Properties

Label 224.3.o.c.207.3
Level $224$
Weight $3$
Character 224.207
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(79,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 207.3
Root \(1.83041 + 0.805972i\) of defining polynomial
Character \(\chi\) \(=\) 224.207
Dual form 224.3.o.c.79.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.717214 + 1.24225i) q^{3} +(-2.27256 + 1.31206i) q^{5} +(5.39122 - 4.46483i) q^{7} +(3.47121 + 6.01231i) q^{9} +O(q^{10})\) \(q+(-0.717214 + 1.24225i) q^{3} +(-2.27256 + 1.31206i) q^{5} +(5.39122 - 4.46483i) q^{7} +(3.47121 + 6.01231i) q^{9} +(-2.75399 + 4.77006i) q^{11} +6.22525i q^{13} -3.76412i q^{15} +(-10.5368 + 18.2502i) q^{17} +(12.6229 + 21.8634i) q^{19} +(1.67978 + 9.89949i) q^{21} +(-12.5463 + 7.24359i) q^{23} +(-9.05697 + 15.6871i) q^{25} -22.8682 q^{27} -29.9514i q^{29} +(23.5249 + 13.5821i) q^{31} +(-3.95041 - 6.84230i) q^{33} +(-6.39374 + 17.2202i) q^{35} +(31.5708 - 18.2274i) q^{37} +(-7.73332 - 4.46483i) q^{39} +37.4338 q^{41} +28.9008 q^{43} +(-15.7771 - 9.10890i) q^{45} +(-51.8379 + 29.9287i) q^{47} +(9.13053 - 48.1418i) q^{49} +(-15.1143 - 26.1787i) q^{51} +(36.5373 + 21.0948i) q^{53} -14.4537i q^{55} -36.2131 q^{57} +(14.6430 - 25.3625i) q^{59} +(-92.2850 + 53.2808i) q^{61} +(45.5580 + 16.9153i) q^{63} +(-8.16793 - 14.1473i) q^{65} +(56.4293 - 97.7384i) q^{67} -20.7808i q^{69} -94.6530i q^{71} +(32.3682 - 56.0634i) q^{73} +(-12.9916 - 22.5021i) q^{75} +(6.45012 + 38.0126i) q^{77} +(46.8433 - 27.0450i) q^{79} +(-14.8395 + 25.7027i) q^{81} -91.8824 q^{83} -55.2997i q^{85} +(37.2071 + 21.4815i) q^{87} +(-36.1385 - 62.5938i) q^{89} +(27.7947 + 33.5617i) q^{91} +(-33.7448 + 19.4826i) q^{93} +(-57.3724 - 33.1240i) q^{95} +97.4646 q^{97} -38.2388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 8 q^{9} + 14 q^{11} - 82 q^{17} + 94 q^{19} + 116 q^{25} + 60 q^{27} + 146 q^{33} - 270 q^{35} + 120 q^{41} - 40 q^{43} - 204 q^{49} + 106 q^{51} - 372 q^{57} - 62 q^{59} - 64 q^{65} + 178 q^{67} + 54 q^{73} - 140 q^{75} + 206 q^{81} + 392 q^{83} - 26 q^{89} + 88 q^{91} - 184 q^{97} - 872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(3\) −0.717214 + 1.24225i −0.239071 + 0.414084i −0.960448 0.278459i \(-0.910176\pi\)
0.721377 + 0.692543i \(0.243510\pi\)
\(4\) 0 0
\(5\) −2.27256 + 1.31206i −0.454513 + 0.262413i −0.709734 0.704470i \(-0.751185\pi\)
0.255222 + 0.966883i \(0.417852\pi\)
\(6\) 0 0
\(7\) 5.39122 4.46483i 0.770174 0.637833i
\(8\) 0 0
\(9\) 3.47121 + 6.01231i 0.385690 + 0.668034i
\(10\) 0 0
\(11\) −2.75399 + 4.77006i −0.250363 + 0.433642i −0.963626 0.267255i \(-0.913883\pi\)
0.713263 + 0.700897i \(0.247217\pi\)
\(12\) 0 0
\(13\) 6.22525i 0.478865i 0.970913 + 0.239433i \(0.0769614\pi\)
−0.970913 + 0.239433i \(0.923039\pi\)
\(14\) 0 0
\(15\) 3.76412i 0.250942i
\(16\) 0 0
\(17\) −10.5368 + 18.2502i −0.619811 + 1.07354i 0.369709 + 0.929147i \(0.379457\pi\)
−0.989520 + 0.144396i \(0.953876\pi\)
\(18\) 0 0
\(19\) 12.6229 + 21.8634i 0.664361 + 1.15071i 0.979458 + 0.201647i \(0.0646294\pi\)
−0.315098 + 0.949059i \(0.602037\pi\)
\(20\) 0 0
\(21\) 1.67978 + 9.89949i 0.0799897 + 0.471404i
\(22\) 0 0
\(23\) −12.5463 + 7.24359i −0.545490 + 0.314939i −0.747301 0.664486i \(-0.768651\pi\)
0.201811 + 0.979424i \(0.435317\pi\)
\(24\) 0 0
\(25\) −9.05697 + 15.6871i −0.362279 + 0.627485i
\(26\) 0 0
\(27\) −22.8682 −0.846972
\(28\) 0 0
\(29\) 29.9514i 1.03281i −0.856346 0.516403i \(-0.827271\pi\)
0.856346 0.516403i \(-0.172729\pi\)
\(30\) 0 0
\(31\) 23.5249 + 13.5821i 0.758868 + 0.438133i 0.828889 0.559413i \(-0.188973\pi\)
−0.0700212 + 0.997546i \(0.522307\pi\)
\(32\) 0 0
\(33\) −3.95041 6.84230i −0.119709 0.207343i
\(34\) 0 0
\(35\) −6.39374 + 17.2202i −0.182678 + 0.492007i
\(36\) 0 0
\(37\) 31.5708 18.2274i 0.853264 0.492632i −0.00848686 0.999964i \(-0.502701\pi\)
0.861751 + 0.507332i \(0.169368\pi\)
\(38\) 0 0
\(39\) −7.73332 4.46483i −0.198290 0.114483i
\(40\) 0 0
\(41\) 37.4338 0.913020 0.456510 0.889718i \(-0.349099\pi\)
0.456510 + 0.889718i \(0.349099\pi\)
\(42\) 0 0
\(43\) 28.9008 0.672112 0.336056 0.941842i \(-0.390907\pi\)
0.336056 + 0.941842i \(0.390907\pi\)
\(44\) 0 0
\(45\) −15.7771 9.10890i −0.350602 0.202420i
\(46\) 0 0
\(47\) −51.8379 + 29.9287i −1.10293 + 0.636780i −0.936990 0.349355i \(-0.886401\pi\)
−0.165945 + 0.986135i \(0.553067\pi\)
\(48\) 0 0
\(49\) 9.13053 48.1418i 0.186337 0.982486i
\(50\) 0 0
\(51\) −15.1143 26.1787i −0.296358 0.513307i
\(52\) 0 0
\(53\) 36.5373 + 21.0948i 0.689383 + 0.398015i 0.803381 0.595466i \(-0.203032\pi\)
−0.113998 + 0.993481i \(0.536366\pi\)
\(54\) 0 0
\(55\) 14.4537i 0.262794i
\(56\) 0 0
\(57\) −36.2131 −0.635318
\(58\) 0 0
\(59\) 14.6430 25.3625i 0.248187 0.429873i −0.714836 0.699292i \(-0.753499\pi\)
0.963023 + 0.269420i \(0.0868319\pi\)
\(60\) 0 0
\(61\) −92.2850 + 53.2808i −1.51287 + 0.873455i −0.512982 + 0.858400i \(0.671459\pi\)
−0.999887 + 0.0150554i \(0.995208\pi\)
\(62\) 0 0
\(63\) 45.5580 + 16.9153i 0.723143 + 0.268497i
\(64\) 0 0
\(65\) −8.16793 14.1473i −0.125660 0.217650i
\(66\) 0 0
\(67\) 56.4293 97.7384i 0.842228 1.45878i −0.0457783 0.998952i \(-0.514577\pi\)
0.888007 0.459831i \(-0.152090\pi\)
\(68\) 0 0
\(69\) 20.7808i 0.301171i
\(70\) 0 0
\(71\) 94.6530i 1.33314i −0.745442 0.666571i \(-0.767761\pi\)
0.745442 0.666571i \(-0.232239\pi\)
\(72\) 0 0
\(73\) 32.3682 56.0634i 0.443401 0.767992i −0.554539 0.832158i \(-0.687105\pi\)
0.997939 + 0.0641656i \(0.0204386\pi\)
\(74\) 0 0
\(75\) −12.9916 22.5021i −0.173221 0.300028i
\(76\) 0 0
\(77\) 6.45012 + 38.0126i 0.0837678 + 0.493670i
\(78\) 0 0
\(79\) 46.8433 27.0450i 0.592953 0.342342i −0.173311 0.984867i \(-0.555447\pi\)
0.766264 + 0.642525i \(0.222113\pi\)
\(80\) 0 0
\(81\) −14.8395 + 25.7027i −0.183203 + 0.317317i
\(82\) 0 0
\(83\) −91.8824 −1.10702 −0.553508 0.832844i \(-0.686711\pi\)
−0.553508 + 0.832844i \(0.686711\pi\)
\(84\) 0 0
\(85\) 55.2997i 0.650585i
\(86\) 0 0
\(87\) 37.2071 + 21.4815i 0.427668 + 0.246914i
\(88\) 0 0
\(89\) −36.1385 62.5938i −0.406051 0.703301i 0.588392 0.808576i \(-0.299761\pi\)
−0.994443 + 0.105275i \(0.966428\pi\)
\(90\) 0 0
\(91\) 27.7947 + 33.5617i 0.305436 + 0.368810i
\(92\) 0 0
\(93\) −33.7448 + 19.4826i −0.362847 + 0.209490i
\(94\) 0 0
\(95\) −57.3724 33.1240i −0.603920 0.348674i
\(96\) 0 0
\(97\) 97.4646 1.00479 0.502395 0.864638i \(-0.332452\pi\)
0.502395 + 0.864638i \(0.332452\pi\)
\(98\) 0 0
\(99\) −38.2388 −0.386250
\(100\) 0 0
\(101\) 167.533 + 96.7250i 1.65874 + 0.957673i 0.973298 + 0.229546i \(0.0737240\pi\)
0.685441 + 0.728128i \(0.259609\pi\)
\(102\) 0 0
\(103\) −79.1012 + 45.6691i −0.767973 + 0.443389i −0.832151 0.554549i \(-0.812891\pi\)
0.0641781 + 0.997938i \(0.479557\pi\)
\(104\) 0 0
\(105\) −16.8062 20.2932i −0.160059 0.193269i
\(106\) 0 0
\(107\) −1.72143 2.98160i −0.0160881 0.0278654i 0.857869 0.513868i \(-0.171788\pi\)
−0.873957 + 0.486003i \(0.838455\pi\)
\(108\) 0 0
\(109\) −167.840 96.9024i −1.53981 0.889012i −0.998849 0.0479679i \(-0.984725\pi\)
−0.540966 0.841045i \(-0.681941\pi\)
\(110\) 0 0
\(111\) 52.2917i 0.471097i
\(112\) 0 0
\(113\) −41.5646 −0.367828 −0.183914 0.982942i \(-0.558877\pi\)
−0.183914 + 0.982942i \(0.558877\pi\)
\(114\) 0 0
\(115\) 19.0081 32.9230i 0.165288 0.286287i
\(116\) 0 0
\(117\) −37.4281 + 21.6091i −0.319898 + 0.184693i
\(118\) 0 0
\(119\) 24.6782 + 145.436i 0.207379 + 1.22215i
\(120\) 0 0
\(121\) 45.3310 + 78.5156i 0.374637 + 0.648890i
\(122\) 0 0
\(123\) −26.8481 + 46.5022i −0.218277 + 0.378067i
\(124\) 0 0
\(125\) 113.137i 0.905093i
\(126\) 0 0
\(127\) 70.9794i 0.558893i −0.960161 0.279446i \(-0.909849\pi\)
0.960161 0.279446i \(-0.0901509\pi\)
\(128\) 0 0
\(129\) −20.7281 + 35.9021i −0.160683 + 0.278311i
\(130\) 0 0
\(131\) 84.6252 + 146.575i 0.645994 + 1.11889i 0.984071 + 0.177776i \(0.0568903\pi\)
−0.338077 + 0.941119i \(0.609776\pi\)
\(132\) 0 0
\(133\) 165.669 + 61.5116i 1.24563 + 0.462493i
\(134\) 0 0
\(135\) 51.9695 30.0046i 0.384959 0.222256i
\(136\) 0 0
\(137\) −48.4716 + 83.9554i −0.353808 + 0.612813i −0.986913 0.161253i \(-0.948447\pi\)
0.633106 + 0.774066i \(0.281780\pi\)
\(138\) 0 0
\(139\) 97.6033 0.702182 0.351091 0.936341i \(-0.385811\pi\)
0.351091 + 0.936341i \(0.385811\pi\)
\(140\) 0 0
\(141\) 85.8610i 0.608943i
\(142\) 0 0
\(143\) −29.6948 17.1443i −0.207656 0.119890i
\(144\) 0 0
\(145\) 39.2981 + 68.0663i 0.271021 + 0.469423i
\(146\) 0 0
\(147\) 53.2557 + 45.8704i 0.362283 + 0.312043i
\(148\) 0 0
\(149\) 118.934 68.6665i 0.798214 0.460849i −0.0446321 0.999003i \(-0.514212\pi\)
0.842846 + 0.538154i \(0.180878\pi\)
\(150\) 0 0
\(151\) −41.5059 23.9635i −0.274874 0.158698i 0.356227 0.934400i \(-0.384063\pi\)
−0.631100 + 0.775701i \(0.717396\pi\)
\(152\) 0 0
\(153\) −146.301 −0.956219
\(154\) 0 0
\(155\) −71.2824 −0.459887
\(156\) 0 0
\(157\) 47.5524 + 27.4544i 0.302882 + 0.174869i 0.643737 0.765247i \(-0.277383\pi\)
−0.340855 + 0.940116i \(0.610717\pi\)
\(158\) 0 0
\(159\) −52.4101 + 30.2590i −0.329623 + 0.190308i
\(160\) 0 0
\(161\) −35.2983 + 95.0688i −0.219244 + 0.590489i
\(162\) 0 0
\(163\) −4.07026 7.04990i −0.0249709 0.0432509i 0.853270 0.521470i \(-0.174616\pi\)
−0.878241 + 0.478219i \(0.841283\pi\)
\(164\) 0 0
\(165\) 17.9551 + 10.3664i 0.108819 + 0.0628265i
\(166\) 0 0
\(167\) 12.1877i 0.0729802i −0.999334 0.0364901i \(-0.988382\pi\)
0.999334 0.0364901i \(-0.0116177\pi\)
\(168\) 0 0
\(169\) 130.246 0.770688
\(170\) 0 0
\(171\) −87.6331 + 151.785i −0.512474 + 0.887631i
\(172\) 0 0
\(173\) −229.411 + 132.450i −1.32607 + 0.765609i −0.984690 0.174314i \(-0.944229\pi\)
−0.341384 + 0.939924i \(0.610896\pi\)
\(174\) 0 0
\(175\) 21.2123 + 125.011i 0.121213 + 0.714347i
\(176\) 0 0
\(177\) 21.0044 + 36.3807i 0.118669 + 0.205540i
\(178\) 0 0
\(179\) −118.385 + 205.049i −0.661369 + 1.14552i 0.318888 + 0.947793i \(0.396691\pi\)
−0.980256 + 0.197731i \(0.936643\pi\)
\(180\) 0 0
\(181\) 19.7435i 0.109080i 0.998512 + 0.0545402i \(0.0173693\pi\)
−0.998512 + 0.0545402i \(0.982631\pi\)
\(182\) 0 0
\(183\) 152.855i 0.835272i
\(184\) 0 0
\(185\) −47.8310 + 82.8458i −0.258546 + 0.447815i
\(186\) 0 0
\(187\) −58.0365 100.522i −0.310355 0.537551i
\(188\) 0 0
\(189\) −123.288 + 102.103i −0.652316 + 0.540227i
\(190\) 0 0
\(191\) 139.204 80.3697i 0.728819 0.420784i −0.0891711 0.996016i \(-0.528422\pi\)
0.817990 + 0.575233i \(0.195088\pi\)
\(192\) 0 0
\(193\) 84.9196 147.085i 0.439998 0.762099i −0.557691 0.830049i \(-0.688312\pi\)
0.997689 + 0.0679499i \(0.0216458\pi\)
\(194\) 0 0
\(195\) 23.4326 0.120167
\(196\) 0 0
\(197\) 47.2927i 0.240064i −0.992770 0.120032i \(-0.961700\pi\)
0.992770 0.120032i \(-0.0382998\pi\)
\(198\) 0 0
\(199\) 243.806 + 140.761i 1.22516 + 0.707344i 0.966013 0.258495i \(-0.0832266\pi\)
0.259143 + 0.965839i \(0.416560\pi\)
\(200\) 0 0
\(201\) 80.9438 + 140.199i 0.402705 + 0.697506i
\(202\) 0 0
\(203\) −133.728 161.474i −0.658758 0.795440i
\(204\) 0 0
\(205\) −85.0707 + 49.1156i −0.414979 + 0.239588i
\(206\) 0 0
\(207\) −87.1014 50.2880i −0.420780 0.242937i
\(208\) 0 0
\(209\) −139.053 −0.665326
\(210\) 0 0
\(211\) 105.459 0.499804 0.249902 0.968271i \(-0.419602\pi\)
0.249902 + 0.968271i \(0.419602\pi\)
\(212\) 0 0
\(213\) 117.583 + 67.8865i 0.552032 + 0.318716i
\(214\) 0 0
\(215\) −65.6789 + 37.9197i −0.305483 + 0.176371i
\(216\) 0 0
\(217\) 187.470 31.8106i 0.863916 0.146593i
\(218\) 0 0
\(219\) 46.4299 + 80.4190i 0.212009 + 0.367210i
\(220\) 0 0
\(221\) −113.612 65.5941i −0.514083 0.296806i
\(222\) 0 0
\(223\) 386.423i 1.73284i −0.499316 0.866420i \(-0.666415\pi\)
0.499316 0.866420i \(-0.333585\pi\)
\(224\) 0 0
\(225\) −125.755 −0.558909
\(226\) 0 0
\(227\) 71.6708 124.137i 0.315730 0.546861i −0.663862 0.747855i \(-0.731084\pi\)
0.979592 + 0.200994i \(0.0644172\pi\)
\(228\) 0 0
\(229\) 154.383 89.1332i 0.674163 0.389228i −0.123489 0.992346i \(-0.539408\pi\)
0.797652 + 0.603118i \(0.206075\pi\)
\(230\) 0 0
\(231\) −51.8473 19.2505i −0.224447 0.0833354i
\(232\) 0 0
\(233\) −172.708 299.139i −0.741235 1.28386i −0.951933 0.306305i \(-0.900907\pi\)
0.210699 0.977551i \(-0.432426\pi\)
\(234\) 0 0
\(235\) 78.5366 136.029i 0.334199 0.578849i
\(236\) 0 0
\(237\) 77.5882i 0.327376i
\(238\) 0 0
\(239\) 140.165i 0.586464i 0.956041 + 0.293232i \(0.0947308\pi\)
−0.956041 + 0.293232i \(0.905269\pi\)
\(240\) 0 0
\(241\) 109.775 190.135i 0.455496 0.788943i −0.543220 0.839590i \(-0.682795\pi\)
0.998717 + 0.0506475i \(0.0161285\pi\)
\(242\) 0 0
\(243\) −124.193 215.109i −0.511083 0.885222i
\(244\) 0 0
\(245\) 42.4154 + 121.385i 0.173124 + 0.495449i
\(246\) 0 0
\(247\) −136.105 + 78.5803i −0.551033 + 0.318139i
\(248\) 0 0
\(249\) 65.8993 114.141i 0.264656 0.458398i
\(250\) 0 0
\(251\) 207.856 0.828113 0.414057 0.910251i \(-0.364112\pi\)
0.414057 + 0.910251i \(0.364112\pi\)
\(252\) 0 0
\(253\) 79.7953i 0.315396i
\(254\) 0 0
\(255\) 68.6962 + 39.6617i 0.269397 + 0.155536i
\(256\) 0 0
\(257\) 14.7596 + 25.5643i 0.0574302 + 0.0994720i 0.893311 0.449439i \(-0.148376\pi\)
−0.835881 + 0.548911i \(0.815043\pi\)
\(258\) 0 0
\(259\) 88.8227 239.226i 0.342945 0.923653i
\(260\) 0 0
\(261\) 180.077 103.967i 0.689950 0.398343i
\(262\) 0 0
\(263\) −2.50134 1.44415i −0.00951079 0.00549106i 0.495237 0.868758i \(-0.335081\pi\)
−0.504748 + 0.863267i \(0.668415\pi\)
\(264\) 0 0
\(265\) −110.711 −0.417778
\(266\) 0 0
\(267\) 103.676 0.388300
\(268\) 0 0
\(269\) −309.676 178.791i −1.15121 0.664652i −0.202029 0.979380i \(-0.564753\pi\)
−0.949182 + 0.314728i \(0.898087\pi\)
\(270\) 0 0
\(271\) −294.743 + 170.170i −1.08761 + 0.627934i −0.932940 0.360031i \(-0.882766\pi\)
−0.154674 + 0.987966i \(0.549433\pi\)
\(272\) 0 0
\(273\) −61.6268 + 10.4571i −0.225739 + 0.0383043i
\(274\) 0 0
\(275\) −49.8857 86.4046i −0.181403 0.314198i
\(276\) 0 0
\(277\) 11.2603 + 6.50111i 0.0406507 + 0.0234697i 0.520188 0.854052i \(-0.325862\pi\)
−0.479537 + 0.877522i \(0.659195\pi\)
\(278\) 0 0
\(279\) 188.585i 0.675933i
\(280\) 0 0
\(281\) 232.300 0.826689 0.413344 0.910575i \(-0.364361\pi\)
0.413344 + 0.910575i \(0.364361\pi\)
\(282\) 0 0
\(283\) 6.07204 10.5171i 0.0214560 0.0371628i −0.855098 0.518466i \(-0.826503\pi\)
0.876554 + 0.481303i \(0.159836\pi\)
\(284\) 0 0
\(285\) 82.2966 47.5140i 0.288760 0.166716i
\(286\) 0 0
\(287\) 201.814 167.136i 0.703185 0.582354i
\(288\) 0 0
\(289\) −77.5475 134.316i −0.268330 0.464762i
\(290\) 0 0
\(291\) −69.9029 + 121.075i −0.240216 + 0.416067i
\(292\) 0 0
\(293\) 84.6048i 0.288753i 0.989523 + 0.144377i \(0.0461177\pi\)
−0.989523 + 0.144377i \(0.953882\pi\)
\(294\) 0 0
\(295\) 76.8505i 0.260510i
\(296\) 0 0
\(297\) 62.9790 109.083i 0.212051 0.367282i
\(298\) 0 0
\(299\) −45.0931 78.1036i −0.150813 0.261216i
\(300\) 0 0
\(301\) 155.811 129.037i 0.517643 0.428695i
\(302\) 0 0
\(303\) −240.313 + 138.745i −0.793114 + 0.457904i
\(304\) 0 0
\(305\) 139.816 242.168i 0.458412 0.793992i
\(306\) 0 0
\(307\) 97.3710 0.317169 0.158585 0.987345i \(-0.449307\pi\)
0.158585 + 0.987345i \(0.449307\pi\)
\(308\) 0 0
\(309\) 131.018i 0.424007i
\(310\) 0 0
\(311\) −86.7666 50.0947i −0.278992 0.161076i 0.353975 0.935255i \(-0.384830\pi\)
−0.632967 + 0.774179i \(0.718163\pi\)
\(312\) 0 0
\(313\) −70.9257 122.847i −0.226600 0.392482i 0.730199 0.683235i \(-0.239428\pi\)
−0.956798 + 0.290753i \(0.906094\pi\)
\(314\) 0 0
\(315\) −125.727 + 21.3339i −0.399135 + 0.0677267i
\(316\) 0 0
\(317\) 24.9036 14.3781i 0.0785604 0.0453569i −0.460205 0.887813i \(-0.652224\pi\)
0.538766 + 0.842456i \(0.318891\pi\)
\(318\) 0 0
\(319\) 142.870 + 82.4859i 0.447867 + 0.258576i
\(320\) 0 0
\(321\) 4.93853 0.0153848
\(322\) 0 0
\(323\) −532.017 −1.64711
\(324\) 0 0
\(325\) −97.6563 56.3819i −0.300481 0.173483i
\(326\) 0 0
\(327\) 240.754 138.999i 0.736251 0.425075i
\(328\) 0 0
\(329\) −145.843 + 392.800i −0.443293 + 1.19392i
\(330\) 0 0
\(331\) 140.896 + 244.039i 0.425668 + 0.737278i 0.996483 0.0838009i \(-0.0267060\pi\)
−0.570815 + 0.821079i \(0.693373\pi\)
\(332\) 0 0
\(333\) 219.177 + 126.542i 0.658190 + 0.380006i
\(334\) 0 0
\(335\) 296.156i 0.884046i
\(336\) 0 0
\(337\) −22.7495 −0.0675058 −0.0337529 0.999430i \(-0.510746\pi\)
−0.0337529 + 0.999430i \(0.510746\pi\)
\(338\) 0 0
\(339\) 29.8107 51.6337i 0.0879372 0.152312i
\(340\) 0 0
\(341\) −129.575 + 74.8101i −0.379985 + 0.219385i
\(342\) 0 0
\(343\) −165.720 300.309i −0.483150 0.875538i
\(344\) 0 0
\(345\) 27.2658 + 47.2257i 0.0790312 + 0.136886i
\(346\) 0 0
\(347\) −40.0334 + 69.3399i −0.115370 + 0.199827i −0.917928 0.396748i \(-0.870139\pi\)
0.802558 + 0.596575i \(0.203472\pi\)
\(348\) 0 0
\(349\) 222.628i 0.637903i 0.947771 + 0.318951i \(0.103331\pi\)
−0.947771 + 0.318951i \(0.896669\pi\)
\(350\) 0 0
\(351\) 142.360i 0.405585i
\(352\) 0 0
\(353\) −162.239 + 281.006i −0.459601 + 0.796052i −0.998940 0.0460367i \(-0.985341\pi\)
0.539339 + 0.842089i \(0.318674\pi\)
\(354\) 0 0
\(355\) 124.191 + 215.105i 0.349834 + 0.605929i
\(356\) 0 0
\(357\) −198.368 73.6523i −0.555651 0.206309i
\(358\) 0 0
\(359\) 114.236 65.9541i 0.318206 0.183716i −0.332387 0.943143i \(-0.607854\pi\)
0.650593 + 0.759427i \(0.274521\pi\)
\(360\) 0 0
\(361\) −138.173 + 239.322i −0.382750 + 0.662942i
\(362\) 0 0
\(363\) −130.048 −0.358259
\(364\) 0 0
\(365\) 169.877i 0.465416i
\(366\) 0 0
\(367\) −165.468 95.5332i −0.450868 0.260309i 0.257329 0.966324i \(-0.417158\pi\)
−0.708197 + 0.706015i \(0.750491\pi\)
\(368\) 0 0
\(369\) 129.941 + 225.064i 0.352143 + 0.609929i
\(370\) 0 0
\(371\) 291.165 49.4061i 0.784813 0.133170i
\(372\) 0 0
\(373\) −385.278 + 222.440i −1.03292 + 0.596354i −0.917819 0.397000i \(-0.870051\pi\)
−0.115098 + 0.993354i \(0.536718\pi\)
\(374\) 0 0
\(375\) 140.544 + 81.1431i 0.374784 + 0.216382i
\(376\) 0 0
\(377\) 186.455 0.494574
\(378\) 0 0
\(379\) 253.564 0.669034 0.334517 0.942390i \(-0.391427\pi\)
0.334517 + 0.942390i \(0.391427\pi\)
\(380\) 0 0
\(381\) 88.1742 + 50.9074i 0.231428 + 0.133615i
\(382\) 0 0
\(383\) 547.948 316.358i 1.43067 0.826000i 0.433501 0.901153i \(-0.357278\pi\)
0.997172 + 0.0751535i \(0.0239447\pi\)
\(384\) 0 0
\(385\) −64.5332 77.9230i −0.167619 0.202397i
\(386\) 0 0
\(387\) 100.321 + 173.761i 0.259227 + 0.448994i
\(388\) 0 0
\(389\) 116.434 + 67.2230i 0.299315 + 0.172810i 0.642135 0.766591i \(-0.278049\pi\)
−0.342820 + 0.939401i \(0.611382\pi\)
\(390\) 0 0
\(391\) 305.297i 0.780810i
\(392\) 0 0
\(393\) −242.778 −0.617755
\(394\) 0 0
\(395\) −70.9696 + 122.923i −0.179670 + 0.311197i
\(396\) 0 0
\(397\) 270.162 155.978i 0.680509 0.392892i −0.119538 0.992830i \(-0.538141\pi\)
0.800047 + 0.599938i \(0.204808\pi\)
\(398\) 0 0
\(399\) −195.233 + 161.686i −0.489306 + 0.405227i
\(400\) 0 0
\(401\) 319.775 + 553.866i 0.797444 + 1.38121i 0.921276 + 0.388910i \(0.127148\pi\)
−0.123832 + 0.992303i \(0.539518\pi\)
\(402\) 0 0
\(403\) −84.5520 + 146.448i −0.209806 + 0.363395i
\(404\) 0 0
\(405\) 77.8813i 0.192299i
\(406\) 0 0
\(407\) 200.793i 0.493348i
\(408\) 0 0
\(409\) −51.6414 + 89.4456i −0.126263 + 0.218693i −0.922226 0.386652i \(-0.873632\pi\)
0.795963 + 0.605345i \(0.206965\pi\)
\(410\) 0 0
\(411\) −69.5291 120.428i −0.169170 0.293012i
\(412\) 0 0
\(413\) −34.2954 202.114i −0.0830397 0.489379i
\(414\) 0 0
\(415\) 208.808 120.556i 0.503153 0.290496i
\(416\) 0 0
\(417\) −70.0024 + 121.248i −0.167871 + 0.290762i
\(418\) 0 0
\(419\) 55.0511 0.131387 0.0656934 0.997840i \(-0.479074\pi\)
0.0656934 + 0.997840i \(0.479074\pi\)
\(420\) 0 0
\(421\) 513.922i 1.22072i 0.792125 + 0.610358i \(0.208975\pi\)
−0.792125 + 0.610358i \(0.791025\pi\)
\(422\) 0 0
\(423\) −359.881 207.777i −0.850782 0.491199i
\(424\) 0 0
\(425\) −190.863 330.584i −0.449089 0.777844i
\(426\) 0 0
\(427\) −259.639 + 699.285i −0.608054 + 1.63767i
\(428\) 0 0
\(429\) 42.5950 24.5923i 0.0992891 0.0573246i
\(430\) 0 0
\(431\) 277.403 + 160.159i 0.643627 + 0.371598i 0.786011 0.618213i \(-0.212143\pi\)
−0.142383 + 0.989812i \(0.545476\pi\)
\(432\) 0 0
\(433\) −37.1029 −0.0856880 −0.0428440 0.999082i \(-0.513642\pi\)
−0.0428440 + 0.999082i \(0.513642\pi\)
\(434\) 0 0
\(435\) −112.741 −0.259174
\(436\) 0 0
\(437\) −316.739 182.870i −0.724804 0.418466i
\(438\) 0 0
\(439\) −88.7426 + 51.2356i −0.202147 + 0.116710i −0.597657 0.801752i \(-0.703901\pi\)
0.395509 + 0.918462i \(0.370568\pi\)
\(440\) 0 0
\(441\) 321.137 112.215i 0.728203 0.254455i
\(442\) 0 0
\(443\) −173.497 300.505i −0.391641 0.678342i 0.601025 0.799230i \(-0.294759\pi\)
−0.992666 + 0.120888i \(0.961426\pi\)
\(444\) 0 0
\(445\) 164.254 + 94.8322i 0.369110 + 0.213106i
\(446\) 0 0
\(447\) 196.994i 0.440703i
\(448\) 0 0
\(449\) 249.728 0.556187 0.278094 0.960554i \(-0.410297\pi\)
0.278094 + 0.960554i \(0.410297\pi\)
\(450\) 0 0
\(451\) −103.093 + 178.562i −0.228587 + 0.395924i
\(452\) 0 0
\(453\) 59.5372 34.3738i 0.131429 0.0758805i
\(454\) 0 0
\(455\) −107.200 39.8026i −0.235605 0.0874782i
\(456\) 0 0
\(457\) 135.736 + 235.101i 0.297015 + 0.514445i 0.975452 0.220214i \(-0.0706756\pi\)
−0.678437 + 0.734659i \(0.737342\pi\)
\(458\) 0 0
\(459\) 240.958 417.351i 0.524962 0.909261i
\(460\) 0 0
\(461\) 276.405i 0.599577i −0.954006 0.299788i \(-0.903084\pi\)
0.954006 0.299788i \(-0.0969161\pi\)
\(462\) 0 0
\(463\) 796.625i 1.72057i 0.509811 + 0.860286i \(0.329715\pi\)
−0.509811 + 0.860286i \(0.670285\pi\)
\(464\) 0 0
\(465\) 51.1247 88.5507i 0.109946 0.190432i
\(466\) 0 0
\(467\) 378.294 + 655.224i 0.810051 + 1.40305i 0.912828 + 0.408345i \(0.133894\pi\)
−0.102776 + 0.994704i \(0.532773\pi\)
\(468\) 0 0
\(469\) −132.163 778.877i −0.281797 1.66072i
\(470\) 0 0
\(471\) −68.2105 + 39.3814i −0.144821 + 0.0836123i
\(472\) 0 0
\(473\) −79.5927 + 137.859i −0.168272 + 0.291456i
\(474\) 0 0
\(475\) −457.299 −0.962735
\(476\) 0 0
\(477\) 292.898i 0.614042i
\(478\) 0 0
\(479\) −446.732 257.921i −0.932635 0.538457i −0.0449907 0.998987i \(-0.514326\pi\)
−0.887644 + 0.460531i \(0.847659\pi\)
\(480\) 0 0
\(481\) 113.470 + 196.536i 0.235904 + 0.408598i
\(482\) 0 0
\(483\) −92.7829 112.034i −0.192097 0.231954i
\(484\) 0 0
\(485\) −221.494 + 127.880i −0.456689 + 0.263670i
\(486\) 0 0
\(487\) 115.410 + 66.6319i 0.236981 + 0.136821i 0.613788 0.789471i \(-0.289645\pi\)
−0.376807 + 0.926292i \(0.622978\pi\)
\(488\) 0 0
\(489\) 11.6770 0.0238793
\(490\) 0 0
\(491\) 595.369 1.21256 0.606282 0.795249i \(-0.292660\pi\)
0.606282 + 0.795249i \(0.292660\pi\)
\(492\) 0 0
\(493\) 546.619 + 315.591i 1.10876 + 0.640144i
\(494\) 0 0
\(495\) 86.9000 50.1717i 0.175556 0.101357i
\(496\) 0 0
\(497\) −422.610 510.295i −0.850322 1.02675i
\(498\) 0 0
\(499\) −191.203 331.173i −0.383172 0.663673i 0.608342 0.793675i \(-0.291835\pi\)
−0.991514 + 0.130002i \(0.958502\pi\)
\(500\) 0 0
\(501\) 15.1402 + 8.74119i 0.0302199 + 0.0174475i
\(502\) 0 0
\(503\) 513.765i 1.02140i 0.859758 + 0.510701i \(0.170614\pi\)
−0.859758 + 0.510701i \(0.829386\pi\)
\(504\) 0 0
\(505\) −507.638 −1.00522
\(506\) 0 0
\(507\) −93.4145 + 161.799i −0.184249 + 0.319129i
\(508\) 0 0
\(509\) 732.796 423.080i 1.43968 0.831199i 0.441851 0.897088i \(-0.354322\pi\)
0.997827 + 0.0658897i \(0.0209886\pi\)
\(510\) 0 0
\(511\) −75.8095 446.769i −0.148355 0.874304i
\(512\) 0 0
\(513\) −288.662 499.978i −0.562695 0.974616i
\(514\) 0 0
\(515\) 119.842 207.572i 0.232702 0.403052i
\(516\) 0 0
\(517\) 329.693i 0.637705i
\(518\) 0 0
\(519\) 379.981i 0.732141i
\(520\) 0 0
\(521\) 141.818 245.636i 0.272204 0.471471i −0.697222 0.716855i \(-0.745581\pi\)
0.969426 + 0.245385i \(0.0789142\pi\)
\(522\) 0 0
\(523\) −448.840 777.415i −0.858204 1.48645i −0.873641 0.486571i \(-0.838247\pi\)
0.0154372 0.999881i \(-0.495086\pi\)
\(524\) 0 0
\(525\) −170.508 63.3084i −0.324778 0.120587i
\(526\) 0 0
\(527\) −495.754 + 286.223i −0.940709 + 0.543118i
\(528\) 0 0
\(529\) −159.561 + 276.367i −0.301627 + 0.522434i
\(530\) 0 0
\(531\) 203.316 0.382893
\(532\) 0 0
\(533\) 233.035i 0.437213i
\(534\) 0 0
\(535\) 7.82410 + 4.51725i 0.0146245 + 0.00844345i
\(536\) 0 0
\(537\) −169.815 294.128i −0.316228 0.547724i
\(538\) 0 0
\(539\) 204.494 + 176.135i 0.379395 + 0.326782i
\(540\) 0 0
\(541\) −104.490 + 60.3276i −0.193143 + 0.111511i −0.593453 0.804869i \(-0.702236\pi\)
0.400310 + 0.916380i \(0.368902\pi\)
\(542\) 0 0
\(543\) −24.5264 14.1603i −0.0451684 0.0260780i
\(544\) 0 0
\(545\) 508.569 0.933153
\(546\) 0 0
\(547\) −65.8090 −0.120309 −0.0601545 0.998189i \(-0.519159\pi\)
−0.0601545 + 0.998189i \(0.519159\pi\)
\(548\) 0 0
\(549\) −640.681 369.897i −1.16700 0.673765i
\(550\) 0 0
\(551\) 654.839 378.071i 1.18846 0.686155i
\(552\) 0 0
\(553\) 131.791 354.953i 0.238321 0.641868i
\(554\) 0 0
\(555\) −68.6102 118.836i −0.123622 0.214119i
\(556\) 0 0
\(557\) 394.697 + 227.878i 0.708612 + 0.409117i 0.810547 0.585674i \(-0.199170\pi\)
−0.101935 + 0.994791i \(0.532503\pi\)
\(558\) 0 0
\(559\) 179.915i 0.321851i
\(560\) 0 0
\(561\) 166.498 0.296788
\(562\) 0 0
\(563\) 413.897 716.891i 0.735164 1.27334i −0.219487 0.975615i \(-0.570438\pi\)
0.954651 0.297726i \(-0.0962283\pi\)
\(564\) 0 0
\(565\) 94.4582 54.5355i 0.167183 0.0965229i
\(566\) 0 0
\(567\) 34.7554 + 204.825i 0.0612971 + 0.361243i
\(568\) 0 0
\(569\) 23.9051 + 41.4049i 0.0420126 + 0.0727679i 0.886267 0.463175i \(-0.153290\pi\)
−0.844255 + 0.535942i \(0.819956\pi\)
\(570\) 0 0
\(571\) 249.237 431.692i 0.436493 0.756028i −0.560923 0.827868i \(-0.689554\pi\)
0.997416 + 0.0718399i \(0.0228871\pi\)
\(572\) 0 0
\(573\) 230.569i 0.402389i
\(574\) 0 0
\(575\) 262.420i 0.456383i
\(576\) 0 0
\(577\) −107.802 + 186.719i −0.186832 + 0.323603i −0.944192 0.329395i \(-0.893155\pi\)
0.757360 + 0.652997i \(0.226489\pi\)
\(578\) 0 0
\(579\) 121.811 + 210.983i 0.210382 + 0.364392i
\(580\) 0 0
\(581\) −495.358 + 410.240i −0.852596 + 0.706092i
\(582\) 0 0
\(583\) −201.247 + 116.190i −0.345192 + 0.199297i
\(584\) 0 0
\(585\) 56.7051 98.2162i 0.0969319 0.167891i
\(586\) 0 0
\(587\) −302.330 −0.515042 −0.257521 0.966273i \(-0.582906\pi\)
−0.257521 + 0.966273i \(0.582906\pi\)
\(588\) 0 0
\(589\) 685.780i 1.16431i
\(590\) 0 0
\(591\) 58.7494 + 33.9190i 0.0994067 + 0.0573925i
\(592\) 0 0
\(593\) −426.339 738.440i −0.718952 1.24526i −0.961415 0.275101i \(-0.911289\pi\)
0.242463 0.970161i \(-0.422045\pi\)
\(594\) 0 0
\(595\) −246.904 298.133i −0.414965 0.501064i
\(596\) 0 0
\(597\) −349.722 + 201.912i −0.585799 + 0.338211i
\(598\) 0 0
\(599\) −275.095 158.826i −0.459257 0.265152i 0.252475 0.967604i \(-0.418756\pi\)
−0.711732 + 0.702451i \(0.752089\pi\)
\(600\) 0 0
\(601\) 279.540 0.465125 0.232562 0.972581i \(-0.425289\pi\)
0.232562 + 0.972581i \(0.425289\pi\)
\(602\) 0 0
\(603\) 783.511 1.29936
\(604\) 0 0
\(605\) −206.035 118.954i −0.340554 0.196619i
\(606\) 0 0
\(607\) 827.056 477.501i 1.36253 0.786657i 0.372570 0.928004i \(-0.378476\pi\)
0.989960 + 0.141346i \(0.0451431\pi\)
\(608\) 0 0
\(609\) 296.503 50.3118i 0.486869 0.0826138i
\(610\) 0 0
\(611\) −186.313 322.704i −0.304932 0.528157i
\(612\) 0 0
\(613\) −915.014 528.284i −1.49268 0.861800i −0.492717 0.870189i \(-0.663996\pi\)
−0.999965 + 0.00838891i \(0.997330\pi\)
\(614\) 0 0
\(615\) 140.906i 0.229115i
\(616\) 0 0
\(617\) 974.782 1.57987 0.789937 0.613188i \(-0.210113\pi\)
0.789937 + 0.613188i \(0.210113\pi\)
\(618\) 0 0
\(619\) −180.385 + 312.435i −0.291413 + 0.504742i −0.974144 0.225928i \(-0.927459\pi\)
0.682731 + 0.730670i \(0.260792\pi\)
\(620\) 0 0
\(621\) 286.911 165.648i 0.462015 0.266744i
\(622\) 0 0
\(623\) −474.301 176.104i −0.761319 0.282671i
\(624\) 0 0
\(625\) −77.9818 135.068i −0.124771 0.216110i
\(626\) 0 0
\(627\) 99.7308 172.739i 0.159060 0.275500i
\(628\) 0 0
\(629\) 768.232i 1.22135i
\(630\) 0 0
\(631\) 1180.39i 1.87067i 0.353760 + 0.935336i \(0.384903\pi\)
−0.353760 + 0.935336i \(0.615097\pi\)
\(632\) 0 0
\(633\) −75.6363 + 131.006i −0.119489 + 0.206961i
\(634\) 0 0
\(635\) 93.1295 + 161.305i 0.146661 + 0.254024i
\(636\) 0 0
\(637\) 299.695 + 56.8398i 0.470478 + 0.0892305i
\(638\) 0 0
\(639\) 569.083 328.560i 0.890584 0.514179i
\(640\) 0 0
\(641\) 238.773 413.567i 0.372501 0.645190i −0.617449 0.786611i \(-0.711834\pi\)
0.989950 + 0.141421i \(0.0451670\pi\)
\(642\) 0 0
\(643\) −1072.27 −1.66760 −0.833801 0.552066i \(-0.813840\pi\)
−0.833801 + 0.552066i \(0.813840\pi\)
\(644\) 0 0
\(645\) 108.786i 0.168661i
\(646\) 0 0
\(647\) −373.550 215.669i −0.577357 0.333337i 0.182725 0.983164i \(-0.441508\pi\)
−0.760082 + 0.649827i \(0.774841\pi\)
\(648\) 0 0
\(649\) 80.6537 + 139.696i 0.124274 + 0.215249i
\(650\) 0 0
\(651\) −94.9392 + 255.700i −0.145836 + 0.392780i
\(652\) 0 0
\(653\) 874.317 504.787i 1.33892 0.773028i 0.352276 0.935896i \(-0.385408\pi\)
0.986648 + 0.162868i \(0.0520745\pi\)
\(654\) 0 0
\(655\) −384.632 222.068i −0.587225 0.339034i
\(656\) 0 0
\(657\) 449.428 0.684060
\(658\) 0 0
\(659\) −986.016 −1.49623 −0.748115 0.663569i \(-0.769041\pi\)
−0.748115 + 0.663569i \(0.769041\pi\)
\(660\) 0 0
\(661\) −264.228 152.552i −0.399740 0.230790i 0.286632 0.958041i \(-0.407464\pi\)
−0.686372 + 0.727251i \(0.740798\pi\)
\(662\) 0 0
\(663\) 162.969 94.0899i 0.245805 0.141915i
\(664\) 0 0
\(665\) −457.201 + 77.5796i −0.687520 + 0.116661i
\(666\) 0 0
\(667\) 216.955 + 375.778i 0.325270 + 0.563385i
\(668\) 0 0
\(669\) 480.035 + 277.148i 0.717540 + 0.414272i
\(670\) 0 0
\(671\) 586.940i 0.874724i
\(672\) 0 0
\(673\) −959.704 −1.42601 −0.713005 0.701159i \(-0.752666\pi\)
−0.713005 + 0.701159i \(0.752666\pi\)
\(674\) 0 0
\(675\) 207.117 358.737i 0.306840 0.531463i
\(676\) 0 0
\(677\) 105.318 60.8054i 0.155566 0.0898160i −0.420196 0.907433i \(-0.638039\pi\)
0.575762 + 0.817617i \(0.304705\pi\)
\(678\) 0 0
\(679\) 525.453 435.163i 0.773863 0.640888i
\(680\) 0 0
\(681\) 102.807 + 178.066i 0.150964 + 0.261478i
\(682\) 0 0
\(683\) −45.6786 + 79.1177i −0.0668794 + 0.115838i −0.897526 0.440961i \(-0.854638\pi\)
0.830647 + 0.556800i \(0.187971\pi\)
\(684\) 0 0
\(685\) 254.392i 0.371375i
\(686\) 0 0
\(687\) 255.710i 0.372213i
\(688\) 0 0
\(689\) −131.320 + 227.454i −0.190596 + 0.330121i
\(690\) 0 0
\(691\) 378.521 + 655.617i 0.547787 + 0.948795i 0.998426 + 0.0560887i \(0.0178630\pi\)
−0.450639 + 0.892706i \(0.648804\pi\)
\(692\) 0 0
\(693\) −206.154 + 170.730i −0.297480 + 0.246363i
\(694\) 0 0
\(695\) −221.809 + 128.062i −0.319150 + 0.184262i
\(696\) 0 0
\(697\) −394.432 + 683.176i −0.565899 + 0.980167i
\(698\) 0 0
\(699\) 495.473 0.708832
\(700\) 0 0
\(701\) 513.435i 0.732433i −0.930530 0.366216i \(-0.880653\pi\)
0.930530 0.366216i \(-0.119347\pi\)
\(702\) 0 0
\(703\) 797.026 + 460.163i 1.13375 + 0.654571i
\(704\) 0 0
\(705\) 112.655 + 195.124i 0.159795 + 0.276772i
\(706\) 0 0
\(707\) 1335.07 226.539i 1.88835 0.320423i
\(708\) 0 0
\(709\) −353.678 + 204.196i −0.498841 + 0.288006i −0.728235 0.685328i \(-0.759659\pi\)
0.229394 + 0.973334i \(0.426326\pi\)
\(710\) 0 0
\(711\) 325.206 + 187.758i 0.457392 + 0.264075i
\(712\) 0 0
\(713\) −393.533 −0.551940
\(714\) 0 0
\(715\) 89.9777 0.125843
\(716\) 0 0
\(717\) −174.120 100.528i −0.242845 0.140207i
\(718\) 0 0
\(719\) −797.802 + 460.611i −1.10960 + 0.640628i −0.938726 0.344665i \(-0.887993\pi\)
−0.170874 + 0.985293i \(0.554659\pi\)
\(720\) 0 0
\(721\) −222.547 + 599.386i −0.308665 + 0.831326i
\(722\) 0 0
\(723\) 157.464 + 272.735i 0.217792 + 0.377227i
\(724\) 0 0
\(725\) 469.851 + 271.269i 0.648070 + 0.374164i
\(726\) 0 0
\(727\) 404.650i 0.556603i −0.960494 0.278301i \(-0.910229\pi\)
0.960494 0.278301i \(-0.0897714\pi\)
\(728\) 0 0
\(729\) 89.1823 0.122335
\(730\) 0 0
\(731\) −304.522 + 527.447i −0.416582 + 0.721541i
\(732\) 0 0
\(733\) −181.337 + 104.695i −0.247390 + 0.142830i −0.618568 0.785731i \(-0.712287\pi\)
0.371179 + 0.928561i \(0.378954\pi\)
\(734\) 0 0
\(735\) −181.212 34.3685i −0.246547 0.0467598i
\(736\) 0 0
\(737\) 310.812 + 538.342i 0.421726 + 0.730451i
\(738\) 0 0
\(739\) −611.765 + 1059.61i −0.827828 + 1.43384i 0.0719108 + 0.997411i \(0.477090\pi\)
−0.899739 + 0.436429i \(0.856243\pi\)
\(740\) 0 0
\(741\) 225.436i 0.304232i
\(742\) 0 0
\(743\) 571.739i 0.769501i 0.923021 + 0.384750i \(0.125712\pi\)
−0.923021 + 0.384750i \(0.874288\pi\)
\(744\) 0 0
\(745\) −180.190 + 312.098i −0.241866 + 0.418923i
\(746\) 0 0
\(747\) −318.943 552.425i −0.426965 0.739525i
\(748\) 0 0
\(749\) −22.5929 8.38858i −0.0301641 0.0111997i
\(750\) 0 0
\(751\) −1108.18 + 639.811i −1.47561 + 0.851945i −0.999622 0.0275052i \(-0.991244\pi\)
−0.475991 + 0.879450i \(0.657910\pi\)
\(752\) 0 0
\(753\) −149.078 + 258.210i −0.197978 + 0.342908i
\(754\) 0 0
\(755\) 125.766 0.166578
\(756\) 0 0
\(757\) 159.036i 0.210087i −0.994468 0.105043i \(-0.966502\pi\)
0.994468 0.105043i \(-0.0334981\pi\)
\(758\) 0 0
\(759\) 99.1257 + 57.2303i 0.130600 + 0.0754022i
\(760\) 0 0
\(761\) 130.214 + 225.538i 0.171110 + 0.296371i 0.938808 0.344441i \(-0.111931\pi\)
−0.767698 + 0.640811i \(0.778598\pi\)
\(762\) 0 0
\(763\) −1337.51 + 226.955i −1.75297 + 0.297450i
\(764\) 0 0
\(765\) 332.479 191.957i 0.434613 0.250924i
\(766\) 0 0
\(767\) 157.888 + 91.1565i 0.205851 + 0.118848i
\(768\) 0 0
\(769\) −556.399 −0.723536 −0.361768 0.932268i \(-0.617827\pi\)
−0.361768 + 0.932268i \(0.617827\pi\)
\(770\) 0 0
\(771\) −42.3430 −0.0549196
\(772\) 0 0
\(773\) 90.2809 + 52.1237i 0.116793 + 0.0674304i 0.557258 0.830339i \(-0.311853\pi\)
−0.440466 + 0.897770i \(0.645187\pi\)
\(774\) 0 0
\(775\) −426.129 + 246.026i −0.549844 + 0.317452i
\(776\) 0 0
\(777\) 233.474 + 281.916i 0.300481 + 0.362827i
\(778\) 0 0
\(779\) 472.521 + 818.431i 0.606574 + 1.05062i
\(780\) 0 0
\(781\) 451.501 + 260.674i 0.578106 + 0.333769i
\(782\) 0 0
\(783\) 684.935i 0.874757i
\(784\) 0 0
\(785\) −144.088 −0.183551
\(786\) 0 0
\(787\) 18.1655 31.4635i 0.0230819 0.0399791i −0.854254 0.519856i \(-0.825985\pi\)
0.877336 + 0.479877i \(0.159319\pi\)
\(788\) 0 0
\(789\) 3.58799 2.07153i 0.00454752 0.00262551i
\(790\) 0 0
\(791\) −224.084 + 185.579i −0.283292 + 0.234613i
\(792\) 0 0
\(793\) −331.686 574.497i −0.418267 0.724460i
\(794\) 0 0
\(795\) 79.4035 137.531i 0.0998786 0.172995i
\(796\) 0 0
\(797\) 17.3808i 0.0218077i 0.999941 + 0.0109039i \(0.00347087\pi\)
−0.999941 + 0.0109039i \(0.996529\pi\)
\(798\) 0 0
\(799\) 1261.41i 1.57873i
\(800\) 0 0
\(801\) 250.889 434.552i 0.313219 0.542512i
\(802\) 0 0
\(803\) 178.284 + 308.797i 0.222022 + 0.384554i
\(804\) 0 0
\(805\) −44.5188 262.363i −0.0553029 0.325917i
\(806\) 0 0
\(807\) 444.207 256.463i 0.550443 0.317798i
\(808\) 0 0
\(809\) 395.110 684.350i 0.488393 0.845921i −0.511518 0.859273i \(-0.670917\pi\)
0.999911 + 0.0133513i \(0.00424997\pi\)
\(810\) 0 0
\(811\) 1088.07 1.34163 0.670817 0.741623i \(-0.265944\pi\)
0.670817 + 0.741623i \(0.265944\pi\)
\(812\) 0 0
\(813\) 488.194i 0.600484i
\(814\) 0 0
\(815\) 18.4999 + 10.6809i 0.0226992 + 0.0131054i
\(816\) 0 0
\(817\) 364.811 + 631.871i 0.446525 + 0.773403i
\(818\) 0 0
\(819\) −105.302 + 283.610i −0.128574 + 0.346288i
\(820\) 0 0
\(821\) 615.199 355.185i 0.749329 0.432625i −0.0761224 0.997098i \(-0.524254\pi\)
0.825451 + 0.564473i \(0.190921\pi\)
\(822\) 0 0
\(823\) −10.5957 6.11742i −0.0128745 0.00743307i 0.493549 0.869718i \(-0.335699\pi\)
−0.506423 + 0.862285i \(0.669033\pi\)
\(824\) 0 0
\(825\) 143.115 0.173473
\(826\) 0 0
\(827\) −50.6323 −0.0612240 −0.0306120 0.999531i \(-0.509746\pi\)
−0.0306120 + 0.999531i \(0.509746\pi\)
\(828\) 0 0
\(829\) 703.863 + 406.375i 0.849050 + 0.490199i 0.860330 0.509737i \(-0.170257\pi\)
−0.0112800 + 0.999936i \(0.503591\pi\)
\(830\) 0 0
\(831\) −16.1520 + 9.32538i −0.0194369 + 0.0112219i
\(832\) 0 0
\(833\) 782.393 + 673.894i 0.939247 + 0.808996i
\(834\) 0 0
\(835\) 15.9910 + 27.6973i 0.0191510 + 0.0331704i
\(836\) 0 0
\(837\) −537.973 310.599i −0.642740 0.371086i
\(838\) 0 0
\(839\) 247.088i 0.294503i −0.989099 0.147252i \(-0.952957\pi\)
0.989099 0.147252i \(-0.0470427\pi\)
\(840\) 0 0
\(841\) −56.0837 −0.0666869
\(842\) 0 0
\(843\) −166.608 + 288.574i −0.197638 + 0.342318i
\(844\) 0 0
\(845\) −295.993 + 170.892i −0.350287 + 0.202239i
\(846\) 0 0
\(847\) 594.949 + 220.900i 0.702419 + 0.260803i
\(848\) 0 0
\(849\) 8.70990 + 15.0860i 0.0102590 + 0.0177691i
\(850\) 0 0
\(851\) −264.064 + 457.371i −0.310298 + 0.537452i
\(852\) 0 0
\(853\) 659.254i 0.772866i −0.922318 0.386433i \(-0.873707\pi\)
0.922318 0.386433i \(-0.126293\pi\)
\(854\) 0 0
\(855\) 459.921i 0.537919i
\(856\) 0 0
\(857\) 52.1741 90.3682i 0.0608799 0.105447i −0.833979 0.551796i \(-0.813943\pi\)
0.894859 + 0.446349i \(0.147276\pi\)
\(858\) 0 0
\(859\) −675.738 1170.41i −0.786657 1.36253i −0.928004 0.372569i \(-0.878477\pi\)
0.141348 0.989960i \(-0.454856\pi\)
\(860\) 0 0
\(861\) 62.8807 + 370.576i 0.0730322 + 0.430401i
\(862\) 0 0
\(863\) 1129.48 652.105i 1.30878 0.755625i 0.326889 0.945063i \(-0.394000\pi\)
0.981893 + 0.189437i \(0.0606664\pi\)
\(864\) 0 0
\(865\) 347.567 602.004i 0.401812 0.695958i
\(866\) 0 0
\(867\) 222.473 0.256600
\(868\) 0 0
\(869\) 297.927i 0.342839i
\(870\) 0 0
\(871\) 608.446 + 351.286i 0.698560 + 0.403314i
\(872\) 0 0
\(873\) 338.320 + 585.987i 0.387537 + 0.671234i
\(874\) 0 0
\(875\) −505.136 609.944i −0.577298 0.697079i
\(876\) 0 0
\(877\) −1043.59 + 602.517i −1.18995 + 0.687021i −0.958296 0.285777i \(-0.907748\pi\)
−0.231658 + 0.972797i \(0.574415\pi\)
\(878\) 0 0
\(879\) −105.100 60.6797i −0.119568 0.0690327i
\(880\) 0 0
\(881\) −485.495 −0.551073 −0.275536 0.961291i \(-0.588855\pi\)
−0.275536 + 0.961291i \(0.588855\pi\)
\(882\) 0 0
\(883\) −355.597 −0.402714 −0.201357 0.979518i \(-0.564535\pi\)
−0.201357 + 0.979518i \(0.564535\pi\)
\(884\) 0 0
\(885\) −95.4676 55.1182i −0.107873 0.0622805i
\(886\) 0 0
\(887\) −596.239 + 344.239i −0.672197 + 0.388093i −0.796909 0.604100i \(-0.793533\pi\)
0.124712 + 0.992193i \(0.460199\pi\)
\(888\) 0 0
\(889\) −316.911 382.666i −0.356480 0.430445i
\(890\) 0 0
\(891\) −81.7356 141.570i −0.0917346 0.158889i
\(892\) 0 0
\(893\) −1308.69 755.570i −1.46549 0.846103i
\(894\) 0 0
\(895\) 621.315i 0.694207i
\(896\) 0 0
\(897\) 129.366 0.144220
\(898\) 0 0
\(899\) 406.803 704.603i 0.452506 0.783763i
\(900\) 0 0
\(901\) −769.971 + 444.543i −0.854574 + 0.493388i
\(902\) 0 0
\(903\) 48.5471 + 286.103i 0.0537620 + 0.316836i
\(904\) 0 0
\(905\) −25.9048 44.8684i −0.0286241 0.0495784i
\(906\) 0 0
\(907\) 101.105 175.119i 0.111472 0.193075i −0.804892 0.593421i \(-0.797777\pi\)
0.916364 + 0.400346i \(0.131110\pi\)
\(908\) 0 0
\(909\) 1343.01i 1.47746i
\(910\) 0 0
\(911\) 402.691i 0.442032i −0.975270 0.221016i \(-0.929063\pi\)
0.975270 0.221016i \(-0.0709373\pi\)
\(912\) 0 0
\(913\) 253.044 438.284i 0.277156 0.480049i
\(914\) 0 0
\(915\) 200.555 + 347.372i 0.219186 + 0.379642i
\(916\) 0 0
\(917\) 1110.67 + 412.382i 1.21120 + 0.449708i
\(918\) 0 0
\(919\) −49.5319 + 28.5973i −0.0538976 + 0.0311178i −0.526707 0.850047i \(-0.676573\pi\)
0.472809 + 0.881165i \(0.343240\pi\)
\(920\) 0 0
\(921\) −69.8358 + 120.959i −0.0758261 + 0.131335i
\(922\) 0 0
\(923\) 589.238 0.638395
\(924\) 0 0
\(925\) 660.340i 0.713881i
\(926\) 0 0
\(927\) −549.154 317.054i −0.592399 0.342022i
\(928\) 0 0
\(929\) −327.576 567.378i −0.352611 0.610740i 0.634095 0.773255i \(-0.281373\pi\)
−0.986706 + 0.162515i \(0.948039\pi\)
\(930\) 0 0
\(931\) 1167.80 408.062i 1.25435 0.438305i
\(932\) 0 0
\(933\) 124.460 71.8573i 0.133398 0.0770174i
\(934\) 0 0
\(935\) 263.783 + 152.295i 0.282121 + 0.162883i
\(936\) 0 0
\(937\) −1490.29 −1.59049 −0.795245 0.606288i \(-0.792658\pi\)
−0.795245 + 0.606288i \(0.792658\pi\)
\(938\) 0 0
\(939\) 203.475 0.216694
\(940\) 0 0
\(941\) −301.240 173.921i −0.320128 0.184826i 0.331322 0.943518i \(-0.392505\pi\)
−0.651450 + 0.758692i \(0.725839\pi\)
\(942\) 0 0
\(943\) −469.655 + 271.155i −0.498043 + 0.287545i
\(944\) 0 0
\(945\) 146.214 393.797i 0.154723 0.416716i
\(946\) 0 0
\(947\) −716.497 1241.01i −0.756596 1.31046i −0.944577 0.328290i \(-0.893528\pi\)
0.187981 0.982173i \(-0.439806\pi\)
\(948\) 0 0
\(949\) 349.009 + 201.500i 0.367765 + 0.212329i
\(950\) 0 0
\(951\) 41.2488i 0.0433741i
\(952\) 0 0
\(953\) −67.7570 −0.0710986 −0.0355493 0.999368i \(-0.511318\pi\)
−0.0355493 + 0.999368i \(0.511318\pi\)
\(954\) 0 0
\(955\) −210.900 + 365.290i −0.220838 + 0.382503i
\(956\) 0 0
\(957\) −204.936 + 118.320i −0.214145 + 0.123636i
\(958\) 0 0
\(959\) 113.525 + 669.040i 0.118379 + 0.697643i
\(960\) 0 0
\(961\) −111.553 193.215i −0.116080 0.201056i
\(962\) 0 0
\(963\) 11.9509 20.6995i 0.0124100 0.0214948i
\(964\) 0 0
\(965\) 445.680i 0.461845i
\(966\) 0 0
\(967\) 1272.02i 1.31543i −0.753267 0.657715i \(-0.771523\pi\)
0.753267 0.657715i \(-0.228477\pi\)
\(968\) 0 0
\(969\) 381.570 660.898i 0.393777 0.682042i
\(970\) 0 0
\(971\) −544.112 942.430i −0.560363 0.970577i −0.997465 0.0711647i \(-0.977328\pi\)
0.437102 0.899412i \(-0.356005\pi\)
\(972\) 0 0
\(973\) 526.201 435.782i 0.540802 0.447875i
\(974\) 0 0
\(975\) 140.081 80.8757i 0.143673 0.0829495i
\(976\) 0 0
\(977\) 222.510 385.398i 0.227748 0.394471i −0.729392 0.684096i \(-0.760197\pi\)
0.957140 + 0.289625i \(0.0935305\pi\)
\(978\) 0 0
\(979\) 398.101 0.406641
\(980\) 0 0
\(981\) 1345.47i 1.37153i
\(982\) 0 0
\(983\) −898.398 518.691i −0.913935 0.527661i −0.0322401 0.999480i \(-0.510264\pi\)
−0.881695 + 0.471819i \(0.843597\pi\)
\(984\) 0 0
\(985\) 62.0511 + 107.476i 0.0629960 + 0.109112i
\(986\) 0 0
\(987\) −383.355 462.896i −0.388404 0.468992i
\(988\) 0 0
\(989\) −362.597 + 209.346i −0.366630 + 0.211674i
\(990\) 0 0
\(991\) −57.7139 33.3211i −0.0582381 0.0336238i 0.470598 0.882348i \(-0.344038\pi\)
−0.528836 + 0.848724i \(0.677371\pi\)
\(992\) 0 0
\(993\) −404.210 −0.407060
\(994\) 0 0
\(995\) −738.752 −0.742465
\(996\) 0 0
\(997\) 1642.43 + 948.258i 1.64737 + 0.951111i 0.978111 + 0.208084i \(0.0667228\pi\)
0.669262 + 0.743027i \(0.266611\pi\)
\(998\) 0 0
\(999\) −721.968 + 416.828i −0.722691 + 0.417246i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 224.3.o.c.207.3 12
4.3 odd 2 56.3.k.c.11.6 yes 12
7.2 even 3 inner 224.3.o.c.79.4 12
7.3 odd 6 1568.3.g.i.687.4 6
7.4 even 3 1568.3.g.k.687.3 6
8.3 odd 2 inner 224.3.o.c.207.4 12
8.5 even 2 56.3.k.c.11.1 12
28.3 even 6 392.3.g.l.99.4 6
28.11 odd 6 392.3.g.k.99.4 6
28.19 even 6 392.3.k.k.275.1 12
28.23 odd 6 56.3.k.c.51.1 yes 12
28.27 even 2 392.3.k.k.67.6 12
56.3 even 6 1568.3.g.i.687.3 6
56.5 odd 6 392.3.k.k.275.6 12
56.11 odd 6 1568.3.g.k.687.4 6
56.13 odd 2 392.3.k.k.67.1 12
56.37 even 6 56.3.k.c.51.6 yes 12
56.45 odd 6 392.3.g.l.99.3 6
56.51 odd 6 inner 224.3.o.c.79.3 12
56.53 even 6 392.3.g.k.99.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.c.11.1 12 8.5 even 2
56.3.k.c.11.6 yes 12 4.3 odd 2
56.3.k.c.51.1 yes 12 28.23 odd 6
56.3.k.c.51.6 yes 12 56.37 even 6
224.3.o.c.79.3 12 56.51 odd 6 inner
224.3.o.c.79.4 12 7.2 even 3 inner
224.3.o.c.207.3 12 1.1 even 1 trivial
224.3.o.c.207.4 12 8.3 odd 2 inner
392.3.g.k.99.3 6 56.53 even 6
392.3.g.k.99.4 6 28.11 odd 6
392.3.g.l.99.3 6 56.45 odd 6
392.3.g.l.99.4 6 28.3 even 6
392.3.k.k.67.1 12 56.13 odd 2
392.3.k.k.67.6 12 28.27 even 2
392.3.k.k.275.1 12 28.19 even 6
392.3.k.k.275.6 12 56.5 odd 6
1568.3.g.i.687.3 6 56.3 even 6
1568.3.g.i.687.4 6 7.3 odd 6
1568.3.g.k.687.3 6 7.4 even 3
1568.3.g.k.687.4 6 56.11 odd 6