# Properties

 Label 7800.2.a.bm.1.3 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.76156 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.76156 q^{7} +1.00000 q^{9} +4.49084 q^{11} +1.00000 q^{13} +3.62620 q^{17} -6.14931 q^{19} -2.76156 q^{21} +5.52311 q^{23} -1.00000 q^{27} +6.25240 q^{29} +8.49084 q^{31} -4.49084 q^{33} -0.896916 q^{37} -1.00000 q^{39} +2.89692 q^{41} +0.270718 q^{43} +3.38776 q^{47} +0.626198 q^{49} -3.62620 q^{51} -3.27072 q^{53} +6.14931 q^{57} -9.53707 q^{59} +11.4200 q^{61} +2.76156 q^{63} -12.4340 q^{67} -5.52311 q^{69} +8.89692 q^{71} +3.25240 q^{73} +12.4017 q^{77} +13.6724 q^{79} +1.00000 q^{81} -3.03228 q^{83} -6.25240 q^{87} -2.20617 q^{89} +2.76156 q^{91} -8.49084 q^{93} -17.7572 q^{97} +4.49084 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + 2 q^{17} + 3 q^{19} - 2 q^{21} + 4 q^{23} - 3 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} + q^{37} - 3 q^{39} + 5 q^{41} + 6 q^{43} - 5 q^{47} - 7 q^{49} - 2 q^{51} - 15 q^{53} - 3 q^{57} + 8 q^{59} + 18 q^{61} + 2 q^{63} + 3 q^{67} - 4 q^{69} + 23 q^{71} - 8 q^{73} - 2 q^{77} + 7 q^{79} + 3 q^{81} - 8 q^{83} - q^{87} - 14 q^{89} + 2 q^{91} - 14 q^{93} + 2 q^{99} + O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.76156 1.04377 0.521885 0.853016i $$-0.325229\pi$$
0.521885 + 0.853016i $$0.325229\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.49084 1.35404 0.677019 0.735965i $$-0.263271\pi$$
0.677019 + 0.735965i $$0.263271\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.62620 0.879482 0.439741 0.898125i $$-0.355070\pi$$
0.439741 + 0.898125i $$0.355070\pi$$
$$18$$ 0 0
$$19$$ −6.14931 −1.41075 −0.705375 0.708835i $$-0.749221\pi$$
−0.705375 + 0.708835i $$0.749221\pi$$
$$20$$ 0 0
$$21$$ −2.76156 −0.602621
$$22$$ 0 0
$$23$$ 5.52311 1.15165 0.575824 0.817573i $$-0.304681\pi$$
0.575824 + 0.817573i $$0.304681\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 6.25240 1.16104 0.580520 0.814246i $$-0.302849\pi$$
0.580520 + 0.814246i $$0.302849\pi$$
$$30$$ 0 0
$$31$$ 8.49084 1.52500 0.762500 0.646988i $$-0.223972\pi$$
0.762500 + 0.646988i $$0.223972\pi$$
$$32$$ 0 0
$$33$$ −4.49084 −0.781755
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.896916 −0.147452 −0.0737261 0.997279i $$-0.523489\pi$$
−0.0737261 + 0.997279i $$0.523489\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 2.89692 0.452422 0.226211 0.974078i $$-0.427366\pi$$
0.226211 + 0.974078i $$0.427366\pi$$
$$42$$ 0 0
$$43$$ 0.270718 0.0412841 0.0206421 0.999787i $$-0.493429\pi$$
0.0206421 + 0.999787i $$0.493429\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.38776 0.494155 0.247077 0.968996i $$-0.420530\pi$$
0.247077 + 0.968996i $$0.420530\pi$$
$$48$$ 0 0
$$49$$ 0.626198 0.0894569
$$50$$ 0 0
$$51$$ −3.62620 −0.507769
$$52$$ 0 0
$$53$$ −3.27072 −0.449268 −0.224634 0.974443i $$-0.572119\pi$$
−0.224634 + 0.974443i $$0.572119\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.14931 0.814496
$$58$$ 0 0
$$59$$ −9.53707 −1.24162 −0.620810 0.783961i $$-0.713196\pi$$
−0.620810 + 0.783961i $$0.713196\pi$$
$$60$$ 0 0
$$61$$ 11.4200 1.46219 0.731093 0.682278i $$-0.239011\pi$$
0.731093 + 0.682278i $$0.239011\pi$$
$$62$$ 0 0
$$63$$ 2.76156 0.347924
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.4340 −1.51905 −0.759526 0.650476i $$-0.774569\pi$$
−0.759526 + 0.650476i $$0.774569\pi$$
$$68$$ 0 0
$$69$$ −5.52311 −0.664905
$$70$$ 0 0
$$71$$ 8.89692 1.05587 0.527935 0.849285i $$-0.322967\pi$$
0.527935 + 0.849285i $$0.322967\pi$$
$$72$$ 0 0
$$73$$ 3.25240 0.380664 0.190332 0.981720i $$-0.439044\pi$$
0.190332 + 0.981720i $$0.439044\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 12.4017 1.41331
$$78$$ 0 0
$$79$$ 13.6724 1.53827 0.769134 0.639087i $$-0.220688\pi$$
0.769134 + 0.639087i $$0.220688\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −3.03228 −0.332835 −0.166418 0.986055i $$-0.553220\pi$$
−0.166418 + 0.986055i $$0.553220\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −6.25240 −0.670327
$$88$$ 0 0
$$89$$ −2.20617 −0.233853 −0.116927 0.993141i $$-0.537304\pi$$
−0.116927 + 0.993141i $$0.537304\pi$$
$$90$$ 0 0
$$91$$ 2.76156 0.289490
$$92$$ 0 0
$$93$$ −8.49084 −0.880459
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −17.7572 −1.80297 −0.901485 0.432811i $$-0.857522\pi$$
−0.901485 + 0.432811i $$0.857522\pi$$
$$98$$ 0 0
$$99$$ 4.49084 0.451346
$$100$$ 0 0
$$101$$ 5.83237 0.580342 0.290171 0.956975i $$-0.406288\pi$$
0.290171 + 0.956975i $$0.406288\pi$$
$$102$$ 0 0
$$103$$ −3.45856 −0.340782 −0.170391 0.985376i $$-0.554503\pi$$
−0.170391 + 0.985376i $$0.554503\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.832365 −0.0804678 −0.0402339 0.999190i $$-0.512810\pi$$
−0.0402339 + 0.999190i $$0.512810\pi$$
$$108$$ 0 0
$$109$$ −12.1493 −1.16369 −0.581847 0.813299i $$-0.697670\pi$$
−0.581847 + 0.813299i $$0.697670\pi$$
$$110$$ 0 0
$$111$$ 0.896916 0.0851315
$$112$$ 0 0
$$113$$ 4.50479 0.423775 0.211888 0.977294i $$-0.432039\pi$$
0.211888 + 0.977294i $$0.432039\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 10.0140 0.917978
$$120$$ 0 0
$$121$$ 9.16763 0.833421
$$122$$ 0 0
$$123$$ −2.89692 −0.261206
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.62620 0.587980 0.293990 0.955808i $$-0.405017\pi$$
0.293990 + 0.955808i $$0.405017\pi$$
$$128$$ 0 0
$$129$$ −0.270718 −0.0238354
$$130$$ 0 0
$$131$$ 1.10308 0.0963769 0.0481884 0.998838i $$-0.484655\pi$$
0.0481884 + 0.998838i $$0.484655\pi$$
$$132$$ 0 0
$$133$$ −16.9817 −1.47250
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −20.4479 −1.74699 −0.873493 0.486837i $$-0.838150\pi$$
−0.873493 + 0.486837i $$0.838150\pi$$
$$138$$ 0 0
$$139$$ 16.5693 1.40539 0.702697 0.711490i $$-0.251979\pi$$
0.702697 + 0.711490i $$0.251979\pi$$
$$140$$ 0 0
$$141$$ −3.38776 −0.285300
$$142$$ 0 0
$$143$$ 4.49084 0.375543
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −0.626198 −0.0516479
$$148$$ 0 0
$$149$$ 3.79383 0.310803 0.155401 0.987851i $$-0.450333\pi$$
0.155401 + 0.987851i $$0.450333\pi$$
$$150$$ 0 0
$$151$$ −5.80779 −0.472631 −0.236315 0.971676i $$-0.575940\pi$$
−0.236315 + 0.971676i $$0.575940\pi$$
$$152$$ 0 0
$$153$$ 3.62620 0.293161
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.6541 1.08972 0.544858 0.838528i $$-0.316584\pi$$
0.544858 + 0.838528i $$0.316584\pi$$
$$158$$ 0 0
$$159$$ 3.27072 0.259385
$$160$$ 0 0
$$161$$ 15.2524 1.20206
$$162$$ 0 0
$$163$$ 7.25240 0.568052 0.284026 0.958817i $$-0.408330\pi$$
0.284026 + 0.958817i $$0.408330\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.14931 −0.166319 −0.0831594 0.996536i $$-0.526501\pi$$
−0.0831594 + 0.996536i $$0.526501\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −6.14931 −0.470250
$$172$$ 0 0
$$173$$ −25.5693 −1.94400 −0.972001 0.234978i $$-0.924498\pi$$
−0.972001 + 0.234978i $$0.924498\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 9.53707 0.716850
$$178$$ 0 0
$$179$$ −21.5510 −1.61080 −0.805399 0.592732i $$-0.798049\pi$$
−0.805399 + 0.592732i $$0.798049\pi$$
$$180$$ 0 0
$$181$$ −0.607876 −0.0451831 −0.0225915 0.999745i $$-0.507192\pi$$
−0.0225915 + 0.999745i $$0.507192\pi$$
$$182$$ 0 0
$$183$$ −11.4200 −0.844193
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.2847 1.19085
$$188$$ 0 0
$$189$$ −2.76156 −0.200874
$$190$$ 0 0
$$191$$ −21.2803 −1.53979 −0.769894 0.638171i $$-0.779691\pi$$
−0.769894 + 0.638171i $$0.779691\pi$$
$$192$$ 0 0
$$193$$ 7.45856 0.536879 0.268440 0.963297i $$-0.413492\pi$$
0.268440 + 0.963297i $$0.413492\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.45856 −0.531401 −0.265700 0.964056i $$-0.585603\pi$$
−0.265700 + 0.964056i $$0.585603\pi$$
$$198$$ 0 0
$$199$$ 19.3372 1.37077 0.685387 0.728179i $$-0.259633\pi$$
0.685387 + 0.728179i $$0.259633\pi$$
$$200$$ 0 0
$$201$$ 12.4340 0.877026
$$202$$ 0 0
$$203$$ 17.2663 1.21186
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 5.52311 0.383883
$$208$$ 0 0
$$209$$ −27.6156 −1.91021
$$210$$ 0 0
$$211$$ −8.02791 −0.552664 −0.276332 0.961062i $$-0.589119\pi$$
−0.276332 + 0.961062i $$0.589119\pi$$
$$212$$ 0 0
$$213$$ −8.89692 −0.609607
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 23.4479 1.59175
$$218$$ 0 0
$$219$$ −3.25240 −0.219777
$$220$$ 0 0
$$221$$ 3.62620 0.243924
$$222$$ 0 0
$$223$$ −16.5048 −1.10524 −0.552621 0.833432i $$-0.686372\pi$$
−0.552621 + 0.833432i $$0.686372\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.69701 0.179007 0.0895033 0.995987i $$-0.471472\pi$$
0.0895033 + 0.995987i $$0.471472\pi$$
$$228$$ 0 0
$$229$$ −10.5616 −0.697933 −0.348967 0.937135i $$-0.613467\pi$$
−0.348967 + 0.937135i $$0.613467\pi$$
$$230$$ 0 0
$$231$$ −12.4017 −0.815973
$$232$$ 0 0
$$233$$ −25.2158 −1.65194 −0.825969 0.563715i $$-0.809372\pi$$
−0.825969 + 0.563715i $$0.809372\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −13.6724 −0.888120
$$238$$ 0 0
$$239$$ −8.51875 −0.551032 −0.275516 0.961297i $$-0.588849\pi$$
−0.275516 + 0.961297i $$0.588849\pi$$
$$240$$ 0 0
$$241$$ 7.45856 0.480448 0.240224 0.970717i $$-0.422779\pi$$
0.240224 + 0.970717i $$0.422779\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.14931 −0.391271
$$248$$ 0 0
$$249$$ 3.03228 0.192163
$$250$$ 0 0
$$251$$ −9.64452 −0.608757 −0.304378 0.952551i $$-0.598449\pi$$
−0.304378 + 0.952551i $$0.598449\pi$$
$$252$$ 0 0
$$253$$ 24.8034 1.55938
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 13.3555 0.833092 0.416546 0.909115i $$-0.363240\pi$$
0.416546 + 0.909115i $$0.363240\pi$$
$$258$$ 0 0
$$259$$ −2.47689 −0.153906
$$260$$ 0 0
$$261$$ 6.25240 0.387014
$$262$$ 0 0
$$263$$ −9.25240 −0.570527 −0.285264 0.958449i $$-0.592081\pi$$
−0.285264 + 0.958449i $$0.592081\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.20617 0.135015
$$268$$ 0 0
$$269$$ 25.2986 1.54248 0.771242 0.636542i $$-0.219636\pi$$
0.771242 + 0.636542i $$0.219636\pi$$
$$270$$ 0 0
$$271$$ 31.3309 1.90322 0.951608 0.307314i $$-0.0994300\pi$$
0.951608 + 0.307314i $$0.0994300\pi$$
$$272$$ 0 0
$$273$$ −2.76156 −0.167137
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.0096 −1.38251 −0.691256 0.722610i $$-0.742942\pi$$
−0.691256 + 0.722610i $$0.742942\pi$$
$$278$$ 0 0
$$279$$ 8.49084 0.508333
$$280$$ 0 0
$$281$$ −9.19554 −0.548560 −0.274280 0.961650i $$-0.588440\pi$$
−0.274280 + 0.961650i $$0.588440\pi$$
$$282$$ 0 0
$$283$$ 9.93545 0.590601 0.295301 0.955404i $$-0.404580\pi$$
0.295301 + 0.955404i $$0.404580\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ −3.85069 −0.226511
$$290$$ 0 0
$$291$$ 17.7572 1.04094
$$292$$ 0 0
$$293$$ −23.7938 −1.39005 −0.695025 0.718985i $$-0.744607\pi$$
−0.695025 + 0.718985i $$0.744607\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −4.49084 −0.260585
$$298$$ 0 0
$$299$$ 5.52311 0.319410
$$300$$ 0 0
$$301$$ 0.747604 0.0430912
$$302$$ 0 0
$$303$$ −5.83237 −0.335061
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 25.0664 1.43062 0.715309 0.698809i $$-0.246286\pi$$
0.715309 + 0.698809i $$0.246286\pi$$
$$308$$ 0 0
$$309$$ 3.45856 0.196751
$$310$$ 0 0
$$311$$ 18.3632 1.04128 0.520640 0.853776i $$-0.325693\pi$$
0.520640 + 0.853776i $$0.325693\pi$$
$$312$$ 0 0
$$313$$ 10.7293 0.606455 0.303227 0.952918i $$-0.401936\pi$$
0.303227 + 0.952918i $$0.401936\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.20617 0.460904 0.230452 0.973084i $$-0.425979\pi$$
0.230452 + 0.973084i $$0.425979\pi$$
$$318$$ 0 0
$$319$$ 28.0785 1.57209
$$320$$ 0 0
$$321$$ 0.832365 0.0464581
$$322$$ 0 0
$$323$$ −22.2986 −1.24073
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 12.1493 0.671859
$$328$$ 0 0
$$329$$ 9.35548 0.515784
$$330$$ 0 0
$$331$$ 30.7110 1.68803 0.844014 0.536322i $$-0.180187\pi$$
0.844014 + 0.536322i $$0.180187\pi$$
$$332$$ 0 0
$$333$$ −0.896916 −0.0491507
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8.13099 −0.442923 −0.221462 0.975169i $$-0.571083\pi$$
−0.221462 + 0.975169i $$0.571083\pi$$
$$338$$ 0 0
$$339$$ −4.50479 −0.244667
$$340$$ 0 0
$$341$$ 38.1310 2.06491
$$342$$ 0 0
$$343$$ −17.6016 −0.950398
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 14.9248 0.801206 0.400603 0.916252i $$-0.368801\pi$$
0.400603 + 0.916252i $$0.368801\pi$$
$$348$$ 0 0
$$349$$ −16.5048 −0.883481 −0.441741 0.897143i $$-0.645639\pi$$
−0.441741 + 0.897143i $$0.645639\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −23.0664 −1.22770 −0.613851 0.789422i $$-0.710381\pi$$
−0.613851 + 0.789422i $$0.710381\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −10.0140 −0.529995
$$358$$ 0 0
$$359$$ 32.6681 1.72415 0.862077 0.506777i $$-0.169163\pi$$
0.862077 + 0.506777i $$0.169163\pi$$
$$360$$ 0 0
$$361$$ 18.8140 0.990213
$$362$$ 0 0
$$363$$ −9.16763 −0.481176
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 32.1493 1.67818 0.839090 0.543992i $$-0.183088\pi$$
0.839090 + 0.543992i $$0.183088\pi$$
$$368$$ 0 0
$$369$$ 2.89692 0.150807
$$370$$ 0 0
$$371$$ −9.03228 −0.468932
$$372$$ 0 0
$$373$$ −6.81404 −0.352818 −0.176409 0.984317i $$-0.556448\pi$$
−0.176409 + 0.984317i $$0.556448\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.25240 0.322015
$$378$$ 0 0
$$379$$ 1.65078 0.0847948 0.0423974 0.999101i $$-0.486500\pi$$
0.0423974 + 0.999101i $$0.486500\pi$$
$$380$$ 0 0
$$381$$ −6.62620 −0.339470
$$382$$ 0 0
$$383$$ −10.0202 −0.512009 −0.256004 0.966676i $$-0.582406\pi$$
−0.256004 + 0.966676i $$0.582406\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.270718 0.0137614
$$388$$ 0 0
$$389$$ 19.4017 0.983706 0.491853 0.870678i $$-0.336320\pi$$
0.491853 + 0.870678i $$0.336320\pi$$
$$390$$ 0 0
$$391$$ 20.0279 1.01285
$$392$$ 0 0
$$393$$ −1.10308 −0.0556432
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2.56165 0.128565 0.0642827 0.997932i $$-0.479524\pi$$
0.0642827 + 0.997932i $$0.479524\pi$$
$$398$$ 0 0
$$399$$ 16.9817 0.850147
$$400$$ 0 0
$$401$$ −3.55102 −0.177330 −0.0886648 0.996062i $$-0.528260\pi$$
−0.0886648 + 0.996062i $$0.528260\pi$$
$$402$$ 0 0
$$403$$ 8.49084 0.422959
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −4.02791 −0.199656
$$408$$ 0 0
$$409$$ 28.0558 1.38727 0.693635 0.720326i $$-0.256008\pi$$
0.693635 + 0.720326i $$0.256008\pi$$
$$410$$ 0 0
$$411$$ 20.4479 1.00862
$$412$$ 0 0
$$413$$ −26.3372 −1.29597
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −16.5693 −0.811404
$$418$$ 0 0
$$419$$ 36.3188 1.77429 0.887146 0.461490i $$-0.152685\pi$$
0.887146 + 0.461490i $$0.152685\pi$$
$$420$$ 0 0
$$421$$ 10.2062 0.497418 0.248709 0.968578i $$-0.419994\pi$$
0.248709 + 0.968578i $$0.419994\pi$$
$$422$$ 0 0
$$423$$ 3.38776 0.164718
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 31.5371 1.52619
$$428$$ 0 0
$$429$$ −4.49084 −0.216820
$$430$$ 0 0
$$431$$ −11.8140 −0.569062 −0.284531 0.958667i $$-0.591838\pi$$
−0.284531 + 0.958667i $$0.591838\pi$$
$$432$$ 0 0
$$433$$ 0.598291 0.0287521 0.0143760 0.999897i $$-0.495424\pi$$
0.0143760 + 0.999897i $$0.495424\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −33.9634 −1.62469
$$438$$ 0 0
$$439$$ −13.6724 −0.652549 −0.326275 0.945275i $$-0.605793\pi$$
−0.326275 + 0.945275i $$0.605793\pi$$
$$440$$ 0 0
$$441$$ 0.626198 0.0298190
$$442$$ 0 0
$$443$$ 32.1772 1.52879 0.764393 0.644751i $$-0.223039\pi$$
0.764393 + 0.644751i $$0.223039\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −3.79383 −0.179442
$$448$$ 0 0
$$449$$ −9.81404 −0.463153 −0.231577 0.972817i $$-0.574388\pi$$
−0.231577 + 0.972817i $$0.574388\pi$$
$$450$$ 0 0
$$451$$ 13.0096 0.612597
$$452$$ 0 0
$$453$$ 5.80779 0.272874
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −19.5877 −0.916272 −0.458136 0.888882i $$-0.651483\pi$$
−0.458136 + 0.888882i $$0.651483\pi$$
$$458$$ 0 0
$$459$$ −3.62620 −0.169256
$$460$$ 0 0
$$461$$ 11.7938 0.549294 0.274647 0.961545i $$-0.411439\pi$$
0.274647 + 0.961545i $$0.411439\pi$$
$$462$$ 0 0
$$463$$ 38.8819 1.80700 0.903498 0.428592i $$-0.140990\pi$$
0.903498 + 0.428592i $$0.140990\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 25.4017 1.17545 0.587725 0.809060i $$-0.300024\pi$$
0.587725 + 0.809060i $$0.300024\pi$$
$$468$$ 0 0
$$469$$ −34.3372 −1.58554
$$470$$ 0 0
$$471$$ −13.6541 −0.629148
$$472$$ 0 0
$$473$$ 1.21575 0.0559003
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −3.27072 −0.149756
$$478$$ 0 0
$$479$$ 34.0987 1.55801 0.779005 0.627018i $$-0.215725\pi$$
0.779005 + 0.627018i $$0.215725\pi$$
$$480$$ 0 0
$$481$$ −0.896916 −0.0408959
$$482$$ 0 0
$$483$$ −15.2524 −0.694008
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.7616 0.759539 0.379769 0.925081i $$-0.376003\pi$$
0.379769 + 0.925081i $$0.376003\pi$$
$$488$$ 0 0
$$489$$ −7.25240 −0.327965
$$490$$ 0 0
$$491$$ 1.65222 0.0745635 0.0372817 0.999305i $$-0.488130\pi$$
0.0372817 + 0.999305i $$0.488130\pi$$
$$492$$ 0 0
$$493$$ 22.6724 1.02111
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.5693 1.10209
$$498$$ 0 0
$$499$$ −22.7326 −1.01765 −0.508826 0.860870i $$-0.669920\pi$$
−0.508826 + 0.860870i $$0.669920\pi$$
$$500$$ 0 0
$$501$$ 2.14931 0.0960242
$$502$$ 0 0
$$503$$ 6.09246 0.271649 0.135825 0.990733i $$-0.456632\pi$$
0.135825 + 0.990733i $$0.456632\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 0 0
$$509$$ 7.87090 0.348871 0.174436 0.984669i $$-0.444190\pi$$
0.174436 + 0.984669i $$0.444190\pi$$
$$510$$ 0 0
$$511$$ 8.98168 0.397326
$$512$$ 0 0
$$513$$ 6.14931 0.271499
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 15.2139 0.669105
$$518$$ 0 0
$$519$$ 25.5693 1.12237
$$520$$ 0 0
$$521$$ 19.7938 0.867184 0.433592 0.901109i $$-0.357246\pi$$
0.433592 + 0.901109i $$0.357246\pi$$
$$522$$ 0 0
$$523$$ 27.1878 1.18884 0.594421 0.804154i $$-0.297381\pi$$
0.594421 + 0.804154i $$0.297381\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 30.7895 1.34121
$$528$$ 0 0
$$529$$ 7.50479 0.326295
$$530$$ 0 0
$$531$$ −9.53707 −0.413873
$$532$$ 0 0
$$533$$ 2.89692 0.125479
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 21.5510 0.929995
$$538$$ 0 0
$$539$$ 2.81215 0.121128
$$540$$ 0 0
$$541$$ −38.0558 −1.63615 −0.818074 0.575114i $$-0.804958\pi$$
−0.818074 + 0.575114i $$0.804958\pi$$
$$542$$ 0 0
$$543$$ 0.607876 0.0260865
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10.3353 0.441904 0.220952 0.975285i $$-0.429084\pi$$
0.220952 + 0.975285i $$0.429084\pi$$
$$548$$ 0 0
$$549$$ 11.4200 0.487395
$$550$$ 0 0
$$551$$ −38.4479 −1.63794
$$552$$ 0 0
$$553$$ 37.7572 1.60560
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1.25240 0.0530657 0.0265329 0.999648i $$-0.491553\pi$$
0.0265329 + 0.999648i $$0.491553\pi$$
$$558$$ 0 0
$$559$$ 0.270718 0.0114502
$$560$$ 0 0
$$561$$ −16.2847 −0.687539
$$562$$ 0 0
$$563$$ −45.1868 −1.90440 −0.952198 0.305480i $$-0.901183\pi$$
−0.952198 + 0.305480i $$0.901183\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 2.76156 0.115975
$$568$$ 0 0
$$569$$ −15.8603 −0.664897 −0.332449 0.943121i $$-0.607875\pi$$
−0.332449 + 0.943121i $$0.607875\pi$$
$$570$$ 0 0
$$571$$ −29.5789 −1.23784 −0.618920 0.785454i $$-0.712429\pi$$
−0.618920 + 0.785454i $$0.712429\pi$$
$$572$$ 0 0
$$573$$ 21.2803 0.888997
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.8401 1.11737 0.558683 0.829381i $$-0.311307\pi$$
0.558683 + 0.829381i $$0.311307\pi$$
$$578$$ 0 0
$$579$$ −7.45856 −0.309967
$$580$$ 0 0
$$581$$ −8.37380 −0.347404
$$582$$ 0 0
$$583$$ −14.6883 −0.608326
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −23.4359 −0.967302 −0.483651 0.875261i $$-0.660690\pi$$
−0.483651 + 0.875261i $$0.660690\pi$$
$$588$$ 0 0
$$589$$ −52.2128 −2.15139
$$590$$ 0 0
$$591$$ 7.45856 0.306804
$$592$$ 0 0
$$593$$ −43.2880 −1.77763 −0.888813 0.458271i $$-0.848469\pi$$
−0.888813 + 0.458271i $$0.848469\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −19.3372 −0.791417
$$598$$ 0 0
$$599$$ 3.58767 0.146588 0.0732940 0.997310i $$-0.476649\pi$$
0.0732940 + 0.997310i $$0.476649\pi$$
$$600$$ 0 0
$$601$$ 31.0646 1.26715 0.633575 0.773681i $$-0.281587\pi$$
0.633575 + 0.773681i $$0.281587\pi$$
$$602$$ 0 0
$$603$$ −12.4340 −0.506351
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −30.2697 −1.22861 −0.614304 0.789069i $$-0.710563\pi$$
−0.614304 + 0.789069i $$0.710563\pi$$
$$608$$ 0 0
$$609$$ −17.2663 −0.699668
$$610$$ 0 0
$$611$$ 3.38776 0.137054
$$612$$ 0 0
$$613$$ −46.0558 −1.86018 −0.930088 0.367336i $$-0.880270\pi$$
−0.930088 + 0.367336i $$0.880270\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1.43835 −0.0579059 −0.0289530 0.999581i $$-0.509217\pi$$
−0.0289530 + 0.999581i $$0.509217\pi$$
$$618$$ 0 0
$$619$$ −36.2620 −1.45749 −0.728746 0.684784i $$-0.759897\pi$$
−0.728746 + 0.684784i $$0.759897\pi$$
$$620$$ 0 0
$$621$$ −5.52311 −0.221635
$$622$$ 0 0
$$623$$ −6.09246 −0.244089
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 27.6156 1.10286
$$628$$ 0 0
$$629$$ −3.25240 −0.129682
$$630$$ 0 0
$$631$$ 34.8401 1.38696 0.693480 0.720475i $$-0.256076\pi$$
0.693480 + 0.720475i $$0.256076\pi$$
$$632$$ 0 0
$$633$$ 8.02791 0.319081
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.626198 0.0248109
$$638$$ 0 0
$$639$$ 8.89692 0.351957
$$640$$ 0 0
$$641$$ −29.8882 −1.18051 −0.590256 0.807216i $$-0.700973\pi$$
−0.590256 + 0.807216i $$0.700973\pi$$
$$642$$ 0 0
$$643$$ 18.8603 0.743777 0.371888 0.928278i $$-0.378710\pi$$
0.371888 + 0.928278i $$0.378710\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −6.50479 −0.255730 −0.127865 0.991792i $$-0.540812\pi$$
−0.127865 + 0.991792i $$0.540812\pi$$
$$648$$ 0 0
$$649$$ −42.8294 −1.68120
$$650$$ 0 0
$$651$$ −23.4479 −0.918997
$$652$$ 0 0
$$653$$ 28.1589 1.10194 0.550971 0.834524i $$-0.314257\pi$$
0.550971 + 0.834524i $$0.314257\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 3.25240 0.126888
$$658$$ 0 0
$$659$$ 30.2139 1.17697 0.588483 0.808510i $$-0.299726\pi$$
0.588483 + 0.808510i $$0.299726\pi$$
$$660$$ 0 0
$$661$$ 36.0356 1.40162 0.700811 0.713347i $$-0.252822\pi$$
0.700811 + 0.713347i $$0.252822\pi$$
$$662$$ 0 0
$$663$$ −3.62620 −0.140830
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 34.5327 1.33711
$$668$$ 0 0
$$669$$ 16.5048 0.638112
$$670$$ 0 0
$$671$$ 51.2855 1.97986
$$672$$ 0 0
$$673$$ 1.68305 0.0648769 0.0324385 0.999474i $$-0.489673\pi$$
0.0324385 + 0.999474i $$0.489673\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.50479 0.173133 0.0865666 0.996246i $$-0.472410\pi$$
0.0865666 + 0.996246i $$0.472410\pi$$
$$678$$ 0 0
$$679$$ −49.0375 −1.88189
$$680$$ 0 0
$$681$$ −2.69701 −0.103350
$$682$$ 0 0
$$683$$ 35.5650 1.36086 0.680428 0.732815i $$-0.261794\pi$$
0.680428 + 0.732815i $$0.261794\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.5616 0.402952
$$688$$ 0 0
$$689$$ −3.27072 −0.124604
$$690$$ 0 0
$$691$$ 6.29237 0.239373 0.119686 0.992812i $$-0.461811\pi$$
0.119686 + 0.992812i $$0.461811\pi$$
$$692$$ 0 0
$$693$$ 12.4017 0.471102
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 10.5048 0.397897
$$698$$ 0 0
$$699$$ 25.2158 0.953747
$$700$$ 0 0
$$701$$ −36.9065 −1.39394 −0.696970 0.717101i $$-0.745469\pi$$
−0.696970 + 0.717101i $$0.745469\pi$$
$$702$$ 0 0
$$703$$ 5.51542 0.208018
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.1064 0.605744
$$708$$ 0 0
$$709$$ −5.25240 −0.197258 −0.0986289 0.995124i $$-0.531446\pi$$
−0.0986289 + 0.995124i $$0.531446\pi$$
$$710$$ 0 0
$$711$$ 13.6724 0.512756
$$712$$ 0 0
$$713$$ 46.8959 1.75626
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.51875 0.318138
$$718$$ 0 0
$$719$$ −22.6339 −0.844102 −0.422051 0.906572i $$-0.638690\pi$$
−0.422051 + 0.906572i $$0.638690\pi$$
$$720$$ 0 0
$$721$$ −9.55102 −0.355699
$$722$$ 0 0
$$723$$ −7.45856 −0.277387
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −34.5606 −1.28178 −0.640891 0.767632i $$-0.721435\pi$$
−0.640891 + 0.767632i $$0.721435\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 0.981678 0.0363087
$$732$$ 0 0
$$733$$ 11.4942 0.424547 0.212273 0.977210i $$-0.431913\pi$$
0.212273 + 0.977210i $$0.431913\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −55.8390 −2.05686
$$738$$ 0 0
$$739$$ 20.4619 0.752703 0.376351 0.926477i $$-0.377179\pi$$
0.376351 + 0.926477i $$0.377179\pi$$
$$740$$ 0 0
$$741$$ 6.14931 0.225901
$$742$$ 0 0
$$743$$ 34.2836 1.25774 0.628872 0.777509i $$-0.283517\pi$$
0.628872 + 0.777509i $$0.283517\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −3.03228 −0.110945
$$748$$ 0 0
$$749$$ −2.29862 −0.0839899
$$750$$ 0 0
$$751$$ −23.6358 −0.862482 −0.431241 0.902237i $$-0.641924\pi$$
−0.431241 + 0.902237i $$0.641924\pi$$
$$752$$ 0 0
$$753$$ 9.64452 0.351466
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −7.54913 −0.274378 −0.137189 0.990545i $$-0.543807\pi$$
−0.137189 + 0.990545i $$0.543807\pi$$
$$758$$ 0 0
$$759$$ −24.8034 −0.900307
$$760$$ 0 0
$$761$$ 48.2418 1.74876 0.874381 0.485239i $$-0.161268\pi$$
0.874381 + 0.485239i $$0.161268\pi$$
$$762$$ 0 0
$$763$$ −33.5510 −1.21463
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −9.53707 −0.344364
$$768$$ 0 0
$$769$$ 24.0558 0.867475 0.433737 0.901039i $$-0.357195\pi$$
0.433737 + 0.901039i $$0.357195\pi$$
$$770$$ 0 0
$$771$$ −13.3555 −0.480986
$$772$$ 0 0
$$773$$ −37.8863 −1.36268 −0.681338 0.731969i $$-0.738601\pi$$
−0.681338 + 0.731969i $$0.738601\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2.47689 0.0888578
$$778$$ 0 0
$$779$$ −17.8140 −0.638254
$$780$$ 0 0
$$781$$ 39.9546 1.42969
$$782$$ 0 0
$$783$$ −6.25240 −0.223442
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14.3213 0.510500 0.255250 0.966875i $$-0.417842\pi$$
0.255250 + 0.966875i $$0.417842\pi$$
$$788$$ 0 0
$$789$$ 9.25240 0.329394
$$790$$ 0 0
$$791$$ 12.4402 0.442324
$$792$$ 0 0
$$793$$ 11.4200 0.405537
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −7.97398 −0.282453 −0.141226 0.989977i $$-0.545105\pi$$
−0.141226 + 0.989977i $$0.545105\pi$$
$$798$$ 0 0
$$799$$ 12.2847 0.434600
$$800$$ 0 0
$$801$$ −2.20617 −0.0779511
$$802$$ 0 0
$$803$$ 14.6060 0.515434
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −25.2986 −0.890554
$$808$$ 0 0
$$809$$ 7.42192 0.260941 0.130470 0.991452i $$-0.458351\pi$$
0.130470 + 0.991452i $$0.458351\pi$$
$$810$$ 0 0
$$811$$ −54.6391 −1.91864 −0.959319 0.282323i $$-0.908895\pi$$
−0.959319 + 0.282323i $$0.908895\pi$$
$$812$$ 0 0
$$813$$ −31.3309 −1.09882
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.66473 −0.0582416
$$818$$ 0 0
$$819$$ 2.76156 0.0964966
$$820$$ 0 0
$$821$$ −2.12910 −0.0743062 −0.0371531 0.999310i $$-0.511829\pi$$
−0.0371531 + 0.999310i $$0.511829\pi$$
$$822$$ 0 0
$$823$$ 36.3188 1.26600 0.632998 0.774154i $$-0.281824\pi$$
0.632998 + 0.774154i $$0.281824\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −22.0419 −0.766471 −0.383235 0.923651i $$-0.625190\pi$$
−0.383235 + 0.923651i $$0.625190\pi$$
$$828$$ 0 0
$$829$$ 48.9344 1.69956 0.849781 0.527136i $$-0.176734\pi$$
0.849781 + 0.527136i $$0.176734\pi$$
$$830$$ 0 0
$$831$$ 23.0096 0.798194
$$832$$ 0 0
$$833$$ 2.27072 0.0786757
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.49084 −0.293486
$$838$$ 0 0
$$839$$ 10.2986 0.355548 0.177774 0.984071i $$-0.443110\pi$$
0.177774 + 0.984071i $$0.443110\pi$$
$$840$$ 0 0
$$841$$ 10.0925 0.348016
$$842$$ 0 0
$$843$$ 9.19554 0.316711
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.3169 0.869901
$$848$$ 0 0
$$849$$ −9.93545 −0.340984
$$850$$ 0 0
$$851$$ −4.95377 −0.169813
$$852$$ 0 0
$$853$$ −24.0356 −0.822963 −0.411482 0.911418i $$-0.634989\pi$$
−0.411482 + 0.911418i $$0.634989\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.0096 0.512718 0.256359 0.966582i $$-0.417477\pi$$
0.256359 + 0.966582i $$0.417477\pi$$
$$858$$ 0 0
$$859$$ −24.2707 −0.828106 −0.414053 0.910253i $$-0.635887\pi$$
−0.414053 + 0.910253i $$0.635887\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ −42.4985 −1.44667 −0.723333 0.690499i $$-0.757391\pi$$
−0.723333 + 0.690499i $$0.757391\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.85069 0.130776
$$868$$ 0 0
$$869$$ 61.4007 2.08287
$$870$$ 0 0
$$871$$ −12.4340 −0.421309
$$872$$ 0 0
$$873$$ −17.7572 −0.600990
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 4.56165 0.154036 0.0770179 0.997030i $$-0.475460\pi$$
0.0770179 + 0.997030i $$0.475460\pi$$
$$878$$ 0 0
$$879$$ 23.7938 0.802546
$$880$$ 0 0
$$881$$ 6.10308 0.205618 0.102809 0.994701i $$-0.467217\pi$$
0.102809 + 0.994701i $$0.467217\pi$$
$$882$$ 0 0
$$883$$ −13.0096 −0.437807 −0.218904 0.975746i $$-0.570248\pi$$
−0.218904 + 0.975746i $$0.570248\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 11.4307 0.383804 0.191902 0.981414i $$-0.438534\pi$$
0.191902 + 0.981414i $$0.438534\pi$$
$$888$$ 0 0
$$889$$ 18.2986 0.613716
$$890$$ 0 0
$$891$$ 4.49084 0.150449
$$892$$ 0 0
$$893$$ −20.8324 −0.697129
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.52311 −0.184411
$$898$$ 0 0
$$899$$ 53.0881 1.77059
$$900$$ 0 0
$$901$$ −11.8603 −0.395123
$$902$$ 0 0
$$903$$ −0.747604 −0.0248787
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 19.8863 0.660313 0.330157 0.943926i $$-0.392898\pi$$
0.330157 + 0.943926i $$0.392898\pi$$
$$908$$ 0 0
$$909$$ 5.83237 0.193447
$$910$$ 0 0
$$911$$ −57.8217 −1.91572 −0.957860 0.287236i $$-0.907264\pi$$
−0.957860 + 0.287236i $$0.907264\pi$$
$$912$$ 0 0
$$913$$ −13.6175 −0.450672
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 3.04623 0.100595
$$918$$ 0 0
$$919$$ −8.17722 −0.269742 −0.134871 0.990863i $$-0.543062\pi$$
−0.134871 + 0.990863i $$0.543062\pi$$
$$920$$ 0 0
$$921$$ −25.0664 −0.825967
$$922$$ 0 0
$$923$$ 8.89692 0.292846
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −3.45856 −0.113594
$$928$$ 0 0
$$929$$ −14.7312 −0.483314 −0.241657 0.970362i $$-0.577691\pi$$
−0.241657 + 0.970362i $$0.577691\pi$$
$$930$$ 0 0
$$931$$ −3.85069 −0.126201
$$932$$ 0 0
$$933$$ −18.3632 −0.601183
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.9431 0.618846 0.309423 0.950925i $$-0.399864\pi$$
0.309423 + 0.950925i $$0.399864\pi$$
$$938$$ 0 0
$$939$$ −10.7293 −0.350137
$$940$$ 0 0
$$941$$ 12.3353 0.402118 0.201059 0.979579i $$-0.435562\pi$$
0.201059 + 0.979579i $$0.435562\pi$$
$$942$$ 0 0
$$943$$ 16.0000 0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −18.1710 −0.590477 −0.295238 0.955424i $$-0.595399\pi$$
−0.295238 + 0.955424i $$0.595399\pi$$
$$948$$ 0 0
$$949$$ 3.25240 0.105577
$$950$$ 0 0
$$951$$ −8.20617 −0.266103
$$952$$ 0 0
$$953$$ 37.4113 1.21187 0.605935 0.795514i $$-0.292799\pi$$
0.605935 + 0.795514i $$0.292799\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −28.0785 −0.907649
$$958$$ 0 0
$$959$$ −56.4681 −1.82345
$$960$$ 0 0
$$961$$ 41.0943 1.32562
$$962$$ 0 0
$$963$$ −0.832365 −0.0268226
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 38.4821 1.23750 0.618750 0.785588i $$-0.287639\pi$$
0.618750 + 0.785588i $$0.287639\pi$$
$$968$$ 0 0
$$969$$ 22.2986 0.716335
$$970$$ 0 0
$$971$$ −44.5896 −1.43095 −0.715473 0.698640i $$-0.753789\pi$$
−0.715473 + 0.698640i $$0.753789\pi$$
$$972$$ 0 0
$$973$$ 45.7572 1.46691
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.2524 −1.06384 −0.531919 0.846795i $$-0.678529\pi$$
−0.531919 + 0.846795i $$0.678529\pi$$
$$978$$ 0 0
$$979$$ −9.90754 −0.316646
$$980$$ 0 0
$$981$$ −12.1493 −0.387898
$$982$$ 0 0
$$983$$ −10.9186 −0.348248 −0.174124 0.984724i $$-0.555709\pi$$
−0.174124 + 0.984724i $$0.555709\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −9.35548 −0.297788
$$988$$ 0 0
$$989$$ 1.49521 0.0475448
$$990$$ 0 0
$$991$$ 17.3738 0.551897 0.275949 0.961172i $$-0.411008\pi$$
0.275949 + 0.961172i $$0.411008\pi$$
$$992$$ 0 0
$$993$$ −30.7110 −0.974583
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −12.0752 −0.382425 −0.191212 0.981549i $$-0.561242\pi$$
−0.191212 + 0.981549i $$0.561242\pi$$
$$998$$ 0 0
$$999$$ 0.896916 0.0283772
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 7800.2.a.bm.1.3 3
5.4 even 2 7800.2.a.bn.1.1 yes 3

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bm.1.3 3 1.1 even 1 trivial
7800.2.a.bn.1.1 yes 3 5.4 even 2