Properties

Label 432.2.be.a.47.6
Level $432$
Weight $2$
Character 432.47
Analytic conductor $3.450$
Analytic rank $0$
Dimension $36$
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] = [432,2,Mod(47,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 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("432.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 432.be (of order \(18\), degree \(6\), 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: \(3.44953736732\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 47.6
Character \(\chi\) \(=\) 432.47
Dual form 432.2.be.a.239.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.72868 - 0.107984i) q^{3} +(0.902279 - 0.159096i) q^{5} +(2.30964 - 2.75253i) q^{7} +(2.97668 - 0.373339i) q^{9} +O(q^{10})\) \(q+(1.72868 - 0.107984i) q^{3} +(0.902279 - 0.159096i) q^{5} +(2.30964 - 2.75253i) q^{7} +(2.97668 - 0.373339i) q^{9} +(-0.316002 + 1.79214i) q^{11} +(-4.18526 - 1.52331i) q^{13} +(1.54257 - 0.372458i) q^{15} +(-3.92200 + 2.26437i) q^{17} +(-0.794277 - 0.458576i) q^{19} +(3.69541 - 5.00764i) q^{21} +(5.68258 - 4.76825i) q^{23} +(-3.90967 + 1.42300i) q^{25} +(5.10542 - 0.966817i) q^{27} +(3.02791 + 8.31913i) q^{29} +(3.84549 + 4.58288i) q^{31} +(-0.352746 + 3.13216i) q^{33} +(1.64603 - 2.85100i) q^{35} +(-3.15380 - 5.46255i) q^{37} +(-7.39948 - 2.18138i) q^{39} +(-1.16283 + 3.19486i) q^{41} +(-0.919062 - 0.162055i) q^{43} +(2.62640 - 0.810434i) q^{45} +(-0.999632 - 0.838791i) q^{47} +(-1.02641 - 5.82105i) q^{49} +(-6.53538 + 4.33789i) q^{51} +2.53299i q^{53} +1.66728i q^{55} +(-1.42257 - 0.706963i) q^{57} +(2.13801 + 12.1253i) q^{59} +(-2.66045 - 2.23238i) q^{61} +(5.84744 - 9.05567i) q^{63} +(-4.01863 - 0.708592i) q^{65} +(-0.986708 + 2.71096i) q^{67} +(9.30847 - 8.85641i) q^{69} +(-4.44194 - 7.69367i) q^{71} +(-0.528554 + 0.915482i) q^{73} +(-6.60491 + 2.88210i) q^{75} +(4.20306 + 5.00901i) q^{77} +(-3.99634 - 10.9798i) q^{79} +(8.72124 - 2.22262i) q^{81} +(-15.7994 + 5.75052i) q^{83} +(-3.17849 + 2.66707i) q^{85} +(6.13263 + 14.0542i) q^{87} +(13.1352 + 7.58359i) q^{89} +(-13.8594 + 8.00174i) q^{91} +(7.14251 + 7.50709i) q^{93} +(-0.789617 - 0.287397i) q^{95} +(1.86037 - 10.5507i) q^{97} +(-0.271563 + 5.45260i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 6 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 6 q^{5} - 12 q^{9} - 36 q^{21} + 18 q^{25} - 24 q^{29} - 36 q^{33} - 18 q^{41} - 18 q^{45} + 42 q^{65} + 54 q^{69} - 18 q^{73} + 90 q^{77} + 36 q^{81} + 72 q^{85} + 72 q^{89} + 54 q^{93} + 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{7}{18}\right)\)

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
\(3\) 1.72868 0.107984i 0.998055 0.0623444i
\(4\) 0 0
\(5\) 0.902279 0.159096i 0.403511 0.0711499i 0.0317906 0.999495i \(-0.489879\pi\)
0.371721 + 0.928345i \(0.378768\pi\)
\(6\) 0 0
\(7\) 2.30964 2.75253i 0.872963 1.04036i −0.125869 0.992047i \(-0.540172\pi\)
0.998832 0.0483102i \(-0.0153836\pi\)
\(8\) 0 0
\(9\) 2.97668 0.373339i 0.992226 0.124446i
\(10\) 0 0
\(11\) −0.316002 + 1.79214i −0.0952783 + 0.540350i 0.899383 + 0.437161i \(0.144016\pi\)
−0.994662 + 0.103189i \(0.967095\pi\)
\(12\) 0 0
\(13\) −4.18526 1.52331i −1.16078 0.422491i −0.311406 0.950277i \(-0.600800\pi\)
−0.849377 + 0.527786i \(0.823022\pi\)
\(14\) 0 0
\(15\) 1.54257 0.372458i 0.398290 0.0961682i
\(16\) 0 0
\(17\) −3.92200 + 2.26437i −0.951226 + 0.549190i −0.893461 0.449140i \(-0.851731\pi\)
−0.0577643 + 0.998330i \(0.518397\pi\)
\(18\) 0 0
\(19\) −0.794277 0.458576i −0.182220 0.105205i 0.406115 0.913822i \(-0.366883\pi\)
−0.588335 + 0.808617i \(0.700216\pi\)
\(20\) 0 0
\(21\) 3.69541 5.00764i 0.806405 1.09276i
\(22\) 0 0
\(23\) 5.68258 4.76825i 1.18490 0.994248i 0.184965 0.982745i \(-0.440783\pi\)
0.999934 0.0115034i \(-0.00366171\pi\)
\(24\) 0 0
\(25\) −3.90967 + 1.42300i −0.781934 + 0.284601i
\(26\) 0 0
\(27\) 5.10542 0.966817i 0.982538 0.186064i
\(28\) 0 0
\(29\) 3.02791 + 8.31913i 0.562270 + 1.54482i 0.816301 + 0.577627i \(0.196021\pi\)
−0.254031 + 0.967196i \(0.581757\pi\)
\(30\) 0 0
\(31\) 3.84549 + 4.58288i 0.690671 + 0.823110i 0.991437 0.130588i \(-0.0416866\pi\)
−0.300766 + 0.953698i \(0.597242\pi\)
\(32\) 0 0
\(33\) −0.352746 + 3.13216i −0.0614052 + 0.545239i
\(34\) 0 0
\(35\) 1.64603 2.85100i 0.278229 0.481907i
\(36\) 0 0
\(37\) −3.15380 5.46255i −0.518482 0.898037i −0.999769 0.0214743i \(-0.993164\pi\)
0.481287 0.876563i \(-0.340169\pi\)
\(38\) 0 0
\(39\) −7.39948 2.18138i −1.18487 0.349300i
\(40\) 0 0
\(41\) −1.16283 + 3.19486i −0.181604 + 0.498953i −0.996773 0.0802698i \(-0.974422\pi\)
0.815169 + 0.579223i \(0.196644\pi\)
\(42\) 0 0
\(43\) −0.919062 0.162055i −0.140156 0.0247132i 0.103130 0.994668i \(-0.467114\pi\)
−0.243286 + 0.969955i \(0.578225\pi\)
\(44\) 0 0
\(45\) 2.62640 0.810434i 0.391520 0.120812i
\(46\) 0 0
\(47\) −0.999632 0.838791i −0.145811 0.122350i 0.566964 0.823742i \(-0.308118\pi\)
−0.712776 + 0.701392i \(0.752562\pi\)
\(48\) 0 0
\(49\) −1.02641 5.82105i −0.146630 0.831579i
\(50\) 0 0
\(51\) −6.53538 + 4.33789i −0.915136 + 0.607426i
\(52\) 0 0
\(53\) 2.53299i 0.347934i 0.984752 + 0.173967i \(0.0556585\pi\)
−0.984752 + 0.173967i \(0.944341\pi\)
\(54\) 0 0
\(55\) 1.66728i 0.224816i
\(56\) 0 0
\(57\) −1.42257 0.706963i −0.188424 0.0936395i
\(58\) 0 0
\(59\) 2.13801 + 12.1253i 0.278346 + 1.57858i 0.728130 + 0.685439i \(0.240390\pi\)
−0.449785 + 0.893137i \(0.648499\pi\)
\(60\) 0 0
\(61\) −2.66045 2.23238i −0.340636 0.285827i 0.456381 0.889784i \(-0.349145\pi\)
−0.797017 + 0.603957i \(0.793590\pi\)
\(62\) 0 0
\(63\) 5.84744 9.05567i 0.736709 1.14091i
\(64\) 0 0
\(65\) −4.01863 0.708592i −0.498449 0.0878901i
\(66\) 0 0
\(67\) −0.986708 + 2.71096i −0.120546 + 0.331196i −0.985259 0.171069i \(-0.945278\pi\)
0.864713 + 0.502266i \(0.167500\pi\)
\(68\) 0 0
\(69\) 9.30847 8.85641i 1.12061 1.06619i
\(70\) 0 0
\(71\) −4.44194 7.69367i −0.527161 0.913070i −0.999499 0.0316527i \(-0.989923\pi\)
0.472337 0.881418i \(-0.343410\pi\)
\(72\) 0 0
\(73\) −0.528554 + 0.915482i −0.0618625 + 0.107149i −0.895298 0.445468i \(-0.853037\pi\)
0.833435 + 0.552617i \(0.186371\pi\)
\(74\) 0 0
\(75\) −6.60491 + 2.88210i −0.762669 + 0.332796i
\(76\) 0 0
\(77\) 4.20306 + 5.00901i 0.478983 + 0.570829i
\(78\) 0 0
\(79\) −3.99634 10.9798i −0.449623 1.23533i −0.932987 0.359910i \(-0.882807\pi\)
0.483364 0.875420i \(-0.339415\pi\)
\(80\) 0 0
\(81\) 8.72124 2.22262i 0.969026 0.246958i
\(82\) 0 0
\(83\) −15.7994 + 5.75052i −1.73421 + 0.631201i −0.998916 0.0465461i \(-0.985179\pi\)
−0.735295 + 0.677747i \(0.762956\pi\)
\(84\) 0 0
\(85\) −3.17849 + 2.66707i −0.344755 + 0.289284i
\(86\) 0 0
\(87\) 6.13263 + 14.0542i 0.657487 + 1.50676i
\(88\) 0 0
\(89\) 13.1352 + 7.58359i 1.39232 + 0.803859i 0.993572 0.113200i \(-0.0361102\pi\)
0.398752 + 0.917059i \(0.369444\pi\)
\(90\) 0 0
\(91\) −13.8594 + 8.00174i −1.45286 + 0.838811i
\(92\) 0 0
\(93\) 7.14251 + 7.50709i 0.740644 + 0.778449i
\(94\) 0 0
\(95\) −0.789617 0.287397i −0.0810130 0.0294863i
\(96\) 0 0
\(97\) 1.86037 10.5507i 0.188892 1.07126i −0.731960 0.681347i \(-0.761394\pi\)
0.920852 0.389912i \(-0.127495\pi\)
\(98\) 0 0
\(99\) −0.271563 + 5.45260i −0.0272931 + 0.548007i
\(100\) 0 0
\(101\) −2.54770 + 3.03623i −0.253506 + 0.302116i −0.877756 0.479108i \(-0.840960\pi\)
0.624250 + 0.781225i \(0.285405\pi\)
\(102\) 0 0
\(103\) 2.32334 0.409667i 0.228925 0.0403657i −0.0580087 0.998316i \(-0.518475\pi\)
0.286934 + 0.957950i \(0.407364\pi\)
\(104\) 0 0
\(105\) 2.53759 5.10622i 0.247644 0.498316i
\(106\) 0 0
\(107\) −7.73225 −0.747504 −0.373752 0.927529i \(-0.621929\pi\)
−0.373752 + 0.927529i \(0.621929\pi\)
\(108\) 0 0
\(109\) −15.5672 −1.49107 −0.745533 0.666469i \(-0.767805\pi\)
−0.745533 + 0.666469i \(0.767805\pi\)
\(110\) 0 0
\(111\) −6.04179 9.10244i −0.573461 0.863966i
\(112\) 0 0
\(113\) −9.66087 + 1.70347i −0.908818 + 0.160249i −0.608467 0.793579i \(-0.708215\pi\)
−0.300352 + 0.953829i \(0.597104\pi\)
\(114\) 0 0
\(115\) 4.36866 5.20636i 0.407379 0.485496i
\(116\) 0 0
\(117\) −13.0269 2.97189i −1.20434 0.274751i
\(118\) 0 0
\(119\) −2.82569 + 16.0253i −0.259031 + 1.46904i
\(120\) 0 0
\(121\) 7.22471 + 2.62958i 0.656792 + 0.239053i
\(122\) 0 0
\(123\) −1.66518 + 5.64846i −0.150144 + 0.509305i
\(124\) 0 0
\(125\) −7.26847 + 4.19645i −0.650112 + 0.375342i
\(126\) 0 0
\(127\) −16.7402 9.66498i −1.48546 0.857628i −0.485593 0.874185i \(-0.661396\pi\)
−0.999863 + 0.0165567i \(0.994730\pi\)
\(128\) 0 0
\(129\) −1.60627 0.180899i −0.141424 0.0159272i
\(130\) 0 0
\(131\) 13.4142 11.2558i 1.17200 0.983427i 0.172004 0.985096i \(-0.444976\pi\)
0.999999 + 0.00166932i \(0.000531362\pi\)
\(132\) 0 0
\(133\) −3.09674 + 1.12712i −0.268521 + 0.0977338i
\(134\) 0 0
\(135\) 4.45269 1.68459i 0.383227 0.144986i
\(136\) 0 0
\(137\) −5.50764 15.1321i −0.470549 1.29282i −0.917312 0.398170i \(-0.869645\pi\)
0.446762 0.894653i \(-0.352577\pi\)
\(138\) 0 0
\(139\) 10.6052 + 12.6388i 0.899525 + 1.07201i 0.997048 + 0.0767785i \(0.0244634\pi\)
−0.0975235 + 0.995233i \(0.531092\pi\)
\(140\) 0 0
\(141\) −1.81862 1.34206i −0.153156 0.113022i
\(142\) 0 0
\(143\) 4.05254 7.01921i 0.338890 0.586975i
\(144\) 0 0
\(145\) 4.05556 + 7.02444i 0.336796 + 0.583348i
\(146\) 0 0
\(147\) −2.40291 9.95191i −0.198189 0.820820i
\(148\) 0 0
\(149\) −2.32731 + 6.39423i −0.190661 + 0.523836i −0.997783 0.0665481i \(-0.978801\pi\)
0.807123 + 0.590384i \(0.201024\pi\)
\(150\) 0 0
\(151\) 14.6770 + 2.58795i 1.19440 + 0.210604i 0.735275 0.677769i \(-0.237053\pi\)
0.459121 + 0.888374i \(0.348164\pi\)
\(152\) 0 0
\(153\) −10.8292 + 8.20454i −0.875487 + 0.663298i
\(154\) 0 0
\(155\) 4.19882 + 3.52323i 0.337258 + 0.282993i
\(156\) 0 0
\(157\) 1.99263 + 11.3007i 0.159029 + 0.901897i 0.955009 + 0.296575i \(0.0958446\pi\)
−0.795981 + 0.605322i \(0.793044\pi\)
\(158\) 0 0
\(159\) 0.273522 + 4.37874i 0.0216917 + 0.347257i
\(160\) 0 0
\(161\) 26.6544i 2.10066i
\(162\) 0 0
\(163\) 3.48604i 0.273047i 0.990637 + 0.136524i \(0.0435930\pi\)
−0.990637 + 0.136524i \(0.956407\pi\)
\(164\) 0 0
\(165\) 0.180039 + 2.88220i 0.0140160 + 0.224379i
\(166\) 0 0
\(167\) 0.759638 + 4.30812i 0.0587826 + 0.333372i 0.999990 0.00445650i \(-0.00141855\pi\)
−0.941208 + 0.337829i \(0.890307\pi\)
\(168\) 0 0
\(169\) 5.23738 + 4.39469i 0.402876 + 0.338053i
\(170\) 0 0
\(171\) −2.53551 1.06850i −0.193895 0.0817102i
\(172\) 0 0
\(173\) 10.3347 + 1.82229i 0.785732 + 0.138546i 0.552098 0.833779i \(-0.313827\pi\)
0.233634 + 0.972325i \(0.424938\pi\)
\(174\) 0 0
\(175\) −5.11309 + 14.0481i −0.386513 + 1.06194i
\(176\) 0 0
\(177\) 5.00527 + 20.7299i 0.376219 + 1.55815i
\(178\) 0 0
\(179\) −12.6163 21.8521i −0.942988 1.63330i −0.759731 0.650238i \(-0.774669\pi\)
−0.183257 0.983065i \(-0.558664\pi\)
\(180\) 0 0
\(181\) 1.92076 3.32686i 0.142769 0.247283i −0.785769 0.618520i \(-0.787733\pi\)
0.928538 + 0.371236i \(0.121066\pi\)
\(182\) 0 0
\(183\) −4.84013 3.57179i −0.357793 0.264034i
\(184\) 0 0
\(185\) −3.71468 4.42698i −0.273109 0.325478i
\(186\) 0 0
\(187\) −2.81870 7.74432i −0.206124 0.566321i
\(188\) 0 0
\(189\) 9.13050 16.2858i 0.664146 1.18462i
\(190\) 0 0
\(191\) 9.28905 3.38094i 0.672132 0.244636i 0.0166663 0.999861i \(-0.494695\pi\)
0.655465 + 0.755225i \(0.272472\pi\)
\(192\) 0 0
\(193\) 7.95376 6.67400i 0.572524 0.480405i −0.309958 0.950750i \(-0.600315\pi\)
0.882482 + 0.470345i \(0.155871\pi\)
\(194\) 0 0
\(195\) −7.02344 0.790984i −0.502959 0.0566436i
\(196\) 0 0
\(197\) 11.0769 + 6.39528i 0.789200 + 0.455645i 0.839681 0.543080i \(-0.182742\pi\)
−0.0504809 + 0.998725i \(0.516075\pi\)
\(198\) 0 0
\(199\) −6.79765 + 3.92463i −0.481873 + 0.278209i −0.721197 0.692730i \(-0.756408\pi\)
0.239324 + 0.970940i \(0.423074\pi\)
\(200\) 0 0
\(201\) −1.41296 + 4.79293i −0.0996628 + 0.338067i
\(202\) 0 0
\(203\) 29.8920 + 10.8798i 2.09801 + 0.763613i
\(204\) 0 0
\(205\) −0.540911 + 3.06766i −0.0377788 + 0.214254i
\(206\) 0 0
\(207\) 15.1350 16.3151i 1.05196 1.13398i
\(208\) 0 0
\(209\) 1.07283 1.27854i 0.0742089 0.0884387i
\(210\) 0 0
\(211\) 28.3624 5.00106i 1.95255 0.344287i 0.953467 0.301497i \(-0.0974862\pi\)
0.999084 0.0427907i \(-0.0136249\pi\)
\(212\) 0 0
\(213\) −8.50949 12.8202i −0.583061 0.878429i
\(214\) 0 0
\(215\) −0.855033 −0.0583127
\(216\) 0 0
\(217\) 21.4962 1.45926
\(218\) 0 0
\(219\) −0.814844 + 1.63965i −0.0550620 + 0.110797i
\(220\) 0 0
\(221\) 19.8640 3.50255i 1.33620 0.235607i
\(222\) 0 0
\(223\) 15.0617 17.9499i 1.00861 1.20201i 0.0293149 0.999570i \(-0.490667\pi\)
0.979294 0.202443i \(-0.0648881\pi\)
\(224\) 0 0
\(225\) −11.1066 + 5.69545i −0.740438 + 0.379697i
\(226\) 0 0
\(227\) −0.871006 + 4.93972i −0.0578107 + 0.327861i −0.999973 0.00729519i \(-0.997678\pi\)
0.942163 + 0.335156i \(0.108789\pi\)
\(228\) 0 0
\(229\) −9.01670 3.28181i −0.595840 0.216868i 0.0264559 0.999650i \(-0.491578\pi\)
−0.622296 + 0.782782i \(0.713800\pi\)
\(230\) 0 0
\(231\) 7.80664 + 8.20512i 0.513639 + 0.539857i
\(232\) 0 0
\(233\) −18.7009 + 10.7969i −1.22513 + 0.707332i −0.966008 0.258512i \(-0.916768\pi\)
−0.259126 + 0.965843i \(0.583435\pi\)
\(234\) 0 0
\(235\) −1.03540 0.597786i −0.0675417 0.0389952i
\(236\) 0 0
\(237\) −8.09404 18.5491i −0.525764 1.20490i
\(238\) 0 0
\(239\) −8.27842 + 6.94642i −0.535486 + 0.449326i −0.869991 0.493068i \(-0.835875\pi\)
0.334505 + 0.942394i \(0.391431\pi\)
\(240\) 0 0
\(241\) 12.1965 4.43917i 0.785646 0.285952i 0.0821214 0.996622i \(-0.473830\pi\)
0.703525 + 0.710670i \(0.251608\pi\)
\(242\) 0 0
\(243\) 14.8362 4.78395i 0.951745 0.306891i
\(244\) 0 0
\(245\) −1.85221 5.08891i −0.118334 0.325119i
\(246\) 0 0
\(247\) 2.62571 + 3.12919i 0.167070 + 0.199106i
\(248\) 0 0
\(249\) −26.6912 + 11.6469i −1.69149 + 0.738092i
\(250\) 0 0
\(251\) 9.36454 16.2199i 0.591085 1.02379i −0.403002 0.915199i \(-0.632033\pi\)
0.994087 0.108590i \(-0.0346335\pi\)
\(252\) 0 0
\(253\) 6.74965 + 11.6907i 0.424347 + 0.734991i
\(254\) 0 0
\(255\) −5.20659 + 4.95374i −0.326050 + 0.310215i
\(256\) 0 0
\(257\) 9.97322 27.4012i 0.622113 1.70924i −0.0796439 0.996823i \(-0.525378\pi\)
0.701756 0.712417i \(-0.252399\pi\)
\(258\) 0 0
\(259\) −22.3200 3.93561i −1.38689 0.244547i
\(260\) 0 0
\(261\) 12.1190 + 23.6329i 0.750146 + 1.46284i
\(262\) 0 0
\(263\) −6.74060 5.65604i −0.415643 0.348766i 0.410859 0.911699i \(-0.365229\pi\)
−0.826503 + 0.562932i \(0.809673\pi\)
\(264\) 0 0
\(265\) 0.402989 + 2.28547i 0.0247554 + 0.140395i
\(266\) 0 0
\(267\) 23.5254 + 11.6912i 1.43973 + 0.715491i
\(268\) 0 0
\(269\) 20.0940i 1.22515i −0.790412 0.612575i \(-0.790134\pi\)
0.790412 0.612575i \(-0.209866\pi\)
\(270\) 0 0
\(271\) 9.14240i 0.555361i −0.960673 0.277681i \(-0.910434\pi\)
0.960673 0.277681i \(-0.0895657\pi\)
\(272\) 0 0
\(273\) −23.0945 + 15.3291i −1.39774 + 0.927757i
\(274\) 0 0
\(275\) −1.31475 7.45634i −0.0792826 0.449634i
\(276\) 0 0
\(277\) 11.2006 + 9.39839i 0.672977 + 0.564694i 0.913945 0.405839i \(-0.133021\pi\)
−0.240968 + 0.970533i \(0.577465\pi\)
\(278\) 0 0
\(279\) 13.1578 + 12.2061i 0.787735 + 0.730760i
\(280\) 0 0
\(281\) 4.33423 + 0.764241i 0.258558 + 0.0455908i 0.301425 0.953490i \(-0.402538\pi\)
−0.0428663 + 0.999081i \(0.513649\pi\)
\(282\) 0 0
\(283\) 4.34262 11.9313i 0.258142 0.709239i −0.741140 0.671351i \(-0.765715\pi\)
0.999282 0.0378888i \(-0.0120633\pi\)
\(284\) 0 0
\(285\) −1.39603 0.411552i −0.0826937 0.0243782i
\(286\) 0 0
\(287\) 6.10820 + 10.5797i 0.360556 + 0.624501i
\(288\) 0 0
\(289\) 1.75474 3.03931i 0.103220 0.178783i
\(290\) 0 0
\(291\) 2.07669 18.4397i 0.121737 1.08095i
\(292\) 0 0
\(293\) −13.6489 16.2661i −0.797376 0.950276i 0.202201 0.979344i \(-0.435190\pi\)
−0.999577 + 0.0290682i \(0.990746\pi\)
\(294\) 0 0
\(295\) 3.85817 + 10.6002i 0.224631 + 0.617169i
\(296\) 0 0
\(297\) 0.119346 + 9.45513i 0.00692517 + 0.548642i
\(298\) 0 0
\(299\) −31.0466 + 11.3000i −1.79547 + 0.653498i
\(300\) 0 0
\(301\) −2.56877 + 2.15545i −0.148061 + 0.124238i
\(302\) 0 0
\(303\) −4.07630 + 5.52379i −0.234177 + 0.317333i
\(304\) 0 0
\(305\) −2.75563 1.59096i −0.157787 0.0910983i
\(306\) 0 0
\(307\) −1.24710 + 0.720015i −0.0711759 + 0.0410934i −0.535166 0.844747i \(-0.679751\pi\)
0.463990 + 0.885841i \(0.346417\pi\)
\(308\) 0 0
\(309\) 3.97207 0.959066i 0.225963 0.0545594i
\(310\) 0 0
\(311\) 8.32511 + 3.03009i 0.472073 + 0.171821i 0.567091 0.823655i \(-0.308069\pi\)
−0.0950180 + 0.995476i \(0.530291\pi\)
\(312\) 0 0
\(313\) −5.38779 + 30.5557i −0.304536 + 1.72711i 0.321145 + 0.947030i \(0.395932\pi\)
−0.625681 + 0.780079i \(0.715179\pi\)
\(314\) 0 0
\(315\) 3.83530 9.10104i 0.216095 0.512785i
\(316\) 0 0
\(317\) 1.05369 1.25574i 0.0591813 0.0705295i −0.735642 0.677371i \(-0.763119\pi\)
0.794823 + 0.606841i \(0.207564\pi\)
\(318\) 0 0
\(319\) −15.8659 + 2.79758i −0.888318 + 0.156634i
\(320\) 0 0
\(321\) −13.3666 + 0.834957i −0.746050 + 0.0466027i
\(322\) 0 0
\(323\) 4.15354 0.231109
\(324\) 0 0
\(325\) 18.5307 1.02790
\(326\) 0 0
\(327\) −26.9107 + 1.68100i −1.48817 + 0.0929597i
\(328\) 0 0
\(329\) −4.61759 + 0.814205i −0.254576 + 0.0448886i
\(330\) 0 0
\(331\) 4.17455 4.97503i 0.229454 0.273452i −0.639017 0.769193i \(-0.720659\pi\)
0.868471 + 0.495740i \(0.165103\pi\)
\(332\) 0 0
\(333\) −11.4272 15.0828i −0.626209 0.826533i
\(334\) 0 0
\(335\) −0.458983 + 2.60302i −0.0250769 + 0.142218i
\(336\) 0 0
\(337\) 19.8793 + 7.23548i 1.08290 + 0.394142i 0.820985 0.570950i \(-0.193425\pi\)
0.261911 + 0.965092i \(0.415647\pi\)
\(338\) 0 0
\(339\) −16.5166 + 3.98798i −0.897060 + 0.216597i
\(340\) 0 0
\(341\) −9.42834 + 5.44346i −0.510573 + 0.294780i
\(342\) 0 0
\(343\) 3.38918 + 1.95675i 0.182999 + 0.105654i
\(344\) 0 0
\(345\) 6.98981 9.47189i 0.376319 0.509949i
\(346\) 0 0
\(347\) −7.60026 + 6.37737i −0.408003 + 0.342355i −0.823577 0.567204i \(-0.808025\pi\)
0.415574 + 0.909559i \(0.363581\pi\)
\(348\) 0 0
\(349\) −10.6407 + 3.87290i −0.569583 + 0.207311i −0.610726 0.791842i \(-0.709122\pi\)
0.0411427 + 0.999153i \(0.486900\pi\)
\(350\) 0 0
\(351\) −22.8403 3.73075i −1.21912 0.199133i
\(352\) 0 0
\(353\) 3.15842 + 8.67769i 0.168106 + 0.461867i 0.994927 0.100598i \(-0.0320755\pi\)
−0.826821 + 0.562464i \(0.809853\pi\)
\(354\) 0 0
\(355\) −5.23190 6.23514i −0.277680 0.330927i
\(356\) 0 0
\(357\) −3.15425 + 28.0078i −0.166941 + 1.48233i
\(358\) 0 0
\(359\) −8.31565 + 14.4031i −0.438883 + 0.760168i −0.997604 0.0691878i \(-0.977959\pi\)
0.558720 + 0.829356i \(0.311293\pi\)
\(360\) 0 0
\(361\) −9.07942 15.7260i −0.477864 0.827685i
\(362\) 0 0
\(363\) 12.7732 + 3.76556i 0.670418 + 0.197640i
\(364\) 0 0
\(365\) −0.331253 + 0.910110i −0.0173386 + 0.0476374i
\(366\) 0 0
\(367\) 15.0549 + 2.65459i 0.785860 + 0.138568i 0.552158 0.833740i \(-0.313805\pi\)
0.233703 + 0.972308i \(0.424916\pi\)
\(368\) 0 0
\(369\) −2.26862 + 9.94420i −0.118100 + 0.517675i
\(370\) 0 0
\(371\) 6.97213 + 5.85032i 0.361975 + 0.303733i
\(372\) 0 0
\(373\) −3.27227 18.5580i −0.169432 0.960896i −0.944376 0.328866i \(-0.893333\pi\)
0.774945 0.632029i \(-0.217778\pi\)
\(374\) 0 0
\(375\) −12.1117 + 8.03920i −0.625446 + 0.415143i
\(376\) 0 0
\(377\) 39.4302i 2.03076i
\(378\) 0 0
\(379\) 27.7620i 1.42604i 0.701144 + 0.713019i \(0.252673\pi\)
−0.701144 + 0.713019i \(0.747327\pi\)
\(380\) 0 0
\(381\) −29.9822 14.9000i −1.53603 0.763350i
\(382\) 0 0
\(383\) −5.46348 30.9849i −0.279171 1.58325i −0.725394 0.688333i \(-0.758343\pi\)
0.446224 0.894921i \(-0.352769\pi\)
\(384\) 0 0
\(385\) 4.58924 + 3.85083i 0.233889 + 0.196256i
\(386\) 0 0
\(387\) −2.79626 0.139265i −0.142142 0.00707926i
\(388\) 0 0
\(389\) −9.58341 1.68981i −0.485898 0.0856769i −0.0746678 0.997208i \(-0.523790\pi\)
−0.411230 + 0.911532i \(0.634901\pi\)
\(390\) 0 0
\(391\) −11.4900 + 31.5685i −0.581075 + 1.59649i
\(392\) 0 0
\(393\) 21.9734 20.9063i 1.10841 1.05458i
\(394\) 0 0
\(395\) −5.35266 9.27108i −0.269322 0.466479i
\(396\) 0 0
\(397\) −8.72266 + 15.1081i −0.437778 + 0.758253i −0.997518 0.0704151i \(-0.977568\pi\)
0.559740 + 0.828668i \(0.310901\pi\)
\(398\) 0 0
\(399\) −5.23156 + 2.28283i −0.261906 + 0.114284i
\(400\) 0 0
\(401\) 6.59597 + 7.86077i 0.329387 + 0.392548i 0.905167 0.425057i \(-0.139746\pi\)
−0.575780 + 0.817605i \(0.695301\pi\)
\(402\) 0 0
\(403\) −9.11325 25.0385i −0.453963 1.24725i
\(404\) 0 0
\(405\) 7.51537 3.39294i 0.373442 0.168596i
\(406\) 0 0
\(407\) 10.7863 3.92587i 0.534655 0.194598i
\(408\) 0 0
\(409\) −5.65802 + 4.74764i −0.279771 + 0.234756i −0.771865 0.635786i \(-0.780676\pi\)
0.492094 + 0.870542i \(0.336232\pi\)
\(410\) 0 0
\(411\) −11.1550 25.5639i −0.550234 1.26097i
\(412\) 0 0
\(413\) 38.3132 + 22.1201i 1.88527 + 1.08846i
\(414\) 0 0
\(415\) −13.3406 + 7.70219i −0.654864 + 0.378086i
\(416\) 0 0
\(417\) 19.6979 + 20.7033i 0.964609 + 1.01385i
\(418\) 0 0
\(419\) 15.7103 + 5.71807i 0.767496 + 0.279346i 0.695949 0.718091i \(-0.254984\pi\)
0.0715474 + 0.997437i \(0.477206\pi\)
\(420\) 0 0
\(421\) 1.99066 11.2896i 0.0970187 0.550221i −0.897091 0.441845i \(-0.854324\pi\)
0.994110 0.108375i \(-0.0345648\pi\)
\(422\) 0 0
\(423\) −3.28874 2.12361i −0.159904 0.103253i
\(424\) 0 0
\(425\) 12.1115 14.4340i 0.587495 0.700150i
\(426\) 0 0
\(427\) −12.2894 + 2.16695i −0.594725 + 0.104866i
\(428\) 0 0
\(429\) 6.24759 12.5716i 0.301637 0.606961i
\(430\) 0 0
\(431\) −16.5555 −0.797449 −0.398724 0.917071i \(-0.630547\pi\)
−0.398724 + 0.917071i \(0.630547\pi\)
\(432\) 0 0
\(433\) −4.88427 −0.234723 −0.117361 0.993089i \(-0.537444\pi\)
−0.117361 + 0.993089i \(0.537444\pi\)
\(434\) 0 0
\(435\) 7.76930 + 11.7051i 0.372509 + 0.561216i
\(436\) 0 0
\(437\) −6.70014 + 1.18142i −0.320511 + 0.0565148i
\(438\) 0 0
\(439\) −19.5417 + 23.2889i −0.932677 + 1.11152i 0.0608757 + 0.998145i \(0.480611\pi\)
−0.993552 + 0.113375i \(0.963834\pi\)
\(440\) 0 0
\(441\) −5.22851 16.9442i −0.248977 0.806867i
\(442\) 0 0
\(443\) 0.340783 1.93268i 0.0161911 0.0918243i −0.975641 0.219372i \(-0.929599\pi\)
0.991832 + 0.127547i \(0.0407104\pi\)
\(444\) 0 0
\(445\) 13.0581 + 4.75276i 0.619013 + 0.225302i
\(446\) 0 0
\(447\) −3.33270 + 11.3049i −0.157631 + 0.534703i
\(448\) 0 0
\(449\) −25.2445 + 14.5749i −1.19136 + 0.687834i −0.958616 0.284704i \(-0.908105\pi\)
−0.232747 + 0.972537i \(0.574771\pi\)
\(450\) 0 0
\(451\) −5.35817 3.09354i −0.252307 0.145669i
\(452\) 0 0
\(453\) 25.6513 + 2.88886i 1.20520 + 0.135731i
\(454\) 0 0
\(455\) −11.2320 + 9.42478i −0.526565 + 0.441841i
\(456\) 0 0
\(457\) 28.2232 10.2724i 1.32023 0.480523i 0.416695 0.909046i \(-0.363188\pi\)
0.903531 + 0.428523i \(0.140966\pi\)
\(458\) 0 0
\(459\) −17.8342 + 15.3524i −0.832431 + 0.716589i
\(460\) 0 0
\(461\) −0.669179 1.83856i −0.0311668 0.0856301i 0.923133 0.384480i \(-0.125619\pi\)
−0.954300 + 0.298850i \(0.903397\pi\)
\(462\) 0 0
\(463\) 13.0605 + 15.5649i 0.606974 + 0.723363i 0.978772 0.204950i \(-0.0657033\pi\)
−0.371799 + 0.928313i \(0.621259\pi\)
\(464\) 0 0
\(465\) 7.63888 + 5.63714i 0.354245 + 0.261416i
\(466\) 0 0
\(467\) −0.632284 + 1.09515i −0.0292586 + 0.0506774i −0.880284 0.474447i \(-0.842648\pi\)
0.851025 + 0.525125i \(0.175981\pi\)
\(468\) 0 0
\(469\) 5.18304 + 8.97729i 0.239330 + 0.414533i
\(470\) 0 0
\(471\) 4.66491 + 19.3202i 0.214948 + 0.890228i
\(472\) 0 0
\(473\) 0.580852 1.59588i 0.0267076 0.0733785i
\(474\) 0 0
\(475\) 3.75791 + 0.662622i 0.172425 + 0.0304032i
\(476\) 0 0
\(477\) 0.945665 + 7.53991i 0.0432990 + 0.345229i
\(478\) 0 0
\(479\) −5.46294 4.58395i −0.249608 0.209446i 0.509395 0.860533i \(-0.329869\pi\)
−0.759004 + 0.651086i \(0.774314\pi\)
\(480\) 0 0
\(481\) 4.87834 + 27.6664i 0.222433 + 1.26148i
\(482\) 0 0
\(483\) −2.87824 46.0770i −0.130964 2.09657i
\(484\) 0 0
\(485\) 9.81563i 0.445705i
\(486\) 0 0
\(487\) 35.3831i 1.60336i −0.597752 0.801681i \(-0.703939\pi\)
0.597752 0.801681i \(-0.296061\pi\)
\(488\) 0 0
\(489\) 0.376435 + 6.02625i 0.0170230 + 0.272516i
\(490\) 0 0
\(491\) 0.202915 + 1.15079i 0.00915741 + 0.0519343i 0.989044 0.147621i \(-0.0471615\pi\)
−0.979887 + 0.199555i \(0.936050\pi\)
\(492\) 0 0
\(493\) −30.7131 25.7713i −1.38325 1.16068i
\(494\) 0 0
\(495\) 0.622462 + 4.96297i 0.0279776 + 0.223069i
\(496\) 0 0
\(497\) −31.4363 5.54307i −1.41011 0.248641i
\(498\) 0 0
\(499\) 9.12958 25.0833i 0.408696 1.12288i −0.549180 0.835704i \(-0.685060\pi\)
0.957877 0.287180i \(-0.0927178\pi\)
\(500\) 0 0
\(501\) 1.77838 + 7.36534i 0.0794521 + 0.329059i
\(502\) 0 0
\(503\) 11.9257 + 20.6560i 0.531742 + 0.921005i 0.999313 + 0.0370494i \(0.0117959\pi\)
−0.467571 + 0.883956i \(0.654871\pi\)
\(504\) 0 0
\(505\) −1.81568 + 3.14486i −0.0807969 + 0.139944i
\(506\) 0 0
\(507\) 9.52832 + 7.03146i 0.423168 + 0.312278i
\(508\) 0 0
\(509\) 13.0480 + 15.5500i 0.578343 + 0.689242i 0.973321 0.229449i \(-0.0736923\pi\)
−0.394978 + 0.918691i \(0.629248\pi\)
\(510\) 0 0
\(511\) 1.29912 + 3.56929i 0.0574695 + 0.157896i
\(512\) 0 0
\(513\) −4.49847 1.57330i −0.198612 0.0694629i
\(514\) 0 0
\(515\) 2.03112 0.739267i 0.0895018 0.0325760i
\(516\) 0 0
\(517\) 1.81912 1.52642i 0.0800046 0.0671319i
\(518\) 0 0
\(519\) 18.0622 + 2.03417i 0.792841 + 0.0892902i
\(520\) 0 0
\(521\) 28.4047 + 16.3995i 1.24443 + 0.718474i 0.969993 0.243131i \(-0.0781745\pi\)
0.274439 + 0.961604i \(0.411508\pi\)
\(522\) 0 0
\(523\) 30.4619 17.5872i 1.33201 0.769034i 0.346399 0.938087i \(-0.387404\pi\)
0.985607 + 0.169053i \(0.0540711\pi\)
\(524\) 0 0
\(525\) −7.32193 + 24.8368i −0.319555 + 1.08397i
\(526\) 0 0
\(527\) −25.4594 9.26646i −1.10903 0.403653i
\(528\) 0 0
\(529\) 5.56158 31.5413i 0.241808 1.37136i
\(530\) 0 0
\(531\) 10.8910 + 35.2948i 0.472630 + 1.53167i
\(532\) 0 0
\(533\) 9.73354 11.6000i 0.421606 0.502451i
\(534\) 0 0
\(535\) −6.97664 + 1.23017i −0.301626 + 0.0531849i
\(536\) 0 0
\(537\) −24.1693 36.4130i −1.04298 1.57134i
\(538\) 0 0
\(539\) 10.7565 0.463315
\(540\) 0 0
\(541\) −35.1137 −1.50965 −0.754827 0.655924i \(-0.772279\pi\)
−0.754827 + 0.655924i \(0.772279\pi\)
\(542\) 0 0
\(543\) 2.96114 5.95849i 0.127075 0.255703i
\(544\) 0 0
\(545\) −14.0459 + 2.47668i −0.601662 + 0.106089i
\(546\) 0 0
\(547\) −9.15157 + 10.9064i −0.391293 + 0.466324i −0.925345 0.379127i \(-0.876224\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(548\) 0 0
\(549\) −8.75274 5.65184i −0.373558 0.241214i
\(550\) 0 0
\(551\) 1.40995 7.99622i 0.0600659 0.340650i
\(552\) 0 0
\(553\) −39.4524 14.3595i −1.67769 0.610629i
\(554\) 0 0
\(555\) −6.89954 7.25172i −0.292869 0.307818i
\(556\) 0 0
\(557\) −21.0603 + 12.1592i −0.892352 + 0.515200i −0.874711 0.484645i \(-0.838949\pi\)
−0.0176411 + 0.999844i \(0.505616\pi\)
\(558\) 0 0
\(559\) 3.59966 + 2.07826i 0.152249 + 0.0879012i
\(560\) 0 0
\(561\) −5.70890 13.0831i −0.241030 0.552369i
\(562\) 0 0
\(563\) −5.33375 + 4.47555i −0.224791 + 0.188622i −0.748227 0.663443i \(-0.769094\pi\)
0.523436 + 0.852065i \(0.324650\pi\)
\(564\) 0 0
\(565\) −8.44578 + 3.07401i −0.355317 + 0.129325i
\(566\) 0 0
\(567\) 14.0251 29.1389i 0.589000 1.22372i
\(568\) 0 0
\(569\) 6.42509 + 17.6528i 0.269354 + 0.740043i 0.998451 + 0.0556334i \(0.0177178\pi\)
−0.729098 + 0.684410i \(0.760060\pi\)
\(570\) 0 0
\(571\) −17.2113 20.5116i −0.720269 0.858383i 0.274388 0.961619i \(-0.411525\pi\)
−0.994657 + 0.103236i \(0.967080\pi\)
\(572\) 0 0
\(573\) 15.6927 6.84763i 0.655573 0.286064i
\(574\) 0 0
\(575\) −15.4318 + 26.7286i −0.643549 + 1.11466i
\(576\) 0 0
\(577\) −4.84104 8.38493i −0.201535 0.349069i 0.747488 0.664275i \(-0.231260\pi\)
−0.949023 + 0.315206i \(0.897926\pi\)
\(578\) 0 0
\(579\) 13.0288 12.3961i 0.541460 0.515164i
\(580\) 0 0
\(581\) −20.6626 + 56.7699i −0.857228 + 2.35521i
\(582\) 0 0
\(583\) −4.53948 0.800433i −0.188006 0.0331505i
\(584\) 0 0
\(585\) −12.2267 0.608942i −0.505512 0.0251767i
\(586\) 0 0
\(587\) −6.78871 5.69641i −0.280200 0.235116i 0.491846 0.870682i \(-0.336322\pi\)
−0.772046 + 0.635566i \(0.780767\pi\)
\(588\) 0 0
\(589\) −0.952788 5.40353i −0.0392589 0.222648i
\(590\) 0 0
\(591\) 19.8391 + 9.85927i 0.816072 + 0.405556i
\(592\) 0 0
\(593\) 6.46963i 0.265676i 0.991138 + 0.132838i \(0.0424090\pi\)
−0.991138 + 0.132838i \(0.957591\pi\)
\(594\) 0 0
\(595\) 14.9088i 0.611203i
\(596\) 0 0
\(597\) −11.3272 + 7.51846i −0.463590 + 0.307710i
\(598\) 0 0
\(599\) 0.995333 + 5.64481i 0.0406682 + 0.230641i 0.998367 0.0571331i \(-0.0181959\pi\)
−0.957698 + 0.287774i \(0.907085\pi\)
\(600\) 0 0
\(601\) −13.9437 11.7002i −0.568777 0.477260i 0.312463 0.949930i \(-0.398846\pi\)
−0.881240 + 0.472670i \(0.843290\pi\)
\(602\) 0 0
\(603\) −1.92501 + 8.43803i −0.0783923 + 0.343623i
\(604\) 0 0
\(605\) 6.93706 + 1.22319i 0.282032 + 0.0497298i
\(606\) 0 0
\(607\) 12.4668 34.2523i 0.506012 1.39026i −0.379307 0.925271i \(-0.623837\pi\)
0.885318 0.464985i \(-0.153940\pi\)
\(608\) 0 0
\(609\) 52.8486 + 15.5799i 2.14153 + 0.631328i
\(610\) 0 0
\(611\) 2.90599 + 5.03331i 0.117564 + 0.203626i
\(612\) 0 0
\(613\) 9.95224 17.2378i 0.401967 0.696228i −0.591996 0.805941i \(-0.701660\pi\)
0.993963 + 0.109713i \(0.0349932\pi\)
\(614\) 0 0
\(615\) −0.603805 + 5.36141i −0.0243478 + 0.216193i
\(616\) 0 0
\(617\) 10.3239 + 12.3036i 0.415625 + 0.495323i 0.932718 0.360607i \(-0.117430\pi\)
−0.517093 + 0.855929i \(0.672986\pi\)
\(618\) 0 0
\(619\) 11.8156 + 32.4631i 0.474909 + 1.30480i 0.913764 + 0.406245i \(0.133162\pi\)
−0.438855 + 0.898558i \(0.644616\pi\)
\(620\) 0 0
\(621\) 24.4019 29.8379i 0.979214 1.19735i
\(622\) 0 0
\(623\) 51.2115 18.6395i 2.05175 0.746775i
\(624\) 0 0
\(625\) 10.0454 8.42910i 0.401816 0.337164i
\(626\) 0 0
\(627\) 1.71651 2.32604i 0.0685509 0.0928932i
\(628\) 0 0
\(629\) 24.7385 + 14.2828i 0.986387 + 0.569491i
\(630\) 0 0
\(631\) 26.9199 15.5422i 1.07166 0.618725i 0.143028 0.989719i \(-0.454316\pi\)
0.928635 + 0.370994i \(0.120983\pi\)
\(632\) 0 0
\(633\) 48.4896 11.7079i 1.92729 0.465348i
\(634\) 0 0
\(635\) −16.6420 6.05720i −0.660418 0.240373i
\(636\) 0 0
\(637\) −4.57149 + 25.9262i −0.181129 + 1.02723i
\(638\) 0 0
\(639\) −16.0946 21.2432i −0.636692 0.840369i
\(640\) 0 0
\(641\) −23.7440 + 28.2970i −0.937831 + 1.11766i 0.0550422 + 0.998484i \(0.482471\pi\)
−0.992873 + 0.119179i \(0.961974\pi\)
\(642\) 0 0
\(643\) 0.700721 0.123556i 0.0276337 0.00487257i −0.159814 0.987147i \(-0.551090\pi\)
0.187448 + 0.982275i \(0.439978\pi\)
\(644\) 0 0
\(645\) −1.47808 + 0.0923296i −0.0581993 + 0.00363547i
\(646\) 0 0
\(647\) 33.2439 1.30695 0.653476 0.756947i \(-0.273310\pi\)
0.653476 + 0.756947i \(0.273310\pi\)
\(648\) 0 0
\(649\) −22.4058 −0.879504
\(650\) 0 0
\(651\) 37.1601 2.32124i 1.45642 0.0909766i
\(652\) 0 0
\(653\) 5.33170 0.940122i 0.208645 0.0367898i −0.0683484 0.997662i \(-0.521773\pi\)
0.276994 + 0.960872i \(0.410662\pi\)
\(654\) 0 0
\(655\) 10.3126 12.2900i 0.402945 0.480212i
\(656\) 0 0
\(657\) −1.23155 + 2.92242i −0.0480473 + 0.114015i
\(658\) 0 0
\(659\) 3.94718 22.3856i 0.153760 0.872017i −0.806150 0.591711i \(-0.798453\pi\)
0.959911 0.280307i \(-0.0904361\pi\)
\(660\) 0 0
\(661\) 1.00411 + 0.365468i 0.0390555 + 0.0142150i 0.361474 0.932382i \(-0.382274\pi\)
−0.322419 + 0.946597i \(0.604496\pi\)
\(662\) 0 0
\(663\) 33.9603 8.19978i 1.31891 0.318453i
\(664\) 0 0
\(665\) −2.61480 + 1.50966i −0.101398 + 0.0585419i
\(666\) 0 0
\(667\) 56.8740 + 32.8362i 2.20217 + 1.27142i
\(668\) 0 0
\(669\) 24.0987 32.6561i 0.931708 1.26256i
\(670\) 0 0
\(671\) 4.84145 4.06246i 0.186902 0.156829i
\(672\) 0 0
\(673\) −16.4390 + 5.98331i −0.633677 + 0.230640i −0.638831 0.769347i \(-0.720582\pi\)
0.00515388 + 0.999987i \(0.498359\pi\)
\(674\) 0 0
\(675\) −18.5847 + 11.0450i −0.715325 + 0.425120i
\(676\) 0 0
\(677\) −0.164509 0.451985i −0.00632259 0.0173712i 0.936491 0.350691i \(-0.114053\pi\)
−0.942814 + 0.333320i \(0.891831\pi\)
\(678\) 0 0
\(679\) −24.7442 29.4890i −0.949597 1.13169i
\(680\) 0 0
\(681\) −0.972282 + 8.63325i −0.0372579 + 0.330827i
\(682\) 0 0
\(683\) 20.5853 35.6548i 0.787675 1.36429i −0.139713 0.990192i \(-0.544618\pi\)
0.927388 0.374101i \(-0.122049\pi\)
\(684\) 0 0
\(685\) −7.37688 12.7771i −0.281856 0.488189i
\(686\) 0 0
\(687\) −15.9414 4.69955i −0.608202 0.179299i
\(688\) 0 0
\(689\) 3.85854 10.6013i 0.146999 0.403876i
\(690\) 0 0
\(691\) −10.6791 1.88301i −0.406251 0.0716330i −0.0332112 0.999448i \(-0.510573\pi\)
−0.373040 + 0.927815i \(0.621685\pi\)
\(692\) 0 0
\(693\) 14.3812 + 13.3410i 0.546297 + 0.506784i
\(694\) 0 0
\(695\) 11.5797 + 9.71650i 0.439242 + 0.368568i
\(696\) 0 0
\(697\) −2.67371 15.1633i −0.101274 0.574353i
\(698\) 0 0
\(699\) −31.1619 + 20.6839i −1.17865 + 0.782336i
\(700\) 0 0
\(701\) 26.1415i 0.987349i −0.869647 0.493675i \(-0.835653\pi\)
0.869647 0.493675i \(-0.164347\pi\)
\(702\) 0 0
\(703\) 5.78503i 0.218187i
\(704\) 0 0
\(705\) −1.85442 0.921575i −0.0698415 0.0347085i
\(706\) 0 0
\(707\) 2.47303 + 14.0252i 0.0930077 + 0.527473i
\(708\) 0 0
\(709\) 1.19849 + 1.00565i 0.0450101 + 0.0377680i 0.665015 0.746830i \(-0.268425\pi\)
−0.620005 + 0.784598i \(0.712869\pi\)
\(710\) 0 0
\(711\) −15.9950 31.1915i −0.599860 1.16977i
\(712\) 0 0
\(713\) 43.7046 + 7.70630i 1.63675 + 0.288603i
\(714\) 0 0
\(715\) 2.53979 6.97802i 0.0949828 0.260963i
\(716\) 0 0
\(717\) −13.5606 + 12.9021i −0.506432 + 0.481837i
\(718\) 0 0
\(719\) 14.4301 + 24.9937i 0.538153 + 0.932109i 0.999004 + 0.0446310i \(0.0142112\pi\)
−0.460850 + 0.887478i \(0.652455\pi\)
\(720\) 0 0
\(721\) 4.23846 7.34123i 0.157848 0.273402i
\(722\) 0 0
\(723\) 20.6045 8.99093i 0.766291 0.334376i
\(724\) 0 0
\(725\) −23.6763 28.2163i −0.879315 1.04793i
\(726\) 0 0
\(727\) 13.9968 + 38.4560i 0.519114 + 1.42625i 0.871499 + 0.490398i \(0.163149\pi\)
−0.352385 + 0.935855i \(0.614629\pi\)
\(728\) 0 0
\(729\) 25.1305 9.87200i 0.930760 0.365630i
\(730\) 0 0
\(731\) 3.97152 1.44552i 0.146892 0.0534643i
\(732\) 0 0
\(733\) 1.89373 1.58903i 0.0699466 0.0586922i −0.607144 0.794592i \(-0.707685\pi\)
0.677091 + 0.735900i \(0.263240\pi\)
\(734\) 0 0
\(735\) −3.75141 8.59710i −0.138373 0.317109i
\(736\) 0 0
\(737\) −4.54661 2.62499i −0.167477 0.0966926i
\(738\) 0 0
\(739\) −9.69724 + 5.59870i −0.356719 + 0.205952i −0.667640 0.744484i \(-0.732696\pi\)
0.310922 + 0.950436i \(0.399362\pi\)
\(740\) 0 0
\(741\) 4.87691 + 5.12585i 0.179158 + 0.188303i
\(742\) 0 0
\(743\) −41.9122 15.2548i −1.53761 0.559645i −0.572140 0.820156i \(-0.693887\pi\)
−0.965471 + 0.260511i \(0.916109\pi\)
\(744\) 0 0
\(745\) −1.08258 + 6.13964i −0.0396628 + 0.224939i
\(746\) 0 0
\(747\) −44.8829 + 23.0160i −1.64218 + 0.842111i
\(748\) 0 0
\(749\) −17.8587 + 21.2832i −0.652544 + 0.777671i
\(750\) 0 0
\(751\) −11.5655 + 2.03931i −0.422032 + 0.0744156i −0.380631 0.924727i \(-0.624293\pi\)
−0.0414009 + 0.999143i \(0.513182\pi\)
\(752\) 0 0
\(753\) 14.4368 29.0502i 0.526107 1.05865i
\(754\) 0 0
\(755\) 13.6545 0.496937
\(756\) 0 0
\(757\) 5.35502 0.194632 0.0973158 0.995254i \(-0.468974\pi\)
0.0973158 + 0.995254i \(0.468974\pi\)
\(758\) 0 0
\(759\) 12.9304 + 19.4807i 0.469344 + 0.707105i
\(760\) 0 0
\(761\) 13.3535 2.35458i 0.484064 0.0853536i 0.0737099 0.997280i \(-0.476516\pi\)
0.410354 + 0.911926i \(0.365405\pi\)
\(762\) 0 0
\(763\) −35.9547 + 42.8491i −1.30165 + 1.55124i
\(764\) 0 0
\(765\) −8.46562 + 9.12566i −0.306075 + 0.329939i
\(766\) 0 0
\(767\) 9.52242 54.0043i 0.343835 1.94998i
\(768\) 0 0
\(769\) −16.2757 5.92388i −0.586917 0.213620i 0.0314556 0.999505i \(-0.489986\pi\)
−0.618373 + 0.785885i \(0.712208\pi\)
\(770\) 0 0
\(771\) 14.2816 48.4449i 0.514341 1.74470i
\(772\) 0 0
\(773\) −43.8577 + 25.3213i −1.57745 + 0.910743i −0.582239 + 0.813018i \(0.697823\pi\)
−0.995213 + 0.0977249i \(0.968843\pi\)
\(774\) 0 0
\(775\) −21.5561 12.4454i −0.774316 0.447052i
\(776\) 0 0
\(777\) −39.0091 4.39323i −1.39944 0.157606i
\(778\) 0 0
\(779\) 2.38870 2.00436i 0.0855840 0.0718135i
\(780\) 0 0
\(781\) 15.1918 5.52936i 0.543605 0.197856i
\(782\) 0 0
\(783\) 23.5018 + 39.5452i 0.839887 + 1.41323i
\(784\) 0 0
\(785\) 3.59581 + 9.87940i 0.128340 + 0.352611i
\(786\) 0 0
\(787\) 3.42817 + 4.08553i 0.122201 + 0.145634i 0.823676 0.567060i \(-0.191919\pi\)
−0.701475 + 0.712694i \(0.747475\pi\)
\(788\) 0 0
\(789\) −12.2631 9.04961i −0.436579 0.322175i
\(790\) 0 0
\(791\) −17.6243 + 30.5262i −0.626649 + 1.08539i
\(792\) 0 0
\(793\) 7.73407 + 13.3958i 0.274645 + 0.475699i
\(794\) 0 0
\(795\) 0.943434 + 3.90733i 0.0334601 + 0.138579i
\(796\) 0 0
\(797\) 7.46871 20.5201i 0.264555 0.726860i −0.734291 0.678835i \(-0.762485\pi\)
0.998846 0.0480248i \(-0.0152927\pi\)
\(798\) 0 0
\(799\) 5.81990 + 1.02620i 0.205893 + 0.0363045i
\(800\) 0 0
\(801\) 41.9304 + 17.6700i 1.48154 + 0.624340i
\(802\) 0 0
\(803\) −1.47365 1.23654i −0.0520038 0.0436364i
\(804\) 0 0
\(805\) −4.24061 24.0497i −0.149462 0.847640i
\(806\) 0 0
\(807\) −2.16982 34.7361i −0.0763813 1.22277i
\(808\) 0 0
\(809\) 35.0378i 1.23186i 0.787799 + 0.615932i \(0.211221\pi\)
−0.787799 + 0.615932i \(0.788779\pi\)
\(810\) 0 0
\(811\) 27.1844i 0.954573i −0.878748 0.477287i \(-0.841620\pi\)
0.878748 0.477287i \(-0.158380\pi\)
\(812\) 0 0
\(813\) −0.987230 15.8043i −0.0346237 0.554281i
\(814\) 0 0
\(815\) 0.554615 + 3.14538i 0.0194273 + 0.110178i
\(816\) 0 0
\(817\) 0.655675 + 0.550177i 0.0229392 + 0.0192483i
\(818\) 0 0
\(819\) −38.2677 + 28.9929i −1.33718 + 1.01309i
\(820\) 0 0
\(821\) −6.62706 1.16853i −0.231286 0.0407820i 0.0568037 0.998385i \(-0.481909\pi\)
−0.288090 + 0.957603i \(0.593020\pi\)
\(822\) 0 0
\(823\) −0.0376566 + 0.103461i −0.00131263 + 0.00360641i −0.940347 0.340216i \(-0.889500\pi\)
0.939035 + 0.343823i \(0.111722\pi\)
\(824\) 0 0
\(825\) −3.07795 12.7477i −0.107161 0.443817i
\(826\) 0 0
\(827\) 10.7113 + 18.5526i 0.372470 + 0.645137i 0.989945 0.141454i \(-0.0451777\pi\)
−0.617475 + 0.786590i \(0.711844\pi\)
\(828\) 0 0
\(829\) 7.92370 13.7242i 0.275201 0.476663i −0.694985 0.719025i \(-0.744589\pi\)
0.970186 + 0.242362i \(0.0779221\pi\)
\(830\) 0 0
\(831\) 20.3771 + 15.0373i 0.706873 + 0.521640i
\(832\) 0 0
\(833\) 17.2066 + 20.5060i 0.596173 + 0.710492i
\(834\) 0 0
\(835\) 1.37081 + 3.76627i 0.0474389 + 0.130337i
\(836\) 0 0
\(837\) 24.0636 + 19.6796i 0.831761 + 0.680227i
\(838\) 0 0
\(839\) 4.96451 1.80693i 0.171394 0.0623822i −0.254898 0.966968i \(-0.582042\pi\)
0.426292 + 0.904586i \(0.359820\pi\)
\(840\) 0 0
\(841\) −37.8243 + 31.7384i −1.30429 + 1.09443i
\(842\) 0 0
\(843\) 7.57502 + 0.853104i 0.260898 + 0.0293825i
\(844\) 0 0
\(845\) 5.42476 + 3.13198i 0.186617 + 0.107744i
\(846\) 0 0
\(847\) 23.9245 13.8128i 0.822056 0.474614i
\(848\) 0 0
\(849\) 6.21863 21.0943i 0.213423 0.723953i
\(850\) 0 0
\(851\) −43.9685 16.0032i −1.50722 0.548584i
\(852\) 0 0
\(853\) 0.220692 1.25161i 0.00755634 0.0428541i −0.980796 0.195034i \(-0.937518\pi\)
0.988353 + 0.152180i \(0.0486293\pi\)
\(854\) 0 0
\(855\) −2.45773 0.560694i −0.0840527 0.0191753i
\(856\) 0 0
\(857\) −15.6352 + 18.6333i −0.534087 + 0.636500i −0.963851 0.266441i \(-0.914152\pi\)
0.429764 + 0.902941i \(0.358597\pi\)
\(858\) 0 0
\(859\) 7.06237 1.24529i 0.240965 0.0424887i −0.0518612 0.998654i \(-0.516515\pi\)
0.292826 + 0.956166i \(0.405404\pi\)
\(860\) 0 0
\(861\) 11.7016 + 17.6294i 0.398789 + 0.600808i
\(862\) 0 0
\(863\) −38.1184 −1.29757 −0.648783 0.760973i \(-0.724722\pi\)
−0.648783 + 0.760973i \(0.724722\pi\)
\(864\) 0 0
\(865\) 9.61469 0.326909
\(866\) 0 0
\(867\) 2.70520 5.44348i 0.0918734 0.184870i
\(868\) 0 0
\(869\) 20.9403 3.69233i 0.710350 0.125254i
\(870\) 0 0
\(871\) 8.25927 9.84301i 0.279855 0.333518i
\(872\) 0 0
\(873\) 1.59874 32.1005i 0.0541093 1.08644i
\(874\) 0 0
\(875\) −5.23673 + 29.6990i −0.177034 + 1.00401i
\(876\) 0 0
\(877\) −4.23247 1.54049i −0.142921 0.0520188i 0.269569 0.962981i \(-0.413119\pi\)
−0.412490 + 0.910962i \(0.635341\pi\)
\(878\) 0 0
\(879\) −25.3510 26.6450i −0.855069 0.898715i
\(880\) 0 0
\(881\) −20.5813 + 11.8826i −0.693401 + 0.400335i −0.804885 0.593431i \(-0.797773\pi\)
0.111484 + 0.993766i \(0.464440\pi\)
\(882\) 0 0
\(883\) 4.93969 + 2.85193i 0.166234 + 0.0959751i 0.580809 0.814040i \(-0.302736\pi\)
−0.414575 + 0.910015i \(0.636070\pi\)
\(884\) 0 0
\(885\) 7.81419 + 17.9078i 0.262671 + 0.601964i
\(886\) 0 0
\(887\) 1.84014 1.54406i 0.0617857 0.0518444i −0.611372 0.791343i \(-0.709382\pi\)
0.673158 + 0.739499i \(0.264938\pi\)
\(888\) 0 0
\(889\) −65.2671 + 23.7553i −2.18899 + 0.796726i
\(890\) 0 0
\(891\) 1.22731 + 16.3320i 0.0411165 + 0.547143i
\(892\) 0 0
\(893\) 0.409335 + 1.12464i 0.0136979 + 0.0376346i
\(894\) 0 0
\(895\) −14.8600 17.7095i −0.496716 0.591963i
\(896\) 0 0
\(897\) −52.4495 + 22.8867i −1.75124 + 0.764165i
\(898\) 0 0
\(899\) −26.4817 + 45.8677i −0.883216 + 1.52977i
\(900\) 0 0
\(901\) −5.73564 9.93442i −0.191082 0.330963i
\(902\) 0 0
\(903\) −4.20783 + 4.00348i −0.140028 + 0.133227i
\(904\) 0 0
\(905\) 1.20377 3.30734i 0.0400147 0.109940i
\(906\) 0 0
\(907\) −45.9266 8.09809i −1.52497 0.268893i −0.652584 0.757717i \(-0.726315\pi\)
−0.872382 + 0.488824i \(0.837426\pi\)
\(908\) 0 0
\(909\) −6.45015 + 9.98905i −0.213938 + 0.331316i
\(910\) 0 0
\(911\) 27.6584 + 23.2082i 0.916365 + 0.768922i 0.973319 0.229455i \(-0.0736944\pi\)
−0.0569539 + 0.998377i \(0.518139\pi\)
\(912\) 0 0
\(913\) −5.31307 30.1319i −0.175837 0.997221i
\(914\) 0 0
\(915\) −4.93540 2.45271i −0.163159 0.0810840i
\(916\) 0 0
\(917\) 62.9199i 2.07780i
\(918\) 0 0
\(919\) 23.9024i 0.788468i −0.919010 0.394234i \(-0.871010\pi\)
0.919010 0.394234i \(-0.128990\pi\)
\(920\) 0 0
\(921\) −2.07809 + 1.37934i −0.0684755 + 0.0454509i
\(922\) 0 0
\(923\) 6.87085 + 38.9665i 0.226157 + 1.28260i
\(924\) 0 0
\(925\) 20.1035 + 16.8689i 0.661000 + 0.554645i
\(926\) 0 0
\(927\) 6.76288 2.08684i 0.222122 0.0685408i
\(928\) 0 0
\(929\) 11.3439 + 2.00024i 0.372181 + 0.0656256i 0.356611 0.934253i \(-0.383932\pi\)
0.0155709 + 0.999879i \(0.495043\pi\)
\(930\) 0 0
\(931\) −1.85414 + 5.09421i −0.0607671 + 0.166956i
\(932\) 0 0
\(933\) 14.7187 + 4.33909i 0.481867 + 0.142055i
\(934\) 0 0
\(935\) −3.77535 6.53909i −0.123467 0.213851i
\(936\) 0 0
\(937\) −17.1278 + 29.6661i −0.559539 + 0.969150i 0.437996 + 0.898977i \(0.355689\pi\)
−0.997535 + 0.0701733i \(0.977645\pi\)
\(938\) 0 0
\(939\) −6.01426 + 53.4028i −0.196268 + 1.74274i
\(940\) 0 0
\(941\) −16.8452 20.0754i −0.549139 0.654439i 0.418071 0.908414i \(-0.362706\pi\)
−0.967211 + 0.253975i \(0.918262\pi\)
\(942\) 0 0
\(943\) 8.62599 + 23.6997i 0.280901 + 0.771769i
\(944\) 0 0
\(945\) 5.64725 16.1469i 0.183705 0.525260i
\(946\) 0 0
\(947\) 7.77710 2.83063i 0.252722 0.0919832i −0.212553 0.977150i \(-0.568178\pi\)
0.465275 + 0.885166i \(0.345956\pi\)
\(948\) 0 0
\(949\) 3.60670 3.02638i 0.117078 0.0982405i
\(950\) 0 0
\(951\) 1.68590 2.28456i 0.0546690 0.0740819i
\(952\) 0 0
\(953\) 25.1053 + 14.4946i 0.813241 + 0.469525i 0.848080 0.529868i \(-0.177759\pi\)
−0.0348391 + 0.999393i \(0.511092\pi\)
\(954\) 0 0
\(955\) 7.84341 4.52840i 0.253807 0.146535i
\(956\) 0 0
\(957\) −27.1249 + 6.54938i −0.876824 + 0.211711i
\(958\) 0 0
\(959\) −54.3722 19.7899i −1.75577 0.639048i
\(960\) 0 0
\(961\) −0.831882 + 4.71784i −0.0268349 + 0.152188i
\(962\) 0 0
\(963\) −23.0164 + 2.88675i −0.741694 + 0.0930241i
\(964\) 0 0
\(965\) 6.11470 7.28721i 0.196839 0.234584i
\(966\) 0 0
\(967\) −37.9337 + 6.68873i −1.21986 + 0.215095i −0.746267 0.665646i \(-0.768156\pi\)
−0.473597 + 0.880741i \(0.657045\pi\)
\(968\) 0 0
\(969\) 7.18015 0.448515i 0.230660 0.0144084i
\(970\) 0 0
\(971\) −36.9077 −1.18442 −0.592212 0.805783i \(-0.701745\pi\)
−0.592212 + 0.805783i \(0.701745\pi\)
\(972\) 0 0
\(973\) 59.2830 1.90053
\(974\) 0 0
\(975\) 32.0336 2.00101i 1.02590 0.0640836i
\(976\) 0 0
\(977\) −43.6838 + 7.70263i −1.39757 + 0.246429i −0.821144 0.570722i \(-0.806663\pi\)
−0.576424 + 0.817151i \(0.695552\pi\)
\(978\) 0 0
\(979\) −17.7416 + 21.1436i −0.567023 + 0.675752i
\(980\) 0 0
\(981\) −46.3385 + 5.81184i −1.47948 + 0.185558i
\(982\) 0 0
\(983\) 6.09294 34.5548i 0.194335 1.10213i −0.719028 0.694981i \(-0.755413\pi\)
0.913363 0.407146i \(-0.133476\pi\)
\(984\) 0 0
\(985\) 11.0120 + 4.00802i 0.350870 + 0.127706i
\(986\) 0 0
\(987\) −7.89442 + 1.90613i −0.251282 + 0.0606727i
\(988\) 0 0
\(989\) −5.99536 + 3.46142i −0.190641 + 0.110067i
\(990\) 0 0
\(991\) −5.19698 3.00048i −0.165088 0.0953134i 0.415180 0.909739i \(-0.363719\pi\)
−0.580267 + 0.814426i \(0.697052\pi\)
\(992\) 0 0
\(993\) 6.67924 9.05103i 0.211959 0.287226i
\(994\) 0 0
\(995\) −5.50898 + 4.62258i −0.174646 + 0.146546i
\(996\) 0 0
\(997\) 51.5630 18.7674i 1.63302 0.594369i 0.647218 0.762305i \(-0.275933\pi\)
0.985798 + 0.167936i \(0.0537103\pi\)
\(998\) 0 0
\(999\) −21.3828 24.8394i −0.676520 0.785885i
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 432.2.be.a.47.6 yes 36
4.3 odd 2 inner 432.2.be.a.47.1 36
27.23 odd 18 inner 432.2.be.a.239.1 yes 36
108.23 even 18 inner 432.2.be.a.239.6 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
432.2.be.a.47.1 36 4.3 odd 2 inner
432.2.be.a.47.6 yes 36 1.1 even 1 trivial
432.2.be.a.239.1 yes 36 27.23 odd 18 inner
432.2.be.a.239.6 yes 36 108.23 even 18 inner