.1130 + 176.510i −0.246172 + 0.594311i
\(298\) 0 0
\(299\) 212.169 317.534i 0.709597 1.06199i
\(300\) 0 0
\(301\) 178.442 + 267.058i 0.592832 + 0.887236i
\(302\) 0 0
\(303\) 644.399 + 128.179i 2.12673 + 0.423033i
\(304\) 0 0
\(305\) 404.021i 1.32466i
\(306\) 0 0
\(307\) 2.94383 0.00958901 0.00479451 0.999989i \(-0.498474\pi\)
0.00479451 + 0.999989i \(0.498474\pi\)
\(308\) 0 0
\(309\) 92.8182 466.628i 0.300382 1.51012i
\(310\) 0 0
\(311\) −422.278 + 282.157i −1.35781 + 0.907257i −0.999651 0.0264056i \(-0.991594\pi\)
−0.358154 + 0.933662i \(0.616594\pi\)
\(312\) 0 0
\(313\) 309.366 + 206.712i 0.988391 + 0.660422i 0.940983 0.338454i \(-0.109904\pi\)
0.0474079 + 0.998876i \(0.484904\pi\)
\(314\) 0 0
\(315\) −905.155 374.928i −2.87351 1.19025i
\(316\) 0 0
\(317\) −141.716 + 28.1890i −0.447053 + 0.0889243i −0.413483 0.910512i \(-0.635688\pi\)
−0.0335701 + 0.999436i \(0.510688\pi\)
\(318\) 0 0
\(319\) −151.196 151.196i −0.473968 0.473968i
\(320\) 0 0
\(321\) −14.9408 36.0703i −0.0465446 0.112369i
\(322\) 0 0
\(323\) −0.824963 + 4.80837i −0.00255407 + 0.0148866i
\(324\) 0 0
\(325\) 449.683 186.265i 1.38364 0.573123i
\(326\) 0 0
\(327\) −274.822 + 274.822i −0.840434 + 0.840434i
\(328\) 0 0
\(329\) −69.5132 349.467i −0.211286 1.06221i
\(330\) 0 0
\(331\) 29.9649 72.3416i 0.0905283 0.218555i −0.872130 0.489274i \(-0.837262\pi\)
0.962658 + 0.270720i \(0.0872617\pi\)
\(332\) 0 0
\(333\) −295.789 + 442.680i −0.888255 + 1.32937i
\(334\) 0 0
\(335\) −223.622 334.674i −0.667529 0.999028i
\(336\) 0 0
\(337\) −477.505 94.9817i −1.41693 0.281845i −0.573553 0.819168i \(-0.694435\pi\)
−0.843376 + 0.537324i \(0.819435\pi\)
\(338\) 0 0
\(339\) 427.758i 1.26182i
\(340\) 0 0
\(341\) −142.164 −0.416903
\(342\) 0 0
\(343\) 33.0945 166.377i 0.0964855 0.485066i
\(344\) 0 0
\(345\) 1269.73 848.406i 3.68038 2.45915i
\(346\) 0 0
\(347\) −210.968 140.964i −0.607976 0.406236i 0.213121 0.977026i \(-0.431637\pi\)
−0.821096 + 0.570789i \(0.806637\pi\)
\(348\) 0 0
\(349\) 309.024 + 128.002i 0.885456 + 0.366768i 0.778610 0.627508i \(-0.215925\pi\)
0.106846 + 0.994276i \(0.465925\pi\)
\(350\) 0 0
\(351\) 177.574 35.3216i 0.505908 0.100631i
\(352\) 0 0
\(353\) 128.231 + 128.231i 0.363260 + 0.363260i 0.865012 0.501752i \(-0.167311\pi\)
−0.501752 + 0.865012i \(0.667311\pi\)
\(354\) 0 0
\(355\) −403.776 974.801i −1.13740 2.74592i
\(356\) 0 0
\(357\) −156.000 688.602i −0.436974 1.92886i
\(358\) 0 0
\(359\) 497.226 205.958i 1.38503 0.573699i 0.439210 0.898385i \(-0.355259\pi\)
0.945822 + 0.324686i \(0.105259\pi\)
\(360\) 0 0
\(361\) −255.207 + 255.207i −0.706945 + 0.706945i
\(362\) 0 0
\(363\) −8.24424 41.4466i −0.0227114 0.114178i
\(364\) 0 0
\(365\) −123.582 + 298.352i −0.338580 + 0.817403i
\(366\) 0 0
\(367\) 265.965 398.045i 0.724701 1.08459i −0.267932 0.963438i \(-0.586340\pi\)
0.992634 0.121155i \(-0.0386597\pi\)
\(368\) 0 0
\(369\) 354.165 + 530.046i 0.959798 + 1.43644i
\(370\) 0 0
\(371\) 734.580 + 146.117i 1.98000 + 0.393847i
\(372\) 0 0
\(373\) 93.1819i 0.249817i −0.992168 0.124909i \(-0.960136\pi\)
0.992168 0.124909i \(-0.0398638\pi\)
\(374\) 0 0
\(375\) 943.901 2.51707
\(376\) 0 0
\(377\) −39.5312 + 198.737i −0.104857 + 0.527153i
\(378\) 0 0
\(379\) −436.932 + 291.949i −1.15286 + 0.770313i −0.976818 0.214070i \(-0.931328\pi\)
−0.176037 + 0.984384i \(0.556328\pi\)
\(380\) 0 0
\(381\) −45.1450 30.1649i −0.118491 0.0791730i
\(382\) 0 0
\(383\) 166.702 + 69.0503i 0.435253 + 0.180288i 0.589542 0.807738i \(-0.299308\pi\)
−0.154288 + 0.988026i \(0.549308\pi\)
\(384\) 0 0
\(385\) −790.531 + 157.246i −2.05333 + 0.408432i
\(386\) 0 0
\(387\) −328.849 328.849i −0.849739 0.849739i
\(388\) 0 0
\(389\) 10.7102 + 25.8567i 0.0275326 + 0.0664696i 0.937048 0.349201i \(-0.113547\pi\)
−0.909515 + 0.415671i \(0.863547\pi\)
\(390\) 0 0
\(391\) 604.508 + 231.872i 1.54606 + 0.593022i
\(392\) 0 0
\(393\) −546.608 + 226.412i −1.39086 + 0.576113i
\(394\) 0 0
\(395\) −273.867 + 273.867i −0.693335 + 0.693335i
\(396\) 0 0
\(397\) −77.3968 389.100i −0.194954 0.980101i −0.947060 0.321058i \(-0.895962\pi\)
0.752105 0.659043i \(-0.229038\pi\)
\(398\) 0 0
\(399\) −4.56116 + 11.0116i −0.0114315 + 0.0275980i
\(400\) 0 0
\(401\) −321.659 + 481.397i −0.802142 + 1.20049i 0.174296 + 0.984693i \(0.444235\pi\)
−0.976438 + 0.215798i \(0.930765\pi\)
\(402\) 0 0
\(403\) 74.8476 + 112.017i 0.185726 + 0.277959i
\(404\) 0 0
\(405\) −263.528 52.4190i −0.650687 0.129430i
\(406\) 0 0
\(407\) 438.006i 1.07618i
\(408\) 0 0
\(409\) 683.623 1.67145 0.835725 0.549149i \(-0.185048\pi\)
0.835725 + 0.549149i \(0.185048\pi\)
\(410\) 0 0
\(411\) 84.2797 423.703i 0.205060 1.03091i
\(412\) 0 0
\(413\) −535.348 + 357.708i −1.29624 + 0.866121i
\(414\) 0 0
\(415\) −309.651 206.902i −0.746147 0.498559i
\(416\) 0 0
\(417\) −2.88155 1.19358i −0.00691019 0.00286230i
\(418\) 0 0
\(419\) −409.937 + 81.5416i −0.978370 + 0.194610i −0.658277 0.752776i \(-0.728714\pi\)
−0.320094 + 0.947386i \(0.603714\pi\)
\(420\) 0 0
\(421\) −36.5361 36.5361i −0.0867840 0.0867840i 0.662382 0.749166i \(-0.269546\pi\)
−0.749166 + 0.662382i \(0.769546\pi\)
\(422\) 0 0
\(423\) 197.435 + 476.650i 0.466749 + 1.12683i
\(424\) 0 0
\(425\) 476.428 + 673.765i 1.12101 + 1.58533i
\(426\) 0 0
\(427\) −386.634 + 160.149i −0.905466 + 0.375056i
\(428\) 0 0
\(429\) 350.790 350.790i 0.817692 0.817692i
\(430\) 0 0
\(431\) −4.56652 22.9574i −0.0105952 0.0532655i 0.975126 0.221651i \(-0.0711446\pi\)
−0.985721 + 0.168386i \(0.946145\pi\)
\(432\) 0 0
\(433\) 135.415 326.920i 0.312736 0.755012i −0.686866 0.726785i \(-0.741014\pi\)
0.999602 0.0282270i \(-0.00898613\pi\)
\(434\) 0 0
\(435\) −450.157 + 673.707i −1.03484 + 1.54875i
\(436\) 0 0
\(437\) −6.07220 9.08769i −0.0138952 0.0207956i
\(438\) 0 0
\(439\) −224.617 44.6790i −0.511655 0.101775i −0.0674908 0.997720i \(-0.521499\pi\)
−0.444164 + 0.895945i \(0.646499\pi\)
\(440\) 0 0
\(441\) 384.598i 0.872103i
\(442\) 0 0
\(443\) −185.983 −0.419827 −0.209914 0.977720i \(-0.567318\pi\)
−0.209914 + 0.977720i \(0.567318\pi\)
\(444\) 0 0
\(445\) 177.289 891.293i 0.398403 2.00291i
\(446\) 0 0
\(447\) −244.461 + 163.343i −0.546892 + 0.365421i
\(448\) 0 0
\(449\) −187.951 125.585i −0.418600 0.279700i 0.328380 0.944546i \(-0.393497\pi\)
−0.746981 + 0.664846i \(0.768497\pi\)
\(450\) 0 0
\(451\) 484.529 + 200.698i 1.07434 + 0.445007i
\(452\) 0 0
\(453\) −33.1383 + 6.59162i −0.0731530 + 0.0145510i
\(454\) 0 0
\(455\) 540.107 + 540.107i 1.18705 + 1.18705i
\(456\) 0 0
\(457\) 291.902 + 704.713i 0.638734 + 1.54204i 0.828367 + 0.560186i \(0.189270\pi\)
−0.189633 + 0.981855i \(0.560730\pi\)
\(458\) 0 0
\(459\) 124.920 + 280.381i 0.272156 + 0.610853i
\(460\) 0 0
\(461\) −610.198 + 252.752i −1.32364 + 0.548269i −0.928834 0.370495i \(-0.879188\pi\)
−0.394805 + 0.918765i \(0.629188\pi\)
\(462\) 0 0
\(463\) 284.548 284.548i 0.614574 0.614574i −0.329561 0.944134i \(-0.606901\pi\)
0.944134 + 0.329561i \(0.106901\pi\)
\(464\) 0 0
\(465\) 105.098 + 528.365i 0.226018 + 1.13627i
\(466\) 0 0
\(467\) −12.9528 + 31.2708i −0.0277362 + 0.0669611i −0.937141 0.348952i \(-0.886538\pi\)
0.909405 + 0.415913i \(0.136538\pi\)
\(468\) 0 0
\(469\) 231.631 346.660i 0.493882 0.739147i
\(470\) 0 0
\(471\) −193.181 289.115i −0.410150 0.613833i
\(472\) 0 0
\(473\) −375.252 74.6422i −0.793344 0.157806i
\(474\) 0 0
\(475\) 13.9301i 0.0293266i
\(476\) 0 0
\(477\) −1084.47 −2.27352
\(478\) 0 0
\(479\) −131.861 + 662.909i −0.275283 + 1.38394i 0.557423 + 0.830229i \(0.311790\pi\)
−0.832707 + 0.553715i \(0.813210\pi\)
\(480\) 0 0
\(481\) 345.125 230.605i 0.717515 0.479428i
\(482\) 0 0
\(483\) 1315.20 + 878.791i 2.72299 + 1.81944i
\(484\) 0 0
\(485\) 1116.80 + 462.595i 2.30269 + 0.953805i
\(486\) 0 0
\(487\) 328.059 65.2551i 0.673633 0.133994i 0.153591 0.988135i \(-0.450916\pi\)
0.520042 + 0.854141i \(0.325916\pi\)
\(488\) 0 0
\(489\) −111.689 111.689i −0.228403 0.228403i
\(490\) 0 0
\(491\) −9.18393 22.1720i −0.0187045 0.0451568i 0.914251 0.405148i \(-0.132780\pi\)
−0.932956 + 0.359991i \(0.882780\pi\)
\(492\) 0 0
\(493\) −343.413 + 9.08040i −0.696577 + 0.0184187i
\(494\) 0 0
\(495\) 1078.23 446.619i 2.17825 0.902261i
\(496\) 0 0
\(497\) 772.800 772.800i 1.55493 1.55493i
\(498\) 0 0
\(499\) −3.25477 16.3628i −0.00652258 0.0327912i 0.977387 0.211459i \(-0.0678214\pi\)
−0.983910 + 0.178667i \(0.942821\pi\)
\(500\) 0 0
\(501\) 169.110 408.269i 0.337546 0.814907i
\(502\) 0 0
\(503\) −513.276 + 768.172i −1.02043 + 1.52718i −0.181168 + 0.983452i \(0.557988\pi\)
−0.839261 + 0.543728i \(0.817012\pi\)
\(504\) 0 0
\(505\) −669.487 1001.96i −1.32572 1.98408i
\(506\) 0 0
\(507\) 313.912 + 62.4409i 0.619155 + 0.123158i
\(508\) 0 0
\(509\) 788.802i 1.54971i −0.632139 0.774855i \(-0.717823\pi\)
0.632139 0.774855i \(-0.282177\pi\)
\(510\) 0 0
\(511\) −334.499 −0.654598
\(512\) 0 0
\(513\) 1.01089 5.08208i 0.00197054 0.00990660i
\(514\) 0 0
\(515\) −725.547 + 484.795i −1.40883 + 0.941350i
\(516\) 0 0
\(517\) 352.913 + 235.809i 0.682618 + 0.456110i
\(518\) 0 0
\(519\) −976.256 404.379i −1.88103 0.779150i
\(520\) 0 0
\(521\) −762.882 + 151.747i −1.46427 + 0.291261i −0.861948 0.506996i \(-0.830756\pi\)
−0.602317 + 0.798257i \(0.705756\pi\)
\(522\) 0 0
\(523\) −429.923 429.923i −0.822032 0.822032i 0.164367 0.986399i \(-0.447442\pi\)
−0.986399 + 0.164367i \(0.947442\pi\)
\(524\) 0 0
\(525\) 771.496 + 1862.56i 1.46952 + 3.54773i
\(526\) 0 0
\(527\) −157.180 + 165.718i −0.298255 + 0.314456i
\(528\) 0 0
\(529\) −851.356 + 352.643i −1.60937 + 0.666622i
\(530\) 0 0
\(531\) 659.215 659.215i 1.24146 1.24146i
\(532\) 0 0
\(533\) −96.9593 487.447i −0.181912 0.914535i
\(534\) 0 0
\(535\) −27.4029 + 66.1565i −0.0512204 + 0.123657i
\(536\) 0 0
\(537\) −257.852 + 385.903i −0.480172 + 0.718628i
\(538\) 0 0
\(539\) −175.785 263.081i −0.326132 0.488092i
\(540\) 0 0
\(541\) 539.441 + 107.301i 0.997118 + 0.198339i 0.666564 0.745448i \(-0.267764\pi\)
0.330554 + 0.943787i \(0.392764\pi\)
\(542\) 0 0
\(543\) 395.905i 0.729106i
\(544\) 0 0
\(545\) 712.834 1.30795
\(546\) 0 0
\(547\) −85.1659 + 428.158i −0.155696 + 0.782739i 0.821468 + 0.570255i \(0.193155\pi\)
−0.977164 + 0.212484i \(0.931845\pi\)
\(548\) 0 0
\(549\) 503.830 336.648i 0.917723 0.613203i
\(550\) 0 0
\(551\) 4.82185 + 3.22186i 0.00875109 + 0.00584729i
\(552\) 0 0
\(553\) −370.640 153.524i −0.670235 0.277620i
\(554\) 0 0
\(555\) 1627.89 323.807i 2.93313 0.583437i
\(556\) 0 0
\(557\) −148.713 148.713i −0.266990 0.266990i 0.560896 0.827886i \(-0.310457\pi\)
−0.827886 + 0.560896i \(0.810457\pi\)
\(558\) 0 0
\(559\) 138.752 + 334.976i 0.248214 + 0.599242i
\(560\) 0 0
\(561\) 711.421 + 448.619i 1.26813 + 0.799678i
\(562\) 0 0
\(563\) 747.065 309.445i 1.32694 0.549635i 0.397157 0.917751i \(-0.369997\pi\)
0.929780 + 0.368116i \(0.119997\pi\)
\(564\) 0 0
\(565\) −554.760 + 554.760i −0.981876 + 0.981876i
\(566\) 0 0
\(567\) −54.2963 272.966i −0.0957607 0.481422i
\(568\) 0 0
\(569\) 250.724 605.300i 0.440639 1.06380i −0.535086 0.844798i \(-0.679721\pi\)
0.975725 0.218999i \(-0.0702792\pi\)
\(570\) 0 0
\(571\) 264.956 396.535i 0.464021 0.694456i −0.523486 0.852034i \(-0.675369\pi\)
0.987507 + 0.157578i \(0.0503686\pi\)
\(572\) 0 0
\(573\) −746.897 1117.81i −1.30349 1.95080i
\(574\) 0 0
\(575\) −1813.17 360.663i −3.15335 0.627240i
\(576\) 0 0
\(577\) 361.492i 0.626503i 0.949670 + 0.313252i \(0.101418\pi\)
−0.949670 + 0.313252i \(0.898582\pi\)
\(578\) 0 0
\(579\) 458.042 0.791091
\(580\) 0 0
\(581\) 75.2564 378.339i 0.129529 0.651186i
\(582\) 0 0
\(583\) −741.825 + 495.672i −1.27243 + 0.850209i
\(584\) 0 0
\(585\) −919.588 614.449i −1.57195 1.05034i
\(586\) 0 0
\(587\) −580.651 240.513i −0.989184 0.409733i −0.171364 0.985208i \(-0.554817\pi\)
−0.817820 + 0.575475i \(0.804817\pi\)
\(588\) 0 0
\(589\) 3.78160 0.752208i 0.00642038 0.00127709i
\(590\) 0 0
\(591\) −689.195 689.195i −1.16615 1.16615i
\(592\) 0 0
\(593\) −240.296 580.125i −0.405220 0.978288i −0.986378 0.164497i \(-0.947400\pi\)
0.581158 0.813791i \(-0.302600\pi\)
\(594\) 0 0
\(595\) −690.733 + 1095.37i −1.16090 + 1.84095i
\(596\) 0 0
\(597\) −291.389 + 120.697i −0.488089 + 0.202173i
\(598\) 0 0
\(599\) 340.674 340.674i 0.568738 0.568738i −0.363037 0.931775i \(-0.618260\pi\)
0.931775 + 0.363037i \(0.118260\pi\)
\(600\) 0 0
\(601\) 187.466 + 942.455i 0.311924 + 1.56815i 0.745173 + 0.666871i \(0.232367\pi\)
−0.433249 + 0.901274i \(0.642633\pi\)
\(602\) 0 0
\(603\) −231.020 + 557.732i −0.383118 + 0.924928i
\(604\) 0 0
\(605\) −43.0602 + 64.4442i −0.0711739 + 0.106519i
\(606\) 0 0
\(607\) −47.5651 71.1862i −0.0783609 0.117275i 0.790219 0.612825i \(-0.209967\pi\)
−0.868580 + 0.495549i \(0.834967\pi\)
\(608\) 0 0
\(609\) −823.153 163.735i −1.35165 0.268859i
\(610\) 0 0
\(611\) 402.227i 0.658309i
\(612\) 0 0
\(613\) 486.131 0.793035 0.396518 0.918027i \(-0.370219\pi\)
0.396518 + 0.918027i \(0.370219\pi\)
\(614\) 0 0
\(615\) 387.713 1949.17i 0.630428 3.16938i
\(616\) 0 0
\(617\) 294.487 196.770i 0.477289 0.318914i −0.293551 0.955943i \(-0.594837\pi\)
0.770839 + 0.637029i \(0.219837\pi\)
\(618\) 0 0
\(619\) −176.878 118.186i −0.285747 0.190930i 0.404433 0.914568i \(-0.367469\pi\)
−0.690180 + 0.723637i \(0.742469\pi\)
\(620\) 0 0
\(621\) −635.322 263.159i −1.02306 0.423767i
\(622\) 0 0
\(623\) 923.214 183.639i 1.48188 0.294765i
\(624\) 0 0
\(625\) −366.060 366.060i −0.585696 0.585696i
\(626\) 0 0
\(627\) −5.43332 13.1172i −0.00866558 0.0209206i
\(628\) 0 0
\(629\) 510.577 + 484.272i 0.811728 + 0.769907i
\(630\) 0 0
\(631\) 194.893 80.7272i 0.308863 0.127935i −0.222868 0.974849i \(-0.571542\pi\)
0.531731 + 0.846913i \(0.321542\pi\)
\(632\) 0 0
\(633\) −104.888 + 104.888i −0.165700 + 0.165700i
\(634\) 0 0
\(635\) 19.4277 + 97.6696i 0.0305948 + 0.153810i
\(636\) 0 0
\(637\) −114.745 + 277.018i −0.180133 + 0.434880i
\(638\) 0 0
\(639\) −879.172 + 1315.77i −1.37586 + 2.05911i
\(640\) 0 0
\(641\) 428.730 + 641.640i 0.668846 + 1.00100i 0.998382 + 0.0568559i \(0.0181076\pi\)
−0.329536 + 0.944143i \(0.606892\pi\)
\(642\) 0 0
\(643\) 765.226 + 152.213i 1.19009 + 0.236723i 0.750126 0.661294i \(-0.229993\pi\)
0.439960 + 0.898017i \(0.354993\pi\)
\(644\) 0 0
\(645\) 1449.84i 2.24781i
\(646\) 0 0
\(647\) 381.431 0.589538 0.294769 0.955568i \(-0.404757\pi\)
0.294769 + 0.955568i \(0.404757\pi\)
\(648\) 0 0
\(649\) 149.629 752.235i 0.230553 1.15907i
\(650\) 0 0
\(651\) −463.968 + 310.013i −0.712700 + 0.476211i
\(652\) 0 0
\(653\) 404.364 + 270.188i 0.619241 + 0.413764i 0.825235 0.564790i \(-0.191043\pi\)
−0.205994 + 0.978553i \(0.566043\pi\)
\(654\) 0 0
\(655\) 1002.53 + 415.262i 1.53058 + 0.633988i
\(656\) 0 0
\(657\) 475.031 94.4895i 0.723030 0.143820i
\(658\) 0 0
\(659\) 343.801 + 343.801i 0.521702 + 0.521702i 0.918085 0.396383i \(-0.129735\pi\)
−0.396383 + 0.918085i \(0.629735\pi\)
\(660\) 0 0
\(661\) −228.858 552.512i −0.346230 0.835873i −0.997058 0.0766480i \(-0.975578\pi\)
0.650828 0.759225i \(-0.274422\pi\)
\(662\) 0 0
\(663\) −21.0675 796.753i −0.0317760 1.20174i
\(664\) 0 0
\(665\) 20.1964 8.36561i 0.0303705 0.0125799i
\(666\) 0 0
\(667\) 544.206 544.206i 0.815900 0.815900i
\(668\) 0 0
\(669\) 181.825 + 914.097i 0.271787 + 1.36636i
\(670\) 0 0
\(671\) 190.772 460.564i 0.284310 0.686384i
\(672\) 0 0
\(673\) 517.949 775.165i 0.769612 1.15181i −0.214926 0.976630i \(-0.568951\pi\)
0.984537 0.175175i \(-0.0560491\pi\)
\(674\) 0 0
\(675\) −486.929 728.740i −0.721376 1.07962i
\(676\) 0 0
\(677\) −1205.69 239.827i −1.78093 0.354250i −0.808693 0.588231i \(-0.799825\pi\)
−0.972241 + 0.233981i \(0.924825\pi\)
\(678\) 0 0
\(679\) 1252.11i 1.84405i
\(680\) 0 0
\(681\) 724.688 1.06415
\(682\) 0 0
\(683\) 234.044 1176.62i 0.342670 1.72272i −0.297707 0.954657i \(-0.596222\pi\)
0.640377 0.768061i \(-0.278778\pi\)
\(684\) 0 0
\(685\) −658.803 + 440.198i −0.961757 + 0.642625i
\(686\) 0 0
\(687\) −1007.14 672.953i −1.46600 0.979553i
\(688\) 0 0
\(689\) 781.124 + 323.552i 1.13371 + 0.469597i
\(690\) 0 0
\(691\) −342.624 + 68.1522i −0.495838 + 0.0986284i −0.436674 0.899620i \(-0.643844\pi\)
−0.0591644 + 0.998248i \(0.518844\pi\)
\(692\) 0 0
\(693\) 854.799 + 854.799i 1.23348 + 1.23348i
\(694\) 0 0
\(695\) 2.18914 + 5.28504i 0.00314984 + 0.00760438i
\(696\) 0 0
\(697\) 769.659 342.910i 1.10425 0.491980i
\(698\) 0 0
\(699\) −407.753 + 168.897i −0.583337 + 0.241626i
\(700\) 0 0
\(701\) −206.308 + 206.308i −0.294306 + 0.294306i −0.838778 0.544473i \(-0.816730\pi\)
0.544473 + 0.838778i \(0.316730\pi\)
\(702\) 0 0
\(703\) −2.31755 11.6511i −0.00329666 0.0165734i
\(704\) 0 0
\(705\) 615.506 1485.96i 0.873057 2.10775i
\(706\) 0 0
\(707\) 693.464 1037.84i 0.980854 1.46795i
\(708\) 0 0
\(709\) 560.408 + 838.709i 0.790420 + 1.18295i 0.979591 + 0.201003i \(0.0644199\pi\)
−0.189171 + 0.981944i \(0.560580\pi\)
\(710\) 0 0
\(711\) 569.722 + 113.325i 0.801297 + 0.159388i
\(712\) 0 0
\(713\) 511.697i 0.717668i
\(714\) 0 0
\(715\) −909.880 −1.27256
\(716\) 0 0
\(717\) −221.875 + 1115.44i −0.309449 + 1.55570i
\(718\) 0 0
\(719\) −156.675 + 104.687i −0.217907 + 0.145601i −0.659733 0.751500i \(-0.729331\pi\)
0.441826 + 0.897101i \(0.354331\pi\)
\(720\) 0 0
\(721\) −751.532 502.157i −1.04235 0.696474i
\(722\) 0 0
\(723\) −1265.30 524.104i −1.75007 0.724902i
\(724\) 0 0
\(725\) 962.054 191.364i 1.32697 0.263951i
\(726\) 0 0
\(727\) 193.501 + 193.501i 0.266164 + 0.266164i 0.827552 0.561388i \(-0.189733\pi\)
−0.561388 + 0.827552i \(0.689733\pi\)
\(728\) 0 0
\(729\) 444.981 + 1074.28i 0.610400 + 1.47363i
\(730\) 0 0
\(731\) −501.898 + 354.899i −0.686591 + 0.485498i
\(732\) 0 0
\(733\) 333.378 138.090i 0.454814 0.188390i −0.143503 0.989650i \(-0.545837\pi\)
0.598316 + 0.801260i \(0.295837\pi\)
\(734\) 0 0
\(735\) −847.812 + 847.812i −1.15349 + 1.15349i
\(736\) 0 0
\(737\) 96.8909 + 487.103i 0.131467 + 0.660927i
\(738\) 0 0
\(739\) −109.129 + 263.461i −0.147672 + 0.356511i −0.980356 0.197237i \(-0.936803\pi\)
0.832684 + 0.553748i \(0.186803\pi\)
\(740\) 0 0
\(741\) −7.47505 + 11.1872i −0.0100878 + 0.0150974i
\(742\) 0 0
\(743\) 371.396 + 555.834i 0.499860 + 0.748094i 0.992514 0.122132i \(-0.0389731\pi\)
−0.492654 + 0.870225i \(0.663973\pi\)
\(744\) 0 0
\(745\) 528.882 + 105.201i 0.709908 + 0.141210i
\(746\) 0 0
\(747\) 558.547i 0.747721i
\(748\) 0 0
\(749\) −74.1717 −0.0990277
\(750\) 0 0
\(751\) −100.375 + 504.621i −0.133655 + 0.671932i 0.854620 + 0.519253i \(0.173790\pi\)
−0.988276 + 0.152678i \(0.951210\pi\)
\(752\) 0 0
\(753\) −1534.83 + 1025.54i −2.03828 + 1.36194i
\(754\) 0 0
\(755\) 51.5258 + 34.4285i 0.0682461 + 0.0456006i
\(756\) 0 0
\(757\) −818.393 338.989i −1.08110 0.447806i −0.230205 0.973142i \(-0.573940\pi\)
−0.850895 + 0.525336i \(0.823940\pi\)
\(758\) 0 0
\(759\) −1848.03 + 367.597i −2.43483 + 0.484317i
\(760\) 0 0
\(761\) −104.490 104.490i −0.137307 0.137307i 0.635113 0.772419i \(-0.280954\pi\)
−0.772419 + 0.635113i \(0.780954\pi\)
\(762\) 0 0
\(763\) 282.559 + 682.159i 0.370327 + 0.894048i
\(764\) 0 0
\(765\) 671.508 1750.67i 0.877788 2.28846i
\(766\) 0 0
\(767\) −671.497 + 278.143i −0.875486 + 0.362638i
\(768\) 0 0
\(769\) 252.375 252.375i 0.328185 0.328185i −0.523711 0.851896i \(-0.675453\pi\)
0.851896 + 0.523711i \(0.175453\pi\)
\(770\) 0 0
\(771\) 200.659 + 1008.78i 0.260258 + 1.30840i
\(772\) 0 0
\(773\) −248.266 + 599.367i −0.321172 + 0.775378i 0.678014 + 0.735049i \(0.262841\pi\)
−0.999186 + 0.0403292i \(0.987159\pi\)
\(774\) 0 0
\(775\) 362.326 542.259i 0.467517 0.699689i
\(776\) 0 0
\(777\) 955.150 + 1429.48i 1.22928 + 1.83975i
\(778\) 0 0
\(779\) −13.9505 2.77494i −0.0179083 0.00356218i
\(780\) 0 0
\(781\) 1301.88i 1.66694i
\(782\) 0 0
\(783\) 364.870 0.465990
\(784\) 0 0
\(785\) −124.418 + 625.491i −0.158494 + 0.796804i
\(786\) 0 0
\(787\) 1062.94 710.234i 1.35062 0.902457i 0.351195 0.936302i \(-0.385775\pi\)
0.999427 + 0.0338454i \(0.0107754\pi\)
\(788\) 0 0
\(789\) 1020.88 + 682.129i 1.29389 + 0.864549i
\(790\) 0 0
\(791\) −750.788 310.986i −0.949163 0.393156i
\(792\) 0 0
\(793\) −463.338 + 92.1637i −0.584285 + 0.116222i
\(794\) 0 0
\(795\) 2390.62 + 2390.62i 3.00707 + 3.00707i
\(796\) 0 0
\(797\) −384.664 928.661i −0.482640 1.16520i −0.958351 0.285594i \(-0.907809\pi\)
0.475711 0.879602i \(-0.342191\pi\)
\(798\) 0 0
\(799\) 665.070 150.669i 0.832378 0.188571i
\(800\) 0 0
\(801\) −1259.20 + 521.579i −1.57204 + 0.651160i
\(802\) 0 0
\(803\) 281.754 281.754i 0.350877 0.350877i
\(804\) 0 0
\(805\) −565.984 2845.39i −0.703086 3.53465i
\(806\) 0 0
\(807\) −629.826 + 1520.53i −0.780453 + 1.88418i
\(808\) 0 0
\(809\) −497.857 + 745.095i −0.615398 + 0.921008i −0.999998 0.00207569i \(-0.999339\pi\)
0.384600 + 0.923083i \(0.374339\pi\)
\(810\) 0 0
\(811\) −12.6263 18.8967i −0.0155689 0.0233005i 0.823604 0.567166i \(-0.191960\pi\)
−0.839172 + 0.543865i \(0.816960\pi\)
\(812\) 0 0
\(813\) −317.357 63.1262i −0.390353 0.0776461i
\(814\) 0 0
\(815\) 289.699i 0.355459i
\(816\) 0 0
\(817\) 10.3768 0.0127011
\(818\) 0 0
\(819\) 223.493 1123.58i 0.272885 1.37189i
\(820\) 0 0
\(821\) 865.321 578.189i 1.05398 0.704250i 0.0972639 0.995259i \(-0.468991\pi\)
0.956720 + 0.291009i \(0.0939909\pi\)
\(822\) 0 0
\(823\) 303.928 + 203.078i 0.369293 + 0.246753i 0.726344 0.687331i \(-0.241218\pi\)
−0.357052 + 0.934085i \(0.616218\pi\)
\(824\) 0 0
\(825\) −2218.70 919.017i −2.68934 1.11396i
\(826\) 0 0
\(827\) 1447.23 287.873i 1.74998 0.348093i 0.786854 0.617139i \(-0.211708\pi\)
0.963127 + 0.269046i \(0.0867085\pi\)
\(828\) 0 0
\(829\) 343.531 + 343.531i 0.414392 + 0.414392i 0.883265 0.468874i \(-0.155340\pi\)
−0.468874 + 0.883265i \(0.655340\pi\)
\(830\) 0 0
\(831\) −444.943 1074.19i −0.535431 1.29264i
\(832\) 0 0
\(833\) −501.023 85.9597i −0.601468 0.103193i
\(834\) 0 0
\(835\) −748.804 + 310.165i −0.896771 + 0.371455i
\(836\) 0 0
\(837\) 171.537 171.537i 0.204943 0.204943i
\(838\) 0 0
\(839\) −209.886 1055.17i −0.250163 1.25765i −0.877755 0.479110i \(-0.840959\pi\)
0.627592 0.778542i \(-0.284041\pi\)
\(840\) 0 0
\(841\) 165.566 399.712i 0.196868 0.475282i
\(842\) 0 0
\(843\) 785.069 1174.94i 0.931280 1.39376i
\(844\) 0 0
\(845\) −326.133 488.092i −0.385956 0.577624i
\(846\) 0 0
\(847\) −78.7395 15.6623i −0.0929628 0.0184915i
\(848\) 0 0
\(849\) 1003.20i 1.18163i
\(850\) 0 0
\(851\) −1576.54 −1.85257
\(852\) 0 0
\(853\) 311.200 1564.51i 0.364830 1.83412i −0.165345 0.986236i \(-0.552874\pi\)
0.530175 0.847888i \(-0.322126\pi\)
\(854\) 0 0
\(855\) −26.3182 + 17.5853i −0.0307816 + 0.0205676i
\(856\) 0 0
\(857\) 542.902 + 362.755i 0.633491 + 0.423285i 0.830424 0.557132i \(-0.188098\pi\)
−0.196933 + 0.980417i \(0.563098\pi\)
\(858\) 0 0
\(859\) 178.719 + 74.0278i 0.208055 + 0.0861791i 0.484277 0.874915i \(-0.339083\pi\)
−0.276222 + 0.961094i \(0.589083\pi\)
\(860\) 0 0
\(861\) 2018.97 401.599i 2.34492 0.466433i
\(862\) 0 0
\(863\) 252.782 + 252.782i 0.292911 + 0.292911i 0.838229 0.545318i \(-0.183591\pi\)
−0.545318 + 0.838229i \(0.683591\pi\)
\(864\) 0 0
\(865\) 741.669 + 1790.55i 0.857421 + 2.07000i
\(866\) 0 0
\(867\) 1309.52 333.287i 1.51040 0.384414i
\(868\) 0 0
\(869\) 441.511 182.880i 0.508068 0.210449i
\(870\) 0 0
\(871\) 332.799 332.799i 0.382088 0.382088i
\(872\) 0 0
\(873\) −353.697 1778.16i −0.405151 2.03683i
\(874\) 0 0
\(875\) 686.230 1656.71i 0.784263 1.89338i
\(876\) 0 0
\(877\) 178.790 267.578i 0.203865 0.305106i −0.715422 0.698692i \(-0.753766\pi\)
0.919288 + 0.393586i \(0.128766\pi\)
\(878\) 0 0
\(879\) −6.20676 9.28908i −0.00706116 0.0105678i
\(880\) 0 0
\(881\) 490.124 + 97.4918i 0.556327 + 0.110660i 0.465243 0.885183i \(-0.345967\pi\)
0.0910839 + 0.995843i \(0.470967\pi\)
\(882\) 0 0
\(883\) 1165.97i 1.32046i 0.751063 + 0.660231i \(0.229541\pi\)
−0.751063 + 0.660231i \(0.770459\pi\)
\(884\) 0 0
\(885\) −2906.36 −3.28403
\(886\) 0 0
\(887\) 226.028 1136.32i 0.254823 1.28108i −0.615317 0.788280i \(-0.710972\pi\)
0.870140 0.492804i \(-0.164028\pi\)
\(888\) 0 0
\(889\) −85.7656 + 57.3067i −0.0964742 + 0.0644620i
\(890\) 0 0
\(891\) 275.658 + 184.189i 0.309381 + 0.206722i
\(892\) 0 0
\(893\) −10.6353 4.40529i −0.0119096 0.00493313i
\(894\) 0 0
\(895\) 834.887 166.069i 0.932835 0.185552i
\(896\) 0 0
\(897\) 1262.61 + 1262.61i 1.40760 + 1.40760i
\(898\) 0 0
\(899\) 103.899 + 250.835i 0.115572 + 0.279016i
\(900\) 0 0
\(901\) −242.385 + 1412.76i −0.269018 + 1.56799i
\(902\) 0 0
\(903\) −1387.45 + 574.699i −1.53649 + 0.636433i
\(904\) 0 0
\(905\) −513.450 + 513.450i −0.567348 + 0.567348i
\(906\) 0 0
\(907\) −228.715 1149.83i −0.252167 1.26773i −0.874519 0.484991i \(-0.838823\pi\)
0.622352 0.782737i \(-0.286177\pi\)
\(908\) 0 0
\(909\) −691.635 + 1669.75i −0.760874 + 1.83691i
\(910\) 0 0
\(911\) 604.823 905.182i 0.663912 0.993614i −0.334771 0.942299i \(-0.608659\pi\)
0.998683 0.0513143i \(-0.0163410\pi\)
\(912\) 0 0
\(913\) 255.291 + 382.071i 0.279618 + 0.418478i
\(914\) 0 0
\(915\) −1852.76 368.537i −2.02487 0.402773i
\(916\) 0 0
\(917\) 1123.99i 1.22573i
\(918\) 0 0
\(919\) −261.705 −0.284771 −0.142386 0.989811i \(-0.545477\pi\)
−0.142386 + 0.989811i \(0.545477\pi\)
\(920\) 0 0
\(921\) −2.68528 + 13.4998i −0.00291562 + 0.0146578i
\(922\) 0 0
\(923\) 1025.81 685.426i 1.11139 0.742606i
\(924\) 0 0
\(925\) −1670.70 1116.32i −1.80616 1.20684i
\(926\) 0 0
\(927\) 1209.12 + 500.833i 1.30433 + 0.540273i
\(928\) 0 0
\(929\) −1410.63 + 280.591i −1.51844 + 0.302036i −0.882725 0.469891i \(-0.844293\pi\)
−0.635712 + 0.771926i \(0.719293\pi\)
\(930\) 0 0
\(931\) 6.06795 + 6.06795i 0.00651767 + 0.00651767i
\(932\) 0 0
\(933\) −908.727 2193.86i −0.973983 2.35140i
\(934\) 0 0
\(935\) −340.828 1504.46i −0.364522 1.60905i
\(936\) 0 0
\(937\) 1265.07 524.009i 1.35013 0.559241i 0.413799 0.910368i \(-0.364202\pi\)
0.936328 + 0.351128i \(0.114202\pi\)
\(938\) 0 0
\(939\) −1230.14 + 1230.14i −1.31005 + 1.31005i
\(940\) 0 0
\(941\) 119.823 + 602.392i 0.127336 + 0.640162i 0.990754 + 0.135672i \(0.0433194\pi\)
−0.863418 + 0.504490i \(0.831681\pi\)
\(942\) 0 0
\(943\) −722.383 + 1743.99i −0.766048 + 1.84940i
\(944\) 0 0
\(945\) 764.132 1143.60i 0.808605 1.21016i
\(946\) 0 0
\(947\) 761.297 + 1139.36i 0.803904 + 1.20313i 0.975941 + 0.218037i \(0.0699652\pi\)
−0.172037 + 0.985091i \(0.555035\pi\)
\(948\) 0 0
\(949\) −370.347 73.6666i −0.390250 0.0776255i
\(950\) 0 0
\(951\) 675.594i 0.710404i
\(952\) 0 0
\(953\) −1182.43 −1.24075 −0.620373 0.784307i \(-0.713019\pi\)
−0.620373 + 0.784307i \(0.713019\pi\)
\(954\) 0 0
\(955\) −481.038 + 2418.34i −0.503705 + 2.53230i
\(956\) 0 0
\(957\) 831.271 555.437i 0.868622 0.580394i
\(958\) 0 0
\(959\) −682.397 455.963i −0.711572 0.475457i
\(960\) 0 0
\(961\) −721.076 298.680i −0.750339 0.310801i
\(962\) 0 0
\(963\) 105.333 20.9521i 0.109380 0.0217571i
\(964\) 0 0
\(965\) −594.035 594.035i −0.615581 0.615581i
\(966\) 0 0
\(967\) −611.666 1476.69i −0.632539 1.52709i −0.836420 0.548089i \(-0.815355\pi\)
0.203880 0.978996i \(-0.434645\pi\)
\(968\) 0 0
\(969\) −21.2977 8.16919i −0.0219791 0.00843054i
\(970\) 0 0
\(971\) −619.842 + 256.747i −0.638354 + 0.264415i −0.678298 0.734787i \(-0.737282\pi\)
0.0399436 + 0.999202i \(0.487282\pi\)
\(972\) 0 0
\(973\) −4.18986 + 4.18986i −0.00430613 + 0.00430613i
\(974\) 0 0
\(975\) 443.986 + 2232.07i 0.455370 + 2.28930i
\(976\) 0 0
\(977\) 625.620 1510.38i 0.640348 1.54594i −0.185864 0.982576i \(-0.559508\pi\)
0.826211 0.563360i \(-0.190492\pi\)
\(978\) 0 0
\(979\) −622.955 + 932.319i −0.636318 + 0.952317i
\(980\) 0 0
\(981\) −593.966 888.933i −0.605470 0.906150i
\(982\) 0 0
\(983\) −569.258 113.232i −0.579103 0.115191i −0.103157 0.994665i \(-0.532894\pi\)
−0.475946 + 0.879474i \(0.657894\pi\)
\(984\) 0 0
\(985\) 1787.64i 1.81486i
\(986\) 0 0
\(987\) 1666.00 1.68794
\(988\) 0 0
\(989\) 268.663 1350.66i 0.271651 1.36568i
\(990\) 0 0
\(991\) 200.153 133.738i 0.201971 0.134953i −0.450472 0.892791i \(-0.648744\pi\)
0.652443 + 0.757838i \(0.273744\pi\)
\(992\) 0 0
\(993\) 304.411 + 203.401i 0.306557 + 0.204835i
\(994\) 0 0
\(995\) 534.436 + 221.371i 0.537122 + 0.222483i
\(996\) 0 0
\(997\) −456.434 + 90.7904i −0.457808 + 0.0910636i −0.418606 0.908168i \(-0.637481\pi\)
−0.0392012 + 0.999231i \(0.512481\pi\)
\(998\) 0 0
\(999\) −528.506 528.506i −0.529035 0.529035i
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 136.3.t.a.65.1 32
4.3 odd 2 272.3.bh.f.65.4 32
17.11 odd 16 inner 136.3.t.a.113.1 yes 32
68.11 even 16 272.3.bh.f.113.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.3.t.a.65.1 32 1.1 even 1 trivial
136.3.t.a.113.1 yes 32 17.11 odd 16 inner
272.3.bh.f.65.4 32 4.3 odd 2
272.3.bh.f.113.4 32 68.11 even 16