Properties

Label 128.4.g.a.81.4
Level $128$
Weight $4$
Character 128.81
Analytic conductor $7.552$
Analytic rank $0$
Dimension $44$
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] = [128,4,Mod(17,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.g (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: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 81.4
Character \(\chi\) \(=\) 128.81
Dual form 128.4.g.a.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.56924 - 1.89264i) q^{3} +(1.37033 + 3.30826i) q^{5} +(6.14642 - 6.14642i) q^{7} +(-1.79604 - 1.79604i) q^{9} +O(q^{10})\) \(q+(-4.56924 - 1.89264i) q^{3} +(1.37033 + 3.30826i) q^{5} +(6.14642 - 6.14642i) q^{7} +(-1.79604 - 1.79604i) q^{9} +(-17.2398 + 7.14095i) q^{11} +(-25.9810 + 62.7238i) q^{13} -17.7098i q^{15} +87.5919i q^{17} +(-48.8193 + 117.860i) q^{19} +(-39.7174 + 16.4515i) q^{21} +(55.7254 + 55.7254i) q^{23} +(79.3216 - 79.3216i) q^{25} +(55.9086 + 134.975i) q^{27} +(-114.139 - 47.2779i) q^{29} -229.997 q^{31} +92.2879 q^{33} +(28.7565 + 11.9113i) q^{35} +(-123.846 - 298.992i) q^{37} +(237.427 - 237.427i) q^{39} +(111.562 + 111.562i) q^{41} +(-76.5574 + 31.7111i) q^{43} +(3.48061 - 8.40293i) q^{45} -367.843i q^{47} +267.443i q^{49} +(165.780 - 400.228i) q^{51} +(-244.994 + 101.480i) q^{53} +(-47.2482 - 47.2482i) q^{55} +(446.134 - 446.134i) q^{57} +(183.495 + 442.997i) q^{59} +(524.604 + 217.298i) q^{61} -22.0784 q^{63} -243.109 q^{65} +(-393.028 - 162.797i) q^{67} +(-149.155 - 360.091i) q^{69} +(354.722 - 354.722i) q^{71} +(-22.2443 - 22.2443i) q^{73} +(-512.566 + 212.312i) q^{75} +(-62.0716 + 149.854i) q^{77} +396.453i q^{79} -653.969i q^{81} +(410.445 - 990.903i) q^{83} +(-289.777 + 120.029i) q^{85} +(432.048 + 432.048i) q^{87} +(170.977 - 170.977i) q^{89} +(225.836 + 545.217i) q^{91} +(1050.91 + 435.301i) q^{93} -456.811 q^{95} -1723.11 q^{97} +(43.7888 + 18.1379i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\)

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
\(3\) −4.56924 1.89264i −0.879350 0.364239i −0.103105 0.994670i \(-0.532878\pi\)
−0.776245 + 0.630432i \(0.782878\pi\)
\(4\) 0 0
\(5\) 1.37033 + 3.30826i 0.122566 + 0.295900i 0.973239 0.229795i \(-0.0738055\pi\)
−0.850673 + 0.525695i \(0.823806\pi\)
\(6\) 0 0
\(7\) 6.14642 6.14642i 0.331875 0.331875i −0.521423 0.853298i \(-0.674599\pi\)
0.853298 + 0.521423i \(0.174599\pi\)
\(8\) 0 0
\(9\) −1.79604 1.79604i −0.0665200 0.0665200i
\(10\) 0 0
\(11\) −17.2398 + 7.14095i −0.472544 + 0.195734i −0.606230 0.795290i \(-0.707319\pi\)
0.133685 + 0.991024i \(0.457319\pi\)
\(12\) 0 0
\(13\) −25.9810 + 62.7238i −0.554296 + 1.33819i 0.359928 + 0.932980i \(0.382801\pi\)
−0.914224 + 0.405209i \(0.867199\pi\)
\(14\) 0 0
\(15\) 17.7098i 0.304843i
\(16\) 0 0
\(17\) 87.5919i 1.24966i 0.780762 + 0.624828i \(0.214831\pi\)
−0.780762 + 0.624828i \(0.785169\pi\)
\(18\) 0 0
\(19\) −48.8193 + 117.860i −0.589470 + 1.42311i 0.294541 + 0.955639i \(0.404833\pi\)
−0.884011 + 0.467467i \(0.845167\pi\)
\(20\) 0 0
\(21\) −39.7174 + 16.4515i −0.412716 + 0.170953i
\(22\) 0 0
\(23\) 55.7254 + 55.7254i 0.505198 + 0.505198i 0.913049 0.407851i \(-0.133722\pi\)
−0.407851 + 0.913049i \(0.633722\pi\)
\(24\) 0 0
\(25\) 79.3216 79.3216i 0.634572 0.634572i
\(26\) 0 0
\(27\) 55.9086 + 134.975i 0.398504 + 0.962074i
\(28\) 0 0
\(29\) −114.139 47.2779i −0.730865 0.302734i −0.0139575 0.999903i \(-0.504443\pi\)
−0.716907 + 0.697168i \(0.754443\pi\)
\(30\) 0 0
\(31\) −229.997 −1.33254 −0.666269 0.745711i \(-0.732110\pi\)
−0.666269 + 0.745711i \(0.732110\pi\)
\(32\) 0 0
\(33\) 92.2879 0.486826
\(34\) 0 0
\(35\) 28.7565 + 11.9113i 0.138878 + 0.0575253i
\(36\) 0 0
\(37\) −123.846 298.992i −0.550277 1.32849i −0.917272 0.398262i \(-0.869613\pi\)
0.366995 0.930223i \(-0.380387\pi\)
\(38\) 0 0
\(39\) 237.427 237.427i 0.974840 0.974840i
\(40\) 0 0
\(41\) 111.562 + 111.562i 0.424952 + 0.424952i 0.886904 0.461953i \(-0.152851\pi\)
−0.461953 + 0.886904i \(0.652851\pi\)
\(42\) 0 0
\(43\) −76.5574 + 31.7111i −0.271509 + 0.112463i −0.514284 0.857620i \(-0.671942\pi\)
0.242775 + 0.970083i \(0.421942\pi\)
\(44\) 0 0
\(45\) 3.48061 8.40293i 0.0115302 0.0278363i
\(46\) 0 0
\(47\) 367.843i 1.14161i −0.821087 0.570803i \(-0.806632\pi\)
0.821087 0.570803i \(-0.193368\pi\)
\(48\) 0 0
\(49\) 267.443i 0.779718i
\(50\) 0 0
\(51\) 165.780 400.228i 0.455173 1.09889i
\(52\) 0 0
\(53\) −244.994 + 101.480i −0.634952 + 0.263006i −0.676855 0.736116i \(-0.736658\pi\)
0.0419036 + 0.999122i \(0.486658\pi\)
\(54\) 0 0
\(55\) −47.2482 47.2482i −0.115835 0.115835i
\(56\) 0 0
\(57\) 446.134 446.134i 1.03670 1.03670i
\(58\) 0 0
\(59\) 183.495 + 442.997i 0.404899 + 0.977513i 0.986459 + 0.164009i \(0.0524427\pi\)
−0.581560 + 0.813504i \(0.697557\pi\)
\(60\) 0 0
\(61\) 524.604 + 217.298i 1.10113 + 0.456101i 0.857873 0.513862i \(-0.171786\pi\)
0.243252 + 0.969963i \(0.421786\pi\)
\(62\) 0 0
\(63\) −22.0784 −0.0441527
\(64\) 0 0
\(65\) −243.109 −0.463907
\(66\) 0 0
\(67\) −393.028 162.797i −0.716656 0.296849i −0.00560040 0.999984i \(-0.501783\pi\)
−0.711056 + 0.703136i \(0.751783\pi\)
\(68\) 0 0
\(69\) −149.155 360.091i −0.260233 0.628259i
\(70\) 0 0
\(71\) 354.722 354.722i 0.592926 0.592926i −0.345495 0.938421i \(-0.612289\pi\)
0.938421 + 0.345495i \(0.112289\pi\)
\(72\) 0 0
\(73\) −22.2443 22.2443i −0.0356644 0.0356644i 0.689050 0.724714i \(-0.258028\pi\)
−0.724714 + 0.689050i \(0.758028\pi\)
\(74\) 0 0
\(75\) −512.566 + 212.312i −0.789147 + 0.326876i
\(76\) 0 0
\(77\) −62.0716 + 149.854i −0.0918664 + 0.221785i
\(78\) 0 0
\(79\) 396.453i 0.564614i 0.959324 + 0.282307i \(0.0910996\pi\)
−0.959324 + 0.282307i \(0.908900\pi\)
\(80\) 0 0
\(81\) 653.969i 0.897077i
\(82\) 0 0
\(83\) 410.445 990.903i 0.542798 1.31043i −0.379943 0.925010i \(-0.624057\pi\)
0.922741 0.385420i \(-0.125943\pi\)
\(84\) 0 0
\(85\) −289.777 + 120.029i −0.369773 + 0.153165i
\(86\) 0 0
\(87\) 432.048 + 432.048i 0.532419 + 0.532419i
\(88\) 0 0
\(89\) 170.977 170.977i 0.203635 0.203635i −0.597920 0.801555i \(-0.704006\pi\)
0.801555 + 0.597920i \(0.204006\pi\)
\(90\) 0 0
\(91\) 225.836 + 545.217i 0.260155 + 0.628069i
\(92\) 0 0
\(93\) 1050.91 + 435.301i 1.17177 + 0.485362i
\(94\) 0 0
\(95\) −456.811 −0.493345
\(96\) 0 0
\(97\) −1723.11 −1.80367 −0.901833 0.432085i \(-0.857778\pi\)
−0.901833 + 0.432085i \(0.857778\pi\)
\(98\) 0 0
\(99\) 43.7888 + 18.1379i 0.0444539 + 0.0184134i
\(100\) 0 0
\(101\) −134.279 324.178i −0.132290 0.319376i 0.843829 0.536612i \(-0.180296\pi\)
−0.976119 + 0.217236i \(0.930296\pi\)
\(102\) 0 0
\(103\) −694.181 + 694.181i −0.664075 + 0.664075i −0.956338 0.292263i \(-0.905592\pi\)
0.292263 + 0.956338i \(0.405592\pi\)
\(104\) 0 0
\(105\) −108.852 108.852i −0.101170 0.101170i
\(106\) 0 0
\(107\) 264.682 109.635i 0.239138 0.0990544i −0.259896 0.965637i \(-0.583688\pi\)
0.499034 + 0.866582i \(0.333688\pi\)
\(108\) 0 0
\(109\) 69.2899 167.281i 0.0608878 0.146996i −0.890507 0.454969i \(-0.849650\pi\)
0.951395 + 0.307973i \(0.0996505\pi\)
\(110\) 0 0
\(111\) 1600.56i 1.36864i
\(112\) 0 0
\(113\) 2198.94i 1.83061i 0.402758 + 0.915306i \(0.368051\pi\)
−0.402758 + 0.915306i \(0.631949\pi\)
\(114\) 0 0
\(115\) −107.992 + 260.716i −0.0875680 + 0.211408i
\(116\) 0 0
\(117\) 159.318 65.9915i 0.125888 0.0521446i
\(118\) 0 0
\(119\) 538.376 + 538.376i 0.414730 + 0.414730i
\(120\) 0 0
\(121\) −694.943 + 694.943i −0.522121 + 0.522121i
\(122\) 0 0
\(123\) −298.606 720.898i −0.218897 0.528465i
\(124\) 0 0
\(125\) 784.645 + 325.011i 0.561446 + 0.232559i
\(126\) 0 0
\(127\) −1982.34 −1.38507 −0.692535 0.721384i \(-0.743506\pi\)
−0.692535 + 0.721384i \(0.743506\pi\)
\(128\) 0 0
\(129\) 409.827 0.279715
\(130\) 0 0
\(131\) −201.609 83.5090i −0.134463 0.0556963i 0.314437 0.949278i \(-0.398184\pi\)
−0.448900 + 0.893582i \(0.648184\pi\)
\(132\) 0 0
\(133\) 424.354 + 1024.48i 0.276663 + 0.667924i
\(134\) 0 0
\(135\) −369.920 + 369.920i −0.235834 + 0.235834i
\(136\) 0 0
\(137\) −332.511 332.511i −0.207360 0.207360i 0.595784 0.803144i \(-0.296841\pi\)
−0.803144 + 0.595784i \(0.796841\pi\)
\(138\) 0 0
\(139\) 917.619 380.090i 0.559938 0.231934i −0.0847201 0.996405i \(-0.527000\pi\)
0.644658 + 0.764471i \(0.277000\pi\)
\(140\) 0 0
\(141\) −696.195 + 1680.76i −0.415817 + 1.00387i
\(142\) 0 0
\(143\) 1266.87i 0.740848i
\(144\) 0 0
\(145\) 442.388i 0.253368i
\(146\) 0 0
\(147\) 506.174 1222.01i 0.284003 0.685645i
\(148\) 0 0
\(149\) 2064.41 855.107i 1.13505 0.470155i 0.265558 0.964095i \(-0.414444\pi\)
0.869496 + 0.493940i \(0.164444\pi\)
\(150\) 0 0
\(151\) 609.506 + 609.506i 0.328482 + 0.328482i 0.852009 0.523527i \(-0.175384\pi\)
−0.523527 + 0.852009i \(0.675384\pi\)
\(152\) 0 0
\(153\) 157.319 157.319i 0.0831272 0.0831272i
\(154\) 0 0
\(155\) −315.171 760.890i −0.163323 0.394298i
\(156\) 0 0
\(157\) −2898.83 1200.73i −1.47358 0.610376i −0.505906 0.862589i \(-0.668842\pi\)
−0.967672 + 0.252213i \(0.918842\pi\)
\(158\) 0 0
\(159\) 1311.50 0.654142
\(160\) 0 0
\(161\) 685.023 0.335325
\(162\) 0 0
\(163\) −112.364 46.5428i −0.0539942 0.0223651i 0.355523 0.934668i \(-0.384303\pi\)
−0.409517 + 0.912302i \(0.634303\pi\)
\(164\) 0 0
\(165\) 126.464 + 305.312i 0.0596681 + 0.144052i
\(166\) 0 0
\(167\) 1557.93 1557.93i 0.721895 0.721895i −0.247096 0.968991i \(-0.579476\pi\)
0.968991 + 0.247096i \(0.0794763\pi\)
\(168\) 0 0
\(169\) −1705.75 1705.75i −0.776398 0.776398i
\(170\) 0 0
\(171\) 299.363 124.000i 0.133877 0.0554535i
\(172\) 0 0
\(173\) 1117.57 2698.06i 0.491141 1.18572i −0.462999 0.886359i \(-0.653227\pi\)
0.954140 0.299361i \(-0.0967734\pi\)
\(174\) 0 0
\(175\) 975.087i 0.421198i
\(176\) 0 0
\(177\) 2371.45i 1.00706i
\(178\) 0 0
\(179\) −1394.97 + 3367.75i −0.582485 + 1.40624i 0.308069 + 0.951364i \(0.400317\pi\)
−0.890553 + 0.454879i \(0.849683\pi\)
\(180\) 0 0
\(181\) −16.5037 + 6.83606i −0.00677741 + 0.00280730i −0.386070 0.922470i \(-0.626168\pi\)
0.379292 + 0.925277i \(0.376168\pi\)
\(182\) 0 0
\(183\) −1985.77 1985.77i −0.802145 0.802145i
\(184\) 0 0
\(185\) 819.433 819.433i 0.325653 0.325653i
\(186\) 0 0
\(187\) −625.489 1510.06i −0.244600 0.590518i
\(188\) 0 0
\(189\) 1173.25 + 485.977i 0.451542 + 0.187035i
\(190\) 0 0
\(191\) 3640.57 1.37917 0.689587 0.724203i \(-0.257792\pi\)
0.689587 + 0.724203i \(0.257792\pi\)
\(192\) 0 0
\(193\) 3234.02 1.20616 0.603082 0.797679i \(-0.293939\pi\)
0.603082 + 0.797679i \(0.293939\pi\)
\(194\) 0 0
\(195\) 1110.82 + 460.118i 0.407937 + 0.168973i
\(196\) 0 0
\(197\) −556.709 1344.01i −0.201339 0.486076i 0.790670 0.612243i \(-0.209733\pi\)
−0.992009 + 0.126167i \(0.959733\pi\)
\(198\) 0 0
\(199\) −1366.73 + 1366.73i −0.486860 + 0.486860i −0.907314 0.420454i \(-0.861871\pi\)
0.420454 + 0.907314i \(0.361871\pi\)
\(200\) 0 0
\(201\) 1487.72 + 1487.72i 0.522068 + 0.522068i
\(202\) 0 0
\(203\) −992.136 + 410.956i −0.343026 + 0.142086i
\(204\) 0 0
\(205\) −216.199 + 521.951i −0.0736586 + 0.177828i
\(206\) 0 0
\(207\) 200.170i 0.0672116i
\(208\) 0 0
\(209\) 2380.50i 0.787860i
\(210\) 0 0
\(211\) −1144.59 + 2763.28i −0.373444 + 0.901574i 0.619717 + 0.784825i \(0.287247\pi\)
−0.993161 + 0.116749i \(0.962753\pi\)
\(212\) 0 0
\(213\) −2292.17 + 949.447i −0.737356 + 0.305423i
\(214\) 0 0
\(215\) −209.817 209.817i −0.0665554 0.0665554i
\(216\) 0 0
\(217\) −1413.66 + 1413.66i −0.442236 + 0.442236i
\(218\) 0 0
\(219\) 59.5391 + 143.740i 0.0183711 + 0.0443519i
\(220\) 0 0
\(221\) −5494.09 2275.73i −1.67228 0.692679i
\(222\) 0 0
\(223\) 1712.42 0.514226 0.257113 0.966381i \(-0.417229\pi\)
0.257113 + 0.966381i \(0.417229\pi\)
\(224\) 0 0
\(225\) −284.930 −0.0844236
\(226\) 0 0
\(227\) 2071.09 + 857.873i 0.605564 + 0.250833i 0.664330 0.747439i \(-0.268717\pi\)
−0.0587668 + 0.998272i \(0.518717\pi\)
\(228\) 0 0
\(229\) 1656.50 + 3999.14i 0.478010 + 1.15402i 0.960541 + 0.278140i \(0.0897179\pi\)
−0.482530 + 0.875879i \(0.660282\pi\)
\(230\) 0 0
\(231\) 567.240 567.240i 0.161565 0.161565i
\(232\) 0 0
\(233\) 4732.54 + 4732.54i 1.33064 + 1.33064i 0.904797 + 0.425844i \(0.140023\pi\)
0.425844 + 0.904797i \(0.359977\pi\)
\(234\) 0 0
\(235\) 1216.92 504.065i 0.337801 0.139922i
\(236\) 0 0
\(237\) 750.343 1811.49i 0.205654 0.496493i
\(238\) 0 0
\(239\) 3120.14i 0.844457i −0.906489 0.422229i \(-0.861248\pi\)
0.906489 0.422229i \(-0.138752\pi\)
\(240\) 0 0
\(241\) 1994.40i 0.533073i 0.963825 + 0.266537i \(0.0858793\pi\)
−0.963825 + 0.266537i \(0.914121\pi\)
\(242\) 0 0
\(243\) 271.803 656.192i 0.0717539 0.173229i
\(244\) 0 0
\(245\) −884.771 + 366.484i −0.230718 + 0.0955666i
\(246\) 0 0
\(247\) −6124.27 6124.27i −1.57764 1.57764i
\(248\) 0 0
\(249\) −3750.84 + 3750.84i −0.954619 + 0.954619i
\(250\) 0 0
\(251\) 759.835 + 1834.40i 0.191077 + 0.461301i 0.990163 0.139915i \(-0.0446830\pi\)
−0.799086 + 0.601216i \(0.794683\pi\)
\(252\) 0 0
\(253\) −1358.63 562.761i −0.337613 0.139844i
\(254\) 0 0
\(255\) 1551.23 0.380948
\(256\) 0 0
\(257\) 5357.21 1.30029 0.650143 0.759812i \(-0.274709\pi\)
0.650143 + 0.759812i \(0.274709\pi\)
\(258\) 0 0
\(259\) −2598.94 1076.52i −0.623515 0.258268i
\(260\) 0 0
\(261\) 120.085 + 289.911i 0.0284793 + 0.0687551i
\(262\) 0 0
\(263\) −3387.07 + 3387.07i −0.794127 + 0.794127i −0.982162 0.188035i \(-0.939788\pi\)
0.188035 + 0.982162i \(0.439788\pi\)
\(264\) 0 0
\(265\) −671.442 671.442i −0.155647 0.155647i
\(266\) 0 0
\(267\) −1104.83 + 457.636i −0.253238 + 0.104895i
\(268\) 0 0
\(269\) −225.824 + 545.188i −0.0511849 + 0.123571i −0.947404 0.320041i \(-0.896303\pi\)
0.896219 + 0.443612i \(0.146303\pi\)
\(270\) 0 0
\(271\) 4177.50i 0.936403i 0.883622 + 0.468201i \(0.155098\pi\)
−0.883622 + 0.468201i \(0.844902\pi\)
\(272\) 0 0
\(273\) 2918.65i 0.647051i
\(274\) 0 0
\(275\) −801.055 + 1933.92i −0.175656 + 0.424071i
\(276\) 0 0
\(277\) −2459.54 + 1018.78i −0.533500 + 0.220983i −0.633136 0.774041i \(-0.718232\pi\)
0.0996351 + 0.995024i \(0.468232\pi\)
\(278\) 0 0
\(279\) 413.084 + 413.084i 0.0886405 + 0.0886405i
\(280\) 0 0
\(281\) −1012.51 + 1012.51i −0.214952 + 0.214952i −0.806367 0.591415i \(-0.798569\pi\)
0.591415 + 0.806367i \(0.298569\pi\)
\(282\) 0 0
\(283\) −567.782 1370.75i −0.119262 0.287924i 0.852963 0.521971i \(-0.174803\pi\)
−0.972225 + 0.234047i \(0.924803\pi\)
\(284\) 0 0
\(285\) 2087.28 + 864.578i 0.433823 + 0.179695i
\(286\) 0 0
\(287\) 1371.41 0.282062
\(288\) 0 0
\(289\) −2759.34 −0.561640
\(290\) 0 0
\(291\) 7873.31 + 3261.23i 1.58605 + 0.656965i
\(292\) 0 0
\(293\) −421.800 1018.32i −0.0841018 0.203040i 0.876234 0.481886i \(-0.160048\pi\)
−0.960336 + 0.278846i \(0.910048\pi\)
\(294\) 0 0
\(295\) −1214.10 + 1214.10i −0.239619 + 0.239619i
\(296\) 0 0
\(297\) −1927.70 1927.70i −0.376622 0.376622i
\(298\) 0 0
\(299\) −4943.12 + 2047.51i −0.956079 + 0.396021i
\(300\) 0 0
\(301\) −275.644 + 665.464i −0.0527836 + 0.127431i
\(302\) 0 0
\(303\) 1735.39i 0.329028i
\(304\) 0 0
\(305\) 2033.29i 0.381725i
\(306\) 0 0
\(307\) −2140.75 + 5168.22i −0.397977 + 0.960800i 0.590169 + 0.807280i \(0.299061\pi\)
−0.988145 + 0.153521i \(0.950939\pi\)
\(308\) 0 0
\(309\) 4485.71 1858.04i 0.825836 0.342072i
\(310\) 0 0
\(311\) 5781.99 + 5781.99i 1.05423 + 1.05423i 0.998443 + 0.0557904i \(0.0177679\pi\)
0.0557904 + 0.998443i \(0.482232\pi\)
\(312\) 0 0
\(313\) 804.942 804.942i 0.145361 0.145361i −0.630681 0.776042i \(-0.717224\pi\)
0.776042 + 0.630681i \(0.217224\pi\)
\(314\) 0 0
\(315\) −30.2546 73.0412i −0.00541161 0.0130648i
\(316\) 0 0
\(317\) −1951.08 808.164i −0.345690 0.143189i 0.203082 0.979162i \(-0.434904\pi\)
−0.548771 + 0.835972i \(0.684904\pi\)
\(318\) 0 0
\(319\) 2305.34 0.404621
\(320\) 0 0
\(321\) −1416.90 −0.246366
\(322\) 0 0
\(323\) −10323.6 4276.18i −1.77839 0.736634i
\(324\) 0 0
\(325\) 2914.49 + 7036.21i 0.497437 + 1.20092i
\(326\) 0 0
\(327\) −633.204 + 633.204i −0.107083 + 0.107083i
\(328\) 0 0
\(329\) −2260.92 2260.92i −0.378871 0.378871i
\(330\) 0 0
\(331\) 7533.21 3120.36i 1.25095 0.518158i 0.343826 0.939033i \(-0.388277\pi\)
0.907119 + 0.420875i \(0.138277\pi\)
\(332\) 0 0
\(333\) −314.568 + 759.435i −0.0517665 + 0.124975i
\(334\) 0 0
\(335\) 1523.32i 0.248442i
\(336\) 0 0
\(337\) 3018.53i 0.487922i −0.969785 0.243961i \(-0.921553\pi\)
0.969785 0.243961i \(-0.0784469\pi\)
\(338\) 0 0
\(339\) 4161.81 10047.5i 0.666780 1.60975i
\(340\) 0 0
\(341\) 3965.10 1642.40i 0.629683 0.260823i
\(342\) 0 0
\(343\) 3752.04 + 3752.04i 0.590644 + 0.590644i
\(344\) 0 0
\(345\) 986.884 986.884i 0.154006 0.154006i
\(346\) 0 0
\(347\) −1631.78 3939.46i −0.252445 0.609456i 0.745955 0.665996i \(-0.231993\pi\)
−0.998400 + 0.0565400i \(0.981993\pi\)
\(348\) 0 0
\(349\) 2555.50 + 1058.52i 0.391956 + 0.162354i 0.569953 0.821678i \(-0.306962\pi\)
−0.177996 + 0.984031i \(0.556962\pi\)
\(350\) 0 0
\(351\) −9918.72 −1.50833
\(352\) 0 0
\(353\) −4332.58 −0.653257 −0.326629 0.945153i \(-0.605913\pi\)
−0.326629 + 0.945153i \(0.605913\pi\)
\(354\) 0 0
\(355\) 1659.60 + 687.427i 0.248119 + 0.102774i
\(356\) 0 0
\(357\) −1441.02 3478.92i −0.213632 0.515753i
\(358\) 0 0
\(359\) 9290.59 9290.59i 1.36585 1.36585i 0.499574 0.866271i \(-0.333490\pi\)
0.866271 0.499574i \(-0.166510\pi\)
\(360\) 0 0
\(361\) −6657.67 6657.67i −0.970648 0.970648i
\(362\) 0 0
\(363\) 4490.63 1860.08i 0.649303 0.268950i
\(364\) 0 0
\(365\) 43.1080 104.072i 0.00618186 0.0149243i
\(366\) 0 0
\(367\) 2140.92i 0.304511i 0.988341 + 0.152255i \(0.0486536\pi\)
−0.988341 + 0.152255i \(0.951346\pi\)
\(368\) 0 0
\(369\) 400.739i 0.0565356i
\(370\) 0 0
\(371\) −882.096 + 2129.57i −0.123440 + 0.298010i
\(372\) 0 0
\(373\) 4123.46 1707.99i 0.572399 0.237095i −0.0776592 0.996980i \(-0.524745\pi\)
0.650058 + 0.759885i \(0.274745\pi\)
\(374\) 0 0
\(375\) −2970.10 2970.10i −0.409001 0.409001i
\(376\) 0 0
\(377\) 5930.90 5930.90i 0.810231 0.810231i
\(378\) 0 0
\(379\) −2989.05 7216.20i −0.405111 0.978025i −0.986405 0.164331i \(-0.947453\pi\)
0.581294 0.813694i \(-0.302547\pi\)
\(380\) 0 0
\(381\) 9057.77 + 3751.85i 1.21796 + 0.504496i
\(382\) 0 0
\(383\) −6358.57 −0.848323 −0.424161 0.905587i \(-0.639431\pi\)
−0.424161 + 0.905587i \(0.639431\pi\)
\(384\) 0 0
\(385\) −580.814 −0.0768858
\(386\) 0 0
\(387\) 194.455 + 80.5458i 0.0255418 + 0.0105798i
\(388\) 0 0
\(389\) −3947.57 9530.27i −0.514523 1.24217i −0.941226 0.337777i \(-0.890325\pi\)
0.426703 0.904392i \(-0.359675\pi\)
\(390\) 0 0
\(391\) −4881.09 + 4881.09i −0.631324 + 0.631324i
\(392\) 0 0
\(393\) 763.145 + 763.145i 0.0979531 + 0.0979531i
\(394\) 0 0
\(395\) −1311.57 + 543.270i −0.167069 + 0.0692022i
\(396\) 0 0
\(397\) 1253.91 3027.20i 0.158519 0.382698i −0.824587 0.565735i \(-0.808593\pi\)
0.983106 + 0.183037i \(0.0585927\pi\)
\(398\) 0 0
\(399\) 5484.25i 0.688110i
\(400\) 0 0
\(401\) 13053.8i 1.62562i 0.582528 + 0.812810i \(0.302063\pi\)
−0.582528 + 0.812810i \(0.697937\pi\)
\(402\) 0 0
\(403\) 5975.56 14426.3i 0.738620 1.78319i
\(404\) 0 0
\(405\) 2163.50 896.150i 0.265445 0.109951i
\(406\) 0 0
\(407\) 4270.17 + 4270.17i 0.520060 + 0.520060i
\(408\) 0 0
\(409\) 652.548 652.548i 0.0788910 0.0788910i −0.666560 0.745451i \(-0.732234\pi\)
0.745451 + 0.666560i \(0.232234\pi\)
\(410\) 0 0
\(411\) 889.998 + 2148.65i 0.106814 + 0.257871i
\(412\) 0 0
\(413\) 3850.68 + 1595.00i 0.458789 + 0.190036i
\(414\) 0 0
\(415\) 3840.61 0.454284
\(416\) 0 0
\(417\) −4912.19 −0.576861
\(418\) 0 0
\(419\) 382.177 + 158.303i 0.0445598 + 0.0184573i 0.404852 0.914382i \(-0.367323\pi\)
−0.360292 + 0.932839i \(0.617323\pi\)
\(420\) 0 0
\(421\) −171.786 414.729i −0.0198868 0.0480111i 0.913624 0.406560i \(-0.133272\pi\)
−0.933511 + 0.358549i \(0.883272\pi\)
\(422\) 0 0
\(423\) −660.662 + 660.662i −0.0759396 + 0.0759396i
\(424\) 0 0
\(425\) 6947.92 + 6947.92i 0.792997 + 0.792997i
\(426\) 0 0
\(427\) 4560.04 1888.83i 0.516805 0.214068i
\(428\) 0 0
\(429\) −2397.73 + 5788.64i −0.269846 + 0.651465i
\(430\) 0 0
\(431\) 3644.76i 0.407336i 0.979040 + 0.203668i \(0.0652863\pi\)
−0.979040 + 0.203668i \(0.934714\pi\)
\(432\) 0 0
\(433\) 361.093i 0.0400763i −0.999799 0.0200382i \(-0.993621\pi\)
0.999799 0.0200382i \(-0.00637877\pi\)
\(434\) 0 0
\(435\) −837.280 + 2021.37i −0.0922863 + 0.222799i
\(436\) 0 0
\(437\) −9288.29 + 3847.34i −1.01675 + 0.421151i
\(438\) 0 0
\(439\) −2187.45 2187.45i −0.237817 0.237817i 0.578129 0.815945i \(-0.303783\pi\)
−0.815945 + 0.578129i \(0.803783\pi\)
\(440\) 0 0
\(441\) 480.339 480.339i 0.0518668 0.0518668i
\(442\) 0 0
\(443\) 6147.99 + 14842.6i 0.659367 + 1.59185i 0.798783 + 0.601620i \(0.205478\pi\)
−0.139415 + 0.990234i \(0.544522\pi\)
\(444\) 0 0
\(445\) 799.930 + 331.342i 0.0852142 + 0.0352969i
\(446\) 0 0
\(447\) −11051.2 −1.16936
\(448\) 0 0
\(449\) −13620.0 −1.43156 −0.715779 0.698326i \(-0.753928\pi\)
−0.715779 + 0.698326i \(0.753928\pi\)
\(450\) 0 0
\(451\) −2719.96 1126.64i −0.283986 0.117631i
\(452\) 0 0
\(453\) −1631.40 3938.55i −0.169205 0.408497i
\(454\) 0 0
\(455\) −1494.25 + 1494.25i −0.153959 + 0.153959i
\(456\) 0 0
\(457\) −3302.76 3302.76i −0.338067 0.338067i 0.517573 0.855639i \(-0.326836\pi\)
−0.855639 + 0.517573i \(0.826836\pi\)
\(458\) 0 0
\(459\) −11822.7 + 4897.14i −1.20226 + 0.497993i
\(460\) 0 0
\(461\) −2319.14 + 5598.89i −0.234301 + 0.565654i −0.996675 0.0814840i \(-0.974034\pi\)
0.762373 + 0.647138i \(0.224034\pi\)
\(462\) 0 0
\(463\) 11103.9i 1.11456i 0.830326 + 0.557279i \(0.188154\pi\)
−0.830326 + 0.557279i \(0.811846\pi\)
\(464\) 0 0
\(465\) 4073.19i 0.406214i
\(466\) 0 0
\(467\) 3094.31 7470.32i 0.306611 0.740225i −0.693199 0.720746i \(-0.743799\pi\)
0.999810 0.0194791i \(-0.00620077\pi\)
\(468\) 0 0
\(469\) −3416.33 + 1415.09i −0.336357 + 0.139324i
\(470\) 0 0
\(471\) 10972.9 + 10972.9i 1.07347 + 1.07347i
\(472\) 0 0
\(473\) 1093.39 1093.39i 0.106287 0.106287i
\(474\) 0 0
\(475\) 5476.44 + 13221.3i 0.529002 + 1.27712i
\(476\) 0 0
\(477\) 622.280 + 257.757i 0.0597322 + 0.0247419i
\(478\) 0 0
\(479\) 2969.80 0.283285 0.141643 0.989918i \(-0.454762\pi\)
0.141643 + 0.989918i \(0.454762\pi\)
\(480\) 0 0
\(481\) 21971.6 2.08278
\(482\) 0 0
\(483\) −3130.03 1296.50i −0.294868 0.122139i
\(484\) 0 0
\(485\) −2361.23 5700.50i −0.221067 0.533704i
\(486\) 0 0
\(487\) 14166.3 14166.3i 1.31814 1.31814i 0.402895 0.915246i \(-0.368004\pi\)
0.915246 0.402895i \(-0.131996\pi\)
\(488\) 0 0
\(489\) 425.331 + 425.331i 0.0393336 + 0.0393336i
\(490\) 0 0
\(491\) −16702.1 + 6918.24i −1.53514 + 0.635877i −0.980555 0.196247i \(-0.937125\pi\)
−0.554589 + 0.832124i \(0.687125\pi\)
\(492\) 0 0
\(493\) 4141.16 9997.65i 0.378314 0.913330i
\(494\) 0 0
\(495\) 169.719i 0.0154108i
\(496\) 0 0
\(497\) 4360.54i 0.393555i
\(498\) 0 0
\(499\) −4361.00 + 10528.4i −0.391232 + 0.944519i 0.598439 + 0.801168i \(0.295788\pi\)
−0.989672 + 0.143351i \(0.954212\pi\)
\(500\) 0 0
\(501\) −10067.2 + 4169.96i −0.897741 + 0.371856i
\(502\) 0 0
\(503\) −9988.91 9988.91i −0.885454 0.885454i 0.108628 0.994082i \(-0.465354\pi\)
−0.994082 + 0.108628i \(0.965354\pi\)
\(504\) 0 0
\(505\) 888.460 888.460i 0.0782890 0.0782890i
\(506\) 0 0
\(507\) 4565.59 + 11022.3i 0.399931 + 0.965520i
\(508\) 0 0
\(509\) −1708.35 707.622i −0.148765 0.0616204i 0.307059 0.951691i \(-0.400655\pi\)
−0.455824 + 0.890070i \(0.650655\pi\)
\(510\) 0 0
\(511\) −273.446 −0.0236723
\(512\) 0 0
\(513\) −18637.6 −1.60404
\(514\) 0 0
\(515\) −3247.78 1345.28i −0.277892 0.115107i
\(516\) 0 0
\(517\) 2626.75 + 6341.53i 0.223451 + 0.539459i
\(518\) 0 0
\(519\) −10212.9 + 10212.9i −0.863770 + 0.863770i
\(520\) 0 0
\(521\) 11373.3 + 11373.3i 0.956377 + 0.956377i 0.999087 0.0427108i \(-0.0135994\pi\)
−0.0427108 + 0.999087i \(0.513599\pi\)
\(522\) 0 0
\(523\) 3452.22 1429.96i 0.288633 0.119556i −0.233669 0.972316i \(-0.575073\pi\)
0.522302 + 0.852760i \(0.325073\pi\)
\(524\) 0 0
\(525\) −1845.49 + 4455.40i −0.153417 + 0.370380i
\(526\) 0 0
\(527\) 20145.9i 1.66521i
\(528\) 0 0
\(529\) 5956.35i 0.489550i
\(530\) 0 0
\(531\) 466.075 1125.21i 0.0380903 0.0919581i
\(532\) 0 0
\(533\) −9896.07 + 4099.08i −0.804214 + 0.333116i
\(534\) 0 0
\(535\) 725.402 + 725.402i 0.0586203 + 0.0586203i
\(536\) 0 0
\(537\) 12747.9 12747.9i 1.02442 1.02442i
\(538\) 0 0
\(539\) −1909.80 4610.66i −0.152617 0.368451i
\(540\) 0 0
\(541\) 13493.1 + 5589.03i 1.07230 + 0.444161i 0.847802 0.530314i \(-0.177926\pi\)
0.224498 + 0.974475i \(0.427926\pi\)
\(542\) 0 0
\(543\) 88.3476 0.00698224
\(544\) 0 0
\(545\) 648.357 0.0509588
\(546\) 0 0
\(547\) 10714.7 + 4438.16i 0.837525 + 0.346914i 0.759877 0.650067i \(-0.225259\pi\)
0.0776476 + 0.996981i \(0.475259\pi\)
\(548\) 0 0
\(549\) −551.934 1332.49i −0.0429070 0.103587i
\(550\) 0 0
\(551\) 11144.4 11144.4i 0.861645 0.861645i
\(552\) 0 0
\(553\) 2436.77 + 2436.77i 0.187381 + 0.187381i
\(554\) 0 0
\(555\) −5295.07 + 2193.29i −0.404979 + 0.167748i
\(556\) 0 0
\(557\) −8959.38 + 21629.9i −0.681546 + 1.64540i 0.0796083 + 0.996826i \(0.474633\pi\)
−0.761154 + 0.648571i \(0.775367\pi\)
\(558\) 0 0
\(559\) 5625.86i 0.425668i
\(560\) 0 0
\(561\) 8083.67i 0.608365i
\(562\) 0 0
\(563\) −144.332 + 348.448i −0.0108044 + 0.0260841i −0.929190 0.369603i \(-0.879494\pi\)
0.918385 + 0.395687i \(0.129494\pi\)
\(564\) 0 0
\(565\) −7274.68 + 3013.27i −0.541678 + 0.224370i
\(566\) 0 0
\(567\) −4019.56 4019.56i −0.297718 0.297718i
\(568\) 0 0
\(569\) −8107.90 + 8107.90i −0.597365 + 0.597365i −0.939611 0.342246i \(-0.888813\pi\)
0.342246 + 0.939611i \(0.388813\pi\)
\(570\) 0 0
\(571\) −3680.55 8885.63i −0.269748 0.651229i 0.729723 0.683743i \(-0.239649\pi\)
−0.999471 + 0.0325132i \(0.989649\pi\)
\(572\) 0 0
\(573\) −16634.6 6890.28i −1.21278 0.502349i
\(574\) 0 0
\(575\) 8840.46 0.641170
\(576\) 0 0
\(577\) 20668.9 1.49126 0.745629 0.666361i \(-0.232149\pi\)
0.745629 + 0.666361i \(0.232149\pi\)
\(578\) 0 0
\(579\) −14777.0 6120.83i −1.06064 0.439331i
\(580\) 0 0
\(581\) −3567.73 8613.27i −0.254758 0.615040i
\(582\) 0 0
\(583\) 3498.97 3498.97i 0.248564 0.248564i
\(584\) 0 0
\(585\) 436.634 + 436.634i 0.0308591 + 0.0308591i
\(586\) 0 0
\(587\) 834.858 345.810i 0.0587024 0.0243153i −0.353139 0.935571i \(-0.614886\pi\)
0.411841 + 0.911256i \(0.364886\pi\)
\(588\) 0 0
\(589\) 11228.3 27107.5i 0.785491 1.89634i
\(590\) 0 0
\(591\) 7194.77i 0.500767i
\(592\) 0 0
\(593\) 8473.46i 0.586785i 0.955992 + 0.293392i \(0.0947842\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(594\) 0 0
\(595\) −1043.34 + 2518.84i −0.0718868 + 0.173550i
\(596\) 0 0
\(597\) 8831.67 3658.20i 0.605454 0.250787i
\(598\) 0 0
\(599\) 4342.08 + 4342.08i 0.296181 + 0.296181i 0.839516 0.543335i \(-0.182839\pi\)
−0.543335 + 0.839516i \(0.682839\pi\)
\(600\) 0 0
\(601\) −6136.50 + 6136.50i −0.416494 + 0.416494i −0.883993 0.467499i \(-0.845155\pi\)
0.467499 + 0.883993i \(0.345155\pi\)
\(602\) 0 0
\(603\) 413.503 + 998.284i 0.0279256 + 0.0674184i
\(604\) 0 0
\(605\) −3251.35 1346.75i −0.218489 0.0905013i
\(606\) 0 0
\(607\) −3622.97 −0.242260 −0.121130 0.992637i \(-0.538652\pi\)
−0.121130 + 0.992637i \(0.538652\pi\)
\(608\) 0 0
\(609\) 5311.10 0.353393
\(610\) 0 0
\(611\) 23072.5 + 9556.95i 1.52768 + 0.632787i
\(612\) 0 0
\(613\) 1515.46 + 3658.63i 0.0998510 + 0.241062i 0.965909 0.258882i \(-0.0833541\pi\)
−0.866058 + 0.499944i \(0.833354\pi\)
\(614\) 0 0
\(615\) 1975.73 1975.73i 0.129543 0.129543i
\(616\) 0 0
\(617\) −5257.77 5257.77i −0.343063 0.343063i 0.514454 0.857518i \(-0.327994\pi\)
−0.857518 + 0.514454i \(0.827994\pi\)
\(618\) 0 0
\(619\) 23417.3 9699.75i 1.52055 0.629831i 0.542846 0.839832i \(-0.317347\pi\)
0.977701 + 0.210000i \(0.0673466\pi\)
\(620\) 0 0
\(621\) −4406.02 + 10637.1i −0.284714 + 0.687361i
\(622\) 0 0
\(623\) 2101.79i 0.135163i
\(624\) 0 0
\(625\) 10981.0i 0.702786i
\(626\) 0 0
\(627\) −4505.43 + 10877.1i −0.286969 + 0.692804i
\(628\) 0 0
\(629\) 26189.3 10847.9i 1.66015 0.687656i
\(630\) 0 0
\(631\) −19776.5 19776.5i −1.24768 1.24768i −0.956740 0.290944i \(-0.906031\pi\)
−0.290944 0.956740i \(-0.593969\pi\)
\(632\) 0 0
\(633\) 10459.8 10459.8i 0.656776 0.656776i
\(634\) 0 0
\(635\) −2716.45 6558.08i −0.169762 0.409842i
\(636\) 0 0
\(637\) −16775.0 6948.45i −1.04341 0.432194i
\(638\) 0 0
\(639\) −1274.19 −0.0788829
\(640\) 0 0
\(641\) 304.324 0.0187521 0.00937603 0.999956i \(-0.497015\pi\)
0.00937603 + 0.999956i \(0.497015\pi\)
\(642\) 0 0
\(643\) −16207.7 6713.43i −0.994040 0.411745i −0.174432 0.984669i \(-0.555809\pi\)
−0.819608 + 0.572924i \(0.805809\pi\)
\(644\) 0 0
\(645\) 561.596 + 1355.81i 0.0342835 + 0.0827676i
\(646\) 0 0
\(647\) −13156.7 + 13156.7i −0.799450 + 0.799450i −0.983009 0.183559i \(-0.941238\pi\)
0.183559 + 0.983009i \(0.441238\pi\)
\(648\) 0 0
\(649\) −6326.83 6326.83i −0.382666 0.382666i
\(650\) 0 0
\(651\) 9134.88 3783.79i 0.549960 0.227801i
\(652\) 0 0
\(653\) 7533.14 18186.6i 0.451447 1.08989i −0.520326 0.853968i \(-0.674189\pi\)
0.971772 0.235921i \(-0.0758106\pi\)
\(654\) 0 0
\(655\) 781.408i 0.0466139i
\(656\) 0 0
\(657\) 79.9035i 0.00474480i
\(658\) 0 0
\(659\) 4387.91 10593.4i 0.259376 0.626189i −0.739521 0.673133i \(-0.764948\pi\)
0.998898 + 0.0469438i \(0.0149482\pi\)
\(660\) 0 0
\(661\) 17601.5 7290.76i 1.03573 0.429013i 0.200953 0.979601i \(-0.435596\pi\)
0.834777 + 0.550588i \(0.185596\pi\)
\(662\) 0 0
\(663\) 20796.7 + 20796.7i 1.21821 + 1.21821i
\(664\) 0 0
\(665\) −2807.75 + 2807.75i −0.163729 + 0.163729i
\(666\) 0 0
\(667\) −3725.86 8995.03i −0.216291 0.522172i
\(668\) 0 0
\(669\) −7824.47 3241.00i −0.452185 0.187301i
\(670\) 0 0
\(671\) −10595.8 −0.609605
\(672\) 0 0
\(673\) 1023.41 0.0586174 0.0293087 0.999570i \(-0.490669\pi\)
0.0293087 + 0.999570i \(0.490669\pi\)
\(674\) 0 0
\(675\) 15141.2 + 6271.69i 0.863385 + 0.357626i
\(676\) 0 0
\(677\) −3769.24 9099.74i −0.213979 0.516590i 0.780049 0.625718i \(-0.215194\pi\)
−0.994028 + 0.109128i \(0.965194\pi\)
\(678\) 0 0
\(679\) −10591.0 + 10591.0i −0.598592 + 0.598592i
\(680\) 0 0
\(681\) −7839.65 7839.65i −0.441139 0.441139i
\(682\) 0 0
\(683\) 2058.49 852.656i 0.115324 0.0477686i −0.324276 0.945963i \(-0.605120\pi\)
0.439599 + 0.898194i \(0.355120\pi\)
\(684\) 0 0
\(685\) 644.384 1555.68i 0.0359426 0.0867731i
\(686\) 0 0
\(687\) 21408.2i 1.18890i
\(688\) 0 0
\(689\) 18003.5i 0.995468i
\(690\) 0 0
\(691\) −11541.0 + 27862.4i −0.635370 + 1.53392i 0.197415 + 0.980320i \(0.436745\pi\)
−0.832784 + 0.553598i \(0.813255\pi\)
\(692\) 0 0
\(693\) 380.627 157.661i 0.0208641 0.00864220i
\(694\) 0 0
\(695\) 2514.87 + 2514.87i 0.137258 + 0.137258i
\(696\) 0 0
\(697\) −9771.90 + 9771.90i −0.531043 + 0.531043i
\(698\) 0 0
\(699\) −12667.1 30581.1i −0.685428 1.65477i
\(700\) 0 0
\(701\) 3783.29 + 1567.09i 0.203841 + 0.0844339i 0.482268 0.876024i \(-0.339813\pi\)
−0.278427 + 0.960457i \(0.589813\pi\)
\(702\) 0 0
\(703\) 41285.4 2.21495
\(704\) 0 0
\(705\) −6514.41 −0.348010
\(706\) 0 0
\(707\) −2817.87 1167.20i −0.149897 0.0620892i
\(708\) 0 0
\(709\) 7928.02 + 19139.9i 0.419948 + 1.01384i 0.982362 + 0.186988i \(0.0598727\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(710\) 0 0
\(711\) 712.046 712.046i 0.0375581 0.0375581i
\(712\) 0 0
\(713\) −12816.7 12816.7i −0.673196 0.673196i
\(714\) 0 0
\(715\) 4191.15 1736.03i 0.219217 0.0908025i
\(716\) 0 0
\(717\) −5905.31 + 14256.7i −0.307584 + 0.742573i
\(718\) 0 0
\(719\) 35836.7i 1.85881i 0.369064 + 0.929404i \(0.379678\pi\)
−0.369064 + 0.929404i \(0.620322\pi\)
\(720\) 0 0
\(721\) 8533.45i 0.440780i
\(722\) 0 0
\(723\) 3774.68 9112.89i 0.194166 0.468758i
\(724\) 0 0
\(725\) −12803.8 + 5303.53i −0.655894 + 0.271680i
\(726\) 0 0
\(727\) −17934.8 17934.8i −0.914946 0.914946i 0.0817100 0.996656i \(-0.473962\pi\)
−0.996656 + 0.0817100i \(0.973962\pi\)
\(728\) 0 0
\(729\) −14969.4 + 14969.4i −0.760523 + 0.760523i
\(730\) 0 0
\(731\) −2777.64 6705.81i −0.140540 0.339293i
\(732\) 0 0
\(733\) 12620.5 + 5227.58i 0.635946 + 0.263418i 0.677277 0.735728i \(-0.263160\pi\)
−0.0413310 + 0.999146i \(0.513160\pi\)
\(734\) 0 0
\(735\) 4736.35 0.237691
\(736\) 0 0
\(737\) 7938.23 0.396755
\(738\) 0 0
\(739\) 12203.1 + 5054.70i 0.607441 + 0.251610i 0.665134 0.746724i \(-0.268374\pi\)
−0.0576928 + 0.998334i \(0.518374\pi\)
\(740\) 0 0
\(741\) 16392.2 + 39574.3i 0.812662 + 1.96194i
\(742\) 0 0
\(743\) −3084.12 + 3084.12i −0.152282 + 0.152282i −0.779136 0.626854i \(-0.784342\pi\)
0.626854 + 0.779136i \(0.284342\pi\)
\(744\) 0 0
\(745\) 5657.83 + 5657.83i 0.278237 + 0.278237i
\(746\) 0 0
\(747\) −2516.88 + 1042.53i −0.123277 + 0.0510629i
\(748\) 0 0
\(749\) 952.985 2300.71i 0.0464904 0.112238i
\(750\) 0 0
\(751\) 30535.2i 1.48368i 0.670575 + 0.741842i \(0.266048\pi\)
−0.670575 + 0.741842i \(0.733952\pi\)
\(752\) 0 0
\(753\) 9819.92i 0.475243i
\(754\) 0 0
\(755\) −1181.18 + 2851.62i −0.0569372 + 0.137459i
\(756\) 0 0
\(757\) −8109.52 + 3359.07i −0.389360 + 0.161278i −0.568771 0.822496i \(-0.692581\pi\)
0.179411 + 0.983774i \(0.442581\pi\)
\(758\) 0 0
\(759\) 5142.78 + 5142.78i 0.245943 + 0.245943i
\(760\) 0 0
\(761\) −13700.0 + 13700.0i −0.652595 + 0.652595i −0.953617 0.301022i \(-0.902672\pi\)
0.301022 + 0.953617i \(0.402672\pi\)
\(762\) 0 0
\(763\) −602.292 1454.06i −0.0285772 0.0689915i
\(764\) 0 0
\(765\) 736.028 + 304.873i 0.0347858 + 0.0144088i
\(766\) 0 0
\(767\) −32553.8 −1.53253
\(768\) 0 0
\(769\) −5576.68 −0.261509 −0.130754 0.991415i \(-0.541740\pi\)
−0.130754 + 0.991415i \(0.541740\pi\)
\(770\) 0 0
\(771\) −24478.4 10139.3i −1.14341 0.473614i
\(772\) 0 0
\(773\) 12414.0 + 29969.9i 0.577618 + 1.39449i 0.894945 + 0.446177i \(0.147215\pi\)
−0.317327 + 0.948316i \(0.602785\pi\)
\(774\) 0 0
\(775\) −18243.7 + 18243.7i −0.845592 + 0.845592i
\(776\) 0 0
\(777\) 9837.72 + 9837.72i 0.454216 + 0.454216i
\(778\) 0 0
\(779\) −18595.1 + 7702.33i −0.855247 + 0.354255i
\(780\) 0 0
\(781\) −3582.27 + 8648.37i −0.164128 + 0.396240i
\(782\) 0 0
\(783\) 18049.2i 0.823787i
\(784\) 0 0
\(785\) 11235.5i 0.510842i
\(786\) 0 0
\(787\) −1888.35 + 4558.87i −0.0855302 + 0.206488i −0.960858 0.277042i \(-0.910646\pi\)
0.875328 + 0.483530i \(0.160646\pi\)
\(788\) 0 0
\(789\) 21886.8 9065.82i 0.987568 0.409064i
\(790\) 0 0
\(791\) 13515.6 + 13515.6i 0.607535 + 0.607535i
\(792\) 0 0
\(793\) −27259.5 + 27259.5i −1.22070 + 1.22070i
\(794\) 0 0
\(795\) 1797.18 + 4338.78i 0.0801753 + 0.193560i
\(796\) 0 0
\(797\) 8355.80 + 3461.09i 0.371365 + 0.153824i 0.560557 0.828116i \(-0.310587\pi\)
−0.189192 + 0.981940i \(0.560587\pi\)
\(798\) 0 0
\(799\) 32220.1 1.42661
\(800\) 0 0
\(801\) −614.163 −0.0270916
\(802\) 0 0
\(803\) 542.333 + 224.642i 0.0238338 + 0.00987227i
\(804\) 0 0
\(805\) 938.705 + 2266.23i 0.0410994 + 0.0992227i
\(806\) 0 0
\(807\) 2063.69 2063.69i 0.0900189 0.0900189i
\(808\) 0 0
\(809\) 9416.77 + 9416.77i 0.409241 + 0.409241i 0.881474 0.472233i \(-0.156552\pi\)
−0.472233 + 0.881474i \(0.656552\pi\)
\(810\) 0 0
\(811\) 4597.28 1904.26i 0.199053 0.0824506i −0.280930 0.959728i \(-0.590643\pi\)
0.479983 + 0.877278i \(0.340643\pi\)
\(812\) 0 0
\(813\) 7906.51 19088.0i 0.341074 0.823426i
\(814\) 0 0
\(815\) 435.509i 0.0187181i
\(816\) 0 0
\(817\) 10571.2i 0.452680i
\(818\) 0 0
\(819\) 573.621 1384.84i 0.0244737 0.0590847i
\(820\) 0 0
\(821\) 12716.7 5267.45i 0.540581 0.223916i −0.0956491 0.995415i \(-0.530493\pi\)
0.636231 + 0.771499i \(0.280493\pi\)
\(822\) 0 0
\(823\) −18764.2 18764.2i −0.794751 0.794751i 0.187511 0.982262i \(-0.439958\pi\)
−0.982262 + 0.187511i \(0.939958\pi\)
\(824\) 0 0
\(825\) 7320.42 7320.42i 0.308926 0.308926i
\(826\) 0 0
\(827\) 6248.37 + 15084.9i 0.262729 + 0.634284i 0.999105 0.0422880i \(-0.0134647\pi\)
−0.736376 + 0.676572i \(0.763465\pi\)
\(828\) 0 0
\(829\) −23591.6 9771.95i −0.988383 0.409401i −0.170858 0.985296i \(-0.554654\pi\)
−0.817524 + 0.575894i \(0.804654\pi\)
\(830\) 0 0
\(831\) 13166.4 0.549624
\(832\) 0 0
\(833\) −23425.8 −0.974379
\(834\) 0 0
\(835\) 7288.92 + 3019.17i 0.302088 + 0.125129i
\(836\) 0 0
\(837\) −12858.8 31043.9i −0.531022 1.28200i
\(838\) 0 0
\(839\) −26103.1 + 26103.1i −1.07411 + 1.07411i −0.0770872 + 0.997024i \(0.524562\pi\)
−0.997024 + 0.0770872i \(0.975438\pi\)
\(840\) 0 0
\(841\) −6453.12 6453.12i −0.264591 0.264591i
\(842\) 0 0
\(843\) 6542.73 2710.09i 0.267312 0.110724i
\(844\) 0 0
\(845\) 3305.62 7980.48i 0.134576 0.324896i
\(846\) 0 0
\(847\) 8542.81i 0.346558i
\(848\) 0 0
\(849\) 7337.88i 0.296626i
\(850\) 0 0
\(851\) 9760.05 23562.8i 0.393150 0.949147i
\(852\) 0 0
\(853\) −44742.9 + 18533.1i −1.79598 + 0.743918i −0.808026 + 0.589147i \(0.799464\pi\)
−0.987950 + 0.154770i \(0.950536\pi\)
\(854\) 0 0
\(855\) 820.451 + 820.451i 0.0328173 + 0.0328173i
\(856\) 0 0
\(857\) 21556.5 21556.5i 0.859225 0.859225i −0.132022 0.991247i \(-0.542147\pi\)
0.991247 + 0.132022i \(0.0421469\pi\)
\(858\) 0 0
\(859\) −10150.3 24505.0i −0.403171 0.973342i −0.986891 0.161387i \(-0.948403\pi\)
0.583720 0.811955i \(-0.301597\pi\)
\(860\) 0 0
\(861\) −6266.30 2595.58i −0.248031 0.102738i
\(862\) 0 0
\(863\) 9799.79 0.386545 0.193273 0.981145i \(-0.438090\pi\)
0.193273 + 0.981145i \(0.438090\pi\)
\(864\) 0 0
\(865\) 10457.3 0.411051
\(866\) 0 0
\(867\) 12608.1 + 5222.43i 0.493878 + 0.204571i
\(868\) 0 0
\(869\) −2831.05 6834.76i −0.110514 0.266805i
\(870\) 0 0
\(871\) 20422.5 20422.5i 0.794479 0.794479i
\(872\) 0 0
\(873\) 3094.78 + 3094.78i 0.119980 + 0.119980i
\(874\) 0 0
\(875\) 6820.41 2825.10i 0.263511 0.109150i
\(876\) 0 0
\(877\) 5253.32 12682.6i 0.202271 0.488326i −0.789896 0.613241i \(-0.789866\pi\)
0.992168 + 0.124914i \(0.0398656\pi\)
\(878\) 0 0
\(879\) 5451.24i 0.209176i
\(880\) 0 0
\(881\) 9710.12i 0.371331i −0.982613 0.185665i \(-0.940556\pi\)
0.982613 0.185665i \(-0.0594440\pi\)
\(882\) 0 0
\(883\) −6324.17 + 15267.9i −0.241025 + 0.581887i −0.997385 0.0722692i \(-0.976976\pi\)
0.756360 + 0.654156i \(0.226976\pi\)
\(884\) 0 0
\(885\) 7845.37 3249.66i 0.297988 0.123431i
\(886\) 0 0
\(887\) −33197.6 33197.6i −1.25667 1.25667i −0.952672 0.303999i \(-0.901678\pi\)
−0.303999 0.952672i \(-0.598322\pi\)
\(888\) 0 0
\(889\) −12184.3 + 12184.3i −0.459671 + 0.459671i
\(890\) 0 0
\(891\) 4669.96 + 11274.3i 0.175589 + 0.423908i
\(892\) 0 0
\(893\) 43354.1 + 17957.9i 1.62462 + 0.672942i
\(894\) 0 0
\(895\) −13053.0 −0.487499
\(896\) 0 0
\(897\) 26461.5 0.984975
\(898\) 0 0
\(899\) 26251.6 + 10873.8i 0.973905 + 0.403405i
\(900\) 0 0
\(901\) −8888.79 21459.4i −0.328667 0.793471i
\(902\) 0 0
\(903\) 2518.97 2518.97i 0.0928305 0.0928305i
\(904\) 0 0
\(905\) −45.2309 45.2309i −0.00166136 0.00166136i
\(906\) 0 0
\(907\) −4347.03 + 1800.60i −0.159141 + 0.0659183i −0.460832 0.887487i \(-0.652449\pi\)
0.301691 + 0.953406i \(0.402449\pi\)
\(908\) 0 0
\(909\) −341.067 + 823.408i −0.0124450 + 0.0300448i
\(910\) 0 0
\(911\) 27597.7i 1.00368i −0.864961 0.501840i \(-0.832657\pi\)
0.864961 0.501840i \(-0.167343\pi\)
\(912\) 0 0
\(913\) 20013.9i 0.725480i
\(914\) 0 0
\(915\) 3848.29 9290.60i 0.139039 0.335670i
\(916\) 0 0
\(917\) −1752.45 + 725.889i −0.0631091 + 0.0261406i
\(918\) 0 0
\(919\) 10437.8 + 10437.8i 0.374658 + 0.374658i 0.869171 0.494512i \(-0.164653\pi\)
−0.494512 + 0.869171i \(0.664653\pi\)
\(920\) 0 0
\(921\) 19563.2 19563.2i 0.699922 0.699922i
\(922\) 0 0
\(923\) 13033.5 + 31465.5i 0.464790 + 1.12210i
\(924\) 0 0
\(925\) −33540.2 13892.8i −1.19221 0.493830i
\(926\) 0 0
\(927\) 2493.56 0.0883485
\(928\) 0 0
\(929\) −692.011 −0.0244393 −0.0122197 0.999925i \(-0.503890\pi\)
−0.0122197 + 0.999925i \(0.503890\pi\)
\(930\) 0 0
\(931\) −31520.9 13056.4i −1.10962 0.459620i
\(932\) 0 0
\(933\) −15476.1 37362.5i −0.543047 1.31103i
\(934\) 0 0
\(935\) 4138.56 4138.56i 0.144754 0.144754i
\(936\) 0 0
\(937\) −7694.36 7694.36i −0.268264 0.268264i 0.560136 0.828401i \(-0.310749\pi\)
−0.828401 + 0.560136i \(0.810749\pi\)
\(938\) 0 0
\(939\) −5201.44 + 2154.51i −0.180769 + 0.0748772i
\(940\) 0 0
\(941\) −7735.22 + 18674.5i −0.267971 + 0.646940i −0.999388 0.0349914i \(-0.988860\pi\)
0.731416 + 0.681931i \(0.238860\pi\)
\(942\) 0 0
\(943\) 12433.6i 0.429369i
\(944\) 0 0
\(945\) 4547.36i 0.156535i
\(946\) 0 0
\(947\) −258.346 + 623.702i −0.00886496 + 0.0214019i −0.928250 0.371957i \(-0.878687\pi\)
0.919385 + 0.393359i \(0.128687\pi\)
\(948\) 0 0
\(949\) 1973.18 817.318i 0.0674943 0.0279571i
\(950\) 0 0
\(951\) 7385.39 + 7385.39i 0.251827 + 0.251827i
\(952\) 0 0
\(953\) 29841.2 29841.2i 1.01433 1.01433i 0.0144302 0.999896i \(-0.495407\pi\)
0.999896 0.0144302i \(-0.00459343\pi\)
\(954\) 0 0
\(955\) 4988.76 + 12043.9i 0.169039 + 0.408097i
\(956\) 0 0
\(957\) −10533.6 4363.18i −0.355804 0.147379i
\(958\) 0 0
\(959\) −4087.50 −0.137635
\(960\) 0 0
\(961\) 23107.6 0.775657
\(962\) 0 0
\(963\) −672.289 278.471i −0.0224966 0.00931840i
\(964\) 0 0
\(965\) 4431.66 + 10699.0i 0.147834 + 0.356903i
\(966\) 0 0
\(967\) 28561.9 28561.9i 0.949834 0.949834i −0.0489665 0.998800i \(-0.515593\pi\)
0.998800 + 0.0489665i \(0.0155928\pi\)
\(968\) 0 0
\(969\) 39077.7 + 39077.7i 1.29552 + 1.29552i
\(970\) 0 0
\(971\) 9671.99 4006.27i 0.319659 0.132407i −0.217084 0.976153i \(-0.569655\pi\)
0.536743 + 0.843746i \(0.319655\pi\)
\(972\) 0 0
\(973\) 3303.88 7976.26i 0.108857 0.262803i
\(974\) 0 0
\(975\) 37666.2i 1.23721i
\(976\) 0 0
\(977\) 35356.3i 1.15778i −0.815407 0.578889i \(-0.803487\pi\)
0.815407 0.578889i \(-0.196513\pi\)
\(978\) 0 0
\(979\) −1726.67 + 4168.54i −0.0563682 + 0.136085i
\(980\) 0 0
\(981\) −424.890 + 175.995i −0.0138284 + 0.00572793i
\(982\) 0 0
\(983\) 26544.3 + 26544.3i 0.861274 + 0.861274i 0.991486 0.130213i \(-0.0415660\pi\)
−0.130213 + 0.991486i \(0.541566\pi\)
\(984\) 0 0
\(985\) 3683.47 3683.47i 0.119153 0.119153i
\(986\) 0 0
\(987\) 6051.57 + 14609.8i 0.195161 + 0.471159i
\(988\) 0 0
\(989\) −6033.31 2499.08i −0.193982 0.0803500i
\(990\) 0 0
\(991\) −40304.4 −1.29194 −0.645969 0.763364i \(-0.723547\pi\)
−0.645969 + 0.763364i \(0.723547\pi\)
\(992\) 0 0
\(993\) −40326.8 −1.28875
\(994\) 0 0
\(995\) −6394.38 2648.64i −0.203734 0.0843895i
\(996\) 0 0
\(997\) 11444.4 + 27629.1i 0.363537 + 0.877656i 0.994777 + 0.102069i \(0.0325461\pi\)
−0.631240 + 0.775587i \(0.717454\pi\)
\(998\) 0 0
\(999\) 33432.4 33432.4i 1.05881 1.05881i
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 128.4.g.a.81.4 44
4.3 odd 2 32.4.g.a.29.9 yes 44
8.3 odd 2 256.4.g.b.161.4 44
8.5 even 2 256.4.g.a.161.8 44
32.5 even 8 256.4.g.a.97.8 44
32.11 odd 8 32.4.g.a.21.9 44
32.21 even 8 inner 128.4.g.a.49.4 44
32.27 odd 8 256.4.g.b.97.4 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.21.9 44 32.11 odd 8
32.4.g.a.29.9 yes 44 4.3 odd 2
128.4.g.a.49.4 44 32.21 even 8 inner
128.4.g.a.81.4 44 1.1 even 1 trivial
256.4.g.a.97.8 44 32.5 even 8
256.4.g.a.161.8 44 8.5 even 2
256.4.g.b.97.4 44 32.27 odd 8
256.4.g.b.161.4 44 8.3 odd 2