# Properties

 Label 2169.2.a.e.1.7 Level $2169$ Weight $2$ Character 2169.1 Self dual yes Analytic conductor $17.320$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2169 = 3^{2} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2169.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3195521984$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: 7.7.31056073.1 Defining polynomial: $$x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 241) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$-1.60363$$ of defining polynomial Character $$\chi$$ $$=$$ 2169.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +O(q^{10})$$ $$q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +4.40364 q^{10} +3.27094 q^{11} +4.30649 q^{13} -3.39998 q^{14} +9.27984 q^{16} +1.02456 q^{17} -7.01250 q^{19} +8.08274 q^{20} +8.51631 q^{22} -0.835873 q^{23} -2.13934 q^{25} +11.2125 q^{26} -6.24054 q^{28} +1.11761 q^{29} -3.97344 q^{31} +9.69098 q^{32} +2.66758 q^{34} -2.20867 q^{35} -11.3098 q^{37} -18.2579 q^{38} +12.2371 q^{40} -1.22869 q^{41} +10.8406 q^{43} +15.6314 q^{44} -2.17630 q^{46} +0.151820 q^{47} -5.29473 q^{49} -5.57003 q^{50} +20.5801 q^{52} -3.02053 q^{53} +5.53231 q^{55} -9.44808 q^{56} +2.90984 q^{58} +4.15373 q^{59} +5.62714 q^{61} -10.3454 q^{62} +6.67199 q^{64} +7.28378 q^{65} +12.9934 q^{67} +4.89626 q^{68} -5.75055 q^{70} +11.2862 q^{71} +11.7148 q^{73} -29.4464 q^{74} -33.5118 q^{76} -4.27140 q^{77} -1.66517 q^{79} +15.6955 q^{80} -3.19906 q^{82} +2.34322 q^{83} +1.73290 q^{85} +28.2250 q^{86} +23.6657 q^{88} -18.1099 q^{89} -5.62368 q^{91} -3.99453 q^{92} +0.395283 q^{94} -11.8606 q^{95} -7.17873 q^{97} -13.7855 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10})$$ $$7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.60363 1.84104 0.920521 0.390694i $$-0.127765\pi$$
0.920521 + 0.390694i $$0.127765\pi$$
$$3$$ 0 0
$$4$$ 4.77887 2.38943
$$5$$ 1.69135 0.756395 0.378197 0.925725i $$-0.376544\pi$$
0.378197 + 0.925725i $$0.376544\pi$$
$$6$$ 0 0
$$7$$ −1.30586 −0.493569 −0.246785 0.969070i $$-0.579374\pi$$
−0.246785 + 0.969070i $$0.579374\pi$$
$$8$$ 7.23513 2.55801
$$9$$ 0 0
$$10$$ 4.40364 1.39255
$$11$$ 3.27094 0.986226 0.493113 0.869965i $$-0.335859\pi$$
0.493113 + 0.869965i $$0.335859\pi$$
$$12$$ 0 0
$$13$$ 4.30649 1.19441 0.597203 0.802090i $$-0.296279\pi$$
0.597203 + 0.802090i $$0.296279\pi$$
$$14$$ −3.39998 −0.908682
$$15$$ 0 0
$$16$$ 9.27984 2.31996
$$17$$ 1.02456 0.248493 0.124247 0.992251i $$-0.460349\pi$$
0.124247 + 0.992251i $$0.460349\pi$$
$$18$$ 0 0
$$19$$ −7.01250 −1.60878 −0.804388 0.594104i $$-0.797507\pi$$
−0.804388 + 0.594104i $$0.797507\pi$$
$$20$$ 8.08274 1.80736
$$21$$ 0 0
$$22$$ 8.51631 1.81568
$$23$$ −0.835873 −0.174292 −0.0871458 0.996196i $$-0.527775\pi$$
−0.0871458 + 0.996196i $$0.527775\pi$$
$$24$$ 0 0
$$25$$ −2.13934 −0.427867
$$26$$ 11.2125 2.19895
$$27$$ 0 0
$$28$$ −6.24054 −1.17935
$$29$$ 1.11761 0.207535 0.103767 0.994602i $$-0.466910\pi$$
0.103767 + 0.994602i $$0.466910\pi$$
$$30$$ 0 0
$$31$$ −3.97344 −0.713652 −0.356826 0.934171i $$-0.616141\pi$$
−0.356826 + 0.934171i $$0.616141\pi$$
$$32$$ 9.69098 1.71314
$$33$$ 0 0
$$34$$ 2.66758 0.457486
$$35$$ −2.20867 −0.373333
$$36$$ 0 0
$$37$$ −11.3098 −1.85932 −0.929658 0.368423i $$-0.879898\pi$$
−0.929658 + 0.368423i $$0.879898\pi$$
$$38$$ −18.2579 −2.96182
$$39$$ 0 0
$$40$$ 12.2371 1.93486
$$41$$ −1.22869 −0.191890 −0.0959448 0.995387i $$-0.530587\pi$$
−0.0959448 + 0.995387i $$0.530587\pi$$
$$42$$ 0 0
$$43$$ 10.8406 1.65318 0.826591 0.562804i $$-0.190277\pi$$
0.826591 + 0.562804i $$0.190277\pi$$
$$44$$ 15.6314 2.35652
$$45$$ 0 0
$$46$$ −2.17630 −0.320878
$$47$$ 0.151820 0.0221453 0.0110726 0.999939i $$-0.496475\pi$$
0.0110726 + 0.999939i $$0.496475\pi$$
$$48$$ 0 0
$$49$$ −5.29473 −0.756389
$$50$$ −5.57003 −0.787721
$$51$$ 0 0
$$52$$ 20.5801 2.85395
$$53$$ −3.02053 −0.414902 −0.207451 0.978245i $$-0.566517\pi$$
−0.207451 + 0.978245i $$0.566517\pi$$
$$54$$ 0 0
$$55$$ 5.53231 0.745976
$$56$$ −9.44808 −1.26255
$$57$$ 0 0
$$58$$ 2.90984 0.382080
$$59$$ 4.15373 0.540769 0.270385 0.962752i $$-0.412849\pi$$
0.270385 + 0.962752i $$0.412849\pi$$
$$60$$ 0 0
$$61$$ 5.62714 0.720482 0.360241 0.932859i $$-0.382694\pi$$
0.360241 + 0.932859i $$0.382694\pi$$
$$62$$ −10.3454 −1.31386
$$63$$ 0 0
$$64$$ 6.67199 0.833999
$$65$$ 7.28378 0.903442
$$66$$ 0 0
$$67$$ 12.9934 1.58740 0.793700 0.608309i $$-0.208152\pi$$
0.793700 + 0.608309i $$0.208152\pi$$
$$68$$ 4.89626 0.593758
$$69$$ 0 0
$$70$$ −5.75055 −0.687322
$$71$$ 11.2862 1.33942 0.669711 0.742622i $$-0.266418\pi$$
0.669711 + 0.742622i $$0.266418\pi$$
$$72$$ 0 0
$$73$$ 11.7148 1.37112 0.685560 0.728017i $$-0.259558\pi$$
0.685560 + 0.728017i $$0.259558\pi$$
$$74$$ −29.4464 −3.42308
$$75$$ 0 0
$$76$$ −33.5118 −3.84407
$$77$$ −4.27140 −0.486771
$$78$$ 0 0
$$79$$ −1.66517 −0.187346 −0.0936732 0.995603i $$-0.529861\pi$$
−0.0936732 + 0.995603i $$0.529861\pi$$
$$80$$ 15.6955 1.75481
$$81$$ 0 0
$$82$$ −3.19906 −0.353277
$$83$$ 2.34322 0.257201 0.128601 0.991696i $$-0.458951\pi$$
0.128601 + 0.991696i $$0.458951\pi$$
$$84$$ 0 0
$$85$$ 1.73290 0.187959
$$86$$ 28.2250 3.04358
$$87$$ 0 0
$$88$$ 23.6657 2.52277
$$89$$ −18.1099 −1.91965 −0.959825 0.280598i $$-0.909467\pi$$
−0.959825 + 0.280598i $$0.909467\pi$$
$$90$$ 0 0
$$91$$ −5.62368 −0.589522
$$92$$ −3.99453 −0.416458
$$93$$ 0 0
$$94$$ 0.395283 0.0407703
$$95$$ −11.8606 −1.21687
$$96$$ 0 0
$$97$$ −7.17873 −0.728890 −0.364445 0.931225i $$-0.618741\pi$$
−0.364445 + 0.931225i $$0.618741\pi$$
$$98$$ −13.7855 −1.39254
$$99$$ 0 0
$$100$$ −10.2236 −1.02236
$$101$$ 5.48113 0.545393 0.272696 0.962100i $$-0.412085\pi$$
0.272696 + 0.962100i $$0.412085\pi$$
$$102$$ 0 0
$$103$$ −15.7371 −1.55062 −0.775311 0.631580i $$-0.782407\pi$$
−0.775311 + 0.631580i $$0.782407\pi$$
$$104$$ 31.1580 3.05530
$$105$$ 0 0
$$106$$ −7.86433 −0.763851
$$107$$ 11.1816 1.08097 0.540483 0.841355i $$-0.318242\pi$$
0.540483 + 0.841355i $$0.318242\pi$$
$$108$$ 0 0
$$109$$ −0.296424 −0.0283922 −0.0141961 0.999899i $$-0.504519\pi$$
−0.0141961 + 0.999899i $$0.504519\pi$$
$$110$$ 14.4041 1.37337
$$111$$ 0 0
$$112$$ −12.1182 −1.14506
$$113$$ −10.5881 −0.996045 −0.498023 0.867164i $$-0.665940\pi$$
−0.498023 + 0.867164i $$0.665940\pi$$
$$114$$ 0 0
$$115$$ −1.41375 −0.131833
$$116$$ 5.34091 0.495891
$$117$$ 0 0
$$118$$ 10.8147 0.995578
$$119$$ −1.33794 −0.122649
$$120$$ 0 0
$$121$$ −0.300940 −0.0273582
$$122$$ 14.6510 1.32644
$$123$$ 0 0
$$124$$ −18.9886 −1.70522
$$125$$ −12.0751 −1.08003
$$126$$ 0 0
$$127$$ 14.6989 1.30432 0.652160 0.758081i $$-0.273863\pi$$
0.652160 + 0.758081i $$0.273863\pi$$
$$128$$ −2.01059 −0.177713
$$129$$ 0 0
$$130$$ 18.9642 1.66327
$$131$$ −9.63853 −0.842122 −0.421061 0.907032i $$-0.638342\pi$$
−0.421061 + 0.907032i $$0.638342\pi$$
$$132$$ 0 0
$$133$$ 9.15735 0.794043
$$134$$ 33.8300 2.92247
$$135$$ 0 0
$$136$$ 7.41286 0.635647
$$137$$ −23.2376 −1.98532 −0.992661 0.120933i $$-0.961411\pi$$
−0.992661 + 0.120933i $$0.961411\pi$$
$$138$$ 0 0
$$139$$ −1.29376 −0.109736 −0.0548678 0.998494i $$-0.517474\pi$$
−0.0548678 + 0.998494i $$0.517474\pi$$
$$140$$ −10.5549 −0.892055
$$141$$ 0 0
$$142$$ 29.3850 2.46593
$$143$$ 14.0863 1.17795
$$144$$ 0 0
$$145$$ 1.89027 0.156978
$$146$$ 30.5011 2.52429
$$147$$ 0 0
$$148$$ −54.0480 −4.44271
$$149$$ −17.5316 −1.43625 −0.718123 0.695916i $$-0.754998\pi$$
−0.718123 + 0.695916i $$0.754998\pi$$
$$150$$ 0 0
$$151$$ 5.26990 0.428858 0.214429 0.976740i $$-0.431211\pi$$
0.214429 + 0.976740i $$0.431211\pi$$
$$152$$ −50.7363 −4.11526
$$153$$ 0 0
$$154$$ −11.1211 −0.896166
$$155$$ −6.72048 −0.539802
$$156$$ 0 0
$$157$$ −14.2165 −1.13460 −0.567299 0.823512i $$-0.692012\pi$$
−0.567299 + 0.823512i $$0.692012\pi$$
$$158$$ −4.33548 −0.344912
$$159$$ 0 0
$$160$$ 16.3908 1.29581
$$161$$ 1.09153 0.0860250
$$162$$ 0 0
$$163$$ −15.9688 −1.25077 −0.625386 0.780315i $$-0.715059\pi$$
−0.625386 + 0.780315i $$0.715059\pi$$
$$164$$ −5.87176 −0.458508
$$165$$ 0 0
$$166$$ 6.10086 0.473518
$$167$$ 21.5275 1.66585 0.832925 0.553386i $$-0.186665\pi$$
0.832925 + 0.553386i $$0.186665\pi$$
$$168$$ 0 0
$$169$$ 5.54585 0.426604
$$170$$ 4.51181 0.346040
$$171$$ 0 0
$$172$$ 51.8060 3.95017
$$173$$ 5.82531 0.442890 0.221445 0.975173i $$-0.428923\pi$$
0.221445 + 0.975173i $$0.428923\pi$$
$$174$$ 0 0
$$175$$ 2.79368 0.211182
$$176$$ 30.3538 2.28801
$$177$$ 0 0
$$178$$ −47.1515 −3.53416
$$179$$ 6.03932 0.451400 0.225700 0.974197i $$-0.427533\pi$$
0.225700 + 0.974197i $$0.427533\pi$$
$$180$$ 0 0
$$181$$ −4.03706 −0.300073 −0.150036 0.988680i $$-0.547939\pi$$
−0.150036 + 0.988680i $$0.547939\pi$$
$$182$$ −14.6420 −1.08533
$$183$$ 0 0
$$184$$ −6.04765 −0.445839
$$185$$ −19.1288 −1.40638
$$186$$ 0 0
$$187$$ 3.35129 0.245071
$$188$$ 0.725529 0.0529146
$$189$$ 0 0
$$190$$ −30.8805 −2.24031
$$191$$ 16.4803 1.19247 0.596235 0.802810i $$-0.296663\pi$$
0.596235 + 0.802810i $$0.296663\pi$$
$$192$$ 0 0
$$193$$ −23.0631 −1.66012 −0.830060 0.557674i $$-0.811694\pi$$
−0.830060 + 0.557674i $$0.811694\pi$$
$$194$$ −18.6907 −1.34192
$$195$$ 0 0
$$196$$ −25.3028 −1.80734
$$197$$ 13.5540 0.965680 0.482840 0.875709i $$-0.339605\pi$$
0.482840 + 0.875709i $$0.339605\pi$$
$$198$$ 0 0
$$199$$ −25.6725 −1.81988 −0.909938 0.414745i $$-0.863871\pi$$
−0.909938 + 0.414745i $$0.863871\pi$$
$$200$$ −15.4784 −1.09449
$$201$$ 0 0
$$202$$ 14.2708 1.00409
$$203$$ −1.45944 −0.102433
$$204$$ 0 0
$$205$$ −2.07815 −0.145144
$$206$$ −40.9735 −2.85476
$$207$$ 0 0
$$208$$ 39.9635 2.77097
$$209$$ −22.9375 −1.58662
$$210$$ 0 0
$$211$$ 5.18301 0.356813 0.178407 0.983957i $$-0.442906\pi$$
0.178407 + 0.983957i $$0.442906\pi$$
$$212$$ −14.4347 −0.991380
$$213$$ 0 0
$$214$$ 29.1127 1.99010
$$215$$ 18.3353 1.25046
$$216$$ 0 0
$$217$$ 5.18877 0.352237
$$218$$ −0.771777 −0.0522713
$$219$$ 0 0
$$220$$ 26.4382 1.78246
$$221$$ 4.41227 0.296802
$$222$$ 0 0
$$223$$ −6.50914 −0.435884 −0.217942 0.975962i $$-0.569934\pi$$
−0.217942 + 0.975962i $$0.569934\pi$$
$$224$$ −12.6551 −0.845553
$$225$$ 0 0
$$226$$ −27.5675 −1.83376
$$227$$ −11.0460 −0.733150 −0.366575 0.930389i $$-0.619470\pi$$
−0.366575 + 0.930389i $$0.619470\pi$$
$$228$$ 0 0
$$229$$ 8.95679 0.591881 0.295941 0.955206i $$-0.404367\pi$$
0.295941 + 0.955206i $$0.404367\pi$$
$$230$$ −3.68089 −0.242710
$$231$$ 0 0
$$232$$ 8.08605 0.530875
$$233$$ 6.46647 0.423632 0.211816 0.977310i $$-0.432062\pi$$
0.211816 + 0.977310i $$0.432062\pi$$
$$234$$ 0 0
$$235$$ 0.256781 0.0167506
$$236$$ 19.8501 1.29213
$$237$$ 0 0
$$238$$ −3.48349 −0.225801
$$239$$ 25.0588 1.62092 0.810460 0.585794i $$-0.199217\pi$$
0.810460 + 0.585794i $$0.199217\pi$$
$$240$$ 0 0
$$241$$ −1.00000 −0.0644157
$$242$$ −0.783536 −0.0503676
$$243$$ 0 0
$$244$$ 26.8914 1.72154
$$245$$ −8.95523 −0.572129
$$246$$ 0 0
$$247$$ −30.1992 −1.92153
$$248$$ −28.7484 −1.82552
$$249$$ 0 0
$$250$$ −31.4391 −1.98838
$$251$$ 12.5611 0.792847 0.396424 0.918068i $$-0.370251\pi$$
0.396424 + 0.918068i $$0.370251\pi$$
$$252$$ 0 0
$$253$$ −2.73409 −0.171891
$$254$$ 38.2706 2.40131
$$255$$ 0 0
$$256$$ −18.5788 −1.16117
$$257$$ −16.0367 −1.00034 −0.500171 0.865927i $$-0.666730\pi$$
−0.500171 + 0.865927i $$0.666730\pi$$
$$258$$ 0 0
$$259$$ 14.7690 0.917702
$$260$$ 34.8082 2.15871
$$261$$ 0 0
$$262$$ −25.0951 −1.55038
$$263$$ −3.33210 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$264$$ 0 0
$$265$$ −5.10877 −0.313829
$$266$$ 23.8423 1.46187
$$267$$ 0 0
$$268$$ 62.0939 3.79299
$$269$$ 7.44101 0.453686 0.226843 0.973931i $$-0.427160\pi$$
0.226843 + 0.973931i $$0.427160\pi$$
$$270$$ 0 0
$$271$$ −20.6031 −1.25155 −0.625774 0.780005i $$-0.715217\pi$$
−0.625774 + 0.780005i $$0.715217\pi$$
$$272$$ 9.50779 0.576495
$$273$$ 0 0
$$274$$ −60.5020 −3.65506
$$275$$ −6.99764 −0.421974
$$276$$ 0 0
$$277$$ 5.17667 0.311036 0.155518 0.987833i $$-0.450295\pi$$
0.155518 + 0.987833i $$0.450295\pi$$
$$278$$ −3.36848 −0.202028
$$279$$ 0 0
$$280$$ −15.9800 −0.954988
$$281$$ 14.1821 0.846031 0.423015 0.906123i $$-0.360972\pi$$
0.423015 + 0.906123i $$0.360972\pi$$
$$282$$ 0 0
$$283$$ −7.05196 −0.419196 −0.209598 0.977788i $$-0.567215\pi$$
−0.209598 + 0.977788i $$0.567215\pi$$
$$284$$ 53.9351 3.20046
$$285$$ 0 0
$$286$$ 36.6754 2.16866
$$287$$ 1.60450 0.0947109
$$288$$ 0 0
$$289$$ −15.9503 −0.938251
$$290$$ 4.92155 0.289004
$$291$$ 0 0
$$292$$ 55.9837 3.27620
$$293$$ 2.78966 0.162974 0.0814869 0.996674i $$-0.474033\pi$$
0.0814869 + 0.996674i $$0.474033\pi$$
$$294$$ 0 0
$$295$$ 7.02540 0.409035
$$296$$ −81.8278 −4.75614
$$297$$ 0 0
$$298$$ −45.6458 −2.64419
$$299$$ −3.59968 −0.208175
$$300$$ 0 0
$$301$$ −14.1564 −0.815960
$$302$$ 13.7208 0.789546
$$303$$ 0 0
$$304$$ −65.0749 −3.73230
$$305$$ 9.51747 0.544969
$$306$$ 0 0
$$307$$ 19.8285 1.13167 0.565837 0.824517i $$-0.308553\pi$$
0.565837 + 0.824517i $$0.308553\pi$$
$$308$$ −20.4124 −1.16311
$$309$$ 0 0
$$310$$ −17.4976 −0.993798
$$311$$ −33.5109 −1.90023 −0.950114 0.311902i $$-0.899034\pi$$
−0.950114 + 0.311902i $$0.899034\pi$$
$$312$$ 0 0
$$313$$ 19.4152 1.09741 0.548705 0.836016i $$-0.315121\pi$$
0.548705 + 0.836016i $$0.315121\pi$$
$$314$$ −37.0144 −2.08884
$$315$$ 0 0
$$316$$ −7.95763 −0.447652
$$317$$ 11.7255 0.658569 0.329284 0.944231i $$-0.393193\pi$$
0.329284 + 0.944231i $$0.393193\pi$$
$$318$$ 0 0
$$319$$ 3.65564 0.204676
$$320$$ 11.2847 0.630832
$$321$$ 0 0
$$322$$ 2.84195 0.158376
$$323$$ −7.18475 −0.399770
$$324$$ 0 0
$$325$$ −9.21303 −0.511047
$$326$$ −41.5768 −2.30272
$$327$$ 0 0
$$328$$ −8.88976 −0.490855
$$329$$ −0.198256 −0.0109302
$$330$$ 0 0
$$331$$ 4.93072 0.271017 0.135509 0.990776i $$-0.456733\pi$$
0.135509 + 0.990776i $$0.456733\pi$$
$$332$$ 11.1979 0.614566
$$333$$ 0 0
$$334$$ 56.0496 3.06690
$$335$$ 21.9764 1.20070
$$336$$ 0 0
$$337$$ 34.4425 1.87620 0.938100 0.346364i $$-0.112584\pi$$
0.938100 + 0.346364i $$0.112584\pi$$
$$338$$ 14.4393 0.785395
$$339$$ 0 0
$$340$$ 8.28128 0.449116
$$341$$ −12.9969 −0.703822
$$342$$ 0 0
$$343$$ 16.0552 0.866900
$$344$$ 78.4334 4.22885
$$345$$ 0 0
$$346$$ 15.1669 0.815378
$$347$$ 23.4553 1.25915 0.629574 0.776940i $$-0.283229\pi$$
0.629574 + 0.776940i $$0.283229\pi$$
$$348$$ 0 0
$$349$$ 15.8727 0.849648 0.424824 0.905276i $$-0.360336\pi$$
0.424824 + 0.905276i $$0.360336\pi$$
$$350$$ 7.27369 0.388795
$$351$$ 0 0
$$352$$ 31.6986 1.68954
$$353$$ 28.4598 1.51476 0.757382 0.652972i $$-0.226478\pi$$
0.757382 + 0.652972i $$0.226478\pi$$
$$354$$ 0 0
$$355$$ 19.0889 1.01313
$$356$$ −86.5450 −4.58688
$$357$$ 0 0
$$358$$ 15.7241 0.831047
$$359$$ 0.772338 0.0407625 0.0203812 0.999792i $$-0.493512\pi$$
0.0203812 + 0.999792i $$0.493512\pi$$
$$360$$ 0 0
$$361$$ 30.1751 1.58816
$$362$$ −10.5110 −0.552446
$$363$$ 0 0
$$364$$ −26.8748 −1.40862
$$365$$ 19.8139 1.03711
$$366$$ 0 0
$$367$$ −23.3631 −1.21954 −0.609772 0.792577i $$-0.708739\pi$$
−0.609772 + 0.792577i $$0.708739\pi$$
$$368$$ −7.75677 −0.404350
$$369$$ 0 0
$$370$$ −49.8042 −2.58920
$$371$$ 3.94439 0.204783
$$372$$ 0 0
$$373$$ 27.7831 1.43855 0.719277 0.694724i $$-0.244473\pi$$
0.719277 + 0.694724i $$0.244473\pi$$
$$374$$ 8.72550 0.451185
$$375$$ 0 0
$$376$$ 1.09844 0.0566477
$$377$$ 4.81297 0.247881
$$378$$ 0 0
$$379$$ 7.33328 0.376685 0.188343 0.982103i $$-0.439688\pi$$
0.188343 + 0.982103i $$0.439688\pi$$
$$380$$ −56.6802 −2.90763
$$381$$ 0 0
$$382$$ 42.9084 2.19539
$$383$$ 33.4931 1.71142 0.855708 0.517459i $$-0.173122\pi$$
0.855708 + 0.517459i $$0.173122\pi$$
$$384$$ 0 0
$$385$$ −7.22443 −0.368191
$$386$$ −60.0477 −3.05635
$$387$$ 0 0
$$388$$ −34.3062 −1.74163
$$389$$ −15.6882 −0.795425 −0.397713 0.917510i $$-0.630196\pi$$
−0.397713 + 0.917510i $$0.630196\pi$$
$$390$$ 0 0
$$391$$ −0.856405 −0.0433103
$$392$$ −38.3080 −1.93485
$$393$$ 0 0
$$394$$ 35.2894 1.77786
$$395$$ −2.81639 −0.141708
$$396$$ 0 0
$$397$$ 8.34483 0.418815 0.209408 0.977828i $$-0.432846\pi$$
0.209408 + 0.977828i $$0.432846\pi$$
$$398$$ −66.8416 −3.35047
$$399$$ 0 0
$$400$$ −19.8527 −0.992635
$$401$$ 38.5766 1.92642 0.963211 0.268747i $$-0.0866094\pi$$
0.963211 + 0.268747i $$0.0866094\pi$$
$$402$$ 0 0
$$403$$ −17.1116 −0.852389
$$404$$ 26.1936 1.30318
$$405$$ 0 0
$$406$$ −3.79985 −0.188583
$$407$$ −36.9936 −1.83371
$$408$$ 0 0
$$409$$ −19.5415 −0.966266 −0.483133 0.875547i $$-0.660501\pi$$
−0.483133 + 0.875547i $$0.660501\pi$$
$$410$$ −5.41073 −0.267217
$$411$$ 0 0
$$412$$ −75.2055 −3.70511
$$413$$ −5.42419 −0.266907
$$414$$ 0 0
$$415$$ 3.96320 0.194546
$$416$$ 41.7341 2.04618
$$417$$ 0 0
$$418$$ −59.7206 −2.92103
$$419$$ 29.7748 1.45459 0.727297 0.686323i $$-0.240776\pi$$
0.727297 + 0.686323i $$0.240776\pi$$
$$420$$ 0 0
$$421$$ −4.78920 −0.233411 −0.116706 0.993167i $$-0.537233\pi$$
−0.116706 + 0.993167i $$0.537233\pi$$
$$422$$ 13.4946 0.656908
$$423$$ 0 0
$$424$$ −21.8539 −1.06132
$$425$$ −2.19189 −0.106322
$$426$$ 0 0
$$427$$ −7.34827 −0.355608
$$428$$ 53.4354 2.58290
$$429$$ 0 0
$$430$$ 47.7383 2.30214
$$431$$ −5.35939 −0.258153 −0.129076 0.991635i $$-0.541201\pi$$
−0.129076 + 0.991635i $$0.541201\pi$$
$$432$$ 0 0
$$433$$ 21.5028 1.03336 0.516680 0.856179i $$-0.327168\pi$$
0.516680 + 0.856179i $$0.327168\pi$$
$$434$$ 13.5096 0.648482
$$435$$ 0 0
$$436$$ −1.41657 −0.0678414
$$437$$ 5.86156 0.280396
$$438$$ 0 0
$$439$$ 4.22306 0.201556 0.100778 0.994909i $$-0.467867\pi$$
0.100778 + 0.994909i $$0.467867\pi$$
$$440$$ 40.0270 1.90821
$$441$$ 0 0
$$442$$ 11.4879 0.546424
$$443$$ −8.34030 −0.396260 −0.198130 0.980176i $$-0.563487\pi$$
−0.198130 + 0.980176i $$0.563487\pi$$
$$444$$ 0 0
$$445$$ −30.6303 −1.45201
$$446$$ −16.9474 −0.802481
$$447$$ 0 0
$$448$$ −8.71269 −0.411636
$$449$$ −15.7796 −0.744684 −0.372342 0.928096i $$-0.621445\pi$$
−0.372342 + 0.928096i $$0.621445\pi$$
$$450$$ 0 0
$$451$$ −4.01898 −0.189247
$$452$$ −50.5992 −2.37998
$$453$$ 0 0
$$454$$ −28.7597 −1.34976
$$455$$ −9.51161 −0.445911
$$456$$ 0 0
$$457$$ −8.71419 −0.407633 −0.203816 0.979009i $$-0.565335\pi$$
−0.203816 + 0.979009i $$0.565335\pi$$
$$458$$ 23.3201 1.08968
$$459$$ 0 0
$$460$$ −6.75614 −0.315007
$$461$$ −24.9140 −1.16036 −0.580181 0.814487i $$-0.697018\pi$$
−0.580181 + 0.814487i $$0.697018\pi$$
$$462$$ 0 0
$$463$$ −2.87272 −0.133507 −0.0667534 0.997770i $$-0.521264\pi$$
−0.0667534 + 0.997770i $$0.521264\pi$$
$$464$$ 10.3712 0.481473
$$465$$ 0 0
$$466$$ 16.8363 0.779925
$$467$$ −4.87474 −0.225576 −0.112788 0.993619i $$-0.535978\pi$$
−0.112788 + 0.993619i $$0.535978\pi$$
$$468$$ 0 0
$$469$$ −16.9676 −0.783492
$$470$$ 0.668562 0.0308385
$$471$$ 0 0
$$472$$ 30.0528 1.38329
$$473$$ 35.4591 1.63041
$$474$$ 0 0
$$475$$ 15.0021 0.688343
$$476$$ −6.39383 −0.293061
$$477$$ 0 0
$$478$$ 65.2438 2.98418
$$479$$ −0.194055 −0.00886661 −0.00443331 0.999990i $$-0.501411\pi$$
−0.00443331 + 0.999990i $$0.501411\pi$$
$$480$$ 0 0
$$481$$ −48.7055 −2.22078
$$482$$ −2.60363 −0.118592
$$483$$ 0 0
$$484$$ −1.43815 −0.0653707
$$485$$ −12.1417 −0.551328
$$486$$ 0 0
$$487$$ −30.7592 −1.39383 −0.696915 0.717154i $$-0.745445\pi$$
−0.696915 + 0.717154i $$0.745445\pi$$
$$488$$ 40.7131 1.84300
$$489$$ 0 0
$$490$$ −23.3161 −1.05331
$$491$$ 21.5528 0.972663 0.486332 0.873774i $$-0.338335\pi$$
0.486332 + 0.873774i $$0.338335\pi$$
$$492$$ 0 0
$$493$$ 1.14506 0.0515710
$$494$$ −78.6275 −3.53762
$$495$$ 0 0
$$496$$ −36.8729 −1.65564
$$497$$ −14.7382 −0.661097
$$498$$ 0 0
$$499$$ 21.8992 0.980341 0.490170 0.871627i $$-0.336935\pi$$
0.490170 + 0.871627i $$0.336935\pi$$
$$500$$ −57.7054 −2.58066
$$501$$ 0 0
$$502$$ 32.7043 1.45966
$$503$$ −4.44114 −0.198021 −0.0990104 0.995086i $$-0.531568\pi$$
−0.0990104 + 0.995086i $$0.531568\pi$$
$$504$$ 0 0
$$505$$ 9.27050 0.412532
$$506$$ −7.11855 −0.316458
$$507$$ 0 0
$$508$$ 70.2443 3.11659
$$509$$ −29.6966 −1.31628 −0.658140 0.752895i $$-0.728657\pi$$
−0.658140 + 0.752895i $$0.728657\pi$$
$$510$$ 0 0
$$511$$ −15.2980 −0.676742
$$512$$ −44.3511 −1.96006
$$513$$ 0 0
$$514$$ −41.7536 −1.84167
$$515$$ −26.6169 −1.17288
$$516$$ 0 0
$$517$$ 0.496595 0.0218402
$$518$$ 38.4530 1.68953
$$519$$ 0 0
$$520$$ 52.6991 2.31101
$$521$$ 32.5336 1.42532 0.712661 0.701509i $$-0.247490\pi$$
0.712661 + 0.701509i $$0.247490\pi$$
$$522$$ 0 0
$$523$$ −21.6812 −0.948053 −0.474027 0.880510i $$-0.657200\pi$$
−0.474027 + 0.880510i $$0.657200\pi$$
$$524$$ −46.0612 −2.01219
$$525$$ 0 0
$$526$$ −8.67555 −0.378272
$$527$$ −4.07105 −0.177338
$$528$$ 0 0
$$529$$ −22.3013 −0.969622
$$530$$ −13.3013 −0.577773
$$531$$ 0 0
$$532$$ 43.7618 1.89731
$$533$$ −5.29135 −0.229194
$$534$$ 0 0
$$535$$ 18.9120 0.817637
$$536$$ 94.0092 4.06058
$$537$$ 0 0
$$538$$ 19.3736 0.835255
$$539$$ −17.3187 −0.745971
$$540$$ 0 0
$$541$$ 6.08311 0.261533 0.130767 0.991413i $$-0.458256\pi$$
0.130767 + 0.991413i $$0.458256\pi$$
$$542$$ −53.6427 −2.30415
$$543$$ 0 0
$$544$$ 9.92903 0.425703
$$545$$ −0.501356 −0.0214757
$$546$$ 0 0
$$547$$ −12.1956 −0.521447 −0.260723 0.965414i $$-0.583961\pi$$
−0.260723 + 0.965414i $$0.583961\pi$$
$$548$$ −111.049 −4.74379
$$549$$ 0 0
$$550$$ −18.2192 −0.776871
$$551$$ −7.83723 −0.333877
$$552$$ 0 0
$$553$$ 2.17448 0.0924684
$$554$$ 13.4781 0.572630
$$555$$ 0 0
$$556$$ −6.18273 −0.262206
$$557$$ 37.5982 1.59309 0.796544 0.604581i $$-0.206659\pi$$
0.796544 + 0.604581i $$0.206659\pi$$
$$558$$ 0 0
$$559$$ 46.6851 1.97457
$$560$$ −20.4961 −0.866118
$$561$$ 0 0
$$562$$ 36.9248 1.55758
$$563$$ −20.6037 −0.868342 −0.434171 0.900830i $$-0.642959\pi$$
−0.434171 + 0.900830i $$0.642959\pi$$
$$564$$ 0 0
$$565$$ −17.9082 −0.753403
$$566$$ −18.3607 −0.771756
$$567$$ 0 0
$$568$$ 81.6569 3.42625
$$569$$ −20.5762 −0.862601 −0.431301 0.902208i $$-0.641945\pi$$
−0.431301 + 0.902208i $$0.641945\pi$$
$$570$$ 0 0
$$571$$ 37.1320 1.55392 0.776962 0.629547i $$-0.216760\pi$$
0.776962 + 0.629547i $$0.216760\pi$$
$$572$$ 67.3165 2.81464
$$573$$ 0 0
$$574$$ 4.17753 0.174367
$$575$$ 1.78821 0.0745736
$$576$$ 0 0
$$577$$ −34.7064 −1.44484 −0.722422 0.691452i $$-0.756971\pi$$
−0.722422 + 0.691452i $$0.756971\pi$$
$$578$$ −41.5285 −1.72736
$$579$$ 0 0
$$580$$ 9.03335 0.375089
$$581$$ −3.05992 −0.126947
$$582$$ 0 0
$$583$$ −9.87998 −0.409187
$$584$$ 84.7585 3.50733
$$585$$ 0 0
$$586$$ 7.26324 0.300042
$$587$$ 1.14455 0.0472405 0.0236203 0.999721i $$-0.492481\pi$$
0.0236203 + 0.999721i $$0.492481\pi$$
$$588$$ 0 0
$$589$$ 27.8638 1.14811
$$590$$ 18.2915 0.753050
$$591$$ 0 0
$$592$$ −104.953 −4.31354
$$593$$ −3.60735 −0.148136 −0.0740680 0.997253i $$-0.523598\pi$$
−0.0740680 + 0.997253i $$0.523598\pi$$
$$594$$ 0 0
$$595$$ −2.26292 −0.0927708
$$596$$ −83.7813 −3.43181
$$597$$ 0 0
$$598$$ −9.37222 −0.383258
$$599$$ 31.6723 1.29410 0.647048 0.762449i $$-0.276003\pi$$
0.647048 + 0.762449i $$0.276003\pi$$
$$600$$ 0 0
$$601$$ 6.55665 0.267451 0.133726 0.991018i $$-0.457306\pi$$
0.133726 + 0.991018i $$0.457306\pi$$
$$602$$ −36.8579 −1.50222
$$603$$ 0 0
$$604$$ 25.1841 1.02473
$$605$$ −0.508996 −0.0206936
$$606$$ 0 0
$$607$$ −9.16648 −0.372056 −0.186028 0.982544i $$-0.559562\pi$$
−0.186028 + 0.982544i $$0.559562\pi$$
$$608$$ −67.9579 −2.75606
$$609$$ 0 0
$$610$$ 24.7799 1.00331
$$611$$ 0.653812 0.0264504
$$612$$ 0 0
$$613$$ 7.92345 0.320025 0.160012 0.987115i $$-0.448847\pi$$
0.160012 + 0.987115i $$0.448847\pi$$
$$614$$ 51.6260 2.08346
$$615$$ 0 0
$$616$$ −30.9041 −1.24516
$$617$$ −16.5959 −0.668126 −0.334063 0.942551i $$-0.608420\pi$$
−0.334063 + 0.942551i $$0.608420\pi$$
$$618$$ 0 0
$$619$$ 4.49086 0.180503 0.0902514 0.995919i $$-0.471233\pi$$
0.0902514 + 0.995919i $$0.471233\pi$$
$$620$$ −32.1163 −1.28982
$$621$$ 0 0
$$622$$ −87.2498 −3.49840
$$623$$ 23.6491 0.947481
$$624$$ 0 0
$$625$$ −9.72656 −0.389063
$$626$$ 50.5499 2.02038
$$627$$ 0 0
$$628$$ −67.9387 −2.71105
$$629$$ −11.5876 −0.462028
$$630$$ 0 0
$$631$$ −5.96553 −0.237484 −0.118742 0.992925i $$-0.537886\pi$$
−0.118742 + 0.992925i $$0.537886\pi$$
$$632$$ −12.0477 −0.479233
$$633$$ 0 0
$$634$$ 30.5288 1.21245
$$635$$ 24.8611 0.986581
$$636$$ 0 0
$$637$$ −22.8017 −0.903435
$$638$$ 9.51791 0.376818
$$639$$ 0 0
$$640$$ −3.40061 −0.134421
$$641$$ 1.73375 0.0684789 0.0342395 0.999414i $$-0.489099\pi$$
0.0342395 + 0.999414i $$0.489099\pi$$
$$642$$ 0 0
$$643$$ 38.3312 1.51163 0.755817 0.654783i $$-0.227240\pi$$
0.755817 + 0.654783i $$0.227240\pi$$
$$644$$ 5.21630 0.205551
$$645$$ 0 0
$$646$$ −18.7064 −0.735994
$$647$$ −29.2098 −1.14836 −0.574178 0.818730i $$-0.694678\pi$$
−0.574178 + 0.818730i $$0.694678\pi$$
$$648$$ 0 0
$$649$$ 13.5866 0.533321
$$650$$ −23.9873 −0.940858
$$651$$ 0 0
$$652$$ −76.3127 −2.98864
$$653$$ 42.7952 1.67471 0.837353 0.546662i $$-0.184102\pi$$
0.837353 + 0.546662i $$0.184102\pi$$
$$654$$ 0 0
$$655$$ −16.3021 −0.636976
$$656$$ −11.4021 −0.445177
$$657$$ 0 0
$$658$$ −0.516185 −0.0201230
$$659$$ 23.7004 0.923238 0.461619 0.887078i $$-0.347269\pi$$
0.461619 + 0.887078i $$0.347269\pi$$
$$660$$ 0 0
$$661$$ 24.1228 0.938268 0.469134 0.883127i $$-0.344566\pi$$
0.469134 + 0.883127i $$0.344566\pi$$
$$662$$ 12.8378 0.498954
$$663$$ 0 0
$$664$$ 16.9535 0.657923
$$665$$ 15.4883 0.600610
$$666$$ 0 0
$$667$$ −0.934180 −0.0361716
$$668$$ 102.877 3.98044
$$669$$ 0 0
$$670$$ 57.2184 2.21054
$$671$$ 18.4061 0.710558
$$672$$ 0 0
$$673$$ −3.67522 −0.141669 −0.0708346 0.997488i $$-0.522566\pi$$
−0.0708346 + 0.997488i $$0.522566\pi$$
$$674$$ 89.6753 3.45416
$$675$$ 0 0
$$676$$ 26.5029 1.01934
$$677$$ −45.7769 −1.75935 −0.879675 0.475575i $$-0.842240\pi$$
−0.879675 + 0.475575i $$0.842240\pi$$
$$678$$ 0 0
$$679$$ 9.37443 0.359758
$$680$$ 12.5377 0.480800
$$681$$ 0 0
$$682$$ −33.8391 −1.29577
$$683$$ 1.52419 0.0583215 0.0291608 0.999575i $$-0.490717\pi$$
0.0291608 + 0.999575i $$0.490717\pi$$
$$684$$ 0 0
$$685$$ −39.3029 −1.50169
$$686$$ 41.8018 1.59600
$$687$$ 0 0
$$688$$ 100.599 3.83532
$$689$$ −13.0079 −0.495561
$$690$$ 0 0
$$691$$ 20.3036 0.772386 0.386193 0.922418i $$-0.373790\pi$$
0.386193 + 0.922418i $$0.373790\pi$$
$$692$$ 27.8384 1.05826
$$693$$ 0 0
$$694$$ 61.0689 2.31814
$$695$$ −2.18821 −0.0830034
$$696$$ 0 0
$$697$$ −1.25888 −0.0476833
$$698$$ 41.3267 1.56424
$$699$$ 0 0
$$700$$ 13.3506 0.504606
$$701$$ 10.3863 0.392285 0.196143 0.980575i $$-0.437158\pi$$
0.196143 + 0.980575i $$0.437158\pi$$
$$702$$ 0 0
$$703$$ 79.3098 2.99123
$$704$$ 21.8237 0.822511
$$705$$ 0 0
$$706$$ 74.0988 2.78874
$$707$$ −7.15759 −0.269189
$$708$$ 0 0
$$709$$ 34.9053 1.31090 0.655448 0.755240i $$-0.272480\pi$$
0.655448 + 0.755240i $$0.272480\pi$$
$$710$$ 49.7002 1.86522
$$711$$ 0 0
$$712$$ −131.028 −4.91048
$$713$$ 3.32129 0.124383
$$714$$ 0 0
$$715$$ 23.8248 0.890998
$$716$$ 28.8611 1.07859
$$717$$ 0 0
$$718$$ 2.01088 0.0750454
$$719$$ −5.17696 −0.193068 −0.0965341 0.995330i $$-0.530776\pi$$
−0.0965341 + 0.995330i $$0.530776\pi$$
$$720$$ 0 0
$$721$$ 20.5505 0.765339
$$722$$ 78.5646 2.92387
$$723$$ 0 0
$$724$$ −19.2926 −0.717003
$$725$$ −2.39094 −0.0887974
$$726$$ 0 0
$$727$$ −7.00384 −0.259758 −0.129879 0.991530i $$-0.541459\pi$$
−0.129879 + 0.991530i $$0.541459\pi$$
$$728$$ −40.6881 −1.50800
$$729$$ 0 0
$$730$$ 51.5880 1.90936
$$731$$ 11.1069 0.410804
$$732$$ 0 0
$$733$$ −1.22518 −0.0452530 −0.0226265 0.999744i $$-0.507203\pi$$
−0.0226265 + 0.999744i $$0.507203\pi$$
$$734$$ −60.8288 −2.24523
$$735$$ 0 0
$$736$$ −8.10042 −0.298586
$$737$$ 42.5008 1.56554
$$738$$ 0 0
$$739$$ −31.4569 −1.15716 −0.578580 0.815625i $$-0.696393\pi$$
−0.578580 + 0.815625i $$0.696393\pi$$
$$740$$ −91.4140 −3.36045
$$741$$ 0 0
$$742$$ 10.2697 0.377014
$$743$$ −17.5176 −0.642659 −0.321330 0.946967i $$-0.604130\pi$$
−0.321330 + 0.946967i $$0.604130\pi$$
$$744$$ 0 0
$$745$$ −29.6521 −1.08637
$$746$$ 72.3368 2.64844
$$747$$ 0 0
$$748$$ 16.0154 0.585580
$$749$$ −14.6016 −0.533532
$$750$$ 0 0
$$751$$ −35.2893 −1.28773 −0.643863 0.765141i $$-0.722669\pi$$
−0.643863 + 0.765141i $$0.722669\pi$$
$$752$$ 1.40887 0.0513761
$$753$$ 0 0
$$754$$ 12.5312 0.456359
$$755$$ 8.91324 0.324386
$$756$$ 0 0
$$757$$ −17.2974 −0.628686 −0.314343 0.949309i $$-0.601784\pi$$
−0.314343 + 0.949309i $$0.601784\pi$$
$$758$$ 19.0931 0.693493
$$759$$ 0 0
$$760$$ −85.8129 −3.11276
$$761$$ −9.53250 −0.345553 −0.172776 0.984961i $$-0.555274\pi$$
−0.172776 + 0.984961i $$0.555274\pi$$
$$762$$ 0 0
$$763$$ 0.387088 0.0140135
$$764$$ 78.7570 2.84933
$$765$$ 0 0
$$766$$ 87.2034 3.15079
$$767$$ 17.8880 0.645897
$$768$$ 0 0
$$769$$ −37.0399 −1.33569 −0.667847 0.744299i $$-0.732784\pi$$
−0.667847 + 0.744299i $$0.732784\pi$$
$$770$$ −18.8097 −0.677855
$$771$$ 0 0
$$772$$ −110.216 −3.96675
$$773$$ 6.97450 0.250855 0.125428 0.992103i $$-0.459970\pi$$
0.125428 + 0.992103i $$0.459970\pi$$
$$774$$ 0 0
$$775$$ 8.50053 0.305348
$$776$$ −51.9391 −1.86450
$$777$$ 0 0
$$778$$ −40.8463 −1.46441
$$779$$ 8.61621 0.308708
$$780$$ 0 0
$$781$$ 36.9164 1.32097
$$782$$ −2.22976 −0.0797360
$$783$$ 0 0
$$784$$ −49.1342 −1.75479
$$785$$ −24.0450 −0.858204
$$786$$ 0 0
$$787$$ 29.9359 1.06710 0.533550 0.845768i $$-0.320857\pi$$
0.533550 + 0.845768i $$0.320857\pi$$
$$788$$ 64.7726 2.30743
$$789$$ 0 0
$$790$$ −7.33281 −0.260890
$$791$$ 13.8266 0.491617
$$792$$ 0 0
$$793$$ 24.2332 0.860547
$$794$$ 21.7268 0.771056
$$795$$ 0 0
$$796$$ −122.685 −4.34847
$$797$$ 10.7103 0.379380 0.189690 0.981844i $$-0.439252\pi$$
0.189690 + 0.981844i $$0.439252\pi$$
$$798$$ 0 0
$$799$$ 0.155550 0.00550295
$$800$$ −20.7323 −0.732996
$$801$$ 0 0
$$802$$ 100.439 3.54662
$$803$$ 38.3186 1.35223
$$804$$ 0 0
$$805$$ 1.84617 0.0650688
$$806$$ −44.5522 −1.56928
$$807$$ 0 0
$$808$$ 39.6567 1.39512
$$809$$ 30.4625 1.07100 0.535502 0.844534i $$-0.320122\pi$$
0.535502 + 0.844534i $$0.320122\pi$$
$$810$$ 0 0
$$811$$ −6.38159 −0.224088 −0.112044 0.993703i $$-0.535740\pi$$
−0.112044 + 0.993703i $$0.535740\pi$$
$$812$$ −6.97449 −0.244757
$$813$$ 0 0
$$814$$ −96.3176 −3.37593
$$815$$ −27.0088 −0.946077
$$816$$ 0 0
$$817$$ −76.0199 −2.65960
$$818$$ −50.8788 −1.77894
$$819$$ 0 0
$$820$$ −9.93121 −0.346813
$$821$$ 31.3814 1.09522 0.547609 0.836734i $$-0.315538\pi$$
0.547609 + 0.836734i $$0.315538\pi$$
$$822$$ 0 0
$$823$$ −8.61720 −0.300377 −0.150188 0.988657i $$-0.547988\pi$$
−0.150188 + 0.988657i $$0.547988\pi$$
$$824$$ −113.860 −3.96650
$$825$$ 0 0
$$826$$ −14.1226 −0.491387
$$827$$ 18.2047 0.633039 0.316520 0.948586i $$-0.397486\pi$$
0.316520 + 0.948586i $$0.397486\pi$$
$$828$$ 0 0
$$829$$ 38.0162 1.32036 0.660179 0.751108i $$-0.270480\pi$$
0.660179 + 0.751108i $$0.270480\pi$$
$$830$$ 10.3187 0.358167
$$831$$ 0 0
$$832$$ 28.7328 0.996132
$$833$$ −5.42479 −0.187958
$$834$$ 0 0
$$835$$ 36.4106 1.26004
$$836$$ −109.615 −3.79112
$$837$$ 0 0
$$838$$ 77.5224 2.67797
$$839$$ −45.6228 −1.57507 −0.787537 0.616267i $$-0.788644\pi$$
−0.787537 + 0.616267i $$0.788644\pi$$
$$840$$ 0 0
$$841$$ −27.7509 −0.956929
$$842$$ −12.4693 −0.429720
$$843$$ 0 0
$$844$$ 24.7689 0.852582
$$845$$ 9.37997 0.322681
$$846$$ 0 0
$$847$$ 0.392987 0.0135032
$$848$$ −28.0300 −0.962556
$$849$$ 0 0
$$850$$ −5.70685 −0.195743
$$851$$ 9.45354 0.324063
$$852$$ 0 0
$$853$$ 17.2406 0.590307 0.295153 0.955450i $$-0.404629\pi$$
0.295153 + 0.955450i $$0.404629\pi$$
$$854$$ −19.1321 −0.654689
$$855$$ 0 0
$$856$$ 80.9003 2.76512
$$857$$ −9.87638 −0.337371 −0.168685 0.985670i $$-0.553952\pi$$
−0.168685 + 0.985670i $$0.553952\pi$$
$$858$$ 0 0
$$859$$ 25.8904 0.883369 0.441685 0.897170i $$-0.354381\pi$$
0.441685 + 0.897170i $$0.354381\pi$$
$$860$$ 87.6220 2.98789
$$861$$ 0 0
$$862$$ −13.9538 −0.475270
$$863$$ 14.8445 0.505311 0.252656 0.967556i $$-0.418696\pi$$
0.252656 + 0.967556i $$0.418696\pi$$
$$864$$ 0 0
$$865$$ 9.85263 0.334999
$$866$$ 55.9853 1.90246
$$867$$ 0 0
$$868$$ 24.7964 0.841646
$$869$$ −5.44668 −0.184766
$$870$$ 0 0
$$871$$ 55.9561 1.89600
$$872$$ −2.14467 −0.0726275
$$873$$ 0 0
$$874$$ 15.2613 0.516221
$$875$$ 15.7684 0.533070
$$876$$ 0 0
$$877$$ 42.0781 1.42088 0.710439 0.703759i $$-0.248497\pi$$
0.710439 + 0.703759i $$0.248497\pi$$
$$878$$ 10.9953 0.371072
$$879$$ 0 0
$$880$$ 51.3389 1.73064
$$881$$ 19.0418 0.641534 0.320767 0.947158i $$-0.396059\pi$$
0.320767 + 0.947158i $$0.396059\pi$$
$$882$$ 0 0
$$883$$ 6.51087 0.219108 0.109554 0.993981i $$-0.465058\pi$$
0.109554 + 0.993981i $$0.465058\pi$$
$$884$$ 21.0857 0.709188
$$885$$ 0 0
$$886$$ −21.7150 −0.729531
$$887$$ −5.59140 −0.187741 −0.0938704 0.995584i $$-0.529924\pi$$
−0.0938704 + 0.995584i $$0.529924\pi$$
$$888$$ 0 0
$$889$$ −19.1948 −0.643773
$$890$$ −79.7497 −2.67322
$$891$$ 0 0
$$892$$ −31.1063 −1.04152
$$893$$ −1.06464 −0.0356268
$$894$$ 0 0
$$895$$ 10.2146 0.341437
$$896$$ 2.62555 0.0877135
$$897$$ 0 0
$$898$$ −41.0841 −1.37099
$$899$$ −4.44076 −0.148108
$$900$$ 0 0
$$901$$ −3.09473 −0.103100
$$902$$ −10.4639 −0.348411
$$903$$ 0 0
$$904$$ −76.6063 −2.54789
$$905$$ −6.82809 −0.226973
$$906$$ 0 0
$$907$$ 18.3915 0.610681 0.305340 0.952243i $$-0.401230\pi$$
0.305340 + 0.952243i $$0.401230\pi$$
$$908$$ −52.7874 −1.75181
$$909$$ 0 0
$$910$$ −24.7647 −0.820941
$$911$$ 38.2665 1.26783 0.633913 0.773404i $$-0.281448\pi$$
0.633913 + 0.773404i $$0.281448\pi$$
$$912$$ 0 0
$$913$$ 7.66452 0.253659
$$914$$ −22.6885 −0.750469
$$915$$ 0 0
$$916$$ 42.8033 1.41426
$$917$$ 12.5866 0.415646
$$918$$ 0 0
$$919$$ −4.89962 −0.161624 −0.0808118 0.996729i $$-0.525751\pi$$
−0.0808118 + 0.996729i $$0.525751\pi$$
$$920$$ −10.2287 −0.337230
$$921$$ 0 0
$$922$$ −64.8669 −2.13628
$$923$$ 48.6038 1.59981
$$924$$ 0 0
$$925$$ 24.1954 0.795541
$$926$$ −7.47949 −0.245791
$$927$$ 0 0
$$928$$ 10.8307 0.355536
$$929$$ −17.9381 −0.588528 −0.294264 0.955724i $$-0.595075\pi$$
−0.294264 + 0.955724i $$0.595075\pi$$
$$930$$ 0 0
$$931$$ 37.1292 1.21686
$$932$$ 30.9024 1.01224
$$933$$ 0 0
$$934$$ −12.6920 −0.415295
$$935$$ 5.66820 0.185370
$$936$$ 0 0
$$937$$ 29.8369 0.974730 0.487365 0.873198i $$-0.337958\pi$$
0.487365 + 0.873198i $$0.337958\pi$$
$$938$$ −44.1773 −1.44244
$$939$$ 0 0
$$940$$ 1.22712 0.0400243
$$941$$ −26.8047 −0.873809 −0.436904 0.899508i $$-0.643925\pi$$
−0.436904 + 0.899508i $$0.643925\pi$$
$$942$$ 0 0
$$943$$ 1.02703 0.0334448
$$944$$ 38.5459 1.25456
$$945$$ 0 0
$$946$$ 92.3222 3.00165
$$947$$ −2.94622 −0.0957393 −0.0478697 0.998854i $$-0.515243\pi$$
−0.0478697 + 0.998854i $$0.515243\pi$$
$$948$$ 0 0
$$949$$ 50.4499 1.63767
$$950$$ 39.0598 1.26727
$$951$$ 0 0
$$952$$ −9.68017 −0.313736
$$953$$ 4.62861 0.149935 0.0749676 0.997186i $$-0.476115\pi$$
0.0749676 + 0.997186i $$0.476115\pi$$
$$954$$ 0 0
$$955$$ 27.8739 0.901978
$$956$$ 119.753 3.87308
$$957$$ 0 0
$$958$$ −0.505247 −0.0163238
$$959$$ 30.3451 0.979894
$$960$$ 0 0
$$961$$ −15.2117 −0.490702
$$962$$ −126.811 −4.08854
$$963$$ 0 0
$$964$$ −4.77887 −0.153917
$$965$$ −39.0078 −1.25571
$$966$$ 0 0
$$967$$ 20.1721 0.648689 0.324345 0.945939i $$-0.394856\pi$$
0.324345 + 0.945939i $$0.394856\pi$$
$$968$$ −2.17734 −0.0699825
$$969$$ 0 0
$$970$$ −31.6126 −1.01502
$$971$$ −30.0251 −0.963550 −0.481775 0.876295i $$-0.660008\pi$$
−0.481775 + 0.876295i $$0.660008\pi$$
$$972$$ 0 0
$$973$$ 1.68948 0.0541621
$$974$$ −80.0853 −2.56610
$$975$$ 0 0
$$976$$ 52.2190 1.67149
$$977$$ −57.7096 −1.84629 −0.923147 0.384448i $$-0.874392\pi$$
−0.923147 + 0.384448i $$0.874392\pi$$
$$978$$ 0 0
$$979$$ −59.2366 −1.89321
$$980$$ −42.7959 −1.36706
$$981$$ 0 0
$$982$$ 56.1154 1.79071
$$983$$ 3.33817 0.106471 0.0532356 0.998582i $$-0.483047\pi$$
0.0532356 + 0.998582i $$0.483047\pi$$
$$984$$ 0 0
$$985$$ 22.9245 0.730435
$$986$$ 2.98132 0.0949444
$$987$$ 0 0
$$988$$ −144.318 −4.59137
$$989$$ −9.06139 −0.288136
$$990$$ 0 0
$$991$$ 4.48757 0.142552 0.0712762 0.997457i $$-0.477293\pi$$
0.0712762 + 0.997457i $$0.477293\pi$$
$$992$$ −38.5065 −1.22258
$$993$$ 0 0
$$994$$ −38.3727 −1.21711
$$995$$ −43.4212 −1.37654
$$996$$ 0 0
$$997$$ −47.4116 −1.50154 −0.750770 0.660564i $$-0.770317\pi$$
−0.750770 + 0.660564i $$0.770317\pi$$
$$998$$ 57.0172 1.80485
$$999$$ 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 2169.2.a.e.1.7 7
3.2 odd 2 241.2.a.a.1.1 7
12.11 even 2 3856.2.a.j.1.2 7
15.14 odd 2 6025.2.a.f.1.7 7

By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.1 7 3.2 odd 2
2169.2.a.e.1.7 7 1.1 even 1 trivial
3856.2.a.j.1.2 7 12.11 even 2
6025.2.a.f.1.7 7 15.14 odd 2