Properties

Label 224.3.n.a.17.4
Level $224$
Weight $3$
Character 224.17
Analytic conductor $6.104$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(17,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.4
Character \(\chi\) \(=\) 224.17
Dual form 224.3.n.a.145.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.70138 + 2.94687i) q^{3} +(-2.15858 - 3.73877i) q^{5} +(1.43197 + 6.85197i) q^{7} +(-1.28938 - 2.23327i) q^{9} +O(q^{10})\) \(q+(-1.70138 + 2.94687i) q^{3} +(-2.15858 - 3.73877i) q^{5} +(1.43197 + 6.85197i) q^{7} +(-1.28938 - 2.23327i) q^{9} +(-15.4899 - 8.94308i) q^{11} -3.25607 q^{13} +14.6903 q^{15} +(-13.6263 - 7.86717i) q^{17} +(0.778522 + 1.34844i) q^{19} +(-22.6282 - 7.43796i) q^{21} +(-20.7069 - 35.8655i) q^{23} +(3.18105 - 5.50975i) q^{25} -21.8499 q^{27} +3.74374i q^{29} +(0.0145172 + 0.00838150i) q^{31} +(52.7082 - 30.4311i) q^{33} +(22.5269 - 20.1443i) q^{35} +(1.16774 - 0.674194i) q^{37} +(5.53981 - 9.59523i) q^{39} +70.3018i q^{41} +13.0380i q^{43} +(-5.56646 + 9.64139i) q^{45} +(-30.9797 + 17.8862i) q^{47} +(-44.8989 + 19.6236i) q^{49} +(46.3671 - 26.7701i) q^{51} +(-39.7989 - 22.9779i) q^{53} +77.2174i q^{55} -5.29824 q^{57} +(-34.3509 + 59.4974i) q^{59} +(48.0386 + 83.2052i) q^{61} +(13.4559 - 12.0328i) q^{63} +(7.02849 + 12.1737i) q^{65} +(12.0808 + 6.97484i) q^{67} +140.921 q^{69} +75.7095 q^{71} +(-46.0282 - 26.5744i) q^{73} +(10.8244 + 18.7483i) q^{75} +(39.0967 - 118.942i) q^{77} +(-11.6744 - 20.2206i) q^{79} +(48.7794 - 84.4884i) q^{81} +102.487 q^{83} +67.9277i q^{85} +(-11.0323 - 6.36952i) q^{87} +(-76.6985 + 44.2819i) q^{89} +(-4.66259 - 22.3105i) q^{91} +(-0.0493984 + 0.0285202i) q^{93} +(3.36100 - 5.82143i) q^{95} -140.869i q^{97} +46.1241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 4 q^{7} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 4 q^{7} - 32 q^{9} - 28 q^{15} - 6 q^{17} - 30 q^{23} - 32 q^{25} + 6 q^{31} - 6 q^{33} + 20 q^{39} + 294 q^{47} - 20 q^{49} + 124 q^{57} - 432 q^{63} - 52 q^{65} + 136 q^{71} + 234 q^{73} + 162 q^{79} - 18 q^{81} - 48 q^{87} - 150 q^{89} - 290 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.70138 + 2.94687i −0.567126 + 0.982291i 0.429722 + 0.902961i \(0.358612\pi\)
−0.996848 + 0.0793303i \(0.974722\pi\)
\(4\) 0 0
\(5\) −2.15858 3.73877i −0.431716 0.747754i 0.565305 0.824882i \(-0.308758\pi\)
−0.997021 + 0.0771275i \(0.975425\pi\)
\(6\) 0 0
\(7\) 1.43197 + 6.85197i 0.204567 + 0.978853i
\(8\) 0 0
\(9\) −1.28938 2.23327i −0.143264 0.248141i
\(10\) 0 0
\(11\) −15.4899 8.94308i −1.40817 0.813007i −0.412958 0.910750i \(-0.635504\pi\)
−0.995212 + 0.0977432i \(0.968838\pi\)
\(12\) 0 0
\(13\) −3.25607 −0.250467 −0.125233 0.992127i \(-0.539968\pi\)
−0.125233 + 0.992127i \(0.539968\pi\)
\(14\) 0 0
\(15\) 14.6903 0.979350
\(16\) 0 0
\(17\) −13.6263 7.86717i −0.801550 0.462775i 0.0424631 0.999098i \(-0.486479\pi\)
−0.844013 + 0.536323i \(0.819813\pi\)
\(18\) 0 0
\(19\) 0.778522 + 1.34844i 0.0409748 + 0.0709705i 0.885786 0.464095i \(-0.153620\pi\)
−0.844811 + 0.535065i \(0.820287\pi\)
\(20\) 0 0
\(21\) −22.6282 7.43796i −1.07753 0.354188i
\(22\) 0 0
\(23\) −20.7069 35.8655i −0.900301 1.55937i −0.827103 0.562050i \(-0.810013\pi\)
−0.0731984 0.997317i \(-0.523321\pi\)
\(24\) 0 0
\(25\) 3.18105 5.50975i 0.127242 0.220390i
\(26\) 0 0
\(27\) −21.8499 −0.809257
\(28\) 0 0
\(29\) 3.74374i 0.129095i 0.997915 + 0.0645473i \(0.0205603\pi\)
−0.997915 + 0.0645473i \(0.979440\pi\)
\(30\) 0 0
\(31\) 0.0145172 + 0.00838150i 0.000468296 + 0.000270371i 0.500234 0.865890i \(-0.333247\pi\)
−0.499766 + 0.866161i \(0.666581\pi\)
\(32\) 0 0
\(33\) 52.7082 30.4311i 1.59722 0.922155i
\(34\) 0 0
\(35\) 22.5269 20.1443i 0.643626 0.575553i
\(36\) 0 0
\(37\) 1.16774 0.674194i 0.0315605 0.0182215i −0.484137 0.874992i \(-0.660866\pi\)
0.515697 + 0.856771i \(0.327533\pi\)
\(38\) 0 0
\(39\) 5.53981 9.59523i 0.142046 0.246031i
\(40\) 0 0
\(41\) 70.3018i 1.71468i 0.514753 + 0.857339i \(0.327884\pi\)
−0.514753 + 0.857339i \(0.672116\pi\)
\(42\) 0 0
\(43\) 13.0380i 0.303210i 0.988441 + 0.151605i \(0.0484442\pi\)
−0.988441 + 0.151605i \(0.951556\pi\)
\(44\) 0 0
\(45\) −5.56646 + 9.64139i −0.123699 + 0.214253i
\(46\) 0 0
\(47\) −30.9797 + 17.8862i −0.659144 + 0.380557i −0.791951 0.610585i \(-0.790934\pi\)
0.132807 + 0.991142i \(0.457601\pi\)
\(48\) 0 0
\(49\) −44.8989 + 19.6236i −0.916305 + 0.400482i
\(50\) 0 0
\(51\) 46.3671 26.7701i 0.909160 0.524904i
\(52\) 0 0
\(53\) −39.7989 22.9779i −0.750923 0.433546i 0.0751042 0.997176i \(-0.476071\pi\)
−0.826027 + 0.563630i \(0.809404\pi\)
\(54\) 0 0
\(55\) 77.2174i 1.40395i
\(56\) 0 0
\(57\) −5.29824 −0.0929516
\(58\) 0 0
\(59\) −34.3509 + 59.4974i −0.582218 + 1.00843i 0.412998 + 0.910732i \(0.364482\pi\)
−0.995216 + 0.0976993i \(0.968852\pi\)
\(60\) 0 0
\(61\) 48.0386 + 83.2052i 0.787517 + 1.36402i 0.927484 + 0.373864i \(0.121967\pi\)
−0.139966 + 0.990156i \(0.544699\pi\)
\(62\) 0 0
\(63\) 13.4559 12.0328i 0.213586 0.190996i
\(64\) 0 0
\(65\) 7.02849 + 12.1737i 0.108131 + 0.187288i
\(66\) 0 0
\(67\) 12.0808 + 6.97484i 0.180310 + 0.104102i 0.587438 0.809269i \(-0.300136\pi\)
−0.407128 + 0.913371i \(0.633470\pi\)
\(68\) 0 0
\(69\) 140.921 2.04234
\(70\) 0 0
\(71\) 75.7095 1.06633 0.533166 0.846011i \(-0.321002\pi\)
0.533166 + 0.846011i \(0.321002\pi\)
\(72\) 0 0
\(73\) −46.0282 26.5744i −0.630523 0.364033i 0.150432 0.988620i \(-0.451934\pi\)
−0.780955 + 0.624588i \(0.785267\pi\)
\(74\) 0 0
\(75\) 10.8244 + 18.7483i 0.144325 + 0.249978i
\(76\) 0 0
\(77\) 39.0967 118.942i 0.507749 1.54470i
\(78\) 0 0
\(79\) −11.6744 20.2206i −0.147777 0.255957i 0.782628 0.622489i \(-0.213878\pi\)
−0.930406 + 0.366532i \(0.880545\pi\)
\(80\) 0 0
\(81\) 48.7794 84.4884i 0.602215 1.04307i
\(82\) 0 0
\(83\) 102.487 1.23479 0.617393 0.786655i \(-0.288189\pi\)
0.617393 + 0.786655i \(0.288189\pi\)
\(84\) 0 0
\(85\) 67.9277i 0.799150i
\(86\) 0 0
\(87\) −11.0323 6.36952i −0.126808 0.0732129i
\(88\) 0 0
\(89\) −76.6985 + 44.2819i −0.861781 + 0.497549i −0.864608 0.502447i \(-0.832433\pi\)
0.00282755 + 0.999996i \(0.499100\pi\)
\(90\) 0 0
\(91\) −4.66259 22.3105i −0.0512373 0.245170i
\(92\) 0 0
\(93\) −0.0493984 + 0.0285202i −0.000531166 + 0.000306669i
\(94\) 0 0
\(95\) 3.36100 5.82143i 0.0353790 0.0612782i
\(96\) 0 0
\(97\) 140.869i 1.45226i −0.687558 0.726130i \(-0.741317\pi\)
0.687558 0.726130i \(-0.258683\pi\)
\(98\) 0 0
\(99\) 46.1241i 0.465900i
\(100\) 0 0
\(101\) −17.6988 + 30.6553i −0.175236 + 0.303518i −0.940243 0.340504i \(-0.889402\pi\)
0.765007 + 0.644022i \(0.222735\pi\)
\(102\) 0 0
\(103\) −87.1651 + 50.3248i −0.846263 + 0.488590i −0.859388 0.511324i \(-0.829155\pi\)
0.0131250 + 0.999914i \(0.495822\pi\)
\(104\) 0 0
\(105\) 21.0360 + 100.657i 0.200343 + 0.958640i
\(106\) 0 0
\(107\) 92.6215 53.4751i 0.865622 0.499767i −0.000269099 1.00000i \(-0.500086\pi\)
0.865891 + 0.500233i \(0.166752\pi\)
\(108\) 0 0
\(109\) −45.5799 26.3156i −0.418165 0.241427i 0.276127 0.961121i \(-0.410949\pi\)
−0.694292 + 0.719694i \(0.744282\pi\)
\(110\) 0 0
\(111\) 4.58824i 0.0413355i
\(112\) 0 0
\(113\) 45.4346 0.402076 0.201038 0.979583i \(-0.435568\pi\)
0.201038 + 0.979583i \(0.435568\pi\)
\(114\) 0 0
\(115\) −89.3952 + 154.837i −0.777350 + 1.34641i
\(116\) 0 0
\(117\) 4.19831 + 7.27168i 0.0358830 + 0.0621511i
\(118\) 0 0
\(119\) 34.3931 104.633i 0.289018 0.879267i
\(120\) 0 0
\(121\) 99.4572 + 172.265i 0.821961 + 1.42368i
\(122\) 0 0
\(123\) −207.171 119.610i −1.68431 0.972439i
\(124\) 0 0
\(125\) −135.395 −1.08316
\(126\) 0 0
\(127\) −125.695 −0.989723 −0.494861 0.868972i \(-0.664781\pi\)
−0.494861 + 0.868972i \(0.664781\pi\)
\(128\) 0 0
\(129\) −38.4215 22.1827i −0.297841 0.171959i
\(130\) 0 0
\(131\) −56.6504 98.1214i −0.432446 0.749018i 0.564638 0.825339i \(-0.309016\pi\)
−0.997083 + 0.0763210i \(0.975683\pi\)
\(132\) 0 0
\(133\) −8.12464 + 7.26533i −0.0610875 + 0.0546265i
\(134\) 0 0
\(135\) 47.1648 + 81.6919i 0.349369 + 0.605125i
\(136\) 0 0
\(137\) −39.1679 + 67.8408i −0.285897 + 0.495188i −0.972826 0.231536i \(-0.925625\pi\)
0.686929 + 0.726724i \(0.258958\pi\)
\(138\) 0 0
\(139\) 149.038 1.07222 0.536109 0.844149i \(-0.319894\pi\)
0.536109 + 0.844149i \(0.319894\pi\)
\(140\) 0 0
\(141\) 121.725i 0.863295i
\(142\) 0 0
\(143\) 50.4361 + 29.1193i 0.352700 + 0.203631i
\(144\) 0 0
\(145\) 13.9970 8.08117i 0.0965310 0.0557322i
\(146\) 0 0
\(147\) 18.5617 165.699i 0.126270 1.12720i
\(148\) 0 0
\(149\) −73.8369 + 42.6298i −0.495550 + 0.286106i −0.726874 0.686771i \(-0.759028\pi\)
0.231324 + 0.972877i \(0.425694\pi\)
\(150\) 0 0
\(151\) 65.9012 114.144i 0.436432 0.755922i −0.560979 0.827830i \(-0.689575\pi\)
0.997411 + 0.0719076i \(0.0229087\pi\)
\(152\) 0 0
\(153\) 40.5751i 0.265197i
\(154\) 0 0
\(155\) 0.0723686i 0.000466894i
\(156\) 0 0
\(157\) −122.552 + 212.267i −0.780589 + 1.35202i 0.151010 + 0.988532i \(0.451747\pi\)
−0.931599 + 0.363487i \(0.881586\pi\)
\(158\) 0 0
\(159\) 135.426 78.1883i 0.851737 0.491750i
\(160\) 0 0
\(161\) 216.097 193.241i 1.34222 1.20026i
\(162\) 0 0
\(163\) −208.089 + 120.140i −1.27662 + 0.737057i −0.976225 0.216758i \(-0.930452\pi\)
−0.300395 + 0.953815i \(0.597118\pi\)
\(164\) 0 0
\(165\) −227.550 131.376i −1.37909 0.796219i
\(166\) 0 0
\(167\) 73.1965i 0.438302i −0.975691 0.219151i \(-0.929671\pi\)
0.975691 0.219151i \(-0.0703288\pi\)
\(168\) 0 0
\(169\) −158.398 −0.937266
\(170\) 0 0
\(171\) 2.00762 3.47730i 0.0117405 0.0203351i
\(172\) 0 0
\(173\) 18.5246 + 32.0855i 0.107078 + 0.185465i 0.914585 0.404393i \(-0.132517\pi\)
−0.807507 + 0.589858i \(0.799184\pi\)
\(174\) 0 0
\(175\) 42.3078 + 13.9067i 0.241759 + 0.0794668i
\(176\) 0 0
\(177\) −116.888 202.455i −0.660382 1.14382i
\(178\) 0 0
\(179\) −205.982 118.924i −1.15074 0.664379i −0.201672 0.979453i \(-0.564637\pi\)
−0.949067 + 0.315074i \(0.897971\pi\)
\(180\) 0 0
\(181\) 292.553 1.61631 0.808157 0.588966i \(-0.200465\pi\)
0.808157 + 0.588966i \(0.200465\pi\)
\(182\) 0 0
\(183\) −326.927 −1.78649
\(184\) 0 0
\(185\) −5.04132 2.91061i −0.0272504 0.0157330i
\(186\) 0 0
\(187\) 140.713 + 243.723i 0.752478 + 1.30333i
\(188\) 0 0
\(189\) −31.2884 149.715i −0.165547 0.792143i
\(190\) 0 0
\(191\) 70.6135 + 122.306i 0.369704 + 0.640346i 0.989519 0.144402i \(-0.0461257\pi\)
−0.619815 + 0.784748i \(0.712792\pi\)
\(192\) 0 0
\(193\) 32.9799 57.1229i 0.170880 0.295973i −0.767848 0.640633i \(-0.778672\pi\)
0.938728 + 0.344659i \(0.112006\pi\)
\(194\) 0 0
\(195\) −47.8325 −0.245295
\(196\) 0 0
\(197\) 199.421i 1.01229i −0.862448 0.506145i \(-0.831070\pi\)
0.862448 0.506145i \(-0.168930\pi\)
\(198\) 0 0
\(199\) −58.6230 33.8460i −0.294588 0.170080i 0.345421 0.938448i \(-0.387736\pi\)
−0.640009 + 0.768367i \(0.721069\pi\)
\(200\) 0 0
\(201\) −41.1079 + 23.7337i −0.204517 + 0.118078i
\(202\) 0 0
\(203\) −25.6520 + 5.36092i −0.126365 + 0.0264085i
\(204\) 0 0
\(205\) 262.842 151.752i 1.28216 0.740254i
\(206\) 0 0
\(207\) −53.3982 + 92.4884i −0.257962 + 0.446804i
\(208\) 0 0
\(209\) 27.8495i 0.133251i
\(210\) 0 0
\(211\) 62.1464i 0.294533i 0.989097 + 0.147266i \(0.0470475\pi\)
−0.989097 + 0.147266i \(0.952953\pi\)
\(212\) 0 0
\(213\) −128.811 + 223.106i −0.604744 + 1.04745i
\(214\) 0 0
\(215\) 48.7463 28.1437i 0.226727 0.130901i
\(216\) 0 0
\(217\) −0.0366416 + 0.111473i −0.000168855 + 0.000513702i
\(218\) 0 0
\(219\) 156.623 90.4261i 0.715172 0.412905i
\(220\) 0 0
\(221\) 44.3683 + 25.6161i 0.200762 + 0.115910i
\(222\) 0 0
\(223\) 115.525i 0.518050i −0.965871 0.259025i \(-0.916599\pi\)
0.965871 0.259025i \(-0.0834012\pi\)
\(224\) 0 0
\(225\) −16.4063 −0.0729171
\(226\) 0 0
\(227\) 28.2532 48.9360i 0.124463 0.215577i −0.797060 0.603901i \(-0.793612\pi\)
0.921523 + 0.388324i \(0.126946\pi\)
\(228\) 0 0
\(229\) −59.1696 102.485i −0.258383 0.447532i 0.707426 0.706787i \(-0.249856\pi\)
−0.965809 + 0.259255i \(0.916523\pi\)
\(230\) 0 0
\(231\) 283.990 + 317.579i 1.22939 + 1.37480i
\(232\) 0 0
\(233\) 12.3403 + 21.3740i 0.0529625 + 0.0917337i 0.891291 0.453431i \(-0.149800\pi\)
−0.838329 + 0.545165i \(0.816467\pi\)
\(234\) 0 0
\(235\) 133.745 + 77.2175i 0.569126 + 0.328585i
\(236\) 0 0
\(237\) 79.4503 0.335233
\(238\) 0 0
\(239\) 251.189 1.05100 0.525499 0.850794i \(-0.323879\pi\)
0.525499 + 0.850794i \(0.323879\pi\)
\(240\) 0 0
\(241\) 97.3782 + 56.2213i 0.404059 + 0.233283i 0.688234 0.725489i \(-0.258386\pi\)
−0.284175 + 0.958772i \(0.591720\pi\)
\(242\) 0 0
\(243\) 67.6598 + 117.190i 0.278436 + 0.482265i
\(244\) 0 0
\(245\) 170.286 + 125.508i 0.695046 + 0.512276i
\(246\) 0 0
\(247\) −2.53492 4.39061i −0.0102628 0.0177758i
\(248\) 0 0
\(249\) −174.370 + 302.017i −0.700280 + 1.21292i
\(250\) 0 0
\(251\) 121.248 0.483059 0.241529 0.970394i \(-0.422351\pi\)
0.241529 + 0.970394i \(0.422351\pi\)
\(252\) 0 0
\(253\) 740.735i 2.92781i
\(254\) 0 0
\(255\) −200.174 115.571i −0.784998 0.453219i
\(256\) 0 0
\(257\) −90.7377 + 52.3874i −0.353065 + 0.203842i −0.666034 0.745921i \(-0.732010\pi\)
0.312969 + 0.949763i \(0.398676\pi\)
\(258\) 0 0
\(259\) 6.29172 + 7.03588i 0.0242924 + 0.0271656i
\(260\) 0 0
\(261\) 8.36079 4.82710i 0.0320337 0.0184946i
\(262\) 0 0
\(263\) −52.3392 + 90.6542i −0.199008 + 0.344693i −0.948207 0.317653i \(-0.897105\pi\)
0.749199 + 0.662345i \(0.230439\pi\)
\(264\) 0 0
\(265\) 198.399i 0.748675i
\(266\) 0 0
\(267\) 301.361i 1.12869i
\(268\) 0 0
\(269\) 152.466 264.079i 0.566789 0.981707i −0.430092 0.902785i \(-0.641519\pi\)
0.996881 0.0789222i \(-0.0251479\pi\)
\(270\) 0 0
\(271\) −88.8942 + 51.3231i −0.328023 + 0.189384i −0.654963 0.755661i \(-0.727316\pi\)
0.326940 + 0.945045i \(0.393982\pi\)
\(272\) 0 0
\(273\) 73.6790 + 24.2185i 0.269886 + 0.0887125i
\(274\) 0 0
\(275\) −98.5482 + 56.8968i −0.358357 + 0.206898i
\(276\) 0 0
\(277\) −14.4235 8.32739i −0.0520703 0.0300628i 0.473739 0.880665i \(-0.342904\pi\)
−0.525809 + 0.850603i \(0.676237\pi\)
\(278\) 0 0
\(279\) 0.0432277i 0.000154938i
\(280\) 0 0
\(281\) 75.8291 0.269855 0.134927 0.990856i \(-0.456920\pi\)
0.134927 + 0.990856i \(0.456920\pi\)
\(282\) 0 0
\(283\) 43.6656 75.6311i 0.154296 0.267248i −0.778507 0.627636i \(-0.784023\pi\)
0.932802 + 0.360389i \(0.117356\pi\)
\(284\) 0 0
\(285\) 11.4367 + 19.8089i 0.0401287 + 0.0695050i
\(286\) 0 0
\(287\) −481.706 + 100.670i −1.67842 + 0.350767i
\(288\) 0 0
\(289\) −20.7152 35.8798i −0.0716789 0.124151i
\(290\) 0 0
\(291\) 415.124 + 239.672i 1.42654 + 0.823614i
\(292\) 0 0
\(293\) −27.5057 −0.0938760 −0.0469380 0.998898i \(-0.514946\pi\)
−0.0469380 + 0.998898i \(0.514946\pi\)
\(294\) 0 0
\(295\) 296.597 1.00541
\(296\) 0 0
\(297\) 338.452 + 195.406i 1.13957 + 0.657931i
\(298\) 0 0
\(299\) 67.4232 + 116.780i 0.225496 + 0.390570i
\(300\) 0 0
\(301\) −89.3363 + 18.6701i −0.296798 + 0.0620269i
\(302\) 0 0
\(303\) −60.2248 104.312i −0.198762 0.344266i
\(304\) 0 0
\(305\) 207.390 359.211i 0.679968 1.17774i
\(306\) 0 0
\(307\) 247.996 0.807805 0.403902 0.914802i \(-0.367654\pi\)
0.403902 + 0.914802i \(0.367654\pi\)
\(308\) 0 0
\(309\) 342.486i 1.10837i
\(310\) 0 0
\(311\) −378.484 218.518i −1.21699 0.702630i −0.252717 0.967540i \(-0.581324\pi\)
−0.964273 + 0.264910i \(0.914658\pi\)
\(312\) 0 0
\(313\) −71.7330 + 41.4151i −0.229179 + 0.132317i −0.610193 0.792253i \(-0.708908\pi\)
0.381014 + 0.924569i \(0.375575\pi\)
\(314\) 0 0
\(315\) −74.0335 24.3350i −0.235027 0.0772540i
\(316\) 0 0
\(317\) 211.775 122.268i 0.668059 0.385704i −0.127282 0.991867i \(-0.540625\pi\)
0.795341 + 0.606162i \(0.207292\pi\)
\(318\) 0 0
\(319\) 33.4806 57.9900i 0.104955 0.181787i
\(320\) 0 0
\(321\) 363.925i 1.13372i
\(322\) 0 0
\(323\) 24.4991i 0.0758485i
\(324\) 0 0
\(325\) −10.3577 + 17.9401i −0.0318699 + 0.0552004i
\(326\) 0 0
\(327\) 155.097 89.5456i 0.474304 0.273840i
\(328\) 0 0
\(329\) −166.917 186.660i −0.507348 0.567355i
\(330\) 0 0
\(331\) −66.2919 + 38.2736i −0.200278 + 0.115630i −0.596785 0.802401i \(-0.703555\pi\)
0.396507 + 0.918032i \(0.370222\pi\)
\(332\) 0 0
\(333\) −3.01132 1.73858i −0.00904299 0.00522097i
\(334\) 0 0
\(335\) 60.2230i 0.179770i
\(336\) 0 0
\(337\) −38.2520 −0.113507 −0.0567537 0.998388i \(-0.518075\pi\)
−0.0567537 + 0.998388i \(0.518075\pi\)
\(338\) 0 0
\(339\) −77.3015 + 133.890i −0.228028 + 0.394956i
\(340\) 0 0
\(341\) −0.149913 0.259656i −0.000439627 0.000761456i
\(342\) 0 0
\(343\) −198.754 279.546i −0.579459 0.815002i
\(344\) 0 0
\(345\) −304.190 526.873i −0.881711 1.52717i
\(346\) 0 0
\(347\) −208.395 120.317i −0.600561 0.346734i 0.168701 0.985667i \(-0.446043\pi\)
−0.769262 + 0.638933i \(0.779376\pi\)
\(348\) 0 0
\(349\) −430.367 −1.23314 −0.616572 0.787298i \(-0.711479\pi\)
−0.616572 + 0.787298i \(0.711479\pi\)
\(350\) 0 0
\(351\) 71.1449 0.202692
\(352\) 0 0
\(353\) −265.950 153.546i −0.753399 0.434975i 0.0735214 0.997294i \(-0.476576\pi\)
−0.826921 + 0.562318i \(0.809910\pi\)
\(354\) 0 0
\(355\) −163.425 283.061i −0.460353 0.797354i
\(356\) 0 0
\(357\) 249.824 + 279.372i 0.699787 + 0.782555i
\(358\) 0 0
\(359\) 230.880 + 399.896i 0.643120 + 1.11392i 0.984732 + 0.174075i \(0.0556935\pi\)
−0.341613 + 0.939841i \(0.610973\pi\)
\(360\) 0 0
\(361\) 179.288 310.536i 0.496642 0.860209i
\(362\) 0 0
\(363\) −676.858 −1.86462
\(364\) 0 0
\(365\) 229.452i 0.628635i
\(366\) 0 0
\(367\) 542.949 + 313.471i 1.47942 + 0.854146i 0.999729 0.0232895i \(-0.00741394\pi\)
0.479695 + 0.877435i \(0.340747\pi\)
\(368\) 0 0
\(369\) 157.003 90.6457i 0.425482 0.245652i
\(370\) 0 0
\(371\) 100.453 305.605i 0.270763 0.823732i
\(372\) 0 0
\(373\) −357.317 + 206.297i −0.957953 + 0.553075i −0.895543 0.444976i \(-0.853212\pi\)
−0.0624108 + 0.998051i \(0.519879\pi\)
\(374\) 0 0
\(375\) 230.359 398.993i 0.614290 1.06398i
\(376\) 0 0
\(377\) 12.1899i 0.0323339i
\(378\) 0 0
\(379\) 327.118i 0.863107i 0.902087 + 0.431554i \(0.142034\pi\)
−0.902087 + 0.431554i \(0.857966\pi\)
\(380\) 0 0
\(381\) 213.854 370.407i 0.561298 0.972196i
\(382\) 0 0
\(383\) 215.523 124.432i 0.562724 0.324889i −0.191514 0.981490i \(-0.561340\pi\)
0.754238 + 0.656601i \(0.228006\pi\)
\(384\) 0 0
\(385\) −529.091 + 110.573i −1.37426 + 0.287203i
\(386\) 0 0
\(387\) 29.1175 16.8110i 0.0752390 0.0434392i
\(388\) 0 0
\(389\) 326.728 + 188.637i 0.839918 + 0.484927i 0.857236 0.514923i \(-0.172179\pi\)
−0.0173181 + 0.999850i \(0.505513\pi\)
\(390\) 0 0
\(391\) 651.620i 1.66655i
\(392\) 0 0
\(393\) 385.535 0.981005
\(394\) 0 0
\(395\) −50.4003 + 87.2958i −0.127596 + 0.221002i
\(396\) 0 0
\(397\) −335.874 581.752i −0.846031 1.46537i −0.884723 0.466118i \(-0.845652\pi\)
0.0386913 0.999251i \(-0.487681\pi\)
\(398\) 0 0
\(399\) −7.58692 36.3034i −0.0190148 0.0909859i
\(400\) 0 0
\(401\) 235.200 + 407.378i 0.586534 + 1.01591i 0.994682 + 0.102991i \(0.0328411\pi\)
−0.408149 + 0.912915i \(0.633826\pi\)
\(402\) 0 0
\(403\) −0.0472689 0.0272907i −0.000117293 6.77189e-5i
\(404\) 0 0
\(405\) −421.177 −1.03994
\(406\) 0 0
\(407\) −24.1175 −0.0592567
\(408\) 0 0
\(409\) −57.7400 33.3362i −0.141174 0.0815067i 0.427750 0.903897i \(-0.359307\pi\)
−0.568923 + 0.822391i \(0.692640\pi\)
\(410\) 0 0
\(411\) −133.279 230.846i −0.324279 0.561669i
\(412\) 0 0
\(413\) −456.864 150.172i −1.10621 0.363614i
\(414\) 0 0
\(415\) −221.227 383.177i −0.533077 0.923317i
\(416\) 0 0
\(417\) −253.571 + 439.197i −0.608083 + 1.05323i
\(418\) 0 0
\(419\) −437.380 −1.04387 −0.521933 0.852986i \(-0.674789\pi\)
−0.521933 + 0.852986i \(0.674789\pi\)
\(420\) 0 0
\(421\) 703.800i 1.67173i −0.548933 0.835867i \(-0.684966\pi\)
0.548933 0.835867i \(-0.315034\pi\)
\(422\) 0 0
\(423\) 79.8893 + 46.1241i 0.188864 + 0.109040i
\(424\) 0 0
\(425\) −86.6923 + 50.0518i −0.203982 + 0.117769i
\(426\) 0 0
\(427\) −501.330 + 448.306i −1.17407 + 1.04990i
\(428\) 0 0
\(429\) −171.622 + 99.0858i −0.400051 + 0.230969i
\(430\) 0 0
\(431\) −274.869 + 476.087i −0.637747 + 1.10461i 0.348178 + 0.937428i \(0.386800\pi\)
−0.985926 + 0.167183i \(0.946533\pi\)
\(432\) 0 0
\(433\) 355.012i 0.819890i −0.912110 0.409945i \(-0.865548\pi\)
0.912110 0.409945i \(-0.134452\pi\)
\(434\) 0 0
\(435\) 54.9965i 0.126429i
\(436\) 0 0
\(437\) 32.2416 55.8441i 0.0737794 0.127790i
\(438\) 0 0
\(439\) −477.032 + 275.415i −1.08663 + 0.627369i −0.932678 0.360709i \(-0.882535\pi\)
−0.153956 + 0.988078i \(0.549201\pi\)
\(440\) 0 0
\(441\) 101.717 + 74.9692i 0.230650 + 0.169998i
\(442\) 0 0
\(443\) −234.027 + 135.116i −0.528278 + 0.305001i −0.740315 0.672260i \(-0.765324\pi\)
0.212037 + 0.977262i \(0.431990\pi\)
\(444\) 0 0
\(445\) 331.120 + 191.172i 0.744089 + 0.429600i
\(446\) 0 0
\(447\) 290.117i 0.649032i
\(448\) 0 0
\(449\) 455.397 1.01425 0.507124 0.861873i \(-0.330709\pi\)
0.507124 + 0.861873i \(0.330709\pi\)
\(450\) 0 0
\(451\) 628.714 1088.96i 1.39404 2.41456i
\(452\) 0 0
\(453\) 224.246 + 388.405i 0.495024 + 0.857406i
\(454\) 0 0
\(455\) −73.3492 + 65.5914i −0.161207 + 0.144157i
\(456\) 0 0
\(457\) 84.3172 + 146.042i 0.184501 + 0.319566i 0.943408 0.331633i \(-0.107600\pi\)
−0.758907 + 0.651199i \(0.774266\pi\)
\(458\) 0 0
\(459\) 297.735 + 171.897i 0.648659 + 0.374504i
\(460\) 0 0
\(461\) −265.062 −0.574971 −0.287485 0.957785i \(-0.592819\pi\)
−0.287485 + 0.957785i \(0.592819\pi\)
\(462\) 0 0
\(463\) −97.4735 −0.210526 −0.105263 0.994444i \(-0.533568\pi\)
−0.105263 + 0.994444i \(0.533568\pi\)
\(464\) 0 0
\(465\) 0.213261 + 0.123126i 0.000458626 + 0.000264788i
\(466\) 0 0
\(467\) −37.0997 64.2586i −0.0794427 0.137599i 0.823567 0.567219i \(-0.191981\pi\)
−0.903010 + 0.429620i \(0.858647\pi\)
\(468\) 0 0
\(469\) −30.4921 + 92.7648i −0.0650150 + 0.197793i
\(470\) 0 0
\(471\) −417.016 722.293i −0.885385 1.53353i
\(472\) 0 0
\(473\) 116.600 201.958i 0.246512 0.426972i
\(474\) 0 0
\(475\) 9.90608 0.0208549
\(476\) 0 0
\(477\) 118.509i 0.248447i
\(478\) 0 0
\(479\) 475.220 + 274.368i 0.992108 + 0.572794i 0.905904 0.423484i \(-0.139193\pi\)
0.0862043 + 0.996277i \(0.472526\pi\)
\(480\) 0 0
\(481\) −3.80224 + 2.19522i −0.00790486 + 0.00456387i
\(482\) 0 0
\(483\) 201.795 + 965.588i 0.417795 + 1.99915i
\(484\) 0 0
\(485\) −526.678 + 304.078i −1.08593 + 0.626964i
\(486\) 0 0
\(487\) 283.938 491.795i 0.583034 1.00985i −0.412083 0.911146i \(-0.635199\pi\)
0.995117 0.0986990i \(-0.0314681\pi\)
\(488\) 0 0
\(489\) 817.617i 1.67202i
\(490\) 0 0
\(491\) 78.8005i 0.160490i −0.996775 0.0802449i \(-0.974430\pi\)
0.996775 0.0802449i \(-0.0255702\pi\)
\(492\) 0 0
\(493\) 29.4527 51.0135i 0.0597417 0.103476i
\(494\) 0 0
\(495\) 172.447 99.5626i 0.348379 0.201136i
\(496\) 0 0
\(497\) 108.414 + 518.759i 0.218136 + 1.04378i
\(498\) 0 0
\(499\) 290.932 167.970i 0.583030 0.336612i −0.179307 0.983793i \(-0.557385\pi\)
0.762337 + 0.647181i \(0.224052\pi\)
\(500\) 0 0
\(501\) 215.701 + 124.535i 0.430541 + 0.248573i
\(502\) 0 0
\(503\) 274.052i 0.544836i −0.962179 0.272418i \(-0.912177\pi\)
0.962179 0.272418i \(-0.0878233\pi\)
\(504\) 0 0
\(505\) 152.817 0.302609
\(506\) 0 0
\(507\) 269.495 466.779i 0.531548 0.920669i
\(508\) 0 0
\(509\) 168.009 + 291.000i 0.330076 + 0.571709i 0.982526 0.186123i \(-0.0595923\pi\)
−0.652450 + 0.757831i \(0.726259\pi\)
\(510\) 0 0
\(511\) 116.176 353.437i 0.227350 0.691658i
\(512\) 0 0
\(513\) −17.0106 29.4633i −0.0331591 0.0574333i
\(514\) 0 0
\(515\) 376.306 + 217.260i 0.730691 + 0.421865i
\(516\) 0 0
\(517\) 639.829 1.23758
\(518\) 0 0
\(519\) −126.069 −0.242908
\(520\) 0 0
\(521\) −547.572 316.141i −1.05100 0.606796i −0.128072 0.991765i \(-0.540879\pi\)
−0.922930 + 0.384969i \(0.874212\pi\)
\(522\) 0 0
\(523\) −389.623 674.847i −0.744977 1.29034i −0.950206 0.311624i \(-0.899127\pi\)
0.205229 0.978714i \(-0.434206\pi\)
\(524\) 0 0
\(525\) −112.963 + 101.015i −0.215167 + 0.192410i
\(526\) 0 0
\(527\) −0.131877 0.228418i −0.000250242 0.000433431i
\(528\) 0 0
\(529\) −593.054 + 1027.20i −1.12109 + 1.94178i
\(530\) 0 0
\(531\) 177.165 0.333644
\(532\) 0 0
\(533\) 228.907i 0.429470i
\(534\) 0 0
\(535\) −399.862 230.861i −0.747406 0.431515i
\(536\) 0 0
\(537\) 700.908 404.669i 1.30523 0.753574i
\(538\) 0 0
\(539\) 870.974 + 97.5673i 1.61591 + 0.181015i
\(540\) 0 0
\(541\) 583.617 336.952i 1.07878 0.622831i 0.148209 0.988956i \(-0.452649\pi\)
0.930566 + 0.366125i \(0.119316\pi\)
\(542\) 0 0
\(543\) −497.743 + 862.117i −0.916655 + 1.58769i
\(544\) 0 0
\(545\) 227.217i 0.416913i
\(546\) 0 0
\(547\) 52.5329i 0.0960382i 0.998846 + 0.0480191i \(0.0152908\pi\)
−0.998846 + 0.0480191i \(0.984709\pi\)
\(548\) 0 0
\(549\) 123.880 214.566i 0.225646 0.390831i
\(550\) 0 0
\(551\) −5.04821 + 2.91458i −0.00916190 + 0.00528963i
\(552\) 0 0
\(553\) 121.834 108.948i 0.220314 0.197012i
\(554\) 0 0
\(555\) 17.1544 9.90409i 0.0309088 0.0178452i
\(556\) 0 0
\(557\) 678.123 + 391.515i 1.21746 + 0.702899i 0.964373 0.264546i \(-0.0852220\pi\)
0.253083 + 0.967445i \(0.418555\pi\)
\(558\) 0 0
\(559\) 42.4528i 0.0759441i
\(560\) 0 0
\(561\) −957.628 −1.70700
\(562\) 0 0
\(563\) −446.202 + 772.844i −0.792543 + 1.37272i 0.131845 + 0.991270i \(0.457910\pi\)
−0.924388 + 0.381454i \(0.875423\pi\)
\(564\) 0 0
\(565\) −98.0743 169.870i −0.173583 0.300654i
\(566\) 0 0
\(567\) 648.763 + 213.250i 1.14420 + 0.376102i
\(568\) 0 0
\(569\) 148.722 + 257.593i 0.261373 + 0.452712i 0.966607 0.256263i \(-0.0824912\pi\)
−0.705234 + 0.708975i \(0.749158\pi\)
\(570\) 0 0
\(571\) 218.885 + 126.373i 0.383335 + 0.221319i 0.679268 0.733890i \(-0.262297\pi\)
−0.295933 + 0.955209i \(0.595631\pi\)
\(572\) 0 0
\(573\) −480.561 −0.838676
\(574\) 0 0
\(575\) −263.479 −0.458225
\(576\) 0 0
\(577\) 764.454 + 441.358i 1.32488 + 0.764918i 0.984502 0.175371i \(-0.0561126\pi\)
0.340375 + 0.940290i \(0.389446\pi\)
\(578\) 0 0
\(579\) 112.223 + 194.375i 0.193821 + 0.335709i
\(580\) 0 0
\(581\) 146.759 + 702.239i 0.252597 + 1.20867i
\(582\) 0 0
\(583\) 410.987 + 711.850i 0.704951 + 1.22101i
\(584\) 0 0
\(585\) 18.1248 31.3930i 0.0309825 0.0536633i
\(586\) 0 0
\(587\) 66.7814 0.113767 0.0568836 0.998381i \(-0.481884\pi\)
0.0568836 + 0.998381i \(0.481884\pi\)
\(588\) 0 0
\(589\) 0.0261007i 4.43136e-5i
\(590\) 0 0
\(591\) 587.670 + 339.291i 0.994365 + 0.574097i
\(592\) 0 0
\(593\) −311.911 + 180.082i −0.525989 + 0.303680i −0.739381 0.673287i \(-0.764882\pi\)
0.213393 + 0.976967i \(0.431549\pi\)
\(594\) 0 0
\(595\) −465.439 + 97.2704i −0.782250 + 0.163480i
\(596\) 0 0
\(597\) 199.480 115.170i 0.334137 0.192914i
\(598\) 0 0
\(599\) −99.0219 + 171.511i −0.165312 + 0.286329i −0.936766 0.349956i \(-0.886196\pi\)
0.771454 + 0.636285i \(0.219530\pi\)
\(600\) 0 0
\(601\) 373.907i 0.622141i −0.950387 0.311071i \(-0.899312\pi\)
0.950387 0.311071i \(-0.100688\pi\)
\(602\) 0 0
\(603\) 35.9728i 0.0596565i
\(604\) 0 0
\(605\) 429.373 743.696i 0.709708 1.22925i
\(606\) 0 0
\(607\) −200.164 + 115.565i −0.329760 + 0.190387i −0.655735 0.754992i \(-0.727641\pi\)
0.325975 + 0.945379i \(0.394308\pi\)
\(608\) 0 0
\(609\) 27.8458 84.7142i 0.0457238 0.139104i
\(610\) 0 0
\(611\) 100.872 58.2386i 0.165094 0.0953168i
\(612\) 0 0
\(613\) −444.718 256.758i −0.725479 0.418855i 0.0912873 0.995825i \(-0.470902\pi\)
−0.816766 + 0.576969i \(0.804235\pi\)
\(614\) 0 0
\(615\) 1032.75i 1.67927i
\(616\) 0 0
\(617\) −1119.01 −1.81363 −0.906815 0.421529i \(-0.861493\pi\)
−0.906815 + 0.421529i \(0.861493\pi\)
\(618\) 0 0
\(619\) −64.1019 + 111.028i −0.103557 + 0.179366i −0.913148 0.407629i \(-0.866356\pi\)
0.809591 + 0.586995i \(0.199689\pi\)
\(620\) 0 0
\(621\) 452.445 + 783.658i 0.728575 + 1.26193i
\(622\) 0 0
\(623\) −413.248 462.125i −0.663319 0.741774i
\(624\) 0 0
\(625\) 212.735 + 368.469i 0.340377 + 0.589550i
\(626\) 0 0
\(627\) 82.0690 + 47.3826i 0.130892 + 0.0755703i
\(628\) 0 0
\(629\) −21.2160 −0.0337297
\(630\) 0 0
\(631\) −313.995 −0.497615 −0.248808 0.968553i \(-0.580039\pi\)
−0.248808 + 0.968553i \(0.580039\pi\)
\(632\) 0 0
\(633\) −183.138 105.735i −0.289317 0.167037i
\(634\) 0 0
\(635\) 271.322 + 469.944i 0.427279 + 0.740070i
\(636\) 0 0
\(637\) 146.194 63.8959i 0.229504 0.100307i
\(638\) 0 0
\(639\) −97.6183 169.080i −0.152767 0.264601i
\(640\) 0 0
\(641\) 115.594 200.215i 0.180334 0.312348i −0.761660 0.647977i \(-0.775615\pi\)
0.941994 + 0.335629i \(0.108949\pi\)
\(642\) 0 0
\(643\) −637.869 −0.992020 −0.496010 0.868317i \(-0.665202\pi\)
−0.496010 + 0.868317i \(0.665202\pi\)
\(644\) 0 0
\(645\) 191.532i 0.296949i
\(646\) 0 0
\(647\) 586.461 + 338.594i 0.906432 + 0.523329i 0.879281 0.476303i \(-0.158023\pi\)
0.0271505 + 0.999631i \(0.491357\pi\)
\(648\) 0 0
\(649\) 1064.18 614.405i 1.63972 0.946695i
\(650\) 0 0
\(651\) −0.266157 0.297636i −0.000408843 0.000457199i
\(652\) 0 0
\(653\) −916.022 + 528.865i −1.40279 + 0.809901i −0.994678 0.103031i \(-0.967146\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(654\) 0 0
\(655\) −244.569 + 423.606i −0.373388 + 0.646726i
\(656\) 0 0
\(657\) 137.058i 0.208612i
\(658\) 0 0
\(659\) 644.502i 0.978000i −0.872284 0.489000i \(-0.837362\pi\)
0.872284 0.489000i \(-0.162638\pi\)
\(660\) 0 0
\(661\) 560.069 970.068i 0.847306 1.46758i −0.0362979 0.999341i \(-0.511557\pi\)
0.883604 0.468236i \(-0.155110\pi\)
\(662\) 0 0
\(663\) −150.975 + 87.1652i −0.227714 + 0.131471i
\(664\) 0 0
\(665\) 44.7011 + 14.6934i 0.0672197 + 0.0220953i
\(666\) 0 0
\(667\) 134.271 77.5214i 0.201306 0.116224i
\(668\) 0 0
\(669\) 340.438 + 196.552i 0.508876 + 0.293800i
\(670\) 0 0
\(671\) 1718.45i 2.56103i
\(672\) 0 0
\(673\) −307.811 −0.457371 −0.228686 0.973500i \(-0.573443\pi\)
−0.228686 + 0.973500i \(0.573443\pi\)
\(674\) 0 0
\(675\) −69.5058 + 120.388i −0.102972 + 0.178352i
\(676\) 0 0
\(677\) −507.773 879.488i −0.750033 1.29910i −0.947806 0.318848i \(-0.896704\pi\)
0.197773 0.980248i \(-0.436629\pi\)
\(678\) 0 0
\(679\) 965.231 201.720i 1.42155 0.297084i
\(680\) 0 0
\(681\) 96.1388 + 166.517i 0.141173 + 0.244519i
\(682\) 0 0
\(683\) 840.220 + 485.102i 1.23019 + 0.710251i 0.967070 0.254512i \(-0.0819147\pi\)
0.263121 + 0.964763i \(0.415248\pi\)
\(684\) 0 0
\(685\) 338.188 0.493706
\(686\) 0 0
\(687\) 402.680 0.586143
\(688\) 0 0
\(689\) 129.588 + 74.8177i 0.188081 + 0.108589i
\(690\) 0 0
\(691\) −274.581 475.588i −0.397367 0.688260i 0.596033 0.802960i \(-0.296743\pi\)
−0.993400 + 0.114700i \(0.963409\pi\)
\(692\) 0 0
\(693\) −316.041 + 66.0483i −0.456047 + 0.0953078i
\(694\) 0 0
\(695\) −321.711 557.221i −0.462894 0.801756i
\(696\) 0 0
\(697\) 553.076 957.956i 0.793510 1.37440i
\(698\) 0 0
\(699\) −83.9818 −0.120146
\(700\) 0 0
\(701\) 452.665i 0.645742i −0.946443 0.322871i \(-0.895352\pi\)
0.946443 0.322871i \(-0.104648\pi\)
\(702\) 0 0
\(703\) 1.81822 + 1.04975i 0.00258637 + 0.00149324i
\(704\) 0 0
\(705\) −455.100 + 262.752i −0.645533 + 0.372698i
\(706\) 0 0
\(707\) −235.393 77.3744i −0.332946 0.109440i
\(708\) 0 0
\(709\) −609.174 + 351.707i −0.859202 + 0.496060i −0.863745 0.503929i \(-0.831887\pi\)
0.00454321 + 0.999990i \(0.498554\pi\)
\(710\) 0 0
\(711\) −30.1054 + 52.1442i −0.0423424 + 0.0733392i
\(712\) 0 0
\(713\) 0.694220i 0.000973661i
\(714\) 0 0
\(715\) 251.425i 0.351644i
\(716\) 0 0
\(717\) −427.367 + 740.221i −0.596049 + 1.03239i
\(718\) 0 0
\(719\) −54.1160 + 31.2439i −0.0752656 + 0.0434546i −0.537161 0.843480i \(-0.680503\pi\)
0.461895 + 0.886935i \(0.347170\pi\)
\(720\) 0 0
\(721\) −469.642 525.189i −0.651376 0.728417i
\(722\) 0 0
\(723\) −331.354 + 191.307i −0.458305 + 0.264602i
\(724\) 0 0
\(725\) 20.6271 + 11.9090i 0.0284511 + 0.0164263i
\(726\) 0 0
\(727\) 889.995i 1.22420i −0.790779 0.612101i \(-0.790324\pi\)
0.790779 0.612101i \(-0.209676\pi\)
\(728\) 0 0
\(729\) 417.569 0.572797
\(730\) 0 0
\(731\) 102.573 177.661i 0.140318 0.243038i
\(732\) 0 0
\(733\) −456.127 790.035i −0.622274 1.07781i −0.989061 0.147505i \(-0.952876\pi\)
0.366787 0.930305i \(-0.380458\pi\)
\(734\) 0 0
\(735\) −659.577 + 288.276i −0.897383 + 0.392212i
\(736\) 0 0
\(737\) −124.753 216.079i −0.169271 0.293187i
\(738\) 0 0
\(739\) −1081.52 624.415i −1.46349 0.844946i −0.464319 0.885668i \(-0.653701\pi\)
−0.999171 + 0.0407224i \(0.987034\pi\)
\(740\) 0 0
\(741\) 17.2514 0.0232813
\(742\) 0 0
\(743\) 305.880 0.411682 0.205841 0.978585i \(-0.434007\pi\)
0.205841 + 0.978585i \(0.434007\pi\)
\(744\) 0 0
\(745\) 318.766 + 184.040i 0.427874 + 0.247033i
\(746\) 0 0
\(747\) −132.145 228.882i −0.176901 0.306401i
\(748\) 0 0
\(749\) 499.041 + 558.065i 0.666276 + 0.745080i
\(750\) 0 0
\(751\) 258.895 + 448.420i 0.344734 + 0.597097i 0.985305 0.170802i \(-0.0546358\pi\)
−0.640571 + 0.767899i \(0.721302\pi\)
\(752\) 0 0
\(753\) −206.288 + 357.302i −0.273955 + 0.474504i
\(754\) 0 0
\(755\) −569.012 −0.753659
\(756\) 0 0
\(757\) 939.898i 1.24161i 0.783965 + 0.620804i \(0.213194\pi\)
−0.783965 + 0.620804i \(0.786806\pi\)
\(758\) 0 0
\(759\) −2182.85 1260.27i −2.87596 1.66044i
\(760\) 0 0
\(761\) −976.757 + 563.931i −1.28352 + 0.741039i −0.977490 0.210983i \(-0.932334\pi\)
−0.306028 + 0.952022i \(0.599000\pi\)
\(762\) 0 0
\(763\) 115.044 349.995i 0.150779 0.458710i
\(764\) 0 0
\(765\) 151.701 87.5846i 0.198302 0.114490i
\(766\) 0 0
\(767\) 111.849 193.728i 0.145826 0.252579i
\(768\) 0 0
\(769\) 300.115i 0.390267i 0.980777 + 0.195133i \(0.0625139\pi\)
−0.980777 + 0.195133i \(0.937486\pi\)
\(770\) 0 0
\(771\) 356.524i 0.462417i
\(772\) 0 0
\(773\) −375.120 + 649.727i −0.485278 + 0.840527i −0.999857 0.0169165i \(-0.994615\pi\)
0.514579 + 0.857443i \(0.327948\pi\)
\(774\) 0 0
\(775\) 0.0923598 0.0533240i 0.000119174 6.88051e-5i
\(776\) 0 0
\(777\) −31.4385 + 6.57022i −0.0404613 + 0.00845588i
\(778\) 0 0
\(779\) −94.7977 + 54.7315i −0.121691 + 0.0702586i
\(780\) 0 0
\(781\) −1172.73 677.076i −1.50158 0.866935i
\(782\) 0 0
\(783\) 81.8005i 0.104471i
\(784\) 0 0
\(785\) 1058.16 1.34797
\(786\) 0 0
\(787\) 144.776 250.760i 0.183960 0.318627i −0.759266 0.650781i \(-0.774442\pi\)
0.943225 + 0.332153i \(0.107775\pi\)
\(788\) 0 0
\(789\) −178.098 308.474i −0.225726 0.390969i
\(790\) 0 0
\(791\) 65.0610 + 311.316i 0.0822516 + 0.393573i
\(792\) 0 0
\(793\) −156.417 270.922i −0.197247 0.341642i
\(794\) 0 0
\(795\) −584.657 337.552i −0.735417 0.424593i
\(796\) 0 0
\(797\) −1086.57 −1.36332 −0.681659 0.731670i \(-0.738741\pi\)
−0.681659 + 0.731670i \(0.738741\pi\)
\(798\) 0 0
\(799\) 562.854 0.704448
\(800\) 0 0
\(801\) 197.787 + 114.192i 0.246925 + 0.142562i
\(802\) 0 0
\(803\) 475.313 + 823.267i 0.591922 + 1.02524i
\(804\) 0 0
\(805\) −1188.95 390.811i −1.47696 0.485480i
\(806\) 0 0
\(807\) 518.806 + 898.598i 0.642882 + 1.11350i
\(808\) 0 0
\(809\) −90.9745 + 157.572i −0.112453 + 0.194774i −0.916759 0.399441i \(-0.869204\pi\)
0.804306 + 0.594216i \(0.202537\pi\)
\(810\) 0 0
\(811\) 1005.31 1.23960 0.619799 0.784760i \(-0.287214\pi\)
0.619799 + 0.784760i \(0.287214\pi\)
\(812\) 0 0
\(813\) 349.280i 0.429619i
\(814\) 0 0
\(815\) 898.355 + 518.665i 1.10228 + 0.636399i
\(816\) 0 0
\(817\) −17.5810 + 10.1504i −0.0215190 + 0.0124240i
\(818\) 0 0
\(819\) −43.8135 + 39.1795i −0.0534963 + 0.0478382i
\(820\) 0 0
\(821\) 851.009 491.330i 1.03655 0.598453i 0.117697 0.993050i \(-0.462449\pi\)
0.918855 + 0.394596i \(0.129115\pi\)
\(822\) 0 0
\(823\) 742.505 1286.06i 0.902194 1.56265i 0.0775532 0.996988i \(-0.475289\pi\)
0.824641 0.565657i \(-0.191377\pi\)
\(824\) 0 0
\(825\) 387.212i 0.469348i
\(826\) 0 0
\(827\) 708.113i 0.856243i 0.903721 + 0.428121i \(0.140824\pi\)
−0.903721 + 0.428121i \(0.859176\pi\)
\(828\) 0 0
\(829\) −75.0164 + 129.932i −0.0904902 + 0.156734i −0.907718 0.419582i \(-0.862177\pi\)
0.817227 + 0.576316i \(0.195510\pi\)
\(830\) 0 0
\(831\) 49.0796 28.3361i 0.0590609 0.0340988i
\(832\) 0 0
\(833\) 766.191 + 85.8294i 0.919797 + 0.103037i
\(834\) 0 0
\(835\) −273.665 + 158.001i −0.327743 + 0.189222i
\(836\) 0 0
\(837\) −0.317199 0.183135i −0.000378972 0.000218799i
\(838\) 0 0
\(839\) 1106.41i 1.31873i 0.751824 + 0.659364i \(0.229174\pi\)
−0.751824 + 0.659364i \(0.770826\pi\)
\(840\) 0 0
\(841\) 826.984 0.983335
\(842\) 0 0
\(843\) −129.014 + 223.459i −0.153042 + 0.265076i
\(844\) 0 0
\(845\) 341.915 + 592.214i 0.404633 + 0.700845i
\(846\) 0 0
\(847\) −1037.93 + 928.156i −1.22542 + 1.09582i
\(848\) 0 0
\(849\) 148.584 + 257.354i 0.175010 + 0.303126i
\(850\) 0 0
\(851\) −48.3606 27.9210i −0.0568279 0.0328096i
\(852\) 0 0
\(853\) 1243.82 1.45817 0.729086 0.684423i \(-0.239946\pi\)
0.729086 + 0.684423i \(0.239946\pi\)
\(854\) 0 0
\(855\) −17.3344 −0.0202742
\(856\) 0 0
\(857\) 245.650 + 141.826i 0.286639 + 0.165491i 0.636425 0.771339i \(-0.280412\pi\)
−0.349786 + 0.936830i \(0.613746\pi\)
\(858\) 0 0
\(859\) −455.900 789.641i −0.530733 0.919256i −0.999357 0.0358586i \(-0.988583\pi\)
0.468624 0.883398i \(-0.344750\pi\)
\(860\) 0 0
\(861\) 522.902 1590.80i 0.607319 1.84762i
\(862\) 0 0
\(863\) −436.908 756.747i −0.506266 0.876879i −0.999974 0.00725099i \(-0.997692\pi\)
0.493707 0.869628i \(-0.335641\pi\)
\(864\) 0 0
\(865\) 79.9735 138.518i 0.0924550 0.160137i
\(866\) 0 0
\(867\) 140.978 0.162604
\(868\) 0 0
\(869\) 417.620i 0.480575i
\(870\) 0 0
\(871\) −39.3358 22.7105i −0.0451617 0.0260741i
\(872\) 0 0
\(873\) −314.599 + 181.634i −0.360365 + 0.208057i
\(874\) 0 0
\(875\) −193.882 927.724i −0.221579 1.06026i
\(876\) 0 0
\(877\) 549.476 317.240i 0.626540 0.361733i −0.152871 0.988246i \(-0.548852\pi\)
0.779411 + 0.626513i \(0.215518\pi\)
\(878\) 0 0
\(879\) 46.7975 81.0557i 0.0532395 0.0922136i
\(880\) 0 0
\(881\) 670.044i 0.760549i 0.924874 + 0.380274i \(0.124170\pi\)
−0.924874 + 0.380274i \(0.875830\pi\)
\(882\) 0 0
\(883\) 875.514i 0.991522i 0.868459 + 0.495761i \(0.165111\pi\)
−0.868459 + 0.495761i \(0.834889\pi\)
\(884\) 0 0
\(885\) −504.623 + 874.033i −0.570196 + 0.987608i
\(886\) 0 0
\(887\) −854.152 + 493.145i −0.962967 + 0.555969i −0.897085 0.441858i \(-0.854319\pi\)
−0.0658820 + 0.997827i \(0.520986\pi\)
\(888\) 0 0
\(889\) −179.991 861.257i −0.202465 0.968793i
\(890\) 0 0
\(891\) −1511.17 + 872.476i −1.69604 + 0.979210i
\(892\) 0 0
\(893\) −48.2368 27.8495i −0.0540166 0.0311865i
\(894\) 0 0
\(895\) 1026.83i 1.14729i
\(896\) 0 0
\(897\) −458.850 −0.511538
\(898\) 0 0
\(899\) −0.0313782 + 0.0543486i −3.49034e−5 + 6.04545e-5i
\(900\) 0 0
\(901\) 361.543 + 626.210i 0.401268 + 0.695017i
\(902\) 0 0
\(903\) 96.9764 295.028i 0.107394 0.326719i
\(904\) 0 0
\(905\) −631.499 1093.79i −0.697789 1.20861i
\(906\) 0 0
\(907\) 750.592 + 433.355i 0.827555 + 0.477789i 0.853015 0.521887i \(-0.174772\pi\)
−0.0254599 + 0.999676i \(0.508105\pi\)
\(908\) 0 0
\(909\) 91.2820 0.100420
\(910\) 0 0
\(911\) 128.713 0.141288 0.0706438 0.997502i \(-0.477495\pi\)
0.0706438 + 0.997502i \(0.477495\pi\)
\(912\) 0 0
\(913\) −1587.51 916.552i −1.73879 1.00389i
\(914\) 0 0
\(915\) 705.699 + 1222.31i 0.771256 + 1.33585i
\(916\) 0 0
\(917\) 591.203 528.673i 0.644714 0.576525i
\(918\) 0 0
\(919\) 430.087 + 744.933i 0.467995 + 0.810591i 0.999331 0.0365701i \(-0.0116432\pi\)
−0.531336 + 0.847161i \(0.678310\pi\)
\(920\) 0 0
\(921\) −421.935 + 730.813i −0.458127 + 0.793500i
\(922\) 0 0
\(923\) −246.515 −0.267081
\(924\) 0 0
\(925\) 8.57859i 0.00927415i
\(926\) 0 0
\(927\) 224.778 + 129.776i 0.242479 + 0.139995i
\(928\) 0 0
\(929\) 202.025 116.639i 0.217465 0.125554i −0.387311 0.921949i \(-0.626596\pi\)
0.604776 + 0.796396i \(0.293263\pi\)
\(930\) 0 0
\(931\) −61.4160 45.2661i −0.0659678 0.0486209i
\(932\) 0 0
\(933\) 1287.89 743.563i 1.38037 0.796960i
\(934\) 0 0
\(935\) 607.483 1052.19i 0.649714 1.12534i
\(936\) 0 0
\(937\) 1426.29i 1.52219i 0.648641 + 0.761095i \(0.275338\pi\)
−0.648641 + 0.761095i \(0.724662\pi\)
\(938\) 0 0
\(939\) 281.851i 0.300161i
\(940\) 0 0
\(941\) −635.425 + 1100.59i −0.675265 + 1.16959i 0.301126 + 0.953584i \(0.402638\pi\)
−0.976391 + 0.216010i \(0.930696\pi\)
\(942\) 0 0
\(943\) 2521.41 1455.73i 2.67381 1.54373i
\(944\) 0 0
\(945\) −492.212 + 440.152i −0.520859 + 0.465770i
\(946\) 0 0
\(947\) 1413.50 816.086i 1.49261 0.861759i 0.492646 0.870230i \(-0.336030\pi\)
0.999964 + 0.00847064i \(0.00269632\pi\)
\(948\) 0 0
\(949\) 149.871 + 86.5280i 0.157925 + 0.0911781i
\(950\) 0 0
\(951\) 832.098i 0.874972i
\(952\) 0 0
\(953\) 95.9158 0.100646 0.0503231 0.998733i \(-0.483975\pi\)
0.0503231 + 0.998733i \(0.483975\pi\)
\(954\) 0 0
\(955\) 304.850 528.015i 0.319215 0.552896i
\(956\) 0 0
\(957\) 113.926 + 197.326i 0.119045 + 0.206192i
\(958\) 0 0
\(959\) −520.930 171.231i −0.543201 0.178552i
\(960\) 0 0
\(961\) −480.500 832.250i −0.500000 0.866025i
\(962\) 0 0
\(963\) −238.849 137.899i −0.248025 0.143198i
\(964\) 0 0
\(965\) −284.759 −0.295087
\(966\) 0 0
\(967\) −1419.97 −1.46843 −0.734216 0.678916i \(-0.762450\pi\)
−0.734216 + 0.678916i \(0.762450\pi\)
\(968\) 0 0
\(969\) 72.1956 + 41.6822i 0.0745053 + 0.0430157i
\(970\) 0 0
\(971\) 329.817 + 571.261i 0.339668 + 0.588322i 0.984370 0.176112i \(-0.0563520\pi\)
−0.644702 + 0.764434i \(0.723019\pi\)
\(972\) 0 0
\(973\) 213.418 + 1021.21i 0.219341 + 1.04954i
\(974\) 0 0
\(975\) −35.2448 61.0459i −0.0361486 0.0626111i
\(976\) 0 0
\(977\) 957.151 1657.83i 0.979683 1.69686i 0.316160 0.948706i \(-0.397606\pi\)
0.663523 0.748156i \(-0.269060\pi\)
\(978\) 0 0
\(979\) 1584.07 1.61804
\(980\) 0 0
\(981\) 135.723i 0.138352i
\(982\) 0 0
\(983\) −193.655 111.806i −0.197004 0.113740i 0.398253 0.917275i \(-0.369616\pi\)
−0.595257 + 0.803535i \(0.702950\pi\)
\(984\) 0 0
\(985\) −745.591 + 430.467i −0.756945 + 0.437022i
\(986\) 0 0
\(987\) 834.053 174.306i 0.845038 0.176602i
\(988\) 0 0
\(989\) 467.616 269.978i 0.472816 0.272981i
\(990\) 0 0
\(991\) −736.371 + 1275.43i −0.743058 + 1.28701i 0.208038 + 0.978121i \(0.433292\pi\)
−0.951096 + 0.308894i \(0.900041\pi\)
\(992\) 0 0
\(993\) 260.472i 0.262308i
\(994\) 0 0
\(995\) 292.237i 0.293706i
\(996\) 0 0
\(997\) −53.4480 + 92.5746i −0.0536088 + 0.0928532i −0.891584 0.452854i \(-0.850406\pi\)
0.837976 + 0.545708i \(0.183739\pi\)
\(998\) 0 0
\(999\) −25.5150 + 14.7311i −0.0255405 + 0.0147458i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 224.3.n.a.17.4 28
4.3 odd 2 56.3.j.a.45.5 yes 28
7.3 odd 6 1568.3.h.a.881.8 28
7.4 even 3 1568.3.h.a.881.22 28
7.5 odd 6 inner 224.3.n.a.145.11 28
8.3 odd 2 56.3.j.a.45.14 yes 28
8.5 even 2 inner 224.3.n.a.17.11 28
28.3 even 6 392.3.h.a.293.10 28
28.11 odd 6 392.3.h.a.293.9 28
28.19 even 6 56.3.j.a.5.14 yes 28
28.23 odd 6 392.3.j.e.117.14 28
28.27 even 2 392.3.j.e.325.5 28
56.3 even 6 392.3.h.a.293.11 28
56.5 odd 6 inner 224.3.n.a.145.4 28
56.11 odd 6 392.3.h.a.293.12 28
56.19 even 6 56.3.j.a.5.5 28
56.27 even 2 392.3.j.e.325.14 28
56.45 odd 6 1568.3.h.a.881.21 28
56.51 odd 6 392.3.j.e.117.5 28
56.53 even 6 1568.3.h.a.881.7 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.j.a.5.5 28 56.19 even 6
56.3.j.a.5.14 yes 28 28.19 even 6
56.3.j.a.45.5 yes 28 4.3 odd 2
56.3.j.a.45.14 yes 28 8.3 odd 2
224.3.n.a.17.4 28 1.1 even 1 trivial
224.3.n.a.17.11 28 8.5 even 2 inner
224.3.n.a.145.4 28 56.5 odd 6 inner
224.3.n.a.145.11 28 7.5 odd 6 inner
392.3.h.a.293.9 28 28.11 odd 6
392.3.h.a.293.10 28 28.3 even 6
392.3.h.a.293.11 28 56.3 even 6
392.3.h.a.293.12 28 56.11 odd 6
392.3.j.e.117.5 28 56.51 odd 6
392.3.j.e.117.14 28 28.23 odd 6
392.3.j.e.325.5 28 28.27 even 2
392.3.j.e.325.14 28 56.27 even 2
1568.3.h.a.881.7 28 56.53 even 6
1568.3.h.a.881.8 28 7.3 odd 6
1568.3.h.a.881.21 28 56.45 odd 6
1568.3.h.a.881.22 28 7.4 even 3