# Properties

 Label 800.2.f.c.49.1 Level $800$ Weight $2$ Character 800.49 Analytic conductor $6.388$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 800.49 Dual form 800.2.f.c.49.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} +O(q^{10})$$ $$q-2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} +2.00000i q^{11} -3.46410 q^{13} +3.46410i q^{17} +0.535898i q^{19} +2.00000i q^{21} -6.19615i q^{23} -4.00000 q^{27} -6.92820i q^{29} +5.46410 q^{31} -5.46410i q^{33} +2.00000 q^{37} +9.46410 q^{39} +1.46410 q^{41} +5.26795 q^{43} -3.26795i q^{47} +6.46410 q^{49} -9.46410i q^{51} +11.4641 q^{53} -1.46410i q^{57} -7.46410i q^{59} -8.92820i q^{61} -3.26795i q^{63} +10.7321 q^{67} +16.9282i q^{69} -5.46410 q^{71} -7.46410i q^{73} +1.46410 q^{77} -1.07180 q^{79} -2.46410 q^{81} -1.26795 q^{83} +18.9282i q^{87} -8.92820 q^{89} +2.53590i q^{91} -14.9282 q^{93} -14.3923i q^{97} +8.92820i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} + 4 q^{9} - 16 q^{27} + 8 q^{31} + 8 q^{37} + 24 q^{39} - 8 q^{41} + 28 q^{43} + 12 q^{49} + 32 q^{53} + 36 q^{67} - 8 q^{71} - 8 q^{77} - 32 q^{79} + 4 q^{81} - 12 q^{83} - 8 q^{89} - 32 q^{93} + O(q^{100})$$

## 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
$$3$$ −2.73205 −1.57735 −0.788675 0.614810i $$-0.789233\pi$$
−0.788675 + 0.614810i $$0.789233\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 0.732051i − 0.276689i −0.990384 0.138345i $$-0.955822\pi$$
0.990384 0.138345i $$-0.0441781\pi$$
$$8$$ 0 0
$$9$$ 4.46410 1.48803
$$10$$ 0 0
$$11$$ 2.00000i 0.603023i 0.953463 + 0.301511i $$0.0974911\pi$$
−0.953463 + 0.301511i $$0.902509\pi$$
$$12$$ 0 0
$$13$$ −3.46410 −0.960769 −0.480384 0.877058i $$-0.659503\pi$$
−0.480384 + 0.877058i $$0.659503\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ 0.535898i 0.122944i 0.998109 + 0.0614718i $$0.0195794\pi$$
−0.998109 + 0.0614718i $$0.980421\pi$$
$$20$$ 0 0
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ − 6.19615i − 1.29199i −0.763343 0.645994i $$-0.776443\pi$$
0.763343 0.645994i $$-0.223557\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ − 6.92820i − 1.28654i −0.765641 0.643268i $$-0.777578\pi$$
0.765641 0.643268i $$-0.222422\pi$$
$$30$$ 0 0
$$31$$ 5.46410 0.981382 0.490691 0.871334i $$-0.336744\pi$$
0.490691 + 0.871334i $$0.336744\pi$$
$$32$$ 0 0
$$33$$ − 5.46410i − 0.951178i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 9.46410 1.51547
$$40$$ 0 0
$$41$$ 1.46410 0.228654 0.114327 0.993443i $$-0.463529\pi$$
0.114327 + 0.993443i $$0.463529\pi$$
$$42$$ 0 0
$$43$$ 5.26795 0.803355 0.401677 0.915781i $$-0.368427\pi$$
0.401677 + 0.915781i $$0.368427\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 3.26795i − 0.476679i −0.971182 0.238340i $$-0.923397\pi$$
0.971182 0.238340i $$-0.0766032\pi$$
$$48$$ 0 0
$$49$$ 6.46410 0.923443
$$50$$ 0 0
$$51$$ − 9.46410i − 1.32524i
$$52$$ 0 0
$$53$$ 11.4641 1.57472 0.787358 0.616496i $$-0.211449\pi$$
0.787358 + 0.616496i $$0.211449\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.46410i − 0.193925i
$$58$$ 0 0
$$59$$ − 7.46410i − 0.971743i −0.874030 0.485872i $$-0.838502\pi$$
0.874030 0.485872i $$-0.161498\pi$$
$$60$$ 0 0
$$61$$ − 8.92820i − 1.14314i −0.820554 0.571570i $$-0.806335\pi$$
0.820554 0.571570i $$-0.193665\pi$$
$$62$$ 0 0
$$63$$ − 3.26795i − 0.411723i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.7321 1.31113 0.655564 0.755139i $$-0.272431\pi$$
0.655564 + 0.755139i $$0.272431\pi$$
$$68$$ 0 0
$$69$$ 16.9282i 2.03792i
$$70$$ 0 0
$$71$$ −5.46410 −0.648470 −0.324235 0.945977i $$-0.605107\pi$$
−0.324235 + 0.945977i $$0.605107\pi$$
$$72$$ 0 0
$$73$$ − 7.46410i − 0.873607i −0.899557 0.436804i $$-0.856111\pi$$
0.899557 0.436804i $$-0.143889\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.46410 0.166850
$$78$$ 0 0
$$79$$ −1.07180 −0.120587 −0.0602933 0.998181i $$-0.519204\pi$$
−0.0602933 + 0.998181i $$0.519204\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ −1.26795 −0.139176 −0.0695878 0.997576i $$-0.522168\pi$$
−0.0695878 + 0.997576i $$0.522168\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 18.9282i 2.02932i
$$88$$ 0 0
$$89$$ −8.92820 −0.946388 −0.473194 0.880958i $$-0.656899\pi$$
−0.473194 + 0.880958i $$0.656899\pi$$
$$90$$ 0 0
$$91$$ 2.53590i 0.265834i
$$92$$ 0 0
$$93$$ −14.9282 −1.54798
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 14.3923i − 1.46132i −0.682743 0.730659i $$-0.739213\pi$$
0.682743 0.730659i $$-0.260787\pi$$
$$98$$ 0 0
$$99$$ 8.92820i 0.897318i
$$100$$ 0 0
$$101$$ 2.92820i 0.291367i 0.989331 + 0.145684i $$0.0465381\pi$$
−0.989331 + 0.145684i $$0.953462\pi$$
$$102$$ 0 0
$$103$$ 15.6603i 1.54305i 0.636199 + 0.771525i $$0.280506\pi$$
−0.636199 + 0.771525i $$0.719494\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.73205 0.264117 0.132059 0.991242i $$-0.457841\pi$$
0.132059 + 0.991242i $$0.457841\pi$$
$$108$$ 0 0
$$109$$ − 16.9282i − 1.62143i −0.585443 0.810714i $$-0.699079\pi$$
0.585443 0.810714i $$-0.300921\pi$$
$$110$$ 0 0
$$111$$ −5.46410 −0.518630
$$112$$ 0 0
$$113$$ 12.9282i 1.21618i 0.793867 + 0.608092i $$0.208065\pi$$
−0.793867 + 0.608092i $$0.791935\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −15.4641 −1.42966
$$118$$ 0 0
$$119$$ 2.53590 0.232465
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ −4.00000 −0.360668
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.7321i − 1.48473i −0.669996 0.742365i $$-0.733704\pi$$
0.669996 0.742365i $$-0.266296\pi$$
$$128$$ 0 0
$$129$$ −14.3923 −1.26717
$$130$$ 0 0
$$131$$ 19.8564i 1.73486i 0.497557 + 0.867431i $$0.334230\pi$$
−0.497557 + 0.867431i $$0.665770\pi$$
$$132$$ 0 0
$$133$$ 0.392305 0.0340171
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.92820i 0.421045i 0.977589 + 0.210522i $$0.0675165\pi$$
−0.977589 + 0.210522i $$0.932484\pi$$
$$138$$ 0 0
$$139$$ − 0.535898i − 0.0454543i −0.999742 0.0227272i $$-0.992765\pi$$
0.999742 0.0227272i $$-0.00723490\pi$$
$$140$$ 0 0
$$141$$ 8.92820i 0.751890i
$$142$$ 0 0
$$143$$ − 6.92820i − 0.579365i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −17.6603 −1.45659
$$148$$ 0 0
$$149$$ 7.85641i 0.643622i 0.946804 + 0.321811i $$0.104292\pi$$
−0.946804 + 0.321811i $$0.895708\pi$$
$$150$$ 0 0
$$151$$ 12.3923 1.00847 0.504236 0.863566i $$-0.331774\pi$$
0.504236 + 0.863566i $$0.331774\pi$$
$$152$$ 0 0
$$153$$ 15.4641i 1.25020i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.07180 0.245156 0.122578 0.992459i $$-0.460884\pi$$
0.122578 + 0.992459i $$0.460884\pi$$
$$158$$ 0 0
$$159$$ −31.3205 −2.48388
$$160$$ 0 0
$$161$$ −4.53590 −0.357479
$$162$$ 0 0
$$163$$ 0.196152 0.0153638 0.00768192 0.999970i $$-0.497555\pi$$
0.00768192 + 0.999970i $$0.497555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 9.80385i 0.758645i 0.925265 + 0.379322i $$0.123843\pi$$
−0.925265 + 0.379322i $$0.876157\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.39230i 0.182944i
$$172$$ 0 0
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 20.3923i 1.53278i
$$178$$ 0 0
$$179$$ − 8.53590i − 0.638003i −0.947754 0.319002i $$-0.896652\pi$$
0.947754 0.319002i $$-0.103348\pi$$
$$180$$ 0 0
$$181$$ 16.0000i 1.18927i 0.803996 + 0.594635i $$0.202704\pi$$
−0.803996 + 0.594635i $$0.797296\pi$$
$$182$$ 0 0
$$183$$ 24.3923i 1.80313i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.92820 −0.506640
$$188$$ 0 0
$$189$$ 2.92820i 0.212995i
$$190$$ 0 0
$$191$$ −15.3205 −1.10855 −0.554277 0.832333i $$-0.687005\pi$$
−0.554277 + 0.832333i $$0.687005\pi$$
$$192$$ 0 0
$$193$$ − 0.535898i − 0.0385748i −0.999814 0.0192874i $$-0.993860\pi$$
0.999814 0.0192874i $$-0.00613975\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 19.4641 1.38676 0.693380 0.720572i $$-0.256121\pi$$
0.693380 + 0.720572i $$0.256121\pi$$
$$198$$ 0 0
$$199$$ −1.85641 −0.131597 −0.0657986 0.997833i $$-0.520959\pi$$
−0.0657986 + 0.997833i $$0.520959\pi$$
$$200$$ 0 0
$$201$$ −29.3205 −2.06811
$$202$$ 0 0
$$203$$ −5.07180 −0.355970
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 27.6603i − 1.92252i
$$208$$ 0 0
$$209$$ −1.07180 −0.0741377
$$210$$ 0 0
$$211$$ − 26.7846i − 1.84393i −0.387275 0.921964i $$-0.626584\pi$$
0.387275 0.921964i $$-0.373416\pi$$
$$212$$ 0 0
$$213$$ 14.9282 1.02286
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 4.00000i − 0.271538i
$$218$$ 0 0
$$219$$ 20.3923i 1.37798i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ − 5.80385i − 0.388654i −0.980937 0.194327i $$-0.937748\pi$$
0.980937 0.194327i $$-0.0622523\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.0526 0.667212 0.333606 0.942713i $$-0.391735\pi$$
0.333606 + 0.942713i $$0.391735\pi$$
$$228$$ 0 0
$$229$$ 4.00000i 0.264327i 0.991228 + 0.132164i $$0.0421925\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ 5.32051i 0.348558i 0.984696 + 0.174279i $$0.0557595\pi$$
−0.984696 + 0.174279i $$0.944241\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.92820 0.190207
$$238$$ 0 0
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ 16.3923 1.05592 0.527961 0.849269i $$-0.322957\pi$$
0.527961 + 0.849269i $$0.322957\pi$$
$$242$$ 0 0
$$243$$ 18.7321 1.20166
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 1.85641i − 0.118120i
$$248$$ 0 0
$$249$$ 3.46410 0.219529
$$250$$ 0 0
$$251$$ − 24.9282i − 1.57345i −0.617301 0.786727i $$-0.711774\pi$$
0.617301 0.786727i $$-0.288226\pi$$
$$252$$ 0 0
$$253$$ 12.3923 0.779098
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 0 0
$$259$$ − 1.46410i − 0.0909748i
$$260$$ 0 0
$$261$$ − 30.9282i − 1.91441i
$$262$$ 0 0
$$263$$ − 11.6603i − 0.719002i −0.933145 0.359501i $$-0.882947\pi$$
0.933145 0.359501i $$-0.117053\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 24.3923 1.49278
$$268$$ 0 0
$$269$$ 8.92820i 0.544362i 0.962246 + 0.272181i $$0.0877450\pi$$
−0.962246 + 0.272181i $$0.912255\pi$$
$$270$$ 0 0
$$271$$ 19.3205 1.17364 0.586819 0.809718i $$-0.300380\pi$$
0.586819 + 0.809718i $$0.300380\pi$$
$$272$$ 0 0
$$273$$ − 6.92820i − 0.419314i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 0 0
$$279$$ 24.3923 1.46033
$$280$$ 0 0
$$281$$ 10.5359 0.628519 0.314260 0.949337i $$-0.398244\pi$$
0.314260 + 0.949337i $$0.398244\pi$$
$$282$$ 0 0
$$283$$ 9.66025 0.574242 0.287121 0.957894i $$-0.407302\pi$$
0.287121 + 0.957894i $$0.407302\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 1.07180i − 0.0632662i
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 39.3205i 2.30501i
$$292$$ 0 0
$$293$$ −15.8564 −0.926341 −0.463171 0.886269i $$-0.653288\pi$$
−0.463171 + 0.886269i $$0.653288\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 8.00000i − 0.464207i
$$298$$ 0 0
$$299$$ 21.4641i 1.24130i
$$300$$ 0 0
$$301$$ − 3.85641i − 0.222280i
$$302$$ 0 0
$$303$$ − 8.00000i − 0.459588i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −24.9808 −1.42573 −0.712864 0.701303i $$-0.752602\pi$$
−0.712864 + 0.701303i $$0.752602\pi$$
$$308$$ 0 0
$$309$$ − 42.7846i − 2.43393i
$$310$$ 0 0
$$311$$ −31.3205 −1.77602 −0.888012 0.459821i $$-0.847914\pi$$
−0.888012 + 0.459821i $$0.847914\pi$$
$$312$$ 0 0
$$313$$ 4.14359i 0.234210i 0.993120 + 0.117105i $$0.0373614\pi$$
−0.993120 + 0.117105i $$0.962639\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.53590 −0.479424 −0.239712 0.970844i $$-0.577053\pi$$
−0.239712 + 0.970844i $$0.577053\pi$$
$$318$$ 0 0
$$319$$ 13.8564 0.775810
$$320$$ 0 0
$$321$$ −7.46410 −0.416606
$$322$$ 0 0
$$323$$ −1.85641 −0.103293
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 46.2487i 2.55756i
$$328$$ 0 0
$$329$$ −2.39230 −0.131892
$$330$$ 0 0
$$331$$ 14.0000i 0.769510i 0.923019 + 0.384755i $$0.125714\pi$$
−0.923019 + 0.384755i $$0.874286\pi$$
$$332$$ 0 0
$$333$$ 8.92820 0.489263
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 19.8564i − 1.08165i −0.841136 0.540824i $$-0.818113\pi$$
0.841136 0.540824i $$-0.181887\pi$$
$$338$$ 0 0
$$339$$ − 35.3205i − 1.91835i
$$340$$ 0 0
$$341$$ 10.9282i 0.591795i
$$342$$ 0 0
$$343$$ − 9.85641i − 0.532196i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.66025 −0.0891271 −0.0445636 0.999007i $$-0.514190\pi$$
−0.0445636 + 0.999007i $$0.514190\pi$$
$$348$$ 0 0
$$349$$ 28.0000i 1.49881i 0.662114 + 0.749403i $$0.269659\pi$$
−0.662114 + 0.749403i $$0.730341\pi$$
$$350$$ 0 0
$$351$$ 13.8564 0.739600
$$352$$ 0 0
$$353$$ − 12.9282i − 0.688099i −0.938952 0.344049i $$-0.888201\pi$$
0.938952 0.344049i $$-0.111799\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.92820 −0.366679
$$358$$ 0 0
$$359$$ 18.9282 0.998992 0.499496 0.866316i $$-0.333518\pi$$
0.499496 + 0.866316i $$0.333518\pi$$
$$360$$ 0 0
$$361$$ 18.7128 0.984885
$$362$$ 0 0
$$363$$ −19.1244 −1.00377
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 2.87564i − 0.150107i −0.997179 0.0750537i $$-0.976087\pi$$
0.997179 0.0750537i $$-0.0239128\pi$$
$$368$$ 0 0
$$369$$ 6.53590 0.340245
$$370$$ 0 0
$$371$$ − 8.39230i − 0.435707i
$$372$$ 0 0
$$373$$ −25.7128 −1.33136 −0.665679 0.746238i $$-0.731858\pi$$
−0.665679 + 0.746238i $$0.731858\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 24.0000i 1.23606i
$$378$$ 0 0
$$379$$ − 36.2487i − 1.86197i −0.365056 0.930986i $$-0.618950\pi$$
0.365056 0.930986i $$-0.381050\pi$$
$$380$$ 0 0
$$381$$ 45.7128i 2.34194i
$$382$$ 0 0
$$383$$ 21.1244i 1.07940i 0.841856 + 0.539702i $$0.181463\pi$$
−0.841856 + 0.539702i $$0.818537\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 23.5167 1.19542
$$388$$ 0 0
$$389$$ − 6.78461i − 0.343993i −0.985098 0.171997i $$-0.944978\pi$$
0.985098 0.171997i $$-0.0550218\pi$$
$$390$$ 0 0
$$391$$ 21.4641 1.08549
$$392$$ 0 0
$$393$$ − 54.2487i − 2.73649i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −32.2487 −1.61852 −0.809258 0.587453i $$-0.800131\pi$$
−0.809258 + 0.587453i $$0.800131\pi$$
$$398$$ 0 0
$$399$$ −1.07180 −0.0536570
$$400$$ 0 0
$$401$$ −7.85641 −0.392330 −0.196165 0.980571i $$-0.562849\pi$$
−0.196165 + 0.980571i $$0.562849\pi$$
$$402$$ 0 0
$$403$$ −18.9282 −0.942881
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.00000i 0.198273i
$$408$$ 0 0
$$409$$ 11.3205 0.559763 0.279882 0.960035i $$-0.409705\pi$$
0.279882 + 0.960035i $$0.409705\pi$$
$$410$$ 0 0
$$411$$ − 13.4641i − 0.664135i
$$412$$ 0 0
$$413$$ −5.46410 −0.268871
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 1.46410i 0.0716974i
$$418$$ 0 0
$$419$$ 18.3923i 0.898523i 0.893400 + 0.449261i $$0.148313\pi$$
−0.893400 + 0.449261i $$0.851687\pi$$
$$420$$ 0 0
$$421$$ 0.143594i 0.00699832i 0.999994 + 0.00349916i $$0.00111382\pi$$
−0.999994 + 0.00349916i $$0.998886\pi$$
$$422$$ 0 0
$$423$$ − 14.5885i − 0.709315i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −6.53590 −0.316294
$$428$$ 0 0
$$429$$ 18.9282i 0.913862i
$$430$$ 0 0
$$431$$ 21.4641 1.03389 0.516945 0.856019i $$-0.327069\pi$$
0.516945 + 0.856019i $$0.327069\pi$$
$$432$$ 0 0
$$433$$ 19.4641i 0.935385i 0.883891 + 0.467693i $$0.154915\pi$$
−0.883891 + 0.467693i $$0.845085\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.32051 0.158841
$$438$$ 0 0
$$439$$ 40.7846 1.94654 0.973272 0.229657i $$-0.0737605\pi$$
0.973272 + 0.229657i $$0.0737605\pi$$
$$440$$ 0 0
$$441$$ 28.8564 1.37411
$$442$$ 0 0
$$443$$ −20.9808 −0.996826 −0.498413 0.866940i $$-0.666084\pi$$
−0.498413 + 0.866940i $$0.666084\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 21.4641i − 1.01522i
$$448$$ 0 0
$$449$$ 23.3205 1.10056 0.550281 0.834979i $$-0.314520\pi$$
0.550281 + 0.834979i $$0.314520\pi$$
$$450$$ 0 0
$$451$$ 2.92820i 0.137884i
$$452$$ 0 0
$$453$$ −33.8564 −1.59071
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 26.7846i − 1.25293i −0.779449 0.626466i $$-0.784501\pi$$
0.779449 0.626466i $$-0.215499\pi$$
$$458$$ 0 0
$$459$$ − 13.8564i − 0.646762i
$$460$$ 0 0
$$461$$ − 10.9282i − 0.508977i −0.967076 0.254489i $$-0.918093\pi$$
0.967076 0.254489i $$-0.0819071\pi$$
$$462$$ 0 0
$$463$$ − 11.2679i − 0.523666i −0.965113 0.261833i $$-0.915673\pi$$
0.965113 0.261833i $$-0.0843270\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −25.6603 −1.18741 −0.593707 0.804681i $$-0.702336\pi$$
−0.593707 + 0.804681i $$0.702336\pi$$
$$468$$ 0 0
$$469$$ − 7.85641i − 0.362775i
$$470$$ 0 0
$$471$$ −8.39230 −0.386697
$$472$$ 0 0
$$473$$ 10.5359i 0.484441i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 51.1769 2.34323
$$478$$ 0 0
$$479$$ 5.85641 0.267586 0.133793 0.991009i $$-0.457284\pi$$
0.133793 + 0.991009i $$0.457284\pi$$
$$480$$ 0 0
$$481$$ −6.92820 −0.315899
$$482$$ 0 0
$$483$$ 12.3923 0.563869
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 6.58846i − 0.298551i −0.988796 0.149276i $$-0.952306\pi$$
0.988796 0.149276i $$-0.0476942\pi$$
$$488$$ 0 0
$$489$$ −0.535898 −0.0242342
$$490$$ 0 0
$$491$$ 16.9282i 0.763959i 0.924171 + 0.381980i $$0.124758\pi$$
−0.924171 + 0.381980i $$0.875242\pi$$
$$492$$ 0 0
$$493$$ 24.0000 1.08091
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.00000i 0.179425i
$$498$$ 0 0
$$499$$ 31.4641i 1.40853i 0.709939 + 0.704263i $$0.248723\pi$$
−0.709939 + 0.704263i $$0.751277\pi$$
$$500$$ 0 0
$$501$$ − 26.7846i − 1.19665i
$$502$$ 0 0
$$503$$ − 0.339746i − 0.0151485i −0.999971 0.00757426i $$-0.997589\pi$$
0.999971 0.00757426i $$-0.00241099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 2.73205 0.121335
$$508$$ 0 0
$$509$$ 1.85641i 0.0822838i 0.999153 + 0.0411419i $$0.0130996\pi$$
−0.999153 + 0.0411419i $$0.986900\pi$$
$$510$$ 0 0
$$511$$ −5.46410 −0.241718
$$512$$ 0 0
$$513$$ − 2.14359i − 0.0946420i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6.53590 0.287448
$$518$$ 0 0
$$519$$ −5.46410 −0.239847
$$520$$ 0 0
$$521$$ −43.8564 −1.92138 −0.960692 0.277616i $$-0.910456\pi$$
−0.960692 + 0.277616i $$0.910456\pi$$
$$522$$ 0 0
$$523$$ 11.8038 0.516146 0.258073 0.966125i $$-0.416912\pi$$
0.258073 + 0.966125i $$0.416912\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 18.9282i 0.824525i
$$528$$ 0 0
$$529$$ −15.3923 −0.669231
$$530$$ 0 0
$$531$$ − 33.3205i − 1.44599i
$$532$$ 0 0
$$533$$ −5.07180 −0.219684
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 23.3205i 1.00635i
$$538$$ 0 0
$$539$$ 12.9282i 0.556857i
$$540$$ 0 0
$$541$$ − 26.9282i − 1.15773i −0.815422 0.578867i $$-0.803495\pi$$
0.815422 0.578867i $$-0.196505\pi$$
$$542$$ 0 0
$$543$$ − 43.7128i − 1.87590i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 33.2679 1.42243 0.711217 0.702972i $$-0.248144\pi$$
0.711217 + 0.702972i $$0.248144\pi$$
$$548$$ 0 0
$$549$$ − 39.8564i − 1.70103i
$$550$$ 0 0
$$551$$ 3.71281 0.158171
$$552$$ 0 0
$$553$$ 0.784610i 0.0333650i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −14.7846 −0.626444 −0.313222 0.949680i $$-0.601408\pi$$
−0.313222 + 0.949680i $$0.601408\pi$$
$$558$$ 0 0
$$559$$ −18.2487 −0.771838
$$560$$ 0 0
$$561$$ 18.9282 0.799149
$$562$$ 0 0
$$563$$ 22.0526 0.929405 0.464702 0.885467i $$-0.346161\pi$$
0.464702 + 0.885467i $$0.346161\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.80385i 0.0757545i
$$568$$ 0 0
$$569$$ 13.4641 0.564445 0.282222 0.959349i $$-0.408928\pi$$
0.282222 + 0.959349i $$0.408928\pi$$
$$570$$ 0 0
$$571$$ − 6.78461i − 0.283927i −0.989872 0.141964i $$-0.954658\pi$$
0.989872 0.141964i $$-0.0453416\pi$$
$$572$$ 0 0
$$573$$ 41.8564 1.74858
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 39.5692i 1.64729i 0.567107 + 0.823644i $$0.308063\pi$$
−0.567107 + 0.823644i $$0.691937\pi$$
$$578$$ 0 0
$$579$$ 1.46410i 0.0608460i
$$580$$ 0 0
$$581$$ 0.928203i 0.0385084i
$$582$$ 0 0
$$583$$ 22.9282i 0.949589i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.80385 −0.157002 −0.0785008 0.996914i $$-0.525013\pi$$
−0.0785008 + 0.996914i $$0.525013\pi$$
$$588$$ 0 0
$$589$$ 2.92820i 0.120655i
$$590$$ 0 0
$$591$$ −53.1769 −2.18741
$$592$$ 0 0
$$593$$ − 32.6410i − 1.34041i −0.742178 0.670203i $$-0.766207\pi$$
0.742178 0.670203i $$-0.233793\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.07180 0.207575
$$598$$ 0 0
$$599$$ −34.6410 −1.41539 −0.707697 0.706516i $$-0.750266\pi$$
−0.707697 + 0.706516i $$0.750266\pi$$
$$600$$ 0 0
$$601$$ 18.5359 0.756095 0.378048 0.925786i $$-0.376596\pi$$
0.378048 + 0.925786i $$0.376596\pi$$
$$602$$ 0 0
$$603$$ 47.9090 1.95100
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 30.9808i − 1.25747i −0.777619 0.628735i $$-0.783573\pi$$
0.777619 0.628735i $$-0.216427\pi$$
$$608$$ 0 0
$$609$$ 13.8564 0.561490
$$610$$ 0 0
$$611$$ 11.3205i 0.457979i
$$612$$ 0 0
$$613$$ 26.3923 1.06598 0.532988 0.846123i $$-0.321069\pi$$
0.532988 + 0.846123i $$0.321069\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 20.5359i − 0.826744i −0.910562 0.413372i $$-0.864351\pi$$
0.910562 0.413372i $$-0.135649\pi$$
$$618$$ 0 0
$$619$$ 1.32051i 0.0530757i 0.999648 + 0.0265379i $$0.00844825\pi$$
−0.999648 + 0.0265379i $$0.991552\pi$$
$$620$$ 0 0
$$621$$ 24.7846i 0.994572i
$$622$$ 0 0
$$623$$ 6.53590i 0.261855i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 2.92820 0.116941
$$628$$ 0 0
$$629$$ 6.92820i 0.276246i
$$630$$ 0 0
$$631$$ 23.3205 0.928375 0.464187 0.885737i $$-0.346346\pi$$
0.464187 + 0.885737i $$0.346346\pi$$
$$632$$ 0 0
$$633$$ 73.1769i 2.90852i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −22.3923 −0.887215
$$638$$ 0 0
$$639$$ −24.3923 −0.964945
$$640$$ 0 0
$$641$$ 0.392305 0.0154951 0.00774755 0.999970i $$-0.497534\pi$$
0.00774755 + 0.999970i $$0.497534\pi$$
$$642$$ 0 0
$$643$$ −39.1244 −1.54291 −0.771457 0.636281i $$-0.780472\pi$$
−0.771457 + 0.636281i $$0.780472\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 16.7321i − 0.657805i −0.944364 0.328902i $$-0.893321\pi$$
0.944364 0.328902i $$-0.106679\pi$$
$$648$$ 0 0
$$649$$ 14.9282 0.585983
$$650$$ 0 0
$$651$$ 10.9282i 0.428310i
$$652$$ 0 0
$$653$$ 12.2487 0.479329 0.239665 0.970856i $$-0.422963\pi$$
0.239665 + 0.970856i $$0.422963\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 33.3205i − 1.29996i
$$658$$ 0 0
$$659$$ − 17.3205i − 0.674711i −0.941377 0.337356i $$-0.890468\pi$$
0.941377 0.337356i $$-0.109532\pi$$
$$660$$ 0 0
$$661$$ − 8.14359i − 0.316749i −0.987379 0.158375i $$-0.949375\pi$$
0.987379 0.158375i $$-0.0506253\pi$$
$$662$$ 0 0
$$663$$ 32.7846i 1.27325i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −42.9282 −1.66219
$$668$$ 0 0
$$669$$ 15.8564i 0.613044i
$$670$$ 0 0
$$671$$ 17.8564 0.689339
$$672$$ 0 0
$$673$$ − 12.5359i − 0.483223i −0.970373 0.241612i $$-0.922324\pi$$
0.970373 0.241612i $$-0.0776760\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −17.6077 −0.676719 −0.338359 0.941017i $$-0.609872\pi$$
−0.338359 + 0.941017i $$0.609872\pi$$
$$678$$ 0 0
$$679$$ −10.5359 −0.404331
$$680$$ 0 0
$$681$$ −27.4641 −1.05243
$$682$$ 0 0
$$683$$ 16.9808 0.649751 0.324875 0.945757i $$-0.394678\pi$$
0.324875 + 0.945757i $$0.394678\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 10.9282i − 0.416937i
$$688$$ 0 0
$$689$$ −39.7128 −1.51294
$$690$$ 0 0
$$691$$ − 18.0000i − 0.684752i −0.939563 0.342376i $$-0.888768\pi$$
0.939563 0.342376i $$-0.111232\pi$$
$$692$$ 0 0
$$693$$ 6.53590 0.248278
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 5.07180i 0.192108i
$$698$$ 0 0
$$699$$ − 14.5359i − 0.549798i
$$700$$ 0 0
$$701$$ 19.0718i 0.720332i 0.932888 + 0.360166i $$0.117280\pi$$
−0.932888 + 0.360166i $$0.882720\pi$$
$$702$$ 0 0
$$703$$ 1.07180i 0.0404236i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 2.14359 0.0806181
$$708$$ 0 0
$$709$$ 12.7846i 0.480136i 0.970756 + 0.240068i $$0.0771698\pi$$
−0.970756 + 0.240068i $$0.922830\pi$$
$$710$$ 0 0
$$711$$ −4.78461 −0.179437
$$712$$ 0 0
$$713$$ − 33.8564i − 1.26793i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 54.6410 2.04061
$$718$$ 0 0
$$719$$ −1.85641 −0.0692323 −0.0346161 0.999401i $$-0.511021\pi$$
−0.0346161 + 0.999401i $$0.511021\pi$$
$$720$$ 0 0
$$721$$ 11.4641 0.426945
$$722$$ 0 0
$$723$$ −44.7846 −1.66556
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24.0526i 0.892060i 0.895018 + 0.446030i $$0.147163\pi$$
−0.895018 + 0.446030i $$0.852837\pi$$
$$728$$ 0 0
$$729$$ −43.7846 −1.62165
$$730$$ 0 0
$$731$$ 18.2487i 0.674953i
$$732$$ 0 0
$$733$$ −35.0718 −1.29541 −0.647703 0.761893i $$-0.724270\pi$$
−0.647703 + 0.761893i $$0.724270\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.4641i 0.790640i
$$738$$ 0 0
$$739$$ 29.3205i 1.07857i 0.842123 + 0.539286i $$0.181306\pi$$
−0.842123 + 0.539286i $$0.818694\pi$$
$$740$$ 0 0
$$741$$ 5.07180i 0.186317i
$$742$$ 0 0
$$743$$ 10.9808i 0.402845i 0.979504 + 0.201423i $$0.0645564\pi$$
−0.979504 + 0.201423i $$0.935444\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −5.66025 −0.207098
$$748$$ 0 0
$$749$$ − 2.00000i − 0.0730784i
$$750$$ 0 0
$$751$$ −26.2487 −0.957829 −0.478915 0.877862i $$-0.658970\pi$$
−0.478915 + 0.877862i $$0.658970\pi$$
$$752$$ 0 0
$$753$$ 68.1051i 2.48189i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.0718 0.693176 0.346588 0.938017i $$-0.387340\pi$$
0.346588 + 0.938017i $$0.387340\pi$$
$$758$$ 0 0
$$759$$ −33.8564 −1.22891
$$760$$ 0 0
$$761$$ −5.71281 −0.207089 −0.103545 0.994625i $$-0.533018\pi$$
−0.103545 + 0.994625i $$0.533018\pi$$
$$762$$ 0 0
$$763$$ −12.3923 −0.448632
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 25.8564i 0.933621i
$$768$$ 0 0
$$769$$ −12.9282 −0.466203 −0.233101 0.972452i $$-0.574887\pi$$
−0.233101 + 0.972452i $$0.574887\pi$$
$$770$$ 0 0
$$771$$ 5.46410i 0.196785i
$$772$$ 0 0
$$773$$ −22.3923 −0.805395 −0.402698 0.915333i $$-0.631927\pi$$
−0.402698 + 0.915333i $$0.631927\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 4.00000i 0.143499i
$$778$$ 0 0
$$779$$ 0.784610i 0.0281116i
$$780$$ 0 0
$$781$$ − 10.9282i − 0.391042i
$$782$$ 0 0
$$783$$ 27.7128i 0.990375i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 16.5885 0.591315 0.295657 0.955294i $$-0.404461\pi$$
0.295657 + 0.955294i $$0.404461\pi$$
$$788$$ 0 0
$$789$$ 31.8564i 1.13412i
$$790$$ 0 0
$$791$$ 9.46410 0.336505
$$792$$ 0 0
$$793$$ 30.9282i 1.09829i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −50.1051 −1.77481 −0.887407 0.460986i $$-0.847496\pi$$
−0.887407 + 0.460986i $$0.847496\pi$$
$$798$$ 0 0
$$799$$ 11.3205 0.400491
$$800$$ 0 0
$$801$$ −39.8564 −1.40826
$$802$$ 0 0
$$803$$ 14.9282 0.526805
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 24.3923i − 0.858650i
$$808$$ 0 0
$$809$$ −23.8564 −0.838747 −0.419373 0.907814i $$-0.637750\pi$$
−0.419373 + 0.907814i $$0.637750\pi$$
$$810$$ 0 0
$$811$$ 28.9282i 1.01581i 0.861414 + 0.507903i $$0.169579\pi$$
−0.861414 + 0.507903i $$0.830421\pi$$
$$812$$ 0 0
$$813$$ −52.7846 −1.85124
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 2.82309i 0.0987673i
$$818$$ 0 0
$$819$$ 11.3205i 0.395571i
$$820$$ 0 0
$$821$$ 34.7846i 1.21399i 0.794705 + 0.606996i $$0.207625\pi$$
−0.794705 + 0.606996i $$0.792375\pi$$
$$822$$ 0 0
$$823$$ − 9.12436i − 0.318055i −0.987274 0.159028i $$-0.949164\pi$$
0.987274 0.159028i $$-0.0508359\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23.1244 0.804113 0.402056 0.915615i $$-0.368296\pi$$
0.402056 + 0.915615i $$0.368296\pi$$
$$828$$ 0 0
$$829$$ − 28.9282i − 1.00472i −0.864659 0.502359i $$-0.832466\pi$$
0.864659 0.502359i $$-0.167534\pi$$
$$830$$ 0 0
$$831$$ 5.46410 0.189548
$$832$$ 0 0
$$833$$ 22.3923i 0.775847i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −21.8564 −0.755468
$$838$$ 0 0
$$839$$ 24.7846 0.855660 0.427830 0.903859i $$-0.359278\pi$$
0.427830 + 0.903859i $$0.359278\pi$$
$$840$$ 0 0
$$841$$ −19.0000 −0.655172
$$842$$ 0 0
$$843$$ −28.7846 −0.991395
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 5.12436i − 0.176075i
$$848$$ 0 0
$$849$$ −26.3923 −0.905782
$$850$$ 0 0
$$851$$ − 12.3923i − 0.424803i
$$852$$ 0 0
$$853$$ −21.6077 −0.739833 −0.369917 0.929065i $$-0.620614\pi$$
−0.369917 + 0.929065i $$0.620614\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 19.8564i − 0.678282i −0.940736 0.339141i $$-0.889864\pi$$
0.940736 0.339141i $$-0.110136\pi$$
$$858$$ 0 0
$$859$$ 28.2487i 0.963834i 0.876217 + 0.481917i $$0.160059\pi$$
−0.876217 + 0.481917i $$0.839941\pi$$
$$860$$ 0 0
$$861$$ 2.92820i 0.0997929i
$$862$$ 0 0
$$863$$ 47.6603i 1.62237i 0.584787 + 0.811187i $$0.301178\pi$$
−0.584787 + 0.811187i $$0.698822\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −13.6603 −0.463927
$$868$$ 0 0
$$869$$ − 2.14359i − 0.0727164i
$$870$$ 0 0
$$871$$ −37.1769 −1.25969
$$872$$ 0 0
$$873$$ − 64.2487i − 2.17449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.71281 0.0578376 0.0289188 0.999582i $$-0.490794\pi$$
0.0289188 + 0.999582i $$0.490794\pi$$
$$878$$ 0 0
$$879$$ 43.3205 1.46116
$$880$$ 0 0
$$881$$ 9.46410 0.318854 0.159427 0.987210i $$-0.449035\pi$$
0.159427 + 0.987210i $$0.449035\pi$$
$$882$$ 0 0
$$883$$ 27.9090 0.939211 0.469606 0.882876i $$-0.344396\pi$$
0.469606 + 0.882876i $$0.344396\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 13.9090i 0.467017i 0.972355 + 0.233509i $$0.0750207\pi$$
−0.972355 + 0.233509i $$0.924979\pi$$
$$888$$ 0 0
$$889$$ −12.2487 −0.410809
$$890$$ 0 0
$$891$$ − 4.92820i − 0.165101i
$$892$$ 0 0
$$893$$ 1.75129 0.0586046
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 58.6410i − 1.95797i
$$898$$ 0 0
$$899$$ − 37.8564i − 1.26258i
$$900$$ 0 0
$$901$$ 39.7128i 1.32303i
$$902$$ 0 0
$$903$$ 10.5359i 0.350613i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.87564 0.161893 0.0809466 0.996718i $$-0.474206\pi$$
0.0809466 + 0.996718i $$0.474206\pi$$
$$908$$ 0 0
$$909$$ 13.0718i 0.433564i
$$910$$ 0 0
$$911$$ 49.1769 1.62930 0.814652 0.579950i $$-0.196928\pi$$
0.814652 + 0.579950i $$0.196928\pi$$
$$912$$ 0 0
$$913$$ − 2.53590i − 0.0839260i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 14.5359 0.480018
$$918$$ 0 0
$$919$$ −38.9282 −1.28412 −0.642061 0.766653i $$-0.721921\pi$$
−0.642061 + 0.766653i $$0.721921\pi$$
$$920$$ 0 0
$$921$$ 68.2487 2.24887
$$922$$ 0 0
$$923$$ 18.9282 0.623029
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 69.9090i 2.29611i
$$928$$ 0 0
$$929$$ 17.4641 0.572979 0.286489 0.958083i $$-0.407512\pi$$
0.286489 + 0.958083i $$0.407512\pi$$
$$930$$ 0 0
$$931$$ 3.46410i 0.113531i
$$932$$ 0 0
$$933$$ 85.5692 2.80141
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 4.24871i 0.138799i 0.997589 + 0.0693997i $$0.0221084\pi$$
−0.997589 + 0.0693997i $$0.977892\pi$$
$$938$$ 0 0
$$939$$ − 11.3205i − 0.369431i
$$940$$ 0 0
$$941$$ − 32.0000i − 1.04317i −0.853199 0.521585i $$-0.825341\pi$$
0.853199 0.521585i $$-0.174659\pi$$
$$942$$ 0 0
$$943$$ − 9.07180i − 0.295418i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −3.12436 −0.101528 −0.0507640 0.998711i $$-0.516166\pi$$
−0.0507640 + 0.998711i $$0.516166\pi$$
$$948$$ 0 0
$$949$$ 25.8564i 0.839334i
$$950$$ 0 0
$$951$$ 23.3205 0.756219
$$952$$ 0 0
$$953$$ 17.2154i 0.557661i 0.960340 + 0.278831i $$0.0899468\pi$$
−0.960340 + 0.278831i $$0.910053\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −37.8564 −1.22372
$$958$$ 0 0
$$959$$ 3.60770 0.116499
$$960$$ 0 0
$$961$$ −1.14359 −0.0368901
$$962$$ 0 0
$$963$$ 12.1962 0.393016
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 16.3397i − 0.525451i −0.964871 0.262725i $$-0.915379\pi$$
0.964871 0.262725i $$-0.0846213\pi$$
$$968$$ 0 0
$$969$$ 5.07180 0.162930
$$970$$ 0 0
$$971$$ − 36.9282i − 1.18508i −0.805540 0.592541i $$-0.798125\pi$$
0.805540 0.592541i $$-0.201875\pi$$
$$972$$ 0 0
$$973$$ −0.392305 −0.0125767
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 24.5359i − 0.784973i −0.919758 0.392486i $$-0.871615\pi$$
0.919758 0.392486i $$-0.128385\pi$$
$$978$$ 0 0
$$979$$ − 17.8564i − 0.570693i
$$980$$ 0 0
$$981$$ − 75.5692i − 2.41274i
$$982$$ 0 0
$$983$$ − 48.7321i − 1.55431i −0.629309 0.777156i $$-0.716662\pi$$
0.629309 0.777156i $$-0.283338\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.53590 0.208040
$$988$$ 0 0
$$989$$ − 32.6410i − 1.03792i
$$990$$ 0 0
$$991$$ −41.4641 −1.31715 −0.658575 0.752515i $$-0.728841\pi$$
−0.658575 + 0.752515i $$0.728841\pi$$
$$992$$ 0 0
$$993$$ − 38.2487i − 1.21379i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 11.1769 0.353976 0.176988 0.984213i $$-0.443365\pi$$
0.176988 + 0.984213i $$0.443365\pi$$
$$998$$ 0 0
$$999$$ −8.00000 −0.253109
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 800.2.f.c.49.1 4
3.2 odd 2 7200.2.d.o.2449.2 4
4.3 odd 2 200.2.f.c.149.3 4
5.2 odd 4 800.2.d.e.401.4 4
5.3 odd 4 160.2.d.a.81.1 4
5.4 even 2 800.2.f.e.49.4 4
8.3 odd 2 200.2.f.e.149.1 4
8.5 even 2 800.2.f.e.49.3 4
12.11 even 2 1800.2.d.p.1549.2 4
15.2 even 4 7200.2.k.j.3601.3 4
15.8 even 4 1440.2.k.e.721.3 4
15.14 odd 2 7200.2.d.n.2449.3 4
20.3 even 4 40.2.d.a.21.1 4
20.7 even 4 200.2.d.f.101.4 4
20.19 odd 2 200.2.f.e.149.2 4
24.5 odd 2 7200.2.d.n.2449.2 4
24.11 even 2 1800.2.d.l.1549.4 4
40.3 even 4 40.2.d.a.21.2 yes 4
40.13 odd 4 160.2.d.a.81.4 4
40.19 odd 2 200.2.f.c.149.4 4
40.27 even 4 200.2.d.f.101.3 4
40.29 even 2 inner 800.2.f.c.49.2 4
40.37 odd 4 800.2.d.e.401.1 4
60.23 odd 4 360.2.k.e.181.4 4
60.47 odd 4 1800.2.k.j.901.1 4
60.59 even 2 1800.2.d.l.1549.3 4
80.3 even 4 1280.2.a.a.1.1 2
80.13 odd 4 1280.2.a.n.1.2 2
80.27 even 4 6400.2.a.z.1.1 2
80.37 odd 4 6400.2.a.cj.1.2 2
80.43 even 4 1280.2.a.o.1.2 2
80.53 odd 4 1280.2.a.d.1.1 2
80.67 even 4 6400.2.a.ce.1.2 2
80.77 odd 4 6400.2.a.be.1.1 2
120.29 odd 2 7200.2.d.o.2449.3 4
120.53 even 4 1440.2.k.e.721.1 4
120.59 even 2 1800.2.d.p.1549.1 4
120.77 even 4 7200.2.k.j.3601.4 4
120.83 odd 4 360.2.k.e.181.3 4
120.107 odd 4 1800.2.k.j.901.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 20.3 even 4
40.2.d.a.21.2 yes 4 40.3 even 4
160.2.d.a.81.1 4 5.3 odd 4
160.2.d.a.81.4 4 40.13 odd 4
200.2.d.f.101.3 4 40.27 even 4
200.2.d.f.101.4 4 20.7 even 4
200.2.f.c.149.3 4 4.3 odd 2
200.2.f.c.149.4 4 40.19 odd 2
200.2.f.e.149.1 4 8.3 odd 2
200.2.f.e.149.2 4 20.19 odd 2
360.2.k.e.181.3 4 120.83 odd 4
360.2.k.e.181.4 4 60.23 odd 4
800.2.d.e.401.1 4 40.37 odd 4
800.2.d.e.401.4 4 5.2 odd 4
800.2.f.c.49.1 4 1.1 even 1 trivial
800.2.f.c.49.2 4 40.29 even 2 inner
800.2.f.e.49.3 4 8.5 even 2
800.2.f.e.49.4 4 5.4 even 2
1280.2.a.a.1.1 2 80.3 even 4
1280.2.a.d.1.1 2 80.53 odd 4
1280.2.a.n.1.2 2 80.13 odd 4
1280.2.a.o.1.2 2 80.43 even 4
1440.2.k.e.721.1 4 120.53 even 4
1440.2.k.e.721.3 4 15.8 even 4
1800.2.d.l.1549.3 4 60.59 even 2
1800.2.d.l.1549.4 4 24.11 even 2
1800.2.d.p.1549.1 4 120.59 even 2
1800.2.d.p.1549.2 4 12.11 even 2
1800.2.k.j.901.1 4 60.47 odd 4
1800.2.k.j.901.2 4 120.107 odd 4
6400.2.a.z.1.1 2 80.27 even 4
6400.2.a.be.1.1 2 80.77 odd 4
6400.2.a.ce.1.2 2 80.67 even 4
6400.2.a.cj.1.2 2 80.37 odd 4
7200.2.d.n.2449.2 4 24.5 odd 2
7200.2.d.n.2449.3 4 15.14 odd 2
7200.2.d.o.2449.2 4 3.2 odd 2
7200.2.d.o.2449.3 4 120.29 odd 2
7200.2.k.j.3601.3 4 15.2 even 4
7200.2.k.j.3601.4 4 120.77 even 4