# Properties

 Label 2352.2.b.g.1567.1 Level $2352$ Weight $2$ Character 2352.1567 Analytic conductor $18.781$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 336) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1567.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1567 Dual form 2352.2.b.g.1567.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.73205i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.73205i q^{5} +1.00000 q^{9} +1.73205i q^{11} -1.73205i q^{15} +3.46410i q^{17} -2.00000 q^{19} +2.00000 q^{25} +1.00000 q^{27} +9.00000 q^{29} +5.00000 q^{31} +1.73205i q^{33} +10.0000 q^{37} -10.3923i q^{41} +3.46410i q^{43} -1.73205i q^{45} +12.0000 q^{47} +3.46410i q^{51} -9.00000 q^{53} +3.00000 q^{55} -2.00000 q^{57} -9.00000 q^{59} -13.8564i q^{67} +13.8564i q^{71} -6.92820i q^{73} +2.00000 q^{75} -5.19615i q^{79} +1.00000 q^{81} +3.00000 q^{83} +6.00000 q^{85} +9.00000 q^{87} +3.46410i q^{89} +5.00000 q^{93} +3.46410i q^{95} -19.0526i q^{97} +1.73205i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} - 4q^{19} + 4q^{25} + 2q^{27} + 18q^{29} + 10q^{31} + 20q^{37} + 24q^{47} - 18q^{53} + 6q^{55} - 4q^{57} - 18q^{59} + 4q^{75} + 2q^{81} + 6q^{83} + 12q^{85} + 18q^{87} + 10q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ − 1.73205i − 0.774597i −0.921954 0.387298i $$-0.873408\pi$$
0.921954 0.387298i $$-0.126592\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.73205i 0.522233i 0.965307 + 0.261116i $$0.0840907\pi$$
−0.965307 + 0.261116i $$0.915909\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ − 1.73205i − 0.447214i
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 2.00000 0.400000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 1.73205i 0.301511i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 10.3923i − 1.62301i −0.584349 0.811503i $$-0.698650\pi$$
0.584349 0.811503i $$-0.301350\pi$$
$$42$$ 0 0
$$43$$ 3.46410i 0.528271i 0.964486 + 0.264135i $$0.0850865\pi$$
−0.964486 + 0.264135i $$0.914913\pi$$
$$44$$ 0 0
$$45$$ − 1.73205i − 0.258199i
$$46$$ 0 0
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.46410i 0.485071i
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 3.00000 0.404520
$$56$$ 0 0
$$57$$ −2.00000 −0.264906
$$58$$ 0 0
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 13.8564i − 1.69283i −0.532524 0.846415i $$-0.678756\pi$$
0.532524 0.846415i $$-0.321244\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.8564i 1.64445i 0.569160 + 0.822226i $$0.307268\pi$$
−0.569160 + 0.822226i $$0.692732\pi$$
$$72$$ 0 0
$$73$$ − 6.92820i − 0.810885i −0.914121 0.405442i $$-0.867117\pi$$
0.914121 0.405442i $$-0.132883\pi$$
$$74$$ 0 0
$$75$$ 2.00000 0.230940
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ − 5.19615i − 0.584613i −0.956325 0.292306i $$-0.905577\pi$$
0.956325 0.292306i $$-0.0944227\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 0 0
$$87$$ 9.00000 0.964901
$$88$$ 0 0
$$89$$ 3.46410i 0.367194i 0.983002 + 0.183597i $$0.0587741\pi$$
−0.983002 + 0.183597i $$0.941226\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 5.00000 0.518476
$$94$$ 0 0
$$95$$ 3.46410i 0.355409i
$$96$$ 0 0
$$97$$ − 19.0526i − 1.93449i −0.253837 0.967247i $$-0.581693\pi$$
0.253837 0.967247i $$-0.418307\pi$$
$$98$$ 0 0
$$99$$ 1.73205i 0.174078i
$$100$$ 0 0
$$101$$ − 13.8564i − 1.37876i −0.724398 0.689382i $$-0.757882\pi$$
0.724398 0.689382i $$-0.242118\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.1244i 1.17211i 0.810273 + 0.586053i $$0.199319\pi$$
−0.810273 + 0.586053i $$0.800681\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 8.00000 0.727273
$$122$$ 0 0
$$123$$ − 10.3923i − 0.937043i
$$124$$ 0 0
$$125$$ − 12.1244i − 1.08444i
$$126$$ 0 0
$$127$$ 5.19615i 0.461084i 0.973062 + 0.230542i $$0.0740499\pi$$
−0.973062 + 0.230542i $$0.925950\pi$$
$$128$$ 0 0
$$129$$ 3.46410i 0.304997i
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ − 1.73205i − 0.149071i
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ − 15.5885i − 1.29455i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 5.19615i 0.422857i 0.977393 + 0.211428i $$0.0678115\pi$$
−0.977393 + 0.211428i $$0.932188\pi$$
$$152$$ 0 0
$$153$$ 3.46410i 0.280056i
$$154$$ 0 0
$$155$$ − 8.66025i − 0.695608i
$$156$$ 0 0
$$157$$ 6.92820i 0.552931i 0.961024 + 0.276465i $$0.0891631\pi$$
−0.961024 + 0.276465i $$0.910837\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 13.8564i 1.08532i 0.839953 + 0.542659i $$0.182582\pi$$
−0.839953 + 0.542659i $$0.817418\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −9.00000 −0.676481
$$178$$ 0 0
$$179$$ 24.2487i 1.81243i 0.422813 + 0.906217i $$0.361043\pi$$
−0.422813 + 0.906217i $$0.638957\pi$$
$$180$$ 0 0
$$181$$ 10.3923i 0.772454i 0.922404 + 0.386227i $$0.126222\pi$$
−0.922404 + 0.386227i $$0.873778\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 17.3205i − 1.27343i
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.8564i 1.00261i 0.865269 + 0.501307i $$0.167147\pi$$
−0.865269 + 0.501307i $$0.832853\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ − 13.8564i − 0.977356i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −18.0000 −1.25717
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 3.46410i − 0.239617i
$$210$$ 0 0
$$211$$ − 6.92820i − 0.476957i −0.971148 0.238479i $$-0.923351\pi$$
0.971148 0.238479i $$-0.0766487\pi$$
$$212$$ 0 0
$$213$$ 13.8564i 0.949425i
$$214$$ 0 0
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ − 6.92820i − 0.468165i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 19.0000 1.27233 0.636167 0.771551i $$-0.280519\pi$$
0.636167 + 0.771551i $$0.280519\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ −21.0000 −1.39382 −0.696909 0.717159i $$-0.745442\pi$$
−0.696909 + 0.717159i $$0.745442\pi$$
$$228$$ 0 0
$$229$$ 3.46410i 0.228914i 0.993428 + 0.114457i $$0.0365129\pi$$
−0.993428 + 0.114457i $$0.963487\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ − 20.7846i − 1.35584i
$$236$$ 0 0
$$237$$ − 5.19615i − 0.337526i
$$238$$ 0 0
$$239$$ 6.92820i 0.448148i 0.974572 + 0.224074i $$0.0719358\pi$$
−0.974572 + 0.224074i $$0.928064\pi$$
$$240$$ 0 0
$$241$$ 22.5167i 1.45043i 0.688525 + 0.725213i $$0.258259\pi$$
−0.688525 + 0.725213i $$0.741741\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 6.00000 0.375735
$$256$$ 0 0
$$257$$ 10.3923i 0.648254i 0.946014 + 0.324127i $$0.105071\pi$$
−0.946014 + 0.324127i $$0.894929\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 9.00000 0.557086
$$262$$ 0 0
$$263$$ 17.3205i 1.06803i 0.845476 + 0.534014i $$0.179317\pi$$
−0.845476 + 0.534014i $$0.820683\pi$$
$$264$$ 0 0
$$265$$ 15.5885i 0.957591i
$$266$$ 0 0
$$267$$ 3.46410i 0.212000i
$$268$$ 0 0
$$269$$ − 15.5885i − 0.950445i −0.879866 0.475223i $$-0.842368\pi$$
0.879866 0.475223i $$-0.157632\pi$$
$$270$$ 0 0
$$271$$ 11.0000 0.668202 0.334101 0.942537i $$-0.391567\pi$$
0.334101 + 0.942537i $$0.391567\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3.46410i 0.208893i
$$276$$ 0 0
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 0 0
$$279$$ 5.00000 0.299342
$$280$$ 0 0
$$281$$ −24.0000 −1.43172 −0.715860 0.698244i $$-0.753965\pi$$
−0.715860 + 0.698244i $$0.753965\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ 3.46410i 0.205196i
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ − 19.0526i − 1.11688i
$$292$$ 0 0
$$293$$ 5.19615i 0.303562i 0.988414 + 0.151781i $$0.0485009\pi$$
−0.988414 + 0.151781i $$0.951499\pi$$
$$294$$ 0 0
$$295$$ 15.5885i 0.907595i
$$296$$ 0 0
$$297$$ 1.73205i 0.100504i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 13.8564i − 0.796030i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ 5.19615i 0.293704i 0.989158 + 0.146852i $$0.0469141\pi$$
−0.989158 + 0.146852i $$0.953086\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.0000 −1.17948 −0.589739 0.807594i $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 0 0
$$319$$ 15.5885i 0.872786i
$$320$$ 0 0
$$321$$ 12.1244i 0.676716i
$$322$$ 0 0
$$323$$ − 6.92820i − 0.385496i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −4.00000 −0.221201
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 13.8564i − 0.761617i −0.924654 0.380808i $$-0.875646\pi$$
0.924654 0.380808i $$-0.124354\pi$$
$$332$$ 0 0
$$333$$ 10.0000 0.547997
$$334$$ 0 0
$$335$$ −24.0000 −1.31126
$$336$$ 0 0
$$337$$ −1.00000 −0.0544735 −0.0272367 0.999629i $$-0.508671\pi$$
−0.0272367 + 0.999629i $$0.508671\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 8.66025i 0.468979i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 17.3205i − 0.929814i −0.885360 0.464907i $$-0.846088\pi$$
0.885360 0.464907i $$-0.153912\pi$$
$$348$$ 0 0
$$349$$ − 6.92820i − 0.370858i −0.982658 0.185429i $$-0.940632\pi$$
0.982658 0.185429i $$-0.0593675\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 6.92820i 0.368751i 0.982856 + 0.184376i $$0.0590263\pi$$
−0.982856 + 0.184376i $$0.940974\pi$$
$$354$$ 0 0
$$355$$ 24.0000 1.27379
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 27.7128i − 1.46263i −0.682042 0.731313i $$-0.738908\pi$$
0.682042 0.731313i $$-0.261092\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 8.00000 0.419891
$$364$$ 0 0
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 19.0000 0.991792 0.495896 0.868382i $$-0.334840\pi$$
0.495896 + 0.868382i $$0.334840\pi$$
$$368$$ 0 0
$$369$$ − 10.3923i − 0.541002i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2.00000 0.103556 0.0517780 0.998659i $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ 0 0
$$375$$ − 12.1244i − 0.626099i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 24.2487i − 1.24557i −0.782392 0.622786i $$-0.786001\pi$$
0.782392 0.622786i $$-0.213999\pi$$
$$380$$ 0 0
$$381$$ 5.19615i 0.266207i
$$382$$ 0 0
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.46410i 0.176090i
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −9.00000 −0.453990
$$394$$ 0 0
$$395$$ −9.00000 −0.452839
$$396$$ 0 0
$$397$$ 20.7846i 1.04315i 0.853206 + 0.521575i $$0.174655\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 1.73205i − 0.0860663i
$$406$$ 0 0
$$407$$ 17.3205i 0.858546i
$$408$$ 0 0
$$409$$ − 22.5167i − 1.11338i −0.830721 0.556689i $$-0.812072\pi$$
0.830721 0.556689i $$-0.187928\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ − 5.19615i − 0.255069i
$$416$$ 0 0
$$417$$ 14.0000 0.685583
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 6.92820i 0.336067i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 38.1051i 1.83546i 0.397206 + 0.917729i $$0.369980\pi$$
−0.397206 + 0.917729i $$0.630020\pi$$
$$432$$ 0 0
$$433$$ 34.6410i 1.66474i 0.554220 + 0.832370i $$0.313017\pi$$
−0.554220 + 0.832370i $$0.686983\pi$$
$$434$$ 0 0
$$435$$ − 15.5885i − 0.747409i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −29.0000 −1.38409 −0.692047 0.721852i $$-0.743291\pi$$
−0.692047 + 0.721852i $$0.743291\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 12.1244i − 0.576046i −0.957624 0.288023i $$-0.907002\pi$$
0.957624 0.288023i $$-0.0929979\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ 0 0
$$451$$ 18.0000 0.847587
$$452$$ 0 0
$$453$$ 5.19615i 0.244137i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −31.0000 −1.45012 −0.725059 0.688686i $$-0.758188\pi$$
−0.725059 + 0.688686i $$0.758188\pi$$
$$458$$ 0 0
$$459$$ 3.46410i 0.161690i
$$460$$ 0 0
$$461$$ − 6.92820i − 0.322679i −0.986899 0.161339i $$-0.948419\pi$$
0.986899 0.161339i $$-0.0515813\pi$$
$$462$$ 0 0
$$463$$ − 38.1051i − 1.77090i −0.464739 0.885448i $$-0.653852\pi$$
0.464739 0.885448i $$-0.346148\pi$$
$$464$$ 0 0
$$465$$ − 8.66025i − 0.401610i
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.92820i 0.319235i
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −9.00000 −0.412082
$$478$$ 0 0
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −33.0000 −1.49845
$$486$$ 0 0
$$487$$ − 8.66025i − 0.392434i −0.980561 0.196217i $$-0.937134\pi$$
0.980561 0.196217i $$-0.0628656\pi$$
$$488$$ 0 0
$$489$$ 13.8564i 0.626608i
$$490$$ 0 0
$$491$$ − 25.9808i − 1.17250i −0.810132 0.586248i $$-0.800605\pi$$
0.810132 0.586248i $$-0.199395\pi$$
$$492$$ 0 0
$$493$$ 31.1769i 1.40414i
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ − 3.46410i − 0.155074i −0.996989 0.0775372i $$-0.975294\pi$$
0.996989 0.0775372i $$-0.0247057\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ −24.0000 −1.06799
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ − 32.9090i − 1.45866i −0.684160 0.729332i $$-0.739831\pi$$
0.684160 0.729332i $$-0.260169\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.00000 −0.0883022
$$514$$ 0 0
$$515$$ 6.92820i 0.305293i
$$516$$ 0 0
$$517$$ 20.7846i 0.914106i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 38.1051i 1.66942i 0.550693 + 0.834708i $$0.314363\pi$$
−0.550693 + 0.834708i $$0.685637\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 17.3205i 0.754493i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −9.00000 −0.390567
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 21.0000 0.907909
$$536$$ 0 0
$$537$$ 24.2487i 1.04641i
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 0 0
$$543$$ 10.3923i 0.445976i
$$544$$ 0 0
$$545$$ 6.92820i 0.296772i
$$546$$ 0 0
$$547$$ − 10.3923i − 0.444343i −0.975008 0.222171i $$-0.928686\pi$$
0.975008 0.222171i $$-0.0713145\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −18.0000 −0.766826
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ − 17.3205i − 0.735215i
$$556$$ 0 0
$$557$$ −27.0000 −1.14403 −0.572013 0.820244i $$-0.693837\pi$$
−0.572013 + 0.820244i $$0.693837\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −6.00000 −0.253320
$$562$$ 0 0
$$563$$ −45.0000 −1.89652 −0.948262 0.317489i $$-0.897160\pi$$
−0.948262 + 0.317489i $$0.897160\pi$$
$$564$$ 0 0
$$565$$ − 10.3923i − 0.437208i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ − 20.7846i − 0.869809i −0.900477 0.434904i $$-0.856782\pi$$
0.900477 0.434904i $$-0.143218\pi$$
$$572$$ 0 0
$$573$$ 13.8564i 0.578860i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 1.73205i − 0.0721062i −0.999350 0.0360531i $$-0.988521\pi$$
0.999350 0.0360531i $$-0.0114785\pi$$
$$578$$ 0 0
$$579$$ −5.00000 −0.207793
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 15.5885i − 0.645608i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −33.0000 −1.36206 −0.681028 0.732257i $$-0.738467\pi$$
−0.681028 + 0.732257i $$0.738467\pi$$
$$588$$ 0 0
$$589$$ −10.0000 −0.412043
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ 0 0
$$593$$ 27.7128i 1.13803i 0.822328 + 0.569014i $$0.192675\pi$$
−0.822328 + 0.569014i $$0.807325\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 27.7128i 1.13231i 0.824297 + 0.566157i $$0.191571\pi$$
−0.824297 + 0.566157i $$0.808429\pi$$
$$600$$ 0 0
$$601$$ − 15.5885i − 0.635866i −0.948113 0.317933i $$-0.897011\pi$$
0.948113 0.317933i $$-0.102989\pi$$
$$602$$ 0 0
$$603$$ − 13.8564i − 0.564276i
$$604$$ 0 0
$$605$$ − 13.8564i − 0.563343i
$$606$$ 0 0
$$607$$ 19.0000 0.771186 0.385593 0.922669i $$-0.373997\pi$$
0.385593 + 0.922669i $$0.373997\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −4.00000 −0.161558 −0.0807792 0.996732i $$-0.525741\pi$$
−0.0807792 + 0.996732i $$0.525741\pi$$
$$614$$ 0 0
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ 12.0000 0.483102 0.241551 0.970388i $$-0.422344\pi$$
0.241551 + 0.970388i $$0.422344\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ − 3.46410i − 0.138343i
$$628$$ 0 0
$$629$$ 34.6410i 1.38123i
$$630$$ 0 0
$$631$$ 32.9090i 1.31009i 0.755592 + 0.655043i $$0.227349\pi$$
−0.755592 + 0.655043i $$0.772651\pi$$
$$632$$ 0 0
$$633$$ − 6.92820i − 0.275371i
$$634$$ 0 0
$$635$$ 9.00000 0.357154
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 13.8564i 0.548151i
$$640$$ 0 0
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 0 0
$$643$$ 16.0000 0.630978 0.315489 0.948929i $$-0.397831\pi$$
0.315489 + 0.948929i $$0.397831\pi$$
$$644$$ 0 0
$$645$$ 6.00000 0.236250
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ − 15.5885i − 0.611900i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −27.0000 −1.05659 −0.528296 0.849060i $$-0.677169\pi$$
−0.528296 + 0.849060i $$0.677169\pi$$
$$654$$ 0 0
$$655$$ 15.5885i 0.609091i
$$656$$ 0 0
$$657$$ − 6.92820i − 0.270295i
$$658$$ 0 0
$$659$$ 10.3923i 0.404827i 0.979300 + 0.202413i $$0.0648785\pi$$
−0.979300 + 0.202413i $$0.935122\pi$$
$$660$$ 0 0
$$661$$ 3.46410i 0.134738i 0.997728 + 0.0673690i $$0.0214605\pi$$
−0.997728 + 0.0673690i $$0.978540\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 19.0000 0.734582
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −35.0000 −1.34915 −0.674575 0.738206i $$-0.735673\pi$$
−0.674575 + 0.738206i $$0.735673\pi$$
$$674$$ 0 0
$$675$$ 2.00000 0.0769800
$$676$$ 0 0
$$677$$ − 29.4449i − 1.13166i −0.824523 0.565829i $$-0.808556\pi$$
0.824523 0.565829i $$-0.191444\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −21.0000 −0.804722
$$682$$ 0 0
$$683$$ − 25.9808i − 0.994126i −0.867714 0.497063i $$-0.834412\pi$$
0.867714 0.497063i $$-0.165588\pi$$
$$684$$ 0 0
$$685$$ 20.7846i 0.794139i
$$686$$ 0 0
$$687$$ 3.46410i 0.132164i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −22.0000 −0.836919 −0.418460 0.908235i $$-0.637430\pi$$
−0.418460 + 0.908235i $$0.637430\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 24.2487i − 0.919806i
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ −45.0000 −1.69963 −0.849813 0.527084i $$-0.823285\pi$$
−0.849813 + 0.527084i $$0.823285\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ 0 0
$$705$$ − 20.7846i − 0.782794i
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ − 5.19615i − 0.194871i
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.92820i 0.258738i
$$718$$ 0 0
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 22.5167i 0.837404i
$$724$$ 0 0
$$725$$ 18.0000 0.668503
$$726$$ 0 0
$$727$$ −37.0000 −1.37225 −0.686127 0.727482i $$-0.740691\pi$$
−0.686127 + 0.727482i $$0.740691\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ 10.3923i 0.383849i 0.981410 + 0.191924i $$0.0614728\pi$$
−0.981410 + 0.191924i $$0.938527\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.0000 0.884051
$$738$$ 0 0
$$739$$ 10.3923i 0.382287i 0.981562 + 0.191144i $$0.0612196\pi$$
−0.981562 + 0.191144i $$0.938780\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 13.8564i − 0.508342i −0.967159 0.254171i $$-0.918197\pi$$
0.967159 0.254171i $$-0.0818026\pi$$
$$744$$ 0 0
$$745$$ − 10.3923i − 0.380745i
$$746$$ 0 0
$$747$$ 3.00000 0.109764
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ − 5.19615i − 0.189610i −0.995496 0.0948051i $$-0.969777\pi$$
0.995496 0.0948051i $$-0.0302228\pi$$
$$752$$ 0 0
$$753$$ −15.0000 −0.546630
$$754$$ 0 0
$$755$$ 9.00000 0.327544
$$756$$ 0 0
$$757$$ −4.00000 −0.145382 −0.0726912 0.997354i $$-0.523159\pi$$
−0.0726912 + 0.997354i $$0.523159\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.4974i 1.75803i 0.476794 + 0.879015i $$0.341799\pi$$
−0.476794 + 0.879015i $$0.658201\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 6.00000 0.216930
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 19.0526i 0.687053i 0.939143 + 0.343526i $$0.111621\pi$$
−0.939143 + 0.343526i $$0.888379\pi$$
$$770$$ 0 0
$$771$$ 10.3923i 0.374270i
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 10.0000 0.359211
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 20.7846i 0.744686i
$$780$$ 0 0
$$781$$ −24.0000 −0.858788
$$782$$ 0 0
$$783$$ 9.00000 0.321634
$$784$$ 0 0
$$785$$ 12.0000 0.428298
$$786$$ 0 0
$$787$$ −28.0000 −0.998092 −0.499046 0.866575i $$-0.666316\pi$$
−0.499046 + 0.866575i $$0.666316\pi$$
$$788$$ 0 0
$$789$$ 17.3205i 0.616626i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 15.5885i 0.552866i
$$796$$ 0 0
$$797$$ 5.19615i 0.184057i 0.995756 + 0.0920286i $$0.0293351\pi$$
−0.995756 + 0.0920286i $$0.970665\pi$$
$$798$$ 0 0
$$799$$ 41.5692i 1.47061i
$$800$$ 0 0
$$801$$ 3.46410i 0.122398i
$$802$$ 0 0
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 15.5885i − 0.548740i
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ 0 0
$$813$$ 11.0000 0.385787
$$814$$ 0 0
$$815$$ 24.0000 0.840683
$$816$$ 0 0
$$817$$ − 6.92820i − 0.242387i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 3.00000 0.104701 0.0523504 0.998629i $$-0.483329\pi$$
0.0523504 + 0.998629i $$0.483329\pi$$
$$822$$ 0 0
$$823$$ − 38.1051i − 1.32826i −0.747617 0.664130i $$-0.768802\pi$$
0.747617 0.664130i $$-0.231198\pi$$
$$824$$ 0 0
$$825$$ 3.46410i 0.120605i
$$826$$ 0 0
$$827$$ 39.8372i 1.38527i 0.721286 + 0.692637i $$0.243551\pi$$
−0.721286 + 0.692637i $$0.756449\pi$$
$$828$$ 0 0
$$829$$ − 17.3205i − 0.601566i −0.953693 0.300783i $$-0.902752\pi$$
0.953693 0.300783i $$-0.0972480\pi$$
$$830$$ 0 0
$$831$$ 28.0000 0.971309
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 20.7846i 0.719281i
$$836$$ 0 0
$$837$$ 5.00000 0.172825
$$838$$ 0 0
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ −24.0000 −0.826604
$$844$$ 0 0
$$845$$ − 22.5167i − 0.774597i
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 48.4974i − 1.66052i −0.557376 0.830260i $$-0.688192\pi$$
0.557376 0.830260i $$-0.311808\pi$$
$$854$$ 0 0
$$855$$ 3.46410i 0.118470i
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ −14.0000 −0.477674 −0.238837 0.971060i $$-0.576766\pi$$
−0.238837 + 0.971060i $$0.576766\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 31.1769i 1.06127i 0.847599 + 0.530637i $$0.178047\pi$$
−0.847599 + 0.530637i $$0.821953\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5.00000 0.169809
$$868$$ 0 0
$$869$$ 9.00000 0.305304
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 19.0526i − 0.644831i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ 0 0
$$879$$ 5.19615i 0.175262i
$$880$$ 0 0
$$881$$ − 13.8564i − 0.466834i −0.972377 0.233417i $$-0.925009\pi$$
0.972377 0.233417i $$-0.0749907\pi$$
$$882$$ 0 0
$$883$$ − 24.2487i − 0.816034i −0.912974 0.408017i $$-0.866220\pi$$
0.912974 0.408017i $$-0.133780\pi$$
$$884$$ 0 0
$$885$$ 15.5885i 0.524000i
$$886$$ 0 0
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 1.73205i 0.0580259i
$$892$$ 0 0
$$893$$ −24.0000 −0.803129
$$894$$ 0 0
$$895$$ 42.0000 1.40391
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 45.0000 1.50083
$$900$$ 0 0
$$901$$ − 31.1769i − 1.03865i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 18.0000 0.598340
$$906$$ 0 0
$$907$$ 20.7846i 0.690142i 0.938577 + 0.345071i $$0.112145\pi$$
−0.938577 + 0.345071i $$0.887855\pi$$
$$908$$ 0 0
$$909$$ − 13.8564i − 0.459588i
$$910$$ 0 0
$$911$$ − 41.5692i − 1.37725i −0.725118 0.688625i $$-0.758215\pi$$
0.725118 0.688625i $$-0.241785\pi$$
$$912$$ 0 0
$$913$$ 5.19615i 0.171968i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 17.3205i − 0.571351i −0.958326 0.285675i $$-0.907782\pi$$
0.958326 0.285675i $$-0.0922179\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ 17.3205i 0.568267i 0.958785 + 0.284134i $$0.0917060\pi$$
−0.958785 + 0.284134i $$0.908294\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −18.0000 −0.589294
$$934$$ 0 0
$$935$$ 10.3923i 0.339865i
$$936$$ 0 0
$$937$$ − 19.0526i − 0.622420i −0.950341 0.311210i $$-0.899266\pi$$
0.950341 0.311210i $$-0.100734\pi$$
$$938$$ 0 0
$$939$$ 5.19615i 0.169570i
$$940$$ 0 0
$$941$$ − 39.8372i − 1.29865i −0.760509 0.649327i $$-0.775051\pi$$
0.760509 0.649327i $$-0.224949\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 38.1051i 1.23825i 0.785292 + 0.619125i $$0.212513\pi$$
−0.785292 + 0.619125i $$0.787487\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −21.0000 −0.680972
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ 0 0
$$957$$ 15.5885i 0.503903i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 12.1244i 0.390702i
$$964$$ 0 0
$$965$$ 8.66025i 0.278783i
$$966$$ 0 0
$$967$$ − 39.8372i − 1.28108i −0.767926 0.640538i $$-0.778711\pi$$
0.767926 0.640538i $$-0.221289\pi$$
$$968$$ 0 0
$$969$$ − 6.92820i − 0.222566i
$$970$$ 0 0
$$971$$ −33.0000 −1.05902 −0.529510 0.848304i $$-0.677624\pi$$
−0.529510 + 0.848304i $$0.677624\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 18.0000 0.574111 0.287055 0.957914i $$-0.407324\pi$$
0.287055 + 0.957914i $$0.407324\pi$$
$$984$$ 0 0
$$985$$ 10.3923i 0.331126i
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 12.1244i 0.385143i 0.981283 + 0.192571i $$0.0616827\pi$$
−0.981283 + 0.192571i $$0.938317\pi$$
$$992$$ 0 0
$$993$$ − 13.8564i − 0.439720i
$$994$$ 0 0
$$995$$ − 27.7128i − 0.878555i
$$996$$ 0 0
$$997$$ 27.7128i 0.877674i 0.898567 + 0.438837i $$0.144609\pi$$
−0.898567 + 0.438837i $$0.855391\pi$$
$$998$$ 0 0
$$999$$ 10.0000 0.316386
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 2352.2.b.g.1567.1 2
3.2 odd 2 7056.2.b.e.1567.2 2
4.3 odd 2 2352.2.b.c.1567.1 2
7.2 even 3 2352.2.bl.b.31.1 2
7.3 odd 6 2352.2.bl.h.607.1 2
7.4 even 3 336.2.bl.c.271.1 yes 2
7.5 odd 6 336.2.bl.g.31.1 yes 2
7.6 odd 2 2352.2.b.c.1567.2 2
12.11 even 2 7056.2.b.i.1567.2 2
21.5 even 6 1008.2.cs.d.703.1 2
21.11 odd 6 1008.2.cs.e.271.1 2
21.20 even 2 7056.2.b.i.1567.1 2
28.3 even 6 2352.2.bl.b.607.1 2
28.11 odd 6 336.2.bl.g.271.1 yes 2
28.19 even 6 336.2.bl.c.31.1 2
28.23 odd 6 2352.2.bl.h.31.1 2
28.27 even 2 inner 2352.2.b.g.1567.2 2
56.5 odd 6 1344.2.bl.b.703.1 2
56.11 odd 6 1344.2.bl.b.1279.1 2
56.19 even 6 1344.2.bl.f.703.1 2
56.53 even 6 1344.2.bl.f.1279.1 2
84.11 even 6 1008.2.cs.d.271.1 2
84.47 odd 6 1008.2.cs.e.703.1 2
84.83 odd 2 7056.2.b.e.1567.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.c.31.1 2 28.19 even 6
336.2.bl.c.271.1 yes 2 7.4 even 3
336.2.bl.g.31.1 yes 2 7.5 odd 6
336.2.bl.g.271.1 yes 2 28.11 odd 6
1008.2.cs.d.271.1 2 84.11 even 6
1008.2.cs.d.703.1 2 21.5 even 6
1008.2.cs.e.271.1 2 21.11 odd 6
1008.2.cs.e.703.1 2 84.47 odd 6
1344.2.bl.b.703.1 2 56.5 odd 6
1344.2.bl.b.1279.1 2 56.11 odd 6
1344.2.bl.f.703.1 2 56.19 even 6
1344.2.bl.f.1279.1 2 56.53 even 6
2352.2.b.c.1567.1 2 4.3 odd 2
2352.2.b.c.1567.2 2 7.6 odd 2
2352.2.b.g.1567.1 2 1.1 even 1 trivial
2352.2.b.g.1567.2 2 28.27 even 2 inner
2352.2.bl.b.31.1 2 7.2 even 3
2352.2.bl.b.607.1 2 28.3 even 6
2352.2.bl.h.31.1 2 28.23 odd 6
2352.2.bl.h.607.1 2 7.3 odd 6
7056.2.b.e.1567.1 2 84.83 odd 2
7056.2.b.e.1567.2 2 3.2 odd 2
7056.2.b.i.1567.1 2 21.20 even 2
7056.2.b.i.1567.2 2 12.11 even 2