# Properties

 Label 320.2.d.a.161.3 Level $320$ Weight $2$ Character 320.161 Analytic conductor $2.555$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 161.3 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 320.161 Dual form 320.2.d.a.161.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.732051i q^{3} +1.00000i q^{5} -1.26795 q^{7} +2.46410 q^{9} +O(q^{10})$$ $$q+0.732051i q^{3} +1.00000i q^{5} -1.26795 q^{7} +2.46410 q^{9} +3.46410i q^{11} +3.46410i q^{13} -0.732051 q^{15} +3.46410 q^{17} +2.00000i q^{19} -0.928203i q^{21} -8.19615 q^{23} -1.00000 q^{25} +4.00000i q^{27} +9.46410 q^{31} -2.53590 q^{33} -1.26795i q^{35} -6.00000i q^{37} -2.53590 q^{39} +2.53590 q^{41} -10.1962i q^{43} +2.46410i q^{45} +8.19615 q^{47} -5.39230 q^{49} +2.53590i q^{51} +10.3923i q^{53} -3.46410 q^{55} -1.46410 q^{57} -6.00000i q^{59} -12.9282i q^{61} -3.12436 q^{63} -3.46410 q^{65} -10.1962i q^{67} -6.00000i q^{69} +4.39230 q^{71} -14.3923 q^{73} -0.732051i q^{75} -4.39230i q^{77} +12.0000 q^{79} +4.46410 q^{81} -4.73205i q^{83} +3.46410i q^{85} +0.928203 q^{89} -4.39230i q^{91} +6.92820i q^{93} -2.00000 q^{95} -6.39230 q^{97} +8.53590i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 12 q^{7} - 4 q^{9} + 4 q^{15} - 12 q^{23} - 4 q^{25} + 24 q^{31} - 24 q^{33} - 24 q^{39} + 24 q^{41} + 12 q^{47} + 20 q^{49} + 8 q^{57} + 36 q^{63} - 24 q^{71} - 16 q^{73} + 48 q^{79} + 4 q^{81} - 24 q^{89} - 8 q^{95} + 16 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0.732051i 0.422650i 0.977416 + 0.211325i $$0.0677778\pi$$
−0.977416 + 0.211325i $$0.932222\pi$$
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ −1.26795 −0.479240 −0.239620 0.970867i $$-0.577023\pi$$
−0.239620 + 0.970867i $$0.577023\pi$$
$$8$$ 0 0
$$9$$ 2.46410 0.821367
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 3.46410i 0.960769i 0.877058 + 0.480384i $$0.159503\pi$$
−0.877058 + 0.480384i $$0.840497\pi$$
$$14$$ 0 0
$$15$$ −0.732051 −0.189015
$$16$$ 0 0
$$17$$ 3.46410 0.840168 0.420084 0.907485i $$-0.362001\pi$$
0.420084 + 0.907485i $$0.362001\pi$$
$$18$$ 0 0
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 0 0
$$21$$ − 0.928203i − 0.202551i
$$22$$ 0 0
$$23$$ −8.19615 −1.70902 −0.854508 0.519438i $$-0.826141\pi$$
−0.854508 + 0.519438i $$0.826141\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 9.46410 1.69980 0.849901 0.526942i $$-0.176661\pi$$
0.849901 + 0.526942i $$0.176661\pi$$
$$32$$ 0 0
$$33$$ −2.53590 −0.441443
$$34$$ 0 0
$$35$$ − 1.26795i − 0.214323i
$$36$$ 0 0
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ −2.53590 −0.406069
$$40$$ 0 0
$$41$$ 2.53590 0.396041 0.198020 0.980198i $$-0.436549\pi$$
0.198020 + 0.980198i $$0.436549\pi$$
$$42$$ 0 0
$$43$$ − 10.1962i − 1.55490i −0.628946 0.777449i $$-0.716513\pi$$
0.628946 0.777449i $$-0.283487\pi$$
$$44$$ 0 0
$$45$$ 2.46410i 0.367327i
$$46$$ 0 0
$$47$$ 8.19615 1.19553 0.597766 0.801671i $$-0.296055\pi$$
0.597766 + 0.801671i $$0.296055\pi$$
$$48$$ 0 0
$$49$$ −5.39230 −0.770329
$$50$$ 0 0
$$51$$ 2.53590i 0.355097i
$$52$$ 0 0
$$53$$ 10.3923i 1.42749i 0.700404 + 0.713746i $$0.253003\pi$$
−0.700404 + 0.713746i $$0.746997\pi$$
$$54$$ 0 0
$$55$$ −3.46410 −0.467099
$$56$$ 0 0
$$57$$ −1.46410 −0.193925
$$58$$ 0 0
$$59$$ − 6.00000i − 0.781133i −0.920575 0.390567i $$-0.872279\pi$$
0.920575 0.390567i $$-0.127721\pi$$
$$60$$ 0 0
$$61$$ − 12.9282i − 1.65529i −0.561254 0.827643i $$-0.689681\pi$$
0.561254 0.827643i $$-0.310319\pi$$
$$62$$ 0 0
$$63$$ −3.12436 −0.393632
$$64$$ 0 0
$$65$$ −3.46410 −0.429669
$$66$$ 0 0
$$67$$ − 10.1962i − 1.24566i −0.782358 0.622829i $$-0.785983\pi$$
0.782358 0.622829i $$-0.214017\pi$$
$$68$$ 0 0
$$69$$ − 6.00000i − 0.722315i
$$70$$ 0 0
$$71$$ 4.39230 0.521271 0.260635 0.965437i $$-0.416068\pi$$
0.260635 + 0.965437i $$0.416068\pi$$
$$72$$ 0 0
$$73$$ −14.3923 −1.68449 −0.842246 0.539093i $$-0.818767\pi$$
−0.842246 + 0.539093i $$0.818767\pi$$
$$74$$ 0 0
$$75$$ − 0.732051i − 0.0845299i
$$76$$ 0 0
$$77$$ − 4.39230i − 0.500550i
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 4.46410 0.496011
$$82$$ 0 0
$$83$$ − 4.73205i − 0.519410i −0.965688 0.259705i $$-0.916375\pi$$
0.965688 0.259705i $$-0.0836253\pi$$
$$84$$ 0 0
$$85$$ 3.46410i 0.375735i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.928203 0.0983893 0.0491947 0.998789i $$-0.484335\pi$$
0.0491947 + 0.998789i $$0.484335\pi$$
$$90$$ 0 0
$$91$$ − 4.39230i − 0.460439i
$$92$$ 0 0
$$93$$ 6.92820i 0.718421i
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 0 0
$$97$$ −6.39230 −0.649040 −0.324520 0.945879i $$-0.605203\pi$$
−0.324520 + 0.945879i $$0.605203\pi$$
$$98$$ 0 0
$$99$$ 8.53590i 0.857890i
$$100$$ 0 0
$$101$$ − 12.0000i − 1.19404i −0.802225 0.597022i $$-0.796350\pi$$
0.802225 0.597022i $$-0.203650\pi$$
$$102$$ 0 0
$$103$$ −8.19615 −0.807591 −0.403795 0.914849i $$-0.632309\pi$$
−0.403795 + 0.914849i $$0.632309\pi$$
$$104$$ 0 0
$$105$$ 0.928203 0.0905834
$$106$$ 0 0
$$107$$ 16.7321i 1.61755i 0.588119 + 0.808774i $$0.299869\pi$$
−0.588119 + 0.808774i $$0.700131\pi$$
$$108$$ 0 0
$$109$$ − 0.928203i − 0.0889057i −0.999011 0.0444529i $$-0.985846\pi$$
0.999011 0.0444529i $$-0.0141545\pi$$
$$110$$ 0 0
$$111$$ 4.39230 0.416899
$$112$$ 0 0
$$113$$ 0.928203 0.0873180 0.0436590 0.999046i $$-0.486098\pi$$
0.0436590 + 0.999046i $$0.486098\pi$$
$$114$$ 0 0
$$115$$ − 8.19615i − 0.764295i
$$116$$ 0 0
$$117$$ 8.53590i 0.789144i
$$118$$ 0 0
$$119$$ −4.39230 −0.402642
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 1.85641i 0.167387i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ 3.80385 0.337537 0.168768 0.985656i $$-0.446021\pi$$
0.168768 + 0.985656i $$0.446021\pi$$
$$128$$ 0 0
$$129$$ 7.46410 0.657178
$$130$$ 0 0
$$131$$ 10.3923i 0.907980i 0.891007 + 0.453990i $$0.150000\pi$$
−0.891007 + 0.453990i $$0.850000\pi$$
$$132$$ 0 0
$$133$$ − 2.53590i − 0.219890i
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ 12.9282 1.10453 0.552265 0.833668i $$-0.313763\pi$$
0.552265 + 0.833668i $$0.313763\pi$$
$$138$$ 0 0
$$139$$ − 10.0000i − 0.848189i −0.905618 0.424094i $$-0.860592\pi$$
0.905618 0.424094i $$-0.139408\pi$$
$$140$$ 0 0
$$141$$ 6.00000i 0.505291i
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 3.94744i − 0.325579i
$$148$$ 0 0
$$149$$ 18.0000i 1.47462i 0.675556 + 0.737309i $$0.263904\pi$$
−0.675556 + 0.737309i $$0.736096\pi$$
$$150$$ 0 0
$$151$$ −2.53590 −0.206368 −0.103184 0.994662i $$-0.532903\pi$$
−0.103184 + 0.994662i $$0.532903\pi$$
$$152$$ 0 0
$$153$$ 8.53590 0.690086
$$154$$ 0 0
$$155$$ 9.46410i 0.760175i
$$156$$ 0 0
$$157$$ 0.928203i 0.0740787i 0.999314 + 0.0370393i $$0.0117927\pi$$
−0.999314 + 0.0370393i $$0.988207\pi$$
$$158$$ 0 0
$$159$$ −7.60770 −0.603329
$$160$$ 0 0
$$161$$ 10.3923 0.819028
$$162$$ 0 0
$$163$$ 5.80385i 0.454592i 0.973826 + 0.227296i $$0.0729886\pi$$
−0.973826 + 0.227296i $$0.927011\pi$$
$$164$$ 0 0
$$165$$ − 2.53590i − 0.197419i
$$166$$ 0 0
$$167$$ 8.19615 0.634237 0.317119 0.948386i $$-0.397285\pi$$
0.317119 + 0.948386i $$0.397285\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 4.92820i 0.376869i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 1.26795 0.0958479
$$176$$ 0 0
$$177$$ 4.39230 0.330146
$$178$$ 0 0
$$179$$ 19.8564i 1.48414i 0.670324 + 0.742069i $$0.266155\pi$$
−0.670324 + 0.742069i $$0.733845\pi$$
$$180$$ 0 0
$$181$$ 6.92820i 0.514969i 0.966282 + 0.257485i $$0.0828937\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 0 0
$$183$$ 9.46410 0.699607
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 12.0000i 0.877527i
$$188$$ 0 0
$$189$$ − 5.07180i − 0.368919i
$$190$$ 0 0
$$191$$ −16.3923 −1.18611 −0.593053 0.805164i $$-0.702077\pi$$
−0.593053 + 0.805164i $$0.702077\pi$$
$$192$$ 0 0
$$193$$ 6.39230 0.460128 0.230064 0.973175i $$-0.426106\pi$$
0.230064 + 0.973175i $$0.426106\pi$$
$$194$$ 0 0
$$195$$ − 2.53590i − 0.181599i
$$196$$ 0 0
$$197$$ − 10.3923i − 0.740421i −0.928948 0.370211i $$-0.879286\pi$$
0.928948 0.370211i $$-0.120714\pi$$
$$198$$ 0 0
$$199$$ 6.92820 0.491127 0.245564 0.969380i $$-0.421027\pi$$
0.245564 + 0.969380i $$0.421027\pi$$
$$200$$ 0 0
$$201$$ 7.46410 0.526477
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.53590i 0.177115i
$$206$$ 0 0
$$207$$ −20.1962 −1.40373
$$208$$ 0 0
$$209$$ −6.92820 −0.479234
$$210$$ 0 0
$$211$$ − 6.39230i − 0.440064i −0.975493 0.220032i $$-0.929384\pi$$
0.975493 0.220032i $$-0.0706162\pi$$
$$212$$ 0 0
$$213$$ 3.21539i 0.220315i
$$214$$ 0 0
$$215$$ 10.1962 0.695372
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ − 10.5359i − 0.711950i
$$220$$ 0 0
$$221$$ 12.0000i 0.807207i
$$222$$ 0 0
$$223$$ 15.1244 1.01280 0.506401 0.862298i $$-0.330976\pi$$
0.506401 + 0.862298i $$0.330976\pi$$
$$224$$ 0 0
$$225$$ −2.46410 −0.164273
$$226$$ 0 0
$$227$$ − 18.5885i − 1.23376i −0.787058 0.616880i $$-0.788397\pi$$
0.787058 0.616880i $$-0.211603\pi$$
$$228$$ 0 0
$$229$$ − 18.9282i − 1.25081i −0.780300 0.625405i $$-0.784934\pi$$
0.780300 0.625405i $$-0.215066\pi$$
$$230$$ 0 0
$$231$$ 3.21539 0.211557
$$232$$ 0 0
$$233$$ −1.60770 −0.105324 −0.0526618 0.998612i $$-0.516771\pi$$
−0.0526618 + 0.998612i $$0.516771\pi$$
$$234$$ 0 0
$$235$$ 8.19615i 0.534658i
$$236$$ 0 0
$$237$$ 8.78461i 0.570622i
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ −20.3923 −1.31358 −0.656792 0.754072i $$-0.728087\pi$$
−0.656792 + 0.754072i $$0.728087\pi$$
$$242$$ 0 0
$$243$$ 15.2679i 0.979439i
$$244$$ 0 0
$$245$$ − 5.39230i − 0.344502i
$$246$$ 0 0
$$247$$ −6.92820 −0.440831
$$248$$ 0 0
$$249$$ 3.46410 0.219529
$$250$$ 0 0
$$251$$ − 15.4641i − 0.976085i −0.872820 0.488043i $$-0.837711\pi$$
0.872820 0.488043i $$-0.162289\pi$$
$$252$$ 0 0
$$253$$ − 28.3923i − 1.78501i
$$254$$ 0 0
$$255$$ −2.53590 −0.158804
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ 7.60770i 0.472719i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 20.1962 1.24535 0.622674 0.782481i $$-0.286046\pi$$
0.622674 + 0.782481i $$0.286046\pi$$
$$264$$ 0 0
$$265$$ −10.3923 −0.638394
$$266$$ 0 0
$$267$$ 0.679492i 0.0415842i
$$268$$ 0 0
$$269$$ 14.7846i 0.901434i 0.892667 + 0.450717i $$0.148832\pi$$
−0.892667 + 0.450717i $$0.851168\pi$$
$$270$$ 0 0
$$271$$ −4.39230 −0.266814 −0.133407 0.991061i $$-0.542592\pi$$
−0.133407 + 0.991061i $$0.542592\pi$$
$$272$$ 0 0
$$273$$ 3.21539 0.194604
$$274$$ 0 0
$$275$$ − 3.46410i − 0.208893i
$$276$$ 0 0
$$277$$ 19.8564i 1.19306i 0.802592 + 0.596528i $$0.203454\pi$$
−0.802592 + 0.596528i $$0.796546\pi$$
$$278$$ 0 0
$$279$$ 23.3205 1.39616
$$280$$ 0 0
$$281$$ −28.3923 −1.69374 −0.846871 0.531798i $$-0.821517\pi$$
−0.846871 + 0.531798i $$0.821517\pi$$
$$282$$ 0 0
$$283$$ − 10.5885i − 0.629418i −0.949188 0.314709i $$-0.898093\pi$$
0.949188 0.314709i $$-0.101907\pi$$
$$284$$ 0 0
$$285$$ − 1.46410i − 0.0867259i
$$286$$ 0 0
$$287$$ −3.21539 −0.189798
$$288$$ 0 0
$$289$$ −5.00000 −0.294118
$$290$$ 0 0
$$291$$ − 4.67949i − 0.274317i
$$292$$ 0 0
$$293$$ − 30.0000i − 1.75262i −0.481749 0.876309i $$-0.659998\pi$$
0.481749 0.876309i $$-0.340002\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ 0 0
$$297$$ −13.8564 −0.804030
$$298$$ 0 0
$$299$$ − 28.3923i − 1.64197i
$$300$$ 0 0
$$301$$ 12.9282i 0.745169i
$$302$$ 0 0
$$303$$ 8.78461 0.504663
$$304$$ 0 0
$$305$$ 12.9282 0.740267
$$306$$ 0 0
$$307$$ − 34.1962i − 1.95168i −0.218492 0.975839i $$-0.570114\pi$$
0.218492 0.975839i $$-0.429886\pi$$
$$308$$ 0 0
$$309$$ − 6.00000i − 0.341328i
$$310$$ 0 0
$$311$$ 7.60770 0.431393 0.215696 0.976460i $$-0.430798\pi$$
0.215696 + 0.976460i $$0.430798\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ − 3.12436i − 0.176037i
$$316$$ 0 0
$$317$$ − 22.3923i − 1.25768i −0.777536 0.628839i $$-0.783531\pi$$
0.777536 0.628839i $$-0.216469\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.2487 −0.683656
$$322$$ 0 0
$$323$$ 6.92820i 0.385496i
$$324$$ 0 0
$$325$$ − 3.46410i − 0.192154i
$$326$$ 0 0
$$327$$ 0.679492 0.0375760
$$328$$ 0 0
$$329$$ −10.3923 −0.572946
$$330$$ 0 0
$$331$$ − 5.60770i − 0.308227i −0.988053 0.154113i $$-0.950748\pi$$
0.988053 0.154113i $$-0.0492521\pi$$
$$332$$ 0 0
$$333$$ − 14.7846i − 0.810192i
$$334$$ 0 0
$$335$$ 10.1962 0.557075
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 0.679492i 0.0369049i
$$340$$ 0 0
$$341$$ 32.7846i 1.77539i
$$342$$ 0 0
$$343$$ 15.7128 0.848412
$$344$$ 0 0
$$345$$ 6.00000 0.323029
$$346$$ 0 0
$$347$$ 28.0526i 1.50594i 0.658055 + 0.752970i $$0.271380\pi$$
−0.658055 + 0.752970i $$0.728620\pi$$
$$348$$ 0 0
$$349$$ − 8.78461i − 0.470229i −0.971968 0.235115i $$-0.924453\pi$$
0.971968 0.235115i $$-0.0755466\pi$$
$$350$$ 0 0
$$351$$ −13.8564 −0.739600
$$352$$ 0 0
$$353$$ −14.7846 −0.786905 −0.393453 0.919345i $$-0.628719\pi$$
−0.393453 + 0.919345i $$0.628719\pi$$
$$354$$ 0 0
$$355$$ 4.39230i 0.233119i
$$356$$ 0 0
$$357$$ − 3.21539i − 0.170177i
$$358$$ 0 0
$$359$$ −8.78461 −0.463634 −0.231817 0.972759i $$-0.574467\pi$$
−0.231817 + 0.972759i $$0.574467\pi$$
$$360$$ 0 0
$$361$$ 15.0000 0.789474
$$362$$ 0 0
$$363$$ − 0.732051i − 0.0384227i
$$364$$ 0 0
$$365$$ − 14.3923i − 0.753328i
$$366$$ 0 0
$$367$$ −10.0526 −0.524739 −0.262370 0.964967i $$-0.584504\pi$$
−0.262370 + 0.964967i $$0.584504\pi$$
$$368$$ 0 0
$$369$$ 6.24871 0.325295
$$370$$ 0 0
$$371$$ − 13.1769i − 0.684111i
$$372$$ 0 0
$$373$$ − 7.85641i − 0.406789i −0.979097 0.203395i $$-0.934803\pi$$
0.979097 0.203395i $$-0.0651974\pi$$
$$374$$ 0 0
$$375$$ 0.732051 0.0378029
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 2.00000i − 0.102733i −0.998680 0.0513665i $$-0.983642\pi$$
0.998680 0.0513665i $$-0.0163577\pi$$
$$380$$ 0 0
$$381$$ 2.78461i 0.142660i
$$382$$ 0 0
$$383$$ −3.80385 −0.194368 −0.0971838 0.995266i $$-0.530983\pi$$
−0.0971838 + 0.995266i $$0.530983\pi$$
$$384$$ 0 0
$$385$$ 4.39230 0.223853
$$386$$ 0 0
$$387$$ − 25.1244i − 1.27714i
$$388$$ 0 0
$$389$$ 26.7846i 1.35803i 0.734123 + 0.679017i $$0.237594\pi$$
−0.734123 + 0.679017i $$0.762406\pi$$
$$390$$ 0 0
$$391$$ −28.3923 −1.43586
$$392$$ 0 0
$$393$$ −7.60770 −0.383757
$$394$$ 0 0
$$395$$ 12.0000i 0.603786i
$$396$$ 0 0
$$397$$ 27.4641i 1.37838i 0.724579 + 0.689192i $$0.242034\pi$$
−0.724579 + 0.689192i $$0.757966\pi$$
$$398$$ 0 0
$$399$$ 1.85641 0.0929366
$$400$$ 0 0
$$401$$ −4.14359 −0.206921 −0.103461 0.994634i $$-0.532992\pi$$
−0.103461 + 0.994634i $$0.532992\pi$$
$$402$$ 0 0
$$403$$ 32.7846i 1.63312i
$$404$$ 0 0
$$405$$ 4.46410i 0.221823i
$$406$$ 0 0
$$407$$ 20.7846 1.03025
$$408$$ 0 0
$$409$$ 3.60770 0.178389 0.0891945 0.996014i $$-0.471571\pi$$
0.0891945 + 0.996014i $$0.471571\pi$$
$$410$$ 0 0
$$411$$ 9.46410i 0.466830i
$$412$$ 0 0
$$413$$ 7.60770i 0.374350i
$$414$$ 0 0
$$415$$ 4.73205 0.232287
$$416$$ 0 0
$$417$$ 7.32051 0.358487
$$418$$ 0 0
$$419$$ 0.928203i 0.0453457i 0.999743 + 0.0226728i $$0.00721761\pi$$
−0.999743 + 0.0226728i $$0.992782\pi$$
$$420$$ 0 0
$$421$$ − 6.00000i − 0.292422i −0.989253 0.146211i $$-0.953292\pi$$
0.989253 0.146211i $$-0.0467079\pi$$
$$422$$ 0 0
$$423$$ 20.1962 0.981971
$$424$$ 0 0
$$425$$ −3.46410 −0.168034
$$426$$ 0 0
$$427$$ 16.3923i 0.793279i
$$428$$ 0 0
$$429$$ − 8.78461i − 0.424125i
$$430$$ 0 0
$$431$$ −28.3923 −1.36761 −0.683805 0.729665i $$-0.739676\pi$$
−0.683805 + 0.729665i $$0.739676\pi$$
$$432$$ 0 0
$$433$$ 26.3923 1.26833 0.634167 0.773196i $$-0.281343\pi$$
0.634167 + 0.773196i $$0.281343\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 16.3923i − 0.784150i
$$438$$ 0 0
$$439$$ −18.9282 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$440$$ 0 0
$$441$$ −13.2872 −0.632723
$$442$$ 0 0
$$443$$ 0.339746i 0.0161418i 0.999967 + 0.00807091i $$0.00256908\pi$$
−0.999967 + 0.00807091i $$0.997431\pi$$
$$444$$ 0 0
$$445$$ 0.928203i 0.0440011i
$$446$$ 0 0
$$447$$ −13.1769 −0.623247
$$448$$ 0 0
$$449$$ −2.53590 −0.119676 −0.0598382 0.998208i $$-0.519058\pi$$
−0.0598382 + 0.998208i $$0.519058\pi$$
$$450$$ 0 0
$$451$$ 8.78461i 0.413651i
$$452$$ 0 0
$$453$$ − 1.85641i − 0.0872216i
$$454$$ 0 0
$$455$$ 4.39230 0.205914
$$456$$ 0 0
$$457$$ 22.7846 1.06582 0.532910 0.846172i $$-0.321099\pi$$
0.532910 + 0.846172i $$0.321099\pi$$
$$458$$ 0 0
$$459$$ 13.8564i 0.646762i
$$460$$ 0 0
$$461$$ − 12.0000i − 0.558896i −0.960161 0.279448i $$-0.909849\pi$$
0.960161 0.279448i $$-0.0901514\pi$$
$$462$$ 0 0
$$463$$ 15.8038 0.734467 0.367234 0.930129i $$-0.380305\pi$$
0.367234 + 0.930129i $$0.380305\pi$$
$$464$$ 0 0
$$465$$ −6.92820 −0.321288
$$466$$ 0 0
$$467$$ 22.9808i 1.06342i 0.846926 + 0.531711i $$0.178451\pi$$
−0.846926 + 0.531711i $$0.821549\pi$$
$$468$$ 0 0
$$469$$ 12.9282i 0.596969i
$$470$$ 0 0
$$471$$ −0.679492 −0.0313093
$$472$$ 0 0
$$473$$ 35.3205 1.62404
$$474$$ 0 0
$$475$$ − 2.00000i − 0.0917663i
$$476$$ 0 0
$$477$$ 25.6077i 1.17250i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.7846 0.947697
$$482$$ 0 0
$$483$$ 7.60770i 0.346162i
$$484$$ 0 0
$$485$$ − 6.39230i − 0.290260i
$$486$$ 0 0
$$487$$ −39.1244 −1.77289 −0.886447 0.462830i $$-0.846834\pi$$
−0.886447 + 0.462830i $$0.846834\pi$$
$$488$$ 0 0
$$489$$ −4.24871 −0.192133
$$490$$ 0 0
$$491$$ − 22.3923i − 1.01055i −0.862958 0.505275i $$-0.831391\pi$$
0.862958 0.505275i $$-0.168609\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ −8.53590 −0.383660
$$496$$ 0 0
$$497$$ −5.56922 −0.249814
$$498$$ 0 0
$$499$$ 39.5692i 1.77136i 0.464295 + 0.885681i $$0.346308\pi$$
−0.464295 + 0.885681i $$0.653692\pi$$
$$500$$ 0 0
$$501$$ 6.00000i 0.268060i
$$502$$ 0 0
$$503$$ −8.19615 −0.365448 −0.182724 0.983164i $$-0.558492\pi$$
−0.182724 + 0.983164i $$0.558492\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 0.732051i 0.0325115i
$$508$$ 0 0
$$509$$ 8.78461i 0.389371i 0.980866 + 0.194685i $$0.0623686\pi$$
−0.980866 + 0.194685i $$0.937631\pi$$
$$510$$ 0 0
$$511$$ 18.2487 0.807275
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ − 8.19615i − 0.361166i
$$516$$ 0 0
$$517$$ 28.3923i 1.24869i
$$518$$ 0 0
$$519$$ −4.39230 −0.192801
$$520$$ 0 0
$$521$$ 31.8564 1.39565 0.697827 0.716266i $$-0.254150\pi$$
0.697827 + 0.716266i $$0.254150\pi$$
$$522$$ 0 0
$$523$$ 14.9808i 0.655063i 0.944840 + 0.327531i $$0.106217\pi$$
−0.944840 + 0.327531i $$0.893783\pi$$
$$524$$ 0 0
$$525$$ 0.928203i 0.0405101i
$$526$$ 0 0
$$527$$ 32.7846 1.42812
$$528$$ 0 0
$$529$$ 44.1769 1.92074
$$530$$ 0 0
$$531$$ − 14.7846i − 0.641597i
$$532$$ 0 0
$$533$$ 8.78461i 0.380504i
$$534$$ 0 0
$$535$$ −16.7321 −0.723390
$$536$$ 0 0
$$537$$ −14.5359 −0.627270
$$538$$ 0 0
$$539$$ − 18.6795i − 0.804583i
$$540$$ 0 0
$$541$$ 15.7128i 0.675547i 0.941227 + 0.337773i $$0.109674\pi$$
−0.941227 + 0.337773i $$0.890326\pi$$
$$542$$ 0 0
$$543$$ −5.07180 −0.217652
$$544$$ 0 0
$$545$$ 0.928203 0.0397599
$$546$$ 0 0
$$547$$ − 1.80385i − 0.0771270i −0.999256 0.0385635i $$-0.987722\pi$$
0.999256 0.0385635i $$-0.0122782\pi$$
$$548$$ 0 0
$$549$$ − 31.8564i − 1.35960i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −15.2154 −0.647024
$$554$$ 0 0
$$555$$ 4.39230i 0.186443i
$$556$$ 0 0
$$557$$ − 14.7846i − 0.626444i −0.949680 0.313222i $$-0.898592\pi$$
0.949680 0.313222i $$-0.101408\pi$$
$$558$$ 0 0
$$559$$ 35.3205 1.49390
$$560$$ 0 0
$$561$$ −8.78461 −0.370887
$$562$$ 0 0
$$563$$ − 26.1962i − 1.10404i −0.833832 0.552018i $$-0.813858\pi$$
0.833832 0.552018i $$-0.186142\pi$$
$$564$$ 0 0
$$565$$ 0.928203i 0.0390498i
$$566$$ 0 0
$$567$$ −5.66025 −0.237708
$$568$$ 0 0
$$569$$ −42.2487 −1.77116 −0.885579 0.464489i $$-0.846238\pi$$
−0.885579 + 0.464489i $$0.846238\pi$$
$$570$$ 0 0
$$571$$ 30.3923i 1.27188i 0.771739 + 0.635939i $$0.219387\pi$$
−0.771739 + 0.635939i $$0.780613\pi$$
$$572$$ 0 0
$$573$$ − 12.0000i − 0.501307i
$$574$$ 0 0
$$575$$ 8.19615 0.341803
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 4.67949i 0.194473i
$$580$$ 0 0
$$581$$ 6.00000i 0.248922i
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ 0 0
$$585$$ −8.53590 −0.352916
$$586$$ 0 0
$$587$$ − 13.5167i − 0.557892i −0.960307 0.278946i $$-0.910015\pi$$
0.960307 0.278946i $$-0.0899851\pi$$
$$588$$ 0 0
$$589$$ 18.9282i 0.779923i
$$590$$ 0 0
$$591$$ 7.60770 0.312939
$$592$$ 0 0
$$593$$ −0.928203 −0.0381167 −0.0190584 0.999818i $$-0.506067\pi$$
−0.0190584 + 0.999818i $$0.506067\pi$$
$$594$$ 0 0
$$595$$ − 4.39230i − 0.180067i
$$596$$ 0 0
$$597$$ 5.07180i 0.207575i
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −20.3923 −0.831819 −0.415910 0.909406i $$-0.636537\pi$$
−0.415910 + 0.909406i $$0.636537\pi$$
$$602$$ 0 0
$$603$$ − 25.1244i − 1.02314i
$$604$$ 0 0
$$605$$ − 1.00000i − 0.0406558i
$$606$$ 0 0
$$607$$ 8.19615 0.332672 0.166336 0.986069i $$-0.446806\pi$$
0.166336 + 0.986069i $$0.446806\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 28.3923i 1.14863i
$$612$$ 0 0
$$613$$ 13.6077i 0.549610i 0.961500 + 0.274805i $$0.0886132\pi$$
−0.961500 + 0.274805i $$0.911387\pi$$
$$614$$ 0 0
$$615$$ −1.85641 −0.0748575
$$616$$ 0 0
$$617$$ 13.6077 0.547825 0.273913 0.961755i $$-0.411682\pi$$
0.273913 + 0.961755i $$0.411682\pi$$
$$618$$ 0 0
$$619$$ − 6.78461i − 0.272696i −0.990661 0.136348i $$-0.956463\pi$$
0.990661 0.136348i $$-0.0435366\pi$$
$$620$$ 0 0
$$621$$ − 32.7846i − 1.31560i
$$622$$ 0 0
$$623$$ −1.17691 −0.0471521
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ − 5.07180i − 0.202548i
$$628$$ 0 0
$$629$$ − 20.7846i − 0.828737i
$$630$$ 0 0
$$631$$ −21.4641 −0.854472 −0.427236 0.904140i $$-0.640513\pi$$
−0.427236 + 0.904140i $$0.640513\pi$$
$$632$$ 0 0
$$633$$ 4.67949 0.185993
$$634$$ 0 0
$$635$$ 3.80385i 0.150951i
$$636$$ 0 0
$$637$$ − 18.6795i − 0.740108i
$$638$$ 0 0
$$639$$ 10.8231 0.428155
$$640$$ 0 0
$$641$$ −4.39230 −0.173486 −0.0867428 0.996231i $$-0.527646\pi$$
−0.0867428 + 0.996231i $$0.527646\pi$$
$$642$$ 0 0
$$643$$ − 10.5885i − 0.417568i −0.977962 0.208784i $$-0.933049\pi$$
0.977962 0.208784i $$-0.0669506\pi$$
$$644$$ 0 0
$$645$$ 7.46410i 0.293899i
$$646$$ 0 0
$$647$$ 36.5885 1.43844 0.719220 0.694782i $$-0.244499\pi$$
0.719220 + 0.694782i $$0.244499\pi$$
$$648$$ 0 0
$$649$$ 20.7846 0.815867
$$650$$ 0 0
$$651$$ − 8.78461i − 0.344296i
$$652$$ 0 0
$$653$$ 19.1769i 0.750451i 0.926934 + 0.375225i $$0.122435\pi$$
−0.926934 + 0.375225i $$0.877565\pi$$
$$654$$ 0 0
$$655$$ −10.3923 −0.406061
$$656$$ 0 0
$$657$$ −35.4641 −1.38359
$$658$$ 0 0
$$659$$ − 40.6410i − 1.58315i −0.611073 0.791575i $$-0.709262\pi$$
0.611073 0.791575i $$-0.290738\pi$$
$$660$$ 0 0
$$661$$ − 35.5692i − 1.38348i −0.722146 0.691741i $$-0.756844\pi$$
0.722146 0.691741i $$-0.243156\pi$$
$$662$$ 0 0
$$663$$ −8.78461 −0.341166
$$664$$ 0 0
$$665$$ 2.53590 0.0983379
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 11.0718i 0.428060i
$$670$$ 0 0
$$671$$ 44.7846 1.72889
$$672$$ 0 0
$$673$$ −39.1769 −1.51016 −0.755080 0.655633i $$-0.772402\pi$$
−0.755080 + 0.655633i $$0.772402\pi$$
$$674$$ 0 0
$$675$$ − 4.00000i − 0.153960i
$$676$$ 0 0
$$677$$ − 43.1769i − 1.65942i −0.558192 0.829712i $$-0.688505\pi$$
0.558192 0.829712i $$-0.311495\pi$$
$$678$$ 0 0
$$679$$ 8.10512 0.311046
$$680$$ 0 0
$$681$$ 13.6077 0.521448
$$682$$ 0 0
$$683$$ 11.6603i 0.446167i 0.974799 + 0.223084i $$0.0716123\pi$$
−0.974799 + 0.223084i $$0.928388\pi$$
$$684$$ 0 0
$$685$$ 12.9282i 0.493961i
$$686$$ 0 0
$$687$$ 13.8564 0.528655
$$688$$ 0 0
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ − 5.60770i − 0.213327i −0.994295 0.106663i $$-0.965983\pi$$
0.994295 0.106663i $$-0.0340167\pi$$
$$692$$ 0 0
$$693$$ − 10.8231i − 0.411135i
$$694$$ 0 0
$$695$$ 10.0000 0.379322
$$696$$ 0 0
$$697$$ 8.78461 0.332741
$$698$$ 0 0
$$699$$ − 1.17691i − 0.0445150i
$$700$$ 0 0
$$701$$ − 14.7846i − 0.558407i −0.960232 0.279204i $$-0.909930\pi$$
0.960232 0.279204i $$-0.0900704\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ 0 0
$$707$$ 15.2154i 0.572234i
$$708$$ 0 0
$$709$$ 10.1436i 0.380951i 0.981692 + 0.190475i $$0.0610029\pi$$
−0.981692 + 0.190475i $$0.938997\pi$$
$$710$$ 0 0
$$711$$ 29.5692 1.10893
$$712$$ 0 0
$$713$$ −77.5692 −2.90499
$$714$$ 0 0
$$715$$ − 12.0000i − 0.448775i
$$716$$ 0 0
$$717$$ − 15.2154i − 0.568229i
$$718$$ 0 0
$$719$$ −44.7846 −1.67018 −0.835092 0.550110i $$-0.814586\pi$$
−0.835092 + 0.550110i $$0.814586\pi$$
$$720$$ 0 0
$$721$$ 10.3923 0.387030
$$722$$ 0 0
$$723$$ − 14.9282i − 0.555186i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −34.0526 −1.26294 −0.631470 0.775401i $$-0.717548\pi$$
−0.631470 + 0.775401i $$0.717548\pi$$
$$728$$ 0 0
$$729$$ 2.21539 0.0820515
$$730$$ 0 0
$$731$$ − 35.3205i − 1.30638i
$$732$$ 0 0
$$733$$ 38.7846i 1.43254i 0.697822 + 0.716271i $$0.254153\pi$$
−0.697822 + 0.716271i $$0.745847\pi$$
$$734$$ 0 0
$$735$$ 3.94744 0.145604
$$736$$ 0 0
$$737$$ 35.3205 1.30105
$$738$$ 0 0
$$739$$ − 2.00000i − 0.0735712i −0.999323 0.0367856i $$-0.988288\pi$$
0.999323 0.0367856i $$-0.0117119\pi$$
$$740$$ 0 0
$$741$$ − 5.07180i − 0.186317i
$$742$$ 0 0
$$743$$ −36.5885 −1.34230 −0.671150 0.741321i $$-0.734199\pi$$
−0.671150 + 0.741321i $$0.734199\pi$$
$$744$$ 0 0
$$745$$ −18.0000 −0.659469
$$746$$ 0 0
$$747$$ − 11.6603i − 0.426626i
$$748$$ 0 0
$$749$$ − 21.2154i − 0.775193i
$$750$$ 0 0
$$751$$ 40.3923 1.47394 0.736968 0.675928i $$-0.236257\pi$$
0.736968 + 0.675928i $$0.236257\pi$$
$$752$$ 0 0
$$753$$ 11.3205 0.412542
$$754$$ 0 0
$$755$$ − 2.53590i − 0.0922908i
$$756$$ 0 0
$$757$$ − 23.0718i − 0.838559i −0.907857 0.419279i $$-0.862283\pi$$
0.907857 0.419279i $$-0.137717\pi$$
$$758$$ 0 0
$$759$$ 20.7846 0.754434
$$760$$ 0 0
$$761$$ −21.7128 −0.787089 −0.393544 0.919306i $$-0.628751\pi$$
−0.393544 + 0.919306i $$0.628751\pi$$
$$762$$ 0 0
$$763$$ 1.17691i 0.0426072i
$$764$$ 0 0
$$765$$ 8.53590i 0.308616i
$$766$$ 0 0
$$767$$ 20.7846 0.750489
$$768$$ 0 0
$$769$$ −6.78461 −0.244659 −0.122330 0.992490i $$-0.539037\pi$$
−0.122330 + 0.992490i $$0.539037\pi$$
$$770$$ 0 0
$$771$$ 4.39230i 0.158185i
$$772$$ 0 0
$$773$$ 46.3923i 1.66862i 0.551299 + 0.834308i $$0.314132\pi$$
−0.551299 + 0.834308i $$0.685868\pi$$
$$774$$ 0 0
$$775$$ −9.46410 −0.339961
$$776$$ 0 0
$$777$$ −5.56922 −0.199795
$$778$$ 0 0
$$779$$ 5.07180i 0.181716i
$$780$$ 0 0
$$781$$ 15.2154i 0.544449i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −0.928203 −0.0331290
$$786$$ 0 0
$$787$$ 5.80385i 0.206885i 0.994635 + 0.103442i $$0.0329857\pi$$
−0.994635 + 0.103442i $$0.967014\pi$$
$$788$$ 0 0
$$789$$ 14.7846i 0.526346i
$$790$$ 0 0
$$791$$ −1.17691 −0.0418463
$$792$$ 0 0
$$793$$ 44.7846 1.59035
$$794$$ 0 0
$$795$$ − 7.60770i − 0.269817i
$$796$$ 0 0
$$797$$ 1.60770i 0.0569475i 0.999595 + 0.0284737i $$0.00906470\pi$$
−0.999595 + 0.0284737i $$0.990935\pi$$
$$798$$ 0 0
$$799$$ 28.3923 1.00445
$$800$$ 0 0
$$801$$ 2.28719 0.0808138
$$802$$ 0 0
$$803$$ − 49.8564i − 1.75939i
$$804$$ 0 0
$$805$$ 10.3923i 0.366281i
$$806$$ 0 0
$$807$$ −10.8231 −0.380991
$$808$$ 0 0
$$809$$ 35.5692 1.25055 0.625274 0.780406i $$-0.284987\pi$$
0.625274 + 0.780406i $$0.284987\pi$$
$$810$$ 0 0
$$811$$ 38.3923i 1.34814i 0.738669 + 0.674068i $$0.235455\pi$$
−0.738669 + 0.674068i $$0.764545\pi$$
$$812$$ 0 0
$$813$$ − 3.21539i − 0.112769i
$$814$$ 0 0
$$815$$ −5.80385 −0.203300
$$816$$ 0 0
$$817$$ 20.3923 0.713436
$$818$$ 0 0
$$819$$ − 10.8231i − 0.378189i
$$820$$ 0 0
$$821$$ − 50.7846i − 1.77240i −0.463308 0.886198i $$-0.653337\pi$$
0.463308 0.886198i $$-0.346663\pi$$
$$822$$ 0 0
$$823$$ −30.8372 −1.07492 −0.537458 0.843290i $$-0.680615\pi$$
−0.537458 + 0.843290i $$0.680615\pi$$
$$824$$ 0 0
$$825$$ 2.53590 0.0882886
$$826$$ 0 0
$$827$$ − 4.73205i − 0.164550i −0.996610 0.0822748i $$-0.973781\pi$$
0.996610 0.0822748i $$-0.0262185\pi$$
$$828$$ 0 0
$$829$$ − 50.7846i − 1.76382i −0.471416 0.881911i $$-0.656257\pi$$
0.471416 0.881911i $$-0.343743\pi$$
$$830$$ 0 0
$$831$$ −14.5359 −0.504245
$$832$$ 0 0
$$833$$ −18.6795 −0.647206
$$834$$ 0 0
$$835$$ 8.19615i 0.283640i
$$836$$ 0 0
$$837$$ 37.8564i 1.30851i
$$838$$ 0 0
$$839$$ −8.78461 −0.303278 −0.151639 0.988436i $$-0.548455\pi$$
−0.151639 + 0.988436i $$0.548455\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ − 20.7846i − 0.715860i
$$844$$ 0 0
$$845$$ 1.00000i 0.0344010i
$$846$$ 0 0
$$847$$ 1.26795 0.0435672
$$848$$ 0 0
$$849$$ 7.75129 0.266024
$$850$$ 0 0
$$851$$ 49.1769i 1.68576i
$$852$$ 0 0
$$853$$ 3.46410i 0.118609i 0.998240 + 0.0593043i $$0.0188882\pi$$
−0.998240 + 0.0593043i $$0.981112\pi$$
$$854$$ 0 0
$$855$$ −4.92820 −0.168541
$$856$$ 0 0
$$857$$ −16.1436 −0.551455 −0.275727 0.961236i $$-0.588919\pi$$
−0.275727 + 0.961236i $$0.588919\pi$$
$$858$$ 0 0
$$859$$ 51.5692i 1.75952i 0.475419 + 0.879760i $$0.342296\pi$$
−0.475419 + 0.879760i $$0.657704\pi$$
$$860$$ 0 0
$$861$$ − 2.35383i − 0.0802183i
$$862$$ 0 0
$$863$$ −40.9808 −1.39500 −0.697501 0.716584i $$-0.745705\pi$$
−0.697501 + 0.716584i $$0.745705\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ − 3.66025i − 0.124309i
$$868$$ 0 0
$$869$$ 41.5692i 1.41014i
$$870$$ 0 0
$$871$$ 35.3205 1.19679
$$872$$ 0 0
$$873$$ −15.7513 −0.533100
$$874$$ 0 0
$$875$$ 1.26795i 0.0428645i
$$876$$ 0 0
$$877$$ − 7.85641i − 0.265292i −0.991163 0.132646i $$-0.957653\pi$$
0.991163 0.132646i $$-0.0423473\pi$$
$$878$$ 0 0
$$879$$ 21.9615 0.740744
$$880$$ 0 0
$$881$$ −7.60770 −0.256310 −0.128155 0.991754i $$-0.540905\pi$$
−0.128155 + 0.991754i $$0.540905\pi$$
$$882$$ 0 0
$$883$$ 5.80385i 0.195315i 0.995220 + 0.0976575i $$0.0311350\pi$$
−0.995220 + 0.0976575i $$0.968865\pi$$
$$884$$ 0 0
$$885$$ 4.39230i 0.147646i
$$886$$ 0 0
$$887$$ 21.3731 0.717637 0.358819 0.933407i $$-0.383180\pi$$
0.358819 + 0.933407i $$0.383180\pi$$
$$888$$ 0 0
$$889$$ −4.82309 −0.161761
$$890$$ 0 0
$$891$$ 15.4641i 0.518067i
$$892$$ 0 0
$$893$$ 16.3923i 0.548548i
$$894$$ 0 0
$$895$$ −19.8564 −0.663726
$$896$$ 0 0
$$897$$ 20.7846 0.693978
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 36.0000i 1.19933i
$$902$$ 0 0
$$903$$ −9.46410 −0.314946
$$904$$ 0 0
$$905$$ −6.92820 −0.230301
$$906$$ 0 0
$$907$$ − 1.41154i − 0.0468695i −0.999725 0.0234348i $$-0.992540\pi$$
0.999725 0.0234348i $$-0.00746020\pi$$
$$908$$ 0 0
$$909$$ − 29.5692i − 0.980749i
$$910$$ 0 0
$$911$$ 37.1769 1.23173 0.615863 0.787853i $$-0.288807\pi$$
0.615863 + 0.787853i $$0.288807\pi$$
$$912$$ 0 0
$$913$$ 16.3923 0.542506
$$914$$ 0 0
$$915$$ 9.46410i 0.312874i
$$916$$ 0 0
$$917$$ − 13.1769i − 0.435140i
$$918$$ 0 0
$$919$$ −39.7128 −1.31000 −0.655002 0.755627i $$-0.727332\pi$$
−0.655002 + 0.755627i $$0.727332\pi$$
$$920$$ 0 0
$$921$$ 25.0333 0.824876
$$922$$ 0 0
$$923$$ 15.2154i 0.500821i
$$924$$ 0 0
$$925$$ 6.00000i 0.197279i
$$926$$ 0 0
$$927$$ −20.1962 −0.663329
$$928$$ 0 0
$$929$$ 7.60770 0.249600 0.124800 0.992182i $$-0.460171\pi$$
0.124800 + 0.992182i $$0.460171\pi$$
$$930$$ 0 0
$$931$$ − 10.7846i − 0.353451i
$$932$$ 0 0
$$933$$ 5.56922i 0.182328i
$$934$$ 0 0
$$935$$ −12.0000 −0.392442
$$936$$ 0 0
$$937$$ −9.60770 −0.313870 −0.156935 0.987609i $$-0.550161\pi$$
−0.156935 + 0.987609i $$0.550161\pi$$
$$938$$ 0 0
$$939$$ 16.1051i 0.525571i
$$940$$ 0 0
$$941$$ 20.7846i 0.677559i 0.940866 + 0.338779i $$0.110014\pi$$
−0.940866 + 0.338779i $$0.889986\pi$$
$$942$$ 0 0
$$943$$ −20.7846 −0.676840
$$944$$ 0 0
$$945$$ 5.07180 0.164986
$$946$$ 0 0
$$947$$ 45.8038i 1.48843i 0.667943 + 0.744213i $$0.267175\pi$$
−0.667943 + 0.744213i $$0.732825\pi$$
$$948$$ 0 0
$$949$$ − 49.8564i − 1.61841i
$$950$$ 0 0
$$951$$ 16.3923 0.531557
$$952$$ 0 0
$$953$$ 24.9282 0.807504 0.403752 0.914869i $$-0.367706\pi$$
0.403752 + 0.914869i $$0.367706\pi$$
$$954$$ 0 0
$$955$$ − 16.3923i − 0.530443i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −16.3923 −0.529335
$$960$$ 0 0
$$961$$ 58.5692 1.88933
$$962$$ 0 0
$$963$$ 41.2295i 1.32860i
$$964$$ 0 0
$$965$$ 6.39230i 0.205776i
$$966$$ 0 0
$$967$$ 8.19615 0.263570 0.131785 0.991278i $$-0.457929\pi$$
0.131785 + 0.991278i $$0.457929\pi$$
$$968$$ 0 0
$$969$$ −5.07180 −0.162930
$$970$$ 0 0
$$971$$ 33.0333i 1.06009i 0.847970 + 0.530045i $$0.177825\pi$$
−0.847970 + 0.530045i $$0.822175\pi$$
$$972$$ 0 0
$$973$$ 12.6795i 0.406486i
$$974$$ 0 0
$$975$$ 2.53590 0.0812137
$$976$$ 0 0
$$977$$ 5.32051 0.170218 0.0851091 0.996372i $$-0.472876\pi$$
0.0851091 + 0.996372i $$0.472876\pi$$
$$978$$ 0 0
$$979$$ 3.21539i 0.102764i
$$980$$ 0 0
$$981$$ − 2.28719i − 0.0730243i
$$982$$ 0 0
$$983$$ 11.4115 0.363972 0.181986 0.983301i $$-0.441747\pi$$
0.181986 + 0.983301i $$0.441747\pi$$
$$984$$ 0 0
$$985$$ 10.3923 0.331126
$$986$$ 0 0
$$987$$ − 7.60770i − 0.242156i
$$988$$ 0 0
$$989$$ 83.5692i 2.65735i
$$990$$ 0 0
$$991$$ −44.1051 −1.40105 −0.700523 0.713630i $$-0.747050\pi$$
−0.700523 + 0.713630i $$0.747050\pi$$
$$992$$ 0 0
$$993$$ 4.10512 0.130272
$$994$$ 0 0
$$995$$ 6.92820i 0.219639i
$$996$$ 0 0
$$997$$ 25.6077i 0.811004i 0.914094 + 0.405502i $$0.132903\pi$$
−0.914094 + 0.405502i $$0.867097\pi$$
$$998$$ 0 0
$$999$$ 24.0000 0.759326
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 320.2.d.a.161.3 yes 4
3.2 odd 2 2880.2.k.e.1441.2 4
4.3 odd 2 320.2.d.b.161.2 yes 4
5.2 odd 4 1600.2.f.e.1249.3 4
5.3 odd 4 1600.2.f.i.1249.2 4
5.4 even 2 1600.2.d.h.801.2 4
8.3 odd 2 320.2.d.b.161.3 yes 4
8.5 even 2 inner 320.2.d.a.161.2 4
12.11 even 2 2880.2.k.l.1441.1 4
16.3 odd 4 1280.2.a.c.1.2 2
16.5 even 4 1280.2.a.b.1.2 2
16.11 odd 4 1280.2.a.m.1.1 2
16.13 even 4 1280.2.a.p.1.1 2
20.3 even 4 1600.2.f.d.1249.3 4
20.7 even 4 1600.2.f.h.1249.2 4
20.19 odd 2 1600.2.d.b.801.3 4
24.5 odd 2 2880.2.k.e.1441.4 4
24.11 even 2 2880.2.k.l.1441.3 4
40.3 even 4 1600.2.f.h.1249.1 4
40.13 odd 4 1600.2.f.e.1249.4 4
40.19 odd 2 1600.2.d.b.801.2 4
40.27 even 4 1600.2.f.d.1249.4 4
40.29 even 2 1600.2.d.h.801.3 4
40.37 odd 4 1600.2.f.i.1249.1 4
80.19 odd 4 6400.2.a.ck.1.1 2
80.29 even 4 6400.2.a.y.1.2 2
80.59 odd 4 6400.2.a.bf.1.2 2
80.69 even 4 6400.2.a.cd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 8.5 even 2 inner
320.2.d.a.161.3 yes 4 1.1 even 1 trivial
320.2.d.b.161.2 yes 4 4.3 odd 2
320.2.d.b.161.3 yes 4 8.3 odd 2
1280.2.a.b.1.2 2 16.5 even 4
1280.2.a.c.1.2 2 16.3 odd 4
1280.2.a.m.1.1 2 16.11 odd 4
1280.2.a.p.1.1 2 16.13 even 4
1600.2.d.b.801.2 4 40.19 odd 2
1600.2.d.b.801.3 4 20.19 odd 2
1600.2.d.h.801.2 4 5.4 even 2
1600.2.d.h.801.3 4 40.29 even 2
1600.2.f.d.1249.3 4 20.3 even 4
1600.2.f.d.1249.4 4 40.27 even 4
1600.2.f.e.1249.3 4 5.2 odd 4
1600.2.f.e.1249.4 4 40.13 odd 4
1600.2.f.h.1249.1 4 40.3 even 4
1600.2.f.h.1249.2 4 20.7 even 4
1600.2.f.i.1249.1 4 40.37 odd 4
1600.2.f.i.1249.2 4 5.3 odd 4
2880.2.k.e.1441.2 4 3.2 odd 2
2880.2.k.e.1441.4 4 24.5 odd 2
2880.2.k.l.1441.1 4 12.11 even 2
2880.2.k.l.1441.3 4 24.11 even 2
6400.2.a.y.1.2 2 80.29 even 4
6400.2.a.bf.1.2 2 80.59 odd 4
6400.2.a.cd.1.1 2 80.69 even 4
6400.2.a.ck.1.1 2 80.19 odd 4