# Properties

 Label 160.2.d.a.81.1 Level $160$ Weight $2$ Character 160.81 Analytic conductor $1.278$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 160.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.27760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 81.1 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 160.81 Dual form 160.2.d.a.81.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.73205i q^{3} -1.00000i q^{5} -0.732051 q^{7} -4.46410 q^{9} +O(q^{10})$$ $$q-2.73205i q^{3} -1.00000i q^{5} -0.732051 q^{7} -4.46410 q^{9} +2.00000i q^{11} -3.46410i q^{13} -2.73205 q^{15} +3.46410 q^{17} -0.535898i q^{19} +2.00000i q^{21} +6.19615 q^{23} -1.00000 q^{25} +4.00000i q^{27} +6.92820i q^{29} +5.46410 q^{31} +5.46410 q^{33} +0.732051i q^{35} -2.00000i q^{37} -9.46410 q^{39} +1.46410 q^{41} +5.26795i q^{43} +4.46410i q^{45} -3.26795 q^{47} -6.46410 q^{49} -9.46410i q^{51} +11.4641i q^{53} +2.00000 q^{55} -1.46410 q^{57} +7.46410i q^{59} -8.92820i q^{61} +3.26795 q^{63} -3.46410 q^{65} -10.7321i q^{67} -16.9282i q^{69} -5.46410 q^{71} +7.46410 q^{73} +2.73205i q^{75} -1.46410i q^{77} +1.07180 q^{79} -2.46410 q^{81} -1.26795i q^{83} -3.46410i q^{85} +18.9282 q^{87} +8.92820 q^{89} +2.53590i q^{91} -14.9282i q^{93} -0.535898 q^{95} -14.3923 q^{97} -8.92820i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{7} - 4 q^{9} - 4 q^{15} + 4 q^{23} - 4 q^{25} + 8 q^{31} + 8 q^{33} - 24 q^{39} - 8 q^{41} - 20 q^{47} - 12 q^{49} + 8 q^{55} + 8 q^{57} + 20 q^{63} - 8 q^{71} + 16 q^{73} + 32 q^{79} + 4 q^{81} + 48 q^{87} + 8 q^{89} - 16 q^{95} - 16 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 2.73205i − 1.57735i −0.614810 0.788675i $$-0.710767\pi$$
0.614810 0.788675i $$-0.289233\pi$$
$$4$$ 0 0
$$5$$ − 1.00000i − 0.447214i
$$6$$ 0 0
$$7$$ −0.732051 −0.276689 −0.138345 0.990384i $$-0.544178\pi$$
−0.138345 + 0.990384i $$0.544178\pi$$
$$8$$ 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
$$13$$ − 3.46410i − 0.960769i −0.877058 0.480384i $$-0.840497\pi$$
0.877058 0.480384i $$-0.159503\pi$$
$$14$$ 0 0
$$15$$ −2.73205 −0.705412
$$16$$ 0 0
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 0 0
$$19$$ − 0.535898i − 0.122944i −0.998109 0.0614718i $$-0.980421\pi$$
0.998109 0.0614718i $$-0.0195794\pi$$
$$20$$ 0 0
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ 6.19615 1.29199 0.645994 0.763343i $$-0.276443\pi$$
0.645994 + 0.763343i $$0.276443\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ 6.92820i 1.28654i 0.765641 + 0.643268i $$0.222422\pi$$
−0.765641 + 0.643268i $$0.777578\pi$$
$$30$$ 0 0
$$31$$ 5.46410 0.981382 0.490691 0.871334i $$-0.336744\pi$$
0.490691 + 0.871334i $$0.336744\pi$$
$$32$$ 0 0
$$33$$ 5.46410 0.951178
$$34$$ 0 0
$$35$$ 0.732051i 0.123739i
$$36$$ 0 0
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 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
$$43$$ 5.26795i 0.803355i 0.915781 + 0.401677i $$0.131573\pi$$
−0.915781 + 0.401677i $$0.868427\pi$$
$$44$$ 0 0
$$45$$ 4.46410i 0.665469i
$$46$$ 0 0
$$47$$ −3.26795 −0.476679 −0.238340 0.971182i $$-0.576603\pi$$
−0.238340 + 0.971182i $$0.576603\pi$$
$$48$$ 0 0
$$49$$ −6.46410 −0.923443
$$50$$ 0 0
$$51$$ − 9.46410i − 1.32524i
$$52$$ 0 0
$$53$$ 11.4641i 1.57472i 0.616496 + 0.787358i $$0.288551\pi$$
−0.616496 + 0.787358i $$0.711449\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ −1.46410 −0.193925
$$58$$ 0 0
$$59$$ 7.46410i 0.971743i 0.874030 + 0.485872i $$0.161498\pi$$
−0.874030 + 0.485872i $$0.838502\pi$$
$$60$$ 0 0
$$61$$ − 8.92820i − 1.14314i −0.820554 0.571570i $$-0.806335\pi$$
0.820554 0.571570i $$-0.193665\pi$$
$$62$$ 0 0
$$63$$ 3.26795 0.411723
$$64$$ 0 0
$$65$$ −3.46410 −0.429669
$$66$$ 0 0
$$67$$ − 10.7321i − 1.31113i −0.755139 0.655564i $$-0.772431\pi$$
0.755139 0.655564i $$-0.227569\pi$$
$$68$$ 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
$$73$$ 7.46410 0.873607 0.436804 0.899557i $$-0.356111\pi$$
0.436804 + 0.899557i $$0.356111\pi$$
$$74$$ 0 0
$$75$$ 2.73205i 0.315470i
$$76$$ 0 0
$$77$$ − 1.46410i − 0.166850i
$$78$$ 0 0
$$79$$ 1.07180 0.120587 0.0602933 0.998181i $$-0.480796\pi$$
0.0602933 + 0.998181i $$0.480796\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ − 1.26795i − 0.139176i −0.997576 0.0695878i $$-0.977832\pi$$
0.997576 0.0695878i $$-0.0221684\pi$$
$$84$$ 0 0
$$85$$ − 3.46410i − 0.375735i
$$86$$ 0 0
$$87$$ 18.9282 2.02932
$$88$$ 0 0
$$89$$ 8.92820 0.946388 0.473194 0.880958i $$-0.343101\pi$$
0.473194 + 0.880958i $$0.343101\pi$$
$$90$$ 0 0
$$91$$ 2.53590i 0.265834i
$$92$$ 0 0
$$93$$ − 14.9282i − 1.54798i
$$94$$ 0 0
$$95$$ −0.535898 −0.0549820
$$96$$ 0 0
$$97$$ −14.3923 −1.46132 −0.730659 0.682743i $$-0.760787\pi$$
−0.730659 + 0.682743i $$0.760787\pi$$
$$98$$ 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
$$103$$ −15.6603 −1.54305 −0.771525 0.636199i $$-0.780506\pi$$
−0.771525 + 0.636199i $$0.780506\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ − 2.73205i − 0.264117i −0.991242 0.132059i $$-0.957841\pi$$
0.991242 0.132059i $$-0.0421587\pi$$
$$108$$ 0 0
$$109$$ 16.9282i 1.62143i 0.585443 + 0.810714i $$0.300921\pi$$
−0.585443 + 0.810714i $$0.699079\pi$$
$$110$$ 0 0
$$111$$ −5.46410 −0.518630
$$112$$ 0 0
$$113$$ −12.9282 −1.21618 −0.608092 0.793867i $$-0.708065\pi$$
−0.608092 + 0.793867i $$0.708065\pi$$
$$114$$ 0 0
$$115$$ − 6.19615i − 0.577794i
$$116$$ 0 0
$$117$$ 15.4641i 1.42966i
$$118$$ 0 0
$$119$$ −2.53590 −0.232465
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ − 4.00000i − 0.360668i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ −16.7321 −1.48473 −0.742365 0.669996i $$-0.766296\pi$$
−0.742365 + 0.669996i $$0.766296\pi$$
$$128$$ 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
$$133$$ 0.392305i 0.0340171i
$$134$$ 0 0
$$135$$ 4.00000 0.344265
$$136$$ 0 0
$$137$$ 4.92820 0.421045 0.210522 0.977589i $$-0.432484\pi$$
0.210522 + 0.977589i $$0.432484\pi$$
$$138$$ 0 0
$$139$$ 0.535898i 0.0454543i 0.999742 + 0.0227272i $$0.00723490\pi$$
−0.999742 + 0.0227272i $$0.992765\pi$$
$$140$$ 0 0
$$141$$ 8.92820i 0.751890i
$$142$$ 0 0
$$143$$ 6.92820 0.579365
$$144$$ 0 0
$$145$$ 6.92820 0.575356
$$146$$ 0 0
$$147$$ 17.6603i 1.45659i
$$148$$ 0 0
$$149$$ − 7.85641i − 0.643622i −0.946804 0.321811i $$-0.895708\pi$$
0.946804 0.321811i $$-0.104292\pi$$
$$150$$ 0 0
$$151$$ 12.3923 1.00847 0.504236 0.863566i $$-0.331774\pi$$
0.504236 + 0.863566i $$0.331774\pi$$
$$152$$ 0 0
$$153$$ −15.4641 −1.25020
$$154$$ 0 0
$$155$$ − 5.46410i − 0.438887i
$$156$$ 0 0
$$157$$ − 3.07180i − 0.245156i −0.992459 0.122578i $$-0.960884\pi$$
0.992459 0.122578i $$-0.0391162\pi$$
$$158$$ 0 0
$$159$$ 31.3205 2.48388
$$160$$ 0 0
$$161$$ −4.53590 −0.357479
$$162$$ 0 0
$$163$$ 0.196152i 0.0153638i 0.999970 + 0.00768192i $$0.00244526\pi$$
−0.999970 + 0.00768192i $$0.997555\pi$$
$$164$$ 0 0
$$165$$ − 5.46410i − 0.425380i
$$166$$ 0 0
$$167$$ 9.80385 0.758645 0.379322 0.925265i $$-0.376157\pi$$
0.379322 + 0.925265i $$0.376157\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 2.39230i 0.182944i
$$172$$ 0 0
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0.732051 0.0553378
$$176$$ 0 0
$$177$$ 20.3923 1.53278
$$178$$ 0 0
$$179$$ 8.53590i 0.638003i 0.947754 + 0.319002i $$0.103348\pi$$
−0.947754 + 0.319002i $$0.896652\pi$$
$$180$$ 0 0
$$181$$ 16.0000i 1.18927i 0.803996 + 0.594635i $$0.202704\pi$$
−0.803996 + 0.594635i $$0.797296\pi$$
$$182$$ 0 0
$$183$$ −24.3923 −1.80313
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 6.92820i 0.506640i
$$188$$ 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
$$193$$ 0.535898 0.0385748 0.0192874 0.999814i $$-0.493860\pi$$
0.0192874 + 0.999814i $$0.493860\pi$$
$$194$$ 0 0
$$195$$ 9.46410i 0.677738i
$$196$$ 0 0
$$197$$ − 19.4641i − 1.38676i −0.720572 0.693380i $$-0.756121\pi$$
0.720572 0.693380i $$-0.243879\pi$$
$$198$$ 0 0
$$199$$ 1.85641 0.131597 0.0657986 0.997833i $$-0.479041\pi$$
0.0657986 + 0.997833i $$0.479041\pi$$
$$200$$ 0 0
$$201$$ −29.3205 −2.06811
$$202$$ 0 0
$$203$$ − 5.07180i − 0.355970i
$$204$$ 0 0
$$205$$ − 1.46410i − 0.102257i
$$206$$ 0 0
$$207$$ −27.6603 −1.92252
$$208$$ 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
$$213$$ 14.9282i 1.02286i
$$214$$ 0 0
$$215$$ 5.26795 0.359271
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 0 0
$$219$$ − 20.3923i − 1.37798i
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 0 0
$$223$$ 5.80385 0.388654 0.194327 0.980937i $$-0.437748\pi$$
0.194327 + 0.980937i $$0.437748\pi$$
$$224$$ 0 0
$$225$$ 4.46410 0.297607
$$226$$ 0 0
$$227$$ − 10.0526i − 0.667212i −0.942713 0.333606i $$-0.891735\pi$$
0.942713 0.333606i $$-0.108265\pi$$
$$228$$ 0 0
$$229$$ − 4.00000i − 0.264327i −0.991228 0.132164i $$-0.957808\pi$$
0.991228 0.132164i $$-0.0421925\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ −5.32051 −0.348558 −0.174279 0.984696i $$-0.555759\pi$$
−0.174279 + 0.984696i $$0.555759\pi$$
$$234$$ 0 0
$$235$$ 3.26795i 0.213177i
$$236$$ 0 0
$$237$$ − 2.92820i − 0.190207i
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ 16.3923 1.05592 0.527961 0.849269i $$-0.322957\pi$$
0.527961 + 0.849269i $$0.322957\pi$$
$$242$$ 0 0
$$243$$ 18.7321i 1.20166i
$$244$$ 0 0
$$245$$ 6.46410i 0.412976i
$$246$$ 0 0
$$247$$ −1.85641 −0.118120
$$248$$ 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
$$253$$ 12.3923i 0.779098i
$$254$$ 0 0
$$255$$ −9.46410 −0.592665
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 1.46410i 0.0909748i
$$260$$ 0 0
$$261$$ − 30.9282i − 1.91441i
$$262$$ 0 0
$$263$$ 11.6603 0.719002 0.359501 0.933145i $$-0.382947\pi$$
0.359501 + 0.933145i $$0.382947\pi$$
$$264$$ 0 0
$$265$$ 11.4641 0.704234
$$266$$ 0 0
$$267$$ − 24.3923i − 1.49278i
$$268$$ 0 0
$$269$$ − 8.92820i − 0.544362i −0.962246 0.272181i $$-0.912255\pi$$
0.962246 0.272181i $$-0.0877450\pi$$
$$270$$ 0 0
$$271$$ 19.3205 1.17364 0.586819 0.809718i $$-0.300380\pi$$
0.586819 + 0.809718i $$0.300380\pi$$
$$272$$ 0 0
$$273$$ 6.92820 0.419314
$$274$$ 0 0
$$275$$ − 2.00000i − 0.120605i
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ 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
$$283$$ 9.66025i 0.574242i 0.957894 + 0.287121i $$0.0926983\pi$$
−0.957894 + 0.287121i $$0.907302\pi$$
$$284$$ 0 0
$$285$$ 1.46410i 0.0867259i
$$286$$ 0 0
$$287$$ −1.07180 −0.0632662
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ 39.3205i 2.30501i
$$292$$ 0 0
$$293$$ − 15.8564i − 0.926341i −0.886269 0.463171i $$-0.846712\pi$$
0.886269 0.463171i $$-0.153288\pi$$
$$294$$ 0 0
$$295$$ 7.46410 0.434577
$$296$$ 0 0
$$297$$ −8.00000 −0.464207
$$298$$ 0 0
$$299$$ − 21.4641i − 1.24130i
$$300$$ 0 0
$$301$$ − 3.85641i − 0.222280i
$$302$$ 0 0
$$303$$ 8.00000 0.459588
$$304$$ 0 0
$$305$$ −8.92820 −0.511227
$$306$$ 0 0
$$307$$ 24.9808i 1.42573i 0.701303 + 0.712864i $$0.252602\pi$$
−0.701303 + 0.712864i $$0.747398\pi$$
$$308$$ 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
$$313$$ −4.14359 −0.234210 −0.117105 0.993120i $$-0.537361\pi$$
−0.117105 + 0.993120i $$0.537361\pi$$
$$314$$ 0 0
$$315$$ − 3.26795i − 0.184128i
$$316$$ 0 0
$$317$$ 8.53590i 0.479424i 0.970844 + 0.239712i $$0.0770530\pi$$
−0.970844 + 0.239712i $$0.922947\pi$$
$$318$$ 0 0
$$319$$ −13.8564 −0.775810
$$320$$ 0 0
$$321$$ −7.46410 −0.416606
$$322$$ 0 0
$$323$$ − 1.85641i − 0.103293i
$$324$$ 0 0
$$325$$ 3.46410i 0.192154i
$$326$$ 0 0
$$327$$ 46.2487 2.55756
$$328$$ 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
$$333$$ 8.92820i 0.489263i
$$334$$ 0 0
$$335$$ −10.7321 −0.586355
$$336$$ 0 0
$$337$$ −19.8564 −1.08165 −0.540824 0.841136i $$-0.681887\pi$$
−0.540824 + 0.841136i $$0.681887\pi$$
$$338$$ 0 0
$$339$$ 35.3205i 1.91835i
$$340$$ 0 0
$$341$$ 10.9282i 0.591795i
$$342$$ 0 0
$$343$$ 9.85641 0.532196
$$344$$ 0 0
$$345$$ −16.9282 −0.911384
$$346$$ 0 0
$$347$$ 1.66025i 0.0891271i 0.999007 + 0.0445636i $$0.0141897\pi$$
−0.999007 + 0.0445636i $$0.985810\pi$$
$$348$$ 0 0
$$349$$ − 28.0000i − 1.49881i −0.662114 0.749403i $$-0.730341\pi$$
0.662114 0.749403i $$-0.269659\pi$$
$$350$$ 0 0
$$351$$ 13.8564 0.739600
$$352$$ 0 0
$$353$$ 12.9282 0.688099 0.344049 0.938952i $$-0.388201\pi$$
0.344049 + 0.938952i $$0.388201\pi$$
$$354$$ 0 0
$$355$$ 5.46410i 0.290004i
$$356$$ 0 0
$$357$$ 6.92820i 0.366679i
$$358$$ 0 0
$$359$$ −18.9282 −0.998992 −0.499496 0.866316i $$-0.666482\pi$$
−0.499496 + 0.866316i $$0.666482\pi$$
$$360$$ 0 0
$$361$$ 18.7128 0.984885
$$362$$ 0 0
$$363$$ − 19.1244i − 1.00377i
$$364$$ 0 0
$$365$$ − 7.46410i − 0.390689i
$$366$$ 0 0
$$367$$ −2.87564 −0.150107 −0.0750537 0.997179i $$-0.523913\pi$$
−0.0750537 + 0.997179i $$0.523913\pi$$
$$368$$ 0 0
$$369$$ −6.53590 −0.340245
$$370$$ 0 0
$$371$$ − 8.39230i − 0.435707i
$$372$$ 0 0
$$373$$ − 25.7128i − 1.33136i −0.746238 0.665679i $$-0.768142\pi$$
0.746238 0.665679i $$-0.231858\pi$$
$$374$$ 0 0
$$375$$ 2.73205 0.141082
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 36.2487i 1.86197i 0.365056 + 0.930986i $$0.381050\pi$$
−0.365056 + 0.930986i $$0.618950\pi$$
$$380$$ 0 0
$$381$$ 45.7128i 2.34194i
$$382$$ 0 0
$$383$$ −21.1244 −1.07940 −0.539702 0.841856i $$-0.681463\pi$$
−0.539702 + 0.841856i $$0.681463\pi$$
$$384$$ 0 0
$$385$$ −1.46410 −0.0746175
$$386$$ 0 0
$$387$$ − 23.5167i − 1.19542i
$$388$$ 0 0
$$389$$ 6.78461i 0.343993i 0.985098 + 0.171997i $$0.0550218\pi$$
−0.985098 + 0.171997i $$0.944978\pi$$
$$390$$ 0 0
$$391$$ 21.4641 1.08549
$$392$$ 0 0
$$393$$ 54.2487 2.73649
$$394$$ 0 0
$$395$$ − 1.07180i − 0.0539279i
$$396$$ 0 0
$$397$$ 32.2487i 1.61852i 0.587453 + 0.809258i $$0.300131\pi$$
−0.587453 + 0.809258i $$0.699869\pi$$
$$398$$ 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
$$403$$ − 18.9282i − 0.942881i
$$404$$ 0 0
$$405$$ 2.46410i 0.122442i
$$406$$ 0 0
$$407$$ 4.00000 0.198273
$$408$$ 0 0
$$409$$ −11.3205 −0.559763 −0.279882 0.960035i $$-0.590295\pi$$
−0.279882 + 0.960035i $$0.590295\pi$$
$$410$$ 0 0
$$411$$ − 13.4641i − 0.664135i
$$412$$ 0 0
$$413$$ − 5.46410i − 0.268871i
$$414$$ 0 0
$$415$$ −1.26795 −0.0622412
$$416$$ 0 0
$$417$$ 1.46410 0.0716974
$$418$$ 0 0
$$419$$ − 18.3923i − 0.898523i −0.893400 0.449261i $$-0.851687\pi$$
0.893400 0.449261i $$-0.148313\pi$$
$$420$$ 0 0
$$421$$ 0.143594i 0.00699832i 0.999994 + 0.00349916i $$0.00111382\pi$$
−0.999994 + 0.00349916i $$0.998886\pi$$
$$422$$ 0 0
$$423$$ 14.5885 0.709315
$$424$$ 0 0
$$425$$ −3.46410 −0.168034
$$426$$ 0 0
$$427$$ 6.53590i 0.316294i
$$428$$ 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
$$433$$ −19.4641 −0.935385 −0.467693 0.883891i $$-0.654915\pi$$
−0.467693 + 0.883891i $$0.654915\pi$$
$$434$$ 0 0
$$435$$ − 18.9282i − 0.907538i
$$436$$ 0 0
$$437$$ − 3.32051i − 0.158841i
$$438$$ 0 0
$$439$$ −40.7846 −1.94654 −0.973272 0.229657i $$-0.926240\pi$$
−0.973272 + 0.229657i $$0.926240\pi$$
$$440$$ 0 0
$$441$$ 28.8564 1.37411
$$442$$ 0 0
$$443$$ − 20.9808i − 0.996826i −0.866940 0.498413i $$-0.833916\pi$$
0.866940 0.498413i $$-0.166084\pi$$
$$444$$ 0 0
$$445$$ − 8.92820i − 0.423237i
$$446$$ 0 0
$$447$$ −21.4641 −1.01522
$$448$$ 0 0
$$449$$ −23.3205 −1.10056 −0.550281 0.834979i $$-0.685480\pi$$
−0.550281 + 0.834979i $$0.685480\pi$$
$$450$$ 0 0
$$451$$ 2.92820i 0.137884i
$$452$$ 0 0
$$453$$ − 33.8564i − 1.59071i
$$454$$ 0 0
$$455$$ 2.53590 0.118885
$$456$$ 0 0
$$457$$ −26.7846 −1.25293 −0.626466 0.779449i $$-0.715499\pi$$
−0.626466 + 0.779449i $$0.715499\pi$$
$$458$$ 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
$$463$$ 11.2679 0.523666 0.261833 0.965113i $$-0.415673\pi$$
0.261833 + 0.965113i $$0.415673\pi$$
$$464$$ 0 0
$$465$$ −14.9282 −0.692279
$$466$$ 0 0
$$467$$ 25.6603i 1.18741i 0.804681 + 0.593707i $$0.202336\pi$$
−0.804681 + 0.593707i $$0.797664\pi$$
$$468$$ 0 0
$$469$$ 7.85641i 0.362775i
$$470$$ 0 0
$$471$$ −8.39230 −0.386697
$$472$$ 0 0
$$473$$ −10.5359 −0.484441
$$474$$ 0 0
$$475$$ 0.535898i 0.0245887i
$$476$$ 0 0
$$477$$ − 51.1769i − 2.34323i
$$478$$ 0 0
$$479$$ −5.85641 −0.267586 −0.133793 0.991009i $$-0.542716\pi$$
−0.133793 + 0.991009i $$0.542716\pi$$
$$480$$ 0 0
$$481$$ −6.92820 −0.315899
$$482$$ 0 0
$$483$$ 12.3923i 0.563869i
$$484$$ 0 0
$$485$$ 14.3923i 0.653521i
$$486$$ 0 0
$$487$$ −6.58846 −0.298551 −0.149276 0.988796i $$-0.547694\pi$$
−0.149276 + 0.988796i $$0.547694\pi$$
$$488$$ 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
$$493$$ 24.0000i 1.08091i
$$494$$ 0 0
$$495$$ −8.92820 −0.401293
$$496$$ 0 0
$$497$$ 4.00000 0.179425
$$498$$ 0 0
$$499$$ − 31.4641i − 1.40853i −0.709939 0.704263i $$-0.751277\pi$$
0.709939 0.704263i $$-0.248723\pi$$
$$500$$ 0 0
$$501$$ − 26.7846i − 1.19665i
$$502$$ 0 0
$$503$$ 0.339746 0.0151485 0.00757426 0.999971i $$-0.497589\pi$$
0.00757426 + 0.999971i $$0.497589\pi$$
$$504$$ 0 0
$$505$$ 2.92820 0.130303
$$506$$ 0 0
$$507$$ − 2.73205i − 0.121335i
$$508$$ 0 0
$$509$$ − 1.85641i − 0.0822838i −0.999153 0.0411419i $$-0.986900\pi$$
0.999153 0.0411419i $$-0.0130996\pi$$
$$510$$ 0 0
$$511$$ −5.46410 −0.241718
$$512$$ 0 0
$$513$$ 2.14359 0.0946420
$$514$$ 0 0
$$515$$ 15.6603i 0.690073i
$$516$$ 0 0
$$517$$ − 6.53590i − 0.287448i
$$518$$ 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
$$523$$ 11.8038i 0.516146i 0.966125 + 0.258073i $$0.0830875\pi$$
−0.966125 + 0.258073i $$0.916912\pi$$
$$524$$ 0 0
$$525$$ − 2.00000i − 0.0872872i
$$526$$ 0 0
$$527$$ 18.9282 0.824525
$$528$$ 0 0
$$529$$ 15.3923 0.669231
$$530$$ 0 0
$$531$$ − 33.3205i − 1.44599i
$$532$$ 0 0
$$533$$ − 5.07180i − 0.219684i
$$534$$ 0 0
$$535$$ −2.73205 −0.118117
$$536$$ 0 0
$$537$$ 23.3205 1.00635
$$538$$ 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
$$543$$ 43.7128 1.87590
$$544$$ 0 0
$$545$$ 16.9282 0.725125
$$546$$ 0 0
$$547$$ − 33.2679i − 1.42243i −0.702972 0.711217i $$-0.748144\pi$$
0.702972 0.711217i $$-0.251856\pi$$
$$548$$ 0 0
$$549$$ 39.8564i 1.70103i
$$550$$ 0 0
$$551$$ 3.71281 0.158171
$$552$$ 0 0
$$553$$ −0.784610 −0.0333650
$$554$$ 0 0
$$555$$ 5.46410i 0.231938i
$$556$$ 0 0
$$557$$ 14.7846i 0.626444i 0.949680 + 0.313222i $$0.101408\pi$$
−0.949680 + 0.313222i $$0.898592\pi$$
$$558$$ 0 0
$$559$$ 18.2487 0.771838
$$560$$ 0 0
$$561$$ 18.9282 0.799149
$$562$$ 0 0
$$563$$ 22.0526i 0.929405i 0.885467 + 0.464702i $$0.153839\pi$$
−0.885467 + 0.464702i $$0.846161\pi$$
$$564$$ 0 0
$$565$$ 12.9282i 0.543894i
$$566$$ 0 0
$$567$$ 1.80385 0.0757545
$$568$$ 0 0
$$569$$ −13.4641 −0.564445 −0.282222 0.959349i $$-0.591072\pi$$
−0.282222 + 0.959349i $$0.591072\pi$$
$$570$$ 0 0
$$571$$ − 6.78461i − 0.283927i −0.989872 0.141964i $$-0.954658\pi$$
0.989872 0.141964i $$-0.0453416\pi$$
$$572$$ 0 0
$$573$$ 41.8564i 1.74858i
$$574$$ 0 0
$$575$$ −6.19615 −0.258397
$$576$$ 0 0
$$577$$ 39.5692 1.64729 0.823644 0.567107i $$-0.191937\pi$$
0.823644 + 0.567107i $$0.191937\pi$$
$$578$$ 0 0
$$579$$ − 1.46410i − 0.0608460i
$$580$$ 0 0
$$581$$ 0.928203i 0.0385084i
$$582$$ 0 0
$$583$$ −22.9282 −0.949589
$$584$$ 0 0
$$585$$ 15.4641 0.639362
$$586$$ 0 0
$$587$$ 3.80385i 0.157002i 0.996914 + 0.0785008i $$0.0250133\pi$$
−0.996914 + 0.0785008i $$0.974987\pi$$
$$588$$ 0 0
$$589$$ − 2.92820i − 0.120655i
$$590$$ 0 0
$$591$$ −53.1769 −2.18741
$$592$$ 0 0
$$593$$ 32.6410 1.34041 0.670203 0.742178i $$-0.266207\pi$$
0.670203 + 0.742178i $$0.266207\pi$$
$$594$$ 0 0
$$595$$ 2.53590i 0.103962i
$$596$$ 0 0
$$597$$ − 5.07180i − 0.207575i
$$598$$ 0 0
$$599$$ 34.6410 1.41539 0.707697 0.706516i $$-0.249734\pi$$
0.707697 + 0.706516i $$0.249734\pi$$
$$600$$ 0 0
$$601$$ 18.5359 0.756095 0.378048 0.925786i $$-0.376596\pi$$
0.378048 + 0.925786i $$0.376596\pi$$
$$602$$ 0 0
$$603$$ 47.9090i 1.95100i
$$604$$ 0 0
$$605$$ − 7.00000i − 0.284590i
$$606$$ 0 0
$$607$$ −30.9808 −1.25747 −0.628735 0.777619i $$-0.716427\pi$$
−0.628735 + 0.777619i $$0.716427\pi$$
$$608$$ 0 0
$$609$$ −13.8564 −0.561490
$$610$$ 0 0
$$611$$ 11.3205i 0.457979i
$$612$$ 0 0
$$613$$ 26.3923i 1.06598i 0.846123 + 0.532988i $$0.178931\pi$$
−0.846123 + 0.532988i $$0.821069\pi$$
$$614$$ 0 0
$$615$$ −4.00000 −0.161296
$$616$$ 0 0
$$617$$ −20.5359 −0.826744 −0.413372 0.910562i $$-0.635649\pi$$
−0.413372 + 0.910562i $$0.635649\pi$$
$$618$$ 0 0
$$619$$ − 1.32051i − 0.0530757i −0.999648 0.0265379i $$-0.991552\pi$$
0.999648 0.0265379i $$-0.00844825\pi$$
$$620$$ 0 0
$$621$$ 24.7846i 0.994572i
$$622$$ 0 0
$$623$$ −6.53590 −0.261855
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ − 2.92820i − 0.116941i
$$628$$ 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
$$633$$ −73.1769 −2.90852
$$634$$ 0 0
$$635$$ 16.7321i 0.663991i
$$636$$ 0 0
$$637$$ 22.3923i 0.887215i
$$638$$ 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
$$643$$ − 39.1244i − 1.54291i −0.636281 0.771457i $$-0.719528\pi$$
0.636281 0.771457i $$-0.280472\pi$$
$$644$$ 0 0
$$645$$ − 14.3923i − 0.566696i
$$646$$ 0 0
$$647$$ −16.7321 −0.657805 −0.328902 0.944364i $$-0.606679\pi$$
−0.328902 + 0.944364i $$0.606679\pi$$
$$648$$ 0 0
$$649$$ −14.9282 −0.585983
$$650$$ 0 0
$$651$$ 10.9282i 0.428310i
$$652$$ 0 0
$$653$$ 12.2487i 0.479329i 0.970856 + 0.239665i $$0.0770375\pi$$
−0.970856 + 0.239665i $$0.922963\pi$$
$$654$$ 0 0
$$655$$ 19.8564 0.775854
$$656$$ 0 0
$$657$$ −33.3205 −1.29996
$$658$$ 0 0
$$659$$ 17.3205i 0.674711i 0.941377 + 0.337356i $$0.109532\pi$$
−0.941377 + 0.337356i $$0.890468\pi$$
$$660$$ 0 0
$$661$$ − 8.14359i − 0.316749i −0.987379 0.158375i $$-0.949375\pi$$
0.987379 0.158375i $$-0.0506253\pi$$
$$662$$ 0 0
$$663$$ −32.7846 −1.27325
$$664$$ 0 0
$$665$$ 0.392305 0.0152129
$$666$$ 0 0
$$667$$ 42.9282i 1.66219i
$$668$$ 0 0
$$669$$ − 15.8564i − 0.613044i
$$670$$ 0 0
$$671$$ 17.8564 0.689339
$$672$$ 0 0
$$673$$ 12.5359 0.483223 0.241612 0.970373i $$-0.422324\pi$$
0.241612 + 0.970373i $$0.422324\pi$$
$$674$$ 0 0
$$675$$ − 4.00000i − 0.153960i
$$676$$ 0 0
$$677$$ 17.6077i 0.676719i 0.941017 + 0.338359i $$0.109872\pi$$
−0.941017 + 0.338359i $$0.890128\pi$$
$$678$$ 0 0
$$679$$ 10.5359 0.404331
$$680$$ 0 0
$$681$$ −27.4641 −1.05243
$$682$$ 0 0
$$683$$ 16.9808i 0.649751i 0.945757 + 0.324875i $$0.105322\pi$$
−0.945757 + 0.324875i $$0.894678\pi$$
$$684$$ 0 0
$$685$$ − 4.92820i − 0.188297i
$$686$$ 0 0
$$687$$ −10.9282 −0.416937
$$688$$ 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
$$693$$ 6.53590i 0.248278i
$$694$$ 0 0
$$695$$ 0.535898 0.0203278
$$696$$ 0 0
$$697$$ 5.07180 0.192108
$$698$$ 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
$$703$$ −1.07180 −0.0404236
$$704$$ 0 0
$$705$$ 8.92820 0.336256
$$706$$ 0 0
$$707$$ − 2.14359i − 0.0806181i
$$708$$ 0 0
$$709$$ − 12.7846i − 0.480136i −0.970756 0.240068i $$-0.922830\pi$$
0.970756 0.240068i $$-0.0771698\pi$$
$$710$$ 0 0
$$711$$ −4.78461 −0.179437
$$712$$ 0 0
$$713$$ 33.8564 1.26793
$$714$$ 0 0
$$715$$ − 6.92820i − 0.259100i
$$716$$ 0 0
$$717$$ − 54.6410i − 2.04061i
$$718$$ 0 0
$$719$$ 1.85641 0.0692323 0.0346161 0.999401i $$-0.488979\pi$$
0.0346161 + 0.999401i $$0.488979\pi$$
$$720$$ 0 0
$$721$$ 11.4641 0.426945
$$722$$ 0 0
$$723$$ − 44.7846i − 1.66556i
$$724$$ 0 0
$$725$$ − 6.92820i − 0.257307i
$$726$$ 0 0
$$727$$ 24.0526 0.892060 0.446030 0.895018i $$-0.352837\pi$$
0.446030 + 0.895018i $$0.352837\pi$$
$$728$$ 0 0
$$729$$ 43.7846 1.62165
$$730$$ 0 0
$$731$$ 18.2487i 0.674953i
$$732$$ 0 0
$$733$$ − 35.0718i − 1.29541i −0.761893 0.647703i $$-0.775730\pi$$
0.761893 0.647703i $$-0.224270\pi$$
$$734$$ 0 0
$$735$$ 17.6603 0.651408
$$736$$ 0 0
$$737$$ 21.4641 0.790640
$$738$$ 0 0
$$739$$ − 29.3205i − 1.07857i −0.842123 0.539286i $$-0.818694\pi$$
0.842123 0.539286i $$-0.181306\pi$$
$$740$$ 0 0
$$741$$ 5.07180i 0.186317i
$$742$$ 0 0
$$743$$ −10.9808 −0.402845 −0.201423 0.979504i $$-0.564556\pi$$
−0.201423 + 0.979504i $$0.564556\pi$$
$$744$$ 0 0
$$745$$ −7.85641 −0.287836
$$746$$ 0 0
$$747$$ 5.66025i 0.207098i
$$748$$ 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
$$753$$ −68.1051 −2.48189
$$754$$ 0 0
$$755$$ − 12.3923i − 0.451002i
$$756$$ 0 0
$$757$$ − 19.0718i − 0.693176i −0.938017 0.346588i $$-0.887340\pi$$
0.938017 0.346588i $$-0.112660\pi$$
$$758$$ 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
$$763$$ − 12.3923i − 0.448632i
$$764$$ 0 0
$$765$$ 15.4641i 0.559106i
$$766$$ 0 0
$$767$$ 25.8564 0.933621
$$768$$ 0 0
$$769$$ 12.9282 0.466203 0.233101 0.972452i $$-0.425113\pi$$
0.233101 + 0.972452i $$0.425113\pi$$
$$770$$ 0 0
$$771$$ 5.46410i 0.196785i
$$772$$ 0 0
$$773$$ − 22.3923i − 0.805395i −0.915333 0.402698i $$-0.868073\pi$$
0.915333 0.402698i $$-0.131927\pi$$
$$774$$ 0 0
$$775$$ −5.46410 −0.196276
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ − 0.784610i − 0.0281116i
$$780$$ 0 0
$$781$$ − 10.9282i − 0.391042i
$$782$$ 0 0
$$783$$ −27.7128 −0.990375
$$784$$ 0 0
$$785$$ −3.07180 −0.109637
$$786$$ 0 0
$$787$$ − 16.5885i − 0.591315i −0.955294 0.295657i $$-0.904461\pi$$
0.955294 0.295657i $$-0.0955387\pi$$
$$788$$ 0 0
$$789$$ − 31.8564i − 1.13412i
$$790$$ 0 0
$$791$$ 9.46410 0.336505
$$792$$ 0 0
$$793$$ −30.9282 −1.09829
$$794$$ 0 0
$$795$$ − 31.3205i − 1.11082i
$$796$$ 0 0
$$797$$ 50.1051i 1.77481i 0.460986 + 0.887407i $$0.347496\pi$$
−0.460986 + 0.887407i $$0.652504\pi$$
$$798$$ 0 0
$$799$$ −11.3205 −0.400491
$$800$$ 0 0
$$801$$ −39.8564 −1.40826
$$802$$ 0 0
$$803$$ 14.9282i 0.526805i
$$804$$ 0 0
$$805$$ 4.53590i 0.159869i
$$806$$ 0 0
$$807$$ −24.3923 −0.858650
$$808$$ 0 0
$$809$$ 23.8564 0.838747 0.419373 0.907814i $$-0.362250\pi$$
0.419373 + 0.907814i $$0.362250\pi$$
$$810$$ 0 0
$$811$$ 28.9282i 1.01581i 0.861414 + 0.507903i $$0.169579\pi$$
−0.861414 + 0.507903i $$0.830421\pi$$
$$812$$ 0 0
$$813$$ − 52.7846i − 1.85124i
$$814$$ 0 0
$$815$$ 0.196152 0.00687092
$$816$$ 0 0
$$817$$ 2.82309 0.0987673
$$818$$ 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
$$823$$ 9.12436 0.318055 0.159028 0.987274i $$-0.449164\pi$$
0.159028 + 0.987274i $$0.449164\pi$$
$$824$$ 0 0
$$825$$ −5.46410 −0.190236
$$826$$ 0 0
$$827$$ − 23.1244i − 0.804113i −0.915615 0.402056i $$-0.868296\pi$$
0.915615 0.402056i $$-0.131704\pi$$
$$828$$ 0 0
$$829$$ 28.9282i 1.00472i 0.864659 + 0.502359i $$0.167534\pi$$
−0.864659 + 0.502359i $$0.832466\pi$$
$$830$$ 0 0
$$831$$ 5.46410 0.189548
$$832$$ 0 0
$$833$$ −22.3923 −0.775847
$$834$$ 0 0
$$835$$ − 9.80385i − 0.339276i
$$836$$ 0 0
$$837$$ 21.8564i 0.755468i
$$838$$ 0 0
$$839$$ −24.7846 −0.855660 −0.427830 0.903859i $$-0.640722\pi$$
−0.427830 + 0.903859i $$0.640722\pi$$
$$840$$ 0 0
$$841$$ −19.0000 −0.655172
$$842$$ 0 0
$$843$$ − 28.7846i − 0.991395i
$$844$$ 0 0
$$845$$ − 1.00000i − 0.0344010i
$$846$$ 0 0
$$847$$ −5.12436 −0.176075
$$848$$ 0 0
$$849$$ 26.3923 0.905782
$$850$$ 0 0
$$851$$ − 12.3923i − 0.424803i
$$852$$ 0 0
$$853$$ − 21.6077i − 0.739833i −0.929065 0.369917i $$-0.879386\pi$$
0.929065 0.369917i $$-0.120614\pi$$
$$854$$ 0 0
$$855$$ 2.39230 0.0818151
$$856$$ 0 0
$$857$$ −19.8564 −0.678282 −0.339141 0.940736i $$-0.610136\pi$$
−0.339141 + 0.940736i $$0.610136\pi$$
$$858$$ 0 0
$$859$$ − 28.2487i − 0.963834i −0.876217 0.481917i $$-0.839941\pi$$
0.876217 0.481917i $$-0.160059\pi$$
$$860$$ 0 0
$$861$$ 2.92820i 0.0997929i
$$862$$ 0 0
$$863$$ −47.6603 −1.62237 −0.811187 0.584787i $$-0.801178\pi$$
−0.811187 + 0.584787i $$0.801178\pi$$
$$864$$ 0 0
$$865$$ 2.00000 0.0680020
$$866$$ 0 0
$$867$$ 13.6603i 0.463927i
$$868$$ 0 0
$$869$$ 2.14359i 0.0727164i
$$870$$ 0 0
$$871$$ −37.1769 −1.25969
$$872$$ 0 0
$$873$$ 64.2487 2.17449
$$874$$ 0 0
$$875$$ − 0.732051i − 0.0247478i
$$876$$ 0 0
$$877$$ − 1.71281i − 0.0578376i −0.999582 0.0289188i $$-0.990794\pi$$
0.999582 0.0289188i $$-0.00920642\pi$$
$$878$$ 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
$$883$$ 27.9090i 0.939211i 0.882876 + 0.469606i $$0.155604\pi$$
−0.882876 + 0.469606i $$0.844396\pi$$
$$884$$ 0 0
$$885$$ − 20.3923i − 0.685480i
$$886$$ 0 0
$$887$$ 13.9090 0.467017 0.233509 0.972355i $$-0.424979\pi$$
0.233509 + 0.972355i $$0.424979\pi$$
$$888$$ 0 0
$$889$$ 12.2487 0.410809
$$890$$ 0 0
$$891$$ − 4.92820i − 0.165101i
$$892$$ 0 0
$$893$$ 1.75129i 0.0586046i
$$894$$ 0 0
$$895$$ 8.53590 0.285324
$$896$$ 0 0
$$897$$ −58.6410 −1.95797
$$898$$ 0 0
$$899$$ 37.8564i 1.26258i
$$900$$ 0 0
$$901$$ 39.7128i 1.32303i
$$902$$ 0 0
$$903$$ −10.5359 −0.350613
$$904$$ 0 0
$$905$$ 16.0000 0.531858
$$906$$ 0 0
$$907$$ − 4.87564i − 0.161893i −0.996718 0.0809466i $$-0.974206\pi$$
0.996718 0.0809466i $$-0.0257943\pi$$
$$908$$ 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
$$913$$ 2.53590 0.0839260
$$914$$ 0 0
$$915$$ 24.3923i 0.806385i
$$916$$ 0 0
$$917$$ − 14.5359i − 0.480018i
$$918$$ 0 0
$$919$$ 38.9282 1.28412 0.642061 0.766653i $$-0.278079\pi$$
0.642061 + 0.766653i $$0.278079\pi$$
$$920$$ 0 0
$$921$$ 68.2487 2.24887
$$922$$ 0 0
$$923$$ 18.9282i 0.623029i
$$924$$ 0 0
$$925$$ 2.00000i 0.0657596i
$$926$$ 0 0
$$927$$ 69.9090 2.29611
$$928$$ 0 0
$$929$$ −17.4641 −0.572979 −0.286489 0.958083i $$-0.592488\pi$$
−0.286489 + 0.958083i $$0.592488\pi$$
$$930$$ 0 0
$$931$$ 3.46410i 0.113531i
$$932$$ 0 0
$$933$$ 85.5692i 2.80141i
$$934$$ 0 0
$$935$$ 6.92820 0.226576
$$936$$ 0 0
$$937$$ 4.24871 0.138799 0.0693997 0.997589i $$-0.477892\pi$$
0.0693997 + 0.997589i $$0.477892\pi$$
$$938$$ 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
$$943$$ 9.07180 0.295418
$$944$$ 0 0
$$945$$ −2.92820 −0.0952545
$$946$$ 0 0
$$947$$ 3.12436i 0.101528i 0.998711 + 0.0507640i $$0.0161656\pi$$
−0.998711 + 0.0507640i $$0.983834\pi$$
$$948$$ 0 0
$$949$$ − 25.8564i − 0.839334i
$$950$$ 0 0
$$951$$ 23.3205 0.756219
$$952$$ 0 0
$$953$$ −17.2154 −0.557661 −0.278831 0.960340i $$-0.589947\pi$$
−0.278831 + 0.960340i $$0.589947\pi$$
$$954$$ 0 0
$$955$$ 15.3205i 0.495760i
$$956$$ 0 0
$$957$$ 37.8564i 1.22372i
$$958$$ 0 0
$$959$$ −3.60770 −0.116499
$$960$$ 0 0
$$961$$ −1.14359 −0.0368901
$$962$$ 0 0
$$963$$ 12.1962i 0.393016i
$$964$$ 0 0
$$965$$ − 0.535898i − 0.0172512i
$$966$$ 0 0
$$967$$ −16.3397 −0.525451 −0.262725 0.964871i $$-0.584621\pi$$
−0.262725 + 0.964871i $$0.584621\pi$$
$$968$$ 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
$$973$$ − 0.392305i − 0.0125767i
$$974$$ 0 0
$$975$$ 9.46410 0.303094
$$976$$ 0 0
$$977$$ −24.5359 −0.784973 −0.392486 0.919758i $$-0.628385\pi$$
−0.392486 + 0.919758i $$0.628385\pi$$
$$978$$ 0 0
$$979$$ 17.8564i 0.570693i
$$980$$ 0 0
$$981$$ − 75.5692i − 2.41274i
$$982$$ 0 0
$$983$$ 48.7321 1.55431 0.777156 0.629309i $$-0.216662\pi$$
0.777156 + 0.629309i $$0.216662\pi$$
$$984$$ 0 0
$$985$$ −19.4641 −0.620178
$$986$$ 0 0
$$987$$ − 6.53590i − 0.208040i
$$988$$ 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
$$993$$ 38.2487 1.21379
$$994$$ 0 0
$$995$$ − 1.85641i − 0.0588520i
$$996$$ 0 0
$$997$$ − 11.1769i − 0.353976i −0.984213 0.176988i $$-0.943365\pi$$
0.984213 0.176988i $$-0.0566354\pi$$
$$998$$ 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 160.2.d.a.81.1 4
3.2 odd 2 1440.2.k.e.721.3 4
4.3 odd 2 40.2.d.a.21.1 4
5.2 odd 4 800.2.f.c.49.1 4
5.3 odd 4 800.2.f.e.49.4 4
5.4 even 2 800.2.d.e.401.4 4
8.3 odd 2 40.2.d.a.21.2 yes 4
8.5 even 2 inner 160.2.d.a.81.4 4
12.11 even 2 360.2.k.e.181.4 4
15.2 even 4 7200.2.d.o.2449.2 4
15.8 even 4 7200.2.d.n.2449.3 4
15.14 odd 2 7200.2.k.j.3601.3 4
16.3 odd 4 1280.2.a.a.1.1 2
16.5 even 4 1280.2.a.d.1.1 2
16.11 odd 4 1280.2.a.o.1.2 2
16.13 even 4 1280.2.a.n.1.2 2
20.3 even 4 200.2.f.e.149.2 4
20.7 even 4 200.2.f.c.149.3 4
20.19 odd 2 200.2.d.f.101.4 4
24.5 odd 2 1440.2.k.e.721.1 4
24.11 even 2 360.2.k.e.181.3 4
40.3 even 4 200.2.f.c.149.4 4
40.13 odd 4 800.2.f.c.49.2 4
40.19 odd 2 200.2.d.f.101.3 4
40.27 even 4 200.2.f.e.149.1 4
40.29 even 2 800.2.d.e.401.1 4
40.37 odd 4 800.2.f.e.49.3 4
60.23 odd 4 1800.2.d.l.1549.3 4
60.47 odd 4 1800.2.d.p.1549.2 4
60.59 even 2 1800.2.k.j.901.1 4
80.19 odd 4 6400.2.a.ce.1.2 2
80.29 even 4 6400.2.a.be.1.1 2
80.59 odd 4 6400.2.a.z.1.1 2
80.69 even 4 6400.2.a.cj.1.2 2
120.29 odd 2 7200.2.k.j.3601.4 4
120.53 even 4 7200.2.d.o.2449.3 4
120.59 even 2 1800.2.k.j.901.2 4
120.77 even 4 7200.2.d.n.2449.2 4
120.83 odd 4 1800.2.d.p.1549.1 4
120.107 odd 4 1800.2.d.l.1549.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 4.3 odd 2
40.2.d.a.21.2 yes 4 8.3 odd 2
160.2.d.a.81.1 4 1.1 even 1 trivial
160.2.d.a.81.4 4 8.5 even 2 inner
200.2.d.f.101.3 4 40.19 odd 2
200.2.d.f.101.4 4 20.19 odd 2
200.2.f.c.149.3 4 20.7 even 4
200.2.f.c.149.4 4 40.3 even 4
200.2.f.e.149.1 4 40.27 even 4
200.2.f.e.149.2 4 20.3 even 4
360.2.k.e.181.3 4 24.11 even 2
360.2.k.e.181.4 4 12.11 even 2
800.2.d.e.401.1 4 40.29 even 2
800.2.d.e.401.4 4 5.4 even 2
800.2.f.c.49.1 4 5.2 odd 4
800.2.f.c.49.2 4 40.13 odd 4
800.2.f.e.49.3 4 40.37 odd 4
800.2.f.e.49.4 4 5.3 odd 4
1280.2.a.a.1.1 2 16.3 odd 4
1280.2.a.d.1.1 2 16.5 even 4
1280.2.a.n.1.2 2 16.13 even 4
1280.2.a.o.1.2 2 16.11 odd 4
1440.2.k.e.721.1 4 24.5 odd 2
1440.2.k.e.721.3 4 3.2 odd 2
1800.2.d.l.1549.3 4 60.23 odd 4
1800.2.d.l.1549.4 4 120.107 odd 4
1800.2.d.p.1549.1 4 120.83 odd 4
1800.2.d.p.1549.2 4 60.47 odd 4
1800.2.k.j.901.1 4 60.59 even 2
1800.2.k.j.901.2 4 120.59 even 2
6400.2.a.z.1.1 2 80.59 odd 4
6400.2.a.be.1.1 2 80.29 even 4
6400.2.a.ce.1.2 2 80.19 odd 4
6400.2.a.cj.1.2 2 80.69 even 4
7200.2.d.n.2449.2 4 120.77 even 4
7200.2.d.n.2449.3 4 15.8 even 4
7200.2.d.o.2449.2 4 15.2 even 4
7200.2.d.o.2449.3 4 120.53 even 4
7200.2.k.j.3601.3 4 15.14 odd 2
7200.2.k.j.3601.4 4 120.29 odd 2