# Properties

 Label 800.4.f.a.49.4 Level $800$ Weight $4$ Character 800.49 Analytic conductor $47.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3x^{2} + 4$$ x^4 - 3*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.4 Root $$-1.32288 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 800.49 Dual form 800.4.f.a.49.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.29150 q^{3} +8.00000i q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+5.29150 q^{3} +8.00000i q^{7} +1.00000 q^{9} -15.8745i q^{11} -52.9150 q^{13} -14.0000i q^{17} +37.0405i q^{19} +42.3320i q^{21} -152.000i q^{23} -137.579 q^{27} -158.745i q^{29} -224.000 q^{31} -84.0000i q^{33} +243.409 q^{37} -280.000 q^{39} -70.0000 q^{41} -439.195 q^{43} -336.000i q^{47} +279.000 q^{49} -74.0810i q^{51} -31.7490 q^{53} +196.000i q^{57} -534.442i q^{59} -95.2470i q^{61} +8.00000i q^{63} -174.620 q^{67} -804.308i q^{69} +72.0000 q^{71} +294.000i q^{73} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} -840.000i q^{87} -266.000 q^{89} -423.320i q^{91} -1185.30 q^{93} +994.000i q^{97} -15.8745i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100})$$ 4 * q + 4 * q^9 - 896 * q^31 - 1120 * q^39 - 280 * q^41 + 1116 * q^49 + 288 * q^71 - 1856 * q^79 - 3020 * q^81 - 1064 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 5.29150 1.01835 0.509175 0.860663i $$-0.329951\pi$$
0.509175 + 0.860663i $$0.329951\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 8.00000i 0.431959i 0.976398 + 0.215980i $$0.0692945\pi$$
−0.976398 + 0.215980i $$0.930705\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.0370370
$$10$$ 0 0
$$11$$ − 15.8745i − 0.435122i −0.976047 0.217561i $$-0.930190\pi$$
0.976047 0.217561i $$-0.0698101\pi$$
$$12$$ 0 0
$$13$$ −52.9150 −1.12892 −0.564461 0.825460i $$-0.690916\pi$$
−0.564461 + 0.825460i $$0.690916\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 14.0000i − 0.199735i −0.995001 0.0998676i $$-0.968158\pi$$
0.995001 0.0998676i $$-0.0318419\pi$$
$$18$$ 0 0
$$19$$ 37.0405i 0.447246i 0.974676 + 0.223623i $$0.0717885\pi$$
−0.974676 + 0.223623i $$0.928212\pi$$
$$20$$ 0 0
$$21$$ 42.3320i 0.439886i
$$22$$ 0 0
$$23$$ − 152.000i − 1.37801i −0.724757 0.689004i $$-0.758048\pi$$
0.724757 0.689004i $$-0.241952\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −137.579 −0.980633
$$28$$ 0 0
$$29$$ − 158.745i − 1.01649i −0.861212 0.508245i $$-0.830294\pi$$
0.861212 0.508245i $$-0.169706\pi$$
$$30$$ 0 0
$$31$$ −224.000 −1.29779 −0.648897 0.760877i $$-0.724769\pi$$
−0.648897 + 0.760877i $$0.724769\pi$$
$$32$$ 0 0
$$33$$ − 84.0000i − 0.443107i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 243.409 1.08152 0.540760 0.841177i $$-0.318137\pi$$
0.540760 + 0.841177i $$0.318137\pi$$
$$38$$ 0 0
$$39$$ −280.000 −1.14964
$$40$$ 0 0
$$41$$ −70.0000 −0.266638 −0.133319 0.991073i $$-0.542564\pi$$
−0.133319 + 0.991073i $$0.542564\pi$$
$$42$$ 0 0
$$43$$ −439.195 −1.55759 −0.778797 0.627276i $$-0.784170\pi$$
−0.778797 + 0.627276i $$0.784170\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 336.000i − 1.04278i −0.853319 0.521390i $$-0.825414\pi$$
0.853319 0.521390i $$-0.174586\pi$$
$$48$$ 0 0
$$49$$ 279.000 0.813411
$$50$$ 0 0
$$51$$ − 74.0810i − 0.203400i
$$52$$ 0 0
$$53$$ −31.7490 −0.0822842 −0.0411421 0.999153i $$-0.513100\pi$$
−0.0411421 + 0.999153i $$0.513100\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 196.000i 0.455453i
$$58$$ 0 0
$$59$$ − 534.442i − 1.17929i −0.807661 0.589647i $$-0.799267\pi$$
0.807661 0.589647i $$-0.200733\pi$$
$$60$$ 0 0
$$61$$ − 95.2470i − 0.199920i −0.994991 0.0999601i $$-0.968128\pi$$
0.994991 0.0999601i $$-0.0318715\pi$$
$$62$$ 0 0
$$63$$ 8.00000i 0.0159985i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −174.620 −0.318406 −0.159203 0.987246i $$-0.550892\pi$$
−0.159203 + 0.987246i $$0.550892\pi$$
$$68$$ 0 0
$$69$$ − 804.308i − 1.40329i
$$70$$ 0 0
$$71$$ 72.0000 0.120350 0.0601748 0.998188i $$-0.480834\pi$$
0.0601748 + 0.998188i $$0.480834\pi$$
$$72$$ 0 0
$$73$$ 294.000i 0.471371i 0.971829 + 0.235686i $$0.0757336\pi$$
−0.971829 + 0.235686i $$0.924266\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 126.996 0.187955
$$78$$ 0 0
$$79$$ −464.000 −0.660811 −0.330406 0.943839i $$-0.607186\pi$$
−0.330406 + 0.943839i $$0.607186\pi$$
$$80$$ 0 0
$$81$$ −755.000 −1.03567
$$82$$ 0 0
$$83$$ −545.025 −0.720774 −0.360387 0.932803i $$-0.617355\pi$$
−0.360387 + 0.932803i $$0.617355\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 840.000i − 1.03514i
$$88$$ 0 0
$$89$$ −266.000 −0.316808 −0.158404 0.987374i $$-0.550635\pi$$
−0.158404 + 0.987374i $$0.550635\pi$$
$$90$$ 0 0
$$91$$ − 423.320i − 0.487649i
$$92$$ 0 0
$$93$$ −1185.30 −1.32161
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 994.000i 1.04047i 0.854024 + 0.520234i $$0.174155\pi$$
−0.854024 + 0.520234i $$0.825845\pi$$
$$98$$ 0 0
$$99$$ − 15.8745i − 0.0161156i
$$100$$ 0 0
$$101$$ − 751.393i − 0.740262i −0.928980 0.370131i $$-0.879313\pi$$
0.928980 0.370131i $$-0.120687\pi$$
$$102$$ 0 0
$$103$$ 1176.00i 1.12500i 0.826798 + 0.562499i $$0.190160\pi$$
−0.826798 + 0.562499i $$0.809840\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 269.867 0.243822 0.121911 0.992541i $$-0.461098\pi$$
0.121911 + 0.992541i $$0.461098\pi$$
$$108$$ 0 0
$$109$$ − 1894.36i − 1.66465i −0.554290 0.832324i $$-0.687010\pi$$
0.554290 0.832324i $$-0.312990\pi$$
$$110$$ 0 0
$$111$$ 1288.00 1.10137
$$112$$ 0 0
$$113$$ 1710.00i 1.42357i 0.702398 + 0.711784i $$0.252113\pi$$
−0.702398 + 0.711784i $$0.747887\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −52.9150 −0.0418119
$$118$$ 0 0
$$119$$ 112.000 0.0862775
$$120$$ 0 0
$$121$$ 1079.00 0.810669
$$122$$ 0 0
$$123$$ −370.405 −0.271531
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1664.00i 1.16265i 0.813673 + 0.581323i $$0.197465\pi$$
−0.813673 + 0.581323i $$0.802535\pi$$
$$128$$ 0 0
$$129$$ −2324.00 −1.58618
$$130$$ 0 0
$$131$$ − 672.021i − 0.448204i −0.974566 0.224102i $$-0.928055\pi$$
0.974566 0.224102i $$-0.0719449\pi$$
$$132$$ 0 0
$$133$$ −296.324 −0.193192
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1062.00i − 0.662283i −0.943581 0.331142i $$-0.892566\pi$$
0.943581 0.331142i $$-0.107434\pi$$
$$138$$ 0 0
$$139$$ − 2693.37i − 1.64352i −0.569835 0.821759i $$-0.692993\pi$$
0.569835 0.821759i $$-0.307007\pi$$
$$140$$ 0 0
$$141$$ − 1777.94i − 1.06191i
$$142$$ 0 0
$$143$$ 840.000i 0.491219i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1476.33 0.828337
$$148$$ 0 0
$$149$$ 793.725i 0.436406i 0.975903 + 0.218203i $$0.0700195\pi$$
−0.975903 + 0.218203i $$0.929980\pi$$
$$150$$ 0 0
$$151$$ −744.000 −0.400966 −0.200483 0.979697i $$-0.564251\pi$$
−0.200483 + 0.979697i $$0.564251\pi$$
$$152$$ 0 0
$$153$$ − 14.0000i − 0.00739760i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 179.911 0.0914552 0.0457276 0.998954i $$-0.485439\pi$$
0.0457276 + 0.998954i $$0.485439\pi$$
$$158$$ 0 0
$$159$$ −168.000 −0.0837941
$$160$$ 0 0
$$161$$ 1216.00 0.595244
$$162$$ 0 0
$$163$$ −1772.65 −0.851809 −0.425905 0.904768i $$-0.640044\pi$$
−0.425905 + 0.904768i $$0.640044\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1960.00i 0.908200i 0.890951 + 0.454100i $$0.150039\pi$$
−0.890951 + 0.454100i $$0.849961\pi$$
$$168$$ 0 0
$$169$$ 603.000 0.274465
$$170$$ 0 0
$$171$$ 37.0405i 0.0165647i
$$172$$ 0 0
$$173$$ −2000.19 −0.879026 −0.439513 0.898236i $$-0.644849\pi$$
−0.439513 + 0.898236i $$0.644849\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 2828.00i − 1.20094i
$$178$$ 0 0
$$179$$ − 3264.86i − 1.36328i −0.731688 0.681639i $$-0.761267\pi$$
0.731688 0.681639i $$-0.238733\pi$$
$$180$$ 0 0
$$181$$ 2338.84i 0.960469i 0.877140 + 0.480235i $$0.159448\pi$$
−0.877140 + 0.480235i $$0.840552\pi$$
$$182$$ 0 0
$$183$$ − 504.000i − 0.203589i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −222.243 −0.0869092
$$188$$ 0 0
$$189$$ − 1100.63i − 0.423594i
$$190$$ 0 0
$$191$$ −3904.00 −1.47897 −0.739486 0.673172i $$-0.764931\pi$$
−0.739486 + 0.673172i $$0.764931\pi$$
$$192$$ 0 0
$$193$$ − 3330.00i − 1.24196i −0.783826 0.620981i $$-0.786734\pi$$
0.783826 0.620981i $$-0.213266\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1195.88 −0.432502 −0.216251 0.976338i $$-0.569383\pi$$
−0.216251 + 0.976338i $$0.569383\pi$$
$$198$$ 0 0
$$199$$ −1736.00 −0.618401 −0.309200 0.950997i $$-0.600061\pi$$
−0.309200 + 0.950997i $$0.600061\pi$$
$$200$$ 0 0
$$201$$ −924.000 −0.324248
$$202$$ 0 0
$$203$$ 1269.96 0.439083
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 152.000i − 0.0510373i
$$208$$ 0 0
$$209$$ 588.000 0.194607
$$210$$ 0 0
$$211$$ − 2915.62i − 0.951277i −0.879641 0.475638i $$-0.842217\pi$$
0.879641 0.475638i $$-0.157783\pi$$
$$212$$ 0 0
$$213$$ 380.988 0.122558
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 1792.00i − 0.560594i
$$218$$ 0 0
$$219$$ 1555.70i 0.480021i
$$220$$ 0 0
$$221$$ 740.810i 0.225486i
$$222$$ 0 0
$$223$$ 1568.00i 0.470857i 0.971892 + 0.235428i $$0.0756493\pi$$
−0.971892 + 0.235428i $$0.924351\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1264.67 0.369775 0.184888 0.982760i $$-0.440808\pi$$
0.184888 + 0.982760i $$0.440808\pi$$
$$228$$ 0 0
$$229$$ 5153.92i 1.48725i 0.668595 + 0.743626i $$0.266896\pi$$
−0.668595 + 0.743626i $$0.733104\pi$$
$$230$$ 0 0
$$231$$ 672.000 0.191404
$$232$$ 0 0
$$233$$ 838.000i 0.235619i 0.993036 + 0.117809i $$0.0375872\pi$$
−0.993036 + 0.117809i $$0.962413\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −2455.26 −0.672937
$$238$$ 0 0
$$239$$ 6288.00 1.70183 0.850914 0.525305i $$-0.176049\pi$$
0.850914 + 0.525305i $$0.176049\pi$$
$$240$$ 0 0
$$241$$ −2926.00 −0.782076 −0.391038 0.920375i $$-0.627884\pi$$
−0.391038 + 0.920375i $$0.627884\pi$$
$$242$$ 0 0
$$243$$ −280.450 −0.0740364
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 1960.00i − 0.504906i
$$248$$ 0 0
$$249$$ −2884.00 −0.734000
$$250$$ 0 0
$$251$$ 5444.96i 1.36925i 0.728894 + 0.684627i $$0.240035\pi$$
−0.728894 + 0.684627i $$0.759965\pi$$
$$252$$ 0 0
$$253$$ −2412.93 −0.599602
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2562.00i 0.621841i 0.950436 + 0.310921i $$0.100637\pi$$
−0.950436 + 0.310921i $$0.899363\pi$$
$$258$$ 0 0
$$259$$ 1947.27i 0.467172i
$$260$$ 0 0
$$261$$ − 158.745i − 0.0376478i
$$262$$ 0 0
$$263$$ − 5896.00i − 1.38237i −0.722679 0.691184i $$-0.757089\pi$$
0.722679 0.691184i $$-0.242911\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1407.54 −0.322622
$$268$$ 0 0
$$269$$ − 5365.58i − 1.21615i −0.793878 0.608077i $$-0.791941\pi$$
0.793878 0.608077i $$-0.208059\pi$$
$$270$$ 0 0
$$271$$ 1680.00 0.376578 0.188289 0.982114i $$-0.439706\pi$$
0.188289 + 0.982114i $$0.439706\pi$$
$$272$$ 0 0
$$273$$ − 2240.00i − 0.496597i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1576.87 −0.342039 −0.171019 0.985268i $$-0.554706\pi$$
−0.171019 + 0.985268i $$0.554706\pi$$
$$278$$ 0 0
$$279$$ −224.000 −0.0480664
$$280$$ 0 0
$$281$$ −2742.00 −0.582114 −0.291057 0.956706i $$-0.594007\pi$$
−0.291057 + 0.956706i $$0.594007\pi$$
$$282$$ 0 0
$$283$$ 2989.70 0.627983 0.313991 0.949426i $$-0.398334\pi$$
0.313991 + 0.949426i $$0.398334\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 560.000i − 0.115177i
$$288$$ 0 0
$$289$$ 4717.00 0.960106
$$290$$ 0 0
$$291$$ 5259.75i 1.05956i
$$292$$ 0 0
$$293$$ 9238.96 1.84214 0.921068 0.389401i $$-0.127318\pi$$
0.921068 + 0.389401i $$0.127318\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 2184.00i 0.426695i
$$298$$ 0 0
$$299$$ 8043.08i 1.55566i
$$300$$ 0 0
$$301$$ − 3513.56i − 0.672818i
$$302$$ 0 0
$$303$$ − 3976.00i − 0.753846i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2587.54 −0.481039 −0.240520 0.970644i $$-0.577318\pi$$
−0.240520 + 0.970644i $$0.577318\pi$$
$$308$$ 0 0
$$309$$ 6222.81i 1.14564i
$$310$$ 0 0
$$311$$ 2744.00 0.500315 0.250157 0.968205i $$-0.419518\pi$$
0.250157 + 0.968205i $$0.419518\pi$$
$$312$$ 0 0
$$313$$ − 2282.00i − 0.412097i −0.978542 0.206048i $$-0.933940\pi$$
0.978542 0.206048i $$-0.0660604\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9577.62 1.69695 0.848474 0.529237i $$-0.177522\pi$$
0.848474 + 0.529237i $$0.177522\pi$$
$$318$$ 0 0
$$319$$ −2520.00 −0.442298
$$320$$ 0 0
$$321$$ 1428.00 0.248297
$$322$$ 0 0
$$323$$ 518.567 0.0893308
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 10024.0i − 1.69519i
$$328$$ 0 0
$$329$$ 2688.00 0.450438
$$330$$ 0 0
$$331$$ − 4249.08i − 0.705590i −0.935701 0.352795i $$-0.885231\pi$$
0.935701 0.352795i $$-0.114769\pi$$
$$332$$ 0 0
$$333$$ 243.409 0.0400563
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6130.00i 0.990868i 0.868646 + 0.495434i $$0.164991\pi$$
−0.868646 + 0.495434i $$0.835009\pi$$
$$338$$ 0 0
$$339$$ 9048.47i 1.44969i
$$340$$ 0 0
$$341$$ 3555.89i 0.564699i
$$342$$ 0 0
$$343$$ 4976.00i 0.783320i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2481.71 −0.383935 −0.191967 0.981401i $$-0.561487\pi$$
−0.191967 + 0.981401i $$0.561487\pi$$
$$348$$ 0 0
$$349$$ − 328.073i − 0.0503191i −0.999683 0.0251595i $$-0.991991\pi$$
0.999683 0.0251595i $$-0.00800937\pi$$
$$350$$ 0 0
$$351$$ 7280.00 1.10706
$$352$$ 0 0
$$353$$ 10206.0i 1.53884i 0.638743 + 0.769420i $$0.279455\pi$$
−0.638743 + 0.769420i $$0.720545\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 592.648 0.0878607
$$358$$ 0 0
$$359$$ −3176.00 −0.466916 −0.233458 0.972367i $$-0.575004\pi$$
−0.233458 + 0.972367i $$0.575004\pi$$
$$360$$ 0 0
$$361$$ 5487.00 0.799971
$$362$$ 0 0
$$363$$ 5709.53 0.825545
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 11760.0i 1.67266i 0.548225 + 0.836331i $$0.315304\pi$$
−0.548225 + 0.836331i $$0.684696\pi$$
$$368$$ 0 0
$$369$$ −70.0000 −0.00987549
$$370$$ 0 0
$$371$$ − 253.992i − 0.0355434i
$$372$$ 0 0
$$373$$ 10974.6 1.52344 0.761719 0.647908i $$-0.224356\pi$$
0.761719 + 0.647908i $$0.224356\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8400.00i 1.14754i
$$378$$ 0 0
$$379$$ − 3074.36i − 0.416674i −0.978057 0.208337i $$-0.933195\pi$$
0.978057 0.208337i $$-0.0668051\pi$$
$$380$$ 0 0
$$381$$ 8805.06i 1.18398i
$$382$$ 0 0
$$383$$ 2688.00i 0.358617i 0.983793 + 0.179309i $$0.0573861\pi$$
−0.983793 + 0.179309i $$0.942614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −439.195 −0.0576887
$$388$$ 0 0
$$389$$ 10487.8i 1.36697i 0.729966 + 0.683484i $$0.239536\pi$$
−0.729966 + 0.683484i $$0.760464\pi$$
$$390$$ 0 0
$$391$$ −2128.00 −0.275237
$$392$$ 0 0
$$393$$ − 3556.00i − 0.456429i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −5704.24 −0.721127 −0.360564 0.932735i $$-0.617416\pi$$
−0.360564 + 0.932735i $$0.617416\pi$$
$$398$$ 0 0
$$399$$ −1568.00 −0.196737
$$400$$ 0 0
$$401$$ 12402.0 1.54445 0.772227 0.635346i $$-0.219143\pi$$
0.772227 + 0.635346i $$0.219143\pi$$
$$402$$ 0 0
$$403$$ 11853.0 1.46511
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 3864.00i − 0.470593i
$$408$$ 0 0
$$409$$ 12278.0 1.48437 0.742186 0.670194i $$-0.233789\pi$$
0.742186 + 0.670194i $$0.233789\pi$$
$$410$$ 0 0
$$411$$ − 5619.58i − 0.674436i
$$412$$ 0 0
$$413$$ 4275.53 0.509407
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 14252.0i − 1.67368i
$$418$$ 0 0
$$419$$ 8207.12i 0.956907i 0.878113 + 0.478454i $$0.158802\pi$$
−0.878113 + 0.478454i $$0.841198\pi$$
$$420$$ 0 0
$$421$$ 1449.87i 0.167844i 0.996472 + 0.0839221i $$0.0267447\pi$$
−0.996472 + 0.0839221i $$0.973255\pi$$
$$422$$ 0 0
$$423$$ − 336.000i − 0.0386215i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 761.976 0.0863574
$$428$$ 0 0
$$429$$ 4444.86i 0.500233i
$$430$$ 0 0
$$431$$ −7632.00 −0.852948 −0.426474 0.904500i $$-0.640244\pi$$
−0.426474 + 0.904500i $$0.640244\pi$$
$$432$$ 0 0
$$433$$ − 3794.00i − 0.421081i −0.977585 0.210540i $$-0.932478\pi$$
0.977585 0.210540i $$-0.0675224\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5630.16 0.616309
$$438$$ 0 0
$$439$$ −1848.00 −0.200912 −0.100456 0.994942i $$-0.532030\pi$$
−0.100456 + 0.994942i $$0.532030\pi$$
$$440$$ 0 0
$$441$$ 279.000 0.0301263
$$442$$ 0 0
$$443$$ −12334.5 −1.32287 −0.661433 0.750004i $$-0.730051\pi$$
−0.661433 + 0.750004i $$0.730051\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4200.00i 0.444414i
$$448$$ 0 0
$$449$$ 3582.00 0.376492 0.188246 0.982122i $$-0.439720\pi$$
0.188246 + 0.982122i $$0.439720\pi$$
$$450$$ 0 0
$$451$$ 1111.22i 0.116020i
$$452$$ 0 0
$$453$$ −3936.88 −0.408324
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2714.00i 0.277802i 0.990306 + 0.138901i $$0.0443570\pi$$
−0.990306 + 0.138901i $$0.955643\pi$$
$$458$$ 0 0
$$459$$ 1926.11i 0.195867i
$$460$$ 0 0
$$461$$ − 8349.99i − 0.843596i −0.906690 0.421798i $$-0.861399\pi$$
0.906690 0.421798i $$-0.138601\pi$$
$$462$$ 0 0
$$463$$ 2224.00i 0.223236i 0.993751 + 0.111618i $$0.0356032\pi$$
−0.993751 + 0.111618i $$0.964397\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10292.0 −1.01982 −0.509910 0.860228i $$-0.670321\pi$$
−0.509910 + 0.860228i $$0.670321\pi$$
$$468$$ 0 0
$$469$$ − 1396.96i − 0.137538i
$$470$$ 0 0
$$471$$ 952.000 0.0931334
$$472$$ 0 0
$$473$$ 6972.00i 0.677744i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −31.7490 −0.00304756
$$478$$ 0 0
$$479$$ 17696.0 1.68800 0.843999 0.536345i $$-0.180195\pi$$
0.843999 + 0.536345i $$0.180195\pi$$
$$480$$ 0 0
$$481$$ −12880.0 −1.22095
$$482$$ 0 0
$$483$$ 6434.47 0.606166
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 1304.00i − 0.121334i −0.998158 0.0606672i $$-0.980677\pi$$
0.998158 0.0606672i $$-0.0193228\pi$$
$$488$$ 0 0
$$489$$ −9380.00 −0.867440
$$490$$ 0 0
$$491$$ 16662.9i 1.53154i 0.643112 + 0.765772i $$0.277643\pi$$
−0.643112 + 0.765772i $$0.722357\pi$$
$$492$$ 0 0
$$493$$ −2222.43 −0.203029
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 576.000i 0.0519862i
$$498$$ 0 0
$$499$$ − 3095.53i − 0.277705i −0.990313 0.138853i $$-0.955659\pi$$
0.990313 0.138853i $$-0.0443414\pi$$
$$500$$ 0 0
$$501$$ 10371.3i 0.924865i
$$502$$ 0 0
$$503$$ − 19320.0i − 1.71260i −0.516481 0.856298i $$-0.672758\pi$$
0.516481 0.856298i $$-0.327242\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3190.78 0.279502
$$508$$ 0 0
$$509$$ − 4476.61i − 0.389828i −0.980820 0.194914i $$-0.937557\pi$$
0.980820 0.194914i $$-0.0624427\pi$$
$$510$$ 0 0
$$511$$ −2352.00 −0.203613
$$512$$ 0 0
$$513$$ − 5096.00i − 0.438585i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −5333.83 −0.453737
$$518$$ 0 0
$$519$$ −10584.0 −0.895156
$$520$$ 0 0
$$521$$ −2982.00 −0.250756 −0.125378 0.992109i $$-0.540014\pi$$
−0.125378 + 0.992109i $$0.540014\pi$$
$$522$$ 0 0
$$523$$ 2016.06 0.168559 0.0842794 0.996442i $$-0.473141\pi$$
0.0842794 + 0.996442i $$0.473141\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 3136.00i 0.259215i
$$528$$ 0 0
$$529$$ −10937.0 −0.898907
$$530$$ 0 0
$$531$$ − 534.442i − 0.0436776i
$$532$$ 0 0
$$533$$ 3704.05 0.301014
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 17276.0i − 1.38830i
$$538$$ 0 0
$$539$$ − 4428.99i − 0.353933i
$$540$$ 0 0
$$541$$ − 15419.4i − 1.22539i −0.790321 0.612693i $$-0.790086\pi$$
0.790321 0.612693i $$-0.209914\pi$$
$$542$$ 0 0
$$543$$ 12376.0i 0.978094i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12609.7 0.985649 0.492824 0.870129i $$-0.335965\pi$$
0.492824 + 0.870129i $$0.335965\pi$$
$$548$$ 0 0
$$549$$ − 95.2470i − 0.00740445i
$$550$$ 0 0
$$551$$ 5880.00 0.454621
$$552$$ 0 0
$$553$$ − 3712.00i − 0.285444i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −7143.53 −0.543413 −0.271706 0.962380i $$-0.587588\pi$$
−0.271706 + 0.962380i $$0.587588\pi$$
$$558$$ 0 0
$$559$$ 23240.0 1.75840
$$560$$ 0 0
$$561$$ −1176.00 −0.0885040
$$562$$ 0 0
$$563$$ −7572.14 −0.566834 −0.283417 0.958997i $$-0.591468\pi$$
−0.283417 + 0.958997i $$0.591468\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 6040.00i − 0.447365i
$$568$$ 0 0
$$569$$ −15594.0 −1.14892 −0.574459 0.818533i $$-0.694788\pi$$
−0.574459 + 0.818533i $$0.694788\pi$$
$$570$$ 0 0
$$571$$ − 16737.0i − 1.22666i −0.789827 0.613330i $$-0.789830\pi$$
0.789827 0.613330i $$-0.210170\pi$$
$$572$$ 0 0
$$573$$ −20658.0 −1.50611
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6594.00i 0.475757i 0.971295 + 0.237879i $$0.0764520\pi$$
−0.971295 + 0.237879i $$0.923548\pi$$
$$578$$ 0 0
$$579$$ − 17620.7i − 1.26475i
$$580$$ 0 0
$$581$$ − 4360.20i − 0.311345i
$$582$$ 0 0
$$583$$ 504.000i 0.0358037i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 23213.8 1.63226 0.816130 0.577868i $$-0.196115\pi$$
0.816130 + 0.577868i $$0.196115\pi$$
$$588$$ 0 0
$$589$$ − 8297.08i − 0.580433i
$$590$$ 0 0
$$591$$ −6328.00 −0.440438
$$592$$ 0 0
$$593$$ − 14322.0i − 0.991794i −0.868381 0.495897i $$-0.834839\pi$$
0.868381 0.495897i $$-0.165161\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −9186.05 −0.629749
$$598$$ 0 0
$$599$$ −16088.0 −1.09739 −0.548696 0.836022i $$-0.684876\pi$$
−0.548696 + 0.836022i $$0.684876\pi$$
$$600$$ 0 0
$$601$$ −21238.0 −1.44146 −0.720729 0.693217i $$-0.756193\pi$$
−0.720729 + 0.693217i $$0.756193\pi$$
$$602$$ 0 0
$$603$$ −174.620 −0.0117928
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 13664.0i 0.913681i 0.889549 + 0.456841i $$0.151019\pi$$
−0.889549 + 0.456841i $$0.848981\pi$$
$$608$$ 0 0
$$609$$ 6720.00 0.447140
$$610$$ 0 0
$$611$$ 17779.4i 1.17722i
$$612$$ 0 0
$$613$$ −20393.5 −1.34369 −0.671846 0.740690i $$-0.734499\pi$$
−0.671846 + 0.740690i $$0.734499\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 3782.00i − 0.246771i −0.992359 0.123385i $$-0.960625\pi$$
0.992359 0.123385i $$-0.0393751\pi$$
$$618$$ 0 0
$$619$$ − 5825.94i − 0.378295i −0.981949 0.189147i $$-0.939428\pi$$
0.981949 0.189147i $$-0.0605724\pi$$
$$620$$ 0 0
$$621$$ 20912.0i 1.35132i
$$622$$ 0 0
$$623$$ − 2128.00i − 0.136848i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3111.40 0.198178
$$628$$ 0 0
$$629$$ − 3407.73i − 0.216017i
$$630$$ 0 0
$$631$$ −2056.00 −0.129712 −0.0648558 0.997895i $$-0.520659\pi$$
−0.0648558 + 0.997895i $$0.520659\pi$$
$$632$$ 0 0
$$633$$ − 15428.0i − 0.968733i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −14763.3 −0.918278
$$638$$ 0 0
$$639$$ 72.0000 0.00445740
$$640$$ 0 0
$$641$$ 11842.0 0.729689 0.364845 0.931068i $$-0.381122\pi$$
0.364845 + 0.931068i $$0.381122\pi$$
$$642$$ 0 0
$$643$$ −16250.2 −0.996649 −0.498325 0.866991i $$-0.666051\pi$$
−0.498325 + 0.866991i $$0.666051\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 19320.0i − 1.17395i −0.809604 0.586976i $$-0.800318\pi$$
0.809604 0.586976i $$-0.199682\pi$$
$$648$$ 0 0
$$649$$ −8484.00 −0.513137
$$650$$ 0 0
$$651$$ − 9482.37i − 0.570881i
$$652$$ 0 0
$$653$$ 2317.68 0.138894 0.0694470 0.997586i $$-0.477877\pi$$
0.0694470 + 0.997586i $$0.477877\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 294.000i 0.0174582i
$$658$$ 0 0
$$659$$ − 27732.8i − 1.63932i −0.572847 0.819662i $$-0.694161\pi$$
0.572847 0.819662i $$-0.305839\pi$$
$$660$$ 0 0
$$661$$ − 22467.7i − 1.32208i −0.750352 0.661039i $$-0.770116\pi$$
0.750352 0.661039i $$-0.229884\pi$$
$$662$$ 0 0
$$663$$ 3920.00i 0.229623i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −24129.3 −1.40073
$$668$$ 0 0
$$669$$ 8297.08i 0.479497i
$$670$$ 0 0
$$671$$ −1512.00 −0.0869897
$$672$$ 0 0
$$673$$ 10078.0i 0.577234i 0.957445 + 0.288617i $$0.0931954\pi$$
−0.957445 + 0.288617i $$0.906805\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 16160.2 0.917413 0.458707 0.888588i $$-0.348313\pi$$
0.458707 + 0.888588i $$0.348313\pi$$
$$678$$ 0 0
$$679$$ −7952.00 −0.449440
$$680$$ 0 0
$$681$$ 6692.00 0.376561
$$682$$ 0 0
$$683$$ −16356.0 −0.916320 −0.458160 0.888870i $$-0.651491\pi$$
−0.458160 + 0.888870i $$0.651491\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 27272.0i 1.51454i
$$688$$ 0 0
$$689$$ 1680.00 0.0928925
$$690$$ 0 0
$$691$$ − 29246.1i − 1.61009i −0.593211 0.805047i $$-0.702140\pi$$
0.593211 0.805047i $$-0.297860\pi$$
$$692$$ 0 0
$$693$$ 126.996 0.00696130
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 980.000i 0.0532570i
$$698$$ 0 0
$$699$$ 4434.28i 0.239943i
$$700$$ 0 0
$$701$$ − 2465.84i − 0.132858i −0.997791 0.0664290i $$-0.978839\pi$$
0.997791 0.0664290i $$-0.0211606\pi$$
$$702$$ 0 0
$$703$$ 9016.00i 0.483705i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 6011.15 0.319763
$$708$$ 0 0
$$709$$ − 31674.9i − 1.67782i −0.544267 0.838912i $$-0.683192\pi$$
0.544267 0.838912i $$-0.316808\pi$$
$$710$$ 0 0
$$711$$ −464.000 −0.0244745
$$712$$ 0 0
$$713$$ 34048.0i 1.78837i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 33273.0 1.73306
$$718$$ 0 0
$$719$$ −9296.00 −0.482173 −0.241086 0.970504i $$-0.577504\pi$$
−0.241086 + 0.970504i $$0.577504\pi$$
$$720$$ 0 0
$$721$$ −9408.00 −0.485953
$$722$$ 0 0
$$723$$ −15482.9 −0.796427
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 21672.0i − 1.10560i −0.833315 0.552799i $$-0.813560\pi$$
0.833315 0.552799i $$-0.186440\pi$$
$$728$$ 0 0
$$729$$ 18901.0 0.960270
$$730$$ 0 0
$$731$$ 6148.73i 0.311106i
$$732$$ 0 0
$$733$$ 9471.79 0.477283 0.238642 0.971108i $$-0.423298\pi$$
0.238642 + 0.971108i $$0.423298\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2772.00i 0.138545i
$$738$$ 0 0
$$739$$ − 6863.08i − 0.341627i −0.985303 0.170814i $$-0.945360\pi$$
0.985303 0.170814i $$-0.0546396\pi$$
$$740$$ 0 0
$$741$$ − 10371.3i − 0.514171i
$$742$$ 0 0
$$743$$ 17432.0i 0.860724i 0.902656 + 0.430362i $$0.141614\pi$$
−0.902656 + 0.430362i $$0.858386\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −545.025 −0.0266953
$$748$$ 0 0
$$749$$ 2158.93i 0.105321i
$$750$$ 0 0
$$751$$ 11632.0 0.565190 0.282595 0.959239i $$-0.408805\pi$$
0.282595 + 0.959239i $$0.408805\pi$$
$$752$$ 0 0
$$753$$ 28812.0i 1.39438i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −16731.7 −0.803336 −0.401668 0.915785i $$-0.631569\pi$$
−0.401668 + 0.915785i $$0.631569\pi$$
$$758$$ 0 0
$$759$$ −12768.0 −0.610605
$$760$$ 0 0
$$761$$ 39466.0 1.87995 0.939975 0.341244i $$-0.110848\pi$$
0.939975 + 0.341244i $$0.110848\pi$$
$$762$$ 0 0
$$763$$ 15154.9 0.719060
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28280.0i 1.33133i
$$768$$ 0 0
$$769$$ −35266.0 −1.65374 −0.826869 0.562395i $$-0.809880\pi$$
−0.826869 + 0.562395i $$0.809880\pi$$
$$770$$ 0 0
$$771$$ 13556.8i 0.633252i
$$772$$ 0 0
$$773$$ −16244.9 −0.755872 −0.377936 0.925832i $$-0.623366\pi$$
−0.377936 + 0.925832i $$0.623366\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 10304.0i 0.475745i
$$778$$ 0 0
$$779$$ − 2592.84i − 0.119253i
$$780$$ 0 0
$$781$$ − 1142.96i − 0.0523668i
$$782$$ 0 0
$$783$$ 21840.0i 0.996805i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −34844.5 −1.57824 −0.789119 0.614240i $$-0.789463\pi$$
−0.789119 + 0.614240i $$0.789463\pi$$
$$788$$ 0 0
$$789$$ − 31198.7i − 1.40774i
$$790$$ 0 0
$$791$$ −13680.0 −0.614924
$$792$$ 0 0
$$793$$ 5040.00i 0.225694i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2550.50 0.113354 0.0566772 0.998393i $$-0.481949\pi$$
0.0566772 + 0.998393i $$0.481949\pi$$
$$798$$ 0 0
$$799$$ −4704.00 −0.208280
$$800$$ 0 0
$$801$$ −266.000 −0.0117336
$$802$$ 0 0
$$803$$ 4667.11 0.205104
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 28392.0i − 1.23847i
$$808$$ 0 0
$$809$$ 24390.0 1.05996 0.529979 0.848010i $$-0.322200\pi$$
0.529979 + 0.848010i $$0.322200\pi$$
$$810$$ 0 0
$$811$$ − 9582.91i − 0.414922i −0.978243 0.207461i $$-0.933480\pi$$
0.978243 0.207461i $$-0.0665200\pi$$
$$812$$ 0 0
$$813$$ 8889.72 0.383489
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 16268.0i − 0.696628i
$$818$$ 0 0
$$819$$ − 423.320i − 0.0180611i
$$820$$ 0 0
$$821$$ 8773.31i 0.372948i 0.982460 + 0.186474i $$0.0597061\pi$$
−0.982460 + 0.186474i $$0.940294\pi$$
$$822$$ 0 0
$$823$$ − 21688.0i − 0.918586i −0.888285 0.459293i $$-0.848103\pi$$
0.888285 0.459293i $$-0.151897\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 19446.3 0.817670 0.408835 0.912608i $$-0.365935\pi$$
0.408835 + 0.912608i $$0.365935\pi$$
$$828$$ 0 0
$$829$$ − 19546.8i − 0.818925i −0.912327 0.409462i $$-0.865716\pi$$
0.912327 0.409462i $$-0.134284\pi$$
$$830$$ 0 0
$$831$$ −8344.00 −0.348315
$$832$$ 0 0
$$833$$ − 3906.00i − 0.162467i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 30817.7 1.27266
$$838$$ 0 0
$$839$$ −18760.0 −0.771951 −0.385976 0.922509i $$-0.626135\pi$$
−0.385976 + 0.922509i $$0.626135\pi$$
$$840$$ 0 0
$$841$$ −811.000 −0.0332527
$$842$$ 0 0
$$843$$ −14509.3 −0.592796
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 8632.00i 0.350176i
$$848$$ 0 0
$$849$$ 15820.0 0.639506
$$850$$ 0 0
$$851$$ − 36998.2i − 1.49034i
$$852$$ 0 0
$$853$$ −28732.9 −1.15333 −0.576667 0.816979i $$-0.695647\pi$$
−0.576667 + 0.816979i $$0.695647\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 8778.00i 0.349884i 0.984579 + 0.174942i $$0.0559738\pi$$
−0.984579 + 0.174942i $$0.944026\pi$$
$$858$$ 0 0
$$859$$ 5646.03i 0.224261i 0.993693 + 0.112130i $$0.0357675\pi$$
−0.993693 + 0.112130i $$0.964233\pi$$
$$860$$ 0 0
$$861$$ − 2963.24i − 0.117290i
$$862$$ 0 0
$$863$$ − 9312.00i − 0.367305i −0.982991 0.183652i $$-0.941208\pi$$
0.982991 0.183652i $$-0.0587921\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 24960.0 0.977724
$$868$$ 0 0
$$869$$ 7365.77i 0.287534i
$$870$$ 0 0
$$871$$ 9240.00 0.359455
$$872$$ 0 0
$$873$$ 994.000i 0.0385359i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 137.579 0.00529728 0.00264864 0.999996i $$-0.499157\pi$$
0.00264864 + 0.999996i $$0.499157\pi$$
$$878$$ 0 0
$$879$$ 48888.0 1.87594
$$880$$ 0 0
$$881$$ −31150.0 −1.19123 −0.595613 0.803272i $$-0.703091\pi$$
−0.595613 + 0.803272i $$0.703091\pi$$
$$882$$ 0 0
$$883$$ 12577.9 0.479366 0.239683 0.970851i $$-0.422957\pi$$
0.239683 + 0.970851i $$0.422957\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 37128.0i − 1.40545i −0.711460 0.702726i $$-0.751966\pi$$
0.711460 0.702726i $$-0.248034\pi$$
$$888$$ 0 0
$$889$$ −13312.0 −0.502216
$$890$$ 0 0
$$891$$ 11985.3i 0.450641i
$$892$$ 0 0
$$893$$ 12445.6 0.466379
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 42560.0i 1.58421i
$$898$$ 0 0
$$899$$ 35558.9i 1.31919i
$$900$$ 0 0
$$901$$ 444.486i 0.0164351i
$$902$$ 0 0
$$903$$ − 18592.0i − 0.685164i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −35204.4 −1.28880 −0.644400 0.764688i $$-0.722893\pi$$
−0.644400 + 0.764688i $$0.722893\pi$$
$$908$$ 0 0
$$909$$ − 751.393i − 0.0274171i
$$910$$ 0 0
$$911$$ 10512.0 0.382303 0.191152 0.981561i $$-0.438778\pi$$
0.191152 + 0.981561i $$0.438778\pi$$
$$912$$ 0 0
$$913$$ 8652.00i 0.313625i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 5376.17 0.193606
$$918$$ 0 0
$$919$$ −46104.0 −1.65488 −0.827438 0.561557i $$-0.810202\pi$$
−0.827438 + 0.561557i $$0.810202\pi$$
$$920$$ 0 0
$$921$$ −13692.0 −0.489866
$$922$$ 0 0
$$923$$ −3809.88 −0.135865
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1176.00i 0.0416666i
$$928$$ 0 0
$$929$$ 5726.00 0.202222 0.101111 0.994875i $$-0.467760\pi$$
0.101111 + 0.994875i $$0.467760\pi$$
$$930$$ 0 0
$$931$$ 10334.3i 0.363795i
$$932$$ 0 0
$$933$$ 14519.9 0.509496
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1274.00i 0.0444181i 0.999753 + 0.0222091i $$0.00706994\pi$$
−0.999753 + 0.0222091i $$0.992930\pi$$
$$938$$ 0 0
$$939$$ − 12075.2i − 0.419659i
$$940$$ 0 0
$$941$$ 26446.9i 0.916201i 0.888900 + 0.458101i $$0.151470\pi$$
−0.888900 + 0.458101i $$0.848530\pi$$
$$942$$ 0 0
$$943$$ 10640.0i 0.367430i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −23922.9 −0.820897 −0.410448 0.911884i $$-0.634628\pi$$
−0.410448 + 0.911884i $$0.634628\pi$$
$$948$$ 0 0
$$949$$ − 15557.0i − 0.532141i
$$950$$ 0 0
$$951$$ 50680.0 1.72809
$$952$$ 0 0
$$953$$ − 38250.0i − 1.30015i −0.759872 0.650073i $$-0.774738\pi$$
0.759872 0.650073i $$-0.225262\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −13334.6 −0.450414
$$958$$ 0 0
$$959$$ 8496.00 0.286079
$$960$$ 0 0
$$961$$ 20385.0 0.684267
$$962$$ 0 0
$$963$$ 269.867 0.00903046
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 4664.00i − 0.155103i −0.996988 0.0775513i $$-0.975290\pi$$
0.996988 0.0775513i $$-0.0247101\pi$$
$$968$$ 0 0
$$969$$ 2744.00 0.0909701
$$970$$ 0 0
$$971$$ 30971.2i 1.02360i 0.859106 + 0.511798i $$0.171020\pi$$
−0.859106 + 0.511798i $$0.828980\pi$$
$$972$$ 0 0
$$973$$ 21547.0 0.709933
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 4814.00i − 0.157639i −0.996889 0.0788196i $$-0.974885\pi$$
0.996889 0.0788196i $$-0.0251151\pi$$
$$978$$ 0 0
$$979$$ 4222.62i 0.137850i
$$980$$ 0 0
$$981$$ − 1894.36i − 0.0616536i
$$982$$ 0 0
$$983$$ − 12376.0i − 0.401560i −0.979636 0.200780i $$-0.935652\pi$$
0.979636 0.200780i $$-0.0643476\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 14223.6 0.458704
$$988$$ 0 0
$$989$$ 66757.6i 2.14638i
$$990$$ 0 0
$$991$$ −45344.0 −1.45348 −0.726740 0.686912i $$-0.758966\pi$$
−0.726740 + 0.686912i $$0.758966\pi$$
$$992$$ 0 0
$$993$$ − 22484.0i − 0.718538i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26002.4 −0.825984 −0.412992 0.910735i $$-0.635516\pi$$
−0.412992 + 0.910735i $$0.635516\pi$$
$$998$$ 0 0
$$999$$ −33488.0 −1.06057
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 800.4.f.a.49.4 4
4.3 odd 2 200.4.f.a.149.1 4
5.2 odd 4 800.4.d.a.401.1 2
5.3 odd 4 32.4.b.a.17.2 2
5.4 even 2 inner 800.4.f.a.49.1 4
8.3 odd 2 200.4.f.a.149.3 4
8.5 even 2 inner 800.4.f.a.49.2 4
15.8 even 4 288.4.d.a.145.1 2
20.3 even 4 8.4.b.a.5.2 yes 2
20.7 even 4 200.4.d.a.101.1 2
20.19 odd 2 200.4.f.a.149.4 4
40.3 even 4 8.4.b.a.5.1 2
40.13 odd 4 32.4.b.a.17.1 2
40.19 odd 2 200.4.f.a.149.2 4
40.27 even 4 200.4.d.a.101.2 2
40.29 even 2 inner 800.4.f.a.49.3 4
40.37 odd 4 800.4.d.a.401.2 2
60.23 odd 4 72.4.d.b.37.1 2
80.3 even 4 256.4.a.l.1.2 2
80.13 odd 4 256.4.a.j.1.1 2
80.43 even 4 256.4.a.l.1.1 2
80.53 odd 4 256.4.a.j.1.2 2
120.53 even 4 288.4.d.a.145.2 2
120.83 odd 4 72.4.d.b.37.2 2
240.53 even 4 2304.4.a.v.1.2 2
240.83 odd 4 2304.4.a.bn.1.1 2
240.173 even 4 2304.4.a.v.1.1 2
240.203 odd 4 2304.4.a.bn.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.b.a.5.1 2 40.3 even 4
8.4.b.a.5.2 yes 2 20.3 even 4
32.4.b.a.17.1 2 40.13 odd 4
32.4.b.a.17.2 2 5.3 odd 4
72.4.d.b.37.1 2 60.23 odd 4
72.4.d.b.37.2 2 120.83 odd 4
200.4.d.a.101.1 2 20.7 even 4
200.4.d.a.101.2 2 40.27 even 4
200.4.f.a.149.1 4 4.3 odd 2
200.4.f.a.149.2 4 40.19 odd 2
200.4.f.a.149.3 4 8.3 odd 2
200.4.f.a.149.4 4 20.19 odd 2
256.4.a.j.1.1 2 80.13 odd 4
256.4.a.j.1.2 2 80.53 odd 4
256.4.a.l.1.1 2 80.43 even 4
256.4.a.l.1.2 2 80.3 even 4
288.4.d.a.145.1 2 15.8 even 4
288.4.d.a.145.2 2 120.53 even 4
800.4.d.a.401.1 2 5.2 odd 4
800.4.d.a.401.2 2 40.37 odd 4
800.4.f.a.49.1 4 5.4 even 2 inner
800.4.f.a.49.2 4 8.5 even 2 inner
800.4.f.a.49.3 4 40.29 even 2 inner
800.4.f.a.49.4 4 1.1 even 1 trivial
2304.4.a.v.1.1 2 240.173 even 4
2304.4.a.v.1.2 2 240.53 even 4
2304.4.a.bn.1.1 2 240.83 odd 4
2304.4.a.bn.1.2 2 240.203 odd 4