# Properties

 Label 200.6.c.a.49.2 Level $200$ Weight $6$ Character 200.49 Analytic conductor $32.077$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0767639626$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 200.49 Dual form 200.6.c.a.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+20.0000i q^{3} +24.0000i q^{7} -157.000 q^{9} +O(q^{10})$$ $$q+20.0000i q^{3} +24.0000i q^{7} -157.000 q^{9} +124.000 q^{11} +478.000i q^{13} +1198.00i q^{17} -3044.00 q^{19} -480.000 q^{21} +184.000i q^{23} +1720.00i q^{27} +3282.00 q^{29} -5728.00 q^{31} +2480.00i q^{33} -10326.0i q^{37} -9560.00 q^{39} -8886.00 q^{41} -9188.00i q^{43} -23664.0i q^{47} +16231.0 q^{49} -23960.0 q^{51} +11686.0i q^{53} -60880.0i q^{57} -16876.0 q^{59} -18482.0 q^{61} -3768.00i q^{63} +15532.0i q^{67} -3680.00 q^{69} -31960.0 q^{71} -4886.00i q^{73} +2976.00i q^{77} -44560.0 q^{79} -72551.0 q^{81} +67364.0i q^{83} +65640.0i q^{87} -71994.0 q^{89} -11472.0 q^{91} -114560. i q^{93} -48866.0i q^{97} -19468.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 314 q^{9} + O(q^{10})$$ $$2 q - 314 q^{9} + 248 q^{11} - 6088 q^{19} - 960 q^{21} + 6564 q^{29} - 11456 q^{31} - 19120 q^{39} - 17772 q^{41} + 32462 q^{49} - 47920 q^{51} - 33752 q^{59} - 36964 q^{61} - 7360 q^{69} - 63920 q^{71} - 89120 q^{79} - 145102 q^{81} - 143988 q^{89} - 22944 q^{91} - 38936 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$0.221688\pi$$
−0.767123 + 0.641500i $$0.778312\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 24.0000i 0.185125i 0.995707 + 0.0925627i $$0.0295059\pi$$
−0.995707 + 0.0925627i $$0.970494\pi$$
$$8$$ 0 0
$$9$$ −157.000 −0.646091
$$10$$ 0 0
$$11$$ 124.000 0.308987 0.154493 0.987994i $$-0.450625\pi$$
0.154493 + 0.987994i $$0.450625\pi$$
$$12$$ 0 0
$$13$$ 478.000i 0.784458i 0.919868 + 0.392229i $$0.128296\pi$$
−0.919868 + 0.392229i $$0.871704\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1198.00i 1.00539i 0.864464 + 0.502695i $$0.167658\pi$$
−0.864464 + 0.502695i $$0.832342\pi$$
$$18$$ 0 0
$$19$$ −3044.00 −1.93446 −0.967232 0.253894i $$-0.918288\pi$$
−0.967232 + 0.253894i $$0.918288\pi$$
$$20$$ 0 0
$$21$$ −480.000 −0.237516
$$22$$ 0 0
$$23$$ 184.000i 0.0725268i 0.999342 + 0.0362634i $$0.0115455\pi$$
−0.999342 + 0.0362634i $$0.988454\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1720.00i 0.454066i
$$28$$ 0 0
$$29$$ 3282.00 0.724676 0.362338 0.932047i $$-0.381979\pi$$
0.362338 + 0.932047i $$0.381979\pi$$
$$30$$ 0 0
$$31$$ −5728.00 −1.07053 −0.535265 0.844684i $$-0.679788\pi$$
−0.535265 + 0.844684i $$0.679788\pi$$
$$32$$ 0 0
$$33$$ 2480.00i 0.396430i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 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 $$-0.635442\pi$$
−0.412778 + 0.910832i $$0.635442\pi$$
$$42$$ 0 0
$$43$$ − 9188.00i − 0.757792i −0.925439 0.378896i $$-0.876304\pi$$
0.925439 0.378896i $$-0.123696\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 23664.0i − 1.56258i −0.624165 0.781292i $$-0.714561\pi$$
0.624165 0.781292i $$-0.285439\pi$$
$$48$$ 0 0
$$49$$ 16231.0 0.965729
$$50$$ 0 0
$$51$$ −23960.0 −1.28992
$$52$$ 0 0
$$53$$ 11686.0i 0.571447i 0.958312 + 0.285724i $$0.0922339\pi$$
−0.958312 + 0.285724i $$0.907766\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 60880.0i − 2.48192i
$$58$$ 0 0
$$59$$ −16876.0 −0.631160 −0.315580 0.948899i $$-0.602199\pi$$
−0.315580 + 0.948899i $$0.602199\pi$$
$$60$$ 0 0
$$61$$ −18482.0 −0.635952 −0.317976 0.948099i $$-0.603003\pi$$
−0.317976 + 0.948099i $$0.603003\pi$$
$$62$$ 0 0
$$63$$ − 3768.00i − 0.119608i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 15532.0i 0.422708i 0.977410 + 0.211354i $$0.0677873\pi$$
−0.977410 + 0.211354i $$0.932213\pi$$
$$68$$ 0 0
$$69$$ −3680.00 −0.0930519
$$70$$ 0 0
$$71$$ −31960.0 −0.752421 −0.376210 0.926534i $$-0.622773\pi$$
−0.376210 + 0.926534i $$0.622773\pi$$
$$72$$ 0 0
$$73$$ − 4886.00i − 0.107312i −0.998559 0.0536558i $$-0.982913\pi$$
0.998559 0.0536558i $$-0.0170874\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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 $$0.180316\pi$$
−0.843796 + 0.536664i $$0.819684\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 65640.0i 0.929759i
$$88$$ 0 0
$$89$$ −71994.0 −0.963432 −0.481716 0.876327i $$-0.659986\pi$$
−0.481716 + 0.876327i $$0.659986\pi$$
$$90$$ 0 0
$$91$$ −11472.0 −0.145223
$$92$$ 0 0
$$93$$ − 114560.i − 1.37349i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 48866.0i − 0.527324i −0.964615 0.263662i $$-0.915070\pi$$
0.964615 0.263662i $$-0.0849303\pi$$
$$98$$ 0 0
$$99$$ −19468.0 −0.199633
$$100$$ 0 0
$$101$$ 51606.0 0.503381 0.251690 0.967808i $$-0.419014\pi$$
0.251690 + 0.967808i $$0.419014\pi$$
$$102$$ 0 0
$$103$$ 180424.i 1.67572i 0.545886 + 0.837860i $$0.316193\pi$$
−0.545886 + 0.837860i $$0.683807\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 65700.0i 0.554761i 0.960760 + 0.277381i $$0.0894663\pi$$
−0.960760 + 0.277381i $$0.910534\pi$$
$$108$$ 0 0
$$109$$ 112706. 0.908617 0.454308 0.890844i $$-0.349886\pi$$
0.454308 + 0.890844i $$0.349886\pi$$
$$110$$ 0 0
$$111$$ 206520. 1.59094
$$112$$ 0 0
$$113$$ − 23502.0i − 0.173145i −0.996246 0.0865723i $$-0.972409\pi$$
0.996246 0.0865723i $$-0.0275913\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 94592.0i 0.520409i 0.965553 + 0.260205i $$0.0837900\pi$$
−0.965553 + 0.260205i $$0.916210\pi$$
$$128$$ 0 0
$$129$$ 183760. 0.972247
$$130$$ 0 0
$$131$$ 70292.0 0.357872 0.178936 0.983861i $$-0.442735\pi$$
0.178936 + 0.983861i $$0.442735\pi$$
$$132$$ 0 0
$$133$$ − 73056.0i − 0.358119i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 277290.i − 1.26221i −0.775696 0.631107i $$-0.782601\pi$$
0.775696 0.631107i $$-0.217399\pi$$
$$138$$ 0 0
$$139$$ 130308. 0.572050 0.286025 0.958222i $$-0.407666\pi$$
0.286025 + 0.958222i $$0.407666\pi$$
$$140$$ 0 0
$$141$$ 473280. 2.00480
$$142$$ 0 0
$$143$$ 59272.0i 0.242387i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 324620.i 1.23903i
$$148$$ 0 0
$$149$$ 401530. 1.48167 0.740836 0.671685i $$-0.234429\pi$$
0.740836 + 0.671685i $$0.234429\pi$$
$$150$$ 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.i − 0.649573i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 394322.i 1.27674i 0.769730 + 0.638369i $$0.220391\pi$$
−0.769730 + 0.638369i $$0.779609\pi$$
$$158$$ 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 $$-0.994499\pi$$
0.999851 0.0172813i $$-0.00550109\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 551928.i 1.53141i 0.643192 + 0.765705i $$0.277610\pi$$
−0.643192 + 0.765705i $$0.722390\pi$$
$$168$$ 0 0
$$169$$ 142809. 0.384626
$$170$$ 0 0
$$171$$ 477908. 1.24984
$$172$$ 0 0
$$173$$ 432894.i 1.09968i 0.835270 + 0.549840i $$0.185311\pi$$
−0.835270 + 0.549840i $$0.814689\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 337520.i − 0.809779i
$$178$$ 0 0
$$179$$ −559620. −1.30545 −0.652726 0.757594i $$-0.726375\pi$$
−0.652726 + 0.757594i $$0.726375\pi$$
$$180$$ 0 0
$$181$$ 604710. 1.37199 0.685995 0.727607i $$-0.259367\pi$$
0.685995 + 0.727607i $$0.259367\pi$$
$$182$$ 0 0
$$183$$ − 369640.i − 0.815927i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 148552.i 0.310652i
$$188$$ 0 0
$$189$$ −41280.0 −0.0840592
$$190$$ 0 0
$$191$$ −409152. −0.811524 −0.405762 0.913979i $$-0.632994\pi$$
−0.405762 + 0.913979i $$0.632994\pi$$
$$192$$ 0 0
$$193$$ 540866.i 1.04519i 0.852580 + 0.522596i $$0.175037\pi$$
−0.852580 + 0.522596i $$0.824963\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 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 $$-0.581528\pi$$
−0.253336 + 0.967378i $$0.581528\pi$$
$$200$$ 0 0
$$201$$ −310640. −0.542335
$$202$$ 0 0
$$203$$ 78768.0i 0.134156i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 28888.0i − 0.0468588i
$$208$$ 0 0
$$209$$ −377456. −0.597724
$$210$$ 0 0
$$211$$ 142756. 0.220744 0.110372 0.993890i $$-0.464796\pi$$
0.110372 + 0.993890i $$0.464796\pi$$
$$212$$ 0 0
$$213$$ − 639200.i − 0.965357i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 137472.i − 0.198182i
$$218$$ 0 0
$$219$$ 97720.0 0.137681
$$220$$ 0 0
$$221$$ −572644. −0.788686
$$222$$ 0 0
$$223$$ 889696.i 1.19806i 0.800726 + 0.599031i $$0.204447\pi$$
−0.800726 + 0.599031i $$0.795553\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 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. 0.876773 0.438386 0.898787i $$-0.355550\pi$$
0.438386 + 0.898787i $$0.355550\pi$$
$$230$$ 0 0
$$231$$ −59520.0 −0.0733893
$$232$$ 0 0
$$233$$ − 347126.i − 0.418887i −0.977821 0.209444i $$-0.932835\pi$$
0.977821 0.209444i $$-0.0671653\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 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
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 1.45503e6i − 1.51751i
$$248$$ 0 0
$$249$$ −1.34728e6 −1.37708
$$250$$ 0 0
$$251$$ −790612. −0.792098 −0.396049 0.918229i $$-0.629619\pi$$
−0.396049 + 0.918229i $$0.629619\pi$$
$$252$$ 0 0
$$253$$ 22816.0i 0.0224098i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 129790.i 0.122577i 0.998120 + 0.0612884i $$0.0195209\pi$$
−0.998120 + 0.0612884i $$0.980479\pi$$
$$258$$ 0 0
$$259$$ 247824. 0.229559
$$260$$ 0 0
$$261$$ −515274. −0.468206
$$262$$ 0 0
$$263$$ 70888.0i 0.0631951i 0.999501 + 0.0315975i $$0.0100595\pi$$
−0.999501 + 0.0315975i $$0.989941\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 1.43988e6i − 1.23608i
$$268$$ 0 0
$$269$$ −1.79017e6 −1.50839 −0.754197 0.656649i $$-0.771973\pi$$
−0.754197 + 0.656649i $$0.771973\pi$$
$$270$$ 0 0
$$271$$ −1.77362e6 −1.46702 −0.733511 0.679678i $$-0.762120\pi$$
−0.733511 + 0.679678i $$0.762120\pi$$
$$272$$ 0 0
$$273$$ − 229440.i − 0.186321i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 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 $$-0.427942\pi$$
0.224448 + 0.974486i $$0.427942\pi$$
$$282$$ 0 0
$$283$$ 1.09243e6i 0.810824i 0.914134 + 0.405412i $$0.132872\pi$$
−0.914134 + 0.405412i $$0.867128\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 213264.i − 0.152831i
$$288$$ 0 0
$$289$$ −15347.0 −0.0108088
$$290$$ 0 0
$$291$$ 977320. 0.676557
$$292$$ 0 0
$$293$$ 333654.i 0.227053i 0.993535 + 0.113527i $$0.0362147\pi$$
−0.993535 + 0.113527i $$0.963785\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 213280.i 0.140300i
$$298$$ 0 0
$$299$$ −87952.0 −0.0568942
$$300$$ 0 0
$$301$$ 220512. 0.140287
$$302$$ 0 0
$$303$$ 1.03212e6i 0.645838i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 1.05997e6i − 0.641872i −0.947101 0.320936i $$-0.896003\pi$$
0.947101 0.320936i $$-0.103997\pi$$
$$308$$ 0 0
$$309$$ −3.60848e6 −2.14995
$$310$$ 0 0
$$311$$ −1.33649e6 −0.783545 −0.391773 0.920062i $$-0.628138\pi$$
−0.391773 + 0.920062i $$0.628138\pi$$
$$312$$ 0 0
$$313$$ 1.64419e6i 0.948615i 0.880359 + 0.474308i $$0.157302\pi$$
−0.880359 + 0.474308i $$0.842698\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.72370e6i 0.963414i 0.876332 + 0.481707i $$0.159983\pi$$
−0.876332 + 0.481707i $$0.840017\pi$$
$$318$$ 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
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.25412e6i 1.16576i
$$328$$ 0 0
$$329$$ 567936. 0.289274
$$330$$ 0 0
$$331$$ 2.74963e6 1.37944 0.689722 0.724074i $$-0.257733\pi$$
0.689722 + 0.724074i $$0.257733\pi$$
$$332$$ 0 0
$$333$$ 1.62118e6i 0.801164i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3.41489e6i 1.63796i 0.573824 + 0.818978i $$0.305459\pi$$
−0.573824 + 0.818978i $$0.694541\pi$$
$$338$$ 0 0
$$339$$ 470040. 0.222145
$$340$$ 0 0
$$341$$ −710272. −0.330780
$$342$$ 0 0
$$343$$ 792912.i 0.363906i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 730764.i − 0.325802i −0.986642 0.162901i $$-0.947915\pi$$
0.986642 0.162901i $$-0.0520851\pi$$
$$348$$ 0 0
$$349$$ 2.29749e6 1.00969 0.504847 0.863209i $$-0.331549\pi$$
0.504847 + 0.863209i $$0.331549\pi$$
$$350$$ 0 0
$$351$$ −822160. −0.356196
$$352$$ 0 0
$$353$$ − 1.17072e6i − 0.500052i −0.968239 0.250026i $$-0.919561\pi$$
0.968239 0.250026i $$-0.0804392\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 575040.i − 0.238796i
$$358$$ 0 0
$$359$$ −3.88654e6 −1.59157 −0.795787 0.605577i $$-0.792942\pi$$
−0.795787 + 0.605577i $$0.792942\pi$$
$$360$$ 0 0
$$361$$ 6.78984e6 2.74215
$$362$$ 0 0
$$363$$ − 2.91350e6i − 1.16051i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 933040.i − 0.361606i −0.983519 0.180803i $$-0.942130\pi$$
0.983519 0.180803i $$-0.0578696\pi$$
$$368$$ 0 0
$$369$$ 1.39510e6 0.533384
$$370$$ 0 0
$$371$$ −280464. −0.105789
$$372$$ 0 0
$$373$$ − 392218.i − 0.145967i −0.997333 0.0729836i $$-0.976748\pi$$
0.997333 0.0729836i $$-0.0232521\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.56880e6i 0.568477i
$$378$$ 0 0
$$379$$ 4.72930e6 1.69122 0.845608 0.533805i $$-0.179238\pi$$
0.845608 + 0.533805i $$0.179238\pi$$
$$380$$ 0 0
$$381$$ −1.89184e6 −0.667686
$$382$$ 0 0
$$383$$ 1.89734e6i 0.660920i 0.943820 + 0.330460i $$0.107204\pi$$
−0.943820 + 0.330460i $$0.892796\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.44252e6i 0.489602i
$$388$$ 0 0
$$389$$ 3.72295e6 1.24742 0.623711 0.781655i $$-0.285624\pi$$
0.623711 + 0.781655i $$0.285624\pi$$
$$390$$ 0 0
$$391$$ −220432. −0.0729177
$$392$$ 0 0
$$393$$ 1.40584e6i 0.459150i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 3.33808e6i − 1.06297i −0.847068 0.531484i $$-0.821635\pi$$
0.847068 0.531484i $$-0.178365\pi$$
$$398$$ 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
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1.28042e6i − 0.383149i
$$408$$ 0 0
$$409$$ 2.57319e6 0.760613 0.380306 0.924861i $$-0.375819\pi$$
0.380306 + 0.924861i $$0.375819\pi$$
$$410$$ 0 0
$$411$$ 5.54580e6 1.61942
$$412$$ 0 0
$$413$$ − 405024.i − 0.116844i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 2.60616e6i 0.733941i
$$418$$ 0 0
$$419$$ −5.26828e6 −1.46600 −0.732999 0.680230i $$-0.761880\pi$$
−0.732999 + 0.680230i $$0.761880\pi$$
$$420$$ 0 0
$$421$$ −973354. −0.267649 −0.133824 0.991005i $$-0.542726\pi$$
−0.133824 + 0.991005i $$0.542726\pi$$
$$422$$ 0 0
$$423$$ 3.71525e6i 1.00957i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 443568.i − 0.117731i
$$428$$ 0 0
$$429$$ −1.18544e6 −0.310983
$$430$$ 0 0
$$431$$ 3.55736e6 0.922433 0.461216 0.887288i $$-0.347413\pi$$
0.461216 + 0.887288i $$0.347413\pi$$
$$432$$ 0 0
$$433$$ − 1.95496e6i − 0.501092i −0.968105 0.250546i $$-0.919390\pi$$
0.968105 0.250546i $$-0.0806102\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 560096.i − 0.140300i
$$438$$ 0 0
$$439$$ 3.29681e6 0.816455 0.408228 0.912880i $$-0.366147\pi$$
0.408228 + 0.912880i $$0.366147\pi$$
$$440$$ 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
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 8.03060e6i 1.90099i
$$448$$ 0 0
$$449$$ −2.12730e6 −0.497981 −0.248990 0.968506i $$-0.580099\pi$$
−0.248990 + 0.968506i $$0.580099\pi$$
$$450$$ 0 0
$$451$$ −1.10186e6 −0.255086
$$452$$ 0 0
$$453$$ − 1.51952e6i − 0.347905i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 289130.i − 0.0647594i −0.999476 0.0323797i $$-0.989691\pi$$
0.999476 0.0323797i $$-0.0103086\pi$$
$$458$$ 0 0
$$459$$ −2.06056e6 −0.456513
$$460$$ 0 0
$$461$$ 2.66870e6 0.584854 0.292427 0.956288i $$-0.405537\pi$$
0.292427 + 0.956288i $$0.405537\pi$$
$$462$$ 0 0
$$463$$ 7.58619e6i 1.64464i 0.569024 + 0.822321i $$0.307321\pi$$
−0.569024 + 0.822321i $$0.692679\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 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. −0.0782540
$$470$$ 0 0
$$471$$ −7.88644e6 −1.63806
$$472$$ 0 0
$$473$$ − 1.13931e6i − 0.234148i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 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
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 6.01388e6i 1.14903i 0.818493 + 0.574516i $$0.194810\pi$$
−0.818493 + 0.574516i $$0.805190\pi$$
$$488$$ 0 0
$$489$$ 234480. 0.0443439
$$490$$ 0 0
$$491$$ 4.29232e6 0.803504 0.401752 0.915749i $$-0.368401\pi$$
0.401752 + 0.915749i $$0.368401\pi$$
$$492$$ 0 0
$$493$$ 3.93184e6i 0.728581i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 767040.i − 0.139292i
$$498$$ 0 0
$$499$$ −1.34509e6 −0.241825 −0.120912 0.992663i $$-0.538582\pi$$
−0.120912 + 0.992663i $$0.538582\pi$$
$$500$$ 0 0
$$501$$ −1.10386e7 −1.96480
$$502$$ 0 0
$$503$$ 202008.i 0.0355999i 0.999842 + 0.0177999i $$0.00566620\pi$$
−0.999842 + 0.0177999i $$0.994334\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 2.85618e6i 0.493476i
$$508$$ 0 0
$$509$$ −9.78344e6 −1.67377 −0.836887 0.547375i $$-0.815627\pi$$
−0.836887 + 0.547375i $$0.815627\pi$$
$$510$$ 0 0
$$511$$ 117264. 0.0198661
$$512$$ 0 0
$$513$$ − 5.23568e6i − 0.878374i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 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 $$-0.820987\pi$$
−0.845985 + 0.533207i $$0.820987\pi$$
$$522$$ 0 0
$$523$$ 6.21017e6i 0.992772i 0.868102 + 0.496386i $$0.165340\pi$$
−0.868102 + 0.496386i $$0.834660\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 6.86214e6i − 1.07630i
$$528$$ 0 0
$$529$$ 6.40249e6 0.994740
$$530$$ 0 0
$$531$$ 2.64953e6 0.407787
$$532$$ 0 0
$$533$$ − 4.24751e6i − 0.647614i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 1.11924e7i − 1.67489i
$$538$$ 0 0
$$539$$ 2.01264e6 0.298397
$$540$$ 0 0
$$541$$ 5.08088e6 0.746355 0.373178 0.927760i $$-0.378268\pi$$
0.373178 + 0.927760i $$0.378268\pi$$
$$542$$ 0 0
$$543$$ 1.20942e7i 1.76026i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 3.34687e6i − 0.478267i −0.970987 0.239133i $$-0.923137\pi$$
0.970987 0.239133i $$-0.0768633\pi$$
$$548$$ 0 0
$$549$$ 2.90167e6 0.410883
$$550$$ 0 0
$$551$$ −9.99041e6 −1.40186
$$552$$ 0 0
$$553$$ − 1.06944e6i − 0.148711i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 7.00377e6i − 0.956520i −0.878218 0.478260i $$-0.841268\pi$$
0.878218 0.478260i $$-0.158732\pi$$
$$558$$ 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 $$-0.668550\pi$$
0.505116 0.863052i $$-0.331450\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1.74122e6i − 0.227456i
$$568$$ 0 0
$$569$$ −1.89942e6 −0.245946 −0.122973 0.992410i $$-0.539243\pi$$
−0.122973 + 0.992410i $$0.539243\pi$$
$$570$$ 0 0
$$571$$ −1.66300e6 −0.213452 −0.106726 0.994288i $$-0.534037\pi$$
−0.106726 + 0.994288i $$0.534037\pi$$
$$572$$ 0 0
$$573$$ − 8.18304e6i − 1.04119i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 8.77344e6i − 1.09706i −0.836131 0.548530i $$-0.815188\pi$$
0.836131 0.548530i $$-0.184812\pi$$
$$578$$ 0 0
$$579$$ −1.08173e7 −1.34098
$$580$$ 0 0
$$581$$ −1.61674e6 −0.198700
$$582$$ 0 0
$$583$$ 1.44906e6i 0.176570i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 5.18393e6i − 0.620961i −0.950580 0.310480i $$-0.899510\pi$$
0.950580 0.310480i $$-0.100490\pi$$
$$588$$ 0 0
$$589$$ 1.74360e7 2.07090
$$590$$ 0 0
$$591$$ −1.25980e7 −1.48365
$$592$$ 0 0
$$593$$ 8.49858e6i 0.992452i 0.868193 + 0.496226i $$0.165281\pi$$
−0.868193 + 0.496226i $$0.834719\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 5.66096e6i − 0.650061i
$$598$$ 0 0
$$599$$ −1.12471e7 −1.28078 −0.640388 0.768051i $$-0.721227\pi$$
−0.640388 + 0.768051i $$0.721227\pi$$
$$600$$ 0 0
$$601$$ −3.46439e6 −0.391238 −0.195619 0.980680i $$-0.562672\pi$$
−0.195619 + 0.980680i $$0.562672\pi$$
$$602$$ 0 0
$$603$$ − 2.43852e6i − 0.273108i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 999712.i 0.110129i 0.998483 + 0.0550647i $$0.0175365\pi$$
−0.998483 + 0.0550647i $$0.982463\pi$$
$$608$$ 0 0
$$609$$ −1.57536e6 −0.172122
$$610$$ 0 0
$$611$$ 1.13114e7 1.22578
$$612$$ 0 0
$$613$$ 9.81340e6i 1.05480i 0.849619 + 0.527398i $$0.176832\pi$$
−0.849619 + 0.527398i $$0.823168\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5.34745e6i 0.565501i 0.959193 + 0.282751i $$0.0912469\pi$$
−0.959193 + 0.282751i $$0.908753\pi$$
$$618$$ 0 0
$$619$$ 6.82768e6 0.716221 0.358110 0.933679i $$-0.383421\pi$$
0.358110 + 0.933679i $$0.383421\pi$$
$$620$$ 0 0
$$621$$ −316480. −0.0329319
$$622$$ 0 0
$$623$$ − 1.72786e6i − 0.178356i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 7.54912e6i − 0.766880i
$$628$$ 0 0
$$629$$ 1.23705e7 1.24670
$$630$$ 0 0
$$631$$ −3.60970e6 −0.360909 −0.180455 0.983583i $$-0.557757\pi$$
−0.180455 + 0.983583i $$0.557757\pi$$
$$632$$ 0 0
$$633$$ 2.85512e6i 0.283214i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 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 $$-0.843287\pi$$
0.881235 0.472679i $$-0.156713\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 1.78359e7i 1.67508i 0.546378 + 0.837539i $$0.316006\pi$$
−0.546378 + 0.837539i $$0.683994\pi$$
$$648$$ 0 0
$$649$$ −2.09262e6 −0.195020
$$650$$ 0 0
$$651$$ 2.74944e6 0.254268
$$652$$ 0 0
$$653$$ − 4.32323e6i − 0.396758i −0.980125 0.198379i $$-0.936432\pi$$
0.980125 0.198379i $$-0.0635677\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 767102.i 0.0693330i
$$658$$ 0 0
$$659$$ −1.97858e7 −1.77476 −0.887382 0.461035i $$-0.847478\pi$$
−0.887382 + 0.461035i $$0.847478\pi$$
$$660$$ 0 0
$$661$$ 1.57772e7 1.40451 0.702255 0.711925i $$-0.252176\pi$$
0.702255 + 0.711925i $$0.252176\pi$$
$$662$$ 0 0
$$663$$ − 1.14529e7i − 1.01188i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 603888.i 0.0525584i
$$668$$ 0 0
$$669$$ −1.77939e7 −1.53711
$$670$$ 0 0
$$671$$ −2.29177e6 −0.196501
$$672$$ 0 0
$$673$$ 6.78762e6i 0.577670i 0.957379 + 0.288835i $$0.0932679\pi$$
−0.957379 + 0.288835i $$0.906732\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 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
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 1.39157e7i 1.12490i
$$688$$ 0 0
$$689$$ −5.58591e6 −0.448276
$$690$$ 0 0
$$691$$ −220156. −0.0175402 −0.00877012 0.999962i $$-0.502792\pi$$
−0.00877012 + 0.999962i $$0.502792\pi$$
$$692$$ 0 0
$$693$$ − 467232.i − 0.0369572i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 1.06454e7i − 0.830006i
$$698$$ 0 0
$$699$$ 6.94252e6 0.537433
$$700$$ 0 0
$$701$$ 4.78933e6 0.368111 0.184056 0.982916i $$-0.441077\pi$$
0.184056 + 0.982916i $$0.441077\pi$$
$$702$$ 0 0
$$703$$ 3.14323e7i 2.39877i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.23854e6i 0.0931886i
$$708$$ 0 0
$$709$$ −4.26892e6 −0.318935 −0.159468 0.987203i $$-0.550978\pi$$
−0.159468 + 0.987203i $$0.550978\pi$$
$$710$$ 0 0
$$711$$ 6.99592e6 0.519004
$$712$$ 0 0
$$713$$ − 1.05395e6i − 0.0776421i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 6.53426e6i − 0.458522i −0.973365 0.229261i $$-0.926369\pi$$
0.973365 0.229261i $$-0.0736310\pi$$
$$728$$ 0 0
$$729$$ 3.03131e6 0.211257
$$730$$ 0 0
$$731$$ 1.10072e7 0.761876
$$732$$ 0 0
$$733$$ 1.31617e7i 0.904800i 0.891815 + 0.452400i $$0.149432\pi$$
−0.891815 + 0.452400i $$0.850568\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.92597e6i 0.130611i
$$738$$ 0 0
$$739$$ 1.42348e7 0.958825 0.479412 0.877590i $$-0.340850\pi$$
0.479412 + 0.877590i $$0.340850\pi$$
$$740$$ 0 0
$$741$$ 2.91006e7 1.94696
$$742$$ 0 0
$$743$$ − 2.15835e7i − 1.43434i −0.696901 0.717168i $$-0.745438\pi$$
0.696901 0.717168i $$-0.254562\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 1.05761e7i − 0.693467i
$$748$$ 0 0
$$749$$ −1.57680e6 −0.102700
$$750$$ 0 0
$$751$$ 1.86594e7 1.20725 0.603625 0.797268i $$-0.293722\pi$$
0.603625 + 0.797268i $$0.293722\pi$$
$$752$$ 0 0
$$753$$ − 1.58122e7i − 1.01626i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 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 $$-0.801859\pi$$
−0.812436 + 0.583051i $$0.801859\pi$$
$$762$$ 0 0
$$763$$ 2.70494e6i 0.168208i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 8.06673e6i − 0.495118i
$$768$$ 0 0
$$769$$ −5.53267e6 −0.337380 −0.168690 0.985669i $$-0.553954\pi$$
−0.168690 + 0.985669i $$0.553954\pi$$
$$770$$ 0 0
$$771$$ −2.59580e6 −0.157266
$$772$$ 0 0
$$773$$ 8.32940e6i 0.501378i 0.968068 + 0.250689i $$0.0806571\pi$$
−0.968068 + 0.250689i $$0.919343\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 4.95648e6i 0.294524i
$$778$$ 0 0
$$779$$ 2.70490e7 1.59701
$$780$$ 0 0
$$781$$ −3.96304e6 −0.232488
$$782$$ 0 0
$$783$$ 5.64504e6i 0.329051i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.36523e7i 0.785719i 0.919598 + 0.392860i $$0.128514\pi$$
−0.919598 + 0.392860i $$0.871486\pi$$
$$788$$ 0 0
$$789$$ −1.41776e6 −0.0810793
$$790$$ 0 0
$$791$$ 564048. 0.0320535
$$792$$ 0 0
$$793$$ − 8.83440e6i − 0.498877i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 8.54626e6i 0.476574i 0.971195 + 0.238287i $$0.0765859\pi$$
−0.971195 + 0.238287i $$0.923414\pi$$
$$798$$ 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
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 3.58035e7i − 1.93527i
$$808$$ 0 0
$$809$$ −7.58484e6 −0.407451 −0.203725 0.979028i $$-0.565305\pi$$
−0.203725 + 0.979028i $$0.565305\pi$$
$$810$$ 0 0
$$811$$ 6.18473e6 0.330194 0.165097 0.986277i $$-0.447206\pi$$
0.165097 + 0.986277i $$0.447206\pi$$
$$812$$ 0 0
$$813$$ − 3.54723e7i − 1.88219i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.79683e7i 1.46592i
$$818$$ 0 0
$$819$$ 1.80110e6 0.0938273
$$820$$ 0 0
$$821$$ −2.78102e6 −0.143995 −0.0719973 0.997405i $$-0.522937\pi$$
−0.0719973 + 0.997405i $$0.522937\pi$$
$$822$$ 0 0
$$823$$ 1.63895e7i 0.843461i 0.906721 + 0.421731i $$0.138577\pi$$
−0.906721 + 0.421731i $$0.861423\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 2.29511e7i − 1.16692i −0.812142 0.583459i $$-0.801699\pi$$
0.812142 0.583459i $$-0.198301\pi$$
$$828$$ 0 0
$$829$$ 3.50136e6 0.176950 0.0884750 0.996078i $$-0.471801\pi$$
0.0884750 + 0.996078i $$0.471801\pi$$
$$830$$ 0 0
$$831$$ −5.50900e6 −0.276739
$$832$$ 0 0
$$833$$ 1.94447e7i 0.970934i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 9.85216e6i − 0.486091i
$$838$$ 0 0
$$839$$ −5.29668e6 −0.259776 −0.129888 0.991529i $$-0.541462\pi$$
−0.129888 + 0.991529i $$0.541462\pi$$
$$840$$ 0 0
$$841$$ −9.73962e6 −0.474845
$$842$$ 0 0
$$843$$ 1.18834e7i 0.575933i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3.49620e6i − 0.167451i
$$848$$ 0 0
$$849$$ −2.18486e7 −1.04029
$$850$$ 0 0
$$851$$ 1.89998e6 0.0899344
$$852$$ 0 0
$$853$$ − 2.02948e7i − 0.955021i −0.878626 0.477511i $$-0.841539\pi$$
0.878626 0.477511i $$-0.158461\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.82785e6i 0.224544i 0.993678 + 0.112272i $$0.0358128\pi$$
−0.993678 + 0.112272i $$0.964187\pi$$
$$858$$ 0 0
$$859$$ 1.30210e7 0.602092 0.301046 0.953610i $$-0.402664\pi$$
0.301046 + 0.953610i $$0.402664\pi$$
$$860$$ 0 0
$$861$$ 4.26528e6 0.196083
$$862$$ 0 0
$$863$$ − 3.92387e7i − 1.79344i −0.442596 0.896721i $$-0.645942\pi$$
0.442596 0.896721i $$-0.354058\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 306940.i − 0.0138677i
$$868$$ 0 0
$$869$$ −5.52544e6 −0.248209
$$870$$ 0 0
$$871$$ −7.42430e6 −0.331596
$$872$$ 0 0
$$873$$ 7.67196e6i 0.340699i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 1.34622e7i − 0.591041i −0.955336 0.295520i $$-0.904507\pi$$
0.955336 0.295520i $$-0.0954930\pi$$
$$878$$ 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 $$0.177727\pi$$
−0.848133 + 0.529784i $$0.822273\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.61463e7i 0.689070i 0.938773 + 0.344535i $$0.111963\pi$$
−0.938773 + 0.344535i $$0.888037\pi$$
$$888$$ 0 0
$$889$$ −2.27021e6 −0.0963410
$$890$$ 0 0
$$891$$ −8.99632e6 −0.379639
$$892$$ 0 0
$$893$$ 7.20332e7i 3.02276i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 1.75904e6i − 0.0729953i
$$898$$ 0 0
$$899$$ −1.87993e7 −0.775787
$$900$$ 0 0
$$901$$ −1.39998e7 −0.574527
$$902$$ 0 0
$$903$$ 4.41024e6i 0.179988i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.03361e7i 0.820824i 0.911900 + 0.410412i $$0.134615\pi$$
−0.911900 + 0.410412i $$0.865385\pi$$
$$908$$ 0 0
$$909$$ −8.10214e6 −0.325230
$$910$$ 0 0
$$911$$ 1.07726e7 0.430054 0.215027 0.976608i $$-0.431016\pi$$
0.215027 + 0.976608i $$0.431016\pi$$
$$912$$ 0 0
$$913$$ 8.35314e6i 0.331644i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1.68701e6i 0.0662512i
$$918$$ 0 0
$$919$$ −4.18566e7 −1.63484 −0.817419 0.576043i $$-0.804596\pi$$
−0.817419 + 0.576043i $$0.804596\pi$$
$$920$$ 0 0
$$921$$ 2.11994e7 0.823522
$$922$$ 0 0
$$923$$ − 1.52769e7i − 0.590242i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 2.83266e7i − 1.08267i
$$928$$ 0 0
$$929$$ −2.99845e7 −1.13988 −0.569939 0.821687i $$-0.693033\pi$$
−0.569939 + 0.821687i $$0.693033\pi$$
$$930$$ 0 0
$$931$$ −4.94072e7 −1.86817
$$932$$ 0 0
$$933$$ − 2.67298e7i − 1.00529i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.42402e7i − 0.529867i −0.964267 0.264934i $$-0.914650\pi$$
0.964267 0.264934i $$-0.0853501\pi$$
$$938$$ 0 0
$$939$$ −3.28837e7 −1.21707
$$940$$ 0 0
$$941$$ −4.14546e7 −1.52615 −0.763077 0.646307i $$-0.776313\pi$$
−0.763077 + 0.646307i $$0.776313\pi$$
$$942$$ 0 0
$$943$$ − 1.63502e6i − 0.0598749i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.54079e7i 0.558300i 0.960248 + 0.279150i $$0.0900527\pi$$
−0.960248 + 0.279150i $$0.909947\pi$$
$$948$$ 0 0
$$949$$ 2.33551e6 0.0841813
$$950$$ 0 0
$$951$$ −3.44740e7 −1.23606
$$952$$ 0 0
$$953$$ − 2.06328e7i − 0.735912i −0.929843 0.367956i $$-0.880058\pi$$
0.929843 0.367956i $$-0.119942\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 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
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 1.18724e7i − 0.408294i −0.978940 0.204147i $$-0.934558\pi$$
0.978940 0.204147i $$-0.0654421\pi$$
$$968$$ 0 0
$$969$$ 7.29342e7 2.49530
$$970$$ 0 0
$$971$$ 1.53222e6 0.0521523 0.0260761 0.999660i $$-0.491699\pi$$
0.0260761 + 0.999660i $$0.491699\pi$$
$$972$$ 0 0
$$973$$ 3.12739e6i 0.105901i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 1.74321e7i − 0.584269i −0.956377 0.292135i $$-0.905635\pi$$
0.956377 0.292135i $$-0.0943655\pi$$
$$978$$ 0 0
$$979$$ −8.92726e6 −0.297688
$$980$$ 0 0
$$981$$ −1.76948e7 −0.587049
$$982$$ 0 0
$$983$$ 2.23270e6i 0.0736963i 0.999321 + 0.0368482i $$0.0117318\pi$$
−0.999321 + 0.0368482i $$0.988268\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 1.13587e7i 0.371139i
$$988$$ 0 0
$$989$$ 1.69059e6 0.0549602
$$990$$ 0 0
$$991$$ 2.22501e7 0.719693 0.359847 0.933011i $$-0.382829\pi$$
0.359847 + 0.933011i $$0.382829\pi$$
$$992$$ 0 0
$$993$$ 5.49926e7i 1.76983i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 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 200.6.c.a.49.2 2
4.3 odd 2 400.6.c.d.49.1 2
5.2 odd 4 8.6.a.a.1.1 1
5.3 odd 4 200.6.a.a.1.1 1
5.4 even 2 inner 200.6.c.a.49.1 2
15.2 even 4 72.6.a.f.1.1 1
20.3 even 4 400.6.a.l.1.1 1
20.7 even 4 16.6.a.a.1.1 1
20.19 odd 2 400.6.c.d.49.2 2
35.2 odd 12 392.6.i.b.361.1 2
35.12 even 12 392.6.i.e.361.1 2
35.17 even 12 392.6.i.e.177.1 2
35.27 even 4 392.6.a.b.1.1 1
35.32 odd 12 392.6.i.b.177.1 2
40.27 even 4 64.6.a.g.1.1 1
40.37 odd 4 64.6.a.a.1.1 1
55.32 even 4 968.6.a.a.1.1 1
60.47 odd 4 144.6.a.k.1.1 1
80.27 even 4 256.6.b.d.129.2 2
80.37 odd 4 256.6.b.f.129.1 2
80.67 even 4 256.6.b.d.129.1 2
80.77 odd 4 256.6.b.f.129.2 2
120.77 even 4 576.6.a.g.1.1 1
120.107 odd 4 576.6.a.h.1.1 1
140.27 odd 4 784.6.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.a.a.1.1 1 5.2 odd 4
16.6.a.a.1.1 1 20.7 even 4
64.6.a.a.1.1 1 40.37 odd 4
64.6.a.g.1.1 1 40.27 even 4
72.6.a.f.1.1 1 15.2 even 4
144.6.a.k.1.1 1 60.47 odd 4
200.6.a.a.1.1 1 5.3 odd 4
200.6.c.a.49.1 2 5.4 even 2 inner
200.6.c.a.49.2 2 1.1 even 1 trivial
256.6.b.d.129.1 2 80.67 even 4
256.6.b.d.129.2 2 80.27 even 4
256.6.b.f.129.1 2 80.37 odd 4
256.6.b.f.129.2 2 80.77 odd 4
392.6.a.b.1.1 1 35.27 even 4
392.6.i.b.177.1 2 35.32 odd 12
392.6.i.b.361.1 2 35.2 odd 12
392.6.i.e.177.1 2 35.17 even 12
392.6.i.e.361.1 2 35.12 even 12
400.6.a.l.1.1 1 20.3 even 4
400.6.c.d.49.1 2 4.3 odd 2
400.6.c.d.49.2 2 20.19 odd 2
576.6.a.g.1.1 1 120.77 even 4
576.6.a.h.1.1 1 120.107 odd 4
784.6.a.l.1.1 1 140.27 odd 4
968.6.a.a.1.1 1 55.32 even 4