# Properties

 Label 9075.2.a.bc.1.1 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -5.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +0.618034 q^{12} +3.23607 q^{13} +8.47214 q^{14} -4.85410 q^{16} +2.00000 q^{17} -1.61803 q^{18} -5.00000 q^{19} -5.23607 q^{21} -4.61803 q^{23} +2.23607 q^{24} -5.23607 q^{26} +1.00000 q^{27} -3.23607 q^{28} -0.854102 q^{29} +7.00000 q^{31} +3.38197 q^{32} -3.23607 q^{34} +0.618034 q^{36} -1.47214 q^{37} +8.09017 q^{38} +3.23607 q^{39} -11.4721 q^{41} +8.47214 q^{42} +9.09017 q^{43} +7.47214 q^{46} +6.09017 q^{47} -4.85410 q^{48} +20.4164 q^{49} +2.00000 q^{51} +2.00000 q^{52} +5.38197 q^{53} -1.61803 q^{54} -11.7082 q^{56} -5.00000 q^{57} +1.38197 q^{58} +6.70820 q^{59} -7.00000 q^{61} -11.3262 q^{62} -5.23607 q^{63} +4.23607 q^{64} +11.6180 q^{67} +1.23607 q^{68} -4.61803 q^{69} -8.00000 q^{71} +2.23607 q^{72} -4.52786 q^{73} +2.38197 q^{74} -3.09017 q^{76} -5.23607 q^{78} -3.09017 q^{79} +1.00000 q^{81} +18.5623 q^{82} +7.70820 q^{83} -3.23607 q^{84} -14.7082 q^{86} -0.854102 q^{87} +4.14590 q^{89} -16.9443 q^{91} -2.85410 q^{92} +7.00000 q^{93} -9.85410 q^{94} +3.38197 q^{96} -2.52786 q^{97} -33.0344 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - 6 * q^7 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} - q^{4} - q^{6} - 6 q^{7} + 2 q^{9} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} + 4 q^{17} - q^{18} - 10 q^{19} - 6 q^{21} - 7 q^{23} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 5 q^{29} + 14 q^{31} + 9 q^{32} - 2 q^{34} - q^{36} + 6 q^{37} + 5 q^{38} + 2 q^{39} - 14 q^{41} + 8 q^{42} + 7 q^{43} + 6 q^{46} + q^{47} - 3 q^{48} + 14 q^{49} + 4 q^{51} + 4 q^{52} + 13 q^{53} - q^{54} - 10 q^{56} - 10 q^{57} + 5 q^{58} - 14 q^{61} - 7 q^{62} - 6 q^{63} + 4 q^{64} + 21 q^{67} - 2 q^{68} - 7 q^{69} - 16 q^{71} - 18 q^{73} + 7 q^{74} + 5 q^{76} - 6 q^{78} + 5 q^{79} + 2 q^{81} + 17 q^{82} + 2 q^{83} - 2 q^{84} - 16 q^{86} + 5 q^{87} + 15 q^{89} - 16 q^{91} + q^{92} + 14 q^{93} - 13 q^{94} + 9 q^{96} - 14 q^{97} - 37 q^{98}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 - q^6 - 6 * q^7 + 2 * q^9 - q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 + 4 * q^17 - q^18 - 10 * q^19 - 6 * q^21 - 7 * q^23 - 6 * q^26 + 2 * q^27 - 2 * q^28 + 5 * q^29 + 14 * q^31 + 9 * q^32 - 2 * q^34 - q^36 + 6 * q^37 + 5 * q^38 + 2 * q^39 - 14 * q^41 + 8 * q^42 + 7 * q^43 + 6 * q^46 + q^47 - 3 * q^48 + 14 * q^49 + 4 * q^51 + 4 * q^52 + 13 * q^53 - q^54 - 10 * q^56 - 10 * q^57 + 5 * q^58 - 14 * q^61 - 7 * q^62 - 6 * q^63 + 4 * q^64 + 21 * q^67 - 2 * q^68 - 7 * q^69 - 16 * q^71 - 18 * q^73 + 7 * q^74 + 5 * q^76 - 6 * q^78 + 5 * q^79 + 2 * q^81 + 17 * q^82 + 2 * q^83 - 2 * q^84 - 16 * q^86 + 5 * q^87 + 15 * q^89 - 16 * q^91 + q^92 + 14 * q^93 - 13 * q^94 + 9 * q^96 - 14 * q^97 - 37 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.61803 −1.14412 −0.572061 0.820211i $$-0.693856\pi$$
−0.572061 + 0.820211i $$0.693856\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ −1.61803 −0.660560
$$7$$ −5.23607 −1.97905 −0.989524 0.144370i $$-0.953885\pi$$
−0.989524 + 0.144370i $$0.953885\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0.618034 0.178411
$$13$$ 3.23607 0.897524 0.448762 0.893651i $$-0.351865\pi$$
0.448762 + 0.893651i $$0.351865\pi$$
$$14$$ 8.47214 2.26427
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ −1.61803 −0.381374
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ −5.23607 −1.14260
$$22$$ 0 0
$$23$$ −4.61803 −0.962927 −0.481463 0.876466i $$-0.659895\pi$$
−0.481463 + 0.876466i $$0.659895\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −5.23607 −1.02688
$$27$$ 1.00000 0.192450
$$28$$ −3.23607 −0.611559
$$29$$ −0.854102 −0.158603 −0.0793014 0.996851i $$-0.525269\pi$$
−0.0793014 + 0.996851i $$0.525269\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 3.38197 0.597853
$$33$$ 0 0
$$34$$ −3.23607 −0.554981
$$35$$ 0 0
$$36$$ 0.618034 0.103006
$$37$$ −1.47214 −0.242018 −0.121009 0.992651i $$-0.538613\pi$$
−0.121009 + 0.992651i $$0.538613\pi$$
$$38$$ 8.09017 1.31240
$$39$$ 3.23607 0.518186
$$40$$ 0 0
$$41$$ −11.4721 −1.79165 −0.895823 0.444410i $$-0.853413\pi$$
−0.895823 + 0.444410i $$0.853413\pi$$
$$42$$ 8.47214 1.30728
$$43$$ 9.09017 1.38624 0.693119 0.720823i $$-0.256236\pi$$
0.693119 + 0.720823i $$0.256236\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 7.47214 1.10171
$$47$$ 6.09017 0.888343 0.444171 0.895942i $$-0.353498\pi$$
0.444171 + 0.895942i $$0.353498\pi$$
$$48$$ −4.85410 −0.700629
$$49$$ 20.4164 2.91663
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 2.00000 0.277350
$$53$$ 5.38197 0.739270 0.369635 0.929177i $$-0.379483\pi$$
0.369635 + 0.929177i $$0.379483\pi$$
$$54$$ −1.61803 −0.220187
$$55$$ 0 0
$$56$$ −11.7082 −1.56457
$$57$$ −5.00000 −0.662266
$$58$$ 1.38197 0.181461
$$59$$ 6.70820 0.873334 0.436667 0.899623i $$-0.356159\pi$$
0.436667 + 0.899623i $$0.356159\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ −11.3262 −1.43843
$$63$$ −5.23607 −0.659683
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.6180 1.41937 0.709684 0.704520i $$-0.248838\pi$$
0.709684 + 0.704520i $$0.248838\pi$$
$$68$$ 1.23607 0.149895
$$69$$ −4.61803 −0.555946
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 2.23607 0.263523
$$73$$ −4.52786 −0.529946 −0.264973 0.964256i $$-0.585363\pi$$
−0.264973 + 0.964256i $$0.585363\pi$$
$$74$$ 2.38197 0.276898
$$75$$ 0 0
$$76$$ −3.09017 −0.354467
$$77$$ 0 0
$$78$$ −5.23607 −0.592868
$$79$$ −3.09017 −0.347671 −0.173836 0.984775i $$-0.555616\pi$$
−0.173836 + 0.984775i $$0.555616\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 18.5623 2.04986
$$83$$ 7.70820 0.846085 0.423043 0.906110i $$-0.360962\pi$$
0.423043 + 0.906110i $$0.360962\pi$$
$$84$$ −3.23607 −0.353084
$$85$$ 0 0
$$86$$ −14.7082 −1.58603
$$87$$ −0.854102 −0.0915693
$$88$$ 0 0
$$89$$ 4.14590 0.439464 0.219732 0.975560i $$-0.429482\pi$$
0.219732 + 0.975560i $$0.429482\pi$$
$$90$$ 0 0
$$91$$ −16.9443 −1.77624
$$92$$ −2.85410 −0.297561
$$93$$ 7.00000 0.725866
$$94$$ −9.85410 −1.01637
$$95$$ 0 0
$$96$$ 3.38197 0.345170
$$97$$ −2.52786 −0.256666 −0.128333 0.991731i $$-0.540963\pi$$
−0.128333 + 0.991731i $$0.540963\pi$$
$$98$$ −33.0344 −3.33698
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −19.5623 −1.94652 −0.973261 0.229702i $$-0.926225\pi$$
−0.973261 + 0.229702i $$0.926225\pi$$
$$102$$ −3.23607 −0.320418
$$103$$ −4.61803 −0.455028 −0.227514 0.973775i $$-0.573060\pi$$
−0.227514 + 0.973775i $$0.573060\pi$$
$$104$$ 7.23607 0.709555
$$105$$ 0 0
$$106$$ −8.70820 −0.845816
$$107$$ −3.85410 −0.372590 −0.186295 0.982494i $$-0.559648\pi$$
−0.186295 + 0.982494i $$0.559648\pi$$
$$108$$ 0.618034 0.0594703
$$109$$ −10.8541 −1.03963 −0.519817 0.854278i $$-0.674000\pi$$
−0.519817 + 0.854278i $$0.674000\pi$$
$$110$$ 0 0
$$111$$ −1.47214 −0.139729
$$112$$ 25.4164 2.40162
$$113$$ 3.47214 0.326631 0.163316 0.986574i $$-0.447781\pi$$
0.163316 + 0.986574i $$0.447781\pi$$
$$114$$ 8.09017 0.757714
$$115$$ 0 0
$$116$$ −0.527864 −0.0490109
$$117$$ 3.23607 0.299175
$$118$$ −10.8541 −0.999201
$$119$$ −10.4721 −0.959979
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 11.3262 1.02543
$$123$$ −11.4721 −1.03441
$$124$$ 4.32624 0.388508
$$125$$ 0 0
$$126$$ 8.47214 0.754758
$$127$$ −1.94427 −0.172526 −0.0862631 0.996272i $$-0.527493\pi$$
−0.0862631 + 0.996272i $$0.527493\pi$$
$$128$$ −13.6180 −1.20368
$$129$$ 9.09017 0.800345
$$130$$ 0 0
$$131$$ 9.18034 0.802090 0.401045 0.916058i $$-0.368647\pi$$
0.401045 + 0.916058i $$0.368647\pi$$
$$132$$ 0 0
$$133$$ 26.1803 2.27012
$$134$$ −18.7984 −1.62393
$$135$$ 0 0
$$136$$ 4.47214 0.383482
$$137$$ −5.29180 −0.452109 −0.226054 0.974115i $$-0.572583\pi$$
−0.226054 + 0.974115i $$0.572583\pi$$
$$138$$ 7.47214 0.636070
$$139$$ 1.18034 0.100115 0.0500576 0.998746i $$-0.484060\pi$$
0.0500576 + 0.998746i $$0.484060\pi$$
$$140$$ 0 0
$$141$$ 6.09017 0.512885
$$142$$ 12.9443 1.08626
$$143$$ 0 0
$$144$$ −4.85410 −0.404508
$$145$$ 0 0
$$146$$ 7.32624 0.606324
$$147$$ 20.4164 1.68392
$$148$$ −0.909830 −0.0747876
$$149$$ 20.1246 1.64867 0.824336 0.566101i $$-0.191549\pi$$
0.824336 + 0.566101i $$0.191549\pi$$
$$150$$ 0 0
$$151$$ 15.2361 1.23989 0.619947 0.784644i $$-0.287154\pi$$
0.619947 + 0.784644i $$0.287154\pi$$
$$152$$ −11.1803 −0.906845
$$153$$ 2.00000 0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −12.3262 −0.983741 −0.491870 0.870668i $$-0.663687\pi$$
−0.491870 + 0.870668i $$0.663687\pi$$
$$158$$ 5.00000 0.397779
$$159$$ 5.38197 0.426818
$$160$$ 0 0
$$161$$ 24.1803 1.90568
$$162$$ −1.61803 −0.127125
$$163$$ 6.56231 0.513999 0.257000 0.966411i $$-0.417266\pi$$
0.257000 + 0.966411i $$0.417266\pi$$
$$164$$ −7.09017 −0.553649
$$165$$ 0 0
$$166$$ −12.4721 −0.968025
$$167$$ 9.03444 0.699106 0.349553 0.936917i $$-0.386333\pi$$
0.349553 + 0.936917i $$0.386333\pi$$
$$168$$ −11.7082 −0.903308
$$169$$ −2.52786 −0.194451
$$170$$ 0 0
$$171$$ −5.00000 −0.382360
$$172$$ 5.61803 0.428371
$$173$$ −16.5623 −1.25921 −0.629604 0.776916i $$-0.716783\pi$$
−0.629604 + 0.776916i $$0.716783\pi$$
$$174$$ 1.38197 0.104767
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.70820 0.504219
$$178$$ −6.70820 −0.502801
$$179$$ −10.5279 −0.786890 −0.393445 0.919348i $$-0.628717\pi$$
−0.393445 + 0.919348i $$0.628717\pi$$
$$180$$ 0 0
$$181$$ −0.763932 −0.0567826 −0.0283913 0.999597i $$-0.509038\pi$$
−0.0283913 + 0.999597i $$0.509038\pi$$
$$182$$ 27.4164 2.03224
$$183$$ −7.00000 −0.517455
$$184$$ −10.3262 −0.761260
$$185$$ 0 0
$$186$$ −11.3262 −0.830480
$$187$$ 0 0
$$188$$ 3.76393 0.274513
$$189$$ −5.23607 −0.380868
$$190$$ 0 0
$$191$$ −16.4164 −1.18785 −0.593925 0.804521i $$-0.702422\pi$$
−0.593925 + 0.804521i $$0.702422\pi$$
$$192$$ 4.23607 0.305712
$$193$$ 8.03444 0.578332 0.289166 0.957279i $$-0.406622\pi$$
0.289166 + 0.957279i $$0.406622\pi$$
$$194$$ 4.09017 0.293657
$$195$$ 0 0
$$196$$ 12.6180 0.901288
$$197$$ 9.76393 0.695651 0.347826 0.937559i $$-0.386920\pi$$
0.347826 + 0.937559i $$0.386920\pi$$
$$198$$ 0 0
$$199$$ −4.79837 −0.340148 −0.170074 0.985431i $$-0.554401\pi$$
−0.170074 + 0.985431i $$0.554401\pi$$
$$200$$ 0 0
$$201$$ 11.6180 0.819473
$$202$$ 31.6525 2.22706
$$203$$ 4.47214 0.313882
$$204$$ 1.23607 0.0865421
$$205$$ 0 0
$$206$$ 7.47214 0.520608
$$207$$ −4.61803 −0.320976
$$208$$ −15.7082 −1.08917
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2.85410 −0.196484 −0.0982422 0.995163i $$-0.531322\pi$$
−0.0982422 + 0.995163i $$0.531322\pi$$
$$212$$ 3.32624 0.228447
$$213$$ −8.00000 −0.548151
$$214$$ 6.23607 0.426289
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ −36.6525 −2.48813
$$218$$ 17.5623 1.18947
$$219$$ −4.52786 −0.305965
$$220$$ 0 0
$$221$$ 6.47214 0.435363
$$222$$ 2.38197 0.159867
$$223$$ 21.0344 1.40857 0.704285 0.709917i $$-0.251268\pi$$
0.704285 + 0.709917i $$0.251268\pi$$
$$224$$ −17.7082 −1.18318
$$225$$ 0 0
$$226$$ −5.61803 −0.373706
$$227$$ 8.05573 0.534677 0.267339 0.963603i $$-0.413856\pi$$
0.267339 + 0.963603i $$0.413856\pi$$
$$228$$ −3.09017 −0.204652
$$229$$ 6.90983 0.456614 0.228307 0.973589i $$-0.426681\pi$$
0.228307 + 0.973589i $$0.426681\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.90983 −0.125386
$$233$$ 10.2705 0.672843 0.336422 0.941711i $$-0.390783\pi$$
0.336422 + 0.941711i $$0.390783\pi$$
$$234$$ −5.23607 −0.342292
$$235$$ 0 0
$$236$$ 4.14590 0.269875
$$237$$ −3.09017 −0.200728
$$238$$ 16.9443 1.09833
$$239$$ −22.0344 −1.42529 −0.712645 0.701525i $$-0.752503\pi$$
−0.712645 + 0.701525i $$0.752503\pi$$
$$240$$ 0 0
$$241$$ −0.618034 −0.0398111 −0.0199055 0.999802i $$-0.506337\pi$$
−0.0199055 + 0.999802i $$0.506337\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −4.32624 −0.276959
$$245$$ 0 0
$$246$$ 18.5623 1.18349
$$247$$ −16.1803 −1.02953
$$248$$ 15.6525 0.993933
$$249$$ 7.70820 0.488488
$$250$$ 0 0
$$251$$ 16.4721 1.03971 0.519856 0.854254i $$-0.325986\pi$$
0.519856 + 0.854254i $$0.325986\pi$$
$$252$$ −3.23607 −0.203853
$$253$$ 0 0
$$254$$ 3.14590 0.197391
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −9.56231 −0.596480 −0.298240 0.954491i $$-0.596400\pi$$
−0.298240 + 0.954491i $$0.596400\pi$$
$$258$$ −14.7082 −0.915693
$$259$$ 7.70820 0.478964
$$260$$ 0 0
$$261$$ −0.854102 −0.0528676
$$262$$ −14.8541 −0.917689
$$263$$ 24.2148 1.49315 0.746574 0.665303i $$-0.231698\pi$$
0.746574 + 0.665303i $$0.231698\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −42.3607 −2.59730
$$267$$ 4.14590 0.253725
$$268$$ 7.18034 0.438609
$$269$$ 0.854102 0.0520755 0.0260378 0.999661i $$-0.491711\pi$$
0.0260378 + 0.999661i $$0.491711\pi$$
$$270$$ 0 0
$$271$$ −18.3820 −1.11662 −0.558312 0.829631i $$-0.688551\pi$$
−0.558312 + 0.829631i $$0.688551\pi$$
$$272$$ −9.70820 −0.588646
$$273$$ −16.9443 −1.02551
$$274$$ 8.56231 0.517268
$$275$$ 0 0
$$276$$ −2.85410 −0.171797
$$277$$ 22.6525 1.36106 0.680528 0.732722i $$-0.261751\pi$$
0.680528 + 0.732722i $$0.261751\pi$$
$$278$$ −1.90983 −0.114544
$$279$$ 7.00000 0.419079
$$280$$ 0 0
$$281$$ 1.09017 0.0650341 0.0325170 0.999471i $$-0.489648\pi$$
0.0325170 + 0.999471i $$0.489648\pi$$
$$282$$ −9.85410 −0.586803
$$283$$ −16.0344 −0.953149 −0.476574 0.879134i $$-0.658122\pi$$
−0.476574 + 0.879134i $$0.658122\pi$$
$$284$$ −4.94427 −0.293389
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 60.0689 3.54575
$$288$$ 3.38197 0.199284
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −2.52786 −0.148186
$$292$$ −2.79837 −0.163762
$$293$$ 25.4721 1.48810 0.744049 0.668125i $$-0.232903\pi$$
0.744049 + 0.668125i $$0.232903\pi$$
$$294$$ −33.0344 −1.92661
$$295$$ 0 0
$$296$$ −3.29180 −0.191332
$$297$$ 0 0
$$298$$ −32.5623 −1.88628
$$299$$ −14.9443 −0.864250
$$300$$ 0 0
$$301$$ −47.5967 −2.74343
$$302$$ −24.6525 −1.41859
$$303$$ −19.5623 −1.12383
$$304$$ 24.2705 1.39201
$$305$$ 0 0
$$306$$ −3.23607 −0.184994
$$307$$ −28.1246 −1.60516 −0.802578 0.596547i $$-0.796539\pi$$
−0.802578 + 0.596547i $$0.796539\pi$$
$$308$$ 0 0
$$309$$ −4.61803 −0.262711
$$310$$ 0 0
$$311$$ 12.5279 0.710390 0.355195 0.934792i $$-0.384414\pi$$
0.355195 + 0.934792i $$0.384414\pi$$
$$312$$ 7.23607 0.409662
$$313$$ 8.47214 0.478873 0.239437 0.970912i $$-0.423037\pi$$
0.239437 + 0.970912i $$0.423037\pi$$
$$314$$ 19.9443 1.12552
$$315$$ 0 0
$$316$$ −1.90983 −0.107436
$$317$$ −3.58359 −0.201275 −0.100637 0.994923i $$-0.532088\pi$$
−0.100637 + 0.994923i $$0.532088\pi$$
$$318$$ −8.70820 −0.488332
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −3.85410 −0.215115
$$322$$ −39.1246 −2.18033
$$323$$ −10.0000 −0.556415
$$324$$ 0.618034 0.0343352
$$325$$ 0 0
$$326$$ −10.6180 −0.588079
$$327$$ −10.8541 −0.600233
$$328$$ −25.6525 −1.41642
$$329$$ −31.8885 −1.75807
$$330$$ 0 0
$$331$$ −27.5967 −1.51685 −0.758427 0.651758i $$-0.774032\pi$$
−0.758427 + 0.651758i $$0.774032\pi$$
$$332$$ 4.76393 0.261455
$$333$$ −1.47214 −0.0806726
$$334$$ −14.6180 −0.799863
$$335$$ 0 0
$$336$$ 25.4164 1.38658
$$337$$ −28.8541 −1.57178 −0.785892 0.618364i $$-0.787796\pi$$
−0.785892 + 0.618364i $$0.787796\pi$$
$$338$$ 4.09017 0.222476
$$339$$ 3.47214 0.188581
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 8.09017 0.437466
$$343$$ −70.2492 −3.79310
$$344$$ 20.3262 1.09592
$$345$$ 0 0
$$346$$ 26.7984 1.44069
$$347$$ 28.1803 1.51280 0.756400 0.654109i $$-0.226956\pi$$
0.756400 + 0.654109i $$0.226956\pi$$
$$348$$ −0.527864 −0.0282965
$$349$$ −32.2361 −1.72556 −0.862779 0.505582i $$-0.831278\pi$$
−0.862779 + 0.505582i $$0.831278\pi$$
$$350$$ 0 0
$$351$$ 3.23607 0.172729
$$352$$ 0 0
$$353$$ −17.8328 −0.949145 −0.474573 0.880216i $$-0.657397\pi$$
−0.474573 + 0.880216i $$0.657397\pi$$
$$354$$ −10.8541 −0.576889
$$355$$ 0 0
$$356$$ 2.56231 0.135802
$$357$$ −10.4721 −0.554244
$$358$$ 17.0344 0.900298
$$359$$ −3.29180 −0.173734 −0.0868672 0.996220i $$-0.527686\pi$$
−0.0868672 + 0.996220i $$0.527686\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 1.23607 0.0649663
$$363$$ 0 0
$$364$$ −10.4721 −0.548889
$$365$$ 0 0
$$366$$ 11.3262 0.592032
$$367$$ −22.1246 −1.15490 −0.577448 0.816428i $$-0.695951\pi$$
−0.577448 + 0.816428i $$0.695951\pi$$
$$368$$ 22.4164 1.16854
$$369$$ −11.4721 −0.597216
$$370$$ 0 0
$$371$$ −28.1803 −1.46305
$$372$$ 4.32624 0.224305
$$373$$ −32.7426 −1.69535 −0.847675 0.530516i $$-0.821998\pi$$
−0.847675 + 0.530516i $$0.821998\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 13.6180 0.702296
$$377$$ −2.76393 −0.142350
$$378$$ 8.47214 0.435760
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ −1.94427 −0.0996081
$$382$$ 26.5623 1.35905
$$383$$ −13.3607 −0.682699 −0.341349 0.939936i $$-0.610884\pi$$
−0.341349 + 0.939936i $$0.610884\pi$$
$$384$$ −13.6180 −0.694942
$$385$$ 0 0
$$386$$ −13.0000 −0.661683
$$387$$ 9.09017 0.462079
$$388$$ −1.56231 −0.0793141
$$389$$ −21.0557 −1.06757 −0.533784 0.845621i $$-0.679230\pi$$
−0.533784 + 0.845621i $$0.679230\pi$$
$$390$$ 0 0
$$391$$ −9.23607 −0.467088
$$392$$ 45.6525 2.30580
$$393$$ 9.18034 0.463087
$$394$$ −15.7984 −0.795911
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.72949 −0.136989 −0.0684946 0.997651i $$-0.521820\pi$$
−0.0684946 + 0.997651i $$0.521820\pi$$
$$398$$ 7.76393 0.389171
$$399$$ 26.1803 1.31066
$$400$$ 0 0
$$401$$ −30.8885 −1.54250 −0.771250 0.636532i $$-0.780368\pi$$
−0.771250 + 0.636532i $$0.780368\pi$$
$$402$$ −18.7984 −0.937578
$$403$$ 22.6525 1.12840
$$404$$ −12.0902 −0.601508
$$405$$ 0 0
$$406$$ −7.23607 −0.359120
$$407$$ 0 0
$$408$$ 4.47214 0.221404
$$409$$ 26.3050 1.30070 0.650348 0.759636i $$-0.274623\pi$$
0.650348 + 0.759636i $$0.274623\pi$$
$$410$$ 0 0
$$411$$ −5.29180 −0.261025
$$412$$ −2.85410 −0.140612
$$413$$ −35.1246 −1.72837
$$414$$ 7.47214 0.367235
$$415$$ 0 0
$$416$$ 10.9443 0.536587
$$417$$ 1.18034 0.0578015
$$418$$ 0 0
$$419$$ −24.5967 −1.20163 −0.600815 0.799388i $$-0.705157\pi$$
−0.600815 + 0.799388i $$0.705157\pi$$
$$420$$ 0 0
$$421$$ −3.85410 −0.187837 −0.0939187 0.995580i $$-0.529939\pi$$
−0.0939187 + 0.995580i $$0.529939\pi$$
$$422$$ 4.61803 0.224802
$$423$$ 6.09017 0.296114
$$424$$ 12.0344 0.584444
$$425$$ 0 0
$$426$$ 12.9443 0.627152
$$427$$ 36.6525 1.77374
$$428$$ −2.38197 −0.115137
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.8885 0.765324 0.382662 0.923888i $$-0.375007\pi$$
0.382662 + 0.923888i $$0.375007\pi$$
$$432$$ −4.85410 −0.233543
$$433$$ 1.03444 0.0497121 0.0248561 0.999691i $$-0.492087\pi$$
0.0248561 + 0.999691i $$0.492087\pi$$
$$434$$ 59.3050 2.84673
$$435$$ 0 0
$$436$$ −6.70820 −0.321265
$$437$$ 23.0902 1.10455
$$438$$ 7.32624 0.350061
$$439$$ −25.3262 −1.20876 −0.604378 0.796698i $$-0.706578\pi$$
−0.604378 + 0.796698i $$0.706578\pi$$
$$440$$ 0 0
$$441$$ 20.4164 0.972210
$$442$$ −10.4721 −0.498109
$$443$$ 8.27051 0.392944 0.196472 0.980509i $$-0.437052\pi$$
0.196472 + 0.980509i $$0.437052\pi$$
$$444$$ −0.909830 −0.0431786
$$445$$ 0 0
$$446$$ −34.0344 −1.61158
$$447$$ 20.1246 0.951861
$$448$$ −22.1803 −1.04792
$$449$$ −34.2705 −1.61733 −0.808663 0.588273i $$-0.799808\pi$$
−0.808663 + 0.588273i $$0.799808\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 2.14590 0.100935
$$453$$ 15.2361 0.715853
$$454$$ −13.0344 −0.611737
$$455$$ 0 0
$$456$$ −11.1803 −0.523567
$$457$$ −7.47214 −0.349532 −0.174766 0.984610i $$-0.555917\pi$$
−0.174766 + 0.984610i $$0.555917\pi$$
$$458$$ −11.1803 −0.522423
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −8.05573 −0.375193 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$462$$ 0 0
$$463$$ −0.270510 −0.0125717 −0.00628583 0.999980i $$-0.502001\pi$$
−0.00628583 + 0.999980i $$0.502001\pi$$
$$464$$ 4.14590 0.192468
$$465$$ 0 0
$$466$$ −16.6180 −0.769816
$$467$$ 30.8885 1.42935 0.714676 0.699456i $$-0.246574\pi$$
0.714676 + 0.699456i $$0.246574\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −60.8328 −2.80900
$$470$$ 0 0
$$471$$ −12.3262 −0.567963
$$472$$ 15.0000 0.690431
$$473$$ 0 0
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ −6.47214 −0.296650
$$477$$ 5.38197 0.246423
$$478$$ 35.6525 1.63071
$$479$$ −6.58359 −0.300812 −0.150406 0.988624i $$-0.548058\pi$$
−0.150406 + 0.988624i $$0.548058\pi$$
$$480$$ 0 0
$$481$$ −4.76393 −0.217217
$$482$$ 1.00000 0.0455488
$$483$$ 24.1803 1.10024
$$484$$ 0 0
$$485$$ 0 0
$$486$$ −1.61803 −0.0733955
$$487$$ −8.18034 −0.370687 −0.185343 0.982674i $$-0.559340\pi$$
−0.185343 + 0.982674i $$0.559340\pi$$
$$488$$ −15.6525 −0.708554
$$489$$ 6.56231 0.296758
$$490$$ 0 0
$$491$$ −7.20163 −0.325005 −0.162502 0.986708i $$-0.551957\pi$$
−0.162502 + 0.986708i $$0.551957\pi$$
$$492$$ −7.09017 −0.319650
$$493$$ −1.70820 −0.0769336
$$494$$ 26.1803 1.17791
$$495$$ 0 0
$$496$$ −33.9787 −1.52569
$$497$$ 41.8885 1.87896
$$498$$ −12.4721 −0.558890
$$499$$ 11.1803 0.500501 0.250250 0.968181i $$-0.419487\pi$$
0.250250 + 0.968181i $$0.419487\pi$$
$$500$$ 0 0
$$501$$ 9.03444 0.403629
$$502$$ −26.6525 −1.18956
$$503$$ 13.8885 0.619260 0.309630 0.950857i $$-0.399795\pi$$
0.309630 + 0.950857i $$0.399795\pi$$
$$504$$ −11.7082 −0.521525
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.52786 −0.112266
$$508$$ −1.20163 −0.0533135
$$509$$ 9.47214 0.419845 0.209923 0.977718i $$-0.432679\pi$$
0.209923 + 0.977718i $$0.432679\pi$$
$$510$$ 0 0
$$511$$ 23.7082 1.04879
$$512$$ 5.29180 0.233867
$$513$$ −5.00000 −0.220755
$$514$$ 15.4721 0.682447
$$515$$ 0 0
$$516$$ 5.61803 0.247320
$$517$$ 0 0
$$518$$ −12.4721 −0.547994
$$519$$ −16.5623 −0.727005
$$520$$ 0 0
$$521$$ 29.3607 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$522$$ 1.38197 0.0604870
$$523$$ 3.76393 0.164585 0.0822926 0.996608i $$-0.473776\pi$$
0.0822926 + 0.996608i $$0.473776\pi$$
$$524$$ 5.67376 0.247859
$$525$$ 0 0
$$526$$ −39.1803 −1.70834
$$527$$ 14.0000 0.609850
$$528$$ 0 0
$$529$$ −1.67376 −0.0727723
$$530$$ 0 0
$$531$$ 6.70820 0.291111
$$532$$ 16.1803 0.701507
$$533$$ −37.1246 −1.60805
$$534$$ −6.70820 −0.290292
$$535$$ 0 0
$$536$$ 25.9787 1.12211
$$537$$ −10.5279 −0.454311
$$538$$ −1.38197 −0.0595808
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.61803 0.0695647 0.0347824 0.999395i $$-0.488926\pi$$
0.0347824 + 0.999395i $$0.488926\pi$$
$$542$$ 29.7426 1.27756
$$543$$ −0.763932 −0.0327835
$$544$$ 6.76393 0.290001
$$545$$ 0 0
$$546$$ 27.4164 1.17331
$$547$$ 19.3607 0.827803 0.413901 0.910322i $$-0.364166\pi$$
0.413901 + 0.910322i $$0.364166\pi$$
$$548$$ −3.27051 −0.139709
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ 4.27051 0.181930
$$552$$ −10.3262 −0.439514
$$553$$ 16.1803 0.688058
$$554$$ −36.6525 −1.55721
$$555$$ 0 0
$$556$$ 0.729490 0.0309373
$$557$$ −42.3951 −1.79634 −0.898169 0.439649i $$-0.855103\pi$$
−0.898169 + 0.439649i $$0.855103\pi$$
$$558$$ −11.3262 −0.479478
$$559$$ 29.4164 1.24418
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −1.76393 −0.0744070
$$563$$ 20.6738 0.871295 0.435648 0.900117i $$-0.356519\pi$$
0.435648 + 0.900117i $$0.356519\pi$$
$$564$$ 3.76393 0.158490
$$565$$ 0 0
$$566$$ 25.9443 1.09052
$$567$$ −5.23607 −0.219894
$$568$$ −17.8885 −0.750587
$$569$$ 22.5623 0.945861 0.472931 0.881100i $$-0.343196\pi$$
0.472931 + 0.881100i $$0.343196\pi$$
$$570$$ 0 0
$$571$$ 11.0902 0.464109 0.232055 0.972703i $$-0.425455\pi$$
0.232055 + 0.972703i $$0.425455\pi$$
$$572$$ 0 0
$$573$$ −16.4164 −0.685805
$$574$$ −97.1935 −4.05678
$$575$$ 0 0
$$576$$ 4.23607 0.176503
$$577$$ −30.0132 −1.24946 −0.624732 0.780839i $$-0.714792\pi$$
−0.624732 + 0.780839i $$0.714792\pi$$
$$578$$ 21.0344 0.874917
$$579$$ 8.03444 0.333900
$$580$$ 0 0
$$581$$ −40.3607 −1.67444
$$582$$ 4.09017 0.169543
$$583$$ 0 0
$$584$$ −10.1246 −0.418959
$$585$$ 0 0
$$586$$ −41.2148 −1.70257
$$587$$ 21.6180 0.892272 0.446136 0.894965i $$-0.352800\pi$$
0.446136 + 0.894965i $$0.352800\pi$$
$$588$$ 12.6180 0.520359
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 9.76393 0.401634
$$592$$ 7.14590 0.293695
$$593$$ 3.11146 0.127772 0.0638861 0.997957i $$-0.479651\pi$$
0.0638861 + 0.997957i $$0.479651\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12.4377 0.509468
$$597$$ −4.79837 −0.196384
$$598$$ 24.1803 0.988808
$$599$$ −17.7639 −0.725815 −0.362907 0.931825i $$-0.618216\pi$$
−0.362907 + 0.931825i $$0.618216\pi$$
$$600$$ 0 0
$$601$$ 10.3607 0.422621 0.211310 0.977419i $$-0.432227\pi$$
0.211310 + 0.977419i $$0.432227\pi$$
$$602$$ 77.0132 3.13882
$$603$$ 11.6180 0.473123
$$604$$ 9.41641 0.383148
$$605$$ 0 0
$$606$$ 31.6525 1.28579
$$607$$ −33.0000 −1.33943 −0.669714 0.742619i $$-0.733583\pi$$
−0.669714 + 0.742619i $$0.733583\pi$$
$$608$$ −16.9098 −0.685784
$$609$$ 4.47214 0.181220
$$610$$ 0 0
$$611$$ 19.7082 0.797309
$$612$$ 1.23607 0.0499651
$$613$$ 3.76393 0.152024 0.0760119 0.997107i $$-0.475781\pi$$
0.0760119 + 0.997107i $$0.475781\pi$$
$$614$$ 45.5066 1.83650
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.2492 −1.70089 −0.850445 0.526064i $$-0.823667\pi$$
−0.850445 + 0.526064i $$0.823667\pi$$
$$618$$ 7.47214 0.300573
$$619$$ 42.8885 1.72384 0.861918 0.507048i $$-0.169263\pi$$
0.861918 + 0.507048i $$0.169263\pi$$
$$620$$ 0 0
$$621$$ −4.61803 −0.185315
$$622$$ −20.2705 −0.812773
$$623$$ −21.7082 −0.869721
$$624$$ −15.7082 −0.628831
$$625$$ 0 0
$$626$$ −13.7082 −0.547890
$$627$$ 0 0
$$628$$ −7.61803 −0.303993
$$629$$ −2.94427 −0.117396
$$630$$ 0 0
$$631$$ −39.6312 −1.57769 −0.788846 0.614590i $$-0.789321\pi$$
−0.788846 + 0.614590i $$0.789321\pi$$
$$632$$ −6.90983 −0.274858
$$633$$ −2.85410 −0.113440
$$634$$ 5.79837 0.230283
$$635$$ 0 0
$$636$$ 3.32624 0.131894
$$637$$ 66.0689 2.61774
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 4.23607 0.167315 0.0836573 0.996495i $$-0.473340\pi$$
0.0836573 + 0.996495i $$0.473340\pi$$
$$642$$ 6.23607 0.246118
$$643$$ 13.6738 0.539241 0.269620 0.962967i $$-0.413102\pi$$
0.269620 + 0.962967i $$0.413102\pi$$
$$644$$ 14.9443 0.588887
$$645$$ 0 0
$$646$$ 16.1803 0.636607
$$647$$ −31.5967 −1.24220 −0.621098 0.783733i $$-0.713313\pi$$
−0.621098 + 0.783733i $$0.713313\pi$$
$$648$$ 2.23607 0.0878410
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −36.6525 −1.43652
$$652$$ 4.05573 0.158835
$$653$$ −43.5623 −1.70472 −0.852362 0.522952i $$-0.824831\pi$$
−0.852362 + 0.522952i $$0.824831\pi$$
$$654$$ 17.5623 0.686741
$$655$$ 0 0
$$656$$ 55.6869 2.17421
$$657$$ −4.52786 −0.176649
$$658$$ 51.5967 2.01145
$$659$$ −42.0344 −1.63743 −0.818715 0.574201i $$-0.805313\pi$$
−0.818715 + 0.574201i $$0.805313\pi$$
$$660$$ 0 0
$$661$$ 15.0902 0.586940 0.293470 0.955968i $$-0.405190\pi$$
0.293470 + 0.955968i $$0.405190\pi$$
$$662$$ 44.6525 1.73547
$$663$$ 6.47214 0.251357
$$664$$ 17.2361 0.668889
$$665$$ 0 0
$$666$$ 2.38197 0.0922993
$$667$$ 3.94427 0.152723
$$668$$ 5.58359 0.216036
$$669$$ 21.0344 0.813239
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −17.7082 −0.683109
$$673$$ −35.3050 −1.36091 −0.680453 0.732792i $$-0.738217\pi$$
−0.680453 + 0.732792i $$0.738217\pi$$
$$674$$ 46.6869 1.79831
$$675$$ 0 0
$$676$$ −1.56231 −0.0600887
$$677$$ 5.94427 0.228457 0.114228 0.993455i $$-0.463560\pi$$
0.114228 + 0.993455i $$0.463560\pi$$
$$678$$ −5.61803 −0.215759
$$679$$ 13.2361 0.507954
$$680$$ 0 0
$$681$$ 8.05573 0.308696
$$682$$ 0 0
$$683$$ 15.7082 0.601058 0.300529 0.953773i $$-0.402837\pi$$
0.300529 + 0.953773i $$0.402837\pi$$
$$684$$ −3.09017 −0.118156
$$685$$ 0 0
$$686$$ 113.666 4.33977
$$687$$ 6.90983 0.263626
$$688$$ −44.1246 −1.68224
$$689$$ 17.4164 0.663512
$$690$$ 0 0
$$691$$ −29.3050 −1.11481 −0.557406 0.830240i $$-0.688203\pi$$
−0.557406 + 0.830240i $$0.688203\pi$$
$$692$$ −10.2361 −0.389117
$$693$$ 0 0
$$694$$ −45.5967 −1.73083
$$695$$ 0 0
$$696$$ −1.90983 −0.0723919
$$697$$ −22.9443 −0.869076
$$698$$ 52.1591 1.97425
$$699$$ 10.2705 0.388466
$$700$$ 0 0
$$701$$ −40.4164 −1.52651 −0.763253 0.646099i $$-0.776399\pi$$
−0.763253 + 0.646099i $$0.776399\pi$$
$$702$$ −5.23607 −0.197623
$$703$$ 7.36068 0.277613
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 28.8541 1.08594
$$707$$ 102.430 3.85226
$$708$$ 4.14590 0.155812
$$709$$ 18.8197 0.706787 0.353394 0.935475i $$-0.385028\pi$$
0.353394 + 0.935475i $$0.385028\pi$$
$$710$$ 0 0
$$711$$ −3.09017 −0.115890
$$712$$ 9.27051 0.347427
$$713$$ −32.3262 −1.21063
$$714$$ 16.9443 0.634123
$$715$$ 0 0
$$716$$ −6.50658 −0.243162
$$717$$ −22.0344 −0.822891
$$718$$ 5.32624 0.198773
$$719$$ 26.9098 1.00357 0.501784 0.864993i $$-0.332677\pi$$
0.501784 + 0.864993i $$0.332677\pi$$
$$720$$ 0 0
$$721$$ 24.1803 0.900523
$$722$$ −9.70820 −0.361302
$$723$$ −0.618034 −0.0229849
$$724$$ −0.472136 −0.0175468
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −21.6738 −0.803835 −0.401918 0.915676i $$-0.631656\pi$$
−0.401918 + 0.915676i $$0.631656\pi$$
$$728$$ −37.8885 −1.40424
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 18.1803 0.672424
$$732$$ −4.32624 −0.159902
$$733$$ −4.85410 −0.179290 −0.0896452 0.995974i $$-0.528573\pi$$
−0.0896452 + 0.995974i $$0.528573\pi$$
$$734$$ 35.7984 1.32134
$$735$$ 0 0
$$736$$ −15.6180 −0.575688
$$737$$ 0 0
$$738$$ 18.5623 0.683288
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −16.1803 −0.594400
$$742$$ 45.5967 1.67391
$$743$$ 31.6525 1.16122 0.580608 0.814183i $$-0.302815\pi$$
0.580608 + 0.814183i $$0.302815\pi$$
$$744$$ 15.6525 0.573848
$$745$$ 0 0
$$746$$ 52.9787 1.93969
$$747$$ 7.70820 0.282028
$$748$$ 0 0
$$749$$ 20.1803 0.737374
$$750$$ 0 0
$$751$$ −10.1115 −0.368972 −0.184486 0.982835i $$-0.559062\pi$$
−0.184486 + 0.982835i $$0.559062\pi$$
$$752$$ −29.5623 −1.07803
$$753$$ 16.4721 0.600278
$$754$$ 4.47214 0.162866
$$755$$ 0 0
$$756$$ −3.23607 −0.117695
$$757$$ 1.81966 0.0661367 0.0330683 0.999453i $$-0.489472\pi$$
0.0330683 + 0.999453i $$0.489472\pi$$
$$758$$ 8.09017 0.293848
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.96556 −0.361251 −0.180626 0.983552i $$-0.557812\pi$$
−0.180626 + 0.983552i $$0.557812\pi$$
$$762$$ 3.14590 0.113964
$$763$$ 56.8328 2.05749
$$764$$ −10.1459 −0.367066
$$765$$ 0 0
$$766$$ 21.6180 0.781091
$$767$$ 21.7082 0.783838
$$768$$ 13.5623 0.489388
$$769$$ −17.2361 −0.621549 −0.310774 0.950484i $$-0.600588\pi$$
−0.310774 + 0.950484i $$0.600588\pi$$
$$770$$ 0 0
$$771$$ −9.56231 −0.344378
$$772$$ 4.96556 0.178714
$$773$$ −27.5066 −0.989343 −0.494671 0.869080i $$-0.664712\pi$$
−0.494671 + 0.869080i $$0.664712\pi$$
$$774$$ −14.7082 −0.528675
$$775$$ 0 0
$$776$$ −5.65248 −0.202912
$$777$$ 7.70820 0.276530
$$778$$ 34.0689 1.22143
$$779$$ 57.3607 2.05516
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 14.9443 0.534406
$$783$$ −0.854102 −0.0305231
$$784$$ −99.1033 −3.53940
$$785$$ 0 0
$$786$$ −14.8541 −0.529828
$$787$$ −32.7984 −1.16914 −0.584568 0.811345i $$-0.698736\pi$$
−0.584568 + 0.811345i $$0.698736\pi$$
$$788$$ 6.03444 0.214968
$$789$$ 24.2148 0.862069
$$790$$ 0 0
$$791$$ −18.1803 −0.646418
$$792$$ 0 0
$$793$$ −22.6525 −0.804413
$$794$$ 4.41641 0.156732
$$795$$ 0 0
$$796$$ −2.96556 −0.105111
$$797$$ −15.2918 −0.541663 −0.270832 0.962627i $$-0.587299\pi$$
−0.270832 + 0.962627i $$0.587299\pi$$
$$798$$ −42.3607 −1.49955
$$799$$ 12.1803 0.430909
$$800$$ 0 0
$$801$$ 4.14590 0.146488
$$802$$ 49.9787 1.76481
$$803$$ 0 0
$$804$$ 7.18034 0.253231
$$805$$ 0 0
$$806$$ −36.6525 −1.29103
$$807$$ 0.854102 0.0300658
$$808$$ −43.7426 −1.53886
$$809$$ −29.6738 −1.04327 −0.521637 0.853168i $$-0.674678\pi$$
−0.521637 + 0.853168i $$0.674678\pi$$
$$810$$ 0 0
$$811$$ −43.6312 −1.53210 −0.766049 0.642782i $$-0.777780\pi$$
−0.766049 + 0.642782i $$0.777780\pi$$
$$812$$ 2.76393 0.0969950
$$813$$ −18.3820 −0.644684
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −9.70820 −0.339855
$$817$$ −45.4508 −1.59012
$$818$$ −42.5623 −1.48816
$$819$$ −16.9443 −0.592081
$$820$$ 0 0
$$821$$ −20.8197 −0.726611 −0.363306 0.931670i $$-0.618352\pi$$
−0.363306 + 0.931670i $$0.618352\pi$$
$$822$$ 8.56231 0.298645
$$823$$ −16.5279 −0.576125 −0.288063 0.957612i $$-0.593011\pi$$
−0.288063 + 0.957612i $$0.593011\pi$$
$$824$$ −10.3262 −0.359732
$$825$$ 0 0
$$826$$ 56.8328 1.97747
$$827$$ −51.4164 −1.78792 −0.893962 0.448143i $$-0.852086\pi$$
−0.893962 + 0.448143i $$0.852086\pi$$
$$828$$ −2.85410 −0.0991869
$$829$$ −32.8885 −1.14227 −0.571133 0.820857i $$-0.693496\pi$$
−0.571133 + 0.820857i $$0.693496\pi$$
$$830$$ 0 0
$$831$$ 22.6525 0.785806
$$832$$ 13.7082 0.475246
$$833$$ 40.8328 1.41477
$$834$$ −1.90983 −0.0661320
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 7.00000 0.241955
$$838$$ 39.7984 1.37481
$$839$$ 36.3050 1.25339 0.626693 0.779266i $$-0.284408\pi$$
0.626693 + 0.779266i $$0.284408\pi$$
$$840$$ 0 0
$$841$$ −28.2705 −0.974845
$$842$$ 6.23607 0.214909
$$843$$ 1.09017 0.0375474
$$844$$ −1.76393 −0.0607170
$$845$$ 0 0
$$846$$ −9.85410 −0.338791
$$847$$ 0 0
$$848$$ −26.1246 −0.897123
$$849$$ −16.0344 −0.550301
$$850$$ 0 0
$$851$$ 6.79837 0.233045
$$852$$ −4.94427 −0.169388
$$853$$ 23.3607 0.799854 0.399927 0.916547i $$-0.369035\pi$$
0.399927 + 0.916547i $$0.369035\pi$$
$$854$$ −59.3050 −2.02937
$$855$$ 0 0
$$856$$ −8.61803 −0.294558
$$857$$ −41.0132 −1.40098 −0.700491 0.713661i $$-0.747036\pi$$
−0.700491 + 0.713661i $$0.747036\pi$$
$$858$$ 0 0
$$859$$ −39.2705 −1.33989 −0.669946 0.742410i $$-0.733683\pi$$
−0.669946 + 0.742410i $$0.733683\pi$$
$$860$$ 0 0
$$861$$ 60.0689 2.04714
$$862$$ −25.7082 −0.875625
$$863$$ −23.0344 −0.784102 −0.392051 0.919944i $$-0.628234\pi$$
−0.392051 + 0.919944i $$0.628234\pi$$
$$864$$ 3.38197 0.115057
$$865$$ 0 0
$$866$$ −1.67376 −0.0568768
$$867$$ −13.0000 −0.441503
$$868$$ −22.6525 −0.768875
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 37.5967 1.27392
$$872$$ −24.2705 −0.821903
$$873$$ −2.52786 −0.0855552
$$874$$ −37.3607 −1.26374
$$875$$ 0 0
$$876$$ −2.79837 −0.0945483
$$877$$ 7.52786 0.254198 0.127099 0.991890i $$-0.459433\pi$$
0.127099 + 0.991890i $$0.459433\pi$$
$$878$$ 40.9787 1.38296
$$879$$ 25.4721 0.859154
$$880$$ 0 0
$$881$$ −17.1459 −0.577660 −0.288830 0.957380i $$-0.593266\pi$$
−0.288830 + 0.957380i $$0.593266\pi$$
$$882$$ −33.0344 −1.11233
$$883$$ −23.2361 −0.781956 −0.390978 0.920400i $$-0.627863\pi$$
−0.390978 + 0.920400i $$0.627863\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −13.3820 −0.449576
$$887$$ 0.167184 0.00561350 0.00280675 0.999996i $$-0.499107\pi$$
0.00280675 + 0.999996i $$0.499107\pi$$
$$888$$ −3.29180 −0.110465
$$889$$ 10.1803 0.341438
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.0000 0.435272
$$893$$ −30.4508 −1.01900
$$894$$ −32.5623 −1.08905
$$895$$ 0 0
$$896$$ 71.3050 2.38213
$$897$$ −14.9443 −0.498975
$$898$$ 55.4508 1.85042
$$899$$ −5.97871 −0.199401
$$900$$ 0 0
$$901$$ 10.7639 0.358599
$$902$$ 0 0
$$903$$ −47.5967 −1.58392
$$904$$ 7.76393 0.258225
$$905$$ 0 0
$$906$$ −24.6525 −0.819024
$$907$$ −44.8885 −1.49050 −0.745250 0.666785i $$-0.767670\pi$$
−0.745250 + 0.666785i $$0.767670\pi$$
$$908$$ 4.97871 0.165224
$$909$$ −19.5623 −0.648841
$$910$$ 0 0
$$911$$ 52.9017 1.75271 0.876356 0.481664i $$-0.159968\pi$$
0.876356 + 0.481664i $$0.159968\pi$$
$$912$$ 24.2705 0.803677
$$913$$ 0 0
$$914$$ 12.0902 0.399907
$$915$$ 0 0
$$916$$ 4.27051 0.141102
$$917$$ −48.0689 −1.58737
$$918$$ −3.23607 −0.106806
$$919$$ 22.2361 0.733500 0.366750 0.930319i $$-0.380470\pi$$
0.366750 + 0.930319i $$0.380470\pi$$
$$920$$ 0 0
$$921$$ −28.1246 −0.926737
$$922$$ 13.0344 0.429266
$$923$$ −25.8885 −0.852132
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0.437694 0.0143835
$$927$$ −4.61803 −0.151676
$$928$$ −2.88854 −0.0948211
$$929$$ −10.8541 −0.356112 −0.178056 0.984020i $$-0.556981\pi$$
−0.178056 + 0.984020i $$0.556981\pi$$
$$930$$ 0 0
$$931$$ −102.082 −3.34560
$$932$$ 6.34752 0.207920
$$933$$ 12.5279 0.410144
$$934$$ −49.9787 −1.63535
$$935$$ 0 0
$$936$$ 7.23607 0.236518
$$937$$ 17.2016 0.561953 0.280976 0.959715i $$-0.409342\pi$$
0.280976 + 0.959715i $$0.409342\pi$$
$$938$$ 98.4296 3.21384
$$939$$ 8.47214 0.276478
$$940$$ 0 0
$$941$$ 9.70820 0.316478 0.158239 0.987401i $$-0.449418\pi$$
0.158239 + 0.987401i $$0.449418\pi$$
$$942$$ 19.9443 0.649819
$$943$$ 52.9787 1.72522
$$944$$ −32.5623 −1.05981
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.56231 −0.148255 −0.0741275 0.997249i $$-0.523617\pi$$
−0.0741275 + 0.997249i $$0.523617\pi$$
$$948$$ −1.90983 −0.0620284
$$949$$ −14.6525 −0.475639
$$950$$ 0 0
$$951$$ −3.58359 −0.116206
$$952$$ −23.4164 −0.758930
$$953$$ 49.0902 1.59019 0.795093 0.606487i $$-0.207422\pi$$
0.795093 + 0.606487i $$0.207422\pi$$
$$954$$ −8.70820 −0.281939
$$955$$ 0 0
$$956$$ −13.6180 −0.440439
$$957$$ 0 0
$$958$$ 10.6525 0.344166
$$959$$ 27.7082 0.894745
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 7.70820 0.248522
$$963$$ −3.85410 −0.124197
$$964$$ −0.381966 −0.0123023
$$965$$ 0 0
$$966$$ −39.1246 −1.25881
$$967$$ −32.6738 −1.05072 −0.525359 0.850881i $$-0.676069\pi$$
−0.525359 + 0.850881i $$0.676069\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ −31.5410 −1.01220 −0.506100 0.862475i $$-0.668913\pi$$
−0.506100 + 0.862475i $$0.668913\pi$$
$$972$$ 0.618034 0.0198234
$$973$$ −6.18034 −0.198133
$$974$$ 13.2361 0.424111
$$975$$ 0 0
$$976$$ 33.9787 1.08763
$$977$$ 46.0902 1.47456 0.737278 0.675590i $$-0.236111\pi$$
0.737278 + 0.675590i $$0.236111\pi$$
$$978$$ −10.6180 −0.339527
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.8541 −0.346545
$$982$$ 11.6525 0.371845
$$983$$ 19.0000 0.606006 0.303003 0.952990i $$-0.402011\pi$$
0.303003 + 0.952990i $$0.402011\pi$$
$$984$$ −25.6525 −0.817771
$$985$$ 0 0
$$986$$ 2.76393 0.0880215
$$987$$ −31.8885 −1.01502
$$988$$ −10.0000 −0.318142
$$989$$ −41.9787 −1.33485
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 23.6738 0.751643
$$993$$ −27.5967 −0.875756
$$994$$ −67.7771 −2.14976
$$995$$ 0 0
$$996$$ 4.76393 0.150951
$$997$$ −11.2918 −0.357615 −0.178807 0.983884i $$-0.557224\pi$$
−0.178807 + 0.983884i $$0.557224\pi$$
$$998$$ −18.0902 −0.572634
$$999$$ −1.47214 −0.0465763
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 9075.2.a.bc.1.1 2
5.4 even 2 9075.2.a.bt.1.2 2
11.7 odd 10 825.2.n.d.676.1 yes 4
11.8 odd 10 825.2.n.d.526.1 yes 4
11.10 odd 2 9075.2.a.by.1.2 2
55.7 even 20 825.2.bx.c.49.1 8
55.8 even 20 825.2.bx.c.724.1 8
55.18 even 20 825.2.bx.c.49.2 8
55.19 odd 10 825.2.n.b.526.1 4
55.29 odd 10 825.2.n.b.676.1 yes 4
55.52 even 20 825.2.bx.c.724.2 8
55.54 odd 2 9075.2.a.z.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.n.b.526.1 4 55.19 odd 10
825.2.n.b.676.1 yes 4 55.29 odd 10
825.2.n.d.526.1 yes 4 11.8 odd 10
825.2.n.d.676.1 yes 4 11.7 odd 10
825.2.bx.c.49.1 8 55.7 even 20
825.2.bx.c.49.2 8 55.18 even 20
825.2.bx.c.724.1 8 55.8 even 20
825.2.bx.c.724.2 8 55.52 even 20
9075.2.a.z.1.1 2 55.54 odd 2
9075.2.a.bc.1.1 2 1.1 even 1 trivial
9075.2.a.bt.1.2 2 5.4 even 2
9075.2.a.by.1.2 2 11.10 odd 2