Properties

Label 256.3.h.b.31.1
Level $256$
Weight $3$
Character 256.31
Analytic conductor $6.975$
Analytic rank $0$
Dimension $28$
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] = [256,3,Mod(31,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 256.h (of order \(8\), degree \(4\), 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: \(6.97549476762\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 31.1
Character \(\chi\) \(=\) 256.31
Dual form 256.3.h.b.223.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.35131 + 1.80237i) q^{3} +(2.81639 + 1.16659i) q^{5} +(-6.23443 + 6.23443i) q^{7} +(9.32143 - 9.32143i) q^{9} +O(q^{10})\) \(q+(-4.35131 + 1.80237i) q^{3} +(2.81639 + 1.16659i) q^{5} +(-6.23443 + 6.23443i) q^{7} +(9.32143 - 9.32143i) q^{9} +(8.06262 + 3.33965i) q^{11} +(-13.3208 + 5.51766i) q^{13} -14.3576 q^{15} -4.56488i q^{17} +(-13.4421 - 32.4522i) q^{19} +(15.8912 - 38.3647i) q^{21} +(-6.75277 - 6.75277i) q^{23} +(-11.1065 - 11.1065i) q^{25} +(-7.53841 + 18.1993i) q^{27} +(0.266504 + 0.643399i) q^{29} -0.326715i q^{31} -41.1023 q^{33} +(-24.8316 + 10.2856i) q^{35} +(-31.5133 - 13.0532i) q^{37} +(48.0182 - 48.0182i) q^{39} +(15.7509 - 15.7509i) q^{41} +(-4.83274 - 2.00179i) q^{43} +(37.1271 - 15.3785i) q^{45} -49.7096 q^{47} -28.7362i q^{49} +(8.22762 + 19.8632i) q^{51} +(-4.45882 + 10.7645i) q^{53} +(18.8115 + 18.8115i) q^{55} +(116.982 + 116.982i) q^{57} +(-13.1268 + 31.6909i) q^{59} +(35.4023 + 85.4687i) q^{61} +116.228i q^{63} -43.9535 q^{65} +(41.3348 - 17.1214i) q^{67} +(41.5545 + 17.2124i) q^{69} +(37.6381 - 37.6381i) q^{71} +(-52.2302 + 52.2302i) q^{73} +(68.3461 + 28.3099i) q^{75} +(-71.0866 + 29.4450i) q^{77} +26.9061 q^{79} +25.8643i q^{81} +(-10.6315 - 25.6667i) q^{83} +(5.32533 - 12.8565i) q^{85} +(-2.31929 - 2.31929i) q^{87} +(-103.292 - 103.292i) q^{89} +(48.6482 - 117.447i) q^{91} +(0.588862 + 1.42164i) q^{93} -107.080i q^{95} +77.9778 q^{97} +(106.285 - 44.0249i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{3} + 4 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 4 q^{3} + 4 q^{5} - 4 q^{7} - 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{19} + 4 q^{21} - 68 q^{23} - 4 q^{25} + 100 q^{27} + 4 q^{29} - 8 q^{33} - 92 q^{35} + 4 q^{37} + 188 q^{39} - 4 q^{41} - 92 q^{43} + 40 q^{45} - 8 q^{47} - 224 q^{51} + 164 q^{53} + 252 q^{55} - 4 q^{57} - 124 q^{59} + 68 q^{61} - 8 q^{65} + 164 q^{67} - 188 q^{69} - 260 q^{71} - 4 q^{73} + 488 q^{75} - 220 q^{77} - 520 q^{79} + 484 q^{83} - 96 q^{85} - 452 q^{87} - 4 q^{89} + 196 q^{91} - 32 q^{93} - 8 q^{97} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(-1\)

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
\(3\) −4.35131 + 1.80237i −1.45044 + 0.600791i −0.962304 0.271977i \(-0.912322\pi\)
−0.488135 + 0.872768i \(0.662322\pi\)
\(4\) 0 0
\(5\) 2.81639 + 1.16659i 0.563278 + 0.233318i 0.646108 0.763246i \(-0.276396\pi\)
−0.0828294 + 0.996564i \(0.526396\pi\)
\(6\) 0 0
\(7\) −6.23443 + 6.23443i −0.890633 + 0.890633i −0.994583 0.103950i \(-0.966852\pi\)
0.103950 + 0.994583i \(0.466852\pi\)
\(8\) 0 0
\(9\) 9.32143 9.32143i 1.03571 1.03571i
\(10\) 0 0
\(11\) 8.06262 + 3.33965i 0.732965 + 0.303604i 0.717770 0.696280i \(-0.245163\pi\)
0.0151955 + 0.999885i \(0.495163\pi\)
\(12\) 0 0
\(13\) −13.3208 + 5.51766i −1.02468 + 0.424436i −0.830789 0.556588i \(-0.812110\pi\)
−0.193889 + 0.981023i \(0.562110\pi\)
\(14\) 0 0
\(15\) −14.3576 −0.957176
\(16\) 0 0
\(17\) 4.56488i 0.268522i −0.990946 0.134261i \(-0.957134\pi\)
0.990946 0.134261i \(-0.0428661\pi\)
\(18\) 0 0
\(19\) −13.4421 32.4522i −0.707481 1.70801i −0.706201 0.708011i \(-0.749593\pi\)
−0.00128016 0.999999i \(-0.500407\pi\)
\(20\) 0 0
\(21\) 15.8912 38.3647i 0.756724 1.82689i
\(22\) 0 0
\(23\) −6.75277 6.75277i −0.293599 0.293599i 0.544901 0.838500i \(-0.316567\pi\)
−0.838500 + 0.544901i \(0.816567\pi\)
\(24\) 0 0
\(25\) −11.1065 11.1065i −0.444261 0.444261i
\(26\) 0 0
\(27\) −7.53841 + 18.1993i −0.279200 + 0.674050i
\(28\) 0 0
\(29\) 0.266504 + 0.643399i 0.00918981 + 0.0221862i 0.928408 0.371563i \(-0.121178\pi\)
−0.919218 + 0.393749i \(0.871178\pi\)
\(30\) 0 0
\(31\) 0.326715i 0.0105392i −0.999986 0.00526959i \(-0.998323\pi\)
0.999986 0.00526959i \(-0.00167737\pi\)
\(32\) 0 0
\(33\) −41.1023 −1.24552
\(34\) 0 0
\(35\) −24.8316 + 10.2856i −0.709475 + 0.293874i
\(36\) 0 0
\(37\) −31.5133 13.0532i −0.851710 0.352790i −0.0862502 0.996274i \(-0.527488\pi\)
−0.765460 + 0.643484i \(0.777488\pi\)
\(38\) 0 0
\(39\) 48.0182 48.0182i 1.23124 1.23124i
\(40\) 0 0
\(41\) 15.7509 15.7509i 0.384169 0.384169i −0.488433 0.872601i \(-0.662431\pi\)
0.872601 + 0.488433i \(0.162431\pi\)
\(42\) 0 0
\(43\) −4.83274 2.00179i −0.112389 0.0465531i 0.325780 0.945446i \(-0.394373\pi\)
−0.438170 + 0.898892i \(0.644373\pi\)
\(44\) 0 0
\(45\) 37.1271 15.3785i 0.825046 0.341745i
\(46\) 0 0
\(47\) −49.7096 −1.05765 −0.528825 0.848731i \(-0.677367\pi\)
−0.528825 + 0.848731i \(0.677367\pi\)
\(48\) 0 0
\(49\) 28.7362i 0.586454i
\(50\) 0 0
\(51\) 8.22762 + 19.8632i 0.161326 + 0.389475i
\(52\) 0 0
\(53\) −4.45882 + 10.7645i −0.0841286 + 0.203105i −0.960346 0.278812i \(-0.910059\pi\)
0.876217 + 0.481917i \(0.160059\pi\)
\(54\) 0 0
\(55\) 18.8115 + 18.8115i 0.342027 + 0.342027i
\(56\) 0 0
\(57\) 116.982 + 116.982i 2.05232 + 2.05232i
\(58\) 0 0
\(59\) −13.1268 + 31.6909i −0.222488 + 0.537134i −0.995227 0.0975907i \(-0.968886\pi\)
0.772739 + 0.634724i \(0.218886\pi\)
\(60\) 0 0
\(61\) 35.4023 + 85.4687i 0.580366 + 1.40113i 0.892482 + 0.451083i \(0.148962\pi\)
−0.312116 + 0.950044i \(0.601038\pi\)
\(62\) 0 0
\(63\) 116.228i 1.84488i
\(64\) 0 0
\(65\) −43.9535 −0.676207
\(66\) 0 0
\(67\) 41.3348 17.1214i 0.616938 0.255544i −0.0522539 0.998634i \(-0.516641\pi\)
0.669192 + 0.743090i \(0.266641\pi\)
\(68\) 0 0
\(69\) 41.5545 + 17.2124i 0.602239 + 0.249455i
\(70\) 0 0
\(71\) 37.6381 37.6381i 0.530114 0.530114i −0.390492 0.920606i \(-0.627695\pi\)
0.920606 + 0.390492i \(0.127695\pi\)
\(72\) 0 0
\(73\) −52.2302 + 52.2302i −0.715482 + 0.715482i −0.967677 0.252195i \(-0.918848\pi\)
0.252195 + 0.967677i \(0.418848\pi\)
\(74\) 0 0
\(75\) 68.3461 + 28.3099i 0.911282 + 0.377465i
\(76\) 0 0
\(77\) −71.0866 + 29.4450i −0.923203 + 0.382403i
\(78\) 0 0
\(79\) 26.9061 0.340583 0.170292 0.985394i \(-0.445529\pi\)
0.170292 + 0.985394i \(0.445529\pi\)
\(80\) 0 0
\(81\) 25.8643i 0.319313i
\(82\) 0 0
\(83\) −10.6315 25.6667i −0.128090 0.309237i 0.846804 0.531905i \(-0.178524\pi\)
−0.974894 + 0.222667i \(0.928524\pi\)
\(84\) 0 0
\(85\) 5.32533 12.8565i 0.0626510 0.151253i
\(86\) 0 0
\(87\) −2.31929 2.31929i −0.0266585 0.0266585i
\(88\) 0 0
\(89\) −103.292 103.292i −1.16058 1.16058i −0.984348 0.176234i \(-0.943609\pi\)
−0.176234 0.984348i \(-0.556391\pi\)
\(90\) 0 0
\(91\) 48.6482 117.447i 0.534596 1.29063i
\(92\) 0 0
\(93\) 0.588862 + 1.42164i 0.00633185 + 0.0152864i
\(94\) 0 0
\(95\) 107.080i 1.12715i
\(96\) 0 0
\(97\) 77.9778 0.803895 0.401948 0.915663i \(-0.368333\pi\)
0.401948 + 0.915663i \(0.368333\pi\)
\(98\) 0 0
\(99\) 106.285 44.0249i 1.07359 0.444696i
\(100\) 0 0
\(101\) −104.064 43.1046i −1.03033 0.426778i −0.197500 0.980303i \(-0.563282\pi\)
−0.832833 + 0.553525i \(0.813282\pi\)
\(102\) 0 0
\(103\) −76.6571 + 76.6571i −0.744243 + 0.744243i −0.973392 0.229148i \(-0.926406\pi\)
0.229148 + 0.973392i \(0.426406\pi\)
\(104\) 0 0
\(105\) 89.5117 89.5117i 0.852492 0.852492i
\(106\) 0 0
\(107\) 40.9728 + 16.9715i 0.382923 + 0.158612i 0.565837 0.824517i \(-0.308553\pi\)
−0.182914 + 0.983129i \(0.558553\pi\)
\(108\) 0 0
\(109\) −102.183 + 42.3255i −0.937456 + 0.388307i −0.798502 0.601992i \(-0.794374\pi\)
−0.138954 + 0.990299i \(0.544374\pi\)
\(110\) 0 0
\(111\) 160.651 1.44731
\(112\) 0 0
\(113\) 123.602i 1.09383i 0.837190 + 0.546913i \(0.184197\pi\)
−0.837190 + 0.546913i \(0.815803\pi\)
\(114\) 0 0
\(115\) −11.1408 26.8962i −0.0968761 0.233880i
\(116\) 0 0
\(117\) −72.7365 + 175.602i −0.621680 + 1.50087i
\(118\) 0 0
\(119\) 28.4594 + 28.4594i 0.239155 + 0.239155i
\(120\) 0 0
\(121\) −31.7073 31.7073i −0.262044 0.262044i
\(122\) 0 0
\(123\) −40.1481 + 96.9262i −0.326408 + 0.788018i
\(124\) 0 0
\(125\) −47.4883 114.647i −0.379906 0.917175i
\(126\) 0 0
\(127\) 133.213i 1.04892i −0.851434 0.524462i \(-0.824266\pi\)
0.851434 0.524462i \(-0.175734\pi\)
\(128\) 0 0
\(129\) 24.6367 0.190982
\(130\) 0 0
\(131\) −228.056 + 94.4640i −1.74089 + 0.721100i −0.742185 + 0.670195i \(0.766210\pi\)
−0.998704 + 0.0509041i \(0.983790\pi\)
\(132\) 0 0
\(133\) 286.125 + 118.517i 2.15132 + 0.891104i
\(134\) 0 0
\(135\) −42.4622 + 42.4622i −0.314535 + 0.314535i
\(136\) 0 0
\(137\) −111.817 + 111.817i −0.816180 + 0.816180i −0.985552 0.169372i \(-0.945826\pi\)
0.169372 + 0.985552i \(0.445826\pi\)
\(138\) 0 0
\(139\) −31.7750 13.1616i −0.228597 0.0946880i 0.265445 0.964126i \(-0.414481\pi\)
−0.494042 + 0.869438i \(0.664481\pi\)
\(140\) 0 0
\(141\) 216.302 89.5952i 1.53406 0.635427i
\(142\) 0 0
\(143\) −125.828 −0.879914
\(144\) 0 0
\(145\) 2.12296i 0.0146411i
\(146\) 0 0
\(147\) 51.7934 + 125.040i 0.352336 + 0.850615i
\(148\) 0 0
\(149\) 108.344 261.565i 0.727140 1.75547i 0.0752422 0.997165i \(-0.476027\pi\)
0.651898 0.758307i \(-0.273973\pi\)
\(150\) 0 0
\(151\) 51.5292 + 51.5292i 0.341253 + 0.341253i 0.856838 0.515585i \(-0.172425\pi\)
−0.515585 + 0.856838i \(0.672425\pi\)
\(152\) 0 0
\(153\) −42.5512 42.5512i −0.278112 0.278112i
\(154\) 0 0
\(155\) 0.381142 0.920157i 0.00245898 0.00593650i
\(156\) 0 0
\(157\) −39.3450 94.9872i −0.250605 0.605014i 0.747648 0.664095i \(-0.231183\pi\)
−0.998253 + 0.0590811i \(0.981183\pi\)
\(158\) 0 0
\(159\) 54.8764i 0.345134i
\(160\) 0 0
\(161\) 84.1994 0.522978
\(162\) 0 0
\(163\) 4.97467 2.06057i 0.0305194 0.0126416i −0.367372 0.930074i \(-0.619742\pi\)
0.397891 + 0.917433i \(0.369742\pi\)
\(164\) 0 0
\(165\) −115.760 47.9494i −0.701577 0.290603i
\(166\) 0 0
\(167\) −165.012 + 165.012i −0.988097 + 0.988097i −0.999930 0.0118333i \(-0.996233\pi\)
0.0118333 + 0.999930i \(0.496233\pi\)
\(168\) 0 0
\(169\) 27.4985 27.4985i 0.162713 0.162713i
\(170\) 0 0
\(171\) −427.801 177.201i −2.50176 1.03626i
\(172\) 0 0
\(173\) 115.760 47.9493i 0.669132 0.277163i −0.0221438 0.999755i \(-0.507049\pi\)
0.691275 + 0.722591i \(0.257049\pi\)
\(174\) 0 0
\(175\) 138.486 0.791348
\(176\) 0 0
\(177\) 161.556i 0.912748i
\(178\) 0 0
\(179\) 44.8368 + 108.246i 0.250485 + 0.604724i 0.998243 0.0592467i \(-0.0188699\pi\)
−0.747758 + 0.663971i \(0.768870\pi\)
\(180\) 0 0
\(181\) −80.3652 + 194.019i −0.444007 + 1.07193i 0.530524 + 0.847670i \(0.321995\pi\)
−0.974530 + 0.224256i \(0.928005\pi\)
\(182\) 0 0
\(183\) −308.093 308.093i −1.68357 1.68357i
\(184\) 0 0
\(185\) −73.5260 73.5260i −0.397438 0.397438i
\(186\) 0 0
\(187\) 15.2451 36.8049i 0.0815245 0.196818i
\(188\) 0 0
\(189\) −66.4648 160.460i −0.351666 0.848996i
\(190\) 0 0
\(191\) 338.117i 1.77025i 0.465357 + 0.885123i \(0.345926\pi\)
−0.465357 + 0.885123i \(0.654074\pi\)
\(192\) 0 0
\(193\) 234.508 1.21507 0.607533 0.794295i \(-0.292159\pi\)
0.607533 + 0.794295i \(0.292159\pi\)
\(194\) 0 0
\(195\) 191.255 79.2206i 0.980797 0.406259i
\(196\) 0 0
\(197\) −237.007 98.1713i −1.20308 0.498332i −0.311086 0.950382i \(-0.600693\pi\)
−0.891993 + 0.452050i \(0.850693\pi\)
\(198\) 0 0
\(199\) 39.2172 39.2172i 0.197071 0.197071i −0.601672 0.798743i \(-0.705499\pi\)
0.798743 + 0.601672i \(0.205499\pi\)
\(200\) 0 0
\(201\) −149.002 + 149.002i −0.741302 + 0.741302i
\(202\) 0 0
\(203\) −5.67273 2.34972i −0.0279445 0.0115750i
\(204\) 0 0
\(205\) 62.7356 25.9859i 0.306027 0.126761i
\(206\) 0 0
\(207\) −125.891 −0.608169
\(208\) 0 0
\(209\) 306.542i 1.46671i
\(210\) 0 0
\(211\) 79.9026 + 192.902i 0.378685 + 0.914227i 0.992213 + 0.124554i \(0.0397499\pi\)
−0.613528 + 0.789673i \(0.710250\pi\)
\(212\) 0 0
\(213\) −95.9373 + 231.613i −0.450410 + 1.08739i
\(214\) 0 0
\(215\) −11.2756 11.2756i −0.0524448 0.0524448i
\(216\) 0 0
\(217\) 2.03688 + 2.03688i 0.00938655 + 0.00938655i
\(218\) 0 0
\(219\) 133.132 321.408i 0.607907 1.46762i
\(220\) 0 0
\(221\) 25.1875 + 60.8079i 0.113970 + 0.275149i
\(222\) 0 0
\(223\) 344.421i 1.54449i −0.635326 0.772244i \(-0.719134\pi\)
0.635326 0.772244i \(-0.280866\pi\)
\(224\) 0 0
\(225\) −207.058 −0.920256
\(226\) 0 0
\(227\) −68.2639 + 28.2758i −0.300722 + 0.124563i −0.527942 0.849280i \(-0.677036\pi\)
0.227220 + 0.973843i \(0.427036\pi\)
\(228\) 0 0
\(229\) 41.8202 + 17.3225i 0.182621 + 0.0756440i 0.472120 0.881534i \(-0.343489\pi\)
−0.289499 + 0.957178i \(0.593489\pi\)
\(230\) 0 0
\(231\) 256.249 256.249i 1.10930 1.10930i
\(232\) 0 0
\(233\) 203.044 203.044i 0.871432 0.871432i −0.121197 0.992629i \(-0.538673\pi\)
0.992629 + 0.121197i \(0.0386731\pi\)
\(234\) 0 0
\(235\) −140.002 57.9906i −0.595752 0.246768i
\(236\) 0 0
\(237\) −117.077 + 48.4948i −0.493995 + 0.204619i
\(238\) 0 0
\(239\) 87.6710 0.366824 0.183412 0.983036i \(-0.441286\pi\)
0.183412 + 0.983036i \(0.441286\pi\)
\(240\) 0 0
\(241\) 15.4754i 0.0642135i −0.999484 0.0321067i \(-0.989778\pi\)
0.999484 0.0321067i \(-0.0102216\pi\)
\(242\) 0 0
\(243\) −114.463 276.338i −0.471041 1.13719i
\(244\) 0 0
\(245\) 33.5233 80.9325i 0.136830 0.330337i
\(246\) 0 0
\(247\) 358.121 + 358.121i 1.44988 + 1.44988i
\(248\) 0 0
\(249\) 92.5220 + 92.5220i 0.371574 + 0.371574i
\(250\) 0 0
\(251\) −95.9530 + 231.651i −0.382283 + 0.922913i 0.609241 + 0.792985i \(0.291474\pi\)
−0.991524 + 0.129927i \(0.958526\pi\)
\(252\) 0 0
\(253\) −31.8932 76.9969i −0.126060 0.304336i
\(254\) 0 0
\(255\) 65.5409i 0.257023i
\(256\) 0 0
\(257\) −131.142 −0.510282 −0.255141 0.966904i \(-0.582122\pi\)
−0.255141 + 0.966904i \(0.582122\pi\)
\(258\) 0 0
\(259\) 277.847 115.088i 1.07277 0.444355i
\(260\) 0 0
\(261\) 8.48160 + 3.51319i 0.0324965 + 0.0134605i
\(262\) 0 0
\(263\) −281.350 + 281.350i −1.06977 + 1.06977i −0.0723955 + 0.997376i \(0.523064\pi\)
−0.997376 + 0.0723955i \(0.976936\pi\)
\(264\) 0 0
\(265\) −25.1156 + 25.1156i −0.0947757 + 0.0947757i
\(266\) 0 0
\(267\) 635.626 + 263.285i 2.38062 + 0.986085i
\(268\) 0 0
\(269\) −133.290 + 55.2107i −0.495503 + 0.205244i −0.616419 0.787419i \(-0.711417\pi\)
0.120915 + 0.992663i \(0.461417\pi\)
\(270\) 0 0
\(271\) −368.673 −1.36042 −0.680208 0.733019i \(-0.738111\pi\)
−0.680208 + 0.733019i \(0.738111\pi\)
\(272\) 0 0
\(273\) 598.732i 2.19316i
\(274\) 0 0
\(275\) −52.4559 126.640i −0.190749 0.460508i
\(276\) 0 0
\(277\) −21.6001 + 52.1472i −0.0779786 + 0.188257i −0.958061 0.286564i \(-0.907487\pi\)
0.880083 + 0.474821i \(0.157487\pi\)
\(278\) 0 0
\(279\) −3.04545 3.04545i −0.0109156 0.0109156i
\(280\) 0 0
\(281\) 85.0605 + 85.0605i 0.302706 + 0.302706i 0.842072 0.539365i \(-0.181336\pi\)
−0.539365 + 0.842072i \(0.681336\pi\)
\(282\) 0 0
\(283\) −27.0948 + 65.4127i −0.0957414 + 0.231140i −0.964493 0.264107i \(-0.914923\pi\)
0.868752 + 0.495248i \(0.164923\pi\)
\(284\) 0 0
\(285\) 192.997 + 465.937i 0.677184 + 1.63487i
\(286\) 0 0
\(287\) 196.396i 0.684306i
\(288\) 0 0
\(289\) 268.162 0.927896
\(290\) 0 0
\(291\) −339.306 + 140.545i −1.16600 + 0.482973i
\(292\) 0 0
\(293\) 321.033 + 132.976i 1.09568 + 0.453844i 0.855983 0.517005i \(-0.172953\pi\)
0.239694 + 0.970849i \(0.422953\pi\)
\(294\) 0 0
\(295\) −73.9404 + 73.9404i −0.250645 + 0.250645i
\(296\) 0 0
\(297\) −121.559 + 121.559i −0.409289 + 0.409289i
\(298\) 0 0
\(299\) 127.212 + 52.6929i 0.425458 + 0.176231i
\(300\) 0 0
\(301\) 42.6094 17.6494i 0.141559 0.0586358i
\(302\) 0 0
\(303\) 530.504 1.75084
\(304\) 0 0
\(305\) 282.013i 0.924634i
\(306\) 0 0
\(307\) 92.6973 + 223.791i 0.301946 + 0.728961i 0.999918 + 0.0128360i \(0.00408593\pi\)
−0.697972 + 0.716125i \(0.745914\pi\)
\(308\) 0 0
\(309\) 195.394 471.724i 0.632344 1.52661i
\(310\) 0 0
\(311\) −44.5768 44.5768i −0.143334 0.143334i 0.631799 0.775133i \(-0.282317\pi\)
−0.775133 + 0.631799i \(0.782317\pi\)
\(312\) 0 0
\(313\) −27.9303 27.9303i −0.0892343 0.0892343i 0.661081 0.750315i \(-0.270098\pi\)
−0.750315 + 0.661081i \(0.770098\pi\)
\(314\) 0 0
\(315\) −135.590 + 327.342i −0.430443 + 1.03918i
\(316\) 0 0
\(317\) 125.850 + 303.829i 0.397003 + 0.958450i 0.988373 + 0.152049i \(0.0485873\pi\)
−0.591370 + 0.806400i \(0.701413\pi\)
\(318\) 0 0
\(319\) 6.07751i 0.0190518i
\(320\) 0 0
\(321\) −208.874 −0.650699
\(322\) 0 0
\(323\) −148.140 + 61.3618i −0.458639 + 0.189975i
\(324\) 0 0
\(325\) 209.230 + 86.6660i 0.643785 + 0.266665i
\(326\) 0 0
\(327\) 368.343 368.343i 1.12643 1.12643i
\(328\) 0 0
\(329\) 309.911 309.911i 0.941978 0.941978i
\(330\) 0 0
\(331\) 169.515 + 70.2155i 0.512131 + 0.212131i 0.623756 0.781619i \(-0.285606\pi\)
−0.111626 + 0.993750i \(0.535606\pi\)
\(332\) 0 0
\(333\) −415.423 + 172.074i −1.24752 + 0.516739i
\(334\) 0 0
\(335\) 136.389 0.407131
\(336\) 0 0
\(337\) 67.3116i 0.199738i 0.995001 + 0.0998689i \(0.0318423\pi\)
−0.995001 + 0.0998689i \(0.968158\pi\)
\(338\) 0 0
\(339\) −222.778 537.833i −0.657161 1.58653i
\(340\) 0 0
\(341\) 1.09111 2.63418i 0.00319974 0.00772486i
\(342\) 0 0
\(343\) −126.333 126.333i −0.368318 0.368318i
\(344\) 0 0
\(345\) 96.9538 + 96.9538i 0.281026 + 0.281026i
\(346\) 0 0
\(347\) 107.157 258.699i 0.308809 0.745530i −0.690935 0.722916i \(-0.742801\pi\)
0.999744 0.0226140i \(-0.00719886\pi\)
\(348\) 0 0
\(349\) −165.558 399.692i −0.474378 1.14525i −0.962209 0.272311i \(-0.912212\pi\)
0.487831 0.872938i \(-0.337788\pi\)
\(350\) 0 0
\(351\) 284.024i 0.809187i
\(352\) 0 0
\(353\) −574.524 −1.62755 −0.813773 0.581183i \(-0.802590\pi\)
−0.813773 + 0.581183i \(0.802590\pi\)
\(354\) 0 0
\(355\) 149.912 62.0955i 0.422287 0.174917i
\(356\) 0 0
\(357\) −175.130 72.5414i −0.490562 0.203197i
\(358\) 0 0
\(359\) 499.243 499.243i 1.39065 1.39065i 0.566782 0.823868i \(-0.308188\pi\)
0.823868 0.566782i \(-0.191812\pi\)
\(360\) 0 0
\(361\) −617.189 + 617.189i −1.70966 + 1.70966i
\(362\) 0 0
\(363\) 195.117 + 80.8201i 0.537512 + 0.222645i
\(364\) 0 0
\(365\) −208.032 + 86.1696i −0.569950 + 0.236081i
\(366\) 0 0
\(367\) 295.566 0.805357 0.402679 0.915341i \(-0.368079\pi\)
0.402679 + 0.915341i \(0.368079\pi\)
\(368\) 0 0
\(369\) 293.642i 0.795778i
\(370\) 0 0
\(371\) −39.3126 94.9090i −0.105964 0.255819i
\(372\) 0 0
\(373\) 133.878 323.209i 0.358921 0.866512i −0.636531 0.771251i \(-0.719631\pi\)
0.995452 0.0952612i \(-0.0303686\pi\)
\(374\) 0 0
\(375\) 413.273 + 413.273i 1.10206 + 1.10206i
\(376\) 0 0
\(377\) −7.10011 7.10011i −0.0188332 0.0188332i
\(378\) 0 0
\(379\) −170.642 + 411.967i −0.450244 + 1.08698i 0.521986 + 0.852954i \(0.325191\pi\)
−0.972229 + 0.234030i \(0.924809\pi\)
\(380\) 0 0
\(381\) 240.100 + 579.654i 0.630185 + 1.52140i
\(382\) 0 0
\(383\) 254.902i 0.665540i 0.943008 + 0.332770i \(0.107983\pi\)
−0.943008 + 0.332770i \(0.892017\pi\)
\(384\) 0 0
\(385\) −234.558 −0.609242
\(386\) 0 0
\(387\) −63.7075 + 26.3885i −0.164619 + 0.0681874i
\(388\) 0 0
\(389\) −687.246 284.667i −1.76670 0.731791i −0.995453 0.0952550i \(-0.969633\pi\)
−0.771247 0.636536i \(-0.780367\pi\)
\(390\) 0 0
\(391\) −30.8256 + 30.8256i −0.0788379 + 0.0788379i
\(392\) 0 0
\(393\) 822.086 822.086i 2.09182 2.09182i
\(394\) 0 0
\(395\) 75.7781 + 31.3883i 0.191843 + 0.0794640i
\(396\) 0 0
\(397\) 56.8981 23.5679i 0.143320 0.0593651i −0.309871 0.950779i \(-0.600286\pi\)
0.453191 + 0.891414i \(0.350286\pi\)
\(398\) 0 0
\(399\) −1458.63 −3.65572
\(400\) 0 0
\(401\) 704.010i 1.75564i −0.478994 0.877818i \(-0.658999\pi\)
0.478994 0.877818i \(-0.341001\pi\)
\(402\) 0 0
\(403\) 1.80270 + 4.35211i 0.00447321 + 0.0107993i
\(404\) 0 0
\(405\) −30.1730 + 72.8441i −0.0745013 + 0.179862i
\(406\) 0 0
\(407\) −210.486 210.486i −0.517166 0.517166i
\(408\) 0 0
\(409\) 528.488 + 528.488i 1.29215 + 1.29215i 0.933457 + 0.358690i \(0.116776\pi\)
0.358690 + 0.933457i \(0.383224\pi\)
\(410\) 0 0
\(411\) 285.014 688.085i 0.693465 1.67417i
\(412\) 0 0
\(413\) −115.737 279.413i −0.280234 0.676544i
\(414\) 0 0
\(415\) 84.6901i 0.204072i
\(416\) 0 0
\(417\) 161.985 0.388453
\(418\) 0 0
\(419\) 153.485 63.5754i 0.366312 0.151731i −0.191932 0.981408i \(-0.561475\pi\)
0.558244 + 0.829677i \(0.311475\pi\)
\(420\) 0 0
\(421\) 240.175 + 99.4838i 0.570487 + 0.236304i 0.649231 0.760591i \(-0.275091\pi\)
−0.0787437 + 0.996895i \(0.525091\pi\)
\(422\) 0 0
\(423\) −463.364 + 463.364i −1.09542 + 1.09542i
\(424\) 0 0
\(425\) −50.7000 + 50.7000i −0.119294 + 0.119294i
\(426\) 0 0
\(427\) −753.562 312.136i −1.76478 0.730997i
\(428\) 0 0
\(429\) 547.516 226.789i 1.27626 0.528645i
\(430\) 0 0
\(431\) −607.318 −1.40909 −0.704546 0.709659i \(-0.748849\pi\)
−0.704546 + 0.709659i \(0.748849\pi\)
\(432\) 0 0
\(433\) 233.380i 0.538984i −0.963003 0.269492i \(-0.913144\pi\)
0.963003 0.269492i \(-0.0868557\pi\)
\(434\) 0 0
\(435\) −3.82637 9.23768i −0.00879626 0.0212361i
\(436\) 0 0
\(437\) −128.371 + 309.914i −0.293754 + 0.709185i
\(438\) 0 0
\(439\) −496.850 496.850i −1.13178 1.13178i −0.989882 0.141896i \(-0.954680\pi\)
−0.141896 0.989882i \(-0.545320\pi\)
\(440\) 0 0
\(441\) −267.863 267.863i −0.607399 0.607399i
\(442\) 0 0
\(443\) −59.8483 + 144.487i −0.135098 + 0.326155i −0.976922 0.213597i \(-0.931482\pi\)
0.841824 + 0.539752i \(0.181482\pi\)
\(444\) 0 0
\(445\) −170.411 411.409i −0.382947 0.924515i
\(446\) 0 0
\(447\) 1333.43i 2.98306i
\(448\) 0 0
\(449\) −15.4530 −0.0344165 −0.0172082 0.999852i \(-0.505478\pi\)
−0.0172082 + 0.999852i \(0.505478\pi\)
\(450\) 0 0
\(451\) 179.596 74.3911i 0.398217 0.164947i
\(452\) 0 0
\(453\) −317.095 131.345i −0.699989 0.289945i
\(454\) 0 0
\(455\) 274.025 274.025i 0.602253 0.602253i
\(456\) 0 0
\(457\) −93.8365 + 93.8365i −0.205332 + 0.205332i −0.802280 0.596948i \(-0.796380\pi\)
0.596948 + 0.802280i \(0.296380\pi\)
\(458\) 0 0
\(459\) 83.0778 + 34.4120i 0.180997 + 0.0749716i
\(460\) 0 0
\(461\) 574.348 237.903i 1.24588 0.516058i 0.340329 0.940306i \(-0.389462\pi\)
0.905546 + 0.424248i \(0.139462\pi\)
\(462\) 0 0
\(463\) 568.089 1.22697 0.613487 0.789705i \(-0.289766\pi\)
0.613487 + 0.789705i \(0.289766\pi\)
\(464\) 0 0
\(465\) 4.69085i 0.0100879i
\(466\) 0 0
\(467\) 156.459 + 377.726i 0.335030 + 0.808835i 0.998178 + 0.0603459i \(0.0192204\pi\)
−0.663147 + 0.748489i \(0.730780\pi\)
\(468\) 0 0
\(469\) −150.957 + 364.442i −0.321869 + 0.777061i
\(470\) 0 0
\(471\) 342.405 + 342.405i 0.726974 + 0.726974i
\(472\) 0 0
\(473\) −32.2793 32.2793i −0.0682437 0.0682437i
\(474\) 0 0
\(475\) −211.136 + 509.727i −0.444497 + 1.07311i
\(476\) 0 0
\(477\) 58.7783 + 141.903i 0.123225 + 0.297492i
\(478\) 0 0
\(479\) 327.880i 0.684509i −0.939607 0.342254i \(-0.888810\pi\)
0.939607 0.342254i \(-0.111190\pi\)
\(480\) 0 0
\(481\) 491.806 1.02247
\(482\) 0 0
\(483\) −366.378 + 151.759i −0.758547 + 0.314200i
\(484\) 0 0
\(485\) 219.616 + 90.9680i 0.452817 + 0.187563i
\(486\) 0 0
\(487\) −147.493 + 147.493i −0.302861 + 0.302861i −0.842132 0.539271i \(-0.818700\pi\)
0.539271 + 0.842132i \(0.318700\pi\)
\(488\) 0 0
\(489\) −17.9324 + 17.9324i −0.0366716 + 0.0366716i
\(490\) 0 0
\(491\) −598.802 248.032i −1.21956 0.505157i −0.322288 0.946642i \(-0.604452\pi\)
−0.897269 + 0.441485i \(0.854452\pi\)
\(492\) 0 0
\(493\) 2.93704 1.21656i 0.00595748 0.00246767i
\(494\) 0 0
\(495\) 350.700 0.708485
\(496\) 0 0
\(497\) 469.304i 0.944274i
\(498\) 0 0
\(499\) −58.1446 140.373i −0.116522 0.281309i 0.854849 0.518877i \(-0.173650\pi\)
−0.971371 + 0.237568i \(0.923650\pi\)
\(500\) 0 0
\(501\) 420.606 1015.43i 0.839533 2.02681i
\(502\) 0 0
\(503\) −256.204 256.204i −0.509351 0.509351i 0.404976 0.914327i \(-0.367280\pi\)
−0.914327 + 0.404976i \(0.867280\pi\)
\(504\) 0 0
\(505\) −242.799 242.799i −0.480789 0.480789i
\(506\) 0 0
\(507\) −70.0921 + 169.217i −0.138249 + 0.333762i
\(508\) 0 0
\(509\) 229.271 + 553.510i 0.450435 + 1.08745i 0.972157 + 0.234331i \(0.0752898\pi\)
−0.521722 + 0.853115i \(0.674710\pi\)
\(510\) 0 0
\(511\) 651.251i 1.27446i
\(512\) 0 0
\(513\) 691.941 1.34881
\(514\) 0 0
\(515\) −305.324 + 126.469i −0.592861 + 0.245571i
\(516\) 0 0
\(517\) −400.789 166.012i −0.775221 0.321107i
\(518\) 0 0
\(519\) −417.285 + 417.285i −0.804017 + 0.804017i
\(520\) 0 0
\(521\) 80.7376 80.7376i 0.154967 0.154967i −0.625365 0.780332i \(-0.715050\pi\)
0.780332 + 0.625365i \(0.215050\pi\)
\(522\) 0 0
\(523\) −123.197 51.0300i −0.235559 0.0975717i 0.261781 0.965127i \(-0.415690\pi\)
−0.497341 + 0.867555i \(0.665690\pi\)
\(524\) 0 0
\(525\) −602.595 + 249.603i −1.14780 + 0.475435i
\(526\) 0 0
\(527\) −1.49141 −0.00283001
\(528\) 0 0
\(529\) 437.800i 0.827599i
\(530\) 0 0
\(531\) 173.044 + 417.765i 0.325883 + 0.786751i
\(532\) 0 0
\(533\) −122.907 + 296.723i −0.230594 + 0.556704i
\(534\) 0 0
\(535\) 95.5966 + 95.5966i 0.178685 + 0.178685i
\(536\) 0 0
\(537\) −390.198 390.198i −0.726626 0.726626i
\(538\) 0 0
\(539\) 95.9689 231.689i 0.178050 0.429850i
\(540\) 0 0
\(541\) −184.993 446.613i −0.341947 0.825532i −0.997519 0.0703996i \(-0.977573\pi\)
0.655572 0.755132i \(-0.272427\pi\)
\(542\) 0 0
\(543\) 989.085i 1.82152i
\(544\) 0 0
\(545\) −337.163 −0.618648
\(546\) 0 0
\(547\) 570.529 236.321i 1.04302 0.432031i 0.205622 0.978632i \(-0.434078\pi\)
0.837394 + 0.546600i \(0.184078\pi\)
\(548\) 0 0
\(549\) 1126.69 + 466.691i 2.05226 + 0.850074i
\(550\) 0 0
\(551\) 17.2973 17.2973i 0.0313926 0.0313926i
\(552\) 0 0
\(553\) −167.744 + 167.744i −0.303335 + 0.303335i
\(554\) 0 0
\(555\) 452.456 + 187.413i 0.815236 + 0.337682i
\(556\) 0 0
\(557\) −340.362 + 140.983i −0.611063 + 0.253111i −0.666683 0.745341i \(-0.732286\pi\)
0.0556199 + 0.998452i \(0.482286\pi\)
\(558\) 0 0
\(559\) 75.4212 0.134922
\(560\) 0 0
\(561\) 187.627i 0.334451i
\(562\) 0 0
\(563\) −241.885 583.963i −0.429637 1.03723i −0.979403 0.201916i \(-0.935283\pi\)
0.549766 0.835319i \(-0.314717\pi\)
\(564\) 0 0
\(565\) −144.193 + 348.113i −0.255209 + 0.616128i
\(566\) 0 0
\(567\) −161.249 161.249i −0.284390 0.284390i
\(568\) 0 0
\(569\) 88.6373 + 88.6373i 0.155777 + 0.155777i 0.780693 0.624915i \(-0.214866\pi\)
−0.624915 + 0.780693i \(0.714866\pi\)
\(570\) 0 0
\(571\) 160.453 387.367i 0.281003 0.678401i −0.718857 0.695158i \(-0.755334\pi\)
0.999860 + 0.0167573i \(0.00533426\pi\)
\(572\) 0 0
\(573\) −609.413 1471.25i −1.06355 2.56763i
\(574\) 0 0
\(575\) 150.000i 0.260869i
\(576\) 0 0
\(577\) −501.285 −0.868778 −0.434389 0.900725i \(-0.643036\pi\)
−0.434389 + 0.900725i \(0.643036\pi\)
\(578\) 0 0
\(579\) −1020.42 + 422.670i −1.76238 + 0.730000i
\(580\) 0 0
\(581\) 226.299 + 93.7360i 0.389498 + 0.161336i
\(582\) 0 0
\(583\) −71.8995 + 71.8995i −0.123327 + 0.123327i
\(584\) 0 0
\(585\) −409.709 + 409.709i −0.700358 + 0.700358i
\(586\) 0 0
\(587\) 805.600 + 333.691i 1.37240 + 0.568468i 0.942439 0.334378i \(-0.108526\pi\)
0.429964 + 0.902846i \(0.358526\pi\)
\(588\) 0 0
\(589\) −10.6026 + 4.39175i −0.0180010 + 0.00745628i
\(590\) 0 0
\(591\) 1208.23 2.04438
\(592\) 0 0
\(593\) 1035.33i 1.74591i 0.487798 + 0.872957i \(0.337800\pi\)
−0.487798 + 0.872957i \(0.662200\pi\)
\(594\) 0 0
\(595\) 46.9525 + 113.353i 0.0789117 + 0.190510i
\(596\) 0 0
\(597\) −99.9623 + 241.330i −0.167441 + 0.404238i
\(598\) 0 0
\(599\) 361.938 + 361.938i 0.604237 + 0.604237i 0.941434 0.337197i \(-0.109479\pi\)
−0.337197 + 0.941434i \(0.609479\pi\)
\(600\) 0 0
\(601\) −221.839 221.839i −0.369116 0.369116i 0.498039 0.867155i \(-0.334054\pi\)
−0.867155 + 0.498039i \(0.834054\pi\)
\(602\) 0 0
\(603\) 225.703 544.896i 0.374301 0.903642i
\(604\) 0 0
\(605\) −52.3109 126.290i −0.0864642 0.208743i
\(606\) 0 0
\(607\) 465.834i 0.767437i −0.923450 0.383719i \(-0.874643\pi\)
0.923450 0.383719i \(-0.125357\pi\)
\(608\) 0 0
\(609\) 28.9189 0.0474859
\(610\) 0 0
\(611\) 662.172 274.281i 1.08375 0.448905i
\(612\) 0 0
\(613\) −292.579 121.190i −0.477290 0.197700i 0.131051 0.991376i \(-0.458165\pi\)
−0.608341 + 0.793676i \(0.708165\pi\)
\(614\) 0 0
\(615\) −226.146 + 226.146i −0.367717 + 0.367717i
\(616\) 0 0
\(617\) −226.657 + 226.657i −0.367354 + 0.367354i −0.866511 0.499158i \(-0.833643\pi\)
0.499158 + 0.866511i \(0.333643\pi\)
\(618\) 0 0
\(619\) −756.530 313.365i −1.22218 0.506244i −0.324078 0.946030i \(-0.605054\pi\)
−0.898102 + 0.439787i \(0.855054\pi\)
\(620\) 0 0
\(621\) 173.801 71.9908i 0.279873 0.115927i
\(622\) 0 0
\(623\) 1287.93 2.06730
\(624\) 0 0
\(625\) 14.3854i 0.0230167i
\(626\) 0 0
\(627\) 552.503 + 1333.86i 0.881185 + 2.12737i
\(628\) 0 0
\(629\) −59.5864 + 143.854i −0.0947320 + 0.228703i
\(630\) 0 0
\(631\) 338.810 + 338.810i 0.536941 + 0.536941i 0.922629 0.385688i \(-0.126036\pi\)
−0.385688 + 0.922629i \(0.626036\pi\)
\(632\) 0 0
\(633\) −695.363 695.363i −1.09852 1.09852i
\(634\) 0 0
\(635\) 155.405 375.181i 0.244733 0.590837i
\(636\) 0 0
\(637\) 158.557 + 382.790i 0.248912 + 0.600927i
\(638\) 0 0
\(639\) 701.682i 1.09809i
\(640\) 0 0
\(641\) 729.839 1.13859 0.569297 0.822132i \(-0.307215\pi\)
0.569297 + 0.822132i \(0.307215\pi\)
\(642\) 0 0
\(643\) 370.578 153.498i 0.576327 0.238722i −0.0754293 0.997151i \(-0.524033\pi\)
0.651756 + 0.758429i \(0.274033\pi\)
\(644\) 0 0
\(645\) 69.3867 + 28.7409i 0.107576 + 0.0445595i
\(646\) 0 0
\(647\) −179.567 + 179.567i −0.277538 + 0.277538i −0.832126 0.554587i \(-0.812876\pi\)
0.554587 + 0.832126i \(0.312876\pi\)
\(648\) 0 0
\(649\) −211.673 + 211.673i −0.326152 + 0.326152i
\(650\) 0 0
\(651\) −12.5343 5.19189i −0.0192540 0.00797525i
\(652\) 0 0
\(653\) 964.894 399.672i 1.47763 0.612056i 0.509048 0.860738i \(-0.329998\pi\)
0.968585 + 0.248683i \(0.0799976\pi\)
\(654\) 0 0
\(655\) −752.497 −1.14885
\(656\) 0 0
\(657\) 973.720i 1.48207i
\(658\) 0 0
\(659\) −233.939 564.778i −0.354990 0.857023i −0.995989 0.0894804i \(-0.971479\pi\)
0.640998 0.767542i \(-0.278521\pi\)
\(660\) 0 0
\(661\) −281.181 + 678.831i −0.425387 + 1.02698i 0.555345 + 0.831620i \(0.312586\pi\)
−0.980732 + 0.195356i \(0.937414\pi\)
\(662\) 0 0
\(663\) −219.197 219.197i −0.330614 0.330614i
\(664\) 0 0
\(665\) 667.580 + 667.580i 1.00388 + 1.00388i
\(666\) 0 0
\(667\) 2.54508 6.14437i 0.00381571 0.00921195i
\(668\) 0 0
\(669\) 620.775 + 1498.68i 0.927915 + 2.24018i
\(670\) 0 0
\(671\) 807.333i 1.20318i
\(672\) 0 0
\(673\) 705.345 1.04806 0.524031 0.851699i \(-0.324428\pi\)
0.524031 + 0.851699i \(0.324428\pi\)
\(674\) 0 0
\(675\) 285.857 118.406i 0.423492 0.175416i
\(676\) 0 0
\(677\) 10.4550 + 4.33061i 0.0154432 + 0.00639676i 0.390392 0.920649i \(-0.372340\pi\)
−0.374948 + 0.927046i \(0.622340\pi\)
\(678\) 0 0
\(679\) −486.147 + 486.147i −0.715976 + 0.715976i
\(680\) 0 0
\(681\) 246.074 246.074i 0.361342 0.361342i
\(682\) 0 0
\(683\) 307.101 + 127.205i 0.449635 + 0.186245i 0.595998 0.802986i \(-0.296757\pi\)
−0.146363 + 0.989231i \(0.546757\pi\)
\(684\) 0 0
\(685\) −445.364 + 184.476i −0.650166 + 0.269308i
\(686\) 0 0
\(687\) −213.194 −0.310327
\(688\) 0 0
\(689\) 167.995i 0.243824i
\(690\) 0 0
\(691\) 243.176 + 587.078i 0.351919 + 0.849607i 0.996383 + 0.0849727i \(0.0270803\pi\)
−0.644465 + 0.764634i \(0.722920\pi\)
\(692\) 0 0
\(693\) −388.159 + 937.099i −0.560114 + 1.35224i
\(694\) 0 0
\(695\) −74.1366 74.1366i −0.106671 0.106671i
\(696\) 0 0
\(697\) −71.9010 71.9010i −0.103158 0.103158i
\(698\) 0 0
\(699\) −517.546 + 1249.47i −0.740410 + 1.78751i
\(700\) 0 0
\(701\) −65.7383 158.706i −0.0937779 0.226400i 0.870029 0.493000i \(-0.164100\pi\)
−0.963807 + 0.266600i \(0.914100\pi\)
\(702\) 0 0
\(703\) 1198.14i 1.70432i
\(704\) 0 0
\(705\) 713.712 1.01236
\(706\) 0 0
\(707\) 917.510 380.045i 1.29775 0.537546i
\(708\) 0 0
\(709\) 1029.00 + 426.228i 1.45135 + 0.601167i 0.962519 0.271214i \(-0.0874250\pi\)
0.488827 + 0.872381i \(0.337425\pi\)
\(710\) 0 0
\(711\) 250.803 250.803i 0.352747 0.352747i
\(712\) 0 0
\(713\) −2.20623 + 2.20623i −0.00309429 + 0.00309429i
\(714\) 0 0
\(715\) −354.380 146.789i −0.495637 0.205299i
\(716\) 0 0
\(717\) −381.484 + 158.016i −0.532056 + 0.220385i
\(718\) 0 0
\(719\) −439.735 −0.611592 −0.305796 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(720\) 0 0
\(721\) 955.826i 1.32570i
\(722\) 0 0
\(723\) 27.8925 + 67.3385i 0.0385789 + 0.0931377i
\(724\) 0 0
\(725\) 4.18599 10.1059i 0.00577378 0.0139391i
\(726\) 0 0
\(727\) 137.308 + 137.308i 0.188869 + 0.188869i 0.795207 0.606338i \(-0.207362\pi\)
−0.606338 + 0.795207i \(0.707362\pi\)
\(728\) 0 0
\(729\) 831.529 + 831.529i 1.14064 + 1.14064i
\(730\) 0 0
\(731\) −9.13791 + 22.0609i −0.0125006 + 0.0301790i
\(732\) 0 0
\(733\) −57.7693 139.467i −0.0788121 0.190269i 0.879562 0.475784i \(-0.157836\pi\)
−0.958374 + 0.285514i \(0.907836\pi\)
\(734\) 0 0
\(735\) 412.584i 0.561339i
\(736\) 0 0
\(737\) 390.447 0.529778
\(738\) 0 0
\(739\) −536.590 + 222.263i −0.726103 + 0.300762i −0.714950 0.699176i \(-0.753550\pi\)
−0.0111538 + 0.999938i \(0.503550\pi\)
\(740\) 0 0
\(741\) −2203.76 912.828i −2.97404 1.23189i
\(742\) 0 0
\(743\) 907.324 907.324i 1.22116 1.22116i 0.253944 0.967219i \(-0.418272\pi\)
0.967219 0.253944i \(-0.0817279\pi\)
\(744\) 0 0
\(745\) 610.278 610.278i 0.819165 0.819165i
\(746\) 0 0
\(747\) −338.351 140.150i −0.452947 0.187617i
\(748\) 0 0
\(749\) −361.249 + 149.634i −0.482309 + 0.199779i
\(750\) 0 0
\(751\) −662.862 −0.882639 −0.441320 0.897350i \(-0.645489\pi\)
−0.441320 + 0.897350i \(0.645489\pi\)
\(752\) 0 0
\(753\) 1180.93i 1.56830i
\(754\) 0 0
\(755\) 85.0132 + 205.240i 0.112600 + 0.271841i
\(756\) 0 0
\(757\) −76.1190 + 183.767i −0.100553 + 0.242757i −0.966148 0.257988i \(-0.916941\pi\)
0.865595 + 0.500745i \(0.166941\pi\)
\(758\) 0 0
\(759\) 277.554 + 277.554i 0.365684 + 0.365684i
\(760\) 0 0
\(761\) 435.811 + 435.811i 0.572683 + 0.572683i 0.932877 0.360195i \(-0.117290\pi\)
−0.360195 + 0.932877i \(0.617290\pi\)
\(762\) 0 0
\(763\) 373.176 900.926i 0.489090 1.18077i
\(764\) 0 0
\(765\) −70.2012 169.481i −0.0917662 0.221543i
\(766\) 0 0
\(767\) 494.578i 0.644821i
\(768\) 0 0
\(769\) −1422.48 −1.84978 −0.924889 0.380237i \(-0.875843\pi\)
−0.924889 + 0.380237i \(0.875843\pi\)
\(770\) 0 0
\(771\) 570.642 236.368i 0.740132 0.306573i
\(772\) 0 0
\(773\) 0.439715 + 0.182136i 0.000568842 + 0.000235622i 0.382968 0.923762i \(-0.374902\pi\)
−0.382399 + 0.923997i \(0.624902\pi\)
\(774\) 0 0
\(775\) −3.62867 + 3.62867i −0.00468215 + 0.00468215i
\(776\) 0 0
\(777\) −1001.57 + 1001.57i −1.28902 + 1.28902i
\(778\) 0 0
\(779\) −722.878 299.426i −0.927956 0.384372i
\(780\) 0 0
\(781\) 429.160 177.764i 0.549500 0.227610i
\(782\) 0 0
\(783\) −13.7184 −0.0175204
\(784\) 0 0
\(785\) 313.420i 0.399262i
\(786\) 0 0
\(787\) −23.0303 55.6002i −0.0292635 0.0706482i 0.908572 0.417728i \(-0.137173\pi\)
−0.937836 + 0.347079i \(0.887173\pi\)
\(788\) 0 0
\(789\) 717.144 1731.34i 0.908928 2.19435i
\(790\) 0 0
\(791\) −770.590 770.590i −0.974197 0.974197i
\(792\) 0 0
\(793\) −943.175 943.175i −1.18938 1.18938i
\(794\) 0 0
\(795\) 64.0181 154.553i 0.0805259 0.194407i
\(796\) 0 0
\(797\) −49.3444 119.128i −0.0619127 0.149470i 0.889895 0.456165i \(-0.150777\pi\)
−0.951808 + 0.306694i \(0.900777\pi\)
\(798\) 0 0
\(799\) 226.918i 0.284003i
\(800\) 0 0
\(801\) −1925.65 −2.40406
\(802\) 0 0
\(803\) −595.542 + 246.682i −0.741647 + 0.307200i
\(804\) 0 0
\(805\) 237.138 + 98.2260i 0.294582 + 0.122020i
\(806\) 0 0
\(807\) 480.478 480.478i 0.595388 0.595388i
\(808\) 0 0
\(809\) 29.6470 29.6470i 0.0366465 0.0366465i −0.688546 0.725193i \(-0.741751\pi\)
0.725193 + 0.688546i \(0.241751\pi\)
\(810\) 0 0
\(811\) 755.209 + 312.818i 0.931207 + 0.385719i 0.796136 0.605117i \(-0.206874\pi\)
0.135071 + 0.990836i \(0.456874\pi\)
\(812\) 0 0
\(813\) 1604.21 664.486i 1.97320 0.817326i
\(814\) 0 0
\(815\) 16.4144 0.0201404
\(816\) 0 0
\(817\) 183.741i 0.224897i
\(818\) 0 0
\(819\) −641.305 1548.25i −0.783034 1.89041i
\(820\) 0 0
\(821\) −120.712 + 291.425i −0.147031 + 0.354964i −0.980187 0.198074i \(-0.936531\pi\)
0.833156 + 0.553038i \(0.186531\pi\)
\(822\) 0 0
\(823\) 173.105 + 173.105i 0.210334 + 0.210334i 0.804409 0.594076i \(-0.202482\pi\)
−0.594076 + 0.804409i \(0.702482\pi\)
\(824\) 0 0
\(825\) 456.504 + 456.504i 0.553338 + 0.553338i
\(826\) 0 0
\(827\) 59.9551 144.744i 0.0724971 0.175023i −0.883477 0.468475i \(-0.844804\pi\)
0.955974 + 0.293451i \(0.0948038\pi\)
\(828\) 0 0
\(829\) −259.829 627.283i −0.313425 0.756674i −0.999573 0.0292124i \(-0.990700\pi\)
0.686149 0.727461i \(-0.259300\pi\)
\(830\) 0 0
\(831\) 265.840i 0.319904i
\(832\) 0 0
\(833\) −131.178 −0.157476
\(834\) 0 0
\(835\) −657.240 + 272.238i −0.787114 + 0.326033i
\(836\) 0 0
\(837\) 5.94599 + 2.46291i 0.00710394 + 0.00294255i
\(838\) 0 0
\(839\) 182.343 182.343i 0.217333 0.217333i −0.590040 0.807374i \(-0.700888\pi\)
0.807374 + 0.590040i \(0.200888\pi\)
\(840\) 0 0
\(841\) 594.334 594.334i 0.706699 0.706699i
\(842\) 0 0
\(843\) −523.436 216.814i −0.620920 0.257194i
\(844\) 0 0
\(845\) 109.526 45.3671i 0.129617 0.0536889i
\(846\) 0 0
\(847\) 395.354 0.466770
\(848\) 0 0
\(849\) 333.466i 0.392775i
\(850\) 0 0
\(851\) 124.657 + 300.947i 0.146482 + 0.353640i
\(852\) 0 0
\(853\) 300.757 726.092i 0.352588 0.851222i −0.643712 0.765268i \(-0.722606\pi\)
0.996299 0.0859535i \(-0.0273936\pi\)
\(854\) 0 0
\(855\) −998.134 998.134i −1.16741 1.16741i
\(856\) 0 0
\(857\) −545.430 545.430i −0.636441 0.636441i 0.313235 0.949676i \(-0.398587\pi\)
−0.949676 + 0.313235i \(0.898587\pi\)
\(858\) 0 0
\(859\) −230.652 + 556.843i −0.268512 + 0.648246i −0.999414 0.0342370i \(-0.989100\pi\)
0.730901 + 0.682483i \(0.239100\pi\)
\(860\) 0 0
\(861\) −353.979 854.580i −0.411125 0.992544i
\(862\) 0 0
\(863\) 980.846i 1.13655i −0.822837 0.568277i \(-0.807610\pi\)
0.822837 0.568277i \(-0.192390\pi\)
\(864\) 0 0
\(865\) 381.962 0.441574
\(866\) 0 0
\(867\) −1166.86 + 483.328i −1.34586 + 0.557472i
\(868\) 0 0
\(869\) 216.933 + 89.8568i 0.249636 + 0.103403i
\(870\) 0 0
\(871\) −456.143 + 456.143i −0.523701 + 0.523701i
\(872\) 0 0
\(873\) 726.865 726.865i 0.832606 0.832606i
\(874\) 0 0
\(875\) 1010.82 + 418.696i 1.15522 + 0.478509i
\(876\) 0 0
\(877\) −1395.56 + 578.059i −1.59129 + 0.659132i −0.990150 0.140014i \(-0.955285\pi\)
−0.601137 + 0.799146i \(0.705285\pi\)
\(878\) 0 0
\(879\) −1636.59 −1.86188
\(880\) 0 0
\(881\) 1513.62i 1.71807i 0.511914 + 0.859037i \(0.328937\pi\)
−0.511914 + 0.859037i \(0.671063\pi\)
\(882\) 0 0
\(883\) 91.3318 + 220.494i 0.103434 + 0.249711i 0.967121 0.254317i \(-0.0818508\pi\)
−0.863687 + 0.504028i \(0.831851\pi\)
\(884\) 0 0
\(885\) 188.470 455.006i 0.212960 0.514131i
\(886\) 0 0
\(887\) 892.828 + 892.828i 1.00657 + 1.00657i 0.999978 + 0.00659168i \(0.00209821\pi\)
0.00659168 + 0.999978i \(0.497902\pi\)
\(888\) 0 0
\(889\) 830.510 + 830.510i 0.934207 + 0.934207i
\(890\) 0 0
\(891\) −86.3777 + 208.534i −0.0969447 + 0.234045i
\(892\) 0 0
\(893\) 668.203 + 1613.19i 0.748268 + 1.80648i
\(894\) 0 0
\(895\) 357.168i 0.399071i
\(896\) 0 0
\(897\) −648.512 −0.722978
\(898\) 0 0
\(899\) 0.210208 0.0870710i 0.000233824 9.68531e-5i
\(900\) 0 0
\(901\) 49.1388 + 20.3540i 0.0545381 + 0.0225904i
\(902\) 0 0
\(903\) −153.596 + 153.596i −0.170095 + 0.170095i
\(904\) 0 0
\(905\) −452.680 + 452.680i −0.500199 + 0.500199i
\(906\) 0 0
\(907\) 469.078 + 194.298i 0.517175 + 0.214221i 0.625976 0.779843i \(-0.284701\pi\)
−0.108800 + 0.994064i \(0.534701\pi\)
\(908\) 0 0
\(909\) −1371.82 + 568.226i −1.50915 + 0.625111i
\(910\) 0 0
\(911\) 469.566 0.515441 0.257720 0.966220i \(-0.417029\pi\)
0.257720 + 0.966220i \(0.417029\pi\)
\(912\) 0 0
\(913\) 242.446i 0.265549i
\(914\) 0 0
\(915\) −508.293 1227.13i −0.555512 1.34112i
\(916\) 0 0
\(917\) 832.872 2010.73i 0.908257 2.19273i
\(918\) 0 0
\(919\) 737.868 + 737.868i 0.802903 + 0.802903i 0.983548 0.180645i \(-0.0578186\pi\)
−0.180645 + 0.983548i \(0.557819\pi\)
\(920\) 0 0
\(921\) −806.710 806.710i −0.875907 0.875907i
\(922\) 0 0
\(923\) −293.696 + 709.045i −0.318197 + 0.768196i
\(924\) 0 0
\(925\) 205.027 + 494.979i 0.221651 + 0.535113i
\(926\) 0 0
\(927\) 1429.11i 1.54165i
\(928\) 0 0
\(929\) 796.539 0.857416 0.428708 0.903443i \(-0.358969\pi\)
0.428708 + 0.903443i \(0.358969\pi\)
\(930\) 0 0
\(931\) −932.554 + 386.277i −1.00167 + 0.414905i
\(932\) 0 0
\(933\) 274.312 + 113.624i 0.294011 + 0.121783i
\(934\) 0 0
\(935\) 85.8723 85.8723i 0.0918420 0.0918420i
\(936\) 0 0
\(937\) 764.262 764.262i 0.815648 0.815648i −0.169826 0.985474i \(-0.554321\pi\)
0.985474 + 0.169826i \(0.0543206\pi\)
\(938\) 0 0
\(939\) 171.875 + 71.1927i 0.183040 + 0.0758176i
\(940\) 0 0
\(941\) −113.549 + 47.0336i −0.120669 + 0.0499826i −0.442201 0.896916i \(-0.645802\pi\)
0.321532 + 0.946899i \(0.395802\pi\)
\(942\) 0 0
\(943\) −212.725 −0.225583
\(944\) 0 0
\(945\) 529.456i 0.560271i
\(946\) 0 0
\(947\) −396.726 957.782i −0.418930 1.01139i −0.982658 0.185425i \(-0.940634\pi\)
0.563729 0.825960i \(-0.309366\pi\)
\(948\) 0 0
\(949\) 407.560 983.937i 0.429463 1.03681i
\(950\) 0 0
\(951\) −1095.23 1095.23i −1.15166 1.15166i
\(952\) 0 0
\(953\) 391.136 + 391.136i 0.410426 + 0.410426i 0.881887 0.471461i \(-0.156273\pi\)
−0.471461 + 0.881887i \(0.656273\pi\)
\(954\) 0 0
\(955\) −394.443 + 952.270i −0.413030 + 0.997142i
\(956\) 0 0
\(957\) −10.9539 26.4452i −0.0114461 0.0276334i
\(958\) 0 0
\(959\) 1394.23i 1.45383i
\(960\) 0 0
\(961\) 960.893 0.999889
\(962\) 0 0
\(963\) 540.123 223.726i 0.560875 0.232322i
\(964\) 0 0
\(965\) 660.465 + 273.574i 0.684420 + 0.283496i
\(966\) 0 0
\(967\) −22.8939 + 22.8939i −0.0236752 + 0.0236752i −0.718845 0.695170i \(-0.755329\pi\)
0.695170 + 0.718845i \(0.255329\pi\)
\(968\) 0 0
\(969\) 534.009 534.009i 0.551093 0.551093i
\(970\) 0 0
\(971\) 619.006 + 256.401i 0.637493 + 0.264058i 0.677933 0.735124i \(-0.262876\pi\)
−0.0404399 + 0.999182i \(0.512876\pi\)
\(972\) 0 0
\(973\) 280.154 116.044i 0.287928 0.119264i
\(974\) 0 0
\(975\) −1066.63 −1.09398
\(976\) 0 0
\(977\) 430.856i 0.440999i −0.975387 0.220500i \(-0.929231\pi\)
0.975387 0.220500i \(-0.0707688\pi\)
\(978\) 0 0
\(979\) −487.844 1177.76i −0.498309 1.20302i
\(980\) 0 0
\(981\) −557.955 + 1347.02i −0.568762 + 1.37311i
\(982\) 0 0
\(983\) 154.209 + 154.209i 0.156876 + 0.156876i 0.781181 0.624305i \(-0.214618\pi\)
−0.624305 + 0.781181i \(0.714618\pi\)
\(984\) 0 0
\(985\) −552.978 552.978i −0.561399 0.561399i
\(986\) 0 0
\(987\) −789.945 + 1907.09i −0.800349 + 1.93221i
\(988\) 0 0
\(989\) 19.1168 + 46.1520i 0.0193294 + 0.0466653i
\(990\) 0 0
\(991\) 966.543i 0.975320i 0.873033 + 0.487660i \(0.162150\pi\)
−0.873033 + 0.487660i \(0.837850\pi\)
\(992\) 0 0
\(993\) −864.169 −0.870260
\(994\) 0 0
\(995\) 156.201 64.7007i 0.156986 0.0650258i
\(996\) 0 0
\(997\) −732.694 303.492i −0.734899 0.304405i −0.0163357 0.999867i \(-0.505200\pi\)
−0.718564 + 0.695461i \(0.755200\pi\)
\(998\) 0 0
\(999\) 475.120 475.120i 0.475596 0.475596i
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 256.3.h.b.31.1 28
4.3 odd 2 256.3.h.a.31.7 28
8.3 odd 2 128.3.h.a.15.1 28
8.5 even 2 32.3.h.a.27.4 yes 28
24.5 odd 2 288.3.u.a.91.4 28
32.3 odd 8 32.3.h.a.19.4 28
32.13 even 8 256.3.h.a.223.7 28
32.19 odd 8 inner 256.3.h.b.223.1 28
32.29 even 8 128.3.h.a.111.1 28
96.35 even 8 288.3.u.a.19.4 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.3.h.a.19.4 28 32.3 odd 8
32.3.h.a.27.4 yes 28 8.5 even 2
128.3.h.a.15.1 28 8.3 odd 2
128.3.h.a.111.1 28 32.29 even 8
256.3.h.a.31.7 28 4.3 odd 2
256.3.h.a.223.7 28 32.13 even 8
256.3.h.b.31.1 28 1.1 even 1 trivial
256.3.h.b.223.1 28 32.19 odd 8 inner
288.3.u.a.19.4 28 96.35 even 8
288.3.u.a.91.4 28 24.5 odd 2