Properties

Label 1500.2.o.a.649.2
Level $1500$
Weight $2$
Character 1500.649
Analytic conductor $11.978$
Analytic rank $0$
Dimension $16$
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] = [1500,2,Mod(49,1500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1500.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.o (of order \(10\), 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: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 649.2
Root \(-0.994522 + 0.104528i\) of defining polynomial
Character \(\chi\) \(=\) 1500.649
Dual form 1500.2.o.a.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.587785 + 0.809017i) q^{3} +0.747238i q^{7} +(-0.309017 - 0.951057i) q^{9} +O(q^{10})\) \(q+(-0.587785 + 0.809017i) q^{3} +0.747238i q^{7} +(-0.309017 - 0.951057i) q^{9} +(0.0646021 - 0.198825i) q^{11} +(-2.38108 + 0.773659i) q^{13} +(4.00842 + 5.51712i) q^{17} +(1.00739 - 0.731913i) q^{19} +(-0.604528 - 0.439216i) q^{21} +(-3.09174 - 1.00457i) q^{23} +(0.951057 + 0.309017i) q^{27} +(-4.19332 - 3.04662i) q^{29} +(-3.02547 + 2.19813i) q^{31} +(0.122881 + 0.169131i) q^{33} +(-1.86924 + 0.607352i) q^{37} +(0.773659 - 2.38108i) q^{39} +(0.993096 + 3.05644i) q^{41} +12.7127i q^{43} +(-3.81017 + 5.24425i) q^{47} +6.44163 q^{49} -6.81953 q^{51} +(-2.43520 + 3.35177i) q^{53} +1.24520i q^{57} +(-3.61882 - 11.1376i) q^{59} +(-3.85634 + 11.8686i) q^{61} +(0.710666 - 0.230909i) q^{63} +(-1.71460 - 2.35995i) q^{67} +(2.62999 - 1.91080i) q^{69} +(-5.29912 - 3.85004i) q^{71} +(2.39603 + 0.778516i) q^{73} +(0.148570 + 0.0482732i) q^{77} +(-8.28621 - 6.02028i) q^{79} +(-0.809017 + 0.587785i) q^{81} +(-3.33658 - 4.59240i) q^{83} +(4.92954 - 1.60171i) q^{87} +(0.284829 - 0.876615i) q^{89} +(-0.578108 - 1.77923i) q^{91} -3.73968i q^{93} +(-9.13679 + 12.5757i) q^{97} -0.209057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{9} - 4 q^{11} - 10 q^{19} - 6 q^{21} - 54 q^{29} - 6 q^{31} + 40 q^{41} + 16 q^{49} - 16 q^{51} - 4 q^{59} - 28 q^{61} - 4 q^{69} - 30 q^{71} - 48 q^{79} - 4 q^{81} + 10 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{10}\right)\) \(1\)

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
\(3\) −0.587785 + 0.809017i −0.339358 + 0.467086i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.747238i 0.282430i 0.989979 + 0.141215i \(0.0451008\pi\)
−0.989979 + 0.141215i \(0.954899\pi\)
\(8\) 0 0
\(9\) −0.309017 0.951057i −0.103006 0.317019i
\(10\) 0 0
\(11\) 0.0646021 0.198825i 0.0194783 0.0599480i −0.940845 0.338837i \(-0.889966\pi\)
0.960323 + 0.278889i \(0.0899663\pi\)
\(12\) 0 0
\(13\) −2.38108 + 0.773659i −0.660392 + 0.214574i −0.619991 0.784609i \(-0.712864\pi\)
−0.0404014 + 0.999184i \(0.512864\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00842 + 5.51712i 0.972185 + 1.33810i 0.940935 + 0.338586i \(0.109949\pi\)
0.0312497 + 0.999512i \(0.490051\pi\)
\(18\) 0 0
\(19\) 1.00739 0.731913i 0.231112 0.167912i −0.466203 0.884678i \(-0.654378\pi\)
0.697314 + 0.716766i \(0.254378\pi\)
\(20\) 0 0
\(21\) −0.604528 0.439216i −0.131919 0.0958447i
\(22\) 0 0
\(23\) −3.09174 1.00457i −0.644673 0.209467i −0.0316092 0.999500i \(-0.510063\pi\)
−0.613064 + 0.790033i \(0.710063\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.951057 + 0.309017i 0.183031 + 0.0594703i
\(28\) 0 0
\(29\) −4.19332 3.04662i −0.778680 0.565744i 0.125903 0.992043i \(-0.459817\pi\)
−0.904582 + 0.426299i \(0.859817\pi\)
\(30\) 0 0
\(31\) −3.02547 + 2.19813i −0.543390 + 0.394796i −0.825342 0.564633i \(-0.809018\pi\)
0.281953 + 0.959428i \(0.409018\pi\)
\(32\) 0 0
\(33\) 0.122881 + 0.169131i 0.0213908 + 0.0294419i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.86924 + 0.607352i −0.307301 + 0.0998480i −0.458608 0.888639i \(-0.651652\pi\)
0.151307 + 0.988487i \(0.451652\pi\)
\(38\) 0 0
\(39\) 0.773659 2.38108i 0.123885 0.381278i
\(40\) 0 0
\(41\) 0.993096 + 3.05644i 0.155096 + 0.477335i 0.998171 0.0604609i \(-0.0192570\pi\)
−0.843075 + 0.537796i \(0.819257\pi\)
\(42\) 0 0
\(43\) 12.7127i 1.93866i 0.245755 + 0.969332i \(0.420964\pi\)
−0.245755 + 0.969332i \(0.579036\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.81017 + 5.24425i −0.555770 + 0.764952i −0.990781 0.135473i \(-0.956745\pi\)
0.435011 + 0.900425i \(0.356745\pi\)
\(48\) 0 0
\(49\) 6.44163 0.920234
\(50\) 0 0
\(51\) −6.81953 −0.954926
\(52\) 0 0
\(53\) −2.43520 + 3.35177i −0.334501 + 0.460401i −0.942825 0.333288i \(-0.891842\pi\)
0.608325 + 0.793688i \(0.291842\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.24520i 0.164931i
\(58\) 0 0
\(59\) −3.61882 11.1376i −0.471131 1.44999i −0.851106 0.524995i \(-0.824067\pi\)
0.379975 0.924997i \(-0.375933\pi\)
\(60\) 0 0
\(61\) −3.85634 + 11.8686i −0.493753 + 1.51962i 0.325138 + 0.945667i \(0.394589\pi\)
−0.818891 + 0.573949i \(0.805411\pi\)
\(62\) 0 0
\(63\) 0.710666 0.230909i 0.0895355 0.0290918i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.71460 2.35995i −0.209472 0.288314i 0.691334 0.722535i \(-0.257023\pi\)
−0.900806 + 0.434222i \(0.857023\pi\)
\(68\) 0 0
\(69\) 2.62999 1.91080i 0.316614 0.230034i
\(70\) 0 0
\(71\) −5.29912 3.85004i −0.628890 0.456916i 0.227125 0.973866i \(-0.427067\pi\)
−0.856016 + 0.516950i \(0.827067\pi\)
\(72\) 0 0
\(73\) 2.39603 + 0.778516i 0.280434 + 0.0911184i 0.445857 0.895104i \(-0.352899\pi\)
−0.165423 + 0.986223i \(0.552899\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.148570 + 0.0482732i 0.0169311 + 0.00550124i
\(78\) 0 0
\(79\) −8.28621 6.02028i −0.932271 0.677335i 0.0142765 0.999898i \(-0.495455\pi\)
−0.946548 + 0.322563i \(0.895455\pi\)
\(80\) 0 0
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 0 0
\(83\) −3.33658 4.59240i −0.366237 0.504082i 0.585636 0.810574i \(-0.300845\pi\)
−0.951873 + 0.306492i \(0.900845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.92954 1.60171i 0.528502 0.171721i
\(88\) 0 0
\(89\) 0.284829 0.876615i 0.0301919 0.0929210i −0.934825 0.355109i \(-0.884444\pi\)
0.965017 + 0.262188i \(0.0844439\pi\)
\(90\) 0 0
\(91\) −0.578108 1.77923i −0.0606022 0.186514i
\(92\) 0 0
\(93\) 3.73968i 0.387787i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.13679 + 12.5757i −0.927700 + 1.27687i 0.0330496 + 0.999454i \(0.489478\pi\)
−0.960750 + 0.277416i \(0.910522\pi\)
\(98\) 0 0
\(99\) −0.209057 −0.0210110
\(100\) 0 0
\(101\) −11.0405 −1.09857 −0.549285 0.835635i \(-0.685100\pi\)
−0.549285 + 0.835635i \(0.685100\pi\)
\(102\) 0 0
\(103\) −8.16097 + 11.2326i −0.804124 + 1.10678i 0.188079 + 0.982154i \(0.439774\pi\)
−0.992203 + 0.124628i \(0.960226\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.02510i 0.0991002i 0.998772 + 0.0495501i \(0.0157788\pi\)
−0.998772 + 0.0495501i \(0.984221\pi\)
\(108\) 0 0
\(109\) 2.07199 + 6.37694i 0.198461 + 0.610800i 0.999919 + 0.0127488i \(0.00405817\pi\)
−0.801458 + 0.598051i \(0.795942\pi\)
\(110\) 0 0
\(111\) 0.607352 1.86924i 0.0576473 0.177420i
\(112\) 0 0
\(113\) 7.91428 2.57151i 0.744513 0.241907i 0.0878944 0.996130i \(-0.471986\pi\)
0.656618 + 0.754223i \(0.271986\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.47159 + 2.02547i 0.136048 + 0.187254i
\(118\) 0 0
\(119\) −4.12260 + 2.99525i −0.377918 + 0.274574i
\(120\) 0 0
\(121\) 8.86383 + 6.43995i 0.805803 + 0.585450i
\(122\) 0 0
\(123\) −3.05644 0.993096i −0.275590 0.0895445i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −13.8248 4.49195i −1.22675 0.398597i −0.377217 0.926125i \(-0.623119\pi\)
−0.849537 + 0.527529i \(0.823119\pi\)
\(128\) 0 0
\(129\) −10.2848 7.47232i −0.905523 0.657901i
\(130\) 0 0
\(131\) −6.00611 + 4.36370i −0.524757 + 0.381258i −0.818393 0.574659i \(-0.805135\pi\)
0.293636 + 0.955917i \(0.405135\pi\)
\(132\) 0 0
\(133\) 0.546913 + 0.752762i 0.0474234 + 0.0652727i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.97636 2.26676i 0.596030 0.193662i 0.00456114 0.999990i \(-0.498548\pi\)
0.591469 + 0.806328i \(0.298548\pi\)
\(138\) 0 0
\(139\) −6.10219 + 18.7806i −0.517581 + 1.59295i 0.260954 + 0.965351i \(0.415963\pi\)
−0.778536 + 0.627600i \(0.784037\pi\)
\(140\) 0 0
\(141\) −2.00313 6.16499i −0.168694 0.519185i
\(142\) 0 0
\(143\) 0.523398i 0.0437687i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.78630 + 5.21139i −0.312289 + 0.429828i
\(148\) 0 0
\(149\) 21.7551 1.78224 0.891122 0.453763i \(-0.149919\pi\)
0.891122 + 0.453763i \(0.149919\pi\)
\(150\) 0 0
\(151\) 10.1308 0.824432 0.412216 0.911086i \(-0.364755\pi\)
0.412216 + 0.911086i \(0.364755\pi\)
\(152\) 0 0
\(153\) 4.00842 5.51712i 0.324062 0.446033i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 9.99520i 0.797704i −0.917015 0.398852i \(-0.869409\pi\)
0.917015 0.398852i \(-0.130591\pi\)
\(158\) 0 0
\(159\) −1.28026 3.94024i −0.101531 0.312481i
\(160\) 0 0
\(161\) 0.750652 2.31027i 0.0591597 0.182075i
\(162\) 0 0
\(163\) −2.43763 + 0.792035i −0.190930 + 0.0620369i −0.402921 0.915235i \(-0.632005\pi\)
0.211991 + 0.977272i \(0.432005\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.2560 + 16.8690i 0.948401 + 1.30536i 0.952234 + 0.305369i \(0.0987797\pi\)
−0.00383355 + 0.999993i \(0.501220\pi\)
\(168\) 0 0
\(169\) −5.44624 + 3.95692i −0.418941 + 0.304379i
\(170\) 0 0
\(171\) −1.00739 0.731913i −0.0770372 0.0559708i
\(172\) 0 0
\(173\) −1.95438 0.635016i −0.148589 0.0482794i 0.233778 0.972290i \(-0.424891\pi\)
−0.382367 + 0.924011i \(0.624891\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.1376 + 3.61882i 0.837153 + 0.272007i
\(178\) 0 0
\(179\) −15.8247 11.4973i −1.18279 0.859349i −0.190309 0.981724i \(-0.560949\pi\)
−0.992484 + 0.122375i \(0.960949\pi\)
\(180\) 0 0
\(181\) 5.96251 4.33202i 0.443190 0.321996i −0.343711 0.939075i \(-0.611684\pi\)
0.786901 + 0.617079i \(0.211684\pi\)
\(182\) 0 0
\(183\) −7.33519 10.0960i −0.542233 0.746319i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.35589 0.440557i 0.0991528 0.0322167i
\(188\) 0 0
\(189\) −0.230909 + 0.710666i −0.0167962 + 0.0516933i
\(190\) 0 0
\(191\) −0.693806 2.13532i −0.0502021 0.154506i 0.922813 0.385249i \(-0.125884\pi\)
−0.973015 + 0.230743i \(0.925884\pi\)
\(192\) 0 0
\(193\) 1.10589i 0.0796035i −0.999208 0.0398018i \(-0.987327\pi\)
0.999208 0.0398018i \(-0.0126726\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.1299 16.6953i 0.864217 1.18949i −0.116331 0.993211i \(-0.537113\pi\)
0.980547 0.196282i \(-0.0628868\pi\)
\(198\) 0 0
\(199\) 12.3822 0.877749 0.438874 0.898548i \(-0.355377\pi\)
0.438874 + 0.898548i \(0.355377\pi\)
\(200\) 0 0
\(201\) 2.91706 0.205753
\(202\) 0 0
\(203\) 2.27655 3.13341i 0.159783 0.219922i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.25085i 0.225950i
\(208\) 0 0
\(209\) −0.0804429 0.247578i −0.00556435 0.0171253i
\(210\) 0 0
\(211\) 6.37422 19.6178i 0.438820 1.35055i −0.450302 0.892876i \(-0.648684\pi\)
0.889121 0.457671i \(-0.151316\pi\)
\(212\) 0 0
\(213\) 6.22949 2.02409i 0.426838 0.138688i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.64253 2.26074i −0.111502 0.153469i
\(218\) 0 0
\(219\) −2.03818 + 1.48083i −0.137728 + 0.100065i
\(220\) 0 0
\(221\) −13.8127 10.0355i −0.929145 0.675063i
\(222\) 0 0
\(223\) 3.95129 + 1.28385i 0.264598 + 0.0859732i 0.438312 0.898823i \(-0.355577\pi\)
−0.173713 + 0.984796i \(0.555577\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.4725 7.30175i −1.49155 0.484634i −0.554010 0.832510i \(-0.686903\pi\)
−0.937541 + 0.347876i \(0.886903\pi\)
\(228\) 0 0
\(229\) 14.7565 + 10.7212i 0.975139 + 0.708480i 0.956617 0.291348i \(-0.0941039\pi\)
0.0185221 + 0.999828i \(0.494104\pi\)
\(230\) 0 0
\(231\) −0.126381 + 0.0918211i −0.00831525 + 0.00604138i
\(232\) 0 0
\(233\) 1.33276 + 1.83438i 0.0873117 + 0.120174i 0.850438 0.526075i \(-0.176337\pi\)
−0.763127 + 0.646249i \(0.776337\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.74102 3.16505i 0.632747 0.205592i
\(238\) 0 0
\(239\) −5.28631 + 16.2696i −0.341943 + 1.05239i 0.621257 + 0.783607i \(0.286622\pi\)
−0.963200 + 0.268786i \(0.913378\pi\)
\(240\) 0 0
\(241\) 8.71435 + 26.8200i 0.561341 + 1.72763i 0.678581 + 0.734526i \(0.262595\pi\)
−0.117240 + 0.993104i \(0.537405\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.83243 + 2.52212i −0.116595 + 0.160479i
\(248\) 0 0
\(249\) 5.67652 0.359735
\(250\) 0 0
\(251\) 23.9575 1.51219 0.756093 0.654464i \(-0.227106\pi\)
0.756093 + 0.654464i \(0.227106\pi\)
\(252\) 0 0
\(253\) −0.399467 + 0.549819i −0.0251142 + 0.0345668i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.5421i 1.78041i −0.455561 0.890204i \(-0.650561\pi\)
0.455561 0.890204i \(-0.349439\pi\)
\(258\) 0 0
\(259\) −0.453837 1.39677i −0.0282000 0.0867908i
\(260\) 0 0
\(261\) −1.60171 + 4.92954i −0.0991431 + 0.305131i
\(262\) 0 0
\(263\) −3.38270 + 1.09911i −0.208586 + 0.0677738i −0.411447 0.911434i \(-0.634976\pi\)
0.202860 + 0.979208i \(0.434976\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.541778 + 0.745693i 0.0331563 + 0.0456357i
\(268\) 0 0
\(269\) 23.6733 17.1997i 1.44339 1.04868i 0.456066 0.889946i \(-0.349258\pi\)
0.987321 0.158736i \(-0.0507419\pi\)
\(270\) 0 0
\(271\) 20.0784 + 14.5878i 1.21968 + 0.886147i 0.996073 0.0885338i \(-0.0282181\pi\)
0.223603 + 0.974680i \(0.428218\pi\)
\(272\) 0 0
\(273\) 1.77923 + 0.578108i 0.107684 + 0.0349887i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.5745 5.38539i −0.995867 0.323577i −0.234654 0.972079i \(-0.575396\pi\)
−0.761213 + 0.648502i \(0.775396\pi\)
\(278\) 0 0
\(279\) 3.02547 + 2.19813i 0.181130 + 0.131599i
\(280\) 0 0
\(281\) −3.03664 + 2.20625i −0.181151 + 0.131614i −0.674665 0.738124i \(-0.735712\pi\)
0.493515 + 0.869738i \(0.335712\pi\)
\(282\) 0 0
\(283\) 9.62409 + 13.2464i 0.572093 + 0.787418i 0.992801 0.119778i \(-0.0382183\pi\)
−0.420708 + 0.907196i \(0.638218\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.28389 + 0.742080i −0.134814 + 0.0438036i
\(288\) 0 0
\(289\) −9.11787 + 28.0619i −0.536345 + 1.65070i
\(290\) 0 0
\(291\) −4.80349 14.7836i −0.281586 0.866632i
\(292\) 0 0
\(293\) 8.70991i 0.508838i −0.967094 0.254419i \(-0.918116\pi\)
0.967094 0.254419i \(-0.0818843\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0.122881 0.169131i 0.00713025 0.00981395i
\(298\) 0 0
\(299\) 8.13888 0.470683
\(300\) 0 0
\(301\) −9.49939 −0.547536
\(302\) 0 0
\(303\) 6.48944 8.93194i 0.372808 0.513127i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.7123i 1.01090i −0.862857 0.505448i \(-0.831327\pi\)
0.862857 0.505448i \(-0.168673\pi\)
\(308\) 0 0
\(309\) −4.29048 13.2047i −0.244077 0.751191i
\(310\) 0 0
\(311\) 6.26921 19.2946i 0.355494 1.09410i −0.600228 0.799829i \(-0.704924\pi\)
0.955722 0.294270i \(-0.0950763\pi\)
\(312\) 0 0
\(313\) 19.7535 6.41831i 1.11654 0.362785i 0.308091 0.951357i \(-0.400310\pi\)
0.808445 + 0.588572i \(0.200310\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.85429 + 13.5633i 0.553472 + 0.761789i 0.990478 0.137670i \(-0.0439613\pi\)
−0.437006 + 0.899458i \(0.643961\pi\)
\(318\) 0 0
\(319\) −0.876642 + 0.636918i −0.0490825 + 0.0356606i
\(320\) 0 0
\(321\) −0.829324 0.602539i −0.0462884 0.0336305i
\(322\) 0 0
\(323\) 8.07610 + 2.62408i 0.449366 + 0.146008i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.37694 2.07199i −0.352646 0.114582i
\(328\) 0 0
\(329\) −3.91870 2.84711i −0.216045 0.156966i
\(330\) 0 0
\(331\) 21.2090 15.4092i 1.16575 0.846968i 0.175257 0.984523i \(-0.443924\pi\)
0.990494 + 0.137555i \(0.0439243\pi\)
\(332\) 0 0
\(333\) 1.15525 + 1.59007i 0.0633074 + 0.0871352i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.7298 8.03519i 1.34712 0.437705i 0.455395 0.890289i \(-0.349498\pi\)
0.891722 + 0.452584i \(0.149498\pi\)
\(338\) 0 0
\(339\) −2.57151 + 7.91428i −0.139665 + 0.429845i
\(340\) 0 0
\(341\) 0.241591 + 0.743542i 0.0130829 + 0.0402651i
\(342\) 0 0
\(343\) 10.0441i 0.542331i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.70460 3.72256i 0.145191 0.199838i −0.730228 0.683204i \(-0.760586\pi\)
0.875418 + 0.483366i \(0.160586\pi\)
\(348\) 0 0
\(349\) 29.2108 1.56362 0.781810 0.623516i \(-0.214297\pi\)
0.781810 + 0.623516i \(0.214297\pi\)
\(350\) 0 0
\(351\) −2.50361 −0.133633
\(352\) 0 0
\(353\) 16.5292 22.7505i 0.879759 1.21088i −0.0967283 0.995311i \(-0.530838\pi\)
0.976488 0.215574i \(-0.0691622\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.09582i 0.269699i
\(358\) 0 0
\(359\) 2.90570 + 8.94282i 0.153357 + 0.471984i 0.997991 0.0633604i \(-0.0201817\pi\)
−0.844634 + 0.535345i \(0.820182\pi\)
\(360\) 0 0
\(361\) −5.39218 + 16.5954i −0.283799 + 0.873444i
\(362\) 0 0
\(363\) −10.4201 + 3.38568i −0.546911 + 0.177702i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −12.9349 17.8033i −0.675195 0.929326i 0.324669 0.945828i \(-0.394747\pi\)
−0.999864 + 0.0165016i \(0.994747\pi\)
\(368\) 0 0
\(369\) 2.59996 1.88898i 0.135349 0.0983365i
\(370\) 0 0
\(371\) −2.50457 1.81968i −0.130031 0.0944728i
\(372\) 0 0
\(373\) −17.7085 5.75384i −0.916911 0.297923i −0.187712 0.982224i \(-0.560107\pi\)
−0.729199 + 0.684302i \(0.760107\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.3417 + 4.01005i 0.635628 + 0.206528i
\(378\) 0 0
\(379\) −13.6515 9.91838i −0.701230 0.509473i 0.179103 0.983830i \(-0.442680\pi\)
−0.880332 + 0.474357i \(0.842680\pi\)
\(380\) 0 0
\(381\) 11.7601 8.54421i 0.602488 0.437733i
\(382\) 0 0
\(383\) −5.32445 7.32847i −0.272066 0.374467i 0.651020 0.759061i \(-0.274342\pi\)
−0.923086 + 0.384594i \(0.874342\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.0905 3.92843i 0.614593 0.199693i
\(388\) 0 0
\(389\) −10.2225 + 31.4616i −0.518301 + 1.59517i 0.258893 + 0.965906i \(0.416642\pi\)
−0.777194 + 0.629261i \(0.783358\pi\)
\(390\) 0 0
\(391\) −6.85069 21.0843i −0.346454 1.06628i
\(392\) 0 0
\(393\) 7.42396i 0.374489i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12.7706 + 17.5773i −0.640939 + 0.882177i −0.998665 0.0516494i \(-0.983552\pi\)
0.357726 + 0.933827i \(0.383552\pi\)
\(398\) 0 0
\(399\) −0.930465 −0.0465815
\(400\) 0 0
\(401\) 20.1663 1.00706 0.503529 0.863978i \(-0.332035\pi\)
0.503529 + 0.863978i \(0.332035\pi\)
\(402\) 0 0
\(403\) 5.50327 7.57460i 0.274137 0.377318i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.410887i 0.0203669i
\(408\) 0 0
\(409\) 4.34139 + 13.3614i 0.214668 + 0.660679i 0.999177 + 0.0405623i \(0.0129149\pi\)
−0.784509 + 0.620117i \(0.787085\pi\)
\(410\) 0 0
\(411\) −2.26676 + 6.97636i −0.111811 + 0.344118i
\(412\) 0 0
\(413\) 8.32244 2.70412i 0.409520 0.133061i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.6071 15.9757i −0.568400 0.782336i
\(418\) 0 0
\(419\) −16.3919 + 11.9094i −0.800794 + 0.581811i −0.911147 0.412081i \(-0.864802\pi\)
0.110353 + 0.993892i \(0.464802\pi\)
\(420\) 0 0
\(421\) −3.54258 2.57384i −0.172655 0.125441i 0.498102 0.867119i \(-0.334031\pi\)
−0.670757 + 0.741677i \(0.734031\pi\)
\(422\) 0 0
\(423\) 6.16499 + 2.00313i 0.299752 + 0.0973953i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.86866 2.88160i −0.429184 0.139450i
\(428\) 0 0
\(429\) −0.423438 0.307645i −0.0204438 0.0148533i
\(430\) 0 0
\(431\) −0.944967 + 0.686559i −0.0455175 + 0.0330704i −0.610311 0.792162i \(-0.708956\pi\)
0.564794 + 0.825232i \(0.308956\pi\)
\(432\) 0 0
\(433\) 1.32187 + 1.81940i 0.0635250 + 0.0874347i 0.839597 0.543209i \(-0.182791\pi\)
−0.776072 + 0.630644i \(0.782791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.84985 + 1.25089i −0.184163 + 0.0598383i
\(438\) 0 0
\(439\) −6.83477 + 21.0353i −0.326206 + 1.00396i 0.644688 + 0.764446i \(0.276987\pi\)
−0.970893 + 0.239512i \(0.923013\pi\)
\(440\) 0 0
\(441\) −1.99057 6.12636i −0.0947893 0.291731i
\(442\) 0 0
\(443\) 28.1534i 1.33761i −0.743439 0.668804i \(-0.766807\pi\)
0.743439 0.668804i \(-0.233193\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −12.7873 + 17.6002i −0.604819 + 0.832462i
\(448\) 0 0
\(449\) −16.5924 −0.783044 −0.391522 0.920169i \(-0.628051\pi\)
−0.391522 + 0.920169i \(0.628051\pi\)
\(450\) 0 0
\(451\) 0.671852 0.0316363
\(452\) 0 0
\(453\) −5.95473 + 8.19598i −0.279777 + 0.385081i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.1599i 0.522039i −0.965334 0.261019i \(-0.915941\pi\)
0.965334 0.261019i \(-0.0840586\pi\)
\(458\) 0 0
\(459\) 2.10735 + 6.48576i 0.0983628 + 0.302729i
\(460\) 0 0
\(461\) 1.93631 5.95935i 0.0901830 0.277555i −0.895785 0.444487i \(-0.853386\pi\)
0.985968 + 0.166932i \(0.0533861\pi\)
\(462\) 0 0
\(463\) −16.1961 + 5.26243i −0.752696 + 0.244566i −0.660141 0.751142i \(-0.729503\pi\)
−0.0925550 + 0.995708i \(0.529503\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.9823 + 15.1159i 0.508202 + 0.699481i 0.983615 0.180282i \(-0.0577012\pi\)
−0.475413 + 0.879763i \(0.657701\pi\)
\(468\) 0 0
\(469\) 1.76344 1.28122i 0.0814283 0.0591611i
\(470\) 0 0
\(471\) 8.08629 + 5.87503i 0.372597 + 0.270707i
\(472\) 0 0
\(473\) 2.52760 + 0.821266i 0.116219 + 0.0377618i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.94024 + 1.28026i 0.180411 + 0.0586191i
\(478\) 0 0
\(479\) 2.38111 + 1.72998i 0.108796 + 0.0790448i 0.640853 0.767664i \(-0.278581\pi\)
−0.532057 + 0.846708i \(0.678581\pi\)
\(480\) 0 0
\(481\) 3.98092 2.89230i 0.181514 0.131878i
\(482\) 0 0
\(483\) 1.42782 + 1.96523i 0.0649683 + 0.0894212i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.3297 4.33109i 0.604028 0.196261i 0.00899199 0.999960i \(-0.497138\pi\)
0.595036 + 0.803699i \(0.297138\pi\)
\(488\) 0 0
\(489\) 0.792035 2.43763i 0.0358170 0.110234i
\(490\) 0 0
\(491\) 2.76416 + 8.50720i 0.124745 + 0.383925i 0.993854 0.110695i \(-0.0353075\pi\)
−0.869110 + 0.494619i \(0.835308\pi\)
\(492\) 0 0
\(493\) 35.3472i 1.59196i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.87690 3.95971i 0.129046 0.177617i
\(498\) 0 0
\(499\) 2.39366 0.107155 0.0535774 0.998564i \(-0.482938\pi\)
0.0535774 + 0.998564i \(0.482938\pi\)
\(500\) 0 0
\(501\) −20.8512 −0.931564
\(502\) 0 0
\(503\) −4.95152 + 6.81519i −0.220777 + 0.303874i −0.905010 0.425389i \(-0.860137\pi\)
0.684233 + 0.729264i \(0.260137\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.73192i 0.298975i
\(508\) 0 0
\(509\) −9.98528 30.7315i −0.442590 1.36215i −0.885106 0.465390i \(-0.845914\pi\)
0.442516 0.896761i \(-0.354086\pi\)
\(510\) 0 0
\(511\) −0.581737 + 1.79040i −0.0257345 + 0.0792027i
\(512\) 0 0
\(513\) 1.18426 0.384789i 0.0522864 0.0169889i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.796543 + 1.09635i 0.0350319 + 0.0482173i
\(518\) 0 0
\(519\) 1.66249 1.20787i 0.0729754 0.0530197i
\(520\) 0 0
\(521\) −2.88427 2.09554i −0.126362 0.0918074i 0.522809 0.852450i \(-0.324884\pi\)
−0.649171 + 0.760642i \(0.724884\pi\)
\(522\) 0 0
\(523\) 30.8511 + 10.0241i 1.34903 + 0.438325i 0.892365 0.451315i \(-0.149045\pi\)
0.456662 + 0.889640i \(0.349045\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.2547 7.88082i −1.05655 0.343294i
\(528\) 0 0
\(529\) −10.0577 7.30732i −0.437290 0.317710i
\(530\) 0 0
\(531\) −9.47420 + 6.88341i −0.411145 + 0.298715i
\(532\) 0 0
\(533\) −4.72928 6.50929i −0.204848 0.281949i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.6030 6.04449i 0.802781 0.260839i
\(538\) 0 0
\(539\) 0.416143 1.28076i 0.0179246 0.0551661i
\(540\) 0 0
\(541\) −3.40378 10.4757i −0.146340 0.450388i 0.850841 0.525423i \(-0.176093\pi\)
−0.997181 + 0.0750356i \(0.976093\pi\)
\(542\) 0 0
\(543\) 7.37007i 0.316280i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.13718 + 4.31795i −0.134136 + 0.184622i −0.870801 0.491635i \(-0.836399\pi\)
0.736665 + 0.676258i \(0.236399\pi\)
\(548\) 0 0
\(549\) 12.4794 0.532606
\(550\) 0 0
\(551\) −6.45418 −0.274957
\(552\) 0 0
\(553\) 4.49859 6.19177i 0.191299 0.263301i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.0652731i 0.00276571i −0.999999 0.00138286i \(-0.999560\pi\)
0.999999 0.00138286i \(-0.000440177\pi\)
\(558\) 0 0
\(559\) −9.83527 30.2699i −0.415988 1.28028i
\(560\) 0 0
\(561\) −0.440557 + 1.35589i −0.0186003 + 0.0572459i
\(562\) 0 0
\(563\) −27.3359 + 8.88197i −1.15207 + 0.374330i −0.821923 0.569599i \(-0.807098\pi\)
−0.330147 + 0.943929i \(0.607098\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.439216 0.604528i −0.0184453 0.0253878i
\(568\) 0 0
\(569\) −6.68501 + 4.85694i −0.280250 + 0.203614i −0.719026 0.694983i \(-0.755412\pi\)
0.438776 + 0.898596i \(0.355412\pi\)
\(570\) 0 0
\(571\) −9.49451 6.89817i −0.397333 0.288679i 0.371121 0.928585i \(-0.378974\pi\)
−0.768454 + 0.639905i \(0.778974\pi\)
\(572\) 0 0
\(573\) 2.13532 + 0.693806i 0.0892041 + 0.0289842i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 20.3418 + 6.60944i 0.846838 + 0.275154i 0.700121 0.714024i \(-0.253129\pi\)
0.146717 + 0.989178i \(0.453129\pi\)
\(578\) 0 0
\(579\) 0.894682 + 0.650024i 0.0371817 + 0.0270141i
\(580\) 0 0
\(581\) 3.43162 2.49322i 0.142368 0.103436i
\(582\) 0 0
\(583\) 0.509096 + 0.700710i 0.0210846 + 0.0290204i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −39.2133 + 12.7412i −1.61851 + 0.525885i −0.971589 0.236674i \(-0.923943\pi\)
−0.646918 + 0.762559i \(0.723943\pi\)
\(588\) 0 0
\(589\) −1.43899 + 4.42876i −0.0592925 + 0.182484i
\(590\) 0 0
\(591\) 6.37705 + 19.6265i 0.262317 + 0.807328i
\(592\) 0 0
\(593\) 23.2238i 0.953685i 0.878989 + 0.476843i \(0.158219\pi\)
−0.878989 + 0.476843i \(0.841781\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.27806 + 10.0174i −0.297871 + 0.409984i
\(598\) 0 0
\(599\) 22.6226 0.924335 0.462168 0.886793i \(-0.347072\pi\)
0.462168 + 0.886793i \(0.347072\pi\)
\(600\) 0 0
\(601\) −12.9540 −0.528405 −0.264203 0.964467i \(-0.585109\pi\)
−0.264203 + 0.964467i \(0.585109\pi\)
\(602\) 0 0
\(603\) −1.71460 + 2.35995i −0.0698240 + 0.0961045i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.54036i 0.184288i 0.995746 + 0.0921439i \(0.0293720\pi\)
−0.995746 + 0.0921439i \(0.970628\pi\)
\(608\) 0 0
\(609\) 1.19686 + 3.68354i 0.0484990 + 0.149265i
\(610\) 0 0
\(611\) 5.01505 15.4347i 0.202887 0.624423i
\(612\) 0 0
\(613\) −24.1128 + 7.83471i −0.973905 + 0.316441i −0.752391 0.658716i \(-0.771100\pi\)
−0.221514 + 0.975157i \(0.571100\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5790 + 36.5829i 1.07003 + 1.47277i 0.870045 + 0.492972i \(0.164090\pi\)
0.199986 + 0.979799i \(0.435910\pi\)
\(618\) 0 0
\(619\) 25.3666 18.4299i 1.01957 0.740762i 0.0533766 0.998574i \(-0.483002\pi\)
0.966195 + 0.257812i \(0.0830016\pi\)
\(620\) 0 0
\(621\) −2.62999 1.91080i −0.105538 0.0766779i
\(622\) 0 0
\(623\) 0.655040 + 0.212835i 0.0262436 + 0.00852707i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.247578 + 0.0804429i 0.00988730 + 0.00321258i
\(628\) 0 0
\(629\) −10.8435 7.87828i −0.432360 0.314128i
\(630\) 0 0
\(631\) −25.8455 + 18.7779i −1.02889 + 0.747535i −0.968087 0.250614i \(-0.919367\pi\)
−0.0608071 + 0.998150i \(0.519367\pi\)
\(632\) 0 0
\(633\) 12.1245 + 16.6879i 0.481905 + 0.663286i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −15.3380 + 4.98363i −0.607715 + 0.197459i
\(638\) 0 0
\(639\) −2.02409 + 6.22949i −0.0800716 + 0.246435i
\(640\) 0 0
\(641\) 12.7727 + 39.3103i 0.504491 + 1.55267i 0.801624 + 0.597829i \(0.203970\pi\)
−0.297132 + 0.954836i \(0.596030\pi\)
\(642\) 0 0
\(643\) 33.2313i 1.31052i 0.755406 + 0.655258i \(0.227440\pi\)
−0.755406 + 0.655258i \(0.772560\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.8034 27.2571i 0.778553 1.07159i −0.216887 0.976197i \(-0.569590\pi\)
0.995440 0.0953892i \(-0.0304096\pi\)
\(648\) 0 0
\(649\) −2.44822 −0.0961009
\(650\) 0 0
\(651\) 2.79443 0.109522
\(652\) 0 0
\(653\) −12.8419 + 17.6754i −0.502544 + 0.691692i −0.982640 0.185524i \(-0.940602\pi\)
0.480096 + 0.877216i \(0.340602\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.51933i 0.0982885i
\(658\) 0 0
\(659\) 3.48829 + 10.7359i 0.135884 + 0.418209i 0.995727 0.0923507i \(-0.0294381\pi\)
−0.859842 + 0.510560i \(0.829438\pi\)
\(660\) 0 0
\(661\) −11.3014 + 34.7821i −0.439573 + 1.35287i 0.448754 + 0.893655i \(0.351868\pi\)
−0.888327 + 0.459211i \(0.848132\pi\)
\(662\) 0 0
\(663\) 16.2378 5.27599i 0.630625 0.204903i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.90412 + 13.6319i 0.383489 + 0.527828i
\(668\) 0 0
\(669\) −3.36117 + 2.44203i −0.129950 + 0.0944145i
\(670\) 0 0
\(671\) 2.11064 + 1.53347i 0.0814804 + 0.0591990i
\(672\) 0 0
\(673\) −7.31143 2.37563i −0.281835 0.0915737i 0.164689 0.986346i \(-0.447338\pi\)
−0.446523 + 0.894772i \(0.647338\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −35.4711 11.5253i −1.36327 0.442952i −0.466133 0.884714i \(-0.654353\pi\)
−0.897132 + 0.441763i \(0.854353\pi\)
\(678\) 0 0
\(679\) −9.39705 6.82736i −0.360626 0.262010i
\(680\) 0 0
\(681\) 19.1162 13.8888i 0.732535 0.532218i
\(682\) 0 0
\(683\) 21.4421 + 29.5125i 0.820458 + 1.12926i 0.989625 + 0.143676i \(0.0458925\pi\)
−0.169166 + 0.985587i \(0.554108\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.3473 + 5.63649i −0.661842 + 0.215046i
\(688\) 0 0
\(689\) 3.20528 9.86483i 0.122111 0.375820i
\(690\) 0 0
\(691\) −9.99734 30.7686i −0.380317 1.17049i −0.939821 0.341667i \(-0.889008\pi\)
0.559504 0.828827i \(-0.310992\pi\)
\(692\) 0 0
\(693\) 0.156215i 0.00593413i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.8820 + 17.7305i −0.487940 + 0.671591i
\(698\) 0 0
\(699\) −2.26742 −0.0857617
\(700\) 0 0
\(701\) 5.27498 0.199233 0.0996166 0.995026i \(-0.468238\pi\)
0.0996166 + 0.995026i \(0.468238\pi\)
\(702\) 0 0
\(703\) −1.43853 + 1.97996i −0.0542550 + 0.0746756i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.24988i 0.310268i
\(708\) 0 0
\(709\) −9.03055 27.7932i −0.339149 1.04379i −0.964642 0.263564i \(-0.915102\pi\)
0.625493 0.780230i \(-0.284898\pi\)
\(710\) 0 0
\(711\) −3.16505 + 9.74102i −0.118699 + 0.365317i
\(712\) 0 0
\(713\) 11.5621 3.75677i 0.433005 0.140692i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.0552 13.8397i −0.375517 0.516855i
\(718\) 0 0
\(719\) 35.5675 25.8413i 1.32644 0.963718i 0.326616 0.945157i \(-0.394092\pi\)
0.999828 0.0185605i \(-0.00590834\pi\)
\(720\) 0 0
\(721\) −8.39344 6.09819i −0.312588 0.227108i
\(722\) 0 0
\(723\) −26.8200 8.71435i −0.997447 0.324090i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.4614 + 9.24768i 1.05558 + 0.342977i 0.784854 0.619680i \(-0.212738\pi\)
0.270721 + 0.962658i \(0.412738\pi\)
\(728\) 0 0
\(729\) 0.809017 + 0.587785i 0.0299636 + 0.0217698i
\(730\) 0 0
\(731\) −70.1373 + 50.9577i −2.59412 + 1.88474i
\(732\) 0 0
\(733\) −9.71402 13.3702i −0.358795 0.493840i 0.591017 0.806659i \(-0.298727\pi\)
−0.949813 + 0.312819i \(0.898727\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.579984 + 0.188448i −0.0213640 + 0.00694158i
\(738\) 0 0
\(739\) 15.7285 48.4073i 0.578581 1.78069i −0.0450666 0.998984i \(-0.514350\pi\)
0.623648 0.781706i \(-0.285650\pi\)
\(740\) 0 0
\(741\) −0.963364 2.96493i −0.0353901 0.108919i
\(742\) 0 0
\(743\) 28.6937i 1.05267i 0.850277 + 0.526336i \(0.176434\pi\)
−0.850277 + 0.526336i \(0.823566\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.33658 + 4.59240i −0.122079 + 0.168027i
\(748\) 0 0
\(749\) −0.765995 −0.0279888
\(750\) 0 0
\(751\) −0.927935 −0.0338608 −0.0169304 0.999857i \(-0.505389\pi\)
−0.0169304 + 0.999857i \(0.505389\pi\)
\(752\) 0 0
\(753\) −14.0819 + 19.3821i −0.513173 + 0.706322i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41.4243i 1.50559i 0.658254 + 0.752796i \(0.271295\pi\)
−0.658254 + 0.752796i \(0.728705\pi\)
\(758\) 0 0
\(759\) −0.210012 0.646350i −0.00762295 0.0234610i
\(760\) 0 0
\(761\) 1.17031 3.60185i 0.0424237 0.130567i −0.927601 0.373572i \(-0.878133\pi\)
0.970025 + 0.243005i \(0.0781330\pi\)
\(762\) 0 0
\(763\) −4.76509 + 1.54827i −0.172508 + 0.0560513i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 17.2334 + 23.7197i 0.622262 + 0.856470i
\(768\) 0 0
\(769\) −1.06014 + 0.770234i −0.0382295 + 0.0277753i −0.606736 0.794903i \(-0.707521\pi\)
0.568506 + 0.822679i \(0.307521\pi\)
\(770\) 0 0
\(771\) 23.0911 + 16.7766i 0.831604 + 0.604196i
\(772\) 0 0
\(773\) −27.7066 9.00241i −0.996536 0.323794i −0.235055 0.971982i \(-0.575527\pi\)
−0.761480 + 0.648188i \(0.775527\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.39677 + 0.453837i 0.0501087 + 0.0162813i
\(778\) 0 0
\(779\) 3.23748 + 2.35217i 0.115995 + 0.0842752i
\(780\) 0 0
\(781\) −1.10782 + 0.804877i −0.0396409 + 0.0288008i
\(782\) 0 0
\(783\) −3.04662 4.19332i −0.108877 0.149857i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 44.3341 14.4050i 1.58034 0.513484i 0.618197 0.786023i \(-0.287864\pi\)
0.962145 + 0.272540i \(0.0878635\pi\)
\(788\) 0 0
\(789\) 1.09911 3.38270i 0.0391292 0.120427i
\(790\) 0 0
\(791\) 1.92153 + 5.91385i 0.0683217 + 0.210272i
\(792\) 0 0
\(793\) 31.2435i 1.10949i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.08703 + 11.1308i −0.286457 + 0.394275i −0.927859 0.372930i \(-0.878353\pi\)
0.641402 + 0.767205i \(0.278353\pi\)
\(798\) 0 0
\(799\) −44.2059 −1.56389
\(800\) 0 0
\(801\) −0.921727 −0.0325676
\(802\) 0 0
\(803\) 0.309577 0.426096i 0.0109247 0.0150366i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 29.2618i 1.03006i
\(808\) 0 0
\(809\) −3.08886 9.50654i −0.108599 0.334232i 0.881960 0.471325i \(-0.156224\pi\)
−0.990558 + 0.137093i \(0.956224\pi\)
\(810\) 0 0
\(811\) 5.72277 17.6129i 0.200954 0.618472i −0.798902 0.601462i \(-0.794585\pi\)
0.999855 0.0170101i \(-0.00541474\pi\)
\(812\) 0 0
\(813\) −23.6036 + 7.66927i −0.827814 + 0.268973i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.30457 + 12.8066i 0.325526 + 0.448048i
\(818\) 0 0
\(819\) −1.51351 + 1.09963i −0.0528862 + 0.0384240i
\(820\) 0 0
\(821\) −22.9081 16.6437i −0.799498 0.580869i 0.111269 0.993790i \(-0.464509\pi\)
−0.910767 + 0.412921i \(0.864509\pi\)
\(822\) 0 0
\(823\) −11.3987 3.70366i −0.397333 0.129101i 0.103533 0.994626i \(-0.466985\pi\)
−0.500866 + 0.865525i \(0.666985\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.7167 + 4.13191i 0.442204 + 0.143681i 0.521652 0.853158i \(-0.325316\pi\)
−0.0794485 + 0.996839i \(0.525316\pi\)
\(828\) 0 0
\(829\) −13.3909 9.72909i −0.465087 0.337905i 0.330437 0.943828i \(-0.392804\pi\)
−0.795523 + 0.605923i \(0.792804\pi\)
\(830\) 0 0
\(831\) 14.0991 10.2436i 0.489094 0.355347i
\(832\) 0 0
\(833\) 25.8208 + 35.5393i 0.894637 + 1.23136i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.55665 + 1.15563i −0.122936 + 0.0399442i
\(838\) 0 0
\(839\) −7.56589 + 23.2854i −0.261204 + 0.803902i 0.731340 + 0.682013i \(0.238895\pi\)
−0.992544 + 0.121889i \(0.961105\pi\)
\(840\) 0 0
\(841\) −0.659493 2.02971i −0.0227411 0.0699900i
\(842\) 0 0
\(843\) 3.75349i 0.129277i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −4.81218 + 6.62339i −0.165348 + 0.227582i
\(848\) 0 0
\(849\) −16.3735 −0.561936
\(850\) 0 0
\(851\) 6.38933 0.219023
\(852\) 0 0
\(853\) −11.2475 + 15.4809i −0.385108 + 0.530055i −0.956929 0.290323i \(-0.906237\pi\)
0.571821 + 0.820379i \(0.306237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.43367i 0.253929i −0.991907 0.126965i \(-0.959477\pi\)
0.991907 0.126965i \(-0.0405235\pi\)
\(858\) 0 0
\(859\) −12.7368 39.1999i −0.434575 1.33748i −0.893522 0.449020i \(-0.851773\pi\)
0.458947 0.888464i \(-0.348227\pi\)
\(860\) 0 0
\(861\) 0.742080 2.28389i 0.0252900 0.0778346i
\(862\) 0 0
\(863\) −36.2532 + 11.7794i −1.23407 + 0.400975i −0.852189 0.523234i \(-0.824725\pi\)
−0.381886 + 0.924210i \(0.624725\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −17.3432 23.8709i −0.589007 0.810698i
\(868\) 0 0
\(869\) −1.73229 + 1.25858i −0.0587639 + 0.0426945i
\(870\) 0 0
\(871\) 5.90840 + 4.29270i 0.200198 + 0.145453i
\(872\) 0 0
\(873\) 14.7836 + 4.80349i 0.500350 + 0.162574i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.6047 + 8.96931i 0.932145 + 0.302872i 0.735439 0.677591i \(-0.236976\pi\)
0.196705 + 0.980463i \(0.436976\pi\)
\(878\) 0 0
\(879\) 7.04647 + 5.11956i 0.237671 + 0.172678i
\(880\) 0 0
\(881\) 27.8503 20.2344i 0.938300 0.681715i −0.00971098 0.999953i \(-0.503091\pi\)
0.948011 + 0.318238i \(0.103091\pi\)
\(882\) 0 0
\(883\) −22.8077 31.3921i −0.767539 1.05643i −0.996549 0.0830028i \(-0.973549\pi\)
0.229010 0.973424i \(-0.426451\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.48715 + 1.78288i −0.184241 + 0.0598634i −0.399684 0.916653i \(-0.630880\pi\)
0.215444 + 0.976516i \(0.430880\pi\)
\(888\) 0 0
\(889\) 3.35656 10.3304i 0.112575 0.346472i
\(890\) 0 0
\(891\) 0.0646021 + 0.198825i 0.00216425 + 0.00666089i
\(892\) 0 0
\(893\) 8.07173i 0.270110i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −4.78391 + 6.58449i −0.159730 + 0.219850i
\(898\) 0 0
\(899\) 19.3836 0.646480
\(900\) 0 0
\(901\) −28.2534 −0.941257
\(902\) 0 0
\(903\) 5.58360 7.68517i 0.185811 0.255746i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 31.7260i 1.05345i 0.850037 + 0.526723i \(0.176579\pi\)
−0.850037 + 0.526723i \(0.823421\pi\)
\(908\) 0 0
\(909\) 3.41170 + 10.5001i 0.113159 + 0.348267i
\(910\) 0 0
\(911\) −4.95450 + 15.2484i −0.164150 + 0.505202i −0.998973 0.0453174i \(-0.985570\pi\)
0.834823 + 0.550519i \(0.185570\pi\)
\(912\) 0 0
\(913\) −1.12863 + 0.366716i −0.0373523 + 0.0121365i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.26072 4.48800i −0.107679 0.148207i
\(918\) 0 0
\(919\) −8.82328 + 6.41049i −0.291053 + 0.211462i −0.723724 0.690089i \(-0.757571\pi\)
0.432671 + 0.901552i \(0.357571\pi\)
\(920\) 0 0
\(921\) 14.3296 + 10.4111i 0.472176 + 0.343056i
\(922\) 0 0
\(923\) 15.5962 + 5.06753i 0.513357 + 0.166800i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 13.2047 + 4.29048i 0.433700 + 0.140918i
\(928\) 0 0
\(929\) 18.3021 + 13.2973i 0.600473 + 0.436269i 0.846047 0.533108i \(-0.178976\pi\)
−0.245573 + 0.969378i \(0.578976\pi\)
\(930\) 0 0
\(931\) 6.48925 4.71472i 0.212677 0.154519i
\(932\) 0 0
\(933\) 11.9247 + 16.4130i 0.390398 + 0.537337i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.90347 1.59323i 0.160189 0.0520487i −0.227824 0.973702i \(-0.573161\pi\)
0.388014 + 0.921654i \(0.373161\pi\)
\(938\) 0 0
\(939\) −6.41831 + 19.7535i −0.209454 + 0.644632i
\(940\) 0 0
\(941\) 3.84942 + 11.8473i 0.125488 + 0.386211i 0.993990 0.109475i \(-0.0349168\pi\)
−0.868502 + 0.495686i \(0.834917\pi\)
\(942\) 0 0
\(943\) 10.4474i 0.340213i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.79345 13.4795i 0.318244 0.438026i −0.619686 0.784850i \(-0.712740\pi\)
0.937930 + 0.346824i \(0.112740\pi\)
\(948\) 0 0
\(949\) −6.30743 −0.204748
\(950\) 0 0
\(951\) −16.7651 −0.543646
\(952\) 0 0
\(953\) −30.4812 + 41.9537i −0.987382 + 1.35901i −0.0546255 + 0.998507i \(0.517396\pi\)
−0.932756 + 0.360508i \(0.882604\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.08359i 0.0350275i
\(958\) 0 0
\(959\) 1.69381 + 5.21300i 0.0546959 + 0.168337i
\(960\) 0 0
\(961\) −5.25786 + 16.1820i −0.169608 + 0.522001i
\(962\) 0 0
\(963\) 0.974929 0.316774i 0.0314166 0.0102079i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −7.50713 10.3327i −0.241413 0.332277i 0.671068 0.741396i \(-0.265836\pi\)
−0.912481 + 0.409119i \(0.865836\pi\)
\(968\) 0 0
\(969\) −6.86994 + 4.99131i −0.220694 + 0.160344i
\(970\) 0 0
\(971\) 12.5679 + 9.13110i 0.403323 + 0.293031i 0.770893 0.636965i \(-0.219810\pi\)
−0.367570 + 0.929996i \(0.619810\pi\)
\(972\) 0 0
\(973\) −14.0336 4.55979i −0.449896 0.146180i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −31.5811 10.2613i −1.01037 0.328288i −0.243367 0.969934i \(-0.578252\pi\)
−0.767001 + 0.641646i \(0.778252\pi\)
\(978\) 0 0
\(979\) −0.155892 0.113262i −0.00498234 0.00361988i
\(980\) 0 0
\(981\) 5.42455 3.94117i 0.173193 0.125832i
\(982\) 0 0
\(983\) −28.8570 39.7183i −0.920395 1.26682i −0.963490 0.267745i \(-0.913722\pi\)
0.0430944 0.999071i \(-0.486278\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.60671 1.49681i 0.146633 0.0476440i
\(988\) 0 0
\(989\) 12.7707 39.3043i 0.406086 1.24980i
\(990\) 0 0
\(991\) −7.91583 24.3624i −0.251455 0.773898i −0.994508 0.104665i \(-0.966623\pi\)
0.743053 0.669233i \(-0.233377\pi\)
\(992\) 0 0
\(993\) 26.2157i 0.831932i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.36565 + 8.76157i −0.201602 + 0.277482i −0.897833 0.440337i \(-0.854859\pi\)
0.696231 + 0.717818i \(0.254859\pi\)
\(998\) 0 0
\(999\) −1.96543 −0.0621835
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 1500.2.o.a.649.2 16
5.2 odd 4 1500.2.m.b.601.1 8
5.3 odd 4 300.2.m.a.121.2 8
5.4 even 2 inner 1500.2.o.a.649.3 16
15.8 even 4 900.2.n.a.721.1 8
25.6 even 5 inner 1500.2.o.a.349.4 16
25.8 odd 20 300.2.m.a.181.2 yes 8
25.9 even 10 7500.2.d.d.1249.5 8
25.12 odd 20 7500.2.a.d.1.1 4
25.13 odd 20 7500.2.a.g.1.4 4
25.16 even 5 7500.2.d.d.1249.4 8
25.17 odd 20 1500.2.m.b.901.1 8
25.19 even 10 inner 1500.2.o.a.349.1 16
75.8 even 20 900.2.n.a.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.m.a.121.2 8 5.3 odd 4
300.2.m.a.181.2 yes 8 25.8 odd 20
900.2.n.a.181.1 8 75.8 even 20
900.2.n.a.721.1 8 15.8 even 4
1500.2.m.b.601.1 8 5.2 odd 4
1500.2.m.b.901.1 8 25.17 odd 20
1500.2.o.a.349.1 16 25.19 even 10 inner
1500.2.o.a.349.4 16 25.6 even 5 inner
1500.2.o.a.649.2 16 1.1 even 1 trivial
1500.2.o.a.649.3 16 5.4 even 2 inner
7500.2.a.d.1.1 4 25.12 odd 20
7500.2.a.g.1.4 4 25.13 odd 20
7500.2.d.d.1249.4 8 25.16 even 5
7500.2.d.d.1249.5 8 25.9 even 10