# Properties

 Label 1536.2.d.d.769.2 Level $1536$ Weight $2$ Character 1536.769 Analytic conductor $12.265$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 769.2 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1536.769 Dual form 1536.2.d.d.769.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} +1.41421i q^{5} -1.41421 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} +1.41421i q^{5} -1.41421 q^{7} -1.00000 q^{9} +2.00000i q^{11} +1.41421 q^{15} +2.00000 q^{17} -4.00000i q^{19} +1.41421i q^{21} -2.82843 q^{23} +3.00000 q^{25} +1.00000i q^{27} +9.89949i q^{29} -7.07107 q^{31} +2.00000 q^{33} -2.00000i q^{35} +8.48528i q^{37} -6.00000 q^{41} +8.00000i q^{43} -1.41421i q^{45} +2.82843 q^{47} -5.00000 q^{49} -2.00000i q^{51} +1.41421i q^{53} -2.82843 q^{55} -4.00000 q^{57} +12.0000i q^{59} +14.1421i q^{61} +1.41421 q^{63} -8.00000i q^{67} +2.82843i q^{69} +14.1421 q^{71} +8.00000 q^{73} -3.00000i q^{75} -2.82843i q^{77} -4.24264 q^{79} +1.00000 q^{81} -6.00000i q^{83} +2.82843i q^{85} +9.89949 q^{87} -2.00000 q^{89} +7.07107i q^{93} +5.65685 q^{95} -14.0000 q^{97} -2.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{9} + 8 q^{17} + 12 q^{25} + 8 q^{33} - 24 q^{41} - 20 q^{49} - 16 q^{57} + 32 q^{73} + 4 q^{81} - 8 q^{89} - 56 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$517$$ $$1025$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 1.41421i 0.632456i 0.948683 + 0.316228i $$0.102416\pi$$
−0.948683 + 0.316228i $$0.897584\pi$$
$$6$$ 0 0
$$7$$ −1.41421 −0.534522 −0.267261 0.963624i $$-0.586119\pi$$
−0.267261 + 0.963624i $$0.586119\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000i 0.603023i 0.953463 + 0.301511i $$0.0974911\pi$$
−0.953463 + 0.301511i $$0.902509\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 1.41421 0.365148
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.917663i −0.888523 0.458831i $$-0.848268\pi$$
0.888523 0.458831i $$-0.151732\pi$$
$$20$$ 0 0
$$21$$ 1.41421i 0.308607i
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 9.89949i 1.83829i 0.393919 + 0.919145i $$0.371119\pi$$
−0.393919 + 0.919145i $$0.628881\pi$$
$$30$$ 0 0
$$31$$ −7.07107 −1.27000 −0.635001 0.772512i $$-0.719000\pi$$
−0.635001 + 0.772512i $$0.719000\pi$$
$$32$$ 0 0
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ − 2.00000i − 0.338062i
$$36$$ 0 0
$$37$$ 8.48528i 1.39497i 0.716599 + 0.697486i $$0.245698\pi$$
−0.716599 + 0.697486i $$0.754302\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ − 1.41421i − 0.210819i
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ −5.00000 −0.714286
$$50$$ 0 0
$$51$$ − 2.00000i − 0.280056i
$$52$$ 0 0
$$53$$ 1.41421i 0.194257i 0.995272 + 0.0971286i $$0.0309658\pi$$
−0.995272 + 0.0971286i $$0.969034\pi$$
$$54$$ 0 0
$$55$$ −2.82843 −0.381385
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ 12.0000i 1.56227i 0.624364 + 0.781133i $$0.285358\pi$$
−0.624364 + 0.781133i $$0.714642\pi$$
$$60$$ 0 0
$$61$$ 14.1421i 1.81071i 0.424650 + 0.905357i $$0.360397\pi$$
−0.424650 + 0.905357i $$0.639603\pi$$
$$62$$ 0 0
$$63$$ 1.41421 0.178174
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ 2.82843i 0.340503i
$$70$$ 0 0
$$71$$ 14.1421 1.67836 0.839181 0.543852i $$-0.183035\pi$$
0.839181 + 0.543852i $$0.183035\pi$$
$$72$$ 0 0
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 0 0
$$75$$ − 3.00000i − 0.346410i
$$76$$ 0 0
$$77$$ − 2.82843i − 0.322329i
$$78$$ 0 0
$$79$$ −4.24264 −0.477334 −0.238667 0.971101i $$-0.576710\pi$$
−0.238667 + 0.971101i $$0.576710\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 2.82843i 0.306786i
$$86$$ 0 0
$$87$$ 9.89949 1.06134
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 7.07107i 0.733236i
$$94$$ 0 0
$$95$$ 5.65685 0.580381
$$96$$ 0 0
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 0 0
$$99$$ − 2.00000i − 0.201008i
$$100$$ 0 0
$$101$$ 9.89949i 0.985037i 0.870302 + 0.492518i $$0.163924\pi$$
−0.870302 + 0.492518i $$0.836076\pi$$
$$102$$ 0 0
$$103$$ −12.7279 −1.25412 −0.627060 0.778971i $$-0.715742\pi$$
−0.627060 + 0.778971i $$0.715742\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ − 11.3137i − 1.08366i −0.840489 0.541828i $$-0.817732\pi$$
0.840489 0.541828i $$-0.182268\pi$$
$$110$$ 0 0
$$111$$ 8.48528 0.805387
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ − 4.00000i − 0.373002i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2.82843 −0.259281
$$120$$ 0 0
$$121$$ 7.00000 0.636364
$$122$$ 0 0
$$123$$ 6.00000i 0.541002i
$$124$$ 0 0
$$125$$ 11.3137i 1.01193i
$$126$$ 0 0
$$127$$ −4.24264 −0.376473 −0.188237 0.982124i $$-0.560277\pi$$
−0.188237 + 0.982124i $$0.560277\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ − 12.0000i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ 5.65685i 0.490511i
$$134$$ 0 0
$$135$$ −1.41421 −0.121716
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ − 2.82843i − 0.238197i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −14.0000 −1.16264
$$146$$ 0 0
$$147$$ 5.00000i 0.412393i
$$148$$ 0 0
$$149$$ − 4.24264i − 0.347571i −0.984784 0.173785i $$-0.944400\pi$$
0.984784 0.173785i $$-0.0555999\pi$$
$$150$$ 0 0
$$151$$ −4.24264 −0.345261 −0.172631 0.984987i $$-0.555227\pi$$
−0.172631 + 0.984987i $$0.555227\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ − 10.0000i − 0.803219i
$$156$$ 0 0
$$157$$ 14.1421i 1.12867i 0.825547 + 0.564333i $$0.190866\pi$$
−0.825547 + 0.564333i $$0.809134\pi$$
$$158$$ 0 0
$$159$$ 1.41421 0.112154
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 0 0
$$165$$ 2.82843i 0.220193i
$$166$$ 0 0
$$167$$ 11.3137 0.875481 0.437741 0.899101i $$-0.355779\pi$$
0.437741 + 0.899101i $$0.355779\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 4.00000i 0.305888i
$$172$$ 0 0
$$173$$ − 15.5563i − 1.18273i −0.806405 0.591364i $$-0.798590\pi$$
0.806405 0.591364i $$-0.201410\pi$$
$$174$$ 0 0
$$175$$ −4.24264 −0.320713
$$176$$ 0 0
$$177$$ 12.0000 0.901975
$$178$$ 0 0
$$179$$ 4.00000i 0.298974i 0.988764 + 0.149487i $$0.0477622\pi$$
−0.988764 + 0.149487i $$0.952238\pi$$
$$180$$ 0 0
$$181$$ − 11.3137i − 0.840941i −0.907306 0.420471i $$-0.861865\pi$$
0.907306 0.420471i $$-0.138135\pi$$
$$182$$ 0 0
$$183$$ 14.1421 1.04542
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 0 0
$$189$$ − 1.41421i − 0.102869i
$$190$$ 0 0
$$191$$ 22.6274 1.63726 0.818631 0.574320i $$-0.194733\pi$$
0.818631 + 0.574320i $$0.194733\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 7.07107i 0.503793i 0.967754 + 0.251896i $$0.0810542\pi$$
−0.967754 + 0.251896i $$0.918946\pi$$
$$198$$ 0 0
$$199$$ −24.0416 −1.70427 −0.852133 0.523325i $$-0.824691\pi$$
−0.852133 + 0.523325i $$0.824691\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ − 14.0000i − 0.982607i
$$204$$ 0 0
$$205$$ − 8.48528i − 0.592638i
$$206$$ 0 0
$$207$$ 2.82843 0.196589
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ − 14.1421i − 0.969003i
$$214$$ 0 0
$$215$$ −11.3137 −0.771589
$$216$$ 0 0
$$217$$ 10.0000 0.678844
$$218$$ 0 0
$$219$$ − 8.00000i − 0.540590i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −12.7279 −0.852325 −0.426162 0.904647i $$-0.640135\pi$$
−0.426162 + 0.904647i $$0.640135\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 22.0000i 1.46019i 0.683345 + 0.730096i $$0.260525\pi$$
−0.683345 + 0.730096i $$0.739475\pi$$
$$228$$ 0 0
$$229$$ − 16.9706i − 1.12145i −0.828003 0.560723i $$-0.810523\pi$$
0.828003 0.560723i $$-0.189477\pi$$
$$230$$ 0 0
$$231$$ −2.82843 −0.186097
$$232$$ 0 0
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 0 0
$$235$$ 4.00000i 0.260931i
$$236$$ 0 0
$$237$$ 4.24264i 0.275589i
$$238$$ 0 0
$$239$$ −22.6274 −1.46365 −0.731823 0.681495i $$-0.761330\pi$$
−0.731823 + 0.681495i $$0.761330\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 0 0
$$245$$ − 7.07107i − 0.451754i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ − 6.00000i − 0.378717i −0.981908 0.189358i $$-0.939359\pi$$
0.981908 0.189358i $$-0.0606408\pi$$
$$252$$ 0 0
$$253$$ − 5.65685i − 0.355643i
$$254$$ 0 0
$$255$$ 2.82843 0.177123
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ − 12.0000i − 0.745644i
$$260$$ 0 0
$$261$$ − 9.89949i − 0.612763i
$$262$$ 0 0
$$263$$ 28.2843 1.74408 0.872041 0.489432i $$-0.162796\pi$$
0.872041 + 0.489432i $$0.162796\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 2.00000i 0.122398i
$$268$$ 0 0
$$269$$ − 21.2132i − 1.29339i −0.762748 0.646696i $$-0.776150\pi$$
0.762748 0.646696i $$-0.223850\pi$$
$$270$$ 0 0
$$271$$ −24.0416 −1.46043 −0.730213 0.683220i $$-0.760579\pi$$
−0.730213 + 0.683220i $$0.760579\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 6.00000i 0.361814i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 7.07107 0.423334
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 0 0
$$285$$ − 5.65685i − 0.335083i
$$286$$ 0 0
$$287$$ 8.48528 0.500870
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000i 0.820695i
$$292$$ 0 0
$$293$$ − 9.89949i − 0.578335i −0.957279 0.289167i $$-0.906622\pi$$
0.957279 0.289167i $$-0.0933784\pi$$
$$294$$ 0 0
$$295$$ −16.9706 −0.988064
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 11.3137i − 0.652111i
$$302$$ 0 0
$$303$$ 9.89949 0.568711
$$304$$ 0 0
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ 0 0
$$309$$ 12.7279i 0.724066i
$$310$$ 0 0
$$311$$ −22.6274 −1.28308 −0.641542 0.767088i $$-0.721705\pi$$
−0.641542 + 0.767088i $$0.721705\pi$$
$$312$$ 0 0
$$313$$ 4.00000 0.226093 0.113047 0.993590i $$-0.463939\pi$$
0.113047 + 0.993590i $$0.463939\pi$$
$$314$$ 0 0
$$315$$ 2.00000i 0.112687i
$$316$$ 0 0
$$317$$ − 9.89949i − 0.556011i −0.960579 0.278006i $$-0.910327\pi$$
0.960579 0.278006i $$-0.0896734\pi$$
$$318$$ 0 0
$$319$$ −19.7990 −1.10853
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −11.3137 −0.625650
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ − 32.0000i − 1.75888i −0.476011 0.879440i $$-0.657918\pi$$
0.476011 0.879440i $$-0.342082\pi$$
$$332$$ 0 0
$$333$$ − 8.48528i − 0.464991i
$$334$$ 0 0
$$335$$ 11.3137 0.618134
$$336$$ 0 0
$$337$$ −8.00000 −0.435788 −0.217894 0.975972i $$-0.569919\pi$$
−0.217894 + 0.975972i $$0.569919\pi$$
$$338$$ 0 0
$$339$$ − 6.00000i − 0.325875i
$$340$$ 0 0
$$341$$ − 14.1421i − 0.765840i
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ −4.00000 −0.215353
$$346$$ 0 0
$$347$$ − 26.0000i − 1.39575i −0.716218 0.697877i $$-0.754128\pi$$
0.716218 0.697877i $$-0.245872\pi$$
$$348$$ 0 0
$$349$$ − 25.4558i − 1.36262i −0.731995 0.681310i $$-0.761411\pi$$
0.731995 0.681310i $$-0.238589\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 20.0000i 1.06149i
$$356$$ 0 0
$$357$$ 2.82843i 0.149696i
$$358$$ 0 0
$$359$$ 19.7990 1.04495 0.522475 0.852654i $$-0.325009\pi$$
0.522475 + 0.852654i $$0.325009\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 11.3137i 0.592187i
$$366$$ 0 0
$$367$$ 4.24264 0.221464 0.110732 0.993850i $$-0.464680\pi$$
0.110732 + 0.993850i $$0.464680\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ − 2.00000i − 0.103835i
$$372$$ 0 0
$$373$$ − 19.7990i − 1.02515i −0.858642 0.512576i $$-0.828691\pi$$
0.858642 0.512576i $$-0.171309\pi$$
$$374$$ 0 0
$$375$$ 11.3137 0.584237
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 20.0000i 1.02733i 0.857991 + 0.513665i $$0.171713\pi$$
−0.857991 + 0.513665i $$0.828287\pi$$
$$380$$ 0 0
$$381$$ 4.24264i 0.217357i
$$382$$ 0 0
$$383$$ −5.65685 −0.289052 −0.144526 0.989501i $$-0.546166\pi$$
−0.144526 + 0.989501i $$0.546166\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ 0 0
$$387$$ − 8.00000i − 0.406663i
$$388$$ 0 0
$$389$$ − 24.0416i − 1.21896i −0.792802 0.609480i $$-0.791378\pi$$
0.792802 0.609480i $$-0.208622\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ − 6.00000i − 0.301893i
$$396$$ 0 0
$$397$$ 2.82843i 0.141955i 0.997478 + 0.0709773i $$0.0226118\pi$$
−0.997478 + 0.0709773i $$0.977388\pi$$
$$398$$ 0 0
$$399$$ 5.65685 0.283197
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 1.41421i 0.0702728i
$$406$$ 0 0
$$407$$ −16.9706 −0.841200
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ − 10.0000i − 0.493264i
$$412$$ 0 0
$$413$$ − 16.9706i − 0.835067i
$$414$$ 0 0
$$415$$ 8.48528 0.416526
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 30.0000i 1.46560i 0.680446 + 0.732798i $$0.261786\pi$$
−0.680446 + 0.732798i $$0.738214\pi$$
$$420$$ 0 0
$$421$$ − 5.65685i − 0.275698i −0.990453 0.137849i $$-0.955981\pi$$
0.990453 0.137849i $$-0.0440189\pi$$
$$422$$ 0 0
$$423$$ −2.82843 −0.137523
$$424$$ 0 0
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ − 20.0000i − 0.967868i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.82843 0.136241 0.0681203 0.997677i $$-0.478300\pi$$
0.0681203 + 0.997677i $$0.478300\pi$$
$$432$$ 0 0
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 0 0
$$435$$ 14.0000i 0.671249i
$$436$$ 0 0
$$437$$ 11.3137i 0.541208i
$$438$$ 0 0
$$439$$ 4.24264 0.202490 0.101245 0.994862i $$-0.467717\pi$$
0.101245 + 0.994862i $$0.467717\pi$$
$$440$$ 0 0
$$441$$ 5.00000 0.238095
$$442$$ 0 0
$$443$$ 30.0000i 1.42534i 0.701498 + 0.712672i $$0.252515\pi$$
−0.701498 + 0.712672i $$0.747485\pi$$
$$444$$ 0 0
$$445$$ − 2.82843i − 0.134080i
$$446$$ 0 0
$$447$$ −4.24264 −0.200670
$$448$$ 0 0
$$449$$ −22.0000 −1.03824 −0.519122 0.854700i $$-0.673741\pi$$
−0.519122 + 0.854700i $$0.673741\pi$$
$$450$$ 0 0
$$451$$ − 12.0000i − 0.565058i
$$452$$ 0 0
$$453$$ 4.24264i 0.199337i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ 0 0
$$459$$ 2.00000i 0.0933520i
$$460$$ 0 0
$$461$$ 12.7279i 0.592798i 0.955064 + 0.296399i $$0.0957859\pi$$
−0.955064 + 0.296399i $$0.904214\pi$$
$$462$$ 0 0
$$463$$ 35.3553 1.64310 0.821551 0.570135i $$-0.193109\pi$$
0.821551 + 0.570135i $$0.193109\pi$$
$$464$$ 0 0
$$465$$ −10.0000 −0.463739
$$466$$ 0 0
$$467$$ 22.0000i 1.01804i 0.860755 + 0.509019i $$0.169992\pi$$
−0.860755 + 0.509019i $$0.830008\pi$$
$$468$$ 0 0
$$469$$ 11.3137i 0.522419i
$$470$$ 0 0
$$471$$ 14.1421 0.651635
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ − 12.0000i − 0.550598i
$$476$$ 0 0
$$477$$ − 1.41421i − 0.0647524i
$$478$$ 0 0
$$479$$ 25.4558 1.16311 0.581554 0.813508i $$-0.302445\pi$$
0.581554 + 0.813508i $$0.302445\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ − 4.00000i − 0.182006i
$$484$$ 0 0
$$485$$ − 19.7990i − 0.899026i
$$486$$ 0 0
$$487$$ 4.24264 0.192252 0.0961262 0.995369i $$-0.469355\pi$$
0.0961262 + 0.995369i $$0.469355\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 28.0000i 1.26362i 0.775122 + 0.631811i $$0.217688\pi$$
−0.775122 + 0.631811i $$0.782312\pi$$
$$492$$ 0 0
$$493$$ 19.7990i 0.891702i
$$494$$ 0 0
$$495$$ 2.82843 0.127128
$$496$$ 0 0
$$497$$ −20.0000 −0.897123
$$498$$ 0 0
$$499$$ − 20.0000i − 0.895323i −0.894203 0.447661i $$-0.852257\pi$$
0.894203 0.447661i $$-0.147743\pi$$
$$500$$ 0 0
$$501$$ − 11.3137i − 0.505459i
$$502$$ 0 0
$$503$$ −2.82843 −0.126113 −0.0630567 0.998010i $$-0.520085\pi$$
−0.0630567 + 0.998010i $$0.520085\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 0 0
$$507$$ − 13.0000i − 0.577350i
$$508$$ 0 0
$$509$$ − 24.0416i − 1.06563i −0.846233 0.532813i $$-0.821135\pi$$
0.846233 0.532813i $$-0.178865\pi$$
$$510$$ 0 0
$$511$$ −11.3137 −0.500489
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ − 18.0000i − 0.793175i
$$516$$ 0 0
$$517$$ 5.65685i 0.248788i
$$518$$ 0 0
$$519$$ −15.5563 −0.682848
$$520$$ 0 0
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ 0 0
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 0 0
$$525$$ 4.24264i 0.185164i
$$526$$ 0 0
$$527$$ −14.1421 −0.616041
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ − 12.0000i − 0.520756i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −5.65685 −0.244567
$$536$$ 0 0
$$537$$ 4.00000 0.172613
$$538$$ 0 0
$$539$$ − 10.0000i − 0.430730i
$$540$$ 0 0
$$541$$ − 28.2843i − 1.21604i −0.793923 0.608018i $$-0.791965\pi$$
0.793923 0.608018i $$-0.208035\pi$$
$$542$$ 0 0
$$543$$ −11.3137 −0.485518
$$544$$ 0 0
$$545$$ 16.0000 0.685365
$$546$$ 0 0
$$547$$ 16.0000i 0.684111i 0.939680 + 0.342055i $$0.111123\pi$$
−0.939680 + 0.342055i $$0.888877\pi$$
$$548$$ 0 0
$$549$$ − 14.1421i − 0.603572i
$$550$$ 0 0
$$551$$ 39.5980 1.68693
$$552$$ 0 0
$$553$$ 6.00000 0.255146
$$554$$ 0 0
$$555$$ 12.0000i 0.509372i
$$556$$ 0 0
$$557$$ 38.1838i 1.61790i 0.587879 + 0.808949i $$0.299963\pi$$
−0.587879 + 0.808949i $$0.700037\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 4.00000 0.168880
$$562$$ 0 0
$$563$$ 6.00000i 0.252870i 0.991975 + 0.126435i $$0.0403535\pi$$
−0.991975 + 0.126435i $$0.959647\pi$$
$$564$$ 0 0
$$565$$ 8.48528i 0.356978i
$$566$$ 0 0
$$567$$ −1.41421 −0.0593914
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ 28.0000i 1.17176i 0.810397 + 0.585882i $$0.199252\pi$$
−0.810397 + 0.585882i $$0.800748\pi$$
$$572$$ 0 0
$$573$$ − 22.6274i − 0.945274i
$$574$$ 0 0
$$575$$ −8.48528 −0.353861
$$576$$ 0 0
$$577$$ −4.00000 −0.166522 −0.0832611 0.996528i $$-0.526534\pi$$
−0.0832611 + 0.996528i $$0.526534\pi$$
$$578$$ 0 0
$$579$$ 4.00000i 0.166234i
$$580$$ 0 0
$$581$$ 8.48528i 0.352029i
$$582$$ 0 0
$$583$$ −2.82843 −0.117141
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 28.0000i − 1.15568i −0.816149 0.577842i $$-0.803895\pi$$
0.816149 0.577842i $$-0.196105\pi$$
$$588$$ 0 0
$$589$$ 28.2843i 1.16543i
$$590$$ 0 0
$$591$$ 7.07107 0.290865
$$592$$ 0 0
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ − 4.00000i − 0.163984i
$$596$$ 0 0
$$597$$ 24.0416i 0.983958i
$$598$$ 0 0
$$599$$ −36.7696 −1.50236 −0.751182 0.660096i $$-0.770516\pi$$
−0.751182 + 0.660096i $$0.770516\pi$$
$$600$$ 0 0
$$601$$ −44.0000 −1.79480 −0.897399 0.441221i $$-0.854546\pi$$
−0.897399 + 0.441221i $$0.854546\pi$$
$$602$$ 0 0
$$603$$ 8.00000i 0.325785i
$$604$$ 0 0
$$605$$ 9.89949i 0.402472i
$$606$$ 0 0
$$607$$ −4.24264 −0.172203 −0.0861017 0.996286i $$-0.527441\pi$$
−0.0861017 + 0.996286i $$0.527441\pi$$
$$608$$ 0 0
$$609$$ −14.0000 −0.567309
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 8.48528i 0.342717i 0.985209 + 0.171359i $$0.0548157\pi$$
−0.985209 + 0.171359i $$0.945184\pi$$
$$614$$ 0 0
$$615$$ −8.48528 −0.342160
$$616$$ 0 0
$$617$$ −46.0000 −1.85189 −0.925945 0.377658i $$-0.876729\pi$$
−0.925945 + 0.377658i $$0.876729\pi$$
$$618$$ 0 0
$$619$$ 36.0000i 1.44696i 0.690344 + 0.723481i $$0.257459\pi$$
−0.690344 + 0.723481i $$0.742541\pi$$
$$620$$ 0 0
$$621$$ − 2.82843i − 0.113501i
$$622$$ 0 0
$$623$$ 2.82843 0.113319
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ − 8.00000i − 0.319489i
$$628$$ 0 0
$$629$$ 16.9706i 0.676661i
$$630$$ 0 0
$$631$$ 15.5563 0.619288 0.309644 0.950852i $$-0.399790\pi$$
0.309644 + 0.950852i $$0.399790\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 6.00000i − 0.238103i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −14.1421 −0.559454
$$640$$ 0 0
$$641$$ −42.0000 −1.65890 −0.829450 0.558581i $$-0.811346\pi$$
−0.829450 + 0.558581i $$0.811346\pi$$
$$642$$ 0 0
$$643$$ − 40.0000i − 1.57745i −0.614749 0.788723i $$-0.710743\pi$$
0.614749 0.788723i $$-0.289257\pi$$
$$644$$ 0 0
$$645$$ 11.3137i 0.445477i
$$646$$ 0 0
$$647$$ 14.1421 0.555985 0.277992 0.960583i $$-0.410331\pi$$
0.277992 + 0.960583i $$0.410331\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ − 10.0000i − 0.391931i
$$652$$ 0 0
$$653$$ 29.6985i 1.16219i 0.813835 + 0.581096i $$0.197376\pi$$
−0.813835 + 0.581096i $$0.802624\pi$$
$$654$$ 0 0
$$655$$ 16.9706 0.663095
$$656$$ 0 0
$$657$$ −8.00000 −0.312110
$$658$$ 0 0
$$659$$ − 44.0000i − 1.71400i −0.515319 0.856998i $$-0.672327\pi$$
0.515319 0.856998i $$-0.327673\pi$$
$$660$$ 0 0
$$661$$ 19.7990i 0.770091i 0.922897 + 0.385046i $$0.125814\pi$$
−0.922897 + 0.385046i $$0.874186\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ − 28.0000i − 1.08416i
$$668$$ 0 0
$$669$$ 12.7279i 0.492090i
$$670$$ 0 0
$$671$$ −28.2843 −1.09190
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 0 0
$$675$$ 3.00000i 0.115470i
$$676$$ 0 0
$$677$$ 24.0416i 0.923995i 0.886881 + 0.461997i $$0.152867\pi$$
−0.886881 + 0.461997i $$0.847133\pi$$
$$678$$ 0 0
$$679$$ 19.7990 0.759815
$$680$$ 0 0
$$681$$ 22.0000 0.843042
$$682$$ 0 0
$$683$$ 26.0000i 0.994862i 0.867503 + 0.497431i $$0.165723\pi$$
−0.867503 + 0.497431i $$0.834277\pi$$
$$684$$ 0 0
$$685$$ 14.1421i 0.540343i
$$686$$ 0 0
$$687$$ −16.9706 −0.647467
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000i 0.304334i 0.988355 + 0.152167i $$0.0486252\pi$$
−0.988355 + 0.152167i $$0.951375\pi$$
$$692$$ 0 0
$$693$$ 2.82843i 0.107443i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 22.0000i 0.832116i
$$700$$ 0 0
$$701$$ − 4.24264i − 0.160242i −0.996785 0.0801212i $$-0.974469\pi$$
0.996785 0.0801212i $$-0.0255307\pi$$
$$702$$ 0 0
$$703$$ 33.9411 1.28011
$$704$$ 0 0
$$705$$ 4.00000 0.150649
$$706$$ 0 0
$$707$$ − 14.0000i − 0.526524i
$$708$$ 0 0
$$709$$ − 33.9411i − 1.27469i −0.770580 0.637343i $$-0.780034\pi$$
0.770580 0.637343i $$-0.219966\pi$$
$$710$$ 0 0
$$711$$ 4.24264 0.159111
$$712$$ 0 0
$$713$$ 20.0000 0.749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 22.6274i 0.845036i
$$718$$ 0 0
$$719$$ −25.4558 −0.949343 −0.474671 0.880163i $$-0.657433\pi$$
−0.474671 + 0.880163i $$0.657433\pi$$
$$720$$ 0 0
$$721$$ 18.0000 0.670355
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 29.6985i 1.10297i
$$726$$ 0 0
$$727$$ 24.0416 0.891655 0.445827 0.895119i $$-0.352910\pi$$
0.445827 + 0.895119i $$0.352910\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 16.0000i 0.591781i
$$732$$ 0 0
$$733$$ − 11.3137i − 0.417881i −0.977928 0.208941i $$-0.932998\pi$$
0.977928 0.208941i $$-0.0670016\pi$$
$$734$$ 0 0
$$735$$ −7.07107 −0.260820
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ − 12.0000i − 0.441427i −0.975339 0.220714i $$-0.929161\pi$$
0.975339 0.220714i $$-0.0708386\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −28.2843 −1.03765 −0.518825 0.854881i $$-0.673630\pi$$
−0.518825 + 0.854881i $$0.673630\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ 6.00000i 0.219529i
$$748$$ 0 0
$$749$$ − 5.65685i − 0.206697i
$$750$$ 0 0
$$751$$ 12.7279 0.464448 0.232224 0.972662i $$-0.425400\pi$$
0.232224 + 0.972662i $$0.425400\pi$$
$$752$$ 0 0
$$753$$ −6.00000 −0.218652
$$754$$ 0 0
$$755$$ − 6.00000i − 0.218362i
$$756$$ 0 0
$$757$$ 50.9117i 1.85042i 0.379459 + 0.925208i $$0.376110\pi$$
−0.379459 + 0.925208i $$0.623890\pi$$
$$758$$ 0 0
$$759$$ −5.65685 −0.205331
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ 16.0000i 0.579239i
$$764$$ 0 0
$$765$$ − 2.82843i − 0.102262i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 20.0000 0.721218 0.360609 0.932717i $$-0.382569\pi$$
0.360609 + 0.932717i $$0.382569\pi$$
$$770$$ 0 0
$$771$$ − 18.0000i − 0.648254i
$$772$$ 0 0
$$773$$ 15.5563i 0.559523i 0.960070 + 0.279761i $$0.0902554\pi$$
−0.960070 + 0.279761i $$0.909745\pi$$
$$774$$ 0 0
$$775$$ −21.2132 −0.762001
$$776$$ 0 0
$$777$$ −12.0000 −0.430498
$$778$$ 0 0
$$779$$ 24.0000i 0.859889i
$$780$$ 0 0
$$781$$ 28.2843i 1.01209i
$$782$$ 0 0
$$783$$ −9.89949 −0.353779
$$784$$ 0 0
$$785$$ −20.0000 −0.713831
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 0 0
$$789$$ − 28.2843i − 1.00695i
$$790$$ 0 0
$$791$$ −8.48528 −0.301702
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 2.00000i 0.0709327i
$$796$$ 0 0
$$797$$ − 1.41421i − 0.0500940i −0.999686 0.0250470i $$-0.992026\pi$$
0.999686 0.0250470i $$-0.00797354\pi$$
$$798$$ 0 0
$$799$$ 5.65685 0.200125
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ 0 0
$$803$$ 16.0000i 0.564628i
$$804$$ 0 0
$$805$$ 5.65685i 0.199378i
$$806$$ 0 0
$$807$$ −21.2132 −0.746740
$$808$$ 0 0
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ 28.0000i 0.983213i 0.870817 + 0.491606i $$0.163590\pi$$
−0.870817 + 0.491606i $$0.836410\pi$$
$$812$$ 0 0
$$813$$ 24.0416i 0.843177i
$$814$$ 0 0
$$815$$ −22.6274 −0.792604
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 9.89949i 0.345495i 0.984966 + 0.172747i $$0.0552644\pi$$
−0.984966 + 0.172747i $$0.944736\pi$$
$$822$$ 0 0
$$823$$ 21.2132 0.739446 0.369723 0.929142i $$-0.379453\pi$$
0.369723 + 0.929142i $$0.379453\pi$$
$$824$$ 0 0
$$825$$ 6.00000 0.208893
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ 0 0
$$829$$ 11.3137i 0.392941i 0.980510 + 0.196471i $$0.0629480\pi$$
−0.980510 + 0.196471i $$0.937052\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10.0000 −0.346479
$$834$$ 0 0
$$835$$ 16.0000i 0.553703i
$$836$$ 0 0
$$837$$ − 7.07107i − 0.244412i
$$838$$ 0 0
$$839$$ −14.1421 −0.488241 −0.244120 0.969745i $$-0.578499\pi$$
−0.244120 + 0.969745i $$0.578499\pi$$
$$840$$ 0 0
$$841$$ −69.0000 −2.37931
$$842$$ 0 0
$$843$$ − 2.00000i − 0.0688837i
$$844$$ 0 0
$$845$$ 18.3848i 0.632456i
$$846$$ 0 0
$$847$$ −9.89949 −0.340151
$$848$$ 0 0
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ − 24.0000i − 0.822709i
$$852$$ 0 0
$$853$$ − 14.1421i − 0.484218i −0.970249 0.242109i $$-0.922161\pi$$
0.970249 0.242109i $$-0.0778391\pi$$
$$854$$ 0 0
$$855$$ −5.65685 −0.193460
$$856$$ 0 0
$$857$$ 26.0000 0.888143 0.444072 0.895991i $$-0.353534\pi$$
0.444072 + 0.895991i $$0.353534\pi$$
$$858$$ 0 0
$$859$$ 8.00000i 0.272956i 0.990643 + 0.136478i $$0.0435784\pi$$
−0.990643 + 0.136478i $$0.956422\pi$$
$$860$$ 0 0
$$861$$ − 8.48528i − 0.289178i
$$862$$ 0 0
$$863$$ 22.6274 0.770246 0.385123 0.922865i $$-0.374159\pi$$
0.385123 + 0.922865i $$0.374159\pi$$
$$864$$ 0 0
$$865$$ 22.0000 0.748022
$$866$$ 0 0
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ − 8.48528i − 0.287843i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ − 16.0000i − 0.540899i
$$876$$ 0 0
$$877$$ 42.4264i 1.43264i 0.697773 + 0.716319i $$0.254174\pi$$
−0.697773 + 0.716319i $$0.745826\pi$$
$$878$$ 0 0
$$879$$ −9.89949 −0.333902
$$880$$ 0 0
$$881$$ −22.0000 −0.741199 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$882$$ 0 0
$$883$$ − 48.0000i − 1.61533i −0.589643 0.807664i $$-0.700731\pi$$
0.589643 0.807664i $$-0.299269\pi$$
$$884$$ 0 0
$$885$$ 16.9706i 0.570459i
$$886$$ 0 0
$$887$$ −50.9117 −1.70945 −0.854724 0.519083i $$-0.826273\pi$$
−0.854724 + 0.519083i $$0.826273\pi$$
$$888$$ 0 0
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ 2.00000i 0.0670025i
$$892$$ 0 0
$$893$$ − 11.3137i − 0.378599i
$$894$$ 0 0
$$895$$ −5.65685 −0.189088
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 70.0000i − 2.33463i
$$900$$ 0 0
$$901$$ 2.82843i 0.0942286i
$$902$$ 0 0
$$903$$ −11.3137 −0.376497
$$904$$ 0 0
$$905$$ 16.0000 0.531858
$$906$$ 0 0
$$907$$ 8.00000i 0.265636i 0.991140 + 0.132818i $$0.0424025\pi$$
−0.991140 + 0.132818i $$0.957597\pi$$
$$908$$ 0 0
$$909$$ − 9.89949i − 0.328346i
$$910$$ 0 0
$$911$$ 5.65685 0.187420 0.0937100 0.995600i $$-0.470127\pi$$
0.0937100 + 0.995600i $$0.470127\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 20.0000i 0.661180i
$$916$$ 0 0
$$917$$ 16.9706i 0.560417i
$$918$$ 0 0
$$919$$ 21.2132 0.699759 0.349880 0.936795i $$-0.386223\pi$$
0.349880 + 0.936795i $$0.386223\pi$$
$$920$$ 0 0
$$921$$ 32.0000 1.05444
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 25.4558i 0.836983i
$$926$$ 0 0
$$927$$ 12.7279 0.418040
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 20.0000i 0.655474i
$$932$$ 0 0
$$933$$ 22.6274i 0.740788i
$$934$$ 0 0
$$935$$ −5.65685 −0.184999
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ 0 0
$$939$$ − 4.00000i − 0.130535i
$$940$$ 0 0
$$941$$ 1.41421i 0.0461020i 0.999734 + 0.0230510i $$0.00733802\pi$$
−0.999734 + 0.0230510i $$0.992662\pi$$
$$942$$ 0 0
$$943$$ 16.9706 0.552638
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −9.89949 −0.321013
$$952$$ 0 0
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ 32.0000i 1.03550i
$$956$$ 0 0
$$957$$ 19.7990i 0.640010i
$$958$$ 0 0
$$959$$ −14.1421 −0.456673
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ 0 0
$$965$$ − 5.65685i − 0.182101i
$$966$$ 0 0
$$967$$ 52.3259 1.68269 0.841344 0.540500i $$-0.181765\pi$$
0.841344 + 0.540500i $$0.181765\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ − 22.0000i − 0.706014i −0.935621 0.353007i $$-0.885159\pi$$
0.935621 0.353007i $$-0.114841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 50.0000 1.59964 0.799821 0.600239i $$-0.204928\pi$$
0.799821 + 0.600239i $$0.204928\pi$$
$$978$$ 0 0
$$979$$ − 4.00000i − 0.127841i
$$980$$ 0 0
$$981$$ 11.3137i 0.361219i
$$982$$ 0 0
$$983$$ 50.9117 1.62383 0.811915 0.583775i $$-0.198425\pi$$
0.811915 + 0.583775i $$0.198425\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 0 0
$$987$$ 4.00000i 0.127321i
$$988$$ 0 0
$$989$$ − 22.6274i − 0.719510i
$$990$$ 0 0
$$991$$ 46.6690 1.48249 0.741246 0.671234i $$-0.234235\pi$$
0.741246 + 0.671234i $$0.234235\pi$$
$$992$$ 0 0
$$993$$ −32.0000 −1.01549
$$994$$ 0 0
$$995$$ − 34.0000i − 1.07787i
$$996$$ 0 0
$$997$$ 31.1127i 0.985349i 0.870214 + 0.492675i $$0.163981\pi$$
−0.870214 + 0.492675i $$0.836019\pi$$
$$998$$ 0 0
$$999$$ −8.48528 −0.268462
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 1536.2.d.d.769.2 4
3.2 odd 2 4608.2.d.g.2305.1 4
4.3 odd 2 inner 1536.2.d.d.769.4 4
8.3 odd 2 inner 1536.2.d.d.769.1 4
8.5 even 2 inner 1536.2.d.d.769.3 4
12.11 even 2 4608.2.d.g.2305.2 4
16.3 odd 4 1536.2.a.c.1.2 yes 2
16.5 even 4 1536.2.a.c.1.1 2
16.11 odd 4 1536.2.a.j.1.1 yes 2
16.13 even 4 1536.2.a.j.1.2 yes 2
24.5 odd 2 4608.2.d.g.2305.3 4
24.11 even 2 4608.2.d.g.2305.4 4
48.5 odd 4 4608.2.a.j.1.2 2
48.11 even 4 4608.2.a.h.1.2 2
48.29 odd 4 4608.2.a.h.1.1 2
48.35 even 4 4608.2.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.c.1.1 2 16.5 even 4
1536.2.a.c.1.2 yes 2 16.3 odd 4
1536.2.a.j.1.1 yes 2 16.11 odd 4
1536.2.a.j.1.2 yes 2 16.13 even 4
1536.2.d.d.769.1 4 8.3 odd 2 inner
1536.2.d.d.769.2 4 1.1 even 1 trivial
1536.2.d.d.769.3 4 8.5 even 2 inner
1536.2.d.d.769.4 4 4.3 odd 2 inner
4608.2.a.h.1.1 2 48.29 odd 4
4608.2.a.h.1.2 2 48.11 even 4
4608.2.a.j.1.1 2 48.35 even 4
4608.2.a.j.1.2 2 48.5 odd 4
4608.2.d.g.2305.1 4 3.2 odd 2
4608.2.d.g.2305.2 4 12.11 even 2
4608.2.d.g.2305.3 4 24.5 odd 2
4608.2.d.g.2305.4 4 24.11 even 2