# Properties

 Label 800.2.d.e.401.4 Level $800$ Weight $2$ Character 800.401 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.d (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 401.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 800.401 Dual form 800.2.d.e.401.1

## $q$-expansion

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

## 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
$$3$$ 2.73205i 1.57735i 0.614810 + 0.788675i $$0.289233\pi$$
−0.614810 + 0.788675i $$0.710767\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.732051 0.276689 0.138345 0.990384i $$-0.455822\pi$$
0.138345 + 0.990384i $$0.455822\pi$$
$$8$$ 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 $$0.159503\pi$$
−0.877058 + 0.480384i $$0.840497\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.46410 −0.840168 −0.420084 0.907485i $$-0.637999\pi$$
−0.420084 + 0.907485i $$0.637999\pi$$
$$18$$ 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 $$-0.723557\pi$$
−0.645994 + 0.763343i $$0.723557\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 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 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 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 $$-0.868427\pi$$
0.915781 0.401677i $$-0.131573\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.26795 0.476679 0.238340 0.971182i $$-0.423397\pi$$
0.238340 + 0.971182i $$0.423397\pi$$
$$48$$ 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 $$-0.711449\pi$$
0.616496 0.787358i $$-0.288551\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.7321i 1.31113i 0.755139 + 0.655564i $$0.227569\pi$$
−0.755139 + 0.655564i $$0.772431\pi$$
$$68$$ 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 $$-0.643889\pi$$
−0.436804 + 0.899557i $$0.643889\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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 $$0.0221684\pi$$
−0.997576 + 0.0695878i $$0.977832\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 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 0
$$96$$ 0 0
$$97$$ 14.3923 1.46132 0.730659 0.682743i $$-0.239213\pi$$
0.730659 + 0.682743i $$0.239213\pi$$
$$98$$ 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 $$-0.219494\pi$$
0.771525 + 0.636199i $$0.219494\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.73205i 0.264117i 0.991242 + 0.132059i $$0.0421587\pi$$
−0.991242 + 0.132059i $$0.957841\pi$$
$$108$$ 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 $$-0.291935\pi$$
0.608092 + 0.793867i $$0.291935\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.7321 1.48473 0.742365 0.669996i $$-0.233704\pi$$
0.742365 + 0.669996i $$0.233704\pi$$
$$128$$ 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
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −4.92820 −0.421045 −0.210522 0.977589i $$-0.567516\pi$$
−0.210522 + 0.977589i $$0.567516\pi$$
$$138$$ 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
$$145$$ 0 0
$$146$$ 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
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3.07180i 0.245156i 0.992459 + 0.122578i $$0.0391162\pi$$
−0.992459 + 0.122578i $$0.960884\pi$$
$$158$$ 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 $$-0.997555\pi$$
0.999970 0.00768192i $$-0.00244526\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −9.80385 −0.758645 −0.379322 0.925265i $$-0.623843\pi$$
−0.379322 + 0.925265i $$0.623843\pi$$
$$168$$ 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 $$-0.975776\pi$$
0.997106 0.0760286i $$-0.0242240\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 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
$$185$$ 0 0
$$186$$ 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 $$-0.506140\pi$$
−0.0192874 + 0.999814i $$0.506140\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 19.4641i 1.38676i 0.720572 + 0.693380i $$0.243879\pi$$
−0.720572 + 0.693380i $$0.756121\pi$$
$$198$$ 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
$$205$$ 0 0
$$206$$ 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
$$215$$ 0 0
$$216$$ 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 $$-0.562252\pi$$
−0.194327 + 0.980937i $$0.562252\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.0526i 0.667212i 0.942713 + 0.333606i $$0.108265\pi$$
−0.942713 + 0.333606i $$0.891735\pi$$
$$228$$ 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 $$-0.444241\pi$$
0.174279 + 0.984696i $$0.444241\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 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
$$245$$ 0 0
$$246$$ 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
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 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 $$-0.617053\pi$$
−0.359501 + 0.933145i $$0.617053\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 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
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 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 $$-0.907302\pi$$
0.957894 0.287121i $$-0.0926983\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 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 $$0.153288\pi$$
−0.886269 + 0.463171i $$0.846712\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 24.9808i − 1.42573i −0.701303 0.712864i $$-0.747398\pi$$
0.701303 0.712864i $$-0.252602\pi$$
$$308$$ 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 $$-0.462639\pi$$
0.117105 + 0.993120i $$0.462639\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8.53590i − 0.479424i −0.970844 0.239712i $$-0.922947\pi$$
0.970844 0.239712i $$-0.0770530\pi$$
$$318$$ 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
$$325$$ 0 0
$$326$$ 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
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 19.8564 1.08165 0.540824 0.841136i $$-0.318113\pi$$
0.540824 + 0.841136i $$0.318113\pi$$
$$338$$ 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
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 1.66025i − 0.0891271i −0.999007 0.0445636i $$-0.985810\pi$$
0.999007 0.0445636i $$-0.0141897\pi$$
$$348$$ 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 $$-0.611799\pi$$
−0.344049 + 0.938952i $$0.611799\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2.87564 0.150107 0.0750537 0.997179i $$-0.476087\pi$$
0.0750537 + 0.997179i $$0.476087\pi$$
$$368$$ 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 $$0.231858\pi$$
−0.746238 + 0.665679i $$0.768142\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 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 $$-0.318537\pi$$
0.539702 + 0.841856i $$0.318537\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 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
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 32.2487i − 1.61852i −0.587453 0.809258i $$-0.699869\pi$$
0.587453 0.809258i $$-0.300131\pi$$
$$398$$ 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
$$405$$ 0 0
$$406$$ 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
$$415$$ 0 0
$$416$$ 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
$$425$$ 0 0
$$426$$ 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 $$-0.345085\pi$$
0.467693 + 0.883891i $$0.345085\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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 $$0.166084\pi$$
−0.866940 + 0.498413i $$0.833916\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 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
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.7846 1.25293 0.626466 0.779449i $$-0.284501\pi$$
0.626466 + 0.779449i $$0.284501\pi$$
$$458$$ 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 $$-0.584327\pi$$
−0.261833 + 0.965113i $$0.584327\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 25.6603i − 1.18741i −0.804681 0.593707i $$-0.797664\pi$$
0.804681 0.593707i $$-0.202336\pi$$
$$468$$ 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 0
$$476$$ 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
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 6.58846 0.298551 0.149276 0.988796i $$-0.452306\pi$$
0.149276 + 0.988796i $$0.452306\pi$$
$$488$$ 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
$$495$$ 0 0
$$496$$ 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 $$-0.502411\pi$$
−0.00757426 + 0.999971i $$0.502411\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 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
$$515$$ 0 0
$$516$$ 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 $$-0.916912\pi$$
0.966125 0.258073i $$-0.0830875\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 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
$$535$$ 0 0
$$536$$ 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
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 33.2679i 1.42243i 0.702972 + 0.711217i $$0.251856\pi$$
−0.702972 + 0.711217i $$0.748144\pi$$
$$548$$ 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
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 14.7846i − 0.626444i −0.949680 0.313222i $$-0.898592\pi$$
0.949680 0.313222i $$-0.101408\pi$$
$$558$$ 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 $$-0.846161\pi$$
0.885467 0.464702i $$-0.153839\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 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
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −39.5692 −1.64729 −0.823644 0.567107i $$-0.808063\pi$$
−0.823644 + 0.567107i $$0.808063\pi$$
$$578$$ 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
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 3.80385i − 0.157002i −0.996914 0.0785008i $$-0.974987\pi$$
0.996914 0.0785008i $$-0.0250133\pi$$
$$588$$ 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 $$-0.733793\pi$$
−0.670203 + 0.742178i $$0.733793\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 30.9808 1.25747 0.628735 0.777619i $$-0.283573\pi$$
0.628735 + 0.777619i $$0.283573\pi$$
$$608$$ 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 $$-0.821069\pi$$
0.846123 0.532988i $$-0.178931\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 20.5359 0.826744 0.413372 0.910562i $$-0.364351\pi$$
0.413372 + 0.910562i $$0.364351\pi$$
$$618$$ 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
$$625$$ 0 0
$$626$$ 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
$$635$$ 0 0
$$636$$ 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 $$0.280472\pi$$
−0.636281 + 0.771457i $$0.719528\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16.7321 0.657805 0.328902 0.944364i $$-0.393321\pi$$
0.328902 + 0.944364i $$0.393321\pi$$
$$648$$ 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 $$-0.922963\pi$$
0.970856 0.239665i $$-0.0770375\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 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 0
$$666$$ 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 $$-0.577676\pi$$
−0.241612 + 0.970373i $$0.577676\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 17.6077i − 0.676719i −0.941017 0.338359i $$-0.890128\pi$$
0.941017 0.338359i $$-0.109872\pi$$
$$678$$ 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 $$-0.894678\pi$$
0.945757 0.324875i $$-0.105322\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 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 0
$$696$$ 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
$$705$$ 0 0
$$706$$ 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
$$715$$ 0 0
$$716$$ 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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −24.0526 −0.892060 −0.446030 0.895018i $$-0.647163\pi$$
−0.446030 + 0.895018i $$0.647163\pi$$
$$728$$ 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 $$0.224270\pi$$
−0.761893 + 0.647703i $$0.775730\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 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 $$-0.435444\pi$$
0.201423 + 0.979504i $$0.435444\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.0718i 0.693176i 0.938017 + 0.346588i $$0.112660\pi$$
−0.938017 + 0.346588i $$0.887340\pi$$
$$758$$ 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
$$765$$ 0 0
$$766$$ 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 $$0.131927\pi$$
−0.915333 + 0.402698i $$0.868073\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 16.5885i 0.591315i 0.955294 + 0.295657i $$0.0955387\pi$$
−0.955294 + 0.295657i $$0.904461\pi$$
$$788$$ 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
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 50.1051i − 1.77481i −0.460986 0.887407i $$-0.652504\pi$$
0.460986 0.887407i $$-0.347496\pi$$
$$798$$ 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
$$805$$ 0 0
$$806$$ 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 0
$$816$$ 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 $$-0.550836\pi$$
−0.159028 + 0.987274i $$0.550836\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23.1244i 0.804113i 0.915615 + 0.402056i $$0.131704\pi$$
−0.915615 + 0.402056i $$0.868296\pi$$
$$828$$ 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
$$835$$ 0 0
$$836$$ 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
$$845$$ 0 0
$$846$$ 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 $$0.120614\pi$$
−0.929065 + 0.369917i $$0.879386\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 19.8564 0.678282 0.339141 0.940736i $$-0.389864\pi$$
0.339141 + 0.940736i $$0.389864\pi$$
$$858$$ 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 $$-0.198822\pi$$
0.811187 + 0.584787i $$0.198822\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 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 0
$$876$$ 0 0
$$877$$ 1.71281i 0.0578376i 0.999582 + 0.0289188i $$0.00920642\pi$$
−0.999582 + 0.0289188i $$0.990794\pi$$
$$878$$ 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 $$-0.844396\pi$$
0.882876 0.469606i $$-0.155604\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −13.9090 −0.467017 −0.233509 0.972355i $$-0.575021\pi$$
−0.233509 + 0.972355i $$0.575021\pi$$
$$888$$ 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
$$895$$ 0 0
$$896$$ 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
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.87564i 0.161893i 0.996718 + 0.0809466i $$0.0257943\pi$$
−0.996718 + 0.0809466i $$0.974206\pi$$
$$908$$ 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
$$915$$ 0 0
$$916$$ 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
$$925$$ 0 0
$$926$$ 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
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −4.24871 −0.138799 −0.0693997 0.997589i $$-0.522108\pi$$
−0.0693997 + 0.997589i $$0.522108\pi$$
$$938$$ 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
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 3.12436i − 0.101528i −0.998711 0.0507640i $$-0.983834\pi$$
0.998711 0.0507640i $$-0.0161656\pi$$
$$948$$ 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 $$-0.410053\pi$$
0.278831 + 0.960340i $$0.410053\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 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 0
$$966$$ 0 0
$$967$$ 16.3397 0.525451 0.262725 0.964871i $$-0.415379\pi$$
0.262725 + 0.964871i $$0.415379\pi$$
$$968$$ 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
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 24.5359 0.784973 0.392486 0.919758i $$-0.371615\pi$$
0.392486 + 0.919758i $$0.371615\pi$$
$$978$$ 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 $$-0.783338\pi$$
−0.777156 + 0.629309i $$0.783338\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 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
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 11.1769i 0.353976i 0.984213 + 0.176988i $$0.0566354\pi$$
−0.984213 + 0.176988i $$0.943365\pi$$
$$998$$ 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.d.e.401.4 4
3.2 odd 2 7200.2.k.j.3601.3 4
4.3 odd 2 200.2.d.f.101.4 4
5.2 odd 4 800.2.f.e.49.4 4
5.3 odd 4 800.2.f.c.49.1 4
5.4 even 2 160.2.d.a.81.1 4
8.3 odd 2 200.2.d.f.101.3 4
8.5 even 2 inner 800.2.d.e.401.1 4
12.11 even 2 1800.2.k.j.901.1 4
15.2 even 4 7200.2.d.n.2449.3 4
15.8 even 4 7200.2.d.o.2449.2 4
15.14 odd 2 1440.2.k.e.721.3 4
16.3 odd 4 6400.2.a.ce.1.2 2
16.5 even 4 6400.2.a.cj.1.2 2
16.11 odd 4 6400.2.a.z.1.1 2
16.13 even 4 6400.2.a.be.1.1 2
20.3 even 4 200.2.f.c.149.3 4
20.7 even 4 200.2.f.e.149.2 4
20.19 odd 2 40.2.d.a.21.1 4
24.5 odd 2 7200.2.k.j.3601.4 4
24.11 even 2 1800.2.k.j.901.2 4
40.3 even 4 200.2.f.e.149.1 4
40.13 odd 4 800.2.f.e.49.3 4
40.19 odd 2 40.2.d.a.21.2 yes 4
40.27 even 4 200.2.f.c.149.4 4
40.29 even 2 160.2.d.a.81.4 4
40.37 odd 4 800.2.f.c.49.2 4
60.23 odd 4 1800.2.d.p.1549.2 4
60.47 odd 4 1800.2.d.l.1549.3 4
60.59 even 2 360.2.k.e.181.4 4
80.19 odd 4 1280.2.a.a.1.1 2
80.29 even 4 1280.2.a.n.1.2 2
80.59 odd 4 1280.2.a.o.1.2 2
80.69 even 4 1280.2.a.d.1.1 2
120.29 odd 2 1440.2.k.e.721.1 4
120.53 even 4 7200.2.d.n.2449.2 4
120.59 even 2 360.2.k.e.181.3 4
120.77 even 4 7200.2.d.o.2449.3 4
120.83 odd 4 1800.2.d.l.1549.4 4
120.107 odd 4 1800.2.d.p.1549.1 4

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