# Properties

 Label 8550.2.a.ck.1.3 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.12489$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +4.12489 q^{7} -1.00000 q^{8} +2.64002 q^{11} +2.51514 q^{13} -4.12489 q^{14} +1.00000 q^{16} +0.515138 q^{17} +1.00000 q^{19} -2.64002 q^{22} -3.09461 q^{23} -2.51514 q^{26} +4.12489 q^{28} +7.79518 q^{29} +3.67030 q^{31} -1.00000 q^{32} -0.515138 q^{34} +10.2498 q^{37} -1.00000 q^{38} -8.88979 q^{41} +8.64002 q^{43} +2.64002 q^{44} +3.09461 q^{46} +4.96972 q^{47} +10.0147 q^{49} +2.51514 q^{52} -5.48486 q^{53} -4.12489 q^{56} -7.79518 q^{58} -3.15516 q^{59} -12.6400 q^{61} -3.67030 q^{62} +1.00000 q^{64} -7.40493 q^{67} +0.515138 q^{68} -11.1396 q^{71} -2.70436 q^{73} -10.2498 q^{74} +1.00000 q^{76} +10.8898 q^{77} +16.7493 q^{79} +8.88979 q^{82} -3.28005 q^{83} -8.64002 q^{86} -2.64002 q^{88} +7.60975 q^{89} +10.3747 q^{91} -3.09461 q^{92} -4.96972 q^{94} +3.93945 q^{97} -10.0147 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8} + 8 q^{13} - 4 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{26} + 4 q^{28} + 8 q^{29} + 4 q^{31} - 3 q^{32} - 2 q^{34} + 14 q^{37} - 3 q^{38} - 2 q^{41} + 18 q^{43} + 14 q^{47} - 3 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{56} - 8 q^{58} - 2 q^{59} - 30 q^{61} - 4 q^{62} + 3 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} + 10 q^{73} - 14 q^{74} + 3 q^{76} + 8 q^{77} + 2 q^{82} + 6 q^{83} - 18 q^{86} + 14 q^{89} + 6 q^{91} - 14 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 4 * q^7 - 3 * q^8 + 8 * q^13 - 4 * q^14 + 3 * q^16 + 2 * q^17 + 3 * q^19 - 8 * q^26 + 4 * q^28 + 8 * q^29 + 4 * q^31 - 3 * q^32 - 2 * q^34 + 14 * q^37 - 3 * q^38 - 2 * q^41 + 18 * q^43 + 14 * q^47 - 3 * q^49 + 8 * q^52 - 16 * q^53 - 4 * q^56 - 8 * q^58 - 2 * q^59 - 30 * q^61 - 4 * q^62 + 3 * q^64 + 2 * q^67 + 2 * q^68 + 8 * q^71 + 10 * q^73 - 14 * q^74 + 3 * q^76 + 8 * q^77 + 2 * q^82 + 6 * q^83 - 18 * q^86 + 14 * q^89 + 6 * q^91 - 14 * q^94 + 10 * q^97 + 3 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.12489 1.55906 0.779530 0.626365i $$-0.215458\pi$$
0.779530 + 0.626365i $$0.215458\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.64002 0.795997 0.397999 0.917386i $$-0.369705\pi$$
0.397999 + 0.917386i $$0.369705\pi$$
$$12$$ 0 0
$$13$$ 2.51514 0.697574 0.348787 0.937202i $$-0.386594\pi$$
0.348787 + 0.937202i $$0.386594\pi$$
$$14$$ −4.12489 −1.10242
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.515138 0.124939 0.0624697 0.998047i $$-0.480102\pi$$
0.0624697 + 0.998047i $$0.480102\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.64002 −0.562855
$$23$$ −3.09461 −0.645271 −0.322635 0.946523i $$-0.604569\pi$$
−0.322635 + 0.946523i $$0.604569\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.51514 −0.493259
$$27$$ 0 0
$$28$$ 4.12489 0.779530
$$29$$ 7.79518 1.44753 0.723765 0.690047i $$-0.242410\pi$$
0.723765 + 0.690047i $$0.242410\pi$$
$$30$$ 0 0
$$31$$ 3.67030 0.659205 0.329603 0.944120i $$-0.393085\pi$$
0.329603 + 0.944120i $$0.393085\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −0.515138 −0.0883454
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.2498 1.68505 0.842526 0.538656i $$-0.181068\pi$$
0.842526 + 0.538656i $$0.181068\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.88979 −1.38835 −0.694176 0.719805i $$-0.744231\pi$$
−0.694176 + 0.719805i $$0.744231\pi$$
$$42$$ 0 0
$$43$$ 8.64002 1.31759 0.658796 0.752322i $$-0.271066\pi$$
0.658796 + 0.752322i $$0.271066\pi$$
$$44$$ 2.64002 0.397999
$$45$$ 0 0
$$46$$ 3.09461 0.456275
$$47$$ 4.96972 0.724909 0.362454 0.932002i $$-0.381939\pi$$
0.362454 + 0.932002i $$0.381939\pi$$
$$48$$ 0 0
$$49$$ 10.0147 1.43067
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.51514 0.348787
$$53$$ −5.48486 −0.753404 −0.376702 0.926335i $$-0.622942\pi$$
−0.376702 + 0.926335i $$0.622942\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −4.12489 −0.551211
$$57$$ 0 0
$$58$$ −7.79518 −1.02356
$$59$$ −3.15516 −0.410767 −0.205384 0.978682i $$-0.565844\pi$$
−0.205384 + 0.978682i $$0.565844\pi$$
$$60$$ 0 0
$$61$$ −12.6400 −1.61839 −0.809195 0.587541i $$-0.800096\pi$$
−0.809195 + 0.587541i $$0.800096\pi$$
$$62$$ −3.67030 −0.466129
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.40493 −0.904656 −0.452328 0.891852i $$-0.649406\pi$$
−0.452328 + 0.891852i $$0.649406\pi$$
$$68$$ 0.515138 0.0624697
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −11.1396 −1.32202 −0.661012 0.750376i $$-0.729873\pi$$
−0.661012 + 0.750376i $$0.729873\pi$$
$$72$$ 0 0
$$73$$ −2.70436 −0.316521 −0.158261 0.987397i $$-0.550589\pi$$
−0.158261 + 0.987397i $$0.550589\pi$$
$$74$$ −10.2498 −1.19151
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 10.8898 1.24101
$$78$$ 0 0
$$79$$ 16.7493 1.88444 0.942222 0.334988i $$-0.108732\pi$$
0.942222 + 0.334988i $$0.108732\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 8.88979 0.981714
$$83$$ −3.28005 −0.360032 −0.180016 0.983664i $$-0.557615\pi$$
−0.180016 + 0.983664i $$0.557615\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.64002 −0.931678
$$87$$ 0 0
$$88$$ −2.64002 −0.281427
$$89$$ 7.60975 0.806632 0.403316 0.915061i $$-0.367858\pi$$
0.403316 + 0.915061i $$0.367858\pi$$
$$90$$ 0 0
$$91$$ 10.3747 1.08756
$$92$$ −3.09461 −0.322635
$$93$$ 0 0
$$94$$ −4.96972 −0.512588
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.93945 0.399990 0.199995 0.979797i $$-0.435907\pi$$
0.199995 + 0.979797i $$0.435907\pi$$
$$98$$ −10.0147 −1.01164
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.7990 1.37305 0.686524 0.727107i $$-0.259136\pi$$
0.686524 + 0.727107i $$0.259136\pi$$
$$102$$ 0 0
$$103$$ 4.57947 0.451229 0.225614 0.974217i $$-0.427561\pi$$
0.225614 + 0.974217i $$0.427561\pi$$
$$104$$ −2.51514 −0.246630
$$105$$ 0 0
$$106$$ 5.48486 0.532737
$$107$$ 10.3747 1.00296 0.501478 0.865170i $$-0.332790\pi$$
0.501478 + 0.865170i $$0.332790\pi$$
$$108$$ 0 0
$$109$$ 3.01468 0.288754 0.144377 0.989523i $$-0.453882\pi$$
0.144377 + 0.989523i $$0.453882\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.12489 0.389765
$$113$$ 19.2001 1.80620 0.903098 0.429435i $$-0.141287\pi$$
0.903098 + 0.429435i $$0.141287\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.79518 0.723765
$$117$$ 0 0
$$118$$ 3.15516 0.290456
$$119$$ 2.12489 0.194788
$$120$$ 0 0
$$121$$ −4.03028 −0.366389
$$122$$ 12.6400 1.14437
$$123$$ 0 0
$$124$$ 3.67030 0.329603
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 14.3103 1.26984 0.634918 0.772580i $$-0.281034\pi$$
0.634918 + 0.772580i $$0.281034\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.56009 0.573158 0.286579 0.958057i $$-0.407482\pi$$
0.286579 + 0.958057i $$0.407482\pi$$
$$132$$ 0 0
$$133$$ 4.12489 0.357673
$$134$$ 7.40493 0.639689
$$135$$ 0 0
$$136$$ −0.515138 −0.0441727
$$137$$ 6.45459 0.551452 0.275726 0.961236i $$-0.411082\pi$$
0.275726 + 0.961236i $$0.411082\pi$$
$$138$$ 0 0
$$139$$ −23.0596 −1.95589 −0.977946 0.208856i $$-0.933026\pi$$
−0.977946 + 0.208856i $$0.933026\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 11.1396 0.934812
$$143$$ 6.64002 0.555267
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2.70436 0.223814
$$147$$ 0 0
$$148$$ 10.2498 0.842526
$$149$$ −16.0294 −1.31318 −0.656588 0.754249i $$-0.728001\pi$$
−0.656588 + 0.754249i $$0.728001\pi$$
$$150$$ 0 0
$$151$$ −14.3103 −1.16456 −0.582279 0.812989i $$-0.697839\pi$$
−0.582279 + 0.812989i $$0.697839\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ 0 0
$$154$$ −10.8898 −0.877525
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0294 −1.43890 −0.719450 0.694544i $$-0.755606\pi$$
−0.719450 + 0.694544i $$0.755606\pi$$
$$158$$ −16.7493 −1.33250
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.7649 −1.00602
$$162$$ 0 0
$$163$$ 2.70058 0.211525 0.105763 0.994391i $$-0.466272\pi$$
0.105763 + 0.994391i $$0.466272\pi$$
$$164$$ −8.88979 −0.694176
$$165$$ 0 0
$$166$$ 3.28005 0.254581
$$167$$ −8.95035 −0.692599 −0.346299 0.938124i $$-0.612562\pi$$
−0.346299 + 0.938124i $$0.612562\pi$$
$$168$$ 0 0
$$169$$ −6.67408 −0.513391
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.64002 0.658796
$$173$$ −0.310323 −0.0235934 −0.0117967 0.999930i $$-0.503755\pi$$
−0.0117967 + 0.999930i $$0.503755\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2.64002 0.198999
$$177$$ 0 0
$$178$$ −7.60975 −0.570375
$$179$$ 1.52982 0.114344 0.0571720 0.998364i $$-0.481792\pi$$
0.0571720 + 0.998364i $$0.481792\pi$$
$$180$$ 0 0
$$181$$ −14.7493 −1.09631 −0.548154 0.836377i $$-0.684669\pi$$
−0.548154 + 0.836377i $$0.684669\pi$$
$$182$$ −10.3747 −0.769021
$$183$$ 0 0
$$184$$ 3.09461 0.228138
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.35998 0.0994513
$$188$$ 4.96972 0.362454
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.1249 −1.31147 −0.655735 0.754991i $$-0.727641\pi$$
−0.655735 + 0.754991i $$0.727641\pi$$
$$192$$ 0 0
$$193$$ −4.56009 −0.328243 −0.164121 0.986440i $$-0.552479\pi$$
−0.164121 + 0.986440i $$0.552479\pi$$
$$194$$ −3.93945 −0.282836
$$195$$ 0 0
$$196$$ 10.0147 0.715334
$$197$$ 2.14048 0.152503 0.0762515 0.997089i $$-0.475705\pi$$
0.0762515 + 0.997089i $$0.475705\pi$$
$$198$$ 0 0
$$199$$ 16.5639 1.17418 0.587091 0.809521i $$-0.300273\pi$$
0.587091 + 0.809521i $$0.300273\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −13.7990 −0.970892
$$203$$ 32.1542 2.25679
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.57947 −0.319067
$$207$$ 0 0
$$208$$ 2.51514 0.174393
$$209$$ 2.64002 0.182614
$$210$$ 0 0
$$211$$ −15.2838 −1.05218 −0.526091 0.850428i $$-0.676343\pi$$
−0.526091 + 0.850428i $$0.676343\pi$$
$$212$$ −5.48486 −0.376702
$$213$$ 0 0
$$214$$ −10.3747 −0.709197
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 15.1396 1.02774
$$218$$ −3.01468 −0.204180
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.29564 0.0871544
$$222$$ 0 0
$$223$$ −3.75023 −0.251134 −0.125567 0.992085i $$-0.540075\pi$$
−0.125567 + 0.992085i $$0.540075\pi$$
$$224$$ −4.12489 −0.275606
$$225$$ 0 0
$$226$$ −19.2001 −1.27717
$$227$$ 24.1542 1.60317 0.801587 0.597878i $$-0.203989\pi$$
0.801587 + 0.597878i $$0.203989\pi$$
$$228$$ 0 0
$$229$$ −13.0596 −0.863005 −0.431503 0.902112i $$-0.642016\pi$$
−0.431503 + 0.902112i $$0.642016\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −7.79518 −0.511779
$$233$$ 6.43899 0.421832 0.210916 0.977504i $$-0.432355\pi$$
0.210916 + 0.977504i $$0.432355\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −3.15516 −0.205384
$$237$$ 0 0
$$238$$ −2.12489 −0.137736
$$239$$ −22.8742 −1.47961 −0.739804 0.672822i $$-0.765082\pi$$
−0.739804 + 0.672822i $$0.765082\pi$$
$$240$$ 0 0
$$241$$ 4.96972 0.320128 0.160064 0.987107i $$-0.448830\pi$$
0.160064 + 0.987107i $$0.448830\pi$$
$$242$$ 4.03028 0.259076
$$243$$ 0 0
$$244$$ −12.6400 −0.809195
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.51514 0.160034
$$248$$ −3.67030 −0.233064
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −15.9201 −1.00487 −0.502433 0.864616i $$-0.667562\pi$$
−0.502433 + 0.864616i $$0.667562\pi$$
$$252$$ 0 0
$$253$$ −8.16984 −0.513634
$$254$$ −14.3103 −0.897910
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 1.04965 0.0654756 0.0327378 0.999464i $$-0.489577\pi$$
0.0327378 + 0.999464i $$0.489577\pi$$
$$258$$ 0 0
$$259$$ 42.2791 2.62710
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.56009 −0.405284
$$263$$ 11.9394 0.736218 0.368109 0.929783i $$-0.380005\pi$$
0.368109 + 0.929783i $$0.380005\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.12489 −0.252913
$$267$$ 0 0
$$268$$ −7.40493 −0.452328
$$269$$ 15.9394 0.971845 0.485923 0.874002i $$-0.338484\pi$$
0.485923 + 0.874002i $$0.338484\pi$$
$$270$$ 0 0
$$271$$ 1.46548 0.0890218 0.0445109 0.999009i $$-0.485827\pi$$
0.0445109 + 0.999009i $$0.485827\pi$$
$$272$$ 0.515138 0.0312348
$$273$$ 0 0
$$274$$ −6.45459 −0.389936
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 32.4995 1.95271 0.976354 0.216177i $$-0.0693590\pi$$
0.976354 + 0.216177i $$0.0693590\pi$$
$$278$$ 23.0596 1.38303
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.04965 −0.0626171 −0.0313085 0.999510i $$-0.509967\pi$$
−0.0313085 + 0.999510i $$0.509967\pi$$
$$282$$ 0 0
$$283$$ −9.34060 −0.555241 −0.277620 0.960691i $$-0.589546\pi$$
−0.277620 + 0.960691i $$0.589546\pi$$
$$284$$ −11.1396 −0.661012
$$285$$ 0 0
$$286$$ −6.64002 −0.392633
$$287$$ −36.6694 −2.16453
$$288$$ 0 0
$$289$$ −16.7346 −0.984390
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −2.70436 −0.158261
$$293$$ −25.5748 −1.49409 −0.747047 0.664771i $$-0.768529\pi$$
−0.747047 + 0.664771i $$0.768529\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.2498 −0.595756
$$297$$ 0 0
$$298$$ 16.0294 0.928556
$$299$$ −7.78337 −0.450124
$$300$$ 0 0
$$301$$ 35.6391 2.05420
$$302$$ 14.3103 0.823467
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9.43991 0.538764 0.269382 0.963033i $$-0.413181\pi$$
0.269382 + 0.963033i $$0.413181\pi$$
$$308$$ 10.8898 0.620504
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 17.4655 0.990377 0.495188 0.868786i $$-0.335099\pi$$
0.495188 + 0.868786i $$0.335099\pi$$
$$312$$ 0 0
$$313$$ −10.4546 −0.590928 −0.295464 0.955354i $$-0.595474\pi$$
−0.295464 + 0.955354i $$0.595474\pi$$
$$314$$ 18.0294 1.01746
$$315$$ 0 0
$$316$$ 16.7493 0.942222
$$317$$ −4.33348 −0.243393 −0.121696 0.992567i $$-0.538833\pi$$
−0.121696 + 0.992567i $$0.538833\pi$$
$$318$$ 0 0
$$319$$ 20.5795 1.15223
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.7649 0.711361
$$323$$ 0.515138 0.0286630
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −2.70058 −0.149571
$$327$$ 0 0
$$328$$ 8.88979 0.490857
$$329$$ 20.4995 1.13018
$$330$$ 0 0
$$331$$ −4.56387 −0.250853 −0.125427 0.992103i $$-0.540030\pi$$
−0.125427 + 0.992103i $$0.540030\pi$$
$$332$$ −3.28005 −0.180016
$$333$$ 0 0
$$334$$ 8.95035 0.489741
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 13.0109 0.708749 0.354374 0.935104i $$-0.384694\pi$$
0.354374 + 0.935104i $$0.384694\pi$$
$$338$$ 6.67408 0.363022
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 9.68968 0.524725
$$342$$ 0 0
$$343$$ 12.4352 0.671438
$$344$$ −8.64002 −0.465839
$$345$$ 0 0
$$346$$ 0.310323 0.0166831
$$347$$ −13.2195 −0.709660 −0.354830 0.934931i $$-0.615461\pi$$
−0.354830 + 0.934931i $$0.615461\pi$$
$$348$$ 0 0
$$349$$ −20.4390 −1.09407 −0.547037 0.837108i $$-0.684244\pi$$
−0.547037 + 0.837108i $$0.684244\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.64002 −0.140714
$$353$$ −10.7044 −0.569735 −0.284868 0.958567i $$-0.591950\pi$$
−0.284868 + 0.958567i $$0.591950\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 7.60975 0.403316
$$357$$ 0 0
$$358$$ −1.52982 −0.0808534
$$359$$ 4.31410 0.227690 0.113845 0.993499i $$-0.463683\pi$$
0.113845 + 0.993499i $$0.463683\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.7493 0.775207
$$363$$ 0 0
$$364$$ 10.3747 0.543780
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −12.8099 −0.668669 −0.334335 0.942454i $$-0.608512\pi$$
−0.334335 + 0.942454i $$0.608512\pi$$
$$368$$ −3.09461 −0.161318
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −22.6244 −1.17460
$$372$$ 0 0
$$373$$ 0.704357 0.0364702 0.0182351 0.999834i $$-0.494195\pi$$
0.0182351 + 0.999834i $$0.494195\pi$$
$$374$$ −1.35998 −0.0703227
$$375$$ 0 0
$$376$$ −4.96972 −0.256294
$$377$$ 19.6060 1.00976
$$378$$ 0 0
$$379$$ 6.12489 0.314614 0.157307 0.987550i $$-0.449719\pi$$
0.157307 + 0.987550i $$0.449719\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.1249 0.927350
$$383$$ −18.6400 −0.952461 −0.476230 0.879321i $$-0.657997\pi$$
−0.476230 + 0.879321i $$0.657997\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 4.56009 0.232103
$$387$$ 0 0
$$388$$ 3.93945 0.199995
$$389$$ −13.9201 −0.705776 −0.352888 0.935666i $$-0.614800\pi$$
−0.352888 + 0.935666i $$0.614800\pi$$
$$390$$ 0 0
$$391$$ −1.59415 −0.0806197
$$392$$ −10.0147 −0.505818
$$393$$ 0 0
$$394$$ −2.14048 −0.107836
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −28.3784 −1.42427 −0.712136 0.702041i $$-0.752272\pi$$
−0.712136 + 0.702041i $$0.752272\pi$$
$$398$$ −16.5639 −0.830272
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.54920 −0.277114 −0.138557 0.990354i $$-0.544246\pi$$
−0.138557 + 0.990354i $$0.544246\pi$$
$$402$$ 0 0
$$403$$ 9.23131 0.459844
$$404$$ 13.7990 0.686524
$$405$$ 0 0
$$406$$ −32.1542 −1.59579
$$407$$ 27.0596 1.34130
$$408$$ 0 0
$$409$$ −5.01090 −0.247773 −0.123886 0.992296i $$-0.539536\pi$$
−0.123886 + 0.992296i $$0.539536\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4.57947 0.225614
$$413$$ −13.0147 −0.640411
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.51514 −0.123315
$$417$$ 0 0
$$418$$ −2.64002 −0.129128
$$419$$ 15.8889 0.776222 0.388111 0.921613i $$-0.373128\pi$$
0.388111 + 0.921613i $$0.373128\pi$$
$$420$$ 0 0
$$421$$ −2.38647 −0.116310 −0.0581548 0.998308i $$-0.518522\pi$$
−0.0581548 + 0.998308i $$0.518522\pi$$
$$422$$ 15.2838 0.744005
$$423$$ 0 0
$$424$$ 5.48486 0.266368
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −52.1386 −2.52317
$$428$$ 10.3747 0.501478
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.35906 0.113632 0.0568160 0.998385i $$-0.481905\pi$$
0.0568160 + 0.998385i $$0.481905\pi$$
$$432$$ 0 0
$$433$$ −0.640023 −0.0307576 −0.0153788 0.999882i $$-0.504895\pi$$
−0.0153788 + 0.999882i $$0.504895\pi$$
$$434$$ −15.1396 −0.726722
$$435$$ 0 0
$$436$$ 3.01468 0.144377
$$437$$ −3.09461 −0.148035
$$438$$ 0 0
$$439$$ −25.4499 −1.21466 −0.607328 0.794451i $$-0.707759\pi$$
−0.607328 + 0.794451i $$0.707759\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −1.29564 −0.0616275
$$443$$ 35.6685 1.69466 0.847330 0.531067i $$-0.178209\pi$$
0.847330 + 0.531067i $$0.178209\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.75023 0.177578
$$447$$ 0 0
$$448$$ 4.12489 0.194883
$$449$$ 11.4399 0.539883 0.269941 0.962877i $$-0.412996\pi$$
0.269941 + 0.962877i $$0.412996\pi$$
$$450$$ 0 0
$$451$$ −23.4693 −1.10512
$$452$$ 19.2001 0.903098
$$453$$ 0 0
$$454$$ −24.1542 −1.13361
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 19.4849 0.911463 0.455732 0.890117i $$-0.349378\pi$$
0.455732 + 0.890117i $$0.349378\pi$$
$$458$$ 13.0596 0.610237
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 29.3893 1.36880 0.684399 0.729108i $$-0.260065\pi$$
0.684399 + 0.729108i $$0.260065\pi$$
$$462$$ 0 0
$$463$$ 12.1892 0.566481 0.283241 0.959049i $$-0.408591\pi$$
0.283241 + 0.959049i $$0.408591\pi$$
$$464$$ 7.79518 0.361882
$$465$$ 0 0
$$466$$ −6.43899 −0.298280
$$467$$ 17.8889 0.827799 0.413899 0.910323i $$-0.364167\pi$$
0.413899 + 0.910323i $$0.364167\pi$$
$$468$$ 0 0
$$469$$ −30.5445 −1.41041
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3.15516 0.145228
$$473$$ 22.8099 1.04880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2.12489 0.0973940
$$477$$ 0 0
$$478$$ 22.8742 1.04624
$$479$$ 1.15894 0.0529534 0.0264767 0.999649i $$-0.491571\pi$$
0.0264767 + 0.999649i $$0.491571\pi$$
$$480$$ 0 0
$$481$$ 25.7796 1.17545
$$482$$ −4.96972 −0.226365
$$483$$ 0 0
$$484$$ −4.03028 −0.183194
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 30.6888 1.39064 0.695320 0.718700i $$-0.255263\pi$$
0.695320 + 0.718700i $$0.255263\pi$$
$$488$$ 12.6400 0.572187
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −3.67030 −0.165638 −0.0828192 0.996565i $$-0.526392\pi$$
−0.0828192 + 0.996565i $$0.526392\pi$$
$$492$$ 0 0
$$493$$ 4.01560 0.180853
$$494$$ −2.51514 −0.113161
$$495$$ 0 0
$$496$$ 3.67030 0.164801
$$497$$ −45.9494 −2.06111
$$498$$ 0 0
$$499$$ 31.3893 1.40518 0.702590 0.711595i $$-0.252027\pi$$
0.702590 + 0.711595i $$0.252027\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 15.9201 0.710548
$$503$$ 18.1542 0.809458 0.404729 0.914437i $$-0.367366\pi$$
0.404729 + 0.914437i $$0.367366\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8.16984 0.363194
$$507$$ 0 0
$$508$$ 14.3103 0.634918
$$509$$ −11.5298 −0.511050 −0.255525 0.966802i $$-0.582248\pi$$
−0.255525 + 0.966802i $$0.582248\pi$$
$$510$$ 0 0
$$511$$ −11.1552 −0.493475
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −1.04965 −0.0462982
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 13.1202 0.577025
$$518$$ −42.2791 −1.85764
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 42.5895 1.86588 0.932939 0.360035i $$-0.117235\pi$$
0.932939 + 0.360035i $$0.117235\pi$$
$$522$$ 0 0
$$523$$ 1.21571 0.0531594 0.0265797 0.999647i $$-0.491538\pi$$
0.0265797 + 0.999647i $$0.491538\pi$$
$$524$$ 6.56009 0.286579
$$525$$ 0 0
$$526$$ −11.9394 −0.520585
$$527$$ 1.89071 0.0823607
$$528$$ 0 0
$$529$$ −13.4234 −0.583626
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.12489 0.178836
$$533$$ −22.3591 −0.968478
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.40493 0.319844
$$537$$ 0 0
$$538$$ −15.9394 −0.687198
$$539$$ 26.4390 1.13881
$$540$$ 0 0
$$541$$ −1.92007 −0.0825503 −0.0412751 0.999148i $$-0.513142\pi$$
−0.0412751 + 0.999148i $$0.513142\pi$$
$$542$$ −1.46548 −0.0629480
$$543$$ 0 0
$$544$$ −0.515138 −0.0220864
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10.9697 0.469032 0.234516 0.972112i $$-0.424650\pi$$
0.234516 + 0.972112i $$0.424650\pi$$
$$548$$ 6.45459 0.275726
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 7.79518 0.332086
$$552$$ 0 0
$$553$$ 69.0890 2.93796
$$554$$ −32.4995 −1.38077
$$555$$ 0 0
$$556$$ −23.0596 −0.977946
$$557$$ −23.5005 −0.995746 −0.497873 0.867250i $$-0.665886\pi$$
−0.497873 + 0.867250i $$0.665886\pi$$
$$558$$ 0 0
$$559$$ 21.7309 0.919117
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.04965 0.0442770
$$563$$ 7.87890 0.332056 0.166028 0.986121i $$-0.446906\pi$$
0.166028 + 0.986121i $$0.446906\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 9.34060 0.392615
$$567$$ 0 0
$$568$$ 11.1396 0.467406
$$569$$ 1.09083 0.0457299 0.0228650 0.999739i $$-0.492721\pi$$
0.0228650 + 0.999739i $$0.492721\pi$$
$$570$$ 0 0
$$571$$ 21.4886 0.899272 0.449636 0.893212i $$-0.351554\pi$$
0.449636 + 0.893212i $$0.351554\pi$$
$$572$$ 6.64002 0.277633
$$573$$ 0 0
$$574$$ 36.6694 1.53055
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −16.1055 −0.670481 −0.335241 0.942133i $$-0.608818\pi$$
−0.335241 + 0.942133i $$0.608818\pi$$
$$578$$ 16.7346 0.696069
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −13.5298 −0.561311
$$582$$ 0 0
$$583$$ −14.4802 −0.599707
$$584$$ 2.70436 0.111907
$$585$$ 0 0
$$586$$ 25.5748 1.05648
$$587$$ −0.480164 −0.0198185 −0.00990925 0.999951i $$-0.503154\pi$$
−0.00990925 + 0.999951i $$0.503154\pi$$
$$588$$ 0 0
$$589$$ 3.67030 0.151232
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 10.2498 0.421263
$$593$$ −40.4683 −1.66184 −0.830918 0.556395i $$-0.812184\pi$$
−0.830918 + 0.556395i $$0.812184\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −16.0294 −0.656588
$$597$$ 0 0
$$598$$ 7.78337 0.318286
$$599$$ 36.1698 1.47786 0.738930 0.673782i $$-0.235331\pi$$
0.738930 + 0.673782i $$0.235331\pi$$
$$600$$ 0 0
$$601$$ 24.3903 0.994899 0.497450 0.867493i $$-0.334270\pi$$
0.497450 + 0.867493i $$0.334270\pi$$
$$602$$ −35.6391 −1.45254
$$603$$ 0 0
$$604$$ −14.3103 −0.582279
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 21.6897 0.880357 0.440178 0.897910i $$-0.354915\pi$$
0.440178 + 0.897910i $$0.354915\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.4995 0.505677
$$612$$ 0 0
$$613$$ −26.2304 −1.05944 −0.529718 0.848174i $$-0.677702\pi$$
−0.529718 + 0.848174i $$0.677702\pi$$
$$614$$ −9.43991 −0.380964
$$615$$ 0 0
$$616$$ −10.8898 −0.438762
$$617$$ 22.7200 0.914671 0.457335 0.889294i $$-0.348804\pi$$
0.457335 + 0.889294i $$0.348804\pi$$
$$618$$ 0 0
$$619$$ −32.9192 −1.32313 −0.661566 0.749887i $$-0.730108\pi$$
−0.661566 + 0.749887i $$0.730108\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −17.4655 −0.700302
$$623$$ 31.3893 1.25759
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.4546 0.417849
$$627$$ 0 0
$$628$$ −18.0294 −0.719450
$$629$$ 5.28005 0.210529
$$630$$ 0 0
$$631$$ 17.2876 0.688209 0.344104 0.938931i $$-0.388183\pi$$
0.344104 + 0.938931i $$0.388183\pi$$
$$632$$ −16.7493 −0.666252
$$633$$ 0 0
$$634$$ 4.33348 0.172105
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 25.1883 0.997997
$$638$$ −20.5795 −0.814749
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.29942 0.367305 0.183653 0.982991i $$-0.441208\pi$$
0.183653 + 0.982991i $$0.441208\pi$$
$$642$$ 0 0
$$643$$ −3.40115 −0.134128 −0.0670642 0.997749i $$-0.521363\pi$$
−0.0670642 + 0.997749i $$0.521363\pi$$
$$644$$ −12.7649 −0.503008
$$645$$ 0 0
$$646$$ −0.515138 −0.0202678
$$647$$ −14.9348 −0.587146 −0.293573 0.955937i $$-0.594844\pi$$
−0.293573 + 0.955937i $$0.594844\pi$$
$$648$$ 0 0
$$649$$ −8.32970 −0.326969
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 2.70058 0.105763
$$653$$ −21.1202 −0.826497 −0.413248 0.910618i $$-0.635606\pi$$
−0.413248 + 0.910618i $$0.635606\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −8.88979 −0.347088
$$657$$ 0 0
$$658$$ −20.4995 −0.799155
$$659$$ −27.6547 −1.07727 −0.538637 0.842538i $$-0.681061\pi$$
−0.538637 + 0.842538i $$0.681061\pi$$
$$660$$ 0 0
$$661$$ 10.2342 0.398063 0.199032 0.979993i $$-0.436220\pi$$
0.199032 + 0.979993i $$0.436220\pi$$
$$662$$ 4.56387 0.177380
$$663$$ 0 0
$$664$$ 3.28005 0.127291
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −24.1231 −0.934048
$$668$$ −8.95035 −0.346299
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −33.3700 −1.28823
$$672$$ 0 0
$$673$$ −13.4693 −0.519202 −0.259601 0.965716i $$-0.583591\pi$$
−0.259601 + 0.965716i $$0.583591\pi$$
$$674$$ −13.0109 −0.501161
$$675$$ 0 0
$$676$$ −6.67408 −0.256695
$$677$$ −15.1433 −0.582006 −0.291003 0.956722i $$-0.593989\pi$$
−0.291003 + 0.956722i $$0.593989\pi$$
$$678$$ 0 0
$$679$$ 16.2498 0.623609
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −9.68968 −0.371037
$$683$$ −15.8789 −0.607589 −0.303795 0.952738i $$-0.598254\pi$$
−0.303795 + 0.952738i $$0.598254\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −12.4352 −0.474778
$$687$$ 0 0
$$688$$ 8.64002 0.329398
$$689$$ −13.7952 −0.525555
$$690$$ 0 0
$$691$$ 27.3094 1.03890 0.519449 0.854501i $$-0.326137\pi$$
0.519449 + 0.854501i $$0.326137\pi$$
$$692$$ −0.310323 −0.0117967
$$693$$ 0 0
$$694$$ 13.2195 0.501805
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.57947 −0.173460
$$698$$ 20.4390 0.773627
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20.9697 0.792016 0.396008 0.918247i $$-0.370395\pi$$
0.396008 + 0.918247i $$0.370395\pi$$
$$702$$ 0 0
$$703$$ 10.2498 0.386577
$$704$$ 2.64002 0.0994996
$$705$$ 0 0
$$706$$ 10.7044 0.402864
$$707$$ 56.9192 2.14067
$$708$$ 0 0
$$709$$ 37.5592 1.41056 0.705282 0.708927i $$-0.250820\pi$$
0.705282 + 0.708927i $$0.250820\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −7.60975 −0.285187
$$713$$ −11.3581 −0.425366
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.52982 0.0571720
$$717$$ 0 0
$$718$$ −4.31410 −0.161001
$$719$$ −3.94323 −0.147058 −0.0735288 0.997293i $$-0.523426\pi$$
−0.0735288 + 0.997293i $$0.523426\pi$$
$$720$$ 0 0
$$721$$ 18.8898 0.703493
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ −14.7493 −0.548154
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 33.2139 1.23183 0.615917 0.787811i $$-0.288786\pi$$
0.615917 + 0.787811i $$0.288786\pi$$
$$728$$ −10.3747 −0.384510
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 4.45080 0.164619
$$732$$ 0 0
$$733$$ −8.62065 −0.318411 −0.159205 0.987245i $$-0.550893\pi$$
−0.159205 + 0.987245i $$0.550893\pi$$
$$734$$ 12.8099 0.472821
$$735$$ 0 0
$$736$$ 3.09461 0.114069
$$737$$ −19.5492 −0.720104
$$738$$ 0 0
$$739$$ −45.1689 −1.66157 −0.830783 0.556597i $$-0.812107\pi$$
−0.830783 + 0.556597i $$0.812107\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 22.6244 0.830569
$$743$$ 24.7905 0.909475 0.454737 0.890626i $$-0.349733\pi$$
0.454737 + 0.890626i $$0.349733\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −0.704357 −0.0257883
$$747$$ 0 0
$$748$$ 1.35998 0.0497257
$$749$$ 42.7943 1.56367
$$750$$ 0 0
$$751$$ −6.76869 −0.246993 −0.123497 0.992345i $$-0.539411\pi$$
−0.123497 + 0.992345i $$0.539411\pi$$
$$752$$ 4.96972 0.181227
$$753$$ 0 0
$$754$$ −19.6060 −0.714007
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 45.2101 1.64319 0.821595 0.570072i $$-0.193085\pi$$
0.821595 + 0.570072i $$0.193085\pi$$
$$758$$ −6.12489 −0.222466
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −19.8851 −0.720834 −0.360417 0.932791i $$-0.617366\pi$$
−0.360417 + 0.932791i $$0.617366\pi$$
$$762$$ 0 0
$$763$$ 12.4352 0.450185
$$764$$ −18.1249 −0.655735
$$765$$ 0 0
$$766$$ 18.6400 0.673491
$$767$$ −7.93567 −0.286540
$$768$$ 0 0
$$769$$ 8.07615 0.291233 0.145617 0.989341i $$-0.453483\pi$$
0.145617 + 0.989341i $$0.453483\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.56009 −0.164121
$$773$$ −45.9456 −1.65255 −0.826275 0.563267i $$-0.809544\pi$$
−0.826275 + 0.563267i $$0.809544\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −3.93945 −0.141418
$$777$$ 0 0
$$778$$ 13.9201 0.499059
$$779$$ −8.88979 −0.318510
$$780$$ 0 0
$$781$$ −29.4087 −1.05233
$$782$$ 1.59415 0.0570067
$$783$$ 0 0
$$784$$ 10.0147 0.357667
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −17.8439 −0.636067 −0.318034 0.948079i $$-0.603022\pi$$
−0.318034 + 0.948079i $$0.603022\pi$$
$$788$$ 2.14048 0.0762515
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 79.1983 2.81597
$$792$$ 0 0
$$793$$ −31.7914 −1.12895
$$794$$ 28.3784 1.00711
$$795$$ 0 0
$$796$$ 16.5639 0.587091
$$797$$ 37.3326 1.32239 0.661194 0.750215i $$-0.270050\pi$$
0.661194 + 0.750215i $$0.270050\pi$$
$$798$$ 0 0
$$799$$ 2.56009 0.0905696
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 5.54920 0.195949
$$803$$ −7.13957 −0.251950
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −9.23131 −0.325159
$$807$$ 0 0
$$808$$ −13.7990 −0.485446
$$809$$ 38.6950 1.36044 0.680221 0.733007i $$-0.261884\pi$$
0.680221 + 0.733007i $$0.261884\pi$$
$$810$$ 0 0
$$811$$ 13.3444 0.468585 0.234292 0.972166i $$-0.424723\pi$$
0.234292 + 0.972166i $$0.424723\pi$$
$$812$$ 32.1542 1.12839
$$813$$ 0 0
$$814$$ −27.0596 −0.948440
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.64002 0.302276
$$818$$ 5.01090 0.175202
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 37.3482 1.30346 0.651730 0.758451i $$-0.274044\pi$$
0.651730 + 0.758451i $$0.274044\pi$$
$$822$$ 0 0
$$823$$ −51.5630 −1.79737 −0.898686 0.438593i $$-0.855477\pi$$
−0.898686 + 0.438593i $$0.855477\pi$$
$$824$$ −4.57947 −0.159533
$$825$$ 0 0
$$826$$ 13.0147 0.452839
$$827$$ 48.5639 1.68873 0.844366 0.535767i $$-0.179978\pi$$
0.844366 + 0.535767i $$0.179978\pi$$
$$828$$ 0 0
$$829$$ 0.325919 0.0113196 0.00565982 0.999984i $$-0.498198\pi$$
0.00565982 + 0.999984i $$0.498198\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.51514 0.0871967
$$833$$ 5.15894 0.178747
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.64002 0.0913071
$$837$$ 0 0
$$838$$ −15.8889 −0.548872
$$839$$ 17.1202 0.591055 0.295527 0.955334i $$-0.404505\pi$$
0.295527 + 0.955334i $$0.404505\pi$$
$$840$$ 0 0
$$841$$ 31.7649 1.09534
$$842$$ 2.38647 0.0822432
$$843$$ 0 0
$$844$$ −15.2838 −0.526091
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −16.6244 −0.571222
$$848$$ −5.48486 −0.188351
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −31.7190 −1.08731
$$852$$ 0 0
$$853$$ −24.9092 −0.852874 −0.426437 0.904517i $$-0.640231\pi$$
−0.426437 + 0.904517i $$0.640231\pi$$
$$854$$ 52.1386 1.78415
$$855$$ 0 0
$$856$$ −10.3747 −0.354598
$$857$$ 11.6509 0.397988 0.198994 0.980001i $$-0.436233\pi$$
0.198994 + 0.980001i $$0.436233\pi$$
$$858$$ 0 0
$$859$$ −5.35998 −0.182880 −0.0914400 0.995811i $$-0.529147\pi$$
−0.0914400 + 0.995811i $$0.529147\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −2.35906 −0.0803499
$$863$$ 41.0790 1.39835 0.699173 0.714953i $$-0.253552\pi$$
0.699173 + 0.714953i $$0.253552\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0.640023 0.0217489
$$867$$ 0 0
$$868$$ 15.1396 0.513870
$$869$$ 44.2186 1.50001
$$870$$ 0 0
$$871$$ −18.6244 −0.631065
$$872$$ −3.01468 −0.102090
$$873$$ 0 0
$$874$$ 3.09461 0.104677
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 45.9532 1.55173 0.775865 0.630899i $$-0.217314\pi$$
0.775865 + 0.630899i $$0.217314\pi$$
$$878$$ 25.4499 0.858892
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 6.90917 0.232776 0.116388 0.993204i $$-0.462868\pi$$
0.116388 + 0.993204i $$0.462868\pi$$
$$882$$ 0 0
$$883$$ 48.5213 1.63287 0.816437 0.577435i $$-0.195946\pi$$
0.816437 + 0.577435i $$0.195946\pi$$
$$884$$ 1.29564 0.0435772
$$885$$ 0 0
$$886$$ −35.6685 −1.19831
$$887$$ −23.7990 −0.799091 −0.399546 0.916713i $$-0.630832\pi$$
−0.399546 + 0.916713i $$0.630832\pi$$
$$888$$ 0 0
$$889$$ 59.0284 1.97975
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −3.75023 −0.125567
$$893$$ 4.96972 0.166305
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −4.12489 −0.137803
$$897$$ 0 0
$$898$$ −11.4399 −0.381755
$$899$$ 28.6107 0.954219
$$900$$ 0 0
$$901$$ −2.82546 −0.0941298
$$902$$ 23.4693 0.781441
$$903$$ 0 0
$$904$$ −19.2001 −0.638586
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 17.9726 0.596770 0.298385 0.954446i $$-0.403552\pi$$
0.298385 + 0.954446i $$0.403552\pi$$
$$908$$ 24.1542 0.801587
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −19.5592 −0.648024 −0.324012 0.946053i $$-0.605032\pi$$
−0.324012 + 0.946053i $$0.605032\pi$$
$$912$$ 0 0
$$913$$ −8.65940 −0.286584
$$914$$ −19.4849 −0.644502
$$915$$ 0 0
$$916$$ −13.0596 −0.431503
$$917$$ 27.0596 0.893588
$$918$$ 0 0
$$919$$ 35.4948 1.17087 0.585433 0.810720i $$-0.300924\pi$$
0.585433 + 0.810720i $$0.300924\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −29.3893 −0.967886
$$923$$ −28.0175 −0.922209
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −12.1892 −0.400563
$$927$$ 0 0
$$928$$ −7.79518 −0.255889
$$929$$ −17.9532 −0.589026 −0.294513 0.955648i $$-0.595157\pi$$
−0.294513 + 0.955648i $$0.595157\pi$$
$$930$$ 0 0
$$931$$ 10.0147 0.328218
$$932$$ 6.43899 0.210916
$$933$$ 0 0
$$934$$ −17.8889 −0.585342
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −5.66652 −0.185117 −0.0925585 0.995707i $$-0.529505\pi$$
−0.0925585 + 0.995707i $$0.529505\pi$$
$$938$$ 30.5445 0.997313
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 24.7044 0.805339 0.402670 0.915345i $$-0.368082\pi$$
0.402670 + 0.915345i $$0.368082\pi$$
$$942$$ 0 0
$$943$$ 27.5104 0.895863
$$944$$ −3.15516 −0.102692
$$945$$ 0 0
$$946$$ −22.8099 −0.741613
$$947$$ −13.5904 −0.441628 −0.220814 0.975316i $$-0.570871\pi$$
−0.220814 + 0.975316i $$0.570871\pi$$
$$948$$ 0 0
$$949$$ −6.80183 −0.220797
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −2.12489 −0.0688679
$$953$$ −24.7375 −0.801326 −0.400663 0.916225i $$-0.631220\pi$$
−0.400663 + 0.916225i $$0.631220\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −22.8742 −0.739804
$$957$$ 0 0
$$958$$ −1.15894 −0.0374437
$$959$$ 26.6244 0.859748
$$960$$ 0 0
$$961$$ −17.5289 −0.565448
$$962$$ −25.7796 −0.831167
$$963$$ 0 0
$$964$$ 4.96972 0.160064
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −35.6897 −1.14770 −0.573851 0.818960i $$-0.694551\pi$$
−0.573851 + 0.818960i $$0.694551\pi$$
$$968$$ 4.03028 0.129538
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −16.4995 −0.529495 −0.264748 0.964318i $$-0.585289\pi$$
−0.264748 + 0.964318i $$0.585289\pi$$
$$972$$ 0 0
$$973$$ −95.1184 −3.04935
$$974$$ −30.6888 −0.983331
$$975$$ 0 0
$$976$$ −12.6400 −0.404597
$$977$$ −49.8501 −1.59485 −0.797423 0.603420i $$-0.793804\pi$$
−0.797423 + 0.603420i $$0.793804\pi$$
$$978$$ 0 0
$$979$$ 20.0899 0.642076
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 3.67030 0.117124
$$983$$ 5.19014 0.165540 0.0827698 0.996569i $$-0.473623\pi$$
0.0827698 + 0.996569i $$0.473623\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −4.01560 −0.127883
$$987$$ 0 0
$$988$$ 2.51514 0.0800172
$$989$$ −26.7375 −0.850203
$$990$$ 0 0
$$991$$ −32.4272 −1.03008 −0.515042 0.857165i $$-0.672224\pi$$
−0.515042 + 0.857165i $$0.672224\pi$$
$$992$$ −3.67030 −0.116532
$$993$$ 0 0
$$994$$ 45.9494 1.45743
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.1992 0.323012 0.161506 0.986872i $$-0.448365\pi$$
0.161506 + 0.986872i $$0.448365\pi$$
$$998$$ −31.3893 −0.993612
$$999$$ 0 0
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 8550.2.a.ck.1.3 3
3.2 odd 2 950.2.a.n.1.1 3
5.2 odd 4 1710.2.d.d.1369.1 6
5.3 odd 4 1710.2.d.d.1369.4 6
5.4 even 2 8550.2.a.cl.1.1 3
12.11 even 2 7600.2.a.bi.1.3 3
15.2 even 4 190.2.b.b.39.6 yes 6
15.8 even 4 190.2.b.b.39.1 6
15.14 odd 2 950.2.a.i.1.3 3
60.23 odd 4 1520.2.d.j.609.5 6
60.47 odd 4 1520.2.d.j.609.2 6
60.59 even 2 7600.2.a.cd.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.1 6 15.8 even 4
190.2.b.b.39.6 yes 6 15.2 even 4
950.2.a.i.1.3 3 15.14 odd 2
950.2.a.n.1.1 3 3.2 odd 2
1520.2.d.j.609.2 6 60.47 odd 4
1520.2.d.j.609.5 6 60.23 odd 4
1710.2.d.d.1369.1 6 5.2 odd 4
1710.2.d.d.1369.4 6 5.3 odd 4
7600.2.a.bi.1.3 3 12.11 even 2
7600.2.a.cd.1.1 3 60.59 even 2
8550.2.a.ck.1.3 3 1.1 even 1 trivial
8550.2.a.cl.1.1 3 5.4 even 2