# Properties

 Label 256.6.b.d.129.1 Level $256$ Weight $6$ Character 256.129 Analytic conductor $41.058$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.0582578721$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 129.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 256.129 Dual form 256.6.b.d.129.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-20.0000i q^{3} +74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} +O(q^{10})$$ $$q-20.0000i q^{3} +74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} +124.000i q^{11} +478.000i q^{13} +1480.00 q^{15} -1198.00 q^{17} -3044.00i q^{19} +480.000i q^{21} +184.000 q^{23} -2351.00 q^{25} -1720.00i q^{27} -3282.00i q^{29} +5728.00 q^{31} +2480.00 q^{33} -1776.00i q^{35} -10326.0i q^{37} +9560.00 q^{39} +8886.00 q^{41} -9188.00i q^{43} -11618.0i q^{45} -23664.0 q^{47} -16231.0 q^{49} +23960.0i q^{51} -11686.0i q^{53} -9176.00 q^{55} -60880.0 q^{57} +16876.0i q^{59} -18482.0i q^{61} +3768.00 q^{63} -35372.0 q^{65} +15532.0i q^{67} -3680.00i q^{69} -31960.0 q^{71} +4886.00 q^{73} +47020.0i q^{75} -2976.00i q^{77} -44560.0 q^{79} -72551.0 q^{81} -67364.0i q^{83} -88652.0i q^{85} -65640.0 q^{87} -71994.0 q^{89} -11472.0i q^{91} -114560. i q^{93} +225256. q^{95} +48866.0 q^{97} -19468.0i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 48 q^{7} - 314 q^{9}+O(q^{10})$$ 2 * q - 48 * q^7 - 314 * q^9 $$2 q - 48 q^{7} - 314 q^{9} + 2960 q^{15} - 2396 q^{17} + 368 q^{23} - 4702 q^{25} + 11456 q^{31} + 4960 q^{33} + 19120 q^{39} + 17772 q^{41} - 47328 q^{47} - 32462 q^{49} - 18352 q^{55} - 121760 q^{57} + 7536 q^{63} - 70744 q^{65} - 63920 q^{71} + 9772 q^{73} - 89120 q^{79} - 145102 q^{81} - 131280 q^{87} - 143988 q^{89} + 450512 q^{95} + 97732 q^{97}+O(q^{100})$$ 2 * q - 48 * q^7 - 314 * q^9 + 2960 * q^15 - 2396 * q^17 + 368 * q^23 - 4702 * q^25 + 11456 * q^31 + 4960 * q^33 + 19120 * q^39 + 17772 * q^41 - 47328 * q^47 - 32462 * q^49 - 18352 * q^55 - 121760 * q^57 + 7536 * q^63 - 70744 * q^65 - 63920 * q^71 + 9772 * q^73 - 89120 * q^79 - 145102 * q^81 - 131280 * q^87 - 143988 * q^89 + 450512 * q^95 + 97732 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 20.0000i − 1.28300i −0.767123 0.641500i $$-0.778312\pi$$
0.767123 0.641500i $$-0.221688\pi$$
$$4$$ 0 0
$$5$$ 74.0000i 1.32375i 0.749613 + 0.661876i $$0.230240\pi$$
−0.749613 + 0.661876i $$0.769760\pi$$
$$6$$ 0 0
$$7$$ −24.0000 −0.185125 −0.0925627 0.995707i $$-0.529506\pi$$
−0.0925627 + 0.995707i $$0.529506\pi$$
$$8$$ 0 0
$$9$$ −157.000 −0.646091
$$10$$ 0 0
$$11$$ 124.000i 0.308987i 0.987994 + 0.154493i $$0.0493745\pi$$
−0.987994 + 0.154493i $$0.950625\pi$$
$$12$$ 0 0
$$13$$ 478.000i 0.784458i 0.919868 + 0.392229i $$0.128296\pi$$
−0.919868 + 0.392229i $$0.871704\pi$$
$$14$$ 0 0
$$15$$ 1480.00 1.69837
$$16$$ 0 0
$$17$$ −1198.00 −1.00539 −0.502695 0.864464i $$-0.667658\pi$$
−0.502695 + 0.864464i $$0.667658\pi$$
$$18$$ 0 0
$$19$$ − 3044.00i − 1.93446i −0.253894 0.967232i $$-0.581712\pi$$
0.253894 0.967232i $$-0.418288\pi$$
$$20$$ 0 0
$$21$$ 480.000i 0.237516i
$$22$$ 0 0
$$23$$ 184.000 0.0725268 0.0362634 0.999342i $$-0.488454\pi$$
0.0362634 + 0.999342i $$0.488454\pi$$
$$24$$ 0 0
$$25$$ −2351.00 −0.752320
$$26$$ 0 0
$$27$$ − 1720.00i − 0.454066i
$$28$$ 0 0
$$29$$ − 3282.00i − 0.724676i −0.932047 0.362338i $$-0.881979\pi$$
0.932047 0.362338i $$-0.118021\pi$$
$$30$$ 0 0
$$31$$ 5728.00 1.07053 0.535265 0.844684i $$-0.320212\pi$$
0.535265 + 0.844684i $$0.320212\pi$$
$$32$$ 0 0
$$33$$ 2480.00 0.396430
$$34$$ 0 0
$$35$$ − 1776.00i − 0.245060i
$$36$$ 0 0
$$37$$ − 10326.0i − 1.24002i −0.784595 0.620009i $$-0.787129\pi$$
0.784595 0.620009i $$-0.212871\pi$$
$$38$$ 0 0
$$39$$ 9560.00 1.00646
$$40$$ 0 0
$$41$$ 8886.00 0.825556 0.412778 0.910832i $$-0.364558\pi$$
0.412778 + 0.910832i $$0.364558\pi$$
$$42$$ 0 0
$$43$$ − 9188.00i − 0.757792i −0.925439 0.378896i $$-0.876304\pi$$
0.925439 0.378896i $$-0.123696\pi$$
$$44$$ 0 0
$$45$$ − 11618.0i − 0.855264i
$$46$$ 0 0
$$47$$ −23664.0 −1.56258 −0.781292 0.624165i $$-0.785439\pi$$
−0.781292 + 0.624165i $$0.785439\pi$$
$$48$$ 0 0
$$49$$ −16231.0 −0.965729
$$50$$ 0 0
$$51$$ 23960.0i 1.28992i
$$52$$ 0 0
$$53$$ − 11686.0i − 0.571447i −0.958312 0.285724i $$-0.907766\pi$$
0.958312 0.285724i $$-0.0922339\pi$$
$$54$$ 0 0
$$55$$ −9176.00 −0.409022
$$56$$ 0 0
$$57$$ −60880.0 −2.48192
$$58$$ 0 0
$$59$$ 16876.0i 0.631160i 0.948899 + 0.315580i $$0.102199\pi$$
−0.948899 + 0.315580i $$0.897801\pi$$
$$60$$ 0 0
$$61$$ − 18482.0i − 0.635952i −0.948099 0.317976i $$-0.896997\pi$$
0.948099 0.317976i $$-0.103003\pi$$
$$62$$ 0 0
$$63$$ 3768.00 0.119608
$$64$$ 0 0
$$65$$ −35372.0 −1.03843
$$66$$ 0 0
$$67$$ 15532.0i 0.422708i 0.977410 + 0.211354i $$0.0677873\pi$$
−0.977410 + 0.211354i $$0.932213\pi$$
$$68$$ 0 0
$$69$$ − 3680.00i − 0.0930519i
$$70$$ 0 0
$$71$$ −31960.0 −0.752421 −0.376210 0.926534i $$-0.622773\pi$$
−0.376210 + 0.926534i $$0.622773\pi$$
$$72$$ 0 0
$$73$$ 4886.00 0.107312 0.0536558 0.998559i $$-0.482913\pi$$
0.0536558 + 0.998559i $$0.482913\pi$$
$$74$$ 0 0
$$75$$ 47020.0i 0.965227i
$$76$$ 0 0
$$77$$ − 2976.00i − 0.0572013i
$$78$$ 0 0
$$79$$ −44560.0 −0.803299 −0.401650 0.915793i $$-0.631563\pi$$
−0.401650 + 0.915793i $$0.631563\pi$$
$$80$$ 0 0
$$81$$ −72551.0 −1.22866
$$82$$ 0 0
$$83$$ − 67364.0i − 1.07333i −0.843796 0.536664i $$-0.819684\pi$$
0.843796 0.536664i $$-0.180316\pi$$
$$84$$ 0 0
$$85$$ − 88652.0i − 1.33089i
$$86$$ 0 0
$$87$$ −65640.0 −0.929759
$$88$$ 0 0
$$89$$ −71994.0 −0.963432 −0.481716 0.876327i $$-0.659986\pi$$
−0.481716 + 0.876327i $$0.659986\pi$$
$$90$$ 0 0
$$91$$ − 11472.0i − 0.145223i
$$92$$ 0 0
$$93$$ − 114560.i − 1.37349i
$$94$$ 0 0
$$95$$ 225256. 2.56075
$$96$$ 0 0
$$97$$ 48866.0 0.527324 0.263662 0.964615i $$-0.415070\pi$$
0.263662 + 0.964615i $$0.415070\pi$$
$$98$$ 0 0
$$99$$ − 19468.0i − 0.199633i
$$100$$ 0 0
$$101$$ − 51606.0i − 0.503381i −0.967808 0.251690i $$-0.919014\pi$$
0.967808 0.251690i $$-0.0809865\pi$$
$$102$$ 0 0
$$103$$ 180424. 1.67572 0.837860 0.545886i $$-0.183807\pi$$
0.837860 + 0.545886i $$0.183807\pi$$
$$104$$ 0 0
$$105$$ −35520.0 −0.314412
$$106$$ 0 0
$$107$$ − 65700.0i − 0.554761i −0.960760 0.277381i $$-0.910534\pi$$
0.960760 0.277381i $$-0.0894663\pi$$
$$108$$ 0 0
$$109$$ − 112706.i − 0.908617i −0.890844 0.454308i $$-0.849886\pi$$
0.890844 0.454308i $$-0.150114\pi$$
$$110$$ 0 0
$$111$$ −206520. −1.59094
$$112$$ 0 0
$$113$$ −23502.0 −0.173145 −0.0865723 0.996246i $$-0.527591\pi$$
−0.0865723 + 0.996246i $$0.527591\pi$$
$$114$$ 0 0
$$115$$ 13616.0i 0.0960075i
$$116$$ 0 0
$$117$$ − 75046.0i − 0.506831i
$$118$$ 0 0
$$119$$ 28752.0 0.186123
$$120$$ 0 0
$$121$$ 145675. 0.904527
$$122$$ 0 0
$$123$$ − 177720.i − 1.05919i
$$124$$ 0 0
$$125$$ 57276.0i 0.327867i
$$126$$ 0 0
$$127$$ 94592.0 0.520409 0.260205 0.965553i $$-0.416210\pi$$
0.260205 + 0.965553i $$0.416210\pi$$
$$128$$ 0 0
$$129$$ −183760. −0.972247
$$130$$ 0 0
$$131$$ − 70292.0i − 0.357872i −0.983861 0.178936i $$-0.942735\pi$$
0.983861 0.178936i $$-0.0572655\pi$$
$$132$$ 0 0
$$133$$ 73056.0i 0.358119i
$$134$$ 0 0
$$135$$ 127280. 0.601071
$$136$$ 0 0
$$137$$ −277290. −1.26221 −0.631107 0.775696i $$-0.717399\pi$$
−0.631107 + 0.775696i $$0.717399\pi$$
$$138$$ 0 0
$$139$$ − 130308.i − 0.572050i −0.958222 0.286025i $$-0.907666\pi$$
0.958222 0.286025i $$-0.0923341\pi$$
$$140$$ 0 0
$$141$$ 473280.i 2.00480i
$$142$$ 0 0
$$143$$ −59272.0 −0.242387
$$144$$ 0 0
$$145$$ 242868. 0.959291
$$146$$ 0 0
$$147$$ 324620.i 1.23903i
$$148$$ 0 0
$$149$$ 401530.i 1.48167i 0.671685 + 0.740836i $$0.265571\pi$$
−0.671685 + 0.740836i $$0.734429\pi$$
$$150$$ 0 0
$$151$$ −75976.0 −0.271165 −0.135583 0.990766i $$-0.543291\pi$$
−0.135583 + 0.990766i $$0.543291\pi$$
$$152$$ 0 0
$$153$$ 188086. 0.649573
$$154$$ 0 0
$$155$$ 423872.i 1.41712i
$$156$$ 0 0
$$157$$ − 394322.i − 1.27674i −0.769730 0.638369i $$-0.779609\pi$$
0.769730 0.638369i $$-0.220391\pi$$
$$158$$ 0 0
$$159$$ −233720. −0.733167
$$160$$ 0 0
$$161$$ −4416.00 −0.0134265
$$162$$ 0 0
$$163$$ 11724.0i 0.0345626i 0.999851 + 0.0172813i $$0.00550109\pi$$
−0.999851 + 0.0172813i $$0.994499\pi$$
$$164$$ 0 0
$$165$$ 183520.i 0.524775i
$$166$$ 0 0
$$167$$ −551928. −1.53141 −0.765705 0.643192i $$-0.777610\pi$$
−0.765705 + 0.643192i $$0.777610\pi$$
$$168$$ 0 0
$$169$$ 142809. 0.384626
$$170$$ 0 0
$$171$$ 477908.i 1.24984i
$$172$$ 0 0
$$173$$ 432894.i 1.09968i 0.835270 + 0.549840i $$0.185311\pi$$
−0.835270 + 0.549840i $$0.814689\pi$$
$$174$$ 0 0
$$175$$ 56424.0 0.139274
$$176$$ 0 0
$$177$$ 337520. 0.809779
$$178$$ 0 0
$$179$$ − 559620.i − 1.30545i −0.757594 0.652726i $$-0.773625\pi$$
0.757594 0.652726i $$-0.226375\pi$$
$$180$$ 0 0
$$181$$ − 604710.i − 1.37199i −0.727607 0.685995i $$-0.759367\pi$$
0.727607 0.685995i $$-0.240633\pi$$
$$182$$ 0 0
$$183$$ −369640. −0.815927
$$184$$ 0 0
$$185$$ 764124. 1.64148
$$186$$ 0 0
$$187$$ − 148552.i − 0.310652i
$$188$$ 0 0
$$189$$ 41280.0i 0.0840592i
$$190$$ 0 0
$$191$$ 409152. 0.811524 0.405762 0.913979i $$-0.367006\pi$$
0.405762 + 0.913979i $$0.367006\pi$$
$$192$$ 0 0
$$193$$ 540866. 1.04519 0.522596 0.852580i $$-0.324963\pi$$
0.522596 + 0.852580i $$0.324963\pi$$
$$194$$ 0 0
$$195$$ 707440.i 1.33230i
$$196$$ 0 0
$$197$$ 629898.i 1.15639i 0.815898 + 0.578195i $$0.196243\pi$$
−0.815898 + 0.578195i $$0.803757\pi$$
$$198$$ 0 0
$$199$$ 283048. 0.506673 0.253336 0.967378i $$-0.418472\pi$$
0.253336 + 0.967378i $$0.418472\pi$$
$$200$$ 0 0
$$201$$ 310640. 0.542335
$$202$$ 0 0
$$203$$ 78768.0i 0.134156i
$$204$$ 0 0
$$205$$ 657564.i 1.09283i
$$206$$ 0 0
$$207$$ −28888.0 −0.0468588
$$208$$ 0 0
$$209$$ 377456. 0.597724
$$210$$ 0 0
$$211$$ − 142756.i − 0.220744i −0.993890 0.110372i $$-0.964796\pi$$
0.993890 0.110372i $$-0.0352042\pi$$
$$212$$ 0 0
$$213$$ 639200.i 0.965357i
$$214$$ 0 0
$$215$$ 679912. 1.00313
$$216$$ 0 0
$$217$$ −137472. −0.198182
$$218$$ 0 0
$$219$$ − 97720.0i − 0.137681i
$$220$$ 0 0
$$221$$ − 572644.i − 0.788686i
$$222$$ 0 0
$$223$$ −889696. −1.19806 −0.599031 0.800726i $$-0.704447\pi$$
−0.599031 + 0.800726i $$0.704447\pi$$
$$224$$ 0 0
$$225$$ 369107. 0.486067
$$226$$ 0 0
$$227$$ − 1.14316e6i − 1.47245i −0.676736 0.736226i $$-0.736606\pi$$
0.676736 0.736226i $$-0.263394\pi$$
$$228$$ 0 0
$$229$$ 695786.i 0.876773i 0.898787 + 0.438386i $$0.144450\pi$$
−0.898787 + 0.438386i $$0.855550\pi$$
$$230$$ 0 0
$$231$$ −59520.0 −0.0733893
$$232$$ 0 0
$$233$$ 347126. 0.418887 0.209444 0.977821i $$-0.432835\pi$$
0.209444 + 0.977821i $$0.432835\pi$$
$$234$$ 0 0
$$235$$ − 1.75114e6i − 2.06847i
$$236$$ 0 0
$$237$$ 891200.i 1.03063i
$$238$$ 0 0
$$239$$ 1.64296e6 1.86051 0.930255 0.366912i $$-0.119585\pi$$
0.930255 + 0.366912i $$0.119585\pi$$
$$240$$ 0 0
$$241$$ −1.16744e6 −1.29477 −0.647383 0.762165i $$-0.724137\pi$$
−0.647383 + 0.762165i $$0.724137\pi$$
$$242$$ 0 0
$$243$$ 1.03306e6i 1.12230i
$$244$$ 0 0
$$245$$ − 1.20109e6i − 1.27839i
$$246$$ 0 0
$$247$$ 1.45503e6 1.51751
$$248$$ 0 0
$$249$$ −1.34728e6 −1.37708
$$250$$ 0 0
$$251$$ − 790612.i − 0.792098i −0.918229 0.396049i $$-0.870381\pi$$
0.918229 0.396049i $$-0.129619\pi$$
$$252$$ 0 0
$$253$$ 22816.0i 0.0224098i
$$254$$ 0 0
$$255$$ −1.77304e6 −1.70753
$$256$$ 0 0
$$257$$ −129790. −0.122577 −0.0612884 0.998120i $$-0.519521\pi$$
−0.0612884 + 0.998120i $$0.519521\pi$$
$$258$$ 0 0
$$259$$ 247824.i 0.229559i
$$260$$ 0 0
$$261$$ 515274.i 0.468206i
$$262$$ 0 0
$$263$$ 70888.0 0.0631951 0.0315975 0.999501i $$-0.489941\pi$$
0.0315975 + 0.999501i $$0.489941\pi$$
$$264$$ 0 0
$$265$$ 864764. 0.756455
$$266$$ 0 0
$$267$$ 1.43988e6i 1.23608i
$$268$$ 0 0
$$269$$ 1.79017e6i 1.50839i 0.656649 + 0.754197i $$0.271973\pi$$
−0.656649 + 0.754197i $$0.728027\pi$$
$$270$$ 0 0
$$271$$ 1.77362e6 1.46702 0.733511 0.679678i $$-0.237880\pi$$
0.733511 + 0.679678i $$0.237880\pi$$
$$272$$ 0 0
$$273$$ −229440. −0.186321
$$274$$ 0 0
$$275$$ − 291524.i − 0.232457i
$$276$$ 0 0
$$277$$ 275450.i 0.215697i 0.994167 + 0.107848i $$0.0343961\pi$$
−0.994167 + 0.107848i $$0.965604\pi$$
$$278$$ 0 0
$$279$$ −899296. −0.691659
$$280$$ 0 0
$$281$$ −594170. −0.448895 −0.224448 0.974486i $$-0.572058\pi$$
−0.224448 + 0.974486i $$0.572058\pi$$
$$282$$ 0 0
$$283$$ 1.09243e6i 0.810824i 0.914134 + 0.405412i $$0.132872\pi$$
−0.914134 + 0.405412i $$0.867128\pi$$
$$284$$ 0 0
$$285$$ − 4.50512e6i − 3.28545i
$$286$$ 0 0
$$287$$ −213264. −0.152831
$$288$$ 0 0
$$289$$ 15347.0 0.0108088
$$290$$ 0 0
$$291$$ − 977320.i − 0.676557i
$$292$$ 0 0
$$293$$ − 333654.i − 0.227053i −0.993535 0.113527i $$-0.963785\pi$$
0.993535 0.113527i $$-0.0362147\pi$$
$$294$$ 0 0
$$295$$ −1.24882e6 −0.835500
$$296$$ 0 0
$$297$$ 213280. 0.140300
$$298$$ 0 0
$$299$$ 87952.0i 0.0568942i
$$300$$ 0 0
$$301$$ 220512.i 0.140287i
$$302$$ 0 0
$$303$$ −1.03212e6 −0.645838
$$304$$ 0 0
$$305$$ 1.36767e6 0.841843
$$306$$ 0 0
$$307$$ − 1.05997e6i − 0.641872i −0.947101 0.320936i $$-0.896003\pi$$
0.947101 0.320936i $$-0.103997\pi$$
$$308$$ 0 0
$$309$$ − 3.60848e6i − 2.14995i
$$310$$ 0 0
$$311$$ −1.33649e6 −0.783545 −0.391773 0.920062i $$-0.628138\pi$$
−0.391773 + 0.920062i $$0.628138\pi$$
$$312$$ 0 0
$$313$$ −1.64419e6 −0.948615 −0.474308 0.880359i $$-0.657302\pi$$
−0.474308 + 0.880359i $$0.657302\pi$$
$$314$$ 0 0
$$315$$ 278832.i 0.158331i
$$316$$ 0 0
$$317$$ − 1.72370e6i − 0.963414i −0.876332 0.481707i $$-0.840017\pi$$
0.876332 0.481707i $$-0.159983\pi$$
$$318$$ 0 0
$$319$$ 406968. 0.223915
$$320$$ 0 0
$$321$$ −1.31400e6 −0.711759
$$322$$ 0 0
$$323$$ 3.64671e6i 1.94489i
$$324$$ 0 0
$$325$$ − 1.12378e6i − 0.590163i
$$326$$ 0 0
$$327$$ −2.25412e6 −1.16576
$$328$$ 0 0
$$329$$ 567936. 0.289274
$$330$$ 0 0
$$331$$ 2.74963e6i 1.37944i 0.724074 + 0.689722i $$0.242267\pi$$
−0.724074 + 0.689722i $$0.757733\pi$$
$$332$$ 0 0
$$333$$ 1.62118e6i 0.801164i
$$334$$ 0 0
$$335$$ −1.14937e6 −0.559561
$$336$$ 0 0
$$337$$ −3.41489e6 −1.63796 −0.818978 0.573824i $$-0.805459\pi$$
−0.818978 + 0.573824i $$0.805459\pi$$
$$338$$ 0 0
$$339$$ 470040.i 0.222145i
$$340$$ 0 0
$$341$$ 710272.i 0.330780i
$$342$$ 0 0
$$343$$ 792912. 0.363906
$$344$$ 0 0
$$345$$ 272320. 0.123178
$$346$$ 0 0
$$347$$ 730764.i 0.325802i 0.986642 + 0.162901i $$0.0520851\pi$$
−0.986642 + 0.162901i $$0.947915\pi$$
$$348$$ 0 0
$$349$$ − 2.29749e6i − 1.00969i −0.863209 0.504847i $$-0.831549\pi$$
0.863209 0.504847i $$-0.168451\pi$$
$$350$$ 0 0
$$351$$ 822160. 0.356196
$$352$$ 0 0
$$353$$ −1.17072e6 −0.500052 −0.250026 0.968239i $$-0.580439\pi$$
−0.250026 + 0.968239i $$0.580439\pi$$
$$354$$ 0 0
$$355$$ − 2.36504e6i − 0.996019i
$$356$$ 0 0
$$357$$ − 575040.i − 0.238796i
$$358$$ 0 0
$$359$$ 3.88654e6 1.59157 0.795787 0.605577i $$-0.207058\pi$$
0.795787 + 0.605577i $$0.207058\pi$$
$$360$$ 0 0
$$361$$ −6.78984e6 −2.74215
$$362$$ 0 0
$$363$$ − 2.91350e6i − 1.16051i
$$364$$ 0 0
$$365$$ 361564.i 0.142054i
$$366$$ 0 0
$$367$$ −933040. −0.361606 −0.180803 0.983519i $$-0.557870\pi$$
−0.180803 + 0.983519i $$0.557870\pi$$
$$368$$ 0 0
$$369$$ −1.39510e6 −0.533384
$$370$$ 0 0
$$371$$ 280464.i 0.105789i
$$372$$ 0 0
$$373$$ 392218.i 0.145967i 0.997333 + 0.0729836i $$0.0232521\pi$$
−0.997333 + 0.0729836i $$0.976748\pi$$
$$374$$ 0 0
$$375$$ 1.14552e6 0.420653
$$376$$ 0 0
$$377$$ 1.56880e6 0.568477
$$378$$ 0 0
$$379$$ − 4.72930e6i − 1.69122i −0.533805 0.845608i $$-0.679238\pi$$
0.533805 0.845608i $$-0.320762\pi$$
$$380$$ 0 0
$$381$$ − 1.89184e6i − 0.667686i
$$382$$ 0 0
$$383$$ −1.89734e6 −0.660920 −0.330460 0.943820i $$-0.607204\pi$$
−0.330460 + 0.943820i $$0.607204\pi$$
$$384$$ 0 0
$$385$$ 220224. 0.0757204
$$386$$ 0 0
$$387$$ 1.44252e6i 0.489602i
$$388$$ 0 0
$$389$$ 3.72295e6i 1.24742i 0.781655 + 0.623711i $$0.214376\pi$$
−0.781655 + 0.623711i $$0.785624\pi$$
$$390$$ 0 0
$$391$$ −220432. −0.0729177
$$392$$ 0 0
$$393$$ −1.40584e6 −0.459150
$$394$$ 0 0
$$395$$ − 3.29744e6i − 1.06337i
$$396$$ 0 0
$$397$$ 3.33808e6i 1.06297i 0.847068 + 0.531484i $$0.178365\pi$$
−0.847068 + 0.531484i $$0.821635\pi$$
$$398$$ 0 0
$$399$$ 1.46112e6 0.459466
$$400$$ 0 0
$$401$$ 4.27490e6 1.32759 0.663796 0.747913i $$-0.268944\pi$$
0.663796 + 0.747913i $$0.268944\pi$$
$$402$$ 0 0
$$403$$ 2.73798e6i 0.839785i
$$404$$ 0 0
$$405$$ − 5.36877e6i − 1.62644i
$$406$$ 0 0
$$407$$ 1.28042e6 0.383149
$$408$$ 0 0
$$409$$ 2.57319e6 0.760613 0.380306 0.924861i $$-0.375819\pi$$
0.380306 + 0.924861i $$0.375819\pi$$
$$410$$ 0 0
$$411$$ 5.54580e6i 1.61942i
$$412$$ 0 0
$$413$$ − 405024.i − 0.116844i
$$414$$ 0 0
$$415$$ 4.98494e6 1.42082
$$416$$ 0 0
$$417$$ −2.60616e6 −0.733941
$$418$$ 0 0
$$419$$ − 5.26828e6i − 1.46600i −0.680230 0.732999i $$-0.738120\pi$$
0.680230 0.732999i $$-0.261880\pi$$
$$420$$ 0 0
$$421$$ 973354.i 0.267649i 0.991005 + 0.133824i $$0.0427258\pi$$
−0.991005 + 0.133824i $$0.957274\pi$$
$$422$$ 0 0
$$423$$ 3.71525e6 1.00957
$$424$$ 0 0
$$425$$ 2.81650e6 0.756375
$$426$$ 0 0
$$427$$ 443568.i 0.117731i
$$428$$ 0 0
$$429$$ 1.18544e6i 0.310983i
$$430$$ 0 0
$$431$$ −3.55736e6 −0.922433 −0.461216 0.887288i $$-0.652587\pi$$
−0.461216 + 0.887288i $$0.652587\pi$$
$$432$$ 0 0
$$433$$ −1.95496e6 −0.501092 −0.250546 0.968105i $$-0.580610\pi$$
−0.250546 + 0.968105i $$0.580610\pi$$
$$434$$ 0 0
$$435$$ − 4.85736e6i − 1.23077i
$$436$$ 0 0
$$437$$ − 560096.i − 0.140300i
$$438$$ 0 0
$$439$$ −3.29681e6 −0.816455 −0.408228 0.912880i $$-0.633853\pi$$
−0.408228 + 0.912880i $$0.633853\pi$$
$$440$$ 0 0
$$441$$ 2.54827e6 0.623948
$$442$$ 0 0
$$443$$ − 5.05820e6i − 1.22458i −0.790634 0.612289i $$-0.790249\pi$$
0.790634 0.612289i $$-0.209751\pi$$
$$444$$ 0 0
$$445$$ − 5.32756e6i − 1.27535i
$$446$$ 0 0
$$447$$ 8.03060e6 1.90099
$$448$$ 0 0
$$449$$ 2.12730e6 0.497981 0.248990 0.968506i $$-0.419901\pi$$
0.248990 + 0.968506i $$0.419901\pi$$
$$450$$ 0 0
$$451$$ 1.10186e6i 0.255086i
$$452$$ 0 0
$$453$$ 1.51952e6i 0.347905i
$$454$$ 0 0
$$455$$ 848928. 0.192239
$$456$$ 0 0
$$457$$ −289130. −0.0647594 −0.0323797 0.999476i $$-0.510309\pi$$
−0.0323797 + 0.999476i $$0.510309\pi$$
$$458$$ 0 0
$$459$$ 2.06056e6i 0.456513i
$$460$$ 0 0
$$461$$ 2.66870e6i 0.584854i 0.956288 + 0.292427i $$0.0944629\pi$$
−0.956288 + 0.292427i $$0.905537\pi$$
$$462$$ 0 0
$$463$$ −7.58619e6 −1.64464 −0.822321 0.569024i $$-0.807321\pi$$
−0.822321 + 0.569024i $$0.807321\pi$$
$$464$$ 0 0
$$465$$ 8.47744e6 1.81816
$$466$$ 0 0
$$467$$ 1.41961e6i 0.301216i 0.988594 + 0.150608i $$0.0481231\pi$$
−0.988594 + 0.150608i $$0.951877\pi$$
$$468$$ 0 0
$$469$$ − 372768.i − 0.0782540i
$$470$$ 0 0
$$471$$ −7.88644e6 −1.63806
$$472$$ 0 0
$$473$$ 1.13931e6 0.234148
$$474$$ 0 0
$$475$$ 7.15644e6i 1.45534i
$$476$$ 0 0
$$477$$ 1.83470e6i 0.369207i
$$478$$ 0 0
$$479$$ 1.88406e6 0.375195 0.187597 0.982246i $$-0.439930\pi$$
0.187597 + 0.982246i $$0.439930\pi$$
$$480$$ 0 0
$$481$$ 4.93583e6 0.972741
$$482$$ 0 0
$$483$$ 88320.0i 0.0172263i
$$484$$ 0 0
$$485$$ 3.61608e6i 0.698046i
$$486$$ 0 0
$$487$$ −6.01388e6 −1.14903 −0.574516 0.818493i $$-0.694810\pi$$
−0.574516 + 0.818493i $$0.694810\pi$$
$$488$$ 0 0
$$489$$ 234480. 0.0443439
$$490$$ 0 0
$$491$$ 4.29232e6i 0.803504i 0.915749 + 0.401752i $$0.131599\pi$$
−0.915749 + 0.401752i $$0.868401\pi$$
$$492$$ 0 0
$$493$$ 3.93184e6i 0.728581i
$$494$$ 0 0
$$495$$ 1.44063e6 0.264265
$$496$$ 0 0
$$497$$ 767040. 0.139292
$$498$$ 0 0
$$499$$ − 1.34509e6i − 0.241825i −0.992663 0.120912i $$-0.961418\pi$$
0.992663 0.120912i $$-0.0385820\pi$$
$$500$$ 0 0
$$501$$ 1.10386e7i 1.96480i
$$502$$ 0 0
$$503$$ 202008. 0.0355999 0.0177999 0.999842i $$-0.494334\pi$$
0.0177999 + 0.999842i $$0.494334\pi$$
$$504$$ 0 0
$$505$$ 3.81884e6 0.666352
$$506$$ 0 0
$$507$$ − 2.85618e6i − 0.493476i
$$508$$ 0 0
$$509$$ 9.78344e6i 1.67377i 0.547375 + 0.836887i $$0.315627\pi$$
−0.547375 + 0.836887i $$0.684373\pi$$
$$510$$ 0 0
$$511$$ −117264. −0.0198661
$$512$$ 0 0
$$513$$ −5.23568e6 −0.878374
$$514$$ 0 0
$$515$$ 1.33514e7i 2.21824i
$$516$$ 0 0
$$517$$ − 2.93434e6i − 0.482818i
$$518$$ 0 0
$$519$$ 8.65788e6 1.41089
$$520$$ 0 0
$$521$$ 1.04830e7 1.69197 0.845985 0.533207i $$-0.179013\pi$$
0.845985 + 0.533207i $$0.179013\pi$$
$$522$$ 0 0
$$523$$ 6.21017e6i 0.992772i 0.868102 + 0.496386i $$0.165340\pi$$
−0.868102 + 0.496386i $$0.834660\pi$$
$$524$$ 0 0
$$525$$ − 1.12848e6i − 0.178688i
$$526$$ 0 0
$$527$$ −6.86214e6 −1.07630
$$528$$ 0 0
$$529$$ −6.40249e6 −0.994740
$$530$$ 0 0
$$531$$ − 2.64953e6i − 0.407787i
$$532$$ 0 0
$$533$$ 4.24751e6i 0.647614i
$$534$$ 0 0
$$535$$ 4.86180e6 0.734366
$$536$$ 0 0
$$537$$ −1.11924e7 −1.67489
$$538$$ 0 0
$$539$$ − 2.01264e6i − 0.298397i
$$540$$ 0 0
$$541$$ 5.08088e6i 0.746355i 0.927760 + 0.373178i $$0.121732\pi$$
−0.927760 + 0.373178i $$0.878268\pi$$
$$542$$ 0 0
$$543$$ −1.20942e7 −1.76026
$$544$$ 0 0
$$545$$ 8.34024e6 1.20278
$$546$$ 0 0
$$547$$ − 3.34687e6i − 0.478267i −0.970987 0.239133i $$-0.923137\pi$$
0.970987 0.239133i $$-0.0768633\pi$$
$$548$$ 0 0
$$549$$ 2.90167e6i 0.410883i
$$550$$ 0 0
$$551$$ −9.99041e6 −1.40186
$$552$$ 0 0
$$553$$ 1.06944e6 0.148711
$$554$$ 0 0
$$555$$ − 1.52825e7i − 2.10601i
$$556$$ 0 0
$$557$$ 7.00377e6i 0.956520i 0.878218 + 0.478260i $$0.158732\pi$$
−0.878218 + 0.478260i $$0.841268\pi$$
$$558$$ 0 0
$$559$$ 4.39186e6 0.594456
$$560$$ 0 0
$$561$$ −2.97104e6 −0.398567
$$562$$ 0 0
$$563$$ 1.29819e7i 1.72610i 0.505116 + 0.863052i $$0.331450\pi$$
−0.505116 + 0.863052i $$0.668550\pi$$
$$564$$ 0 0
$$565$$ − 1.73915e6i − 0.229200i
$$566$$ 0 0
$$567$$ 1.74122e6 0.227456
$$568$$ 0 0
$$569$$ −1.89942e6 −0.245946 −0.122973 0.992410i $$-0.539243\pi$$
−0.122973 + 0.992410i $$0.539243\pi$$
$$570$$ 0 0
$$571$$ − 1.66300e6i − 0.213452i −0.994288 0.106726i $$-0.965963\pi$$
0.994288 0.106726i $$-0.0340368\pi$$
$$572$$ 0 0
$$573$$ − 8.18304e6i − 1.04119i
$$574$$ 0 0
$$575$$ −432584. −0.0545633
$$576$$ 0 0
$$577$$ 8.77344e6 1.09706 0.548530 0.836131i $$-0.315188\pi$$
0.548530 + 0.836131i $$0.315188\pi$$
$$578$$ 0 0
$$579$$ − 1.08173e7i − 1.34098i
$$580$$ 0 0
$$581$$ 1.61674e6i 0.198700i
$$582$$ 0 0
$$583$$ 1.44906e6 0.176570
$$584$$ 0 0
$$585$$ 5.55340e6 0.670918
$$586$$ 0 0
$$587$$ 5.18393e6i 0.620961i 0.950580 + 0.310480i $$0.100490\pi$$
−0.950580 + 0.310480i $$0.899510\pi$$
$$588$$ 0 0
$$589$$ − 1.74360e7i − 2.07090i
$$590$$ 0 0
$$591$$ 1.25980e7 1.48365
$$592$$ 0 0
$$593$$ 8.49858e6 0.992452 0.496226 0.868193i $$-0.334719\pi$$
0.496226 + 0.868193i $$0.334719\pi$$
$$594$$ 0 0
$$595$$ 2.12765e6i 0.246381i
$$596$$ 0 0
$$597$$ − 5.66096e6i − 0.650061i
$$598$$ 0 0
$$599$$ 1.12471e7 1.28078 0.640388 0.768051i $$-0.278773\pi$$
0.640388 + 0.768051i $$0.278773\pi$$
$$600$$ 0 0
$$601$$ 3.46439e6 0.391238 0.195619 0.980680i $$-0.437328\pi$$
0.195619 + 0.980680i $$0.437328\pi$$
$$602$$ 0 0
$$603$$ − 2.43852e6i − 0.273108i
$$604$$ 0 0
$$605$$ 1.07799e7i 1.19737i
$$606$$ 0 0
$$607$$ 999712. 0.110129 0.0550647 0.998483i $$-0.482463\pi$$
0.0550647 + 0.998483i $$0.482463\pi$$
$$608$$ 0 0
$$609$$ 1.57536e6 0.172122
$$610$$ 0 0
$$611$$ − 1.13114e7i − 1.22578i
$$612$$ 0 0
$$613$$ − 9.81340e6i − 1.05480i −0.849619 0.527398i $$-0.823168\pi$$
0.849619 0.527398i $$-0.176832\pi$$
$$614$$ 0 0
$$615$$ 1.31513e7 1.40210
$$616$$ 0 0
$$617$$ 5.34745e6 0.565501 0.282751 0.959193i $$-0.408753\pi$$
0.282751 + 0.959193i $$0.408753\pi$$
$$618$$ 0 0
$$619$$ − 6.82768e6i − 0.716221i −0.933679 0.358110i $$-0.883421\pi$$
0.933679 0.358110i $$-0.116579\pi$$
$$620$$ 0 0
$$621$$ − 316480.i − 0.0329319i
$$622$$ 0 0
$$623$$ 1.72786e6 0.178356
$$624$$ 0 0
$$625$$ −1.15853e7 −1.18633
$$626$$ 0 0
$$627$$ − 7.54912e6i − 0.766880i
$$628$$ 0 0
$$629$$ 1.23705e7i 1.24670i
$$630$$ 0 0
$$631$$ −3.60970e6 −0.360909 −0.180455 0.983583i $$-0.557757\pi$$
−0.180455 + 0.983583i $$0.557757\pi$$
$$632$$ 0 0
$$633$$ −2.85512e6 −0.283214
$$634$$ 0 0
$$635$$ 6.99981e6i 0.688893i
$$636$$ 0 0
$$637$$ − 7.75842e6i − 0.757573i
$$638$$ 0 0
$$639$$ 5.01772e6 0.486132
$$640$$ 0 0
$$641$$ −1.33853e7 −1.28672 −0.643361 0.765563i $$-0.722460\pi$$
−0.643361 + 0.765563i $$0.722460\pi$$
$$642$$ 0 0
$$643$$ 9.91115e6i 0.945358i 0.881235 + 0.472679i $$0.156713\pi$$
−0.881235 + 0.472679i $$0.843287\pi$$
$$644$$ 0 0
$$645$$ − 1.35982e7i − 1.28701i
$$646$$ 0 0
$$647$$ −1.78359e7 −1.67508 −0.837539 0.546378i $$-0.816006\pi$$
−0.837539 + 0.546378i $$0.816006\pi$$
$$648$$ 0 0
$$649$$ −2.09262e6 −0.195020
$$650$$ 0 0
$$651$$ 2.74944e6i 0.254268i
$$652$$ 0 0
$$653$$ − 4.32323e6i − 0.396758i −0.980125 0.198379i $$-0.936432\pi$$
0.980125 0.198379i $$-0.0635677\pi$$
$$654$$ 0 0
$$655$$ 5.20161e6 0.473734
$$656$$ 0 0
$$657$$ −767102. −0.0693330
$$658$$ 0 0
$$659$$ − 1.97858e7i − 1.77476i −0.461035 0.887382i $$-0.652522\pi$$
0.461035 0.887382i $$-0.347478\pi$$
$$660$$ 0 0
$$661$$ − 1.57772e7i − 1.40451i −0.711925 0.702255i $$-0.752176\pi$$
0.711925 0.702255i $$-0.247824\pi$$
$$662$$ 0 0
$$663$$ −1.14529e7 −1.01188
$$664$$ 0 0
$$665$$ −5.40614e6 −0.474060
$$666$$ 0 0
$$667$$ − 603888.i − 0.0525584i
$$668$$ 0 0
$$669$$ 1.77939e7i 1.53711i
$$670$$ 0 0
$$671$$ 2.29177e6 0.196501
$$672$$ 0 0
$$673$$ 6.78762e6 0.577670 0.288835 0.957379i $$-0.406732\pi$$
0.288835 + 0.957379i $$0.406732\pi$$
$$674$$ 0 0
$$675$$ 4.04372e6i 0.341603i
$$676$$ 0 0
$$677$$ 1.49942e7i 1.25734i 0.777673 + 0.628669i $$0.216400\pi$$
−0.777673 + 0.628669i $$0.783600\pi$$
$$678$$ 0 0
$$679$$ −1.17278e6 −0.0976211
$$680$$ 0 0
$$681$$ −2.28631e7 −1.88916
$$682$$ 0 0
$$683$$ 1.15580e7i 0.948053i 0.880511 + 0.474026i $$0.157200\pi$$
−0.880511 + 0.474026i $$0.842800\pi$$
$$684$$ 0 0
$$685$$ − 2.05195e7i − 1.67086i
$$686$$ 0 0
$$687$$ 1.39157e7 1.12490
$$688$$ 0 0
$$689$$ 5.58591e6 0.448276
$$690$$ 0 0
$$691$$ 220156.i 0.0175402i 0.999962 + 0.00877012i $$0.00279165\pi$$
−0.999962 + 0.00877012i $$0.997208\pi$$
$$692$$ 0 0
$$693$$ 467232.i 0.0369572i
$$694$$ 0 0
$$695$$ 9.64279e6 0.757253
$$696$$ 0 0
$$697$$ −1.06454e7 −0.830006
$$698$$ 0 0
$$699$$ − 6.94252e6i − 0.537433i
$$700$$ 0 0
$$701$$ 4.78933e6i 0.368111i 0.982916 + 0.184056i $$0.0589227\pi$$
−0.982916 + 0.184056i $$0.941077\pi$$
$$702$$ 0 0
$$703$$ −3.14323e7 −2.39877
$$704$$ 0 0
$$705$$ −3.50227e7 −2.65385
$$706$$ 0 0
$$707$$ 1.23854e6i 0.0931886i
$$708$$ 0 0
$$709$$ − 4.26892e6i − 0.318935i −0.987203 0.159468i $$-0.949022\pi$$
0.987203 0.159468i $$-0.0509777\pi$$
$$710$$ 0 0
$$711$$ 6.99592e6 0.519004
$$712$$ 0 0
$$713$$ 1.05395e6 0.0776421
$$714$$ 0 0
$$715$$ − 4.38613e6i − 0.320860i
$$716$$ 0 0
$$717$$ − 3.28592e7i − 2.38704i
$$718$$ 0 0
$$719$$ 1.61960e7 1.16838 0.584190 0.811617i $$-0.301412\pi$$
0.584190 + 0.811617i $$0.301412\pi$$
$$720$$ 0 0
$$721$$ −4.33018e6 −0.310218
$$722$$ 0 0
$$723$$ 2.33488e7i 1.66119i
$$724$$ 0 0
$$725$$ 7.71598e6i 0.545188i
$$726$$ 0 0
$$727$$ 6.53426e6 0.458522 0.229261 0.973365i $$-0.426369\pi$$
0.229261 + 0.973365i $$0.426369\pi$$
$$728$$ 0 0
$$729$$ 3.03131e6 0.211257
$$730$$ 0 0
$$731$$ 1.10072e7i 0.761876i
$$732$$ 0 0
$$733$$ 1.31617e7i 0.904800i 0.891815 + 0.452400i $$0.149432\pi$$
−0.891815 + 0.452400i $$0.850568\pi$$
$$734$$ 0 0
$$735$$ −2.40219e7 −1.64017
$$736$$ 0 0
$$737$$ −1.92597e6 −0.130611
$$738$$ 0 0
$$739$$ 1.42348e7i 0.958825i 0.877590 + 0.479412i $$0.159150\pi$$
−0.877590 + 0.479412i $$0.840850\pi$$
$$740$$ 0 0
$$741$$ − 2.91006e7i − 1.94696i
$$742$$ 0 0
$$743$$ −2.15835e7 −1.43434 −0.717168 0.696901i $$-0.754562\pi$$
−0.717168 + 0.696901i $$0.754562\pi$$
$$744$$ 0 0
$$745$$ −2.97132e7 −1.96137
$$746$$ 0 0
$$747$$ 1.05761e7i 0.693467i
$$748$$ 0 0
$$749$$ 1.57680e6i 0.102700i
$$750$$ 0 0
$$751$$ −1.86594e7 −1.20725 −0.603625 0.797268i $$-0.706278\pi$$
−0.603625 + 0.797268i $$0.706278\pi$$
$$752$$ 0 0
$$753$$ −1.58122e7 −1.01626
$$754$$ 0 0
$$755$$ − 5.62222e6i − 0.358956i
$$756$$ 0 0
$$757$$ 2.56681e6i 0.162800i 0.996682 + 0.0813999i $$0.0259391\pi$$
−0.996682 + 0.0813999i $$0.974061\pi$$
$$758$$ 0 0
$$759$$ 456320. 0.0287518
$$760$$ 0 0
$$761$$ 2.59586e7 1.62487 0.812436 0.583051i $$-0.198141\pi$$
0.812436 + 0.583051i $$0.198141\pi$$
$$762$$ 0 0
$$763$$ 2.70494e6i 0.168208i
$$764$$ 0 0
$$765$$ 1.39184e7i 0.859874i
$$766$$ 0 0
$$767$$ −8.06673e6 −0.495118
$$768$$ 0 0
$$769$$ 5.53267e6 0.337380 0.168690 0.985669i $$-0.446046\pi$$
0.168690 + 0.985669i $$0.446046\pi$$
$$770$$ 0 0
$$771$$ 2.59580e6i 0.157266i
$$772$$ 0 0
$$773$$ − 8.32940e6i − 0.501378i −0.968068 0.250689i $$-0.919343\pi$$
0.968068 0.250689i $$-0.0806571\pi$$
$$774$$ 0 0
$$775$$ −1.34665e7 −0.805381
$$776$$ 0 0
$$777$$ 4.95648e6 0.294524
$$778$$ 0 0
$$779$$ − 2.70490e7i − 1.59701i
$$780$$ 0 0
$$781$$ − 3.96304e6i − 0.232488i
$$782$$ 0 0
$$783$$ −5.64504e6 −0.329051
$$784$$ 0 0
$$785$$ 2.91798e7 1.69009
$$786$$ 0 0
$$787$$ 1.36523e7i 0.785719i 0.919598 + 0.392860i $$0.128514\pi$$
−0.919598 + 0.392860i $$0.871486\pi$$
$$788$$ 0 0
$$789$$ − 1.41776e6i − 0.0810793i
$$790$$ 0 0
$$791$$ 564048. 0.0320535
$$792$$ 0 0
$$793$$ 8.83440e6 0.498877
$$794$$ 0 0
$$795$$ − 1.72953e7i − 0.970532i
$$796$$ 0 0
$$797$$ − 8.54626e6i − 0.476574i −0.971195 0.238287i $$-0.923414\pi$$
0.971195 0.238287i $$-0.0765859\pi$$
$$798$$ 0 0
$$799$$ 2.83495e7 1.57101
$$800$$ 0 0
$$801$$ 1.13031e7 0.622465
$$802$$ 0 0
$$803$$ 605864.i 0.0331578i
$$804$$ 0 0
$$805$$ − 326784.i − 0.0177734i
$$806$$ 0 0
$$807$$ 3.58035e7 1.93527
$$808$$ 0 0
$$809$$ −7.58484e6 −0.407451 −0.203725 0.979028i $$-0.565305\pi$$
−0.203725 + 0.979028i $$0.565305\pi$$
$$810$$ 0 0
$$811$$ 6.18473e6i 0.330194i 0.986277 + 0.165097i $$0.0527937\pi$$
−0.986277 + 0.165097i $$0.947206\pi$$
$$812$$ 0 0
$$813$$ − 3.54723e7i − 1.88219i
$$814$$ 0 0
$$815$$ −867576. −0.0457524
$$816$$ 0 0
$$817$$ −2.79683e7 −1.46592
$$818$$ 0 0
$$819$$ 1.80110e6i 0.0938273i
$$820$$ 0 0
$$821$$ 2.78102e6i 0.143995i 0.997405 + 0.0719973i $$0.0229373\pi$$
−0.997405 + 0.0719973i $$0.977063\pi$$
$$822$$ 0 0
$$823$$ 1.63895e7 0.843461 0.421731 0.906721i $$-0.361423\pi$$
0.421731 + 0.906721i $$0.361423\pi$$
$$824$$ 0 0
$$825$$ −5.83048e6 −0.298242
$$826$$ 0 0
$$827$$ 2.29511e7i 1.16692i 0.812142 + 0.583459i $$0.198301\pi$$
−0.812142 + 0.583459i $$0.801699\pi$$
$$828$$ 0 0
$$829$$ − 3.50136e6i − 0.176950i −0.996078 0.0884750i $$-0.971801\pi$$
0.996078 0.0884750i $$-0.0281993\pi$$
$$830$$ 0 0
$$831$$ 5.50900e6 0.276739
$$832$$ 0 0
$$833$$ 1.94447e7 0.970934
$$834$$ 0 0
$$835$$ − 4.08427e7i − 2.02721i
$$836$$ 0 0
$$837$$ − 9.85216e6i − 0.486091i
$$838$$ 0 0
$$839$$ 5.29668e6 0.259776 0.129888 0.991529i $$-0.458538\pi$$
0.129888 + 0.991529i $$0.458538\pi$$
$$840$$ 0 0
$$841$$ 9.73962e6 0.474845
$$842$$ 0 0
$$843$$ 1.18834e7i 0.575933i
$$844$$ 0 0
$$845$$ 1.05679e7i 0.509150i
$$846$$ 0 0
$$847$$ −3.49620e6 −0.167451
$$848$$ 0 0
$$849$$ 2.18486e7 1.04029
$$850$$ 0 0
$$851$$ − 1.89998e6i − 0.0899344i
$$852$$ 0 0
$$853$$ 2.02948e7i 0.955021i 0.878626 + 0.477511i $$0.158461\pi$$
−0.878626 + 0.477511i $$0.841539\pi$$
$$854$$ 0 0
$$855$$ −3.53652e7 −1.65448
$$856$$ 0 0
$$857$$ 4.82785e6 0.224544 0.112272 0.993678i $$-0.464187\pi$$
0.112272 + 0.993678i $$0.464187\pi$$
$$858$$ 0 0
$$859$$ − 1.30210e7i − 0.602092i −0.953610 0.301046i $$-0.902664\pi$$
0.953610 0.301046i $$-0.0973358\pi$$
$$860$$ 0 0
$$861$$ 4.26528e6i 0.196083i
$$862$$ 0 0
$$863$$ 3.92387e7 1.79344 0.896721 0.442596i $$-0.145942\pi$$
0.896721 + 0.442596i $$0.145942\pi$$
$$864$$ 0 0
$$865$$ −3.20342e7 −1.45570
$$866$$ 0 0
$$867$$ − 306940.i − 0.0138677i
$$868$$ 0 0
$$869$$ − 5.52544e6i − 0.248209i
$$870$$ 0 0
$$871$$ −7.42430e6 −0.331596
$$872$$ 0 0
$$873$$ −7.67196e6 −0.340699
$$874$$ 0 0
$$875$$ − 1.37462e6i − 0.0606965i
$$876$$ 0 0
$$877$$ 1.34622e7i 0.591041i 0.955336 + 0.295520i $$0.0954930\pi$$
−0.955336 + 0.295520i $$0.904507\pi$$
$$878$$ 0 0
$$879$$ −6.67308e6 −0.291309
$$880$$ 0 0
$$881$$ −917710. −0.0398351 −0.0199175 0.999802i $$-0.506340\pi$$
−0.0199175 + 0.999802i $$0.506340\pi$$
$$882$$ 0 0
$$883$$ − 2.45488e7i − 1.05957i −0.848133 0.529784i $$-0.822273\pi$$
0.848133 0.529784i $$-0.177727\pi$$
$$884$$ 0 0
$$885$$ 2.49765e7i 1.07195i
$$886$$ 0 0
$$887$$ −1.61463e7 −0.689070 −0.344535 0.938773i $$-0.611963\pi$$
−0.344535 + 0.938773i $$0.611963\pi$$
$$888$$ 0 0
$$889$$ −2.27021e6 −0.0963410
$$890$$ 0 0
$$891$$ − 8.99632e6i − 0.379639i
$$892$$ 0 0
$$893$$ 7.20332e7i 3.02276i
$$894$$ 0 0
$$895$$ 4.14119e7 1.72809
$$896$$ 0 0
$$897$$ 1.75904e6 0.0729953
$$898$$ 0 0
$$899$$ − 1.87993e7i − 0.775787i
$$900$$ 0 0
$$901$$ 1.39998e7i 0.574527i
$$902$$ 0 0
$$903$$ 4.41024e6 0.179988
$$904$$ 0 0
$$905$$ 4.47485e7 1.81617
$$906$$ 0 0
$$907$$ − 2.03361e7i − 0.820824i −0.911900 0.410412i $$-0.865385\pi$$
0.911900 0.410412i $$-0.134615\pi$$
$$908$$ 0 0
$$909$$ 8.10214e6i 0.325230i
$$910$$ 0 0
$$911$$ −1.07726e7 −0.430054 −0.215027 0.976608i $$-0.568984\pi$$
−0.215027 + 0.976608i $$0.568984\pi$$
$$912$$ 0 0
$$913$$ 8.35314e6 0.331644
$$914$$ 0 0
$$915$$ − 2.73534e7i − 1.08009i
$$916$$ 0 0
$$917$$ 1.68701e6i 0.0662512i
$$918$$ 0 0
$$919$$ 4.18566e7 1.63484 0.817419 0.576043i $$-0.195404\pi$$
0.817419 + 0.576043i $$0.195404\pi$$
$$920$$ 0 0
$$921$$ −2.11994e7 −0.823522
$$922$$ 0 0
$$923$$ − 1.52769e7i − 0.590242i
$$924$$ 0 0
$$925$$ 2.42764e7i 0.932890i
$$926$$ 0 0
$$927$$ −2.83266e7 −1.08267
$$928$$ 0 0
$$929$$ 2.99845e7 1.13988 0.569939 0.821687i $$-0.306967\pi$$
0.569939 + 0.821687i $$0.306967\pi$$
$$930$$ 0 0
$$931$$ 4.94072e7i 1.86817i
$$932$$ 0 0
$$933$$ 2.67298e7i 1.00529i
$$934$$ 0 0
$$935$$ 1.09928e7 0.411227
$$936$$ 0 0
$$937$$ −1.42402e7 −0.529867 −0.264934 0.964267i $$-0.585350\pi$$
−0.264934 + 0.964267i $$0.585350\pi$$
$$938$$ 0 0
$$939$$ 3.28837e7i 1.21707i
$$940$$ 0 0
$$941$$ − 4.14546e7i − 1.52615i −0.646307 0.763077i $$-0.723687\pi$$
0.646307 0.763077i $$-0.276313\pi$$
$$942$$ 0 0
$$943$$ 1.63502e6 0.0598749
$$944$$ 0 0
$$945$$ −3.05472e6 −0.111274
$$946$$ 0 0
$$947$$ 1.54079e7i 0.558300i 0.960248 + 0.279150i $$0.0900527\pi$$
−0.960248 + 0.279150i $$0.909947\pi$$
$$948$$ 0 0
$$949$$ 2.33551e6i 0.0841813i
$$950$$ 0 0
$$951$$ −3.44740e7 −1.23606
$$952$$ 0 0
$$953$$ 2.06328e7 0.735912 0.367956 0.929843i $$-0.380058\pi$$
0.367956 + 0.929843i $$0.380058\pi$$
$$954$$ 0 0
$$955$$ 3.02772e7i 1.07426i
$$956$$ 0 0
$$957$$ − 8.13936e6i − 0.287283i
$$958$$ 0 0
$$959$$ 6.65496e6 0.233668
$$960$$ 0 0
$$961$$ 4.18083e6 0.146034
$$962$$ 0 0
$$963$$ 1.03149e7i 0.358426i
$$964$$ 0 0
$$965$$ 4.00241e7i 1.38358i
$$966$$ 0 0
$$967$$ 1.18724e7 0.408294 0.204147 0.978940i $$-0.434558\pi$$
0.204147 + 0.978940i $$0.434558\pi$$
$$968$$ 0 0
$$969$$ 7.29342e7 2.49530
$$970$$ 0 0
$$971$$ 1.53222e6i 0.0521523i 0.999660 + 0.0260761i $$0.00830123\pi$$
−0.999660 + 0.0260761i $$0.991699\pi$$
$$972$$ 0 0
$$973$$ 3.12739e6i 0.105901i
$$974$$ 0 0
$$975$$ −2.24756e7 −0.757180
$$976$$ 0 0
$$977$$ 1.74321e7 0.584269 0.292135 0.956377i $$-0.405635\pi$$
0.292135 + 0.956377i $$0.405635\pi$$
$$978$$ 0 0
$$979$$ − 8.92726e6i − 0.297688i
$$980$$ 0 0
$$981$$ 1.76948e7i 0.587049i
$$982$$ 0 0
$$983$$ 2.23270e6 0.0736963 0.0368482 0.999321i $$-0.488268\pi$$
0.0368482 + 0.999321i $$0.488268\pi$$
$$984$$ 0 0
$$985$$ −4.66125e7 −1.53078
$$986$$ 0 0
$$987$$ − 1.13587e7i − 0.371139i
$$988$$ 0 0
$$989$$ − 1.69059e6i − 0.0549602i
$$990$$ 0 0
$$991$$ −2.22501e7 −0.719693 −0.359847 0.933011i $$-0.617171\pi$$
−0.359847 + 0.933011i $$0.617171\pi$$
$$992$$ 0 0
$$993$$ 5.49926e7 1.76983
$$994$$ 0 0
$$995$$ 2.09456e7i 0.670709i
$$996$$ 0 0
$$997$$ − 5.32662e7i − 1.69712i −0.529095 0.848562i $$-0.677469\pi$$
0.529095 0.848562i $$-0.322531\pi$$
$$998$$ 0 0
$$999$$ −1.77607e7 −0.563050
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 256.6.b.d.129.1 2
4.3 odd 2 256.6.b.f.129.2 2
8.3 odd 2 256.6.b.f.129.1 2
8.5 even 2 inner 256.6.b.d.129.2 2
16.3 odd 4 64.6.a.a.1.1 1
16.5 even 4 16.6.a.a.1.1 1
16.11 odd 4 8.6.a.a.1.1 1
16.13 even 4 64.6.a.g.1.1 1
48.5 odd 4 144.6.a.k.1.1 1
48.11 even 4 72.6.a.f.1.1 1
48.29 odd 4 576.6.a.h.1.1 1
48.35 even 4 576.6.a.g.1.1 1
80.27 even 4 200.6.c.a.49.1 2
80.37 odd 4 400.6.c.d.49.2 2
80.43 even 4 200.6.c.a.49.2 2
80.53 odd 4 400.6.c.d.49.1 2
80.59 odd 4 200.6.a.a.1.1 1
80.69 even 4 400.6.a.l.1.1 1
112.11 odd 12 392.6.i.b.177.1 2
112.27 even 4 392.6.a.b.1.1 1
112.59 even 12 392.6.i.e.177.1 2
112.69 odd 4 784.6.a.l.1.1 1
112.75 even 12 392.6.i.e.361.1 2
112.107 odd 12 392.6.i.b.361.1 2
176.43 even 4 968.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.a.a.1.1 1 16.11 odd 4
16.6.a.a.1.1 1 16.5 even 4
64.6.a.a.1.1 1 16.3 odd 4
64.6.a.g.1.1 1 16.13 even 4
72.6.a.f.1.1 1 48.11 even 4
144.6.a.k.1.1 1 48.5 odd 4
200.6.a.a.1.1 1 80.59 odd 4
200.6.c.a.49.1 2 80.27 even 4
200.6.c.a.49.2 2 80.43 even 4
256.6.b.d.129.1 2 1.1 even 1 trivial
256.6.b.d.129.2 2 8.5 even 2 inner
256.6.b.f.129.1 2 8.3 odd 2
256.6.b.f.129.2 2 4.3 odd 2
392.6.a.b.1.1 1 112.27 even 4
392.6.i.b.177.1 2 112.11 odd 12
392.6.i.b.361.1 2 112.107 odd 12
392.6.i.e.177.1 2 112.59 even 12
392.6.i.e.361.1 2 112.75 even 12
400.6.a.l.1.1 1 80.69 even 4
400.6.c.d.49.1 2 80.53 odd 4
400.6.c.d.49.2 2 80.37 odd 4
576.6.a.g.1.1 1 48.35 even 4
576.6.a.h.1.1 1 48.29 odd 4
784.6.a.l.1.1 1 112.69 odd 4
968.6.a.a.1.1 1 176.43 even 4