# Properties

 Label 9200.2.a.cw.1.5 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.791953.1 Defining polynomial: $$x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2$$ x^5 - 2*x^4 - 7*x^3 + 7*x^2 + 9*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-2.08344$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.08344 q^{3} -0.555022 q^{7} +6.50759 q^{9} +O(q^{10})$$ $$q+3.08344 q^{3} -0.555022 q^{7} +6.50759 q^{9} -4.65190 q^{11} -5.02256 q^{13} -1.32867 q^{17} +0.196402 q^{19} -1.71138 q^{21} +1.00000 q^{23} +10.8154 q^{27} -0.812298 q^{29} +2.11145 q^{31} -14.3439 q^{33} +5.64564 q^{37} -15.4868 q^{39} -4.89714 q^{41} +1.66507 q^{43} -9.89310 q^{47} -6.69195 q^{49} -4.09688 q^{51} -2.23261 q^{53} +0.605593 q^{57} -2.43488 q^{59} -5.71138 q^{61} -3.61185 q^{63} +6.91830 q^{67} +3.08344 q^{69} -0.0120411 q^{71} -15.2989 q^{73} +2.58191 q^{77} -10.6351 q^{79} +13.8260 q^{81} -5.64020 q^{83} -2.50467 q^{87} +9.06693 q^{89} +2.78763 q^{91} +6.51051 q^{93} +14.0720 q^{97} -30.2727 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} - q^{7} + 4 q^{9}+O(q^{10})$$ 5 * q + 3 * q^3 - q^7 + 4 * q^9 $$5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} + 5 q^{51} - 18 q^{53} - 27 q^{57} - 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} + 2 q^{71} - 4 q^{73} - 15 q^{77} - 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} + 19 q^{91} + 15 q^{93} - 8 q^{97} - 37 q^{99}+O(q^{100})$$ 5 * q + 3 * q^3 - q^7 + 4 * q^9 + 4 * q^11 + q^13 - 5 * q^17 - 4 * q^19 - 6 * q^21 + 5 * q^23 + 6 * q^27 - 11 * q^29 - 4 * q^31 - 13 * q^33 - 6 * q^37 - 31 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^47 - 2 * q^49 + 5 * q^51 - 18 * q^53 - 27 * q^57 - 23 * q^59 - 26 * q^61 + 5 * q^63 + 3 * q^67 + 3 * q^69 + 2 * q^71 - 4 * q^73 - 15 * q^77 - 43 * q^79 - 3 * q^81 + 30 * q^83 + 27 * q^87 + 15 * q^89 + 19 * q^91 + 15 * q^93 - 8 * q^97 - 37 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.08344 1.78022 0.890112 0.455742i $$-0.150626\pi$$
0.890112 + 0.455742i $$0.150626\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.555022 −0.209779 −0.104889 0.994484i $$-0.533449\pi$$
−0.104889 + 0.994484i $$0.533449\pi$$
$$8$$ 0 0
$$9$$ 6.50759 2.16920
$$10$$ 0 0
$$11$$ −4.65190 −1.40260 −0.701301 0.712866i $$-0.747397\pi$$
−0.701301 + 0.712866i $$0.747397\pi$$
$$12$$ 0 0
$$13$$ −5.02256 −1.39301 −0.696504 0.717553i $$-0.745262\pi$$
−0.696504 + 0.717553i $$0.745262\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.32867 −0.322251 −0.161125 0.986934i $$-0.551512\pi$$
−0.161125 + 0.986934i $$0.551512\pi$$
$$18$$ 0 0
$$19$$ 0.196402 0.0450577 0.0225288 0.999746i $$-0.492828\pi$$
0.0225288 + 0.999746i $$0.492828\pi$$
$$20$$ 0 0
$$21$$ −1.71138 −0.373453
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 10.8154 2.08143
$$28$$ 0 0
$$29$$ −0.812298 −0.150840 −0.0754200 0.997152i $$-0.524030\pi$$
−0.0754200 + 0.997152i $$0.524030\pi$$
$$30$$ 0 0
$$31$$ 2.11145 0.379227 0.189613 0.981859i $$-0.439277\pi$$
0.189613 + 0.981859i $$0.439277\pi$$
$$32$$ 0 0
$$33$$ −14.3439 −2.49694
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.64564 0.928138 0.464069 0.885799i $$-0.346389\pi$$
0.464069 + 0.885799i $$0.346389\pi$$
$$38$$ 0 0
$$39$$ −15.4868 −2.47987
$$40$$ 0 0
$$41$$ −4.89714 −0.764804 −0.382402 0.923996i $$-0.624903\pi$$
−0.382402 + 0.923996i $$0.624903\pi$$
$$42$$ 0 0
$$43$$ 1.66507 0.253920 0.126960 0.991908i $$-0.459478\pi$$
0.126960 + 0.991908i $$0.459478\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.89310 −1.44306 −0.721528 0.692385i $$-0.756560\pi$$
−0.721528 + 0.692385i $$0.756560\pi$$
$$48$$ 0 0
$$49$$ −6.69195 −0.955993
$$50$$ 0 0
$$51$$ −4.09688 −0.573678
$$52$$ 0 0
$$53$$ −2.23261 −0.306673 −0.153336 0.988174i $$-0.549002\pi$$
−0.153336 + 0.988174i $$0.549002\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.605593 0.0802127
$$58$$ 0 0
$$59$$ −2.43488 −0.316994 −0.158497 0.987359i $$-0.550665\pi$$
−0.158497 + 0.987359i $$0.550665\pi$$
$$60$$ 0 0
$$61$$ −5.71138 −0.731267 −0.365633 0.930759i $$-0.619148\pi$$
−0.365633 + 0.930759i $$0.619148\pi$$
$$62$$ 0 0
$$63$$ −3.61185 −0.455051
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 6.91830 0.845205 0.422602 0.906315i $$-0.361117\pi$$
0.422602 + 0.906315i $$0.361117\pi$$
$$68$$ 0 0
$$69$$ 3.08344 0.371202
$$70$$ 0 0
$$71$$ −0.0120411 −0.00142902 −0.000714510 1.00000i $$-0.500227\pi$$
−0.000714510 1.00000i $$0.500227\pi$$
$$72$$ 0 0
$$73$$ −15.2989 −1.79061 −0.895303 0.445458i $$-0.853041\pi$$
−0.895303 + 0.445458i $$0.853041\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.58191 0.294236
$$78$$ 0 0
$$79$$ −10.6351 −1.19654 −0.598272 0.801293i $$-0.704146\pi$$
−0.598272 + 0.801293i $$0.704146\pi$$
$$80$$ 0 0
$$81$$ 13.8260 1.53622
$$82$$ 0 0
$$83$$ −5.64020 −0.619092 −0.309546 0.950884i $$-0.600177\pi$$
−0.309546 + 0.950884i $$0.600177\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −2.50467 −0.268529
$$88$$ 0 0
$$89$$ 9.06693 0.961093 0.480547 0.876969i $$-0.340438\pi$$
0.480547 + 0.876969i $$0.340438\pi$$
$$90$$ 0 0
$$91$$ 2.78763 0.292223
$$92$$ 0 0
$$93$$ 6.51051 0.675108
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0720 1.42880 0.714398 0.699739i $$-0.246701\pi$$
0.714398 + 0.699739i $$0.246701\pi$$
$$98$$ 0 0
$$99$$ −30.2727 −3.04252
$$100$$ 0 0
$$101$$ 7.53015 0.749278 0.374639 0.927171i $$-0.377767\pi$$
0.374639 + 0.927171i $$0.377767\pi$$
$$102$$ 0 0
$$103$$ −14.4476 −1.42357 −0.711784 0.702399i $$-0.752112\pi$$
−0.711784 + 0.702399i $$0.752112\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.35862 0.228016 0.114008 0.993480i $$-0.463631\pi$$
0.114008 + 0.993480i $$0.463631\pi$$
$$108$$ 0 0
$$109$$ −10.3578 −0.992101 −0.496050 0.868294i $$-0.665217\pi$$
−0.496050 + 0.868294i $$0.665217\pi$$
$$110$$ 0 0
$$111$$ 17.4080 1.65229
$$112$$ 0 0
$$113$$ 19.2723 1.81298 0.906491 0.422226i $$-0.138751\pi$$
0.906491 + 0.422226i $$0.138751\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −32.6848 −3.02171
$$118$$ 0 0
$$119$$ 0.737442 0.0676012
$$120$$ 0 0
$$121$$ 10.6402 0.967291
$$122$$ 0 0
$$123$$ −15.1000 −1.36152
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −18.4934 −1.64102 −0.820510 0.571632i $$-0.806311\pi$$
−0.820510 + 0.571632i $$0.806311\pi$$
$$128$$ 0 0
$$129$$ 5.13413 0.452035
$$130$$ 0 0
$$131$$ 14.7727 1.29070 0.645351 0.763887i $$-0.276711\pi$$
0.645351 + 0.763887i $$0.276711\pi$$
$$132$$ 0 0
$$133$$ −0.109007 −0.00945213
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −11.6173 −0.992533 −0.496266 0.868170i $$-0.665296\pi$$
−0.496266 + 0.868170i $$0.665296\pi$$
$$138$$ 0 0
$$139$$ −11.1389 −0.944787 −0.472393 0.881388i $$-0.656610\pi$$
−0.472393 + 0.881388i $$0.656610\pi$$
$$140$$ 0 0
$$141$$ −30.5047 −2.56896
$$142$$ 0 0
$$143$$ 23.3645 1.95384
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −20.6342 −1.70188
$$148$$ 0 0
$$149$$ −12.4796 −1.02237 −0.511184 0.859472i $$-0.670793\pi$$
−0.511184 + 0.859472i $$0.670793\pi$$
$$150$$ 0 0
$$151$$ −1.32673 −0.107968 −0.0539840 0.998542i $$-0.517192\pi$$
−0.0539840 + 0.998542i $$0.517192\pi$$
$$152$$ 0 0
$$153$$ −8.64646 −0.699025
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −23.1754 −1.84960 −0.924798 0.380458i $$-0.875766\pi$$
−0.924798 + 0.380458i $$0.875766\pi$$
$$158$$ 0 0
$$159$$ −6.88412 −0.545946
$$160$$ 0 0
$$161$$ −0.555022 −0.0437418
$$162$$ 0 0
$$163$$ 25.0682 1.96349 0.981746 0.190196i $$-0.0609125\pi$$
0.981746 + 0.190196i $$0.0609125\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −4.39603 −0.340175 −0.170087 0.985429i $$-0.554405\pi$$
−0.170087 + 0.985429i $$0.554405\pi$$
$$168$$ 0 0
$$169$$ 12.2261 0.940473
$$170$$ 0 0
$$171$$ 1.27810 0.0977389
$$172$$ 0 0
$$173$$ −19.4211 −1.47656 −0.738281 0.674493i $$-0.764362\pi$$
−0.738281 + 0.674493i $$0.764362\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −7.50779 −0.564320
$$178$$ 0 0
$$179$$ −10.4169 −0.778592 −0.389296 0.921113i $$-0.627282\pi$$
−0.389296 + 0.921113i $$0.627282\pi$$
$$180$$ 0 0
$$181$$ −9.13693 −0.679143 −0.339571 0.940580i $$-0.610282\pi$$
−0.339571 + 0.940580i $$0.610282\pi$$
$$182$$ 0 0
$$183$$ −17.6107 −1.30182
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.18086 0.451989
$$188$$ 0 0
$$189$$ −6.00280 −0.436640
$$190$$ 0 0
$$191$$ 1.49819 0.108405 0.0542026 0.998530i $$-0.482738\pi$$
0.0542026 + 0.998530i $$0.482738\pi$$
$$192$$ 0 0
$$193$$ −14.4113 −1.03735 −0.518675 0.854972i $$-0.673575\pi$$
−0.518675 + 0.854972i $$0.673575\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −24.4686 −1.74331 −0.871657 0.490116i $$-0.836954\pi$$
−0.871657 + 0.490116i $$0.836954\pi$$
$$198$$ 0 0
$$199$$ 1.61185 0.114261 0.0571307 0.998367i $$-0.481805\pi$$
0.0571307 + 0.998367i $$0.481805\pi$$
$$200$$ 0 0
$$201$$ 21.3321 1.50465
$$202$$ 0 0
$$203$$ 0.450843 0.0316430
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.50759 0.452309
$$208$$ 0 0
$$209$$ −0.913642 −0.0631979
$$210$$ 0 0
$$211$$ −6.25081 −0.430324 −0.215162 0.976578i $$-0.569028\pi$$
−0.215162 + 0.976578i $$0.569028\pi$$
$$212$$ 0 0
$$213$$ −0.0371281 −0.00254398
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −1.17190 −0.0795536
$$218$$ 0 0
$$219$$ −47.1733 −3.18768
$$220$$ 0 0
$$221$$ 6.67334 0.448898
$$222$$ 0 0
$$223$$ 21.2953 1.42604 0.713019 0.701144i $$-0.247327\pi$$
0.713019 + 0.701144i $$0.247327\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 10.5174 0.698061 0.349031 0.937111i $$-0.386511\pi$$
0.349031 + 0.937111i $$0.386511\pi$$
$$228$$ 0 0
$$229$$ −24.0063 −1.58638 −0.793190 0.608975i $$-0.791581\pi$$
−0.793190 + 0.608975i $$0.791581\pi$$
$$230$$ 0 0
$$231$$ 7.96115 0.523805
$$232$$ 0 0
$$233$$ 11.3112 0.741021 0.370510 0.928828i $$-0.379183\pi$$
0.370510 + 0.928828i $$0.379183\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −32.7927 −2.13012
$$238$$ 0 0
$$239$$ 25.5592 1.65329 0.826644 0.562726i $$-0.190247\pi$$
0.826644 + 0.562726i $$0.190247\pi$$
$$240$$ 0 0
$$241$$ −16.1617 −1.04106 −0.520532 0.853842i $$-0.674266\pi$$
−0.520532 + 0.853842i $$0.674266\pi$$
$$242$$ 0 0
$$243$$ 10.1852 0.653380
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.986440 −0.0627657
$$248$$ 0 0
$$249$$ −17.3912 −1.10212
$$250$$ 0 0
$$251$$ 13.3497 0.842627 0.421313 0.906915i $$-0.361569\pi$$
0.421313 + 0.906915i $$0.361569\pi$$
$$252$$ 0 0
$$253$$ −4.65190 −0.292463
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −10.1834 −0.635222 −0.317611 0.948221i $$-0.602881\pi$$
−0.317611 + 0.948221i $$0.602881\pi$$
$$258$$ 0 0
$$259$$ −3.13345 −0.194703
$$260$$ 0 0
$$261$$ −5.28610 −0.327201
$$262$$ 0 0
$$263$$ −23.3073 −1.43719 −0.718594 0.695430i $$-0.755214\pi$$
−0.718594 + 0.695430i $$0.755214\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 27.9573 1.71096
$$268$$ 0 0
$$269$$ 9.80817 0.598015 0.299007 0.954251i $$-0.403344\pi$$
0.299007 + 0.954251i $$0.403344\pi$$
$$270$$ 0 0
$$271$$ 6.08332 0.369535 0.184768 0.982782i $$-0.440847\pi$$
0.184768 + 0.982782i $$0.440847\pi$$
$$272$$ 0 0
$$273$$ 8.59549 0.520223
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −0.0445722 −0.00267809 −0.00133904 0.999999i $$-0.500426\pi$$
−0.00133904 + 0.999999i $$0.500426\pi$$
$$278$$ 0 0
$$279$$ 13.7404 0.822617
$$280$$ 0 0
$$281$$ 27.9996 1.67032 0.835158 0.550010i $$-0.185376\pi$$
0.835158 + 0.550010i $$0.185376\pi$$
$$282$$ 0 0
$$283$$ −12.1798 −0.724013 −0.362006 0.932176i $$-0.617908\pi$$
−0.362006 + 0.932176i $$0.617908\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.71802 0.160440
$$288$$ 0 0
$$289$$ −15.2346 −0.896155
$$290$$ 0 0
$$291$$ 43.3902 2.54358
$$292$$ 0 0
$$293$$ −2.71056 −0.158352 −0.0791762 0.996861i $$-0.525229\pi$$
−0.0791762 + 0.996861i $$0.525229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −50.3124 −2.91942
$$298$$ 0 0
$$299$$ −5.02256 −0.290462
$$300$$ 0 0
$$301$$ −0.924148 −0.0532670
$$302$$ 0 0
$$303$$ 23.2188 1.33388
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.1631 1.83564 0.917821 0.396994i $$-0.129947\pi$$
0.917821 + 0.396994i $$0.129947\pi$$
$$308$$ 0 0
$$309$$ −44.5484 −2.53427
$$310$$ 0 0
$$311$$ 4.07951 0.231328 0.115664 0.993288i $$-0.463100\pi$$
0.115664 + 0.993288i $$0.463100\pi$$
$$312$$ 0 0
$$313$$ 13.7806 0.778925 0.389462 0.921042i $$-0.372661\pi$$
0.389462 + 0.921042i $$0.372661\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.4630 0.587658 0.293829 0.955858i $$-0.405070\pi$$
0.293829 + 0.955858i $$0.405070\pi$$
$$318$$ 0 0
$$319$$ 3.77873 0.211568
$$320$$ 0 0
$$321$$ 7.27266 0.405920
$$322$$ 0 0
$$323$$ −0.260954 −0.0145199
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −31.9377 −1.76616
$$328$$ 0 0
$$329$$ 5.49088 0.302722
$$330$$ 0 0
$$331$$ 12.6451 0.695038 0.347519 0.937673i $$-0.387024\pi$$
0.347519 + 0.937673i $$0.387024\pi$$
$$332$$ 0 0
$$333$$ 36.7395 2.01331
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.5733 1.33859 0.669295 0.742997i $$-0.266596\pi$$
0.669295 + 0.742997i $$0.266596\pi$$
$$338$$ 0 0
$$339$$ 59.4248 3.22751
$$340$$ 0 0
$$341$$ −9.82224 −0.531904
$$342$$ 0 0
$$343$$ 7.59933 0.410325
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.50569 0.188195 0.0940977 0.995563i $$-0.470003\pi$$
0.0940977 + 0.995563i $$0.470003\pi$$
$$348$$ 0 0
$$349$$ −18.1085 −0.969327 −0.484664 0.874701i $$-0.661058\pi$$
−0.484664 + 0.874701i $$0.661058\pi$$
$$350$$ 0 0
$$351$$ −54.3212 −2.89945
$$352$$ 0 0
$$353$$ −20.1915 −1.07469 −0.537343 0.843364i $$-0.680572\pi$$
−0.537343 + 0.843364i $$0.680572\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 2.27386 0.120345
$$358$$ 0 0
$$359$$ 9.43409 0.497912 0.248956 0.968515i $$-0.419912\pi$$
0.248956 + 0.968515i $$0.419912\pi$$
$$360$$ 0 0
$$361$$ −18.9614 −0.997970
$$362$$ 0 0
$$363$$ 32.8084 1.72199
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 30.3412 1.58380 0.791899 0.610652i $$-0.209092\pi$$
0.791899 + 0.610652i $$0.209092\pi$$
$$368$$ 0 0
$$369$$ −31.8686 −1.65901
$$370$$ 0 0
$$371$$ 1.23915 0.0643333
$$372$$ 0 0
$$373$$ 14.7893 0.765758 0.382879 0.923798i $$-0.374933\pi$$
0.382879 + 0.923798i $$0.374933\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.07982 0.210121
$$378$$ 0 0
$$379$$ 11.2405 0.577385 0.288693 0.957422i $$-0.406779\pi$$
0.288693 + 0.957422i $$0.406779\pi$$
$$380$$ 0 0
$$381$$ −57.0231 −2.92138
$$382$$ 0 0
$$383$$ 30.4602 1.55644 0.778221 0.627991i $$-0.216122\pi$$
0.778221 + 0.627991i $$0.216122\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.8356 0.550803
$$388$$ 0 0
$$389$$ 1.94011 0.0983673 0.0491836 0.998790i $$-0.484338\pi$$
0.0491836 + 0.998790i $$0.484338\pi$$
$$390$$ 0 0
$$391$$ −1.32867 −0.0671939
$$392$$ 0 0
$$393$$ 45.5509 2.29774
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 18.3206 0.919487 0.459743 0.888052i $$-0.347941\pi$$
0.459743 + 0.888052i $$0.347941\pi$$
$$398$$ 0 0
$$399$$ −0.336117 −0.0168269
$$400$$ 0 0
$$401$$ −1.32199 −0.0660170 −0.0330085 0.999455i $$-0.510509\pi$$
−0.0330085 + 0.999455i $$0.510509\pi$$
$$402$$ 0 0
$$403$$ −10.6049 −0.528266
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −26.2630 −1.30181
$$408$$ 0 0
$$409$$ 21.9963 1.08765 0.543823 0.839200i $$-0.316976\pi$$
0.543823 + 0.839200i $$0.316976\pi$$
$$410$$ 0 0
$$411$$ −35.8212 −1.76693
$$412$$ 0 0
$$413$$ 1.35141 0.0664985
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −34.3460 −1.68193
$$418$$ 0 0
$$419$$ −28.3194 −1.38349 −0.691746 0.722141i $$-0.743158\pi$$
−0.691746 + 0.722141i $$0.743158\pi$$
$$420$$ 0 0
$$421$$ 14.7650 0.719602 0.359801 0.933029i $$-0.382845\pi$$
0.359801 + 0.933029i $$0.382845\pi$$
$$422$$ 0 0
$$423$$ −64.3802 −3.13027
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.16994 0.153404
$$428$$ 0 0
$$429$$ 72.0429 3.47826
$$430$$ 0 0
$$431$$ 27.0514 1.30302 0.651510 0.758640i $$-0.274136\pi$$
0.651510 + 0.758640i $$0.274136\pi$$
$$432$$ 0 0
$$433$$ 38.2700 1.83914 0.919570 0.392926i $$-0.128537\pi$$
0.919570 + 0.392926i $$0.128537\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0.196402 0.00939517
$$438$$ 0 0
$$439$$ 2.39959 0.114526 0.0572630 0.998359i $$-0.481763\pi$$
0.0572630 + 0.998359i $$0.481763\pi$$
$$440$$ 0 0
$$441$$ −43.5485 −2.07374
$$442$$ 0 0
$$443$$ 10.6542 0.506198 0.253099 0.967440i $$-0.418550\pi$$
0.253099 + 0.967440i $$0.418550\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −38.4800 −1.82004
$$448$$ 0 0
$$449$$ −28.9533 −1.36639 −0.683195 0.730236i $$-0.739410\pi$$
−0.683195 + 0.730236i $$0.739410\pi$$
$$450$$ 0 0
$$451$$ 22.7810 1.07272
$$452$$ 0 0
$$453$$ −4.09090 −0.192207
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.5855 1.80496 0.902478 0.430736i $$-0.141746\pi$$
0.902478 + 0.430736i $$0.141746\pi$$
$$458$$ 0 0
$$459$$ −14.3702 −0.670743
$$460$$ 0 0
$$461$$ −8.07659 −0.376164 −0.188082 0.982153i $$-0.560227\pi$$
−0.188082 + 0.982153i $$0.560227\pi$$
$$462$$ 0 0
$$463$$ 6.07771 0.282455 0.141228 0.989977i $$-0.454895\pi$$
0.141228 + 0.989977i $$0.454895\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −10.2961 −0.476446 −0.238223 0.971210i $$-0.576565\pi$$
−0.238223 + 0.971210i $$0.576565\pi$$
$$468$$ 0 0
$$469$$ −3.83981 −0.177306
$$470$$ 0 0
$$471$$ −71.4598 −3.29270
$$472$$ 0 0
$$473$$ −7.74572 −0.356149
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −14.5289 −0.665233
$$478$$ 0 0
$$479$$ 33.9760 1.55240 0.776201 0.630486i $$-0.217144\pi$$
0.776201 + 0.630486i $$0.217144\pi$$
$$480$$ 0 0
$$481$$ −28.3556 −1.29290
$$482$$ 0 0
$$483$$ −1.71138 −0.0778703
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.4951 0.747464 0.373732 0.927537i $$-0.378078\pi$$
0.373732 + 0.927537i $$0.378078\pi$$
$$488$$ 0 0
$$489$$ 77.2962 3.49546
$$490$$ 0 0
$$491$$ −27.4876 −1.24050 −0.620249 0.784405i $$-0.712968\pi$$
−0.620249 + 0.784405i $$0.712968\pi$$
$$492$$ 0 0
$$493$$ 1.07928 0.0486082
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0.00668310 0.000299778 0
$$498$$ 0 0
$$499$$ −18.2961 −0.819045 −0.409523 0.912300i $$-0.634305\pi$$
−0.409523 + 0.912300i $$0.634305\pi$$
$$500$$ 0 0
$$501$$ −13.5549 −0.605587
$$502$$ 0 0
$$503$$ −1.97389 −0.0880115 −0.0440058 0.999031i $$-0.514012\pi$$
−0.0440058 + 0.999031i $$0.514012\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 37.6986 1.67425
$$508$$ 0 0
$$509$$ −38.3441 −1.69957 −0.849787 0.527126i $$-0.823269\pi$$
−0.849787 + 0.527126i $$0.823269\pi$$
$$510$$ 0 0
$$511$$ 8.49125 0.375631
$$512$$ 0 0
$$513$$ 2.12417 0.0937844
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 46.0217 2.02403
$$518$$ 0 0
$$519$$ −59.8839 −2.62861
$$520$$ 0 0
$$521$$ −23.1814 −1.01560 −0.507799 0.861476i $$-0.669541\pi$$
−0.507799 + 0.861476i $$0.669541\pi$$
$$522$$ 0 0
$$523$$ −43.5123 −1.90266 −0.951331 0.308172i $$-0.900283\pi$$
−0.951331 + 0.308172i $$0.900283\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.80542 −0.122206
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −15.8452 −0.687622
$$532$$ 0 0
$$533$$ 24.5962 1.06538
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −32.1197 −1.38607
$$538$$ 0 0
$$539$$ 31.1303 1.34088
$$540$$ 0 0
$$541$$ 18.6112 0.800158 0.400079 0.916481i $$-0.368983\pi$$
0.400079 + 0.916481i $$0.368983\pi$$
$$542$$ 0 0
$$543$$ −28.1732 −1.20903
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6.78956 0.290300 0.145150 0.989410i $$-0.453633\pi$$
0.145150 + 0.989410i $$0.453633\pi$$
$$548$$ 0 0
$$549$$ −37.1673 −1.58626
$$550$$ 0 0
$$551$$ −0.159537 −0.00679649
$$552$$ 0 0
$$553$$ 5.90272 0.251009
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24.9869 1.05873 0.529364 0.848395i $$-0.322430\pi$$
0.529364 + 0.848395i $$0.322430\pi$$
$$558$$ 0 0
$$559$$ −8.36290 −0.353713
$$560$$ 0 0
$$561$$ 19.0583 0.804642
$$562$$ 0 0
$$563$$ 44.7551 1.88620 0.943100 0.332508i $$-0.107895\pi$$
0.943100 + 0.332508i $$0.107895\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −7.67371 −0.322265
$$568$$ 0 0
$$569$$ −27.4895 −1.15242 −0.576211 0.817301i $$-0.695469\pi$$
−0.576211 + 0.817301i $$0.695469\pi$$
$$570$$ 0 0
$$571$$ 1.40495 0.0587952 0.0293976 0.999568i $$-0.490641\pi$$
0.0293976 + 0.999568i $$0.490641\pi$$
$$572$$ 0 0
$$573$$ 4.61957 0.192985
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −19.1987 −0.799253 −0.399627 0.916678i $$-0.630860\pi$$
−0.399627 + 0.916678i $$0.630860\pi$$
$$578$$ 0 0
$$579$$ −44.4364 −1.84671
$$580$$ 0 0
$$581$$ 3.13043 0.129872
$$582$$ 0 0
$$583$$ 10.3859 0.430139
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 22.8565 0.943390 0.471695 0.881762i $$-0.343642\pi$$
0.471695 + 0.881762i $$0.343642\pi$$
$$588$$ 0 0
$$589$$ 0.414692 0.0170871
$$590$$ 0 0
$$591$$ −75.4473 −3.10349
$$592$$ 0 0
$$593$$ −26.5387 −1.08981 −0.544906 0.838497i $$-0.683435\pi$$
−0.544906 + 0.838497i $$0.683435\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.97005 0.203411
$$598$$ 0 0
$$599$$ 8.82768 0.360689 0.180345 0.983603i $$-0.442279\pi$$
0.180345 + 0.983603i $$0.442279\pi$$
$$600$$ 0 0
$$601$$ −30.3072 −1.23626 −0.618129 0.786077i $$-0.712109\pi$$
−0.618129 + 0.786077i $$0.712109\pi$$
$$602$$ 0 0
$$603$$ 45.0215 1.83342
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −38.0102 −1.54278 −0.771392 0.636360i $$-0.780439\pi$$
−0.771392 + 0.636360i $$0.780439\pi$$
$$608$$ 0 0
$$609$$ 1.39015 0.0563316
$$610$$ 0 0
$$611$$ 49.6887 2.01019
$$612$$ 0 0
$$613$$ −31.7417 −1.28204 −0.641018 0.767526i $$-0.721488\pi$$
−0.641018 + 0.767526i $$0.721488\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 46.6699 1.87886 0.939430 0.342741i $$-0.111355\pi$$
0.939430 + 0.342741i $$0.111355\pi$$
$$618$$ 0 0
$$619$$ −23.2223 −0.933383 −0.466692 0.884420i $$-0.654554\pi$$
−0.466692 + 0.884420i $$0.654554\pi$$
$$620$$ 0 0
$$621$$ 10.8154 0.434009
$$622$$ 0 0
$$623$$ −5.03235 −0.201617
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −2.81716 −0.112506
$$628$$ 0 0
$$629$$ −7.50121 −0.299093
$$630$$ 0 0
$$631$$ 13.0906 0.521129 0.260565 0.965456i $$-0.416091\pi$$
0.260565 + 0.965456i $$0.416091\pi$$
$$632$$ 0 0
$$633$$ −19.2740 −0.766072
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 33.6107 1.33171
$$638$$ 0 0
$$639$$ −0.0783588 −0.00309983
$$640$$ 0 0
$$641$$ 22.8746 0.903494 0.451747 0.892146i $$-0.350801\pi$$
0.451747 + 0.892146i $$0.350801\pi$$
$$642$$ 0 0
$$643$$ 30.6333 1.20806 0.604030 0.796962i $$-0.293561\pi$$
0.604030 + 0.796962i $$0.293561\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.74454 0.343783 0.171892 0.985116i $$-0.445012\pi$$
0.171892 + 0.985116i $$0.445012\pi$$
$$648$$ 0 0
$$649$$ 11.3268 0.444616
$$650$$ 0 0
$$651$$ −3.61348 −0.141623
$$652$$ 0 0
$$653$$ −24.7443 −0.968320 −0.484160 0.874979i $$-0.660875\pi$$
−0.484160 + 0.874979i $$0.660875\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −99.5593 −3.88418
$$658$$ 0 0
$$659$$ −8.30726 −0.323605 −0.161803 0.986823i $$-0.551731\pi$$
−0.161803 + 0.986823i $$0.551731\pi$$
$$660$$ 0 0
$$661$$ −14.1755 −0.551364 −0.275682 0.961249i $$-0.588904\pi$$
−0.275682 + 0.961249i $$0.588904\pi$$
$$662$$ 0 0
$$663$$ 20.5768 0.799138
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −0.812298 −0.0314523
$$668$$ 0 0
$$669$$ 65.6627 2.53867
$$670$$ 0 0
$$671$$ 26.5688 1.02568
$$672$$ 0 0
$$673$$ 7.30229 0.281482 0.140741 0.990046i $$-0.455051\pi$$
0.140741 + 0.990046i $$0.455051\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 39.7090 1.52614 0.763071 0.646315i $$-0.223691\pi$$
0.763071 + 0.646315i $$0.223691\pi$$
$$678$$ 0 0
$$679$$ −7.81027 −0.299731
$$680$$ 0 0
$$681$$ 32.4296 1.24271
$$682$$ 0 0
$$683$$ 38.5993 1.47696 0.738481 0.674274i $$-0.235543\pi$$
0.738481 + 0.674274i $$0.235543\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −74.0219 −2.82411
$$688$$ 0 0
$$689$$ 11.2134 0.427198
$$690$$ 0 0
$$691$$ 21.4163 0.814713 0.407357 0.913269i $$-0.366451\pi$$
0.407357 + 0.913269i $$0.366451\pi$$
$$692$$ 0 0
$$693$$ 16.8020 0.638255
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.50669 0.246459
$$698$$ 0 0
$$699$$ 34.8773 1.31918
$$700$$ 0 0
$$701$$ −5.40147 −0.204011 −0.102005 0.994784i $$-0.532526\pi$$
−0.102005 + 0.994784i $$0.532526\pi$$
$$702$$ 0 0
$$703$$ 1.10881 0.0418197
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −4.17940 −0.157182
$$708$$ 0 0
$$709$$ −15.3161 −0.575208 −0.287604 0.957749i $$-0.592859\pi$$
−0.287604 + 0.957749i $$0.592859\pi$$
$$710$$ 0 0
$$711$$ −69.2090 −2.59554
$$712$$ 0 0
$$713$$ 2.11145 0.0790742
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 78.8102 2.94322
$$718$$ 0 0
$$719$$ −11.6505 −0.434491 −0.217245 0.976117i $$-0.569707\pi$$
−0.217245 + 0.976117i $$0.569707\pi$$
$$720$$ 0 0
$$721$$ 8.01875 0.298634
$$722$$ 0 0
$$723$$ −49.8334 −1.85333
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.8188 1.06883 0.534415 0.845223i $$-0.320532\pi$$
0.534415 + 0.845223i $$0.320532\pi$$
$$728$$ 0 0
$$729$$ −10.0725 −0.373056
$$730$$ 0 0
$$731$$ −2.21233 −0.0818259
$$732$$ 0 0
$$733$$ 25.2304 0.931906 0.465953 0.884810i $$-0.345712\pi$$
0.465953 + 0.884810i $$0.345712\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −32.1833 −1.18549
$$738$$ 0 0
$$739$$ 1.22800 0.0451728 0.0225864 0.999745i $$-0.492810\pi$$
0.0225864 + 0.999745i $$0.492810\pi$$
$$740$$ 0 0
$$741$$ −3.04163 −0.111737
$$742$$ 0 0
$$743$$ 35.1117 1.28812 0.644061 0.764974i $$-0.277248\pi$$
0.644061 + 0.764974i $$0.277248\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −36.7041 −1.34293
$$748$$ 0 0
$$749$$ −1.30909 −0.0478329
$$750$$ 0 0
$$751$$ −31.7367 −1.15809 −0.579044 0.815296i $$-0.696574\pi$$
−0.579044 + 0.815296i $$0.696574\pi$$
$$752$$ 0 0
$$753$$ 41.1630 1.50006
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.22287 0.117137 0.0585685 0.998283i $$-0.481346\pi$$
0.0585685 + 0.998283i $$0.481346\pi$$
$$758$$ 0 0
$$759$$ −14.3439 −0.520649
$$760$$ 0 0
$$761$$ −9.60077 −0.348027 −0.174014 0.984743i $$-0.555674\pi$$
−0.174014 + 0.984743i $$0.555674\pi$$
$$762$$ 0 0
$$763$$ 5.74882 0.208121
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.2293 0.441575
$$768$$ 0 0
$$769$$ 14.9778 0.540113 0.270056 0.962844i $$-0.412958\pi$$
0.270056 + 0.962844i $$0.412958\pi$$
$$770$$ 0 0
$$771$$ −31.3998 −1.13084
$$772$$ 0 0
$$773$$ −28.7430 −1.03381 −0.516906 0.856042i $$-0.672916\pi$$
−0.516906 + 0.856042i $$0.672916\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −9.66181 −0.346616
$$778$$ 0 0
$$779$$ −0.961806 −0.0344603
$$780$$ 0 0
$$781$$ 0.0560142 0.00200435
$$782$$ 0 0
$$783$$ −8.78536 −0.313963
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −34.1965 −1.21897 −0.609486 0.792797i $$-0.708624\pi$$
−0.609486 + 0.792797i $$0.708624\pi$$
$$788$$ 0 0
$$789$$ −71.8666 −2.55852
$$790$$ 0 0
$$791$$ −10.6965 −0.380324
$$792$$ 0 0
$$793$$ 28.6857 1.01866
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.303391 0.0107467 0.00537334 0.999986i $$-0.498290\pi$$
0.00537334 + 0.999986i $$0.498290\pi$$
$$798$$ 0 0
$$799$$ 13.1447 0.465026
$$800$$ 0 0
$$801$$ 59.0039 2.08480
$$802$$ 0 0
$$803$$ 71.1692 2.51151
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 30.2429 1.06460
$$808$$ 0 0
$$809$$ −13.1865 −0.463611 −0.231806 0.972762i $$-0.574463\pi$$
−0.231806 + 0.972762i $$0.574463\pi$$
$$810$$ 0 0
$$811$$ −14.7284 −0.517185 −0.258593 0.965986i $$-0.583259\pi$$
−0.258593 + 0.965986i $$0.583259\pi$$
$$812$$ 0 0
$$813$$ 18.7575 0.657856
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0.327022 0.0114410
$$818$$ 0 0
$$819$$ 18.1408 0.633890
$$820$$ 0 0
$$821$$ 36.6177 1.27797 0.638984 0.769220i $$-0.279355\pi$$
0.638984 + 0.769220i $$0.279355\pi$$
$$822$$ 0 0
$$823$$ 0.420867 0.0146705 0.00733525 0.999973i $$-0.497665\pi$$
0.00733525 + 0.999973i $$0.497665\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 35.1118 1.22096 0.610479 0.792032i $$-0.290977\pi$$
0.610479 + 0.792032i $$0.290977\pi$$
$$828$$ 0 0
$$829$$ −35.6234 −1.23725 −0.618626 0.785685i $$-0.712311\pi$$
−0.618626 + 0.785685i $$0.712311\pi$$
$$830$$ 0 0
$$831$$ −0.137436 −0.00476759
$$832$$ 0 0
$$833$$ 8.89141 0.308069
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 22.8362 0.789335
$$838$$ 0 0
$$839$$ 19.9998 0.690469 0.345235 0.938516i $$-0.387799\pi$$
0.345235 + 0.938516i $$0.387799\pi$$
$$840$$ 0 0
$$841$$ −28.3402 −0.977247
$$842$$ 0 0
$$843$$ 86.3350 2.97354
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −5.90554 −0.202917
$$848$$ 0 0
$$849$$ −37.5556 −1.28890
$$850$$ 0 0
$$851$$ 5.64564 0.193530
$$852$$ 0 0
$$853$$ −42.4994 −1.45515 −0.727577 0.686026i $$-0.759353\pi$$
−0.727577 + 0.686026i $$0.759353\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 18.0100 0.615208 0.307604 0.951514i $$-0.400473\pi$$
0.307604 + 0.951514i $$0.400473\pi$$
$$858$$ 0 0
$$859$$ −22.3869 −0.763832 −0.381916 0.924197i $$-0.624736\pi$$
−0.381916 + 0.924197i $$0.624736\pi$$
$$860$$ 0 0
$$861$$ 8.38084 0.285618
$$862$$ 0 0
$$863$$ 18.7613 0.638642 0.319321 0.947647i $$-0.396545\pi$$
0.319321 + 0.947647i $$0.396545\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −46.9750 −1.59536
$$868$$ 0 0
$$869$$ 49.4735 1.67827
$$870$$ 0 0
$$871$$ −34.7476 −1.17738
$$872$$ 0 0
$$873$$ 91.5749 3.09934
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31.6345 1.06822 0.534111 0.845415i $$-0.320647\pi$$
0.534111 + 0.845415i $$0.320647\pi$$
$$878$$ 0 0
$$879$$ −8.35783 −0.281903
$$880$$ 0 0
$$881$$ 39.6850 1.33702 0.668512 0.743702i $$-0.266932\pi$$
0.668512 + 0.743702i $$0.266932\pi$$
$$882$$ 0 0
$$883$$ −10.3778 −0.349242 −0.174621 0.984636i $$-0.555870\pi$$
−0.174621 + 0.984636i $$0.555870\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0199 0.806508 0.403254 0.915088i $$-0.367879\pi$$
0.403254 + 0.915088i $$0.367879\pi$$
$$888$$ 0 0
$$889$$ 10.2642 0.344251
$$890$$ 0 0
$$891$$ −64.3170 −2.15470
$$892$$ 0 0
$$893$$ −1.94302 −0.0650207
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −15.4868 −0.517088
$$898$$ 0 0
$$899$$ −1.71512 −0.0572025
$$900$$ 0 0
$$901$$ 2.96641 0.0988254
$$902$$ 0 0
$$903$$ −2.84955 −0.0948271
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −18.5181 −0.614885 −0.307442 0.951567i $$-0.599473\pi$$
−0.307442 + 0.951567i $$0.599473\pi$$
$$908$$ 0 0
$$909$$ 49.0032 1.62533
$$910$$ 0 0
$$911$$ 3.34956 0.110976 0.0554879 0.998459i $$-0.482329\pi$$
0.0554879 + 0.998459i $$0.482329\pi$$
$$912$$ 0 0
$$913$$ 26.2376 0.868339
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −8.19920 −0.270761
$$918$$ 0 0
$$919$$ 32.9754 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$920$$ 0 0
$$921$$ 99.1728 3.26785
$$922$$ 0 0
$$923$$ 0.0604774 0.00199064
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −94.0193 −3.08800
$$928$$ 0 0
$$929$$ −36.5008 −1.19755 −0.598777 0.800916i $$-0.704346\pi$$
−0.598777 + 0.800916i $$0.704346\pi$$
$$930$$ 0 0
$$931$$ −1.31431 −0.0430748
$$932$$ 0 0
$$933$$ 12.5789 0.411816
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 12.4598 0.407043 0.203521 0.979071i $$-0.434761\pi$$
0.203521 + 0.979071i $$0.434761\pi$$
$$938$$ 0 0
$$939$$ 42.4916 1.38666
$$940$$ 0 0
$$941$$ −45.2938 −1.47653 −0.738267 0.674508i $$-0.764356\pi$$
−0.738267 + 0.674508i $$0.764356\pi$$
$$942$$ 0 0
$$943$$ −4.89714 −0.159473
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −41.4016 −1.34537 −0.672685 0.739929i $$-0.734859\pi$$
−0.672685 + 0.739929i $$0.734859\pi$$
$$948$$ 0 0
$$949$$ 76.8399 2.49433
$$950$$ 0 0
$$951$$ 32.2619 1.04616
$$952$$ 0 0
$$953$$ −15.3738 −0.498006 −0.249003 0.968503i $$-0.580103\pi$$
−0.249003 + 0.968503i $$0.580103\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 11.6515 0.376639
$$958$$ 0 0
$$959$$ 6.44785 0.208212
$$960$$ 0 0
$$961$$ −26.5418 −0.856187
$$962$$ 0 0
$$963$$ 15.3489 0.494612
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −53.4925 −1.72020 −0.860101 0.510124i $$-0.829600\pi$$
−0.860101 + 0.510124i $$0.829600\pi$$
$$968$$ 0 0
$$969$$ −0.804635 −0.0258486
$$970$$ 0 0
$$971$$ −19.8881 −0.638241 −0.319120 0.947714i $$-0.603387\pi$$
−0.319120 + 0.947714i $$0.603387\pi$$
$$972$$ 0 0
$$973$$ 6.18231 0.198196
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −14.8758 −0.475919 −0.237960 0.971275i $$-0.576479\pi$$
−0.237960 + 0.971275i $$0.576479\pi$$
$$978$$ 0 0
$$979$$ −42.1785 −1.34803
$$980$$ 0 0
$$981$$ −67.4045 −2.15206
$$982$$ 0 0
$$983$$ 1.16403 0.0371269 0.0185634 0.999828i $$-0.494091\pi$$
0.0185634 + 0.999828i $$0.494091\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.9308 0.538913
$$988$$ 0 0
$$989$$ 1.66507 0.0529460
$$990$$ 0 0
$$991$$ 47.8693 1.52062 0.760310 0.649561i $$-0.225047\pi$$
0.760310 + 0.649561i $$0.225047\pi$$
$$992$$ 0 0
$$993$$ 38.9904 1.23732
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −11.4353 −0.362159 −0.181079 0.983468i $$-0.557959\pi$$
−0.181079 + 0.983468i $$0.557959\pi$$
$$998$$ 0 0
$$999$$ 61.0601 1.93186
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 9200.2.a.cw.1.5 5
4.3 odd 2 4600.2.a.bc.1.1 5
5.4 even 2 9200.2.a.cs.1.1 5
20.3 even 4 4600.2.e.v.4049.1 10
20.7 even 4 4600.2.e.v.4049.10 10
20.19 odd 2 4600.2.a.bg.1.5 yes 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.1 5 4.3 odd 2
4600.2.a.bg.1.5 yes 5 20.19 odd 2
4600.2.e.v.4049.1 10 20.3 even 4
4600.2.e.v.4049.10 10 20.7 even 4
9200.2.a.cs.1.1 5 5.4 even 2
9200.2.a.cw.1.5 5 1.1 even 1 trivial