# Properties

 Label 576.3.e.g.449.2 Level $576$ Weight $3$ Character 576.449 Analytic conductor $15.695$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 576.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6948632272$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 576.449 Dual form 576.3.e.g.449.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.41421i q^{5} +8.00000 q^{7} +O(q^{10})$$ $$q+1.41421i q^{5} +8.00000 q^{7} +11.3137i q^{11} +8.00000 q^{13} +12.7279i q^{17} -32.0000 q^{19} -33.9411i q^{23} +23.0000 q^{25} +43.8406i q^{29} +40.0000 q^{31} +11.3137i q^{35} +26.0000 q^{37} +66.4680i q^{41} -16.0000 q^{43} +11.3137i q^{47} +15.0000 q^{49} +32.5269i q^{53} -16.0000 q^{55} +22.6274i q^{59} +54.0000 q^{61} +11.3137i q^{65} +80.0000 q^{67} +79.1960i q^{71} +96.0000 q^{73} +90.5097i q^{77} -104.000 q^{79} -101.823i q^{83} -18.0000 q^{85} +77.7817i q^{89} +64.0000 q^{91} -45.2548i q^{95} -80.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{7} + O(q^{10})$$ $$2q + 16q^{7} + 16q^{13} - 64q^{19} + 46q^{25} + 80q^{31} + 52q^{37} - 32q^{43} + 30q^{49} - 32q^{55} + 108q^{61} + 160q^{67} + 192q^{73} - 208q^{79} - 36q^{85} + 128q^{91} - 160q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.41421i 0.282843i 0.989949 + 0.141421i $$0.0451672\pi$$
−0.989949 + 0.141421i $$0.954833\pi$$
$$6$$ 0 0
$$7$$ 8.00000 1.14286 0.571429 0.820652i $$-0.306389\pi$$
0.571429 + 0.820652i $$0.306389\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.3137i 1.02852i 0.857635 + 0.514259i $$0.171933\pi$$
−0.857635 + 0.514259i $$0.828067\pi$$
$$12$$ 0 0
$$13$$ 8.00000 0.615385 0.307692 0.951486i $$-0.400443\pi$$
0.307692 + 0.951486i $$0.400443\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 12.7279i 0.748701i 0.927287 + 0.374351i $$0.122134\pi$$
−0.927287 + 0.374351i $$0.877866\pi$$
$$18$$ 0 0
$$19$$ −32.0000 −1.68421 −0.842105 0.539313i $$-0.818684\pi$$
−0.842105 + 0.539313i $$0.818684\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 33.9411i − 1.47570i −0.674964 0.737851i $$-0.735841\pi$$
0.674964 0.737851i $$-0.264159\pi$$
$$24$$ 0 0
$$25$$ 23.0000 0.920000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 43.8406i 1.51175i 0.654719 + 0.755873i $$0.272787\pi$$
−0.654719 + 0.755873i $$0.727213\pi$$
$$30$$ 0 0
$$31$$ 40.0000 1.29032 0.645161 0.764046i $$-0.276790\pi$$
0.645161 + 0.764046i $$0.276790\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 11.3137i 0.323249i
$$36$$ 0 0
$$37$$ 26.0000 0.702703 0.351351 0.936244i $$-0.385722\pi$$
0.351351 + 0.936244i $$0.385722\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 66.4680i 1.62117i 0.585620 + 0.810586i $$0.300851\pi$$
−0.585620 + 0.810586i $$0.699149\pi$$
$$42$$ 0 0
$$43$$ −16.0000 −0.372093 −0.186047 0.982541i $$-0.559568\pi$$
−0.186047 + 0.982541i $$0.559568\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.3137i 0.240717i 0.992730 + 0.120359i $$0.0384044\pi$$
−0.992730 + 0.120359i $$0.961596\pi$$
$$48$$ 0 0
$$49$$ 15.0000 0.306122
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 32.5269i 0.613715i 0.951755 + 0.306858i $$0.0992776\pi$$
−0.951755 + 0.306858i $$0.900722\pi$$
$$54$$ 0 0
$$55$$ −16.0000 −0.290909
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 22.6274i 0.383516i 0.981442 + 0.191758i $$0.0614188\pi$$
−0.981442 + 0.191758i $$0.938581\pi$$
$$60$$ 0 0
$$61$$ 54.0000 0.885246 0.442623 0.896708i $$-0.354048\pi$$
0.442623 + 0.896708i $$0.354048\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 11.3137i 0.174057i
$$66$$ 0 0
$$67$$ 80.0000 1.19403 0.597015 0.802230i $$-0.296353\pi$$
0.597015 + 0.802230i $$0.296353\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 79.1960i 1.11544i 0.830030 + 0.557718i $$0.188323\pi$$
−0.830030 + 0.557718i $$0.811677\pi$$
$$72$$ 0 0
$$73$$ 96.0000 1.31507 0.657534 0.753425i $$-0.271599\pi$$
0.657534 + 0.753425i $$0.271599\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 90.5097i 1.17545i
$$78$$ 0 0
$$79$$ −104.000 −1.31646 −0.658228 0.752819i $$-0.728694\pi$$
−0.658228 + 0.752819i $$0.728694\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 101.823i − 1.22679i −0.789777 0.613394i $$-0.789804\pi$$
0.789777 0.613394i $$-0.210196\pi$$
$$84$$ 0 0
$$85$$ −18.0000 −0.211765
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 77.7817i 0.873952i 0.899473 + 0.436976i $$0.143951\pi$$
−0.899473 + 0.436976i $$0.856049\pi$$
$$90$$ 0 0
$$91$$ 64.0000 0.703297
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 45.2548i − 0.476367i
$$96$$ 0 0
$$97$$ −80.0000 −0.824742 −0.412371 0.911016i $$-0.635299\pi$$
−0.412371 + 0.911016i $$0.635299\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 123.037i − 1.21818i −0.793100 0.609092i $$-0.791534\pi$$
0.793100 0.609092i $$-0.208466\pi$$
$$102$$ 0 0
$$103$$ −72.0000 −0.699029 −0.349515 0.936931i $$-0.613653\pi$$
−0.349515 + 0.936931i $$0.613653\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 181.019i − 1.69177i −0.533366 0.845885i $$-0.679073\pi$$
0.533366 0.845885i $$-0.320927\pi$$
$$108$$ 0 0
$$109$$ 88.0000 0.807339 0.403670 0.914905i $$-0.367734\pi$$
0.403670 + 0.914905i $$0.367734\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 137.179i − 1.21397i −0.794713 0.606985i $$-0.792379\pi$$
0.794713 0.606985i $$-0.207621\pi$$
$$114$$ 0 0
$$115$$ 48.0000 0.417391
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 101.823i 0.855659i
$$120$$ 0 0
$$121$$ −7.00000 −0.0578512
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 67.8823i 0.543058i
$$126$$ 0 0
$$127$$ −56.0000 −0.440945 −0.220472 0.975393i $$-0.570760\pi$$
−0.220472 + 0.975393i $$0.570760\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 248.902i 1.90001i 0.312231 + 0.950006i $$0.398924\pi$$
−0.312231 + 0.950006i $$0.601076\pi$$
$$132$$ 0 0
$$133$$ −256.000 −1.92481
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 46.6690i 0.340650i 0.985388 + 0.170325i $$0.0544818\pi$$
−0.985388 + 0.170325i $$0.945518\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −0.115108 −0.0575540 0.998342i $$-0.518330\pi$$
−0.0575540 + 0.998342i $$0.518330\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 90.5097i 0.632935i
$$144$$ 0 0
$$145$$ −62.0000 −0.427586
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 182.434i − 1.22439i −0.790708 0.612193i $$-0.790288\pi$$
0.790708 0.612193i $$-0.209712\pi$$
$$150$$ 0 0
$$151$$ 168.000 1.11258 0.556291 0.830987i $$-0.312224\pi$$
0.556291 + 0.830987i $$0.312224\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 56.5685i 0.364958i
$$156$$ 0 0
$$157$$ 10.0000 0.0636943 0.0318471 0.999493i $$-0.489861\pi$$
0.0318471 + 0.999493i $$0.489861\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 271.529i − 1.68652i
$$162$$ 0 0
$$163$$ −80.0000 −0.490798 −0.245399 0.969422i $$-0.578919\pi$$
−0.245399 + 0.969422i $$0.578919\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 294.156i − 1.76142i −0.473660 0.880708i $$-0.657067\pi$$
0.473660 0.880708i $$-0.342933\pi$$
$$168$$ 0 0
$$169$$ −105.000 −0.621302
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 55.1543i − 0.318811i −0.987213 0.159406i $$-0.949042\pi$$
0.987213 0.159406i $$-0.0509578\pi$$
$$174$$ 0 0
$$175$$ 184.000 1.05143
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 135.765i − 0.758461i −0.925302 0.379230i $$-0.876189\pi$$
0.925302 0.379230i $$-0.123811\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.0441989 −0.0220994 0.999756i $$-0.507035\pi$$
−0.0220994 + 0.999756i $$0.507035\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 36.7696i 0.198754i
$$186$$ 0 0
$$187$$ −144.000 −0.770053
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 67.8823i − 0.355404i −0.984084 0.177702i $$-0.943134\pi$$
0.984084 0.177702i $$-0.0568664\pi$$
$$192$$ 0 0
$$193$$ −258.000 −1.33679 −0.668394 0.743808i $$-0.733018\pi$$
−0.668394 + 0.743808i $$0.733018\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 371.938i − 1.88801i −0.329930 0.944005i $$-0.607025\pi$$
0.329930 0.944005i $$-0.392975\pi$$
$$198$$ 0 0
$$199$$ 88.0000 0.442211 0.221106 0.975250i $$-0.429033\pi$$
0.221106 + 0.975250i $$0.429033\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 350.725i 1.72771i
$$204$$ 0 0
$$205$$ −94.0000 −0.458537
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 362.039i − 1.73224i
$$210$$ 0 0
$$211$$ −368.000 −1.74408 −0.872038 0.489438i $$-0.837202\pi$$
−0.872038 + 0.489438i $$0.837202\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ − 22.6274i − 0.105244i
$$216$$ 0 0
$$217$$ 320.000 1.47465
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 101.823i 0.460739i
$$222$$ 0 0
$$223$$ 104.000 0.466368 0.233184 0.972433i $$-0.425086\pi$$
0.233184 + 0.972433i $$0.425086\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 169.706i − 0.747602i −0.927509 0.373801i $$-0.878054\pi$$
0.927509 0.373801i $$-0.121946\pi$$
$$228$$ 0 0
$$229$$ −344.000 −1.50218 −0.751092 0.660198i $$-0.770472\pi$$
−0.751092 + 0.660198i $$0.770472\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 80.6102i 0.345966i 0.984925 + 0.172983i $$0.0553406\pi$$
−0.984925 + 0.172983i $$0.944659\pi$$
$$234$$ 0 0
$$235$$ −16.0000 −0.0680851
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 271.529i − 1.13610i −0.822992 0.568052i $$-0.807697\pi$$
0.822992 0.568052i $$-0.192303\pi$$
$$240$$ 0 0
$$241$$ −272.000 −1.12863 −0.564315 0.825559i $$-0.690860\pi$$
−0.564315 + 0.825559i $$0.690860\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 21.2132i 0.0865845i
$$246$$ 0 0
$$247$$ −256.000 −1.03644
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 33.9411i 0.135224i 0.997712 + 0.0676118i $$0.0215379\pi$$
−0.997712 + 0.0676118i $$0.978462\pi$$
$$252$$ 0 0
$$253$$ 384.000 1.51779
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 43.8406i − 0.170586i −0.996356 0.0852930i $$-0.972817\pi$$
0.996356 0.0852930i $$-0.0271826\pi$$
$$258$$ 0 0
$$259$$ 208.000 0.803089
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 203.647i 0.774322i 0.922012 + 0.387161i $$0.126544\pi$$
−0.922012 + 0.387161i $$0.873456\pi$$
$$264$$ 0 0
$$265$$ −46.0000 −0.173585
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 46.6690i 0.173491i 0.996231 + 0.0867454i $$0.0276467\pi$$
−0.996231 + 0.0867454i $$0.972353\pi$$
$$270$$ 0 0
$$271$$ 264.000 0.974170 0.487085 0.873355i $$-0.338060\pi$$
0.487085 + 0.873355i $$0.338060\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 260.215i 0.946237i
$$276$$ 0 0
$$277$$ −40.0000 −0.144404 −0.0722022 0.997390i $$-0.523003\pi$$
−0.0722022 + 0.997390i $$0.523003\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 190.919i 0.679426i 0.940529 + 0.339713i $$0.110330\pi$$
−0.940529 + 0.339713i $$0.889670\pi$$
$$282$$ 0 0
$$283$$ 224.000 0.791519 0.395760 0.918354i $$-0.370481\pi$$
0.395760 + 0.918354i $$0.370481\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 531.744i 1.85277i
$$288$$ 0 0
$$289$$ 127.000 0.439446
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 352.139i − 1.20184i −0.799309 0.600920i $$-0.794801\pi$$
0.799309 0.600920i $$-0.205199\pi$$
$$294$$ 0 0
$$295$$ −32.0000 −0.108475
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 271.529i − 0.908124i
$$300$$ 0 0
$$301$$ −128.000 −0.425249
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 76.3675i 0.250385i
$$306$$ 0 0
$$307$$ 432.000 1.40717 0.703583 0.710613i $$-0.251582\pi$$
0.703583 + 0.710613i $$0.251582\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 203.647i − 0.654813i −0.944884 0.327406i $$-0.893825\pi$$
0.944884 0.327406i $$-0.106175\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.0447284 −0.0223642 0.999750i $$-0.507119\pi$$
−0.0223642 + 0.999750i $$0.507119\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 589.727i 1.86034i 0.367132 + 0.930169i $$0.380340\pi$$
−0.367132 + 0.930169i $$0.619660\pi$$
$$318$$ 0 0
$$319$$ −496.000 −1.55486
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 407.294i − 1.26097i
$$324$$ 0 0
$$325$$ 184.000 0.566154
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 90.5097i 0.275105i
$$330$$ 0 0
$$331$$ −16.0000 −0.0483384 −0.0241692 0.999708i $$-0.507694\pi$$
−0.0241692 + 0.999708i $$0.507694\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 113.137i 0.337723i
$$336$$ 0 0
$$337$$ 128.000 0.379822 0.189911 0.981801i $$-0.439180\pi$$
0.189911 + 0.981801i $$0.439180\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 452.548i 1.32712i
$$342$$ 0 0
$$343$$ −272.000 −0.793003
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 350.725i − 1.01073i −0.862904 0.505367i $$-0.831357\pi$$
0.862904 0.505367i $$-0.168643\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.0286533 −0.0143266 0.999897i $$-0.504560\pi$$
−0.0143266 + 0.999897i $$0.504560\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 247.487i 0.701097i 0.936545 + 0.350549i $$0.114005\pi$$
−0.936545 + 0.350549i $$0.885995\pi$$
$$354$$ 0 0
$$355$$ −112.000 −0.315493
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 328.098i 0.913921i 0.889487 + 0.456960i $$0.151062\pi$$
−0.889487 + 0.456960i $$0.848938\pi$$
$$360$$ 0 0
$$361$$ 663.000 1.83657
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 135.765i 0.371958i
$$366$$ 0 0
$$367$$ −696.000 −1.89646 −0.948229 0.317588i $$-0.897127\pi$$
−0.948229 + 0.317588i $$0.897127\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 260.215i 0.701389i
$$372$$ 0 0
$$373$$ 454.000 1.21716 0.608579 0.793493i $$-0.291740\pi$$
0.608579 + 0.793493i $$0.291740\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 350.725i 0.930305i
$$378$$ 0 0
$$379$$ 64.0000 0.168865 0.0844327 0.996429i $$-0.473092\pi$$
0.0844327 + 0.996429i $$0.473092\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 362.039i − 0.945271i −0.881258 0.472635i $$-0.843303\pi$$
0.881258 0.472635i $$-0.156697\pi$$
$$384$$ 0 0
$$385$$ −128.000 −0.332468
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 386.080i − 0.992494i −0.868181 0.496247i $$-0.834711\pi$$
0.868181 0.496247i $$-0.165289\pi$$
$$390$$ 0 0
$$391$$ 432.000 1.10486
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 147.078i − 0.372350i
$$396$$ 0 0
$$397$$ 662.000 1.66751 0.833753 0.552137i $$-0.186188\pi$$
0.833753 + 0.552137i $$0.186188\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 171.120i 0.426733i 0.976972 + 0.213366i $$0.0684428\pi$$
−0.976972 + 0.213366i $$0.931557\pi$$
$$402$$ 0 0
$$403$$ 320.000 0.794045
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 294.156i 0.722743i
$$408$$ 0 0
$$409$$ −176.000 −0.430318 −0.215159 0.976579i $$-0.569027\pi$$
−0.215159 + 0.976579i $$0.569027\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 181.019i 0.438303i
$$414$$ 0 0
$$415$$ 144.000 0.346988
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 79.1960i 0.189012i 0.995524 + 0.0945059i $$0.0301271\pi$$
−0.995524 + 0.0945059i $$0.969873\pi$$
$$420$$ 0 0
$$421$$ −488.000 −1.15914 −0.579572 0.814921i $$-0.696780\pi$$
−0.579572 + 0.814921i $$0.696780\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 292.742i 0.688805i
$$426$$ 0 0
$$427$$ 432.000 1.01171
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 305.470i − 0.708747i −0.935104 0.354374i $$-0.884694\pi$$
0.935104 0.354374i $$-0.115306\pi$$
$$432$$ 0 0
$$433$$ 478.000 1.10393 0.551963 0.833869i $$-0.313879\pi$$
0.551963 + 0.833869i $$0.313879\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1086.12i 2.48539i
$$438$$ 0 0
$$439$$ 392.000 0.892938 0.446469 0.894799i $$-0.352681\pi$$
0.446469 + 0.894799i $$0.352681\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 33.9411i 0.0766165i 0.999266 + 0.0383083i $$0.0121969\pi$$
−0.999266 + 0.0383083i $$0.987803\pi$$
$$444$$ 0 0
$$445$$ −110.000 −0.247191
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 125.865i − 0.280323i −0.990129 0.140161i $$-0.955238\pi$$
0.990129 0.140161i $$-0.0447622\pi$$
$$450$$ 0 0
$$451$$ −752.000 −1.66741
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 90.5097i 0.198922i
$$456$$ 0 0
$$457$$ −16.0000 −0.0350109 −0.0175055 0.999847i $$-0.505572\pi$$
−0.0175055 + 0.999847i $$0.505572\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 284.257i − 0.616609i −0.951288 0.308305i $$-0.900238\pi$$
0.951288 0.308305i $$-0.0997616\pi$$
$$462$$ 0 0
$$463$$ 568.000 1.22678 0.613391 0.789779i $$-0.289805\pi$$
0.613391 + 0.789779i $$0.289805\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 418.607i 0.896375i 0.893940 + 0.448188i $$0.147930\pi$$
−0.893940 + 0.448188i $$0.852070\pi$$
$$468$$ 0 0
$$469$$ 640.000 1.36461
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 181.019i − 0.382705i
$$474$$ 0 0
$$475$$ −736.000 −1.54947
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 33.9411i − 0.0708583i −0.999372 0.0354291i $$-0.988720\pi$$
0.999372 0.0354291i $$-0.0112798\pi$$
$$480$$ 0 0
$$481$$ 208.000 0.432432
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 113.137i − 0.233272i
$$486$$ 0 0
$$487$$ −424.000 −0.870637 −0.435318 0.900277i $$-0.643364\pi$$
−0.435318 + 0.900277i $$0.643364\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 724.077i 1.47470i 0.675511 + 0.737350i $$0.263923\pi$$
−0.675511 + 0.737350i $$0.736077\pi$$
$$492$$ 0 0
$$493$$ −558.000 −1.13185
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 633.568i 1.27478i
$$498$$ 0 0
$$499$$ 192.000 0.384770 0.192385 0.981320i $$-0.438378\pi$$
0.192385 + 0.981320i $$0.438378\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 441.235i − 0.877206i −0.898681 0.438603i $$-0.855473\pi$$
0.898681 0.438603i $$-0.144527\pi$$
$$504$$ 0 0
$$505$$ 174.000 0.344554
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 250.316i − 0.491780i −0.969298 0.245890i $$-0.920920\pi$$
0.969298 0.245890i $$-0.0790801\pi$$
$$510$$ 0 0
$$511$$ 768.000 1.50294
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 101.823i − 0.197715i
$$516$$ 0 0
$$517$$ −128.000 −0.247582
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 156.978i − 0.301301i −0.988587 0.150650i $$-0.951863\pi$$
0.988587 0.150650i $$-0.0481368\pi$$
$$522$$ 0 0
$$523$$ −576.000 −1.10134 −0.550669 0.834724i $$-0.685627\pi$$
−0.550669 + 0.834724i $$0.685627\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 509.117i 0.966066i
$$528$$ 0 0
$$529$$ −623.000 −1.17769
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 531.744i 0.997644i
$$534$$ 0 0
$$535$$ 256.000 0.478505
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 169.706i 0.314853i
$$540$$ 0 0
$$541$$ 536.000 0.990758 0.495379 0.868677i $$-0.335029\pi$$
0.495379 + 0.868677i $$0.335029\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 124.451i 0.228350i
$$546$$ 0 0
$$547$$ −144.000 −0.263254 −0.131627 0.991299i $$-0.542020\pi$$
−0.131627 + 0.991299i $$0.542020\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 1402.90i − 2.54610i
$$552$$ 0 0
$$553$$ −832.000 −1.50452
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 533.159i − 0.957197i −0.878034 0.478598i $$-0.841145\pi$$
0.878034 0.478598i $$-0.158855\pi$$
$$558$$ 0 0
$$559$$ −128.000 −0.228980
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 576.999i − 1.02487i −0.858727 0.512433i $$-0.828744\pi$$
0.858727 0.512433i $$-0.171256\pi$$
$$564$$ 0 0
$$565$$ 194.000 0.343363
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 994.192i − 1.74726i −0.486589 0.873631i $$-0.661759\pi$$
0.486589 0.873631i $$-0.338241\pi$$
$$570$$ 0 0
$$571$$ −416.000 −0.728546 −0.364273 0.931292i $$-0.618683\pi$$
−0.364273 + 0.931292i $$0.618683\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 780.646i − 1.35765i
$$576$$ 0 0
$$577$$ 834.000 1.44541 0.722704 0.691158i $$-0.242899\pi$$
0.722704 + 0.691158i $$0.242899\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 814.587i − 1.40204i
$$582$$ 0 0
$$583$$ −368.000 −0.631218
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 67.8823i 0.115643i 0.998327 + 0.0578213i $$0.0184154\pi$$
−0.998327 + 0.0578213i $$0.981585\pi$$
$$588$$ 0 0
$$589$$ −1280.00 −2.17317
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 224.860i − 0.379190i −0.981862 0.189595i $$-0.939282\pi$$
0.981862 0.189595i $$-0.0607176\pi$$
$$594$$ 0 0
$$595$$ −144.000 −0.242017
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 690.136i 1.15215i 0.817398 + 0.576074i $$0.195416\pi$$
−0.817398 + 0.576074i $$0.804584\pi$$
$$600$$ 0 0
$$601$$ −626.000 −1.04160 −0.520799 0.853680i $$-0.674366\pi$$
−0.520799 + 0.853680i $$0.674366\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 9.89949i − 0.0163628i
$$606$$ 0 0
$$607$$ 232.000 0.382208 0.191104 0.981570i $$-0.438793\pi$$
0.191104 + 0.981570i $$0.438793\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 90.5097i 0.148134i
$$612$$ 0 0
$$613$$ −666.000 −1.08646 −0.543230 0.839584i $$-0.682799\pi$$
−0.543230 + 0.839584i $$0.682799\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 284.257i − 0.460708i −0.973107 0.230354i $$-0.926012\pi$$
0.973107 0.230354i $$-0.0739884\pi$$
$$618$$ 0 0
$$619$$ 768.000 1.24071 0.620355 0.784321i $$-0.286988\pi$$
0.620355 + 0.784321i $$0.286988\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 622.254i 0.998803i
$$624$$ 0 0
$$625$$ 479.000 0.766400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 330.926i 0.526114i
$$630$$ 0 0
$$631$$ 472.000 0.748019 0.374010 0.927425i $$-0.377983\pi$$
0.374010 + 0.927425i $$0.377983\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 79.1960i − 0.124718i
$$636$$ 0 0
$$637$$ 120.000 0.188383
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 371.938i − 0.580247i −0.956989 0.290123i $$-0.906304\pi$$
0.956989 0.290123i $$-0.0936963\pi$$
$$642$$ 0 0
$$643$$ 240.000 0.373250 0.186625 0.982431i $$-0.440245\pi$$
0.186625 + 0.982431i $$0.440245\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 712.764i 1.10164i 0.834623 + 0.550822i $$0.185686\pi$$
−0.834623 + 0.550822i $$0.814314\pi$$
$$648$$ 0 0
$$649$$ −256.000 −0.394453
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 1163.90i − 1.78239i −0.453625 0.891193i $$-0.649869\pi$$
0.453625 0.891193i $$-0.350131\pi$$
$$654$$ 0 0
$$655$$ −352.000 −0.537405
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 203.647i 0.309024i 0.987991 + 0.154512i $$0.0493805\pi$$
−0.987991 + 0.154512i $$0.950619\pi$$
$$660$$ 0 0
$$661$$ 1018.00 1.54009 0.770045 0.637989i $$-0.220234\pi$$
0.770045 + 0.637989i $$0.220234\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 362.039i − 0.544419i
$$666$$ 0 0
$$667$$ 1488.00 2.23088
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 610.940i 0.910492i
$$672$$ 0 0
$$673$$ −382.000 −0.567608 −0.283804 0.958882i $$-0.591596\pi$$
−0.283804 + 0.958882i $$0.591596\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 247.487i − 0.365565i −0.983153 0.182782i $$-0.941490\pi$$
0.983153 0.182782i $$-0.0585104\pi$$
$$678$$ 0 0
$$679$$ −640.000 −0.942563
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 169.706i 0.248471i 0.992253 + 0.124235i $$0.0396478\pi$$
−0.992253 + 0.124235i $$0.960352\pi$$
$$684$$ 0 0
$$685$$ −66.0000 −0.0963504
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 260.215i 0.377671i
$$690$$ 0 0
$$691$$ −1040.00 −1.50507 −0.752533 0.658555i $$-0.771168\pi$$
−0.752533 + 0.658555i $$0.771168\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 22.6274i − 0.0325574i
$$696$$ 0 0
$$697$$ −846.000 −1.21377
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 657.609i − 0.938102i −0.883171 0.469051i $$-0.844596\pi$$
0.883171 0.469051i $$-0.155404\pi$$
$$702$$ 0 0
$$703$$ −832.000 −1.18350
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 984.293i − 1.39221i
$$708$$ 0 0
$$709$$ −952.000 −1.34274 −0.671368 0.741124i $$-0.734293\pi$$
−0.671368 + 0.741124i $$0.734293\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 1357.65i − 1.90413i
$$714$$ 0 0
$$715$$ −128.000 −0.179021
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 192.333i − 0.267501i −0.991015 0.133750i $$-0.957298\pi$$
0.991015 0.133750i $$-0.0427020\pi$$
$$720$$ 0 0
$$721$$ −576.000 −0.798890
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1008.33i 1.39081i
$$726$$ 0 0
$$727$$ −648.000 −0.891334 −0.445667 0.895199i $$-0.647033\pi$$
−0.445667 + 0.895199i $$0.647033\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 203.647i − 0.278587i
$$732$$ 0 0
$$733$$ −1208.00 −1.64802 −0.824011 0.566574i $$-0.808269\pi$$
−0.824011 + 0.566574i $$0.808269\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 905.097i 1.22808i
$$738$$ 0 0
$$739$$ 1312.00 1.77537 0.887686 0.460449i $$-0.152312\pi$$
0.887686 + 0.460449i $$0.152312\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 610.940i 0.822261i 0.911576 + 0.411131i $$0.134866\pi$$
−0.911576 + 0.411131i $$0.865134\pi$$
$$744$$ 0 0
$$745$$ 258.000 0.346309
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 1448.15i − 1.93345i
$$750$$ 0 0
$$751$$ −632.000 −0.841545 −0.420772 0.907166i $$-0.638241\pi$$
−0.420772 + 0.907166i $$0.638241\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 237.588i 0.314686i
$$756$$ 0 0
$$757$$ 840.000 1.10964 0.554822 0.831969i $$-0.312786\pi$$
0.554822 + 0.831969i $$0.312786\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 521.845i − 0.685736i −0.939384 0.342868i $$-0.888602\pi$$
0.939384 0.342868i $$-0.111398\pi$$
$$762$$ 0 0
$$763$$ 704.000 0.922674
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 181.019i 0.236010i
$$768$$ 0 0
$$769$$ 130.000 0.169051 0.0845254 0.996421i $$-0.473063\pi$$
0.0845254 + 0.996421i $$0.473063\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 284.257i − 0.367732i −0.982951 0.183866i $$-0.941139\pi$$
0.982951 0.183866i $$-0.0588613\pi$$
$$774$$ 0 0
$$775$$ 920.000 1.18710
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 2126.98i − 2.73039i
$$780$$ 0 0
$$781$$ −896.000 −1.14725
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 14.1421i 0.0180155i
$$786$$ 0 0
$$787$$ 864.000 1.09784 0.548920 0.835875i $$-0.315039\pi$$
0.548920 + 0.835875i $$0.315039\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ − 1097.43i − 1.38740i
$$792$$ 0 0
$$793$$ 432.000 0.544767
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 759.433i 0.952864i 0.879211 + 0.476432i $$0.158070\pi$$
−0.879211 + 0.476432i $$0.841930\pi$$
$$798$$ 0 0
$$799$$ −144.000 −0.180225
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 1086.12i 1.35257i
$$804$$ 0 0
$$805$$ 384.000 0.477019
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 906.511i − 1.12053i −0.828313 0.560266i $$-0.810699\pi$$
0.828313 0.560266i $$-0.189301\pi$$
$$810$$ 0 0
$$811$$ −1472.00 −1.81504 −0.907522 0.420005i $$-0.862028\pi$$
−0.907522 + 0.420005i $$0.862028\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 113.137i − 0.138819i
$$816$$ 0 0
$$817$$ 512.000 0.626683
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1189.35i 1.44866i 0.689451 + 0.724332i $$0.257852\pi$$
−0.689451 + 0.724332i $$0.742148\pi$$
$$822$$ 0 0
$$823$$ −664.000 −0.806804 −0.403402 0.915023i $$-0.632172\pi$$
−0.403402 + 0.915023i $$0.632172\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 248.902i 0.300969i 0.988612 + 0.150485i $$0.0480834\pi$$
−0.988612 + 0.150485i $$0.951917\pi$$
$$828$$ 0 0
$$829$$ 280.000 0.337756 0.168878 0.985637i $$-0.445986\pi$$
0.168878 + 0.985637i $$0.445986\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 190.919i 0.229194i
$$834$$ 0 0
$$835$$ 416.000 0.498204
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 554.372i 0.660753i 0.943849 + 0.330376i $$0.107176\pi$$
−0.943849 + 0.330376i $$0.892824\pi$$
$$840$$ 0 0
$$841$$ −1081.00 −1.28537
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 148.492i − 0.175731i
$$846$$ 0 0
$$847$$ −56.0000 −0.0661157
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 882.469i − 1.03698i
$$852$$ 0 0
$$853$$ 762.000 0.893318 0.446659 0.894704i $$-0.352614\pi$$
0.446659 + 0.894704i $$0.352614\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1537.25i 1.79376i 0.442277 + 0.896879i $$0.354171\pi$$
−0.442277 + 0.896879i $$0.645829\pi$$
$$858$$ 0 0
$$859$$ −1552.00 −1.80675 −0.903376 0.428849i $$-0.858919\pi$$
−0.903376 + 0.428849i $$0.858919\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 22.6274i − 0.0262195i −0.999914 0.0131097i $$-0.995827\pi$$
0.999914 0.0131097i $$-0.00417308\pi$$
$$864$$ 0 0
$$865$$ 78.0000 0.0901734
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 1176.63i − 1.35400i
$$870$$ 0 0
$$871$$ 640.000 0.734788
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 543.058i 0.620638i
$$876$$ 0 0
$$877$$ 86.0000 0.0980616 0.0490308 0.998797i $$-0.484387\pi$$
0.0490308 + 0.998797i $$0.484387\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 657.609i 0.746435i 0.927744 + 0.373218i $$0.121745\pi$$
−0.927744 + 0.373218i $$0.878255\pi$$
$$882$$ 0 0
$$883$$ −496.000 −0.561721 −0.280861 0.959749i $$-0.590620\pi$$
−0.280861 + 0.959749i $$0.590620\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1018.23i 1.14795i 0.818872 + 0.573976i $$0.194600\pi$$
−0.818872 + 0.573976i $$0.805400\pi$$
$$888$$ 0 0
$$889$$ −448.000 −0.503937
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 362.039i − 0.405418i
$$894$$ 0 0
$$895$$ 192.000 0.214525
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1753.62i 1.95064i
$$900$$ 0 0
$$901$$ −414.000 −0.459489
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 11.3137i − 0.0125013i
$$906$$ 0 0
$$907$$ 880.000 0.970232 0.485116 0.874450i $$-0.338777\pi$$
0.485116 + 0.874450i $$0.338777\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 1561.29i − 1.71382i −0.515464 0.856911i $$-0.672381\pi$$
0.515464 0.856911i $$-0.327619\pi$$
$$912$$ 0 0
$$913$$ 1152.00 1.26177
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1991.21i 2.17144i
$$918$$ 0 0
$$919$$ 264.000 0.287269 0.143634 0.989631i $$-0.454121\pi$$
0.143634 + 0.989631i $$0.454121\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 633.568i 0.686422i
$$924$$ 0 0
$$925$$ 598.000 0.646486
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 284.257i 0.305982i 0.988228 + 0.152991i $$0.0488905\pi$$
−0.988228 + 0.152991i $$0.951110\pi$$
$$930$$ 0 0
$$931$$ −480.000 −0.515575
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 203.647i − 0.217804i
$$936$$ 0 0
$$937$$ 1070.00 1.14194 0.570971 0.820970i $$-0.306567\pi$$
0.570971 + 0.820970i $$0.306567\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 575.585i 0.611674i 0.952084 + 0.305837i $$0.0989362\pi$$
−0.952084 + 0.305837i $$0.901064\pi$$
$$942$$ 0 0
$$943$$ 2256.00 2.39236
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 610.940i − 0.645132i −0.946547 0.322566i $$-0.895455\pi$$
0.946547 0.322566i $$-0.104545\pi$$
$$948$$ 0 0
$$949$$ 768.000 0.809273
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 1039.45i − 1.09071i −0.838205 0.545355i $$-0.816395\pi$$
0.838205 0.545355i $$-0.183605\pi$$
$$954$$ 0 0
$$955$$ 96.0000 0.100524
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 373.352i 0.389314i
$$960$$ 0 0
$$961$$ 639.000 0.664932
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 364.867i − 0.378101i
$$966$$ 0 0
$$967$$ 696.000 0.719752 0.359876 0.933000i $$-0.382819\pi$$
0.359876 + 0.933000i $$0.382819\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1572.61i 1.61957i 0.586725 + 0.809787i $$0.300417\pi$$
−0.586725 + 0.809787i $$0.699583\pi$$
$$972$$ 0 0
$$973$$ −128.000 −0.131552
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 507.703i 0.519655i 0.965655 + 0.259827i $$0.0836657\pi$$
−0.965655 + 0.259827i $$0.916334\pi$$
$$978$$ 0 0
$$979$$ −880.000 −0.898876
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 475.176i − 0.483393i −0.970352 0.241697i $$-0.922296\pi$$
0.970352 0.241697i $$-0.0777039\pi$$
$$984$$ 0 0
$$985$$ 526.000 0.534010
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 543.058i 0.549098i
$$990$$ 0 0
$$991$$ 520.000 0.524723 0.262361 0.964970i $$-0.415499\pi$$
0.262361 + 0.964970i $$0.415499\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 124.451i 0.125076i
$$996$$ 0 0
$$997$$ 486.000 0.487462 0.243731 0.969843i $$-0.421629\pi$$
0.243731 + 0.969843i $$0.421629\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 576.3.e.g.449.2 2
3.2 odd 2 inner 576.3.e.g.449.1 2
4.3 odd 2 576.3.e.b.449.2 2
8.3 odd 2 288.3.e.a.161.1 2
8.5 even 2 288.3.e.d.161.1 yes 2
12.11 even 2 576.3.e.b.449.1 2
16.3 odd 4 2304.3.h.g.2177.4 4
16.5 even 4 2304.3.h.b.2177.1 4
16.11 odd 4 2304.3.h.g.2177.1 4
16.13 even 4 2304.3.h.b.2177.4 4
24.5 odd 2 288.3.e.d.161.2 yes 2
24.11 even 2 288.3.e.a.161.2 yes 2
48.5 odd 4 2304.3.h.b.2177.3 4
48.11 even 4 2304.3.h.g.2177.3 4
48.29 odd 4 2304.3.h.b.2177.2 4
48.35 even 4 2304.3.h.g.2177.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.a.161.1 2 8.3 odd 2
288.3.e.a.161.2 yes 2 24.11 even 2
288.3.e.d.161.1 yes 2 8.5 even 2
288.3.e.d.161.2 yes 2 24.5 odd 2
576.3.e.b.449.1 2 12.11 even 2
576.3.e.b.449.2 2 4.3 odd 2
576.3.e.g.449.1 2 3.2 odd 2 inner
576.3.e.g.449.2 2 1.1 even 1 trivial
2304.3.h.b.2177.1 4 16.5 even 4
2304.3.h.b.2177.2 4 48.29 odd 4
2304.3.h.b.2177.3 4 48.5 odd 4
2304.3.h.b.2177.4 4 16.13 even 4
2304.3.h.g.2177.1 4 16.11 odd 4
2304.3.h.g.2177.2 4 48.35 even 4
2304.3.h.g.2177.3 4 48.11 even 4
2304.3.h.g.2177.4 4 16.3 odd 4