# Properties

 Label 1728.3.e.u.1025.2 Level $1728$ Weight $3$ Character 1728.1025 Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1025.2 Root $$-0.965926 + 0.258819i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1025 Dual form 1728.3.e.u.1025.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-6.61037i q^{5} +9.43879 q^{7} +O(q^{10})$$ $$q-6.61037i q^{5} +9.43879 q^{7} +17.6969i q^{11} -20.6969 q^{13} -16.0492i q^{17} +12.2993 q^{19} -26.6969i q^{23} -18.6969 q^{25} -0.921404i q^{29} +53.7722 q^{31} -62.3939i q^{35} +64.0908 q^{37} -19.7990i q^{41} -69.7893 q^{43} -0.606123i q^{47} +40.0908 q^{49} -45.3512i q^{53} +116.983 q^{55} -71.3939i q^{59} +4.00000 q^{61} +136.814i q^{65} -7.43545 q^{67} -27.3031i q^{71} +6.30306 q^{73} +167.038i q^{77} -42.6191 q^{79} -72.3031i q^{83} -106.091 q^{85} -126.486i q^{89} -195.354 q^{91} -81.3031i q^{95} +27.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100})$$ 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ − 6.61037i − 1.32207i −0.750354 0.661037i $$-0.770117\pi$$
0.750354 0.661037i $$-0.229883\pi$$
$$6$$ 0 0
$$7$$ 9.43879 1.34840 0.674200 0.738549i $$-0.264489\pi$$
0.674200 + 0.738549i $$0.264489\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 17.6969i 1.60881i 0.594080 + 0.804406i $$0.297516\pi$$
−0.594080 + 0.804406i $$0.702484\pi$$
$$12$$ 0 0
$$13$$ −20.6969 −1.59207 −0.796036 0.605249i $$-0.793073\pi$$
−0.796036 + 0.605249i $$0.793073\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 16.0492i − 0.944068i −0.881580 0.472034i $$-0.843520\pi$$
0.881580 0.472034i $$-0.156480\pi$$
$$18$$ 0 0
$$19$$ 12.2993 0.647333 0.323667 0.946171i $$-0.395084\pi$$
0.323667 + 0.946171i $$0.395084\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 26.6969i − 1.16074i −0.814354 0.580368i $$-0.802909\pi$$
0.814354 0.580368i $$-0.197091\pi$$
$$24$$ 0 0
$$25$$ −18.6969 −0.747878
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 0.921404i − 0.0317725i −0.999874 0.0158863i $$-0.994943\pi$$
0.999874 0.0158863i $$-0.00505697\pi$$
$$30$$ 0 0
$$31$$ 53.7722 1.73459 0.867294 0.497796i $$-0.165857\pi$$
0.867294 + 0.497796i $$0.165857\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 62.3939i − 1.78268i
$$36$$ 0 0
$$37$$ 64.0908 1.73218 0.866092 0.499884i $$-0.166624\pi$$
0.866092 + 0.499884i $$0.166624\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 19.7990i − 0.482902i −0.970413 0.241451i $$-0.922377\pi$$
0.970413 0.241451i $$-0.0776233\pi$$
$$42$$ 0 0
$$43$$ −69.7893 −1.62301 −0.811503 0.584348i $$-0.801350\pi$$
−0.811503 + 0.584348i $$0.801350\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 0.606123i − 0.0128962i −0.999979 0.00644812i $$-0.997947\pi$$
0.999979 0.00644812i $$-0.00205251\pi$$
$$48$$ 0 0
$$49$$ 40.0908 0.818180
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 45.3512i − 0.855682i −0.903854 0.427841i $$-0.859274\pi$$
0.903854 0.427841i $$-0.140726\pi$$
$$54$$ 0 0
$$55$$ 116.983 2.12697
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 71.3939i − 1.21007i −0.796201 0.605033i $$-0.793160\pi$$
0.796201 0.605033i $$-0.206840\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.0655738 0.0327869 0.999462i $$-0.489562\pi$$
0.0327869 + 0.999462i $$0.489562\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 136.814i 2.10484i
$$66$$ 0 0
$$67$$ −7.43545 −0.110977 −0.0554884 0.998459i $$-0.517672\pi$$
−0.0554884 + 0.998459i $$0.517672\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 27.3031i − 0.384550i −0.981341 0.192275i $$-0.938413\pi$$
0.981341 0.192275i $$-0.0615866\pi$$
$$72$$ 0 0
$$73$$ 6.30306 0.0863433 0.0431717 0.999068i $$-0.486254\pi$$
0.0431717 + 0.999068i $$0.486254\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 167.038i 2.16932i
$$78$$ 0 0
$$79$$ −42.6191 −0.539482 −0.269741 0.962933i $$-0.586938\pi$$
−0.269741 + 0.962933i $$0.586938\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ − 72.3031i − 0.871121i −0.900159 0.435561i $$-0.856550\pi$$
0.900159 0.435561i $$-0.143450\pi$$
$$84$$ 0 0
$$85$$ −106.091 −1.24813
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 126.486i − 1.42119i −0.703599 0.710597i $$-0.748425\pi$$
0.703599 0.710597i $$-0.251575\pi$$
$$90$$ 0 0
$$91$$ −195.354 −2.14675
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 81.3031i − 0.855822i
$$96$$ 0 0
$$97$$ 27.0000 0.278351 0.139175 0.990268i $$-0.455555\pi$$
0.139175 + 0.990268i $$0.455555\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 33.7806i − 0.334461i −0.985918 0.167231i $$-0.946518\pi$$
0.985918 0.167231i $$-0.0534824\pi$$
$$102$$ 0 0
$$103$$ 95.2451 0.924710 0.462355 0.886695i $$-0.347005\pi$$
0.462355 + 0.886695i $$0.347005\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 96.5755i 0.902575i 0.892379 + 0.451287i $$0.149035\pi$$
−0.892379 + 0.451287i $$0.850965\pi$$
$$108$$ 0 0
$$109$$ −90.8786 −0.833748 −0.416874 0.908964i $$-0.636874\pi$$
−0.416874 + 0.908964i $$0.636874\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 50.1829i 0.444097i 0.975036 + 0.222048i $$0.0712743\pi$$
−0.975036 + 0.222048i $$0.928726\pi$$
$$114$$ 0 0
$$115$$ −176.477 −1.53458
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 151.485i − 1.27298i
$$120$$ 0 0
$$121$$ −192.182 −1.58828
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 41.6655i − 0.333324i
$$126$$ 0 0
$$127$$ −134.146 −1.05627 −0.528136 0.849160i $$-0.677109\pi$$
−0.528136 + 0.849160i $$0.677109\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 46.8184i − 0.357392i −0.983904 0.178696i $$-0.942812\pi$$
0.983904 0.178696i $$-0.0571879\pi$$
$$132$$ 0 0
$$133$$ 116.091 0.872863
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 182.926i 1.33523i 0.744507 + 0.667614i $$0.232685\pi$$
−0.744507 + 0.667614i $$0.767315\pi$$
$$138$$ 0 0
$$139$$ −137.864 −0.991829 −0.495914 0.868371i $$-0.665167\pi$$
−0.495914 + 0.868371i $$0.665167\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 366.272i − 2.56135i
$$144$$ 0 0
$$145$$ −6.09082 −0.0420056
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 43.6368i − 0.292864i −0.989221 0.146432i $$-0.953221\pi$$
0.989221 0.146432i $$-0.0467790\pi$$
$$150$$ 0 0
$$151$$ 84.0920 0.556900 0.278450 0.960451i $$-0.410179\pi$$
0.278450 + 0.960451i $$0.410179\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 355.454i − 2.29325i
$$156$$ 0 0
$$157$$ 236.545 1.50666 0.753328 0.657645i $$-0.228447\pi$$
0.753328 + 0.657645i $$0.228447\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 251.987i − 1.56514i
$$162$$ 0 0
$$163$$ 282.307 1.73194 0.865971 0.500094i $$-0.166701\pi$$
0.865971 + 0.500094i $$0.166701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 70.1816i 0.420249i 0.977675 + 0.210125i $$0.0673870\pi$$
−0.977675 + 0.210125i $$0.932613\pi$$
$$168$$ 0 0
$$169$$ 259.363 1.53469
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 338.586i − 1.95715i −0.205901 0.978573i $$-0.566013\pi$$
0.205901 0.978573i $$-0.433987\pi$$
$$174$$ 0 0
$$175$$ −176.477 −1.00844
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 348.576i − 1.94735i −0.227942 0.973675i $$-0.573200\pi$$
0.227942 0.973675i $$-0.426800\pi$$
$$180$$ 0 0
$$181$$ −16.1816 −0.0894013 −0.0447006 0.999000i $$-0.514233\pi$$
−0.0447006 + 0.999000i $$0.514233\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 423.664i − 2.29007i
$$186$$ 0 0
$$187$$ 284.021 1.51883
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 171.303i − 0.896875i −0.893814 0.448437i $$-0.851981\pi$$
0.893814 0.448437i $$-0.148019\pi$$
$$192$$ 0 0
$$193$$ −241.091 −1.24918 −0.624588 0.780955i $$-0.714733\pi$$
−0.624588 + 0.780955i $$0.714733\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 182.165i 0.924698i 0.886698 + 0.462349i $$0.152993\pi$$
−0.886698 + 0.462349i $$0.847007\pi$$
$$198$$ 0 0
$$199$$ 70.0782 0.352152 0.176076 0.984377i $$-0.443660\pi$$
0.176076 + 0.984377i $$0.443660\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 8.69694i − 0.0428421i
$$204$$ 0 0
$$205$$ −130.879 −0.638432
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 217.660i 1.04144i
$$210$$ 0 0
$$211$$ 102.681 0.486638 0.243319 0.969946i $$-0.421764\pi$$
0.243319 + 0.969946i $$0.421764\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 461.333i 2.14573i
$$216$$ 0 0
$$217$$ 507.545 2.33892
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 332.168i 1.50302i
$$222$$ 0 0
$$223$$ 205.082 0.919650 0.459825 0.888010i $$-0.347912\pi$$
0.459825 + 0.888010i $$0.347912\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 103.757i 0.457080i 0.973535 + 0.228540i $$0.0733952\pi$$
−0.973535 + 0.228540i $$0.926605\pi$$
$$228$$ 0 0
$$229$$ −435.576 −1.90208 −0.951038 0.309073i $$-0.899981\pi$$
−0.951038 + 0.309073i $$0.899981\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 387.344i 1.66242i 0.555959 + 0.831210i $$0.312351\pi$$
−0.555959 + 0.831210i $$0.687649\pi$$
$$234$$ 0 0
$$235$$ −4.00670 −0.0170498
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 225.303i − 0.942691i −0.881949 0.471345i $$-0.843769\pi$$
0.881949 0.471345i $$-0.156231\pi$$
$$240$$ 0 0
$$241$$ −127.757 −0.530113 −0.265056 0.964233i $$-0.585391\pi$$
−0.265056 + 0.964233i $$0.585391\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 265.015i − 1.08169i
$$246$$ 0 0
$$247$$ −254.558 −1.03060
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 54.0000i 0.215139i 0.994198 + 0.107570i $$0.0343069\pi$$
−0.994198 + 0.107570i $$0.965693\pi$$
$$252$$ 0 0
$$253$$ 472.454 1.86741
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 181.941i − 0.707941i −0.935257 0.353970i $$-0.884831\pi$$
0.935257 0.353970i $$-0.115169\pi$$
$$258$$ 0 0
$$259$$ 604.940 2.33568
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 42.8786i − 0.163036i −0.996672 0.0815182i $$-0.974023\pi$$
0.996672 0.0815182i $$-0.0259769\pi$$
$$264$$ 0 0
$$265$$ −299.788 −1.13127
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 140.500i 0.522305i 0.965298 + 0.261152i $$0.0841025\pi$$
−0.965298 + 0.261152i $$0.915898\pi$$
$$270$$ 0 0
$$271$$ −264.854 −0.977323 −0.488661 0.872474i $$-0.662515\pi$$
−0.488661 + 0.872474i $$0.662515\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 330.879i − 1.20319i
$$276$$ 0 0
$$277$$ 91.8184 0.331474 0.165737 0.986170i $$-0.447000\pi$$
0.165737 + 0.986170i $$0.447000\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 417.814i 1.48688i 0.668801 + 0.743442i $$0.266808\pi$$
−0.668801 + 0.743442i $$0.733192\pi$$
$$282$$ 0 0
$$283$$ 389.851 1.37757 0.688783 0.724968i $$-0.258145\pi$$
0.688783 + 0.724968i $$0.258145\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 186.879i − 0.651145i
$$288$$ 0 0
$$289$$ 31.4245 0.108735
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 108.145i 0.369095i 0.982824 + 0.184547i $$0.0590819\pi$$
−0.982824 + 0.184547i $$0.940918\pi$$
$$294$$ 0 0
$$295$$ −471.940 −1.59980
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 552.545i 1.84798i
$$300$$ 0 0
$$301$$ −658.727 −2.18846
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ − 26.4415i − 0.0866933i
$$306$$ 0 0
$$307$$ −52.3468 −0.170511 −0.0852554 0.996359i $$-0.527171\pi$$
−0.0852554 + 0.996359i $$0.527171\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 412.999i − 1.32797i −0.747745 0.663985i $$-0.768864\pi$$
0.747745 0.663985i $$-0.231136\pi$$
$$312$$ 0 0
$$313$$ −447.636 −1.43015 −0.715073 0.699050i $$-0.753607\pi$$
−0.715073 + 0.699050i $$0.753607\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 422.582i − 1.33307i −0.745476 0.666533i $$-0.767778\pi$$
0.745476 0.666533i $$-0.232222\pi$$
$$318$$ 0 0
$$319$$ 16.3060 0.0511161
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 197.394i − 0.611127i
$$324$$ 0 0
$$325$$ 386.969 1.19068
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 5.72107i − 0.0173893i
$$330$$ 0 0
$$331$$ 211.082 0.637711 0.318855 0.947803i $$-0.396702\pi$$
0.318855 + 0.947803i $$0.396702\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 49.1510i 0.146719i
$$336$$ 0 0
$$337$$ −113.455 −0.336662 −0.168331 0.985731i $$-0.553838\pi$$
−0.168331 + 0.985731i $$0.553838\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 951.604i 2.79063i
$$342$$ 0 0
$$343$$ −84.0920 −0.245166
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 448.485i 1.29246i 0.763141 + 0.646232i $$0.223656\pi$$
−0.763141 + 0.646232i $$0.776344\pi$$
$$348$$ 0 0
$$349$$ 260.454 0.746287 0.373143 0.927774i $$-0.378280\pi$$
0.373143 + 0.927774i $$0.378280\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 541.129i − 1.53294i −0.642279 0.766471i $$-0.722011\pi$$
0.642279 0.766471i $$-0.277989\pi$$
$$354$$ 0 0
$$355$$ −180.483 −0.508403
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 314.908i − 0.877181i −0.898687 0.438591i $$-0.855478\pi$$
0.898687 0.438591i $$-0.144522\pi$$
$$360$$ 0 0
$$361$$ −209.727 −0.580960
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 41.6655i − 0.114152i
$$366$$ 0 0
$$367$$ 227.677 0.620374 0.310187 0.950676i $$-0.399608\pi$$
0.310187 + 0.950676i $$0.399608\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 428.060i − 1.15380i
$$372$$ 0 0
$$373$$ −165.546 −0.443823 −0.221911 0.975067i $$-0.571230\pi$$
−0.221911 + 0.975067i $$0.571230\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 19.0702i 0.0505842i
$$378$$ 0 0
$$379$$ 124.708 0.329044 0.164522 0.986373i $$-0.447392\pi$$
0.164522 + 0.986373i $$0.447392\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 382.454i 0.998575i 0.866436 + 0.499287i $$0.166405\pi$$
−0.866436 + 0.499287i $$0.833595\pi$$
$$384$$ 0 0
$$385$$ 1104.18 2.86800
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 215.057i − 0.552845i −0.961036 0.276423i $$-0.910851\pi$$
0.961036 0.276423i $$-0.0891489\pi$$
$$390$$ 0 0
$$391$$ −428.463 −1.09581
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 281.728i 0.713234i
$$396$$ 0 0
$$397$$ 241.909 0.609343 0.304672 0.952457i $$-0.401453\pi$$
0.304672 + 0.952457i $$0.401453\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 325.183i 0.810929i 0.914111 + 0.405464i $$0.132890\pi$$
−0.914111 + 0.405464i $$0.867110\pi$$
$$402$$ 0 0
$$403$$ −1112.92 −2.76159
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1134.21i 2.78676i
$$408$$ 0 0
$$409$$ 419.302 1.02519 0.512594 0.858631i $$-0.328685\pi$$
0.512594 + 0.858631i $$0.328685\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 673.872i − 1.63165i
$$414$$ 0 0
$$415$$ −477.950 −1.15169
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 337.151i − 0.804656i −0.915496 0.402328i $$-0.868201\pi$$
0.915496 0.402328i $$-0.131799\pi$$
$$420$$ 0 0
$$421$$ 422.969 1.00468 0.502339 0.864671i $$-0.332473\pi$$
0.502339 + 0.864671i $$0.332473\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 300.070i 0.706047i
$$426$$ 0 0
$$427$$ 37.7552 0.0884196
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 137.333i 0.318637i 0.987227 + 0.159319i $$0.0509297\pi$$
−0.987227 + 0.159319i $$0.949070\pi$$
$$432$$ 0 0
$$433$$ 523.545 1.20911 0.604555 0.796563i $$-0.293351\pi$$
0.604555 + 0.796563i $$0.293351\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 328.354i − 0.751383i
$$438$$ 0 0
$$439$$ −566.617 −1.29070 −0.645349 0.763888i $$-0.723288\pi$$
−0.645349 + 0.763888i $$0.723288\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 592.788i 1.33812i 0.743208 + 0.669061i $$0.233303\pi$$
−0.743208 + 0.669061i $$0.766697\pi$$
$$444$$ 0 0
$$445$$ −836.120 −1.87892
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 127.151i − 0.283187i −0.989925 0.141593i $$-0.954777\pi$$
0.989925 0.141593i $$-0.0452225\pi$$
$$450$$ 0 0
$$451$$ 350.382 0.776899
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1291.36i 2.83816i
$$456$$ 0 0
$$457$$ −192.576 −0.421391 −0.210695 0.977552i $$-0.567573\pi$$
−0.210695 + 0.977552i $$0.567573\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 227.077i − 0.492575i −0.969197 0.246287i $$-0.920789\pi$$
0.969197 0.246287i $$-0.0792107\pi$$
$$462$$ 0 0
$$463$$ 638.399 1.37883 0.689416 0.724365i $$-0.257867\pi$$
0.689416 + 0.724365i $$0.257867\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 604.423i 1.29427i 0.762376 + 0.647134i $$0.224033\pi$$
−0.762376 + 0.647134i $$0.775967\pi$$
$$468$$ 0 0
$$469$$ −70.1816 −0.149641
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 1235.06i − 2.61111i
$$474$$ 0 0
$$475$$ −229.960 −0.484126
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 368.302i − 0.768898i −0.923146 0.384449i $$-0.874391\pi$$
0.923146 0.384449i $$-0.125609\pi$$
$$480$$ 0 0
$$481$$ −1326.48 −2.75776
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 178.480i − 0.368000i
$$486$$ 0 0
$$487$$ 536.865 1.10239 0.551196 0.834376i $$-0.314172\pi$$
0.551196 + 0.834376i $$0.314172\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 405.606i 0.826082i 0.910712 + 0.413041i $$0.135533\pi$$
−0.910712 + 0.413041i $$0.864467\pi$$
$$492$$ 0 0
$$493$$ −14.7878 −0.0299954
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 257.708i − 0.518527i
$$498$$ 0 0
$$499$$ −874.090 −1.75168 −0.875842 0.482598i $$-0.839693\pi$$
−0.875842 + 0.482598i $$0.839693\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 586.515i 1.16603i 0.812460 + 0.583017i $$0.198128\pi$$
−0.812460 + 0.583017i $$0.801872\pi$$
$$504$$ 0 0
$$505$$ −223.302 −0.442182
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 541.096i 1.06306i 0.847040 + 0.531529i $$0.178382\pi$$
−0.847040 + 0.531529i $$0.821618\pi$$
$$510$$ 0 0
$$511$$ 59.4933 0.116425
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 629.605i − 1.22253i
$$516$$ 0 0
$$517$$ 10.7265 0.0207476
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 563.050i 1.08071i 0.841437 + 0.540355i $$0.181710\pi$$
−0.841437 + 0.540355i $$0.818290\pi$$
$$522$$ 0 0
$$523$$ −353.232 −0.675396 −0.337698 0.941254i $$-0.609648\pi$$
−0.337698 + 0.941254i $$0.609648\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 862.999i − 1.63757i
$$528$$ 0 0
$$529$$ −183.727 −0.347309
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 409.778i 0.768815i
$$534$$ 0 0
$$535$$ 638.399 1.19327
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 709.485i 1.31630i
$$540$$ 0 0
$$541$$ −122.302 −0.226067 −0.113033 0.993591i $$-0.536057\pi$$
−0.113033 + 0.993591i $$0.536057\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 600.741i 1.10228i
$$546$$ 0 0
$$547$$ −189.055 −0.345622 −0.172811 0.984955i $$-0.555285\pi$$
−0.172811 + 0.984955i $$0.555285\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 11.3326i − 0.0205674i
$$552$$ 0 0
$$553$$ −402.272 −0.727437
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 522.883i 0.938749i 0.882999 + 0.469375i $$0.155521\pi$$
−0.882999 + 0.469375i $$0.844479\pi$$
$$558$$ 0 0
$$559$$ 1444.42 2.58394
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 743.574i 1.32074i 0.750942 + 0.660368i $$0.229600\pi$$
−0.750942 + 0.660368i $$0.770400\pi$$
$$564$$ 0 0
$$565$$ 331.728 0.587128
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 317.984i − 0.558848i −0.960168 0.279424i $$-0.909857\pi$$
0.960168 0.279424i $$-0.0901435\pi$$
$$570$$ 0 0
$$571$$ 1032.27 1.80782 0.903911 0.427720i $$-0.140683\pi$$
0.903911 + 0.427720i $$0.140683\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 499.151i 0.868089i
$$576$$ 0 0
$$577$$ 374.545 0.649125 0.324562 0.945864i $$-0.394783\pi$$
0.324562 + 0.945864i $$0.394783\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 682.454i − 1.17462i
$$582$$ 0 0
$$583$$ 802.577 1.37663
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 338.455i 0.576585i 0.957542 + 0.288292i $$0.0930875\pi$$
−0.957542 + 0.288292i $$0.906913\pi$$
$$588$$ 0 0
$$589$$ 661.362 1.12286
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 134.371i − 0.226596i −0.993561 0.113298i $$-0.963859\pi$$
0.993561 0.113298i $$-0.0361414\pi$$
$$594$$ 0 0
$$595$$ −1001.37 −1.68297
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 615.031i 1.02676i 0.858161 + 0.513381i $$0.171607\pi$$
−0.858161 + 0.513381i $$0.828393\pi$$
$$600$$ 0 0
$$601$$ 373.817 0.621992 0.310996 0.950411i $$-0.399337\pi$$
0.310996 + 0.950411i $$0.399337\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1270.39i 2.09982i
$$606$$ 0 0
$$607$$ 471.082 0.776083 0.388042 0.921642i $$-0.373152\pi$$
0.388042 + 0.921642i $$0.373152\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.5449i 0.0205317i
$$612$$ 0 0
$$613$$ −210.727 −0.343763 −0.171881 0.985118i $$-0.554985\pi$$
−0.171881 + 0.985118i $$0.554985\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1037.73i 1.68190i 0.541114 + 0.840949i $$0.318003\pi$$
−0.541114 + 0.840949i $$0.681997\pi$$
$$618$$ 0 0
$$619$$ −75.7896 −0.122439 −0.0612194 0.998124i $$-0.519499\pi$$
−0.0612194 + 0.998124i $$0.519499\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 1193.88i − 1.91634i
$$624$$ 0 0
$$625$$ −742.848 −1.18856
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 1028.60i − 1.63530i
$$630$$ 0 0
$$631$$ −289.174 −0.458279 −0.229139 0.973394i $$-0.573591\pi$$
−0.229139 + 0.973394i $$0.573591\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 886.757i 1.39647i
$$636$$ 0 0
$$637$$ −829.757 −1.30260
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 985.234i 1.53703i 0.639834 + 0.768513i $$0.279003\pi$$
−0.639834 + 0.768513i $$0.720997\pi$$
$$642$$ 0 0
$$643$$ 743.103 1.15568 0.577840 0.816150i $$-0.303896\pi$$
0.577840 + 0.816150i $$0.303896\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 328.849i 0.508267i 0.967169 + 0.254134i $$0.0817903\pi$$
−0.967169 + 0.254134i $$0.918210\pi$$
$$648$$ 0 0
$$649$$ 1263.45 1.94677
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 205.865i − 0.315261i −0.987498 0.157630i $$-0.949615\pi$$
0.987498 0.157630i $$-0.0503854\pi$$
$$654$$ 0 0
$$655$$ −309.487 −0.472499
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 364.151i 0.552581i 0.961074 + 0.276291i $$0.0891052\pi$$
−0.961074 + 0.276291i $$0.910895\pi$$
$$660$$ 0 0
$$661$$ −507.728 −0.768120 −0.384060 0.923308i $$-0.625474\pi$$
−0.384060 + 0.923308i $$0.625474\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 767.403i − 1.15399i
$$666$$ 0 0
$$667$$ −24.5987 −0.0368795
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 70.7878i 0.105496i
$$672$$ 0 0
$$673$$ 847.363 1.25908 0.629542 0.776967i $$-0.283243\pi$$
0.629542 + 0.776967i $$0.283243\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 902.076i 1.33246i 0.745746 + 0.666230i $$0.232093\pi$$
−0.745746 + 0.666230i $$0.767907\pi$$
$$678$$ 0 0
$$679$$ 254.847 0.375328
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 754.604i − 1.10484i −0.833567 0.552419i $$-0.813705\pi$$
0.833567 0.552419i $$-0.186295\pi$$
$$684$$ 0 0
$$685$$ 1209.21 1.76527
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 938.630i 1.36231i
$$690$$ 0 0
$$691$$ 420.450 0.608466 0.304233 0.952598i $$-0.401600\pi$$
0.304233 + 0.952598i $$0.401600\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 911.333i 1.31127i
$$696$$ 0 0
$$697$$ −317.757 −0.455893
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 423.953i − 0.604783i −0.953184 0.302391i $$-0.902215\pi$$
0.953184 0.302391i $$-0.0977850\pi$$
$$702$$ 0 0
$$703$$ 788.274 1.12130
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 318.848i − 0.450987i
$$708$$ 0 0
$$709$$ −14.3020 −0.0201721 −0.0100861 0.999949i $$-0.503211\pi$$
−0.0100861 + 0.999949i $$0.503211\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 1435.55i − 2.01340i
$$714$$ 0 0
$$715$$ −2421.19 −3.38629
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 453.453i 0.630672i 0.948980 + 0.315336i $$0.102117\pi$$
−0.948980 + 0.315336i $$0.897883\pi$$
$$720$$ 0 0
$$721$$ 898.999 1.24688
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 17.2274i 0.0237620i
$$726$$ 0 0
$$727$$ 818.324 1.12562 0.562809 0.826587i $$-0.309721\pi$$
0.562809 + 0.826587i $$0.309721\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1120.06i 1.53223i
$$732$$ 0 0
$$733$$ −164.636 −0.224605 −0.112303 0.993674i $$-0.535823\pi$$
−0.112303 + 0.993674i $$0.535823\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 131.585i − 0.178541i
$$738$$ 0 0
$$739$$ 392.143 0.530641 0.265320 0.964160i $$-0.414522\pi$$
0.265320 + 0.964160i $$0.414522\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1341.12i 1.80501i 0.430684 + 0.902503i $$0.358272\pi$$
−0.430684 + 0.902503i $$0.641728\pi$$
$$744$$ 0 0
$$745$$ −288.455 −0.387188
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 911.556i 1.21703i
$$750$$ 0 0
$$751$$ −330.637 −0.440262 −0.220131 0.975470i $$-0.570649\pi$$
−0.220131 + 0.975470i $$0.570649\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 555.879i − 0.736263i
$$756$$ 0 0
$$757$$ 70.1816 0.0927102 0.0463551 0.998925i $$-0.485239\pi$$
0.0463551 + 0.998925i $$0.485239\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 761.704i 1.00093i 0.865758 + 0.500463i $$0.166837\pi$$
−0.865758 + 0.500463i $$0.833163\pi$$
$$762$$ 0 0
$$763$$ −857.784 −1.12423
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1477.63i 1.92651i
$$768$$ 0 0
$$769$$ −853.272 −1.10959 −0.554794 0.831988i $$-0.687203\pi$$
−0.554794 + 0.831988i $$0.687203\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 487.925i − 0.631209i −0.948891 0.315605i $$-0.897793\pi$$
0.948891 0.315605i $$-0.102207\pi$$
$$774$$ 0 0
$$775$$ −1005.38 −1.29726
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 243.514i − 0.312599i
$$780$$ 0 0
$$781$$ 483.181 0.618669
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 1563.65i − 1.99191i
$$786$$ 0 0
$$787$$ 492.532 0.625834 0.312917 0.949780i $$-0.398694\pi$$
0.312917 + 0.949780i $$0.398694\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 473.666i 0.598820i
$$792$$ 0 0
$$793$$ −82.7878 −0.104398
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 361.085i − 0.453055i −0.974005 0.226528i $$-0.927263\pi$$
0.974005 0.226528i $$-0.0727374\pi$$
$$798$$ 0 0
$$799$$ −9.72777 −0.0121749
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 111.545i 0.138910i
$$804$$ 0 0
$$805$$ −1665.73 −2.06922
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 579.548i − 0.716376i −0.933649 0.358188i $$-0.883395\pi$$
0.933649 0.358188i $$-0.116605\pi$$
$$810$$ 0 0
$$811$$ 1406.37 1.73412 0.867059 0.498205i $$-0.166007\pi$$
0.867059 + 0.498205i $$0.166007\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 1866.15i − 2.28975i
$$816$$ 0 0
$$817$$ −858.361 −1.05063
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 111.702i 0.136056i 0.997683 + 0.0680280i $$0.0216707\pi$$
−0.997683 + 0.0680280i $$0.978329\pi$$
$$822$$ 0 0
$$823$$ −161.895 −0.196713 −0.0983564 0.995151i $$-0.531359\pi$$
−0.0983564 + 0.995151i $$0.531359\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 592.788i 0.716793i 0.933569 + 0.358396i $$0.116676\pi$$
−0.933569 + 0.358396i $$0.883324\pi$$
$$828$$ 0 0
$$829$$ 950.363 1.14640 0.573199 0.819416i $$-0.305702\pi$$
0.573199 + 0.819416i $$0.305702\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 643.424i − 0.772418i
$$834$$ 0 0
$$835$$ 463.926 0.555600
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 326.636i 0.389316i 0.980871 + 0.194658i $$0.0623596\pi$$
−0.980871 + 0.194658i $$0.937640\pi$$
$$840$$ 0 0
$$841$$ 840.151 0.998991
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 1714.49i − 2.02898i
$$846$$ 0 0
$$847$$ −1813.96 −2.14163
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 1711.03i − 2.01061i
$$852$$ 0 0
$$853$$ 906.727 1.06299 0.531493 0.847063i $$-0.321631\pi$$
0.531493 + 0.847063i $$0.321631\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 715.483i 0.834869i 0.908707 + 0.417435i $$0.137071\pi$$
−0.908707 + 0.417435i $$0.862929\pi$$
$$858$$ 0 0
$$859$$ 49.7558 0.0579229 0.0289615 0.999581i $$-0.490780\pi$$
0.0289615 + 0.999581i $$0.490780\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 1595.76i − 1.84908i −0.381086 0.924539i $$-0.624450\pi$$
0.381086 0.924539i $$-0.375550\pi$$
$$864$$ 0 0
$$865$$ −2238.18 −2.58749
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 754.227i − 0.867925i
$$870$$ 0 0
$$871$$ 153.891 0.176683
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 393.272i − 0.449454i
$$876$$ 0 0
$$877$$ 494.182 0.563491 0.281746 0.959489i $$-0.409087\pi$$
0.281746 + 0.959489i $$0.409087\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 728.383i 0.826768i 0.910557 + 0.413384i $$0.135653\pi$$
−0.910557 + 0.413384i $$0.864347\pi$$
$$882$$ 0 0
$$883$$ −1378.34 −1.56098 −0.780489 0.625170i $$-0.785030\pi$$
−0.780489 + 0.625170i $$0.785030\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1329.00i − 1.49831i −0.662397 0.749153i $$-0.730461\pi$$
0.662397 0.749153i $$-0.269539\pi$$
$$888$$ 0 0
$$889$$ −1266.18 −1.42428
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 7.45491i − 0.00834816i
$$894$$ 0 0
$$895$$ −2304.21 −2.57454
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 49.5459i − 0.0551123i
$$900$$ 0 0
$$901$$ −727.848 −0.807822
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 106.967i 0.118195i
$$906$$ 0 0
$$907$$ 940.152 1.03655 0.518276 0.855214i $$-0.326574\pi$$
0.518276 + 0.855214i $$0.326574\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 962.486i − 1.05652i −0.849084 0.528258i $$-0.822845\pi$$
0.849084 0.528258i $$-0.177155\pi$$
$$912$$ 0 0
$$913$$ 1279.54 1.40147
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 441.909i − 0.481907i
$$918$$ 0 0
$$919$$ −441.610 −0.480533 −0.240267 0.970707i $$-0.577235\pi$$
−0.240267 + 0.970707i $$0.577235\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 565.090i 0.612232i
$$924$$ 0 0
$$925$$ −1198.30 −1.29546
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 401.078i 0.431731i 0.976423 + 0.215866i $$0.0692573\pi$$
−0.976423 + 0.215866i $$0.930743\pi$$
$$930$$ 0 0
$$931$$ 493.090 0.529635
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 1877.48i − 2.00800i
$$936$$ 0 0
$$937$$ 493.000 0.526147 0.263074 0.964776i $$-0.415264\pi$$
0.263074 + 0.964776i $$0.415264\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 556.439i 0.591328i 0.955292 + 0.295664i $$0.0955408\pi$$
−0.955292 + 0.295664i $$0.904459\pi$$
$$942$$ 0 0
$$943$$ −528.572 −0.560522
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 1295.91i − 1.36844i −0.729278 0.684218i $$-0.760144\pi$$
0.729278 0.684218i $$-0.239856\pi$$
$$948$$ 0 0
$$949$$ −130.454 −0.137465
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 838.672i 0.880034i 0.897990 + 0.440017i $$0.145028\pi$$
−0.897990 + 0.440017i $$0.854972\pi$$
$$954$$ 0 0
$$955$$ −1132.38 −1.18573
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1726.60i 1.80042i
$$960$$ 0 0
$$961$$ 1930.45 2.00880
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1593.70i 1.65150i
$$966$$ 0 0
$$967$$ −96.6705 −0.0999695 −0.0499848 0.998750i $$-0.515917\pi$$
−0.0499848 + 0.998750i $$0.515917\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1218.66i 1.25506i 0.778592 + 0.627531i $$0.215934\pi$$
−0.778592 + 0.627531i $$0.784066\pi$$
$$972$$ 0 0
$$973$$ −1301.27 −1.33738
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 315.692i 0.323124i 0.986863 + 0.161562i $$0.0516532\pi$$
−0.986863 + 0.161562i $$0.948347\pi$$
$$978$$ 0 0
$$979$$ 2238.42 2.28643
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 40.2429i 0.0409388i 0.999790 + 0.0204694i $$0.00651607\pi$$
−0.999790 + 0.0204694i $$0.993484\pi$$
$$984$$ 0 0
$$985$$ 1204.18 1.22252
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1863.16i 1.88388i
$$990$$ 0 0
$$991$$ 173.635 0.175212 0.0876062 0.996155i $$-0.472078\pi$$
0.0876062 + 0.996155i $$0.472078\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 463.243i − 0.465571i
$$996$$ 0 0
$$997$$ 818.454 0.820917 0.410458 0.911879i $$-0.365369\pi$$
0.410458 + 0.911879i $$0.365369\pi$$
$$998$$ 0 0
$$999$$ 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 1728.3.e.u.1025.2 8
3.2 odd 2 inner 1728.3.e.u.1025.8 8
4.3 odd 2 inner 1728.3.e.u.1025.1 8
8.3 odd 2 864.3.e.f.161.7 yes 8
8.5 even 2 864.3.e.f.161.8 yes 8
12.11 even 2 inner 1728.3.e.u.1025.7 8
24.5 odd 2 864.3.e.f.161.2 yes 8
24.11 even 2 864.3.e.f.161.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.f.161.1 8 24.11 even 2
864.3.e.f.161.2 yes 8 24.5 odd 2
864.3.e.f.161.7 yes 8 8.3 odd 2
864.3.e.f.161.8 yes 8 8.5 even 2
1728.3.e.u.1025.1 8 4.3 odd 2 inner
1728.3.e.u.1025.2 8 1.1 even 1 trivial
1728.3.e.u.1025.7 8 12.11 even 2 inner
1728.3.e.u.1025.8 8 3.2 odd 2 inner