# Properties

 Label 2352.2.b.c.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.c.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 $$-0.915909\pi$$
0.965307 0.261116i $$-0.0840907\pi$$
$$12$$ 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 $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 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 $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 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 $$-0.914913\pi$$
0.964486 0.264135i $$-0.0850865\pi$$
$$44$$ 0 0
$$45$$ − 1.73205i − 0.258199i
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 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 $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 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 $$0.321244\pi$$
−0.532524 + 0.846415i $$0.678756\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 13.8564i − 1.64445i −0.569160 0.822226i $$-0.692732\pi$$
0.569160 0.822226i $$-0.307268\pi$$
$$72$$ 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 $$0.0944227\pi$$
−0.956325 + 0.292306i $$0.905577\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 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 $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 12.1244i − 1.17211i −0.810273 0.586053i $$-0.800681\pi$$
0.810273 0.586053i $$-0.199319\pi$$
$$108$$ 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 $$-0.925950\pi$$
0.973062 0.230542i $$-0.0740499\pi$$
$$128$$ 0 0
$$129$$ 3.46410i 0.304997i
$$130$$ 0 0
$$131$$ 9.00000 0.786334 0.393167 0.919467i $$-0.371379\pi$$
0.393167 + 0.919467i $$0.371379\pi$$
$$132$$ 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 $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 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 $$-0.932188\pi$$
0.977393 0.211428i $$-0.0678115\pi$$
$$152$$ 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 $$-0.817418\pi$$
0.839953 0.542659i $$-0.182582\pi$$
$$164$$ 0 0
$$165$$ 3.00000 0.233550
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 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 $$-0.638957\pi$$
0.422813 0.906217i $$-0.361043\pi$$
$$180$$ 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 $$-0.832853\pi$$
0.865269 0.501307i $$-0.167147\pi$$
$$192$$ 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 $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 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 $$0.0766487\pi$$
−0.971148 + 0.238479i $$0.923351\pi$$
$$212$$ 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 $$-0.719481\pi$$
−0.636167 + 0.771551i $$0.719481\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ 21.0000 1.39382 0.696909 0.717159i $$-0.254558\pi$$
0.696909 + 0.717159i $$0.254558\pi$$
$$228$$ 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 $$-0.928064\pi$$
0.974572 0.224074i $$-0.0719358\pi$$
$$240$$ 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 $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 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 $$-0.820683\pi$$
0.845476 0.534014i $$-0.179317\pi$$
$$264$$ 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 $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ 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 $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 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 $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 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 $$0.124354\pi$$
−0.924654 + 0.380808i $$0.875646\pi$$
$$332$$ 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 $$0.153912\pi$$
−0.885360 + 0.464907i $$0.846088\pi$$
$$348$$ 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 $$0.261092\pi$$
−0.682042 + 0.731313i $$0.738908\pi$$
$$360$$ 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 $$-0.665160\pi$$
−0.495896 + 0.868382i $$0.665160\pi$$
$$368$$ 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 $$0.213999\pi$$
−0.782392 + 0.622786i $$0.786001\pi$$
$$380$$ 0 0
$$381$$ 5.19615i 0.266207i
$$382$$ 0 0
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 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 $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 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 $$-0.630020\pi$$
0.397206 0.917729i $$-0.369980\pi$$
$$432$$ 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 $$-0.256709\pi$$
0.692047 + 0.721852i $$0.256709\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.1244i 0.576046i 0.957624 + 0.288023i $$0.0929979\pi$$
−0.957624 + 0.288023i $$0.907002\pi$$
$$444$$ 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 $$0.346148\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ − 8.66025i − 0.401610i
$$466$$ 0 0
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 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 $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 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 $$0.0628656\pi$$
−0.980561 + 0.196217i $$0.937134\pi$$
$$488$$ 0 0
$$489$$ 13.8564i 0.626608i
$$490$$ 0 0
$$491$$ 25.9808i 1.17250i 0.810132 + 0.586248i $$0.199395\pi$$
−0.810132 + 0.586248i $$0.800605\pi$$
$$492$$ 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 $$0.0247057\pi$$
−0.996989 + 0.0775372i $$0.975294\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 0 0
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 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 $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 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 $$0.0713145\pi$$
−0.975008 + 0.222171i $$0.928686\pi$$
$$548$$ 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 $$-0.102840\pi$$
0.948262 + 0.317489i $$0.102840\pi$$
$$564$$ 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 $$0.143218\pi$$
−0.900477 + 0.434904i $$0.856782\pi$$
$$572$$ 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 $$-0.261533\pi$$
0.681028 + 0.732257i $$0.261533\pi$$
$$588$$ 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 $$-0.808429\pi$$
0.824297 0.566157i $$-0.191571\pi$$
$$600$$ 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 $$-0.626003\pi$$
−0.385593 + 0.922669i $$0.626003\pi$$
$$608$$ 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 $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 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 $$-0.772651\pi$$
0.755592 0.655043i $$-0.227349\pi$$
$$632$$ 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 $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 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 $$-0.935122\pi$$
0.979300 0.202413i $$-0.0648785\pi$$
$$660$$ 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 $$0.165588\pi$$
−0.867714 + 0.497063i $$0.834412\pi$$
$$684$$ 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 $$-0.362570\pi$$
0.418460 + 0.908235i $$0.362570\pi$$
$$692$$ 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 $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 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 $$-0.259309\pi$$
0.686127 + 0.727482i $$0.259309\pi$$
$$728$$ 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 $$-0.938780\pi$$
0.981562 0.191144i $$-0.0612196\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 13.8564i 0.508342i 0.967159 + 0.254171i $$0.0818026\pi$$
−0.967159 + 0.254171i $$0.918197\pi$$
$$744$$ 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 $$0.0302228\pi$$
−0.995496 + 0.0948051i $$0.969777\pi$$
$$752$$ 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 $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 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 $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 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 $$0.231198\pi$$
−0.747617 + 0.664130i $$0.768802\pi$$
$$824$$ 0 0
$$825$$ 3.46410i 0.120605i
$$826$$ 0 0
$$827$$ − 39.8372i − 1.38527i −0.721286 0.692637i $$-0.756449\pi$$
0.721286 0.692637i $$-0.243551\pi$$
$$828$$ 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 $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 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 $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 31.1769i − 1.06127i −0.847599 0.530637i $$-0.821953\pi$$
0.847599 0.530637i $$-0.178047\pi$$
$$864$$ 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 $$0.133780\pi$$
−0.912974 + 0.408017i $$0.866220\pi$$
$$884$$ 0 0
$$885$$ 15.5885i 0.524000i
$$886$$ 0 0
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 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 $$-0.887855\pi$$
0.938577 0.345071i $$-0.112145\pi$$
$$908$$ 0 0
$$909$$ − 13.8564i − 0.459588i
$$910$$ 0 0
$$911$$ 41.5692i 1.37725i 0.725118 + 0.688625i $$0.241785\pi$$
−0.725118 + 0.688625i $$0.758215\pi$$
$$912$$ 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 $$0.0922179\pi$$
−0.958326 + 0.285675i $$0.907782\pi$$
$$920$$ 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 $$-0.787487\pi$$
0.785292 0.619125i $$-0.212513\pi$$
$$948$$ 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 $$0.221289\pi$$
−0.767926 + 0.640538i $$0.778711\pi$$
$$968$$ 0 0
$$969$$ − 6.92820i − 0.222566i
$$970$$ 0 0
$$971$$ 33.0000 1.05902 0.529510 0.848304i $$-0.322376\pi$$
0.529510 + 0.848304i $$0.322376\pi$$
$$972$$ 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 $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ 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 $$-0.938317\pi$$
0.981283 0.192571i $$-0.0616827\pi$$
$$992$$ 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.c.1567.1 2
3.2 odd 2 7056.2.b.i.1567.2 2
4.3 odd 2 2352.2.b.g.1567.1 2
7.2 even 3 2352.2.bl.h.31.1 2
7.3 odd 6 2352.2.bl.b.607.1 2
7.4 even 3 336.2.bl.g.271.1 yes 2
7.5 odd 6 336.2.bl.c.31.1 2
7.6 odd 2 2352.2.b.g.1567.2 2
12.11 even 2 7056.2.b.e.1567.2 2
21.5 even 6 1008.2.cs.e.703.1 2
21.11 odd 6 1008.2.cs.d.271.1 2
21.20 even 2 7056.2.b.e.1567.1 2
28.3 even 6 2352.2.bl.h.607.1 2
28.11 odd 6 336.2.bl.c.271.1 yes 2
28.19 even 6 336.2.bl.g.31.1 yes 2
28.23 odd 6 2352.2.bl.b.31.1 2
28.27 even 2 inner 2352.2.b.c.1567.2 2
56.5 odd 6 1344.2.bl.f.703.1 2
56.11 odd 6 1344.2.bl.f.1279.1 2
56.19 even 6 1344.2.bl.b.703.1 2
56.53 even 6 1344.2.bl.b.1279.1 2
84.11 even 6 1008.2.cs.e.271.1 2
84.47 odd 6 1008.2.cs.d.703.1 2
84.83 odd 2 7056.2.b.i.1567.1 2

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